Search Results for measure*


  1. Pierre Méchain (1744-1804)
    • We now quote a story from [',' K Alder, The measure of all things (London, 2002).','2] which is based on Mechain's own account:- .
    • [See Borda's biography for a description of the repeating circle.] Dominique Cassini, with Mechain as his assistant, made the measurements with Borda's repeating circle on the French side.
    • Initially Mechain used older surveying equipment to check Dominique Cassini's measurements with the repeating circle, but later in the project he took over measurements with the repeating circle.
    • The Commission of Weights and Measures, which had as its members Condorcet, Lavoisier, Laplace and Legendre, was set up by the Academie des Sciences in 1790 to bring in a uniform system of measurement.
    • It was decided to measure, using the method of triangulation with sightings made with the Borda repeating circle, that part of the meridian between Dunkerque and Barcelona.
    • The northern part was much the longer since it had been accurately measured by Cassini de Thury in 1740.
    • He then went to Barcelona for the winter months to take accurate measurements of its latitude.
    • Mechain was fanatical about accuracy and, as we shall soon see, these measurements would cause him anguish for the rest of his life.
    • Back in Barcelona for the winter he could not return to Mont-Jouy because of the war but made latitude measurements from the inn he was living in.
    • By March of 1794 Mechain had discovered that the latitude measurements he had taken from the inn did not agree with those from Mont-Jouy, yet he could not return to the castle to take further measurements.
    • He remained there until late August when he went to Perpignan to resume his measurements.
    • Taking more and more measurements would only lead to greater accuracy if the errors were random.
    • This put him in a good position to get a project approved to extend the measurement of the meridian line to the Balearic Islands.
    • He argued that accuracy would be improved by measuring the longer line, but his real reason was that the extension would make the latitude measurements at Barcelona which so tormented him, unnecessary.
    • Despite what the Spanish had told him he could not see his Montsia station and he wrote to Delambre (see for example [',' K Alder, The measure of all things (London, 2002).','2]):- .
    • Delambre wrote a fine eulogy for Mechain describing his meridian measurements [',' K Alder, The measure of all things (London, 2002).','2]:- .
    • We should make it clear however that Delambre wanted to preserve Mechain's high reputation which, in Delambre's words [',' K Alder, The measure of all things (London, 2002).','2]:- .
    • History Topics: The history of measurement .

  2. John Greaves (1602-1652)
    • His main scientific aim was the [',' Z Shalev, Measurer of all things : John Greaves (16-2-1652), the Great Pyramid, and early modern metrology, J.
    • practical and sober project of standardising and synchronising the weights and measures of all ancient and modern nations.
    • His desire to find out about measurements in the ancient world led him to plan visits to Italy and Egypt, where he wanted to make measurements of the pyramids.
    • As Shalev puts it [',' Z Shalev, Measurer of all things : John Greaves (16-2-1652), the Great Pyramid, and early modern metrology, J.
    • Well, it is not completely true to say that he took all the instruments he required for in the spring of 1638 he wrote from Istanbul (Constantinople) to Peter Turner at Merton College, Oxford [',' Z Shalev, Measurer of all things : John Greaves (16-2-1652), the Great Pyramid, and early modern metrology, J.
    • He made two visits to Italy, visiting there on both his outward and on his return journey [',' Z Shalev, Measurer of all things : John Greaves (16-2-1652), the Great Pyramid, and early modern metrology, J.
    • In Rome he measured, among many other ancient structures, Cestius's Pyramid and St Peter's basilica.
    • One of these was to measure the latitudes of Istanbul, Rhodes (which he visted on his way from Istanbul to Alexandria on his way to the Pyramids) and Alexandria.
    • The most important aspect of his travels, in Greaves's way of thinking, was to make precise measurements of the pyramids in an attempt to study the weights and measures of the ancients [',' Z Shalev, Measurer of all things : John Greaves (16-2-1652), the Great Pyramid, and early modern metrology, J.
    • He gave his own [measurements], obtained "by experience and by diligent calculation," using "an exquisite radius of ten feet," "most accurately divided".
    • in the second gallery he took his measures as precisely as he could, "judging this to be the fittest place for the fixing of measures for posterity.
    • This was a scholarly work which contained the measurements he had made of the pyramids.
    • He [',' Z Shalev, Measurer of all things : John Greaves (16-2-1652), the Great Pyramid, and early modern metrology, J.
    • In 1649 he published A Discourse of the Roman Foot, and Denarius; from whence, as from two Principles, the Measures and Weights used by the Ancients may be deduced.

  3. Jean-Baptiste-Joseph Delambre biography
    • The Academie had already set up a Commission of Weights and Measures in 1790 consisting of Borda, Condorcet, Laplace, Legendre and Lavoisier to advise on a metric system of weights and measures.
    • It was decided to measure, using the method of triangulation with sightings made with the Borda repeating circle which was an extremely accurate new instrument, that part of the meridian between Dunkerque and Barcelona.
    • The northern part was much the longer since it had been accurately measured by Cassini de Thury in 1740.
    • Although Dominique Cassini was keen to take charge of the project, he refused to personally measure one sector and, on 5 May 1792 Delambre was given charge of the Dunkerque to Rodez sector and Mechain the Rodez to Barcelona sector.
    • In December 1793, however, he was removed from the meridian measuring task by the Committee of Public Safety who decreed that (see for example [',' K Alder, The measure of all things (London, 2002).','3]):- .
    • Also required was an accurate baseline measurement so that the scale of the triangulation could be fixed precisely.
    • Delambre made accurate baseline measurements in Melun, near Paris, in April 1798.
    • An International Commission for Weights and Measures was set up and Delambre reported his results to it in February 1799.
    • The first of the three volumes, containing the history of measurement of the Earth and the project's triangulation data, was published in 1806.
    • ','1] or [',' K Alder, The measure of all things (London, 2002).','3]):- .
    • We have already noted that in the first volume of Base du systeme metrique Ⓣ he presented a history of measurement of the Earth.
    • History Topics: The history of measurement .

  4. Jean-Charles de Borda biography
    • One of his instruments, the Borda repeating circle, was used during the time of the French Revolution to measure an arc of a meridian as part of a project to introduce the decimal system.
    • To measure the angle between two points A and B, the instrument was set up so that the plane of the rotating rings was in the plane of A, B and the observer.
    • Of course the first scope was now an angle of 2θ from the angle determined by B so, decoupling the rings on which the scopes turned and moving it back to sight B one measured the angle 2θ.
    • When Borda was made Chairman of the Commission of Weights and Measures, which had as its members Condorcet, Lavoisier, Laplace and Legendre, he soon put his accurate surveying instrument to good use.
    • The Commission was set up in 1790 to bring in a uniform system of measurement.
    • This proposal had found favour with Britain and the United States who considered it a truly international measure.
    • Under Borda's leadership the project to accurately measure the distance from the North Pole to the equator using the Borda repeating circle was carried out.
    • As Chairman of the Commission of Weights and Measures, Borda made other important proposals.
    • Substitution weighing had its advent and found wider application during the French Revolution, when, under the direction of Jean-Charles de Borda, new standards for measures and weights, the metre scale and the kilogram, were introduced.
    • An International Commission was set up to try to make the metre an international measure, but Borda argued against this on the grounds that the measurements were based on the Earth and should therefore be equally acceptable to every nation on Earth.
    • His funeral is described in [',' K Alder, The measure of all things (London, 2002).','3]:- .
    • History Topics: The history of measurement .

  5. Jean Richer (1630-1696)
    • Richer and Meurisse, following their luggage and instruments, went to La Rochelle, from where the ship was to depart, and while there Richer measured the height of the tides at the vernal equinox.
    • Arriving in Canada, Richer measured the latitude of the French fort at Pentagouet as 44° 23' 20".
    • This observation is remarkable, first because Richer had the confidence to give the reading in degrees, minutes and seconds where all measurements at the time were only given in degrees and minutes, second because it is remarkably close to the correct value of 44° 23' 25".
    • Some historians have claimed that Richer must have been lucky to get all his readings as close as he did, but since his measurements are consistently good it is only fair to assume that he was an observer of extraordinary high ability.
    • His first task there was to measure the parallax of Mars and the observations were to be compared with those taken at other sites in order to compute the distance to the planet.
    • He measured the meridian altitude of the planet and of near-by stars as well as the precise time when the planet made its meridian transit.
    • For the same measurement marked on an iron rod in the former place in accordance with the length found necessary to make a seconds pendulum was transported to France and compared with the Paris measurement.
    • The difference between them was found to be 11/4 lines, by which the Cayenne measurement falls short of the Paris measurement, which is 3 feet, 183/5 lines [a line is 1/144 part of a foot].
    • Richer measured the positions of many southern hemisphere stars not visible from Paris.
    • In Cayenne, as he had done on his earlier expedition, he made measurements of the height of the tides.

  6. John Bell biography
    • All one can say is that if a measurement of sx, for example, is performed, the probabilities of the result obtained being either ℏ/2 or -ℏ/2 are both 1/2.
    • If, on the other hand, the initial state-vector has the general form of c+α++ c-α-, then all we can say is that in a measurement of sz, the probability of obtaining the value of ℏ/2 is |c+2 |, and that of obtaining the value of -ℏ/2 is |c-2 |.
    • Before any measurement, sz just does not have a value.
    • Realism and determinism would both be restored; sz would have a value at all times, and, with full knowledge of the state of the system, including the value of the hidden variable, we can predict the result of the measurement of sz .
    • Bohr put forward his (perhaps rather obscure) framework of complementarity, which attempted to explain why one should not expect to measure sx and sy (or x and p) simultaneously.
    • If we get +v/2, we know that an immediate measurement of s2z is bound to yield -ℏ/2, and vice-versa, although, at least according to Copenhagen, before any measurement, no component of either spin has a particular value.
    • First he constructed his own hidden variable account of a measurement of any component of spin.
    • He demonstrated this by extending the EPR argument, allowing measurements in each wing of the apparatus of any component of spin, not just sz.
    • He found that, even when hidden variables are allowed, in some cases the result obtained in one wing must depend on which component of spin is measured in the other; this violates locality.
    • Except in the simplest cases, the result you obtained when measuring a variable must depend on which other quantities are measured simultaneously.
    • Thus hidden variables cannot be thought of as saying what value a quantity 'has', only what value we will get if we measure it.
    • For the rest of his life, Bell continued to criticise the usual theories of measurement in quantum theory.

  7. Igor Kluvánek biography
    • For example: On systems of sets closed with respect to certain set operations (Slovak) (1955); Abstract integral as a positive functional and the theorem on extension of measure (Czech) (1956); On vector measure (Slovak) (1957); and On the theory of vector measures (Russian) (1961).
    • Some of the important papers he published while working in Australia are: Fourier transforms of vector-valued functions and measures (1970); On the product of vector measures (1973); The range of a vector-valued measure (1973); (with Greg Knowles) Liapunov decomposition of a vector measure (1974); and (with Greg Knowles) Attainable sets in infinite-dimensional spaces (1974).
    • In 1975, in collaboration with Greg Knowles, Kluvanek published the monograph Vector measures and control systems.
    • This is a monograph on the geometry of the range of a vector measure and applications to control systems governed by partial differential equations.
    • Completely self-contained, the book starts with a thorough discussion of integration with respect to a vector measure.
    • Then the authors employ a time-honoured method for the study of the properties of a vector measure by studying the properties of the mapping induced by a vector measure as an operator on the space of its integrable function.
    • After this, they introduce and study the concept of a closed vector measure with its value in a locally convex space.
    • Every vector measure whose range is in a Banach space is a closed measure, and the authors show that closed vector measures have a structure rich enough for intrinsic interest and applications.
    • This splendid book gives much more extensive treatment of the range of a vector measure than other available books ..
    • Moreover, it opens the new area of applications of vector measures to control systems governed by partial differential equations.
    • It is a must for anyone interested in the theory of vector measures.
    • The review [',' J J Uhl, Jr, Review: Vector measures and control systems by Igor Kluvanek and Greg Knowles, Bull.
    • In addition to bridging the gap between pure and applied mathematics, it is the definitive work on the range of a vector measure.
    • Indeed, to express the required solutions in integral form one may have to integrate with respect to a vector-valued measure of infinite total variation.

  8. César-François Cassini de Thury biography
    • Cassini gained extremely valuable experience in assisting his father Jacques Cassini on his project to measure the perpendicular to the meridian from Saint-Malo to Strasbourg.
    • Measurements by Jacques Cassini, however, had supported the elongation theory, but there were others in France who supported the Newtonian view.
    • In 1733 Jacques Cassini and his son, assisted by other scientists, measured the perpendicular to the Paris meridian from Paris west to Saint-Malo.
    • Cesar-Francois Cassini, although only nineteen years old at the time, addressed the Academie des Sciences in 1733 on the importance of the geodesic measurements he was carrying out with his father.
    • The results of the survey seemed to support the views of Jacques Cassini but, after a while, the opponents to his theory in the Academy planned expeditions to Peru, led by Bouguer and La Condamine, in 1735, and Lapland, under Maupertuis in 1736, to measure the length of a meridian degree and to settle the argument.
    • Cesar-Francois Cassini on the other hand looked to extend the scope of the measurements which had been made in France and, in 1735-36, he directed observations for a more general survey of the country.
    • It relied on the fact that if one built up a system of triangles, each successive one standing on one side of the previous one, then measuring the angles and finding an accurate distance for the length of just one side of one triangle was sufficient to give accurate measurements of the sides of all the triangles.
    • The stations from which the observer took angular measurement of adjacent stations were the vertices of the triangles.
    • He began with a preliminary survey in 1740 when he reported that he had set up 400 triangles on eighteen accurately measured bases and would use these to produce his first map of France.
    • History Topics: The history of measurement .

  9. Dominique Cassini biography
    • Cassini had Mechain as his assistant but he took control and made the measurements with the repeating circle while Mechain was given the task of checking the results with older equipment.
    • At this time the Academie des Sciences was setting up its project to accurately measure the meridian from Dunkerque to Barcelona in order to obtain an accurate value for the metre which was to be defined as one ten millionth of the distance from the North Pole to the equator.
    • Although Cassini de Thury had surveyed almost exactly this in 1740, it was the invention of the Borda repeating circle which made the Academy confident that a new much more accurate measurement could be achieved.
    • Will you again measure the meridian your father and grandfather measured before you? Do you think you can do better than they? .
    • My father and grandfather's instruments could but measure to within fifteen seconds; the instrument of M Borda here can measure to within one second.
    • He was supposed to set out to measure the northern part of the meridian, but he was reluctant to do so now that he had to bring up his five young children himself.
    • How can I recognise myself in the changes they have wrought in our old ways of calculating, our old measures, when we had not ten hours in a day, but twenty-four, and no circles of four hundred degrees ..

  10. Nicolas-Louis de Lacaille (1713-1762)
    • Their objective was to revise earlier measures by Dominique Cassini (1690) and Jean Picard (1669-70).
    • The measurements of the meridian confirmed Newton's idea that the Earth was a sphere flattened at the poles, as did the results from the expeditions sent to Peru and Lapland in 1735 and 1736.
    • More than any other astronomer of his time, [Lacaille] saw the value to science of making accurate measurements of the stars and planets.
    • In 1752, he performed a measure of the arc of the meridian between Klipfontein and Cape Town.
    • The latitudes of the end stations were measured using a zenith sector.
    • Sir Thomas Mclear later revised this measure between 1838 and 1847 and was able to confirm that the error was not due to measurement errors by Lacaille for his baseline and latitudes were remarkably accurate.

  11. Jean Picard (1620-1682)
    • Picard devised a micrometer to measure the diameters of celestial objects such as the Sun, Moon and planets.
    • Picard greatly increased the accuracy of measurements of the Earth, using Snell's method of triangulation.
    • He measured the length of the arc of the meridian; the measurements appear in Mesure de la Terre Ⓣ (1671).
    • He began by accurately measuring his baseline from Villejuif to Juvisy-sur-Orge then, using thirteen triangles, measured by triangulation one degree of latitude along the Paris Meridian from Malvoisine, in the southern suburbs of Paris, to the clock-tower of Sourdon near Amiens.
    • Finally he measured the altitudes of stars.
    • Picard was also involved with the measurement of the length of the second pendulum.
    • Also at the Paris Observatory, Picard tried to measure the parallax of nearby stars and so verify the fact that the Earth orbits the sun.

  12. Eratosthenes (276 BC-194 BC)
    • If, good friend, thou mindest to obtain from any small cube a cube the double of it, and duly to change any solid figure into another, this is in thy power; thou canst find the measure of a fold, a pit, or the broad basin of a hollow well, by this method, that is, if thou thus catch between two rulers two means with their extreme ends converging.
    • In the field of geodesy, however, Eratosthenes will always be remembered for his measurements of the Earth.
    • Eratosthenes made a surprisingly accurate measurement of the circumference of the Earth.
    • Details were given in his treatise On the measurement of the Earth which is now lost.
    • Several of the papers referenced, for example [',' B R Goldstein, Eratosthenes on the ’measurement’ of the earth, Historia Math.
    • In it Rawlins argues convincingly that the only measurement which Eratosthenes made himself in his calculations was the zenith distance on the summer solstice at Alexandria, and that he obtained the value of 7°12'.
    • Eratosthenes also measured the distance to the sun as 804,000,000 stadia and the distance to the Moon as 780,000 stadia.
    • Ptolemy tells us that Eratosthenes measured the tilt of the Earth's axis with great accuracy obtaining the value of 11/83 of 180°, namely 23° 51' 15".

  13. Herbert Federer biography
    • He made important contributions to both measure theory and to the foundations of mathematics.
    • In 1914 Caratheodory defined a k-dimensional measure in Rn in which he proved that the length of a rectifiable curve coincides with its one-dimensional measure.
    • In 1919 Hausdorff, developing Caratheodory's ideas, constructed a continuous scale of measures.
    • After this, it became obvious that area should be regarded as a two-dimensional measure and should establish the well-known integral formulas associated with area.
    • It asks for the type of multiplicity function that, when integrated over the range of f with respect to Hausdorff measure, will yield the Lebesgue area of f.
    • Federer is most famous, however, for his book Geometric measure theory (1969).
    • During the last three decades the subject of geometric measure theory has developed from a collection of isolated special results into a cohesive body of basic knowledge with an ample natural structure of its own, and with strong ties to many other parts of mathematics.
    • Recently the methods of geometric measure theory have led to very substantial progress in the study of quite general elliptic variational problems, including the multidimensional problem of least area.
    • This book aims to fill the need for a comprehensive treatise on geometric measure theory.
    • Casper Goffman reviews the book in [',' C Goffman, Review: Geometric measure theory, by Herbert Federer, Bull.
    • 'Geometric measure theory' is a great book which should have a profound influence on the development of mathematics in the next few decades.
    • Federer's timely and beautiful book indeed fills the need for a comprehensive treatise on geometric measure theory, and his detailed exposition leads from the foundations of the theory to the most recent discoveries.
    • All these characteristics are evident in his seminal book 'Geometric Measure Theory'.
    • Its appearance in 1969 was timely, as it brought together earlier studies of geometric Hausdorff-type measures, work on rectifiability of sets and measures of general dimension, and the fast developing theory of geometric higher-dimensional calculus of variations.

  14. Giovanni Cassini biography
    • In July 1664 he measured the period of rotation of Jupiter on its axis, discovered the bands and spots on the planet, and saw that the planet was flattened at its poles.
    • In 1666 he measured the period of rotation of Mars on its axis, getting a value within three minutes of the correct one, and observed surface features.
    • While French expeditions measured the longitudes of numerous places, Cassini remained in Paris coordinating their data and making his own measurements.
    • In 1672 Jean Richer made measurements of Mars from Cayenne, French Guyana, while Jean Picard and Cassini made measurements in Paris.
    • Another measurement made by Jean Richer, namely that a pendulum with a period of one second is shorter in Cayenne than Paris, led him to explain this by suggesting that the Earth was flattened at the poles.
    • The project was begun in 1683 with Cassini making measurements from Paris towards the south, while Philippe de La Hire began making measurements north from Paris.
    • They made measurements of the meridian from Paris to Perpignan, which is 13 km west of the Mediterranean coast.

  15. Heron of Alexandria (about 10-about 75)
    • gave measurements of plane figures which agree with the formulas used by Heron, notably those for the equilateral triangle, the regular hexagon (in this case not only the formula but the actual figures agree with Heron's) and the segment of a circle which is less than a semicircle ..
    • Metrica which gives methods of measurement.
    • Stereometrica measures three-dimensional objects and is at least in part based on the second chapter of the Metrica again based on examples.
    • Mensurae measures a whole variety of different objects and is connected with parts of Stereometrica and Metrica although it must be mainly the work of a later author; .
    • In Book II of Metrica, Heron considers the measurement of volumes of various three dimensional figures such as spheres, cylinders, cones, prisms, pyramids etc.
    • After the measurement of surfaces, rectilinear or not, it is proper to proceed to solid bodies, the surfaces of which we have already measured in the preceding book, surfaces plane and spherical, conical and cylindrical, and irregular surfaces as well.

  16. Jérôme Lalande (1732-1807)
    • A project had been set up to measure the parallax of the moon and Mars and hence determine their distances.
    • Again observations taken from different places on the Earth of the precise timing of the passage of the planet in front of the sun would allow parallax measurements to be made and the distance from the Earth to the sun to be calculated.
    • He gave a speech which allowed him a platform to express his atheist views and he also took the opportunity to try to moderate the extreme form of patriotism which was gripping France (quoted in [',' K Alder, The measure of all things (London, 2002).','2]):- .
    • First his appearance is described in [',' K Alder, The measure of all things (London, 2002).','2] as follows:- .
    • Many laughed at his atheist views saying that they were his revenge against God who had made him so ugly [',' K Alder, The measure of all things (London, 2002).','2]:- .
    • He was in the late 1790s [',' K Alder, The measure of all things (London, 2002).','2]:- .
    • In this work he said (quoted in [',' K Alder, The measure of all things (London, 2002).','2], see also [',' T L Hankins, Biography in Dictionary of Scientific Biography (New York 1970-1990).
    • Michel Lefrancais assisted Delambre in his measurements of the meridian in the 1790s and played an important role in that project.
    • History Topics: The history of measurement .

  17. Nevil Maskelyne biography
    • Maskelyne had earlier proposed that the same expedition should try to measure the parallax of the star Sirius.
    • This Venus transit was important since accurate measurements would allow the distance from the Earth to the Sun to be accurately measured and the scale of the solar system determined.
    • Sadly, the 6 June was cloudy and he was unable to make measurements of the transit.
    • They agreed a compromise in terms of who made the measurements and the trials were carried out.
    • Although he did not visit North America, Maskelyne was very much involved in astronomical and surveying measurements made by Charles Mason and Jeremiah Dixon in Pennsylvania during 1765-68 (see [',' T D Cope, ’A Clock Sent Thither by the Royal Society’, Proc.
    • Nevil Maskelyne measures the Earth's density .

  18. François Arago biography
    • Delambre and Mechain measured the meridian from Dunkerque (Dunkirk) to Barcelona between 1792 and 1798.
    • In September 1801 the Bureau des longitudes requested that an expedition be sent to extend the meridian measurements south of Barcelona.
    • In fact they were asked to carry out a second scientific task at the same time for they were asked to use a pendulum to measure the force of gravity at the various locations on their travels so that the data might be used for a more accurate estimate of the exact shape of the Earth.
    • Biot fled back to France but Arago remained on Mallorca, disguised as a Spaniard, trying to complete his measurements which he had recorded in a logbook.
    • Arriving back in Paris with his logbook containing the measurements he was treated as a hero.
    • Working with Biot, Arago made measurements of arc length on the Earth which led to the standardisation of the metric system of lengths.
    • We described above the adventures he had in taking readings in the south, but later, in 1821, the two extended their results to the north making measurements of the force of gravity using a pendulum in Scotland at Leith, near Edinburgh, and in the Shetland Islands.
    • Other contributions made by Arago include work on the polarization of light, investigations of the solar corona and chromosphere, and measurements of the diameters of the planets.

  19. Pierre Bouguer biography
    • In April 1735 Bouguer set out on an expedition, organised by the Academie Royale des Sciences, to Peru to measure the length of a degree of meridian at the equator.
    • Bouguer was the first to attempt to measure the density of the Earth using the deflection of a plumb line due to the attraction of a mountain.
    • Together with La Condamine, he made measurements in Peru in 1740 publishing his results in La Figure de la terre Ⓣ (1749).
    • In 1741 Bouguer discovered a small error in the joint measurements made with La Condamine to determine the length of a degree of meridian.
    • All three made independent measurements, the work being completed in 1743.
    • Bouguer wrote on naval manoeuvres and navigation and, in ship design, derived a formula for calculating the metacentric radius (a measure of ship stability).
    • Starting in 1721 he made some of the earliest measurements in astronomical photometry.
    • By moving the candle and using Kepler's inverse square law he was able to measure brightness.

  20. Ambrose Rogers biography
    • At University College, Rogers was promoted to reader before he moved to Birmingham University in 1954 as the Mason Professor of Pure Mathematics [',' S Abbott, Review: Hausdorff Measures, by C A Rogers, The Mathematical Gazette 83 (497) (1999), 362.','1]:- .
    • In collaboration with Geoffrey Shephard and James Taylor during that period his interest in convex geometry and Hausdorff Measure Theory widened.
    • His later work covered a wide range of different topics in geometry and analysis including Borel functions, Hausdorff measure and local measure, topological properties of Banach spaces and upper semicontinuous functions.
    • Rogers has written three important books, Packing and Covering in 1964, Hausdorff Measures in 1970, and (with John E Jayne) Selectors (2002).
    • Edwin Hewitt, reviewing Hausdorff Measures writes:- .
    • Mature mathematicians who need facts about Hausdorff measures for their own purposes - surface theory, harmonic analysis, and so on - will find clear statements, clear proofs, and abundant references for further pursuit.
    • Since about 1958, Professor Rogers' main research interests moved to the theory of Hausdorff measures, of analytic sets and of general convex bodies, to all of which he has made important contributions.

  21. Gabriel Mouton (1618-1694)
    • His most famous work Observationes diametrorum solis et lunae apparentium Ⓣ published in 1670 studied interpolation and a standard of measurement based on the pendulum.
    • This volume contains interesting memoirs on interpolations and on the project of a universal standard of measurement based on the pendulum.
    • In this work Mouton became the first to propose the decimal system of measurement based on the size of the earth.
    • He also suggested a standard linear measurement, which he called the mille, based on the length of the arc of one second of longitude at the equator on the Earth's surface and divided decimally.
    • The virga was quite close to the ancient French measure of a toise or 6 pieds (feet).
    • Certainly one could not measure the circumference of the earth, so he proposed a standard based on the length of a pendulum.
    • Mouton stated that there was a marvellous regularity in nature which made a metric system of measurement based on nature fit in with human activity.
    • Mouton's proposed standard of measurement was taken seriously, at least at the theoretical level, and Jean Picard strongly supported him, as did Huygens in 1673.
    • History Topics: The history of measurement .

  22. Gustave Choquet biography
    • Here he gained his first introduction to measure theory and to basic general topology.
    • He made important contributions to a variety of fields: topology, measure theory, descriptive set theory, potential theory, and functional analysis.
    • Professor Gustave Choquet, who was born in 1915, has made major contributions to a wide range of topics in analysis, potential theory, functional analysis, measure theory and infinite dimensional convexity.
    • Whereas Lebesgue's classical integration theory is based on s-additive measures, there have been made, during the second half of the 20th century and often independently, many attempts to depart from additivity of the measure, guided by requirements of different applications.
    • a monotone non-additive measure is called Choquet integral.
    • It contains the essentials of non-additive measure theory, especially the theory of infinity-alternating set functions and their dual, totally monotone ones, which later have been called belief functions as well.
    • In this context he investigated, in its dual version, the Mobius transform of a non-additive measure under topological assumptions.
    • Infinite dimensional measures and problem solutions.

  23. Jacques Cassini (1677-1756)
    • In 1700 Cassini's father undertook a project to measure the meridian from Paris to Perpignan, which is 13 km west of the Mediterranean coast.
    • He undertook the measurement of the Paris meridian north to Dunkerque in 1718.
    • However, those like Maupertuis who believed that the Earth was flattened at the poles argued ever more strongly against Cassini's theory and, in an attempt to gain further evidence to support his case, Cassini organised another project in 1733, this time to measure the perpendicular to the meridian from Saint-Malo to Strasbourg.
    • By 1738 the geodesic measurements carried out in Peru by Bouguer and La Condamine in 1735 and Lapland by Maupertuis in 1736 to measure the length of a meridian degree had produced very strong evidence for the flattening at the poles.
    • Here for the first time the ancient belief in the unchanging sphere of the stars had been shown to be incorrect by direct measurement.

  24. Claude-Louis Mathieu biography
    • Later he was sent with Jean-Baptiste Biot to continue the measurements of the arc of the meridian which had been carried out by Delambre and Pierre Mechain until the death of the latter.
    • After Arago and Biot left Paris in 1806 to begin their measurements in Spain, Mathieu was appointed as secretary of the Paris Observatory to replace Arago.
    • In 1808 Mathieu, together with the astronomers Alexis Bouvard and Jean Burckhardt, worked on reducing the data obtained by Arago and Biot in their extension of the meridian measurements to the Balearic Islands.
    • In the same year, together with Biot, he embarked on a series of measurements of the length of the seconds pendulum at different points on the meridian, in particular at Bordeaux and at Dunkirk.
    • Between August 1812 and November 1813, Mathieu and Arago carried out numerous observations on the star 61 Cygni, attempting to measure its parallax.
    • There was much resistance from the French people, however, and the old measures were still used in some parts of France into the 20th century.

  25. Andrew Gray (1847-1925)
    • showed them how to measure the distance of conspicuous objects out of doors by means of a measured baseline.
    • In this way Gray, when a boy, measured the distance of Nelson's monument on the Calton Hill, the lighthouse on the island of Inchkeith, the Martello tower at Leith Harbour, North Berwick Law, and other objects that can be seen from Burntisland.
    • Among his books are Absolute Measurements in Electricity and Magnetism (1883), Theory and Practice of Absolute Measurements in Electricity and Magnetism (Vol.

  26. Mollie Orshansky biography
    • Yet there is an underlying disquietude reflected in our current social literature, an uncomfortable realization that an expanding economy has not brought gains to all in equal measure.
    • It was at this time she did the work for which she is best known, in particular devising in 1963 the Orshansky index, which is the official measure of poverty used by the U.S.
    • First she made very clear that it was a threshold to measure inadequate incomes, not adequate ones.
    • Over the following years until her retirement in 1982, Orshansky continued to apply statistics to measures of poverty.
    • For example she wrote Children of the Poor (1963), Counting the Poor: Another Look at the Poverty Profile (1965), Who's Who Among the Poor: A Demographic View of Poverty (1965), How poverty is measured (1969), History of the Poverty Line (1970), and was a joint author of Measuring Poverty: A Debate (1978) and Improving the Poverty Definition (1979).

  27. Charles-Marie de La Condamine (1701-1774)
    • In April 1735 La Condamine set out on the expedition to Peru to measure the length of a degree of meridian at the equator.
    • specially suited to fix the endpoints of the measurement that had been the foundation of all our geographical and astronomical observations.
    • In 1741 Bouguer discovered a small error in their joint measurements and the two fell out when Bouguer refused to allow La Condamine to recheck the results.
    • All three made independent measurements, the work being completed in 1743.
    • Neil Safier writes in [',' N Safier, Myths and measurements, in Jordana Dym and Karl Offen (eds.), Mapping Latin America: A Cartographic Reader (University of Chicago Press, 2011), 107-109.','17]:- .

  28. Anders Celsius biography
    • The Academie des Sciences had decided to send two expeditions, one to Peru near the Equator and one to the Arctic Circle, to measure the length of one degree of the meridian arc.
    • There, they measured a network of triangles along the river Torne (oriented north to south).
    • In it, Celsius shows that, in his previous measures of the meridian in France, Cassini had committed bigger mistakes than the ones he was accusing them of having made.
    • This was confirmed by a later measurement of the French meridian and the results of the Peru expedition confirmed the findings of the Lapland expedition.

  29. Banu Musa brothers biography
    • The treatise considers problems similar to those considered in the two texts by Archimedes, namely On the measurement of the circle and On the sphere and the cylinder.
    • They were instructed by al-Ma'mun to measure a degree of latitude and they made their measurements in the desert in northern Mesopotamia.
    • Muhammad and Ahmad measured the length of the year, obtaining the value of 365 days and 6 hours.

  30. Al-Biruni biography
    • Al-Biruni was amazingly well read, having knowledge of Sanskrit literature on topics such as astrology, astronomy, chronology, geography, grammar, mathematics, medicine, philosophy, religion, and weights and measures.
    • He introduced techniques to measure the earth and distances on it using triangulation.
    • Not all, however, were measured by al-Biruni himself, some being taken from a similar table given by al-Khwarizmi.
    • Topics in physics that were studied by al-Biruni included hydrostatics and made very accurate measurements of specific weights.

  31. Marina Ratner biography
    • During her years in Jerusalem she continued to undertake research on geometrical dynamical systems, corresponding with Rufus Bowen at the University of California at Berkeley who was producing innovative results on measures associated with dynamical systems.
    • This structure carries along with it some measure-theoretic structure, but on the surface it would seem that the measure-theoretic description of these horocycle flows contains little information about them.
    • The surprise is that, in fact, the measure-theoretic description contains everything! ..
    • The latter actions through random and chaotic from a dynamical point of view, seem to be rigidly linked to the algebraic structure of the underlying homogeneous space: their ergodic invariant measures and orbit closures have an algebraic nature.
    • Based on her previous work on horocycle flows, she completely proved the "topological Raghunathan conjecture" through the "measure-theoretic Raghunathan conjecture".

  32. Heinz Bauer biography
    • His main interests at this time were in measure and integration, and in the work submitted for his habilitation he studied an abstract Riemann integral, introduced by L H Loomis, from the point of view of the theory of Radon measure.
    • Chatterji [',' S D Chatterji, The work of Heinz Bauer in measure and integration, Selecta (de Gruyter, Berlin, 2003), 1-10.','2] writes:- .
    • The first part is a standard development of measure theory, containing three chapters dealing with measure theory, integration theory, and product measure spaces in that order.
    • Generally speaking, only probability theory as it pertains to product measure spaces is discussed.
    • The first half of this volume consists of the 1964 text, and it is followed by a section on measure in topological spaces and on the Fourier transform.
    • An English version Probability Theory and Elements of Measure Theory was published in 1972.
    • Because of the great popularity the book enjoyed, an extensive reworking and expansion of the sections on probability appeared in English translation as Probability theory in 1996, with the same treatment was given to the sections of measure theory, published in English translation as Measure and integration theory in 2001.
    • This was Mass- und Integrationstheorie Ⓣ (1990) which provided an introduction to measure theory and the theory of integration.

  33. Wilhelm Weber (1804-1891)
    • In 1832 Weber and Gauss published a joint paper which introduced absolute units of measurement of magnetism for the first time.
    • Before this major advance, measurements were made with a pre-calibrated magnetic instrument and were not properly reproducible.
    • Equally important was Weber's later work extending these ideas on magnetic measurements to electrical measurements which we mention again below.
    • Gauss and Weber jointly published Atlas Des Erdmagnetismus: Nach Den Elementen Der Theorie Entworfen Ⓣ in 1840 which contains magnetic maps constructed using a network of magnetic observatories which they had organized from 1836 onwards to correlate measurements of terrestrial magnetism around the world.
    • Until Weber's work there had been no such thing as electrical measurements.

  34. Edgar Raymond Lorch biography
    • While I lived in Szeged, I published a paper on functions of self-adjoint transformations in which I replaced the previous definition by bilinear forms and Lebesgue-Stieltjes integration with a theory of measure determined by a resolution of the identity, where the measure of a set is a closed linear manifold.
    • This measure has the virtues that the measure of the intersection and union of sets is the intersection and union of the measures, and the measure of a set essentially determines the set.

  35. Jean-Baptiste Biot biography
    • They achieved a height of 4000 metres and measured magnetic, electrical, and chemical properties of the atmosphere at various heights.
    • On 3 September of that year he set out with Francois Arago to Formentera, in the Balearic Islands, to complete earlier work begun there on calculating the measure of the arc of the meridian.
    • They were still undertaking measurements when, in May 1808, Napoleon declared his brother Joseph Bonaparte as Spanish ruler and the War of Independence began.
    • Later in 1808, together with Claude-Louis Mathieu, he embarked on a series of measurements of the length of the seconds pendulum at different points on the meridian, in particular at Bordeaux and at Dunkirk.

  36. Kazimierz Urbanik biography
    • Urbanik then began to mix an interest in topology with measure theory and probability and his 1954 papers show this mix: Sur un probleme de J F Pal sur les courbes continues Ⓣ; Limit properties of homogeneous Markov processes with a denumerable set of states; Sur la structure non-topologique du corps des operateurs Ⓣ; and Quelques theoremes sur les measures Ⓣ.
    • This paper is motivated by the measure-theoretic formulation of the Steinhaus cake problem.
    • In the author's mathematization of the problem it is assumed that the n normalized measures that enter are non-atomic, have the same null-sets, and are not all identical.
    • ','3] divides Urbanik's research into five different major areas: topology, measure theory and analysis; probability theory; stochastic processes; information theory and theoretical physics; and general algebras.

  37. Hans Hahn biography
    • He also studied the theory of ordered abelian groups and ordered fields, initiating the theory in 1907 (see for example [',' P Ehrlich, Hahn’s ’Uber die nichtarchimedischen Grossensysteme’ and the development of the modern theory of magnitudes and numbers to measure them, in From Dedekind to Godel, Boston, MA, 1992 (Kluwer Acad.
    • Another area on which Hahn did research was measure theory.
    • In this area he studied a construction of the Lebesgue integral as a limit of Riemann sums, an integral proposed by Borel around 1910, and worked on the theory of abstract measures, in particular product measures.
    • The book is divided into an introduction (containing an exposition of the relevant parts of set theory and point set topology) and five chapters, entitled (I) Additive and totally additive set functions, (II) Measure, (III) Measurable functions, (IV) Integration and (V) Differentiation.

  38. Willebrord Snell (1580-1626)
    • made a great advance over the methods used by his predecessors by introducing trigonometrical methods in the measurement of distances across country.
    • In this work Snell attempted to measure the circumference of the earth and so required a considerable number of measurements.
    • His measurements were surprising accurate allowing him to deduce a good value for the radius of the earth.

  39. Francis Galton (1822-1911)
    • In 1884-85 the International Health Exhibition was held and in connection with this Galton set up a laboratory to measure human statistics.
    • He collected data such as height, weight, and strength of a large number of people devising himself the apparatus used to make the measurements.
    • He defined an index of correlation as a measure of the degree to which the two were related.
    • However, when there are more than two measures which were correlated, he failed to understand the complexity of the mathematics involved.

  40. Arthur Eddington (1882-1944)
    • He remained on Principe Island to develop the photographs and to try to measure the deviation in the stellar positions.
    • The cloud made the plates of poor quality and hard to measure.
    • one plate I measured gave a result agreeing with Einstein.
    • Oh leave the Wise our measures to collate .

  41. Walter Shewhart (1891-1967)
    • His writings were on statistical control of industrial processes and applications to measurement processes in science.
    • Chapter III explores the presentation of measurements of physical properties and constants.
    • Among the topics considered are measurements presented as original data, characteristics of original data, summarizing original data (both by symmetric functions and by Chebyshev's theorem), measurement presented as meaningful predictions, and measurement presented as knowledge.

  42. Tycho Brahe biography
    • He published his account in De mundi aetherei recentioribus phaenomenis (1588) where he draws cosmological conclusions from the fact that his measurements show that the comet is not closer to Earth than the Moon, contradicting Aristotle's model of the cosmos.
    • The problem was, of course, that in the Sun centred model of Copernicus a parallax shift should be observed but despite his attempts to measure such a shift, Tycho could detect none.
    • The first measurement of the parallax of a star was in 1838 by Bessel who found 0.3" for the parallax of 61 Cygni.
    • Despite the quality of Tycho's measurements, this value in about 100 times smaller that Tycho's observational errors.

  43. Samuel Molyneux (1689-1728)
    • After this the two scientists set about trying to measure the parallax of a star, the one final step which was required to prove that the earth orbited the sun and was also a necessary step in deducing a scale for the universe.
    • Hooke had claimed to have measured the parallax of the star Gamma Draconis and their aim was to verify Hooke's result.
    • In fact they showed that Hooke was wrong and, having failed to measure the parallax of Gamma Draconis, they at least had shown that Hooke's value for the parallax was incorrect.
    • In fact it would be more than 100 years later before Wilhelm Bessel made the first successful measurement of the parallax of a star.

  44. Andre-Louis Cholesky biography
    • Delambre had completed his baseline measurements in the spring of 1798 as part of his contribution to defining the metre.
    • In 1882 French experts returned to Delambre's baseline but did not re-measure it, preferring to check Delambre's calculations indirectly by triangulation.
    • He measured a baseline in the Kavousi plain and used astronomical measurements to determine the precise position of the southern end of his baseline.

  45. Charles-Étienne Camus biography
    • Jean Picard had measured the length of the arc of the meridian, the measurements appear in Mesure de la Terre Ⓣ (1671).
    • With Bouguer, Pingre and Cassini de Thury, Camus was involved in another Academie des Sciences project relating to further work on the measurement of the earth.
    • treatment of toothed wheels and their use in clocks, studies of the raising of water from wells by buckets and pumps, an evaluation of an alleged solution to the problem of perpetual motion, and works on devices and standards of measurement.

  46. Charles Boys (1855-1944)
    • One application he made of [quartz fibres] was to the suspension of the moving system of his radiomicrometer for the measurement of radiant heat, an instrument so sensitive that, aided by a reflecting telescope to bring the heat to a focus, it could detect the differences in radiation from different parts of the moon's disc and would respond to the heat of a candle at a distance of more than a mile ..
    • He was then able to use attracting masses much larger in proportion than Cavendish had been able to use, thus enabling him to make accurate measurements of the small forces involved.
    • To avoid these problems he conducted his experiments in Oxford, rather than London, but he still had to make his measurements at times when no shunting was going on in the railway yards more than a mile from his laboratory.
    • It was in this role that he worked for many years to improve the instruments used to measure the calorific value of gas.

  47. Naum Il'ich Feldman biography
    • Feldman proved in his thesis Borel type results (called the measure of transcendence) for logarithms of algebraic numbers, obtaining estimates for the lower bound depending (as did Gelfond) on both the degree of P and the maximum modulus of its coefficients.
    • In addition to his work on the measure of transcendence of numbers, Feldman also produced many results strengthening Liouville's theorem on the rational approximation of algebraic numbers.
    • For example in 1960 Feldman published two papers The measure of transcendency of the number π and Approximation by algebraic numbers to logarithms of algebraic numbers which were reviewed together by Mahler:- .
    • In these two important papers, which are closely connected, measures of transcendency for π and log α (α algebraic) are found which are far better than any obtained before.
    • Further applications of these methods to the arithmetic properties of elliptic functions, transcendence measures and algebraic independence are also given.

  48. George Udny Yule (1871-1951)
    • The correlation of lengths or measurements on portions of the body form examples of the first kind; of numbers of children in families, petals or other parts of flowers, are examples of the second.
    • Certain practical cases arise, however, where either no variation is thinkable at all, or else is not measured or possibly measured.
    • Amongst the measures of prophylaxis which need to be discussed, that of preventative inoculation is clearly of exceptional interest ..

  49. Harry Pitt (1914-2005)
    • However he took the opportunity to write two classic texts publishing Tauberian theorems in 1958 and Integration, measure and probability in 1963.
    • As the author states in the preface, the purpose of this book is to provide an introduction to the modern theory of probability and the fundamental ideas and techniques on which it is based, namely, those of measure and integration.
    • The book Integration, measure and probability appeared after he had spent the year 1962-63 as a visiting professor at Yale University.
    • Also in 1985 he published the third of his texts, namely Measure and integration for use.
    • Although of unquestioned power and practical utility, the Lebesgue Theory of measure and integration tends to be avoided by mathematicians, due to the difficulty of obtaining detailed proofs of a few crucial theorems.
    • Postgraduates in mathematics and science who need integration and measure theory as a working tool, as well as statisticians and other scientists, will find this practical work invaluable.
    • As the title implies, this book deals with measure and integration.
    • For reasons of brevity, the author derives the concepts of measure and measurability from that of the integral.
    • The exposition, however, is rigorous, clear and precise and the three chapters in the first part of the book cover the main results of measure and integration, such as the theorems of Lebesgue, Fubini and Radon-Nikodym.

  50. William Kruskal biography
    • In 1954 Kruskal began a collaboration with Leo Goodman on measures of association.
    • In 1954, prior to the era of modern high speed computers, the present authors published the first of a series of four landmark papers on measures of association for cross classifications.
    • By describing each of several cross classifications using one or more interpretable measures, they aimed to guide other investigators in the use of sensible data summaries.
    • It suggests criteria for judging measures of association and introduces several new measures with specific contextual meanings.
    • The 1963 paper derives large-sample standard errors for the sample analogues of population measures of association and presents some numerical results about the adequacy of large-sample normal approximations.
    • The 1972 paper presents a new look at the asymptotics, and provides a more unified way to derive large-sample variances for those measures of association that can be expressed as ratios of functions of the cell probabilities.
    • Thus the techniques can be used for tried and true measures, and also for ones not yet invented.
    • There he taught a new course which resulted in his paper Ordinal measures of association (1958).

