gives the application of arithmetic to geometry, and rules for the extraction of the square root and the cube root.

The fifth chapter of the Lilavati treats of progressions, arithmetical and geometrical. Rules are given for finding the sum, the first term, the common difference, and the number of terms of an arithmetic series.

The following rule given for finding the number of terms, re quires the solution of a quadratic equation:

"From the sum of the progression multiplied by twice the common increase, and added to the square of the difference between the first term and half that increase, the square root being extracted, this root less the first term and added to the (above-mentioned) portion of the increase, being divided by the increase, is pronounced to be the period."

One of the commentators is quoted for a method of finding the sum of an arithmetic progression, grounded on placing the numbers of the series in the reversed order under the direct one; where it becomes obvious, that each pair of terms, multiplied by the number of terms, is twice the sum of the progression.

The following rules are also given for finding the sum of the squares, and the sum of the cubes of any series of the natural numbers.

Twice the number of terms added to one and divided by three, being multiplied by the sum of the natural numbers, is the sum of the squares.

The sum of the cubes of the numbers one, two, &c., is pronounced by the ancients equal to the square of the sum of the series, one, two, &c.

The section on geometrical progressions is restricted to progressions increasing by a multiplier, and omits altogether any notice of decreasing geometrical progressions.

The rules for variations, permutations, and combinations are given

several of the elder boys have made considerable and nearly equal progress, they are seated together in one line, and receive their instructions directly from the master.

"The plan of getting the older boys, and those who are more advanced, to assist those who are less advanced and younger, greatly lessens the burden imposed upon the master, whose duty, according to this system, is not to furnish instruction to each individual scholar, but to superintend the whole, and see that every one does his duty. If the younger boy does not learn his lessons with sufficient promptitude and exactness, his instructor reports him to the master, who inquires into the case, orders the pupil to repeat before him what he has learned, and punishes him if he has been idle or negligent. As the master usually gives lessons to the elder scholars only, he has sufficient leisure to exercise a vigilant superintendence over the whole school, and by casting his eyes about continually, or walking up and down, and inquiring into the progress made by each pupil under his instructor, he maintains strict discipline, and keeps every one on the alert through expectation of being called upon to repeat his lesson."

in the sixth section of the fifth chapter, and in the thirteenth chapter of the Lilavati.

The following rule for combinations is taken from the Persian version, as rendered in English by Mr. Strachey :

66

First, write them all with one, in order, and below write one, the last opposite to the first in order. Then divide the first term of the first line by the number which is opposite to it in the second line. The quotient will be the number of combinations of that thing. Multiply this quotient by the second term of the first line, and divide the product by the number which is opposite to it in the second line, the quotient will be the number of combinations of that thing; and multiply this quotient by the third term, and divide by that which is below it, and add together whatever is obtained below each term, the sum will be the amount of all the combinations of these things.1

The first section of the first chapter of the Vija Ganita opens with the following invocation, as translated by Mr. Colebrooke :

"I revere the unapparent primary matter, which the Sanchyas 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. 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. 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. Since the arithmetic of apparent [or known] quantity, which has been already propounded in a former treatise [Lilavati], is founded on that of unapparent [or unknown] quantity; and since questions to be solved

1 The following is one of the examples given (ch. v., sect. 6) to illustrate the rule. How many are the combinations in one composition, with ingredients of six different tastes, sweet, pungent, astringent, sour, salt, and bitter, taking them one by one, by twos, by threes, &c.? 1 2 3 4 5 6 6 5 4 3 2 1

6

6X5

Write them thus,

6 combinations when taken singly.

15, when taken two and two together.

2

15X4

=20, when taken three and three.

3

20X3

=15, whon taken four and four.

4

15X2

=6, when taken five and five.

5

6×1

=1, when taken all six together.

6

And the whole number is 63.

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."

In these oldest treatises known to be extant of the Arithmetic and Algebra of the Hindus, their writers describe the analytic art as "a method of calculation attended with an exhibition of its principles, and aided by literal signs and other devices."

The initial syllables of words and of the names of colours, as well as single letters, were assumed to denote unknown numbers. A negative quantity was distinguished by a dot or point placed over the syllable or number, and a positive quantity was known by the absence of the dot.

There is no account given of the origin of this convention. The Sanscrit words translated affirmative quantity and negative quantity, have the literal meanings of property or wealth, and debt or loss, and imply a contrariety with respect to the possessor. Likewise two distances, one towards the right hand and the other towards the left hand, have also a contrariety of direction with respect to a person in a fixed position. If it be admitted that symbols, without any additional mark, denote affirmative quantities, then any mark, as a dot, may be employed to denote negative quantities: and this convention is sufficient, as an adjective in ordinary language, to distinguish that the symbol so marked is to be considered of a contrary character to that which is not so marked.

