427 
geometry. 
p 68. upon agivtn line - A B, to de- 
fcribe a/quare.—Dn^'i B C perpen. 
dicular, and equal, to AB. On A 
and C, with the radius A B, de- 
fcribe arcs cutting in D. Then 
draw DC, dX; and the figure 
*A B CD is the fquare required.— 
■ This depends on articles lOi, 104, 
9 J> 30- 
6 c). To dcfcribea reSangk whofe length Jhall be equal to a 
given line, £ F, and breadth equal to .pj- ^ 
another given line G.—At E and h — 
ereiSt two perpendiculars, E H and 
FI, each equal to the given line G. 
Draw HI, and the figure EFIH 
will be the reiStangle required.— 
This depends on articles 29, loi, 
104, 95. 
70. To find the centre of a circle. 
—Draw any chord AB. Bilect 
A B with the chord C D. Bifetl 
CD with the chord E F, and 
their interl'edlion G will be tlie 
centre required.—This depends 
on article 105. 
71. To divide the circumference of a circle into two, four, 
eight, fixteen, thirty-two, &c. equal parts. —A diameter A B 
divides the circle into two equal 
parts. A diameter DE, per- " 
pendicular to AB, divides the _ 
circumference into four equal 
parts. On A, D, Bjdefcribearcs 
cutting in a, b •, then by the in- 
tej fedfions a, b, and the centre, 
the diameters F G, HI,, being 
efrawn, divide the circumference 
into eight equal parts ; and fo 
on by continual bifeftion.—For at each operation, the 
intercepted arcs are bifedted, and the parts doubled. 
72. Todeferibe a circle, the circumference of which fall pafs 
through three given points ABC, 
provided they do not lie in one right 
line, —Bifect thediflanceC Bwith 
the line D E. Bifedl the diftance 
A B with the line F G. On H, 
the interfedtion of thefe lines, 
with the dillance to either of the 
given points, deferibe the circle 
required. This depends on ar¬ 
ticle J25.. 
t'icTo draw a tangent to a given cir- g 
cle, that flail pafs through a given point —^ 
A.—Cafe I. When A is in the circum¬ 
ference of the circle, h'rom the centre 
C, draw the radius C A. Through 
A draw B D perpendicular to C A, 
(58,) and B D is the tangent re¬ 
quired.—This depends on article 
X26. 
74. Cafell. When the given point A-. 
is.without the given circle. —From the 
centre C, draw C A, which bifeft 
in B. On B,. with the radius B A, 
cut the given circumference in D. 
Through D, the line A E being 
drawn, wfill be the tangent re¬ 
quired.— This depends on articles 
J3P, 126.. 
75. Totwogiven right lines, A, 
B, to find a third proportional .— 
Draw two right lines making 
any angle, and meeting in a. 
In thefe lines, take abzviiirA tz 
term, and ac, ad, each equal 
to the fecond term. Dr.aw 
bd, and through c, draw ce, 
parallel to ^o'; then ae is the 
third proportional fought. 
And a b ; ac :: a d : ae,— 
This depends on article 165 
To three given right lines, A, B, C, to find a fourth pro¬ 
portional. —Draw two riglit 
lines making any angle, and 
m-eting in a. In thefe lines 
take a ^ fir ft term,ac= fe¬ 
cond term, and a lin^third 
term. Draw be, and par- 
rallel to it, through rf, draw 
de-, then ae is the fourth 
proportional required. And 
ab •. ac :: ad : ae. —Thi^ 
depends on article 165, 
lines,'A, B, to find a mem 
B- 
A- 
Between two given rig 
proportional. —Draw a right line, 
in which take acr::: A, a^=B 
Bifedt in F (55) ; and on F, 
with the radius F b, deferibe a- 
femicircle bee. From a draw ae 
perpendicular to (57) ; then 
ae is the mean proportional re¬ 
quired. And ac •. a eae •. ab ,— 
This depends on article 171. 
78. To divide a given line B C, in the fame proportion as a 
given line A is divided. —From one end B of B G draw 
1111 B D, making any angle with B C , 
^ (g In BD apply from B the feveral 
divifions of A ; fo B D will be 
equal to A, and alike divided. 
Draw G D ; then lines drawn pa¬ 
rallel to C D through the fevera! 
'’'B divifions of B D, will divide the 
line B C in the manner required.—This depends on ar. 
tide 165. 
79. To divide a given right line A B into a propofed number 
of equal parts-, (fiuppofe q.) —From. _ • 
A,.one end of AB, draw AE to - ^ 
make any angle with A B ; and 
from B, the other end, draw B F, 
making the angle ABF equal to 
the angle BAE. Begin at A 
and B, and fet off, on thefe lines, 
as many equal parts, except one, 
as AB is to be divided into, 
viz. I, 2, 3, 4, 5, 6. Then lines 
drawn from i to 6, 2 to 5, 3 to 4, 
&c. will divide A B, as was re¬ 
quired.—^I'his depends on article 165. 
80. Hence is derived the method of making diagonal 
fcales. Let a fcale be conftruded to i2ths of the line 
A Bi—Having divided AB into 
three equal parts, draw two pd- 
rallel lines A H, B K,. making 
any convenient angles with AB : 
in thofe lines take four equal 
diftances, fupp.ole from A to D, 
and from B to R ; and through 
the points of divifioii draw four 
lines parallel to AB ; next di¬ 
vide D R into three equal parts; 
then if the poigis ®f divifton in A B and D R are joined 
diagonallyp 
