Diffusion of Matter through Space—Gravitation—Its Velocity—Simplicity of
its Laws—Gravitation independent of the Magnitude and Distances of the
Bodies—Not impeded by the intervention of any Substance—Its Intensity
invariable—General Laws—Recapitulation and Conclusion.
THE known quantity of matter bears a very small proportion to the
immensity of space. Large as the bodies are, the distances which
separate them are immeasurably greater; but, as design is manifest in
every part of creation, it is probable that, if the various systems in
the universe had been nearer to one another, their mutual disturbances
would have been inconsistent with the harmony and stability of the
whole. It is clear that space is not pervaded by atmospheric air of such
density as that we breathe, since its resistance would long ere this
have arrested the motion of the planets: it certainly is not a void, but
replete with a medium possibly in itself electric or magnetic, but at
all events capable of transmitting light, heat, magnetism, gravity, and
probably influences of which we can form no idea.
Whatever the laws may be that obtain in the more distant regions of
creation, we are assured that one alone regulates the motions, not only
of our own system, but also of the binary systems of the fixed stars;
and, as general laws form the ultimate object of philosophical research,
we cannot conclude these remarks without considering the nature of
gravitation—that extraordinary power whose effects we have been
endeavouring to trace through some of their mazes. It was at one time
imagined that the acceleration in the moon’s mean motion was occasioned
by the successive transmission of the gravitating force. It has been
proved that, in order to produce this effect, its velocity must be about
fifty millions of times greater than that of light, which flies at the
rate of 192,000 miles in a second. Its action, even at the distance of
the sun, may therefore be regarded as instantaneous; yet, remote as the
fixed stars are, the solar gravitation must have some influence on the
nearest of them, as, for example, α Centauri, which is only 20,602 times
the radius of the earth’s orbit from the sun, while La Place has
computed that the solar gravitation extends a hundred millions of times
farther than the semidiameter of the terrestrial orbit. Possibly the
star dust in the Milky Way may be beyond, or on the verge of, that
enormous limit; yet it is very unlikely that either the sun, or any of
the stars which form the great cluster to which we belong, should be
unconnected bodies.
The curves in which the celestial bodies move by the force of
gravitation are only lines of the second order. The attraction of
spheroids, according to any other law of force than that of gravitation,
would be much more complicated; and, as it is easy to prove that matter
might have been moved according to an infinite variety of laws, it may
be concluded that gravitation must have been selected by Divine Wisdom
out of an infinity of others, as being the most simple, and that which
gives the greatest stability to the celestial motions.
It is a singular result of the simplicity of the laws of nature, which
admit only of the observation and comparison of ratios, that the
gravitation and theory of the motions of the celestial bodies are
independent of their absolute magnitudes and distances. Consequently, if
all the bodies of the solar system, their mutual distances, and their
velocities, were to diminish proportionally, they would describe curves
in all respects similar to those in which they now move; and the system
might be successively reduced to the smallest sensible dimensions, and
still exhibit the same appearances.
The action of the gravitating force is not impeded by the intervention
even of the densest substances. If the attraction of the sun for the
centre of the earth, and of the hemisphere diametrically opposite to
him, were diminished by a difficulty in penetrating the interposed
matter, the tides would be more obviously affected. Its attraction is
the same also, whatever the substances of the celestial bodies may be;
for, if the action of the sun upon the earth differed by a millionth
part from his action upon the moon, the difference would occasion a
periodical variation in the moon’s parallax, whose maximum would be the
1/15 of a second, and also a variation in her longitude amounting to
several seconds—a supposition proved to be impossible by the agreement
of theory with observation. Thus all matter is pervious to gravitation,
and is equally attracted by it.
Gravitation is a feeble force, vastly inferior to electric action,
chemical affinity, and cohesion; yet, as far as human knowledge extends,
the intensity of gravitation has never varied within the limits of the
solar system; nor does even analogy lead us to expect that it should: on
the contrary, there is every reason to be assured that the great laws of
the universe are immutable, like their Author. Nor can we suppose the
structure of the globe alone to be exempt from the universal fiat of
general laws, though ages may pass before the changes it has undergone,
or that are now in progress, can be referred to existing causes with the
same certainty with which the motions of the planets, and all their
periodic and secular variations, are referable to the law of
gravitation. The traces of extreme antiquity perpetually occurring to
the geologist give that information, as to the origin of things, in vain
looked for in the other parts of the universe. They date the beginning
of time with regard to our system, since there is ground to believe that
the formation of the earth was contemporaneous with that of the rest of
the planets; but they show that creation is the work of Him with whom “a
thousand years are as one day, and one day as a thousand years.”
In the work now brought to a conclusion, it has been necessary to select
from the whole circle of the sciences a few of the most obvious of those
proximate links which connect them together, and to pass over
innumerable cases both of evident and occult alliance. Any one branch
traced through its ramifications would alone have occupied a volume; it
is hoped, nevertheless, that the view here given will suffice to show
the extent to which a consideration of the reciprocal influence of even
a few of these subjects may ultimately lead. It thus appears that the
theory of dynamics, founded upon terrestrial phenomena, is indispensable
for acquiring a knowledge of the revolutions of the celestial bodies and
their reciprocal influences. The motions of the satellites are affected
by the forms of their primaries, and the figures of the planets
themselves depend upon their rotations. The symmetry of their internal
structure proves the stability of these rotatory motions, and the
immutability of the length of the day, which furnishes an invariable
standard of time; and the actual size of the terrestrial spheroid
affords the means of ascertaining the dimensions of the solar system,
and provides an invariable foundation for a system of weights and
measures. The mutual attraction of the celestial bodies disturbs the
fluids at their surfaces, whence the theory of the tides and of the
oscillations of the atmosphere. The density and elasticity of the air,
varying with every alternation of temperature, lead to the consideration
of barometrical changes, the measurement of heights, and capillary
attraction; and the doctrine of sound, including the theory of music, is
to be referred to the small undulations of the aërial medium. A
knowledge of the action of matter upon light is requisite for tracing
the curved path of its rays through the atmosphere, by which the true
places of distant objects are determined, whether in the heavens or on
the earth. By this we learn the nature and properties of the sunbeam,
the mode of its propagation through the ethereal medium, or in the
interior of material bodies, and the origin of colour. By the eclipses
of Jupiter’s satellites the velocity of light is ascertained; and that
velocity, in the aberration of the fixed stars, furnishes a direct proof
of the real motion of the earth (N. 237). The effects of the invisible
rays of the spectrum are immediately connected with chemical action; and
heat, forming a part of the solar ray, so essential to animated and
inanimated existence, is too important an agent in the economy of
creation not to hold a principal place in the connexion of physical
sciences; whence follows its distribution in the interior and over the
surface of the globe, its power on the geological convulsions of our
planet, its influence on the atmosphere and on climate, and its effects
on vegetable and animal life, evinced in the localities of organized
beings on the earth, in the waters, and in the air. The correlation
between molecular and chemical action, light, heat, electricity, and
magnetism, is continually becoming more perfect, and there is every
reason to believe that these different modes of force, as well as
gravity itself, will ultimately be found to merge in one great and
universal power. Many more instances might be given in illustration of
the immediate connexion of the physical sciences, most of which are
united still more closely by the common bond of analysis, which is daily
extending its empire, and will ultimately embrace almost every subject
in nature in its formulæ.
These formulæ, emblematic of Omniscience, condense into a few symbols
the immutable laws of the universe. This mighty instrument of human
power itself originates in the primitive constitution of the human mind,
and rests upon a few fundamental axioms, which have eternally existed in
Him who implanted them in the breast of man when He created him after
His own image.
NOTES.
NOTE 1, page 2. _Diameter._ A straight line passing through the centre,
and terminated both ways by the sides or surface of a figure, such as of
a circle or sphere. In fig. 1, q Q, N S, are diameters.
NOTE 2, p. 2. _Mathematical and mechanical sciences._ Mathematics teach
the laws of number and quantity; mechanics treat of the equilibrium and
motion of bodies.
NOTE 3, p. 2. _Analysis_ is a series of reasoning conducted by signs or
symbols of the quantities whose relations form the subject of inquiry.
NOTE 4, p. 3. _Oscillations_ are movements to and fro, like the swinging
of the pendulum of a clock, or waves in water. The tides are
oscillations of the sea.
NOTE 5, p. 3. _Gravitation._ _Gravity_ is the reciprocal attraction of
matter on matter; _gravitation_ is the difference between gravity and
the centrifugal force induced by the velocity of rotation or revolution.
Sensible gravity, or weight, is a particular instance of gravitation. It
is the force which causes substances to fall to the surface of the
earth, and which retains the celestial bodies in their orbits. Its
intensity increases as the squares of the distance decrease.
NOTE 6, p. 4. _Particles of matter_ are the indefinitely small or
ultimate atoms into which matter is believed to be divisible. Their form
is unknown; but, though too small to be visible, they must have
magnitude.
NOTE 7, p. 4. _A hollow sphere._ A hollow ball, like a bomb-shell. A
sphere is a ball or solid body, such, that all lines drawn from its
centre to its surface are equal. They are called radii, and every line
passing through the centre and terminated both ways by the surface is a
diameter, which is consequently equal to twice the radius. In fig. 3, Q
q or N S is a diameter, and C Q, C N are radii. A great circle of the
sphere has the same centre with the sphere as the circles Q E q d and Q
N q S. The circle A B is a lesser circle of the sphere.
NOTE 8, p. 4. _Concentric hollow spheres._ Shells, or hollow spheres,
having the same centre, like the coats of an onion.
[Illustration: _Fig. 1._]
NOTE 9, p. 4. _Spheroid._ A solid body, which sometimes has the shape of
an orange, as in fig. 1; it is then called an oblate spheroid, because
it is flattened at the poles N and S. Such is the form of the earth and
planets. When, on the contrary, it is drawn out at the poles like an
egg, as in fig. 2, it is called a prolate spheroid. It is evident that
in both these solids the radii C q, C a, C N, &c., are generally
unequal; whereas in the sphere they are all equal.
[Illustration: _Fig. 2._]
NOTE 10, p. 4. _Centre of gravity._ A point in every body, which if
supported, the body will remain at rest in whatever position it may be
placed. About that point all the parts exactly balance one another. The
celestial bodies attract each other as if each were condensed into a
single particle situate in the centre of gravity, or the particle
situate in the centre of gravity of each may be regarded as possessing
the resultant power of the innumerable oblique forces which constitute
the whole attraction of the body.
NOTE 11, pp. 4, 6. _Poles and equator._ Let fig. 1 or 3 represent the
earth, C its centre, N C S the axis of rotation, or the imaginary line
about which it performs its daily revolution. Then N and S are the north
and south poles, and the great circle q E Q, which divides the earth
into two equal parts, is the equator. The earth is flattened at the
poles, fig. 1, the equatorial diameter, q Q, exceeding the polar
diameter, N S, by about 26-1/2 miles. Lesser circles, A B G, which are
parallel to the equator, are circles or parallels of latitude, which is
estimated in degrees, minutes, and seconds, north and south of the
equator, every place in the same parallel having the same latitude.
Greenwich is in the parallel of 51° 28ʹ 40ʺ. Thus terrestrial latitude
is the angular distance between the direction of a plumb-line at any
place and the plane of the equator. Lines such as N Q S, N G E S, fig.
3, are called meridians; all the places in any one of these lines have
noon at the same instant. The meridian of Greenwich has been chosen by
the British as the origin of terrestrial longitude, which is estimated
in degrees, minutes, and seconds, east and west of that line. If N G E S
be the meridian of Greenwich, the position of any place, B, is
determined, when its latitude, Q C B, and its longitude, E C Q, are
known.
[Illustration: _Fig. 3._]
NOTE 12, p. 4. _Mean quantities_ are such as are intermediate between
others that are greater and less. The mean of any number of unequal
quantities is equal to their sum divided by their number. For instance,
the mean between two unequal quantities is equal to half their sum.
NOTE 13, p. 4. _A certain mean latitude._ The attraction of a sphere on
an external body is the same as if its mass were collected into one
heavy particle in its centre of gravity, and the intensity of its
attraction diminishes as the square of its distance from the external
body increases. But the attraction of a spheroid, fig. 1, on an external
body at m in the plane of its equator, E Q, is greater, and its
attraction on the same body when at mʹ in the axis N S less, than if it
were a sphere. Therefore, in both cases, the force deviates from the
exact law of gravity. This deviation arises from the protuberant matter
at the equator; and, as it diminishes towards the poles, so does the
attractive force of the spheroid. But there is one mean latitude, where
the attraction of a spheroid is the same as if it were a sphere. It is a
part of the spheroid intermediate between the equator and the pole. In
that latitude the square of the sine is equal to 1/3 of the equatorial
radius.
NOTE 14, p. 4. _Mean distance._ The mean distance of a planet from the
centre of the sun, or of a satellite from the centre of its planet, is
equal to half the sum of its greatest and least distances, and,
consequently, is equal to half the major axis of its orbit. For example,
let P Q A D, fig. 6, be the orbit or path of the moon or of a planet;
then P A is the major axis, C the centre, and C S is equal to C F. Now,
since the earth or the sun is supposed to be in the point S according as
P D A Q is regarded as the orbit of the moon or that of a planet, S A, S
P are the greatest and least distances. But half the sum of S A and S P
is equal to half of A P, the major axis of the orbit. When the body is
at Q or D, it is at its mean distance from S, for S Q, S D, are each
equal to C P, half the major axis by the nature of the curve.
NOTE 15, p. 4. _Mean radius of the earth._ The distance from the centre
to the surface of the earth, regarded as a sphere. It is intermediate
between the distances of the centre of the earth from the pole and from
the equator.
NOTE 16, p. 5. _Ratio._ The relation which one quantity bears to
another.
