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the same, so much the quotient is increased, and 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."
Like quantities are denoted by the same colour, unlike by different colours. The multiplication of two like quantities produces the square, three the cube, four the biquadrate, &c. But when unlike quantities are multiplied, the result was called bhavita (to be) their factum, the initial syllable bha being annexed to the statement. Thus the product of two unknown quantities is denoted by three syllables (or three initial letters), as ya ca bha is equivalent to xxy or xy in modern algebraic notation. But if one or more of the quantities be of a higher power than the first, other initial syllables are employed, as va gha &c., for the square, cube, &c. Thus, ya va ca gha bha will signify the square of the first unknown quantity, multiplied by the cube of the second, xx y3, or x3y3.
In some Sanscrit texts, and in some commentaries, where two or more factors are to be multiplied, a point or dot is found interposed between the factors, without any directions given for this notation.
The terms of compound quantities are arranged according to the descending powers of the unknown quantity, the numerical coefficients are placed after the symbols which denote the powers of the unknown quantities, and the negative mark is placed over the coefficient, and not over the power of the unknown quantity; also a known number is always placed last, thus: ya vv 1 ya gh 6 ya v 5 ya 12 ru 4; in modern notation x+6x3+5x3- 12x+4.
As the Hindus had no sign to denote either equality or relative magnitude, greater than and less than, the two sides of an equation were placed one under the other.' It was the general practice to arrange the terms in order according to the powers of the unknown quantity, and where a term on either side was wanting, to write the literal character with zero for its coefficient.
The following is an example of a simple equation:One person has 300 of known species and six horses. like price; but he owes a debt of 100 of known species. What is the price of a horse?
The statement of the equation is thus made:
ya 6 ru 300
ya 10 ru 100
In modern notation 10x-100=6x+300.
Another has ten horses of They are both equally rich.
The following is an example of an indeterminate problem of the first degree:The quantity of rubies without flaw, sapphires, and pearls, belonging to one person, is 5, 8, and 7 respectively; the like gems appertaining to another is 7, 9, and 6. One has 90, the other 62, known species. They are equally rich. Tell me
If a number consist of two parts, the square of the number is twice the product of the parts added to the sum of the squares of the parts. In the same manner is stated the cube of a number consisting of two parts. The cube of the second part is to be first set down; next, the square of the second part multiplied by three times the first; thirdly, the square of the first part multiplied by three times the second; and lastly, the cube of the first part. All these added together make the cube of the number.
The same process might have been begun with the first instead of the second part of the number.
The cube of a number consisting of two parts, is also expressed in the following manner :— "Three times the proposed number multiplied by its two parts, added to the sum of the cubes of these parts gives the cube of the number."
Methods of finding the square root and the cube root are given by reversing the direct process. Also rules are given for the addition and subtraction, multiplication and division, of quadratic surd numbers, as also for their involution and evolution.
It is further noted that the square of an affirmative or of a negative quantity is affirmative; but that the square root of an affirmative quantity is sometimes affirmative, and sometimes negative, according to difference of circumstances which the process may require. As, for instance, the square of 3 affirmative or of 3 negative is 9 affirmative. Hence, the square root of 9 affirmative may be 3 affirmative or 3 negative. There is no square root of a negative quantity, for it is not a square. One of the commentators remarks:-" If it be maintained that a negative quantity may be a square, it must be shown of what it can be the square. Now, it cannot be the square of an affirmative quantity, for a square is the product of two like quantities; and if an affirmative quantity be multiplied by an affirmative one, the product is affirmative. Nor can it be the square of a negative quantity; for a negative quantity multiplied by a negative quantity is also positive. Therefore, it is clear that there is no quantity such, that its square can be negative.
One of the commentators has remarked that some have pretended to have found the root of a surd, and that this might be effected by the Cutaca Ganita; but whether or not this could have been possible, I shall relate what Bhascara and others have omitted to explain.
quickly, then, intelligent friend, who art conversant with algebra, the prices of each sort of gem.
The statement of the problem
ya 5 ca 8
which expressed in modern notation is 7x+9y+6x+62=5x+8y+7x+90, and when reduced becomes 2x+y-z=28.
A root is of two kinds; one a line, the other a number. And the root of a square formed by a line expressing 5 may be found, though the root of 5 cannot be numerically expressed; but the numbers 1, 4, 9, &c., may be expressed both ways. The square roots of the numbers 2, 3, 5, &c., are surds, and can have their roots expressed only by lines. Thus the root of a surd quantity can be shown geometrically with exactness, but only approximately by numbers.
Quadratic equations,' or equations of the second degree, both pure and adfected, are discussed in the treatise of Brahmegupta, and in the Vija Ganita of Bhascara. In the solution of quadratic equations, when the square of the unknown quantity is affected with an integral coefficient different from unity, Bhascara gives this rule of Sridhara : "Multiply both sides of the equation by a number equal to four times the coefficient of the square, and add to each side a number equal to the original coefficient of the first power of the unknown quantity." He also gives an example in which fractions are avoided by the application of the rule.2
Brahmegupta has given this rule, and another which is thus expressed :
"To the number multiplied by the coefficient of the square of the unknown quantity, add the square of half the coefficient of the first power of the unknown quantity; the square root of the sum less half the coefficient of the unknown quantity being divided by the coefficient of the square is the value of the unknown quantity."
