379 



VIQA GANITA. 



VIGA GANIIA. 



333 



everything by words at length. The description of the Hindoo nota- 

 tion always led us to suspect that there was some communication with 

 Hindoo algebra over and above that which was marie through the 

 Arabs; and the preceding account, with that which follows, will lead 

 every one who knows the history of algebra to wish that there had 

 been more of it. 



The Yiga Ganita contains aa follows, it being presumed that the pre- 

 ceding aocount of Hindoo notation will prevent the reader from 

 imagining that the algebraical symbols which we here employ are con- 

 tained in the work : The rules for addition, subtraction, multiplica- 

 tion, and division of positive and negative quantities : the rules for the 

 square and square roots of the same, it being distinctly specified 

 that the square root of a negative quantity is imaginary. Rules for 

 the cipher, as in the Liliwati; but here it is more distinctly stated that 

 " the fraction of which the denominator is cipher is termed an infinite 

 quantity." The commentator Christina is well worth quoting on this 

 poiut : "As much as the divisor is diminished, so much is the quo- 

 tient increased. If the divisor be reduced to the utmost, the quotient 

 is to the utmost increased. But if it can be specified that the amount 

 of the quotient is so much, it has not been raised to the utmost, for a 

 quantity greater than that can be assigned. The quotient therefore is 

 indefinitely great, and is rightly termed infinite." Then follow arith- 

 metical operations on unknown quantities, and combinations of 

 them. Surds, the usual operations on them, the rationalization of 

 surd denominators, and the extraction of square roots. The rule for 

 the extraction of such a surd as the square root of a + V b + Vc + V d 

 is worth citing as a proof of the decided character of their knowledge 

 of this part of algebra. Let V (a? b c) e, ^ (a + e) = f, 

 ^ ( a e) = g, */ (f' d ) = A ; then the square root required is 



The Cuttaca, or pulverizer, is the rule for the solution, in integers, 

 of ax j; by = c ; a, b, and c being integers. There is no need to 

 describe it, as it is the rule which' is now found in every European 

 book on the theory of numbers, and which proceeds by resolving 

 a _:_ J into a continued fraction. The Hindoos give no use of con- 

 tinued fractions except in this rule, though it is obvious, from the skill 

 with which they manage the reduction of fractions to nearly equal 

 fractions of more simple terms, that they must have applied continued 

 fractions, directly or indirectly, probably by means of this very rule. We 

 do not mean to say that they had continued fractions, but only the pro- 

 cesses involved in the use of them, and power of attaining their results. 



The Varga-pracriti, or principle of the square, is a rule which is 

 remarkable, as the whole of it was not used in Europe till after the 

 middle of the last century. It consists in a rule for finding an indefi- 

 nite number of solutions of y* = ax'+l (a being an integer which is 

 not a square) by means of one solution given or found, and of feeling 

 for one solution by making a solution of 7/ 2 = ax-\-b give a solution of 

 y 2 = ax z + b-. It amounts to the following theorem : If p and q be one 

 set of values of x and y in y* = ax- + b, andp' and q' the same or another 

 set, then qp + pq' and app' -f qq' are values of x and y in y 2 = oa^+ 

 From this it is obvious that one solution of y == ax 2 + 1 may be made 

 to give any number, and that if, taking b at pleasure, y' 2 = ax 2 + b- can 

 be solved so that x and y are divisible by 6, then one preliminary 

 solution of y" = ax"' + 1 can be found. Another mode of trying for solu- 

 tions is the combination of the preceding with the Cuttaca, as follows : 

 Let y = q, x = p, satisfy y^ = ax- + b : then solve pz + q = bw, and 



and will be a square. It is then said that y" = ax- 1 is impossible 

 unless a be the sum of two squares; and some miscellaneous pro- 

 visions are then given. 



The chapter on simple equations requires no particular description ; 

 many of the examples are geometrical, as Given the sides of a triangle 

 to find the perpendicular. In the chapter on quadratic equations the 

 well-known rules are given, and some cubic and biquadratic equations 

 (special cases of course) are solved by completion of the cubes and 

 squares. The two roots are mentioned, when positive, and it is said, 

 " people do not approve, an absolute negative number," on which the 

 commentators speak as if the negative roots were seen, but not ad- 

 mitted. The property of the right-angled triangle is proved in a 

 twofold way : first, by the similarity of the right-angled triangles 

 formed by the perpendicular on the hypothenuse to the whole and 

 to one another : next, by the method called Indian. Various of the 

 propositions in Euclid's second book are proved. In the chapter on 

 equations of more than one unknown quantity questions both of the 

 determinate and indeterminate kind are considered. 



In the next chapter are considered the equations ax + bx* = < 



w 3 : "in what period is the sum of a progression continued to a certain 

 period tripled, its first term being three, and the common difference 

 two;" ax- + by" v- and ax* by- + 1 =w 2 ; x 3 + y" = v- and x 



x = p 3 , and v + w + 1 + u + p + 2 = s 2 ; x- + y 2 + 1 = t-, x- y + 1 it-, 

 + y- 1 = v- and x- y' 2 1 =w- ; Sx + l = v- and ox + l = w-; 

 x + 1 = t> : ' and 3t> 2 + 1 w- ; 2ar 2 2y- + 3 = v- and 3x- + 3 y- + 3 



y; vwxy = 20 (v+w + x + y) ; x + y + x- + y- + xy = (23 x y)-; 



x + 3y + 2 = xy; 2xy=5& lOa; 14 y. 



