377 



VIGA GANITA. 



VIGA GANITA. 



378 



Why not recognise it in this multiform earth? As heat is in the sun 

 and fire, coldness in the moon, fluidity in water, hardness in iron ; so 

 mobility is in air, and immobility in the earth, by nature. How 

 wonderful are the implanted faculties ! The earth possessing an 

 attractive force " (like the attraction of the loadstone for iron, adds a 

 commentator), " draws towards itself any heavy substance situated in 

 the surrounding atmosphere, and that substance appears as if it fell. 

 But whither can the earth fall in ethereal space, which is equal and 

 alike on every side ? Observing the revolution of the star?, the Baudd- 

 has (Jains) acknowledge that the earth has no support, but as nothing 

 heavy is seen to remain iu the atmosphere, they thence conclude that 

 it falls in ethereal space. Whence dost thou deduce, Bauddha, this 

 idle notion?" &c. He adds in his notes, "For if the earth were 

 falling, an arrow shot into the air would not return to it, since both 

 would descend. Nor can it be said that it moves slower and is over- 

 taken by the arrow, for heaviest bodies fall quickest, and the earth is 

 heaviest.'" 



As to the observations and instruments, it is sufficiently evident 

 from the differences between the Hindoo system and that of the Greeks, 

 that they must have had both. Their system is more accurate than 

 that of Hipparchus or Ptolemy, precisely in the three fundamental 

 results of widely separated observations the tropical year, the synodic 

 month, and the precession of the equinoxes. But no observations 

 have been preserved, except indirectly in results ; Bhaskara describes 

 nine instruments, including the quadrant, semicircle, circle, armillary 

 sphere, horary ring, gnomon, and clepsydra. 



The periods of the Hindoos, which were of interest as long as it was 

 a question whether the beginning of the Cali Yug was or was not to 

 be considered as an epoch of actual observation, may now be returned 

 into the hands of the mythologists, warranted as long as ever. A 

 Yug, or age, is 432,000 years ; a Maha-Yug, ten Yugs, or 4,320,000 

 years ; a Calpa, or day of Brahma, is 1000 Maha-Yugs, or 4,320 

 millions of years; and Brahma's life is 100 years of such days and 

 nights, of which about one-half is past. Various attempts have been 

 made to expound these periods by combinations of astronomical 

 cycles; and considering that the number of years in a Calpa has 382 

 distinct divisors, it is not wonderful that various modes of putting 

 astronomical periods together should seem equally effective in this 

 respect. It is just as well to leave these speculations, and to remark 

 what a power of expressing large numbers was given by the Indian 

 numeration, now universally diffused. Archimedes wrote a book (the 

 ' Arenarius ') merely to prove that it was possible to express such 

 numbers as the Brahmins played with in their astronomical computa- 

 tions, and spoke of to the people in the common mythological stories. 



The astronomy of the Hindoos would have had little interest, but 

 for their arithmetic and algebra. In leaving the former to turn to 

 the two latter, we shall soon cease to feel any surprise at the respect 

 with which the astronomy has been treated, coupled as it is with an 

 arithmetic which is greatly superior to any which the Greeks had, 

 and an algebra which no other nation ever had, except those who 

 derived it from the Hindoos. For even supposing Diophantus to 

 have been an original inventor, which we greatly doubt, his work is 

 hardly algebraical, in any sense in which that term can be applied to 

 the science of .India. 



We shall begin by describing the Liliwali and Viga, Ganita, the 

 proper subject of this article, presuming the reader to be aware that 

 the Indian arithmetic is that which we now use, and that both this 

 arithmetic and algebra were introduced among the Arabs from India 

 (as the Mohammedan writers themselves inform us), through whom 

 they were transmitted to Europe. [VlETA.] Bhascara Acharya (A.D. 

 1150, as already mentioned) was the author of the Liliwati (called 

 after his daughter), and the Viga Gauita (or casual calculus : viya, cause ; 

 ganita, computation). These two works form the preliminary chapters 

 of the Siddhantasiromani, an astronomical work of the same writer. 



The Liliwati opens with a salutation to Ganesa, the god of wisdom, 

 and then proceeds to describe the system of weights and measures. 

 Then follows decimal numeration, briefly described ; and the eight 

 operations of arithmetic, addition, subtraction, multiplication, division, 

 square, cube, square-root, cube-root. Reduction of fractions to a 

 common denominator, fractions of fractions, mixed numbers, the eight 

 rules applied to fractions. Cipher ; * a = a ; 2 =0, /O = 0, &c., 

 a 0, the submultiple of 0, called infinite by the commentator ; 

 u x = 0. Inversion of processes, the solution of such an equation as 

 |(a;-f-a) + 6} c d = e, which is made a rule of arithmetic. Rule of 

 false position. Rule of concurrence, to solve x + y = a, x y = 6, and 

 30^2 y = <*"> x<i y* 1 = b. A problem concerning squares, finding pairs of 

 fractions the sum and difference of whose squares, diminished by 1, 

 are both squares. Solution of x" ^ ax = 6. Rule of three. Compound 

 rule of three, various cases. Interest, discount, partnership. Time 

 of filling a cistern by several fountains (a practical matter to those 

 who used the clepsydra). Barter. Presents of gems. Alligation. 