The chapter on quadratic equations includes other questions involving the cube and the biquadrate, besides the square and the first power
1 The following is their mode of writing a quadratic equation:
The square root of half the number of a swarm of bees is gone to a shrub of jasmine; and so are eight-ninths of the whole swarm; a female is buzzing to one remaining male, that is humming within a lotus, in which he is confined, having been allured to it by its fragrance at night. Say, lovely woman, the number of bees. Employing the modern notation
Let Sa2 denote the number of bees; then the question reduced to an equation is 642
2x+ -+2=82: which gives the values of x, 3 and-}.
The positive value gives the answer to the question, and 82=8×9=72, the number of the swarm of bees.
3 Of the two following examples in this chapter, one leads to a cubic, and the other to a biquadratic equation.
"What is the number, learned man, which being multiplied by 12 and added to the cube of the number, is equal to six times the square added to 13?"
"If thou be conversant with the operations of algebra, tell the numbers of
of the unknown quantity. With respect to such equations, the author observes, if the solutions be not accomplished by the methods given, the value of the unknown quantity must be elicited by the sagacity of the intelligent analyst. Intelligent calculators commence the work of solution, sometimes from the beginning of the conditions of the problem, sometimes from the middle, sometimes from the end, so as the solution may be best effected.
It has been supposed that the Binomial Theorem in its general form, when the index is an integer, was not unknown to the Hindus. The Lilavati and the Vija Ganita afford no direct evidence of such knowledge beyond the second and third powers of a binomial, which also appears in the older treatises of Brahmegupta. The law for the formation of the number of combinations of a number of different things taken, 1, 2, 3, &c., at a time was well known; but these treatises do not exhibit any connection of, or identity of, this law with the coefficients of the terms of an expanded binomial.
At the end of the second volume of the "Asiatic Researches," an Appendix has been added by Mr. Reuben Burrow, entitled, "A Proof that the Hindoos had the Binomial Theorem." He produces a question and its solution in proof of his statement, and adds:"The demonstration is evident to mathematicians; for as the second term's coefficient in a general equation shows the sum of the roots, therefore in the nth power of 1+1, where every root is unity, the coefficient shows the different ones that can be taken in n things; also, because the third term's coefficient is the sum of the products of all the different twos of the roots, therefore when each root is unity, the product of each two roots will be unity, and therefore the number of units, or the coefficient itself, shows the number of different twos that can be taken in n things. Again, because the fourth term is the sum of the products of the different threes that can be taken among the roots, therefore, when each root is unity, the product of each three will be unity, and therefore every unit in the fourth will show a product of three different roots, and consequently the coefficient itself shows all the different threes that can be taken in n things; and so for the rest."
which the biquadrate, less double the sum of the square and of 200 times the simple number, is a myriad less one.
Expressed in the modern algebraic notation, the two 12x+x=6x+35, and x1 - (2x2+200x)=10000-1.
1 A rajah's palace had eight doors: now these doors may either be opened one at a time, or by two at a time, or by three at a time, and so through the whole, till at last all are opened together; it is required to tell the number of times that this can be done. The question and the solution itself appears to exhibit an example of forming the combinations of a certain number of things when taken 1, 2, 3, &c., at a time, and the total number of such combinations; but no reference is made to a series of terms arising from the expansion of the integral power of a binomial expression.
It may reasonably be doubted if this mode of demonstration which he gives, arising from the formation of an equation, was known to the Hindu mathematicians. Their books show that they understood the forms of the second and third powers of a binomial, but exhibit no instances of a higher power, or of a generalised form.
Mr. Strachey makes the following comparison of the Beej Gunnit [Vija Ganita] and the Arithmetics of Diophantus :—
"The Beej Gunnit will be found to differ much from Diophantus's work. It contains a great deal of knowledge which the Greeks had not, such as the use of the indefinite number of unknown quantities and the use of arbitrary marks to express them; a good arithmetic of surds; a perfect theory of indeterminate problems of the first degree; a very extensive and general knowledge of those of the second degree; a knowledge of quadratic equations, &c. The arrangement and manner of the two works will be found as essentially different as their substance. The one constitutes a body of science, the other does not. The Beej Gunnit is well digested and well connected, and is full of general rules, which suppose great learning; the rules are illustrated by examples, and the solutions are performed with skill. Diophantus, though not entirely without method, gives very few general propositions, and is chiefly remarkable for the ability with which he makes assumptions in view to the solution of his questions. The former teaches algebra as a science by treating it systematically; the latter sharpens the wit by solving a variety of abstruse and complicated problems in an ingenious manner. The author of the Beej Gunnit goes deeper into his subject, and treats it more abstractedly, and more methodically, though not more acutely than Diophantus. The former has every characteristic of an assiduous and learned compiler; the latter of a man of genius in the infancy of science."
1 Asiatic Researches, vol. xii., 1816, "On the Early History of Algebra," by Edward Strachey, Esq. In this essay Mr. Strachey refers to a tract "On the Mathematical Science of the Hindoos," which he printed at Calcutta, in the year 1812, pp. 17, 4to.