ON THE OPTICAL ANGLE, AND THE APPARENT MAGNITUDE OF OBJECTS.
In order to understand the principle on which telescopes represent
distant objects as magnified, it may be expedient to explain what is
meant by the angle of vision, and the apparent magnitudes under which
different objects appear, and the same object, when placed at different
distances.
[Illustration: _figure 40_*.]
The optical angle is the angle contained under two right lines drawn
from the extreme points of an object to the eye. Thus AEB or CED (fig.
40.) is the optical or visual angle, or the angle under which the
object AB or CD, appears to the eye at E. These two objects, being at
different distances, are seen under the same angle, although CD is
evidently larger than AB. On the retina of the eye, their images are
exactly of the same size, and so is the still larger object FG.
[Illustration: _figure 41._]
The _apparent magnitude_ of objects denotes their magnitude as they
appear to us, in contradistinction from their real or true magnitude,
and it is measured by the visual angle; for whatever objects are seen
under the same or equal angles _appear_ equal, however different their
real magnitudes. If a half-crown or half-dollar be placed at about 120
yards from the eye, it is just perceptible as a visible point, and its
apparent magnitude, or the angle under which it is seen, is very small.
At the distance of thirty or forty yards, its bulk appears sensibly
increased, and we perceive it to be a round body; at the distance
of six or eight yards, we can see the king or queen’s head engraved
upon it; and at the distance of eight or ten inches from the eye it
will appear so large, that it will seem to cover a large building
placed within the distance of a quarter of a mile, in other words, the
apparent magnitude of the half-crown held at such a distance, will more
than equal that of such a building, in the picture on the retina, owing
to the increase of the optical angle. If we suppose A (fig. 41.) to
represent the apparent size of the half-crown at nine yards distance,
then we say it is seen under the small angle FED. B will represent its
apparent magnitude at 4-1/2 yards distant under the angle HEG, and the
circle C, its apparent magnitude at 3 yards distant, under the large
angle KEI.
[Illustration: _figure 42._]
This may be otherwise illustrated by the following figure. Let AB (fig.
42.) be an object viewed directly by the eye QR. From each extremity A
and B draw the lines AN,BM, intersecting each other in the crystalline
humour in I: then is AIB the optical angle which is the measure of the
apparent magnitude or length of the object AB. From an inspection of
this figure, it will evidently appear that the apparent magnitudes of
objects will vary according to their distances. Thus AB, CD, EF, the
real magnitudes of which are unequal, may be situated at such distances
from the eye, as to have their _apparent_ magnitudes all equal, and
occupying the same space on the retina MN, as here represented. In like
manner, objects of equal magnitude, placed at unequal distances, will
appear unequal. The objects AB and GH which are equal, being situated
at different distances from the eye, GH will appear under the large
angle TIV, or as large as an object TV, situated at the same place as
the object AB, while AB appears under the smaller angle AIB. Therefore
the object GH is _apparently_ greater than the object AB, though it is
only equal to it. Hence it appears that we have no certain standard of
the _true magnitude_ of objects, by our visual perception abstractly
considered, but only of the _proportions_ of magnitude.
In reference to apparent magnitudes, we scarcely ever judge any object
to be so great or so small as it appears to be, or that there is so
great a disparity in the visible magnitude of two equal bodies at
different distances from the eye. Thus, for example, suppose two men,
each six feet 3 inches high, to stand directly before us, one at the
distance of a pole, or 5-1/2 yards, and the other at the distance of
100 poles, or 550 yards--we should observe a considerable difference in
their apparent size, but we should scarcely suppose, at first sight,
that the one nearest the eye appeared a hundred times greater than the
other, or that, while the nearest one appeared 6 feet 3 inches high,
the remote one appeared only about _three fourths of an inch_. Yet
such is in reality the case; and not only so, but the visible bulk
or area of the one is to that of the other, as the square of these
numbers, namely as 10,000 to 1; the man nearest us presenting to the
eye a magnitude or surface ten thousand times greater than that of the
other. Again, suppose two chairs standing in a large room, the one
21 feet distance from us, and the other 3 feet--the one nearest us
will appear 7 times larger both in length and breadth, than the more
distant one, and consequently, its visible area 49 times greater. If
I hold up my finger at 9 inches distant from my eye, it seems to cover
a large town a mile and a half in extent, situated at 3 miles distant;
consequently, the apparent magnitude of my finger, at 9 inches distant
from the organ of vision, is greater than that of the large town at 3
miles distance, and forms a larger picture on the retina of the eye.
