66 INVESTIGATION of fome 
2m am wee 1.3-5-7- : ght yim a 
Therefore a” -+- 4" + &c. =X aot x nate 
ER I ae er - 
ge 1.2+3-4+ +3” 2 
Ir m=1,¢7+0+&.=2X a 4 2 it, and the 
diameter, (or 27) X @ +5 +c + &c. = fum of the cubes of 
perpendiculars drawn from any point in the circumference, to 
the fides of a regular circumfcribing polygon of 7 number of 
fides, and a’ + 4° + &c. is to the fum of the fquares of thefe 
perpendiculars as 5 to 6; and if the perpendiculars to the fides 
of the polygon correfponding to the chords A, B, C, D, &c. and 
drawn from the fame point in the circumference that thefe 
chords are drawn from, be denoted by P,Q. R,S, &c. a+ 5+ 
Loam xP 
CxR , D = S 
e+ &e. 27 x or r fr 
a +P tom 4 &e. = P+ Qa" + R" + &c. = the fum 
of the 3 m powers of thefe perpendiculars, = 2 X ear . ; = 
x 73, 
THEOREM ®. From any pointG, (Pl. III. Fig. 8.), let the chord 
GA bedrawn ; let GAF bea tangent tothe circleat A; and let AD 
be perpendicular to the diameter BC, and CF, BG toGF. The 
right line which has to BC (27), the ratioof AC” to BC’, or the 
triplicate ratio of the chord of the arc AC, to the diameter, is 
are or sy =a fourth proportional to the diameter, 
the chord and the perpendicular drawn from one extremity of 
the 
