VOL. X.XXXIV.] PHILOSOPHICAL TRANSACTIONS. 351 



A. Now take away the weights a and c, and put a weight at b equal to their 



sum ; and then the weight at b being equally distant from q and p, the prop and 



fulcrum must sustain equal parts of the whole weight, and therefore the prop 



will now also sustain a weight equal to a. Hence if the prop q be taken away, 



the moving force to turn the lever about p in both cases must evidently be the 



same ; therefore the effects of a and c on the lever to turn it about any point, are 



the same as when they are both placed in the middle point between them. And 



the same is manifestly true if a and c be placed without the fulcrum and prop. 



If therefore ac be a cylindrical lever of uniform density, its effect to turn itself 



about any point will be the same as if the whole were collected into the middle 



point B ; which follows from what has been already proved, by conceiving the 



whole cylinder to be divided into an infinite number of laminas perpendicular to 



its axis, of equal thicknesses. 



The principle therefore assumed by Archimedes is thus established on the 



most self-evident principle, that is, that equal bodies at equal distances must 



produce equal effects ; which is manifest from this consideration, that when all 



the circumstances in the cause are equal, the effects must be equal. Thus the 



whole demonstration of Archimedes is rendered perfectly complete, and at the 



same time it is very short and simple. The other part of the demonstration we 



shall here insert for the use of those who may not be acquainted with it. 



Let XY be a cylinder, which bisect in a, on which point 



R A 7 C 

 it would manifestly rest. Take any point z, and bisect v « ., r i - Y 



zx in B, and zy in c ; then, from what has been proved, 



the effects of the two parts zx, zy to turn the lever about a, is the same as if 



the weight of each part were collected into b and c respectively, which weights 



are manifestly as zx, zy, and which therefore conceive to be placed at b and c. 



Now ab = ax — xb = ixY — -^xz = -i-Yz ; and ac = ay — yc = ^xy — ^zy 



= i xz ; consequently ab : ac :: 4yz : ixz :: yz : xz :: the weight at c : the 



weight at b. 



Tlje property of the straight lever being thus established, every thing relative 



to the bent lever immediately follows. 



FI. Of some Particulars Observed during the late Eclipse of the Sun. By Wm. 

 Herschel, LL.D., F.R.S., p. 39. 



It will be proper to remark, says Dr. H., that my attention, in observing 

 this eclipse, was not directed to the time of the several particulars which are 

 usually noticed in phaenomena of this kind ; such as the beginning, the end, and 

 the digits eclipsed. I was very well assured that the care of other astronomers 

 would render my endeavours in that respect perfectly unnecessary. The only view 

 I had was, to avail myself of the power and distinctness of my telescopes, in 



