ANGLES AND MEASUREMENT OF ANGLES.
THE QUADRANT—TRANSIT INSTRUMENT—CLOCKS—STELLAR TIME—SOLAR TIME—“MEAN”
TIME.
We must say a few words respecting the various instruments and aids
to astronomical observation before proceeding, for astronomy requires
very accurate calculations; and though we do not propose to be very
scientific in our descriptions, some little idea of the manner in which
observations may be made is necessary. The first thing to see about is
the ANGLE.
Suppose we draw four lines on a piece of paper, _ab_ and _cd_. These
intersect at a point, _m_. We have then four spaces marked out, and
called _angles_. The four angles are in the diagram all the same size,
and are termed _right angles_, and the lines containing them are
perpendicular to each other.
[Illustration: Fig. 503.—Right angles.]
But by altering the position of the lines (_see_ fig. 504), we have two
pairs of angles quite different from right angles; one angle, _a´ m´
c´_, is smaller, while _a´ m´ d´_ is much larger than the right angle.
The former kind are called _acute_, the latter _obtuse_ angles. We can
therefore obtain a great number of acute angles, but only three obtuse,
and four right angles around a given point, _m_.
[Illustration: Fig. 504.—Obtuse and acute angles.]
The length of the sides of an angle have no effect on its magnitude,
which is determined by the inclination of the lines towards each other.
We now may consider the magnitude of angles, and the way to determine
them. For this purpose we must describe a circle, which is figured in
the diagram. But what is a circle?—A circle is a curved line which
always is at the same distance from a certain fixed point, and the ends
of this line meet at the point from which the line started.
[Illustration: Fig. 505.—The circle, etc.]
If we fasten a nail or hold a pencil on the table, and tie a thread to
it, and to the other end of the thread another pencil, we can describe
a line around the first pencil by keeping the thread tightly stretched.
This line is at all points at equal distance from the centre point.
Any line from the centre to the circumference is called a _radius_,
and a line through the centre to each side of the circumference is the
diameter, or double the radius. The circumference is three (3·14) times
the diameter. Any portion, say _k i l_, is an _arc_, and the line, _k
l_, is the _chord_ of that arc. A line like _m n_ is a _secant_, and _o
p_ is a _tangent_, or a line touching at one point only.
We may now resume our consideration of the angles by means of the
circle. Let us recur to our previous figure of the right angles,
around which we will describe a circle. We see that the portion of the
circumference contained between the sides of the right angle is exactly
one-fourth of the whole. This is termed a _quadrant_, and is divided
into 90°—the fourth of 360 equal parts or degrees into which the whole
circumference is divided. The angle of 45° so often quoted as an angle
of inclination is half a right angle. To measure angles an instrument
called a _Protractor_ is used.
[Illustration: Fig. 506.—Circle and angles.]
[Illustration: Fig. 507.—The Protractor.]
The Protractor, as will be seen from the accompanying illustration
(fig. 507), is a semi-circle containing 180°. The lower portion is
a _diagonal scale_, the use of which will be explained presently.
The Protractor measures any actual angle with accuracy. If we put
the vertical point of the angle and the centre point of the circle
together, we can arrive at the dimensions of the angle by producing the
lines containing it to the circumference. An angle instrument, figured
herewith, may be assumed as the basis of most apparatus for measuring
angles. An index hand, R R, moves round a dial like the hand of a
clock, and the instrument is used by gazing first at one of the two
objects, between which the angle we wish to determine is made—like the
church steeples (fig. 508) for instance. The centre of the instrument
is placed upon the spot where lines, if drawn from the eye to each of
the objects, would intersect. The index hand is then put at 0°, and in
a line between the observer and the object, A. Then the index is moved
into a similar position towards B, and when in line with it the numbers
of degrees passed over (in this imaginary case 20), shows the magnitude
of the angle.
[Illustration: Fig. 508.—Determination of distance.]
[Illustration: Fig. 509.—Measuring angles.]
The simple quadrant is shown in the cut (fig. 510). This was so
arranged that when any object in the horizon was being looked at
through the telescope attached, a plummet line is at 0°. But if the
telescope be raised to C S, the quadrant will move, and the line will
mark a certain number of degrees of the angle which a line if drawn
from the star makes with the line of the horizon. The “Astronomical
Quadrants” are as shown in fig. 516, and consist of a quadrant of wood
strengthened and fitted with a telescope. The circle is graduated on
the outer edge, and a “vernier” is attached. The time is determined
by the observation of the altitude of a star, and then by calculation
finding out at what time the star would have the observed altitude. The
quadrant is now superseded by circular instruments.
