differences. The time required may however be ascertained within certain limits; and if such observations be taken at great intervals of time, any inaccuracy in the estimate will be of less importance. There are however better methods of making the computation. The principle on which they depend is very simple. We have seen that the radius vector of the sun describes equal areas in equal times. Now the only straight line which can be drawn through the focus of an ellipse, so as to divide the ellipse into two equal parts, is the transverse axis. The sun's position at one extremity of any line passing through the earth is distant by 180° from its position at the other extremity; if, therefore, he be observed at any two points 180° distant from each other, he is then at the two extremities of a line passing through the focus of the ellipse, and the portions of the ellipse on each side of that line must be unequal, unless the line be the transverse axis which passes through the apogee and perigee. If the portions are unequal, his time of passing through them is unequal also: if the times, therefore, are found to be equal, the instant of observation is the instant of his being in perigee or apogee: if unequal, it is not so, but the instant of his being in perigee or apogee may be ascertained from the observations made, by calculations of which it is not necessary here to enter into any detail. The general result is that the perigee or apogee (or the apsides of the sun's orbit, as they are also termed) have a progressive motion on the ecliptic of about 11". 8 annually. The longitude of the perigee and apogee therefore increases at the rate of about 62" annually, for it increases by the actual motion forwards of those points themselves, and also by the whole amount of the retrocession of the equinox, from which it is measured; or by the sum of the two quantities 11". 8 and

50". 1.

The principle above referred to, that any inaccuracy of observation may be rendered less material by comparing observations made at distant periods, is one of very great and general importance. If we suppose ourselves unable to discover the exact situation of the equinox or the perigee within a minute, it is plain, assuming the amount we have already assigned to their motions to be correct, that we cannot, by observing two successive positions of the sun in either of these points, form any

estimate of the precession of the equinox or the progression of the apogee on which we can at all rely; for the motion of the point in the interval between the two observations will be less than the probable error of the observations themselves, and we shall be unable to tell whether the difference between the observed positions is the effect of a motion in the object observed, or merely the result of the inaccuracy of observation. If however there be any continuing motion, the distance between the places of the point observed will be increased if the distance between the times of observing be so; and it may therefore become greater than the amount of any probable inaccuracy of observation. For instance, taking the case of precession, in twelve years the equinox would have receded about 10', a quantity much greater than that which we have supposed to be the limit of the errors of observation; and therefore, from observations made at this interval of time, we should be able to pronounce with certainty that a retrograde motion existed. Still we should be unable to determine its amount with any great accuracy. Having supposed the probable error of observation to be l', we could not say whether the real retrogression might not be either 9' or 11', quantities respectively 1' less, and greater than the observed retrogression of 10'; and as this would be the retrogression of twelve years, the annual retrogression might be as little as 45", or as great as 55". This would be a great and important uncertainty; but it may be very greatly diminished by a further extension of the same principle, that is to say, by taking still longer intervals of observation. For instance, let us suppose the interval 400 years, and the observed precession 5° 34', or 20,040". If we suppose, as before, that the amount of probable error is 1', the least possible amount of precession will be 5° 33', or 19,980"; and the greatest 5° 35', or 20,100"; and the least possible annual amount 49". 95, and the greatest 50". 25; two quantities approaching very near to each other, and the mean between them, or 50" 1, would probably differ very little indeed from the true value.

The principle on which this power of approximating to a correct value depends is obvious. The error of observation, whatever its amount may be, occurs only at the times of observation.

If the element whose value is sought occurs only once during the interval, it is affected by the whole error; if it occurs oftener, the whole result is affected by the whole error, but the whole result being the sum of so many repetitions of the element, the value of the element itself is only affected by a proportional part of the error, the whole error being divided among all the repetitions. If therefore the element be sufficiently often repeated, the proportion of the error involved in its value will become exceedingly small; and this may be the case, even when the actual amount of the probable error of observation is increased, if the number of repetitions among which the error is to be distributed is increased in a greater proportion.

SECTION 9. Of the different years Equinoctial (or tropical) year-Sidereal year-Anomalistic year-Construction of the calendar-Julian correction - Julian year-Leap yearGregorian correction-Persian correction.

WE now see that there are three different periods at which the sun may, in different senses, be said to return to the same position: when he returns to the same equinox at which he was before; when he returns to the same spot in his orbit; and when, having been in perigee or apogee, he returns to it again; or, which is the same thing, when having been at a given distance from any of these points, he returns to the same point with respect to them. Each of these may be said to be the completion of a revolution of the sun; and a revolution of the sun is called a year. The year from equinox to equinox is called the equinoctial year, or sometimes the tropical year; for his time of returning from tropic to tropic, they being situations always holding the same relation to the equinox for the time being, is obviously the same as that from equinox to equinox. The year from any point in the ecliptic to the same point again is the sidereal year, for the sun is then in the same position as before, with relation to the stars. The sun's angular distance from the apogee is called the true anomaly, and the period between his leaving and returning to a given situation with respect to the apogee is therefore called the anomalistic year.

