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The Philosophical transactions of the Royal society of London, from their commencement in 1665, in the year 1800 (Volume 8) online

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Royal Society's brass rod, O rev. 16.8 div. = .021 = xt'.-bt-

Now, as to the weights, those in the Chamberlain's Office in his Majesty's
Exchequer, and which are esteemed the standards, are a pile, or box, of hollow
brass Troy weights, from 256 ounces downwards, to the l6th part of one
ounce, all severally marked with a crowned e: but they have no penny- weights,
nor grain weights, that are any ways considered as standards.

The weight mentioned in all our old acts of parliament, from the time of
King Edward the First, is universally allowed to be the Troy weight, its pound
consisting of 12 ounces, each containing 20 penny- weights : and as the pound
is the weight of the largest single denomination commonly mentioned in those
acts, 12 ounces taken from the pile of Troy weights abovementioned, as there
is no single Troy pound weight, must now be reputed the true standard of the
Troy pound, used at this day in England.

Besides which Troy standards, there are also kept in the Exchequer the fol-
lowing standards for Avoirdupois weights: that is to say, a 14 pound bell weight
of brass, marked with a crowned e, and inscribed

XIIII. POVNDE AVERDEPOIZ.
ELIZABETH. REGINA.

1582.
As also a 7 pound, a 4 pound, a 2 pound, and a single pound, like Avoirdupois
bell-weights, and severally marked as follows, excepting the variations for the
number of pounds in each respective weight.



704



PHILOSOPHICAl, TRANSACTIONS.

VII. A.

AN° -



I^ANNO 1743.




E. L.

1588.

A° KEG. XXX.

With which are also kept a pile of flat Avoirdupois weights, from J 4 pounds
down to the 64th part of a pound.

When the Avoirdupois weight came first to be esteemed a lawful weight, does
not appear ; but by these standards it is plain it has been used as such ever since
the reign of Queen Elizabeth. And as the weight of 15 pounds Avoirdupois
has before been made use of, in determimng the proportion between the weight
of this pound and that of the pound Troy, we shall begin by giving the
counterpoise of the said 15 pound Avoirdupois, as it was found in Troy weight:
from whence we shall deduce the proportions of those pounds, and afterwards
compare the same with the like proportions, deduced from the 7 pounds, and
single pound bell-weights, and the single pound flat weight abovementioned :
all which weights were taken in the presence of the above-named noblemen and
gentlemen, by Mr. Samuel Read, scale and weight-maker near Aldersgate, who
brought to the Exchequer a large balance of his own for that purpose, and
which, when loaded with 15 pounds at each end, was very readily turned with
6 grains; as a less one, he brought also for examining the single pound weights,
was with half a grain. He also brought with him what he called his own
standard penny and grain weights, 10 supply what was necessary to make the
counterpoise of the Exchequer weights : with all which the result was, that

The standard 14 pound, and single pound Avoirdupois weights, taken to-
gether, were, on a medium of 4 trials, alter counterchanging the weights in
each basin, changing the basins, and then again counterchanging the weights,
found to be counterpoised by 218 Troy ounces, 13 penny-weight, 23 grains
and 4th. from whence the Avoirdupois pound is deduced equal to 6998.35 of
such grains as the Troy ounce is reputed to contain 480 of; and the Avoirdu-
pois ounce, of which 16 are supposed to make a pound, is found equal to 437.4
like grains.

Again : the 7 pound bell Avoirdupois weight, with the same scales, and on a
medium of 4 like experiments, counterchanging, as before, both vveights and
basins, was found to be counterpoised by 102 Troy ounces 1 penny-weight,
and 21 grains. According to which, the Avoirdupois pound conies out equal to
7000.7, and the ounce to 437-54 Troy grains.



VOL. XLII.] PHILOSOPHICAL TRANSACTIONS. 705

Again : the single bell Avoirdupois pound, with the less scales, on the mediuui
of two experiments, counterchanging the weights, the basins not being move-
able, was found to weigh 14 Troy ounces 11 penny-weight and IS grains; or
was equal in weight to 7002, and the ounce to 437.62 Troy grains.

The single Avoirdupois bell pound, against tne flat Avoirdupois pound weight,
was found, on a medium of two like experiments, to be heavier by two Troy
grains and a half: whence the fiat Avoirdupois single pound weight weighs only
699Q.5, and the ounce 437-46 Troy grains.

The Royal Society's Avoirdupois pound was, in like manner, found to be
lighter than the Exchequer single bell pound weight, by one grain.

And their Troy pound weight to be lighter than the 8 and 4 ounce Troy
weights at the Exchequer, taken together, by half a grain.

