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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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the sides of the bodice, 'and between these 2 sides a notch is cut out, to make



662 PHILOSOPHICAL TRANSACTIONS. [aNNO 1742-3.

room not only for the foot, or for the point it rests on, which may meet there,
but also for a part of the lever, called its spur, which always moves towards
that notch when the lever is lowered. The figures and the use of the machine
will show the necessity of this construction much better than any description.
From the bodice come out 2 broad straps, of the strongest leather, with their
buckles. One of those straps is to go about the back of the chair, and round
the body of the patient : the other goes over the shoulder, very near the arti-
culation, and keeps the scapula and the clavicula in their situation against the
efforts of the lever.

That part of the machine, which may properly be called the ambe, is com-
posed, like that of Hippocrates, of 2 pieces ; one vertical, which he called the
foot of the ambe; and the other horizontal, which forms the lever. It is
chiefly in these 2 pieces that this ambe differs from that of Hippocrates.

The foot is a piece made either of wood, pi. l6. fig. 2, or of iron, fig. 3.
Its upper extremity is split into a sort of mortise, which receives the spur or
tenant t of the lever ab, fig. 8. It is pierced by several holes, which answer
to as many others on the spur. Below this mortise the foot becomes more
slender and cylindrical ; by this part it enters into a round hole in the arm of
the chair ; this slender part of the foot is pierced by several holes, in order to
run an iron pin through, which lies flat on the arm of the chair, and keeps the
foot raised to a height proper for the person that undergoes the operation. For
the greater security, 2 pins may be run through; one which rests on the arm of
the chair, and the other on the seat itself, through which the foot also passes.
The iron foot, fig. 3, may be provided with a sort of large ring c, under the
pin, which will render its rotation the easier. If an iron foot should be pre-
ferred, the hole for it in the arm of the chair must be made narrower, either by
filling up the old one with an iron box or clout, which may be taken away, if a
wooden foot be used ; or we may even at first fit those holes for the iron foot,
setting the wooden one quite aside.

The lever abhd, fig. 3, pi. 15, is the most compound piece of all, and also
the most important. It is made of a real lever ab, and of a piece fitted to it
DG. The lever properly so called, ab, is made round on its inferior surface ;
the upper surface is fiat; and all along on the middle of it there runs a rod,
forked at the end, which fits to a groove of the same figure in the inferior surface
of the sliding-piece fg, fig. Q, pi. l6. This lever diminishes towards the extre-
mity a, where the moving power is to be applied ; the other extremity, b, is
somewhat rounded off at the end, the better to insinuate itself under the arm-
pit. On this larger extremity is a kind of spur or tenant, t, the upper part of
which is joined to the lever by 2 iron pins ; so that, on taking out the pins, the



VOh. XLII.] PHILOSOPHICAL TRANSACTIONS. 663

spur comes out, and separates itself from the lever, as appears by fig. 4. It
was necessary to make this spur moveable, and give it the figure of a square
rule, in order to bring it quite close to the end of the lever, or set it back, as
it may be necessary. For this reason the upper part of this spur ab, slides
along in a mortise, or groove, of the length of one foot, contrived under the
lever, beginning from its extremity b, to which answers the shoulder b, of
the spur.

The rest of the tenant, or its principal parte, is fitted to enter into the mortise d,
which is the uppermost part of the foot, fig. 2, 3. They are both pierced with a
row of holes, through one of which must be run an iron pin, to unite them, and to
form the point of rest, or the hinge of the lever. Towards the other extremity a
of the lever, there is a piece of iron c, fig. 8, pi. l6, made arch-wise, under which
passes the elastic tailDf,of the rod, fastened to the sliding-pieccFG, and into which
catch teeth, made on the said tail, as in fig. 3, pi. 15, or fig. 6 and Q, pi. l6. This
iron arch ought to be very solid, because it keeps down the arm, and supports all
the effort of the lever. He gives to the sliding-piece fg, which is fitted to the
lever, the name of the bracer ; it is a groove made of one piece of wood, repre-
sented in its situation in fig. Q, pi. l6. This piece is hollow in the upper surface,
as above said, to place the luxated arm into ; this cavity is quilted, and has 3
girths, H, of strong leather, with buckles, to tie the arm fast and conveniently.
It has on its inferior surface a groove with a dove-tail, kk, to lay hold of the
rod of the lever, and to slide in it without being separated from it, unless it be
in sliding beyond the extremity b, of the lever, where it pulls out like a drawer,
which is easily done, if the bracer has nothing to stop it on the lever. The
extremity of the bracer, which answers to the thick end of the lever, is rounded, "
in order to enter jointly with it under the arm-pit ; the other gives hold to the
piece of iron de, called above by the name of the elastic tail of the bracer. This
latter consists of 4 parts : the fork f, which attaches itself to the inferior lateral
surfaces of the bracer; the spring f, which is the piece that follows next, the
longest and slenderest of all ; the teeth e, and the handle d.

