302 



Application of the Fluxional Ratio, &j-c. 



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 letwixt 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 

 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 Yp, FP, are 

 made to coincide, and any 



right lines FD, FG are drawn a" B F H i K 



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

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

 Yp : FP, thatisFG : FD::FE : FG::Fp : FP,andFC : FD:: 

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

 upon the lines FD, FG; and ahhough the curved lines BpCY, 

 APDG are not parallel, but continually approximate toward each 

 other, yet the homologous right lines HC, ID 5 KE, LG, are par- 

 allel. 



In trigonometry, let AC=a (Fig. 

 2.) be the base, and CB=J 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 D A 



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

 =B6, FC=D«, and CG=DJ. By the principles of trigonometry 



Bh m b 



Ba * B6: :Da : D6. Hence ^^t' =Tr =~j which is the ratio, and 



s5a Da a' ' 



BDab 



p =D6 the fourth term. It is most convenient to find this term, 



Fig. 2. 



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



