second power, or the square of A; AAA, or A3, the third power, or cube of A; AAAA, or A4, its fourth power, &c.

In relation to these powers, A is called their root. Thus, A is the second or square root of Aạ, the third or cube root of A3, the fourth root of A4, &c. In like manner, the second or square root of A is a number which, when multiplied by itself, produces A; the third or cube root of A is such a number, that if it be multiplied by itself, and the product by the same root again, the final product will be A. The square root of A is denoted by VA or At, its cube root by yA or At, its fourth root by Ał, &c.

6. The quotient arising from dividing one number by another, is denoted by writing the dividend as the numerator of a fraction,

A and the divisor as its denominator. Thus, denotes the quotient

B obtained by dividing the number represented by A, by the one represented by B, or it is the number which shows how often B is contained in A.

7. The signs =, >, < signify respectively equal to, greater than, less than. Thus A=B denotes that A is equal to B; A > B, that A is greater than B; and A<B, that A is less than B: the opening of the angle in the last two being turned towards the greater magnitude.

The principles employed in this Supplement, in addition to the axioms of the first and fifth books, are the following.

8. A fraction is multiplied by a number, by multiplying its numerator by that number. Thus, if be multiplied by 3, the product is g.

9. A fraction is divided by a number, by multiplying its denominator by that number. Thus, & is evidently twice as great as &; the one being a third of 8, and the other only a sixth of it.

io. The value of a fraction is not changed by multiplying or dividing both the numerator and denominator by the same number. Thus, is the same as 18.

11. Fractions having a common denominator, are added or subtracted by adding or subtracting their numerators, and retaining the common denominator. Thus, the sum of 11 and i is and their difference is :

12. To find the product of any number of fractions, multiply their numerators together for the numerator of the required fraction, and their denominators for its denominator. Thus, the product of f and is in In this operation, since is the same as the fifth part of 4, the fraction is multiplied by 4 (according to No. 8.), and divided by 5 (according to No. 9.).

13. To divide one fraction by another, multiply the numerator of the dividend by the denominator of the divisor, to find the numerator of the quotient; and, to find its denominator, multiply the denominator of the dividend by the numerator of the divisor.

* This rule is usually expressed briefly thus :--Invert the divisor, and proceed as in multiplication.

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Thus, if be divided by the quotient is 3? It will be seen that this operation is the same as dividing (No. 9.) the dividend by the numerator of the divisor, and multiplying (No. 8.) by the denominator. If be divided by 7, the quotient is. But, being only a ninth part of 7, the quotient obtained by dividing by s will be nine times as great as that resulting from the division by 7 : and therefore is multiplied by 9.

14. It follows from the nature of a fraction, that if it be multiplied by its denominator, the product is its numerator. Thus, five times is 3 : for means a fifth part of 3 ; and five times the fifth part of any quantity is the quantity itself.


If there be four numbers such, that the quotients obtained by dividing the first by the second, and the third by the fourth, are equal ; the first has to the second the same ratio that the third has to the fourth.




A C Let A, B, C, D be four magnitudes such that


: then A: B::C:D. For, let m and n be any whole numbers, and multiply (No.8.) the A

с fractions and

by m, and (No. 9.) divide them by n; then B D mc

Now, if mA be greater than nB, mC will also be greater nB nDo

mC than nD; for if this were not so,

would not be equal to

nB In like manner it might be shown, that if mA be equal to nB, mC will be equal to nD; and that if mA be less than nB, mC will be less than nD. But mA, mC are any like multiples wbatever of A, C; and nB, n D any whatever of B, D: and therefore (V. def. 5.) A : B:: C:D. Therefore, if there be four numbers, &c. Schol. This proposition comprehends in it proposition C. of this


1 book, being the same as it, when Ô or ö = p or p being a

р whole number.

Cor. By multiplying (Nos. 14. and 3.) the equal fractions А С

and B D'

by B and D, we obtain (V. ax. 1.) AD= BC; and it is proved in this proposition, that A:B::C:D. Therefore, if the product of two numbers be equal to that of two others, the one pair may be taken as the extremes and the other as the means of an analogy.

PROP. II. THEOR. If any four numbers be proportional, and if the first be divided by the second, and the third by the fourth, the quotients are equal.

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so that



A с Let A:B::C:D; then

B D If A and B be whole numbers, let the first and third terms be multiplied by B, and the second and fourth by A, and the products are AB, AB, BC, AD. Now, since the first and second of these are the same, the third and fourth are (V. def. 5.) equal; that is, AD = BC; and by dividing these (Nos. 4. and 6.) by B and D,

A C we find (V. ax. 2.) B =D


F If A and B be fractions, let A==

and B== (Nos. 3. and 14.) mA = E, and nB = F; the numerators and denominators E, F, m, n being whole numbers. Then (hyp.) E F

:::C: D. Multiply the first and third of these by mF, and the second and fourth by nE, and the products are EF, EF, mFC, and nED. Now, the first and second of these being the same, the third and fourth (V. def. 5.) are equal : that is, nED = mFC, or mnAD = mnBC, since E= mA, and F = nB. Hence, by dividing these (Nos. 4. and 6.) by m, n, B, and D, we get (V. ax. 2.)

