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### Irish Logarithms

Ideas for speeding up multiplication on mechanical calculators.

### Multiplication.

The only thing most mechanical calculators can really do is adding and subtracting.

Multiplication is achieved by repeatedly adding the multiplicand to itself, as many times as the multiplier is large. To speed up the process the multiplicand is shifted one position for each higher order of the multiplier, so that a multiplication by 345 does not need 345 additions, but only 3+4+5 (Figure 1).

The machines of Leibniz (1673) and Thomas de Colmar (1820) already worked on this principle. But this had the inconvenience that the number of additions, and the duration of the calculation, depended on the value of the multiplier.

### Direct multiplication

In 1878, Ramon Verea, a Spaniard from Cuba who had ended up in New York, proposed to put the tables of multiplication in "hardware" in order to carry out immediately the multiplication of a multiplicand with a one-digit multiplier. He made a decagonal prism with holes whose "shallowness" represented the simple products. (Figure 2) A pin probes the depth of a hole and rotates a counting wheel proportional to the depth of the hole. For each digit in a multiplicand the corresponding prism is turned so that the face representing the value of the digit is looking at the pin. The pins are set at a height that corresponds to a digit of the multiplier. Then the prisms are pushed towards the pins, and thereby move the pins over a distance required by the multiplication tables. When multiplying with a multi-digit multiplier, this process must be repeated for each digit. If the shallowness of the holes would represent directly the simple products, the deepest hole should have been 9×9 = 81 times deeper  than the shallowest hole. This required a probing accuracy that, at Verea's time, was not achieved without an inconveniently large and slow mechanism. Therefore Verea choose to represent the simple products by two holes: one hole for the units and one hole for the tens. The holes are sensed by two pins that control two successive orders in the adder. For 9×9 one pin would move one step, and add one to the units-counter, and the other pin would move 8 steps and add 8 to the tens counter. Another reason to use two holes is that the normal tens transfer mechanism  in the adder can be used: if the units wheel rotates past 9, the tens wheel should be rotated 1 step further. If a single hole was used, a transfer of 8 might occur, and the usual tens-transfer mechanisms can not do thatt.

In 1872, before Verea patented his machine, Edmund Barbour had already received a patent on a multiplier, in which the simple products are represented as short racks on large cylinders.

In 1896, the Frenchman Leon Bollée developed a multiplier by a similar principle. Instead of prisms with holes, he used sticks on a block to represent the tables (Figure 4). Bollee clearly describes in his patent the operation of his machine: after setting the multiplicand and one digit of the multiplier, first all units of the simple products are added to the counting mechanism, then the possible tens-transfers are performed, then the tens of simple products are added to the counting mechanism, which is shifted one position, and the resulting tens-transfers are performed, and finally the counting mechanism slides back. Bollée only needs one probe per resulting digit, whereas Verea needs two. It is not clear how two probes, the units-probe of one order and the tens-probe of a lower order, could simultaneously operate one and the same counting wheel.

Multiplication tables similar to Bollée's design have been successfully employed by Otto Steiger and Hans Egli in the "Millionaire" calculator. There were several other direct multipliers on the market, but most of the mechanical calculators continued multiplying by repeated addition.

### Notes

1. Ramón Verea, “Improvement in calculating machines”, US Patent 207918, September 10, 1878.
2. Shallowness = the distance between the bottom of the hole and the axis of the prism.
3. Simple products: 1×1, 1×2, … 9×8, 9×9
4. Edmund Barbour, “Improvements in Calculating Machines”, US Patent 130404, August 13, 1872.
5. Leon Bollée, “Calculating Machine”, US Patent 556720, March 17, 1896.
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