LESSONS IN GEOMETRY. 



213 



the point* A and B an centre*, and the radii A B and B A 

 respectively, describe the semicircles A D, B c E, intersecting 

 each other in the point c. Bisect A B in F, and through the 

 point i- draw F a perpendicular to A B or x Y. Next trisect the 

 are A c in the points H and K, and the arc B c in the points 

 i. and M ; and from the point A, through the points if, L, and c 

 in the arc B c, draw the straight lines A N, A o, A P of indefinite 

 Ifir.'th ; and from B, through the points H, K, and c, in the arc 

 A c, draw the straight lines B s, B R, and B Q, also of indefinite 

 length. From the points A and B draw the straight lines A T, 

 B u to the points, T, u, in which the straight linos B s, A N 

 cut the semicircles B c K, A c D ; bisect AT, B u in the points 

 v and w respectively, and through the points v and w draw 

 the perpendicular lines v I, w z of indefinite length, intersecting 

 each other and the straight line F o in the point a. This 

 point is the centre of a circle circumscribing the required nona- 

 gon. From the point a as centre, with the distance a A or a B, 

 describe the circle A d B. This circle passes through the extremi- 

 ties of the given straight line A B, the points T and u in which 

 the straight lines B B, A N respectively intersect the semicircles 

 B c E, A c D, and the points c and e in which the straight lines 

 B Q, w z and A P, v i intersect each other : it also cuts the 

 straight lines B R, F o, and A o in the points I, d, and /. Join 

 T 6, b c, e d, d e, ef, and / u : the figure Aifccde/usisa 

 nonagon, and it is described, as required, upon the given straight 

 line A B. 



The construction of the uneven-sided polygons, the heptagon 

 and nonagon, by the aid of the ruler and compasses, have been 

 given to show the learner that there is no regular polygon of any 

 number of sides that could not be constructed without having 

 recourse to the measurement of the angle of the polygon or the 

 angle at the centre of the circumscribing circle. The construction 

 of the decagon and dodecagon on any given line by the ruler and 

 compasses alone we do not give, because either figure may be con- 

 structed by learners, if they will exercise a little thought, and it 



will afford them two 

 useful exercises to 

 do so. We shall 

 therefore conclude 

 our problems on the 

 construction of the 

 regular polygons 

 with the method of 

 constructing an un- 

 decagon or eleven- 

 sided figure on a 

 given straight line, 

 and then bring our 

 Lessons on Geome- 

 try to an end with 

 a brief description 

 of the methods used 

 for drawing the el- 

 lipse, parabola, and 

 hyperbola, curves made by the section of a right cone in parti- 

 cular directions ; the mode of tracing a spiral ; and one or two 

 other things, such as the connection of two curves by a straight 

 line, etc., which may be of practical use to our students. 



PROBLEM LVII. To construct an undecagon on any given 

 straight line. 



Let A B (Fig. 81) be the given straight line on which it is re- 

 quired to construct an undecagon. First bisect A B in c, and pro- 

 duce A B indefinitely both ways to x and Y. Then along c x set 

 off a line, c D, equal to three times A B, or six times c A, and 

 along c Y set off a line, c E, equal to c D. From the point A as 

 centre, with the distance A E, describe the arc E T, and from the 

 point B as centre, with the distance B D, describe the arc D z, 

 and let the arcs E T, D z intersect each other in the point F. 

 This point is the apex of the undecagon, the straight lino A B on 

 which it is constructed being considered as its base. From the 

 point F as centre, with a radius equal to A B, draw small arcs 

 cutting the larger arcs D z, E T in G and H, and draw the chords 

 F a, F H. Join c F : the straight line drawn from c, through F, is 

 perpendicular to A B, and the centre of the circle circumscribing 

 the required nndecagon will be in c F. To find the centre, bisect 

 F O, F H in the points K and L, and join A L, B K. The straight 

 lines A L, B K intersect each other and the straight line c F in the 

 point M, which is the centre of the circumscribing circle. From 



the point M M centre, with toe distance v A or M B, describe 

 the circle A B r. This circle puce* through A, B, r, a, H, tht 

 extremities of three nide of the required nndooafon that have 

 been already determined, and the vertices of the remaining 

 angles of the polygon will be found in itn circumference. To 



Fig. 80. 



Fig. 81. 



determine them, with an opening of the compasses equal to A B, 

 set off from A, along the arc A o, the arcs AN, NO, OP, and 

 along tho arc B H, from B, set off the arcs B Q, Q R, R s. Joiii 

 the chords AN, NO, OP, P o, B Q, Q R, R s, s H. The figure 

 ABQESHFQPONisan undecagon, and it is described on 

 the given straight line A B, as required. 



The reader will have noticed, doubtless, that the method of 

 constructing an undecagon on a given straight line by a purely 

 geometrical process, as given above, is similar in all essential 

 details to the process used for constructing a heptagon on a 

 given straight line, and it is based in both cases on the 

 numerical relation of the straight line on which either is to be 

 constructed, to the sides of an isosceles triangle whose vertex is 

 the apex of the polygon, and whose base is the given straight 

 line. In the case of the heptagon, the proportion of the base to 

 the sides of the isosceles, whose vertex is the apex of the polygon, 

 is as 1 to 21 or 2'25 ; and to construct a heptagon on any given 

 straight line, we have only to produce it indefinitely both ways, 

 and find points on either side of each extremity at a distance 

 equal to 1* of the given line, or to bisect the given line and set 

 off on either side of the perpendicular section straight lines 

 equal to 1? of the given line. In the case of the undecagon, 

 the proportion of the base to the side of the isosceles triangle, 

 whose apex is the vertex of the polygon, is as 1 to 3| or 3*5 ; and 

 to construct an undecagon on any given straight line, we have 

 only to produce the given straight line indefinitely both ways, 

 and set off from either extremity lines equal to 2^ of the given 

 straight line, or to bisect the given line, and from the point of 

 bisection to set off on either side of it, along the given line pro- 

 duced indefinitely, lines equal to 3 times the given straight line. 

 We have added these remarks on the geometrical constructions 

 that we have given of the heptagon and undecagon, in the hope 

 that they may give the student a clue to other geometrical con- 

 structions of a similar character. Wo also recommend to hil 

 notice the geometrical construction of the nonagon, based on the 

 preliminary construction of an equilateral triangle on the given 

 straight line on which it is required to construct the nonagon, 

 and the trisection of the angles on either side of the base, or 

 the arcs that are described opposite to them by drawing semi- 

 circles from either extremity of the base as centres, with a radius 

 equal to the base. 



In drawing figures to exhibit the methods of constructing the 

 different polygons, from the pentagon to the nndecagon, that 

 have been given in detail in this and preceding lessons, the 

 student is advised, for the sake of accuracy, to make them on 

 a large scale ; as, if he attempt to construct his figures in the 

 limited space in which are drawn the figures that are used to 

 illustrate our Lessons in Geometry, he may fail to complete 

 them to his satisfaction, in consequence of not being able to 

 draw the straight lines and arcs, of which the figures are com- 

 posed, of suitable fineness, and to subdivide the arcs, whenever 

 it is necessary to do so, with sufficient accuracy. In all cases, 

 for the sake of good practice, the straight line on which a poly 

 gon is to be constructed, should never be taken lees tham an inch 

 in length. 



