CHAP. I. was compared with a chronometer in the cabin, which was too fast for mean Greenwich time, 19 m. 12.5 s., according to estimation from its rate of motion. The chronometer was fast of watch by 3 h. 56 m. 39 s. What was the longitude of the ship?

How to decide from the

observations

east or west.

mark out a

meridian

line.

The longitude is west, because it is later in the day, at Greenwich, than at the ship. This example explains all the whether the details of finding the longitude by a chronometer. longitude is By taking advantage of the observations for time on shore, How to de- We may draw a meridian line with considerable exactness; termine and for instance, in the last observation (if it had been on land), in 4h. 11 m. 24 s. after the observation was taken, the sun would be exactly on the meridian; and if the watch could be. depended upon to measure that interval with tolerable accuracy, the direction from any point toward the sun's center, at the end of that interval, would be a meridian line. Several such meridians, drawn from the same point, from time to time, and the mean of them taken, will give as true a meridian as it is practical to find; although, for such a purpose, a prominent fixed star would be better than the sun.

The problem of time includes that of longitude, and finding the difference of longitude between two places always resolves itself into the comparison of the local times, at the same instant of absolute time. When any definite thing occurs, wherever it may be, that is absolute time. For instance, the explosion of a cannon is at a certain instant of absolute time, wherever the cannon may be, or whoever may note the event; but if the instant of its occurrence could be known at far distant places, the local clocks would mark very diffe

diffe

СНАР. І.

Absolute

time defined by means of

But events.

A clock is a noter of

rent hours and minutes of time; but such difference would be occasioned entirely by difference of longitude: the event is the same for all places — it is a point in absolute time. Thus any single event marks a point in absolute time. If the same event is observed from different localities, the rence in local time will give the difference in longitude. a perfect clock is a noter of events, it marks the event of noon, the event of sunrise, the event of one hour after events, when noon, &c.; and if we could have perfect confidence in this it runs true, marker of events, nothing more would be necessary to give us wise. the local time at a distant place. The time, at the place where we are, can be determined by the altitude of the sun, or a star, as we have just seen. But, unfortunately, we cannot have perfect confidence in any chronometer or clock; and therefore we must look for some event that distant observers can see at the same time.

but not otner

events,

common pur.

value.

The passage of the moon into the earth's shadow is such Eclipses are an event, but it occurs so seldom as to amount to no practical which mark value. The eclipses of Jupiter's satellites are such events, absolute but they cannot be observed without a telescope of consider- time, but for able power, and a large telescope cannot be used at sea. poses they Hence these events are serviceable to the local astronomer are of little only; the sailor and the practical traveler can be little benefited by them. The moon has comparatively a rapid motion among the stars (about 13° in a day), and when it comes to any definite distance to or from any particular star, that circumstance may be called an event, and it is an event that can be observed from half the globe at once.

of the moon among the

considered

as an index

Thus, if we observe that the moon is 30° from a particular The motion star, that event must correspond to some instant of absolute time; and if we are sufficiently acquainted with the moon, stars, may be and its motion, so as to know exactly how far it will be from certain definite points (stars) at the times, when it is noon, moving 3, 6, 9, &c., hours at Greenwich, then, if we observe these events from any other meridian, we thereby know the Green- solute time. wich time, and, of course, our longitude.

Finding the Greenwich time by means of the moon's angular distance from the sun or stars, is called taking a lunar;

round a circle

marking ab

U*

CHAP. I. and it is probably the only reliable method for long voyages

Lunar observations il

at sea.

If the motion of our moon had been very slow, or if the earth had not been blessed with a moon, then the only methods, for sea purposes, would have been chronometers and dead reckoning. For a practical illustration of the theory of lunars, we mention the following facts.

In a sea journal of 1823, it is stated that the distance of lustrated by the moon from the star Antares was found to be 66° 37′ 8′′, an example. when the observation was properly reduced to the center of the

Observed distances, and tances

feen

dis

as

from

the center of the earth.

