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Book IV.

See N.

2. 5. 1.

b. 32. 1.

PROP. XV. PRO В.

10 inscribe an equilateral and equiangular hexagon in a given circle.

T

Let ABCDEF be the given circle; it is required to inscribe an equilateral and equiangular hexagon in it.

Find the center G of the circle ABCDEF, and draw the diameter AGD; and from Das a center, at the distance DG describe the circle EGCH, join EG, CG and produce them to the points B, F; and join AB, BC, CD, DE, EF, FA. the hexagon ABCDEF is equilateral and equiangular.

Because G is the center of the circle ABCDEF, GE is equal to GD. and because D is the center of the circle EGCH, DE is equal to DG; wherefore GE is equal to ED, and the triangle EGD is equilateral, and therefore its three angles EGD, GDE, DEG are equal to one another, because the angles at the base of an isosceles triangle are equal a. and the three angles of a triangle are equal to two right angles; therefore the angle EGD is the third part of two right angles. in the same manner it may be demonstrated that the angle DGC is also the third part of two right angles. and because the straight line GC makes F with EB the adjacent angles EGC, CGB C. 13. 1. equal to two right angles; the remaining angle CGB is the third part of. two right angles; therefore the angles E

A

B

G

C

d. 15. 1.

EGD, DGC, CGB are equal to one
another. and to these are equal d the
vertical opposite angles BGA, AGF,
FGE. therefore the fix angles EGD,

D

DGC, CGB, BGA, AGF, FGE, are

equal to one another. but equal angles

H

e. 26. 3.

f. 29. 3.

stand upon equal circumferences; therefore the fix circumferences AB, BC, CD, DE, EF, FA are equal to one another. and equal circumferences are fubtended by equal f straight lines; therefore the fix straight lines are equal to one another, and the hexagon ABCDEF is equilateral. It is also equiangular; for since the circumference AF is equal to ED, to each of these add the circum

ference ABCD; therefore the whole circumference FABCD shall

be equal to the whole EDCBA. and the angle FED stands upon

the circumference FABCD, and the angle AFE upon EDCBA; Book IV. therefore the angle AFE is equal to FED. in the fame manner it may be demonstrated that the other angles of the hexagon ABCDEF are each of them equal to the angle AFE or FED. therefore the hexagon is equiangular. and it is equilateral, as was shewn; and it is infcribed in the given circle ABCDEF. Which was to be done. COR. From this it is manifest, that the side of the hexagon is equal to the straight line from the center, that is, to the semidiameter of the circle.

And if thro' the points A, B, C, D, E, F there be drawn straight lines touching the circle, an equilateral and equiangular hexagon shall be described about it, which may be demonstrated from what has been said of the pentagon; and likewise a circle may be infcribed in a given equilateral and equiangular hexagon, and circumscribed about it, by a method like to that used for the pentagon.

To infcril

PROP. XVI. PRO B.

10 inscribe an equilateral and equiangular quinde- See N. cagon in a given circle.

Let ABCD be the given circle; it is required to inscribe an equilateral and equiangular quindecagon in the circle ABCD.

Let AC be the fide of an equilateral triangle inscribed a in the 8. 2. 4. circle, and AB the side of an equilateral and equiangular pentagon inscribed in the same; therefore of such equal parts as the b. 11. 4. whole circumference ABCDF contains fifteen, the circumference

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tains two of the fame parts. bifect F BC in E; therefore BE, EC are, each

of them, the fifteenth part of the C

whole circumference ABCD. there

fore if the straight lines BE, EC be

A

F

C. 30. 3.

D

drawn, and straight lines equal to them be placed d around in the d. 1. 4. whole circle, an equilateral and equiangular quindecagon shall be

inscribed in it. Which was to be done.

Book IV.

And in the same manner as was done in the pentagon, if thro the points of divifion made by inscribing the quindecagon, straight lines be drawn touching the circle, an equilateral and equiangular quindecagon shall be described about it. and likewife, as in the pentagon, a circle may be inscribed in a given equilateral and equiangular quindecagon, and circumfcribed about it

Book V.

THE

ELEMENTS

OF

A

EUCLID.

BOOK V.

DEFINITIONS.

I.

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 faid to be a multiple of a less, when the greater is measured by the less, that is, when the greater con'tains the less a certain number of times exactly.'

III.

" Ratio is a mutual relation of two magnitudes of the fame kind See N. " to one another, in respect of quantity."

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Magnitudes are said to have a ratio to one another, when the less can be multiplied so as to exceed the other.

V.

The first of four magnitudes is faid to have the fame ratio to the second, which the third has to the fourth, when any equimultiples whatsoever of the first and third being taken, and any equimultiples whatsoever of the second and fourth; if the multiple of the first be less than that of the second, the multiple of the third is also less than that of the fourth; or, if the multiple of the first be equal to that of the second, the multiple of the third is also equal to that of the fourth; or, if the multiple of the first be greater

Book V.

See N.

than that of the second, the multiple of the third is also greater than that of the fourth.

VI.

Magnitudes which have the fame ratio are called proportionals. N. B. When four magnitudes are proportionals, it is usually • expressed by saying, the first is to the second, as the third to • the fourth.'

VII.

When of the equimultiples of four magnitudes (taken as in the 5th 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.

"Analogy, or proportion, is the similitude of ratios."

IX.

Proportion confists in three terms at least.

Χ.

When three magnitudes are proportionals, the first is faid to have to the third the duplicate ratio of that which it has to the fecond.

XI.

When four magnitudes are continual proportionals, the first is faid
to have to the fourth the Triplicate ratio of that which it has
to the second, and so on Quadruplicate, &c. increasing the de-
nomination still by unity, in any number of proportionals.
Definition A, to wit, of Compound ratio.

When there are any number of magnitudes of the fame kind, the
first is faid to have to the last of them the ratio compounded 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 fame kind,
the first A is faid 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 faid to be compounded
of the ratios of A to B, B to C, and C to D.

And if A has to B, the fame ratio which E has to F; and B to C,
the fame ratio that G has to H; and C to D, the fame that K

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