al-'Abbas b. Sa'id al-Jauhari 85
"Abthiniathus" (or "Anthisathus") 203
Abū 'l 'Abbas al-Faḍl b. Hatim, see an-
Nairizi

Abū 'Abdallah Muḥ. b. Mu'adh al-Jayyāni 90
Abu 'Ali al-Başri 88

Abū 'Ali al-Hasan b. al-Hasan b. al-Haitham
88, 89

Abū Da'ūd Sulaiman b. 'Uqba 85, 90
Abu Ja'far al-Khazin 77, 85

Abu Ja'far Muḥ. b. Muḥ. b. al-Hasan
Naşiraddin at-Tusi, see Nașiraddin

Abū Muḥ. b. Abdalbaqi al-Bagdadi al-Faraḍi

8 n., 90

Abu Muḥ. al-Hasan b. 'Ubaidallāh b. Sulai-
man b. Wahb 87

Abū Nașr Gars al-Na'ma 90

Abū Nasr Mansur b. 'Ali b. 'Iraq 90
Abū Nasr Muḥ. b. Muḥ. b. Tarkhan b.
Uzlag al-Farabi 88

Abu Sahl Wijan b. Rustam al-Kūhi 88
Abu Sa'id Sinan b. Thābit b. Qurra 88
Abū 'Uthman ad-Dimashqi 25, 77
Abū 'I Wafa al-Būzjāni 77, 85, 86
Abu Yusuf Yaqub b. Ishaq b. aṣ-Ṣabbāḥ al-
Kindi 86

Abu Yusuf Yaqub b. Muḥ. ar-Rāzi 86
Adjacent (èpens), meaning 181

Aenaeas (or Aigeias) of Hierapolis 28, 311
Aganis 27-8, 191

Aḥmad b. al-Husain al-Ahwazi al-Kātib 89
Aḥmad b. 'Umar al-Karābisi 85
al-Ahwazi 89

Aigeias (?Aenaeas) of Hierapolis 28, 311
Alexander Aphrodisiensis 7., 29
Algebra, geometrical, 372-4: classical method
was that of Eucl. 11. (cf. Apollonius) 373:
preferable to semi-algebraical method 377-
8: semi-algebraical method due to Heron
373, and favoured by Pappus 373: geome-
trical equivalents of algebraical operations
374: algebraical equivalents of propositions
in Book II. 372-3

'Ali b. Aḥmad Abū 'l Qāsim al-Anṭāki 86
Allman, G. J. 135 m., 318, 352
Alternate (angles) 308

Alternative proofs, interpolated, 58, 59
Amaldi 175, 179-80, 193, 201, 313, 328
Ambiguous case 306-7

Amphinomus 125, 128, 150 n.
Amyclas of Heraclea 117

Analysis (and synthesis) 18: alternative
proofs of XIII. 1-5 by, 137: definitions of,

interpolated, 138: described by Pappus
138-9: modern studies of Greek analysis
139: theoretical and problematical analysis
138: Treasury of analysis (TóπOS ȧvaλvó-
μevos) 8, 10, 11, 138: method of analysis
and precautions necessary to 139-40:
analysis and synthesis of problems 140-2:
two parts of analysis (a) transformation,
(b) resolution, and two parts of synthesis,
(a) construction, (b) demonstration 141:
example from Pappus 141-2: analysis
should also reveal dioptouós (conditions of
possibility) 142

Analytical method 36: supposed discovery
of, by Plato 134, 137
Anaximander 370
Anchor-ring 163
Andron 126