NOTE 17, p. 5. _Square of moon’s distance._ In order to avoid large
numbers, the mean radius of the earth is taken for unity: then the mean
distance of the moon is expressed by 60; and the square of that number
is 3600, or 60 times 60.
[Illustration: _Fig. 4_]
NOTE 18, p. 5. _Centrifugal force._ The force with which a revolving
body tends to fly from the centre of motion: a sling tends to fly from
the hand in consequence of the centrifugal force. A tangent is a
straight line touching a curved line in one point without cutting it, as
m T, fig. 4. The direction of the centrifugal force is in the tangent to
the curved line or path in which the body revolves, and its intensity
increases with the angular swing of the body, and with its distance from
the centre of motion. As the orbit of the moon does not differ much from
a circle, let it be represented by m d g h, fig. 4, the earth being in
C. The centrifugal force arising from the velocity of the moon in her
orbit balances the attraction of the earth. By their joint action, the
moon moves through the arc m n during the time that she would fly off in
the tangent m T by the action of the centrifugal force alone, or fall
through m p by the earth’s attraction alone. T n, the deflection from
the tangent, is parallel and equal to m p, the versed sine of the arc m
n, supposed to be moved over by the moon in a second, and therefore so
very small that it may be regarded as a straight line. T n, or m p, is
the space the moon would fall through in the first second of her descent
to the earth, were she not retained in her orbit by her centrifugal
force.
NOTE 19, p. 5. _Action and reaction._ When motion is communicated by
collision or pressure, the action of the body which strikes is returned
with equal force by the body which receives the blow. The pressure of a
hand on a table is resisted with an equal and contrary force. This
necessarily follows from the impenetrability of matter, a property by
which no two particles of matter can occupy the same identical portion
of space at the same time. When motion is communicated without apparent
contact, as in gravitation, attraction, and repulsion, the quantity of
motion gained by the one body is exactly equal to that lost by the
other, but in a contrary direction; a circumstance known by experience
only.
NOTE 20, p. 5. _Projected._ A body is projected when it is thrown: a
ball fired from a gun is projected; it is therefore called a projectile.
But the word has also another meaning. A line, surface, or solid body,
is said to be projected upon a plane, when parallel straight lines are
drawn from every point of it to the plane. The figure so traced upon a
plane is a projection. The projection of a terrestrial object is
therefore its daylight shadow, since the sun’s rays are sensibly
parallel.
NOTE 21, p. 5. _Space._ The boundless region which contains all
creation.
[Illustration: _Fig. 5._]
[Illustration: _Fig. 6._]
NOTE 22, pp. 5, 11. _Conic sections._ Lines formed by any plane cutting
a cone. A cone is a solid figure, like a sugar-loaf, fig. 5, of which A
is the apex, A D the axis, and the plane B E C F the base. The axis may
or may not be perpendicular to the base, and the base may be a circle,
or any other curved line. When the axis is perpendicular to the base,
the solid is a right cone. If a right cone with a circular base be cut
at right angles to the base by a plane passing through the apex, the
section will be a triangle. If the cone be cut through both sides by a
plane parallel to the base, the section will be a circle. If the cone be
cut slanting quite through both sides, the section will be an ellipse,
fig. 6. If the cone be cut parallel to one of the sloping sides as A B,
the section will be a parabola, fig. 7. And if the plane cut only one
side of the cone, and be not parallel to the other, the section will be
a hyperbola, fig. 8. Thus there are five conic sections.
[Illustration: _Fig. 7._]
[Illustration: _Fig. 8._]
NOTE 23, p. 5. _Inverse square of distance._ The attraction of one body
for another at the distance of two miles is four times less than at the
distance of one mile; at three miles, it is nine times less than at one;
at four miles, it is sixteen times less, and so on. That is, the
gravitating force decreases in intensity as the squares of the distance
increase.
NOTE 24, p. 5. _Ellipse._ One of the conic sections, fig. 6. An ellipse
may be drawn by fixing the ends of a string to two points, S and F, in a
sheet of paper, and then carrying the point of a pencil round in the
loop of the string kept stretched, the length of the string being
greater than the distance between the two points. The points S and F are
called the foci, C the centre, S C or C F the excentricity, A P the
major axis, Q D the minor axis, and P S the focal distance. It is
evident that, the less the excentricity C S, the nearer does the ellipse
approach to a circle; and from the construction it is clear that the
length of the string S m F is equal to the major axis P A. If T t be a
tangent to the ellipse at m, then the angle T m S is equal to the angle
t m F; and, as this is true for every point in the ellipse, it follows
that, in an elliptical reflecting surface, rays of light or sound coming
from one focus S will be reflected by the surface to the other focus F,
since the angle of incidence is equal to the angle of reflection by the
theories of light and sound.
NOTE 25, p. 5. _Periodic time._ The time in which a planet or comet
performs a revolution round the sun, or a satellite about its planet.
NOTE 26, p. 5. Kepler discovered three laws in the planetary motions by
which the principle of gravitation is established:—1st law, That the
radii vectores of the planets and comets describe areas proportional to
the time.—Let fig. 9 be the orbit of a planet; then, supposing the
spaces or areas P S p, p S a, a S b, &c., equal to one another, the
radius vector S P, which is the line joining the centres of the sun and
planet, passes over these equal spaces in equal times; that is, if the
line S P passes to S p in one day, it will come to S a in two days, to S
b in three days, and so on. 2nd law, That the orbits or paths of the
planets and comets are conic sections, having the sun in one of their
foci. The orbits of the planets and satellites are curves like fig. 6 or
9, called ellipses, having the sun in the focus S. Several comets are
known to move in ellipses; but the greater part seem to move in
parabolas, fig. 7, having the sun in S, though it is probable that they
really move in very long flat ellipses; others appear to move in
hyperbolas, like fig. 8. The third law is, that the squares of the
periodic times of the planets are proportional to the cubes of their
mean distances from the sun. The square of a number is that number
multiplied by itself, and the cube of a number is that number twice
multiplied by itself. For example, the squares of the numbers 2, 3, 4,
&c., are 4, 9, 16, &c., but their cubes are 8, 27, 64, &c. Then the
squares of the numbers representing the periodic times of two planets
are to one another as the cubes of the numbers representing their mean
distances from the sun. So that, three of these quantities being known,
the other may be found by the rule of three. The mean distances are
measured in miles or terrestrial radii, and the periodic times are
estimated in years, days, and parts of a day. Kepler’s laws extend to
the satellites.
[Illustration: _Fig. 9._]
NOTE 27, p. 5. _Mass._ The quantity of matter in a given bulk. It is
proportional to the density and volume or bulk conjointly.
NOTE 28, p. 5. _Gravitation proportional to mass._ But for the
resistance of the air, all bodies would fall to the ground in equal
times. In fact, a hundred equal particles of matter at equal distances
from the surface of the earth would fall to the ground in parallel
straight lines with equal rapidity, and no change whatever would take
place in the circumstances of their descent, if 99 of them were united
in one solid mass; for the solid mass and the single particle would
touch the ground at the same instant, were it not for the resistance of
the air.
NOTE 29, p. 5. _Primary_ signifies, in astronomy, the planet about which
a satellite revolves. The earth is primary to the moon.
NOTE 30, p. 6. _Rotation._ Motion round an axis, real or imaginary.
NOTE 31, p. 7. _Compression of a spheroid._ The flattening at the poles.
It is equal to the difference between the greatest and least diameters,
divided by the greatest, these quantities being expressed in some
standard measure, as miles.
NOTE 32, p. 7. SATELLITES. Small bodies revolving about some of the
planets. The moon is a satellite to the earth.
NOTE 33, p. 7. _Nutation._ A nodding motion in the earth’s axis while in
rotation, similar to that observed in the spinning of a top. It is
produced by the attraction of the sun and moon on the protuberant matter
at the terrestrial equator.
NOTE 34, p. 7. _Axis of rotation._ The line, real or imaginary, about
which a body revolves. The axis of the earth’s rotation is that
diameter, or imaginary line, passing through the centre and both poles.
Fig. 1 being the earth, N S is the axis of rotation.
NOTE 35, p. 7. _Nutation of lunar orbit._ The action of the bulging
matter at the earth’s equator on the moon occasions a variation in the
inclination of the lunar orbit to the plane of the ecliptic. Suppose the
plane N p n, fig. 13, to be the orbit of the moon, and N m n the plane
of the ecliptic, the earth’s action on the moon causes the angle p N m
to become less or greater than its mean state. The nutation in the lunar
orbit is the reaction of the nutation in the earth’s axis.
NOTE 36, p. 7. _Translated._ Carried forward in space.
NOTE 37, p. 7. _Force proportional to velocity._ Since a force is
measured by its effect, the motions of the bodies of the solar system
among themselves would be the same whether the system be at rest or not.
The real motion of a person walking the deck of a ship at sea is
compounded of his own motion and that of the ship, yet each takes place
independently of the other. We walk about as if the earth were at rest,
though it has the double motion of rotation on its axis and revolution
round the sun.
NOTE 38, p. 8. _Tangent._ A straight line which touches a curved line in
one point without cutting it. In fig. 4, m T is tangent to the curve in
the point m. In a circle the tangent is at right angles to the radius, C
m.
NOTE 39, p. 8. _Motion in an elliptical orbit._ A planet m, fig. 6,
moves round the sun at S in an ellipse P D A Q, in consequence of two
forces, one urging it in the direction of the tangent m T, and another
pulling it towards the sun in the direction m S. Its velocity, which is
greatest at P, decreases throughout the arc to P D A to A, where it is
least, and increases continually as it moves along the arc A Q P till it
comes to P again. The whole force producing the elliptical motion varies
inversely as the square of the distance. See note 23.
NOTE 40, p. 8. _Radii vectores._ Imaginary lines adjoining the centre of
the sun and the centre of a planet or comet, or the centres of a planet
and its satellite. In the circle, the radii are all equal; but in an
ellipse, fig. 6, the radius vector S A is greater, and S P less than all
the others. The radii vectores S Q, S D, are equal to C A or C P, half
the major axis P A, and consequently equal to the mean distance. A
planet is at its mean distance from the sun when in the points Q and D.
NOTE 41, p. 8. _Equal areas in equal times._ See Kepler’s 1st law, in
note 26, p. 5.
NOTE 42, p. 8. _Major axis._ The line P A, fig. 6 or 10.
NOTE 43, p. 8. _If the planet described a circle, &c._ The motion of a
planet about the sun, in a circle A B P, fig. 10, whose radius C A is
equal to the planet’s mean distance from him, would be equable, that is,
its velocity, or speed, would always be the same. Whereas, if it moved
in the ellipse A Q P, its speed would be continually varying, by note
39; but its motion is such, that the time elapsing between its departure
from P and its return to that point again would be the same whether it
moved in the circle or in the ellipse; for these curves coincide in the
points P and A.
NOTE 44, p. 8. _True motion._ The motion of a body in its real orbit P D
A Q, fig. 10.
[Illustration: _Fig. 10._]
NOTE 45, p. 9. _Mean motion._ Equable motion in a circle P E A B, fig.
10, at the mean distance C P or C m, in the time that the body would
accomplish a revolution in its elliptical orbit P D A Q.
NOTE 46, p. 9. _The equinox._ Fig. 11 represents the celestial sphere,
and C its centre, where the earth is supposed to be. q ♈ Q ♎ is the
equinoctial or great circle, traced in the starry heavens by an
imaginary extension of the plane of the terrestrial equator, and E ♈ e ♎
is the ecliptic, or apparent path of the sun round the earth. ♈ ♎, the
intersection of these two planes, is the line of the equinoxes; ♈ is the
vernal equinox, and ♎ the autumnal. When the sun is in these points, the
days and nights are equal. They are distant from one another by a
semicircle, or two right angles. The points E and e are the solstices,
where the sun is at his greatest distance from the equinoctial. The
equinoctial is everywhere ninety degrees distant from its poles N and S,
which are two points diametrically opposite to one another, where the
axis of the earth’s rotation, if prolonged, would meet the heavens. The
northern celestial pole N is within 1° 24ʹ of the pole star. As the
latitude of any place on the surface of the earth is equal to the height
of the pole above the horizon, it is easily determined by observation.
The ecliptic E ♈ e ♎ is also everywhere ninety degrees distant from its
poles P and p. The angle P C N, between the poles P and N of the
equinoctial and ecliptic, is equal to the angle e C Q, called the
obliquity of the ecliptic.
[Illustration: _Fig. 11._]
NOTE 47, p. 9. _Longitude._ The vernal equinox, ♈, fig. 11, is the zero
point in the heavens whence celestial longitudes, or the angular motions
of the celestial bodies, are estimated from west to east, the direction
in which they all revolve. The vernal equinox is generally called the
first point of Aries, though these two points have not coincided since
the early ages of astronomy, about 2233 years ago, on account of a
motion in the equinoctial points, to be explained hereafter. If S ♈,
fig. 10, be the line of the equinoxes, and ♈ the vernal equinox, the
true longitude of a planet p is the angle ♈ S p, and its mean longitude
is the angle ♈ C m, the sun being in S. Celestial longitude is the
angular distance of a heavenly body from the vernal equinox; whereas
terrestrial longitude is the angular distance of a place on the surface
of the earth from a meridian arbitrarily chosen, as that of Greenwich.
NOTE 48, pp. 9, 58. _Equation of the centre._ The difference between ♈ C
m and ♈ S p, fig. 10; that is, the difference between the true and mean
longitudes of a planet or satellite. The true and mean places only
coincide in the points P and A; in every other point of the orbit, the
true place is either before or behind the mean place. In moving from A
through the arc A Q P, the true place p is behind the mean place m; and
through the arc P D A the true place is before the mean place. At its
maximum, the equation of the centre measures C S, the excentricity of
the orbit, since it is the difference between the motion of a body in an
ellipse and in a circle whose diameter A P is the major axis of the
ellipse.
