al-Karkhi, 5, 41 m.

Arabian scale of powers of unknown
compared with that of Diophantus,

40, 41

Arabic versions and commentaries, 19
Archimedes, 11, 13, 35, 278, 279, 390;
Codex Paris. of, 48; Cattle-Problem,
121-124, 279; Arenarius, 35, 122
Arenarius of Archimedes, 35, 122
Arithmetica of Diophantus: different titles
by which known, 4-5; lost Books, 5-12;
division into Books, 5, 17-18; notation
in, 31-53; conspectus of probleins in,
260-266

Arithmetical progression, summation of,
248-249

Ars rei et census, 20
Aryabhata, 281

MS. of Euclid, 35

Cracow MS. of Dioph., 5., 14, 18
Cube: Vieta's formulae for transforming
the sum of two cubes into a difference
of two cubes and vice verså, 101-103;
Fermat's extensions, ibid.; a cube cannot
be the sum of two cubes, 144 ".; Euler's
solution of problem of finding all sets of
three cubes having a cube for their sum,
329-334; sign for cube of unknown or
x3, 38, 129

"Cube-cube" (=sixth power of unknown,

or x), sign for, 38, 129
Cubic equation, sinple case of, 66-67,

Diophantus: spelling of name, 1; date,
1-7; epigram on, 3; works, 13; in
Arabia, 5-6, 19; "Pseudepigraphus,"
12, 31; MSS. of, 14-18; commentators
and editors, 18-31; notation of, 32-53;
methods of solution, 54-98; porisms
of, 3, 8-10, 99-101; other assumptions,
103 sqq.; theorems in theory of numbers,
105-110; on numbers which are the
sum of two squares, 105-106; numbers
which are not the sum of two squares,
107-108; numbers not sum of three
squares, 108-109; numbers as sums of
four squares, 110; Dioph. not inventor
of algebra, 111-116; nor of indeter-
minate analysis, 115-124; his work
a collection in best sense, 124; his ex-
tensions of theory of polygonal numbers,
127

Division, how represented by Dioph.,
44-47

Doppelmayer, 2015.

Double-equations (for making two ex-
pressions in x simultaneously squares),
II, 73--87, 91-92; two expressions of
first degree, 73-80, 80-81 m.; two ex-
pressions of second degree or one of first
and one of second, 81-37; general rule
for solving, 73, 146; double equations
for making one expression a square and
another a cube, 91-92

Dudicius Sbardellatus, Andreas, 17, 25

Egyptians: hau, sign for, 37; names for
successive powers, 41; beginnings of alge-
bra, hau-calculations, 111-111; method
of writing fractions, 113
Eisenlohr, 113 M.

Eneström, 63 n., 286 n.
Epanthema of Thymaridas, 114-116
Epigrams, arithmetical, in Anthology, 113-
114; on Diophantus, 3; one in Dio-
phantus (V. 30), 124

Equality: abbreviation for, 47-48; sign in
Xylander, 48; the sign = due to Recorde,
50 n.

Equations, see Determinate, Indeterminate,
Double, Triple, etc.

Eratosthenes, 121

Euclid, 8, 11, 12, 19, 63, 117, 124, 132 N.,
144 m., 191
Eudoxus, 124

Euler, 56, 71-72 n., 83-85'n., 86 n., 90 m.,

"False supposition," use of, in Egypt,
112-113

Fermat, 28, 29, 30, 38, 78, 90, 101,
101, 103, 106, 107, 108, 109, 110,
144-145 m., 163 m., 173 n., 179–180 n.,
182 m., 183 m., 184 N., 188 m., 190-
191 m., 197 m., 202 m., 204 m., 205 m.,
213-214 m., 218 m., 220 N., 223 m.,
229 m., 230 m., 231 m., 232 m., 233 m.,
235 m., 236 n., 239 m., 240 m., 241 M.,
243 m., 246, 254 m., Supplement, 267-
328 passim, 364; "great theorem of
Fermat," 144-145 #.; Fermat on num.
bers which are, or are not, the sums of
two, three, or four squares respectively,
100-110, 267-275; on numbers of form
x3-ay3 or 2.x3-y3, 2;6–277, of form
*3+373, 275, and of form x2+5oa, 276,
277; on equation x3 – Aμ3= 1, 385–287;

- cannot be solved in integers,
224, 293-297; problems on right-angled
triangles, 204–205 m., 218-219 n., 220 n.,
229 n., 230 n., 231–233 n., 235 n., 236 n.,
239-240 m., 297-318; Fermat's "triple-
equations," 321-328

Fractions: representation of, in Diophantus,
44-47; sign for, 45; for 4, 45;
sign for submultiple, 45-47
Frénicle, 102 m., 276, 277, 285, 287,
295-297, 309, 310, 313, 314

Gardthausen, 35, 36

Geminus, 4

Georg v. Peurbach, 20

Georgius Pachymeres, 18, 19, 31, 37

Girard, Albert, 30, 106 n.

Gnomons, 125

Gollob, 14, 18

Hérigone, 50 n.

Heron, 12, 13, 35, 36, 43, 44, 45, 63,
129 n.