B = D.

Therefore, if any four numbers, &c.
Schol. 1. This proposition comprehends proposition D. of this


1 book, being the same as it, when

= p or p being a whole

р number.

Schol. 2. From this proposition and the foregoing, it appears that if two fractions be equal, the numerator of the one is to its




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• If either A or B be a whole number, the proof is included in the second part of the demonstration given above. Thus, if A be a whole number, we have simply E= A and m= 1, and every thing will proceed as above. The proof would also be readily obtained by substituting for B as before, but retaining A unchanged.

If A and B be incommensurable, such as the numbers expressing the lengths of the diagonal and side of a square, the lengths of the diameter and circumference of a circle, &c., their ratios may be approximated as nearly as we please. Thus, the diagonal of a square is to its side, as 1% : 1, nearly ; as 147:1, more nearly; as 1404:1, still more nearly, &c. Hence, in such cases we cannot hesitate to admit the truth of the proposition, as we see that it holds with respect to numbers the ratio of which differs from that of the proposed numbers by a quantity, which may be rendered as small as we please,-smaller in fact, than any thing that can be assigned.

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denominator, as the numerator of the other to its denominator ; and that if the first and second of four proportional numbers be made the numerator and denominator of one fraction, and the third and fourth those of another, the two fractions are equal. This is the same in substance as that the two expressions, A:B::C:D,

А C and are equivalent, and may be used for one another.

B D Cor. 1. It appears in the demonstration of this proposition, that AD = BC; that is, if four numbers be proportionals, the product of the extremes is equal to the product of the means.

Cor. 2. It is evident that if A be greater than B, C must be greater than D; if equal, equal; and if less, less; as otherwise A C

and could not be equal. This is the same as Prop. A. B D

A C Cor. 3. If A:B::C:D, and consequently

by multi


mC plying these fractions by

we get

or mA:nB ::

nB nD' mC:nD; which is the same as the fourth proposition of this book.

А Cor. 4. If A be greater than B, the fraction

is evidently



C greater than

and the fraction less than : that is, of two C


B unequal numbers, the greater has a greater ratio to a third than the less has ; and a third number has a greater ratio to the less than it has to the greater ; which is the eighth proposition of this book.


B if

A is greater C

C' с

С than B; and, if be less than A

A is also greater than B;

B which is the 10th proposition of this book.

PROP. III. THEOR. If four numbers be proportionals, they are proportionals also when taken inversely.

If A:B::C:D; then, inversely, B:A::D:C.
For (Sup. 2. cor. 1.) BC = AD; and hence by dividing (Nos.

B D 4 and 6.) by A and C, we obtain

or (Sup. 2. schol. 2.) Ā=ē

A C'
B:A::D:C. Therefore, if four numbers, &c.

PROP. IV. THEOR. If four numbers be proportionals, they are also proportionals when taken alternately.

If A : B::C:D; then, alternately, A :C::B: D.


For (Sup. 2. cor. 1.) AD = BC; whence by dividing by C and

A B D we get c =D

Therefore, if four numbers, &c.

are also

PROP. V. THEOR. EQUAL numbers have the same ratio to the same number; and the same has the same ratio to equal numbers.

Let A and B be equal numbers, and C a third : then A:C:: B: C, and C:A::C:B.

A B For, A and B being equal, the fractions and

с 0 equal, or, which is the same, A:C::B:C; and, by inversion, (Sup. 3.) C:A::C: B. Therefore, equal numbers, &c.

PROP. VI. THEOR. NUMBERS which have the same ratio to the same number are equal; and those to which the same has the same ratio are equal. If A:C::B:C, or if C : A ::C:B, A is equal to B.

A B For, since A and B must be equal, or these fractions С C

C с could not be equal. In like manner, if

A and B must

A B' be equal, as otherwise the fractions would be unequal. Therefore, numbers, &c.

PROP. VII. THEOR. Ratios that are equal to the same ratio, are equal to one another. If A:B::C:D, and E:F::C:D; then A:B::E:F. A C E C

A E For, since


therefore (I. ax. 1.)


F that is, (Sup. 2. schol. 2.) A:B::E:F. Therefore, ratios, &c.

PROP. VIII. THEOR. Of numbers which are proportionals, as any one of the antecedents is to its consequent, so are all the antecedents taken together, to all the consequents. If A:B::C:D::E:F; then A:B:: A+C+E:B+D+F.

A C E Since

put each fraction equal to q, and multiply B D F

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