as the weight and the square of the time of oscillation and inversely as the length.

Prop. 25. The bob of a cycloidal pendulum, moving under gravity in a cycloid whose axis is vertical, and in a medium whose resistance is constant, will perform its oscillations in the same time as it would in a non-resisting medium of the same density, and proportional parts of the arcs are described simultaneously.

Prop. 26. Cycloidal pendulums, moving under gravity in cycloids whose axes are vertical, and in a medium whose resistance is proportional to the velocity, are isochronous.

Prop. 27. The difference between the time of oscillation of a pendulum moving in a medium whose resistance is proportional to the square of the velocity, and the time of oscillation of a similar pendulum moving in a non-resisting medium of the same density is approximately proportional to the arc described.

Prop. 28. If a cycloidal pendulum of length L move in a medium whose resistance is constant, then the ratio of the resistance to gravity is equal to the ratio of the excess of the whole arc described in the descent of the pendulum above the whole arc described in the subsequent ascent to 2L.

Prop. 29. A cycloidal pendulum oscillates under gravity in a cycloid whose axis is vertical, and in a medium whose resistance varies as the square of the velocity; to find the resistance at each point ; and therefore (cor. 3), the velocity at each point.

Prop. 30. On constrained motion in a cycloid under any law of resistance; from which the result of prop. 28 is deduced, also approximate solutions of prop. 29, and of the corresponding proposition when the resistance varies as the velocity. The proof was somewhat simplified in the second edition.

Prop. 31. A pendulum oscillates in a resisting medium. If the resistance in each of the proportional parts of the arc described be altered in any ratio, then the difference between the arc of descent and the arc of subsequent ascent is altered in the same ratio.

General Scholium. On Newton's pendulum experiments. In the first edition, this was printed at the end of section vii.

SECTION VII.-On the motion of fluids, and the resistance to projectiles.

The problems treated in this section are far from easy, and in general their treatment here is incomplete, but there is much that is interesting in studying the way in which Newton attacked questions which seemed to be beyond the analysis at his command.

Prop. 32. If two similar systems of bodies consist of an equal number of particles, and if the corresponding particles (each in one system to each in the other) be similar, proportional, and similarly situated among themselves, and their densities have to each other the same given ratio; and if they begin to move among themselves in proportional times, and with similar motions (that is, those in one system among one another, and those in the other among one another); and if the particles that are in the same system do not touch one another, except at the instants of reflexion; nor attract, nor repel each other, except with accelerations that are inversely as the diameters of the corresponding particles, and directly as the squares of the velocities; then the particles of these systems will continue to move among themselves with like motions and in proportional times.

Prop. 33. Under the same hypotheses the resistance offered by finite parts of the systems varies at any point as the square of the velocity of the particles there, as the squares of their diameters, and as the density of the part of the system there.

To which were added corollaries in which the proposition is applied to give the law of resistance to motion in the air and other fluids.

[Propositions 34-40 inclusive, and the corollaries, lemmas, and scholiums thereto attached, were rewritten in the second edition. I have here followed the order of the second edition. In the first edition prop. 34 is on an extension of the results of props. 32, 33 to cases where the particles of the systems are contiguous but frictionless; prop. 35 was what is here printed as 34; prop. 36 was on the resistance experienced by a sphere moving in a rare and elastic fluid; prop. 37 (of which the argument is erroneous) was on the motion of water passing through a hole in a vessel; prop. 38 was on the resistance experienced by the front of a sphere moving in a fluid; lemma 4 and prop. 39 were on the effect of acceleration impressed on a vessel containing fluid and floating bodies in relative equilibrium; prop. 40 was on the resistance experienced by a sphere moving in a fluid of given density; with a general scholium the substance of which was subsequently transferred to the end of section vi, see above, p. 99.]

Prop. 34. A rare medium consists of small quiescent particles of equal magnitude and freely disposed at equal distances. If a globe and a cylinder of equal diameters move in such a medium with equal velocities in the direction of the axis of the cylinder, then the resistance to the motion of the globe is half that to the motion of the cylinder.

Scholium. On similar propositions concerning the motion of other figures.

Newton commences by determining the conical frustrum of given

base and altitude which will meet with least resistance when moving in the direction of its axis.

Next consider a solid generated by the revolution of the oval ADFB about AB and moving in the direction AB. Then he shows that, if at B we draw the tangent HBG, and if we take points H and G on

it so that the tangents GF and HI make the angles FGB and IHB each equal to 135°, then the solid generated by the revolution of the figure ADFGBHIE about AB will encounter less resistance than the original solid.

Also, if N be any point on the generating curve, and if NM be drawn perpendicular to AB, and if a line through G parallel to the tangent at N cut AB produced in R, then the solid described by a curve such that 4 MN·BR.GB2 GR4

will encounter less resistance than any other solid of revolution of the same length and breadth.

