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A department was instituted for "Philosophical Instruments, and processes depending on their use," known as Class X. In this department was included calculating machines. The jury appointed to report upon this class contained the following among its 12 names -Sir David Brewster, Sir John Herschel, Mr. James Glaisher. Neither Babbage's nor Scheutz's machines were exhibited-indeed the latter was not then finally completed. Regarding Babbage's machine, then for several years past deposited at King's College, it is impossible to account for its omission: it may have arisen from official indifference; from jealousy; or from shame at its incomplete condition. Confining themselves therefore simply to the machines which were exhibited, the jury gave the first place to Staffel's, "which on examination seems to combine accuracy with economy of time, and works easily and directly." Again, "upon the whole it must be considered that Mr. Staffel has made an instrument possessed of considerable powers, and that great praise is due to him. The double motion of the handle, as well as the warning bell, are important improvements." Prize medal awarded. M. Thomas (de Colmar) "exhibits the next best calculating machine in the Exhibition, and has combined the two essentials of economy of time and accuracy of results." Again, on trying the machine, the number I was almost instantaneously taken from 10,000, giving the difference 9999 accurately; the performance of this operation is generally a severe test to these machines." Prize medal awarded. Regarding Herr Wertheimer, the jury say, "The machines are ingenious, but they are much wanting in the essentials of such machines, economy of time, and unerring accuracy.' They were awarded Honourable Mention. The like was awarded to M. Schilt, and to M. Lalanne. Dr. Roget's sliding scale (as improved by Brooker) was also exhibited.
In 1852 Mr. Willich, author of the well-known tables bearing his name, said, "We have but little hope that the splendid prospect opened by Mr. Babbage's calculating machine will ever be realized, the aid of Gov. having been withdrawn." He then adds : Is there no chivalrous millionnaire in this country anxious to immortalize his name by being associated with the successful carrying out of Mr. Babbage's calculating machine? A slice of a million would be well bestowed to obtain such a result." No one has yet come forward.
By 1853 the machines of the MESSRS. SCHEUTZ was finally completed, on the principles of the working model which had been prepared 10 years previously; and which we have already described. The difficulties in the way of its completion had been very considerable. In 1851, after a further inspection by a Committee of the Royal Academy, Herr Geo. Scheutz had applied to his Gov. for means to construct a larger and more perfect machine. This application was refused. Afterwards, however, the Diet for Stockholm did grant a sum of £280, upon condition that the machine should be completed before the end of 1853. It was so completed; after which a further £280 was voted, making in all £560.
In 1854 the inventors brought their machine to England, and it was first exhibited at Bermondsey, at the manufactory of Messrs. Bryan Donkin & Co. In 1855 the machine was exhibited in the Paris International Exhibition, and the inventors obtained a gold medal. Mr. W. Gravatt, an English engineer, and F.R.S., took great interest in the invention, and introduced it to the notice of the Royal So. The machine was brought back to Lond., and was set to work to print tables. The following is an authentic description of it, and its capabilities:
The size of the whole machine when on its proper stand and protected by its cover is about that of a small square pianoforte. The calculating portion of the machine, which appears in the front of the drawing, consists of a series of 15 upright steel axes passing down the middle of 5 horizontal rows of silver-coated numbering rings, 15 in each row. Each ring being supported by and turning concentrically on its own brass shelf, having with it a hole rather less than the largest diameter of the ring. Round the cylindrical surface of each ring are engraved the ordinary numerals from o to 9, one of which, in each position of the ring, appears in front, so that the successive numbers shown in any horizontal row of rings may be read from left to right, as in ordinary writing. The upper row exhibits the number or answer resulting from the calculation of 15 places of figures, the first 8 of which the machine stereotypes. The numbers seen in the 2nd row of rings constitute the first order of differences, also to 15 places of figures, if that number be required; and the 3rd, 4th, and 5th row of rings in like manner exhibit the 2nd, 3rd, and 4th order of differences. Any row can be set by hand, so as to present to the eye any number expressed according to the decimal scale of rotation-such as the number 987654321056789, the first 8 figures of which, if in the uppermost row, would, on being calculated by the machine, be immediately stereotyped. But by simply changing a ring in each of two of the vertical columns, the machine can be made to exhibit and to calculate numbers expressed in the mixed senary system of notation, as in that of degrees, minutes, seconds, and decimals of a second. Thus, for instance, if the result 874324687356402 were indicated, in the upper row of rings, it would be stereotyped 87 degrees, 43 minutes, 24'69 seconds.
