MODE OF COMPUTING INTEREST

The mode of computing interest among the ancient Greeks appears to have been in many respects the same as that now prevailing in India, which has probably undergone no change from a very remote period. Precisely the same term, too, is used to denote the rate of interest, namely, τοκος in Greek and taka or tuka in the languages of Western India. Τοκοε επιδεκατοι in Greek, and dus také in Hindostanee, respectively denote ten per cent. At Athens, the rate of interest might be calculated either by the month or by the year—each being expressed by different terms (Böckh. Pub. Econ. of Athens, i. 165.). Precisely the same system prevails here. Pono taka, that is, three quarters of a taka, denotes ¾ per cent. per month. Nau také, that is, nine také, denotes nine per cent. per annum. For the Greek mode of reckoning interest by the month, see Smith's Dictionary of Greek and Roman Antiquities, p. 524. At Athens, the year, in calculating interest, was reckoned at 360 days (Böckh, i. 183.). Here also, in all native accounts-current, the year is reckoned at 360 days.

The word τοκος, as applied to interest, was understood by the Greeks themselves to be derived from τικτω, "to produce," i.e. money begetting money; the offspring or produce of money lent out. Whether its identity may not be established with the word in current use for thousands of years in this country to express precisely the same meaning, is a question I should like to see discussed by some of your correspondents. The word taka signifies any thing pressed or stamped, anything on which an impression is made hence a coin; and is derived from the Sanscrit root , tak, to press, to stamp, to coin: whence, , tank, a small coin; and tank-sala, a mint; and (query) the English word token, a piece of stamped metal given to communicants. Many of your readers will remember that it used to be a common practice in England for copper coins, representing a half-penny, penny, &c., stamped with the name of the issuer, and denominated "tokens," to be issued in large quantities by shopkeepers as a subsidiary currency, and received at their shop in payment of goods, &c. May not ticket, defined by Johnson, "a token of any right or debt upon the delivery of which admission is granted, or a claim acknowledged," and tick, score or trust, (to go on tick), proceed from the same root?

J.S.
Bombay.

ON THE CULTIVATION OF GEOMETRY IN LANCASHIRE

If our Queries on this subject be productive of no other result than that of eliciting the able and judicious analysis subsequently given by MR. WILKINSON (Vol. ii., p. 57.), they will have been of no ordinary utility. The silent early progress of any strong, moral, social, or intellectual phenomenon amongst a large mass of people, is always difficult to trace: for it is not thought worthy of record at the time, and before it becomes so distinctly marked as to attract attention, even tradition has for the most part died away. It then becomes a work of great difficulty, from the few scattered indications in print (the books themselves being often so rare¹ that "money will not purchase them"), with perhaps here and there a stray letter, or a metamorphosed tradition, to offer even a probable account of the circumstances. It requires not only an intimate knowledge of the subject-matter which forms the groundwork of the inquiry, both in its antecedent and cotemporary states, and likewise in its most improved state at the present time; it also requires an analytical mind of no ordinary powers, to separate the necessary from the probable; and these again from the irrelevant and merely collateral.

MR. WILKINSON has shown himself to possess so many of the qualities essential to the historian of mathematical science, that we trust he will continue his valuable researches in this direction still further.

It cannot be doubted that MR. WILKINSON has traced with singular acumen the manner in which the spirit of geometrical research was diffused amongst the operative classes, and the class immediately above them—the exciseman and the country schoolmaster. Still it is not to be inferred, that even these classes did not contain a considerable number of able geometers anterior to the period embraced in his discussion. The Mathematical Society of Spitalfields existed more than half a century before the Oldham Society was formed. The sameness of pursuit, combined with the sameness of employment, would rather lead us to infer that geometry was transplanted from Spitalfields to Manchester or Oldham. Simpson found his way from the country to London; and some other Simpson as great as Thomas (though less favourably looked upon by fortune in furnishing stimulus and opportunity) might have migrated from London to Oldham. Or, again, some Lancashire weaver might have adventured to London (a very common case with country artisans after the expiration of apprenticeship); and, there having acquired a taste for mathematics, as well as improvement in his mechanical skill, have returned into the country, and diffused the knowledge and the tastes he took home with him amongst his fellows. The very name betokens Jeremiah Ainsworth to have been of a Lancashire family.

