1. 90

Abu Ali al-Başri 1. 88

Abū 'Ali al-Hasan b. al-Hasan b. al-Haitham
1. 88, 89

Abū Da'úd Sulaiman b. 'Uqba 1. 85, 90
Abu Ja'far al-Khazin 1. 77, 85
Abu Ja'far Muh. b. Muh. b. al-Hasan
Naşiraddin at-Tusi, see Naṣiraddin
Abu Muḥ. b. Abdalbāqi al-Baģdādi al-Faraḍī
I. 8 n., 90

Abu Muḥ. al-Hasan b. 'Ubaidallah b. Sulai-
man b. Wahb 1. 87

Abū Naṣr Gars al-Na'ma 1. 90

Abu Naṣr Mansur b. 'Ali b. Iraq I. 90
Abu Nasr Muh. b. Muḥ. b. Tarkhan b.
Uzlag al-Farabi 1. 88

Abu Sahl Wijan b. Rustam al-Kūhi 1. 88
Abū Sa'id Sinan b. Thābit b. Qurra 1. 88
Abū 'Uthman ad-Dimashqi 1. 25, 77
Abū 'I Wafa al-Būzjānī 1. 77, 85, 86
Abū Yusuf Ya'qūb b. Isḥāq b. aṣ-Ṣabbāḥ al-
Kindi 1. 86

Abu Yusuf Ya'qub b. Muḥ. ar-Razi 1. 86
Adjacent (épens), meaning 1. 181

Adrastus II. 292

Aenaeas (or Aigeias) of Hierapolis 1. 28, 311
Aganis 1. 27-8, 191

Aḥmad b. al-Husain al-Ahwāzī al-Kātib 1. 89
Aḥmad b. 'Umar al-Karābisi 1. 85
al-Ahwazi I. 89

Aigeias (? Aenaeas) of Hierapolis 1. 28, 311
Alcinous II. 98

Alexander Aphrodisiensis I. 7 n., 29, II. 120
Algebra, geometrical 1. 372-4: classical

method was that of Eucl. 11. (cf. Apol-
lonius) I. 373: preferable to semi-alge-
braical method 1. 377-8: semi-algebraical
method due to Heron 1. 373, and favoured
by Pappus 1. 373: geometrical equivalents
of algebraical operations 1. 374: algebraical
equivalents of propositions in Book II., I.
372-3: equivalents in Book x. of pro-
positions in algebra, √ √ cannot be

equal to k', III. 58-60: if a±√b=x±√y,
then a=x, b=y, III. 93-4, 167-8

'Ali b. Aḥmad Abū 'l Qāsim al-Antaki 1. 86
Allman, G. J. 1. 135 n., 318, 352, III. 18-
9, 439

Alternate: (of angles) 1. 308: (of ratios),
alternately II. 134

Alternative proofs, interpolated 1. 58, 59:
cf. III. 9 and following II. 22: that in
III. 10 claimed by Heron II. 23-4
Amaldi, Ugo I. 175, 179-80, 193, 201, 313,
328, 11. 30, 126

Ambiguous case 1. 306-7: in VI. 7, II. 208-9
Amphinomus I. 125, 128, 150 n.

Amyclas of Heraclea 1. 117

Analysis (and synthesis) 1. 18: definitions
of, interpolated, I. 138, III. 442: described
by Pappus I. 138-9: mystery of Greek
analysis III. 246: modern studies of Greek
analysis 1. 139: theoretical and problem-
atical analysis 1. 138: Treasury of Analy
sis (τόπος ἀναλυόμενος) 1. 8, 10, 11, 138:
method of analysis and precautions neces-
sary to, I. 139-40: analysis and synthesis
of problems 1. 140-2: two parts of analysis
(a) transformation, (b) resolution, and two
parts of synthesis, (a) construction, (b)
demonstration I. 141: example from
Pappus 1. 141-2: analysis should also
reveal dioptoubs (conditions of possibility)
I. 142: interpolated alternative proofs of
XIII. 1-5 by analysis and synthesis I. 137,
III. 442-3

Analytical method 1. 36: supposed discovery
of, by Plato 1. 134, 137
Anaximander I. 370, II. III
Anaximenes II. III

