THE OF \AJ 1 310 A ft OCIETY S TREATISE ON ASTRONOMY THEORETICAL AND PRACTICAL, BY ROBERT WOODHOUSE, A.M. F.R.S. FELLOW OF GONVILLE AND CAIUS COLLEGE, AND LUCASIAN PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF CAMBRIDGE* PART I. VOL. I. CONTAINING THE THEORIES OF THE FIXED STARS* A NEW EDITION. CAMBRIDGE: PRINTED BY J. SMITH, PRINTER TO THE UNIVERSITY; FOR J. DEIGHTON & SONS, AND G. & W. B. \VH1TI AKER, LONDON. 1821 / ASJftONOMY LIBRARY WC5 /, i ; ASTRO CONTENTS. . CHAP. I. Page CERTAIN Phenomena of the Heavens explained by the Rotation of the Earth I CHAP. II. On the proper Motions of the Earth, Moon, and Planets 15 CHAP. III. On the Vicissitude of Seasons, and of Day and Night 23 CHAP. IV. On the Phases and Eclipses of the Moon 33 CHAP. V. On the Earth; Its Figure and Dimensions 37 CHAP. V. On Astronomical Instruments 46* CHAP. VI. Sun's Motion; Path; Ecliptic; Obliquity of Ecliptic 1*24 CHAP. VII. On the Methods of finding the Right Ascensions of Stars from equal Altitudes, near to the Equinoxes, and from the Obliquity and Declination; Latitudes and Longitudes of Stars. Angles of Position .. , 144 By mistake Chap. V. repeated. a II CONTENTS. CHAP. VIII. Page Comparison of the Catalogues of Stars made for different Epochs. The annual Increments of the Longitudes of all Stars nearly the same. The Precession of the Equinoxes : Comparison of the Latitudes of Stars, subject only to slight variations. Comparison of the North Polar Distances and Right Ascensions of Stars. Suggested Formulae of their Variations : Consequences respecting the Length of the Year, &c. that follow from the fact of the Precession ., .183 THEORIES OF CORRECTIONS. CHAP. IX. On the Corrections necessary to be made to the observed Right As- censions and Declinations of Stars ; in order to reduce them to the mean Refraction ; Parallax ; Aberration ; Precession ; Ine- quality of Precession; Nutation 201 CHAP. X. Refraction. General explanation of its Effects. Computation of its Effects on the supposition of the Earth's Surface being plane, and the Laminae of the Atmosphere parallel to it. Error produced by that Supposition at 80 Zenith Distance. Tycho Brahe's, and Bradley's Methods of determining the Refraction. Method of determining the Refraction by Observations of circumpolar Stars. Different Formulae of Refraction. Dependence of the Value of the Latitude of the Observatory on the Mean Refrac- tion at 45 of Zenith Distance. Corrections of Refraction due to the Thermometer and Barometer. Instances and uses of the Formulae and Tables of Refraction. Explanation of certain Phe- nomena arising from Refraction 212 ABERRATION. CHAP. XI. Its Principle. Illustration of it. Roemer's Discovery of the Pro- gressive Motion of Light. The general Effect of Aberration is CONTENTS. HI the Apparent Translation of a Star's Place towards the Path of the Earth's Motion. The partial Effects of Aberration on the Right Ascension and Declination of a Star: On its Latitude and Longitude. The Effects of Aberration on a Star situated on the Solstitial Colure at the Seasons of the Equinoxes and Solstices. Formulae for the Aberration in Right Ascension ; In Declination ; In Latitude ; In Longitude. Application and Use of such Formula? . . 252 ON PARALLAX. CHAP. XII. Its Formulas similar to the Formulas of Aberration. Values of the Parallaxes of particular Stars deduced from the Tables of Aberration. Table of Parallaxes similar to the Table of Aberra- tions. Expressions for the Parallax in Right Ascension: North Polar Distance ; In Longitude ; In Latitude. Attempts made to establish the Existence of Parallax. Parallax of a Planet. Method of determining it. Its Use and Exemplification 311 ON PRECESSION. CHAP. XIII. Formula for computing the Precession in North Polar Distance and Right Ascension. Uses of the Formulae in correcting Obser- vations 339 ON SOLAR AND LUNAR NUTATION. CHAP. XIV. The Origin of the Nutations in the Inequable Generation of the Precession. Formulae of the Lunar Nutation in Right Ascension and North Polar Distance, made similar to the Formulas of Aberration and Parallax. Formulae of the Solar, similar to those of the Lunar Nutation. History of the Discovery of Nutation . .353 IV CONTENTS. CHAP. XV. On the Means by which Bradley separated Nutation from the Inequalities of Precession and Aberration; On the successive Corrections applied to the Apparent Place of a Star; On the Secular Diminution of the Obliquity CHAP. XVI. On the proper Motions of Stars ; The Means of discovering such ; Their ambiguous nature ; arising from the Means used for deter- mining the Precession. Instances of the Methods used for finding the proper Motions of a Star in Right Ascension and Declination . ON THE SOLAR THEORY. CHAP. XVII. Inequable Motion of the Sun in Right Ascension and Longitude. The Obliquity of the Ecliptic determined from Observations made near to the Solstices. The Reduction of Zenith Distances, to the Solstitial Zenith Distance. Formula of such Reduction. Its Application. Investigation of the Form of the Solar Orbit. Kepler's Discoveries. The Computation of the relative Values of the Sun's Distances, and of the Angles described round the Earth. The Solar Orbit an Ellipse. The Objects of the Ellip- tical Theory , . . 429 CHAP. XVIII. On the Solution of Kepler's Problem by which a Body's place is found in an Elliptical Orbit. Definitions of the Anomalies .... 457 CHAP. XIX. On the Place and Motion of the Aphelion of an Orbit; Duration of Seasons; Application of Kepler's Problem to the deter- mination of the Sun's Place 476 CHAP. XX. On the Inequalities of the Earth's Orbit and Motion, caused by the Disturbing Forces of the Moon and the Planets. On the Methods of determining the Coefficients of the Arguments of the several Equations of Perturbation 492 CONTENTS. V CHAP. XXI. Pae On the Methods of correcting the Solar Tables. The Formula of the Reduction of the Ecliptic to the Equator 500 CHAP. XXII. On Mean Solar and Apparent Solar Time. The Methods of mutually converting into each other Solar and Sidereal Time. The Lengths of the several kinds of Years deduced. On the Equation of Time 525 THEORY OF THE PLANETS. CHAP. XXIII. On the general Phenomena of the Planets; Their Phases, Points of Station, Retrogradations, &c 547 CHAP. XXIV. On the Method of reducing Observations made at the Earth to Observations that would, at the same time, be made by a Spectator situated in the Sun : Or, on the Methods of extricating from the Geocentric Observations of a Planet's Place, the Elements of its Orbit 562 CHAP. XXV. On the Formation of Tables of the Planets. The Variations of the Elements of their Motions. The Processes for deducing the Heliocentric Places of Planets from Tables t . . . 597 CHAP. XXVI. On the Deduction of Geocentric Longitudes and Latitudes from Heliocentric. Examples of the same : the Method of correct- ing the Tables of Planets 603 CHAP. XXVII. On the Synodical Revolutions of Planets ; On the Method of com- puting the Returns of Planets to the same Points of their Orbits ; Tables of the Elements of the Orbits of Planets ,. 6'10 ; VI CONTENTS. CHAP. XXVIII. Page On the Satellites of the Planets; On Saturn's Ring 6*27 THEORY OF THE MOON. CHAP. XXIX. On the Phases of the Moon ; Its Disk ; Its Librations in Longitude, iu Latitude, and Diurnal 631 CHAP. XXX. On the Method of deducing from Observations, the Moon's Parallax : the Moon's true Zenith Distance, &c 639 CHAP. XXXI. On the Elements of the Lunar Orbit; Nodes; Inclination; Mean Distance ; Eccentricity ; Mean Motion ; Apogee ; Mean Longi- tude at a given Epoch 658 CHAP. XXXII. On the Secular Equations that affect the Elements of the Lunar Orbit 672 CHAP. XXXIII. On the Inequalities affecting the Moon's Orbit. The Evection. Variation. Annual Equation, &t. The Inequalities of Latitude and Parallax 6'74 CHAP. XXXIV. On the Methods of Finding the Errors and Corrections of the Lunar Tables 691 CHAP. XXXV. On Eclipses of the Moon 713 CHAP. XXXVI. On Solar Eclipses 731 CONTENTS. Vll CHAP. XXXVII. Page On the Occultation of fixed Stars by the Moon 738 CHAP. XXXVIII. On the Transits of Venus and Mercury over the Sun's Disk 754 CHAP. XXXIX. On Comets , 767 ON THE APPLICATION OF ASTRONOMICAL ELEMENTS AND RESULTS, DEDUCED FROM MERIDIONAL OBSERVATIONS, TO OBSERVATIONS MADE OUT OF THE MERIDIAN. CHAP. XL. On Hadley's Quadrant, and the Sextant 770 CHAP. XLI. On the Mode of computing Time and the Hour of the Day ; By the Sun ; By the Transit of Stars ; By equal Altitudes ; By the Alti- tude of the Sun, of a Star, &c 777 CHAP..XLII. On Geographical Latitude 809 CHAP. XLIII. On Geographical Longitude 836 CHAP. XLIV. On the Calendar . , 871 PREFACE. XT may be necessary briefly to state the arrangement of the present Treatise. In the first Chapters, I have explained, in a general way, certain of the obvious Phenomena of the Heavens : then, with a view of affording the Student the means of distinctly apprehending the methods, by which, those Phenomena are observed, and their quantities and laws ascertained, I have described, although not minutely, some of the principal instruments of an Observatory. By an attentive consideration of the means, by which, in practice, right ascensions and latitudes are estimated and computed, a more precise notion of those quantities may, perhaps, be obtained, than either from the terms of a definition, or from their representation in a geome- trical diagram. But, an observation expressed by the graduations of a quadrant, or the seconds of a sidereal clock, cannot be immediately used for Astronomical purposes. It must b X PREFACE. previously be reduced or corrected. To the theories, then, of the necessary corrections, I have very soon called the attention of the Student: since, without a knowledge of them, he would be unable to understand the common process of regulating a sidereal clock, or that, by which, the difference of the latitudes of two places is usually determined. The corrections are five ; Refraction, Parallax, Aber- ration, Precession, and Nutation. The two latter, al- though they may be investigated on the principles of Physical Astronomy, are yet, in the ordinary processes of Plane Astronomy, equally necessary with the pre- ceding. To the Theory of the fixed Stars, which includes, as subordinate ones, the theories of the corrections that have been enumerated, succeed, the Solar, Planetary, and Lunar Theories. Of these, the last is, by many degrees, the most difficult. And, since, in its present improved state, it is not made to rest solely on obser- vation, I have been compelled, in endeavouring to elucidate it, slightly to trespass on the province of Phy- sical Astronomy. . % The Equation of Time, which, essentially, depends on the Sun's motion, is placed immediately after its Theory. PREFACE. XI On the same principle of arrangement, Eclipses are made to succeed the Solar and Lunar Theories. The method of computing them is that, which M. Biot has, in the last Edition of his Physical Astronomy, adopted, probably, from a Memoir of Delambre's* on the passage of Mercury over the Sun's disk. The traces of this method, may be discerned in a Posthumous workf, of the celebrated Tobias Mayer, on Solar Eclipses. The method just noticed is as extensive as it is simple. For, it equally applies to Eclipses, Occultations of fixed Stars by the Moon, and the Transits of inferior Planets over the Sun's disk. And this circumstance has determined the places of the two latter subjects, which are immediately after that of the former. In the last Chapters are discussed, the methods of computing Time, Geographical Latitude and Longitude, and the Calendar. Such is the arrangement of the present Treatise. And, since it could not be entirely regulated by the necessary connexion of the subjects, it has, occasionally, been so, by certain views, of what seemed, their proper and natural sequence. It so happens, therefore, that * Mem. Inst. torn. III. p. 392. (1802). t Mayer, Opera Incrlita. vol. I. p. 23. Xii PREFACE. the more difficult investigations are not invariably pre- ceded by the more easy. The methods, for instance, of computing the Time, Geographical Latitude and Lon- gitude, follow the Lunar Inequalities, Eclipses, Occul- tations, and Transits; but, since they do not follow by strict consequence, the latter, if it so suits the convenience of the Student, may, in a first perusal, be omitted. I have been solicitous to supply every part of the Treatise with suitable Examples. For, they are found to be in Astronomy, more than in any other science, the means of explanation. They become the means of explanation for reasons different from those which operate in other cases. For, Astronomical Examples are not always the mere trans- lations of a rule, or of an algebraical formula, or of a geometrical construction, into arithmetical results. But, frequently, they are of a different description, and require the aid of certain subsidiary departments of Astronomical Science not then the subjects of con- sideration. For instance, the difference of the latitudes of two places is equal to the sum or the difference of the zenith distances of the same Star. This rule cannot be applied according to its strict letter; for, when we descend into its detail, we may be obliged to reduce the observed zenith distances by four corrections. Consequently, we PREFACE. Xlll ought either to have previously established, or we must proceed to investigate, the theories of those corrections. This instance will also serve to shew, what frequently happens, that a rule shall possess a seeming facility in its general enunciation,, which vanishes when we become minute and are in quest of actual results. There is, in fact, scarcely any thing in Astronomical science single, or produced, at first, perfect by its processes. No series of propositions, as in Geometry, originating from a simple principle and terminating in exactness of result. But, every thing is in connexion ; when first disengaged, imperfect, and advanced towards accuracy only by successive approximation. Consider, for instance, the Sun's Parallax. That essential element is determined by no simple process, but is, as it were, extricated by laborious calculations from a phenomenon in which, at first sight, it does not seem involved. Again, the common method of deter- mining the Longitude at Sea rests on whatever is most refined in theory and exact in practice : on Newton's system in its most improved state, and on the most accurate of Maskelyne's observations. The preceding remarks, besides their proper purpose, may perhaps serve to shew that an Astronomical Treatise, with any pretensions to utility, cannot be contained XIV PREFACE. within a small compass. It ought to teach the Principles of Astronomy; but it cannot well do that, except by detailing and explaining its best methods : that is, by explaining methods such as are practised, and as they are practised. Now, the methods of Astronomy are very numerous, and the details of several of them very tedious. Some methods are merely speculative ; such as cannot be practised, although founded precisely on the same principle as other methods that are practised. For in- stance, the separation of the Sun from a Star, in a given time, is equally certain and of the same kind, as the separation of the Moon from a Star, but since, in practice, it is not so ascertainable, it cannot be made the basis, as the latter is, of a method of finding the Longitude. The exclusion then of methods merely curious, and of no practical utility, has been one mean of contracting the bulk of this Treatise. Another I have found, in omitting to explain the systems of Ptolemy and of Tycho Brahe. These do not now, as formerly, require confu- tation. The spirit of defending them is extinct. They are not only exploded but forgotten. And, were they not, it would be right to divert the attention of the Student, from what is foreign, fanciful, and antiquated, to real inventions and discoveries of more modern date, and purely of English origin. PREFACE. XV The present Treatise is not intended to explain Physical Astronomy and the system of Newton. But, the discoveries and inventions of Bradley and Halley are within its scope. Their numerous and accurate obser- vations and their various Astronomical methods, would alone place them in the first rank of illustrious Astro- nomers. But, they have an higher title to pre-eminence. In point of genius,, they are, after Newton, unrivalled. The first, for his two Theories of Aberration and Nutation : the last, for his invention of the methods of determining the Sun's Parallax from the transit of Venus, and the Longitude from the Lunar motions. This Lunar method of determining the Longitude was not reduced to practice by its author. That it has been since, is owing to Hadley and Maskelyne. The first, by his Quadrant, furnishing the instrumental, the latter, by the Nautical Almanack, the mathematical means. This last-mentioned Astronomical Work, for such it is, and the most useful one ever published, is alone a sufficient basis for the fame of its author. Besides its results, it contains many valuable remarks and precepts. It is a collection of most convenient Astronomical Tables, and should be in the hands of every Student who is de- sirous of learning Astronomy ; and who, for that end, must be conversant with Examples and Tables. XVI PREFACE. But, mere precepts and instances will not effect every thing. In order to remove the imperfection necessarily attached to knowledge acquired solely in the closet, in- struments must be used and observations made. The means of doing this, however,, 'are not easily had ; and, it is to be regretted, they are not afforded to the Students of this University. An Observatory is still wanting to its utility and splendor. PREFACE TO THE SECOND EDITION. _L HE present Edition is, in its plan, like the former. In matter and manner, however, it is so different that the Author,, instead of calling it a new Edition, might have called it a new Work. It is not worth the while to point out the changes which the Work has undergone. Few of its readers will trouble themselves on that point. The fact worth enquiring about is whether the Work be a good Work, not whether it be better than that it comes after. That it is better may be presumed from the very circumstance of its coming after. Nor can there be any arrogance in attributing its improved state to the change that, during the two Editions, has taken place in the Author's knowledge. The usual effect of time, in this respect, has not been counteracted. The other cause which ought to improve a treatise, namely, the improved state of the science treated of, has, of late, but slowly operated. Astronomical Science is now, nearly, the same as it was ten years ago. Having reached a c XVIll PREFACE. kind of maximum state of excellence, its changes are minute and must continue to be so. All great changes ended with Bradley. He swept the ground of discovery, and left little to be gathered by those that follow him. Yet, during the 60 years that have elapsed since Bradley, it cannot be said, but that Astronomy has greatly advanced, although not by the aid of discoveries, such as those of Aberration and Nutation. The aid has come partly, indeed, from the Observatory, but principally from Physical Astronomy : which, originating with Newton, has, under his successors, Mayer, Clairaut, Euler and Laplace, grown up into an exceedingly great science. Of the benefits thence accruing to Astronomy, the most excellent, by many degrees, are the Lunar Theories of Mayer and Laplace ; or, as it may be stated, the Lunar Tables deduced from those Theories, and the Observations of Bradley and Maskelyne. If we go back to Halley's time, the improvement in such Tables will appear most striking. Halley states that, in his time, the differences between Observations and the results of Newton's Theory amounted frequently to 5 minutes, which differences now (if we speak of their mean states) do not much exceed as many seconds. Navigation has been made more safe by means of these Lunar Tables : which, perhaps, is the only prac- PREFACE. XIX tical good that Astronomy has conferred on Society. Its other benefits are philosophical and intellectual. Should these be held to be of no moment, we might, perhaps, at the present time, shut up our Observatories, and live upon the hoards of Astronomical Science. We are now possessed of sufficient means, as far as Astro- nomy is concerned *, for determining the place of a vessel at sea ; and if we would enable the mariner on the Atlantic or Indian Ocean to determine his place, to within less than 10 miles, we must provide him with better means of observation : with an Instrument more excellent than the Sextant. But, such is the present ardour for philosophical pursuits, the duties of an Observatory, instead of ceasing, are likely to become more arduous. Within a few years from the present date, an Astronomical Society has been formed in the Metropolis, and an Observatory nearly established here. These Institutions indicative, as we have said, of the spirit of the times, can hardly fail to augment Science : they will do some good although perhaps not all the good that is intended to be done by them. * The words of the commission that appoints the Astronomer Royal of Greenwich enjoin him, to apply himself with the utmost care and diligence to the rectifying the Tables of the Motions of the Heavens, and the places of the Fixed Stars, in order to find out the so much desired Longitude at sea, for the perfecting the Art of Navigation.' XX PREFACE. As the latter of these Institutions may, in future times, become one not merely of local interest, we shall be excused if we say something farther concerning it. The good resulting from Observatories, whatever it may be, practical or intellectual, the founders (if we may so call them) of the present Observatory are anxious to secure. Their first and chief object is to have Obser- vations made as good as they can be made. The second, to have as many as possible of such Obser- vations. In order to obtain the first, the best Instru- ments that Europe can furnish are ordered to be made. To secure the second object, houses are attached to the Observatory for the constant residence of the Observers. Another object of the Institution is, the instruction of Academical Students in the use of Instruments, and in practical Astronomy: an object, it 'should seem, not incompatible with the former, but secondary and sub- ordinate. Instruction alone could have been imparted by means much more simple than those which are now- put into action. But good Observations* will not necessarily be made, * Two circumstances (there may be more) are unfavorable to the Observatory we are speaking of. One is, the not sufficient vicinity to the Artists of London : the other, common to our Island, and the same as that of which Lacaille complains, ' Constans nimis Parisiis tempore hiberno nebularum, imbrium et nubium mora, eorumque tempore aestivo frequens reditus, &c/ PREFACE. XXI because he, who ought to make them, is obliged to reside in an Observatory furnished with good Instru- ments. Something else remains to be done : some regulation to be made, or motive supplied, to compel (as it were) Observers to employ, in the duties of an Observatory, the time they must spend there. To effect this, there will not be found, perhaps, any means so simple and efficacious as that of some absolute rule for printing and publishing annually the Observations, and for sending copies thereof to the principal Observatories of Europe. Other Regulations may be suggested to counteract the proneness of Institutions, like the one spoken of, to become worse. But they should be simple and few. Regulations may, indeed, prevent much wrong from being done; but they rarely create a zeal for the performance of duties. The minute detail of the hours, modes, and objects of Observation, would never supply motives to him who should be insensible to his own personal reputation, and the honor of his Country and University. The augmentations of Astronomical Science have, with scarcely any exception, come from publick Ob- servatories : which fact is to be accounted for, from the excellence of the Observers' Instruments, the constant discharge of their duties, and, above all, the zealous discharge of those duties by the influence of publick opinion. A like moral controul will, probably, operate here, and serve to carry into effect the enlightened in- XX11 PREFACE. tentions of the munificent Patrons of our Observatory. It has not been built merely to prevent its being said that an University, famous for its science,, was without such an Institution : nor to add to the title and emolument of an individual; nor to be used as a kind of Astronomical toy, and to become the mere resort of leisurely amateurs and random star-gazers : nor, which is indeed a better but still a subordinate object, to confirm or correct results elsewhere obtained, to see, for instance, that Observations have been rightly made at Paris and Palermo. The chief object of the Observatory is, by its own means, to enlarge the boundaries of Science ; to extend the fame of the University that founds it, is a secondary one, or rather, will be a sure consequence, if the first shall be obtained. Caius College, Dec. 27, 1822. TREATISE ON ASTRONOMY, IN TWO VOLUMES. AN ELEMENTARY TREATISE ASTRONOMY. < CHAP. I. Certain Phenomena of the Heavens explained by the Rotation of the Earth. IN an Elementary Treatise on Plane Astronomy, two objects are required to be accomplished: 1st, The description and gene- ral explanation of the heavenly phenomena. 