Proofs ( - 499 AD)

"It is evidently equally foolish to accept probable reasoning from a mathematician and to demand from a rhetorician demonstrative proofs." (Aristotle, cca. 4th century BC)

"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." (Eudoxus, cca. 4th century BC)

"Analysis is a method where one assumes that which is sought, and from this, through a series of implications, arrives at something which is agreed upon on the basis of synthesis; because in analysis, one assumes that which is sought to be known, proved, or constructed, and examines what this is a consequence of and from what this latter follows, so that by backtracking we end up with something that is already known or is part of the starting points of the theory; we call such a method analysis; it is, in a sense, a solution in reversed direction. In synthesis we work in the opposite direction: we assume the last result of the analysis to be true. Then we put the causes from analysis in their natural order, as consequences, and by putting these together we obtain the proof or the construction of that which is sought. We call this synthesis." (Pappus of Alexandria, cca. 4th century BC)

"In analysis, we start from what is required, we take it for granted, and we draw consequences from it, and consequences from the consequences, till we reach a point that we can use as starting point in synthesis. For in analysis we assume what is required to be done as already done (what is sought as already found, what we have to prove as true). We inquire from what antecedent the desired result could be derived; then we inquire again what could be the antecedent of that antecedent, and so on, until passing from antecedent to antecedent, we come eventually upon something already known or admittedly true. This procedure we call analysis, or solution backwards, or regressive reasoning." (Pappus of Alexandria, cca. 4th century BC)

"Now analysis is of two kinds; the one is the analysis of the ‘problems to prove’ and aims at establishing true theorems; the other is the analysis of the ‘problems to find’ and aims at finding the unknown."  (Pappus of Alexandria, cca. 4th century BC)

"It is of course easier to supply the proof when we have previously acquired some knowledge of the questions by the method, than it is to find it without any previous knowledge." (Archimedes, cca. 3rd century BC)

"It is not possible to find in all geometry more difficult and more intricate questions or more simple and lucid explanations [than those given by Archimedes]. Some ascribe this to his natural genius; while others think that incredible effort and toil produced these, to all appearance, easy and unlaboured results. No amount of investigation of yours would succeed in attaining the proof, and yet, once seen, you immediately believe you would have discovered it; by so smooth and so rapid a path he leads you to the conclusion required." (Plutarch, cca. 1st century)