ENGLISH INDEX.

Adrastus 292 Alcinous 98 Alternate and alternately (of ratios) 134 Alternative proofs, interpolated (cf. III. 9 and following) 22: that in III. 10 claimed by Heron 23-4 Amaldi, Ugo, 30, 126 Ambiguous case of VI. 7, 208-9 Anaximander 111 Anaximenes 111 Angle: angles not less than two right angles not recognised as angles (cf. Heron, Proclus, Zenodorus) 47-9: hollow-angled figure (the re-entrant angle was exterior) 48: did Euclid extend “angle” to angles greater than two right angles in VI. 33? 275-6: “angle of semicircle” and “of segment” 4: hornlike angle 4, 39, 40: controversies about “angle of semicircle” and hornlike angle 39-42 (see also Hornlike) Antecedents (leading terms in proportion) 134 Antiparallels: may be used for construction of VI. 12, 215 Apollonius: Plane νεύσεις, problem from, 81, lemma by Pappus on, 64-5: Plane Loci, theorem from (arising out of Eucl. VI. 3), also found in Aristotle 198-200: 75, 190, 258 Application of areas (including exceeding and falling short) corresponding to solution of quadratic equations 187, 258-60, 263-5, 266-7 Approximations: 7/5 as approximation to $\sqrt{2}$ (Pythagoreans and Plato) 119: approximations to $\sqrt{3}$ in Archimedes and (in sexagesimal fractions) in Ptolemy 119: to $\pi$ (Archimedes) 119: to $\sqrt{4500}$ (Theon of Alexandria) 119 Archimedes: Method of, 40: Liber assumptorum, proposition from, 65: approximations to $\sqrt{3}$, square roots of large numbers, and to $\pi$ 119: extension of a proportion in commensurables to cover incommensurables 193: 136, 190 Archytas: proof that there is no numerical geometric mean between n and n + 1 295 Aristotle: indicates proof (pre-Euclidean) that angle in semicircle is right 63: on def. of same ratio (=same ἀνταναίρεσις) 120-1: on proportion as “equality of ratios” 119: on theorem in proportion not proved generally till his time 113: on proportion in three terms (συνεχής, continuous), and in four terms (διῃρημένη, discrete) 131, 293: on alternate ratios 134: on inverse ratio 134, 149: on similar rectilinear figures 188: has locus-theorem (arising out of Eucl. VI. 3) also given in Apollonius’ Plane Loci 198-200: on unit 279: on number 280: on non-applicability of arithmetical proofs to magnitudes if these are not numbers 113: on definitions of odd and even by one another 281: on prime numbers 284-5: on composite numbers as plane and solid 286, 288, 290: on representation of numbers by pebbles forming figures 288 Arithmetic, Elements of, anterior to Euclid 295 August, E. F. 23, 25, 149, 238, 256, 412 Austin, W. 172, 188, 211, 259 Axioms tacitly assumed: in Book V. 137: in Book VII. 294

Babylonians 112 Baermann, G. F. 213 Baltzer, R. 30 Barrow: on Eucl. V. Def. 3, 117: on V. Def. 5, 121: 56, 186, 238 Billingsley, H. 56, 238 Boëthius 295 Borelli, G. A. 2, 84 Breadth (of numbers) = second dimension or factor 288 Briggs, H. 143

Camerer, J. G. 22, 25, 28, 33, 34, 40, 67, 121, 131, 189, 213, 244 Campanus 28, 41, 56, 90, 116, 119, 121, 146, 189, 211, 234, 235, 253, 275, 320, 322, 328 Candalla 189 Cantor, Moritz 5, 40, 97 Cardano, Hieronimo 41 Case: Greeks did not infer limiting cases, but proved them separately 75 Casey, J. 227 Catalan 426 Cesàro, E. 426 “Chance equimultiples” in phrase “other, chance, equimultiples” 143-4 Circle: definition of equal circles 2: circles 432 ENGLISH INDEX

