Royal Society (Great Britain).

# 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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Font size invariable forces, it is shown, that the figure of the fluid must be that of an
oblong spheroid, and that those two centres must be the foci of the generating
ellipse. The nature of the figure is also shown, when the fluid gravitates to-
wards several centres, or when it revolves on its axis; but these are mentioned
briefly, because such theories are of little or no use for discovering the figures
of the planets.

In the 12th chapter, the author proceeds to consider the more concise me-
thods, by which the fluxions of quantities are usually determined, and to de-
duce general theorems more immediately applicable to the resolution of geome-
trical and philosophical problems, In the method of infinitesimals, the ele-
ment, by which any quantity increases or decreases, is supposed to become in-
finitely small, and is generally expressed by two or more terms, some of which
become infinitely less than the rest, and therefore being neglected as of no im-
portance, the remaining terms form, what is called the difi^erence of the quan-
tity proposed. The terms that are neglected in this manner, are the very same
which arise in consequence of the acceleration or retardation of the generating
motion, during the infinitely small time in which the element is generated;
and therefore these differences are in the same ratio to each other as the gene-
rating motions or fluxions. Hence the conclusions in this method are accu-
rately true, without even an infinitely small error, and agree with those that are
deduced by the method of fluxions.

It is usual in this method to consider a curve as a polygon of an infinite
number of sides, which, being produced, give the tangents of the curve, and,
by their inclination to each other, measure its curvature. But it is necessary in
some cases, if we would avoid error, to resolve the element of the curve into
several infinitely small parts, or even sometimes into infinitesimals of the second
order; and errors that might otherwise arise in its application, may, with due
care, be corrected by a proper use of this method itself, of which some in-
stances are given. If we were to suppose, for example, the least arc that can
be described by a pendulum to coincide with its chord, the time of the vibra-
tion derived from this supposition will be found erroneous; but by resolving
that arc into more and more infinitely small parts, we approach to the true time
in which it is described. By supposing the tangent of the curve to be the pro-
duction of the rectilineal element of the curve, the subtense of the angle of
contact is found equal to the second dift'erence or fluxion of the ordinate; but

VOL. XLII.] PHILOSOPHICAL TRANSACTIONS. 647

in this inquiry, the tangent ought to be supposed to be equally inclined to the
two elements of the curve that terminate at the point of contact: and then the
subtense of the angle of contact will be found equal to half the second differ-
ence of the ordinate, which is its true value.

Sir Isaac Newton, however, investigates the fluxions of quantities in a more
unexceptionable manner. He first determines the finite simultaneous incre-
ments of the fiuents, and, by comparing them, investigates the ratio that is
the limit of the various proportions which they bear to each other, while he
supposes them to decrease together till they vanish. When the generating
motions are variable, the ratio of the simultaneous increments that are gene-
rated from any term, is expressed by several quantities, some of which arise
from the ratio of the generating motions at that term, and others from the
subsequent acceleration or retardation of these motions. While the increments
are supposed to be diminished, the former remain invariable, but the latter de-
crease continually, and vanish with the increments; and hence the limit of the
variable ratio of the increments, or their ultimate ratio, gives the precise ratio
of the generating motions or fluxions. Most of the propositions in the pre-
ceding chapters may be more briefly demonstrated by this method, of which
several examples are given, and the author makes always use of it in the sequel
of this book.

