*Translated into English by Brian Stone, Australia, 2007

"Description of a calculating machine named arithmometer,
the invention of Mr le chevalier Thomas, de Colmar "

by Mr Hoyau.

"It is through the quest to replace skill or intelligence by mechanical processes that mankind has succeeded in perfecting the useful arts, increasing the volume of goods produced, reducing their price, and spreading over all classes of society the benefits of industry; but although some manual operations have been successfully supplanted by ingenious mechanisms, it seems very difficult to substitute purely mechanical methods for reasoning based on scientific theorems and obtain identical results.

However, of all sciences created by human intelligence, the mathematical ones are those which offer the most likelihood of success, since it is by calculation that mechanisms are designed, and hence in justice those mechanisms should make available the results of calculation : moreover philosophers through all the ages have laboured to facilitate the application of mathematics to practical life,  either in simplifying scientific reasoning, or in replacing it by manual operations to give relief to the intelligence or at least the memory. Thus the Orientals used to calculate with beads threaded on parallel sticks ; the Romans used counting tokens ; as a last resort men have counted on their fingers, and all these methods aided [arithmetic] operations in greater or lesser degree ; but of all time-saving techniques, the most powerful and fruitful is, without doubt, that which follows from the invention of logarithms.

By their use may be performed operations of a higher order than those whose answers can be most easily found ; calculations which are painstaking, long, and subject to frequent errors due to the fatigue they induce, have been replaced by methods so simple that errors have become almost impossible, provided one pays the slightest attention ; finally, representation of logarithms along a line has given birth to portable instruments of admirable simplicity, so easy to use that we may rightly be astonished not to see straight and circular slide-rules spreading more rapidly in France, when throughout England there is scarcely a single construction firm in which the slide-rule is unfamiliar.

But all these methods still demand scientific understanding or only give results for specific operations : thus the logarithmic [slide-]rule can not be used for addition or subtraction.

At last Pascal endeavoured to replace the various arithmetic operations by mechanical means ; but the machine which he put together was extremely complicated, and still lacked the simplicity or promptness desirable for certain operations.

If limits could be assigned to our intellectual faculties, it would appear that so many methods [already] discovered with the aim of calculating mechanically have exhausted this genre of research, and that there is no more left to be done now that the famous and wise in all countries have busied themselves with this goal.

However M. le chevalier Thomas, de Colmar, has succeeded in overcoming all the difficulties, putting together a machine which he demonstrated to the Société d’Encouragement [the BSEIN] last February, with which the four arithmetic operations can be performed : it is with this ingenious dicovery that we shall now concern ourselves.

Before going into details about the mechanism, we shall recall some mathematical principles on which its construction is based :

1° Multiplication is a shortened form of adding one quantity to itself

2° Division is a shortened form of subtraction for the purpose of finding how many times one quantity is contained in another. "

" The arithmometer or calculating machine of M. Thomas is built up of two plates A, B (Fig. 1), held apart by four pillars C, and providing pivot points for a certain number of gear wheels and drums or cylinders, which constitute the mechanism.

The first cylinder D carries nine notches positioned in a spiral around its circumference, increasing uniformly in length from each one to the next ; they appear almost like a stairway with very short steps, wrapped around a column. One end of this cylinder carries a barrel E containing a strong spring which always tends to turn it in the direction marked by the arrow. The second cylinder F rotates freely on its axis ; at one end it has an integrally made pinion G, of 27 teeth, and at the other a ratchet wheel H, which is visible, in plan, detached, Fig. 9 : its arbor I has a gear wheel K of 54 teeth. The latter wheel drives the mechanism ; it carries a pin r’, Fig. 4, which arrests its motion. The cylinder F communicates its movement to cylinder D with the aid of the pinion G, which is engaged with gear wheel L. This wheel of 81 teeth is nothing but a return [reaction?] wheel, with a pinion M, of 27 teeth, on its arbor, in engagement with the gear wheel N,  of 81 teeth, mounted on the arbor of the first cylinder D.