The Founders' Company of London are, by their charter from King James
the First, authorized and directed to have the sizing and marking of all manner
of brass weights, to be made or virought, or to be uttered, or kept for sale,
within the city of London, or 3 miles from the same. And the weights de-
livered to them from his Majesty's Exchequer, and now kept in their hall, as
their standards for the uses abovementioned, are a pile of flat brass Troy
weights, from 256 ounces, down to the l6th part of an ounce, all sealed with
the Exchequer seal, and marked with c. r. crowned l684, and a stamp of the
initial letters of the maker's name: as also a set of bell brass Avoirdupois weights,
sealed and marked in like manner. And here the following trials were made,
before the abovenamed gentlemen, by Mr. Read, but with a large balance, com-
monly used for trials at the hall, in their office for that purpose ; and found to
turn with about the same weight as the former ; and also with a less one, turn-
ing in like manner under these circumstances, with about half a grain, which
balance belonged likewise to the hall, as did also the penny and grain weights
made use of, but which were not kept by them as standard weights.

And here it was found on a medium of 4 trials, made in like manner as be-
fore, at the Exchequer, that 15 pounds Avoirdupois, being their 14 pounds,
and single pound standard weights, were counterpoised by 218 Troy ounces, 15
penny-weight and 23 grains : whence the Avoirdupois pound is deduced equal to
7001.53, and the ounce to 437.59 Troy grains.

Again : the single Avoirdupois standard pound weighed, on a medium of 2
experiments, counterchanging the weights, as before, 14 Troy ounces, ll
penny-weight, 164 grains : or was equal to 7000.5, and the ounce to 437.53
Troy grains.

Again : this standard Avoirdupois pound, at a medium as before, outweighed
the Royal Society's Avoirdupois pound, by 2 grains and -J^th : and the Troj

VOL. VIII. 4 X Q



706 FHXLOSOPHICAL TKANSACTION8. [aNNO 1743.

Standards of 8 and 4 ounces, taken together, outweighed the Royal Society's
single Troy pound weight, by 1 grains and -i-th, at a like medium.

At the Mint in the Tower of London, their standard weights are only a pile
of Troy hollow weights, from 256 ounces, down to the l6th part of one ounce,
without any penny or grain weights. The larger of these weights, as low as the
8 ounce weight, are marked with a. r. crowned, and inscribed primo maii, a®
DNi. 1707, a° REGNi vi°. The 4 and the 1 ounce weights are only marked
A. R. crowned, without the date; and the less have only the Exchequer seal, and
the rose and crown, being the mark of his Majesty's mint, as all the larger
ones have also. And here it was found by Mr. Joseph Harris, one of the
assay-masters of the mint, with a very curious balance of his own, fixed in a
glass lantern, and which he was well assured might in such circumstances be
depended on to less than half a grain ; and with the addition of so many penny
jmd grain weights belonging to bis office as were necessary : that

The Royal Society's whole Troy pound weight weighed, at a medium, less than
the 8 ounces and 4 ounces of these standards, taken together, by 2 grains and
iths.

That the Royal Society's Avoirdupois pound weighed in Troy weight, by these
standards, 14 ounces 11 penny weight l6 grains and -l-ths ; or 700O.87 grains.

That the Royal Society's pile of ]6 ounces Troy, was lighter than l6 ounces
of these standard weights, by 4 grains and ^ths.

And lastly. That the Royal Society's 8 ounces and 4 ounces together, taken
from their pile, weighed lighter than their single Troy pound weight, by ^ths
of a grain.

An Instrument for Reducing a Dislocated Shoulder ; invented by Mr. John Freke,
Surgeon of St. Bartholomew's Hospital, and F. R. S. N°470, p. 556.

Mr. F.'s object in the present communication, is to show within how small a
compass the whole power which can be made use of, in reducing a dislocated
shoulder, may be contracted. If a machine for this purpose be not portable, it
matters but little to an afflicted patient 10 miles off, how good an instrument is
out of his reach.

This machine, fig. 2, pi. 20, which consists of 2 boxes a, joined at the ends
by 1 hinges, contains, when folded together, every thing that can possibly be
wanted in the operation beforementioned ; and it may so easily be made use of,
without the assistance of any other operator than the surgeon employed, that
he ventures to affirm, a patient may be set down, the instrument applied, and
the shoulder reduced, in 1 minute, ordinarily speaking.

