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Book IV.

PROP. XIV. PROB.

O describe a circle about a given equilateral and
equiangular pentagon.

Let ABCDE be the given equilateral and equiangular pentagon; it is required to describe a circle about it.

Bifect a the angles BCD, CDE by the straight lines, CF, a 9. I. FD, and from the point F, in which they meet, draw the straight lines FB, FA, FE to the points B, A, E. It may be demonstrated, in the same manner as in the preceding proposition, that the angles CBA, BAE, AED

E

B are bifected by the straight lines

F FB, FA, FE: and because the angle BCD is equal to the angle CDE, and that FCD is the half of the angle BCD, and CDF the half of ČDE ; the angle FCD is equal to FDC; wherefore the fide CF is equal to the 6. I. fide FD : In like manner it may be demonstrated, that FB, FA, FE are each of them equal to FC or FD: therefore the five straight lines FA, FB, FC, FD, FE are equal to one another; and the circle described from the centre F, at the distance of one of them, shall pass through the extremities of the other four, and be described about the equilateral and equiangular pentagon ABCDE. Which was to be done.

PROP. XV. PROB.

T.
O inscribe an equilateral and equiangular hex-

agon in a given circle.

Let

di. 4.

Book IV. BC in E; therefore BE, EC are, each of them, the fifteenth

part of the whole circumference ABCD: therefore if the straight lines BE, EC be drawn, and straight lines equal to them be placed d around in the whole circle, an equilateral and equiangular quindecagon will be inscribed in it. Which was to be done.

And in the same manner as was done in the pentagon, if through the points of division made by infcribing the quindecagon, fraight lines be drawn touching the circle, an equilateral and equiangular quindecagon may be described about it: And likewise, as in the pentagon, a circle may be inscribed in a given equilateral and equiangular quindecagon, and cir. cumscribed about it.

ELE

E L E M E N T S

OF

G É O M E T RY.

BOOK V.

"IN

N the demonstrations of this book there are certain hgns Book V.

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 numbers only.

2. The fign + (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 fign - (minus), written between two letters, fignifies 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.

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4. When a number, or a letter denoting a number, is written close to another letter denoting a magnitude of any kind, K

'it

Book V. it fignifies that the magnitude is multiplied by the number,

Thus, 3A fignifies three times A ; mB, m times B, or s 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 fignifies 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 fign = fignifies the equality of the magnitudes de noted by the letters that stand on the opposite sides of it; A=B fignifies that A is equal to B: A+B=C-D fignifies that the sum of A and B is equal to the excess of C above D.

6. The fign> is used to signify the inequality of the mag. nitudes between which it is placed, and that the magnitude to which the opening of the lines is turned is greater than the other. Thus AB fignifies that A is greater than B; and A<B signifies that A is less than B.”

DEFINITION $.

I.
A.

Less magnitude is said to be a part of a greater magni

tude, 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.

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. V. If there be four magnitudes, and if any equimultiples what. See N.

foever be taken of the first and third, and any equimultiples whatsoever of the second and fourth, and if, according as the multiple of the firft 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

fame 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 fixth, 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 magni-
tude 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
fecond.

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.

IX.
When three magnitudes are continual proportionals, the fe-

cond is said to be a mean proportional between the other
two.

X.

K 2

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