In order to get an intersection of two lines of position and thereby ascertain the latitude and longitude at once it is assumed that the observer took an observation of another star bearing S. 45° E., simultaneously with Arcturus.
When ordinary A.M. time sights are taken the resulting longitude establishes a north and south Sumner line but the latitude is by D. R.; at noon the latitude by meridian altitude establishes an east and west line but the longitude is by D. R. So it is with a Sumner line a position is established upon it but the position along it is by D. R. The latitude and longitude, however, can be obtained by a slight calculation without drawing the lines on the chart; that is, the most probable position. The altitude difference having been determined enter Table 2, Bowditch, using the azimuth, or its reciprocal as the case may be, as the course, and with the altitude difference as the distance, pick out the difference of latitude and the departure and apply them to the dead reckoning latitude and longitude as is the usual practice. The result is the most probable position (according to the D. R.) on the Sumner line.
CHAPTER X
The Moon
The moon is the most interesting of the heavenly bodies not only from a romantic viewpoint, but from the astronomical as well. Looking at the practical side, it is due mostly to the moon’s influence of attraction on the waters of the earth that we have the highly important phenomena of the tides. The moon is our nearest neighbor in the heavens; in fact, she is a satellite, that is, revolves around the earth. This movement is from west to east at an average rate of 51 minutes each day. The moon’s orbit is elliptical with the earth lying a little out of center, not unlike the situation of the sun in the earth’s orbital ellipse but more pronounced. When the moon is at the nearest point to the earth she is said to be at “perigee” and the point where she is most remote is called the “apogee.”
The moon is a non-luminous body and gives off nothing but reflected sunlight. The lunar hemisphere facing the sun is therefore the only illuminated portion of the body, and as she turns on her axis precisely as the earth does, the same side is always towards us. The astronomers have seen but one side of our satellite. This solar illumination accounts for the various interesting phases of the moon which we see each month. When this body in her monthly revolution around the earth passes between us and the sun, the illuminated side is towards the sun and the dark side towards us. We see no moon at this time and call it New Moon. Two weeks later, she has completed one-half of her revolution and is now on the other side of the earth and we are between the moon and the sun. The illuminated face of the moon is now directly towards us and we call it Full Moon. At the time of new moon, the eastward movement quickly brings her out of range with the sun and in a couple of days we are able to see a fine crescent in the western sky. This is the very edge of the illuminated face—we can see around the corner just that much. Day after day the moon’s lighted surface becomes larger and larger until in about a week she is near our meridian at sunset and therefore at, roughly (depending on the time of the year), 90° from the bearing of the sun. The moon now presents to us a face one-half dark and one-half light. This is called the quadrature. This term also applies to the similar condition occurring a week after full moon when she is again bearing at right angles to the sun. These occasions are also called the first and last quarter, respectively.
The movements of the moon are very rapid. She makes her revolution around the earth in 27⅓ days, making a change in right ascension of 360° or 24 hours in this interval, a change of over two minutes each hour. The declination passes through its whole cycle of change from north to south and return also in 27⅓ days; the sun requires a year to pass through its extremes of declination and return. The change of the moon’s declination averages about 9´ per hour. These facts demand careful attention when employing the moon in navigation.
It is a very curious and happy circumstance that in the higher latitudes when the short days of the winter sun occur, the moon at full rides its highest declinations, and consequently gives extra long nights of moonlight; and that in the summer, when the sun is in higher declination and the days are long, the moon at full is in low declination and there is less moonlight when it is least needed. The reason of these conditions is that the full moon occurs when on the opposite side of the earth from the sun and at the winter solstice when the earth’s north pole is inclined away from the sun she must be inclined towards the moon passing that body in high declination. The reverse conditions exist at the summer solstice.
Another fortunate provision for lovers of moonlight nights is the fact that the plane of the moon’s orbit is not in the same plane as that of the earth’s orbit, for if such were the case each time the three bodies, the earth, sun and moon, came in range there would be an eclipse. The new moon coming in between the earth and the sun would cause an eclipse of the sun, and at full moon when the earth is between the sun and moon, there would be an eclipse of the moon. Therefore, there would be an eclipse twice a month. This fortunately is avoided by the angle of 5° that the plane of the moon’s orbit takes with that of the earth. As a result they only come in exact range occasionally when the moon at new and full happens to be on the ecliptic—the earth’s orbit. If, to repeat, this occurs at Full there is an eclipse of the moon, if it occurs at New, there is an eclipse of the sun. The moon moves eastward through the heavens on her monthly course of revolution; it then becomes apparent that she must return to the meridian later and later each day the amount of “retardation,” as it is called. This retardation is a variable quantity dependent upon the moon’s irregular change in right ascension. It is caused by the moon’s motion in her elliptical orbit and at the inclination which her orbit takes with the celestial equator. These causes are precisely the same in character as those producing the equation of time in the conditions relative to the sun and the earth’s orbit, but those of the moon are much greater. The errors causing a variation in the right ascension of the sun requiring a year where the similar conditions in the moon are brought about in a month, which accounts for the marked changes in the moon’s rate of eastward motion. The average daily retardation, or average later time in arriving at the meridian, is very close to 51 minutes. Yet the extremes of retardation range from 38 to 66 minutes. The average of 51 minutes daily retardation is also noticed in the later rising and setting of the moon. The extreme times between successive risings or settings during the year, while they average 51 minutes like the crossing of the meridian, they do not maintain the same extremes, changing on account of the latitude of the observer as well as upon her own motions. At 41° north the retardation on successive risings and settings ranges between 23 minutes and 1 hour and 17 minutes. As the vessel proceeds farther north the range is greater until near 66° north when the moon is in her average greatest declination north she does not set at all becoming circumpolar for a certain time each month. In the duration of a month the moon changes her right ascension 24 hours, where the sun takes a year to accomplish this amount as it (apparently for navigational purposes) moves eastward around the earth. This shows the much more rapidly increasing change in right ascension in the case of our satellite. Thus again the moon’s rapid motions are accounted for.
The moon’s orbit around the earth is not coincident—does not lie in the same plane as the earth’s orbit (the ecliptic) but takes an angle of about 5° 8´ with it. The point of intersection between the moon’s orbit and the ecliptic are called nodes (corresponding with the equinoxes). The point crossed by the moon as it passes from southern to the northern side of the ecliptic is called the ascending and the other the descending node. The moon’s axis is very slowly describing a circle in the heavens similar to that of the earth; and in consequence the nodes are slowly moving westward along the ecliptic year by year. Just as is the equinox by the movement of precession, but at a much greater rate (see remarks on precession elsewhere). The moon’s axis completes its revolution in about 19 years, while the earth requires 26,000 years. This is called the lunar cycle. At the time in the lunar cycle when the ascending node of the moon’s orbit is in range with the vernal equinox the moon has her greatest range of declination—about 57° from extreme north to extreme south. She is then 23° north, the amount that the ecliptic is from the equator and 5° more, the amount that the moon’s orbit is above the ecliptic. About 9½ years later when the moon’s axis has listed in the opposite direction and the descending node coincides with the vernal equinox, the moon’s maximum declination equals 23° minus 5° or 18° north or south, a range of only about 26°.