De Morgan's Preface to Ramchundra's book

In 1850 Ramchandra published in Calcutta, at his own expense, A Treatise of Problems on Maxima And Minima, Solved by Algebra. The book was sent to Augustus De Morgan who republished the book in London in 1859. De Morgan added an Editor's Preface which contained De Morgan's thoughts on Ramchandra's book, Indian mathematics, the Hindu in India, and other related topics. This Editor's Preface also contained autobiographical notes that Ramchundra had sent De Morgan. This is contained on a separate page.

For Ramchundra's autobiographical notes see THIS LINK.

For Ramchundra's Preface to his 1850 edition, see THIS LINK.

For The Calcutta Review of the 1850 book, see THIS LINK.

For The Calcutta Review of the 1859 book, see THIS LINK.

Editor's Preface

In the year 1850 my friend the late J E Drinkwater-Bethune forwarded to England a number of copies of a work on Maxima and Minima, by Ramchundra, teacher of science, Delhi College, with directions to present copies to various persons, and among others to myself. On examining this work I saw in it, not merely merit worthy of encouragement, but merit of a peculiar kind, the encouragement of which, as it appeared to me, was likely to promote native effort towards the restoration of the native mind in India. Mr Drinkwater-Bethune's lamented death, which took place shortly after he had dispatched the books, prevented my knowing whether he also entertained any opinion similar to mine as to the distinctive character of Ramchundra's work; but, from his own knowledge of the history of mathematics, I think it highly probable. I addressed my thanks for the present to his successor, Mr Colvile, with some remarks on the subject. Having taken further time to think of it, I determined to call the attention of the Court of Directors to Ramchundra's work, in the hope that it would lead to acknowledgment of his deserts. I accordingly addressed a letter (24 July 1856) to Colonel Sykes, the Chairman, to whom I had previously mentioned the matter at a casual meeting. This letter was at once forwarded to the Lieutenant-Governor of the North West Provinces, with instructions to procure a report on the case, and to suggest, on the supposition of a public reward being approved of, the kind of reward which should be given. Answers were received by the Court, which were communicated to me on the 3rd of March 1858. They contained various replies to the questions proposed, by H Stewart Reid, Esq., Director of Public Instruction in the North-West Provinces, and other gentlemen connected with the same department. With some difference of opinion as to the mode of acknowledgment, there was unanimous appreciation of Ramchundra's services to his country, and admission of the desirableness of encouraging his efforts. The Court accompanied the communication of these answers to me with a request that I would point out how to bring Ramchundra under the notice of scientific men in Europe. In my reply (18 March), assuming distinctly that I conceived the question to be, not merely how Ramchundra could be rewarded, but how his work might be made most effective in the development of Hindu talent, I recommended the circulation of the work in Europe, with a distinct account of the grounds on which the step was taken. I entered at some length into my own view of those grounds, and volunteered to draw up the statement which should accompany the publication. After some correspondence on details, the Court (1 July), expressing entire satisfaction with my views, and characterizing them as "deserving of the most attentive consideration by all who are charged with the superintendence of education in India in its higher grades," accepted my offer to superintend the present reprint for circulation in Europe and in India.

I shall at once proceed to a short account of these views after which I shall give some account of Ramchundra the author of the work. Of course it will be remembered that the late Court of Directors is in no way answerable for the details of my exposition though their decided approbation was bestowed on the general sketch which I laid before them.

There are many persons, even among those who seriously turn their thoughts to the improvement of India, who look upon the native races as men to be dealt with in the same manner as Caffres or New Zealanders. Judging by the lower races of the Peninsula, and judging even these more by the grosser parts of their mythology than by the state of domestic life and hereditary institutions, they presume that the Indian question resolves itself into an inquiry how to create a mind in the country, and that mind fashioned on the English standard. They forget that at this very moment there still exists among the higher castes of the country - castes which exercise vast influence over the rest - a body of literature and science which might well be the nucleus of a new civilization, though every trace of Christian and Mohamedan civilization were blotted out of existence. They forget that there exists in India, under circumstances which prove a very high antiquity, a philosophical language which is one of the wonders of the world, and which is a near collateral of the Greek, if not its parent form. From those who wrote in this language we derive our system of arithmetic, and the algebra which is the most powerful instrument of modern analysis. In this language we find a system of logic and of metaphysics: an astronomy worthy of comparison with that of Greece in its best days; above comparison, if some books of Ptolemy's Syntaxis be removed. We find also a geometry, of a kind which proves that the Hindu was below the Greek as a geometer, but not in that degree in which he was above the Greek as an arithmetician. Of the literature, poetry, drama, &c., which flourished in union with this science, I have not here to speak.

