Hauff, J. K. F. 108

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

Helix, cylindrical 161, 162, 329, 330

Helmholtz 226, 227

Henrici and Treutlein 313, 404

Henrion, Denis 108

Hérigone, Pierre 108

Herlin, Christian 100

Hermotimus of Colophon I

Herodotus 37 n., 370
"Heromides" 158

Heron of Alexandria, mechanicus, date of
20-1: Heron and Vitruvius 20-1: com-
mentary on Euclid's Elements 20-4:
direct proof of 1. 25, 301: comparison of
areas of triangles in 1. 24, 334-5: addi-
tion to 1. 47, 366-8: apparently originated
semi-algebraical method of proving theo-
rems of Book 11. 373, 378: 137 m., 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
Heron, Proclus' instructor 29
"Herundes" 156

Hieronymus of Rhodes 305

Hilbert 157, 193, 201, 228-31, 249, 313, 328
Hipparchus 4., 30n.

Hippias of Elis 42, 265-6

Hippocrates of Chios 8n., 29, 35, 38, 116,
135, 136 n., 386-7

Hippopede (ITOV Téon), a certain curve used
by Eudoxus 162-3, 176
Hoffmann, Heinrich 107

Hoffmann, John Jos. Ign. 108, 365
Holtzmann, Wilhelm (Xylander) 107
Homoeomeric (uniform) lines 40, 161, 162
Hoppe, E. 21

Hornlike (angle), kepaтoeidhs 177, 178, 182,
265

Horsley, Samuel 106

Hotel, J. 219

Hudson, John 102

Hultsch, F. 20, 329, 400

Hunain b. Ishaq al-Ibādi 75

Hypotheses, in Plato 122: in Aristotle 118-
20: confused by Proclus with definitions
121-2: geometer's hypotheses not false
(Aristotle) 119

Hypothetical construction 199

Hypsicles 5: author of Book XIV. 5, 6

Iamblichus 63, 83

Ibn al-'Amid 86
Ibn al-Haitham 88, 89
Ibn al-Lubudi 90

Ibn Rāhawaihi al-Arjāni 86

Ibn Sina (Avicenna) 77, 89

"Iflaton" 88

Incomposite (of lines) 160-1, (of surfaces) 170
Indivisible lines (aтоμο yраμμal), theory of,
rebutted 268

Infinite, Aristotle on the, 232-4: infinite
division not assumed, but proved, by geo-
meters 268

Infinity, parallels meeting at, 192-3
Ingrami, G. 175, 193, 195, 201, 227-8
Interior and exterior (of angles) 263, 280:
interior and opposite angle 280
Interpolations in the Elements before Theon's
time 58-63 by Theon 46, 55-6: I. 40
interpolated 338

Irrational: discovered with reference to √2
351 claim of India to priority of dis-
covery 363-4: "irrational diameter of 5"
(Pythagoreans and Plato) 399-400: ap-
proximation to 2 by means of "side."
and "diagonal-" numbers 399-401: Indian
approximation to 2 361, 363-4: un-
ordered irrationals (Apollonius) 42, 115:
irrational ratio (άрpητos λóyos) 137
Isaacus Monachus (or Argyrus) 73-4, 407
Ishaq b. Hunain b. Ishaq al-Ibādī, Abū
Yaqub, translation of Elements by, 75-80,
83-4

Isma'il b. Bulbul 88

Leiden Ms. 399, 1 of al-Hajjāj and an-
Nairizi 22

Lemma 114: meaning 133-4: lemmas inter-
polated 59-60, especially from Pappus 67
Leodamas of Thasos 36, 134
Leon 116

Linderup, H. C. 113

Line Platonic definition 158: objection of
Aristotle 158: "magnitude extended one
way' (Aristotle, "Heromides") 158:
"divisible or continuous one way" (Aris-
totle) 158-9: "flux of point" 159: Apol-
lonius on, 159: classification of lines, Plato
and Aristotle 159-60, Heron 159-60,
Geminus, first classification 160-1, second
161 straight (eveîa), curved (xaμπúλŋ),
circular (Teppeрns), spiral-shaped (éλiko-
ειδής), bent (κεκαμμένη), broken (κεκλα-
quévn), round (OTрOYYÚλos) 159, composite
(σύνθετος), incomposite (ἀσύνθετος), “form-
ing a figure" (oxnuarоrolovσa), determinate
(ὡρισμένη), indeterminate (ἀόριστος) 16ο:
"asymptotic" or non-secant (áoÚμπTWTOS),
secant (ovurTwтós) 161: simple, "mixed"
161-2 homoeomeric (uniform) 161-2:
Proclus on lines without extremities 165:
loci on lines 329, 330

Linear, loci 330: problems 330
Lionardo da Vinci, proof of I. 47 365-6
Lippert 88 n.

