BOOK II. ON THE MOTION OF BODIES IN RESISTING MEDIUMS.

The second book is devoted to the discussion of the motion of bodies in resisting mediums. It is divided into nine sections as follows:

SECTION I.-On the motion of bodies in a medium whose resistance varies directly as the velocity.

Prop. 1. If a body be resisted in the ratio of its velocity the momentum lost by the resistance is proportional to the space traversed. Lemma 1. Quantities proportional to their differences are continually proportional. That is, if a : a-b-bb-c-c:c-d, etc., then a:b=b:c=c: d, etc.: see lemma 2 of the tract De Motu quoted above (p. 36).

Prop. 2. If a body move under the action of no external force in a homogeneous medium whose resistance varies as the velocity, and the time of motion be divided into a number of equal intervals, then the velocities at the beginnings of those intervals are in a geometrical progression, and the spaces described in each of those intervals are as those velocities.

Prop. 3. A body moves in a straight line under gravity (supposed uniform) in a homogeneous medium whose resistance varies as the velocity; to find the motion.

Prop. 4. A body is projected under gravity (supposed uniform and constant in direction) in a homogeneous medium whose resistance varies as the velocity; to find the motion.

The first and second of the seven corollaries attached to this proposition were added in the second edition.

Scholium. The above law of resistance is to be regarded as a mathematical hypothesis rather than a physical one. In mediums void of tenacity the resistance varies as the square of the velocity, and to the consideration of motion under that law the next section is devoted.

SECTION II.-On the motion of bodies in a medium whose resistance varies as the square of the velocity.

Prop. 5. If a body move under the action of no external force in a homogeneous medium whose resistance varies as the square of the velocity, and if the time of motion [reckoned from a certain era] be divided into a number of intervals in a geometrical progression whose ratio is greater than unity, then the velocities at the beginnings of those

intervals are as the reciprocals of the corresponding terms of that geometrical progression, and the spaces described in each of those intervals are equal.

Prop. 6. Homogeneous and equal spheres moving under no external force in a medium whose resistance varies as the square of the velocity will, in times which are reciprocally as their velocities at the beginnings of those times, describe equal spaces and lose parts of their velocities proportional to the wholes.

Prop. 7. Under the same hypotheses any homogeneous spheres will, in times which are directly as their momenta and inversely as the squares of their velocities at the beginnings of those times, describe spaces proportional to those times and the velocities at the beginnings of those times conjointly, and lose parts of their momenta proportional to the wholes.

Lemma 2. On the rule for forming the fluxions (or moments) of products, quotients, and powers of simple algebraical quantities. Scholium. To this lemma a scholium was added.

In the first edition this scholium was to the following effect: In some letters which passed about ten years ago between that most skilful geometrician, G. G. Leibnitz, and myself, I informed him that I possessed a method of finding maxima and minima, of drawing tangents, and of performing similar operations, which was applicable to both rational and irrational quantities, and I concealed this method in transposed letters involving this sentence [Data aequatione quotcunque fluentes quantitates involvente, fluxiones invenire, et vice versa]: that illustrious man replied that he also had lighted on a method of the same kind, and he communicated his method which hardly differed from my own except in the language and notation (and in the idea of the generation of quantities). The fundamental principle of both is contained in this lemma.

Leibnitz appealed to this as evidence of his invention of the calculus independently of Newton, but Newton asserted that it was not written in that sense, and was merely a statement of an historical fact.

In the second edition the scholium remained unaltered, save that in the last line but one the words in brackets () were inserted.

In the third edition this scholium was replaced by another to the following effect: In a certain letter of mine to Mr. J. Collins, dated December 10, 1672, having described a method of tangents which I suspected to be the same as that of Slusius, at that time not yet published, I added these words: "Hoc est unum particulare vel corollarium potius “methodi generalis, quae extendit se citra molestum ullum calculum, non "modo ad ducendum tangentes ad quavis curvas sive geometricas sive

"mechanicas vel quomodocunque rectas lineas aliasve curvas respicientes, verum etiam ad resolvendum alia abstrusiora problematum genera de "curvitatibus, areis, longitudinibus, centris gravitatis curvarum, &c. neque (quemadmodum Huddenii methodus de maximis et minimis) ad "solas restringitur aequationes illas quae quantitatibus surdis sunt im(( munes. Hanc methodum intertexui alteri isti qua aequationum exegesin "instituo reducendo eas ad series infinitas." Thus far that letter. And these last words relate to a tract which I had written on these matters in the year 1671. The fundamental principle of that general method is contained in the preceding lemma.

Prop. 8. If a body move in a straight line under the action of gravity (supposed uniform) in a homogeneous medium, and the space described be divided into equal parts, then the resultants of gravity and the resistances at the beginnings of those spaces are in a geometrical progression.

Prop. 9. Under the hypotheses of prop. 8, to find the time of ascent to the highest point and the time of descent to any point.

