Casiri, I. 4., 9n.

Cassiodorius, Magnus Aurelius 1. 92
Catalan III. 527

Cataldi, Pietro Antonio I. 106

Catoptrica, attributed to Euclid, probably
Theon's I. 17: Catoptrica of Heron I. 21,
253

Cauchy III. 267: proof of Eucl. XI. 4, III. 280
"Cause": consideration of, omitted by com-

mentators I. 19, 45: definition should state
cause (Aristotle) 1. 149: cause = middle
term (Aristotle) 1. 149: question whether
geometry should investigate cause (Gemi-
nus), I. 45, 150n.
Censorinus I. 91
Centre, κέντρον 1. 184-5
Ceria Aristotelica 1. 35
Cesaro, E. III. 527

"Chance equimultiples" in phrase “other,
chance, equimultiples II. 143-4

Chasles on Porisms of Euclid 1. 10, 11, 14, 15
Chinese, knowledge of triangle 3, 4, 5,

I. 352: "Tcheou pei" 1. 355

Christensen III. 8

Chrysippus 1. 330

Chrystal, G. III. 19

Cicero I. 91, 351

=

=

round,
περιφερό

Circle: definition of, 1. 183-5:
στρογγύλον (Plato), 1. 184:
Ypauμov (Aristotle) 1. 184: a plane figure
I. 183-4: exceptionally in sense of "cir-
cumference" 11. 23: centre of, I. 184-5:
pole of, 1. 185: bisected by diameter
(Thales) 1. 185, (Saccheri) 1. 185-6: inter-
sections with straight line I. 237-8, 272-4,
with another circle I. 238-40, 242-3,
293-4 definition of "equal circles" 11. 2:
circles touching, meaning of definition,
II. 3: circles intersecting and touching,
difficulties in Euclid's treatment of, II.
25-7, 28-9, modern treatment of, II. 30-2
Circumference, περιφέρεια, Ι. 184
Cissoid, 1. 161, 164, 176, 330
Clairaut I. 328

Clavius (Christoph Schlüssel) 1. 103, 105, 194,

232, 381, 391, 407, II. 2, 41, 42, 47, 49,
53, 56, 67, 70, 73, 130, 170, 190, 231, 238,
244, 271, III. 273, 331, 341, 350, 359, 433
Claymundus, Joan. 1. 101

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

Commandinus 1. 4, 102, 103, 104-5, 106,

110, 111, 407, II. 47, 130, 190: scholia
included in translation of Elements 1. 73:
edited (with Dee) De divisionibus 1. 8,
9, 110
Commensurable: defined III. 10: com-
mensurable in length, commensurable in
square, and commensurable in square only
defined III. 10, 11: symbols used in notes
for these terms III. 34
Commentators on Eucl. criticised by Proclus
I. 19, 26, 45

539

Common Notions: axioms 1. 62, 120-1,
221-2: which are genuine? I. 221 sq.:
meaning and appropriation of term 1. 221:
called "axioms" by Proclus I. 221
Complement, Tараτλýрwμа: meaning of, I.
341: "about diameter " I. 341: not
necessarily parallelograms I. 341: use for
application of areas I. 342-3

Componendo (ovvlévτ), denoting "composi
tion of ratios q.v.: componendo and
separando used relatively to each other
II. 168, 170

Composite, dúv@ETOS: (of lines) 1. 160: (of
surfaces) 1. 170: (of numbers) II. 286:
with Eucl. and Theon of Smyrna composite
numbers may be even, but with Nicom.
and Iamblichus are a subdivision of odd 11.
286, plane and solid numbers are species
of, II. 286

66

Composite to one another" (of numbers)
II. 286-7
Composition of ratio (ovvdeos λóyou), de-
noted by componendo (ovv0évтT), distinct
from compounding ratios II. 134-5
Compound ratio: explanation of, II. 132-3:
(interpolated?) definition of, II. 189–90, III.
526: compounded ratios in V. 20-3, II.
176-8