Angle. Curvilineal and rectilineal, Euclid's
definition of, 176 sq.: definition criticised
by Syrianus 176: Aristotle's notion of
angle as kλáois 176: Apollonius' view of,
as contraction 176, 177: Plutarch and
Carpus on, 177: to which category does it
belong? quantum, Plutarch, Carpus, "A-
ganis" 177, Euclid 178; quale, Aristotle
and Eudemus 177-8: relation, Euclid 178:
Syrianus' compromise 178: treatise on the
Angle by Eudemus 34, 38, 177-8: classifi-
cation of angles (Geminus) 178-9: curvi-
lineal and "mixed ""
angles 26, 178-9,
horn-like (Keparоeidńs) 177, 178, 182, 265,
lune-like (unvoeidhs) 26, 178-9, scraper-like
(EVσTPOELDŃS) 178: angle of a segment 253:
angle of a semicircle 182, 253: definitions
of angle classified 179: recent Italian views
179-81: angle as cluster of straight lines
or rays 180-1, defined by Veronese 180:
as part of a plane ("angular sector") 179-
80: flat angle (Veronese etc.) 180-1, 269:
three kinds of angles, which is prior
(Aristotle)? 181-2: adjacent angles 181:
alternate 308: similar (=equal) 178, 182,
252: vertical 278: exterior and interior
(to a figure) 263, 280: exterior when re-
entrant 263: interior and opposite 280:
construction by Apollonius of angle equal
to angle 296: angle in a semicircle, theorem
of, 317-19: trisection of angle, by conchoid
of Nicomedes 265-6, by quadratrix of
Hippias 266, by spiral of Archimedes 267
al-Antāki 86

Antiphon 7 n., 35

"Anthisathus" (or "Abthiniathus") 203
Apastamba-Sulba-Sutra 352: evidence in, as
to early discovery of Eucl. I. 47 and use
of gnomon 360-4: Bürk's claim that
Indians had discovered the irrational 363-
4: approximation to 2 and Thibaut's
explanation 361, 363-4: inaccurate values
of in, 364

Apollodorus "Logisticus" 37, 319, 351
Apollonius: disparaged by Pappus in com-
parison with Euclid 3: supposed by some
Arabians to be author of the Elements 5:
a "carpenter" 5: on elementary geometry
42: on the line 159: on the angle 176:
general definition of diameter 325: tried to
prove axioms 42, 62, 222-3: his "general
treatise" 42: constructions by, for bisec-
tion of straight line 268, for a perpendicular
270, for an angle equal to an angle 296:
on parallel-axiom (?) 42-3: adaptation to
conics of theory of application of areas
344-5: geometrical algebra in, 373: Plane'
Loci 14, 259, 330: Plane vevoeis 151: com-
parison of dodecahedron and icosahedron
6: on the cochlias 34, 42, 162: on unordered
irrationals 42, 115: 138, 188, 221, 222, 246,
259,370, 373

Application of areas 36, 343-5: contrasted
with exceeding and falling-short 343:
complete method equivalent to geometric
solution of mixed quadratic equation 344-5,
383-5, 386-8: adaptation to conics (Apol-
lonius) 344-5: application contrasted with
construction (Proclus) 343

66

'Agaton" 88

Arabian editors and commentators 75-90
Arabic numerals in scholia to Book X.,
12th C., 71

Archimedes 116, 142: “postulates" in, 120,
123: famous "lemma" (assumption) known
as Postulate of Archimedes 234: "Porisms"
in, 11., 13: spiral of, 26, 267: on straight
line 166: on plane 171−2: 225, 370
Archytas 20

Areskong, M. E. 113

Arethas, Bishop of Caesarea 48: owned

Bodleian Ms. (B) 47-8: had famous Plato
MS. of Patmos (Cod. Clarkianus) written 48
Argyrus, Isaak 74

Aristaeus 138: on conics 3: Solid Loci 16,
329: comparison of five (regular solid)
figures 6

