Book VI

. contains problems of finding rational _right-angled triangles_ such that different functions of their parts (the sides and the area) are squares. A word is necessary on Diophantus' notation. He has only one symbol (written somewhat like a final sigma) for an unknown quantity, which he calls [Greek: arithmos] (defined as "an undefined number of units"); the symbol may be a contraction of the initial letters [alpha][rho], as [Delta]^[Upsilon], [Kappa]^[Upsilon], [Delta]^[Upsilon][Delta], &c., are for the powers of the unknown ([Greek: dynamis], square; [Greek: kubos], cube; [Greek: dynamodynamis], fourth power, &c.). The only other algebraical symbol is [graphic: /|\] for minus; plus being expressed by merely writing terms one after another. With one symbol for an unknown, it will easily be understood what scope there is for adroit assumptions, for the required numbers, of expressions in the one unknown which are at once seen to satisfy some of the conditions, leaving only one or two to be satisfied by the particular value of x to be determined. Often assumptions are made which lead to equations in x which cannot be solved "rationally," i.e. would give negative, surd or imaginary values; Diophantus then traces how each element of the equation has arisen, and formulates the auxiliary problem of determining how the assumptions must be corrected so as to lead to an equation (in place of the "impossible" one) which can be solved rationally. Sometimes his x has to do duty twice, for different unknowns, in one problem. In general his object is to reduce the final equation to a simple one by making such an assumption for the side of the square or cube to which the expression in x is to be equal as will make the necessary number of coefficients vanish. The book is valuable also for the propositions in the theory of numbers, other than the "porisms," stated or assumed in it. Thus Diophantus knew that _no number of the form 8n + 7 can be the sum of three squares_. He also says that, if 2n + 1 is to be the sum of two squares, "n must not be odd" (i.e. _no number of the form 4n + 3, or 4n - 1, can be the sum of two squares_), and goes on to add, practically, the condition stated by Fermat, "and the double of it [n] increased by one, when divided by the greatest square which measures it, must not be divisible by a prime number of the form 4n - 1," except for the omission of the words "when divided ... measures it."

# AUTHORITIES.--The first to publish anything on Diophantus in Europe was Rafael Bombelli, who embodied in his Algebra (1572) all the problems of Books I.-IV. and some of