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nydus/A History of MathematicsPublic

This text examines the transition from the Middle Ages to the Modern era, highlighting how the fall of Constantinople and the invention of the printing press catalyzed a revival of classical learning. It traces the shift toward scientific inquiry through the rise of pure mathematics and astronomy, detailing the intellectual struggle against established scholastic and ecclesiastical authority.

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Table of Contents

Euler, Lagrange, and Laplace

had used it still earlier, but did not publish it until 1809. The first deduction of the law of probability of error that appeared in print was given in 1808 by Robert Adrain in the

Analyst, a journal published by himself in Philadelphia.2 Proofs of this law have since been given by Gauss, Ivory,

Herschel, Hagen, and others; but all proofs contain some

point of difficulty. Laplace's proof is perhaps the most satisfactory.

Laplace's work on probability is very difficult reading, particularly the part on the method of least squares. The analytical processes are by no means clearly established or free from error. "No one was more sure of giving the result of analytical processes correctly, and no one ever took so little care to point out the various small considerations on which correctness depends" (De Morgan).

Of Laplace's papers on the attraction of ellipsoids, the most

important is the one published in 1785, and to a great extent reprinted in the third volume of the Mécanique Céleste. It gives an exhaustive treatment of the general problem of attraction of any ellipsoid upon a particle situated outside

or upon its surface. Spherical harmonics, or the so-called "Laplace's coefficients," constitute a powerful analytic engine

in the theory of attraction, in electricity, and magnetism. The theory of spherical harmonics for two dimensions had been previously given by Legendre. Laplace failed to make due acknowledgment of this, and there existed, in consequence, between the two great men, "a feeling more than coldness." The potential function, V, is much used by

Laplace, and is shown by him to satisfy the partial differential equation ∂2V∂x2+∂2V∂y2+∂2V∂z2=0. This is known as Laplace's equation, and was first given by him in the more complicated form which it assumes in polar co-ordinates. The notion of potential was, however, not introduced into analysis by Laplace. The honour of that achievement belongs to Lagrange.49

Among the minor discoveries of Laplace are his method of solving equations of the second, third, and fourth degrees,

his memoir on singular solutions of differential equations, his

researches in finite differences and in determinants, the establishment

of the expansion theorem in determinants which had been previously given by Vandermonde for a special case, the

determination of the complete integral of the linear differential equation of the second order. In the Mécanique Céleste he made a generalisation of Lagrange's theorem on the development

of functions in series known as Laplace's theorem.

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