:

Lobachewsky, N. I. 174-5, 213, 219
Locus-theorems (TоTIKа Oewρημаra) and loci
(TÓTO) locus defined by Proclus 329:
loci likened by Chrysippus to Platonic
ideas 330-1: locus-theorems and loci (1) on
lines (a) plane loci (straight lines and
circles) (b) solid loci (conics), (2) on sur-
faces 329: corresponding distinction be-
tween plane and solid problems, to which
Pappus adds linear problems 330: further
distinction in Pappus between (1) ¿PEKTIKOL
(2) διεξοδικοί (3) αναστροφικοί τόποι 330:
Proclus regards locus in 1. 35, III. 21, 31
as an area which is locus of area (parallelo-
gram or triangle) 330

Logical conversion, distinct from geometrical
256

Logical deductions 256, 284-5, 300: logical

equivalents 309, 314-5

Lorenz, J. F. 107-8

Loria, Gino 7 n., 10 n., 11 N., 12 N.

Luca Paciuolo 98-9, 100

Lundgren, F. A. A. 113

Machir, Jakob b. 76

Magni, Domenico 106

Magnitude: common definition vicious 148

al-Māhāni 85

al-Ma'mun, Caliph 75

Mansion, P. 219

al-Manşür, Caliph 75

Paciuolo, Luca 98-9, 100
Pamphile 317, 319
Pappus contrasts Euclid and Apollonius 3:
on Euclid's Porisms 10-14, Surface-loci
15, 16, Data 8: on Treasury of Analysis
8, 10, 11, 138: commentary on Elements
24-7, partly preserved in scholia 66:
evidence of scholia as to Pappus' text
66-7 lemmas in Book x. interpolated
from, 67: on Analysis and Synthesis 138–9,
141-2: additional axioms by, 25, 223, 224,
232 on converse of Post. 4 25, 201:
proof of 1. 5 by, 254: extension of I. 47
366: semi-algebraical methods in 373,
378 on loci 329, 330: on conchoids 161,
266: on quadratrix 266: on isoperimetric
figures 26, 27, 333: on paradoxes of
Erycinus 27, 290: 17, 39, 133 m., 137,
151, 225, 388, 391, 401

Papyrus, Herculanensis No. 1061 50, 184:
Oxyrhynchus 50: Fayům 51, 337, 338:
Rhind 304

Paradoxes, in geometry 188: of Erycinus
27, 290, 329: an ancient "Budget of
Paradoxes" 329

Parallelogram (= parallelogrammic area),
first introduced 325: rectangular parallelo-
gram 370

Parallels: Aristotle on, 190, 191-2: defini-

tions, by "Aganis" 191, by Geminus 191,
Posidonius 190, Simplicius 190: as equi-
distants 190-1, 194: direction-theory of,
191-2, 194: definitions classified 192-4:
Veronese's definition and postulate 194:
Parallel Postulate, see Postulate 5:
Legendre's attempt to establish theory of
213-9

Paris Mss. of Elements, (p) 49: (q) 50
Pasch, M. 157, 228, 250

"Peacock's tail," name for 111. 8 99
Pediasimus, Joannes 72-3
Peithon 203

Peletarius Jacques Peletier) 103, 104, 249,

pretation of Euclid's def. 171: possible
origin of Euclid's def. 171: Archimedes'
assumption 171, 172: other ancient defini-
tions of, in Proclus, Heron, Theon of
Smyrna, an-Nairizi 171-2: "Simson's'
definition and Gauss on 172-3: Crelle's
tract on, 172-4: other definitions by
Fourier 173, Deahna 174, J. K. Becker
174, Leibniz 176, Beez 176: evolution of,
by Bolyai and Lobachewsky 174-5:
Enriques and Amaldi, Ingrami, Veronese
and Hilbert on, 175

