СНАР ПІ.

Mean and

true

maly.

equal times; but the angles will be unequal. Conceive the two suns to depart, at the same time, from the point B, and, after a certain interval of time, one is at s, the other at s'. Then we must have

area oBs area mBs' :: area ellipse: area circle.

The angle Bms' is the angle the sun would make, or its ano- increase in longitude from the apogee; provided the angular motion of the sun was uniform. The angle Bos is its true increase of longitude; the difference between these two angles is the angle m n o.

The equa tion of the center.

The angle Bms' is always known by the time; and if to every degree of the angle Bms' we knew the corresponding angle mno, the two would give us the angle Bos; for,

Bms'-mno=mon, or Bos.

The angle Bms' is called the mean anomaly, and the angle Bos is called the true anomaly.

way

The angle Bms' is greater than the angle Bos, all the from the apogee to the perigee; but from the perigee to the apogee, the true sun, on the ellipse, is in advance of the imaginary sun on the circle.

The angle mno is called the equation of the center; that is, it is the angle to be applied to the angle about the center m, to make it equal to the true anomaly.

The angle mno depends on the eccentricity of the ellipse; and its amount is put in a table corresponding to every

degree of the mean anomaly; subtractive, from the apogee to CHAP. III. the perigee, and additive from the perigee to the apogee.*

*

The great

of the center

the orbit.

(81.) Again: conceive the two suns to set out from the same point, B; and as the angle B ms' increases uniformly, it will est equation increase and become greater and greater than the angle Bos, gives the ecuntil the true sun attains its mean angular motion, and no centricity of longer. Then the angle mno attains its greatest value, and, at that time the side mn=no, and the point n is perpendicular over om, and os is a mean proportional between o B and o A. That is, when the sun, or any planet, attains the greatest equation of the center, the true sun is very near the extremity of the shorter axis of the ellipse: o, the greatest equation of the center, can be determined by observation; and, from the greatest equation, we have the most accurate method of computing the eccentricity of the ellipse, as we may see by the note below.†

Then, the area of the circle is a2; the area of the ellipse is

α

wab; that of the sector is (GF), and of the triangle

eab

2

* By a mere mechanical contrivance, the modern astronomical tables are so arranged, that all corrections are rendered additive; so that the mechanical operator cannot make a mistake, as to signs, and he may continue to work without stopping to think. These arrangements have their advantages, but they cover up and obscure principles.

CHAP. III.

When once the eccentricity of any planetary ellipse becomes known, the equation of the center, corresponding to all degrees of the mean anomaly, can be computed and put into a table for future use; but this labor of constructing tables belongs exclusively to the mathematician.

the

Method of deducing the

eccentricity from

greatest equation

the center.

Consequently, GF=ea, and FG-om; which shows that the

of angle o Cm is nearly equal to F'm G, unless it is a very eccentric ellipse. Now we must compute the number of degrees in the arc FG. The whole circumference is 2a.

But the angle onm=nm C+n C'm=2nm C, nearly;

But the greatest equation of the center, for the solar orbit, is, by observation, 1° 55′ 30′′; and as the sun has not quite its greatest equation of the center, when at the point C, it will be more accurate to put

From this equation, it is true, we have only the approximate value of e; but it is a very approximate value, and sufficiently accurate.

Reducing both members to seconds, and we have,

3600 360 e 6924, and e=0.0167842.

The greatest equation of the center is at present diminishing at the rate of 17".17 in one hundred years: this corresponds to a diminution of eccentricity by 0.00004166, which is determined by a solution of the following equation:

CHAPTER IV.

THE CAUSES OF THE CHANGE OF SEASONS.

(82.) THE annual revolution of the earth in its orbit, CHAP. IV. combined with the position of the earth's axis to the plane of its orbit, produces the change of the seasons.

The cause

If the axis were perpendicular to the plane of its orbit, there would be no change of seasons, and the sun would then of the change be all the while in the celestial equator.

This will be understood by Fig. 21. Conceive the plane of the paper to be the plane of the earth's orbit, and conceive the several representations of the earth's axis, NS, to be inclined to the paper at an angle of 66° 32'.

Fig. 21.

of seasons.

In all representations of VS, one half of it is supposed to be above the paper, the other half below it.

NS is always parallel to itself; that is, it is always in the same position*- always at the same inclination to the plane

* Except minute variations, which it would be improper to notice in this part of the work.

CHAP. IV. of its orbit always directed to the same point in the hea

Importance of inspecting the figure.

cause

change

no

--

vens, in whatever part of the orbit it may

be.

The plane of the equator, represented by Eq, is inclined to the plane of the orbit by an angle of 23° 28'.

By inspecting the figure, the reader will gather a clearer view of the subject than by whole pages of description: he will perceive the reason why the sun must shine over the north pole, in one part of its orbit, and fall as far short of that point when in the opposite part of its orbit; and the number of degrees of this variation depends, of course, on the position of the axis to the plane of the orbit.

Position of Now conceive the line NS to stand perpendicular to the the axis to plane of the paper, and continue so; then Eq would lie on. of the paper, and the sun would at all times be in the plane of the equator, and there would be no change of seasons. If NS were more inclined from the perpendicular than it now is, then we should have a greater change of seasons.

seasons.

The equi.

By inspecting the figure, we perceive, also, that when it is summer in the northern hemisphere, it is winter in the southern; and conversely, when it is winter in the northern, it is summer in the southern.

When a line from the sun makes a right angle with the earth's axis, as it must do in two opposite points of its orbit, the sun will shine equally on both poles, and it is then in the plane of the equator; which gives equal day and night the world over.

Equal days and nights, for all places, happen on the 20th of March of each year, and on the 22d or 23d of September. At these times the sun crosses the celestial equator, and is said to be in the equinox.

The longitude of the sun, at the vernal equinox, is 0°; and noctial and at the autumnal equinox, its longitude is 180o.

solstitial

points.

The time of the greatest north declination is the 20th of June; the sun's longitude is then 90°, and is said to be at the summer solstice.

The time of the greatest south declination is the 22d of December; the sun's longitude, at that time, is 270°, and is said to be at the winter solstice.