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Indian Science and Technology in the Eighteenth Century: Some Contemporary European Accounts c12: V. A Proof that the Hindoos had the Binomial Theorem

Indian Science and Technology in the Eighteenth Century: Some Contemporary European Accounts c12

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English

V. A Proof that the Hindoos had the Binomial Theorem

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(By Reuben Burrow (Published 1790))

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The Islands in the Bay of Bengal are many of them covered with shells and marine productions to a great height, and there are beds of large smooth pebbles near the Herdwar some hundreds of feet above the present level of the Ganges; the sea has therefore gradually been retiring, and consequently the position of the Equator was formerly farther north than it is at present in this part of the earth; and if a few similar observations were made in other countries, it is evident that the ancient situation of the Pole upon the surface of the earth might be determined sufficiently near for explaining many difficulties and paradoxes in geographical antiquities, for this purpose also it would be advisable to have permanent meridian lines drawn in high northern latitudes, to be compared in succeeding ages, and also to have marks cut upon rocks in the sea to show the proper level of the water.

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In the aforesaid position of the Equator the sands of Tartary were inhabitable and the Siberian climates temperate; the deserts of the lesser Bukharia were then part of the seat of the Paradise of Moses; and the four sacred rivers of Eden went through India, China, Siberia, and into the Caspian Sea, respectively: this appears from a Bramin map of the world in the Sanscrit language which I met with about two years ago in the higher parts of India, together with a valuable Treatise of Geography upon the system of Boodh; both of which I communicated with my idea on the subject to Mr Wilford, of the Bengal Engineers; and from him the world may expect shortly to be favoured with the first true representation of Scriptural and Hindoo Geography.

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From the aforesaid country the Hindoo religion probably spread over the whole earth: there are signs of it in every northern country, and in almost every system of worship: in England it is obvious; Stonehenge is evidently one of the temples of Boodh; and the Arithmetic, the Astrology, the Holidays, Games, names of the Stars and figures of the Constellations; the ancient Monuments, Laws, and even the languages of the different nations have the strongest marks of the same original. The worship of the sun and fire; human and animal sacrifices, & c. have apparently once been universal: the religious ceremonies of the papists seem in many parts to be a mere servile copy of those of the Goseigns and Fakeers; the Christian Ascetics were very little different from their filthy original the Byraggys, & c.; even the hell of the northern nations is not at all like the hell of the scripture, except in some few particulars; but it is so striking a likeness of the hell of the Hindoos that I should not at all be surprised if the story of the soldier that saw it in Saint Patrick’s purgatory, described in Matthew Paris’s history, should hereafter turn out to be merely a translation from the Sanscrit with the names changed. The different tenets of Popery and Deism have a great similarity to the two doctrines of Brahma and Boodh, and as the Bramins were the authors of the Ptolemaic system, so the Boodhists appear to have been the inventors of the ancient Philolaic or Copernican, as well as of the doctrine of attraction; and probably too the established religion of the Greeks and the Eleusinian mysteries may only be varieties of the two different sects. That the Druids of Britain were Bramins is beyond the least shadow of a doubt; but that they were all murdered and their sciences lost, is out of the bounds of probability; it is is much more likely that they turned school-masters and freemasons and fortune tellers, and in this way part of their sciences might easily descend posterity, as we find they have done; an old paper said to have been found by Locke bears a considerable degree of internal evidence both of its own antiquity and of this idea; and on this hypotheses it will be easy to account for many difficult matters that perhaps cannot so clearly be done on any other, and particularly of the great similarity between the Hindoo sciences and ours; a comparison between our oldest scientific writers and those of the Hindoos will set the matter beyond dispute, and fortunately the works of Bede carry us twelve hundred years back, which is near enough to the times of the Druids to give hopes of finding there some of their remains: I should have made the comparison myself but Bede is not an author to be met within this country; however, I compared an Astrolabe in the Nagry character (brought by Dr Mackinnon from Jynagur) with Chaucer’s description, and found them to agree most minutely, even the centre pin which Chaucer calls ‘the horse’ has a horse’s head upon it in the instrument; therefore if Chaucer’s description should happen to be a translation from Bede it will be a strong argument in favour of the hypothesis, for we then could have nothing from the Arabians: what Bungey and Swisset may contain, will also deserve enquiry, and that the comparison may be the readier made, where the books are procurable, I mean very shortly to publish translations of the Lilavati and Beej Ganeta, or the Arithmetic and Algebra of the Hindoos.

