Cataldi wrote around thirty books on mathematics, and some on other topics. He wrote on arithmetic publishing Practica aritmetica in four parts between 1602 and 1617. This work was dedicated to the Senate of Bologna, but it is believed that he published it at his own expense. Carruccio writes :-
Cataldi showed his benevolence by giving the superiors of various Franciscan monasteries the task of distributiong free copies of his 'Practica aritmetica' to monasteries, seminaries, and poor children.He is, however, best known for his work on perfect numbers and on continued fractions. His contributions to perfect numbers were made in 1603. Euclid knew that if 2n - 1 is prime, then 2n-1( 2n - 1) is a perfect number. This gives the perfect numbers 6, 28, 496 and 8128 by taking n = 2, 3, 5, 7 respectively. These were known to the ancient Greeks, and the next perfect number had been found in 1536 by Hudalrichus Regius who showed that 213 - 1 is prime giving 33350336 as the next perfect number (this had been discovered earlier by a number of mathematicians but their discoveries only became common knowledge comparatively recently). Cataldi, in 1603, showed that if n is composite then 2n - 1 is composite, and he also showed that 2n - 1 is prime for n = 17 and n = 19. He used no clever tricks, merely checked that these numbers were prime by dividing each by all primes up to their square roots. Of course, to do this he required a list of primes up to 724 (the approximate root of 219 - 1). In fact Cataldi calculated a list of all primes up to 750 and a list of the factorisation of all numbers up to 800. He published these lists separately. By showing that 217 - 1 = 131071 and 219 - 1 = 524287 were prime, Cataldi had, in fact, found the sixth and seventh perfect numbers 8589869056 and 137438691328. He also conjectured that 2n - 1 was prime for n = 23, 29, 31 and 37 but all of these turned out to be false except for n = 31. Fermat showed that 223 - 1 = 8388607 = 47 × 178481 and 237 - 1 = 137438953471 = 223 × 616318177) were composite in 1640. Euler showed that 231 - 1 was prime in 1732; it gave rise to the first discovery of a perfect number since those of Cataldi about 130 years earlier. Euler also disproved the last part of Cataldi's conjecture in 1738 when he showed that 229 - 1 = 536870911 = 233 × 1103 × 2089) is composite.
Cataldi found square roots of numbers by use of an infinite series leading to an early investigation into continued fractions. This work on continued fractions appears in Trattato del modo brevissimo di trovar la radice quadra delli numeri (1613) although he announced them in Operetta delle linee rette equidistanti et non equidistanti (1603). His methods make precise some ideas which went back to Heron :-
In this work the square root of a number is found through the use of infinite series and unlimited continued fractions. It represents a notable contribution to the development of infinite algorithms.Cataldi calculates the square root of a number N by first taking the integer a such that a2 < N < (a+1)2. The remainder is then r = N - a2. His first approximation to √N is
(N - a2) / 2a and (N - a2) / (2a + 1) or r / 2a and r / (2a + 1).He puts
a1 = (N - a2) / 2a, r1 = a12 - N;Let us compute this sequence for N = 18 by putting a = 4 (so 42 < 18 < (4+1)2).
a2 = (N - a12) / 2a1, r2 = a22 - N;
a3 = (N - a22) / 2a2, r3 = a32 - N;
a = 4,Note that after 5 iterations, the result is already correct to 47 places.
