994 
Regular © 
Polygons. 
—_—y—_ 
Pig. 137. 
Hence it appears that the inscribed and circumseri- 
bing polygons of 32768 sides agree in the first seven 
decimal places of, the numeral expressions for their 
value ; therefore the numeral expression for the circle 
itself, which is between’ these, will be correct in all 
these figures ; so that the radius of a circle bemg one, 
its area is 3.1415926 nearly, and this divided by the 
radius gives 3.14159926. for half the circumference 
nearly, the radius being one ; hence the diameter is to 
the circumference nearly as 1 to 3.1415926 nearly. 
Scitotrum. By this method, the ratio of the diameter 
to the circumference may be found to any LY aed 
degree of accuracy. Archimedes, by means of inscribed 
and circumscribed polygons of 96 sides, shewed that 
the diameter is to the circumference nearly as 7 to 22; 
and Metius found, by polygons of a greater number of 
sides, that the diameter is to the circumference as 113 
to 355. The manner of deducing these from the ratio 
found in the proposition, is explained in ALG@EsRa, art. 
364, * 
Pror. XI. 
To find a straight line nearly equal to any given are 
of a circle. 
Pros, 
Let AB be any are of a circle, of which C is the 
centre. Draw the radii CA, CB, and draw AH per- 
pendicular to AC, and CD perpendicular to CB, meet- 
ing HA in D. Bisect the angle ACB by the straight 
“Tine CH ; again bisect the angle ACH by the line CI, 
Of Planes 
and solid 
Angles. 
—_—\— 
Definitions. 
and bisect the angle ACI by the line CK, and bisect 
the angle ACK by the line CL, and proceed in this 
manner with any number of bisections ; the greater the 
number, the more accurate will be the result. 
Let H, I, K, L, &c. be the points in which the bisect- 
ing lines meet the line DA. In AD produced, take 
DP=3 of AH, PQ=} of AI, QR=+ of AK, RS=+¥, 
of AL, and so on, if there were more lines intercepted 
Parr Il. THE GEOMETRY OF SOLIDS. 
SECT. I. 
Or PLaNnes anp Soittp ANGLES. 
Definitions. 
1. A straight line is perpendicular or at right angles 
to a plane, when it makes right angles with every 
straight line, meeting it in that plane. On the other 
hand, the plane is perpendicular to the line. 
2. A straight line is parallel to a plane when they 
do not meet, to whatever distance both are produced. 
The plane is also parallel to the line. ' 
3. Two,planes are parallel to each ether, when they 
do not meet although produced. 
_ 4, Admitting what will be afterward demonstrated, 
(in Prop. 3.) that the common sections of two planes is 
a straight line, the angle or the inclination of two planes 
is the angle contained by two straight lines, drawn from 
the same point of their. common section at right angles 
to it; the one in the one plane, and the other in the 
other. This angle may be either acute, right, or obtuse. 
5. If this angle is a right angle, the pl er- 
pendicular, 4 8 Sa planes are p 
* The ratio of 113 and 355.is easily remembered, by observing that the figures of the numbers are 1, 1, 3, 3, 5, 5, viz. the first thrée 
- 
edd numbers each repeated. 
: GEOMETRY. 
&: FAQGIBE, 365 spp ee Peder SE 
between A and the bis lines, 
‘last, then take ST=t of RS. 
‘now from C to T, and draw Cd 
meeting AH in d, and the straight 
nearly equal to the are AB.of the circle. 
This construction is derived from an eleg nt formula 
which we have investigated in the. conclusion of the 
Anitumeiic oF Sings, also in Conic Sections, ar 
VIL) It is this, putting a for any are, 
A . : 
—<= Cot, 6 
a 
+ ftom ja+ ptm dat $tan p04 
&e. ! d , 
a 
the sum of the remaining terms of the series; and as | 
TC 6 is a right-angled triangle, of which C is the right 
angle, CA®’= AT x Ab (21. 4.) and 
CA* erty ab) 
Ap = AT, that BAGS cot. @ ob hes d aq; be. 
Hence it is evident that AUmarea . . 
Scuo.tum. If we suppose the are AB a quadrant, — 
then AD = 0, and by calculation, (see ArrrumeTic 
or Sryas, Art. 33), supposing CA=1, it will be found 
that DP = 5000000 PQ = .1035534, QR = .0248640, 
RS = .0061557, ST = .0020519; hence DT or AT 
= .6366250; and since AT: AC = 1::AC:A4, we 
find A éor AB=1.570784,the approximate value of the 
quadrant. The more correct value is 1.570796, &e. by 
which it appears that the error is little more than the 
=o0000 part of the radius. 
6. A solid angle is that which is made b: ‘the meet- or 
ing of more than two plane angles, whic! are not in 
the same plane, in one point. . Ms the: 
Prop. I. Teor, 
A straight line cannot be partly on a plane, and 
partly above it. : 
For, according to, the ‘definition of a 
straight line has two Common points wi 
entirely on that plane. 
Bute ies 
a plane, itis 
t Eee 
Prov. II. Teor. OR 
Two straight lines, which cut each other ina plane — 
determine its position ; that is, a plane which passes — 
through two straight lines, that cut each other, can 
have only one position. s ie ae 
Let AB, AC be two dtraight lines which ‘cut each Wi 
other in A ; suppose a plane to pass through AB, and 
to turn on that line, until it pass through C ; then the — 
‘ 
