Philosophical transactions, Vol. L. Part I. For the year 1757. · Various
GENERAL PROPOSITION.
_Supposing an homogeneous sphere_ OABCD (Fig. 1.) _revolving uniformly about its centre, to be acted on at the extremity_ A _of the radius_ OA, _in a direction_ AL _perpendicular to the plane of the equator_ ABCD, _and parallel to the axis of rotation_ Pp, _by a given force, tending to generate a new motion of rotation at right angles to the former; It is proposed to determine the change, that will arise in the direction of the rotation in consequence of the said force._ Let _F_ denote the given force, whereby the motion about the axis P_p_ is disturbed, supposing _f_ to represent the centrifugal force of a small particle of matter in the circumference of the equator, arising from the sphere’s rotation; and let the whole number of such particles, or the content of the sphere, be denoted by _c_: let also the momentum of rotation of the whole sphere, or of all the particles, be supposed, in proportion to the momentum of an equal number of particles, revolving at the distance OA of the remotest point A, as _n_ is to _unity_.
It is well known, that the centripetal force, whereby any body is made to revolve in the circumference of a circle, is such, as is sufficient to generate all the motion in the body, in a time equal to _that_, wherein the body describes an arch of the circumference, equal in length to the radius. Therefore, if we here take the arch AR = OA, and assume _m_ to express the time, in which that arch would be uniformly described by the point A, the _motion_ of a particle of matter at A (whose central force is represented by _f_) will be equal to _that_, which might be uniformly generated by the force _f_, in the time _m_; and the motion of as many particles (revolving, all, at the same distance) as are expressed by _cn_ (which, by hypothesis, is equal to the momentum of the whole body), will, consequently, be equal to the momentum, that might be generated by the force _f_ × _cn_, in the same time _m_.
Whence it appears, that the momentum of the whole body about its axe P_p_ is in proportion to the momentum generated in a given particle of time _m’_, by the given force _F_ in the direction AL, as _ncf_ × _m_ is to _F_ × _m’_, or, as _unity_ to (_F_/_ncf_) × (_m’_/_m_) (because the quantities of motion produced by unequal forces, in unequal times, are in the ratio of the forces and of the times, conjunctly). Let, therefore, AL be taken in proportion to AM, as (_F_/_ncf_) × (_m’_/_m_) is to _unity_ (supposing AM to be a tangent to the circle ABCD in A), and let the parallelogram AMNL be compleated; drawing also the diagonal AN; then, by the composition of forces, the angle NAM (whose tangent, to the radius OA, is expressed by OA × (_F_/_ncf_) × (_m’_/_m_)) will be the change of the direction of the rotation, at the end of the aforesaid time (_m’_). But, this angle being exceeding small, the tangent may be taken to represent the measure of the angle itself; and, if Z be assumed to represent the arch described by A, in the same time (_m’_) about the center O, we shall also have (_m’_/_m_) = (Z/AR) = (Z/AO), and consequently OA × (_F_/_ncf_) x (_m_/_m’_) = Z × (_F_/_ncf_). From whence it appears, that the angle expressing the change of the direction of the rotation, during any small particle of time, will be in proportion to the angle described about the axe of rotation in the same time, as _F_/_ncf_ is to _unity_. _Q.E.I._
Altho’, in the preceding proposition, the body is supposed to be a perfect sphere, the solution, nevertheless, holds equally true in every other species of figures, as is manifest from the investigation. It is true, indeed, that the value of _n_ will not be the same in these cases, even supposing those of _c_, _f_ and _F_ to remain unchanged; except in the spheroid only, where, as well as in the sphere, _n_ will be = ⅖; the momentum of any spheroid about its axis being 2-5ths of the momentum of an equal quantity of matter placed in the circumference of the equator, as is very easy to demonstrate. But to shew now the use and application of the general proportion here derived, in determining the regress of the equinoctial points of the terrestrial spheroid, let AE_a_F (_Fig. 2._) be the equator, and P_p_ the axis of the spheroid: also let HECF represent the plane of the ecliptic, S the place of the sun, and HAPNH the plane of the sun’s declination, making right-angles with the plane of the equator AE_a_F: then, if AK be supposed parallel, and OKM perpendicular, to OS, and there be assumed _T_ and _t_ to express the respective times of the annual and diurnal revolutions of the earth, it will appear (from the _Principia_, B. III. prop. xxv.) that the force, with which a particle of matter at A tends to recede from the line OM in consequence of the sun’s attraction, will be expressed by (_3tt_/_TT_) × (AK/OA) × _f_; _f_ denoting the centrifugal force of the same particle, arising from the diurnal rotation. Hence, by the resolution of forces, (_3tt_/_TT_) × (AK/OA) × (OK/OA) × _f_ will be the effect of that particle, in a direction perpendicular to OA, to turn the earth about its center O.