While this process is going on, the argument proper to each result is at the same time also stereotyped in its proper place; nothing more being required for that purpose than to set each row of figure rings to differences previously calculated from the proper formula, and to place a strip of sheet lead on the slide of the printing apparatus; then by turning the handle (to do which requires no greater power than that which is exerted in turning that of a small barrel organ), the whole T. required is calculated and stereo-moulded in the lead. By this expression is meant that the strip of lead is made into a beautiful stereotype mould, from which any number of sharp stereotype plates can be produced ready for the working of an ordinary printing press.
At the average rate of working the machine, 120 lines per hour of arguments and results are calculated, and actually stereotyped ready for the press. It is found on trial that the machine calculates and stereotypes, without chance of error, 24 pages of figures in the same time that a skilful compositor would take merely to set up the types for one single page.
The machine was shortly afterwards purchased for £1000 by Mr. Rathbone, an American merchant, and by him presented to the Dudley Observatory at Albany, N.Y., where it still remains. Mr. Babbage was not slow to bear his testimony to the perfection and in some respects the originality of the machine.
In 1854 Mr. W. T. Thomson read before the Inst. of Act. a paper on Decimal Numeration and Decimal Coinage. The paper, which is printed in vol. iv. of Assu. Mag., exhibits much quaint scholarship, of which we have been glad to make some use. It contains a division, "Aids to Calculation," coming within the scope of our present art.
The Paris Exposition Universal of 1855 was remarkable for the number and ingenuity of the machines which performed arithmetical operations. Pre-eminently above all others stood the Swedish machine. It is honourable to France (says Mr. Babbage), that its highest reward was deservedly given to the inventor of that machine; whilst it is somewhat remarkable that the English Commissioners appointed to report upon the French Exhibition omitted all notice of these Calculating Machines.
In the Journal of the So. of Arts, 27 April, 1855, there appeared a brief description of Scheutz's machine, which developes one feature not previously noted:
The machine calculates to 16 figures, but prints to 8 only; and by a singularly ingenious, and at the same time simple, contrivance, the 8th figure in the T. is printed, not in all cases as calculated, but with a correction when required, for the 9th and subsequent figures omitted in the T. Thus whenever the 9th figure as calculated amounts to 5 or more, it is more accurate that the 8th or final figure in the T. should be printed by the addition of one: this the machine accomplishes.
In 1856 there were pub.: (1) Observations by Charles Babbage, on the Mechanical Notation of Scheutz's Difference Engine, prepared and drawn up by his Son, Major Henry Prevost Babbage, addressed to the Inst. of Civil Engineers. The same is pub. in the Minutes of Proceedings of that So. vol. 15. (2) Observations addressed to the Royal So. on the Swedish Tabulating Machine of Geo. Scheutz. This latter paper we believe was not by Mr. Babbage.
In the Journal of the So. of Arts, 3rd July, 1857, there was an account of Scheutz's machine from which we have drawn some of the preceding details.
It had been a matter of deep regret on the part of many of our leading scientific men that Scheutz's machine had ever been allowed to leave these shores. The more so as there was at the time work required to be done, which it could have accomplished. We may here quote Dr. Farr:
At that time it appeared to be desirable to construct a new life T. from the materials accumulated at the Gen.-Reg, office by the regis. of births and deaths in 17 years (1838-54), and by the two enumerations of the pop. of E. and W. in 1841 and 1851. The T. for single lives, and the various combinations for joint lives, male and female, involve a great deal of numerical computation; and as it was found that the calculations of the series, thrown into a form which is described elsewhere [paper read before Royal So., 7th April, 1859], could be performed by the machine, the Reg.-Gen. was pleased to bring the matter under the notice of Sir Geo. Lewis, then Sec. of State for the Home Depart., and in doing so he pointed out the importance which had been justly attached by the most scientific men of the country, by H.M. Gov., and by Parl., to the machine of Mr. Babbage, for which, though it had not been completed, £17,000 of money had been granted, besides the money expended by Mr. Babbage himself. The Astronomer-Royal concurred with the Reg.-Gen. in advising H.M. Gov. to order a new machine.
The request was granted. A contract was entered into with MR. SCHEUTZ and the MESSRS. DONKIN, who agreed to construct a new machine, with several improvements, for the sum of £1200. In the work several new tools were required, and workmen had to be specially instructed. From this expense the contractors did not shrink. The machine was completed, and was reported to be superior to the first.
