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 this rule does not hold, for the ordinate is in that case neither a maximum nor
minimum. If the first, second, and third fluxions vanish, and the fourth
fluxion be real, the ordinate is a maximum or minimum. The general rule

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demonstrated in this chapter, and again in the last chapter of the second book,
is, that when the first fluxion of the ordinate, with its fluxions of any subse-
quent successive orders, vanish, and the number of all these fluxions that
vanish is odd, then the ordinate is a maximum or minimum, according as the
fluxion of the next order to these is negative or positive. The ordinate passes
through a point of contrary flexure, when its fluxion becomes a maximum or
minimum, supposing the curve to be continued on both sides of the ordinate.
Hence the cotnmon rule for finding the points of contrary flexure is corrected
in a similar manner. Such a point is not always formed when the second fluxion
of the ordinate vanishes ; for if its third fluxion likewise vanishes, and its fourth
fluxion be real, the curve may have its cavity turned all one way. The s^me
is to be said, when its fluxions of the subsequent successive orders vanish, if
the number of all those that vanish be even. Other theorems are subjoined re-
lating to this subject.

The J 0th chapter treats of the asymptotes of lines, the areas bounded by
them and the curves, the solids generated by these areas, of spiral lines, and
the limits of the sums of progressions. The analogy between these subjects,
induced the author to treat of them in one chapter, and illustrate them by one
another. He begins with three of the most simple instances of figures that
have asymptotes. In the common hyperbola, the ordinate is reciprocally as
the base, and therefore decreases while the base increases, but never vanishes,
because the rectangle contained by it and the base is always a given area, and it
is assignable at any assignable distance, how great soever. Tlie points of the
conchoid are determined by drawing right lines from a given centre, and on
these produced from the asymptote, taking always a given right line ; so that
the curve never meets the asymptote, but continually approaches to it, because
of the greater and greater obliquity of this right line. The third is the loga-
rithmic curve, wherein the ordinates, at equal distances, decrease in geometri-
cal proportion, but never vanish, because each ordinate is in a given ratio to
the preceding ordinate. Geometrical magnitude is always understood to consist
of parts ; and to have no parts, or to have no magnitude, are considered as
equivalent in this science.* There is, however, no necessity for considering
magnitude as made up of an infinite number of small parts ; it is sufficient, that
no quantity can be supposed to be so small, but it may be conceived to be di-
minished further ; and it is obvious, that we are not to estimate the number of
parts that may be conceived in a given magnitude, by those which in particular

â€˘ See Euclid's Elements, Def. 1, lib. i. â€” Orig.

VOL. XLII.] PHILOSOPHICAL TRANSACTIONS. 63p

determinate circumstances may be actually perceived in it by sense ; since a
greater number of parts become visible in it by varying the circumstances in
which it is perceived.

It is hardly possible to give a tolerable extract of this or the following chap-
ters, without diagrams and computations : we shall therefore observe only, that
after giving some plain and obvious instances, wherein a quantity is always in
creasing, and yet never amounts to a certain finite magnitude (as, while the
tangent increases, the arc increases, but never amounts to a quadrant); this is
applied successively to the several subjects mentioned in the title of the chap-
ter. Let the figure be concave towards the base, and suppose it to have an
asymptote parallel to the base ; in this case the ordinate always increases while
the base is produced, but never amounts to the distance between the asymptote
and the base. In like manner a curvilineal area, in a second figure, may in-
crease, while the base is produced, and approach continually to a certain finite
space, but never amount to it : this is always the case, when the ordinate of
this latter figure is to a given right line, as the fluxion of the ordinate of the
former is to the fluxion of the base ; and of this various examples are given.
A solid may increase in the same manner, and yet never amount to a given
cube or cylinder, when the square of the ordinate of the latter figure is to a
given square, as the fluxion of the ordinate of the first figure is to the fluxion
of the base. A spiral may in like manner approach to a point continually, and
yet in any number of revolutions never arrive at it ; and there are progressions
of fractions that may be continued at pleasure, and yet the sum of the terms
may be always less than a given number. Various rules are demonstrated, and
illustrated by examples, for determining when a figure has an asymptote parallel
or oblique to the base ; when the area terminated by the curve and the asymp-
tote has a limit which it never exceeds, or may be produced till it surpass any
assignable space ; when the solid generated by that area, the surface generated
by the perimeter of the curve, the spiral area generated by the revolving ray,
the spiral line itself, or the sum of the terms of a progression, have such limits
or not; and for measuring those limits. The author insists on these subjects,
the rather that they are commonly described in very mysterious terms, and have
been the most fertile of paradoxes of any parts of the higher geometry. These
paradoxes, however, amount to no more than this : that a line or number may
be continually acquiring increments, and those increments may decrease in such
a manner, that the whole line or number shall never amount to a given line or
number. The necessity of admitting this is obvious enough, and is here shown
from the nature of the most common geometrical figures in Art. 292, 293, &c.
and from any series of fractions that decrease continually, in Art. 354, 355, &c.

