CHAP. VI. But this is a digression. With the history of astronomy, as interesting as it may be, we design to have little to do, and to proceed only with the science itself.

Distinction

CHAPTER VII.

FIRST APPROXIMATIONS TO THE RELATIVE DISTANCES OF THE
PLANETS FROM THE SUN. HOW THE RESULTS ARE OBTAINED.

(97.) BEING Convinced of the truth of the Copernican system, the next step seems to be, to find the periodical times of the revolutions of the planets, and at least their relative distances from the sun.

Mercury and Venus, never coming in opposition to the sun, between in- but revolving around that body in orbits that are within that ferior and su- of the earth, are called inferior planets.

perior plan

ets.

Those that come in opposition, and thereby show that their orbits are outside of the earth, are called superior planets.

We shall show how to investigate and determine the position of one inferior planet; and the same principles will be sufficient to determine the position of any inferior planet.

It will be sufficient, also, to investigate and determine the orbit of one superior planet; and if that is understood, it may be considered as substantially determining the orbits of all the superior planets; and after that, it will be sufficient to state results.

For materials to operate with, we give the following table of the planetary irregularities (so called) drawn from obser

word MEAN should be

used.

In the preceding table, the word mean is used at the head Why the of several columns, because these elements are variablesometimes more and sometimes less, than the numbers here given - which indicates that the planets do not revolve in circles round the sun, but most probably in ellipses, like the orbit of the earth.

On the supposition, however, that the planets revolve in circles (which is not far from the truth), the greatest and least apparent diameters furnish us with sufficient data to compute the distances of the planets from the sun in relation to the distance of the earth, taken as unity.

(98.) In addition to the facts presented in the preceding The elonga. table, we must not fail to note the important element of the tions of Merelongations of Mercury and Venus. This term can be applied us. to no other planets.

cury and Ve.

variable and

It is very variable in regard to Mercury-showing that This element the orbit of that planet is quite elliptical. The variation is what it much less in regard to Venus, showing that Venus moves shows. round the sun more nearly in a circle.

The least extreme elongation of Mercury is 17° 37'.

46° 30'.

The least extreme elongation of Venus is
The greatest

66

66

The mean (or at mean distances), is

The least extremes must happen when the planet is in its perigee and the earth in its apogee, and the greatest when the earth is in perigee and the planet in apogee; but it is

CHAP. VII. very seldom that these two circumstances take place at the

How to

find the comparative magnitudes

of the orbits

of Mercury, Venus, and

the earth

same time.

Relying on these facts as established by observations, we can easily deduce the relative orbits of Mercury and Venus.

Let (Fig. 23) represent the sun, E the earth, V Venus.

Conceive the planet

to pass round the sun in the direction of A V B.

The earth moves also in the same direction. but not so rapidly as Venus.

Now it is clearly evident, from inspection, that when the planet is passing by the earth, as at B, it will appear to pass along in the heavens in the direction of

m to n. But when the planet is passing along in its orbit, at
A, and the earth about the position of E, the planet will
appear to pass in the direction of n to m.
When the planet
is at V, as represented in the figure, its absolute motion is
nearly toward the earth, and, of course, its appearance is
nearly stationary.

What to It is absolutely stationary only at one point, and even then understand but for a moment; and that point is where its apparent mo

by

ary.

station

tion changes from direct to retrograde, and from retrograde to direct; which takes place when the angle SEV is about 29 degrees on each side of the line SE.

When the line EV touches the circumference A VB, the angle SE V, or angle of elongation, is then greatest; and the triangle SE V is right angled at V; and if SE is made radius, SV will be the sine of the angle SE V.

But the line SE is assumed equal to unity, and then SV

will be the natural sine of 46° 20', and can be taken out of any table of natural sines; or it can be computed by logarithms, and the result is .72336.

For the planet Mercury, the mean of the same angle is 22° 46'; and the natural sine of that angle, or the mean radius of the planet's orbit, is .38698.

Thus we have found the relative mean distances of three

planets from the sun, to stand as follows:

CHAP. VII.

of Mercury Venus

(99.) If the orbits were perfect circles, then the angle The orbits SEV, of greatest elongation, would always be the same; and but it is an observed fact that it is not always the same; not circles. therefore the orbits are not circles; and when SV is least, and SE greatest, then the angle of elongation is least; and conversely, when SV is greatest and SE least, then the angle of elongation is the greatest possible; and by observing in what parts of the heavens the greatest and least elongations take place, we can approximate to the positions of the longer axis of the orbits.

Computa. tion of orbits

from appa

(100.) By means of the apparent diameters, we can also find the approximate relations of their orbits. For instance, when the planet Venus is at B, and appears on the sun's rent diame. disc, its apparent diameter is 59′′.6; and when it is at A, or as near A as can be seen by a telescope, its apparent diameter is 9".6. Now put

SB=x; then EB-1-x, and AE-1+x. By Art. 66, 1-x : 1+x :: 96 : 596;

By a like computation, the mean distance of Mercury from the sun is 0.3864.

(101.) To determine the mean relative distances of the superior planets from the sun, we proceed as follows:

Let S (Fig. 24) represent the sun, E the earth, and Mone of the superior planets, say Mars. It is easy to decide, from observation, when the planet is in opposition to the sun.

ters

The relative

planet from

This gives the position of S, E, and M, in one right line, in respect to longitude. Now by knowing the true angular motion of the earth about the sun (73), and the mean angular motion of the planet,* we can determine the angle m Se, corresponding to any definite future time; for, by the motion of the earth round the sun, we can determine the angle ESe; and by the motion of the planet in the

same time, we can determine the angle MS m; and the dif

By means of apparent diameters, we can determine the distance of a values of the orbit. When the planet is in opposition to the the sun de- sun, at E (Fig. 24), measure its apparent diameter; and, termined by after a definite time, when the earth is at e, measure the aption in its parent diameter again, and observe the angle Sem. Proapparent dia duce Se to n. Then, by the apparent diameters, we have

the varia

meter

the proportion of e m and en (en is the same as E M, brought to this position); and in the triangle emn we have the proportion between the two sides and the included angle men. These are sufficient data to determine the angles en m and emn; and their difference is the angle Sme. Now we can determine the side Sm, of the triangle Sme, and the triangle Sem is completely known. Subtract the angle e Sm from the whole angle e SM, and the angle M Sm is left. That is, while the earth is describing the angle E Se, the planet describes the angle MSm. Put P for the periodical revo

*Here we anticipate a little; for we have not shown how to determine the periodical time of revolution from observation: but this is shown in a future chapter, and in the above text note