32 INTRODUCTION. 



The fluxion of the arc is equal to that of the tafi- 

 gcnt, since their evanescent incremeuts coincide 

 (141). Let AB be the sine, AC the cosine, BD the 

 increment of the tangent, DE that of the sine: then 

 ^ABCizEBD (16), and tlie triangles ABC, EBD, 

 are similar, and BD is to DE as BC to AC ; but tlie 

 ultimate ratio of the increments is that of the fluxions, therefore the 

 fluxion of the tangent, or of the arc, is to that of the sine as the radius 

 to the cosine. The same may easily be inferred from the tlieorera 

 for finding the sine of the sum of two arcs (140). 



143. Theorem. The area of a circle is equal to half 

 the rectangle contained by the radius and a line equal to 

 the circumference. 



Suppose the circle to be described by the revolution of the radius : 

 the elementary triangle, to which the fluxion of the circle is propor- 

 tional (141), is equal to the contemporaneous increment of the rect- 

 angle, of which the base is equal to the circumference, and the height 

 to half tlie radius : consequently the whole areas are equal (47). 



144. Theorem. The circumferences of circles are in 

 the ratio of their diameters. 



Supposing the circles to be concentric, and to be described by the 

 revolution of different points of the same right line, the ratio of the 

 fluxions, and consequently tliatof the whole circumferences, will be 

 the ratio of the radii, or of the diameters (47). 



Scholium. The diameter of a circle is to its circumference 

 nearly as 7 to 22, and more nearly as 113 : 355, or 1 : 3.14159265359; 

 hence the radius is equal to 57.29578''=3437.7467'=i206264.8" ; and, 

 the radius being unity, 1"=:.017453293, rzi.000290888, and 1"= 

 .000004848. 



145. Definition. A straight line is perpendicular ta 

 a plane, when it is perpendicular to every straight line 

 meeting it in that plane. 



146. Definition. A plane is perpendicular to a 

 plane, when all the straight lines, drawn in one of the planes. 



