The Calcutta Review/Series 1/Number 4/Article 5
Art. V.—1. Lilawati, or a Treatise on Arithmetic and Geometry, by Bhascara Acharja. Translated from the original Sanscrit, by John Taylor, M.D., of the Hon’ble East India Company’s Bombay Medical Establishment. Bombay, 1816.
2. Algebra, with Arithmetic and Mensuration from the Sanscrit of Brahmegupta and Bhascara. Translated by Henry Thomas Colebrooke, Esq. F.R.S., &c. London, 1817.
3. History of Algebra in all Nations, by Charles Hutton, LL D., (Mathematical and Philosophical Tracts, vol. ii.) London, 1812.
4. Lectures on the Principles of Demonstrative Mathematics, by the Rev. Philip Kelland, A.M., F.R.S.S.L. and E., Professor of Mathematics in the University of Edinburgh. Late Fellow and Tutor of Queen’s College, Cambridge. Edinburgh, 1843.
Herodotus informs us that Geometry took its origin in Egypt, and supposes that necessity was the mother of the invention;—that the purpose of it was to delineate the boundaries of the fields on their emerging from the waters of the Nile. We see no reason to reject the venerable historian’s statement of the fact, nor to accept his theory in regard to it. As to the fact, it seems to be incontrovertible that Geometry as a Science was unknown in Greece before the time of Thales the Milesian, and there seems no reason to question the uniform tradition that he imported the knowledge of it from Egypt. But as to the theory of Herodotus regarding the necessity that gave rise to the invention,[1] we can suppose no foundation on which it can rest, except the etymology of the Greek name of the Science ((Greek characters), or measure of the earth) and this is useless as a foundation for the hypothesis, unless it can be shown that this name is an exact rendering of the Egyptian name of the Science. Moreover, we should suppose that (Greek characters) is not the term that would have been employed to signify the mensuration of land, since we know of no instance in which the term (Greek characters) or (Greek characters) is employed in such a sense. If we might be allowed to conjecture, we would venture to suggest that this name was given to the Science only when it reached such a stage of advancement that mathematicians began to apply it to the determination of the size of the earth. In fact the whole amount of Geometry that would be required for the purpose indicated by Herodotus, (the rather that he tells us the Egyptian estates were all squares) would be the problem to draw a straight line between two points, and this problem we presume it was not left to the Egyptians to be the first to solve.
This consideration suggests to us a fact that seems to have been strangely overlooked by writers on the history of the Mathematical Sciences;—viz. that, speaking strictly, the mathematical sciences could have no beginning apart from the original creation of the human race, for their first elements are bound up in the very constitution of the mind of man. We believe there has never been a man capable of exercising his faculties, who did not know that things which are equal to the same thing are equal to one another, yet no one who knows this can properly be said to be wholly ignorant of mathematical science. From this initial point a line continuous and unbroken stretches upwards and onwards to all that modern mathematicians know of the properties and relation of space and figure, and is destined to be prolonged to all that their successors shall ever know. And what is true of Geometry is equally true in regard to Algebra—the other great branch of mathematical science. An utter ignorance of number and quantity seems to be scarce compatible with rationality. We scarcely know how thought can be exercised apart from a knowledge that there is a difference between one and two. Yet this is the foundation of Algebra, the first step on the ladder that stretches continuously upward to that lofty eminence from which Lagrange looked down. If in this statement we be in error at all, it is, we apprehend, in speaking of the ascent from the lowest degree of knowledge that is compatible with rationality to the highest attainment in this department that is permitted to man in his present state of being, as accomplished by a series of successive steps: it is rather by a continuous plane of the gentlest and scarcely perceptible elevation, so that of those engaged in the ascent it is often difficult to determine who has attained the greatest height. There is no break in the whole ascent; and this, we may state in passing, is one of the grand advantages of mathematical study as a mental exercise. There is no man who is incapable of the study, neither is there any man who does not find in it full employment for all his faculties. No man is incapable of taking his place on the bottom of the plane, and beginning the ascent; no man, on the other hand, has ever reached, or will ever reach, the summit.
