LOGARITHMS. 
division of the two geometrical progres- point P sets out from a, describing a £, c <i, 
sionals belonging to the two first assumed d e, ef,fg-, in equal parts of the time ; and 
arithmetical progressionals ; and the double, let the space described by P in any given 
triple, &c. of any one of the arithmetical time, be always in the same ratio to the dis- 
progressionals will be the arithmetical tanceofPfrom c at the beginning of that 
progressional standing over the square, time, then will the right line ao decrease 
cube, &c. of that geometrical progression proportionally. 
which the assumed arithmetrical progres- 
sional stands over, as well as the one-half, 
one-third, &c. of that arithmetical progres- 
sional, will be the geometrical progres- 
sional ansvvering to the square root, cube 
root, &c. of the arithmetical progressional 
over it ; and from hence arises the following 
common, though imperfect definition of lo- 
garithms ; viz. 
That they are so many arithmetical pro- 
gressionals, answering to the same num- 
ber of geometrical ones. Whereas, if any 
one looks into the tables of logarithms, he 
will find, that these do not all run on in an 
arithmetical progression, nor the numbers 
they answer to in a geometrical one ; these 
last being themselves arithmetical progres- 
sionals, Dr. Wallis, in his history of algebra, 
calls logarithms, the indexes of the ratios of 
numbers to one another. Dr. Halley, in 
the Philosophical Transactions, Number 
216, says, they are the exponents of the 
ratios of unity to numbers. So, also 
Mr. Cotes, in his “ Harmonia Mensura- 
rum,” says, they are the numerical mea- 
sures of ratios : but all these definitions 
convey but a very confused notion of loga- 
rithms. Mr. Maclaurin, in his “ Treatise 
of Fluxions,” has explained the natural and 
genesis of logarithms, agreeably to the no- 
tion of their first inventor. Lord Neper. 
Logarithms then, and the quantities to which 
they correspond, may be supposed to be 
generated by the motion of a point : and 
if this point moves over equal spaces in 
eqdal tiroes, the line described by it in- 
creases equally. 
Again, a line decreases proportionably, 
when the point that moves over it describes 
such parts in equal times as are always 
in the same constant ratio to the lines from 
which they are subducted,, or to the dis- 
tances of that point, at the beginning of 
those lines, from a given terra in that line. 
In like manner, a line may increase propor- 
tionably, if in equal times the moving point 
describes spaces proportional to its dis- 
tances from a certain term at the beginning 
of each time. Thus, in the first case, let a c 
(Plate IX.Miscel. fig. 1 and 2. ) be to a o, c d to 
c 0 , d e to d 0 , ef to e o,fg to f o, always in 
the same ratio of Q R to Q S ; and suppose the 
In like manner, the line o a (fig. 3.) in- 
creases proportionally, if the point p, in 
equal times, describes spaces ac, cd, d e, 
/§■> so that ac is to a 0 , c d to 
CO, d e to do, &c. in a constant ratio. If we 
now suppose a point P describing the line 
A G (fig. 4) with an uniform motion, while 
the point p describes a line increasing or 
decreasing proportionally, the line A P, de- 
scribed by P, with this uniform motion, in 
the same time that oa, by increasing or de- 
creasing proportionally, becomes equal to 
0 p, is the logarithm of op. Thus A C, A D, 
A E, &c. are the logarithms of oc, o d, 
0 e, &c. respectively ; and o a is the quan- 
tity whose logarithm is supposed equal to 
nothing. 
We have here abstracted from numbers, 
that the doctrine may be the more genera! ; 
but it is plain, that if A C, A D, A E, &c. 
be supposed, 1, 2, 3, &c. in arithmetic pro- 
gression ; oc, od, oe, &c. will be in geo- 
metric progression ; and that the logarithm 
of 0 a, which may be taken for unity, is no- 
thing. 
Lord Neper, in his first scheme of loga- 
rithms, supposes, that while op increases 
or decreases proportionally, the uniform 
motion of tlie point P, by which the loga- 
rithm of op is generated, is equal to the ve- 
locity ofp at a ; that is, at the term of time 
when the logarithms begin to be generated. 
Hence logarithms, formed after this model, 
are called Neper’s Logarithms, and some- 
times Natural Logarithms. 
When a ratio is given, the point p de- 
scribes the difference of the terms of the 
ratio in the same time. When a ratio is 
duplicate of another ratio, the point p de- 
-scribes the difference of the terms in a 
double time. When a ratio is triplicate 
of another, it describes the difference of the 
terms in a triple time j and so on. Also, 
when a ratio is compounded of two or 
more ratios, the point p describes the dif- 
ference of the terms of that ratio in a time 
equal to the sum of the times, in which it 
describes the difference of the terms of 
the simple ratios of which it is compound- 
ed. And what is here said of the times 
of the motion of p when op increases pro- 
portionally, is to be applied to the spaces 
L 2 