The use of the new Ambe. — ^The patient, being undressed down to the waist,
is placed in the arm-chair, as in fig. 6, pi. 1 6. Next, the lever, furnished with
its bracer, is raised and kept in a horizontal position, taking great care, as
Hippocrates recommends, to push this bracer as far as may be under the arm-
pit to the end of the bone of the arm, and even beyond, if possible, so that the
humerus, supported by the bracer in all its length, may be secure against all
the power of this machine, and that its violence may only act on those muscles
which keep this bone out of its place. Besides the quilting, which the bracer
is lined with, a small cushion is put on its extremity, in order to lodge stilt



664 rHlLOSOPHICAL TRANSACTIONS. [aNNO 1742-3.

more conveniently the head and the neck of the humerus, and to preserve the
soft parts from any contusion, which the impulse of the machine might pro-
duce, by its greatest forces acting on that part.

The arm being thus placed, and well stretched out on the bracer, you tie
about it 1 sliding knots, one above the elbow, and the other over the wrist,
after having guarded those parts with a very thick and soft compress ; the 2
sliding knots are fastened to the fork of the elastic tail of the bracer ; after
which you complete the fixing of the arm with the 3 girths of the bracer, under
which are also put compresses like those just mentioned.

The arm being thus well adjusted, you endeavour to give to the body, and
to the hollow of the articulation of the luxated bone, the proper situation and
steadiness necessary for the success of the operation, which is easily executed
with this machine, by the girths of the bodice, of which the horizontal one
keeps the patient's breast closely applied against this piece, and the vertical
girth retains the scapula, the clavicula, in short, all the parts where the bone
is to be pushed back, in a situation proper for receiving it, and for not deviat-
ing by yielding to the efforts of the machine.

Every thing being thus disposed, the surgeon places himself behind the pa-
tient, mounted on something that raises him high enough to inspect the effects
of the process ; to examine by the touch where it operates : in short, to con-
duct the whole both by feeling and by the eye. The surgeon being placed,
the assistant who is to conduct the extremity of the lever, works it according
to his directions, but exceedingly slow, that the extensions may be made with
less pain, and more effectually.

When the luxation is below, it is sufficient for its reduction to lower the ex-
tremity of the lever, as is done with the ambe of Hippocrates. But here ap-
pears a great difference between the working or playing of these 2 sorts of
levers. The ambe of Hippocrates is a plain lever ab, fig. 7, pi. l6, the motion of
which is from a to a, and consequently has for its extension only the space ca,
when it is brought to its last term of becoming perpendicular, ab, while it has
all AC, or ia, for its elevation. The ambe of Hippocrates therefore only raises
the bone of the arm, without scarcely stretching it ; and this is the defect
which M. Petit, with reason, blames it for ; and which is still more sensible,
if we take the action of the lever in d, the point whereabouts it must meet the
edge of the cavity, and may cause those mischiefs that are apprehended from
it; but instead of placing the fixed point of that lever in 1, lower it to 2, by
means of the tenant 1 2 ; then the direction of the end of the lever becomes
VE : its elevation is but ih ; and the extension it produces is ae, or de : if you
lower still the lever's point of rest, as in 3, by a longer spur, the elevation of



VOL. XLII.] PHILOSOPHICAL TRANSACTIOKS. - 665

its extremity is reduced to ik ; and the extension it produces, reaches from a
to F, if we carry those levers as far as they will go, which is never necessary
In short, it will be in your power to give to this lever an extension as great as
you please, joined to a very small elevation. To this end you need only set
more backward the lever's point of rest, along the perpendicular marked in
fig. 7. Now this is precisely what the spur does, which we have added to our
ambe ; the holes it is pierced with, as well as the mortise of the foot, are
placed in different degrees, as the points 1, 2, 3 ; and these holes, as has
been said, are the places of the pin which forms the lever's hinge or point
of rest.