NOTE 49, p. 9. _Apsides._ The points P and A, fig. 10, at the
extremities of the major axis of an orbit. P is commonly called the
perihelion, a Greek term signifying _round the sun_; and the point A is
called the aphelion, a Greek term signifying _at a distance from the
sun_.
NOTE 50, p. 9. _Ninety degrees._ A circle is divided into 360 equal
parts, or degrees; each degree into 60 equal parts, called minutes; and
each minute into 60 equal parts, called seconds. It is usual to write
these quantities thus, 15° 16ʹ 10ʺ, which means fifteen degrees, sixteen
minutes, and ten seconds. It is clear that an arc m n, fig. 4, measures
the angle m C n; hence we may say, an arc of so many degrees, or an
angle of so many degrees; for, if there be ten degrees in the angle m C
n, there will be ten degrees in the arc m n. It is evident that there
are 90° in a right angle, m C d, or quadrant, since it is the fourth
part of 360°.
NOTE 51, p. 9. _Quadratures._ A celestial body is said to be in
quadrature when it is 90 degrees distant from the sun. For example, in
fig. 14, if d be the sun, S the earth, and p the moon, then the moon is
said to be in quadrature when she is in either of the points Q or D,
because the angles Q S d and D S d, which measure her apparent distance
from the sun, are right angles.
NOTE 52, p. 9. _Excentricity._ Deviation from circular form. In fig. 6,
C S is the excentricity of the orbit P Q A D. The less C S, the more
nearly does the orbit or ellipse approach the circular form; and, when C
S is zero, the ellipse becomes a circle.
NOTE 53, p. 9. _Inclination of an orbit._ Let S, fig. 12, be the centre
of the sun, P N A n the orbit of a planet moving from west to east in
the direction N p. Let E N m e n be the shadow or projection of the
orbit on the plane of the ecliptic, then N S n is the intersection of
these two planes, for the orbit rises above the plane of the ecliptic
towards N p, and sinks below it at N P. The angle p N m, which these two
planes make with one another, is the inclination of the orbit P N p A to
the plane of the ecliptic.
NOTE 54, p. 9. _Latitude of a planet._ The angle p S m, fig. 12, or the
height of the planet p above the ecliptic E N m. In this case the
latitude is north. Thus, celestial latitude is the angular distance of a
celestial body from the plane of the ecliptic, whereas terrestrial
latitude is the angular distance of a place on the surface of the earth
from the equator.
[Illustration: _Fig. 12._]
NOTE 55, p. 9. _Nodes._ The two points N and n, fig. 12, in which the
orbit N A n P of a planet or comet intersects the plane of the ecliptic
e N E n. The part N A n of the orbit lies above the plane of the
ecliptic, and the part n P N below it. The ascending node N is the point
through which the body passes in rising above the plane of the ecliptic,
and the descending node n is the point in which the body sinks below it.
The nodes of a satellite’s orbit are the points in which it intersects
the plane of the orbit of the planet.
NOTE 56, p. 10. _Distance from the sun._ S p in fig. 12. If ♈ be the
vernal equinox, then ♈ S p is the longitude of the planet p, m S p is
its latitude, and S p its distance from the sun. When these three
quantities are known, the place of the planet p is determined in space.
NOTE 57, pp. 10, 59. _Elements of an orbit._ Of these there are seven.
Let P N A n, fig. 12, be the elliptical orbit of a planet, C its centre,
S the sun in one of the foci, ♈ the point of Aries, and E N e n the
plane of the ecliptic. The elements are—the major axis A P; the
excentricity C S; the periodic time, that is, the time of a complete
revolution of the body in its orbit; and the fourth is the longitude of
the body at any given instant—for example, that at which it passes
through the perihelion P, the point of its orbit nearest to the sun.
That instant is assumed as the origin of time, whence all preceding and
succeeding periods are estimated. These four quantities are sufficient
to determine the form of the orbit, and the motion of the body in it.
Three other elements are requisite for determining the position of the
orbit in space. These are, the angle ♈ S P, the longitude of the
perihelion; the angle A N e, which is the inclination of the orbit to
the plane of the ecliptic; and, lastly, the angle ♈ S N, the longitude
of N the ascending node.
NOTE 58, p. 10. _Whose planes, &c._ The planes of the orbits, as P N A
n, fig. 12, in which the planets move, are inclined or make small angles
e N A with the plane of the ecliptic E N e n, and cut it in straight
lines, N S n passing through S, the centre of the sun.
NOTE 59, p. 11. _Momentum._ Force measured by the weight of a body and
its speed, or simple velocity, conjointly. The primitive momentum of the
planets is, therefore, the quantity of motion which was impressed upon
them when they were first thrown into space.
NOTE 60, p. 11. _Unstable equilibrium._ A body is said to be in
equilibrium when it is so balanced as to remain at rest. But there are
two kinds of equilibrium, _stable_ and _unstable_. If a body balanced in
stable equilibrium be slightly disturbed, it will endeavour to return to
rest by a number of movements to and fro, which will continually
decrease till they cease altogether, and then the body will be restored
to its original state of repose. But, if the equilibrium be unstable,
these movements to and fro, or oscillations, will become greater and
greater till the equilibrium is destroyed.
NOTE 61, p. 14. _Retrograde._ Going backwards, as from east to west,
contrary to the motion of the planets.
NOTE 62, p. 14. _Parallel directions._ Such as never meet, though
prolonged ever so far.
[Illustration: _Fig. 13._]
[Illustration: _Fig. 14._]
NOTE 63, pp. 14, 16. _The whole force, &c._ Let S, fig. 13, be the sun,
N m n the plane of the ecliptic, p the disturbed planet moving in its
orbit n p N, and d the disturbing planet. Now, d attracts the sun and
the planet p with different intensities in the directions d S, d p: the
difference only of these forces disturbs the motion of p; it is
therefore called the disturbing force. But this whole disturbing force
may be regarded as equivalent to three forces, acting in the directions
p S, p T, and p m. The force acting in the radius vector p S, joining
the centres of the sun and planet, is called the _radial force_. It
sometimes draws the disturbed planet p from the sun, and sometimes
brings it nearer to him. The force which acts in the direction of the
tangent p T is called the _tangential force_. It disturbs the motion of
p in longitude, that is, it accelerates its motion in some parts of its
orbit and retards it in others, so that the radius vector S p does not
move over equal areas in equal times. (See note 26.) For example, in the
position of the bodies in fig. 14, it is evident that, in consequence of
the attraction of d, the planet p will have its motion accelerated from
Q to C, retarded from C to D, again accelerated from D to O, and lastly
retarded from O to Q. The disturbing body is here supposed to be at
rest, and the orbit circular; but, as both bodies are perpetually moving
with different velocities in ellipses, the perturbations or changes in
the motions of p are very numerous. Lastly, that part of the disturbing
force which acts in the direction of a line p m, fig. 13, at right
angles to the plane of the orbit N p n, may be called the perpendicular
force. It sometimes causes the body to approach nearer, and sometimes to
recede farther from, the plane of the ecliptic N m n, than it would
otherwise do. The action of the disturbing forces is admirably explained
in a work on gravitation, by Mr. Airy, the Astronomer Royal.
NOTE 64, pp. 16, 74. _Perihelion._ Fig. 10, P, the point of an orbit
nearest the sun.
NOTE 65, p. 16. _Aphelion._ Fig. 10, A, the point of an orbit farthest
from the sun.
NOTE 66, pp. 16, 17. In fig. 15 the central force is greater than the
exact law of gravity; therefore the curvature P p a is greater than P p
A the real ellipse; hence the planet p comes to the point a, called the
aphelion, sooner than if it moved in the orbit P p A, which makes the
line P S A advance to a. In fig. 16, on the contrary, the curvature P p
a is less than in the true ellipse, so that the planet p must move
through more than the arc P p A, or 180°, before it comes to the
aphelion a, which causes the greater axis P S A to recede to a.
[Illustration: _Fig. 15._]
[Illustration: _Fig. 16._]
NOTE 67, pp. 16, 17. _Motion of apsides._ Let P S A, fig. 17, be the
position of the elliptical orbit of a planet, at any time; then, by the
action of the disturbing forces, it successively takes the position Pʹ S
Aʹ, Pʺ S Aʺ, &c., till by this direct motion it has accomplished a
revolution, and then it begins again; so that the motion is perpetual.
[Illustration: _Fig. 17._]
NOTE 68, p. 17. _Sidereal revolution._ The consecutive return of an
object to the same star.
NOTE 69, p. 17. _Tropical revolution._ The consecutive return of an
object to the same tropic or equinox.
NOTE 70, p. 17. _The orbit only bulges, &c._ In fig. 18 the effect of
the variation in the excentricity is shown where P p A is the elliptical
orbit at any given instant; after a time it will take the form P pʹ A,
in consequence of the decrease in the excentricity C S; then the forms P
pʺ A, P pʹʹʹ A, &c., consecutively from the same cause; and, as the
major axis P A always retains the same length, the orbit approaches more
and more nearly to the circular form. But, after this has gone on for
some thousands of years, the orbit contracts again, and becomes more and
more elliptical.
[Illustration: _Fig. 18._]
NOTE 71, pp. 18, 19. _The ecliptic_ is the apparent path of the sun in
the heavens. See note 46.
NOTE 72, p. 18. _This force tends to pull, &c._ The force in question,
acting in the direction p m, fig. 13, pulls the planet p towards the
plane N m n, or pushes it farther above it, giving the planet a tendency
to move in an orbit above or below its undisturbed orbit N p n, which
alters the angle p N m, and makes the node N and the line of nodes N n
change their positions.
[Illustration: _Fig. 19._]
NOTE 73, p. 18. _Motion of the nodes._ Let S, fig. 19, be the sun; S N n
the plane of the ecliptic; P the disturbing body; and p a planet moving
in its orbit p n, of which p n is so small a part that it is represented
as a straight line. The plane S n p of this orbit cuts the plane of the
ecliptic in the straight line S n. Suppose the disturbing force begins
to act on p, so as to draw the planet into the arc p pʹ; then, instead
of moving in the orbit p n, it will tend to move in the orbit p pʹ nʹ,
whose plane cuts the ecliptic in the straight line S nʹ. If the
disturbing force acts again upon the body when at pʹ, so as to draw it
into the arc pʹ pʺ, the planet will now tend to move in the orbit pʹ pʺ
nʺ, whose plane cuts the ecliptic in the straight line S nʺ. The action
of the disturbing force on the planet when at pʺ will bring the node to
nʹʹʹ, and so on. In this manner the node goes backwards through the
successive points n, nʹ, nʺ, nʹʹʹ, &c., and the line of nodes S n has a
perpetual retrograde motion about S, the centre of the sun. The
disturbing force has been represented as acting at intervals for the
sake of illustration: in nature it is continuous, so that the motion of
the node is continuous also; though it is sometimes rapid and sometimes
slow, now retrograde and now direct; but, on the whole, the motion is
slowly retrograde.
NOTE 74, p. 18. _When the disturbing planet_ is anywhere in the line S
N, fig. 19, or in its prolongation, it is in the same plane with the
disturbed planet; and, however much it may affect its motions in that
plane, it can have no tendency to draw it out of it. But when the
disturbing planet is in P, at right angles to the line S N, and not in
the plane of the orbit, it has a powerful effect on the motion of the
nodes: between these two positions there is great variety of action.
NOTE 75, p. 19. _The changes in the inclination_ are extremely minute
when compared with the motion of the node, as evidently appears from
fig. 19, where the angles n p nʹ, nʹ pʹ nʺ, &c., are much smaller than
the corresponding angles n S nʹ, S nʺ, &c.
NOTE 76, p. 20. _Sines and cosines._ Figure 4 is a circle; n p is the
sine, and C p is the cosine of an arc m n. Suppose the radius C m to
begin to revolve at m, in the direction m n a; then at the point m the
sine is zero, and the cosine is equal to the radius C m. As the line C m
revolves and takes the successive positions C n, C a, C b, &c., the
sines n p, a q, b r, &c., of the arcs m n, m a, m h, &c., increase,
while the corresponding cosines C p, C q, C r, &c., decrease; and when
the revolving radius takes the position C d, at right angles to the
diameter g m, the sine becomes equal to the radius C d, and the cosine
is zero. After passing the point d, the contrary happens; for the sines
e K, l V, &c., diminish, and the cosines C K, C V, &c., go on
increasing, till at g the sine is zero, and the cosine is equal to the
radius C g. The same alternation takes place through the remaining parts
g h, h m, of the circle, so that a sine or cosine never can exceed the
radius. As the rotation of the earth is invariable, each point of its
surface passes through a complete circle, or 360 degrees, in twenty-four
hours, at a rate of 15 degrees in an hour. Time, therefore, becomes a
measure of angular motion, and _vice versâ_, the arcs of a circle a
measure of time, since these two quantities vary simultaneously and
equably; and, as the sines and cosines of the arcs are expressed in
terms of the time, they vary with it. Therefore, however long the time
may be, and how often soever the radius may revolve round the circle,
the sines and cosines never can exceed the radius; and, as the radius is
assumed to be equal to unity, their values oscillate between unity and
zero.
NOTE 77, p. 20. The small excentricities and inclinations of the
planetary orbits, and the revolutions of all the bodies in the same
direction, were proved by Euler, La Grange, and La Place, to be
conditions necessary for the stability of the solar system.
Subsequently, however, the periodicity of the terms of the series
expressing the perturbations was supposed to be sufficient _alone_, but
M. Poisson has shown that to be a mistake; that these three conditions
are requisite for the necessary convergence of the series, and that
therefore the stability of the system depends on them _conjointly_ with
the periodicity of the sines and cosines of each term. The author is
aware that this note can only be intelligible to the analyst, but she is
desirous of correcting an error, and the more so as the conditions of
stability afford one of the most striking instances of design in the
original construction of our system, and of the foresight and supreme
wisdom of the Divine Architect.
