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But in triangle APZ, as S,AP : (S,PZA=)

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Now the value of the angle APZ may be calculated from this value of it by a table of common logarithms; but if we desire the value immediately in time, or in parts of a circle, by a table of logistical or proportional logarithms, we must use the reciprocal of each term in the above theorem, according to the rule which we have already given.

TO FIND THE TIME OF THE DAY OR NIGHT, BY THE MERÍ DIAN TELESCOPE.

When the telescope is fixed in the meridian, and moves only in a vertical circle, observe the appulse of the sun's limbs to the perpendicular wires of the telescope, noting the time by the clock. Then half the time which the body of the sun takes to pass a wire, being added to the time of the western limb's appulse to the middle wire, gives the time of his center passing the meridian, or apparent noon by a dial; which is converted into the mean noon of the clock, by the equation of time being added or subtracted.

At night, observe when a known star passes the meridian, and note the time; then from the right ascen

sion of the star, subtract the right ascension of the sun, computed up to the time of the star's passage, and the remainder, when less than 12 hours; is the time of the star's culmination after noon; but when more, its overplus above 12 is the time after midnight. The dif ference between this calculated time and the observed time, gives the error of the clock, by which it may be regulated.

OTHER METHODS OF FINDING THE APPARENT TIME.

Observe the altitude of the sun in a place whose latitude is known: and correct the observation for refraction and parallax, so as to obtain the correct altitude. To the logarithm sine of half the sum of the zenith distance, co-latitude and polar distance, add the logarithm sine of the said half sum lessened by the zenith distance, and also the arithmetical complement of sines of the co-latitude and polar distance; then half the sum of these logarithms will be the logarithm co-sine of half the hour angle from noon, which convert into time, at the rate of fifteen degrees for every hour.

The distance of a star from the meridian is found in the same manner as the distance of the sun, from the observed altitude, and consequently the time of its culmination, which is the difference between its right ascension and the right ascension of the sun, by which the time of the night may be found. These altitudes should be taken when the sun or star alters its altitude fastest, that is, when due east or west, or as near to that position as possible, as the result will thereby be the more accurate.

But the method by equal altitudes is much more accurate, and is as follows. Let the altitude of the sun or star be observed by a Hadley's quadrant, or equal

altitude instrument, while the object is on the east side of the meridian; and, the instrument remaining set to the same altitude, observe when the object has the same altitude again on the west side of the meridian, and note both the times; then half the sum of these times increased by six hours, will be the time when the star was on the meridian; but there must be an allowance made for the change of the sun's declination in the interval of the observed altitudes. This correction in the latitude of Philadelphia may amount to near a quarter of a minute, and is found ready calculated among astro. nomical tables.

The method of observing the sun is this. Note the time when the preceding limb touches each of the horizontal wires, when the observation is made by the transit instrument, while he is on the east side of the meridian; and in the afternoon, when the sun descends to the same altitude, having turned the instrument round on its axis, still preserving the altitude of the telescope, observe the appulse of the preceding limb of the sun to the first horizontal wire, and write down the time opposite to the last observed time in the forenoon; also the time of its appulse to the second wire, write opposite to the last observation but one in the forenoon, and so on to the last. Half the sum of any two corresponding times, increased by six hours, gives the time of apparent noon nearly, to be corrected, as above, by the equation of corresponding altitudes; and the mean of the six sets of such observations will give the apparent noon within a second.

When the observation is to be made with the com. mon reflecting quadrant, stand in such a position, that you may see the image of the sun reflected from a bason of oil or mercury, and bring the image, formed by the double reflexion of the instrument, a little below

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the image seen in the bason, and screw the index fast, without altering until the afternoon observations are over. In a little time the limbs of the two images will be in contact, which time note down as the first observation; when the centers are coincident, note the time also for the second; and finally, when the limbs are in contact before the separation of the images, for the third and last observation. In like manner make the afternoon observations, and write them down in an inverted order, as before directed, and deduce the time of apparent noon from them, correcting the mean by the equation of corresponding altitudes. The elements of solar tables being determined by the preceding methods, the tables themselves may be easily constructed in the following manner.

If 360° be divided by the length of a solar revolution, the quotient will be 0° 59 8" for the motion of one day, and its multiples show the mean motion for any number of days. The twenty-fourth part of it gives the mean motion for an hour, and in like manner for any other time.

Let the time of his entering the first point of Aries be well ascertained by observation, when his mean motion from Aries is nothing; his mean motion therefore, for as many days, hours, and minutes, as have elapsed from the noon of the first day of January to that time, subtracted from 360°, will be the mean motion for the beginning of that year; for astronomers begin the year at that time. To this add the motion for 365 days, and the sum will give the mean motion for the succeeding year. Thus you proceed for three years; but because the fourth is bissextile, containing 366 days, the mean motion of one day more must be added, to have the motion for the beginning of it. In like manner, the radical mean motion for any year

being lessened by the multiples of the yearly motions, observing to allow for leap-years, will give the mean motions of any preceding years.

By the same method, the mean motion of the apogee from Aries being 66 per annum, its multiple will give the mean motion of the apogee for any given time.

Let the place of the sun's apogee, and the time of his passage through it, be accurately determined by observation, when the motion of the apogee is nothing; then the distance of the apogee from Aries is known. Let this distance be lessened by the apogeon motion from the beginning of the year to the time of passing the apogee, and you will have the motion of the apogee at the beginning of the year. Then that radical mean motion being increased or diminished by the multiples of the yearly motions, will give its mean motions for any succeeding or preceding years. And the sun's mean motion for any time being lessened by the motion of his apogee, gives his mean anomaly for that time, to which applying the equation of his center, his longitude will be obtained.

The table of equation of the center is thus constructed. To every degree of mean anomaly assumed, find the true anomaly, and the difference between these will be the equation of the center. If these equations be computed for six signs, the equations for the other six signs are the same, only placed in an inverted order; because equal anomalies are at equal distances, on each side of either apsis.

THE CIVIL YEAR.

In order to record past transactions, mankind soon found it necessary to have some large measure of time, such as a year, which they naturally deduced

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