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Again, since the point D is the centre of the circle GCH,

DE is equal to DG.

But GE was proved equal to GD;

therefore GE is also equal to ED;

therefore the triangle EGD is equilateral ;

and therefore its three angles EGD, GDE, DEG are equal to one another, inasmuch as, in isosceles triangles, the angles at the base are equal to one another. [1.5] And the three angles of the triangle are equal to two right angles; [1. 32] therefore the angle EGD is one-third of two right angles. Similarly, the angle DGC can also be proved to be onethird of two right angles.

And, since the straight line CG standing on EB makes the adjacent angles EGC, CGB equal to two right angles, therefore the remaining angle CGB is also one-third of two right angles.

Therefore the angles EGD, DGC, CGB are equal to one another;

so that the angles vertical to them, the angles BGA, AGF, FGE are equal. [1. 15]

Therefore the six angles EGD, DGC, CGB, BGA, AGF, FGE are equal to one another.

But equal angles stand on equal circumferences; [111. 26] therefore the six circumferences AB, BC, CD, DE, EF, FA are equal to one another.

And equal circumferences are subtended by equal straight lines;

[III. 29] therefore the six straight lines are equal to one another; therefore the hexagon ABCDEF is equilateral.

I say next that it is also equiangular.

For, since the circumference FA is equal to the circumference ED,

let the circumference ABCD be added to each

therefore the whole FABCD is equal to the whole EDCBA;

and the angle FED stands on the circumference FABCD, and the angle AFE on the circumference EDCBA;

therefore the angle AFE is equal to the angle DEF.

[111. 27] Similarly it can be proved that the remaining angles of the hexagon ABCDEF are also severally equal to each of the angles AFE, FED;

therefore the hexagon ABCDEF is equiangular.

But it was also proved equilateral;

and it has been inscribed in the circle ABCDEF.

Therefore in the given circle an equilateral and equiangular hexagon has been inscribed.

Q. E. F.

PORISM. From this it is manifest that the side of the hexagon is equal to the radius of the circle.

And, in like manner as in the case of the pentagon, if through the points of division on the circle we draw tangents to the circle, there will be circumscribed about the circle an equilateral and equiangular hexagon in conformity with what was explained in the case of the pentagon.

And further by means similar to those explained in the case of the pentagon we can both inscribe a circle in a given hexagon and circumscribe one about it.

Q. E. F.

Heiberg, I think with good reason, considers the Porism to this proposition to be referred to in the instance which Proclus (p. 304, 2) gives of a porism following a problem. As the text of Proclus stands, "the (porism) found in the second Book (τὸ δὲ ἐν τῷ δευτέρῳ βιβλίω κείμενον) is a porism to a problem"; but this is not true of the only porism that we find in the second Book, namely the porism to II. 4. Hence Heiberg thinks that for T δευτέρῳ βιβλίῳ should be read τῷ δ' βιβλίῳ, i.e. the fourth Book. Moreover Proclus speaks of the porism in the particular Book, from which we gather that there was only one porism in Book Iv. as he knew it, and therefore that he did not regard as a porism the addition to IV. 5. Cf. note on that proposition.

It appears that Theon substituted for the first words of the Porism to IV. 15 "And in like manner as in the case of the pentagon" (óμoíws dè Tois èπì TOû TEνTaywvov) the simple word "and" or "also" (kai), apparently thinking that the words had the same meaning as the similar words lower down. This is however not the case, the meaning being that "if, as in the case of the pentagon, we draw tangents, we can prove, also as was done in the case of the pentagon, that the figure so formed is a circumscribed regular hexagon."

PROPOSITION 16.

In a given circle to inscribe a fifteen-angled figure which shall be both equilateral and equiangular.

Let ABCD be the given circle;

thus it is required to inscribe in the circle ABCD a fifteenangled figure which shall be

both equilateral and equiangular.

In the circle ABCD let there be inscribed a side AC of the equilateral triangle inscribed in it, and a side AB of an equilateral pentagon; therefore, of the equal segments of which there are fifteen in the circle ABCD, there will be five in the circumference ABC which is one-third of the circle, and

there will be three in the cir

E

B

cumference AB which is one-fifth of the circle;

A

therefore in the remainder BC there will be two of the

equal segments.

Let BC be bisected at E;

[III. 30]

therefore each of the circumferences BE, EC is a fifteenth of the circle ABCD.

If therefore we join BE, EC and fit into the circle ABCD straight lines equal to them and in contiguity, a fifteen-angled figure which is both equilateral and equiangular will have been inscribed in it.