640 PHILOSOPHICAL TRANSACTIONS. [aNNO 1742-3.

The 1 1th chapter treats of the curvature of lines, its variation, the degrees
of contact of the curve and circle of curvature, and of various problems that
depend on the curvature of lines. This subject is treated fully, because of its
extensive usefulness, and because in this consists one of the greatest advan-
tages of the modern geometry above that of the ancients. The author on this,
as on former occasions, begins by premising the necessary definitions. Curve
lines touch each other in a point, when the same right line is their common
tangent at that point ; and that which has the closest contact with the tangent,
or passes between it and the other curve through the angle of contact formed
by them, being less inflected from the tangent, is therefore less curve. Thus
a greater circle has a less curvature than a less circle ; and since the curvature
of circles may be varied indefinitely, by enlarging or diminishing their diame-
ters, they afford a scale by which the curvature of other lines may be measured.
As the tangent is the right line which touches the arc so closely, that no other
right line can be drawn between them ; so the circle of curvature is that which
touches the curve so closely, that no other circle can be drawn through the
point of contact between them. As the curve is separated from its tangent in
consequence of its flexure or curvature, so it is separated from the circle of cur-
vature in consequence of the variation of its curvature ; which is greater or less,
according as its flexure from that circle is greater or less.

The tangent of the figure being considered as the base, a new figure is ima-
gined, whose ordinate is a third proportional to the ordinate and base of the
first. This new figure determines the chord of the circle of curvature by its in-
tersection with the ordinate at the point of contact, and by the tangent of the
angle in which it cuts that circle, measures the variation of curvature. The
less this angle is, the closer is the contact of the curve and circle of curvature,
of which there may be indefinite degrees. When the figure proposed is a conic
section, the new figure is likewise a conic section ; and it is a right line when
the first figure is a parabola, and the ordinates are parallel to the axis ; or when
the first figure is an hyperbola, and the ordinates are parallel to either asymp-
tote. Hence the curvature and its variation in a conic section are determined
by several constructions ; and among other theorems, it is shown, that the
variation of curvature at any point of a conic section, is as the tangent of the
angle contained by the diameter which passes through that point, and by the
perpendicular to the curve.

When the ordinate at the point of contact is an asymptote to the new figure,
the curvature is less than in any circle; and this is the case in which it is said
to be infinitely little, or the ray of curvature is said to be infinitely great. Of
this kind is the curvature at the points of contrary flexure in the lines of the
third order. When the new figure passes through the point of contact, the

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curvature is greater than in any circle^ or the ray of curvature vanishes; and in
this case the curvature is said to be infinitely great. Of this kind is the curva-
ture at the cuspids of the Hnes of the third order.

