The Construction of the Wonderful Canon of Logarithms/Appendix

by John Napier

APPENDIX
On the Construction of another and
better kind of Logarithms, namely
one in which the Logarithm of unity is 0.

Mong the various improvements of Logarithms, the more important is that which adopts a cypher as the Logarithm of unity, and 10,000,000,000 as the Logarithm of either one tenth of unity or ten times unity. Then, these being once fixed, the Logarithms of all other numbers necessarily follow. But the methods of finding them are various, of which the first ws as follows:—

Divide the given Logarithm of a tenth, or of ten, namely 10,000,000,000, by 5 ten times successively, and thereby the following numbers it will be produced, 200000000 400000000, 400000000, 800000000, 1600000000, 3200000000, 640000, 128000, 25600, 5120, 1024. Also divide the last of these by 2, ten times successively, and there will be produced 512, 256, 128, 64, 32, 16, 8, 4, 2, 1. Moreover all these numbers are logarithms.

Thereupon let us seek for the common numbers which correspond to each of them in order. Accordingly, between a tenth and unity, or between ten and unity (adding for the purpose of calculation as many cyphers as you wish, say twelve), find four mean proportionals, or rather the least of them, by extracting the fifth root, which for ease in demonstration call A. Similarly, between A and unity, find the least of four mean proportionals, which call B. Between B and unity find four means, or the least of them, which call C. And thus proceed, by the extraction of the fifth root, dividing the interval between that last found and unity into five proportional intervals, or into four means, of all which let the fourth or least be always noted down, until you come to the tenth least mean; and let them be denoted by the letters D, E, F, G, H, I, K.

When these proportionals have been accurately computed, proceed also to find the mean proportional between K and unity, which call L. Then find the mean proportional between L. and unity, which call M. Then in like manner a mean between M and unity, which call N. In the same way, by extraction of the square root, may be formed between each last found number and unity, the rest of the intermediate proportionals, to be denoted by the letters O, P, Q, R, S, T, V.

To each of these proportionals in order corresponds its Logarithm of the first series. Whence 1 will be the Logarithm of the number V, whatever it may turn out to be, and 2 will be the Logarithm of the number T, and 4 of the number S, and 8 of the number R, 16 of the number Q, 32 of the number P, 64 of the number O, 128 of the number N, 256 of the number M, 512 of the number L, 1024 of the number K; all of which is manifest from the above construction.

From these, once computed, there may then be formed both the proportionals of other Logarithms and the Logarithms of other proportionals.

For as in statics, from weights of 1, of 2, of 4, of 8, and of other like numbers of pounds in the same proportion, every number of pounds weight, which to us now are Logarithms, may be formed by addition; so, from the proportionals V, T, S, R, &c., which correspond to them, and from others also to be formed in duplicate ratio, the proportionals corresponding to every proposed Logarithm may be formed by corresponding multiplication of them among themselves, as experience will show.

The special difficulty of this method, however, is in finding the ten proportionals to twelve places by extraction of the fifth root from sixty places, but though this method is considerably more difficult, it is correspondingly more exact for finding both the Logarithms of proportionals and the proportionals of Logarithms.

Another method for the easy construction
of the Logarithms of composite numbers, when
the Logarithms of their primes are known.

IF two numbers with known Logarithms be multiplied together, forming a third; the sum of their Logarithms will be the Logarithm of the third.

Also if one number be divided by another number, producing a third; the Logarithm of the second subtracted from the Logarithm of the first, leaves the Logarithm of the third,

If from a number raised to the second power, to the third power, to the fifth power, &c., certain other numbers be produced; from the Logarithm of the first multiplied by two, three, five, &c., the Logarithms of the others are produced.

Also if from a given number there be extracted the second, third, fifth, &c., roots; and the Logarithm of the given number be divided by two, three, five, &c., there will be produced the Logarithms of these roots.

Finally any common number being formed from other common numbers by multiplication, division, [raising to a power] or extraction [of a root]; its Logarithm is correspondingly formed from their Logarithms by addition, subtraction, multiplication, by 2, 3, &c. [or division by 2, 3, &c.]: whence the only difficulty is in finding the Logarithms of the prime numbers; and these may be found by the following general method.

