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in the Southern than the Northern part of his orbit, the days during which he is longer below than above the horizon, are not, in places North of the equator, quite so many as those when he is longer above it. In the same manner, if the circle eq is entirely above the horizon H, N., the circle e' q' is entirely below it; and thus the period, during which the sun never sinks below the horizon, is counterbalanced by a period during which he never rises
tions, that they divided the whole earth
Taking the extreme case, when the pole is in the zenith, these periods are each of them from equinox to equinox, or they comprehend each half the sun's course, and they only differ therefore, like the others, by the small inequality occasioned by his variable rate of motion. When the pole is in the horizon we have already seen that the day and night are always equal. Neglecting therefore the slight inequality we have mentioned, (which makes the whole period during which, in the course of a year, the sun is above the horizon, somewhat longer when the North Pole is so, and shorter when the South Pole is, and thus tends to render the heat of the former greater than that of the latter climates) we may consider that everywhere the sun is half the year above, and half below the horizon; and his influence to produce heat throughout the year, and consequently the average heat, as far as he occasions it, will depend upon his average elevation above the horizon, and be greatest where that is greatest, and least where that is least. It would however lead to very complicated investigation, if we were to endeavour to determine the manner in which this average elevation differs at different places. It is sufficient to state generally that it is greatest when the pole is in the horizon, and continually diminishes as the elevation of the pole increases; or the influence of the sun for the whole year is greatest when the latitude is nothing, it gradually diminishes with the increase either of North or South latitude, and is least where the latitude is of 90°. And it is familiarly known to every one that these deductions actually correspond with the general results of observation; that, generally, the countries near the equinoctial line (for so the line where the latitude is nothing is called, the day and night being there always equal) are the hottest, those in high lati tudes the coldest. The ancients indeed put so much faith in these considera
*Our intercourse with our North American possessions furnishes a still more striking example.
The mouth of the river St. Lawrence is in the latitude of about 49° N., more than 2° south of London; and this is the most Northerly point of the river; yet all access to Canada is stopped by the
frost which closes the river at an early period of the winter, and the ice does not break up, and navigation recommence, until the month of May.
SECTION 7.-On the equation of Time
Inequality arising from unequal mo-
A GREAT number of the most important elements involved in astronomical researches are variable in their amount. Their variations, however, generally succeed each other in a certain order, and are confined within certain limits; and when these limits, and all the varying values are ascertained, it is of course possible to take an average among them, and this average value is termed a mean value.
It is indeed always possible to take an average between any number of observations, or of ascertained values of a particular element; but unless the observations are so taken that the whole course or cycle of the variation is included, it is not usual to call the average a mean value; or rather it is not the absolute mean value of the thing itself, it is only its average or mean value for a certain time. For instance, we have already seen that the length of the day, considered as the interval from sunrise to sunset is continually varying, but that it goes through all its changes in the interval from one solstice to another. Its average duration for this whole time, then, is its mean value: it is a little more than twelve hours, and it is very nearly the same everywhere; it would be everywhere exactly twelve hours, if the sun always moved at the same rate, and there were no parallax or refraction. An average, however, might be taken of the lengths of this day for a portion only of this interval, for instance, from the vernal equinox to the summer solstice; in this case, the shortest length of the day would be a little more than twelve hours, and the average length at London about fourteen hours and three quarters. This would be a correct average of the lengths observed; but as the time of observation would not comprehend all the variations of the element in question, it would not be the mean length of the day absolutely, though it might be called the mean length for the period of observation.
In the same manner as we have taken an instance of mean duration, we might have an instance of mean motion; that is to say, if a body moves with a variable motion; but if the whole course
of its variation is ascertained, its average rate of motion during this whole course may be found, and this will be called its this mean motion, and of course moving mean motion. A body moving with uniformly, for the whole time occupied by the whole series of the real motions, would move through the same space as the real body, but its place at many, or all intermediate periods, would be different from the place of the real body, on account of the difference between the real and mean motions. The place of a body so moving, or the place which the real body would occupy on the supposition that it moved uniformly, and described in the time occupied by the whole series of its real motions the same spaces which it actually does, is called the mean place of the body. In the same manner an event which happens at various intervals which succeed each other in a certain and recurring order, will have a mean time of occurrence.
