640 PHILOSOPHICAL TRANSACTIONS. [aNNO 1742-3. 



The 1 1th chapter treats of the curvature of lines, its variation, the degrees 

 of contact of the curve and circle of curvature, and of various problems that 

 depend on the curvature of lines. This subject is treated fully, because of its 

 extensive usefulness, and because in this consists one of the greatest advan- 

 tages of the modern geometry above that of the ancients. The author on this, 

 as on former occasions, begins by premising the necessary definitions. Curve 

 lines touch each other in a point, when the same right line is their common 

 tangent at that point ; and that which has the closest contact with the tangent, 

 or passes between it and the other curve through the angle of contact formed 

 by them, being less inflected from the tangent, is therefore less curve. Thus 

 a greater circle has a less curvature than a less circle ; and since the curvature 

 of circles may be varied indefinitely, by enlarging or diminishing their diame- 

 ters, they afford a scale by which the curvature of other lines may be measured. 

 As the tangent is the right line which touches the arc so closely, that no other 

 right line can be drawn between them ; so the circle of curvature is that which 

 touches the curve so closely, that no other circle can be drawn through the 

 point of contact between them. As the curve is separated from its tangent in 

 consequence of its flexure or curvature, so it is separated from the circle of cur- 

 vature in consequence of the variation of its curvature ; which is greater or less, 

 according as its flexure from that circle is greater or less. 



The tangent of the figure being considered as the base, a new figure is ima- 

 gined, whose ordinate is a third proportional to the ordinate and base of the 

 first. This new figure determines the chord of the circle of curvature by its in- 

 tersection with the ordinate at the point of contact, and by the tangent of the 

 angle in which it cuts that circle, measures the variation of curvature. The 

 less this angle is, the closer is the contact of the curve and circle of curvature, 

 of which there may be indefinite degrees. When the figure proposed is a conic 

 section, the new figure is likewise a conic section ; and it is a right line when 

 the first figure is a parabola, and the ordinates are parallel to the axis ; or when 

 the first figure is an hyperbola, and the ordinates are parallel to either asymp- 

 tote. Hence the curvature and its variation in a conic section are determined 

 by several constructions ; and among other theorems, it is shown, that the 

 variation of curvature at any point of a conic section, is as the tangent of the 

 angle contained by the diameter which passes through that point, and by the 

 perpendicular to the curve. 



When the ordinate at the point of contact is an asymptote to the new figure, 

 the curvature is less than in any circle; and this is the case in which it is said 

 to be infinitely little, or the ray of curvature is said to be infinitely great. Of 

 this kind is the curvature at the points of contrary flexure in the lines of the 

 third order. When the new figure passes through the point of contact, the 



