VOL. XVII.] PHILOSOPHICAL TRANSACTIONS. 5Q3 



little of this spirit, until it shot into salts before my eyes, which it did in an 

 instant, like lightning ; but the figures by this means made, were like little irre- 

 gular pipes, so that I am not certain concerning their shape in this spirit. I 

 mixed some of this spirit with an equal quantity of blood, and at first could 

 see no alteration ; but in about 4- of an hour several of the blood-globules were 

 dissolved, and the spirit mixed with the serum looked reddish ; in another 

 quarter there were but few globules undissolved, and the serum looked redder 

 than before. 



y/n Instance of the Excellence of the Modern Algebra, in the Resolution of the 

 Problem of priding the Foci of Optic Glasses universally. By E. Halley, 

 S. R. S. N° 205, p. 960. 



The excellence of the modern geometry is in nothing more evident, than in 

 those full and adequate solutions it gives to problems ; representing all the pos- 

 sible cases at one view, and in one general theorem often comprehending whole 

 sciences ; which deduced at length into propositions, and demonstrated after 

 the manner of the ancients, might well become the subjects of large treatises : 

 for whatever theorem solves the most complicated problem of the kind, does 

 with a due reduction reach all the subordinate cases. Of this I now design to 

 give an instance in the doctrine of dioptrics. 



This dioptric problem, is that of finding the focus of any sort of lens, ex- 

 posed either to converging, diverging, or parallel rays of light, proceeding from, 

 or tending to, a given point, in the axis of the lens, be the ratio of refraction 

 what it will, according to the nature of the transparent material the lens is 

 formed of, and also with allowance for the thickness of the lens between the 

 vertices of the two spherical segments. This problem being solved in one 

 case, mutatis mutandis, will exhibit theorems for all the possible cases, whether 

 the lens be double-convex or double- concave, plano-convex or plano-concave, 

 or convexo-concave, which sort is usually lalled a meniscus. But this is only 

 to be understood of those rays which are nearest to the axis of the lens, so as 

 to occasion no sensible difference by their inclination to it ; and the focus here 

 formed is by dioptric writers commonly called the principal focus, being that of 

 use in telescopes and microscopes. 



Let then, in fig. 6, pi. 1 4, B E A be a double convex lens, C the centre of the 

 segment E B, and K the centre of the segment Eb, Bb the thickness of the 

 lens, D a point in the axis of the lens ; and it is required to find the point F, 

 where the beams, proceeding from the point D, are collected, the ratio of re- 

 fraction being as m to n. Let the distance of the object D B = D A = t/, (the 

 point A being supposed the same with B, but taken at a distance from it, to 



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