When I stand at the distance of a foot from my window, and look through
one of the panes to a village less than a quarter of a mile distant,
I see, through that pane, nearly the whole extent of the village,
comprehending two or three hundred houses; consequently, the apparent
magnitude of the pane is equal to nearly the extent of the village,
and all the buildings it contains do not appear larger than the pane
of glass in the window, otherwise, the houses and other objects which
compose the village could not be seen through that single pane. For, if
we suppose a line drawn from one end of the village, passing through
the one side of the pane, and another line drawn from the other end,
and passing through the other side of the pane to the eye, these lines
would form the optical angle under which the pane of glass and the
village appears. If the pane of glass be fourteen inches broad, and
the length of the village 2640 yards, or half a mile--this last lineal
extent is 6,788 times greater than the other, and yet they have the
same _apparent_ magnitude in the case supposed.
Hence we may learn the absurdity and futility of attempting to
describe the extent of spaces in the heavens, by saying, that a
certain phenomenon was two or three feet or yards distant from
another, or that the tail of a comet appeared several yards in length.
Such representations can convey no definite ideas in relation to
such magnitudes, unless it be specified at what distance from the
eye, the foot or yard is supposed to be placed. If a rod, a yard
in length, be held at nine inches from the eye, it will subtend an
angle, or cover a space in the heavens, equal to more than one fourth
of the circumference of the sky, or about one hundred degrees. If
it be eighteen inches from the eye, it will cover a space equal to
fifty degrees; if at three feet, twenty-five degrees, and so on in
proportion to the distance from the eye; so that we can form no correct
conceptions of apparent spaces or distances in the heavens, when we
are merely told that two stars, for example, appear to be three yards
distant from each other. The only definite measure we can use, in such
cases, is that of degrees. The sun and moon are about half a degree
in apparent diameter, and the distance between the extreme stars in
_Orion’s belt_, three degrees, which measures being made familiar
to the eye, may be applied to other spaces of the heavens, and an
approximate idea conveyed of the relative distances of objects in the
sky.
From what has been stated above, it is evident that the magnitude
of objects may be considered in different points of view. The true
dimensions of an object, considered in itself, give what is called its
_real_ or _absolute magnitude_; and the opening of the visual angle
determines the _apparent magnitude_. The real magnitude, therefore, is
a constant quantity; but the apparent magnitude varies continually with
the distance, real or imaginary; and therefore, if we always judged
of the dimensions of an object from its apparent magnitude, every
thing around us would, in this respect, be undergoing very sensible
variations, which might lead us into strange and serious mistakes. A
fly, near enough to the eye, might appear under an angle as great as
an elephant at the distance of twenty feet, and the one be mistaken for
the other. A giant eight feet high, seen at the distance of twenty-four
feet, would not appear taller than a child two feet in height, at the
distance of six feet; for both would be seen nearly under the same
angle. But our experience generally prevents us from being deceived by
such illusions. By the help of touch, and by making allowance for the
different distances at which we see particular objects, we learn to
correct the ideas we might otherwise form from attending to the optical
angle alone, especially in the case of objects that are near us. By the
sense of touch we acquire an impression of the distance of an object;
this impression combines itself with that of the apparent magnitude,
so that the impression which represents to us the real magnitude is
the product of these two elements. When the objects, however, are at
a great distance, it is more difficult to form a correct estimate
of their true magnitudes. The visual angles are so small, that they
prevent comparison; and the estimated bulks of the objects depend in
a great measure upon the _apparent_ magnitudes; and thus an object
situated at a great distance, appears to us much smaller than it is
in reality. We also estimate objects to be nearer or farther distant
according as they are more or less clear, and our perception of them
more or less distinct and well defined; and likewise, when several
objects intervene between us and the object we are particularly
observing. We make a sort of addition of all the estimated distances
of intermediate objects, in order to form a total distance of the
remote object, which in this case appears to be farther off than if the
intervening space were unoccupied. It is generally estimated that no
terrestrial object can be distinctly perceived, if the visual angle it
subtends be less than _one minute of a degree_; and that most objects
become indistinct, when the angle they subtend at the pupil of the eye
is less than six minutes.
We have deemed it expedient to introduce the above remarks on the
apparent magnitude of objects, because the principal use of a telescope
is to increase the angle of vision, or to represent objects under a
larger angle than that under which they appear to the naked eye, so as
to render the view of distant objects more distinct, and to exhibit to
the organ of vision those objects which would otherwise be invisible.
A telescope may be said to enlarge an object just as many times as the
angle under which the instrument represents it, is greater than that
under which it appears to the unassisted eye. Thus the moon appears to
the naked eye under an angle of about half a degree; consequently a
telescope magnifies 60 times if it represents that orb under an angle
of 30 degrees; and if it magnified 180 times, it would exhibit the moon
under an angle of 90 degrees, which would make her appear to fill half
of the visible heavens, or the space which intervenes from the horizon
to the zenith.