Art. VII.—On certain Decimal and Metrical Fallacies.

Transactions and Proceedings of the Royal Society of New Zealand, Volume 36, 1903, Page 85

Art. VII.—On certain Decimal and Metrical Fallacies. By R. Coupland Hardinq. [Read before the Wellington Philosophical Society, 5th August and 7th October, 1903.] If there be one quality more than another that distinguishes the scientific from the unscientific mind is might well be defined as the ability, or, perhaps more correctly, the habit, of discriminating clearly between the symbol and the thing signified. Symbols, indispensable though they be, are always more or less defective representives of truth, and are full of dancer, when, instead of servants, they become masters and dominate the thoughts of men. They give rise to a multitudinous brood of fallacies, and fallacies as they gradually develop ripen into positive evil. In theology the exaltation of the symbol becomes idolatry, with intellectual, darkness, moral corruption, and spiritual death: in science it may be found elevating the latest convenient working-hypothesis into an established law of nature, when that which was at one time helpful becomes injurious. Truths appeal to the higher intellectual faculties, and can be appreciated only by application and study; symbols may be memorised by rote and acquired with slight trouble—they are nothing more than counters, though only too easily mistaken for genuine currency. Truths grow by accretion; symbols remain unchanged, and when outgrown they are a cheek on progress. Of this fact schemes of notation afford sufficient proof. Our English tongue, the world-language of the future, is cruelly hampered in its conquering course by a defective notation—not only inadequate, but the type of all that is etymologically and phonetically misleading. A thousand years ago our Saxon ancestors had something like a scientific and consistent orthography—a living system, adapted to its purpose. To-day we find a living language imprisoned in a dead alphabet. The symbol is outgrown. Musical notation affords another, though less extreme, instance. A very slight change would convert a system at present perplexing and inconsistent into one consistent and helpful; but again progress is fettered by the outworn symbol. Our arithmetical notation, happily, is free from the absurdities attaching to musical and orthographic notation. If, for example, the symbol 5 might under certain arbitrary rules signify the same as 32, 41, or 55, if it was sometimes to be read as 7, and occasionally, being “silent,” had to be ignored altogether—if the interpolation of another symbol having sometimes a negative and sometimes a positive value made it equal to 6—then our

arithmetical notation would be fairly comparable with our orthography, and computation would be as difficult as English spelling. But it is quite easy to be in bondage to arithmetical notation. Thousands of folk skilled in computation, using figures daily, confound them with the things they represent, never realising that after all they are merely counters — symbols, perfect and efficient within their own very limited range, but representing in reality only a convention, and no solid fact. The mathematician knows their place and their limitations; the mere computer, finding them ready to his hand, imagines them to be rooted in the very nature of things. He knows, for example, some of the curious properties of the number 9, but would be unable to discriminate between the inherent properties of that number—those, for example, which are peculiar to the square of 3, and which no notational system could affect—and those other qualities which are accidental and notational, and would be transferred to another number were any other radix substituted for that of ten. This is a point of basic importance in the consideration of the subject of any proposed change in measures and weights, and especially when a change of standard is involved. The subject is of practical importance now, when the foreign system, already permissive, has advanced another stage in our Parliament, and will, unless resistance is offered, displace in a few years—legally if not actually—our Imperial standards. There has been no popular demand for the change; there is absolutely no popular discontent with our present standards: but an active minority has carried its point so far, as agitators can—without public sympathy or approval, but in the face of that massive apathy and general indifference to any change that does not threaten some immediate loss or disability. And it is noteworthy that the arguments urged in support of the change appeal to popular notational fallacies rather than to mathematical facts or scientific truths. Of late years we have seen, at annual meetings of Institutes of Accountants, Chambers of Commerce, and similar bodies, formal resolutions passed on the subject as casually and perfunctorily as the vote of thanks to the chairman. In our own Parliament during the current session a Bill, further-reaching in its effects (should it ever become effective) and more revolutionary in its scope than any legislation ever before proposed in this colony, passed its preliminary stages with less notice than is sometimes given to a fifty-pound item on the estimates. It does not seem unreasonable to infer that those who deal thus lightly with grave matters have neither studied them nor realised their importance. They know that our system of money-computation is defective and causes unneces

sary waste of time, and when they are told that “the decimal system” will set everything right they do not investigate for themselves. The onus of proof lies with the aggressive minority, and that proof is not forthcoming. They offer instead of proof many assertions which will not bear examination. The initial fallacy is to call the French scheme “the” decimal system. It is a decimal scheme, as are also other and better schemes. Its one distinguishing feature is its unit—the meter. It is therefore not “the decimal system,” but the “metric” system. The misnomer gives rise to the inference—and the notion is widely held—that decimalisation involves a change of standards, which is not the case. There can be no warrant for so radical a change except it be plainly shown— (1) That the standard proposed is scientifically or practically better intrinsically or extrinsically than the existing one; (2) That the advantage is so great as to outweigh the disturbance and inconvenience occasioned by the change. I do not think that these propositions can be seriously disputed. If, then, it can be proved— (1) That the meter is arbitrary, possessing no scientific value nor any ascertainable relation to anything in creation; (2) That the national standards possess these qualities in a high degree; (3) That for practical purposes the existing standards are essentially better and more convenient— what excuse is there for the change? And there is overwhelming proof in support of these propositions. I have spoken of the resolutions of certain bodies in favour of the change as perfunctory. It is significant that almost immediately after the last formal resolution of the Wellington Chamber of Commerce the chairman wrote to the press pointing out that the change, if made, would have serious and unlooked-for consequences. Yet these consequences must surely suggest themselves to any one who gives the matter any consideration. As for the decimalisation of the national coinage, it has long been a desideratum. A suggestion was lately made to divide the pound sterling into four hundred parts. It was put forward as “decimalisation of the coinage,” which it is not. The subject has been fully dealt with by a commission of leading mathematicians in Britain, and a complete and consistent scheme was long ago formulated. The figure representing the number of pounds would be followed by three figures separated by a space or decimal point, and

representing respectively “florins,” “cents,” and “mils.” The first step was actually taken in 1849, when the florin was coined, and the complete reform would undoubtedly have been carried out and extended to weights and measures also as far as practicable but for the opposition of the “metric” party, who have regularly blocked any systematic reform of the national measures. They have realised throughout that with a rational coinage system the crusade against the national standards would be almost hopeless. They invariably lump together the present coinage system, which is indefensible, with the standards of weight and measure, and draw a supposed parallel. But the imaginary parallel is a fallacy. Weights and measures deal with entities and qualities outside our own control. Whether we weigh a load of ballast or compute the distance of a star we are engaged in investigating realities—relations and interrelations between ourselves and the universe without—and must adapt our methods, as conveniently as we may, to things as we find them. Coinage, on the other hand, is entirely artificial. From first to last the form it takes is under our own control; it should be adapted to the radix of computation. Our coinage is not so adapted, and to that extent is irrational. Further, it conforms only very imperfectly to weight and measure standards. We need not go further than to America for a practical example. In the United States and Canada the coinage has been decimalised, the weights and measures remaining unaltered. We come now to perhaps the greatest and most audacious of the fallacies propounded by the advocates of the meter. We find it seated that it would save so much of the time at present given to the study of arithmetic as would amount to a complete revolution. Last year it was asserted by an Australian writer that “compound calculations would be no longer necessary, and need not be taught in schools.” Children, he said, were “kept at school learning arithmetic from one to two years unnecessarily because of archaic and antiquated rules and clumsy and involved methods.” Every mathematician, every qualified teacher, knows that this is not the truth. Every one engaged in any kind of calculation has necessarily to deal with varying ratios; one of the chief purposes of the study of arithmetic is to qualify us to equate them, and it is to assist us in this work that artificially fixed standards are required. Fallacious as the assertion is, it is the stock plea of the metrists. It is the argument of the spelling-reformers borrowed and misapplied. Years of school life are wasted in learning by rote archaic and outworn forms of spelling—forms which misrepresent and caricature our speech; but what chance would a Bill for reforming ortho-