2dly, The establish- ment of methods for exactly ascertaining and computing such phenomena. Our attention will be first directed to the former of these two objects. If, on a clear night, we observe the Heavens *, they will appear to undergo a continual change. Some stars will be seen ascending from a quarter called the East, or rising; others descending towards the opposite quarter the West, or setting. In some intermediate point, between the East and West, each star will reach its greatest height, or, will culminate : The greatest heights of the several stars will be different, but they will all appear to be attained towards the same part of the Heavens ; which part is called the South. If we now turn our backs to the South and observe the North, the opposite quarter, new phenomena will present them- selves. Some stars will appear, as before, rising, reaching their * Exposition du Systeme du Monde, p. 2. A greatest heights, and setting; but, besides these phenomena, other stars will be seen that never set, and that move with different degrees of velocity ; and there are some stars that, to appearance, are nearly stationary. About one of these stationary stars, the other stars that never set appear to revolve, and to describe circles. Such stationary star is called the Polar Star: and the stars revolving round it, Circumpolar. The Polar Star, that which is usually so denominated, is not, when accurately observed, or observed by means of instruments, strictly stationary. It is not, therefore, to be held as the place of the Pole, which is indeed an imaginary point, always, however, as we shall hereafter see, ascertainable by theory and observation. In such point or pole, a star, if we suppose it there placed, would appear stationary. Almost all the stars in the Heavens retain towards each other the same relative position ; no mutual approach or recess takes place between them : and accordingly they are called Fixed Stars. There are, however, certain stars, called Planets, not under the above conditions, and which continually change their places. The Sun and Moon also, the two celestial objects of the greatest interest, are from day to day changing their places in the Heavens. A spectator at sea, or placed in a level country, may imagine himself in the centre of a plane, extended equally on all sides, and bounded by a circular or curved line apparently separating the sky and sea, or the sky and land. The plane so extended and bounded is called the spectator's Horizon, and sometimes the sensible Horizon. It is the boundary of the spectator's view, and when stars first appear just above it, they are said to rise : when they sink beneath it, they are said to set. On this imaginary plane the concave heavens, or the hemisphere of the heavens, may be fancied to rest. The surface of the sea is not strictly plane; a few simple observations are sufficient to shew that it is a convex surface, the convexity being towards the heavens, and the spectator being placed on its summit, The preceding definition, therefore, of the horizon, must be slightly altered : it must now be defined to be a plane which, at the summit just mentioned, (the place indeed of the spectator) is a tangent plane to the earth's convex surface, extended on all sides till it is bounded by the sky. The convex surface of the Earth is nearly spherical; more nearly spheroidical : the Earth being, (as it appears probable from various reasons) a spheroid of small eccentricity. The plane of the horizon, therefore, is a tangent plane to the spheroid at the place of the spectator, and a perpendicular to the plane at such place passes very nearly through the centre of the Earth. The perpendicular line just mentioned tends, if produced upwards, to a point in the Heavens called the Zenith. The opposite point in the line's direction continued downwards is called the Nadir. If the eye of the spectator were in that plane which has been defined to be the plane of the horizon, stars would not appear to have risen whilst they were beneath that plane. But it is other- wise, if the spectator be elevated above the horizon either by being on a tower, or eminence, or on a ship; indeed, as mere elevation above the horizon is the circumstance that modifies the preceding statement (see p. 2. 1. 28.) his own stature will cause stars to appear to have risen before they are above the horizontal plane. Thus if, on the Earth's surface ABD, A be the place of the spectator's feet, or the bottom of a tower on which he is, A a drawn a tangent to the surface at the point A> may represent his sensible horizon. If the eye be supposed to be at A, it cannot see an object till it is level with or above A a. But if the place of the eye be transferred to O } an object may be seen if it be level with or above the line OBb. The line O B b is a line drawn from the spectator's eye at O and touching the earth's surface at JB; and the horizon, were it supposed to be composed of such lines as OBb would be a conical surface having its apex in O. The depression of b below a, measured by the angle amb is technically denominated, the Dip : which, from O A, the eye's elevation, and the radius of the Earth, may easily be computed. The tangent plane in which aAb lies, has been called by Astronomers (as we have seen), the Sensible Horizon : but they have also imagined, for the purposes of calculation, another horizon the plane of which, parallel to the former, passes through the Earth's centre, and is denominated the Rational Horizon. HCh t parallel to a A b, may represent this latter plane. It is plain that both the Sensible and the Rational horizon are merely relative: in other words, they must change with a change in the spectator's place. Of a spectator at A 3 a b perpendicular to CAZ is the sensible, and Hh, parallel to a b, the rational horizon ; and Z is his zenith. Of a spectator at J5, ed perpendicular to CBZ' is the sensible, and H'fi, parallel to ed, the rational horizon: and Z' is his zenith. Let us consider a little farther the appearances that would take place, were a spectator stationed at sea, or in the midst of a level country. Suppose then O (see tig. p. 5.) to represent his station and SENW the imaginary circular boundary of a plane extended beneath his feet to be his horizon. If a star rose at A it would describe a curve above the horizontal plane, and sink beneath it, or set, at some point a. In like manner another star rising at B would describe a curve above the plane of horizon, and set at some point b. But this circumstance also, wherever the stars A, B were, would always take place ; namely, the equality of a b, the distance of the points of setting with AB with the distance of the points of rising. If the arc AB equals the arc a b, then the chord A a is parallel to the chord JBb: and a diameter such as SON drawn perpendicularly to A a, and consequently bisecting it, will be perpendicular to and will bisect all other chords such as Bb: and will moreover bisect the arcs AS a, BSb, &c. The points A^ and N determined after the preceding manner are the South and North points of the horizon, or (as it is called) the Azimuth circle SENfV. EOW drawn perpendicularly to SON determines E and W 9 the East and West points, which together with the two preceding form the four Cardinal points. SENJVhas been called the Azimuth circle, and azimuth distances are measured from the South and North points. SA is the azimuth of the star rising at A, Sa of its setting at a. The complement of the azimuth of a star is its Amplitude: and amplitude is accordingly measured from the East and West points. Thus the Amplitude of the Star's rising at A is EA-, the Amplitude of its setting at a is Wa. A star rising at A will gradually ascend above the plane of the horizon till it attains its greatest height; it will then decline, by like degrees, until it sets or disappears at a. If we conceive a plane passing through S and N and perpendicular to the plane of the horizon, then a star rising at A and ascending after the manner just described will be at its greatest height above the hori- zontal plane when it reaches the perpendicular plane. The same will happen to every other star. The greatest heights of different stars will be different, but they will all be attained to in that plane which, passing* through S and N, is perpendicular to the plane of the horizon. The perpendicular plane above described is called the plane of the Meridian ; because, the middle of the day happens when the Sun in his ascent above the horizon reaches it. It is usual to suppose this plane bounded by a circle passing through S and N, and having therefore the same radius as the horizon or azimuth circle SENW: which, in fact, is to suppose these circles to be the great circles of the same sphere. The meridian intersects (see 1. 2.) the horizon in S and Nthe South and North points : it must also pass through the zenith (see p. 4.) and through the pole (see II. ], 2 &c.) Every circle, the plane of which is perpendicular to the plane of the horizon, is denominated a Vertical circle. The meridian, therefore, is a vertical circle. The vertical circle, which passes through E and W the East and West points, is distinguished by the name of the Prime Vertical. We have spoken of the risings and settings of stars, such as they will appear to be to a spectator placed at C the centre of the plane of the horizon, but, hitherto, we have said nothing, of the intervals of time elapsed between the respective risings and settings. Now a spectator in our northern climate, looking towards 5 the south, cannot fail to remark that a star between its rising at F and setting aty' is lopger above the horizon than a star which rises at A and sets at a : which kind of inequality takes place, and in a greater degree, with every star successively placed between A and S. But he may also note that every star takes the same time in passing from its rising through its setting to its rising again. A star therefore at A is longer below the horizon than a star at F, and still much longer than a star at E. But a star rising at E the East point has this peculiarity : namely, that it is above the horizon exactly as long as it is below. On this account the great circle in which such star moves is called the Equator. The phenomena that have been described may be explained by supposing the concave Heavens, in form like an hollow sphere, to revolve round an axis passing through the pole and the centre of the Earth, and in a time equal the interval between two suc- cessive risings of a star. Thus, let PCp be the axis, HCh the rational horizon: then CZ drawn perpendicularly to Hh (see p. 4. 1. 10.) determines Zthe spectator's zenith*. EQ is perpendicular to Pp and vti, Hg,kl, m H, h r, (representing the projections of circles to the planes of which Pp is perpendicular) parallel to EQ. If we conceive the plane of this diagram to be placed perpendicularly on the plane of the former (see p. 6.) which was meant to represent the horizon, so as to be adapted to northern latitudes,, then the plane PEpl will be the plane of the meridian., E will be the point of the greatest ascent of a star rising at C, and if we suppose t, s to be the ortho- graphical projections on the plane of the meridian of the points A, By then n and / will be the points of the greatest ascents of stars rising at A and JB. Now suppose the figure to revolve round Pp: then tn will be proportional to the star's ascent from t, the place of rising, to n its greater elevation, and tm, every point of which is below the horizon, will be proportional to the time from the star's greatest depression (at m) beneath the horizon to its rising at t, and tm, as it is evident, is greater than tn: again, since CE= CQ, the time that a star is above the horizon is exactly equal to the time of its depression beneath that plane (p. 6. 1. 29). A star rising at .$ will be above the horizon during a time proportional to 2 S /, and below it during a time proportional to 2 5 k: and, as it is evident, ZSl is greater than QSk (p. 6.) Suppose a star to be exactly at H, then it can never set, but it will be a circumpolar star (see p. 2.) : and such will be all stars * For the rational horizon is parallel to the sensible. * 8 situated between H and P. A star placed at P would appear to be at rest. Each of the stars of which we have spoken must (from the very nature of the scheme intended to explain their phenomena,) consume an equal portion between two of its successive risings : which portion of time may be called a Sidereal day, and which it is usual to divide into 24 equal parts, or hours of Sidereal time. The preceding scheme is intended to shew, that the hypothesis of the revolution of the sphere of the Heavens round an axis passing through the poles, will adequately account for all those common phenomena relative to the risings, settings, ascents^ &c. of stars which will present themselves to a spectator situated as we have described him to be. The hypothesis, therefore, is, at the least, a probable one. There is, however, another hypothesis equally probable with the former or rather more so, as being more simple, which hypothesis makes the concave Heavens to be at rest, but the globe of the Earth to revolve within them, round an axis, and in a direction from West to East. Each hypothesis equally explains such phenomena as have been already described : and since also to each hypothesis the same mathe- matical explanations and reasonings are applicable, we will adhere to the one already made use of and its connected diagram, and deduce some farther results. The line EQ is intended to represent the Equator, Ik, nm, vu,&c. which, from the supposition of the revolution of the figure round Pp, must be parallel to EQ, are called Parallels of Declination. The decimation of a star is its angular distance from the Equator. The declination, therefore, of a star, which appears to move in the parallel k I is kQ (which is the measure of the angle subtended by kQ at the centre of the sphere); the declination of a star whose parallel is mn, is mQ or n E: of the circumpolar star at v, vQ is the declination ; v P is its distance from the pole, or, as it is called (P being the north pole) its north polar distance: mp is the south polar distance of a star at m, the complement, as it is plain, of m Q the star's south declination. A secondary is a great circle passing through the poles of that other great circle to which it is a secondary. Thus HphP, the meridian, is a secondary to the horizon Hh. The circle P sp&c. is a secondary to the Equator JEQ. The prime Vertical (see p, 6.) a secondary to the horizon, as indeed is every great circle passing through Z and a point in Hh : a great circle, however, of this latter description, is farther distinguished by being called a Vertical Circle, since its plane, perpendicular to that of the horizon, is, in other words, vertical. What Declination and its complement Polar distance are with respect to the Equator, Altitude and its complement, Zenith dis- tance, are with regard to the horizon. The former is the star's angular distance from the spectator's horizon measured on a ver- tical circle : the latter is the distance from the zenith of the same spectator. The altitude, for instance, of the Equator, or of a star therein situated, is Eh: its zenith distance is ZE : the altitude of a star at ^^, is nh ; its zenith distance is Zn. Since the sphere, with all the stars supposed to be fixed in its surface, revolves in 24 hours of sidereal time, the stars situated in different parallels will appear to more with different velocities. A star near to P will appear scarcely to move : the velocity of a star describing v u will be as much less than the velocity of a star situ- ated in the Equator, as u v is less than EQ: but uv has to EQ the same proportion as its radius has to the radius of the Equator : or that proportion which the sine of the angle PCu has to CM: but sin. PCw=sin. Pw=sin. North polar distance, or =cos. declination, If therefore we call V the velocity of a star or point in the Equator, the velocity of any other star= V. cos. star's declination. The Hour-angles are those angles at the Pole which, contained between two secondaries to the Equator, intercept the space passed B 10 over by a star, in any assigned time, either on the Equator or on a parallel. Thus if a star move from 5 to /, the hour-angle is said to be sPs or bPb, which is measured by ab, ab being an arc of the equator. Now ab y or the angle b Pa, must be proportional to the time, for since the point b is, by reason of the sphere's revolution, transferred from C to Q or through an arc of 90, in 6 hours, it must be transferred from C to b and from b to a, by reason of the sphere's uniform revolution, in times which bear, respectively, that proportion to 6 hours which C b, a b, estimated in degrees, bear to 90 degrees. If a b, therefore, contains 1, the time through b a, or the hour-angle aPb = ^th of 6 hours, or jLths of an hour, or the value of the horary angle aPb, or sPs is O h . 06666 &c. or 4 m . The Poles and the Equator, that have hitherto been described, belong to the celestial sphere ; but the Earth also has its Equator, Poles and Axis. Conceive an interior sphere, in the figure of p. 7, described round C to represent the Earth, then the plane of its Equator and axis will be such parts of the Equator EQ and axis Pp as are contained within the sphere representing the Earth and are terminated by its surface. Or, we may reverse the process and give to the Celestial Sphere its Equator and Axis, by extending to \he Heavens the Earth's Equator and Axis. Places situated on the Earth's surface are said to have Latitude, which is to be defined, distance from the Earth's Equator. But the Latitude of a place in its astronomical meaning, or with refer- ence to its astronomical measure, is an arc of the meridian inter- cepted between the zenith of the place and the celestial Equator, or, which is the same thing, it is the complement of the arc which lies between the zenith of the place and the pole : which latter arc, therefore, may be called the Co-latitude of the place. If the Pole Star, that which is usually so called, were exactly situated in the Pole, the method of determining the latitude of a place, by means of that arc which is its complement, would be a very simple one : since the plumb-line determines the zenith. But the Pole Star being, in fact, a circumpolar star, its angular distance from the zenith will vary with the time of observation. Its distance, therefore, cannot give the true value of the co-latitude, or its distance requires a correction in order to give the co-latitude 11 truly. Such method then of determining the latitude cannot be a very simple one^ and can only be practised subsequently to, and by the aid of, an improved or refined state of astronomical science. It is now mentioned for the purpose of giving the student some general notion of latitude and of the means of measuring it. If a circumpolar star can be made subservient to the finding the latitude, any other known star may, and for like reasons. It is sufficient to measure the angular distance from such star, when on the meridian, and the zenith, which latter point, as we have already said, the plumb-line determines ; that is, it is that point in the heavens to which the plumb-line, if we imagine it to be continued upwards, is directed. The polar distance of the star is known (since that condition is implied in the expression of known star), therefore the co-latitude of the place (PZ in the figure of p. Q.) is either the star's polar distance minus the meridional distance of the star from the zenith, or the star's polar distance plus the star's distance from the zenith. For instance, if the star be at /. PZ=Pl-Zl if situ, PZ = Pu+Zu. It is easy to shew, on grounds like those that have been laid down, that the difference of the latitudes of two places may be de- termined simply from the distances of the same star from their repective zeniths. Thus, if the star *y (fig. p. 9.) should lie between Z t Z', the two zeniths, = (90 P#) - (90 PZ) = lat. of Z' ~ lat. of Z in which operation it is not necessary to know the declination of the star y. Suppose the star 7 Draconis should be 2' 4".9 North of the Greenwich Observatory, and 19' 23".3 South of the Observatory at Blenheim, then ZZ' = Zy + Z'y = 2' 4".9 + 19' 23".3 = 2l' 28".2 the difference between the latitudes of the two Observatories: consequently if the latitude of one Observatory were known, that of the other might be determined : for instance, if the latitude of Greenwich be taken at 51 28' 40", that of Blenhein must equal 51 28' 40" + 21' 28".2 = 51 50' 8".2. As a second instance, if the zenith distance of y Draconis from the Dublin Observatory on January 1, 1818, be 1 52' 20".7, then the difference of latitudes between the two Observatories of Green- wich and Dublin is 1 52'20".7 + 2' 14".9= 1 54' 35".6, supposing the distance of y Draconis from the zenith of Greenwich to be, at the same time, 2' 14".9. * There are other methods explicable, as to their general nature, even in this early stage of our progress, that may be used in deter- * We have taken what were, nearly, the mean or reduced zenith distances of 7 Draconis from the two Observatories at the beginning of 1818. It will appear, and fully, during the progress of the work, why the zenith distance of a star does not always remain the same at the same place. The star 7 Draconis is continually approaching the zenith of Greenwich, and receding, by equal quantities, from that of Dublin. 