16 GENERAL THEOREMS, CHIEFLY PORI8MS, 



Scholium. This proposition points out, in a very striking 

 manner, the connexion between all parabolas and hyperbolas, 

 and their common connexion with the conic hyperbola. The 

 demonstration here given is much abridged ; and, to avoid 

 circumlocution, algebraic symbols, and even ideas, have been 

 introduced : but by attending to the several steps, any one 

 will easily perceive that it may be translated into geometrical 

 language, and conducted on purely geometrical principles, if 

 any numbers be substituted for m, n, p, and q; or if these 

 letters be made representatives of lines, and if conciseness be 

 less rigidly studied, 



PROP. 14. Theorem. A common logarithmic being given ; 

 if from a given point, as origin, a parabola, or hyperbola, of 

 any order whatever be described, cutting in a given ratio a 

 given area of the logarithmic ; the point where this curve 

 meets the logarithmic is always situated in a conic hyperbola, 

 which may be found. 



Scholium. This proposition is, properly speaking, neither a 

 porism, a theorem, nor a problem. It is not a theorem, be- 

 cause something is left to be found, or, as Pappus expresses 

 it, there is a deficiency in the hypothesis : neither is it a 

 porisrn; for the theorem, from which the deficiency dis- 

 tinguishes it, is not local. 



PROP. ',15. Porism. Fig. 15. A conic hyperbola being 

 given ; two points may be found, from which if straight lines 

 IYtf.1.5. ^6 i R flected, to the innumerable intersec- 

 tions of the given curve with parabolas or 

 hyperbolas, of any given order whatever, 

 described between given straight lines ; 

 and if co-ordinaces be drawn to the inter- 

 sections of these curves with another conic hyperbola, which 

 may be found ; the lines inflected shall always cut off areas 

 that have to one another a given ratio, from the areas con- 

 tained by the co-ordinates. Let x and Y be the points found ; 

 H D the given hyperbola, F E the one to be found ; A D c one of 

 the curves lying between A B and A G, intersecting H D and F E ; 

 join x D, YD; then the area AYD:xDCBina given ratio. 