NOTE 78, p. 22. _Resisting medium._ A fluid which resists the motions of
bodies, such as atmospheric air, or the highly elastic fluid called
ether, with which space is filled.
NOTE 79, p. 23. _Obliquity of the ecliptic._ The angle e ♈ q, fig. 11,
between the plane of the terrestrial equator q ♈ Q, and the plane of the
ecliptic E ♈ e. The obliquity is variable.
NOTE 80, p. 23. _Invariable plane._ In the earth the equator is the
invariable plane which nearly maintains a parallel position with regard
to itself while revolving about the sun, as in fig. 20, where E Q
represents it. The two hemispheres balance one another on each side of
this plane, and would still do so if all the particles of which they
consist were moveable among themselves, provided the earth were not
disturbed by the action of the sun and moon, which alters the
parallelism of the equator by the small variation called nutation, to be
explained hereafter.
[Illustration: _Fig. 20._]
[Illustration: _Fig. 21._]
NOTE 81, p. 24. _If each particle, &c._ Let P, Pʹ, Pʺ, &c., fig. 21, be
planets moving in their orbits about the centre of gravity of the
system. Let P S M, Pʹ S Mʹ, &c., be portions of these orbits moved over
by the radii vectores S P, S Pʹ, &c., in a given time, and let p S m, pʹ
S mʹ, &c., be their shadows or projections on the invariable plane.
Then, if the numbers which represent the masses of the planets P, Pʹ,
&c., be respectively multiplied by the numbers representing the areas or
spaces p S m, pʹ S mʹ, &c., the sum of the whole will be greater for the
invariable plane than it would be for any plane that could pass through
S, the centre of gravity of the system.
NOTE 82, p. 24. _The centre of gravity_ of the solar system lies within
the body of the sun, because his mass is much greater than the masses of
all the planets and satellites added together.
NOTE 83, pp. 25, 36. _Conjunction._ A planet is said to be in
conjunction when it has the same longitude with the sun, and in
opposition when its longitude differs from that of the sun by 180
degrees. Thus two bodies are said to be in conjunction when they are
seen exactly in the same part of the heavens, and in opposition when
diametrically opposite to one another. Mercury and Venus, which are
nearer to the sun than the earth, are called inferior planets; while all
the others, being farther from the sun than the earth, are said to be
superior planets. Suppose the earth to be at E, fig. 24; then a superior
planet will be in conjunction with the sun at C, and in opposition to
him when at O. Again, suppose the earth to be in O, then an inferior
planet will be in conjunction when at E, and in opposition when at F.
NOTE 84, p. 26. _The periodic inequalities_ are computed for a given
time; and consequently for a given form and position of the orbits of
the disturbed and disturbing bodies. Although the elements of the orbits
vary so slowly that no sensible effect is produced on inequalities of a
short period, yet, in the course of time, the secular variations of the
elements change the forms and relative positions of the orbits so much,
that Jupiter and Saturn, which would have come to the same relative
positions with regard to the sun and to one another after 850 years, do
not arrive at the same relative positions till after 918 years.
NOTE 85, p. 26. _Configuration._ The relative position of the planets
with regard to one another, to the sun, and to the plane of the
ecliptic.
NOTE 86, p. 27. In the same manner that the excentricity of an
elliptical orbit may be increased or diminished by the action of the
disturbing forces, so a circular orbit may acquire less or more
ellipticity from the same cause. It is thus that the forms of the orbits
of the first and second satellites of Jupiter oscillate between circles
and ellipses differing very little from circles.
[Illustration: _Fig. 22._]
NOTE 87, p. 28. _The plane of Jupiter’s equator_ is the imaginary plane
passing through his centre at right angles to his axis of rotation, and
corresponds to the plane q E Q e, in fig. 1. The satellites move very
nearly in the plane of Jupiter’s equator; for, if J be Jupiter, fig. 22,
P p his axis of rotation, e Q his equatorial diameter, which is 6000
miles longer than P p, and if J O and J E be the planes of his orbit and
equator seen edgewise, then the orbits of his four satellites seen
edgewise will have the positions J1, J2, J3, J4. These are extremely
near to one another, for the angle E J O is only 3° 5ʹ 30ʺ.
NOTE 88, p. 28. In consequence of the satellites moving so nearly in the
plane of Jupiter’s equator, when seen from the earth, they appear to be
always very nearly in a straight line, however much they may change
their positions with regard to one another and to their primary. For
example, on the evenings of the 3rd, 4th, 5th, and 6th of January, 1835,
the satellites had the configurations given in fig. 23, where O is
Jupiter, and 1, 2, 3, 4, are the first, second, third, and fourth
satellites. The satellite is supposed to be moving in a direction from
the figure towards the point. On the sixth evening the second satellite
was seen on the disc of the planet.
[Illustration: _Fig. 23._]
NOTE 89, p. 28. _Angular motion or velocity_ is the swiftness with which
a body revolves—a sling, for example; or the speed with which the
surface of the earth performs its daily rotation about its axis.
NOTE 90, p. 29. _Displacement of Jupiter’s orbit._ The action of the
planets occasions secular variations in the position of Jupiter’s orbit
J O, fig. 22, without affecting the plane of his equator J E. Again, the
sun and satellites themselves, by attracting the protuberant matter at
Jupiter’s equator, change the position of the plane J E without
affecting J O. Both of these cause perturbations in the motions of the
satellites.
NOTE 91, p. 29. _Precession_, with regard to Jupiter, is a retrograde
motion of the point where the lines J O, J E, intersect fig. 22.
NOTE 92, p. 30. _Synodic motion of a satellite._ Its motion during the
interval between two of its consecutive eclipses.
[Illustration: _Fig. 24._]
NOTE 93, p. 30. _Opposition._ A body is said to be in opposition when
its longitude differs from that of the sun by 180°. If S, fig. 24, be
the sun, and E the earth, then Jupiter is in opposition when at O, and
in conjunction when at C. In these positions the three bodies are in the
same straight line.
NOTE 94, p. 30. _Eclipses of the satellites._ Let S, fig. 25, be the
sun, J Jupiter, and a B b his shadow. Let the earth be moving in its
orbit, in the direction E A R T H, and the third satellite in the
direction a b m n. When the earth is at E, the satellite, in moving
through the arc a b, will vanish at a, and reappear at b, on the same
side of Jupiter. If the earth be in R, Jupiter will be in opposition;
and then the satellite, in moving through the arc a b, will vanish close
to the disc of the planet, and will reappear on the other side of it.
But, if the satellite be moving through the arc m n, it will appear to
pass over the disc, and eclipse the planet.
[Illustration: _Fig. 25._]
NOTE 95, pp. 30, 43. _Meridian._ A terrestrial meridian is a line
passing round the earth and through both poles. In every part of it noon
happens at the same instant. In figures 1 and 3, the lines N Q S and N G
S are meridians, C being the centre of the earth, and N S its axis of
rotation. The meridian passing through the Observatory at Greenwich is
assumed by the British as a fixed origin from whence terrestrial
longitudes are measured. And as each point on the surface of the earth
passes through 360°, or a complete circle, in twenty-four hours, at the
rate of 15° in an hour, time becomes a representative of angular motion.
Hence, if the eclipse of a satellite happens at any place at eight
o’clock in the evening, and the Nautical Almanac shows that the same
phenomenon will take place at Greenwich at nine, the place of
observation will be in the 15° of west longitude.
NOTE 96, p. 31. _Conjunction._ Let S be the sun, fig. 24, E the earth,
and J O Jʹ Cʹ the orbit of Jupiter. Then the eclipses which happen when
Jupiter is in O are seen 16^m 26^s sooner than those which take place
when the planet is in C. Jupiter is in conjunction when at C, and in
opposition when in O.
[Illustration: _Fig. 26._]
NOTE 97, p. 31. _In the diagonal, &c._ Were the line A S, fig. 26,
100,000 times longer than A B, Jupiter’s true place would be in the
direction A Sʹ, the diagonal of the figure A B Sʹ S, which is, of
course, out of proportion.
NOTE 98, p. 31. _Aberration of light._ The celestial bodies are so
distant that the rays of light coming from them may be reckoned
parallel. Therefore, let S A, Sʹ B, fig. 26, be two rays of light coming
from the sun, or a planet, to the earth moving in its orbit in the
direction A B. If a telescope be held in the direction A S, the ray S A,
instead of going down the tube, will impinge on its side, and be lost in
consequence of the telescope being carried with the earth in the
direction A B. But, if the tube be held in the position A E, so that A B
is to A S as the velocity of the earth to the velocity of light, the ray
will pass through Sʹ E A. The star appears to be in the direction A Sʹ,
when it really is in the direction A S; hence the angle S A Sʹ is the
angle of aberration.
NOTE 99, p. 32. _Density proportional to elasticity._ The more a fluid,
such as atmospheric air, is reduced in dimensions by pressure, the more
it resists the pressure.
NOTE 100, p. 32. _Oscillations of pendulum retarded._ If a clock be
carried from the pole to the equator, its rate will be gradually
diminished, that is, it will go slower and slower: because the
centrifugal force, which increases from the pole to the equator,
diminishes the force of gravity.
NOTE 101, p. 34. _Disturbing action._ The disturbing force acts here in
the very same manner as in note 63; only that the disturbing body d,
fig. 14, is the sun, S the earth, and p the moon.
NOTE 102, pp. 35, 36, 86. _Perigee._ A Greek word, signifying round the
earth. The perigee of the lunar orbit is the point P, fig. 6, where the
moon is nearest to the earth. It corresponds to the perihelion of a
planet. Sometimes the word is used to denote the point where the sun is
nearest to the earth.
NOTE 103, p. 35. _Evection._ The evection is produced by the action of
the radial force in the direction S p, fig. 14, which sometimes
increases and sometimes diminishes the earth’s attraction to the moon.
It produces a corresponding temporary change in the excentricity, which
varies with the position of the major axis of the lunar orbit in respect
of the line S d, joining the centres of the earth and sun.
NOTE 104, p. 35. _Variation._ The lunar perturbation called the
variation is the alternate acceleration and retardation of the moon in
longitude, from the action of the tangential force. She is accelerated
in going from quadratures in Q and D, fig. 14, to the points C and O,
called syzygies, and is retarded in going from the syzygies C and O to Q
and D again.
NOTE 105, p. 36. _Square of time._ If the times increase at the rate of
1, 2, 3, 4, &c., years or hundreds of years, the squares of the times
will be 1, 4, 9, 16, &c., years or hundreds of years.
NOTE 106, p. 37. In all investigations hitherto made with regard to the
acceleration, it was tacitly assumed that the areas described by the
radius vector of the moon were not permanently altered; that is to say,
that the tangential disturbing force produced no permanent effect. But
Mr. Adams has discovered that, in consequence of the constant decrease
in the excentricity of the earth’s orbit, there is a gradual change in
the central disturbing force which affects the aërial velocity, and
consequently it alters the amount of the acceleration by a very small
quantity, as well as the variation and other periodical inequalities of
the moon. On the latter, however, it has no permanent effect, because it
affects them in opposite directions in very moderate intervals of time,
whereas a very small error in the amount of the acceleration goes on
increasing as long as the excentricity of the earth’s orbit diminishes,
so that it would ultimately vitiate calculations of the moon’s place for
distant periods of time. This shows how complicated the moon’s motions
are, and what rigorous accuracy is required in their determination.
To give an idea of the labour requisite _merely_ to _perfect_ or
_correct_ the lunar tables, the moon’s place was determined by
observation at the Greenwich Observatory in 6000 different points of her
orbit, each of which was compared with the same points calculated from
Baron Plana’s formulæ, and to do that _sixteen computers_ were
constantly employed for _eight years_. Since the longitude is determined
by the motions of the moon, the lunar tables are of the greatest
importance.
NOTE 107, p. 37. _Mean anomaly._ The mean anomaly of a planet is its
angular distance from the perihelion, supposing it to move in a circle.
The true anomaly is its angular distance from the perihelion in its
elliptical orbit. For example, in fig. 10, the mean anomaly is P C m,
and the true anomaly is P S p.
NOTE 108, pp. 38, 68. _Many circumferences._ There are 360 degrees or
1,296,000 seconds in a circumference; and, as the acceleration of the
moon only increases at the rate of eleven seconds in a century, it must
be a prodigious number of ages before it accumulates to many
circumferences.
NOTE 109, p. 39. _Phases of the moon._ The periodical changes in the
enlightened part of her disc, from a crescent to a circle, depending
upon her position with regard to the sun and earth.
NOTE 110, p. 39. _Lunar eclipse._ Let S, fig. 27, be the sun, E the
earth, and m the moon. The space a A b is a section of the shadow, which
has the form of a cone or sugar-loaf, and the spaces A a c, A b d, are
the penumbra. The axis of the cone passes through A, and through E and
S, the centres of the sun and earth, and n m nʹ is the path of the moon
through the shadow.
[Illustration: _Fig. 27._]
NOTE 111, p. 39. _Apparent diameter._ The diameter of a celestial body
as seen from the earth.
NOTE 112, p. 40. _Penumbra._ The shadow or imperfect darkness which
precedes and follows an eclipse.
NOTE 113, p. 40. _Synodic revolution of the moon._ The time between two
consecutive new or full moons.
NOTE 114, p. 40. _Horizontal refraction._ The light, in coming from a
celestial object, is bent into a curve as soon as it enters our
atmosphere; and that bending is greatest when the object is in the
horizon.