At the top of cylinder F is positioned an axle O, of square cross-section for its whole length : it carries at one end a fixed lever arm P, which we shall call the ' end stop ' . On this same axle slides a ring Q, in which is machined a groove [lit. throat] like that of a pulley ; this ring is provided with a lever arm R, the tip of which reaches to the bottom of the notches of cylinder D. The groove of the ring Q is there to receive a fork T which can move it along for various distances. Near the left-hand corner of Fig. 4 may be seen a small stop U, against which the pawl V comes and presses, and which is mounted on wheel N. The purpose of this stop is to hold back cylinder D and exercise control over the spring enclosed in barrel E.

In the absence of this restraint, the spring would unwind completely, and the various parts of the mechanism would not return to convenient positions. The end of stop U is hinged, in order to allow the pin V  to pass, when the spring needs to be wound while the machine is assembled.

All that part of the mechanism which we have just described is in a sense independent of the calculating part ; it provides it with motive power and controls the number of turns neccessary for cylinder F, in order to carry out the desired operation.

The remainder of the mechanism is composed of three systems which are entirely similar to one another, and could be more numerous if desired. Each of these systems represents a range of figures [decimal digit], such that with the machine as drawn, Fig.6, results can only be obtained for which the factors have three or fewer digits. We need only describe one of these systems in order to make all the others understood.

The chief component is a cylinder X which is grooved over half of its circumference in such a way as to form 27 triangular teeth. It may be seen in plan and in elevation, Fig. 7 and 8. The longest tooth, which is a single tooth, extends from one end of the cylinder to the other ; the others are cut in pairs, to lengths which form one ninth, two-ninths, three-ninths,  etc., of the length of the cylinder. At one extremity of the arbor of the cylinder is fixed a gear wheel W having the same number of teeth as the wheel K, and receiving its movement from the latter via the intermediate gear Y, which also has 54 teeth. At the other extremity of the cylinder X there are two small lever arms, of which one, Z, has a pointed end, and the other, a, has a short inclined plane [ramp] exposed at its end. To the upper right of the cylinder is shown a square-section shaft b, which carries three toothed wheels, of which two are movable : one of these, c, has a square centre hole through which the shaft passes [sliding on the shaft] ; it is made as a unit with a small pulley d which accepts a fork e (Fig. 4), which serves to move it to the required position on the cylinder ; another of these wheels f, constructed the same as the first, has only a very small movement, under the action of the levers Z and a fixed on the arbor of cylinder X ; the third is a conical wheel [bevel gear] j, with 20 teeth, fixed at the extremity of shaft b [and not moving relative to it].

Near this shaft is a round arbor g, of which one end h passes through the upper plate : this end is retained almost flush with the plate by a ratchet l’ opposing the force from the coil spring i, which would tend to lift the end of this arbor above the plate until it pressed on the bearing k. The arbor g is provided with a fork l which engages in the pulley groove of wheel f and also an arm m, which has a ramp at its end.

The three systems such as we have just described communicate among themselves by means of intermediate gear wheels n, turning on [pivot] screws fixed in the lower plate.

The first intermediate gear Y, which transmits [driving force] from cylinder F to the first cylinder X, instead of being mounted, like the gear wheels n, on a screw pivot, is fastened on an arbor p which passes through the upper plate ; the end of this arbor extending above the plate carries a gear wheel q, of 45 teeth, engaged with a pinion q’ of 15 teeth, which is fastened to an upright control wheel r : this control wheel is fixed on an arbor, of which one pivot end rotates in a hole drilled through the plate, and the other in a hole machined at the end of a leaf spring s : this section is not strictly necessary, having no other object than to regulate the force needed for the movement.

The mechanism shown, (Fig. 1), has a cover plate t, Fig. 6, through which pass the shafts of the forks c and that of fork T ; each one carries a knob u, by means of which they may be slid in the longitudinal slots v  : a small pointer u  indicates the digit value[s] to which one may wish to make position of the fork correspond. The knob at left, Fig. 6, moves fork T, Fig. 4, which belongs to the first part of the machine. Near the slot for this knob may be seen another small non-sliding knob x, fastened to a short lever arm which is part of the end stop P, Fig. 4; it is under pressure from a spring y which moves the end stop out of its stop position. The author calls this part the regulator, because its purpose is to set the first part of the machine appropriately for operation.