  51. Sergei Fomin biography
    • He was awarded his habilitation for a dissertation On dynamical systems with invariant measure in 1951.
    • In this area he examined the theory of differentiable measures in infinite dimensional spaces and the theory of distributions.
    • He worked with a number of collaborators from 1973 on the writing of a monograph on measure theory and differential equations.
    • Some of the mathematical interests of Sergei Vasilovich were always close to some of mine (measure and ergodic theory); he supervised the translation of a couple of my books into Russian.
    • The second volume on measure, the Lebesgue integral and Hilbert space appeared in 1960.

  52. Lóránd Eötvös (1848-1919)
    • Kirchhoff taught him the importance of accurate measurements.
    • At this time Eotvos had devised an instrument to measure the constant of surface tension.
    • The first field measurements with his torsion balance were carried out on Sag Hill in Hungary in 1891.
    • One reason why his interests moved towards gravitation must have been the fact that the Hungarian Society for Natural Sciences requested, in 1881, that measurements be made to ascertain the values for the acceleration due to gravity in different locations throughout Hungary [',' L Marton, Biography in Dictionary of Scientific Biography (New York 1970-1990).','1]:- .
    • He took many photographs while on mountaineering trips, but also took photographs while on field trips making scientific measurements.

  53. John Lewis biography
    • After a revolutionary insight on how to measure Internet traffic, John persuaded visionaries from the Computer Laboratory in Cambridge and the Swedish operator Telia to join him in a three-year research contract funded by the European Commission.
    • Instead of making elaborate models of tele-traffic on broadband and calculating the rate function, I decided to measure it.
    • In 1999 the company Measure Technology Ireland was set up.
    • I will be thrilled if the company [Measure Technology Ireland] develops a successful product which embodied the original idea.
    • John made significant contributions to theoretical chemistry, quantum measurement, quantum stochastic processes, the Ising model, boson condensation and telecommunications.

  54. Donald Ornstein biography
    • Two further publications appeared in 1960, namely The differentiability of transition functions and On invariant measure.
    • Ergodic theory, the study of measure-preserving transformations or flows, arose from the study of the long-term statistical behaviour of dynamical systems.
    • The Lebesgue measure of a set does not change as it evolves and can be identified with its probability.
    • One can abstract the statistical properties (e.g., ignoring sets of probability 0) and regard the state-space as an abstract measure space.
    • Equivalently, one says that two flows are isomorphic if there is a one-to-one measure-preserving (probability-preserving) correspondence between their state spaces so that corresponding sets evolve in the same way (i.e., the correspondence is maintained for all time).
    • Measure-preserving transformations (or flows) also arise from the study of stationary processes.
    • A N Kolmogorov and Ya G Sinai solved this problem by introducing a new invariant for measure-preserving transformations: the entropy, which they took from Shannon's theory of information.
    • This immediately led to many new results about measure-preserving transformations including proofs that many transformations of physical and mathematical interest are just disguised versions of Bernoulli shifts.

  55. Chrysippus biography
    • In fact 'one' was considered as that by which things are measured.
    • a measure is not the things measured, but the measure or the One is the beginning of number.

  56. Juha Heinonen biography
    • He completed his thesis on non-linear potential theory in 1987 and in the same year his first publication appeared, namely Estimates for F-harmonic measures and Oksendal's theorem for quasiconformal mappings written jointly with his thesis advisor Olli Martio.
    • The others were the single author publications Boundary accessibility and elliptic harmonic measures and Asymptotic paths for subsolutions of quasilinear elliptic equations, and the paper On quasiconformal rigidity in plane and space written with K Astala.
    • Teichmuller used quasiconformal mappings to measure a distance between two conformally inequivalent compact Riemann surfaces, starting what is now called Teichmuller theory.
    • In the past ten years, it has become known that a full-fledged quasiconformal mapping theory exists in rather general metric measure spaces.

  57. Hippolyte Fizeau (1819-1896)
    • Fizeau, however, predicted that subtle displacements of the absorbsion lines in stellar spectra could be used to measure much smaller [than suggested by Doppler] celestial velocities, and the motion of the terrestrial observer, and this correct prediction underpins much of modern astrophysical inquiry.
    • He was thus the first to make a successful terrestrial measurement of the velocity of light.
    • We know today that the value he found was almost as accurate as Romer's astronomical measurement having an error of about 5%.
    • In 1851 Fizeau tried to measure the passage of the earth through the ether and achieved a negative result.

  58. Abraham Plessner biography
    • Plessner's theorem states that if a trigonometric series converges everywhere in a set E of positive measure, then its conjugate series converges almost everywhere in E.
    • In Eine Kennzeichunung der totalstetigen Funktionen Ⓣ (1929), Plessner characterised the absolutely continuous measures among the class of Borel measures.
    • The book is divided into six chapters, whose titles are self-explanatory: (1) The fundamental concepts of the theory of sets; (2) The measure of sets of points; (3) Functions of real variables; (4) The Lebesgue integral; (5) Functions of one and two variables; and (6) Fourier series.

  59. John von Neumann (1903-1957)
    • Polya said [',' P R Halmos, Von Neumann on measure and ergodic theory, Bull.
    • Now married to Klara the parties continued [',' P R Halmos, Von Neumann on measure and ergodic theory, Bull.
    • In his youthful work, he was concerned not only with mathematical logic and the axiomatics of set theory, but, simultaneously, with the substance of set theory itself, obtaining interesting results in measure theory and the theory of real variables.
    • It was in this period also that he began his classical work on quantum theory, the mathematical foundation of the theory of measurement in quantum theory and the new statistical mechanics.
    • Haar's construction of measure in groups provided the inspiration for his wonderful partial solution of Hilbert's fifth problem, in which he proved the possibility of introducing analytical parameters in compact groups.
    • He advanced the theory of cellular automata, advocated the adoption of the bit as a measurement of computer memory, and solved problems in obtaining reliable answers from unreliable computer components.
    • Eugene Wigner wrote of von Neumann's death [',' P R Halmos, Von Neumann on measure and ergodic theory, Bull.

  60. Fischer Black biography
    • Indeed Black's 76-formula if slightly modified, by replacing e-rT by PT(t) (the price of a T-bond), enabled the pricing of European options on anything that had a log-normal distribution for the forward price under the appropriate measure, even if interest rates were stochastic.
    • However advances in financial mathematics explored the effect of taking expectations with respect to special probabilities - the so-called forward-measure probabilities - under which the forward-price of assets (the ratio of the price of the asset to the price of the T-bond) were martingales.
    • The expected value of the forward-price at time T, was shown to be equal to the current forward-price and this enabled Black's formula to be shown to be exact under log-normal assumptions for the forward-price (of the underlying asset) under the forward-measure.
    • an upper bound for the applicable forward interest rates) on interest rates (assumed to be log-normal under the 'forward measure') and swaptions (option on the swap rate, being the coupon rate on a par bond commencing at the exercise date of the option) assuming the forward swap rate is log-normal under the so-called forward annuity measure.
    • It was recognised that, in a complete market with no free lunch, there had to be a unique martingale measure, corresponding to the so-called traded numeraire portfolio, under which the process formed by the ratio of all traded portfolios to the numeraire portfolio, was a martingale process.
    • For example, if the traded portfolio was taken as the T-bond, price PT(t), then, if a traded portfolio could replicate the option, then the ratio of the option price O(t) to PT(t) had to be a martingale process under the probability measure induced by PT(t) [Note 13: THIS LINK], if there was to be no arbitrage in financial markets.

  61. Abram Samoilovitch Besicovitch biography
    • At Cambridge Besicovitch lectured on analysis in most years but he also gave an advanced course on a topic which was directly connected with his research interests such as almost periodic functions, Hausdorff measure, or the geometry of plane sets.
    • Hausdorff, in 1918, had extended Caratheodory's theory of measure to sets having finite measure of non-integral order.
    • Besicovitch, around 1930, extended his density properties of sets to those of finite Hausdorff measure.
    • Other areas on which Besicovitch worked included geometric measure theory, Hausdorff measure, real function theory, and complex function theory.
    • in recognition of his outstanding work on almost-periodic functions, the theory of measure and integration and many other topics of the theory of functions.

  62. George Airy (1801-1892)
    • I had now in some measure taken science as my line (but not irrevocably) and I thought it best to work it well for a time at least and wait for accidents.
    • Airy was made chairman of the Commission set up to construct Standard Weights and Measures in 1834.
    • History Topics: The history of measurement .

  63. Florence Nightingale David biography
    • For example in 1972 David published Measurement of diversity and we present her own summary of the work:- .
    • The several measurements used by ecologists to measure diversity in plant and animal populations have been summarized by E C Pielou.

  64. Frederick Mosteller biography
    • In his second year he took courses on calculus, mechanics, French, quantitative analysis, and physical measurements.
    • A Second Course in Statistics by Fredrick Mosteller and John W Tukey, Journal of Educational Measurement 16 (1) (1979), 60-61.','23]:- .
    • In the course of presenting these techniques, they offer new perspectives on the formulation of statistical and data analytical problems, the nature of uncertainty and the need for cross-validation, the need for multiple analyses of the same data, ways in which the data can guide the analysis, the importance of robust and resistant measures, and similar practical and philosophical problems.

  65. Yurii Vasilevich Prokhorov biography
    • This course covered the foundations of functional analysis, measure theory and the theory of orthogonal series.
    • He then began to look at Probability distributions in functional spaces, publishing a paper with this title in 1953 which examined sequences of probability measures on the Banach space of continuous functions on a compact interval, and in 1956 he submitted his dissertation for his Doctorate in the Physical and Mathematical Sciences.
    • He proposed new methods for studying limit theorems for random processes; these methods were based on studying the convergence of measures in function space.
    • Chapter headings (slightly truncated) are as follows: (I) Basic concepts of the elementary theory; (II) Spaces and measures; (III) Foundations; (IV) Limit theorems; (V) Markov processes; (VI) Stationary processes.

  66. Lai-Sang Young biography
    • They show that orbits from a subset of the basin of attraction of positive measure have a common distribution in the limit ..
    • This suggests the existence of a natural invariant measure, one that governs the asymptotic distribution of almost all points in the basin of attraction.
    • After giving details of how this measure was discovered, and her own contributions, but she explained that there had been an embarrassing lack of examples.
    • measurements of dynamical complexity, including entropy, Lyapunov exponents and fractal dimension; .

  67. Robert Woodward biography
    • These photographs were deposited with the Naval Observatory, where they were carefully measured, and the data of measurement were used in computations to obtain a more nearly correct value of parallax than had hitherto been possible.
    • For example he published Geodesy: On the measurement of the base lines at Holton, Indiana and at St Albans, West Virginia 1891 and 1892 (1893) which illustrates the type of work he was undertaking at this time.
    • Among his publications during his time at Columbia University we mention: The century's progress in applied mathematics: Presidential address to the American Mathematical Society (1900), Mathematical Theories of the Earth (1900), Observation and experiment: New York Academy of Sciences (1901), Measurement and calculation: Presidential address to the New York Academy of Sciences (1902), The Unity of Physical Science (1904), and Academic ideals: Address at the opening of Columbia University (1904).

  68. Joseph Doob biography
    • Doob's work was in probability and measure theory, in particular he studied the relations between probability and potential theory.
    • probability theory is simply a branch of measure theory, with its own special emphasis and field of application ..
    • Doob is also the author of a well known book on measure theory published in 1994 when he was 84 years old.
    • what measure theory every would-be analyst should learn.
    • this text, written by one of the most illustrious probabilists alive, is an interesting addition to the textbook literature in measure theory; every serious mathematical library should acquire it and teachers of measure theory - especially those who are analysts by profession - should not fail to consult it for their future courses.

  69. Alfréd Haar (1885-1933)
    • In 1932 he introduced an invariant measure on locally compact groups, now called the Haar measure, which allows an analogue of Lebesgue integrals to be defined on locally compact topological groups.
    • At first, however, von Neumann tried to discourage Haar in seeking such a measure since he felt certain that no such measure could exist.
    • They call it Haar measure - .

  70. Francis Edgeworth biography
    • They were applied to the measure of utility, the measure of ethical value, the measure of evidence, the measure of probability, the measure of economic value, and the determination of economic equilibria.

  71. Li Zhi (1192-1279)
    • It was here, in 1248, that he completed his most famous work the Ce yuan hai jing (Sea mirror of circle measurements).
    • We are lucky to have any works by Li Zhi other than the Sea mirror of circle measurements for he told his son to burn all his books except the Sea mirror of circle measurements on his death since this was the only work of which he was proud.
    • By any standards the Sea mirror of circle measurements is a most remarkable work.
    • It is thought by many historians to have been written because people found understanding the Sea mirror of circle measurements was beyond them.

  72. Zoárd Geöcze biography
    • He spent 1908 in Paris where he learnt of the effective theory of the measure of sets of points being developed by Borel, Baire and Lebesgue.
    • See [',' B Szenassy, Zoard Geocze’s mathematical life-work and recent results of surface measurement, A Szent Istvan Akademia Ertesitoje (1943), 118-142.','4] for a description of Geocze's work on surface measurement and a discussion of how his ideas have been taken forward.

  73. Pierre-Louis Moreau de Maupertuis (1698-1759)
    • In May 1735 the Paris Academy sent an expedition to Peru to make measurements of the Earth.
    • A second expedition was sent to Lapland headed by Maupertuis, also to measure the length of a degree along the meridian.
    • They set up base in Tornio in northern Finland and managed to make their measurements despite the problems of being attacked by insects in summer and suffering unbearably cold weather during the winter.

  74. John Pratt (1809-1871)
    • Pratt's next contribution was motivated by measurements made by George Everest published in 1847 (Mount Everest was renamed in his honour in 1865).
    • He noted that there was a small difference in this measurement compared with that made by astronomical methods, but put this down to errors in the measurements.

  75. Pierre-Simon Laplace (1749-1827)
    • Laplace was made a member of the committee of the Academie des Sciences to standardise weights and measures in May 1790.
    • The weights and measures commission was the only one allowed to continue but soon Laplace, together with Lavoisier, Borda, Coulomb, Brisson and Delambre were thrown off the commission since all those on the committee had to be worthy:- .
    • History Topics: The history of measurement .

  76. John Brinkley (1766-1835)
    • Once the instrument was installed, Brinkley began the major research of his career, namely his attempts to measure the parallax of stars.
    • He then attempted to measure their apparent movement against the faint distant stars using as a base line the diameter of the Earth's orbit round the sun.
    • However, his work and methods on this topic made an important contribution and led eventually to the measurement of the parallax of 61 Cygni by Wilhelm Bessel in 1838.

  77. Finlay Freundlich (1885-1964)
    • Freundlich worked with Einstein in 1911 attempting to make the measurements of Mercury's orbit required to confirm the general theory of relativity.
    • The only way to make such measurements at this time was during an eclipse and Freundlich wanted to journey to somewhere within the path of totality of the eclipse which would happen in 1914.
    • During this period Freundlich planned three further expeditions to observe an eclipse and measure the deflection of light passing close to the sun.

  78. Charles S Peirce (1839-1914)
    • Coast Survey was to measure the force of gravity at various sites both in the United States and abroad.
    • to work on data on gravity measurements which had been returned from the Artic.
    • He wrote on probability arguing against De Morgan's ideas that probability is a measure of confidence and also arguing against the ideas of Bayes.

  79. Edwin Hubble biography
    • In 1924 Hubble measured the distance to the Andromeda nebula, a faint patch of light with about the same apparent diameter as the moon, and showed it was about a hundred thousand times as far away as the nearest stars.
    • Hubble was able to measure the distances to only a handful of other galaxies, but he realised that as a rough guide he could take their apparent brightness as an indication of their distance.
    • The speed with which a galaxy was moving toward or away from us was relatively easy to measure due to the Doppler shift of their light.

  80. Archimedes biography
    • On plane equilibriums (two books), Quadrature of the parabola, On the sphere and cylinder (two books), On spirals, On conoids and spheroids, On floating bodies (two books), Measurement of a circle, and The Sandreckoner.
    • In Measurement of the Circle Archimedes shows that the exact value of π lies between the values 310/71 and 31/7.
    • as Archimedes was carrying to Marcellus mathematical instruments, dials, spheres, and angles, by which the magnitude of the sun might be measured to the sight, some soldiers seeing him, and thinking that he carried gold in a vessel, slew him.

  81. Vitaly Vitalievich Fedorchuk biography
    • The scientific work of Vitaly Fedorchuk covers various areas of topology and has found applications in functional analysis, differential geometry, measure theory and probability theory.
    • It starts leisurely with the notion of topological metric spaces and their hyperspaces consisting of closed subsets (with the Hausdorff distance or Vietoris topology), but then gathers momentum as it runs through function spaces (metric of uniform convergence, compact-open topology, linear spaces, norm), exponential functors (including parts on connectedness and symmetric products), multivalued mappings (with semicontinuities, selections, retractions), probability measures, spaces of partial mappings, axiomatization of solution spaces for an ordinary differential equation (with a digression on optimal control), and autonomous spaces.
    • As a result, a mature mathematician can perhaps read this text with some pleasure as a sort of a survey, but one may doubt what profit can be gained by a beginner reading about, e.g., the Riesz theorem on a measure representation of a functional, or the Poincare-Bendixson theorem on curves without self-intersections - without much explanation and no indication of where to turn for further details.

  82. John Walsh biography
    • Failing in every effort of this nature, he published at his own expense a large number of tracts, in which he endeavoured to establish his views, and denounced in no measured terms the unjust and selfish opposition which he thought that he had met with.
    • Providence seems to have in some measure vindicated the equality of its dispensations by assigning to them a double measure of hope, which serves them in the stead both of ability and of success.

  83. Levi ben Gerson (1288-1344)
    • He invented Jacob's staff, an instrument to measure the angular distance between celestial objects.
    • of a staff of 4½ feet long and about one inch wide, with six or seven perforated tablets which could slide along the staff, each tablet being an integral fraction of the staff length to facilitate calculation, used to measure the distance between stars or planets, and the altitudes and diameters of the Sun, Moon and stars.
    • We note that while Levi's method for constructing the scale is theoretically correct, it requires making measurements that seem extremely difficult, so perhaps the theory was never put into practice.

  84. Errett Bishop biography
    • Examples are extensions of Mergelyan's approximation theorem and the theorem of Frigyes Riesz and Marcel Riesz concerning measures on the unit circle orthogonal to polynomials.
    • Here Bishop worked on uniform algebras (commutative Banach algebras with unit whose norms are the spectral norms) proving results such as antisymmetric decomposition of a uniform algebra, the Bishop-DeLeeuw theorem, and the proof of existence of Jensen measures.
    • In 1972 he published Constructive measure theory written in collaboration with Henry Cheng.

  85. Joseph-Louis Lagrange (1736-1813)
    • Lagrange was made a member of the committee of the Academie des Sciences to standardise weights and measures in May 1790.
    • The weights and measures commission was the only one allowed to continue and Lagrange became its chairman when others such as the chemist Lavoisier, Borda, Laplace, Coulomb, Brisson and Delambre were thrown off the commission.
    • History Topics: The history of measurement .

  86. Mikhail Vasilevich Ostrogradski (1801-1862)
    • Not only did he measure the dimensions of his toys but he also measured the depth of wells and lengths of fields.
    • He always carried a stone in his pocket which had a long piece of string tied round it so that he could measure the depth of any well he came across.

  87. Aristarchus biography
    • He knew that the moon shines by reflected sunlight, so he argued, if one measured the angle between the moon and sun when the moon is exactly half illuminated then one could compute the ratio of their distances.
    • One has to assume Aristarchus was able to develop instruments to make accurate astronomical measurements later in his career.
    • He was showing that such measurements could be made and, since he succeeds in showing this, his work is of major importance.

  88. Tobias Mayer biography
    • It was the first map of the moon which used accurately measured positions of the craters.
    • In fact Mayer measured the positions of 24 craters, which he included in the map, using a micrometer to obtain an accuracy of 1' in latitude and longitude.
    • In fact Tobias Mayer's improvements to the reflecting circle were further developed by Jean Charle de Borda and used in the measurements of the arc of the meridian by Jean Baptiste Delambre and Pierre Mechain in their efforts to define the metre.

  89. Adolphe Quetelet (1796-1874)
    • In Sur l'homme et le developpement de ses facultes, essai d'une physique sociale Ⓣ he presented his conception of the average man as the central value about which measurements of a human trait are grouped according to the normal curve.
    • This probability may be considered as giving, in cities, the measure of the apparent tendency to marriage of a Belgian aged 25 to 30.
    • As a footnote let us mention that the internationally used measure of obesity is the Body Mass Index or Quetelet Index.

  90. Evgenii B Dynkin biography
    • He created the fundamental concept of a Markov process as a family of measures corresponding to various initial times and states and he defined time homogeneous processes in terms of the shift operators ..
    • a class of measure-valued Markov processes [which] can be used to give probabilistic solutions to certain nonlinear PDE's in a way which is analogous to the classical solution of the Dirichlet problem by means of Brownian motion.
    • This is a class of measure-valued Markov processes, which in many cases can be constructed as a suitable scaled limit of branching processes.

  91. Karl Schwarzschild (1873-1916)
    • First he saw that the photographic magnitudes which he measured differed from the visual magnitudes which had been tabulated.
    • Choosing 367 stars to measure from the Von Kuffner Observatory, he included two variable stars.
    • The range of magnitude change as measured by his photographic methods was much greater than the range of change in visual magnitude.

  92. Anaximander biography
    • We have discussed above both his attempts to measure the scale of the heavens and also to measure distances on earth with his attempts at maps.
    • These are shown in [',' A Sabo, The measurement of angles and the start of trigonometry (Bulgarian), Fiz.-Mat.

  93. Émile Borel biography
    • the theory of measure, Borel's theory of divergent series, his theory of non-analytic continuation and the theory of quasi-analytic functions all derive from ideas which make their first appearance in this paper.
    • In [',' M Frechet, La vie et l’oeuvre d’Emile Borel, Enseignement mathematique 11 (1965), 1-95.','8] Borel's mathematical work is divided into the following topics: Arithmetic; Numerical series; Set theory; Measure of sets; Rarefaction of a set of measure zero; Real functions of real variables; Complex functions of complex variables; Differential equations; Geometry; Calculus of probabilities; and Mathematical physics.
    • Borel created the first effective theory of the measure of sets of points.

  94. Raphael Weldon (1860-1906)
    • His work involved studying correlation coefficients for the relation between measurements of organs in animals and is important for the beginnings of biometry.
    • He studied shrimps making measurements of different features on shrimps from different locations.
    • He soon found that the measurements he took lay on a normal distribution.
    • He published two papers on this topic, the second investigating correlations between measurements of certain organs in the shrimps.

  95. Robert Edward Bowen biography
    • Bowen extended Gibbs's work on invariant measures associated with dynamical systems.
    • The second field of applications concerns the investigation of properties of the set of all the invariant measures for a given dynamical system and the finding of some remarkable measures in this set.
    • His pioneering studies of topological entropy, symbolic dynamics, Markov partitions, and invariant measures are of lasting importance; much of today's research is inspired by his ideas.

  96. Hugo Steinhaus biography
    • During one such walk I overheard the words "Lebesgue measure".
    • There is no doubt that none of these theories would have achieved today's level of prominence without an essential understanding of the Lebesgue measure and integral.
    • On the other hand, the ideas of Lebesgue measure and integral found their most striking and fruitful applications there in Lvov.
    • In 1923 he published in Fundamenta Mathematicae the first rigorous account of the theory of tossing coins based on measure theory.

  97. William Feller (1906-1970)
    • The calculus of probabilities is constructed axiomatically, with no gaps and in the greatest generality, and for the first time systematically integrated, fully and naturally, with abstract measure theory.
    • Measure theory is assumed.
    • Feller worked on mathematical probability using Kolmogorov's measure theoretic formulation.
    • outlines some new results and open problems concerning diffusion theory where we find an intimate interplay between differential equations and measure theory in function space.

  98. Stan Ulam biography
    • He investigated a problem which originated with Lebesgue in 1902 to find a measure on [0,1] with certain properties.
    • Banach in 1929 had solved a related measure problem, but assuming the Generalised Continuum Hypothesis.
    • It is too technical to describe: measurable cardinals, measure in set theory, abstract measure.

  99. Adriaan Cornelis Zaanen biography
    • Measure and integral, Banach and Hilbert space, linear integral equations (1953) which contained much of his own research as well as material from a lecture course by N G de Bruijn.
    • Measure and integral, Banach and Hilbert space, linear integral equations (1953), we have already mentioned above.
    • This book is a presentation in textbook form of the modern approach to the theory of integration, not however after the manner of Bourbaki, but based on measure theory of abstract sets in the Caratheodory manner, combined with the general integral concepts due in the first place to P J Daniell, and elaborated and extended by M H Stone.
    • I have sailed an intermediate course between the measure approach and the linear functional approach, fully realizing the danger that the attempt to do so may not find favour in the eyes of the extreme adherents of either school.

  100. Felix Hausdorff (1868-1942)
    • Within the mathematical work of Hausdorff the two publications devoted explicitly to measure theory occupy a significant place: they are not only important for measure theory but have also contributed fundamentally to its development.
    • It is not well known that throughout his life Hausdorff had been interested in various fundamental problems of measure and integration theory and had made important contributions at different times.
    • He studied the Gaussian law of errors, limit theorems and problems of moments, and set theory and the strong law of large numbers, which he based on measure theory.

  101. Vladimir Drinfeld biography
    • He worked on differential geometry, particularly on measure and integration.
    • He also gave a one page proof of the fact that any rotation invariant finitely additive measure on the two or three dimensional sphere is proportional to Lebesgue measure by using a clever combination of known results.

  102. Bruno de Finetti biography
    • Although the idea of probability as a measure of the observer's belief that an event will happen had already been conceived by F P Ramsey in 1926, Bruno de Finetti was unaware of Ramsey's work and, moreover, his chief interest was for coherent probability assessments and not for rational decisions; see the obituary by D V Lindley [',' D V Lindley, Obituary : Bruno de Finetti, 1906-1985, Journal of the Royal Statistical Society, Series A 149 (1986), 252.','7] for more information.
    • However, his contributions to probability and statistics do not reduce to his subjective approach and in fact they include important results on finitely additive measures, processes with independent increments, sequences of exchangeable variables and associative means; see the review by M D Cifarelli and E Regazzini [',' D M Cifarelli and E Regazzini, De Finetti’s contribution to probability and statistics, Statistical Science 11 (1996), 253-282.','4] for details on these.

  103. Wilhelm Bessel biography
    • He came to use Bradley's and Maskelyne's eighteenth-century Greenwich observations because these two astronomers were the first to provide exhaustive analyses of their own instrumental errors, along with temperature and pressure of the atmosphere through which the measurements had been made.
    • Bessel, using a Fraunhofer heliometer to make the measurements, announced his value of 0.314" which given the diameter of the Earth's orbit, gave a distance of about 10 light years.
    • During 1831-32 he directed geodetical measurements of meridian arcs in East Prussia, and in 1841 he deduced a value of 1/299 for the ellipticity of the Earth, the amount of elliptical distortion by which the Earth's shape departs from a perfect sphere.

  104. Niels Bohr biography
    • He proposed complementarity of perceptions and pictures, particle-wave, conjugate variables, quantum evolution - classical measurements etc.
    • Humanity will be confronted with dangers of unprecedented character unless, in due time, measures can be taken to forestall a disastrous competition in such formidable armaments and to establish an international control of the manufacture and use of powerful materials.

  105. Calogero Vinti biography
    • Among his most important publications during his years in Perugia, we mention On the Weierstrass integrals of the calculus of variations over BV varieties: recent results of the mathematical seminar in Perugia (1989) and Problems associated with the theory of finitely additive measures: some recent results of the Scuola Matematica Perugina (1990).
    • The scientific interests of Calogero Vinti covered several areas of Mathematical Analysis, from Calculus of Variations to Differential Equations, from Approximation Theory to Real Analysis and Measure Theory.

  106. Frederick Justin Almgren biography
    • At Brown University an exciting phase had just started with Wendell Fleming collaborating with Herbert Federer in geometric measure theory.
    • By contrast, for an m-dimensional mass-minimizing surface of codimension greater than one, the singular set was not even known to have m-measure 0.
    • Around 1974, Almgren started on what would become his most massive project, culminating ten years later in a three-volume, 1700-page preprint containing a proof that the singular set not only has m-dimensional measure 0, but in fact has dimension at most (m - 2).

  107. Roman Sikorski biography
    • The following eight papers appeared in 1947 and 1948: On the Cartesian product of metric spaces (1947); Sur les corps de Boole topologiques Ⓣ (1948); Sur la convergence des suites d'homomorphies Ⓣ (1948); (with Edward Marczewski) Measures in non-separable metric spaces (1948); On a generalization of theorems of Banach and Cantor-Bernstein (1948); On the representation of Boolean algebras as fields of sets (1948); Remarks on a problem of Banach (1948); A theorem on extension of homomorphisms (1948); and On an ordered algebraic field (1948).
    • Still more remarkable is the fact that the book could serve very well to introduce a serious student to a wide range of topics in set theory, topology, measure theory, logic, and, of course, the theory of Boolean algebras.

  108. Ptolemy biography
    • Ptolemy, whose intention was to develop a comprehensive theory of celestial phenomena, had no access to the methods of data evaluation using arithmetical means with which modern astronomers can derive from a set of varying measurement results, the one representative value needed to test a hypothesis.
    • For methodological reason, then, Ptolemy was forced to choose from a set of measurements the one value corresponding best to what he had to consider as the most reliable data.

  109. Carl Friedrich Gauss (1777-1855)
    • Gauss was pleased to accept and took personal charge of the survey, making measurements during the day and reducing them at night, using his extraordinary mental capacity for calculations.
    • These papers all dealt with the current theories on terrestrial magnetism, including Poisson's ideas, absolute measure for magnetic force and an empirical definition of terrestrial magnetism.

  110. Al-Sijzi biography
    • A treatise on spheres by al-Sijzi Book of the measurement of spheres by spheres is of considerable interest.
    • The authors of [',' B A Rozenfel’d and R S Safarov (trs.), Abu Said Akhmad as-Sidzhizi, Book of measurement of spheres by spheres (Russian), Istor.-Mat.
    • However J P Hogendijk reviewing [',' B A Rozenfel’d and R S Safarov (trs.), Abu Said Akhmad as-Sidzhizi, Book of measurement of spheres by spheres (Russian), Istor.-Mat.

  111. Samuil Osipovich Shatunovsky (1859-1929)
    • He published a book On the measurement of straight line segments and constructing numbers using ruler and compass on geometrical constructions with ruler and compass where he gave an algebraic interpretation but did not introduce Galois theory.
    • In his book Methods of solving linear trigonometry he proposed a general principle of solving trigonometric problems based on the consideration of homogeneous functions of measure zero.

  112. Hideo Tanaka biography
    • Chapter 2 covers the fundamentals of possibility theory, such as possibility distributions and the associated operations, possibility and necessity measures and possibilistic linear systems.
    • Chapter 7 addresses discriminant analysis based on possibility distributions where the discriminant function separates the data into proper groups according to the possibility measure.

  113. Jacques Bertillon biography
    • To you, Monsieur, I say marry and you will do well even from a selfish standpoint; but watch carefully over your wife's health as, even from this egotistical point of view, her loss will be a terrible misfortune; for your life depends in a great measure on her own.
    • Together they developed anthropometry, an identification system based on physical measurements which was used by police to identify criminals.

  114. Ennio De Giorgi biography
    • De Giorgi attended lectures by Caccioppoli on geometric measure theory, but already by this time he had his own ideas about how to attack problems of minimal surfaces.
    • Influenced by methods which Caccioppoli had developed, De Giorgi went on to develop new techniques in geometric measure theory and he applied his results to the calculus of variations proving his regularity theorem for almost all minimal surfaces.
    • The authors of this paper are all students of De Giorgi and they describe his contributions to geometric measure theory, the solution of Hilbert's XIXth problem in any dimension, the solution of the n-dimensional Plateau problem, the solution of the n-dimensional Bernstein problem, some results on partial differential equations in Gevrey spaces, convergence problems for functionals and operators, free boundary problems, semicontinuity and relaxation problems, minimum problems with free discontinuity set, and motion by mean curvature.

  115. Niels Norlund biography
    • Norlund's first publication was in 1905 when he published a paper on a known double star in Ursa Major which, with careful measurements of their orbits, he was able to deduce was actually a triple system with a third star which was too faint to observe.
    • Besides the necessary updating of the ordnance maps of Denmark, he put into operation new more precise triangulations and measurement of the matching new baselines and new astronomical determinations.

  116. Félix Tisserand biography
    • Tisserand decided to undertake a programme of measurement of the separation of binary stars, but he discovered that the micrometer used for such measurements did not work.

  117. Constantin Carathéodory biography
    • He also visited the Cheops pyramid and made measurements which he wrote up and published in 1901.
    • Caratheodory made significant contributions to the calculus of variations, the theory of point set measure, and the theory of functions of a real variable.

  118. David Foulis biography
    • The study of measures on these logical models is a burgeoning new field called noncommutative measure theory.

  119. Otton Nikodym biography
    • Nikodym's name is mostly known in measure theory (e.
    • Let μ be a σ-finite measure on a σ-algebra Σ of subsets of Ω and ν a countably additive set function on Ω.
    • In 1965 he was invited for a semester by the University of Naples, Italy, to lecture on measure theory.

  120. Tjalling Charles Koopmans biography
    • The paper On the existence of a subinvariant measure (1964), written with R E Williamson, is an interesting work on measure theory which gives a construction analogous to that of the Haar measure for compact sets.

  121. John Herschel (1792-1871)
    • The work on double stars had been undertaken as a continuation of his father's work which attempted to measure the parallax of a star.
    • History Topics: The history of measurement .

  122. Tadeusz Banachiewicz biography
    • He graduated from the University of Warsaw with a bachelor's degree in physical and mathematical sciences in the following year, 1904, having submitted a dissertation discussing certain heliometric measurements that had been made at Pulkovo Observatory with a heliometer made in the workshop of the famous German company Repsold & Sohnes.
    • The instrument was specially designed to allow accurate measurements to be made of the distance from the earth to the sun during a transit of Venus.
    • During these five years, Banachiewicz took part in a scientific expedition to the Volga River Basin to make accurate measurements of terrestrial gravity.

  123. Herbert Robbins biography
    • Nevertheless, his work on the naval officers' problem led to the fundamental papers ['On the measure of a random set' (1944) and 'On the measure of a random set II' (1944)] in the field of geometric probability.
    • He didn't need me as a statistician; he wanted me to teach measure theory, probability, analytic methods, etc.

  124. Posidonius (135 BC-51 BC)
    • We should note, however, that Taisbak in [',' C M Taisbak, Posidonius vindicated at all costs? Modern scholarship versus the Stoic earth measurer, Centaurus 18 (1973/74), 253-269.','11] attempts to prove that attributing this far too small value of 180000 stadia to Posidonius is unfounded.
    • His measurements of the moon are inaccurate partly because he assumes a cylindrical rather than conical shadow.

  125. Moshe Carmeli biography
    • Re-envisioning Einstein's theories of Special and General Relativity and building on the work of Edwin Hubble, Dr Carmeli has suggested that the universe's expansion must be constantly accelerating, and that time is therefore relative; in other words, it can only be measured relative to the position and velocity of the measurer.

  126. Robert Hooke (1635-1703)
    • He then invented a helioscope to attempt to measure the rotation of the sun using sunspots.
    • In 1666 he proposed that gravity could be measured using a pendulum.

  127. Christopher Wren (1632-1723)
    • His mind leapt from one topic to another as he came up with ideas such as: an instrument to measure angles, instruments for surveying, machines to lift water, ways to find longitude and distance at sea, military devices for defending cities, and means for fortifying ports.
    • History Topics: The history of measurement .

  128. Federico Cafiero biography
    • We have already seen that Cafiero made contributions to the theory of ordinary differential equations and to the theory of measure and integration.
    • The fact is that the book, in spite of its size, covers only a part of the theory of measure and integration.
    • Nor is there any mention of the deeper questions about measure, for instance, those related to the continuum hypothesis, and generally there are only the barest references to historical and philosophical matters, and to the motivation of the concepts.

  129. Agner Erlang biography
    • The quantity thus measured is of course dimensionless, and the erlang is to be compared with the octave, the stellar magnitude and the decibel in describing the mode of calculation rather than the unit of measurement in the usual sense of physics.

  130. Johann Werner (1468-1522)
    • Werner suggested using for these measurements a cross-staff of the type which sailors already used for determining the latitude of a ship at sea by measuring the height of the pole star.
    • The geographer will be in one of these places and will measure with a cross-staff the distance of the Moon from a star on the Ecliptic.

  131. Charles Hutton (1737-1823)
    • He also computed the mean density of the Earth based on Maskelyne's data from the mountain Schiehallion in An Account of the Calculations made from the Survey and Measures taken at Schiehallion in order to ascertain the mean density of the Earth (1779).
    • The Compendious Measurer appeared in 1784, The Elements of Conic Sections in 1787 and, in 1795, his most famous work The Mathematical and Philosophical Dictionary in two volumes.

  132. Vincenzo Viviani (1622-1703)
    • Two days later, at the palace in Florence, he measured the velocity of sound by timing, again using a pendulum, the difference between the flash and the sound of a cannon fired at the villa of Petraia.
    • Geometry alone teaches how to achieve knowledge and even reminds the human intellect - which is a spark of the divine one - that as a knower through the principles most known with the light of nature it can, if it so wishes, without deceiving itself or others, know the existence and properties of all things concerning the created universe and the order disposed by God, in number, weight, and measure.

  133. Mikhail Iosiphovich Kadets biography
    • A new chapter devoted to non-norm topologies presents the authors' results showing that a sum range can be non-convex for the weak topology and for convergence in measure, and the theorem of Banaszczyk stating that the Steinitz property holds in nuclear metrizable spaces.
    • His interests are very similar to those of his father, namely Banach space theory, in particular the study of sequences and series, bases, vector-valued measures and integration, isomorphic and isometric structures of Banach spaces, and operators in Banach spaces.

  134. Guidobaldo del Monte (1545-1607)
    • It consisted of a pair of dividers with the addition of a number of sharp points that could be slid up and down the arms to provide a device capable of giving measurements in fixed proportion to how far the legs of the dividers were opened.
    • It had points at both ends of the legs and, depending on the position of the hinge, a fixed proportion was achieved between distances measures with the points at one end and those at the other.

  135. Patrick Moran biography
    • While studying mathematics at Cambridge it had been Besicovitch who had the most influence and, during the war years, Moran undertook research on Hausdorff measure publishing several papers on the topic, including The measure of plane sets (1943), Measuring the surface area of a convex body (1944), and The measure of product and cylinder sets (1945) which was a joint publication with Besicovitch.

  136. Hendrik van Heuraet (1634-1660)
    • When van Heuraet learned that I had measured the surface of the parabolic conoid and had determined the length of the parabola equal to a given quadrature of the hyperbola (concerning both of which I wrote you previously), he found not only both of them by his own technique but, in addition, he rectified completely all other curves of those genera that we allow in geometry.
    • Indeed, it was not difficult for this man of very keen ability to deduce that the surface of that conoid is associated with the measure of the parabolic curve itself.

  137. Kiyosi Ito biography
    • In 1923, against this scientific background, Wiener defined probability measures in path spaces, and used the concept of Lebesgue integrals to lay the mathematical foundations of stochastic analysis.
    • In this book, Ito develops the theory on a probability space using terms and tools from measure theory.

  138. George Birkhoff biography
    • His ergodic theorem transformed the Maxwell-Boltzmann kinetic theory of gases into a rigorous principle through the use of Lebesgue measure.
    • Before writing Aesthetic measure he spent a year travelling round the world studying art, music and poetry and various countries.
    • Among his works, some of which we have already mentioned above, are Relativity and Modern Physics (1923), Dynamical Systems (1928), Aesthetic Measure (1933), and Basic Geometry (1941).

  139. Proclus (411-485)
    • In his astronomical writings, Proclus described how the water clock invented by Heron could be used to measure the apparent diameter of the Sun.
    • As soon as the Sun has risen the water is collected in another container and this measurement continues until sunrise the following day.

  140. Gregori Margulis biography
    • for his monumental contributions to algebra, in particular to the theory of lattices in semi-simple Lie groups, and striking applications of this to ergodic theory, representation theory, number theory, combinatorics and measure theory.
    • One was the solution to a problem posed by Rusiewicz, about finitely additive measures on spheres and Euclidean spaces.

  141. Adrien Auzout biography
    • The turns and fractions of a turn of a micrometer screw could be converted into angular measurements after the instrument had been calibrated..
    • The interval between the transits of two stars measures their difference of right ascension; and upon Picard's procedure is based the standard method of determining, on the one hand, the absolute right ascensions of stars, and, on the other, the local sidereal time of the place of observation.

  142. Sydney Chapman biography
    • At first he supervised the installation of new instruments to measure magnetism, but was disappointed to find that the scientists were only interested in collecting data and were making no real attempts to interpret it.
    • His success is measured by the award of the first Smith's prize in 1913, and with this topic he began a research interest that he would continue through the rest of his life.

  143. Vladimir Abramovich Rokhlin biography
    • He was studying measure theory and dynamical systems and had done a lot of work bringing a major set of lectures by Plessner on the spectral theory of operators to publishable form.
    • He had written a prize-winning paper On the fundamental ideas of measure theory influenced by Kolmogorov.
    • After recovering from typhoid he spent his time working on mathematics recording his ideas on measure theory in a notebook.

  144. Ole Jacob Broch biography
    • The efforts during the 19th century to standardise weights and measures in various areas was a key instrument in promoting technological development as well as scientific communication.
    • In 1879, he was employed by the International Bureau of Weights and Measures in Sevres, and was asked to devise an international scientific weights and measures system.

  145. Yakov Grigorevich Sinai biography
    • These systems can be accurately measured in the short term (short term being relative to the issue at hand); but when analyzed in the long term, the systems are difficult to understand and predict.
    • Working in the tradition of the Kolmogorov school, he first formulated the rigorous definition of the invariant entropy for an arbitrary measure-preserving map.

  146. Guillaume Bigourdan biography
    • He was awarded the degree for his thesis Sur l'equation personnelle dans les mesures d'etoiles doubles Ⓣ in which he studied measurements of 2800 double stars.
    • Many of the observations that had been recorded had no pretensions to precision in measurement.

  147. Edward Marczewski biography
    • These were Mazurkiewicz and Sierpiński who interested Marczewski in measure theory and related topics.
    • His main work was in set theory, general topology, and measure theory.
    • He obtained particularly interesting and frequently applied results on the duality between the notions of a set of the first category and a set of measure zero; and similarly - between a set with Baire's property and a measurable set.

  148. J Presper Eckert biography
    • eventually involved with work on ultraviolet light and the development of the means to measure metal fatigue.
    • It contained roughly 18000 vacuum tubes and measured about 2.5 metres in height and 24 metres in length.

  149. Nicholas Kryffs biography
    • Whatever is not truth cannot measure truth precisely.
    • Finally, when he has made a complete representation of the perceptible world in his own city, he compiles it into a well-ordered and proportionately measured map lest it be lost.

  150. Norbert Wiener (1894-1964)
    • He introduced a measure in the space of one dimensional paths which brings in probability concepts in a natural way.
    • Even measured by Wiener's standards "Cybernetics" is a badly organised work -- a collection of misprints, wrong mathematical statements, mistaken formulas, splendid but unrelated ideas, and logical absurdities.

  151. Stefan Banach biography
    • During one such walk I overheard the words "Lebesgue measure".
    • In 1922 the Jan Kazimierz University in Lvov awarded Banach his habilitation for a thesis on measure theory.
    • In addition, he contributed to measure theory, integration, the theory of sets, and orthogonal series.

  152. Richard Schoen biography
    • Not only did their work employ serious tools of geometric analysis, including partial differential equations and geometric measure theory, to resolve a question motivated by gravitational physics, but they also established a link between the positivity of the mass of an isolated gravitational system and the relationship between positive scalar curvature and topology, a topic of interest to a broad range of mathematicians.
    • The mathematical influence of Richard M Schoen can be measured in many ways.

  153. Léon Foucault biography
    • He next suggested that Foucault and Fizeau try to measure the speed of light in water.
    • Foucault now devised his own methods to approach the problem of measurement, building a steam engine to drive a spinning mirror.

  154. Paul Cohen biography
    • In On a conjecture of Littlewood and idempotent measures (1960) Cohen made a significant breakthrough in solving the Littlewood Conjecture.
    • He was an invited speaker giving the address Idempotent measures and homomorphisms of group algebras.
    • .for his paper, On a conjecture of Littlewood and idempotent measures, American Journal of Mathematics 82 (1960), 191-212.

  155. Pedro Nunes (1502-1578)
    • Nunes devised a system to allow fractional parts of a degree to be measured.
    • In 1572 Nunes was called to court to preside over the reform of weights and measures.

  156. Bhaskara I biography
    • He regretted that an exact measure of the circumference of a circle in terms of diameter was not available and he clearly believed that π was not rational.
    • ','12], [',' K S Shukla, Hindu mathematics in the seventh century as found in Bhaskara I’s commentary on the Aryabhatiya III, Ganita 23 (1) (1972), 57-79','13] and [',' K S Shukla, Hindu mathematics in the seventh century as found in Bhaskara I’s commentary on the Aryabhatiya IV, Ganita 23 (2) (1972), 41-50.','14] Shukla discusses some features of Bhaskara's mathematics such as: numbers and symbolism, the classification of mathematics, the names and solution methods of equations of the first degree, quadratic equations, cubic equations and equations with more than one unknown, symbolic algebra, unusual and special terms in Bhaskara's work, weights and measures, the Euclidean algorithm method of solving linear indeterminate equations, examples given by Bhaskara I illustrating Aryabhata I's rules, certain tables for solving an equation occurring in astronomy, and reference made by Bhaskara I to the works of earlier Indian mathematicians.

