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3. Each polygon is equal to the polygon immediately above it in the diagram plus the triangle with 1 less in its side, i.e. the triangle in the preceding column.

The vertical columns are arithmetical progressions, the common difference of which is the triangle in the preceding column.

But Plutarch, a contemporary of Nicomachus, mentioned another method of transforming triangles into squares: Every triangular number taken eight times and then increased by I gives a square'. That is,

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Diophantus generalised this proposition into his theorem for transforming any polygonal number into a square. If P be a polygonal number, a the number of angles,

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He deduces rules for finding a polygonal number when the side and the number of angles are given, and for finding the side when the number and the number of its angles are given. These fine results and the fragment of the difficult problem of finding the number of ways in which any given number can be a polygonal number no doubt represent part of the original contributions by Diophantus to the theory of that class of numbers.

1 Plat. quaest. V. 2, 4, 1003 F.





"Knowing, my most esteemed friend Dionysius, that you are anxious to learn how to investigate problems in numbers, I have tried, beginning from the foundations on which the science is built up, to set forth to you the nature and power subsisting in numbers.

"Perhaps the subject will appear rather difficult, inasmuch as it is not yet familiar (beginners are, as a rule, too ready to despair of success); but you, with the impulse of your enthusiasm and the benefit of my teaching, will find it easy to master; for eagerness to learn, when seconded by instruction, ensures rapid progress."

After the remark that "all numbers are made up of some multitude of units, so that it is manifest that their formation is subject to no limit," Diophantus proceeds to define what he calls the different "species" of numbers, and to describe the abbreviative signs used to denote them. These "species" are, in the first place, the various powers of the unknown quantity from the second to the sixth inclusive, the unknown quantity itself, and units.


A square (=x1) is dúvaμs ("power"), and its sign is a 4 with Y superposed, thus 4".

A cube (=x3) is xúßos, and its sign K".

A square-square (= x1) is dvvaμodúvaμis', and its sign is 44. A square-cube (= x3) is dvvaμóxußos, and its sign 4K”.

A cube-cube (= x) is Kußóκußos, and its sign KK.

1 The term duvaμodóvaμis was already used by Heron (Metrica, ed. Schöne, p. 48, 11, 19) for the fourth power of a side of a triangle.

H. D.


"It is," Diophantus observes, "from the addition, subtraction or multiplication of these numbers or from the ratios which they bear to one another or to their own sides respectively that most arithmetical problems are formed"; and "each of these numbers... is recognised as an element in arithmetical inquiry."

"But the number which has none of these characteristics, but merely has in it an indeterminate multitude of units (λños μоvádov άópioтov) is called ȧpiðμós, 'number, and its sign is $ [= x}."

"And there is also another sign denoting that which is invariable in determinate numbers, namely the unit, the sign being M with o superposed, thus M."

Next follow the definitions of the reciprocals, the names of which are derived from the names of the corresponding species themselves.

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and each of these has the same sign as the corresponding original species, but with a distinguishing mark which Tannery writes in the form above the line to the right.

Thus 4x=1/x2, just as y* = }.

Sign of Subtraction (minus).

“A minus_multiplied by a minus makes a plus'; a minus multiplied by a plus makes a minus; and the sign of a minus is a truncated turned upside down, thus ."

Diophantus proceeds: "It is well that one who is beginning this study should have acquired practice in the addition, subtraction and multiplication of the various species. He should know how to add positive and negative terms with different coefficients to

The literal rendering would be "A wanting multiplied by a wanting makes a forthcoming." The word corresponding to minus is Xeîpis (“wanting "): when it is used exactly as our minur is, it is in the dative Melyet, but there is some doubt whether Diophantus himself used this form (cf. p. 44 above). For the probable explanation of the sign, see pp. 43-44. The word for "forthcoming" is Grapßit, from &rdpxw, to exist. Negative terms are λείποντα είδη, and positive υπάρχοντα.

other terms', themselves either positive or likewise partly positive and partly negative, and how to subtract from a combination of positive and negative terms other terms either positive or likewise partly positive and partly negative.

"Next, if a problem leads to an equation in which certain terms are equal to terms of the same species but with different coefficients, it will be necessary to subtract like from like on both sides, until one term is found equal to one term. If by chance there are on either side or on both sides any negative terms, it will be necessary to add the negative terms on both sides, until the terms on both sides are positive, and then again to subtract like from like until one term only is left on each side.

"This should be the object aimed at in framing the hypotheses of propositions, that is to say, to reduce the equations, if possible, until one term is left equal to one term; but I will show you later how, in the case also where two terms are left equal to one term, such a problem is solved."

Diophantus concludes by explaining that, in arranging the mass of material at his disposal, he tried to distinguish, so far as possible, the different types of problems, and, especially in the elementary portion at the beginning, to make the more simple lead up to the more complex, in due order, such an arrangement being calculated to make the beginner's course easier and to fix what he learns in his memory. The treatise, he adds, has been divided into thirteen Books.


1. To divide a given number into two having a given difference.

Given number 100, given difference 40.

Lesser number required x. Therefore

2x+40= 100,
x = 30.

The required numbers are 70, 30.

2. To divide a given number into two having a given ratio. Given number 60, given ratio 3: 1.

Two numbers x, 3r. Therefore x = 15.

The numbers are 45, 15.

1 eldos, "species," is the word used by Diophantus throughout.

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