and therefore the right lines subtending these arches are also equal, (by Schol. Prop. 29, B. 3), therefore the hexagon is equilateral; and therefore, since it is inscribed in a circle, it is also equiangular (by Schol. Prop. 11, B. 4).

COR. 1.-The side of a hexagon is equal to the radius of the circle in which it is inscribed.

COR. 2.-By bisecting the arches AB, BC, a figure of twelve sides might be inscribed, and again bisecting these, a figure of twenty-four sides, and so on.

COR. 3.-An equilateral and equiangular hexagon can be described upon a given line BA, by describing an equilateral triangle BGA upon it, and from the centre G describing a circle through A, and inscribing a hexagon in it.

COR. 4.-Drawing AC, AE, and CE, an equilateral triangle will be inscribed in the given circle, whose sides bisect the radii perpendicular to them. For let GD be perpendicular to any side CE, and in the triangles DCK, GCK, the angles at K are right, and the angles DCK and GCK standing on equal arches DE and EF are equal (by Schol. Prop. 29, B. 3), and the side CK between the equal angles common, therefore the sides DK and GK are equal (by Prop. 26, B. 1), and therefore GD is bisected.

COR. 5.-Hence it follows that the square of the side of an equilateral triangle is triple the square of the radius of the circle in which it is inscribed; for if AC be drawn, its square will be equal to the squares of AG and GC together with twice the rectangle under AG and GK (by Prop. 12, B. 2); but since GK is half of DG (by Cor. pr.), twice the rectangle under AG and GK is equal to the square of AG, therefore the square of AC is equal to twice the square of AG together with the square of GC, that is, to triple the square of the radius.

PROPOSITION XVI. PROBLEM.

To inscribe an equilateral and equiangular quindecagon in a given circle (CAD).

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Let CD be the side of an equilateral triangle inscribed in the circle CAD, and also let CA be the side of an equilateral and equiangular pentagon inscribed in the circle CAD, bisect AD, and A AB will be the side of the required quindecagon. For if the whole circumference be divided into fif

teen parts, because the arch CD is one third part of the whole circumference, it contains five of these parts in it, and similarly the arch CA contains three of them, and therefore AD contains two, therefore the arch AB is the fifteenth part of the whole circumference, and therefore AB is the side of the quindecagon.

END OF FOURTH BOOK.

FIFTH BOOK.

DEFINITIONS.

1. A less magnitude is called an aliquot part, or a submultiple of a greater, when the less measures the greater.

2. A greater is called a multiple of a less, when the greater is measured by the less.

3. Ratio is a mutual relation of two magnitudes of the same kind, in respect of quantity.

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

5. Magnitudes are said to be in the same ratio, the first to the second, and the third to the fourth, when any submultiple whatsoever of the first is contained in the second, as often as an equi-submultiple of the third is contained in the fourth.

6. Magnitudes which have the same ratio are called proportionals.

7. When a submultiple of the first is contained oftener in the second, than an equi-submultiple of the third is contained in the fourth, then the first is said to have a less ratio to the second, than the third has to the fourth; and conversely, the third is said to have a greater ratio to the fourth, than the first to the second. 8. Proportion is similitude of ratios.

9. And proportion consists in three terms, at least.

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10. When three magnitudes are proportional (A to B as B to C), the first is said to have to the third (A to C) a duplicate ratio of that which it has to the second, namely, of the ratio of A to B.

11. When four magnitudes are proportional, one after the other (A to B as B to C, and B to C as C to D), the first is said to have to the fourth (A to D) a triplicate ratio of that which it has to the second (namely, of the ratio of A to B). And so on, increasing by unity, whatsoever be the number of proportionals.

12. If there be any number of magnitudes of the same kind (A, D, C, and F), the first is said to have to the last (A to F) a ratio compounded of the ratios which the first has to the second, and the second to the third, and the third to the fourth (A to D, D to C, and C to F), and so on to the last.

13. In proportionals, the antecedents are called homologous to the antecedents, and the consequents to the consequents.

Either the order or the magnitude of proportionals can be changed, so that they remain still proportionals; the various modes of changing, which are made use of by Geometers, are designated by the following names :—

14. By Permutation, when it is inferred that if there be four magnitudes, of the same kind, proportional, the first is to the third as the second to the fourth; as is shown in Prop. 33, B. 5. Ex. gr.-A to B as C to Do, becomes A to C as B to Do.

15. By Inversion, when it is inferred that if four magnitudes be proportional, the second is to the first as the fourth to the third; as is demonstrated in Prop. 20, namely A to B as CD to E, becomes B to A as E to CD.

16. By Composition, when it is inferred, if four magnitudes be proportional, that the first together with the second is to the second, as the third together with

the fourth is to the fourth; as shown in Cor. Prop. 21. Thus-A to B as C to D, will be A + B to B as C + D to D.

17. By Division, when it is inferred, if there be four magnitudes proportional, that the difference between the first and the second is to the second as the difference between the third and the fourth to the fourth; as is proved in Cor. 2, Prop. 25. Ex. gr.-A to B as C to D, becomes A B to B as C D to D.

18. By Conversion, when it is inferred, that if there be four magnitudes proportional, the first is to the sum or difference of the first and second, as the third is to the sum or difference of the third and fourth. See Props. 21 and 25, also Cor. 1, Prop. 25. Thus-A is to B as C is to D, becomes A to (A + B) as C to (C +D), or A to (AB) as C to (CD).

19. Ex æquali vel ex æquo (or from equality), when it is inferred, that if there be many magnitudes, and an equal number of others, which, if taken two by two, are in the same ratio, the first is to the last in the first series, as the first to the last in the next series..

Of this there are the two following species:— 20. Ex æquo ordinate (from orderly equality), when the first magnitude will be to the second in the first series, as the first to the second in the next series, and the second to the third in the first series, as the second to the third in the next, and so on; and it is inferred, as in Def. 19, that the first is to the last in the first series, as the first to the last in the next series; as shown in Prop. 34. Ex. gr.-Let A, B, and CO, be three magnitudes in the first series, and D, E, and F, be three in the second series; then, according to the Definition, we have

And so on, if there were four or more magnitudes.