15 2 Mechanic“ Powers. Bock VIII.plane CE be equal to the arch of the quadrant CH, ſo thatWhile it is turned upon the plane CE, each point of the archanſwers each point of the plane; When therefore the leſſerWpcel in this motion is carried away by force by the greater,ſurely the point D will come to F, vhen the point A will cometo G, that is, the radius AD will meet with G FP.Therefore the whole difficulty will depend on this iſſue, thatB F will be double the arch B D, if ſo be the points of thearch B D ought ſucceſſively to correſpond with the points ofthe plane, or of the line B E, or each point of the arch B Panſwerseach point of B F, or each point of B C anſwers two of B P, orthe altern points B by skips remain wholy untoucht; for neitherdoth any thing elſe ſeem to remain which can be ſaid in truth noneof cheſe ought tobe ſaid: For if it be ſaid in the firſt place, thatthere is ſo many points in the arch B D, as there are in the line B F,which cannot be ſaid, ſince this is doubly greater than the arch:But if it be ſaid ſecondly, that each point of the arch B D cor-reſponds to two points of the line B F, it follows that the planeE, is double the arch CH. when notwithſtanding they areſuppoſed equal; for when the points B C, are in the ſameradius A C, if the point C touch the point of the plane nextfollowing towards E, it will not be perpendicular to the planeCE. Surely AB when it touches the point of the plane B Fin B. and is perpendicular to the plane, alſo A C which is ſup-poſcd a right line will be perpendicular to the plane C E, whichis parallel to the plane BF: Therefore it toucheth not tlieplane in any other point than in Cwbich if it ſhould: touch itwould be alſo in the other next following point; thereforeevery point of the arch C N, touches two points of the planeCE, therefore the plane CE will be double the arch CH.RNloreover it cannot be ſaid that the altern points, or every o-ther point of the plane B F, are toucht as it were by leaps, andnot the other intermediate points, for at the ſame time in whichſome point of the plane C E is toucht ànſwering the untouchtpoint ot the piane E P, ſome point of the arch B D without doubttouches the plane E, therefore it touches the point of the planeE F,which correſponds to the point toucht by the greater Wheclin dhe plane CE, for if it ſnould not touch the Centre A, oughtto be elevated above the line A G, and conſequently the greaterWitecl will not touch the plane CE, which is againſt the ſuppo-ſition; therefore no point of the plane B F remains untoucht.2 N More-