1880.] Mr C. Taylor, On a Section of N'ewtons Principia. 359 



(3) C. Taylor, M.A. On a section of Newtons Principia in 

 relation to Modern Geometry. 



The following notes on the fifth section of the first book of 

 Newton's Principia comprise inter alia some particulars (§ 3) 

 which I have not seen stated before. 



1. The section commences with what may be called the 

 Tetragram of Pappus, i.e. the theorem (handed down by him 

 without solution) that the product of the distances of any point 

 on a conic from two opposite sides of a fixed inscribed quadrilateral 

 varies as the product of its distances from the other two sides. 

 Descartes proved this only by his new method of co-ordinates : 

 Newton here proves it by the most elementary geometry. It is 

 well known that the anharmonic property of four poiiits on a conic 

 is merely another way of stating the Tetragram of Pappus. 



2. From this theorem Newton deduces his own organic de- 

 scription of a conic by means of two rotating angles of given 

 magnitudes. It is easy to see that we have here again, under 

 another form, the same anharmonic property. 



3. The anharmonic projjerty of tangents to a conic. 



Newton shews in a corollary to Lemma 25 that if the sides 

 IM, ML, LK, KI of a fixed parallelogram circumscribed to a given 

 conic be met by any fifth tangent in points EFHQ, then 



KQ . ME = a constant = KH . MF. 



a. Hence (K and M being fixed points on the tangents from a 

 fixed point /) any ttuo fixed tangents to a conic are divided homo- 

 graphically by a variable tangent. 



b. It is likewise evident that the intercepts IE and IQ are 

 connected by a relation of the form, 



a.IE.IQ-^h.IE + c.IQ + d^O, 



which is a "tangential equation" to the conic referred to any given 

 pair of tangents. 



c. From the constancy of the rectangle KQ.ME, and in like 



FE 



manner of QI . LH, it follows that the ratio of the ratios ^, and 



QE . 



Y-= IS constant. 