Conchoids 1. 160-1, 265-6, 330
Conclusion, σvμπéраoμа: necessary part of a
proposition I. 129-30: particular con-
clusion immediately made general I. 131:
definition merely stating conclusion 1. 149
Cone: definitions of, by Euclid III. 262, 270,
by Apollonius III. 270: distinction between
right-angled, obtuse-angled and acute-
angled cones a relic of old theory of
conics III. 270: similar cones, definition
of, III. 262, 271

Congruence-Axioms or Postulates: Common
Notion 4 in Euclid I. 224-5: modern
systems of (Pasch, Veronese, Hilbert) 1.
228-31

Congruence theorems for triangles, recapitula-
tion of, 1. 305-6

Conics, of Euclid, 1. 3, 16: of Aristaeus, I. 3,
16: of Apollonius 1. 3, 16: fundamental
property as proved by Apollonius equi-
valent to Cartesian equation I. 344-5: focus-
directrix property proved by Pappus I. 15
Consequents ("following" terms in a pro-
portion) II. 134

Constantinus Lascaris 1. 3
Construct (συνίστασθαι)

contrasted with
describe on I. 348, with apply to I. 343:
special connotation I. 259, 289
Construction, KATAσKEVÝ, Oпе of formal
divisions of a proposition 1. 129: some-
times unnecessary 1. 130: turns nominal
into real definition 1. 146: mechanical
constructions 1. 151, 387

Continuity, Principle of, 1. 234 sq., 242, 272,
294
Continuous proportion (συνεχής οι συνημμένη
ávaλoyía) in three terms II. 131

540

66

Conversion, geometrical: distinct from logical
I. 256: leading" and partial varieties
of, 1. 256-7, 337
Conversion of ratio (ἀναστροφὴ λόγου), de-
noted by convertendo (avaorрÉĻAVTI) II.
135 convertendo theorem not established
by v. 19, Por. II. 174-5, but proved by
Simson's Prop. E. 11. 175, 111. 38: Euclid's
roundabout substitute III. 38
Convertendo denoting "conversion" of ratios,

q.v.

Copernicus 1. 101

Cordonis, Mattheus 1. 97

Corresponding magnitudes 11. 134
Cossali III. 8

Cratistus I. 133

Crelle, on the plane 1. 172-4, III. 263
Ctesibius I. 20, 21, 391.

Cube defined III. 262: problem of in-
cribing in sphere, Euclid's solution III.
478-80, Pappus' solution III. 480: duplica-
tion of cube reduced by Hippocrates of
Chios to problem of two mean propor-
tionals 1. 135, 11. 133: cube number, de-
fined II. 291: two mean proportionals
between two cube numbers 11. 294, 364-5
Cunn, Samuel 1. 111

Curtze, Maximilian, editor of an-Nairizi
I. 22, 78, 92, 94, 96, 97 n.
Curves, classification of: see line

Cyclic, of a particular kind of square number
II. 291

Cyclomathia of Leotaud II. 42

Cylinder: definition of, III. 262: similar
cylinders defined III. 262
Cylindrical helix 1. 161, 162, 329, 330
Czecha, Jo. 1. 113

Dasypodius (Rauchfuss) Conrad 1. 73, 102
Data of Euclid: 1. 8, 132, 141, 385, 391:
Def. 2, II. 248: Prop. 8, II. 249-50:
Prop. 24, II. 246-7: Prop. 55, II. 254:
Props. 56 and 68, 11. 249: Prop. 58, 11.
263-5 Props. 59 and 84, 11. 266–7:
Prop. 67 assumes part of converse of
Simson's Prop. B (Book VI.) II. 224:
Prop. 70, 11. 250: Prop. 85, 11. 264:
Prop. 87, II. 228: Prop. 93, II. 227
Deahna I. 174

Dechales, Claude François Milliet 1. 106,
107, 108, 110, II. 259
Dedekind's theory of irrational numbers
corresponds exactly to Eucl. v. Def. 5,
11. 124-6; Dedekind's Postulate and
applications of, I. 235-40, III. 16
Dee, John I. 109, 110; discovered De
divisionibus 1. 8, 9