  157. Guo Shoujing biography
    • The simplest astronomical instruments was the gnomon, nothing other than a stick which was erected and the length of its shadow measured.
    • One has to believe that Guo chose π = 3 because he knew that the answers that he then found for different sizes of the angle at O more closely approximated values he found by direct measurement.

  158. Iossif Vladimirovich Ostrovskii biography
    • In the late 1970s he studied complex-valued Borel measures on the real axis.
    • He continued to work on this topic with his student Alexandr M Ulanovskii and they surveyed their results in this area in the joint paper Classes of complex-valued Borel measures than can be uniquely determined by restrictions (1989).
    • In the 1980s Ostrovskii wrote a number of papers on the asymptotic behaviour of entire functions that are characteristic functions of probability measures.

  159. Czesaw Olech biography
    • It includes a number of other branches of mathematics, and his works are classified by Mathematical Reviews as including, among others, the area of linear and multilinear algebra, measure and integration theory, calculus of variations, convex and discrete geometry, operations research and general systems theory.
    • He obtained significant results for vector measures and their applications in the theory of differential equations and the theory of optimal control.

  160. Frank Ramsey (1903-1930)
    • Ramsey, proposing a probability measure based on strength of belief, [',' D H Mellor, Frank Plumpton Ramsey, Routledge Encyclopedia of Philosophy 8 (London, New York, 1998), 44-49.','11]:- .
    • derives measures both of desires (subjective utilities) and of beliefs (subjective probabilities), thereby founding the now standard use of these concepts.

  161. al-Khazin biography
    • We know that in 959/960 al-Khazin was required by the vizier of Rayy, who was appointed by Adud ad-Dawlah, to measure the obliquity of the ecliptic (the angle which the plane in which the sun appears to move makes with the equator of the earth).
    • He is said to have made the measurement:- .

  162. Frederico Commandino biography
    • It consisted of a pair of dividers with the addition of a number of sharp points that could be slid up and down the arms to provide a device capable of giving measurements in fixed proportion to how far the legs of the dividers were opened.
    • It had points at both ends of the legs and, depending on the position of the hinge, a fixed proportion was achieved between distances measures with the points at one end and those at the other.

  163. James Bradley (1693-1762)
    • Bessel would go on in 1838 to measure annual parallax for the star 61 Cygni, further adding to the Copernican argument.
    • He was very interested in the accurate measurement of time.

  164. Gaspard de Prony biography
    • In 1785 de Prony visited England on a project to obtain an accurate measurement of the relative positions of the Greenwich Observatory and the Paris Observatory.
    • One of de Prony's most important scientific inventions was the 'de Prony brake' which he invented in 1821 to measure the performance of machines and engines.

  165. Mei Wending biography
    • Yang Guangxian and Adam Schall von Bell measured the length of the sun's shadow in front of the Wu Men gate.
    • In 1700, in the Qiandu celiang (The Measurement of a Prism with Two Right Triangular Bases), he gave a trigonometric interpretation of the celestial coordinate transformation introduced by Guo Shoujing in 1280.

  166. Renato Caccioppoli biography
    • the principles of a theory of measure of plane and curved surfaces, and more generally of two or more dimensional varieties embedded in a linear space.
    • The most successful approach to measure was, at this time, that proposed by Lebesgue.
    • The image of a net triangulating D is a polyhedral plane surface, and by considering the lower limit of total variation of the pair (f, g) a measure for S can be defined.

  167. Leon Simon biography
    • Simon published Lectures on geometric measure theory in 1983.
    • Many years ago H Federer's book on geometric measure theory appeared and immediately became a reference for those working in and interested in the field.
    • The book under review, which the author says is a preliminary version of a more complete book he hopes to write, is both an introduction to geometric measure theory and related variational problems and to the regularity theory.

  168. Philippe de la Hire (1640-1718)
    • He also studied instruments to measure climatic conditions such as temperature, pressure and wind speed, making measurements with such instruments at the Paris Observatory.

  169. Nicomachus (about 60-about 120)
    • And in the case of those that are found between the too much and the too little, that is in equality, is produced virtue, just measure, propriety, beauty and things of that sort - of which the most exemplary form is that type of number which is called perfect.
    • But, unlike Euclid, who attempts to prove musical propositions through mathematical theorems, Nicomachus seeks to show their validity by measurement of the lengths of strings.

  170. Bronius Grigelionis biography
    • Bronius was interested in studying the theory of probability, measure and integral theory and mathematical statistics.
    • Using the independently scattered random measures, generated by the bivariate centred Student-Levy process, and stochastic integration theory with respect to them, it is defined as an univariate strictly stationary process with the centred Student's t marginals and the arbitrary correlation structure.

  171. Paul Halmos biography
    • This was awarded in 1938 for his thesis on measure-theoretic probability Invariants of Certain Stochastic Transformation: The Mathematical Theory of Gambling Systems.
    • These include Finite dimensional vector spaces (1942), Measure theory (1950), Introduction to Hilbert space and theory of spectral multiplicity (1951), Lectures on ergodic theory (1956), Entropy in ergodic theory (1959), Naive set theory, Algebraic logic (1962), A Hilbert space problem book (1967) and Lectures on Boolean algebras (1974).
    • The award for a book or substantial survey or research-expository paper is made to Paul R Halmos for his many graduate texts in mathematics, dealing with finite dimensional vector spaces, measure theory, ergodic theory and Hilbert space.

  172. Adrien-Marie Legendre (1752-1833)
    • He became an associe in 1785 and then in 1787 he was a member of the team whose task it was to work with the Royal Observatory at Greenwich in London on measurements of the Earth involving a triangulation survey between the Paris and Greenwich observatories.
    • On 13 May 1791 Legendre became a member of the committee of the Academie des Sciences with the task to standardise weights and measures.

  173. Elias Stein biography
    • He held this position for two years during which time a whole series of his papers appeared in print: Interpolation of linear operators (1956), Functions of exponential type (1957), Interpolation in polynomial classes and Markoff's inequality (1957), Note on singular integrals (1957), (with G Weiss) On the inerpolation of analytic families of operators action on Hpspaces (1957), (with E H Ostrow) A generalization of lemmas of Marcinkiewicz and Fine with applications to singular integrals (1957), A maximal function with applications to Fourier series (1958), (with G Weiss) Fractional integrals on n-dimensional Euclidean space (1958), (with G Weiss) Interpolation of operators with change of measures (1958), Localization and summability of multiple Fourier series (1958), and On the functions of Littlewood-Paley, Luzin, Marcinkiewicz (1958).
    • The second in this three volume series Complex analysis was published in 2003 and the third volume Real analysis: Measure theory, integration, and Hilbert spaces in 2005.

  174. Dmitrii Menshov biography
    • It was expected that Vallee Poussin's result would still hold if the countable set E was replaced by a set E of measure zero.
    • The remarkable, and unexpected, result that Menshov discovered in 1916 was that this was not so, for he constructed a trigonometric series which converges to 0 for all x in [0, 2π] - E, for a set E of measure zero, yet not all the coefficients of the trigonometric series are zero.

  175. Thomas Graham biography
    • His research was undertaken with E Taylor Jones as his advisor, but Graham never intended this to be anything more than an interim measure.
    • He continued this work after his retirement in 1970 and this compensated in a small measure for his regret at giving up his work of teaching and advising students, which he had so much enjoyed and carried out so well and sympathetically.

  176. Francesco Grimaldi (1618-1663)
    • First, Grimaldi and Riccioli calibrated a pendulum by getting it to swing for 24 hours (measured by the star Arcturus crossing the meridian line).
    • He placed a thin rod in the path of the light and measured the size of the shadow on the screen.

  177. João Baptista Lavanha (1555-1624)
    • He began work on a map of Aragon in about 1611 by making a series of geodesic measurements [',' L de Alburquerque, Biography in Dictionary of Scientific Biography (New York 1970-1990).','1]:- .
    • In addition to the map, he had produced a political description of the country in Itenerario do reyno de Aragao (1611-12) which consists of the notes he made while travelling round the district making measurements for the map.

  178. Henry Dye biography
    • In 1959 Dye published On groups of measure preserving transformation I in the American Journal of Mathematics.
    • This important paper studies the classification of automorphism groups of a finite non-atomic measure algebra.

  179. Jacob Bronowski (1908-1974)
    • He published a statistics paper with Neyman The variance of the measure of a two-dimensional random set in 1945.
    • Could I combine a measure of the size of the Taung child's teeth with their shape, so as to discriminate them from the teeth of apes? .

  180. Joseph Liouville (1809-1882)
    • He proved a major theorem concerning the measure preserving property of Hamiltonian dynamics.
    • The result is of fundamental importance in statistical mechanics and measure theory.

  181. Filippo Brunelleschi biography
    • 4 (2) (1989), 31-118.','17] made measurements of the fresco, painted by Masaccio in 1425, which hangs in Santa Maria Novella, Florence.
    • Measurements were made directly from the fresco, using scaffolding.

  182. Daniel Rudolph biography
    • The complementary style of ergodic theory in the Ornstein school was measure theory with deep combinatorial insight and barehanded invention.
    • The ergodic theory of measure-preserving transformations is a subject which has come of age only in the last thirty years.

  183. Shizuo Kakutani biography
    • Kakutani not only took great interest in the work of Weyl's group at Princeton but also the group of mathematicians working with von Neumann on measure theory and ergodic theory.
    • Among the areas on which he has written papers we must mention: complex analysis, topological groups, fixed point theorems, Banach spaces and Hilbert spaces, Markov processes, measure theory, flows, Brownian motion, and ergodic theory.

  184. Jan Mikusinski biography
    • 89 (1) (1988), ii-xi.','8] lists 26 books including: (with Stanisław Hartman) Theory of measure and Lebesgue integral (Polish) (1957), English edition published under the title The theory of Lebesgue measure and integration (1961); (with Roman Sikorski) Theorie elementaire des distributions Ⓣ (1964); (with Piotr Antosik and Roman Sikorski) Theory of distributions.

  185. Andrea Tacquet (1612-1660)
    • Two noteworthy contributions : ’Cuts of rational numbers’ by the Galilean G A Borelli and ’Classes of measures’ by the Jesuit A Tacquet (Italian), Nuncius Ann.
    • 6 (2) (1991), 33-81.','5] about Tacquet's Classes of measures:- .

  186. Yativrsabha (about 500-about 570)
    • Yativrsabha's work Tiloyapannatti gives various units for measuring distances and time and also describes the system of infinite time measures.
    • It led them to consider different measures of infinity, and in this respect the Jaina mathematicians would appear to be the only ones before the time when Cantor developed the theory of infinite cardinals to envisage different magnitudes of infinity.

  187. Emilie Martin biography
    • Most of the methods depend upon measurements of some kind, and so are subject to error.
    • There is small wonder that at present we can only say that some measurements indicate that our space constant is very large and positive, while other theories indicate that we live in an elliptic space.

  188. Yvonne Bruhat biography
    • He made important contributions to a variety of fields: topology, measure theory, descriptive set theory, potential theory, and functional analysis.
    • The original book already had a number of interesting applications, such as the Schrodinger equation, soap bubbles, electromagnetism, shocks, gravity, Hamiltonian systems, monopoles, spinors, degree theory applied to PDE, Wiener measure, etc.

  189. William Parry biography
    • In 1963 he published An ergodic theorem of information theory without invariant measure generalising the individual version of McMillan's ergodic theorem of information theory without the hypothesis of an invariant probability function.
    • There he worked on entropy theory showing, amongst other things, that each aperiodic measure-preserving transformation could be viewed as the shift on the realisation space of a stationary, countable state, stochastic process indexed by the integers or the natural numbers.

  190. Walter Rudin biography
    • The author deals with the classical elements of real variable theory: Dedekind real numbers, elementary set-theory, convergence, continuity, differentiation, the Riemann-Stieltjes integral, uniform convergence, functions of several variables, Lebesgue measure and integrals.
    • In general, the spaces considered are Euclidean, but the reader is introduced to more abstract ideas: for example, general metric spaces and measure spaces.

  191. Giuseppe Vitali (1875-1932)
    • Vitali's most significant output took place in the first eight years of the twentieth century when Lebesgue's measure and integration were revolutionising the principles of the theory of functions of real variables.
    • This period saw the emergence of some of his most important general and profound results in that field: the theorem on discontinuity points of Riemann integrable functions (1903), the theorem of the quasi-continuity of measurable functions (1905), the first example of a nonmeasurable set for Lebesgue measure (1905), the characterisation of absolutely continuous functions as antidervatives of Lebesgue integrable functions (1905), the covering theorem (1908).

  192. Konrad Zuse biography
    • His S1 and S2 computers were used for computing the precise measurements necessary for the production of aircraft.
    • For the S2 the computer included measuring devices to make measurements of the planes in production and to feed these directly into the calculations.

  193. Hillel Furstenberg biography
    • Part II carries the title "Recurrence in measure preserving systems".
    • After a short introduction to the relevant part of measure-theoretic ergodic theory, this section is devoted to a proof of the multiple recurrence theorem ..

  194. Kazimierz Kuratowski (1896-1980)
    • At Lvov, however, Kuratowski worked with Banach and they answered some fundamental problems on measure theory.
    • He also considered the topology of the continuum, the theory of connectivity, dimension theory, and answered measure theory questions.

  195. Gilbert Hunt biography
    • In Semigroups of measures on Lie groups (1956), he explains: .
    • We shall characterise those families of finite positive measures on a Lie group G which are weakly continuous and form a semigroup under convolution.

  196. Aryeh Dvoretzky biography
    • His best known fundamental result in this field is the Dvoretzky theorem, which was related by Vitali Milman to Paul Levy's measure concentration phenomena and served as a starting point to modern Banach space theory.
    • Its proof, which depends on some intricate measure-theoretic arguments, is sketched in the paper A theorem on convex bodies and applications to Banach spaces and discussed in greater detail in Some results on convex bodies and Banach spaces (1960).

  197. Johann Heinrich Lambert (1728-1777)
    • Lambert tried to build up geometry from two new principles: measurement and extent, which occurred in his version as definite building blocks of a more general metatheory.
    • The lines drawn by Jacob Bernoulli in the "Ars conjectandi" were taken up and developed in a decisive way by Lambert, whose fundamental contributions to the theory of errors in measurement have been re-evaluated in recent years.

  198. Anzelm Iwanik biography
    • an analogue of the classical result of Halmos on the residuality of ergodic transformations in the measure preserving invertible ones.
    • Iwanik began working on another topic during the years that he was also working on topological dynamics, namely the spectral theory of measure-preserving transformations.

  199. René Eugène Gateaux biography
    • In his autobiography ([',' Paul Levy, Quelques aspects de la pensee d’un mathematicien (Albert Blanchard, Paris, 1970).','14]), Levy has observed how in [',' Paul Levy, Lecons d’Analyse fonctionnelle (Gauthier-Villars, 1922).','12], he was close to Wiener measure.
    • He was indeed so close that Wiener, when speaking with him in 1922, will immediately see that Levy's considerations to define the integral over the infinite dimensional sphere are precisely what he could use to define his Differential-space and construct the Wiener measure of Brownian motion.

  200. Heinrich Friedrich Weber biography
    • However, 'he and his institute at least helped further sensitise Einstein to the importance of measurement for testing theory and for finding the best fit between theory and empirical reality' [',' D Cahan, The Young Einstein’s Physics Education: H.F.
    • Polytechnikum vor der Jahrhundertwende, Schweizerische Bauzeitung 76 (52), 1958, 787-788','4] he helped to establish a system of units of measurement, together with physicists such as Lord Rayleigh, Silvanus Thompson, Friedrich Kohlrausch, Eletuhere Mascart, and Lord Kelvin.

  201. Giovanni Alfonso Borelli biography
    • His small publication about the comet Del movimento della cometa apparsa il mese di Dicembre 1664 Ⓣ, written in the form of a letter to Stefano degli Angeli, he argued that his parallax measurements showed that the comet changed its distance from the earth.
    • In a letter about the comet to Prince Leopold, he argued that his measurements indicated that it followed a parabolic orbit.

  202. Philipp Furtwängler biography
    • In this post he carried out, in collaboration with Friedrich Kuhnen, measurements of absolute masses and measurements of the gravitational constant at various locations in Silesia.

  203. Saharon Shelah biography
    • His staggering output of 700 papers and half a dozen monographs includes the creation of several entirely new theories that changed the course of model theory and modern set theory and also provided the tools to settle old problems from many other branches of mathematics, including group theory, topology, measure theory, Banach spaces, and combinatorics.
    • Shelah’s Theorem on the Measure Problem and Related Results by Jean Raisonnier, J.

  204. Gaspard Monge (1746-1818)
    • By this time he was on the major Academie Commission on Weights and Measures.
    • He continued to serve on the Commission on Weights and Measures which survived despite ending the Academie des Sciences.

  205. Otto Hölder (1859-1937)
    • Joel Michell writes [',' J Michell and C Ernst, The Axioms of Quantity and the Theory of Measurement, Journal of Mathematical Psychology 40 (1996), 235-252.','12]:- .
    • His paper is a watershed in measurement theory, dividing the classical (stretching from Euclid) and the modern (stretching to Luce et al ., 1990) eras.

  206. Richard von Mises (1883-1953)
    • After the measure theory approach by Kolmogorov had become favoured by almost all statisticians over von Mises' limiting frequency theory approach, there was a return to von Mises ideas and there was an attempt to incorporate them into the measure theoretic approach of Kolmogorov who wrote himself in 1963:- .

  207. Arnljot Høyland biography
    • The engineers were making all sorts of measurements, and they thought that they really knew what kind of size measurements represent.

  208. Woolsey Johnson biography
    • from Johns Hopkins in 1896 for his thesis Musical Pitch and the Measurement of Intervals Among the Ancient Greeks.
    • I am glad to say he is his mother's boy, but he has mathematics enough in him to have written his thesis on the mathematical theory of music as it appears in the old Greek philosophers, Musical Pitch and the Measurement of Intervals among the Ancient Greeks.

  209. Simion Stoilow biography
    • The second volume has the following chapter headings: The Dirichlet problem; Local properties of harmonic functions; The Dirichlet problem for multiply-connected domains; The Dirichlet integral and the minimum principle; Green's function, Lindelof's principle, the principle of harmonic measure; Harmonic measure; Riemann surfaces; Analytic functions on closed Riemann surfaces; Analytic functions on open Riemann surfaces; Regularly and normally exhaustible Riemann surfaces.

  210. al-Kashi biography
    • (eds), Amphora : Festschrift for Hans Wussing on the occasion of his 65th birthday (Basel- Boston- Berlin, 1992), 171-181.','11], [',' Y Dold-Samplonius, Practical Arabic mathematics : measuring the muqarnas by al-Kashi, Centaurus 35 (3-4) (1992), 193-242.','12], and [',' Y Dold-Samplonius, al-Kashi’s measurement of Muqarnas, in Deuxieme Colloque Maghrebin sur l’Histoire des Mathematiques Arabes (Tunis, 1990), 74-84.','13].
    • For example the measurement of the muqarnas refers to a type of decoration used to hide the edges and joints in buildings such as mosques and palaces.

  211. Dmitrii Konstantinovich Faddeev biography
    • According to Faddeev's own words, it was difficult to find a professional job upon graduation and he had to work with various organizations, including the Weights and Measures Department, where he became addicted to smoking because of long breaks between instrumental observations.
    • Leaving the Weights and Measures Department in 1930, Faddeev taught at various Leningrad schools and also for a time at the Polytechnic Institute and the Engineering Institute.

  212. Maryam Mirzakhani biography
    • These papers were: Weil-Petersson volumes and intersection theory on the moduli space of curves (2007); Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces (2007); Random hyperbolic surfaces and measured laminations (2007); Growth of the number of simple closed geodesics on hyperbolic surfaces (2008); Ergodic theory of the earthquake flow (2008); and (with Elon Lindenstrauss) Ergodic theory of the space of measured laminations (2008).

  213. Donald Knuth biography
    • Nobody, to our knowledge, has tried to measure the impact of TeX on the level of mathematical production, and indeed this would be a very difficult thing to measure, but nevertheless I [EFR] am certain that the added ease of production and communication of mathematics using TeX has had a major impact on the subject over the last ten years, say.

  214. Al-Kindi biography
    • For example al-Kindi's commentary on Archimedes' The measurement of the circle has only received careful attention as recently as the 1993 publication [',' R Rashed, al-Kindi’s commentary on Archimedes’ The measurement of the circle, Arabic Sci.

  215. Antoni Zygmund biography
    • Together with Richard L Wheeden, Zygmund wrote Measure and integral (1977).
    • This textbook has been conceived with the explicit intention of providing an easy and quick access to the most useful techniques of measure and integration in the modern analysis of real variables.

  216. Anders Wiman biography
    • In his own paper, which was a second attempt to put Gylden's work on a rigorous basis, Wiman applied measure theory to the probabilistic problem.
    • Torsten Broden, who we mentioned above in connection with Wiman's use of measure theory in questions of probability, was appointed to the chair at Lund.

  217. Gaspard-Gustave de Coriolis biography
    • The unit represents 1000 kilogram-metres and was proposed by Coriolis as a measure which could provide a sensible unit with which to measure the work which a person might do, a horse, or a steam engine.

  218. George Peacock (1791-1858)
    • These were The tutor's assistant modernised (1791) and The Practical Measurer (1798).
    • The Practical Measurer .

  219. Heinrich Bruns biography
    • He was interested in the figure of the Earth, as were a large number of scientists at the time when he worked, and he asserted that geodetic measurements can only give the true figure of the Earth when they are taken in conjunction with other types of measurements.

  220. Patrick d'Arcy (1725-1779)
    • The setting for d'Arcy's investigation was a darkened building in which he constructed a revolving cross, the speed of which he could measure.
    • Le Roy and d'Arcy had jointly worked on measuring electricity in 1748 and had invented a floating repulsion electrometer to measure electrostatic repulsion.

  221. René de Possel biography
    • He published an article in 1935 which was the first to present a general theory of differentiation in abstract measure spaces and in the following year he published another article in which he gave complete proofs.
    • We mention Sur l'indetermination de la puissance d'un torseur reparti Ⓣ in which he gave proofs of some formulas of use in the mechanics of continuous media, where the differential elements are subjected to couples as well as forces per unit volume; Les principes mathematiques de la mecanique classique Ⓣ which was based in ideas due to Brelot; Sur la definition d'un torseur reparti et sur l'evaluation de sa puissance Ⓣ which examines when external forces on part of a body are equivalent to couples alone; Initiation a la topologie Ⓣ resulting from work carried out in Portugal; Sur les systemes derivants et l'extension du theoreme de Lebesgue relatif a la derivation d'une fonction a variation bornee Ⓣ extending the classical theorem for linear Lebesgue measure; and La notion physique d'energie vis-a-vis des definitions du travail et de la force Ⓣ which considers the formulation of classical mechanics given by Brelot.

  222. Gábor Szeg biography
    • Szegő went to von Neumann's home once or twice a week and, over tea, discussed set theory, measure theory and other topics.
    • I got a letter from Szegő in the beginning of January; although no official measure was taken against him [until the beginning of January] and no direct collision happened with the students, I cannot see how it would go on indefinitely under those circumstances.

  223. Ivor Grattan-Guinness biography
    • Offord was an expert on measure theory and Grattan-Guinness was tempted to change to undertake research on the history of measure theory but he decided to stay with Fourier and related ideas.

  224. Eberhard Hopf biography
    • In particular he studied the theory of measure and invariant integrals in ergodic theory and his paper On time average theorem in dynamics which appeared in the Proceedings of the National Academy of Sciences is considered by many as the first readable paper in modern ergodic theory.
    • The measure-theoretical point of view became dominant in the later thirties after the advent of ergodic theory, and the papers of G A Hedlund and E Hopf on the ergodic character of the geodesic flow came into being.

  225. Eduard Stiefel biography
    • David Lide writes [',' D R Lide, A Century of Excellence in Measurements, Standards, and Technology (CRC Press, 2001).','1]:- .
    • David Lide explains what some of the contributions of Stiefel's team were [',' D R Lide, A Century of Excellence in Measurements, Standards, and Technology (CRC Press, 2001).','1]:- .

  226. Philip Stein biography
    • enjoyed some measure of independence in that ..
    • Roseveare wrote a number of papers including: A chapter on algebra (1903), On convergence of series (1905), On 'Circular Measure' and the product forms of the sine and cosine (1905), Expansions of trigonometrical functions (1905), and Expansions of functions in general (1905).

  227. George FitzGerald (1851-1901)
    • The cultivation and training of the practical ability to do things and to learn from observation, experiment and measurement, is a part of education which the clergyman and the lawyer can maybe neglect, because they have to deal with emotions and words, but which the doctor and the engineer can only neglect at their own peril and that of those who employ them.
    • FitzGerald certainly showed that he had acquired the ability to learn from observation, experiment and measurement.

  228. Euclid (about 325 BC-about 265 BC)
    • most fair and well disposed towards all who were able in any measure to advance mathematics, careful in no way to give offence, and although an exact scholar not vaunting himself.

  229. Alexander Dinghas biography
    • His work is in many areas of mathematics including differential equations, functions of a complex variable, functions of several complex variables, measure theory and differential geometry.

  230. Leslie Fox (1918-1992)
    • It contains chapters on: Matrix algebra; Elimination methods of Gauss, Jordan, and Aitken; Compact elimination methods of Doolittle, Crout, Banachiewicz and Cholesky; Orthogonalization methods; Condition, accuracy and precision; Comparison of methods, measure of work; Iterative and gradient methods; Iterative methods for latent roots and vectors; and Notes on error analysis for latent roots and vectors.

  231. George Batchelor biography
    • As is well-known in this difficult subject of turbulent diffusion, most of the theoretical developments merely help to provide a framework for the interpretation of measurements, and do not provide definite deductions.

  232. William Brouncker biography
    • The Lord Viscount Brouncker moved, that the experiments concerning the measure of the first velocity of bodies might be presented, that is what force is required to raise, for instance, one pound weight, one yard high in one second of time.

  233. Christoff Rudolff (1499-1543)
    • tables of measurements for many regions, a list of symbols used in gauging, and numerous hints for solving problems.

  234. Richard Rado (1906-1989)
    • He studied inequalities, geometry and measure theory, particularly working in this area with Besicovitch.

  235. Konrad Knopp (1882-1957)
    • Friedrich Losch added a fourth volume in 1980 to cover more modern material: set theory, Lebesgue measure and integral, topological spaces, vector spaces, functional analysis, integral equations.

  236. Salomon Bochner biography
    • Among much else the book contains Bochner's most famous theorem, characterising the Fourier-Stieltjes transforms of positive measures as positive-definite functions ..

  237. Vitruvius (about 85 BC-about 20 BC)
    • It is true that it is by arithmetic that the cost of buildings are calculated and measurements are computed, but difficult questions involving symmetry are solved by means of geometrical theories and methods.

  238. Alexander Aleksandrovich Kirillov biography
    • In particular in the 1960s these were: On certain division algebras over the field of rational functions (1967), Plancherel measure for nilpotent Lie groups (1967), The method of orbits in the theory of unitary representations of Lie groups (1968), Characters of unitary representations of Lie groups (1968), The structure of the Lie field associated with a semisimple decomposable Lie algebra (1969), and Characters of unitary representations of Lie groups: Reduction theorems (1969).

  239. Pierre-Simon Girard (1765-1836)
    • Also in 1793, working with Jacques-Elie Lamblardie, he developed a method to measure the resistance of timbers to bending and buckling.

  240. Egnatio Danti biography
    • Danti's description of a machine he made to measure the wind appears in Anemographia.

  241. Barry Johnson biography
    • His mathematical publications started in 1964 with a series of papers on topological algebras, measure algebras and Banach algebras.

  242. Ehrenfried Walter von Tschirnhaus (1651-1708)
    • During his university years he lacked the guidance of a kind, experienced, yet strict teacher, who could have restrained his exuberant temperament, moderate his excessive enthusiasm for Descartes' ideas, and instilled in him a greater measure of self-criticism.

  243. Félix Savary biography
    • The relative positions of the two stars were first accurately measured in 1826 by Friedrich Georg Wilhelm von Struve, who was at that time director of Russia's Dorpat Observatory.

  244. Humphrey Lloyd (1800-1881)
    • The task given to the committee was to establish stations across the world to make simultaneous measurements of geo-magnetism.

  245. Thales of Miletus (about 624 BC-about 547 BC)
    • There are several accounts of how Thales measured the height of pyramids.

  246. George Green (1793-1841)
    • This takes me in some measure from those pursuits which ought to be my proper business, but I hope on my return to lay aside my freshnesses and become a regular Second Year Man.

  247. Nikolai Ivanovich Lobachevsky (1792-1856)
    • Owing to resolute and reasonable measures taken by Lobachevsky the damage to the University was reduced to a minimum.

  248. Eutocius biography
    • One has to assume that indeed Ammonius did approve, for Eutocius went on to write commentaries on other works by Archimedes, namely Measurement of the circle and On plane equilibria.

  249. John Kerr biography
    • I was fortunate beyond measure.

  250. Antonio Bordoni biography
    • There was widespread discontent, and severe measures were imposed to keep control.

  251. John Mauchly biography
    • It contained roughly 18000 vacuum tubes and measured about 2.5 metres in height and 24 metres in length.

  252. Albert Girard (1595-1632)
    • and the rest on the measure of the superficies of spherical triangles and polygons, by him then lately discovered.

  253. Leopold Vietoris biography
    • In the role of a Gletscherknecht, he carried the heavy instruments for geological measurements and set up experiments in countless scientific alpine excursions.

  254. Johannes Kepler (1571-1630)
    • Kepler concluded that the orbit of Mars was an ellipse with the Sun in one of its foci (a result which when extended to all the planets is now called "Kepler's First Law"), and that a line joining the planet to the Sun swept out equal areas in equal times as the planet described its orbit ("Kepler's Second Law"), that is the area is used as a measure of time.

  255. Pierre-Louis Lions biography
    • He introduced certain measures to handle the concentrations.

  256. Elizabeth Fennema biography
    • They published Fennema-Sherman Mathematics Attitude Scales: Instruments designed to measure attitudes towards the learning of mathematics by females and males (1976), Sex-related differences in mathematics achievement, spatial visualization, and affective factors (1977), The study of mathematics among High School girls and boys: Related factors (1977) and Sex-related differences in mathematics achievement and related factors: A further study (1978).

  257. Irving John Good (1916-2009)
    • We mention two further books: (jointly with David B Osteyee) Information, weight of evidence, the singularity between probability measures and signal detection (1974) and Good thinking.

  258. Franz Neumann (1798-1895)
    • He used least squares methods of error analysis of instruments giving new precision to measurements.

  259. Raphaël Salem (1898-1963)
    • We should also note that Salem introduced the idea of a random measure into harmonic analysis.

  260. Mikhael Leonidovich Gromov biography
    • Among the most substantial additions, each taking over a hundred pages, there is a chapter on convergence of metric spaces with measures, and an appendix on analysis on metric spaces written by Semmes.

  261. Édouard Benjamin Baillaud biography
    • It was the responsibility of the Bureau to combine different measurements of Universal Time.

  262. Cora Sadosky biography
    • After a gap in her publication record for the reasons we have explained, Sadosky began publishing again with three papers written jointly with Mischa Cotlar, namely A moment theory approach to the Riesz theorem on the conjugate function with general measures (1975) and Transformee de Hilbert, theoreme de Bochner et le probleme des moments Ⓣ (1977).

  263. Pierre Deligne biography
    • Because of the conciseness of his style and of his habit of never writing the same thing twice (in fact, quite a few of his best ideas have never been written!), the volume of his publications is a true measure of the richness of his scientific production.

  264. Lucien Le Cam biography
    • He wrote on the blackboard seventeen different types of convergence of probability measures.

  265. Jozéf Hoëné Wronski (1778-1853)
    • For good measure, it contains a summary of the "general solution of the fifth degree equation".

  266. Basil Rennie biography
    • he offers a solution to the tomography problem (that is, to find out about the inside of a system through measurements on the outside) for electrical circuits with two-terminal linear components such as resistors.

  267. F F P Bisacre biography
    • The critical (optimum) length of the grating for automatic focusing is determined by the condition that the quadratic term in the expansion for the optical path in powers of distance measured along the grating face from its centre must be three-eighths of a wave-length.

  268. Allen Shields biography
    • Shields worked on a wide range of mathematical topics including measure theory, complex functions, functional analysis and operator theory.

  269. Yuri Ivanovich Manin biography
    • For example, a mathematician represents the motion of planets of the solar system by a flow line of an incompressible fluid in a 54-dimensional phase space, whose volume is given by the Liouville measure ..

  270. Florence Nightingale (1820-1910)
    • The area of each coloured wedge, measured from the centre as a common point, is in proportion to the statistic it represents.

  271. Albert Tucker biography
    • Because you, Sir, embody in extraordinary measure both your profession's love of precision and man's need for conscientious leadership, mathematics in America at all levels is today higher than it was and tomorrow will be higher.

  272. Henry Whitehead (1904-1960)
    • His influence on the development of mathematics during his active lifetime can be partly measured by the innumerable references, implicit and explicit, in current mathematical literature on algebraic and geometric topology; but it could not have been so great without the generosity and enthusiasm which he poured into every mathematical enterprise and which inspired such deep affection in all who knew him well.

  273. Luis Caffarelli biography
    • A standing area of research we pursued was the three-dimensional Navier-Stokes flows (a model for the evolution of viscous, incompressible fluid flows) where we showed that the speed of the flow could become infinite at most on a set of zero one-dimensional measure (that is less than a curve) in space and in time.

  274. al-Karaji biography
    • This occurs in a chapter entitled On measurement and balances for measuring of buildings and structures.

  275. Ernest Hobson (1856-1933)
    • His book Theory of Functions of a Real Variable published in 1907 was the first English book on the measure and integration developed by Baire, Borel and Lebesgue.

  276. Giovanni Battista Riccioli biography
    • First, Grimaldi and Riccioli calibrated a pendulum by getting it to swing for 24 hours (measured by the star Arcturus crossing the meridian line).

  277. John Kingman biography
    • Two important books by Kingman were published in that year, namely Introduction to Measure and Probability (written jointly with S J Taylor) and The Algebra of Queues.

  278. Yakov Davydovich Tamarkin biography
    • Both he and Friedmann were student leaders of strikes at the school in protest at the government's repressive measures against schools.

  279. Fritz Noether biography
    • He clearly satisfied the exemption clause but, on 26 April 1933, a group of students complained to the Rektor of the University of Breslau that having Noether on the staff "in large measure contradict the Aryan principle." The students suggested that Noether, as a Jew, would never work in the national interest.

  280. Sigekatu Kuroda biography
    • He was a mathematician whose role in the mathematical community and whose influence on its development cannot fully be measured by the amount of his published work done.

  281. Zhang Heng biography
    • As Mo notes, the significance here is that all earlier attempts to calculate were based on practical measurement, whereas the work by Zhang was based on a theoretical calculation.

  282. Al-Baghdadi biography
    • It is concerned with the measurement of lengths, areas and volumes.

  283. Karl Reinhardt biography
    • For the German schoolboy has made determinations of area - such as the quadrature of the circle - in his secondary school training long before he has taken up the measurement of the slope of tangent lines.

  284. Wilbur Knorr biography
    • It narrowly avoided bankruptcy with risky measures such as the teachers' union investing $150 million of their pension funds in City securities.

  285. Adam Ries biography
    • Three of the sons, Adam, Abraham and Jacob, became mathematicians working in Annaberg, while Isaac, one of other two sons, became a 'Visierer' (weights and measures master) in Leipzig.

  286. Christian Doppler (1803-1853)
    • It is almost to be accepted with certainty that this will in the not too distant future offer astronomers a welcome means to determine the movements and distances of such stars which, because of their unmeasurable distances from us and the consequent smallness of the parallactic angles, until this moment hardly presented the hope of such measurements and determinations.

  287. George Pólya biography
    • In probability Polya looked at the Fourier transform of a probability measure, showing in 1923 that it was a characteristic function.

  288. Julian Schwinger biography
    • Schwinger's calculation was indeed earlier than and very important for the proper interpretation of these measurements.

  289. Al-Battani biography
    • 53 (1) (1998), 1-49.','5] there is a discussion on how al-Battani managed to produce more accurate measurements of the motion of the sun than did Copernicus.

  290. Yves Rocard biography
    • Rocard had made measurements of EMP (electromagnetic pulse) generated by the first French atomic test, a shot on a tower in the Sahara in what is now Algeria.

  291. Benjamin Osgood Peirce (1854-1914)
    • Big and powerful of body, and ambidextrous, he was in mind also capable and proficient far beyond the ordinary measure of his fellows.

  292. Bruce Kellogg biography
    • It is shown that, measured in an e-weighted energy norm, the Galerkin finite element solution attains the same order of accuracy as the bilinear nodal interpolant.

  293. Theodorus (465 BC-398 BC)
    • If, when the lesser of two unequal magnitudes is continually subtracted in turn from the greater, that which is left never measures the one before it, the magnitudes will be incommensurable.

  294. al-Hasan Banu Musa biography
    • One objective was to measure a curved area while the other was to study the geometric properties of curves.

  295. Jen Egerváry biography
    • In geometry, he examined the important and deep question about the curvature of metric curves, which is related to measurement and applications.

  296. Edwin Abbott Abbott (1838-1926)
    • Suppose a person of the Fourth Dimension, condescending to visit you, were to say, 'Whenever you open your eyes, you see a Plane (which is of Two Dimensions) and you infer a Solid (which is of Three); but in reality you also see (though you do not recognise) a Fourth Dimension, which is not colour nor brightness nor anything of the kind, but a true Dimension, although I cannot point out to you its direction, nor can you possibly measure it.' What would you say to such a visitor? Would not you have him locked up? Well, that is my fate: and it is as natural for us Flatlanders to lock up a Square for preaching the Third Dimension, as it is for you Spacelanders to lock up a Cube for preaching the Fourth.

  297. William Whiston (1667-1752)
    • He also anticipated Graham's 1723 measurement of the ratio of horizontal and vertical magnetic intensities, and he understood their relationship to magnetic inclination.

  298. Max Dehn biography
    • I fear it would, once again, not resist an unjust measure coming from outside.

  299. Cleomedes biography
    • Of course not everyone believes that the story of Eratosthenes' measurement is authentic but, despite this, it is widely accepted.

  300. Louis de Branges biography
    • The condition appears, for example, in the structure theory of plane measures with respect to which the Newton polynomials form an orthogonal set.

  301. Howard Percy Robertson biography
    • He also discusses the geometry of the thermal field in terms of measurements made by rods after they have been allowed to come into thermal equilibrium with a heated medium.

  302. Dunham Jackson biography
    • measures up fully to the expectation of the reader who is acquainted (and who is not?) with the lucid and elegant presentation - oral and written - of Dunham Jackson.

  303. Hermann Hankel (1839-1873)
    • Hankel looked at Riemann's integration theory and restated it in terms of measure theoretic concepts.

  304. Bernhard Riemann (1826-1866)
    • We considered it our duty to turn the attention of the Academy to our colleague whom we recommend not as a young talent which gives great hope, but rather as a fully mature and independent investigator in our area of science, whose progress he in significant measure has promoted.

  305. Vladimir Maz'ya biography
    • The book is in two parts, the first is on the higher-dimensional potential theory and the solution of the boundary problems for regions with irregular boundaries while the second part is on the space of functions whose derivatives are measures.

  306. Lennart Carleson biography
    • As so often in his work, not only did he solve the problem but in doing so he introduced what are today called 'Carleson measures' which went on to become a fundamental tool in complex analysis and harmonic analysis.

  307. Joseph Ritt biography
    • The Observatory measured the position of celestial objects for purposes of timekeeping and navigation and in 1904, six years before Ritt took a job there, the Observatory broadcast the world's first radio time signals.

  308. William Cochran biography
    • it was a measure of good sense that he accepted my argument that a PhD, even from Cambridge, was little evidence of research ability, and that Cambridge had at that time little to teach him in statistics that could not be much better learnt from practical work in a research institute.

  309. Farkas Bolyai biography
    • I have measured that bottomless night, and all the light and all the joy of my life went out there.

  310. Tartaglia (1500-1557)
    • For good measure, he added a few malicious personal insults directed against Cardan.

  311. Samarendra Nath Roy biography
    • The monograph deals with samples of fixed size, and the main emphasis is on obtaining confidence bounds on certain parametric functions that are a set of natural measures of deviation from a null hypothesis.

  312. Aleksandr Aleksandrov biography
    • those properties that appear as a result of measurements carried out on the surface) of an arbitrary convex surface had to be studied, and methods found for the proof of theorems on the connection between intrinsic and exterior properties of convex surfaces.

  313. Liu Hong biography
    • His measurements of the length of the shadow of a pole at the summer and at the winter solstices give results which are accurate to within 1% of their true value.

  314. Thomas Clausen biography
    • Clausen became an assistant at Altona Observatory in 1824 and in October of that year he met Carl Friedrich Gauss, who was conducting geodesic measurements nearby, for the first time.

  315. Emilio Baiada biography
    • We have mentioned some of Baiada's publications above but we note that his output totals 60 scientific publications on a wide range of different fields in analysis: ordinary and partial differential equations, Fourier series and the series expansion of orthonormal functions, topology, real analysis, functional analysis, calculus of variations, measure and integration, optimisation, and the theory of functions.

  316. Joseph Boussinesq biography
    • He was aware of Russell's observations, and also of the more precise measurements of solitary waves performed by the French hydraulician Henry Bazin.

  317. Geoffrey Taylor (1886-1975)
    • He took the opportunity to take a whole range of measurements of pressure, humidity and temperature on which he was later to base his theoretical model of turbulent mixing of the air.

  318. Wadysaw Orlicz (1903-1990)
    • Orlicz's contribution is important in the following areas in mathematics: function spaces (mainly Orlicz spaces), orthogonal series, unconditional convergence in Banach spaces, summability, vector-valued functions, metric locally convex spaces, Saks spaces, real functions, measure theory and integration, polynomial operators and modular spaces.

  319. William Rankine (1820-1872)
    • History Topics: The history of measurement .

  320. Nicolaus Mercator (1620-1687)
    • He also made measurements of air pressure for the Royal Society and got to know Hooke with whom he shared many common interests.

  321. Christiaan Huygens (1629-1695)
    • In 1656 he patented the first pendulum clock, which greatly increased the accuracy of time measurement.

  322. T J J See biography
    • He published the result in The Astronomical Journal and it was only after the Hipparchus satellite measured the binary in 1997 at 303.0° and 0.226" that See's old paper was noticed and the accuracy of his observations (which had been essentially ignored) were realised.

  323. Fritz Ursell biography
    • The waves are measured by the fluctuating pressure which they produce upon an instrument laid on the sea bed in shallow water near the coast.

  324. Filadelfo Insolera biography
    • This small volume deals with a large variety of topics, including approximate computation, averages, measures of dispersion, permutations, combinations, probability, binomial distribution of frequency, interpolation by the formulas of Newton and Lagrange, graduation of data, least squares, moments, correlation and contingency.

  325. Michael Mästlin (1550-1631)
    • For example he attempted to measure the parallax of a supernova and, having failed to find any, deduced that it was as far away as the "fixed stars".

  326. James Stirling (1692-1770)
    • And hitherto I have not been able to reconcile the measurements made in the north to the theory..

  327. Edmund Gunter (1581-1626)
    • He assumed an error in Borough's measurements, but this was in fact the first observation of temporal change in magnetic variation, a contribution acknowledged by his successor, Henry Gellibrand, who discovered the phenomenon.

  328. Jean d'Alembert biography
    • History Topics: The history of measurement .

  329. Anatolii Volodymyrovych Skorokhod biography
    • All of the most important fundamental concepts and principles are briefly and expertly presented with uncompromising measure-theoretic rigour.

  330. Abigail Thompson biography
    • One tenet of this movement is that standardized tests are an inappropriate way to measure mathematical skills.

  331. George Mackey biography
    • Early in his career Mackey worked on the duality theory of locally convex spaces publishing papers which include On infinite dimensional linear spaces (1943), On convex topological linear spaces (1943), Equivalence of a problem in measure theory to a problem in the theory of vector lattices (1944), (with Shizuo Kakutani) Ring and lattice characterization of complex Hilbert space (1946), On convex topological linear spaces (1946).

  332. Jakob Rebstein biography
    • Moreover, he also chaired the commission of geometers who measured the perimeter, i.e.

  333. Horst Sachs biography
    • Whereas most of us would enjoy a beautiful proof but consider a problem to be solved if there is just any correct proof, Sachs took a step further and considered a problem to be finally settled only if it comes with a beautiful solution, in terms of transparency and precision (and other, more subjective measures).

  334. Jerzy Neyman (1894-1981)
    • In Paris for session 1926-27 Neyman attended lectures by Borel, Lebesgue (whose lectures he particularly enjoyed) and Hadamard and his interests began to move back towards sets, measure and integration.

  335. Gavin Brown biography
    • To give an idea of the topics he now works on we give the titles of the four papers he published in 2004: The maximal Riesz, Fejer, and Cesaro operators on real Hardy spaces; Lebesgue measure of sum sets - the basic result for coin-tossing; The maximal Fejer operator on real Hardy spaces; and Approximation on two-point homogeneous spaces.

  336. Vera Nikolaevna Faddeeva biography
    • Following her graduation, Faddeeva worked at the Leningrad Board of Weights and Measures (as her husband had done in the previous couple of years).

  337. Claude Mydorge (1585-1647)
    • One of Mydorge's most famous results was an extremely accurate measurement of the latitude of Paris.