As lines which pass through the same point have the same tangent when the
first fluxions of the ordinate are equal, so they have the same curvature when
the second fluxions of the ordinate are likewise equal ; and half the chord of
the circle of curvature that is intercepted between the points wherein it inter-
sects the ordinate, is a third proportional to the right lines that measure the
second fluxion of the ordinate, and first fluxion of the curve, the base being
supposed to flow uniformly. When a ray revolving about a given point, and
terminated by the curve, becomes perpendicular to it, the first fluxion of the
ray vanishes; and if its second fluxion vanishes at the same time, that point
must be the centre of curvature. The same is to be said when the angular mo-
tion of the ray about that point is equal to the angular motion of the tangent
of the curve; as the angular motion of the radius of a circle about its centre
is always equal to the angular motion of the tangent of the circle. Thus the
various properties of the circle suggest various theorems for determining the
centre of the curvature.

Because figures are often supposed to be described by the intersections of
right lines revolving about given poles, three theorems are given in prop. 18,
26, and 35, for determining the tangents, asymptotes, and curvature of such
lines, from the description, which are illustrated by examples. A new pro-
perty of lines of the third order is subjoined to prop. 35. The evolution of
lines is considered in prop. 36. The tangents of the evoluta are the rays of
curvature of the line which is described by its evolution; and the variation of
curvature in the latter is measured by the ratio of the ray of curvature of the
former to the ray of curvature of the latter.

Sir Isaac Newton, in a treatise lately published, measures the variation of
the curvature by the ratio of the fluxion of the ray of curvature to the fluxion
of the curve; and is followed by the author, to avoid the perplexity which a
difference in definitions occasions to readers, though he hints in art. 386, tiiat
this ratio gives rather the variation of the ray of curvature, and that it might
have been proper to have measured the variation of curvature rather by the ratio
of the fluxion of the curvature itself to the fluxion of the curve; so that the
curvature being inversely as the ray of curvature, and consequently its fluxion
as the fluxion of the ray itself directly, and the square of the ray inversely, its
variation would have been directly as the measure of it, according to Sir Isaac
Newton's definition, and inversely as the square of the ray of curvature; ac-
cording to this explication, it would have been measured by the angle of con-

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tact contained by the curve and circle of curvature, in the same manner as the
curvature itself is measured by the angle of contact contained by the curve and
tangent. The ground of this remark will better appear from an example: ac-
cording to Sir Isaac Newton's explication, the variation of curvature is uniform
in the logarithmic spiral, the fluxion of the ray of curvature in this figure being
always in the same ratio to the fluxion of the curve; and yet while the spiral is
produced, though its curvature decreases, it never vanishes; which must appear
strange to such as do not attend to the import of his definition. â€” It is easy,
however, to derive one of these measures of this variation from the other, and
because Sir Isaac Newton's is, generally speaking, assigned by more simple ex-
pressions, the author has the rather conformed to it in this treatise, but thought
it necessary to give the caution we have mentioned.

The greatest part of this chapter is employed in treating of useful problems,
that have a dependence on the curvature of lines. First, the properties of the
cycloid are briefly demonstrated, with the application of this doctrine to the
motion of pendulums, by showing that when the motion of the generating
circle along the base is uniform, and therefore may measure the time, the mo-
tion of the point that describes the cycloid, is such as would be acquired by a
heavy body descending along the cycloidal arc, the axis of the figure being sup-
posed perpendicular to the horizon. In the next place, the caustics, by re-
flexion and refraction, are determined. If perpendiculars be always drawn from
the radiating point to the tangents of the curve, and a new curve be supposed
to be the locus of the intersections of the perpendiculars and tangents, then
the line, by the evolution of which that new curve can be described, is similar
and similarly situated to the caustic by reflexion. The doctrine of centripetal
forces is treated at length from art. 4l6 to 493.

