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Now analysis is of two kinds, the one directed to searching for the truth and called theoretical, the other directed to finding what we are told to f… - Pappus of Alexandria
" "Now analysis is of two kinds, the one directed to searching for the truth and called theoretical, the other directed to finding what we are told to find and called problematical. (1) In the theoretical kind we assume what is sought as if it were existent and true, after which we pass through its successive consequences, as if they too were true and established by virtue of our hypothesis, to something admitted: then (a), if that something admitted is true, that which is sought will also be true and the proof will correspond in the reverse order to the analysis, but (b), if we come upon something admittedly false, that which is sought will also be false. (2) In the problematical kind we assume that which is propounded as if it were known, after which we pass through its successive consequences, taking them as true, up to something admitted: if then (a) what is admitted is possible and obtainable, that is, what mathematicians call given, what was originally proposed will also be possible, and the proof will again correspond in reverse order to the analysis, but if (b) we come upon something admittedly impossible, the problem will also be impossible.
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About Pappus of Alexandria
Pappus of Alexandria (c. 290 - c. 350 AD) was one of the last great Greek mathematicians of Antiquity, known for his Synagoge (Συναγωγή) or Collection (c. 340), and for Pappus's hexagon theorem in projective geometry. Nothing is known of his life, except (from his own writings) that he had a son named Hermodorus, and was a teacher in Alexandria. Collection, his best-known work, is a compendium of mathematics in eight volumes, the bulk of which survives. It covers a wide range of topics, including geometry, recreational mathematics, doubling the cube, polygons and polyhedra.
Also Known As
Native Name:
Πάππος ὁ Ἀλεξανδρεύς
Alternative Names:
Pappus
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Pappos of Alexandria
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Additional quotes by Pappus of Alexandria
Analysis... takes that which is sought as if it were admitted and passes from it through its successive consequences to something which is admitted as the result of synthesis: for in analysis we assume that which is sought as if it were (already) done (ɣϵɣονός) and we inquire what it is from which this results, and again what is the antecedent cause of the latter, and so on, until by so retracing our steps we come upon something already known or belonging to the class of first principles, and such a method we call analysis as being solution backwards (άνάπαλɩν λὐσɩν).
But in synthesis, reversing the process, we take as already done that which was last arrived at in the analysis and, by arranging in their natural order as consequences what were before antecedents, and successively connecting them one with another, we arrive finally at the construction of what was sought; and this we call synthesis.
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