The gradation of those holes therefore enables us to augment at will the ex-
tension, while the elevation diminishes in the same proportion ; but if we want
the elevation to diminish more or less than in the foresaid proportion, for in-
stance, to make a great extension, and a very small elevation, there is nothing
easier for it than this machine. We need only push the spur 1 3, which is
moveable, towards the end of the lever to l, and stop it there : then the end
of the lever al, being very short, it has but little room to play ; on the con-
trary, to have a great elevation, we need only bring back the said spur to m,
or I, or still farther ; the farther we remove from the end of the lever, the more
it will have room to play, and the more considerable will be its elevation. It
is true, the power of the lever will decrease in the same proportion ; but this
power is so great, that losses like this ought to be reckoned for nothing.

We have it therefore in our power with this sort of ambe, to make, as occa-
sion requires, such extensions and counter-extensions as we please ; and we
may also vary all the degrees of the elevation, which shall be necessary to give
to the bone to be reduced ; and these are the perfections which have been
hitherto required in this machine.

Commonly, when the bone of the arm is sufficiently stretched and raised, so
as to be on a level with the cavity of the articulation, those bones replace them -
selves as it were, because this level is not always exact ; on the contrary, the
extension and counter-extension being never regular enough to hinder the sca-
pula, which is a moveable part, from following a little the head of the bone, or
its extension, it mostly happens that this head bears pretty strongly against
the edge of the cavity, and consequently does not fail to fall into the said cavity,
as soon as it has only passed its edge, and even before it has met the level, or
the axis of the hollow of the articulation ; but it is otherwise after an extension,
a counter extension, and an elevation so regular as those which may be per-
formed by this machine ; it may happen, that after the 3 preceding operations,
the head of the bone, without having touched the edge of the cavity, will be

VOL. VIII. 4 Q



9S6 PHILOSOPHICAL TBANSACTIONS, [aNWO 1742-3.

placed over- against this cavity, and on a level with its axis, without being able
to enter into it, by reason of the firmness and exactness of the powers for re-
taining the opposite parts in this state of regular extension ; and in this case
there will remain for us, in order to finish the operation, to conduct the head
of the bone into its cavity, or to let it go into it : but what shall we do then ?
If we slacken the extremity of the lever, or if we lift the same up, we
bring the head back to the same place where we took it up ; that is, the luxa-
tion to its former state. If we resolve to relax the running knots, the opera-
tion will be long, and the patient will have time enough to cry out.

In order to avoid these inconvenients, M. le Cat mounted the bracer on
the lever in a groove, and he stopped it in this state by the teeth of its elastic
tail ; by means of this construction, when the surgeon perceives that the bone
is over-against its cavity, he directs the assistant, who attends the extremity
of the lever, to press upon the handle d, fig. 6, of the elastic tail of the bracer,
that the teeth placed under the arch c, near the said handle, may quit their
hold, and that the whole bracer, which is now no longer stopped, may slide on
the lever towards the patient, and by this means let the head of the bone enter
into its cavity.

The necessity of this management with our ambe, is a demonstration that
it is far from having that capital fault, with which M. Petit reproaches the ambe
of Hippocrates ; viz. " that it pushes the head of the bone into its cavity, be-
fore the extension and counter-extension are made."

If it be feared that the re-entering of the head of the bone might be too
sudden, and occasion a shock, that might hurt the bones, it will be easy to
prevent it, by substituting to the stop, into which the teeth of the bracer catch,
a toothed wheel a, fig. 5, having in its centre a handle bd; which handle
during the operation will be stopped by the iron c, fixed on this piece by the
screw p ; the said handle will also stop the teeth e, that catch into the toothed
wheel ; and when the bracer is to be loosened, the assistant, who holds the
lever with one hand, will take the handle with the other, and having got the
screw p taken off, he will remove from the piece c, that stops it, the part db
of the handle, by means of its moveable arbor d, so that the handle will come
at a right angle, as it is represented by dots : then the assistant's hand, sustain-
ing all the effort of the handle and of the bracer, will moderate by the handle
the sliding of the bracer, and the entering of the head of the bone into its ca-
vity, with all the slowness he shall think proper for this operation.