13 mining the latitudes of places. For instance, if we determine the respective zenith distances of two known stars at two places, we may deduce the difference of latitude of those places. In point of theory it matters not where the stars, relatively to the zeniths of the places of observation, are situated : but the excellence of the practical method depends on this circumstance, that the star observed should be near the zenith of the place of observation : for, in such a case, one great cause of inequality, namely, the refraction of the air, would be nearly rescinded, and the accuracy of determining the difference of latitudes would rest on the ascer- tained or ascertainable difference of the declinations of the two stars. In this first chapter we have advanced, very little beyond the general description of the ordinary appearances of the Heavens, and their explanation on the hypothesis of the revolution of the starry sphere. The revolution of that sphere (the Primum Mobile as it was called) from East to West, with the supposed quiescence of the Earth, will account for the risings, settings, durations of ascent and descent of the stars equally well (and we may add, on the same principle), as the rotation of the Earth round its axis from West to East, the Heavens being supposed quiescent. The first is the most obvious hypothesis, the latter, when more closely viewed, the most simple hypothesis. The stars seem to move round us ; but when we consider the prodigious velocity with which, by reason of their immense distance (a point easily made out) they must revolve, we are disposed to search out for and to adopt some other hypothesis that is free of so revolting a circumstance. There is, indeed, no summary proof to be given of the truth or falsehood of either of the hypotheses. One, for several reasons that will hereafter appear, is much more probable than the other. Indeed the hypothesis of the revolution of the sphere is inadequate, as astronomical science now stands, to solve all the phenomena. We must, however, be content, at present, to take for granted the truth of the hypothesis of the Earth's rotation. If it continues to explain simply and satisfactorily, other astronomical phenomena than those already noted, the probability of its being a true hy- pothesis will go on increasing. We shall never indeed arrive at a term when we shall be able to pronounce it absolutely proved to be true. The nature of the subject excludes such a possibility. 14 We will now proceed to notice some other phenomena different from those that have preceded and not explicable solely on the hypothesis of the Earth's rotation. They need not, however, be considered as overturning that hypothesis. It will be more simple to consider that hypothesis to be established, and the new phenomena as indicating the necessity of some additional hypothesis, or the existence of certain circumstances of motion and translation that take place contemporaneously with the Earth's rotation and consistently with it. CHAP. II. On the proper Motions of the Earth, Moon, and Planets. IN the preceding Chapter the phenomena described and ex- plained are chiefly phenomena of stars called, from their pre- serving the same invariable distance from each other, Fixed Stars. Their risings, settings, the times of their elevation above the horizon, of their depression beneath it, are easily explicable, as \ve have seen, on the hypothesis of the Earth's rotation round an axis inclined, in our latitude and in every habitable latitude, to the horizon. There are other heavenly bodies, the Sun, the Moon, and the planets, that assume only in part, or nearly, those appearances that belong to the fixed stars. The Sun, for instance, if he should rise at the same point in the horizon, which a fixed star rises in, would set in the evening, nearly where the star sets. The length of day would not seem to differ from the time of the star's ascent above the horizon : and his meridian height, would, to common observation, appear to be the same as that of the greatest elevation of the star above the horizon. The same circumstances would appear to take place with the Moon and Planets. But minute differences are not to be detected by common observation. The Sun and star, if they rose exactly at the same point of the horizon, would not pass the meridian exactly at the same point. On any day between the middle of winter and the middle of summer, the Sun rising where the star rises would pass the meridian in some point above the star's passage : during the other half year, in some point below. But in order to distinguish these circumstances some nicety of observation is requisite. If, however, we examine a star and the Sun, or a star and one of the planets for a longer interval than a day, their separation or their approach, which is perpetually taking place, will become manifest even without the aid of instruments. Suppose, for instance, at the beginning of March, that we observed the Sun and a star to rise at the same point F (fig. p. 5.) of the horizon : they would set nearly at the same pointy* and 16 cross the meridian nearly at the same point. The next day the star would still rise at the same point P, but the Sun would rise at some point between F and E, would set at some point between f and W t and would pass the meridian above the point of the star's passage. The like would happen on each succeeding day. The Sun would rise nearer and nearer to the east, would set nearer to the west, and pass the meridian more and more above the Star. In about C 20 days from the time of the first observation, the Sun would rise in the east (at E) set in the west at W, and reach a meridional height equal to the co-latitude of the place of observation. After that time the Sun would rise between the east and north points of horizon (E and IV) and set between the west and north ( W and N) till about the end of June, at which time, having, reached his extreme intermediate point of rising between E and N, and his greatest meridional height, he will begin to reiterate his course of risings and meridional heights, and passing the term from which we began (see 1. 1.) to date them, he will reach, between E and S, his farthest point of rising from E, will ascend to his least meridional height*, and again begin to regress. If we take a line MM' and erect on it perpendiculars ME, me, m'e 1 , &c. to represent the Sun's meridional heights on suc- cessive days, ME representing the height on the day when the E' Sun rose in the east, n S his greatest height on the day when he rose on a point of the horizon nearest the north, &c. then the curve passing through the meridian Sun, during the year, will be of the form ESk WE, the part KWE' being similar to ESK. * Either on that day or on the preceding. 17 If we go no farther than the preceding instance, it is clear, if the stars be supposed to be fixed and their phenomena ac- counted for by the rotation of the Earth, that the phenomena just described as appertaining to the Sun cannot be so accounted for ; they plainly indicate the Sun to have a proper and peculiar motion, or, which we shall find to be the same thing, the place of observation (meaning thereby the Earth) to have a proper motion, or, one distinct from the conversion of the Heavens. But it is easy to make other observations that shall plainly indicate a proper motion in the Sun, and shew the necessity, if we would explain the phenomena, of correcting or of adding to the hypo- thesis of the Earth's rotation : which cannot be the sole hypothesis. As a second instance, leading to the same inference as the former, let us take that of the Sun and a star when they set nearly together. Suppose, on a particular day, that we observe a certain star to set a little after the Sun. On the following day and on each successive day, the star's setting will follow more closely that of the Sun : till their proximity will become so close as to cause the light of the former to fade away and to be ex- tinguished by the effulgence of the latter : the star, therefore, for some time, will disappear ; but, if, after a few days, we direct our view to the rising Sun, we shall perceive the star emerging, as it were, from its beams, and, after this, on succeeding mornings, preceding, by still greater and greater intervals, the Sun in its rising. The latter part of the phenomenon, which we have just noticed, namely, that of the star's rising just before the Sun, is technically called the Heliacal rising of the star. There are only certain stars that can so rise, and that only at particular times of the year. Their heliacal risings, therefore, must be indicative of those times. It was by such observations that the rude notions of antiquity recognised the seasons, and regulated the labours of the year *. The phenomenon which we have last described indicates, like the former, the Sun to have a proper motion among the fixed * The Egyptians looked for the inundation of the Nile at the time of the heliacal rising of Sinus, or, as they called it, of Thoth the Wateh- Dog. 18 stars : towards those stars that set after him mud from- those stars that rise before him : which are circumstances of the same kind, or indicate the same direction of the Sun's motion. The Sun's motion, however, although, as it has been described, is first towards a certain star, and then, having passed it, from it, is not made in a direction either the same as that of the star's parallel (see p. 8, 1. 26, &c.) or parallel to it, but in some oblique direction : which indeed may easily be collected from those circumstances which were described in pages 15, 16. as belonging to the first phenomenon. For it was there shewn, by noting the points of the horizon at which the Sun rose on successive days, that the Sun has an horizontal motion, or, as it is technically called (see p. 5.) a motion in azimuth; and, also, by noting his me- ridional heights on those days, that the^ Sun has a motion per- pendicularly to the plane of the horizon : which two motions so detected must be the parts of a compound oblique motion. The apparent motion of the fixed stars is from east to west : the real motion of the Earth (according to the preceding supposition, (see p. 13.) which causes the former apparent one, from west to east, and, in our hemisphere, to a spectator looking towards the south, from the right hand to the left : and in the same direction, that is, from the right towards the left, or from the west towards the east, is the Sun's proper motion. The fact of a motion of the Sun from the west to the east is sufficient to explain why certain remarkable stars and groups of stars, called Constellations,, are seen in the south at different hours of the night during the year. For, the hour depends solely on the Sun : it is noon, when he is in the south. Stars .directly opposite to him are, therefore, by the rotation of the Earth, brought on the meridian at midnight. But the stars on the meridian at 12 one night, cannot again be there, at the same hour, on the succeeding night : for, the Sun having shifted his place a little to the east, the stars before opposite to him are now opposite a part of the Heavens to the west of the Sun : that is, they must come on the meridian a little before midnight : and on succeeding nights more and more before midnight. It thus happens then that every star is, during the year, on the meridian at all the hours of the four and twenty. There are some stars indeed that may be on the meridian, and yet, by reason of the Sun's brightness, may not be discerned there. 19 If we apply to the Moon the same kind of observations that have been described to be used for detecting the Sun's motion, we shall find the Moon to move, by a proper motion, amongst the fixed stars towards the same parts, (that is, in a general way of speaking, from west to east) as the Sun, but with greater rapidity and not by similar and regular changes of place, whether we consider the azimuthal or the meridional changes, (see p. 18.) For instance, the Sun's annual path traced out in p. 16. will be nearly the same every year. But a path so traced out for the Moon, during one of her revolutions, would not be her path in her next revolution round the Earth. The Moon, therefore, has a proper motion of her own and not similar to the Sun's : we may go farther and state that, as far as we can judge from common observations, the two motions are unconnected, or there is no single principle which will account both for the one and the other. Besides the Sun and Moon there are certain other stars which have their proper motions : and motions so peculiar and irregular as to have procured to the stars possessing them the denomination of Planets. They sometimes appear to move, like the Moon, towards the east : at other times, however, towards the west ; and there are conjunctures, when, during several successive nights, they appear nearly stationary. It will be seen hereafter that there is no real difference between the direction of the planets' motions and that of the Earth. If the spectator be supposed to have taken his stand at the Sun, he will view the Earth as one of the planets, and, then, all the planets constantly moving in the same direction. That they sometimes appear stationary, and, at other times, retrograde (that is, moving in a direction contrary to their usual one) is to be attributed to the motion of the Earth, which motion combined with that of the planets, causes them, under certain circum- stances, to appear to move otherwise than they are really moving. The retrogradation of a planet is a phenomenon partaking some- what of the nature of an illusion. The motions from west to east that we have spoken of, take place and must be combined with that diurnal motion from east to west, which arises from the rotation of the Earth. This latter 20 motion is so great, that, as it were, it overpowers the former, and, with an inattentive spectator, prevents it from being observed. Even the Moon, which of all the planets has the swiftest proper motion towards the east, shifts her place in the course of a day by not more than 13; whilst, by the rotation of the Earth, she is seemingly carried in the same time through 360. There are, however, conjunctures when we cannot but recognise her proper motion ; when, for instance, the Moon is near a star previously to an occultation : for moving over a space equal to her diameter in an hour she then visibly approaches the star. As the stars which are fixed seem to move by reason of the Earth's rotation, so the Sun, which is, in fact, stationary, seems to move by reason of the Earth's revolution round him. But it makes no difference either in the explanation of phenomena, or in the deduction of such results as belong to the subject ; whether we suppose the Earth to move round the Sun, or the Sun to move round the Earth. A spectator at E sees the Sun S in the heavens at the place X . Transferred to =*= he sees the Sun in : the same appearance as that of a real translation of S from X to T Of the Solar System, composed of the Earth, the Moon, the Planets, their Satellites, and certain stars more erratic than the planets, and called Comets, the Sun, the chief body, occupies the centre. Round the Sun, in their order, at different distances, and in different periods, revolve Mercury, Venus, the Earth, Mars, Vesta, Juno, Ceres, Pallas, Jupiter, Saturn, the Georgium Sidus. These planets Astronomers have distinguished (as they have also the Sun and Moon) by appropriate symbols : thus The Sun ........ Mercury ........ $ Venus .......... $ The Earth ....... Mars ...... . . , . . Vesta .......... ] Juno .......... f Ceres Pallas Jupiter Saturn The Georgium Sidusl orHerschel. j The Moon . . Mercury, Venus, Mars, Jupiter and Saturn, are what are called the old Planets, discernible by the naked eye, and conse- quently known to the antients *. The Georgium Sidus, (or in order to give it what the others have, a mythological denomina- tion, Uranus) was discovered in 1778 by Dr. Herschel, and therefore, it is frequently called by Foreigners, the Herschel. The other four planets Vesta, Juno, Ceres, Pallas, (at first fan- tastically called Asteroids) have been discovered since 1801, the first and fourth by Olbers, the second by Harding, and the third by Piazzi. The latter new planets are extremely small and cannot be seen without a telescope, which is the case also with the Georgium Sidus, not indeed by reason of his small size, but of his great distance. The system which has been briefly described is sometimes called, from its author Copernicus, the Copernican. The charac- teristical point, it must be noted, in his system is the placing the Sun, as an immoveable and the chief body, in the centre of it. In the next Chapter we will consider whether, on the pro- posed hypotheses and the established facts, we are able to account for the vicissitudes of seasons and the different durations of day and night. The only thing aimed at will be something of * Maxume vero sunt admirabiles motus earum quinque stellarum, quae falso vocantur errantes, nihil enim errat, quod in orani aeternitate conservat progressus et regressus reliquosque motus constantes et ratos. Cic. de Nat. Deorum, Lib. II. 19, 20. 22 the nature of a popular explanation, probably accounting for the phenomena, on hypotheses that are simple and consistent with themselves. Independent and rigorous demonstrations belong not to the present subject of enquiry : as far indeed as the establish- ment of systems and the verification of hypotheses are concerned. The purely mathematical demonstrations which are subsidiary are, indeed, as true in Astronomy as in any other science : but the theory they have acted in aid of, they may have vainly propped, and it may be false. A theory if false may be proved to be so by one instance : whereas the truth of a theory can hardly ever be easily or soon established. CHAP. III. On the Vicissitude of Seasons, and of Day and Night. THE daily rotation of the Earth round its axis, and the annuaF revolution of the Earth round the Sun, are the two hypotheses "which, in the preceding Chapters, have been found adequate to explain several of the ordinary phenomena of the Heavens. A condition attending the former hypothesis is that the axis of the Earth always preserves its parallelism. For the polar star is always (to common observation at least) quiescent, and the cir- cumpolar stars always describe circles of the same magnitude. A condition attending the second hypothesis is that the path of the Earth's circuit, or its orbit, lies in one plane : since the points of the Heaven in which the Sun, during the year, is successively seen, lie in one or the same plane. If the Earth's axis of rotation were perpendicular to the plane of its orbit, the planes of the equator and of the orbit would be coincident. The Sun would always describe the^ same parallel of declination ; if he rose once at the east point E, (see fig. pp. 5 and 7.) he would always rise there, his apparent diurnal course would be always in the equator, and his annual course would be amongst those fixed stars which lie in the celestial equator. But we have seen (pp. 15, &c.) that this is not the case ; his annual course is made obliquely to the equator, or, since it is made in the same plane, the plane of his orbit is in- clined to the plane of the equator, and (which is only to repeat the same thing in different words) the axis of the Earth is inclined to the plane of the Earth's orbit. This point enables us at once to explain the vicissitudes of the seasons, and the different durations of day and night, as depen- dent on the combined circumstances of the time of the year and of the latitude of the place. Let S be the Sun, E the Earth in three positions 1, 2, 3, of her orbit ; let also Pp be the Earth's axis, EQ the equator, and PA Qp must be conceived to be a section of the Earth perpen- dicular to the plane passing through the orbit EEE ; so that PAQp will be opposite to the Sun, and to a spectator at A will be a meridian *. The axes Pp are drawn parallel to each other in the three positions. Let us first consider the position marked 1, in which the angle SEP is the greatest possible. The spectator, situated in a northern latitude, is supposed to be at A : his zenith at Z, and his rational horizon will be Hh, Hh being perpendicular to EZ. The Sun in this position, as it is plain, is the most below the equator EQ, and least above the horizon when on the meridian ; * Diagrams in Astronomy are not only imperfect representations, since solids are to be represented in piano, but, with regard to proportion, preposterous representations of the things they ought to exhibit. The first is a real evil, the latter a very slight one ; for, the demonstration in the text is equally clear whether E2 be the half or the double of what it is in the above Figure, it is in fact, independent of the represented relative proportion of E2 to SE; yet, the former is to the latter, in fact, as 1 to 22984, and not, as in the Figure, as 1 to about 3 or 4. The first evil, however, if we do not recur to schemes of solid representation, admits of no remedy, except from the student's attention. The orbit 1, 2, 3, must be conceived as viewed obliquely, and then PAHp to be perpendi- cular to it. Or, if the orbit be conceived coincident with the plane of the paper on which it is drawn, then the plane passing through PAHp is per- pendicular to the paper. and, according to the construction of the diagram, the Sun is on the meridian of the spectator A. The position 1, corresponds to that case of p. 16, in which the Sun rose between the east and south points at his farthest point from the east. In this position of the Earth, a plane drawn perpendicular to SE, at the point E, would divide the Earth into two hemispheres, one illumined, the other in darkness as it is represented in Fig. of p. 29 ' the south pole ( p) being in the former, the north pole (P) in the latter. In this case, since the boundary (df) of light and darkness falls between A and P, it is clear that the spectator at A would, by the rotation of the Earth round P/>, be transferred from A to c in a less time than he would be transferred from c to a: but 2 Ac is proportional to his day, Qca to his night. Again, since 2 A k is proportional to 12 hours, the duration of the day (2 Ac) would be less than 12 hours, and the duration of the night (2 cfl) greater than 12 hours, and the difference would be measured by 2cAr. This difference is easily computed in any given latitude : through c draw PCM, a quadrant of a secondary to the equator, then, by similar figures, mE bears to QE the same proportion as ck bears to Ak : now, in the right-angled spherical triangle cEm, we have cm = QA = (see p. 10.) the latitude of the place, /.cEm = 90 SEQ = co-declination of the Sun, whence by Naper's Rule, (see Trig. ed. 3. p. 146.) rad. x sin. mE = co-tan. cEm x tan. cm =s tan. 's dec. X tan. lat. Suppose, for instance, the latitude of the place to be 51 52' and the Sun's declination (which must be his greatest south de- clination) to be taken equal to 23 28', then we shall have log. tan. lat. (51 52') 10.10510 log. tan. dec. (23 28') 9.63761 .'. 10 -f log. sin. mE = 19.742? 