Those who consult Colebrooke's translation of the Vija Ganita, or the account given of it in the Penny Cyclopaedia, will see that I have not exaggerated the point most connected with this preface. For others I will quote the impression made, five and thirty years ago, upon the mind of a mathematician whose subsequent career and present position will give that weight to an extract from his opinions which would have been given to any reader of the whole article by the article itself, even had it been anonymous. Sir John Herschel, in the historical article Mathematics in Brewster's Cyclopaedia, after some general account of the Hindu algebra, proceeds as follows:-

"The Brahma Sidd'hanta, the work of Brahmagupta, an Indian astronomer at the beginning of the seventh century, contains a general method for the resolution of indeterminate problems of the second degree; an investigation which actually baffled the skill of every modern analyst till the time of Lagrange's solution, not excepting the all inventive Euler himself. This is matter of a deeper dye. The Greeks cannot for a moment be thought of as the authors of this capital discovery; and centuries of patient thought, and many successive efforts of invention, must have prepared the way to it in the country where it did originate. It marks the maturity and vigour of mathematical knowledge, while the very work of Brahmagupta, in which it is delivered, contains internal evidence that in his time geometry at least was on the decline. For example, he mentions several properties of quadrilaterals as general, which are only true of quadrilaterals inscribed in a circle. The discoverer of these properties (which are of considerable difficulty) could not have been ignorant of this limitation, which enters as an essential element in their demonstration. Brahmagupta then, in this instance, retailed, without fully comprehending, the knowledge of his predecessors. When the stationary character of Hindu intelligence is taken into the account, we shall see reason to conclude, that all we now possess of Indian science is but part of a system, perhaps of much greater extent, which existed at a very remote period, even antecedent to the earliest dawn of science among the Greeks, and might authorize as well the visits of sages as the curiosity of conquerors."

Greece and India stand out, in ancient times, as the countries of indigenous speculation. But the intellectual fate of the two nations was very different. Among the Greeks, the power of speculation remained active during their whole existence as a nation, even down to the taking of Constantinople: it declined, indeed, but it was never extinguished. Their latest knowledge was inquisitive, as well as their earliest. They preserved their great writers unabridged and unaltered: and Euclid did not degenerate into what are called practical rules.

In India, speculation died a natural death. A taste for routine - a thing to which inaccurate thinkers give the name of practical - converted their system into a collection of rules and results. Of this character are all the mathematical books which have been translated into English; perhaps all which still exist. That they must have had an extensive body of demonstrated truths is obvious; that they lost the power and the wish to demonstrate is certain. The Hindu became, to speak of the highest and best class, the teacher of results which he could not explain, the retailer of propositions on which he could not found thought. He had the remains of ancestors who had investigated for him, and he lived on such comprehension of his ancestors as his own small grasp of mind would allow him to obtain. He fed himself and his pupils upon the chaff of obsolete civilization, out of which Europeans had thrashed the grain for their own use.

But the mind thus degenerated is still a mind; and the means of restoring it to activity differ greatly from those by which a barbarous race is to be gifted with its first steps of progress. No man alive can, on sufficient data, reason out the restoration of a decayed national intellect, possessed of a system of letters and science which has left nothing but dry results, inveterate habits of routine, great reverence for old teachers, and small power of comprehending the very teaching which is held in traditional respect. And this because the question is now tried for the first time. Many friends of education have proposed that Hindus should be fully instructed in English ideas and methods, and made the media through which the mass of their countrymen might receive the results in their own languages. Some trial has been given to this plan but the results have not been very encouraging, in any of the higher branches of knowledge. My conviction is, that the Hindu mind must work out its own problem; and that all we can do is to set it to work; that is, to promote independent speculation on all subjects by previous encouragement and subsequent reward. This is the true plan; all others are neither fish nor flesh.

