On a Properly of the Hyperbola. 54-7 



tween weights?) I can no longer bear it should be thought 

 that I have made a wrong balance, or consented to an unequal 

 division of property. 



All who have made me their study are generally aware of 

 the impartial manner in which I have managed that every pro- 

 perty of the ellipse should be accompanied by another of pre- 

 cisely the same character, belonging to the hyperbola. Why 

 then should they allow that cases of palpable non- symmetry 

 do sometimes occur : and, which is even worse, why do they 

 suppose that I have allowed the ellipse, in certain cases, to rob 

 the hyperbola of its birthright without any compensation? 

 For an example of the first: in the hyperbola, the diagonal of 

 the parallelogram described on semiconjugate diameters has, 

 for the locus of its fourth point, the asymptotes, an extreme 

 case of the hyperbola itself; while, in the ellipse, the corre- 

 sponding point has an ordinary ellipse for its locus, having 

 the semiaxes of the former increased in the ratio of V2 to 1. 

 How can any one imagine there is not a word to be said upon 

 this? Again, the circle on the major axis is, in the ellipse, 

 the locus of the intersection of the perpendicular from a focus 

 upon the tangent; and this same circle is still the locus, when 

 an hyperbola on the same major axis is substituted for the 

 ellipse. Now symmetry would require that, in this second case, 

 the equilateral hyperbola should take the place of the circle : 

 why did not the mathematicians, when they discovered that 

 such was not the case, bestow no pains on the clearing of my 

 character? Did they suppose that I would allow the ellipse 

 to take and hold the share of the hyperbola in any property, 

 without restoring the equilibrium by giving to the hyperbola 

 the share of the ellipse in some other ? 1 do not go for da- 

 mages: but I think I have a right to such reparation as can 

 be made by inserting demonstration of the following proper- 

 ties in future works on conic sections: — 



1. Every ellipse, and every hyperbola, has, the first two 

 ellipses, the second two hyperbolas, related to it. One of them 

 is an extreme case, having vanishing axes: it is the centre in 

 the ellipse, the asymptotes in the hyperbola. The other is 

 ordinary, having axes increased in the proportion of V2 to 1. 

 Each of either pair has this property, that if a chord P Q of 

 the original cut either of the related curves in R, the rectangle 

 under RP and RQ is equal to the square on the semidiameter 

 parallel to that chord. The extreme case in the ellipse corre- 

 sponds to the ordinary case in the hyperbola, and vice versa : 

 and thus the fourth point of the parallelogram above mentioned, 

 has the asymptotes for its locus in the hyperbola, and the other 

 sort of ellipse for its locus in the ellipse. 



