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Theo Smyrnaeus on arithmetic Macadam, Joseph Duncan 1969

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THEO SMYRNAEUS ON ARITHMETIC by JOSEPH DUNCAN MACADAM B.A., London U n i v e r s i t y , 1948 B.Ed., U n i v e r s i t y o f B r i t i s h Columbia, 1958 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n t he Department of CLASSICS We accept t h i s t h e s i s as conforming t o the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1969 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h C olumbia, I a g r e e t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and Study. I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u rposes may be g r a n t e d by the Head o f my Department or by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f CLASSICS The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date 25th August 1969 ABSTRACT The purpose of t h i s t h e s i s i s t o examine the a r i t h -m e t i c a l p o r t i o n of Theon of Smyrna's work e n t i t l e d : TCC Kara TO ya6r|p,aTiKOV xpno'iu.a ei<j rrjv IIAcrrajvoc, ava^fvaxJiv and to determine i t s s i g n i f i c a n c e i n the study of the t h e o r y of numbers. The t h e s i s comprises three main p a r t s . The f i r s t i s a b r i e f i n t r o d u c t o r y d i s c u s s i o n of the biography of Theon w i t h an attempt t o e s t a b l i s h h i s i d e n t i t y and works. Very l i t t l e s c h o l a r l y work has been devoted t o Theon; what l i t t l e c o u l d be found was dated f o r the most p a r t i n the second h a l f of t h e n i n e t e e n t h century, and i n Greek mathematical works h i s a c t i v i t i e s have a t t r a c t e d l i t t l e more than a pas-s i n g mention. I n my i n t r o d u c t o r y chapter I have drawn e x c l u s -i v e l y from t h i s secondary m a t e r i a l . The second p a r t of the study i s a l i t e r a l t r a n s l a t i o n of the a p p r o p r i a t e a r i t h m e t i c a l s e c t i o n of the work. The t h i r d p a r t c o n s i s t s of a commentary amounting t o a simple e x p o s i t i o n of the mathematical content. In a c o n c l u d i n g c hapter I have attempted to a s s i g n to Theon h i s p l a c e i n the h i s t o r y of a r i t h m e t i c and have g i v e n some i n d i c a t i o n s of the reasons f o r h i s r e l a t i v e unimportance. i v ABBREVIATIONS Dupuis : Theon de Smyrne, ph i l o s o p h e P l a t o n i c i e n . E x p o s i t i o n  des connalssances mathematiques u t i l e s pour l a l e c t -ure de P l a t o n . par J . Dupuis. Heath, H.G.M. : S i r Thomas L. Heath, A H i s t o r y of Greek  Mathematics. H i l l e r : Theoni3 Smyrnaei, p h i l o s o p h i P l a t o n i c ! E x p o s i t i o rerum mathematicarum ad legendum Platonem u t i l l u m . r e c e n s u i t Eduardus H i l l e r . O.C.D. : The Oxford C l a s s i c a l D i c t i o n a r y . PW : Pauly-Wissowa, R e a l e n c y c l o p a e d i e der k l a s s i s c h e n A l t e r t u m s w l s s e n s c h a f t . TABLE OF CONTENTS Page ABBREVIATIONS . i v CHAPTERS I. INTRODUCTION 1 I I . ON ARITHMETIC: TEXT AND APPARATUS . . . 21 I I I . ON ARITHMETIC: TRANSLATION 69 IV. MATHEMATICAL COMMENTARY . 121 V. CONCLUSION 146 BIBLIOGRAPHY 155 ACKNOWLEG DMENT I would l i k e t o express my thanks t o P r o f e s s o r E. Bongie and P r o f e s s o r H.A.Thurston f o r t h e i r h e l p f u l s u g g estions, t o Mr H.G.Radunz and Dr G.A.Lieben of Vancouver C i t y C o l l e g e f o r t h e i r a i d w i t h German t r a n s l a t i o n , and t o P r o f e s s o r J . R u s s e l l f o r h i s c o n s i d e r a t e guidance and a s s i s t -ance throughout. CHAPTER ONE INTRODUCTION I CHRONOLOGY AND IDENTITY We have v e r y l i t t l e p r e c i s e i n f o r m a t i o n about the date and i d e n t i t y of Theon o f Smyrna, P l a t o n i c p h i l o s o p h e r and astronomer,^ but some c l u e s may be e l i c i t e d from c e r t a i n a u t h o r i t i e s he mentions i n the course of h i s w r i t i n g s . On 2 f i v e o c c a s i o n s , f o r i n s t a n c e , he quotes T h r a s y l l u s , the c o u r t astronomer of T i b e r i u s , w h i l e he a l s o makes e x t e n s i v e use of the P e r i p a t e t i c p h i l o s o p h e r Adrastus i n the second p o r t i o n of h i s work, quoting him on a v a r i e t y of p o i n t s i n music and astronomy. On the oth e r hand, h i s f a i l u r e t o men-F r i t z , i n PW s.v. "Theon" 14), VA2, 2067, 1 8 f f . 2 H i l l e r , 47.17, 85.8, 93.8, 205.5. T h r a s y l l u s (d. 36 A.D.) made T i b e r i u s ' acquaintance i n Rhodes and remained i n c l o s e c o n t a c t w i t h him u n t i l h i s death. He wrote two s e r i o u s works on astronomy, and was a l s o r e s p o n s i b l e w i t h D e r c y l l i d e s f o r the d i v i s i o n o f P l a t o ' s works i n t o t e t r a l o g i e s (PW .s.v". " T h r a s y l l u s " ) . 3 Nothing very c e r t a i n i s known about Adrastus of Aph-r o d i s i a s . He was a P e r i p a t e t i c of t h e middle of the second century A.D., a tea c h e r o f L u c i u s Yerus, the c o l l e a g u e of Marcus A u r e l i u s i n the p r i n c i p a t e . He i s mentioned by Galen t o g e t h e r w i t h A s p a s i u s , another P e r i p a t e t i c dated i n the f i r s t h a l f of t h e second c e n t u r y . (PW rs.v. " A d r a s t u s " ) . 2 t i o n the Almagest o f Ptolemy — a strange omission f o r anyone w r i t i n g on an a s t r o n o m i c a l t o p i c a f t e r the date of i t s p u b l i -c a t i o n — s u g g e s t s t h a t Theon's a s t r o n o m i c a l t r e a t i s e at l e a s t antedates t h a t famous work. T h i s argumentum ex s l l e n t i o , moreover, appears t o f i n d some c o n f i r m a t i o n i n the d i s c o v e r y of a b u s t 5 of Theon of Smyrna, found i n Smyrna b e a r i n g the i n s c r i p t i o n eecova HXaTa>vtKOV qpiXocrotpov 6 tepeu<; 6ea)v T O V 7tarepa and c l e a r l y , dated by s t y l e t o the r e i g n of Hadrian (117 - 138 A.D.). Any attempt t o be more s p e c i f i c , however, encounters d i f f i c u l t i e s . The f i r s t c o m p l i c a t i o n a r i s e s i n Ptolemy's mention of , ft a Theon, whom he terms pa0TiuaTiKO<; i n the Almagest i n r e f e r -ence t o some a s t r o n o m i c a l o b s e r v a t i o n s made i n the years 127, 129, 130 and 132 A.D. Theon of A l e x a n d r i a , 7 i n h i s commentary on Ptolemy, r e f e r s t o t h i s Theon as T O V 7raXaiov 6ea>va on C l a u d i u s Ptolemaeus has a f l o r u i t ^between 121 an,d 151 A.jD. H i s g r e a t e s t work, the uaefiytaTiKf} cuvTa£i<; or iieYaXq a u v r a ^ i ? , termed a l Ma.jlsti by the Arabs (and hence Almagest) has no f i r m date. 5 T h i s bust, made out of one p i e c e t o g e t h e r w i t h i t s p e d e s t a l , i s now kept i n the " H a l l of P h i l o s o p h e r s " i n the C a p i t o l i n e Museum (N.25). F o r photographs see The Museo C a p i t o l i n o , (Oxford 1912) P l a t e 57; B e r n o u i l l i , Grjbech. Ikonographie, I I , P l a t e 29; Sc h u s t e r , Uber d i e E r h a l t e n e n  P o r t r a t s g r l e e h . P h l l o s o p h e n , P l a t e 2, 6; G i s e l a M. A. R i c h t e r , The P o r t r a i t s o f the Greeks. (London 1965), I I I , p.285, f i g . 2038. 6Almagest. IX, 9; X, 1, 2. 7PW, l o c . c i t . , 2067, 50. Theon of A l e x a n d r i a l i v e d towards the end of the f o u r t h c e n t u r y A.D. and wrote a comm-entary on Ptolemy's Syntax!s i n eleven books i n a d d i t i o n t o e d i t i n g E u c l i d ' s Elements (Heath. H.G.M., i i , 526-7). 3 s e v e r a l o c c a s i o n s , but i n one i n s t a n c e he adds the words "the mathematician". T h i s r a i s e s the awkward q u e s t i o n of i d e n t -i f y i n g the Theons. I s Theon the mathematician t o be i d e n t -i f i e d w i t h Theon of Smyrna (perhaps T O V 7raXatov 6ecova) or are they d i s t i n c t persons? M a r t i n , i n h i s e d i t i o n of the second ( a s t r o n o m i c a l ) p a r t of Theon of Smyrna's work^ argues a g a i n s t the two Theons*' having a common i d e n t i t y . He p o i n t s out that Theon of Smyrna g i v e s the g r e a t e s t angular d i s t a n c e of Mercury from t h e sun as 20°, a f i g u r e t h a t agrees w i t h t h a t of Cleomedes, whereas Ptolemy, q u o t i n g Theon"the mathematician", g i v e s 26°15', a much more a c c u r a t e f i g u r e .'L0 The e x i s t e n c e of two d i f f e r e n t o b s e r v a t i o n s each championed by an a u t h o r i t y named Theon c e r t -a i n l y suggests t h a t the Theons concerned were d i f f e r e n t persons. Nonetheless, an a l t e r n a t i v e e x p l a n a t i o n may w e l l p r o-v i d e the grounds f o r c o n s i d e r i n g the two Theons mentioned t o be i n f a c t the same person. To p o s t u l a t e a common i d e n t i t y f o r the Theon of Smyrna and h i s mathematical namesake one must assume t h a t h i s preserved work had been w r i t t e n and p u b l i s h e d before the date of the a s t r o n o m i c a l o b s e r v a t i o n s mentioned by Ptolemy, and so has r e t a i n e d t h e e a r l i e r and l e s s a c c u r a t e Q T.H.Martin, Theonis P l a t o n i c i l i b e r de astronomia ( P a r i s , 1849) p. 9. 9Cleomedes ( p r o b a b l y ca. 150 - 200 A . D . a s t r o n o m e r , was an author of t h e p o p u l a r work KUHXIHTI Qecopia u-eTecopcov (De Motu C i r c u l a r ! Corporum Caelestium) l a r g e l y founded upon P o s i d o n i u s (O.C.D. s.v. "Cleomedes"). 1 0 P t o l e m y , Almagest, IX, 9. 4 f i g u r e . F u r t h e r grounds f o r t h i s t h e o r y of a common i d e n t i t y may be sought i n the i n t e r n a l c o n t r a d i c t i o n s w i t h i n the work i t s e l f . "^^The w r i t e r indeed, d e s p i t e h i s o r i g i n a l c l a i m s , seems more e n t h u s i a s t i c about the P l a t o n i c i d e a l s of e d u c a t i o n than about the fundamentals of mathematics, a study i n which he appears t o be too s i n g u l a r l y i n e p t and u n r e l i a b l e to be con-s i d e r e d a t r u e mathematician. On the other hand, i t i s gener-a l l y agreed t h a t t h e a s t r o n o m i c a l p a r t of the work i s f a r b e t -12 t e r t h an the mathematical p a r t , a q u a l i t y amply emphasised by the b e a u t i f u l l y simple proof he advances to supersede the f a u l t y proof of A d r a s t u s , showing t h a t the e p i c y c l i c movement of the i n n e r p l a n e t s must n e c e s s a r i l y be c o n s i d e r e d e c c e n t r i c . With t h i s i n mind, i t i s p o s s i b l e t o argue t h a t Theon may have undergone a change of i n t e r e s t s , beginning from p h i l o s o p h y and ending w i t h exact mathematics and astronomy. Such a change, i f c ontinued a f t e r the p u b l i c a t i o n of the extant work, would at l e a s t account f o r t h e d u a l c h a r a c t e r of Theon and r e c o n c i l e the v a r i a n t a s t r o n o m i c a l o b s e r v a t i o n s . As f o r the Theons mentioned by Ptolemy and Theon of A l e x a n d r i a , one may j u s t i f i a b l y suppose i n the absence of any more s p e c i f i c d i s t i n g u i s h i n g f a c t o r s t h a t they a l l r e f e r t o the same person and t h a t t h a t person i s Theon of Smyrna. 1:LPW, l o e . c i t . , 2068, 5. 1 2PW, l o c . . c i t . , 2068, 9. 5 I f t h i s h y p o thesis i s c o r r e c t i t may be u s e f u l t o con-s i d e r a more s p e c i f i c date f o r Theon of Smyrna w i t h i n the per-i o d a l r e a d y proposed. The f a i l u r e t o take account of Ptolemy's a s t r o n o m i c a l w r i t i n g s i n h i s extant work suggests a date before the e a r l i e s t p o s s i b l e date of p u b l i c a t i o n f o r the Almagest (no l a t e r than 127 A.D.). The more ac c u r a t e a s t r o n o m i c a l f i g u r e s a t t r i b u t e d by Ptolemy to Theon "the mathematician" would of n e c e s s i t y date t o a l a t e r p e r i o d of Theon of Smyrna's c a r e e r , perhaps i n d i c a t i n g c ontinued a c t i v i t y f o r a f u r t h e r decade or two. There i s , however, one s e r i o u s o b s t a c l e to an e a r l y dat-i n g f o r t h e extant t r e a t i s e , v i z . the f r e q u e n t r e f e r e n c e s t o Adrastus i n the work. U n f o r t u n a t e l y the l i t t l e t h a t i s known of Adrastus seems t o i n d i c a t e a date i n the middle of the sec-ond century A.D. T h e r e f o r e , i n the f a c e of t h i s , a f l o r u i t date f o r Theon of Smyrna ca. 125 - 150 A.D. i s as c l o s e as c o n f l i c t i n g evidence p e r m i t s . II' LOST WORKS There are two works not now extant that have been a s s i g n e d by t r a d i t i o n t o Theon of Smyrna. 1. A commentary on P l a t O c ' s R e p u b l i c , u7rop,vT]iaaTa TT)C, TTOX-i r e t ' a q , quoted by Theon h i m s e l f i n h i s x p T i c i u a , 1 3 but not mentioned elsewhere. 2. A second work d e a l i n g w i t h t h e order i n which one should read P l a t o ' s works, d i s c u s s i n g t h e i r t i t l e s , and known o n l y 1 3 H i l l e r , 146.4 6 from A r a b i c t r a d i t i o n . I n the l a t t e r work Theon presumes a t e t r a l o g i c a l order 14 f o r P l a t o ' s w r i t i n g s , a c c o r d i n g to a c e r t a i n an-Nadim, an e i g h t h c e n t u r y A r a b i a n w r i t e r . But the catalogue of w r i t i n g s he l i s t s as Theon's bears l i t t l e resemblance t o the c a n o n i c a l 15 t e t r a l o g i e s g i v e n by T h r a s y l l u s . T h i s may be c o n s i d e r e d a strange anomaly i n view of the s t r o n g dependence of Theon 16 upon T h r a s y l l u s elsewhere i n h i s work. L i p p e r t b e l i e v e s t h a t Theon i n h i s work f i r s t l i s t e d the d i a l o g u e s of P l a t o i n a n o n - T h r a s y l l i a n catalogue and subsequently re - o r d e r e d them 17 mainly a c c o r d i n g t o T h r a s y l l u s . A f u r t h e r p o s s i b l e r e f e r e n c e should be mentioned t h a t may have some b e a r i n g on one or mother of these l o s t works of 18 Theon. The i s s u e a r i s e s from a d i s c u s s i o n by P r o c l u s of 19 P l a t o ' s genealogy. I n h i s Commentary on P l a t o ' s Timaeus 1 4 A b i Ja'kub an-Nadim, K i t a b a l - F l h r l s t . L e i p z i g , F.C. -W.Vogel, 1871-2, 255, 12. 1 5an-Nadim, op. c i t . . 246, 4 f f . 1 6PW, l o c . c i t . , 2069, 24. 17 J . L i p p e r t , S t u d i e n auf dem Gebiet der A r a b i s c h e n Uber-s e t z u n g s l i t e r a t u r , Braunschweig, 1894, 2, 4 5 f f . The s l i g h t d i v e r g e n c i e s from T h r a s y l l u s ' o r d e r i n g are a l s o found i n anoth-er Greek o r d e r i n g , t h a t of Diogenes L a e r t i u s (Diog. L a e r t . , I l l , 62.). 18 P r o c l u s (410-485 A.D.) was a competent and i n d u s t r i o u s mathematician and t e a c h e r of the N e o p l a t o n i c s c h o o l , even a poet. He wrote a famous commentary on E u c l i d (Book 1) and a l s o commentaries on t h e R e p u b l i c , the Timaeus and o t h e r d i a l o g u e s of P l a t o . (Heath, H.G.M. i i , 529-37] 19 P r o c l u s , Comm. on P l a t o ' s Timaeus, 26A. 7 P r o c l u s r e p o r t s t h a t Theon c o n s i d e r e d the ( o l d e r ) Glaucon t o be the son of C a l l a i s c h r u s and the b r o t h e r of the famous C r i t -i a s , w h i l e he h i m s e l f c o n s i d e r s him to be a brother of C a l l -a i s c h r u s and the son of the o l d e r C r i t i a s . T h i s i n f o r m a t i o n undoubtedly belongs t o an e x t e n s i v e genealogy o f P l a t o which, on the s i d e of the f a t h e r , reaches back t o the kings Codrus and Melanthus and, on the s i d e of the mother, to Solon. H i l l -e r a s signs the genealogy t o Theon's Commentary on the R e p u b l i c , whereas L i p p e r t c o n s i d e r s t h a t i t must be a s s i g n e d t o the t r e a t -20 i s e upon the s e r i a l sequence of the P l a t o n i c w r i t i n g s . But as no c o n f i r m a t i o n of t h i s appears i n Nadim, s o l e a u t h o r -i t y f o r the l a t t e r work a t t r i b u t e d to Theon, i t i s not poss-i b l e t o a s s i g n t h i s i n f o r m a t i o n w i t h any c e r t a i n t y t o any s p e c i f i c work. I l l EXTANT WORKS i ) The xpnoiya of Theon The s o l e extant work of Theon of Smyrna i s e n t i t l e d : to. Kara TO u.a0T])iaTiHOV xPncip - a TT)V nXaTcuvo^ avdyvcoaiv. T h i s work i s a somewhat c u r i o u s medley, v a l u a b l e not p a r t i c u -l a r l y f o r i t s own i n t r i n s i c worth as a s c i e n t i f i c t r e a t i s e , but r a t h e r f o r the l a r g e number of h i s t o r i c a l n o t i c e s i t con-21 t a i n s c h i e f l y i n the second p a r t . The t i t l e w i t h i t s c l a i m 20 L i p p e r t , l o c . c i t . 2 1 H e a t h , H.G-.M. , i i , 243. 8 t o p r e s e n t the mathematics u s e f u l f o r the r e a d i n g of P l a t o i s indeed a l i t t l e too p r e t e n t i o u s . I t was no doubt an element-ary i n t r o d u c t i o n or vade mecum f o r students of p h i l o s o p h y , but l i t t l e i n i t has p a r t i c u l a r r e f e r e n c e t o the mathematical que s t i o n s posed by P l a t o . There i s a l o n g i n t r o d u c t i o n p o i n t i n g out the para-mount importence of mathematics i n the t r a i n i n g of the p h i l o -sopher and demonstrating the r e l a t i o n s h i p between the f i v e branches of mathematics, a r i t h m e t i c , geometry, stereometry (or s o l i d geometry), astronomy and music. Theon promises t o present the mathematical theorems i n a r i t h m e t i c , music and geometry and the a p p l i c a t i o n s of s t e r e o -metry and astronomy most necessary f o r students of P l a t o , but the promise i s by no means f u l f i l l e d as regards geometry and stereometry. The work i n e f f e c t comprises two separate h a l v e s : P a r t 1 a) I n t r o d u c t i o n ( H i l l e r 1. 1 - 16.24) b) A r i t h m e t i c ( H i l l e r 16.24 - 46.19) c) Music i ) i n s t r u m e n t a l ( H i l l e r 46.20 - 72.20) i i ) based on numbers ( H i l l e r 72.21 - 119.21) P a r t 2 Astronomy ( H i l l e r 120. 1 - 205. 6) i i ) Summary of content of Theon's x p T l c t i i a a) A r i t h m e t i c T h i s s e c t i o n comprises b r o a d l y what i s termed Pythagorean a r i t h m e t i c . I t de a l s w i t h the c l a s s i f i c -a t i o n of numbers i n t o odd and even and t h e i r s u b d i v i s i o n f u r -t h e r i n t o prime numbers, composite numbers with equal and un-equal f a c t o r s , i n v o k i n g a d i s t i n c t i o n between those w i t h a 9 d i f f e r e n c e of 1, and of 2 and more than 2. P l a n e numbers are d i v i d e d i n t o square, oblong, t r i -a n gular and p o l y g o n a l numbers, w i t h t h e i r r e s p e c t i v e "gnomons" and t h e i r p r o p e r t i e s as the sums of s u c c e s s i v e terms of a r i t h -m e t i c a l p r o g r e s s i o n s beginning w i t h 1 as the f i r s t term. C i r c u l a r numbers are d e f i n e d , as are s p h e r i c a l numbers, s o l i d numbers w i t h t h r e e f a c t o r s , pyramidal numbers and t r u n c a t e d pyramidal numbers, p e r f e c t numbers and the o v e r - p e r f e c t and d e f e c t i v e k i n d s . A l l t h i s i s found i n the I n t r o d u c t i o A r i t h -m e t i c a of Nicomachus of G-erasa ( f l o r u i t ca. 100 A.D.) where i t 22 i s t r e a t e d i n s t i l l g r e a t e r d e t a i l . Of p a r t i c u l a r i n t e r e s t however, i s Theon's e x p o s i t i o n of s i d e and d i a g o n a l numbers, a treatment which does not have an e q u i v a l e n t i n Nicomachus, f o r i t demonstrates without p r o d u c i n g a proof a method f o r f i n d i n g an approximation of the v a l u e of VIT t o any r e q u i r e d de-gree of accuracy. b) Music Theon f i r s t d eals w i t h i n s t r u m e n t a l music (p,oucriHfji ev opyavoic,) i . e . , the t e a c h i n g of the sounds and the i n t e r v a l s i n music. The i n t e r v a l s which provide harmony, notes and s c a l e s , the tones and semitones and the octave are d i s c u s s e d , and the w r i t e r f u r n i s h e s s u b s t a n t i a l q u o t a t i o n s from T h r a s y l l u s and Adrastus, and i n a d d i t i o n r e f e r s t o the views of A r i s t o x e n u s , Hippasus, A r c h y t a s , Eudoxus and P l a t o . 2 2Nicomachus, I n t r o d u c t i o A r i t h m e t i o a , Ricardus Hoche, L e i p z i g , 1866. A f u l l e r d i s c u s s i o n of Nicomachus and the mer-i t s . o f ^ t n i s work appears i n Chapter 5 (p.149). 2 3 H i l l e r , 46.20 - 72.20 10 The second s e c t i o n , music based on numbers, (UOUCUHT] 24 ev api6u.oTc,), developes i n t o a g e n e r a l d i s c u s s i o n of r a t i o s , p r o p o r t i o n s and means. There are f u r t h e r q u o t a t i o n s from P l a t o , f o l l o w e d by a p r e s e n t a t i o n of T h r a s y l l u s ' d l v l s l o oanonls. Although Theon promises t o apply t h i s m u s i c a l a n a l y s i s l a t e r t o the "harmony of the u n i v e r s e " , he proceeds without apology t o a d i s c u s s i o n of the benac, (decad) and the TeTpaxTuc, (quat-ernary) w i t h i t s e l e v e n a p p l i c a t i o n s and the mystic and c u r -ious p r o p e r t i e s of t h e numbers from 2 t o 10, p a r t of the 25 theologumena of a r i t h m e t i c . c) Astronomy The most voluminous p a r t of Theon's work i s t h a t d e a l i n g with, astronomy. Here Theon i s mainly depend-ent on the work of Adrastus whom he quotes r e g a r d i n g the spher-i c i t y of the e a r t h . By u s i n g E r a t o s t h e n e s ' f i g u r e of 252,000 stades f o r the circumference of the e a r t h and the Archimedean value of 22/7 f o r ix he ob t a i n s a f i g u r e of 80,182 stades f o r i t s diameter. He then d e s c r i b e s the p r i n c i p a l a s t r o n o m i c a l d i v i s i o n s of the heavens and g i v e s the maximum d e v i a t i o n s i n l a t i t u d e of the p l a n e t s and the sun. T h i s i s f o l l o w e d by an explan-a t i o n of the o r b i t s of the sun, moon and p l a n e t s on the hypoth-e s i s of a g e o c e n t r i c u n i v e r s e . He observes t h a t "some of the Pythagoreans" made the o r d e r , . r e c k o n i n g outward from the e a r t h , 2 4 f l i l l e r , 72.21 - 119.21 2^Theologumena A r i t h m e t i c a e . A c c e d l t Nicomaohi G e r a s e n i  I n s t l t u t i o a r i t h m e t i c s , ed. As t , L e i p z i g , 1817. Heath b e l i e v e s t h i s work i s c e r t a i n l y not by Nicomachus (H.G.M.. 1, 97). 11 to be: the moon, Mercury, Venus, the sun, Mars, J u p i t e r and S a t u r n . Now t h i s i s the "Chaldaean" order not found i n Greece u n t i l the second c e n t u r y B.C. so t h a t "some of the Pythagor-eans" w i l l r e f e r t o the l a t e r Pythagoreans, of whom Nicomachus was one. P l a t o and the e a r l i e r Pythagoreans p l a c e d the sun next t o the moon and interchanged the p o s i t i o n s - of Venus and Mercury. Then a s s i g n i n g t o each of these p l a n e t s , t o the e a r t h , and t o the sphere of the f i x e d s t a r s , one note each he proceeds t o arrange the nine t o form an octave by comparing the d i s t -ances s e p a r a t i n g them to m u s i c a l i n t e r v a l s . Thus by t h e i r r o t a t i o n he demonstrates t h a t the "symphony of the s t a r s " i s produced. A c c o r d i n g to Heath, the whole of t h i s passage i s p o s s i b l y intended as the promised account of the "harmony of t h e u n i v e r s e " , although at the end of the work Theon i m p l i e s t h a t he has s t i l l t o g i v e a b r i e f account of what he and T h r a s -27 y l l u s t h i n k of t h i s s u b j e c t . Next Theon d e a l s with the movements and apparent s t a t -i o n a r y p o i n t s and r e t r o g r a d a t i o n s of the f i v e p l a n e t s , and w i t h the "saving of the phenomena" by the hypotheses of e c c e n t -r i c c i r c l e s and e p i c y c l e s . There f o l l o w s an a l l u s i o n t o the 28 29 systems of Eudoxus, C a l l i p p u s , and A r i s t o t l e and a d e s c r i p -2 6 H e a t h , H.G.M., i i , 2 4 3 . c H i l l e r , 2 0 5 . 28 Eudoxus ( 4 0 8 - 3 5 5 B.C.), a mathematician of the h i g h -e s t rank; to C i c e r o '_in a s t r o l o g i a i u d i c i o doctisslmorum homin-um f a c i l e p r i n c e p s U He was famous f o r h i s Ingenious " t h e o r y of c o n c e n t r i c spheres" and h i s b r i l l i a n t use of "the method of exhaustion." t o o b t a i n the volume of a pyramid. 12 t i o n of a system i n which " c a r r y i n g " spheres, termed "hollow" have " s o l i d " spheres attached t o them which, by t h e i r motion r o l l t he " c a r r y i n g " spheres i n the opposite d i r e c t i o n . The f i g u r e s of 20° and 50° g i v e n by Theon f o r the maximum ar c s 31 s e p a r a t i n g Venus and Mercury agree w i t h Cleomedes' observ-a t i o n s . The f i n a l pages, quoted from A d r a s t u s , d e a l w i t h con-j u n c t i o n s , t r a n s i t s , o c c u l t a t i o n s and e c l i p s e s . The work con-32 eludes w i t h a c o n s i d e r a b l e e x t r a c t from D e r c y l l i d e s , a P l a t -o n i s t w i t h Pythagorean l e a n i n g s . i i i ) Theon's "harmony of the u n i v e r s e " The manuscript has come down t o us i n two separate h a l v e s and i n view of Theon's vague and at times c o n t r a d i c t -o r y statements r e g a r d i n g the scope of the work we may w e l l wonder whether the work as we have i t i s complete. 33 R i g h t at the begi n n i n g Theon d e f i n e s the scope of h i s work t o be a d e s c r i p t i o n of a r i t h m e t i c , music, geometry, 29 C a l l i p p u s of Gyzicus ( c a . 370-300 B.C.) was perhaps the g r e a t e s t astronomer of h i s time. He c o r r e c t e d and added t o Eudoxus* t h e o r y of c o n c e n t r i c spheres designed t o account f o r the movements of the sun, moon and the p l a n e t s ( S i m p l i c i u s , on A r i s t . De Caelo. 493, 5-8, Heib.) 3 0 H i l l e r , 181.12ff. 3 1 H i l l e r , 187.10-13 32 See Note 2. D e r c y l l i d e s ' date i s d i f f i c u l t ; he wrote a book on P l a t o ' s p h i l o s o p h y , a c c o r d i n g t o Heath before T i b e r -i u s o r perhaps even before V a r r o (116-27 B.C.), (H.G.M. i i , 244) 33 H i l l e r , 1.15 13 stereometry end music i n so f a r as they are important f o r the understanding o f the P l a t o n i c w r i t i n g s , an e n t e r p r i s e which he t r i e s t o j u s t i f y by quoting numerous passages from P l a t o t h a t emphasise the importance of mathematics i n the p u r s u i t of ph i l o s o p h y . Then, as i f t o l i m i t the scope of h i s work, he ple a d s t h a t i t i s not r e a l l y h i s o b j e c t i v e t o make p e r f e c t mathematicians. P l a t o d i d not teach t h a t one should waste time upon mathematics u n t i l one's o l d age; r a t h e r should math-ematics be c o n s i d e r e d a p r e p a r a t o r y t r a i n i n g f o r phi l o s o p h y . Furthermore, t o compound h i s i n c o n s i s t e n c y he c o n s i d e r s i t des-i r a b l e t h a t h i s readers should have some elementary knowledge of geometry a l r e a d y , i n order t o make the understanding of h i s d i s s e r t a t i o n the e a s i e r ; y e t h i s s t y l e and e x p o s i t i o n are to be such t h a t even a mind completely u n t r a i n e d i n mathematics 34 w i l l understand him. There would appear t o be from the very outset some i n c o n s i s t e n c y o f o b j e c t i v e s here. On s p e c i f i c i s s u e s too, Theon may be c o n v i c t e d of c o n t r a d i c t i o n . F o r i n s t a n c e , he s t a t e s t h a t he i n t e n d s t o des-c r i b e a r i t h m e t i c and t h a t t h i s w i l l be f o l l o w e d o n l y by "music 1 based on numbers"; " i n s t r u m e n t a l music" need not be presented 35 at a l l s i n c e P l a t o h i m s e l f d e c l a r e s t h i s t o be s u p e r f l u o u s . Yet he c o n t r a d i c t s h i m s e l f i n t h i s r e s o l v e f o r , at the c o n c l -u s i o n of the s e c t i o n on a r i t h m e t i c he now s t a t e s i n h i s i n t r o -3 4 H i l l e r , 16.21 3 5 f i i l l e r , 1 6 . 2 6 . P l a t o p r e f e r r e d t h e o r e t i c a l music (see Note 15,p. 74.) 14 d u c t i o n to the second p a r t , t h a t he does wish a f t e r a l l t o speak about pouainf) ev o p y a v o i c ; a g w e l l as youCTinfi ev.dpiOuotc, and at the end w i l l d e a l w i t h yet a t h i r d category, dpp,ovict ^ 36 ev Hoap*. And w h i l e the former two are d u l y presented as promised, the t h i r d , "harmony of the u n i v e r s e " , i s apparent-l y m i s s i n g . Now i f we examine h i s i n t r o d u c t o r y remarks t o t h e s e c t -37 i o n on a r i t h m e t i c , Theon appears t o take great pains t o j u s t -i f y h i s i n t e n t i o n of t r e a t i n g "music based on numbers" next 38 a f t e r a r i t h m e t i c , " f o r i n t h e n a t u r a l o r d e r should have come geometry f o l l o w e d by stereometry and astronomy w i t h music f i f t h " . Then he p o i n t s out t h a t P l a t o ' s music was the "harm-ony of the u n i v e r s e " and cannot be understood u n l e s s one 39 f i r s t l e a r n s "music based on numbers"; wherefore, he i s p r e -pared t o take the d r a s t i c s t e p of p r e s e n t i n g i t next a f t e r a r i t h m e t i c . Yet the most cogent reason f o r "music based on numbers" to f o l l o w a r i t h m e t i c — t h a t i t i s c l o s e l y a l l i f e d t o the study of numbers "pure and s i m p l e " — i s added as an a f t e r -40 thought. The r e a s o n i n g i s a l t o g e t h e r vague and unconvinc-i n g . f 2 C H i l l e r , 47.6 H i l l e r , 17.14 38 The " n a t u r a l " o r d e r presumably a c c o r d i n g t o P l a t o , who g i v e s t h i s o r d e r f o r the branches of mathematics i n Rep., 522C, 526C, 527D, 528B and 530D. 3 9 H i l l e r , 17.8 4 < ? H i l l e r , 17.11 15 Now d e s p i t e a l l t h i s e l a b o r a t e p r e s e n t a t i o n of ex-cuses f o r d e p a r t i n g from P l a t o ' s r e c o g n i s e d order of p r i o r i t y i n t h e branches of mathematics, what we a c t u a l l y f i n d a f t e r t h e treatment of a r i t h m e t i c i s a d i g r e s s i o n on " i n s t r u m e n t a l music?' a f t e r a l l , a t o p i c t h a t Theon had a p p a r e n t l y a l r e a d y 41 decided t o r e j e c t on P l a t o ' s a u t h o r i t y . And moreover "music based on numbers", when i t e v e n t u a l l y does appear, i s a con-f u s i n g complex of q u i t e u n r e l a t e d t o p i c s . There i s a s e c t i o n 42 43 on "means", f o l l o w e d by one o n " p r o p o r t i o n s " ; then, a 44 treatment of the d i v i s i o canonis of T h r a s y l l u s which would s u r e l y more f i t t i n g l y have been presented along w i t h " i n s t r -umental music"; next, the s u b j e c t of "means" i s resumed o n l y t o be i n t e r r u p t e d by two o t h e r t o p i c s , the 6eKac;and the TETpaHTuc,, 4 5 and the treatment of f i g u r e s (7repi cooiparcuv) . 4 6 The i n s e r t i o n of these two s u b j e c t s s e r i o u s l y mars the c o n t i n -u i t y of t h e d i s c u s s i o n of "means" and c o u l d have been i n c o r -porated w i t h much g r e a t e r r e l e v a n c e i n other s e c t i o n s of the work. The Senaqand T e r p a K T u c , f o r i n s t a n c e , i s r e l a t e d o n l y s u p e r f i c i a l l y t o music and perhaps would have found a more 4 1 H i l l e r , 16.26 4 2 H i l l e r , ;7.2.21ff. 4 3 H i l l e r , 8 5 . 6 f f . 4 4 H i l l e r , 8 7 . 4 f f . 4 5 H i l l e r , 93.17ff. 4 6 H i l l e r , 111.13ff. 16 n a t u r a l context i n the d i s c u s s i o n of the f o u r dimensions i . e . , p o i n t , l i n e , s u r f a c e amd s o l i d , and t h a t of the f o u r r e g u l a r s o l i d s . The same might be s a i d f o r the p o s i t i o n i n g of the 7tepi cx^P-ctTcov, f o r i t c o n t a i n s a few elementary d e f -i n i t i o n s of plane and s p a t i a l f i g u r e s . The f i r s t h a l f of the work concludes w i t h a proof of the "golden mean"—the 47 o n l y proof i n the f i r s t h a l f of the work. In r e t r o s p e c t we are conscious of a f a i l u r e t o f u l f i l the p l a n proposed at the beginning of the s e c t i o n . In no sense can Theon's piecemeal treatment of u n r e l a t e d t o p i c s be regarded as the promised d e s c r i p t i o n of geometry and s t e r e o -metry. A l l the more astounding t h e r e f o r e are the words w i t h which the author concludes P a r t One of h i s work: r a u r a ^p-ev r d a v a y H a t O T d T a xP^ciP^TaTaiv e v TOTC; yrpo-^ e i pripevoic, p,a0T]p.a0iv abc; ev xe^cpaXaiaj&e t 7tapa6oaei 7rpoc, TT)V TCBV HXaTcuviHcuv avdyvaxsiv. keinexat 6e uAfTipoveuo"ai crroixs i<^ 6a)<g TCUV n a T * daTpovopi ' av .^^ — j u s t as i f t h e r e were nothing amiss and he were checking o f f t h e l i s t o f t o p i c s so e l a b o r a t e l y enumerated e a r l i e r . A l s o m i s s i n g i s the "harmony of t h e universe,'; another t o p i c prom-49 i s e d not o n l y i n the g e n e r a l i n t r o d u c t i o n , but a g a i n i n the 50 i n t r o d u c t i o n t o music, and yet again at the b e g i n n i n g of , 51 the e x p o s i t i o n of the T e T p a n r u q . 4 7 H i l l e r , 117.12 4 8 H i l l e r , 119.16 4 9 H i l l e r , 17.24 5 0 H i l l e r , 4 7 . 8 f f . 5 1 H i l l e r , 93.9 17 Moreover the treatment of "harmony of the u n i v e r s e " s t i l l remains unpresented i n P a r t Two, devoted i n i t s e n t i r e t y t o astronomy, where i t c o u l d w e l l have been d i s c u s s e d i n a s u i t a b l e context as we l e a r n from the c l o s i n g words of the L t r e a t i s e : £7rei 6 ' ecpayiev etvat uouainfiv nat dpjaovi'av TT\V iiev ev opyavoic,, TT)V 6e ev dp^'ieuoTc;, TT\V 6e ev ^noauxp, nat nepi Tr[c, ev noauxp T^dvayHaTa 7ravTa e§?5<5 ercTjrYYf iXap-e©a perd rrjv rrepi daTpoXoYiaq 7 r a p a 6 q a i v v . a nat rtepi TOUTO>V ev necpaXatoic^ 7tapa6e I'KVUCJIV Q 6'paauXXoc, avv oic, nat a u r o i Ttpoe^etpYacru.eea 6T)X(0Te'ov. ^ 2 From t h i s i t would appear t h a t Theon*s i n t e n t i o n remains un-shaken, to f i n i s h the work w i t h the "harmony of the u n i v e r s e " . I t s absence l e a v e s a gap of some importance. Regarding the m i s s i n g s e c t i o n s on geometry and s t e r e -ometry, i t i s not easy to make a d e f i n i t i v e judgment. 53 54 Tannery and, i n p a r t i a l agreement wi t h him, Heath are of t h e o p i n i o n t h a t the work of Theon was not m u t i l a t e d at the time of i t s d i v i s i o n i n t o two p a r t s . Tannery b e l i e v e s t h a t the s e c t i o n on t r i a n g u l a r numbers, square numbers, py r a m i d a l numbers and the l i k e i n the i n i t i a l p a r t of the work t h a t d e a l s with number theory r e p r e s e n t s a l l t h a t Theon intended t o o f f e r as geometry and stereometry, f o r Theon's c o n c e p t i o n of r e a l geometry i s one of a p u r e l y a b s t r a c t s c i e n c e d i f f e r -i n g from E u c l i d ' s geometry of s p a t i a l f i g u r e s . And he supp-5 2 H i l l e r , 204.23ff. P.Tannery, Memoires S c i e n t i f i q u e s , J". L. Heiberg and H.G. Zeuthen, (Toulouse, 1912), I I , 453ff. 5 4 H § a t h ; , H.G.M. , i i , 240. 