First, a body is supposed to descend freely by its gravity in a vertical line,
and because the gravity is the power which accelerates the motion of the body,
it must be measured by the fluxion of its velocity, or the second fluxion of the
space described by it. When the vertical line is supposed to move parallel to
itself with an uniform motion, the body will descend in it in the same manner
as before; and the gravity will be still measured by the second fluxion of the
descent, or the second fluxion of the ordinate of the curve that is traced in this
case by the body on an immoveable plane, and therefore is as the square of the
velocity, which is measured by the fluxion of the curve, directly, and the chord
of the circle of curvature that is in the direction of the gravity inversely, by a
proposition mentioned above. When the gravity acts uniformly, and in parallel
lines, the projectile, in describing any arc, falls below the tangent drawn at the
beginning of the arc, as much as if it had fallen perpendicularly in the ver-

VOL. XLII.] PHILOSOPHICAL TRANSACTIONS. 643

tical ; and, the time being given, the gravity may be measured by the space
which is the subtense of the angle of contact. In other cases, when the gra-
vity varies, or its direction changes, it may be measured at any point by the
subtense of the angle of contact, that would have been generated in a given
time, if the gravity had continued to act uniformly in parallel lines from that
term, that is, by the subtense of the angle of contact in the parabola that has
its diameter in the direction of the force, and has the closest contact with the
curve; which leads us to the same theorem as before.

In general, let the gravity, that results from the composition of any number
of centripetal forces, which are supposed to act on the body in one plane, be
resolved into a force parallel to the ordinates, and a force parallel to the base;
then the former shall be measured by the second fluxion of the ordinate, and
the latter by the second fluxion of the base, the time being supposed to flow
uniformly, so that the velocity of the body may be measured by the fluxion of
the curve. When the trajectory is not in one plane, the force is resolved in a
similar manner into three forces, which are measured by three second fluxions
analogous to them.

Whether the body move in a void, or in a medium that resists its motion;
the gravity that results from the composition of the centripetal forces which
act upon the body, is always as the square of its velocity directly, and the
chord of the circle of curvature that is in the direction of the gravity in-
versely.

When a body describes any trajectory in a void or in a medium, by a force
directed to one given centre, the velocity at any point of the trajectory, is to
the velocity by which a circle could be described in a void about the same centre,
at the same distance, by the same gravity, in the subduplicate ratio of the
angular motion of the ray drawn always from the body to the centre, to the
angular motion of the tangent of the trajectory; and, if there be no resistance,
the velocity in the trajectory at any point, is the same that would be acquired
by the body, if it was to fall from that point through one-fourth of the chord
of the circle of curvature that is in the direction of the gravity, and the gra-
vity at that point was to be continued uniformly during its descent.

If the centripetal force be inversely as any power of the distance whose expo-
nent is any number m greater than unit, there is a certain velocity (viz. that
which is to the velocity in a circle at the same distance as \/ 2 to Vm â€” 1) which
would be just sufficient to carry off' the body upwards in a vertical line, so as
that it should continue to ascend for ever, and never return towards the centre.
If the body be projecte<i in any other direction with the same velocity, it will
describe a trajectory which is here constructed; it is a parabola when m=1, a.

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logarithmic spiral when m = 3, an epicyloid when m = 4, a circle that passes
through the centre of the forces when m = 5, and the lemniscata when m-=.'] .
In general, it is constructed by drawinjg a perpendicular from the centre of the
forces to a right line given in position, and any other ray to the same right line,
then increasing or diminishing the angle contained by this ray and the perpen-
dicular in the given ratio of 2 to the difference between 3 and m, and increasing
or diminishing the logarithm of the ray in the same given ratio. The trajec-
tories described in analogous cases by centrifugal forces, are constructed in a
similar manner. These are the figures in which the perpendicular, from a given
centre on the tangent, is always as some power of the ray drawn from the same
centre to the point of contact, which are afterwards found to arise in the reso-
lution of the most simple cases of problems of various kinds.

