425 
GEOMETRY. 
■;'c 
A 
E 
B 
% 
;;d 
foot of a pair of compafTes, the other foot reding on 
the point which is taken for the centre ; and the dif- 
tance of the feet, or points, of the compaiTes -is taken as 
the radius: alfo, that the point marked out by a letter 
is to be underftood, when the reference is made to that 
letter. 
54. ft is alfo taken for granted, that a line or diftance 
can be taken between the compaffes, and may be traitf- 
ferred or applied from one place to another. Alfo, that 
one figure can be applied to or laid upon another, or 
conceived to he fo applied.—In any problem, when a 
line, angle, or figure, is faid to be given; that line, an¬ 
gle, or figure,.mud be made, before any part of the oper¬ 
ation is performed. 
PROBLEMS. 
55. To hifeB, or divide into two equal parti, a given line 
AB. —Operation. From the 
ends A and B, with one and the 
fame radius, greater than half A B, 
defcribe arcs cutting in C and D, 
as in article 45.—Then a line 
drawn from C to D, gives E, the 
middle of A B, as required.— The 
proof of this operation depends on 
ar-ticles loi, 99. 
,56. To hifeEl a givert right-lined angle 
ABC.—From B, defcribe an arc 
AC. From A and C, with one and 
the fame radius, defcribe arcs cut¬ 
ting in D. Then a right line drawn 
from B to D will divide the angle 
into two equal parts, as required.-— 
The proot depends on article 101. 
57. From a given point B, in a given right line A F, to draw 
a right line perpendicular to the given 
ifne, when B is near the middle of the 
line. —On cacli fide of B, take the 
equal didances BC and Bfi.*" 
On C and E defcribe, with the 
fame radius, arcs cutting in D. 
Then a right line drawn through 
B and D will be the perpendicu¬ 
lar required.—The proof de¬ 
pends on article 103. 
^8. When B ts at or near the end of the given line. —On any 
_ Convenient point C, taken at 
pleal'ure, with the didance, or 
radius, CB, defcribe an arc 
C,-'' I DBE, cutting A F in D and B. 
; A line drawn through D and C 
T>.-- eL- will cut this arc in E. Then a 
,.-'E right line drawn tltrough B and 
E will be the perpendicular re¬ 
quired.—Tliis depends on arti¬ 
cle 130. 
59. To draw a tine perpendicular to a given right line AB, 
from a point C without that line. 
Cafe I. When the point is nearly 
eppofte to the middle of the given 
line. —On C, with one radius, cut 
AB in D and E. On D and E, 
a right angle 
C 
61. To trifeB,, or divide into three equal parts, 
ABC.—From B, with any radius 
B A, defcribe the arc AC, cutting 
the legs B A, B C, in A, C. From 
A, with the radius AB, cut the 
arc A C in-E, and from C, with the 
fame radius, cut AC in D. Draw 
B E, B D, and the angle ABC will 
be divided into three equal parts.— 
This depends on article 193. 
62. At a given point D, to make a right-lined angle equal to 
a given right-lined angle h. B C.—From 
D and B, with the fame radius, de¬ 
fcribe the arcs E F and A C, cutr 
ting the legs of the given angle in 
the points A, C. Transfer the dif¬ 
tance A C to the arc E F, from F 
to E. Then lines drawn from D-, 
through E and F,wili form the angle 
EDF equal to the angle ABC.— 
This depends on article 101. 
'' 63. To draw a line parallel to a given right line A B.— 
Cafe I. When the parallel line is to pafs through a given point, 
C.—From C, vrith any conveni- 
eat radius, defcribe an arc D F, JE _ . 
cutting AB in D. Apply the ra- t 
dius C D from D to E ; and from \ 
E, with the fame radius, cut the 
arc D F. Then a line drawn 
through F and C will be paral- 
lei to AB.—This depends on articles lOi, 95. 
D 
"I)"' 
A 
C 
This operation 
E 
B 
tance C from AB.—From the 
points A and B, with the radius 
C, defcribes arcs D and E. Then 
lay a ruler to touch the arcs D 
and E, and a line drawn in that 
pofition is the parallel required.—* 
A. G 
B 
EE 
65. Upon a given line A B, to 
make an equilateral triangle. —From 
the points A and B, with the ra¬ 
dius A B, defcribe arcs cutting in 
C. Draw C A, CB,-and the figure 
A B C is the triangle required.— 
The truth of this operation is felf 
evident; for the fides are radii of 
equal circles. A. 
G- E.-' 
A 
66. By a like opera¬ 
tion, an. ifofteles tri¬ 
angle D E F may be 
conftrufled on a given 
bafe D E, with the 
E given equal legs D F, 
E F, either greater or lefs than the bafe D E. 
ting in F. Then a line drawn from 
C to F will be the perpendicular 
required.—This depends on arti¬ 
cles lOI, 99. 
60, Cale II. When C is nearly oppofte to one end of the 
given line A B.—To any point D in 
AB, draw the line CD. Bifeft 
•. the line C D in E. Oh E, with the 
En;''" 1 radius EC, cut AB in G. Then 
C G being dr.awn, will be the per¬ 
pendicular required. This de¬ 
pends on article 139. 
the fides of which fhadl 
C 
D 
A 
"B 
67. To make a right lined triangle, 
be refpeEiively equal, either to thqfe of 
a given triangle ABC, or to three 
given lines, provided any two of them 
taken togetker are greater than the _ 
third. —Draw a line D E equal to 
the line A B. On D, with a ra¬ 
dius equal to AC, defcribe an 
arc F. OnE, witharadiusequal 
to B C, defcribe another arc, cut- _ 
ting the former arc in F. Then B 
draw F D, F E, and the triangle D F E will be that 
required, " 
. 48. Upast. 
