

to bo bisected. 



(ho tv. .'1 i:, with a radius of any ! 



thun half of tin- lim-, d.-i-riin- or draw arcs of > rsecting 



. .u-h otluT ut tin' point, c, above tho straight 1. 



i>, lidnw il . Thru, 



from tlio point of C 



draw a straight linn to tlm ioii 

 iutorspction n ; a-id !!,, M r.u^ht lino 

 A B will l)< lii-ivti-d l.y thn .t might 

 lino C D, at tho point K ; that K 

 is divided into two equal parta, A E, A 

 B B, at tho point 



By this method of construction, a 

 straight lino may bo divided into any 

 numhiT <>f f.[unl ].:irt-, denoted by 

 tho scries 2, 4, 8, 10, 3J, Gl, 128, etc. 



It is not necessary in tho above 

 ruction that tho two arcs at D 



Fig. C. 



Fig. 



bo drawn with tho same radius as tho two arcs at c ; but it is 

 .iry that r<>r/i, ;/i'r bo drawn with tho samo radius ; that 

 is, practically speaking, without shifting tho legs of the com- 

 passes. 



It is self-evident that in Fig. 6 the straight lino c D is bisected 

 by tho straight lino A B at tho point E ; and that A B and c D 

 intersect each other at right angles. The problem therefore 

 teaches us how to draw two straight lines at right angles to 

 each other. 



PROBLEM II. To draw a perpendicular to a straight line from 



a point in it. 



Let A F (Fig. 7) be the straight lino 



to which tho perpendicular is to bo 

 drawn, and B tho point in it. From 

 tho point B, with any convenient 

 radius, less than B A or B F, cut off, 

 or measure off equal parts of the 

 straight lines B A, B F namely, B c, 

 B E ; and from the points c, E, with 

 any radius greater than c B or E B, 

 describe arcs of circles intersecting 

 each other at the point D. Then 

 join D B, that is, draw a straight lino 

 from the point D to tho point B, and B D will bo perpendicular 

 to A v. 



PROBLEM III. To draw a perpendicular to a straight line 

 from one of its extremities. 



Let A B (Fig. 8) bo the straight line, and B one of its 

 extremities, from which tho perpendicular is to be drawn. Take 

 any point c, at a convenient distance from B, and nearly over 

 the middle of the straight line A B ; 

 then with c as a centre, at the dis- 

 tance c B as radius, describe the 

 arc D B E, so that it shall bo greater 

 than a semicircle ; from the point D, 

 draw through the point c, the 

 straight line D c E, to meet tho aro D 

 in the point E; and join E B, that is, 

 draw a straight line from the point E 

 to the point B, and B E will be per- 

 pendicular to A B, at the extremity of 

 B, as required. 



The demonstration of this proposition is founded on the fact 

 that tho angle contained in a semicircle is a right angle. This 

 fact, indeed, is well known to intelligent workmen, who aro 

 accustomed to make use of the F or the T square ; for they try tho 

 accuracy of that instrument by this 

 property of the circle. Thus, if in 

 Fig. 9 A o c were an angle drawn by 

 means of an F or T square, in order 

 to test its accuracy, and consequently 

 that of tho instrument, they -join any 

 two points in the legs of the angle, 

 fay l> c, by drawing the straight line 

 D c ; they bisect it in E by means of 

 the arcs shown in the figure on either 

 side of the straight lino c D, and 

 drawn by the method explained in 

 Problem I. ; and then, with radius E c or E D, they describe tho 

 semicircle D a c ; if tho arc of this Hemicircle passes exactly 



Fig. 8.. 



Fig. &. 



through tho : angle and tho instrument are correct ; 



if not, thoy are incorrect, and the instrument must be adjoited. 



Mi.EM IV. To draw a perpendicular toaitraiijHt liiufrom 



<tt it. 



Lot A ; bo the straight line, and c tho point from 



\vhirli tlio perpendicular is to be ^ 



drawn. From the point c an ac*i.' 

 with iiny radiur, nuffi'-it.-ut to extend 

 beyond the straight liuo A n, describe 

 an aro of a circle D E, intersecting 

 tho straight lino A n in tho points 

 D, E ; then, from those point* an 

 centres, with any radius greater than 

 half tho straight lino D E, d 

 arcs intersecting each other in tho 

 point K; then join c *; that is, draw 

 a straight line from c to r, cutting 

 A B in the point G ; then c a is per- 



i 



Tig. 10. 



pendicnlar to A B, and is drawn from tbo point c, as required. 



PROBLEM V. To draw a perpendicular to a tlraiyht line at or 

 near one of Us extremities, from a point without it. 



Let A o (Fig. 9) be the straight line, o ono of its extremities, 

 and c the point without it, from which the perpendicular is to 

 bo drawn. Take any point D in A o, and join D c ; bisect it 

 in K ; and from tho pom 1 ; E, as a centre, with radios E D or K c, 

 describe the semicircle DOC; then join o c, and it will be per- 

 pendicular to A o. It is evident, from the remarks made on 

 Problem III., that c o is perpendicular to A o, and it ia drawn 

 from the point c, as required. 



Observe, that unless tho point happens fb be exactly in the 

 vertical line above tho point o, the semicircle will not pass 

 exactly through o, but will pass through a point either nearer 

 to or farther from tho point A. In tho latter case, the straight 

 lino A G must be produced till it meets the arc of the semi- 

 circle. This problem is considered as merely a case of the 

 preceding problem, although tho construction bo different. 



HISTORIC SKETCHES. V. 



THE RISING OF THE LABOURERS UNDER RICHARD II. 



ON Whit Monday, 1382, Sir Simon Burley, who is called by one 

 historian " a favourite of King Richard II.," and by another, " > 

 Knight of the King's Household," rode into Gravesend, and 

 seeing one of the townsmen, claimed him as his slave. There 

 was great dissatisfaction and open murmuring among the 

 people, with whom the man was a favourite, and they protested 

 against his removal. The townsman himself loudly declared 

 that he never was slave to any one, to Sir Simon or another, 

 and seeing the sympathy the crowd had with him, he appealed 

 to them for help. Sir Simon claimed the man as the son of one 

 of his female slaves, called nicfs, and disregarding the earnest 

 entreaty of the crowd, would not abate his claim unless he were 

 paid three hundred pounds ef silver a price he well knew the 

 friends of the bondman could not possibly raise. Some disorder 

 ensuing, Sir Simon, who was attended by two Serjeants of law 

 and a following of armed men, pushed on through the crowd, 

 and gave orders that the prisoner should be taken to Rochester 

 Castle. 



As soon as the great man's train had left, the awe inspired by . 

 its presence died away, and the people, whom tho seizure of 

 their fellow had taken completely by surprise, and had also 

 deprived of their power to act, recovered their self-possession, and ; 

 began to cry out with one voice, " Down with the tyrants ! Let 

 us go to Rochester ! Let us join our brethren of Essex ! " 



Tho Essex men had already risen in arms, and were vowing 

 vengeance on all the lords and owners of land, and especially 

 against lawyers, whom they hated as the ministers of the law 

 that crushed them. Norfolk, Suffolk, Cambridgeshire, and some 

 of the other home counties, had been infected with the same 

 spirit. In them the bubbles of rebellion were beginning to rise 

 to tho surface and to break, though as yet there was nothing 

 liko united action. The above-mentioned claim of Sir Simon 

 Burley, made in spite of tho ferment which was going on only on 

 the opposite bank of the river, was the spark which fired tho 

 train of the Kentish men's anger. 



Before time enough had elapsed to throw cold water on the 

 fire, another and more serious offence had been given to th. 