Crishna, a commentator on the Vija Ganita, gives the following illustrations of affirmative and negative quantities. Here negation is of three sorts, according to place, time, and things. It is, in short, contrariety. Wherefore the Lilavati, Sect. 166, expresses "the segment as negative, that is to say, is in the contrary direction." Thus of two countries, east and west; if one be taken as positive, the other is relatively negative. As an example, the situation of Patna and Allahabad relatively to Benares. Patna, on the Ganges, bears east of Benares, distant 15 yojanas, and Allahabad bears west of the same place 8 yojanas. The interval or difference between Patna and Allahabad is 23 yojanas, and is not obtained but by the addition of the numbers. Therefore, if the difference between two contrary quantities be required, their sum must be taken. Next, so when motion to the east is assumed to be positive; if a planet's motion be westward, then the number of degrees equivalent to the planet's motion is negative. In like manner, if a revolution westward be affirmative, so much as a planet moves eastward, is, in respect of a western revolution, negative. The same may be understood in regard to south and north, &c. That prior and subsequent times [reckoned from a fixed epoch] are relatively to each other as positive and negative, is familiarly understood in the reckoning of days. Thirdly. So in respect of chattels; that to which

a man bears the relation of owner, is considered positive with regard to him; and the contrary [or negative quantity] is that to which another person has the relation of owner. Hence, so much as belongs to Yajnyadatta in the wealth possessed by Devadatta, is negative in respect of Devadatta.

A known quantity which is always a number, when employed as a coefficient, is written after the character which denotes the unknown quantity; but when it is not so used, but simply as a number, it is denoted by ru, the first syllable of rupa, form or species.

quantum,'
"" as much as," 99 66

The first syllable of ya of yavat-tavat, the Sanscrit term for "tantum as many as," is used to denote an unknown quantity, when only one is employed. Thus ya 2 is denoted by 2x in modern notation.

If there be more than one unknown quantity required, the initial syllables ca, ni, pi, lo, &c., of the names of the colours, black, blue, yellow, red, &c., are used to denote the second, third, &c., unknown quantities, as ya 6 ca 4 ni 2 ru 2 signifies - 6x

modern notation.

4y

23 + 2 in

The syllables va, gha, being the initial syllables of the Sanscrit of square and cube, are employed to denote the second and third powers, and these are combined to denote the higher powers, which are reckoned by the products, and not by the sums of the lower powers, as in the modern algebra. A surd root is in the same manner denoted by an initial syllable.

Sometimes the initial letters of the words are employed instead of the initial syllables.

The Hindus had also a figurative method of expressing numbers by the names of the objects, of which a certain number was generally known; as, for instance, the name of the sacred fire stands for 3, the arrow of Kamedeva for 5, the treasure of Kuvera for 9, the sun in reference to the month of the year for 12, the lunar mansion for 27, &c.; and larger numbers were made up by combining these names, as may be seen in the Lilavati.

They had no symbols to denote the operations of addition and subtraction. They simply declared that in the addition of two negative or two affirmative quantities, the sum must be taken; but the sum of an affirmative and negative quantity, the difference is their addition. And in subtraction, the quantity to be subtracted being affirmative, becomes negative; or being negative, becomes affirmative; and the addition of the quantities then taken, constitutes their difference.

As the Hindu algebraists employed no symbols to denote the operations of addition and subtraction, so likewise they had no marks to indicate multiplication and division. They exhibit the following directions for determining the nature of products and quotients as affirmative or negative.

The product of a negative quantity and an affirmative is negative; of two negative quantities is positive; of two affirmative quantities is affirmative. The product of cipher and a negative quantity, or of cipher and an affirmative quantity, is zero.

And a positive quantity divided by a positive, or a negative quantity by a negative, is affirmative. A positive quantity divided. by a negative is negative; and a negative quantity divided by an affirmative is negative."

The division of one number by another was indicated by writing the dividend above the divisor without any line or mark of separation. The same notation was adopted to denote fractions by placing the denominator under the numerator. The rule for division, the reverse of multiplication, is thus stated:-"Those colours or unknown quantities and absolute numbers, by which the divisor being multiplied, the products in their several places subtracted from the dividend exactly balance it, are here the quotients in division."

The product of any quantity by cipher is cipher, and the product of cipher by any quantity is cipher. And the quotient of cipher divided by any quantity is cipher; but any quantity divided by cipher becomes a fraction, the denominator of which is cipher. On this subject one of the commentators remarks:- "If the dividend be diminished, the divisor remaining the same, the quotient is reduced, and if the dividend be reduced to cipher, the quotient becomes cipher. Again, as much as the divisor is diminished, the dividend remaining

1 The following note is subjoined by Mr. Colebrooke :

Multiplication, as explained by the commentators, is a sort of addition resting on repetition of the multiplicand as many times as is the number of the multiplier. Now a multiplier is of two sorts, positive or negative. If the multiplier be positive, the repetition of the multiplicand, which is affirmative or negative, will give correspondently an affirmative or negative product. The multiplication, then, of positive quantities is positive; and that of a negative multiplicand by a positive multiplier is negative. The question for disquisition concerns a negative multiplier. It has been before observed that negation is contrariety. A negative multiplier, therefore, is a contrary one; that is, it makes a contrary repetition of the multiplicand. Such being the case, if the multiplicand be positive, and the multiplier be negative, the product will be negative; if the multiplicand be negative and the multiplier be negative, the product will be affirmative. In the latter case, the multiplication of two negative quantities gives an affirmative product. Multiplication was exhibited as in this following example ::

In modern notation, the statement of the operation is equivalent to (5x-1)× (3x+2), and the product to 15x2+7x-2.

* Division being the reverse operation of multiplication, if the product ya v 15 ya 7 ru 2 be taken as the dividend, and the multiplier ya 3 ru 2 be taken as the divisor; division being made, the quotient is ya 5 rui, the original multiplicand.