148

HISTORY OF ALGEBRA.

TRACT 35.

unknown quantity he calls simply a-pdpo;, numerus, thenumber ; and in the solutions he commonly marks it by thefinal thus ; ; also he denotes the monades, or indefinite' unit,by yw.°. Diophantus then remarks on the multiplication anddivision of simple species together, showing what powers orspecies they produce; declares that minus (Aenf'ij) multipliedby minus produces plus (yitapfyv ); but that minus multipliedby plus, produces minus; and that the mark used for minusis /j», namely the $ inverted and curtailed; but he uses nomark for plus, but a word or conjunction copulative.

As to the operations, viz, of addition, subtraction, mul-tiplication, and division of compound species, or those con-nected by plus and minus, Diophantus does not teach, butsupposes his reader to know them. He then remarks on thepreparation or simplifying the equations that are derivedfrom the questions, which we call reduction of equations,by collecting like quantities together, adding quantities thatare minus, and subtracting such as are plus; called by themoderns transposition, so as to bring the equation to simpleterms, and then depressing it to a lower degree by equal di-vision; when the powers of the unknown quantity are inevery term: which preparation, or reduction of the com-plex equation, being now made, or reduced to what wecall a final equation, Diophantus goes no further, but barelysays what the root or res ignota is, without giving any rulesfor finding it, or for the resolution of equations; therebyintimating that such rules were to be found in some otherwork, done either by himself or others.

Of the body of the work, lib. 1 contains forty-three ques-tions, concerning one, two, three, or four unknown num-bers, having certain relations to each other, viz, concern-ing their sums, differences, ratios, products, squares, sumsand differences of squares, &c, &c ; but none of them con-cerning either square or cubic numbers. Lib. 2 contains .thirty-six questions. The first five questions are concerningtwo numbers, though only one condition is given in each