touching, meaning of definition, 3: “circle” in sense of “circumference” 23: circles intersecting and touching, difficulties in Euclid’s treatment of, 25-7, 28-9, modern treatment of, 30-2 Clavius 2, 41, 42, 47, 49, 53, 56, 67, 70, 73, 130, 170, 190, 231, 238, 244, 271 Commandinus 47, 130, 190 Componendo (συνθέντι), denoting “composition” of ratios q.v.: componendo and separando used relatively to each other 168, 170 Composite numbers, in Euclid 286: with Eucl. and Theon of Smyrna may be even, but with Nicom. and Iambl. are a sub-division of odd 286: plane and solid numbers species of, 286 “Composite to one another” (of numbers) 286-7 Composition of ratios (σύνθεσις λόγου), denoted by componendo (συνθέντι), distinct from compounding ratios 134-5 Compound ratio: explanation of, 132-3: questionable definition of, 189-90: compounded ratios in V. 20-23, 176-8 Consequents (“following” terms in proportion) 134, 238 Continuous proportion (συνεχὴς or συνημμένη ἀναλογία) in three terms 131 Conversion of ratio (ἀναστροφὴ λόγου), denoted by convertendo (ἀναστρέψαντι) 135: convertendo theorem not established by V. 19, Por. 174-5, but proved by Simson’s Prop. E 175 Convertendo denoting “conversion” of ratios, q.v. Corresponding magnitudes 134 Cube: duplication of, reduced by Hippocrates to problem of two mean proportionals 133: cube number, def. of, 291: two mean proportionals between two cube numbers, 294, 364-5 Curtze, M. 426 Cyclic, of a particular kind of square number 291 Cyclomathia of Leotaud 42

Data of Euclid: Def. 2, 248: Prop. 8, 249-50: Prop. 24, 246-7: Prop. 55, 254: Props. 56 and 68, 249: Prop. 58, 263, 265: Props. 59 and 84, 266-7: Prop. 67 assumes part of converse of Simson’s Prop. B (Book VI.) 224: Prop. 70, 250: Prop. 85, 264: Prop. 87, 228: Prop. 93, 227 Dechales 259 Dedekind’s theory of irrational numbers corresponds exactly to Eucl. V. Def. 5, 124-6 Democritus: “On difference of gnomon” etc. (? on “angle of contact”) 40: on parallel and infinitely near sections of cone 40: stated, without proving, propositions about volumes of cone and pyramid 40 De Morgan, A.: on definition of ratio 116-7: on extension of meaning of ratio to cover incommensurables 118: means of expressing ratios between incommensurables by approximation to any extent 118-9: defence and explanation of V. Def. 5, 122-4: on necessity of proof that tests for greater and less, or greater and equal, ratios cannot coexist 130-1, 157: on compound ratio 132-3, 234: sketch of proof of existence of fourth proportional (assumed in V. 18) 171: proposed lemma about duplicate ratios as alternative means of proving VI. 22, 246-7: 5, 7, 9-10, 11, 15, 20, 22, 29, 56, 76-7, 83, 101, 104, 116-9, 120, 130, 139, 145, 197, 202, 217-8, 232, 233, 234, 272, 275 Dercyllides 111 Dickson, L. E. 426 Diorismus for solution of a quadratic 259 Discrete proportion, διῃρημένη or διεζευγμένη ἀναλογία, in four terms 131, 293 “Dissimilarly ordered” proportion (ἀνομοίως τεταγμένων τῶν λόγων) in Archimedes = “perturbed proportion” 136 Dividendo (of ratios), see Separation, separando “Division (of ratios),” see Separation Divisions (of figures), On, treatise by Euclid, proposition from, 5 Dodecahedron: decomposition of faces into elementary triangles, 98 Dodgson, C. L. 48, 275 Duplicate ratio: διπλασίων, duplicate, distinct from διπλάσιος, double (=ratio 2 : 1), though use of terms not uniform 133: “duplicate” of given ratio found by VI. 11, 214: lemma on duplicate ratio as alternative to method of VI. 22 (De Morgan and others) 242-7 Duplication of cube: reduction of, by Hippocrates, to problem of finding two mean proportionals 133: wrongly supposed to be alluded to in Timaeus 32 A, B, 294-5 n.