1 /. (by the Tables) mE = 33 34' 20", nearly, and (15 being equal 1 hour) in time. . . = 2 h 14 m 17% nearly; D 26 consequently, the length of the day = 12 h 4 h 28* 34* = 7 h 31 m 26 s , and the length of the night = 16 28 34. We may also arrive at this result by means of a diagram like the present. Thus, let Z be the zenith of the place, take ZQ = 51 52' and draw QJE, representing the equator, through the centre; also Q7kf=23 28', and draw Mvm parallel to QE : then Mvm (see p. 8.) is the Sun's parallel of declination : and (Hh being the horizon drawn perpendicularly to ZE) QMv measures the time the Sun is above the horizon (Mm mea- suring 12 hours), or QQt measures the length of the day, and 2QE + 2tE the length of the night, 2QE measuring 12 hours : in order to find 2 E, the difference of day and night, we have by Naper's Rule, rad. x sin. t E = cotan. L tEv x tan. tv = tan. ZEQ X tan. QM = tan. lat. x tan. 's dec. as before (see p. 25.) At the point v, which is the intersection of the parallel of declination Mvm and of the horizon H h, the Sun rises. In the case which we have taken, it is the point of the Sun's rising that is farthest from the east towards the south (see p. 16.). Its distance from E may be computed by solving the right-angled triangle t Ev thus: 27 By Naper*, r X sin. tv = cos. lat. x sin. Ev, log. r-h log. sin. 23 28' 19.6001 1 log. cos. 51 52' 9-79063 log. sin. Ev = .w/|. s?u- . . 9.80948 .'. Efl= 40 9' 25". - In the position (3), which is diametrically opposite to (1), the Sun, (since the axes Pp, Pp are parallel to each other) is as much above the equator as he was below in the position (1). If there- fore we were to draw, as before, a plane passing through E and perpendicular to SE, it would separate the Earth into two hemi- spheres, one illumined by the Sun, the other deprived of his light : but, in this latter case, the north pole P would be as much within the illumined part as the southern pole p was in the posi- tion (1). The length of the day, therefore, will be what the length of the night was in the position (1), and vice versa : and the Sun in rising will now rise between the east and the north points, and as much towards the north, as in the position (1) it rose towards the south. This scarcely needs any proof; a proof, however, if required, might easily be had by the aid of the diagram already used. Thus take NQ equal the Sun's greatest northern decli- nation, and draw Nun parallel to the equator QE: then the Sun will rise at M, and, in order to find Eu, we have (supposing a secondary to the equator to pass through u\ rad. x sin. O *s dec = sin. Eu X sin. QEH, or rad. X sin. 's dec. = sin. Eu x cos. lat. the same equation as that given above, for determining Ev ; .'. since NQ = QM, Eu = Ev, and consequently the arc Eu, or the Sun's amplitude, (see p. 5.) equals 40 9' 25". The instances taken have been those, in which the Sun is most below and most above the equator : but the scheme will serve for other positions of the Earth : and, the computations for the lengths of day and night, and for the distance from the east, * Trigonometry, p. 146*. 28 will be similar : since, instead of 23 28', we have only to sub- stitute some other number of degrees, representing the decli- nation. In the position (2), the Sun is neither above nor below the equator, but in its plane produced. In the preceding diagram, Q would be the Sun's place : and the parallel described in 12 hours would be Qq, and, EQ being = Eq*, the days and nights would be equal. The position (2) represents the Earth in spring. In the preceding instances we have supposed the spectator situated in some northern latitude between P the north pole and Q the equator. If we suppose him transferred from A (see fig. of p. 2Q.) towards Q, the zenith Z, which is always in EA produced,, will descend towards the equator, and the point h (Hh being always perpendicular to EA) will approach to P. When A reaches Q, or when the spectator is at the equator, h and P will coincide, and the axis of the Earth will lie in the spectator's horizon. The diagram, therefore, of p. 26, will now assume the following appearance, i which the parallels of declination Mm, Nn, always bisected by Pp, are now bisected by H h. In other words, the Sun (if Mm, -Nn represent his parallels of de- clination) will, whatever be his declination, remain as long above as below the horizon : or the days and the nights of a spec- * g is omitted (Fig. p. 26.) in the point where QE produced cuts the circle 29 tator at the equator consist, whatever be the season, each, of 12 hours. If Mm, Nn represent the parallels of decimation be- longing to stars, then the inference is that every star is as long above as below the horizon, and that there are no circumpolar stars. If the spectator, instead of moving towards Q, move towards P, the arc A c which represents, or relatively measures, half his day, will decrease : At the point d, the spectator will be in darkness during the 24 hours * : but, since the figure is con- structed for the greatest southern declination of the Sun, the above circumstance, namely, that of a night's duration of 24 hours, cannot take place either on a preceding or a following day : since, in either case, the Sun's declination, being less than his greatest declination, will cause the boundary of light and dark- ness to fall a little within the point A (the place of the spectator) and P. Between d and P the spectator will be always within the darkened hemisphere, an&, at P, the zenith and pole will coincide, as will the equator and horizon : the following diagram will represent the circumstances of the spectator's situation, which it will represent not only when it corresponds to fig. 1, (see p. 24,) that is, for the greatest southern declination of the Sun, but for any other declination. Thus, it must be continual night whilst * IfEQ=23 28', PEd=23 e 28', which angle, if the spectator be at c?, is the complement of his latitude. Consequently, in latitude 66 32', on the shortest day, there is no direct light from the Sun. He would at noon just appear on the south point of the horizon. 30 the Sun describes any parallel beneath QHh, or whilst the Sun is QH to the south of the equator ; and continual night, when the Sun's declination is northern. A spectator then, if we imagine him in such an extreme situation, would, during one half of the year, experience continual day, and, during the other half, continual night. We have spoken (see p. 26.) of the Sun's describing a parallel of declination, which expression is not strictly correct : since the Sun's declination, which is perpetually changing, will be a little different at the end of 24 hours from what it was at their beginning. If the Sun is ascending from the equator towards the north, he will be higher above the horizon of the spectator at the north pole at the end of 24 hours than at the beginning. Instead, therefore, of describing a parallel to the horizon (the horizon and equator in this instance are coincident) he will describe a spiral, and, in such a curve, he will appear continually ascending above the horizon till he has reached his greatest northern declination. From that summit he will, by like steps, descend, during a quarter of a year, or thereabouts, to the horizon and equator. But if the Sun does not describe an exact parallel to the horizon of a spectator situated at the pole, a fixed star does. Every star, in fact, that is then visible, is a circumpolar star : equally elevated above the horizon wherever viewed ; a spectator in fact, placed exactly in the pole has neither a meridian nor any cast and west points. 31 Whatever be the circumstances relating to the durations of light and darkness which a spectator experiences in a northern latitude, when the Sun has a south declination, the same will a spectator, situated in a corresponding southern latitude, experience when the Sun has a corresponding north declination. Or the durations of day and night, when the Sun has a certain declina- tion, will become reciprocally the durations of night and day when the Sun has an equal contrary declination. Thus, the Earth occupying the position (3) (in which the Sun is supposed to be at his greatest northern declination) the length of the day to a spectator in north latitude 66 3%' (see Note to p. 2Q.) would, on his longest day, be just 24 hours. The Sun, at midnight, would just cease to be visible on the north point of the horizon. It has appeared (see p. 26.) that PEd = SEQ =23 28' when the Sun is at his greatest northern declination. Draw from d (fig. p. 29-) a parallel db to the equator, and also a similar parallel from the pointy*: the parallels or small circles thus determined are denominated respectively the Arctic and Antarctic circles, or generally the Polar Circles. The distance of the former from the north pole, and of the latter from the south pole, is equal to the Sun's greatest declination. The vicissitudes of seasons, inasmuch as they depend on the durations of day and night, have been explained from the revo- lution of the Earth round the Sun, and from the rotation of the Earth round an axis constantly inclined at the same angle to the plane of the Earth's orbit. If the Sun be the source of heat as well as of light, then heat will be imparted to an inhabitant of a northern latitude, during a less time in the position (l) than in the position (3). But, besides this circumstance, the Sun's rays fall more obliquely on A in the position (1) than in the position (3), for in (I) z SEA = L AEQ + z SEQ, and in (3) L SEA = z AEQ -SEQ. This, in some degree, will account for the differences of tempera- ture experienced by the same spectator at different seasons of the year; and one of the causes previously assigned, namely, the degree of obliquity of the Sun's rays, will explain why the regions near the equator are, cateris paribus, hotter than the more remote. The distinction of the Earth's surface into climates and zones has 32 been long made. Within two parallels of declination, each distant from the equator 23 28', and called Tropics, the Torrid Zones lie : the Frigid Zones lie within the arctic circle and north pole, and the antarctic circle and south pole. The Temperate Zones are included within the tropics and the polar circles. The above must be viewed merely as general and arbitrary divisions. We cannot affirm a place not to be cold solely because it is within the temperate zone. Local causes have vast influence. The temperature of the air at a place is not proportional solely to the place's latitude and the Sun's declination and distance*. * We have not supposed hitherto the Sun's distance to be variable, which it is. CHAP. IV. On the Phases and Eclipses of the Moon. IF, in arranging the heavenly phenomena, we had purposed to give precedence to those which were either more obvious or which excited greater curiosity, we ought to have considered the Moon previously to the Sun and the planets. The proper motions of the latter, and their other phenomena, do not obtrude themselves so forcibly on our notice, as those of the Moon. Venus, to unassisted vision, always appears to shine with a full orb : but viewed through a telescope she assumes, like the Moon, her several Phases, and shines with an orb more or less deficient. The Earth, as it was stated in p. 20, moves round the Sun. The Moon also (such is the doctrine to be laid down) moves round the Earth but, in an orbit, the plane of which is not co- incident with, or parallel to, the plane of the Earth's orbit. If to these we add another condition, namely, that the Sun illuminates the Moon, and that the inhabitants of the Earth perceive the effects of such illumination, we shall possess the means of ex- plaining why, at some times, the whole face or disk of the Moon is luminous, whilst, at other times, only portions of it are : we shall, in other words, be able to explain the Moon's Phases. Let M, M', M" be three different positions of the Moon in M her orbit, and let the dotted curve line represent the outline of a E 34 portion of the plane of the ecliptic, which plane we must sup- pose inclined to that of the Moon's orbit. JE is meant to represent the Earth, and the Sun is supposed to be so far distant that lines from it to M, M', M", &c. and E may, for small portions near those points, be held as parallel. Nn is the line of the nodes, that is, the intersection of the plane of the Moon's orbit with the plane of the ecliptic, or the plane of the Earth's orbit round the Sun. Now Ss is the direction of light issuing from the Sun to illuminate the Moon : suppose the Moon to be a sphere ; then a plane, passing through its centre and perpendi- cularly to Ss, would divide the Moon into two hemispheres, the convex surface of the one being bright, that of the other dark. But, except in certain positions, a spectator at E will see only part of the illumined hemisphere. Divide the Moon into two hemi- spheres by a plane passing through the Moon's centre, and drawn perpendicularly to a line joining that centre and the spectator, then the hemisphere, which is towards the spectator, is the one he views. Mm (in the figure of p. 33.) perpendicular to S s is the projected boundary of light and darkness : ab, perpendicular to, a line drawn from JE to the centre of the Moon, is the pro- jected boundary of vision : a spectator at JE, therefore, views only that illumined part of the Moon's disk, of which mb and two lines, drawn from the Moon's centre to m and 6, form the projected boundary. If the Moon, therefore, were at c between the Sun and Earth, a b, and Mm coinciding, no portion of her illumined disk would be visible : but, at M ", the whole illumined disk would be visible, (supposing the planes of the Earth's orbit and of the Moon's to be so inclined, that the Earth impede no light from falling on the Moon) ; at M', (in which position it is intended that the lines M' m 1 , a b' should be perpendicular to each other) the Moon will shine with half a face. There are several technical denominations given to the Moon in the above positions. At c, the Moon is a new Moon; at M", a full Moon; at M 1 , supposing half of her disk to be luminous, the Moon is said to be dichotomized. In the course of her circuit, which occupies a period of about 29 days, the Moon must, it is plain, exhibit all her Phases : the narrowest crescent near to d : an half Moon at M ', a full orb at M." : past that state, her orb becomes deficient, and the Moon wanes, till reach- 35 big a line joining the Earth and the Sun she turns her dark side entirely to the spectator. In the position c when the Moon is new, she passes the me- ridian at the same time the Sun does, or, in other words, she is on the meridian at noontide. In the position M, she must, since the Earth's rotation is from west to east, pass the meridian after the Sun, and it is her western limb which appears illuminated. At M", the Moon, at her full, comes on the meridian at midnight : and past M" and beginning to wane, she becomes deficient on her western side. The Moon's orbit, as it has been already remarked, is in- clined to the ecliptic. The line Nn is meant to represent the intersection of their two planes. Now the line Nn, technically denominated the line of the nodes, is found to be continually changing its position. If during these changes it should occupy the position M"Ec, whilst the Moon were either at c or at M", then the Moon, Earth and Sun would be situated in the same right line, and give occasion to the phenomenon of an eclipse. Suppose, in the first place, the Moon to be at c, and the Sun to be in the line EC produced. Then a spectator at E would either see the Moon as a dark spot, or dark circle, con- centric with the Sun's disk and within it, or, if we choose to conceive the Moon sufficiently large, the spectator would be unable to see the Sun by reason of the Moon's interference. The phenomenon, in the first of these predicaments, is called an Annular Solar Eclipse, in the latter, a Total Solar Eclipse. In the second place, if the Moon be at M", the Earth, being interposed between the Moon and Sun, must intercept some of the Sun's light in its passage to the Moon. It may (if we argue the matter independently of the actual magnitudes of the Sun and Earth) intercept the whole ; and, under any consideration, it must cause the Moon to be less illuminated than it would be, did it not intervene. In fact, the Earth being a sphere or nearly so, its shadow will be conical and towards the Moon. We may, hypothetically, assign such dimensions to the Earth that the vertex of its shadow shall fall within the Earth and the Moon, in which case the Moon's disk would be only dimmed but not eclipsed ; but, according to the actual dimensions of the Earth and its distance from the Moon, the shadow of the former always extends beyond the latter and causes it to be eclipsed 36 From the preceding account of the causes of eclipses, we may easily infer a material distinction between a lunar and a solar eclipse. When the forme; happens, the Moon is deprived of the Sun's light, and is darkened by the Earth's shadow ; and every spectator on the Earth, that can see the Moon, sees her eclipsed. In the case of a solar eclipse, the Sun is not darkened but con- cealed, either entirely or partially, by the intervention of the Moon. The Sun may appear, on its rising, eclipsed to one in- habitant of the Earth, whilst, at the same time, to another inhabitant, in a different region, he may set with a full and bright orb. It will require the aid of computations to point out the exact circumstances of eclipses : that matter is reserved for a future Chapter. We will close the present by observing that the Earth's shadow, at the Moon, is sufficiently large to eclipse the whole of the latter body. The section of the Moon's shadow, on the contrary, at the Earth, is a round spot, of no great dimen- sions, that rapidly passes over the parts of the Earth's surface which it successively eclipses. We have, in the present Chapter, supposed the Earth to be either spherical, or nearly so, and to cast a conical shadow. In the next Chapter we will briefly examine the grounds on which such supposition is built. CHAP. V. On the Earth ; its Figure and Dimensions. ONE of the proofs of the spherical form of the Earth is drawn from the phenomena of the preceding Chapter. In all lunar eclipses, the boundary of the Earth's shadow on the Moon's disk is apparently circular : such as ought to be the section of a conical shadow of a sphere. A considerable defect of sphericity might, however, exist in the Earth's figure without its being detected by this phenomenon. There are, besides, other circumstances that render probable, and with like nature and degree of evidence, the globular form of the Earth. A ship, viewed as it approaches us, first comes in sight by shewing us the tops of her masts : next, more and more of the masts are seen, and, lastly, the hull. And, this pheno- menon is the same, whatever be the quarter, be it the east, west, north, or south, that the ship approaches from. Again, on a rock or mountain surrounded by the sea, such as is the Peak of Teneriffe, the sea appears, as it were, depressed, and equally on all sides of the spectator. On the mountain just alluded to, the angular distance between the zenith and any point of the horizon is nearly Q2 degress. The Sun, therefore, must there rise sooner and set later (by about 12 m in the case before us) than to an observer on the plain : and, which is the same phenomenon or one immediately following from it, the summit of the mountain will be illuminated 12 minutes before Sun-rise and 12 minutes after Sun-set. The same phenomenon, modified solely with regard to time, and consistently with the hypothesis of the sea's spherical surface, is always found to take place in mountains of less or greater height. The preceding circumstances shew that the Earth is round, and that it is neither flat like a plane, nor concave like a bowl : but they will not serve, not being of a sufficiently precise nature, to found thereon a proof of the Earth's sphericity. That the 38 Earth cannot be a perfect sphere it is indeed easy to shew, although it is not easy to shew what is precisely its figure. The disposition of mankind to believe in the existence of simple and regular bodies first suggested a sphere, and then a spheroid, as the Earth's figure. And the labours of mathematicians have been directed, these last hundred years, to ascertain the truth of the latter suggestion. It is a matter, no unworthy of notice, that the Moon which, by one of the circumstances of her eclipses, (see p. 37, 1. 4.) proves the roundness of the Earth, in another way (by one of her inequalities) proves its non-sphericity and the degree thereof. We have not yet mentioned an argument, an analogical one, indeed, and not a very strong one, by which it is inferred that the Earth, one of the planets, is round, because Venus, Jupiter, &c. appear to be so. If we argue similarly with respect to the nature of the Earth's deviation from a spherical form, we ought to infer that the Earth resembles an oblate spheroid bulging out at its equator and flattened at its poles, because Jupiter is so formed. Indeed, if the Earth be not a rigid mass, such ought to be its figure. It is easy to see, on mechanical principles, that a fluid globe revolving like the Earth round an axis would become protuberant in its equatoreal parts. What has preceded relates to the figure of the Earth ; but its dimensions are an object of enquiry. If the Earth be a sphere, what is its radius? if a spheroid, what is (as it technically is called) its Ellipticity ? These are questions about which Astro- nomers have been busied from the earliest times. If we look to all the curious apparatus of methods, instru- mental as well as computative, by which modem science has attempted to measure the Earth, there cannot well be a wider interval than that which exists between the rude Essay of Erastothenes made more than 2000 years ago, and what is now practised. The methods, however, rest on a common ground. At Syene, in the Thebais, the Sun on the meridian, at the time of the solstice, was vertical. It illuminated the bottoms of wells, and the highest buildings cast no shadow. On the same day the Sun's distance from the zenith of Alexandria was observed to be 7 12'. Let C be the centre of the Earth, s the Sun vertical to 39 E (Syene), and let the angle s AZ, the angular distance at A (Alexandria) of s from Z the zenith of A, be 7 12' : then we have the Z EGA = z ZAs z CsA = 7 12' if we neglect, by reason of its smallness, the Z CsA, the angle which the Earth's radius, in this case, subtends at the Sun. Hence the Earth's circumference equals 360 7 12' X EA, and is known in linear dimensions when EA is so expressed. If the distance of Syene and Alexandria be assumed equal to 5000 Stadia * and those places be supposed to lie (which they do not) in the same meridian, then 360 the Earth's circumference = , x 5000^=50 x 5000* =250000 stadia, and the number of stadia contained in 1 degree would be * According to Lalande, the Egyptian Stade= 114.13 toises, and a French toise = 6 ft 4 in .34 . 