Mr. Colebrooke has also given the algebra of Brahinegupta, being a 

 chapter of the Brahmc-sphuta-siddhanta. It contains the operations 

 of algebra, barter, interest, progression, plane geometrical questions 

 [the ratio of the circumference to the diameter is called 3 for practice, 

 and A/ 10 for more accuracy), and many of the more practical applica- 

 tions of arithmetic, as in the Liliwati. Also the Cuttaca, simple and 

 quadratic equations, the indeterminate equation y- ax- + b, and 

 miscellaneous problems. The whole of this algebra is contained in 



olebrooke's 'Algebra, with Arithmetic and Mensuration, from the 

 Sanscrit of Brahinegupta and Bhascara,' London, 1817. Dr. John 

 Taylor, in 1816, published at Bombay a translation of the Liliwati 

 from the Persian, with an appendix on the mode in which arithmetic 

 is now taught in Hindoo schools; and (London, 1813) Mr. Edward 

 Strachey published a great part of the Viga Ganita, also from the 

 Persian, with Mr. Davis' s notes. It remains to mention that, by the 

 extracts which were made from the Surya Siddhanta, it appears that 

 the Hindoo arithmetic of sines was more perfect than could be 

 gathered from what is said of the mode of finding chords in the 

 Liliwati. They had a table of sines, calculated by the method of 

 second differences for every 3|" from to 90 ; and among their 

 astronomical uses of this table is one which is equivalent to the equa- 

 tion d (sin a) = cosa da. (Delambre, 'Astron. Anc.,' i. 45G.) The 

 minimum of trigonometrical formula) which Delambre allows them 

 (and he never grants them more than the barest minimum) amounts to 



sin 1 ^ + cos 2 a; = 1, sin 30 = \ t sin 60 = ^ ^3 

 sin 2 7^ A = \ (1 cos A) ; 



but how they were to find out a theorem equivalent to A 2 am a: = 

 4 sin'- ^ A x sin x, with only this amount of formula), he does not 

 say. 



The Mohammedans brought but a small part of this splendid body 

 of algebra into Europe. The work of Mohammed-ben-Musa, which is 

 ufficiently shown by Dr. Rosen hi his translation to have had an 

 Indian origin (and indeed no one now questions that origin), contains 

 merely simple and quadratic equations of the determinate kind, applied 

 to various questions connected with pecuniary transactions. The 

 algebra of Diophantus is more Indian in its character, as it treats 

 entirely of those problems which are therefore called Diophantt/"., 

 namely, integer solutions of indeterminate equations. It is, to all 

 appearance, a part of the Indian algebra, similar in its contents to 

 some of the classes of problems which fill the two last chapters of the 

 Viga Ganita, translated into that strict and consecutive mode of 

 demonstration which the Greek mathematicians (fortunately for us) 

 never dispensed with. But, while granting to the first European 

 algebraist full credit "for the superior completeness of his mode of 

 exposition, every comparison confirms us more and more in the 

 impression that the Hindoo was his teacher : whether we consider the 

 probable era of the older Indian algebraists, or the contents of the 

 book itself, it is difficult to come to any other conclusion. The extra- 

 vagant mania of Bailly, and the reaction caused by the writings of 

 Delambre, have left no medium opinion upon Hindoo antiquity; 

 and conclusions founded on the most sober views of history, and the 

 most usual modes of chronological reasoning, have been entirely kept 

 out of sight. In both our suspicions with respect to ancient inter- 

 course between the two nations, namely, that the Indians received 

 some astronomy between the time of Hipparchus and Ptolemy, and 

 communicated some algebra, which was finally systematised by Dio- 

 phantus, we think we derive some support from the period at which 

 the Grecian kingdom of Bactria was in existence. That principality 

 was governed and partly colonised by Greeks at a time when the 

 discoveries of Hipparchus must have been in the hands of Greek 

 astronomers, if of those of any country ; and to put a difficulty in the 

 way of Bactrian Greeks knowing of Hipparchus, is to put a much 

 stronger one in the way of Hindoos having the same information. 

 Again, though it is possible that Hindoos might have taught algebra 

 to Greeks in Bactria, it is impossible that the latter could have com- 

 municated it to the former, since Bactria ceased to be a Grecian 

 kingdom about B.C. 140; and Diophantus, though his time is not 

 known, has never been supposed to have lived till two or three 

 centuries after the Christian era. Granting, which is likely enough, 

 that Greeks remained in Bactria after their government was over- 

 thrown by the Scythians, and that they retained the knowledge of 

 Grecian arts ; granting also that the descendants of these same Greeks 

 became in time incorporated with the Hindoo race after Vicramaditya 

 had checked the advance of the Scythians, and established a govern- 

 ment which was likely enough to attract the remaining Greeks of 

 Bactria, and more particularly the learned among them this, though 

 a reasonable account of the transmission from Greece to India of the 

 astronomy of Hipparchus, gives no clue whatever to that of the 

 algebra. Colebrooke's researches give a chain of algebraical writers 

 who are cited, each by his successor, and who begin (even upon his 



