72 NEWTON S PRINCIPIA. 



admitting of an expression for its area, or the area of 

 its sectors, in finite terms of any product of straight lines, 

 the problem becomes incapable of a definite solution. 

 Newton accordingly begins his investigation by a lemma, 

 in which he endeavours to demonstrate that no figure of 

 an oval form, no curve returning into itself and without 

 touching any infinite arch, is capable of definite quadrature. 

 It is rarely, indeed, that the expression ff endeavour,&quot; can 

 be applied to Sir Isaac Newton. But some have ques 

 tioned the conclusiveness of his reasoning in this instance. 

 The demonstration consists in supposing a straight line to 

 revolve round a point within the oval, while another point 

 moves along it with a velocity as the square of the portion 

 of the revolving line between the given centre and the 

 oval, that is, as the radius vector of the oval from the 

 given centre. It is certainly shown, that the moving 

 point describes a spiral of infinite revolutions ; and, also, 

 that its radius is always as the area of the oval at the 

 point where that radius meets the oval. If then the relation 

 between the area and any two ordinates from the oval to 

 any axis is such as can be expressed by a finite equation, 

 so can the relation between the radius of the spiral and 

 co-ordinates drawn parallel to the former, or the co 

 ordinates to the same axis. Therefore it w r ill follow, 

 that the spiral can be cut only in a finite number of points 

 by a straight line, contrary to the nature of that curve. 

 Indeed, its co-ordinates being related to each other by an 

 algebraical equation is equally contrary to its nature ; 

 consequently the possibility of expressing the relation be 

 tween the area of the oval and the co-ordinates leads to 

 this absurd conclusion, and therefore that possibility cannot 

 exist ; and hence it is inferred that the oval is not quadrable. 

 Sir Isaac Newton himself observes that this demon 

 stration does not apply to ovals which form parts of curves, 