1 8 o s e s t h a t t h e s e c o n d s e c t i o n d e v o t e d t o " m u s i c b a s e d o n n u m b e r s " c o n t a i n s s e v e r a l l a t e r a d d i t i o n s a s , f o r i n s t a n c e , t h e p o r t i o n c o n c e r n i n g f i g u r e s (n-epi cxtip-arcuv) a n d t h a t o n " m e a n s " (]ieaoxr]re<;); t h e s e h e b e l i e v e s t o b e B y z a n t i n e i n t e r p o l a t i o n s . T h e p a s s a g e a b o u t t h e e e n a q a n d t h e rerpaKTuq w h i c h r a t h e r f o r c i b l y i n t e r r u p t s t h e d i s c u s s i o n a b o u t " m e a n s " i s c o n s i d e r e d b y T a n n e r y t o b e a p a r t o u t o f t h e s e c t i o n o n " h a r m o n y o f t h e u n i v e r s e " , w h i c h h e s u p p o s e s t o h a v e f o r m e d t h e l a s t p a r t o f t h e w o r k a n d f r o m w h i c h h e b e l i e v e s s e v e r a l p o r t i o n s w e r e r e m o v e d a n d r e p l a c e d i n d i f f e r e n t p o s i t i o n s e l s e w h e r e i n t h e w o r k . T h i s w a s : T a n n e r y ' s o r i g i n a l e x p l a n -a t i o n o f t h e l o c a t i o n o f " h a r m o n y o f t h e u n i v e r s e " i n t h e w o r k a l t h o u g h h e h i m s e l f l a t e r a b a n d o n e d h i s h y p o t h e s i s a b o u t t h e o r i g i n o f t h e bendq a n d t h e TETPCIHTUC, a n d d e c l a r e d t h i s 57 p o r t i o n t o o t o b e a B y z a n t i n e i n t e r p o l a t i o n . H o w e v e r , i f we s u r v e y t h e w h o l e w o r k a n d c o n s i d e r t h e d e t a i l e d s t a t e m e n t s o f T h e o n a b o u t t h e s e q u e n c e t o b e f o l l o w -e d i n h i s w o r k , i t i s i n d e e d d i f f i c u l t t o b e l i e v e t h a t t h e s e c t i o n o n p o l y g o n a l n u m b e r s a n d p y r a m i d a l n u m b e r s a p p e a r i n g i n t h e s e c t i o n d e a l i n g w i t h n u m b e r t h e o r y , a s T a n n e r y b e l i e v e s , o r t h e c h a p t e r " c o n c e r n i n g f i g u r e s " , a s H e a t h s u p p o s e s , w e r e a l l t h a t T h e o n i n t e n d e d t o o f f e r i n t h e f i e l d o f g e o m e t r y a n d 55 H i l l e r , 1 1 1 . 1 3 f f . 5 6 H i l l e r , 1 1 3 . 9 - 1 1 9 . 2 1 57 T a n n e r y , op_. c i t . . 1 2 6 f f . 19 stereometry and t h a t , t h e r e f o r e , o n l y the p o r t i o n d e a l i n g w i t h "harmony of thei:universe" at the end i s m i s s i n g . I t i s t r u e t h a t Theon cannot have intended to present a treatment of pure geometry and s o l i d geometry a f t e r the manner of E u c l i d , as i s shown most c l e a r l y by h i s treatment of the t h e o r y of 58 p r o p o r t i o n s . H i s p r e s e n t a t i o n would be expected t o c o n s i s t of d e f i n i t i o n s , e x p l a n a t i o n s and examples and not t o comprise r i g o r o u s p r o o f s and c o n s t r u c t i o n s . N e v e r t h e l e s s , an explan-a t i o n i s s t i l l r e q u i r e d f o r the f a c t t h a t , when b r i n g i n g t o a c l o s e h i s treatment of "means", Theon announces astronomy as the next s u b j e c t t o be d i s c u s s e d i n s t e a d of geometry and stereometry as h i s o r i g i n a l arrangement r e q u i r e d . But i f one c o n s i d e r s the nature and content of the s e c t i o n s which i n t e r r u p t the c o n t i n u i t y of the f i r s t p a r t , 59 and f u r t h e r i f one notes t h a t a s e c t i o n i n the second p a r t of the work may best be e x p l a i n e d as h a ving some c o n n e c t i o n w i t h the "harmony of the u n i v e r s e , " then an expansion of Tan-nery's e a r l i e r h y p o t h e s i s seems to o f f e r the most l i k e l y explan-a t i o n of the c o n d i t i o n of the work as i t has come down to us. I t seems, then, t h a t the work was e d i t e d i n two h a l v e s and t h a t i n both cases on l y p a r t s were p u b l i s h e d complete, on the one hand, a r i t h m e t i c and music, and on the o t h e r , a s t r o -5 8PW, l o c . c i t . , 2074, 26. 59 H i l l e r , 138 - 1 4 7 % T h i s p o r t i o n f i r s t quotes the v e r -ses of Alexander ( 6 AITCDAO'C;) and develops i n t o a d e s c r i p t i o n of the "sphere of the s t a r s " ( c f . P l a t o , Rep., 614B, where the myth of the Pamphylian i s p r e s e n t e d ) . 20 nomy, and t h a t s e c t i o n s of the p a r t s omitted were then, e i t h e r immediately or l a t e r , i n s e r t e d i n t o the p u b l i s h e d p a r t s i n a 60 r a t h e r u n r e l a t e d f a s h i o n . Nonetheless, none of these con-c l u s i o n s accounts f o r the p a r t i c u l a r l y i n v o l v e d statements made by Theon r e g a r d i n g the arrangement of h i s work, and the q u e s t i o n whether the work ever d i d appear i n the f i n a l comp-l e t e form intended by the author c e r t a i n l y remains open. 6 0PW, l o c . c i t . , 2074, 57. CHAPTER TWO ON ARITHMETIC: TEXT AND APPARATUS Theon*s work has come down t o us d i v i d e d i n t o two p a r t s , which have been d e r i v e d r e s p e c t i v e l y from two manu-s c r i p t s , d e s i g n a t e d A and B. Codex A i s an l l t h / 1 2 t h c e n t -ury parchment manuscript (307) of the S t . Mark's L i b r a r y , V e n i c e . Codex B i s a 14th/15th c e n t u r y paper manuscript of the same l i b r a r y . A l l o t h e r manuscripts of e i t h e r both halves of the work or of one complete h a l f are e i t h e r d i r e c t l y or i n d i r e c t l y dependent upon the above-mentioned manuscripts. The f o l l o w i n g are the e d i t i o n s of the work which have been p u b l i s h e d t o date: 1. I . B u l l i a l d u s , f i r s t p a r t o n l y , P a r i s , 1644. 2. J . J . de G e l d e r , f i r s t p a r t o n l y , Lyons, 1827. 3. T. H. M a r t i n , second p a r t only, P a r i s , 1849. 4. E. H i l l e r , both p a r t s , Teubner, L e i p z i g , 1878. 5. J . Dupuis, both p a r t s , w i t h French t r a n s l a t i o n , Hachette, P a r i s , 1892. E d i t i o n 4 by H i l l e r , noted above, c o n t a i n s a l l the Greek t e x t of what we have of Theon's work and i s c a r e f u l l y annotated w i t h the numerous v a r i a n t r e a d i n g s of the s e v e r a l manuscripts. T h i s t e x t has been used f o r the t r a n s l a t i o n and the a p p r o p r i a t e p o r t i o n (pp. 1-46) i s here presented, 22 preceded by a summary of H i l l e r ' s accompanying notes. 1. A l l manuscripts have been d e r i v e d from those des-i g n a t e d A and B. 2. To o b v i a t e an undue burdening of the notes, manu-s c r i p t s d e r i v e d from A and B are g i v e n the common d e s i g n a t i o n 'apograph! 1. 3. The passage 46.20 - 57. 6 has an archetype (Z) d i f f e r e n t from A. 4. Ms. 203 Venet. Marc, used f o r the c o r r e c t i o n of the "verses of Alexander" i s designated C. 5. <C > brackets e n c l o s e suggested a d d i t i o n s t o f i l l c l e a r l y i n d i c a t e d lacunae. 6. £ ^ brackets enclose what should be erased. l p 7. A i s used to mark readings i n A changed by A . 8. ^ i n d i c a t e s M a r t i n ' s c o r r e c t i o n s of B. 9. C o r r e c t i o n s not a s s i g n e d above are H i l l e r ' s own. "Oxi (tlv ov% ocov xe GvvEtvai xav fia^rj^azLxag Xeyopivav aaga W.dxavi (IT) xal aixbv ^Gxrjt,iivov iv TJJ &eagiu xavzt], nag av nov biioXoyr^Geuv ag de ovde zu uXXtt uvaq>EXr\g ovde dvovqzog r\ itegl xavxa ijiTteigia, dice TtoXXav avzbg ipcpaviXsLV ioixe. TO [IEV 5 ovv Gvtindarjg yeapexgCag xal tfujutacr^g (iovGixi}g y.ccl aaxQovofitag eunetgov yevopEvov xoig W.dxavog Gvy-ygduftaGtv IvxvyydvEtv paxagiGzbv (ifv st xa yivoixo, ov (ii)v evxogov ovde gddiov d).Xk ndvv jzoXXov xov ix icaidav jzovov deopevov. aGze de zovg dirjpagziixo-xag xov iv xotg iia&rj[iu6iv aGxr^yivai, ogeyoiiivovg de zijg yvaGscog xav Gvyyoafifidxav avtov fit) auvzdzccGiv av no&ovGt, dLKpugzEtv, xstpaXaiadrj xal Gvvzopov TioitjGoned'u xav dvayxaiav xal av del pdXiGza xoTg ivxev%o(iivoigIIXdxavi pad-quuxixav &EaQrtiidzcov nagd- 15 doGiv, dgi&iirjzixav xe xal {lovGixav xal yEutuxgixuv xav xe xaza GzegEopezgiav xal dGxgovofiCav, av %aglg Inscr. 0£a>vog Savqvatav nXazajviKOV t<ov nazd to (to supra vs.) fiafrijfiartxoi' xQx\aCyi,av clg xr\v IIXuxcovos uvdyviocLV A 1 inscr. o n dvctyxaia ra (ic<9iqpazu A 2 ijexquivov: ov corr. es'cov A 4 ovSs zee ci] ovis re u. A 5 to corr. ex zov A 6 ystufttrgiaj: xal aot9'(ii;n-/C<jj add. recentior manus in apogr. fort, recte 10 exoaros tow QiQXiov mg. A Theo Smjrn. I 2 DE UTILITATK oi'x oiov ze elvui cprtGi TVXCLV zov clgi'Grov fii'ov, dia zol-kav Ttavv drj.adag ag ov XQ>) uccd->tu.cizuv duekitv. 'EouxoGxjivr^ plv yug iv TO tziygccrfo^iiva W.a-zavixa cprfitv Zxi, drjAioig zov {her 'igrficcvzog izl 5 tntakfocyft XOLUOV (Saiwv zov ovzog diz/.ccGiovu y.uuc-GxevuGca, 3CO/J.ytu ug%izixzoGiv iii-tGe'v uxogtav roiGiv oxag %q>j Gzegebv 6zegeov yeveGQca dizluGiov, cccfixaGQ-ca ze xevGoiievovg negl zovzou W.cizavog. TOI/ de cpuvcu cevzo'g, ag agu ov dizkuGiov @a{iov 6 Qebg io de6(ievog Torro sJ>ikioig iuccvzcvGazo, xgocpigav de xcd oveidc^av zoig "El/.t;6ii' CIUIQVGI iLc&^aclxav xcd yea-(lazQiag all} cooiy/.oGtv. uxolov&cog de zjj zov Tlvd-iov zcgcuveGet Ttollcc xcd ccvzbg dii^eiGiv vzeg zov iv zotg {iKu/jUaGi %gj;Gtuov. i'v is re yccg xi} Ezuvjiidi xgozgi~av izl zu ucdi]uaxci qrtGiV ov yccg civev zovicov note zig iv zio/.ei eidounovav yevrtGezca epvGig, c\)X ovzog 6 rgozog, avzt\ rj T(>o<p;, xavza ta ^ .c^ruaxa, eize %c'J.cTca ein ocldic, dia zuv-trjg izioV ciu.el)\Gcu 6s ov Q-euixov ; c r : Qeav. xcd iv 20 xolg icpe%y]g t o v zoiovzov cprfiiv ix zc/.lav i'l'cc yeyovoza SvSciipovcc Tc iGcG&ut xcd Gocpazazov c.ucc y.cd [laxdgiov. iv de T/J Tlolizeicc cftjaiv' ix zcov xe izav ot zgo-XQiQ-evzeg ziuug ze zav akhov peiZovg oiGovzcu, zd ze 3 'EnaiocS-hr^: Bernhardy Eratosthenica p. 168. cf. Phi-lol. XXX p. C7 12 aliyoQ^noaiv A, em. apogr. 15 Epin. p. 992 A ov yoo uvsv ye {ys om. Nicom. introd. arithni. I 3, 5) zovzmv (tTjrrori zii iv TCO/.SCIV evSuiauv (ivduiuoudv codd. duo, tvSaifiovmv nnns) yivr,tat (pvcis, ul).' ouroj 6 ZQOTZOS, avr»j (rj add. Ast) xoo(frt, zavta tu p.u9i\uuTa, sirs jjcdfnrti eizl gaSta, zavxrj nogivzicv (triov Kicom.) • diisl^ccti ci ov ftepizov iezi •Oscov ivSuincvav A 17 qpr's A, turn dnau litt. erasae 20 Epin. p. 932 B 22 Civ. VII p. S37 B x f j sC*ociv Plato., cf. Schneider 23 {m'jjoi'j zov aV.tav Plato MATHEJIATICAE. 3 XVSTJV pad-quata itaCiv iv xrj •zaiSatcc yevvuavu zovzoig Gvvaxziov aig Gvvo^-iv oixaiozrjzog re akfaj),avxav . iicc&^aurcjv xai zilg zov ovxog cpvGaag. jiagaival rs Ttgazov {ilv epxaigov yavaGd-at, dgid-^yjzixrig, l%nza yaa-{itzgixiig, xgizov Sa Gzagao^axgucg, xaxaoxov aGzgovo- 5 (ii'ug, i\v (pr\<Siv tlvcti ftaagCay <pago[iEvov Gzagaov, naii-JZZOV ds [lovGixijg. zo zs %gtJGiuov Ttccgudaixvvg xav [la&TjUUzav rp^Giv' rjdvg at, ozi ioixag dadtavai, pr) axQ-rjGza xa (lu&tfpaxa xgoGxazzotiu. ro d' aCziv ov Ttavv cpavkoig, d?.?.a nuGi %aXazbv TCcGzev&rjvat, oxi iv 10 xovxoig xo'g {la&rjiiaGiv ixaGrov olov ogydvbig TO ipv-%r]g ixxa&aigazca y.al dva%coTCvgalxai oufia xvcpXovfisvov xaX ccTCOOfisivvuavov v-ru xov cilXav ixiztidavp.dxuv, xgaizzov ov Goi&tjvca uvgiav otxpdxav' [lova yccg ccvxco airftEiu ugdztxi. 15 iv da to ifidoaa ztjg TlokixeCag rcagl txgi&itrjzixrjg Xiyuv ag ioziv ccvayxccioxdzr] jtaGav rpTjGiv, anaiza r\g 1 TiuiGiv Plato, cf. H. Heller curae crit. in Plat, do repobl. libre 3 p. 16 sialicic: corn ex TZOHSIC: A (-xaiSia Platonia codd. tres) 2 oiv.it6xr,zos uXXr]Xa>v Plato (oUsiorr/tog rs uXXrjXtov codd. tres) 3 cf. Civ. p. 525 s^q. nctQcttvioft A, em. apegr. ^ 8 Ciy.^  p. 527 D i}Sv; ti, T]V S syd, on eW.ag Se-Sion rovg TIOV.OV;, prj Soy.fjs (on toiv.ag StSiivui, {ir} agu Ni-com. I 3, 7) cixnr^ta uaS-^ uarrc (ra fiaSijaaza cod. Par. K, T K u r a T « u.uxfr,u.azu Nicom.) 7Tooczcczzeiv {ngoazazzoiai Xicom.). T O 8 i'ativ ov navv rpavXov dXXa jaXinav niaztvcai (nayx&Xt- . rzov Nicom.), on Iv xovxoig ZOL$ iic.9qu.aaiv tv.uoxov oqyuvav XL ili>ixr)q sxxa&aiQtzctC xs xal ava^caizvqiizai (o^fia rijj ilivx'.S exstat etiam apud Niconi., ubi haec contracta sunt, cf. Akin. .. 27 p. 180 Herm., Iaiabl.de vita Pyth. § 70, Boeth. Inat. arithm. I 1 p. 10 Friedl.) uizolXvy.zvov Y.al rvgr.'.oiJtuvov V J T O v.zX. 9 ra r.al 7tnnntartoiui fort. add. ab A* •nnoaxarzoiat A 10 cpcevXoig: i in ra?. A 11 cf. Cobet, Mnemos. XI p. 177. Wex, Jabrb. f. Philol. 1863 p. 692 sqi}. ISC 1 p. 331 12 OUJIU in ras. A 1G Civ. p. 522 C V . . . . . . . 1* 26 4 ' DE UTI LIT ATE 6ei itdoaig {ilv xi%vaig, Ttdoaig de Siuvolaig xcd ijr-.aztj-[iaig xcd zrj 3toJ.su.Lxf]. ztayyiloiov yovv Gzgcirijybv ^Ayaiii^vova iv zoig xgayaduug JJulatirid^g ixdazoxe aitocpalvEi. tpr]Gl yccg dgi&ubv Evnav rag ZE rd\ztg 5 xazaGzfJGca ta Gzgazonida iv 'Ilia xcd i^ugi&^irjGca vavg XE xcd zee dlla ndvza, ag Jtgb rov dvaQi&iLr\zav ovrav xcd rov 'Ayapipvovog ag EOIXEV ovde baovg ei%e TtoSag aidozog, ei'ye pr} r^xiaxaxo dgi&uEiv. xivdvvevEi ovv zav ngbg voriGiv dyovzav cpvGei eivat, xcd ovdelg io ctvza xgtjtai ilxxixa ovzi regbg ovGi'uv xcd vorfiiag %agaxh\zcxa. oGa pev yap axlag XIVEI XT)V uiGd'qGiv, ovx iazcv inEyegzixu xcd jtagccxlrjzixd vort6eag, oiov ozi 6 bgapEvog Sdxzv?.6g iozi, xcd Zzi na%vg 7} lETtxbg [liyag 77 [iixgog. oGa 8' ivccvriag xtvel aiG&qGiv, i s iiteyegxixcc xcd ztugaxlrtzixd EGzc'Siccvoiccg, oiov Zxav zb avxb cpuivr\ztti [liya xcd [iixgov, xovepov xcd /3tvpv, ev xcd stolid, xcd zb 'iv ovv xcd 6 c':gi&[ibg zagaxlrjzt.xu xcd eneyegzixd iazt diccvoi'ag, ixel zb EV XOZE stolla cpaivzxca' loyiGzixtj de xcd ciQi&atiriy.t) b/.xbg xcd dya-20 yog Ttgbg alifoeiuv. dnziov 8s loycGrixtjg pr) idiazixag, 2 itayyilotov: yy ex *y A (nuvyjlotov Platonts cod. Yind. F) 3 naXccaCS^i A 4 dizoifaivn. TJ OVX ivrtvorjy.ag, on cf^oiv ctQi&aov tvnav rag ZE zdiiig ra crprercrr/ijro y.azucz^aca iv 'ilia Plato 7 oaovg TI. li'/.iv SI'S, {info do. fir/ 7;.T. Plato (irj i'ni-czazo corr. es ziczazo A Civ. p. 52o A y.iiSvvevn zav 3t(jos TJJV vorfiiv ayovzav rpvcn stvai u>v fazoiytv, xgrtcdui 8' ovdets avza doS'cGj, sl-itf/.ro ovzi ztavraTzcci zznbg ovaiav 10 vor)Cctcg 7ianccY.?.r;zLY.c~>: cf. Civ. p. 523 D 19 6sj Sr) Stall-baum ad Pkt. Civ. p. 520 A 20 Civ. p. 525 B Ttgoe^y.ov Sr; to fid&rtu.a av si'/;, cd riavy.av, touo&ez^cc;: y.ai 7iii9siv zov; piHovzag iv zf[ Tti'/.n ziav fiiyicxwv (ls&ilfiv, irrl !.oy.aziy.rtv iivat nai ccv$u7rz-e&cti avr?;g urj ifiitiiztxig, d'X' Ea; af i~\ &iav r r : ; z-vv r'.^: ?;i.-:-y rj'^-t:;; c::j •'•/.:::T :O))CJI c.iii^ o i ~ arris ovSs Tindacco; IOCOLV dg itiTzooovg rj xcn-i;ioi'j (IEZSIVVZCI;, 27 MATHEMATICAE. 5 a)X ag av ixl &iav xijg xav ugi&^iav cpvGeag uyiv.av-xat xrj voijGei, ovde ngdaeag %C(QLV iyxtogav t) xa%v\-Xov jteXexavxag, KAA' evexa 4'vz*}s rijg in aXq&Huv xal ovoioiv odov. xovxo yag ava ayei xr]v ipvffiv xal seal avxav xav dgt&pav dvayxd%ei diuXeyeG&ui, oix 5 dxod£x6(iEVpv, av xig avxa Gapuxa tj av tec cgaza iiovxa dgi&povg 3tgoGtpeg6(ievog diaXiyqxat. xal ita7.iv iv xa avxa <pr\Giv' exi ol XoyiGxcxol eig aaavxa xa (ic&rjttaxa 6%etg (pvovxai, oi xe figadclg eig xo o^vxe-goi avxol avxav ytveG&at. ext. iv xa avxa <pr[Gi' xal 10 iv aoXi(ia <J' av xgr'\Giu.ov agog xdg GxgazortedevGetg xal xaxaXrjrpeig ^wot'ov xal %vvayayag xal i^exaGeig Gxguxidg. ev xe xotg i^yjg ixaivav xr)v itegl xa xoiavxa pcfrijuara ffrroffljjV, yeauixgCa ftfV, tftjGiv, iGxl negl Tt)v xov iTcmiSov fteagiav, uGxgovoy.ia de xegl xr\v xov 15 ttxegeov qpogdv avxr\ d' dvayxd^et eig xb ava bgdv xal cab xav iv&evde ixetGe ayet. xal [ilv Si} Ttegl povGi-xrtg iv xa avxa cprjGiv, oxi dvelv deixai f\ xav ovxav o i l * £f£x« izoXiuov TS xal avzfjg rjjs ipi'X^i? qaczmvrg TS (ZB om. codd. rnulti) u,exuOTqo<pr4; ano yeviaicog trz' e'lftudv TI {ts is' al. codd. complures) xal oveiav 4 Civ. r 525 D rovzo 71 — cos atpodqa avco JTOI ctyn zrtv tfivz^v y.al z >l avtdv zwv agi^fiav ai<KyxaJ«i diaXiytc&ai, ovSuuij ctTtoSi;jc,u;j'OV, lav rig a r x j oqata JJ aJtt<i (rj and zu codd. duo, !] zovg tec unus) eoojiara ijovzag (fzovzee codd. ties) aoiOjioug Trootfiroaevog dialf'yijrat 8 Civ. p. 526 B zoie i^irj tneOY.tipa), oog 01 ze tfvasi loytenxol c'i Ttttvza ta fiudrifiuzct wg inog tfattv o^eig cpvovzcti, 01 XB jicaSetg, av ev rouroj naiitvd'caai xui yvuvdcaixui, v.dv u^itv cXXo catptXrftuioiv, oucog ci; ye TO d|u'rEgoi «i!rol avzav yiyvt-aftcti itavztg smSiSoaaiv; 10 Civ. p. 526 D oco»> fifv, icprj, -qo; zu izoXeptxu avzov zit'rti, dfjXov ort 7cqoaqxsf xoog yao •zz; erqaro7isSivaeig xal.xcTal ^ i i t i s xmniav xal cvvaycoyag x a l txtcOEt; croariaj — 10 Civ. p. 529 A rcnvzl ydq fioi Sor.ti dijlov, on a v t ^ ye dvayy.d^et yi'xfjv fig zo ayo) bqav y.al c'.-ri t&v lift ivi s t x i f c E ayet 18 iveiv coir, es SvoCv A G D E U T i L I T A T E Q-eaoi'u, aoznovouiug y.cd ciouovi'ag' y.cd avrcu uSelfcd ul iziOiiiuca, ag OL Tlv^uyogiy.oC. ol (ilv ovv rug clxovoue'vag Gvacpaviug civ y.cd cp&oyyovg u).h\).Qig v.va-^.ezgovvzeg uvt\vvzu rcovovGi. xe?.eiag xc.Qujjul/.ovTcg 5 zu cozu, oiov ty. yetzovav cfavi]v ^^gaaevoi. ol uiv cpuGiv uzoveiv iv (isGa zivcc r\%ov xul ^ .ly.nozuzov eivca SiccGzrjuu zovxo, a ^.ezQrjziov, ol Ss uyufiGfazovGiv cog oaoiov T\S)] rpdeyyouivov, zu aza zov VOL zigoGzrfiu-(tevoi. zcdg %og}cdg TtQayfiaza •xaotyovGiv irtl rav y.oV.u-10 ficov Grosfilovvreg. ol Se clyaQ-ol d.oiQa^ny.ol ^zovGiv ixiGxonovvzeg, ziveg Gvucpavoc doi&iiol uoid'iioTg y.cd ziveg ov. y.al zovxo %Q)'jGi[iov rcgog rr]v zov uyo.doV 1 Civ. p. 530 D — y.u\ avzai dXXr,Xuv dSiX'fui zirsg a' ini-azrjiiai eivai, dig oi ze TTv&ciyooEwL fuai — 2 Civ. p. 531 A zccg yag av.ovoiiivag av ovufoniag y.al q^oyyovg dXXrJ.oig cha-fiezgovvzeg dvrtvvza aamo ol dotgoiouoi ziovovoi. VT] zovg Qeovg, t'cp>], y.al yeXoioig yi, nvv.vuiu.az' uzxu 6ioud*ovze; v.ui TiugufidXXovztg zd uza, oiov t'x yeizovar yuv^v ti^gcvouevot, ol fiiv traciv ezi (fn om. codd. tres) xaraxovtiv iv u-iaoi zivd JJZfjv xal cu.iy.goza.zov elvai zovzo Siaozrua, o> uir.gr.zioi1, of de durpia'S^zoiizeg cog ouoiov iirt cffttyyouivcav {irQ tyyoiisvov codd. duo), durfozsgoi. loza zov iov crgoBz^udiifvoi. C'J uiv, rtv S eyat, zovg xgKozov; Xiyzi; zov; zaig x,jgSa'; -gdyiiara ~.\g-tjovra; *oi (SacaviZovza:, t.Ti t a c v.a'u.or.uv (xoXd jar, Y.UXXQ-• ncov, xoXXoxoiv a l i i : cf. l i m . les.) ozgipXovvzctg 4 av^vvza v corr, ex oi A 5 fx corr. ex iy A 7 ufro^rf'or: r> corr. ex t A 11 Civ. p. 531 C zoig ydg iv zrvraig zaig cvpcpcovicug zctig dy.ovouivcug dgi&iiovg frjroruiv, dXX' ot'x ii'g TigojiXrjiiaza uvittoiv iTZIG-/.O-.* iv, zh-ig |i:urf oroi dgiduo] vr-i ziveg ov, y.al Sid ZL ixdzigni. Satuoviov ydg, t ' y ; . trgxy:::; . 7-iytig. ^o/;ui;ioy iiev ovv, r)v d' iyoi, ngog zr]v zov xa/.ov ze xal ayaSov i,<\zrfiiv, dXl.ot; dl uizaSmjy.6y.il ov &xQ"erav. eixog y , t.j.fj. olftai Si ye, rtv S iyoi, -/.al i) tovzav zzavzav op 5teX7;Xv&au?r ui3oSo; idv fisv ir.i zf,v dXXrXav x o . j i j -vtav cirf'y.^zai Mat |•.•;•}•*>>(««»•, xr.-i £vV.oy:c$tj zavza r\ icnv , aXXrt\oig c'y.zia, epic,sir ri avzav eig a fiovXaui&c zrtv Tzgccyua-zei'av. cf. Sc'oDeider M A T H E M A T I C A E . .7 xal y.aloZ £IIT)]6LV, aV.ag Se u%grtGtov. xal xovxav •xuvxav i) pi&odog civ t.ilv ixl xi\v uXXi\Xav ittpixt\xai xoivaviav xal %vkXoyiG\}i~] rj iGxiv.uX?.rj?.oig olxEla, rpe-osi avxav rj noaypaxela xagzov. ol Se xuvta Seivol diaXexxixol' ov yag (ir) dvvavxai X.afieiv xe xal dito- 5 dc'zaG&ai Xoyov. ov% oiov xe Se xovro (it) Si' ixeivav iX&ovta xav (la&rjixurav' 6Sbg yag iGxi Si avxav iid xrtv xav ovxav &£av iv xa SiaXiyeG&ai. TtaXiv xe iv xa 'Eiuvouuo xoXX.a nev xal dX.Xa vrteg agid,(irlxr/.rjg diezegxexcu, d-sov Scogov p.vxt]v Xeyav, xal-10 ov% oiov TE avev xavttig GrtovSaiov yEvsG&ai xivd. vzto-(3a j SE avxixgvg <prtGtv' s'faeg yag dgi&nbv ix xrtg dv-9Qaxivrtg cpvGeug i^eXot-fiev, ovx av xov ETC (pgoviuoi yEvoi'us&a, ovd' av ext TCOTE XOVXOV XOV £aov, gprjGiv, ij ir'V%ii .Tidcav r'p£Ti ; i ' Xdpoi' G%eSbv 6 xovxov Xoyog is sit). £aov Sa O TI ft»j yivaGxot Svo xal xgia (itjSe Tte-Qirrbv U7t5l dgxiov, uyvooT de xb Tcagdxav dgi%[iov, ovx dv TtoxE Sidovai Xoyov, Ttegl av alGd-qGaig xal (ivrjjias 4 C i v . p . 5 3 1 D o u ydg xov Sov.ovat ye cot o f xuvta Ssi-toi $Lc:lexT!y.ol eivc-:i. ov ad tov dC\ i-'fr,, si fir] ad/.a yi xiveg oli'yoi ov jyd tITETU/a).).' rj6r>, ecrrov, urj Sovatai xiveg avxeg Sovvxi xs xal dnoSi^aa^ui Xoyov ei'aecd^ai note xi av tpaulv Selv eiSh-ai; 6 tn' avxa p o s t ixu'vcov A , s o d d e -l e n d a h a e c e^se p u u c t i s s i j j i i i f i c a i u i i i es t 8 c f . C i v . p . 532 C 10 c f . E i j i n . p . 9 7 6 D E 11 vxopdg c o r r . e x vneqpds A 12 E p i n . p . 9 7 7 C si'neq dqiOuov f x r i j j dvO-nam'vrig cpvcecog e^eXotuzv, ovx av noxe xi qjqoviuoL yevoi'u.f&a. . ou yap dv txt •note tyv%rj xovtov xov JMOU nacav dnexi)v Xd^ot cx^Sov, oxov Xoyog ccist'ij. £nov Si, 6 r t fti j yiyiaay.oi Svo y.al xqia urjSi mqixxov fxrjSe dqxiov, ayvooi Si to naqdnav dqi'duov, ovx dv note SiSovtu Xoyov txoi r r j p t mv uicdi'josis y.al fivruag uo'vov fi(] xexz^uivav TIJV di dXXrtv dqtri'v. dvSqsi'av xal cwij-qoav-vr,v, ovdev unov.iaXvei- cxeqofitvog Si dXr^ovs Xoyov cotfog ovx-av Ttoxe yivoLzo 15 ydn p o i t cxeSov s u p r a TS. a d d . A 2 8 D E U T 1 L I T A T E fiovov eirj xexzrk<xivog' Gzagoitevog 6e c\?.rt9ovg Xoyov o~o<pbg ovx uv note yevoizo. ov {ii)v ovde tu zav uk?.cov xeyvov keyopeva, l\ vvv dir/f.&otiav, ovSt'Ttoxe xovxav ovSev (idvai, stuvxu Sa unoP.eixui to xuguzuv, ozuv 5 ciQi%ii.7ixiv.~]s teg u{ieXjj. Sonata 6' uv iGag xiol j3ga%iag ccgi&pov SeTo~Tj~ca xb xav ccv&gazav yivog, ag tig xccg xExvug aitop?.e'rl'UQ"L' xui'zoi (is'ya fiev xcA zovxo. at Si xig iSot xb &ei"ov zt'ig yeveGeag xul xb dv^xov, iv a y.al xb fteoGefies yvcogiG&rJGezat. xul 6 ugi&fibg ovzag, ovx io uv izt Ttcig uccvzig yvoiq Gvuituvza c'.gi&iiov, oGrtg rtaiv Svvc'cueag uiziog uv ei't] Gvyyivoixevog, inel xul [IOVGL-xr)v jtuGccv St ugi&^iov pazu xcvi]Gedg xa xul cp&oyyav SrjXov ozi Sal. xul xb fiiyiGzov, dyuxtbv ag nclvxav uE-XLOV oxi Se xuxav ovSevog iGzi, xovxo yvaGziov. G%t-15 Sbv Se dXoyiGxog, uzuxzog, uGyj\u.av xa xul c'ggv&iiog dvugpoGzog xa GcpoSga xul jtuvd'' oGu xuxov xexoivd-vrtxi zivog, oGzig XiX.eiztzui Ttuvzbg dgi&uoi. iv Si zoig icpe^ijg <prtGi,v e'Gziv eyov iir]Selg fjacig zoxe xei&ira xtjg evGefieiag elvai xa &vr{za yevei. ix yccg tovzov °o tpveG&ui xcd xccg aXXug (.gsrug zip pudovzi y.uzu zgozov. 2 E p i n . p . 977 D xal 6 vvv (Xoyog) O'OJK,- orfijoerai, ozi y.al tct TIOV uXXoiv re%vdiv Xeyofieva, a vvv 8r> SntX&opev idtvtes ttvai jraoorj rag zexvag, ovSe zovxcov 'iv ovSev uivet (fitviC Stephaiuis) , Ttdvxa 6' dizoXtlxai ro nagdrtav, ozav dgidurtrix^v ng aviXzj. So'^eie 8' av ZxavtZg not figuxiw iviY.a dgi&uov SeLO&ai T O rav av&gmnav yivcg, e'g rag ri^vag dTtofiXiipaoi' xaixoi v.rX. 9 6 n o n exstat ap. PI . 11 iiztl r.al rd xard UOVGI-xr,v iiaoav 8iagi9u.ovu.ivuv v.ivi,aewg rl xzX. Tl. 13 dei'(fivai') cj. Bullialdus dyu&dv P I . 1-t o n 81 XKKCJI' ovSevog, ev rovro yvacziov, 8 Y.al rd%a yivoir'_ uv, dXX' fj oxeSov dXoyicxog re xul azuy.rog v.xX. PI. 1G C[68ga] tzogd P I . hitoaa P I . XE-XOLVIOVIKC A 17 o c r i j XiXeizczai] imXiXeixzai P I . E p i n . p . 989 B fiei^ovjiev ydg dgeriis iii]8ilg rjfias note izeiftrj Ttjs tvosfleias elvat za> ©r>;rc5 yivet 19 cf. E p i n . p. 939 D AIATHEUATICAE. 9 en SITU zianciSciy.vvdi &soc£(icic(v ore) xgoxa tig (ladrJGe-xi i. Xt'yu de 6clv (la&ztv TTQCJZOV dazgovouiav. el ydg zb xaTaipevdeGdai xal dv&gcoxuv deivov, xoX.v deivoze-gov &ecov' xazavevSoizo 8' dv o ipevSeig £'/,av S6%ag zegl&edJV ibevdeig d' dv do*ag .e%6i itegl •d'eav b urjde xijv xav aiG&}]Tcav &eav cpvGiv exEGxeuuevog, xovxeGziv daxgovduiav. dyvoetG&ai de <fi]Gi xoig noXXoig, bzi Go-(paTaTOv didyy.q xov uX.rfoag uGroovouov elvai, kur] TOV y.a& 'HGIOSOV dozgovouovvza, oiov dvGiidg xe'xal dva-xoXug ineGxeuuivov, dXJ.cc xdg Tizgiodovg xcov iztzd, o Hi) gadi'ag xoxe TiaGa (fvGig ixavr) yivoixo Q-eagrJGai. xov 6' eicl xavxa nagaGv.evd'tpvxa tpvGeig o"ag dvvazbv xoXX.dg rtgi-StdaGxeiv. xgela eGzlv i&i%ovzu naida ovza xal veaviG<ov did pa&iiudxav' dv xb uiyiGrov elvai 1 Epin. p. 9?9 E — &soat(Stiag dszivi rpoirrj Tig Tiva fiad-rj-atzai 7 Epin. p. 9'JO A dyvotizs Sti corpriratov dvdyxr, TOV af.rfiiag daz<, •jvouav fivsi, urj TOV y.a&' 'HaioSov dazqovoiiovvzu y.al zzdvzag :oi; zoiovzovg, oiov Svaudg ts xal dvazoXdf E~s-exsuui.vov, I'.'.d tov rcjv oy.zdi ntqioSmv zd; txzd ntqioSov;, 6ic^iovart; T avzov xvxXov txdazr4gt ovzmg dig ovx dv qaSi'cog TCOZE ndca i. vci; tv.avfj yiiotzo &togt'Cai 12 Epin. p. 900 C tzu. Si tavT'i zzaqaaxivd^ovzag cfvastg, Si eg (o"ag et Si otag alii) dvvazov tivai ^oftiv zzoV.u ngoSiSdaxovza y.al i&i'Sovzci Sii Starzovr^ac&ai zzaiSa ovza y.al vtavi'axov. Stb (ia9rjudxaiv 8tov av £i'V" TO Si (ityiazov zt xal jrocfirov dqiOutov avTaiv, dl/. ov aoiuaza C%OVTIOV, dXXd oXr^g Tr]g TOV izzqiTTOv TE xal aqTiov ytvteicog r? xal 8-vvuiitiag, oar4v Ttaqsxezai nqbg zrjv TCOV OVTCCV rpvaiv. zavza Si [ia9dvza zovzoig £rps£r;g iaziv o xaXovai uzv cr-oSqa ytXoiov ovoua ystouzzqittv, zdtv ovx ovriov 8i ouotwv aXXrtXoig yvazi aqi&fid>v ouoimaig zzqdg zr)v zaiv tTtiztiSav uoi-qav ytyovvia tezi Siarpavr4g- 6 Sf] &avua ovx dvd'qcoTZivov dXXd ytyovog diwv ccavsgnv dv yiyvoizo roi Svvaiisva f-vvvosiv. fitzd Stzovzr,v TOV; zqiig (rplj Ijekker) ^vl^uivovg xal TTJ aztqta tyvau ouoLOvg, zovg Si dvoiioiovg av ytyovozag izsqa rf'^vj ouoi'a zavzj], rtv dr) ytuutzni'av (erfo,-cntzqiav cod. Z) ixd-Xtcav oi TTooazvxc.ig uvzrj yeyovozzg' o Si fttiov r' taxi xal davjicazov — 10 DE UTILITATE dgi&uav ixiGTi'iaova amuv, d?J.' ov Guuc.ru ^o'j^rMi', xcd uvxi'jg zijg roi XEOIZTOV zs xul c/.gziov yEVEGECOg re xal SwccuEag, oGov TiagtyEzuc xobg zt)v zav ovrciv cpvGiv. rovzoig d\ irpei,~jg uud-y'iuc'.Ta UEV xukovGi, f^oi, 5 Gcpodga '/EXOLOV bvoua yEauErgiav' sGzi dl zav ovx ovzav buoicov d?J.yj?.oig cpvGEi cxgi&uav ouoi'aGig r r p o j zr)v rav ETtMidcov [iotgav. l.tyEL bi nvu xul izioccv iuTtELQi'av xcd zsyv^v, yjv d-q GzEgsouEroiav xc.J.si, EL zig, cpiqGi, rovg zgsig ccgi&uovg f'| av zu EXLXESU E'IVC.I uv\r\-10 Qivrclg opoiovg xcd dvouoLOvg ovzug, dg vzgoei-xov, GXE-QECC TtoiEt Gcoacczu' zovco dl&Eiov ZE xcd &uvuaOr6v EGzi. xal iv Tlofozsla dl tragi Gvpycoviag zijg xuzu uov-GIXIJV cp>]Gim xaUJazy] xcd usycGTi] zav zsgl rcolaav Gt'iirpcovtcov EGTCV )) Oocpicc, y)g 6 uev xuzu s.oyov ^cov "i5 iiEzoyog, 6 SE uTtolaizouEvog oixocfdooog xul TCEOI xohv ovSuufi GCIIZ)~OIQg, axE zee iit'yiGrr. dur.&uLvav. xal EV za rgc'ra da zy]g IloXiZELug, SiduGxav ozi [.iuvog iiovGixhg 6 cf.iJ.oGocf.og, cf-rfiiv' i.g' oiv ztgng &EO~)V ovzag ovde [covGtzol rrgozsgov to6ued~('., OVTE cvzol OVZE 20 oig tpc.UcV TjiicTg xccidavtiov ELVUC zovg cf-v/.uxug, r.glv 12 L e g . I l l p. CS9 D tdV r] xaXXi'cri; y.al uvy'ozr] zav ^vurpcovitSv uiyt'ozij Stxaiizax'. civ Xiyoixo Gorpi'a, i.g 6 ulv v.azd Xoyov £av uizoxog, 6 S d-Loleirzoutiog o'y.otf&ogog y.al TCIQL •JZOXLV ovSaurj CCJT/W) alXa rtdv zovtaizi'ov aua&cii rov ft'c zavra ixdezoze tfaveizcu IS Civ. Ill p. 102 1) ag' oiV, 6 Xiyn, HQOf &£cov, ovxcog ovdl uovaiy.ol Tzgozegov icouz&a, ovis avro'i our? ovs <pauev ijtity nc-tSivxior s'nai zovg yvXaxag, nglv civ za x>js ocorpnoci'v^g iiii; xal d'rSgeiag xal fljvO'f t)idrr (t-oj xal ILtyaXoTznzTTSLag y.al oca ZOVTOJV ad~Xnfd y.al za zovzmv av' irc:v-zi.a rravrayov 7ztgrrf.eg6ac-i'a y vwoI'^V a 5 v ^cf. p. 12, 1) xal tvovxu iv olg iptoxtv lifc&ccruutQG xcd avza xcd tixovai ctv-T I O V y.al ft'/'rs iv outy.goig U(T,- {>• ueydXoig ctznin'ioniv, clXXcc xijg avzr,g o'au.c9a zixviyg sivcci xal [ifXizt^g- cin corr. ex 00 A M A T H E M A T ^ A E . 11 dv drcavxu tec T)~:g GaygoGvv: ii'dr] xal dvdgei'ag xal HcyaJ.EioTijTog xal (iEyaXo~n-.r.Eiag xal oGa xovxav ddE?.<pd xal xd xovxav vzeiir.-zui nevxaxu TtEgicpsgo-(tiva xGJsnJcdim' xal ivovxa iv cig EGXIV aiG&avaut&a xal avxa xal slxovag avxav y~\ UI'JXE iv (iiy.goTg (itjtE 5 iv (isydXoig dxi(id^a(iEV, dX.X.d x~tgavx>}g O/BUE&H xt'xv>]S elvai xal {itXeTrjg; did ydg XOITCJV xal xav xgb avxav xi XE ocpeX.og ix (lovGixrjg dr;/.:', xal oxi fiovog ovxag 110vG1y.bg 6 tpiXoGoyog, dtiovGog dl 6 xaxbg. x\} (tlv ydg Bvtjd'sici ovxag, rtxtg iGxlv dcizq xo ev xd "ifrtj XUXE- it GxEvaGixEva IXSIVJ EasG&ai <+i;~-v EvXoyiav, XOVXEGXL XO EV X.oya XQrJGfrat, xjj dl svX.oy::: xrtv EvGxt]uoGvvnv xal Evgvd'uu'.v y.al cvaguoGxiav' cvGxf};ioGvv>jv ydg nsgl ps'Xog, tvaguoGu'av de xeol coiioviav, svgv&(iiav de zegl gvtuov' x7j dl y.axorftvl:;.. XOVXEGXC xa xaxa ijd-st, 15 tp^olv e: cGfrai xaxoXoyi'av, xcvxioxi xaxov Xoyov XQ*}' Giv, xfi <VE xaxoXoyia dGyriuz'j\iriv xal dggv^fiiav xal „dvaguoGziav Ttegl navxt: xd y-ixusva xal (UUOVUEVW aGxE fio'j og dv Ei'rj (iovuiy.bg c i-.vgiag svjjd-rjg, oGxig sii] dv 6 (piXGocfog. drjXot dl y.c.) r i £ig)]u£vu. ixsl ydg 1) 20 HovGixi) xb Evgv&pov xal c-'::oaoGxov xal EVGX>]UOV ifixoist xT] 4>vxfj ix viov EIG*I :u/r>; did xi xfi atpeXiiu liEiiiyii£in]v EXEIV djiP.ccji!i >;<3o: advvaxov cprjGi xiX.sov (tovGixbv yeveG&ai fiij Ei'doxa r : iv itavxl EVGXW0V xa xrjg EVGxriiioGvvrjg xal i/.-r r;gioxtjxog xal GarpgoGv- 25 l a y supra vs. A 5 EV::": :~!ra vs. A 9 cf. Civ. p. 4 0 0 D — 401 A 12 iv loyn] f:' :r; A 13 ad svQv9(u'ap in mg. A adnotatum erat TO (Jipi/cj igvSiii'cv, quae verba deleta sunt 17 anv&iii'uv A 2') ni t'grtaha: p. 10, 18 sqq. an scr. zu ngoftg>]u.iva ? 12 I>E L ' T I L I T A T E vr\g iidi\ fir) yvagC%ovza, xovxtGxi xdg idi'c.g. aptki iru-cftott.' £v xai'zl TtsQMfcOOUcVU — xovxiGxi ri'. eidrj •— y.al ui) dxiud^av avxa \ii\z1 iv Gu.iy.goig (ITJZ' iv utydXoig. TJ dl xav idcav yvaGig negl XOV (piX.oGocpeV ovdl ydg 5 iiScit} xig dv xo xoGjiiov y.al Ga<poov xal EiG^uov avxbg av dGxij^av xal dxoXaGzog' xb d' iv fiia tvGfj^ [lov xal EVQV&UOV xal Evdg^ioGzov tixoveg xrjg oixug ivGx^uoGvvqg y.al evugiioGxiag xal evgv&uiag, XOVXEGTI xav vorjxav xal idsav sixoveg xd aiG\}i\zd. vs xal o i IJv&ayooiy.ol di, oig xoXXazy ETtexui ffl.dxuv, xr)v (iovGixr]v (paGiv ivavxiav GvvaQuoyrtv xal xav izoXXav svaGiv xal xav dixa rpgovovvxav Gvurfg6vrtG'.v ov yag gvd-uav pvvov xal piXovg Gvvxay.xiy.i\v, eAA' dnX.ag xavxbg GvGzylucxog' xtXog ydg avxt^g xb iiovv 15 XB xal Gvvaguolsiv. xal ydg b debg GvvugfioGZ))g rJv diacpavovvzav, y.al zovzo [liyiGzov igyov &tov xr.rc [lovGixtjv XE xal y.axd iazgixy)v xd ix&Qd tpiX.u xoulv. iv iiovGixt), tpaGiv, t) buovoia xav xgayiidzav, tzt y.al dgiGzoxgazCa zov xavxog' xal ydg avztj iv xoGua ulv 20 dg(.iovia, iv xo/.si 6' tvvouia, iv oixoig ds Gacfgciii JJ yCvEGdai 7t£(Tvy.s' GvGzaziy.)) ydg iazi xc-il irate/.)) x5v xoX.XaV ij dl ivigyeia y.al r) xQ^Gig, f^oi. zijg iziGz}]-[irjg xavxrtg ircl zsGGagcov yi'vsxai xav dv&gcorth av, 4'vxrjg, Gapaxog, ot'y.ov, xoXeag' xgoGdzixai ydg xc.iza 25 xa xtGGaga Gvvaguoyy'g y.c\ Gvvzd%tag. iv dl zij JIOXLXCIC: HX.dzav vxig zav f/fi-!b;iiK7e:r _ 2 cf. p. 11, 3 sn . (Cir. p. 402 A C ) 10 cf. P.c;-:ka Philolaos Lebren p. (U 12 dtiofgovovvTov Ast aJ p. 290 18 <xci) iv pci'cjxj? ' 20 ef. p. 47, 2 21 ivvor^i-Afj (o>j es corr. ut \id.) A 22 fttoil M A T H E M A T I C A L . 13 xal tclds icprt' uya&bg de ccvijo Znzig diuGoi&i rt)v 6n&i)v dotccv rav ix ncudeiag uvza iyysvotiivav iv tc Xvzutg xul rtdovuTg xul izi^v^uag xul cpopoig xul at] ixjiuXXei. a di fiot doxel o/to<oi/ eivui, d'iXa uzeixuGai. ol vvv (taepaig, iaetSuv jSouAjj&coffi pdipcu iota act' sivui 5 aXovgyd, ngazov (isv ixXeyovzui ix toGovzav %gaud-xav piav tpvGiv rr)v rav X.evxdv, litutu zgoxuzu-OxsvulovGiv ovx oXiyrj TZagaGxevrj frsgaizevCavzag, onag Si%rtzui o ti puXiGzu zb c'.v&og, xal ovzag (iuzzovOi' xal 0 [uv civ tovza za rgozcp fiucpj], b[xov rt to (iucplv xul 10 1 Civ. p. 4 2 9 C (cf. ctiam antecedentia) Sid xavTog Ss iXfyov avzr)v amz^giav zh iv zi Xvrzaig ovza 8iaou>$£G&ai avtrtv xcd it r)8ovuig xal iv (iv om. codices complurea Platoni3 et Stobaei Flor. X X X X I I 1 97) i7zi9vu.icug • xal iv {iv om. Stob. ed. T: i u c a v . ) rzofloig xcd ur, ixfiuXXriv. co Si uoi Soxsi Ofioiov t h a i , i&iXa> dzztixt'iGai, tt (lovXti. dXXd (iovXouai. OVAOVV olc&a, rv 6' iyia, o n of ( J a t j c E t j , txuSdv flovXrftciat flayjai igia a>cz eivai ciXovnyd, izgazov uiv ixXiyovzai ix ZOGOVZIOV xga/idtiav fiUiv fvisiv zr)v zav XSVACOV, imiza Tzgo^agaaxcvct^ovaiv ovx oXiyrj izagaov.tvfi 9ega-zivoavzsg, onto; Si^szat [Si^rtzai Stobaei codex A et complares Platonis) 0 zi udXierct to dv&og, xal ovza Sr] fia^zzoVOL' xal o ulv uv zovzo) zco zgoTtm fiarprj, StvGor.mov yiyiezai TO (3c.'rr.tV, xcd 7] TtXvoig our dviv gvuudzcov ovzs n;zd gvutidzaiv Svvazai avzdiv to dv&og acpai-giio9c:i.' a S dv ixr], oio&u oia Sr) yiyvszui, idv xi tig aX'i.a xgcouaza (Sdzzzjj idv ZB xal xavxa ur) ngo&sgancvaag. o i d a , icprj, on ixzzXvzct xal ytXola. zoiovzov (rotouro codd. duo) zoCvvv, rtv d" iyci, vxoXafie xazd Svvccfitv igyd&o&ai xal T^idg, or? i^sXty6us9a zovg ozgazidzag xcd inaiSsvoasv (tv add. codd. Stobaei) uovaixj} xal yvuiaezixij' UTJSIV 0100 aXXo ixr^avda^ai, 7] onag 7'IILV 0 ti xdXXtcza zovg vouovg zTHodivzig Si^oivzo o>G7zsg prrri-v, iva Ssvaonoiog avzdiv ^ So^a yiyvoizo xal jtfgl Sstrdiv y.al zzsgl zav aXXwv, Sid TO zrtv rf <Z-VGIV y.cd zr)v zoorr^v ir:izrtSsiav ioiryr.ivai, xal fir avzdiv e/.r:Xviai zr)v @aq;i;v zd u-'unaza tavza, Sftvd ovra ixxiv ' f tv , ij zf f;ioirj, Travrog %c~*.SGznai'ov Ssivozigtt OVBi zovzo Sndv xcd zone ; , P.V.TJJ r>= y.;d -yofJo; y.al iirtO-vuia, aayzog aXXov gvp.auzo; :36 14 DK L'TIIilTATK t) efiGig, xcd ovze civev QVJXII:'.ZU>I< OVZE UEZU gvum'.xbiv Svvuzuc avziov zb dv&og df c.igEiOdca' a d' c.v a>'t, vta&a oiu d>i yivezui, di> y>) rcgodcgaxevGcg jlc'.nrr,, exziXvzu xal i$izr}?.a xal ov SevGozoid. zoiovzo dl xuzci 5 Svvay.iv igyu^EGd-cu r)yEi~G<)ai XQH xcd r)uug' Tccidivousv ycig zovg zcudug iv ixovGixj] r£ yM* yvuiuGri/J; xcd ygduiicGt. xal yeayErgicc xal iv dgi&u>tzixtj, ovdlv uXX.o yy-jXavcoiuvot, i) ozag >)uEig •ngoExxcddguvzEg xcd ~go-&Egu7cevGuvzeg coGXcQ ZLGI Givzziy.otg zoig ucfriluuGt, 10 zovzoig, zoig ztsgl u7tc'cGrtg doaztjg i\v civ ixyavQ-dvuGiv VGZEQOV Xoyovg ivSEi'^oivzo SGKEQ ^ acf^v, ivu SEVGO-•xoibg aizdv r\ So^a yivoizo, Sid zb zi)v q vGiv xcd zgocfi)v ixizijSsiav iGpyxivai, xcd [it) ixrcXvi't] uvxuv zi)v fia<prtv za gvunazu ruvza, SEIVCC ovza ixx/.v^eiv, 15 1] ZE r'jSov)'], Ttavzbg GzoEJi/.ov SEIVOZE'QCC OVUU xcd xoiva-vutg, XiTti] ZE xcd cpo^og xcd ixi&vyi'u, xavzbg uX?.ov gvtiiiazog. xcd yccg av z>)v cfiXoGo(fLav [u'>i<jiv (fca'tj rig uv dX.tj&ovg ZElEztlg xcd rc3v urrni' cig c.Xrftcog uvGzijgiav •io xugdSooiv. [ivi'iGEcog dl «c'p>; ~EIXE. ro UEV zgo>}yov-UEi'OV xa&ugiiog' ovza ydg uzuGi zoig fiovXouivoig UEZ-ovGt'u iivGzrtgu3v iGziv, dl).' EIGIV org avrcov Eigys-G&ut xgoayogevezat, oiov zovg X£~Qc:* xu&ugcg xcd (favrjv d^vvErov ixovzcg, xcd uvxovg ds zovg in) tigyo-25 uivovg uvdyxt\ xuftagicov rivog xgazegov zvxetv. [IETCI de n)v xdd-ugGiv dsvzigcc iozlv r) zi]g ZE?.Ery]g -ccgddoJ:g' 9 oru.-rnx.of;: x in raj. triuni aut rtuattuor Iitterarurn A i 12 yivotzo A IS cf. Hat. Fkaeii. p. 69 D 20 a A in margine 23 cf. Bt-mhanly Grumlriss tier griecb. Litt. I p. 22. Schoeraann opu=c. II p. 351 26 ji m<j. A MATIIE1IATICAE. 