Thus much concerning the reduction of a luxation of the arm below, which
is the only kind of luxation in which the ambe of Hippocrates can be used.
He has succeeded in remedying this defect by the simplest thing in the world.



VOL. XLII.3 PHILOSOPHICAL TRANSACTIONS. QQj

viz. by giving to the foot, tiiat enters into the arm of the chair, a cylindrical
shape, by which means it can turn all manner of ways; so that if the luxation
be forwards, we need only turn the extremity of the lever accordingly, lower-
ing it at the same time enough to make the necessary extension and elevation;
by this turn of the extremity of the lever forward, the head of the bone is of
necessity carried backward, and replaced into its cavity. It is easily conceived,
that we must go to work in the opposite way when the luxation is backward,
and so on as for the rest ; all according to the directions of the surgeon placed
at the articulation, who is to be attentive to examine the state of the parts,
and to order in what direction, and how much is necessary to be done.

Continuation of the Account of a Treatise of Fluxions, &c. Book 2. By Colin
Maclaurin, F. R. S* N" 469, p. 403.

In the first book, the author described the method of fluxions, and its appli-
cation to problems of different kinds, without making use of any particular
signs or characters, by geometrical demonstrations, that its evidence might ap-
pear in the most simple and plain form. In the second book, he treats of the
method of computation, or the algebraic part ; to the facility, conciseness, and
great extent of which, the improvements that have been made by this method
are in great measure to be ascribed. In order to obtain these advantages, it was
necessary to admit various symbols into the algebra; but the number and com-
plication of those signs must occasion some obscurity in this art, unless care be
taken to define their use and import clearly, with the nature of the several
operations. An example of this is given by an illustration of one of the first
rules in algebra. As it is the nature of quantity to be capable of augmentation
and diminution, so addition and subtraction are the primary operations in the
sciences that treat of it. The positive sign implies an increment, or a quantity
to be added. The negative sign implies a decrement, or quantity to be sub
tracted; and these serve to keep in our view what elements enter into the com-
position of quantities, and in which manner, whether as increments or decre-
ments. It is the same thing to subtract a decrement as to add an equal incre-
ment. As the multiplication of a quantity by a positive number implies a re-
peated addition of the quantity, so the multiplication by a negative number
implies a repeated subtraction: and hence to multiply a negative quantity, or
decrement, by a negative number, is to subtract the decrement as often as there
are units in this number, and therefore is equivalent to adding the equal incre-
ment the same number of times; or, when a negative quantity is multiplied by

• See the beginning of this account, N" 46'8, p. 6"32 of thit vol.
4Q 2



668 PHILOSOPHICAL TRANSACTIONS. [anNO 1742-3.

a negative number, the product is positive. When we inquire into the pro-
portion of lines in geonnetry, we have no regard to their position or form; and
there is no ground for imagining any other proportion between a positive and
negative quantity in algebra, or between an increment and a decrement, than
that of the absolute quantities or numbers themselves. The algebraic expres-
sions, however, are chiefly useful, as they serve to represent the effects of the
operations ; and such expressions are not to be supposed equal that involve equal
quantities, unless the operations denoted by the signs are the same, or have
the same effect. Nor is every expression to be supposed to represent a certain
quantity ; for if the -/ — 1 should be said to represent a certain quantity, it
must be allowed to be imaginary, and yet to have a real square; a way of speak-
ing which it is better to avoid. It denotes only, that an operation is supposed
to be performed on the quantity that is under the radical sign. The operation
is indeed in this case imaginary, or cannot succeed ; but the quantity that is
under the radical sign, is not less real on that account. The author mentions
those things briefly, because they belong rather to a treatise of algebra than of
fluxions, wherein the common algebra is admitted.