40 694 8 .444, &c. ( = It is not necessary to stop here to shew the various sources of inaccuracy, in the above method. Let us attend to the modern way of proceeding. If we advance towards the north, the pole star approaches our zenith, or, if we proceed along the same meridian, the star which we at first observed in our zenith, recedes from it. Suppose between two stations of our progress that the pole star has become 1 nearer to the zenith, or (which is the same thing) that the star, which was vertical at the first station, is 1 distant from the zenith of the second station ; then, if the actual distance between the two stations should be 69i miles, the Earth's circumference, which contains 360 degrees or 360 such differences of latitude as are equal to 1, would equal 3 60 X 69 J and would be about 25020 miles : and its diameter would be about 7960 miles. This method, it is plain, is founded on the same principle as that of the Astronomer of Alexandria ; and, if it be pursued, it must needs furnish a proof of the Earth's ellipticity, or rather, of the defect of its figure from perfect sphericity. For, were the Earth a perfect sphere, the same linear distance (69 \ miles for instance) ought always to be found between any two places on the same meridian and differing in their latitude by 1. This, however, is not the case. In latitude 66 the linear distance between two places, under the above predicaments, is found to be 122457 yards. But, near the equator, such distance is found to be 121027 yards. The former distance being 69^ miles -H37 yards, the latter 69j miles 1293 yards. And it is established as a fact, by means of observations and measurements, that degrees (by which we mean their linear values) increase as we move from the equator to the pole. If the Earth be supposed to be a spheroid, its measurement is to be conducted, as in the hypothesis of its being a sphere, by finding the difference of latitudes between two places, and by measuring and computing the linear distance between them. The axes of the spheroid cannot, it is plain, be determined by so simple a process as that which gives the radius of the sphere. It is a question of pure mathematics to assign, from two degrees, one measured at the equator, the other near the pole (or any two other places), the eccentricity of that ellipse, which, by revolving round its minor axis, shall generate the spheroid to which it is believed the Earth is like. If all meridians were similar, and all measurements equally to be relied on, the same eccentricity ought to result, wherever the two degrees, the data of the problem, should have been measured. But the case is otherwise. One mathematician by comparing a degree measured in Lapland, with a degree measured in France, assigns for the 307.405 Earth's oblateness; results from Col. Lambton's measure- 320 ments in India : who compared (for so may the problem be mathematically solved) a degree of the meridian with a degree perpendicular to it. Lalande thinks , Delambre to be its 300 309 true value. In fact the question, whether we look to its the- oretical or to its practical part, is a very difficult one, and likely, for many years, to remain doubtful, and to be the subject of discussion. There is another method of determining the Earth's oblate- ness, founded on the different times of vibration of the same pendulum in different latitudes, or rather, on those differences of vibration which depend solely on an augmented or diminished gravity. The variation of gravity, or of the weight of a body, arises from two causes : the non-spherical form of the Earth and its rotation. From the first cause, the attraction is not, as in the case of an attracting sphere, the same as if all the matter of the spheroid were collected into the centre, and the resulting force directed to that centre. Two plumb-lines (and the directions of gravity are no other than the directions of such lines) containing, at the pole, an angle equal to 1, will meet in a point of the polar diameter beyond the centre of the spheroid. At the equator two such lines, so conditioned, would meet in a point of the equa- toreal diameter short of the centre. Jn other situations the point of concourse will not be in a diameter passing through one of the extremities of the arc. 42 The second cause, the Earth's rotation, gives rise to a cen- trifugal force, a resolved part of which, acts in the direction of gravity and diminishes it. This centrifugal force is nothing at the poles and greatest at the equator, and that resolved part of it, which counteracts gravity, varies as the square of the cosine of latitude. This enquiry, like the former one, is not easy, and, whatever be the mathematical skill bestowed upon it, must always terminate in doubtful results. For it rests on two hypotheses very difficult to be verified, 1 st , the spheroidal form of the Earth, and 2 dly , an assumed regularity and law in the disposition of its materials. If we refer to p. 4, we shall find that the rational and sensible horizons are parallel to each other, and distant from each other by an interval equal the Earth's radius. Now that radius, as we have just seen, is about 4000 miles. It is, however, a distance, compared with that of a fixed star from the Earth, of no relative value : from which it follows that, in what regards the fixed stars, we may suppose the two horizons coincident : or, which amounts to the same thing, any calculation, made with respect to a fixed star by a spectator on the surface of the Earth, is precisely the same as if the spectator had been placed in the Earth's centre, to which point, on other occasions, that is, when the Moon or a planet is concerned, it is usual to refer or reduce Astronomical computations. In order to prove what has been just asserted, let S represent the Sun, 5 a fixed star, and E, E' two positions of the Earth in opposite points of her orbit. At these two positions the angles sEL, sE'L can be determined by observation and calculation, 43 and, on comparing them, they are found to be equal ; but sEL = tsE'L+ tE'sE, consequently, the angle EsEf has no value, or, the distance sE is so immense that the diameter of the Earth's orbit subtends no angle at s. There is no assign- able proportion, therefore, between sE and EE' ' , and, a fortiori, none between the Earth's radius and sE: since EE 1 is to the Earth's radius as 45968 to 1 *. We have in this, as in each preliminary Chapter, treated its subject in a popular manner. The explanation has been general, and consequently vague, and indeed it is scarcely worth any thing if it were not preparatory to discussions of greater pre- cision. We have spoken (see pp. 38, &c.) of the antient mea- surement of the Earth as of a rude method : but that which is afterwards described as the modern method may, notwithstanding any thing contained in that description, be equally so. In fact, the superiority of one method over another, cannot be shewn except by entering into their respective details. Those of the first may be comprised, as they have been, in a few lines : the details of the latter are sufficient to fill a large volume. We have spoken of the zenith distance of the Sun at Alex- andria, in the time of the solstice, as being 7 12', and of two places differing from each other in latitude by 1; and a student, in the outset of his Astronomical career, may imagine that nothing is easier than to form a notion of these angular distances. It is not likely, indeed, that he should anticipate (for he can only know them till after trial) the difficulties that await him. The angular distance of a star on the meridian from the zenith is the angle contained between a straight line drawn from the star to the spectator, and a line vertical to the spectator (the direction, in fact, of the plumb-line.) Now the first point of enquiry (which Erastothenes did not enter into) is, whether the star is really in the direction of the former line, or whether the direction of the ray of light when it enters the eye coincides with that of the former line. If it does not, then is the angle we see and measure, not * If * were near the pole of the ecliptic, and Es= 200000 ES, the angle EsE' would be about 2": but since no such angle can be detected, or at the utmost, an angle not exceeding 2", the ratio of Es to the Earth's radius must be at the least, that of 4569800000 to 1. 44 the angle we are in search of. We may be able to correct the former angle, and thence find the latter, but then there comes a second point of enquiry, whether or not the correction known for one case will suit all others ; whether, for instance, the same quantity of correction which reduced the observed zenith distance (7 12') of the Sun at Alexandria, would truly reduce an observed distance at another place, at Rhodes for instance, where, at the solstice, the Sun's zenith distance would be about 13 degrees. If we would answer these questions we must enter into an inves- tigation, which is no other than that of the Laws of Refraction. But the enquiry would not terminate with the settling of those laws. Suppose we knew how much the light of a star would be made to deviate, by reason of the atmosphere, from a line joining the star and the spectator, would the deviation of the same star, to the spectator at the same place, be the same at whatever hour the star passed the meridian ? The student, it is probable, would here also feel no hesitation in answering that the star's apparent angular distance must be independent of the time of its transit over the meridian, and that, if refraction were away, a star would always pass the meridian of Greenwich at the same distance from the zenith of Greenwich (such distance being de- termined by an instrument) whether the hour of transit were 9 in the morning or noon. The fact, however, is otherwise, and, as it will be shewn here- after, there is, besides refraction/ a cause of inequality which makes the instrumental zenith distance different from, if we may so call it, the true zenith distance : which cause of inequality is connected with the time of the star's transit over the meridian. But the process of correction would not cease here ; there are, at the least, six causes of inequality, each of which will render the observed angle, whether it be an angle between two stars, or, between the zenith of the observer and a star on the meridian, unequal to the true one. So hard to be understood then, not- withstanding its apparent simplicity, is the expression, of the difference of the latitude of two places being 1. Erastothenes if he had possessed the most perfect of modern instruments, had he possessed them without modern science, could not have ascer- tained the Earth's dimensions. But although this be the case : although it is essential to know 45 the quantities and the laws of those corrections, by which we are to derive the latitude of a place, or any other angle, which may be the object of research, from the observed or instrumental angle ; still, it is plain, this latter angle is of primary importance. If we are unable to determine that exactly, the corrections pro- vided by Astronomical Science may be of little or no use. Their sum may be less than the error of observation, which error, in such a case, would vitiate all subsequent processes founded on the observation. It will, therefore, be following something like a natural order, to describe the instruments by which angular distances are mea- sured, previously to the investigation of methods for correcting such distances. And, in pursuit of this plan, we shall not digress into a description of antient instruments nor (however in- structive in itself such an enquiry may be) into the history of their successive improvements. We shall be content to describe the instruments which are essentially necessary to determine the places of the heavenly bodies ; those instruments which are called, for distinction's sake, the Capital Instruments of an Observatory, which, indeed, are few in number, and simple in their construction, each being appropriated to one class only of observations. The tendency (if we may so describe it) of im- provement in Astronomical Instruments has been towards sim- plicity in their construction. In former times Astronomers endeavoured, in their instruments, to imitate the celestial sphere : which were formed in call effigiem ; hence came their Astrolabes and Armillary Spheres. According to modern practice, all im- portant observations are made on stars on the meridian. It is there that Astronomers, with fixed instruments, wait for a star instead of attending on its course from east to west. CHAP. V. On Astronomical Instruments. THE position of a point in a plane may be determined, by means of two rectangular co-ordinates (as they are called), that is, of two lines perpendicular to each other and measured from the same point. In like manner, the position of a star on the celestial sphere may be determined by portions of two great circles, perpendicular to each other, and measured from the same point. Thus, let P be the pole, AmQ, a portion of the equator, s a star, and Psm a circle of declination: then, if A should be a known point or known star in the equator, the position of s on the sphere will be determined from A m and A t ( = wzs) : since we have only to set off Am, on AmQ the equa- tor, to draw the quadrant m P and to set off, onmP, ms equal to At. Now, the right ascension of a star is its distance measured on the equator from a fixed point in the equator. If that point be Ay Am will be the right ascension of the star s, ms its decli- nation, Ps its polar distance, and Zt its zenith distance, when s is on the meridian, the position of which is represented by PZA. We must consider what are the means of measuring A t and A m when the star s is on the meridian. With regard to the first point ; we have only (by which term, however, we do not mean to signify the great facility of the 47 operation) to divide a quadrantal arch such as aeb into a number of equal parts, to place it in the plane of the meridian, and to direct a telescope in the direction eC to the star t : then eb will express, by a certain number of the above-mentioned equal parts, the zenith distance of the star s on the meridian, and ea will express the altitude. Besides the conditions mentioned, it is clear that Cb must be vertical, which it will become by being made coincident with, or parallel to, the direction of the plumb- line. With regard to the second point : there are no obvious means, and certainly no simple ones, of instrumentally measuring the angular distance between A (even supposing it to be a star) and m the point where the secondary passes through 5. Other means, than those of instruments giving angular distances, must be resorted to : and Astronomers have called in time to express, intermediately , the right ascension of a star : which plan may be thus explained. Suppose (for the sake of simplicity) A to be a star, and the point m to be carried, by the rotation of the sphere, in the direc- tion m A : then m and S would be on the meridian at A and t at the same instant, and the arc Am, the measure of the angle s PA, would bear to the whole circle, or to 360 degrees, that proportion which the time elapsed, between the transits of A and s over the meridian, bears to the whole time of the sphere's rotation ; and contrariwise, an observed or noted time between the transits would, in terms of time, be the right ascension Am, which, if 24 hours be assumed as the time of the sphere's rotation (or of the Earth's diurnal rotation) would equal x 360 . To enable us, then, to find the right ascension of stars, there are, according to the above plan, two instruments necessary : a telescope in the plane of the meridian to observe A and the star s when on the meridian, and a clock to note the respective times of their being there. The instant of a star's passage cross the meridian being denominated its transit, the telescope used for observing the star, at that instant, is denominated the Transit Instrument. The three capital instruments then of an Observatory, are the Astronomical Quadrant, the Astronomical Clock, and the Transit Instrument. 48 In point of theory, or if we regard solely the mere purpose of explanation, the two former instruments are the only ones essen- tially necessary, because no reason, not suggested by actual ex- periment, can be assigned why the office of the third instrument should not be performed by the quadrant, which is supposed to be placed in the plane of the meridian, and to be furnished with a telescope capable of being pointed to any part of the meridian. The special use, or the practical convenience of the transit in- strument, depends on reasons altogether practical and not yet explained. We will now proceed to a more particular description of the Astronomical Quadrant, which may be considered as repre- senting a class of instruments, known by the names of Declina- tion Circles, and of Mural Circles, and designed for the measur- ing of zenith distances and polar distances. The annexed figure is meant to represent a mural quadrant, or one fixed in the plane of the meridian. TV is a telescope T moveable about a centre at V : and Vv is a plumb-line, which is, in general, a fine thread or wire with a weight attached to it, and, for the sake of steadiness, plunged in water. The first point to be considered is the division of the quad- rantal arc A B. The most usual graduation of the arc consists of 90 degrees : but many quadrants (the two 8 feet mural circles of Greenwich, for instance,) have, besides this usual graduation, a second one, 49 consisting of 96 equal parts*. An observation is to be read off (as the phrase is) on each scale, and then, by means of a com- puted Table, the divisions of the ninety- six scale are to be reduced to those of the ninety. But the graduation is not limited to 90 or 96 parts or degrees. Each degree is itself divided into a number of equal parts, each part containing a certain number of minutes : the number of minutes being the less (we are speaking practically) the greater the instrument. In quadrants and circles of nine inch radius, the smallest division on the limb of the instrument contains generally 30 or 20 minutes. Quadrant of 18 inches are divided to 15 minutes. The 8 feet mural quadrants of Greenwich, and the 6 feet mural circle, are divided into equal parts of 5 minutes each. There are, however, certain little and subsidiary instruments, called Verniers, attached to that end of the telescope which moves along the arc of the quadrant, that enable us to read off the observations to a greater nicety, and that (if we may so express ourselves) stand in the stead of a minuter graduation of the limb of the instrument. We will now explain the principle and use of the Vernier. Let A B represent part of the limb of a quadrant (of that, for instance, which was represented in p. 48.), Tt part of the telescope which moves along the limb, and Mm- a thin plate of * The graduation of ninety-six degrees was adopted on principles of mechanical convenience; and for the purpose of lessening the great dif- ficulty which attends the graduation of instruments. A chord of 60 degrees, in the common division of the circle, being equal to the radius, a chord of 64 degrees, will be equal to radius, when the quadrant is divided into 96 equal parts, or degrees. Hence, by means of a line equal to the radius of the quadrant, two points can be determined on its arc, containing 64 out of 96 equal parts ; and, by the continual bisection of 64 ( = 2 . 2 . 2 . 2 . 2 . 2) a, division equal to one of those equal parts is obtained. It is very easy to conceive a circle divided into 360 equal parts or to describe it as so divided : but the practical effecting of the graduation requires a great deal more than mere dexterity of hand, as artists will testify, or, as any one who will make the trial, will soon experience. 50 metal (the Vernier) attached to the telescope at w, and together with the telescope, moveable along the limb of the quadrant. A In the present scheme the vernier is divided into five equal parts, the sum of which is equal to the sum of four equal divisions of the quadrant : and this equality is represented in the figure : in which the lozenge, that mark on the vernier to which would correspond, coincides * with the division or mark on the quadrant marked 41, whilst the mark 5 of the vernier coincides with the mark 45 of the instrument. In reading off we must first look to * Instead of coincides with, we ought, perhaps to say, is opposite to, or in the same right line with, the mark 41 of the instrument. The engraver having separated the boundary of the vernier from the circular line on which the division-marks of the limb of the instrument abut prevents a coincidence from taking place. We may farther note, that one boundary of the vernier is the fourth concentrical circular line, reckoned from the left hand : the other is the seventh, reckoned in the same way. 51 the lozenge for the position that is intended by the instrument maker to mark the altitude of the observed star or other object. Thus, as the figure is drawn, if the telescope were properly directed to a star, the altitude of such star would be 41 : and in such a simple case the vernier is of no use. But suppose the telescope were directed to a star a little higher than the former, then the lozenge would be moved from the division 41 towards 45, and let us suppose it just so far moved that the second mark (1) of the vernier coincides with the division of the quadrant next succeeding the 41 st (the 42 d ) *. In this case it is clear the lozenge (to which we are to look in noting the altitude) has been moved through a space equal to the difference between one division of the instrument and one of the vernier. The altitude of the star then is 41 + this difference : which difference must now be estimated. In the figure to which we are at present referring, the divisions of the instrument are intended to represent divisions of one degree each, and, since four of these divisions, or 4, are equal to five divisions of the vernier, the difference between a division of the quadrant and the vernier is 4 1 1 1 = = 12', 5 5 so that the altitude of the star is to be read off equal to 41 12', and this is the most simple illustration of the use and property (for such it is) of the vernier. If a star still higher be supposed to be observed, and the telescope and its attached vernier be so moved, that the mark 2 of the vernier coincides with the 43 d of the instrument, then the index or lozenge has been moved from its original place, opposite to 41, through a space equal to twice the difference of the divisions of the quadrant and vernier, and consequently, the altitude must now be 41 + 2 X 12', or 41* 24'. * To avoid confusion, and to lessen the difficulties of the engraver, the divisions which lie between 41 and 45, namely, 42, 43, 44, are not figured. 52 If the mark 3 of the vernier coincided with the 44 th of the in- strument the altitude would be 41 + 3 X 12', or 41 3&. If the marks 4 and 45 should coincide, the vernier and the lozenge must have moved through a space equal to five times the difference of a division of the instrument and of the vernier, or through a space equal to one division of the vernier ; and in such case the altitude would be 4 41 -f 4 x 12', or 41 -f- 1, o each of which equals 41 48'. If the mark 5 should be found to coincide with the mark next to the 45 th of the quadrant (which mark would be 46) ; then, it is plain, the vernier and every mark on it and, of course, the lozenge, must have been moved through a space equal to one division of the instrument, or through 1 ; and the altitude of the star, if such were the object observed, would be 42 . In this situation, the vernier would have returned to a position precisely similar to its original one (that in which the lozenge coincided with 41), and any subsequent translations or movements of the vernier, producing exact coincidences (or coincidences seem- ingly such) between any two marks or lines of the vernier and instrument, will be precisely similar to those that have been just explained. But it is obvious that the motions or translations of the telescope and attached vernier may be less, in degree, than those which have hitherto been spoken of. The spaces through which the telescope moves, may be less than the difference between a division of the instrument and a division of the vernier, in which case, there would be no exact coincidence between any two marks or lines of the respective divisions. If, for instance, the telescope should be moved from the position in which o of the vernier coincided with 41 of the instrument, and through a space less than the difference of a division of the instrument and the vernier ; the mark 1 would not reach 42 of the instrument, and the altitude to be noted would be something between 41 and 41 12', and which the observer, should there be no other mechanism belong- ing to the vernier than what we have described, must estimate by guess and according to the best of his judgment. 