This too it is that renders the history of mathematical discovery often a work of great difficulty. Of this the most notable example is furnished by the long-agitated, and still undecided question, as to the invention of the differential or fluxional Calculus. All mathematicians, and multitudes who are mathematicians are familiar with the details of this celebrated contest, in which it was most warmly disputed whether Newton or Leibnitz should be regarded as the inventor of the Calculus. To some it might appear that this would be a matter of very easy determination; but the fact is that neither the one nor the other advanced more than a scarcely measurable step above the point reached by several of their predecessors. Roberval and Fermat and Wallis and Pascal are constantly mentioned as having approached indefinitely near to the method; and we believe it might be shown that one who is never mentioned at all in connection with this subject approached at an earlier period nearer to it than any of them. We mean the famous Napier of Mercheston, the inventor of the most useful of all mathematical instruments, the logarithnic Calculus. These facts show how difficult it is to trace the progress of mathematical discovery. But if it be so in regard to those lofty elevations which so few can reach, and where each man stands prominently out to the view of all who are capable of seeing so far aloft, and where there is comparatively little chance of the favoured ones jostling each other, how much more may we expect to find difficulties in ascertaining the precedence of those who throng the lower regions, where progress is comparatively easy. We are told, for example, that Pythagoras discovered the proposition that is now universally known as Euclid’s forty-seventh, and we do not doubt the fact; but we can pronounce no judgment as to the merit of the discovery, since we know not, and now can never know, what propositions were known to him and his predecessors before—or, to keep up the figure that we have hitherto employed,—since we cannot tell how far in taking this step he left his contemporaries behind. It is scarcely possible to construct a square and draw its diagonals without perceiving by inspection that in an isosceles right-angled triangle the squares on the sides are equal to the square on the hypothenuse, since it is evident that the square of half the diagonal is equal to half the original square, and therefore the square of the whole diagonal equal to double the original squares, or equal to the sum of the squares of the two sides of the square, that is the sum of the squares of the sides of an isosceles right-angled triangle of which the diagonal of the original square is the hypothenuse. Thus then a particular case of this celebrated proposition is almost self evident; and it does not seem to us now that it could ever be a matter of difficulty to generalize the theorem; but then we know that it is one thing to invent new methods of proving a truth, and another and very different thing to discover the truth itself. If, however, it be true that Thales brought the knowledge of Geometry into Greece, and if it be true that Pythagoras, who was born sixty years after, was overjoyed at the discovery of the equality of the squares on the sides of a right-angled triangle to the square on the hypothenuse, we may safely conclude both that the amount of knowledge introduced by Thales was not great, and that the progress of the Greeks in Geometry was not rapid,—that no one of them shot far ahead of his predecessors or compeers.
In the history of mathematical science we have no very marked exception to our theory of the gradual progress of invention and discovery, unless it be in the case of Euclid: and it is not improbable that his merits as a discoverer may be very considerably over-estimated from an ignorance of the attainments of his immediate predecessors. It is difficult indeed on the one hand to suppose that one man could have so far surpassed all others as Euclid must have done, had he been the discoverer of any considerable number of the propositions in his Elements, as well as the arranger of the whole; and equally difficult to suppose on the other hand, that a man possessed of so pure and exquisite a mathematical taste should not have been himself a great discoverer. For ourselves we are disposed to believe (since the matter must ever remain one of mere conjecture) that the Elements were chiefly or entirely composed out of previously existing materials. There is that about them which indicates that the materials existed in his mind in their totality from the outset, and fell naturally into their proper places. At all events it is the exquisiteness of its arrangement that makes the Elements such an incomparable book, for incomparable it unquestionably is. There is not a more singular fact in the whole history of man than this—that in the most progressive of all the pure Sciences, a work composed in the comparative infancy of the Science should hold its place to this day as the very best elementary work that has yet been produced. That it will still hold that place we think may be safely asserted, for we can confidently appeal to every mathematician whether he does not feel a degree of revulsion from the very idea of intermeddling with Euclid,—a revulsion similar in kind to that which the pious Christian feels when neological criticism lays its impious hand on the inspired record of his faith.
