VOt. XVI.] PHILOSOPHICAL TRANSACTIONS. 355 



terms of the first column taken backward. For a is the last, m a the next be- 

 fore it; and so of the rest. And the aggregate of all the numerators is so 

 many times 1, as is the number of terms, suppose t, wanting the first column ; 



that is t , or ; & this again divided by the common de- 

 nominator n, becomes '^—^ -. Therefore "—^ — —g, is the line of descent 



nn ran »' 



by its own gravity. 



33. If therefore this be added to a projecting force perpendicularly down- 

 ward, or subducted from such projecting force upward ; that is, to or from 



-—-^f\ the descent in the first case will be — ^/ + ^—^ — ~-z \ and the 



ascent in the other case ^-^^J ^—^ — —&• And in this latter case, when 



« "^ f»» ° ' 



the negative part becomes equal to the positive part, the ascent is at the highest; 



after which it will descend. 



34. In an horizontal or oblique projection, having taken -^^f in the line of 



projection, and thence, at the angle given, — — g in the line of descent ; 



the point in the curve answering to these, is the place of the project answer- 

 ing to that moment. 



35. I am aware of some objections to be made both to some points of the 

 process, and to some of the suppositions. But I saw not well how to avoid 

 them without making the computation much more perplexed. And in a matter 

 so nice, and which much depend upon physical observations, it will be hard to 

 attain such accuracy as not to stand in need of some allowances. Somewhat 

 migtit have been further added to direct the experiments above suggested. But 

 that may be done at leisure, after deliberation had, which way to attempt the 

 experiment. The like is to be said of the different resistance which different 

 bodies may meet with in the same medium, according to their different gravities, 

 extensively or intensively considered, and their different figures and positions in 

 motion. ^ 



3'6. As to the question proposed whether the resistance of the medium do 

 not always take off such a proportional part of the force moving through it, as 

 is the specific gravity of the medium to that of the body moved in it. I think 

 this can by no means be admitted. For there are many other things of consi- 

 deration herein, besides the intensive or specific gravity of the medium. As 

 for instance, a viscous medium will ,resist more than one more fluid, though of 

 like intensive gravity ; and a sharp arrow will make its way more easily through 

 a medium than a blunt headed bolt, though of equal weight ; and the same py- 



z z 2 