[Illustration: _Fig. 28._]
NOTE 115, p. 40. _Solar eclipse._ Let S, fig. 28, be the sun, m the
moon, and E the earth. Then a E b is the moon’s shadow, which sometimes
eclipses a small portion of the earth’s surface at e, and sometimes
falls short of it. To a person at e, in the centre of the shadow, the
eclipse may be total or annular; to a person not in the centre of the
shadow a part of the sun will be eclipsed; and to one at the edge of the
shadow there will be no eclipse at all. The spaces P b E, Pʹ a E, are
the penumbra.
NOTE 116, p. 43. _From the extremities, &c._ If the length of the line a
b, fig. 29, be measured, in feet or fathoms, the angles S b a, S a b,
can be measured, and then the angle a S b is known, whence the length of
the line S C may be computed. a S b is the parallax of the object S; and
it is clear that, the greater the distance of S, the less the base a b
will appear, because the angle a Sʹ b is less than a S b.
[Illustration: _Fig. 29._]
NOTE 117, p. 44. _Every particle will describe a circle, &c._ If N S,
fig. 3, be the axis about which the body revolves, then particles at B,
Q, &c., will whirl in the circles B G A a, Q E q d, whose centres are in
the axis N S, and their planes parallel to one another. They are, in
fact, parallels of latitude, Q E q d being the equator.
NOTE 118, p. 44. _The force of gravity, &c._ Gravity at the equator acts
in the direction Q C, fig. 30. Whereas the direction of the centrifugal
force is exactly contrary, being in the direction C Q; hence the
difference of the two is the force called gravitation, which makes
bodies fall to the surface of the earth. At any point, m, not at the
equator, the direction of gravity is m b, perpendicular to the surface,
but the centrifugal force acts perpendicularly to N S, the axis of
rotation. Now the effect of the centrifugal force is the same as if it
were two forces, one of which acting in the direction b m, diminishes
the force of gravity, and another which, acting in the direction m t,
tangent to the surface at m, urges the particles towards Q, and tends to
swell out the earth at the equator.
[Illustration: _Fig. 30._]
NOTE 119, p. 45. _Homogeneous mass._ A quantity of matter, everywhere of
the same density.
NOTE 120, p. 45. _Ellipsoid of revolution._ A solid formed by the
revolution of an ellipse about its axis. If the ellipse revolve about
its minor axis Q D, fig. 6, the ellipsoid will be _oblate_, or flattened
at the poles like an orange. If the revolution be about the greater axis
A P, the ellipsoid will be prolate, like an egg.
NOTE 121, p. 45. _Concentric elliptical strata._ Strata, or layers,
having an elliptical form and the same centre.
NOTE 122, p. 46. _On the whole, &c._ The line N Q S q, fig. 1,
represents the ellipse in question, its major axis being Q q, its minor
axis N S.
NOTE 123, p. 46. _Increase in the length of the radii, &c._ The radii
gradually increase from the polar radius C N, fig. 30, which is least,
to the equatorial radius C Q, which is greatest. There is also an
increase in the lengths of the arcs corresponding to the same number of
degrees from the equator to the poles; for, the angle N C r being equal
to q C d, the elliptical arc N r is less than q d.
NOTE 124, p. 46. _Cosine of latitude._ The angles m C a, m C b, fig. 4,
being the latitudes of the points a, b, &c., the cosines are C q, C r,
&c.
NOTE 125, p. 47. _An arc of the meridian._ Let N Q S q, fig. 30, be the
meridian, and m n the arc to be measured. Then, if Zʹ m, Z n, be
verticals, or lines perpendicular to the surface of the earth, at the
extremities of the arc m n they will meet in p. Q a n, Q b m, are the
latitudes of the points m and n, and their difference is the angle m p
n. Since the latitudes are equal to the height of the pole of the
equinoctial above the horizon of the places m and n, the angle m p n may
be found by observation. When the distance m n is measured in feet or
fathoms, and divided by the number of degrees and parts of a degree
contained in the angle m p n, the length of an arc of one degree is
obtained.
NOTE 126, p. 47. _A series of triangles._ Let M Mʹ, fig. 31, be the
meridian of any place. A line A B is measured with rods, on level
ground, of any number of fathoms, C being some point seen from both ends
of it. As two of the angles of the triangle A B C can be measured, the
lengths of the sides A C, B C, can be computed; and if the angle m A B,
which the base A B makes with the meridian, be measured, the length of
the sides B m, A m, may be obtained by computation, so that A m, a small
part of the meridian, is determined. Again, if D be a point visible from
the extremities of the known line B C, two of the angles of the triangle
B C D may be measured, and the length of the sides C D, B D, computed.
Then, if the angle B m mʹ be measured, all the angles and the side B m
of the triangle B m mʹ are known, whence the length of the line m mʹ may
be computed, so that the portion A mʹ of the meridian is determined, and
in the same manner it may be prolonged indefinitely.
[Illustration: _Fig. 31._]
NOTE 127, pp. 47, 49. _The square of the sine of the latitude._ Q b m,
fig. 30, being the latitude of m, e m is the sine and b e the cosine.
Then the number expressing the length of e m, multiplied by itself, is
the square of the sine of the latitude; and the number expressing the
length of b e, multiplied by itself, is the square of the cosine of the
latitude.
NOTE 128, p. 48. The polar diameter of the earth determined by the
survey of Great Britain is 7900 miles; the equatorial is 7926, which
gives a compression of 1/299·33.
NOTE 129, p. 50. _A pendulum_ is that part of a clock which swings to
and fro.
[Illustration: _Fig. 32._]
NOTE 130, p. 52. _Parallax._ The angle a S b, fig. 29, under which we
view an object a b: it therefore diminishes as the distance increases.
The parallax of a celestial object is the angle which the radius of the
earth would be seen under, if viewed from that object. Let E, fig. 32,
be the centre of the earth, E H its radius, and m H O the horizon of an
observer at H. Then H m E is the parallax of a body m, the moon for
example. As m rises higher and higher in the heavens to the points mʹ,
mʺ, &c., the parallax H mʹ E, H mʺ E, &c., decreases. At Z, the zenith,
or point immediately above the head of the observer, it is zero; and at
m, where the body is in the horizon, the angle H m E is the greatest
possible, and is called the horizontal parallax. It is clear that with
regard to celestial bodies the whole effect of parallax is in the
vertical, or in the direction m mʹ Z; and as a person at H sees mʹ in
the direction H mʹ A, when it really is in the direction E mʹ B, it
makes celestial objects appear to be lower than they really are. The
distance of the moon from the earth has been determined from her
horizontal parallax. The angle E m H can be measured. E H m is a right
angle, and E H, the radius of the earth, is known in miles; whence the
distance of the moon E m is easily found. Annual parallax is the angle
under which the diameter of the earth’s orbit would be seen if viewed
from a star.
NOTE 131, p. 52. _The radii_ n B, n G, &c., fig. 3, are equal in any one
parallel of latitude, A a B G; therefore a change in the parallax
observed in that parallel can only arise from a change in the moon’s
distance from the earth; and when the moon is at her mean distance,
which is a constant quantity equal to half the major axis of her orbit,
a change in the parallax observed in different latitudes, G and E, must
arise from the difference in the lengths of the radii n G and C E.
NOTE 132, p. 52. _When Venus is in her nodes._ She must be in the line N
S n where her orbit P N A n cuts the plane of the ecliptic E N e n, fig.
12.
NOTE 133, p. 53. _The line described, &c._ Let E, fig. 33, be the earth,
S the centre of the sun, and V the planet Venus. The real transit of the
planet, seen from E the centre of the earth, would be in the direction A
B. A person at W would see it pass over the sun in the line v a, and a
person at O would see it move across him in the direction vʹ aʹ.
[Illustration: _Fig. 33._]
NOTE 134, p. 54. _Kepler’s law._ Suppose it were required to find the
distance of Jupiter from the sun. The periodic times of Jupiter and
Venus are given by observation, and the mean distance of Venus from the
centre of the sun is known in miles or terrestrial radii; therefore, by
the rule of three, the square root of the periodic time of Venus is to
the square root of the periodic time of Jupiter as the cube root of the
mean distance of Venus from the sun to the cube root of the mean
distance of Jupiter from the sun, which is thus obtained in miles or
terrestrial radii. The root of a number is that number which, once
multiplied by itself, gives its square; twice multiplied by itself,
gives its cube, &c. For example, twice 2 are 4, and twice 4 are 8; 2 is
therefore the square root of 4, and the cube root of 8. In the same
manner 3 times 3 are 9, and 3 times 9 are 27; 3 is therefore the square
root of 9, and the cube root of 27.
NOTE 135, p. 55. _Inversely, &c._ The quantities of matter in any two
primary planets are greater in proportion as the cubes of the numbers
representing the mean distances of their satellites are greater, and
also in proportion as the squares of their periodic times are less.
NOTE 136, p. 55. As hardly anything appears more impossible than that
man should have been able to weigh the sun as it were in scales and the
earth in a balance, the method of doing so may have some interest. The
attraction of the sun is to the attraction of the earth as the quantity
of matter in the sun to the quantity of matter in the earth; and, as the
force of this reciprocal attraction is measured by its effects, the
space the earth would fall through in a second by the sun’s attraction
is to the space which the sun would fall through by the earth’s
attraction as the mass of the sun to the mass of the earth. Hence, as
many times as the fall of the earth to the sun in a second exceeds the
fall of the sun to the earth in the same time, so many times does the
mass of the sun exceed the mass of the earth. Thus the weight of the sun
will be known if the length of these two spaces can be found in miles or
parts of a mile. Nothing can be easier. A heavy body falls through
16·0697 feet in a second at the surface of the earth by the earth’s
attraction; and, as the force of gravity is inversely as the square of
the distance, it is clear that 16·0697 feet are to the space a body
would fall through at the distance of the sun by the earth’s attraction,
as the square of the distance of the sun from the earth to the square of
the distance of the centre of the earth from its surface; that is, as
the square of 95,000,000 miles to the square of 4000 miles. And thus, by
a simple question in the rule of three, the space which the sun would
fall through in a second by the attraction of the earth may be found in
parts of a mile. The space the earth would fall through in a second, by
the attraction of the sun, must now be found in miles also. Suppose m n,
fig. 4, to be the arc which the earth describes round the sun in C, in a
second of time, by the joint action of the sun and the centrifugal
force. By the centrifugal force alone the earth would move from m to T
in a second, and by the sun’s attraction alone it would fall through T n
in the same time. Hence the length of T n, in miles, is the space the
earth would fall through in a second by the sun’s attraction. Now, as
the earth’s orbit is very nearly a circle, if 360 degrees be divided by
the number of seconds in a sidereal year of 365-1/4 days, it will give m
n, the arc which the earth moves through in a second, and then the
tables will give the length of the line C T in numbers corresponding to
that angle; but, as the radius C n is assumed to be unity in the tables,
if 1 be subtracted from the number representing C T, the length of T n
will be obtained; and, when multiplied by 95,000,000, to reduce it to
miles, the space which the earth falls through, by the sun’s attraction,
will be obtained in miles. By this simple process it is found that, if
the sun were placed in one scale of a balance, it would require 354,936
earths to form a counterpoise.
NOTE 137, p. 59. The sum of the greatest and least distances S P, S A,
fig. 12, is equal to P A, the major axis; and their difference is equal
to twice the excentricity C S. The longitude ♈ S P of the planet, when
in the point P, at its least distance from the sun, is the longitude of
the perihelion. The greatest height of the planet above the plane of the
ecliptic E N e n, is equal to the inclination of the orbit P N A n to
that plane. The longitude of the planet, when in the plane of the
ecliptic, can only be the longitude of one of the points N or n; and,
when one of these points is known, the other is given, being 180°
distant from it. Lastly, the time included between two consecutive
passages of the planet through the same node N or n, is its periodic
time, allowance being made for the recess of the node in the interval.
NOTE 138, p. 60. Suppose that it were required to find the position of a
point in space, as of a planet, and that one observation places it in n,
fig. 34, another observation places it in nʹ, another in nʺ, and so on;
all the points n, nʹ, nʺ, nʹʹʹ, &c., being very near to one another. The
true place of the planet P will not differ much from any of these
positions. It is evident, from this view of the subject, that P n, P nʹ,
P nʺ, &c., are the errors of observation. The true position of the
planet P is found by this property, that the squares of the numbers
representing the lines P n, P nʹ, &c., when added together, is the least
possible. Each line P n, P nʹ, &c., being the whole error in the place
of the planet, is made up of the errors of all the elements; and, when
compared with the errors obtained from theory, it affords the means of
finding each. The principle of least squares is of very general
application; its demonstration cannot find a place here; but the reader
is referred to Biot’s Astronomy, vol. ii. p. 203.
[Illustration: _Fig. 34._]
NOTE 139, p. 61. The true longitude of Uranus was in advance of the
tables previous to 1795, and continued to advance till 1822, after which
it diminished rapidly till 1830-1, when the observed and calculated
longitudes agreed, but then the planet fell behind the calculated place
so rapidly that it was clear the tables could no longer represent its
motion.
NOTE 140, p. 65. _An axis that, &c._ Fig. 20 represents the earth
revolving in its orbit about the sun in S, the axis of rotation P p
being everywhere parallel to itself.
NOTE 141, p. 65. _Angular velocities that are sensibly uniform._ The
earth and planets revolve about their axis with an equable motion, which
is never either faster or slower. For example, the length of the day is
never more nor less than twenty-four hours.
NOTE 142, p. 68. If fig. 1 be the moon, her polar diameter N S is the
shortest; and of those in the plane of the equator, Q E q, that which
points to the earth is greater than all the others.
NOTE 143, p. 73. _Inversely proportional, &c._ That is, the total amount
of solar radiation becomes less as the minor axis C Cʹ, fig. 20, of the
earth’s orbit becomes greater.
