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given P Z, the co-latitude; PE, the polar distance of the pole of the ecliptic, 23° 28', and ihe angle ZPE; from which we may find, 1st, the side ZE, which is easily seen to be equal to the altitude of the nonagesimal point sought; and, 2dly, the angle P ZE, which is the azimuth of the pole of the ecliptic, and which, therefore, being added to and subtracted from 90°, gives the azimuths of the eastern and western intersections of the ecliptic with the horizon. Lastly, the longitude of the nonagesimal point may be had, by calculating in the same triangle the angle PEZ, which is its complement.

(287.) The angle of situation of a star is the angle included at the star between circles of latitude and of declination passing through it. To determine it in any proposed case, we must resolve the triangle PSE, in which are given PS, PE, and the angle SPE, which is the difference between the star's right ascension and 18 hours; from which it is easy to find the angle PSE required. This angle is of use in many enquiries in physical astronomy. It is called in most books on astronomy the angle of position ; but the latter expression has become otherwise, and more conveniently, appropriated.

(288.) From these instances, the manner of treating such questions in uranography as depend on spherical trigonometry will be evident, and will, for the most part, offer little difficulty, if the student will bear in mind, as a practical maxim, rather to consider the poles of the great circles which his question refers to, than the circles themselves.





(289.) In the foregoing chapters, it has been shown that the apparent path of the sun is a great circle of the sphere, which it performs in a period of one sidereal year. From this it follows, that the line joining the earth and sun lies constantly in one plane; and that, therefore, whatever be the real motion from which this apparent motion arises, it must be confined to one plane, which is called the plane of the ecliptic.

(290.) We have already seen (art. 118.) that the sun's motion in right ascension among the stars is not uniform. This is partly accounted for by the obliquity of the ecliptic, in consequence of which equal variations in longitude do not correspond to equal changes of right ascension. But if we observe the place of the sun daily throughout the year, by the transit and circle, and from these calculate the longitude for each day, it will still be found that, even in its own proper path, its apparent angular motion is far from uniform. The change of longitude in twenty-four mean solar hours averages 0° 59' 8".33; but about the 31st of December it amounts to 1° 1' 9":9, and about the 1st of July is only 0° 57' 11".5. Such are the extreme limits, and such the mean value of the sun's apparent angular velocity in its annual orbit.

(291.) This variation of its angular velocity is accompanied with a corresponding change of its distance from

us. The change of distance is recognized by a variation observed to take place in its apparent diameter, when mea. sured at different seasons of the year, with an instrument adapted for that purpose, called the heliometer *, or, by calculating from the time which its disk takes to traverse the meridian in the transit instrument. The greatest apparent diameter corresponds to the 31st of December, or to the greatest angular velocity, and measures 32' 35.6; the least is 31' 31":0, and corresponds to the 1st of July; at which epochs, as we have seen, the angular motion is also at its extreme limit either way. Now, as we cannot suppose the sun to alter its real size periodically, the observed change of its apparent size can only arise from an actual change of distance, And the sines or tangents of such small arcs being proportional to the arcs themselves, its distances from us, at the above-named epoch, must be in the inverse proportion of the apparent diameters. It appears, therefore, that the greatest, the mean, and the least distances of the sun from us are in the respective proportions of the numbers 1.01679, 1.00000, and 0.98321; and that its apparent angular velocity diminishes as the distance increases, and vice versa.

(292.) It follows from this, that the real orbit of the sun, as referred to the earth supposed at rest, is not a circle with the earth in the centre. The situation of the earth within it is excentric, the excentricity amount

ing to 0·01679 of the mean distance, which may be regarded as our unit of measure in this enquiry. But besides this, the form of the orbit is not circular, but

* 'Hilos, the sun, and uitgev, to measure.

elliptic. If from any point o, taken to represent the earth, we draw a line, O A, in some fixed direction, from which we then set off a series of angles, A O B, AOG, &c. equal to the observed longitudes of the sun throughout the year, and in these respective directions measure off from 0 the distances 0 A, OB, O C, &c. representing the distances deduced from the observed diameter, and then connect all the extremities A, B, C, &c. of these lines by a continuous curve, it is evident this will be a correct representation of the relative orbit of the sun about the earth. Now, when this is done, a deviation from the circular figure in the resulting curve becomes apparent; it is found to be evidently longer than it is broad —that is to say, elliptic, and the point O to occupy not the centre, but one of the foci of the ellipse. The graphical process here described is suffi. cient to point out the general figure of the curve in question ; but for the purposes of exact verification, it is necessary to recur to the properties of the ellipse*, and to express the distance of any one of its points in terms of the angular situation of that point with respect to the longer axis, or diameter of the ellipse. This, however, is readily done; and when numerically calculated, on the supposition of the excentricity being such as above stated, a perfect coincidence is found to subsist between the distances thus computed, and those Jerived from the measurement of the apparent diameter.

(293.) The mean distance of the earth and sun being taken for unity, the extremes are 1.01679 and 0.98321. But if we compare, in like manner, the mean or average angular velocity with the extremes, greatest and least, we shall find these to be in the proportions of 1.03386, 1.00000, and 0.96614. The variation of the sun's angular velocity, then, is much greater in proportion than that of its distance - fully twice as great; and if we examine its numerical expressions at different periods, comparing them with the mean value, and also with the corresponding distances, it will be found, that, by what

• See Conic Sections, by the Rev. H. P. Hamilton

ever fraction of its mean value the distance exceeds the mean, the angular velocity will fall short of its mean or average quantity by very nearly twice as great a fraction of the latter, and vice versa. Hence we are led to conclude that the angular velocity is in the inverse proportion, not of the distance simply, but of its square; so that, to compare the daily motion in longitude of the sun, at one point, A, of its path, with that at B, we must state the proportion thus:

O B 2 : 0 A 2 :: dailymotion at A : daily motion at B. And this is found to be exactly verified in every part of the orbit.

(294.) Hence we deduce another remarkable conclusion - viz. that if the sun be supposed really to move around the circumference of this ellipse, its actual speed cannot be uniform, but must be greatest at its least distance, and less at its greatest. For, were it uniform, the apparent angular velocity would be, of course, inversely proportional to the distance; simply because the same linear change of place, being produced in the same time at different distances from the eye, must, by the laws of perspective, correspond to apparent angular displacements inversely as those distances. Since then, observation indicates a more rapid law of variation in the angular velocities, it is evident that mere change of distance, unaccompanied with a change of actual speed, is insufficient to account for it; and that the increased proximity of the sun to the earth must be accompanied with an actual increase of its real velocity of motion along its path.

(295.) This elliptic form of the sun's path, the excen. tric position of the earth within it, and the unequal speed with which it is actually traversed by the sun itself, all tend to render the calculation of its longitude from theory (i. e. from a knowledge of the causes and nature of its motion) difficult; and indeed impossible, so long as the law of its actual velocity continues unknown. This law, however, is not immediately apparent. It does not come forward, as it were, and present itself at

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