HK or KL; and that the angles KHG, HGM, and GML are each equal to the angle HKL or KLM. Therefore the pentagon GHKLM is both equilateral and equiangular. Wherefore a regular pentagon GHKLM has been described about the circle. Q. E. F.

PROP. XIII. (PROBLEM.)—To inscribe a circle in a given regular pentagon (ABCDE).

Bisect the angles BCD and CDE by the straight lines CF and DF (I. 9). From the point F, in which they meet, draw the straight lines FB, FA, and FE. Draw also the perpendiculars FG, FH, FK, FL, and FM to the sides of the pentagon. From centre F, with distance FH, describe the circle GHKLM, and it is inscribed in the pentagon.

B

H

M

E.

K D

Because the angles FCD and FDC (I. Ax. 7) are equal, FC is equal to FD. Because in the two triangles BCF and DCF, the side BC is equal to the side DC (Hyp.), CF is common. and the angle BCF is equal to the angle DCF (Const.) Therefore BF is equal to DF, and also to CF. The angle CBF is also equal to the angle CDF, and is the half of the angle ABC. In the same manner it may be shown that the angles A and E are bisected by the straight lines AF and EF. Because the two angles FCH and FCK are equal (Const.), and the two angles FHC and FKC are also equal, being right angles (Const.); therefore in the two triangles FHC, FKC, two angles of the one are equal to two angles of the other, and the side FC is common to both. Wherefore, the two triangles are equal (I. 26), and FH is equal to FK. In the same manner, it may shown that FL, FM, and FG are each of them equal to FH, or FK. Therefore the five straight lines FG, FH, FK, FL, and FM are equal to one another, and the circle described from the centre F, at the distance of one of them FH, will pass through the extremities of the other four, and touch the straight lines AB, BC, CD, DE, and EA. Because the angles at the points G, H, K, L, and M are right angles, and a straight line drawn through the extremity of the diameter of a circle at right angles to it (III. 16) touches the circle; therefore each of the straight lines AB, BC, CD, DE, and EA touches the circle. Wherefore the circle GHKLM is inscribed in the pentagon ABCDE. Q. E. F.

PROP. XIV. (PROBLEM.)-To describe a circle about a given regular pentagon (ABCDE).

Bisect the angles BCD and CDE by the straight lines CF and FD (I. 9). From the point F, in which they meet, draw the straight lines FB, FA, and FE. With centre F, and distance FC, describe the circle ABCDE, and it is described about the pentagon.

Because it may be shown, as in the preceding proposition, that the side CF is equal to the side FD, and that FB, FA, and FE are each of them equal to FC or FI; therefore the five straight lines FA, FB, FC, FD, and F E are equal to one another. And the circle described from the

B

E

centre F, at the distance of one of them, will pass through the extremities of the other four. Wherefore the circle ABCDE is described about the pentagon ABCDE. Q. E. F.

PROP. XV. (PROBLEM.)-To inscribe a regular hexagon in a given circle

(ABCDEF).

Find the centre G of the circle ABCDEF (III. 1), and draw the diameter AGD. From D, as a centre, at the distance DG, describe the circle EGCH, join EG and CG, and produce them to the points B and F. Join AB, BC, CD, DE, EF, and FA. The hexagon ABCDEF is a regular hexagon.

F

A

D

H

B

Because G is the centre of the circle ABCDEF, GE is equal to GD. Because D is the centre of the circle EGCH, DE is equal to DG. Therefore GE E is equal to ED (I. Ax. 1), and the triangle EGD is equilateral. Because the three angles EGD, GDE, and DEG, are equal to one another (I. 5, Cor.) But the three angles of a triangle are equal to two right angles (I. 32); therefore the angle EGD is the third part of two right angles. In the same manner it may be shown that the angle DGC is also the third part of two right angles. Because the straight line GC makes with EB the adjacent angles EGC, CGB equal to two right angles (I. 13), the remaining angle CGB is the third part of two right angles. Therefore the angles EGD, DGC, and CGB are equal to one another. And the vertical angles BGA, AGF, and FGE (I. 15) are equal to these angles, each to each. Therefore the six angles EGD, DGC, CGB, BGA, AGF, and FGE are equal to one another. But equal angles stand upon equal arcs (III. 26); therefore the six arcs AB, BC, CD, DE, EF, and FA are equal to one another. And equal arcs are subtended by equal straight lines (III. 29); therefore the six straight lines AB, BC, CD, DE. EF, and FA are equal to one another, 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; therefore the whole arc FABCD is equal to the whole arc EDCBA. But the angle FED stands upon the arc FABCD, and the angle AFE upon the arc EDCBA. Therefore the angle AFE is equal to the angle FED (III. 27). In the same manner it may be shown that the other angles of the hexagon ABCDEF are each equal to the angle AFE or FED. Therefore the hexagon is equiangular; and it was shown to be equilateral; therefore the regular hexagon ABCDEF is inscribed in the given circle ABCDEF. Q. E. F.

Cor. From this it is manifest, that the side of the hexagon is equal to the straight line drawn from the centre to the circumference, that is, to the radius or semidiameter of the circle.

