A Treatise on Algebra

speaking, no axioms, since the propositions, immediately deducible from the definitions and assumptions, must be considered rather as the necessary and immediate consequences of defined operations, than the necessary and selfevident results of reasoning.

required to

sciences

149. It is only, therefore, in the sciences subordinate Axioms not to Algebra, such as Arithmetic and Arithmetical Algebra be formally and Geometry, that we must look for the application of stated in axioms: if stated formally, therefore, they must be spe- subordinate cially adapted to those sciences and the series of them to Algebra. which correspond to one science, would only correspond to another so far as the definitions from which they were deduced were the same: under such circumstances, it seems unnecessary to encumber our demonstrations by references to them, as they may in all cases be supplied by the understanding of the reader: a deficiency in form, which will be much less observable in demonstrations in most of the sciences subordinate to Algebra, than in that of Geometry, inasmuch as they are less complete in the mutual dependence of all their parts upon each other, in consequence of the more miscellaneous nature of the quantities which are considered, and the algebraical form under which they are exhibited.

CHAP. IV.

consider

ation of numerical and algebraical fractions.

ON THE APPLICATION OF ALGEBRA TO THE THEORY OF
NUMERICAL FRACTIONS.

150. THERE are some advantages attending the separate consideration of numerical and algebraic fracReasons for tions: for in the first place, the processes for the reduction the separate of numerical and compound algebraical fractions to their most simple terms, are merely connected by analogy, and are not founded upon the same principles: in the second place, there are some propositions which are intimately connected with the theory of numerical fractions, which have no application with respect to those which are algebraical in the third and last place, the demonstrations of the rules of numerical fractions, merely require the aid of that Arithmetical Algebra, the nature and use of which we have very fully considered in the preceding chapter.

The frac

tion

may

b have a

common measure.

151. The fraction where a and b are whole numbers,

be reduced may be reduced to an equivalent fraction in lower terms, when a and when a and b have a common divisor: and the process for finding the greatest common divisor or greatest common measure, as it is called, of the numerator and denominator of a fraction, or of two numbers generally, one of the most common and most useful in the arithmetic of fractions, is expressed in the following rule.

Rule for finding the greatest

common measure of

two num

bors.

152. Divide the greater of the two numbers by the less, and the last divisor by the last remainder, repeating the process until there is no remainder: the last divisor is the greatest common measure required.

the oper

153. The form of the operation, expressed in symbols, Form of may be exhibited as follows:

ation.

b) a (p

c) b (q

d) c (r
rd

The interpretation of this form of the process is very easy: bis contained in a, p times, with a remainder c; e is contained in b, q times, with a remainder d; d is contained in c, r times and there is no remainder.

We have supposed the process to terminate after three divisions: the demonstration which follows would be equally applicable, if it had proceeded to a greater number of them: for this purpose, it is convenient for us to premise the two following Lemmas,

154.

Its interpretation.

LEMMA I. If a number measure another, it Lemma. will measure any multiple of that number.

If a=cx, then ma=mcx: or a is contained c times in a, and mc times in ma.

155. LEMMA II. If a number measure each of two Lemma. others, it will measure their sum and difference.

For if a=cx and bdx, we have

a+b=cx + dx = (c+ d) x;

and a-b-cx-dx=(c-d) x:

and since c and d are whole numbers, c+d and c-d are whole numbers; and therefore a, which measures a and b, measures a + b and a-b.

Proof that

d is a measure of a and b.

Every mea-
sure of a
and b is a
measure
of d.

Proof that d is the greatest

measure of a and b.

Greatest

common

three or

156. In the first place, we shall prove that d is a measure of a and b.

Since d measures c by the units in r, it measures qc (Art. 154): it measures qc+d or b, since it measures qc and d (Art. 155): it measures b, and therefore pb: it measures pb and c, and therefore pb+c or a: it appears, therefore, that d is a measure both of a and b.

157. In the second place, we shall prove that every measure of a and b is a measure of d.

For if a number measures a and b, it measures a and pb, and therefore their difference (Art. 155) a—pb or c: it measures b and c, and therefore b and qc, and consequently b-qc or d.

158.

Since every number which measures a and b, measures d, the greatest number which measures a and b measures d: therefore d, which measures a and b, is their greatest common measure; for no number greater than d can measure d.

159. If the greatest common measure of three nummeasure of bers a, b and c be required, we must find d the greatest common measure of a and b, and then the greatest common measure of d and c is the greatest common measure of a, b and c.

more numbers.

If a be prime to b, there is no

other fraction equal

to, whose

For every common measure of a and b is a measure of d (Art. 157), and therefore the greatest common measure of d and c is the greatest common measure of a, b and c.

The same principle may easily be extended to find the greatest common measure of four or more numbers.

160. If a and b have no common measure except unity, they are said to be prime to each other, and the

fraction is in its lowest terms: for in that case there b

is no other fraction equal to equimultiples of a and b.

whose terms are not terms are

In order to prove this proposition, let us suppose