It appears then that even among the ancient Greeks,—yea even among the English and German mathematicians of the eighteenth century, the progress of mathematical discovery cannot be accurately traced. Now if the question as to the discovery of the Calculus be destined to remain for ever undecided, as it apparently is, how much more may we expect to find it impossible to trace the history of discoveries far less important, in times far more distant, in a country where there was no printing to record discoveries, nor transactions of societies at once to identify and treasure them? The History of the Hindu mathematics is accordingly, as might well be expected, hopelessly obscure. Whether the Hindus derived their first knowledge of mathematics as a science from Egypt, or whether the Egyptains on the other hand derived theirs from India, or whether both drew from a common fountain, or whether each nation set forth independently from those first principles, which, as we have attempted to show, are inseparable from the existence of rationality—will probably never be determined. One thing, however, which we may notice as deserving attention, both as interesting in itself and as pointing to an origin of the science distinct from the Egyptian source from which the stream of Greek mathematics took its rise, is the very different direction that the mathematical pursuit seems to have taken in the two countries. If we assume that the Grecian mathematics are the development of that science whose rudiments were found in Egypt in the days of Thales, then we must infer that Geometry, or the science which treats of figure and space, was far more cultivated there than Algebra, or the science which treats of numerable quantity. Now the very opposite seems to have been the case in India. The Greek Algebra was as nothing in comparison with the Greek Geometry; the Hindu Geometry was as little worthy of comparison with the Hindu Algebra. How far this difference might be accounted for, on the supposition of a common origin, by the difference of climate and of the habits of the people, we must leave to others who may be better qualified and disposed than we to speculate upon. It seems certain that the countrymen of Phidias and Praxiteles and Appelles must have been more capable than the Hindus of appreciating the beauties of mathematical figure; while the mysteries of number, a thing undefinable but real, a thing regarding which men may reason all their days without making the matter a whit plainer, must have been equally grateful to the taste of the abstruse Hindu sages. As for the common notion (adopted by Professor Kelland in the work whose title stands at the head of this article) that the attempts at the quadrature and the publication of the cube had a great influence in promoting the progress of Geometry among the Greeks, we set nothing by it. Every one knows the application made by Lord Bacon of the fable of the youths being made to dig up their vineyard under the delusive belief that there was a treasure buried in the soil, but with all deference we must express a doubt whether such results have really followed the vain pursuits of men in geometry, alchemy, and astrology as are generally supposed to have resulted from them. At all events, we think it very certain that the questions as to the duplication of the cube and the quadration of the circle must have arisen, not in the infancy of geometrical science, but after it had made considerable progress, and thus if the spirit of geometrical science had not taken this direction, it must have taken some other, and probably a far better one. Any one who thinks otherwise may as well suppose that an apple falling before a peasant would have led him to the theory of universal gravitation, or that if no apple had fallen before Newton, he would never have solved that sublime theory.
As with the heroes who fought before Agamemnon, so has it fared with the Hindu algebraists who studied before Brahmegupta. This Brahmegupta, who lived in the seventh century of the Christian era, and Bhascara who flourished in the twelfth century, are the authors of the works translated by Mr. Colebrooke,—the standard works of Hindu mathematics. All subsequent algebraists have been content merely to illustrate and simplify the works of Brahmegupta and Bhascara. An account of these works is, therefore, to all intents and purposes, an account of the Hindu algebra. Such an account we purpose in the present article to give. Some of the commentators indeed mention several algebraists who seem to have preceded Brahmegupta, but of them no record remains but the mere mention of their names. Arya Bhatta, indeed, who lived, as is supposed, in the fourth century, was certainly an algebraist; and is mentioned, by a commentator on Bhascara, as the founder of the science; but what was the amount of his attainments, it is impossible to discover. Nor may we suppose that Algebra was wholly unknown before his time. In fact, we find allusion in very ancient Hindu books which leave no doubt on our minds that algebra was cultivated at a very early period among the Hindus. One came under our observation very recently, which it may be well to quote as a Specimen of many that we have met with from time to time. In the Nalodaya, recently translated by the Rev. Dr. Yates, of Calcutta, we find that when its hero Naloh, in the days of his humiliation, was serving as charioteer to Ritipurna, his master astonished him by telling at sight the number of leaves and fruit on a particular tree; Ritipurna’s power of doing this the poet ascribes to his familiarity with dice. The passage, as translated by Dr. Yates, is as follows:—
And to astonish and delight the mind
Of his expert and pious charioteer
With calculations of immense extent.
Such knowledge had he gained by means of dice,
That when a tree was full of leaves and fruit
He could, at sight, of each the number tell.
Descending from the car he marked a tree
And told in sums exact its whole contents.
When Nala counted all the leaves and fruit
And found the sum of both and each agree
With what had been declared, he was surprised
And wished to understand the wondrous art
By which such calculations could be made.
The king as ardently desired to know
By what mysterious art the charioteer
All other men in horsemanship excelled.
So they agreed their secrets to reveal
And from each other mutual aid derive.
But when these heroes, famed for martial deeds,
Had thus their art consented to transfer;
The transfer they confirmed by solemn oath
That neither to a third should e’er disclose
The science which they both now understood.
Nalodaya, book iv.
