3. At what time of the year does Arcturus rise heliacally at Jerusalem, and at what time does it set heliacally?

4. At what time of the year does Cor Hydræ rise and set heliacally at London?

5. At what time of the year does Procyon rise and set heliacally at London?

6. If the precession of the equinoxes be 50 seconds in a year, how many years will elapse, from 1808, before Sirius, the Dog Star, will rise heliacally at Christmas, at Cairo in Egypt? When this period happens, Sirius will perhaps no longer be accused of bringing sultry weather.

Hesiod informs us, that the hottest season of the year (dog days) ended about 50 days after the summer solstice We have determined in the note of Example 1, Prob. LXXIII. (though perhaps not very accurately), that the winter solstice, in the time of Hesiod, was in the 9th deg. of Aquarius; consequently, the summer solstice was in the 9th degree of Leo: now, it appears from the above, that Sirius rises heliacally at Alexandria when the sun is in the 12th degree of Leo; and, as a degree nearly answers to a day, Sirius rose heliacally, in the time of Hesiod, about four days after the summer solstice; and, if the dog days continued forty days, they ended about forty-four days after the summer solstice. The dog days, in our almanacs, begin on the third of July, which is twelve days after the summer solstice, and end on the eleventh of August, which is fifty. one days after the summer solstice; and their continuance is thirty, nine days. Hence it is plain, that the dog days of the moderns have no reference whatever to the rising of Sirius, for this star rises heliacally at London on the 25th of August, and, as well as the rest of the stars, varies in its rising and setting according to the variation of the latitudes of places, and therefore it could have no influence whatever on the temperature of the atmosphere; yet, as the dog star rose heliacally at the commencement of the hottest season in Egypt, Greece, &c. in the earlier ages of the world, it was very natural for the ancients to imagine that the heat, &c. was the effect of this star. A few years ago, the dog days in our almanacs began at the cosmical rising of Procyon, viz. on the 30th of July, and continued to the 7th of September; but they are now, very properly, altered, and made not to depend on the variable rising of any particu Jar star, but on the summer solstice.

PROBLEM LXXVI.

The latitude of a place and day of the month being given, to find all those stars that rise and set achronically, cosmically, and heliacally.*

Rule. Elevate the pole so many degrees above the horizon as are equal to the latitude of the given place.

Then,

1. For the achronical rising and setting, find the sun's place in the ecliptic, and bring it to the western edge of the horizon, and all the stars along the eastern edge of the horizon will rise achronically, while those along the western edge will set achronically.

2. For the cosmical rising and setting, bring the sun's place to the eastern edge of the horizon, and all the stars along that edge of the horizon will rise cosmically, while those along the western edge will set cosmically.

3. For the heliacal rising and setting, screw the quadrant of altitude over the latitude, turn the globe eastward on its axis till the sun's place cuts the quadrant twelve degrees below the horizon; then all stars of the first magnitude, along the eastern edge of the horizon, will rise heliacally; and by continuing the motion of the globe eastward till the sun's place intersects the quadrant in 13, 14, 15, &c. degrees below the

This problem is the reverse of the three preceding problems. Their principal use is to illustrate several passages in the ancient writers, such as Hesiod, Virgil, Columella, Ovid, Pliny, &c. See Definition 64, page 15. The knowledge of these poetical risings and settings of the stars was held in great esteem among the ancients, and was very useful to them in adjusting the times set apart for their religious and civil duties, and for marking the seasons proper for the several parts of husbandry; for, the knowledge which the ancients had of the motions of the heavenly bodies was not suffi cient to adjust the true length of the year; and, as the returns of the seasons depend upon the approach of the sun to the tropical and equinoctial points, so they made use of these risings and settings to determine the commencement of the different seasons, the time of the overflowing of the Nile, &c. The knowledge which the moderns have acquired of the motions of the heavenly bodies, renders such observations as the ancients attended to in a great measure useless, and, instead of watching the rising and setting of particular stars for any remarkable season, they can sit by the fire-side and consult an almanac.

horizon, you will find all the stars of the second, third, fourth, &c. magnitudes, which rise heliacally on that day. By turning the globe westward on its axis, in a similar manner, and bringing the quadrant to the western edge of the horizon, you will find all the stars that set heliacally.

Examples. 1. What stars rise and set cosmically at Edinburgh, on the 11th of June ?

Answer. The bright star in Castor, Aldebaran in Taurus, Fomalhaut in the Southern Fish, &c. rise cosmically: those stars in the body of Leo Minor, the arm of Virgo, the right foot of Boötes, part of the Centaur, &c. set cosmically.