A third section of the machine consists of a set of dials z, Fig. 1 and 2. The plate on which they are mounted is depicted from below, Fig. 1.

Each dial has a bevel gear a', of 40 teeth, driven by one of the bevel gears j ; on these bevel gears a' are short ramps b', whose use we shall explain later. The springs c' which are visible, Fig. 1, are for the purpose of causing friction against the circumference of the dials to prevent them passing beyond where they should stop. These dials, of which one is drawn separately, Fig. 10, are divided on two concentric circles, each of which has the 10 digits around each half of its circumference : those in the outer circle are distinguished from the others by a different colour.

The first [digits] which run [increasing] from left to right are viewed through the lenses d', Fig. 6 ; the second [set], running in the other direction, are seen through the [lenses in the] openings e' : a small sliding ribbon shutter, fixed below the plate and movable by the claw t', is pierced by holes which may be made to align with lenses d' or e', so that when the holes d' are closed, the others are open and vice-versa. The dials z have knobs f' at their centres, and may be turned by these with the fingers ; finally, the plate g' rotates around a rod h', Fig. 1, carried on three supports i' : this rod passes through holes machined at the ends of two arms k fastened to plate A; all this forms a sort of hinge around which the system can turn, and even move lengthwise.

The plate shown in Fig.2 is the one which carries the control wheel and the pawls that arrest movement around arbors g : these pawls l' are held against the dial wheels by springs m' ; the two pawls n' n' are merely two springs which halt the dials in their zero position.

Around cylinder F is wound a silk ribbon o' making at least ten turns around the cylinder : this ribbon passes through a small tube p' fixed to one of the plate spacer pillars ; this directs it towards the middle of the cylinder around which it is to wrap. One of its ends is attached to a point on the cylinder, and there is a small knob s' on the other, by which it can be pulled to turn the cylinder and provide movement to the mechanism.

Such is the machine invented by M. le chevalier Thomas : it may appear very complicated, because there are many pieces which go to make it up ; but in reality it is very simple : for the same pieces are repeated in it numerous times, which was inevitable, as we shall soon see : the inventor proposes to simplify it still further.

Now it remains for us to explain the working and the outputs of the machine, and it is through the details that we shall examine that one may judge the difficulties which M. Thomas has encountered, and the ingenious means by which he has been able to overcome them.

The machine, in its structure, imitates perfectly the operations of arithmetic, and its internal movements seem to illustrate all the logical arguments needed in order to arrive at the answer.

The dials may be made quite independent of the [rest of the] mechanism, and to achieve this it is only necessary to move the knobs u so that the pointer w indicates zero : then the gear wheels c are no longer in engagement with any part of the cylinders X, and can move without any external effect. If at the same time the pointer of the knob u at left in Fig. 6 is at the point marked I, the cord can be pulled, which will cause cylinder F to make one turn, and move the pin r', Fig. 4 to the point where it encounters part P, which we called the 'end stop'. Cylinder F will in fact be unable to make more than one rotation ; in this position the lever arm R, attached to the sliding box [ring on square shaft] Q, will be pushed by the first notch of cylinder D, and, encountering part P, will halt the gear K of cylinder F : then the [spring in the] barrel E will align all the parts, and the machine will be ready to calculate.

In order to explain the working of the machine more easily, we shall indicate the movements it undergoes to carry out the four arithmetic operations.

Supposing that we wish to add 4 to 2, with all the knobs u at zero and the sliding shutter uncovering windows d’, we turn the first dial to the right until it shows the figure 2 through window d’ ; then we move the indicator of the first knob u to the right, to the mark 4, whereupon the right-hand gear wheel c Fig. 1, will have been moved to that point on the first cylinder X, also at right, where the fourth grooved portion is located, that is, to where that cylinder, if turned, will rotate gear wheel c by 8 of its 20 teeth ; if we pull cord o’, the cylinder will make one turn and it [gear wheel c] will make four tenths of a turn. Gear j meshes with a’, which is mounted on the first dial ; and since the latter gear has twice as many teeth as gear j, 40 teeth, it will make four-tenths of a half-turn, which will turn the dial through four digits : it was showing 2, and so will now indicate four units more, that is 6.