"Plane loci" 329-30: Plane Loci of Apol-
lonius 14, 259, 330
"Plane problems" 329
Planudes, Maximus 72

Plato: 1, 2, 3, 137, 155-6, 159, 184, 187,
203, 221: supposed invention of Analysis
by, 134: def. of straight line 165-6: def.
of plane surface 171: generation of cosmic
figures by putting together triangles 226:
rule for rational right-angled triangles 356,
357, 359, 360, 385: "rational diameter
of 5" 399
"Platonic" figures 2
Playfair, John 103, III: 'Playfair's "
Axiom 220: used to prove I. 29, 312, and
Eucl. Post. 5, 313: comparison of Axiom
with Post. 5, 313-4

Pliny 20, 333

Plutarch 21, 29, 37, 177, 343, 351
Point: Pythagorean definition of, 155: inter-
pretation of Euclid's definition 155: Plato's
view of, and Aristotle's criticism 155-6:
attributes of, according to Aristotle 156:
terms for (στιγμή, σημεῖον) 156: other
definitions by "Herundes," Posidonius
156, Simplicius 157: negative character of
Euclid's def. 156: is it sufficient? 156:
motion of, produces line 157: an-Nairizi
on, 157: modern explanations by abstrac-
tion 157
Polybius 331

-

Polygon: sum of interior angles (Proclus'
proof) 322: sum of exterior angles 322
Porism: two senses 13: (1) corollary 134,
278-9: interpolated Porisms (corollaries)
60-1, 381: (2) as used in Porisms of Euclid,
distinguished from theorems and problems
10, 11: account of the Porisms given by
Pappus 10-13 modern restorations by
Simson and Chasles 14: views of Heiberg
II, 14, and Zeuthen 15

Porphyry 17 commentary on Euclid 24:
Symmikta 24, 34, 44: 136, 277, 283, 287
Posidonius, the Stoic 20, 21, 27, 28 m., 189,
197: book directed against the Epicurean
Zeno 34, 43: on parallels 40, 190: defini-
tion of figure 41, 183

Postulate, distinguished from axiom, by
Aristotle 118-9, by Proclus (Geminus
and "others") 121-3: from hypothesis,
by Aristotle 120-1, by Proclus 121-2:
postulates in Archimedes 120, 123:
Euclid's view of, reconcileable with

Aristotle's 119-20, 124: postulates do not
confine us to rule and compass 124: Postu-
lates 1, 2, significance of, 195-6: famous
"Postulate of Archimedes" 234
Postulate 4 significance of, 200: proofs of,
resting on other postulates 200-1, 231:
converse true only when angles rectilineal
(Pappus) 201

Postulate 5: due to Euclid himself 202:
Proclus on, 202-3: attempts to prove,
Ptolemy 204-6, Proclus 206-8, Naşiraddin
at-Tusi 208-10, Wallis 210-1, Saccheri
211-2, Lambert 212-3: substitutes for,
"Playfair's" axiom (in Proclus) 220, others
by Proclus 207, 220, Posidonius and
Geminus 220, Legendre 213, 214, 220,
Wallis 220, Carnot, Laplace, Lorenz,
W. Bolyai, Gauss, Worpitzky, Clairaut,
Veronese, Ingrami 220: Post. 5 proved
from, and compared with, "Playfair's '
axiom 313-4: i. 30 is logical equivalent
of, 220

Potts, Robert 112, 246

Prime (of numbers): two senses of, 146
Principles, First 117-124

Problem, distinguished from theorem 124-8:
problems classified according to number of
solutions (a) one solution, ordered (TETαy-
μéva) (b) a definite number, intermediate
(uéoa) (c) an infinite number of solutions,
unordered (ärakтa) 128: in widest sense
anything propounded (possible or not) but
generally a construction which is possible
128-9: another classification (1) problem
in excess (λeováɲov), asking too much 129,
(2) deficient problem (Xinès æpóßλnμa),
giving too little 129