53 In the scheme illustrating the use of the vernier, we have chosen to consider each division of the instrument to be equal to 1, in which case the vernier will not note smaller angles than twelve minutes : but if each division, instead of 1, were l', the accuracy of the vernier would then extend to twelve seconds : and, generally, when five divisions of the vernier are equal four of the quadrant, the difference between a division of the one and the other will always be equal to ~ , o since L V = L , o L being a division of the quadrant, and V of the vernier. It is clear then, the smaller the divisions of the instrument are, the more minutely (with regard to degrees and parts of degrees) will an observed angle be noted by means of the vernier. But supposing, in an instrument of a given size, the magnitude of each division to be settled, (and there are practical and mechanical reasons that prevent the instrument from being subdivided beyond a certain point) a question will then arise concerning the length of the vernier, or, as the case is stated, concerning the number of its divisions. Instead of five of its divisions being equal to four of the instrument, will it not be better to make ten of its divisions equal to nine of the instrument? or twenty equal to nineteen, or sixty equal to fifty-nine ? In fine, if n divisions of the vernier are to equal n 1 of the instrument, what is the value which it is most commodious to assign to n ? Let, as before, L denote the value of a division of the instru- ment, and V that of one of the vernier, then since (it 1)L = nV, consequently, L being given, L V is less, the greater n is. But n cannot exceed a certain limit, for the magnitude of each division being (see p. 49-) supposed to be assigned, and each division being an aliquot part of a circle, the arc of the quad- rant can only contain a certain number of such divisions ; for 54 instance, if each division contains fifteen minutes, the quadrant contains 4 x 90, or 360 of such divisions, and, in such a case, the limiting value of n is 360, and the difference between a division of the quadrant and one of the vernier, with such extreme value of n, would equal j 15 x so" . 1JL --!., " 360 = 360 6 ' ' 2 ' = ^ * 5 ' But it is plain that a vernier extending along the whole limb of the instrument would be very incommodious (to say the least) : and a like objection would lie against verniers either half or a quarter of the arc of a quadrant : so that there are (in this as in every other case relative to the construction of instruments) certain practical considerations that limit, in a quadrant of a given radius and given number of divisions, the length of the vernier. It is proper then now to state what are usually the propor- tions between the length of the vernier and the radius of the in- struments. Quadrants and circular instruments of 9 inches radius, are frequently divided into equal parts, each consisting of 20 minutes, and 59 of such equal parts are made equal to 60 divisions of their verniers. In this case 60 60 so that, with such instruments, you can read off, by the aid of their verniers, to an accuracy of 20 seconds. In this case, the vernier must occupy on the limb of the instrument a space, at the least, equal to 19 40'. There are quadrants, of 18 inches radius, divided to every 15 minutes, and in which 14 of such divisions are equal to 15 of the vernier. In these instruments then , 15 15 and the space occupied by the vernier, is, at the least, equal to 3 30'. It would appear then that, in this case, we are not able to read off' so accurately as before, although the instrument is twice the 55 size of the former. The fact is, that that happens here which we before alluded to in p. 52. The divisions of the instrument and vernier differ so much, that, in taking an altitude, the telescope, will probably occupy a position, in which there is no exact coin- cidence between a dividing mark of the vernier and one of the quadrant. But, instead of guessing what the defect between the two nearest coincidences is,*the observer is assisted by a piece of mechanism attached to the instrument, which enables him to compute that defect. This we will now briefly explain. The part E can be fastened to the limb of the quadrant by means of a screw. jPG a screw, (Fig. p. 50.) with a milled head at F, works in a collar fixed in the under part of E, and in a female screw fixed in the under part of the telescope Tt. When the part E, then, is fixed or clamped, and the screw is turned round by its milled head at F, it must communicate a direct motion to the female screw (and, consequently, to the telescope and vernier) in the direction of FG. Attached to the male screw, or to the small cylinder on which it is formed, is an index D moveable together with the screw and on a thin graduated im- moveable plate, the profile only of which is shewn in the Figure of p. 50. It is more fully exhibited in the above figure, in which F, D, E, represent the same parts as in the former figure. 56 Suppose now the screw to be of that fineness that, whilst it is turned round, or whilst the screw-head and the index D make one complete revolution, the vernier is so far advanced on the limb of the quadrant, that the mark 1 of the vernier is brought into coincidence with 42 of the limb : then, in our scheme of illus- tration, one revolution of the screw is equal to 12'. If the cir- cumference of the thin plate then (see Fig. p. 55.) be divided into 60 equal parts, one of such equal parts must be equal to 12" : and if, in order to make a coincidence between the lozenge of the vernier and any division of the limb, it were necessary so to turn the screw that the index D should be moved from D to d, and 15 graduations should be contained between Dd, then the space moved through by the vernier on the limb would be equal to 15 X 12", or 3'. Similar results will take place, if the instrument and vernier be differently divided : thus, if each degree of the quadrant be divided into 4 equal parts, and 14 of such parts be equal to 15 of the vernier, the difference between the respective divisions being l', one graduation of the brass plate would equal 3", supposing, now, that three revolutions of the screw move the vernier through a space equal l'. In the annexed Figure, in which a degree is divided into four equal parts, the lozenge or index of the vernier occupies a position between 41 15' and 41 30'. The dividing mark 2 of the vernier very nearly coincides with the mark of the quadrant which denotes 41 45'. If it exactly coincided, then 57 *he lozenge,, or index, being advanced beyond the mark next to 41 , (the mark denoting 41 15') by a space exactly double the dif- ference of a division of the instrument and one of the vernier, the altitude or angular distance denoted by the instrument would be 41 15' + 2 x l', or 41 if, but the angular distance is, obviously, of somewhat greater value. Suppose, in order to carry the vernier so far back as to make its division 2 coincide exactly with that division of the instrument, which is just behind it (the division 41 45'), that we must so much turn the screw F (see Figure of p. 55,) that the index: D should be advanced from o to d, or through a quarter of the circumference, then this quarter, which is 15 x 3 rf or 45 ", is the value of the space through which the vernier has been moved, or of the distance between 2 of the vernier and 41 45' of the instrument : it measures, therefore, the excess of the altitude, which the in- strument ought to denote, above 41 1?'; in other words, the altitude is now to be estimated equal to 4 1 1 7' 45". By this contrivance, then, without any inconvenient minuteness of division of the limb of the instrument, or of inconvenient length of vernier, we are enabled to read off angles to as great an exactness as that of 3 seconds. In the Greenwich mural quadrants, by a similar contrivance, the angles may be read off to one second. That part of the vernier which we have been just describing, and which enables us to measure minute differences, is called a Micrometer. The two Greenwich mural quadrants, of 8 feet jadii, are, as we have said, furnished with such. But the mural circle is furnished with a micrometer of a different construction. Having now examined the methods of reading off the altitude to which the index of the vernier, in a fixed position, points, we will next consider by what means the vernier is brought to such fixed position. The vernier is attached to the telescope, and the telescope is moved, till the star (the altitude of which we are seeking) is seen through it. But, as the field of view is not a mere point, there is not one certain position of the telescope in which only we can see the star. If the star should appear to be nearly in the middle of the view, we may move the telescope, a little upwards and a little downwards, and still see the star. It is evident then, since the altitude we are seeking for is a certain and H 58 determinate quantity, that we require some rule for stopping and fixing the telescope. We cannot say that the telescope is in its just position when the star appears in the centre of the field of view, because the eye cannot judge of that circumstance with suf- ficient precision. We must therefore place some fixed point in the field of view, and in the focus of the eye-glass, which fixed point is to be the centre of the field of view, or to be considered as such, and the telescope is to be judged to be in its proper position, when the fixed point and the star appear to be coincident, or when, as the technical phrase is, the point bisects the star. The intersection of two fixed wires placed in the focus of the object-glass of the telescope, will furnish us with such a fixed point ; and one wire may be vertical, the other horizontal, de may represent the former,y*A the latter, and then a would be the inter- section, or their centre. These wires, as we have said, are fixed in the principal focus of the object-glass, and then must be viewed with the eye-glass : or, if they are attached to the tube containing the eye-glasses, that tube must be moved so that the wires shall be in the above principal focus : in either of these cases the eye sees distinctly, at the same time, the wires and the image of a star : and the observation is to be held as made when the star is upon, or is bisected by, the point a. We gain, at the least, this advantage by the above method, that all stars are observed according to it, and that any error attached to it must equally affect all stars : in other words, that the error must be a common one, and consequently all observations may be immediately corrected should the quantity of that error be 59 once detected. We will now consider by what means that error may be detected and valued. Let the subjoined Figure represent (in part) the Astronomical Quadrant, placed in the plane of the meridian, and with its gra- duated face opposite the east, and suppose the telescope to be directed to a star the altitude of which is 85. If A be the intersection, or centre of the cross wires (what answers to a in the Figure of p. 58,) and OA be the direction of a ray of light passing through O the object-glass and coming to its focus at A, then, the image of the star and the centre of the wires being coincident, the observation (see p. 58, 1. 6, from bottom) is properly made, and the index of the vernier, being made to coincide with 85 of the quadrant, will properly denote the star's altitude, and also, (the instrument being supposed to be truly graduated) the vernier, in other positions of the telescope, directed to other stars, will justly note their altitudes. Suppose now from some accident, or, purposely, the system of cross wires to be deranged, so that their centre, instead of 60 being at A, is moved, through a little space, to C, so that A is be- tween C and/) TFthe plumb-line, the line passing through A, C, being supposed to be in the plane of the meridian. In this new position of the cross-wires (the telescope retaining its position) the star is no longer bisected by their centre, but will be seen in the field of view, a little to the south of that centre, or towards the plumb-line. In order then to bring the star on the centre, that end of the telescope in which A, C, are, (the telescope being moveable about a pivot or centre of motion situated near its other end) must be pushed a little to the south and towards the plumb-line, 23" for instance, in which case the index of the vernier, moving with the telescope, will point to 85 C)' 23". We have now then to enquire (putting aside the supposition of the star's altitude being exactly 85) why the altitude, in this case, is not justly indicated. Suppose we are able to turn the quadrant half round, or that we possess some means or other of placing its graduated face \vhich, in the Figure of p. 59, is opposite to the east, opposite to* the west, and let the above Figure represent the quadrant in this 61 latter position : in which, OA would be directed as db is, not, as before, to the south of the zenith, but to the north. In order then to bring the star into the field of view, the telescope must be moved past the graduation of 90, to that of 95. In this latter position, the image of the star and the point A would be coincident, but C being now the centre of the cross-wires, in order to bring the star upon C 5 the end of the telescope which contains the eye-glasses and cross-wires must be pushed towards the plumb-line (as the Figure is constructed) or from the division of 90. It must be pushed also, since the distance AC is supposed to remain invariable, just as much as it was in the former position of the quadrant (the position of p. 590 that is, through 23". The index of the vernier now then will point to a graduation of 95 0' 23", or, which is the same thing, will indicate a zenith distance equal to 5 0' 23", whereas, the altitude in the first position of the quadrant being 85 0' 23", the zenith distance will be 4 59' 37". Half the sum of these two zenith distances is 5, the true zenith distance, and half their difference (46") is the error caused by the derangement of the cross-wires after they had been once adjusted. This error or derangement has a technical denomination : the line between O and A, A being the centre of the cross-wires, or the line between O and C, C being the centre of the cross-wires, is called the Line of Collimation, and the error, of which we have treated, and shewn the method of detecting and valuing, is called the Error of the Line of Collimqtion, or, more briefly, the Error of Collimation *. * This error may be corrected by moving and adjusting the cross- wires, so that C (in the Figure we have used) may be -replaced in A- But it is plain we may leave the system of cross-wires untouched, and so alter the index of the vernier, that it shall, the telescope being directed to the star, note its true altitude. On this account the error of collimation is frequently called the Index Error. 62 We have then, in all cases, in which we are able to turn the instrument half way round in azimuth, or through 180 of azimuth, this simple rule for finding the zenith distance of an observed star : add the zenith distance, or the mean of several zenith distances, taken with the face of the instrument to the east, to the zenith distance, or the mean of several zenith distances, taken with the face of the instrument to the west ; half this sum is the star's zenith distance : and, half the difference of the above observed zenith distances is the error of the line of collimation. The rule is the same, if, instead of the zenith distances of stars, we seek to determine their altitudes. We subjoin an instance or two, in which the instrument used, instead of a quad- rant, is a circle. Altitudes. 6th Sept. Star Rigel, position E.* .... 30 2l' 36".25 position W 30 20 22.05 sum = 60 41 58.30 true altitude = 30 20 59.15 difference = 1 14.20 error of collimation = 37. 1O Again, $ Sagittarii W. . . . 8 56' 45". 8 E. ...8 58 7.1 sum 17 54 52.9 true altitude. ... 8 57 26.45 difference. ... 1 21.3 error of collimation. . . . 40.65. If great accuracy be required, the above operations arc re- peated with several stars, and the mean of the whole taken for the error of collimation : thus, * Position E, position W, denote respectively the graduated side of the circle turned towards the east and west. Rigel, Sirius, a, 0, y, c. Capellae, are the names of certain known stars. 63 Error of Collimation. Rigel 37" . 05 Sirius 40 . 05 S Sagittarii 40 . 06 X 42 . a Capellae 39-45 9 37 . 12 7 37 . 35 i 37 . 87 8)310.95 Mean error of collimation 38 . 87 That process, then, of turning the quadrant, or the circle, half way round in azimuth, which finds the altitude and zenith distance, finds also the error of the line of collimation ;' but it is unim- portant to know this latter, if, every time that an altitude is to be determined, the above-mentioned process be resorted to. We may, however, as it is plain, having once determined the quantity of the error of the line of collimation, employ it as a correction either additive or subtractive, to the zenith distances of stars deter- mined from one position of the quadrant only, that is, when its face is constantly turned either towards the east, or towards the west. Thus, suppose that by the mean of twenty observations made at Greenwich, the quadrant facing the east, the north zenith distance of y Draconis = '' 2l".76. By the mean of 30 observations the quadrant facing the west, the zenith distance. . = 2' 1 5".48 0' 6".2S .". error of collimation = 3". 14. This is the error deduced from one star, y Draconis, which star is to the north of the zenith of the Greenwich Observatory. When, therefore, the face of the quadrant is to the west, the above correction (3 f/ . 14) must be added to the north zenith of stars, but subtracted from the distances of those stars which are observed to the south of the zenith * : for, since the instrument, its face * When the quadrant faces the west, a few stars only, those which are near the zenith, can be observed to the south of the zenith (see p. 64, 65.) . 64 being towards the west, gives the north zenith distance of y Draconis too little by 3". 14, (since instead of being 2' 15".48 it ought to be 2' 18". 62) it must also give the north zenith distances of all stars too small by the same quantity : and if a star were to the north of the zenith by an angular distance equal to 3". 14, it would, by the instrument, seem to be on the zenith ; conse- quently, a star on the zenith would by the index of the instrument appear to be 3". 14 to the south of the zenith : and a star 1 to the south of the zenith would appear to be, by the instrument, 1 3 7/ . 14 to the south. The contrary will happen if the obser- vations are made with the face of the instrument to the east ; for, then, the error of the line of collimation must be subtracted from all north zenith distances, and added to south zenith dis- tances ; for instance, if we had the following observations : Zenith distance of a Andromedse 23 24' 56". 