That sound judgment which gives men well to know what is best for them, as well as that faculty of invention which leads to development of resources and to the increase of wealth and comfort, are both materially advanced, perhaps cannot rapidly be advanced without, a great taste for pure speculation among the general mass of the people, down to the lowest of those who can read and write. England is a marked example. Many persons will be surprised at this assertion. They imagine that our country is the great instance of the refusal of all unpractical knowledge in favour of what is useful. I affirm, on the contrary, that there is no country in Europe in which there has been so wide a diffusion of speculation, theory, or what other unpractical word the reader pleases. In our country, the scientific society is always formed and maintained by the people; in every other, the scientific academy - most aptly named - has been the creation of the government, of which it has never ceased to be the nursling. In all the parts of England in which manufacturing pursuits have given the artisan some command of time, the cultivation of mathematics and other speculative studies has been, as is well known, a very frequent occupation. In no other country has the weaver at his loom bent over the Principia of Newton; in no other country has the man of weekly wages maintained his own scientific periodical. With us, since the beginning of the last century, scores upon scores - perhaps hundreds, for I am far from knowing all - of annuals have run, some their ten years, some their half-century some their century and a half, containing questions to be answered, from which many of our examiners in the universities have culled materials for the academical contests. And these questions have always been answered, and in eases without number by the lower order of purchasers, the mechanics, the weavers, and the printers' workmen. I cannot here digress to point out the manner in which the concentration of manufactures, and the general diffusion of education, have affected the state of things; I speak of the time during which the present system took its rise, and of the circumstances under which many of its most effective promoters were trained. In all this there is nothing which stands out, like the state-nourished academy, with its few great names and brilliant single achievements. This country has differed from all others in the wide diffusion of the disposition to speculate, which disposition has found its place among the ordinary habits of life, moderate in its action, healthy in its amount.

The history of England, as well as of other countries, having impressed me with a strong conviction that pure speculation is a powerful instrument in the progress of a nation, and my own birth and descent having always given me a lively interest in all that relates to India, I took up the work of Ramchundra with a mingled feeling of satisfaction and curiosity: a few minutes of perusal added much to both. I found in this dawn of the revival of Hindu speculation two points of character belonging peculiarly to the Greek mind as distinguished from the Hindu; one of which may have been fostered by the author's European teachers, but certainly not the other.

The first point is a leaning towards geometry. Persons who are not mathematicians imagine that all mathematicians are for all mathematics. Nothing can be more erroneous. Not merely have the two great branches, geometry and algebra, their schools of disciples, each of which looks coldly upon the other; but even geometry itself, and algebra itself, have subdivisions of which the same thing may be said. For example, Mr Drinkwater-Bethune, above mentioned, was by taste an algebraist; as a practised eye would at once detect from his unfinished work on equations. Business brought him to my house one morning, nearly thirty years ago, at a time when I happened to be studying some of the geometrical developments of the school of Monge. On my pointing out to him some of the most remarkable of the conclusions, he said with a smile, "I see that sort of thing has charms for you." Now the Hindu was also an algebraist, as decidedly as the Greek was a geometer: the first sought refuge from geometry in algebra, the second sought refuge from arithmetic in geometry. The greatness of Hindu invention is in algebra; the greatness of Greek invention is in geometry. But Ramchundra has a much stronger leaning towards geometry than could have been expected by a person acquainted with the Vija Ganita; but he has not the power in geometry which he has in algebra. I have left one or two failures - one very remarkable - unnoticed, for the reader to find out. Should this preface - as I hope it will - fall into the hands of some young Hindus who are systematic students of mathematics, I beg of them to consider well my assertion that their weak point must be strengthened by the cultivation of pure geometry. Euclid must be to them what Bhaskara, or some other algebraist has been to Europe.

The second point is yet more remarkable. Greek geometry, as all who have read Euclid may guess, gained its strength by striving against self imposed difficulties. It was not permitted to take instruments from every conception which the human mind could form; definite limitation of means was imposed as a condition of thought, and it was sternly required that every feat of progress should be achieved by those means, and no more. Just as the Greek architecture studied the production of rich and varied effect out of the simplest elements of form, so the Greek geometry aimed at the demonstration of all the relations of figure on the smallest amount of postulated basis. The great problem of squaring the circle, now with good reason held in low esteem, was the struggle of centuries to bring under the dominion of the prescribed means what might with the utmost ease have been conquered by a very small additional allowance. The attempt was unsuccessful; so was that of Columbus to discover India from the west. But Columbus commenced the addition of America to the known world; and in like manner the squarers of the circle, and their refuters, added field on field to the extent of geometry, and aided largely in the preparation for the modern form of mathematics. Very few of these additions would have been made, at or near the time when they were made, if it had satisfied the Greek mind to meet each difficulty, as it occurred, by permission to use additional assumptions in geometry.

