days. From a comparison of this period with that of the moon itself a very remarkable result is obtained. If the moon revolved actually in the plane of the ecliptic, then the centre of the moon would in each revolution pass across the centre of the sun, and the moment of this occurrence is called the time of new moon. Owing, however, to the circumstance that the orbit of the moon is inclined to the plane of the ecliptic, the moon will not usually pass over the surface of the sun. It is therefore necessary to modify the definition of new moon accordingly. We define the time of new moon to be the moment when the longitude of the centre of the moon is equal to the longitude of the centre of the sun. The interval between two successive new moons is termed a lunation, and it is by this period that the successive phases of the moon are regulated. The length of the lunation is such that 223 lunations make 6585.32 days. Thus 19 periods of the revolution of the sun with respect to the nodes of the moon coincide very nearly with 223 lunations. This remarkable period, amounting to about 18 years 11 days, is of service in the prediction of eclipses. It is known as the Saros.

"Another very remarkable period arises from the circumstance that 235 lunations form 6939.69 days, while 19 years of 365.25 days amount to 6939.75 days. We therefore conclude that 19 years are nearly identical with 235 lunations. This is the Cycle of Meton. If the dates of new moon and full moon are known for a period of 19 years, they can be predicted indefinitely, for in each subsequent 19 years the dates are reproduced in the same manner. The number which each year bears in the Cycle of Meton is called the golden number. In 1888 the golden number is 8. In 1881 the golden number was 1, being the commencement of a new cycle.

The period called the Solar Cycle is founded upon the recurrence of the day of the week upon the same day of the month. Owing to the complication produced by leap year, this period is 28 years. In the year 1888 the Solar Cycle is said to be 21. This signifies that 1888 is the twenty-first of one of these groups of 28 years. The cycle known as the Roman Indiction is a period of 15 years. Though this cycle is not connected with any astronomical phenomenon, it is still retained. Thus the year 1888 is the first year of the Roman Indiction and the commencement of a new cycle.

"In the Almanacs it is usual to find a certain number stated as the Julian Period. Thus, for example, 1888 is the 6601st year of the Julian Period. This cycle arises from the three numbers 19, 29, 15, which represent the entire periods of the Cycle of Meton, the Solar Cycle, and the Roman Indiction respectively. It appears that in a period of 19 × 28 × 15 7,980 consecutive years there are not two years which have the same Golden Number accompanied with the same Solar Cycle and the same Roman Indiction. There is thus a new period, called the Julian Period, consisting of 7,980 years. The first year of this period is 4713 B.C., which has been adopted, because each of the three other cycles had the value I on that year. This period will continue till the year A.D. 3267."-R. S. Ball's Elements of Astronomy.

It remains now to illustrate by Diagrams, as far as it is possible to do so on a flat surface, some of the terms in Nautical Astronomy that have been defined in words; for this purpose Fig. 1, plate VII. is taken as a projection of the CELESTIAL SPHERE ON THE PLANE OF THE OBSERVER'S MERIDIAN in lat. 49° N

The outer circle of Fig. I is the celestial meridian passing through the north and south points of the horizon, the zenith, and the pole. Other important great circles, as the horizon, the equinoctial, and the prime vertical are of necessity represented by straight lines. The observer is supposed to be standing in the centre of the projection with the various circles around him meeting in their respective poles; then

H H is the rational horizon, with H (N) the north point, and (S) H the South point; E (in the centre of the diagram) will be the east point, and must also stand for the west point (which is represented only in imagination). The great circle of the horizon is divided into four quadrants by a plane passing through the north and south points, and another passing through the east and west points.

P (N) is the north pole, and (S) P the south pole of the celestial sphere, towards which the great circles called hour-circles and the circles of declination trend.

Z is the observer's zenith, or point directly overhead; N is his nadir, or the point in the opposite hemisphere, beneath his feet.

