The Genaille-Lucas financial calculating rods

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The Genaille-Lucas financial rods

This set of eleven square rods, each about 17.5 cm long, would serve quite well to entertain a child: you could use them to build all sorts of structures. Nevertheless, it is not a toy, but a sophisticated and powerful calculation device.

Click photo to enlarge

These are Genaille-Lucas rods, which are an improvement on the much older rods invented in 1617 by John Napier, the Baron of Merchiston and the inventor of logarithms. Where “Napier’s bones” could multiply and divide but required intermediate steps of addition or subtraction to be performed by the user, Genaille’s rods give the answer directly.
These rods result from a collaboration between two talented Frenchmen. Henri Genaille (18??–1903) was a civil engineer employed by the state railroad company, who had already developed a number of mathematical devices; Édouard Lucas (1842–1891) was a mathematician who has left his mark on numbers theory (and, I was surprised to learn, has invented the famous “Towers of Hanoi” puzzle). Genaille had the idea for the principle underlying these rods while solving a problem posed by Lucas at a meeting of the French Association for the Advancement of Science; Lucas subsequently improved this idea, and the two proceeded to develop the rods into a commercial product. They also did their best at marketing, by presenting their invention to the said association and by describing it in a number of books. In his Recreations Mathematiques (Vol. 3), published posthumously in 1893, Lucas gives glowing praise to co-inventor and invention alike:

“An engineer at the State Railways in Tours, Mr. Henri Genaille, obscure yesterday, illustrious tomorrow, had the exceedingly remarkable and ingenious idea of replacing these additions with very simple drawings which allow all these partial products to be instantly read. The maneuvering of these rods is as easy as that which consists in following a path through a labyrinth, by means of indicator arrows on posts placed at the crossroads; that is to say that we learn to use these rods in a minute at most”.

He then waxes lyrical about the benefits of the system, though he does get carried away -- there can’t have been many customers interested in multiplying 20-digit numbers!

“Suppose you have two boxes of Genaille rods … You have in these two boxes the partial products of all the numbers up to twenty digits; Now, if we wanted to catalog all these results in volumes of 1000 pages at 100 lines per page, it would take to house these volumes a hundred million libraries such as the National Library, assuming that it contains 10 million volumes! This is the economy of this system”.

Genaille and Lucas marketed four boxed sets. The first was for multiplication; the second, for division; the third, the one shown here, for financial calculations; and the last was a set of the classic Napier rods. The financial set is actually a special derivative of the division set, optimized for the single task of figuring daily interest for a given initial sum and annual interest rate. Here is the box:

Genaille-Lucas financial rods - instructions in lid

Click a photo to enlarge

The box contains ten four-sided rods, each with a digit at the top that is used to assemble the number to be divided (in the financial scenario, the initial capital) and a one-sided index rod marked with annual percentage rates. To help identify the faces of the rods without turning them over, each face has at the bottom the four digits represented on this rod.

Click photo to enlarge

   Instructions for using the set are on the inside of the box’s lid; you can see them, with my English translation, here. Although comprehensive, they suffer from the lack of illustrations, which I correct with the example below.
    So -- imagine that you’re a millionaire, and you’ve invested $2,095,425 at an annual return rate of 9%. What is the daily interest?
    To find out, you assemble 7 rods to represent your initial investment (as the digits at the top), and add the index rod at the right, as in this photo. You then focus on the row marked 9% on the index rod. You start at the leftmost rod, noting the top digit on it, which is 0. You now follow the lines from rod to rod, adding digits as you go: 0–5–2–3–8–5–6. And this is your result for the daily interest: $523.856. Lucas was quite right in likening the process to “following a path through a labyrinth, by means of indicator arrows on posts placed at the crossroads”; and he was right that “we learn to use these rods in a minute at most”. What’s more, the actual calculation is pretty much “exact and instantaneous”, as stated on the box -- just compare it to long division with pencil and paper!

Genaille-Lucas financial rods - example calculation

Click photo to enlarge

You will note that the calculation in the example is actually a division by 4, which is why the index rod says 1/4 above the 9%. In fact, these rods allow division by 2, 4, 6, 8, 9 and 12 (though not 3, as erroneously stated in the instructions in the lid). The regular division rods cover all the integer divisors from 2 to 9, and this financial set covers a subset of them plus the 12. The financial connection is that the daily interest at 9% annual interest can be derived by dividing the initial sum (given in thousands) by 4; at 6%, by 6; at 4 1/2%, by 8; and so on. This is an approximation that assumes no compound interest and a business year of 360 days, which is evidently what French businesses were assuming at the time. You can check the math: it works.

And if you’re a billionaire, here is a case using the entire set of 11 rods. You can click the photo at the right and calculate your daily income from $8,693,095,624 at various interest rates. Have fun!

Of course the Genaille-Lucas rods all have the limitation that they can only multiply or divide by a single-digit number. This problem did not go unnoticed by the inventors, and they had plans to do better, as Lucas tells us in his book Récréations mathématiques (see link below):

Click photo to enlarge

“But, you will say, the products are only obtained by a single number; if one could have immediately the products by the numbers of two, three, four digits! We have foreseen the case, and I will show you another arrangement, a little larger, of the rods, but with other designs. With these, but by sliding movements parallel to the fixed rod, we can obtain all the products of ten digits by ten digits. Thus the principle is found; but the operation is still rather painful, rather delicate; however we hope to shortly show you a calculating machine, giving the products of two ten-digit numbers; this machine will be really popular, because the price will not exceed 20 fr”.

It is unfortunate that soon after writing these words Mr. Lucas died, at the age of 49, taking that wonderful plan to his grave.

		Exhibit provenance: Purchased from a fellow collector at an IM meeting. More info: - A detailed analysis of Napier and Genaille-Lucas rods by Denis Roegel. - A scan of Lucas’s book Récréations mathématiques, Volume 3. (1893). Pages 81-83 describe the Genaille rods. - A scan of the textbook Nouvelles annales de mathématiques (1885). Pages 516-517 describe the Genaille rods. - Instructions for the financial rods.

The Genaille-Lucas financial rods