Clearing the distance.

earth, and the time of observation at ship was September 16th, at 7h. 24m. 44s. P. M. apparent time.

By comparing this with the Nautical Almanac, it was found that at 9 P. M., apparent time at Greenwich, the distance between the moon and Antares was 66° 5′ 2′′, and at midnight it was 67° 35′ 31′′; but the observed distance was between these two distances, therefore the Greenwich time was between 9 and 12 P. M., and the time must fall between 9 and 12 hours in the same proportion as 66° 37′ 8′′ falls between the distances in the Nautical Almanac; and thus an observer, with a good instrument, can at any moment determine the Greenwich time, whenever the moon and stars are in full view before him.

The moon, in connection with the stars in the heavens, may be considered a public clock (quite an enlargement of the town-clock), by which certain persons, who understand the dial plate and the motion of the index, and who have the necessary instrument, can read the Greenwich time, or the time corresponding to any other meridian to which the computations may be adapted.

The angular distances from the moon to the sun, stars, and planets, as put down in the Nautical Almanac, corresponding to every third hour, are distances as seen from the center of the earth, and when observations are taken on the surface the distance is a little different; the position of the moon is affected by parallax and refraction, the sun or star by refraction alone; and therefore a reduction is necessary, which is called clearing the distance. This is done by spheri

cal trigonometry. The distance between the moon and star is observed, the altitudes of the two bodies are also observed. The co-altitudes come to the zenith, and the co-altitudes, with the distance, form three sides of a spherical triangle, from which the angle at the zenith can be computed. Then correct the altitude of the moon for parallax and refraction, and the star for refraction, and find the true altitudes and coaltitudes, and the true co-altitudes and angle at the zenith give two sides and the included angle of a spherical triangle, and the third side, computed, is the true distance.

An immense amount of labor has been expended by mathematicians, to bring in artifices to abbreviate the computation of clearing lunar distances; and they have been in a measure successful, and many special rules have been given, but they would be out of place in a work of this kind.

PROPORTIONAL LOGARITHMS.

CHAP. I.

an explana

In every part of practical astronomy there are many pro- Proportional portional problems to be resolved, and as the terms are logarithms mostly incommensurable, it would be very tedious, in most tion of the cases, to proceed arithmetically, we therefore resort to loga- construction rithms, and to a prepared scale of logarithms, very appropri- given. ately called proportional logarithms.

The tables of proportional logarithms commonly correspond to one hour of time, or 60′ of arc, or to three hours of time. The table in this book corresponds to one hour of time, or 3600 seconds of either time or arc. To explain the construction and use of a table of proportional logarithms, we propose the following problem :

At a certain time, the moon's hourly motion in longitude was 33′17′′; how much would it change its longitude in 13m. 23s.? Put to represent the required result, then we have the following proportion :

Divide the first and second terms of this proportion by the

of the table

CHAP. I. second, and the third and fourth by the third, then we have

Divide the third and fourth terms by x, and multiply the

same terms by 3600, and the proportion becomes

Multiplying extremes and means, using logarithms, and remembering that the addition of logarithms performs multiplication,

By the construction of the table, the proportional logarithm

hence the proportional logarithm of 3600 is 0.

We now work the problem:

Examples

given to il

lustrate the

EXAMPLES FOR PRACTICE.

1. When the sun's hourly motion in longitude is 2′ 29′′, practical uti- what is its change of longitude in 37 m. 12 s.?

lity of propor

tional logar. ithms.

Ans. 1' 32".5.

2. When the moon's declination changes 57′′.2 in one hour, what will it change in 17 m. 31 s.? Ans. 16".8.

3. When the moon changes longitude 27′ 31′′ in an hour, how much will it change in 7 m. 19 s.? Ans. 3' 21".

4. When the moon changes her right ascension 1 m. 58 s. in one hour, how much will it change in 13 m. 7 s.?

Ans. 25".8.