Aristotelian Problems 166, 182, 187
Aristotle: on nature of elements 116: on
first principles 117 sqq.: on definitions 117,
119-20, 143-4, 146-50: on distinction he-
tween hypotheses and definitions 119, 120,
between hypotheses and postulates 118,
119, between hypotheses and axioms 120:
on axioms 119-21: axioms indemon-
strable 121: on definition by negation
156-7: on points 155-6, 165: on lines,
definitions of 158-9, classification of 159-
60: quotes Plato's definition of straight
line 166: on definitions of surface 170:

on the angle 176-8: on priority as between
right and acute angles 181-2: on figure
and definition of 182-3: definitions of
"squaring" 149–50, 410: on parallels 190-
2, 308-9: on gnomon 351, 355, 359: on
attributes κατά παντός and πρῶτον καθόλου
319, 320, 325: on the objection 135: on
reduction 135: on reductio ad absurdum
136: on the infinite 232-4: supposed pos-
tulate or axiom about divergent lines taken
by Proclus from, 45, 207: gives pre-Eucli-
dean proof of 1. 5 252-3: on theorem of
angle in a semicircle 149: on sum of angles
of triangle 319-21: on sum of exterior
angles of polygon 322: 38, 45, 117, 150n.,
181, 184, 185, 187, 188, 195, 202, 203,
221, 222, 223, 226, 259, 262-3, 283
al-Arjāni, Ibn Rahawaihi 86

Ashkal at-ta'sis 5 n.

Ashraf Shamsaddin as-Samarqandi, Muḥ. b.
5 n., 89

Astaroff, Ivan 113

Asymptotic (non-secant) lines 40, 161, 203
Athelhard of Bath 78, 93-6
Athenaeus of Cyzicus 117
August, E. F. 103
Austin, W. 103, 111

Autolycus, On the moving sphere 17
Avicenna 77, 89

Axioms, distinguished from postulates by
Aristotle 118-9, by Proclus (Geminus and
"others") 40, 121-3: Proclus on diffi-
culties in distinctions 123-4: distinguished
from hypotheses, by Aristotle 120-1, by
Proclus 121-2: indemonstrable 121: at-
tempt by Apollonius to prove 222-3:

common (things)" or "common
opinions" in Aristotle 120, 221: common
to all sciences 119, 120: called "common
notions" in Euclid 121, 221: which are
genuine? 221 sqq.: Proclus recognises five
222, Heron three 222: interpolated axioms
224, 232: Pappus' additions to axioms
25, 223, 224, 232: axioms of congruence,
(1) Euclid's Common Notion 4, 224-7;
(2) modern systems (Pasch, Veronese and
Hilbert) 228-31: "axiom" with Stoics=
every simple declaratory statement 41, 221
Babylonians, knowledge of triangle 3, 4, 5,
352

Bacon, Roger 94

Balbus, de mensuris 91
Barbarin 219

Barlaam, arithmetical commentary on Eucl. .
74

Barrow 103, 105, 110, 111
Base, meaning 248-9

Basel, editio princeps of Eucl. 100-1
Basilides of Tyre 5, 6
Baudhāyana Sulba-Sūtra 360
Bayfius (Baif, Lazare) 100
Becker, J. K. 174
Beez 176
Beltrami, E. 219

al-Biruni 90

Björnbo, Axel Anthon 17., 93
Boccaccio 96

Bodleian Ms. (B) 47, 48

Boeckh 351, 371

Boethius 92, 95, 184

Bologna Ms. (b) 49
Bolyai, J. 219

Bolyai, W. 174-5, 219, 328
Bolzano 167

Boncompagni 93 n., 104 n.

Bonola, R. 202, 219, 237

Borelli, Giacomo Alfonso 106, 194
Boundary (opos) 182, 183

Bråkenhjelm, P. R. 113

Breitkopf, Joh. Gottlieb Immanuel 97
Bretschneider 136 n., 137, 295, 304, 344, 354,
358

Briconnet, François 100

Briggs, Henry 102

Brit. Mus. palimpsest, 7th-8th c., 50
Bryson 8 n.