Mr. Bryan Donkin reported that the machine consists of about 4320 pieces, of which 2054 were screws; 364 compose the chain; and 902 constituted the other parts of the mechanism. The weight, exclusive of the case, is about 10 cwt. The machine was shaken out of order on its way from the factory to Somerset House. It was, however, soon set in order, and put to work. [See 1862 and 1864.]
In January, 1860, MR. JARDINE HENRY brought before the Royal So. of Arts, in Edin. an instrument which he had invented for the construction of Life Annu. T. without the use of logarithms. His instrument accomplished by a single movement all multiplications necessary in forming L. Annu. Commutation T. We can only give a brief description of the invention. A right-angled triangle with two equal sides is divided into 10,000 equal parts. Lines are drawn from the left hand, where zero stands to the numbers in the perpendicular side, representing the numbers alive at the end of each year from birth to death, say out of 10,000 persons born. A vertical sliding scale on T, also divided into 10,000 equal parts, is adjusted so as to slide upon the triangle. The principal use of the instrument was said to be to furnish Joint Life Tables. It was not said to arrive at perfect accuracy; but its results were sufficiently correct for all practical purposes." From 400 to 600 values could be read off the instrument with complete ease; and the labour of constructing T. compared with the use of logarithms reduced to one-fourth. [See 1867.] In the International Exhibition in London, in 1862, there were exhibited [Class XIII.] several calculating machines, or arithmetical instruments, of which we have not previously made any mention, and concerning which we have very few accurate details.
The first of these is the invention of HERR WIBERG, of Malmö, Sweden. It is a small difference engine of cylindrical form, in which the difference axes are arranged round the
circumference of a circle, in place of lying in one plane, as in the machines of Messrs. Babbage and Scheutz. This machine, however, was so much deranged in its transit to the Exhibition, that the jury had no means of testing its mode of action. Herr Wiberg also exhibited a smaller calculating machine, like the other of a circular form, very compactly arranged, and very neatly executed. It was presumed to be for the purposes of multiplication and division; but no precise information could be obtained.
M. C. X. THOMAS exhibited several well-constructed machines for multiplying to the extent of 7 figures by 7. These seem to be the same we have spoken of as the invention of M. Thomas de Colmar. Medal awarded.
SIGNOR T. GONNELLA, of Florence, exhibited a calculating machine of very simple form and construction, but the jury considered "probably of little practical utility." Mr. Babbage's Difference Engine No. 1 (as far as completed) was also exhibited. The jury, consisting of 13 members, among whom were Sir David Brewster, Mr. James Glaisher, and Prof. Wheatstone, say in their report:
These eminently ingenious and practically useful contrivances have undergone considerable development since the former Exhibition of 1851; but that progress, as it will presently appear, is not completely represented in the present Exhibition. Calculating machines, as they at present exist, are essentially of two kinds. In the simpler form the operation of addition is performed by causing a figure wheel to advance a given number of unit spaces by moving through the same number of spaces or wheel with which it is in gear. The process of multiplication is merely the repetition of successive additions; and subtraction and division are merely the inverse processes of the former.
Then follows a short description of Babbage's and Scheutz's machines, in which occurs the following, "It must suffice to observe that in the engine of Messrs. Scheutz these arrangements [by which the number on any given wheel is transferred to the wheel above it] are more simple, and apparently not less efficacious than in that of Mr. Babbage."
Scheutz and Donkin's machine was not exhibited. It was indeed in use at the time, calculating and printing portions of the English Life T. No. 3. A description of it and examples of its work were distributed, and many scientific men visited and saw it in operation during the Exhibition. It may be be seen at Somerset House now, by any properly accredited person. In the description so circulated occur the following passages : The machine has been extensively tried, and it has upon the whole answered every expectation. But it is a delicate instrument, and requires considerable skill in the manipulation. It approaches infallibility in certain respects, but it is not infallible, except in very skilful hands. The weakest point of the machine is the printing apparatus, and that admits of evident improvement. The machine calculates and print series of a particular kind; and to the execution of these operations its utility is therefore limited. Its scope is less ambitious than the new analytical machine for which Mr. Babbage abandoned his first invention, as that machine seeks to embrace the whole field of analysis. [See 1864.] In 1864 there was announced for pub. by the late Mr. Babbage: Hist. of the Analytical Engine. The vol. was to contain a reprint of chapters 5-8 of the Passages from the Life of a Philosopher. Also a reprint of translation of General Menabrea's Sketch, etc. [See 1842.] We believe this work never was pub.
