F«. 27.

IFg. 28.

F^. 27.

Fi£. 28.

'F«. 27.

(50

thc Semicircle B K A, is equal to a Prifm or Semicylinder, whose Base is theSemi circle B K A, and height the Circumference of a Circle whose DiameterisBC.

Having E isecfed B A in R, and B C in P, the Sursace generatedby the Conversion of the Arc H M about the Axis BC, is equal to

c ------

— x BPxHM + BRxDE (supposing the Ratio of the Radius to thc Cir-cumference to be as r to c) and the Sursace generated by the Semicircumtc-rence B K A is equal to a Rectangle, whole Base is the Summ of that Semi-circumference ana Diameter B A, and height the Circumference of a Circle,whole Diameter is B C. As for the Sursace generated by the Arc G F, tiswell known, that it is equal to a Rectangle, whose Base is the Circumferenceof a Circle whose Radius is B C, and height D E; therefore the Sursace ge-nerated by the Conversion of the Portion M H F G, is known.

lf upon B A you take any two Points I, L, and draw IN, L V, Perpen-dicul&r to it, eutttng the Quadrant in O and T, and the Circumference in Nand V, the Solid generated by the Conversion of the Portion O N V T aboutthe Axis B A, is equal to a Prisen, whose Base is IO T L, and height theCircumference of a Circle whose Diameter is B A.

Having Bisected B A in R, and drawn C R, meeting the Quadrantin G, the Sursace generated by the Conversion of thc Arc OT about B A,

is equal to — x CGxIL — CRxOT.

r

Bisect D E in Y, thro’ the Center R draw S 0,1 Parallel to B C, meetingthe Circumference B K A in S, B K Parallel to A C in V, and the Lines D H,E M,inNandQjthe Solid generated by the Conversion of the PortionFGM H

about the Axis AC, is — x j-MOi — 7 N H 3 +PCxNOMH +

r

CY x DNOE—7 E G 5 -t-j D F ?, and the Solid generated by the Segment

KBS is — x f V K 5 -f PCx B V K S. Therefore the Solid generated byr

the Semicirclc B K A about AC is — xPCxVQ.AK + PCx BCQ.V

r

— j A C J H- f V K 3 -+ P C x B V*K S, which by due Reductiou will bcfourtd'equal tö the Solid generated by the Conversion os the seine Semicirclcabout the Axis B C.

The Solid generated by thc. Portion ON V T, about the Axis CP, Bequal to — x ‘f L V’ — f I N? — f Q_T 3 -f 7 PO 3 -t-~CSx PQJL~

r

From the Pöihts M, H, drop the two Perpendiculars M Z, H W,tisön C A prolbng’d if need be , the Susface generated by the Conver.

sion