A characteristic feature of the calculus is that a meaning can be attached to a symbol of this kind by adopting a new rule, called that of regressive multiplication, as distinguished from the foregoing, which is progressive.
But his account of the first is imperfect, because in ancient analysis the more general propositions, with which it concludes, are not mere consequences, but the real grounds of the given proposition; while his addition of the second reduces the nature of analysis to the utmost confusion, because hypothetical deduction is progressive from hypothesis to consequent facts whereas analysis is regressive from consequent facts to real ground.
Deduction is analysis when it is regressive from consequence to real ground, as when we start from the proposition that the angles of a triangle are equal to two right angles and deduce analytically that therefore (i) they are equal to equal angles made by a straight line standing on another straight line, and (2) such equal angles are two right angles.
Much of the Principia consists of synthetical deductions from definitions and axioms. But the discovery of the centripetal force of the planets to the sun is an analytic deduction from the facts of their motion discovered by Kepler to their real ground, and is so stated by Newton in the first regressive order of Aristotle - P-M, S-P, S-M.
He re-defines analysis in the very opposite way to the ancients; whereas they defined it as a regressive process from consequence to ground, according to Wundt it is a progressive process of taking for granted a proposition and deducing a consequence, which being true verifies the proposition.
Semon called this stage the Pentactula, and supposed that, in its early history, the class had passed through a similar stage, which he called the Pentactaea, and regarded as the ancestor of all Echinoderms. It has since been proved that the five tentacles with their canals are interradial, so that one can scarcely look on the Pentactula as a primitive stage, while the apparent simplicity of the Synaptidae, at least as compared with other holothurians, is now believed to be the result of regressive vlu.
For instance, if n= 4, E r = e l e 3, Es= e 2 e 3 e 4, we have IErIE8 = (-e2e4) (- e 1) = ele2e4 = l e3, consequently, by the rule of regressive multiplication, eie3ï¿½e2e3e4 = e3.
Applying the distributive law, we obtain, when r+s>n, ArB 8 = EaErE i 3E 8 = E(a(3)ErEs, where the regressive products E r E B are to be reduced to units of species (r+s-n) by the foregoing rule.