In the same year there was published the English Life Table: Tables of the Lifetimes, Annu. and Prems. ; with an Intro. by Wm. Farr, M.D., etc. In a letter to the Reg.-Gen., forming such Introduction, Dr. Farr says: "Several of the series were calculated by Scheutz's machine. . . It gave us an opportunity of testing its working powers in England, where Mr. Babbage explained the principles, and first demonstrated the practicability of performing certain calculations and printing the results by machinery." machine was employed to introduce the element of int. into the various T. In an appendix [p. cxxxix] the learned Doctor enters upon further details, while also replying to a complaint of Mr. Babbage regarding the non-exhibition of the machine, by which circumstance the latter considered the constructors lost a medal, and many scientific men were disappointed. Dr. Farr says that the work upon which the machine was employed was much required; and that quietude was essential to the proper conduct of its operations : The machine required incessant attention. The differences had to be inserted at the proper terms of the various series; checking was required; and when the machine got out of order it had to be set right. The idea had been as beautifully embodied in metal by Mr. Bryan Donkin as it had been conceived by the genius of its inventors; but it was untried. So its work had to be watched with anxiety, and its arithmetical music had to be elicited by frequent tuning and skilful handling in the quiet most congenial to such productions. This vol. is the result; and thus-if I may use the expressionthe soul of the machine is exhibited in a series of T. which are submitted to the criticism of consummate judges of this kind of work in England and in the world. If their approving testimony be won, it will be some compensation to the English workmen-the firm of Messrs. Donkin, and the Messrs. Scheutz, for the loss of a medal at the Exhibition of 1862.
It will be generally admitted that this beautifully printed and useful vol. of between 700 and 800 pages, mainly of T.—although not all the produce of the machine-must remain a very enduring monument of its usefulness; and it ought not to be forgotten that it is prob. to Dr. Farr that we really owe the existence of this particular machine.
In 1864 also, Dr. Farr read a paper before the Brit. Asso. at Bath, on Life Tables, by the Swedish Calculating Machine-with Photographs by A. Claudet; and at the same meeting Major-Gen. Hannyngton read: Some Remarks on the French Calculating Machine [M. Thomas de Colmar's]. Abstracts of these papers were not printed in the Official Rep. In the Companion to the (Brit.) Almanack, 1866, there is a paper by Mr. F. J. Williams, The Swedish Calculating Machine at the General Regis. Office, Somerset House; wherein is a very clearly written hist. of the machine, and of its performances. He also points out
that the first adaptation of this machine to the computation of L. T. was made by Dr. Farr.
In 1867 there was read before the Inst. of Act. a Memoir on Instrument for furnishing the D. numbers to four figures each, in Two Joint Life Annu. T. on any basis. The paper was prepared by Mr. Jardine Henry; and an abstract of it appears in the Assu. Mag. [vol. 14, p. 212]. We have already given some account of this instrument under
In (or about) 1868 MR. C. H. WEBB invented and patented in the U.S. a very simple and durable arithmetical instrument, called an "Adding Machine." It consists of two disks in the same plane, moved by an iron-pointed pencil or style. One disk counts units up to 100; the other counts hundreds up to 5000. It is used by practical actuaries on that side of the Atlantic.
In 1869, or prob. earlier, PROF. ELIZUR WRIGHT invented a Calculating Machine having the advantage of great simplicity of construction. Its chief functions are multiplication and division. The machine consists of two wheels or cylinders mounted on one axisthe outer surface of the cylinder being covered with figures. There are pointers attached. If two numbers are to be multiplied, one pointer is placed opposite one number, and one opposite the other. A slide is moved, one wheel turned, and the product appears opposite the pointer. Division is accomplished with the same ease. It gives correct results to 5 places. The Ins. Times of N.Y. says:
For all practical computations of prems., reserves, and dividends, as well as for interest calculations, it seems to us really invaluable. It has been tested by long and constant use, and has proved reliable in every sense.. In every life ins. office in the country there may be found a well-thumbed T. of logarithms. These T. are extremely useful, but Mr. Wright's machine is as much superior to them as they are to the child's multiplication table.