  338. Edwin Olds biography
    • A little statistics and probability entered the physical-measurements course, and somewhere along the way we were asked to compute the probability of casting a total of 9 and of casting a total of 10 using three ordinary dice rather than two.

  339. Harish-Chandra (1923-1983)
    • Some major contributions by Harish-Chandra's work may be singled out: the explicit determination of the Plancherel measure for semisimple groups, the determination of the discrete series representations, his results on Eisenstein series and in the theory of automorphic forms, his "philosophy of cusp forms", as he called it, as a guiding principle to have a common view of certain phenomena in the representation theory of reductive groups in a rather broad sense, including not only the real Lie groups, but p-adic groups or groups over adele rings.

  340. Egbert van Kampen biography
    • Five papers appeared in 1940, one of them a major article over 30 pages in length in the American Journal of Mathematics with the title Infinite product measures and infinite convolutions.

  341. Regiomontanus (1436-1476)
    • These books were reprinted many times and had great influence, for example both Christopher Columbus and Amerigo Vespucci used Regiomontanus's Ephemerides to measure longitudes in the New World.

  342. Beniamino Segre biography
    • Article 4 of the Royal Decree Law of 5 September 1938 was titled 'Measures for the defence of race in fascist schools' and, after Segre had been identified as Jewish by the University of Bologna, he was expelled from the University on 16 October 1938.

  343. Robert Geary biography
    • Economics must therefore be converted with all due speed into an experimental science and the essence of science is measurement.

  344. Edward Witten biography
    • It is a measure of Witten's mastery of the field that he has been able to make intelligent and skilful use of this difficult point of view in much of his subsequent work.

  345. Kathleen Ollerenshaw (1912-2014)
    • It was here she learned of the Stanford research project that was being established to measure the standards of mathematics teaching in countries across the world, precisely Kathleen's area of interest.

  346. Marquis de Condorcet biography
    • History Topics: The history of measurement .

  347. Guido Grandi (1671-1742)
    • Grandi gave the curve the name Scala, the scale curve, because it can serve as a measure of light intensity.

  348. Giuseppe Pompilj biography
    • He did not assume the 'degree of confidence in the occurrence of an event' as a definition of probability, but only as a measure of the probability itself [',' G Dall’Aglio, Giuseppe Pompilj (Italian), in Studi di probabilita, statistica e ricerca operativa in onore di Giuseppe Pompilj (Edizioni Oderisi, Gubbio, 1971), 1-16.','6]:- .

  349. Robert Rankin biography
    • The problem is to devise a mathematical theory which will, after experimental measurement of suitable constants, predict position, velocity, angular position, and angular velocity of a rocket at the end of burning from a knowledge of these and other physical data at the beginning of burning.

  350. Ibrahim ibn Sinan (908-946)
    • In On the measurement of the parabola Ibrahim ibn Sinan gives a beautiful proof that the area of a segment of the parabola is four-thirds of the area of the inscribed triangle.

  351. Simon Stevin (1548-1620)
    • Stevin states that the universal introduction of decimal coinage, measures and weights would only be a matter of time (but he probably would be amazed to know that in the 21st century some countries still resist adopting decimal systems).

  352. Benedetto Castelli (1578-1643)
    • This second notion has not yet been fully acknowledged because Castelli only hinted at it by criticizing Giovanni Fontana's measurements of the Tiber and his use of the term 'acqua premuta'.

  353. Omar Khayyam (1048-1131)
    • Khayyam measured the length of the year as 365.24219858156 days.

  354. Prasanta Chandra Mahalanobis biography
    • He saw that statistics was a new science connected with measurements and their analysis, and as such capable of wide application.

  355. Wilhelm Lexis biography
    • As Stigler writes in [',' S M Stigler, The measurement of uncertainty in nineteenth-century social science, in The probabilistic revolution Vol.

  356. Aldo Andreotti biography
    • Then barely twenty-seven years old, Andreotti had reached an absolutely stunning maturity, coming in large measure from long months spent in the United States close to remarkable mathematicians ..

  357. Oliver Byrne biography
    • Byrne tried to get a paper "How to measure the Earth with the assistance of Railroads" accepted for publication.

  358. Kamalakara (about 1616-about 1700)
    • It deals with the topics of: units of time measurement; mean motions of the planets; true longitudes of the planets; the three problems of diurnal rotation; diameters and distances of the planets; the earth's shadow; the moon's crescent; risings and settings; syzygies; lunar eclipses, solar eclipses; planetary transits across the sun's disk; the patas of the moon and sun; the "great problems"; and a final chapter which forms a conclusion.

  359. Roger Cotes (1682-1716)
    • Cotes discovered an important theorem on the nth roots of unity, gave the continued fraction expansion of e, invented radian measure of angles, anticipated the method of least squares, published graphs of tangents and secants, and discovered a method of integrating rational fractions with binomial denominators.

  360. Guglielmo Righini biography
    • Not only the place of long residence was in common between Galileo and Guglielmo Righini; also their interest in applying and devising technological devices to improve astronomical observations and measurements.

  361. George Chrystal (1851-1911)
    • He supervised the whole of the work and designed special instruments to carry out the measurements.

  362. Mauro Picone biography
    • university life was painfully interrupted for seven years, from 1938 to 1945, because of those senseless racial measures which deprived Italy, in that long and difficult period, of the precious work of citizens of very high moral, spiritual and intellectual standing.

  363. Hannes Alfvén biography
    • In 1940 he was appointed as professor of electromagnetic theory and electric measurements at the Royal Institute of Technology in Stockholm.

  364. Nina Bari biography
    • The fifteen chapters of the book are: Basic concepts and theorems; Fourier coefficients; Convergence of a Fourier series at a point; Fourier series of continuous functions; Convergence and divergence of a Fourier series on a set; "Correcting" a function on a set of small measure; Summability of Fourier series; Conjugate trigonometric series; Absolute convergence of Fourier series; Sine and cosine series with decreasing coefficients; Lacunary series; Convergence and divergence of general trigonometric series; Absolute convergence of general trigonometric series; The problem of uniqueness of the expansion of a function in a trigonometric series; and Representation of functions by trigonometric series.

  365. Baha' al-Din al-Amili biography
    • In 1587 he was still in Isfahan for he completed a work on weights and measures there in that year [',' D J Stewart, The lost biography of Baha’ al-Din al-’Amili and the reign of Shah Isma’il II in Safavid historiography’, Iranian Studies 31 (2) (1998), 177-205.','10]:- .

  366. Alexander Friedmann (1888-1925)
    • Friedmann and Tamarkin were student leaders of strikes at the school in protest at the government's repressive measures against schools.

  367. Vivienne Malone-Mayes biography
    • it took a faith in scholarship almost beyond measure to endure the stress of earning a Ph.D.

  368. Gustav Herglotz biography
    • In this last paper Herglotz solved Abel's integral equation which results from the inversion of measured seismic travel times into a velocity-depth function.

  369. Christopher Clavius biography
    • Clavius also produced a number of instruments, perhaps the most interesting being an instrument to measure fractions of angles.

  370. Luca Pacioli (1445-1517)
    • An encyclopaedic work (600 pages of close print, in folio) written in Italian, it contains a general treatise on theoretical and practical arithmetic; the elements of algebra; a table of moneys, weights and measures used in the various Italian states; a treatise on double-entry bookkeeping; and a summary of Euclid's geometry.

  371. Mstislav Vsevolodovich Keldysh biography
    • Chapter IV contains an exposition of the harmonic measure and integral representation of the generalized solution.

  372. Cheng Dawei biography
    • In problems of piles on the ground, against a wall, at an inner corner or an outer corner, the ancients always measured their altitude and then calculated.

  373. Guido Ascoli biography
    • Article 4 of the Royal Decree Law of 5 September 1938 was titled 'Measures for the defence of race in fascist schools' and, after Ascoli had been identified as Jewish by the University of Milan, he was expelled from the University.

  374. Alexander Andreevich Samarskii biography
    • Beginning in 1928 special measures were introduced against peasants as the country moved to industrialisation.

  375. Alexander Oppenheim biography
    • But it is also due in great measure to the active co-operation and goodwill of individuals, such as Dr Alexander Oppenheim, Vice-Chancellor of the University of Malaya, whom we gladly welcome into our society today.

  376. Lewis Fry Richardson (1881-1953)
    • In particular, neither armed might nor collective security measures (contrary to widespread opinion) emerge as significant war-preventing influences.

  377. Adolf Lindenbaum biography
    • Courses taught by Lindenbaum were on set theory, measure theory, algebra, actuarial mathematics and the foundations of mathematics.

  378. Giuseppe Basso biography
    • However, in 1872 Gilberto Govi, the professor of experimental physics, was appointed to the International Commission of Weights and Measures, meaning that he had to spend long periods of time in Paris.

  379. Aleksei Krylov (1863-1945)
    • In a paper on forced vibrations of fixed-section pivots (1905), he presented an original development of Fourier's method for solving boundary value problems, pointing out its applicability to a series of important questions: for example, the theory of steam-driven machine indicators, the measurement of gas pressure in the conduit of an instrument, and the twisting vibrations of a roller with a flywheel on its end.

  380. Charles Fefferman biography
    • These papers were not his first publications for, in addition to the paper mentioned above, he had already published A Radon-Nikodym theorem for finitely additive set functions (1967) and Lp spaces over finitely additive measures (1968), both appearing in the Pacific Journal of Mathematics.

  381. Louis Bachelier biography
    • measure theory and axiomatic probability) although, his results were basically correct.

  382. Thomas Craig biography
    • Undoubtedly the intense ardour with which he engaged in this work contributed in large measure to that impairment of the nervous system from which he had recently suffered.

  383. Dorothy Wrinch biography
    • A measure of her research activity during this period is that from 1918 to 1932 Wrinch published twenty papers on pure and applied mathematics, and sixteen papers on scientific methodology and the philosophy of science.

  384. Athanase Dupré (1808-1869)
    • A statistical method gives for one part of common soap in 5000 of water a surface tension about one-half as great as for pure water, but if the tension be measured on a jet close to the orifice, the value (for the same solution) is sensibly identical with that of pure water.

  385. Shen Kua (1031-1095)
    • He set up a programme which would measure the positions of the moon and planets, plotting exact coordinates three times a night for five years.

  386. Henry More biography
    • I have measured myself from the height to the depth; and what I can do, and what I ought to do, and I do it.

  387. John William Strutt (1842-1919)
    • During the last few years much interest has been felt in the reduction to an absolute standard of measurements of electromotive force, current, resistance, etc.

  388. Georges Reeb biography
    • He made important contributions to a variety of fields: topology, measure theory, descriptive set theory, potential theory, and functional analysis.

  389. Archibald Milne biography
    • Where formerly a businessman would engage a boy, largely as the result of a personal interview, he now not infrequently demanded of him that he should show a measure of scholastic success.

  390. William Youden (1900-1971)
    • In fact over the first few years that he worked at the Institute Youden became more and more disillusioned with the way that measurements were made in biology.

  391. Bartel van der Waerden biography
    • In itself I have nothing against German citizenship, however, at this moment, since Germany has occupied my homeland, I would not gladly give up my previous neutrality and throw myself in a certain measure publicly on the German side.

  392. Andrei Yuryevich Okounkov biography
    • Okounkov gave the first proof of the celebrated Baik-Deift-Johansson conjecture, which states that the asymptotics of random partitions distributed according to the Plancherel measure coincides with that of the eigenvalues of large Hermitian matrices.

  393. Frank Adams (1930-1989)
    • After taking his first degree he started graduate work at Cambridge with Besicovitch on geometric measure theory.

  394. Yaroslav Borisovich Lopatynsky biography
    • Lopatynsky's first papers were on analysis: On uniform convergence (1929); Embedding of a Riemannian space in a Euclidean space (1934); (with L P Fridolina) The justification of mathematics, a critical situation (1934); and Limiting values of an analytic function on a set of singular points of measure greater than zero on a rectified curve (1935).

  395. Michael Freedman biography
    • Then he published papers such as: Quantum computation and the localization of modular functors (2001); Projective plane and planar quantum codes (2001); Poly-locality in quantum computing (2002); Simulation of topological field theories by quantum computers (2002); Topological quantum computation (2003); Approximate Counting and Quantum Computation (2005); Topological quantum computation (2006); Topological quantum computing with only one mobile quasiparticle (2006); Interacting anyons in topological quantum liquids: The golden chain (2008); Measurement-Only Topological Quantum Computation (2008); and Topological Phase in a Quantum Gravity Model (2008).

  396. James Jeans biography
    • Of course Jeans' paper can be seen as a mathematical "proof" that classical physics does not suffice, but it is interesting to note that his pre-quantum ideas concerning the very long time required for systems to come into equilibrium and the observed breakdown of equipartition in specific heat measurements on molecular gases have been used again in relatively recent times more than 80 years after Jeans introduced them.

  397. Paolo Frisi (1728-1784)
    • This work was remarkable since it discussed the figure of the earth and whether there was a equatorial bulge, not just from geodetic measurements but from a discussion of the movement of the Earth and the effect that had on its shape.

  398. Carl Boyer biography
    • The first of these papers are: Early estimates of the velocity of light (1939), A vestige of Babylonian influence in thermometry (1942), Cavalieri, limits and discarded infinitesimals (1942), Early principles in the calibration of thermometers (1942), An early reference to division by zero (1943), Fractional indices, exponents, and powers (1943), History of the measurement of heat (1943), and Pascal's formula for the sums of powers of the integers (1943).

  399. David Bates (1916-1994)
    • In his studies, Bates combined physical insight with mathematical formulations constructed so that numerical calculations could be carried out to enable quantitative comparisons to be made of theory and measurement.

  400. Emma Castelnuovo biography
    • The problems he treats are the measurement of lands.

  401. John Hadley biography
    • This motivated Hadley to tackle the problem and in 1730 he invented the reflecting octant which measured the altitude of the sun or of a star.

  402. Luther Eisenhart (1876-1965)
    • The intimate atmosphere which surrounded him, its very serenity, was due in large measure to the care and devotion which he received from Mrs Eisenhart.

  403. Abraham bar Hiyya (1070-1136)
    • Abraham bar Hiyya is famed for his book Hibbur ha-Meshihah ve-ha-Tishboret (Treatise on Measurement and Calculation), translated into Latin by Plato of Tivoli as Liber embadorum in 1145.

  404. John Jackson biography
    • Jackson retired in 1950, having measured the distances of over 1600 southern stars with greatly improved accuracy and having advanced or completed all the major projects which had been going on at the time of his appointment.

  405. Julius Schauder (1899-1943)
    • After obtaining his doctorate in 1923 with a thesis The theory of surface measure, he taught both in a secondary school and worked for an insurance firm.

  406. Lochlainn O'Raifeartaigh biography
    • The articles illustrate that the reassessment of gauge-theory, due in a large measure to Weyl, constituted a major philosophical as well as technical advance.

  407. Giovanni Magini biography
    • In 1592 Magini published De Planis Triangulis which explains the use of quadrants in astronomy and in surveying, in particular describing details of calculations and measurements which could be performed with a quadrant.

  408. Andrey Kolmogorov biography
    • Almost simultaneously [Kolmogorov] exhibited his interest in a number of other areas of classical analysis: in problems of differentiation and integration, in measures of sets etc.

  409. Duro Kurepa biography
    • Chapter 4 is on topological and metric spaces, with the fifth and final chapter on limiting processes in analysis, measure theory, Borel and Souslin sets.

  410. Karl Pearson (1857-1936)
    • Pearson used large samples which he measured and the tried to deduce correlations in the data.

  411. Siméon-Denis Poisson (1781-1840)
    • It is quite unlikely that he ever attempted an experimental measurement, nor did he try his hand at drafting experimental designs.

  412. Pierre Samuel biography
    • Unlike the first, however, the second volume is concerned in large measure with those parts of commutative algebra that are the fruits of its union with algebraic geometry ..

  413. Charles Sims biography
    • It is a measure of how far the subject has progressed in the past 20 years that it would now take at least four substantial books to cover the field, not including the necessary background material on group theory and the design and analysis of algorithms.

  414. David Cariolaro biography
    • Carlo Bertoluzza and Cariolaro published the joint paper On the measure of a fuzzy set based on continuous t-co-norms in 1997.

  415. Jacques Le Tenneur (1610-1660)
    • Le Tenneur then argues that Galileo's theory is superior to that of Fabri since it did not depend of the unit of time used for measurements.

  416. Dionysodorus (about 250 BC-about 190 BC)
    • In Natural history Pliny mentions a certain Dionysodorus who measured the earth's radius and gave the value 42000 stades.

  417. Otto Haupt biography
    • In their early semesters his students learned very quickly that half-measures and superficialities have no place in the work of a mathematician.

  418. Abu'l-Wafa biography
    • comprises all that an experienced or novice, subordinate or chief in arithmetic needs to know, the art of civil servants, the employment of land taxes and all kinds of business needed in administrations, proportions, multiplication, division, measurements, land taxes, distribution, exchange and all other practices used by various categories of men for doing business and which are useful to them in their daily life.

  419. Griffith Evans (1887-1973)
    • He pioneered the use of general notions of integration and measure theory in this area, and his interests lay in application and development of new techniques rather than in deep structural theorems.

  420. Christoph Scheiner (1573-1650)
    • They are not terribly exact, but rather are hand-drawn on paper as they appeared to the eye without certain and precise measurement, which could not be done sometimes due to the inclement and inconstant weather, sometimes due to the lack of time, and at other times due to other impediments.

  421. Madan Lal Puri biography
    • Professor Puri's research has lead to his being considered one of the most versatile and prolific researchers in the world in mathematical statistics in the areas of nonparametric statistics, order statistics, limit theory under mixing, time series, splines, tests of normality, generalized inverses of matrices and related topics, stochastic processes, statistics of directional data, random sets, and fuzzy sets and fuzzy measures.

  422. Edward Copson biography
    • His influence in and beyond St Andrews can be measured by the number of members of university departments, not all in mathematics, who were his pupils.

  423. James MacCullagh biography
    • Franz Neumann's paper is very elaborate, and supersedes, in a great measure, the design which I had formed of treating the subject more fully at my leisure..

  424. Andrew Gleason biography
    • Let us also mention his papers Measures on the closed subspaces of a Hilbert space (1957) and Projective topological spaces (1958) as further important contributions.

  425. Srinivasa Varadhan biography
    • It addresses a fundamental question: what is the qualitative behaviour of a stochastic system if it deviates from the ergodic behaviour predicted by some law of large numbers or if it arises as a small perturbation of a deterministic system? The key to the answer is a powerful variational principle that describes the unexpected behaviour in terms of a new probabilistic model minimizing a suitable entropy distance to the initial probability measure.

  426. Jaroslav Hájek biography
    • Hajek developed the property of sequences of pairs of probability measures from ideas due to de la Vallee Poussin.

  427. Richard Hamming biography
    • His interests were in analysis, particularly measure theory, integration and differential equations.

  428. Charles Augustin Coulomb (1736-1806)
    • At about the same time that the Academie des Sciences was abolished in August 1783, he was removed from his role in charge of the water supply and, in December 1793, the weights and measures committee on which he was serving was also disbanded.

  429. Ferdinand Joachimsthal (1818-1861)
    • Bessel was an astronomer who was the first to measure to parallax of a star.

  430. Dimitrie Pompeiu biography
    • However, Pompeiu's doctoral thesis written in the same year proved the existence of certain analytic functions which could be extended continuously on their set of singularities even though this set had positive measure.

  431. Robion Kirby biography
    • He got a B in measure theory, given by Halmos, and a C in algebraic topology, given by Eldon Dyer.

  432. Nicolas-François Canard biography
    • Different (unmeasurable) qualities of labour, however, render labour quantity an unsatisfactory measure.

  433. Nathaniel Bliss (1700-1764)
    • He had a great interest in improving clocks, which he saw as important to improve astronomical measurements.

  434. Karl Mollweide biography
    • To correct these defects, Mollweide drew his elliptical projection; but in preserving the correct relation between the areas he was compelled to sacrifice configuration and angular measurement.

  435. Alessandro Faedo biography
    • We have already seen that he made contributions to a wide variety of areas such as the calculus of variations, the theory of linear ordinary differential equations, the theory of partial differential equations, measure theory, the Laplace transform for functions of several variables, questions relating to existence for linear equations in Banach spaces, and foundational problems such as his work on Zermelo's principle in infinite-dimensional function spaces.

  436. Vladimir Aleksandrovich Marchenko biography
    • Also in the 1950s he studied the asymptotic behaviour of the spectral measure and of the spectral function for the Sturm-Liouville equation.

  437. Theodore Samuel Motzkin biography
    • In many publications on this topic Motzkin examined a wide variety of different ideas, including new measure of closeness of approximation.

  438. Aurel Wintner (1903-1958)
    • These were Lectures on asymptotic distributions and infinite convolutions (1938), Analytical foundations of celestial mechanics (1941), Eratosthenian averages (1943), Theory of measure in arithmetical semigroups (1944), The Fourier transforms of probability distributions (1947), and An arithmetical approach to ordinary Fourier series (1945).

  439. Clement Durell (1882-1968)
    • Each subject - including the velocity of light, the measurement of time and distance, and the properties of mass and momentum - is illustrated with diagrams, formulas, and examples.

  440. Zhao Youqin biography
    • He also describes an instrument he has designed to measure the difference in right ascension between objects on the celestial sphere.

  441. R A Fisher (1890-1962)
    • A strong advocate of measures to counter this trend, he proposed that family allowances should be proportional to income to support the well-adapted healthy members of society.

  442. al-Khujandi biography
    • He described his measurements in detail in a treatise On the obliquity of the ecliptic and the latitudes of the cities.

  443. Vijayanandi (about 940-about 1010)
    • It deals with the topics of: units of time measurement; mean and true longitudes of the sun and moon; the length of daylight; mean longitudes of the five planets; true longitudes of the five planets; the three problems of diurnal rotation; lunar eclipses, solar eclipses; the projection of eclipses; first visibility of the planets; conjunctions of the planets with each other and with fixed stars; the moon's crescent; and the patas of the moon and sun.

  444. Henry Gellibrand (1597-1637)
    • He achieved this by comparing measurement he took in Deptford with similar ones taken by Gunter twelve years earlier.

  445. Hermann Bondi biography
    • Bondi, among other things, led a research team with all its cumbersome equipment to the top of Snowdon (Hoyle's idea), to make systematic measurements of these effects.

  446. Johann Benedict Listing biography
    • By topology we mean the doctrine of the modal features of objects, or of the laws of connection, of relative position and of succession of points, lines, surfaces, bodies and their parts, or aggregates in space, always without regard to matters of measure or quantity.

  447. Stanley Jevons (1835-1882)
    • This involved determining the characteristics such as weight, measure, or quality of the coinage, and Jevons was offered the post because of his already impressive abilities at chemistry.

  448. Beppo Levi biography
    • To this already broad range of work, he added contributions to subjects such as number theory, electrical engineering, the theory of physical measurements, and theoretical physics.

  449. Kunihiko Kodaira biography
    • At this time Kodaira was interested in topology, Hilbert spaces, Haar measure, Lie groups and almost periodic functions.

  450. Klaus Fuchs biography
    • The conductivity of thin films of the alkali metals has recently been measured in the H W Wills Physical Laboratory, Bristol.

  451. Nathan Fine biography
    • It is a measure of the breadth of this field that after the respective first chapters there is virtually no overlap between these books.

  452. Ruggero Giuseppe Boscovich biography
    • Boscovich, who attended meetings of the Academy of Sciences during his stay in Paris, was known in France for his studies on astronomy, the aurora borealis, and the measurement of the arch of the meridian through Rome and Rimini which he had carried out in 1739.

  453. Denis Papin (1647-1712)
    • According to Boyle's own testimony, this is what Papin did: he designed and constructed the particular instrument used in these experiments; he operated the instrument, either by himself or with the assistance of other technicians; he measured and recorded almost all of the experimental phenomena; and he planned and organized a great part of the experiments to be per formed.

  454. Aryabhata II biography
    • Aryabhata II constructed a sine table correct up to five decimal places when measured in decimal parts of the radius, see [',' S K Jha and V N Jha, Computation of sine-table based on the Mahasiddhanta of Aryabhata II, J.

  455. Duncan Sommerville (1879-1934)
    • the classification of all types on non-euclidean geometry (including those usually excluded as bizarre), the extension, involving the measurement of generalised angles in higher space, of Euler's Theorem on polyhedra, space filling figures, the classification of polytopes (i.e.

  456. Benjamin Peirce (1809-1880)
    • Peirce examined the thirty downward lines in the signature and measured the angle of each.

  457. Ralph Fox biography
    • The influence of a great teacher and a superb mathematician is measured by his published works, the published works of his students, and perhaps foremost, the mathematical environment he fostered and helped to maintain.

  458. Cheryl Praeger biography
    • But how smart are mathematicians? How can we measure what makes a mathematician "good"? .

  459. Gerard Murphy biography
    • The reader is assumed to have a good background in real and complex analysis, point set topology, measure theory, and elementary functional analysis.

  460. Joseph Pérès (1890-1962)
    • In addition to experimental work on propeller blades in the new low-turbulence wind tunnel facility, Peres hoped to extend to three dimensions the analog voltage and current equivalent measurements that are analogous to speed voltages in incompressible fluids surrounding subsonic aircraft in motion.

  461. Fabian Franklin biography
    • This is in a great measure unavoidable from the nature of the case; and yet the contrast I have just mentioned seems to give appropriateness to the subject to which I shall venture to ask your attention for a few minutes - the need of exact thinking in the discussions of actual life.

  462. Georg Peurbach (1423-1461)
    • He measured the duration of the eclipse and then found the time of the midpoint.

  463. Charles Galton Darwin (1887-1962)
    • Darwin was openly torn between two beliefs - that eugenic measures were a crucial necessity as natural selection became largely suspended for much of mankind; and the desperate conclusion that all attempts to create a eugenically-determined world were almost certainly doomed to failure.

  464. Pappus (about 290-about 350)
    • Certainly an instrument to measure liquids is attributed to him.

  465. Vyacheslaw Vassilievich Stepanov (1889-1950)
    • In works published in 1923 and 1925 Stepanov established the necessary and sufficient conditions under which a function of two variables, defined on a measurable plane set of finite measure greater than zero, possesses a total differential almost everywhere on that set.

  466. John Airey biography
    • John R Airey, Sines and cosines of angles in circular measure, British Association for the Advancement of Science Report 1916, 60-91.

  467. Martha Betz Shapley biography
    • continued with measures of the positions of photographic meteors, and computations of their orbits, to provide readings of atmospheric density in the 50- to 100-mile-up area.

  468. Isaac Barrow biography
    • The final lectures cover measurement, proportion and ratio.

  469. Giorgio Bidone biography
    • It had a tower seven metres high that allowed observations and measurements of phenomena related to the control of irrigation.

  470. Wilhelm Schickard biography
    • He also undertook land surveying of the duchy of Wurttemberg which involved the first use of Willebrord Snell's triangulation method in geodesic measurements; see [',' G Betsch, Sudwestdeutsche Mathematici aus dem Kreis um Michael Mastlin, in Der ’mathematicus’: Zur Entwicklung und Bedeutung einer neuen Berufsgruppe in der Zeit Gerhard Mercators, Schloss Krickenbeck, 1995 (Brockmeyer, Bochum, 1996), 121-150.','6] for further details.

  471. Mark Kac biography
    • When I announced that I was going to write my own derivation, my father offered me a reward of five Polish zlotys (a large sum and no doubt the measure of his scepticism).

  472. Henry Ernest Dudeney (1857-1930)
    • How could he have done it? There is no necessity to give measurements, for if the smaller piece (which is half a square) be made a little too large or small, it will not effect the method of solution.

  473. Albert of Saxony biography
    • Pope Gregory XI took vigorous measures against heresies, particularly in Germany, and he contacted the German Inquisitors in 1372 asking that they investigate a charge of determinism (a belief that man has no free will and therefore is not responsible for his actions) which had been made against Albert.

  474. David van Dantzig biography
    • The government rapidly appointed a committee, consisting of prominent hydraulic engineers under the chairmanship of A G Maris, in order to design measures for preventing similar disasters in the future.

  475. Roger Apéry biography
    • It is some measure of Apery's achievement that these questions have been considered by mathematicians of the top rank over the past few centuries without much success being achieved.

  476. Annie Scott Dill Maunder (1868-1947)
    • Part of her job was to examine and measure the daily sunspot photographs.

  477. Lorenzo Mascheroni (1750-1800)
    • The French had been working on the introduction of the metric system of weights and measures and on 7 April 1795 the National Convention had passed a law introducing the metric system putting Legendre in charge of the transition to the new system.

  478. Oenopides (about 490 BC-about 420 BC)
    • It does not appear that Oenopides made any measurement of the obliquity of the ecliptic.

  479. Alexis Clairaut biography
    • From 20 April 1736 to 20 August 1737 Clairaut had taken part in an expedition to Lapland, led by Maupertuis, to measure a degree of longitude.

  480. Thomas Hobbes (1588-1679)
    • His approach is certainly consistently materialistic, denying abstract ideas; for Hobbes mathematics is the study of quantity, and quantities are the measures of 3-dimensional bodies.

  481. Monteiro da Rocha biography
    • Thus he dealt with the problem of parabolic orbits of comets and gave the first practical solution to this problem; he took up the problem of predicting eclipses and gave an easier method to solve it than the other processes employed in his time; he took care of the measure of casks and gave a solution that exceeds in its approach, and is not inferior in its simplicity, to the best than had been given previously; he dealt with Fontaine's quadrature rule and gave, for the first time, conditions that could be applied with confidence.

  482. Willem de Sitter (1872-1934)
    • De Sitter's work led directly to Arthur Eddington's 1919 expedition to measure the gravitational deflection of light rays passing near the Sun, results which, at that time, could only be obtained during an eclipse.

  483. Dennis Sullivan biography
    • The main character in algebraic topology is the nilpotent operator or boundary operator while in quantum field theory an important role is played by the nilpotent operators called Q and "delta" which encode whatever symmetry is present in the action of the particular theory and measure the obstruction to invariantly assign meaning to the integral over all paths.

  484. Alice Bache Gould biography
    • Her mother Mary died in 1883 while the family were on a visit to Boston where Benjamin Gould was arranging to have measurements made of the photographic plates he had taken in Cordoba.

  485. Liu Hui (about 220-about 280)
    • It shows how to use the Gougu theorem (Pythagoras theorem) to calculate heights of objects and distances to objects which cannot be measured directly.

  486. Corrado Gini biography
    • Some of Gini's early work was associated with the problem of how to measure inequalities in income and wealth in different countries.

  487. Lloyd Williams biography
    • It was an education to see how easily a genuine rapport with industry was made and to marvel at the effective measures employed to gain financial support for the Congress.

  488. Wilhelm Wien (1864-1928)
    • In 1912 [Einstein] turned by letter to W Wien with the request to measure the difference between the periods of oscillation of pendulums made of uranium and lead, as well as the proportionality of inertial and gravitational masses of a uranium and a lead weight, respectively, namely with a torsion balance.

  489. Zenodorus (about 200 BC-about 140 BC)
    • In On isometric figures Zenodorus himself follows the style of Euclid and Archimedes quite closely and he refers to results of Archimedes from his treatise Measurement of a circle.

  490. Lars Ahlfors biography
    • The award contributed in great measure to the confidence I felt in my work.

  491. James Gray biography
    • Among his publications were Absolute Measurements in Electricity and Magnetism (1883), (with G B Mathews) Treatise on Bessel Functions (1895), and The Scientific Work of Lord Kelvin.

  492. Heinrich Behnke biography
    • You have certainly already heard of the new measures against half-Jews.

  493. Rudolf Clausius biography
    • Still without giving the concept a name Clausius formulated, in a memoir of 1854, the rudiments of the theory of the concept of the measure of transformation equivalence he later called entropy.

  494. Alexis Petit (1791-1820)
    • Competitors were asked to measure the expansion of mercury in a thermometer between 0°C and 200°C and then to determine the rate at which a body cooled both in a vacuum and in certain specified gases (air, hydrogen and carbon dioxide) at different temperatures and pressures.

  495. Eustachio Manfredi (1674-1739)
    • He then carried out a series of observations on stars, attempting to measure parallax.

  496. Ibn Yunus (950-1009)
    • Perhaps it is worth mentioning that, contrary to claims which are often made, there is no evidence to suggest that ibn Yunus used a pendulum for time measurements.

  497. William Young (1863-1942)
    • His definitions of measure and integration were quite different from those which Lebesgue had given but were shown to be essentially equivalent.

  498. Jean-Marie Duhamel (1797-1872)
    • There had been other measures imposed on the Ecole which had also angered the students.

  499. Isaac Newton (1643-1727)
    • He led it through the difficult period of recoinage and he was particularly active in measures to prevent counterfeiting of the coinage.

  500. Adriaan van Roomen (1561-1615)
    • The first part contains a Latin translation by van Roomen of the Greek text of Archimedes' On the measurement of the circle.

  501. Ivan Ivanovich Privalov (1891-1941)
    • Privalov, often in collaboration with Luzin, studied analytic functions in the vicinity of singular points by means of measure theory and Lebesgue integrals.

  502. Alfred Tarski (1901-1983)
    • Tarski made important contributions in many areas of mathematics: set theory, measure theory, topology, geometry, classical and universal algebra, algebraic logic, various branches of formal logic and metamathematics.

  503. Honoré Fabri (1608-1688)
    • Although when Fabri was sent to Rome it was intended to be a temporary measure, he was soon assigned to St Peter's Penitentiary College which is better known as the Inquisition.

  504. Paddy Kennedy biography
    • Heini Halberstam (1926-2014) and I and some of our other colleagues benefited from a course on probability and measure theory given by Paddy.

  505. René de Sluze (1622-1685)
    • Sluse was later to reply fully to all these requests, satisfying the Society's mathematical interests in good measure as well as Oldenburg's expressed interest in natural history and the history of trades.

  506. Enrico Bombieri biography
    • He began to become interested in problems that De Giorgi and his school of geometric measure theory were working on at the Scuola Normale Superiore in Pisa.

  507. William Rowan Hamilton (1805-1865)
    • Again I do venture to submit to your consideration, whether the poetical parts of your nature would not find a field more favourable to their nature in the regions of prose, not because those regions are humbler, but because they may be gracefully and profitably trod, with footsteps less careful and in measures less elaborate.

  508. Thomas Young (1773-1829)
    • This second paper contained further details of the process of accommodation of the eye and measured astigmatism for the first time.

  509. J C Burkill (1900-1993)
    • This was a particularly active area of research in the early decades of this century after the pioneering work of Lebesgue, Borel and their contemporaries in establishing the concepts of measure and the Lebesgue integral associated with it.

  510. Edward Van Vleck biography
    • A brief description of the evolution of the link between measure theory and probability theory is given.

  511. Jerzy Supecki biography
    • Słupecki and his friends firmly opposed the anti-Semitic measures; Jerzy, because he was considerably older than his fellow-students, was in a sense an authority among his friends .

  512. Frantisek Wolf biography
    • Chapter IV deals with trigonometric integrals summable over sets of positive measure and extends Kuttner's theorem and some similar results of Marcinkiewicz and Zygmund.

  513. Pierre Petit (1594-1677)
    • In particular, late in his life, Petit devised a filar micrometer to measure the diameters of celestial objects such as the Sun, Moon and planets.

  514. Henri Lebesgue (1875-1941)
    • Building on the work of others, including that of Emile Borel and Camille Jordan, Lebesgue formulated the theory of measure in 1901 and in his famous paper Sur une generalisation de l'integrale definie Ⓣ, which appeared in the Comptes Rendus on 29 April 1901, he gave the definition of the Lebesgue integral that generalises the notion of the Riemann integral by extending the concept of the area below a curve to include many discontinuous functions.

  515. Hans Rademacher (1892-1969)
    • In addition to the significant contributions to real analysis and measure theory which we have briefly mentioned above, he contributed to complex analysis, geometry, and numerical analysis.

  516. Magnus Wenninger biography
    • Completing the task of constructing the remaining ten uniform polyhedra proved difficult and R Buckley of Oxford University assisted Father Magnus by supplying the precise measurements.

  517. François Bruhat biography
    • The first covers elementary properties of Lie groups, the second covers the general theory of measures on a locally compact group and representation in general, while the final part contains a construction of the continuous sum of Hilbert spaces, decomposition of a unitary representation into a continuous sum of irreducible representations and the Plancherel formula.

  518. Philippa Fawcett (1868-1948)
    • There is no doubt that Philippa Fawcett's dazzling academic honours assured her a large measure of confidence from British sections of the population.

  519. Ivar Fredholm (1866-1927)
    • Fredholm also served on many government committees and he also served on the International Committee for Weights and Measures.

  520. Rolf Nevanlinna (1895-1980)
    • They are set out in detail in his the book Eindeutige analytische Funktionen Ⓣ (1936) which discusses ideas related to his invention of harmonic measure.

  521. Luigi Brusotti biography
    • These are on "measure theory" (II, p.

  522. Benjamin Robins (1707-1751)
    • Robins invented the ballistic pendulum which allowed precise measurements of the velocity of projectiles fired from guns.

  523. Sewall Green Wright biography
    • Therefore the change of this frequency (called briefly the gene frequency q) is a measure of the evolution of the population.

  524. Jurjen Koksma biography
    • The properties of continued fractions are then discussed together with the neighboring theories, namely, of irrationality and transcendency and their measures.

  525. Herman Goldstine biography
    • I (1940), written jointly with R P McKeon, and Linear functionals and integrals in abstract spaces (1942) in which he shows that the Daniell integral and the integral similarly defined by Banach in his addendum to Saks's "Theory of the Integral" are Lebesgue integrals with respect to regular Caratheodory outer measures.

  526. Michel Plancherel biography
    • In his work he achieved fundamental results, one of them is the famous Plancherel theorem in harmonic analysis and which is now known in many generalizations (Plancherel measures).

  527. Torsten Carleman biography
    • In complex analysis there are Carleman formulae (proved already in 1926) which, unlike the Cauchy formula, reconstruct a function holomorphic in a domain D from its values on a part M of the boundary ∂D of a positive Lebesgue measure.

  528. Elliott Montroll biography
    • Some of his later work appeared in Social dynamics and the quantifying of social forces (1978) in which he proposed a general framework in which social and technological change can be modelled and measured.

  529. Henrietta Swan Leavitt biography
    • Leavitt needed to continue to work on making further measurements.

  530. Ruth Michler biography
    • After the district attorney decided not to prosecute those responsible [for the accident], Michler's family chose to invest her estate in legal proceedings that might offer a measure of justice for her death, which they believe resulted from negligence.

  531. Bertrand Russell (1872-1970)
    • In Principia Mathematica, Whitehead and Russell were able to provide detailed derivations of many major theorems in set theory, finite and transfinite arithmetic, and elementary measure theory.

  532. Harlow Shapley (1885-1972)
    • Shapley had been influenced by the wrong measurements of his friend the Dutch astronomer Adriaan Van Mannen, who also worked at Mount Wilson Observatory.

  533. Al-Nayrizi biography
    • Al-Nayrizi must have worked for al-Mu'tadid during his ten year of rule, for he wrote works for the caliph on meteorological phenomena and on instruments to measure the distance to objects.

  534. Edward Lorenz biography
    • In measured tones, Lorenz starts slowly and builds up gradually a complete understanding of chaos.

  535. Nikolai Luzin (1883-1950)
    • Luzin's main contributions are in the area of foundations of mathematics and measure theory.

  536. Kurt Mahler biography
    • Other major themes of his work were rational approximations of algebraic numbers, p-adic numbers, p-adic Diophantine approximation, geometry of numbers (a term coined by Minkowski to describe the mathematics of packings and coverings) and measure on polynomials.

  537. Cathleen Morawetz biography
    • She also provided a remarkable measure of service to the mathematical community through her membership on many AMS committees and through her term as AMS President (1995-1997).

  538. Josef Stefan (1835-1893)
    • Tyndall measured the radiation from a platinum wire heated by an electric current.

  539. Sharaf al-Din al-Tusi biography
    • It was furnished with a plumb line and a double chord for making angular measurements and bore a perforated pointer.

  540. Gladys Mackenzie biography
    • G I Harper and E Salaman, Measurements on the Ranges of Alpha-Particles, Proc.

  541. Tom Whiteside (1932-2008)
    • Every reply to a question from friends and teachers alike, was measured, deliberate, and carefully considered.

  542. Arne Beurling biography
    • Beurling's leading idea was to find new estimates for the harmonic measure by introducing concepts, and problems, which are inherently invariant under conformal mapping.

  543. Wilhelm Kutta biography
    • Kutta made measurements of glaciers working from photographs taken in the East Alps and also worked with others in constructing maps of the area covered by glaciers.

  544. Loo-Keng Hua biography
    • If many Chinese mathematicians nowadays are making distinguished contributions at the frontiers of science and if mathematics in China enjoys high popularity in public esteem, that is due in large measure to the leadership Hua gave his country, as scholar and teacher, for 50 years.

  545. Alan Baker biography
    • The fact that this is now "a fertile and extensive theory, enriching wide-spread branches of mathematics" is due in large measure to the author himself, who was awarded in 1970 a Fields Medal (the Nobel Prize of mathematics) for his contributions.

  546. Theodor Estermann biography
    • Now although Rademacher was by this time turning towards number theory and Estermann's interests were in that area after attending Edmund Landau's lectures in Gottingen, Rademacher still did not feel confident enough to suggest a number theory topic to Estermann; instead, he suggested a problem in measure theory.

  547. Werner Rogosinski biography
    • This new and welcome addition to a well known collection provides a clear and, in view of its price and size, a surprisingly comprehensive introduction to the theories of measure and integration in Euclidean space Rn of n dimensions.

  548. Friedrich Hartogs biography
    • On 15 September 1935, at a convention in Nurnberg, two measures were approved by the Nazi Party which removed rights from Jews.

  549. Jacqueline Ferrand biography
    • Volume II covered series, elementary functions of a complex variable, and elementary measure and integration.

