Part 10

The persons, from whose own mouths he can authenticate the fact, are James Martin, Adam Killick, Mrs. Jewell, and Mr. Chapman: and he has no less doubt as to Mr. Orgles and Mrs. Pigott (who was waked and much frighted by it), tho’ he did not indeed personally converse with either of the two last.

These two, and Mrs. Jewell, all inhabit quite close to Lingfield church-yard, on different sides of it: and Chapman lives within a quarter of a mile of it, to the south-west.

James Martin lives within a bow-shot of Starborough-castle, at the eastern edge of the parish of Lingfield, where it joins to that of Edenbridge; and Adam Killick’s habitation is three miles north-east of Starborough, at the north-western point of the parish of Edenbridge.

All these four, with whom Mr. Burrow personally conversed, agreed as to the _time_ of the concussion; _viz._ between one and two in the morning: and they all agreed as to the _shaking_ of their beds and windows; and all of them described the _continuance_ of the shock as not much more than instantaneous: but they did _not_ all hear the _noise_, which _some_ of them observed it to conclude with; particularly Adam Killick heard NO _noise_ at all; and yet, he says, he was broad awake when it first began: and it shook his house and bed, and made his windows rattle so much, that he was apprehensive of their being broken; and even caused one pane of glass (which was indeed loose before) actually to drop out. But James Martin, who was likewise fully awake (as was his wife too), _did hear_ the noise distinctly. He says, he felt his house and bed shake, heard his windows rattle, and some earthen ware clatter upon a chest of drawers; and also heard a noise, like the distant discharge of a cannon: whereupon he immediately said to his wife, “Lord! what is _that_?” but she happening, at that very instant, either to cough or sneeze (she cannot recollect which of the two), did not, tho’ quite awake, perceive any thing at all of the matter. However, she confirmed her husband’s asking her this question under an apparent surprize.

Mr. Burrow had a very particular conversation with these two separately: and he had also a very minute detail from Adam Killick (who works for him as a sort of gardener at Starborough); who further added, “that the shock waked and frighted his wife, tho’ she was fast asleep before.”

6th April, 1758.

James Burrow.

LXXXIII. _An Account of the Case of the First Joint of the Thumb torn off, with the Flexor Tendon in its whole Extent torn out. By_ Robert Home, _late Surgeon to the Thirtieth Regiment of Foot, and Surgeon at_ Kingston upon Hull. _In a Letter to_ John Pringle, _M. D. F.R.S._

[Read April 6, 1758.]

SIR,

I Take the liberty of inclosing to you a case in surgery, which I imagine is not very common. Marchetis indeed has an observation of the same kind; and there are several others collected together by Mons. Morand, in the second volume of the Memoires of the Royal Academy of Surgery at Paris: but as I have not heard of that volume’s being translated into English, and believe there is no observation of a similar nature in the Philosophical Transactions, I beg the favour of you to communicate it to the Royal Society, of which you are a Fellow; and at the same time to make them an offer of the joint of the thumb, with its adherent tendon, which you will receive at the same time with this; hoping they will do me the honour of accepting it, as a testimony (tho’ trifling) of my great esteem and respect for the most learned Society in Europe. Your Friend Dr. Knox saw the patient dressed oftener than once; and Mr. Thornhill, late Surgeon and Manmidwife in Bristol, saw it when near healed.

I beg you will believe me to be, with great truth,

SIR, Your most obedient, and most humble Servant, Robert Home.

Hull, March 17th, 1758.

JAnuary 2d, 1758, William Taylor, 17 years of age, an apprentice to a white-smith in this place, in endeavouring to make his escape from one, who was going to correct him, opened the door of a cellar, and threw himself into it; but in his hurry so intangled his right thumb with the latch, that the whole weight of his body was suspended by it, until it gave way, and was torn off at the first articulation; the flexor tendon being at the same time pulled out in its whole length, having broke when it became muscular. I was immediately sent for, found little or no hæmorrhage, and the bone of the second phalanx safe, and covered with its cartilage, but protruding considerably, occasioned by part of the skin belonging to it being irregularly torn off with the first joint.

I was doubtful, whether or not I should be obliged, at last, to make a circular incision, and saw the bone even with the skin; but thought it proper to give him a chance for the use of the whole phalanx.