The remains of the Hindu algebra and geometry show to us no vestige of any attempt to gain force of thought by struggling against limitation of means: this, of course, because their mode of demonstration does not appear in the works which are left, or at least in those which have become known to Englishmen. But we have here a native of India who turns aside, at no suggestion but that of his own mind, and applies himself to a problem which has hitherto been assigned to the differential calculus, under the condition that none but purely algebraical process shall be used. He did not learn this course of proceeding from his European guides, whose aim it has long been to push their readers into the differential calculus with injurious speed, that they may reach the full application of mathematics to physics; and who often allow their pupils to read Euclid with eyes shut to his limitations. Ramchundra proposed to himself a problem which a beginner in the differential calculus masters with a few strokes of the pen in a month's study, but which might have been thought hardly within the possibilities of pure algebra. His victory over the theory of the difficulty is complete. Many mathematicians of sufficient power to have done as much would have told him, when he first began, that the end proposed was perhaps unattainable by any amount of thought; next, that when attained, it would be of no use. But he found in the demands of his own spirit an impulse towards speculation of a character more fitted to the state of his own community than the imported science of his teachers. He applied to the branch of mathematics which is indigenous in India, the mode of thought under which science made its greatest advances in Greece. My own strong suspicion that it was the want of this mode of thought which allowed the decline of algebra in ancient India, coupled with my thorough conviction that, whether or no, this mode of thought yields the proper nutriment for mathematical science in its early and feeble life, produced the recommendation to the Court of Directors to which this reprint owes its existence.

Ramchundra's problem - and I think it ought to go by that name, for I cannot find that it was ever current as an exercise of ingenuity in Europe - is to find the value of a variable which will make an algebraical function a maximum or a minimum, under the following conditions. Not only is the differential calculus to be excluded, but even that germ of it which, as given by Fermat in his treatment of this very problem, made some think that he was entitled to claim the invention. The values of Δx\Delta x and of Δ(x+h)\Delta (x + h) are not to be compared; and no process is to be allowed which immediately points out the relation of Δx\Delta x to the derived function Δx\Delta 'x. A mathematician to whom I stated the conditioned problem made it, very naturally, his first remark, that he could not see how on earth I was to find out when it would be biggest, if I would not let it grow. The mathematician will at last see that the question resolves itself into the following:- Required a constant, rr, such that Δxr\Delta x - r shall have a pair of equal roots, without assuming the development of Δ(x+h)\Delta (x + h) or any of its consequences.

It will readily be seen that a short paper, with a few examples, would have sufficed to put the whole matter before a scientific society. But it was Ramchundra's object to found an elementary work upon his theorem, for the use of beginners, with a large store of examples. As to the method which he has adopted, Europeans must remember that his purpose is to teach Hindus, and that probably he knows better how to do this than they could tell him. The excessive reiteration of details, and the extreme minuteness of the algebraical manipulations, are excellent examples of that patience of routine which is held to be a part of the Hindu character. I may make two remarks on matters which would strike the most casual observer.

First, the constant occurrence of "the same solved without impossible roots," and the transformation by which it is effected, will remind the English mathematician who has his half-century over his head, of the old "pure quadratic" and the victory which was supposed to be gained when the "adfected quadratic" was evaded by attention to the structure of the given equation. Ramchundra and Dr Miles Bland, &c. &c., are here precisely on the same scent, both making much of the same little.

Secondly in the confusion of terms which sometimes appears, in language implying that an equation is a factor of an equation, instead of an expression a factor of an expression, we have the same incorrectness which appears in more than one edition of Waring's Meditationes Algebraicae and which occasioned some amount of objection to the whole theory from those who could not see the inaccuracy and its correction. As in
"Scribatur xα=0,xβ=0,xγ=0,xδ=0x - \alpha = 0, x - \beta = 0, x - \gamma = 0, x - \delta = 0, etc. et per aequationem ex horum factorum continua multiplicatione

(xα)×(xβ)×(xγ)×(xδ)×etc.=0(x - \alpha) \times (x - \beta) \times (x - \gamma) \times (x - \delta) \times etc. = 0
generatam dividatur data sequatio."

I believe that selections from Ramchundra's work might advantageously be introduced into elementary instruction in this country. The exercise in quadratic equations which it would afford, applied as it is to real problems, would advantageously supersede some of the conundrums which are manufactured under the name of problems producing equations.

In the printing I have followed the original in every point, altering nothing except obvious errata, including the restoration of the numeral symbol 0 which in the original is always the letter o. This again is a mistake into which Waring allowed his printer to fall in almost all his writings. I thought that the European reader would be more curious to look at the way in which the Calcutta printer treated mathematical manuscript when his author was no nearer than Delhi, than to see the manner in which I could mend it. My printers, Messrs Cox and Wyman, have entered fully into the plan, and have produced as nearly a facsimile as possible. I may add that the Calcutta printer has acquitted himself in a manner entitled to especial notice and high praise.

University College, London
17 January, 1859.

Last Updated April 2016