Definition, in sense of "closer statement

(diopioμós), one of formal divisions of a pro-
position 1. 129: may be unnecessary 1. 130
Definitions: Aristotle on, I. 117, 119, 120,
143: a class of thesis (Aristotle) I. 120:
distinguished from hypotheses 1. 119, but
confused therewith by Proclus 1. 121-2:
must be assumed 1. 117-9, but say nothing

about existence (except in the case of a few
primary things) 1. 119, 143: terms for, öpos
and opouós 1. 143: real and nominal
definitions (real nominal plus postulate
or proof), Mill anticipated by Aristotle,
Saccheri and Leibniz I. 143-5: Aristotle's
requirements in, 1. 146-50, exceptions
I. 148: should state cause or middle term
and be genetic I. 149-50: Aristotle on un-
scientific definitions (ἐκ μὴ προτέρων) 1.
148-9: Euclid's definitions agree generally
with Aristotle's doctrine I. 146: inter-
polated definitions 1. 61, 62: definitions
of technical terms in Aristotle and Heron,
not in Euclid 1. 150

De levi et ponderoso, tract 1. 18
Demetrius Cydonius 1. 72
Democritus I. 38: On difference of gnomon
etc. (? on "angle of contact ") 11. 40: on
parallel and infinitely near sections of cone,
II. 40, III. 368: stated, without proving,
propositions about volumes of cone and
pyramid, II. 40, III. 366: was evidently
on the track of the infinitesimal calculus
III. 368: treatise on irrationals (wepì áλóywv
γραμμῶν καὶ ναστῶν β') 111. 4

De Morgan, A.: 1. 246, 260, 269, 284, 291,
298, 300, 309, 313, 314, 315, 369, 376,
II. 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:
on definition of ratio II. 116-7: on ex-
tension of meaning of ratio to cover
incommensurables II. 118: means of ex-
pressing ratios between incommensurables
by approximation to any extent 11. 118-9:
defence and explanation of v. Def. 5, II.
121-4 on necessity of proof that tests for
greater and less, or greater and equal,
ratios cannot coexist II. 130-1, 157: on
compound ratio II. 132-3, 234: sketch of
proof of existence of fourth proportional
(assumed in v. 18) 11. 171; proposed
lemma about duplicate ratios as alternative
means of proving VI. 22, II. 246-7: on
Book X., III. 8
Dercyllides II. III
Desargues I. 193

Describe on (ἀναγράφειν ἀπό) contrasted with
construct 1. 348

De Zolt 1. 328

Diagonal (diayúvios) 1. 185

66

Diagonal" numbers: see

diagonal-" numbers

"Side-" and

Diameter (diáμerpos), of circle or parallelogram
I. 185 of sphere 111. 261, 269, 270:
as applied to figures generally 1. 325;
"rational" and "irrational" diameter of
5 (Plato) 1. 399, taken from Pythagoreans
I. 399-400, III. 12, 525
Dihedral angle = inclination of plane to plane,
measured by a plane angle III. 264-5
Dimensions (cf. diaσráσeis), 1. 157, 158:
Aristotle's view of, 1. 158-9, III. 262-3,
speaks of six 111. 263

Dinostratus 1. 117, 266
Diocles 1. 164
Diodorus I. 203

Diogenes Laertius 1. 37, 305, 317, 351, III. 4
Diophantus 1. 86

Diorismus (dopioμós) = (a) "definition" or
"specification,' a formal division of a
proposition 1. 129: (b) condition of possi-
bility 1. 128, determines how far solution
possible and in how many ways I. 130-1,
243: diorismi said to have been discovered
by Leon 1. 116; revealed by analysis
1. 142: introduced by deî dŋ 1. 293: first
instances in Elements 1. 234, 293: for
solution of quadratic II. 259
Dippe 1. 108

Direction, as primary notion, discussed 1.
179: direction-theory of parallels 1. 191-2
Discrete proportion, διηρημένη oι διεζευγμένη
ávaλoyia, in four terms, II. 131, 293

Dissimilarly ordered" proportion (dvouolws
τεταγμένων τῶν λόγων) in Archimedes
="perturbed proportion" 11. 136
Distance, διάστημα : = radius 1. 199: in

