Q* 



OF THE PROPERTIES OF CURVES. 



gAE.E F 

 CH 



It has been shown 

 thatABq:CDq-:AE: 

 EF (199), therefore 

 III ABq : AEq : : CDq : 

 AE.EF;butthe chord 

 of curvature EG is 



, and AE.EF=iEG.CH, therefore ABq : AEq : : 



CDq : 1 EG.CH : : 2^ : EG. - 



Scholium. It may easily be demonstrated that a per- 

 pendicular to the normal of the curve, or to the line perpen- 

 dicular to its tangent, passing through the point where it 

 meets the axis, bisects the focal chord of curvature, and 

 that a perpendicular falling from the same point on the 

 chord, cuts off a constant portion from it, equal to_the third 

 proportional to the semiaxes. 



202. Theorem. The square of any or- 

 dinate of an ellipsis parallel to the lesser axis, 

 is to die rectangle contained by the segments 

 of the greater axis, as the square of the lesser 

 axis to the square of the greater. 



On the centre A de- 

 scribe the circle BODE 

 passing through the 

 jtHj focus B ; then EF : 

 BF: : OF: DF ()38). 

 CallHT,a,HB,i.,AB,T, 

 GH,«, then EFr=2a, BF=2J', CF=2BH — 2BG=2GH= 

 21, DFz;EF— ED=2a— 2x, and 2a : 2i : : 2z : 2a— 2i', 

 a:l::z: a—x, a : a+l : : z : z + a—x : : a+z : 2tt— x 

 +i+x (32) ; also a : a— I : : z : z—{a—x) ■■ : a—z : la 

 _x— (fc-fit), and by multiplying the terras, aa : aa— lb : : 

 (a+»).(a-») : (2a— x)*— (i+^^jS or Hlq.HKq : : IG. 

 GL •. AFq— GFq, or AGq. 



203. Theorem. The area of an elhpsis 

 is to that of its circumscribing circle, as the 

 lesser axis to the greater. 



For since the square of the ordinate is to the rectangle 

 contained by the segments of the axis, or to the square of 

 the correspondingordinateof the circle (13"), as the square 

 of the lesser axis to that of the greater, the ordinate itself 

 is to that of the circle in the constant ratio of the lesser 

 axis to the greater. For if four quantities are proportional, 

 their squares are proportional, and the reverse. But the 

 fluxions of the areas are equal to the rectangles contained 

 by these ordinates and the same fluxion of the absciss 

 (190), they are therefore in the constant ratio of the ordi- 



nates, and the correspondfng areas are also in the same 

 ratio (47).~ 



204. Definition. If the square of the 

 absciss is equal to the rectangle contained 

 by the ordinate and a given quantity, the 

 curve is a parabola, and the given quantity 

 its parameter. 



Scholium. Thus 

 ABq=:P.BC. If the 

 axes of an ellipsis are 

 supposed infinite it be- 

 comes a parabola, for A JB B B 



, if a becomes infinite, xx vanishes in 



smce •—'iz — — 



comparison of flo", and , , — r~v', and — is the pa- 



^ a' ax a a 



rameter of the parabola ; and the distance from the focus is 



in a constant ratio to the square of the perpendicular falling 



on the tangent. 



205. Definition. When the ordinate 

 is as any other power of the absciss than the 

 second, the curve is still a parabola of a dif- 

 ferent order. 



Thus when the ordinate is as the third power of the ab- 

 sciss the curve is a cubic parabola. 



206. Theorem. If any figure be sup- 

 posed to roll on another, and any point in 

 its plane to describe a curve, that curve will 

 always be jjerpendicular to the right line 

 joining the describing point and the point of 

 contact. 



Suppose the figures rectilinear polygons ; then the point 

 of contact will always be the centre of motion, and the 

 figure described will consist of portions of circles meeting 

 each other in finite angles, so that each portion will be 

 always perpendicular to the radius, though no two radii 

 meet in the point of contact. And if the number of sides 

 be increased without limit, the polygons will approach in- 

 finitely near to curves, and each portion of the curve de- 

 scribed will still be perpendicular to the line passing through 

 the point of contact. 



207. Definition. A circle being sup- 

 posed to roll on a straight line, the curve 

 described by a point in the circumference is 

 called a cycloid. 



208. Theorem. The evolute of a cycloid 