QQ' is the equinoctial, or celestial equator, coincident with which great circle is the sun's path on the 20th of March and 23rd of September, giving equal day and night to all parts of the earth. The plane of the equinoctial passes through the E. and W. points of the horizon (represented at the centre of the diagram).

ZEN is the prime vertical, which passes through the E. and W. points of the horizon, and whose plane is perpendicular to the meridian, the latter passing through the N. and S. points of the horizon.

PEP is the six o'clock hour-circle.

H (N) P (N) is the arc of the meridian representing the altitude of the elevated pole, which is also the latitude of the observer's station—here projected for lat. 49° N.

P (N) Z is the co-latitude (i.e., the complement of the latitude).

The line joining D D'"', cutting the plane of the equinoctial at E, represents the plane of the ecliptic whose obliquity in respect of the equinoctial is 231°.

D D' is the small circle representing the parallel of the sun's greatest northern declination (23° 27′ N.) on the 21st of June.

(S) H D is the arc of the meridian representing the sun's altitude (i.e., the meridian altitude) at noon on the 21st of June.

Z D is the arc of the meridian representing the sun's meridian zenith distance on June 21st.

H (N) D' is the distance to which the sun descends below the horizon on June 21st.

r is the place of the sun's rising and setting, on June 21st, and Er is the sun's Amplitude, reckoned from the east point at rising, and from the west point at setting, towards north, because the declination is (in this case) N.

D is the semi-diurnal arc representing half the length of the longest day, in lat. 49° N., and—

D' is the semi-nocturnal arc representing half the length of the shortest night for the same position.

Similarly, D" D"" is the parallel of the sun's greatest southern declination (23° 27' S.), on the 21st of December.

(S) H D" is the sun's meridian altitude on December 21st in lat. 49° N. Z D" is the sun's meridian zenith distance on December 21st.

H (N) D"" is the distance to which the sun descends below the horizon on December 21st.

is the place of the sun's rising and setting on December 21st; and E. r is his amplitude, reckoned from the east point at rising, and from the west point at setting, towards south because the declination is (in this case) S.

'D" is the semi-diurnal arc, representing half the length of the shortest day, and

'D''' is the semi-nocturnal arc, representing half the length of the longest night.

TT' is the small circle, parallel with the horizon H H, and 18° below it, indicating the extent of twilight. The sun is on this circle before rising, at dawn, at the beginning of twilight, and on it again after setting, at the end of twilight.

It has been stated above that the projection of the diagram (Fig. 1, plate VII.) is for an observer's station in lat. 49° N.; it is further projected to illustrate an observation of the sun when the declination is 17° N., as on the 7th of May or 4th of August, 1888.

O is the sun, and dd' is the parallel of his declination (17° N.) on either day.

r" will be the place of the sun's rising and setting; and Er" is the rising and setting amplitude, reckoned from East at rising, but from west at setting, toward north as the declination is N.

S is the sun's place on the six o'clock hour-circle, and V is his place on the prime vertical.

Taking as the place of the sun at between 8h. and 9h. A.M. or between 3h. and 4h. P.M.; then AO is his altitude, and Zo his zenith distance. The dotted line a a' is the sun's parallel of altitude.

C is the arc of the sun's circle of declination representing 17° N., and P (N) is the sun's north polar distance, 73°.

Three sides of a spherical triangle (viz., P (N) Z the co-lat., ZO the zenith distance of the sun, and P (N) the Sun's N. polar distance) are given to find the hour-angle and azimuth; it is customary, however, to use the latitude and altitude instead of their complements.

OP (N) Z is the sun's hour-angle, or meridian distance measured on the equinoctial by the arc CQ; O Z P (N) is the sun's azimuth from the north measured by the arc of the horizon H (N) A; and Z (S) H is his azimuth from the south, measured by the arc of the horizon (S) H A.

(N) POP (S) is the sun's hour-circle, and ZOA N his vertical circle for altitude and azimuth.

