340 A Theory of Fluxions. 



Demonstration. 



From the similarity of figures, CW : CQ: :WR : QU: : 

 NL : MK::WR-NL : QU-MK(5.E.i9.): :Wm : Qe:: 

 HI :FG::HI-WR:FG-QU::H?i:FA. Also WY : 

 QS::Wm : Qe::HO : FP::Hw : FA. HenceWY-Wm: 

 QS • Qe : : HO • H^ : FP • FA (6. E. C.) : : WmT Y : QeXS : : 

 H»YO : FASP. 



Secondly, bisect the arcs NW, WH, and MQ, QF, and 

 suppose parallelograms to be drawn corresponding to the 

 bisections. Let the parallelograms within the arc NW be 

 represented by A, B ; those within the arc WH by C, D ; 

 those within the arc MQ. by N, O; and those within the Q,F 

 by P, Q,. Then proceeding as before we obtain A : N; :B 



: 0::C : P::D : Q. Hence A : N::A+B :N+0::C+ 



D : p.f Q (5. E. 12.) : : the polygon within the arc NW : the 

 polygon within the arc Q,M : 3 the polygon within the arc 

 HW J the polygon within the Q,F. 



For the same reason, if ever so many polygons be formed 

 by bisecting the arcs, they will be in the same constant ratio, 

 and at the same time will converge towards the curvilineal 

 spaces WNTY, QMXS, HWYO, FQSP, as their limits. 

 When the number of bisections is greater than any assigna- 

 ble number, the series is supposed to have run through an 

 infinite number of terms ; and the little spaces, lying between 

 the polygons and the curves, are exhausted. On this account 

 the process is called the method of exhaustions. The peri- 

 meters of the polygons have now undergone a change into 

 curves, and the polygons become the curvilineal spaces. 

 Hence WNTY : QMXS : ■ HWYO : FQSP. Q. E. D. 



To investigate the fluxions of variable quantities, let the 

 similar curvilineal spaces AHNBC, DFMEC, (Fig. 6,) be 

 divided into any proposed number of parts NBCT=a, 

 HNTO=6, &c, and MECX=n, FMXP=o, &c, by draw- 

 ing the lines CM, CF, &c, from the point C to the curve 

 DFME, and drawing the ordinates NT, MX, HO, FP, from 

 the points where the lines CM, CF, &c, intersect the curves. 

 Then, by the foregoing lemma these curvilineal spaces are 

 proportional, that is, a I n \ 3 6 : o \ '. c i p : 3 , &c. Hence a : 

 n::a+ & +c -i-,&c. : w+o+p-f,&c. : : AHNBC ; DFMEC. 

 (5. E. 12.) 



