Early modern methods of analysis

Abstract

Two kinds of analysis were distinguished in early modern geometry: the classical and the algebraic.¹ The former method was known from examples in classical mathematical texts² in which the constructions of problems were preceded by an argument referred to as “analysis;” in those cases the constructions were analysis called “synthesis.” Moreover, a few classical sources³ spoke in general about this arrangement. The most important of these texts was the opening of the seventh book of Pappus’ Collection; I quote this passage here in full:

Now, analysis is the path from what one is seeking, as if it were established, by way of its consequences, to something that is established by synthesis. That is to say, in analysis we assume what is sought as if it has been achieved, and look for the thing from which it follows, and again what comes before that, until by regressing in this way we come upon some one of the things that are already known, or that occupy the rank of a first principle. We call this kind of method “analysis,” as if to say anapalin lysis (reduction backward). In synthesis, by reversal, we assume what was obtained last in the analysis to have been achieved already, and, setting now in natural order, as precedents, what before were following, and fitting them to each other, we attain the end of the construction of what was sought. This is what we call “synthesis.”

Thus in Book V, Ch. IV, pp. 330–343 of [Ghetaldi 1630] Ghetaldi treated a number of problems that he considered beyond the force of algebra and that he therefore solved“ by the method which the ancients used in analysing and synthesizing all problems”(“Methodo, quaveteres in resolvendis et componendis omnibus problematibus utebantur” p. 330).

Notably: Book II of Archimedes’ Sphere and Cylinder together with the commentaries of Eutociuson Propositions II–1 and II–4, and Pappus’ Collection. Furthermore, Euclid’s Data was recognized as a collection of theorems useful in the analysis of plane problems.

Primarily the passages in Pappus’ Collection discussed below; also a scholium to Elements XIII-1-5, which in the Renaissance was attributed toTheon(cf. [Euclid 1956] vol. 3pp. 442–443).In his Isagoge Viète referred to the definition of analysis in the scholium and attributed it toTheon ([Viète 1591] p. 1, [Viète 1983] p. 11).