536

Euclid I. 178: Syrianus' compromise
1. 178: treatise on the Angle by Eudemus
I. 34, 38, 177-8: classification of angles
(Geminus) 1. 178-9: curvilineal and
"mixed" angles 1. 26, 178-9, horn-like
(KEрATOELDS) I. 177, 178, 182, 265, II. 4,
39, 40, lune-like (unvoeidńs) 1. 26, 178–9,
scraper-like (Evoтpoeldńs) I. 178: angle of a
segment 1. 253, II. 4: angle of a semi-
circle 1. 182, 253, II. 4: controversies about

angle of semicircle" and hornlike angle
II. 39-42: definitions of angle classified
I. 179: recent Italian views I. 179-81:
angle as cluster of straight lines or rays
1. 180-1, defined by Veronese 1. 180: as
part of a plane ("angular sector") 1. 179-
80: flat angle (Veronese etc.) 1. 180-1,
269: three kinds of angles, which is prior
(Aristotle)? 1. 181-2: angles not less than
two right angles not recognised as angles
(cf. Heron, Proclus, Zenodorus) II. 47-9:
did Euclid extend "angle" to angles
greater than two right angles in vi. 33?
11. 275-6: adjacent angles 1. 181: alternate
1. 308: similar (= equal) 1. 178, 182, 252:
vertical 1. 278: exterior and interior
(to a figure) 1. 263, 280: exterior when
re-entrant 1. 263, in which case we have a
hollow-angled figure 1. 27, 188, 11. 48:
interior and opposite 1. 280: construction
by Apollonius of angle equal to angle
1. 296: angle in a semicircle, theorem of,
I. 317-9 trisection of angle, by con-
choid of Nicomedes 1. 265-6, by quadratrix
of Hippias 1. 266, by spiral of Archimedes
1. 267: dihedral angle III. 264-5: solid
angle III. 261, 267-8

Annex (πpooаpubfovoa) = the straight line
which, when added to a compound ir-
rational straight line formed by subtraction,
makes up the greater term, "i.e. the
negative """term III. 159

al-Antāki I. 86

Antecedents (leading terms in proportion) II.
134

"Anthisathus" (or "Abthiniathus") 1. 203
Antiparallels: may be used for construction
of VI. 12, II. 215
Antiphon I. 7., 35

Apastamba-Sulba-Sutra 1. 352: evidence in,
as to early discovery of Eucl. 1. 47 and use
of gnomon 1. 360-4: Bürk's claim that
Indians had discovered the irrational 1.
363-4: approximation to /2 and Thibaut's
explanation 1. 361, 363-4: inaccurate
values of in, 1. 364
Apollodorus "Logisticus I. 37, 319, 351
Apollonius: disparaged by Pappus in com-
parison with Euclid 1. 3: supposed by
some Arabians to be author of the Ele-
ments 1. 5: a "carpenter" I. 5: on ele-
mentary geometry I. 42: on the line 1.
159: on the angle 1. 176: general defini-
tion of diameter 1. 325: tried to prove
axioms I. 42, 62, 222-3: his "general

treatise" I. 42: constructions by, for
bisection of straight line 1. 268, for a
perpendicular I. 270, for an angle equal to
an angle 1. 296: on parallel-axiom (?)
1. 42-3 adaptation to conics of theory of
application of areas 1. 344-5: geometrical
algebra in, 1.373: Plane Loci, I. 14, 259, 330,
theorem from (arising out of Eucl. vi. 3),
also found in Aristotle 11. 198-200: Plane
veúσeis 1. 151, problem from, 11. 81, lemma
by Pappus on, II. 64-5: comparison of do-
decahedron and icosahedron I. 6, III. 439,
512, 513 on the cochlias 1. 34, 42, 162:
on "unordered" irrationals 1. 42, 115, III.
3, 10, 246, 255-9: general definition of ob-
lique (circular) cone III. 270: 1. 138, 188,
221, 222, 246, 259, 370, 373, II. 75, 190,
258, 111. 264, 267

