286 MR TALBOT ON FAGXANl'S THEOREM. 



Consequently, 7 lies between 4 and 5 ; that is, the deviations 1, 4, and 4, 7, are in 

 the same direction. And the total deviation tends to increase without limit, be- 

 cause it is plain, from the nature of the problem, that the deviations are quantities 

 of the same order (though sometimes larger, sometimes smaller.) 



Theorem I. (see fig. 1) If a triangle ABC is inscribed in one circle and 

 circumscribed about another, then if any point D be taken in the outer circle, and 

 three tangents, DE, EF, FD, be drawn, the extremity of the last will coincide 

 with the original point D. 



For if not, let its extremity be D'. It is immaterial on which side of D we 

 suppose D' to lie. Let it be between D and C. Then calling D the origin, and 

 D' the terminus, of the three tangents, it follows from lemma 1, that both D 

 and D' lie between A and C. For however near to C we take an origin, 

 the terminus and origin cannot lie on opjiosite sides of C. Now to return to the 

 case represented in our figure. We have supposed D the origin, D' the terminus. 

 Take D' as a new origin, and D" will be the new terminus. Continue the 

 process, and successive termini D'", D"", &c., will be obtained. All the devia- 

 tions DD', D'D'', &c., are in the same direction, and the total deviation ultimately 

 surpasses any given arc, and therefore surpasses the arc DC (by lemma 2). 

 But this result involves an absurdity, since it implies that a certain origin D„ 

 and terminus D„+i are on opposite sides of C. Therefore the supposition from 

 which we started, viz., that D and D' are different points, leads to a false con- 

 clusion. Consequently those points are the same. Q.E.D. 



Moreover, the theorem is true when two ellipses, one lying within the other, 

 are substituted for circles. For, by Salomon's Conic Sections, p. 308, two ellipses 

 can be so projected that both will be circles, and of course the projection of the 

 triangle inscribed to the one and circumscribed to the other ellipse, will be a 

 triangle inscribed and circumscribed to the two circles. Such a triangle has the 

 property (as we have just shown) of remaining always c/os^c? or complete, however 

 we vary the point of origin, which may be taken in any part of the outer 

 circle. 



And therefore all triangles inscribed and circumscribed to the two given 

 ellipses are closed and complete. For otherwise their projections could not 

 be so. 



Theorem on the Ellipse. 



In Salmon's Conic Sections, p. 297, we find the following beautiful theorem, 

 due to Dr Graves. 



If two ellipses have the same centre and focus, and a pair of tangents be 

 drawn from any point of the outer to the inner ellipse, the difference between 

 the sum of the tangents and the intercepted elliptic arc is a constant quantity. 



