planet with the moon, the same as if it were Jupiter or Saturn, according as it may suit the convenience of the observer. But, it must be borne in mind, that each of the above-named planets is susceptible of two contacts on the same limb of the moon, viz., an external and an internal contact on the east limb, and an internal and an external contact on the west limb of the moon. The external contact on the east side of the moon takes place the instant that the moon's eastern limb appears to touch the west limb of the planet; the internal contact, on the same side, takes place the moment that the east limb of the planet touches the east limb of the moon; at which moment the disc of the planet will be immersed behind the body of the moon. The converse of this takes place on the west limb of the moon; that is, the internal contact takes place the moment that the west limb of the planet appears to emerge from behind the west limb of the moon; and the external contact, the moment that the planet's eastern limb is about to be separated from the moon's west limb.-Hence, it is manifest that, at an external contact, it is the sum of the semidiameters that expresses the apparent central distance between the objects; but, at an internal contact, as the body of the planet is behind the moon, it is the difference of their semidiameters that will express the apparent central distance, or the value of the semidiameter for calculation :-this is general for Mars, Jupiter, and Saturn.

2. Since the disc of Venus is never seen full, or never forms a complete circle; it will be advisable to make use of a little precaution in observing the moment of an external or of an internal contact:—As thus, when Venus is an evening star; that is, when her longitude is greater than the sun's, it is her western limb that will be enlightened. In this case the external contact will take place betwixt the nearest limbs of the objects, which is to be noted the instant that the western limb of Venus touches the moon's eastern limb; and the internal contact, the moment that the same limb appears from behind the moon's western limb. The converse of this takes place when Venus is a morning star; that is, when her longitude is less than the sun's; because, then, it is her eastern limb that will be illuminated :-in this case, it is the internal contact of the planet's farthest or east limb that is to be noted (which amounts to an immersion of her whole disc); and the external contact of the same limb at the precise moment of its appearing to emerge from behind the body of the moon. At an external contact, the sum of the semidiameters is to be taken; but, at an internal contact, the difference of the semidiameters :-in either case, the result will express the apparent central distance, or the semidia

meter for calculation, with which, and the horizontal parallax for calculation, proceed as if it were a fixed star that were under consi-deration.

Since writing the two last Problems, a friend has kindly supplied me with a copy of the Nautical Almanac for 1837, in the Appendix to which I perceive that Mr. W. S. B. Woolhouse has turned his attention to "the determination of the longitude from an observed solar eclipse or an occultation."-As I am, in common with all lovers of astronomy, already much indebted to the writings of that learned gentleman for a great deal of valuable information relative to eclipses (see "Appendix to the Ephemeris for 1836"); I shall add another link to the chain of obligation, by making use of one of his examples (given in page 181 of the Appendix), for the purpose of showing with what manifest ease and conciseness the longitude may be determined by either of my methods of calculation.

Example.

Suppose, at Bedford, on January 7th, 1836, in latitude 52:8:28% north, and longitude by account 0:28. west, the immersion of i Leonis to be observed at 10:39 22:4, apparent time; required the true longitude?

Since the apparent time of observation was 10:39 22:4, the mean time was 10:45:53:3:-the moment of an immersion, or an emersion, should be always noted in mean time, and not in apparent

time.

Remark. The Greenwich time thus found, differs 1:2 from the time as determined in page 182 of the Appendix to the Nautical Almanac for 1837:-this difference is owing to the moon's hourly motion

in declination having been taken in the Appendix, last line, page 181, as 11:415 instead of 11:39:1-as thus,-the occultation took place between the hours of 10 and 11; at the former hour the moon's declination was 15:58:501, and at the latter 15:47:110; the difference of these is 11:39:1; which, therefore, is the true variation of the moon's declination during the intervening hour: and, if this hourly motion be made use of in that part of the calculation which stands in the right hand compartment of page 182 of the Appendix, it will cause the final correction to amount to 19" 43:7, instead of 19":44:; and thus the mean time at Greenwich will be 10:47" 46:2, being in almost. perfect accordance with the mean time at Greenwich, found as above.

Having thus shown with what evident facility the longitude may be deduced from an occultation, by adopting the familiar modes of calculation laid down in pages 546 and 553, I shall now close this subject by recommending the young navigator to turn back and give the Articles between pages 537 and 546 another patient perusal.

SOLUTION OF PROBLEMS RELATIVE TO THE VARIATION OF THE COMPASS.

Definitions.

1. The variation of the compass is the deviation of the points of the mariner's compass from the corresponding points of the horizon, and is denominated east or west variation accordingly.

2. East variation is, when the north point of the compass is to the eastward of the true north point of the horizon; west variation is, when the north point of the compass is to the westward of the true north point of the horizon.

The variation of the compass may be found by various methods, such as amplitudes, azimuths, transits, equal altitudes, rising and setting of the celestial objects, &c.

3. The true amplitude of any celestial object is, an arch of the horizon intercepted between the true east or west point thereof, and the object's centre at the time of its rising or setting.

4. The magnetic amplitude of an object is, the arch of the horizon that is intercepted between its centre, and the cast or west point of the compass, at the time of its rising or setting; or, it is the compass bearing of the object when in the horizon of the eastern or western hemisphere.

The true amplitude of a celestial object is found by calculation; and the magnetic amplitude is found by an azimuth compass.

5. The true azimuth of a celestial object is, the angle contained between the true meridian and the vertical circle passing through the object's centre.

6. The magnetic azimuth is, the angle contained between the magnetic meridian and the azimuth, or vertical circle passing through the centre of the object; or, in other words, it is the compass bearing of the object, at any given elevation above the horizon.

The true azimuth of a celestial object is found by calculation; and the magnetic azimuth by an azimuth compass.

PROBLEM I.

Given the Latitude of a place, the Sun's Declination, and his Magnetic Amplitude; to find the true Amplitude, and the Variation of the Compass.

The computation of an amplitude is involved in a right angled spherical triangle, the principles of which may be familiarly illustrated; as thus:

In the annexed diagram let

H be an arc of the horizon equal to 90 degrees, in which the point represents the prime vertical, or the true east, or west, point of the horizon. Let P represent the elevated pole of the heavens, and HP an arc of the meridian equal to the latitude of

the place of observation: and,

S

P

H

let PS be the polar distance of a celestial object, and S the point of the horizon in which it rises or sets: then, the arc S represents the true amplitude, and the arc S H the complement of the amplitude of the celestial object at the moment of its rising or setting.-Now, in the right angled spherical triangle P H S, given the leg H P= the latitude, and the hypothenuse P S the polar distance of the object; to find the leg HS the complement of the true amplitude; which leg is to be found by spherical trigonometry, Problem I., page 184, reading HS for BC in the operation for finding the leg in the upper part of