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It is much to be feared, however, that many of the best treatises of the Hindoos are lost, and that many of those that remain are imperfect; by the help of a Pundit I translated part of the Beej Ganeta near six years ago when no European but myself, I believe, even suspected that the Hindoos had any Algebra, but finding that my copy was imperfect I deferred completing the translation in hopes of procuring the remainder; I have since found a small part more, and have seen many copies, but from the plan of the work (which in my opinion is the best way of judging) they still seem all to be imperfect, though the copier generally takes care to put at the end of them that they are complete. I have the same opinion of the Lilavati, and for the same reason; indeed it is obvious that there must have been treatises existing where Algebra was carried much farther; because many of their rules in Astronomy are approximations deduced from infinite series; or at least have every appearance of it; such for instance as finding the sine from the arc, and the contrary; and finding the angles of a right angled triangle from the hypothenuse and sides, independent of tables of sines; and several others of a similar nature much more complicated. I have been informed by one of their Pundits, that some time ago, there were other treatises of Algebra, besides that just mentioned, and much more difficult, though he had not seen them; and therefore, as it is possible they may still be existing, and yet be in danger of perishing very soon, it is much to be wished that people would collect as many of the books of science as possible; (their poetry is in no danger, and particularly those of the doctrine of Boodh which perhaps may be met with towards Tibet. That many of their best books are depraved and lost is evident, because there is not now a single book of geometrical elements to be met with; and yet that they had elements not long ago, and apparently more extensive than those of Euclid is obvious from some of their works of no great antiquity: the same remarks are applicable to their cosmographical remains, in some of which there are indications of an Astronomy superior to that of the Soorya Siddhant, and such popular treatises.

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Till we can therefore find some of their more superior works, it must be rather from the form and construction of their astronomical tables and rules, and the properties implied in their accidental solutions of questions, & c., that we can judge what they formally knew, than otherwise; that they were acquainted with a differential method similar to Newton’s, I shall give many reasons for believing, in a treatise on the principles of the Hindoo Astronomy, which I began more than three years ago, but was prevented from finishing, by a troublesome and laborious employment that for two years gave me no leisure whatever; and which (though the short time I had to spare since has been employed in writing a comment on the works of Newton, and explaining them to a very ingenious native who is translating them into Arabic) I hope ere long to have an opportunity of completing: at present I shall only give an extract of a paper explaining the construction of some tables, which first led me to the idea of their having a differential method; it is part of one out of a number of papers that were written in the latter part of the year 1783, and the beginning of 1784, and of which several copies were taken by different people, and some of them sent to England: this particular extract, was to investigate the rules at pages 253, 254 and 255, of Mons. Gentil’s Voyage, of which the author says, ‘Je n’ai pu savoir sur quels principes cette table est fondee, & c.’ and is as follows:

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‘Now by proceeding in the manner explained in the aforesaid paper to calculate the right ascension and ascensional difference for Tirvalour, and afterwards taking the differences algebraically, and reducing them to *puls* of a *gurry* as in the following table, the principles of the method will be evident. (Table on the following page)

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The fifth and sixth columns sufficiently explain the tables in pages 253 and 254 of Mons. Gentil, but there remains a part more difficult, namely, why in calculating the *Bauja,* or the doubles of the first differences of the ascensional difference ‘20/60 of the length of the shadow is taken for the first; 4/5 of the first term for the second, and 1/3 of the first term for the third.’ The primary reason of taking differences here, seems to be that the chords may be nearly equal to the arcs, and that, by adding of the differences, the arcs themselves may be found nearly; the reason will appear from the following investigation. Let N be the equatorial shadow of the Bramins in *bingles*; then 720 the length of the Gnomon, or twelve *ongles*, will be to N the shadow, as radius to the tangent of the latitude; and radius to the tangent of the latitude as the tangent of the declination to the sine of the ascensional difference; consequently 720 is to N as the tangent of declination to the sine of the ascensional difference. Now if the declinations for one, two, and three signs be substituted in the last proportion, we get the sines of the three ascensional differences in terms of N and known quantities; and, if these values be substituted in the Newtonian form for finding the arc from the sine, we get the arcs in parts of the radius; and if each of these be multiplied by 3600 and divided by 6,28318, the values come out in *puls* of a *gurry* if N be in *bingles*, but in parts of a *gurry* if N be in *ongles*; and by taking the doubles we get the values nearly as follows:

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‘Now because the values in the first column are doubles of the ascensional differences for one, two and three signs, their halves are the ascensional differences in parts of a *gurry*, supposing N to be in *ongles*; and if each of these halves be multiplied by sixty, the products, namely, 9,9168 N; 17,9784 N; and 21,2580 N will be the same in *puls* of a *gurry*; and if to get each of these nearly, in round numbers, the whole be multiplied by three, and afterwards divided by three, the three products will be 29.75 N; 53.94 N, and 63.77 N which are nearly equal to thirty N; fifty-four N, and sixty-four N respectively; and hence the foundation of the Bramin rule is evident, which directs to multiply the equatorial shadow by thirty, fifty-four, and sixty-four, respectively; and to divide the products by three for the Chorardo in *puls*: and these parts answer to one, two, and three signs of longitude from the true Equinox and therefore the *Ayanongsh*, or Bramin precession of the Equinox, must be added to find the intermediate Chorardo by proportion.’