NOTE 144, p. 75. Fig. 35 represents the position of the apparent orbit
of the sun as it is at present, the earth being in E. The sun is nearer
to the earth in moving through ♎ P ♈ than in moving through ♈ A ♎, but
its motion through ♎ P ♈ is more rapid than its motion through ♈ A ♎;
and, as the swiftness of the motion and the quantity of heat received
vary in the same proportion, a compensation takes place.
[Illustration: _Fig. 35._]
NOTE 145, p. 76. _In an ellipsoid of revolution_, fig. 1, the polar
diameter N S, and every diameter in the equator q E Q e, are permanent
axes of rotation, but the rotation would be unstable about any other.
Were the earth to begin to rotate about C a, the angular distance from a
to the equator at q would no longer be ninety degrees, which would be
immediately detected by the change it would occasion in the latitudes.
NOTE 146, pp. 50, 80. Let q ♈ Q, and E ♎ e, fig. 11, be the planes of
the equator and ecliptic. The angle e ♈ Q, which separates them, called
the obliquity of the ecliptic, varies in consequence of the action of
the sun and moon upon the protuberant matter at the earth’s equator.
That action brings the point Q towards e, and tends to make the plane q
♈ Q coincide with the ecliptic E ♈ e, which causes the equinoctial
points ♈ and ♎ to move slowly backwards on the plane e ♈ E, at the rate
of 50ʺ·41 annually. This part of the motion, which depends upon the form
of the earth, is called luni-solar precession. Another part, totally
independent of the form of the earth, arises from the mutual action of
the earth, planets, and sun, which, altering the position of the plane
of the ecliptic e ♈ E, causes the equinoctial points ♈ and ♎ to advance
at the rate of Oʺ·31 annually; but, as this motion is much less than the
former, the equinoctial points recede on the plane of the ecliptic at
the rate of 50ʺ·1 annually. This motion is called the precession of the
equinoxes.
NOTE 147, p. 81. Let q ♈ Q, e ♈ E, fig. 36, be the planes of the
equinoctial or celestial equator and ecliptic, and p, P, their poles.
Then suppose p, the pole of the equator, to revolve with a tremulous or
wavy motion in the little ellipse p c d b in about 19 years, both
motions being very small, while the point a is carried round in the
circle a A B in 25,868 years. The tremulous motion may represent the
half-yearly variation, the motion in the ellipse gives an idea of the
nutation discovered by Bradley, and the motion in the circle a A B
arises from the precession of the equinoxes. The greater axis p d of the
small ellipse is 18ʺ·5, its minor axis b c is 13ʺ·74. These motions are
so small that they have very little effect on the parallelism of the
axis of the earth’s rotation during its revolution round the sun, as
represented in fig. 20. As the stars are fixed, this real motion in the
pole of the earth must cause an apparent change in their places.
[Illustration: _Fig. 36._]
[Illustration: figure: equidistant wires in an eye-piece]
NOTE 148, p. 83. By means of a transit instrument, which is a telescope
mounted so as to revolve only in the plane of the meridian, the instant
of the transit or passage of a celestial body across the meridian can be
determined. The transits of the principal stars are used to ascertain
the time, or, which is the same thing, the amount of the error of
clocks. A system of equidistant wires, as represented in the figure, is
placed in the focus of the eye-piece, so that the middle wire is
perpendicular and at right angles to the axis of the telescope. It
consequently represents a portion of the celestial meridian; and when a
star is seen to cross that wire it then crosses the celestial meridian
of the place of observation. A clock beating seconds being close at
hand, the duty of an observer is to note the exact second and part of a
second at which a star crosses each wire successively in consequence of
the rotation of the earth. Then the mean of all these observations will
give the time at which the star crosses the celestial meridian of the
place of observation to the tenth of a second, provided the observations
are accurate. Now it happens that the simultaneous impression on the eye
and ear is estimated differently by different observers, so that one
person will note the transit of a star, for example, as happening the
fraction of a second sooner or later than another person; and as that is
the case in every observation he makes, it is called his _personal
equation_, that is to say, it is a correction that must be applied to
all the observations of the individual, and a curious instance of
individuality it is. For instance, M. Otto Struve notes every
observation Oʺ·11 too soon, M. Peters Oʺ·13 too late; M. Struve noted
every observation one second later than M. Bessel, and M. Argelander
estimated the transit of a star 1ʺ·2 later than M. Bessel. All these
gentlemen were or are first-rate observers; and when the personal
equation is known it is easy to correct the observations. However, to
avoid that inconvenience Mr. Bond has introduced a method in the
Observatory at Cambridge in the United States in which touch is combined
with sight instead of hearing, which is now used also at Greenwich. The
observer at the moment of the observation presses his fingers on a
machine which by means of a galvanic battery conveys the impression to
where time is measured and marked, so that the observation is at once
recorded and the personal equation avoided.
NOTE 149, p. 84. _Let_ N be the pole, fig. 11, e E the ecliptic, and Q q
the equator. Then, N n m S being a meridian, and at right angles to the
equator, the arc ♈ m is less than the arc ♈ n.
NOTE 150, p. 85. _Heliacal rising of Sirius._ When the star appears in
the morning, in the horizon, a little before the rising of the sun.
NOTE 151, p. 87. Let P ♈ A ♎, fig. 35, be the apparent orbit or path of
the sun, the earth being in E. Its major axis, A P, is at present
situate as in the figure, where the solar perigee P is between the
solstice of winter and the equinox of spring. So that the time of the
sun’s passage through the arc ♈ A ♎ is greater than the time he takes to
go through the arc ♎ P ♈. The major axis A P coincided with ♎ ♈, the
line of the equinoxes, 4000 years before the Christian era; at that time
P was in the point ♈. In 6468 of the Christian era the perigee P will
coincide with ♎. In 1234 A.D. the major axis was perpendicular to ♈ ♎,
and then P was in the winter solstice.
NOTE 152, p. 88. _At the solstices, &c._ Since the declination of a
celestial object is its angular distance from the equinoctial, the
declination of the sun at the solstice is equal to the arc Q e, fig. 11,
which measures the obliquity of the ecliptic, or angular distance of the
plane ♈ e ♎ from the plane ♈ Q ♎.
NOTE 153, p. 88. _Zenith distance_ is the angular distance of a
celestial object from the point immediately over the head of an
observer.
NOTE 154, p. 89. _Reduced to the level of the sea._ The force of
gravitation decreases as the square of the height above the surface of
the earth increases, so that a pendulum vibrates slower on high ground;
and, in order to have a standard independent of local circumstances, it
is necessary to reduce it to the length that would exactly make 86,400
vibrations in a mean solar day at the level of the sea.
NOTE 155, p. 90. _A quadrant of the meridian_ is a fourth part of a
meridian, or an arc of a meridian containing 90°, as N Q, fig. 11.
NOTE 156, p. 93. _Moon’s southing._ The time when the moon is on the
meridian of any place, which happens about forty-eight minutes later
every day.
NOTE 157, p. 96. _The angular velocity of the earth’s rotation_ is at
the rate of 180° in twelve hours, which is the time included between the
passages of the moon at the upper and under meridian.
NOTE 158, p. 96. If S be the earth, fig. 14, d the sun, and C Q O D the
orbit of the moon, then C and O are the syzygies. When the moon is new,
she is at C, and when full she is at O; and, as both sun and moon are
then on the same meridian, it occasions the spring-tides, it being high
water at places under C and O, while it is low water at those under Q
and D. The neap-tides happen when the moon is in quadrature at Q or D,
for then she is distant from the sun by the angle d S Q, or d S D, each
of which is 90°.
NOTE 159, p. 97. _Declination._ If the earth be in C, fig. 11, and if q
♈ Q be the equinoctial, and N m S a meridian, then m C n is the
declination of a body at n. Therefore the cosine of that angle is the
cosine of the declination.
NOTE 160, pp. 99, 131. Fig 37 shows the propagation of waves from two
points C and Cʹ, where stones are supposed to have fallen. Those points
in which the waves cross each other are the places where they counteract
each other’s effects, so that the water is smooth there, while it is
agitated in the intermediate spaces.
NOTE 161, p. 100. _The centrifugal force may, &c._ The centrifugal force
acts in a direction at right angles to N S, the axis of rotation, fig.
30. Its effects are equivalent to two forces, one of which is in the
direction b m perpendicular to the surface Q m n of the earth, and
diminishes the force of gravity at m. The other acts in the direction of
the tangent m T, which makes the fluid particles tend towards the
equator.
[Illustration: _Fig. 37._]
NOTE 162, p. 106. _Analytical formula or expression._ A combination of
symbols or signs expressing or representing a series of calculation, and
including every particular case that can arise from a general law.
NOTE 163, p. 106. _Fig. 38 is a perfect octahedron._ Sometimes its
angles, A, X, a, a, &c., are truncated, or cut off. Sometimes a slice is
cut off its edges A a, X a, a a, &c. Occasionally both these
modifications take place.
[Illustration: _Fig. 38._]
NOTE 164, p. 107. Prismatic crystals of sulphate of nickel are somewhat
like fig. 62, only that they are thin, like a hair.
NOTE 165, p. 108. _Zinc_, a metal either found as an ore or mixed with
other metals. It is used in making brass.
NOTE 166, p. 108. _A cube_ is a solid contained by six plane square
surfaces, as fig. 39.
[Illustration: _Fig. 39._]
NOTE 167, p. 108. _A tetrahedron_ is a solid contained by four
triangular surfaces, as fig. 40: of this solid there are many varieties.
[Illustration: _Fig. 40._]
NOTE 168, p. 108. There are many varieties of the octahedron. In that
mentioned in the text, the base a a a a, fig. 38, is a square, but the
base may be a rhomb; this solid may also be elongated in the direction
of its axis A X, or it may be depressed.
NOTE 169, pp. 109, 192, 273. _A rhombohedron_ is a solid contained by
six plane surfaces, as in fig. 63, the opposite planes being equal and
similar rhombs parallel to one another; but all the planes are not
necessarily equal or similar, nor are its angles right angles. In
carbonate of lime the angle C A B is 105°·55, and the angle B or C is
75°·05.
NOTE 170, p. 109. _Sublimation._ Bodies raised into vapour which is
again condensed into a solid state.
NOTE 171, p. 112. _Platinum._ The heaviest of metals; its colour is
between that of silver and lead.
NOTE 172, p. 113. The surface of a column of water, or spirit of wine,
in a capillary tube, is hollow; and that of a column of quicksilver is
convex, or rounded, as in fig. 41.
NOTE 173, p. 113. _Inverse ratio, &c._ The elevation of the liquid is
greater in proportion as the internal diameter of the tube is less.
NOTE 174, p. 114. In fig. 41 the line c d shows the direction of the
resulting force in the two cases.
[Illustration: _Fig. 41._]
NOTE 175, p. 115. When two plates of glass are brought near to one
another in water, the liquid rises between them; and, if the plates
touch each other at one of their upright edges, the outline of the water
will become an hyperbola.
NOTE 176, p. 115. Let A Aʹ, fig. 42, be two plates, both of which are
wet, and B Bʹ two that are dry. When partly immersed in a liquid, its
surface will be curved close to them, but will be of its usual level for
the rest of the distance. At such a distance they will neither attract
nor repel one another. But, as soon as they are brought near enough to
have the whole of the liquid surface between them curved, as in a aʹ, b
bʹ, they will rush together. If one be wet and another dry, as C Cʹ,
they will repel one another at a certain distance; but, as soon as they
are brought very near, they will rush together, as in the former cases.
[Illustration: _Fig. 42._]
NOTE 177, p. 123. In a paper on the atmospheric changes that produce
rain and wind, by Thomas Hopkins, Esq., in the Geographical Journal, it
is shown that, when vapour is condensed and falls in rain, a partial
vacuum is formed, and that heavier air presses in as a current of wind.
Thus the vacuum arising from the great precipitation at the tropics
causes the polar winds to descend from the upper regions of the
atmosphere and blow along the surface to the equator as trade winds to
supply the place of the hot currents that are continually raising them
into the higher regions. This circumstance removes the only difficulty
in Lieutenant Maury’s theory of the winds.
NOTE 178, p. 134. _Latent or absorbed heat._ There is a certain quantity
of heat in all bodies, which cannot be detected by the thermometer, but
which may become sensible by compression.
NOTE 179, p. 137. _Reflected waves._ A series of waves of light, sound,
or water, diverge in all directions from their origin I, fig. 43, as
from a centre. When they meet with an obstacle S S, they strike against
it, and are reflected or turned back by it in the same form as if they
had proceeded from the centre C, at an equal distance on the other side
of the surface S S.
[Illustration: _Fig. 43._]
NOTE 180, p. 138. _Elliptical shell._ If fig. 6 be a section of an
elliptical shell, then all sounds coming from the focus S to different
points on the surface, as m, are reflected back to F, because the angle
T m S is equal to t m F. In a spherical hollow shell, a sound diverging
from the centre is reflected back to the centre again.
NOTE 181, p. 142. Fig. 44 represents musical strings in vibration; the
straight lines are the strings when at rest. The first figure of the
four would give the fundamental note, as, for example, the low C. The
second and third figures would give the first and second harmonics; that
is, the octave and the 12th above C, n n n being the points at rest; the
fourth figure shows the real motion when compounded of all three.
[Illustration: _Fig. 44._]
NOTE 182, p. 143. Fig. 45 represents sections of an open and of a shut
pipe, and of a pipe open at one end. When sounded, the air spontaneously
divides itself into segments. It remains at rest in the divisions or
nodes n nʹ, &c., but vibrates between them in the direction of the
arrow-heads. The undulations of the whole column of air give the
fundamental note, while the vibrations of the divisions give the
harmonics.
[Illustration: _Fig. 45._]
NOTE 183, p. 144. Fig. 1, plate 1, shows the vibrating surface when the
sand divides it into squares, and fig. 2 represents the same when the
nodal lines divide it into triangles. The portions marked a a are in
different states of vibration from those marked b b.
