the third part of two right angles. Hence the whole angle EGC is equal to two thirds of two right angles; but because CG makes with EGB on the same side of it the adjacent angles EGC, CGB, these two angles are equal (i. 13) to two right angles, and one of them EGC is equal to two thirds of two right angles: therefore the other angle CGB is equal to one third of two right angles. Hence the three angles EGD, DGC, CGB are equal; and to these are equal the opposite vertical angles BGA, AGF, FGE (i. 15): therefore the six angles EGD, DGC, CGB, BGA, AGF, FGE are all equal. But in the same circle equal angles at the centre stand on equal arcs (iii. 26); therefore the six arcs AB, BC, CD, DE, EF, FA are all equal: and in the same circle equal arcs are subtended by equal straight lines (iii. 29), therefore the six straight lines AB, BC, CD, DE, EF, FA are all equal, and the hexagon ABCDEF is equilateral. Again because the arc AF is equal to the arc ED; to each of these equals add the arc ABCD: then the whole arc FABCD is equal (Ax. 2) to the whole arc EDCBA. But in the same circle the angles at the circumference standing on equal arcs are equal (iii. 27); therefore the angle AFE is equal to the angle FED. In like manner it may be shewn that each other adjacent pair of the angles of the figure are equal: therefore the six angles are all equal and the hexagon equiangular. Hence the hexagon ABCDEF is both equilateral and equiangular, and is therefore a regular hexagon; and since all its angular points A, B, C, D, E, F are in the circumference of the given circle ABC, it is inscribed (iv. Def. 3) in it. Which was to be done.

COR.-The side of a regular hexagon inscribed in a circle shall be equal to the radius of the circle.

For it was shewn in the proof of the prop that GE is equal to ED, and ED is the side of the inscribed regular hexagon, and GE the radius of the circle. Therefore the side of the in

F

E

B

D

H

C

scribed regular hexagon is equal to the radius of the circle. Which was to be proved.

OBS.-If through A, B, C, D, E, F, the angular points of the inscribed regular hexagon, there be drawn straight lines touching the circle, these shall form a regular hexagon circumscribing the circle; and also if two adjacent angles of a regular hexagon be bisected, and from the point where the bisecting line meets perpendiculars be drawn to the sides of the hexagon, then the circle described with this point as centre and any one of the perpendiculars as radius, shall be the circle inscribed in the regular hexagon.

The proof of these is precisely similar to those of Propns 12 and 13. In fact, the three Propas 12, 13, 14 may be generalized so as to apply to any regular polygon; care being taken to substitute polygon for pentagon throughout, and the propositions being in other respects proved alike. The general form of enunciation is as follows:

(1) A regular polygon may be circumscribed about a given circle by drawing straight lines touching the circle through the angular points of the regular inscribed polygon of the same number of sides.

(2) A circle may be inscribed in a regular polygon by bisecting two of its adjacent angles, drawing perpendiculars on its sides from the point where the bisecting lines meet, and describing a circle with this point as centre and any one of the perpendiculars as radius.

(3) A circle may be circumscribed about a regular polygon by bisecting two of its adjacent angles, joining the point where the bisecting lines meet with the angular points of the polygon, and describing a circle with this point as centre and any one of the joining lines as radius.

PROP. XVI. PROB.

To inscribe a regular polygon of fifteen sides, or quindecagon in a given circle.

Let ABC be the given circle. It is required to describe a regular quindecagon in the circle ABC.

Describe an equilateral and therefore an equiangular (i. 5, Cor.) triangle, and inscribe (iv. 2) in the circle ABC a triangle ACD equiangular to it; then the triangle ACD will be an equiangular triangle, and

B

E

H

F

therefore equilateral (i. 6, Cor.). Inscribe (iv. 11) also in the circle a regular pentagon ABGHF, having one of its angular points A coinciding with one of those of the triangle ACD.

Since in the same circle equal straight lines cut off equal arcs (iii. 28), the three arcs AC, CD, DA are all equal, and therefore the arc AC is a third part of the whole circumference of the circle: similarly it may be shewn that the arc AB is a fifth part of it. Hence of the fifteen equal arcs into which the whole circumference may be divided, the arc AC contains five, and the arc AB three, and consequently their difference, the arc BC, contains two. Bisect (iii. 30) the arc BC in E; therefore each of the arcs BE, EC is a fifteenth part of the whole circumference. Hence if the straight lines BE, EC be joined, and straight lines equal to them be placed (iv. 1) around in the whole circle, contiguous to one another, and in number fifteen, they will form one equilateral quindecagon inscribed in the circle ABC; and this equilateral quindecagon may be shewn to be equiangular, and therefore regular, as the equilateral pentagon was in the concluding part of the proof of Prop. XI. Hence in the given circle ABC has been inscribed a regular quindecagon. Which was to be done.

THE

ELEMENTS OF EUCLID.

BOOK V.

DEFINITIONS.

I.

A LESS magnitude is defined to be a measure or submultiple of a greater magnitude of the same kind, when the less magnitude is contained a certain number of times exactly in the greater.

66

OBS. Such a less magnitude is sometimes called a part" of the greater; but when part is thus used as equivalent to measure or submultiple, we must be careful not to confound this restricted signification of part with the general one employed in Book I. Ax. 9.

II.

A greater magnitude is defined to be a multiple of a less magnitude of the same kind, when the greater magnitude contains the less a certain number of times exactly.

OBS. When two or more greater magnitudes contain two or more less magnitudes of corresponding kinds the same number of times exactly, the former magnitudes are called "equimultiples" of the latter.

III.

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

OBS. It appears that for one magnitude to have a ratio to another, they must both be of the same kind. Although in this and the following defn Euclid lays down the conditions for two magnitudes to have a ratio to one another, he nowhere gives a mode of estimating a single ratio; he defines only when one ratio is the same as (Def. 5), greater than, or less than (Def. 7) another ratio.

IV.

Two unequal magnitudes of the same kind are said to have a ratio to one another, provided that the less can be multiplied a sufficient number of times for its multiple to exceed the greater; and two equal magnitudes of the same kind are said to have a ratio to one another, and it is called a ratio of equality. OBS. This def excludes from our present consideration all ratios between (1) two magnitudes, both of which are infinitely great or infinitely small; (2) two magnitudes, one of which is finite, and the other infinitely great or infinitely small.

V.

The first of four magnitudes is defined to have the same ratio to the second which the third has to the fourth, when, there having been taken of the first and third any equimultiples whatever, and of the second and fourth any equimultiples whatever, the multiple of the third is greater than, equal to, or less than the multiple of the fourth, according as the multiple of the first is greater than, equal to, or less than the multiple of the second.

OBS. 1. The word "any " in the def" must be specially noted. In the application of this def" to test the proportionality (Def. 6) of four magnitudes, the criterion must be satisfied in every instance for them to be proportional, whereas a single failure would shew their non-proportionality.

OBS. 2. It appears from Def. 3, that the first and second of the four magnitudes must be of the same kind, and also that the third and fourth must be of the same kind. But there is no necessity for all four to be of the same kind.

OBS. 3. When the first of four magnitudes has the same ratio to the second which the third has to the fourth, the third clearly has the same ratio to the fourth which the first has to the second. Such will appear also from the def".

OBS. 4. When this defn is referred to, it is cited as the "definition of proportion."