Proclus: details of career 29-30: remarks

on earlier commentators 19, 33, 45: com-
mentary on Eucl. 1, sources of, 29-45,
object and character of, 31-2: com-
mentary probably not continued, though
continuation intended 32-3: books
quoted by name in, 34: famous "sum-
mary" 37-8: list of writers quoted 44:
his own contributions 44-5: character of
MS. used by, 62, 63: on the nature of
elements and things elementary 114-6: on
advantages of Euclid's Elements, and
their object 115-6: on first principles,
hypotheses, postulates, axioms 121-4: on
difficulties in three distinctions between
postulates and axioms 123: on theorems
and problems 124-9: attempt to prove
Postulate 5 206-8: commentary on Plato's
Republic, allusion in to "side-" and
diagonal." numbers in connexion with
Eucl. 11. 9, 10 399-400

66

Proof (dwbdelis), necessary part of pro-
position 129-30

Proposition, formal divisions of, 129-131
Protarchus 5

Psellus, Michael, scholia by, 70, 71

Pseudaria of Euclid 7: Pseudographemata 7 n.
Pseudoboethius 92

Ptolemy I.: 1, 2: story of Euclid and
Ptolemy I

Ptolemy, Claudius 30 n.: Harmonica of, and

commentary on 17: on Parallel-Postulate
28 n., 34, 43, 45: attempt to prove it 204-6
Pythagoras 4 n., 36: supposed discoverer of
the irrational 351, of application of areas
343-4, of theorem of 1. 47 343-4, 350-4;
story of sacrifice 37, 343, 350: probable
method of discovery of 1. 47 and proof of,
352-5 suggestions by Bretschneider and
Hankel 354, by Zeuthen 355-6: rule for
forming right-angled triangles in rational
numbers 351, 356-9, 385

Pythagoreans 19, 36, 155, 188, 279: term
for surface (xpotá) 169: angles of triangle
equal to two right angles, theorem and
proof 317-20: three polygons which in
contact fill space round point 318: method
of application of areas (including exceeding
and falling-short) 343, 384, 403: gnomon
Pythagorean 351: "rational" and "ir-
rational diameter of 5" 399-400

Radius, no Greek word for, 199

Ramus, Petrus (Pierre de la Ramée) 104
Ratdolt, Erhard 78, 97

Rational (pnbs): (of ratios) 137:
"rational
diameter of 5 "2 399-400 rational right-
angled triangles, see right-angled triangles
Rauchfuss, see Dasypodius
Rausenberger, O. 157, 175, 313
ar-Razi, Abū Yusuf Yaqub b. Muh. 86
Rectangle : = rectangular parallelogram
370: "rectangle contained by" 370
Rectilineal angle: definitions classified 179-
81: rectilineal figure 187: "rectilineal
segment" 196

Reductio ad absurdum 134: described by
Aristotle and Proclus 136: synonyms for,
in Aristotle 136: a variety of Analysis
140: by exhaustion 285, 293: nominal
avoidance of 369

Reduction (draywy), technical term, ex-
plained by Aristotle and Proclus 135:
first "reduction " of a difficult construction
due to Hippocrates 135

Regiomontanus (Johannes Müller of Königs-
berg) 93, 96, 100

Reyher, Samuel 107
Rhaeticus IOI

Rhomboid 189

Rhombus, meaning and derivation 189
Riccardi, P. 96, 112, 202
Riemann, B. 219, 273, 274, 280
Right angle: definition 181: drawing straight
line at right angles to another, Apollonius'
construction for, 270: construction when
drawn at extremity of second line (Heron)
270 ·

Right-angled triangles, rational: rule for
finding, by Pythagoras 356-9, by Plato
356, 357, 359, 360, 385: discovery of rules
by means of gnomons 358-60: connexion
of rules with Eucl. 11. 4, 8, 360: rational
right-angled triangles in Apastamba 361,
363

Röth 357-8

Rouche and de Comberousse 313
Rudd, Capt. Thos. 110

Ruellius, Joan. (Jean Ruel) 100
Russell, Bertrand 227, 249

Saccheri, Gerolamo 106, 144-5, 167-8,
185-6, 194, 197-8, 200-1
Sa'id b. Masud b. al-Qass 90
Sathapatha-Brāhmaṇa 362