NOTE 184, p. 145. Plates 1 and 2 contain a few of Chladni’s figures. The
white lines are the forms assumed by the sand, from different modes of
vibration, corresponding to musical notes of different degrees of pitch.
Plate 3 contains six of Chladni’s circular figures.
NOTE 185, p. 145. Mr. Wheatstone’s principle is, that when vibrations
producing the forms of figs. 1 and 2, plate 3, are united in the same
surface, they make the sand assume the form of fig. 3. In the same
manner, the vibrations which would separately cause the sand to take the
forms of figs. 4 and 5, would make it assume the form in fig. 6 when
united. The figure 9 results from the modes of vibration of 7 and 8
combined. The parts marked a a are in different states of vibration from
those marked b b. Figs. 1, 2, and 3, plate 4, represent forms which the
sand takes in consequence of simple modes of vibration; 4 and 5 are
those arising from two combined modes of vibration; and the last six
figures arise from four superimposed simple modes of vibration. These
complicated figures are determined by computation independent of
experiment.
NOTE 186, p. 146. The long cross-lines of fig. 46 show the two systems
of nodal lines given by M. Savart’s laminæ.
[Illustration: _Fig. 46._]
NOTE 187, p. 146. The short lines on fig. 46 show the positions of the
nodal lines on the other sides of the same laminæ.
NOTE 188, p. 146. Fig. 47 gives the nodal lines on a cylinder, with the
paper rings that mark the quiescent points.
[Illustration: _Fig. 47._]
[Illustration: _Fig. 48._]
NOTE 189, pp. 138, 153, 156. _Reflection and Refraction._ Let P C p,
fig. 48, be perpendicular to a surface of glass or water A B. When a ray
of light, passing through the air, falls on this surface in any
direction I C, part of it is reflected in the direction C S, and the
other part is bent at C, and passes through the glass or water in the
direction C R. I C is called the incident ray, and I C P the angle of
incidence; C S is the reflected ray, and P C S the angle of reflection;
C R is the refracted ray, and p C R the angle of refraction. The plane
passing through S C and I C is the plane of reflection, and the plane
passing through I C and C R is the plane of refraction. In ordinary
cases, C I, C S, C R, are all in the same plane. We see the surface by
means of the reflected light, which would otherwise be invisible.
Whatever the reflecting surface may be, and however obliquely the light
may fall upon it, the angle of reflection is always equal to the angle
of incidence. Thus I C, Iʹ C, being rays incident on the surface at C,
they will be reflected into C S, C Sʹ, so that the angle S C P will be
equal to the angle I C P, and Sʹ C P equal to Iʹ C P. That is by no
means the case with the refracted rays. The incident rays I C, Iʹ C, are
bent at C towards the perpendicular, in the direction C R, C Rʹ; and the
law of refraction is such, that the sine of the angle of incidence has a
constant ratio to the sine of the angle of refraction; that is to say,
the number expressing the length of I m, the sine of I C P, divided by
the number expressing the length of R n, the sine of R C p, is the same
for all the rays of light that can fall upon the surface of any one
substance, and is called its index of refraction. Though the index of
refraction be the same for any one substance, it is not the same for all
substances. For water it is 1·336; for crown-glass it is 1·535; for
flint-glass, 1·6; for diamond, 2·487; and for chromate of lead it is 3,
which substance has a higher refractive power than any other known.
Light falling perpendicularly on a surface passes through it without
being refracted. If the light be now supposed to pass from a dense into
a rare medium, as from glass or water into air, then R C, Rʹ C, become
the incident rays; and in this case the refracted rays, C I, C Iʹ, are
bent from the perpendicular instead of towards it. When the incidence is
very oblique, as r C, the light never passes into the air at all, but it
is _totally_ reflected in the direction C rʹ, so that the angle p C r is
equal to p C rʹ; that frequently happens at the second surface of glass.
When a ray I C falls from air upon a piece of glass A B, it is in
general refracted at each surface. At C it is bent towards the
perpendicular, and at R from it, and the ray emerges parallel to I C;
but, when the ray is very oblique to the second surface, it is totally
reflected. An object seen by total reflection is nearly as vivid as when
seen by direct vision, because no part of the light is refracted. When
light falls upon a plate of crown-glass, at an angle of 4° 32ʹ counted
from the surface, the glass reflects 4 times more light than it
transmits. At an angle of 7° 1ʹ the reflected light is double of the
transmitted; at an angle of 11° 8ʹ the light reflected is equal to that
transmitted; at 17° 17ʹ the reflected is equal to 1/2 the transmitted
light; at 26° 38ʹ it is equal to 1/4, the variation, according to Arago,
being as the square of the cosine.
NOTE 189, p. 154. _Atmospheric refraction._ Let a b, a b, &c., fig. 49,
be strata, or extremely thin layers, of the atmosphere, which increase
in density towards m n, the surface of the earth. A ray coming from a
star meeting the surface of the atmosphere at S would be refracted at
the surface of each layer, and would consequently move in the curved
line S v v v A; and as an object is seen in the direction of the ray
that meets the eye, the star, which really is in the direction A S,
would seem to a person at A to be in s. So that refraction, which always
acts in a vertical direction, raises objects above their true place. For
that reason, a body at Sʹ, below the horizon H A O, would be raised, and
would be seen in sʹ. The sun is frequently visible by refraction after
he is set, or before he is risen. There is no refraction in the zenith
at Z. It increases all the way to the horizon, where it is greatest, the
variation being proportional to the tangent of the angles Z A S, Z A Sʹ,
the distances of the bodies S Sʹ from the zenith. The more obliquely the
rays fall, the greater the refraction.
[Illustration: _Fig. 49._]
[Illustration: _Fig. 50._]
NOTE 190, p. 154. _Bradley’s method of ascertaining the amount of
refraction._ Let Z, fig. 50, be the zenith or point immediately above an
observer at A; let H O be his horizon, and P the pole of the equinoctial
A Q. Hence P A Q is a right angle. A star as near to the pole as s would
appear to revolve about it, in consequence of the rotation of the earth.
At noon, for example, it would be at s above the pole, and at midnight
it would be in sʹ below it. The sum of the true zenith distances, Z A s,
Z A sʹ, is equal to twice the angle Z A P. Again, S and Sʹ being the sun
at his greatest distances from the equinoctial A Q when in the
solstices, the sum of his true zenith distances, Z A S, Z A Sʹ, is equal
to twice the angle Z A Q. Consequently, the four true zenith distances,
when added together, are equal to twice the right angle Q A P; that is,
they are equal to 180°. But the observed or apparent zenith distances
are less than the true on account of refraction; therefore the sum of
the four apparent zenith distances is less than 180° by the whole amount
of the four refractions.
NOTE 191, p. 155. _Terrestrial refraction._ Let C, fig. 51, be the
centre of the earth, A an observer at its surface, A H his horizon, and
B some distant point, as the top of a hill. Let the arc B A be the path
of a ray coming from B to A; E B, E A, tangents to its extremities; and
A G, B F, perpendicular to C B. However high the hill B may be, it is
nothing when compared with C A, the radius of the earth; consequently, A
B differs so little from A D that the angles A E B and A C B are
supplementary to one another; that is, the two taken together are equal
to 180°. A C B is called the horizontal angle. Now B A H is the real
height of B, and E A H its apparent height; hence refraction raises the
object B, by the angle E A B, above its real place. Again, the real
depression of A, when viewed from B, is F B A, whereas its apparent
depression is F B E, so E B A is due to refraction. The angle F B A is
equal to the sum of the angles B A H and A C B; that is, the true
elevation is equal to the true depression and the horizontal angle. But
the true elevation is equal to the apparent elevation diminished by the
refraction; and the true depression is equal to the apparent depression
increased by refraction. Hence twice the refraction is equal to the
horizontal angle augmented by the difference between the apparent
elevation and the apparent depression.
[Illustration: _Fig. 51._]
NOTE 192, p. 155. Fig. 52 represents the phenomenon in question. S P is
the real ship, with its inverted and direct images seen in the air. Were
there no refraction, the rays would come from the ship S P to the eye E
in the direction of the straight lines; but, on account of the variable
density of the inferior strata of the atmosphere, the rays are bent in
the curved lines P c E, P d E, S m E, S n E. Since an object is seen in
the direction of the tangent to that point of the ray which meets the
eye, the point P of the real ship is seen at p and pʹ, and the point S
seems to be in s and sʹ; and, as all the other points are transferred in
the same manner, direct and inverted images of the ship are formed in
the air above it.
[Illustration: _Fig. 52._]
NOTE 193, p. 156. Fig. 53 represents the section of a poker, with the
refraction produced by the hot air surrounding it.
[Illustration: _Fig. 53._]
NOTE 194, p. 156. _The solar spectrum._ A ray from the sun at S, fig.
54, admitted into a dark room, through a small round hole H in a
window-shutter, proceeds in a straight line to a screen D, on which it
forms a bright circular spot of white light, of nearly the same diameter
with the hole H. But when the refracting angle B A C of a glass prism is
interposed, so that the sunbeam falls on A C the first surface of the
prism, and emerges from the second surface A B at equal angles, it
causes the rays to deviate from the straight path S D, and bends them to
the screen M N, where they form a coloured image V R of the sun, of the
same breadth with the diameter of the hole H, but much longer. The space
V R consists of seven colours—violet, indigo, blue, green, yellow,
orange, and red. The violet and red, being the most and least
refrangible rays, are at the extremities, and the green occupy the
middle part at G. The angle D g G is called the mean _deviation_, and
the spreading of the coloured rays over the angle V g R the
_dispersion_. The deviation and dispersion vary with the refracting
angle B A C of the prism, and with the substance of which it is made.
[Illustration: _Fig. 54._]
NOTE 195, pp. 159, 164. Under the same circumstances, and where the
refracting angles of the two prisms are equal, the angles D g G and V g
R, fig. 54, are greater for flint-glass than for crown-glass. But, as
they vary with the angle of the prism, it is only necessary to augment
the refracting angle of the crown-glass prism by a certain quantity, to
produce nearly the same deviation and dispersion with the flint-glass
prism. Hence, when the two prisms are placed with their refracting
angles in opposite directions, as in fig. 54, they nearly neutralize
each other’s effects, and refract a beam of light without resolving it
into its elementary coloured rays. Sir David Brewster has come to the
conclusion that there may be refraction without colour by means of two
prisms, or two lenses, when properly adjusted, even though they be made
of the same kind of glass.
NOTE 196, p. 165. The object glass of the achromatic telescope consists
of a convex lens A B, fig. 55, of crown-glass placed on the outside,
towards the object, and of a concave-convex lens C D of flint-glass,
placed towards the eye. The focal length of a lens is the distance of
its centre from the point in which the rays converge, as F, fig. 60. If,
then, the lenses A B and C D be so constructed that their focal lengths
are in the same proportion as their dispersive powers, they will refract
rays of light without colour.
[Illustration: _Fig. 55._]
NOTE 197, p. 165. If the mean refracting angle of the prism D g G, fig.
54, were the same for all substances, then the difference D g V - D g R
would be the dispersion. But the angle of the prism being the same, all
these angles are different in each substance, so that in order to obtain
the dispersion of any substance the angle D g V - D g R must be divided
by the angle D g G or its excess above unity, to which the mean
refraction is always proportional. According to Mr. Fraunhofer the
refraction of the extreme violet and red rays in crown-glass is 1·5466
and 1·5258; so D g V - D g R = 1·5466 - 1·5258 = ·0208, and half the sum
of the excess of each above unity is = ·5362; consequently
(D g V - D g R)/D g G = ·0208/·5362 = 0·03879; for diamond
(D g V - D g R)/D g G = (2·467 - 2·411)/1·439 = 0·0389;
so that the dispersive power of diamond is a little less than that of
crown-glass; hence the splendid refracted colours which distinguish
diamond from every other precious stone are not owing to its high
dispersive power, but to its great mean refraction.—SIR DAVID BREWSTER.
NOTE 198, p. 168. When a sunbeam, after having passed through a coloured
glass V Vʹ, fig. 56, enters a dark room by two small slits O Oʹ in a
card, or piece of tin, they produce alternate bright and black bands on
a screen S Sʹ at a little distance. When either one or other of the
slits O or Oʹ is stopped, the dark bands vanish, and the screen is
illuminated by a uniform light, proving that the dark bands are produced
by the interference of the two sets of rays. Again, let H m, fig. 57, be
a beam of white light passing through a hole at H, made with a fine
needle in a piece of lead or a card, and received on a screen S Sʹ. When
a hair, or a small slip of card h hʹ, about the 30th of an inch in
breadth, is held in the beam, the rays bend round on each side of it,
and, arriving at the screen in different states of vibration, interfere
and form a series of coloured fringes on each side of a central white
band m. When a piece of card is interposed at C, so as to intercept the
light which passes on one side of the hair, the coloured fringes vanish.
When homogeneous light is used, the fringes are broadest in red, and
become narrower for each colour of the spectrum progressively to the
violet, which gives the narrowest and most crowded fringes. These very
elegant experiments are due to Dr. Thomas Young.
[Illustration: _Fig. 56._]
[Illustration: _Fig. 57._]
[Illustration: _Fig. 58._]
NOTE 199, pp. 171, 200. Fig. 58 shows Newton’s rings, of which there are
seven, formed by screwing two lenses of glass together. Provided the
incident light be white, they always succeed each other in the following
order:—
1st ring, or 1st order of colours: Black, very faint blue, brilliant
white, yellow, orange, red.
2nd ring: Dark purple, or rather violet, blue, a very imperfect yellow
green, vivid yellow, crimson red.
3rd ring: Purple, blue, rich grass green, fine yellow, pink, crimson.
4th ring: Dull blueish green, pale yellowish pink, red.
