678 PHILOSOPHICAL TRANSACTIONS. [aNNO 1712. 



dotes. Hist. Ox. chap. 5, par. §5, and in each dimple a small circle; and in the 

 centre of each circle a little stud like a pin's head. This is the only curiosity of 

 the kind I have seen ; and is not referrible to any thing I can think of, either in 

 the animal or vegetable kingdom. Among the iron oars of the same hills we 

 found some new spars, and several specimens of oars shot into a constant and 

 regular figure, though not reducible to any animal or vegetable bodies. 



We had at Pontipool, on the 6th instant, an extraordinary shower of hail ; 

 which extended about a mile, and lasted near half an hour. It broke down the 

 stalks of all the beans and wheat within that circumference ; and ruined as much 

 glass at Major Hanbury's house, as cost 4l. the repairing. Some of the stones 

 were 8 inches about ; as to their figure, very irregular and unconstant ; several 

 of the hail-stones being compounded. 



Concerning the Proportion of Mathematical Points to each other. By the Hon, 

 Francis Robartes, Esq. V. P. R. S. N° 334, p. 470. 



It has hitherto passed for a current maxim, that all infinites are equal. 

 Divines and metaphysicians have not scrupled to ground many of their argu- 

 ments on that foundation. Yet the position is certainly erroneous, as Dr. 

 Halley has abundantly shown in the Phil. Trans, for Oct. 1696. He there gives 

 several instances of infinite quantities which are in a determinate finite propor- 

 tion to each other, and some infinitely greater than others. The like may be 

 observed of infinitely small quantities, viz. matheumtical points, as the follow ■ 

 ing propositions will make appear. 



Prop. I. The Points of contact between Circles and their Tangents are in 

 Subduplicate proportion to the Diameters of the Circles, — Let two circles adch, 

 AFBG, fig. 8, pi. 17, touch each other internally, at the point a. Draw the 

 tangent PAa, and parallel to it the line mn ; also draw the diameter ac. Let 

 AC the diameter of the greater circle be called r ; and ab, the diameter of the 

 lesser circle, 5; dh, the chord of the arch dah = z; and pg, the chord of the 

 arch fag, be = 3/ ; and the absciss ak = x. 



If the line mn be supposed to move, till it becomes coincident with the 

 tangent PAa,, the nature of a circle will always give the following equations, 



zz = 4rx — Axx, 

 yy = Asx — Axx, 



When the line is arrived at the tangent, z and y will become the two points 

 of contact, and then zz = Arx, and yy = Asx. (Axx being rejected as hetero 

 geneous to the rest of the equation, by reason of x being become infinitely 

 little). Therefore 



zz : yy :: Arx : Asx v. r ', s. 

 Therefore z '. y '.', Vr : /*. o. e. d. 