History Topics

  1. Measurement
    • The history of measurement .
    • This article looks at the problems surrounding systems of measurement which grew up over many centuries, and looks at the introduction of the metric system.
    • Let us first comment on what, in broad terms, is the meaning of measurement.
    • It is associating numbers with physical quantities and so the earliest forms of measurement constitute the first steps towards mathematics.
    • Ancient measurement of length was based on the human body, for example the length of a foot, the length of a stride, the span of a hand, and the breadth of a thumb.
    • There were unbelievably many different measurement systems developed in early times, most of them only being used in a small locality.
    • To measure smaller lengths required subdivisions of the royal cubit.
    • There were 28 digits in a cubit, 4 digits in a palm, 5 digits in a hand, 3 palms (so 12 digits) in a small span, 14 digits (or a half cubit) in a large span, 24 digits in a small cubit, and several other similar measurements.
    • Now one might want measures smaller than a digit, and for this the Egyptians used measures composed of unit fractions.
    • It is not surprising that the earliest mathematics which comes down to us is concerned with problems about weights and measures for this indeed must have been one of the earliest reasons to develop the subject.
    • Egyptian papyri, for example, contain methods for solving equations which arise from problems about weights and measures.
    • A later civilisation whose weights and measures had a wide influence was that of the Babylonians around 1700 BC.
    • This presents a problem as we look at developing systems of measures.
    • Most such systems were not positional systems, so the reason to use multiples of ten in measurement subdivision was less strong.
    • Also ten is an unfortunate number into which to divide a unit of measurement since it only divides naturally into 1/2 , 1/5 , 1/10 .
    • However, since most measuring systems seem to have grown up as a combination of different "natural" measures, no decision about a number to subdivide by would arise.
    • One exception, and the earliest known decimal system of weights and measures, is the Harappan system.
    • The Harappans appear to have adopted a uniform system of weights and measures.
    • Several scales for the measurement of length were also discovered during excavations.
    • One was a decimal scale based on a unit of measurement of 1.32 inches (3.35 centimetres) which has been called the "Indus inch".
    • Of course ten units is then 13.2 inches (33.5 centimetres) which is quite believable as the measure of a "foot", although this suggests the Harappans had rather large feet! Another scale was discovered when a bronze rod was found to have marks in lengths of 0.367 inches.
    • Now 100 units of this measure is 36.7 inches (93 centimetres) which is about the length of a stride.
    • Measurements of the ruins of the buildings which have been excavated show that these units of length were accurately used by the Harappans in their construction.
    • European systems of measurement were originally based on Roman measures, which in turn were based on those of Greece.
    • The Greeks used as their basic measure of length the breadth of a finger (about 19.3 mm), with 16 fingers in a foot, and 24 fingers in a Greek cubit.
    • Trade, of course, was the main reason why units of measurement were spread more widely than their local areas.
    • Most disputes would arise over the weights and measures of the goods being traded, and there a standard set of measures kept in order that such disputes might be settled fairly.
    • The size of a container to measure nuts, dates, beans, and other such items, had been laid down by law and if a container were found which did not conform to the standard, its contents were confiscated and the container destroyed.
    • The Romans did not use the cubit but, perhaps because most of the longer measurements were derived from marching, they had five feet equal to one pace (which was a double step, that is the distance between two consecutive positions of where the right foot lands as one walks).
    • Then 1,000 paces measured a Roman mile which is reasonably close to the British mile as used today.
    • However, if one looks at a country like England, it was invaded at different times by many peoples bringing their own measures.
    • The Angles, Saxons, and Jutes brought measures such as the perch, rod and furlong.
    • The fathom has a Danish origin, and was the distance from fingertip to fingertip of outstretched arms while the ell was originally a German measure of woollen cloth.
    • In England and France measures developed in rather different ways.
    • We have seen above how the problem of standardisation of measures always presented problems, and in early 13th century England a royal ordinance Assize of Weights and Measures gave a long list of definitions of measurement to be used.
    • Locally, however, these standards were not always adhered to and districts still retained their own measures.
    • Of course, although an attempt had been made to standardise measures, no attempt had been made to rationalise them and Great Britain retained a bewildering array of measures which were defined by the ordinance as rather strange subdivisions of each other.
    • Scientists had long seen the benefits of rationalising measures and those such as Wren had proposed a new system based on the yard defined as the length of a pendulum beating at the rate of one second in the Tower of London.
    • In France the infinite perplexity of the measures exceeds all comprehension.
    • In fact it has been estimated that France had about 800 different names for measures at this time, and taking into account their different values in different towns, around 250,000 differently sized units.
    • To a certain extent this reflected the powers which resided in the hands of local nobles who had resisted all attempts by the French King over centuries to standardise measures.
    • Gabriel Mouton, in 1670, had suggested that the world should adopt a uniform scale of measurement based on the mille, which he defined as the length of one minute of the Earth's arc.
    • Lalande, in April 1789, proposed that the measures used in Paris should become national ones, an attempt at standardisation but not rationalisation.
    • Talleyrand put to the National Assembly a proposal due to Condorcet, namely that a new measurement system be adopted based on a length from nature.
    • The system should have decimal subdivisions, all measures of area, volume, weight etc should be linked to the fundamental unit of length.
    • This proposal was not designed to bring in a French system of measurement but to design an international system of measurement, so agreement was sought from other countries.
    • Diplomatic wording allowed an international agreement to be reached, but in March 1791 Borda, as chairman of the Commission of Weights and Measures, proposed using instead of the length of a pendulum, the length of 1/10,000,000 of the distance from the pole to the equator of the Earth.
    • The Royal Society in London declared this was based on a measurement of France, the Americans were not prepared to accept the word of the French mathematicians for its length and even in France it was claimed that the whole project was really proposed in order to gain information on the shape of the Earth.
    • Delambre and Mechain measured the meridian from Dunkerque and Barcelona between 1792 and 1798.
    • The metric system was passed into law by the National Assembly and a metre bar together with a kilogram weight were dispatched to the United States in the expectation that they would adopt the new measures.
    • However, as one might expect, introducing the new measure was easier said than done.
    • In November 1800 an attempt was made to make the system more acceptable by dropping the Greek and Latin prefixes and reinstating the older names for measures but with new metric values.
    • In September of the following year it became illegal to use any other system of weights and measures anywhere in France but it was largely ignored.
    • In 1830 Belgium became independent of Holland and made the metric system, together with its former Greek and Latin prefixes, the only legal measurement system.
    • In 1840 the French government reintroduced the metric system but it took many years before use of the old measures died out.
    • And some of decilitres, to measure beer and drams; .
    • The outcome was the setting up of the International Bureau of Weights and Measures, to be situated in Paris, and the Convention of the Metre of 1875 which was signed by seventeen nations.
    • In 1889 the International Bureau of Weights and Measures replaced the original metre bar in Paris by a new one and at the same time had copies of the bar sent to every country which had signed up to the Convention of the Metre.
    • This remained the standard until 1960 when the International Bureau of Weights and Measures adopted a more accurate standard for international science when it defined the metre in terms of the wavelength of light emitted by the krypton-86 atom, namely 1,650,763.73 wavelengths of the orange-red line in the spectrum of the atom in a vacuum.
    • Indeed the second, then defined as 1/86,400 of the mean solar day, does change but a fixed definition was introduced in 1956 by the International Bureau of Weights and Measures, as 1/31,556,925.9747 of the length of the tropical year 1900.
    • Although this fixed the value, it was seen as an unsatisfactory definition since the length of the year 1900 could never be measured after 1900.
    • .

  2. Size of the Universe
    • We shall return to this event later in the article, but we should begin with the earliest attempts to measure the size of the universe.
    • Knowing that the Moon took 27 days, Mercury 88 days, Venus 225 days, Mars 2 years, Jupiter 12 years, and Saturn 29 years, this was taken as a measure of their distances from Earth.
    • However, all that was now needed, as Kepler had pointed out, was an accurate measurement of the distance to Mars and the scale was fixed.
    • Halley, in 1718, noted that three stars, Sirius, Procyon and Arcturus, had moved relative to the ecliptic (the apparent line of the Sun through the stars) since Hipparchus had measured their positions.
    • Many attempts to measure the parallax of a star were made, but none were successful.
    • William Herschel believed that the brightness of a star could be taken as a measure of its distance.
    • The first person to measure a stellar parallax was Bessel.
    • Bessel, using a Fraunhofer heliometer to make the measurements, announced his value of 0.314" which, given the diameter of the Earth's orbit, gave a distance of about 10 light years.
    • Although there had been many years of failed attempts to measure a stellar parallax, once Bessel made his announcement distances to several others were measured.
    • Thomas Henderson measured the parallax of Alpha Centuari in 1839, showing it had a parallax around three times greater that 61 Cygni, so was much closer.
    • Henderson had indeed measured the distance to the nearest star.
    • By 1912 this had been refined so that it was a reliable measure of the distance of a Cepheid variable star, for one only needed to determine the period, which is the time between two occurrences of maximum brightness, to obtain a value for the absolute brightness of the star.
    • Harlow Shapley, working at the Mount Wilson Observatory in the United States, began to try to work out the shape and size of the Milky Way using Cepheids as a measure of distance.
    • This was at the time the greatest velocity measured for an astronomical object.
    • Now to measure distance to a distant galaxy one only had to find its redshift and use Hubble's Law to find its distance.
    • We now knew the distances to galaxies, could measure their brightness, so could find their size.
    • The answer came out to somewhere between 55 and 60 so it this argument was right then Hubble's measurements gave a constant 10 times too large.
    • He had measured the distance to the Andromeda galaxy pretty accurately but his value for the Hubble constant still left it open to Eddington's objection.

  3. Cartography
    • It is hardly surprising that cartography should be considered as a mathematical discipline in early times since cartography measures positions of places (mathematics was the science of measurement) and represents a the surface of a sphere on a two dimensional map.
    • In Egypt geometry was used from very early times to help measure land.
    • The annual flooding by the Nile meant that without such measurements it was impossible to reconstruct the boundaries that had existed before the flood.
    • Such measurements, however, seem only to have been of local use and there is no evidence that the Egyptians integrated the measurements into maps of large areas.
    • He measured the circumference of the Earth with great accuracy.
    • That the Romans made few contributions is slightly strange given their skills at road building which required accurate surveying measurements.
    • He introduced techniques to measure the Earth and distances on it using triangulation.
    • His Masudic canon contains a table giving the coordinates of six hundred places, some of which were measured by al-Biruni himself, some being taken al-Khwarizmi's work referred to above.
    • Positions of places were fixed as the point of intersection of two lines and, as Frisius pointed out, only one accurate measurement of actual distance was required to fix the scale.
    • Of course the Mercator projection has the property that distances near the poles are greatly distorted so it was not easy to use the map to measure distances.

  4. Fair book
    • Suppose that from one corner, a measurer cannot see all the other corners of a field, but takes his observations from a point of rising ground at A, and that its angles are as follows; BAC = 105°; CAD = 59°30; DAE = 129° and EAB = 66°30'; and the lines AB = 480, AC = 550, AD = 665, AE = 730.
    • There follow a number of problems which involve fields which are drawn with given measurements (all lengths of lines).
    • These contain up to 18 measured lines.
    • The length of a base line within a field curvilinear on the other side is 315 links and 11 equidistant ordinates erected thereon measure 70, 86, 96, 104, 109, 110, 108, 105, 99, 90 and 85 links, respectively, what is the area of the space between the base line and the curvilinear side of the field.
    • He presumably has tables which allow him to scale up from the measurements of solids of unit side.
    • What freight, at the rate of 2 / 6 per barrel bulk, of 5 cubic feet, should be charged for a package of the lower part of which, the height, breadth, and depth, respectively measure 3 feet 6 inches, 4 ft 3 in and 1 ft 9 in; and the upper part, the height, breadth, and depth, respectively measures 4 ft 3 in, 4 ft 2 in, and 1 ft 3 ins.
    • The imperial gallon is not a measure on area! Walker computes A = 82.4 + 82.4, B = 96.5 + 119.2 + 122.4 + 122.6 + 118.9 + 96.3, C = 112 + 121.3 + 123 + 121.2 + 112.
    • drip -- By measure 10 gallons .
    • 43 Crown by measure 38 gallons .
    • A roof which measures 30 feet 8 in by 16 feet 6 inches, is to be covered with lead, at 8 lbs per square foot, find the expense of the lead @ 41/6 a cwt.
    • A perpendicular drawn from one of the angles of an equilateral triangle to the opposite side measures 12 feet, find the length of a side of the triangle.
    • A straight line 330 links long, drawn from the right angle of a right-angled triangle, divides the hypotenuse into two segments which respectively measure 217 and 480.12 links, what is the area of the triangle.

  5. Longitude1
    • Werner proposed using a cross-staff, an instrument derived from the Jacob staff of Levi ben Gerson, to measure the distance of the Moon from the chosen stars.
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    • The method is theoretically correct but Werner had not solved the longitude problem since the cross-staff could not make accurate enough measurements, and more seriously there was no mathematical theory of the Moons orbit (and even when Newton gave his theory of gravitation 150 years later the Moon's motion, a three body problem, was beyond solution).
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    • He wrote to the Spanish Court in 1616 proposing that the way to measure absolute time, which could be measured at any point on the Earth, was to use the moons of Jupiter.
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    • Graindorge wrote to Colbert claiming that his method allowed mariners to determine lines of longitude by direct measurement as easily as they could calculate their latitude.
    • The size of the Earth was still not known with sufficient accuracy to allow precise conversion between linear distance on the surface and angular measures provided by comparing local and absolute times.
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    • In 1669 Picard was assigned the task of making precise measurements of the size of the Earth.
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    • Observations of Jupiter's moons were made with three telescopes and Picard used two pendulum clocks to measure time, one with a pendulum beating once per second, the other clock beating every half second.
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    • After the measurements had all been taken and the results of the survey had been studied it was announced that the diameter of the Earth was about 12554 km, a good result compared with the equatorial diameter now known to be 12756 km.
    • Having completed his measurements of the size of the Earth, Picard was sent on an expedition to Cayenne in 1672.
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    • Other expeditions which set out from Paris on longitude measurements were all told to watch out for any unexpected variations in the performance of their pendulum clocks.
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    • Varin and des Hayes were chosen to lead it and they were trained by Cassini in Paris before leaving so that they might perfect their skills in obtaining precise longitude measurements.
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    • This was an important task for there were few reliable longitude measurements from that part of the World.
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    • Cassini wrote a detailed description of precisely how the longitude measurements were to be carried out.
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  6. 20th century time
    • There had been remarkable progress towards more and more accurate measurement and at the beginning of the century pendulum clocks had been perfected to the extent that they recorded time to an accuracy of less than 1/100 of a second error in a day.
    • He did assume that positions of particles in different places could be measured at the same instant so he in effect used absolute time to define absolute space.
    • It is utterly beyond our power to measure the changes of things by time.
    • What does this mean? We must think about this statement clearly for, with no absolute space to measure velocity against, how can a body move at close to the velocity of light? Let us be more precise.
    • In fact the difference in the rate at which clocks run at the top of a high building compared with at the bottom has now been measured.
    • We began this article by noting that at the beginning of the 20th century time could be measured to an accuracy of around 1/1000 of a second in an hour.
    • This states, in its best known form, that there is a lower limit to the product of the uncertainty in a particle's position and the uncertainty in its momentum so that the more accurately one is able to measure the position of a particle, the more uncertainty there is in the knowledge of its momentum.
    • For example the uncertainty in the energy of a particle and the time at which this energy is measured cannot both be determined to an arbitrary degree of accuracy.
    • The more precisely one determines the time at which the energy is measured, the less accurately one can know that energy.
    • Before we describe it, however, we should stress that the Uncertainty Principle is not about practical problems of measurement but about theoretical uncertainty.
    • There is a scale beside the box and a pointer attached to the box to measure its height.
    • But, claimed Einstein, we can measure the energy of the particle as accurately as we want for its energy is determined by its mass and so we measure the mass by attaching a weight to the bottom of the box to bring the pointer back to its original position.
    • To weigh the particle one must measure the position of the pointer at rest on the scale.
    • If we cannot measure the height of the box to arbitrary precision, we cannot measure the height of the clock inside the box with arbitrary precision, so we do not know the rate of the clock with arbitrary precision (by Einstein's own general relativity results).
    • In quantum theory the particle will have the properties of both possible states until we measure it when it collapses into one of the two states.
    • However, when we measure one particle and it collapses into one state, the other particle must instantly have the complementary property.
    • His argument against this is that to measure how the present moves forward one would need a second "time" against which to measure the progress our standard time.
    • Again to measure this second time's flow one would need a third time and so on.

  7. Alcuin's book
    • He ordered them to be given 20 measures of corn as follows.
    • The men must receive three measures, the women must receive two measures, and the children half a measure each.
    • A head of household had 30 servants whom he ordered to be given 30 measures of corn as follows.
    • The men should each receive three measures, the women should each receive two measures, and the children should receive a half measure each.
    • A gentleman has a household of 90 persons and ordered that they be given 90 measures of grain.
    • He directed that each man should receive three measures, each woman two measures, and each child half a measure.
    • He ordered that they be given 100 measures of corn as follows.
    • The men should receive three measures, the women should receive two measures, and the children should receive half a measure each.
    • How many pints do 100 measures of wine contain, and how many cups do 100 measures contain? .
    • [A measure contains 48 pints, and a pint contains 6 cups.] .
    • There are 4800 pints and 28800 cups in 100 measures.
    • In the first flask there were 40 measures of wine; in the second there were 30 measures of wine, in the third there were 20 measures of wine, and in the fourth there were 10 measures of wine.
    • The total amount of wine is 40 + 30 + 20 + 10 = 100 measures.
    • Hence each son must receive 25 measures of wine.
    • The servant has, therefore, to find a way of getting 25 measures into each flask without being able to measure it.
    • He takes the 2 flasks, one containing 40 measures and the other containing 10 measures.
    • These now contain 25 measures each.
    • Similarly he take the 2 flasks containing 30 measures and 20 measures, and tips wine from one to the other until each has an equal amount of wine.
    • These now contain 25 measures each and the problem is solved.
    • A certain head of household ordered that 90 measures of grain be taken from one of his houses to another 30 leagues away.
    • Given that the camel eats one measure of grain for each league it goes (the camel only eats when carrying a load), how many measures were left after the grain was transported to the second house? .
    • The camel can carry 30 measures of grain per trip and will eat all 30 measures by the time it reaches the second house.

  8. References for Measurement
    • References for Measurement .
    • K Alder, The measure of all things (London, 2002).
    • R D Connor, The weights and measures of England (London, 1987).
    • H A Klein, The science of measurement : A historical survey (New York, 1988).
    • R Zupko, Revolution in measurement : western European weights and measures since the age of science (Philadelphia, 1990).
    • L L Kulvecas, Two dates in the history of the development of the metric system of measures (Russian), in Problems in the history of mathematics and mechanics (Kiev, 1977), 109-115; 133.

  9. Classical time
    • A huge effort has been put into making devices to measure time with ever increasing accuracy from the beginnings of recorded history to the present day.
    • Now units of time require some way of measurement and, not surprisingly, because of their astronomical definitions the early devices to measure time used the sun.
    • There was high religious significance in time measurement in these early civilisations.
    • Of course the importance of successful crop management to the survival of a civilisation meant that time gained a religious significance, and the astronomical way that time was measured emphasised this.
    • In a sense this is reasonable since to Aristotle time was measured by the motions of the heavenly bodies so a period of time was represented by the movement of the sun across the sky.
    • Surely only the present actually exists and this is instantaneous, only measured by its passing.
    • There was some progress in clocks to measure periods of time going in the period when St Augustine was contemplating the puzzle.
    • It is a fascinating discussion which essentially asks if time as measured by the sun and the moon is the "same" time.
    • Of course one might reasonably ask how he discovered this since in Galileo's time there was no device to accurately measure small intervals of time.
    • This invention brought a new accuracy to the measurement of time, with his early versions achieving errors of less than 1 minute a day.
    • No longer was time determined by the universe, but rather Newton postulated an absolute clock, external to the universe, which measured time independently of the universe itself.
    • He defined entropy which originally measured the amount energy in the form of work that can be extracted from a hot gas but later came to represent a more general measure of the randomness of a system.

  10. Kepler's Laws
    • Law I (the Ellipse Law) - the curve or path of a planet is an ellipse whose radius vector is measured from the Sun which is fixed at one focus.
    • (The time is measured by the fraction of the total time taken for the planet to complete one whole circuit, that being called its period T.
    • Kepler followed the ancients in always starting to measure at the point furthest from the Sun.) Almost certainly Kepler was responsible for introducing the term 'orbit', in Astronomia Nova Ⓣ Ch.1, and on his behalf we shall precisely define an orbit as possessing a pair of independent constituents: the path or curve, together with a (geometrical) way of representing time.
    • In their day - and indeed until comparatively recently - the aim of astronomers was to achieve accurate observations of angles, simply because no other feature could be measured directly.
    • In order to transpose the observations from a geocentric to a heliocentric basis, he applied triangulation to ensure that each Mars-distance was measured as if from the fixed Sun.
    • Bearing in mind that the observations contained no distance-measurements (as explained in Section 2), this involved expressing all the Mars-Sun distances in terms of the Earth-Sun distance, regarded as a standard unit or 'baseline' (since the path of the Earth is very nearly circular, this approximation happened to be accurate enough for Kepler's purpose); .
    • It is well-known that a pair of (mathematically-defined) directed quantities are mutually independent if and only if they are at right angles: using Euclid's term 'orthogonal' for mutually perpendicular (it was defined in Elements Book I), this will be named the Principle of Orthogonal Independence; it will, with hindsight, justify our separate treatment of the path and the time-measure.
    • (This usage was authenticated by tradition, since in ancient astronomy motions consisted of combinations of rotations which were measured by the angles at the centres of their respective circles.) Then we have corresponding angles from the parallels AZ and BQ, so that: .
    • In Ch.40, at the first of the three stages set out in Section 6, Kepler put this into practice, by citing Archimedes, Measurement of a Circle, Prop.3, to justify him in taking a sum of distances to be equivalent to the area of a sector of a circle.
    • Thus the area representation was altogether satisfactory in providing an inherently quantifiable measure of time.
    • radial motion which is measured by the linear variation in distance from the Sun; .
    • transradial motion which is measured by the variation in the area swept out: this motion is defined to be circular round the Sun, and thus precisely at right angles to the radial motion.

  11. Weather forecasting
    • From the 17th century onwards, scientists were able to measure factors related to weather such as pressure and temperature.
    • By the end of the Renaissance, scientists realised that it would be much easier to observe weather changes if they had instruments to measure fundamental quantities in the atmosphere such as temperature, pressure and moisture.
    • His student Evangelista Torricelli (1608-1647) invented the barometer in 1643, which allowed people to measure atmospheric pressure.
    • In the 1920s, weather maps became much more detailed due to the invention of the radiosonde: a small lightweight box containing measurement equipment and a radio transmitter.
    • The radiosonde transmits humidity, pressure, temperature, and wind speed measurements to a ground station.
    • For this he used data collected on 20th May 1910, which had been an international balloon day where European observatories had taken measurements in the upper air, and that were provided by Vilhelm Bjerknes.
    • It took Richardson weeks to produce a six-hour forecast, and when he compared his results to the actual measurements, he discovered that his results were not only highly inaccurate, but also absolutely unrealistic.
    • This involves the use of enhanced observation and measurement techniques as well as refining the mathematical models.
    • The accuracy of a medium-range forecast is measured in terms of the anomaly correlation coefficient (ACC), which has to be above 60% in order for a forecast to be considered accurate.

  12. Greek astronomy
    • Some scholars accept that he discovered that the ecliptic was at an angle but doubt that he measured the angle.
    • 67">The order of accuracy is an essential measure for the development of natural sciences.
    • We know that Aristarchus measured the ratio of the distances to the moon and to the sun and, although his methods could never yield accurate results, they did show that the sun was much further from the earth than was the moon.
    • His results also showed that the sun was much larger than the earth, although again his measurements were very inaccurate.
    • Now Goldstein and Bowen in [',' B R Goldstein and A C Bowen, The introduction of dated observations and precise measurement in Greek astronomy, Arch.
    • Goldstein and Bowen in [',' B R Goldstein and A C Bowen, The introduction of dated observations and precise measurement in Greek astronomy, Arch.
    • Archimedes measured the apparent diameter of the sun and also is said to have designed a planetarium.
    • Eratosthenes made important measurements of the size of the earth, accurately measured the angle of the ecliptic and improved the calendar.

  13. Real numbers 1
    • The submultiple, which is by its nature the smaller, is the number which when compared with the greater can measure it more times than one so as to fill it out exactly.
    • Definition V.1nnn A magnitude is a part of a magnitude, the less of the greater, when it measures the greater.
    • Again the term "measures" here is undefined but clearly Euclid means that (in modern symbols) the smaller magnitude x is a part of the greater magnitude y if nx = y for some natural number n > 1.
    • Definition V.2nnn The greater is a multiple of the less when it is measured by the less.
    • Definition X.1nnnThose magnitudes are said to be commensurable which are measured by the same measure, and those incommensurable which cannot have any common measure.
    • Proposition X.2nnnIf, when two unequal magnitudes are set out and the lesser is always subtracted in turn from the greater, the remainder never measures the magnitude before it, then the magnitudes will be incommensurable.
    • One can measure the properties of physical circles, he claims, but one cannot measure a mathematical circle with physical instruments.

  14. Indian mathematics
    • We do know that the Harappans had adopted a uniform system of weights and measures.
    • Several scales for the measurement of length were also discovered during excavations.
    • One was a decimal scale based on a unit of measurement of 1.32 inches (3.35 centimetres) which has been called the "Indus inch".
    • Of course ten units is then 13.2 inches which is quite believable as the measure of a "foot".
    • A similar measure based on the length of a foot is present in other parts of Asia and beyond.
    • Now 100 units of this measure is 36.7 inches which is the measure of a stride.
    • Measurements of the ruins of the buildings which have been excavated show that these units of length were accurately used by the Harappans in construction.

  15. Planetary motion
    • Before that breakthrough, planetary motion involved merely a path (a curve), together with a measure of time, represented geometrically: that is, a strictly kinematical treatment.
    • Law I (the Ellipse Law) - the curve or path of a planet is an ellipse whose radius vector is measured from the Sun which is fixed at one focus.
    • Law II (the Area Law) - the time taken by a planet to reach a particular position is measured by the area swept out by the radius vector drawn from the fixed Sun.
    • The figure shows an ellipse with its major auxiliary circle diameter CD, centre B, whose given measures will be denoted by BC = BD = a, the major semiaxis of the ellipse, and BF = b, its minor semiaxis.
    • Since both the focal distance and the polar distance are measured from the centre B of the ellipse, it is only when these two distances coincide (aε = ae), uniquely, that we obtain the simplest possible equation -- as expressed in (5).
    • The constant of proportionality involved (1/2h is standard usage) is expressed mathematically by the following relationship, in which r represents the radius vector measured from the source of motion at the Sun, still taken as the origin of coordinates, again with reference to the figure: .
    • This is Law II: the time expressed in angular measure.

  16. Mayan mathematics
    • Perhaps they measured 260 days and 105 days as the successive periods between the sun being directly overhead (the fact that this is true for the Yucatan peninsular cannot be taken to prove this theory).
    • It was an absolute timescale which was based on a creation date and time was measured forward from this.
    • Now one might expect that this measurement of time would either give the number of ritual calendar years since creation or the number of civil calendar years since creation.
    • The Mayans appear to have had no concept of a fraction but, as we shall see below, they were still able to make remarkably accurate astronomical measurements.
    • The Mayans carried out astronomical measurements with remarkable accuracy yet they had no instruments other than sticks.

  17. Chinese overview
    • It was very much problem based, motivated by problems of the calendar, trade, land measurement, architecture, government records and taxes.
    • It is an astronomy text, showing how to measure the positions of the heavenly bodies using shadow gauges which are also called gnomons, but it contains important sections on mathematics.
    • In it Liu uses Pythagoras's theorem to calculate heights of objects and distances to objects which cannot be measured directly.

  18. Ten classics
    • The text measures the positions of the heavenly bodies using shadow gauges which are also called gnomons.
    • When the gnomon is turned up, it can measure height; when it is turned over, it can measure depth and when it lies horizontally it can measure distance.
    • The Haidao suanjing shows how to use the Gougu theorem (Pythagoras theorem) to calculate heights of objects and distances to objects which cannot be measured directly.

  19. Real numbers 2
    • All could be measured by real numbers.
    • It was clear to Hankel (see the quote above) that the new ideas of number had suddenly totally changed a concept which had been motivated by measurement and quantity.
    • Similarly Cantor realised that if he wants the line to represent the real numbers then he has to introduce an axiom to recover the connection between the way the real numbers are now being defined and the old concept of measurement.
    • If this distance has a rational relation to the unit of measure, then it is expressed by a rational quantity in the domain of rational numbers; otherwise, if the point is one known through a construction, it is always possible to give a sequence of rationals a1 , a2 , a3 , ..
    • By the beginning of the 20th century, then, the concept of a real number had moved away completely from the concept of a number which had existed from the most ancient times to the beginning of the 19th century, namely its connection with measurement and quantity.

  20. Measurement
    • Measurement .

  21. Decimal time
    • The metric system succeeded because it was the simplest and it put an end to a veritable incoherence in local measures; the decimalisation of time and circumference failed because the whole world employed the same measures and the proposals sinned precisely because of their lack of unity.
    • Poincare made a table which showed the factors required to convert three different quantities: angle to time; old angular measure to new angular measure; and fractions of a circle into grads.

  22. Black holes
    • The difference between a ball and light, is that Einstein said light would always, no matter what frame of reference the observers were in, always be measured with the same velocity, c.
    • With the train analogy, the difference in measured velocities of the ball from the two frames of reference is 100 km/h, however, with light, there can be no difference in measurement, even if it were measured from a frame of reference travelling at almost the speed of light.

  23. Modern light
    • Maxwell made the 'obvious' assumption that each beam would travel at the same speed through the aether so, due to the earth's motion, one should return slightly before the other and measuring the interference fringes would let the earth's speed through the aether be measured.
    • In 1919 Eddington made an expedition to Principe Island off the west coast of Africa to observe a solar eclipse and to measure the apparent position of stars observed close to the disk of the eclipsed sun.
    • Lasers are today in common use in applications such as laser printers, CD and DVD players, computer storage systems, guiding military hardware, accurate astronomical measurements, surgery and other medical applications and many others.

  24. Babylonian numerals
    • On the other hand many measures do involve 12, for example it occurs frequently in weights, money and length subdivisions.
    • For example in old British measures there were twelve inches in a foot, twelve pennies in a shilling etc.
    • Neugebauer proposed a theory based on the weights and measures that the Sumerians used.
    • His idea basically is that a decimal counting system was modified to base 60 to allow for dividing weights and measures into thirds.
    • Certainly we know that the system of weights and measures of the Sumerians do use 1/3 and 2/3 as basic fractions.
    • However although Neugebauer may be correct, the counter argument would be that the system of weights and measures was a consequence of the number system rather than visa versa.

  25. Classical light
    • Through accurate measurements of positions of stars, he realised that light is refracted by the atmosphere.
    • Descartes and Fermat carried on a discussion after this publication (see [',' W Tobin, Toothed wheels and rotating mirrors : Parisian astronomy and mid-nineteenth century experimental measurements of the speed of light, Vistas Astronom.
    • Dark lines in the spectrum of light had first been observed in 1802 by William Wollaston but the correct explanation of them had to wait a few years until a more thorough investigation by Joseph von Fraunhofer who measured the exact positions of over 500 such lines.

  26. General relativity
    • How much they learnt from each other is hard to measure but the fact that they both discovered the same final form of the gravitational field equations within days of each other must indicate that their exchange of ideas was helpful.
    • From 1911 Einstein had realised the importance of astronomical observations to his theories and he had worked with Freundlich to make measurements of Mercury's orbit required to confirm the general theory of relativity.
      Go directly to this paragraph
    • In fact after many failed attempts (due to cloud, war, incompetence etc.) to measure the deflection, two British expeditions in 1919 were to confirm Einstein's prediction by obtaining 1.98" ± 0.30" and 1.61" ± 0.30".

  27. Jaina mathematics
    • There was a fascination with large numbers in Indian thought over a long period and this again almost required them to consider infinitely large measures.
    • The first point worth making is that they had different infinite measures which they did not define in a rigorous mathematical fashion, but nevertheless are quite sophisticated.
    • Despite this some of the astronomical measurements were fairly good.

  28. Infinity
    • An actual infinite set requires a measure, and no such measure seemed possible to Aquinas.
    • We have to move forward to Cantor near the end of the 19th Century before a satisfactory measure for infinite sets was found.
    • Newton rejected indivisibles in favour of his fluxion which was a measure of the instantaneous variation of a quantity.

  29. Matrices and determinants
    • There are three types of corn, of which three bundles of the first, two of the second, and one of the third make 39 measures.
    • Two of the first, three of the second and one of the third make 34 measures.
    • And one of the first, two of the second and three of the third make 26 measures.
    • How many measures of corn are contained of one bundle of each type? .

  30. Longitude2
    • The Colledge will the whole world measure, .
    • (Of course we now know that with more accurate measurements we can detect that the Earth's rate of rotation does change and leap seconds are added on various occasion to correct for this.) .

  31. Maxwell's House
    • Their speed you measure.
    • Might be measured too.

  32. Braids arithmetic
    • The size of a hogshead is one of various measures for beer, wine, dry goods, ..
    • To find the greatest common measure of two numbers.

  33. Doubling the cube
    • 12">The story goes that one of the ancient tragic poets represented Minos having a tomb built for Glaucus, and that when Minos found that the tomb measured a hundred feet on every side, he said "Too small is the tomb you have marked out as the royal resting place.
    • 94">If, good friend, thou mindest to obtain from any small cube a cube the double of it, and duly to change any solid figure into another, this is in thy power; thou canst find the measure of a fold, a pit, or the broad basin of a hollow well, by this method, that is, if thou thus catch between two rulers two means with their extreme ends converging.

  34. Mathematics and Architecture
    • Much has been written on the measurements of this pyramid and many coincidences have been found with , the golden number and its square root.
    • There are at least nine theories which claim to explain the shape of the Pyramid and at least half of these theories agree with the observed measurements to one decimal place.
    • Berger took the greatest common denominator of these measurements to arrive at the ratios .

  35. Newton poetry
    • Measureless distance, unconceiv'd by thought! .
    • Roll endless measures through th' etherial plain, .

  36. Pi history
    • The earliest values of π including the 'Biblical' value of 3, were almost certainly found by measurement.
    • Kendall and Moran comment that you would do better to cut out a large circle of wood and use a tape measure to find its circumference and diameter.

  37. Newton's bucket
    • There is nothing to measure rotation with respect to.
    • Newton deduced from this thought experiment that there had to be something to measure rotation with respect to, and that something had to be space itself.

  38. Quantum mechanics history

  39. Greek numbers
    • This meant that they each had their own currency, weights and measures etc.
    • A very similar system was also used in dealing with weights and measures which is not surprising since the value of money would certainly have evolved from a system of weights.

  40. Topology history
    • The next step in freeing mathematics from being a subject about measurement was also due to Euler.
      Go directly to this paragraph
    • Again the reason must be that to everyone before Euler, it had been impossible to think of geometrical properties without measurement being involved.
      Go directly to this paragraph

  41. References for Classical light
    • The measurements for absolute electromagnetic units and the velocity of light, Scientia (Milano) 113 (5-8) (1978), 469-480.
    • W Tobin, Toothed wheels and rotating mirrors : Parisian astronomy and mid-nineteenth century experimental measurements of the speed of light, Vistas Astronom.

  42. Squaring the circle
    • Now it may not be clear that this is solved the problem of squaring the circle but Archimedes had already proved as the first proposition of Measurement of the circle that the area of a circle is equal to a right-angled triangle having the two shorter sides equal to the radius of the circle and the circumference of the circle.

  43. Science in the 17th century
    • By the 16th and 17th centuries, the paradigm started to shift as some natural philosophers were rejecting unproven theories and using precise tools to obtain exact measurements to base their discoveries on observation and experimentation [Hakim 2005, 19].

  44. Egyptian mathematics
    • Joseph [',' G G Joseph, The crest of the peacock (London, 1991).','8] and many other authors gives some of the measurements of the Great Pyramid which make some people believe that it was built with certain mathematical constants in mind.

  45. Arabic mathematics
    • Of Archimedes' works only two - Sphere and Cylinder and Measurement of the Circle - are known to have been translated, but these were sufficient to stimulate independent researches from the 9th to the 15th century.

  46. Set theory

  47. References for Decimal time
    • K Alder, The measure of all things (Little Brown, London, 2002).

  48. Tait's scrapbook
    • The many experiments extended over several months, during which the Professor used to "bunker" himself intentionally and drive the ball against a wall of sand in order to obtain a measure of the force employed.

  49. References for Nine chapters
    • J M Li, A textual criticism on the "art of milu" in ring measurement in Nine chapters on arithmetic (Chinese), J.

  50. Special relativity
    • He proposed the existence of a single ether and the article tells of a failed attempt by Maxwell to measure the effect of the ether drag on the earth's motion.
      Go directly to this paragraph

  51. Trigonometric functions
    • The first known tables of shadows were produced by the Arabs around 860 and used two measures translated into Latin as umbra recta and umbra versa.
      Go directly to this paragraph

  52. Non-Euclidean geometry
    • Riemann, who wrote his doctoral dissertation under Gauss's supervision, gave an inaugural lecture on 10 June 1854 in which he reformulated the whole concept of geometry which he saw as a space with enough extra structure to be able to measure things like length.
      Go directly to this paragraph

  53. Fractal Geometry
    • The Cantor set has a Lebesgue measure of zero; however, it is also uncountably infinite.

  54. Bourbaki 2
    • One often needs these concepts in cases when the multiplication of scalars is not commutative; this appropriate measure of generality is here systematically carried out.

  55. ETH history
    • The Polytechnic developed two strategies: negotiations with secondary schools and, as an interim measure, establishing a Preparatory Course in 1859.

  56. Nine chapters
    • Is there other evidence for dating parts of the Nine Chapters on the Mathematical Art other than units of measurement? Yes, there are.

  57. Cosmology
    • It was not until in the nineteenth century that the astronomer and mathematician Bessel finally measured the distance to the stars by parallax.
      Go directly to this paragraph

  58. Sundials
    • Time in the ancient world was first measured by naturally occurring events, such as sunrise, sunset, and meal times [',' F W Cousins, Sundials: a simplified approach by means of the equatorial dial / by Frank W Cousins ; with illustrations by Malcolm Chandler ; foreword by J G Porter, ed.

  59. Indian Sulbasutras
    • If the ritual sacrifice was to be successful then the altar had to conform to very precise measurements.

  60. Real numbers 3
    • His proof of this involved showing that the non-normal numbers formed a subset of the reals of measure zero.

  61. Perfect numbers
    • And in the case of those that are found between the too much and the too little, that is in equality, is produced virtue, just measure, propriety, beauty and things of that sort - of which the most exemplary form is that type of number which is called perfect.

  62. History overview
    • A more precise formulation of concepts led to the realisation that the rational numbers did not suffice to measure all lengths.
      Go directly to this paragraph

  63. Mathematics and Art
    • architectural settings in the relation with which the eye measures them, and real to such a degree that ..

  64. References for Greek astronomy
    • B R Goldstein and A C Bowen, The introduction of dated observations and precise measurement in Greek astronomy, Arch.

  65. References for Golden ratio
    • A P Stakhov, The golden section in the measurement theory, in Symmetry 2: unifying human understanding, Part 2, Comput.

  66. Fair book insert
    • The sides of all the resulting triangles of quadrilaterals have been measured so that basically the whole back yard has been triangulated with triangles with known base and height.

  67. Calculus history
    • The method of exhaustion is so called because one thinks of the areas measured expanding so that they account for more and more of the required area.
      Go directly to this paragraph

  68. Orbits
    • Cassini made a measurement of an arc of longitude in 1712 but obtained a result which wrongly suggested that the Earth was elongated at the poles.
      Go directly to this paragraph

  69. Poincaré - Inspector of mines
    • Poincare then gave a precise description of the lengths of the three galleries and the accurate measurements of the underground passages.

Societies etc

  1. Swiss Academy of Science
    • The "Central European degree measurement" was one of the first international scientific co-operations.
    • When the Prussian state invited the Swiss Federal Council to participate in the "Central European degree measurement," it immediately pledged to establish a geodetic commission within the Swiss Academy of Natural Sciences.
    • This commission participated in the process of reaching an agreement between the participating States on the standardization of methods and measures, which was negotiated between 1864 and 1912 at 17 conferences.
    • Soon the cooperation extended across the whole of Europe, which is why it changed its name to "European degree measurement".
    • In 1886, when Mexico, Chile, Argentina, the USA and Japan joined, it became the "International Earth Measurement".

  2. Sinica Academia
    • In order to fulfil these goals, Academia Sinica has adopted various measures to promote the internal integration of research activities in the three research disciplines of mathematics and physical sciences, life sciences, and humanities and social sciences; to improve the planning, implementation, and evaluation of long-term projects in order to enhance the impact of the research activities; to harness basic research results for applications and technology transfer; to engage the entire academic and research community in Taiwan in a modern and forward-looking collective academic vision; to cultivate an intellectual environment that is conducive to the nurturing of young scholars and the recognition of outstanding scholarship in Taiwan; and to promote international cooperation and scholarly exchanges that will accelerate the overall development of academic research in Academia Sinica and the Republic of China.

  3. Royal Astronomical Society
    • Mathematics were at the last gasp, and astronomy nearly so -- I mean in those members of its frame which depend upon precise measurement and systematic calculation.

  4. Malaysian Mathematical Society
    • To take such measures as may be expedient to advance the views of the Society on any question affecting the study and teaching of mathematics.

  5. Armenian Academy of Sciences
    • Probability Theory: integral and stochastic geometry; combinatorial integral geometry; point processes; sections of convex bodies by random planes and lines; measures generation by finite additive functionals; mathematical problems of statistical physics; limit theorems for random Gibbs processes and fields; statistics of stationary Gaussian processes.

  6. Japan Academy
    • Moreover, at least some members of the Academy realized that they were themselves seen as being superannuated folk who had in large measure lost touch with the world of active scholars; accordingly, plans were initiated to enlarge the Academy's membership in order to bring in a substantial number of younger, more active, scholars.

  7. Royal Statistical Society
    • And whereas it has been represented to Us that the same Society has, since its establishment, sedulously pursued such its proposed objects, and by its publications (including those of its transactions), and by promoting the discussion of legislative and other public measures from the statistical point of view, has greatly contributed to the progress of statistical and economic science.

  8. Indian Mathematical Society
    • If the Society succeeds its success will in no small measure be due to the 'Mathematical Gazette', which is the chief source of mathematical enlightenment in India.

  9. Norwegian Academy of Sciences
    • The general excitement may be measured by the fact that the grant proposal from the government to support the meeting was increased by the national assembly (Stortinget), an unprecedented action from a political body traditionally extremely restrictive on state expenditure.

  10. Nepal Mathematical Society
    • From 1968, with Ram Chandra Chaudhery as head of the mathematics department, they introduced courses on topology, groups rings and fields, measure theory, relativity theory, quantum mechanics, and differential geometry.

  11. Lisbon Academy of Sciences
    • It is stated that it was founded at a time when science had progressed qualitatively, when people were experiencing a unique time of discovery, and this in all layers of the Universe, both in the depths of the Earth, on its surface, and in its atmosphere, and in the faraway parts of the Cosmos, a time when the philosopher wanted not only to know the causes of properties and laws, but also "to measure and number their quantity." The founding of "literary corporations" is one of the most efficient means to increase all kinds of knowledge, and this is highlighted as the main reason for the establishment of the Academy, together with the patriotic duty of contributing to the development of Portugal.

  12. International Astronomical Union
    • Many astronomers considered the exit of China a big loss, since astronomy had been present in China for longer than in Europe, but the Executive Committee was not willing to go back on its acceptance of the Taiwanese (a measure other international scientific unions had taken when faced with a similar dilemma).

  13. New Zealand Royal Society
    • He also assumes responsibility for the custody of the standard weights and measures and the Patent Office library.

  14. Polish Mathematical Society
    • During one such walk I overheard the words "Lebesgue measure".

  15. Turkish Mathematical Society
    • The magazine has been published since 1991 and, to illustrate the contents, we note that the first issue in 1991 contains articles on the following: Plane Geometric Angles and Measures; Four Colour Problems; Drawings that cannot be made with ruler and compass; The Extraordinary Features of Infinite Cardinals; 1, 2, 3, Endless Or Rapid Disaster!; Mathematics Teaching in the World.

  16. National Academy of Sciences
    • the Academy has indorsed the French decimal system of weights and measures as against the anomalous and puzzling lack of system now in use in this country, and recommended its adoption.


  1. European Mathematical Society Prize
    • Recently he proved a remarkable "Optional decomposition of supermartingales" which is an extension of the fundamental Doob-Meyer decomposition for the case of many probability measures.
    • pioneered the use of measure-transportation techniques (due to Kantorovich, Brenier, Caffarelli, Mc Cann and others) in geometric inequalities of harmonic and functional analysis with striking applications to geometry of convex bodies.
    • Okounkov gave the first proof of the celebrated Baik-Deift-Johansson conjecture, which states that the asymptotics of random partitions distributed according to the Plancherel measure coincides with that of the eigenvalues of large Hermitian matrices.

  2. Clay Award
    • for their solutions of the Marden Tameness Conjecture, and, by implication through the work of Thurston and Canary, of the Ahlfors Measure Conjecture.
    • for their spectacular work on stationary measures and orbit closures for actions of non-abelian groups on homogeneous spaces .

  3. AMS Steele Prize
    • for his paper "Measure algebras".

  4. Balaguer Prize
    • Discrete Groups, Expanding Graphs and Invariant Measures.

  5. The Moran Medal
    • Citation: Dr Delaigle's has made influential contributions to contemporary statistical problems, including deconvolution, regression with measurement errors, functional data analysis, analysis of high dimensional data, group testing, and a wide variety of contributions to function estimation.

  6. Haim Nessyahu Prize
    • 2008 Adi Shraibman, Complexity Measures of Sign Matrices, Nati Linial, The Hebrew University .

  7. Ostrowski Prize
    • is undoubtedly the leading researcher in the world in geometric measure theory.

  8. SeMA Award
    • Regularity of solutions of elliptic or parabolic problems with Dirac measures as data, SeMA Journal 73 (4) (2016), 379-426.

  9. Bowen Lecturer
    • His pioneering studies of topological entropy, symbolic dynamics, Markov partitions, and invariant measures are of lasting importance; much of today's research is inspired by his ideas.

  10. Wolf Prize
    • for his monumental contributions to algebra, in particular to the theory of lattices in semi-simple Lie groups, and striking applications of this to ergodic theory, representation theory, number theory, combinatorics, and measure theory.

  11. AMS Bôcher Prize
    • for his paper "On a conjecture of Littlewood and idempotent measures".

  12. International Congress Speaker
    • Albert Nikolayevich Shiryaev, Absolute Continuity and Singularity of Probability Measures in Functional Spaces.

  13. Fermat Prize
    • for his impressive contributions to the Calculus of Variations and Geometric Measure Theory, and their link with partial differential equations.

  14. Young Mathematician prize
    • for works on Poisson-Dirichlet measures.

  15. Sylvester Medal
    • for his outstanding work on almost-periodic functions, the theory of measure and integration and many other topics of theory of functions.

  16. Copley Medal
    • for his various inventions and improvements in the construction of the Instruments for the Trigonometrical measurements carried on by the late Major General Roy, and by Lieut.


  1. References for Galileo Galilei
    • S Drake, Galileo's physical measurements, Amer.
    • A T Grigorian, Measure, proportion and mathematical structure of Galileo's mechanics, in Nature mathematized I (Dordrecht-Boston, Mass., 1983), 61-65.
    • A T Grigorian, Measure, proportion and mathematical structure of Galileo's mechanics, Organon 7 (1970), 285-289.

  2. References for Archimedes
    • W R Knorr, Archimedes and the measurement of the circle : a new interpretation, Arch.
    • G M Kozhukhova, The Arabic version of Archimedes' 'Measurement of a circle' (Russian), Istor.-Mat.
    • T Sato, Archimedes' 'On the measurement of a circle', Proposition 1 : an attempt at reconstruction, Japan.
    • A E Shapiro, Archimedes's measurement of the sun's apparent diameter, J.

  3. References for Paul Halmos
    • J L B Cooper, Review: Measure theory, by Paul R Halmos, The Mathematical Gazette 35 (312) (1951), 142.
    • H M Gehman, Review: Measure theory, by Paul R Halmos, Mathematics Magazine 26 (3) (1953), 173-174.
    • J C Oxtoby, Review: Measure theory, by Paul R Halmos, Bull.

  4. References for John von Neumann
    • H Araki, Some of the legacy of John von Neumann in physics: theory of measurement, quantum logic, and von Neumann algebras in physics, The legacy of John von Neumann (Providence, R.I., 1990), 119-136.
    • P R Halmos, Von Neumann on measure and ergodic theory, Bull.