graphy have in Parliament? What New Zealand Chamber of Commerce has ever urged the necessity of rational spelling? Yet this colony has as much right to lead in the one direction as in the other, and the gain orthographic reform would bring would be incalculable. In this respect, however, prejudice is so strong that the Education Department insists on retaining corrupt forms that have largely died out in common use. It clings tenaciously to the intrusive “u” in “honour,” “labour,” and even in “neighbour.” It was not without surprise that I saw in the Weights and Measures Bill of 1903 even such a concession to systematic spelling as “meter.” By the way, it seems curious that this little colony should strain the “silken thread” by endeavouring to initiate a change that should begin at the centre of the nation's commerce. It can scarcely expect to force the hand of the Empire in a matter like this. In fact, should our Parliament attempt to give effect to the proposal it would have enough to do for many years to come in forcing its acceptance upon the people of the colony. We have been persistently told that the metric system is scientific. It professes to be, but the claim is based on fallacy. Not only is it found on examination to break down in this respect at nearly every point, but on purely scientific grounds as on practical it compares unfavourably with the national standards. It was itself the outcome of a period of social and scientific delirium, and its history to the present day is a record of practical inadequacy and of bureaucratic coercion. “Up to the last decade of the eighteenth century European weights and measures, though exhibiting appreciable variation in standard and still encumbered with obsolescent tables applying only to specific industries or localities, were practically uniform in principle. The standard of measure was the foot, duodecimally divided. In England the inch was also divided into twelve, and the twelfth-inch, known as the “line,” was used chiefly in scientific measurements. It is the unit known to printers as “nonpareil,” or in present nomenclature “six-point.” The third-inch, equal to four lines, was the lowest unit popularly recognised, and was known as the “barleycorn,” as it was supposed to be fairly represented by the length of a grain of barley from the centre of the ear. Old table-books still in use in my childhood began, not as might reasonably have been expected, “Twelve lines equal one inch,” but “Three barleycorns, one inch.” Some idle jocosity has been indulged in on the assumption that an actual barleycorn was the ultimate basis of Saxon measurement, but this, is a fallacy. The popular name of every measure, without exception, refers the standard to some supposed natural equivalent. In weight, for instance, we have the “grain.” No

people were more strict in the matter of standards than the Saxons. They had a practical and scientific system; their monarchs made it a matter of duty to see that no corruptions or deviations took place, and the British measures have been so closely guarded that after a thousand years the divergence is so slight as only to be detected by very accurate measurements. In France, as in England, the foot was divided into twelve inches, and the inch into twelve lines, and the line again into six, and this last subdivision, of 1/72-inch, was called the “point.” Whether the point was also recognised in Britain I do not know; it is now, and as it is the basis of all printers' measurements it is usually known as the “typographical point.” As it is not minute enough for all purposes, the half-point difference is recognised in the smaller types, so that in systematic type-measurement, which is exactly the same in principle (though not in standard) all the world over, the old national measures are retained, in which the duodecimal subdivision is consistently followed throughout:— 12 half-points = 1 line. 12 lines = 1 inch. 12 inches = 1 foot. The French standard was larger than the English in the proportion approximately of 555 to 517, and the Continental and British type-standards differ respectively to-day in the same proportion.* Theoretically. The systematization of type was seriously taken in hand in the United States Some years ago, and is now in general use. At first the national standard was taken, but the vested interests of large houses working on an inaccurate system prevailed, leading to a departure from the true standard, which, though infinitesimal and ignored in all ordinary reckoning, is greatly to be regretted. The precise divergence between the American type-standard and the British Imperial standard is 0.005 inch in the foot, the typographical inch as at present denned therefore equalling 0.999583 of the standard inch. A like agreement in principle with diversity of standard prevailed in regard to weights for ordinary purposes, the pound being divided by the simplest series of all—2, 4, 8, 16. The larger measures and weights were varying multiples, more or less systematic, some local, others adapted to special purposes only, and many of limited application, obsolete, or nearly so. A uniform principle underlying so much divergence in detail might have suggested a reason, and possibly a good reason, for its retention. Scientific reformers would have inquired if such reason existed; for with uniformity of principle already to their hand nothing more was required to establish an international system than the harmonizing and unification of the standards of measure and weight, which could have been accomplished with a minimum disturbance of

local prejudices, for the changes as they applied to any given locality would scarcely have been noticed, while the advantages would speedily have made themselves manifest. But no such inquiry was made. The accidental discrepancies were magnified, the underlying principles ignored, just as they are by the metrists of to-day, and a bran-new scheme must be devised-one which, as it was to supersede all others and last for all time, must be nothing less than perfect. In that strange period of unrest the judicial faculty seemed to be completely suspended; the scientific spirit, which breathes only in an atmosphere of humility, was dead. Truth-seekers there were none, for there remained no truth, to seek. Carlyle, in his trenchant style, has pictured the utter intellectual barrenness of the time, and the absence of the creative faculty—the scientists who made no discovery, the ingenious men who brought forth no invention. In every department, theoretical or practical—in science, philosophy, polities, or morals—empty symbols took the place of realities, fallacies of facts. Miss Clerke, in a late article in Knowledge, has eloquently described the self-sufficient “science” of the time:— There were no more worlds to conquer…. Nature for the moment submitted readily to the trammels put upon her by human thought; her intricacies no longer seemed to defy unravelment; her modes of procedure looked straightforward and intelligible…. It was an epoch of peremptory renewals. The formula of equality promised to regenerate society; a political panacea had been found by the creation of a republic “one and indivisible,” and the success of the guillotine in securing its supremacy was almost outdone by the triumphs of the calculus in vindicating the unimpeded sway of gravitation. In this spirit—the antithesis of the scientific spirit—the task was undertaken. The result was a comedy of blunders to which the history of science can scarcely furnish a parallel. The unit of measurement was necessarily the foundation of the entire system. The reformers had to their hand the ancient foot of France with its authoritative standards; they had access to the corresponding measures of Europe, from which, had they chosen, they might have deduced an average. But their unit must be new. Destined to be universal, it must be earth-commensurable. It must at the same time be unmistakably and indisputably French. So at great expense and with enormous labour they measured an arc of the meridian passing through France, divided the quadrant of the meridian thus deduced into ten million parts—that is, a forty—millionth of the entire circle—and this unit is the mètre. Note, first, the initial blunder—denounced by Herschel as “a scientific sin”—the choice of a curve as the basis of rectilinear measurement. The fact that the curve was on so large a scale that its true form was inappreciable to the senses does

