860 Mr C. Taylor, On a Section of Newton's Principia. [Mar. 8, 



This anharmonic property of four fixed tangents, being pro- 

 jective, is true for any form of the quadrilateral determined by 

 the fixed tangents, and not for a parallelogram only. 



d. It follows also that the ratio of the ratios { \-r' -r^-r ] in 



-^ \QI.FLJ 



which the variable tangent divides either pair of opposite sides of 



IMLK is constant. And this ratio of ratios will still be constant 



(although not of the same magnitude as before) if the figure be 



projected from any vertex upon any plane, so that IMLK becomes 



a quadrilateral of any other form. The theorem in question may 



also be stated as follows : 



The ratios of the products of the distances of any tangeyit to a 

 conic from the three pair^s of opposite summits of a given circum- 

 scribed quadrilateral are constant. 



4, In the same Lemma 25, Cor. 3, it is proved that the centres 

 of all conies inscribed in a given quadrilateral are collinear. This 

 important and suggestive theorem served as a starting point for 

 the investigation of the properties of systems of conies subject to 

 four conditions. Notice the use made of it by Brianchon and 

 Poncelet in Gergonne's Annales, xi. 219. 



5. It is well known that a general method of transforming 

 curves is briefly but adequately laid down in Lemma 22. 



Thus the section under consideration contains the fundamental 

 propositions and methods on which so much of the modern geo- 

 metry is founded. The subject deserves to be treated at greater 

 length ; but sufiice it for the present to call attention to Newton's 

 proof of the important general property, § 3, of tangents to a conic, 

 which Chasles appears to have overlooked in his note 8ur la 

 propriete anharnionique des tangentes cVune conique (Apergu 

 Historique, Note xvi., pp. 341 — 344, edit. 2). 



I may remark in conclusion, that Lambert's solution of the 

 problem^ — sometimes called Sir Christopher Wren's problem — to 

 draw a straight line whose segments by four given st7'aight lines 

 shall be in given ratios, contains the anharmonic property in 

 question. Lambert shews (Insigniores orbitce cometarum proprie- 

 tates, Sect. i. Lemma 18) that the envelope of the line so divided 

 is the parabola which touches the four given lines. This evidently 

 includes the anharmonic property of four tangents to a parabola, 

 which may be at once extended to the general conic by projection. 



1 He refers to Newton for another solution, and Newton (sect. v. lemma 27, cor.) 

 refers to Wren and Wallis. 