MR. JOHN SANG has invented an instrument of calculation called a "Platometer." It consists of a cone mounted upon two wheels. Mr. Edward Sang gives us the following description, and of the operation and uses of the instrument :
Thus let us take Mr. Sang's Platometer, and instead of making it to roll like a carriage on wheels, let us fix the frame, leaving the wheels and cone free to turn; and seek to apply it to such a solution as this: "to find the value of an assu." We shall divide along the periphery of one of the wheels parts to represent £1, and in the direction of the slope of the cone, other parts to represent the decreasing values of 1, payable, 1, 2, 3, 44 years hence. The product of £1 payable for each person who dies in the year, into the value of 1 for that year, is obtained by bringing the index wheel to the proper distance from the apex of the cone, and then turning the cone round by as many divisions as there are deaths in the mort. And if we perform this operation regularly for each year during the whole of the T., the index wheel will at once record the sum total of all these products.
MR. BEVERLEY, of Dunedin, New Zealand, has since invented a modification of the Platometer of Mr. John Sang. It exhibits the same principle brought out in a different way. Mr. Edward Sang gives us the following illustration of its use :
If a cylinder be dragged along a surface endwise, it does not turn; if it be moved in a direction at right angles to its axis, it rolls; in other positions it partly slides and partly rolls, and in general the quantity of turning is proportional to the sine of the angle between the direction of the axis and the direction of the motion. Taking advantage of this law, Mr. Beverley arranges his instrument so that the quantity of angular motion is proportional to the distance of the tracer from a certain straight line; it is at the same time proportional to the extent of the actual displacement, and hence the indication is proportional to the area of the rectangle.
Mr. Edward Sang adds:
Both of these instruments however, and indeed all machines of this kind, depend on a combination of sliding and rolling, and are thus liable to considerable inexactitude in their indications. They fall far short of the certainty which attends the use of toothed wheels, and would be unfit for such calculations as come under the notice of the actuary.
In 1871 Major-Gen. Hannyngton read before the Inst. of Act. a paper, On the Use of M. Thomas de Colmar's Arithmometer in Actuarial and other Computations. The paper is for the most part technical; but we take the following passages from it as supplemental to the description we have already given of the instrument:
Division is less tractable than multiplication, but so long as the divisor contains no more than 8 figures, the dividend and the quotient may be unlimited; for the remainders can be trans. to the left or highest place on the slide; the partial quotient be recorded, then effaced, and the operation be carried on to any extent. But a divisor of more than 8 figures offers difficulties. Among the powers of the machine may to some extent be included that of the Difference Engine; for a second difference can often be supplied by the operator. For instance, a table of square numbers, having the second difference constant, requires merely that the operator should continually add 2 to the difference on the face of the machine. Thus, in fact, any quadratic form could easily be tabulated. Hence, also, two or more operators, working together on separate machines, might compute T. requiring differences of the third or higher orders. Such an application of these machines might have important uses. Survivorship assu. may be computed. What is here shown brings out one of the most useful powers of the machine, namely, that of giving the sum of a series of products, without exhibiting the several quantities. . . . A simple and striking example of continuous calculation is afforded in the construction of temporary annu.. In conclusion I may say that, having had considerable experience in act. computations, I have never found the machine fail to afford help; though it may happen that the right process is not always indicated by the seemingly most appropriate formula. The machine asks for peculiar methods, and such as are not easily to be described.
The paper is printed in the Assu. Mag. [vol. xvi. p. 244]. In the same vol. there is also a short paper of obs. on this Hancock, Act. and Sec of the Patriotic Assu. Co. of Dublin.
instrument by Mr. W. J. His remarks are in the
main directed to answer some remarks of Mr. Sang, in his Lecture to be presently quoted. Mr. Hancock, after giving various examples of its use, says:
I think it is therefore manifest that the machine is far superior to the long multiplication by hand, when 8 figures by 8 figures are involved; and that it saves mental labour, time, and risk of error. Of course the value of the machine diminishes with the [diminishing] number of figures required to be dealt with, and is not, perhaps, marked when the figures in multiplicand or multiplier do not exceed 2. If a man has not half a mile to travel, there is not much difference between walking and going in a railway train; but when the distance is one or two hundred miles, the advantage of going by train instead of walking becomes evident.