Bürk, A. 352, 360-4

Bürklen 179

Buteo (Borrel), Johannes 104

Cabasilas, Nicolaus and Theodorus 72
Caiani, Angelo 101
Camerarius, Joachim 101
Camerer, J. G. 103, 293
Camorano, Rodrigo 112

Čampanus, Johannes 3, 78, 94-96, 104, 106,

Chinese, knowledge of triangle 3, 4, 5, 352:
"Tcheou pei 355

Chrysippus 330

Cicero 91, 351

Circle: definition of, 183-5: =round, σтpoy.
yúlov (Plato) 184: = περιφερόγραμμον
(Aristotle) 184: a plane figure 183-4:
centre of, 184-5: pole of, 185: bisected by
diameter (Thales) 185, (Saccheri) 185-6:
intersections with straight line 237-8,
272-4, with another circle 238-40, 242-3,
293-4

Circumference, περιφέρεια 184
Cissoid 161, 164, 176, 330
Clairaut 328

Claymundus, Joan. 101

Clavius (Christoph Schlüssel) 103, 105, 194,
232, 381, 391, 407

Cleonides, Introduction to Harmony 17
Cochlias or cochlion (cylindrical helix) 162
Codex Leidensis 399, 1: 22, 27 n., 79 n.
Coets, Hendrik 109

Commandinus 4, 102, 103, 104–5, 106, 110,

III, 407: scholia included in translation
of Elements 73: edited (with Dee) De
divisionibus 8, 9, 110

Commentators on Eucl. criticised by Proclus
19, 26, 45

Common Notions: = axioms 62, 120-1, 221-2:
which are genuine? 221 sq.: meaning and
appropriation of term 221: called "axioms"
by Proclus 221

Complement, aparλńpwμa: meaning of, 341:
"about diameter" 341: not necessarily
parallelograms 341: use for application of
areas 342-3

Composite, ovveros, (of lines) 160, (of sur-
faces) 170

Conchoids 160-1, 265-6, 330
Conclusion, ovμrépaσua: necessary part of a
proposition 129-30: particular conclusion
immediately made general 131: definition
merely stating conclusion 149
Congruence-Axioms or Postulates: Common
Notion 4 in Euclid 224-5: modern systems
of (Pasch, Veronese, Hilbert), 228-31
Congruence theorems for triangles, recapitula
tion of, 305-6

Conics, of Euclid 3, 16: of Aristaeus 3, 16:
of Apollonius 3, 16: fundamental property
as proved by Apollonius equivalent to
Cartesian equation 344-5: focus-directrix
property proved by Pappus 15
Constantinus Lascaris 3
Construct (ovviorao@ai), contrasted
describe on 348, with apply to 343: special
connotation 259, 289

with

Construction, KaтaσKevý, one of formal di-
visions of a proposition 129: sometimes
unnecessary 130: turns nominal into real
- definition 146: mechanical, 151, 387
Continuity, Principle of, 234 sq., 242, 272, 294
Conversion, geometrical: distinct from logical
256: “leading" and partial varieties 256-7,
337

78, 92, 94, 96, 97 n.

Curves, classification of: see line
Cylindrical helix 161, 162, 329, 330
Czecha, Jo. 113

Dasypodius (Rauchfuss), Conrad 73, 102
Data of Euclid 8, 132, 141, 385, 391
Deahna 174

Dechales, Claude François Milliet 106, 107,
108, 110

Dedekind's Postulate, and applications 235-40
Dee, John 109, 110: discovered De divisi
onibus 8, 9

Definition, in sense of "closer statement
(dopio μós), one of formal divisions of a
proposition 129: may be unnecessary 130
Definitions: Aristotle on, 117, 119, 120, 143:
a class of thesis (Aristotle) 120: distin-
guished from hypotheses 119, but confused
therewith by Proclus 121-2: must be
assumed 117-9, but say nothing about
existence (except in the case of a few
primary things) 119, 143: terms for, öpos
and opiombs 143: real and nominal defi-
nitions (real nominal plus postulate or
proof), Mill anticipated by Aristotle, Sac-
cheri and Leibniz 143-5: Aristotle's re-
quirements in, 146-50, exceptions 148:
should state cause or middle term and be
genetic 149-50: Aristotle on unscientific
definitions (ex un «potéρwv) 148–9: Euclid's
definitions agree generally with Aristotle's
doctrine 146: interpolated definitions 61,
62: definitions of technical terms in Aris-
totle and Heron, not in Euclid 150
De levi et ponderoso, tract 18
Demetrius Cydonius 72
Democritus 38