If now we should wish to add 7 to this 6, with the first dial showing 6, we move the first pointer u to graduation 7, where it will be ready to be acted upon by the part of cylinder X where the number of grooves will cause a movement of seven divisions on the dial. We pull the cord o’, the cylinder makes one turn, and the dial will turn seven divisions and indicate 3 ; but immediately after the zero passes behind the lens, the small ramp b’ of the first dial presses on the first ratchet l’ ; then the end of the first small shaft g, which is under pressure from coil spring i, escapes, being no longer retained by the ratchet, and the first gear wheel f is lifted up into the same plane as the small arm Z of the second cylinder ; the latter makes one turn, and arm Z meets gear wheel f and rotates it through two teeth, that is, by one division on the second dial : hence this dial which was showing zero will now show 1 ; together with 3 from the first dial, that will give 13, the sum of 7 and 6.

This extremely simple example shows the method used to indicate the carries, and what we are about to say now will be merely a consequence of what has gone before.

Suppose that all the dials show zero, that the left-hand pointer, which we shall call the multiplier (as engraved near the slider), is at 1, and that we set the number 237 on the three sliders at right, that is, 7 on the first, 3 on the second, and 2 on the third, as indicated by the words units, tens, and hundreds, engraved alongside each of these sliders : if we pull the cord until we feel it halted by the end stop P, we find [this first] number shown on the first three dials at the left ; if now we set the new number 394 on the three sliders, by pulling the cord this number is added to the other, and we can read through the lenses d’ the result, which is 632. As the units were added to each other, they caused one unit more than half a turn of the dial ; the pawl allowed the small shaft carrying fork l to escape, the gear wheel f was raised to the level of the arm Z of the second cylinder X, and the movement of the latter caused the former to rotate one extra division. This second dial, on the same reasoning as the first, had turned at the same time and indicated nine units more than the three tens which it initially indicated, with the effect that it made more than one half turn, and hence acted so on the gear wheel of the following cylinder as to make it indicate one unit carried ; finally the third dial moved on through 3 units, which, with the 2 it was already showing, made 5, and moved on one further division to indicate the carry out from the two preceding digit positions.

Taking notice of the manner in which the cylinders X are grooved, we see that only one half of their circumference has the grooves, so that the gears with which they engage remain stationary during movement through [the other] half of the circumference ; it is in just this period that the carries take effect, and before the second half circumference totally passes, ramp a’, mounted on the cylinder, returns the small shaft g to its normal position, that is, makes its end return to below the ratchet l’, which, due to its spring m’, now moves above shaft g, and returns the wheel or wheels f to where the arms Z can not reach them.

Now it will not be hard to understand multiplication : so, supposing that we have to multiply 25 by 6, we set the dials to zero, and enter 25 by moving pointers u to the figures 2 et 5 on the tens and units sliders v ; we move the multiplier pointer w to the figure 6 : then if the cord is pulled, cylinder F makes six turns, because arm R will be caught by the sixth notch of cylinder D, so that the arm will only be raised after the sixth rotation, and the stop P will not oppose rotation of gear wheel K until the sixth rotation is complete. Now, we have already seen that one turn will display on the dials the number which was entered on the sliders, and another turn will add that number to what was already displayed : thus, after the second turn, we will have twice the number 25, at the third three times the number 25, and eventually at the sixth, six times this number or 150.

Now consider the multiplication of 643 by 237 ; enter 643 on the three sliders, with the dials at zero : set the multiplier to 7, pull the cord, and the first partial product, 643 x 7 = 4501, appears on the dials : then raise plate g’, thus disengaging the dial gears d’ from those ( j ) of the mechanism ; slide this plate to the right until a little part [projecting from] under it drops into a notch, which halts it.

As a result of this re-alignment, the first dial at the right ceases to be driven by the mechanism, and the second dial [gear] meshes with the units gear : so all the tens which resulted from the first partial product remain in the dials. Noting now that the second partial product provided by the tens digit can only give [a result in] tens, we can see that if we set the multiplier pointer w to the figure 3, and pull the cord, we shall obtain not only the second partial product 643 x 3 = 1929, but also the sum of that product and the tens from the product obtained in the units multiplication, that is, [as an overall result so far,] 23791.

