OF SPACE. 



31 



Let AB and BC be tlic sines of any two angles, 

 ACB, BAG, then AC will be the sine of their sura 

 CBD, orofABC. Now making BE perpendicular 

 to AC, AC=zAE+EC, and rad. : cos. BAG; *. AB : ^ ^'' ^ ^' 



AE, and rad. ; cos. ACB::BG ; CE (139). Again, make EF=: 

 EC ; then it is plain tliat AF will represent the sine of ABF, the 

 difference of ACB or CFB and BAG (108). 



141. Theorem. The ratio of the evanescent tangent, 

 arc, chord, and sine, is that of equality. 



Let AB be the tangent, and CD the sine of 

 the arc AD. Let AE be taken at pleasure 

 in the tangent, and EF be always parallel to 

 DG, the radius of AD, and on the centre F, 

 draw the circle AH; join AH, then since 

 Z.EADzr|AGD=|AFH, tlie chord AH will 

 coincide with the chord AD (133, 134). And 

 when DA vanishes, DG coinciding with AG, 

 EF will be parallel to AF, and the angle 

 EAH will vanish, therefore AH will coincide 

 with AE, and with IH parallel to the sine 

 CD ; and by similar triangles the ratio of AB, 



AD, and CD, is the same as that of AE, AH, and IH, and is ulti- 

 mately that of equality. But the arc AD is nearer to the chord AD 

 than the figure ABD, and it has no contrary flexure, therefore it is 

 longer than the line AD (79), and shorter than ABD (80), until their 

 difference vanishes, and it coincides with both. 



Scholium. The same is obviously true of any curve coinciding at 

 a given point with any circle ; and all the elements agree as well in 

 position as in length. 



141, B. Theorem. The fluxion of the area of any 

 figure is equal to the parallelogram contained by the ordi- 

 nate and the fluxion of the absciss. See 190. 



142. Theorem. The fluxion of the arc being con- 

 stant, the fluxion of the sine varies as the cosine. 