15 XQlXr] dl EZOVOUulouivTl ixOJTZEltC XezaQXT] d£, 8 dr) xal xe'Xog xrjg izoxzeiag, dvddsGtg xal Gxe^itidxav ixi&eGig, aGxe xal exigoig, dg tig TtageXajie xelexds, nqgadovvai dvvaG&ai, dadovix_tagxv%o'vxu Jj tegocpavxiag rj nvog aX.Xyg teoaovvr\g' Ttt'uzirj de r) ii avxav szegi- S yevouivr] xaxu to freotpileg xal fteoig Gvvdiaizov evdai-[lovia. xazd xavxd dr) xal -r) zav TD.azavixav X.oyav rcagd-doGtg to n\v ztgaxov e%ei xaSc.g(wv xiva, oiov zr)v iv zotg 7tgoGi]XOvGi iiad-tjiiaGiv ix naidav GvyyvfivaGi'av. 6 }iev ydg'EurtedoyJ.tjgy.grjvdavdxb nivz' uvi^iavzdgprfGiv dzei- io oil %al-/.a deiv dxoggvzzeGdai' 6 de ffl.dzav dzb icivze ua&r^idTaiv deiv cprtGi TtoietG&ai zt)v xdfrccgGiV zavxa d' iGtlr agi&[ir]xixr], yeapexgia, Gxegeo^iexgLa, ixovGixrj, aGzgovcuia. zrj de zeXezfj e'oixev t) xav xazd (piXoGotpiuv dzagtiiidzav zagc'SoGig, zav xe Xoyiy.av xal noXixixav v> xal tfVGiy.av. inozxeiav de 6vo^.dt,ei xr)v negl xd vor,xk xal xd ovxag ovxa xal xd xav ideav itgay-{icixsiav. dvddeGiv de xal zaxciGxeipiv qyijxiov xb i$ av avxvg xig xaxiiia%-ev oiov xe yei'EG&ai xal exegov; tig xr,v avxr;v frzagiav xuxaGxijGat. neiizxov d' dv ei"r} :« xal xeX.edxaxov -r) ix xovzav aegiyevov-ivrj evdaipovia 1 7 et * p g . A _ r) add. Lobeck Aglaoph. p. 39 5 jif/ujrrTj Si fj] fj Ss s' A 6 tvSataovCav A , em. Bullialdus 7 xavxa A TH.axmvixavJ itohxiY.av A 8 a mg. A 9 xr,v— cvyyvuvaaiav] ij — ovyyvfivctc'cc A 10 'EfinsSoxXfjs: vs. 422 Kar=tun, 412 Stein, 452 Mullach. cf. Aristot. Poet p. 1457 b avtUbivxu: tcv et ca es corr. 1 in raa. A dxtigh corr. es aY.qgii, inter (JE et t una lit. er. A 11 %u\x.(i> Siiv a.noQQvxxte$ui: vii dtlv et pr. o in r a v A' 13 cxeqioiiinyia: a corr. ex a A jf mg. A 16 cf. Phaedrus p. 250 C y mg. A 17 ti ttav Ilultsch] rijv xmv A 18 if tog. A 20 e mg. A IC) DE CONSILIO SCRIPTORIS.; xal XHT avrbv rov W.urava ouoiaGig &sa xaxa ro Svvaxov. noXXa (iiv ovv xal dk).u £%oi rig uv J.iyEiv xugu-Seixvvg ro rav fia&rtuuxav %QI\GIUOV xul avuyxulov. 5 rov 8s [LJ] Soxslv unsigoxaXag SiuxgifSstv (iv") xa rav pufrrjuurav iauiva xgsTtxiov "}8r\ rcgbg r>)v zcoudoGiv rav uvuyxuiav xaxa ru iiafri'juuxa dsagtjuurav, ovx oGu 6vvui.ro av xbv ivxvyxuvovxa r\ ugi&[ii]xr/.bv xsXiag 7} ysafiEtQYjv ij fiovGixbv r) uGxgovouov dzocprjvui' ovde io yug sGxi rovro xgoriyovuEvov ij TtgoxEiusvov ccxc.Gi rolg IlkuravL Evrvy/avovGi' iiova ds xuvxu xugudaGo-[isv, oGa i^ugxEi Ttgbg xb Svvrftijvui GVVEIVUI rav Gvyygufifiurav avrov. ovds yccg avrbg d^tcl Eig EGxurov yrjgug dcpixEG&ca Staygutiuuzu ygufovxa xul us).a8iav. 15 u).ka nuiSixa olsrca ruvxu rd (iu&tjuaru, Ttgozuga-GxsvuGxixu xul xuUugxixu ovxu il'vxqg Eig rb EXIXIISEIOV avxijv xgcg cpiXoGocpiuv ysviG^at,. yiuXiGru yisv ovV %gr) xbv JIE'XXOVXU org XE rjusig TtagaScoGouev clg XE JlXuxav Gwiygat'Ev EVXEI%EG9UI Sice yovv xijg rrgdv^g 20 yguuuixrjg C r o t ^ c i o f f c W j XE/ugry/.ivai' gaov yc'.g av %VVE'XOIZO olg jcugudaGoyEv. EGXUI 8' oiiag xoiuvxu xul xa. 7iug' r\u.dv, dg xul xa •z.uvxuTtuGiv uavryta xcov (tud-riyuxav yvdgqiu ysvsG&ai.j Tcgdrov ds iivrtu.ovsvGoaEV xdv dgi&ayjrtxcov trEagtj-25 [idrav, olg Gvvit,Evxxui xul ruxijgsv ugi9iioi~g uovGr/.yjg' rrjg (lev yug ivogyuvoig ovuuvxuna.Gi xgoGSsi'iiE&u, xa&u xal avrbg 6 JJluxav ccfrjyEixui Xiyav dg ov xg>) aGzsg 1 Theaet. p. 176 B 3 civ liyciv apogr.] dvaXiynv A 21 inscr. T:SQI cl QI V U rt r iv. rt ; A,. § in mg. 27 Cir. TH p. 631 A, cf. p. 6, 5 sqq. DE DISCIPLINARY! MATH. ORDINE. 17 ix yeixovav (pavrjv frygevoiievovg Ttgdyuaza Tiagiyeiv xatg xogdatg" ogEyopE&a ds x>)v iv xoGua dguoviav xal ri)v iv xovxa fiovGixr)v xaxavorjGai' xavxr\v dl ovx oiov TE xaxidsiv pr) xrjg iv dgi&potg ngi'xegov deagrjxi-xovg ysvoue'vovg. Sib xal TZEJITZZTJV 6 ffl.dxav tprfilv s sivai xi)v fiovGtxrjv, xt)v iv xoGtia ?.£yav, "jxig idzlv iv rfj xivrJGei xal xa%Ei xal Gvpcpavia xav iv avxa xivov-ytevav aGxgav. r^iv 3' uvayxaiov devxegav avxi\v xdxxsiv (lExa agi&(ir}xixi]v xal xaz' avxbv xov Illdzava, ineidr) ovd't] iv xoGpa fiovGixr) J.rjnzr) avev xijg jfcptfr- i o (tovpivrig xal voov^uivrjg fiovGixrjg. aGre si uev Gvve-&vxxai xrj Ttegl ipilovg dgid;uovg freagiu ?; iv dgi&uotg (IOVCIXI'I, Sevxiga av xax&eiri rtgbg xr)v xrjg ituexegag d'sagiag EVfidgetav. ftgbg ds xr)v tpvGixr)v xdtiv ngaxr] (IEV dv sit] x\ itsgl dgi&povg Q-sagia, xa7.ovu£vrt dgt9- is [tqxixrj' dsvxiga de ij itegl xa iniTtsdu, xulovuivj] yea-(isxgi'a' xgixrj de i) negl xd Oxeged, t}xig iGzl ezegeo}ie-xgi'a' Xcxdgxr] (Sly 1) negl xd v.ivov\xsva Gxegeti, Jjrtj loxlv dcxgovoiiia'. r\ Se xrjg xav xivrjGeav xal dtaGxrj-{tdxav Ttoid GxeGig iGxl povGixtj, rjxig oi'x oict xi iGxi 20 Arjtp&ijvai u>) ngoxegov fjiiav avxr)v iv dgt&uo'g xaza-vorjGavxav Stb 7tgbg xrjv r){ieze'gav freagiav uex' dgi9-pijxixr)v xezdx&a rj iv dgi^^ioig povGr/.rj, ag dl zgbg rijv tpvGiv zi£ii.7izi\ (fi} xrjg xov xoGuov dgpoviag &eagt}-Tixij (tovGixrj. xaxd dr) xovg Tlvftayogixovg ngeGJlEvxea 25 4 TT]$ corr. ex rot's A 5 TlXdrwv: cf. Civ. p. 530 D 6 xr\v £*» xo'<xut>> Xiyav] TOJV iv *6oua> loycov A, cf. T S . 2 et 10 7 avza Bull] avzrj A 10 s'crib. vid. avsv zr,i iv egid-uot j (vel /| dgifruav) xanxvooufiEinj ; , cf. va. 21 1^  8s add. Bull. 19 scr. vid. nitmtri SI 77 r/jj zcav niv. Y.al Sizar. sroo; all^la extotvii tff copjJTtxij aovainr] 21 avzr-v] zrtv} 25 uf'zpi zovzov mg. A ngtapivzdi A Thto Sm/TD. » 18 UXO rcc rav ugi&iidv dg ugx7) xul 7trtyr) y.al gitu rav rtc'cvrav. dgi&nos isri GvGztjuu uovudav, rj TtgonoSiGubg jifafoovg uTtb uovccdog ug%6u.Evog y.cd uvcao8iGp.bg tig 5 povudu y.uruh'iyav. uovug 8£ iGzc TcegaivovGa ttoGozrkg [dgxr) y.cd Gzoi%eiov rav ugi&nav], i]ztg ueiovuevov rov Tch'ftovg xuzcc xr)v vcpuigeGiv rov xuvzbg ugi&uov 6zegrfoeiGu [lovi'iv xs xal GzuGiv Xuufidvei. ov yccg oiov re Ttegaiziga yeveGQca ZIJV zoyi'jW xul yug iuv eig uogiu 10 diuigduev zb 'iv iv ulGdrtzoTg, eu.ttcJ.iv 7tli]&og yevr)-Gezcu zb ev y.cd 7to)J.d, xal xuzulrfeei eig ev xuza zr)v vcpuigeGiv ixccGzov zcov ucgiav' xuv ixeivo Ttc'chv eig uogiu diuigduev, nXii&og ze zu yogia yevrJGezai xcd )} xcczciXrfeig xu&' vcpuigeGiv ixuGrov zav uogiav eig ev. is coGze uuigiGzov xcd uSiuigezov zb ev dg ev. xcd yccg 6 fief c'M.og ccgi&ubg dia.igouu.evog ilcczzovrui xcd diui-geczuc eig i).uzrovu avzov uogiu, oiov zee g' eig rd y xul y 7] o xui p t] e XUL a. ro de ev av uev ev ulG&riroLg diuigijzcu, cog iuv Gcoucc iXccrrovzcci y.cd d'.c.i-20 getrue eig it.dzzova uvzov uooiu z>~;g zou7jg yivouivr^g, dg Se ugiftiibg av^ezc.t' uvzl yccg evbg yivetca rroAAr.'. aGze xcd xazci zovzo e'euegeg zb ev. ovSev yccg dtccigoi-[ievov eig tieit,ova eavzov uogiu dtcctgetrcci' zb de <Ti')> 3 inscr. nsgl 11 o g y.al aovuSog A , y in n i g . S t o t . ee ] . I 1, 8 fx T co v HloSindrov II v 9 a y o g f ;' o v. tart 6' dgi9uo; (jg zvTcm tinsiv ovBTrjuc! uoraScov, rj ^goTToSiaaog n''.i9ovg a r r d u o r t i -80S agyoutiog y.cd c'raxofitGuog tig uorc'Sct xarali]yav, uorc'cSccg 8i Tregaivovaa j r o c d r / ; ; , i':rig uaavuivov rov rth'idovg xara. trv vtftiigEGiv rtavxog agiijiiov GTtg\?i:oa uovr'v rs y.al Gtciciv ?.ccu-flavti. zzigtarigco */c':o i; unriig ti]g rr<>edr>;tOf (ser. r f c uoiddcg r) 3t06OTr,s) ot'X ioxvci'dic-rrmh'^-iv. cf . schol. D i o n . T b r . }i. SC'>, 14 9 7Tsganigco: a coir , ex s A 14 p o s t ly.darov rn» . triur.'. fere litt. A 2'-} s'g <(i"<7« icevxep rj ilg)> ufi'Jorc / n v r o i ' ? ET UNIT ATE. . 19 dicagovftevov xal eig fiei^ova xov oXov fiogia cog iv dgitiuoZg diaigelxai xal (elg~) iGa xa oXa' oiov xb ev xb iv aiG&rjxoig dv eig ?| diaiQE&ji} et'g iGa fiev xa oXa dg dgiirubg diaige&tiGezca a' a' a' a' a' a', eig fiei^ova de xov oXov dg dgtd-fibg eig 8' xal /T - xd ydg /S' xal s 8' ag dgid-jiol TtXelova xov evog. dSiaigexog dga rj fiovdg ag dgi&uog. xaXeixai de fiovdg yxoi drib xov fiiveiv dxgenxog xal fir) i%iGxuG\}ui xrjg iavxtjg tpvGeag' badxig ydg dv icp' iavxrjv TtoXXazXaGidGaiiEv xr)v fio-vc'ida, aivei fiovdg' xal ydg urcat, Hv ev, xal ue'xgig 10 ccTteigov idv TtolXaizXaGidZfOfiev xr)v fiovdda, fiivei fiovdg. JJ dnb xov diaxexgiGd'ai xal ueuovaGirai drib xov Xoizoi itXtjfrovg xav dgi&ucov xaXeixai fiovdg. JJ de di'Vi]vo%ev dgi&fibg xal agi%iii]x6v, xavxt] xal fiovdg xal iv. agi&fiog fiev ydg ioxi xb ev voi\xoZg ttoGoy, 15 oiov avxa e' y.al avxa 1, ov Gauaxd xiva ovde ala&rjxd, dXXd vorjxd' dgid'arjxbv de xo iv alc&qxoig TCQGOV, dg innoi e , (ioeg e , dv&gazoi e'. xal fiovdg xoivvv iGxlv r] roi evbg Idia-fj vorjxtj, rj eCxiv axouog' ev de xb iv alG&^xoTg xad' iavxb Xeyofievov, oiov eig ixnog, eig 20 avfrgi ixog. Sax' eiij av dg%r) xav fiev dgi&udv r] fiovdg, xdv fie dgcd-urtxdv ro ev xal xb Hv dg iv aiG&rjxoZg 1 pergit Stob. 1. c. coats uovdg TIZOI dtto tov satdvai y.al . xata tavtd diaai'zmg atgsTctog uivtiv, r) dno zov Siay.tY.gi'a&di xal navzeXcjig jiluovinolrca. tov nlr'/d'ovg svXoymg ly.Xq9ri 11 fort. add. itp'^iavziiv 19 r{\ o A 21 inscr. zlg dgxv agiiruov A, 3 in mg. 22 Stob. eel. I 1, 0 ztveg zd>v dgi9-ficav a.jxrjv arcsrf^vavto tqv uovdSa, zav Ss dgi&aqzmv zb ev, T o u t o ri £ aware ztuvoiicvov tig urtsigov mats za agi&[ir}zd zav ugiduMV zavzr] SiaXXdtzsiv »/ Stvpinei td acauaza zav daoiud-Z'.ov. tlSivai Ss xal zovto x?h ° z l rdtv dgi&u-uiv tlcqyyjaccvto tag agx<-g 01 usv vsatsgoi ti\v ts txovdScc y.al ZTJV SvdSa, of St rivirciyoqzioi 7tdac:g Tzanie to t$',g tdg ztav ogcov ix^ictig\ • ' 2» 20 DE L\N*0 xipveG&ui' cpuGiv eig dzeigov, oi>x ugid-tibv ovde dg ccgxr)v «9 t9',u o'"'» f '^ - ' ®S ul6&t}xov. dote j] uev uovug vo7}tij OVGU dSiuigerog, TO de'iv dgulG9rtrbveig uzetgov xyi]t6v. xul xcc dgi&fitjxd xdv dgibrydv f f i j uv diacpi-5 govta rd rcc uev cdyuxa elvui, ret de uGdpuxu. uzldg 8e aQxicg ugi&ixdv oi pev vaxegov cpudi xi]v re uovudu xul xryv SvuSu, oi de ciitb TTvd'uyogov •xucsc.g xaxa xb e%r}g rug rdv ogav ix&eGeig, oV dv ugxioi xe xul negix-rol voovvxui, oiov xdv iv aiG&rjxoig xgidv ugxw xhv 10 xgiuSa xul xdv iv aiGdiixoig xeGGclgav ztuvxav ugx>)v xr)v texgdSa xul izl xdv u).).cov dgi&udv xuxu xuvxu. oi 81 xal avxdv xovxcov dgxyv XJ)V iiovuda cpuGl xul xb 'ev ttuGr\g antjXf.ayy.evov Siucpogug dg iv dgi&ycig, yovov avrb ev, ov xb ev, xovxiGxiv oi xoSe xb zoibv xul Siatpoguv xiva rtgbg eregov ev TtgoGeiXtjcpcg, af.X' avrb xu&' uvxb ev. ovxa yccg av ug%r] xq xul yixgov eit} xdv vcf iavxb ovtav, xaQb exuGxov rdv ovxtov ev Xiyetuc, y.eruG%bv xrjg ngcortjg rov ivbg ovGi'ug re xul iSiag. 'Agxvrug de xul cI>iX6?.c:og ddiuifogcog r o ev xc.l 20 uovuSa xuXoi'Gi xcd n)v yovdSa ev. of 8e zleiGzoi zgoGrid'iuGi. rd yovc'i8a uvxt\v xitv Ttgdrijv uordSa, dg ovGtjg xivbg ov zgdrtig uovddog, i] icri xoivoxegov xul aiixt) (lovug xcd ev — XiyovGi 8>) xcd rb ev — , rorr-8i a>v dgzioi TE xal itigntol voovvxai. cf. Phot. Bibl. p. 433b 34. Zeller die Philos. d. Gr. I4 p. 313. 335, 1. 339, 4 5 arr/.co; corr. ex aaXav A 11 zuvza A 14 uoiov <(5v)> cvro tv? avzb 'iv corr. in avzolv A TCOLOV ('iv)? 10 v.a9' OVTO TO IV Bull . 17 iavtb: 6 corr. es to A 19 /in^rra-j: Mullach fragm. pliilos. Gr. II p. 117 <Prt6laoi: Boeckh Phi-lolaos Lehren p ; 147. Mullach II p. 5. cf. Zeller I p. 320, 1 21 TC3 uavdda] scrib. vid. aut zfj uoidSt aut : u uoidSx <£ivat> 22 povudos] fi" A 23 o:vzrt A xal TO' ir] ov TO iv? cf. vs.. 14 43 ET UNIT A T E . 21 E'GTIV t) TtQcoTKj y.al vor^xr) ovGia xov evog, ixaGxov xdv ngayudxav zugi'yovGa sv' fiExoyrj yag avx7]g ExaGxov \~v xaX.sixai. dtb xal xovvoua uvxov ovdlv TtagEfiipalvEi ri ? f xal livog yivovg, xaxd Tidvxav de xaxyyogeixai, \aGxs xal T] fiovag y.al EV a r r t , ] xav Ta ulv vor\xd xal 5 zagudEi'yuaxa urjdlv aXXtjXa v diarpe'govxa, xd dl aiG&rtxd. EVIOI dl Ext'gav diarpogdv xrjg fiovddog xal xov ivig xagidoGav. xb filv ydg £ V OVXE xax' ovGi'av aXXoiovxai, OVXE xrj fiovddi xal xoig Ttsgixxoig aixiov EGXI XOV fir) dXXoiovG&ai xax' ovGi'av, OVXE xaxd %oi6xr]xa, avrb 10 ydg uovdg EGXI xal ot>x donag al fiovudsg noXXai, OVXE xaxd xb ZOGOV' oids ydg Gvvxid'sxai aGxeg al fiovddsg aXX.rj fiovddi' ev ydg EGXI xal ov TioXXd, dib xal evixdg XUXEITUI IV. xal ydg si nana TTXdxavi ivddsg Eigrjvxai iv Q>iXrj,3a, ov xagd xb EV eXi'/^i]Gav, uXXd Ttagd xr\v is evuda, rjxig ioxl fiovdg fiEToyrj xov ivog. xaxd xdvxa dr) dfiExd^X^xov xb EV xb dgiGuEvov xovxo ivxr) fiovddi. CJGXE SiacfE'goi dv xb EV xrjg fiovddog, oxi xb fiiv EGXIV dgtGuivov xal xt'oug, aids uovddsg drcaiooi xal dogiGxoi. xdv dl dgi&udv noiovvxai xr)v xgaxqv xofirjv slg 20 di'o' xovg ulv ydg avxav dgTi'ovg, xovg dl xsgixxovg (faGi. xal dgxioi UE'V EIGIV ol EjtidsxofiEvot xtjv eig iGa diaigEGiv, dg 1) dvdg, -r) XEjgdg' nsgiGGol dl ol eig dviGu diaigovfisvoi, oiov 6 e', 6 %'. ngdxr]v dl xdv itsgiGGav EVIOI EcfaGcv xr)v fiovdda. xi ydg ugxiov xa nsgiGGa 25 ivavxiov r) dl fiovdg r)roi TTEOLXXOV EGTIV TJ dgxiov' xal 1 ixdczip? 9 x a l zoig (C'QZI'OI; v.aiy j r f o i r r o f j ? fir] d e l . B u l l . 12 c r n ^ u t i A li il c o r r . e s 01 A 1 5 iv * [ . ' . I I ( 3 M : p. 15 A 10 ucrox>l c o r r . es fiEtozr) A 18 16 , u t » (tv)? 20 i r . ; c r . r r f p i ugzi'ov y.al rz i gixxov A , f i n m g . 2- thilo tea G e l c h r , s o d cf. p. 25, 21 sem. 70, 16 . 19. 7 1 , 3. 72, 20 22 DE NUM. PARIBUS ET IMPARIBUS. dgxiov p.lv OVA uv eft}' ov yug oTiag eig iGu, a ? . ? . ' ovdl o?.ag Siaigeixai' zegirz)) aw. r\ uovdg. y.uv ugxia dl dgxiov TtQoOd'rlg, TO HO.V yivexui 'dgxiov' uovug dl dgzia ngoGxi&euivr} xb rcuv zegirxbv noiel' ovv. agu ugriov 5 ij uovdg uXP.d Ttegirxov. 'AgiGxoxi?.i;g dl iv xa Tlv&u-yogv/.d xb ev <pr]Giv durporioav uexixeiv xrjg epvGeag'. doxia uev ydo TcgoGxed'lv zeoixxbv xoiei, Tteoixxd dl dgxiov, o ovy. uv rjdvvuro, el ui) ccucpoh' xoi~v cpvGeoiv [lexers' dib y.cd ugxioTtigixrov y.a).hi~6&ui xb ev. Gvu-io cpigexui dl xovxoig y.cd 'Agyvzccg. ztegixxov uev ovv Jtgaxrj Idicc iGxlv >) uovug, y.ud-c'cTceg y.cd iv xoGua rd dgiGyiva y.cd xexccyuiva xb rcegixxbv rcgoGuguo^ovGiv' dgxiov dl jxgdxij Idea r] dogiGxog dvdg, y.u&u y.cd iv xoGua xa dogiGxa y.cd uyvdaxco y.cd uxdy.xa xb dgxiov \$ TigoGi.guoxxovGi. dio y.ul dogiGxog y.a?.eTxui i] dvug, izeidr) ovy. eGxiv aGrceg i] uoi'dg dgiGuivij. oi b' eirg enoiievoi xovxoig ogoi r'.-ro uovc'cdog iy.xi^riuevoi xu uixu uvt,ovxui iilv rlj i'Gi] V7tegqj7\' uovdSi yug ey.uGxog c:v-xdv xav Ttgoxigov Tcleovd^ei'-av^cuevoi dl xovg P.oyovg 20 rijg T t g b g d).h'i?.ovg Gyioeag avrdv ueiovGiv. oior iy.xe-frevxav agid'udv cc' j l ' y 6' e' g ' 6 uev xijg dvddog P.oyog T i g b g rt)v uovdScc iGxl diTth'cGiog. 6 dl xijg rgtdbog nge; TT)V Svddu ))uiohog, 6 de Tfjg xexgddog zgbg Tr-vrgidSa ixixgixog, 6 dl xijg xevrdbog T i g b g xrtv xexgddcc ixiri-25 rapro^, 6 Se xijg e^c'cdog T t g b g rijv Tcevxccba i^lrceuTcxog. EGXL 8' D.dxxav ?.6yog b ulv ixi'xeuxxog rov ixixexdgxov, 5 'Agictozilr,;: ed. Berol. fr. 194. Rose Arist. psoudepi-rr: fr. 184. Heitz fragm. Aristot. 115. cf. Zeller I p. 3GS, 4. " i l 2- p. 43 Tzv&ayogiy.a: a corr. ex av ut vid. A S rfvGicw mut. in tfvctoiv A 17 ret avid del. esse cj. Hultscli IS {lovd ydg corr. CI jioYoi A DE NCMERIS PUIMIS. 23 © dl ixiXc'xagxog xov ixixgixov, 6 dl ixixgtxog (xov) tlfiioXiov, 6 dl i}fii6?.iog xov dixXaGioV xal iitl xdv loixdv dl dgt&udv 6 avxbg loyog. iva).).d% d' siGiv cXlt'i/Loig OL xe dgtioi y.al of Ttegixtol nag' eva &ea-QQVUZVOI.. s xdv SI dgt&udv OL fiev xgdxoi xa).ovvxai dx).dg xal aGvv&exoi, of dl ngbg dXhjXovg xgdtOL xal ov% dxldg, of dl Ovv&exoi d-xXdg, OL dl xgbg avxovg Gvv-%exoi. xgdxoi filv dzf.dg xal aGvvd'exoi OL vxb u,i]Se-vog ftev dgi&fiov, vxl fiovtjg SI fiovddog uexgovuevoi, io dg 6 y e' £' ia' iy it,' xal of xovxoig ouoioi. keyovxai dl oi avxol ovxoi ygatiuixol xal ev&vfiexgixol Sid xb xal xd urjxrj xal xdg ygauudg xaxd uiav diaGxaGiv %ea-geiGd'ei' xalovvxai Sexal xegiGGaxigxegiGGoi' aGxe ovo-(idleG&ai avxovg xevxuydg, xgdxovg, aGvv&exovg, ygau- is fiixovg, ev&vuexQiy.ovg, negiGGuxig xegiGGovg. fiovov SI ovxag xaxaaexgovvxca. xd ydg xgia ovx dv vx d?.?.ov xaxaiiexgtiLrei'ri dgiiriiov aGxe yevvt]ift]vai ex xov xoM.a-xlaGiaGfiov avxdv, rj vxb ficvrjg (lovddog' dxa\ yag rgCa xgia. buoiag Si xal ana% e e', xal dxa% %' t,', xal 20 axa% ia' ia'. Sib xal xegiGGaxig xegiGGol xixhyvxai' oi xe ydg xaxafiexgovfievoi xegiGGol 77 xe xaxapexgovGa avxovg uovdg xegiGGrj. dib xal xgdxoi xal aGvv9etai liovoi of xegiGGoi. of ydg dgxioi ovxe xgdxoi ovxe aGvv-Sexoi ovxe vxb fiovijg fiovddog uexgovfievoi, aXXaxal vx* ss 1 TOV^ add. apogr. 2 xal supra vs. add. A 6 inscr. srtpi BQWZOV xal davv&izov A, s ac max ad significanda ^naUaor genera a § y S in mg. 8 scr. vid. 01 Si avvdszot arzXdig xal nQog avxovg, of Si ngog dXXqXovg cvrftezoi 9 a Eg. A It aj] ovzcog A 16 jidvoi A 13 noXXaaiaaiiov A, em. apogr. 24 NUM. C03iru£ril.< &7.).av dgi%\idv oiov xaxgdg ulv vxb dvddcg' dig ydg fi' 6'' i|«S dl vxb dvddog xal xgiddog' dig ydg y y.al xglg $ g" xal of koixol dgxtoi xazd zu avxa vxo xivav uzi-£6vav xrjg uovddog dgid-udv xaxauazgovvxui, x7.r)v n~g & dvddog. xavxtj ydg uovrj Gvufitfitixev, bxeg xal ivioig xdv xagiGGav, xb vxb uovddog uaxgeiG&ai uovov' uxc.% ydg p" /5'* deb xal xegiGGoeidt)g si"grtxai xavxb xotg xiOi.0-GoigxExovQvia. xgbg d7J.tj7.ovg dl liyovxai xgdxoi dg:%-uol xal ov xa&' avxovg of xoiva uixga uaxgovuavoi xrj 10 ftovacTt, xdv ix' dkkav xivdv dgidudv dg xgbg iav-xovg xaxauezgdvzai. oiov c r{ uazgalzai ul.' xal vxb xav p xai o , xai o # vxo xav y , xai o i x.xo xav p xal E'' l%ovGi dl xal xoivbv utxgov xal xgbg d7J.t\7.ovg xal xgbg xovg xa&' iavxovg xgdxovg xt)v uovdd.c y.cu ydg is dxa% y y xal dxa% rf r{ xal axu.% &' xal ixat, i i. Gvv&axoi d£ aiGi xgbg iavxovg of vxo xtvog ih'.z-xovog dgid'uov uixgoviuvot, dg b g vxb d >ddog xc:l rgiddog. xgbg (tX7.ij7.ovg dl Gvv&azoi of y.oivc, dxivioiv ue'xga uaxgoviuvoi' dg 6 if xal 6 [xal 6 # ]" xotrcv 20 ydg i'yovGi utxgov dvddc [y.al xgiddci]' dig yd-) y y.al dig d' rf [xal xglg y %'\ (xal b g' xal 6 ir'')> xonxv ydg aizdv uizgov r) xgiag' xal ydg xglg /3' g' xal xg\g •y ovxe dl r] fiovdg dgi&ubg, d7.7.d dgx>) dgi&uov. OVXE i] dcgiGxog dvdg, ngdxt] oiiGa ixEgoxtjg uovddog 25 xal uqdlv avxijg iv dgxi'oig dgyixaxagov exovGu. xdv dl Gvv&e'xav xovg ulv vxb dvo dgi&udv xegisxouivovg xaXovGiv ixixidovg, dg xaxd dvo diaGxuGitg &£agov-4 d@iduuv corr. es c'oi9udv A 8 jl rag. A_ 14 y.ci] «os A 15 inscr. itsgi evv&irov dgi9 aoi A, £ et J in eg. 18 8 mg. A 23 sqq. ovzs 8\ — ijotoa fort. del. DL" Nl-M. r.UUU.M GENEIUUUS. 25 ue'vovg xcd oiov vxb utjxovg xul xXc'crovg itegiexoye'vovg, XOVg 8l 1.10 tgiCOV OTcQCOig, dg Xul X))v XQlt)]V SlUGXU-Giv xgoGeiXiiepozug. neg:.ox*iv Si xuXovGiv ugi%y.dv xbv SI' ccXXijXcov avtdv Xoki ::XuGiuGu6v. rdv Si ugxiav oi IUV eiGiv ugxiuxig dgxioi, of Si 5 xegixtuxig ugxioi, oi SI ugtiOTCegixxoi. ugxic'cxig uev ug-XIOL [ro Gi]ueTov xovxo eGtiv] olg xgicc Gvufie'jhjxev, e'v x6 t'-to Svo ugricov izt uXXrjXovg itoXvxXuGiuG%svtcov yeyevrJG&cu, Sevxegov xb auvxa ugxia e'xeiv xcc ue'gy us'xgc Ttjg eig uovuSa xuxuXrjlecog, xgixov xb yr]Slv uv- 10 xdv us'gog budvviiov eivui xegixxd' bitotoi eiGiv 6 Xp1' |<5' gxrf xcd oi clrtb xovxcov i^ijg xuxcc xb SmXccGiov Xuufiuvouevoi. xcc yccg A/3' ye'yove ulv ex te $' xul t}\ a sGxiv UOTICC' UE'O»J Se uvrdv Tttlvxa ugxiu, ijiuGv ig' , xe'xugxov 6 >/, oySoov 6 S'~ uvxu xe xcc uogiu oudvvua 15 dgxioig, ro xe rjuiGv dg ev SvuSt, Q-eagovyevov xul xexugtov xul oySoov. 6 Se avtbg Xoyog xul ixl xdv Xoixdv buoiag ugi&udv. dgtiOTcegittoi Se eiGiv oi izb SvuSog xal rcegcxtov oixivoGovv uexgovuevoi, oftiveg ex zavtbg Ttegixxcc ue'gyj 20 tXovGi xcc rjuiGeu xuxcc xt)v etg iGu SiuigeGiV dg xa Slg % iS'. dgtiuxig ulv yccg oi/tot xuXovvxcci aegixxoi, izel V7tb xijg SvuSog ugticcg ovGrjg uexgovvxai xcd stegiGGov xivog, 6 ulv Svo tov evog, 6 Se g' roi> y', 6 Se 1 tov 1, 6 Ss iS' tov Siuigovvtuc 81 ovxoi xi\v itgdxryv 25 6 inscr. nsgl tfjs zd)v dgzioiv Siacpogdg A , ij in mg. 6 inscr. nt gI rw v dgzKIKIJ dgzicov A . cf. Zeller I p. 366 (lev Sgzioi A] u-iv dgziov apogr. 7 ofj] to A 9 to apogr.] zov A 11 6u.avv[iov: 01 corr. ex 0 A negizzd: a corr. ex ow A 14 15 corr. ex iy A 18 inscr. itegl agtiOTtegCzzmv A , I in mg. 20 fiegrj del. Haltsch 26 DE NUM. UUADUATI5 SiaigeGiv sig negixtov, uetd 5s xt)v ztgdrrtv eig iGa 8iaigeGiv ovx eti Siuigovviui. xdv yug g' xa ulv y rjuiGv, rd Ss y ovx eti eig i'Ga Siaigeitui' uovdg yug ddiaigsxog. 5 JtegiGGaxig Se dgxioi EIGIV dv 6 noV.uTtXaGiuGubg ix Svelv dvxtvavovv xsgiGGov xal dgxiov yivexai, xal TtoXlanla6taGd,evxeg eig i'Ga uev ugxia uigi] Si%u Siui-Qovvrui, xaxa Ss xdg xleiovg SiaigeGeig a uev ugtia ue'gri, a Se TtegiGGu e^ovGiV dg 6 t / 3 ' xul x' xglg yag io 8' t/3', xal nevxdxig 8' x' xal rd uev i[i' Si%i~] Siuigei-xai (tig) S xal g', xgi%rj Se eig 8' xul 8' xul 8', tetguyjj Ss eig texguxig y' xa Se x Siyji uev eig t, xsxguyjj 81 eig s', 7ievxayjt\ 81 eig S'. ixi xdv Gvv&ixav dgi&udv oi uev ladxig i'Goi eial 15 xal xexguyavoi'xal iziiteSoi, ixeiSdv tGog izl iGov 7to}.XuxkuGiuG&elg yevviJG)] xivd dgi&uov, [6 yevvtj&elg iGuxig xe iGog xul xetgdyovog iaxiv] dg 6 S', sGxi yug Slg fi, xul 6 sGxi yug xglg y' oi Se dviGuxig uvi-Goi, ixeiSuv dviGoi dgid'iiol ix' d}.h\).ovg xoD.anla-20 GiaG&dGiv, dg 6 g'' sGti yug Slg y q. xovxav 8e ixsgouijxeig uev eiGiv oi xt)v exeguv nksvguv xi)g exigag uovuSi ueifrvu exovxeg. sGxi Se 6 xov negiGGov uoiduov uovuSi itkeovutov xul uoxiog' 5 inscr. n s p l neQteadxig dgxtav A , tor in mg. 11 inscr. Wept lodxig i'atov A, t(? in mg. 15 xal {'m'sscToi fort. del. yap post intiSdv add. A 3 1G yivvijcr]: arj corr. es os A 18 inscr. J l f p l xwv avicaxi; aviacov A , ty in mg. 21 sqq. cf. Cantor matheuiat. Beitr. znni Culturleben der Volker p. 105 sqq. inscr. rrfpl I rt QOU T^Y.WV (corr. es f t j -poujjxcoJ') A 23 tov nsgiaaov doiduov mut. in rc5 rtsgiaa^ dgi9[ia A ET ALTERA PARTI L0NC.I0RIBU3. 27 Sib uovov dgxioi ol EXEgo^/.Eig. r\ ydg dgyjl xdv igi&udv, XOVXEGXIV r\ uovdg. zsgiGOr ovGa xr)v ixsgo-Ttjxa fyxovGa xr)v SvdSa izcQou,i\xr} rd avxtjg SixX.a-GiaGud Ixoirfis, xal Sid roT-ro ij Svdg xr}g uovdSog ixsgourjx^g ovGa xal uovdii ixsgsxovGa xovg dgxi'ovg 3 agttruovg xdv rcsgiGGdv EXcCoiirjxEig noisi fiovdSi vittg-tyovxag. ysvvdvxai ds Sixdg. ix xs JtoXXaxXaGiaGuov xal imGvv&sGEag. ix ulv iz^Gvv&sGscog of dgxioi xolg iq>E£,i]g iziGvvxi&iusvoi xovg cxoysvvauivovg noiovGiv ixsgourjxsig. oiov ixxsiGitaczv dgxioi xaxd xb i^tjg ft 10 6' g r[ i t p " i$' is' trf' yh-ovxai Ss xax' ixiGvv&sGiv p xai o g, g xal g ip, i? xai ij x, x xai i A aGxs tlsv av ol ysysvvriuivoi ixEgouijxstg g' t / J ' x A ' . 6 Si avxbg Xoyog xal ixl xdv ilrtg. xaxd SI noXXanXaGia-Gubv olavxol ixsgour\xEigyEivdvxai xdv iapE^rjg dgxiav 15 ts xal Ttsgixxdv xov xgdxov ixl xov Qrjg noXXaxXaGia-tflusvov oiov a p y S s z ^ r i & i ajrag usv yag §>' p " , Sis Si y g, xglg (Sly S' if}', xsxgdxig Ss s' x', nsvxdxig Ss g' A ' " xal istl xdv Hijg 6 avxbg Xoyog. ixs-Qouijxsig Ss of xoiovxoi xix".7:vxai, ixsiSr) xgdxtjv ixs- 20 QOtrjXtt xdv xXsvgdv x] xgccd-ijxt] xfj sxiga xXsvga TTJS y.ovdSog XOIEI. nuQaXXrjXoygauuoi Si zlGiv agi&pol of SvdSt iq xal psi^ovi dgi&fid xt)v higav itX.Evgdv xijg ixsgag 1 Sto corr. ut vid.'ex Svo A 3 JrEoofi/fxTj: 0 corr. er ca A _5 ttsooftjjxijs: ourjxrj in ra.s_, too corr. ut vid. ex ega A 6 tcov izcgteacov nrut. in za xcc'-vaa A 9 ditoyii>vo}[iivovs: a corr. ex o A 11 g* ijl « I : ^ »S oS "i supra numerorura sen'em add. A 13 ytyevrjuirct A, em. Bull. 16 xov apogr.] xav A 18 Si add. Bull. 20 srotatijv corr. ex izgazov A 23 inscr. TCIQI nagaXX^lc-rgduucov dgtQ'y.av A, tl in tag. figura3 inutiles add. A* 28 DE NUM. QUADRATTS vnioexovCca1 ixovxsg, dg 6 Slg 6' xcd, 6 XExgdxig g' xcd 6 i^ccxig if xcd 6 oxxdxig i, oixivig EIGIV 6 rf xd' urf TI . TErgdycovoi EIGIV oi ix rdv xccrd ro ilgrjg xioiaadv iniGwri^Euivav aX).r\koig yEvvdiiEvoi. oiov ixxEiG&u-5 Gav icpelgrig nsgiGGol a y e' § ' &' id' iv xcd y 6', og iGti TErgdyavog, iGaxig ydg iGnv iGog, rovxicn dig fi 6'' & xal e 6g xal avrbg TErgdyavog' EGzi ydg rglg y & xcd £ i$', og xal avrbg TErgdyavog ion' TErgdxig yhg ig'' ig' xcd XE', og xal airbg TErgdyavog iGn xcd io iGaxig iGog' EGXI ydg TtEvxdxig s XE'' xal uixgig c'.ZEigov 6 avrbg Xoyog. xc.rd idv ovv iziGvv%EGiv ovxag yevvdv-xai oi TErgdycovoi, rdv itpE$rtg xegiGGav rd yswauiva ditb uovdSog XExgaydva TtgoGriv'Euivav' xard zo?J.uz?.a-GiuGubv Si, izuSdv bariGovv dgt&ubg icp iuvxbvzo/./.cc-15 7t?.uGiccG&>l, oiov Slg fi S', xglg y' &', tEzgdxig cY ig'. ot ulv ovv TErgdycovoi zdvxsg xovg ixsgouijxeig TCEgiXaufidvovGi xaxa n)v yEauExgixrtv dvaXoyiav xcd uiGovg avrovg noiovGi \TOVTEGTI rovg uovdSi UEi'^cvag TI\V ixigav nlEvgdv rijg irigccg vZEgixoiTug]' oi SE 20 ixEgoutjxsig ovx ixi rovg XExgaydvovg ZEgi?.c:ufidvovGiv dg uiGovg eivai xard dvaXoyiav. oiov a' fi y S' e. ovroi rd ulv ISicp Tth'ftEi zoXXazlccGialouEVOi zoiovGi TEtgaydvovg' axa% XE ydg a' a' xcd, Slg fi S' xal rglg y %' xal TErgdxig S' ig xcd ZEvxdxig E' XE'' xal ovx 25 (xflaivovGi xdv idiav cgav i] XE ydg Svdg iavxi]v 3 inscr. nzg\ zszgayiovcov dot9iiav A, is in mg. r f -Toaycovot (8iy ilaiv? 5 8 & ts X E /.? s u r r a n u m e r o r u m s c -riem add. A oj apogr.] 3 A 16 inscr. O T I of zizQciya-vot (lioovs zovg tzegou rtxt is Xaupuvovaiv A, tj i n r u g . 18 fiEt'Joiot Gelder 19 vxtpjxovza;] t^01'1'-* apogr. 22 ovrot] ourcaj oi A 25 rov idiov ogov A 2 , ET A L T E R A P A R T E L J N O I O R I S U S . 29 idvciCi xal »; xgidg fcrrjjj ' ixgiaGiv, OGXE EIEV dv XE-xgdyavot of ci d' &' iz xe'. ueoovg di e%ovGi xoig ixtgour-xeig ovxag. XExgdyavoi dvo i^slrjs o XE a xal d'~ xovxav UEGog tXEgourp^g 6 fi' XEiG%aGav dr) a' fi d'' uiGog yivtxai b fi, r a avrd t.oya xdv uv.gav TOV 5 ulv i'XEgiyav. vtp ov di vxEoey/uevog' xov ulv ydg ivbg TO. fi dix/.aGic, rav dl fi ra d'. xaXtv xExgdya-voi alv 0 d' xal usGog dl avxav ixsgourx^g 6 xiiG^aGav dt) d' z' •& ' uiGog 6 z, xa avxa ).6ya xdv axgav xov ulv [ydg'] vxEgi'xav, i'<p' ov dl vxEgeyouE- n vog' xai u\v ydg d' xd z' rtuir'J.ia, xdv dl :' xd fr'. 6 dl avxbg /6yog xal E.TI xdv i%rtg. 01 dl ixBgoutjXcig. vxb xdv x~; uovudi vxcg?%6vTav xo).?.cx).aGiut6u.£voi. OVXE uivoi'iv iv xo'g idi'oig ogoig OVTE XEQiiyovGi xovg xtxgayd 10 -g. oiov xd dig y yivvd xov z' xal rd xglg is d' ycvva T'JV ifi xal rd xexgdxig B yevvd xhv x, xal oidelg avx ~n> UE'VEI iv rd iavxov ogp, d/./.d UEXUXI'XXBI iv xd zi'A'i xx/.CiG:aCfia. oiov dvdg ixl xgu'.dc: xal xgidg ixl xixgdd : xal rirgag ixl xivxc'.da' 01 XB yEvvduzvoi vxb xdv itBQou^xav ov XEQi/.aiifidvii'Gi xovg xsxgayd- 20 vovg agi&iiovg' oiov e ' e r ^ 3 ingou^y.Big fi UExa%v dc avxav iGxt xrt xdzei Texgdyavog b d'' d?./.d xax' ovdcuiuv dvc.7.cyiuv Xcgi).uu~)dvEtci vrr' avxav aGxa iv xd avxa i.oya xgbg xd dy.ga BXVUI. iy.y.EiG%a ydg fi m d 5'' JJ XBxgdg iv diatpogoig ?.6yoig xgbg xd dxga 23 yevrjGBxai' xdv ulv ydg fi xd 6' dtx?.daia, xdv dl d' 9_ f i t o o ; o] ret A 1 0 y a p o m . apogr . 2 0 r r o zav iriQoiir*ai- del . v i d . 23 ovii ui'av A , em. apogr . 24 a v T O . s ' l p n vs. a d d . A ixv.iioda yap] i•A.y.iiodajnc-.v apogr . 2'> zu d n'.daia A , e m . ap'jgT. ;}0 DE NUM. OJiLONGIS. tic g' rju.io7.ia. Tvu Si dvu7.6yag UE'GOV dii aire ovtcog UEGOV sivui, coGts ov EXEC f.oyov to rtgdxov rrgbg to ue'tiov, tovrov to UEGOV ztgbg to tgirov. ztu7.iv rdv g' xul 1$ irEgoiuyxcov UEGog rjj tc'\ei tixgc'r/covog I &\ 5 c\)X Oi'X EVQcU/jGsrui iv rd avrd 7.6yco xgbg tec UXqW g' &' ifi' tdv ulv ydg g' rcc &' ?; in o'Atr:, rdv 61 ©•' rcc ifi izitgita. 6 61 avrbg xul inl tcov i^ijg 7.6yog. 7iQOUi'jxrtg Si iariv ugiQubg 6 i'7tb Svo uviGav dgiQydv cc7toT£?.oviievog dvrivcovovv, rj uovdSi y SvuSi 10 i] xal 7t?.ciovi roi' txigov tov 'ixigov VTtsgi'xovxog, dg 6 xS', sen ydg ilgdxig 6', xcd oi rocovxoi. ioti 61 rgiu uig>) tdv 7iQOii)']Xcov. xcd yao rtug ixegoiu'ixtjg Ttgout)-xrtg, xcc&b usi&vu t>)v ixiguv 7t7.Evgdv xijg irigecg exu-date EI iiiv rig ixcgoutjxtig, ovxog xcd Trgouiyx/jg' oi-ls urtv uvv.Ttcc7.iv' b yug usi^ovee 7t7.iov >j uovuSi T))v EXE-guv E"XCOV 7t7.Evguv ngoury/.^g UEV, or in\v irEgoiiiy/.^g' )}v ydg irEgoui'fXtjg 6 uovuSi UEI&VC. x>,v ixigcv Eycov n?.Evgdv, dg 6 g'' EGXC ydg 6\g y g\ in rcgouijx^g xcd b xccrd Sicccfoguv 7to7.7.ux7.c:Gic:Guvv TTOXI UEV uovddi 20 Ucitpva ttjv itiguv 7t7.cvgdv ^.iyjovy, TTOXE 61 TC7.EIOV ij fioi'f'.di* cog 6 «/3'* EGXI yug xcd xglg 6' xcd 6lg g', dors xard uh' to rglg 6' siti av irfgoiny/.^g. xard St rb Slg g' Ttgou>]xi]g. in Tigouijxtjg iGxlv 6 xc.xd ztc'iGccg rdg „ GyiGEig rdv 7io7.7.cc7t7.aGic.Gudv stP.iov >}' uovdSi usit'oi'u 25 tyv irigeev iycor x7.cvgc'cv' cog 6 u' xcd yug TErgdxig 1 3 T O TQt'tov a p o g r . ] TOI- zgiror A c< 9' iff f o r t . c\A. S inscr. J i tpl rcooiir^/.Cor ( c o r r . cx S P I N J'ZWI-) dqi9uu>r A , Hj in mg. QSvdSt a p o g v . l tin' A 12 uiot-] - , 'f 'n,? 11 oi-ro; corr. ex OU'TOJJ A 10 xoXlcizluoiwriics A, XCTU Suln 00 or lroXXarzXccoiczaiiov Hull . 20 t'^ tuv add. a p o ^ r . 25 xngdy.i A . em. apogr. D E N U M . P L A X o I t L M G E N E I J A T I u X E . 31 xcd xsirdxig ij' xal dig x' oGng>xcd uovog dv EI">] xgo-urjxtjg. £TiQOur,xTig ydg iGxiv 6 ix xdv i'Gcov dgid-tidv rr)v xgdxtjv Aaupdvav ixsgoxrixa' i] Se xijg uovddog xa ixiga dgi&ua xgoG&tjxtj xgdxyyv XOIEI ixsgoxrixcc' Sib oi ix xovxav xvgi'ag axb xijg xgdxrjg xdv xXsvgdv 5 ixsgoxyxog Exsgout'jXEig. ot Ss x).£ov rj uovdSi xi)v £x£-gctv xXsvgdv usi'lova syovxsg Sid xbv ixl x).£ov xgo-fiifiaGubv xov urjy.ovg xgourjxsig xixhpxai. EI'GI SE xdv dgi&udv of usv ixixsSoi, oGoi vxb Svo dgi&udv> xoD.axfoiGidtpvxui, oiov firjxovg xal xXd- io xovg, xovxav Ss of usv xgiyavoi, of Ss xsxgdyoovoi, of Ss xsvxdyewoi xal xaxd xb i^rjg xoXvyavoi. ysvvdrxat Ss of xgiyavoi xbv xgoxov xovxov. [aGxig] of itps^tjg dgxioi aV.tjXoig ixiGvvri&t'usvoi xaxd xb i^tjg ixsgout'jy.sig dgifruovg xoiovGiv. oiov 6 i> fi xgdxog dgxiog' xcd sGxiv ixEgou)jy.rjg'. sGxi ydg dxu% fi. eira xofg fi dv xgoG&ijg 8', yivsxca s', og xal avxbg -c"rcpoa>('z);j" tGxi ydg Slg y. y.al fic'xgig dxsigov c avxbg ?.6yo:: ivagysGxegov Si, dcxs xuGiv svGvvoxxov sivc.i xb Xsydusvov, Set'y.vvxai xcd X^SE. xgdxrj Svdg 20 EGxa cP.cfa ixxEiuEva Svo xdSs' a a xb G%ijuc< avxav sGxai ixsgoiirjy.Eg' xaxd usi' ydg xb ui]x6g iGxiv ixl Svo, xaxd Ss xb xP.dxog icp' EV. UEXU xd Svo iazlv dgxiog 6 S'' a idv xgoaDditsv xoig xgdxoig 25 1 fio'yov? 3 Ictu^ivov A, em. apogr. 4 3roco'r>jv coir, ex Tzqiotov ut vid. , ar.tea una 1 It tr. era-a A 9 inscr. mql imrcidiov dni&anv A, i& iu mg. 13 inscr. 71 to I roe-y o) v to v a Q 10 u co v n co £ ytwdivtat x a l TTSQI t<ov t £ ij 5 TcoXvyo'ivwv A , •/. in mg. 23 post iarai compendium eius-dem voculae erasum in A 24 itp'] if' A 25 JJv A 32 DE NUM. I'L AN. fn'.NKRATIOX'K. di'o uLqa [cc' a'J xcd jitgiftduLV rcc 6' T O T ^ yivexcci £ragoutjxeg T O rcjv g' e^ijiu." zc:n\ u;!' y .^p T O ui]xog yivexai- irxl rgic-:, xccxd dl ro rr/.f'r c-'.T! /3. /|>j.j c'cny dgxiog uixd 8' Q g'- «i> rrpoffO,^ -; rcrr ; : roTg ttgdrag 5 yivexcci 6 t/3', v.uv TtegiQ^g ui'xd roTg rtgdroig, e'Gxca G-j\\uu iteguyiy/.eg' dg i'/jiv rain: xc.xd r6 uiy/.og ulv 8', xard nldxog 81 y. xcd I'-'y^ig dzcigov b uvxbg P.uyog xccrd T))v rdv dgxiov imoiYf>fju: a a a a ci ct a a a tt a a a r: a a a u nd7.iv 81 oi t^g lUgiGool u7J.i't7.oig eTriGvvTi&e'uevoi io rexguydvovg TCOIOVGLV dgi&uovg. aiol 81 oi ecpe^tig jcegtGGol cc y e' f id. xccvrcc da ecfc^ijg Gvvri&alg nonJCfts tcTQC.ydvovg dgi&uovg. oiov rb av Ttgdxov rargdycovov eGxi ydg v.crcc% h> av. airu TtagiGGcg 6 y' rovxov uv rfgocd fig rbv yvduovc. rd avi, Tton'iGetg la rargdycivov iGc'cxig i'oov' (Greet- ydg xccrd u'yxog fi' xc-.i xccrd :r?.c'rog fi'. i'pe^g TtegiGGog 6 a'' rovvov dv Ttegc-&>,g rbv yvduovee tco 8' zaxguyuvoi, yavrficxci zc'J.iv rargdycovog 6 xcc] xccrd ury/.og eypv y xul xccrd Ttldxog y. itpe%i\g ztagiGGv-g 6 roixov uv zgoJdrtg so rd %•', TtotaTg rbv ig , xcd xc-.ru uiyxog 8' xcd xccrd 7t7.d-rog 8'. b 8a avrbg 7.6yog ue'xgig UXEL'QOV. a a a it ct a a a a a a a a a a a a a a a a a a a a tt u a u xard ravxu de uv f t > ) uovov rovg iye^tg dgriwg 2 yep supra vs. A 5 ittni^f^ Bull.] rooeftfj; A xo?s jrpco'rojs] tof; =''{ cf. vs. 1 G ta t ra d«! . vid. S fig. semper lineis circumscr. A 14 jrjgi&jji Gelder 22 tavxU corr. es xavxec A DE N U M . TKJANG. GEXEKATl'jNE. 33 ur,dl unvov xovg irf£Z>]g xioiGGnvg, dU.d y.al dgxiovg y.al xcgiGGovg cJJjjloig ixiGvvxi&duiv, xgiyavoi Jjiuv f!.)!lhio! yiVi]G0VTKi. iy./.ciG&aGav ydg iqe^ijg XEQLGGOI y.al dgxioi, a fi y d' e g' tf rt & yivovxai xaxd Tt)i' rovrav GvvxTEGiv oi xgiyavoi. xgdxyj ulv y) uovdg' 5 avxr\ ydo, EL y.al uy) ivxc7.E-/da, dvvdusi zed via E'GTI'V, dnpi xdvxav dgi&udv ovGa. xrtg dl £\t]g KIT/} dvddog TIQOOT*V-n'ji;g yivixui xgiyavog 6 y'' Eire xguGftig y', yCvixai g' % ilxa xgoGftsg d', yi'vovrc.i i' Eixa xgoGdeg £, yivovxai IE' aixa xgoGd-Eg g', yivovxai y.a' Eixa 10 xgoGdeg %, yivovxai xtf' eixa xgoG&Eg if, yivovxai ?.g'' ELXU TtooG&Eg {>', yivovxai us'' Eixa xgoG&sg 1, yivov-xai VE'' y.al Uc'zQig dxn'gov 6 avxbg Xoyog. dijXov dl oxi xgiyavoi o t r o ; 01 dgi&uol xaxd xbv Gx>iuaxiGii6v, xotg xgdroig doirnioig xov i(fE^y]g yvduovog TCQOGU&E- IS UEVQV' xal EicV dv 01 ix ti]g ixiGvvd'tGEag dzoyevvd-UEVOI xgiyavoi aids' y' g' 1 IE' xa xtf ?.g' UE' VE'. xal ovxag ixl xdv tl^g xdv UE' y.al VE'. a a u a u a a a ct rt a ft a a a a a ct •-: a a a a a ct a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a x i j ' i s ' a a a a a a a u a a a a a ct a a a a a ct a a a a a a a a a a a a a a cc a a ct ct a a cc a a a a a ct a a a a a a ct \t a c t c t a a u u e t 4 y i Y r n T K i A 1 G ivzsXsxsi'a corr. cx tith'/.kxiia A 10 yiVovtc : ! y.a: yi'vmxai coinpondio ser. A ut in B0 'iU'?ntibu3 IS TWV us' y.al ri' de l . Til. ' 34 DE NUM. MULTIAXG. GENHUATIONE. OL de TETQcr/orot yevvdvxai uev, dg Ttooeigjjxcu, ty. rcov icpetyjg ft.ro uorc'.dog T t e g i x x d v cc?.h]?.oig ixiGvvxi-& e u i v c o v ' Gvufie'jitjy.E 6} uvxoig aGxe ivu7.).cc% rtccg' eva icgxioig eivcci y.u\' zczgixxoig, aGxtg 6 nag ccgidubg TCCO' 5 i'vcc ccQXiog eGtiv y] Ttegirxog' oiov cc 6' &' ig y.e ).g u&' Ttcc g'. xi} dl; tczb [iovuSog xuru xb i\yjg iy.QeGet, rcov u g x i c o v re y.cd T t e g i x x d v icgi&udv Gvufjiftry/.e, rovg y v d i i o v c c g rovg Svc'cSi ulh'J.av V T t t g i ' / o r x u g ev xf] Gvv-xTeGei x e r g u y c o v o v g ccTtoxekeiv, ag e z t c i v a uTtobiSeixrui' io vTtegi%ovGi- yccg Svc'tdi c'.k).y\).av v.ztb uovuSog c;gi<>iievoi <^of> TtenixxoC. b u o i c a g de OL xgtddi c'.t.).y\).tov vixegi'iov-reg iv rj] Gvv&tGei ccTtb uovc'cdog Ttevxccycivovg ccTtore-AovOiv, Quydvovg ds OL xeroc'di, c c i e i re y) vrtegoyjj " •. rdv' y v a u o v a v i t , av ccTtoxe/.ovvxui oi n o l v y a v o c dvadc 15 J.eiztixccL xov TC?.y}9ovg t c o v ccTtorekovuivcov yavid'v.. exigee de ziccXiv iGzl rcc^ig i v roTg not.vydvoig xdv ctTtb uovuSog ztoV.c.TtXaGtav ccgi&udv. rcov yug r ' r ro uovuSog TTO?.?.CCTI).UG{COV , liyn de SiTt/'ccaiav rgtTtluoiav xcd rdv itijg, oi nev eve: Ttccg' tvee d'tcJ.eirrovzeg cgtOtwl 20 retguyai'Oi Tfc'vreg eiGt'v, of 61 dvo diulshovreg xi'tJot Ttc'cvteg. oi Se nivre SicckeiTtovreg xvpoi uiuc y.cd rexgd-ycovoi eiGt y.cd rug uev nXevgdg e%ovGi rerguydvovg 1 inscr. jrfpl nor fi' j? rtclvycovcov A, YAC in nig. 4 9 (J At _ . u y E £ 9 ta in nig. A rpofi'p^rr.i: p. 2S, 3. 32, 9 5 in nig. sup. cod. A.hace scripta sunt: (J & ig rt u$ySiZ£r]9i 8 9 t~ y.t rfrpa'- ;cojoi a (I y 8 f ? J r> 3- t ice if ty ( if If xi-.ruyiuroi a /? y 6 f s J I J 9 i ta i§ ty iS if i ; i j s if x7j £if i^ctyavoi 12 artvzaymvovc corr. ei ztrQuycci ov; A 13 j) supra VF. A DF. NUM. Q l ' A D I ! AT!S E T CCRICIS. 