mathematics is, that the symbols denote the numbers, which represent the number of times that the assumed unit of measure is contained in the quantities; and that these quantities, whether lines or surfaces, or time or motion, &c., are proportional to these numbers. This proposition is rigorously proved for all quantities, for the first time, in the Additional Fifth Book in the Plane Geometry of Chambers's Educational Course; and thus the same rigorous accuracy is, by means of that proposition, and the Algebraical Principles at the end of the Geometry, introduced into the analytical branches of this science, that has so long prevailed in the geometrical department.

NOTE C.

To find the limit of the terms of a decreasing equirational series. Let a be the first term, the common ratio, and l any very small number; then a value of n may be found To find this value of n, let

such that a

n

Now, qp, and the terms of the series diminish more slowly the nearer the value of q is to that of p; let there

fore

r

1 =1+ where is a small fraction. In finding the

Р

n

P

product 7 (2)", it is evident that 12 =1+

Ρ

Р

'; and hence at each mul

r

a quantity not less than -l is added to l;

p

but the sum of these additive quantities must, for a sufficiently great value of n, exceed any given quantity; and hence for

n

that value of n, 1 () isa, or a Zl. But a

12

may represent any term of the series; and hence the error produced by assuming the last term of the series = 0, is less than any assignable quantity; that is, it is nothing.

NOTE D.

The following are some additional new theorems regarding different systems of logarithms:

Let L and L' denote logarithms in the systems whose bases are a and a', and let У be any number, and let

a'x'=y, then x'La' = Ly, or x' = Ly

1

La
1

or as x' = L'y, therefore L'y Ly.
= La'

1

'The quantity may be called the modulus of the

La'

system whose base is a' in reference to that whose base is a; or the relative modulus of the system (a') to (a);' as it is the quantity by which Ly must be multiplied to give L'y. Again, let axy, then x L'a=

or

and hence

L'y = Ly. L'a;

L'y

So that L'a is also the modulus of the system whose base is a' relatively to that whose base is a.

1

Also (509) since Ly=ly. Ta and L'y=ly · Ta

k

1

la'

la

therefore

and hence

1 La'

= L'a= =
k kk

Hence- The modulus of one system relative to another is equal to the ratio of the natural modulus of the former to that of the latter system; it is also equal to the logarithm of the base of the latter system taken in the former; or it is the reciprocal of the logarithm of the base of the former system taken in the latter system.

The logarithm of a number in one system is equal to its logarithm in another multiplied by the modulus of the former relative to the latter system.

When one system of logarithms is formed, those of another system will be obtained, by multiplying the logarithms of the former system by the modulus of the latter in reference to the former."

Let M and M' be the natural moduli of the systems whose bases are a and a', and M" the modulus of the latter system relative to the former, then is