[Illustration: Fig. 510.—The quadrant.]
[Illustration: Fig. 511.—Ellipse.]
An ellipse is a flattened circle, or oval, and will be understood from
the diagrams. Let us fix two pegs upon a sheet of paper, and take a
thread longer than the distance between the pegs; draw with the pencil
controlled by the thread a figure, keeping the thread tight. We shall
thus describe an oval, or ellipse. The orbit of nearly all the heavenly
bodies is an ellipse. The _parabola_ is another curved line, but
its ends never meet; they become more and more distant as they are
continued. The comets move in parabolic curves, and consequently do not
again come within our vision unless their direction be altered.
[Illustration: Fig. 512.—Ellipse.]
This figure has a long axis, _ab_ (fig. 512); and perpendicular to
this a short axis, _de_, passing through the centre, _c_. The two
points, SS′, are called the _foci_ of the ellipse; also, as is evident
from the construction of the figure, any two lines drawn from the two
foci, to any point of the circumference, for instance, S and S′_m_, or
S_m′_ and S′_m′_, etc., which represent the thread when the pencil is
at _m_ or _m′_, are together equal to the larger axis of the ellipse.
These lines, and we may imagine an infinite number of such, are called
_radii vectores_. The distance of the foci, S or S′, from the centre,
_c_, is called the _eccentricity_ of the ellipse. It is evident that
the smaller the eccentricity is, the nearer the figure approaches
to that of the circle. The superficies of the ellipse is found by
multiplying the two half axes, _ac_ and _dc_, by each other, and this
product by the number 3·14.
[Illustration: Fig. 513.—Diagonal scale.]
The _Diagonal Scale_ is shown in the margin. It is used to make
diagrams so as to bring the relative distances before the eye. The
larger divisions represent, it may be, miles, or any given distance;
the figures on the left side tenths, and the upper range hundredths of
a mile. So a measurement from Z to Z′ will represent two miles, we may
say, with so many tenths and hundredths.
[Illustration: Fig. 514.—Transit instrument.]
The Transit instrument is due to Roemer, a Danish astronomer. It
consists of a telescope so constructed as always to point to the
meridian, and rotates upon a hollow axis, directed east and west. At
one end is a graduated circle. The optical axis of the telescope must
be at exactly right angles to the axis of the instrument; it will then
move on the meridian. There is an eye-piece filled with two horizontal
and five vertical wires, very fine, the latter at equal distances
apart. The star appears, and the time it takes to cross is noted as it
passes between each wire, and the mean of all the transits will be the
transit on the meridian. For if we add the times of all the transits
across the wires, and divide by five the number of them we shall get at
a true result.
[Illustration: Fig. 515.—The eyepiece of transit instrument.]
A good clock is also a necessary adjunct for astronomical observations,
and the astronomical clocks and chronometers now in use record the time
with almost perfect accuracy. The improvement in telescopes, the use
of micrometers, etc., have greatly facilitated observations. In the
transit clock we have a most useful timekeeper, for the ordinary clocks
are not sufficiently accurate for very close observations. The sidereal
time differs from solar time, and the twenty-four hours’ period is
calculated from the moment a star passes the meridian until it passes
it again. The sidereal day is nearly four minutes shorter than the
solar day, and the sidereal clock marks twenty-four hours instead of
twelve, like the old dial at Hampton Court Palace over the inner gate.
The Chronograph has also been useful to astronomers, for by “pricking
off” the seconds on a roller by itself, the observer can mark on the
same cylinder the actual moment of transit across each wire of the
instrument, and on inspection the exact moment of transit may be noted.
The _Equitorial_ is another useful instrument, and by its means the
whole progress of a star can be traced. The Equitorial consists of a
telescope fixed so that when it has been pointed at a certain star a
clock-work movement can be set in motion, which exactly corresponds
with the motion of the star across the heavens, and so while the
star moves from its rising to setting it is under observation. Thus
continuous observations maybe made of that particular star or comet
without any jerking or irregular movement.
We can thus see the uniform motion of the stars which go on in greater
or lesser circles as they are nearer to or farther from the pole;
and with the exception of the polar star, which, so far as we are
concerned, may be considered stationary, every star moves round from
east to west—that is, from the east of the polar star to the west of
it, in an oblique direction. Therefore, as Professor Airy remarks,
“Either the heavens are solid, and go all of a piece, or the heavens
may be assumed to be fixed or immovable, and that we and the earth are
turning instead of them.”
[Illustration: Fig. 516.—Astronomical quadrant.]