He complained only for the first day of a pretty sharp pain in the course of the tendon; to which compresses, wrung out of warm brandy, were applied: but his arm was never swelled; there was no _ecchymosis_; nor had he so much fever, as to require bleeding even once. The cure proceeded happily, no symptoms arising from the extracted tendon. At the third dressing the bone was covered; and no other application but dry lint was necessary during the whole time. No exfoliation happened; yet it was twelve weeks before it was intirely cicatrised, owing to the loss of skin: and he seems to enjoy the use of the stump as completely, as if that tendon was not lost.

LXXXIV. _An Account of the late Discoveries of Antiquities at_ Herculaneum, _and of an Earthquake there; in a Letter from_ Camillo Paderni, _Keeper of the Museum at_ Herculaneum, _and F.R.S. to_ Tho. Hollis, _Esq; F.R.S. dated_ Portici, Feb. 1. 1758.

[Read April 6, 1758.]

WE have been working continually at Herculaneum, Pompeii, and Stabiæ, since my last of Dec. 16, 1756. The most remarkable discoveries made there are these, which follow.

February 1757, was found a small and most beautiful figure of a naked Venus in bronze, the height of which is six Neapolitan inches. She has silver eyes, bracelets of gold on her arms, and chains of the same metal above her feet; and appears in the attitude of loosening one of her sandals. The base is of bronze inlaid with foliage of silver, on one side of which is placed a dolphin.

In July we met with an inscription, about twelve Neapolitan palms in length, which I have here copied.

IMP·CAESAR·VESPASIANVS·AVG·PONTIF·MAX

TRIB·POT·̅VĪĪ·IMP·X̅VĪĪ·P·P·COS·̅VĪĪ·DESIGN·̅VĪĪĪ

TEMPLVM·MATRIS·DEVM·TERRAE·MOTV·CONLAPSVM·RESTITVIT

After having found a great number of volumes of papirus in Herculaneum; many pugillaries, styles, and stands with ink in them, as formerly mentioned; at length, in the month of August, upon opening a small box, we also found, to our exceeding great joy, the instrument, with which they used to write their manuscripts. It is made of wood, of an oblong form, but petrified, and broke into two pieces. There is no slit in it, that being unnecessary, as the ancients did not join their letters in the manner we do, but wrote them separate.

In September were discovered eight marble busts, in the form of terms. One of these represents Vitellius, another Archimedes; and both are of the finest workmanship. The following characters, in a black tint, are still legible on the latter, namely, ΑΡΧΙΜΕΔ which is all the inscription that now remains.

In October was dug up a curious bust of a young person, who has a helmet on his head, adorned with a civic crown, and cheek-pieces fastened under his chin. Also another very fine bust of a philosopher, with a beard, and short thick hair, having a slight drapery on his left shoulder. Likewise two female busts; one unknown, in a veil; the other Minerva, with a helmet; both of middling workmanship.

In November we met with two busts of philosophers, of excellent workmanship, and, as may be easily perceived, of the same artist; but unfortunately, like many others, without names.

In January was found a small, but most beautiful eagle, in bronze. It hath silver eyes, perches on a _praefericulum_, and holds a fawn between its talons.

In the same month we discovered, at Stabiæ, a term six palms high, on which is a head of Plato, in the finest preservation, and performed in a very masterly manner. Also divers vases, instruments for sacrificing, scales, balances, weights, and other implements for domestic uses, all in bronze.

At length I have finished, with much labour, the examination and arrangement of the scales, balances, and weights, which are very numerous in this museum; and, what is remarkable, many of the former, with all the weights, exactly answer those now in use at Naples. At present I am considering the liquid measures; and also engaged in disposing the paintings in the new apartment allotted for them. These affairs, with my usual province of inspecting the workmen, who are busied in digging; my being obliged to keep an exact register of every thing, that is discovered; besides other daily and accidental occurrences; employ my time so intirely, that I have not a moment’s repose, but in my bed.

