Popular Science Monthly/Volume 3/July 1873/Early Hindoo Mathematics
By Prof. EDWARD S. HOLDEN,
OF THE NATIONAL OBSERVATORY, WASHINGTON.
THERE is a certain fascination in our scanty knowledge of the elder nations of the earth, which is due quite as much to their chronological position as to the intrinsic interest of their doings and sayings; and it owes not a little of its keenness to the very scantiness of that knowledge.
We are continually told that this is a practical century; that we are utilitarians in the strictest sense; that there is no romantic faculty left to us; that we are apt to scorn all knowledge which has not a direct practical bearing on the daily life and interests of us all. How can we believe this when we would so eagerly hear of the autonomy of the Aztecs, and while we care so little for modern Chili, for example?
We can speak with more interest of Karnac than of Bogotá, and a mummy is dearer to us than a Mongolian. We require our thoughts to be suggested sometimes by an age of old and quaint habits, of strange people with stranger gods. In our busy life, it is a relief to turn to the Hindoo, who could spare the time "to sit beneath the tree and contemplate his own perfections," or to the Egyptian who evolved pyramids, and obelisks, and avenues of sphinxes, out of his infinite leisure.
There are always "the complaining ones," for whom the times are stale, who would lament with Sir Thomas Browne that "mummy is become merchandise, Mizraim cures wounds, and Pharaoh is sold for balsams;" but they forget that the great nineteenth century buys its mummies in order to have a good look at them, and that it studies the Rosetta Stone out of pure interest, and to make no money.
But the real interest of former ages is the study of their manner of thought. We study what they thought to determine how they thought it. We have an immense and vague curiosity to connect our minds with the minds of long ages ago. Half the fascination of Darwin, Tylor, Lubbock, and Wilson, is from this cause.
It piques us to know that, sixteen hundred years before our era, there was a poet who sang:
"Like as a plank of drift-wood
Tossed on the watery main,
This surely is not the verse of a primitive people; these are not the feeble lispings of the infants of our race; did it not require time to accustom the Hindoo mind to similes as complex as these? This verse would not seem childish if Tennyson had written it; it appeals to as deep a consciousness as Coleridge's "Hymn in the Vale of Chamounix," and would even bear comparison with the "Peter Bell" of the great Lake poet.
If this people was so old thirty-four hundred years ago, when was it young? We begin to believe, with Bailly, in the existence of "ce peuple ancien qui nous a tout appris, excepté son nom et son existence."
It may, then, be interesting for us to glance at the state of science among these predecessors of ours. But let us remember that we are applying a severe test, when we compare their progress with the science of to-day. Let us remember that it is only within a hundred years that the return of comets has been predicted; that our knowledge of the constitution of the sun has been gained since 1859; that Newton has been dead only 147 years, and that Lagrange and Laplace both lived and worked in our own century. When we consider what astronomy would be without these three great men—that is, what it was only so few years ago—we are better prepared to appreciate the studies which laid the remote foundations of their triumphs.
It would be impossible, within moderate limits, to determine the value of Hindoo astronomy, however interesting the effort might be, since we should enter at once into debateable ground, and come among great authorities in conflict.
Bailly, Delambre, Bentley, Davis, Hunter, Sir William Jones, and others, have various, often contradictory, beliefs to maintain. Some are partisans of the Greek, some of the Arab, others of the Hindoo scientists of long ago. But, fortunately, some of the original manuscript books of the Hindoos have come down to us: among others various complete treatises on mathematics, and these are authentic and of great age. Precisely of how great age it is difficult to ascertain. Bailly, a Hindoo partisan, accepts the largest estimate; Delambre, a detractor of Hindoo science, and an advocate of the Greek, believes the most important of them to have been written about a. d. 1114; while the translator of this manuscript, Colebrooke, a distinguished Sanscrit scholar, places the date of writing, in a. d. 1150.
This treatise, the "Lílívatí" of Bháscara Achárya, is supposed to have been a compilation, and there are reasons for believing a portion of it to have been written about a. d. 628. However this may be, it is of the greatest interest, and its date is sufficiently remote to give to Hindoo mathematics a respectable antiquity.
The "Lílívatí," according to Delambre, was written to console the daughter of its author for her ill-success in obtaining a husband, and it speaks well for the Hindoo gentlewoman that such a means could be considered worth the attempting. It was called by her name, and many of the questions are addressed to her, as we shall see.
