ABSTRACT
THE ANALYTICAL ART IN THE TENTH TO ELEVENTH CENTURIES
1. The rebirth of a subject Among the many writings that mathematicians dedicated to ‘analysis
and synthesis’ before the mid seventeenth century, that is among those that have come down to us, two undoubtedly stand out: a treatise by Ibrāhīm ibn Sinān (296/909-335/946), called On the Method of Analysis and Synthesis in Geometrical Problems, and a treatise by Ibn al-Haytham also called Analysis and Synthesis, whose text is presented in translation here. In form and in content, both these texts differ from all the other writings we know on the subject. Whereas the Greek philosophers, mathematicians and physicians who discussed the matter, from the fourth century BC onwards, did so only briefly, and have left us only some fragments, Ibrāhīm ibn Sinān and Ibn al-Haytham each composed a substantial work entirely devoted to analysis and synthesis. In fact, the Greek mathematicians who discussed the matter can be counted on the fingers of one hand: there are some lines in pseudo-Euclid,1 a short fragment in Pappus2 and another in Proclus.3 Not that the terms ‘analysis’ and ‘synthesis’ were unknown to Greek mathematicians – Archimedes, Apollonius, Diophantus and others – but none of them felt the need to discuss the meaning of the
132 CHAPTER II: THE ANALYTICAL ART terms. It is one thing, as we know, to employ a procedure, to adopt a certain approach; but it is another, quite different matter to set out the ideas that underlie a subject, as constituting a method or forming something as large as an area of research. One possibility is to follow the example set by Archimedes and do no more than identify the stages of the procedure; a second possibility is to explain briefly what underlies the procedure, and then indicate how it can be used and the conditions for its applicability: this is what Pappus and Proclus do for analysis and synthesis. And Pappus, in fact, in a short text, takes the trouble to describe the approach taken by Euclid, Aristaeus the Elder and Apollonius, to remind us of the direction of reasoning in analysis and synthesis, of their reversibility, and to distinguish between theoretical analysis and problematical analysis, and finally to refer to the conditions for their applicability. Pappus took no more than a page to deal with all these explanations. This immediately accounts for the conflicting interpretations that sprang up around the text of the Alexandrian mathematician.4