It is evident that the equinoctial is the shortest, the anomalistic the longest

of these years. When the sun starts from the equinox, it is a given point of his orbit; before he returns to it, the equinox has receded on the ecliptic, and he therefore meets it again sooner than he returns to the same spot in his orbit. The effect therefore of the retrograde motion of the equinoctial point on the ecliptic is to bring forward the time of the equinox (or the instant at which the sun is upon the equator); and hence, as we have already mentioned, the phenomenon is known by the name of the precession of the equinoxes. In the mean time however, the apogee has moved forward on the ecliptic; and the sun therefore, after returning to the same spot in his orbit where he was at the former equinox, has still a further arc to describe before he arrives at his original position with respect to the apogee, and the time of his doing so is of course later.

The mean length of the equinoctial year is 365d 5h48m 51.6, (or, expresssolar time. After this, the sun has ing it decimally, 3654.242264) of mean to describe 50".1 to return to the same point of his orbit at which he was at the commencement of the year, or to complete the sidereal year; and the mean length of the sidereal year is thus made 365d 6h 9m 11.5, or 365d 256383. He then has to describe a further arc of 11.8 to arrive at his original position with respect to the apogee, and the length of the anomalistic year is thus made 365d 6h 13m 58.8, or 365d 259708. In ascertaining all these lengths, it is of course important to resort to the principle explained at the end of the last section, and to deduce their value from observations made at long intervals, so that any error of observation may be distributed among many periods of the length required.

The lengths assigned to the equinoctial and sidereal years are only mean lengths; that given to the anomalistic year is a true one. We shall hereafter shew, from other considerations, that the length of the anomalistic year does not vary. For the present, we will assume that fact, and then it is obvious that the length of the equinoctial and sidereal years must continually vary; for each of these years is shorter than the anomalistic year by the time which the sun takes to describe a given angle of his orbit; in one case 62", in the other 11.8. Now the rate of the sun's motion is different in different parts of his orbit, faster as he is further from

the apogee, slower as he approaches it; and, consequently, his times of describing these spaces of 62" and 11.8 continually vary, as they are differently situated with respect to the apogee. The times therefore which are to be subtracted from the uniform length of the anomalistic year, to ascertain those of the equinoctial and sidereal years respectively, are themselves of variable duration; and the lengths of the equinoctial and sidereal year are necessarily so too. The variation however is very small, and the mean differs from the true length at any period by a very inconsiderable quantity.

It is obviously necessary, for many purposes, not only of chronology and history, but even of personal and domestic convenience, that we should have the means of dividing time into definite periods of considerable length; and the most obvious and natural period to adopt, is that which includes all the various operations and appearances which succeed each other in regular order, which comprehends seed-time and harvest, summer and winter. All these are included in the space of an equinoctial year. The position of the apogee, as we have already seen, has some effect on the length of the seasons; but it is the position of the sun with respect to the equinox, which determines what the season is, and in his passage through his whole round from one equinox till he returns to the same again, he occasions the whole variety and succession of spring, summer, autumn, and winter. The length of this revolution therefore has been adopted as the unit of long duration; and a period, assigned with more or less accuracy, but intended to represent this duration, has been uniformly adopted by all civilised nations, and called by the name of year, or the civil year.

It would be productive of great in convenience, if the beginning of the year did not correspond with the beginning of a day; and this would be the case, if we took the exact period of 365 5h 48m 51.6 for the length of the civil year. If, for instance, at any given time, the beginning of the year exactly corresponded with the beginning of a day, the following year would begin 5 48 51.6 after the beginning of the 366th day; the year after that would begin 11h 37m 43 2. after the beginning of the 731st day, and so on. We should therefore continually have days belonging in part to two different years; and

our chronology would become confused and inaccurate. It is obviously better to fix some length for the civil year which shall be free from these inconveniences.

On the other hand, it is evident that if the length assigned to the civil year were to differ materially from the true length of the astronomical year, great confusion would before long be produced. Thus if it were shorter by a whole day, in 100 years its commencement would precede that of the astronomical year by 100 days, more than a quarter of the whole year; or if it were shorter by a quarter of a day, the same error would be produced in 400 years. Thus the commencement of the civil year would, at these intervals of time, correspond to completely different stages of the seasons; if, at the one period, it were at mid-winter, at the other it would be early in the autumn; if at one it were seed-time, at the other it would be the depth of winter. This would obviously be inconvenient. We want to know, when we hear that an event took place at a particular period of the year, whether it took place in spring, summer, autumn, or winter; if a battle, whether at the opening or the close of a campaign; if the discovery of a country, whether during the season of abundance or scarcity.

