. The first three sections of Newton's Principia; with an appendix, and the ninth and eleventh sections. Edited by John H. Evans . gles Ahr^ Acbr, yl rfr are always finite; hut the same thingwill be true if RBD revolve round R fixed, in which case also,though r moves off to an infinite distance and the trianglesAbr, Acbr, Adr increase indefinitely, they will be ultimatelynimilar and equal to each other. IX. If the right line AE and the arc ABC, given in position,cut each other in a finite angle at A, and the ordinate^ BD,CK be drawn, ttiakin^- any other g^iven angle icith AK; whenBD,
. The first three sections of Newton's Principia; with an appendix, and the ninth and eleventh sections. Edited by John H. Evans . gles Ahr^ Acbr, yl rfr are always finite; hut the same thingwill be true if RBD revolve round R fixed, in which case also,though r moves off to an infinite distance and the trianglesAbr, Acbr, Adr increase indefinitely, they will be ultimatelynimilar and equal to each other. IX. If the right line AE and the arc ABC, given in position,cut each other in a finite angle at A, and the ordinate^ BD,CK be drawn, ttiakin^- any other g^iven angle icith AK; whenBD, CE more parallel to themselves up to A, the limiting ratioof area ABD : area ACE equals that of AD : AE\ Produce AE to a fixedpoint e, and take Ad in AesiithAhnt Ad: Ae = (lb, ec parallel to DB,or EC, meeting the chordsAB, AC produced in b, c;and on Ac describe an arcsimilar to ABC: fhisarc shallpass through b, for by similar trianglcs^and by construction, AB : Ab - AD : Ad = AE : Ac - AC : Ar. and therefore (Cor. Lemma v.) b is a point in the are. As Hand C move iqi to A, let (he curve Abr so alter its form as to. 11 be always similar to ABC, then the area ABD will be alwayssimilar to Abd, and ACE to Ace. Hence area ABD : area Ahd = AD : Ad = AE- : Ae= area ACE : area Ace,.-. area ABD : area ACE = area Aid : area Ace. Also the two arcs being similar have a common tangentat A, let this be AFGfg ; and let BD, CE move parallelto themselves up to A, then the angle cAg continually di-minishes and ultimately vanishes, and therefore * area Ahd : area Ace = A Afd : A Age= Ad : area ^52) : area ACE = area Abd: area Ace = Ad : Ae= ^7>2 : AE. Lemma X. J%e spaces, described from rest by a body acted on byany Jinite force, are in the beginning of the motion as thesquares of the times, in which they are described. Def. a finite accelerating or retarding force is such,that the ratio of the time to the velocity generated or de-stroyed
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