The very ingenious M. Silvabelle, in his essay on this subject, inserted in the 48th volume of the Philosophical Transactions, makes the quantity of the annual precession of the equinox, caused by the sun, to be the half, only, of what is here determined. But this gentleman appears to have fallen into a twofold mistake. First, in finding the _momenta of rotation_ of the terrestrial spheroid, and of a very slender ring, at the equator thereof; which _momenta_ he refers to an axis perpendicular to the plane of the sun’s declination, instead of the proper axe of rotation, standing at right angles to the plane of the equator. The difference, indeed, arising from thence, with respect to the spheroid (by reason of its near approach to a sphere) will be inconsiderable; but, in the ring, the case will be quite otherwise; the equinoctial points thereof being made to recede just twice as fast as they ought to do. This may seem the more strange, if regard be had to the conclusions, relating to the nodes of a satellite, derived from this very assumption. But, that these conclusions are true, is owing to a second, or subsequent mistake, at Art. 27; where the measure of the sun’s force is taken the half, only, of the true value; by means whereof the motion of the equinoctial points of the ring is reduced to its proper quantity, and the motion of the equinoctial points of the terrestrial spheroid, to the half of what it ought to be.
That expert geometrician M. Cha. Walmsley, in his Essay on the Precession of the Equinox, printed in the last volume of the Philosophical Transactions, has judiciously avoided all mistakes of this last kind, respecting the sun’s force, by pursuing the method, pointed out by Sir Isaac Newton; but, in determining the effect of that force, has fallen into others, not less considerable than those above adverted to. In his third Lemma, the momentum of the whole Earth, about its diameter, is computed on a supposition, that the momentum or force of each particle is proportional to its distance from the axis of motion, or barely as the quantity of motion in such particle, considered abstractedly. No regard is, therefore, had to the lengths of the unequal levers, whereby the particles are supposed to receive and communicate their motion: which, without doubt, ought to have been included in the consideration. In his first proposition, he determines, in a very ingenious and concise manner, the true annual motion of the nodes of a ring (or of a single satellite) at the earth’s equator, revolving with the earth itself, about its center, in the time of one siderial day. This motion he finds to be = (3co-s. 23° 29´/4 rad.) × (⅟366¼) × 360°. Then, in order to infer from thence, the motion of the equinoctial points of the earth itself, he, first, diminishes that quantity, in the ratio of 2 to 5: Because (as is demonstrated by Sir Isaac Newton in his 2d Lemma) the whole force of all the particles situated without the surface of a sphere, inscribed in the spheroid, to turn the body about its center, will be only 2-5ths of the force of an equal number of particles uniformly disposed round the whole circumference of the equator, in the fashion of a ring.
