PROBLEM I. n. ] 



MATHEMATICS. PRACTICAL GEOMETRY. 



601 



PROBLEM II. To prolong or produce the straight line 

 (A B) to any lenyth towards (C), in a straight line. 



Place one edge of the ruler upon the point B at the 

 extremity of the line Fig. 6. 



A B ; take any point A a B C 



in A B, such as a, so ' 



far from B that the distance may be less than the 

 length of the ruler ; make the ruler coincide with that 

 point a, and draw as before with the pencil or pen from 

 B towards C, which will leave the line produced as 

 required. This operation may be repeated as often as 

 wished, so as to prolong a line to any extent. 



PROBLEM III. To describe a circle 

 from a centre (O), and at any dis- 

 tance (O A) from the centre (O). 



Place the fixed leg or point of the 

 compasses in the centre O, opening 

 them until they take in the given 

 distance O A or radius of the circle 

 to be drawn ; then move them round, describing the 

 Fig. 8. circumference of the required 



circle. 



After performing the three 

 preceding problems, it will be 

 useful, in order to acquire a 

 free use of the instruments in 

 describing circles, to draw lines 

 through points of intersection, 

 join points by straight lines, 

 and thus to produce the accom- 

 panying figure several times. 

 The studen* may thereafter proceed to the construction 

 of the following problems. 



PROBLEM L 



To bisect or divide into two equal parts a given straight 

 line (A B). (Euclid, Book 1, Prop. X )* 



IST METHOD. When there w not sufficient space in the 

 drawing on either side of the given line. By means of 

 the compasses, place a leg at one of the extremities of 

 the line, open them until the other leg is at any other 

 point near the middle of the line ; mark this point ; 

 transfer the leg of the compasses to the other extremity 

 of the line, and mark the distance of the other leg from 

 this point ; if the two points so marked do not coincide, 

 open the compasses to as near the middle of these two 

 points as the eye will direct : proceed as before ; the 

 distance between the new points will be smaller than 

 before, and a trial or two will enable us to bisect the 

 line accurately. 



2.VD METHOD, f When the given line (A B) occurs in 

 Fig. 9. or near the centre of the draw- 



ing, and there is sufficient 

 room on both sides of the given 

 line (A B). With the centre 

 A (Fig. 9), and any radius 

 greater than the half of A B, 

 describe an arc ; with the 

 centre B, and the same radius, 

 describe another arc, cutting 

 the former in D and E ; join 

 the points of intersection D 

 and E, cutting the line 

 A B in C, . which will 

 bisect A B in the point C. 



SRD METHOD. When the given line (A B) occurs near 



* References are made to the propositions in Euclid, where several of 

 the Problems are demonstrated, in order that the student, should he H i-h 

 to investigate the Problems further, may do BO by examining the methods 

 of construction there adopted. 



i This is the method most frequently used to bisect a line ; but in prac- 

 tice it is not necessary to draw the whole arcs, but merely the intersec- 

 tions, and by means of the ruler placed on those points, to mark the 

 centre point C. The line D C or D E, in this and the following Problem, 

 when drawn, will also be found to be perpendicular to A B. 

 VOL. L. 



Fig. 10. 



f N 



Fig. 11. 



\, 



the bottom of the drawing, and 

 sufficient room is only left on 

 one side of the given line (A B). 

 With centres A and B (Fig. 

 10), and any equal radii 

 greater than the half of A B 

 (this radius should be as long 

 as can be conveniently got), 

 describe arcs cutting each 

 other in the point D. With 

 the same centres, and less 

 equal radii, also greater than 

 the half of A B, describe arcs 

 cutting each other in the point E ; join D E, and pro- 

 duce the line until it cuts A B in C, which will be the 

 point of bisection of A B. 



PROBLEM II. 



To draw a perpendicular or a straight line at right angles 

 to a given straight line (A B) from a given point (C). 

 (Euclid, Book I., Prop. XI. and XII.) 

 Isr METHOD. When the given point (C) is within or 

 without the given line 

 (A B) and near the middle 

 of it. With the centre 

 C (Fig. 11), and any 

 radius, describe an arc 

 cutting the line AB in 

 the points E and F. 

 With the centres EF, and 

 B the same, or any other 

 equal radii, describe arcs 

 cutting each other in the 

 point D ; join D C, then 

 the line D C is perpen- 

 dicular to the line A B 

 as required. 



2ND METHOD. When 

 / the given point (C) is 

 _ within and at or near the 

 B extremity of the given 

 line (AB). Take any 

 point F (Fig. 12) above 

 the line A B, and with the 

 radius F C describe the 

 arc BCD, cutting AB in 



the point E ; join E F, and produce it until it cuts the 



arc E C D in the point D, and join D C, which will be 



perpendicular to A B, as required. 

 3RD METHOD. When the given point (C) is without 



and opposite the extremity Fig. 12. 



of the given line (A B). Jn 



the line AB (Fig 13) take 



any two convenient points 



E, F. With the centre E, 



and radius E C, describe 



the arc C G D, and with 



the centre F, and radius 



F C, describe an arc cut- 

 ting the former arc C G D 



in the point C and D ; join 



C D ; the line C D will be perpendicular to the given 



line A B, as required. 

 A very simple method of 



setting off a perpendicular 



on the ground may be per- 

 formed with a tape line in 



the following manner : 



Suppose it were required to 



set out a straight line of 



road (C D) (Fig. 12) at right 



angles, or perpendicular to 



another straight line of road 



(A B) from any point (C) 



in A B- Put in a pin at 



the point C, and also one 



at E, along one side of the 



line of road A B, and three 



feet distant from C ; put the end of the tape line at E, 



4 H 



\ 



Fig. IS. 



