**William Niven**'s parents were Charles Niven (born around 1802), a ginger beer brewer and spirit dealer, and Barbara Davidson (born 1810 or 1811) of Monquhitter, Aberdeenshire. Charles and Barbara married on 28 November 1833 and had nine children: John Gibb Niven (born 1834); Thomas Niven (born 1836); William Davidson Niven (born 1842); Charles Niven (born 1845); Barbara Niven (born 1848); James Niven (born 1851); Alexander Niven (born 1854); Thomas Niven (born 1855); and George Niven (born 1859). William was one of three distinguished mathematical brothers, Charles and James also being Cambridge Wranglers. Charles Niven, who was Professor of Natural Philosophy at Aberdeen University for 41 years, has a biography in this archive. James Niven studied mathematics at Queens' College, Cambridge and was bracketed eighth wrangler in the mathematical tripos of 1874. He went on to study medicine and became medical officer of health for Manchester.

William Niven's secondary schooling was at the Grammar School, Aberdeen, from where he entered King's College of the University of Aberdeen. In 1861 he graduated with an M.A. from the University of Aberdeen which, of course, was his local university as Peterhead is not far from Aberdeen in Scotland. From there, as was the tradition of the Scottish Universities at that time, Niven went to study at the University of Cambridge matriculating in 1862.

At Cambridge Niven studied mathematics at Trinity College, where he graduated as third Wrangler in the mathematical tripos of 1866 (the means that he was ranked third among the First Class students). The following year he was elected to a fellowship at Trinity College but after holding the fellowship for some time he left Cambridge to take up an appointment as Professor of Mathematics at the Royal Indian Engineering College, Cooper's Hill, Surrey. This College, explicitly set up to train engineers for work in India, was opened on 5 August 1872 and Niven became its first mathematics professor.

Soon, however, Niven left the Engineering College to take up an appointment as professor at the Royal Arsenal in Woolwich. There he undertook research on gunnery and ballistics, and we quote below from his paper *On the Calculation of the Trajectories of Shot* which describes important work he undertook at this time. However the attraction of Cambridge was great and he returned to Trinity College as a College Lecturer in 1874. He soon became a firm friend of James Clerk Maxwell but this friendship was tragically cut short when Maxwell died in 1879. After Maxwell's death, Niven looked after Maxwell's affairs and, most importantly, helped to edit the second edition of Maxwell's *Electricity and Magnetism *and began the task of editing Maxwell's scientific papers. Inspired by Maxwell and his mathematics, Niven's research interests turned increasingly towards the study of spherical and ellipsoidal harmonics [2]:-

We note the titles of some of the papers referred to in the above quote:Maxwell's new developments in the theory of electricity excited keen interest; Niven's public lectures on the subject were attended by a great many Cambridge mathematicians of the day. From1877he devoted sustained attention to the development of Maxwell's method of defining the general spherical harmonic in terms of its poles. In1879he wrote a memoir exhibiting the scope of this method by various applications, especially to the evaluation of integrals over a sphere of products of three or more harmonics. A further memoir in1890on elliptical harmonics extended and generalised his earlier work. His work in the later1890s was on electrostatics.

*On the Theory of Electric Images, and its application to the case of two Charged Spherical Conductors*(1876);

*On the Electrical Capacity of a Conductor bounded by Two Spherical Surfaces cutting at any angle*(1880);

*On the Harmonics of a Ring*(1892); and

*Note on the Electric Capacity of a Conductor in the form of Two Intersecting Spheres*(1896). These four papers were all published in the

*Proceedings*of the London Mathematical Society. He also published

*On Certain Definite Integrals Occurring in Spherical Harmonic Analysis, and on the Expansion in Series of the Potentials of the Ellipsoid and of the Ellipse*(1879),

*On Ellipsoidal Harmonics*(1891),

*On the Calculation of the Trajectories of Shot*(1877), and

*The Calculation of Ellipsoidal Harmonics*(1906) in the

*Philosophical Transactions*or the

*Proceedings*of the Royal Society. As we mentioned above, he edited

*The scientific papers of James Clerk Maxwell*(2 volumes) (1890) published by

*Cambridge University Press*.

The paper *On the Calculation of the Trajectories of Shot *(1877), already mentioned as illustrating work he did at the Royal Arsenal, was communicated to the Royal Society by Maxwell. We quote from Niven's Introduction to explain the nature of the work:-

In 1882 Niven was appointed as Director of Studies of the Royal Naval College in Greenwich. He was able to encourage some of the best Cambridge mathematicians such as William Burnside to accept posts at Greenwich greatly enhancing its reputation [1]:-In the present state of our knowledge of the resistance of the air to shot, the problem of integrating the equations of motion of the shot and of plotting-out a representation of the curve described by it is peculiar, because, according to the best experiments we possess, the law of the retardation cannot be expressed by a single exact formula which is available for the solution. We are therefore compelled to give a solution adapted to Tables, the magnitudes of the retardation being set down in those Tables for velocities which are common in practice. ... The amount of labour, however, in calculating all the quantities for a single component arc, even with the aid of copious tables, is so great that I was led to examine whether any thing could be done towards simplifying the solution and reducing the amount of calculation. It will appear in the sequel that rules of comparatively easy application can be employed, and that the tables necessary for their use are already existing or can be easily formed.

In the same year of 1882 that he was appointed to Greenwich, on 8 June, he was honoured by being elected a fellow of the Royal Society. He served on the Council of the Society in 1892-94 and, after he retired from Greenwich in 1903, he served as vice-president of the Royal Society in 1904-06. On his retirement in 1903 he was honoured for his services by being made KCB - at this point he became Sir William Davidson Niven.His talent for friendship and quiet social amenities made his beautiful residence, on the river front of the Palace of Greenwich, a centre of the life of the staff; and here also he won and retained through life the friendship of successive admirals who came for a term of office as Governors of the College. He induced some of his Cambridge friends of high mathematical eminence to undertake posts on the staff: by his hospitality he made the position of external examiner so attractive that he could command the services of the best qualified men of the time.

Niven was an active member and staunch supporter of the London Mathematical Society from his election to the Society on 8 May 1873. He served on the Council of the Society from the year of his election to the Society, continuing to serve for fourteen years. He served as its President from 1908 until 1910. His Presidential address was "The relations of mathematics to experimental science" but [1]:-

Shortly before his death, his professional friends and colleagues from Cambridge, London and Aberdeen put up money to have his portrait painted. It was presented to the University of Aberdeen and it was hung in King's Buildings.... with characteristic modesty he seems to have withdrawn it from publication.

**Article by:** *J J O'Connor* and *E F Robertson*

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