328 PHILOSOPHICAL TRANSACTIONS. [aNNO l66g. 



Curvas. Accessit pars altera de Analysi, et Miscellanea. Leodii Eburonum 

 1668, in thin 4to. 



This problem is declared to be the same with that in the geometry of the 

 famous Descartes, viz. That ancient problem of finding two means, or of 

 doubling the cube, which troubled all Greece. The solution of which problem 

 in geometry may be compared to that of finding the cube root of any number 

 proposed in arithmetic : For in arithmetic, the first of two continual propor- 

 tionals between an unit and any number proposed, is the cube root of that 

 number; and the unit in arithmetic is represented by a line in geometry, which 

 is one of the extremes. Concerning this problem, the author declares himself 

 to be none of those that search for that which cannot be found, to wit, to per- 

 form it by right lines and a circle. The author observes, that amongst those 

 that solve this problem by the conic sections, very few have done it by aid of 

 a circle and an hyperbola or parabola; by a circle and ellipsis none, that he could 

 observe to have been published. But that he has found out not only one, but 

 infinite such efFectioiis, and that not in one method but many ; following the 

 guidance of which methods, by the like felicity, he has constructed all solid pro- 

 blems infinite ways, by a circle and an ellipsis or hyperbola. — 1. His general 

 methods for finding two means, by a circle and either an hyperbola or ellipsis, 

 are laid down in Prop. 1,2, 16, and in this prop. 16, he shows how to do it with 

 any ellipsis and a circle. — 2. Particular efFections for finding out one or both of 

 the means, and doubling the cube in prop. 3 to 6. — 3. And though all cubic equa- 

 tions may be solved, either by the finding of two means, or the trisection of an 

 angle, yet he shows the extent of his method, in finding out other infinite ways 

 for the doing thereof, from prop. 7 to 12. — 4. The trisection of an angle by a 

 circle and hyperbola, prop. 13, and by a parabola instead thereof, prop. 15. 

 And the finding of two means by a circle and parabola, prop. 14. 



In the second part of his book De Analysi, the author first gives the analysis 

 or algebra, whereby all his general methods of finding two means were invented. 

 And afterwards, for the advancement of geometry, gives the analysis that re- 

 lates to his particular methods. After that he comes to show how the efFections 

 or delineations for cubic equations were invented ; and then how those con- 

 structions for the trisection of an angle were found out. — Lastly, he comes to 

 treat of general constructions for the resolving of all solid problems, without 

 reduction of the equations proposed ; and shoAVS a general construction for all 

 cubic and biquadratic equations, by means of a circle and a parabola, letting 

 ordinates fall from the points of intersection on some diameter of the parabola, 

 which is always parallel to the axis, whereas Descartes, letting those ordinates 

 always fall upon the axis, was forced to prepare and alter the equations by taking 