When the area described about the centre of an ellipse is given, the subtense
of the angle of contact, drawn through one extremity of the arc parallel to the
semidiameter drawn to the other extremity, is in a given ratio to this semidia-
meter; and therefore, when an ellipse is described by a force directed towards
the centre, that force is always as the distance from the centre. When the
force is directed toward the focus, it is inversely as the square of the distance.
And these two cases are considered particularly, because of their usefulness in
the true theory of gravity. To illustrate which, the laws of centripetal forces
that would cause a body to descend continually toward the centre, or ascend
from it, are distinguished from those which cause the body to approach towards
the centre, and recede from it by turns. A body approaches from the higher
apsid toward the centre, when its velocity is less than what is requisite to carry
it in a circle; and if its velocity increase, while it descends, in a higher pro-
portion than the velocities requisite to carry bodies in circles about the same
centre, the "velocity in the lower part of the curve may exceed the velocity in a
circle at the same distance, and thereby become sufficient to carry off the body
again. But while the distance decreases, if the velocities in circles increase
in the same, or in a higher proportion, than the velocity in a trajectory can in-
crease, the body must either continually approach toward the centre, if it once
begin to approach to it, or recede continually from the centre, if it once begin
to ascend from it; and this is the case, when the centripetal force increases as
the cube of the distance decreases, or in a higher proportion. But though, in
such cases, the body approaches continually towards the centre, we are not to
conclude, that it will always approach to it till it fall into it, or come within
any given distance ; for it is demonstrated afterwards, in art. 879 and 880, that
it may approach to the centre for ever, in a spiral that never descends to a
given circle described in the same plane, and that it may recede from it for ever

VOL. XLII.J PHILOSOPHICAL TRANSACTIONS. 645

in a spiral that never arises to a given altitude. An example of each case is
given when the centripetal force is inversely as the fifth power of the dis-
tance.

When the trajectory is described in a medium, let z be to a given magnitude
as the centripetal force is to the force by which the same trajectory could be
described in a void; and if the area be supposed to flow uniformly, the resist-
ance will be in the compound ratio of the fluxion of z, and of the fluxion of
the curve; and the density of the medium, supposing the resistance to be in
the compound ratio of the density and of the square of the velocity, shall be
as the fluxion of the logarithm of z directly, and the fluxion of the curve in-
versely. Hence, when any figure that can be described in a void by a force
that varies according to any power of the distance from the centre, is described
in a medium, the density of the medium must be inversely as the tangent of
the figure bounded by a perpendicular at the centre to the ray drawn from it to
the point of contact.

After giving some properties of the trajectories that are described by a body
when it gravitates in right lines perpendicular to a given surface, and their ap-
plication to optical uses, the author proceeds to consider the motion of a body
that gravitates towards several centres. In such cases, that surface is said to be
horizontal, which is always perpendicular to the direction of the gravity that
results from the composition of the several forces; and it is shown, that the
velocity which is acquired by descending from one horizontal surface to another
is always the same, whether the body move in right lines, or in any curves;
the square of which is measured by the aggregate of several areas which have
the distances from the respective centres for their bases, and right lines propor-
tional to the forces at these distances for their ordinates.

The force which acts on the moon is resolved into a force perpendicular to
the plane of the ecliptic, and a force parallel to it. This last is again resolved
into that which is parallel to the line of the syzigies, and that which is parallel
to the line joining the quadratures. The first measures the second fluxion of
the distance of the moon from that plane, the second and third measure the
second fluxions of her distances from the line of the quadratures, and from the
line of the syzigies, respectively. Hence a construction is derived of the tra-
jectory which would be described by the moon about the earth, in consequence
of their unequal gravitation towards the sun, if the gravity of the moon to-
wards the earth was as her distance from it. From this a computation is deduced
of the motion of the nodes of the moon, and of the variation of the inclina-
tion of the plane of her orbit. It is sufficient here to observe, that these mo-

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tions are found to agree nearly with those which have been deduced from other
theories, and from astronomical observations.

A fluid being supposed to gravitate towards two given centres with equal and