The length of this instrument, when shut up, is 1 foot 8 inches, its breadth
9 inches and thickness 3 inches and a quarter. When it is opened, it is kept



VOL. XLII.] PHILOSOPHICAL TRANSACTIONS. ' 70/

SO by 2 hooks fixed on the backside of it ; and when one end of it stands on
the ground, the other stands high enough to become a fulcrum, or support of
a lever bb, which is fixed on a roller b, by a large wood screw, which turning
sideways as well as with the roller, it obtains a circumrotatory motion; so that
it will serve to reduce a luxation either backward, forward, or downward.

The roller on which the lever is fixed, is just the diameter of the depth of
one of the boxes, into which are driven 2 iron pins, the ends of which are re-
ceived by the 2 sides of the box, which are 1 inch thick.

The lever is 2 feet 4 inches, and is cut off and joined again by 2 hinges c, to
fold up so as to be contained in the boxes. On the backside of it is a hook, to
keep it straight ; the other end is to hang over the roller about 14. inch, which
is to be excavated and covered with buflP leather, for the more easy reception of
the head of the os humeri.

Two iron cheeks dd are screwed on each side of the lever, to receive through
them an iron roller e, which has 2 holes through it, to receive 2 cords coming
from a brace f, fixed on the lower head of the os humeri ; for on no other part
of the arm above the cubit can a bandage for this purpose be useful ; for, if the
surgeon applies it on the muscular part of the arm, it never fails slipping down
to the joint, before you can extend the limb.

The iron roller has a square end, on which is fixed a wheel g, within the
cheek, notched round, which works as a rotchet on a spring ketch under the
lever, by which it is stopped, as you wind it with a winch ; and may at pleasure
be let loose, as there shall be occasion for it, by discharging the ketch.

He next describes the brace f, which, compared with common bandages, is
of more consequence than can easily be imagined by unexperienced persons. It
consists of a large piece of buff leather, sufficient to embrace the arm, sewed
on 2 pieces of strong iron curved plates, riveted together, one of them having
an eye at each end, to fasten 2 cords in ; the other is bent at the ends into 2
hooks, which are to receive the cords, after they have crossed over the arm
above.

In order to keep the patient steady in his chair from coming forward, or
letting the scapula rise up, on depressing the lever, after the limb is drawn for-
ward by the winch, there must be fixed over the shoulder a girth with 2 hooks
at the ends of it, long enough to reach to the ground on the other side,
where it must be hooked into a ring i, to be screwed into the floor, for that
purpose.



4x 2



708 PHILOSOPHICAL TRANSACTIONS. [anNO 1743.

Concerning a Person who made Bloody Urine in the Small-Pox, and recovered.
By Pierce Dodd, M. D. N° 470, p. 559.

Making bloody water, Dr. D. observes, is universally esteemed as terrible a
symptom as any that can happen in the small-pox; and all who have written
concerning that distemper, unanimously agree that it is a certain forerunner of
approaching death. Dr. Cade indeed says, in his letters to Dr. Freind, con-
cerning purging in that distemper, that he has sometimes cured this symptom,
by the help of camphire, and a copious quantity of acids ; but then he adds,
that this relief was only temporary ; and that, to confess the truth, he never
knew any body, that made that sort of urine, who ever survived the 1 6th day
from the eruption: and there is nobody he knows that has been conversant with
this distemper, but has constantly experienced, sooner or later, the like fatality
in consequence of it. He means when this sort of urine has proceeded from a
broken crasis and contexture, or, as it were, a thorough dissolution of the
whole mass of blood : for he knows very well, that we now and then have
streaks, and sometimes larger quantities of blood in the urine, from the acri-
mony of the Spanish flies, on the application of blisters, which are frequently
used, and so frequently likewise absolutely necessary, in one or other of the
stages of this distemper, and yet the patient does well. And Dr. Browne,
physician in St. Bartholomew's Hospital, gives an account of a gentlewoman,
who lived in Dean's-yard, Westminster, who made bloody urine in the small-
pox, 4 or 5 days together ; which made Dr. Needham, who attended her, to
forsake her ; and yet she recovered: but they found afterwards, that this bloody
water was not occasioned by the malignancy of the distemper, but by a sharp
stone, which was at that time descending from one of the kidneys through the
ureters into the bladder, and which she afterwards voided.

The case here given is that of a young man, about 1 5 years of age, son to a
gentleman of a very considerable fortune in Jamaica. He was taken with a
fever, and great pain in his head, April 20th last, and had the small-pox come
out upon him the day following, notwithstanding which the same symptoms
still continued, and nothing almost would stay on his stomach, and his head
likewise was very delirious: he was obliged therefore to be blooded, and to take
a vomit, and to have blisters applied to his neck and his arms ; and had testace-
ous and nitrous n)edicines given him.