2. What stars rise and set archronically at Drontheim in Norway, latitude 63° 26' N. on the 18th of May?

Answer: Altair in the Eagle, the head of the Dolphin, &e. rise achronically; and Aldebaran in Taurus, Betelguese in Orion, &e. set archronically

3. What star of the first magnitude rises heliacally at London, on the 7th of October?

Answer. Arcturus in Boötes.

4. What star of the first magnitude sets heliacally at London, on the 5th of May?

Answer. Sirius the Dog Star.

5. What stars rise and set archronically at London, on the 26th of September ?

6. What stars rise and set cosmically at London, on the 23d of March?

PROBLEM LXXVII.

To illustrate the precession of the equinoxes.

1

Observations. All the stars in the different constel lations continually increase in longitude; consequently, either the whole starry heavens has a slow motion from west to east, or the equinoctial points have a slow mo tion from east to west. In the time of Meton, the first star in the constellation Aries, now marked 8, passed

* Meton was a famous mathematician of Athens, who flourished about 430 years before Christ. In a book called Enneadecaterides, or cycle of 19 years, he endeavoured to adjust the course of the sun and of the moon; and attempted to shew that the solar and lunar years could regularly begin from the same point in the heavens.

through the vernal equinox, whereas it is now upwards of 30 degrees to the eastward of it.

Illustration. Elevate the north pole 90 degrees above the horizon, then will the equinoctial coincide with the horizon; bring the pole of the ecliptic † to that part of the brass meridian which is numbered from the north pole towards the equinoctial, and make a mark upon the brass meridian above it; let this mark be considered as the pole of the world, let the equinoctial represent the ecliptic, and let the ecliptic be con sidered as the equinoctial; then count 38 degrees, the complement of the latitude of London, from this pole upwards, and mark where the reckoning ends, which will be at 75 degrees, on the brass meridian, from the southern point of the horizon; this mark will stand over the latitude of London.

Now, turn the globe gently on its axis from east to west, and the equinoctial points will move the same way, while at the same time, the pole of the world will describe a circle round the pole of the ecliptic§ of 46° 56' in diameter; this circle will be completed in a Platonic year, consisting of 25,791 years, at the rate of 50 seconds in a year, and the pole of the heavens will vary its situation a small matter every year. When 12,8951 years, being half the platonic year, are come pleted (which may be known by turning the globe half round, or till the point Aries coincides with the east

If the precession of the equinoxes be 501" in a year, and if the equinoctial colure passed through 3 Arietis, 430 years before Christ, the longitude of this star ought now (1804) to be 31° 10′ 58"; for, 1 year: 504": 2234 years (430 + 1804): 31° 10′ 58′′, and this longitude is not far from the truth.

The pole of the ecliptic is that point on the globe where the circular lines meet.

Let it be remembered that the pole of the ecliptic on the globe here represents the pole of the world.

§ Take notice that the extremity of the globe's axis here represents the pole of the ecliptic.

A Platonic year is a period of time determined by the revolution of the equinoxes: this period being once completed, the ancients were of opinion that the world was to begin anew, and the same series of things to return over again. See the 64th Definition, page 14.

ern point of the which is now 8 will be the north

horizon,) that point of the heavens degrees south of the zenith of London pole, as may be seen by referring to the mark which was made over 75 degrees on the meridian.

PROBLEM LXXVIII.

To find the distances of the stars from each other in degrees.

Rule. Lay the quadrant of altitude over any two stars, so that the division marked o may be on one of the stars; the degrees between them will shew their distance, or the angle which these stars subtend, as seen by a spectator on the earth.

Examples. 1. What is the distance between Vega in Lyra, and Altair in the Eagle?

Answer. 34 degrees.

2. Required the distance between 8 in the Bull's Horn, and y Bellatrix in Orion's shoulder?

3. What is the distance between ß in Pollux, and in Procyon?

4. What is the difference between, the brightest of the Pleiades, and in the Great Dog's Foot?

Ε

5. What is the distance between in Orions girdle, and in Cetus?.

6. What is the distance between Arcturus in Bootes, and ẞ in the right shoulder of Serpentarius ?

PROBLEM LXXIX.

To find what stars lie in or near the moon's path, or what stars the moon can eclipse, or make a near approach to.

Rule. Find the moon's longitude and latitude, or her right ascension and declination, in an ephemeris, for several days, and mark the moon's places on the globe (as directed in Problems LXVII and LXVIII ;) · then by laying a thread or the quadrant of altitude,