We cannot hesitate for a moment to conclude that this passage is intended to indicate that Ritapurna was an algebraist. The poet was either himself ignorant, or deemed it inconsistent with his poetical design, to inform us of the data on which the calculations proceeded; but his allusion to dice is quite sufficient, independently of every thing else, to show us that it was an algebraical process that he has thus, either from ignorance or from choice, invested with mystery. There is no connection between anything that can be indicated by means of dice, and the number of leaves and fruit on a particular tree: but the same algebraist calculated, as algebraists calculate now, the chances of the throws of dice, and calculated also the number of leaves on a tree from so small data that the uninitiated supposed that it was done by mere intuition. The question that Ritapurna actually solved might be such an one as those that occur in our ordinary school-books—as, for example, the number of leaves on the tree is to the number of fruit in a given proportion, say as 2 to 1, and 3 times the number of leaves added to 6 times the number of fruit make 20,000. All those who are acquainted with the elements of algebra know that such questions as these may be endlessly multiplied. We can have no doubt that it was the solution of such a question as this that Kalidas intended to eulogise on the part of Ritapurna, though as to its actual difficulty it is of course impossible for us even to conjecture. But whether the question actually solved were difficult or not, what we have at present to do with is this, that at the period in question the algebraist was on a footing with the “horse-whisperer” as an object of admiration. The science was therefore precisely in the same state in regard to its advancement and diffusion that it had attained in Europe in the days of our own Baron Napier of Merchiston, who astonishes the minds of his superstitious countrymen by similarly divulging the results, and concealing the processes of his calculations. Now Kalidas lived in the days of Vikramaditya, a little before the commencement of the Christian era. At this period, therefore, we conclude that the science of algebra was so far known in Hindustan, that its professors were able to solve such equations as those given in our ordinary school-books, and so far unknown that such solutions were regarded as amazing, and almost miraculous, not by the vulgar alone, but even by the generally intelligent but unmathematical portion of the community. This is all that we know regarding the history of algebra till the time of Brahmegupta. Probably more might be inferred from allusions in the poets similar to that quoted above; and we may be permitted to observe that it is a subject well worthy the attention of the oriental scholar.
We have before us the principal, or rather the only, algebraical works of the Hindus, viz. the Ganita (arithmetic) and Cuttaca (algebra) of Brahmegupta, and the Lilavati and Bija Ginata of Bhaskara Acharya. These are all translated (in a manner that requires not our praise) by the late Mr. Colebrooke; and the Lilavati also by Dr. Taylor of Bombay. From Dr. Taylor’s and Mr. Colebrooke’s works having been published in consecutive years, the one in Bombay and the other in London, we suppose that the translators laboured without any knowledge of each other’s intentions. It is well that they did so, as not only is Mr. Colebrooke’s work far more complete than it would have been without the Lilavati, but the possession of a two-fold rendering is the most satisfactory guarantee to the student ignorant of the original Sanscrit, that the renderings are faithful.[2] As it is, it is perfectly satisfactory to find that, with great variation in the mode of rendering, the Lilavati, as translated by Colebrooke and by Taylor, is substantially the same. Taylor’s is the more literal, and therefore to us the more valuable translation; Colebrooke’s is the more elegant, and is moreover enriched by selections from the annotations of the principal Hindu commentators, some of which contain explanations without which the text were well nigh unintelligible.
One striking and important fact we ought to notice at the outset. That it appears certain that we were indebted to the Hindus for our numerals, though, having derived them through the medium of the Arabs, we have appropriated to them the name of that people. Although the numerals of our modern typographers differ very much from the Sanscrit characters so that it were scarcely possible to recognise them, yet it is not difficult to trace the process by which they were transmuted from the original Sanscrit form into their present elegant figures. Dr. Hutton in his mathematical tracts gives several figures by way of showing the transition, and it appears to us clear on inspection that the case is made out. In fact the figures actually used in Europe up till four hundred years ago, were almost as like the Sanscrit characters as they were like the modern numerals. If it be granted then that the decimal notation originated with the Hindus, it will be difficult to deny them the highest place in the scale of Algebraical eminence, for there is unquestionably nothing in the whole range of mathematical science that combines elegance with utility to a greater extent than the decimal notation. But, then, while we believe that the decimal characters were derived from the Hindus, we believe that the decimal scale, properly so called, is of a far more ancient date. In fact it seems altogether a catholic system, common to the whole human race; and must, as we think, have been in use wherever there were men with two hands, and five fingers on each. But if we be indebted to the Hindus for the decimal notation alone, our obligation to them is sufficiently great. Those who know most of numbers will be most willing to admit this, for they will be best able to tell at once how admirable the decimal system of notation is, and how important to the of the Page:Calcutta Review Vol. II (Oct. - Dec. 1844).pdf/550 Page:Calcutta Review Vol. II (Oct. - Dec. 1844).pdf/551 This question if solved in our manner would stand thus:—
Let x y and z be the three numbers; then by the question:
52 a 2la D
A a a 1 2](a +!) in which a is an in- 57a tl + p= a "4 determinate num- í a 20 ber.
97 ge CHP +2)
20 ~ + “30° ~~ 20
These equations, if multiplied by 20, become severally
5x2=2la 7y=2l@+lh. y 9z2=21 (a + 2)..32
= 3(a+ 1) = 7(a+ 2)
We have now to determine a such that x, y, and z may be all whole numbers. It is evident that in order to give x a whole number, a must be a multiple of 5, since 5 and 21 are incommensurable numbers. If therefore we try a = 5 we find that x and y are whole numbers, but z is not. We therefore try a = 10 and find x = 42, y = 33, and z = 28.
