to move with an angular velocity, equal to the difference of the two.

When the planet arrives at v, an observer at A would see the planet projected on the sun, making a dent at v'.

But an observer at G would not see the same thing until after the planet had passed over the small are vq, with a velocity equal to the diference between the angular motion of the two bodies; and as this will require quite an interval of absolute time, it can be detected; and it measures the angle Av G; an angle under which a definite,portion of the earth appears as seen from the sun.

(121.) To have a more definite idea of the practicability of this method, let us suppose the parallactic angle, A v′ G, equal to 10", and inquire how long Venus would be in passing the relative arc v q.

The relative, or excess motion of Venus for a mean solar day is then 37'.

Now, as 37' is to 24h. so is 10" to a fourth term; or, as 2220" 1440m. :: 10": 6 m. 29 s.

Now if observation gave more than 6 minutes and 29 seconds, we shall conclude that the parallactic angle was more than 10"; if less, less. But this is an abstract proposition. When treating of an actual case in place of the mean motion, we must take the actual angular motions of the earth and Venus at that time, and we must know the actual position of the observers A and G in respect to each other, and the position of each in relation to a line joining the center of the

earth and the center of the sun; and then by comparing the CHAP. IX. local time of observation made at A, with the time at G, and referring both to one and the same meridian, we shall have the interval of time occupied by the planet in passing from v to 7, from which we deduce the parallactic angle A v' G, and from thence the horizontal parallax.

The same observations can be made when the planet passes off the sun, and a great many stations can be compared with A, as well as the station G. In this way, the mean result of a great many stations was found in 1761, and in 1769, and the mean of all cannot materially differ from the truth.

A combination of many

observations

thod of dedu

(122.) There is another method of considering this whole Another mesubject, which is in some respects more simple and preferable cing the proto the one just explained. It is for the observers at every blem. station to keep the track of the transit all the way across the sun's disc, and take every precaution to measure the length of chord upon the disc, which can be done by carefully noting the times of external and internal contacts, and the beginning and end of the transit, and at short intervals carefully measuring the distance of the planet to the nearest edge of the sun by a micrometer.

different ob.

servers.

If the parallax is sensible, it is evident that two observers, Situation of situated in different hemispheres, will not obtain the same chord. For example, an observer in the northern hemisphere, as in Sweden or Norway, will see Venus traversing a more southern chord than an observer in the southern hemisphere.

Now if each observer gives us the length of the chord as observed by himself, and, knowing the angular diameter of the sun, we can compute the distance of each chord from the sun's center, and of course we then have the angular breadth of the zone on the sun's disc between them. But as this zone is formed by straight lines passing through the same point, the center of Venus, its absolute breadth will depend on its distance from the point v; that is, the two triangles A B v and a b v (Fig. 27) will be proportional, and we have

Av av AB : ab.

But the first three of these terms are known; therefore the fourth, a b, is known also; and if any definite angular space

K *

The result.

CHAP. IX.

Under what

circumstances this method should

not be used.

Transits of Mercury not important.

Revolutions

and the earth

บ

Fig. 27.

B

on the sun becomes known, the whole semidiameter becomes known, and from thence the horizontal parallax is immediately deduced.*

(123.) The accuracy of this method should be questioned when Venus passes near the sun's center, for the two chords are never more than 30" asunder, and hence they will not perceptibly differ in length when passing near the sun's center, and Venus will be upon the sun nearly the same length of time to all observers.

(124.) The apparent diameter of Mercury and Venus can be very accurately measured when passing the sun's disc. In 1769 the diameter of Venus was observed to be 59".

(125.) The same general principles apply to the transits of Mercury and Venus; but those of Mercury are not important, on account of the smaller parallax and smaller size of that planet; but owing to the more rapid revolution of Mercury, its transits occur more frequently. The frequent appearance of this planet on the face of the sun, gives to astronomers fine opportunities to determine the position of its node and the inclination of its orbit.

In 1779, M. Delambre, from observations on the transit of of Mercury May 7, placed the ascending node, as seen from the sun, in compared. longitude 45° 57' 3". From the transit of the 8th of May, 1845, as observed at Cincinnati, it must have been in longitude 46° 31' 10"; this gives it a progressive motion of about 1° 10' in a century. The inclination of the orbit is 7° 0′ 13′′. The periodical time of revolution is 87.96925 days; that of the earth is 365.25638 days, and by making a fraction of these numbers, and reducing as in the last text note, we find

* That is, as the real diameter of the sun, is to the real diameter of the earth, so is the sun's angular semidiameter to its horizontal parallax. (See 66).

that 6, 7, 13, 33, 46, 79, and 520 years, or revolutions of the CHAP, IX. earth nearly correspond to complete revolutions of Mercury. Hence we may look for a transit in 6, 7, 13, 33, 46, &c., years, or at the expiration of any combination of these years after any transit has been observed to take place; and by examining the following table, the years will be found to fol- Intervals be low each other by some combination of these numbers.

The following is a list of all the transits of Mercury that have occurred, or will occur, between the years 1800 and 1900:

tween tran.

sits.

CHAPTER X.

THE HORIZONTAL PARALLAXES OF THE PLANETS COMPUTED, AND
FROM THENCE THEIR REAL DIAMETERS AND MAGNITUDES.

Real mag.

(126.) HAVING found the real distance to the sun, and the CHAP. X. sun's horizontal parallax, we have now sufficient data to find the real distance, diameter, and magnitude, of every planet nitudes and in the solar system.

distances can now be de

In Art. 60 we have explained, or rather defined, the hori- termined zontal parallax of any body to be the angle under which the semidiameter of the earth appears, as seen from that body; and if the earth were as large as the body, the apparent diameter of the body, and its horizontal parallax, would have the same value. And, in general, the diameter of the earth is to the diameter of any other planetary body, as the horizontal parallax of that body is to its apparent semidiameter.

The mean horizontal parallax of the sun, as determined in

CHAP. X. the last chapter, is 8".6; the semidiameter of the sun, at the Real dia- corresponding mean distance, is 16′ 1′′, or 961". Now let d meter of the represent the real diameter of the earth, and D that of the sun, then we shall have the following proportion:

sun

mined.

deter

Real dis

tance

tween

d : D :: 8".6 : 961".0.

But d is 7912 miles; and the ratio of the last two. terms is 111.66; therefore D=(111.66)(7912)=883454 miles.

(127.) The sun's horizontal parallax is the angle at the be base of a right angled triangle; and the side opposite to it is the the radius of the earth (which, for the sake of convenience, determined. We now call unity). Let a represent the radius of the earth's orbit; then, by trigonometry,

earth and sun

Distance in

sin. 8.6 : 1 :: sin. 90° : x;

Therefore, x=

sin. 90°

sin.8".6

=log. 10.00000-log. 5.620073.*

That is, the log. of x=4.379927, or x=23984; which is the distance between the earth and sun, when the semidiameter of the earth is taken for the unit of measure; but, for general reference, and to aid the memory, we may say the distance is 24000 times the earth's semidiameter.

(128.) Now let us change the unit from the semidiameter of the earth to an English mile; and then the distance between the earth and sun is

(3956)(23984)=94880706; bers. and, in round numbers, we say 95 millions of miles.

round

num

By Kepler's third law, we know the relative distances of

* Students generally would be unable to find the sine of 8".6, or the sine of any other very small arc; for the directions given in common works of trigonometry are too gross, and, indeed, inaccurate, to meet the demands of astronomy.

On the principle that the sines of small arcs vary as the arcs themselves, we can find the sine of any small arc as follows:

In the same manner, find the sine of any other small arc.