346 PHILOSOPHICAL TRANSACTIONS. [ANNO 17Q8. 



tainty and case; and they are, very probably, in the doctrine of curve lines, what 

 the ancients appear to have prized so much in plain geometry; though unfortunately 

 all that remains to us of that treasure, is the knowledge of its high value. I have 

 not added the demonstrations, which are all purely geometrical, granting the 

 methods of tangents and quadratures : I have given an example, in the abridged 

 synthesis, of what I consider as one of the most intricate. It is unnecessary to apolo- 

 gize any further for the conciseness of this tract. Let it be remembered, that were 

 each proposition followed by its analysis and composition, and the corollaries, scholia, 

 limitations, and problems, immediately suggested by it, without any trouble on the 

 reader's part, the whole would form a large volume, in the style of the ancient 

 geometers ; containing the investigation of a series of connected truths, in one 

 branch of the mathematics, all arising from varying the combinations of certain 

 data enumerated in a general enunciation.* 



As a collection of curious general truths, of a nature, so far as I know, hitherto 

 quite unknown, I am persuaded that this paper, with all its defects, may not be 

 unacceptable to those who feel pleasure in contemplating the varied and beautiful 

 relations between abstract quantities, the wonderful and extensive analogies which 

 every step of our progress in the higher parts of geometry opens to our view. 



Prop. 1. Porism. PI. 5, fig. 10. — A conic hyperbola being given, a point may 

 be found, such, that every straight line drawn from it to the curve, shall cut, in 

 a given ratio, that part of a straight line passing through a given point which is in- 

 tercepted between a point in the curve not given, but which may be found, and the 

 ordinate to the point where the first mentioned line meets the curve. — Let x be the 

 point to be found, na the line passing through the given point n, and m any point 

 whatever in the curve ; join xm, and draw the ordinate mp ; then ac is to cp in a 

 given ratio. 



Corol. This property suggests a very simple and accurate method of describing a 

 conic hyperbola, and then finding its centre, asymptotes, and axes ; or, any of 

 these being given, of finding the curve, and the remaining parts. 



Prop. 2. Porism. — A conic hyperbola being given, a point may be found, such, 

 that if from it there be drawn straight lines to all the intersections of the given 

 curve, with an infinite number of parabolas, or hyperbolas, of any given order what- 

 ever, lying between straight lines, of which one passes through a given point, and 

 the other may be found ; the straight lines so drawn, from the point found, shall 

 be tangents to the parabolas, or hyperbolas. — This is in fact 2 propositions ; there 

 being a construction for the case of parabolas, and another for that of hyperbolas. 



Prop. 3. Porism. — If, through any point whatever of a given ellipse, a straight 

 line be drawn parallel to the conjugate axis, and cutting the ellipse in another point; 

 and if at the first point a perpendicular be drawn to the parallel ; a point may be 

 found, such, that if from it there be drawn straight lines, to the innumerable in- 

 tersections of the ellipse with all the parabolas of orders not given, but which may 

 • See the celebrated account of ancient geometrical works, in the 11th book of Pappus.— >Orig. 



