INDEX OF GREEK WORDS AND FORMS.

ἄκρος, extreme (of numbers in a series) 328, 367: ἄκρον καὶ μέσον λόγον τετμῆσθαι, “to be cut in extreme and mean ratio” 189

ἄλογος, irrational 117–8

ἀναλογία, proportion: definitions of, interpolated 119

ἀνάλογον = ἀνὰ λόγον, proportional or in proportion: used as indeclinable adj. and as adv. 129, 165: μέση ἀνάλογον, mean proportional (of straight line) 129, similarly μέσος ἀνάλογον of numbers 295, 363 etc.: τρίτη (τρίτος) ἀνάλογον, third proportional 214, 407–8: τετάρτη (τέταρτος) ἀνάλογον, fourth proportional 215, 409: ἑξῆς ἀνάλογον in continued proportion 346

ἀνάπαλιν (λόγος), inverse (ratio), inversely 134

ἀναστρέψαντι, convertendo 135

ἀναστροφὴ λόγου, conversion of a ratio 135

ἀνισάκις ἀνισάκις ἴσος, unequal by unequal by equal (of solid numbers) = scalene, σφηνίσκος, σφηκίσκος or βωμίσκος 290

ἀνομοίως τεταγμένων τῶν λόγων (of perturbed proportion) in Archimedes 136

ἀνταναίρεσις, ἡ αὐτὴ, definition of same ratio in Aristotle (ἀνθυφαίρεσις Alexander) 120: terms explained 121

ἀντιπεπονθότα σχήματα, reciprocal (=reciprocally related) figures, interpolated def. of, 189

ἀπλατής, breadthless (of prime numbers) 285

ἀποκαταστατικός, recurrent (=*spherical*), of numbers 291

ἅπτεσθαι, to meet, occasionally to touch (instead of ἐφάπτεσθαι) 2: also = to pass through, to lie on 79

ἀριθμός, number, definitions of, 280

ἀρτιάκις ἀρτιοδύναμον (Nicomachus) 282

ἀρτιάκις ἄρτιος, even-times even 281–2

ἀρτιάκις περισσός, even-times odd 282–4

ἀρτιοπέριττος, even-odd (Nicomachus etc.) 282

ἄρτιος (ἀριθμός), even (number) 281

ἀσύνθετος, (prime and) incomposite (of numbers) 284

βεβηκέναι, to stand (of angle standing on circumference) 4

βωμίσκος, altar-shaped (of “scalene” solid numbers) 290

γεγονέτω (in constructions), “let it be made” 248

γεγονὸς ἂν εἴη τὸ ἐπιταχθέν, “what was enjoined will have been done” 80, 261

γενόμενος, ὁ ἐξ αὐτῶν, “their product” 316, 326 etc.: ὁ ἐκ τοῦ ἑνὸς γενόμενος = “the square of the one” 327

γνώμων, gnomon: Democritus περὶ διαφορῆς γνώμονος (γνώμης or γωνίης?) ἢ περὶ ψαύσιος κύκλου καὶ σφαίρης 40: (of numbers) 289

γραμμικός, linear (of numbers in one dimension) 287: (of prime numbers) 285

γράφεσθαι, “to be proved” (Aristotle) 120

δεύτερος, secondary (of numbers): in Nicomachus and Iamblichus a subdivision of odd 286, 287

δεχόμενον, “admitting” (of segment of circle admitting or containing an angle) 5

διαιρεῖσθαι (used of “separation” of ratios): διαιρεθέντα, separando, opp. to συγκείμενα, componendo 168

διαίρεσις λόγου, separation, literally division, of ratio 135

διεζευγμένη (ἀναλογία), disjoined, = discrete (proportion) 293

διελόντι, separando, literally dividendo (of proportions) 135

διῃρημένη (ἀναλογία), discrete (proportion), i.e. in four terms, as distinct from continuous (συνεχής, συνημμένη) in three terms 131, 293

διήχθω (διάγειν), “let it be drawn through” or “across” 7

διʼ ἴσου, ex aequali (of ratios) 136: διʼ ἴσου ἐν τεταραγμένῃ ἀναλογίᾳ, “ex aequali in perturbed proportion” 136

δικόλουρος, twice-truncated (of pyramidal numbers) 291

διπλάσιος λόγος, double ratio: διπλασίων λόγος, duplicate ratio, contrasted with, 133

δύναμις, power: = actual value of a submultiple in units (Nicomachus) 282: = side of number not a complete square (i.e. root or surd) in Plato 288, 290: = square in Plato 294–5