Savile, Henry 105, 166, 245, 250, 262

Scalene (oxanvbs or σkaλnvns) 187-8: of
numbers (=odd) 188: of cone (Apollonius)
188

Schessler, Chr. 107

Scheubel, Joan. 101, 107

Schiaparelli, G. V. 163

Schlüssel, Christoph, see Clavius
Schmidt, Max C. P. 304, 319

Schmidt, W., editor of Heron, on Heron's
date 20-1

Scholia to Elements and MSS. of 64-74:
historical information in, 64: evidence in,
as to text 64-5, 66-7: sometimes inter-
polated in text 67: classes of, "Schol.
Vat." 65-9, "Schol. Vind." 69-70: miscel-
laneous 71-4: "Schol. Vat." partly derived
from Pappus' commentary 66: many
scholia partly extracted from Proclus on
Bk. 1. 66, 69, 72: numerical illustrations
in, in Greek and Arabic numerals 71:
scholia by Psellus 70-1, by Maximus
Planudes 72, Joannes Pediasimus 72-3:
scholia in Latin published by G. Valla,
Commandinus, Conrad Dasypodius 73:
scholia on Eucl. 11. 13 407
Schooten, Franz van 108
Schopenhauer 227, 354

Schotten, H. 167, 174, 179, 192-3, 202
Schumacher 321

Schur, F. 328
Schweikart, F. K. 219
Scipio Vegius 99

Sectio Canonis by Euclid 17

Section (Toun):=point of section 170, 171,
383: "the section"="golden section" q.v.
Segment of circle, angle of, 253: segment
less than semicircle called is 187
Semicircle 186: centre of, 186: angle of,
182, 253

Seqt 304

Serenus of Antinoeia 203
Serle, George 110

Setting-out (exeσis), one of formal divisions
of a proposition 129: may be omitted 130
Sextus Empiricus 62, 63, 184
Shamsaddin as-Samarqandi 5 n., 89
"Side-" and "diagonal." numbers, described
398-400: due to Pythagoreans 400: con-
nexion with Eucl. 11. 9, 10 398-400: use
for approximation to √2 399
Sigboto 94

"Similar" (equal) angles 182, 252: "simi-
lar" numbers 357

Simon, Max 108, 155, 157-8, 167, 202,
328
Simplicius: commentary on Euclid 27-8:
on lunes of Hippocrates 29, 35, 386-7:
on Eudemus' style 35, 38: on parallels
190-1: 22, 167, 171, 184, 185, 197, 203,
223, 224

Simson, Robert: on Euclid's Porisms 14:
on "vitiations" in Elements due to Theon
46, 103, 104, 106, 111, 148: definition
of plane 172-3: 185, 186, 255, 259, 287,
293, 296, 322, 328, 384, 387, 403
Sind b. Ali Abū t-Taiyib 86

Smith and Bryant 404

"Solid loci" 329, 330: Solid Loci of Aris-
taeus 16, 329

"Solid problems" 329, 330

Speusippus 125

Sphaerica, early treatise on, 17

Spiral, "single-turn" 122-3., 164-5: in
Pappus cylindrical helix 165
Spiral of Archimedes 26, 267

Spire (tore) or Spiric surface 163, 170;
varieties of 163

Spiric curves or sections, discovered by
Perseus 161, 162-4
Steenstra, Pybo 109
Steiner, Jakob 193
Steinmetz, Moritz 101
Steinschneider, M. 8n., 76 sqq.
Stephanus Gracilis 101-2
Stephen Clericus 47
Stobaeus 3

Stoic "axioms" 41, 221: illustrations (dely-
ματα) 329 .
Stolz, O. 328
Stone, E. 105

Straight line: pre-Euclidean (Platonic) de-
finition 165-6: Archimedes' assumption
respecting, 166: Euclid's definition, inter-
preted by Proclus and Simplicius 166–7:
language and construction of, 167, and
conjecture as to origin 168: other defi-
nitions 168-9, in Heron 168, by Leib-
niz 169, by Legendre 169: two straight
lines cannot enclose a space 195-6, can-
not have a common segment 196-9: one
or two cannot make a figure 169, 183:
division of straight line into any number
of equal parts (an-Nairizi) 326
Strömer, Mårten 113