Page:A History of Mathematics (1893).djvu/278

 developed in an infinite series the root $$\scriptstyle{x}$$ of the equation $\scriptstyle{x^m+px=q}$. Since each equation of the form $$\scriptstyle{ax^r+bx^s=d}$$ can be reduced to $$\scriptstyle{x^m+px=q}$$ in two ways, one or the other of the two resulting series was always found to be convergent, and to give a value of $\scriptstyle{x}$. 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 $$\scriptstyle{\alpha-x+\phi(x)=0}$$ can be expressed by the series bearing his name. In 1761 Lambert communicated to the Berlin Academy a memoir, in which he proves that $$\scriptstyle{\pi}$$ is irrational. This proof is given in Note IV. of Legendre's Géometrie, where it is extended to $\scriptstyle{\pi^2}$. To the genius of Lambert we owe the introduction into trigonometry of hyperbolic functions, which he designated by $\scriptstyle{sinh~x}$, $\scriptstyle{cosh~x}$, 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 [sic], 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