Smith, at the request of a member of the commission by which the prize was proposed, undertook in 1882 to write out the demonstration of his general theorems so far as was required to prove the results for the special case of five squares.
Fourteen years later the Academie Frangaise, in ignorance of Smith's work, set the demonstration and completion of Eisenstein's theorems for five squares as the subject of their "Grand Prix des Sciences Mathematiques."
As a geometer he is classed by Eudemus, the greatest ancient authority, among those who "have enriched the science with original theorems, and given it a really sound arrangement."
In 1709 he entered the university of Glasgow, where he exhibited a decided genius for mathematics, more especially for geometry; it is said that before the end of his sixteenth year he had discovered many of the theorems afterwards published in his Geometria organica.
In it Maclaurin developed several theorems due to Newton, and introduced the method of generating conics which bears his name, and showed that many curves of the third and fourth degrees can be described by the intersection of two movable angles.