The _Mural Circle_ is another very useful instrument, and is used by
calling to aid the powers of reflection of quicksilver, in which a
bright star will appear below the horizon at the same angle as the real
star above the horizon, and thus the angular distance from the pole or
the horizon of any star can be calculated when we know the inclination
of the telescope. The Transit Circle is also used for this purpose,
and is a combination of the transit instrument with the circle. In all
calculations allowance must be made for refraction, for which a “Table
of Refractions” has been compiled. From the zenith to the horizon
refraction increases. The effect of refraction can be imagined, for
when we see the sun apparently _touching_ the horizon the orb is really
below it, for the refraction of the rays by the air apparently raises
the disc.
The clock and chronometer are both very useful as well as very common
objects, but a brief description of the pendulum and the clock may
fitly close our remarks upon astronomical apparatus and instruments.
The telescope has been already described in a previous portion of this
work, so no more than a passing reference to it has been considered
necessary. We therefore pass on to a consideration of the measurement
of time, so important to all astronomers and to the public generally.
Time was measured by the ancients by dividing the day and night into
twelve hours each, then by sun-dials and water-clocks, or _clepsydra_,
and sand-clocks. The stars were the timekeepers for night before any
mechanical means of measurement were invented.
“What is the star now passing?—
The Pleiades show themselves in the east,
The eagle is soaring in the summit of heaven.”—EURIPIDES.
Sun-dials were in use in Elijah’s time, and the reference to the
miracle of the sun’s shadow going back on the dial as a guarantee
to Hezekiah, will be recalled at once by our readers. These dials
were universal, and till sunset answered the purpose. But the hours
must have been very varying, and on cloudy days the sun-dials were
practically useless.
The water-clocks measured time by the dripping or flow of water, and
they were used to determine the duration of speeches, for orators were
each allotted a certain time if a number of debaters were present.
This method might perhaps be adapted to the House of Commons, and
speaking by the clock might supersede _clôture_. We find allusions to
these practices in the orations of Demosthenes. Even this system was
open to objection, for the vases were frequently tampered with, and
an illiberal or objectionable person was mulcted of a portion of the
water, while a generous or popular adversary had his clepsydra brimming
full. Some of these water-clocks were of elegant design, and a Cupid
marked the time with arrows on the column of the clock of Ctesibius,
while another weeping kept up the supply of water. The motive power was
the water, which filled a wheel-trough in a certain time, and when full
this trough turned over, and another was filled. The wheel revolved
once in six days; and by a series of pinions and wheels the movements
were communicated more slowly to the pillar on which the time was
marked for 360 days, or with other arrangements for twenty-four hours.
The repeating of psalms by monks also marked the time, for by practice
a monk could tell pretty accurately how many paternosters or other
prayer he could repeat in sixty seconds. At the appointed hour he then
awoke the monastery to matins.
Nature also marks time for us—as, for example, the age of trees by
means of rings—one for each year; and horses’ teeth will guide the
initiated to a guess at the ages of the animals, while the horns of
deer or cattle serve a like purpose. But man required accuracy and
minute divisions of time. He had recourse therefore, to machinery and
toothed wheels. Till the mechanical measurement of time was adopted,
the sunrise and sunset only marked the day, and the Italians as well
as Jews counted twenty-four hours from sunset to sunset. This was a
manifestly irregular method. To this day we have marked differences of
time in various places, and at Geneva we have Swiss and French clocks
keeping different hours according to Paris or Berne “time.” This, of
course, is easily accounted for, and will be referred to subsequently.
[Illustration: Fig. 517.—Clock movement.]
We have read that the first clock in England was put up in Old Palace
yard in 1288, and the first application of the toothed-wheel clock to
astronomical purposes was in 1484, by Waltherus, of Nuremberg. Tycho
Brahé had a clock which marked the minutes and seconds. If we had had
any force independent of gravitation which would act with perfect
uniformity, so that it would measure an equal distance in equal spaces
of time, all the various appliances for chronometers would have been
rendered useless. In the supposed case the simple mechanism, as shown
in the margin (fig. 517), would have sufficed. The same effect would
be produced by the spring, were it possible that the spring by itself
would always uncoil with the same force. But it will not do so: we
therefore have to check the unwinding of the cord and weight, for left
to itself it would rapidly increase in velocity; and if we likewise
make an arrangement of wheels whereby the spring shall uncoil with even
pressure all the time, we shall have the principle of the watch.
