Numbers of a mathematic "sounding board" for approximate multiplication

Question

Totten's "Strategos" Volume 2, page 16, plate VIII contains a depiction of a "sounding board for approximate multiplication".

This is made from a disk with a pointer that is spun around and "clicks" as it passes the numbers on the outer edge. Armchairdragoons cites Totten when describing the function:

• Suppose, for instance, the arrow pointed at 112; to multiply that by 2, click forward 6 pins and the arrow will be found pointing to 224.
• Conversely, to divide 112 by 2, click backward 6 pins, and the index will be found opposite 56…”

Sadly, the google-scan does not have a legible outer rim everywhere. But the click-advancement in the book is legible:

1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	30	35	40	45	50	60	70	80	90	100
0	6	9	12	14	16	17	18	19	20	21	22	22	23	23	24	24	25	25	26	26	27	27	28	28	30	31	32	33	34	36	37	28	39	40

The main problem with the depicted tool is, that, not all numbers are legible in the circle of 106 entries. 70 are larger than 1, 35 are smaller than 1. For convenience, I would put the value 1 (which is also at the top of the depicted tool) at index 0, making a natural transition. That turns the multipliers above to a good degree onto the same indexes where they are multiples of: 2 is, as expected, at index 6, 3 at 9, 4 at 12 and so on. However, the 13 is absent from the depicted dial, as are the 15, 17, and 19. 20 is present, but after that, the numbers are too blurry. Index 40 appears to be 100, and 60 seems to be 1000.

Allegedly the numbers are based on logarithm to one another, as von Totten described in Volume 1, page 168-169. Do note that no special typeface was used for formulas:

• The series of numbers arranged upon the inner arc is derived from the equation y = (2 log. x)10, in which x = any number used as a factor or a divisor, and y = the corresponding number of pins.
• Thus, let x = 2, then the log. x = taken to two places = 30; 2 log. x = 60; and y =(2 log. x) 10 = 6. Therefore, to multiply by 2 upon such a board click forward the pointer 6 pins.

The problem now is: what are those numbers, and how to calculate them properly, so one could re-create the tool?

Answer 1

I can’t read the formulas in your post (please consult the MathJax basic tutorial and quick reference), but from the positions of the powers of $10$ I suspect that what they’re saying is that the number at index $k$ is approximately $10^{k/20}$, or equivalently that the index of $n$ is approximately $20\cdot\log_{10}n$.

Here’s a table of the numbers $\ge1$. The first column is the index $k$; the second column is $10^{k/20}$ to two decimal places; the third column is $10^{k/20}$ rounded to the nearest integer; and the last column is what it seems the image shows, where this differs from the previous column. I don’t know why some of the numbers seem to be $1$ less than the third column predicts.

\begin{array}{crrl} 0&1.00&1\\ 1&1.12&1&1\frac1{10}\\ 2&1.26&1&1\frac3{10}\\ 3&1.41&1&1\frac25\\ 4&1.58&2&1\frac35\\ 5&1.78&2&1\frac45\\ 6&2.00&2\\ 7&2.24&2&2\frac14\\ 8&2.51&3&2\frac12\\ 9&2.82&3&\\ 10&3.16&3&3\frac16\\ 11&3.55&4&3\frac12\\ 12&3.98&4\\ 13&4.47&4&4\frac12\\ 14&5.01&5\\ 15&5.62&6&5\frac12\\ 16&6.31&6\\ 17&7.08&7\\ 18&7.94&8\\ 19&8.91&9\\ 20&10.00&10\\ 21&11.22&11\\ 22&12.59&13&12\\ 23&14.13&14\\ 24&15.85&16\\ 25&17.78&18\\ 26&19.95&20\\ 27&22.39&22\\ 28&25.12&25\\ 29&28.18&28\\ 30&31.62&32\\ 31&35.48&35\\ 32&39.81&40\\ 33&44.67&45\\ 34&50.12&50\\ 35&56.23&56&\\ 36&63.10&63&\\ 37&70.79&71&\\ 38&79.43&79&\\ 39&89.13&89&\\ 40&100.00&100&\\ 41&112.20&112&\\ 42&125.89&126&\\ 43&141.25&141&\\ 44&158.49&158&\\ 45&177.83&178&\\ 46&199.53&200&\\ 47&223.87&224&\\ 48&251.19&251&\\ 49&281.84&282&\\ 50&316.23&316&\\ 51&354.81&355&\\ 52&398.11&398&\\ 53&446.68&447&\\ 54&501.19&501&\\ 55&562.34&562&\\ 56&630.96&631&\\ 57&707.95&708&\\ 58&794.33&794&\\ 59&891.25&891&\\ 60&1000.00&1000&\\ 61&1122.02&1122&1121\\ 62&1258.93&1259&1258\\ 63&1412.54&1413&\\ 64&1584.89&1585&1584\\ 65&1778.28&1778&\\ 66&1995.26&1995&\\ 67&2238.72&2239&2238\\ 68&2511.89&2512&2511\\ 69&2818.38&2818&\\ 70&3162.28&3162&\\ \end{array}

The numbers with negative index are the reciprocals of the numbers with the corresponding positive index, e.g. for $k=-10$, $\left(3\frac16\right)^{-1}=\frac6{19}$.

Answer 2

Using the example in the question ...

$112 \times 2 = 112 \oplus 6$ clicks

Our job is to find out what $\oplus$ is. $1$ click = $112 \div 6 = 18 \frac{2}{3}$. This looks like a dead end.

Since multiplication has been converted to addition, some form of logarithmic function is involved.

$a \times b$, if $a = 10^n$ and $b = 10^m$, is equivalent to $10^{n + m}$. $\log (a \times b) = n + m$. A good design would be if we start at $a$ and translate (slide rule) to $b$ i.e. $a \times b$, the pointer should point to $a \times b$ (the initial position of the pointer = $10^n = a$, slide by $m$ i.e. $m + n$, and the final position of the pointer = $10^{n + m} = ab$). The scale the pointer is moving should be logarithmic/exponential, oui? For $a \times b = 1 \times 10$ should be $+1$ for $10^{0 + 1}$.

This particular circular, mechanical calculator requires that we look up the "function conversion" e.g. $\times 2$ = $6$ clicks, in a table (as provided in the OP). Bad design.