Now it very generally happens in astronomy that it is less inconvenient first to compute the mean place of a body, or the mean time of an event, and then to ascertain the difference between the mean and the true, than to go through the computations necessary to find the true time and place in the first instance.
When once the mean values have been ascertained, the mean motion of a body during a known period, its mean place at a known time, the mean time of the occurrence of a given event, are easily found; for the intervals of the mean time, and the rate of the mean motion being always the same, we only want to know how often the event has occurred, or how long the motion has been continued. If, from consideration of the manner in which the difference between the true and mean values arises, we can ascertain the amount of that difference in each particular instance," we can find what is to be added to or subtracted from the mean value to arrive at the true; and the quantity so added or subtracted is called an equation. The mean value thus leads to the true value, and of course it furnishes an approximation to it; and as the subjects of astronomical inquiry generally have their variations confined within narrow limits, so that the difference between the true and mean motion's times and places is not very great, the approximation is not very distant.
We shall find several instances of the application of the terms above explained,
and of the use made of these mean values and results in treating of the equation of time, of which we have still to speak, and then the more obvious appearances of the sun, and their principal effects, will be for the present sufficiently explained.
The solar day is longer than the sidereal day in consequence of the motion of the sun Eastward in his orbit. It is evident that the degree of its excess above the sidereal day must be affected by the quantity of that motion, and must, when other circumstances are the same, be greatest when that motion is greatest, or when the sun is in his perigee, and least when that motion is least, or the sun is in his apogee. The motion of the sun goes through all its variations in the course of one revolution of the sun in his orbit; it admits, therefore, on the principles already explained, of a mean value. Let us call the sun S, and let us suppose a fictitious body, which we call S1, to move uniformly in the ecliptic, and to perform a complete revolution in the same time as S: the motion of S1, therefore, will be the mean motion, and its place, the mean place, of S. The difference between the places of S and S, will be an equation; it is called the equation of the centre. Let us suppose the two bodies, S, and S1, to be together when the sun is in apogee. As the revolution of the supposed body S1, is completed at the same time as that of S, they will be again together at the end of the year, or when the sun returns to his apogee. Besides this, the times of the sun's motion from apogee to perigee, and from perigee back to apogee, are equal; they are therefore each equal to half the time of his whole revolution, or to half the time of the revolution of S, or to the time taken by S1, which moves uniformly, to pass through half the ecliptic. But the sun's apogee and perigee are at the distance of half the ecliptic from each other; consequently, as S and S1, were together at the apogee, they are so also at the perigee, each taking the same time (half of the year) to pass through half of their orbits. We find therefore that the real and mean places of the sun coincide at the apogee and perigee.
It is also plain that they correspond nowhere else. The sun's distance from the earth continually decreases from apogee to perigee, and his angular velocity continually increases during the same period. It is evident then that as
his whole real angular motion for that riod is equal to his mean angular motion for the same time, the real motion will at first be less, and afterwards greater than the mean motion. His real place therefore will at first fall behind his mean place, and the distance between them will increase day by day until his real motion becomes equal to his mean motion; the distance will then diminish as the real motion becomes greater than the mean motion, and this excess will finally bring them together again. It will not however do this until they reach the perigee, for we have already seen that they are then together, and this could not be the case if they had been so before; for as they are only brought together by the real motion exceeding the mean motion, and as the real motion continually increases from apogee to perigee, if at any period before the perigee S had come up with S1, at the following instant S would have passed S, by the excess of its motion, and would have continued from day to day to increase the distance between them by the continuing and growing excess of the real above the mean motion; and the consequence would be, contrary to the fact, that S would arrive at the perigee before S1. We arrive therefore at this conclusion, that the real place of the sun is behind his mean place, as he passes from apogee to perigee, the distance between them continually increasing for a certain time, then continually diminishing till the two places again coincide at the perigee. Exactly in the same manner we find that the real and mean place never coincide from perigee to apogee, only with this difference, that as in this half of the orbit the real motion is at first greater, and afterwards less, than the mean motion, the real place is always before the mean place, until, on the return to the apogee, they both again correspond. The greatest difference in the time of the approach of S and S, to the meridian cannot exceed 8m 248.