If through the points A, B, C, D, E, F there be drawn straight lines touching the circle, a regular hexagon will be described about it and a circle may be inscribed in a given regular hexagon, and circumscribed about it in the same manner as was done in the case of the regular pentagon.

PROP. XVI. (PROBLEM.)-To inscribe a regular quindecagon in a given circle (ABCD).

Find AC the side of an equilateral triangle inscribed in the circle (IV. 2), and AB the side of a regular pentagon inscribed in the same (IV. 11). Bisect BC in E (III. 30). Join BE and EC, and place (IV. 1) in the circumference straight lines equal to these, and contiguous to each other, all round the circle. The figure ABCDF is a regular quindecagon.

Because of such equal parts as the whole circumference ABCDF contains fifteen, the are ABC, which is the third part of the whole, contains five; and the arc AB, which

B

E

is the fifth part of the whole, contains three. Therefore their difference BC contains two of the same parts, and BE, EC are, each of them, the fifteenth part of the whole circumference ABCD, and the figure ABECDF is equilateral. Because each of its angles stands upon thirteen-fifteenths of the circumference, it is also equiangular (III. 27). Therefore a regular quindecagon is inscribed in the circle ABC. Q. E. F.

In the same manner as was done in the pentagon, if through the angular points of the inscribed quindecagon, straight lines be drawn touching the circle, a regular quindecagon will be described about it; and likewise, as in the case of the pentagon, a circle may be inscribed in a given regular quindecagon, and circumscribed about it.

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 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.

The mutual relation of two magnitudes of the same kind to one another, in respect of quantity, is called their ratio.

IV.

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 said to have the same 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 than that of the second, the multiple of the third is also greater than that of the fourth.

VI.

Magnitudes which have the same ratio are called proportionals. N.B." When four magnitudes are proportionals, this property 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 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 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 consists in three terms at least.

X.

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

XI.

When four magnitudes are continual proportionals, the first is said to have to the fourth the triplicate ratio of that which it has to the second, and so on; quadruplicate, &c., increasing the denomination still by unity in any number of proportionals.

A.

When there are any number of magnitudes of the same kind, the first is said 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 to the last magnitude.

XII.

In proportionals, the antecedent terms are called homologous to one another, as also the consequents to one another.

"Geometers make use of the following technical words or phrases to signify certain ways of changing either the order or magnitude of proportionals, so that they continue still to be proportionals."

XIII.

Permutando, or alternando, by permutation or alternately. This phrase is used when there are four proportionals, and it is inferred that the first has the same ratio to the third which the second has to the fourth; or that the first is to the third as the second to the fourth: as is shown in Prop. 16 of this Fifth Book.

XIV.

Invertendo, by inversion: when there are four proportionals, and it is inferred that the second is to the first as the fourth to the third.

XV.

Componendo, by composition: when there are four proportionals, and it is inferred that the first together with the second, is to the second, as the third, together with the fourth, is to the fourth.-Prop. 18, Book V. XVI.

Dividendo, by division: when there are four proportionals, and it is inferred that the excess of the first above the second, is to the second, as the excess of the third above the fourth is to the fourth.-Prop 17, Book V.

XVII.

Convertendo, by conversion: when there are four proportionals, and it is inferred that the first is to its excess above the second, as the third to its excess above the fourth.

XVIII.

Ex æquali (sc. distantiâ), or ex æquo, from equality of distance: when there is any number of magnitudes more than two, and as many others, such that they are proportionals when taken two and two of each rank, and it is inferred that the first is to the last of the first rank of magnitudes, as the first is to the last of the others. "Of this there are the two following kinds, which arise from the different order in which the magnitudes are taken, two and two."

XIX.

Ex æquali, from equality. This term is used simply by itself when the first magnitude is to the second of the first rank, as the first to the second of the other rank; and as the second is to the third of the first rank, so is the second to the third of the other; and so on in order: and the inference is as mentioned in the preceding definition; whence this is called ordinate proportion. It is demonstrated in Prop. 22, Bock V.

XX.

Ex æquali in proportione perturbatâ seu inordinatâ, from equality in perturbate or disorderly proportion (Prop 4, Lib. II. Archimedis de sphæra et cylindro). The term is used when the first magnitude is to the second of the first rank, as the last but one is to the last of the second rank; and as the second is to the third of the first rank, so is the last but two to the last but one of the second rank; and as the third is to the fourth of the first rank, so is the third from the last to the last but two of the second rank; and so on in a cross order: and the inference is as in the 18th definition. It is demonstrated in Prop. 23, Book V.

AXIOMS.

1. Equimultiples of the same, or of equal magnitudes, are equal to one another.

2. Those magnitudes, of which the same or equal magnitudes are equimultiples, are equal to one another.

3. A multiple of a greater magnitude is greater than the same multiple of a less.

4. Of two magnitudes, that one of which a multiple is greater than the same multiple of the other is the greater.

5. A part or submultiple of a greater magnitude is greater than the same part or submultiple of a less magnitude.

6. Of two magnitudes, that one of which a part or submultiple is greater than the same part or submultiple of the other is the greater of

the two.