GEOMETRY. 
■ Now Fc''*, the fiipplementhl cliord of being 
known, the chord Ad'^ of ■j-irTS'> ff'ibcir- 
cuniference, or of whole circunife. 
fence, is ;ilfo known. That is, Ad'^=z 
(t/rt.!'' —e3)|/4—2 + Fi/''‘“ 1=0,0020453073606764 
194, Confequently, the fide of a regular polygon of 
3072 fides, infcribed in a circle whofe diameter is '2, is 
0,0020453073606764 
V T T. 195- The fidf pf a fimilar polygon cir- 
cumjeribing the fame circle, the cecUre of 
/q which is C, may be thus found .—Let 
B E be the fide of the circuml'cribed 
polygon; and draw BC, EC, cutting 
the circle in D and A. Draw DA, 
and it w ill be thedide of the infcribed 
polygon ; and is parallel to BE (184). 
Draw C I bifedting the angle BCE, 
and it will bifedt BE and DA at right 
angles (103). And DGr:2(^DA—) ^Ad'^. 
G A 
Then CG = t/t:D"—DG i~AdM\ 
But ^Ad'"‘— 0,001022653680338 
And its fVjuare is 0,00000104582055 
Which fuhtracled from i, leaves 0,99999895417945 
Whofe fquare root, or CG, is eqiial to 0,99999947708959 
Now the triangles C B I, CD G, are fimilar. 
Then C G : C I :: 2D G ; 2B I. Therefore 
(2BI=2) BE=:(—-—^(164)=) forIC=i. 
Or BEr 
0,0020453073606764 
:o,0020453084301895, 
o»999999+77o895883' 
whicli is the fide of a regular polygon of 3072 fides, cir. 
cumferibing a circle the diameter of wliich is 2. 
196. Scholium. The fide of a regular polygon of 3072 
fides, infcribed in a circle, the dia¬ 
meter of vvliich is 2, is ,0,0020453073606764, 
Which multiplied- by 3072, will 
give the perimeter of that polygon, 
which is 6,2831842119979622. 
The fide of a fimilar polygon, cir- 
cumferibing the fame circle, is 0,0020453084301895. 
Which multiplied by 3072,will give 
for the perimeter of tliat polygon 6,2831874973420925. 
Tite fum of thefe perimeters is 12,5663717093400547. 
The half fum is 6,28318585, &c. ^ 
Which is very nearly equal to the circumference of a 
circle, the diameter of w’hich is 2 (187), the difiference 
between Jt 
and the infcribed polygon being only 0,00000164,&-c. 
and the circumferibedpolygon beingonly 0,00000164,«Stc. 
197. Now the circumferences of circles being in the 
fame proportion as their diameters (188); 
Therefore the diameter of a circle being r. 
The circumference will be- 3,141592, &c. which agrees 
with the circumference as found by other methods. 
Of planes and SOLIDS. 
Definitions and Principles. 
198. A line is faid to be in a plane, when it pafTes 
through two or more points in that plane, and the com¬ 
mon fedlion of two planes is a line which is in both of 
•them. 
199. The inclination of two meet¬ 
ing planes A B, C D,. is meafured by 
an acute angle G F H, made by two 
right lines F G, F H, one in each 
plane, and both drawn perpendicular 
to the common feilioa D E, of thefe 
planes from F, fome point in it. 
Vgl.VIII, No. 515. 
200. A right line DE interfe£ling 
two fides AC, BC, of a triangle ABC, 
fo as to make angles C D E, C li D, 
within the figure, equal to the angles 
CBA, CAB, at the bafe AB, but 
with contrary fides of the triangle, is 
faid to be in a fubcontrary pofition to 
the bafe. A. 
201. If a circle in an oblique pofition be viewed, 
it will appear of an oval form, as g 
ABCD; that is, it will feem to 
be longer ofle way, as AC, than _jp; 
another, as DB; neverthelefs the ^ 
radii EA, E B, are to be efteemed 
as equal. And the fame mull be D 
underftood in viewing any regular figure, when placed 
obliquely to the eye. 
202. If a line be fixed to any point C above the plane 
of a circle ADBE, and this line while firetched be 
moved round the circle, fo as 
always to touch it; then a fo- 
lid which would fill the fpace 
palled over by the line, between 
the circle and the point C, is 
called a cone. 
203. If the figure ADBE 
had been a polygon, and the 
llretched line had moved along 
its fides, the figure which would 
then have been deferibed, is 
called a pyramid. So that cones 
and pyramids are folids which 
regularly taper from a circle, or polygon, to a point. 
The circle or polygon is called the bafe ; and the point 
C the vertex. 
When the vertex is perpendicularly over tlie middle 
or centre of the bafe, then the folid is called a right 
cone, or a right pyramid j otherwife an oblique cone, or oblique 
pyramid. 
204. If a cone or pyramid be cut by a plane palling 
through the vertex C, and centre of the bafe F, the 
fection ABC, or E DC, is a triangle. 
205. A right line AB, is perpendicular to a piano 
CD, when it makes right angles ABE, 
ABF, ABG, with all the right lines 
BE, BF, BG, drawn in that plane to 
touch the faid right line AB. 
206. So that from the fame point B, 
in a plane, only one perpendicular can 
be drawn to that plane on the fame fide. 
207. A plane AB, is perpendicular to a plane CD, 
when the right lines EF, GH, drawn ^ 
in one plane A B, at right angles to A 
F B, the common fefilion of the two 
planes, are alfo at right angles to the C 
other plane CD. V” 
208. So that a line E F, perpeiidi- ' 
cular to a plane CP, is in anotlier 
plane AB, and at right angles to B F, 
the common fefition of the two planes, 
209. If two planes A B, B C, cut eac/^ 
other, their common feElion B D will be a right 
line. —For if it be not, draw a right line 
DEB in the plane AB, from the point 
D to the point B ; alfo draw a right line 
D FB in the plane B C. Tb?n two right _ 
lines DEB and DFB, have the faille * iJ" 
5 T terms, 
D 