This is a very simple example, and therefore the superiority of our notation is not so clearly seen as it would be in a more complicated example. This will however suffice for the present, as an illustration of the inelegant manner of stating the operations, as compared with our modern beautiful system. While on this subject we may be permitted to say again, that we know nothing more thoroughly elegant than the modern system of algebraic notation, when employed in its complete form, with fractional and negative indexes. It has been gradually developed during three centuries by a process somewhat similar to the rise and progress of the Sofa so humorously sketched by its bard. As was “the rugged rock washed by the sea” for a place of repose, so was the notation of the first Italian “Cossists.” The “joint stool on three legs upborne” may represent the notation of Carden. Tartalea may take place with the introducer of the “four legs, with twisted form vermicular;” while Stevinus did that for algebra which was done for upholstery by him who transformed the stool into a chair. The notation of Des Cartes may represent the “settee.” Lagrange must take rank with him who “in happier days” introduced the “sofa,” while the spring cushions and all the improvements which make the sofas of these days far more luxurious than Cowper ever dreamt of, have their parallel in the improvements introduced by Peacock and the rest of our Cambridge mathematicians.
Convenience next suggested elbow-chairs,
And luxury the accomplished sofa last.
But to return to the Indian notation. It will not be difficult to explain to our readers the connection between it and ours. Corresponding to our a is the term ya 1, our a + l is represented by ya 1 ru 1, and our a + 2 by ya 1 ru 2. Then the numbers sought are ca 1, ni 1 and pi 1 corresponding to our x, y and z. Thus ca 5 — ya 20 represents 5x — 20a, and so on. The process of the pulverizer we shall return to ere long; at present we have only to do with the notation; which this short explanation will we trust render in some degree intelligible. It will be observed that a dot over a number or quantity indicates that it is to be subtracted,[3] while ru, the initial syllable of rupa (form) placed before a quantity marks it as an absolute number, and when it is positive or additive no sign is employed to mark the addition. Yavat tavat, (as much or as many as) is contracted into ya, and is generally employed, just as we employ any letter of the alphabet, to express a quantity either known or unknown: ca, ni and pi, are the initial syllables of the Sanscrit words signifying black, blue, and yellow. These are generally used as the representatives of known quantities, as we use the first letters of the alphabet. Sometimes instead of the colours, the initial letters of the names of the things signified are employed, as we might make p stand for the price of a pearl and d for that of a diamond, if required to solve a question relating to the prices of jewellery.
The works of Brahmegupta translated by Mr. Colebrooke are the twelfth and [[eighteenth] chapters of the Brahma-Sphuta Siddhanta, a treatise on astronomy. This accounts for the great brevity with which the rules are expressed, a brevity which sometimes renders them scarcely intelligible. He sets out by informing us that “He who distinctly and severally knows addition and the rest of the twenty logistics, and the eight determinations, including measurement by shadow, is a mathematician.” The twenty logistics are addition, subtraction, multiplication, division, square, square-root, cube, cube-root, five rules of reduction of fractions, rule of three terms (direct and inverse), of five terms, of seven terms, of nine terms, and of eleven terms, and barter. The eight determinations are, mixture, progression, plane figure, excavation, stack, saw, mound and shadow. The first section consists of rules for the logistics, and the succeeding 8 for the determinations. Mixture corresponds with what in our books of arithmetic is called partnership or fellowship. Under progression is treated only what we call arithmetical progression. One rule is ingenious though a little complicated, and clearly indicates a considerable knowledge of the nature of series. It is when translated into our ways of expression as follows: Given the first term, the common difference and the sum of a series in arithmetical progression, to find the number of terms. The rule is as follows:“Add the square of the difference between twice the initial term and the common increase to the product of the sum of the progression by 8 times the increase; the square root, less the foregoing remainder, divided by twice the common increase, is the period.” This rule when translated into our algebraic language will be as follows: a being the first term, d the common difference, s the sum of the series, and n the number of terms: thus n = {(2 a—d)} + 8 ds} 3—(2 a—d)
2d This rule we do not find in any of our books, but it is easily deduced from the ordinary formula 2 s = n {2 a + (n—l)d from which, by the solution of a quadratic equation we find
n= (2 axa? + 8d s}? —2 a—d), 2d
The section on “plane figure” teaches the methods of finding the areas of trigons and tetragons, both approximate and exact, as well as the lengths of various lines in the circle. It appears that Brahmegupta was familiar with the propositions now known as Euc. I. 47. & III. 35. Beyond this it does not appear that his geometrical knowledge extended. It is remarkable that there is no allusion whatever to angles, triangles being merely distinguished as equilateral, isosceles and scalene. Mr. Colebrooke indeed introduces the term right-angled triangle, but explains that it is never viewed by the Hindu mathematicians with reference to its having a right-angle, but is spoken of only as the formal or elementary trigon, because every other trigon may be divided into two such by a perpendicular drawn from one of the angles on the opposite side. A point of more interest is the ratio between the diameter and circumference of a circle. The author states that the diameter being multiplied by 3 gives the “practical” circumference, but that the square root of ten times the square of the diameter is the “neat” circumference. In other words the diameter is to the circumference as 1: ¥ 10, or 1:3.162, &c. a very clumsy approximation, correct only in one decimal place, whereas the proportion of 7:22 is correct to two places, and very nearly correct in the third. From the usually strict manner in which Brahmegupta distinguishes between “gross” or approximate methods and correct ones, we should suppose that he believed this to be the accurate rectification of the circle, and if this be so, it indicates a very meagre acquaintance with geometry.
