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stars, in his apparent annual revolution, caused by the real revolution of the earth, is called by astronomers, the ecliptic. A broad circle of the heavens, about 8 degrees on each side of the ecliptic, is called the zodiac, from the Greek zo-on, an animal, because most of the twelve parts into which it was divided by the ancient Chaldeans or Egyptians were named after animals. The following are the divisions of the ecliptio or circle of the zodiac:
1 sign, s. 12 signs, or 360°, the whole circle of the zodiac. Astronomers apply the above measure in calculating the motions and angular distances of the planets and other celestial bodies.
301. A circle is a round figure in a plane, (104,) generated by the revolution of a finite straight line in that plane, about one of its extremities, which remains fixed; and this fixed point is called the centre. The line described by the motion of the other extremity is called the circumference, which is, of course, in all its parts, equally distant from the centre ; and hence all straight lines drawn from the centre to the circumference are equal. These straight lines are called radië of the circle, and any one of them a radius.
If two straight lines cut one another at right angles, (103) they may be considered as four straight lines which meet in one point, making equal angles with each other. Now, if round the point in which they meet, called the point of intersection, we describe a circle, its circumference will be divided into 4 equal parts, called arcs.
Thus, in the circle abde, the angles at the point c, being right angles, are equal to one another, and the arcs ae, ed, db, and ba, upon which they stand, are also equal.
d First, it is evident that a straight line drawn through the centre, from side to side, divides the circle into two equal parts. For, in describing the circle, as the distance from the
centre c is always the same, the distance passed through on one side of the line must be the same as that passed through on the other. Therefore the arc bae is equal to the arc bde. Now, if bde is supposed to revolve on be, till it meets the arc bae, it will coincide with it; that is, the two arcs will form but one, because they are equal; and the point d will coincide with the point a, because the angles are right angles. But, if we suppose the arc abd to revolve on ad, the point b will coincide with the point e, for the same reason, and the arc ab with the arc ae.
Wherefore the four arcs are equal to one another. Each of these arcs, being the fourth part of a circle, is called a quadrant.
The straight line ad or be, passing through the centre, and terminated both ways by the circumference, is called a diameter of the circle; and the arc bae or bde is a semicircle, or half circle.
302. From the above, it is plain that if two straight lines, meeting in one point, make one and the same straight line, this line produced (continued) will be a diameter of all the circles which can be described about that point. Also, that all the angles formed by any number of straight lines meeting in one point, as at c, on both sides of a diameter, must be equal to 4 right angles, because they will be comprised in the four right angles bca, ace, ecd, and dcb; and their number will be equal to the number of lines meeting in the point c. If formed on one side of a diameter, they will
, of course, be equal to 2 right angles, and their number will be one less than the number of lines meeting in c.
303. If two points be taken in the circumference of a circle, and we suppose a radius to revolve from one to the other, it is plain that the radius will pass over the arc and the angle which it subtends (stretches under) at one and the same time. Wherefore, if two arcs be equal, the angles which they subtend must also be equal : and, vice versa, if the angles be equal, the arcs must also be equal.
Now, if 360 lines radiate from the point c, on both sides of a diameter, so as to make equal angles with each other, there will be 360 angles. Then, if any number of circles be described about the centre c, the circumference of each circle will be divided, by the radiating lines, into 360 equal aros. These equal angles and equal arcs are called degrees. Hence the circumference of every circle, however great or small, is
said to consist of 360 degrees, and a quadrant, or right angle, of 90.
304. The arc of a circle intercepted between the extreme points of any two radii, is called the measure of the angle which they form with each other at the centre. Thus a quadrant, or 90 degrees, is the measure of a right angle. Also it is evident that every arc of a circle is the same part of the whole circumference of that circle, that the angle which it subtends is of four right angles.
Geographical Measure. 305. Geography is derived from two Greek words, Gé, the earth, and grapho, I write; and means the delineation and description of the earth's surface.
The exact measurement of the globe we inhabit, is a subject of peculiar interest to the astronomer, geographer, navigator, and engineer, because many important calculations are founded upon it. It has, therefore, been attempted by skilful men of different nations, and though the results have been various, they have differed so little from each other, as to leave no doubt of their near approximation to the truth.
The circumference of the earth in the direction of the meridian, from measures taken in France, is supposed to be about 24856 miles, and the polar diameter, or axis of the earth, nearly 7912. The equatorial circumference about 24896 miles, and its diameter nearly 7925.
