root a square turn the negative into positive and the positive into negative. If a quantity was diminished by its own proportional part, let the denominator, being increased or diminished by its numerator, become the denominator, and the numerator remain unchanged, and then proceed with the other operations of inversion, as above directed.

From all that is here said, one can hardly guess either at what is required, or what is directed to be done. We learn something more precise, however, from the question that the instructor proposes to his fair pupil.

Pretty girl with tremulous eyes, if thou know the correct method of inversion, tell me what is the number which, multiplied by 3, and added to three quarters of the product, and divided by 7, and reduced by subtraction of a third part of the quotient, and then multiplied into itself, and having 52 subtracted from the product, and the square root of the remainder extracted, and 8 added, and the sum divided by 10, yields 2?'

The numerical statement is next given, but not with much precision; and it is added, that, by applying the rule, 28 will be found to be the number sought. This is true; and if we put the question into an equation, according to the preceding enunciation, we will find a pure quadratic, from which the number sought comes out equal to 28; the steps of the calculation being nearly the same that are enjoined in the preceding rule. *

The next section relates to what is called Supposition, and is in fact the same with our rule of False Position. A number is assumed at pleasure, and is treated as specified in the question proposed, so that a numerical result is obtained; then the given number in the question being multiplied by the assumed number, and divided by the result before mentioned, yields the number sought. This is exactly our rule. In a note subjoined to DR TAYLOR's translation from one of the commentators on the Lilavati, it appears that they were aware, that when the question involved the square, or any higher power of the unknown.

quantity, this method of assumption could not be applied. The following question is resolved in this way.

'Out of a swarm of bees one-fifth part settled on a blossom of Cadamba, and one third on a flower of the Silind'hri; three times the difference of these numbers flew to the bloom of the Cutaja: one bee which remained hovered about in the air, allured by the fragrance of the Jasmin and Pandanus :-Tell me, charming woman, the number of bees?'

The lady is supposed to assume 30 for the number of bees; and the value of the absolute number deduced from that supposition is 2. Had the supposition been right, the result would 30 X 1 have been 1; therefore, or 15, is the true number. 2

This question reminds us, that though the abstractions of one mathematician may very much resemble those of another, they acquire a wonderful diversity of form when embodied in the material substances most familiar to the imagination of their authors. A question concerning a number, of which the 3d and the 5th part, added to three times the difference of those parts, and to 1, may be equal to the number itself, might occur to two arithmeticians of any age or of any country; but the circumstances of the bees, the blossoms of the Cadamba, the Silind❜hri, and the Cutaja, were likely to come into the mind only of an inhabitant of India.

After this, the rule of proportion is treated of at considerable length, and applied to questions of interest, barter, mixture, &c. nearly in the same manner as with us. The author seems fully sensible of the value of the doctrine he delivers under this head; and considers every problem that can be resolved by multiplication and division alone, as belonging to the Rule of Three. He introduces his remark with that mixture of poetry and metaphysicks that belongs so much to the Oriental genius.

As the Being, who relieves the minds of his worshippers from suffering, and who is the sole cause of the production of this universe, pervades the whole, and does so with his various manifestations, as worlds, paradises, mountains, rivers, gods, dæmons, men, trees, and cities, so is all this collection of instructions for computation pervaded by the rule of Three Terms. Whatever is computed, cither in algebra or in arithmetic, may be comprehended by the sagacious learned as belonging to this rule. p. 111.

Under the head of Combinations, we find rules given that are almost exactly the same with those which we employ, and deduce from the coefficients of the Binomial theorem. Thus, a palace being supposed to have eight sides, and a door in each side, it is required to tell how many ways the palace may be open

ed, taking the doors one and one, two and two, three and three, &c. The rule is exactly the same that we use, setting down the numbers in their natural order, beginning with 8, and placing under them the same progression in the reverse order, thus, 8, 7, 6, 5, 4, 3, 2, 1, and 1, 2, 3, 4, 5, 6, 7, 8.

again,

8

Then or 8, is the number of ways of opening one door only;

8 x 7

1 x 2

יד

=28, is the number of ways in which the doors may be

opened by twos; 28 x

5

4

opened by threes; 56 ×, that in which they may be opened by fours, &c.; the total number of changes, or the sum of all these numbers, being 246. This problem seems to be well known in India; it was mentioned long ago by MR BURROWS, who did not fail to remark the very curious coincidence between the Indian and the European process of calculation,

But in the midst of these curious results, there is a subject of regret that almost continually presents itself. When such rules are laid down as the preceding, they are usually given without any analysis whatever, and even without any synthetic demonstration, so that the means by which the knowledge was obtained, remains quite unknown. Analysis is indeed not to be looked for in the Lilavati, which professes only to be a body of arithmetical precepts and examples. But, even in the Vija Ganita, where the analytical investigation of unknown quantities is the object proposed, the rules which are most general, and most difficult to be discovered, are accompanied with no analysis. In consequence of this, a mystery still hangs over the mathematical knowledge of the East; and it is much to be feared that the means of removing it no longer exist.