Aristotle has usual general sense and
= dimension I. 199

Dividendo (of ratios): see Separation,
separando

Division (method of), Plato's I. 134
Divisions (of figures), treatise by Euclid, I.
8, 9: translated by Muhammad al-Bagdādī
1. 8: found by Woepcke in Arabic 1. 9,
and by Dee in Latin translation 1. 8, 9:
1. 110: proposition from, II. 5
Dodecahedron: decomposition of faces into
elementary triangles 11. 98: definition of,
III. 262: dodecahedra found, apparently
dating from centuries before Pythagoras
III. 438, though said to have been dis-
covered by Pythagoreans ibid.: problem
of inscribing in sphere, Euclid's solution
III. 493, Pappus' solution III. 501-3
Dodgson, C. L. I. 194, 254, 261, 313, 11.48, 275
Dou, Jan Pieterszoon 1. 108
Duhamel, J. M. C. 1. 139, 328
Duplicate ratio II. 133; dimλaoiwv, duplicate,
distinct from dirλários, double (= ratio
21), though use of terms not uniform
II. 133: 'duplicate" of given ratio found
by VI. 11, 11. 214: lemma on duplicate
ratio as alternative to method of VI. 22
(De Morgan and others) 11. 242-7
Duplication of cube; reduction of, by Hippo-
crates, to problem of finding two mean
proportionals 1. 135, 11.133: wrongly sup-
posed to be alluded to in Timaeus 32 A, B,
II. 294-5 n.

Egyptians II. 112: knowledge of right-angled
triangles 1. 352: view of number 11. 280
Elements: pre-Euclidean Elements, by Hip-
pocrates of Chios, Leon 1. 116, Theudius
1. 117: contributions to, by Eudoxus 1. 1,
37, II. 112, III. 15, 365-6, 374, 441, The-
aetetus I. 1, 37, III. 3, 438, Hermotimus of

541

Colophon 1. 117; Euclid's Elements, ulti-
mate aims of, 1. 2, 115-6; commentators
on, I. 19-45, Proclus 1. 19, 29-45 and
passim, Heron I. 20-24, an-Nairīzi I. 21-
4, Porphyry I. 24, Pappus I. 24-7,
Simplicius 1. 28, Aenaeas (Aigeias) 1. 28:
MSS. of, I. 46-51: Theon's changes in text
I. 54-8: means of comparing Theonine with
ante-Theonine text 1. 51-3: interpolations
before Theon's time 1. 58-63: scholia 1.
64-74, III. 521-3: external sources
throwing light on text, Heron, Taurus,
Sextus Empiricus, Proclus, Iamblichus 1.
62-3: Arabic translations (1) by al-Hajjāj
1. 75, 76, 79, 80, 83-4, (2) by Ishaq and
Thabit b. Qurra 1. 75-80, 83-4, (3) Naşirad-
din at-Tusi I. 77-80, 84: Hebrew transla-
tion by Moses b. Tibbon or Jakob b.
Machir 1. 76: Arabian versions compared
with Greek text 1. 79-83, with one another
1. 83, 84: translation by Boethius 1. 92:
old translation of 10th c. 1. 92; translations
by Athelhard 1. 93–6, Gherard of Cremona
1. 93-4, Campanus 1. 94-6, 97-100 etc.,
Zamberti 1. 98-100, Commandinus 1. 104-
5: introduction into England, 10th c.,
I. 95 translation by Billingsley 1. 109-10:
Greek texts, editio princeps 1. 100-1,
Gregory's 1. 102-3, Peyrard's 1. 103,
August's 1. 103, Heiberg's passim: trans-
lations and editions generally 1. 97-113:
writers on Book X., III. 8-9: on the nature
of elements (Proclus) 1. 114-6, (Menaech-
mus) 1. 114, (Aristotle) 1. 116: Proclus on
advantages of Euclid's Elements 1. 115:
immediate recognition of, 1. 116: first
principles of, definitions, postulates, com-
mon notions (axioms) 1. 117-24: technical
terms in connexion with, 1. 125-42: no
definitions of such technical terms 1. 150:
sections of Book 1., 1. 308
Elinuam 1. 95