It is not however only by the variation in the sun's rate of motion, that the length of the solar day, or the interval between his successive appearances on the meridian, is affected; it is also influenced by the inclination of the ecliptic to the equator. The motion of rotation of the heavens is uniform; and consequently, if we suppose another fictitious body, S2, to be at the point at the same time with the supposed body S, to
90° also, or Z Y, the arc of a great circle which joins the points ZY, is perpendicular to the equator op Y at Y, and therefore is part of a meridian, PZY, passing through those points. Z therefore, being a point upon the same meridian as Y, comes upon the meridian of the place at the same time with Y, and, as the bodies S, and S. are at Z and Y at the same time, they will there come on the meridian of the place together. But Z, being the middle point of the arc Z, is the sun's place when in the tropic.
move uniformly Eastward in the equator, and to pass completely round the heavens in that circle in the course of a solar year, as this body will every day have moved Eastward by an equal portion, it will always come to the meridian later by an equal period than the point which it occupied on the preceding day; and as this point returns to the meridian in a sidereal day, which we have seen to be always of the same length, and the excess of time above this sidereal day before S returns to the meridian is also equal, the intervals be tween the successive arrivals of S on the meridian are themselves equal; and each of them, as S, performs its whole revolution in a solar year, must be equal to a mean solar day. We now then have two fictitious bodies, S, and S2, each moving uniformly, the first in the ecliptic, the second in the equator; and, as these circles are equal, the actual motions of each are equal. Still we have to inquire whether their periods of arriving at the meridian are the same. For this purpose let us suppose (in fig. 6) that P represents the pole of the heavens, Yhalf the equator, Zs, then S, will come on the meridian of half the ecliptic, (for these circles bisect each other in the points and,) and let Y and Z be the points of bisection of the arcs Y, Z, respectively. Each of the arcs, Y, Z, therefore is 90°, or they are equal; and, consequently, the fictitious bodies, S, and S2, which were together at op, and whose
rates of motion are equal, would arrive at the same time at Z and Y. Besides this, each of the arcs, op Y, Z, being 90°, the angles op ZY, Y Z (by a wellknown property of the sphere) are each
From the time when the body S, leaves the point, until its arrival at Z, we shall find that it will always come upon the meridian of the place before S. To show this, let S, represent any intermediate position of that body, and let PS1s be a meridian drawn through it; of course therefore the points $1,8, come upon the meridian of the place together. If therefore s be the place of the supposed body S, at that time, the bodies S1,S2, will then come on the meridian of the place together; but if the body S, have advanced beyond the point
the place later than s, and consequently than S, for s and S, arrive there together. But the actual motion of S, is equal to that of S1, or to S1. Whenever therefore S, is greater than ∞s, the body S, is more advanced than the point s, or comes on the meridian later than S1. Now it is a general property of all spherical triangles, as well as plane ones, that the greater side is opposite to the greater angle; and in the spherical triangle S, s, as long as the side ops is less than 90° or than op Y, the right angle S, s is greater than the angle S1s. Until S, and S2 therefore arrive at the points Z,Y, S1, which is opposite to the right angles S1, is always greater than ops, and conse
[The arcs Z, Y, being each 90°, the point is the pole of the circle Z Y, and consequently the arcs Z, Y, are secondaries to Z Y, or perpendicular to it.]
cos.s, by Napier's
s is less than 90°, Z or Y, cos. Y s is
+[Cos. S, s=sin. S, s, rules. Whenever, therefore, or at all points between Y and positive, and consequently cos. S, s is so likewise, ors, is less than 90°; for the other factor, sin. S, s, is positive, since S, s is 23° 28'. When s is 90°, or Y, cos. s = 0, consequently cos. Y S1 s=0, or YS,s (in this case When s
2Y)=90°, as we have already seen.
is greater than 90°, cos. Y s is negative; therefore cos. S, s is so, or PS, s is greater than 90° or
than s S, and consequently S, is less than Cs, or the body S, comes later on the meridian than S, as is afterwards independently shown in
quently S, always comes to the meridian the meridian of the place continually after S, in the course of their passage increases from the equinox or tropic till from op to the points Y and Z, or during the sun's passage from the equinox to the tropic.
