The Arnault-Paineau "Calculateur a disque mobile" slide rule

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Pretty as a picture!
The Arnault-Paineau calculator

When this one arrived, I thought the wooden frame must’ve been added by a previous collector who had wanted to hang the calculator on a wall for decoration. But I soon learned that the frame is original, and must’ve been included to provide the thin device some sturdiness – for protection and to facilitate its handling by the user. That settled, I did proceed to hang it on a free spot on my den’s wall – I’ve run out of drawer space, and besides, the thing certainly is pretty as a picture!

Click photo to enlarge

    This calculator has no maker name or model on it, but some inquiry placed it as the “Calculateur a Disque Mobile” – calculator with a moveable disk – invented in France by Jules Arnault and Louis Paineau and patented by them in 1921. The manual bears a copyright by Mathieu et Lefevre, Montrouge (a suburb of Paris), the firm that fabricated the device.
    The calculator, whose frame measures 26.5 cm on a side, is a circular slide rule assembled on a square metal plate. As such, it has a disk with various scales – the “Disque Mobile” – that rotates on a pivot; a stationary plate bearing a set of concentric fixed scales surrounding the disk; and a radial celluloid cursor (with three hairlines). It is very well-built, and after a century still looks great. The sturdy metal plate and disk look like aluminum, but aren’t – a test with a magnet shows them to be ferrous, probably a steel alloy plated with a silvery-white coating. This coating gives the lithographed black scales an excellent contrast usually missing in metal rules. The cursor is made of celluloid that has yellowed with time as celluloid is wont to do, but hasn’t cracked or shrunk. To help moving them, the cursor and disk have knurled brass knobs attached (the flexible cursor can be moved over the knobs on the disk by pulling it up away from the surface). The only imperfection in my exemplar is that the frame had warped a bit, so it doesn’t lay quite flat on the table as it had when it was made.
    At first glance this looks like your usual circular slide rule, but when you look more closely you notice the differences. The most unusual design feature is that the two juxtaposed scales at the interface between disk and the plate, named C and D, are not identical (as they are on a standard slide rule). They are both one-cycle logarithmic scales, but they run in opposite directions – the one on the disk increments clockwise and the other counter-clockwise. They are, in effect, equivalent to the C and CI scales of a standard slide rule – but here they slide next to each other, a fact that is the key to some rapid calculation techniques unique to this calculator. This unusual setup provides the ability to multiply two numbers with one movement of the disk, and to simultaneously multiply or divide the product by any third number (see example below).
    The calculator has seven scales, marked (from the center outward) with the letters A through G, which serve the following functions:

Scales of the Arnault-Paineau calculator

Click photo to enlarge

C and D are the main single-cycle logarithmic scales, used together in multiplication and division. As I said, they run in opposite directions: C runs clockwise and D counter-clockwise. Each scale has at its origin (the 1.0 point) a black arrow pointing at the other scale. Each also has two additional arrows, marked “Circonférence” and “Arc”.
A and B are square roots of the numbers on scale C. The inner scale A gives roots of numbers with an “even number of digits” (i.e., from 10 to 100); scale B is for the roots of 1 to 10.
E is a logarithmic scale identical to D, but runs clockwise; hence it gives the reciprocals of the numbers on D.
F is a linear scale marked 1 to 1000, and gives the logarithms of the numbers on E.
G is a scale of angles from 0 to 90 degrees, marked for both sines and cosines, whose values are read on scale F (an interesting twist – on most slide rules they would be read on the logarithmic scale, E in this case).

    Operation of the calculator can be understood from the scale descriptions above, except for multiplication (and, conversely, division) with the C and D scales. To multiply two numbers, you turn the disk to align the two multiplicands with each other, one on the C scale and one on the D scale. That done, the result will be found on either of these scale opposite the arrow of the other scale. The situation is fully symmetrical – it doesn’t matter which scale has which of the two multiplicands, nor which scale you use to read the result – the arrow of either scale will point it out on the opposite scale.
    You see an example of this in the image on the left below, where the numbers 7.90 and 1.56 (indicated by the pink arrows) are juxtaposed, giving the result 12.32 (indicated by a green arrow) on the D scale, pointed at by the arrow on the C scale.
    A unique advantage of this device is that you can proceed to multiply the result by a third number (without moving the disk) by finding this number on scale E and using the cursor to read the result on scale C. The image on the right multiplies the result of the left hand image by 1.45, and the result of 17.87 is seen marked at the long green arrow.

The Arnault-Paineau calculator - example: 7.90x1.56=12.32

The Arnault-Paineau calculator - example: (7.90x1.56)x1.45=17.87

Click a photo to enlarge

    Division is executed by pointing the arrow of either the C or D scale at the dividend on the other scale, and reading the result on either scale across from the divisor on the opposite scale.
    In addition to squares and square roots, the manual points out an easy process for deriving cubes and cubic roots – for the former, for example, you take the number on the A or B scale and turn the disk so it aligns (via the cursor hairline) with the same number on the D scale; the cube is now indicated by the arrow of either C or D.
    There are of course more details. The “Circonférence” and “Arc” arrows, notably, are useful in converting diameter to circle circumference, and radius and angle to arc length, respectively. The manual is full of other handy calculations, with examples.
    So, I can conclude by saying that this device is much more than pretty; it is sophisticated, useful, and quite ingeniously designed. Kudos to MM. Arnault and Paineau!

Exhibit provenance:
eBay, from a seller in Czechoslovakia.

More info:
The man who sold me this gem had no documentation; indeed, he had mistakenly advertised the device as an “antique aviation slide rule”! Fortunately Fletcher Wallis, an antique merchant at the Portobello Road market, had a manual and was kind enough to share it with me and with you (If you’re a collector of antique tech, you will definitely enjoy Fletcher’s online store).

Here then is the instruction manual and an advertising brochure.
Here is French patent no. 516480.
An article about this device and its various models: About a French Circular Slide Rule by Arnault-Paineau, by Willy Robbrecht, Journal of the Oughtred Society, Vol. 21 No. 2, Fall 2012, pp. 52-56.
A simpler model of this device can be found on the Photocalcul site, with the scan of its manual.

Pretty as a picture!

The Arnault-Paineau calculator