Eneström, G. III. 521

Engel and Stäckel 1. 219, 321
Enriques, F. I. 157, 175, 193, 195, 201, 313,
II. 30, 126

Enunciation (póτaois), one of formal di
visions of a proposition 1. 129-30
Epicureans, objection to Eucl. 1. 20, I. 41,
287: Savile on, I. 287

66

Equality, in sense different from that of
congruence (= "equivalent," Legendre) 1.
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)
1. 328: modern definition of, 1. 228
Equimultiples: "any equimultiples what-
ever,” ισάκις πολλαπλάσια καθ ̓ ὁποιονοῦν
πολλαπλασιασμόν 11. 120: stereotyped
phrase "other, chance, equimultiples
II. 143-4 should include once each magni-
tude 11. 145
Eratosthenes: I. 1, 162: contemporary with
Archimedes I. 1, 2: Archimedes" "Method"

I. 2: taught at Alexandria I. 2: Pappus
on personality of, 1. 3: story of (in
Stobaeus) 1. 3: not "of Megara" 1. 3, 4:
supposed to have been born at Gela I. 4:
Arabian traditions about, I. 4, 5: "of Tyre'
1. 4-6: "of Tūs" I. 4, 5 n.: Arabian
derivation of name ("key of geometry")
1.6: Elements, ultimate aim of, 1. 2, 115-6:
other works, Conics 1. 16, Pseudaria 1. 7,
Data 1. 8, 132, 141, 385, 391, On divisions
(of figures) 1. 8, 9, Porisms 1. 10-15,
Surface-loci 1. 15, 16, Phaenomena 1. 16,
17, Optics 1. 17, Elements of Music or
Sectio Canonis 1. 17, II. 294-5: on "three-
and four-line locus" 1. 3: Arabian list of
works 1. 17, 18: bibliography 1. 91–113
Eudemus 1. 29: On the Angle 1. 34, 38,
177-8: History of Geometry I. 34, 35-8,
278, 295, 304, 317, 320, 387, 11. 99, III,
III. 3, 366, 524

Eudoxus 1. 1, 37, 116, II. 40, 99, 280, 295:
discoverer of theory of proportion covering
incommensurables as expounded generally
in Bks. V., VI., I. 137, 351, II. 112: on the
golden section 1. 137: discoverer of method
of exhaustion 1. 234, III. 365-6, 374: used
"Axiom of Archimedes 111. 15: first to
prove theorems about volume of pyramid
(Eucl. XII. 7 Por.) and cone (Eucl. XII. 10),
also theorem of Eucl. XII. 2, III. 15:
theorems of Eucl. XIII. 1-5 probably due
to, III. 441: inventor of a certain curve,
the hippopede, horse-fetter 1. 163: possibly
wrote Sphaerica 1. 17: III. 442, 522, 523, 526
Euler, Leonhard 1. 401

Eutocius: 1. 25, 35, 39, 142, 161, 164, 259,
317, 329, 330, 373: on "vi. Def. 5" and
meaning of nλXIKÓтns II. 116, 132, 189–90:
gives locus-theorem from Apollonius' Plane
Loci 11. 198-200

Even (number): definitions by Pythagoreans
and in Nicomachus II. 281: definitions of
odd and even by one another unscientific
(Aristotle) 1. 148-9, II. 281: Nicom.
divides even into three classes (1) even-
times even and (2) even-times odd as ex-
tremes, and (3) odd-times even as interme-
diate 11. 282-3

Even-times even: Euclid's use differs from
use by Nicomachus, Theon of Smyrna and
Iamblichus II. 281-2

Even-times odd in Euclid different from even-

odd of Nicomachus and the rest II. 282-4
Ex aequali, of ratios, II. 136: ex aequali pro-
positions (v. 20, 22), and ex aequali "in
perturbed proportion" (V. 21, 23) 11. 176–8
Exhaustion, method of: discovered by

Faifofer II. 126
Falk, H. I. 113
al-Faradi 1. 8 n., 90
Fermat 111. 526-7

Figure, as viewed by Plato I. 182,
Aristotle 1. 182-3, by Euclid 1. 1
according to Posidonius is confir
boundary only 1. 41, 183: figures boun
by two lines classified 1. 187: angle
(άywvov) figure 1. 187