Q. E. F.

And, in like manner as in the case of the pentagon, if through the points of division on the circle we draw tangents to the circle, there will be circumscribed about the circle a fifteen-angled figure which is equilateral and equiangular.

And further, by proofs similar to those in the case of the pentagon, we can both inscribe a circle in the given fifteenangled figure and circumscribe one about it.

Q. E. F.

Here, as in III. 10, we have the term "circle" used by Euclid in its exceptional sense of the circumference of a circle, instead of the "plane figure contained by one line" of 1. Def. 15. Cf. the note on that definition (Vol. 1. pp. 184-5).

Proclus (p. 269) refers to this proposition in illustration of his statement that Euclid gave proofs of a number of propositions with an eye to their use in astronomy. "With regard to the last proposition in the fourth Book in which he inscribes the side of the fifteen-angled figure in a circle, for what object does anyone assert that he propounds it except for the reference of this problem to astronomy? For, when we have inscribed the fifteen-angled figure in the circle through the poles, we have the distance from the poles both of the equator and the zodiac, since they are distant from one another by the side of the fifteen-angled figure." This agrees with what we know from other sources, namely that up to the time of Eratosthenes (circa 275-194 B.C.) 24° was generally accepted as the correct measurement of the obliquity of the ecliptic. This measurement, and the construction of the fifteen-angled figure, were probably due to the Pythagoreans, though it would appear that the former was not known to Oenopides of Chios (fl. circa 460 B.C.), as we learn from Theon of Smyrna (pp. 198-9, ed. Hiller), who gives Dercyllides as his authority, that Eudemus (fl. circa 320 B.C.) stated in his dorpoλoyíaι that, while Oenopides discovered certain things, and Thales, Anaximander and Anaximenes others, it was the rest (oi λoroí) who added other discoveries to these and, among them, that "the axes of the fixed stars and of the planets respectively are distant from one another by the side of a fifteen-angled figure." Eratosthenes evaluated the angle to rds of 180°, i.e. about 23° 51′ 20′′, which measurement was apparently not improved upon in antiquity (cf. Ptolemy, Syntaxis, ed. Heiberg, p. 68).

Euclid has now shown how to describe regular polygons with 3, 4, 5, 6 and 15 sides. Now, when any regular polygon is given, we can construct a regular polygon with twice the number of sides by first describing a circle about the given polygon and then bisecting all the smaller arcs subtended by the sides. Applying this process any number of times, we see that we can by Euclid's methods construct regular polygons with 3.2", 4.2", 5.2′′, 15.2" sides, where n is zero or any positive integer.

BOOK V.

INTRODUCTORY NOTE.

The anonymous author of a scholium to Book v. (Euclid, ed. Heiberg, Vol. v. p. 280), who is perhaps Proclus, tells us that "some say" this Book, containing the general theory of proportion which is equally applicable to geometry, arithmetic, music, and all mathematical science, "is the discovery of Eudoxus, the teacher of Plato." Not that there had been no theory of proportion developed before his time; on the contrary, it is certain that the Pythagoreans had worked out such a theory with regard to numbers, by which must be understood commensurable and even whole numbers (a number being a "multitude made up of units," as defined in Eucl. vII). Thus we are told that the Pythagoreans distinguished three sorts of means, the arithmetic, the geometric and the harmonic mean, the geometric mean being called proportion (avaλoyia) par excellence; and further Iamblichus speaks of the "most perfect proportion consisting of four terms and specially called harmonic," in other words, the proportion

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which was said to be a discovery of the Babylonians and to have been first introduced into Greece by Pythagoras (Iamblichus, Comm. on Nicomachus, p. 118). Now the principle of similitude is one which is presupposed by all the arts of design from their very beginnings; it was certainly known to the Egyptians, and it must certainly have been thoroughly familiar to Pythagoras and his school. This consideration, together with the evidence of the employment by him of the geometric proportion, makes it indubitable that the Pythagoreans used the theory of proportion, in the form in which it was known to them, i.e. as applicable to commensurables only, in their geometry. But the discovery, also due to Pythagoras, of the incommensurable would of course be seen to render the proofs which depended on the theory of proportion as then understood inconclusive; as Tannery observes (La Géométrie grecque, p. 98), "the discovery of incommensurability must have caused a veritable logical scandal in geometry and, in order to avoid it, they were obliged to restrict as far as possible the use of the principle of similitude, pending the discovery of a means of establishing it on the basis of a theory of proportion independent of commensurability." The glory of the latter discovery belongs then most probably to Eudoxus. Certain it is that the complete theory was already familiar to Aristotle, as we shall see later.

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