not affect the question. To treat it as a right line was inexcusable on the part of men professing to institute a scientific reform in a scientific way. They based their whole fabric on a fundamental fallacy, consciously and deliberately, without even the excuse of ignorance. Note, too, that an arc of the meridian—unless it be one like that of Greenwich Observatory, selected by common consent as a starting-point—has no significance, and even in such an exceptional case its significance is only arbitrary and factitious. The meridians of a sphere are infinite in number, and unless the sphere is geometrically perfect are not uniform. The degrees of the terrestrial meridians vary in length from 66 91 to 69.40 miles. Note, too, that the standard, if lost, could never be restored by a repetition of the original process. Variations in the result inevitably appear on remeasurement, however exact, and this particular measurement was exceptionally difficult. As it was, the result was soon found to be appreciably wrong. The error is now stated to be 4,008 ft. So that the new universal earth-commensurable and French unit proved, after all the trouble and expense lavished upon it, to be arbitrary—to possess no more intrinsic significance than the length of a random straw picked up in the harvest-field. It may be said—it has been said, since the insignificance of the standard cannot be disputed—“No matter, so long as the standard is there and is recognised.”* An old article in the Athenæum, referring to the erroneous measurements on which the standard was based, said, “An error of 1/208 inch in the determination of the mètre is more than counterbalanced by the extreme simplicity, symmetry, and convenience of the metric system. Professor Bessel observed in respect to the mètre that, in the measurement of a length between two points on the surface of the earth, there is no advantage at all in proving the relation of the measured distance to a quadrant of the meridian. Professor Miller, of Cambridge, who quotes this remark, deems the error in the relation, of the mètre to the quadrant of the meridian to be of no consequence; and he mentions another slight error in the metric system, discovered by recent research, and relating to the density of water, which he gives in the following words of Bessel. The kilogramme is not exactly the weight of a cubic decimètre of water. Many of the late weighings show that water at its maximum density has different density from that assumed by the French philosophers who prepared the original standard of the kilogramme; but nobody wishes to alter the standard of the gramme on that account.” So that while the defects, real or imagined, of the British system are proclaimed and paraded by metrists, the acknowledged errors in the French scheme, fundamental or co-ordinate, are “of no consequence.” But it is evident that a really scientific standard must have significance. The reformers must have realised this, else why all this costly and elaborate preliminary parade of earth - measurement? Why inconvenience a whole nation for a hundred years when the ancient toise, divided and multiplied by ten, would have

answered every purpose of the mètre, and have saved, in the full literal sense of the words, a world of trouble? In a pamphlet published seventeen years ago Mr. Christopher Giles, of Adelaide, thus summed up the “scientific” foundation of the whole system:— Based on a curve—therefore unscientific in theory. Arbitrary—therefore wanting in scientific significance. Non—earth commensurable. Devoid of universality, in origin, genius, and fact. It does not seem to be as generally known as it should be that our own much-criticized scheme of lineal measurement—the origin of which is too remote to be traced—is scientifically sound on all the points where its modern rival is so conspicuously defective. Some may imagine that the yoke imposed on France was a light one. On the contrary, it was not the least oppressive of the many terrible sequelæ of the Revolution, and it was imposed only against stout resistance and with much utterly unnecessary agony. The system is permissive in the British dominions, and so long as it is only permissive and the national standards are in free and general use no very active opposition is likely to be offered. In India, though an Act was passed in 1870 substituting the metric for the national standard, and the law still stands unrepealed, it is a dead letter. In the United States a strong organization has been formed “for preserving and protecting our Anglo-Saxon weights and measures.” It keeps the subject before the public by means of literature and lectures, and sees that the metrist minority are met with counter-agitation. Some such institution appears to be needed in New Zealand. In Great Britain, where Commission after Commission has investigated the subject, expert and scientific testimony has been strongly in opposition to the “reform.” From astronomers and mathematicians of the highest repute it has met with unqualified opposition. I do not know of one who has pronounced in its favour. These are men of unquestioned scientific standing: men, moreover, trained to consider realities rather than symbols, and the relations of things rather than the mere notation by which such relations are expressed. The leading British daily paper is a consistent opponent of the metric change, and the Times, in dealing editorially with a scientific question, may be trusted to have the best scientific advice available. By whom is the change advocated? Chiefly by computers, who, as is well known, are not necessarily mathematicians. Computation is not an intellectual but a mechanical operation. There is no kind of computation that cannot be performed more efficiently and quickly by a machine than by a human

brain. As regards money-reckoning, the computer has a genuine grievance and a good case; but a coinage reform need not disturb the standard weights and measures. There is one important department of science—that embracing chemistry and physics—the practitioners of which are inclined to look favourably on the proposed change. Their work involves electrical measurements, and the only standard available is affiliated to the meter. It is not that any intrinsic merit can be claimed for the system itself—its nomenclature is a linguistic horror; but those engaged in branches of science where these measures are in use naturally prefer one system instead of two. It must be remembered, however, that the convenience of this limited class of workers would be gained at the cost of the inconvenience of other branches of scientific men, especially those concerned with celestial or terrestrial measurements, as well as of the disturbance of all existing systematic work by land and sea. As against the Times, we find in the Scotsman an able newspaper advocate of the system. But the Scotsman admits that two generations would not suffice to carry the change into effect. The stock arguments in favour of the meter, when examined, still further add to the monumental pile of fallacies. We are told, for example, that in our insular prejudice we are holding out against the peoples of the world—that our commerce is crippled by our unintelligible system. Statistics are adduced to show how many millions have adopted the meter, and how many Governments have legalised it as their standard. These statistics look formidable enough, but it shows a curious lack of proportion when insignificant principalities whose commerce is a negligible quantity are balanced against Powers such as Great Britain, the United States, and the Russian Empire. Comparative - population statistics are wonderfully inflated when wild tribes within European “spheres of influence” are counted in. Figures thus built up may seem imposing, but on applying reasonable tests we find fallacy once again. The truth is the exact contrary. Among the educated, civilised, and progressive peoples of the world—and these alone are concerned in the question—the British-American standards are in use in the proportion of two to one, and a still larger proportion of the world's commerce is in the hands of the people who adhere to these standards. Two-thirds of the world's printed matter—two-thirds of the correspondence passing through the world's post-offices—is produced by English-speaking folk. This gives something like a fair test of the proportions involved. The suppression of the old European national standards leaves the British weights and

