382 C: E O M meet in a point, and this is the centre of the circle inscribed in the triangle. The solutions of most of the other problems contain also theorems. Of these we shall state those which are of special interest ; Euclid does not state any one of them. § 41. Prop. 5. To circumscribe a circle about a giren triangle. The one sol11tion which always exists contains the following :— 'l'uI-:o1:I-:.t.—The three straight lines qchich bisect the sides of a triangle at -right angles meet in a. point, and this point is the centre of the circle circumscrilu-d about the triangle. Euclid adds in a corollary the following property :— 'l‘he centre of the circle circumscribed about a triangle lies within, on a side of, or without the triangle, according as the triangle is acute-angled, right-angled, or obtuse-angled. §-42.. Vhilst it is always possible to draw a circle which is inscribed in or circmnscribed about a given triangle, this is not the case with quadrilatcrals or polygons of more sides. Of those for which this is possible the regular polygons are the most interesting. In each of them a circle may be inscribed, and another may be circumscribed about it. Euclid does not use the word regular, but he describes the polygons i11 question as eyuiangular and equilateral. Ve shall use the name regular polygon. The regular triangle is equi- lateral, the regular quadrilateral is the square. Euclid considers the regular polygons of 4, 5, 6, and 15 sidcs. For each of the first three he solves the problems—(_ 1) to inscribe such a polygon in a given circle; ('2) to circumscribe it about a given circle; (3) to inscribe a circle in, and (4) to circumscribe a circle about, such a polygon. For the regular triangle the problems are not repeated, because more general problems have been solved. Props. 6, 7, 8, and 9 solve these problems for the square. The general problem of inscribing in a given circle a regular polygon of n Slt cs depends upon the problem of dividing the cir- cumference of a circle i11to 12 equal parts, or what comes to the same thing, of drawing froin the centre of the circle -71. radii such that the angles between consecutive radii are equal, that is, to divide the space about the centre into n equal angles. Thus, if it is required to inscribe a square in a circle, we have to draw four lines fro111 the centre, making the four angles equal. This is done by drawing two diameters at right angles to one another‘. The c11ds of these diameters are the vertices of the required square. If, on the other hand, tangents be dravn at these ends, we obtain a square circum- scribed about the circle. § 43. To construct a regular pentagon, we find it convenient first to construct a regular decagou. This requires to divide the space about the centre into tc11 equal angles. Each will be {nth of a right angle, or %th of two right angles. If we suppose the decagon co11- structed, and if we join the centre to the end of one side, we get an isoceles triangle, where the angle at the centre equals %th of two right angles ; hence each of the a11glcs at the base will be §ths of two right angles, as all three angles together equal two right angles. Thus we have to construct an isoceles triangle, having the angle at the vertex equal to half an angle at the base. This is solved in Prop. 10, by aid of the problem in Prop. 11 of the second book. If we make the sides of this triangle equal to the radius of the given circle, then the base will be the side of the reuular dccagon inscribed in the circle. This side being known the decagon can be constructed, and if the vertices are joined alternately, leaving out half their number, we obtain the regular pentagon. Euclid does not proceed thus. He wants the pentagon before the decagou. This, however, does 11ot change the real nature of his solution, 1101‘ does his solution become simpler by not mention- ing the decagou. Once the regular pentagon is inscribed, it is easy to circumscribe another by drawing tangents at thevcrtices of the inscribed pen- tagon. This is shown in Prop. 12. Prop. 13 and 14 teach how a circle may be inscribed in or cir- cumscribed about any given regular pentagon. § 44. The regular he:eagon is more easily constructed, as shown in Prop. 15. The result is that the side of the regular hexagon inscribed in a circle is equal to the radius of the circle. For this polygon the other three problems mentioned are not solved. § 45. The book closes with Prop. 16. To inscribe a regular qumdecagon in a given circle. That this may be do11e is easily seen. If we inseribe a regular pentagon and a regular hexagon in the circle, having one vertex in connnon, then the are from the common vertex to the next vertex of the pentagon is _‘,th of the cir- cumference, and to the next veitex of the liexagon is %,th of the circumference. The difference between these arcs is, therefore, %— ,‘,=._.