  5. References for Ibn al-Haytham
    • A Al-Ayib, Precise measurements in ibn al-Haytham's treatise on geography (Arabic), in Deuxieme Colloque Maghrebin sur l'Histoire des Mathematiques Arabes (Tunis, 1990), A68-A86, 202.
    • R Rashed, ibn al-Haytham and the measurement of the paraboloid (Arabic), J.

  6. References for Charles-Marie de La Condamine
    • L D Ferreiro, Measure of the Earth: The Enlightenment Expedition That Reshaped Our World (Basic Books, New York, 2011).
    • N Safier, Myths and measurements, in Jordana Dym and Karl Offen (eds.), Mapping Latin America: A Cartographic Reader (University of Chicago Press, 2011), 107-109.

  7. References for Anaximander
    • D O'Brien, Anaximander's Measurements, The Classical Quarterly 17 (1967), 423-432.
    • A Sabo, The measurement of angles and the start of trigonometry (Bulgarian), Fiz.-Mat.

  8. References for Corrado Gini
    • C Dagum, Decomposition and Interpretation of Gini and the Generalized Entropy Inequality Measures, Proceedings of the American Statistical Association, Business and Economic Statistics Section (1997), 200-205.
    • A Forcina and G M Giorgi, Early Gini's contributions to inequality measurement and statistical inference, J.

  9. References for Otto Hölder
    • P Cantu, Geometry and Measurement in Otto Holder's Epistemology, Philos.
    • J Michell, The origins of the representational theory of measurement: Helmholtz, Holder, and Russell, Stud.
    • J Michell and C Ernst, The Axioms of Quantity and the Theory of Measurement, Journal of Mathematical Psychology 40 (1996), 235-252.

  10. References for Henrietta Swan Leavitt
    • G Johnson, Miss Leavitt's Stars : The Untold Story of the Woman Who Discovered How To Measure the Universe (W W Norton, New York, 2005).
    • E Howell, Henrietta Swan Leavitt: Discovered How to Measure Stellar Distances, (11 November 2016).

  11. References for Wilhelm Lexis
    • S M Stigler, The measurement of uncertainty in nineteenth-century social science, in The probabilistic revolution Vol.
    • The Measurement of Uncertainty before 1900 (Cambridge, Mass.-London, 1986), 222-.

  12. References for Eratosthenes
    • J Dutka, Eratosthenes' measurement of the Earth reconsidered, Arch.
    • B R Goldstein, Eratosthenes on the 'measurement' of the earth, Historia Math.

  13. References for Renato Caccioppoli
    • P De Lucia, Measure theory in Naples : Renato Caccioppoli, (Italian), International Symposium in honor of Renato Caccioppoli, Naples, 1989, Ricerche Mat.
    • M Miranda, Renato Caccioppoli and geometric measure theory (Italian), International Symposium in honor of Renato Caccioppoli, Naples, 1989, Ricerche Mat.

  14. References for Cecilia Payne-Gaposchkin
    • D Sobel, The Glass Universe: How the Ladies of the Harvard Observatory Took the Measure of the Stars (Penguin, 2016).

  15. References for Jacob Bernoulli
    • The Measurement of Uncertainty before 1900 (Cambridge, Mass.-London, 1986), 63-.

  16. References for Eduard Stiefel
    • D R Lide, A Century of Excellence in Measurements, Standards, and Technology (CRC Press, 2001).

  17. References for Thomas Simpson
    • The Measurement of Uncertainty before 1900 (Cambridge, Mass.-London, 1986), 88-.

  18. References for Posidonius
    • C M Taisbak, Posidonius vindicated at all costs? Modern scholarship versus the Stoic earth measurer, Centaurus 18 (1973/74), 253-269.

  19. References for Ladislaus Bortkiewicz
    • The Measurement of Uncertainty before 1900 (Cambridge, Mass.-London, 1986), 375-376.

  20. References for Anders Celsius
    • H C Stempels, Anders Celsius' contributions to meridian arc measurements and the establishment of an astronomical observatory in Uppsala, Baltic Astronomy 20 (2011), 179-85.

  21. References for George Airy
    • A Chapman, The pit and the pendulum : George Biddell Airy and his measurements of gravity, Astronomy Now 5 (1991), 18-20.

  22. References for Henri Lebesgue
    • K O May, Biographical Sketch of Henri Lebesgue, in Henri Lebesgue, Measure and the Integral (San Francisco, 1966), 1-7.

  23. References for Jean-Charles de Borda
    • K Alder, The measure of all things (London, 2002).

  24. References for Pierre-Simon Laplace
    • The Measurement of Uncertainty before 1900 (Cambridge, Mass.-London, 1986), 100-.

  25. References for Andrea Tacquet
    • Two noteworthy contributions : 'Cuts of rational numbers' by the Galilean G A Borelli and 'Classes of measures' by the Jesuit A Tacquet (Italian), Nuncius Ann.

  26. References for Abraham de Moivre
    • The Measurement of Uncertainty before 1900 (Cambridge, Mass.-London, 1986), 70-.

  27. References for Jérôme Lalande
    • K Alder, The measure of all things (London, 2002).

  28. References for Alfréd Haar
    • B Szokefalvi-Nagy, Alfred Haar (1885-1933): invariant measure of mathematical excellence, A Haar memorial conference I, II (Amsterdam-New York, 1987), 17-24.

  29. References for Zoárd Geöcze
    • B Szenassy, Zoard Geocze's mathematical life-work and recent results of surface measurement, A Szent Istvan Akademia Ertesitoje (1943), 118-142.

  30. References for Heinz Bauer
    • S D Chatterji, The work of Heinz Bauer in measure and integration, Selecta (de Gruyter, Berlin, 2003), 1-10.

  31. References for John Greaves
    • Z Shalev, Measurer of all things : John Greaves (16-2-1652), the Great Pyramid, and early modern metrology, J.

  32. References for Kazimierz Kuratowski
    • E Marczewski, On the papers of Kazimierz Kuratowski in set theory and measure theory (Polish), Wiadomosci matematyczne (2) 3 (1960), 239-244.

  33. References for Axel Harnack
    • G Letta, Riemann integrability conditions and their influence on the origin of the concept of measure (Italian), Rend.

  34. References for Pierre Méchain
    • K Alder, The measure of all things (London, 2002).

  35. References for Francis Edgeworth
    • The Measurement of Uncertainty before 1900 (Cambridge, Mass.-London, 1986), 305-.

  36. References for Igor Kluvánek
    • J J Uhl, Jr, Review: Vector measures and control systems by Igor Kluvanek and Greg Knowles, Bull.

  37. References for Hermann von Helmholtz
    • J Michell, The origins of the representational theory of measurement : Helmholtz, Holder, and Russell, Stud.

  38. References for Ambrose Rogers
    • S Abbott, Review: Hausdorff Measures, by C A Rogers, The Mathematical Gazette 83 (497) (1999), 362.

  39. References for Erastus De Forest
    • S M Stigler, The history of statistics : The measurement of uncertainty before 1900 (The Belknap Press of Harvard University Press, Cambridge, MA, 1986).

  40. References for Karl Pearson
    • The Measurement of Uncertainty before 1900 (Cambridge, Mass.-London, 1986), 326-.

  41. References for Herbert Federer
    • C Goffman, Review: Geometric measure theory, by Herbert Federer, Bull.

  42. References for Giovanni Alfonso Borelli
    • Two noteworthy contributions: Cuts of rational numbers by the Galilean G A Borelli and Classes of measures by the Jesuit A Tacquet (Italian), Nuncius Ann.

  43. References for Francis Galton
    • The Measurement of Uncertainty before 1900 (Cambridge, Mass.-London, 1986), 265-.

  44. References for Saharon Shelah
    • Shelah's Theorem on the Measure Problem and Related Results by Jean Raisonnier, J.

  45. References for Émile Borel
    • O Onicescu, Emile Borel (1871-1956), the creator of the theory of measure (Romanian), Gaz.

  46. References for Dominique Cassini
    • K Alder, The measure of all things (London, 2002).

  47. References for Hans Hahn
    • P Ehrlich, Hahn's 'Uber die nichtarchimedischen Grossensysteme' and the development of the modern theory of magnitudes and numbers to measure them, in From Dedekind to Godel, Boston, MA, 1992 (Kluwer Acad.

  48. References for Felix Hausdorff
    • L Olsen, Review of Integral, probability, and fractal measures, by G Edgar (New York, 1998), Bull.

  49. References for William Feller
    • H Fischer, Feller's Early Work on Measure Theory and Mathematical .

  50. References for John Flamsteed
    • M E W Williams, Flamsteed's alleged measurement of annual parallax for the Pole Star, J.

  51. References for Giovanni Battista Riccioli
    • C M Graney, Riccioli Measures the Stars: Observations of the telescopic disks of stars as evidence against Copernicus and Galileo in the middle of the 17th century.

  52. References for William Rankine
    • B Marsden, Engineering science in Glasgow : economy, efficiency and measurement as prime movers in the differentiation .

  53. References for Al-Kindi
    • R Rashed, al-Kindi's commentary on Archimedes' The measurement of the circle, Arabic Sci.

  54. References for Roman Sikorski
    • E Grzegorek and C Ryll-Nardzewski, The papers of Roman Sikorski in measure theory (Polish), Wiadom.

  55. References for Claude-Louis Mathieu
    • K Alder, The measure of all things (London, 2002).

  56. References for al-Kashi
    • Y Dold-Samplonius, al-Kashi's measurement of Muqarnas, in Deuxieme Colloque Maghrebin sur l'Histoire des Mathematiques Arabes (Tunis, 1990), 74-84.

  57. References for Ida Rhodes
    • .

  58. References for George Udny Yule
    • The Measurement of Uncertainty before 1900 (Cambridge, Mass.-London, 1986), 345-.

  59. References for Frederick Mosteller
    • A Second Course in Statistics by Fredrick Mosteller and John W Tukey, Journal of Educational Measurement 16 (1) (1979), 60-61.

  60. References for Carl Friedrich Gauss
    • The Measurement of Uncertainty before 1900 (Cambridge, Mass.-London, 1986), 140-.

  61. References for Johannes Kepler
    • N M Swerdlow, Shadow measurement : the 'Sciametria' from Kepler's 'Hipparchus' - a translation with commentary, in The investigation of difficult things (Cambridge, 1992), 19-70.

  62. References for Al-Sijzi
    • B A Rozenfel'd and R S Safarov (trs.), Abu Said Akhmad as-Sidzhizi, Book of measurement of spheres by spheres (Russian), Istor.-Mat.

  63. References for Jean-Baptiste-Joseph Delambre
    • K Alder, The measure of all things (London, 2002).

  64. References for Adolphe Quetelet
    • The Measurement of Uncertainty before 1900 (Cambridge, Mass.-London, 1986), 161-.

  65. References for Otton Nikodym
    • J Diestel and J J Uhl, Jr., Vector Measures (Providence 1977).

  66. References for Adrien-Marie Legendre
    • C D Hellman, Legendre and the French reform of weights and measures, Osiris 1 (1936), 314-340.

  67. References for Thomas Bayes
    • The Measurement of Uncertainty before 1900 (Cambridge, Mass.-London, 1986), 88-.

  68. References for James Bradley
    • M Williams, James Bradley and the Eighteenth Century 'Gap' in Attempts to Measure Annual Stellar Parallax, Notes and Records of the Royal Society of London 37 (1) (1982), 83-100.

Additional material

  1. Eddington: 'Mathematical Theory of Relativity' Introduction
    • The parallax of a star is found by a well-known series of operations and calculations; the distance across the room is found by operations with a tape-measure.
    • Distance, parallax and cubic parallax have the same kind of potential existence even when the operations of measurement are not actually made - if you will move sideways you will be able to determine the angular shift, if you will lay measuring-rods in a line to the object you will be able to count their number.
    • The connection of manufactured physical quantities with the existent world-condition can be expressed by saying that the physical quantities are measure-numbers of the world-condition.
    • Measure-numbers may be assigned according to any code, the only requirement being that the same measure-number always indicates the same world-condition and that different world-conditions receive different measure-numbers.
    • Two or more physical quantities may thus be measure-numbers of the same world-condition, but in different codes, e.g.
    • But in admitting that physical quantities can be used as measure-numbers of world-conditions existing independently of our operations, we do not alter their status as manufactured quantities.
    • (Differences of world-conditions which do not influence the results of experiment and observation are ipso facto excluded from the domain of physical knowledge.) The size to which a crystal grows may be a measure-number of the temperature of the mother-liquor; but it is none the less a manufactured size, and we do not conclude that the true nature of size is caloric.
    • Or, what comes to the same thing, we must contemplate its measures according to all possible measure-codes - of course, without confusing the different codes.
    • A tensor expresses simultaneously the whole group of measure-numbers associated with any world-condition; and machinery is provided for keeping the various codes distinct.
    • For example, until recently the practical man was never confronted with problems of non-Euclidean space, and it might be suggested that he would be uncertain how to construct a straight line when so confronted; but as a matter of fact he showed no hesitation, and the eclipse observers measured without ambiguity the bending of light from the "straight line." The appropriate practical definition was so obvious that there was never any danger of different people meaning different loci by this term.
    • The physical quantity is the measure-number of a world-condition in some code; we cannot assert that a code is right or wrong, or that a measure-number is real or unreal; what we require is that the code should be the accepted code, and the measure-number the number in current use.
    • This condition is so accidental and parochial that we are reluctant to insist on it in our definition of time; yet it so happens that the motion of the apparatus makes an important difference in the measurement, and without this restriction the operations lead to no definite result and cannot define anything.
    • Any operation of measurement involves a comparison between a measuring-appliance and the thing measured.
    • Both play an equal part in the comparison and are theoretically, and indeed often practically, interchangeable; for example, the result of an observation with the meridian circle gives the right ascension of the star or the error of the clock indifferently, and we can regard either the clock or the star as the instrument or the object of measurement.
    • It is true that we standardise the measuring appliance as far as possible (the method of standardisation being explained or implied in the definition of the physical quantity) so that in general the variability of the measurement can only indicate a variability of the object measured.
    • To that extent there is no great practical harm in regarding the measurement as belonging solely to the second partner in the relation.
    • Moreover, we have seen that the standardisation of the measuring-appliance is usually left incomplete, as regards the specification of its motion; and rather than complete it in a way which would be arbitrary and pernicious, we prefer to recognise explicitly that our physical quantities belong not solely to the objects measured but have reference also to the particular frame of motion that we choose.
    • We have seen that the world-condition or object which is surveyed can only be apprehended in our knowledge as the sum total of all the measurements in which it can be concerned; any intrinsic property of the object must appear as a uniformity or law in these measures.
    • When one partner in the comparison is fixed and the other partner varied widely, whatever is common to all the measurements may be ascribed exclusively to the first partner and regarded as an intrinsic property of it.
    • Let us apply this to the converse comparison; that is to say, keep the measuring-appliance constant or standardised, and vary as widely as possible the objects measured - or, in simpler terms, make a particular kind of measurement in all parts of the field.
    • Intrinsic properties of the measuring-appliance should appear as uniformities or laws in these measures.
    • The development of physics is progressive, and as the theories of the external world become crystallised, we often tend to replace the elementary physical quantities defined through operations of measurement by theoretical quantities believed to have a more fundamental significance in the external world.
    • Physical quantities defined by operations of measurement are independent of theory, and form the proper starting-point for any new theoretical development.

  2. Coulson: 'Electricity
    • The smallest negative charge which it is possible to obtain is that of the ordinary electron, discovered and measured for the first time by J J Thomson in 1897.
    • All charges are integral multiples of these fundamental units; but each unit is so very small that in any common electrical measurement the discreteness of electric charge will not affect us, and we may suppose that a given charge may be allowed any arbitrary numerical value.
    • The smallness of the electronic unit in relation to ordinary measurements may be shown by the fact that in a 60-watt lamp at 200 volts approximately 2 × 1018 electronic units of charge flow along the filament per second.
    • Since we cannot measure any distances as small as this, it will be quite in order for us to regard our charges as points, and we shall therefore refer, when necessary, to point charges.
    • In the theory of electricity, in contrast with atomic physics, we are not primarily concerned with the forces exerted by one atom, or one electron, on another atom, since the forces and distances involved are far too small for us to measure individually in the laboratory.
    • Thus if the smallest mass we can conveniently measure is taken to be 1/10 milligram, this would represent no less than 1023 electrons, or between 1018 and 1020 atoms, depending on the substance we are using.
    • It makes no difference to our formulation of the laws of current flow, as we develop it in Chapter V, which type of carrier is bearing the charge, for in all cases the current is measured by the rate at which the charge flows, i.e.
    • The distinction which we made in § 1 between microscopic and macroscopic measurement is important here.
    • For on the microscopic point of view the charges are moving in all directions with all possible speeds; but on the macroscopic point of view, in which we consider merely the average motion of the charges within a very small volume, we determine a mean drift velocity, the magnitude and direction of which measure the electric current.
    • We have seen that a current may be measured by the quantity of charge flowing in unit time.
    • This counting of charge, which is made with an electrometer, provides us with an electrostatic measure of current, and the result would naturally be expressed in electrostatic units, generally abbreviated to e.s.u.; we shall have more to say about these units later.
    • In actual practice it is these magnetic, or more properly electromagnetic, effects that are most commonly used to measure currents, and in such cases our result will be expressed in electromagnetic units and written e.m.u.
    • The same current may be measured in both units, and the relation between them, or, which is the same, the ratio of the corresponding units, is a matter of prime importance.

  3. Nevil Maskelyne measures the Earth's density
    • Nevil Maskelyne measures the Earth's density .
    • But the first attempt of this kind was made by the French academicians, who measured 3 degrees of the meridian near Quito in Peru, and who endeavoured to find the effect of the attraction of Chimboraco, a mountain in that neighbourhood, which is elevated near 4 miles above the sea, though only about 2 miles above the general level of the province of Quito.
    • It had also the advantage, by its steepness, of having but a small base from north to south; which circumstance, at the same time that it increases the effect of attraction, brings the two stations on the north and south sides of the hill, at which the sum of the two contrary attractions is to be found by the, experiment, nearer together; so that the necessary allowance of the number of seconds, for the difference of latitude due to the measured horizontal distance of the two stations, in the direction of the meridian, would be very small, and consequently not subject to sensible error from any probable uncertainty of the length of a degree of latitude in this parallel.
    • The quantity of attraction of the hill, the grand point to be determined, is measured by the deviation of the plumb-line from the perpendicular, occasioned by the attraction of the hill, or by the angle contained between the actual perpendicular and that which would have obtained if the hill had been away.
    • Thus the less latitude appearing too small by the attraction on the south side, and the greater latitude appearing too great by the attraction on the north side, the difference of the latitudes will appear too great by the sum of the two contrary attractions; if, therefore, there is an attraction of the hill, the difference of latitude by the celestial observations ought to come out greater than what answers to the distance of the two stations measured trigonometrically, according to the length of a degree of latitude in that parallel, and the observed difference of latitude subtracted from the difference of latitude inferred from the terrestrial operations, will give the sum of the two contrary attractions of the hill.
    • To ascertain the distance between the parallels of latitude passing through the two stations on contrary sides of the hill, a base line must be measured in some level spot near the hill, and connected with the two stations by a chain of triangles, the direction of whose sides, with respect to the meridian, should be settled by astronomical observations.
    • But were that the case, the attraction of mountains, and even smaller inequalities in the earth's surface, would be very great, contrary to experiment, and would affect the measures of the degrees of the meridian much more than we find they do; and the variation of gravity in different latitudes, in going from the equator to the poles, as found by pendulums, would not he near so regular as it has been found by experiment to be.
    • The density of the superficial parts of the earth, being, however, sufficient to produce sensible deflections in the plumb-lines of astronomical instruments, will thus cause apparent inequalities in the mensurations of degrees in the meridian; and, therefore, it becomes a matter of great importance to choose those places for measuring degrees, where the irregular attractions of the elevated parts may he small, or in some measure compensate one another; or else it will he necessary to make allowance for their effects, which cannot but be a work of great difficulty, and perhaps liable to great uncertainty.

  4. German syllabus
    • Geometrical Drawing and Measurement.
    • Measurement of segments and angles.
    • Extension of geometrical considerations and measurements to space.
    • Geometrical Drawing and Measurement.
    • Measure of segments and angles.
    • Geometrical Drawing and Measurement.
    • Measurement of segments, angles, and areas.
    • Geometrical Drawing and Measurement.
    • More exact measurements (nonius).
    • Geometrical Drawing and Measurement.
    • Geometrical Drawing and Measurement.
    • Simple astronomical observations with measurements and calculations.

  5. Rota's lecture on 'Mathematical Snapshots
    • It is obtained by analyzing the everyday notion of volume, or, in abstract terms, measure.
    • We will see that volume is characterized by four axioms, and we will find a new measure that fits these axioms, after a slight twist.
    • Measure is defined by two axioms.
    • A measure v on a family of subsets, for example, subsets of ordinary space, is a real number which is assigned to subsets A, B, ..
    • The picture shows that this axiom states that measure is additive.
    • The best example of measure is the volume v(A) of a solid A in space.
    • Axioms 1 and 2 do not single out volume among all measures.
    • The new measure a satisfies axioms 1,2, and 3, with v changed to a, but instead of satisfying axiom 4 it satisfies axiom 4'.
    • Does the measure a make any sense? Of course it does.
    • We try to define a new measure that satisfies axioms 1, 2, and 3, together with .
    • If the measure w(P) is to be consistent, then by axiom 2 we must have .
    • Again, continuity considerations enable us to compute the measure w(A) when A is any reasonable solid in ordinary space.
    • What is the meaning of the new measure w ? .
    • Since the new measure w is well defined, common sense will have to adjust to reality.
    • The measure w is called the mean width, a misnomer that has been kept for historical reasons.
    • The mean width is a new measure on three dimensional solids that enjoys equal rights with volume and surface area.
    • We thereby obtain n invariant measures in n-dimensional space.
    • These measures are called the intrinsic volumes.
    • It is likely that when scientists become aware of the existence of the mean width, they will find interpretations and applications of this measure.

  6. Max Planck: 'Quantum Theory
    • Some of these investigations could be proved by comparison with available observations, particularly the damping measurements of Vilhelm Bjerknes, and this is a verification of the results.
    • According to Boltzmann, entropy is a measure of physical probability, and the essence of the second law of thermo-dynamics is that in Nature, the more often a condition occurs, the more probable it is.
    • In Nature, entropy itself is never measured, but only the difference of entropy, and to this extent one cannot talk of absolute entropy without a certain arbitrariness.
    • Energy itself cannot be measured, but only a difference of energy.
    • With the conventional measure of temperature, however, this constant has an extremely small value, which is naturally closely dependent upon the energy of a single molecule, and an exact knowledge of it leads, therefore, to the calculation of the mass of a molecule and the quantities depending upon it.
    • A Eucken's measurements of the specific heat of hydrogen verified this deduction, and the fact that the calculations of A Einstein and O Stern, P Ehrenfest, and others have not yet been in satisfactory agreement can be ascribed to our incomplete knowledge of the form of the hydrogen molecule.
    • In the face of these numerous verifications (which could be considered as very strong proofs in view of the great accuracy of spectroscopic measurements), those who had looked on the problem as a game of chance were finally compelled to throw away all doubt when A Sommerfeld showed that - by extending the laws of distribution of quanta to systems with several degrees of freedom (and bearing in mind the variability of mass according to the theory of relativity) - an elegant formula follows which must, so far as can be determined by the most delicate measurements now possible (those of F Paschen), solve the riddle of the structure of hydrogen and helium spectra.
    • Just as Rudolf Clausius introduced, as a basis for the measure of entropy, the theorem that any two conditions of a material system are transformable one to the other by reversible processes, so Bohr's new ideas showed the corresponding way to explore the problems opened up by him.

  7. Heinrich Tietze on Numbers, Part 2
    • But the fraction 5/3 can also be interpreted as the measurement of a length (by dividing the length into three equal parts and then considering five such parts side by side).
    • Negative and positive numbers can also be used to measure lengths by taking a fixed point on a line and specifying that measurements to the right of the point are positive and to the left negative.
    • In other words, AC and AB have a common measure, AE (of which they are both multiples).
    • The Greeks perceived that there are segments which are not in the ratio of two whole numbers, and therefore have no common measure or are incommensurable.
    • For if AC and AB have a common measure, it is a segment which can be marked off exactly m times on AC and n times on AB, with m and n suitable positive whole numbers; then the mth part of AC is equal to the nth part of AB and AC : AB as m : n.
    • If AB is divided into 1000 parts, measurement of AC will yield 1414 of these parts so that AC : AB is as 1414 : 1000, or 707 : 500.
    • Only a rigorously exact mathematics could make the distinction between commensurable and incommensurable segments - a distinction which would make no sense in practical measurements.] .
    • The extension of the number system was required not only for geometric measurements, but also for the solution of algebraic equations.

  8. Santalo honorary doctorate
    • When man began to feel the need or the curiosity to know his surroundings and to understand the world in which he lived, two fundamental activities arose: to count and to measure.
    • With measurement the shapes and figures were outlined: it was geometry.
    • Less than a century after the Republic, Eratosthenes appears (284 BC, 192 BC) who uses mathematics to measure the radius of the Earth and also Archimedes (287 BC, 212 BC) who at the same time make fine speculations about pure mathematics (the method of exhaustion) applied the same to concrete problems of statistics or hydrostatics.
    • As we see, this is a problem for which it is not possible to "count" favourable and possible cases, as in discrete games of chance, such as those based on dice or coins, but must "measure" those cases.
    • Although there have always been measured sets of points (length of curves or areas of domain) however the measure of sets of straight lines had not been considered.
    • He then defined a density to measure sets of lines and found the curious fact that the measure of the lines that cut a convex set is equal to the length of its contour.
    • That is to say, just as the measure of the points of a convex domain is equal to its area, the measure of the lines that cut it (always in the plane) is equal to its perimeter, a curious duality that has since been extended to great generality by R V Ambartzumian (Combinatorial Integral Geometry, 1982).
    • The theory was also linked with the classic harmonic analysis and the measurement in groups.

  9. Johnson pre1900 books
    • The distinction between the view of the differential calculus here presented, and that found in most of the standard works on the subject hitherto published, may be stated thus: - The derivative dy/dx is usually defined as the limit which the ratio of the finite quantities Δy and Δx approaches when these quantities are indefinitely diminished: when this definition is employed, it is necessary, before proceeding to kinematical applications, to prove that this limit is the measure of the relative rates of x and y.
    • A quantity of which the magnitude is to be determined is either directly measured, or, as in the more usual case, deduced by calculation from quantities which are directly measured.
    • The result of a direct measurement is called an observation.
    • An increase in the nicety of the observations, and the precision of the instrument, may decrease the discrepancies in actual magnitude; but at the same time, by diminishing the least count, their numerical measures will generally be increased; so that, with the most refined instruments, the discrepancies may amount to many times the least count.
    • It is one of the objects of the theory of errors to deduce from a number of discordant observations (supposed to be already individually corrected, so far as possible) the best attainable result, together with a measure of its accuracy; that is to say, of the degree of confidence we are entitled to place in it.
    • We may now state more generally the object of the theory of errors to be, when given more than n observation equations involving n unknown quantities, the equations being somewhat inconsistent, to derive from them the best determination of the values of the several unknown quantities, together with a measure of the degree of accuracy obtained.

  10. Halmos books 1
    • Measure theory (1950), by Paul R Halmos.
    • My main purpose in this book is to present a unified treatment of that part of measure theory which in recent years has shown itself to be most useful for its applications in modern analysis.
    • In the few places where my nomenclature differs from that in the existing literature of measure theory, I was motivated by an attempt to harmonize with the usage of other parts of mathematics.
    • The more advanced reader, on the other hand, might be pleased at the interplay between measure theory and other parts of mathematics which it is the purpose of such exercises to exhibit.
    • [The book] gives a comprehensive account of those aspects of the theory of measure and integration which are important in general measure spaces and in topological spaces and groups.
    • [The book] is intended for use as a textbook in graduate courses in Measure Theory.
    • This book presents a unified theory of the general theory of measure and is intended to serve both as a text book for students and as a reference book for advanced mathematicians.
    • In this book Professor Halmos presents an account of the modern theory of measure and integration in the generality required for the study of measure in groups.
    • It can hardly fail to exert a stimulating influence on the development of measure theory.
    • The development of the theory of these last operators is made to rest on measure theory: as a result, and by making full use of the latest work on the subject, the author gives a treatment in which the geometry of the space plays a large part.
    • However, it is clearly written for readers with a training in the abstract modes of mathematical reasoning and, particularly, in measure theory.
    • The main purpose of this book is to present the so-called multiplicity theory and the theory of unitary equivalence, for arbitrary spectral measures, in separable or not separable Hilbert space.
    • For this introduction to Hilbert space, one has to be an expert in measure theory.
    • As a matter of fact it is best to have read the author's book on measure theory or its equivalent.
    • As a prerequisite, some familiarity with the ideas of the general, theory of measure is assumed; Halmos's Measure theory (1950) is an adequate reference.
    • On the one hand, Boolean algebras arise naturally in such diverse fields as logic, measure theory, topology, and ring theory, so that the study of these objects is motivated by important applications.

  11. Kepler's Planetary Laws
    • Law I (the Ellipse Law) - the curve or path of a planet is an ellipse whose radius vector is measured from the Sun which is fixed at one focus.
    • (The time is measured by the fraction of the total time taken for the planet to complete one whole circuit, that being called its period T.
    • Kepler followed the ancients in always starting to measure at the point furthest from the Sun.) Almost certainly Kepler was responsible for introducing the term 'orbit', in Astronomia Nova Ch.1, and on his behalf we shall precisely define an orbit as possessing a pair of independent constituents: the path or curve, together with a (geometrical) way of representing time.
    • In their day - and indeed until comparatively recently - the aim of astronomers was to achieve accurate observations of angles, simply because no other feature could be measured directly.
    • In order to transpose the observations from a geocentric to a heliocentric basis, he applied triangulation to ensure that each Mars-distance was measured as if from the fixed Sun.
    • Bearing in mind that the observations contained no distance-measurements (as explained in Section 2), this involved expressing all the Mars-Sun distances in terms of the Earth-Sun distance, regarded as a standard unit or 'baseline' (since the path of the Earth is very nearly circular, this approximation happened to be accurate enough for Kepler's purpose); .
    • It is well-known that a pair of (mathematically-defined) directed quantities are mutually independent if and only if they are at right angles: using Euclid's term 'orthogonal' for mutually perpendicular (it was defined in Elements Book I), this will be named the Principle of Orthogonal Independence; it will, with hindsight, justify our separate treatment of the path and the time-measure.
    • (This usage was authenticated by tradition, since in ancient astronomy motions consisted of combinations of rotations which were measured by the angles at the centres of their respective circles.) Then we have corresponding angles from the parallels AZ and BQ, so that: .

  12. Haupt calculus textbooks
    • Measure and content, for the most part in general fields of sets are dealt with in the first half.
    • In the second half there is considered first, integration associated with content and second, integration associated with measure.
    • This third volume of the new edition is a thoroughly modern textbook on measure theory and integration.
    • The five parts, which include a total of eleven chapters, deal with, first, content, measure, and their extensions; second, partition integrals, s-additive functions, and linear functionals; third, measure and integration in topological spaces; fourth, primitive functions and the indefinite integral; and fifth, applications.
    • Spaces of measurement; 2.
    • Content and measure; 3.
    • Integral and real functions with respect to a measure; 4.
    • Product measures.

  13. Arthur Eddington's 1927 Gifford Lectures
    • Whether we are studying a material object, a magnetic field, a geometrical figure, or a duration of time, our scientific information is summed up in measures; neither the apparatus of measurement nor the mode of using it suggests that there is anything essentially different in these problems.
    • The measures themselves afford no ground for a classification by categories.
    • We feel it necessary to concede some background to the measures - an external world; but the attributes of this world, except in so far as they are reflected in the measures, are outside scientific scrutiny.
    • Science has at last revolted against attaching the exact knowledge contained in these measurements to a traditional picture gallery of conceptions which convey no authentic information of the background and obtrude irrelevancies into the scheme of knowledge.
    • I am convinced that a just appreciation of the physical world as it is understood today carries with it a feeling of open-mindedness towards a wider significance transcending scientific measurement, which might have seemed illogical a generation ago; and in the later lectures I shall try to focus that feeling and make inexpert efforts to find where it leads.

  14. Aitken: 'Statistical Mathematics
    • On the other side, the corroborative part of the science consists in interpreting the abstract functions, formulae, equations, constants, invariants and the like, which occur in the pure formulation, as measures and measurable relations of actual phenomena, or numbers constructed from those measures in a definite way.
    • Probability as Measure of a Sub-Aggregate.
    • Now the question of assigning a measure to such aggregates has been deeply studied in modern pure mathematics, the guiding idea being that of extending as widely as possible the scope of a concept familiar in simple cases, namely the cardinal number of a finite set of objects, the length of a line, the area of a surface, the volume of a solid.
    • If M is the measure of the whole aggregate S of possible phases, and pM the measure of the aggregate of E-phases contained in it, then p is the probability p(E ; S).
    • For example, if the aggregate were of points on a continuous line segment, and the measure were ordinary length, then we have implied in this description that all points in the segment are equally likely.
    • (iv) a measure M can be given to the whole set S, and if pM is the measure of the subset favourable to E, then p is the probability p(E ; S) of E with respect to S; .
    • (v) the question of equal likeliness of phases is the same as the question of specifying the aggregate and its measure, and in practical applications this must be determined by the circumstances of the particular problem.

  15. Cochran: 'Sampling Techniques' Introduction
    • In a new area of research, where the collection of data presents perplexing problems of measurement, it may be decided to concentrate the resources on this aspect of the survey, choosing a population that is compact and easy to sample, although this is not the broader population about which information is really wanted.
    • Methods of measurement.
    • When the kinds of data that are needed have been decided, there may be a choice as to the methods of measurement to be employed.
    • The personnel must receive training in the purpose of the survey and in the methods of measurement to be employed and must be adequately supervised in their work.
    • Any completed sample is potentially a guide to improved future sampling, through the data which it supplies about the means, standard deviations, and nature of the variability of the principal measurements, and about the costs involved in getting the data.
    • In some of the steps-the definition of the population, the determination of the data to be collected and of the methods of measurement, and the organization of the field work-sampling theory plays at most a minor role.
    • Consequently the sampling variance of the estimate is used to provide, in inverse terms, a measure of its precision.

  16. Gender and Mathematics
    • that females should learn exactly the same mathematics as do males, be able to perform the same on various measures of mathematical learning, and have the same personal feelings toward oneself and mathematics.
    • Under this definition when equity is achieved, girls will be as confident about learning mathematics as are boys, girls will believe that they have as much control of their mathematics learning as do boys, etc., and there would be no differences found on such tests as the SAT or local, state, national, or international measures of achievement.
    • While other papers will address gender differences in attitudes toward and learning mathematics and so I won't be expansive about it, I read the literature to indicate that whenever higher level cognitive skills are measured, girls are still not performing as well as boys, nor do they hold as positive an attitude toward mathematics.
    • Since items that measure spatial visualization are so logically related to mathematics, it has always appeared reasonable to believe that spatial skills contributed to gender differences in mathematics.
    • It also seemed that when tests measured problem solving at the most complex cognitive level, the more apt there were to be gender differences in mathematics in favour of males.
    • What mathematics is being measured in tests where gender differences are either shown or not shown? How was the information about values obtained? Were females' voices a part of the data gathering procedures? Too often research dealing with these issues provides an incomplete picture at best and only helps to perpetuate the belief that females are somehow inadequate in relation to learning and doing mathematics.

  17. Born Inaugural
    • Occupied by his tedious work of routine measurement and calculation, the physicist remembers that all this is done for a higher task: the foundation of a philosophy of nature.
    • Gauss has frankly expressed his opinion that the axioms of geometry have no superior position as compared with the laws of physics, both being formulations of experience, the former stating the general rules of the mobility of rigid bodies and giving the conditions for measurements in space.
    • Of course every measurement is a disturbance of the phenomenon observed; but it was assumed that by skilful arrangement this disturbance can be reduced to a negligible amount.
    • Now it is evident and trivial that not every grammatically correct question is reasonable; take, for instance, the well-known conundrum: Given the length, beam, and horse-power of a steamer, how old is the captain? - or the remark of a listener to a popular astronomical lecture: "I think I grasp everything, how to measure the distances of the stars and so on, but how did they find out that the name of this star is Sirius?" Primitive people are convinced that knowing the "correct" name of a thing is real knowledge, giving mystical power over it, and there are many instances of the survival of such word-fetishism in our modern world.
    • The evidence provided there consists of photographic plates, and of tables and curves representing measurements.

  18. Eddington on the Expanding Universe
    • In recent years the line-of-sight velocities of about 90 of the spiral nebulae have been measured.
    • So also the cosmical constant has an interpretation not only as a repulsive force but as a measure of curvature.
    • Thus measurement in terms of the metre is equivalent to measurement in terms of the world-radius, since the two standards are always in a constant ratio.
    • I think that some day, when electrons and protons have come to order, we shall look back and see that the key which unlocked the mystery was lying somewhere in intergalactic space and was picked up by astronomers who measured the velocities and distances of nebulae ten million light-years away.

  19. Horace Lamb addresses the British Association in 1904, Part 2
    • It is recognised indeed that all our measurements are necessarily to some degree uncertain, but this is usually attributed to our own limitations and those of our instruments rather than to the ultimate vagueness of the entity which it is sought to measure.
    • It is, at any rate, not verified by the experience of those who actually undertake physical measurements.
    • A practical measurement is in fact a classification; we assign a magnitude to a certain category, which may be narrowly limited, but which has in any case a certain breadth.
    • And the progress of science consists in a great measure in the improvement, the development, and the simplification of these artificial conceptions, so that their scope may be wider and the representation more complete.

  20. Cariolaro's papers
    • (with Carlo Bertoluzza) On the measure of a fuzzy set based on continuous t-conorms, Fuzzy Sets and Systems 88 (1997), 355-36.
    • The original aim of this research was the introduction of a compositive measure of information on a class (an algebra) of fuzzy subsets of an universe W.
    • However, there is a bijection between the class of compositive informations and the class of decomposable measures.
    • Therefore, we restated our problem as the one of the construction of a decomposable measure over the space (W, A*), starting from a crisp measure defined on (W, A).
    • This problem has been analyzed by many authors in some special cases; in particular if the crisp measure is a possibility (Sugeno) or an archimedean decomposable measure.
    • Here we present an approach which permits us to construct such a fuzzy measure in the case where the crisp one has an arbitrary continuous composition law.

  21. Einstein: 'Geometry and Experience
    • But there is another reason for the high repute of mathematics, in that it is mathematics which affords the exact natural sciences a certain measure of security, to which without mathematics they could not attain.
    • All linear measurement in physics is practical geometry in this sense, so too is geodetic and astronomical linear measurement, if we call to our help the law of experience that light is propagated in a straight line, and indeed in a straight line in the sense of practical geometry.
    • Now if the actual velocities of the stars, which can, of course, be measured, were smaller than the calculated velocities, we should have a proof that the actual attractions at great distances are smaller than by Newton's law.
    • Their increase in size as they depart from S is not to be detected by measuring with measuring-rods, any more than in the case of the disc-shadows on E, because the standards of measurement behave in the same way as the spheres.

  22. Biography of Mathematics
    • The latter is also expressed by saying that the measure of AB with the CD unit is an irrational number.
    • The most important thing is to observe that the basic problems of Mathematics are perfectly posed in the Elements, that is, linear and multilinear problems, number theory, problems of local approximation and problems of measurement.
    • With this, and taking as a canonical family of infinitesimal powers of the variable, we can assign to each infinitesimal a number, which was called its order, and in this way we obtained a measure of the local approximation of two functions, which is the essential point of the theory of this approach.
    • Problem of measurement.
    • Its complete formulation gives rise to the theory of measurement.
    • On the other hand, within this problem of measurement is the Pythagorean theorem, metric spaces, and so on.

  23. Feller Prefaces
    • The fact that laymen are not deterred by passages which proved difficult to students of mathematics shows that the level of difficulty cannot be measured objectively; it depends on the type of information one seeks and the details one is prepared to skip.
    • The handling of measure theory may illustrate this point.
    • Chapter IV contains an informal introduction to the basic ideas of measure theory and the conceptual foundations of probability.
    • The same chapter lists the few facts of measure theory used in the subsequent chapters to formulate analytical theorems in their simplest form and to avoid futile discussions of regularity conditions.
    • The main function of measure theory in this connection is to justify formal operations and passages to the limit that would never be questioned by a non-mathematician.
    • Readers interested primarily in practical results will therefore not feel any need for measure theory.

  24. Harold Jeffreys on Logic and Scientific Inference
    • The conclusion is free from the difficulties of that of the classical syllogism; it is perfectly possible to measure Brown's height.
    • But how do we know the general proposition? If it is known by experience, we have already measured the heights of all English policemen, and therefore we have measured P.C.
    • But suppose that we have not made extensive measurements of the heights of policemen, but that we know of the official regulation that no man is appointed to be a policeman unless his height is at least five feet nine.

  25. Eulogy to Euler by Fuss
    • That is to say a great deal since Johann Bernoulli was still alive Barely embarked on his career, it is only a truly independent genius who bursts out so rapidly and to be sized-up next to a man resplendent in glory from so many discoveries which were done at the expenses of the English and French mathematicians who had dared to measure up to him.
    • It is there that we find remarkably the full measure of the theory of curves: tautochrones, brachistochrone, trajectories and the very deep research in integral calculus, on the nature of numbers, concerning series, the motion of heavenly bodies, the attraction of spheroid-elliptical bodies and on an infinity of subjects of which one hundredth part would suffice in making the reputation of anyone else.
    • He showed a new algorithm which he found for circular quantities, for which its introduction provided for an entire revolution in the science of calculations, and after having found the utility in the calculus of sine, for which he is truly the author, and the recurrent series, he provides for in the second part the general theory of curves with their divisions and sub-divisions and in a supplement the theory of solids and their surfaces while showing how their measurement leads to the equations with three variables and he ends finally this important work by developing the idea of curves with double curvature which provides for the consideration of the intersection of curved lined surfaces.
    • He felt that everything which possessed a relation was within his capacity to be measured and to be submitted to calculations.

  26. Edward Sang on his tables
    • Mathematicians were then engaged in the introduction of the decimal system into every branch of calculation and measurement; but for the introduction of this new system into the measurement of angles, it was necessary to have a new trigonometrical canon.
    • That again rendered it necessary to calculate the sines measured in parts of the quadrant as a unit, instead of in parts of the radius, as usual.
    • While artisans and physicists are using the ten-millionth part of the earth's quadrant as their unit of linear measure, astronomers are still subdividing the quadrant into 90, 60, 60, and 100 parts.

  27. Proclus on pure and applied mathematics
    • They do not think, as some others do, that military science should be considered as one of the branches of mathematics, though, to be sure, it sometimes uses logistics, for example in the enumeration of companies, and at other times geodesy, for example in the division and measurement of areas.
    • For it is not the business of geodesy to measure the cylinder or the cone, but to measure mounds as if they were cones, or wells as if they were cylinders.
    • Hence he gives to numbers a name after the objects that are being computed and thus speaks of melites [number of apples or sheep] and phialites [weight of liquid measures].

  28. Champernowne reviews
    • In part this can be explained by the fact that in the intervening period relatively few economists have addressed themselves to the subject with which it deals - the personal distribution of income - but, most importantly, it is a measure of the originality of Professor Champernowne's thinking.
    • In the wake of the substantial increase in income inequality in the United States and the United Kingdom that started in the 1980s, interest in economic inequality surged, at least measured by the volume of research papers devoted to documenting and explaining the phenomenon.
    • It covers basically all the topics one can think of - inequality measurement, models of the distribution of earnings, models of the distribution of wealth, social welfare measurement, stochastic models of income distribution, policy, etc.

  29. Max Planck: 'The Nature of Light
    • His theory of electricity led him to the conclusion that every electrical disturbance moved from its source through space in waves with a velocity of 300,000 kilometres per second, and the coincidence of this figure, obtained from purely electrical measurements, with the magnitude of the velocity of light, led him to consider light as an electro-magnetic disturbance.
    • to the results of measurements.
    • For this purpose, in addition to a training in physics and the requisite mathematical ability, it is necessary to have a discriminating judgment of the measure of the reliability that can be placed on the accuracy of the measurements; for the effects sought for are mostly of the same order as the errors of observation.
    • When, on the other hand, we consider that these hypotheses help us to elucidate the mysterious structure of the spectra of the different chemical elements and, in particular, the complicated laws governing the spectral lines, not only as a whole but, as Arnold Sommerfeld first showed, partly even in minute details, with an exactness equal to, and even surpassing, that of the most accurate measurements-when we consider this we must, for good or ill, make up our minds to assign a real existence to these light quanta, at least at the instant of their origin.

  30. Flatland' Second Edition Preface
    • Dimension implied direction, implies measurement, implies the more and the less.
    • No 'delicate micrometer' - as has been suggested by one too hasty Spaceland critic - would in the least avail us; for we should not know what to measure, nor in what direction.
    • Hence, all my Flatland friends - when I talk to them about the unrecognised Dimension which is somehow visible in a Line - say, 'Ah, you mean brightness': and when I reply, 'No, I mean a real Dimension,' they at once retort, 'Then measure it, or tell us in what direction it extends'; and this silences me, for I can do neither.
    • But what was his reply? 'You say I am "high"; measure my "high-ness" and I will believe you.' What could I do? How could I meet his challenge? I was crushed; and he left the room triumphant.
    • Suppose a person of the Fourth Dimension, condescending to visit you, were to say, 'Whenever you open your eyes, you see a Plane (which is of Two Dimensions) and you infer a Solid (which is of Three); but in reality you also see (though you do not recognize) a Fourth Dimension, which is not colour nor brightness nor anything of the kind, but a true Dimension, although I cannot point out to you its direction, nor can you possibly measure it.' What would you say to such a visitor? Would not you have him locked up? Well, that is my fate: and it is as natural for us Flatlanders to lock up a Square for preaching the Third Dimension, as it is for you Spacelanders to lock up a Cube for preaching the Fourth.

