Euler, Lagrange, and Laplace

in Alsace, was the son of a poor tailor. While working at his father's trade, he acquired through his own unaided efforts a knowledge of elementary mathematics. At the age of thirty he became tutor in a Swiss family and secured leisure to continue his studies. In his travels with his pupils through Europe he became acquainted with the leading mathematicians. In 1764 he settled in Berlin, where he became member of the Academy, and enjoyed the society of Euler and Lagrange. He received a small pension, and later became editor of the Berlin Ephemeris. His many-sided scholarship reminds one of Leibniz. In his Cosmological Letters he made some remarkable prophecies regarding the stellar system. In mathematics he made several discoveries which were extended and overshadowed by his great contemporaries. His first research on pure mathematics

developed in an infinite series the root $x$ of the equation

$x^m+px=q$. Since each equation of the form $a{x}^r+b{x}^s=d$ can be reduced to $x^m+px=q$ in two ways, one or the other of the two resulting series was always found to be convergent, and to give a value of $x$. Lambert's results stimulated Euler,

who extended the method to an equation of four terms, and particularly Lagrange, who found that a function of a root of

$a-x+\varphi (x)=0$ can be expressed by the series bearing his name. In 1761 Lambert communicated to the Berlin Academy a memoir, in which he proves that $\pi$ is irrational. This proof

is given in Note IV. of Legendre's Géometrie, where it is

extended to ${\pi }^2$. To the genius of Lambert we owe the introduction

into trigonometry of hyperbolic functions, which he

designated by $\mathit{\text{sinh}}x$, $\mathit{\text{cosh}}x$, etc. His Freye Perspective, 1759 and 1773, contains researches on descriptive geometry, and entitle him to the honour of being the forerunner of Monge.

In his effort to simplify the calculation of cometary orbits, he was led geometrically to some remarkable theorems on conics, for instance this: "If in two ellipses having a common major axis we take two such arcs that their chords are equal, and that also the sums of the radii vectores, drawn respectively from the foci to the extremities of these arcs, are equal to each other, then the sectors formed in each ellipse by the arc and the two radii vectores are to each other as the square roots of the parameters of the ellipses."13

John Landen (1719–1790) was an English mathematician

whose writings served as the starting-point of investigations by Euler, Lagrange, and Legendre. Landen's capital discovery, contained in a memoir of 1755, was that every arc of the hyperbola is immediately rectified by means of two arcs of an ellipse. In his "residual analysis" he attempted to obviate the metaphysical difficulties of fluxions by adopting a purely algebraic method. Lagrange's Calcul des Fonctions is based

upon this idea. Landen showed how the algebraic expression for the roots of a cubic equation could be derived by application of the differential and integral calculus. Most of the time of this suggestive writer was spent in the pursuits of active life.

Étienne Bézout (1730–1783) was a French writer of popular