}Uth of the circumference. The latter may, therefore, be divided into thirty, and hence also i11 fifteen equal parts, and the regular qnindccagon be described. ,8 46. We conclude with a few theorems about regular polygons which are not given by liuclid. TU1-20I:l.'.‘:I.—T he straight lines perjicnzlicular to and bisccting the E T R Y sides of any regular polygon meet in a point. The straight liars bisecting the angles in the regular polygon meet in the same point. This point is the centre of the circles ci-rcumscribetl about and inscribecl in the regular polygon. The proof, which is easy, is left to the reader. We can bisect any given are (Prop. 30, III.). Hence we can divide a circumference into 2n equal parts as soon as it. has been divided into 71. equal parts, or as soon as a regular polygon of IL sides has been constructed. IIence— Tnnolu-:M.—If a regular polygon Q/‘n sides has been cnnstructcn’, then a regular polygon of ‘Zn sides, of 4n, of 811 sitlcs, «('1-., -mrt_u also be constructed. Euclid shows how to construct 1'(-gulnr poly- gons of 3, 4, 5, and 15 sides. It follows that we can construct regular polygons of [EUeL11)IAN. 3, 6, 1-2, 24...Sl(l(‘S 4, 8, 16, 32. . ,, 5,10, 20, 40... ,, 15,30, 60, 120... ,, The construction of any new regular polygon not included in one of these series will give rise to a new series. Till the beginning of this century nothing was added to the knowledge of regular polygons as given by Euclid. Then Gauss, in his celebrated -trithmet2'c, proved that every regular polgon of 2" +1 sides may be constructed if this number 2" +1 be prime, and that no others ea11 be constructed by elementary methods. This shows that regular polygons of 7, 9, 13 sidcs cannot thus be constructed, but that a regular polygon of 17 sides is possible ; for 17=— 2‘ 1. The next polygon is one of 257 sides. The construction becomes already rather complicated for 17 sides. Boon Y. §47. The fifth book of the Elements is 11ot L-.'clu;~ively gcon1c- trical. It contains the theory of ratios and proportion of quantities in general. The treatment, as here given, is admirable, and in every respect superior to the algcbraical method by which Euclid's theory is now generally replaccd. It has, however, the rcputation of being too ditlicult for schools, and is therefore very seldom read. We shall try to make the subject clear, and to show why the usual algcbraical treatment of proportion is not really sound. Ve begin by quoting those definitions at the bcginnin g of Book V. which are most important. These definitions have given rise to much discussion. The only definitions which are essential for the fifth book are Dcfs. 1, 2, 4, 5, 6, and 7. Of the remainder 3, 8, and 9 are more than useless, and probably not Euclid’s, but additions of lattrl‘ editors, of whom Thcon of Alexandria was the most prominent. Dcfs. 10 a11d 11 belong rather to the sixth Look, whilst all the others are merely nominal. The really important ones are 4, 5, 6, and 7. § 48. To define a magnitude is not attempted by I-Euclid. The first two definitions state what is meant by a “part,” that is, a submultiple or measure, and by a “multiple” of a given magni- tudc. The meaning of Def. 4 is that two given quantities can have a ratio to one another only in case that they are comparable as to their magnitude, that is, if they are of the same kind. I)cf. 3, which is prob. bly due to Thcon, professes to define a ratio, but is as meaningless as it is uncalled for, for all that is wanted is given in Dcfs. 5 a11d 7. In Def. 5 it is explained what is meant by saying that two magnitudes have the same ratio to one another as two other magnitudes, and in Def. 7 what we have to mulcrstaml by a greater or a less ratio. The 6th definition is only nominal, c.'plaining the meaning of the word proportional. Euclid represents magnitudes by lines, and often denotes them either by single letters or, like lines, by two letters. We shall use only single letters for the purpose. If a and (2 denote two magni- tudes of the same kind, their ratio will be denoted by a : b; if c and (Z are two other magnitudes of the same kind, b11t possibly of a different kind from a and l), then if c and d have the same ratio to one another as a and I), this will be expressed by writing—— a : l) : : c : tl. Further, if m is a (whole) number, ma shall denote the multiple of a which is obtained by taking it on times. § 49. The whole theory of ratios is based on Def. 5. Def. 6. The first of four n_1agm'tu¢les is said to ha re the same ratio to the second that the third has to the fourth ’t('lt/”)I, any cgutmultiplrs whaterer of the jirst and the third being taken, (nut any rquimul- tiplcs'u'lmte1'cr of the second and the fourth, if the mllltipln if the _/l)‘.5l- be less than that of the second, the multiple of the third is also less than that of the fourth; and if the multiple of the first is equal to that of the second, the multiple of the third is also equal to that of the fourth ; and if the -multiple of the first is greater than that (If the st.-cenrl, the multiple of the third is also greater than that of the
fourth.