ELEMENTS

OF

GEOMETRY.

IN

BOOK V.

N the demonstrations of this book there are certain Book V. signs or characters which it has been found convenient to employ.

"1. The letters A, B, C, &c. are used to denote mag"nitudes of any kind. The letters m, n, p, q, are used "to denote numbers only.

"2. The sign + (plus), written between two letters, "that denote magnitudes or numbers, signifies the sum "of those magnitudes or numbers. Thus, A + B is the "sum of the two magnitudes denoted by the letters A "and B; m + n is the sum of the numbers denoted by 6< m and n.

"3. The sign (minus), written between two letters, "signifies the excess of the magnitude denoted by the "first of these letters, which is supposed the greatest, "above that which is denoted by the other. Thus, "AB signifies the excess of the magnitude A above "the magnitude B.

"4. When a number, or a letter denoting a number, "is written close to another letter denoting a magnitude

K

Book V.

"of any kind, it signifies that the magnitude is multiplied by the number. Thus, 3 A signifies three times "A; mB, m times B, or a multiple of B by m. When "the number is intended to multiply two or more mag"nitudes that follow, it is written thus, m (A + B) "which signifies the sum of A and B taken m times; m (A—B) is m times the excess of A above B.

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"Also, when two letters that denote numbers are "written close to one another, they denote the product "of those numbers, when multiplied into one another. "Thus, mn is the product of m into n; and mn A is A multiplied by the product of m into n.

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"5. The sign signifies the equality of the magni"tudes denoted by the letters that stand on the oppo"site sides of it; A B signifies that A is equal to B, "A+B=C D signifies that the sum of A and B is equal to the excess of C above D.

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“6. The sign is used to signify that the magni"tudes between which it is placed are unequal, and that "the magnitude to which the opening of the lines is "turned is greater than the other, Thus AB signi"fies that A is greater than B; and AB signifies "that A is less than B."

DEFINITIONS.

I.

that is,

A less magnitude is said to be a part of a greater magnitude, when the less measures the greater,

when the less is contained a certain number of times exactly in the greater.

II.

A greater magnitude is said to be a multiple of a less, when the greater is measured by the less, that is, when the greater contains the less a certain number of times exactly.

III.

Ratio is a mutual relation of two magnitudes, of the same kind, to one another in respect of quantity.

IV.

Magnitudes are said to be of the same kind, when the less can be multiplied so as to exceed the greater; and it is only such magnitudes that are said to have a ratio to one another.

V.

Book V.

If there be four magnitudes, and if any equimultiples See N. whatsoever be taken of the first and third, and any equimultiples whatsoever of the second and fourth, and if, according as the multiple of the first is greater than the multiple of the second, equal to it, or less, the multiple of the third is also greater than the multiple of the fourth, equal to it, or less; then the first of the magnitudes is said to have to the second the same ratio that the third has to the fourth.

VI.

Magnitudes are said to be proportionals, when the first has the same ratio to the second that the third has to the fourth; and the third to the fourth the same ratio which the fifth has to the sixth, and so on, whatever be their number.

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“When four magnitudes, A, B, C, D are proportionals, it is usual to say that A is to B as C to D, and "to write them thus, A: B:: C: D, or thus, A: B = C: D."

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VII.

When of the equimultiples of four magnitudes, taken as in the fifth definition, the multiple of the first is greater than that of the second, but the multiple of the third is not greater than the multiple of the fourth; then the first is said to have to the second a greater ratio than the third magnitude has to the fourth; and, on the contrary, the third is said to have to the fourth a less ratio than the first has to the second.

VIII.

When there is any number of magnitudes greater than two, of which the first has to the second the same ratio that the second has to the third, and the second to K 2

Book V.

N.

the third the same ratio which the third has to the fourth, and so on, the magnitudes are said to be continual proportionals.

IX.

When three magnitudes are continual proportionals, the second is said to be a mean proportional between the

other two.

X.

When there is any number of magnitudes of the same
kind, the first is said to have to the last the ratio com-
pounded of the ratio which the first has to the second,
and of the ratio which the second has to the third, and
of the ratio which the third has to the fourth, and so
on unto the last magnitude.
For example, if A, B, C, D be four magnitudes of the
same kind, the first A is said to have to the last D,
the ratio compounded of the ratio of A to B, and of
the ratio of B to C, and of the ratio of C to D; or,
the ratio of A to D is said to be compounded of the
ratios of A to B, B to C, and C to D.

And if A: B::E: F; and B: C::G: H; and C :D ::
K: L, then, since by this definition, A has to D the
ratio compounded of the ratios of A to B, B to C, C
to D; A may also be said to have to D the ratio com-
pounded of the ratios which are the same with the
ratio of E to F, G to H, and K to L.

In like manner, the same things being supposed, if M has to N the same ratio which A has to D, then, for shortness' sake, M is said to have to N a ratio compounded of the same ratios, which compound the ratio of A to D; that is, a ratio compounded of the ratios of E to F, G to H, and K to L.

XI.

If three magnitudes are continual proportionals, the ratio of the first to the third is said to be duplicate of the ratio of the first to the second.

"Thus, if A be to B as B to C, the ratio of A to C is "said to be duplicate of the ratio of A to B. Hence, "since by the last definition, the ratio of A to C is "compounded of the ratios of A to B, and B to C, a

"ratio, which is compounded of two equal ratios, is Book V. "duplicate of either of these ratios.”

XII.

If four magnitudes are continual proportionals, the ratio
of the first to the fourth is said to be triplicate of the
ratio of the first to the second, or of the ratio of the
second to the third, &c.
"So also, if there are five continual proportionals; the
"ratio of the first to the fifth is called quadruplicate
"of the ratio of the first to the second; and so on,
according to the number of ratios. Hence, a ratio
"compounded of three equal ratios is triplicate of any
"one of those ratios; a ratio compounded of four
equal ratios quadruplicate," &c.

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XIII.

In proportionals, the antecedent terms are called homologous to one another, as also the consequents to one another.

Geometers make use of the following technical words to signify certain ways of changing either the order or magnitude of proportionals, so as that they continue still to be proportionals.

XIV.

Permutando, or alternando, by permutation, or alternately; this word is used when there are four proportionals, and it is inferred, that the first has the same ratio to the third which the second has to the fourth; or that the first is to the third as the second to the fourth See Prop. 16. of this Book.

XV.

Invertendo, by inversion: When there are four proportionals, and it is inferred, that the second is to the first as the fourth to the third. Prop. A. Book 5.