A potome: compound irrational straight line
(difference between two "terms ") 111. 7:
defined III. 158-9: connected by Theae-
tetus with harmonic mean III. 3, 4:
biquadratic from which it arises III. 7:
uniquely formed 111. 167-8: first, second,
third, fourth, fifth and sixth apotomes,
quadratics from which arising 111. 5-6,
defined III. 177, and found respectively
(x. 85-90) III. 178-90: apotome equivalent
to square root of first apotome 111. 190-4:
first, second, third, fourth, fifth and sixth
apotomes equivalent to squares of apotome,
first apotome of a medial etc. III. 212-29:
apotome cannot be binomial also III. 240-2:
different from medial (straight line) and
from other irrationals of same series with
itself III. 242: used to rationalise binomial
with proportional terms III. 243-8, 252-4,
Apotome of a medial (straight line): first and
second, and biquadratics of which they are
roots III. 7: first apotome of a medial
defined III. 159-60, uniquely formed III.
168-9, equivalent to square root of second
apotome III. 194-8: second apotome of a
medial, defined III. 161-2, uniquely formed
III. 170-2, equivalent to square root of
third apotome III. 199-202
Application of areas 1. 36, 343-5: contrasted
with exceeding and falling-short 1. 343:
complete method equivalent to geometrical
solution of mixed quadratic equation 1.
344-5, 383-5, 386-8, 11. 187, 258-60,
263-5, 266-7: adaptation to conics (Apol-
lonius) I. 344-5: application contrasted
with construction (Proclus) I. 343
Approximations: 7/5 as approximation to√2
(Pythagoreans and Plato) II. 119: approxi-
mations to 3 in Archimedes and (in
sexagesimal fractions) in Ptolemy II. 119:
to T (Archimedes) 11. 119: to
√4500
(Theon of Alexandria) 11. 119: remarkably
close approximations (stated in sexagesimal
fractions) in scholia to Book X., III. 523
Aqaton I. 88

Arabian editors and commentators I. 75-
90

a proportion between commensurables to
cover incommensurables II. 193: "Axiom"
of (called however "lemma," assumption,
by A. himself) 1. 234: relation of "Axiom"
to X. I, III. 15-6: "Axiom" already
used by Eudoxus and mentioned by
Aristotle III. 16: proved by means of
Dedekind's Postulate (Stolz) III. 16: on
discovery by Eudoxus of method of ex-
haustion III. 365-6, 374: new fragment
of," method (ěpodos) of Archimedes about
mechanical theorems," or épóôtov, dis-
covered by Heiberg and published and
annotated by him and Zeuthen II. 40, III.
366-8, adds new chapter to history of
integral calculus, which the method actually
is, III. 366-7: application to area of para-
bolic segment, ibid.: spiral of Archimedes
I. 26, 267: I. 116, 142, 225, 370, II. 136,
190, III. 246, 270, 375, 521

Archytas 1. 20: proof that there is no
numerical geometric mean between ʼn and
n+1 II. 295

Areskong, M. E. 1. 113

Arethas, Bishop of Caesarea 1. 48: owned
Bodleian MS. (B) 1. 47-8: had famous
Plato Ms. of Patmos (Cod. Clarkianus)
written I. 48

Argyrus, Isaak I. 74

Aristaeus 1. 138: on conics 1. 3: Solid Loci
1. 16, 329: comparison of five (regular
solid) figures 1. 6, 111. 438-9, 513
Aristotelian Problems 1. 166, 182, 187
Aristotle: on nature of elements 1. 116: on
first principles 1. 177 sqq.: on definitions
1. 117, 119-20, 143-4, 146-50: on distinc-
tion between hypotheses and definitions
I. 119, 120, between hypotheses and
postulates 1. 118, 119, between hypotheses
and axioms I. 120: on axioms 1. 119-21:
axioms indemonstrable 1. 121: on defini-
tion by negation 1. 156-7: on points 1.
155-6, 165: on lines, definitions of, I.
158-9, classification of, 1. 159-60: quotes
Plato's definition of straight line 1. 166:
on definitions of surface 1. 170: definition
of "body" as that which has three
dimensions or as "depth" III. 262: body
"bounded by surfaces" (émɩmédois) III.
263: speaks of six "dimensions" III. 263:
definition of sphere III. 269: on the angle
1. 176-8: on priority as between right and
acute angles 1. 181-2: on figure and
definition of, 1. 182-3: definitions of
"squaring" I. 149-50, 410: on parallels
1. 190-2, 308-9: on gnomon 1. 351, 355,
359: on attributes κατὰ παντός and πρῶτον