The next day every thing was more quiet, and so again the 3d day from the
eruption ; but the small-pox were very numerous all over him, and of a small,
rank, angry sort ; as they generally are, he thinks on the West India constitu-
tions : but this young gentleman had besides over-heated himself a little before,
by performing a part at the Montem, near Eton, where he was a scholar.



VOL. XLII.] PHILOSOPHICAL TRANSACTIONS. JOQ

Things continued in much the same state the 4th day, but towards the even-
ing there were a few streaks of blood mixed with, and subsiding in his water ;
which did not much alarm Dr. D. because he did not know but it might be
caused by the blisters : he had but one reason to doubt the contrary, and that
was, he had had little or no strangury : but as certain persons do aver, there is
sometimes such, or even a more bloody sort of waters, occasioned by the flies,
even where there shall be no strangury at all, he was willing to hope the best,
and so made no other alteration in his process, than to direct a very free use of
sp. of vitriol.

What was ordered, happened to succeed : we had no more of that sort of
water, either that night, or the next day, or the morning following : but he
was sent for in a great hurry that day, viz. the 6th, in the afternoon, and found
his friends in the most terrible consternation ; not only because it returned, but
began to increase upon them, and was pouring off in a free manner. It was
necessary therefore to proceed in another method, and he accordingly ordered
some gum arable, olibanum, and pulvis amyli, and alum, together with a mix-
ture of black cherry-water and small cinnamon, and treacle-vvater, with some
tinctura antiphthisica and terra japonica in it, and the tincture of roses, strongly
acidulated and sweetened with diacodium ; on the use of which it began to abate,
and the next day the urine returned to its usual state and colour. There was
nothing further observable in the course of this case, except that the distemper
was of the coherent kind, and accordingly attended with many other dubious
symptoms likewise : for though it is generally thought, that the coherent sort
is not so formidable as the confluent; yet, as Dr. Freind has judiciously ob-
served, and Moreton before him, there is not so much diflTerence between them,
but they are almost always attended with much the same appearances, and the
same fevers plainly at the time of maturation : for that the danger does not
arise so much from the sort, as from the number of the pustules; which if it be
great, there is the like reason to be fearful of the event, whether they flux, or
whether they only cohere : all which notwithstanding, this young gentleman
had the good fortune to escape.

Of the Bases of the Cells where the Bees deposile their Honey,. By Mr^
Maclaurin, F.R.S. N°471, p. 565.

The sagacity of the bees in making their cells of an hexagonal form, has
been admired of old ; and that figure has been taken notice of, as the best they
could have pitched on for their purposes : but a yet more surprising instance of
the geometry of these little insects, is seen in the form of the bases of those
cells, discovered in the late accurate observations of Mons. Maraldi and Mons.



710



PHILOSOPHICAL TRANSACTIONS.



[anno 1743.

de Reaumur, who have found those bases to be of that pyramidal figure, that
requires the least wax for containing the same quantity of honey, and which
has at the same time a very remarkable regularity and beauty, connected of
necessity with its frugality.

These bases are formed from three equal rhombuses, the obtuse angles of
which are found to be the doubles of an angle that often offers itself to mathe-
maticians in questions relating to maxima and minima ; that is, the angle,
whose tangent is to the radius, as the diagonal is to to the side of the square.
By this construction, of the 6 solid angles at the base, that correspond to the
angles of the hexagon, 3 are equal as well to each other, as to the solid angle
at the apex of the figure, each of which solid angles is respectively formed from
3 equal plane obtuse angles: and the other 3 solid angles are also equal to each
other, but severally formed each from 4 equal plane acute angles, supplements
to the former obtuse ones.

By this form the utmost improvement is made of their wax, of which they are
on all occasions very saving, the greatest regularity is obtained in the construc-
tion, and with a particular facility in the execution ; as there is one sort of
angle only with its supplement, that is required in the structure of the whole
figure.

^iM. Maraldi* had found by mensuration, that the obtuse angles of the
rhombuses were of 110 degrees nearly; on which he observed, that if the 3
obtuse angles, which formed the solid angles abovementioned, were supposed
equal to each other, they must each be of 109° 28'; whence it has been inferred,
that this last was really the true and just measure of them : and lately M. de
Reaumur-^ has informed us, that Mr. Koenig having, at his desire, sought
what should be the quantity to be given to this angle, in order to employ the
least wax possible in a cell of the same capacity, that gentleman had found, by
a higher geometry than was known to the ancients, by the method of infinitesi-
mals, that the angle in question ought in this case to be of 109° 26'. And we
shall now make it appear, from the principles of common geometry, that the
most advantageous angle for these rhombuses, is indeed, on that account also,
the same which results from the supposed equality of the 3 plane angles that
form the abovementioned solid ones.

