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XV. Of the Penetration of a Hemisphere by an indefinite Number of equal and similar Cylinders. By Thomas Knight, Esq. Communicated by Sir Humphry Davy, LL. D. Sec. R. S.


THE well known theorems of VIVIANI and BossUT, respect


ing certain portions of the surface and solidity of a hemisphere, form, together, a single case of the following problem; which is one of b the most remarkable, for generality and simplicity of result, in the whole compass of geometry.



Read March 19, 1812.




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To pierce a hemisphere, perpendicularly on the plane of its base, with any number of equal and similar cylinders; of such a kind, that, if we take away from the hemisphere those portions of the cylinders that are within it, the remaining part shall admit of an exact cubature: and if we take away, from the surface of the hemisphere, those portions cut out by the cylinders, the remaining surface shall admit of an exact quadrature.

Let fig. 1 represent the nearest half of the hemisphere, where a is the pole, bdcf a quadrant of the great circle form

ing its base. From every point d, on this side of b, draw the radius dC to the centre of the hemisphere, and (if the number of cylinders is to be 27*) take the arc bs equal to n times the arc bd, draw se perpendicular to Cb, and with the centre C and radius Ce describe the arc er cutting Cd in r. Through all the points (r) thus found, draw the curve line brC, terminated at b and C, and it shall be half the base of one of the required cylinders.

It is, in the first place, evident, from the construction, that the half cylinder, whose base is beCrb, is contained between two planes CabC, CacC, making with each other the angle 90° bCc = ; consequently the whole base of the hemisphere may be pierced by 2n such cylinders as this is the half of.


Let atmb be the intersection of the surfaces of the half cylinder and hemisphere; amd a great circle passing through a and d, and meeting atmb at m. Call the radius of the sphere r, Cr is the cosine of the arc bs to the radius r, by construction; it is also the cosine of the arc md to the same radius; therefore md = bsn x bd.

Put bd = 4; md = n x p; dr= 4. Moreover, put A for the spherical space atmbdcna contained by the arcs anc, cdb and the curve atmb; and let S be the solidity of the portion of the hemisphere contained between the quadrant ancC and the surface (brCatmb) of the half cylinder. It is easy to see that A=rss& cos. & × 4,

S=S& cos. & × cos. ↓ × cos. ↓ ↓


- ESS¢ cos. 14 × × cos. ↓ ↓.

* I do not intend zu to represent an even number only, n may be, or, or 1⁄2, &c. and an express any number whatever.


The fluents to be taken, first from 0 to=no, and then from 4 = 0, to


= 20. The first operation gives



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Arf sin. no,

{sin. no+sin. 3nq}, — Sø sin. no cos. *nq,


and by the second we get



s = = {

Scos. 3no



cos. gnq} + C, which fluents being taken from noo, to no 90°, are A =; S = x; and if these are multiplied by în, we 3;



A = C — —

Cos. no

3 cos. no



A = 4o; S = = r2


for the whole that remains of the surface and solidity of the hemisphere after the subduction of the 2n cylinders. Thus A and S (for the whole hemisphere) do not depend on the number of the cylinders with which the penetration is made; a most remarkable circumstance, seeing that amongst the bases of those cylinders are curves of an infinity of different kinds and orders.

Let fig. 2 represent half the base of one of the cylinders; p b Cb the radius of the hemisphere, C the centre. From r, any point in the curve, let fall the perpendicular rp rp, y.


on the axis; call Cp, x;

By construction, Cr = √x + y2=r cos. n. bCr; now the cosine of the simple arc bCr is which being put in the √x2 + y2 trigonometrical expression for the cosine of the multiple arc

in terms of the cosine of the simple one, we have, for the equation of the curve brC.

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XVI. On the Motions of the Tendrils of Plants. By Thomas Andrew Knight, Esq. F. R. S. In a Letter to the Right Hon. Sir Joseph Banks, Bart. K. B. P. R. S.

Read May 4, 1812.


THE motions of the tendrils of plants, and the efforts they apparently make to approach and attach themselves to contiguous objects, have been supposed by many naturalists to originate in some degrees of sensation and perception: and though other naturalists have rejected this hypothesis, few, or no experiments have been made by them to ascertain with what propriety the various motions of tendrils, of different kinds, can be attributed to peculiarity of organization, and the operation of external causes. I was consequently induced, during the last summer, to employ a considerable portion of time to watch the motions of the tendrils of different species of plants; and I have now the pleasure to address to you an account of the observations I was enabled to make.

The plants selected were the Virginia creeper (the ampelopsis quinquefolia of MICHAUX,) the ivy, and the common vine and pea.

A plant of the ampelopsis, which grew in a garden pot, was removed to a forcing house in the end of May, and a single shoot from it was made to grow perpendicularly upwards, by being supported in that position by a very slender bar of wood, to which it was bound. The plant was placed in the middle

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