Geometry, Topology and Global Analysis

The terms synthesis and analysis are used in mathematics in a more special sense than in logic. In ancient mathematics they had a different meaning from what they now have. The oldest definition of mathematical analysis as opposed to synthesis is that given in Euclid, XIII. 5, which in all probability was framed by Eudoxus: "Analysis is the obtaining of the thing sought by assuming it and so reasoning up to an admitted truth; synthesis is the obtaining of the thing sought by reasoning up to the inference and proof of it."

The word Analysis signifies the general and particular heads of a discourse, with their mutual connections, both co-ordinate and subordinate, drawn out into one or more tables.

Philosophers hasten too much from the analytic to the synthetic method ; that is, they draw general conclusions from too small a number of particular observations and experiments.