The section on “excavation” treats of what the Hindu Pandits are perpetually discussing under the title of “tank arithmetic.” We learn from it that Brahmegupta knew that a cone or pyramid is one-third of a cylinder or parallelopiped having the same base and altitude, a property which is generally understood to have been discovered by Archimedes. The section on “stacks” (of bricks) is merely a repetition of that on excavation, height only being substituted for depth. The like may be said of the section on “saw” or mensuration of timber, with the exception that the author introduces the calculation of the sawyers’ wages, according to the different kinds of wood, precisely as if the rate were fixed as unalterably as the rule for finding the solid contents of a plank.
The section on “mounds of grain” assumes that the ninth part of the circumference of the heap is the height in the case of bearded corn, the tenth part in that of coarse grain and the eleventh part in that of fine grain; and then lays down this rule, that the height multiplied by the square of the sixth part of the circumference is the content. This rule proceeds upon the assumption formerly noticed, that the circumference of a circle is 3 times the diameter.
The section on “measure by shadow” is more properly astronomical than algebraical. It consists of three rules, the first of which is as follows:—“The half day being divided by the shadow (measured in lengths of the gnomon) added to one, the quotient is the elapsed or the remaining portion of day, morning or evening.” This is a very rude approximation, which, as rightly stated by the commentator, does not answer for finding the time even in an equatorial position. As we are denied the use of figures it is impossible for us to render the process intelligible by which the rule has been evolved. There are only two cases in which it will be strictly applicable, and these are, first, mid-day when and where the sun is vertical, for then the shadow being nothing, the time will be the half day, and second, every day and in every place when the sun is horizontal, that is at rising and setting, for then the shadow being infinite, the time elapsed since sunrise, or to elapse before sunset, will be nothing. In all cases between these it will give a result completely erroneous. On this subject we may be allowed to make two remarks. The first is, that it indicates the origin of the Hindu astronomy in a place within the tropics, and militates strongly against M. Bailly’s notion (which we combated in a previous volume of this Review) of its having been derived from a hyperborean race. Our second remark is, that this rule may throw some light on a passage in Herodotus, the meaning of which has been much disputed. The passage we refer to is that in which he states that the Greeks derived their knowledge of the pole, the gnomon and the division of the day from the Babylonians.[4] Salmasius, as we learn from the notes to Beloe’s translation of Herodotus, denies that the pole and the gnomon have any reference to horclogy. We venture to think differently, and to suggest that Herodotus refers to just such a problem as this of Brahmegupta, which, by the way, his commentator tells us that he copied from earlier writers.
To this work is added a supplement, containing various rules for the abbreviation of the processes of calculation, and some other matters which do not require any particular remark.
The other work of Brahmegupta, entitled Cuttacad’ haya, or treatise on the Cutta or pulverizer is, as we have stated, the eighteenth chapter of the Brahma-sphuta-Siddhanta. The prefatory paragraphs are as follows:—“Since questions can scarcely be solved without the pulverizer, therefore will I propound the investigation of it, together with problems. By the pulverizer, cypher, negative, and affirmative, unknown quantity, elevation of the middle term, colours, and factum, well understood, a man becomes a teacher among the learned, and by the affected square.” This same pulverizer plays a most conspicuous part in the Hindu algebra and astronomy, and we must endeavour clearly to explain what it is. In its simplest form it may be stated thus,—given a divisor and remainder, to find the quotient. In this form it is, of course, an indeterminate problem, but if one or more other divisors and the corresponding remainders be given, the problem may become determinate. The rule for such a case is the following: —“Rule for the investigation of the pulverizer.—The divisor which yields the greatest remainder is divided by that which yields the least: the residue is reciprocally divided, and the quotients are severally set down one under the other. The residue [of the reciprocal division] is multiplied by an assumed number, such that the product, having added to it the difference of the remainders, may be exactly divisible [by the residue’s divisor]. That multiplier is to be set down [underneath] and the quotient last. The penultimate is taken into the term next above it, and the product added to the ultimate term is the agranta. This is divided by the divisor yielding least remainder, and the residue multiplied by the divisor yielding greatest remainder and added to the greatest remainder is a remainder of [division by] the product of the divisors.”
