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of 2, 5, 32, 193, 18, 10, and 100, added together; and the remainder when their sum is subtracted from 10,000."

He then rapidly plunges into multiplication as follows: "Example. Beautiful and dear Lílívatí, whose eyes are like a fawn's! tell me what are the numbers resulting from 135 taken into 12? . . . . Tell me, auspicious woman, what is the quotient of the product divided by the same multiplier?"

The treatise continues rapidly through the usual rules, but pauses at the reduction of fractions to hold up the avaricious man to scorn: "The quarter of a sixteenth of the fifth of three-quarters of two-thirds of a moiety of a dramma was given to a beggar by a person from whom he asked alms; tell me how many cowry-shells the miser gave if thou be conversant in arithmetic with the reduction termed subdivision of fractions."

The "venerable preceptor," as Bháscara calls himself, illustrates what he terms the rule of supposition by the following example: "Out of a swarm of bees, one-fifth part settled on a blossom of Cadamba; and one-third on a flower of Silind'hri; three times the difference of those numbers flew to the bloom of a Cutaja. One bee which remained, hovered and flew about in the air, allured at the same moment by the pleasing fragrance of a jasmin and pandanus. Tell me, charming woman, the number of bees."

This example is sufficiently poetical, but there is given a section on interest, and one on purchase and sale for merchants. It is easily seen that this arithmetic varies but little from that taught in our common schools to-day. The processes are nearly the same, and the advance of the Hindoos in this science is due largely to their admirable system of notation, viz., that called the Arabic, which, however, was undoubtedly derived by the Arabs from Hindoo teachers, as is admitted by the best authorities.

The next section of the book is occupied with a kind of arithmetical geometry, which has for its basis the relation between the squares of the sides of a right-angled triangle. The demonstration of this celebrated theorem is given both geometrically and algebraically by one of the commentators. This algebraic demonstration is so short and so direct that it will be given: If C and D are the greater and less sides of a right-angled triangle, and B the hypothenuse whose greater and less segments are c and d, then—

$B:C=C:c{\mbox{ or }}c={\frac {C^{2}}{B}}$

Also$B:D=D:d{\mbox{ or }}c={\frac {D^{2}}{B}}$

Therefore $B=c+d={\frac {C^{2}}{B}}+{\frac {D^{2}}{B}}{\mbox{ and }}B^{2}=C^{2}+D^{2}$

It is noteworthy that Wallis, in his "Treatise on Angular Sections,"