De Morgan 246, 260, 269, 284, 291, 298, 300,
309, 313, 314, 315, 369, 376

Desargues 193

Describe on (αναγράφειν ἀπό) contrasted with
construct 348

De Zolt 328

Diophantus 86

Diorismus (dopio μós)=(a) “definition" or
"specification," a formal division of a
proposition 129: (b) condition of possibility
128, determines how far solution possible
and in how many ways 130-1, 243: dio-
rismi said to have been discovered by
Leon 116: revealed by analysis 142: in-
troduced by deî dh 293: first instances in
Elements 234, 293

Dippe 108

Direction, as primary notion, discussed 179:
direction-theory of parallels 191-2
Distance, diáornua:= radius 199: in Aristotle
has usual general sense and = dimension 199
Division (method of), Plato's 134
Divisions (of figures) by Euclid 8, 9: trans-
lated by Muhammad al-Bagdadi 8: found
(by Woepcke) in Arabic 9, and (by Dee)
in Latin translation 8, 9: 110
Dodgson, C. L. 194, 254, 261, 313
Dou, Jan Pieterszoon 108
Duhamel 139, 328

Egyptians, knowledge of right-angled triangles
352

Elements: pre-Euclidean Elements, by Hip-
pocrates of Chios, Leon 116, Theudius 117:
contributions to, by Eudoxus 1, 37, Theae-
tetus 1, 37, Hermotimus of Colophon
117: Euclid's Elements, ultimate aims of 2,
115-6: commentators on 19-45, Proclus
19, 29-45 and passim, Heron 20-24, an-
Nairizi 21-24, Porphyry 24, Pappus 24-
27, Simplicius 28, Aenaeas (Aigeias) 28:
MSS. of 46-51: Theon's changes in text
54-58: means of comparing Theonine with
ante-Theonine text 51-53: interpolations
before Theon's time 58-63: scholia 64-74:
external sources throwing light on text,
Heron, Taurus, Sextus Empiricus, Proclus,
Iamblichus 62-3: Arabic translations (1)
by al-Hajjaj 75, 76, 79, 80, 83-4, (2) by
Ishaq and Thabit b. Qurra 75-80, 83-4,
(3) Naşiraddin at-Tusi 77-80, 84: Hebrew
translation by Moses b. Tibbon or Jakob
b. Machir 76: Arabian versions compared
with Greek text 79-83, with one another
83, 84: translation by Boethius 92: old
translation of 10th c. 92: translation by
Athelhard 93-6, Gherard of Cremona 93-4,
Campanus 94-6, 97-100 etc., Zamberti
98-100, Commandinus 104-5: introduc-
tion into England, 10th c., 95: translation
by Billingsley 109-10: Greek texts, editio
princeps 100-1, Gregory's 102-3, Peyrard's
103, August's 103, Heiberg's passim: trans-
lations and editions generally 97-113: on
the nature of elements (Proclus) 114-6,
(Menaechmus) 114, (Aristotle) 116: Proclus
on advantages of Euclid's Elements 115:
immediate recognition of, 116: first princi-
ples of, definitions, postulates, common
notions (axioms) 117-24: technical terms
in connexion with, 125-42: no definitions

Flauti, Vincenzo 107

of such technical terms 150: sections of
Book 1. 308

Elinuam 95

Engel and Stäckel 219, 321

Enriques, F. 157, 175, 193, 195, 201, 313
Enunciation (póraσis), one of formal di-
visions of a proposition 129-30
Epicureans, objection to 1. 20 41, 287:
Savile on, 287

Equality, in sense different from that of
congruence (= "equivalent," Legendre)
327-8: two senses of equal (1) "divisibly-
equal" (Hilbert) or "equivalent by sum
(Amaldi), (2) “equal in content" (Hilbert)
or "equivalent by difference" (Amaldi)
328 modern definition of, 228
Eratosthenes : contemporary with Archi-
medes 1, 2 162

:

Errard, Jean, de Bar-le-Duc 108
Erycinus 27, 290, 329

:

