The main work of Descartes, so far as algebra was concerned, was the establishment of a relation between **arithmetical** and geometrical measurement.

He was thus led to conclude that chemistry is a branch of applied mathematics and to endeavour to trace a law according to which the quantities of different bases required to saturate a given acid formed an **arithmetical**, and the quantities of acids saturating a given base a geometrical, progression.

He took a passionate delight in the pursuit of knowledge from his very infancy, and is reported to have worked out long **arithmetical** sums by means of pebbles and biscuit crumbs before he knew the figures.

In mathematics, he was the first to draw up a methodical treatment of mechanics with the aid of geometry; he first distinguished harmonic progression from **arithmetical** and geometrical progressions.

**Arithmetical** groups, connected with the theory of quadratic forms and other branches of the theory of numbers, which are termed "discontinuous," and infinite groups connected with differential forms and equations, came into existence, and also particular linear and higher transformations connected with analysis and geometry.