22 



OF THE PROPERTIES OF CURVES. 



188. Theorem. The stereograpliic pro- 

 jection of any circle of a sphere, seen from 

 a point in its surface, on a plane perpendicular 

 to the diameter passing through that point, 

 is a circle. 



Let ABC be a great circle 

 of the sphere passing through 

 the point A and the centre of 

 the circle to be projected, then 

 the angle ACB=BAD=BEF, 

 and ABC=CAGzzCHI, and 

 the triangle AHE is similar to 

 CV ~K D ABC, and the plane ABC is 



perpendicular to the plane BC and the plane HE, there- 

 fore HE is a subcontrary section of the cone ABC, and is 

 consequently a circle. 



SECTION IV. 



OF THE PROPERTIES OF 

 CURVES. 



189. Definition. Any parallel right 

 lines intercepted between a curve and a 

 given right line, are called ordinates, and 

 each part of that line intercepted between an 

 ordinate and the curve, is the absciss corres- 

 ponding to that ordinate. 



190. Theorem. The fluxion of the area 

 of any figure is equal to the parallelogram 

 contained by the ordinate and the fluxion of 

 the absciss. 



Let AB be the absciss, and BC the 

 ordinate, through C draw DCE [ | AB, 

 and take DC::DE=:half the incre- 

 ment of AB, then the simultaneous 

 increment of the figure ABC will ul- 

 timately coincide with the figure 

 FCGEB, since the curve ultimately 

 coincides with its tangent (l4l), but the triangles CDF, 

 CEG, are equal, therefore the parallelogram DBE is ulti- 

 mately equal to the increment of ABC. And if any other 

 line than DE represent the fluxion of AB, as DE is to this 

 line, so is the parallelogram DBE to the parallelogram con- 

 tained by BC and this line ; therefore that parallelogram is 

 the fluxion of ABC (46). 



Scholium, Those who prefer the geometrical mode of 



representation, may deduce from this proposition a demon- 

 stration of the theorem for determining the fluxion of the 

 product of two quantities (48) ; for every rectangle may be 

 diagonally divided into two such figures as are here consi- 

 dered, and the sum of their fluxions, according to this pro- 

 position, will be the same with the fluxion of the rectangle" 

 determined by that theorem. 



191. Definition. A flexible line being 

 supposed to be applied to any curve, and to 

 be gradually unbent, the curve described by 

 its extremity is called the involute of the first 

 curve, and that curve the evolute of the se- 

 cond. 



192. Definition. The radius of cur- 

 vature of the involute is that portion of the 

 flexible line which is unbent, when any part 

 of it is described. 



193. 1 heorem. The radius of curvature 

 always touches the evolute, and is perpendi- 

 cular to the involute. 



If the radius of curvature did not touch the evolute, it 

 would make an angle with it, and would therefore not be 

 unbent ; and if the evolute were a polygon composed of 

 right lines, each part of the involute would be a portion of 

 a circle, and its tangent therefore perpendicular to the ra- 

 dius : but the number of sides is of no consequence, and if 

 it became infinite, the curvature would be continued, and 

 the curve would still at each point be perpendicular to the 

 radius of curvature. 



194. Theorem. The chord cut off" in the 

 ordinate by the circle of curvature, is directly 

 as the square of the fluxion of the curve, and 

 inversely as the second fluxion of the ordi- 

 nate, that is, as the fluxion of its fluxion. 



The constant fluxion of the absciss 

 being equal to AB, the fluxion of the or- 

 dinate at A , is BC, at D, DE, consequent- 

 ly its increment is CD-I-BE, or CD+AF, 

 twice the sagitta of the arc AD : and the chord is equal to 

 the square of AC divided by CD, and it is therefore always 

 in the direct ratio of the square of the fluxion of the curve, 

 and the inverse ratio of the second fluxion of the ordinate. 



195. Theorem. When the curve ap- 

 proaches infinitely near to the absciss, the cur- 