Figures, printing of, 1. 97
Fihrist I. 4 n., 5 N., 17, 21, 24, 25, 27;
of Euclid's works in, I. 17, 18
Finaeus, Orontius (Oronce Fine) I. 101,
Flauti, Vincenzo I. 107
Florence Ms. Laurent. XXVIII. 3 (F) 1.
Flussates, see Candalla
Forcadel, Pierre 1. 108

Fourier: definition of plane based on F
XI. 4, I. 173-4, III. 263
Fourth proportional: assumption of exist
of, in v. 18, and alternative methods
avoiding (Saccheri, De Morgan, Sim
Smith and Bryant) 11. 170-4: Clavius n
the assumption an axiom II. 170: skeid
proof of assumption by De Morgan II.
condition for existence of number w
is a fourth proportional to three num
II. 409-11

Frankland, W. B. 1. 173, 199
Frischauf, J. 1. 174

Galileo Galilei: on angle of contact 11.
Gartz I. 9n.

Gauss I. 172, 193, 194, 202, 219, 321
Geminus: name not Latin 1. 38-9: titl
work (piλoxaλia) quoted from by Pro
I. 39, and by Schol., III. 522: elem
of astronomy 1. 38: comm. on Posido
1. 39: Proclus' obligations to, I. 39
on postulates and axioms I. 122-3,
522 on theorems and problems 1.
two classifications of lines (or curve
160-2: on homoeomeric (uniform)
1. 162: on "mixed" lines (curves)
surfaces I. 162: classification of sur
I. 170, of angles 1. 178-9: on para
I. 191: on Postulate 4, 1. 200: on st
of proof of theorem of I. 32, I. 317
I. 21, 27-8, 37, 44, 45, 133n., 203, 265,
Geometrical algebra 1. 372-4: Euc
method in Book 11. evidently the clas
method 1. 373: preferable to semi-
braical method I. 377-8
Geometrical progression II. 346 sqq.: sun
tion of n terms of (IX. 35) II. 420-1

Geometric means 11. 357 sqq.: one mean
between square numbers II. 294, 363, or be-
tween similar plane numbers II. 371-2: two
means between cube numbers II. 294, 364-5,
or between similar solid numbers II. 373-5
Gherard of Cremona, translator of Elements
I. 93-4 of an-Nairizi's commentary I. 22,
94, II. 47: of tract De divisionibus 1. 9
Giordano, Vitale 1. 106, 176

Given, dedoμévos, different senses, 1. 132-3
Gnomon: literally "that enabling (something)
to be known" 1. 64, 370: successive senses
of, (1) upright marker of sundial, I. 181,
185, 271-2, introduced into Greece by
Anaximander 1. 370, (2) carpenter's square
for drawing right angles 1. 371, (3) figure
placed round square to make larger square
1. 351, 371, Indian use of gnomon in this
sense 1. 362, (4) use extended by Euclid to
parallelograms 1. 371, (5) by Heron and
Theon to any figures 1. 371-2: Euclid's
method of denoting in figure 1. 383: arith-
metical use of, 1. 358-60, 371, II. 289
"Gnomon-wise" (κaтà vάμova), old name
for perpendicular (xáðeros) 1. 36, 181, 272
Görland, A. I. 233, 234

Golden section (section in extreme and mean
ratio), discovered by Pythagoreans 1. 137,
403, II. 99: connexion with theory of irra-
tionals 1. 137, III. 19: theory carried further
by Plato and Eudoxus II. 99: theorems of
Eucl. XIII. 1-5 on, probably due to Eu-
doxus III. 441

"Goose's foot" (pes anseris), name for Eucl.
III. 7, I. 99

Gow, James I. 135 n.

