SCIENCE. 



[Vol. XVIII. No. 452 



writtea out, while at other times the case-termination is 

 written above to the right, thus f"", the symbol being also 

 generally doubled when the signification is plural. 



Diophantos indicated addition merely by juxtaposition, hav- 

 ing uo sign for plus; for minus, however, he used the sign qi. 

 As a consequence, in order to avoid confusion, he was obliged 

 to do two things; first, to designate the absolute term as so 

 many lAOva Se?, or units, abbreviated into /.i", and second, to 

 write all the negative terms together after the positive. Thus 

 the quantity x^ — 5x^ -\-8x — 1 would be written in Dio- 

 phantos's notation. 



}i" a ?? »' ?7 qi. 6^ e ja° a. 



This may be rendered more expressive if we change it by 

 substituting Arabic numerals, and putting U for units. N for 

 number or unknown, S for square, and C for cube: thus it 

 becomes C1N9, — S5U1. 



It is to be noted that Diophantos and his successors up to 

 comparatively recent times had no conception whatever of 

 an intrinsically negative quantity as possible. Whatever 

 sign may have been used for minus was considered as simply 

 indicating that one number was to be subtracted from an- 

 other, and if the subtrahend were larger than the minuend 

 uo meaning was attached to the expression. 



It is possible that Diophantos might have been able to 

 escape from the limitations of his system if the letters of the 

 Greek alphabet had not been already appropriated for the 

 representation of particular numbers, thus precluding their 

 use as symbols of quantity in general. 



It may be of interest to give at this point specimens of the 

 purely rhetorical and of the syncopated methods of solution. 

 They are given by Nesselmann, and are verbatim transla- 

 tions from the original toilgues. The first is a solution of a 

 quadratic equation by Mohammed ibn Musa, and the second 

 the solution of a problem by Diophantos. 



A square and ten of its roots are equal to nine-and-thirty 

 units, that is, if you add ten roots to one square, the sum is 

 equal to nine-and-thirty. The solution is as follows: halve 

 the number of roots, that is, in this case, five; then multiply 

 this by itself, and the result is five-and-twenty. Add this to 

 the nine-and-thirty, which gives four and sixty ; take the 

 square root, or eight, and subtract from it half the number 

 of roots, namely, five, and there remains three: this is the 

 root of the square which was required and the .square itself 

 is nine. 



(S = square, N =^ number, Z7=unit, as above.) 



To divide the proposed square iato two squares: Let it be 

 proposed, then, to divide 16 into two squares; and let the 

 ifirst be supposed to be one square. Thus 16 minus one square 

 must be equal to a square. I form the square from any 

 number of iV's minus as many ?7's as there are in the side 

 of 16 U's. Suppose this to be 2 iV's minus 4 U's. Thus the 

 square itself will be 4 squares 16 Vs minus 16 N's. These 

 are equal to 16 units minus 1 square. Add to each the nega- 

 tive term, and take equals from equals. Thus 5 squares are 

 equal to 16 numbers. One (square) will be 256 twenty-fifths, 

 and the other 144 twenty-fifths, and the sum of the two makes 

 up 400 twenty-fifths, or 16 units, and each is a square. 



Compare these long-drawn-out statements with their equiv- 

 alents in modern notation: 



First. 



x^ -\- IQx = 39 

 a;3-(-i0x + 3.5 = 64 

 .-. x + 5=8 

 .•. a; = 3 



Second. 



16 — a;- = D =(2x — 4)^ 

 = 4x3 -{- 16 — 16a; 

 .". 16a; ^DX^ 

 .-. x = ^ 



The example from Diophantos evidently does not take full 

 advantage of his notation, for the symbol for minus is not 

 used, and in several cases the words are written out in full 

 where abbreviations might have been employed. Further, 

 no use is made of the symbol for equality, viz., i, an abbre- 

 viation for iGoi, which is elsewhere used by the author. If 

 the fullest use of the syncopated notation had been made, 

 the solution would have been somewhat comparable for con- 

 ciseness and brevity with the modern method, only about 

 twice as many characters and marks being required. Solu- 

 tions in this abbreviated form appear on the margins of 

 Diophantos's manuscripts, but they are believed to have been 

 added by some one else, and not to be due to the author him- 

 self. 



The works of Diophantos, called by him "Arithmetics," 

 deal largely with indeterminate equations and the theory of 

 numbers. Quadratic equations are constantly solved, but 

 only real positive results are recognized or considered; and 

 even when there are two positive roots, only one is taken 

 account of. One very simple case of an equation of the third 

 degree is found. 



We will turn next to the consideration of the ancient 

 algebra of India. There lived at Patna, in India, some time 

 in the sixth century of our era, a mathematician named 

 Arya-Bhatta, who wrote a work treating of arithmetic, 

 algebra, geometry, trigonometry, and astronomy. It consists 

 in the enunciation of rules and propositions in verse. The 

 author gives, of course in a purely rhetorical manner, the 

 sums of the first, second, and third powers of the first »i natural 

 numbers, the general solution of a quadratic equation, and 

 the solution in integers of some indeterminate equations of 

 the first degree. 



The only other ancient Indian mathematician of promi- 

 nence is Brahmagupta, who lived in the seventh century of 

 our era. His work is also written in verse, and is called 

 " Brahma-Sphuta-Siddhauta," or the " System of Brahma in 

 Astronomy." Two chapters of this work deal with arith- 

 metic, algebra, and geometry. The treatment of algebra is 

 purely rhetorical, and includes a discussion of arithmetical 

 progressions, quadratic equations (only the positive roots 

 being considered), and indeterminate equations of the first 

 degree, together with one of the second degree. 



These Indian writings are of special interest as being the 

 sources from which the Arabs derived their first knowledge 

 of algebra. They obtained from the Greeks before a.d. 900, 

 thorough translations of Euclid, Apollonius, Archimedes, 

 and others, a knowledge of geometry, mechanics, and astron- 

 omy, but had no translation of Diophantos till a hundred 

 and fifty years later, when they had themselves already 

 made considerable progress in algebraic analysis. From the 

 Arabians in turn western Europe obtained, not only the deci- 

 mal notation of arithmetic, but also its first knowledge of 

 other branches of mathematics. 



The first great mathematician among the Arabs is gener- 

 ally known by the name of Alkarismi, though this is an in- 

 correct transliteration of only one of his names. From the 

 title of his work, " .\l-gebr we'l Mukabala," we have the 

 name of that branch of mathematics under consideration, 

 al-gebr signifying that the same quantity may be added to 

 or subtracted from both sides of an equation. 



Alkarismi treats the qitadratic, giving geometric proofs of 

 rules for the solution of different cases, and recognizing the 

 existence of two roots, though he only considers such as are 

 real and positive. He treats only numerical equations, and 

 no distinction is made between arithmetic and algebra. This 



