6. To represent the multiplication of complex quantities, they are included by a parenthesis. Thus, A(B+C—D) denotes that the amount of B+C-D, considered as a single quantity, is multiplied into A.

7. Ratios and analogies are expressed, by inserting points in pairs between the terms. Thus A: B denotes the ratio of A to B ; and the compound symbols A: B:: C:D, signify that the ratio of A to B is the same as that of C to D, or that A is to B as C to D.

PROP. I. THEOR.

The product of a number into the sum or dif ference of two numbers, is equal to the sum or difference of its products by those numbers.

Let A, B, and C be three numbers; the product of the sum or difference of B and C by the number A, is equal to the sum or difference of the separate products AB and AC.

For the product AB is the same as each unit contained in B repeated A times, and the product AC is the same as the units in C likewise repeated A times; whence the sum of the products AB and AC is equal to the units contained in both B and C, all repeated A times, or it is equal to the sum of the numbers B and C multiplied by A.

Again, for the same reason, the difference between the products AB and AC must be equal to the difference between the units contained in B and in C, repeated A times; that is, it must be equal to the difference between the numbers B and C multiplied by A.

Cor. 1. Hence a number which measures any two numbers, will measure also their sum and their difference.

Cor. 2. It is hence manifest, that the first part of the proposition may be extended to more numbers than two; or that AB+AC+AD+, &c.=A(B+C+D+, &c.)

PROP. II. THEOR.

The product which arises from the continued multiplication of any numbers, is the same in whatever order this operation be performed.

Let A and B be two numbers; the product AB is equal to BA.

For the product AB is the same as each unit in B added together A times, that is, the same as A itself repeated B times, or BA.

Next, let there be three numbers A, B, and C; the products ABC, ACB, BAC, BCA, CAB, and CBA are all equal.

For put D=AB or BA; then DC-CD, that is, ABC CAB, and BAC=CBA.

Again, put E=AC or CA; then EB-BE, that is, ACB=BAC, and CAB=BCA.

Lastly, put F= BC or CB; then FA=AF, that is, BCA ABC, and CBA=ACB.

And thus the several products are all mutually equal. It is also manifest, that the same mode of reasoning might be extended to the products of any multitude of numbers.

PROP. III. THEOR.

Homogeneous quantities are proportional to their like multiples or submultiples.

Let A, B be two quantities of the same kind, and pA, pB their like multiples; then A: B:: pA: pB.

For, since A and B are capable of being measured to any required degree of precision, suppose a to be the measure which A and B contain m and n times, or that A m.a and Bn.a; consequently pA=p.ma, and pB=p.na. But (V. 2.) p.mam.pa, and p.nan.pa. Wherefore a and pa are like submultiples of A and of pA, which contain them respectively m times; and these like submultiples are both contained equally, or n times, in B and in pB. Consequently (V. def. 10.) the quantities A, B, and pA, PB are proportional; and A, pA are the antecedents, and B, pB, the consequents, of the analogy.

Again, because the ratio of pA to pB is thus the same as that of A to B, which, in reference to pA and pB, are only like submultiples, it follows that homogeneous quantities are also proportional to their like submultiples.

PROP. IV. THEOR.

In proportional quantities, according as the first term is greater, equal, or less than the second, the third term is greater, equal, or less than the fourth.

:

Let A B C : D; if AB, then CD; if A=B, then C=D; or if AB, then CD.

For, if A be greater than B, then the measure or submultiple of A must be contained oftener in B, and hence the like submultiple of C will be contained oftener in D; wherefore C is greater than D.

If A be equal to B, the measure of A is contained equally in B, and hence that of C in D, or C is equal to D.

But, if A be less than B, the measure of A is not contained so often in B, and hence the measure of C is not contained so often in D, or C is less than D.

Scholium. On this proposition is grounded the mode of stating a proportion in the Rule of Three, while the arithmetical operation will be found to depend on Prop. VI.

PROP. V. THEOR.

Of four proportionals, if the first be a multiple or submultiple of the second, the third is a like multiple or submultiple of the fourth.

Let A B C : D; if A=pB, then C=pD.

For, suppose the approximate measures of A and C to be a and c, and let Amp.a, and C=mp.c. It is evident, from the hypothesis, that A=pB=mp.a, or B=m.a; but the consequents B and D must contain their measures equally (V. def. 10.), and therefore D=m.c.

=mp.c (V. 2.) p.mc=pD.

=

Again, if qA=B; then will qC=D.

Whence C

For, let A=na, and C=nc; therefore B=qA=qna= (V. 2.) nq.a, and, from the definition of proportion, D= nq.c

(V. 2.) q.ne=qC.