that angle in semicircle is right 11. 63:
on sum of angles of triangle 1. 319-21:
on sum of exterior angles of polygon 1.
322: on def. of same ratio (= same
ávтavalpeois) II. 120-1: on proportion as
"equality of ratios" 11. 119: on theorem in
proportion (alternando) not proved generally
till his time II. 113: on proportion in three
terms (ovvexns, continuous), and in four
terms (diŋpnuévn, discrete) II. 131, 293: on
alternate ratios II. 134: on inverse ratio II.
134, 149: on similar rectilineal figures 11.
188 has locus-theorem (arising out of
Eucl. VI. 3) also given in Apollonius'
Piane Loci II. 198-200: on unit 11. 279:
on number II. 280: on non-applicability of
arithmetical proofs to magnitudes if these
are not numbers II. 113: on definitions of
odd and even by one another II. 281: on
prime numbers II. 284-5 on composite
numbers as plane and solid II. 286, 288,
290: on representation of numbers by
pebbles forming figures II. 288: gives
proof (no doubt Pythagorean) of incom-
mensurability of 2, III. 2: I. 38, 45, 117,
150 n., 181, 184, 185, 187, 188, 195, 202,
203, 221, 222, 223, 226, 259, 262-3, 283,
II. 2, 4, 22, 79, 112, 135, 149, 159, 160,
165, 184, 188, 189, III. 4
Arithmetic, Elements of, anterior to Euclid
II. 295

al-Arjānī, Ibn Rāhawaihi 1. 86
Ashkal at-ta'sis I. 5 n.

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

Astaroff, Ivan 1. 113

Asymptotic (non-secant): of lines 1. 40, 161,
203: of parallel planes 111. 265
Atheihard of Bath 1. 78, 93-6
Athenaeus of Cyzicus 1. 117

August, E. F. 1. 103, 11. 23, 25, 149, 238,
256, 412, III. 2, 48

Austin, W. I. 103, 111, II. 172, 188, 211, 259
Autolycus, On the moving sphere, 1. 17
Avicenna, I. 77, 89

"Axiom of Archimedes" III. 15-6: already
used by Eudoxus, III. 15, and mentioned by
Aristotle, III. 16: relation of, to Eucl. x.
I, III. 15-6
Axioms, distinguished from postulates by
Aristotle 1. 118-9, by Proclus (Geminus
and "others") I. 40, 121-3: Proclus on
difficulties in distinctions I. 123-4: distin-
guished from hypotheses, by Aristotle 1.
120-1, by Proclus 1. 121-2: indemonstrable
1. 121: attempt by Apollonius to prove I.
222-3: "common (things)" or "common

538

opinions" in Aristotle 1. 120, 221: com-
mon to all sciences I. 119, 120: called
66 common notions" in Euclid 1. 121, 221:
which are genuine? 1. 221 sqq.: Proclus
recognises five 1. 222, Heron three 1. 222:
interpolated axioms 1. 224, 232: Pappus'
additions to axioms 1. 25, 223, 224, 232:
axioms of congruence, (1) Euclid's Common
Notion 4, 1. 224-7, (2) modern systems
(Pasch, Veronese and Hilbert) I. 228-31:

axiom" with Stoics every simple
declaratory statement 1. 41, 221: axioms
tacitly assumed, in Book v., II. 137, in
Book VI., II. 294

Axis of sphere III. 261, 269: of cone III.
261, 271: of cylinder 111. 262, 271

Babylonians: knowledge of triangle 3, 4, 5,
1. 352: supposed discoverers of "harmonic
proportion

II. 112

Bacon, Roger 1. 94
Baermann, G. F. II. 213
Balbus, de mensuris 1. 91
Baltzer, R. 11. 30

Barbarin I. 219

Barlaam, arithmetical commentary on Eucl.
II., I. 74

Barrow on Eucl. v. Def. 3, II. 117: on
v. Def. 5, 11. 121: I. 103, 105, 110, 111,
II. 56, 186, 238

