LESSONS IN GEOMETRY. 



protractor, or circle and soalo of chorda in the second placo ; 

 or, lastly, by a regular process of geometrical construction, 

 .-lion of the triangle, or trigon, which stands first in 

 ;lar polygons, already referred to, and which 

 is iii.' .-.1'iiliitoral triangle, has boon explained in Problem XVII. 

 i \ ..;. I., page 209), and that of the square, or tetragon, in Problem 

 \\lll. (Vol. I., page 255). As the remarks which will bo made on 

 the construction of the pentagon by the different methods above 

 -miiiu>rated will apply equally to those, due regard being had to 

 the difference in the number of the sides and the opening of the 

 . nothing more need be said about them except in availing 

 ourselves of them in the construction of figures whose Rides are 

 multiples of 3 and 4 ; as the hexagon, the number of whose 

 sides is 3 x 2, or 6 ; and the octagon, the number of whoso sides 

 is 4 x 2, or 8 ; any number being a multiple of a lesser number 

 when it can be divided without a remainder by that number. 

 This brings us to 



PUOBLKM XLVI. To construct a regular pentagon on a given 

 straight line by the aid of the protractor only. 



Let A B (Fig. G5) be the given straight lino on which it is re- 

 quired to construct a 

 regular pentagon. Pro- 

 duce A B indefinitely 

 both ways to x and T, 

 and apply the base, or 

 straight edge of the 

 protractor, to the 

 straight line x Y, so 

 that the mark or notch 

 in the centre of the 

 base line shall rest on 

 the point A. On re- 

 ferring to the table 

 given in the last lesson, 

 we find that tho "angle 



Fig. 65. 



of the polygon " for tho pentagon, or, in other words, the 

 angle formed by any two sides of the pentagon, is 108 deg. The 

 degrees on a protractor are usually marked by tens ; look there- 

 fore to the left of 90 for 110, and make a mark on your paper 

 at the extremity of the second line to the right of 110 on the 

 protractor, which will of course indicate 108. This is represented 

 by the point z in Fig. 65, the extremity of the fine line on tho 

 semi-circular arc of the protractor marked 108. Join A z and 

 produce it indefinitely to P, and along A p sot off A c equal to 

 A B. Apply the protractor to the straight line A p, so that tho 

 mark in the centre of its base line may rest on c, as indicated 

 by the dotted lines in the diagram, and determine the direction 

 of c Q, and set off along it c D equal to c A or A B. Repeat tho 

 process to find the point E, and then join E B. The figure 

 A c D E B is a regular pentagon, and it is described on the given 

 straight line A B as required. 



The process above described applies equally to the construc- 

 tion of any regular polygon having tho angle of the polygon 

 expressed in degrees only ; but when the angle of tho polygon 

 is expressed in degrees and fractional parts of a degree, this 

 mode of construction would be attended with difficulty and 

 much uncertainty, unless the protractor were sufficiently large to 

 admit of minutes being marked along its edge as well as degrees. 

 For the sake of practice, which alone can ensure accuracy as 

 well as neatness and nicety in geometrical drawing, the reader 

 may attempt the construction, by means of the above process, 

 of all the regular polygons in the table given at the end of the 

 last lesson, whose angles are expressed in degrees only. 



PROBLEM XL VII. To construct a regular pentagon on a given 

 straight line by the aid of the circle and protractor. 



Let A B (Fig. 66) be the given straight line on which it is 

 required to construct a regular pentagon. Bisect A B in F, and 

 at the point F draw the straight line F a of indefinite length 

 at right angles to A B. On referring to the table we find that 

 each of the angles at the base of the triangles into which tho pen- 

 tagon may be divided by drawing straight lines from each of its 

 angles to the centre, or, in other words, half the angle of the 

 polygon, is 54 degrees. Apply the protractor to the straight 

 line A B, BO that the mark in the centre of the base line or 

 chord may rest on A, and set off the straight line A H of inde- 

 finite length, making on angle of 54 deg. with A B. From the 

 point K, the point of intersection of tho straight lines A H, F o, 

 with the radius K A. describe the circle A E D c B, and opening 



; 



the oompaues to tho extent of the straight line A B, set off the 

 point** K, D, c, in the circumference of the circle A E DC B. Join 

 A B, E D, D c and c B. The figure A K D o B U a regular pentagon, 

 and in described on tho straight line ABM required. 



If it be required merely to describe a regular pentagon, 

 nothing being raid about it* 

 construction on a given straight 

 lino, we may proceed in theJJJJ 

 following manner : From any "" 

 point, K (Fig. 66), aa centre, 1 '^ 

 with iiny distance, K A, as 

 radius, describe tho circle ADD. 

 Join A K, and at the point K, *- 

 in the straight lino A K, make, 

 by tho aid of the protractor, 

 tho angle A K B equal to u.\\ir 

 angle of 72 deg. This is done 

 because tho angle at the apex . . 

 of tho triangles into w.tich a 

 pentagon may be divided by 

 drawing straight lines from each of its angles to its centre, 

 is on angle of 72 deg., as may be seen from the table. Join 

 A B, and opening the compasses to the extent of A B, set off 

 tho points E, D, c in the circumference of the circle A D B, aa 

 before, and join the chords A B, B D, D c, c B, to complete the 

 pentagon. 



This gives us the key to the method of inscribing any regu- 

 lar polygon in a given circle, or, in other words, the method of 

 dividing the circumference of a given circle into any number of 

 equal parts. This method may be generally expressed for any 

 polygon as follows : 



At the centre of tho given circle make an angle containing 

 the number of degrees in the angle at the apex of the equi- 

 angular and similar triangles into which the polygon may be 

 divided by drawing straight lines from each of its angles to it* 

 centre. Join the points in which the circumference of the 

 circle is intersected by the legs of the angle, or, in other words, 

 draw the chord of the arc of the circumference of the circle 

 intercepted between the legs of the angle. The straight line 

 thus obtained will be one side of the required polygon, which 

 may be completed by setting off arcs along the circumference 

 of the circle, by opening tho compasses to the extent of the 

 chord of the arc intercepted between the legs of the angle made 

 at the centre of the given circle, to determine the length of the 

 side of the required polygon. The polygon is of course finally 

 finished by drawing tho chords of the arcs thus set off along 

 tho circumference of the given circle. 



The student is recommended to attempt the construction of 

 all tho polygons named in tho table whose angles are expressed 

 in terms involving degrees only. 



PROBLEM XL VIII. To construct a regular pentagon on a given 

 straight line without the aid of the protractor and the circle ; that 

 is, by a regular process of geometrical construction, 



Let A B bo tho given straight line on which it is required t 

 construct a regular pentagon without tho aid of the protractoi 

 and circle. Produce A B indefinitely both ways to x and T, and 

 bisect it in c. At the extremity A draw the straight line A r, 

 of indefinite length, 

 at right angles to 

 A B, and along it set 

 off A E, equal to A B, 

 am 1 , join c E. Then 

 from c as centre, 

 with the distance 

 C B, describe tho 

 semicircle F B a, 

 meeting the straight 

 lino x Y in the points 

 F and a. From A 

 as centre, with tho 

 distance A o, de- 

 scribe the arc o B, 

 and from B as cen- 

 tre, with the distance B F, describe the arc T K, and let the 

 arcs OH, F K, intersect in tho point t. Bisect the arcs r L, 

 o L, in the pointa M and N, and join AM, ML, B X, N L. The 

 figure A B N L M is a regular pentagon, and it is described on the 

 given straight line A B as required. 



Fig. 67, 



