70 PHILOSOPHICAL TRANSACTIONS. [aNNO J 763, 



AFC and EFB together, as the sum ofAF and fc to g, which is equal to the sum 

 of AB and cb; whence by division the ratio of the rectangle under afc to that 

 under efb, when the problem is limited to this single solution, will be that of 

 the sum of af and cp to the excess of pb above ef. 



Thus directly do these lemmas correspond with Apollonius's first mode of so- 

 lution, and lead to the general principle of applying to a given line a rectangle 

 exceeding or deficient by a square, which shall be equal to a space given. This 

 being a simple case of the 28th and 29th propositions of the 6th book of Euclid's 

 elements, admits of a compendious solution. Such a one is exhibited by Snellius 

 in his treatise on these problems (in Apollon. Batav.) and Des Cartes has ex- 

 hibited another, more contracted in its terms, but not therefore more useful. 

 It may also be performed thus. If upon a given line ab any triangle acb be 

 erected at pleasure; then if the legs ca, cb, fig. 17, whether equal or unequal, 

 be continued to d and e, so that the rectangles under cad and cbe be each equal 

 to the given space, and a circle be described through c, d, e, cutting ab extended 

 in p and g, the rectangle under bfa and bga will each be equal to the space 

 given. Also if in the legs ca, cb, fig. 18, the rectangles under cad and cbe 

 be each taken equal to the space given, and a circle in like manner be described 

 through c, D, E, cutting ab in f and g, the rectangles under apb and agb will 

 each be equal to the given space. Here it is evident, that the space given must 

 not exceed the square of half ab ; when equal, the circle will touch ab in its 

 middle point. 



POSTSCRIPT. 



As this application to a given line of a rectangle exceeding or deficient by a 

 square, or the more general problem treated of in the 6th book of the elements 

 of applying a space to a liwe so as to exceed or be deficient by a parallelogram 

 given in species, is the most obvious result, to which the analysis of plane pro- 

 blems, not too simple to require this construction, leads; so the descriptions of 

 the conic sections here treated of, stand in the like stead in regard to the higher 

 order of problems, styled solid from the use of the conic sections deemed neces- 

 sary for their genuine solution. And these are the only modes of solution, the 

 modern algebra, which grounds its operations on one or two elementary propo- 

 sitions only, naturally leads to. But as the form of analysis among the ancients 

 by expatiating through a larger field, often was found to arrive at conclusions 

 much more concise and elegant, than could offer themselves in a more confined 

 track; the ancient sages in geometry, that the solid order of problems might not 

 want this advantage, sought out that copious and judicious collection of proper- 

 ties attending the conic sections, which, with some useful additions from later 

 writers, have come down to us. 



And as the advantages of this ancient system of analysis cannot be too much 



