SKC. -IV'.] OF NATUEAL PHILOSOPHY. 2S? SECTION IV. Of the circular motion of bodies in resisting' mediums. LEMMA III. Let PQR be a spiral rutting all the radii SP, SO, SR, i// />e £/ie ya/io o/" equality. For from the right angles OPQ, OQR, sub duct the equal angles SPQ, SQR, and there will remain the equal angles OPS, OQS. Therefore a circle which passes through the points OSP will pass also through the point Q. Let the points P and Q, coincide, and this circle will touch the spiral in the place of coincidence PQ, and will therefore cut the right line OP perpendicularly. Therefore OP will become a diameter of this circle, and the angle OSP, being in a semi-circle, becomes a right one. Q.E.1). Draw Q,D, SE perpendicular to OP, and the ultimate ratios of the lines will be as follows : TO to PD as TS or PS to PE, or 2PO to 2PS • and PD to PO as PO to 2PO ; and, ex cequo pertorbatt, to TO to PO as PO to 2PS. Whence PO2 becomes equal to TO X 2PS. O.E.D. PROPOSITION XV. THEOREM XII. Tf the density of a medium in each place thereof be recipr on iJl y as the distance of the places from an immovable centre, aud the centripetal force be in the duplicate ratio of the density ; I say, that a body mny revolve in a spiral which cuts all the radii drawn from that centre in a given angle. Suppose every thing to be as in the forego ing Lemma, and produce SO to V so that SV may be equal to SP. In any time let a body, in a resisting medium, describe the least arc PO, and in double the time the least arc PR : and the decrements of those arcs arising from the resistance, or their differences from the arcs which would be described in a non-resist ing medium in the same times, will be to each other as the squares of the times in which they are generated ; therefore the decrement of the