In 1871 also Mr. Edward Sang delivered a Lecture before the Act. So. of Edinburgh, On Mechanical Aids to Calculation. The paper is full of interest, and throws light upon several of the contrivances here discussed. For instance, he gives a simple illustration of the first element in machine-counting:
By arranging a second wheel for tens, and so placing it as that at each turn of the units wheel it shall be moved one step, we are enabled to count to 100; and by continuing this arrangement, we can carry the numerations to any desired extent. Such is the construction of ordinary counting machinery. In some of these machines the movements are continuously connected by means of toothed wheels, as in the cases of clockwork and of the ordinary gas meter index. In others the motions are by jerks; the tens wheel remaining at rest until the units are passing from 9 to o, at which time the tens are advanced one step; and the indication changes say from 69 to 70. Some of these latter machines are so contrived as that the wheels are locked until the carrying takes place. Such are the machines used for regis. the number of operations performed by the bank-note printing press. From their very construction continuously connected machines are always locked. Such counting machines are of great use in many situations, as at loading and landing wharves, in large warehouses, at turnpike gates, and in general wherever extensive tale has to be made, or wherever a check is required upon operations. They receive various forms according to the purpose for which they are intended; and by modifications and extensions, they become the calculating machines to which our attention is principally to be directed. He speaks well of Thomas's machine. Thus, when noting the time required for operations under certain mechanical methods, he continues :
The contrivance for economizing that time is exceedingly ingenious in Thomas's machine. He causes all the cylinders to be actuated at once by the working handle, so that all the additions go on nearly at once. This however would cause confusion in the carrying. He therefore makes the index wheels entirely independent of each other, and arranges the carrying in another way. On each index wheel there is fixed a stud to come in contact with a lever whenever the indication passes from 9 to 0, and so to push this lever aside; and a detent is provided to keep this lever back after the stud has passed onwards. This lever brings into action the carrier fixed on the axis, and this carrier only acts after the add. by the bars has been completed. Thus by arranging the cylinders one tooth in arrear at each step, the carrying from one rank to another is completely effected, and the whole add. completed by one turn of the handle. By help of this machine, then, we can perform add.; and by turning the handle repeatedly, we can perform successive add., and so form an equi-different progression.
Mr. Sang takes a wide range of the subject. We have already quoted his obs. in several parts of this paper; we cannot follow him further. We are disappointed with some of his conclusions. He says, indeed :
The great benefit to be conferred upon us by calculating machines has always been looked for in the compilation of T.; and that benefit is expected in two ways, one in rapidity, and one in certainty of operation.
But he concludes his learned paper as follows:
Thus, on the whole, arithmeticians have not much to expect from the aid of calculating machines. A few T., otherwise very easily made, comprise the whole extent of our expected benefits; and we must fall back upon the wholesome truth that we cannot delegate our intellectual functions, and say to a machine, to a formula, to a rule, or to a dogma, I am too lazy to think; do, please, think for me. The Editor of the Assu. Mag. [in vol. xvi. of which the Lecture is printed] says, by way of appended note, that he thinks Mr. Sang overlooks the advantage of the arithmometer, "when used for a very long series of calculations," viz. that the work is almost entirely mechanical, "and in consequence much less fatiguing, after a moderate degree of use, than direct calculation, which requires a greater mental strain." We go a step further, and express a belief that the "arithmometer," clever as it is admitted to be by all familiar with its use, has not attained the ultima thule of our calculating appliances. We believe those great mechanical powers, foreshadowed and demonstrated by Babbage, will be yet brought into practical use; and it is in that view, and in that belief, that we have gone so fully into the details of his contrivances in the present art.
[Since the earlier part of the preceding art. has been printed we have been informed, by Dr. Farr, that Col. Babbage has now completed another portion of his late father's Analytical Engine; so that after all there seems a prospect of realizing that which many scientific men, both in Europe and America, had long regarded as almost hopeless. More exact information on this point will very soon be made public. We also learn from Mr. Edward Sang, of Edin., that the full description of Mr. John Sang's Platometer appears in the 4th vol. of the Trans. of the Royal Scottish So. of Arts; and that of Mr. Beverley's in the 7th vol. of same Trans. In vol. iv. of these Trans. there is a description of another Platometer by Mr. James Clerk Maxwell; but this was never constructed.
Mr. R. P. Hardy announces some Valuation T., deduced from Experience T. No. 2, "calculated by means of the Arithmometer of M. Thomas de Colmar" (Laytons).] CALCULATION.-From the Latin, Calculus, a small pebble—the Romans having frequently made use of pebbles in casting up accounts. [CALCULUS.]
CALCULATION, AIDS TO.-In a paper, by Mr. W. T. Thomson, on Decimal Numeration and Decimal Coinage, which was read before the Inst. of Act. in 1854, and is printed in