Hence a degree on the meridian is 69,04 or 694 English or common miles, nearly; and a degree on the equator about
306. The degree on the equator, called a degree of longitude, from the Latin longus, long, (the circumference being longer in that direction than on a meridian,) is divided into 60 equal parts called minutes, and, by some, geographical miles.
Hence, (288,) a common mile is to a geographical mile as 21600 to 24896, or very nearly as 72 to 83.
Longitude is either east or west, as it is measured on the equator, beginning at the point in which the meridian of some remarkable place cuts that line, which meridian is in longitude 0°, and proceeding east or west round the globe to the point
* Some suppose the mean diameter of the earth to be about 7920 miles, which makes the degree at a mean for the whole surface 694 miles, nearly.
where the same meridian cuts the equator again; which last point, being the farthest possible, or opposite extremity of the semicircle, is in longitude 180°, or 0°, because all places through which the same meridian passes, are said to be in the same longitude.
307. The degree on a meridian is called a degree of latitude, from the Latin, latus, wide.
Latitude is north or south, as it is measured on a meridian towards the North or South pole, beginning at the equator, which is in latitude 0°, and ending at the pole, which is in lat. 90°.
Lines passing round the globe, in a direction exactly east and west, and, consequently, parallel to the equator, are called parallels of latitude ; and all places intersected by any one of these parallels, are said to be in the same latitude.
308. Mariners determine the position of their vessel at sea, as well as that of any place on the globe, by its latitude and longitude.
Now, though all meridians are equal, the case is very different with parallels of latitude, which diminish as they recede from the equator towards the pole. Hence, if one of these is at a considerable distance from the equator, the degree measured on it is very different from the equatorial degree. For example, on the parallel of Philadelphia, a degree is about 46 nautical, or 53,03 common miles.
309. The meridian from which the longitude is reckoned, or first meridian, is arbitrary. The French reckon from that of the Observatory at Paris; the English and Americans from that of the Royal Observatory at Greenwich, near London; and several other nations each from that of its own capital. But this is of little consequence, as it is easy to change one reckoning to the other. For example, the difference of longitude between the French and English meridians is 2° 20' 15"; and, as the French meridian is east of the English, we say that it is in longitude 2° 20' 15" E., while the French consider the English meridian in longitude 2° 20' 15'' W. Wherefore, to change French longitude to English, add 2° 20'15" to the east longitude, and subtract it from the west.
310. As the equator, as well as every circle or parallel of latitude, is divided into 360 degrees, and all these are, by the revolution of the earth from west to east, brought successively to the mean solar noon exactly in 24 hours, it is plain
that the noon of each degree in any of those circles, (where day and night alternate in one revolution,) will precede the noon of the next degree to the westward, by the 360th part of 24 hours; that is, by 4 minutes of solar time.
Hence, if we know the longitude of two places, we have only to multiply their difference of longitude* in degrees by 4, which will give the difference of time by the clock at those places, in minutes. This time, subtracted from the time at the easternmost place, will show that of the westernmost; or, added to that of the last, will give that of the first. ample, suppose that it is noon at the Observatory at Greenwich, and that we would know the time at that of Paris :
First, 2° 20' 15"=2810=1879; then, 1887-X=-3=9 m. 21 sec.
Hence, as Paris is E. of Greenwich, it is 9 m. 21 sec. past noon at Paris. When it is noon at the Paris Observatory, it will, of course, be 23 h. 50 m. 39 sec. of the preceding day at Greenwich; for it is thus that astronomers reckon the hours, from noon of one day to noon of the next. In common phrase, we should
say 39 sec. past eleven A.M., of the same day.
Examples. 1. When it is noon at Philadelphia, in longitude 75° 16' W. of Greenwich, what is the time at Paris ?
Answer, 5 h. 10 m. 25 sec. past noon. 2. Being noon at Philadelphia, in long. 75° 16' W., what is the difference of time, and what o'clock is it at each of the following places, taking them in an easterly order round the globe, from Philadelphia to Philadelphia again : Figueira, in Portugal, in long. 8° 52' W.; Lemnos, in the Grecian Archipelago, long. 25° 15' E.; Pekin, long. 116° 27' 30" E., and Cape Mendocino, in long. 124° 7' W. Answer: Between Philadelphia and Figueira...... 4 25 36 Figueira and Lemnos..
2 16 28 Lemnos and Pekin .....
6 4 50 Pekin and Cape Mendocino....... 7 57 42 Cape Mendocino and Philadelphia 3 15 24
* When the longitudes are both E. or_both W., their difference is the difference of longitude; but, when one is E. and the other W., their sum is the difference of longitude. When this sum exceeds 180°, subtract it from 360°, and the remainder will be the difference of longitude.