Base: meaning 1. 248-9: of cone III. 262:
of cylinder III. 262

Basel editio princeps of Eucl., 1. 100–1
Basilides of Tyre 1. 5, 6, III. 512
Baudhayana Sulba-Sutra 1. 360°
Bayfius (Baïf, Lazare) 1. 100
Becker, J. K. 1. 174

Beez 1. 176

Beltrami, E. I. 219

Benjamin of Lesbos I. 113

Bergh, P. I. 400-1

Bernard, Edward I. 102

Besthorn and Heiberg, edition of al-Hajjaj's
translation and an-Nairizi's commentary

I. 22, 27 n., 79 n.
Bhaskara 1. 355

Billingsley, Sir Henry, 1. 109-10, 11. 56, 238,
III. 48

Bimedial (straight line): first and second,
and biquadratic equations of which they
are roots III. 7: first bimedial defined III.
84-5, equivalent to square root of second
binomial III. 84, 120-3, uniquely divided
III. 94-5 second bimedial defined III.
85-7, equivalent to square root of third
binomial III. 84, 124-5, uniquely divided
III. 95-7

Binomial (straight line): compound ir-
rational straight line (sum of two "terms")
III. 7 defined III. 83, 84: connected by
Theaetetus with arithmetic mean III. 3, 4:
biquadratic of which binomial is a positive
root III. 7: first, second, third, fourth,
fifth and sixth binomials, quadratics from
which arising III. 5-6, defined III. 101-2,

and found respectively (x. 48-53) III. 102-
15, are equivalent to squares of binomial,
first bimedial etc. III. 132-45: binomial
equivalent to square root of first binomial
III. 116-20: binomial uniquely divided,
and algebraical equivalent of this fact III.
92-4: cannot be apotome also III. 240-2:
different from medial (straight line) and
from other irrationals (first bimedial etc.)
of same series with itself III. 242: used to
rationalise apotome with proportional terms
III. 248-52, 252-4
al-Biruni I. 90

Björnbo, Axel Anthon I. 17 n., 93
Boccaccio 1. 96

Bodleian MS. (B) 1. 47, 48, III. 521
Boeckh 1. 351, 371

Boethius 1. 92, 95, 184, II. 295
Bologna Ms. (b) I. 49
Bolyai, I. 1. 219

Bolyai, W. 1. 174-5, 219, 328
Bolzano I. 167

Boncompagni 1. 93 n., 104 n.
Bonola, R. I. 202, 219, 237

Borelli, Giacomo Alfonso 1. 106, 194, II. 2, 84
Boundary (8pos) 1. 182, 183
Bråkenhjelm, P. R. 1. 113

Breadth (of numbers) = second dimension or
factor II. 288

Breitkopf, Joh. Gottlieb Immanuel 1. 97
Bretschneider 1. 136 n., 137, 295, 304, 344,
354, 358, III. 439, 442
Briconnet, François I. 100

Briggs, Henry 1. 102, II. 143

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

Bürk, A. 1. 352, 360-4

Bürklen 1. 179

Buteo (Borrel), Johannes 1. 104.

Cabasilas, Nicolaus and Theodorus 1. 72
Caiani, Angelo I. 101

Camerarius, Joachim 1. 101, III. 523
Camerer, J. G. I. 103, 293, II. 22, 25, 28,
33, 34, 40, 67, 121, 131, 189, 213, 244
Çamorano, Rodrigo, 1. 112
Campanus, Johannes 1. 3, 78, 94-6, 104,
106, 110, 407, 11. 28, 41, 56, 90, 116, 119,
121, 146, 189, 211, 234, 235, 253, 275,
320, 322, 328
Candalla, Franciscus Flussates (François de
Foix, Comte de Candale) 1. 3, 104, 110,
11. 189

Cantor, Moritz I. 7 n., 20, 272, 304, 318,
320, 333, 352, 355, 357-8, 360, 401, II. 5,
40, 97, III. 8, 15, 438

Cardano, Hieronimo II. 41, III. 8
Carduchi, L. I. 112

Carpus, on Astronomy, I. 34, 43: 45, 127,
128, 177

Case, technical term 1. 134: cases inter-
polated 1. 58, 59: Greeks did not infer
limiting cases but proved them separately

11.75
Casey, J. II. 227