measures (with perhaps the single exception of the typographic standard, the decimalization of which is impossible) the sole rival of the meter, the sole representative of a system which has survived the vicissitudes of thousands of years, and which now is in a majority of more than two to one. Again to quote Mr. Giles, the inch-and-foot scale represents— The greatest amount of territory by land and sea. The highest rate of increase in population. The greatest wealth. The pre-eminence in commerce. The first place in extent and power of colonisation. The greatest freedom for its individuals. The first in philanthropy and improvement in the condition of subject races. It is significant that no country has ever voluntarily adopted the meter. It has always been arbitrarily enforced upon the people—in Sicily literally at the bayonet-point. And even under the most rigidly repressive rule the old system often lingers. A writer in the London Engineering World only a few months ago stated that in Prussia, Hungary, Hamburg, and Hanover the “fuss” is still used in measurement, and that, he could give other instances of the survival of the old standards. In France, it is needless to say, there was no attempt to consult the people as to the change. It is instructive to read what Rees's Cyclopedia (1819) placed on record regarding the introduction of the scheme into that country:— That beautiful and scientific theory [this authority says] has not been found unexceptionable in practice. On the contrary, it met with such opposition on account of the Greek and Latin terms and the decimal division that in 1801 the Government allowed the people to use, for a limited time, their own vocabulary of names, applying them to the new standards, which are still retained. And in 1812 a further concession was made by the Imperial Government to the prejudices and habits of the people. They were allowed to continue the ancient vocabulary applied to the new standards with the word usuel added to each: thus, two mètres are the toise usuelle; half a kilogramme, the livre usuelle, &.; and these units are not divided decimally, but into halves, quarters, and eighths. The long measures are also divided duodecimally. Besides the binary divison of weights, the livre usuelle is divided into ounces, gros, and grains, like the ancient livre, poids de marc. Hence the new ounce and its divisions depart so widely from the gramme that the proportion cannot be ascertained without a troublesome calculation. Thus, after more than twenty years of troublesome experiment and trial of the metrical system the only advantage that has been gained is that of establishing one common standard, the mètre; but uniformity might as well have been obtained by making the ancient toise (so universally known) their standard. “Uniformity” being the avowed object, the change of the standard, with all the perplexities it involved, was, as the critic of the Cyclopedia recognised, unnecessary. The one remaining recommendation, then, of the metric system is its

decimal radix of weights and measures, universally applied. That certain advantages accrue from this unity of method is too obvious to admit of dispute; but they are far from making the system the perfect one its advocates represent it to be. Its perfection we are expected to take for granted. But, in view of what we are required to abandon in its favour, it' is not only reasonable but necessary to inquire, Is it not possible to overrate the advantages of a universal decimal system? This question brings us to first principles, to the base of the whole fabric—the system of notation and numeration in world-wide use. Is this system itself perfect, either practically or scientifically? It is as much an “accidental inheritance”—to quote Mr. Mattieu “Williams's contemptuous designation of our national standards—as the standards themselves; but the reformers did not go to the root of the matter—they made no attempt to reform nor even, apparently, to investigate it. The fact that our numeral signs are called “the ten digits,” and the further fact that in many languages the words “five” and “hand” are related, or even identical, associate our system with the primitive method of finger-tally as unmistakably as the verb “calculate” embodies the fact that pebbles were used in bygone days as instruments of reckoning. But, widespread, almost universal, as decimal calculation is and has been, ten has never been the sole radix in common use. In practical concerns it is the geometrical or tangible qualities of numbers with which we have to deal, and the geometrical defects of ten as a radix are a standing disqualification, with all its factitious advantages as the root of our notation. Were it possible for the world to start afresh—as the Frenchmen a hundred years ago dreamed of doing—mathematicians would certainly discard ten as a root-number and seek another, and that number would almost certainly be twelve. Had such a proposition come from the French scientists when the change was undertaken it would, even if ultimately found impracticable, have had good scientific reasons in its favour; in fact, the difficulties, great though they are, might not then have proved insurmountable. Our arithmetical notation expresses actual numbers only from 1 to 9. Thenceforth, both in speech and writing, all numbers are indicated not as they are In themselves, but in their relation to the radix ten. So with fractions. Only those with a single numerator and denominator express actual relations to unity; beyond these the artificial element makes its appearance. Mr. R. T. Barbour, writing last year in an Australian review, remarked that, instead of vulgar and decimal fractions, we should say vulgar and natural. This is a good instance of a notational convention obscuring

genuine relations, for the suggestion that there is anything at all “natural” in decimal fractions, especially as contrasted with vulgar fractions, is a complete inversion of fact. Take that convenient school-book illustration the apple; halve and quartet it. Which form most correctly and naturally expresses the proportion the sections bear to the whole? The “half” or “quarter” represents the actual fact accurately expressed by the vulgar fraction, and by that alone: To describe the fruit as divided into sections of 0.5 or 0.25 is to introduce an artificial convention remote from the fact, warrantable only when, for purposes of calculation, we are considering the pieces in association with others actually divided into fifths, tenths, or twenty-fifths. The “Autocrat” has told how, after defending his landlady's pie, “I took more of it than was good for me—as much as 85°, I should think—and had an indigestion in consequence.” This does not strike an appreciative reader an intended for the “natural” mode of expression, hut rather as a piece of playful pedantry on the author's part. Vulgar fractions are so universally and unmistakably natural that no convention or notation, however ingenious, can dislodge them. Therefore the German typefounder of to-day, being a man of common-sense, catalogues his smaller wares by the “half-kilo.” (his nearest approach to the discarded pfund), the American coin's his half- and quarter-dollar, and so to the end of the chapter will weights and measures necessarily be popularly divided, thought of, and spoken of in halves and quarters, whatever official system may be in vogue, or by whatever written symbol these values are expressed. The radix of 2 is one of the most important in practical use, and in many cases it is the only practicable one. It governs not only our own, but the other traditional systems of weights; it is the obvious corollary of any system of balancing. In book-folding it is in all ways the most convenient, and its sole rival, the sextuple fold, is almost extinct. The names “twelves” and “twenty-fours” are still in use to indicate certain shapes and sizes, but as a matter of practice such are nearly always now printed and folded as eights, or sixteens. A “decimo” sheet would be an obvious impossibility. Two of the multiples of 2—eight and sixteen—have been suggested as notational substitutes for ten; but as an arithmetical radix either of them would be little, if any, better, apart from the consideration that one is inconveniently small and the other just as inconveniently large. The latter would further involve a cumbrous notation and an extensive nomenclature, as an American would-be reformer, who boldly selected sixteen as his radix, discovered many years ago. De Morgan, in his “Budget of Paradoxes,” has made merry over

the uncouth terms devised by this innovator; but if there had been any sound basis for the suggestion itself the professor would have treated the paradoxer more seriously. A duodecimal scale has had numerous advocates, and a large amount of work has been done in the way of compiling tables, &c., in appropriate notation by isolated workers, many completely ignorant of the fact that others have gone and are going over the same ground. I have seen more than one publication on the subject. One of these, by Mr. Henry M. Parkhurst, an American mathematician, contains elaborate tables of logarithms, primary and secondary multiples, least divisors, &c., all calculated on the radix of twelve, and I understand that the author has much more matter of the same kind in manuscript. He uses × and A, contracted to the width of the other digits, to represent ten and eleven, and these two special characters give his tables a peculiar appearance. There is, or was, a Duodecimal Society in England, formed to bring about organized and united action, but I have never seen any of its publications, nor do I know its address. The most prominent duodecimalist, however, was the late Sir Isaac Pitman, who took up the subject with characteristic energy. His attention was directed to the matter through the agitation in and out of Parliament in favour of the metric system, among the defects of which he held that its decimalism was not the least. “With a view to carrying his proposal into practice,” his biographer writes, * “A Biography of Isaac Pitman, Inventor of Phonography” By Thomas Allan Reed. London: Griffith, Farran, Okeden, and Welsh. 1890. “new types were ordered (2, ten, and 3, eleven) in four sizes, and it was Mr. Pitman's intention to employ the new notation in his Journal, and to recommend it for general adoption. During 1857-58 he counted everything as far as possible by dozens and grosses, with a view of paving the way for the new numeration; but he was unequal to the task of undertaking a reform of this magnitude in addition to the writing and spelling reform, and after a series of trials he reluctantly abandoned the project, but not without hope of seeing it inaugurated at some future period.” For two years he used the notation in his Journal, and kept it up in his private accounts till 1862. His scheme included a coinage system — pence, shillings (a gold twelve-shilling-piece to take the place of the half-sovereign), and “bancos” = a gross of shillings. The figures “723” with currency symbol prefixed, would thus represent seven “bancos,” ten shillings, three pence. He probably proposed to reform current weights and measures on the same plan, but, not having access to the old Journal files,