Hippocrates of Chios, 63, 124
Holzmann, Wilhelm, see Xylander
Hultsch, 2., 3, 4, 9, 10, 11, 12, 19.g.
35, 36, 37, 47 m., 63 m., 118 n., 123, 253 m.
Hydruntinus, Ioannes, 16
Hypatia, 5, 6, 14, 18
Hypsicles, a; on polygonal numbers,
125-126, 252, 253

Iamblichus, 2, 3, 37 m., 49, 50, 115-116, 126
Ibn abl Uşaibi'a, 19
Ibn al-Haitham, 19

Identical formulae in Diophantus, 104, 105
Indeterminate equations: single, of second
degree, 67-73; of higher degrees, 87-
91; how to find fresh solutions when one
is known, 68--10; double-equations for
making two expressions simultaneously
squares, 11, (1) two expressions of first
degree, 73-80, 80-82 n., (2) two of second
degree, or one of second and one of first,
81-87; double-equations for making one
expression a square and another a cube,
91-92; rule for solving double-equations
in which two expressions are to be made
squares, 73, 146; indeterminate equations
in Anthology, 114; other Greek ex-
amples, 118-121; 2x2-y2= ± 1 solved
by Pythagoreans, 117-118, 278, 310
"Indian method," 12-13, 21 n.
Indian solution of x-A-1, 281-285,
190, 292

Inventum Novum of J. de Billy, 28,
165 m., 184 m., 198 n., 204 n., 205 n.,
221 n., 230 n., 231 n., 239 n., Supple
ment, 267-328 passim

Ioannes Hydruntinus, 16

Ishaq b. Yūnis, 19

Italian scale of powers, 40, 41

50 n.

Greater and less, signs for, 50 n.

Günther, 6, 278n., 279 n.

Ka'b, Arabic term for cube of unknown,

41 m.

Negative quantities not recognised by
Diophantus as real, 52-53
Nesselmann, 6-10, 21 m., 25, 26 m., 29,

33, 34, 49-51, 55-58, 67, 87, 89, 93,
108 m., 140 m., 173 m., 204 m., 207 m.,
252 m,, 329 n.

Nicomachus, 2, 126, 127

Notation, algebraic: three stages, 49–51;
Diophantus' notation, 32-49, 51-52
Numbers which are the sum of two squares,
105-107, 268-271; numbers which are
not, 107-108, 271-272; numbers which
are the sum of three squares, 272-273;
numbers which are not, 108-109, 273;
numbers not square are the sum of two,
three or four squares, 110, 273, 2743
corresponding theorem for triangles,
pentagons, etc., 188, 273

Numerus, numero, term for unknown quan.
tity, 38, 40

Nuñez, 23

Oughtred, 50 n.

Ozanam, 288

Pachymeres, Georgius, 18, 19, 31, 37
Paciuolo, Luca, 21, 40

Pappus, 11, 13

Papyrus Rhind, 113; Berlin papyrus
6619, 112

Paris MSS. of Diophantus, 15, 16, 18
Pazzi, A. M., 31

Pell, John, 31, 286 m., 288

"Pellian" equation, origin of this er
roneous term, 286
Peurbach, G. von, 20

Philippus of Opus on polygonal numbers,
125

Planudes, Maximus, 13, 14, 19, 21, 31,
43, 44, 45, 46, 48

Plato, 4, 38 m., 111, 113, 116, 125
Plus, signs for, 22, 49 m.; expressed in
Diophantus by juxtaposition, 39
Plutarch, 127

Polygonal Numbers, treatise on, 3, 11-12,
247-259; sketch of history of subject,
124-127; began with Pythagoreans, 134–
125; figured by arrangement of dots,
125; Hypsicles on, 125-126, 252, 253;
Diophantus' extensions, 137

Porisms of Diophantus, 3, 8-10, 99–101,
201, 202, 214
Poselger, 30, 98

Powers of unknown quantity and signs

Rhind Papyrus, 112

Right-angled triangles in rational numbers

in Diophantus, 93–94, 105-106; method
of "forming," 93-94; other methods of
forming attributed to Pythagoras, 116-
117, and to Plato, 116-117; Euclid's for-
mula for, 117, 120; Pythagorean formula
once used by Diophantus, 242; Greek
indeterminate problems on, other than
thuse of Dioph., 119-121; Fermat's
theorems and problems on, 204-205 m.,
218-219 n., 220 n., 229 n., 230 m., 231–
· 232 m., 235 n., 236 n., 239–240 m., 293–

318, 364-371

Rodet, 34, 35

Rosen, 50

Rudio, 63 m.

Rudolff, Christoff, 23, 50%.

Salmasius, Claudius, 17

Sand-reckoner of Archimedes, 123
Saunderson, N, 87 m.

Schaewen, P. v., 327, 328
Schmeisser, 31

Schöne, 43, 45, 118

Schreiber, H., see Grammateus
Schuler, Wolfgang, 24

Schulz, 9, 11, 18, 30, 31, 108 #.. 140#.,
219.

Sebastian Theodoric, 24
Serenus, 12

"Side"=square root, 65 #.

"Side-" and "diagonal-" numbers, Py-
thagorean solution of 2-3±1 by
means of, 117-118, 278, 310
Simon Simonius Lucensis, 25
Simplicius, 63 n.
Sirmondus, J., 27

Smith, H. J. S., 292

"Species" (elon) of algebraical quantities,

Squares: numbers as sum of two, three,
or four, 110, 273, 274; of two, 105-107.
268-271; not of two, 107-108, 271–272;
of three, 272-273; not of three, 108,
109, 273

Stevia, Simon, 29, 30 n.

Stifel, M., 23, 49 n., 50n.
Submultiples, sign for, 45-47; decom-
position of fractions into, 46, 112;
submultiples of unknown and powers,

47
Subtraction, symbol for, 41-44
Suidas, 1, 18, 22

Surdesolides, sursolida or supersolida, 41
Surds, 23-24
Suter, H., 19#.