Euclid account of, in Proclus' summary 1 ;
date 1-2: allusions to in Archimedes 1:
(according to Proclus) a Platonist 2: taught
at Alexandria 2: Pappus on personality
of, 3 story of (in Stobaeus) 3: not "of
Megara" 3, 4: supposed to have been
born at Gela 4: Arabian traditions about,
4, 5: "of Tyre" 4-6: "of Tūs" 4, 5 n.:
Arabian derivation of name ("key of
geometry") 6: Elements, ultimate aim of,
2, 115-6: other works, Conics 16, Pseu-
daria 7, Data 8, 132, 141, 385, 391, On
divisions (of figures) 8, 9, Porisms 10-15,
Surface-loci 15, 16, Phaenomena 16, 17,
Optics 17, Elements of Music or Sectio
Canonis 17: on "three- and four-line
locus "
3: Arabian list of works 17, 18:
bibliography 91-113
Eudemus 29: Ón the Angle 34, 38, 177-8:
History of Geometry 34, 35-8, 278, 295,
304, 317, 320, 387

Eudoxus 1, 37, 116: discoverer of theory

of proportion as expounded generally in
Bks. V., VI. 137, 351: on the golden
section 137: founder of method of ex-
haustion 234: inventor of a certain curve,
the hippopede, horse-fetter 163: possibly
wrote Sphaerica 17
Euler, Leonhard 401

Eutocius 25, 35, 39, 142, 161, 164, 259, 317,

Florence Ms. Laurent. XXVIII. 3, (F) 47
Flussates, see Candalla

Forcadel, Pierre 108
Fourier 173-4

Frankland, W. B. 173, 199
Frischauf 174

Gartz 9n.

Gauss 172, 193, 194, 202, 219, 321
Geminus: name not Latin 38-9: title of work
(piλoxalla) quoted from by Proclus_39:
elements of astronomy 38: comm. on Posi-
donius 39: Proclus' obligations to, 39-42:
on postulates and axioms 122-3: on theo-
rems and problems 128: two classifications
of lines (or curves) 160-2: on homoeo-
meric (uniform) lines 162: on "mixed"
lines (curves) and surfaces 162: classifica-
tion of surfaces 170, of angles 178-9:
on parallels 191: on Postulate 4, 200:
on stages of proof of theorem of 1. 32, 317-
20: 21, 27-8, 37, 44, 45, 133 m., 203,
265, 330

Geometrical algebra 372-4: Euclid's method
in Book II. evidently the classical method
373: preferable to semi-algebraical method
377-8

Gherard of Cremona, translator of Elements
93-4: of an-Nairizi's commentary 22, 94:
of tract De divisionibus 9
Giordano, Vitale 106, 176

Given, dedouévos, different senses, 132-3
Gnomon: literally "that enabling (something)
to be known" 64, 370: successive senses of,
(1) upright marker of sundial 181, 185, 271-
2, introduced into Greece by Anaximander
370, (2) carpenter's square for drawing
right angles 371, (3) figure placed round
square to make larger square 351, 371,
Indian use of gnomon in this sense 362,
(4) use extended by Euclid to parallelograms
371, (5) by Heron and Theon to any figures
371-2: Euclid's method of denoting in
figure 383: arithmetical use of, 358-60, 371
"Gnomon-wise" (kaтà yvwμova), old name
for perpendicular (xd0eтos) 36, 181, 272
Görland, A. 233, 234

"Golden section" section in extreme and
mean ratio 137: connexion with theory of
irrationals 137

"Goose's foot" (pes anseris), name for
Eucl. III. 7, 99
Gow, James 135 n.

Gracilis, Stephanus 101-2
Grandi, Guido 107

Gregory, David 102-3

Gregory of St Vincent 401, 404
Gromatici 91 n., 95
Grynaeus 100-1

al-Haitham 88, 89

al-Hajjaj b. Yusuf b. Maṭar, translator of the
Elements 22, 75, 76, 79, 80, 83, 84
Halifax, William 108, 110

27