In 1878, at the age of 24, Henri Poincaré took up Lazarus Fuchs's study of a differential equation that Fuchs had been unable to solve. Between 1880 and 1884, he wrote numerous papers drawing on methods from differential equations, complex analysis, transformation groups and hyperbolic geometry.
Henri Poincaré in 1872, aged 18.
Lobachevsky's geometry
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In 1882, a survey article examined a specific class of geometric transformations. After giving some of their properties, Poincaré wrote: "I cannot pass over in silence the link connecting the preceding notions with Lobatchewski's non-Euclidean geometry [sic.]. Suppose we agree to strip the words 'line', 'length', 'distance' and 'area' of their usual meanings; to call any circle whose center lies on X a 'line'; to call the quantity we have just denoted by L the 'length of a curve'; to call the L of the circular arc joining two points and having its center on X the 'distance between those two points'; and finally, to call the quantity we denote by S the 'area of a plane region'. Suppose, moreover, that we retain the usual meanings of the words 'angle' and 'circle', while agreeing to call the point at a constant distance from every point on a circle (in the new sense of the word 'distance') its 'center', and this constant distance its 'radius'.