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Though the agreement of this investigation with the Bramin results, is no proof that the Hindoos had either the differential method, or Algebra; it gave me at the time a strong suspicion of both; and yet for want of knowing the name that Algebra went by in Sanscrit, it was near two years before I found a treatise on it, and even then I should not have known what to enquire for, if it had not come into my mind to ask how they investigated their rules: of the differential method I have yet met with no regular treatise, but have no doubt whatever that there were such, for the reasons I before hinted at, and I hope others will be more fortunate in their enquiries after it than myself.

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With respect to the Binomial Theorem, the application of it to fractional indices will perhaps remain for ever the exclusive property of Newton; but the following question and its solution evidently show that the Hindoos understood it in whole numbers to the full as well as Briggs, and much better than Pascal. Dr Hutton, in a valuable edition of Sherwin’s tables, has lately done justice to Briggs; but Mr Witchell, who some years before pointed out Briggs as the undoubted inventor of the differential method, said he had found some indications of the Binomial Theorem in much older authors: the method however by which that great man investigated the powers independent of each other, is exactly the same as that in the following translation from the Sanscrit.

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‘A Raja’s palace had eight doors; now these doors may either be opened by one at a time; or by two at a time; or by three at a time; and so on through the whole, till at last all are opened together: it is required to tell the numbers of times that this can be done?

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‘Set down the number of the doors, and proceed in order gradually decreasing by one to unity and then in a contrary order as follows:

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‘Divide the first number eight by the unit beneath it, and the quotient eight shows the number of times that the doors can be opened by one at a time: multiply this last eight by the next term seven, and divide the product by the two beneath it, and the result twenty-eight is the number of times that two different doors may be opened: multiply the last found twenty-eight by the next figure six, and divide the product by the three beneath it, and the quotient fifty-six, shows the number of times that three different doors may be opened: again this fifty-six multiplied by the next five, and divided by the four beneath it, is seventy, the number of times that four different doors may be opened: in the same manner fifty-six is the number of fives that can be opened: twenty-eight the number of times that six can be opened: eight the number of times that seven can be opened: and lastly, one is the number of times the whole may be opened together, and the sum of all the different times is 255.’

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The demonstration is evident to mathematicians; for as the second term’s coefficient in a general equation shows the sum of the roots, therefore in the *n* power of 1+1 where every root is unity the coefficient shows the different *ones* that can be taken in *n* things: also because the third term’s coefficient is the sum of the products of all the different twos of the roots, therefore when each root is unity the product of each two roots will be unity, and therefore the number of units or the coefficient itself shows the number of different *twos* that can be taken in *n* things. Again because the fourth term is the sum of the products of the different threes that can be taken among the roots, therefore when each root is unity, the product of each three will be unity and therefore every unit in the fourth, will show a product of three different roots, and consequently the coefficient itself shows all the different *threes* that can be taken in *n* things; and so for the rest. I should not have added this, but that I do not know well where to refer to it.

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P.S. There is an observation perhaps worth remarking with respect to the change of the Poles; namely, that the small rock oysters are generally all dead within about a foot above high water mark; now possibly naturalists may be able to tell the age of such shells nearly by their appearance, and if so, a pretty good estimate may be formed of the rate of alteration of the level of the sea in such places where they are; for I made some astronomical observations on a rock in the sea near an island about seven miles to the south of the island of Cheduba on the Aracan coast, whose top was eighteen feet above high water mark, and the whole rock covered with those shells fast grown to it, but all of them dead, except those which were a foot above the high water mark of that day, which was February 2, 1788: the shells were evidently altered a little in proportion to their height above the water, but by no means so much as to induce one to believe that the rock had been many years out of it: all the adjacent islands and the coast showed similar appearances, and therefore it was evidently no partial elevation by subterranean fires, or any thing of that sort: this is also apparent from the island of Cheduba itself, in which there is a regular succession of sea beaches and shells more and more decayed to a great height. By a kind of vague estimation from the trees and the coasts and shells, & c. (on which however there is not the least dependence) I supposed that the sea might be subsiding at the rate of about three inches in a year.

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