a1 = 4.25,
a2 = 4.2426470,
a3 = 4.2426406871240,
a4 = 4.24264068711928514640506887,
a5 = 4.2426406871192851464050661726290942357090156261317
√18 = 4.2426406871192851464050661726290942357090156261307Cataldi calculated the convergents of √18 and realised that the convergents are alternately greater than and smaller that √18. We note that he does not use what are called today 'simple continued fractions' and his method will always give a continued fraction for a square root which has period 1. He writes:-
Let us now proceed to the consideration of another method of finding roots continuing by adding row on row to the denominator of the fraction of the preceding rule. But for greater convenience, I shall assume a number whose root may be easily taken and I shall assume that the first part of the root is an integer. Then let 18 be the proposed number, and if I assume that the first root is 4. & 2/8, that is 41/4, this will be in excess by 1/16 which is the square of the fraction 1/4.Here the 4. & 2/8 is a1 and the excess of 1/16 is r1 . The first convergent is therefore 17/4. He finds x such that 42/(8+x) has a rounded minimum value of √18 which he observes is when (and only when) a + x has a rounded minimum value of √18, so x = 2/8. Cataldi continues:-
The second root will be found by the above mentioned method to be 4. & 2/8. & 1/4 which is 4. & 8/33, which is 2/1089 too small. This arises from multiplying the entire fraction 8/33 by 1/132 in which the whole fraction is less than the 1/4 which is the added fraction.The second convergent is therefore 140/33. Cataldi continues to calculate the third convergent to be 577/136. He then computes (577/136)2-18 = 1/18496 and states that the third convergent is too large by 11/18496. He continues the calculation until he reaches the fifteenth convergent. Note that the third convergent gives √18 correct to 5 decimal places, and the fifteenth convergent 886731088897/209004522016 gives √18 correct to 23 decimal places. Let us remark that Cataldi is using a notation quite similar to modern notation for he writes
4. & 2/8. & 2/8. & 2/8. & 2/8where & is just our + and the . indicates that what follows is to be added to the denominator. Cataldi only worked with numerical examples, yet he clearly understood that if the convergents are written as p1 / q1 , p2 / q2 , p3 / q3 , ... then they satisfy the fundamental relation
pn . qn-1 - pn-1 . qn = (-1)n . rn.Among his other works were Transformatione geometrica (1611), dedicated to the Grand Duke Cosimo II, and a book which studied problems of the range of artillery which included tables on the rising of the sun and the time of midday for Bologna (1613). In 1618 he published Operetta di ordinanze quadre which studied military applications of algebra. In 1612 he published a book on squaring the circle Trattato della quadratura del cerchio dove si esamina un nuovo modo di quadrarlo per numeri. Et insieme si mostra come, Dato un rettilineo, si formi un curvilineo equale ad esso Dato. Et di più alcune transformationi di curvilinei misti fra loro. Cataldi begins this work with a commentary on a recently published work on the same topic, namely Pellegrino Borello's Regola e modo facilissimo di quadrare il cerchio (1609). Augustus De Morgan writes in A Budget of paradoxes:-
[Trattato della quadratura del cerchio by Pietro Antonio Cataldi. Bologna, 1612] claims a place as beginning with the quadrature of Pellegrino Borello of Reggio, who will have the circle to be exactly 3 diameters and 69/484 of a diameter. Cataldi, taking Van Ceulen's approximation, works hard at the finding of integers which nearly represent the ratio. He had not then the 'continued fraction', a mode of representation which he gave the next year in his work on the square root. He has but twenty of Van Ceulen's thirty places, which he takes from Clavius ...Cataldi then looks at the more general question of constructing squares and rectangles with an area equal to that of a number of given shapes with curved edges. He illustrated these shapes with diagrams collected at the end of the book.
Cataldi also published an edition of Euclid's Elements. He worked on Euclid's fifth postulate, attempting to prove the postulate was a consequence of the others in Operetta delle linee rette equidistanti et non equidistanti (1603). His approach was the following. He defined equidistant straight lines as follows:
A given straight line is said to be equidistant from another straight line in the same plane when the two shortest lines that are drawn from any two different points on the first line to the second line are equal.He then proceeded to correctly deduce the Fifth Postulate. But wait a minute, we know that the Fifth Postulate cannot be deduced from the first four, for that is how non-Euclidean geometries come about. There must be an error somewhere, so where is it? In fact it is in the definition which Cataldi gave of equidistant for this definition, which we quoted above, assumes that the given line is the locus of points at a constant distance from the second line and it also assumes that it is a straight line. The compatibility of these two assumptions is (of course) equivalent to the Fifth Postulate. If you are still finding this a little difficult to understand, just consider a great circle on a sphere (this is the shortest distance between two points and so the straight line in this geometry). Now consider a line equidistant from the given great circle. This will not be a great circle, so is not the shortest distance between two points and so is not the analogue of a straight line in this geometry.
Cataldi tried, without success, to set up an academy of mathematics in Bologna. Despite the failure he left money in his will to set up a school in his own house but this also seems not to have happened.
Article by: J J O'Connor and E F Robertson
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