GEOMETRY. 
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 con- 
tain a space. 6. All the angles within a 
circle cannot amount to more nor less than 
360 degrees, nor in a semi-circle to more 
nor less than 180 degrees. 7. The value, 
or measure, of an angle is not affected or 
changed by the lines whereby 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 position of the entire figure in 
which they meet, but will still be mutually 
perpendicular. 
After thus much preparation, we may 
conclude the student to be ready to pro- 
ceed in the solution of problems, which we 
shall study to exhibit in the most simple, 
as well as in a progressive manner. 
PROBLEM i. 
To describe an equilateral triangle upon a 
given line. Let AB (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 II. 
To divide an angle equally. Fig. 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 alternately 
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 in- 
tersect above at C, and below at D : draw 
the line C D, passing through the points of 
intersection, and the line AB will be di- 
vided into two equal parts. Observe, this 
is an easy mode of erecting a perpendicular 
upon any given 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 circle 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 perpendicular 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, pass- 
ing 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 ; (fig. 6.) as is shown in this 
figure : o, o, are equal; p, p, are equal; s, s, 
are equal. 
PROBLEM VII. 
To describe a triangle with 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 segment from the point B of 
the base, and with the opening CD 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 angle 
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 AC, and 
from O measure towards D with an opening 
equal to A B : then from O make a segment 
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 front O will 
make an angle with O D equal to the angle 
ABC. 
THEOREM IX. 
All right lines severally parallel to any 
given line are mutually parallel, as shown in 
fig. 9, where AB, CD, EF, and GH, 
being all parallel to I K, are all parallels to 
each other severally. 
N. B. They all make equal angles with 
the oblique line O P. 
PROBLEM X. 
To draw a parallel through a given point.' 
Fig. 10. From the end, on any part of the 
given line A B, draw an oblique line to the 