It can easily be seen that to obtain the hundreds product, it will be necessary to shift the plate g’ carrying dials d’ one further division to the right : then the first two dials will be made independent of the mechanism, and the hundreds dial will be acted upon by the units of the multiplicand ; so finally, setting the multiplier pointer to the figure 2, and pulling the cord, the complete product, 152391, will be displayed on the dials.

In the machine represented by the drawing, only a multiplier of three digits can be used ; but if a product of greater magnitude were required, a machine with a larger number of dials would be possible. M. Thomas is planning to construct some machines like this, with which products of factors having five digits or even more will be obtainable.

It remains for us to explain how subtraction and division are carried out : there could be nothing simpler than the method devised by the inventor ; it consists of dividing the dials in a[nother] concentric circle with the numbers increasing in the opposite direction to the others : by this arrangement, movement of the dial, instead of adding to the numbers displayed as many units as the divisions through which it has moved, takes back this number of units, and the dial displays this difference.

To subtract 4 from 6, firstly pull slider t’ and so uncover openings e’, at the same time closing openings d’ : then the part of the dial divided in the reverse sense will be seen through these openings. With all the pointers w at zero and the multiplier at 1, set the figure 6 on the first dial, move the pointer w at the right to the figure 4, pull the cord, and 2 will be seen to appear on this first dial : this effect will occasion no surprise if we consider that the first movable gear e’ has been moved to the point on the cylinder X which turns the dial through four divisions, and that the markings on this dial decrease in the reverse sense to its direction of movement, from which it follows that this dial will display 4 units less.

By analogy it will easily be concluded that if the numbers in the operation have multiple digits, it suffices to set the larger number on the dials, and the smaller in the sliders w, whereupon on pulling the cord the hundreds will be subtracted from the hundreds, the tens will be subtracted from the tens,and the units from the units. As for the carries in the operation, these will function exactly as for addition ; but the dial, instead of displaying one ten or one hundred more, will show one ten or one hundred less.

As for division, this will function in a manner which is the inverse of multiplication : to divide, say, 43627 by 329, we set the number 43627 on the dials and the divisor 329 on the sliders ; we bring the first section, 436, of the dividend, or the first partial dividend, above the 329, by so positioning the plate carrying the dials that the first two [less significant] dials are at the right, out of engagement with the mechanism : thus we modify the operation as if we wished to subtract 329 from 436. With the multiplier dial showing 1, we pull the cord, and find the difference 107 showing on the dials : if this remainder still contained the divisor, we would pull the cord as many times as necessary to make the remainder less than 329, and this number of times would be the number of units of the first digit at the left of the quotient. The number remaining would therefore be 10727 : then we move the plate g’ with the dials one division further to the left, and the figure 2 of the dividend will correspond [be aligned with] the 9 of the divisor. If we leave the multiplier at unity, it will be necessary to pull the cord as many times as 329 is contained in 1072 ; but it is visible at the first glance that this is three times : in consequence, rather than pull the cord three times, we set the multiplier to the figure 3, and when we pull the cord, the cylinder makes three turns ; this forms the product of 329 and 3, and at the same time subtracts it from 1072 : if the remainder still exceeded 329, we would reset the multiplier to unity, and pull [the cord] as many times as necessary to make the remainder smaller than 329, thus adding to 3 as many units as the times we pulled the cord ; but in the present example we find that the remainder obtained is less than 329 : so the first operation has been enough and we have obtained the digits 1 and 3 of the quotient ; finally, positioning the first dial opposite the units of the divisor, and operating as before, we obtain the units digit of the quotient.

It is easy to conclude from the above that all the problems of arithmetic can be solved with this machine ; and that it will bring, to complex calculations, rigorous exactness and great speed.

It seems to us that the invention of M. le chevalier Thomas must be numbered among those discoveries which bring honour to those who conceive of them, and are glorious for the era which produces them. " 

*Document transcribed into HTML format by Valéry Monnier, 2005
*Translated into English by Brian Stone, Australia, 2007