66. The Three Sections, Tayigencies, 



an inscribed quadiilateral in cii'cle A having the side DD 

 infinitely small^ it follows that the straight line OPST is cut 

 in "involution" by the circle A and the pairs of opposite sides 

 of the quadrilateral ; but the circle A is known^ as also the 

 points O, P and S ; therefore the point T can be found as 

 follows (see Chasles' Geometrie Superieure, page 150) : — 

 Assume any point U on the circumference A_, and describe 

 the circle OPU; through U, Sj and the other point V in 

 which the circles A and OPU intersect^ describe a cii'cle 

 UVS : then will this circle UVS give the point T in its other 

 intersection "with OPS. Hence we know the tangent TD to 

 circle A, &c. 



2. We can find the tangent NH in a similar manner. 

 Thus : — Through W any point in the circumference A 

 describe the circle PWS ; through O^ W, and the point X in 

 which this cii'cle cuts the circle A describe a cii'cle : then will 

 this circle give N in its other intersection with the line SPO. 

 Hence the tangent NH^ &c. 



3. Or^ since the required circle DEF evidently cuts OPQ in 

 known points^ real or imaginary, it is obvious that if through 

 any point D' we draw two straight lines OD' and PD', and on 

 them take D'E' and D'F' such that OD'.OE' = OD.OE, and 

 PD'.PF' = PD.PF, then will the circles D'E'F' and DEF 

 have OPQ as radical axis ; and we can find the tangent DT 

 as follows : — Describe the circle OPD', and through Q, D', 

 and the other point K of intersection of the circles OPD' and 

 D'E'F' describe a circle : then Avill the other point in which 

 the circle D'KQ, cuts OPQ be the required point T from 

 which to di^aw the tangent TD to the circle A. 



Montucla gives a very ciu'ious history of this problem. He 

 says: — "Vieta, in a dispute with Adrian Romanus, proposed 

 this problem. The solution given by Romanus, though 

 obvious, was very indifferent, viz., by determining the centre 

 of the required circle by the point of intersection of two 

 hyperbolas. Vieta solved it very elegantly in his Jpollonius 

 Gallus, printed at Paris in 1600: his solution is the same as 

 that given in Neivton's Universal Arithmetic. Another solu- 

 tion may be seen in Lemma 16, Book I., of the Principia 

 (this question being there necessary for some determinations 

 in Physical Astronomy), where Newton, by a remarkable 

 dexterity, reduced the two higher loci of Romanus to the 

 intersection of two straight lines. Moreover, Descartes 

 attempted to solve this problem by algebraic analysis, but 

 without success; for, of the two solutions which he derived 