In passing from Z and Y to the other equinox the result will be exactly contrary: for in this case, exactly in the same manner as in the other, if a meridian be drawn through the sun's place on any day, the arc of the equator intercepted between it and the equinox will be less than that of the ecliptic so intercepted; and consequently, the point of the equator which comes on the meridian of the place at the same time with S1 will be nearer than S,, and consequently than S 2, to the equinox, towards which the bodies are moving. The interval of time therefore between the arrival of S. and the point on the meridian of the place, will be greater than that between the arrival of S, and the point
Taking then S, T, to represent this uniform motion-
st (the effect in right ascension) =
S, V (for cos. S,s they are similar arcs of parallel circles, of which the radii are to each other
as 1 : cos. S, s.)
S, T. sin. S.TV
(for the very small tri
angle S, T, V, may be considered as a plane triangle, and the angle S, VT, is a right angle.)
S. T. sin.Ss
(for the angles S, 8,
TV as the meridians, Ps, Pt are very close to each other, may be considered as equal.)
But cos. Ss=sin.YS, s cos. S, s, or sin.s, s=
; and, consequently, st=S, T
cos. obliquity of ecliptic
-, a quantity which continually increases, as the declination does so.
At the equinox, cos. declination = 1, and st= S, T, cos. obliquity of the ecliptic. At the solstice, the declination of the ecliptic: and then, therefore, cos. obliquity of the ecliptic cos. obliquity of the ecliptic S. T
cos. obliquity of the ecliptic.
there; and as comes on the meridian of the place after them, of course S, which precedes it by the longer interval, will arrive there before S. We see then that in proceeding from the tropic to the equinox, Se always comes on the meridian of the place before S1. The same conclusions may be deduced in exactly the same way for the remaining two quadrants of the ecliptic and equator: and we draw the general conclusion, that when the sun is either at an equinox or a tropic, S, and S2 come on the meridian of the Hence log.cos.declination= place together: but that while he is moving from equinox to tropic, S, is always earlier on the meridian of the place than S, and always later while he is moving from tropic to equinox.
To find the point where the effect of the motion of S, in right ascension is equal to the motion of S itself, or st S, T, we have this equationcos. obliquity of ecliptic cos, declination cos. obliquity of ecliptic
tan. S, s tan.S.Y
Again, sin. 8- cot. S, stan. S1s
It is further found, by a process of computation sufficiently easy, and the details of which will be found in the ..log. sin s=10+ log. tan. decl.log. tan. obl. note, that the difference between the times of the appearance of S, and S2 on
[It is of some importance to investigate the ratio which the motion in right ascension bears to the motion in longitude at different periods.
For this purpose, let S, T, represent the motion in longitude for a very short period, and let PT,Vt be a meridian passing through T, and S, V, an are of a small circle parallel to the equator. Of course s t is the difference of right ascension corresponding to the difference of longitude S, T. Now the motion of S is entirely in right ascension and uniform; and whenever the effect of the motion S, T, on the right ascension is equal to the uniform motion in right ascension of S2, then, at whatever distance the point s was from the corresponding situation of S2, the point t will be at the same distance from the situation of S2 cor. responding to it; and they will gain or lose on Sa as the effect of the motion S, T, estimated in right ascension, is greater or less than the uniform motion of S2, or of S.
= log. cos. 43° 45'49"+ log. cos. 16° 42" 49-
or S, 46° 14′ 10′′.
or 9m. 5348. in time.