NEWTON S PRINCIPIA. 27 



subtangent and subnormal. Thus in the common parabola, 



whose equation is y 1 = a z, the subtangent , = - 



y 



y 2 W 2 



x ~- = -*- or 2 x ; and in the hyperbola, whose equa- 

 cl y ci 



tion is xy a 1 , the subtangent is x. So in the circle 

 y 2 = 2 r x x 2 , y^JL (t ae subnormal) = r x (r being 



the radius); all which we know from geometrical demon 

 stration to be true. 



Next, it is evident that when a quantity increasing has 

 attained its maximum, it can have no further increment ; 

 or when decreasing it has attained its minimum, it can 

 have no further decrement; consequently in such cases 

 the differential of the quantity is equal to nothing.* Hence 

 a ready solution is afforded of problems of maxima and 

 minima. Thus would we know the proportion which two 

 sides of a rectangle must have to each other, in order 

 that, their sum being given, they may form a rectangle con 

 taining the greatest space possible ; the differential of the 

 rectangle must be put equal to nothing. Thus their sum 

 being = a, the quantities are x and a or, and their rectangle 

 is a x x 1 , its differential adx 2 x d x, and this being 



put = 0, we have adx = 2zdz, or # = - ; therefore the 



figure must be a square. So would we know the point of 

 the parabola (bx)- = a(y c) where the curve comes 

 nearest the line b, the ordinate y must be a minimum, and 



x- (b-x) 2 , , 2(b-x} 



d y = 0. rs o w y = - - - -f c, and d y = - - - x 



* Sir I. Newton s own statement of the method is here followed. Me- 

 thodus Fluxionum Opuscula, torn. i. p. 86. edit. Geneva, 1744. It has, 

 however, been since universally admitted that the more accurate view is 

 to regard the change of the sign as the criterion, both as to maximum and 

 minimum values, and as to points of contrary flexure. 