The square belonging to the palace, in which the museum is deposited, will be finished, and completely ornamented, by Easter. In the center of it I have placed the bronze horse, which was broken in many pieces, and restored by me, as mentioned in my last. In the walls of the colonades are affixed all the inscriptions hitherto discovered: and I shall yet adorn them with altars, curule chairs, and other antiquities proper for such places. The principal entrance into the museum hath been made to correspond with the grand stair-case. On the right side of it stands the consular statue of Marcus Nonius Balbus, the father; and on the left, that of Marcus Nonius Balbus, the son; with two inscriptions relating to, and found near them. Upon the stair-case are placed eight antique statues in bronze, on beautiful pedestals of polished marble. In an opening in the center of the right hand colonade is fixed the statue of the wife of the elder Balbus, with the antique inscription belonging to it. At the entrance of the square, a magnificent pair of iron gates, with palisades, are just put up, ornamented with many bronzes, which are gilt; and on the sides of these gates are two other consular statues of persons unknown.

The whole day and night of the 24th of last month it seemed as if Mount Vesuvius would again have swallowed up this country. On that day it suffered two internal fractures, which intirely changed its appearance within the crater, destroying the little mountain, that had been forming within it for some years, and was risen above the sides; and throwing up, by violent explosions, immense quantities of stones, lava, ashes, and fire. At night the flames burst out with greater vehemence, the explosions were more frequent and horrible, and our houses shook continually. Many fled to Naples, and the boldest persons trembled. For my own part, I resolved to abide the event here at Portici, on account of my family, consisting of eight children, and a very weak and aged mother, whose life must have been lost by a removal in such circumstances, and so rigorous a season. But it pleased God to preserve us; for the mountain having vented itself that night and the succeeding day, is since become calm, and throws out only a few ashes.

LXXXV. _A further Attempt to facilitate the Resolution of Isoperimetrical Problems. By Mr._ Thomas Simpson, _F.R.S._

[Read April 13, 1758.]

ABOUT three years ago I had the honour to lay before the Royal Society the investigation of a general rule for the resolution of isoperimetrical problems of that kind, wherein one, only, of the two indeterminate quantities enters along with the fluxions, into the equations expressing the conditions of the problem. Under which kind are included the determination of the greatest figures under given bounds, lines of the swiftest descent, solids of the least resistance, with innumerable other cases. But altho’ cases of this sort do, indeed, most frequently occur, and have therefore been chiefly attended to by mathematicians, others may nevertheless be proposed, such as actually arise in inquiries into nature, wherein _both_ the flowing quantities, together with their fluxions, are jointly concerned. The investigation of a _rule_ for the resolution of these, is what I shall in this paper attempt, by means of the following

GENERAL PROPOSITION.

_Let_ Q, R, S, T, &c. _represent any variable quantities, expressed in terms of_ x _and_ y (_with given coefficients_), _and let_ q, r, s, t, &c. _denote as many other quantities, expressed in terms of_ ẋ _and_ ẏ; _It is proposed to find an equation for the relation of_ x _and_ y, _so that the fluent of_ Qq + Rr + Ss + Tt, &c. _corresponding to a given value of_ x (_or_ y), _may be a_ maximum _or_ minimum.