It opens most auspiciously with an invocation to Ganesa, as follows: "Having bowed to the Deity whose head is like an elephant's; whose feet are adored by gods; who, when called to mind, relieves his votaries from embarrassment, and bestows happiness upon his worshippers; I propound this easy process of computation, delightful by its elegance, perspicuous with words concise, soft, and correct, and pleasing to the learned."
Thus fairly launched, the author gives various tables of Hindoo moneys, weights, etc., and proceeds to business, not without another invocation, however, shorter this time: "Salutation to Ganesa, resplendent as a blue and spotless lotus; and delighting in the tremulous motion of the dark serpent, which is perpetually twining within his throat."
The principles of numeration and addition are then stated concisely, and he affably propounds his first question: "Dear, intelligent Lîlîvatî, if thou be skilled in addition and subtraction, tell me the sum 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—
It is noteworthy that Wallis, in his "Treatise on Angular Sections," (Chapter VI.), gives this demonstration, and supposes it to be given for the first time.
The Hindoos, however, were not skilled in geometry. One of their authors even chides another for attempting to prove geometrically what can be seen by experience. One of the aphorisms of the present treatise is as follows: "That figure, though rectilinear, of which sides are proposed by some presumptuous person, wherein one side equals or exceeds the sum of the other sides, may be known to be no figure;" and the proof of this is thus given, "Let straight rods, of the length of the proposed sides, be placed on the ground, and the incongruity will be apparent."
The geometry of the circle in "Lílívatí" is the best feature of the book on plane figures. The "rule" of the text is that the ratio of the diameter to the circumference is 1250 or 3.1416 exactly.
This is given in the text without demonstration, but one of the commentators thus establishes it: the side of the inscribed hexagon is first found to be equal to the radius; the side of the dodecagon is derived from this; "from which, in like manner, may be found the side of a polygon with twenty-four sides; and so on, doubling the number of sides in the polygon until the side be near to the arc. The sum of such sides will be the circumference of the circle, nearly." The side of the polygon of three hundred and eighty-four sides is found, and the ratio given above is deduced.
The explanation of the method of finding the area of the circle is somewhat indirect, and is likewise ingenious. The circle is divided into two semicircles by a diameter: if this diameter is 14, the semi-circumference is equal to 211250. Suppose a number of radii drawn, and the semi-circumference developed into a right line; each half of the circle will become a saw-shaped figure (Fig. 1); placing these
together, we should have a rectangle, Fig. 2, of equal area with the circle. This, of course, leads to the formula, , area circle
To find the surface of the sphere, and its contents, similar methods are employed.
The following sections are concerned with some practical questions, as the determination of the number of boards which can be cut from a prism of wood, the number of measures of grain in a mound, and formulæ for the length of the shadows of gnomons. Sections on the subjects of permutations follow which are sufficiently obscure, and the treatise concludes with the neat sentiment that "joy and happiness is indeed ever increasing in this world for those who have Lílívatí clasped to their throats. . . ."
Next follows the "Vija-Ganita," a treatise on algebra, of which science the author observes: "Neither is algebra consisting in symbols, nor are the several sorts of it, analysis. Sagacity alone is the chief analysis: for vast is inference."
The methods of Hindoo algebra are rude. Positive quantities have no sign, while negative ones are distinguished by a dot. For the unknown quantities the different colors are used, and the initial letters of their names are placed in an equation. Equality must be expressed in words, for the sign was first used by Robert Recorde, who says, "No two things can be more equal than a pair of parallel lines."—(Hutton.)
Equations of the first and second degree are treated of, but with obscurity.
It is noteworthy that at least two references are made in this treatise to older authors, which deserve quotation as showing the nature of problems previously proposed.
"Example, by ancient authors. Five doves are to be had for three drammas; seven cranes for five; nine geese for seven; and three peacocks for nine: being a hundred of these birds for a hundred drammas for the prince's gratification."
"Example by an ancient author. What number multiplied by three and having one added to the product becomes a cube: and the cube root squared and multiplied by three and having one added, becomes a square?"
Enough has been given to show that the Hindoo mind was apt at mathematical logic, and to exhibit the characteristic grace of fancy with which it regarded science.
Arithmetic, when the world was young, was not inconsistent with fancy and with enjoyment. Algebra was regarded with a certain awe. We cannot better illustrate this than by one more quotation from the translation by Colebrooke of the "Vija-Ganita:"
"There is no end of instances, and therefore a few only are exhibited. Since the wide ocean of science is difficultly traversed by men of little understanding, and, on the other hand, the intelligent have no occasion for copious instruction, a particle of tuition conveys science to a comprehensive mind, and, having reached it, expands of its own impulse. . . . The rule-of-three terms constitute arithmetic; and sagacity, algebra."