I have no details. The disturbance of measures by complete duodecimalisation would amount to no more than systematization, and would be comparatively slight, but it would have been otherwise with weights. It will be noted that a single radix would have governed arithmetical notation, measurement, and coinage. To those who have any knowledge of the life and work of Isaac Pitman it is needless to say that he was the reverse of an idle dreamer. He was one of the most industrious, methodical, and practical of men, possessing extensive knowledge and an inventive mind. Without these qualities he would never have made the practical success he did of his system of phonography, which, on its own merits alone, dislodged every previous system of shorthand, and still holds its ground against all rivals. When he received his somewhat belated honours from royalty, they were generally approved as having been bestowed on a public benefactor. Any man who looks forward in advance of his time must bear the penalty of being ignorantly esteemed a paradoxer; but even in the field of spelling-reform, to which he devoted years of apparently fruitless toil and expense, his work has not been lost, for he has familiarised the millions who write shorthand with the idea of a rational alphabet, of which they make daily use. But the reform of arithmetical notation, however desirable in theory, seems to be too large a contract for any man, or even any nation, to undertake, and its foremost advocate was well advised to let it drop. As notational signs the Pitman figures could scarcely be improved upon. While conforming in character to the familiar numerals, they can also be read as the initial letters of the words “ten” and “eleven” respectively. The following table of various numbers compared in the two notations gives an example of the symbols as they appear in actual use:- Decimal. Duodecimal. 12 = 10 22 = 12 100 = 84 107 = 83 144 = 100 287 = 133 432 = 300 999 = 633 1570 = 222 1728 = 1000 Apart altogether from any theory of reform, a little practice in this notation is a remarkably illuminating exercise in arithmetic, showing, as it does, that the decimal system conceals more than it discloses of the properties of numbers. Awkward fractions disappear, while numerical relationships come out in a simple and beautiful manner. One important series after another, broken and marred by decimal misrepresentation, falls into regular and harmonious sequence, and the

geometrical relations, which to the worker in any branch of science or art are of essential importance, in a larger measure than is possible under any other radix, are represented in their real significance. This is in itself no slight advantage, for geometry is of all sciences the most tangible and practical. How comes it to pass that the dozen holds its ground so persistently in commerce, notwithstanding the convention of reckoning by tens? Very much because there are so many ways of making a convenient and compact parcel of twelve equal units, while tens and hundreds pack very badly indeed, making misshapen parcels and wasting space. To those who have to pay freight charges—as all must do, directly or indirectly—this is a consideration of importance. The two ratios with which the practical work of life brings us into constant contact are graphically represented respectively on the clock-dial and the compass-card. With both of these decimals are in constant discord. The old Babylonian division of the circle into 360 degrees is comprehensive enough to take in the decimal, but its place is subordinated to the more important and significant geometrical angles, which only duodecimal division can give. It may be regretted that a numerical system so nearly perfect as the duodecimal—simplifying as t would all the practical mathematical work of the world to an amazing extent—can never, unless humanity develops an unforeseen capacity for the acceptance of great reforms, be adopted. If it were merely a matter of relative merit there could be no doubt of the result, for mathematicians are unanimous as to its superiority. But one of the great arguments against the metric system applies with equal force to duodecimalism. The decimal radix is so strongly entrenched in tradition, in notation, in thought, speech, and literature, that its dislodgment may be reasonably assumed to be impossible. We find a tacit recognition of the place that twelve should occupy of right in the significant facts that our popular multiplication-table extends to 12 × 12, and that children are taught simple multiplication and division up to twelve, not ten. And hampered though we are in all directions by our defective arithmetical radix, we still are free to use, in weighing and measuring, the divisions that the experience of many ages has proved to be the best adapted to our needs. That freedom it is the avowed object of the metrists to destroy. They would widen still further the gulf that unfortunately divides arithmetic from geometry, and make our bondage to the decimal complete. In any notation two radices are required — a major as well as a minor—and it is important that there should be

due correlation between the two. There is a notable difference in this respect between the ancient system, universal in the British Empire, and the modern French scale. While our own system does not appear to be the best possible, this correlation is duly observed; in the French scheme it is neglected, with the result that the nomenclature is unscientific and misleading. It is the usual convention to break any long series of figures into groups of three for the sole purpose of facilitating reading. The comma, usually employed for this purpose, is no more a mark of grammatical punctuation than the period when used as a decimal point. But French numeration is so glaringly anti-arithmetical that it can be explained only on the theory that in their inveterate habit of mistaking arbitrary symbols for scientific facts the devisers of the scheme attached some occult mathematic significance to the triple grouping. In fixing the major radix the question arises at which stage to abandon ten and substitute a multiple. Theoretically, as it is easy to show, the second radix should be a power of ten which is (1) a square number, and (2) the roots of which are also squares till we reach the square of ten. Such numbers, are successively one hundred, ten thousand, and one hundred millions. Either of these, radically subdivided, brings us in the end to ten; any other multiple of ten will yield as its root number an interminable decimal. For the larger radix one hundred is obviously too small; multiplied by a million it is inconveniently large. The point naturally indicated for the break, therefore, is the myriad, in which case the ciphers would properly be grouped not in triplets but in fours. By this arrangement the numeration table would stand thus, the square numbers being indicated by small capitals:- One 1 First Series (Minor Radix). Ten 10 Hundred (102) 100 Thousand (108) 1000 Myriad (104 =1002) 1,0000 Second Series (Major Radix). Decamyriad (105) 10,0000 Million (106 = 10002) 100,0000 Milliard (107) 1000,0000 Billion (108 = 1004 = 1,00002) 1,0000,0000 Trillion (1,00003 = 100,00002) 1,0000,0000,0000 Quadrillion (1,00004 = 1,0000,00002) 1,0000,0000,0000,0000 In this scheme the first and second series are consistent and complementary; the powers of the minor radix are in-