But was Ainsworth really the earliest mathematician of his district? Or, was he merely the first that made any figure in print as a correspondent of the mathematical periodicals of that day? This question is worthy of MR. WILKINSON's further inquiry; and probably some light may be thrown upon it by a careful examination of the original Ladies' and Gentleman's Diaries of the period. In the reprints of these works, only the names, real or assumed, of those whose contributions were actually printed, are inserted—not the list of all correspondents.

Now one would be led to suppose that the study of mathematics was peculiarly suited to the daily mode of life and occupation of these men. Their employment was monotonous; their life sedentary; and their minds were left perfectly free from any contemplative purpose they might choose. Algebraic investigation required writing: but the weaver's hands being engaged he could not write. A diagram, on the contrary, might lie before him, and be carefully studied, whilst his hands and feet may be performing their functions with an accuracy almost instinctive. Nay more: an exceedingly complicated diagram which has grown up gradually as the result of investigations successively made, may be carried in the memory and become the subject of successful peripatetic contemplation. On this point a decided experimental opinion is here expressed: but were further instances asked for, they may be found in Stewart, Monge, and Chasles, all of whom possessed this power in an eminent degree. Indeed, without it, all attempts to study the geometry of space (even the very elements of descriptive geometry, to say nothing of the more recondite investigations of the science) would be entirely unproductive. It is, moreover, a power capable of being acquired by men of average intellect without extreme difficulty; and that even to the extent of "mentally seeing" the constituent parts of figures which have never been exhibited to the eye either by drawings or models.

That such men, if once imbued with a love for geometry, and having once got over the drudgery of elementary acquisition, should be favourably situated for its cultivation, follows as a matter of course. The great difficulty lay in finding sufficient stimulus for their ambition, good models for their imitation, and adequate facilities for publishing the results at which they had arrived. The admirable history of the contents of their scanty libraries, given by MR. WILKINSON, leaves nothing more to be said on that head; except, perhaps, that he attributes rather more to the influences of Emerson's writings than I am able to do.² As regards their facilities for publication, these were few, the periods of publication being rarely shorter than annual; and amongst so many competitors, the space which could be allotted to each (even to "the best men") was extremely limited. Yet, contracted as the means of publication were, the spirit of emulation did something; from the belief that insertion was an admitted test of superiority, it was as much an object of ambition amongst these men to solve the "prize question" as it was by philosophers of higher social standing to gain the "prize" conferred by the Académie des Sciences, or any other continental society under the wing of Royalty, at the same period. The prize (half a dozen or a dozen copies of the work itself) was not less an object of triumph, than a Copley or a Royal medal is in our own time amongst the philosophers of the Royal Society.

These men, from similarity of employment and inevitable contiguity of position, were brought into intercourse almost of necessity, and the formation of a little society (such as the "Oldham") the natural result—the older and more experienced men taking the lead in it. At the same time, there can be little doubt that the Spitalfields Society was the pattern after which it was formed; and there can be as little doubt that one or more of its founders had resided in London, and "wrought" in the metropolitan workshops. Could the records of the "Mathematical Society of London" (now in the archives of the Royal Astronomical Society) be carefully examined, some light might be thrown upon this question. A list of members attending every weekly meeting, as well as of visitors, was always kept; and these lists (I have been informed) have been carefully preserved. No doubt any one interested in the question would, upon application to the secretary (Professor De Morgan), obtain ready access to these documents.

The preceding remarks will, in some degree, furnish the elements of an answer to the inquiry, "Why did geometrical speculation take so much deeper root amongst the Lancashire weavers, than amongst any other classes of artisans?" The subject was better adapted to the weaver's mechanical life than any other that could be named; for even the other favourite subjects, botany and entomology, required the suspension of their proper employment at the loom. The formation of the Oldham Society was calculated to keep alive the aspiration for distinction, as well as to introduce novices into the arcanium of geometry. There was generous co-operation, and there was keen competition,—the sure stimulants to eminent success. The unadulterated love of any intellectual pursuit, apart from the love of fame or the hope of emolument, is a rare quality in all stages of society. Few men, however, seem to have realised Basil Montagu's idea of being governed by "a love of excellence rather than the pride of excelling," so closely as the Lancashire geometers of that period—uncultivated as was the age in which they lived, rude as was the society in which their lives were passed, and selfish as the brutal treatment received in those days by mechanics from their employers, was calculated to render them. They were surrounded, enveloped, by the worst social and moral influences; yet, so far as can now be gathered from isolated remarks in the periodicals of the time, they may be held up as a pattern worthy of the imitation of the philosophers of our own time in respect to the generosity and strict honour which marked their intercourse with one another.