36 S. 7 Pegasi 37 19 32.46 S. a Ceti 48 6 55.56, S. then the zenith distances, corrected for the error of the line of collimation, (and for that only) would be respectively, 23 24'' 59".5 37 19 35.6, 48 6 58.7. It appears then, by what has preceded, that, in all quadrants that can be turned round in azimuth, the altitudes and zenith distances of stars can correctly be found as far as the line of colli- mation is concerned. These, however, must generally be found by applying to their quantities, determined by the quadrant, the error of the line of collimation as a correction of such quantities. They cannot be found, except for stars situated near the zenith, by taking the half sum of zenith distances observed respectively, with the face of the quadrant towards the east and west. The reason is obvious from the inspection of the diagrams (see pp. 59, 60.) If AVB (see the following Figure,) should be in the plane of the meridian, and A should be to the south of VB, the zenith distances of those stars only that are to the north of the zenith 65 could be determined by such an instrument. If the quadrant were reversed, and the graduated rim now opposite the west were made to face the east, the zenith distances of those stars only, that are to the south of the zenith, could be observed. In such a case, the reversion of the instrument would be useless, since, not being able to observe the same star in the two positions of the quadrant, we should be unable to deduce the error of the line of collimation. To remedy this inconvenience, or rather to enable us to avail ourselves of the azimuth motion of the instru- ment, the arc of the instrument is made to exceed a quadrant, and the graduation, as it is represented in Fig. of p. 59, is ex- tended beyond 90 to 95 or 96. By this contrivance, the zenith distance of the same star, which is not distant more than 5 or 6 from the zenith, may be observed in the two opposite positions of the instrument, and the error of the line of collimation thence deduced. The star 7 Draconis, for instance, which, when it passes the meridian at London, is nearly vertical, would serve the above purpose in every part of England. But in circular instruments, or declination circles, and en- dowed with an azimuth motion, any star, either near to, or distant from the zenith, will serve to determine the error of the line of collimation, and with such instruments the method given in pp. 61, 62, &c. may always be practised ; that is, we may add the mean of zenith distances observed when the instrument faces the east, to the mean of zenith distances observed in the instrument's reversed position, and then (the error of the line of collimation I 66 being, in fact, compensated for) half the sum will be the zenith distance required. Thus, suppose the telescope Tt to be directed to a star in the south (so directed, as it must be always understood, that the image Z of the star and the middle of the cross-wires are seen, through the eye-tube, in distinct coincidence) the face of the instrument being towards the east : then, if the instrument be turned through 180 of azimuth, so that the face, before opposite to the east, be now opposite to the west, T't 1 will be the position of the telescope. In order, then, that it may be again directed to the star, and that its position may be parallel to its former one Tt, it must be turned through an angle equal to twice its zenith distance : and, con- sequently, half the difference of the number of degrees indicated by the vernier in its two positions (which difference is no other than the number of degrees intercepted between the two positions of the telescope and vernier) is the star's zenith distance. It appears then, from what has preceded, that, in all quad- rants and circles, used for taking altitudes and endowed with azimuth motions, the altitudes so taken can be freed from the error of collimation. But they are instruments of a limited size only * (we are speaking of the practical convenience of the thing) that admit of an azimuth motion ; instruments, for instance, of two * The radii of astronomical quadrants and circles that have an azimuth motion, and are portable, rarely exceed three feet: those of portable zenith sectors may be somewhat larger. The radius of the stationary circle of the Dublin Observatory, which has an azimuth motion, is four feet, and the radius of the quadrant at Blenheim, made by Ramsden, and with an azimuth motion, is six feet. 67 or three feet radii. It would be, almost, an impracticable operation to move, from day to day, such quadrants as the mural quadrants of Greenwich are, of 8 feet radii, and which are very ponderous. Such quadrants when once fixed must so remain, and, consequently, such quadrants are inadequate, from their own properties, to determine the errors of collimation of their telescopes. It is, however, essential to determine those errors. Some subsidiary instrument then must be called in for that purpose. Those circles and quadrants that possess an azimuth motion will not answer that purpose, since, by reason of their small dimensions, they cannot, in the determination of angles, be relied on beyond a certain degree. The error which we seek to investigate in the large instrument (an eight feet mural quadrant for instance) may be within the limits of inexactness (if we may so express ourselves) of the smaller. For instance, a quadrant of two feet radius is not to be relied on beyond 8 or 10 seconds : but the sought for error of the line of collimation, of the mural quadrant of 8 feet radius, may not exceed 4 seconds ; a quantity of moment in this latter instrument, by which it is purposed to determine angles to within I or 2 seconds. It is in vain then we seek for an angle of 4 seconds in an instrument on which we cannot rely to 8 seconds : and, indeed, the error of the line of collimation of a mural quadrant can only be determined by an instrument, of, at least, equal accuracy in the measuring of observed angles, and which, therefore, probably requires, in its essential parts, equal dimen- sions. We have already, in explaining the principle of determining the line of collimation, represented the parts or fragments of the Astronomical Quadrant. If we still farther contract the dimensions of the Fig. of p. 60. and suppose the extremities of the graduated arc to be at n and r, the graduation on each side of the lowest point not exceeding 8 or 10 degrees, we shall have, what is, in fact and principle, a Zenith Sector, an instrument for measuring small angular distances from the zenith, and, (which is the essential point,) capable of being reversed ; which reversion in small in- struments is effected by means of an azimuthal movement, and, in large instruments, by removing the sector from an eastern to a western wall. The reason is obvious why these sectors can be moved whilst the quadrants of equal radius cannot. The graduated arc, instead 68 of containing 90 degrees, contains not more than 10 or 12 : some- times much fewer degrees. The sector, therefore, can be made much less ponderous and unwieldy than the quadrant. The fixed mural quadrants at Greenwich are 8 feet, but the zenith sector's radius exceeds 12 feet. A sector then of these latter dimensions must, to the extent of what it is able to perform^ be more accurate than the mural quadrants. It is capable, for instance, of determining the zenith distance of y Draconis, more exactly, than the mural quadrant. But it is capable also of determining the zenith distance of that star truly by taking the half sum of its zenith distances observed on the eastern and western walls. The dif- ference of that half sum and of the zenith distance of the star, in one of the positions of the sector, is the error of the line of collimation of the sector: the difference of that same half sum, and of the zenith distance of y Draconis observed with the mural quadrant, is the error of the line of collimation of the mural quadrant. For instance, by observations of y Draconis made at Greenwich in 1812 with the zenith sector. Sector on the eastern wall, mean zenith distance = 2' 14 /7 .6l Sector on the western wall, mean zenith distance = 2 22 . 63 2)4 37.24 Mean of eastern and western . . , 2 18.62 Error of line of collimation of the sector , 4.01 But by observations made the same year, on the same star, with the, brass quadrant, the mean zenith distance = 9! 14". 52 but (see 5th line above) mean of eastern and western = 2 18.62 error of line of collimation of the quadrant 4.1 which error (so it happens in this instance) is very nearly the same as the former (see ?th line above,) whereas it might have been different by 2, 3, &c. or more seconds. By these means, then, the error of collimation of a mural quadrant may be corrected, and, if we use such a quadrant, 69 we must also be possessed of a zenith sector*. But the uses of this latter instrument are not merely subsidiary and subordinate ones. Its peculiar utility consists in finding, to a great degree of accu- racy (we refer to a sector of a large radius, such as Bradley's or the Greenwich one is) the zenith distances of stars situated near the zenith : such, for instance, are, with respect to Greenwich, j8 and 7 Draconis, Capella, a Cygni, a Persei, a Cassiopea*, *i Ursae Majoris. What are the inferences to be drawn from zenith distances, so circumstanced and so minutely observed, will be hereafter explained. Having now explained the constructions of the Astronomical Quadrant and of the zenith sector, and shewn the method of freeing them from one error, namely, that of collimation, we ought not to dismiss the subject without explaining, in its prin- ciple at least, the method of placing these instruments in the plane of the meridian. We will confine our attention, in the first instance, to a quadrant endowed with an azimuth motion. A star (see pp. 4, 5,) rising from the horizon, attains its greatest height in the plane of the meridian, and, quitting the meridian, declines, by degrees like those by which it rose, towards the horizon. At equal altitudes to the east and west of the meridian, it is equally distant from its plane. The star so circumstanced, and referred to the plane of the horizon by vertical circles passing through 'it, is equally distant from the south point of the horizon, or equally distant from the north. In other words, it (see p. 5.) has equal azimuths. In the same positions *also, namely, those of the star's equal altitudes, the star, with regard to the timey is equally distant from the meridian. Draw two de- clination circles (see p. 8,) one passing through the eastern, the other through the western position of the star ; then, each circle makes an equal angle with the great circle of the meridian. But such angle, in the terms of sidereal time, expresses how much time will elapse between the star's eastern and meridional * We are speaking here, as it is plain, of fixed mural quadrants and circles. A quadrant or circle, capable of being' re versed, is able to find its own error of collimation. Such is, and perhaps the first of its kind with regard to size and accuracy, the Dublin Circle of 8 feet diameter made by Ramsden and Berge. 70 altitudes, and also between its meridional and western. Two methods then present themselves, by which the meridian may be found. Half the difference of degrees, &c. on the azimuth circle of the instrument, between any two equal altitudes of a star, is the angular distance of the south or north point, from the eastern or western azimuth of the star: or, half the difference of times elapsed between any two equal altitudes of a star, is the time that the star is on the meridian. In each case, we are able to direct the telescope (to the line of the collimation of which the face of the instrument is parallel) towards the meridian : and as, in the course of a day, we may take several pairs of equal alti- tudes, we are, by taking the mean of the azimuths, or the mean of the times, able to determine the direction of the plane of the meridian to a considerable accuracy *. By either of the above methods, or by the aid of both, Astro- nomical quadrants and circles, such as are furnished with azimuth circles, may be placed, nearly, in the plane of the meridian. By means of such instruments, and by other helps, mural quad- rants and mural circles may also be placed in the plane of the meridian. The operation is one of some nicety and is most accu- rately performed by the aid of the Transit Instrument, previously adjusted to move in the plane of the meridian. We will now, then, proceed to explain the Transit Telescope, or Transit Instrument f. Let AD represent a telescope fixed, as it is represented in the figure, to an horizontal axis formed of two cones. The two small ends of these cones are ground into two perfectly equal cylinders : which cylindrical ends are called Pivots. These pivots rest on two angular bearings, in form like the upper part of a Y, and de- nominated Y's. The Y's are placed in two dove-tailed brass * We may, for the above purposes, use the Sun and observe his equal altitudes and azimuths. As we cannot pretend to bisect his centre, by a wire of the telescope, we must make our times of observation, those in which the limbs of the Sun are in contact with the wires of the instru- ment. Since the Sun does not, like a star, describe a parallel of declina- tion, there must be some small correction made, for his changes of decli- nation, during the intervals of observing either equal pairs of azimuths or equal pairs of altitudes. f Instrument des Passages. 71 grooves fastened in two stone pillars JE and W, so erected as to be perfectly steady. One of the grooves is horizontal, the other vertical, so that, by means of screws, one end of the axis may be pushed a little forwards or backwards, and the other end may be either slightly depressed or elevated. Which two small* move- ments are necessary, as it will be soon explained, for two adjust- ments of the telescope. Let jE be called the eastern pillar, W the western. On the eastern end of the axis is fixed (so that it revolves with the axis) an index , the upper part of which, when the telescope revolves, nearly slides along the graduated face of a circle, attached, as it is shewn in the figure, to the eastern pillar. The use of this part of the apparatus is to adjust the telescope to the zenith or polar distance (for the one is as easily done as the other) of a star the * The movements are of small extent since they are only subservient minute adjustments. 72 transit of which is to be observed. Thus, suppose the index of n to be at o (in the upper part of the circle) when the telescope is horizontal : then, by elevating the telescope, the index of n is moved downwards : suppose the position to be that represented in the figure, then the number of degrees between o and what the index of n marks, is the altitude of the telescope : or, we may so graduate the circle that the index shall mark the telescope's zenith distance : or, if we make the o, the beginning of the graduation, to belong to that position of the telescope in which it is directed to the pole, the number of degrees, &c. between o and any other position of the index, will mark either the telescope's polar distance, or, if we please, may be made to mark the telescope's declination ; the telescope, in all these cases, being supposed to move in the plane of the meridian. There are several other parts and contrivances, belonging to the instrument, not shewn in the Figure : for instance, one of the cones is hollowed, and, opposite the orifice, there is placed, in the pillar, a lamp which, throwing its light on a plane speculum, placed in the axis of the telescope and inclined at an angle of 45, illuminates the cross-wires. It is usual, also, in large transits to have counterpoises by which the pressure of the pivots of the axis on the Y's is relieved. We will now explain the three prin- cipal adjustments of the transit. 1 st , To make the axis, on which the telescope moves, hori- zontal. 2 d , To make the line of collimation move in a great vertical circle, or, which is the same thing^ to make it perpendicular to the horizontal axis. 3 d , To make it move in that vertical circle which is the me- ridian. The first adjustment is effected by means of a level ; and in the figure of p. 71, it is intended to represent the level (JL) as hanging, by means of its upright arms, (bent, however, in their upper extremities) on the two pivots of the axis. The principle, however, and mode of rendering any axis horizontal, by means of a level, may be best explained by the subjoined Figure. In this Figure, the spirit-level (including in that term, the brass tube that partly envelopes it, the horizontal bar to which it is affixed, and the two vertical arms by which it is hung on any cylinder or rod) is represented as hanging on a straight cylinder a b, the end towards a lying on a crotchet which is capable of nk f being raised or lowered by a screw B. The end A of the tube AD, which contains the level, is also capable of being lowered or raised by means of a screw at A, as it is shewn in the Figure. If a b were horizontal, and the tube of the spirit-level were parallel to a b, then the bubble would occupy the middle, or, the two extremities of the bubble would be equidistant from the centre, and would be, for instance, aty and e. The same thing would happen if the level were reversed, that is, if it were taken off the rod, turned round, and again hung on, so that D in the second position, should occupy the place that A did in the first, or should be to the right hand. But, if ab should not be horizontal, the above circumstances cannot take place. Suppose the end a to be lower than the end b, then if the level should not be parallel to a b, the bubble might still stand in the middle, by the end at A being, by a certain quantity, higher than the end at .B. But on reversing the level, the bubble cannot occupy its middle, since then the lower part of the rod a b and the lower part of the level would both be situated to the right hand. The bubble, however, may not stand in the middle from two causes, the want of horizontality in a b, and the want of parallelism to it in the tube contained between AD. If the level were parallel to a b : and the extremity of the bubble, instead of being at e, should be at h, on reversing the level, the other extremity of the bubble (which by the reversion would be towards a) would be at k, fk being equal to eh. But suppose this is found not to be the case, and that the extremity of the bubble, on reversing the level, is at n, then the circum- stance of the bubble not standing at the two points e K 74 cannot arise solely from the end a being higher than b, but the level cannot be parallel to a b, and, in the case we have put, the end at A must be lower than the end at D : when the level then is in the second or the reversed position, so elevate the end at A, by means of the screw A, that the extremity of the bubble shall descend from n and occupy a place intermediate to n and k f and then the level is made parallel to ab ; this is the first adjust- ment. Next, by means of the screw B, so depress the end a *, that the extremities of the bubble shall be, (as they ought to be, ef being the length of the bubble) at e and f\ then is ab adjusted or made horizontal : this second adjustment completes the operation. In the preceding reasonings, a b has been considered, (the whole of it,) as cylindrical. But this is not necessary : it is suf- ficient if its extremities at a and b (the pivots), on which the level is hung, be equal cylinders, the axes of which lie in the same straight line. The intermediate parts of the axis of the transit between the pivots, may be of any form : they may be formed, as they generally are, of two cones. The preceding process, then, will render the axis of the transit horizontal ; the level, whether in its primary or in its reversed position, being supposed to be hung on the equally cylindrical pivots. The axis being now horizontal, the next operation is to make the line of collimation describe a great vertical circle, or, which is now the same thing, to make the line of collimation perpendicular to the axis of the transit. The telescope AD (p. 71.) is furnished, like the telescope of the quadrant, with a system of cross-wires placed in the principal focus of the object-glass. Suppose the wires so placed that the line of collimation (see p. 61,) is perpendicular to the axis of the transit. If then a small and well-defined object be bisected by the centre of the cross-wires, it will still be bisected when the transit is lifted off its angular bearings, reversed and directed to the object ; that is, illustrating our meaning by the Figure of * The end of the cylinder at a rests upon an angular bearing (it might have been a F), placed in a groove, and capable of being moved vertically by the screw at E. This part is, in fact, the same as that which is men- tioned in the brief description of page 7 1 75 p. 71, if the end of the axis carrying the index n which is placed on the eastern Y should be placed on the western. Let now the wires be deranged, so that their intersection is moved, not, as in the former case, in the plane of the meridian*, but in a direction perpendicular to that plane, and suppose it moved a little towards the east. In this case, the object before bisected is no longer so, but will be seen in the field of view a little to the west of the present centre of the cross- wires. Reverse the telescope, then the centre will be towards the west and the original object will be seen a little to the east of the centre : as much towards the east as it was before towards the west. If therefore there should be two objects or marks (on the horizon, for instance,) bisected by the centre of the wires in the two positions of the transit, the correction or adjustment of the line of collimation would consist in moving the centre of the cross-wires half way towards that object which is not on the centre. But the moving the centre of the cross-wires, half way towards an object, is a matter of guess and not of certainty. In order to ascertain whether, in moving the centre, we have adjusted it rightly, we may avail ourselves of that angular bearing, or Y 9 which, (see p. 71,) by means of an horizontal groove and screw, we can move, together with the pivot of the axis, in azimuth. So move these then, that the object, to which we have already made the centre to approach half way, may be exactly bisected by that centre. Reverse the transit, and the object and centre are either coincident, or very nearly so. If the latter be the case, again, by their proper motion, move the centre of the wires half way towards * We have supposed, in the quadrant, the derangement of the centre of the cross-wires to be made in the plane of a vertical circle ; or, in the plane of the meridian, if meridional altitudes are to be taken : for such derange- ment is the essential one : a small deviation or derangement to the east or west would very slightly affect the determination of the altitude. But in the transit instrument the reverse is the case : the essential derangement is that which moves the centre of the cross-wires to the east or west of the meridian, and which makes the star to appear to pass the meridian too late or too soon. A small derangement of the cross-wires in the direction of the meridian, is of no consequence, since such derangement will neither accelerate nor retard the star's transit. 76 the object and move it the other half way by the screw that acts on the axis *. Reverse the instrument, and again, if it be necessary, repeat the above operations. By these means, after a few trials, we are sure of making the line of collimation, or axis of vision, perpendicular to the axis of the transit ; and, when that is effected, the cross-wires are no longer to be meddled with, although we must continue to use the above horizontal movement of the axis (see pp. 71, Sec.) for the purpose of placing the line of collimation in the plane of the meridian. That line now moves in a vertical circle, and produced passes through the zenith : it is farther necessary to make it pass through the pole. The transit instrument, (that which in the preceding pages we have spoken of) is supported between two fixed pillars. It must be supposed to be nearly in the meridian (the direction of the meridian being known, to a tolerable degree of accuracy, by some of the methods described in pp. 69, &c.) and to need only some slight adjustments to place it there exactly. It would be easy to effect this were the pole star exactly in the pole ; for, then, it would be only requisite to bisect that star by the middle vertical cross-wire. But the pole star being, in fact, a circumpolar one, we must compute, by means of existing Tables and observations, (for the question is not now concerning the independent derivation of all Astronomical Elements from first principles) the time of its transit, and, at that computed time, bisect the star by the middle vertical wire. By these methods we may place the transit very nearly in the plane of the meridian. We will now shew how to place it there more exactly by means either of the polar, or of any other circumpolar star. The axis being horizontal, the optical axis perpendicular to it passes through the zenith : let ZPH be the true meridian and * It is plain that the horizontal or azimuthal motion given to the Y and pivot, has nothing to do in the adjustment of the line of collimation. The adjustment is solely effected by the screw (or other contrivance) that gives motion to the cross-wires. The motion we can give to the axis only enables us to ascertain whether the last adjustment we have made with the cross-wires be sufficiently exact, or whether a farther one be necessary. 