35 dgt&ttovg y.vfioi bin;, xtxguyavoi dl ovxig dgiiJuol xvfiixdg tfrovGi rag rc/.svgdg. oxi dl xdv xoXJ.axXuaiav itgiQudv of ulv nag' i'va dnb uovddog xrtgityavol tiGiv, of dl Tcagd fi xvfioi, of dl nugd e' xvpoi due. xcd xt-xgdyavoi eiGi, di\Xov ovxag. iv ulv xoig dinXuoioig, 5 xnuivcov altiovav dgi&udv oiov a fi y d' a g' f r\ tr' i ' ice ifi iy id' ie ig t£ ' i i j ' tfr' x' y.a' xfi xy xd' xe xgdxog dinXc'cGiog 6 fi' tixcc 6 d', bg ion xtxgdyavog' ilxct 6 7} , og iGxi y.vjlog' tixcc is', og £GXI xtxgc'cyavog' tixcc 6 ).fi' ut&' ov 6 lid', bg iGxi xixgdyavog ccua xal io xvfiog' tixcc gy.i\' pad-' ov Gvg', og iGxi xaxgdyavog' xcd ui'xgig dxeigov 6 avxbg Xoyog. xcd iv xa xgirtXuGia tVQe&)'tGovxai oi nag' era ttxgccyavot, xcd iv xa mv-xcc-Xaoicp, xcd y.axic xovg i^tjg xoM.axXuGi'ovg. ouoiag dl cvgc&i'iGcvxai xcd of (Jt'o diaX.eixovxig iv xoig TCOX7.CC- IS nXc.Gioig xvfloi ndvtag, xal of t diaXtixovxig xi'fiot "ciicc xcd xtxgdyavoi. idio}g dl xoig xaxgaydvotg Gvufiifri/.ev i\xoi xg'xov a'xciv rj uovddog chfccigad'aiGijg xgixov tx*iv Ttdvxag. i] xdX.iv xixagxov e'xeiv rj uovddog dcf.aigaQaiG7ig xixagxov e"%£iv xc'cvxag' xcd xbv [ilv uovddog dcfutge- 20 dtiaqg xgixov i'fovxa iyiiv xcd xixagxov ndvxag, dg 6 d', tov dl uovddog dcfaiga&aiGrjg xixagxov £%ovxa i'xaiv xgixov Ttdvxag, dg 6 &', ij tbv ccvxbv 7cdX.1v xcd xgixov txtiv xal* xixagxov, dg 6 Xg' [ij urjdt'ragov xovxav tXovxa xovxov uovddog dcpatga&tiGtjg xgixov txttv ndv- 25 5 iv fiiv xoig 6ixXtxc/ois SftXov ovxcoqf 6 pro hac numero-rum serie Gelclcrus hanc posuit: a' §' <$' rf J S ' X$' |<J' Q-ATJ cv;' 13 iv ttu atvxunlucuo A 1 ] 0 1 «orp* tva ittvxcinXaeioi A 1 15 dialiinovxtg: 11 corr. ex t A 17 sq<|. cf. Nesselmann die AlgeLra der Griechen p. 227 sq. 24 jj — Tiavxag del. Ball . 25 xovxov corr. ex xovxav A 3* 3(5 DE XLWi. Si.MlUUU.<. TC3,-], »J lL1'tTi XQIXOV U>,X{ XiXCCgXOV IfOVZCi (J 1 C'.() 1 1 g l(Cfa<Qcd'£l<S>i3 V.Cfd Xgi'tOV i ) j : V y.CU Xc'xCOl OV, dg o y.i . exi xdv dgidudv ot tuv ioclxig foot* xergdyiJi-oi li'oiv, oi dl dviodxtg aviGm LIegoiiiyxng xa\ xgGur,xttg, 5 y.al ux?.dg ct di%dg no"'./.:.:.7.aoic:^duevoi ixixidoi. of de xgr^dg Gxegtoi. )Jyoixai dl ixixedoi c'gtOiun y.r.l xgiyavoi xcd xexgayavoi y.cd Gugeal y.al xdu.cc ov xv-gicog dl/.d ; . i . i > ' vuoioi^zc: idv %cogie)v cc xaxc.uergtiv-Oiv' b ydg d', ixel xexgi'cyavov yagiov xcxaiuxoci. C.X' iu ccvxov xcrf.iTxai xexguyavog, xcd 6 % dice xic avxa ixego-bnoini 6 liclv clijidu'd iv ulv ixixtdoig TE Tor-'VM-rot tu' xt'.vug xuGiv, ixigoiiiy/.eig dl oGav c:' xP.evgc.i, xovxeGxiv oi xegiiyovxtg avxovg dgi&uoi, dvdXoyov ia etGiv. oiov exigo[x\\yi] i)v xd g'" xlevgcd dl avxov u>~r y.og y, rhlxog p'' Ixigog r.c'.hv ixixedog 6 xd'' x?.evgcl dl ccixov (t^y.ig u:v x).d~og dl d. xcd eGxiv dg TO (n,y.og ~,j(xg xb i'.}]xog. ovxng % b xldiog xgbg xb xhlxog' icg yag g xgb; y, ovxag 6' xgbg j3'. oiinioi ovv dgil-;:oi txixedo: o Xi g' y.cd b xd'. oy^uaxi^ovici dl of e m u t'c;0i:u! oil ulv eig x?.evgdg cog uiy/.)} xal xgbg exegav GVGXUGIV Icqipan'iirfji, bxl dl eig ixirct'dovg, exeev iv. xo).).a:c't.i'.Gic:Guov dvo dgi$udv yevvrftdGiv, bxl 1 7i snjira vs. A 3 in-cv. TZIQI loi'/.tg i'ciov r.al ctii-cdnig civ io to i- A , XJ} in mg. 1 cxiozol corr. ex utigoiol A 9 xiioocjiM-av apogr.] • (i. e. xixgu-jcnvav) A 12 inscr. srepl Ojioicov a Q t & i t i i i v A , xy in nig., figuras add. A ' 14 avxovg] avzdg A, cf. p. 21, 2G 19 S nodi § apogr. /J ~po£ S A 21 aig ttifxq xc! (r.l.dxri y.al vOrty? 23 cU'o apogr.] p* Svo A DE N U M . TRIAXGULIS. 37 Se eig Gxegeovg, oxr.v ix ~o?.?.a~XuGtuGuov rgtdv X.rr cp&aGiv dgi&udv. iv 8e toig Gregeotg ndXiv ol uev xv-fioi ~uixeg TXUGI'V eiGiv onoioi, rdv de uXXav ol rug TiXevgug exovreg uvuXoyov' dg i\ rov utjy.ovg rrgbg rr\v rov utjxovg, ovrag JJ rov TtXdxovg ngbg n)v rov nXd- 5 rovg y.al <^ )> rot" i'^ovg stgbg rjjv rov vi^-ovg. rdv de im~iSav y.al itoXvydvav c.gi&udv ngdrog 6 rgi'ycovog, dg xal rdv ijiixe'Sav evQvyguuuav C-/r\-udrav ngdrov iari rb xgiyavov. ztdg 81 yevvdvrui Ttgoeigrjxui, on rd ~gara dgi&ud rov el-ijg dgxiov xal 10 negirrov TtgoGxiQeuevov. •ndvreg Se ol icpe%rjg dgidiioi, UTtoyevvdvxeg rgiydvovg 7] rerguydvovg 1) rcoXvyavovg, yvduoveg xaX.ovvxai. xoGovzcov 8e uovdSav exuGxov xgiyavov ixei rrXevgdg Ttuvxag, oGav xul uovog iGrlv 0 7igoG?.un(iu: ouevog yvduav. oiov eGra ngdxov r\ 15 uovdg, keyou'vi] rgiyavov ov xux' ivreXe'xeiuv, dg Ttgoeigi'iy.aiiev, dXXd xaxa SvvuuiV irtel yug avrr, oiov Grtiguu rtdvmv iGrlv dgid-udv, i'^ at iv avxij xal rgi-yavoeidt] Sviuuiv. TtgoGXayfidvovGa yovv xi]v dvdda u-oreXii rg(y:ovov, e'xov -Xevgug xoGovrav ucruSav, 20 O O ' O J I ' c'ffrit' 6 xgoGXijcpiJelg yvduav rjjg SvuSog. rb Se oXov rgiyavov xoGovxav iGxl uovdSav, oGav xul ol Gvvre&ivreg yvduoveg. o re yug rov ivbg xal <(o)> rdv SveTv yvduav rd y' i~oirkGuv, aGre xal rb rgiyavov 7 inBcr. TICQI rgiyoivcov UQi9[iiov A , x5 in rug. . 10 itQ<in\)r,t:uf. p. 3.°., 1 11 cf. Ncsselmann p. 203 14 I«J ante jr'.trcwj a i d . A * nuvxtog corr. ex -dvnov A 17 nrpo-tiorjv.ciULii: p. 3'5, C 24 xoaovtiiv, <ii,ci, uovciSwv ierlv p nltuija rov TQiydvov, ooeov [loiuSiov icr'iy 6 ^OOOT:9H; yvra-uaV Tocuvzcii ii ticiv ui TOV yvionovo; fioMnSf;, ocoi riciv of y»co'fio»*s oi lit; to rpt}oivov evvi"."touts xng. A 38 B E N U M . CYCLICTS. i~6tca yev tgidv povuSov, eqei 5' £xdGxi\v nXevguv rcov Sveiv, 0G01 xul oi yvduoveg Gvvexi&yGav. elxa ro y xgiyavov ztgoGXcufidvei rbv xdv y yvdixovu, eg uovddi V7ttgi%£i ri^g SvuSog, xal yiverai ro uev oiov rgi-6 yavov g'" ztXevgug 6' e$ei xoGoincov uovdSav xal xovro xb xgiyavov, baoi yvduoveg Gwrifteivruf ix ydg rov ivbg xal fi' xal y GvvExi&i] 6 ;'. a a a a a a I a a a lira 6 g' ztgoGluufidvei rov 6'" yiverai rb rov i rgiya-vov, Bxaaxijv TtXevgdv eyov S'' uovdoav' 6 ydg ~goGX.ii-lo qp&elg yvcniav t)v b 6', xal ix 5' da yvoitoiav rkv rb oXov, rov n ivbg xc.l fi xcd y xcd d'. an 6 t TtgoG-Xuufidvei rbv a', xcd yiverai <(r6 rov ie"} xgiyavov^ TtXevgdv eyov axdcniv uovudav a', xcd ix rdv a' yva-uovcov GvvicTtj. biioiag xcd OL'ii, yvduoveg is rovg yvauovixovg c'.giVuovg dxoreXoiat. Xiyovrci di riveg xs.l xvxXoeiSeTg xcd GtpuigoeidzTg xcd. a-oxuruGTctixol dgtduoi' ovxoi b' eiGiv oiriveg iv rd 7to?.f.c.zla6ic':le6Dc:i .>'; £~iziSag ij Giegsdg, rovr-tGn xard Svo SiaGrc'.Geig i] xc.rd rgeTg, dtp' ov av 20 ug^avxai c.giduov irtl xovzov uzoxafriGrdusro:. roiov-rov Si ioxi xc.l b xvxXog' u<p ov uv ug^tjrut Gijueiov, 2 yviiuoycc: o c o r r . e x co A ilia c o r r . e x f t ' ; A y'] zgizov A, o r a . a p o g r . 3 o; c o r r . ex oi A 4 trjg iSvddo; c o r r . e x zr,y 8vr.Su A 5 T « J ante nltvods a - M . A5 9 i%ov c o r r . ex ixmv 10 yrcouav A2] y i o V i A1 12 to T O U ii a d d . a p o g r . 13 yvunovco A1 14 f | ] f|>"» apogr. 1*J i n s c r . T Z I Q I xvxloeiStav xul cppatgottdiiv v.ccl aitov.c'rociaii*")! ' ttgiftuur A, x8 in rr.g. 1" oT-i i i t j —. uzcoAuQiaziiusiot] s e r . of — asoxafhcrausi oi a u t oT-nvfj — arco-ACiih 'ciraivrca "21 xvxloi "(o;]>? t i l DE NUM. QUADKATIS ET PENTAGONIS. 3*J ixl XOVTO dxoxaxtiGxaxai' vxb ydg uidg yguuurjg j rfpt-exouevog axb TOV avxov dgyexcci y.ccl eig xuvxb xaxc.Xrf' yei. xoiavxr] de xal iv Gxeged i] Gcpatga' xvxX.ov y&4> xaxd x7.evgdv xegiayouivov t] axb xov avxov ixl avxb dxoxaxciGxaGig Gcpcdgav ygdcpei. xal dgi&uol d^ £ of iv xa xo7.7.ax7.aGiaGud icp' iavxovg xaxuh]yoi r . j j xvy.XixoC xe xaX.ovvxai xal Gcpaigoeideig' av etaiv Z T t s' xal 6 g' xevxixig ydg e' xe', xevxdxig xe gxe', xig g' 7.g, xal ii-dxtg 7.- Gig'. xdv de xexgaydvav v\ uev yivtGig, dg elxov, in \Q xdv negtGGav d>.7.rt7.oig ixiGvvxiQeutvow, xovxeGn ride dxb uovddog diddi a7.7.t]7.cov vxegexivxav' ev ydo y d', xcd d' xai e' &, xcd &' xcd £' ig, xcd ig' xcd &' ?:b.. a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a u a ix a a a a a a xevxdyavoi de eiGiv dgi&uol of ix xdv dxb uoid-dog xaxd xb i*\g xgiddi (d?.7.i'}7.avy vxegtxovxav GvV-xi&t'uevoi. dv i'Giv of iizv yvouioveg a'd' £' i iy ig' i?'' avxol de of x, vxdyavoi a' e' ifi xfi 7.e va' xcd i^tfe buoi'ag. Gxmi<:'-':ii,ovxai de xevxayavr/.dg ovxag' a t' tp" xp" Xs' a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a M a a « a a a a a a a a a a a a a a a a a a a a a a a a a 10 inacr. nt^l r s x ^ a y w v w v ccQt&iLcov A tlnov: j*. 23, 3. 32, 9. 31, 1 11 inscr. nsgl TTtvtaycovv>v tt$i&ni"iv A , x i in mg. 40 DE NUM. MULTIAXGULI5. ildyavoi di EIGIV aQi&uol ol ix xdv xaxd to i%t\g dxb uovddog xExgudi cl7J.tp.cov ixEge%6vxav GWXI&E'UE-vof dv ol yvduovtg elGiv a E' &' ty 1% y.a' V.E' ol de ix xovxav iilclyavoi o"de' a g ' IE' xt] UE' £g' xict. Gyri-5 iiaxitpvxca SE ovxag' 0 s ts xi] fit |s a a a a tt a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a ' a a a a a a a a a a a a a a ct a a a a a a a a a a a a a a a a a a at a a a a a a a a a a it a a a a a a a a a a a a a a a a a a a a a a a ct a a a a a a a a it a a a ti a a a a a a a u a a a a a ixxdyavot di et'oiv ol dxb uovddog xevxddi a/.h\-Xav ixeQEyovxav GvviGxdutvoi' dv yvduoveg uev a' g' tu ig' y.n y.z'' ol de ix xovxav GVVXL^E'UEVOI a £' it] }.d' ve' xa. buolag ds xcd oxxclyavoi (of) dxb 10 uovddog i^c'.dc c'JJ.tjlav V7CcQty6vxnr Gvvxid-iuevoi, iv-vedyavoi de ol dxb uovc'.dog ifidoy.ddi u)J.rt\av vxeg-e%6vrav GvviGxduevoi, dexciyavoi de ol axb uovc'idog oydoddi d)J.t]).av vxegeiovrav GvvxtQiuivoi. ixl xc'.v-xav ds xdv xo/.vydvav xu&6).ov oGclyavog dv l.iyrytca 15 doi&uog, dvEtv deovGcav uovddav xov xhfiovg xdi' 1 inscr. TZSQI tluyiovcov c* 51 (r ft to A G ir.scr. O U O I ' E Si ij ovv&iGif v.ctl tTcl zwv Xoindyy rcoXvyciiav A jx zmv hie et in iis quae sequuntur neglegenter omibsura 11 povuSos A 2 ] >e(Jo; A 1 13 xnt ante i;cl add. A * 15 Svsiv Stovacnv (loiccSiov corr. ex Svo Si ovauii tio'vci ut vid. A DE NUM. S0LID1S. 41 ycovidv r) V-EDOX*) rdv ugi&udv Xuupdvazui, fj; dv oi noXvycovoi Gvvxi&evxai. ix Svo xotydvav U-OXEXEIXUI rsrodyavov' a' xal y o , y xai s v, g xai i ig , i xai is XE , is xai xa Ag , xa xal xrj' fi9-', xri' xal Xg' |6", Xg' xal ue' na', xal oi" fjjijs 5 ouoicog Gvv8va%6uEVot roiyavoi tErguydvovg uxoreXov-Giv, dg xal irtl rdv ygauutxdv rgtydvcov Gvv&soig rs-rgdiyavov Gxrjuu x01^-irt rdv GrsQEnv dgi&udv ol UEV laug nX.Evgug ixovGiv, [dg uQi&uovg TQEig iGovg. irtl i'Govg -oXXanXa-Giu%eo§ui^\ ol SE dviGovg. rovrav S' oi pev TtciGug uv- 10 iGovg ixovoiv, ol SE rag Svo ioug xul ri)v uiav JJTTOVK. ~dJ.iv rs xdv rug Svo iGug exovrav ol UEV UECZJOVU riyv T p i T t ; y ixovtiiv, ol SE iXdrxova. ol UEV ovv i'oug exov-xig TC/.Evodg, tGuy.ig i'Goi iGaxig ovxEg, xi'fioi xuXovvraf oi SI -uGug dvlaovg rag —Xevgug, dviGaxtg uviGoi uv- 15 todxi:, (lauiGxoi xaX.ovvxai.' oi SE SVO UEV i'Gccg, xitv Se xgirrjv ixurigug xdv SVEIV iX.uGGova, iGaxig iGot iXuxrovdixig, -Xiv&iSEg ixXi'i&rtGav oi Se Svo UEV iGug, 1 ytoviwv Bull.] uQtdiitov in ras. A apiSiuGi'] yavmv in ras. vocis agi&udyv A 2 ~oXvywvou 1 post v er. A 3 inscr. oti in Svo rgiytavav TO TE tgdycorov A, x? in mg. t in: inter E et x complures literae erasae in A Si post Svo add. A 1 7 i-l] r)? 8 inscr. ncgl oziqtdiv CP£O>UCJV A, y.* in mf.' 11 aut delenda sunt verba xod tr)v fiiuv 7,'TTOVC (sic Bullia dus) aut scribendum xol trtv (u'ctv un-co* 16 ray post c5t add. A 5 teas corr. ex foot A 18 -Xw&idts corr. ex -IrpdiSej A raj post 3'e add. A* 42 D E N U M . PYItAMIDALIIJU3. TJ)V dl Toittjv ixuTc'Qccg TCOI> dvslv /ta!;ot/K, lodxig taut (iii^ovdxtg, doxcdsg >:c:Xovvtcu. P s y S 0 if 13 — ? I ? — 1/3 £io*l «?£ xal nvncqiOEideig ccgi&iioi xvQcqiidccg xccra-fiSTQOVvtsg xcd xoXovQoxvQC'.ui'dc<g. xol.ovqog 61 Tcvga-utg tGziv r] z))v xoQvcpi)v cc7zorsTiii]iitv)]. nvlg 61 [x6?.0VD01>] TO TOIOVTOI' TQC'.xtllOV TtQOCjty/OQEl'GCCV CiXO tcov ircixidav TQCCTC£%(COV' TQccxeiiov ydg Xayerca, OTUV TQtycovov i) xoQvcfi) vxb xc.ocJ.hjlov rtj jidaei Bv^siag dxoTiirfttj. dctxeg dl xgiymnxohg xcd TETQccyanxovg xcd xiv-2 ad figirras, quae sr-.tn ncglegenter descriptae sunt, perti-net haec adnotatio marginis A : T O in&i-to ar^itTov iaziv tcov, T O vitoxuro} uit^ov 3 inscr. rrspl nv q a u o F t SU v aqt9 aciv A, xrj in mg. nvqcqu'da A 4 xolornozi oau'da;: orj corr. ex f j A 10 iuscr. Ttsql nlsvqtxav xal S i a u £ T qi xa v •xoi&jitov A, x9 in mg. cf. Nesselniann p. 223 sqq. DL' NUM. LATERALIBUS ET DIAGONIIS. 43 Tayavtxovg xul xuru ru loinu Gxijuuru ).6yovg eyovGi Svvduei oi dgi&iioi, ovrag xul nlevgixovg y.al Siuue-rgixovg Aoyovg evgoiuev dv y.ard rovg Gnegiiurixovg }.6yov; HitpavilouBvovg toig dgi&uoig. ix ydo rovrav t}vQiiit,trai rd GX'juura. aGrceg OVV ndvxav rav ( J ^ J I K ' -rav xard rbv uvaxura xal Grteguurixbv Xoyov r) uovdg ugxei, ovrcog'xal rijg Siuyergov xal ry]g nh-vgug loyog iv rji uovuSi evgiGxerui. oiov ixtid-evrca Svo uovudeg, av rt)v ulv T}COUSV elvui Siuiiexgov, rr)v 81 nt.evgdv, i~eid>) r>)v uovccdc, ndvrav ovGuv dgx>jv, Sei Svvduei y.al ~/.evgdv eivut y.al Stduergov. xal TtgoGri&exui rjjj ulv ~).cVQa diunzrgog, rjj 81 Siuuerga Svo itkevgai, ineiSi) oGov 1) ~?.evgd Slg Svvarai, ij Sidyergog unulg. iyivero ovv uei^av ulv T) Siduergog, ildrrav Si J) nlevgu. y.al inl uev ri]g 3tgdrt}g itt.evgug re xal Sia-uirgov eF>i dv rb ano T)js povdSog Siaiiirgov rergdya-vov [lovccSt 11 id elarrov 1; Smf.uGiov rov dnb xijg uovu-Sog nlevgdg rergaydvov' iv (Wrj-ri ydg ai uovdSeg' rb 8' iv rov ei'bg uovd8i D.urrov 1] SinhcGiov. stgoG-&duev 8>\ r7, uev nf.evgd Siduergov, rovritin rjj uovctSt uovdSw eGrui 1) n).evgu ugu Svo uovdcSav ry 81 Siu-uirga TtgoG&diiev Svo TtXevgccg, rovriGri rij uovuSi Svo uovdSug' iarui 1) Sidiiergog uovdSav rgidv' xal rb ita „• 4 (ivy tof; otQi&uotsr' 12 Si fit'rpra A 15 y" povu-Sir.fti supra TZQonr^; add. A' 16 fiovdSo; SiantTQOv] uoro-—0S0; (ex corr.) Sv A 17 tXaztov JJ corr. ex iXuvrori A fioidSo; apogr.] [lovo-oSog A 20 StctuiTnov'] 8" A 21 fiovcidov apogr.] fi^A, uovdSft Bull. 23 fioruda;] fi?" A, eia. apogr. iczai] nota vocabuli cfpa in ras. notae voc. larai A fiovdScov corr. ex uovdSt A 44 DE NUM. LATEKALIUUS ET DIAGONIIS. uev dxb xrjg Svddog xXevgdg xexgc'.ycovov 8', xl 6' dxo zijg xgiaSog Stuue'xgov xexgdycrvov fr'* TO fr' agu uovd.Si jisfgov tj SixXdoiov zov dxb T;]S fi xXevgdg. nuhv xgoG&duev tfj uev fi xX.evgd Siduexgov xr)v xgi'aSa' 5 IGXUI fj xXtvgu e'' xrj Se xgiuSi Siuuexga fi xXevgdg, xovxtGxi dig zu fi' e'Gxui eGxai to uev dxb xrjg (e") xXevgug xexgdyavov xe', xb Se dxb xrjg £• (Sttcut'xgovy /ifr'" uovdSt, eXuGGov rj SixXuGiov xov xe' agu xb ftfr'. xdXiv dv xrj (t") xXevnu xgoG^rtg xt]v £' bidutxgov, io eGxai ifi' xav xr~j £' Siuus'xgco xgoadr^g Slg xr]v e' xXev-gdv, IGXUI I%' xal xov drib xr]g tfi xexgaydvov xb dxb rr]s <£' povdSb xXe'ov rj SixX.dGiov. xal xuxd xb i*ijg xrjg xgoG&rjxrjg biiciag yiyi'oue'vrtg, i'Gzia xb dvd.Xoyov evaX.Xd%' xoxe uev uovd'h eXaxxov, xoxe Se iiovd.Si xX.iov is rj SixXd.Giov xb dxb x>jg Siuiitxgov xirgdycorov xov dxb xrjg xX.evgug' xcd grtxcd cd xoiavxcu xul xXevgcd xcd O'.C.'.lfQOl. fr xe ut KI Se Sic'nexgot xdv xX.evqdv ivuXXdz, xagd uiav xoxe 1 SvciSos i n r a s . A 2 Siaiiiroot1] SL% A {)•'• TO 9' a p a : TO e t n o t a v o c i s npa s u p r a v s . , 9 to i n ras. A 4 |3'] 8vo A Sid-pfTpov] 6'v A 5 ferret A'J uou'tSwv ccpu A'- Siauitgy] Svrdfiit. A 6 iczui ft"'' s u p r a n o t a r a v o c i s iatai a d d . A 5 lezcei T O : n o t a v o c . iatca m u t . i n IVOK A S T O U y.r UQX to f l f r apOgT.] TO KS ftp.".' XOV lift A 9 ( W l l c T p C i ' ] Sviiuitt A 10 n o l a v o c i s i'otcu mut. i n A 11 n o t a v o c i s fcrai mut. i n fioyuSiov A (tocc post xr.i a d d . A 2 17 d u o quadrat a cum n u m e r i s cpO- e t and- a d d . A 2 18 at it c o r r . e x rt S.m A DE NTM. PEIIFKCTI55. 45 ulv uovddi utt'tovg 1} dixXc'cGiai dvvduei, xorl r?t iiovddi i?.dxTovg ?] dixXc'cGua baaXng' xaGci ovv cd d:duexgoi xccGcov rcov xX.evgdv yevt'jGovxai dvvduet dixXdtf.ai, TOV ivaXXd* xX.eiovog xcd iXdxxovog t j j avx>i u-zvddi iv xccGaig buaXdg XL&euivr} ta6xr]xa xoiovvxog eig T O uijxe 5 iXXeixeiv [ii]xe vxegfidXX.eiv iv dxd.Gaig TO b.r.'f.daiov' TO ydg rjj xgoxegcc diauixgo) X.eixov dvvduet t g fVyrajjjjjg vxeg^c'cX.Xet. exi xe TCOV dgi&ucov 01 uev tiveg xe'P.eioi Xiyovxai, 01 d' ixegxileioi, o f d' iXXixeig. xcd xiXeioi I±£V eiGiv 10 of Tofj avxav uigeGiv Foot, cog 6 tcov g'w f t e o i j ydg avxov j)uiGv y, xgixov fi, i'xxov cc, ccxiva Gv:~.^iuevu xotet TOV g'. yevvdvxca de of xeX.eioi xovxov z]v rpo'-Ttov. idv £x& ouedu xovg dxb uovddog dixX.c.T.'cvg xcd Gvvxidduiv Kuzovg, iit'xQ'g ov dv y£vi]xcci XQ::zog xcd 15 cloiv&ezog dg ftuog, xal xov ex xi\g Gvv&iGei:; ixl xbv eGyaxov xav cvvxi&euivcov xoXJ.axX.uGiuGaui:-. 6 dxo-yevvii&elg eGxcci xs'Xeiog. oiov ixxet'G&aGav (J.xXdGioi a' fi d' if ig'. Gvv&auev ovv cc xcd fi' yiver:..: y'~ xcd xbv y ixl roi voxegov xbv ix xi]g GvvQiGci:$ xoXXa- 20 xXc.Gic'.Gauev, xovxeGxiv ixl xbv fi' yivexai z . og icSzi xgcozog xiXeiog. dv xdXiv xgcig xovg eyeing ?.xxaGiovg GvvQcouev, a' xal fi xcd d', eGxai tf' xcd xov:-.: ixl xbv eG%uzov xcov xi]g Gvv&tGecog xoXXc xX«GiuGa::r.. xbv ff 1 rav nlivqcov post Svictfisi er. A C t i ' - . ' u i r : let corr. ex Xi A 9 inser. negl xsleiav X K I v z t :zsXs itov xnl §XXt7ttov (corr. ex IXXizzoizcav) t'cqiO-uav A, i iz mg. 15 ftf'^ ni A, em. apogrr. 19 ovv add. fort. A 2 J.'V-TCI corr. ex nota vocis icrai uU'sid. A 20 zov i/. z>_; :-:v9ieiasi irnmo zC)V Giizr9i;zcov 22 Toffj: ti cx 1 A 23 et r. i'., 1 idzat A ' J yivizc-i ut vid. A 2 fort, recte 21 znv ex T ; > A zrjt Gvtfriattoi: irnmo ovvzi 9ivzcov TtoXciztXacii'.caut v A, apogr. 40 UK NL-M. SITUKFLL'IS KT DKMINUTI.-i. t~l TOV 8'' EGXCU o xi\, og iozi J c t ' T c p o j zilEiog' Gvy/.n-Tia ix rov rtuiGEog rov •<$', TETC'QTOV TOV t, {(IduUOV TOV 6', TEGGugccxuidExcczov TOV /3', EIXOGXOV 6y86ov TOV a. V~EQTE?.CIOI 8i EiGiv av TU uig>] Gvvze&ii'TU uEiiovu D iGzi zav Zlav, oiov 6 TCOV ip"' TOVZOV yug tjuiGv iGxiv g, zgizov d', rizugzov y, EXZOV p", 6a5ixuxfV c:'. uzivu GvvzEdivza yivczca ug, og iGzi Ucit,av zoii c's ugyitg, TOVZEGXL xdv t(i'. ilhrcEig di EiGiv tov zu uigi] GVVZEQEVZU ih'.zzovu io TOV ugii}ubv TCOIEL TOV £•'{; ugxi]g ngozEQiizog ugiduov, oiov 6 zav )/' rouroi" yug "jutGv d', ZEZugxov /3', by8oov EV. TO uvrb 6E xul za i Gvufiifiry/.EV, ov xr.9-' i'xEgov ?.6yov TEJ.EIOV itpuGuv oi Ilv&ayogixoi, ~Egl ot xaxa TijV oixEiuv xC^QKV urcoSaGouEV. XiyErca 61 xcd 6 y is TEkEiog, irtEidt) Ttgcozog c\gyj\v xcd iitGu xcd. T.igug EXEC' 6 d' uvzbg xu\ ygu\iui\ iczc xcd irci~E8ov, xgiyavov yug iGonJ.Evgov EXuGxi]v TiJ.Evguv 6VELV uovdSav iyov, xcd Tfgdxog dEGubg xul GrsgEov 6vvc:uig' iv yug zgiGi 8iu-GZUGEGI TO Gxcg-bv i'ozi"G&ca. 20 isttl 6c xcd Gviicpdvovg zivc'cg tyuGiv ugt&uovg. xcd 6 TCEgl Gvticfaviug ?.6yog ovx ilv EvgEirei^ UVEV c.gi-1 cJfJrf jo;] (3 A Gvyxsirr . i — rov K ' in nig. A •Jcri. h a e c c contextu vcrborum removenda), yao po.-t eiyv.ti-ut apc^r. 2 t{l86nov~l J A 3 TfGCaQay.aiSlxdrov] id A fixoarcv dycTooi'] xij A 4 evvxeSiizu: T E corr. ex rt A 5 tnv i/J AJ] tov i(? A 1 7 o; (?) A1] xal A 2 8 xicv corr. e x T O V A 12 T(» covr. ex T O A 14 r'noXuioiiFr: p . 00, IS. 10G, 7 y : TPi'tr corr. ex T P I ' T O ; A 15 uiGov p.; ->jr. 10 yep (Jerivyi 1* tH'vKfiij corr. ex Sviciiuti A 10 vefiif.'t apO<rr.; fort. ex. " d i t tyaoi'y ^ .T tale quid 20 ir.?cr. P u u o ; TJXctravixov avyxttfuXa tacts tftl evvoipti T ^ J O 1 r( s •ioveixi~ts Z, afpl iiovetxljs A' a A CHAPTER THREE TRANSLATION A p r e s e n t a t i o n by the P l a t o n i c p h i l o s o p h e r Theon of mathematics u s e f u l f o r the r e a d i n g of Plato" 1" 1. 1 Everyone would agree, I suppose, t h a t i t i s not p o s s i b l e t o understand the mathematical d i s c o u r s e s o f P l a t o , u n l e s s one i s o n e s e l f t h o r o u g h l y p r a c t i s e d i n t h i s branch of s p e c u l a t i o n . That s k i l l i n t h i s s u b j e c t i s not u s e l e s s nor without p r o f i t i n o t h e r r e s p e c t s a l s o he seems to make c l e a r by many remarks. F o r t u n a t e , ; t h e n , i s any man i f i t happens t h a t he s t u d i e s the w r i t i n g s of P l a t o a f t e r becoming con-versant w i t h the whole of geometry and the whole of music and a s t r o n o m y — a s k i l l which i s not r e a d i l y or e a s i l y a c q u i r e d , but one demanding a very g r e a t d e a l of t o i l from youth onwards. 1.11 To ensure t h a t those who have missed a t r a i n i n g i n mathematics y e t a s p i r e t o an understanding of h i s w r i t i n g s may not e n t i r e l y f a i l i n t h e i r o b j e c t i v e , I s h a l l present a c o n c i s e summary of the r e q u i s i t e f i e l d s of knowledge and an e x p o s i t i o n of those mathematical theorems e s p e c i a l l y needed by those who would become acquainted w i t h P l a t o , namely a r i t h -metic, music and geometry as w e l l as s o l i d geometry and ^6ecjDvocj^ Epupva tou EXa-rouvt HOU TCDV K a r a TO p,a6rip.aTiKOv Xpr\ai]xa ei<-, TTIV nXaTaavo-- avayvaxJiv i s the i n s c r i p t i o n on A. 70 astronomy. Without these, a c c o r d i n g to him, i t i s not poss-i b l e t o a t t a i n the best l i f e , f o r he has demonstrated on many occasions t h a t one ought not t o d i s r e g a r d mathematics. g 2. 3 E r a t o s t h e n e s , i n h i s work e n t i t l e d P l a t o n i c u s , says t h a t the god, i n an o r a c l e on the q u e s t i o n of t h e i r being f r e e d from a p e s t i l e n c e , d i r e c t e d the D e l i a n s t o set up an a l t a r double the s i z e of the e x i s t i n g one. Whereupon, the b u i l d e r s were a f f l i c t e d w i t h g r e a t p e r p l e x i t y , as they sought the way a s o l i d c o u l d become doubled; that was how they came t o con-s u l t J^lato concerning t h i s problem; and he t o l d them t h a t the god had proclaimed t h i s o r a c l e t o them , not r e a l l y because he wanted a d o u b l e - s i z e d a l t a r , but as an o b j e c t i o n and a reproach t o the Greeks f o r t h e i r d i s r e g a r d of mathematics and t h e i r contempt f o r geometry. 2.13 In c o n f o r m i t y w i t h the advice of the P y t h i a n o r a c l e P l a t o h i m s e l f a l s o d i s c o u r s e s i n great d e t a i l upon the u s e f u l -ness of mathematics; f o r i n s t a n c e , i n the Epinomis, u r g i n g man t o mathematical s t u d i e s , he says: Never, without these s t u d i e s , w i l l any nature be happy i n the S t a t e ; no, t h i s i s the way, t h i s the n u r t u r e ; t h e s e are the s t u d i e s , whether they be d i f f -i c u l t or easy, and by t h i s way must we go, f o r i t i s not l a w f u l t o d i s r e g a r d the gods. g T h i s book, not now extant, was maybe a s o r t of comm-entar y on the Timaeus of P l a t o , and e v i d e n t l y d e a l t w i t h t h e fundamental n o t i o n s of mathematics u n d e r l y i n g P l a t o 1 s p h i l o -sophy. A c c o r d i n g l y , i t was p r o b a b l y an important source f o r Theon 1s w r i t i n g and Theon c i t e s the book twice by name. 3 See Mathematical Note 1 (p. 122) 4 c f . P l a t o , E p i n . , 992A. P l a t o has euSaipxuv, Theon 71 Subsequently, he says: Such a man (the mathematician) alone of many w i l l be the one t o become b l e s t by f o r t u n e and at the same time most wise and happyP 2.22 In the R e p u b l i c , he says: Those who are g i v e n p r e f e r e n c e from the twenty-f i v e y e a r - o l d age-group w i l l r e c e i v e g r e a t e r honours than the o t h e r s and they must b r i n g t o g e t h e r the s t u d i e s haphazardly pursued i n t h e i r education as c h i l d r e n t o p r o v i d e a comprehensive p i c t u r e of the i n t e r r e l a t i o n s of these s t u d i e s and of the nature of t h i n g s . 6 He a d v i s e s men f i r s t t o become p r a c t i s e d i n a r i t h m e t i c , then i n geometry, t h i r d l y i n s o l i d geometry, f o u r t h l y i n astronomy, a s u b j e c t which a c c o r d i n g t o him i s the study of the s o l i d i n motion, and f i f t h l y i n music. eu6aip,ova>v. euSaipcov seems e a s i e r t o t r a n s l a t e and has the a u t h o r i t y of one ms, and t h i s i s the r e a d i n g I have assumed. T h i s i s the f i r s t of many q u o t a t i o n s from P l a t o t h a t Theon makes i n h i s proem emphasising t h e importance of math-ematics i n education. L i k e many of the N e o p l a t o n i s t s of h i s day, Theon had a vast knowledge of P l a t o , but h i s q u o t a t i o n s are c a r e l e s s l y p r e s e n t e d and are r a r e l y t e x t u a l , being quoted no doubt l a r g e l y from memory. 5cf. P l a t o , E p i n . , 992B. c f . P l a t o , Rep. . 537B',Cjwhere P l a t o o u t l i n e s h i s c u r r i c u l u m f o r education. EH TCOV HE ETCW may be a c o p y i s t ' s e r r o r f o r kinoai ETCDV, i . e . , the twenty y e a r - o l d age-group. P r e p a r a t o r y s t u d i e s were t o take u n t i l the seventeenth or e i g h t e e n t h year; then was to f o l l o w t h r e e years of m i l i t a r y s e r v i c e , which would delay h i g h e r s t u d i e s proper u n t i l the age of twenty or twenty-one when the p u p i l entered the Academy. The d i r e c t i o n of the mind from the more concre t e t o the more a b s t r a c t — and t h e p u r s u i t of the nature of Being — was r e s e r v e d f o r the years of d i a l e c t i c . The study of mathematics prepared the way f o r such enquiry and i t was a l s o necessary t o c o n s i d e r the importance of the other d i s -c i p l i n e s and t h e i r r e l a t i o n s h i p t o one another. 72 Demonstrating the u t i l i t y of mathematics, he says: You are naive i n appearing t o f e a r t h a t I would p r e s c r i b e a u s e l e s s study. That the eye of each man's s o u l , b l i n d e d and dimmed by other p u r s u i t s , i s c l e a n s e d and k i n d l e d anew by these s t u d i e s as i f by instruments, i s a c o n v i c t i o n t h a t i s w i t h d i f f i c u l t y brought home not o n l y t o second-rate minds but to a l l men; y e t i t i s a f a c u l t y whose p r e s e r v a t i o n i s f a r mote p r e c i o u s than a thousand eyes, f o r through i t alone i s t r u t h p e r c e i v e d . 3.16 In the seventh book of the R e p u b l i c , speaking about a r i t h m e t i c , he says: Of a l l the branches of knowledge, i t i s the most necessary and i t i s t h e r e f o r e needed by a l l the a r t s , a l l forms of thought and a l l s c i e n c e s , and by the a r t of war i t s e l f . Palamedes, at any r a t e , i n tragedy i s always d e p i c t i n g Agamemnon as a comical commander; f o r he says t h a t , a f t e r he had d i s c o v e r e d numbers, he marsh-a l l e d h i s army i n camp at Troy and counted h i s s h i p s and e v e r y t h i n g e l s e as i f , p r i o r to t h i s , they had not been counted, and Agamemnon app a r e n t l y d i d not even know how many 8feet he had, i f indeed he d i d not know how to count. 4. 8 I t seems l i k e l y then t h a t a r i t h m e t i c i s one of those s t u d i e s t h a t i s n a t u r a l l y conducive to thought, but none use i t to l e a d them t o the p u r s u i t of t r u e Being and to summon u them to t h i n k . A l l those o b j e c t s t h a t merely set our sense ' c f . P l a t o , Rep., 527D, E. 8 c f . P l a t o , Rep., 522C, D. Palamedes, l i k e Prometheus, was a " c u l t u r e " hero, who p e r s o n i f i e s i n Greek tragedy the i n -v e n t i o n s and d i s c o v e r i e s t h a t produced c i v i l i s a t i o n . See P. Shorey, P l a t o , The R e p u b l i c , i i , p.l51(note) i n the Loeb s e r i e s . Q c f . P l a t o , Rep., 523A. P l a t o b e l i e v e d i n mathematics f o r i t s own sake, as a p u z z l e i n a b s t r a c t l o g i c ; but Jowett i n d i c a t e s t h a t P l a t o a n t i c i p a t e s modern e d u c a t i o n a l theory by advocating c o n c r e t e a i d s (tiT)A.a)v r e 6iavo*aac; Hai arecpavcuv, Laws, 819B); the modern d o c t r i n e " l e a r n i n g i s f u n " seems to echo t h i s passage. 73 of p e r c e p t i o n i n motion as, f o r i n s t a n c e , the s i g h t of a f i n g e r seen as t h i c k or t h i n , l o n g or s h o r t , are not capable of s t i m u l a t i n g or a r o u s i n g the thought pro c e s s e s . On the o t h e r hand, those o b j e c t s t h a t set our sense of p e r c e p t i o n going i n o p p o s i t e d i r e c t i o n s are capable of s t i m u l a t i n g and a r o u s i n g our mental processes, as when the same obj e c t appears t o us l a r g e and s m a l l , l i g h t and heavy, s i n g l e and m a n i f o l d . 4.17 U n i t y then and number are capable of a r o u s i n g and s t i m u l a t i n g thought, f o r u n i t y sometimes appears to be many. The s c i e n c e of c a l c u l a t i o n and a r i t h m e t i c i s what a t t r a c t s and l e a d s us t o t r u t h . And one must come to g r i p s w i t h the a r t of c a l c u l a t i o n i n no amateur f a s h i o n , but by pure thought s t r i v e t o a t t a i n t o the contemplation of the nature of numb-e r s , p r a c t i s i n g i t not f o r business reasons as do the d e a l e r s and merchants, but i n order t o a s s i s t the s o u l i n i t s journey towards t r u t h and r e a l i t y . For i t i s t h i s t h a t d i r e c t s the s o u l upward and compels one t o d i s c o u r s e concerning pure numbers without a c c e p t i n g any person's r e f e r e n c e i n the d i s -course to numbered o b j e c t s t h a t are t a n g i b l e or v i s i b l e ? " 0 5. 7 And a g a i n i n the same book, he says: Furthermore, those who are good reckoners are n a t u r a l l y quick at a l l t h e i r s t u d i e s , and those who are slow themselves become q u i c k e r than they were befo r e . 1 0 c f . P l a t o , Rep., 525B,D. i : L c f . P l a t o , Rep., 526B. 74 He says f u r t h e r i n the same p l a c e : I n war too the a r t of c a l c u l a t i o n i s u s e f u l f o r encampments, f o r c a p t u r i n g p l a c e s and f o r the assembling and d i s p o s i t i o n of the a r m y . 1 2 Furthermore, i n p r a i s i n g the earnest study o f such s c i e n c e s , he says: While geometry deal s w i t h the study of the plane s u r f a c e , astronomy d e a l s w i t h the movement of the s o l i d , and t h i s compels one t o look upward and l e a d s one away from t h i n g s here t o those h i g h e r v i s i o n s . 1 3 I n t h e same work, on the sub j e c t of music, he says: The contemplation of the u n i v e r s e r e q u i r e s two s c i e n c e s , astronomy and harmony, and these are s i s t e r s c i e n c e s a c c o r d i n g t o the P y t h a g o r e a n s . 1 4 6. 2 Some make a f u t i l e e f f o r t of measuring the harmony of sounds they hear by measuring them a g a i n s t one another. By a s s i d u o u s l y l a y i n g the ear a l o n g s i d e as i f they were t r y i n g t o c a t c h a sound i n the neighbourhood, some c l a i m t h a t they can d e t e c t an i n t e r m e d i a t e note and t h a t t h i s i s the s m a l l e s t i n t e r v a l and should be the u n i t of measurement, w h i l e o t h e r s d i s a g r e e and c l a i m t h a t i t i s the same as the note a l r e a d y sounded, s e t t i n g a g r e a t e r v a l u e on the judgment of ear than of mind and p e r s e c u t i n g the s t r i n g s as they rack them on the 15 pegs. 12 c f . P l a t o , Rep., 526D. 13 c f . P l a t o , Reg., 529A. 14 c f . P l a t o , Rejj. , 530D. 15 c f . P l a t o , Re;p., 531A. P l a t o s a t i r i s e d the e m p i r i c a l methods of the p o u c r i H o i who made the quarte r - t o n e t h e i r u n i t , w h i l e he was a l l f o r s i m p l i c i t y i n music, as J . Adam contends, 75 6.10 But the good a r i t h m e t i c i a n s by t h e i r r e f l e c t i o n s seek to f i n d out which numbers are harmonious wi t h o t h e r numbers 1 ft and which are not. And t h i s search i s u s e f u l f o r the pur-s u i t of the good and the b e a u t i f u l ; f o r other purposes the search i s u s e l e s s . And i f a l l t h i s l i n e of enquiry l e a d s t o the mutual connections of these numbers and an assessment of t h e i r r e l a t i o n s h i p s t o each o t h e r , the study of them proves f r u i t f u l . Those who are c l e v e r i n these matters are the d i a l e c t i c i a n s , e l s e would they be unable to exact and render an account of t h e i r o p i n i o n s i n d i s c u s s i o n and i t i s i m p o s s i b l e t o do t h i s u n l e s s one has proceeded by way of those s t u d i e s , f o r the path to the contemplation of the u n i v e r s e l i e s by way 17 of them i n reasoned d i a l e c t i c . 7 . 9 A g a i n , i n the Epinomis, P l a t o has many other p o i n t s 18 t o make concerning a r i t h m e t i c , c a l l i n g i t a g i f t of God, R e p u b l i c of P l a t o , (Cambridge, Cambridge U n i v e r s i t y P r e s s , 1921), i i , p . l 3 4 ( n o t e ) , drawing a t t e n t i o n t o Laws, 812C. The imagery, taken from P l a t o , i s of the t o r t u r i n g of s l a v e s t o produce evidence and the passage purports t o show the u n r e l i a b i l i t y of the p r a c t i c a l method as compared w i t h the t h e o r e t i c a l . Some c r i t i c s have u n f a i r l y used t h i s passage t o i n f e r P l a t o ' s o p p o s i t i o n t o the a p p l i c a t i o n of experiment t o a l l s c i e n c e , but a c c o r d i n g t o Shorey they overlook h i s avowed e d u c a t i o n a l purpose of s o l v i n g the problem without d i s t r a c t i o n s . Note Rousseau's " e c a r t e r tous l e s f a i t s " and see Shorey's notes on Rep., 529B, 531A (Loeb). The l a t t e r b e l i e v e s t h a t i n t h i s passage P l a t o makes one of h i s r a r e mistakes f o r , although t h e r e may be i n some sense a pure mechanics of astronomy, there can h a r d l y be a s c i e n c e of a c o u s t i c s d i v o r c e d from d i r e c t experience. 1 6 c f . P l a t o , Rep.., 531C. 1 7 c f . P l a t o , Rep., 531D. c f . P l a t o , E p i n . , 976D,E. 76 without which i t would not be p o s s i b l e f o r anyone t o become e x c e l l e n t . Then, moving f u r t h e r on, he says: I f we were t o remove number from human nature we would no l o n g e r have a n y ' i n t e l l i g e n c e at a l l , and moreover the s o u l of t h a t l i v i n g c r e a t u r e would no l o n g e r be capable of a c h i e v i n g complete v i r t u e , f o r h i s reason would s c a r c e l y e x i s t . The c r e a t u r e which d i d not know Two and Three nor Odd and Even but was t o t a l l y i g n o r a n t of number would never be able t o t e l l of those t h i n g s c o n c e r n i n g which he had a c q u i r e d merely s e n s a t i o n s and memories; d e p r i v e d of t r u e r e a s o n i n g he would never become w i s e . 1 9 8. 2 And y e t , w i t h r e g a r d t o the a t t r i b u t e s of the other a r t s which we j u s t now reviewed, not a s i n g l e one can abide but a l l w i l l be u t t e r l y d e s t r o y e d whenever the s c i e n c e of numb-ers i s n e g l e c t e d . And perhaps i t might seem t o some who merely gl a n c e at the a r t s t h a t the r a c e of mankind has but s m a l l need of number, and yet even t h a t need i s a v e r y g r e a t matter. F o r i f a person were t o p e r c e i v e the d i v i n e and the m o r t a l element i n man's g e n e r a t i o n , wherein w i l l be observed reverence f o r the gods and number i n i t s r e a l sense, s t i l l no s eer c o u l d understand the magnitude of the power t h a t number i n i t s e n t i r e t y produces f o r us, s i n c e i t i s c l e a r t h a t a l l music, through i t s a s s o c i a t i o n w i t h notes and movement, i s 20 c r e a t e d w i t h the a i d of numbers; and, most important of a l l , as a b l e s s i n g i t i s the cause of a l l good t h i n g s and i t should c f . P l a t o , E p i n . t 977C. Theon uses P l a t o ' s e x p r e s s i o n , 6 i 6 o v a i Xoyov here, where Lamb, E p i n . , (Loeb), p o i n t s t o t h e c u r i o u s p l a y on the two meanings of KOJOC,, " r e c k o n i n g " and " d e s c r i p t i o n " ; the E n g l i s h " t a l e " and "account" are s i m i l a r . 2 0 c f . E p i n . . 