MATHEMATICS. PLANE GEOMETRY. 



[REMARKS ON BOOK IT. 



PROPOSITION XVL PKOKLKJC 



l\i iiuenbt an eauilaterai and tOHiangitlr quintlragon, or 



JSflren-tiJtJjIgwrt, in a yim. circle (A li C U). 

 Let A C be a side of an equilateral triangle inscribed 



i iv. in the circle,* and A H a tide of an cquil.-it.nl 

 and equiangular pentagon inscribed in the samc:t 



ll iv. then of lucli 

 part* as the whole cir- 

 cumference contains />/- 

 lent, the arc A It C, U-ing 

 the third part of th. 

 cu inference, contains five; 

 and the arc A It, which U 

 the fifth part of the 

 circumference, contains 

 ttrw; .*. BC, their dif- 

 ference, contains two of 

 the same parts. Bisect 



oin. B C in E,* .-. 



BE, E C are each the fifteenth part of the whole circum- 

 ference ; .-.if the straight lines BE, E C, be drawn, 

 and straight lines equal to them be placed round, in the 

 whole circle, an equilateral quindecagon will be inscribed 

 in it ; and that it is equiangular is plain, because each 

 angle stands upon an arc, equal to the whole circumfer- 

 ence diminished by the two arcs which its sides subtend, 

 and which arcs are by construction equal, . . an equi- 

 lateral and equiangular quindecagon is inscribed in tiie 

 circle. Which was to be done. 



And in the same manner as was done in the pentagon, 

 if through the points of division made by inscribing the 

 quindecagon, straight lilies be drawn touching the circle, 

 an equilateral and equiangular quindecagon will be 

 described about it And likewise, as in the pentagon, a 

 circle may be inscribed in a given equilateral and equi- 

 angular quindecagon, and circumscribed about it. 



REMARKS OK BOOK IV. 



The propositions in this fourth book are all problems, 

 relating chiefly to the construction of regular polygons 

 in and about the circle. From the fourth we learn this 

 theoretical truth; namely, that the straight lines bisecting 

 the three angles of a triangle all meet in a point the 

 centre of the inscribed circle ; and from the fifth, that 

 the straight lines bisecting the three sides of a triangle 

 also meet in a point the centre of the circumscribing 

 circle. 



As the second proposition shows that a triangle equi- 

 angular to any proposed triangle may be inscribed in a 

 circle, we know that an equilateral triangle may be 

 inscribed ; and we thus infer that the circumference of a 

 circle may be geometrically divided into three equal 

 parts. By bisecting each of these three equal arcs (Pro- 

 position XXX., Book III.), we may divide the circum- 

 ference into six equal parts, as otherwise shown in 

 Proposition XV. ; and by another series of bisections, 

 into twelve equal parts, and so on. The division of the 

 whole circumference into six equal parts is obviously the 

 division of the semi-circumference into three : the divi- 

 sion of the whole into twelve equal parts is the division 

 of the fourth of it that is, of a quadrant, into three 

 equal parts, and so on. We thus see that an arc equal to 

 a fourth part, or to an eighth part, a sixteenth part, A-c. , 

 of the entire circumference, may be divided geometrically 

 into three equal parts. But the general problem to 

 divide any arc of a circle into three equal parts, or, 

 which amounts to the same thing, to trisect an angle, is a 

 problem that all the geometry in Euclid lias been found 

 hitherto inadequate to accomplish. It is one of the 

 great outstanding problems of antiquity ; and more 

 thought and labour have been expended upon it, during 

 the last two thousand years, than perhaps any other 

 problem has ever called into exercise. If you now learn 

 this fact for the first time, you will be surprised that a 

 thing apparently so simple as to divide an angle or an 

 arc of a circle into three equal parts, should have been 

 fuund to be a matter of such surpassing difficulty as to 

 bare baffled the efforts of the greatest geometers from 



the time of Euclid to the present day. We draw atten- 

 tion to it here mainly for the purpose of discouraging 

 any fresh attempts by students iu geometry to trisect an 

 angle. Novices are very apt to enter upon such attempts 

 from a vague notion that a successful solution of the 

 problem would, in some way or other, advance science. 

