4'ia 
GEOMETRY. 
diagonally, the fcale is confti‘u6led.—:For by fimilar tri¬ 
angles, R-B : B S :: R O : O P ; therefore R O being ^ of 
R B ; OP will be ^ of B S, or-^- of ^ (or ■^) of B A ; 
and the next divifion n ji is -’j, See. 
If Q R =: C B B A is the fcale for a foot, OP is an 
inch, n =: 2 inches, I P 2= 13 inches, dec. 'But if we 
divide A B into four equal -parts, only tliree :nu(t be 
taken in AH and B K to make laths of AB (becaufe 
•4X3=1:). 
Generally, refolve tlte number to which the div'ifions 
are to be extended, into tv/o factors, then divide tlie 
given line (A B) into as many equal parts as there are 
units in one factor, and take as'many equal parts in the 
oti.er lines (AH, B K,} as tliere are units in the other, 
'lihus if A B is divided into three equal parts, and five 
ate taken in AH, B K ; or if A B is divided into five, 
and tliree are taken in A H, B K, in either cafe the fcale 
gives i5ths of AB. On the common plain fcales, the 
equal parts in each line are ten, tvhich give the divifions 
in looths.—A'line divided into equal parts, and one of 
the parts fubdivided, is called a litis or fcale of equal parts. 
A variety are to be found on tlie common plain fcale be¬ 
longing to a cafe of inllruments. 
A line of chords is made by transferring the divi¬ 
fions on the arc of a qua¬ 
drant to its chord. Thus, 
fuppofe ABC is a quadrant, 
and the right line BA the 
chord of its arc BDA. Let 
this arc be divided into 90 
equal parts or degrees: then 
if one foot of a pair of com- 
palfes be kept on the point 
B, and arcs fucceflively de- 
feribed with the other, from 
each of the 90 divifions in 
33 B D A to meet BA, thofe 
arcs will divide it into a line of chords. 
82. To make an angle of a prqpqfed number of degrees .— 
Take the firfi, 60° from the fcale of chords, and with 
this radius deferibe an arc B C, the 
centre of wliich is A. Take the 
cliord of the propofed number of 
degrees from the fcale, reckoning 
from its beginning, and apply this 
dillance to tlie arc B C, from B to C. 
Then lines drawn from A, through 
the points B and C, ivill form an 
angle B A C, the meafure of which is the degrees pro- 
poled.—If the angle exceed 90°, take half the number 
of degrees, and let it off twice, from B to C, and from 
C to C. 
83. To meafure-a given angle B A C.—From the angular 
point A, with the chord of 60°, deferibe the arc B C, 
cutting the legs in the points B and C. Then the dif- 
tance B C being applied to the chords, from the begin¬ 
ning, will Ihew the degrees wliich meafure the angle 
B A C. 
84. In a given circle to inferibe a regular polygon, the number 
of jidts being given. —Divide 36,0° by the number of lides, 
and the quotient will be the de¬ 
grees which meafure the angle 
at the centre of the circle iub- 
tended by a fide o.f the poly¬ 
gon. For there will be as many 
equal angles at the centre as 
there are equal fides. And the 
whole circumference ineafures 
_ the fimi of all the angles at the 
c Jy centre.—.Draw the radius .C B, 
make an angle BCD equal to thofe degrees, and draw 
.»he chord D B ; then will D B be the fide of the poly- 
, '- • ■'f 'tviiich applied to the circumference from 
B to (7, a to b, b io c, &c. will give the points to which 
the Tides of the polygon are to be drawn. 
83. If the polygon has an even number of fides, draw 
tlie diameter A B ; and divide half the cipetuiiference, 
as before; then lines drawn from thefe points through 
the centre, as E D, will give the remaining points, in 
the other femicircumference. 
86. On a given right line A B, to cottJlruEl a regular polygon 
of any ajjigned number offides. —Divide 360° by the number 
of fides; fubtrafl the quotient 
from 180°, the remainder will be 
the degrees whicli meafure the 
angle made by any two adjoin¬ 
ing fides of that polygon, and is 
called the angle of tlie polygon. 
For each fide of the polygon, 
witJi radii drawn to its ends, form 
an ifoceles triangle. Then the 
angl of tlie polygon is derived 
from articles 83, 96, 104.—At the ends A, B, of the line 
A B, make angles ABC, BAD, equal to tlie angle of 
the polygon. Make AD, B C, each equal to AB. 
H hen at the points C, D, make angles equal to that of 
tlie polygon as before ; and let the fides including thofe 
angles be each equal to A B ; and tlius proceed until 
the polygon is conftruCted. 
In figures of any number of fides, the two lafi: fides 
D E, C E ; or E G, H G ; are 
readied found by deferibing arcs 
from C and D, or from E and H, 
witli tlie radius A B, interfered y. 
in E, or in G.—Or, in figures of 
an even number of lides, liaving 
drawn half the number, A D, 
A B, B C, C F, by means of the jf 
angles; tlie remaining fides may 
be tound, by drawing through 
the points D and F the lines 
D 1 ', I' H, parallel and equal to their oppofite fides C F, 
A D ; and fo of the red. 
87. About a given regular polygon, to circvmfcribe a circle: 
or xoithin that polygon to inferibe a circle. —:Bife£l: any two 
angles FAB, CBA, with the 
lines A G, B G; and the point G, 
where they interl'eci one another, 
will be tlie centre of the polygon. 
A circle deferibed from G, with 
the radius G A, will circiunfcribe Fjl 
the given polygon.—This de¬ 
pends on articles 85, 96, 104. 
88. Again, bifeClany two ddes, 
F E, ED, in the points H and 1; 
and draw H G, I G, at right'angles to F E, ED; then 
the point G, where they interfect each otlier, will be 
the centre ot the polygon.—A circle deferibed from G, 
with the radius G H, will be inferibed in the given po¬ 
lygon.—This depends on article 126. 
89. On a given right line AD, to deferibe a fegment of a 
circle, that fiall contain an angle equal to 
a given right-lined angle C.—Make an 
angle DAF equal to the given angle 
C. From H, tlie midd le of AD, draw 
H I at right angles to A B, and from 
A draw A I at right angles to A F, 
cutting H I In I. From I, with the 
radius I A, deferibe a circle. Then 
will the fegment A G D contain aij 
a,ngle AG D equal to the given angle 
C.—This depends on articles 125, 126, 132. 
90. To divide a right line in continued proportion, in the ratio 
of two given right lines A B, A C.—From B, with the ra¬ 
dius A B (the antecedent) deferibe an arc Ac. In that 
arc 