978A. The Greek i s somewhat d i f f i c u l t here. As Theon i s o b v i o u s l y t r y i n g t o quote P l a t o , some such word as yeveo-eai may perhaps be understood t o present P l a t o ' s o p i n i o n on the s u b j e c t . 77 be understood t h a t i t i s the cause of no e v i l t h i n g . V i r t -u a l l y d e v o i d of reason, without order and without grace, wholly b e r e f t of harmony and rhythm and u t t e r l y possessed of the q u a l i t i e s of an e v i l person i s t h a t man who i s a l t o g e t h e r b e r e f t of number. 8.18 F u r t h e r on, he adds: L e t nobody t r y to persuade us t h a t t h e r e i s any g r e a t -er p a r t of v i r t u e f o r the human race than p i e t y , f o r i t i s from t h i s t h a t the other v i r t u e s are engendered i n the man who has s t u d i e d i n a methodical way.21 9. 1 Next he demonstrates how a man may l e a r n reverence f o r 22 the gods. He says t h a t one must f i r s t l e a r n astronomy, f o r i f i t i s a d r e a d f u l c o n d i t i o n t o be i n e r r o r concerning human a f f a i r s , i t i s much more d r e a d f u l t o be so i n r e l a t i o n t o the D i v i n e . And the man i n e r r o r would be the one who holds f a l s e o p i n i o n s concerning the gods, and the man who holds f a l s e o p i n i o n s concerning the gods i s the man who has never examined the nature of the observable gods, i . e . , astronomy. He says: I t i s a f a c t not known by the m a j o r i t y of men t h a t the man who i s t r u l y an astronomer must n e c e s s a r i l y be the w i s e s t , not he who p r a c t i s e s h i s astronomy i n the sense understood by Hesiod, i . e . , the s o r t of person who has merely s t u d i e d the r i s i n g s and s e t t i n g s of the s t a r s , but the man who has s t u d i e d the o r b i t s of the seven p l a n e t s , something t h a t a person of common d i s p o s i t i o n would never r e a d i l y be capable of c o n t e m p l a t i n g . 2 3 21 c f . P l a t o , E p i n . , 989B. ;Theon i s here u n t r a n s l a t a b l e . I have read p,eT£ov TI f o r eaxiv e'xov. ^  P l a t o reads peT£ov ^tev yap apeTTK ripaq nore Tteioxi TT^C, euae pe t'aq e i v a t rep evrjT<p y e v e i . PP c f . P l a t o , JpJLn. , 989E. P"*> c f . P l a t o , E p i n . , 990A. 78 and a l s o : The person who i s p r e p a r i n g natures f o r these p u r s u i t s must g i v e p r e l i m i n a r y i n s t r u c t i o n t o as many as p o s s i b l e , t r a i n i n g them i n c h i l d h o o d and youth i n the mathematical s c i e n c e s . The g r e a t e s t of these s t u d i e s i s t o be s c i e n t -i f i c a l l y v e r s e d i n pure numbers, without r e f e r e n c e to b o d i l y substance, and i n the a b s o l u t e o r i g i n of Odd and Even and the magnitude of t h e i r i n f l u e n c e on the nature of R e a l i t y . 2 4 10. 4 He proceeds: Next a f t e r t h i s comes the study t o which they g i v e the u t t e r l y r i d i c u l o u s name of geometry, but i t r a t h e r c o n s i s t s i n e s t a b l i s h i n g resemblances between numbers not n a t \ i r a l l y a l i k e by having recourse t o the p r o v i n c e of p l a n e s . 2 ^ He mentions a l s o another s c i e n t i f i c s k i l l , which he c a l l s s o l i d geometry: whereby a person, by m u l t i p l y i n g t o g e t h e r three numbers comprising plane s u r f a c e s , a l b e i t l i k e and un-l i k e as mentioned before, produces a s o l i d body — and t h i s i s a d i v i n e and wondrous t h i n g . 2 6 10.12 In the R e p u b l i c , he says, speaking of musical harmony: The f i n e s t and g r e a t e s t form of p o l i t i c a l harmony i s wisdom, and he who l e a d s a r a t i o n a l l i f e has a share i n t h i s wisdom, whereas the man l a c k i n g i t i s a r u i n t o h i s house and no s a v i o u r of the S t a t e at a l l , i n as much as he i s ignorant of t h i n g s of the g r e a t e s t i m p o r t . 2 7 10.17 And i n the t h i r d book of the R e p u b l i c , where he t e a c h -es t h a t the p h i l o s o p h e r alone i s m u s i c a l , he asks: 24 c f . P l a t o , E p i n . , 990G. i b i d . 2 6 i b i d . 2 7 c f . P l a t o , Laws, 689D. Theon m i s t a k e n l y givest.fc.he source of t h i s q u o t a t i o n as the R e p u b l i c . 79 Then, by heaven, s h a l l we thus never be t r u e music-i a n s , n e i t h e r we nor those whom we say we must i n s t r u c t as guardians, u n t i l we can d i f f e r e n t i a t e between a l l the forms of temperance and bravery, g e n e r o s i t y and highmind-edness and a l l those q u a l i t i e s k i n d r e d w i t h them and opp-o s i t e t o them i n a l l t h e i r combinations everywhere, and u n t i l we can r e c o g n i s e the presence of these q u a l i t i e s i n c e r t a i n cases, both the q u a l i t i e s themselves and t h e i r l i k e n e s s e s , d i s r e g a r d i n g them n e i t h e r i n s m a l l a f f a i r s nor i n g r e a t , but b e l i e v i n g them t o form p a r t of the same a r t and d i s c i p l i n e ? 2 8 shows what b e n e f i t i s d e r i v e d from music and demonstrates t h a t the p h i l o s o p h e r alone i s t r u l y ^ m u s i c a l , whereas the e v i l man has no communion w i t h the Muses. For , a c c o r d i n g t o him, t r u e goodness of h e a r t , namely the v i r t u e t h a t c o n s i s t s of having a w e l l - o r d e r e d c h a r a c t e r , i s attended by good speech, i . e . , the a b i l i t y t o employ words p r o p e r l y , and t h i s good speech, i n t u r n , i s attended by elegance, rhythm and concord, elegance i n the tune, rhythm i n the measure and concord i n 29 harmony. 11.15 By c o n t r a s t , e v i l temper, i . e . , the e v i l c h a r a c t e r i s , a c c o r d i n g t o him, attended by e v i l speech, i . e . , the use of the e v i l word and t h i s e v i l speech i s attended by an absence of grace and a l a c k of rhythm and harmony i n connection w i t h 3 0 a l l t h a t one does or i m i t a t e s . Consequently, on l y t h a t man c o u l d be m u s i c a l who i s a b s o l u t e l y good at h e a r t , and he would be our p h i l o s o p h e r . T h i s has been shown by what has 11. 7 By arguments such as these and the e a r l i e r ones, he 28 c c f . P l a t o , Rep., 402B. 29 c f . P l a t o , Rep 400D,E. 30 c f . P l a t o , Rep 401A. 80 been s a i d . For s i n c e music enters the so u l from an e a r l y age because of the harmless p l e a s u r e i t a f f o r d s blended w i t h i t s u s e f u l n e s s , and implants t h e r e rhythm, harmony and elegance i t i s i m p o s s i b l e , a c c o r d i n g t o P l a t o , t o become an accomplish-ed m u s i c i a n i f one has no comprehension of t h a t which i s seem-l y i n e v e r y t h i n g and i f one does not r e c o g n i s e the forme of elegance, of l i b e r a l i t y and temperance, i . e . , t h e i r Ideas. At any r a t e , he contends they are prese n t everywhere, i . e . , t h e i r forms, and one must not d i s r e g a r d them e i t h e r i n smal l t h i n g s or i n l a r g e t h i n g s , f o r the knowledge of Ideas i s the concern of the p h i l o s o p h e r . F o r nobody co u l d p o s s i b l y under-stand p r o p r i e t y , temperance and elegance who i s h i m s e l f un-seemly and intemperate. The elements of l i f e which are elegant and endowed with rhythm and harmony are'images of elegance and of rhythm and harmony i n the a b s o l u t e , t h a t i s t o say, o b j e c t s of p e r c e p t i o n are r e f l e c t i o n s of thoughts and ideas. 3" 1' .10 , Now the Pythagoreans, whom P l a t o f o l l o w s i n many r e s -p e c t s , c l a i m t h a t music i s a combination of o p p o s i t e s , a oneness of many and a concord of d i s c o r d a n t s , f o r music does not merely compose rhythm and tune but harmonises completely every i n t e r v a l ; f o r i t s o b j e c t i s t o u n i f y and b r i n g i n t o harmony. F o r God i s a l s o a b r i n g e r of harmony t o the d i s c o r d -ant and h e r e i n l i e s H i s g r e a t e s t t a s k , by means of music and 3 1 c f . P l a t o , Rep., 40SA,C. 81 by-means of medicine t o r e c o n c i l e t h i n g s which are h o s t i l e t o one another. Upon music, they say, depends the harmony of t h i n g s and moreover, the e x c e l l e n t governing of the un i v e r s e ; f o r t h i s n a t u r a l l y takes the form of harmony i n the u n i v e r s e , good order i n the S t a t e , and moderation i n the home; f o r i t has the power t o b r i n g together and u n i f y the many. The opera-t i o n and p r a c t i c e of t h i s s c i e n c e , a c c o r d i n g t o P l a t o , occurs i n f o u r human a t t r i b u t e s , t h e s o u l , body, home and S t a t e ; f o r these f o u r spheres have need of o r d e r i n g and arrangement. .26 In the Rep u b l i c P l a t o a l s o spoke as f o l l o w s on the 32 s u b j e c t of mathematics: The good man i s he who i n the f a c e of pa i n s and p l e a s -ures, d e s i r e s and f e a r s , p r e s e r v e s and does not r e j e c t the r i g h t b e l i e f of those at h i s d i s p o s a l as a r e s u l t of h i s education. I would l i k e t o i l l u s t r a t e my meaning w i t h a s i m i l e . Now when dyers of the present day wish t o dye wool p u r p l e , f i r s t of a l l they choose out from among the v a r i -ous c o l o u r s the one nature of the white; then, they make t h e i r p r e l i m i n a r y p r e p a r a t i o n s w i t h no small care i n order t h a t the m a t e r i a l may take the hue i n the best way. I n such f a s h i o n do they dye i t ; i f one dyes anything by t h i s method, the n a t u r a l and the dyed c o l o u r become one, and n e i t h e r without nor w i t h washing-soaps can the c o l o u r be removed. I f they do not f o l l o w t h i s method and do not take p r e c a u t i o n s i n the dyeing, you know what t r a n s p i r e s ; the wools have a washed-out appearance, l o s e t h e i r c o l o u r and a re not f a s t - d y e d . And you must b e l i e v e t h a t t h i s i s the k i n d of t h i n g t h a t we too are doing t o the best of our a b i l i t y . We t r a i n our c h i l d r e n i n music and gymnastics, l e t t e r s , c g e o m e t r y and a r i t h m e t i c w i t h the express o b j e c t of g i v i n g them a p r e l i m i n a r y c l e a n s i n g and p r e p a r a t i o n w i t h these s t u d i e s a c t i n g as a s t r i n g e n t s . And the purpose of t h i s i s t h a t they may accept l i k e a dye the arguments concern-i n g v i r t u e i n g e n e r a l which they may l a t e r l e a r n , so t h a t t h e i r o p i n i o n s may be f a s t - d y e d through having had a s u i t -a b l e n u r t u r i n g o f t h e i r i n n a t e c a p a c i t y and t h a t the dye of t h e i r o p i n i o n s may not be washed out of them by such c f . P l a t o , Rep., 429D 82 soaps as these, which are d r e a d f u l washing-agents, namely p l e a s u r e , which i s more deadly than any c l o t h e s - p r e s s or running of the c o l o u r s , or p a i n , f e e r and d e s i r e , which ere more deadly than a l l o t h e r d e t e r g e n t s . 3 14.18 On the oth e r hand, one might d e s c r i b e p h i l o s o p h y as the i n i t i a t i o n i n t o a mystic r i t e and as a t r a n s m i s s i o n of t r u l y genuine m y s t e r i e s . There are f i v e stages t o the i n i t -i a t i o n : 1) The f i r s t i s the p r e l i m i n a r y p u r i f i c a t i o n ; f o r a share i n the mys t e r i e s i s not open t o a l l who d e s i r e i t , but t h e r e are some who are p u b l i c l y f o r b i d d e n access, f o r i n s t -ance, those w i t h unclean hands and a v o i c e devoid of under-st a n d i n g ; and those not f o r b i d d e n access must undergo some p r e l i m i n a r y p u r i f i c a t i o n . 2) A f t e r the p u r i f i c a t i o n comes the t r a n s m i s s i o n o f the mystery. 3) The t h i r d stage i s what i s c a l l e d the mystic v i s i o n . 4) The f o u r t h stage, which i n f a c t completes the mystic v i s i o n , i s the bi n d i n g of the head and the l a y i n g on of the g a r l a n d , so as to be able to hand on t o ot h e r s the mys t e r i e s t h a t one has r e c e i v e d , e i t h e r i n the r o l e of t o r c h b e a r e r or hierophant or some other p r i e s t . 5) The f i f t h stage, f a r s u r p a s s i n g the p r e v i o u s ones, i s the complete happiness which a r i s e s from being l o v e d by the gods and having f e l l o w s h i p w i t h them. 15. 7 In e x a c t l y the same way, the t r a n s m i s s i o n of P l a t o - s thoughts i n v o l v e s i n the f i r s t p l a c e a c e r t a i n p u r i f i c a t i o n , 33 K ~ There i s ,some d i f f i c u l t y here, -rav-roc- arpepxou 6 e i v -oTepa nat HOIvcoviac- as hendiadys may mean "more deadly than any wicked p e r v e r s i o n ; " but I have taken P r o f e s s o r R u s s e l l • s s u g g e s t i o n of arpepXri = c l o t h e s - p r e s s and HOtvcovia - ru n n i n g of K the q o l o u r s . Again, noivcovi'aq may be a copying e r r o r f o r nai HOvia<" ( l y e ) . 83 namely a t r a i n i n g from youth i n the a p p r o p r i a t e mathematical s t u d i e s . F o r , a c c o r d i n g t o Empedocles: He who seeks t o draw water from the f i v e f o u n t a i n s must c l e a n s e h i m s e l f w i t h the t i r e l e s s b r o n z e . 3 4 But P l a t o says the p u r i f i c a t i o n must be obtained from the f i v e branches of mathematics; and these are a r i t h m e t i c , geo-metry, s o l i d geometry, music and astronomy. The t r a n s m i s s i o n of the v a r i o u s p h i l o s o p h i c a l t h e o r i e s , namely l o g i c , p o l i t i c s and p h y s i c s , resembles the process of i n i t i a t i o n . The mystic v i s i o n i s the name he g i v e s t o the d i l i g e n t concern w i t h the 35 i n t e l l i g i b l e , w i t h t r u e e x i s t e n c e and the world of ideas. The b i n d i n g of the head and the crowning one must i n t e r p r e t as the a b i l i t y t o b e n e f i t from one's own s t u d i e s and set up others i n the same contemplation. The f i f t h and most p e r f e c t stage would be the complete, s u r p a s s i n g happiness which acc-I t i s not c e r t a i n whether the "bronze" i s a "cup" or a "blade", but the passage r e f e r s t o Empedocles" r i t u a l of p u r i f y i n g the s p i r i t . The exact form of the verse of Emped-o c l e s i s perhaps i r r e c o v e r a b l e . p i e l s / K r a n z , Fragmente der  V o r s o k r a t i k e r , i , B143 has: nprivdurv arto nevre r a p o v r a <6v> areipei xaAn$ ...but i t i s not d i f f i c u l t t o see, as p o i n t e d out by Bywater (Ingram Bywater, A r i s t o t l e and the A r t of Poe-t r y . O xford, 1909, p. 283), that' Theon's avip-cuvra " [ H i l l e r 15, 10) i s a p r o s a i c s u b s t i t u t e f o r ra]xovra, which i s pre s e r v e d by A r i s t o t l e i n P o e t i c s , 1457b, 10. Here A r i s t o t l e appears t o use Empedocles' q u o t a t i o n t o i l l u s t r a t e what he c a l l s the t r a n s f e r e n c e of meaning from one species' of metaphor t o an-ot h e r . ^Thus as ^ examples he g i v e s y,a\K<$ Q-n0 tyuxtiv aputfac; and rapcQV a T E i p e t xaAncp, the f i r s t of which must be t r a n s l a t -ed "drawing o f f h i s l i f e w i t h the bronze" and the second, " s e v e r i n g w i t h the t i r e l e s s bronze", where "bronze" i n the f i r s t i n s t a n c e w i l l be a "blade" and i n the second a "cup". 35 c f . P l a t o , Phaedrus, 250C. 8 4 o r d i n g t o P l a t o h i m s e l f c o n s i s t s i n becoming as much l i k e 36 God as p o s s i b l e . 1 6 . 3 There are many other t h i n g s one might say to demon-s t r a t e the u s e f u l n e s s of and n e c e s s i t y f o r mathematics. L e s t I should seem t o d i s p l a y a l a c k of good t a s t e i n wasting time upon a eulogy of mathematics I must t u r n f o r t h w i t h t o the p r e s e n t a t i o n of the r e q u i s i t e theorems i n mathematics, not every one which c o u l d make the average person the p e r f e c t a r i t h m e t i c i a n or g e o m e t r i c i a n or m u s i c i a n or astronomer, f o r t h i s i s not the p r e s c r i b e d o b j e c t i v e of those who would become acquainted w i t h P l a t o , but I s h a l l present o n l y as much as-? s u f f i c e s t o enable the reader to understand h i s w r i t i n g s . For not even P l a t o h i m s e l f r e q u i r e s t h a t one should continue i n t o extreme o l d age drawing diagrams and w r i t i n g songs, but he regards these as the s t u d i e s of youth t h a t are c a l c u l a t e d f i r s t to prepare and p u r i f y the s o u l f o r the express purpose of r e n d e r i n g i t capable of a s s i m i l a t i n g p h i l o s o p h y . T h e r e f o r e i t i s e s p e c i a l l y necessary f o r the man who i n t e n d s to study both my p r e s e n t a t i o n and P l a t o ' s w r i t i n g s to have worked through at l e a s t the: elementary steps of geometry; f o r then would he more e a s i l y f o l l o w my p r e s e n t a t i o n . N e v e r t h e l e s s , what I have t o say w i l l be of such a k i n d as to be i n t e l -l i g i b l e even to a person completely unversed i n mathematics. 1 6 . 2 4 F i r s t I s h a l l mention the theorems of a r i t h m e t i c t h a t are c l o s e l y a s s o c i a t e d a l s o w i t h those theorems of music c f . P l a t o , Theaetetus. 176B. 85 expressed i n numbers. Fo r we have no need at a l l of i n s t r -umental music , even as P l a t o h i m s e l f i m p l i e s when he says t h a t we must not t r o u b l e the s t r i n g s ( s t r a i n i n g our ears as i f we were) t r y i n g t o c a t c h a sound i n the neighbourhood. We yearn t o understand the harmony of the u n i v e r s e and t h e music t h e r e i n , but i t i s not p o s s i b l e t o p e r c e i v e t h i s music u n l e s s we f i r s t contemplate the music expressed i n numbers. And t h i s i s the reason P l a t o says music should take f i f t h 3 8 p l a c e , f o r he i s r e f e r r i n g t o the music of the u n i v e r s e t h a t d e a l s w i t h the movement, order and symphony of the s t a r s which move i n the u n i v e r s e . But i n my o p i n i o n i t i s n e c e s s a r y to p l a c e the music based on numbers next a f t e r a r i t h m e t i c a l s o i n accordance w i t h P l a t o ' s o p i n i o n , s i n c e the music of the u n i v e r s e i s not to be comprehended without the music based on numbers and thought. So then, s i n c e the p r i n c i p l e s of music based upon numbers are a s s o c i a t e d c l o s e l y w i t h the study of numbers pure and simple, they should be p l a c e d second f o r the convenience of our study. .14 F o l l o w i n g the n a t u r a l order on the other hand, f i r s t t h e r e should be the study of numbers which i s c a l l e d a r i t h -metic, second the study of s u r f a c e s c a l l e d g eometry,third 37 » See H i l l e r 6. 5. P l a t o i n R e p u b l i c , 531A has n a t -rapaPaXXovTec- r a arra and some such phrase must be understood t o make sense of t h i s passage. 3 8 c f . P l a t o , Rep., 530D. P l a t o ' s order of p r i o r i t y g i v e n i n the Republic t o the branches of mathematics was a r i t h m e t i c ( 5 2 2 C ) , geometry(526C), s o l i d geometry(527D), astronomy(528B) and music(530D). 86 the study of s o l i d s c a l l e d s o l i d geometry, f o u r t h the study of moving bodies which i s astronomy. As f o r the music t h a t de a l s w i t h movements and i n t e r v a l s and t h e i r mutual r e l a t i o n s , t h i s cannot be apprehended without our f i r s t understanding the music t h a t i s based on numbers. 17. 22 Wherefore, f o r the purpose of t h i s study of mine, the music based upon numbers should f o l l o w a r i t h m e t i c ; but, f o l l -owing the n a t u r a l order, f i f t h p l a c e should go to t h a t music-a l study which d e a l s w i t h the harmony of the u n i v e r s e . Indeed, a c c o r d i n g to the Pythagoreans, t h e study of numbers should take f i r s t p l a c e as being the s t a r t i n g - p o i n t , the fount and ro o t of a l l t h i n g s . 18. 3 A number i s a c o l l e c t i o n of u n i t s or a p r o g r e s s i o n of many u n i t s beginning with the number One and f i n i s h i n g o f f 39 by r e t u r n i n g t o One. As f o r U n i t y , i t i s the t e r m i n a t i n g 40 q u a n t i t y — ( t h e i n i t i a l element of the numbers) which, when a l a r g e number i s decreased by s u b t r a c t i o n and i s d e p r i v e d of a l l o t h e r numbers, s t i l l m aintains i t s permanent and f i x e d p o s i t i o n f o r , you see, i t i s impossible f o r the d i s s e c t i o n of the number to proceed f u r t h e r . 18. 9 F o r i f , i n the case of m a t e r i a l o b j e c t s , we p a r t i t i o n the One i n t o p a r t s , then the One on the c o n t r a r y w i l l become a number of many p a r t s and m a n i f o l d and, by the process of s u b t r a c t i n g each of i t s component p a r t s we w i l l f i n i s h up 39 Stobaeus c r e d i t s Moderatus wi t h t h i s d e f i n i t i o n . Stobaeus, Eclogues, 1, p r . , 8. 40 H i l l e r b e l i e v e s these words should be d e l e t e d . 87 w i t h One. And i f t h i s One we a g a i n p a r t i t i o n i n t o p a r t s , i t w i l l become a number of p a r t s and the r e s u l t of the s u b t r a c t -i o n of each of these p a r t s w i l l be One; consequently, One i n 41 r e s p e c t of i t s oneness has no p a r t s and i s i n d i v i s i b l e . 18.16 Now every other number, when p a r t i t i o n e d , i s dimin-i s h e d and d i s s e c t e d i n t o p a r t s s m a l l e r than i t s e l f ; as, f o r example, 6 i n t o 3 and 3, o r 4 and 2, or 5 and 1. But the One, whenever i t i s p a r t i t i o n e d i n the realm of the senses, i s di m i n i s h e d i f regarded as a body and, as a r e s u l t of the process of d i s s e c t i o n , i s p a r t i t i o n e d i n t o p a r t s s m a l l e r than i t s e l f ^ but on the other hand, i f regarded as a number, i t increases., f o r i n p l a c e of one t h e r e are many; so i t i s i n t h i s r e s p e c t t h a t U n i t y i s i n d i v i s i b l e . F o r nothing can be p a r t i t i o n e d and i n the process change i n t o p a r t s g r e a t e r then i t s e l f . 18.23 J u s t as One, when p a r t i t i o n e d , i s d i s s e c t e d i n t o a g r e a t e r number of p a r t s than has the whole, so i t i s p a r t i t -i o n e d a f t e r the manner of numbers i n t o p a r t s which are i n sum equal t o the whole. For example, whenever one m a t e r i a l ob-j e c t i s p a r t i t i o n e d i n t o s i x , as a number i t w i l l d i v i d e i n t o x x x x x x, each p a r t having the same number of p a r t s as the whole ( i . e . , one), but i f p a r t i t i o n e d as a number i n t o 4 and 2,., each w i l l have more p a r t s than the whole; f o r 2 and 4 as numbers are g r e a t e r than one. U n i t y then, c o n s i d e r e d as a number, i s without p a r t s and indeed i t i s c a l l e d U n i t y from i t s a b i d i n g unmoved and not d e s e r t i n g i t s i n h e r e n t nature. 41 See Mathematical Note 2 (p. 125) 88 F o r , however many times we m u l t i p l y U n i t y by i t s e l f , i t r e -mains One and one times one always g i v e s One, and i f we keep on m u l t i p l y i n g one t o i n f i n i t y we s t i l l getfOne. A s s u r e d l y , i t i s c a l l e d U n i t y from i t s having been s i n g l e d ^ o u t and sep-a r a t e d from the r e s t of a l l the numbers. 19.13 J u s t as a number d i f f e r s even from t h a t which i s num-bered, i n the same way does U n i t y d i f f e r from One. The numb-er i s , i n f a c t , a q u a n t i t y conceived i n the mind as, f o r i n s t a n c e , are the numbers F i v e and Ten, which, by themselves, have no p e r c e p t i b l e substance but are concepts. A q u a n t i t y t h a t may be p e r c e i v e d can be numbered, as, f o r i n s t a n c e , 5 horses, 5 oxen, 5 men but U n i t y , however, i s the c o n c e p t u a l form of One and i s i n d i v i s i b l e . As f o r the One which may be p e r c e i v e d such as one horse or one man, i t i s termed One abs-o l u t e l y . 19.21 U n i t y then w i l l . b e the p r i n c i p l e of the numbers and One the f i r s t of the numbered a r t i c l e s . And One, they say, i n so f a r as i t i s the o b j e c t of p e r c e p t i o n , can be d i s s e c t e d t o i n f i n i t y , not i n as much as i t i s a number or the p r i n c i p l e of number but i n so f a r as i t i s p e r c e p t i b l e . Consequently, U n i t y , being a concept, i s not d i v i s i b l e w h i l e One, being-p e r c e p t i b l e , can be d i v i d e d t o i n f i n i t y . And t h i n g s numbered w i l l d i f f e r from numbers i n as much as the former are c o r p o r -e a l bodies while the l a t t e r have no substance. To put i t g e n e r a l l y , men of l a t e r times term U n i t y and the number Two the p r i n c i p a l elements of the numbers, but the f o l l o w e r s of 89 Pythagoras c l a i m t h a t the p r i n c i p a l elements c o n s i s t of the s e r i e s of s u c c e s s i v e terms by which the Odd and Even numbers 4? are conceived; f o r example, the prime q u a l i t y of Three i n the realm of p e r c e p t i o n i s the T r i a d and t h a t of Four i n every i n s t a n c e i s the T e t r a d ; and so on f o r a l l other 43 numbers. F u r t h e r , they a l s o s t a t e t h a t the u n i t i s the p r i n c i p l e of a l l these very numbers, and t h a t the One i s f r e e of a l l the d i s t i n c t i o n s t h a t other numbers a p p a r e n t l y have, being o n l y One i t s e l f , not a p a r t i c u l a r one; that i s , not being the s p e c i f i c one which denotes t h i s q u a l i t y and admits a c e r t a i n d i f f e r e n c e as compared t o another, but being simply One, c o n s i d e r e d i n i t s own r i g h t . For thus i t would become the p r i n c i p l e and measure of those o b j e c t s s u b j e c t to i t s e l f , i n so f a r as each of the t h i n g s t h a t e x i s t i s c a l l e d one and has a share i n the p r i n c i p a l essence and form of One. 44 19 A r c h y t a s and P h i l o l a u s employ the terms u n i t and One without d i s t i n c t i o n and term the u n i t , one. The m a j o r i t y 4 2 c f . Stobaeus, op_. c i t . , i , 1, 9 and Z e l l e r , Die P h i l -osophie der Griechen, 7th ed., L e i p z i g , 1923, I 4 318, 335, 339. 4 3 I n c o n n e c t i o n w i t h the d e f i n i t i o n of the p r i n c i p l e of the numbers we may d e t e c t a p r o g r e s s i v e narrowing of the d e f i n i t i o n by t h r e e stages. At f i r s t , a c c o r d i n g to Stobaeus, a l l numbers were c o n s i d e r e d t o be p r i n c i p l e s ; then the Pyth-agoreans ( o i p,ev ucr-repov) c o nsidered One as the p r i n c i p l e of numbers and Two as the p r i n c i p l e of the Even numbers. Heath (H.G.M. i . 71) b e l i e v e s t h a t t h i s view of Two i s i m p l i c i t i n Nicomachus, I n t r o . A r i t h . , i , 7, 4. By the time of P l a t o , Two i s c o n s i d e r e d t o be an even number, c f . Parmenides, 143D. 44 Owing t o the s e c r e c y which bound the e a r l y Pythagor-eans, some of the fragments a s c r i b e d t o P h i l o l a u s (second h a l f of f i f t h century) assume a g r e a t e r s i g n i f i c a n c e . Thus 9 0 g i v e to t h e u n i t i t s e l f t h e e p i t h e t "prime" on the under-st a n d i n g t h a t t h e r e i s a c e r t a i n u n i t which i s not prime but a more ge n e r a l term, embracing both the u n i t i t s e l f and one; f o r they a l s o c a l l i t One, t h a t i s , the prime i n t e l l i g i b l e essence of the one, which g i v e s oneness t o every s i n g l e t h i n g . By s h a r i n g i n t h i s essence, every s i n g l e t h i n g i s c a l l e d One. Wherefore the e p i t h e t "one" g i v e s no i n d i c a t i o n as t o which one or of what k i n d of one, but i s a p p l i e d to a l l t h i n g s . 21. 5 Although i t i s not the i n t e l l i g i b l e concepts and p a t t e r n s t h a t would d i f f e r the one from the other; i t i s the p e r c e p t i b l e o b j e c t s t h a t d i f f e r . But some a s s i g n another d i s t i n c t i o n between the u n i t and One; f o r One does not change i n i t s essence n e i t h e r i s i t r e s p o n s i b l e f o r the f a i l u r e of the u n i t and the odd numbers t o change i n t h e i r essence, and a l s o i t does not change i n i t s q u a l i t y , f o r i t i s the u n i t i n i t s e l f and not l i k e a m u l t i t u d e of u n i t s as i t were; nor does i t change i n q u a n t i t y e i t h e r , f o r i t i s not composed l i k e u n i t s to which another i s added, being one and not many u n i t s , and f o r t h a t reason i t i s c a l l e d One u n i q u e l y . F o r i f , i n P l a t o , r e f e r e n c e has been made t o the u n i t s i n the P h i l e b u s , they have not been c a l l e d a f t e r the One but a f t e r the number Stobaeus preserves the f o l l o w i n g fragment; " A l l t h i n g s t h a t can be known have number; f o r i t i s impossible f o r a t h i n g t o be known or c o n c e i v e d without number" ( D i e l s , Vors, 44B, 111). A r c h y t a s ( f i r s t h a l f of f o u r t h century) g r e a t l y i n f l -uenced P l a t o and was l a r g e l y i n s t r u m e n t a l i n e s t a b l i s h i n g mathematics i n the c u r r i c u l u m of the Academy. c f . P l a t o , P h i l e b u s 15A. 91 One which i s the u n i t by v i r t u e of i t s share i n the one. So t h i s One which i s d e f i n e d by the bounds of the u n i t i s i n a l l r e s p e c t s unchangeable, so t h a t t h e One would d i f f e r from the u n i t i n so f a r as i t i s d e f i n e d and l i m i t e d , whereas the u n i t s are boundless and undefined. 21.20 Numbers may f i r s t be c l a s s i f i e d i n t o two k i n d s , c a l l -ed Even and Odd. The even numbers such as the number Two and the number Four are those which admit of d i v i s i o n i n t o equal p a r t s , w h i l e the odd numbers such as F i v e and Seven are those which may be d i v i d e d o n l y i n t o unequal p a r t s . Some c a l l e d the f i r s t of the odd numbers U n i t y , f o r even i s the op p o s i t e of odd and U n i t y i s a s s u r e d l y e i t h e r odd or even. Now, i t co u l d not be even, f o r i t cannot be d i v i d e d i n t o equal p a r t s and indeed i t cannot be p a r t i t i o n e d at a l l ; t h e r e f o r e , U n i t y i s odd. I f you add an even number t o an even number, the t o t a l i s even; but U n i t y added t o an even number g i v e s a t o t -a l t h a t i s odd, so a s s u r e d l y U n i t y i s not even but odd. 46 22. 5 A r i s t o t l e in..the Pythagorean s t a t e s t h a t One par-takes of the nature of both odd and even f o r , when added t o an even number, i t makes i t odd, and when added t o an odd number, i t makes i t even, which i t co u l d not do i f i t d i d not share i n the nature of both. Wherefore, One i s c a l l e d 47 "odd-even" and A r c h y t a s agrees w i t h t h i s . 46 Probably one of A r i s t o t l e ' s l o s t s c i e n t i f i c t r e a t i s e s . 4 7 c f . A r i s t . , Metaph.. A5, 986a, 19. A r i s t o t l e ' s r e a s - ' oning f o r t h i s d e f i n i t i o n does not seem c o n v i n c i n g , f o r any odd number would meet such c o n d i t i o n s . Thus, take 7; then 92 22.10 The f i r s t form of the odd then i s U n i t y , j u s t as i n the u n i v e r s e men apply the term odd to th a t which i s d e f i n e d and org a n i s e d ; but the f i r s t form of the even i s the u n d e f i n -ed number Two, j u s t as a l s o i n the u n i v e r s e men apply t h e term even t o t h a t which i s unbounded, unknown and not ordered. T h e r e f o r e the number Two i s a l s o c a l l e d i n d e f i n i t e s i n c e i t 48 i s not, as i s U n i t y , d e f i n e d . As f o r the numbers which f o l -low, set out i n s u c c e s s i o n a f t e r U n i t y , these i n c r e a s e by an equal amount, f o r each of them i s g r e a t e r than the p r e c e d i n g by one; but as they i n c r e a s e the r a t i o of the one t o the 49 other decreases. 22.20 Take, f o r example, the numbers 1, 2, 3, 4, 5, 6; t h e r a t i o of the number 2 t o u n i t y i s double; that of the number 3 t o the number 2 i s one and one-half t o one; that of the number 4 to the number 3 i s one and o n e - t h i r d t o one; t h a t of the number 5 t o the number 4 i s one and one-fourth t o one; and the r a t i o of the number 6 t o the number 5 i s one and one-f i f t h t o one. Now the r a t i o of one and o n e - f i f t h t o one i s 7 + 6 = dd, 7+9 = "ev n, and 7 can h r l y be termed "odd-even". But Heath(H.G . M . i , 71) p o i n t s t o a fragment of P h i l o l a u s which mentions many forms of the even and the odd and a t h i r d form w e v e n - o d d n which r e f e r s t o a number which i s the product of an even number and an odd number, so t h a t "even" i n the same passage must r e f e r t o numbers which are powers of 2, i . e . , of form 2 n. Theon i s here being i n f l u e n c e d by P h i l o l a u s - - h i s f i r s t two even numbers are 2, 4 ( H i l l e r 21.23). 4^The term a o p i c r r o c *(undefined) i s a p p l i e d here and again i n 24.24 t o the number Two; f o r whereas the odd numbers by add-i t i o n , ^ 1+3+5+.. form squares w i t h constant s i d e r a t i o s {r\ \iovaq cLptau-evri), t h e even numbers s t a r t i n g from Two, i . e . , 2+4+6 + .. form r e c t a n g l e s (termed heteromecic) and the r a t i o of t h e i r s i d e s c o n s t a n t l y changes (dopicr-roc; r| 6uac;). 4%-hus 6/5 ^ 5 / 4 <4/3 < 3/2 < 2/1 93 l e s s than the r a t i o of one and one-fourth t o one; t h a t of one and one-fourth t o one l e s s than t h a t of one and o n e - t h i r d t o one; the r a t i o of one and o n e - t h i r d t o one i s l e s s than t h a t of one and one-half t o one;-and the r a t i o of one and o n e - h a l f to one i s l e s s than double. And f o r . t h e r e s t of the numbers the r a t i o s f o l l o w the same p a t t e r n . And the numbers are seen 50 to be a l t e r n a t i v e l y even and odd w i t h the e x c e p t i o n of One. 23. 6 Among numbers some are c a l l e d a b s o l u t e l y prime or i n -composite, some are c a l l e d prime w i t h r e s p e c t to each o t h e r but not a b s o l u t e l y so, some are a b s o l u t e l y composite and others composite w i t h r e s p e c t to each other. 23. 