302 



Application of the Fluxional Ratio, fyc. 



H I 



K L 



the ratio, the fourth may be found. Figures are similar, when they 

 may be supposed to be placed in such a manner, that any right line 

 being drawn from a determined point to the terms that bound them, 

 the parts of the right line intercepted betwixt that point and those 

 terms, are always in one constant ratio to each other ; and when, if 

 all the parts, which the nature of the case admits of, are made to 

 coincide, all their homologous lines, that are rectilineal, either lie 

 one upon another, or are parallel.* This similarity may be exem- 

 plified in the Parabola. Let Fig. l. 



BpCE, APDG, (Fig. 1.) be 



two parabolas, that are simi- 



lar. By the definition, if the 

 foci are both placed at F, and 



the parameters Fp, FP, are 



made to coincide, and any 



right lines FD, FG are drawn A b F 



from the point F to the terms C, E, and D, G, that bound them ; 



then the lines FC, FD, and FE, FG, are in the invariable ratio of 



Fp : FP, that is FC : FD: :FE : FG: :Fp : FP, and FC : FD: : 



HC : ID, and FE : FG: :KE : LG. Here the lines FC, FE lie 



upon the lines FD, FG ; and although the curved lines BpCE, 



APDG are not parallel, but continually approximate toward each 



other, yet the homologous right lines HC, ID ; KE, LG, are par- 

 allel. 



In trigonometry, let AC=a (Fig. Fig. 2. 



2.) be the base, and CB=& be the 

 perpendicular of a right angled trian- 

 gle, taken from a table of sines, tan- 

 gents, and secants. Also let DC be 

 the base, and CE be the perpendicu- 

 lar of a second triangle; FC the base, 

 and CG the perpendicular of a third F 

 triangle ; and suppose they are all similar. Suppose DC=Ba, CE 



Bb, FC=Da, and CG=D6. By the principles of trigonometry 



BalBb::Da:Bb. Hence 



BBab 

 Ba 



BbDb b 

 Ba ~Da ~a 



, which is the ratio, and 



~Db the fourth term. It is most convenient to find this term, 



'Maclaurin's Flux. Vol. I. Art. 122. 



<r 



