ELEMENTS OF GEOMETRY.

BOOK V.

"IN 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 magnitudes of any kind. The letters m, n, p, q are used to denote num

bers 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 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, A-B 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 of any kind,

R

Book V. it signifies that the magnitude is multiplied by the number. Thus, 3A 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 magnitudes 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.

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 mnA is A multiplied by the product of m into n.

=

5. The sign signifies the equality of the magnitudes denoted by the letters that stand on the opposite sides of it. Thus, 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.

6. The sign is used to signify the inequality of the magnitudes between which it is placed, and that the magnitude to which the opening of the lines is turned is greater than the other. Thus, A>B signifies that A is greater than B; and A<B signifies that A is less than B."

DEFINITIONS.

I.

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

Book V.

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.

If there be four magnitudes, and if any equimultiples what- See N. soever 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.
"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."

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

Book V.

IX.

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

two.

X.

N. When there is any number of magnitudes of the same kind,
the first is said to have to the last of them 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 com-
pounded 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 com-
pounded of the ratios of A to B, B to C, C to D; A may
also be said to have to D the ratio compounded of the ratios
which are the same with the ratios 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 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.

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. 16th of this book.

Book V.

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.

XVI.

Componendo, by composition: when there are four proportionals, and it is inferred, that the first, together with the second, is to the second, as the third, together with the fourth, is to the fourth. 18th prop. book 5.