dicated by the number of ciphers and those of the major radix by the number of groups, the index-prefixes “bi,” “tri,” &c., agreeing with the power. By this plan, and this alone, the comma dividing the groups becomes a significant arithmetical sign as well as an aid to legibility. Both the British system and the French are defective in starting from the million, which, though itself a square number, has not a square for its radix. In the British scheme, however, the terms “billion,” “trillion,” &c., representing as they do the successive powers of the greater radix, are correctly used; but the French nomenclature, in which a thousand millions is called a “billion” and a thousand “billions” a “trillion,” is worse than meaningless—it is misleading. If we seek the root of the British billion, the cube-root of the trillion, and so on, we shall always come back to the starting-point—the million, or 106. Applying the same process to the French table, we have a series of anomalous results—the root of the “billion” is 31622777 with an endless decimal fraction; the cube-root of the “trillion” is 10,000. The fact that the French have their own specific word, “milliard,” for their modern “billion” seems to indicate that the numeration scheme has undergone alteration; and, in fact, I find it explicitly stated by one authority that in France the “older writers” use the same system as our own. I have been unable to ascertain whether the change was made as recently as the revolution at the close of the eighteenth century, so few writers concern themselves with the historic aspects of the question; but I am inclined to think it belongs to a remoter period. Unfortunately for the English-speaking world, the French method has become prevalent in the United States, causing such confusion in the interchange of newspaper items and literature in general that one meeting with a reference to a “billion” in print, unless assisted by the context, can never be sure whether a thousand millions or a million millions is intended. Stranger still, the French notation was most unwarrantably introduced into at least one important English text-book—the Sandhurst Military College Arithmetic —but appeared, I am informed, in one edition only. Its temporary adoption in this academy, however, has led to serious confusion. When the present Education Act came into force the Department found both systems being taught to the children of New Zealand, certain provincial Inspectors apparently preferring the French method, and it became necessary to issue a special direction insisting on the uniform use of the British notation in all the public schools. Apart from scientific defects, a low base of computation, especially when combined with a low unit of currency, is not in keeping with the dignity of a great people. Mark Twain

has humorously described the dismay of a small dinner-party at the Azores when the landlord presented his bill, amounting to “21,700 reis,” which amount was found after much inquiry to equal 21 dollars 70 cents. But the small billion, in conjunction with a dollar unit, is a boon to “yellow” journalists and others addicted to “tall talk.” The “billions” so glibly paraded, even if genuine, which is not always the case, represent about two hundred millions sterling in British currency, and it requires five times as much wealth to constitute a millionaire in Britain as it does in the United States. Not the least instructive chapter in the history of the French reformers is the record of their failures—failures which shattered the boasted unity of their system even at the outset, and which the metrists of to-day keep judiciously in the background. But it is not quite forgotten that the scheme of these same “reformers” included the consummation of the Christian era, and the institution of the new order in which mankind should recognise one object of adoration only—the Goddess of Reason. In October, 1793, the new calendar, designed for all time, was promulgated. Books dated in the year 1 are still extant. The week was abolished and the decade substituted. The whole scheme of time was decimalised, but the solar system persisted in pursuing its incommensurable movements as before. Doubtless it was this perverse conservatism on the part of bodies celestial and terrestrial that caused a certain astronomer to wish that he had been present at the creation to give the Almighty the benefit of his counsel. The new era was inaugurated with much ceremony and indecorum— for was it not to abide for ever? All books of chronology record the date of its institutution, but few give the date of its disappearance. In a very few years it had vanished like a wreath of mist—imperceptibly but effectually. The geometric circle, like the arc of the meridian, was regraded into 400 instead of 360 degrees. As under the reformed arrangement many of the most important angles could no longer be expressed in degrees, this was one of the first points in which the scheme broke down. In any case, the preliminary quartering of the circle was in itself a silent but none the less eloquent admission of the essential inadequacy of decimalism. In fact, examination of the pretentious scheme, no matter at what point that examination begins, reveals fallacy piled upon fallacy. Even if our own ancient system had no particular scientific value—if it were really as defective as its assailants assert— we might well hesitate to exchange it for a substitute so ill-considered and so imperfect, quite apart from the incalculable loss and inconvenience such a change would impose upon

us. “History,” as a writer of a past generation has well said, “shows that, while Governments change with great facility their money systems, their constitution, and even their religion, weights and measures seem immovable. They are, indeed, so mixed and, as it were, matted with every concern of property that they cannot be essentially altered without violence and confusion. Nor are these evils of a temporary nature. The habits, customs, and prejudices of the multitude are not to be speedily changed.” The experience of every country where such a change has been decreed sufficiently confirms the statement. But it is not in the interests of “the multitude” that the change is made, nor are their “habits, customs, and prejudices,” it such happen to conflict with the prevailing fashion in science (so called), deemed worthy of consideration. Happily, fashions in science, as in costume, have a way of becoming unfashionable, while despised “habits, customs, and prejudices” —often the intuitive wisdom of the many, or embodying the concrete results of the experience of a distant past —have a perennial vitality extremely irritating to the exponents of the latest theories. After all, our national measures, which —in these days of rampant “Imperialism,” too!—it has become the unpatriotic fashion to contemn, are fundamentally more scientific, as well as more generally convenient, than their foreign rivals. They are no mere “accidental” inheritance, nor do they show any signs of haphazard origin. Whence the Saxons derived them, and how, is not known; but they can be shown to possess a venerable antiquity, and to have passed down the ages practically unchanged. A jealous regard for accurate standards is an ancient characteristic of our race. Were there such a thing in nature as an immutable standard, convenient and everywhere accessible, no doubt it would have been accepted, but no such natural unit has ever been found. The wise men of old everywhere selected the nearest approach which has yet been found to such a unit. They did not take a random terrestrial measurement with neither scientific nor practical value. They recognised that measures were subordinate to man and not man to measures, and from first to last, therefore, their standard was “the measure of a man.” Proof of this fact is built into the very structure of language. There is not an ancient term of measurement in our tongue, save those denoting infinitesimals, that cannot be referred to the human frame. * The fathom is the height of a well-developed man, and also the stretch of his arms, which, in fact, the word itself signifies. Philologists refer it to the root “fat,” to extend. The Saxon word “fæom “signifies “the space reached by the extended arms—reach, embrace.” In Danish, an allied language, the expression for “to embrace” is at tage i favn, literally” to take in fathom,” the two nouns being identical. The cubit, yard, and ell refer to arm-measurements; the spaa, the hand, the foot need no interpretation. “Inch” and “thumb” are convertible terms in more than one living language, and the old “finger-breadth” was two-thirds of an inch. The “pace” is the unit of the longer measures; “mile,” a numerical term, literally means a thousand paces. Measures were also calculated by days' or hours' journeys and variously subdivided. Naturally among different peoples these have diverged more widely than the smaller measures; hence we have the modern mile varying from 11,700 yards in Sweden to 1,165 yards in Russia. But from the finger-breadth to the league the man himself is always the ultimate standard.