9 Those numbers are a b s o l u t e l y prime and incomposite which cannot be d i v i d e d by any other number except u n i t y a l -51 one, such as 3, 5, 7, 11, 13, 17 and numbers such as these. These same numbers are c a l l e d l i n e a r and euthymetric by reason of the f a c t t h a t the l e n g t h s and the l i n e s can be v i s u a l i s e d i n only one dimension; they are thus c a l l e d "oddly-odd". Con-se q u e n t l y they are named i n f i v e ways, prime, incomposite, l i n -ear, euthymetric and "oddly-odd"; t h i s i s the only way i n which they can be d i v i d e d , f o r 3 c o u l d not be d i v i d e d by any o t h e r number so as to be-the r e s u l t of the m u l t i p l i c a t i o n of 5 C F o r One has been termed "odd-even" ( H i l l e r 22. 9) ^^The word used here f o r " d i v i d e d by" i s p,eTpoup,evoi t and i t i s i n t e r e s t i n g to note t h a t t h i s word s t i l l s u r v i v e s i n E n g l i s h i n G.CM.--"greatest common measure", t h e l a r g e s t number t h a t can be d i v i d e d i n t o two or more numbers. A more modern mathematical term i s modulus, from L a t i n modulus, a sma l l measure. See Mathematical Note 9 (p. 134). 94 the f a c t o r s except by u n i t y alone; f o r one times 3 i s 3. L i k e w i s e , one times 5 i s 5 , one times 7 i s 7 and one times 11 i s 11. T h i s i s the reason these numbers are c a l l e d "oddly-odd"; f o r they are odd measures and the u n i t y which d i v i d e s them i s odd a l s o . Wherefore, o n l y odd numbers can be prime and incomposite. 23.24 Indeed, even numbers are n e i t h e r prime nor incomposite and are not d i v i s i b l e by u n i t y alone but by other numbers; as, f o r i n s t a n c e , the number 4 i s d i v i s i b l e by 2, f o r 2 times 2 i s 4; the number 6 i s d i v i s i b l e by 2 and 3, f o r 2 times 3 and 3 times 2 are 6. And, i n the same way, a l l the r e s t of the even numbers w i t h the exception of the number 2 are d i v i s i b l e by numbers g r e a t e r than u n i t y . T h i s number alone of the even numbers i s i n e x a c t l y the same case as are a l s o a number of the odd numbers-, namely being d i v i s i b l e by u n i t y alone. F o r one times 2 i s 2; wherefore, i t i s s a i d to"have an odd-form" and i s t r e a t e d the same as the odd numbers. 24. 8 Numbers are d e f i n e d as prime t o each other but not ab^ s o l u t e l y so, when they have u n i t y as a common f a c t o r yet are a l s o d i v i s i b l e by c e r t a i n other numbers when taken by them-s e l v e s ; as, f o r i n s t a n c e 8 which i s d i v i s i b l e by 2 and 4, 9 which i s d i v i s i b l e by 3^and 10 which i s d i v i s i b l e by 2 and 5. But they a l s o have u n i t y as a common f a c t o r , both w i t h r e s p e c t to each other and t o t h e i r prime f a c t o r s ; f o r one times 3 i s 3 and one times 8 i s 8, one times 9 i s 9 and one times 10 i s 52 10. X 95 84.16 Wholly composite numbers are those d i v i s i b l e by a number l e s s than themselves, such as 6 which has f a c t o r s 2 and 3. Numbers composite with r e s p e c t t o each other are those d i v i s i b l e by some common f a c t o r , as are 8 and 6 which have a common f a c t o r 2, f o r 2 times 3 i s 6 and 2 times 4 i s 8. A l s o composite w i t h r e s p e c t t o each other are 6 and 9, t h e i r common f a c t o r being 3; f o r 3 times 2 i s 6 and 3 times^3 i s 9. Now U n i t y i s not a number but the " p r i n c i p l e of number" and n e i t h e r i s the undefined number Two, f o r i t i s the f i r s t number d i f f -e rent from U n i t y and c o n t a i n s n o t h i n g more b a s i c than U n i t y among the even numbers,^ 3 24.25 Of the composite numbers, those produced by two numb-ers are c a l l e d "plane", being envisaged as having two dimens-i o n s , i . e . , being the product of a l e n g t h and a width; w h i l e those composite numbers made from t h r e e numbers are termed : 54 " s o l i d " , f o r they have a t h i r d dimension. The product i s the name g i v e n to the r e s u l t of the m u l t i p l i c a t i o n of two numbers w i t h each other. 25. 5 Among even numbers, some are c a l l e d "evenly-even", others "oddly-even" and others "even-odd". Those numbers are 5 2 T h u s 3 and 7 would be a b s o l u t e l y prime; 9 and 8 prime w i t h r e s p e c t t o each other, but not a b s o l u t e l y prime, f o r 9 -» 3-3 and 8 = 4*2, "when taken by themselves", but 3 and 7 are 3*1 and 7-1, a b s o l u t e l y prime, "when taken by themselves". Dupuis d e l e t e s artac; y ' y ' H a i j f o r 3 i s " a b s o l u t e l y prime" (Dup-u i s , 38.15). ^"nothing more b a s i c " s c . as a f a c t o r . I t has no even f a c t o r s . 5 4 S e e Mathematical Note 4 (p. 129) 96 "evenly-even" which meet t h r e e c o n d i t i o n s , f i r s t they are formed by the m u l t i p l i c a t i o n of two even numbers; second, they have a l l t h e i r p a r t s even as f a r as the f i n a l f a c t o r u n i -t y ; and t h i r d , none of t h e i r f a c t o r s c ould go by the name of "odd" number. Such numbers are 32, 64, 128 and those o b t a i n -55 ed next a f t e r these by the process of doubling,. For 32 r e s -u l t s from the f a c t o r s 4 and 8 which are even, and a l l i t s f r a c t i o n s are even; one-half of i t i s 16, one-fourth i s 8, one-eighth i s 4--so the f r a c t i o n s themselves are c a l l e d even. For o n e - h a l f i s c o n s i d e r e d as belonging t o the number Two, and so are o n e - f o u r t h and one-eighth, and the same reas o n i n g app-l i e s e q u a l l y t o the r e s t of these numbers. 25.19 "Even-odd" numbers are those numbers which are the products of 2 and some other number which i s odd; a l l such numbers without e x c e p t i o n upon d i v i s i o n i n t o equal p a r t s have h a l v e s which are odd. Such a number i s twice 7, or 14. "Evenly-odd" they are then c a l l e d because they are d i v i s i b l e by the even number Two and by some odd number; as i s 2 d i v i s i b l e by the number One, 6 by 3, 10 by the number 5 and 14 by the numb-er 7. By the f i r s t d i v i s i o n these numbers are rendered odd, and a f t e r the f i r s t d i v i s i o n they are no l o n g e r d i v i s i b l e i n -t o equal p a r t s . F o r the h a l f of 6 i s 3, but 3 i s d i v i s i b l e no f u r t h e r i n t o equal p a r t s - - f o r U n i t y i s i n d i v i s i b l e . 26. 5 The numbers termed "oddly-even" are those r e s u l t i n g from the m u l t i p l i c a t i o n of two other numbers, an odd number and an even number, and when m u l t i p l i e d t h e i r products are 55 n Thus the "evenly-even" numbers are of form 2 . See note 47. 97 d i v i s i b l e i n t o two equal p a r t s t h a t are even; but, u s i n g l a r -ger d i v i s o r s , they produce q u o t i e n t s which are sometimes even and sometimes odd. Such numbers are 12 and 20; f o r 3 times 4 i s 12, and 5 times 4 i s 20; and 12 d i v i d e s i n t o two p a r t s 6 and 6, and 3 p a r t s 4 and 4 and 4, and i n t o 4 p a r t s of 3. I n the same way 20 d i v i d e s i n t o two p a r t s of 10, and f o u r p a r t s of 5, and f i v e p a r t s of 4. 26.14 F u r t h e r , of the composite numbers some are " e q u a l l y -e q u a l " and so square and plane, when some number i s formed by 57 the m u l t i p l i c a t i o n of an equal by an equal; as, f o r i n s t a n c e , 4 which i s 2 times 2, and 9 which i s l 3 times 3. 26.18 On t h e other hand, numbers are "unequally-unequal" when unequal numbers are m u l t i p l i e d t o g e t h e r , as i n the case of 6 which i s 2 times 3. 26.21 Those of the numbers t h a t have one s i d e l a r g e r than 58 the other by u n i t y are heteromecic. Now a number l a r g e r than an odd number by u n i t y i s a l s o even; wherefore, heteromecic numbers can o n l y be even, f o r the p r i n c i p l e of the numbers, i . e . , U n i t y i s odd and, by i t s tendency to e f f e c t a change by 5^Theon has c l a s s i f i e d numbers: (1) "even", of form 2 n (note 47), (2) "even-odd", of form 2x (x i s odd), (3) "oddly-even", of form 2 nx (x i s odd). Type (3) y i e l d s a " quotient t h a t i s odd" upon d i v i s i o n o n l y by 2 n. ^ H i l l e r suggests the d e l e t i o n of 6 y e v v r i Q e i s i a a H i ^ re iao% nat xexpax(X)vo<i koxiv, p r o b a b l y on the grounds of redund-ancy, but i n the liglrt6f the t e d i o u s r e p e t i t i v e d e t a i l of t h i s p o r t i o n of the work, t h i s measure h a r d l y seems j u s t i f i e d . 58£xepO]xr\yiT\c, i n P l a t o and A r i s t o t l e has the more gener-a l sense of "any number w i t h two unequal f a c t o r s " , i . e . , o b l -ong. Nicomachus however and, here, Theon use the word f o r the 98 the d o u b l i n g of i t s e l f produces the number Two which i s het-eromecic. And f o r t h i s reason the number Two, being h e t e r o -mecic and exceeding U n i t y by one u n i t , makes heteromecic num-bers out of the even numbers and the odd numbers t h a t they 59 exceed by one. 27. 7 They are formed i n two ways: by m u l t i p l i c a t i o n and by a d d i t i o n — b y a d d i t i o n , when even numbers are added t o the succeeding even numbers taken i n order and the numbers prod-uced are heteromecic. Take, f o r i n s t a n c e , i n order the even numbers 2, 4, 6, 8, 10, 12, 14, 16, 18. The r e s u l t of the a d d i t i o n of 2 p l u s 4 i s 6, of 6 p l u s 6 i s 12, of 12 p l u s 8 i s 20, and of 20 p l u s 10 i s 30; so the sums r e s u l t i n g , namely 6, 12, 20, and 30 would be heteromecic and the same a p p l i e s t o the sums of succeeding even numbers. 27.14 The same heteromecic numbers are produced by the mult-i p l i c a t i o n of s u c c e s s i v e even and odd numbers, by m u l t i p l y i n g the f i r s t number by the one f o l l o w i n g i t . Thus, take 1, 2, s p e c i a l case of a number whose f a c t o r s d i f f e r by one; and they use - t p o p r i H T i q f o r the g e n e r a l case of "oblong" tnumbers (See H i l l e r 30.8). Thus, f o r t r a n s l a t i n g Theon's ere popriHT-<-, Dupuis uses heteromeque, w h i l e some mathematical w r i t e r s i n E n g l i s h have coined the term "heteromecic". On the other hand, where they use "oblong" f o r ere popqKT--- they use "pro-l a t e " f o r 7rpoy,nHriq. 59 The reasoning i s : Two i s the f i r s t heteromecic number, f o r i t exceeds u n i t y by one; Two makes a l l even numbers even; t h e r e f o r e , Two makes the even numbers, when taken w i t h t h e i r odd c o u n t e r - p a r t s (the numbers p r e c e d i n g them), i n t o h e t e r o -mecic numbers. See Mathematical Note 6 (p. 132) 99 3, 4, 5 , 6, 7, 8, 9, 10. Now 1 times 2 i s 2, 2 times 3 i s 6, 3 times 4 i s 12, 4 times 5 i s 20, 5 times 6 i s 30, and so on f o r the succeeding numbers. Such numbers are termed h e t e r o - . mecic immediately the a d d i t i o n of u n i t y t o one of the s i d e s causes a d i f f e r e n c e between the s i d e s . 27.23 Parallelogram-numbers are those numbers which have one s i d e g r e a t e r than the other by 2, as have the numbers 8, 24, 48, and 80, which are 2 times 4, 4 times 6, 6 times 8, 62 and 8 times 10. 28. 3 The square numbers are those produced by the a d d i t i o n of the odd numbers i n s u c c e s s i o n . Now, take i n order the odd numbers 1, 3, 5 , 7, 9, 11. One p l u s 3 makes 4, which i s a Or p o s s i b l y , "causes a change i n the s i d e s " , namely from b e i n g both odd or both even, to one odd and one even. The Greek here, I t h i n k , g i v e s an i n d i c a t i o n of the p r a c t i c a l way i n which numbers were s t u d i e d , i . e . , by diagrams composed of pebbles or counters. Thus: . . . . . . . • • • =. 9, and the f i r s t change i n one s i d e • • • • produces the heteromecic number 12; c f . TTJV e-repo-rriTa. £T]Toucra } the same idiom i n H i l l e r 27. 3. ^^T have ^followed Dupuis (44.23 note) i n d e l e t i n g the words r\' Hai yaei'^ovi api6yacp; otherwise, j>arallelogram-numbers would hot d i f f e r from the oblong (7tpop,TiHr]c;) numbers a l r e a d y d e f i n e d . I n a d d i t i o n , Theon quotes as examples,, numbers whose f a c t o r s d i f f e r only by 2. I t i s apparent t h a t Theon i s attempting t o c l a s s i f y numbers a c c o r d i n g to t h e i r f a c t o r s as follows:, a) TerpaYCDvoc; ( s q u a r e ) : f a c t o r s equal, of form n«n = n 2 b) 7cpop,TiKT).q (oblong or p r o l a t e ) : f a c t o r s d i f f e r i n g by 1, 2, or more, of form n( n + a} c) eTepotirixTiq ( h e t e r o m e c i c ) : f a c t o r s d i f f e r i n g by 1, of form n(n - j - 1) d) TrapaAAnAoYpawioq ( p a r a l l e l o g r a m ) : f a c t o r s d i f f e r i n g by 2, of form n(n -f- 2) 100 square f o r i t i s " e q u a l l y - e q u a l " , i . e . , 2 times 2 i s 4; 4 p l u s 5 makes 9, which a g a i n i s a l s o a square, f o r 3 times 3 i s 9; 9 p l u s 7 i s 16, which i s a l s o a square f o r 4 times 4 i s 16; 16 p l u s 9 makes 25, which i s a square and " e q u a l l y - e q u a l " , b e i ng 5 times 5; and one c o u l d continue the same procedure t o i n f i n i t y . Such, then, i s the method of producing square num-bers by the process of a d d i t i o n , the next odd number being added to the square o b t a i n e d by summing the p r e c e d i n g odd 6 3 numbers s t a r t i n g from u n i t y . 28.13 And square numbers are a l s o produced by means of mult i p l i c a t i o n , when any number at a l l i s m u l t i p l i e d by i t s e l f ; as, f o r example, 2 times 2 g i v e s 4, 3 times 3 g i v e s 9, and 4 times 4 g i v e s 16. 28.16 Now a l l the square numbers have heteromecic numbers as means i n a geometric p r o p o r t i o n , but heteromecic numbers, f o r t h e i r p a r t do not e n c l o s e square numbers as means i n a 64 p r o p o r t i o n . Thus, take the numbers 1, 2, 3, 4, 5. Each of 6 3 S e e Mathematical Note 5 (p. 137) 64 "Consecutive" or " a d j a c e n t " square numbers are meant. Thus, 4^(16) and 5^(25) have the heteromecic number 20 as a geometric mean, f o r 16:20 - 20:25. The Pythagorean p a t t e r n s would demonstrate t h i s r e l a t i o n s h i p w i t h c l a s s i c s i m p l i c i t y . ill]] ^ ..... Heteromecic numbers do not e n c l o s e square numbers as means i n a"geometric p r o p o r t i o n " , but they do have square numbers as " a r i t h m e t i c a l " means. See Mathematical Note 7 (p. 132) 101 these m u l t i p l i e d by i t s e l f g i v e s a square number; f o r 1 times 1 i s 1, 2 times 2 i s 4, 3 times 3 i s 9, 4 times 4 i s 16, 5 times 5 i s 25. Now these numbers 8 r e produced by i d e n t i c a l 65 f a c t o r s , f o r the 2 o n l y doubles i t s e l f , t h e 3 t r i p l e s i t s e l f , so the square numbers would be i n order 1, 4, 9, 16, 25 and they have heteromecic numbers f o r means i n the f o l l o w i n g way. Two s u c c e s s i v e squares are 1 and 4, and t h e i r mean i s the heteromecic number 2. Now set down the numbers 1, 2, 4, and the mean becomes 2, exceeding the one extreme i n the same r a t i o as i t i s exceeded by the other, f o r 2 i s the double of 1, and 4 i s the double of 2. Again, take the square numbers 4 and 9; t h e i r mean i s the heteromecic number 6. Set down 4, 6, 9; then the mean 6 exceeds the former extreme i n the same r a t i o as i t i s i t s e l f exceeded by the l a t t e r extreme, f o r the r a t i o of 6 t o 4 i s one and one-half t o one, as i s the r a t i o of 9 t o 6. The same re a s o n i n g a p p l i e s t o the r e s t of the s u c c e s s i v e square numbers. .12 On the other hand, the heteromecic numbers, products of f a c t o r s which d i f f e r by u n i t y , do not keep the same f a c t -ors nor e n c l o s e square numbers i n a p r o p o r t i o n . Thus 2 times 3 g i v e s 6, 3 times 4 g i v e s 12, and 4 times 5 g i v e s 20, none of the numbers keeping the same f a c t o r s but changing i t i n the 65«/ opo<- can mean the "term" of a p r o p o r t i o n , but I t h i n k i t i s more l i k e l y t o have i t s o r i g i n a l meaning of " l i m i t " here; hence the l i t e r a l meaning i s : "do not change t h e i r own l i m i t s " , c f . ovb£\$tavx<s>v u-e'vei ev T$ eau-rou opcp ( H i l l e r , 29.17) But A r i s t o t l e uses opoc- i n the sense of Theon*s yvo)]iojv i n Phys, i i i , 203a, 13; t h i s meaning would not however seem p o s s i b l e here, f o r Theon has j u s t expressed the squares as "products". 10 2 process of m u l t i p l i c a t i o n f o r , you see, 2 i s m u l t i p l i e d by 3, and 3 by 4, and 4 by 5. But the r e s u l t i n g heteromecic numbers do not e n c l o s e the square numbers i n a geometric p r o p o r t i o n ; f o r t a k e the s u c c e s s i v e heteromecic numbers 2 and 6; i n the p o s i t i o n between them i s the square number 4, but i t i s not enclosed by them i n any p r o p o r t i o n which has the same r a t i o t o i t s extremes. Set down 2, 4, 6 and 4 w i l l have a d i f f e r e n t r a t i o t o the extremes; f o r t h e r a t i o of 4 t o 2 i s double but the r a t i o of 6 t o 4 i s one and o n e - h a l f t o one. For the mean to form a g e o m e t r i c a l p r o p o r t i o n , i t would be necessary f o r i t t o be such t h a t the r a t i o of the f i r s t term t o the mean sh o u l d be equal t o t h a t of the mean to the t h i r d term. Again, a l t h -ough the'square number 9 i s s e t i n p o s i t i o n between the h e t e r o -mecic numbers 6 and 12, i t w i l l not be found t o bear the same r a t i o t o t h e s e extremes; f o r the r a t i o of 9 t o 6 i s one and one-half t o one, but the r a t i o of 12 to 9 i s one and one-66 t h i r d t o one. The same a p p l i e s t o the r e s t of the s u c c e s s -i v e heteromecic numbers. 30. 8 An oblong number i s a number produced by any two un-equal nunibers, where one exceeds the o t h e r perhaps by 1, per-67 haps by 2 or even more; as, f o r i n s t a n c e , the number 24 and 66 Theon i s somewhat i n c o n s i s t e n t , i n t e r p r e t i n g a geometric p r o p o r t i o n d i f f e r e n t l y , here and i n 30. 1 above. 67 See note 57. Theon has shown t h a t square numbers r e -s u l t from the a d d i t i o n of the odd numbers i n s e r i e s (see Math-e m a t i c a l Note 5 ) , and heteromecic numbers from the a d d i t i o n of the even numbers (see Mathematical Note 6 ) . Heath (H.G.M. i , 82) suggests t h a t we p o s s i b l y have here an e x p l a n a t i o n f o r : t h e 103 such numbers as i t , f o r i t i s 6 times 4. There are t h r e e . c l a s s e s of oblong number. Every heteromecic number i s oblong i n as much as i t has one s i d e l o n g e r than the ot h e r , so i f a number i s heteromecic i t i s a l s o oblong; but the converse i s not t r u e , f o r a number which has one s i d e l a r g e r than the other by more than u n i t y i s oblong, but a s s u r e d l y not heteromecic, f o r the heteromecic number i s t h a t having one s i d e g r e a t e r than the other by u n i t y , as does 6; f o r i t i s 2 times 3. 30.18 Furthermore, a number i s a l s o oblong when i t can be ob-t a i n e d by a v a r i e t y of p o s s i b l e m u l t i p l i c a t i o n s ; i n the f i r s t i n s t a n c e , w i t h one s i d e g r e a t e r than the o t h e r by u n i t y and i n the second, g r e a t e r by more than u n i t y ; as, f o r i n s t a n c e , the number 12, f o r i t i s 3 times 4, and 2 times 6, so t h a t i t would be heteromecic on the f i r s t count and oblong on the second. 30.23 Again, a number i s oblong i f , as a r e s u l t of r e s o r t i n g to a l l p o s s i b l e combinations of m u l t i p l i c a t i o n the one s i d e i s s t i l l , g r e a t e r than the other by more than u n i t y ; as, f o r i n s t a n c e , i s the number 40, f o r i t i s 4 times 10, and 5 times 8, and 2 times 20. Whatever number f o l l o w s t h i s p a t t e r n can Pythagorean i d e n t i f i c a t i o n of odd w i t h " l i m i t " o r " l i m i t e d " and even w i t h " u n l i m i t e d " ( c f . A r i s t . , Metaph., A5, 986a, 17). He a l s o draws a t t e n t i o n to. the Pythagorean scheme of t e n p a i r s of o p p o s i t e s , where odd, l i m i t , square i n one set are opposed t o even, u n l i m i t e d and oblong r e s p e c t i v e l y i n the other ( c f . A r i s t . , i b i d . , 23-26). Thus, the s e r i e s of odd numbers pro-duce square numbers, and a l l squares are s i m i l a r f i g u r e s , the r a t i o of the s i d e s being always equal ( 1 : 1 ) , whereas the s e r -i e s of even numbers produce heteromecic numbers , i . e . , 2*3, 3*4, 4'5 e t c . and the f i g u r e s produced are c o n t i n u a l l y d i s s i m -i l a r oblongs; wherefore, odd i s a s s o c i a t e d w i t h " l i m i t e d " and even w i t h " u n l i m i t e d " o r "undefined". See note 48. - 104 o n l y be oblong, ^ o r the heteromecic number i s t h a t number which f i r s t r e s u l t s from changing a number formed of equal s i d e s — a n d the a d d i t i o n of u n i t y t o one of these s i d e s f i r s t e f f e c t s t h i s change. Wherefore- numbers r e s u l t i n g from t h i s f i r s t change i n a s i d e are r i g h t l y termed heteromecic; but those numbers t h a t have one s i d e g r e a t e r than the ot h e r by more than u n i t y are c a l l e d oblong by reason of the g r e a t e r extent i n the l e n g t h of one s i d e . 31. 9 Those numbers are plane numbers t h a t are produced by the m u l t i p l i c a t i o n of two numbers r e p r e s e n t i n g , as i t were, a l e n g t h and a width. Of these numbers, some are t r i a n g u -l a r , some are square, some pentagonal and of succeeding p o l y g o n a l forms. 31.13 T r i a n g u l a r numbers are produced i n the f o l l o w i n g way. S u c c e s s i v e even numbers added t o one another i n s e r i e s make heteromecic numbers. F o r i n s t a n c e , the number 2 i s the f i r s t even number and i t i s a l s o heteromecic, f o r i t i s one times 2. Then, i f you add 4 t o 2 the r e s u l t i s 6, which again i s h e t e r o -mecic, f o r i t i s 2 times 3. The same process may be repeated t o i n f i n i t y and, t o put i t more p l a i n l y t h a t my p r o p o s i t i o n may be manifest t o a l l , i t may a l s o be demonstrated i n t h i s way. F i r s t l e t t h e number 2 be represented by these two x's x x The f i g u r e t h a t they make w i l l be heteromecic, f o r i t i s 2 i n l e n g t h and 1 i n width. A f t e r the number 2 comes the even 68 See Mathematical Note 4 (p. 129) 105 number 4; i f we should add these f o u r u n i t s t o the f i r s t two u n i t s , and i f we p l a c e the 4 around the 2, the r e s u l t i s a heteromecic f i g u r e of 6 u n i t s , w i t h a l e n g t h of 3 and a width of 2. Next a f t e r 4 comes the even number 6; i f we add t h i s t o the f i r s t 6, the r e s u l t i s 12 and the f i g u r e w i l l be hetero-mecic i f we p l a c e these u n i t s around the f i r s t 6 u n i t s , f o r i t w i l l have a l e n g t h of 4 u n i t s and a width of 3. And the same p a t t e r n may be obtained w i t h the a d d i t i o n of s u c c e s s i v e even numbers to i n f i n i t y . x x I x x x I x X X X X X X X X X X X X 32. 9 Again, t h e a d d i t i o n of s u c c e s s i v e odd numbers p r o -69 duces square numbers. The odd numbers i n o r d e r are 1, 3, 5, 7, 9, 11. I f you add these i n order you w i l l c o n s t r u c t square numbers. Thus the number One i s the f i r s t square num-ber, f o r 1 times 1 i s 1. Then comes the odd number 3; i f you add t h i s as a gnomon t o 1, yoU'Will make a square " e q u a l l y -e q u a l " , f o r i t w i l l be 2 i n l e n g t h and 2 i n width. The next odd number i s 5; i f you should, take t h i s as gnomon and add i t t o your square, once aga i n a square number 9 w i l l be produced, w i t h a l e n g t h of 3 and a width of 3. The next odd number i s 7; i f you add t h i s t o 9 you w i l l get 16, which has a l e n g t h of 4 and a width of 4. The same procedure can be f o l l o w e d t o i n f i n i t y . 69 See Mathematical Note 5 (p.137) 106 x |x X X x {x X X X X X X X X J X X x X X X X X X X X X X X X 32.22 And i n t h e same way, i f we should add to g e t h e r not s o l e l y the s e r i e s of even numbers nor s o l e l y the s e r i e s of odd numbers, but both the even numbers and the odd numbers, 70 we s h a l l have t r i a n g u l a r numbers. L e t us take the odd num-bers and the even numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 1 0 T h e t r i a n g u l a r numbers are formed by the a d d i t i o n of these numbers. The f i r s t t r i a n g u l a r number i s U n i t y ; which, i f not t r i a n g u -l a r i n a c t u a l i t y , i s so by reason of the u n i v e r s a l i t y of i t s power, f o r i t i s the " p r i n c i p l e " of a l l the numbers. I f the number 2 i s added t o i t , t h e t r i a n g u l a r number 3 i s produced. Then by adding 3, 6 i s produced, and by adding 4, 10 i s p r o -duced; then by adding 5, 15 i s produced; by adding 6, 21 i s produced; by adding 7, 28 i s produced; then 8 g i v e s 36, 9 g i v e s 45, 10 g i v e s 55, and so on t o i n f i n i t y . 33.13 Now i t i s obvious t h a t these numbers are t r i a n g u l a r i n accordance w i t h the f i g u r e s o b t a i n e d by the a d d i t i o n of s u c c e s s i v e gnomons t o the numbers a l r e a d y o b t a i n e d . Thus the t r i a n g u l a r numbers r e s u l t i n g from these a d d i t i o n s w i l l be 3, 6, 10, 15, 21, 28, 36, 45, 55 and i n t h i s way do they f o l l o w a f t e r 45 and 55. 70 • , See Mathematical Note 10 (p. 135) 167 1 3 6 10 15 21 x x x x x x x x x x x x x x x x X X X X X X X X X X X X x x x x x x x x x x x x x x x x x x x x x x X X X X X X 28 36 X X X X X X X X X X X X " x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 34. 1 The square numbers are produced, as has been s a i d be-f o r e , by t h e a d d i t i o n of the s u c c e s s i v e odd numbers s t a r t i n g 71 from u n i t y and, w i t h the e x c e p t i o n of one, they happen to be a l t e r n a t i v e l y even and odd, j u s t as are the n a t u r a l numbers, v i z . 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. 34. 6 By s e t t i n g out i n order from one the even numbers and the odd numbers, i t happens t h a t t h e gnomons exceeding each other by 2 produce by the process of a d d i t i o n the square num-bers as has been shown above, f o r the numbers that i n c r e a s e by 72 2 from u n i t y are the odd numbers. 34.10 I n the same way the numbers i n c r e a s i n g by 3 from u n i t y by a d d i t i o n form pentagons; those i n c r e a s i n g by 4 produce hexagons; and furthermore, the i n c r e a s e i n the gnomons from which the p o l y g o n a l numbers are produced i s always l e s s by 2 7 " b n e i s excepted, because i t has a l r e a d y been termed "odd-even" c f . H i l l e r 18.25. 7 23ee Mathematical Note 3 (p. 127) 108 73 than the number of angles i n the f i g u r e produced. 34.16 Among the p o l y g o n a l numbers, t h e r e i s another s e t of numbers formed from m u l t i p l i c a t i o n by a c e r t a i n number s t a r t -i n g from u n i t y . From these numbers t h a t are a r e s u l t of mult-i p l i c a t i o n s t a r t i n g at u n i t y , namely by m u l t i p l y i n g by; 2, or by 3 or the numbers t h a t f o l l o w , a l l those form squares which have an i n t e r v a l of 1, a l l those form cubes t h a t have ;an i n t e r -v a l of 2, w h i l e those with an i n t e r v a l of 5 form cubes and squares at the same time; f o r they are e i t h e r cubes t h a t have s i d e s which are square numbers or squares t h a t have s i d e s 74 which are cubic numbers. T h i s w i l l demonstrate how, of the numbers t h a t are powers s t a r t i n g from 1, every other one i s a square, every t h i r d one i s a cube, and every s i x t h i s both a square and a cube at the same time. In the s e r i e s of numbers o b t a i n e d by m u l t i p l y i n g by 2, some go l i k e t h i s : 1, 8, 4, 8, 75 16, 32, 64, 128, 256. The f i r s t double i s 2; then comes 4 which i s a square; then 8 which i s a cube; then 16 which i s a square; then 32, and a f t e r t h a t 64 which i s at the same time a square and a cube. Then there i s 128, and a f t e r i t 256 which i s a square; and so on, t o i n f i n i t y . 7 3 S e e Mathematical Note 11 (p. 136) 74 See Mathematical Note 8 (p. 133) 7 5 I have read a ' p' 6' n ' i q ' Xp ' J;6 ' pxr i ' avgr' the s e r i e s of numbers c o n j e c t u r e d by Gelder, as t h i s i s the o n l y s e r i e s ,that seems to make sense f o l l o w i n g ev u-ev TOT-J 6i7rXaaioic- j . n 35. 5. 109 35.12 I n the s e r i e s of numbers ob t a i n e d by m u l t i p l y i n g by 3, i t w i l l a l s o be found t h a t every other one i s a square num-ber, and i n the s e r i e s of powers of 5 and i n those d e r i v e d from o t h e r numbers, the same w i l l be the case. And i n t h e seme way too among thes e powers, a l l those t h a t jump two terms w i l l be found to be cubes, and those t h a t jump f i v e terms w i l l a l s o be found t o be cubes and squares at the same time. 35.17 I n the case of the square numbers, they have the pec-u l i a r p r o p e r t y of having the f a c t o r 3, or of being e x a c t l y d i v i s i b l e by 3 a f t e r the s u b t r a c t i o n of 1; or a g a i n , they have the f a c t o r 4 or are e x a c t l y d i v i s i b l e by 4 upon the sub-76 t r a c t i o n of 1. Again, the even square t h a t has 3 as a f a c t -o r a f t e r the s u b t r a c t i o n of 1, i s a l s o e x a c t l y d i v i s i b l e by 4, as i n the case of the number 4; w h i l e the square t h a t i s d i v i s i b l e by 4 upon the s u b t r a c t i o n of 1, i s e x a c t l y d i v i s i b l e by 3 as, f o r i n s t a n c e , i s the number 9; or a square may be ex-a c t l y d i v i s i b l e by both 3 and 4 , as i s 36; or the square t h a t has n e i t h e r 3 nor 4 as a f a c t o r becomes d i v i s i b l e by these numbers a f t e r the s u b t r a c t i o n of 1, as i s the case w i t h 25. 76 See Mathematical Note 9 (p.134). I have taken Dupuis' c o n j e c t u r e < a p r i o v > t o f o l l o w nai T O V VLEV i n 35.20; f o r t h e r e are two types of "square numbers d i v i s i b l e by 3 a f t e r the s u b t r a c t i o n of 1", of t y p e s (A) and (D), but of t h e s e o n l y the even one, of form ( A ) , i s e x a c t l y d i v i s i b l e by 4. There i s a f u r t h e r d i f f i c u l t y ; f o r w h i l e . t h e r e are two types of "squares d i v i s i b l e by 4 a f t e r the s u b t r a c t i o n of 1", of forms (B) and (D), o n l y t h a t of form (B) i s e x a c t l y d i v i s i b l e by 3. Theon l a c k s a method t h a t a f f o r d s t h e g e n e r a l i t y o b t a i n e d by a l g e b r a . 110 36. 3 Ag a i n among the numbers, those " e q u a l l y - e q u a l " are squares, those "unequally-unequal" are heteromecic or oblong; or, t o put i t simply, those t h a t are the product of two f a c t -o r s are plane and those t h a t are the product of t h r e e f a c t o r s are s o l i d . 36. 6 Numbers are c a l l e d p l a n e , and t r i a n g u l a r , and square, s o l i d and other s i m i l a r names , not i n the proper sense but by reason of t h e i r s i m i l a r i t y t o the spaces t h a t they measure. Thus 4, s i n c e i t measures a square space, i s c a l l e d a "square" 77 number, and 6 i s c a l l e d "heteromecic" f o r the same reason. 36.IE Among the plane numbers, the square numbers are a l l s i m i l a r t o each other; and those heteromecic numbers are a l s o s i m i l a r whose s i d e s , i . e . , whose numerical f a c t o r s are i n p r o p o r t i o n . Take, f o r i n s t a n c e , the heteromecic number 6; i t s s i d e s are of l e n g t h 3 and of width 2. Again another plane number i s 24, w i t h s i d e s of l e n g t h 6 and of width 4. Now t h e r a t i o of the l e n g t h of one t o the l e n g t h of the other i s equal t o the r a t i o of the width of the one t o the width of the other; f o r the r a t i o of 6 t o 3 i s equal to the r a t i o of 4 to 2; wherefore, the plane numbers 6 and 24 are s i m i l a r . 36.20 Sometimes the same numbers may be r e p r e s e n t e d i n t h r e e d i f f e r e n t ways; f i r s t , as s i d e s when taken as l e n g t h s f o r the composition of ot h e r numbers and, secondly, as plane numbers, when they are the product of the m u l t i p l i c a t i o n of two other 77 i . e . , because i t measures a "heteromecic" space. I l l numbers; end t h i r d l y , when they are taken as s o l i d numbers, produced as a r e s u l t of the m u l t i p l i c a t i o n of t h r e e numbers. 37. 2 Among the s o l i d numbers again, a l l the cubi c numbers are s i m i l a r t o each other; of the r e s t , those r e c t a n g u l a r s o l i d s are s i m i l a r t h a t have t h e i r s i d e s i n p r o p o r t i o n , f o r the r a t i o s of the l e n g t h of one t o the l e n g t h of the other, the w i d t h of one to the width of the other, and the height of the one t o the height of the other, are equal. 37. 7 Gf the plane p o l y g o n a l numbers, the f i r s t i s the t r i -a n g u l a r number, even as the t r i a n g l e i s the f i r s t of the r e c t -i l i n e a r f i g u r e s . The method of f o r m a t i o n of the t r i a n g u l a r numbers has been t o l d a l r e a d y , namely by adding t o the f i r s t number the even numbers and odd numbers i n s u c c e s s i o n . 37.11 A l l the numbers i n s e r i e s which produce t r i a n g u l a r , 78 square and p o l y g o n a l numbers are c a l l e d gnomons, and f o r the t r i a n g u l a r numbers each t r i a n g l e has s i d e s of e x a c t l y as many u n i t s as has the s i n g l e gnomon l a s t added. Take f i r s t U n i t y , which i s con s i d e r e d as a t r i a n g u l a r number, not i n the proper sense, as we have a l r e a d y s a i d , but p o t e n t i a l l y . S i n c e i t i s , as i t were, the germ of a l l numbers i t holds w i t h i n i t s e l f i n a d d i t i o n the c a p a c i t y t o assume the t r i a n g u l a r form. At any r a t e , when i t takes the number 2, i t completes a t r i a n g l e w i t h s i d e s of as many u n i t s as has the gnomon of 2 j u s t added. Then the whole t r i a n g l e c o n s i s t s of as many u n i t s as the sum of the gnomons, f o r the sum of the gnomons 1 and 2 i s 3 so t h a t the 78 . See Mathematical Note 3 (p. 127) 112 t r i a n g l e w i l l c o n s i s t of 3 u n i t s and w i l l have each s i d e con-s i s t i n g of 2 u n i t s , i . e . , of as many u n i t s as the number of gnomons added. 38. 2 Nextj the t r i a n g l e 3 r e c e i v e s i n a d d i t i o n the gnomon 3, which exceeds the number 2 by 1 and the whole t r i a n g l e produces 6. T h i s t r i a n g l e too, w i l l have s i d e s of as many u n i t s as the number of gnomons added, f o r 6 i s the sum of 1, and 2 and 3. x x X X X X X X X 38. 8 Then the number 6 takes the number 4, and a t r i a n g l e of 10 i s formed w i t h each s i d e of 4 u n i t s , f o r the gnomon 4 has been added and the t o t a l i s now made up of 4 gnomons, i . e . , 1, 2, 3, and 4. 