This rule will probably puzzle our readers. We believe we shall best make it plain by proposing a question and solving it in our own way, and then showing the identity of the steps with those indicated by the rule. Let it be, for example, required to determine a number, which, being divided by 8, gives the remainder 3, and being divided by 7 gives the remainder 6.
Let x be the number required then by the question.
== a+ fre 8a+3 2 6 Hence 7b + 6 =8a +43 8a —3 _. b= 7 — which must be a whole number. 8a—3
i kt a—3 Again, since z is a whole number, B must also be a whole number.
This condition will be fulfilled by making a any number in the arithmetical series 3, 10, 17, &c., which will give the number required 27, or 83, or 139, &c.
Now by Brahmegupta’s rule we have first to divide 7, the number which yields the greater remainder, by 8, the number that yields the less, the quotient is 0 and the remainder 7. This is then reciprocally divided, that is $ = 14. It ought to be observed that the quotient in this division is what we shall afterwards quote as the first quotient. This remainder 1 is to be multiplied by such a number that the product having —3 added to it may be divisible by 7. This number is as before any one of the series 3, 10, 17, &c. Take 10, and the quotient is 1, which we call the last quotient, and then arrange the terms thus:
1 first quotient.
10 assumed number.
1 last quotient.
Then multiplying the penultimate into the preceding, and Page:Calcutta Review Vol. II (Oct. - Dec. 1844).pdf/558 of the first degree. It is of most important application in the Hindu astronomy, for from it can be found, as is evident, the number of revolutions that have taken place since the conjunction of any number of heavenly bodies, by knowing their periodic times and the fraction of a revolution elapsed at any given time. Suppose, for example, the periodic times of three heavenly bodies to be 50 days, 120 days, and 365 days, respectively, and that at a given period they were 4 days, 3 days, and 2 days, respectively, past a particular point in the heavens, then it is evident that by the pulverizer could be found the number of revolutions that they must severally have made since they were all in conjunction in that point. This is unquestionably the process by which the Hindu astronomers fixed the commencement of the Kali-yug, which occupied so much of our attention in a previous article. We have caught the coiners with the very mould in their hands; and M. Bailly, if he were alive, could not by any possible advocacy counterbalance the weight of this circumstantial evidence.
The other sections of this work do not require particular notice. They only give more complicated cases of the pulverizer with astronomical applications.
We have dwelt thus long on the works of Brahmegupta, and have therefore now only to notice a very few improvements introduced after his time, as manifested in the works of Bhascara. These works are two, the Lilavati and the Bija Ganita. The meaning of the former of these titles is curious. We find it given by Dr. Hutton from the preface to a Persian translation of the work. It indicates a degree of indelicacy that will astonish mere European readers, but which is quite in keeping with the manners of the Hindus to the present day.
“It is said that the composing the Lilawati was occasioned by the following circumstance, Lilawati was the name of the author’s (Bhascara) daughter, concerning whom it appeared, from the qualities of the Ascendant at her birth, that she was destined to pass her life unmarried, and to remain without children. The father ascertained a lucky hour for contracting her in marriage, that she might be firmly connected, and have children. It is said that when that hour approached, he brought his daughter and his intended son near him. He left the hour-cup on the vessel of water, and kept in attendance a time-knowing astrologer, in order that when the cup should subside in the water, those two precious jewels should be united. But, as the intended arrangement was not according to destiny, it happened that the girl, from a curiosity natural to children, looked into the cup, to observe the water coming in at the hole; when by chance a pearl separated from her bridal dress, fell into the cup, and, rolling down to the hole, stopped the influx of the water. So the astrologer waited in expectation of the promised hour. When the operation of the cup had thus been delayed beyond all moderate time, the father was in consternation, and examining, he found that a small pearl had stopped the course of the water, and that the long-expected hour was passed. In short, the father, thus disappointed, said to his unfortunate daughter, I will write a book of your name, which shall remain to the latest times—for good name is a second life, and the ground-work of eternal existence.”