Let _A E_, _A F_, and _A G_, denote any three values of the quantity _x_, having indefinitely small _equi-differences E F_, _F G_; and let _E L_, _F M_, and _G N_, (perpendicular to _A G_) be the respective values of _y_, corresponding thereto; and, supposing _EF_ (= _FG_ = _ẋ_) to be denoted by _e_, let _c M_ and _d N_ (the successive values of _ẏ_) be represented by _u_ and _w_. Moreover, supposing _P´p´_ and _P´´p´´_ to be ordinates at the middle points _P´ P´´_, between _E_, _F_ and _F_, _G_, let the former (_P´p´_) be denoted α, and the latter (_P´´p´´_) by β; putting _A P´_ = _a_ and _A P´´_ = _b_. Then, if _a_ and α (the mean values of _x_ and _y_, between the ordinates _E L_ and _F M_) be supposed to be substituted for _x_ and _y_, in the given quantity _Qq_ + _Rr_ + _Ss_ + _Tt_, _&c._ and if, instead of _ẋ_ and _ẏ_, their equals _e_ and _u_ be also substituted, and the said (given) quantity, after such substitution, be denoted by _Q´q´_ + _R´r´_ + _S´s´_ + _T´t´_, _&c._ it is then evident, that this quantity _Q´q´_ + _R´r´_ + _S´s´_ + _T´t´_, _&c._ will express so much of the whole required fluent, as is comprehended between the ordinates _E L_ and _F M_, or as answers to an increase of _E F_ in the value of _x_. And thus, if _b_ and β be conceived to be wrote for _x_ and _y_, _e_ for _ẋ_, and _w_ for _ẏ_, and the quantity resulting be denoted by _Q´´q´´_ + _R´´r´´_ + _S´´s´´_ + _T´´t´´_, _&c._ this quantity will, in like manner, express the part of the required fluent corresponding to the interval _F G_. Whence that part answering to the interval _E G_ will consequently be equal to _Q´q´_ + _R´r´ &c._ + _Q´´q´´_ + _R´´r´´ &c._ But it is manifest, that the whole required fluent cannot be a _maximum_ or _minimum_, unless this part, supposing the bounding ordinates _E L_, _G N_ to remain the same, is also a _maximum_ or _minimum_. Hence, in order to determine the fluxion of this expression (_Q´q´_ + _R´r´ &c. Q´´q´´_ + _R´´r´´ &c._) which must, of consequence, be equal to nothing, let the fluxions of _Q´_ and _q´_ (taking α and _u_ as variable) be denoted by _̅Q_̇α and _̅qu⋅_; also let _̅R_̇α and _̅ru⋅_ denote the respective fluxions of _R´_ and _r´_; and let, in like manner, the fluxions of _Q´´, q´´, R´´, r´´, &c._ be represented by _̿Q_̇β, _̿q͘w_, _̿R_͘͘β͘ _̿rẇ_, _&c._ respectively. Then, by the common rule for finding the fluxion of a rectangle, the fluxion of our whole expression (_Q´q´_ + _R´r´ &c._ + _Q´´q´´_ + _R´´r´´ &c._) will be given equal to _Q´ ̅qu⋅_ + _q´ ̅Q_̇α + _R´ ̅ru⋅_ + _r´ ̅R_̇α _&c._ + _Q´´̿qẇ_ + _q´´ ̿Q_̇͘͘͘β + _R´´ ̿r͘w_ + _r´´ ̿R_̇β _&c._ = 0.

But _u_ + _w_ being = _GN_ - _EL_, and β - α = (_GN_ - _EL_) ⁄ 2 (a constant quantity), we therefore have _ẇ_ = -_u͘_, and ̇β = ̇α: also _u_ being (= 2_rp´_) = 2α - 2_EL_, thence will _u͘_ = 2̇α: which values being substituted above, our equation, after the whole is divided by ̇α, will become

2_Q´ ̅q_ + _q´ ̅Q_ + 2_R´ ̅r_ + _r´ ̅R, &c._ - 2_Q´´ ̿q_ + _q´´ ̿Q_ - 2_R´´ ̿r_ + _r´ ̿R, &c._ = 0;

or, _Q´´ ̿q_ - _Q´ ̅q_ + _R´´ ̿r_ - _R´ ̅r &c._ = (_q´ ̅Q_ + _q´´ ̿Q_) ⁄ 2 + (_r´ ̅R_ + _r´´ ̿R_) ⁄ 2, _&c._

But _Q´´ ̿q_ - _Q´ ̅q_, the excess of _Q´´ ̿q_ above _Q´ ̅q_, is the increment or fluxion (answering to the increment, or fluxion, _ẋ_) arising by substituting _b_ for _a_, β for α, and _w_ for _u_. Moreover, with regard to the quantities on the other side of the equation, it is plain, seeing the difference of _q´ ̅Q_ and _q´´ ̿Q_ is indefinitely little in comparison of their sum, that _q´ ̅Q_ may be substituted in the room of (_q´ ̅Q_ + _q´´ ̿Q_) ⁄ 2, _&c._ which being done, our equation will stand thus:

_Flux. Q´ ̅q_ + _R´ ̅r &c._ = _q´ ̅Q_ + _r´ ̅R &c._

But _q´ ̅Q_ + _r´ ̅R &c._ represents (by the preceding notation) the fluxion of _q´Q´_ + _r´R´ &c._ (or of _Qq_ + _Rr &c._) arising by substituting α for _y_, making α alone variable, and casting off ̇α. If, therefore, that fluxion be denoted by ̇υ, we shall have _flux. Q´ ̅q_ + _R´ ̅r &c._ = ̇υ, and consequently _Q´ ̅q_ + _R´ ̅r &c._ = υ. But _Q´ ̅q_ + _R´ ̅r &c._ (by the same notation) appears to be the fluxion of _Q´q´_ + _R´r´ &c._ (or of _Qq_ + _Rr &c._) arising by substituting _u_ for _ẏ_, making _u_ alone variable, and casting off _̇u_. Whence the following

GENERAL RULE.

