just been said that the equations expressing these conditions may be reduced to the form

an equation involving but a single unknown quantity, which may therefore be solved as above. The method may obviously be extended to three or more unknown quantities, embraced in three or more independent equations-equations not convertible into each other; and it will become manifest, by making the elimination, that the number of independent equations must equal the number of unknown quantities to be determined.

.. From several equations of the first degree to eliminate (31) one or more quantities :

10. Make the coefficients of the quantity to be eliminated +1 in each equation;

2o. Take the differences of the equations thus formed.

Example. Given equations (a) to find x.

After a little practice it will be found unnecessary to write the equations (b) and (d), it being equally easy to pass at once from (a) to (c) and from (c) to (e). This method will generally produce the greatest amount of cancelation, and, therefore, be the most expeditious.

The exercises marked II. at the end of the book may be here introduced.

SECTION SECOND.

Exponents-Proportion-Variation.

PROPOSITION I.

A factor may be carried from one term to the other of a (32) fraction by changing the sign of its exponent.

As in subtracting coëfficients, we must be guided by the same rule, whatever may be their relative values, so, by a like extension in exponents, from (15) we must always have a": a" = am—", m-n, whether m> or < n;

Cor. 1. The rules for multiplication and division, and, con- (33) sequently, for involution and evolution, are the same for minus as for plus exponents, both being supposed integral.

.. (a) a", whatever integers p and q may be,+ or -,

and ..

=

√a3 = √(a3)1 = a2 = a (m) : 9 ̧

Cor. 2. It will be observed that any quantity affected by the (34) exponent zero [0] is equal to unity [1]; since a": am gives either ao or 1.

PROPOSITION II.

The root of a power is to be taken by dividing its expo- (35) nent by the index of the root.

For from (14) we have "a" (a")" = aTM

=

=

= a(mn) : n

which becomes a a", putting r = mn; and the rule thus established must evidently be extended, for the sake of consistency, to the case in which r is not a multiple of n—just as, when we cannot execute the division of 3 by 5 we express it as a fraction, 3, and seek appropriate rules for its management.

Cor. 1. An exponent may be reduced to a given denomi- (36) nator like any other quantity.

rn

As a is equivalent to "/a", so a must be regarded as "/a”, in order to be consistent; and we are to show that

then, raising both sides to the mth power, observing that the mth power of the mth root is the quantity itself, we have

(a+x)" = (aTM)TM, which involved again

rn

m

gives

[(@+2)'] = [(a =)"];
[(a+x)"]=

or (14)

(a+x)TM = (aTM)TM

= aTM ;

therefore by evolution, a + x = a, or x = 0;

and the demonstration is finished, since (33) m, n, r may be either plus or minus.

Cor. 2. All real exponents, whether plus or minus, whole (37) or fractional, are to be managed by the same rules; addition corresponding to multiplication, subtraction to division, multiplication to involution, and division to evolution.

Let a, a be any fractional powers, the exponents reduced to the same denominator, being either + or: then will

then (a+x)"+" = (a" • a ̄)'= (a ̄ˆ• a) · (a7 • a ̄) · (a7• a2) ... [1]

α

Note. It follows that am may be regarded either as the mth root

of the nth power of a,[(a")"], or as the nth power of the mth root

of a,[(a)].

Def. 1. If arc, r is called the ratio of a to c, whatever r may be, whole or fractional, positive or negative, whether capable of being exactly expressed in rational terms or not. Thus, the ratio of 6 to 3 is 2, of 3 to 6 is; of 2 to 3 is + TOO + 1000 +..., of 5.1 to 5 is

-

1.

Def. 2. If of four quantities, as a, c, a', c', a has the same ratio to c that a' has to c',

then, a, c; a', c', are said to be proportional, or a is said to be to c as a' to c'; and we write

Def. 3. We call a, c, belonging to the same ratio, homologous* terms, while a, a', as well as c, c', are denominated analogous terms; a, c, constitute the first, a', c' the second couplet. Inverting the order of (1°), (2°),

=rd },

from a = rc we Sa' arc' have a = rc

}

a' = rc'

c'

=

1

T

then ca: c': a'.

}

(40)

Dividing (10) by (2°), we have a : a' = rc : rc' c: c'; whence, if the ratio of a to a' be r', that of c to c' will be the same;

. from a = rc ? we { a=r'a', } or {then a :a::c:c,

arc' have c=r'c', S

.. (30) and (4°)

a:c:: a': c',
a'

} (50)

cca: a', (60) a' acc. (70)

Whence, by comparing (10) and (2°), (3°), (4°), (5°), (6°), (7°), there results,

* 'Opòs = homos

like, Móyos = logos = ratio=comparison.