εἶδος, figure 234: = form 254

ἕκαστος, each: curious use of, 79

ἔλλειμμα, defect (in application of areas) 262

ἐλλείπειν, “fall short” (in application of areas) 262

428 INDEX OF GREEK WORDS AND FORMS

ἐμπίπτειν, fall in (=be interpolated) 358

ἕνα πλείω, “several ones” (def. of number) 280

ἐναλλάξ (λόγος), alternate (ratio): alternately, alternando 134

ἐναρμόζειν, to fit in (active), IV. Def. 7 and IV. 1, 79, 80, 81

ἐντός, within (of internal contact of circles) 13

ἑξῆς ἀνάλογον, in continued proportion (of terms in geometrical progression) 346

ἐπιμόριος λόγος, superparticularis ratio, = the ratio $(n + 1) : n$, 295

ἐπίπεδος (ἀριθμός), plane (number) 287–8

ἑπόμενα, consequents (=“following” terms) in a proportion 134, 238

ἑτερομήκης, oblong (of numbers): in Plato = προμήκης, which however is distinguished from ἑτερομήκης by Nicomachus etc. 289–90, 293

εὐθυγραμμικός, rectilinear (term for prime numbers) 285

εὐθυμετρικός, euthymetric (of primes) 285

ἡγούμενα, antecedents (“leading” terms) in a proportion 134

ἤπερ, than: construction after διπλασίων etc. 133

ἰδιομήκης, of square number (Iamblichus) 293

ἰσάκις ἰσάκις ἴσος, equal multiplied by equal and again by equal (of a cube number) 290, 291

ἰσάκις ἴσος, equal multiplied by equal (of a square number) 291

ἰσάκις ἴσος ἐλαττονάκις (μειζονάκις), species of solid numbers, = πλινθίς (δοκίς or στηλίς) 291

καλείσθω, “let it be called,” indicating originality of a definition 129

καταμετρεῖν, measure 115: without remainder, completely (πληρούντως) 280

κατασκευάζω, construct: τῶν αὐτῶν κατασκευασθέντων, “with the same construction” 11

κατατομὴ κανόνος, Sectio canonis of Euclid 295

κέντρον, centre: ἡ ἐκ τοῦ κ. = radius 2

κερατοειδὴς γωνία, hornlike angle 4, 39, 40

κλᾶν, to break off, inflect: κεκλάσθω δὴ πάλιν 47: κεκλάσθαι, def. of, alluded to by Aristotle 47

κόλουρος, truncated (of pyramidal number minus vertex) 291

κυκλικός, cyclic, a particular species of square number 291

λόγος, ratio: meaning 117: definition of, 116–9: original meaning (of something expressed) accounts for use of ἄλογος, having no ratio, irrational 117

μεμονῶσθαι, to be isolated, of μονάς (Theon of Smyrna) 279

μέρος, part: two meanings 115: generally = submultiple 280: μέρη, parts (= proper fraction) 115, 280

μέση ἀνάλογον (εὐθεῖα), μέσος ἀνάλογον (ἀριθμός), mean proportional (straight line or number) 129, 295, 363 etc.

μὴ γάρ, “suppose it is not” 7

μῆκος, length (of number in one dimension): = side of complete square in Plato 288

μονάς, unit, monad: supposed etymological connexion with μόνος, solitary, μονή, rest 279

ὅμοιος, similar: (of rectilineal figures) 188: (of plane and solid numbers) 293

ὁμοιότης λόγων, “similarity of ratios” (interpolated def. of proportion) 119

ὁμόλογος, homologous, corresponding 134: exceptionally “in the same ratio with” 238

ὅρος, term, in a proportion 131

παραβάλλειν ἀπό, used, exceptionally, instead of παραβάλλειν παρὰ or ἀναγράφειν ἀπό 262

παραλλάττω, “fall sideways” or “awry” 54

πεντάγραμμον 99

περαίνουσα ποσότης, “limiting quantity” (Thymaridas’ definition of unit) 279

περισσάκις ἄρτιος, odd-times even 282–4

περισσάκις περισσός, odd-times odd 284

περισσάρτιος, odd-even (Nicomachus etc.) 283

περισσός, odd 281

πηλίκος, how great: refers to continuous (geometrical) magnitude as ποσός to discrete (multitude) 116–7