  31. Studies presented to Richard von Mises' Introduction
    • In drawing such a picture, the central task is to understand the relation between the direct sense observation of the experimental physicist and the conceptual system of science, which consists of expressions such as "increase of entropy" or "principle of relativity." Most physicists are inclined to say that the picture drawn and the principles devised by our inductive ability are eventually checked by actual measurement of physical quantities like length, weight, electric charge, etc., but they use the expression "measurement of a length" in a perfunctory way, forgetting that no numerical value can ever be assigned to a length by a single measurement.
    • In fact, a long series of measurements is needed from which eventually "the value of the length" can be computed.

  32. Johnson post1900 books
    • Hence the introduction by Prof James Thomson of the poundal, which serves this purpose when the English system of weights and measures is used, has been of very great value.
    • The force of its gravity has indeed changed, but it is (when we use gravitation units) the unit of this force, and not its numerical measure, which has changed.
    • These have, however, been introduced rather as diagrammatic aids to the comprehension of general principles, and to the calculation of numerical results, than as methods of obtaining results by measurement from accurately constructed diagrams - the latter belonging rather to the province of Applied Mechanics.

  33. Ernest Hobson addresses the British Association in 1910, Part 3
    • In each case the whole is something more than merely the sum of its parts; it has a unity of its own, and that unity must be, in some measure at least, discerned by its creator before the parts fall completely into their places.
    • In that division of the subject known as metric geometry, for example, axioms of congruency are assumed which, by their purely abstract character, avoid the very real difficulties that arise in this regard in reducing perceptual space-relations of measurements to a purely conceptual form.
    • Of late years a new spirit has come over the mathematical teaching in many of our institutions, due in no small measure to the reforming zeal of our General Treasurer, Professor John Perry.
    • I draw, then, the conclusion that a mixed treatment of geometry, as of mechanics, must prevail in the future, as it has done in the past, but that the proportion of the observational or intuitional factor to the logical one must vary in accordance with the needs and intellectual attainments of the students, and that a large measure of freedom of judgment in this regard should be left to the teacher.

  34. R A Fisher: 'Statistical Methods' Introduction
    • If an observation, such as a simple measurement, be repeated indefinitely, the aggregate of the results is a population of measurements.
    • Yet, from the modern point of view, the study of the causes of variation of any variable phenomenon, from the yield of wheat to the intellect of man, should be begun by the examination and measurement of the variation which presents itself.
    • The study of variation has led not merely to measurement of the amount of variation present, but to the study of the qualitative problems of the type, or form, of the variation.

  35. Rydberg's application
    • But whereas the two first mentioned have, themselves, enriched science with a great number of measurements of the spectra of the elements and then have used these fully comparable determinations for their calculations, Docent Rydberg has, here, also, chiefly used the line determinations of others for his calculations.
    • Docent Rydberg's works on Spectrum Analysis are undoubtedly of a great scientific value and prove also the author's great diligence and interest for the treatment of an often ungrateful problem, but these works cannot completely establish his competency for the appointment in question, as they are not based on his own measurements and researches.
    • You have taken the quickest way in using already-existing measurements.
    • This admiration is to a large measure based on the fact that you discovered the regularities in the spectra of elements through an examination of the then-existing, many-fold deficient observations.

  36. Horace Lamb addresses the British Association in 1904
    • The same thing holds in a measure of the problems of ordinary Dynamics, where some practical knowledge of the subject-matter is within the reach of everyone.
    • These appealed to us all; but some of us, I am afraid, under the influence of the academic ideas of the time, thought it a little unnecessary to show practically that the height of the lecture-room could be measured by the barometer, or to verify the calculated period of oscillation of water in a tank by actually timing the waves with the help of the image of a candle-flame reflected at the surface.
    • An argument which asks us to leave out of account such things as the investigation of Fluorescence, the discovery of Spectrum Analysis, and the measurement of the Viscosity of Gases, may well seem audacious; but a survey of the collected works of these writers will show how much, of the very highest quality and import, would remain.

  37. W H Young addresses ICM 1928
    • From what I have said you will understand that the Mathematical Method is not confined to the employment of the theory of Measurement, extraordinarily fertile though this branch of our subject has certainly been.
    • The work of Pure Mathematicians of the 19th Century was partly applied to convince mathematicians themselves that the idea of measurement did not enter even into Geometry to the extent that was supposed.
    • I need hardly say that the Theory of Groups does not involve measurement, and the same is true of the notion of Function, which, indeed, so little depends in itself on the idea of measurement, that, when the occurrence, or non-occurrence, of an event depends on the occurrence, or non-occurrence, of various combinations of other events, we are patently in presence of a functional relationship of a two-valued function of several variables, each assuming only two values, say zero when the event in question does not occur, and unity when it does.
    • Here the idea of measurement is wholly absent, and this though the final decision as to whether an event has happened, or not, may depend in practice on an act of measurement, such, for instance, as the doctor has to carry out with his clinical thermometer.
    • Measurement, as such, represents the persistent attempt to replace Quality by Quantity.

  38. Bessel and the Royal Astronomical Society
    • In fact Henderson was the first to correctly measure a stellar parallax but Bessel published first.
    • In a conversation I had with M Bessel, he expressed his wish that Alpha Centauri were observed with a heliometer, or good equatorial, capable of precise micrometrical measurement; he said he had doubts of the results derived from meridian instruments.
    • this was, of course, because he had measured the distance to the nearest star to the sun, as was later confirmed.

  39. Cafaro's papers
    • However, this argument appears weak: in fact, one can always measure the thermodynamic temperature in a given point of a solid, even where hyperbolic conduction models predict a negative entropy production.
    • Furthermore, at present nobody knows whether the concept itself of generalized temperature has a physical meaning, which is a necessary condition to allow its measurement.
    • Because temperature can be measured only when it is larger than the instrument sensitivity threshold, if one attempts detecting temperature changes at a given point in a solid consequent to a local temperature variation in another point, no change will be detected until a certain amount of time has elapsed.

  40. Poincaré on non-Euclidean geometry
    • According to Lobachevsky, the difference is proportional to the area of the triangle, and will not this become sensible when we operate on much larger triangles, and when our measurements become more accurate? Euclid's geometry would thus be a provisory geometry.
    • We might as well ask if the metric system is true, and if the old weights and measures are false; if Cartesian co-ordinates are true and polar coordinates false.
    • Now, Euclidean geometry is, and will remain, the most convenient: 1st, because it is the simplest, and it is not so only because of our mental habits or because of the kind of direct intuition that we have of Euclidean space; it is the simplest in itself, just as a polynomial of the first degree is simpler than a polynomial of the second degree; 2nd, because it sufficiently agrees with the properties of natural solids, those bodies which we can compare and measure by means of our senses.

  41. Rios's books
    • Measure of sets.
    • An expository survey of the concepts of measure and integral, the Kolmogorov identification of probability with measure theory, probability distributions, and the Fisher-Neyman-Pearson theories of estimation and testing hypotheses.
    • Applications to the measurement of physical magnitudes; theory of errors; quality control.

  42. Feller Reviews 4
    • The atmosphere of the text turns more austere in Chapters 4 and 5 which introduce measure on Euclidean space in fair detail, and on an abstract probability space in elegant outline (the usual extension theorems, as well as the results of Fubini and Radon-Nikodym are stated without proof).
    • Along with the times, basic measure theory is introduced and employed to provide a more solid foundation for proofs of probability theorems, etc., although as Feller says "..
    • The main function of measure theory ..
    • not feel any need for measure theory." Indeed, the chapters are rather self-contained so that many readers in probability could select and gain a more competent view of their own particular interests in probability theory and its applications.
    • Inevitably this means measure theory which is liable to induce mathematical paralysis in many of those brought up before the days of the "new mathematics".
    • The reason was obvious: the mathematics of continuous systems involves more sophisticated material than discrete ones, and the whole apparatus of measure theory has to be invoked.

  43. Cheney books
    • This has made it unnecessary to presuppose the Lebesgue integral or measure theory.
    • The book contains two additional chapters having important material not included in the course: Chapter 8, on measure and integration, is for the benefit of readers who want a concise presentation of that subject, and Chapter 7 contains some topics closely allied, but peripheral, to the principal thrust of the course.
    • One advantage of this strategy is that they will see the necessity for topology, measure theory, and other topics - thus becoming better motivated to study them.
    • Measure and integration.
    • The first six chapters contain enough material for a year course, and the final two chapters contain related material (such as topology, measure theory and integration theory) which are mentioned in the earlier portions.

  44. Finlay Freundlich's Inaugural Address, Part 2
    • Until only about 40 years ago it was believed that the laws according to which distances in space are measured were given by Euclid's laws of geometry, the first mathematical system systematically and consistently built up from axioms, i.e.
    • It was still taken for granted that the laws, according to which distances in physical space have to be measured, remained those given by Euclid's geometry.
    • Alone, the mass, measured on the Earth by the weight, in celestial space by the properties of orbital motion, determines the degree by which the free mobility of another body is impeded.
    • This deflection depends on the solar mass and on the distance at which the light beam passes the Sun, and is large enough to be accurately measured.
    • The value of science has been too frequently measured with the yardstick of successful exploitation of the riches of nature for the benefit of man.

  45. Variable Stars catalogue
    • Sequences of comparison stars are selected from the photographs and measured by Mrs Fleming for all variable stars discovered here, except those in clusters or special regions.
    • Measures of the variables are made by Miss S E Breslin, comparing them by Argelander's method with two of a sequence whose magnitude has been determined.
    • For a large number of the stars discovered here, new elements have been derived, by a method of Least Squares, from all the available dates of maxima obtained from photographic measures, or from both photographic and visual observations.

  46. James Jeans: 'Physics and Philosophy' II
    • This must be objective as regards both quantity and quality, so that its measure will always be the same, whatever means of 'measurement are employed to measure it just as a real object must always weigh the same whether it is weighed on a spring balance or on a weighing-beam.

  47. Carl Runge: 'Graphical Methods
    • wide are substituted for the area so that, measured in square centimetres, it is equal to half the sum of the lengths of the strips measured in centimetres.
    • 2), to measure the area between them by the method described above and to estimate the two segments separately.

  48. Kingman autobiography
    • I got to the 16-times table before graduating to the wonderfully complex weights and measures of that pre-metric age.
    • James and I were encouraged to write a book suitable for final year undergraduates, basing probability firmly on measure theory.
    • (The Oxford don asked to support a measure of reform: "Reform, aren't things bad enough already?") In Sussex there were no precedents, and one always had a blank sheet of paper.

  49. Emma Castelnuovo Geometry
    • He bases his work on examples taken from agriculture, the economy of his country being essentially agricultural: the problem of how to measure fields of different shapes, ..
    • Without any doubt the book was used because of a strange coincidence: The governors of Lombardia thought that Clairaut-s book would be very suitable for the technical-agricultural schools because many examples were connected with the measurement of fields.
    • The examples on fields of different shapes and their measurement were replaced by up-to-date examples and problems, and geometry was strictly connected to the other branches of mathematics in order to investigate practical and theoretical problems.

  50. Rudio's Euler talk
    • The laws that govern the motion of the falling stone have first been laid down by Galilei and can been summarised as follows: If you measure the time that the stone needs in order to cover a certain distance, it will cover the exact same distance in the same amount of time, no matter how often you repeat the experiment.
    • Thus, if you have measured that the stone covers a distance of 5 metres in one second, you are now able to calculate the distance that the stone will cover in an arbitrary amount of time.
    • Now if you measure the sheet's brightness at a given distance, then the brightness will decrease by a factor of 2×2 or 4 as the distance is doubled, by a factor of 3×3 or 9 if the distance is three times as big, by a factor of 10×10 or 100 if the distance is ten times as big, etc.

  51. D'Arcy Thompson on Greek irrationals
    • If the side of the triangle measure One, One must represent the diagonal also, as its nearest rational number or equivalent.
    • This property of the side- and- diagonal series, that not merely is the square of the one in alternate excess and defect as compared with twice the square on the other, but that this alternate excess and defect is in every case measured by one unit, is expressly stated by Theon and by Proclus.
    • [One], in short, is the word used both for that unit with which the series on either side begins, and for that unit which is at every successive stage the measure of excess or defect.

  52. Ernest Hobson addresses the British Association in 1910, Part 2
    • Measurement is regarded as one of the applications, but as no part of the basis, of mathematical analysis.
    • Except in certain very simple cases no process of measurement, such as the determination of an area or a volume, can be carried out with exactitude by a finite number of applications of the operations of arithmetic.
    • It is maintained by M Poincare, for example, that the question which is the true scheme has no meaning; that it is, in fact, entirely a matter of convention and convenience which of these geometries is actually employed in connection with spatial measurements.

  53. Pack wartime papers
    • The time taken by the fastest pulse to penetrate various lengths of steel and of lead has been measured experimentally, and the results confirm that the plane elastic waves move more quickly for steel; while for lead the shock wave before damping has a velocity well in excess of that of the elastic waves.
    • They measure the velocity of the plastic wave - a stress wave that results in the irreversible repositioning of atoms relative to their neighbours, and a type of shock wave - and find that its velocity in steel is less than (and in lead greater than)the velocity of planar elastic waves.
    • The large penetrations measured in lead targets are shown to result from the flow which takes place in the metal after the jet itself has been consumed For a given Jet at a given stand-off it is possible to predict the penetration into a combination of targets from the results of a very small number of standard experiments.

  54. James Jeans addresses the British Association in 1934
    • Physical science obtains its knowledge of the external world by a series of exact measurements, or, more precisely, by comparisons of measurements.
    • All that is essential is the relative blackness of the smear at different places-a ratio of numbers which measures the relative chance of electrons being at different points of space.

  55. Ford - Mathematics for Field Artillery
    • In the section on arithmetic the most interesting features emphasized are the conversion of metric into English units and vice versa, various units of angular measure, mil, grad, degree, the process of extracting the square root, and interpolation.
    • A scout measures the angle between a line to an object C and the straight road along which he is passing.
    • He measures the angle between the two points and draws lines through A' and B', the map positions of the two points, to meet at the angle found at some point C' on the map.

  56. Sikorski books
    • The second part develops Lebesgue's integration, together with a detailed treatment of abstract measure theory and the Stieltjes integral.
    • The treatment of all this material is outstanding by its great clarity and in showing how the deeper results of set theory and the abstract theory of measure find applications in functional analysis in general, and the theory of orthogonal series.
    • Of its eleven chapters, the first two are an introductory exposition of sets, functions, classes of sets, and metric spaces; the next three chapters deal with continuity and convergence, and the remaining six are devoted to measure, integration, and differentiation.
    • Still more remarkable is the fact that the book could serve very well to introduce a serious student to a wide range of topics in set theory, topology, measure theory, logic, and, of course, the theory of Boolean algebras.

  57. Rouche and de Comberousse
    • 3 - Measurement of angles.
    • 8 - Measurement of the circumference.
    • 1 - Measurement of polygon areas.
    • 1 - Measurement of quantities.

  58. Edinburgh Mathematics Examinations
    • Define the radian or unit of circular measure; and find the formula for the number of degrees in a given number of radians, and vice versa.
    • By taking the decimetre as equal to four inches, what percentage of error is introduced first in linear measure, second, in square measure, third, in cubic measure.

  59. Cotlar publications
    • Mischa Cotlar and Cora Sadosky, A Moment Theory Approach to the Riezs Theorem on the Conjugate Function with General measures, Studia Math.
    • Mischa Cotlar and Cora Sadosky, Characterization of Two Measures Satisfying Riesz Inequality for the Hilbert Transform in L2, Acta Cient.
    • Rodrigo Arocena and Mischa Cotlar, A Generalized Herglotz-Bochner Theorem and L2-Weighted Inequalities with Finite Measures, Harmonic Analysis Conf.
    • Mischa Cotlar and Cora Sadosky, Weakly positive matrix measures, generalized Toeplitz forms, and their applications to Hankel and Hilbert transform operators, Continuous and discrete Fourier transforms, extension problems and Wiener-Hopf equations, Oper.

  60. Rudio's talk
    • Unconcerned by the superstition that had been linked to the comets as dreaded celestial phenomena since time immemorial and inspired by the comet of 1472, Regiomontanus came up with the for his time completely novel idea of treating it like any other celestial body for once and carry out astronomical measurements.
    • 4 Morgen -- unit of measurement used in Germany and some other states, used until the 20th century.

  61. Weil reviews
    • The author discusses those problems in the theory of topological groups which center around the notion of Haar measure.
    • The first part - Chapters I and II - deals with the geometry and measure on the space of adelic points of an algebraic variety.
    • The author states in the foreword that he has tried "to draw the conclusions from the developments of the last thirty years, whereby locally compact groups, measure and integration have been seen to play an increasingly important role in classical number theory" and to show that from the point of view which he has adopted one could give a coherent account, logically and aesthetically satisfying, of the topics he was dealing with.

  62. Apostol Project
    • A problem is proposed: measurements must be taken through an obstacle.
    • Intrigue and drama are injected into the story when alternative theories are pro posed, for example, Did Eupalinos physically measure around the mountain or over the mountain? Site exploration, simple mathematics, and common sense sup ply the answer.

  63. Peirce publications
    • (with Robert Wheeler Willson) On the measurement of internal resistance of batteries, Amer.
    • The resistivity of hardened cast iron as a measure of its temper and of its fitness for use in permanent magnets, Proc.

  64. Art Mathematics Music.html
    • As an example our hands, feet and the rest of our body feel warmth and cold providing a subjective estimate of temperature although temperature is accurately measured by thermometers using, for example, the expansion and contraction of liquid mercury in a fine tube.
    • A mathematical theory of heat was developed by Sadi Carnot who introduced the heat engine cycle in 1824 and Rudolf Clausius who in 1850 introduced the second law of thermodynamics and then in 1865 introduced the concept of entropy that may be regarded as a measure of disorder.

  65. Gyula König Prize
    • In 1906 the French mathematician Pierre Fatou, after showing in his famous doctoral dissertation that every function that is bounded and holomorphic inside a circle has a limiting value almost everywhere, that is, with the exception of a set of measure 0, raised the following question: since this limiting function cannot be constant on the whole arc, as was stated above, how large can the set be on which it is constant; or, and this amounts to the same, how large can the set be on which it vanishes? After showing that this set cannot fill out "almost" entirely an arc, he formulated the conjecture, which he believed was difficult to prove, that this set has measure 0.
    • Therefore, the logarithm may be equal to negative infinity, that is, the boundary limit function itself may vanish, only on a set of measure 0.

  66. 21st Century mathematics
    • Lewis Fry Richardson, the British Quaker mathematician and meteorologist, asked the curious question, in the 1920s, on the top of an open London bus: "Does the wind have a velocity?" In other words, does the measured wind always vary smoothly or could it suddenly jump from one value to another? .
    • In this case mathematics helps solve a particular problem, but more often its main use is in providing methods of analysis, concepts, computation and measurement that specialists in other disciplines can use.

  67. The St Andrews Schmidt-Cassegrain Telescope
    • The photographic method provides, in a relatively short time, a permanent record from which measurements can subsequently be made.
    • The area available for accurate measurement is, in fact, more than five hundred times as great as the useful area on a photograph taken with a Newtonian telescope of comparable size.

  68. Leonardo quotes
    • It has measured the distances and sizes of the stars, it has discovered the elements and their locations ..
    • Linear perspective deals with the action of the lines of sight, in proving by measurement how much smaller the second object is than the first, and how much the third is smaller than the second, and so on by degrees to the limit of things visible.

  69. Dixmier reviews
    • The main part of the book and the part properly described by the title consists of the lectures on 'Volumes of Polyhedra' by Henri Cartan, on 'Measure of Angles' by Jacques Dixmier and on the 'Theory of Integration' by Andre Revuz.
    • The second lecture on 'Measure of Angles' ..
    • This is a set of five expository articles, two of which have nothing to do with measure.

  70. Cheltenham exams
    • Arithmetic and geometric progressions also feature heavily, question 13 in the 1875 paper asks, "deduce an expression for the sum of n terms of a geometric series." Factors are called common measures and question 8 of the same paper instructs the pupil to, "find the greatest common measure of 3a3 - 3a2b + ab2 - b3 and 4a2 - 5ab + b2 ".

  71. R A Fisher: the life of a scientist' Preface
    • Fisher advocated practical measures by which to reduce this deleterious selection in contemporary society and thus save Western Civilization from the fate of its predecessors.
    • Combining genetical work in the laboratory with evolutionary field work, in collaboration with E B Ford, he demonstrated the reality of the effects of natural selection and developed statistical methods by which they could be measured.

  72. James Jeans: 'Physics and Philosophy' I
    • Physics gives us exact knowledge because it is based on exact measurements.
    • The wave-lengths of these lines can be measured, and are found to be related with one another in a very simple way which can be expressed by a quite simple mathematical formula.

  73. Slaught on Mathematics and Teaching
    • This assumption seems to find justification in the large number of papers on this subject which have been read in recent years before conferences and associations, in the reports which various committees have worked out and presented before representative bodies of teachers, and in the perennial agitation over the failure of the schools to make their pupils measure up to the college-entrance requirements.
    • It is abundantly evident to the impartial observer that the teaching of college freshmen and sophomores is not so widely different from the teaching of high school juniors and seniors; and that, on the whole, there is as large a proportion of poor teaching done in the colleges as in the high schools; with the certainty that this proportion will rapidly increase unless remedial measures for the colleges are soon undertaken comparable with those now in operation for the high schools.

  74. Born on wave mechanics
    • Yet it is asserted that - apart from the factor ℏ, which serves to transform the units of measurement - energy and frequency are identified, and also momentum and wave-number.
    • This is the celebrated Uncertainty Principle of Heisenberg, which interprets the irrationality of the quantum laws as a limitation of the accuracy with which various quantities can be measured.

  75. Review of du Bois-Reymond's 'Die allgemeine Functionentheorie
    • Among these are the basic geometric concepts: point, line, surface; the perfect (or complete) straight line and the exact measure, and finally also the infinitely large and the infinitely small.
    • In his view, only finite magnitudes exist, among them those that grow beyond any measure, however large, or that shrink below any measure, however small.

  76. Edmund Whittaker: 'Physics and Philosophy
    • In three-dimensional Euclidean geometry there is a fixed relation between the ten mutual distances of any five points, and such a relation may be verified by measurement for the ten mutual distances of five material particles.
    • It is shown in thermodynamics that entropy, which is a measure of the disorder of a system, cannot decrease.

  77. Analysis of Variance
    • The following rough definition of our subject may serve tentatively: The analysis of variance is a statistical technique for analysing measurements depending on several kinds of effects operating simultaneously, to decide which kinds of effects are important and to estimate the effects.
    • The measurements or observations may be in an experimental science like genetics or a non-experimental one like astronomy.
    • A theory of analysing measurements naturally has implications about how the experiment should be planned or the observations should be taken, i.e., experimental design.

  78. Dingle books
    • Historical subjects range from 'Nicolaus Copernicus' to 'Modern Theories of the Origin of the Universe,' and the philosophical selections proceed from the general 'Philosophical Viewpoint of a Scientist' to the more technical 'Theory of Measurement.' The diversity of the collection is at least partially relieved by the consistency of the author's attempt to exhibit science, in both its historical and philosophical ramifications, as a human achievement arising from man's eternal effort to order the data of his senses.
    • It discusses, among other things, astronomy in the 16th and 17th century, physics in the 18th century, cosmological theories, time measurement, the laws of nature, the rational and empirical elements in physics, the relation of science to ethics and religion.
    • Eddington restricted these consequences to the parts of science dealing with measurement, and he made this include the whole of physics, but only the metrical parts of such sciences as deal with life and mind and religious experience (if there is a part of theology which can be called theophysics).

    • Euler that he still felt its full measure when, in 1741 the year after Biren's fall from power that the tyrannical period was replaced by a more moderate and humane government.
    • He knew how to best present it, through the deep understanding of the theory which provided him the way in which to solve a great number of these equations, to distinguish the forms of the orders of integrals, for the different number of variables, to reduce these equations when they attain a certain form and become ordinary integrations, to provide for a way in which to remember these forms, through substitutions after which they vanish; in one word, to discover within the nature of these partial differential equations most of these singular properties which render the general theory so difficult and thorny, qualities which are nearly inseparable in Geometry where the degree of difficulty is so often the measurement of interest that one takes in a question and the honor that one attaches to a discovery.

  80. Proclus and the history of geometry as far as Euclid
    • It owed its discovery to the practice of land measurement.
    • For the Egyptians had to perform such measurements because the overflow of the Nile would cause the boundary of each person's land to disappear.

  81. Teixeira on Rocha
    • The barrel can be considered as a geometrically indefinable solid of revolution, and to measure its capacity, it is replaced by a geometrically defined solid of revolution approximately equal in volume.
    • For this reason Monteiro da Rocha, in order to facilitate its application, considered it necessary to calculate a table which makes it very practical, both in the case that one wants to measure the total capacity of the barrel and that of only a part of it.
    • Thus he dealt with the problem of parabolic orbits of comets and gave the first practical solution to this problem; he took up the problem of predicting eclipses and gave an easier method to solve it than the other processes employed in his time; he took care of the measure of casks and gave a solution that exceeds in its approach, and is not inferior in its simplicity, to the best than had been given previously; he dealt with Fontaine's quadrature rule and gave, for the first time, conditions that could be applied with confidence." .

  82. Knorr's papers
    • Archimedes and the measurement of the circle: a new interpretation.
    • Of all the works in the Archimedean corpus, none has been more widely studied from ancient and medieval times to the present day than the short tract on the measurement of the circle.
    • Its second proposition, for instance, employs a result proved in the third, in violation of the formal deductive ordering of demonstrations, and fails to distinguish between exact and approximate values for the constants of measurement.

  83. Warga abstract
    • A limit solution is then obtained by imbedding the controls in the class of relaxed controls which are functions with values that are probability measures.
    • In more complicated problems (in which the opposing controls are non-additively coupled) one must resort to functions whose values are joint probability measures for both sides that give rise to ordinary relaxed controls for the defense and hyperrelaxed controls (related to conditional probabilities) for the offense.
    • In the third area under discussion, one dealing with modeling of discontinuities in state functions, most researchers used finite measures to describe the problem but in so doing lost much information contained in the physical approximations.

  84. Byrne: Doctrine of Proportion
    • Professor Young will not deny (for they are his own words) that "the term in reality denotes the quotient arising from the division of one magnitude or quantity by another of the same kind (or the multiple or submultiple which an antecedent is of its consequent); it is accurately assignable (in numbers) when the magnitudes are commensurable, but unassignable (in numbers) when they are incommensurable." When this simple fact is known, what is to be understood by the term cannot be misconstrued, although we do allow that in many cases the exact ratio of one magnitude to another of the same kind cannot be expressed by numbers; this may be a fault in our present system of notation, or in the plan adopted for finding a common measure, and not in our geometrical notion of that which is to be conveyed by the term.
    • But the symbols used in geometry must be considered not only as appropriate emblems of the quantities themselves, but also as expressive; and not as any measures or numerical values of them.

  85. William Herschel discoveries
    • He writes, "When I found that the poles of Mars were distinguished with remarkable luminous spots, it occurred to me, that we might obtain a good theory for settling the inclination and the nodes of that planet's axis, by measures taken of the situation of those spots." He also writes in this paper, "I have often noticed occasional changes of partial bright belts ..
    • He realised that in reducing the brightness with a filter, the heat measure by a thermometer was unchanged.

  86. Valdivia aspects of maths
    • To find the answer, he measured near Gottingen a triangle whose vertices were mountain summits and whose sides were about fifty kilometres long: if he could prove that the sum of the angles of this triangle was less than two right angles, then ordinary space would not be Euclidean.
    • He made the measurements and the calculations and concluded that the difference between two right angles and the sum of the angles of said triangle could be due to the errors of the measuring instruments.

  87. Halsted Beltrami
    • As Gino Loria says in his notice of Beltrami, on which we draw here, "This measure - perhaps not absolutely without grounds, but certainly too rigorous - had disastrous consequences for its victim." .
    • In the exordium of a memoir dated Pisa, 31 May 1866, Beltrami remarks that in treating of a map destined to serve for measurements of distance it would be most convenient to determine, that to the geodetics of the surface should correspond the straights of the plane, because, such a representation accomplished, the questions concerning geodetic triangles would be reduced to simple questions of plane trigonometry.

  88. NAS founders
    • Superintendent of the Coast Survey and a inspector of standard weights and measures for the United States .
    • Rutherfurd continued to evolve his telescope designs, engineering models that took increasingly clear pictures, were calibrated for measurement, and captured on film stars that had previously been too faint to record.

  89. Max Planck and the quanta of energy
    • The fruit of this long series of investigations, of which some, by comparison with existing observations, mainly the vapour measurements by Vilhelm Bjerknes, were susceptible to checking, and were thereby confirmed, was the establishment of the general connection between the energy of a resonator of specific natural period of vibration and the energy radiation of the corresponding spectral region in the surrounding field under conditions of stationary energy exchange.
    • This concept could not be maintained for long in the face of fresh measurements.
    • Whilst for small values of the energy and for short waves, Wien's law was satisfactorily confirmed, noteworthy deviations for larger wavelengths were found, first by O Lummer and E Pringsheim, and finally by H Rubens and F Kurlbaum, whose measurements on the infrared residual rays of fluorite and rock salt revealed a totally different, though still extremely simple relationship, characterized by the fact that the quantity R is not proportional to the energy, but to the square of the energy, and in fact this holds with increasing accuracy for greater energies and wavelengths.

  90. Alfred Tarski: 'Cardinal Algebras
    • This book is an axiomatic investigation of the novel types of algebraic systems which arise from three sources: the arithmetic of cardinal numbers; the formal properties of the direct product decompositions of algebraic systems; the algebraic aspects of invariant measures, regarded as functions on a field of sets.
    • By analyzing their proofs we usually arrive at more general formulations which belong to the general theory of one-to-one transformations, and which have found some interesting applications outside the domain of abstract set theory - for instance, in the theory of measure.

  91. E W Hobson: 'Mathematical Education
    • Geometry, the science of spatial relations, is introduced by the observational and experimental study of the simplest spatial relations, verification by actual measurement playing an important part; the abstract treatment in accordance with the deductive method being relegated to a later stage.
    • Owing in large measure to the activities of the Mathematical Association, a considerable transformation in the methods and in the spirit of Mathematical teaching has already taken place in many of our schools, and the changes in the direction indicated by the newer ideals are no doubt destined to have even more far-reaching effects than at present.

  92. Edinburgh Physics Examinations
    • Assuming 32.2 as the foot-second measure of the acceleration produced by gravity, express the same quantity numerically in yard-minute, and in mile-hour, units.
    • What is the amount of kinetic energy in a train of twenty-five tons when moving with a velocity of twenty miles per hour? What force (measured in foot-pound-second units) acting for ten seconds is sufficient to stop the train? .

  93. W H Young addresses ICM 1928 Part 2
    • But once the order-numbers are assimilated -to cardinal numbers, whose values represent, and even to some extent measure, the various degrees of the quality considered, processes valid for cardinal numbers will suggest combinations of the seriated objects, which may, or may not, prove legitimate.
    • Let those who can do so, use it! They will be judged by the results they obtain, by their power to predict the Future, by the plausibility and fertility of the guesses they are enabled to make as to the Past and to the Unseen, by the progress they may initiate in the mastery of Nature, and not merely by the measure of success they may have in giving concrete form to their ideas, or in realising them materially.

  94. R A Fisher: 'History of Statistics
    • The first of the distributions characteristic of modern tests of significance, though originating with Helmert, was rediscovered by K Pearson in 1900, for the measure of discrepancy between observation and hypothesis, known as c2.
    • It supplies an exact and objective measure of the joint discrepancy from their expectations of a number of normally distributed, and mutually correlated, variates.

  95. Gini Eugenics address
    • As I see it, they are indissolubly connected not only because in practice it is difficult to think of a measure affecting the number of inhabitants which does not also affect their qualitative distribution, or of a measure hindering or encouraging the reproduction of certain categories of people which does not also modify, directly or indirectly, the number of the population, but also and above all because population is a biological whole, subject, as such, to biological laws which show us that mass, structure, metabolism, psychic phenomena, the reproduction of organic life are all indissolubly connected, both in their static conditions and in their evolution, so that it would be vain to try to modify some of these characters without taking into account the stage of development attained by the others.

  96. Thomas Peacock preface
    • He wrote two books, both mathematics texts, namely The tutor's assistant modernised (1791) and The Practical Measurer (1798).
    • We give below the Preface to The Practical Measurer.

  97. Cafaro's publications
    • V Bertola and E Cafaro, On the Measurement of Slug Frequency in the Horizontal Gas-Liquid Flow, AIAA Paper (2001), 2001-3035.
    • E Cafaro and V Bertola, Slug Frequency Measurement Techniques in Horizontal Gas-Liquid Flow, AIAA Journal 40 (2002), 1010-1012.

  98. Maclaurin life
    • On the Construction and Measure of Curves, vol.
    • When the Earl of Morton went, in 1739, to visit his estates in Orkney and Shetland, he requested Mr Maclaurin to assist him in settling the geography of those countries, which is very erroneous in all our maps; to examine their natural history, to survey the coasts, and to take the measure of a degree of the meridian.

  99. Percy MacMahon addresses the British Association in 1901, Part 2
    • It is to a certain extent a science of enumeration, of measurement by means of integers, as opposed to measurement of quantities which vary by infinitesimal increments.

  100. Value of Mathematics
    • If the student were accustomed to constantly project the data and results of the problems into the realm of reality, absurdities of this nature would be avoided, the pupil would become accustomed to keep in mind this simple and yet so often forgotten truth that all data translating a measure of the physical world is necessarily approximate, and that, therefore, the alleged accuracy in the results is not only a pure chimera but a grotesque falsification of reality.
    • If we could measure the amount of psychic energy that comes to us every day through the doors of our Centres of Education in the minds of our students, we would be astonished by it.

  101. Mathematics in Aberdeen
    • There are two classes of Natural Philosophy - a Junior Class, attendance on which is required for the Degree of M.A., a Senior Class, and a class for practical instruction in Physical measurements.
    • In the Practical Class, students receive a course of instruction in practical measurements in the subjects of the Junior and Senior Courses.

  102. A de Lapparent: 'Wantzel
    • It was in these works that he achieved the exact measure of what was expected of him.
    • The profoundly regrettable measure with which he performed his duties as examiner was the final blow, and he died in 1848, consoled by hopes of a profoundly religious soul, but leaving with his friends, and with science, irreparable regrets.

  103. Young Researchers
    • Title: Counting, measure, and metrics .
    • Title: Scaling laws, emergence and statistical descriptions of systems that are out of equilibrium: what we can model and measure .

  104. Herschel William papers
    • The principal reason why this has not been looked into, is probably the difficulty of finding a proper standard to measure it by; since it is itself used as the standard by which we measure all the other motions.

  105. Gentry Berlin
    • This attempt at an article is made, therefore, not merely to show appreciation of the honour done me by the editors of The Lantern, but also in the hope to correct in some measure these false impressions.
    • So long as the situation remains as it is, I should be inclined to say to those who might hold me responsible for the result, "You have good opportunities elsewhere than in Germany: let well enough alone." To those who are willing to run all risk and not hold me responsible for the advice, I might offer the advice so often given me this year, "Versuchen Sie es doch 'mal! Schaden kann es jedenfalls nicht." [Try it sometime! It can't hurt you] - To all who may follow this latter piece of advice, I can wish nothing better than that the measure of kindness they receive may be "heaped up, pressed down, and running over," as mine has been.

  106. Groups St Andrews proceedings
    • As in previous volumes, the present editors have endeavoured to produce a measure of uniformity - hopefully without distorting individual styles.
    • It is a measure of the success of this conference series and this subject that mathematical libraries around the world are collecting the series of St Andrews Conference Proceedings.

  107. N S Krylov's monograph - Introduction
    • Most of the questions touched on in the third and fourth chapters deal with obtaining relaxation characteristics, the H-theorem, and so on, which are macroscopic statements, that is, statements that are not, as it may seem, related to the problem of the possibility of measurements.
    • Therefore, lest it appear odd that these questions should arise when solving the problem set in the work, we shall emphasize at this point that the very essence of the problem lies in establishing the connection between macroscopic statements and micromechanics; and, as is known, physical sense can be imparted to macromechanical equations only where measurements are possible.

  108. Moran reviews
    • Probability and Measure Theory.
    • This book is intended as an intermediate-level text from a somewhat applied viewpoint, although measure-theoretic probability is introduced in Chapter 4.

  109. Cusa: On informed ignorance
    • What is itself not true can no more measure the truth than what is not a circle can measure a circle; whose being is indivisible.

  110. Gibson History 4 - John Napier
    • If C is the point P has reached, moving with velocity V, when Q, moving in the way described, has reached D then the number which measures AC is the logarithm of the sine (or number) which measures DY.

  111. Feller Reviews 1
    • In order to avoid measure theory difficulties, only discrete sample spaces are considered.
    • To avoid advanced mathematical concepts (measure theory, etc.) and to make the work useful to beginners, the author limits it to questions which involve only a countable sample space; but about these simple questions it addresses the most advanced problems of probability theory, many of which have not until now been exposed in a book, so that the work is of the highest interest for specialists.

  112. The South-Troughton quarrel
    • micrometric measures of double stars.
    • Measures if several pairs were successfully taken by Airy and others.

  113. Halmos Set Theory
    • Those of us who have been so pleasantly introduced into the intricacies of linear algebra and measure theory by Paul Halmos will not be disappointed by his new excursion into the realm of set theory.
    • In any event the two books form a natural contrasting pair, Suppes's for the careful logician investigating (say) the independence of the axiom of choice, Halmos's for the mathematician-in-the-street who just wants to stay out of trouble when he does (say) measure theory.

  114. Mathematics in Chile
    • In its 2005 study, the Chilean Academy of Science tried to give a rough measure of mathematical activity in Chile.
    • The corresponding figure for Uruguay, its nearest Latin American competitor under this measure, was nine.

  115. Ernest Hobson addresses the British Association in 1910
    • These times must have been preceded by still earlier ages in which the mental evolution of man led him to the use of the tally, and of simple modes of measurement, long before the notions of number and of magnitude appeared in an explicit form.
    • Much of the skill of the true mathematical physicist and of the mathematical astronomer consists in the power of adapting methods and results carried out on an exact mathematical basis to obtain approximations sufficient for the purposes of physical measurement.

  116. Graf theory
    • The brass tub in Solomon's temple was a thick-sided vessel, and the measurement .
    • of ten cubits referred to the outer diameter, while the measurement of thirty cubits referred to the inner circumference.

  117. Dubreil Books
    • The main part of the book and the part properly described by the title consists of the lectures on 'Volumes of Polyhedra' by Henri Cartan, on 'Measure of Angles' by J Dixmier and on the 'Theory of Integration' by A Revuz.
    • This is a set of five expository articles, two of which have nothing to do with measure.

  118. Feller Reviews 3
    • The contemporary efflorescence of probability theory is due, in large measure, to Professor Feller's book.

  119. Taylor versus Continental mathematicians
    • It is reasonably measured compared with what he wrote privately to Leibniz:- .

  120. Boole-Thomson correspondence
    • A letter which I yesterday received from Mr Cayley contains, after a due measure of linear transformations and elliptic functions, an extract from a letter of yours in which you are so good as to suggest that l might possibly succeed in obtaining one of the new Irish professorships.

  121. St Andrews Mathematics Examinations
    • Define the two common units of angular measurement.

  122. Olds' teaching articles
    • Some people would define mathematics as a tool for measurement, others as a mode of thought by which logical conclusions can be reached.

  123. Valdivia aesthetic maths
    • Well, a Pythagorean of the fifth century BC, Hippasus of Metapontum, felt obliged to study the geometrical properties of this symbol, and thus discovered that a side and a diagonal in the regular pentagon are two immeasurable segments, that is to say, that it is impossible to find a third segment that is taken as a unit, so that both the side and the diagonal have whole measures.

  124. Semple and Kneebone: 'Algebraic Projective Geometry
    • If, in fact, we turn back once again to Greek geometry, we may recall that the geometrical knowledge with which the Greeks began was derived ultimately from measurements made upon rigid bodies, and was therefore essentially a knowledge of shapes.

  125. Laplace: 'Essay on probabilities
    • The ratio of this number to that of all the cases possible is the measure of this probability, which is thus simply a fraction whose numerator is the number of favourable cases and whose denominator is the number of all cases possible.

  126. Bell books
    • As the author does not quote his sources it is difficult to measure his accuracy, and his book can be used only for reading, not for reference or study.

  127. Kurosh: 'The theory of groups' 2nd edition
    • The first edition of the present book has also contributed in some measure to the development of the group-theoretical studies - it might be mentioned that a typewritten copy was deposited in 1940 at the Institute for Mathematics and Mechanics of the University of Moscow and was accessible for study.

  128. Andrew Forsyth addresses the British Association in 1905, Part 2
    • Estimates of relative importance are often little more than half-concealed expressions of individual preferences or personal enthusiasms; and though each enthusiastic worker, if quite frank in expressing his opinion, would declare his own subject to be of supreme importance, he would agree to a compromise that the divergence between the different subjects is now so wide as to have destroyed any common measure of comparison.

  129. Wolfgang Pauli and the Exclusion Principle
    • On the other hand, my earlier doubts as well as the cautious expression classically non-describable two-valuedness experienced a certain verification during later developments, since Bohr was able to show on the basis of wave mechanics that the electron spin cannot be measured by classically describable experiments (as, for instance, deflection of molecular beams in external electromagnetic fields) and must therefore be considered as an essentially quantum-mechanical property of the electron .

  130. Henry Baker addresses the British Association in 1913
    • It is a constantly recurring need of science to reconsider the exact implication of the terms employed; and as numbers and functions are inevitable in all measurement, the precise meaning of number, of continuity, of infinity, of limit, and so on, are fundamental questions, those who will receive the evidence can easily convince themselves that these notions have many pitfalls.

  131. Poincaré on intuition in mathematics
    • We well know these lines have no width; we try to imagine them narrower and narrower and thus to approach the limit; so we do in a certain measure, but we shall never attain this limit.

  132. Raphson books
    • (2) Every finite line, to be repeated Infinitely cannot Measure it.

  133. Milnor's books
    • One can also find some maybe less popularized but nonetheless quite interesting other contributions, like his example of a measurable subset of the square of full measure which intersects each leaf of a foliation by analytic curves at most once.

  134. Jacobson: 'Structure of Rings
    • This is an ideal which measures the departure of a ring from semi-simplicity.

  135. Calcutta Review 1859
    • The wily Mussulman had not properly measured his man, and the deistical Hindu, as truly ignorant of that in which true religion consists, and quite as greedy of praise as his opponent, replied 'that for his part he considered not only Christianity, but also Mahommedanism, and all bookish religions, as absurd and false, "Upon this" he says "all Hindus and Mahonamedans present paid me the compliment of being a philosopher, and departed with marks of approbation and good-will." This narrative is not uninstructive, and as a genuine specimen of native inner life, is not without its value in helping us to understand the light in which Englishmen are regarded as Christians by natives, even when so highly educated as to renounce the absurdities of the Hindu mythology.

  136. Feller Reviews 2
    • The reader who is familiar with measure theory might feel that the reluctance of the author to discuss anything more complicated than enumerably infinite probability fields is an unnecessary obstacle.

  137. Sommerfeld: 'Atomic Structure
    • After Maxwell had already surmised that light was an alternating electromagnetic field (he succeeded in calculating the velocity of light from purely electrical measurements made by Kohlrausch), Hertz produced his "rays of electric force," which, just like light, are reflected, refracted, and brought to a focus by appropriate mirrors, and which are propagated in space with the velocity of light.

  138. Ernesto Pascal's books
    • By reason of the arrangement of the material and the style of exposition, it is probable that most mathematicians will find in this section of the Repertorium not only appreciable stimulation but also a measure of downright enjoyment.

  139. Schools Inquiry
    • Beale was now considered to be an educational expert; organisations such as the Schoolmistresses' Associations had helped to foster this new measure of authority and assurance.

  140. David Hilbert: 'Mathematical Problems
    • Experience teaches, explains Bernoulli in the public announcement of this problem, that lofty minds are led to strive for the advance of science by nothing more than by laying before them difficult and at the same time useful problems, and he therefore hopes to earn the thanks of the mathematical world by following the example of men like Mersenne, Pascal, Fermat, Viviani and others and laying before the distinguished analysts of his time a problem by which, as a touchstone, they may test the value of their methods and measure their strength.

  141. Kuratowski: 'Introduction to Set Theory
    • In the course of years, however, when set theory showed its usefulness in many branches of mathematics such as the theory of analytic functions or theory of measure, and when it became an indispensable basis for new mathematical disciplines (such as topology, the theory of functions of a real variable, the foundations of mathematics), it became an especially important branch and tool of modern mathematics.

  142. Galois book review
    • Here the commentary gives accurate measurements of the paper on which it is written and precise details of differences between various versions published later are all noted.

  143. Sansone books
    • The five chapters deal, respectively, with sets and transfinite numbers; measure of linear sets; measurable functions, functions of bounded variation; integration of measurable functions, integration of series; differentiation of integrals and of functions of bounded variation.

