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This Elibron Classics variation is a facsimile reprint of a 1905 version by means of Macmillan and Co. , Ltd. , London.

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4J, 1. 1^-, The sum ^, and to this Hence the quantity of these the to be sum is is 23 | J Add f. added -J, to the given fraction is ^ ^. By what must ^ -^ be mon denominator O = 9' then taken "J Again Q Ql = oi' multiplied to give | ? The com then T^ = -TO^J TTO = i, their -^^ 11^ 2o Since 2 + H- i = 3^, take first A, is 28, -*-^ Hence ^, then half of that' /o and we have ft ^o ^^co^es on iV dhr |multiplication by 1| J. These examples disclose methods quite foreign to modern mathematics. 2 One process, however, found extensive appli- J.

His directions are " If now in any problem the same powers of the unknown : occur on both sides of the equation, but with different coeffi cients, we must subtract equals from equals until we have one term^equal to one term. If there are on one side, or on both sides, terms with negative added on both tive terms. " 1 Thus nowadays achieved by transposing, simplifying, and dividing by the coefficient of x was accomplished by Diophan It is to be observed that in tus by addition and subtraction. is } Diophantus, and in fact in all writings of antiquity, the con ception of a quotient is wanting.

Thus, i J. As Thus, pt/3" WW = with the Egyptians; unit-fractions in juxtaposition are to be added. Like the Eastern nations, the Egyptians and Greeks employ two aids is not to computation, the abacus known what and finger symbolism. If the signs used in the latter were, but by the study of ancient statuary, bas-reliefs, and paintings this secret may yet be unravelled. Of the abacus there existed many forms at different times and among the various nations. In all cases a plane was divided into regions and a pebble or other We object represented a different value in different regions.