38.11 Next, t o . t h e number 10 i s added 5 and a t r i a n g l e of 15, c omprising the sum of 5 gnomons, i s produced w i t h each s i d e of 5 u n i t s ; and i n l i k e manner do the s u c c e s s i v e gnomons 79 produce the t r i a n g u l a r numbers. 38.16 Some numbers are c a l l e d c i r c u l a r , s p h e r i c a l or r e c u r r -ent. These are those which, i n being squared or cubed, t h a t i s , i n being m u l t i p l i e d i n two or t h r e e dimensions, r e t u r n t o the number from which they s t a r t e d . A ' c i r c l e a l s o d e s c r i b e s such a f i g u r e f o r i t r e t u r n s t o i t s s t a r t i n g - p o i n t and com-p r i s e s a s i n g l e l i n e s t a r t i n g and f i n i s h i n g at the same p o i n t . I have read o-ioiaxj^Mai o i e^rj-- yv-jDuovec; roue; TpiYtt>vouc; apiO-iouc; d-roTeAouqi f o r t h i s passage, as c o n j e c t u r e d by Dupuis (64.6) 113 39. 3 Among the s o l i d s the sphere has the same p r o p e r t y , f o r wbren a c i r c l e i s r o t a t e d about i t s diameter i t s r e t u r n from one p o s i t i o n t o the same p o s i t i o n d e s c r i b e s a sphere. And so we see t h a t those numbers which, by being m u l t i p l i e d , f i n i s h w i t h themselves are c a l l e d c i r c u l a r or s p h e r i c a l . Both 5 and 6 produce such numbers. F o r 5 times 5 i s 25, and 5 times 25 i s 125; and 6 times 6 i s 36, and 6 times 36 i s 216. 8 0 39.10 Now as to the f o r m a t i o n of the square numbers, as I have s a i d , they r e s u l t from the a d d i t i o n of the odd numbers, t h a t i s , those i n c r e a s i n g by 2 from u n i t y ; f o r 1 p l u s 3 i s 4, and 4 p l u s 5 i s 9, 9 p l u s 7 i s 16, and 16 p l u s 9 i s 25. x x x x x x x X X X X X X x x x x x x x x x x x x x x x x x x x x x x x x x x . x x x x x X X X X X . x x x x x 39.14 The pentagonal numbers are those which are the r e s u l t of the a d d i t i o n of the s e r i e s of numbers t h a t s t a r t from u n i t y and i n c r e a s e by 3; so the terms on the one hand are 1, 4, 7, 10, 13, 16, 19 while the pentagonal numbers themselves are 1, 5, 12, 22, 35, 51 and so on. They are sketched i n pentagonal form as f o l l o w s : 80 Theon says 5*5 (25) i s c i r c u l a r , 5*5*5 (125) i s spher-i c a l because they end i n 5, the number they s t a r t e d from. T h i s d e f i n i t i o n i s o b v i o u s l y based on n o t a t i o n , i . e . , the dec-imal n otation,and t h e r e f o r e d e p a r t s from the p r i n c i p l e s of number c l a s s i f i c a t i o n we have seen so f a r , depending as they do upon a g r a p h i c a l arrangement of u n i t s . I would suspect 114 1 x x X X X X 12 X X X X X X X X X X X X 22 35 x X X X X X x x x x x x x x x x x x x x x x X X X X X X x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 40. 1 Hexagonal numbers are those which are a r e s u l t of the a d d i t i o n of the s e r i e s of numbers t h a t i n c r e a s e by 4, s t a r t i n g from u n i t y . T h e i r gnomons are 1, 5, 9, 13, 17, 21, 25, from which are d e r i v e d these hexagonal numbers: 1, 6, 15, 28, 45, 66, 91. They are sketched i n t h i s way: 1 6 15 28 45 66 X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X x x x x x x x x x x x x X X X x x x x x x x x x x x x x x X X X X X X X X X X X X X X X X X X x x x x x x x x x x x x x x x X X X x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x X X X x x x x x x X X x x x x x X x x x x X X X X X X t h a t t h i s d e f i n i t i o n i s not Pythagorean at a l l , but a l a t e r a d d i t i o n . Note t h a t 1 would a l s o g i v e c i r c u l a r and s p h e r i c a l numbers, and 0 too i f permitted; i n s h o r t , a l l numbers ending i n 0, 1, 5 or 6 would. Theon*s f a i l u r e t o quote some such numbers as 15*15 (225 - c i r c u l a r ) , 15*15*15 (3375 - s p h e r i c a l ) or 11*11 (121 - c i r c u l a r ) , 11*11*11 (1331 - s p h e r i c a l ) seems to be s t r o n g evidence t h a t Theon does not take a •'mathematic-a l l y " c r i t i c a l approach t o h i s p r e s e n t a t i o n . 115 40. 6 The heptagonal numbers are those t h a t are composed from numbers t h a t i n c r e a s e i n f i v e s from u n i t y ; t h e i r gnomons are 1, 6, 11, 16, 21, 26, and the heptagonal numbers t h a t are formed from them are 1, 7, 18, 34, 55, 81. 40. 9 The o c t a g o n a l numbers are a l s o formed i n l i k e manner by the a d d i t i o n of numbers t h a t i n c r e a s e i n s i x e s from u n i t y ; t h e nonagonal numbers are composed from numbers t h a t i n c r e a s e i n sevens from u n i t y , while decagonal numbers are formed by the a d d i t i o n of numbers t h a t i n c r e a s e from u n i t y i n e i g h t s . Thus i n g e n e r a l f o r a l l p o l y g o n a l numbers, the i n c r e a s e o f the numbers i n the s e r i e s from which they are formed i s o b t a i n -ed by s u b t r a c t i n g two u n i t s from the number of angles t h a t g i v e s the p o l y g o n a l number i t s name. 41. 3 From two t r i a n g u l a r numbers a square number i s prod-uced. Thus 1 p l u s 3 equals 4, 3 p l u s 6 e'quals 9, 6 p l u s 10 equals 16, 10 p l u s 15 equals 25, 15 p l u s 21 equals 36, 21 p l u s 28 equals 49, 28 p l u s 36 equals 64, 36 p l u s 45 equals 81; and i n l i k e manner succeeding t r i a n g u l a r numbers taken i n order two at a time produce square numbers, even as the j o i n i n g t o -get h e r of the sketches of t h e i r t r i a n g l e s makes a square f i g u r e . 81 8 1 H i l l er ( p r a e f . v i . ) c r i t i c i s e s the diagrams of the ms. as " f i g u r i s n e g l e g e n t i s s i m e f a o t i s " . Here the diagram appears to i n d i c a t e t h a t a square number may be d i v i d e d i n t o two " e q u a l " t r i a n g u l a r numbers. T h i s i s c l e a r l y i n c o r r e c t ; see diagram i l l u s t r a t i n g 6 + 3 - 9 . I......_.,.] 116 41. 9 Moreover among the s o l i d numbers some have equal s i d e s and some have unequal s i d e s . In the l a t t e r case, some have a l l s i d e s unequal and some have two s i d e s equal and one 82 unequal. Again, i n the case of those having two s i d e s equal, some have the t h i r d s i d e l o n g e r , and some have the t h i r d s i d e s h o r t e r . 41.13 Now those having equal s i d e s , being equal i n t h r e e dimensions, are c a l l e d cubes; those having a l l s i d e s unequal, being unequal i n t h r e e dimensions are c a l l e d " b l o c k s " ; those t h a t have two s i d e s equal and the t h i r d l e s s than the other two, being " e q u a l l y - e q u a l - l e s s e r " are c a l l e d " t i l e s " ; and those t h a t have two s i d e s equal and the t h i r d s i d e g r e a t e r than the other two, being " e q u a l l y - e q u a l - g r e a t e r " are c a l l e d "rods". 83 x-4 ~8 8 32. Z lo Z A. 8 2 I have read HOU TTIV p,idv a v i c o v f o r T JTTOVQ, as c o n j -ectured by B u l l i a l d u s (see H i l l e r 41.11, n o t e ) . 117 42. 3 Pyramidal numbers are those t h a t measure e x a c t l y 84 pyramids and t r u n c a t e d pyramids. A t r u n c a t e d pyramid i s a pyramid w i t h i t s top cut o f f . Some have c a l l e d such a t r u n c a -t e d s o l i d a t r a p e z o i d a l s o l i d a f t e r the plane trapezium; a t r i a n g l e i s c a l l e d a trapezium.?, whenever i t s top i s cut o f f by a s t r a i g h t l i n e drawn p a r a l l e l t o t h e base. 42.-10 J u s t as some numbers are i n v e s t e d w i t h power to make t r i a n g l e s , squares, pentagons and the other f i g u r e s , so a l s o we f i n d the s i d e and d i a g o n a l r a t i o s b e i n g r e v e a l e d by num-bers i n accordance w i t h the g e n e r a t i v e p r i n c i p l e s , f o r from these r a t i o s do the f i g u r e s r e c e i v e t h e i r p r o p o r t i o n s . F o r j u s t as the u n i t , a c c o r d i n g t o the supreme g e n e r a t i v e p r i n c -i p l e , i s the s t a r t i n g - p o i n t of a l l f i g u r e s , so a l s o i n the The terms used by Theon t o d e f i n e h i s c l a s s i f i c a t i o n of s o l i d numbers are PCDVU'CHOC- ( l i t t l e a l t a r ) , nXivQiq ( p l i n t h , wedge), and 6OHIC; (beam) and these have been t r a n s l a t e d b l o c k , t i l e and rod r e s p e c t i v e l y . However h i s c r i t e r i a f o r the c l a s s -i f i c a t i o n are unconvincing. I t would appear at any r a t e t h a t one s h o u l d p o s t u l a t e i n a d d i t i o n t h a t the t h r e e dimensions should be prime numbers and the f a c t o r s remain i n v a r i a b l e . F o r c o n s i d e r : 60 = 3-4*5 ( b l o c k ) » 15-2 2 (rod) 72 = 8*3 2 (rod) = 2*6 2 ( t i l e ) 147 » 3-7 2 ( t i l e ) i n v a r i a b l e ; prime f a c t o r s . Yet the diagrams g i v e n by Theon r e p r e s e n t i n g a block, a t i l e and a rod d e p i c t the composite numbers 64, 32 and 40 r e s p e c t -i v e l y and these have a l t e r n a t i v e forms: 64 ( r o d , 1 6 « 2 2 ) , 32 ( r o d 8 » 2 2 ) and 40 ( b l o c k , 2*4-5). See Mathematical Note 12 (p. 140) 118 u n i t w i l l be found the r a t i o of d i a g o n a l t o s i d e . For exam-p l e , l e t two u n i t s be taken, of which we s e t one to be a d i a -gonal and the other a s i d e s i n c e the u n i t , as the beginning of a l l t h i n g s , must p o t e n t i a l l y be both s i d e and d i a g o n a l . Now a d i a g o n a l i s added to the s i d e and t o the d i a g o n a l two s i d e s ; f o r as o f t e n as the square on the d i a g o n a l i s taken once, so o f t e n i s the square on the s i d e taken t w i c e . Then the diago-n a l has become the g r e a t e r and the s i d e the l e s s e r . Now i n the case of the f i r s t s i d e and d i a g o n a l , the square on the un-i t d i a g o n a l w i l l be l e s s by one u n i t than t w i c e the square on the u n i t s i d e ; f o r u n i t s are equal, and 1 i s l e s s by one u n i t than t w i c e 1. Indeed l e t us add t o the s i d e a d i a g o n a l , i . e . , t o the one u n i t l e t us add one u n i t ; t h e r e f o r e the (seconcj) s i d e w i l l be two u n i t s . To the d i a g o n a l l e t us now add two s i d e s , i . e . , t o the one u n i t l e t us add two u n i t s ; the (second) d i a -gonal w i l l t h e r e f o r e be t h r e e u n i t s . Now the square on the s i d e of two u n i t s w i l l be 4, w h ile the square on the d i a g o n a l of t h r e e u n i t s w i l l be 9; and 9 i s g r e a t e r by one u n i t than twice the square on the s i d e 2. 8^ 44. 3 I n the same way, l e t us add t o the s i d e 2 the d i a g o n a l 3; the ( t h i r d ) s i d e w i l l be 5. To the d i a g o n a l 3 l e t us add two s i d e s , i . e . , twice 2; the t h i r d d i a g o n a l w i l l be 7. Now the square from the s i d e 5 w i l l be 25, w h i l e t h a t from the d i a -gonal 7 w i l l be 49; and 49 i s l e s s by one u n i t than twice 25. A g a i n i f you add t o the s i d e 5 the d i a g o n a l 7, the r e s u l t w i l l be 12; and i f you add twice the s i d e 5 t o the d i a g o n a l 7, the 8 % e e Mathematical Note 13 (p. 142) 119 r e s u l t w i l l be 17. And the square of 17 i s g r e a t e r by one u n i t than twice the square of 12. And i n l i k e manner f o r sub-sequent c a l c u l a t i o n s t h e r e w i l l be the same a l t e r n a t i n g r e l a t -i o n s h i p ; the square on the d i a g o n a l f i r s t b eing g r e a t e r by one u n i t , and then l e s s by one u n i t than twice the square on the s i d e ; and such s i d e s and d i a g o n a l s are a l i k e r a t i o n a l . 4 Z5- 86 44.18 The squares on the d i a g o n a l s compared w i t h double the squares on the s i d e s , are a l t e r n a t e l y g r e a t e r by one and then l e s s by one. Thus the square on each d i a g o n a l w i l l become double the square on each s i d e , as the a d d i t i o n of the v e r y same u n i t a l t e r n a t e l y r e s t o r e s the e q u a l i t y , p r e v e n t i n g the double square on the s i d e being i n excess or d e f i c i e n t i n each case. F o r the amount l a c k i n g i n the p r e c e d i n g d i a g o n a l i s found i n excess i n the subsequent one. 87 45. 9 Moreover some numbers are c a l l e d p e r f e c t , some over-p e r f e c t , and some d e f e c t i v e . P e r f e c t numbers are those t h a t are equal to t h e i r own p a r t s , as i s the number 6; f o r i t s h a l f p a r t i s 3, i t s t h i r d p a r t i s 2 and i t s s i x t h i s 1, and these added t o g e t h e r g i v e 6. 45.13 The p e r f e c t numbers are produced i n the f o l l o w i n g way. 86 7r(80) and 6(4) g i v e n f o r the f i r s t d i a g o n a l and s i d e i n the diagram are probably copying e r r o r s f o r a ( l ) . 87 See Mathematical Note 14 (p.144) 120 I f we take the double numbers s t a r t i n g from u n i t y and i f we add them u n t i l a prime and incomposite number r e s u l t s , and i f we then m u l t i p l y the sum o b t a i n e d by the l a s t term which was added, the r e s u l t of the m u l t i p l i c a t i o n w i l l be a p e r f e c t number. F o r example, l e t the double numbers be 1, 2, 4, 8, 16; then l e t us add 1 and 2, which g i v e 3; next we must mult-i p l y i t by the l a s t of the terms added, t h a t i s , by 2 and the 88 r e s u l t i s 6, which i s the f i r s t p e r f e c t number. 45.22 Again, i f we add th r e e of the s e r i e s of doubles, 1 and 2 and 4, the r e s u l t w i l l be 7. Then must we m u l t i p l y i t by the l a s t of the numbers added, t h a t i s , we m u l t i p l y 7 by 4; the r e s u l t w i l l be 28, which i s the second p e r f e c t number. I t i s composed of i t s h a l f 14, i t s f o u r t h p a r t 7, i t s seventh p a r t 4, i t s f o u r t e e n t h p a r t 2, and i t s twenty-eighth p a r t 1. 46. 4 O v e r - p e r f e c t numbers are those whose f r a c t i o n a l p a r t s added t o g e t h e r are g r e a t e r than t h e i r wholes as, f o r example, 12: f o r t h e h a l f of t h i s number i s 6, the t h i r d i s 4, the f o u r t h i s 3, the s i x t h i s 2, and t h e t w e l f t h i s 1, a l l of which added t o g e t h e r produce 16, which i s l a r g e r than t h e o r i -g i n a l number 12. 46. 9 D e f e c t i v e numbers are those whose f r a c t i o n a l p a r t s , when added, produce a number l e s s than the number o r i g i n a l l y given,, such as 8; f o r the h a l f of t h i s number i s 4, the f o u r t h 2, and the e i g h t h i s 1. The same p r o p e r t y a p p l i e s t o the number 10, a number t h a t the Pythagoreans c a l l p e r f e c t f o r a 8 8 S e e Mathematical Note 14 (p. 144) 121 d i f f e r e n t reason, and t h i s we s h a l l d i s c u s s i n the proper p l a c e . .14 The number 3 i s a l s o c a l l e d p e r f e c t , because i t i s the f i r s t number which has a beginning, a middle and an end. I t i s , moreover, both a l i n e and a s u r f a c e f o r i t i s an e q u i l a t e r a l t r i a n g l e i n which each s i d e i s two u n i t s , and i t i s t h e f i r s t bond and power of the s o l i d , f o r i t i s i n t h r e e dimensions t h a t the concept of the s o l i d l i e s . Theon omits t o mention the " f r i e n d l y " or "amicable" numbers i n h i s treatment of p e r f e c t numbers. Two numbers are f r i e n d l y when each i s the sum of the a l i q u o t p a r t s of the other, e.g. 284 and 220 ( f o r 284 • 1 + 2 + 4 + 5 + 1 0 + 1 1 +• 20 + 22 + 44 +55 +110, while 220 = 1 + 2 + 4 + 7 1 + 1 4 2 ) . Iamblichus ( i n Nicpm., p.33, 20-23) i n f e r s from a s t o r y he recounts about Pythagoras t h a t the l a t t e r knew of these numbers and, consequ-e n t l y , of p e r f e c t numbers too. 90 For a treatment of the m y s t i c a l p r o p e r t i e s a s s i g n e d by the Pythagoreans t o p a r t i c u l a r numbers, see J . Gow, A Short H i s t o r y o f Greek Mathematics (Cambridge 1884) p.69. CHAPTER FOUR MATHEMATICAL NOTES Note 1 - Doubling the cube ( c f . H i l l e r 2. 5) Theon's s t o r y about the D e l i a n s c o n s u l t i n g the o r a c l e at D e l p h i and being t o l d to c o n s t r u c t an a l t a r of double s i z e i s p robably a r e f e r e n c e t o the i n t r i c a t e problem of d o u b l i n g t h e cube which taxed the i n g e n u i t y of the Greek mathematic-ia n s of the s i x t h century B.C. Of course, i t must be presumed t h a t the a l t a r c a l l e d f o r was to be cu b i c i n shape w i t h l e n g t h , width and height a l l equal; far t o c o n s t r u c t a s i m i l a r p a r a l l e l e p i p e d of double s i z e , p r e s e r v i n g the r a t i o s between corresponding s i d e s , would be an imp o s s i b l e t a s k . E u t o c i u s g i v e s a l i s t of the authors of s o l u t i o n s t o t h i s problem of do u b l i n g the cube as f o l l o w s : P l a t o , Heron, P h i l o n , A p o l l o n i u s , D i o c l e s , Pappus, Sporus, Menaechmus (two s o l u t i o n s ) , A r c h y t a s , Eratosthenes and Nicomedes. He i d e n t i f i e s H i p p o c r a t e s of Chios 1 as the f i r s t t o see t h a t , i f two mean p r o p o r t i o n a l s c o u l d be c o n s t r u c t e d i n continued- p r o p o r t i o n between two s t r a i g h t l i n e s , one of which was double the other, the s m a l l e r of the p r o p o r t i o n a l s found would produce a cube double the volume of t h a t cube d e r i v e d ^Eutocius, Comm. on Archimedes' Sphere and C y l i n d e r i i . Archim. ed. H e i b e r g i i i , 88, 4 f f . 123 from the s m a l l e r of the two s t r a i g h t l i n e s . T h i s may be shown by simple a l g e b r a as f o l l o w s : L e t the two s t r a i g h t l i n e s have l e n g t h s a and 2a, so t h a t we need t o f i n d a l e n g t h x, such t h a t x 3 = 2 a 3 . Suppose two means, x and y, are i n s e r t e d between a and 2a, so as t o form the continued p r e p o r t i o n a, x, y, 2a. Then _5_ = 2L = X x y 2a 2 Thus JL — whence v = 2L x y , a And — y_ whence y — x ~~ 2a, x Hence x 2 2 a 2 a x Th e r e f o r e x 3 =-r 2 a 3 Thus the f i r s t of'"the two mean p r o p o r t i o n a l s found (x) w i l l produce a cube of volume twice t h a t of a cube of the s m a l l e r (a) of the i n i t i a l two s t r a i g h t l i n e s . E u t o c i u s a s c r i b e s t o P l a t o a b e a u t i f u l l y simple p r a c t i c a l s o l u t i o n f o r f i n d i n g the l e n g t h s of two mean pro-2 p o r t i o n a l s between any two s t r a i g h t l i n e s , although i t i s / v i r t u a l l y c e r t a i n t h a t t h i s s o l u t i o n i s wrongly a s c r i b e d t o P l a t o . Thomas p o i n t s out t h a t E u t o c i u s alone mentions i t , and t h a t i f i t had been known to Eratosthenes he c o u l d h a r d l y have f a i l e d t o quote i t along w i t h the s o l u t i o n s of Arc h y t a s , Menaechmus and Eudoxus. Furthermore, P l a t o a c c o r d i n g t o 2 E u t o c i u s , i b i d . , 56, 13. I v o r Thomas, Greek Mathematical Works, Loeb, 1939, i , 262(note). 124 P l u t a r c h t o l d the D e l i a n s t h a t Eudoxus or H e l i c o n of C y z i c u s would s o l v e the problem f o r them; he d i d not a p p a r e n t l y p r o -4 pose t o t a c k l e i t h i m s e l f . And P l u t a r c h twice says t h a t P l a t o o b j e c t e d t o mechanical s o l u t i o n s on the grounds t h a t they 5 destroyed the good of geometry, a statement which i s c o n s i s t -ent w i t h h i s known a t t i t u d e towards mathematics. The s o l -u t i o n i s here e x p l a i n e d , and g i v e s an i n s i g h t i n t o the f a c i l . -i t y w i t h which the Greeks, although l a c k i n g the a n a l y t i c a l ; methods of a l g e b r a , brought geometry i n t o p l a y . A r e c t a n g u l a r framework POJiS i s c o n s t r u c t e d w i t h j grooves which permit the l e n g t h TU t o move i n such a way as to be always p a r a l l e l w i t h QJR, w h i l e an a d d i t i o n a l p i e c e comprising two s t r a i g h t arms f i x e d t ogether at r i g h t - a n g l e s may be clamped on the framework i n any p o s i t i o n . F i g . l - Apparatus f o r f i n d i n g two mean p r o p o r t i o n a l s Suppose a and b are the two l e n g t h s between which the two means are t o be i n s e r t e d i n continued p r o p o r t i o n ; 4 P l u t a r c h , de genio S o c r a t i s , c.7, 579 C,D. 5 P l u t a r c h , Quaest. Conviv.. 8. 2. 1., 718 E,F; V i t a M a r c e l l i , c.14.5. 125 then take BC of l e n g t h a and BA of l e n g t h b, and adjust the p o s i t i o n of the arm-piece i n such a way t h a t A l i e s on QR, C l i e s on TU, and AB produced and CB produced pass through U and R r e s p e c t i v e l j r (See F i g . l ) . There i s only one unique p o s i t i o n i n which t h i s can be e f f e c t e d . Then, i f BC = a, AB = b, BU - x, BR - y the two r e q u i r e d mean p r o p o r t i o n a l s are of le n g t h s x and y. F o r i n the r i g h t - a n g l e d t r i a n g l e s CBU, UBR, RBA l_C\JB = [_ URB = J_ RAB T h e r e f o r e , the t h r e e t r i a n g l e s are equiangular and t h e i r s i d e s are p r o p o r t i o n a l . And a _ x. _y x y b So t h a t , i f b = 2a, — X ' x y 2a and, by proof on p. 123 x 3 = 2 a 3 Thus, i f BC i s marked o f f equal t o a and BA equal t o 2a, BU w i l l g i v e the l e n g t h of the s i d e of the double cube. Note 2 - One i s without p a r t s and i n d i v i s i b l e ( c f . H i l l e r 18.15) Theon's r e a s o n i n g of course appears f a l l a c i o u s . He argues t h a t , i f the One i s d i v i d e d i n t o many p a r t s and i f each of these i s taken away one at a time, we f i n a l l y have one l e f t ; i f then, t h a t one i s d i v i d e d i n t o many p a r t s and each of these i s a g a i n s u b t r a c t e d , then only one w i l l remain again; and, t h i s r e s u l t w i l l always be obtained however many times the d i v i s i o n and s u b t r a c t i o n take p l a c e . One i s always 126 l e f t cucrTe ap,e'p t tfTov nai aBi a t p e r o v TO ev cbq ev ( i n so f a r as i t s oneness i s concerned, One has no p a r t s and i s i n d i v i s i i b l e ) . The Greeks of the s i x t h century B.C. were g r e a t l y p u z z l e d by f r a c t i o n s , and P l a t o was c o n t i n u a l l y f a s c i n a t e d by the c h a r a c t e r of One. The Parmenides contains phrases c o r r -esponding to'what we f i n d i n E u c l i d ' s p r e l i m i n a r y matter. Thus P l a t o speaks of something which i s a " p a r t " but not "parts'* of t h e One. Heath observes t h a t t h i s reminds one of E u c l i d ' s d i s t i n c t i o n between a f r a c t i o n which i s a " p a r t " , i . e . , an a l i q u o t p a r t and one which i s " p a r t s " , i . e . , some number more than one of such p a r t s , e.g., 3/7. Indeed, P l a t o p o i n t s out t h a t Zeno of E l e a wrote a book whose o b j e c t was t o defend the system of Parmenides by 8 a t t a c k i n g the common conception of t h i n g s . Parmenides h e l d t h a t o n l y the One e x i s t s , whereupon i t had been p o i n t e d out t h a t many c o n t r a d i c t i o n s and a b s u r d i t i e s would f o l l o w i f t h i s were admitted. Zeno r e p l i e d t h a t , i f the popular view t h a t the Many e x i s t be accepted, s t i l l more absurd r e s u l t s would f o l l o w and he a c c o r d i n g l y advanced the paradoxes f o r which he i s w e l l known. Heath b e l i e v e s t h a t t h e r e i s no j u s t i f i c a t i o n f o r Tannery's c o n t e n t i o n t h a t the arguments of Zeno r e f u t i n g the 6 c f . P l a t o , Parmenides, 153C 7Heath, H.G.M., i , 294. Q c f . P l a t o , Parmenides, 128C,E. 127 d i v i s i b i l i t y of magnitudes and times were e s p e c i a l l y d i r e c t e d a g a i n s t the Pythagorean view t h a t bodies, s u r f a c e s and l i n e s 9 are made up of mathematical p o i n t s . The Pythagorean d e f i n -i t i o n of a p o i n t was "a u n i t having p o s i t i o n " (u-ovdc; ©eciv e'xouca) but a c c o r d i n g t o A r i s t o t l e the Pythagoreans m a i n t a i n -ed t h a t the u n i t s and numbers do have magnitude. 1 0 Theon i s here, I b e l i e v e , r a t h e r p o i n t i n g out the uniqueness of the number One. A r i s t o t l e observes t h a t the One i s reasonably regarded as not being i t s e l f a number, be-cause a measure i s not the t h i n g s measured, but the measure o r the One i s the be g i n n i n g or " p r i n c i p l e " (apxri) of number. 1 1 T h i s d o c t r i n e may be Pythagorean i n o r i g i n ; Nicomachus has i l ^ " 2 and Theon h i m s e l f re-echoes i t i n h i s treatment of the p-Ovac-. Note 5 - The term yvajp-jov I t i s c l e a r from A r i s t o t l e ' s a l l u s i o n s t o "gnomons" p l a c e d around 1, which "now produce d i f f e r e n t f i g u r e s every time" (oblong f i g u r e s each d i s s i m i l a r from the preceding one), "now p r e s e r v e one and the same f i g u r e " (as i s the case w i t h 9 Heath, H.G.M., i , 283. Heath a l s o draws a t t e n t i o n t o a most comprehensive s e r i e s of papers by F l o r i a n C a j o r i , The H i s t o r y of Zeno's arguments on Motion, American Mathemat-i c a l Monthly, 1915; w h i l e most h e l p f u l of the vast l i t e r a t u r e on Zeno's paradoxes may be recommended Y i f.D. Ross, A r i s t o t l e ' s  P h y s i c s , pp. 655 -666, H.P.D. Lee, Zeno of E l e a , and Heath, H.G.M. , i , 271 - 283. 1 0 A r i s t , Metaph., M 6, 1088 b 19, 32. 1 : L i b i d . , N 1, 1088 a 6. icomachus, I n t r o d . arithm.. i i , 6.3, 7.3. 128 t r i a n g u l a r and square numbers), t h a t these gnomons are the s u c c e s s i v e terms which are added t o produce the patt e r n e d numbers. Heath gives some i n t e r e s t i n g h i s t o r i c a l i n f o r m a t i o n 14 concerning the term YVCOU-CUV. a) I t was o r i g i n a l l y an as t r o n o m i c a l instrument f o r measuring time, and c o n s i s t e d of an u p r i g h t s t i c k which c a s t shadows on a plane or h e m i s p h e r i c a l s u r f a c e . F o l l o w i n g t h i s use of the word, gnomon becomes "marker" or " p o i n t e r " -- a means of marking o f f or "knowing" something, and we f i n d Oenopides d e f i n i n g a s t r a i g h t l i n e drawn from an e x t e r n a l p o i n t , p e r p e n d i c u l a r to a s t r a i g h t l i n e , as drawn Kara yvcijiiova (gnomon-wise). b) Next the term was used t o denote an instrument used f o r drawing r i g h t - a n g l e s , as i s shown i n F i g . 2 F i g 2 - Gnomon f o r drawing r i g h t - a n g l e s Subsequently, gnomon was used t o denote the f i g u r e which remained of a square a f t e r a s m a l l e r square had been A r i s t . Phys., i i i , 4, 203a 13-15. 14 Heath, H.G.M., i , 78. 189 cut out of i t — or the f i g u r e which, as A r i s t o t l e says, "when added t o a square, pre s e r v e s the shape and makes up a l a r g e r square". .Probably, as Boeckh says, t h e connection between the gnomon and the square t o which i t i s added was regarded as s y m b o l i c a l of union and agreement, and P h i l o l a u s used the i d e a t o e x p l a i n the knowledge of t h i n g s , making the "knowing" embrace the "known" as the gnomon does the square. ( I I Def.2) t o cover the figur.e which serves the same compl-ementary f u n c t i o n t o a p a r a l l e l o g r a m , as the f i g u r e i n c) 15 serves f o r the square. F i g 5 - Gnomon of p a r a l l e l o g r a m e ) L a t e r s t i l l , Heron d e f i n e s the "gnomon" i n g e n e r a l as t h a t which, when added t o a n y t h i n g , number or f i g u r e , makes the whole s i m i l a r t o t h a t t o which i t i s added. Note 4 - F i g u r a t e (or f i g u r e d ) numbers i s admirably i l l u s t r a t e d by the manner i n which the Greeks a p p l i e d i t to the study of the composition of numbers and t h e i r p r o p e r t i e s . The theory of f i g u r e d numbers seems to go d) I n E u c l i d the meaning i s s t i l l f u r t h e r expanded The p a r t which geometry played i n Greek mathematics 15 c f . Scholium 1 1 , Bk I I i n E u c l i d , ed. Heib > v, 225 16 Heron, Def. 58 (Heron, i v , Heib., p.225) 130 back t o Pythagoras h i m s e l f . Nicomachus enunciates the p r i n c i p l e s upon which the Greeks s t u d i e d the composition o f numbers a s f o l l o w s : % " E C T t v o u y ai^y,£Tov dpxn 6 laCTqpaToc-, ^ou 6tacj'Trj-aa^ 6e, TO S ' C X S T O Kai dpxn YPau-p,?]--,, o u YP,au.priN6e' nai YPau-u-ri dp'XT) e-Ticpave("a-;, O U H e-ricpaveia^be^ n a t apxr) T O U Sixrj 6«.a-c r T a T o U j o'u^&ixn 6 i a a T a T o v . v n a i e iK O T O-q^ri e7ttcpayeia dpxT} P-ev aiuyxtToc-, o u coou-a: be, nai J\ apTri dpxfl V&Y T 0 ^ T p t x ^ i 6 i a o"TaTou, o u T p i x * ] S i a C T a T o v . OUT-JDC- 6T) n a i e v roxq dpi©"j,oTc- u-ev p-ovaq d p ^ n 7 r a v T o q sdpi© p,ou ecp'ev 6 i a -oxr\}xa. H a T q p,ova6a 7tpof3i (3a£op,evou, q be YPap-P-* HOC- dpt©p,oc-d p ^ n erct7re '6ou dfnQu-ou ecp'eTepov SiacrTT-yta e7ri7te'6coc; - r A a T u v -o p e v o u , 6 ^6e e-riVrESoc; cipiepoc- dpxt] orepeou dpi©iaou kn\ T p t T o v 6iaaTrip,a npoc, xa k\\ d p x f l ? (3a©o<-; T I TrpoaHTcpp-e'vou-o i ' o v na©'u - r o 6 i a i p e a i v Ypap/pinoi p,e'y e i o ' i v dpi©y.oi d7rAa)c-a7ravTec- o i ^ d - r o SuaSoc^dpxop,evoi n a i H a T a ^xovaSoc; -TQOO'-©eaiv e r c i e v n a i TO auTO -TpoxcupouvTe <- 6iaCTr|p,a, e-Ti'-reboi 6e o i ano Tpia6oc- dpxopevqt dpxiHODTCXTT)C, pi£r]c; v t a i N 6 i a TCDV e^fj-" a u v e x y v dpi@iicDy"7TpqiovTec- , ^ A a p p a v o v T e c ; n q i xr\v e7tcp-vuyuay H a T a TTJV ^auTi-jv r a ^ t v -rpa-TiOTOi Yap TPI 'YCDVOI 5 e i T a y.eir'auTou'- TeT^aY^voiy e i T a } i e T ' a u T o u $ 7revTaYcuvoi, e t T a em T O U T O I C; er;aYcuvot n a i e7tTaYcuvoi n a i en' ane i p o v . 1 7 P o i n t i s t h e r e f o r e the p r i n c i p l e o f dimension, but i s not dimension, w h i l e i t i s a l s o the p r i n c i p l e of l i n e , but i s not l i n e ; and l i n e i s the p r i n c i p l e o f s u r f a c e , but i s not s u r f a c e , and i s the p r i n c i p l e o f the two-dimensional, but i s not two-dimensional. N a t u r a l l y a l s o s u r f a c e i s the p r i n c i p l e o f body, but i s not body, while i t i s the p r i n c -i p l e o f the t h r e e - d i m e n s i o n a l , but i s not three-dimension-a l . S i m i l a r l y among numbers the u n i t i s the p r i n c i p l e o f every number set but by u n i t s i n one dimension, while l i n e a r number i s the p r i n c i p l e o f plane number broadened but i n another dimension i n the manner of a s u r f a c e , and plane number i s the p r i n c i p l e of s o l i d number, which ac-q u i r e s a c e r t a i n depth i n a t h i r d dimension (at r i g h t angles) t o the dimensions o f the s u r f a c e . For example, by s u b d i v i s i o n l i n e a r numbers are a l l numbers without except-i o n beginning from two and proceeding by the a d d i t i o n o f a u n i t i n one and the same dimension, w h i l e plane numbers begin from t h r e e a s t h e i r fundamental r o o t and advance through an o r d e r l y s e r i e s of numbers, t a k i n g t h e i r d e s i g -n a t i o n a c c o r d i n g to t h e i r o rder. For f i r s t come t r i a n g l e s , then a f t e r them are squares, then a f t e r these are penta-gons, then succeeding these are hexagons and heptagons and s o on t o i n f i n i t y . cf.Nicomachus, I n t r o . Arithm., i i , 7 1-3, (ed. Hoche, 86.9 - 87.6) 131 T h i s attempt t o present a r a t i o n a l e x p l a n a t i o n of the t h r e e dimensions i s i n every r e s p e c t comparable t o the modern concept. What i s of more i n t e r e s t t o us i s the manner i n which the p o i n t becomes, a s s o c i a t e d w i t h the u n i t (uovac;). Most of the Greeks' d i s c o v e r i e s i n the realm of apt6p,T)TiHT). are based upon the f i g u r e s which these p o i n t s can be made t o r e p r e s e n t . We may remark upon the importance of the graph-i c a l approach as an a i d i n most branches of mathematics even today. Thus, a p o i n t or dot i s used- t o r e p r e s e n t 1; two dots are used t o re p r e s e n t the number 2, which a l s o r e p r e s e n t s a s t r a i g h t l i n e ; three dots represent the number 3 and form the f i r s t plane number (a t r i a n g l e ) ; f o u r dots r e p r e s e n t the number 4 and d e f i n e the f i r s t s o l i d number (a t e t r a h e d r o n ) . Theon f o l l o w s Nicomachus of Gerasa i n h i s p r e s e n t a t i o n of the composition and p r o p e r t i e s of p o l y g o n a l numbers. Note 5 - Formation of square numbers ( c f . H i l l e r 32.10) I f the odd numbers are taken i n s e r i e s and added, square numbers r e s u l t , as i s shown by The'on's diagrams.!A g e n e r a l proof t h a t the sum of n of t h e s e odd numbers w i l l be n 2 may be demonstrated as f o l l o w s : The nth odd number may be denoted ( 2 n - l ) , so t h a t i f S denotes the sum of these n terms, we have S = 1 - f 3 4- 5 4 + (2n-3)+(2n - l ) a l s o S = (2n - l H(2n-,3 ) - f(2n-5)+ + 3 -f- 1 132" whence 2S » 2n + 2n + 2n+ -2n + 2n = 2n • n T h e r e f o r e S = n • n n 2 Note 6 - Formation of heteromecic numbers ( c f . H i l l e r 27.7) Theon p o i n t s out t h a t the sums of the even numbers taken i n order produce heteromecic numbers, i . e . , numbers having one s i d e l a r g e r than the o t h e r by u n i t y . A l g e b r a i c a l l y , we may denote the nth even number by 2n, and denoting the sum of the n even numbers by S, we have S - 2 + 4 4- 6 + + (2n-2) + 2n a l s o S - 2n + ( 2n-2) + (2n-4)+ + 4 + 2 Thus 2S =(2n+-2)+(2n-|-2) + (2n-i-2)-f- -j- ( 2n-h2) + ( 2n+2) and S = (n-HU-1- ( n + l J - f (n+l)-f- (n+1) +- (n+1) - the sum of n terms equal to (n+-l) - n(h-HL) and t h i s i s the g e n e r a l form of the heteromecic number and, as p o i n t e d out by Theon, i t may be obtained e i t h e r by the a d d i t i o n of the even numbers i n s u c c e s s i o n or by the m u l t i p -l i c a t i o n of two adjacent numbers. Note .7 - Means between heteromecic and square numbers ( c f . H i l l e r 28.20) I f we c o n s i d e r two c o n s e c u t i v e heteromecic numbers n ( n - l ) and n ( n + l ) , i . e . , (n 2-n) and (n2-)-n), i t w i l l be obv-ious t h a t n 2 w i l l be t h e i r a r i t h m e t i c mean, f o r (n2—n), n 2 133 and (n 2-fn) form an a r i t h m e t i c p r o g r e s s i o n . Note t h a t the geometric mean between these two numb-er s w i l l be l e s s than t h e i r a r i t h m e t i c mean. Fo r , i f x denotes t h e i r geometric mean, we have n(n-1) _ x x -" n(n+lj x 2 - n 2 ( n 2 - l ) x - n / n S - T Thus x < n 2 , f o r Jn2-1 < n. Note 8 - Powers which are squares and cubes ( c f . H i l l e r 34.16) While Theon c o n s i d e r s t h e s e numbers as being obtained from the number 1, by m u l t i p l y i n g by 2 and by 3 n a t TGQV e ^ r j we n a t u r a l l y r e c o g n i s e these numbers as powers of 2, 3 etc.. and may c o n s i d e r the g e n e r a l case as f o l l o w s : o rz 4 5 A 7 fi 9 Powers of a 1 a or a° a a a a a a Squares (S) S S S S S Cubes (C) C C C C Squares/Cubes SC SC I t i s c l e a r t h a t those numbers w i l l be squares and cubes which are powers of a , i . e . , a , a , a , a ... or, i n g e n e r a l , those of the form a which i s e q u i v a l e n t t o e i t h e r ( a 2 n ) denoting cubes having squares as s i d e s or (a ) denoting squares having cubes as s i d e s . 