Egyptians 112: Egyptian view of number 280 Enriques (F.) and Amaldi (U.) 30, 126 Equimultiples: “any equimultiples whatever,” ἰσάκις πολλαπλάσια καθ’ ὁποιονοῦν πολλαπλασιασμόν 120: stereotyped phrase “other, chance, equimultiples” 143-4: should include once each magnitude 145 Eratosthenes: measurement of obliquity of ecliptic (23° 51’ 20”) 111 Escribed circles of triangle 85, 86-7 Eudemus 99, 111 Eudoxus 99, 280, 295: discovered general theory of proportionals covering incommensurables 112-3: was first to prove scientifically the propositions about volumes of cone and pyramid 40 Euler 426 Eutocius: on “VI. Def. 5” and meaning of πηλικότης 116, 132, 189-90: gives locus-theorem from Apollonius’ Plane Loci 198-200 Even (number): definitions by Pythagoreans and in Nicomachus 281: definitions of odd ENGLISH INDEX 433

and even by one another unscientific (Aristotle) 281: Nicom. divides even into three classes (1) even-times even and (2) even-times odd as extremes, and (3) odd-times even as intermediate 282-3 Even-times even: Euclid’s use differs from use by Nicomachus, Theon of Smyrna and Iamblichus 281-2 Even-times odd in Euclid different from even-odd of Nicomachus and the rest 282-4 Ex aequali, of ratios, 136: ex aequali propositions (V. 20, 22), and ex aequali “in perturbed proportion” (V. 21, 23) 176-8

Faifofer 126 Fauquembergue, E. 426 Fermat, 425, 426 Fourth proportional: assumption of existence of, in V. 18, and alternative methods for avoiding (Saccheri, De Morgan, Simson, Smith and Bryant) 170-4: Clavius made the assumption an axiom 170: sketch of proof of assumption by De Morgan 171: condition for existence of number which is fourth proportional to three numbers 409-11

Galileo Galilei: on angle of contact 42 Geometric means 357 sqq.: one mean between square numbers 294, 363, or between similar plane numbers 371-2: two means between cube numbers 294, 364-5, or between similar solid numbers 373-5 Geometrical progression 346 sqq.: summation of n terms of (IX. 35) 420-1 Gherard of Cremona 47 Gnomon (of numbers) 289 Golden section (section in extreme and mean ratio), discovered by Pythagoreans 99: theory carried further by Plato and Eudoxus 99 Greater ratio: Euclid’s criterion not the only one 130: arguments from greater to less ratios etc. unsafe unless they go back to original definitions (Simson on V. 10) 156-7: test for, cannot coexist with test for equal or less ratio 130-1 Greatest common measure: Euclid’s method of finding corresponds exactly to ours 118, 299: Nicomachus gives the same method 300 Gregory, D. 116, 143

Häbler, Th. 294 n. Hankel, H. 116, 117 Hauber, C. F. 244 Heiberg, J. L. passim Henrici and Treutlein 30 Heron of Alexandria: Eucl. III. 12 interpolated from, 28: extends III. 20, 21 to angles in segments less than semicircles 47-8: does not recognise angles equal to or greater than two right angles 47-8: proof of formula for area of triangle, $\Delta = \sqrt{s(s-a)(s-b)(s-c)}$ 87-8: 5, 16-17, 24, 28, 34, 36, 44, 116, 189, 302, 320, 383, 395 Hippasus 97 Hippocrates of Chios 133 Hornlike angle (κερατοειδὴς γωνία) 4, 39, 40: hornlike angle and angle of semicircle, controversies on, 39-42: Proclus on, 39-40: Democritus may have written on hornlike angle 40: Campanus (“not angles in same sense”) 41: Cardano (quantities of different orders or kinds): Peletier (hornlike angle no angle, no quantity, nothing; angles of all semicircles right angles and equal) 41: Clavius 42: Vieta and Galileo (“angle of contact no angle”) 42: Wallis (angle of contact not inclination at all but degree of curvature) 42 Hultsch, F. 133, 190

