Natural Philosophy and Astronomy under John Pringle Nichol, Chemistry under Thomas Thomson, a very advanced teacher and investigator, Natural History under William Cowper, were, as I can testify by my experience, all made interesting and valuable to the students of Glasgow University in the thirties and forties of the nineteenth century....

"My predecessor in the Natural Philosophy chair, Dr. Meikleham, taught his students reverence for the great French mathematicians Legendre, Lagrange, and Laplace. His immediate successor in the teaching of the Natural Philosophy Class,5 Dr. Nichol, added Fresnel and Fourier to this list of scientific nobles: and by his own inspiring enthusiasm for the great French school of mathematical physics, continually manifested in his experimental and theoretical teaching of the wave theory of light and of practical astronomy, he largely promoted scientific study and thorough appreciation of science in the University of Glasgow....

"As far back as 1818 to 1830 Thomas Thomson, the first Professor of Chemistry in the University of Glasgow, began the systematic teaching of practical chemistry to students, and, aided by the Faculty of Glasgow College, which gave the site and the money for the building, realised a well-equipped laboratory, which preceded, I believe, by some years Liebig's famous laboratory of Giessen, and was, I believe, the first established of all the laboratories in the world for chemical research and the practical instruction of University students in chemistry. That was at a time when an imperfectly informed public used to regard the University of Glasgow as a stagnant survival of mediævalism, and used to call its professors the 'Monks of the Molendinar'!

"The University of Adam Smith, James Watt, and Thomas Reid was never stagnant. For two centuries and a half it has been very progressive. Nearly two centuries ago it had a laboratory of human anatomy. Seventy-five years ago it had the first chemical students' laboratory. Sixty-five years ago it had the first Professorship of Engineering of the British Empire. Fifty years ago it had the first physical students' laboratory—a deserted wine-cellar of an old professorial house, enlarged a few years later by the annexation of a deserted examination-room. Thirty-four years ago, when it migrated from its four-hundred-years-old site off the High Street of Glasgow to this brighter and airier hill-top, it acquired laboratories of physiology and zoology; but too small and too meagrely equipped."

In the summer of 1840 Professor James Thomson and his two sons went for a tour in Germany. It was stipulated that German should be the chief, if not the only, subject of study during the holidays. But William had just begun to study Fourier's famous book, La Théorie Analytique de la Chaleur, and took it with him. He read that great work, full as it was of new theorems and processes of mathematics, with the greatest delight, and finished it in a fortnight. The result was his first original paper "On Fourier's Expansions of Functions in Trigonometrical Series," which is dated "Frankfort, July 1840, and Glasgow, April 1841," and was published in the Cambridge Mathematical Journal (vol. ii, May 1841). The object of the paper is to show in what cases a function f(x), which is to have certain arbitrary values between certain values of x, can be expanded in a series of sines and when in a series of cosines. The conclusion come to is that, for assigned limits of x, between 0 and a, say, and for the assigned values of the function, f(x) can be expressed either as a series of sines or as a series of cosines. If, however, the function is to be calculated for any value of x, which lies outside the limits of that variable between which the values of the function are assigned, the values of f(x) there are to be found from the expansion adopted, by rules which are laid down in the paper.

Fourier used sine-expansions or cosine-expansions as it suited him for the function between the limits, and his results had been pronounced to be "nearly all erroneous." From this charge of error, which was brought by a distinguished and experienced mathematician, the young analyst of sixteen successfully vindicated Fourier's work. Fourier was incontestably right in holding, though he nowhere directly proved, that a function given for any value of x between certain limits, could be expressed either by a sine-series or by a cosine-series. The divergence of the values of the two expressions takes place outside these limits, as has been stated above.

The next paper is of the same final date, but appeared in the Cambridge Mathematical Journal of the following November. In his treatment of the problem of the cooling of a sphere, given with an arbitrary initial distribution of temperature symmetrical about the centre, Fourier assumes that the arbitrary function F(x), which expresses the temperature at distance x from the centre, can be expanded in an infinite series of the form

a₁ sin n₁x + a₂ sin n₂x + ...

where a₁, a₂, ... are multipliers to be determined and n₁, n₂, ... are the roots, infinite in number, of the transcendental equation (tan nX) ⁄ nX = 1 − hX.

This equation expresses, according to a particular solution of the differential equation of the flow of heat in the sphere, the condition fulfilled at the surface, that the heat reaching the surface by conduction from the interior in any time is radiated in that time to the surroundings. Thomson dealt in this second paper with the possibility of the expansion. He showed that, inasmuch as the first of the roots of the transcendental equation lies between 0 and ¹⁄₂, the second between 1 and ³⁄₂, the third between 2 and ⁵⁄₂, and so on, with very close approach to the upper limit as the roots become of high order, the series assumed as possible has between the given limits of x the same value as the series

A₁ sin ¹⁄₂ x + A₂ sin ³⁄₂ x + ...

where A₁, A₂, ... are known in terms of a₁, a₂, ... Conversely, any series of this form is capable of being replaced by a series of the form assumed. Further, a series of the form just written can be made to represent any arbitrary system of values between the given limits, and so the possibility of the expansion is demonstrated.

The next ten papers, with two exceptions, are all on the motion of heat, and appeared in the Cambridge Mathematical Journal between 1841 and 1843, and deal with important topics suggested by Fourier's treatise. Of the ideas contained in one or two of them some account will be given presently.

Fourier's book was called by Clerk Maxwell, himself a man of much spirituality of feeling, and no mean poet, a great mathematical poem. Thomson often referred to it in similar terms. The idea of the mathematician as poet may seem strange to some; but the genius of the greatest mathematicians is akin to that of the true creative artist, who is veritably inspired. For such a book was a work of the imagination as well as of the reason. It contained a new method of analysis