It is to Huygens that the employment of the pendulum in clocks is due,
and the escapement action subsequently rendered the pendulum available
in simple clocks, while the manner of making pendulums self-regulating
by using different metals, has rendered timepieces very exact. Of
course the length of a pendulum determines the movement, fast or slow;
a long pendulum will cause the hands of the clock to go slower, for the
swing will be a fraction longer. A common pendulum with the escapement
is shown (fig. 518). Each movement liberates a tooth of the escapement.
The arrangement of wheels sets the clock going. The forms of pendulum
are now very varied.
But in watches the pendulum cannot be used. The watch was invented by
Peter Hele, and his watches were called “Nuremburg eggs” from their
shape. The weight cannot be introduced into a watch, and so the spring
and _fusee_ are used. The latter is so arranged that immediately the
watch is wound and the spring at its greatest tension, the chain is
upon the smallest diameter of the fusee, and the most difficult to
move. But as the spring is relaxed the lever arm becomes longer,
and the necessary compensating power is retained. Watches without a
“_fusee_” have a toothed arrangement beneath the spring.
The _Pendulum_. A “simple” pendulum is impossible to make, for we
cannot put the connecting line between the “bob” and the clock-work
out of consideration, so “simple pendulums” are looked upon as
“mathematical abstractions.” The most modern clocks have what is called
a “deadbeat” escapement, and a compensating pendulum. Clocks are liable
to alter by reason of the state of the air and varying temperature,
and until all our clocks are placed _in vacuo_ we must be content to
have them lose or gain a little. There is a magnet arrangement by
which the Greenwich Observatory clocks keep time by compensation,
corresponding with the fall or rise of the syphon barometer attached
to it. The description need not be added. We may here state that
detailed descriptions of all the instruments used in the Observatory,
together with full information as to their use, will be found in a very
interesting work by Mr. Lockyer, entitled “Stargazing,” to which we are
indebted for some corrections in our summary.
[Illustration: Fig. 518.—Pendulum and escapement.]
[Illustration: Fig. 519.—Balance.]
[Illustration: Fig. 520—Regulator.]
We have spoken of solar time and sidereal time, and no doubt someone
will inquire what is meant by mean time—an expression so constantly
applied to the Greenwich clock time. Stellar time, we have seen,
corresponds to the daily revolution of a star or stars. Solar time
is regulated by the sun, and this is the astronomical time generally
observed, except for sidereal investigations. But the sun is not always
regular; the orbit of the earth causes this irregularity partly. The
earth moves faster in winter than in summer, so the sun is sometimes a
little fast and sometimes a little slow. Astronomers therefore strike
an average, and calculate upon a MEAN SUN, or uniform timekeeper. Mean
time and true (apparent) time are at some periods the same—viz., on
the 15th of April, on the 14th of June, on the 31st of August, and on
the 24th of December. Twice it is after, and twice before it. The time
occupied by this “mean” sun passing from the meridian and its return to
it, is a mean solar day, and clocks and chronometers are adjusted by it.
[Illustration: Fig. 521.—Fusee and spring.]
Twenty-four hours represents a complete revolution of the heavenly
bodies. The mean solar time is 23^h 56´ 4·091´´, while twenty-four
hours of mean time are equal to 24^h 3´ 56·55´´ of sidereal time. The
difference between the times is given by Dr. Newcomb as follows, and is
called the _Equation of Time_:—
DIFFERENCES BETWEEN MEAN AND APPARENT TIME.
February 10th True Sun 15 minutes slow.
April 15th “ Correct.
May 14th “ 4 minutes fast.
June 14th “ Correct.
July 25th “ 6 minutes slow.
August 31st “ Correct.
November 2nd “ 16 minutes fast.
December 24th “ Correct.
MEASUREMENT OF DISTANCES.
Before passing to consider the planetary system we must say a few words
respecting the manner of ascertaining the distances of inaccessible
objects, and by so doing, we shall arrive at an idea how the immense
distances between the sun (and the planets) and the earth have been so
accurately arrived at. To do this we must speak of _parallax_, a very
unmeaning word to the general reader.
[Illustration: Fig. 522.—Works of a clock.]
Parallax is simply the difference between the directions of an object
when seen from two different positions. Now we can illustrate this by a
very simple method, which we have often tried as a “trick,” but which
has been very happily used by Professor Airy to illustrate the doctrine
of parallax. We give the extract in his own words:—
“If you place your head in a corner of a room, or on a high-backed
chair, and if you close one eye and allow another person to put a
lighted candle upon a table, and if you then try to snuff the candle
with one eye shut, you will find you cannot do it.... You will hold
the snuffers too near or too distant—you cannot form any idea of the
distance. But if you open the other eye, or if you move your head
sensibly you are enabled to judge of the distance.” The difference of
direction between the eyes, which is so well known to all, is ready a
parallax. It can also be illustrated by the diagram herewith.
