Euler, Lagrange, and Laplace

mathematical physics. "Fourier's series" constitutes its

gem. By this research a long controversy was brought to a close, and the fact established that any arbitrary function can be represented by a trigonometric series. The first

announcement of this great discovery was made by Fourier in 1807, before the French Academy. The trigonometric series $\sum _{n=0}^{n=\infty }(a_n\sin nx+b_n\cos nx)$ represents the function $\varphi (x)$ for every value of $x$, if the coefficients $a_n={\displaystyle \frac{1}{\pi }{\int }_{-\pi }^{\pi }\varphi (x)\sin nxdx}$, and $b_n$ be equal to a similar integral. The weak point in Fourier's analysis lies in his failure to prove generally that the trigonometric series actually converges to the value of the function. In 1827 Fourier succeeded Laplace as president of the council of the Polytechnic School.

Before proceeding to the origin of modern geometry we shall speak briefly of the introduction of higher analysis into Great Britain. This took place during the first quarter of this century. The British began to deplore the very small progress that science was making in England as compared with its racing progress on the Continent. In 1813 the "Analytical

Society" was formed at Cambridge. This was a small club established by George Peacock, John Herschel, Charles Babbage,

and a few other Cambridge students, to promote, as it was humorously expressed, the principles of pure "$D$-ism," that is, the Leibnizian notation in the calculus against those

of "dot-age," or of the Newtonian notation. This struggle ended in the introduction into Cambridge of the notation ${\displaystyle \frac{dy}{dx}}$, to the exclusion of the fluxional notation $\dot{y}$. This was a great step in advance, not on account of any great superiority of the Leibnizian over the Newtonian notation, but because the adoption of the former opened up to English students the vast storehouses of continental discoveries. Sir William Thomson, Tait, and some other modern writers find

it frequently convenient to use both notations. Herschel,

Peacock, and Babbage translated, in 1816, from the French,

Lacroix's treatise on the differential and integral calculus, and

added in 1820 two volumes of examples. Lacroix's was one of the best and most extensive works on the calculus of that time. Of the three founders of the "Analytical Society," Peacock afterwards did most work in pure mathematics. Babbage became famous for his invention of a calculating engine superior to Pascal's. It was never finished, owing

to a misunderstanding with the government, and a consequent failure to secure funds. John Herschel, the eminent astronomer, displayed his mastery over higher analysis in memoirs communicated to the Royal Society on new applications of mathematical analysis, and in articles contributed to cyclopædias on light, on meteorology, and on the history of mathematics.