The quantity ((3co-s. 23° 29´/4 rad.) × ⅖ × (⅟366¼) × 360°) thus arising, will, therefore, express the true motion of the equinoctial points of a ring, equal in quantity of matter to the excess of the whole earth above the inscribed sphere, when the force whereby the ring tends to turn about its diameter is supposed equal to the force whereby the earth itself tends to turn about the same diameter, in consequence of the sun’s attraction. Thus far our author agrees with Sir Isaac Newton; but, in deriving from hence the motion of the equinoctial points of the earth itself, he differs from him; and, in the corollary to his third Lemma, assigns the reasons, why he thinks Sir Isaac Newton, in this particular, has _wandered a little from the truth_. Instead of diminishing the quantity above exhibited (as Sir Isaac has done) in the ratio of all the motion in the ring to the motion in the whole earth, he diminishes it in the ratio of the motion of all the matter above the surface of the inscribed sphere to the motion of the whole earth: which matter, tho’ equal to that of the ring, has nevertheless a different momentum, arising from the different situation of the particles in respect to the axis of motion.
But since the aforesaid quantity, from whence the motion of the earth’s equinox is derived, as well by this gentleman, as by Sir Isaac Newton, expresses truly the annual regress of the equinoctial points of the ring (and not of the hollow figure formed by the said matter, which is greater, in the ratio of 5 to 4) it seems, at least, as reasonable to suppose, that the said quantity, to obtain from thence the true regress of the equinoctial points of the earth, ought to be diminished in the former of the two ratios above specified, as that it should be diminished in the latter. But, indeed, both these ways are defective, even supposing the momenta to have been truly computed; the ratio, that ought to be used here, being that of the momenta of the ring and earth about the proper axe of rotation of the two figures, standing at right-angles to the plane of the ring and of the equator. Now this ratio, by a very easy computation, is found to be as 230²-229² to ⅖ of 230²; whence the quantity sought comes out = (3co-s. 23° 29´/4 rad.) × (⅟366¼) × (230²-229²)/230² × 360° = 21´´ 6´´´: which is the same that we before found it to be, and the double of what this author makes it.
What has been said hitherto, relates to that part of the motion only, arising from the force of the sun. It will be but justice to observe here, that the effect of the moon, and the inequalities depending on the position of her nodes, are truly assigned by both the gentlemen above-named; the ratio of the diameters of the earth, and the density of the moon being so assumed, as to give the maxima of those inequalities, such as the observations require: in consequence whereof, and from the law of the increase and decrease (which is rightly determined by theory, tho’ the absolute quantity is not) a true solution, in every other circumstance, is obtained. The freedom, with which I have expressed myself, and the liberty I have here taken, to animadvert on the works of men, who, in many places, have given incontestible proofs of skill and genius, may, I fear, stand in need of some apology. ’Tis possible I may be thought too peremptory. Indeed, I might have delivered my sentiments with more caution and address: but, had not I imagined myself quite clear in what has been advanced, from a multitude of concurrent reasons, I should have thought it too great a presumption to have said any thing at all here, on this subject. The great regard I have for this Society, of which I have the honour to be a member, will, I hope, be considered as the motive for my having attempted to rectify some oversights, that have occurred in the works of this learned body.
LIV. _Remarks upon the Heat of the Air in_ July 1757. _in an Extract of a Letter from_ John Huxham, _M.D. F.R.S. to_ William Watson, _M.D. F.R.S. dated at_ Plymouth _19th of that Month. With additional Remarks by Dr._ Watson. [Read Dec. 22, 1757.] “FROM the beginning of June last we have had a very dry season, generally very warm, and sometimes excessively hot. From the 7th to the 14th of this month the heat was violent; greater, indeed, than has been known here in the memory of man. I have talked with several persons, who have lived a considerable time in Jamaica, Gibraltar, and Minorca; and they severally assert, that they never felt such intense heat in any of those places. Upon the 11th, 12th, and 13th of this month, Fahrenheit’s thermometer, in the shade, about three o’clock in the afternoon, was at 87; nay, upon the 12th it was even above 88. Abundance of people have suffered very severely from these excessive heats: putrid, bilious, petechial, nervous fevers, are exceedingly common every-where. Dysenteries, hæmorrhages, most profuse sweats, affect not only those in fevers, but a vast many others. The days and nights were so intolerably hot, that little or no sleep was to be gotten day or night. The wind we had, like the Campsin, actually blew hot, tho’ strong.