  144. Cosmic Parade
    • But brave hypotheses are necessary to guide, transiently; the observations stand for a while; the good measures, in fact, are essentially permanent; and the enthusiasm back of it all, the will to know and the willingness to fumble as we learn to know - they are eternal.

  145. Herschel Museum
    • kept me employed when my Brother was at the telescope at night; for when I found that a hand sometimes was wanted when any particular measures were to be made with the lamp micrometer etc and a fire kept in, and a dish of coffee necessary during a long night's watching; I undertook with pleasure what others might have thought a hardship.

  146. Slaught's books
    • The elementary course measured up in a large degree to its avowed purpose, but it was obviously insufficient.

  147. St Andrews Physics Examinations
    • How was the velocity of light first measured? .

  148. Godement's reviews
    • This second volume of Godement's Analyse mathematique is devoted to integral calculus (Riemann integral with glimpses on Lebesgue integral, Radon measure and Schwartz distributions), asymptotic expansions, harmonic analysis and holomorphic functions.

  149. Ward Cheney Memory
    • The experience was unusually memorable because Mom and Dad forgot that elevations on European topographical maps were measured in meters rather than feet.

  150. Atiyah on beauty
    • So I think beauty is a measure of significance.

  151. Finkel's Solution Book
    • Hoping that the work will, in a measure, meet the object for which it is written, I respectfully submit it to the use of my fellow teachers and co-labourers in the field of mathematics.

  152. Tait graduates address.html
    • It may be said, indeed, that the framers of the now measure have entirely ignored many of the chief arguments aud conclusions of, that Report: -- a document which in its thoroughness, its ripe wisdom, and its calm impartiality, stands in irreconcilable disaccord with much or the contents of the present hasty, and therefore slipshod and one-sided, Bill it should be a sad reflection that the steady labour of two years ; cheerfully given by a group of men of the highest eminence, chosen from all parties and from almost every field of knowledge; a group including Froude, Huxley, and Stirling-Maxwell; and presided over by our illustrious Chancellor, wbo is not only the greatest of all benefactors of the Scottish Universities, but of all men the most conversant with their position and their wants; that such hearty, honest, and valuable labour has been spent absolutely in vain.

  153. Smith Teaching Papers
    • Why is Mathematics Studied? Ever since man came to think in the abstract, to think of the number two as distinct from two objects, to create for himself units of measure that possess some approach to uniformity, to think of time, and to be aware of such concepts as lines and angles, mathematics has been an object of his study and the basis of most of the natural sciences of antiquity, becoming the very essence of all science of the present day.

  154. Von Neumann: 'The Mathematician
    • Kepler's first attempts at integration were formulated as "dolichometry" - measurement of kegs - that is, volumetry for bodies with curved surfaces.

  155. Charles Bossut on Leibniz and Newton Part 2
    • From that time Johann Bernoulli kept no measures with him; and he published under the name of one Burcard, a schoolmaster in Basle, an answer to Taylor which was full of insult and ridicule, among which however we meet with some useful truths.

  156. Napier Tercentenary
    • The method consisted essentially in using the tangent instead of the angle, and laying off the angle so measured from the cardinal points up to 45 degs.

  157. Miller graduation address
    • The great humanitarian movement which has been sweeping over the civilized world from the middle of the eighteenth century to the present time, manifesting itself in political revolutions, in social and moral reforms, and in works of love and mercy, affords the amplest assurance that all worthy elements of the population will ultimately be admitted to share in the privileges and blessings of civilization according to the measure of their merit.

  158. Mathematics at Aberdeen 3
    • (A pedal curve is one traced by the feet of the perpendiculars from a fixed point to the tangents to a given curve.) He published early results in two papers in the Philosophical Transactions: Of the construction and measure of curves in 1718 and A new method of constructing all kinds of curves in 1719.

  159. Dahlin Introduction
    • Only after the last-mentioned year did the academic activities become more lively, as a consequence of the measures taken by the university to reorganise and become a more solid institute.

  160. Comments by Charlotte Angas Scott
    • But while reading this brilliant exposition it is difficult to avoid cherishing a lurking regret, which is possibly very ungracious, that Klein could not himself spare time to arrange his work for publication; for though we have here in full measure the incisive thought and cultured presentation which together make even strict logic seem intuitive, yet at times we miss the minute finish and careful proportion of parts that we feel justified in expecting from him.

  161. Descartes' Method
    • I learned not to entertain too decided a belief, in regard to anything of the truth of which I had been persuaded merely by example and custom, and thus I gradually extricated myself from many errors powerful enough to darken our natural intelligence, and incapacitate us in great measure from listening to reason.

  162. Rose's Greek mathematical literature
    • At all events, their observations were, at least in some measure, accessible to Greek astronomers after Alexander.

  163. Poincaré on the future of mathematics
    • Thus the importance of a fact is measured by the return it gives - that is, by the amount of thought it enables us to economize.

  164. University of Glasgow Examinations
    • Explain what measurements are necessary for specifying the position of a celestial object, the plane of the ecliptic being taken as the plane of reference.

  165. Selected papers of Edward Marczewski' Preface
    • Until the late fifties his main fields of interest were measure theory, descriptive set theory, general topology and probability theory.

  166. Mathematical and Physical Journal for Secondary Schools
    • In 1890, Lorand (Roland) Eotvos, professor of physics, mailed an invitation to his lecture on Terrestrial Gravitation and its Measurement.

  167. Aitchison books
    • The book appears to be quite suitable for use as a text for a general course in statistical prediction analysis at a pre-measure-theory level.

  168. Kelvin on the sun
    • Thus Langley's measurement of solar radiation corresponds to 133,000 horse-power per square metre, instead of the 78,000 horse-power which we have taken, and diminishes each of our times in the ratio of 1 to 1.7.

  169. Rios Honorary Degree
    • In addition to Algebra and Analysis, probability plays a fundamental role in pattern theory, specifically the probability measure over the space of regular configurations serves to discover the relative frequency of some types and what is called biological variability (e.g.

  170. R L Wilder: 'Cultural Basis of Mathematics III
    • For the so-called "international character" of mathematics is due in large measure to the standardization of symbols that it has achieved, thereby stimulating diffusion.

  171. Booth Education
    • These, it is hoped, may be alleged as in some measure an apology for the many blemishes which will, doubtless, present themselves to the reader.

  172. Condamine letter
    • The measures of the Abbe de la Caille, and those of the Father Maire and Father Boscovich, whose book must now be in the hands of the Royal Society, do not agree with the elliptical curve of the meridian, or with the circularity of the parallels.

  173. Brinkley Copley Medal
    • Attempts have been made to measure the almost inconceivable distances of the fixed stars: and, with this, what sublime, practical, and moral results! The pathless ocean navigated, and in unknown seas, the exact point of distance from known lands ascertained.

  174. John Maynard Keynes: 'Newton, the Man
    • Another large section is concerned with all branches of apocalyptic writings from which he sought to deduce the secret truths of the Universe - the measurements of Solomon's Temple, the Book of David, the Book of Revelations, an enormous volume of work of which some part was published in his later days.

  175. Dahlin Extracts
    • It was not until 1620 that considerable measures were taken for the recovery of its teaching.

  176. Mercer's papers
    • The probability that a component fails is assumed to be linearly dependent on a physical wear variable, which can be measured as well as on the age of the component.

  177. Plato on Mathematics
    • They put their ears close alongside of the strings like people trying to hear a sound through their neighbour's wall - some of them declaring that they can distinguish an intermediate note and have found the least interval which should be the unit of measurement, while the others insist that there is no difference between the two notes - both lots are putting their ears before their understanding.' .

  178. Women mathematicians by Dubreil-Jacotin
    • Then we shall see in what measure she can, as the equal of man, emerge from the role of the excellent pupil or the perfect collaborator, and join those of our scientists whose work has opened new paths and bears the mark of genius.

  179. Mathematicians and Music
    • Lagrange welcomed music at a reception because he could by the fourth measure become oblivious to his surroundings and thus work out mathematical problems; for him the most beautiful musical work was that to which he owed the happiest mathematical inspirations.

  180. De Coste on Mersenne
    • But in the immortal sciences we have today two men who know precisely all that was known by Eudoxus and Hipparchus, those two famous antagonists, who became both the rivals of Euclid and the legitimate successors of Ptolemy; I am speaking of the Reverend Father Mersenne, Religious of the Minim Order, and Pierre Gassendi, two intellects who, despite the ignorance of the age, represent us in some measure, these two live monuments who, in spite of the waters of the universal deluge, preserve for the world all the arts and all the sciences, in which they excel, the one vying with the other.

  181. Louis Auslander books
    • This book departs refreshingly from "classical" introductions to differential geometry by treating the subject consistently from the group theoretic point of view, essentially in the spirit of E Cartan and the method of "moving frames." The gist of this method is that if three orthogonal unit vectors ("frame") are attached to a point constrained to move along a surface so that two of the frame vectors are always tangent to the surface, then the direction changes of each of the frame vectors measure geometric properties of the surface (formally expressed by the "structure equations").

  182. Mathematicians and Music 3
    • But what is to be the future of the almost untried vast rhythmic possibilities so intimately bound up with mathematical relations? Practically all of our music is modulo 2, 3, 4, 6, 8, 9, 12; but why not have modulo 5, 7, 10, 11, 13, for example, or combinations of these moduli in the same measures? .

  183. Gregory's Observatory
    • I desire to know if this Steeple being exactly measured with a chain may not serve for one Place of a very Exact and Large Quadrant, looking exactly with a Telescope." .

  184. Hiawatha Designs an Experiment
    • of a set of measure zero.

  185. Apostol books
    • The development of Lebesgue integration follows the Riesz-Nagy approach which focuses directly on functions and their integrals and does not depend on measure theory.

  186. Hedrick papers
    • Not only must applied mathematics serve the war effort, as it is now doing in increasing measure, but, looking ahead to the post-war period, it appears essential that America should be strong in the whole field of Science in order that it serve as a world centre for advanced instruction and research.

  187. Gregory seeks Flamsteed's advice
    • We have in Saint Andrews a Steeple above 100 foot high, plain Square Work, without any Pricket: I desire to know, if this Steeple beinge exactly measured with a Chaine, may not serve for one Place of a verye Exact, and large Quadrant, lookinge Exactlye with a Telescope When a Starr is seen over any sight placed on Top of it, etc.

  188. Hardy on the Tripos
    • A "b*" in the Tripos, or a first in Greats, is taken to be, and in a measure is, an indication of a man quite outside the common run.

  189. The Samarkand Observatory
    • In large measure, that is why we have available to us the works of so many 11th- to 15th-century Islamic scholars.

  190. Zariski and Samuel: 'Commutative Algebra
    • Unlike the first, however, the second volume is concerned in large measure with those parts of commutative algebra that are the fruits of its union with algebraic geometry ..

  191. James Jeans addresses the British Association in 1934, Part 2
    • If there is an avenue of escape, it does not, as I see it, lie in the direction of less science, but of more science - psychology, which holds out hopes that, for the first time in his long history, man may be enabled to obey the command 'Know thyself'; to which I, for one, would like to see adjoined a morality and, if possible, even a religion, consistent with our new psychological knowledge and the established facts of science; scientific and constructive measures of eugenics and birth control; scientific research in agriculture and industry, sufficient at least to defeat the gloomy prophecies of Malthus and enable ever larger populations to live in comfort and contentment on the same limited area of land.

  192. Students in 1711
    • This vacation Mr MacDonald has gone out with them [Kenneth and Thomas] and me to make trial of some geometrical practices, relating to heights and we measured a piece of ground with a chain, and we took up the angles of an irregular field with a graphometer, whose figure being projected on paper we found the content thereof ..

  193. Samuel Wilks' books
    • It is a book for the mathematics graduate with some familiarity with the Lebesgue-Stieltjes integral and related measure-theoretic notions and, to this extent, is not self-contained.

  194. Kingdom of Naples
    • In cases of minor importance, therefore, armed with the authority granted me by your Majesties, I have acted on my own responsibility; and thus, all severe measures are attributed to me, and clemency to the king.

  195. Al-Biruni: 'Coordinates of Cities
    • While proceeding on their paths, they measured the distances they had traversed, and planted arrows at different stages of their paths (to mark their courses).

  196. Mathematicians and Music 2.2
    • For all Harmony consists in Concord, and Concord is all the World over fixed according to the same invariable Measure and Proportion so that in all Nations the Difference and Distance of Notes is the same, whether they be in a continued gradual Progression, or the Voice skips over one to the next.

  197. Howie Committee
    • We conclude that, when measured against the characteristics which high quality upper school secondary education should display, our system is seriously wanting in many respects.

  198. Berge books
    • The problems treated are still somewhat heterogeneous and, whilst the basic subject matter and method give the book a natural unity, one gets the impression that the emphases on the different subdivisions of the subject are still to some extent the result of historical accident rather than measures of their importance to the theory as a whole.

  199. Vidav bibliography
    • (with Anton Suhadolc) Linearized Boltzmann equation in spaces of measures.

  200. Feller Reviews 5
    • The restriction was deliberately imposed in order to waste no time on dull generalities (often indiscriminately referred to as "measure theory") and to proceed at once to significant results.

  201. Sadosky problems
    • Some of those emigrating have expressed doubts about promises made by the government to respect academic freedom and to restore a large measure of the autonomy enjoyed by nationally chartered universities before they were seized by the military regime July 29.

  202. More Smith History books
    • To his scholarship and indefatigable labours I am indebted for more material than could be used in this work, and whatever praise our efforts may merit should be awarded in large measure to him.

  203. Finlay Freundlich's Inaugural Address
    • Now celestial bodies cannot be placed and weighed on scales; their masses, therefore, in Astronomy have to be measured according to other principles, for which Kepler laid the foundation.

  204. Kelvin on the sun, Part 2
    • This exceedingly small density is nearly six times the density of the oxygen and nitrogen left in some of the receivers exhausted by Bottomley in his experimental measurements of the amount of heat emitted by pure radiation from highly heated bodies.

  205. Rhode Island College
    • A one-term course in solid geometry is given to the mechanical students, in which are studied the point, the line and the plane in space, the familiar polyhedrons, the cylinder, cone and sphere, including the measurement of these solid figures.

  206. Diaconis papers
    • They employ a mathematical framework for the study of fractals proposed by Hutchinson and make interesting connections with stochastic processes and classical problems concerning singular measures.

  207. Clifford's books
    • One cannot ask of a man to give all his measure in thirty-three years.

  208. Scholar and the World.html
    • The world is the subject matter of these artists, as it is our own, but the rules and technique how different! We should never dream of criticising a symphony or a poem because it had not been produced by a process of accurate measurement, or a discussion by the newest methods of mathematical physics.

  209. Bertrand Russell on Euclid
    • This axiom is the basis of the measurement of angles by distances, and is required for proving that if D be on AB, and BD be less than BA, the triangle DBC is less than the triangle ABC.

  210. Christiaan Huygens' article on Saturn's Ring
    • But I had only the ordinary form of telescope, which measured five or six feet in length.

  211. Reviews of Shafarevich's books
    • can be "coordinatised" or "measured".

  212. Dehn on Aristotle
    • In deriving time measurement from movement, the concept of number take on a significant role.

  213. Science at St Andrews
    • He was a versatile geologist and had been the first to set foot on the summit rocks of Sgurr nan Gillean and again to detect and to measure the movements of glacier flow.

  214. Smith's Teaching Books
    • It would not be easy to get together three men who would mean more in the mathematical world than do these authors of "Exercises and Tests in Algebra," and no one has prepared a more wholesome series of tests or measurements.

  215. Kline's books
    • Important questions of sciences such as earth measure, cosmology, gravitation, and electromagnetism, and of arts, such as perspective drawing and musical composition, are described in detail sufficient to motivate discussions of mathematical notions they generate: geometry, algebra, trigonometry, calculus, and so on.

  216. Gregory's Astronomical Clock
    • The hood of the clock-case is square, surmounted by a triangular pediment, and measures over all 15 inches in height.

  217. A D Aleksandrov's view of Mathematics
    • We could measure the angles at the base of a thousand isosceles triangles with extreme accuracy, but such a procedure would never provide us with a mathematical proof of the theorem that the angles at the base of an isosceles triangle are equal.

  218. Mathematics in Glasgow
    • A Treatise on Natural Philosophy, by Professors Sir William Thomson and P G Tait (Cambridge University Press); Elements of Natural Philosophy, by the same authors (Cambridge University Press) ; A Lecture on Navigation, by Professor Sir William Thomson (W Collins & Sons); Dynamics and Hydrostatics, by J T Bottomley (W Collins & Sons) ; Heat, and Elasticity, by Sir W Thomson, reprinted from the Encyclopaedia Britannica (A & C Black); Ganot, Experimental Physics, translated by Atkinson (Longmans & Co.) Electrical Measurements, by A Gray (Macmillan & Co.) Mathematical Tables, J T Bottomley (W Collins & Sons).

  219. Riesz descriptions
    • During the academic year, Riesz would lecture on measure theory and functional analysis.

  220. Atlas de la Lune
    • Changes that early observers thought present had to be put down to errors of measurement or drawing.

  221. Bayes on Fluxions
    • In my opinion this is in some measure the case with respect to his proofs of the first principles of Fluxions, and therefore I don't wonder persons differ in their sentiments about them.

  222. Newcomb Elements of Geometry
    • The additions to the old system of angular measurement are the following two: .

  223. H Weyl: 'Theory of groups and quantum mechanics'Preface to First Edition
    • The continuum of real numbers has retained its ancient prerogative in physics for the expression of physical measurements, but it can justly be maintained that the essence of the new Heisenberg-Schrodinger-Dirac quantum mechanics is to be found in the fact that there is associated with each physical system a set of quantities, constituting a non-commutative algebra in the technical mathematical sense, the elements of which are the physical quantities themselves.

  224. Milnor awards
    • The current state of the classification of topological, piecewise linear, and differentiable manifolds rests in large measure on his work in topology and algebra.

  225. Mary Boole writing
    • Many a life of intellectual muddle and intellectual dishonesty begins at the point where some teacher explains the rule for Greatest Common Measure to a child who has not had the proper basis of sub-conscious knowledge laid in actual experiences.

  226. Roth Family
    • Nothing was pre-packed in those days, so a yard of tape or elastic would be measured out and cut.

  227. Landau and Lifshitz Prefaces
    • Its importance is in fact measured not so much by even the remarkable phenomena that occur in the liquid isotopes of helium as by the fact that the concepts of a quantum liquid and its spectrum are essentially the foundation for the quantum description of macroscopic bodies.

  228. Mathematical Works of Colin Maclaurin
    • Of the Construction and Measure of Curves, No.

  229. Laplace: 'Méchanique Céleste
    • I shall adopt the decimal division of the right angle, and of the day, and shall refer the linear measures to the length of the metre, determined by the are of the terrestrial meridian comprised between Dunkirk and Barcelona.

  230. Leonard J Savage: 'Foundations of Statistics
    • It was for a long time generally believed that all uncertainties could be measured by probabilities, and a few of us today believe that this view, which has recently been very unpopular, must soon again come into its own.

  231. Serre reviews
    • The author goes on to present results on group representations that are often difficult to find in the literature, such as the technique of computing Haar measures in a compact l-adic group by performing a similar computation in a real compact Lie group.

  232. Maclaurin preface
    • His Objections seemed to have been occasioned, in a great measure, by the concise manner in which the Elements of this Method have been usually described; and their having been so much misunderstood by a person of his abilities appeared to me a sufficient proof that a fuller Account of the Grounds of them was requisite.

  233. Godement's preface
    • As for content, I did not hesitate to introduce, sometimes very early, subjects considered relatively advanced - multiple series and unconditional convergence, analytical functions, the definition and immediate properties of Radon measures and distributions, the integrals of semi-continuous functions, Weierstrass elliptic functions, etc.

  234. history of reliability
    • The reliability was measured as the number of accidents per hour of flight time.

  235. O'Brien Calculus
    • VIII contains certain very important Lemmas upon which the use and application of the Differential Calculus in a great measure depends.

  236. Wall's Creative mathematics
    • may it not be possible that the only really important objective in our teaching of mathematics is something that we will never be able to measure satisfactorily by any kind of test or examination?" I go so far as to give no examinations whatsoever.

  237. Isaac Todhunter: 'Euclid' Preface
    • It cannot be denied that defects and difficulties occur in the Elements of Euclid, and that these become more obvious as we examine the work more closely; but probably during such examination the conviction will grow deeper that these defects and difficulties are due in a great measure to the nature of the subject itself, and to the place which it occupies in a course of education; and it may be readily believed that an equally minute criticism of any other work on Geometry would reveal more and graver blemishes.

  238. Boyer's books
    • The present work is an attempt to supply, in some measure, this deficiency.

  239. Harnack calculus book
    • It thus appears that Mathematics are of fundamental importance to all our knowledge of Nature: for our representations of space contain the simplest properties which are common to all things in the surrounding world; and accurate comparison or measurement of quantities leads always to concrete numbers of the units employed: in order to understand the result, we require a knowledge of numbers and of their combinations.

  240. Craig books
    • This should, however, be taken as an unintentional oversight, or due to the fact that the author has not been able to trace back to its original source the solution of process in question, and not in any case to a desire to withhold from any other author his full measure of credit.

  241. Élie Cartan reviews
    • The author works through his subject informally, enriching it with his unified point of view and his unsurpassable geometric insight, finding new approaches leading to a better understanding, and giving here and there a new result for good measure.


  1. Quotations by Halmos
    • The paper was by Dorothy Maharam, and it was a perfectly sound contribution to abstract measure theory.
    • The domains of the underlying measures were not sets but elements of more general Boolean algebras, and their range consisted not of positive numbers but of certain abstract equivalence classes.
    • My proposed first sentence was: "The author discusses valueless measures in pointless spaces." .

  2. Quotations by Yule
    • Measurement does not necessarily mean progress.
    • Failing the possibility of measuring that which you desire, the lust for measurement may, for example, merely result in your measuring something else - and perhaps forgetting the difference - or in your ignoring some things because they cannot be measured.

  3. Quotations by Born
    • Therefore nobody will object to an ardent experimentalist boasting of his measurements and rather looking down on the 'paper and ink' physics of his theoretical friend, who on his part is proud of his lofty ideas and despises the dirty fingers of the other.
    • The problem of physics is how the actual phenomena, as observed with the help of our sense organs aided by instruments, can be reduced to simple notions which are suited for precise measurement and used of the formulation of quantitative laws.

  4. A quotation by Abraham
    • Who wishes correctly to learn the ways to measure surfaces and to divide them, must necessarily thoroughly understand the general theorems of geometry and arithmetic, on which the teaching of measurement ..

  5. Quotations by Borel
    • Probabilities must be regarded as analogous to the measurement of physical magnitudes; that is to say, they can never be known exactly, but only within certain approximation.
    • Geometry measures the surface of a field without bothering to find if the soil is good or bad.

  6. Quotations by Einstein
    • The relativity principle in connection with the basic Maxwellian equations demands that the mass should be a direct measure of the energy contained in a body; light transfers mass.
    • But there is another reason for the high repute of mathematics: it is mathematics that offers the exact natural sciences a certain measure of security which, withut mathematics, they could not attain.
    • A hundred times every day I remind myself that my inner and outer life depend on the labours of other men, living and dead, and that I must exert myself in order to give in the same measure as I have received.

  7. A quotation by Diophantus
    • This tomb hold Diophantus Ah, what a marvel! And the tomb tells scientifically the measure of his life.
    • Alas! late-begotten and miserable child, when he had reached the measure of half his father's life, the chill grave took him.

  8. Quotations by Durer
    • The Art of Measurement.
    • Course in the Art of Measurement .

  9. Quotations by Kepler
    • I used to measure the Heavens, now I measure the shadows of Earth.

  10. Quotations by Nightingale
    • But to understand God's thoughts, she held we must study statistics, for these are the measure of His purpose.

  11. Quotations by Proclus
    • According to most accounts, geometry was first discovered among the Egyptians, taking its origin from the measurement of areas.

  12. Quotations by Thomson
    • When you measure what you are speaking about and express it in numbers, you know something about it, but when you cannot express it in numbers your knowledge about is of a meagre and unsatisfactory kind.

  13. Quotations by Hawking
    • Disorder increases with time because we measure time in the direction in which disorder increases.

  14. Quotations by Klein
    • always united theory and applications in equal measure.

  15. Quotations by Condorcet
    • uniformity of measures can only displease those lawyers who fear to see the number of lawsuits diminished, and those traders who fear a loss of profit from anything which renders commercial transactions easy and simple ..

  16. Quotations by Poincare
    • We might as well ask if the metric system is true and if the old weights and measures are false; if Cartesian coordinates are true and polar coordinates are false.

  17. Quotations by Caccioppoli
    • If you are afraid of something, measure it, and you will realize it is a mere trifle.

  18. Quotations by Bondi
    • We find no sense in talking about something unless we specify how we measure it; a definition by the method of measuring a quantity is the one sure way of avoiding talking nonsense..

  19. Quotations by Newton
    • God created everything by number, weight and measure.

  20. A quotation by Bohr Harald
    • Therefore, as a safety measure, .

  21. Quotations by Maxwell
    • that, in a few years, all great physical constants will have been approximately estimated, and that the only occupation which will be left to men of science will be to carry these measurements to another place of decimals.

  22. Quotations by Hilbert
    • One can measure the importance of a scientific work by the number of earlier publications rendered superfluous by it.

  23. Quotations by Galileo
    • Measure what is measurable, and make measurable what is not so.

Famous Curves

  1. Pursuit
    • Pierre Bouguer was a French scientist who was the first to attempt to measure the density of the Earth using the deflection of a plumb line due to the attraction of a mountain.
    • He made measurements in Peru in 1740.

  2. Epitrochoid
    • An example of an epitrochoid appears in Durer's work Instruction in measurement with compasses and straight edge(1525).

  3. Equiangular
    • In fact, from the point P which is at distance d from the origin measured along a radius vector, the distance from P to the pole is d sec b.

  4. Durers
    • These curves appear in Durer's work Instruction in measurement with compasses and straight edge(1525).

  5. Tricuspoid
    • The length of the tangent to the tricuspoid, measured between the two points P, Q in which it cuts the curve again is constant and equal to 4a.


  1. Mathematical Chronology
    • Harappans adopt a uniform decimal system of weights and measures.
    • In Measurement of the Circle he gives an approximation of the value of π with a method which will allow improved approximations.
    • Heron of Alexandria writes Metrica (Measurements).
    • Abu'l-Wafa invents the wall quadrant for the accurate measurement of the declination of stars in the sky.
    • He measures the length of the year to be 365.24219858156 days, a remarkably accurate result.
    • Pacioli publishes Summa de arithmetica, geometria, proportioni et proportionalita which is a review of the whole of mathematics covering arithmetic, trigonometry, algebra, tables of moneys, weights and measures, games of chance, double-entry book-keeping and a summary of Euclid's geometry.
    • Snell makes the first attempt to measure a degree of the meridian arc on the Earth's surface, and so determine the size of the Earth.
    • Snell publishes his technique of trigonometrical triangulation which improves the accuracy of cartographic measurements.
    • He gives an extremely accurate measurement of the latitude of Paris.
    • Viviani measures the velocity of sound.
    • He presents his conception of the "average man" as the central value about which measurements of a human trait are grouped according to the normal curve.
    • Bessel measures the parallax of the star 61 Cygni, the first star for which this is calculated.
    • Borel introduces "Borel measure".
    • Lebesgue formulates the theory of measure.
    • They were able to provide detailed derivations of many major theorems in set theory, finite and transfinite arithmetic, and elementary measure theory.
    • Banach is awarded his habilitation for a thesis on measure theory.
    • This will transform the Maxwell-Boltzmann kinetic theory of gases into a rigorous principle through the use of Lebesgue measure.
    • Haar introduces the "Haar measure" on groups.

  2. Chronology for 1600 to 1625
    • Snell makes the first attempt to measure a degree of the meridian arc on the Earth's surface, and so determine the size of the Earth.
    • Snell publishes his technique of trigonometrical triangulation which improves the accuracy of cartographic measurements.

  3. Chronology for 1830 to 1840
    • He presents his conception of the "average man" as the central value about which measurements of a human trait are grouped according to the normal curve.
    • Bessel measures the parallax of the star 61 Cygni, the first star for which this is calculated.

  4. Chronology for 900 to 1100
    • Abu'l-Wafa invents the wall quadrant for the accurate measurement of the declination of stars in the sky.
    • He measures the length of the year to be 365.24219858156 days, a remarkably accurate result.

  5. Chronology for 1900 to 1910
    • Lebesgue formulates the theory of measure.
    • They were able to provide detailed derivations of many major theorems in set theory, finite and transfinite arithmetic, and elementary measure theory.

  6. Chronology for 1930 to 1940
    • This will transform the Maxwell-Boltzmann kinetic theory of gases into a rigorous principle through the use of Lebesgue measure.
    • Haar introduces the "Haar measure" on groups.

  7. Chronology for 1650 to 1675
    • Viviani measures the velocity of sound.

  8. Chronology for 1920 to 1930
    • Banach is awarded his habilitation for a thesis on measure theory.

  9. Chronology for 1625 to 1650
    • He gives an extremely accurate measurement of the latitude of Paris.

  10. Chronology for 1AD to 500
    • Heron of Alexandria writes Metrica (Measurements).

  11. Chronology for 30000BC to 500BC
    • Harappans adopt a uniform decimal system of weights and measures.

  12. Chronology for 1300 to 1500
    • Pacioli publishes Summa de arithmetica, geometria, proportioni et proportionalita which is a review of the whole of mathematics covering arithmetic, trigonometry, algebra, tables of moneys, weights and measures, games of chance, double-entry book-keeping and a summary of Euclid's geometry.

  13. Chronology for 1890 to 1900
    • Borel introduces "Borel measure".

  14. Chronology for 500BC to 1AD
    • In Measurement of the Circle he gives an approximation of the value of π with a method which will allow improved approximations.

  15. Chronology for 1910 to 1920
    • They were able to provide detailed derivations of many major theorems in set theory, finite and transfinite arithmetic, and elementary measure theory.

EMS Archive

  1. Edinburgh Mathematical Society Lecturers 1883-2016
    • (The Academy, Edinburgh) Proposal with regard to British tables of linear and square measure: On a necessary reform in arithmetic .
    • (University College, Dundee) A suggested measure of relationship .
    • (Aberdeen) Measure and integral .
    • (Dundee) The measure of sum-sets of p-adic numbers .
    • (Manchester) Some aspects of measure theory .
    • (Aberdeen) The structure semigroup of a measure algebra .
    • (New South Wales) Convolutions of measures .
    • (University College, London) New results in classical geometric measure theory .

  2. EMS 2003 Colloquium
    • Participants were provided with duplicated notes surveying basic ideas of fractal geometry (dimensions, iterated function systems, fractal measures, etc) prior to the course.
    • 'Multifractal Measures' .
    • Other equivalent constructions of the restriction measures.
    • Intersection exponents between restriction measure samples.

  3. EMS 1913 Colloquium
    • In the second part of the lecture Dr Sommerville considered the question from the point of view of geometry on a curved surface, and showed how concrete representations of the Non-Euclidean geometries were obtained by means of certain surfaces which possessed constant measure of curvature.
    • In the new laboratory the members under Professor Whittaker's direction continued the consideration of the example on the magnitude (i.e., a measure of the brightness) of a variable star, and as a result of their periodogram analysis decided that there were two principal periods of 24 and 29 days.
    • The Newtonian measure of momentum - mass multiplied by velocity - has in this system to be divided by the square root of the excess of unity over the squared ratio of the velocity of the particle to the velocity of light.

  4. EMS 1934 Colloquium
    • Derivation of the Lorentz formulae using only clock-measures: the cosmological problem: the cosmological principle: the expansion phenomenon: the time-zero: Hubble's Law for uniform velocities: the invariant velocity-distribution: corresponding density-distribution: world-picture and world-map: time-relations: the observable universe as an open set of points: bearing on evolution and "creation": an invariant statistical distribution of matter in motion as a second approximation: the distribution of acceleration: its physical interpretation: the so called "cosmological constant": the Boltzman equation and its integration: average properties of the statistical distribution: integration of the equations of motion: classification of trajectories according to constants of integration: separation into expanding sub-systems: absence of interaction between interior and exterior of the expanding light-sphere: density distribution in a sub-system: the "K-effect": nebulae and field-stars: universes of discrete condensations: tidal forces: the final picture.
    • As fundamental hypotheses it employs, firstly, the idea that events are separated only by time intervals and that "space is a construct of time measurements".

  5. EMS/SCM
    • Dindos, Martin (University of Edinburgh) The equivalence of BMO solvability of the Dirichlet problem for parabolic equations with A∞ condition for the parabolic measure .
    • Martin, Joaquim (Universitat Autonoma de Barcelona) Isoperimetric weights and generalized uncertainty inequalities in metric measure spaces .

  6. 1894-95 Mar meeting
    • Duthie, George: "Proposal with regard to British tables of linear and square measure", {Title in minutes: "On a necessary reform in arithmetic"} .

  7. EMS 125th Anniversary booklet
    • Finlay Freundlich was a distinguished German astronomer who worked with Einstein on measurements of the orbit of Mercury to confirm the general theory of relativity.

  8. EMS 125th Anniversary booklet
    • Finlay Freundlich was a distinguished German astronomer who worked with Einstein on measurements of the orbit of Mercury to confirm the general theory of relativity.

  9. 1915-16 Nov meeting
    • Steggall, John Edward Aloysius: "A suggested measure of relationship", [Title] .

  10. Zagier Problems
    • If the angles at all other vertices are known to be rational (when measured in degrees), show that the angles at A and B are also rational.

  11. Napier Tercentenary
    • The method consisted essentially in using the tangent instead of the angle, and laying off the angle so measured from the cardinal points up to 45 degs.

  12. EMS 1968 Colloquium
    • (a) Professor Patrick Billingsley (Chicago) - "Weak Convergence of Probability Measures." .

BMC Archive

  1. BMC 2011
    • Jacob, NMetric Spaces Isometric to Hilbert Spaces, Metric Measure Spaces, and the Transition Densities of Levy Processes .
    • Sharp, SUniversity of Manchester Spectral triples and Gibbs measures on Cantor sets .
    • Maleva, MUniversal differentiability sets and geometric measure theory .

  2. BMC 2016
    • Cavalletti, FLevy-Gromov Isoperimetric inequality for metric measure spaces .
    • Le, EBesicovitch Covering Property on graded groups and applications to measure differentiation .
    • Gozlan, NGeneralized transport costs and applications to concentration of measure .
    • Kosloff, ZConservative Anosov diffeomorphisms of the two torus without an absolut(y continuous invariant measure .

  3. BMC 2018
    • Morris, ITowards the construction of high-dimensional measures on self-affine sets .
    • Sohinger, VGibbs measures of nonlinear Schrodinger equations as limits of many-body quantum states in dimension d ≤ 3 .
    • Azzam, JWasserstein distance and rectifiability of measures .
    • Jurga, NA dimension gap for Bernoulli measures for the Gauss map .

  4. Minutes for 1974
    • (a) In view of the financial position, the following measures were agreed: .
    • (i) The convention was confirmed under which members of the committee should be invited to give talks only as an emergency measure.

  5. BMC 1975
    • Cartier, P A survey of measure theory in the past ten years .
    • Moran, WInfinite convolution measures .

  6. BMC 1991
    • Goldie, C M Renewal theory for measures satisfying convolution equations .
    • Preiss, D Rectifiability of measures .

  7. BMC 1973
    • Brown, GBanach algebras of measures .

  8. BMC 1960
    • Eggleston, H GHausdorff's measure in Euclidean space .

  9. BMC 1961
    • Marstrand, J MHausdorff measure in Euclidean space .

  10. BMC 2002
    • Dales, H G The amenability of measure algebras .

  11. BMC 2004
    • Carroll, T Harmonic measure in parabola-shaped regions in Rn .

  12. Report2015.html
    • Robert Calderbank (Duke, The art of measurement) .

  13. BMC 1984
    • Drake, F RLarge cardinals as a measure of the power of assumptions .

  14. BMC 2015
    • Calderbank, RThe Art of Measurement .

  15. BMC 2007
    • Wisewell, LPolygons with holes and tube measure .

  16. Minutes for 2010
    • Hunton noted that the Leicester speakers were in large measure senior.

  17. BMC 1962
    • Taylor, S JProperties of finite measures in Euclidean spaces .

  18. BMC 2009
    • Csornyei, MDifferentiability of Lipschitz functions and other problems in geometric measure theory .

  19. Minutes for 1976
    • This would provide a useful measure of continuity of financial know-how.

Gazetteer of the British Isles

  1. London Museums
    • Case L "Mensuration" displays weights and measures, including some measures from c-1300 and item 6025, a measure of length 2 cubits (1050 mm = 41.34 in) divided into 14 palms, but of uncertain date.
    • I have read that the oldest(?) known measure - a 41.46 inch measure divided into cubits, left in a temple at Karnak, c-1400 - is in the British Museum, but there is no record of it; this may be a garbled description of this last measure.
    • Case 12 "Secular Life" contains several astrolabes, quadrants and measures.
    • The Time Measurement Gallery has portraits of Huygens, John Harrison and Thomas Earnshaw.
    • Some of the material from the Astronomy Gallery can be found in other galleries: Geophysics & Oceanography, George III, Navigation, Optics, Time Measurement, Surveying.

  2. London individuals N-R
    • This was the first systematic survey of an extended area, using angle measurement and a surveyor's chain, requiring 1000 assistants, and being the forerunner of the Ordnance Survey.
    • For the triangulation of southeast England, Ramsden designed and built the cased glass tubing and a coffered steel chain, used for two of the three measurements of the base line in 1784.
    • Starting on 16 Apr 1784 and continuing through the summer, a length of 27,404.72 feet was set out and measured three times - using cased glass tubing made by Ramsden, seasoned deal rods and a coffered steel chain made by Ramsden.
    • The glass tubing was considered to be the most accurate, but the three measurements agreed to within three inches and the steel rods were later adopted as standard.
    • In 1787, the survey had reached the coast and a line was measured on Romney Marsh using Ramsden's steel chain and the measured length agreed with the calculated length to within a foot.
    • The Hounslow Heath base line was resurveyed in 1791 by Captain Williams, Mudge and Dalby, obtaining a value only 2 inches different and the average was accepted as the basic measurement.

  3. Teddington, Middlesex
    • He then adapted the devices to measure the speed of light to greater accuracy than ever done before, getting 299,792 km/sec in 1946 and 299,792.5 km/sec in 1950 (the current value is 299,792.458 km/sec and is now used to define the metre).
    • This clock is now in the Science Museum [Time Measurement.
    • Essen's booklet, see below, describes how observations of time signals over three years (1955-1958 ?) determined atomic time in terms of Universal Time at the US Naval Observatory in Washington which was then related to Ephemeris Time, leading to the definition of the second by the 1967 General Conference of Weights and Measures in Paris as "9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom." By 1973, time standards accurate to about 1 part in 1012 were readily available.
    • [Split-second decisions, The Guardian (4 Sep 1997) 15',9)">Tucker]; [The Measurement of Frequency and Time Interval',10)">Essen]; [HMSO] .
    • Angles can be measured with an accuracy of 0.1".

  4. Oxford Institutions and Colleges
    • Above, in case 65, is a neo-Sumerian hexagonal prism with tables of measures and square roots.
    • Boys came here to measure the universal gravitational constant G - like Cavendish before him, he had to get away from vibration.
    • The adjacent meeting/class room is called the Moseley Room and has a display case with one of Moseley's X-ray tubes and the original graph showing the linear relationship of his measurements with the sequence of elements - the discovery of atomic numbers! .
    • The Museum has the fine late 16C Flemish painting The Measurers depicting a mathematical instrument maker and numerous applications of simple measuring instruments.

  5. London individuals S-Z
    • Inventor of the Spearman rank correlation measure - Pearson violently disagreed with this idea, but it has endured.
    • In 1716-1718, he produced six issues of the first scientific journal in Swedish; in 1716-1747, he served on Sweden's Board of Mines, working as a mining engineer, civil engineer, geologist and a promoter of new techniques, also writing on these topics and cosmology (showing that the earth had slowed down and advancing a nebular hypothesis which influenced Buffon, Kant and Laplace, etc.), chemistry (suggesting that crystals were like lattices of spheres), mineralogy, metallurgy, anatomy and physiology, etc.; in 1718, he wrote a booklet on the octal number system; his Regel-Konsten of 1718 was the first Swedish algebra book; in 1718, he published a method of finding longitude from the moon which was revised and extended several times, then submitted to the Board of Longitude in 1766, when Maskelyne rejected it; in 1719, he advocated the use of decimal measures and coinage and proposed a water tank for testing stability of model ships.
    • The idea had already been used by Appleton and others to measure the height of atmospheric layers.
    • At this time, he set up seven miles of wire in the basement of KCL in order to measure the speed of signal transmission and later laid some of it across the Thames [Electricians and their Marvels, S W Partridge & Co, London, c 1890, pp.52-53 & 71-72',39)">Jerrold].

  6. London individuals D-G
    • [Descriptive Catalogue of the Collection Illustrating Time Measurement Science Museum.
    • 6-7.]',68)">Tremayne] [Time Measurement.
    • HMSO, 3rd ed., 1947, p.25.',69)">Ward] Graham made the first temperature compensating pendulum in 1721, attaching a small vessel of mercury to a pendulum [Time Measurement.
    • In 1725, he invented the cylinder escapement for balance spring watches [Time Measurement.

  7. Manchester
    • Joule was the first to measure electric current, introduced the notion of measuring resistance, stated that P = I2.R in 1840 and made several measurements of the mechanical equivalent of heat in 1843-1878, leading to the principle of conservation of energy.

  8. Cambridge Individuals
    • His measurement of the universal gravitational constant G was based on an idea and device of John Michell, whom he may have met while at Cambridge.
    • He also made the first estimates of the distance of stars--which were pretty close to the first measurements made by Bessel and others about 70 years later.

  9. London Scientific Institutions
    • He also introduced the Spearman rank correlation measure, which Pearson violently disagreed with.
    • Maskelyne's measurements at Schiehallion, leading to a value of G.

  10. Oxford individuals
    • He then went to Manchester to work with Rutherford, but returned to live with his mother in 1913 and work in the Electrical Laboratory, where he carried out the measurements which established the concept of atomic number in 1913-1914.
    • His original graph of elements versus his measured values is in the Clarendon Laboratory.

  11. London Learned Societies
    • The leaflet for their 1993 exhibition on timekeeping says the RS also has: Kater's invertible pendulum, used to measure gravity at various places in the early 19C; a Tompion clock; a chronometer used by Cook on his 2nd and 3rd voyages and another used on the 2nd voyage; the sundial carved into the wall of Woolsthorpe Manor by Newton as a boy.
    • The result of the expedition to measure the curvature of light was presented at a joint meeting of the Royal Society and the Royal Astronomical Society on 6 Nov 1919 - presumably in Burlington House.

  12. References
    • Time Measurement.
    • (2) Descriptive Catalogue of the Collection Illustrating Time Measurement Science Museum.

  13. London individuals A-C
    • It was at the latter that he measured the universal gravitational constant G by use of a torsion balance, working at night so traffic did not jiggle the balance, obtaining a mean density of the earth of 5.48, which he reported in 1798.

  14. Dorchester
    • an 18C pocket scales, along with a number of other weights, measures and scales; .

  15. Cambridge Colleges
    • Newton used the arcade from the Library door to the Hall to measure the speed of sound.

  16. York
    • The Farmhouse Kitchen and Dairy has a number of standard measures including a set of egg weights! In Kirkgate is the shop of a Toyman, but it has only a few rather dull toys in the window.

  17. Haslemere, Surrey
    • Her father Henry William Chisholm, a civil servant, was from 1867-1877 the first and only Warden of the Standards, responsible for erecting the standard measures in Trafalgar Square.

Astronomy section

  1. The Reaches of the Milky Way
    • William's son, John Herschel, continued his father's work and attempted to measure the parallax of the stars.
    • In the 16th century, Brahe, after numerous failed attempts to measure parallax concluded that the earth must therefore be stationary.
    • This was not resolved until 1838 when John Herschel's contemporary, Bessel became the first person to measure parallax successfully, long after Newton had boosted the plausibility of the Keplerian model with his universal theory of gravitation [See: Dynamics of the Solar System].
    • After many attempts to measure the distance to the sun, Le Verrier in 1872 was the first to do so with a value close to our current approximation.
    • Fizeau in 1860 came to more accurate conclusions and the effect was first measured in the years around 1890 using spectroscopy by Vogel, Scheiner and Keeler in Potsdam and California after unsuccessful attempts by Huggins and Secchi.
    • When put together with measurements of the distance of closer variable stars using secular parallax by Herzsprung (1913) and then Shapley (more accurately in 1918), Leavitt's period-luminosity relationship provided a way of measuring the distances away of Cepheids by comparing their absolute to their apparent magnitudes.
    • Hubble subsequently measured the distance to Andromeda, our nearest galaxy.

  2. A Brief History of Time and Calendars
    • Meanwhile, Persian calendar reforms, initiated by Mailk Shah, sultan of the Seljuq Empire in 1079, had prompted Omar Khayyam to an outstandingly accurate measurement of the year - by far the most accurate of the pre-telescope era.
    • For example in the late 18th and early 19th century, there was a calendar reform following the French Revolution  this calendar was based on a 10-day week aimed to remove any trace of religion from the calendar, as well as bringing it into line with the broader decimilisation of measurements.

  3. The Infinite Universe
    • It was first measured by Penzias and Wilson in 1965 and forms part of the static you hear on the radio.

This search was performed by Kevin Hughes' SWISH and Ben Soares' HistorySearch Perl script

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JOC/BS August 2001