It is one of the great advantages of this method, that it suggests general
theorems for the resolution of problems, which may be readily applied as there
is occasion for them. Our author proceeds to treat of these, and first of such
as relate to the centre of gravity and its motion. In any system of bodies, the
sum of their motions, estimated in a given direction, is the same as if all the
bodies were united in their common centre of gravity. If the motion of all
the bodies is uniform and rectilineal, the centre of gravity is either quiescent,
or its motion is uniform and rectilineal. When action is equal to reaction, the
state of the centre of gravity is never affected by the collisions of the bodies,
or by their attracting or repelling each other mutually. It is not, however, the
sum of the absolute motions of the bodies that is preserved invariable in conse-
quence of the equality of the action and reaction, as they seem to imagine,
who tell us, that this sum is unalterable by the collisions of bodies, and that
this follows so evidently from the equality of action and reaction, that to en-
deavour to demonstrate it would serve only to render it more obscure. On this
occasion the author illustrates an argument which he had proposed in a piece
that obtained the prize proposed by the Royal Academy of Sciences at Paris in
1724, against the mensuration of the forces of bodies by the square of the ve-
locities, showing that if this doctrine was admitted, the same power or agent,

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exerting the same effort, would produce more force in the same body when in
a space carried uniformly forwards, than if the space was at rest, or that springs
acting equally on two equal bodies in such a space, would produce unequal
changes in the forces of those bodies.

Various problems concerning the collision of bodies are resolved in a more
general manner than usual. Mr. Bernouilli had determined the motions when
the elasticity is perfect, and one body strikes two equal bodies in directions that
form equal angles with its direction ; or when there are any number of bodies
impelled by it on one side in various directions, providing equal bodies be im-
pelled by it on the other side, in directions equally inclined to its own direction.
But the problem is resolved here without these limitations; some others of this
kind are subjoined, and this doctrine is applied for determining the motions of
bodies that act on each other while they descend by their gravity.

The general principle derived from these inquiries is, that if there be no colli-
sion, or sudden communication of motion from one body to another, while
they descend together, and in any case, if the elasticity be perfect, the sum of
the products, when each body is multiplied by the square of the velocity ac-
quired by it, is the same as if all the bodies had descended freely from the same
> respective altitudes to their several places; only in collecting that sum, if any
body is made to ascend, the product of it multiplied by the square of its velo-
city is to be subducted; and if the bodies be supposed to ascend from their
places with the respective velocities acquired by them, then their common
centre of gravity will rise to the same level from which it descended. In other
cases, however, the ascent of the centre of gravity will be less than its descent,
but is never greater.

After demonstrating the usual rule for finding the centre of oscillation, the
author treats of the motion of water issuing from a cylindric vessel. The
effect of the gravitation of the whole mass of water is considered as threefold.
It accelerates, for some time at least, the motion with which the water in the
vessel descends ; it generates the excess of the motion with which the water
issues at the orifice above the motion which it had in common with the rest of
the water; and it acts on the bottom of the vessel at the same time. Then
supposing the last two parts of the force to be in any invariable ratio to each
other, when the diameters of the base and orifice are given, he determines by
logarithms the velocity with which the water issues at the orifice; and shows
that this velocity will approach very nearly to its utmost limit in an exceedingly
small time. When the water is supposed to be supplied in a cylinder, so as to
stand always at the same altitude above the orifice, there is an analogy between
the acceleration of the motion of the water that issues at the orifice, and the

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acceleration of a body that descends by its gravity in a medium which resists in
the duplicate ratio of the velocity. For when the utmost velocities, or limits,
are equal in those two cases, the time in which the issuing water acquires any
less velocity, is to the time in which the descending body acquires the same
velocity, as the area of the orifice to the area of the base; and if a cylindric
column be supposed to be erected on the orifice equal to the quantity of water
that issues at the orifice in the former of those times, the height of this column
will be to the space described by the descending body in the latter time, in the
same ratio as the orifice to the area of the base. The ratio of the force that
acts on the bottom of the vessel to the force that generates the motion of the
water issuing at the orifice, is deduced from Sir Isaac Newton's cataract, and is
the same that follows from the principle concerning the equality of the ascent
and descent of the centre of gravity, which was first applied to this inquiry by
Mr. Daniel Bernouilli, comment. Acad. Petrop. torn. 1. But there are several
precautions to be taken in applying this doctrine.