 But geometricians know better ; and by them the inquiry 

 has long been abandoned, not only as a hopeless one, 

 but also as a comparatively useless one. The theorist 

 has no need of this solution, because he is never stopped 

 in any mathematical investigation from the want of it ; 

 the practical man has no need of it, for he can avail 

 himself of mechanical methods which effect the trisection 

 of an angle that departs from strict geometrical accuracy, 

 by an amount of error too minute to bo detected by his 

 senses, though aided by the most finished instruments 

 of measurement ; and these methods he would continue 

 to employ even were the problem ever to be brought 

 within the scope of elementary geometry. You see. 

 therefore, that the inquiry is one of pure curiosity, and 

 nothing more ; since the successful issue of it could 

 supply no want either in theory or in practice. 



By aid of certain curves beyond the limits of Euclidean 

 geometry, the trisection of an angle may be readily 

 effected, as will be hereafter shown ; but what geome- 

 tricians have been in quest of, is the accomplishment of 

 this trisection by aid merely of the straight line and the 

 circle the only lines recognised by Euclid. Some have 

 ventured to say that, with this limitation as to materials, 

 the problem is impossible ; but who can prate this 1 

 The third part of an angle of course exists : and there is 

 therefore no just grounds for affirming that it cannot be 

 found, except by the aid of machinery external to 

 geometry. Till the beginning of the present century, 

 there were the very same grounds for affirming that the 

 circumference of a circle could not be divided into . 

 teen equal parts by common geometry ; but in 1801, 

 Gauss, a distinguished German mathematician, showed 

 how this could be effected without going beyond the 

 limits of elementary geometry. The discovery, though 

 of no theoretical or practical value, made a good deal of 

 noise at the time. The work containing this, and several 

 analogous problems, was translated into French by 

 Delisle, under the title of Recherches A rithm'tique ; but 

 the division of the circumference into seven equal parts ; 

 or, which is the same thing, the problem to inscril>e a 

 regular heptagon, or seven-sided figure, in a circle, like 

 the trisection of an arc, is not yet accomplished. 



From what has now been said, you perceive that 

 Euclid enables us (Proposition II.) to divide the circum- 

 ference of a circle into 3, 6, 12, <tc., equal parts ; as also 

 (Proposition VI.) into 4, 8, 16, etc., equal parts; and 

 again (Proposition XI.) into 5, 10, 20, <tc., equal parts ; 

 and finally (Proposition XVI.), into 15, 30, 60, <tc., 

 equal parts ; all these subdivisions of the circumference 

 being obtained by the propositions here referred to, and 

 the repeated application of Proposition XXX,, Book 

 III. As just observed, Gauss has extended Euclid's 

 instructions, and taught vis how to divide the circum- 

 ference into 17, 34, 68, <tc. , equal parts ; as also into 

 other parts, in like manner excluded from Euclid's 

 divisions namely, into 257,65637, itc. ; each part, by 

 repeated bisections, giving rise to a new series of sub- 

 divisions as above. And by combining some of these 

 with the subdivisions of Euclid as Euclid himself has 

 combined the divisions into three parts and into five 

 parts, in Proposition XVI., to get the division into 

 fifteen parts other sections of the circumference may 

 be obtained. But the actual constructions, even in the 

 simplest of these additional cases the division, namely, 

 into seventeen equal parts, are so very complicate"! as to 

 be of no avail in actual practice ; they are interesting 

 merely as showing that this part of the Euclidean 

 geometry is really susceptible of extension ; and we have 

 occupied your attention in this brief account of the 

 speculations of geometers in reference to the division of 

 the circumference, chiefly that you might see suflicient 

 >u why so few of these divisions are accoin]ilishe<l l>y 

 Euclid, and why he should pass at once from the six- 