It is therefore desirable so to fix the length of the civil year, that its commencement shall always be at the beginning of a day; and at the same time to adopt some mode which shall prevent it from ever being far distant from the commencement of the astronomical year. In this consists the adjustment of the calendar; and many attempts have been made, at different times, and in different places, to establish an accurate and complete one. We need not here enter into the history of all these attempts; but there is one so famous, both from the celebrity of the man under whose auspices it was made, and as the ground-work of the adjustment now used, that it may be well to mention it, especially as the simplicity of the numbers involved in it makes it the easiest example which can be given of the principle of such a correction.

The Roman Calendar had fallen into great confusion from the causes already explained, when it was determined to remedy the inconvenience which resulted from its condition, and to rectify it for the future. This was done under the auspices of Julius Cæsar, and it is in

consequence called the Julian correction; Sosigenes of Alexandria was the astronomer who made the calculations. The length of the equinoctial year was not then accurately known; it was however supposed to consist of 3656", (an amount, as we have seen, not very far from the truth), and that time was assumed as its accurate length. Now four years, each of 365 6h, together make up a period of 1461 days; and these 1461 days may be divided into three periods of 365 days, and one of 366. If therefore these successive civil years were made to consist each of 365 days, and then a fourth were made to consist of 366, the length of the four would be equal to that of four astronomical years; and if at the beginning of the first year the commencement of the civil and astronomical years coincided, at the end of the fourth they would do so again. It is true that the civil years would be of unequal length, and that none of them would accurately correspond in duration with the astronomical year, three of them being each six hours shorter, the fourth eighteen hours longer than it; and it is true also that the commencement of each, except the first of the four years, would not coincide with that of the astronomical year, that of the second being six, of the third twelve, and of the fourth eighteen hours before it. None of these inconveniences however were material; the first would not be felt, as soon as the people became familiar with the order in which the longer and shorter years succeeded each other; the second would only be felt by astronomers, and they would have abundant means of removing any difficulty which it occasioned; and the third, although of serious importance if the error accumulated so as to produce a great interval between the commencement of the different years, was of no moment, when, as it seemed, the error would never exceed eighteen hours. On these grounds the ordinary length of the civil year was fixed at 365 days, but it was ordained that every fourth year should consist of 366. The fourth year was called bissextile, from the manner in which the additional, or intercalary day was inserted*. It corresponds to our leap year.

The Romans called the first day of each month the Calends (Calenda) of that month. Hence the word, Calendar. They then reckoned the latter days of the preceding month, by their distance from the Calends of the following one. Thus the 1st of March being the Calends of March, the 28th of February was the day before the Calends

If the length of the astronomical year had been accurately 365d 6, this adjustment of the calendar would have been sufficient. But the true length of the equinoctial year is only 365d 5h 48m 51.6; or it falls short of 365d 6h by 11m 8.54. Four equinoctial years therefore would fall short of four years of 365d 6h each, or of four Julian years, three of 365 and one of 366 days, by the space of 44m 335.6; and 100 equinoctial years would fall short of 100 Julian years by 25 times that space of time, or by 18h 34m; and 400 equinoctial years would fall short of 400 Julian years by 74h 16m, or by a little more than three days. In consequence of this inaccuracy a further correction was required; and this was carried into effect by Pope Gregory XIII. in the year 1582, at which time the vernal equinox fell ten days earlier in the civil year than it had done in the year 325, at the Council of Nice, the period which he chose for the correct standard of the commencement of the civil year. At the Council of Nice, the vernal equinox had been on the 21st March; at the time of Pope Gregory, it was on the 11th; but he omitted ten days in the current year, and made the 15th of October in that year immediately succeed the 4th, so that in the next year the vernal equinox again took place, as it had done in 325, on the 21st of March. In this manner the error which had already taken place was rectified. To prevent it from again occuring, the following additional adjustment was devised.

We have seen that the Julian correction made the length of 400 years too great by a little more than three days. This inaccuracy then would very nearly be removed, by omitting the additional day in three years, taken in the course of the 400. It is of great practical convenience to have these corrections easily remembered, and the three days were in consequence omitted in three of the four years which completed centuries. It was determined that, dividing time into portions of 400 years each, every fourth year, except those which terminated the first three centuries of such a period, should be of 366 days, but that those three, like the common years, should each be of 365 days only. Thus the years 1600,

of March, the 27th, the third day of the Calends of March, and so on. The 24th of February, therefore was the sixth (sextus) day of the Calends of March. The fourth year was lengthened by adding an additional day; at this time there were two days reckoned as the sixth day of the Calends of March, and hence the year was called Dinsextile, as having a double sixth day of these Calends.