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Theudius, 124

Thompson, D'Arcy W., 37

Thymaridas, Epanthema of, 114-116
"Triple-equations" of Fermat, 163 #.,
179 n., 182 n., 202 n., 223 n., 224 n.,
246, 321-328

“Units" (μováões)=absolute term, 39-40;
abbreviation for, 39, 130
Unknown quantity (=x), called in Dio-
phantus ápoμós, "number": definition
of, 32, 115, 130; symbol for, 32-37,
130; signs for powers of, 38, 129;
signs for submultiples of unknown and
powers, 47, 130; Italian-Arabian and
Diophantine scales of powers, 40, 41;
Egyptian scale, 41; * first used by
Descartes, 50 m.; other signs for, 1 used
by Bombelli, 22, 38, N (for Numerus)
by Xylander, Bachet, Fermat and others,
38, R (Radix or Res), 38, Radice, Lato,
Cosa, 40 n.

Vacca, G., 106 n.

Valla, Georgius, 48

Vieta, 27, 38-39, 49, 50 m., 101, 102,
214 m., 285, 329, 331

Vossius, 31

Wallis, 40 n., 286, 287, 288, 289
Weber, Heinrich, 3 n.

Weber and Wellstein, 107 n., 145 n.
Wertheim, 30, 110 m., 137 m., 138 n.,
209 n., 211 n.,
254 n., 256, 257,

145 m., 151 N., 161 n.,
212 n., 216 n., 217 n.,
286 m., 294, 295
Westermann, 125 n.
Widman, 49 n.
Wieferich, 145 n.
Woepcke, 5 n.
"Wurm's problem," 123

x for unknown quantity, originated with
Descartes, 50 n.

Xylander, 17, 22–26, 27, 28, 29, 35, 38,
107-108 n., 140 n.; Xylander's MS. of
Diophantus, 17, 25, 36

Zensus (=Censo), term for square of un-
known quantity, 38

Vatican MSS. of Diophantus, 5 n., 15, 16, Zetetica of Vieta, 27, 101, 285

17

Vergetius, Angelus, 16

Zeuthen, 118-121, 205 m., 278, 281 m.,
290, 294-295

CAMBRIDGE: PRINTED BY JOHN CLAY, M.A. AT THE UNIVERSITY PRESS.

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which fraction has therefore to be made a square.

One obvious case is obtained by putting kab, for then

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whence y = a, x = 1, so that p = a, q = ab, r = ab, s = a, and the result is

only the obvious case where p =s, q = r.

Following up this case, however, let us put kab (1 + 2).

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therefore, multiplying numerator and denominator by 1-s and extracting the square root, we obtain

y

x

a √{(b2 − 1)2 + ( 36a − 1) (ba — 1) 3 + 362 (ba − 2) s2 + b2 (b2 — 4) s3 — b2s‘}

To make the expression under the radical a square, equate it to

{(b2 − 1) + ƒz + gz2}2,

and assume ƒ, g such that the terms in 5, s2 vanish.

In order that the term in s may vanish, ƒ = } (362 – 1), and, in order that the term in may disappear,

whence

362 (62-2)=2(b2 − 1 ) g + ƒ2 = 2 (b2 − 1) g + † (9b1 — 6b2 + 1),

36-1862-1

g= 8 (62-1)

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Now b can be chosen arbitrarily; and, when we have chosen it and thence determined s, we can put

x=b-1-2, y = a (b2 − 1 + fs + gs3),

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If x, y have a common factor, we may suppose this eliminated before P, q, r, s are determined.

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Ex 1. Lei=z fær è cannot be 1, since then g would be )
Therefore

As a does not enter into the calculation, we may write 1 for it;

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Another solution in smaller numbers.

In the second of the papers quoted Euler says that, while investigating quite different matters, he accidentally came across four much smaller numbers satisfying the conditions, namely,

A = 542, B = 103, C=359, D= 514,

which are such that A+ B1 = Ca + Da.

He then develops two methods of analysis leading to this particular solution; but, while they illustrate the extraordinary ingenuity which he brought to bear on such problems, they are perhaps of less general interest than the above.

ádúvatos, "impossible," 53 ἄλογος ( = "undescribed

INDEX.

[The references are to pages.]

I. GREEK.

apparently), Egyptian name for certain powers, 41 αόριστος, indeterminate: πλῆθος μονάδων αόριστον, an undetermined number of units=the unknown, åpɩ@μós, i.e. x, 32, 115, 130; ἐν τῷ ἀορίστῳ, indeterminately, or in terms of an unknown; 177 ἀριθμητική distinguished from λογιστική, 4 åpiðμós, number, used by Dioph. as techni

cal term for unknown quantity (=x), 32, 115, 130; symbol for, 32-37, 130 ȧрilμоσтby (=1/x) and sign for, 47, 130 ATоTOS, "absurd," 53

διπλή ισότης οι διπλοϊσότης, double-equation,

q.v.

dúvams, "square," used for square of unknown (=x2): distinguished from Teтpáγωνος, 37-38 ; sign for, 38, 129; τετραπλῆ dúvaus, "quadruple-square," Egyptian name for eighth power, 41 δυναμοδύναμις, fourth power of unknown (x), sign for, 38, 129 δυναμοδυναμιστόν, submultiple of δυναμοδύvajus ( = 1/x1) and sign, 47, 130 duvaμbкußos, "square-cube" (=x3), sign for, 38, 129

δυναμοκυβοστόν, submultiple of δυναμόκυβος (=1/6) and sign, 47, 130 δυναμοστόν, submultiple of δύναμις (= 1/x2) and sign, 47, 130

eldos, "species," used for the different terms in an algebraic equation, 7, 130, 131 ἔλλειψις, “deficiency”: ἐν ἐλλείψεσί τινα elon, "any terms in deficiency," i.e. “any negative terms,” 7, 131

évváρxorra, "existent," used for positive terms, 7, 130

émávonua ("flower" or "bloom") of Thymaridas, 114-116

loos, equal, abbreviation for, 47-48

kʊßbкußos, “cube-cube," or sixth power of unknown (=x), and sign for, 38, 129 KUBOKUBOσTÓν (= 1/x) and sign, 47, 130 kúßos, cube, and symbol for cube of un-· known, 38, 129; kúßos ¿žeλikтós, Egyptian term for ninth power (x), 41 KUBOσTOV (=1/x) and sign, 47, 130

Xelwew, to be wanting: parts of verb used to express subtraction, 44; λelwovra elồn, negative terms, 130

Xeys, "wanting," term for subtraction or negation, 130; \elyei (dat.)=minus, sign common to this and parts of verb λείπειν, 41-44 MoyaTikh, the science of calculation, 111; distinguished from ἀριθμητική, 4

μείζων ἢ ἐν λόγῳ, 133π., 144 κ. μépos, "part," an aliquot part or submultiple; μépn, "parts," used to describe any other proper fraction, 191 μηλίτης ἀριθμός (from μῆλον, an apple), 4, 113

μovás, "unit," abbreviation for, 39, 130 Mopiaσriká, supposed work by Diophantus, 3-4

μορίου, οι ἐν μορίω, expressing division or a fraction, 46, 47 μυριὰς πρώτη, δευτέρα, 47-48

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