[Newton's determination of the solid of least resistance is deducible from the differential equation of the generating curve, but in the Principia he gave no proof. The problem may be solved by the calculus of variations, but it long remained a puzzle to know how Newton had arrived at the result. A letter contained in the Portsmouth Collection* has set the matter at rest, and as all my readers may not have seen it, I reproduce here the part dealing with this scholium. It is also interesting as showing how freely Newton made use of fluxions in estab. lishing results given in the Principia. The letter seems to have been written to David Gregory shortly after his visit to Cambridge in May, 1694 (see below, p. 122), and after alluding to that visit and some other matters Newton proceeds:

The figure which feels the least resistance in the Schol. of Prop. xxxv. Lib. ii. is demonstrable by these steps.

* Catalogue, pp. xxi-xxiii.

1. If upon BM be erected infinitely narrow parallelograms BGhb and MNom and their distance Mb and altitudes MN, BG be given, and Mm+ Bb be also given and called s and

the semi sum of their bases

their semi difference.

Mm — Bb
2

2

be called and if the lines BG, bh, MN,

mo, butt upon the curve nNgG in the points n, N, g, and G, and the infinitely little lines on and hg be equal to one another and called c, and the figure mnNg GB be turned about its axis BM to generate a solid, and this solid move uniformly in water from M to B according to the direction of its axis BM: the summ of the resistances of the two

surfaces generated by the infinitely little lines Gg, Nn shall be least when gG¶¶ is to nÑ1a as BG × Bb to MN × Mm.

For the resistances of the surfaces generated by the revolution of

BG
Ggquad Nn quad'

MN

that is, if Ggquad and Nnquad be

BG

MN
q

called p and q, as and and their summ + is least when

BG. MN

P q
BG x p

is nothing, or

= +

MN × ġ. qq

Now p Ggquad

Bbquad + għquad = ss

2sxxx+cc and therefore

p=−2sx+2xx, and by the same argument ġ=2s*+2æc and therefore

pp is to qq as BG ×s-a to MN× s+x, that is, gG¶¶ to nNo as BG × Bb to MN × Mm.

2. If the curve line DnNgG be such that the surface of the solid generated by its revolution feels the least resistance of any solid with the same top and bottom BG and CD, then the resistance of the two narrow annular surfaces generated by the revolution of the [infinitely

little lines nN] and Gg is less than if the intermediate solid bg NM be removed [along CB without altering Mb, until bg comes [to BG], supposing as before that on is equal to hg,] and by consequence it is the least that can be, and therefore gG¶¶ is to nNo as BG × Bb [is to MN × Mm].

[Also if] gh be equal to he so that the angle [gGh is 45degr] then will 4 Bb¶¶ be [to nNo as BG × Bb is to] MN × Mm, and by consequence 4BG¶¶ is to GR¶¶ as BG is to MN × BR or 4BGa × BR is to GRcub [as GR to MN].

Whence the proposition to be demonstrated easily follows.]

Prop. 35. To find the resistance of a sphere moving uniformly forward in a medium such as that described in prop. 34. If the sphere and the particles of the medium be perfectly elastic, the resistance is to the force by which the motion could be destroyed in the time in which the sphere describes two-thirds of its diameter as the density of the medium to the density of the sphere. Also (cor. 1) if the sphere and the particles be perfectly inelastic, the resistance is diminished onehalf.

Scholium.

On motion in continuous mediums such as water. Prop. 36. On the motion of water running out of a cylindrical vessel through a circular hole in the bottom.

Lemma 4. The resistance to the motion through a fluid of a cylinder in the direction of its axis is independent of its length.

Prop. 37. If a cylinder move uniformly in the direction of its length through a compressed infinite inelastic fluid, the ratio of the resistance (due to its transverse section) to the force by which motion may be destroyed in the time it takes to move four times its length, is approximately equal to the ratio of the density of the fluid to the density of the cylinder.

Scholium. In the above proposition the stream lines of the fluid are assumed to be parallel to the axis of the cylinder. This is not accurately true.

Lemmas 5, 6, 7, and Scholium. All smooth convex solids of revo lution (such as cylinders, spheres, spheroids) with their axes along the axis of a canal containing inelastic frictionless fluid (such as water) will equally hinder and be equally acted on by the fluid flowing through the canal. Consideration of the circumstances under which this is true.

Prop. 38. If a sphere move uniformly through a compressed infinite inelastic fluid, the ratio of its resistance to the force by which its motion may be destroyed in the time it takes to move eight-thirds of its diameter is approximately equal to the ratio of the density of the fluid to the density of the sphere.