5th ring: Pale blueish green, white, pink.
6th ring: Pale blue green, pale pink.
7th ring: Very pale blueish green, very pale pink.
After the seventh order the colours become too faint to be
distinguished. The rings decrease in breadth, and the colours become
more crowded together, as they recede from the centre. When the light is
homogeneous, the rings are broadest in the red, and decrease in breadth
with every successive colour of the spectrum to the violet.
NOTE 200, p. 172. The absolute thickness of the film of air between the
glasses is found as follows:—Let A F B C, fig. 59, be the section of a
lens lying on a plane surface or plate of glass P Pʹ, seen edgewise, and
let E C be the diameter of the sphere of which the lens is a segment. If
A B be the diameter of any one of Newton’s rings, and B D parallel to C
E, then B D or C F is the thickness of the air producing it. E C is a
known quantity; and when A B, the diameter, is measured with compasses,
B D or F C can be computed. Newton found that the length of B D,
corresponding to the darkest part of the first ring, is the 98,000th
part of an inch when the rays fall perpendicularly on the lens, and from
this he deduced the thickness corresponding to each colour in the system
of rings. By passing each colour of the solar spectrum in succession
over the lenses, Newton also determined the thickness of the film of air
corresponding to each colour, from the breadth of the rings, which are
always of the same colour with the homogeneous light.
[Illustration: _Fig. 59._]
NOTE 201, p. 174. The focal length or distance of a lens is the distance
from its centre to the point F, fig. 60, in which the refracted rays
meet. Let L Lʹ be a lens of very short focal distance fixed in the
window-shutter of a dark room. A sunbeam S L Lʹ passing through the lens
will be brought to a focus in F, whence it will diverge in lines F C, F
D, and will form a circular image of light on the opposite wall. Suppose
a sheet of lead, having a small pin-hole pierced through it, to be
placed in this beam; when the pin-hole is viewed from behind with a lens
at E, it is surrounded with a series of coloured rings, which vary in
appearance with the relative positions of the pin-hole and eye with
regard to the point F. When the hole is the 30th of an inch in diameter
and at the distance of 6-1/2 feet from F, when viewed at the distance of
24 inches, there are seven rings of the following colours:—
1st order: White, pale yellow, yellow, orange, dull red.
2nd order: Violet, blue, whitish, greenish yellow, fine yellow, orange
red.
3rd order: Purple, indigo blue, greenish blue, brilliant green, yellow
green, red.
4th order: Blueish green, blueish white, red.
5th order: Dull green, faint blueish white, faint red.
6th order: Very faint green, very faint red.
7th order: A trace of green and red.
[Illustration: _Fig. 60._]
[Illustration: _Fig. 61._]
[Illustration: _Fig. 62._]
NOTE 202, p. 175. Let L Lʹ, fig. 61, be the section of a lens placed in
a window-shutter, through which a very small beam of light S L Lʹ passes
into a dark room, and comes to a focus in F. If the edge of a knife K N
be held in the beam, the rays bend away from it in hyperbolic curves K
r, K rʹ, &c., instead of coming directly to the screen in the straight
line K E, which is the boundary of the shadow. As these bending rays
arrive at the screen in different states of undulation, they interfere,
and form a series of coloured fringes, r rʹ, &c., along the edge of the
shadow K E S N of the knife. The fringes vary in breadth with the
relative distances of the knife-edge and screen from F.
NOTE 203, p. 177. Fig. 43 represents the phenomena in question, where S
S is the surface, and I the centre of incident waves. The reflected
waves are the dark lines returning towards I, which are the same as if
they had originated in C on the other side of the surface.
NOTE 204, p. 180. Fig. 62 represents a prismatic crystal of tourmaline,
whose axis is A X. The slices that are used for polarising light are cut
parallel to A X.
NOTE 205, p. 181. _Double refraction._ If a pencil of light R r, fig.
63, falls upon a rhombohedron of Iceland spar A B X C, it is separated
into two equal pencils of light at r, which are refracted in the
directions r O, r E: when these arrive at O and E they are again
refracted, and pass into the air in the directions O o, E o, parallel to
one another and to the incident ray R r. The ray r O is refracted
according to the ordinary law, which is, that the sines of the angles of
incidence and refraction bear a constant ratio to one another (see Note
184), and the rays R r, r O, O o, are all in the same plane. The pencil
r E, on the contrary, is bent aside out of that plane, and its
refraction does not follow the constant ratio of the sines; r E is
therefore called the extraordinary ray, and r O the ordinary ray. In
consequence of this bisection of the light, a spot of ink at O is seen
double at O and E, when viewed from r I; and when the crystal is turned
round, the image E revolves about O, which remains stationary.
[Illustration: _Fig. 63._]
NOTE 206, p. 182. Both of the parallel rays O o and E o, fig. 63, are
polarised on leaving the doubly refracting crystal, and in both the
particles of light make their vibrations at right angles to the lines O
o, E o. In the one, however, these vibrations lie, for example, in the
plane of the horizon, while the vibrations of the other lie in the
vertical plane perpendicular to the horizon.
NOTE 207, p. 183. If light be made to fall in various directions on the
natural faces of a crystal of Iceland spar, or on faces cut and polished
artificially, one direction A X, fig. 63, will be found, along which the
light passes without being separated into two pencils. A X is the optic
axis. In some substances there are two optic axes forming an angle with
each other. The optic axis is not a fixed line, it only has a fixed
direction; for if a crystal of Iceland spar be divided into smaller
crystals, each will have its optic axis; but if all these pieces be put
together again, their optic axes will be parallel to A X. Every line,
therefore, within the crystal parallel to A X is an optic axis; but as
these lines have all the same direction, the crystal is still said to
have but one optic axis.
NOTE 208, p. 184. If I C, fig. 48, be the incident and C S the reflected
rays, then the particles of polarised light make their vibrations at
right angles to the plane of the paper.
NOTE 209, p. 184. Let A A, fig. 48, be the surface of the reflector, I C
the incident and C S the reflected rays; then, when the angle S C B is
57°, and consequently the angle P C S equal to 33°, the black spot will
be seen at C by an eye at S.
NOTE 210, p. 185. Let A B, fig. 48, be a reflecting surface, I C the
incident and C S the reflected rays; then, if the surface be
plate-glass, the angle S C B must be 57°, in order that C S may be
polarised. If the surface be crown-glass or water, the angle S C B must
be 56° 55ʹ for the first, and 53° 11ʹ for the second, in order to give a
polarised ray.
NOTE 211, p. 186. A polarising apparatus is represented in fig. 64,
where R r is a ray of light falling on a piece of glass r at an angle of
57°: the reflected ray r s is then polarised, and may be viewed through
a piece of tourmaline in s, or it may be received on another plate of
glass, B, whose surface is at right angles to the surface of r. The ray
r s is again reflected in s, and comes to the eye in the direction s E.
The plate of mica, M I, or of any substance that is to be examined, is
placed between the points r and s.
[Illustration: _Fig. 64._]
NOTE 212, p. 187. In order to see these figures, the polarised ray r s,
fig. 64, must pass through the optic axis of the crystal, which must be
held as near as possible to s on one side, and the eye placed as near as
possible to s on the other. Fig. 65 shows the image formed by a crystal
of Iceland spar which has one optic axis. The colours in the rings are
exactly the same with those of Newton’s rings given in Note 199, and the
cross is black. If the spar be turned round its axis, the rings suffer
no change; but if the tourmaline through which it is viewed, or the
plate of glass, B, be turned round, this figure will be seen at the
angles 0°, 90°, 180°, and 270° of its revolution. But in the
intermediate points, that is, at the angles 45°, 135°, 225°, and 315°,
another system will appear, such as represented in fig. 66, where all
the colours of the rings are complementary to those of fig. 65, and the
cross is white. The two systems of rings, if superposed, would produce
white light.
[Illustration: _Fig. 65._]
[Illustration: _Fig. 66._]
NOTE 213, p. 188. Saltpetre, or nitre, crystallises in six-sided prisms
having two optic axes inclined to one another at an angle of 5°. A slice
of this substance about the 6th or 8th of an inch thick, cut
perpendicularly to the axis of the prism, and placed very near to s,
fig. 64, so that the polarised ray r s may pass through it, exhibits the
system of rings represented in fig. 67, where the points C and C mark
the position of the optic axes. When the plate B, fig. 64, is turned
round, the image changes successively to those given in figs. 68, 69,
and 70. The colours of the rings are the same with those of thin plates,
but they vary with the thickness of the nitre. Their breadth enlarges or
diminishes also with the colour, when homogeneous light is used.
[Illustration: _Fig. 67._]
[Illustration: _Fig. 68._]
[Illustration: _Fig. 69._]
[Illustration: _Fig. 70._]
[Illustration: _Fig. 71._]
NOTE 214, p. 189. Fig. 71 represents the appearance produced by placing
a slice of rock crystal in the polarised ray r s, fig. 64. The uniform
colour in the interior of the image depends upon the thickness of the
slice; but whatever that colour may be, it will alternately attain a
maximum brightness and vanish with the revolution of the glass B. It may
be observed, that the two kinds of quartz, or rock crystal, mentioned in
the text, are combined in the amethyst, which consists of alternate
layers of right-handed and left-handed quartz, whose planes are parallel
to the axis of the crystal.
NOTE 215, p. 193. Suppose the major axis A P of an ellipse, fig. 18, to
be invariable, but the excentricity C S continually to diminish, the
ellipse would bulge more and more; and when C S vanished, it would
become a circle whose diameter is A P. Again, if the excentricity were
continually to increase, the ellipse would be more and more flattened
till C S was equal to C P, when it would become a straight line A P. The
circle and straight line are therefore the limits of the ellipse.
NOTE 216, p. 194. The coloured rings are produced by the interference of
two polarised rays in different states of undulation, on the principle
explained for common light.
NOTE 217, p. 225. According to Mr. Joule, that heat is produced by
motion, and that it is equivalent to it, Mr. Thompson of Glasgow
investigates from whence the sun derives his heat, since he shows that
neither combustion nor his primitive heat could have supplied the waste
during 6000 years. He concludes that the solar heat is maintained by
myriads of minute bodies that are revolving at the edge of his dense
nebulosity or atmosphere, some of which are often seen by us as falling
stars. These, vaporized by his heat, and drawn by his attraction, meet
with intense resistance on entering the solar atmosphere as a shower of
meteoric rain; through it they descend in spiral lines to the sun’s
surface, producing enormous heat by friction during their fall, and
serving for fuel on their arrival.
NOTE 218, p. 252. The class Cryptogamia contains the ferns, mosses,
funguses, and sea-weeds; in all of which the parts of the flowers are in
general too minute to be evident.
NOTE 219, p. 254. Zoophytes are the animals which form madrepores,
corals, sponges, &c.
NOTE 220, p. 254. The Saurian tribe are creatures of the crocodile and
lizard kind.
NOTE 221, p. 266. If heat from a non-luminous source be polarised by
reflection or refraction at r, fig. 64, the polarised ray r s will be
stopped or transmitted by a plate of mica M I, under the same
circumstances that it would stop or transmit light; and if heat were
visible, images analogous to those of figs. 65, 67, &c., would be seen
at the point s.
NOTE 222, pp. 275, 329, 357. The foot-pound, or unit of mechanical force
established by Mr. Joule, is the force that would raise one pound weight
of matter to the height of one foot; or it is the impetus or force
generated by a body of one pound weight falling by its gravitation
through the height of one foot.
Impetus, vis viva, or living force, is equal to the mass of a body
multiplied by the square of the velocity with which it is moving, and is
the true measure of work or labour. For if a weight be raised 10 feet,
it will require four times the labour to raise an equal weight 40 feet.
If both these weights be allowed to descend freely by their gravitation,
at the end of their fall their velocities will be as 1 to 2; that is, as
the square roots of their heights; but the _effect produced_ will be as
their masses multiplied by 1 and 4; but these are the squares of their
velocities: hence the impetus or vis viva is as the mass into the square
of the velocity.
Thus impetus is the true measure of the labour employed to raise the
weights, and of the _effect_ of their descent, and is entirely
independent of time. Now heat is proportional to impetus, and impetus is
the true measure of labour. In percussion the heat evolved is in
proportion to the force of the impetus, and is thus measured by labour.
Travail is a word used in mechanics, to express that _work done_ is
equal to the labouring force employed. The work done may be resistance
overcome or any other effect produced, while the labouring force may be
a horse, a steam-engine, wind, falling water, &c.
NOTE 223, p. 313. When a stream of positive electricity descends from P
to n, fig. 72, in a vertical wire at right angles to the plane of the
horizontal circle A B, the negative electricity ascends from n to P, and
the force exerted by the current makes the north pole of a magnet
revolve about the wire in the direction of the arrow-heads in the
circumference, and it makes the south pole revolve in the opposite
direction. When the current of positive electricity flows upwards from n
to P, these effects are reversed.
[Illustration: _Fig. 72._]
[Illustration: _Fig. 73._]
NOTE 224, p. 314. Fig. 73 represents a helix or coil of copper wire,
terminated by two cups containing a little quicksilver. When the
positive wire of a Voltaic battery is immersed in the cup p, and the
negative wire in the cup n, the circuit is completed. The quicksilver
ensures the connection between the battery and the helix, by conveying
the electricity from the one to the other. While the electricity flows
through the helix, the magnet S N remains suspended within it, but falls
down the moment it ceases. The magnet always turns its south pole S
towards P, the positive wire of the battery, and its north pole towards
the negative wire.
NOTE 225, p. 316. A copper wire coiled in the form represented in fig.
73 was the first and most simple form of the electro-dynamic cylinder.
When its extremities P and n are connected with the positive and
negative poles of a Voltaic battery, it becomes a perfect magnet during
the time that a current of electricity is flowing through it, P and n
being its north and south poles.