Can creation furnish a better unit? I do not think so. All men are not six feet, high, nor are their feet usually twelve inches in length, and it is obviously true that any one who should in his own person combine all the precise measurements which derive their names from the human frame would not be a model for an artist. The necessity that each measure should bear an aliquot relation to all the others—the variations in human stature and build—make exact correspondence with any of them the exception rather than the rule. They are, as it were, the rough draft, from which, by comparisons and averages, the actual standards of reference have been derived. They are, however, near enough for most of the practical needs of daily life; they are universally accessible as no external standard can possibly be; and the man who takes the trouble to ascertain and make due allowance for his own “personal equation” may still serve as his own standard, and be to a great extent independent of external aid. I have seen a tall Maori measuring off a fabric by the “faddom,” using his arms as a gauge, and he did not give himself short measure, as a tape afterwards proved. I have seen women measure off yards with great correctness by hand and eye alone; and the accuracy with which distances can be “paced” does not need to be pointed out. The so-called “patriotic” exercises in our schools, instead of taking the questionable form of homage to a flag, would be better devoted to explaining the beauty and value of our far more ancient standards. If the children were exercised in weighing and measuring by the eye and hand, their work being afterwards tested—in drawing and subdividing six- or twelve-inch scales from memory—they might in after-life be to a great extent independent of artificial standard's, except where commercial or scientific accuracy was required. The antiquity of our measures may be inferred from the fact—which is abundantly demonstrated—that they are built into the Great Pyramid of Egypt, the oldest of all the pyramids, the others being very indifferent imitations—sepulchral

monuments embodying no mathematical science. * So many strange and fanciful theories have been associated with this structure—specially built to embody the astronomical and mathematical science of its founders, and to afford an enduring record of their standard of weights and measures—that it is almost necessary in referring to the subject to disclaim the religious theories which— unfortunately, I think—have been associated with it. Such, for example, as that its builders were divinely inspired, and that it is “a prophecy in stone.” To the late Professor Piazzi Smyth, Astronomer Royal of Scotland, belongs the credit of making the first exact measurements and of interpreting the mathematical and astronomical significance of the venerable monument; but I cannot but think the usefulness of his work was marred, though its accuracy was not affected, by his theological bias. He disliked the division of the circle into 360 degrees not on scientific grounds, but because that method was used by the idolatrous Babylonians. He accepted as fact the Rabbinical tradition recorded by Josephus, that Cain added to his iniquities by devising weights and measures that he might defraud those with whom he had dealings—a story in much the same category as that in the apocryphal Book of Enoch, whence we learn that an evil demon, Penemue by name, taught the children of men writing and the use of ink and paper, with every secret of wisdom, whereby many have gone astray from every period of the world, and by their knowledge they perish. In the Pyramid we find most unmistakably the inch and the cubit of 25 inches; all the measures of weight and capacity are based upon the inch, and the central chamber where they are deposited is perhaps the most perfect contrivance for securing uniformity of temperature ever devised by the skill of man. In the Pyramid standard measure of capacity; equalling four British “quarters,” we find the origin of the term, the larger measure having been so long disused that the fractional name has been somewhat of a puzzle. † A remarkable feature of the Pyramid standard is that it deals only with the concrete, avoiding notation of any kind. It cannot be said to be binary, decimal, or ducdecimal—it is, like number itself, independent of them all, and its interpreter must find and apply his own radix. In this respect it is consistent with, its plan, for it is unique among Egyptian monuments in that it is absolutely without inscription, bearing neither hieroglyph nor alphabetic symbol. Piazzi Smyth was as thorough a decimalist as he was an opponent of the “atheistic mètre,” but he could find no decimal or other radix in the structure. In form it is a five-sided crystal, and if it has a key-number (which is doubtful) that number would seem to be 5, for its cubit is 5 × 5 inches. A further example of the little change which a traditional standard may sustain when carefully preserved is found in the fact that our inch differs from its prototype in the Pyramid by only the thousandth part (minus). If the basis of our measure be the noblest of all—man himself, so its popular divisions are in practice more adapted to his daily needs than the new ones. Only last year a photographic journal complained that the French system provided no convenient-sized storage-bottles, and said that photographers who, perforce, did laboratory work on the metric

system kept Winchester quart bottles for storage, and in preparing solutions thought and worked in the old standards. Yet another worker in the same field, criticizing the nomenclature, said it was, according to all the laws of thought, a defect. A separate characteristic name for each successive unit gives it individuality, whereas the cumbrous and indistinctive names in the metric system are found in practice to be a source of error. As we have seen, the French scientists went to great pains and expense to have some kind of cosmic basis for their system. Our own system has not only the microcosmic basis supplied by man himself, but according to the late Sir John Herschel (who, by the way, did not take the Pyramid into account at all) has relations to earth and water more striking and harmonious than any the rival system can boast. Sir John could speak on this subject, if any man could, with authority. He was the ablest and most learned member of the Standards Commission, and his letter to the Times, written more than thirty years ago, is so much to the point that I quote it in full:- As Mr. Ewart's Bill for the compulsory abolition of our whole system of British weights and measures, and the introduction in its place of the French metrical system, comes on for its second reading on the 13th proximo, I cannot help thinking that a brief statement of the comparative de facto claims of our British units and of the French on abstract scientific grounds may, by its insertion in your pages, tend to disabuse the minds of such, if any, of our legislators who may be under the impression (I believe, a very common one among all classes) that our system is devoid of a natural or rational basis, and as such can advance no à priori claim to maintain its ground De facto, then, though not de jure (i.e., by no legal definition existing in an Act of Parliament, but yet practically verified in our parliamentary standards of length, weight, and capacity as they now exist), our British units refer themselves as well and as naturally to the length of the earth's polar axis as do the French actually existing standards to that of a quadrant of the meridian passing through Paris, and even in itself better, while the former basis is in itself a preferable one. To show this I shall assume as our British unit of length the Imperial foot, of weight the Imperial ounce, and of capacity the Imperial half-pint, and shall proceed to state how they stand related to certain prototypes, which I shall call the geometrical ounce, foot, and half-pint; and shall then institute a similar comparison between the French legally authenticated mèitre, gramme, and litre in common use with their (equally ideal, because nowhere really existing) prototypes, supposed to be derived from the Paris meridian quadrant, distinguishing the former as the practical, the latter as the theoretical, French units. Conceive the length of the earth's axis as divided into five hundred million equal parts or geometrical inches. Then we will define:- (1.) A geometrical foot as twelve such geometrical inches; (2.) A geometrical half-pint as the exact hundredth part of a geometrical cubic foot; and (3.) A geometrical ounce as the weight of one exact thousandth part of a geometrical cubic foot of distilled water, the weighing being performed, as our Imperial system prescribes, in air of 62° Fahr. under a barometric pressure of 30 inches.

In like manner the theoretical kilogramme and litre of the French are decimally referred to their theoretical mètre on their own peculiar conventions as to the mode of weighing. This premised—(1) The Imperial foot is to the geometrical in the exact proportion of 999 to 1,000, a relation numerically so exact that it may fairly be considered as mathematical; and (2) and (3) the Imperial half-pint and ounce are, each of them, to its geometrical prototype as 2,600 to 2,601. Turn we now to the practical deviations from their theoretical ideals in the case of the French units. Here again (1) the practical mètre is shorter than its theoretical ideal. The approximation is, indeed, closer, but the point of real importance is the extreme numerical simplicity of the relation in our case, more easily borne in mind and more readily calculated on in any proposed case. (2 and 3) Any error in the practical value of the mètre entails a triple amount of aliquot error on the practical kilogramme and litre, so that in the cases of these units the proportion between their practical and theoretical values is not that of 6,400 to 6,401, but of 2,133 to 2,134. Here, then, the greater degree of approximation is in our favour; and it is to be observed that in our case this triplication of error does not hold good, since by a happy accident our standard pound has been fixed quite independently of our standard yard, and our gallon is defined as 10 pounds of water. Like all other terrestrial things, the earth's axis is doubtless subject to minute secular changes; but it is at least a right line, and a line of unique significance, as it is used by astronomers as the unit-measure for distances within the solar system. It is therefore very convenient that it should harmonize with our ordinary standards. Moreover, its length —five hundred million inches—is a consistent decimal relation, which the forty million mètres of the French system is not. Was it, however, by a mere “happy accident” that such remarkable concordances occurred? I have more faith in both the knowledge and wisdom of the men who devised them of old than to think so. Astronomically and mathematically, the self-complacent French scientists of a hundred years ago were as babes compared with the architect of the Pyramid. Sir John Herschel's casual reference to the “peculiar conventions” of the French method brings out another instance of the practical nature of our system as contrasted with the artificial and doctrinaire methods of the metric system. British observations and tests are made under reasonable and normal conditions of atmospheric pressure and temperature. The French observations have to be corrected to the practically impossible conditions of sea-level and the freezing-point of water. The British people are free to use, and do habitually use, decimal divisions of their own standards wherever such division is found to be right and convenient. Any instrument-dealer will supply rules graded into eighths, tenths, or twelfths, as desired. Foreigners are not so ignorant of our standards as the metric advocates would have us believe.