Prop. 10. To find the density of a medium which shall make a body move in a given curve, it being supposed that gravity is uniform and constant in direction, and that the resistance of the medium varies jointly as its density and the square of the velocity: also to find the velocity of the body at any point.

Several necessary corrections were introduced in the demonstration given in the second edition (see passim the Cotes Correspondence, letters lxviii, lxxxvii.) The problem is solved by the use of fluxions (moments); is illustrated by applying the results to the cases of (i) a semicircle, (ii) a parabola, (iii) a hyperbola, (iv) the curve xym am+1; and the results were extended by a scholium to the case where the resistance varies as the nth power of the velocity with numerous illustrations; but Newton failed to find a curve which made the density constant.

SECTION III.-On the motion of bodies in a medium whose resistance consists of two terms, one varying as the velocity and the other as the square of the velocity.

Prop. 11. If a body move in such a medium under no external forces, and a series of times be taken in arithmetical progression, then the sums of a constant and quantities inversely proportional to the velocities at the beginnings of these times are in geometrical progression.

Prop. 12. Under the same hypotheses, if a series of spaces described be taken in arithmetical progression, then the sums of a

constant and quantities proportional to the velocities at the beginnings of those spaces are in geometrical progression.

Prop. 13. A body moves in a straight line under gravity (supposed uniform) in such a medium; if the velocity be represented graphically by a line in a certain way, then the time of motion can be represented graphically by an area; and conversely.

A short scholium on this was added in the third edition.

Prop. 14. Under the same hypotheses, and if the resultants of the resistance and gravity be taken in geometrical progression, then the space described in a certain time can be represented graphically by the difference between two areas.

Scholium. On the resistance of fluids as caused partly by the tenacity, partly by the attrition, and partly by the density of fluids; and on the extension of the foregoing propositions. This was added in the third edition.

SECTION IV.-On the spiral motion of bodies in a resisting medium.

Lemma 3. On the radius of curvature at any point of an equiangular spiral.

Prop. 15. A body moves in a resisting medium under a central force which varies inversely as the square of the distance; if the density of the medium (to which, other things being equal, the resistance is proportional) vary inversely as the distance from the centre of force, the body may revolve in an equiangular spiral. The argument was somewhat altered in the second edition.

To this were appended nine corollaries on motion in such an orbit, and when such motion is possible.

Prop. 16. The result of prop. 15 is true also when the central force varies inversely as the nth power of the distance.

Scholium. On motion in mediums such as those discussed in props. 15, 16.

Prop. 17. To find the central force and the resistance of a medium in order that a body may move in a given spiral, the law of velocity being given.

Prop. 18. To find the density of a medium in order that a body may move in a given spiral, the law of force being given.

SECTION V.—On the density and pressure of fluids, and on Hydrostatics.

Definition. A fluid is a body whose parts yield to any force acting on it, and in yielding are easily moved among themselves.

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Prop. 19. The pressure at every point of a homogeneous fluid at rest, contained in a vessel at rest, is the same and is equal in all directions the consideration of condensation, gravity, and centripetal forces being neglected-and the parts of the fluid remain at rest unmoved by such pressure.

Prop. 20. To find the pressure on the surface of a sphere covered by a mass of fluid which gravitates to the centre of the sphere, the strata of equal density being concentric spheres; with corollaries on the theory of floating bodies.

Prop. 21. An elastic fluid in which the density is proportional to the pressure is attracted by a central force which varies inversely as the distance; if a series of distances from the centre be taken in geometrical progression, then the densities at those distances will be also in geometrical progression.

Prop. 22. An elastic fluid in which the density is proportional to the pressure is attracted by a central force which varies inversely as the square of the distance; if a series of distances from the centre be taken in harmonical proportion, then the densities at those distances will be in geometrical progression.

Scholium. On the theorems analogous to props. 21 and 22 under other laws of centripetal force or other laws connecting the density and pressure. Newton, however, says that to discuss all these cases would be tedious, and it would be but of little use, since (he adds in the third edition) experiments show that the density of our atmosphere is either accurately, or at least extremely nearly, proportional to the pressure.

Prop. 23. A fluid is composed of particles which are mutually repulsive; if the density be proportional to the pressure, the force of repulsion must vary inversely as the distance; and vice versa.

Scholium. If the force of repulsion vary inversely as the nth power of the distance, the pressure will vary as the (n+2)th power of the density; and vice versa. But in the above propositions it must be assumed that the repulsion does not extend indefinitely. Also it must be remembered that whether elastic fluids do consist of mutually repulsive particles is a physical question on which no opinion is expressed.

SECTION VI.-On the motion of pendulums in resisting mediums.

Prop. 24. The masses of pendulums, such that the distances between their centres of oscillation and of suspension are constant, vary jointly as the weights and the squares of the times of oscillation in vacuo. And (cor. 5) universally the mass of a pendulum varies directly