In accordance with this origin of the work, it is throughout addressed to Lilavati, in a way that, to our apprehensions, is sometimes very grotesque. Take a few of the examples at random. “Ten times the square root of a flock of geese seeing the clouds collect flew to the Manus lake, one-eighth of the whole flew from the edge of the water amongst a multitude of water-lilies, and three couples were observed playing in the water. Tell me, my young girl with beautiful locks, what were the whole number of geese?” Again—“Beautiful and dear Lilaviti, whose eyes are like a fawn’s! tell me what is the number resulting from one hundred and thirty-five taken into twelve? if thou be skilled in multiplication by whole or by parts, whether by sub-division of form or separation of digits. Tell me, auspicious woman, what is the quotient of the product divided by the same multiplier.”
One of the most interesting processes we find in the work of Bhascara is his method of completing the square in quadratic equations. This method was introduced into the European world by Dr. Hutton, in the work whose title stands at the head of the present article, and has now, we observe, found its way into some of our elementary works on Algebra, under the designation of the Hindu method. As, however, it may still be unknown to many of our readers, and as it is in some cases very decidedly preferable to the ordinary method, it may be as well to present it here at length. Let the equation be $ax^2 + b x= c$, the ordinary method of completing the square would be by dividing both sides by a, and then adding to each side the square of —3 instead of this, however, Bhascara directs us to multiply both sides of the equation by four times the co-efficient of the second power, and to add to each side the square of the co-efficient of the simple power. Thus the complete equation will be this —4a?2?+4abxr+b2=4ac+ b? the first side of which is a complete square. This method, with a simplification of our own which it is not necessary to specify here, we have for a long time used in our practice whenever the second power of the unknown quantity has been effected by a co-efficient. It frequently saves a considerable amount of labour. Bhascara informs us that he takes this rule from Srid’hara.Page:Calcutta Review Vol. II (Oct. - Dec. 1844).pdf/561 Page:Calcutta Review Vol. II (Oct. - Dec. 1844).pdf/562 Page:Calcutta Review Vol. II (Oct. - Dec. 1844).pdf/563 Page:Calcutta Review Vol. II (Oct. - Dec. 1844).pdf/564 Among the Hindus, six hundred, and even twelve hundred years ago, there were men who were as profoundly versed in this branch of mathematical science, as were our fathers a hundred years ago. Their representatives in these days are the miserable drivellers, whose whole knowledge amounts to a few scraps of “tank arithmetic,” and that generally known only by rote. Shall it be then that the successors of Peacock and Airy, and Whewell and De Morgan shall ever become such pigmies in intellect? The history of Hindu science very clearly points out to us that there is nothing in the nature of man to prevent such a degeneracy. Yet we are not without hope, though it comes from another quarter. Since European science is not the property of a man but of the community, its continuance is not dependent on the accident of individual talents, (though its extension may be,) but is secured on a basis as broad and firm as any thing human can rest upon. It was with the Hindu science, as with the monstrous empires that rose by the prowess of a single hero and passed away along with him. But European science is now so deeply rooted and so widely spread, so amalgamated with all the institutions of government and all the arts of life, that nothing short of an entire revulsion of all that is human can ever eradicate it. This advantage, be it remembered, is one of the thousand unthought of blessings which we derive from that blessed book, which has taught us far more clearly than men were ever taught before, the duties and privileges of mankind. Little as some of our philosophers may dream of it, the Bible is the palladium of our science, as well as of those blessings and privileges which are more directly traceable to it. Its absence, and the prevalence of all those barriers to improvement which it alone can wholly banish, have reduced the science of India to its present despicable state; its suppression produced the dark ages in Europe, and its restoration to its due place gave the impulse to that vast movement which has for three centuries been going on; and its dissemination, with all attendant blessings, is the means appointed by the Lord and Ruler of all for introducing light and liberty and joy into all the dwellings of men.
- ↑ It is to be observed, that Herodotus states it merely as an opinion of his own, not as an historical fact. His words are these,—(Greek characters)—Euterpe. 109.
- ↑ We have learned caution by a somewhat ludicrous experience in regard to translations of these same books. A few years ago, being anxious to attain some knowledge of the Hindu mathematics, and not being even aware of the existence of either of the books now before us, we procured the services of a Pundit, an alumnus Sanscrit College in Calcutta, who engaged to translate a portion of the Lilavati into Bengali every day. This he brought to us on the following morning, and we read it together. In this way we had accomplished nearly the whole book, when one morning we were astonished by finding an example relating to the Gobernor Janiral Sahib’s elephant. We soon found that he had thought it his duty to accommodate the work of the sage old Bhascara to European comprehension, not by translating it, but by diluting it with the mixture of what little he knew of European mathematics. Thus did we throw away a good deal of money, and more precious time, but gained withal some profitable experience.
- ↑ We have substituted the sign minus for the convenience of the printers.
- ↑ (Greek characters)