Iamblichus 97, 116, 279, 280, 281, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 419, 425, 426 Icosahedron 98 Incommensurables: method of testing incommensurability (process of finding G.C.M.) 118: means of expression consist in power of approximation without limit (De Morgan) 119: approximations to $\sqrt{2}$ (by means of side- and diagonal-numbers) 119, to $\sqrt{3}$ and to $\pi$, 119: to $\sqrt{4500}$ by means of sexagesimal fractions 119 Incomposite (of number) = prime 284 Ingrami, G. 30, 126 Inverse (ratio), inversely (ἀνάπαλιν) 134: inversion is subject of V. 4, Por. (Theon) 144, and of V. 7, Por. 149, but is not properly put in either place 149: Simson’s Prop. B on, directly deducible from V. Def. 5, 144 Isosceles triangle of IV. 10: construction of, by Pythagoreans 97-9

Jacobi, C. F. A. 188

Lachlan, R. 226, 227, 245-6, 247, 256, 272 Lardner, D. 58, 259, 271 Least common multiple 336-41 Legendre 30: proves VI. 1 and similar propositions in two parts (1) for commensurables, (2) for incommensurables 193-4 Lemma assumed in VI. 22, 242-3: alternative propositions on duplicate ratios and ratios of which they are duplicate (De Morgan and others) 242-7 Length, μῆκος (of numbers in one dimension) 287: Plato restricts term to side of integral square number 287 Leotaud, Vincent 42 Linear (of numbers) = (1) in one dimension 287, (2) prime 285 Logical inferences, not made by Euclid 22, 29 Loria, G. 425 Lucas, E. 426 Lucian 99

Means: three kinds, arithmetic, geometric and harmonic 292-3: geometric mean is 434 ENGLISH INDEX

“proportion par excellence” (κυρίως) 292-3: one geometric mean between two square numbers, two between two cube numbers (Plato) 294, 363-5: one geometric mean between similar plane numbers, two between similar solid numbers 371-5: no numerical geometric mean between n and n + 1 (Archytas and Euclid) 295 Moderatus, a Pythagorean 280 Multiplication, definition of 287

an-Nairīzī 5, 16, 28, 34, 36, 44, 47, 302, 320, 383 Naṣīraddin aṭ-Ṭūsī 28 Nesselmann, G. H. F. 287, 293 Nicomachus 116, 119, 131, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 300, 363, 425 Nixon, R. C. J. 16 Number: defined by Thales, Eudoxus, Moderatus, Aristotle, Euclid 280: Nicomachus and Iamblichus on, 280: represented by lines 288, and by points or dots 288-9

Oblong (of number): in Plato either προμήκης or ἑτερομήκης 288: but these terms denote two distinct divisions of plane numbers in Nicomachus, Theon of Smyrna and Iamblichus 289-90 Octahedron 98 Odd (number): defs. of in Nicomachus 281: Pythagorean definition 281: def. of odd and even by one another unscientific (Aristotle) 281: Nicom. and Iambl. distinguish three classes of odd numbers (1) prime and incomposite, (2) secondary and composite, as extremes, (3) secondary and composite in itself but prime and incomposite to one another, which is intermediate 287 Odd-times even (number): definition in Eucl. spurious 283-4, and differs from definitions by Nicomachus etc. ibid. Odd-times odd (number): defined in Eucl. but not in Nicom. and Iambl. 284: Theon of Smyrna applies term to prime numbers 284 Oenopides of Chios 111 “Ordered” proportion (τεταγμένη ἀναλογία), interpolated definition of, 137