 Arithmetical progression; sums of squares and cubes. Geometrical 

 progression. Right-angled triangles; given two sides to find the third: 



* The reader -will easily understand that, to save room, we put down a sort of 

 table of contents, brief, but we hope intelligible. When we state a result alge- 

 braically, we mean the statement for a European abbreviation, not for a tran- 

 script from the work. \Ve have not put down some things of minor importance, 

 nor have we taken anything from the commentators without mention. 



also to find sides in rational numbers, to a given side or hypothenuse : 

 segments of the base of a given triangle ; perpendicular and area, the 

 sides being given. Four-sided figures, areas, &c., sides and a diagonal 

 or perpendicular being given. Many problems relative to four-sided 

 figures. Circumference of a circle is diameter x 3927 ~ 1250, very 

 nearly ; but x 22 -f- 7 is adapted to practice (the first answers to 

 3'1416) : area is \ diameter x circumference : the surface of the 

 sphere is four times that of the great circle : the solidity of the sphere 

 is surface x diameter -7- 6. Versed sine found from chord of twice the 

 arc and diameter, and the two converses. 'By 103923, 84853, 70534, 

 60000, 52055, 45922, 41031, multiply the diameter, and divide the 

 products by 120000, the quotients are severally the sides of polygons, 

 from the triaugla to the enneagou, within the circle.' To determine 

 roughly the chord of an arc, a rule is used which amounts to the 

 following : 



Bine of 2 ri g bt ap g le a _ _J 



5_\ 



-lJ 



or co-secant of Bright angka = 



n 16 



For 1 this last gives 56'3 instead of 57*3, and the relative error 

 diminishes up to 90. A corresponding rule is given for the arc of a 

 chord. The solid contents of a cone, pyramid, cylinder, prism, and 

 truncated cone or cylinder, are then given, and rules for estimating the 

 contents of mounds of different kinds of grain, derived from experi- 

 ment, the height being greater or less according as the grain is coarser 

 or finer. Various rules on shadows are then given, derived from the 

 geometrical properties of a right-angled triangle, and this is followed 

 by a chapter on the Cuttaca, or pulverizer, presently noticed. The 

 work ends with a chapter on combinations, containing questions of 

 this kind : any number of digits being given, as 5, 5, 7, 8, 6, required 

 the number of different arrangements, as 55786, 57865, 78565, &c., 

 and a rule for the sum of all the numbers thus formed. 



The Viga Ganita commences with a curiosity of the Sanskrit 

 language a sentence in which each of the leading words is threefold 

 in meaning; so that it will bear, and is intended to bear, three 

 different translations, which are as follows : 



1. I revere the unapparent primary matter, which sages conversant 

 with theology declare to be productive of the intelligent principle, 

 being directed to that production by the sentient being : for it is the 

 sole element of all which is apparent. 



2. I adore the ruling power, which sages conversant with the nature 

 of soul pronounce to be the cause of knowledge, being so explained by 

 a holy person : for it is the one element of all which is apparent. 



3. I venerate that unapparent computation, which calculators 

 affirm to be the means of comprehension, being expounded by a 

 fit person : for it is the single element of all which is apparent. 



Bhascara then proceeds thus: 'Since the arithmetic of apparent 

 (known) quantity, which has been already propounded in a former 

 treatise, is founded on that of unapparent (unknown) quantity, and 

 since questions to be solved can hardly be understood by any, and not 

 at all by such as have dull apprehensions, without the application of 

 unapparent quantity : therefore I now propound the operations of 

 analysis (Vija-crya, elemental solution.)' 



According to Colebrooke, whose words we abridge, the algebraic 

 notation of the Hindoos is as follows: Abbreviations and initials for 

 symbols; negative quantities with a dot; no mark for positive, 

 except the absence of negative. No symbol for addition, multiplica- 

 tion, equality, greater or less. A product denoted by the first 

 syllable of a word subjoined to the factors, between which a dot ia 

 sometimes placed. In fractions, divisors under dividend without line 

 of separation. The two sides of an equation are one under the other, 

 confusion being prevented by the recital of the steps in words which 

 always accompanies the operation. Symbols of unknown quantity are 

 various, usually initials of names of colours, except the first, which is 

 the initial of yavat-tavat, 'as much as :' Bombelli used tanto in the same 

 sense. Colour means unknown quantity, but its Sanskrit also signifies 

 a letter, and letters are also used, either from the alphabet, or from 

 initial syllables of subjects of the problem. Symbols are also used for 

 variable and arbitrary quantities, and sometimes for both given and 

 sought quantities. Initials of square and solid denote those powers, 

 and combined, the higher powers, reckoned * not by sums of powers, 

 but by their products. An initial syllable also marks a surd root. 

 Polynomials are arranged in powers, the absolute quantity being 

 always last, distinguished by an initial syllable denoting known 

 quantity. Numeral co-efficients are employed, integer and fractional, 

 unity being always noted : fractional co-efficients preferred to division 

 of unknown quantities, and the negative dot always over the numeral, 

 not over the literal character. The numeral co-efficient always after 

 the unknown quantity. Positive or negative terms indiscriminately 

 allowed to come first : and every power repeated on both sides of an 

 equation, with nought for the co-efficient, when wanted. 



The Arabian algebraists have no symbols, arbitrary or abbreviated, 

 either for quantities known or unknown, positive or negative, or for 

 the steps and operations of an algebraic process ; but they express 



* In the old times of European algebra, some would call, for instance, the 

 sixth power the ' cubo-cube," as being a 3 x 3 ; others would call the ninta 

 power by the same name, as being the cube of the cube. 