For finding all Logarithms, it is necessary as the basis of the work that the Logarithms of some two common numbers be given or at least assumed; thus in the fore-going first method of construction, 0 or a cypher was assumed as the Logarithm of the common number one, and 10,000,000,000 as the Logarithm of one-tenth or of ten. These therefore being given, the Logarithm of the number 5 (which is a prime number) may be sought by the following method. Find the mean proportional between 10 and 1, namely $\scriptstyle {\frac {316227766017}{100000000000}}$, also the arithmetical mean between 10,000,000,000 and 0, namely 5,000,000,000; then jind the geometrical mean between 10 and $\scriptstyle {\frac {316227766017}{100000000000}}$, namely $\scriptstyle {\frac {562341325191}{100000000000}}$, also the arithmetical mean between 10,000,000,000 and 5,000,000,000, namely 7,500,000,000;.....

In all continuous proportionals.

AS the sum of the means and one or other of the extremes to the same extreme; so ts the difference of the extremes to the difference of the same extreme and the nearest mean.

A saving of half the Table of Loga-
rithms.

OF two ares making up a quadrant, as the sine of the greater ts to the sine of double its arc, so ts the sine of 30 degrees to the sine of the less. Whence the Logarithm of the double arc being added to the Logarithm of 30 degrees, and the Logarithm of the greater being subtracted from the sum, there remains the Logarithm of the less.

The relations of Logarithm &
their natural numbers
to each other.

[A] 1.L Et two sines and their Logarithms be given. If as many numbers equal to the less sine be multiplied together as there are units in the Logarithm of the greater; and on the other hand, as many numbers equal to the greater sine be multiplied together as there are units in the Logarithm of the less; two equal numbers will be produced, and the Logarithm of the sine so produced will be the product of the two Logarithms.

2.As the greater sine ts to the less, so ts the velocity of increase or decrease of the Logarithms at the less, to the velocity of increase or decrease of the Logarithms at the greater.

3.Two sines in duplicate, triplicate, quadruplicate, or other ratio, have their Logarithms in double, triple, quadruple, or other ratio.

4.And two sines in the ratio of one order to another order, as for instance the triplicate to the quintuplicate, or the cube to the fifth, have their Logarithms in the ratio of the indices of their orders, that is of 3 to 5.

5.If a first sine be multiplied into a second producing a third, the Logarithm of the first added to the Logarithm of the second produces the Logarithm of the third. So in division, the Logarithm of the divisor subtracted from the Logarithm of the dividend leaves the Logarithm of the quotient.

6.And if any number of equals to a first sine be multiplied together producing a second, just so many equals to the Logarithm of the first added together produce the Logarithm of the second.

7.Any desired geometrical mean between two sines has for its Logarithm the corresponding arithmetical mean between the Logarithms of the sines.

[B]

8.If a first sine divide a third as many times successively as there are units in A; and if a second sine divides the same third as many times successively as there are units in B; also if the same first divide a fourth as many times successively as there are units in C; and of the same second divide the same fourth as many times successively as there are units in D: I say that the ratio of A to B is the same as that of C to D, and as that of the Logarithm of the second to the Logarithm of the first.

[C]

9.Hence it follows that the Logarithm of any given number ts the number of places or figures which are contained in the result obtained by raising the given number to the 10,000,000,000^th power.

10.Also if the index of the power be the Logarithm of 10, the number of places, less one, in the power or multiple, will be the Logarithm of the root.

Suppose it is asked what number is the Logarithm of 2, I reply, the number of places in the result obtained by multiplying together 10,000,000,000 of the number 2.

But, you will say, the number obtained by multiplying together 10,000,000,000 of the number 2 is innumerable. I reply, still the number of places in it, which I seek, is numerable.

Therefore, with 2 as the given root, and 10,000,000,000 as the index, seek for the number of places in the multiple, and not for the multiple itself; and by our rule you will find 301029995 &c. to be the number of places sought, and the Logarithm of the number 2.

FINIS