Pappus: lemma on Apollonius’ Plane νεύσεις 64-5: problem from same work 81: assumes case of VI. 3 where external angle bisected (Simson’s VI. Prop. A) 197: theorem from Apollonius’ Plane Loci 198: theorem that ratio compounded of ratios of sides is equal to ratio of rectangles contained by sides 250: 4, 27, 29, 67, 79, 81, 113, 133, 211, 250, 251, 292 “Parallelepipedal” (solid) numbers: two of the three factors differ by unity (Nicomachus) 290 Peletarius (Peletier): on angle of contact and angle of semicircle 41: 47, 56, 84, 146, 190 Pentagon: decomposition of regular pentagon into 30 elementary triangles 98: relation to pentagram 99 Pentagonal numbers 289 “Perfect” (of a class of numbers) 293-4, 421-6: Pythagoreans applied term to 10, 294: 3 also called “perfect” 294 Perturbed proportion (τεταραγμένη ἀναλογία) 136, 176-7 Pfleiderer, C. F. 2 Philoponus 234, 282 Plane numbers, product of two factors (“sides” or “length” and “breadth”) 287-8: in Plato either square or oblong 287-8: similar plane numbers 293: one mean proportional between similar plane numbers 371-2 Plato: construction of regular solids from triangles 97-8: on golden section 99: 7/5 as approximation to $\sqrt{2}$, 119: on square and oblong numbers 288, 293: on δυνάμεις (square roots or surds) 288, 290: theorem that between square numbers one mean suffices, between cube numbers two means necessary 294, 364 Playfair, John 2 Plutarch 98, 254 Porism (corollary) to proposition precedes “Q.E.D.” or “Q.E.F.” 8, 64: Porism to IV. 15 mentioned by Proclus 109: Porism to VI. 19, 234 Polygonal numbers 289 Powers, R. E. 426 Prestet, Jean 426 Prime (number): definitions of, 284-5: Aristotle on two senses of “prime” 285: 2 admitted as prime by Eucl. and Aristotle, but excluded by Nicomachus, Theon of Smyrna and Iamblichus, who make prime a subdivision of odd 284-5: “prime and incomposite (ἀσύνθετος)” 284: different names for prime, “odd-times odd” (Theon), “linear” (Theon), “rectilinear” (Thymaridas), “euthymetric” (Iamblichus) 285: prime absolutely or in themselves as distinct from prime to one another (Theon) 285: definitions of “prime to one another” 285-6 Proclus: on absence of formal divisions of proposition in certain cases, e.g. IV. 10, 100: on use of “quindecagon” for astronomy 111: 4, 39, 40, 193, 247, 269 Proportion: complete theory applicable to incommensurables as well as commensurables is due to Eudoxus 112: old (Pythagorean) theory practically represented by arithmetical theory of Eucl. VII. 113. in giving older theory as well Euclid simply followed tradition 113: Aristotle on general proof (new in his time) of theorem (alternando) in proportion 113: X. 5 as connecting two theories 113: De Morgan on extension of meaning of ratio ENGLISH INDEX 435

to cover incommensurables 118: power of expressing incommensurable ratio is power of approximation without limit 119: interpolated definitions of proportion as “sameness” or “similarity of ratios” 119: definition in V. Def. 5 substituted for that of VII. Def. 20 because latter found inadequate, not vice versa 121: De Morgan’s defence of V. Def. 5 as necessary and sufficient 122-4: V. Def. 5 corresponds to Weierstrass’ conception of number in general and to Dedekind’s theory of irrationals 124-6: alternatives for V. Def. 5 by a geometer-friend of Saccheri, by Faifofer, Ingrami, Veronese, Enriques and Amaldi 126: proportionals of VII. Def. 20 (numbers) a particular case of those of V. Def. 5 (Simson’s Props. C, D and notes) 126-9: proportion in three terms (Aristotle makes it four) the “least” 131: “continuous” proportion (συνεχὴς or συνημμένη ἀναλογία, in Euclid ἑξῆς ἀνάλογον) 131, 293: three “proportions” 292, but proportion par excellence or primary is continuous or geometric 292-3: “discrete” or “disjoined” (διῃρημένη, διεζευγμένη) 131, 293: “ordered” proportion (τεταγμένη), interpolated definition of, 137: “perturbed” proportion (τεταραγμένη) 136, 176-7: extensive use of proportions in Greek geometry 187: proportions enable any quadratic equation with real roots to be solved 187: supposed use of propositions of Book V. in arithmetical Books 314, 320 Psellus 234 Ptolemy, Claudius: lemma about quadrilateral in circle (Simson’s VI. Prop. D) 225-7: 111, 117, 119 Pyramidal numbers 290: pyramids truncated, twice-truncated etc. 291 Pythagoras: reputed discoverer of construction of five regular solids 97: introduced “the most perfect proportion in four terms and specially called ‘harmonic’” into Greece 112: construction of figure equal to one and similar to another rectilinear figure 254 Pythagoreans: construction of dodecahedron in sphere 97: construction of isosceles triangle of IV. 10 and of regular pentagon due to, 97-8: possible method of discovery of latter 97-9: theorem about only three regular polygons filling space round a point 98: distinguished three sorts of means, arithmetic, geometric, harmonic 112: had theory of proportion applicable to commensurables only 112: 7/5 as approximation to $\sqrt{2}$, 119: definitions of unit 279: of even and odd 281: called 10 “perfect” 294

