The ternary calculating machine of Thomas Fowler
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Thomas Fowler

Balanced ternary arithmetic

Fowler's binary and ternary tables

DeMorgan's description

The reconstruction

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Fowler's machine as described by Augustus DeMorgan

Augustus DeMorgan (1806-1871), first chair of Mathematics at University College, London, cofounder of the London Mathematical Society and Fellow of the Royal Astronomical Society, was one of the most prominent mathematicians of his day. He was a prolific author and is credited with key contributions to the then nascent field of symbolic logic.

DeMorgan saw a demonstration of Fowler’s machine in 1840, and the following text written by him was the basis for the reconstruction of Fowler’s machine.


The machine consists of four essentially distinct parts. The first, second and third exhibit the multiplicand, multiplier and product, or quotient, divisor and dividend, according as the question to be worked is one of multiplication or division. The fourth is a carrying apparatus, which though at present detached, and employed to reduce the result to its simplest form after the main operation has been performed, might without much difficulty be attached to the multiplier or divisor, and work with it.

Let us now suppose a question of multiplication, both multiplier and multiplicand being exhibited in the ternary system. The multiplicand consists of nothing but a number of rods, each bearing an index, and each movable backwards and forwards. When the indices are all arranged in line, and in one particular line, this multiplicand is 000.... But if any one of the rods be advanced by a certain space forwards, the digit +1 is indicated as occupying the numeral column which that rod represents, and if it be moved the same space backwards, -1 is the digit indicated. This, which we may call the frame of the multiplicand, is thus a collection of rods, not itself connected with any machinery, but only useful as indicating the manner in which the frame of the multiplier is to act.

The multiplier is a frame movable in the direction perpendicular to the rods of the multiplicand and product, and situated between the planes of the two, in such manner that its extremity can be brought by a sliding motion over each rod of the multiplicand in succession. This multiplier consists of a number of rods in a common system, each furnished with two teeth, one at each extremity, the tooth by which it is acted on being a continuation of the rod, that by which it acts being perpendicular to the axis of the rod. The first set of teeth are dispersed/disposed so as to rest in a frame which has a slight motion round an axis; and each rod can be moved so that its teeth shall touch the frame above, on or below the axis. Those rods, then, which have their teeth on the axis do not receive motion from the frame, while the others receive motion in one direction or another, according as the teeth touch the frame above or below the axis. The perpendicular teeth at the other extremities may thus be made to move in either direction, or to remain stationery: and these last mentioned teeth act upon the rods which make up the frame of the product. This last frame precisely resembles the frame of the multiplicand, with the addition of the connecting part by which the multiplier acts upon it.

The process of multiplication is then as follows; the frame of the multiplier having been set, and also that of the multiplicand, the extremity of the multiplier frame is brought over the first rod of the multiplicand. To this extremity is attached a tooth which acts upon the rod of the multiplicand over which it comes, giving it a motion in one or the other direction, according as the slightly revolving frame of the multiplier is made to move in one or the other direction. The rule is, to move the revolving frame in such a way as to bring the rod of the multiplicand to its zero position; and this one motion multiplies the figure of the multiplicand by the whole of the multiplier, and by the action of the perpendicular teeth, exhibits the result upon the product frame. The lateral motion is then given to the whole of the multiplier apparatus, until the tooth comes upon the next figure rod of the multiplicand, and the revolving frame being then made to bring the new multiplicand rod to zero, the effect upon the product frame is that the new figure of the multiplicand is multiplied by the whole of the multiplier, and the result added to that of the preceding figure. This process is continued until the whole of the figures of the multiplicand are exhausted.

The result is then completely exhibited on the frame of the product, but not in its simplest form, For, whereas +1 or -1 should be the only digits in the final result, this intermediate result may exhibit +2 or -2, +3 or -3 etc on any rod. The carrying frame is a simple apparatus which, like the multiplier, has a lateral motion, and can be brought on any pair of consecutive rods. By one motion of the hand, it advances the left of two rods by a unit, and throws back the right hand rod by 3 units, or vice versa. Some little expertness is here necessary in making the carriages properly, with reference to the simplicity of the result: but there is no possibility of absolute error being introduced, since each process can only consist in altering a lower column by 3 units while the next column is altered in the contrary way by one unit.

The method of performing division is precisely the reverse of the preceding, and will hardly need description.


De Morgan, Augustus. "Description of a calculating machine, invented by Mr Thomas Fowler of Torrington in Devonshire." AP.23.24. London: The Royal Society, June 1840.