For British buyers they are quite willing to manufacture to British measures. Beautiful and accurate scales to English feet and inches are “made in Germany.” Time has been decimalised for the navigator's convenience, as the Nautical Almanac tables show; but it would be an intolerable inconvenience to have to exchange our present clocks for ten-hour dials. That our foreign trade would benefit by the change is more than doubtful. While we were painfully and laboriously discarding all our patterns and making new tools—probably we should find it necessary to purchase most of them abroad —our foreign rivals, with all their scales ready to hand, would have an unprecedented opportunity of invading all our markets, and during the time of transition, at all events, would have an overwhelming advantage. Just as free trade and exchange throughout the United States has brought vast prosperity, so, high authorities maintain, British, success in trade has been in great measure due to the uniform system of standards existing throughout the British-speaking world. Is this colony desirous to strike the first blow to break it up? France was once kind enough to hint that she would consent to reckon from the Greenwich meridian if Britain would adopt the French measures. Was the suggestion made for Britain's advantage? Scarcely. France would be the gainer in such a compact by both changes. The question is a vital one. Even if the change could be shown to be for the better, would it be worth the price we should have to pay? Every individual in the community would be injuriously affected. Every map, from the magnificent British Ordnance Survey to the diagrams on the margins of deeds, would require recalculation on an incommensurable scale. In the case of city frontages particularly, where fractions of inches are precious, the recalculation would be a fruitful source of dissatisfaction and dispute. All contractors' calculations of quantities, all measurements of bricks, timber, and other building material, would be affected. All graded instruments, from the artisan's two-foot rule to the costly and delicate apparatus of the engineer, would have to be replaced, all patterns superannuated, all calculations translated. If workmen only realised what it meant to them the trades-unions would make inflexible opposition to the change the foremost item on their programme. They will protest loudly enough, we may be sure, the day that it is sprung upon them. Then it will be too late. Those concerned are not so supine in the United States. I read, only a few days ago, * Engineer, 7th August, 1903. that the American Society of Civil

Engineers sent out a ballot-paper with a series of questions on the subject. There were 514 responses, and the adverse votes were three to one. Our textile goods would be affected. The housewife would no longer buy fabrics by the yard, nor could she do her marketing by the pound. All familiar landmarks, such as milestones and railway distance-pegs, would be obsolete. No doubt the State, to set an example, would at once set about pulling them up and grading in kilometers. “A frightful waste,” says a correspondent of the Engineering World (London), “and for what use?” To the astronomer it closes the ledger which, as Proctor told us, has been kept posted for three thousand years. The navigator's tables and books of reference will be obsolete, and the shipowner will have to forget his tonnages. The occupier of land—owner or tenant—must recalculate boundaries and recompute superficial areas. The Land and Survey Department would have the same costly task, and on a truly magnificent scale. Two-thirds of the world's trade would be disorganized while the change—necessarily slow—was in progress. It would play cruel havoc with our national inheritance of literature. The nomenclature is barbarous and unconformable to the genius of our tongue, while our classics, if the present standards ever became obsolete, would require continual annotation. Let us open an edition of Shakespeare, say, of A.D. 2003:— A merry heart goes all the day, Your sad tires in a mile-a.* Mile.—An ancient lineal measure 1.609 kilometers. This bond is forfeit And lawfully by this the Jew may claim A pound [= 0.454 kilo.] of flesh. Full 9.144 meters thy father lies; Of his bones are coral made. How the substituted nomenclature could be worked into English verse (other than burlesque) is a question that poets would have to settle. On patriotic and commercial grounds alike we should resist the change. The widest charity does not require us to commit commercial suicide that France and Germany may enjoy increased prosperity. I have already referred to the impossibility of restoring the metric standard by repeating the original measurements; also to the fact that after the lapse of a century the old standards are not extinct in France. Will it be credited that a German advocate of the meter urges the necessity for careful preserva-

tion of the neglected ancient standard in order that it may remain as an authoritative check by which the accuracy of the meter may be tested? It reads like comic opera, but it is true. In the “History and Review of the Toise Measure Standard,” published by Ferdinand Dummler, Berlin, Professor Dr. W. Foerster, who writes the introduction, says,— The old French system, whose unit was the toise graduated in six Parisian feet (= 72 Paris inches = 864 Paris lines) is also at present, next to the universal and exclusively recognised (legal) metrical system, of great scientific and practical value, since the same has not only been the starting-point for the fixing of the basis of the metrical system, but even at the present time has found employment in pendulum observations, and, by reason of its inclusion in numerous good measure scales and basis-measure apparatus, in land-surveying, especially in Germany. In order to secure a fixed and accurate transition from all the old French units of measurement to measurements according to metric units, and thereby to take the last step to overcome the old system, and finally to attain a homogeneous basis for all measurements, it is of great importance that there should be made as soon as possible in the International Weights and Measures Office a new comparison between the unit-lengths of the old French system and of the metric system. The whole passage, though not so intended, is a powerful indictment of the pseudo-science which wantonly cast aside an established and useful standard for a newly devised and nondescript substitute. Professor Foerster, it may be observed in passing, does not speak disparagingly or contemptuously of the old inch-and-foot scale; but, on the contrary, acknowledges its “great scientific and practical value” as British folk will begin to do after they have lost it, if the metrists have their way. It may be as well to explain, however, how this dependence of the accuracy of the metric standard on the ancient Paris standards arose. Of course, the arc of the meridian could not be measured in “mètres,” because that measurement was a preliminary to its establishment; and the surveyors necessarily made use of the authoritative scale which it was their purpose to supersede. The meter is usually described as equal to 36.9413 French or 39.3708 English inches. Its precise length, however, was reduced to French lines, of which it equals 443.296, and to this standard it would require to be referred should any question hereafter arise. Evidently there is considerable vitality in the old inch-and-foot scale still, even where it has been most rigorously suppressed; and in the coming Battle of the Standards for the supremacy of the world it may yet come off victor. By clearing the field of all inch-and-foot systems save the British it has greatly simplified the issue. What the result of the conflict will be it would be idle to predict; but there can be no question that the fittest will survive if the English-speaking world, awakening to the importance of its trust, remains united, and “England to herself be true.”