_Take the fluxion of the given expression_ (_whose fluent is required to be a_ maximum _or_ minimum) _making_ ẏ _alone variable; and, having divided by_ ÿ, _let the quotient be denoted by_ υ: _Then take, again, the fluxion of the same expression, making_ y _alone variable, which divide by_ ẏ; _and then this last quotient will be_ = ̇υ.

When _ẏ_ is not found in the quantity given, υ will then be = 0; and, consequently, the expression for ̇υ, equal to nothing also. But if _y_ be absent, then will ̇υ = 0, and consequently the value of υ = a constant quantity. It is also easy to comprehend, that, instead of _ẏ_ and _y, ẋ_ and _x_ may be made successively variable. Moreover, should the case to be resolved be confined to other restrictions, besides that of the _maximum_ or _minimum_, such as, having a certain number of other fluents, at the same time, equal to given quantities, still the same method of solution may be applied, and that with equal advantage, if from the particular expressions exhibiting all the several conditions, one general expression composed of them all, with unknown (but determinate) coefficients, be made use of.

In order to render this matter quite clear, let _A, B, C, D, &c._ be supposed to represent any quantities expressed in terms of _x, y_, and their fluxions, and let it be required to determine the relation of _x_ and _y_, so that the fluent of _Aẋ_ shall be a _maximum_, or _minimum_, when the cotemporary fluents of _Bẋ, Cẋ, Dẋ, &c._ are, all of them, equal to given quantities.

It is evident, in the first place, that the fluent of _Aẋ_ + _bBẋ_ + _cCẋ_ + _dDẋ, &c._ (_b, c, d, &c._ being any constant quantities whatever) must be a _maximum_, or _minimum_, in the proposed circumstance: and, if the relation of _x_ and _y_ be determined (_by the rule_), so as to answer this single condition (under all possible values of _b, c, d, &c._) it will also appear evident, that such relation will likewise answer and include all the other conditions propounded. For, there being in the general expression, thus derived, as many unknown quantities _b, c, d, &c._ (to be determined) as there are equations, by making the fluents of _Bẋ, Cẋ, Dẋ, &c._ equal to the values given; those quantities may be so assigned, or conceived to be such, as to answer all the conditions of the said equations. And then, to see clearly that the fluent of the first expression, _Aẋ_, cannot be greater than arises from hence (other things remaining the same) let there be supposed some other different relation of _x_ and _y_, whereby the conditions of all the other fluents of _Bẋ, Cẋ, Dẋ, &c._ can be fulfilled; and let, _if possible_, this new relation give a greater fluent of _Aẋ_ than the relation above assigned. Then, because the fluents _bBẋ, cCẋ, dDẋ, &c._ are given, and the same in both cases, it follows, according to this supposition, that this new relation must give a greater fluent of _Aẋ_ + _bBẋ_ + _cCẋ_ + _dDẋ, &c._ (under all possible values of _b, c, d, &c._) than the former relation gives: _which is impossible_; because (whatever values are assigned to _b, c, d, &c._) _that_ fluent will, it is demonstrated, be the greatest possible, when the relation of _x_ and _y_ is that above determined, by the General Rule.

