74 NEWTON S PRINCIPIA. 



given it. That the ellipse cannot be squared might per 

 haps be sufficiently proved from this consideration, founded 

 upon a reasoning analogous to that on which the lemma 

 in question proceeds. If a curve be described such that 

 its co-ordinates, or the rectangle contained by the co-or 

 dinates, shall always bear a given proportion to the areas 

 of the ellipse on the same axis, this curve cannot be alge 

 braical, not merely because of its equation involving quan 

 tities not integrable (for that may be said to be the ques 

 tion), but because it will stop short at a given line, which 

 no algebraical curve can do. It will have no branch ex 

 tending beyond the perpendicular at the end of the axis : 

 and moreover its equation is known to be that of a tran 

 scendental curve. This reason cannot be applied to all 

 curves returning into themselves ; because, as we have seen 

 in one class, the equation to the curve, whose co-ordinates 

 should express their areas, is algebraical ; and also because, 

 in that class, the secondary curve is found to have two 

 branches which meet in cusps, and so do not stop short. 

 If described by the proportion of areas they would seem 

 to stop short, that property only belonging to one of their 

 branches ; but their equation discloses the second branch. 

 It is one of many instances of a truth perhaps not suf 

 ficiently remarked by geometricians, that curves sometimes 

 have particular portions to which certain properties belong 

 exclusively, no other part of the curve having them. 



As the area of the ellipse cannot be found by alge 

 braical quantities, or by the description of algebraical 

 curves, the problem of Kepler cannot be solved otherwise 

 than by transcendental curves, logarithms, circular arcs, 

 or approximation. Sir Isaac Newton gives a solution by 

 means of the cycloid described on an axis at right angles 

 to the transverse axis of the ellipse, at a distance from 

 its vertex which is a fourth proportional to half the trans- 



