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PRELIMINARIES As I Discovered to My Embarrassment When It Was Too

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PRELIMINARIES As I Discovered to My Embarrassment When It Was Too CHAPTER ONE PRELIMINARIES As I discovered to my embarrassment when it was too late, I failed to include most of the rich evidence available in the fields of ancient mathematics, both pure and applied, and mathematical astronomy, in my study of the so-called isagogical questions and some further, related issues in ancient commentaries, introduc­ tions, autobibliographies, and similar literature.1 (It should be kept in mind that astrology, not always rigorously distinguished from astronomy in the modern way,2 was viewed as a mathematical subdiscipline.) 3 However this omission-which as far as I know 1 Mansfeld (1994), though I mentioned in passing Theon of Smyrna's Expositio rerum mathematicarum ad legendum Platonem utilium, and discussed at some length Proclus' Commentary on Euclid Elements I and the traditions concerned with Aratus (including Hipparchus). On Proclus on Euclid I have little to add, and on the Aratea nothing. No mathematical or mathematico­ astronomical literature is listed in the apparatus superior of the first pages of the edition of Stephanus by Westerink (1985) or mentioned in Hadot (1990a). Though much has been lost, what has been preserved is impressive, and without doubt I have missed some things. Diophantus has been excluded because he has nothing to offer in our present context. Sm;cinct and very informative (though naturally not up-to-date) overview of ancient authors and modern editions at Devreesse ( 1954) 233-43 (mathematics, mechanics, astronomy), 244-5 (canonics), 252-4 (astrology). Apart from Euclid and Heron of Alexandria the mathematicians and astronomers are not yet available in the TLG. 2 Ptolemy for instance in the introduction to the Apotelesmatica argues that these are equally scientific disciplines concerned with foreknowledge in relation to the heavenly bodies; see below, Ch. IX 2. See further e.g. Lloyd (1987) 43. Yet it is not my intention to include more than a few samples from the vast astrological literature. 3 It is of some interest to quote Simpl. in Phys. 293.11-6 Diels: 'the ancients applied the term 'astrology' to what is now called 'astronomy', because it would seem that the art of fortune-telling had not yet arrived in Greece. Later generations made a terminological distinction, applying the name 'astrono­ my' to the discipline which studies the motions of the heavenly bodies, and giving the specific name 'astrology' to the art which busies itself with the effects of these motions on human destiny' (to til~ acrtpoA.oy{a~ OVOI!<X oi. ~LEV nal..awl. ~~~7tCo totE til~ anotilicr~L<Xttlci\~ Ei~ tou~ "EUTJV<X~, ~ eotKEv, V..SouO"TJ~ £nl. til~ vuv lC<XM\lllEVTJ~ acrtpOVOill<X~ E<pEpov, oi. OE VEcOtEpot OtEMVtE~ tOUVOil<X ti]v JlEV tO.~ lCtV~O"Et~ t&v oupav{rov E1tt0"1C01tOUO"<XV acrtpOVOJll<XV lC<XAOUO"t, ti]v of: 7tEpt tO. anotEI..ou)lEV<X E~ aut&v Otatp\poucrav acrtpol..oy{av io{~ E7tOVOJla1;;oucrt.) 2 CHAPTER ONE has not been noticed by reviewers4-allows me to play Jekyll to my own Hyde, since one of the aims of my earlier study was to try and find antecedents in earlier (even very much earlier) works for the explicit scholastic introductory scheme, the accessus ad auctores as it was called in medieval times, of the late Neoplaton­ ist commentators. As is well known, mathematics and astronomy were taught in the philosophical establishments of late antiquity; names that come to mind are Hypatia, Proclus, Ammonius Hermiae, Marinus of Neapolis, and Simplicius. An investigation of the various kinds of mathematical literature that are involved not only enables one to include the evidence in these fields relating to late antiquity, but also to look for earlier antecedents. As it is, insofar as the isagogical questions are concerned these other traditions (if that is what they may be called) provide a number of excellent parallels to those in the fields of philosophy, belles-lettres, medicine, biblical studies, rhetoric,5 and grammar. The evidence that is available shows that the study and teaching of mathematics, from the Hellenistic period onwards at least, was not an isolated affair but is to be under­ stood as being a part of the same cultural traditions as the study and teaching of these other disciplines. With two exceptions6 the mathematical traditions have not been studied from the vantage point of the present enquiry. I shall attempt to deal with original authors such as the great mathema­ tician Apollonius of Perga (3rd/2nd cent. BCE), and the astrono­ mical works of another great man, the philosophically inclined mathematical polymath Ptolemy of Alxandria (2nd cent. CE), both of whom make use of isagogical questions in an implicit way that is nevertheless unmistakable. Heron of Alexandria (mid-1st cent. CE) was a prolific and technically very competent author in several fields of applied mathematics, and an author of introduc­ tory treatises;7 in these capacities he, too, raises isagogical issues. 4 Chiaradonna (1997) in his review points out important passages in Plo­ tinus and Porphyry which had escaped me, and so corrects another mistake by clarifying the position of the latter. 5 Rabe's Prolegomenon Sylloge with its important introduction has been reprinted in 1995. See forther below, p. 122, complementary note 5. 6 Schissel von Fleschenberg (1930), though to a certain extent only, see below, nn. 202 and 250; Mogenet (1956) is almost entirely correct, see below, Ch. X 3. 7 For another work, viz. his Commentary, or comments, on Euclid's Elements see below, Ch. III 1. .
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