TRANSACTIONS OF SECTION A. 645 



portion of its length gets segmented crosswise so as to show somewhat like the 

 rattles of a snake. 



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

 thinnish vapour-laden stratum, if the wedge does not push inwards in increased 

 volume, the visible vapour will gradually disappear, diffused in the neighbouring 

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

 increased volume, the cloudlets will be effaced ioto the formation of large con- 

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

 cumstances. 



TUESDAY, SEPTEMBER 2. 



Subsection of Mathematics. 



The following Papers and Report were read : — 



1. Note on Norton's Theory of Astronomical Refraction, and on Ms Expla- 

 nation of the Motion of the Moon's Apogee. By Professor J. C. Adams, 

 F.B.S. 



2. Historical Note on Continuity. By the Rev. C. Taylor, D.D. 



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



It is the recognition of this principle which differentiates the modern from the 

 ancient geometry, and in the department of geometry it asserts itself in the most 

 complete and striking way in relation to the so-called circular points at infinity 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 pi'actfcal effect in the formation and spread of 

 scientific ideas. The doctrine even of the circular points at infinity is not to be re- 

 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 

 in Kepler's ' Ad Vitellionem paralipomena,' cap. IV., §4 ( IG04). 



In this passage, speaking of the foci of conies as points which then had no 

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

 focus at infinity, that lines radiating from this ' csecus focus' are parallel, and that 

 it may be regarded as lying either within or without the curve. 



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

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

 same 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 the line-pair, laying down clearly and decisively the 

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



How profoundly he was impressed with the depth and range of this principle 

 will be gathered from his great saying : — 



. . . plurimum namque amo analogias, Jidelissimos meos magistros, omnium 

 natures arcanorum conscios. 



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

 infinity. 



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

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

 ference, and therefore at x and »/. Through x draw two straight lines at random. 

 These may be regarded as parallel to x A and .rB respectively, because x is on the 

 line at infinity, and therefore as containing an angle equal to ArB, that is to say, 