(S) H d is the sun's meridian altitude on the day, and Z d his Meridian Zenith distance.

r" d is the semi-diurnal arc representing half the length of the day, and, as d' would be the sun's position at midnight below the horizon, r" d' is the semi-nocturnal arc representing half the length of the night.

"t would be the limit of the duration of twilight, and "s the ascensional difference.

Having in Fig. 1, plate VII. illustrated the various circles of the sphere by means of a diagram drawn on the plane of the observer's meridian, it remains to depict them from another aspect-viz., on a projection in which the observer is supposed to be surveying the entire plane of the horizon from the zenith as his standpoint—hence he sees the whole hemisphere which is above the horizon.

In such a projection (see Fig. 2, plate VII.) the horizon is the bounding circle, with the zenith as its centre; hence all circles that pass through the zenith, as the meridian, prime vertical, and other vertical circles, must appear as straight lines.

NES WN is the horizon, with Z (the zenith) at the centre: Z, and its opposite point at the nadir, are the poles of the horizon.

The cardinal points of the horizon are: N, the north; E, the east ; S, the south; and W, the west.

P is the elevated pole; and N P the elevation of the pole-equal to the latitude of the observer's station.

NZS is the meridian of the observer's station, cutting the horizon in the N. and S. points; E and W are the poles of the meridian, and 90° from every point on that circle. Also, all circles that cut the meridian at right angles meet at the points E and W.

EZ W is the prime vertical, at right angles to the meridian, cutting the horizon in the E. and W. points; N and S are the poles of the prime vertical.

EP W is the six o'clock hour-circle.

EQ W is the equinoctial or celestial equator.

N

Supposing to be a heavenly body; then P is its polar distance; © A is its altitude, east of the meridian; and o A'its altitude, west of the meridian; Z Po (on the right of the diagram) is its easterly hour-angle or meridian distance, and Z Po (on the left of the diagram) is its westerly hour-angle, or meridian distance; SZ A (on the right of the diagram) is its azimuth between south and east, and S ZA' (on the left of the diagram) is its azimuth between south and west; supposing it to have risen at r, it would set at s; having Er for its amplitude at rising, and W s for its amplitude at setting. Again, let the circle represent the equinoctial with the pole as its centre, and all the celestial meridians must appear as straight lines. PQ is the meridian of the place, P S B is the meridian passing through the object S, and PS' C is the meridian through S'. Let A be the position of the first point of Aries, M that of the mean sun, S that of true sun, and S' that of a star; then Q M is the mean time at place, Q B is the apparent time at place, M B is the equation of time, Q M A is the sidereal time at place or right ascension of the meridian, A M is the right ascension of the mean sun, A B is the right ascension of the true sun,

M

S'

A MQ C is the right ascension of the star, Q BAC is the westerly hourangle, and QC is the easterly hour-angle of the star S'.

PREPARATORY PROBLEMS

LONGITUDE AND TIME

The circumference of every circle is divided into 360 equal parts, called degrees; now the earth, whose circumference as a sphere is 360°, turns on its axis in the direction of this circumference once in 24 hours; therefore the 360° of circular measure are the equivalent of 24 hours; dividing 360 by 24 we get 15, and hence 15° of circular measure are the equivalent of 1 hour. We say, astronomically, that as a complete rotation of the earth on its axis is performed in 24 hours, meridians 15° as under are thus brought to the sun at regular intervals of one hour. This gives us the following Table :

This Table furnishes the following rules for converting longitude into time, or time into longitude :

I. TO CONVERT LONGITUDE (OR ARC) INTO TIME, the shortest method is to multiply the degrees and minutes (° and ') of arc by 4, then the minutes (') of arc become seconds of time, and the degrees become minutes of time, which reduce to hours by dividing by 60.

Or the conversion may be made at sight by Table (Norie's Tables, p. 198).

Example.-Convert 137° 26' into its equivalent time.