After some other theorems concerning the centre of gravity, and several ob-
servations concerning the curvature of lines, and the angles of contact; the
author represents 4 general propositions in one view, that the analogy between
them may appear. The first gives the property of the trajectories that are de-
scribed by any centripetal forces, how variable soever these forces, or their
directions, may be. The second gives a like general property of the lines of
swiftest descent. The third gives the property of the line that is described in
less time than any other of an equal perimeter. And the fourth gives the pro-
perty of the figure that is assumed by a flexible line or chain, in consequence
of any such forces acting upon it. If we suppose k body to set out from any
point in the trajectory, or in the line of swiftest descent, with the velocity which
it has acquired there, and to move in the right line which is the direction of
the gravity, that results from the composition of the centripetal forces, then
shall its velocity, and its distance from the point where the perpendicular from
the centre of curvature meets that right line, flow proportionally, i. e. the
fluxion of the velocity, or of the right line that measures it, shall be to the
velocity, as the fluxion of that distance is to the distance. When the velocity
and direction of the motion is the same in the line of swiftest descent as in the
trajectory, their curvature is the same. Thus in the common hypothesis of
gravity, the curvature in the cycloid, the line of swiftest descent, is the same
as the parabola described by a projectile, if the velocities in those lines be equal,
and their tangents be equally inclined to the horizon. In order to find the na-
ture of the catenaria in any hypothesis of gravity, suppose the gravity to be in-
creased or diminished in the same proportion as the thickness of the chain

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varies, and to have its direction changed into the opposite direction ; then ima-
gine a body to set out with a just velocity from a given point in the chain, and
to describe the curve. The tension of the chain at any point will be always
as the square of the velocity acquired at that point, and if a body be projected
with this velocity in the direction of the tangent, the curvature of the trajectory
described by it will be -f of the curvature of the chain at that point. We must
refer to the book for a fuller account of these and of other theorems.

In the 13th chapter, the problems concerning the lines of swiftest descent,
the figures which among all those that have equal perimeters produce maxima
or minima, and the solid of least resistance, are resolved without computations,
from the first fluxions only. There are also easy synthetic demonstrations sub-
joined, because this theory is commonly esteemed of an abstruse nature, and
mistakes have been more frequently committed in the prosecution of it, than
of any other relating to fluxions. To give some idea of the author's method,
suppose the gravity to act in parallel lines, a to denote the velocity acquired at
the lowest point of the curve, and u the velocity acquired at any other point of
the curve. Suppose the element of the curve to be described by this velocity u,
but the element of the base to be always described by the constant velocity a.
Then it is easily demonstrated, without any computation, that the element of
the ordinate being given, the difference of the times in which the elements of
the curve and base are thus described is a minimum, when the ratio of those
elements is that of a to m ; i. e. when the sine of the angle, in which the ordi-
nate intersects the curve, is to the radius in this ratio. Supposing therefore
this property to take place over all the curve, the excess of the time in which it
is described by the body descending along it, above the time in which the base
is described uniformly with the velocity a, must be a minimum ; and this latter
time being given, it follows that the time of descent in this curve is a minimum.
When the gravity tends to a given centre, substitute an arc of a circle described
from that centre through the lowest point of the curve in the place of the base
in the former case ; and the property of the line of swiftest descent will be dis-
covered in the same manner. The nature of the line that among all those of
the same perimeter is described in the least time, is discovered with great faci-
lity, by determining from the former case the property of the figure when the
sum or difi^erence of the time in which it is described by the descending body,
and of the time in which it would be described by any given uniform motion, is
a minimum ; for the latter time being the same in all curves of the same length,
it follows that the figure, which has this property, must be described in less
time than any other of an equal perimeter. The general isoperi metrical pro-
blems are resoked, and the solutions are rendered more general, with like

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facility by the same method; which is also applied for determining the property
of the solid of least resistance, and serves for resolving the problem, when
limitations are added concerning the capacity of the solid, or the surface that
bounds it.