Gracilis, Stephanus I. 101-2

Grandi, Guido I. 107

Greater ratio: Euclid's criterion not the only
one II. 130: arguments from greater to less
ratios etc. unsafe unless they go back to
original definitions (Simson on V. 10) II.
156-7 test for, cannot coexist with test
for equal or less ratio 11. 130-1
Greatest common measure: Euclid's method
of finding corresponds exactly to ours 11.
118, 299, III. 18, 21-2: Nicomachus gives
the same method 11. 300: method used to
prove incommensurability 111. 18-9; for
this purpose often unnecessary to carry it
far (cases of extreme and mean ratio and
of √2) III. 18-9

Gregory, David 1. 102–3, 11. 116, 143, III. 32
Gregory of St Vincent 1. 401, 404

Gromatici 1. 91 n., 95

Grynaeus 1. 100-1

Häbler, Th. II. 294 n.
al-Haitham I. 88, 89

al-Hajjāj b. Yusuf b. Maṭar, translator of the
Elements 1. 22, 75, 76, 79, 80, 83, 84
Halifax, William 1. 108, 110
Halliwell (Phillips) 1. 95 n.

Hankel, H. 1. 139, 141, 232, 234, 344, 354,
II. 116, 117, III. 8

543

Harmonica of Ptolemy, Comm. on, I. 17
Harmony, Introduction to, not by Euclid, 1. 17
Harun ar-Rashid 1. 75

al-Hasan b. 'Ubaidallah b. Sulaiman b.
Wahb I. 87

Hauber, C. F. II. 244
Hauff, J. K. F. 1. 108

66 'Heavy and Light," tract on, I. 18
Heiberg, J. L. passim

Helix, cylindrical 1. 161, 162, 329, 330
Helmholtz, 1. 226, 227

Henrici and Treutlein 1. 313, 404, II. 30
Henrion, Denis I. 108
Hérigone, Pierre 1. 108
Herlin, Christian I. 100
Hermotimus of Colophon I. I
Herodotus I. 37 n., 370
"Heromides" I. 158

of

Heron of Alexandria, mechanicus, date of,
I. 20-1, III. 521: Heron and Vitruvius
I. 20-1; commentary on Euclid's Elements
I. 20-4: direct proof of 1. 25, 1. 301: com-
parison of areas of triangles in I. 24, I. 334-
5: addition to 1. 47, I. 366-8: apparently
originated semi-algebraical method
proving theorems of Book II., I. 373, 378:
Eucl. III. 12 interpolated from, II. 28:
extends III. 20, 21 to angles in segments
less than semicircles II. 47-8: does not
recognise angles equal to or greater than two
right angles 11. 47-8: proof of formula for
area of triangle, ▲ =√s(s − a) (s − b) (s − c),
II. 87-8: 1. 137 n., 159, 163, 168, 170,
171-2, 176, 183, 184, 185, 188, 189, 222,
223, 243, 253, 285, 287, 299, 351, 369,
371, 405, 407, 408, 11. 5, 16-7, 24, 28,
33, 34, 36, 44, 47, 48, 116, 189, 302, 320,
383, 395, III. 24, 263, 265, 267, 268, 269,
270, 366, 404, 442

Heron, Proclus' instructor 1. 29
"Herundes I. 156

Hieronymus of Rhodes 1. 305

Hilbert, D. 1. 157, 193, 201, 228-31, 249,
313, 328

Hipparchus I. 4 n., 30 n., III. 523
Hippasus II. 97, III. 438
Hippias of Elis 1. 42, 265-6
Hippocrates of Chios 1. 8 n., 29, 35, 38, 116,
135, 136 n., 386-7, II. 133: first proved
that circles (and similar segments of circles)
are to one another as the squares on their
diameters III. 366, 374

Hippopede (TOυ Téon), a certain curve used
by Eudoxus I. 162-3, 176

Hoffmann, Heinrich I. 107

Hoffmann, Joh. Jos. Ign. 1. 108, 365
Holgate, T. F. III. 284, 303, 331
Holtzmann, Wilhelm (Xylander) 1. 107
Homoeomeric (uniform) lines I. 40, 161, 162
Hoppe, E. 1. 21, III. 521

Hornlike angle (κερατοειδής γωνία) 1. 177,
178, 182, 265, II. 39, 40: hornlike angle
and angle of semicircle, controversies on,
II. 39-42: Proclus on, 11. 39-40: Demo-
critus may have written on hornlike angle