Mathematicians seldom grow up solitarily in any locality. When one arises, the absence of all external and social incentives to the study can only betoken an inherent propensity and constitutional fitness for it. Such a man is too much in earnest to keep his knowledge to himself, or to wish to stand alone. He makes disciples,—he aids, encourages, guides them. His own researches are fully communicated; and this with a prodigality proportioned to his own great resources. He feels no jealousy of competition, and is always gratified by seeing others successful. Thus such bodies of men are created in wonderfully short periods by the magnanimous labours of one ardent spirit. These are the men that found societies, schools, sects; wherever one unselfish and earnest man settles down, there we invariably find a cluster of students of his subject, that often lasts for ages. Take, for instance, Leeds. There we see that John Ryley created, at a later period, the Yorkshire school of geometers; comprising amongst its members such men as Swale, Whitley, Ryley ("Sam"), Gawthorp, Settle, and John Baines. This, too, was in a district in many respects very analogous to Lancashire, but especially in the one to which the argument more immediately relates:—it was a district of weavers, only substituting wool for cotton, as cotton had in the other case been substituted for the silk of Spitalfields.

We see nothing like this in the agricultural districts; neither do we in those districts where the ordinary manufacturing operations themselves require the employment of the head as well as the hands and feet. With the exception, indeed, of the schoolmaster, and the exciseman, and the surveyor, there are comparatively few instances of persons whose employment was not strictly sedentary having devoted their intellectual energies to mathematics, independent of early cultivation. To them the subject was more or less professional, and their devotion to it was to be expected—indeed far more than has been realised. It is professional now to a larger and more varied class of men, and of course there is a stronger body of non-academic mathematicians now than at any former period. At the same time it may be doubted whether there be even as many really able men devoted to science purely and for its own sake in this country as there were a century ago, when science wore a more humble guise.

Combining what is here said with the masterly analysis which MR. WILKINSON has given of the books which were accessible to these men, it appears that we shall be able to form a correct view on the subject of the Lancashire geometers. Of course documentary evidence would be desirable—it would certainly be interesting too.

To such of your readers as have not seen the mathematical periodicals of that period, the materials for which were furnished by these men, it may be sufficient to state that the "NOTES AND QUERIES" is conceived in the exact spirit of those works. The chief difference, besides the usual subject-matter, consists in the greater formality and "stiffness" of those than of this; arising, however, of necessity out of the specific and rigid character of mathematical research in itself, and the more limited range of subjects that were open to discussion.

The one great defect of the researches of those men was, that they were conducted in a manner so desultory, and that the subjects themselves were often so isolated, that there can seldom be made out more than a few dislocated fragments of any one subject of inquiry whatever. Special inquiries are prosecuted with great vigour and acumen; but we look in vain for system, classification, or general principles. This, however, is not to be charged to them as a scientific vice, peculiarly:—for, in truth, it must be confessed to be a vice, not only too common, but almost universal amongst English geometers; and even in the geometry of the Greeks themselves, the great object appears to have been "problem-solving" rather than the deduction and arrangement of scientific truths. The modern French geometers have, however, broken this spell; and it is not too much too hope that we shall not be long ere we join them in the development of the systems they have already opened; and, moreover, add to the list some independent topics of our own. The chief dangers to which we are in this case exposed are, classification with incomplete data, and drawing inferences upon trust. It cannot be denied, at all events, that some of our French cotemporaries have fallen into both these errors; but the abuse of a principle is no argument for our not using it, though its existence (or even possible existence) should be a strong incentive to caution.

These remarks have taken a more general form than it is usual to give in your pages. As, however, it is probable that many of your readers may feel an interest in a general statement of a very curious intellectual phenomenon, I am not without a hope that, though so far removed from the usual topics discussed in the work, they will not be altogether unacceptable or useless.