Upon the 15th, about seven at night, at Falmouth, Penryn, Truro, and thereabouts, a pretty smart shock of an earthquake was felt, attended with a hollow rumbling noise, throwing down pewter, china-ware, and such-like. The tinners felt it eighty fathom under ground. No great damage however was done. The day before we had, about eleven o’clock before noon, a most violent hurricane, which lasted five or six minutes, attended with a heavy shower.” Thus far Dr. Huxham. The heat of the air at London, during the period above-mentioned, was much greater than has been usually observed in these high latitudes; tho’ it was never quite so severe here as at Plymouth. The following table exhibits the degrees of the heat, taken here upon the respective days, about four o’clock in the afternoon, by a Fahrenheit’s thermometer. The instrument was placed in the shade; and the accuracy of the observer, who favoured me with his minutes, is not to be questioned. 1757. July 5 75 6 78 7 75½ 8 78 10 80¼ 11 83¼ 12 80¼ 13 80 14 85 15 81 16 73 From hence it appears, that the air at London was, upon several days, hotter than it had been observed at Madeira for ten years together: for, by Dr. Thomas Heberden’s observations, mentioned in the Philosophical Transactions, the heat of the air at Madeira, during that period, was never but once at 80.
William Watson. LV. _Remarks upon the Letter of Mr._ John Ellis, _F.R.S. to_ Philip Carteret Webb, _Esq; F.R.S. printed in the_ Philosophical Transactions, _Vol._ xlix. _Part_ ii. _p._ 806. _By Mr._ Philip Miller, _F.R.S._ [Read Dec. 15, 1757.] THE paper of mine, which was read before the Royal Society on the 8th of May 1755, and afterward printed in the xlixth volume of the Philosophical Transactions[208], was written at the request of Mr. Watson; who informed me, that a letter from the Abbé Mazeas to the reverend Dr. Hales had been communicated to the Royal Society, in which it was mentioned, that the Abbé Sauvages had made a discovery of the juice of the Carolina Toxicodendron staining linen of a permanent black. But Mr. Watson said, that the letter, he thought, required a careful perusal before it was printed; and he wished I would confirm it. I told him, if the letter was put into my hands, I would look it over, and deliver my opinion of it. Accordingly Dr. Birch delivered the letter to me; and, upon reading it, I found, that tho’ this might be a discovery to those two gentlemen; yet, as it had been mentioned in several printed books long before, I thought it might not be for the reputation of the Royal Society to have it printed as such in their Transactions.
This was my motive for writing that paper: in which I have not endeavoured to depreciate the discovery of the Abbé Sauvages, but have only mentioned what had occurred to me in those books of botany, where that shrub is taken notice of. And as the knowlege of it, and the method of collecting the varnish, might be of service to the inhabitants of the British colonies in America, I took the liberty of adding the account given of it by Dr. Kœmpfer. Mr. Ellis, in his letter to Mr. Webb, asserts, that the American _Toxicodendron_ is not the same with Kœmpfer’s _Arbor vernicifera legitima_. This assertion of his makes it necessary to lay before the Society the authorities, upon which I have grounded my belief, that they are the same. But it may not be amiss first to take notice, that the shrub mentioned by the Abbé Sauvages is the same with that, which the gardeners about London call the Poison-ash. The title of it, mentioned by the Abbé Sauvages, was given by myself to that shrub, in a catalogue of trees and shrubs, which was printed in the year 1730; before which it had no generical title applied to it. And about the same time I sent several of the plants to Paris and Holland with that title, which I had raised a few years before from seeds, which were sent by Mr. Catesby from Carolina.