The last chapter of the first book treats chiefly of gravitation towards
spheroids, of the figure of the planets, and of the tides. The author,
having occasion in those inquiries for several new properties of the ellipse, be-
gins this chapter by deriving its properties from those of the circle, by consider-
ing it as the oblique section of a cylinder, or as the projection of the circle by
parallel rays on a plane oblique to the circle. In this manner the properties are
briefly transferred from the one to the other, because by this projection the
centre of the circle gives the centre of the ellipse ; diameters perpendicular to
each other in the circle with their ordinates, and the circumscribed square, give
conjugate diameters of the ellipse with their ordinates, and the circumscribed
parallelogram ; parallel lines in the plane of this circle are projected by parallels
in the plane of the ellipse that are in the same ratio ; any area in the former is
projected by an area in the latter, which is in an invariable ratio to it; and con-
centric circles give similar concentric ellipses. It is likewise shown how pro-
perties of a certain kind are briefly transferred from the circle to any conic
section with the same facility.

After demonstrating the properties of the ellipse, it is shown, that if the
gravity of any particle of a spheroid being resolved into two forces, one per-
pendicular to the axis of the solid, the other perpendicular to the plane of its
equator, then all particles, equally distant from the axis, must tend towards it
with equal forces ; and all particles at equal distances from the plane of the
equator, gravitate equally towards this plane ; but that the forces with which
particles at different distances from the axis tend towards it, are as the distances;
and that the same is to be said of the forces with which they tend towards the
plane of the equator.

From this it is demonstrated, that when the particles of a fluid spheroid, of
a uniform density, gravitate towards each other, with forces that are inversely
as the squares of their distances, and at the same time any other powers act on
the particles, either in right lines perpendicular to the axis, that vary in the
same proportion as the distances from the axis, or in right lines perpendicular
to the plane of the equator, that vary as their distances from it, or when any
powers act on the particles of the spheroid, that niay be resolved into forces of
this kind ; then the fluid will be every where in equilibrio, if the whole force
that acts at the pole be to the whole force that acts at the circumference of the
equator; as the semidiameter of the equator to the semiaxis of the spheroid ;

4u 2

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and that the forces with which equal particles at the surface tend towards the
spheroid, will be in the same proportion as perpendiculars to its surface, termi-
nated either by the plane of the equator, or by the axis. Because the centri-
fugal force with which any particle of the spheroid endeavours to recede from
its axis, in consequence of the diurnal rotation, is as the distance from the
axis, it appears that if the earth, or any other planet, was fluid, and of a uni-
form density, the figure which it would assume would be accurately that of an
oblate spheroid generated by an ellipsis revolving about its 2d axis.

Afterwards the gravity towards an oblate spheroid is accurately measured by
circular arcs, not only at the pole, but also at the equator, and in any inter-
mediate places ; and the gravity towards an oblong spheroid is measured by
logarithms. The gravity at any distance in the axis of the spheroid, or in the
plane of the equator produced, is likewise accurately determined by like mea-
sures, without any new computation or quadrature, by showing that when two
spheroids have the same centre and focus, and are of a uniform density, the
gravities towards them at the same point in the axis or plane of the equator pro-
duced, are as the quantities of matter in the solids.

