Fig. 4.
At the equinoxes, where the ecliptic and equator cross and the solstices—the vertices of the ecliptic—that is, four times a year, the true and mean sun are together, but departing from these points they do not travel with the same right ascension, remembering that right ascension is measured on the equator. Taking, for example, the earth in that quadrant of the orbit comprised between the vernal equinox and summer solstice, the apparent sun in the heavens would be by cause of obliquity alone, to the right or to the westward of the mean sun, and thus it will be seen that with the earth rotating from right to left the apparent sun will cross the meridian first; consequently between March 21st and June 21st that part of the equation of time due to the obliquity of the orbit bears a minus sign when mean time is desired from the apparent time. This correction reaches its maximum half-way or 45° from the equinox, amounting at that point to nearly 10 minutes.
Fig. 5.
Now the reason for this difference between the mean and apparent sun when each (so far as this problem is concerned) moves along its respective path—the equator and the ecliptic—at the same rate, is this: suppose the equator between the equinox and the solstice is divided into an equal number of parts and an hour circle drawn through each point of division. Beginning from the equinox (the common apex of the triangle) the arc of each hour circle between the equator and the ecliptic, forms the altitude of a right-angled triangle, while the equator and the ecliptic are base and hypothenuse respectively. Thus it will be seen that each portion of the equator (base) is shorter than the corresponding part of the ecliptic as defined the hour circle, to the extent of the ratio of the base to the hypothenuse.
This amount increases with the increasing size of the triangle, but a new element enters to counteract its effect. With the increasing divergence of the ecliptic and equator, the divergence of the hour circles becomes a factor and as the solstice is approached the divisions on the equator, are represented on the ecliptic by gradually decreasing spaces between the hour circles.
The combination of these effects produces the error due to the obliquity of the orbit. The error has the opposite effect in the next quadrant, that is, from June to September; and in opposite quadrants it is the same.
So it will be seen that error due to the eccentricity causes the apparent sun to lead the mean sun from December 31st to June 30th, reaching a maximum of 8 minutes about April 2d. This sun then falls astern until December, again attaining a maximum of -8 minutes about October 1st. The error due to the obliquity of the orbit accumulates between the equinoxes and solstices for at these points the two suns are together and there is no error, but about the 6th of May, August, November, and February, it reaches a maximum of 10 minutes; in August and February, the mean sun takes the lead.
These two errors of equation of time combined algebraically will result in the plain line of the diagram.
Calendar
The ancients, in order to keep track of past events and to anticipate the future, devised a calendar, which, while not identical with the one now in use, was of itself a remarkable production. They chose the revolutions of the moon as their basis of measuring the passage of time, and, as a lunar month contains only 27⅓ days, the 12 months used in this early calendar comprised a year of about 354 days.
It became apparent as the world progressed that use of the moon was very unsatisfactory for this purpose, as the calendar became complicated and confusion resulted, owing to the difference between the lunar and solar years. This condition remained until the reign of Caesar, when that monarch determined to establish a more satisfactory method of reckoning time. With the aid of an eminent astronomer, he completely revised the calendar, using the revolution of the earth around the sun as the standard for the length of a year. The time required for this is 365¼ days, approximately, and as it was inconvenient to include the ¼ day in the year, it was allowed to accumulate for 4 years, when as a whole day, it was added to the end of February.
Caesar, evidently proud of this accomplishment, honored himself by naming his astronomer’s invention the Julian Calendar and in order to further immortalize himself he changed the name of the seventh month to July. Augustus, his successor, apparently envious of the honor his predecessor derived from this source, and determined not to be thus outdone in perpetuating his name, changed the month Sextilis to August.
The commonly accepted 365¼ days as the length of year is only an approximation, however, and the small difference between this and the actual length of a year began to accumulate until this weak point in the Julian Calendar became a matter of moment. The exact length of a year from vernal equinox to vernal equinox is 365 days 5 hours 48 minutes and 46 seconds, which lacks just 11 minutes 14 seconds of 365¼ days. This caused the dates of the equinoxes and solstices to keep slipping back 11 minutes each year and when considerable time had passed the difference became large enough to cause inconvenience; the date of the vernal equinox having dropped back to March 11th in 1582. In this year Pope Gregory, acting under the advice of an astronomer by the name of Clavius, modified the Julian Calendar. He first added 10 days to restore the date, and then to forestall a further retrogressing of the calendar, provided that only those century years divisible by 400 should be considered leap years. In this way the 11 minutes 14 seconds was prevented from causing further mischief. This calendar known as the Gregorian Calendar is now in almost universal use, though at first it was adopted only by Catholic countries.
It is interesting to note that the time consumed by the sun in making his apparent yearly revolution from a certain star back to that star again is a sidereal year of 365 days 6 hours 9 minutes 9 seconds. The tropical year—the one in common use—is shorter, being the time required for the sun to leave and return to the vernal equinox, or First Point of Aries. This point, it will be remembered, is moving westward about 50´´ annually, and it will be seen that while the sun starts its eastward revolution among the stars, the equinox is very slowly moving westward to meet him, thus making the tropical year about 20 minutes shorter than the sidereal year.
While discussing the calendar it is an opportune time to explain a matter concerning the dates of the equinoxes and solstices. It has of course been noticed by everyone that the vernal equinox occurs one year on March 20th and another on March 21st, or the summer solstice on June 22d and yet another year on June 21st and so on.
Aside from the slight change due to the dropping back of the seasons in the orbit by the precession of the equinoxes, the actual time of the equinoxes and solstices may be considered as constant, yet the dates vary a few hours.
The year in common use—the tropical year—contains approximately 365¼ days, yet we take account of only 365 days, the extra ¼ day being laid aside for future reckoning. During the next year this 6 hours will be augmented by 6 more; the next by another 6, making 18 hours ahead of the calendar. The fourth year this amount reaches 24 hours, and a full day, the 29th of February, is added to the calendar for that year, and we are square again. But the different equinoxes and solstices occur at just 364¼ days; taking for example the vernal equinox, it occurs on the following (approximate) dates, which it will be noticed are 6 hours later each year:
It is evident by the above that the insertion of the extra day just previous to the equinox in the leap years, sets the date of the equinox back a day by the calendar. Juggling the ¼ day as shown above causes the change in the calendar dates of these phenomena.