Let GN and nm, fig. 1 and 2, pi. 21, represent any two adjoining sides of
the hexagon, that is, the section of the cell perpendicular to its length. The
sides of the cell are not complete parallelograms as cgnk, bmnk, but trapezia
cgne, bmne, to which a rhombus cebc, is fitted at e, and that has the opposite

* Memoires de I'Acad. Royale des Sciences, 1712. — Orig.
t Memoires sur les Insectes, torn. r. — Orig.



VOL. XLII^] PHILOSOPHICAL TRANSACTIONS. 711

point e in the apex of the figure, so that 3 rhombuses of this kind, with 6
trapezia, may complete the figure of the cell. Let o be the centre of the
hexagon, of which ok and kb are adjoining sides ; join cb and ko, intersect-
ing it in A ; and, because cob is equal to ckb, and ke equal to oe, the solid
EBCK is equal to the solid cbco ; from which it is obvious, that the solid con-
tent of the cell will be the same wherever the point e is taken in the right line
KN, the points c, k, b, g, n, and m, being given. We are therefore to inquire
where the point e is to be taken in kn, so that the area of the rhombus cebc,
together with that of the 2 trapezia cgne, enmb, may form the least superficies.
Because ec is perpendicular to bc in a, the area of the rhombus is ae X BC,
that of the trapezia cgne, enmb, is cg + en X kg ; these, added to the
rhombus, amount to ae X bc + 2kn X kc — ke X kc ; and because 2kn
X KC is invariable, we are to inquire, when ae X bc — ke X kc is a
minimum ?

Suppose the point l to be so taken on kn, that kl may be to al as kc is to
bc. From the centre a describe, in the plane akb with the radius ae, an arc
of a circle ek meeting al, produced if necessary, in r; let ev be perpendicular to
AR in V, and kh be perpendicular to the same in h ; then the triangles lev,
LKH, LAK, being similar, we have lv : le :: lh : lk :: lk : la :: (by the sup-
position last made) kc : bc Hence, when e is between l and n, we have
LH -f LV (= vh) : lk + le (= ke) :: kc : bc; and when e is between Kand
L, we have lh — lt (= vh) : lk — le (= ke) :: kc : bc ; that is, in both
cases we have ke X kc = vh X bc ; and cons equently ae X bc — ke X

kc = AE X BC — VH X BC = AE — VH X BC = AR — VH X BC = AH + VR

X BC ; which, because ah and bc do not vary, is evidently least when vr
vanishes, that is, when e is on l. Therefore clbI is the rhombus of the most
advantageous form in respect of frugality, when kl is to al as kc is to bc.
This is the same method by which we have elsewhere determined the maxima
and minima, in the resolution^of several problems that have usually been
treated in a more abstruse manner. See Treatise of Fluxions, Art. 572, &c.

Now because ok is bisected in a, kc" = ok'^ = 4ak^; and ac^ = 3ak% or
BC = 2ac = 2^/3 X ak; consequently kc : bc :; 2ak : 2/3 X ak :: ] : \/3,
and KL : al :: (kc : bc) :: 1 : ^3, or al : ak :: y 3 : ^2 ; and (because ak :
AC :: 1 : v/3) AL : AC :: 1 : ^^2 ; that is, the angle cla is that, whose tangent
is to the radius as \/2 is to 1, or as 14142135 to 10000000; and therefore is
of 54° 44' 8", and consequently the angle of the rhombus of the best form is
that of 109° 28' ^^"•

By this solution it is further easy to estimate, what their savings may amount
to on this article, in consequence of this construction. Had they made the



712 PHILOSOPHICAL TRANSACTIONS. [aNNO J 743.

base flat, and not of the pyramidal form described above, then, besides com-
pleting the parallelograms cgnk and bmnk, the surface of the base had been
3cB X AK ; what they really do form amounts in surface to the same parallelo-
grams, and 3CB X ah : the savings therefore amount to 3cb x ak — ah

= 3cB X AH X ^ — , which is almost a 4th part of the pains and expence

of wax, they bestow above what was necessary for completing the parallelogram
sides of the cells : and at the same time they seem also to have other advan-
tages from this form, besides the saving of their wax ; such as a greater strength
of the work, and more convenience for moving in these larger solid angles.

It remains that we should show, that the plane angles clb, cln, and bln, are
equal to each other. We before found, that kl : al :: kc : BC :: ka (= -J-kc):
AC ; consequently kl : ka :: al : ac, and the triangles lka, l^c, are similar:
therefore lk : al :: al : lc :: kc : bc :: 1 : v^3, and lc = 3lk. With the



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