Novum organon renovatum / Being the second part of the philosophy of the inductive sciences

by William Whewell

XL.

Geometrical deduction (and deduction in general) is called Synthesis, because we introduce, at successive steps, the 12 results of new principles. But in reasoning on the relations of space, we sometimes go on separating truths into their component truths, and these into other component truths; and so on: and this is geometrical Analysis. (ii. 11.)

XLI.

Among the foundations of the Higher Mathematics, is the Idea of Symbols considered as general Signs of Quantity. This idea of a Sign is distinct from, and independent of other ideas. The Axiom to which we refer in reasoning by means of Symbols of quantity is this:—The interpretation of such symbols must be perfectly general. This Idea and Axiom are the bases of Algebra in its most general form. (ii. 12.)

XLII.

Among the foundations of the Higher Mathematics is also the Idea of a Limit. The Idea of a Limit cannot be superseded by any other definitions or Hypotheses, The Axiom which we employ in introducing this Idea into our reasoning is this:—What is true up to the Limit is true at the Limit. This Idea and Axiom are the bases of all Methods of Limits, Fluxions, Differentials, Variations, and the like. (ii. 12.)

XLIII.

There is a pure Science of Motion, which does not depend upon observed facts, but upon the Idea of motion. It may also be termed Pure Mechanism, in opposition to Mechanics Proper, or Machinery, which involves the mechanical conceptions of force and matter. It has been proposed to name this Pure Science of Motion, Kinematics. (ii. 13.)

XLIV.

The pure Mathematical Sciences must be successfully cultivated, in order that the progress of the principal Inductive Sciences may take place. This appears in the case of Astronomy, in which Science, both in ancient and in modern times, each advance of the theory has depended upon the 13 previous solution of problems in pure mathematics. It appears also inversely in the Science of the Tides, in which, at present, we cannot advance in the theory, because we cannot solve the requisite problems in the Integral Calculus. (ii. 14.)