SAM Reyer invented, at what time I have not been able to learn, a kind of fexagenal rods in imitation of Napier's, by which fexagenary arithmetic is eafily performed *.

I have an arithmetical machine which came into my poffeffion from my uncle George Lewis Erfkine who, though born deaf, by the affif tance of the learned Henry Baker of the Royal Society at London, acquired not only the use of speech and the learned languages but a deep acquaintance with useful literature. This machine confifts of a small fquare box furnished with fix cylinders moveable round their axes. Upon each of thefe cylinders, which are only Napier's rods, are engraven the ten digits, and their multiples. From a perpetual almanac on the out fide of the box, it would appear that this machine was conftructed in the year 1679.

*See Chamber's Diction. Article Arithmetic.

SECTION

SECTION III.

NAPIER'S THEORY OF THE LOGARITHMS * : NEWTON'S IDEAS OF FLUXIONS, BORROWED FROM NAPIER.

I Shall now proceed to unfold the Logarithms, the discovery of which has justly entitled Napier to the name of the greatest Mathematician of his Country. Let two points, the one in N, and the other in L, (Fig. XI.) having at first a similar velocity, move along the indefinite ftraight lines CND and KL ▲; the first increasing its velocity or diminishing it according to its distance from a fixed point C, and the second preferving its velocity without augmentation or diminution. Let the former, in a certain time, arrive at any point N' or n', and the latter in the fame time at the point L'or l': the space L L' or Ll' described by the fecond moveable point is faid to be the Logarithm of the distance CN' or Cn' of the first from the fixed point C.

મેં

1. THE Logarithm of CN or unity is zero: for the first moveable point not having left N, the fecond has had no time to describe any fpace.

2.

The term Logarithm was first used by Napier after the publication of the Canon in which he afes the term of numerus artificialis.

2. THE Logarithms of the terms of a geometrical feries are in arithmetical progreffion: for let N N', N'N", N'N'", &c. or Nn', n'n", n'n", &c. be continual proportionals, they will be described by the first moveable in equal times, and the equal spaces LL', L'L” L′′L”, &c. or L1', 1'1" 1"1", &c. will be defcribed by the fecond moveable in the fame times. Now it is easily demonftrated that CN, CN', CN", &c. or Cn, Cn' Cn", &c. are in geometrical progreffion, and it is evident that their refpective logarithms o, LL', LL”, &c. or o, L L',

2 L L' &c. and o, L1, LI", &c. or o, Ll', 2 LI', &c. are in arithmetical progreffion.

3. THE logarithms of quantities lefs than CN are negative, if those of quantities greater than CN are pofitive; and converfely: for if Cn" Cn', CN CN', CN" are continual proportionals, in order that their logarithms 2 LI', LI', 0, LL', 2 LL', &c. may be in arithmetical progreffion it is neceffary that the terms on different fides of zero fhould have oppofite figns. Hence,

4. THE logarithm of any quantity is the fame with that of its reciprocal, the fign excepted.

5. THE number of fyftems of logarithms is infinite: for the ratio of CN to CN' and L L' are indeterminate.

6. THE logarithms of any one fyftem, are to the correspondent ones of any other, as the value of LL' in the first fyftem, is to its value in

the

the fecond. From the 2d propofition the four following, expreffed in the language of arithmetic, are easily deduced.

7. THE logarithm of a product is equal to the fum of the logarithms of its factors. Thus the logarithm of CN'X CN" is LL'+ LL"=LL": for CN X CN"= CN””.

8. THE logarithm of a quotient is equal to the difference of the loCN"" garithms of the divifor and dividend. Thus the logarithm of is CN' CN"" =CN". CN'

LL"-LL' LL": for

9. THE logarithm of the power of a quantity is equal to the product of the logarithm of that quantity by the index of its power. Thus the logarithm of CN' is 3 LL'=LL"": for CN=CN".

3

-3

10. THE logarithm of the root of a quantity is equal to the quotient of the logarithm of that quantity by the index of its root. Thus the logarithm of /CN"" is LL': for CN"=CN'.

FROM the 7th and 8th propofitions the two following are evident.

II. THE logarithm of an extreme or mean term of a geometrical proportion, is equal to the difference of the fum of the logarithms of the means or extremes and the logarithm of the other extreme or mean.

12. If the logarithms of all the primary numbers are known, those of all the compofite numbers may be found by fimple addition; and if all the latter are known, all the former may be known by fimple fubtraction.

FROM the 2nd or the 9th and 10th propofitions.

13. The logarithms may be thus defined, Numerorum proportionalium æquidifferentes comites; or more properly (as their name, hoywv àgilμóo, imports) Numeri rationem exponentes; because they denote the rank, order, or distance, with regard to unity, of every number in a feries of continued proportionals of an indefinite number of terms.

14. THE logarithm Ll' of any quantity Cn' is greater than the difference Nn' between CN or unity and that quantity, and less than that difference, increased in the proportion of CN to the faid quantity: for the velocity of the second moveable defcribing Ll' being greater than that of the first describing Nn' during the fame time, Ll' is greater than Nn' or CN-C'; and the velocity with which NN' is described, being greater than that with which Ll' is defcribed, in an equal time, Ll' is lefs than NN' or CN-CN or [fince Cn': CN:: CN'], [CN-Cn']X CN

Cn'

Hence,

15. Ir a quantity Cn' differs infinitely little from CN or unity, its logarithm Ll' will be equal to

[CN+Cn'][CN-Cn] the arith

2 Cn'

metical,