In order to avoid the frequent repetition of figurative expressions in the alge-
braic part, the fluxions of quantities are here defined to be any measures of
their respective rates of increase or decrease, while they are supposed to vary
(or flow) together. These may be determined by comparing the velocities of
points that always describe lines proportional to the quantities, as in the first
book; but they may be likewise determined, without having recourse to such
suppositions, by a just reasoning from the simultaneous increments or decre-
ments themselves. While the quantity a increases by differences equal to a,
2a increases by differences equal to la, and, supposing m and n to be invariable,
— increases by differences equal to — , and therefore at a greater or less rate
than a, in proportion as m is greater or less than n. Thus a quantity may be
always assigned that shall increase at a greater or less rate than a, i. e. shall have
its fluxion greater or less than the fluxion of a, in any proportion; and a scale
of fluxions may be easily conceived, by which the fluxions of any other quan-
tities of the same kind may be measured.

Let B be any other quantity whose relation to a can be expressed by any alge-
braic form ; and while a increases by equal successive differences, suppose b to
increase by differences that are always varying. In this case, b cannot be sup-
posed to increase at any one constant rate; but it is evident, that if b increase
by differences that are always greater than the equal successive differences by
which — increases at the same time, then b cannot be said to increase at a less



VOL. XLII.] VHILOSOPHICAL TRANSACTIONS. 06Q

rate than — ; or if the fluxion of a be represented by a, the fluxion of b can-
not be less than — . And if the successive differences of b be always less than



n



those of — , then surely b cannot be said to increase at a greater rate than — ;

or the fluxion of b cannot be said to be greater in this case than — .

From those principles the primary propositions in the method of fluxions,
and the rules of the direct method, with the fundamental rules of the inverse
method, are demonstrated. We must be brief in our account of the remainder
of this book. The rule for finding the fluxion of a power is not deduced, as
usually, from the binomial theorem, but from one that admits of a much easier
demonstration from the first algebraic elements, viz. that when n is any integer
positive number, if the terms e"-', e'-^f, e'-^p*, e'-4f3, . . . . p"-', (wherein
the index of e constantly decreases, and that of f increases by the same differ-
ence unit) be multiplied by e — p, the sum of the products is e" — p"; from
which it is obvious, that when e is greater than f, then e" — f" is less than
WE"-' X E — F but greater than ne"-' X e — p.

The rules are sometimes proposed in a form somewhat different from the
usual manner of describing them, with a view to facilitate the computations
both in the direct and inverse method. Thus, when a fraction is proposed,
and the numerator and denominator are resolved into any factors, it is demon-
strated, that the fluxion of the fraction divided by the fraction, is equal to the
sum of the quotients, when the fluxion of each factor of the numerator is
divided by the factor itself, diminished by the quotients that arise by dividing
in like manner the fluxion of each factor of the denominator by the factor.

The notation of fluxions is described in chap, 2, with the rules of the direct
method, and the fundamental rules of the inverse method. The latter are
comprehended in 7 propositions, 6 of which relate to fluents that are assignable
in finite algebraic terms, and the 7th to such as are assigned by infinite series.
It is in this place the author treats of the binomial and multinomial theorems,
because of their use on this occasion, and they are investigated by the direct
method of fluxions. The same method is applied for demonstrating other
theorems, by which an ordinate of a figure being given, and its fluxions deter-
mined, any other ordinate and area of the figure may be computed. The most
useful examples are described in this chapter, by computing the series that
serve for determining the arc from its sine or tangent, and the logarithm from
its number, and conversely the sine, tangent, or secant, from the arc, and the
number from its logarithm.
The inverse method is prosecuted farther in the 3d chapter, by reducing



670 PHILOSOPHICAL TRANSACTIONS. [aNNO J 742-3.

fluents to others of a more simple form, when they are not assignable by a
finite number of algebraic terms. When a fluent can be assigned by the qua-
drature of the conic sections, and consequently by circular arcs or logarithms,
this is considered as the 2d degree of resolution; and this subject is treated at
length. An illustration is premised of the analogy between elliptic and hyper-
bolic sectors formed by rays drawn from the centres of the figures; the pro-
perties of the latter are sometimes more easily discovered, because of their rela-
tion to logarithms, and lead us in a brief manner to the analogous properties



Online LibraryRoyal Society (Great Britain)The Philosophical transactions of the Royal society of London, from their commencement in 1665, in the year 1800 (Volume 8) → online text (page 77 of 85)