[Chap. IV.]
Of the RESISTANCE of FLUIDS.
BEFORE the cause can be discovered, which keeps the planets in motion, it is necessary first to know, whether the space, wherein they move, is empty and void, or filled with any quantity of matter. It has been a prevailing opinion, that all space contains in it matter of some kind or other; so that where no sensible matter is found, there was yet a subtle fluid substance by which the space was filled up; even so as to make an absolute plenitude. In order to examine this opinion, Sir Isaac Newton has largely considered the effects of fluids upon bodies moving in them.
2. These effects he has reduced under these three heads. In the first place he shews how to determine in what manner the resistance, which bodies suffer, when moving in a fluid, gradually increases in proportion to the space, they describe in any fluid; to the velocity, with which they describe it; and to the time they have been in motion. Under the second head he considers what degree of resistance different bodies moving in the same fluid undergo, according to the different proportion between the density of the fluid and the density of the body. The densities of bodies, whether fluid or solid, are measured by the quantity of matter, which is comprehended under the same magnitude; that body being the most dense or compact, which under the same bulk contains the greatest quantity of solid matter, or which weighs most, the weight of every body being observed above to be proportional to the quantity of matter in it[95]. Thus water is more dense than cork or wood, iron more dense than water, and gold than iron. The third particular Sir Is. Newton considers concerning the resistance of fluids is the influence, which the diversity of figure in the resisted body has upon its resistance.
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].
5. Our author has also handled distinctly that particular sort of motion, which is described by pendulums[105]; and has likewise considered some few cases of bodies moving in resisting fluids round a center, to which they are impelled by a centripetal force, in order to give an idea of those kinds of motions[106].
6. The treating of the resistance of pendulums has given him an opportunity of inserting into another part of his work some speculations upon the motions of them without resistance, which have a very peculiar elegance; where in he treats of them as moved by a gravitation acting in the law, which he shews to belong to the earth below its surface[107]; performing in this kind of gravitation, where the force is proportional to the distance from the center, all that Huygens had before done in the common supposition of its being uniform, and acting in parallel lines[108].
7. Huygens at the end of his treatise of the cause of gravity[109] informs us, that he likewise had carried his speculations on the first of these suppositions, of the resistance in fluids being proportional to the velocity of the body, as far as our author. But finding by experiment that the second was more conformable to nature, he afterwards made some progress in that, till he was stopt, by not being able to execute to his wish what related to the perpendicular descent of bodies; not observing that the measurement of the curve line, he made use of to explain it by, depended on the hyperbola. Which oversight may well be pardoned in that great man, considering that our author had not been pleased at that time to communicate to the publick his admirable discourse of the quadrature or measurement of curve lines, with which he has since obliged the world: for without the use of that treatise, it is I think no injury even to our author’s unparalleled abilities to believe, it would not have been easy for himself to have succeeded so happily in this and many other parts of his writings.