Gaetano Badalamenti's factorization slide rule

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Badalamenti’s factorization slide rule

Here is a very unusual device – the only mechanical calculator I’ve seen that tackles the problem of factoring a number. That is, given a number, it tells you its prime factors – the unique set of prime numbers whose product it is (unless it is itself prime, which the device then points out). The number being factorized can be up to 3199, so this tool is no danger to today’s cryptographic algorithms, which rely on the difficulty of factorizing numbers with hundreds of digits; still, it’s a cool tool!

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The calculator consists of a small rule of white plastic 16.1 x 5.4 cm, with a complicated cursor sliding along it. The cursor has a “bull’s eye” that can be moved up and down it vertically, while the cursor itself moves right and left; thus the bull’s eye can be made to point at any location on the rule. Specifically, you can make it point at any of the points on a two dimensional grid, each of which represents a single odd number (for even numbers the user is expected to first manually divide by 2 until an odd number is attained).
The calculator actually has two usable sides – one for factoring numbers from 3 to 1599 and the other for the range 1601 to 3199. You need to slide off the cursor, reverse it and slide it onto the other side to change from one range to the other.

Badalamenti's factorization slide rule, front and back

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Here’s how this works: You move the cursor until its bull’s eye is centered on the grid point representing the number to be factored. This number is indicated by the sum of a y coordinate (indicated by the arrow on the left of the cursor, and read off the scale on the left of the rule, indicating a multiple of 100) and an x coordinate (read under the black hairlines at the top or bottom of the rule). Next you examine what happens at the selected grid point. If it shows a black dot, it is a prime and you’re done. If it is crossed by one or more lines of varying slopes, you look their meaning up on the cursor; the slope of each line corresponds to a prime factor between 3 and 23, and you can see which it is by aligning the line on the grid with the red line segments on the cursor. Prime factors above 23 are indicated by tiny geometrical figures surrounding the grid point, whose legend is seen at the end of the rule: a circle means 29, a square means 31, and so on for 37, 41 and 43.
For illustration, here is an example of factoring 1023, into3x11x31:

Badalamenti's factorization slide rule - example

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Here are a couple more examples:

On the left, 1333=31x43; on the right, 513=19x3x3x3

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    As the last example shows, the rule doesn’t always show all factors explicitly – here it shows “3” once and lets you figure that it appears three times in the decomposition. At other times, say for 2279, it shows only 43 and lets you calculate by yourself the other factor, 53, which exceeds its display capability. The instructions cite this as a worthy feature – by not giving all the factors, they claim, the student is required to think about the factorization and thus learns better...
    This fascinating calculator was invented in Italy by Gaetano Roberto Badalamenti of Bergamo. The instructions say he was a mathematician who had discovered the underlying theory in 1957. Badalamenti applied for a patent in Italy in 1958, and in the US in 1959. The US patent was granted on May 8, 1962.
    The motivation behind this invention seems to have been primarily didactic: to facilitate the study of mathematics in the secondary school system. This is reflected in the instructions, and in the fact that a wall-mounted version (98 x 33 cm) was available for classroom instruction. This last was available under the name Scomposit from a small publishing house – Edizioni Beta, of Rome – that was founded in 1944 by Mr. Badalamenti and was managed by him for decades. The email address on the company’s site – which looks quite dated – is dead, and one can assume that the company (and possibly its founder) is no longer around.

The calculator comes in a vinyl sleeve along with an instructions sheet. The latter informs us that the calculator had won a gold medal at the XI international salon of inventions at Brussels in 1962 -- and that its inventor was interviewed on radio, TV, and in the international press.

Badalamenti's factorization slide rule with sleeve and instructions

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The calculator evidently had at least two pocket models as well as the classroom version (below, right). My friend Nicola Marras has a version(below, left) that is shorter than mine, because the legend has been moved to the corners of the bull’s eye cursor -- an excellent idea.

Scomposit classroom factorization slide rule
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		Exhibit provenance: eBay, from a seller in Italy. More info: - A scan of the instructions sheet. - Here is Badalamenti's US Patent. - The Scomposit page at Edizioni Beta.

Badalamenti’s factorization slide rule