πηλικότης, used in v. Def. 3, and spurious Def. 5 of VI.: = size (not quantuplicity as it is translated by De Morgan) 116–7, 189–90: supposed multiplication of πηλικότητες (VI. Def. 5) 132: distinction between πηλικότης and μέγεθος 117

πλάτος, breadth: (of numbers) 288

πλευρά, side: (of factors of “plane” and “solid” numbers) 288

πλῆθος ὡρισμένον or πεπερασμένον, defined or finite multitude (definition of number) 280: ἐκ μονάδων συγκείμενον πλῆθος (Euclid’s def.) 280

πολλαπλασιάζειν, multiply: defined 287

πολλαπλασιασμός, multiplication: καθʼ ὁποιονοῦν πολλαπλασιασμόν “(arising) from any multiplication whatever” 120

πολλαπλάσιος, multiple: ἰσάκις πολλαπλάσια, equimultiples 120 etc.

πολύπλευρον, multilateral: excludes τετράπλευρον, quadrilateral 239

πορίσασθαι, to find 248

ποσάκις ποσάκις ποσοί, “so many times so many times so many” (of solid numbers, in Aristotle) 286, 290

ποσάκις ποσοί, “so many times so many” (of plane numbers, in Aristotle) 286

ποσόν, quantity, in Aristotle 115: refers to multitude as πηλίκον to magnitude 116–7

INDEX OF GREEK WORDS AND FORMS 429

προμήκης, oblong (of numbers): in Plato = ἑτερομήκης, but distinguished by Nicomachus etc. 289–90, 293

προσαναγράψαι, to draw on to: (of a circle) to complete, when segment is given 56

προσευρεῖν, to find in addition (of finding third and fourth proportionals) 214

πρῶτος, prime 284–5

πρῶτοι πρὸς ἀλλήλους, (numbers) prime to one another 285–6

ῥητός, rational (literally “expressible”) 117

συνεχής, continuous: συνεχὴς ἀναλογία, “continuous proportion” (in three terms) 131

συνημμένη ἀναλογία, connected (i.e. continuous) proportion 131, 293: συνημμένος, of compound ratio in Archimedes 133

συνθέντι, componendo 134–5

σύνθεσις λόγου, composition of a ratio, distinct from compounding of ratios 134–5

σύνθετος, composite (of numbers): in Nicomachus and Iamblichus a subdivision of odd 286

συνίστασθαι, construct: οὐ συσταθήσεται 53

συντίθημι, σύγκειμαι (of ratios) 135, 189–90: συγκείμενα and διαιρεθέντα (componendo and separando) used relatively to one another 168, 170

σύστημα μονάδων, “collection of units” (def. of number) 280

συστηματικός, collective 279

σφαιρικός, spherical (of a particular species of cube number) 291

σφηνίσκος, or σφηκίσκος, of solid number with all three sides unequal (= “scalene”) 290

σχέσις, “relation”: ποιὰ σχέσις, “a sort of relation” (in def. of ratio) 116–7

ταυτομήκης, of square number (Nicom.) 293

ταυτότης λόγων, “sameness of ratios” 119

τέλειος, perfect (of a class of numbers) 293–4

τεταγμένη (ἀναλογία), “ordered (proportion)” 137

τεταραγμένη ἀναλογία, perturbed proportion 136

τετράπλευρον, quadrilateral, not a “polygon” 239

τμῆμα (κύκλου), segment (of circle): τμήματος γωνία, angle of a segment 4: ἐν τμήματι γωνία, angle in a segment 4

τομεὺς (κύκλου), sector (of circle): σκυτοτομικὸς τομεύς, “shoemaker’s knife” 5

τομοειδής (of figure), sector-like 5

τοσαυταπλάσιον, “the same multiple” 146

τρίγωνον: τὸ τριπλοῦν, τὸ διʼ ἀλλήλων, triple, interwoven triangle, = pentagram 99

τριπλάσιος, triple, τριπλασίων, triplicate (of ratios) 133

τυγχάνειν, happen: ἄλλα, ἃ ἔτυχεν, ἰσάκις πολλαπλάσια, “other, chance, equimultiples” 143–4: τυχοῦσα γωνία, “any angle” 212

ὑπερτελής or ὑπερτέλειος, “over-perfect” (of a class of numbers) 293–4

ὑποδιπλάσιος, sub-duplicate, = half (Nicomachus) 280

ὑποπολλαπλάσιος, submultiple (Nicomachus) 280

ὕψος, height 189

χωρίον, area 254 [This page is intentionally blank]