And altho’ this shrub had not been reduced to any genus before, yet it had been some years growing in the gardens of the Bishop of London at Fulham, at Mr. Reynardson’s at Hillenden, Mr. Darby’s at Hoxton, and in the Chelsea garden, which were raised from seeds sent by Mr. Banister from Virginia; two of which were growing at Chelsea in the year 1722, when the care of that Garden was intrusted to me. The first intimation I had of the American shrub being the same with Dr. Kœmpfer’s true varnish-tree, was from the late Dr. William Sherard, in the year 1726, when that gentleman desired me to bring him a specimen of the American Toxicodendron from the Chelsea garden; which I accordingly did: and then the Doctor, and Dr. Dillenius, compared it with a dried specimen in the collection of the former, which was gathered in Japan, and which, if I remember right, he told me he received from Dr. Kœmpfer some years before. It appeared to those two gentlemen, that they were the same; and their skill in the science of botany was never doubted. About a year after this, I carried a specimen of the American Toxicodendron to an annual meeting of some botanists at Sir Hans Sloane’s in Bloomsbury; where there were present Mr. Dale of Braintree, Mr. Joseph Miller, Mr. Rand, and some others; which was then compared with Dr. Kœmpfer’s specimen, whose collection Sir Hans Sloane had purchased: and it was the opinion of every one present, that they were the same. Nor has any one doubted of their being so, who has compared the American shrub with Kœmpfer’s figure and description of his true varnish-tree, but Mr. Ellis.
And now give me leave to examine his reasons for differing in opinion from every late botanist, who has mentioned this shrub. He says, that the midrib, which supports the lobe leaves, is quite smooth in the poison-ash, as is also the under side of the leaves; whereas Dr. Kœmpfer, in his description of the midrib of the true varnish-tree, calls it _læviter lanuginoso_; and in his description of the lobes or _pinnæ_ he says, they are _basi inequaliter rotunda_; whereas those of the poison-ash come to a point at their footstalks nearly equal to that at the top. These characters, Mr. Ellis thinks, are sufficient to prove, that they are different plants: and he blames Dr. Dillenius for having omitted these necessary characters in his description of it; and supposes this must have misled the accurate Linnæus, who quotes his synonyma. But as Dr. Linnæus is possessed of Kœmpfer’s book, he would little have deserved the appellation of accurate in this particular, had he not consulted the original, but trusted to a copy. But this I know he has done, and is as well assured, that the plants in question are the same, as Mr. Ellis can be of the contrary.
But here I must observe, that the branch, from which Dr. Kœmpfer’s figure is taken, is produced from the lower part of a stem, which seems to have been cut down, and not from a flowering branch; and it is not improbable, that his description may have been taken from the same branch: and if this be the case, it is easy to account for the minute differences mentioned by Mr. Ellis; for it would not be difficult to produce instances of hundreds of different trees and shrubs, whose lower and upper branches differ much more in the particulars mentioned by Mr. Ellis, than the figure and description given by Kœmpfer do from the American Toxicodendron. I will only mention two of the most obvious: the first is the white poplar, whose shoots from the lower part of the stem, and the suckers from the root, are garnished with leaves very different in form and size from those on the upper branches, and are covered on both sides in the spring with a woolly down. The next is the willow with smooth leaves, which, if a standard, and the head lopped off, as is usual, the young shoots are garnished with leaves much broader, and of different forms from those on the older branches; and these have frequently a hairy down on their under surface, which does not appear on those of the older. So that a person unacquainted with these differences in the same tree would suppose they were different. And the American Toxicodendron has varied in these particulars much more, in different seasons, than what Mr. Ellis has mentioned.
Mr. Ellis next says, that the Toxicodendron mentioned by Mr. Catesby, in his Natural History of Carolina, is not the same with that, which is now called by the gardeners poison-ash: but I am very positive of the contrary; for most of the plants in the nursery-gardens about London were first raised from the seeds, which were sent by Mr. Catesby from Carolina; part of which were sent to the late Dr. Sherard, as is mentioned by him in the Philosophical Transactions, Nº. 367; and another part came to my hands, from which I raised a great many of the plants, which were distributed, and some of them are now growing in the Chelsea garden. And that this shrub grows naturally in Carolina, I can have no doubt, having received the seeds of it two or three times from the late Dr. Dale, who gathered them in the woods of that country.
Ideas discussed in this chapter
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