? I. I.

CORO L.

. Hence the Side of a Hexagon inferibedin a Circle, is equal to the Semidiameter.

2. Hence it is eafy to infcribe an equilateral Triangle ACE in a Circle.

.SCHOL. Prob.

3

You may thus make an Hexagon upon a given right Line CD. Make the equilateral Triangle CGD upon the given right Line CD, and from the Centre D defcribe a Circle thro' C and D, which fhall contain an Hexagon upon the given Line CD.

PROP. XVI. Prob. 16..

To infcribe an equilateral and equiangular Quin-. decagon in a given Circle AEBC.

H

Infcribe an equilateral Pentagon AEFGH in the given Circle; as likewife an equilateral Triangle ABC. Then fhall BF be the Side of the Quindecagon fought.

C

3

For the Arc AB is or of the Periphery, and AF isor of it. Therefore their Dif

6

ference

ference BF is of the Periphery. Whence a Quindecagon, whofe Side is BF, is equilateral But it is alfo equiangular, fince all its Angles do ftand upon equal Arches, every one of which 27.3. is of the whole Circumference. Therefore

&c.

d

SCHOLIU M.

A Circle (4,8,16, &c. by 6, 4, and 9,1. may be Ge-3,6,12, c. by 15,4, and 9,1. ometrically 5,10,20, c. by 11,4, and 9,1. divided into 15,30,60, &c. by 16,4 and 9,1.

Parts

But the Divifion of the Circumference into any given Number of equal Parts cannot be done by help of a ftreight Line and Circle; and for effecting the fame, there are feveral mechanical ways in Books of Practical Geometry.

End of the Fourth Book.

EU

EUCLID's

ELEMENTS.

1.

BOOK V.

DEFINITIONS.

Part is a leffer Magnitude in respect of a greater, when the leffer one measures the greater.

II. A Multiple is a great Magnitude in refpect of a lefs, when the lefs measures the greater. III. Ratio is a certain mutual Habitude, or Refpect of two Magnitudes of the fame kind according to Quantity.

That Quantity, in every Ratio, that is refer❜d to the other, is call'd the Antecedent of the Ratio; and that to which the other is refer'd, is call'd the Confequent of the Ratio; as in the Ratio of 6 to 4, the Antecedent is 6, and Confequent 4.

The Quantity of every Ratio is found, by dividing the Antecedent by the Confequent, as the Ratio of 12 to 5 is exprefs'd by; alfo the Quantity of the Ratio

of

of A to B (A expreffing any Line, Surface, Solid,&c:

(and B the like) is expressed by

A

B

Whence very often for brevity fake, the Quantities of Ratios are denoted thusor, or CD; that is, the Ratio of A to B is greater than the Ratio of C to D, or equal to it, or elfe lefs. And this I would have well obferv'd by every one that defigns to read what follows. IV. Proportion is a Similitude or Likeness of Ratio's.

Note, This Word would be better exprefs'd, by Proportionality, or Analogy, because by many it fignifies the fame as Ratio.

V. Magnitudes are faid to have a Ratio to one another, which being multiplied can exceed the one the ether.

the fame Ratio the firft A to the second B, and the third C to the fourth D, when the Equimu!tiples E, F of the firft A and third C compared with the Equimultiples G, H of the fecond B, and fourth D, according to any Multiplication foever, either both together (E and F,) are less than G, H, or both together equal to them, or both together greater, if those Magnitudes E, G, and F, H be taken, which answer to each other: that is, when G is less than H, E is lefs than F; when G exceeds H, E exceeds F; and when GH, E fhall be F. And this fhall always happen.

When Quantities are in the fame Ratio, for brevity fake they are exprefs'd thus; A': B:: C: D. that is, as A to B, fo is C to D. And

VII. Quantities that have the fame Ratio (A: B:: C:D) are call'd Proportionals.

VIII. But E, 30. A, 6. B, 4. B, 28. of Equimul| F, 60. C, 12. D, 9. | H, 63. tiples when E the Multiple of the first Magnitude A fhall be greater than G, the Multiple of the second B; but F the Multiple of the third C does not exceed H, that of the fourth D: then the firft A is faid to have a greater Ratio to the fecond B, than the third C to the fourth D.

C

D

If be it is not necessary from this Definition that E always exceeds G, when F is iefs than H; but that it is poffible for it to do fo.

IX. Analogy at leaft confifts of three Terms. X. When three Magnitudes A, B, C, are proportional, the firft A is faid to have to the third C, a duplicate Ratio to what it has to the fecond B; and when four Magnitudes A, B, C, D, are proportional, the firft A fhall have a triplicate Ratio to the fourth D of what it has to the fecond B; and fo always one more in order, according as the Proportion fhall be extended.

A

A

A

Duplicate Ratio is thus express'd twice. That is, the Ratio of A to C is the Duplicate of the Ratio of A to B, and triplicate Ratio'thus, D = 1, thrice. That is, the Ratio of A to D, is triplicate of the Ratio of A to B.

XI. Homologous Magnitudes, or Magnitudes of a like Ratio, are faid to be such whose Ante❤ cedents are to the Antecedents, and Confequents to the Confequents.

As if A: B::C: D; then A and C, and B, D, are call'd Homologous Magnitudes.

XII.