This theory is applied for determining the figure of the earth, by comparing
the force of gravity in any given latitude, derived from the length of a pendu-
lum that vibrates there in a second of time, with the centrifugal force at the
equator, deduced from the periodic time of the diurnal rotation, and the ampli-
tude of a degree of the meridian ; or by comparing the lengths of pendulums
that vibrate in equal times in given unequal latitudes; or by comparing different
degrees measured on the meridian. By the best observations it would seem,
that there is a greater increase of gravitation, and of the degrees of the meridian
from the equator towards the poles, than ought to arise from the supposition of
a uniform density. Therefore the author supposes the density to vary from the
surface towards the centre ; and, in several cases he has considered, he finds
that a greater density towards the centre would account for a greater increase of
gravitation towards the poles, but not for a greater increase of the degrees of
the meridian ; and that the hypothesis of a less density towards the centre
would account for the latter, but not for the former, supposing (after Sir Isaac
Newton) the columns of the fliuid to extend from the surface to the centre, and
there to sustain each other. On this account he determines the gravitation to-
wards the earth, when it is supposed to be hollow with a nucleus included, ac-
cording to the hypothesis advanced by Dr. Halley, with the difference of the
semidiameters that might arise from such a disposition of the internal parts. But
in this case, and when the density is supposed variable, the spheroidical figure
is only assumed as an hypothesis. He adds, that by imagining the density to

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be greater in the axis than in the plane of the equator at equal distances from
the centre, an hypothesis perhaps might be found, that would account for most
of the phenomena ; but that a series of many exact observations is requisite, be-
fore we can examine with any certainty the various suppositions that may be
imagined, concerning the internal constitution of the earth. This doctrine is
likewise applied for determining the figure of Jupiter.

It follows from the same theorem, that if we suppose the earth to be fluid,
and abstract from its motion on its axis, and the inclination of the right lines in
which its particles gravitate towards the sun or moon, the figure which it would
assume, in consequence of the unequal gravitation of its particles towards either
of those bodies, would be accurately that of an oblong spheroid, having its axis
directed towards that body. The ascent of the water, deduced from this
theorem, agrees nearly with that which Sir Isaac Newton found, by computing
it briefly from what he had demonstrated concerning the figure of the earth.
Several observations are subjoined concerning the tides, and the causes which
may contribute to increase or diminish them, particularly the inequality of the
velocities with which bodies revolve about the axis of the earth in different
latitudes.

This chapter concludes by demonstrating briefly, that if the attraction of the
particles decreased as the cube of their distance increases, or in any higher pro-
portion, then any particle would tend towards the least portion of matter in
contact with it, with a greater force than towards the greatest body at any dis-
tance, how small soever from it. The true law of gravity is better adapted for
holding the parts of each body in a proper union, while it perpetuates the mo
tions in the great system about the sun, and preserves the revolutions in the less
systems nearly regular; and the author concludes with observing, that a re-
markable geometrical sin)plicity is often found in the conclusions that are derived
from it.

Of a very large Calculus voided by a Woman with her Urine. By Antonio
Leprotti, Physician to the Pope, and F. R. S. NÂ° 468, p. 363. From the
Latin.

A poor woman, aged 50, and who had laboured under a strangury between 3
and 4 years before, one night was seized with a discharge of bloody urine, to
the quantity of about 3 lb. during which she voided a stone, which, after being
dried, weighed Â§ij gr. 2g, or 1 oz. 17 dwts. 4 gr. avoirdupois.

654 VHILOSOPHICAL TRANSACTIONS. [aNNO 1742-3.

A Machine for dressing and curing Patients, luho are very unwieldy, and are
under the Surgeon's Hands for some complaint on the Back, the Os Sacrum, &c.
or are apprehensive of it. By M. le Cat, F.R.S. Abstracted from the French
by P. H. Z. F. R. S. N" 468, p. 364.

Finding the usual methods for supporting unwieldy patients, who required
surgical assistance for some ailment in the back, insufficient, M. le Cat was led
to contrive a sort of hanging cradle or hammock, as represented in the figure
hereto annexed.

PI, 14, fig. 1 1, represents the bed without the bedding. On which lies a sort of
boat of Turkey leather, full as long as the bed, with very strong hems all round,
and eilet-holes for receiving hooks, that serve to lift up this hammock. The
hooks are fastened to several ropes, all which depend on as many cross-beams
of very solid wood. The cross-beams consist of one beam of the length of the
whole bed, running lengthwise over the middle of it, and 4 transverse beams,
the 2 middle ones somewhat longer than the others. The ropes on which the