134 Note 9 - D i v i s i b i l i t y of Squares ( o f . H i l l e r 35.17 f f . ) Theon contents h i m s e l f w i t h simple i l l u s t r a t i o n s of 1 h i s p r o p o s i t i o n . A proof may be d e r i v e d simply by a l g e b r a . Any n a t u r a l number may be w r i t t e n i n the form 6n, 6£+l, 6n+2 6n+3, 6n+4, 6n+5. But 6n+5 • 6n+6-l - 6(n-fl) - 1, i . e . , of form (6n -1) and 6n+4 = 6n+6-2 - 6(n+l) - 2. i . e . , of form (6n -2) Hence, the n a t u r a l numbers may be r e p r e s e n t e d by 6n, fin t 1, 6n ± 2, 6 n 3 and t h e i r squares by 36n 2 3 6 n 2 ± : 12n •+ 1 36n 2 ± 24n -+• 4 36n 2 ± 36ri -V- 9 (C) (D) (A) (B) and the f a c t o r s of t h e s e types may be t a b u l a t e d as f o l l o w s : Square D i v i s i b i l i t y A D i v i s i b l e by 4, but not by 3; but the s u b t r a c t i o n of 1 l e a v e s a remainder ( 3 6 n 2 j : 24n +• 3) d i v i s i b l e by 3. B D i v i s i b l e by 3 but not by 4; but s u b t r a c t i o n of 1 l e a v e s a remainder ( 3 6 n 2 — 36n+8) d i v i s i b l e by 4. G D i v i s i b l e by 3 and 4; but:. s u b t r a c t i o n of 1 l e a v e s a remainder ( 3 6 n 2 — 1) d i v i s i b l e n e i t h e r by 3 nor 4. D D i v i s i b l e by n e i t h e r 3 nor 4; but s u b t r a c t i o n of 1 l e a v e s a remainder ( 3 6 n 2 ± l 2 n ) d i v i s i b l e by 3 and 4. I t has been noted by Thomas t h a t t h i s may be expressed i n modern mathematical terminology as " a l l square numbers are congruent t o 0 o r 1, modulus 3 or congruent t o 0 or 1, mod-u l u s 4". 1 8 I v o r Thomas, Greek Mathematical Works, (Loeb, 1939) i , p.105. He observes that t h e above i s the f i r s t appearance of any work on congruence which i s fundamental i n the modern theory of numbers, 135 Note 10 - T r i a n g u l a r numbers ( c f . H i l l e r 32.22, 37.11) I t was probably Pythagoras who d i s c o v e r e d t h a t the sum of any number of s u c c e s s i v e terms of the s e r i e s of n a t -u r a l numbers forms a t r i a n g u l a r number. T h i s f o l l o w s r e a d i -l y from the Pythagorean r e p r e s e n t a t i o n of numbers by means of dots which form e q u i l a t e r a l t r i a n g l e s . Theon sketches these t r i a n g u l a r numbers up to 36 but, t o c o n s i d e r the gen-e r a l case, the nth number may be denoted n and then the sum of these n numbers (S) may be found thus: S = 1 4- 2 + 3 + + ( n T l ) - f - n and S = n 4- (n-1) 4" (n-2) 4- . . . +2 + 1 then 2S = (n+1) 4- (n+1) 4- (n+1) 4 4~(n+l) 4 (n+1) = n terms of (n+-l) = n ( n f l ) whence S = n ^ n f l J 2 From the above r e s u l t i t may be observed t h a t any t r i a n g u l a r number i s equal t o one-half of the corresponding heteromecic number. Theon, i n a r a t h e r l a b o r i o u s , and c i r c u i t o u s e x p o s i -t i o n of these numbers, has two p o i n t s t o emphasise, namely t h a t the number of u n i t s comprising the s i d e of the e q u i l a t -e r a l t r i a n g l e i s equal to a) the number of u n i t s i n the l a s t term ( o r gnomon) added, and to b) the number of terms ( o r gnomons) comprising the number. 136 Note 11 - P o l y g o n a l numbers ( c f . H i l l e r 37.11:'ff.) Two methods of r e p r e s e n t i n g , numbers g e o m e t r i c a l l y were used by the Greeks. The method used by P l a t o was to d e p i c t numbers by l i n e s p r o p o r t i o n a l i n l e n g t h to t h e i r mag-n i t u d e . The Pythagorean method was to use dots or alphas p a t t e r n e d i n v a r i o u s forms to r e p r e s e n t numbers. As the Pythagoreans have been c r e d i t e d a l s o w i t h the d i s c o v e r y of the " i r r a t i o n a l " , i t i s no doubt l i k e l y t h a t they a l s o used the former method. "The u n i t " , as Theon says, " i s the fount and source of a l l number", hence the " p r i n c i p l e " (ctpxn) of number, and i t i s i n s t r u c t i v e to observe the ingenious method by which s e r i e s of numbers c o u l d be obtained, based upon r e g u l a r p o l y -gons developed axo -j,ova6o^. By means of a l g e b r a i t i s p o s s i b l e t o - g e n e r a l i s e upon the method of f o r m a t i o n of these p o l y g o n a l numbers and i n v e s t i g a t e some of the common p r o p e r t i e s which they have and the r e l a t i o n s h i p s which e x i s t between them. In F i g . 4 are shown the square number 16^ . the t r i a n g -u l a r number 10, the pentagonal number 12 and the hexagonal number 15, t o g e t h e r w i t h the gnomons which are used i n t h e i r composition. I t may be observed t h a t the i n c r e a s e which occurs i n the gnomon and which serves t o p r e s e r v e the o r i g i n -a l p a t t e r n of the number a l s o serves to i n d i c a t e the number of angles composing the polygon, f o r i t i s l e s s by'2 i n - e a c h case. 137 T*» *j <A Pentagonal (12) Hexagonal (15) 14 4 + 7 14549 Gnomon 3, Angles 5 Gnomon 4, Angles 6 F i g 4 - P o l y g o n a l numbers To c o n s i d e r i t a l g e b r a i c a l l y , l e t the gnomons be 1, 1 + d , l + 2d, l + 3d 1 4 ( n - l ) d ( i ) Then the po l y g o n a l numbers w i l l be the sums of 1, 2, 3 ... n terms of t h i s s e r i e s . Thus, adding the s u c c e s s i v e gnomons, the po l y g o n a l numbers w i l l be g i v e n by 1, 2 4 d , 3+3d, 4 4 6d nth. p o l y g o n a l number ( i i ) and the nth p o l y g o n a l number w i l l be g i v e n by 1 + (1 4 d) 4 ( 1 4 2d) 4 (1 4 3d)4 +- (1 +'nTLd) n 4 (1 + 2 4 3 4 + n - l ) d - n 4 If iri ls d ( i i i ) 138 where d g i v e s the i n c r e a s e i n the gnomon and (d + 2) denotes the order of the polygon number. Thus, a p p l y i n g d i f f e r e n t values of d i n ( i ) we get d = 2 d = 3 gnomons are 1, 2, 3, 4 . g i v i n g 1, 3, 6 , 10 gnomons are 1, 3, 5, 7 . g i v i n g 1, 4, 9, 16 gnomons are 1, 4, 7, 10 g i v i n g 1, 5, 12, 22 l gnomons are 1, 5, 9, 13 g i v i n g 1, 6, 15, 28 ( t r i a n g u l a r numbers) (square numbers) (pentagonal numbers) (hexagonal numbers) From ( i i i ) above, we have nth p o l y g o n a l number = n -f- (n-1 )n ^ 19 2 but, as Dupuis p o i n t s out ^ n ~ 2 ^ n = (n-Dth t r i a n g u l a r number Hence, n t h p o l y g o n a l number - n -\- d times ( n - l ) t h t r i a n g u l a r number. Wherefore d = 2 g i v e s polygons of (d+2) s i d e s (square numbers) and nth square number = n-fn times ( n - l ) t h t r i a n g u l a r number Thus f i r s t square number = 14 2(0) = 1 second " " = 2-+2(1) - 4 t h i r d » *» - 3 + 2(3) = 9 f o u r t h »' " - 4 + 2(6) » 16 — a p r o p e r t y which may be admirably i l l u s t r a t e d by the 19. Dupuis, 338. 4 139 25 d i s s e c t i o n of the sqxiare numbers 1, 4, 9, 16 and 25 shown i n F i g . 5. 16 9 4 -A. -* • »- -* «- • • ft 9-1+2(0) 2+2(1) 3 + 2(3) 4 + 2(6) 5 + 2 ( 1 0 ) F i g 5 - D i s s e c t i o n of square numbers The p r i n c i p l e i s f u r t h e r demonstrated i n F i g . 6, where d = 4 g i v e s hexagonal numbers, and nth hexagonal number = n + 4X ( n - l ) t h t r i a n g u l a r number 28 6 1 + 4 ( 0 ) 2 + 4 ( 1 ) 15 V 3 +4(3) 4 +4(6) F i g 6 - D i s s e c t i o n of hexagonal numbers 140 I t may f u r t h e r be noted t h a t the t r i a n g u l a r numbers are 1, 3, 6, 10, 15, 21, 28 and t h a t the hexagonal numbers 1, 6, 15, 28 are the odd-numbered t r i a n g u l a r numbers. In e f f e c t , nth hexagonal number « ( 2 n - l ) t h t r i a n g u l a r number f o r , u s i n g formula ( i i i ) , nth hexagonal number = n+MUz-LllL = 2 n 2 - n and ( 2 n - l ) t h t r i a n g u l a r number = ( 2 n - l j 2 n 2 = 2 n 2 - n T h e r e f o r e , a l l hexagonal numbers are a l s o t r i a n g u l a r numbers Note 12 - Pyramid and t r u n c a t e d pyramid numbers ( c f . H i l l e r 42.3) Pyramid numbers are formed upon t r i a n g u l a r or square bases and c o n s i s t of the sums of the s e r i e s of t r i a n g u l a r or square numbers s t a r t i n g from 1. Thus, the t r i a n g u l a r numbers are 1, 3, 6, 10, 15, * < | ± l i and the t r i a n g u l a r pyramid numbers are then 1, 4, 10, 20, 35 nth number So, nth t r i a n g u l a r pyramid number = X n ( n + 1 ) Z— ^ 2 = t 7 n 2 + iy n 2n°+- 4n 3 n ( n 2 + 2) 3 141 F i g 7 - T r i a n g u l a r pyramid number (10) Now the square numbers are 1, 4, 9, 16, 2 5 . . . n 2 Thus, the square pyramid numbers w i l l be 1, 5, 14, 30 ..... nth number Thus, nth square pyramid number = ^ n 2 . n(n+-l)( 2n+-lJ 1 • 2 • 3 Theon makes r e f e r e n c e a l s o to t r u n c a t e d pyramid num-ber s . These are the numbers which comprise the u n i t s remain-i n g a f t e r the top p o r t i o n of a g i v e n pyramid has been remov-ed. These numbers may t h e r e f o r e be obtained by f i n d i n g the d i f f e r e n c e between any two pyramid numbers, or by summing any a d j a c e n t terms of the s e r i e s of t r i a n g u l a r o r square numbers. Thus, take the f o l l o w i n g t r i a n g u l a r numbers B 1, f, 6^  l p ; 15, 21 A C Then, examples of t r u n c a t e d t r i a n g u l a r pyramid numbers a r e A ( 9 ) , B ( 34 ) , C ( 46 ) 142 And, t a k i n g the square numbers 1, 4, 9, 16, 25, 36 the f o l l o w i n g w i l l be t r u n c a t e d square pyramid numbers -D ( 13 ), E ( 54 ), F ( 77 ) and they are composed d i a g r a m m a t i c a l l y as shown i n F i g . 8. F i g 8 - Truncated square pyramid number - D (15) Note 13 - S i d e and d i a g o n a l numbers ( c f . H i l l e r 43.2) The d i s c o v e r y of the r i g h t - a n g l e d p r o p e r t y of the 3-4-5 t r i a n g l e p r o b a b l y o r i g i n a t e d i n Egypt, although Pyth-agoras or the Pythagoreans are u s u a l l y c r e d i t e d w i t h d i s c o v -e r i n g a proof of i t . A simple c o n s t r u c t i o n would show t h a t t h i s r e l a t i o n s h i p a l s o e x i s t e d f o r i s o s c e l e s r i g h t - a n g l e d t r i a n g l e s but, as no i n t e g r a l values were r e a d i l y at hand, the Greeks were here confronted w i t h the incommensurability of the l e n g t h of the d i a g o n a l of a square with i t s s i d e and with the i r r a t i o n a l i t y of J~2. With h i s e x p o s i t i o n of s i d e and d i a g o n a l numbers, Theon r e v e a l s the method by which the Pythagoreans o b t a i n e d an i n f i n i t e s e r i e s of r a t i o s , which approximated c l o s e l y t o 143 t h e v a l u e of\/~2. The r e a s o n i n g went as f o l l o w s . The u n i t , b e i n g t h e p r i n c i p l e of a l l number, i s f i r s t t a k e n as t h e s i d e and d i a -g o n a l o f t h e f i r s t i s o s c e l e s r i g h t - t r i a n g l e , and t h e f i r s t c r ude ( i n t e g r a l ) a p p r o x i m a t i o n ( l ) i s o b t a i n e d f o r Jz. Then, a p p l y i n g t h e f i n d i n g s o f P y t h a g o r a s ' Theorem, a d i a g o n a l i s added t o t h e s i d e but two s i d e s a r e added t o t h e d i a g o n a l , and a second t r i a n g l e i s o b t a i n e d ; and we have 3 as t h e next i n t e g r a l a p p r o x i m a t i o n f o r s/8. The p r o c e s s i s r e p e a t e d f o r t h e next t r i a n g l e and 7 i s o b t a i n e d as an a p p r o x i m a t i o n f o r v^O, and a g a i n r e p e a t e d f o r subsequent t r i a n g l e s as i s shown i n F i g . 9. 5 +7 F i g 9 •- A p p r o x i m a t i o n s for\/~2 By t h i s method, i n t e g r a l v a l u e s of t h e s i d e and d i a -g o n a l numbers a r e o b t a i n e d as f o l l o w s : '1 , 1, 1 2, '2, 3 5, 5, 7 12, 12, 17 and t h e r a t i o s of d i a g o n a l t o s i d e f u r n i s h v a l u e s w h i c h be-come p r o g r e s s i v e l y more a c c u r a t e a p p r o x i m a t i o n s o f \/2. Thus s/Z 1/1 «•* 3/2 «/ 17/12 & 41/29 .... 144 Now i t may be observed, t h a t the square on the d i a -g o n a l , compared w i t h the double of the square on the s i d e , i s f i r s t 1 l e s s than i t ( 1 ) , then 1 more than i t ( 9 ) , then 1 l e s s than i t (49), then 1 more than i t (289) and so on, so 20 t h a t the p r o c e s s , as Heath i n d i c a t e s , amounts to f i n d i n g a l l the i n t e g r a l s o l u t i o n s of the indeterminate equations 2 x 2 - y 2 - ± 1 1 f o r the formula, i f t r u e , y i e l d s two l a r g e r numbers (x+y) and (2x+y) such t h a t 2 ( x+y) 2 - (2x + y ) 2 = + 1 i . e . , ( 2 x - » r y ) 2 - 2 ( x + y ) 2 which i s demonstrably t r u e , f o r (2x+y) 2 - 2 ( x + y ) 2 = 4 x 2 -+- 4xy + y 2 - 2 x 2 - 4xy - 2 y 2 = 2x 2 - y 2 - ± 1 Note 14 - P e r f e c t numbers ( c f . H i l l e r 45.9) Theon g i v e s the same d e f i n i t i o n of a p e r f e c t number 21 as does E u c l i d , "one which i s equal t o (the sum of) i t s p a r t s " , i . e . , equal t o the sum of a l l i t s f a c t o r s i n c l u d i n g 1. :. Thus 6 - 1 4 - 2 4 - 3 28 - 1 4 - 2 4 - 4 4-7 4 14 and 496 - 1 + 2 +4 + 8 4 16 + 31 4 62 + 124 4- 248 and these numbers are a c c o r d i n g l y p e r f e c t . 20 Heath, H.G.M., i , 9 1 f f . PI E u c l i d , V I I , Def. 22 145 E u c l i d a l s o g i v e s the law of f o r m a t i o n f o r these num-bers t o g e t h e r w i t h a proof and, i n e f f e c t , i t s t a t e s t h a t , i f any number of the terms of the s e r i e s 1, 2, 2 2, 2 3 ... 2 n - 1 ( - S n ) i s prime, i 22 then, S n • 2 n i s a " p e r f e c t " number. 23 The a l g e b r a i c p roof i s r a t h e r l o n g , but i s g i v e n by Heath. But the e a r l y Pythagoreans c a l l e d 10 the p e r f e c t num-24 ber, and Theon draws a t t e n t i o n to t h i s a l s o . They c a l l e d 10 the TETpctK-ruc; as i t was composed of the sum of the numbers 1, 2, 3 and 4. F o r them i t was " t h e i r g r e a t e s t oath", symbol-i s i n g a l l t h a t e x i s t e d and m y s t i c a l l y embracing th e p o i n t ( 1 ) , the l i n e ( 2 ) , the plane (3) and the s o l i d pyramid ( 4 ) , w h i l e these numbers a l s o i n c l u d e d the r a t i o s corresponding t o the m u s i c a l i n t e r v a l s d i s c o v e r e d by Pythagoras, namely 4:3 (the f o u r t h ) , 3:2 (the f i f t h ) and 2:1 (the o c t a v e ) . pp E u c l i d , Elements IX, Prop. 36. 2 3 H e a t h , The T h i r t e e n Books of E u c l i d ' s Elements, i i , 424-5. ?4 H i l l e r , 94. I f f . CHAPTER FIVE CONCLUSION In c o n c l u d i n g t h i s study of Theon of Smyrna's work on a r i t h m e t i c some a p p r a i s a l of h i s c o n t r i b u t i o n t o the study of number i n the a b s t r a c t (apt6p,r)Tixr|) may perhaps be c o n s i d e r e d a p p r o p r i a t e . S i n c e i t i s i m p o s s i b l e t o o f f e r t h i s e v a l u a t i o n without t a k i n g some account of Theon's a r i t h m e t i c a l anteced-e n t s , I s h a l l p r e f a c e my remarks by g i v i n g a b r i e f resume o f the development of Greek a r i t h m e t i c as i t p e r t a i n s to the top-i c s examined by Theon. The s c i e n t i f i c study of number by the Greeks begins w i t h Pythagoras. The e n q u i r i e s g e n e r a l l y a t t r i b u t e d t o him or t o h i s s c h o o l r e p r e s e n t the f i r s t s w i t ch i n the focus of p h i l o s o p h y from the study of matter t o the study of the s t r u c -t u r e of t h i n g s — a n a t t i t u d e s u c c i n c t l y summarised i n the e s t -a b l i s h e d dictum of the Pythagoreans, "Number r u l e s the U n i -v e r s e " . 1 Despite the d i s p a r a g i n g remarks of D a n t z i g and g others t h a t t h e i r . l o r e was e s s e n t i a l l y m y s t i c , few s c h o l a r s now s e r i o u s l y d i s p u t e the c l a i m of Pythagoras and h i s f o l l o w -ers t o be the f i r s t t o have s t u d i e d "numbers pure and simple". T h i s o p i n i o n moreover accords with'the c o n v i c t i o n of the anc-i e n t s . A r i s t o t l e , f o r i n s t a n c e , c r e d i t s Pythagoras with being 1 S e e note 44 (p. 89). 2 T o b i a s D a n t z i g , Bequest of the Greeks (London 1955) p. 32. 147 the f i r s t t o work on mathematics and a r i t h m e t i c , w h i l e , a c c -o r d i n g t o Stobaeus, he a n t i c i p a t e d P l a t o i n d i v o r c i n g a r i t h -4 metic from the realm of commercial u t i l i t y . One of the main d i f f i c u l t i e s i n a p p r a i s i n g the work of Pythagoras i s the l a c k of w r i t t e n evidence, probably as a r e s u l t of the o r a l t r a n s m i s s i o n i n the t e a c h i n g of the scho o l r a t h e r than of the a l l e g e d oath of s e c r e c y t h a t bound the mem-bers of the s c h o o l . Nonetheless d e s p i t e t h i s absence of def-i n i t e evidence some i d e a of the s t a t e of Pythagorean a r i t h -metic may be deduced from l a t e r w r i t e r s . From an excerpt of Speusippus' work On the Pythagorean Numbers, f o r i n s t a n c e , i t would appear t h a t p o l y g o n a l numbers were known and by i n f -erence many of the a s s o c i a t e d ideas on the number, the u n i t , odd and even numbers, composite and prime numbers. Square, t r i a n g u l a r and oblong numbers would a l s o have been adopted and, c e r t a i n l y by P l a t o ' s time i f not be f o r e , these concepts were commonplace. While Heath b e l i e v e s t h a t Pythagoras had not prob a b l y developed any theory of i r r a t i o n a l s , he contends t h a t he would have been aware of the incommensurability o f the diag o n a l of a square w i t h i t s s i d e and have developed a method of approx-imating the value of ./2, a f t e r the manner d e s c r i b e d by Theon. In a l o s t work, On the Pythagoreans, A p o l l o n i u s , H i s t . m i r a b l l . 6 (Vors, i 3 , p.25, 5 ) . 4 S t o b a e u s , E e l . . i , proem., 6 (Vors, i 3 , p.346, 12). Theologumena A r i t h m e t i c a , author unknown, ed A s t , ( L e i p z i g 1817), p. 61. 6 H e a t h , H.G.M., i , 91. 148 Three kinds of p r o p o r t i o n , the a r i t h m e t i c a l , g e o m e t r i c a l and 7 harmonic were c e r t a i n l y known. Yet how much of t h i s was d i s c o v e r e d by Pythagoras h i m s e l f , or what p r e c i s e l y must be a s c r i b e d t o h i s f o l l o w e r s P h i l o l a u s , A r c h y t a s and P l a t o i s obscure and the absence of c e r t a i n b a s i c concepts should be assumed. There i s no t r a c e , f o r example, i n the fragments of P h i l o l a u s , P l a t o or A r i s t o t l e of an e x p o s i t i o n of the p e r -f e c t number ( r e X e i o c ; ) ; f o r the Pythagoreans, a c c o r d i n g t o Q Theon, 10 was the p e r f e c t number. By the end of the f o u r t h century the study of a r i t h -metic had advanced c o n s i d e r a b l y from i t s Pythagorean begin-n i n g s . T h i s p r o g r e s s i s c o n v e n i e n t l y r e c o r d e d i n the p r o -p o s i t i o n s gathered t o g e t h e r and p u b l i s h e d about 300 B.C. by E u c l i d as a p o r t i o n of h i s famous Elements. Prime numbers, composite numbers, square, plane and s o l i d numbers are a l l d e a l t with; the nomenclature i s Pythagorean, but E u c l i d uses a d e d u c t i v e method of proof and proceeds f u r t h e r i n the realm of g e n e r a l i t y . During the succeeding g e n e r a t i o n s geometry enjoyed some amazing successes under the d i r e c t i o n of Archimedes and A p o l l o n i u s , but the study of numbers l a n g u i s h e d . Indeed v i r t u a l l y n o t h i n g was done i n the f i e l d of a r i t h m e t i c f o r n e a r l y f o u r c e n t u r i e s although the s t u d i e s of the astronomers H y p s i c l e s ( f l o r u i t c i r c a 150 B.C.) and Eratosthenes ( f l o r u i t 7 Iamblichus, i n Nicom., p.100, 19-24. 8 H i l l e r , 46,13. 149 230 B.C.) are of some r e l e v a n c e to the development of a r i t h -metic and had a c o n s i d e r a b l e i n f l u e n c e on Theon i n p a r t i c -u l a r . The former i s the reputed author of an a s t r o n o m i c a l t r a c t e n t i t l e d De A s c e n s i o n i b u s , s t i l l extant, besides o t h e r works not s u r v i v i n g , d e a l i n g w i t h the harmony of the u n i v e r s e and p o l y g o n a l numbers. Eratosthenes i s commonly remembered f o r h i s " s i e v e " which comprised a method of " s i f t i n g out" comp-o s i t e numbers i n order t o o b t a i n the s u c c e s s i v e primes; indeed he was a mathematician of great a b i l i t y — A r c h i m e d e s h i m s e l f d e d i c a t e d The Method t o him—and was a l s o author of a work e n t i t l e d P l a t o n i c u s , not now extant, but one which may w e l l Q have been one of Theon's major sources. There are remarkable p a r a l l e l s one c o u l d p o i n t to i n the two works, such as^.their opening w i t h the " D e l i a n problem", the theme of "mathematics f o r the r e a d i n g of P l a t o " , the d i s q u i s i t i o n s on p r o p o r t i o n , the treatment of g e o m e t r i c a l and a r i t h m e t i c a l d e f i n i t i o n s and the d i s c u s s i o n of the p r i n c i p l e s of music. With the f i r s t s t i r r i n g s of Nepplatonism at the begin-nings of the second century a t t e n t i o n was once more focu s e d on the study of numbers and we f i n d a r e v i v a l of i n t e r e s t i n the n e g l e c t e d systems of Pythagoras and P l a t o . I t i s at t h i s p o i n t t h a t Theon wrote h i s d i s s e r t a t i o n upon "mathematics u s e f u l f o r the r e a d i n g of P l a t o " and the a r i t h m e t i c a l s e c t -i o n w i t h which we are concerned. To t h i s p e r i o d a l s o belongs the work of Nicomachus 9 Heath, H.G.M., i i , 104. 150 of Gerasa whom Theon f o l l o w s c l o s e l y i n h i s w r i t i n g s . N i c o -machus' t r e a t i s e e n t i t l e d I n t r o d u c t i o A r i t h m e t i c a l 0 enjoyed a high r e p u t a t i o n , i f one may judge from the number of comment-a r i e s w r i t t e n upon i t , the most important being t h a t of Iamb-l i c h u s . T h i s i s s u r p r i s i n g when one c o n s i d e r s t h a t Nicomach-us does not appear t o be a genuine mathematician and t h a t l i t t l e of h i s p r e s e n t a t i o n appears t o be o r i g i n a l . The work b r o a d l y comprises a p u b l i c a t i o n of the o l d Pythagorean a r i t h m e t i c i n the form i n which i t had become e s t a b l i s h e d by about E u c l i d ' s time. There i s a p h i l o s o p h i c a l i n t r o d u c t i o n and a c l a s s i f i c a t i o n of numbers i n t o odd and even, prime and incomposite, p e r f e c t , o v e r - p e r f e c t and d e f e c t i v e , a f u l l treatment of p o l y g o n a l numbers, pyramidal numbers, c i r c u l a r and s p h e r i c a l numbers. The treatment, however, i s not i n the E u c l i d e a n manner by means of deductive p r o o f s ; Nicomachus w r i t e s a continuous n a r r a t i v e w i t h some attempt at r h e t o r i c and a l l u s i o n s to p h i l o s o p h y and history.^" 1' l ? Heath's judgments are c r i t i c a l i n the extreme: "no p r o o f s i n the proper sense of the word", "cumbrous c i r c u m l o c -u t i o n s " , "not r e a l l y a mathematician", " a popular treatment c a l c u l a t e d t o awaken i n the beginner an i n t e r e s t i n the t h e o r y of numbers by making him acquainted w i t h the most noteworthy 1 0 N i c o m a c h u s , I n t r o d u c t i o A r i t h m e t l c a , H o c h e ^ L e i p z i f g !i*8V6.-J". Gow, A S h o r t H i s t o r y of Greek Mathematics (Cambr-idge 1884) p. 95. 1 2 H e a t h , H.G.M., i , 98-99. 151 r e s u l t s o b tained up-to-date", "the p r o p e r t i e s of numbers app-appear marvellous or mira c u l o u s " , " l i t t l e i s o r i g i n a l " , " i n essence i t e v i d e n t l y goes back t o the e a r l y Pythagoreans". He supposes t h a t the work was adequate as a mathematical comp-endium f o r the p h i l o s o p h e r s of the day, e s p e c i a l l y as the r e were few mathematicians of any s t a t u r e at t h a t time. Theon o f Smyrna was contemporary w i t h , o r c e r t a i n l y not much l a t e r than, Nicomachus. The m a t e r i a l he presents on a r i t h m e t i c i s p r a c t i c a l l y the same as; t h a t g i v e n by Nicomach-us, and i n h i s treatment and arrangement Theon appears equ-a l l y remiss. Heath indeed c o n s i d e r s Theon t o be even l e s s 13 s y s t e m a t i c than Nicomachus. C e r t a i n l y h i s work does not seem w e l l i n t e g r a t e d ; s i m i l a r t o p i c s are d e a l t w i t h i n w i d e l y separated c h a p t e r s ; much of h i s thought i s r e p e t i t i o u s , and much of h i s language redundant. Nonetheless i n s p i t e o f h i s c l o s e dependence on N i c -omachus, c e r t a i n t o p i c s seem t o belong e x c l u s i v e l y t o Theon's t r e a t i s e . I n Nicomachus, f o r i n s t a n c e , we f i n d no treatment of i ) the t h e o r y of s i d e and d i a g o n a l numbers, or of i i ) the d i v i s i b i l i t y of squares. On the o t h e r hand Theon can h a r d l y be c r e d i t e d w i t h e i t h e r d i s c o v e r y . The former was c l e a r l y 14 f a m i l i a r t o the e a r l y Pythagoreans,,, and the treatment of squares, though a s s u r e d l y not found elsewhere u n t i l Iambi-i c h u s d i s c u s s e d the s u b j e c t , i s a l s o b e l i e v e d t o be an 1 3 H e a t h , H.G.M., i , 112. 1 4 H e a t h , H.G.M., i , 91. element i n Pythagorean a r i t h m e t i c . B a s i c a l l y , however, the a r i t h m e t i c a l w r i t i n g s of Theon and Nicomachus s u f f e r from the same weakness. Each l a b o u r s under a c r u s h i n g handicap of a language i l l - s u i t e d f o r the p r e s e n t a t i o n of number theory. F u r t h e r p r o g r e s s c o u l d not be made without the i n v e n t i o n of an a p p r o p r i a t e mathematical symbolism. An a n a l y t i c a l method i n v o l v i n g an 15 "Unknown" had been used as e a r l y as P l a t o , t o be sure, but i t was l e f t f o r Diophantus t o p r o v i d e a developed symbolism and l a y the f o u n d a t i o n s of a l g e b r a . "With Diophantus", w r i t e s Gow, "the h i s t o r y of Greek a r i t h m e t i c comes t o an end; 17 no o r i g i n a l work was done by anyone a f t e r him". Was Theon then p h i l o s o p h e r , mathematician or a s t r o -nomer? H i s i n t r o d u c t i o n t o the work, c o p i o u s l y i l l u s t r a t e d w i t h passages from P l a t o a p p a r e n t l y quoted from memory, would c e r t a i n l y i n d i c a t e a man w i t h an a v i d i n t e r e s t i n 1 5 c f . P l a t o , Gharmides, 168E. 16 T h i s b r i l l i a n t mathematician i s g i v e n a f l o r u i t of 250 B.C. by Heath and by 0.C.D. Hi s most important work, the A r l t h m e t l c a , d e a l s with the s o l u t i o n of equations u s i n g a l g e b r a i c symbols. H i s c h i e f problems were ind e t e r m i n a t e and semi-indeterminate equations and the method i s s t r i c t l y ana-l y t i c a l , a l though a pamphlet On P o l y g o n a l Numbers was a l s o w r i t t e n a f t e r the manner of E u c l i d . 17 Gow, op_. c i t . , p. 121. But t h i s a s s e r t i o n i s not completely a c c u r a t e . Pappus (ca. ,300 A.D.) and Iamblichus ( p r o b a b l y a l i t t l e l a t e r ) made some f u r t h e r c o n t r i b u t i o n s t o the t h e o r y of numbers, the former i n t r e a t i n g A p o l l o n i u s ' " t e t r a d s " and e s t a b l i s h i n g t e n means, and the l a t t e r on the p r o p e r t i e s of numbers of the form 3n-2, 3 n - l , 3n. 153 P l a t o n i c p h i l o s o p h y . The n a r r a t i v e moreover i s n o t i c e a b l y more a l i v e here than i n many of h i s d i s q u i s i t i o n s on numbers where h i s s t y l e i s t u r g i d and the content r e p e t i t i v e . But Theon never d i s p l a y s the l e a s t i n c l i n a t i o n to handle h i s m a t e r i a l w i t h any o r i g i n a l i t y , being content merely to use P l a t o as an a u t h o r i t y f o r a s u c c e s s i o n of p l a t i t u d e s on the value of mathematics. I n no sense can he be c o n s i d e r e d any-t h i n g more than an amateur p h i l o s o p h e r . The p l a c e of Theon amongst a f i e l d . o f mathematical g i a n t s such as Pythagoras, E u c l i d , E r a t o s t h e n e s and Diophan-t u s i s even l e s s d i f f i c u l t t o assess. I f he c o u l d be c r e d i t e d a t : l e a s t w i t h the o r i g i n a l d i s c o v e r y of: the p r o p e r t i e s of squares mentioned e a r l i e r , h i s r e p u t a t i o n would be redeemed. But h i s work i n l a r g e measure comprises what had been known f o r s i x hundred and f i f t y y e a r s , namely Pythagorean a r i t h m e t -i c . Furthermore, as has been noted, h i s work c l o s e l y p a r a l l e l s t h a t of Nicomachus, even to the p h i l o s o p h i c a l i n t r o d u c t i o n , and i s b e l i e v e d to have a s i m i l a r form and content as had the P l a t o n i c u s of E r a t o s t h e n e s . H i s c o n t r i b u t i o n can h a r d l y have been g r e a t e r than t h a t of Nicomachus, and t h a t was l i t t l e enough. I t i s then d i f f i c u l t not to accept h i s own c o n f e s s i o n t h a t he i s p r e s e n t i n g a r i t h m e t i c " f o r the layman". The v e r d i c t of modern mathematical w r i t e r s on the o r i g i n a l i t y of Theon of Smyrna has been c o n s i s t e n t l y unfav-o r a b l e . Heath b e l i e v e s h i s work i s of value not p r i m a r i l y f o r i t s i n t r i n s i c worth, but by v i r t u e of the numerous h i s t -154 18 o r i c a l n o t i c e s t h a t i t c o n t a i n s . To C a j o r i h i s work i s i l l -19 arranged and has l i t t l e m e r i t . I t i s g e n e r a l l y agreed t h a t 20 h i s work i s a c o m p i l a t i o n and Dupuis i n the p r e f a c e of h i s e d i t i o n of the work attaches importance t o i t o n l y i n the r e a l m of the h i s t o r y of s c i e n c e : S i l e s mathematiques n'ont r i e n a gagner a l a p u b l i c -a t i o n de c e t t e t r a d u c t i o n , l ' h i s t o i r e des s c i e n c e s peut y t r o u v e r du moins quelques renseignements u t i l e s . 1 T h i s l e a v e s o n l y h i s c l a i m t o be an astronomer. Here alone perhaps i s t h e r e some j u s t i f i c a t i o n f o r h i s r e p u t a t i o n . H i s s e c t i o n on astronomy i s g e n e r a l l y c o n s i d e r e d to be super-22 i o r t o the s e c t i o n on a r i t h m e t i c . I f o n l y we c o u l d be sure 23 t h a t the Theon r e f e r r e d t o w i t h r e s p e c t by Ptolemy f o r h i s a s t r o n o m i c a l o b s e r v a t i o n s was i n f a c t Theon of Smyrna, we might with c o n f i d e n c e r e s e r v e at l e a s t t h i s d i s t i n c t i o n f o r him. l ft Heath, H.G.M.. i i , 239. 19 C a j o r i , H i s t o r y of Greek Mathematics, p.59. 20 F. E. Robbins, P o s i d o n i u s and Pythagorean Arithmology, C l a s s . P h i l , v o l . 15 (1920), p.309. 2 1 D u p u i s , P r e f . v i i i . 2 2 F r i t z , i n PW s.v. "Theon". 2 3 p . 2 f f . i n my I n t r o d u c t i o n . BIBLIOGRAPHY I . A n c i e n t Authors and Texts Archimedes. Opera Omnia (with the commentaries of E u t o c i u s ) . E d i t e d by J". L. Heiberg. L e i p z i g 1880-81 1910-15. A r i s t o t l e . De Caelo. E d i t e d and t r a n s l a t e d by W. K. C. G u t h r i e . Loeb C l a s s i c a l L i b r a r y . London and Camb-r i d g e , Mass. 1939. . P h y s i c s . ( 1 ) E d i t e d by W. D. Ross. Oxford 1950. (2) E d i t e d and t r a n s l a t e d by P. H. Wicksteed and F. M. Cornf o r d . 2 volumes. Loeb C l a s s i c a l L i b r a r y . London and Cambridge, Mass. 1963. . Metaphysics. E d i t e d by Hugh Tredennick. 2 volumes. Loeb C l a s s i c a l L i b r a r y . London and Cambridge, Mass. 1961. Diophantus. Opera Omnia cum g r a e c i s commentariis. E d i t e d by P a u l Tannery. L e i p z i g 1893-95. Iambiichus. I n Nicomachi i n t r o d u c t i o n e m arithmeticam comm-e n t a r l u s . E d i t e d by H. P i s t e l l i . L e i p z i g 1894. Nicomachus. I n t r o d u c t i o A r i t h m e t i c a . E d i t e d by Richardus Hoche. L e i p z i g 1866. P l a t o . P l a t o n i s Opera. E d i t e d by John Burnet. Oxford 1962. . The R e p u b l i c . (1) E d i t e d by J . Adam. 2 volumes. Cambridge 1921. (2) E d i t e d by B. Jowett and L. Camp-b e l l . 3 volumes. Oxford 1894. (3) E d i t e d and t r a n s l -ated by P a u l Shorey. Loeb C l a s s i c a l L i b r a r y . London 156 and Cambridge, Mass. 1939. P l u t a r c h . Moralia.volume V I I . E d i t e d and t r a n s l a t e d by P h i l l i p H. de Lacey and Benedict E i n a r s o n . volume IX. E d i t e d and t r a n s l a t e d by Edwin L.Minar J r , F. H. Sandbach and W. C. Helmbold. Loeb C l a s s i c a l L i b r a r y . London and Cambridge, Mass. 1961. P r o c l u s Diodochus. In P l a t o n i s Tlmaeum oommentarli. E d i t e d by E. D i e h l . L e i p z i g 1903. Ptolemy. Opera quae extant omnia. E d i t e d by J . L. H e i b e r g . L e i p z i g 1898-1919. S i m p l i c i u s . In A r i s t o t e l l s de c a e l o l i b r o s commentaria. E d i t e d by J . L. H e i b e r g . B e r l i n 1894. Stobaeus, Joannes. Eclogarum physlcarum et ethicarum l l b r i duo. E d i t e d by Augustus Meineke. L e i p z i g 1860-64. Theon. Theonis P l a t o n i c i l i b e r de astronomia. E d i t e d by T. H. M a r t i n . P a r i s 1849. . E x p o s i t l o rerum mathematioarum ad legendum Platonem u t i l i u m . (1) E d i t e d by Eduardus H i l l e r . L e i p z i g 1878. (2) E d i t e d by J . Dupuis w i t h French t r a n s l a t i o n . P a r i s 1892. I I . Modern Authors C a j o r i , F. A H i s t o r y of Greek Mathematics. New York 1893. Da n t z i g , T. The Bequest of the Greeks. London 1955. D i e l s , H. Die Fragmente der V o r s o k r a t l k e r . 3 volumes. Z u r i c h / - B e r l i n 1964. 157 F r i t z , K. "Theon ( 1 4 ) " R e a l e n c y c l o p a e d i e der k l a s s i s c h e n  A l t e r t u m s w i s 3 e n s c h a f t . E d i t e d by G. Wissowa and W. K r o l l . Volume V A, P a r t 2, 2067-75. Gow, J . A Short H i s t o r y of Greek Mathematics. Cambridge 1884. Heath, S i r T. L. The T h i r t e e n Books of E u c l i d ' s Elements. Cambridge 1926. . A H i s t o r y of Greek Mathematics. 2 volumes. Oxford 1921. L i p p e r t , J . S t u d i e n auf dem Gebiet der Arabischen U b e r s e t z -u n g s l i t e r a t u r . Brunswick 1894. R u s s e l l , B e r t r a n d . The P r i n c i p l e s of Mathematics. Cambridge 1903. Tannery, P. Memoires S c i e n t i f i q u e s . P u b l i s h e d by J . L. H e i b e r g and H. G. Zeuthen. Toulouse 1913. Thomas, I. Greek Mathematical Works. S e l e c t i o n s of Greek mathematics w i t h t r a n s l a t i o n . 2 volumes. Loeb C l a s s -i c a l L i b r a r y . London and Cambridge, Mass. 1939. Z e l l e r , Eduard. Die P h i l o s o p h i e der G r i e c h e n . T r a n s l a t e d by Sarah Frances A l l e y n e and E v e l y n Abbott. New York 1890. 

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