Quadratic equations: solution by means of proportions 187, 263-5, 266-7: διορισμός or condition of possibility of solving equation of Eucl. VI. 28, 259: one solution only given, for obvious reasons 260, 264, 267: but method gives both roots if real 258: exact correspondence of geometrical to algebraical solution 263-4, 266-7 Quadrilateral: inscribing in circle of quadrilateral equiangular to another 91-2: condition for inscribing circle in, 93, 95: quadrilateral in circle, Ptolemy’s lemma on (Simson’s VI. Prop. D), 225-7: quadrilateral not a “polygon” 239 “Quindecagon” (fifteen-angled figure): useful for astronomy 111

Radius: no Greek word for, 2 Ramus, P. 121 Ratio: definition of, 116-9, no sufficient ground for regarding it as spurious 117, Barrow’s defence of it 117: method of transition from arithmetical to more general sense covering incommensurables 118: means of expressing ratio of incommensurables is by approximation to any degree of accuracy 119: def. of greater ratio only one criterion (there are others) 130: tests for greater equal and less ratios mutually exclusive 130-1: test for greater ratio easier to apply than that for equal ratio 129-30: arguments about greater and less ratios unsafe unless they go back to original definitions (Simson on V. 10) 156-7: compound ratio 132-3, 189-90, 234: operation of compounding ratios 234: “ratio compounded of their sides” (careless expression) 248: duplicate, triplicate etc. ratio as distinct from double, triple etc. 133: alternate ratio, alternando 134: inverse ratio, inversely 134: composition of ratio, componendo, different from compounding ratios 134-5: separation of ratio, separando (commonly dividendo) 135: conversion of ratio, convertendo 135: ratio ex aequali 136, ex aequali in perturbed proportion 136: division of ratios used in Data as general method alternative to compounding 249-50: names for particular arithmetical ratios 292 Reciprocal or reciprocally related figures: definition spurious 189 Reductio ad absurdum, the only possible method of proving III. 1, 8 “Rule of three”: VI. 12 equivalent to, 215

Saccheri, Gerolamo 126, 130: proof of existence of fourth proportional by VI. 1, 2, 12, 170 Savile, H. 190 Scalene, a class of solid numbers 290 Scholia: IV. No. 2 ascribes Book IV. to Pythagoreans 97: V. No. 1 attributes Book V. to Eudoxus 112 Scholiast to Clouds of Aristophanes 99 Sectio canonis attributed to Euclid 295 Sector (of circle): explanation of name: two kinds (1) with vertex at centre, (2) with vertex at circumference 5 436 ENGLISH INDEX

