At the age of nineteen he communicated to Leonhard Euler his idea of a general method of dealing with "isoperimetrical" problems, known later as the Calculus of Variations.
The calculus of variations lay undeveloped in Euler's mode of treating isoperimetrical problems. The fruitful method, again, of the variation of elements was introduced by Euler, but adopted and perfected by Lagrange, who first recognized its supreme importance to the analytical investigation of the planetary movements.
The calculus of variations is indissolubly associated with his name.
1875); Examples of Analytical Geometry of Three Dimensions (1858, 3rd ed., 1873); Mechanics (1867), History of the Mathematical Theory of Probability from the Time of Pascal to that of Lagrange (1865); Researches in the Calculus of Variations (1871); History of the Mathematical Theories of Attraction and Figure of the Earth from Newton to Laplace (1873); Elementary Treatise on Laplace's, Lame's and Bessel's Functions (1875); Natural Philosophy for Beginners (1877).
It is a classical problem in the calculus of variations to deduce the equation (9) from the condition that the depth of the centre of gravity of a, chain of given length hanging I I between fixed points must be catenary; it determines the scale of the curve, all cate } stationary (~ 9).
Furthermore it can be shown by the application of the calculus of variations that the condition for a minimum value of the function W, is that vV = o.
== Treatise on the Differential Calculus and the Elements of the Integral Calculus (1852, 6th ed., 1873), Treatise on Analytical Statics (1853, 4th ed., 1874); Treatise on the Integral Calculus (1857, 4th ed., 1874); Treatise on Algebra (1858, 6th ed., 1871); Treatise on Plane Coordinate Geometry (1858, 3rd ed., 1861); Plane Trigonometry (1859, 4th ed., 1869); Spherical Trigonometry (1859); History of the Calculus of Variations (1861); Theory of Equations (1861, 2nd ed.
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