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14

Hints toward the Invention of a Scale-Table

c. 1890

Houghton Library

[Version 1]

§1. A system of logarithms is a system of numbers corresponding, one-to-one, to natural numbers in such a way that pairs of natural numbers which are in the same ratio to one another have logarithms which differ from one another by the same amount. Thus, since

10:15 = 14:21

it follows that

log 10 − log 15 = log 14 − log 21.

Logarithms were invented by Napier, 1614.

§2. A logarithmic scale is a scale on which natural numbers are set down at distances from the origin measured by their logarithms. If we apply a piece of paper to such a scale and mark off the distance of 15 from 10 and measure this on from 14 we shall find 21; thus solving the proportion.

The logarithmic scale was invented by Edmund Gunter, 1624.

§3. All linear logarithmic scales are similar. Consequently, different systems of logarithms are only different scales of measurement along a logarithmic scale.

§4. Suppose I convert the edge of this sheet into a rude logarithmic scale, using the spaces between the lines as units of measurement. If I have no means at hand of subdividing them, except that of writing the numbers regularly, the proper subdivision of the scale may be treated in the use of it as a separate problem.

Practice with this scale will suggest several patentable inventions. You will see the scale is at the same time a table of antilogarithms, that is of numbers corresponding to given logarithms.

§5. How long must the slip of paper be for use with this scale?

§6. With how many places of figures can this scale be usefully inscribed? If too many are used the subdivisions of one space will not be in equal proportion throughout the space. Thus, if the last numbers were instead of 670, 741, 819, 905

6700190

7405684

8185466

9047342

the differences would be

705494

779782

861876

952658

and the second differences would be

74288

82094

90782.

The intervals evidently could not be subdivided proportionally.

But if not enough places were inscribed, the table would lose very much of its utility.

§7. You will notice that the first differences are very nearly (though not exactly) tenths of the means between successive pairs of numbers. This can evidently be put to use in subdividing the intervals.

Then what should be the number of spaces on the scale-table?

§8. Suppose you have the problem As 25 is to 55 so is 28 to the answer, how do you proceed? The roughest use of the scale gives 61; but how to get the next figure? We have

The second sheet (R 221:3) of the first version of “Hints toward the Invention of a Scale-Table.” Peirce used the vertical space between the lines (present but invisible in the image above) to define a unit of measurement for the three-place logarithmic scale inscribed along the left-hand margin of the sheet. (By permission of the Houghton Library, Harvard University.)

This is very nearly

This is precisely right.

Required to multiply 23 by .0434. This is

The true answer is 0.9982.

Had we inscribed the scale with an additional figure, we should have had

and the answer would have been .9981.

[Version 2]

A system of logarithms is a set of numbers in a table one for each natural number, such that different pairs of natural numbers having the same ratio correspond to pairs of logarithms having the same difference. Thus, since 14: 21 = 22: 33, log 14 − log 21 = log 22 − log 33.

A table of antilogarithms is a table in which the numbers are entered for constant intervals of the logarithms. In other words, it is a geometrical progression.

A logarithmic scale is one having numbers set down at distances from the origin measured by their logarithms, so that the operation of addition is replaced by measurement along the scale.

Logarithms were invented in 1614 by Napier, and the logarithmic scale in 1624 by Edmund Gunter. Things patentable are combinations of ideas. What is the difference between a table of antilogarithms and a logarithmic scale?

In a table of antilogarithms, what use are the logarithms anyway?

When you make a scale of anything is it not simpler to have the divisions equal even if the numbers attached are irregular (so that the measurement is speedy and precise), rather than have regular numbers at irregular distances the measurement of which requires infinite refinement?

Here make invention No 1.

Now make a model of the thing you have invented, whether you call it scale or table. To do so you want a geometrical progression from 1 to 10. For your convenience I set down several such.

Rules of logic are unfortunately not patentable and therefore the following will have to be used without protection. Rule for Invention. Take any useful thing. Make an accurate logical analysis of the good it does, and of the means whereby that is effected. Consider each element by itself and generally, and compare it with analogous processes; tabulate all possible methods of reaching the given result; consider the precise advantages and essential disadvantages of each, and select the combination which answers best in the case considered.

We will apply this rule to Invention No 1, bearing in mind the aphorism: “Anything is applicable to anything.”

Your invention, what does it do? It works the rule of three, (of which multiplication and division are special cases). Good, but take a broader view of it so as to make it as useful as possible to the persons for whom it is intended. Say, it computes a number from 3 given numbers, (including computation from 2 numbers as a special case).

How does it do this? By the transference of one part of a scale to another. The essence of this is the transference. The thing transferred only has to have its parts immovable or capable of being moved into a certain right position. No reason why it should be stretched out into a line. If 100 points can be measurably distinguished on a line, 10 000 can be so on a sheet, and 1 000 000 in a book of a hundred sheets or pages.

Here make general invention No 2.

Now as to transference. Who constantly practice accurate transference? Draughtsmen. What means do they find the most convenient and speedy?

Here make patentable application No 3.

It is to be remarked in passing that photography is the most marvellous record of transfer. Note side-patent No 4. Also when a record is not required a camera-lucida is very accurate. Note side-patent No 5. These we won’t use at present.

Old things may be patentable very properly as new inventions. For seven hundred years all the world, even the most ignorant, have been in daily use of logarithms,—nay, of a logarithmic scale,—nay, of the very system of measuring down to a unit of measurement and then using computation, which is precisely what we propose to patent in our scaletable. I mean the decimal places of the Arabic system of numeration.

In crowding our numbers into the scale-table of course we shall use the decimal places of these numbers and the spaces between them as units of the scale . This will subdivide the interval into ten parts. Patentable feature No 6.

In this way there is no difficulty in making numerical and trigonometrical scale-tables of a million divisions. It is perfectly practicable. Such a thing, equivalent to a 6-place table of logarithms would be used with extreme facility. Anybody who wanted to measure on it (merely adding and subtracting the numbers of the pages) could do so. Anybody who preferred to set down the numbers and use arithmetic (though this would be folly) could do so. Patent this No 7.

We have now to consider the management of the interpolation. Suppose we wish to calculate

where A, B, C, are the tabular numbers nearest to A + a, B + b, and C + c, respectively. We have

where and .

In the case of extensive tables where the terms on the last line must be exceedingly small, this formula may be useful. The expressions

can be calculated by a scale-table.

In other cases, this formula is inconvenient. We may then use

In this case, we may insert into the table opposite or under each number, say n, the value of

or whatever may be convenient. In other cases the fractions , etc. may be calculated in the head, when it will be convenient to use such a system of logarithms that

nearly. This system must be nearly the natural system or some system where the logs are multiples of the natural system. That is the logarithm of 10 must be nearly 2.3026 or 23.026 or some such number. But to get the advantage of the ordinary system the log of ten must be a whole number.

Here invent the most convenient system for the scale-table. Invention No 8.

I will now put a simple scale-table on the length of this paper.

This scale is carried two significant figures too far. A scale-table for 4 significant figures should have 230 divisions. One for 6 figures should have 2300 divisions, occupying 10 pages. The advantage of the scaletable will only be decidedly great when 6 places is reached. For the present table, it is quite as good as a 3-place table and only slightly inferior.