264 Mr G. Taylor, On the geometrical . [May 19, 



be cut off by a chord qq' at right angles to the axis; but the 

 restriction may be removed by supposing the first ellipse to change 

 its form whilst the form and dimensions of the second remain (for 

 the time being) invariable, and by comparing the several forms of 

 the first ellipse with the second, and thus with one another. 



Take a length ;S^X equal to AA\ and on SX take a length 

 RR' equal to QQ', and such that SR + SR' = SQ + 8Q'. Then, SX 

 being regarded as a flat ellipse*, the time in the arc QQ' is equal to 

 the time in which a comet falling into the sun 8 from rest at X 

 would traverse the distance RR''f. 



The above, which is a simplification of Lambert's own proof, 

 applies to the Hyperbola as well as to the Ellipse |. 



The case of the Parabola is treated § separately in Sectio li. theor. 

 4, where it is remarked by anticipation: "Insignis hsec motus 

 cometarum parabolici proprietas, si debite limitetur, ceteris quo- 

 que Sectionibus conicis adplicabilis est" (p. 45). In the last page 

 of the Preface he speaks expressly of the Hyperbola. 



II. 



The construction might also have been made as follows : 



Draw two separate ellipses on equal major axes, and take 



equal focal radii SP in the one and sj) in the other. Let any 



pair of chords QQ' and qq which make equal intercepts PE and 



pe on SP and sp be ordinately applied to the diameters through 



* The line SPX equal to A A' is a limiting form of the first ellipse, whose further 

 focus may be at any point on the circle drawn with P as centre and radius equal 

 to AA' - SP. 



t Otherwise thus : write dowTi the expression for the time in the are qq' of the 

 auxiliary elhpse, and deduce the expression for the time in QQ' in terms of AA', 

 QQ', and SQ + SQ'. 



J I find that Lexell gave the proof of- the property in question geometrically 

 (making however some slight use of trigonometrical ratios) in 1784; but the 

 complete proof as he gave it is so elaborated {Nova Acta Academics Scientiarum Im- 

 perialis Petropolitance, tom i. pp. (141)— (146), (149), (150), 1787), that it makes the 

 theorem appear less simple geometrically than it really is. He failed to appreciate 

 the generality of Lambert's proof through not observing that the first ellipse was 

 to be regarded as variable in relation to the second, and he was under the im- 

 pression that he had extended the theorem by remarking that it was applicable to 

 the hyperbola. See pp. (141), (149). For the reference to Lexell I am indebted to 

 Chasles, Apercu Historique, p. 187 (ed. 2, 1875), where it is said: "La propriete de 

 I'ellipse qui est le fondement de ce thdoreme couvient aussi aux secteiu-s de I'hyper- 

 bole, ainsi que I'a demontre par de simples considerations de Geometric le c^lebre 

 LexeU." 



§ Euler expressed the area of a parabolic sector ASP in terms of tan J ASP, 

 and thus determined the time. See his Theoria motmim Planetaram et Gometanim, 

 Prob. IV. p. 16 and Prob. vii. p. 29, to which work Lambert refers at the end of his 

 Preface. I learn from Professor Adams that Euler also gave the equation com- 

 monly referred to as Lambert's in the case of the Parabola, viz. in the Miscellanea 

 Berolineima, Tomus vii. pp. 19, 20 (1743). 