Sector-like (figure) 5: bisection of such a figure by straight line 5 Seelhoff, P. 426 Segment of circle: angle of, 4: similar segments 5 Semicircle: angle of, 4, 39-41 (see Angle): angle in semicircle a right angle, pre-Euclidean proof 63 Separation of ratio, διαίρεσις λόγου, and separando (διελόντι) 135: separando and componendo used relatively to one another, not to original ratio 168, 170 Servais, C. 426 Sides of plane and solid numbers 287-8 Similar plane and solid numbers 293: one mean between two similar plane numbers 371-2, two means between two similar solid numbers 294, 373-5 Similar rectilinear figures: def. of, given in Aristotle 188: def. gives at once too little and too much 188: similar figures on straight lines which are proportional are themselves proportional and conversely (VI. 22), alternatives for proposition 242-7 Similar segments of circles 5 Simon, Max 124, 134 Simpson, Thomas 121 Simson, R.: Props. C, D (Book V.) connecting proportionals of VII. Def. 20 as particular case with those of V. Def. 5, 126-9: Axioms to Book V. 137: Prop. B (inversion) 144: Prop. E (convertendo) 175: shortens V. 8 by compressing two cases into one 152-3: important note showing flaw in V. 10 and giving alternative 156-7: Book VI. Prop. A extending VI. 3 to case where external angle bisected 197: Props. B, C, D 222-7: remarks on VI. 27-9, 258-9: 2, 3, 8, 22, 23, 33, 34, 37, 43, 49, 53, 70, 73, 79, 90, 117, 131, 132, 140, 143-4, 145, 146, 148, 154, 161, 162, 163, 165, 170-2, 177, 179, 180, 182, 183, 184, 185, 186, 189, 193, 195, 209, 211, 212, 230-1, 238, 252, 269, 270, 272-3 Size, proper translation of πηλικότης in V. Def. 3, 116-7, 189-90 Smith and Bryant, alternative proofs of V. 16, 17, 18 by means of VI. 1, where magnitudes are straight lines or rectilinear areas 165-6, 169, 173-4 Solid numbers, three varieties according to relative length of sides 290-1 Spherical number, a particular kind of cube number 291 Square number, product of equal numbers 289, 291: one mean between square numbers 294, 363-4 Stobaeus 280

Subduplicate of any ratio found by VI. 13, 216 Swinden, J. H. van 188 Sylvester, J. 426

Tacquet, A. 121, 258 Tannery, P. 112, 113 Tartaglia, Niccolò 2, 47 Taylor, H. M. 16, 22, 29, 56, 75, 102, 227, 244, 247, 272 Tetrahedron 98 Thales 111, 280 Theodosius 37 Theon of Alexandria: interpolation in V. 13 and Porism 144: interpolated Porism to VI. 20, 239: additions to VI. 33 (about sectors) 274-6: 43, 109, 117, 119, 149, 152, 161, 186, 190, 234, 235, 240, 242, 256, 262, 311, 322, 412 Theon of Smyrna: 111, 119, 279, 280, 281, 284, 285, 286, 288, 289, 290, 291, 292, 293, 294, 425 Thrasyllus 292 Thymaridas 279, 285 Timaeus of Plato 97-8, 294-5, 363 Todhunter, I., 3, 7, 22, 49, 51, 52, 67, 73, 90, 99, 172, 195, 202, 204, 208, 259, 271, 272, 300 Trapezium: name applied to truncated pyramidal numbers (Theon of Smyrna) 291 Triangle: Heron’s proof of expression for area in terms of sides, $\sqrt{s(s-a)(s-b)(s-c)}$ 87-8: right-angled triangle which is half of equilateral triangle used for construction of tetrahedron, octahedron and icosahedron (Timaeus of Plato) 98 Triangular numbers 289 Triplicate, distinct from triple, ratio 133 aṭ-Ṭūsī, see Naṣīraddin

Unit: definitions of, by Thymaridas, “some Pythagoreans,” Chrysippus, Aristotle and others 279: Euclid’s definition was that of the “more recent” writers 279: μονάς connected etymologically by Theon of Smyrna and Nicomachus with μόνος (solitary) or μονή (rest) 279

Veronese, G. 30, 126 Vieta: on angle of contact 42

Walker 204, 208, 259 Wallis, John: on angle of contact (“degree of curvature”) 42 Weierstrass 124 Woepcke 5

Zenodorus 276