To exemplify, now, by a particular case, the method of operation above pointed out, let there be proposed the fluxionary quantity (_xⁿ yᵐ ẏᵖ_) ⁄ _ẋ⁽ᵖ ⁻ ¹⁾_; wherein the relation of _x_ and _y_ is so required, that the fluent, corresponding to given values of _x_ and _y_, shall be a _maximum_, or _minimum_. Here, by taking the fluxion, making _ẏ_ alone variable (_according to the rule_) and dividing by _ÿ_, we shall have (_pxⁿ yᵐ ẏ⁽ᵖ ⁻ ¹⁾_) ⁄ _ẋ⁽ᵖ ⁻ ¹⁾_ = υ. And, by taking the fluxion a second time, making _y_ alone variable, and dividing by _ẏ_, will be had (_mxⁿ y⁽ᵐ ⁻ ¹⁾ ẏᵖ_) ⁄ _ẋ⁽ᵖ ⁻ ¹⁾_ = ̇υ. Now from these equations to exterminate υ, let the latter be divided by the former; so shall _mẏ_ ⁄ _py_ = ̇υ ⁄ υ; and therefore _ay⁽ᵐ ⁄ ᵖ⁾_ = υ (_a_ being a constant quantity). From whence _y⁽ᵐ ⁄ ᵖ⁾ẏ_ = _(a ⁄ p)⁽¹ ⁄ ⁽ᵖ ⁻ ¹⁾⁾_ × _ẋx⁽⁻⁽ⁿ ⁄ ⁽ᵖ ⁻ ¹⁾⁾⁾_; and consequently (_p_ ⁄ (_m + p_)) × _y⁽⁽ᵐ + ᵖ⁾ ⁄ ᵖ⁾_ = _(a ⁄ p)⁽¹ ⁄ ⁽ᵖ ⁻ ¹⁾⁾_ × (_p_ - 1) ⁄ (_p_ - _n_ - 1) × _x_⁽⁽_ᵖ_ ⁻ _ⁿ_ ⁻ ¹⁾ ⁄ ⁽_ᵖ_ ⁻ ¹⁾⁾.

Let there be now proposed the two fluxions _xⁿyᵐẋ_ and _xᵖy⒬ẏ_, the fluent of the former being required to be a _maximum_, or _minimum_, and that of the latter, at the same time, equal to a given quantity. Then the latter, with the general coefficient _b_ prefixed, being joined to the former, we shall here have _xⁿyᵐẋ_ + _bxᵖy⒬ẏ_. From whence, by proceeding as before, _bxᵖy⒬_ = υ, and _mxⁿy⁽ᵐ ⁻ ¹⁾ẋ_ + _qbxᵖy⁽ᵖ ⁻¹⁾ẏ_ = ̇υ. From the former of which equations, by taking the fluxions on both sides, will be had _pbx⁽ᵖ ⁻¹⁾y⒬ẋ_ + _qbxᵖy⁽⒬ ⁻ ¹⁾ẏ_ (= ̇υ) = _mxⁿy⁽ᵐ ⁻ ¹⁾ẋ_ + _qbxᵖy⁽⒬ ⁻ ¹⁾ẏ_. Whence _pbx⁽ᵖ ⁻ ¹⁾y⒬_ = _mxⁿy⁽ᵐ ⁻ ¹⁾_; and therefore _pby⁽⒬ ⁻ ᵐ ⁺ ¹⁾_ = _mx⁽ⁿ ⁻ ᵖ ⁺ ¹⁾_. And in the same manner proper equations, to express the relation of _x_ and _y_, may be derived, in any other case, and under any number of limitations.

LXXXVI. _Observations on the_ Alga Marina latifolia; _The Sea Alga with broad Leaves. By_ John Andrew Peyssonel, _M.D. F.R.S. Translated from the_ French.

[Read April 13, 1758.]

HAVING cast anchor at Verdun, the road at the entrance of the river of Bourdeaux, I was fishing with a kind of drag-net upon a bank of sand, which was very fine and muddy. We collected a number of sea-plants, and among them the great broad-leaved Alga, which I did not know: and as the root or pedicle of this plant appeared to be very particular, I observed it with attention. The following is its description, and the detail of my observations.

From a pedicle, which is sometimes flat, and sometimes round (for they vary in these plants, and might be about three lines in diameter, and an inch high, of a blackish colour, and coriaceous substance, approaching to the nature of the bodies of lithophyta), a single flat leaf arises, about an inch or an inch and half broad, thick in its middle to about three lines, ending at the sides in a kind of edge, like a two-edged sabre, almost like the common Alga, formed of longitudinal fibres interlaced with other very delicates ones, and the whole filled with a thick juice, like the _parenchyma_ of succulent plants, such as the Sedum, Aloes, and the like, of a clear yellowish green, and transparent. This first leaf is always single, and serves instead of a trunk or stem to the whole plant.