GEOMETRY. 



bers) the whole will be equal. 3. If from 

 equal things we take equal things, the 

 remainder will be equal, and the reverse 

 in respect to unequal things. 4. The 

 whole is greater than any of its parts. 

 5. Two right lines do not contain a 

 space. 6. All the angles within a circle 

 cannot amount to more nor less than 

 360 degrees, nor in a semicircle to 

 more nor less than 180 degrees. 7. The 

 value, or measure, of an angle is not 

 affected or changed by the lines where- 

 by it is formed being either lengthened 

 or shortened. 8. Two lines standing at 

 an angle of 90 degrees from each other 

 will not be affected by any change of po- 

 sition of the entire figure in which they 

 meet, but will still be mutually perpen- 

 dicular. 



After thus much preparation, we may 

 conclude the student to be ready to 

 proceed in the solution of problems, 

 which we shall study to exhibit in the 

 most simple, as well as in a progressive 

 manner. 



PROBLEM I. 



fo describe an equilateral triangle upon a 

 given line. Let A B (fig. 1.) be the given 

 line, with an opening of your compasses 

 equal to its length : from each end, A and 

 B, draw the arcs C D and E F, to whose 

 point of intersection at C draw the lines 

 A C and B C. 



PROBLEM 11. 



To divide an angle equally. Pig. 2. Let 

 B A C be the given angle, measure off 

 equal distances from A to B, and from A 

 to C ; then with the opening B C draw al- 

 ternately from B and from C the arcs 

 which intersect at D : a line drawn from 

 A to D will bisect the angle B A C. 



PROBLEM III. 



To bisect a given line. Fig. 3. Let A B 

 be the given line ; from each end (or 

 nearer, if space be wanting,) with an 

 opening of your compasses rather more 

 than half the length of A B, describe the 

 arcs which intersect above at C, and 

 below at D : draw the line C D, passing 

 through the points of intersection, and 

 the line A B will be divided into two equal 

 parts. Observe, this is an easy mode 

 of erecting a perpendicular upon any gi- 

 ven line. 



PROBLEM IV, 



To raise a perpendicular on a given point 

 in a Line. Fig. 4. With a moderate open- 

 ing of your compasses, and placing one of 

 its legs a little above or below the given 

 line, describe a circle passing through 

 the given point A on the line B C ; then 

 draw a line from the place where the cir- 

 cle cuts at D, so as to pass through E, 

 the centre to F on the opposite side of the 

 circle : the line F A will be the perpendi- 

 cular required. 



PROBLEM v. 



From a given point to let fall a perpendi- 

 cular on a given line. Fig. 5. From the 

 given point A draw the segment B C, 

 passing under the line D E ; bisect B C in 

 F, and draw the perpendicular A F. 



THEOREM VI. 



The opposite angles made by intersecting 

 lines are equal; (tig. 6.) as is shown in this 

 figure : o, o, are equal ; p, p, are equal - r 

 s, s, are equal. 



PROBLEM VII. 



To describe a triangle ivith three given 

 lines. Fig. 7. Let A B, B C, and C D, 

 be the three given lines ; assume either of 

 them, say A B, for a base ; then with an 

 opening equal to B C, draw the seg- 

 ment from the point B of the base, and 

 with the opening C D make a segment 

 from C : the intersection of the two 

 segments will determine the lengths of 

 the two lines B C and C D, and of the an- 

 gle ABC. 



PROBLEM VIII. 



To imitate a given angle at a given point. 

 Fig. 8. Let A B C be the given angle, and 

 O the point on the line O D whereon it is 

 to be imitated. Draw the line A C, and 

 from O measure towards D with an open- 

 ing equal to A B : then from O make a seg- 

 ment with an opening equal to B C, and 

 from K make a segment with an opening 

 equal to A C ; their intersection at E will 

 give the point through which a line from 

 O will make an angle with O D equal to the 

 angle ABC. 



THEOREM IX. 



Ml right lines severally parallel to any 

 given lii\e are mutually parallel, as show-n in 



