Hi': 



TIIANSACTIOXS OF SECTION A. 



645 



port ion of its lengtli gets segmented crosswise so as to show soiuewLat like the 

 nittli's 111" a snake. 



Assuming that the foregoing is the nsnal mode of formation of this sky from a 

 thiiiiiish vapom'-laden stratum, if tiie wedge does not ])Ush inwards in increased 

 voliinu', tlio visihle vapour will graduallv disappear, diffnsed in tlie neighbouring 

 drier air, and fair weather probably may follow. If, liowever, it drive in in largely 

 iiu reasi d volume, the cloudlets will be ellaced into llie formation of large con- 

 tinuous cloud — a likely harbinger of rain, as we Icnow is often the case in such cir- 

 uumstances. 



TUJJSnAY, SEPTJJMnni '2. 



SUBSKCTIOX CiF MATUrMATICS. 



Tiie following Papers and Tleport were read : — 



1. Soie on Newton^ s Thcori/ of Asfronoimcal Iicfrarffnii, ami on his ]-J.rph(. 

 iinflov, of the Moticii of the Moon's Ai'iorjec. JUj rrofes.soi' J. C. Adams, 

 F.li.S, 



2. Historical Note on Continuity, liij the Rev. C. Taylok, D.B. 



I. A vital principle of all science is expressed by tlie term continuity. 



It is tiie recognition of this principle wliicii dilierentiates tlie modern from the 

 inicient geometry, and in the department of geonu^try it asserts itself in tlie most 

 cDiuplete and striking way in relation to tlie so-called circular points at intinily in 

 any plane. 



The study of mathematics from age to age has contributed directly and in- 

 directly to the advancement of science in general, and even such parts of it as are 

 most abstract have had their full jn'actical ell'ect in the formal ion and sprea'i if 

 scientific ideas. The doctrine even of the circular points at infinity is not to ? .■ 

 garded as barren or unpractical. 



II. A passage of the utmost importance for the history of modern geometry, 

 which has nevertheless escaped the notice of writers on that subject, is to be found 

 ill Kepler's ' Ad Vitellionem paralipomena,' cap. IV., §4 ( l(i04). 



In this passage, speaking of tlu> foci of conies as jioiuts wiiicb then had no 

 name, he liiniself proposes to call them foci. He shows that the parabola has a 

 locus at infinity, that lines radiating from this 'caucus f iciis' are parallel, and that 

 ii may be regarded as lying either within or without t!ie curve. 



Tims he regards every straight line or system of parallels as having one point 

 only at iniiuity. From this we deduce that all the points at infinity in one and the 

 snine plane constitute a quasi-rectilinear locus, since a straight line drawn at 

 random therein meets this locus in one point only. 



He also shows how to pass by insensible gradations from the circle through the 

 three normal forms of conies to tlie line-pair, laying down clearly and decisively the 

 principle of continuity, not indeed under that name, but under the head of analogy. 



How ])rofoundly he was impressed with the depth and range of this principle 

 will be gathered from his great sayiuji: : — 



. . . plurinmm nanique amo analoijias, JldvUssimos mens niaijistros, omnium 

 naiuirfl arcanonim conscios. 



HI. I conclude with three proofs of the existence of the circular points at 

 inQnity. 



(1.) In a given plane draw a circle and let it meet the line at infinity in 

 points .1' and y. Take an arc AB of the circle subtending any angle at the circum- 

 ference, and therefore at .i- and ;/. Througli x draw two straight lines at random. 

 These may be regarded as parallel to ,rA and .)IJ respectively, because x is on the 

 line at infinity, and therefore as containing an angle equal to A.rB, that is to say, 



