I shall so often have to speak of the distances of the celestial bodies
that I may once for all explain how it is that we have been able to
discover what these distances are. This would be a very puzzling matter
if we were to try and describe it fully, but the principle of the
method is not at all difficult. Do you know why you have been provided
with two eyes? It is undoubted that one of the reasons is to aid you in
estimating distances. You see this boy (Fig. 4) judges of the distance
of his finger by the inclination of his two eyes when directed at it.
In a similar way we judge of the distance of a heavenly body by making
observations on it from two different stations.
[Illustration: FIG. 4.--Two Eyes are better than One.]
[Illustration: FIG. 5.--How we measured the Height of the Ball.]
I shall illustrate our method of measuring the actual distance of a
body in the heavens by showing you how we can find the height of that
large india-rubber ball which is hanging from the ceiling. Of course,
I do not intend to have a measuring tape from the ball itself, because
I want to solve the problem on the same principle as that by which we
measure the distance of the sun or of any other celestial body which we
cannot reach. I will ask the aid of a boy and a girl, who will please
stand one at each end of the lecture table. The apparatus we shall want
is very simple; it consists of two cards and a pair of scissors. The
boy will kindly shape his card to such an angle that when he holds it
to his eye one side of the angle shall point straight at the little
girl, and the other side shall point straight at the ball, just as you
see in the picture (Fig. 5). The girl will also please do the same
with her card, so that along one side she just sees the little boy’s
face, while the other side points up to the ball. It will be necessary
to cut these angles properly. If the angle be too big, then when one
side points to the boy’s face, the other will be directed above the
ball. If the angle on the card be too small, then one side will be
directed below the ball, while the other is pointed to the boy. The
whole accuracy of our little observations depends upon cutting the
card angles properly. When they have been truly shaped it will be easy
to find the distance of the ball. We first take a foot rule and measure
the length of our table from one of our young friends to the other.
That length is twelve feet, and to discover the distance of the ball
we must make a drawing. We get a sheet of paper, and first rule a line
twelve inches long. That will represent the length of the table, it
being understood that each inch of the drawing is to correspond to a
foot of the actual table. Let the end where the girl stood be marked
B, and that of the boy, A, and now bring the cards and place them on
the line just as shown in the figure. The card the girl has shaped is
to be put so that the corner of it lies at B, and one edge along B A.
Then the boy’s card is to be so put that its corner is at A and one
edge along A B. Next with a pencil we rule lines on the other edges of
the cards, taking care that they are kept all the time in their proper
positions. These two lines carried on will meet at C; and this must be
the position of the ball on the scale of our little sketch. It only now
remains to take the foot rule and measure on the drawing the length
from A to C. I find it to be twenty inches, and I have so arranged it
that the distance from B to C is the same.
[Illustration: FIG. 6.--This is what we wanted the Cards for.]
I do not intend to trouble you much with Euclid in these lectures, but
as many of my young friends have learned the sixth book, I will just
refer to the well-known proposition, which tells us that the lengths
of the corresponding sides of two similar triangles are proportional.
We have here two similar triangles. There is the big one with the boy
at one corner, the girl at the other, and the ball overhead. Here
is the small triangle which we have just drawn. These triangles are
similar because they have got the same angles, and it was to insure
that they should have the same angles that we were so careful in
shaping the cards. As these two triangles are similar, their sides must
be proportional. We have agreed that the line A B, which is twelve
inches long, is to represent the length of the table between the little
boy and girl. Hence the distance, A C, must, on the same scale, be the
interval between the ball and the boy at the end. This is twenty inches
on the drawing, and therefore the actual distance from the end of the
table to the ball is twenty feet.
Hence you see that without going up to the ball or having a string from
it, or in any other way making direct communication with it, we have
been able to ascertain how far up in the air the ball is actually hung.
This simple illustration explains the principle of the method by which
astronomers are able to learn the distances of the different celestial
bodies from the earth. You must think of the sun, the moon, and the
stars as globes supported in some manner over our heads, and we seek to
discover their distances from measurements of angles made at the ends
of a base-line.
[Illustration: FIG. 7.--This would be our Base-line when finding the
Sun’s Distance.]
Of course, astronomers must choose two stations which are far more
widely separated than are those in our little experiment. In fact, the
greater the interval between the two stations, the better. Astronomers
require a much longer distance than from one side of this room to
the other, or from one side of London to the other side. If it were
merely a balloon at which we were looking, then, when one observer
at one side of London and another at the opposite side shaped their
cards carefully, we should be able to tell the height of the balloon
very easily. But as the sun is so much further off than any balloon
could ever be, we must separate the observers much more widely.
Even the breadth of England would not be enough, so we have to make
them separate more and more until they are as widely divided as it
is possible for any two people on this earth to be. One astronomer
takes up his position at A (Fig. 7), and the other at the opposite
side at B, so that they can both see the sun. They are obliged to
use a much more accurate way of measuring the angles than by cutting
out cards with pairs of scissors; and as the astronomer at A is not
able to see his friend at B, it becomes no easy matter to measure the
angles accurately. However, we shall not now trouble ourselves about
such difficulties. It may suffice for the present to know that the
angles are measured by delicate and very accurate instruments used
by astronomers. They will not, indeed, make a little sketch such as
sufficed for our purpose. They make a calculation which is a much more
accurate way of effecting true measurement. The astronomers know the
size of the earth, and thus they know how many thousands of miles lie
between the two stations where the observations are made. This distance
means in their calculation just what the length of the table did in our
sketch. From each end of the line they set off an angle just as we did,
and the astronomer must use the principle of similar triangles which he
finds in Euclid, just we had to do. At last, when they have calculated
the sides of their triangle, they obtain the distance of the sun.