3. FOR the more perfect illustration of the first of these heads, he distinctly shews the relation between all the particulars specified upon three different suppositions. The first is, that the same body be resisted more or less in the simple proportion to its velocity; so that if its velocity be doubled, its resistance shall become threefold. The second is of the resistance increasing in the duplicate proportion of the velocity; so that, if the velocity of a body be doubled, its resistance shall be rendered four times; and if the velocity be trebled, nine times as great as at first. But what is to be understood by duplicate proportion has been already explained[96]. The third supposition is, that the resistance increases partly in the single proportion of the velocity, and partly in the duplicate proportion thereof. 4. IN all these suppositions, bodies are considered under two respects, either as moving, and opposing themselves against the fluid by that power alone, which is essential to them, of resisting to the change of their state from rest to motion, or from motion to rest, which we have above called their power of inactivity; or else, as descending or ascending, and so having the power of gravity combined with that other power. Thus our author has shewn in all those three suppositions, in what manner bodies are resisted in an uniform fluid, when they move with the aforesaid progressive motion[97]; and what the resistance is, when they ascend or descend perpendicularly[98]. And if a body ascend or descend obliquely, and the resistance be singly proportional to the velocity, it is shewn how the body is resisted in a fluid of an uniform density, and what line it will describe[99], which is determined by the measurement of the hyperbola, and appears to be no other than that line, first considered in particular by Dr. ~BARROW~[100], which is now commonly known by the name of the logarithmical curve. In the supposition that the resistance increases in the duplicate proportion of the velocity, our author has not given us the line which would be described in an uniform fluid; but has instead thereof discussed a problem, which is in some sort the reverse; to find the density of the fluid at all altitudes, by which any given curve line may be described; which problem is so treated by him, as to be applicable to any kind of resistance whatever[101]. But here not unmindful of practice, he shews that a body in a fluid of uniform density, like the air, will describe a line, which approaches towards an hyperbola; that is, its motion will be nearer to that curve line than to the parabola. And consequent upon this remark, he shews how to determine this hyperbola by experiment, and briefly resolves the chief of those problems relating to projectiles, which are in use in the art of gunnery, in this curve[102]; as ~TORRICELLI~ and others have done in the parabola[103], whose inventions have been explained at large above[104].