All these observations are exemplified in the Cuttaca, or, as it is here translated the Pulveriser, a process which makes a great figure both in the Algebra and Arithmetic of the Hindu Astronomers. It is a general rule for the resolution of indeterminate Problems of the first degree; and by its universal application, and the simple and easy calculus to which it leads, very well merits all the eulogies which the Indian Algebraists are inclined to bestow on it. It will be regarded as no small confirmation of those eulogies, to observe, that a method of the same extent and import was not known in Europe till it was published by Bachet de Meziriac about the year 1624, and that the pro

cess for resolving those problems given by BHASCARA and BRAHMEGUPTA, are virtually the same that is explained in EULER'S Algebra, Vol. II. chap. 1st. But we must explain the rule and the Sanscrit name, which, it seems, is literally translated by the word Pulveriser, a term that we might expect to find in mechanicks, but hardly in such sciences as Arithmetic and Algebra. The verb cutt, in Sanscrit, we are informed, signifies to grind or pulverise, also to multiply; all verbs importing tendency to destruction, also signifying multiplication. This is stated on the authority of GANESA; and indeed the thing sought for by the rule of the pulveriser, is a multiplier having the property, that, when it multiplies a certain given number, and when another given number is added to the product, the sum may be divisible by a number which is also given. Thus, 17x+5 if the question be to find x, so that

15

may be an integer,

the number a is the pulveriser; or the method by which a is found is called pulveriser; for we confess that we are not certain which of the two is the fact. This rule is treated of by BHASCARA both in the Lilavati and the Vija Ganita, and by the more ancient author BRAHMEGUPTA, in a distinct treatise; and in all these the pulveriser is found nearly in the same way. The computation is easy; though it requires consideration to apply the general rule, which, in the text, is given with too much conciseness and too little precision. We have not room to enter on it here; but must recommend to those who would make themselves masters of it, to look into the notes in Dr Taylor's translation, p. 114, 115, where the whole process of calculation is distinctly explained and exemplified.

This species of indeterminate problems appears to have been particularly interesting to the astronomers of the East, from its connexion with those cycles, or periodic revolutions, by which they endeavoured to represent the motions of the heavenly bodies. Their application to such cycles often leads to unexpected results, of which DR TAYLOR has given an example, page 132, note.

"Suppose that in a certain unknown period of years, a planet has performed a certain number of revolutions, with a certain number of signs, degrees, minutes, seconds, all unknown, with 10-13th parts of a second over and above. From the fraction all the rest may

10 13'

be found; that is, the number of revolutions, the signs, degrees, &c.

is a fraction of a second, therefore 60x-10 is divisible by

is a whole number; hence a' is found 11, and

4

whole number, and equal to the minutes passed over; which are thus found to be 13, and ' to be 3. Proceeding in the same way, the whole number of degrees is found = 9, and the fraction of a sign Hence the number of signs is 8, and lastly, of revolutions, 1; so that the whole collected together is 1 Rev. 8°. 9°. 13'. 50′′.

10

13

13

The possibility of ascending in this manner to all the quotients in succession, is by no means obvious. The Indian astronomers seem to have been particularly pleased with the subtlety of the investigation.

On the subject of Indeterminate Problems, we must remark, that the Indian algebraists had gone much further than this, and had resolved those of the 2d degree, or such as are properly called Diophantine, where the question is how to render a certain quantity rational, either in fractions or in integers. Diophantus resolved this problem in certain cases, or with certain restrictions in the data of the problem, beyond which his solutions did not extend. After the revival of science in Europe, the same class of problems became an object of attention with some of the first mathematicians of the 17th century. Among these, the most distinguished were Brounker, Fermat and Wailis, who extended their investigations far beyond those of the Greek Mathematician, though still subject to great limitations; and it is not a little remarkable that we find, as MR COLEBROOKE has observed, that one of the solutions of this Problem given by BHASCARA (Vija Ganita 380, 81) is exactly the same that LORD BROUNKER devised to answer a question proposed as a kind of challenge by FERMAT in 1657. This is a fact which we think cannot be controverted, and one that we strongly recommend to the attention of those mathematicians who see nothing commendable or original in the science of the East. This is not all. BHASCARA'S solution just referred to, is not general, but restricted to certain cases, as every solution of the same problem was in Europe before the publication of an Essay on that subject by LA GRANGE, in the Berlin Memoirs for 1767. Even EULER, with a genius so powerful and inventive as hardly to be paralleled in the history of mathematical science, though he had treated this question with great success, yet had not been able to remove all limitation. Now, a solution that appears quite general, is given in the most antient of the treatises un