[Illustration: Fig. 523.—Parallax.]
If two persons, A and C (fig. 523), from different stations, observe
the same point, M, the visual lines naturally meet in the point, M, and
form an angle, which is called the angle of _parallax_. If the eye were
at M, this angle would be the angle of vision, or the angle under which
the base line, A C, of the two observers appears to the eye. The angle
at M also expresses the apparent magnitude of A C when viewed from M,
and this apparent magnitude is called the _parallax_ of M.
Let M represent the moon, C the centre of the earth represented by the
circle, then A C is the parallax of the moon; that is, the apparent
magnitude the semi-diameter of the earth would have if seen from the
moon. If the moon be observed at the same time from A, being then on
the horizon, and from the point B, being then in the zenith, and the
visual line of which when extended passes through the centre of the
earth, we obtain, by uniting the points, A C M, by lines, the triangle,
A C M.
Therefore, as A M, the tangent of the circle stands at right angles to
the radius, A C, the angle at A is a right angle, and the magnitude of
the angle at C is found by means of the arc, A B, the distance of the
two observers from each other. As soon, however, as we are acquainted
with the magnitude of two angles of a triangle, we arrive at that of
the third, because we know that all the angles of a triangle together
equal two right angles (180°). The angle at M, generally called the
moon’s parallax, is thus found to be fifty-six minutes and fifty-eight
seconds. We know that in the right-angled triangle M C A, the measure
of the angle, M = 56´ 58´´, and also that A C, the semi-diameter of
the earth = 3,964 miles. This is sufficient, in order by trigonometry,
to obtain the length of the side, M C; that is, to find the moon’s
distance from the earth. A C is the sine of the angle, M, and by the
table the sine of an angle of 56´ 58´´ is equal to 1652/100000; or, in
other words, if we divide the constant, M C, the distance of the moon,
into 100,000 equal parts, the sine, A C, the earth’s semi-diameter =
1,652 of these parts. And this last quantity being contained 60 times
in 100,000, the distance of the moon from the earth is equal to 60
semi-diameters of the earth, or 60 × 3964 = 237,840 miles.
[Illustration: Fig. 524.—Parallax explained.]
In a similar way the parallax of the sun has been found = 8´´·6, and
the distance of the sun from the earth to be 91,000,000 miles.
Let us first see how we can obtain the distance of any inaccessible
or distant object. We have already mentioned an experiment, but this
method is by a calculation of angles. The three angles of a triangle,
we know, are equal to two right angles; that is an axiom which cannot
be explained away. We first establish a base line; that is, we plant a
pole at one point, A, and take up our position at another point, B,
at some distance in a straight line, and measure that distance very
carefully. By means of the theodolite we can calculate the angles which
our eye, or a supposed line drawn from our eye to the top of the object
(C) we wish to find the distance of, makes with that object. We now
have an imaginary triangle with the length of one side, A B, known, and
all the angles known; for if all three angles are equal to 180°, and we
have calculated the angles at the base, we can easily find the other.
We can then complete our triangle on paper to scale, and find out the
length of the side of the triangle by measurement; that is the distance
between our first position, A, and the object, C. It is of course
necessary that all measurements should be _exact_, and the line we
adopt for a base should bear some relative proportion to the distance
at which we may guess the object to be.
In celestial measurements two observers go to different points of
the earth, and their distance in a straight line is known, and the
difference of the latitudes. By calling the line between the observers
a base line, a figure may be constructed and angles measured; then
by some abstruse calculation the distance between the centre of the
earth and the centre of the moon may be ascertained. The mean distance
is sixty times the radius of the earth. The measurement of the sun’s
distance is calculated by the observations of the transit of Venus
across his disc, a phenomenon which will again occur on 6th December,
1882, and on 8th June, 2004, the next transit will take place; there
will be no others for a long time after 2004.
All astronomical observations are referred to the centre of the earth,
but of course can only be viewed from the surface, and correction
is made. In the cut above, let E be the earth and B a point on the
surface. From B the stars, _a_ _b_ _c_ _d_, will be seen in the
direction of the dotted lines, and be projected to _e_ _i_ _k_ _l_
respectively. But from the centre of the earth they would appear at
_e_ _f_ _g_ _h_ correctly. The angles formed by the lines at _b_ _c_
_d_ are the parallactic angles, _f_ _i_ _g_ _h_ and _h_ _l_ show the
parallax. An object on the zenith thus has no parallax. (_See_ fig.
524.)
[Illustration: Fig. 525.—Halo Nebulæ.]