77 Zsm the vertical circle described by the optical axis or line of collimation : then Hm, which is the measure of the angle at Z, is the deviation of the Transit from the meridian. Let ss's'cr represent the circle described by a circumpolar star, which is seen, through the transit telescope, at cr its inferior passage, and at s its superior. Now, when the Transit is not in the meridian, the time from cr to s cannot equal the time from s through s and s" to 80 -f ^ sPs + Z crPs", Hence,, if the interval between the inferior and superior passage should be less than the interval between the superior and inferior, the plane in which the Transit moves from the zenith to the north of the horizon (P being the north pole) is to the east- ward of the true meridian. But, in order to estimate the quantity of deviation from the observed difference of intervals between the passages, we must compute the angles sPs or sPZ, and which will be the value of the deviation of the line of collimation from the plane of the meridian. Nothing, however, is gained (if we look, in the present case, to the practical convenience of the thing) by this conversion of a measure in time into an angular measure : for the approach of the plane, in which the line of collimation is, to the plane of the meridian is effected (see p. 71.) by means of a screw: suppose, for the sake of illustration, the head of this screw to be graduated like that in the figure of p. 55. Let the time of the transit of an equatoreal star over the middle vertical wire be noted on a par- ticular day. Alter the inclination of the plane, in which the line of collimation moves, to the plane of the meridian, by turning the screw once round, and observe, the next day, the time of the star's transit: suppose the difference of the times of transit, on the two successive days, to amount to two seconds, then will one revolution of the adjusting screw be equal to two seconds, half a revolution to one second, one eighth of a revolution to one quarter of a second, and so on : so that, having thus once obtained the value of the motion of the adjusting screw we may immediately 80 apply it to the result of Z, expressed in time, (see p. 79-)., and correct, accordingly, the Transit's deviation. It appears then, from the preceding computation, that a deviation of about 10 seconds of time, in the transit telescope from the plane of the meridian causes the time, between the inferior and superior transit of the pole star, to differ, from the time between the superior and inferior transit, by about 7 minutes. The difference, it is probable, will not be the same in another circumpolar star. Let us examine what it will be in Capella, the north polar distance of which in January 1, 1819? was 44 19' 53", and which, consequently, passes the meridians of Greenwich and Cambridge to the south of their zeniths. In this case (esti* mating separately the angles sPZ, o-P-fiT), we have COS. (L + 7T) = At . cos. (Li-f-Tr) co-sec. TT, sin. TT L = 52 12' 36" TT = 44 19 53 .... log. co-sec. =10.1556425 96 32 29 .... log. cos. . . = 9.0566035 A = 10.195 .... log ...... = 1.0084017 20.220Q477 /. log. sPZ = .2206477, and sPZ = l".662: for the inferior passage of Capella, 1 L, }) 2L. 97 ~ (28 h 85 m 2 s .6), or, 14 h 42 m 31'. 3, which differs from the mean time only by s . 02. Again, by observations made at Greenwich, Nov. 8, 1816, with the new transit and with five of its seven wires, Meridional Wire. II. in. IV. v. VI. 51 m 29 8 .4 51 m 48 8 .5 14 h 52 m 7 8 .6 52 m 26 s .? 52 m 46 8 1L 53 45 54 4.3 14 54 23.4 54 42.5 55 1.6 02L The sum of all the times is 10 X 14 h + 532 m + 35% the number of contacts is 10; therefore, the mean transit of the Sun's centre is 14 h 53 ra 15 8 .5, which is the same result as the half of the times at the middle wire. We cannot use exactly the same method in finding the transit of the Moon's centre ; because the Moon shines only once with a full orb during her revolution round the Earth. At all other times, amongst which will almost always be found the times of observation, either her western or eastern limb is more or less deficient : so that, on the deficient side, either no contact, or an imperfect one, takes place. On this account the contact of one limb only, that which is turned towards the Sun, is observed. Thus amongst the Greenwich Observations, Jan. 15, 1816, we find the following : II. in. Meridional Wire. IV. v. VI. 49 m 38 s . 6 49 m 59 s 8 h 50 m 18 S .8 50 m 39 s 51 m s D 2L N 98 The sum of these times is 5 X (8 h 50 m ) + 95 9 .4. The number of contacts is 5, and consequently, the mean time of contact, of the Moon's second limb with the meridional wire, is 8 h 50 m 19 8 .08, from which, deducting the time which the Moon takes in passing over a space equal to her semi-diameter, we shall have the transit of the Moon's centre over the meridian. We must proceed in a like manner, when we wish to deter- mine the altitude of the Sun's, or of the Moon's centre by the quadrant or circle. The altitude of the upper (0 U. L.), or lower limb ( L. L.) must be found by bringing it into contact with the horizontal wire. The Sun's semi-diameter deducted or added will give a result equal to the altitude of the Sun's centre. Or, half the sum of the altitudes of the upper and lower limbs will give the altitude of the centre. Thus, by observations made in 18 1 6, with the Greenwich mural circle. Thermometer. Barometer. In. Out. June 3. 29.89 59 64 June 4. 9.86 58 64 U. L N. P. r>. 67 23' 8" L. L 67 47 34.1 and by observations made in 1 787, with the south mural quad- rant of Greenwich, June 4. 29.83 55 58 Z.D. L. L.:73;'fr. ..29 15' 5o".3 U. L. . .28 44 13.5 From the last of these observations the Sun's diameter, as it simply results from the difference of the two zenith distances, is 31' 36". S. The zenith distance of an heavenly body means the zenith distance of its centre. Now the planets possess disks of sensible 99 magnitude. Dr. Maskelyne appears to have taken their zenith distances with the mural quadrant by making the middle hori- zontal wire of its telescope bisect the planet's disk. Thus we find in the Greenwich Observations of 1775, Oct. 17 U centrum 72 36' 24" Dec, 3 . . T? centrum 31 16 11.6. In the observations made with the present mural circle of Greenwich, the practice seems to be, to bring the upper or lower limb into contact with the middle horizontal wire, and, by means of a screw, with a graduated head^ to move another wire (which always keeps a direction parallel the horizontal wire) till" it comes into contact with the lower or upper illuminated part of the planet. Thus, by the Greenwich Observations of 1813, July 25, $ L.L. a Arietis . , , . , 1 56 55.96 4 40 11.36. If the clock, therefore, should, between the transits of a Serpentis and of Sirius, note an interval of time equal to 8 h 58 m 8 s , instead of 8 h 58 m 11 s . 14, it ought, on the supposition of an equable motion, to note between the transits of Sirius and of a Arietis a time equal to 4 h 40 m 11 8 .36 X fa 8 5 * ^ . 8 b 58 m ll s .14 The practical method of determining the clock's daily rate, that is, its gain or loss during two successive transits of a star, is 103 Jan. 23. Jo subtract the mean meridional passages of certain stars on one day (as shewn by the clock) from the passages of the same stars on the next, or on some following day. The sum of the differ- ences divided by the number of days intervening between the observations, and by the number of stars, is the clock's mean daily rate ; to which quotient, or result, should the clock gain, the sign + is affixed ; should it lose, the sign . Thus, by the Greenwich Observations of 1798, the mean transits of the following stars were Stars. ri h 56 m 55 8 .10 a Arietis. Jan. 25, < 5 5 57.46 Rigel. 37.57 (5 14 39.32 ft Tauri. Here the several differences are 1.78, 1.76, 1.75, their sum 5.29 divided by 2, the number of intervening days, is 2.645, and again divided by 3, the number of stars, is .881; and, since the clock gains, the mean daily rate is thus to be expressed, +0 8 .88. In practice, a clock is adjusted very nearly to sidereal time. Its daily gain or loss seldom exceeds three or four seconds. In computing its rate then, we need not concern ourselves with the degrees and minutes of the star's right ascension ; it is sufficient to attend solely to the seconds, and to those, which, in the Registers of Observations, are inserted in a column entitled the Reduction of the Wires, (see p, 88.) Thus, in the Greenwich Observations of 1816, we find |h 56 m 53 S .32 5 55.70 14 Reduction of Wires. 25.58 Number of Days. Daily Rate of Clock. Names of Stars. a Orionis. Aug. 7 . 56.32 a Lyrse. Aug. 8 ....! 55.28 23.18 1 2 - 1.04 -1.2 a Lyrae. a Orconis. The difference between the reductions of the wires, in the in- terval of two days, for a Orionis is 2.4, and half, that is, 1 .2, is to be written 1 .2, since the clock loses, or its pendulum made between the transits of a Orionis., on the sixth and eighth day, only 172797.6 (=2 x 86400-2.4) beats. 104 The daily rate of the clock from two successive transits of a Lyrae is 1 .04, and therefore, the mean daily rate from the two stars is -|-(2.24), or 1.12. It is part of the regular daily business of an Observatory to determine the rate of the clock. But the weather may prevent this practice, so that an observer, in order to determine the rate of his clock, may be obliged to compare observations distant from each other by intervals of four or five days. The greater, however, the number of intervening days, the less accurate is the method (that which has been explained and exemplified) of determining the clock's rate. Indeed, if the number of days be considerable, the method, as we will hereafter shew, is erroneous. The rate of the clock being determined, there remains another point to be settled, which is the error of the clock de- pendent partly on the rate and, under certain considerations, caused entirely by it. There are certain circumstances (circumstances of convention) that require previously to be explained, in order that we may know what the error of the clock is, or what it consists in. The posi- tion of a star (as it has been explained in p. 46.) depends, or is made to depend, on the arcs of two great circles, one measured from the pole, the other along the equator and from some point in it. The pole is not marked by any star, but is a point variable with respect to the stars, ascertainable, however, at any given period, by observation and comp.utation. The point from which Astronomers have agreed to measure the right ascension is, like the former, variable from time to time, but capable of being ascertained at any assigned time. This point (a point of con- vention) is the intersection of the equator and ecliptic : it is not, and cannot be, permanently marked by any star, but still it is a determinable point. All right ascensions are to be measured from it. When such point is on the meridian, the clock, which is adjusted to sidereal time, ought to mark O h . The right ascen- sion of a star passing the meridian an hour after would be l h ; of a second star, passing 2^- hours, 2 h 30 s ; and so on. Suppose then on Feb. 3, that the clock rightly noted, the right ascension of a Arietis, and that it was l h 56 m 55', 105 ten days after, on Feb. 13, if the daily rate of the clock were -f- .88, the gain of the clock would be 8 s . 8 : consequently, at the passage of a Arietis over the meridian, the clock would denote l h 57 3 8 .8, and, if the right ascension of the star remained the same, the clock's error would be 8 9 .8. In twelve days the rate having in- creased to 1 8 .02, the clock's error would be 8",8 + 2.04, or 10\84. In what manner the right ascension of a star is computed will be hereafter explained. But admitting, for the present, that we are able to find it, from Catalogues and subsidiary Tables, it is easy to shew that the error of the clock, and the rate of the clock may both be found by the same process. Thus, suppose, on March 11, the catalogued apparent right ascension of Sirius to be 6 h 37 m 19 8 .4 whilst the clock denoted .... 6 37 10.3 9.1 The clock then would, on March 10, be absolutely too slow by 9 s . 1, or its error would be 9*.l. Again, on March 16, let the star's apparent R. A. 6 h 37 m 19 9 .3 the clock denoting. 6 37 14.1 5.2 On March l6th, then, the clock's error is 5 S .2, too slow. The clock's gain in five days is 9 8 .1 5.2 = 3.9, and conse- quently, (see p. 103.) its mean daily rate, so estimated, is -4- \ (3.9) = + ,78. o This latter result is the true daily rate : the daily rate, estimated from the difference of the transits as shewn by the clock, would be + 1 (3.8) = -f .76. o 106 Now the results for the daily rate do not agree. The ques- tion, then, is which is the right result ; and this immediately leads us to the point to which, in p. 100, we promised to advert, namely, the principle and ground of the practical method of determining the rate of a clock. Let the telescope be directed to a star (a Aquilae for instance,) on some day, the 7th of March, and note the index of the clock when the star is bisected by the centre of the cross-wires. If the two events, the index at the same division, and the bisection of the star, are contemporaneous on the 8th of March, on the Qth of March, &c. the clock is said to be duly adjusted to sidereal time, and its mean motion in twenty-four hours is said to be uniform. Now this depends on the supposition, that the same abso- lute time is always absolved between each successive transit of a star over the meridian. And this latter supposition, the equality of time between successive transits, is founded on another, which is the uniformity of the Earth's rotation round its axis. This supposition, then, is completely compatible with the above rule. It remains now to examine, whether the time be- tween two successive transits of the same star, depends solely on the time of the Earth's rotation, and, if it should not solely depend, whether the impeding circumstances are of magnitude sufficient to vitiate the practical rule. If the Earth's rotation were .uniform, and its axis produced were always directed to the same point of the Heavens, and if, besides, no cause, dependent on the relative position of the Earth and a star, made the latter, at one time, appear on the meridian before its real passage, at another time, after it, then would all the several intervals between the successive transits be equal. And this would also take place, if the deranging causes to which we have alluded, altered equably, and the same way, the star's right ascension. But, as it will be shewn in the succeeding Chapters, the deranging causes not only exist, but are variable, both as to degree and direction, in their effects. It is true their effects are very small : so small as not to be ascertainable, in the short intervals of two or three days, by our measures and reck- onings. But still they exist, and become perceptible in their accumulations. 107 Although the Earth, then, should complete her diurnal rotations in equal portions of absolute time, it does not thence follow that a star will always return to the same point (the wire, for instance, of a fixed telescope) after equal intervals of absolute time. It may seem to do so when we compare one interval with another that succeeds it : but it may seem to do so, only be- cause we have no means, either by our eye or our ear, of dis- tinguishing the hundredths of a second of time. What then shall we define a sidereal day to be ? We may define it to be the portion of time between two successive transits of a star over the meridian : but then, if the preceding statements be admitted to be true, all sidereal days would not be equal. The definition, then, would not be a good one. If we define a sidereal day to be the portion of time absolved whilst the Earth makes a complete rotation round its axis, then, on the hypothesis of an uniform rotation, all sidereal days would be equal. It is no valid objection against this definition, that a sidereal day, not being identical with the interval between two successive transits of a star, and, therefore, not immediately ascer- tainable by observation, would thus become a quantity to be determined by calculation. A sidereal year must be so deter- mined. This is not the place to state the physical causes that prevent the time of the recurrence of a star to the meridional wire of a Transit from being solely dependent on the Earth's rotation : but, if we wanted a practical proof of the fact, we could easily find one in the instance of the pole star. That star is about 1 40' distant from the pole: but, if the times of the transits of stars over the meridian arose solely from the Earth's uniform rotation round a fixed axis, the several intervals between the successive transits of the same star would all be exactly equal, wherever that star were situated, whether near the equator or near the pole. In such case, if on the first of next January, (1 822), Polaris should be (as he will be) on the meridian at O h 57 m 20 8 .3 of sidereal time, he ought to be again there on January 2, at the same sidereal time ; whereas, on this latter day, the time of the transit will be, nearly, O h o7 m 19*. 7, 108 and the succeeding day, January 3, at O h 57 m 19 ^ and the apparent motion of Polaris will so increase that, after ten days, he will be on the meridian at O h 57 m 13 8 .3, and on January 20th, at O h 57 ra 6 s . 3, the apparent mean daily acceleration of the star being, during the above period, about ths of a second. In the above case, the real differences of the intervals of successive transits become discernible from the peculiar situation of the star. But, with other stars, the case is different. The star Procyon (the lesser Dog Star)., for instance, which is near to the equator, will be on the meridian, at the latter period, (January 20, 1822), at 7 h 50 m 0^.9, and the real differences, between the intervals of its transits for the next twenty days, are so minute as completely to baffle detection, with whatever instrument the eye and ear be assisted. The same circumstance takes place, very nearly, with other stars that are not near the pole. It takes place with all those stars which are used in determining the clock's daily rate. With stars, then, such as the last, the rule for finding the clock's daily rate, from the difference of two successive transits, is sufficiently exact for all practical purposes. It can never, so applied, lead into error ; which it would do, were Polaris the star. The latter star may indeed be used for finding either the clock's error, or the clock's rate, but then we must have recourse to operations less simple than those of merely noting the times of its transits *. * We have, on the preceding subjects, somewhat dilated, and been digressive. But the subjects are those on which students (we are speaking in general terms) have no precise notions, nor, through books in ordinary use, any means of acquiring such. It 109 The three capital instruments of an Observatory, it has been said, are the quadrant, the clock, and the Transit. But this is not to be taken literally. In Observatories, where, generally, the instruments are large, the quadrant is fixed, and is, what is called, a Mural Quadrant. But then there must be two quadrants ; one for stars north of the zenith, the other for stars south of the zenith ; and, beside these, there must be introduced a fourth instrument, called a Zenith Sector, subsidiary indeed to the quad- rant in determining the error of its line of collimation, but, more- over, of peculiar and great usefulness. We may, however, should it be our object to have as few instruments as possible, instead of two mural quadrants, use a mural circle; and, since this instrument, according to the present mode of constructing it, would be very loosely and im- perfectly described, by saying, that it is formed by the putting together of four quadrants, we will proceed to give a brief description of it. The circle, with its attached telescope, is made to revolve by means of an horizontal axis ; which axis works in collars fixed in the stone wall, represented in the Figure. The wall faces the east. The plane of the circle, as it is shewn in the Figure, is parallel the wall, but the graduations are made on the outer rim of the instrument, which rim is perpendicular to the wall. It has been said, that art and science render each other mutual assist- ance, and are contemporaneously progressive. In the subject which has been under discussion, namely, that of the instrumental means of mea- suring time, a refined state of science is absolutely necessary to enable us to pronounce on the quality of such means. If the antients had invented exact time-keepers, could they have verified their exactness ? Suppose, for instance, a Watchmaker of Alexandria had constructed a perfect clock, the Astronomer of Alexandria would have found it faulty, since the clock would have indicated an inequality in the revolution of the primum mobile. There seem to be no other means than Astronomical ones of verifying time-keepers ; and these means, if they are to be exact, cannot be made so, except with great difficulty, nor without the results and formulae of refined science. 110 These graduations are viewed and read off, by six microscopes fixed to the wall, one of which microscopes is represented at A, and the places of the five others (precisely similar to the former) are marked by the letters JB, C, D, E, F. The microscopes are distant from each other sixty degrees, or so placed, asmearly as can be, by the instrument-maker. The circle's diameter is six feet. Its rim is divided into equal parts of five minutes each, and the readings off to a less number of minutes and to single seconds, are effected by the Micrometer Microscopes, A, B, &c. The construction of which is as follows. The microscope A, or micrometer microscope A is directed, as it is shewn in the Figure, to the rim on which the graduations are made. Consider the object to the microscope to be one graduation of the instrument, or the space occupied by five minutes. The image of this space will be formed in the conjugate focus of the object-glass, and will be. seen distinctly Ill through the eye-glass of the microscope, when the above-men- tioned image is in its focus. In this latter focus (the focus of the eye-glass) are placed, a thin indented slip of metal and a wire * capable of being moved, in a parallel direction, from one mark of division to another by means of a screw. The revo- lutions of the screw, and parts of its revolution, are noted by means of a screw-head aud graduated plate, similar in the prin- ciple of its construction to the one of p. 55. Now it is desirable, for the more convenient noting of the results of observations, that, by five revolutions of the screw, the wire should be trans- lated through the space occupied by five minutes : in which case, one revolution would answer to one minute, and one-sixtieth to one second. The mode of effecting this may be thus ex- plained. Suppose, the object-glass of the microscope being at a certain distance from the graduated rim, and there being distinct vision, that the moveable wire appears to be translated through the five minutes, by 5^- revolutions of the screw. In such case, the image of the five minutes is too small. It will be increased by moving the object-glass towards the graduated rim. But, if the whole microscope be moved, there will no longer be distinct vision, since the object being nearer to the object- glass, its image will be formed at a greater distance from the object-glass, and beyond the focus of the eye-glass. The eye- glass, therefore, with its wire, &c. must, by a separate move- ment, be withdrawn from the object glass till distinct vision ensues. In this second position, a second trial must be made to ascertain whether five revolutions of the screw are equal, or not, to the translation of the wire over the image of that portion of the divided limb which contains five minutes. Should there be no equality, the adjustments must be made both of the object- glass and of the eye-glass, by their peculiar movements, till five revolutions of the screw shall correspond to the translation of the wire over five minutes. * Instead of one wire moveable, in a direction parallel to the marks of graduation, two wires crossing each other, at an acute angle, are sub- stituted. These wires, in measuring the distance from the index to a graduation, are to be stopped when the mark of the graduation bisects the angle of their intersection. The adjustment, which we have described, is merely a matter of convenience : it saves the observer the trouble of reducing the graduations of the screw-head to their values in minutes and seconds. If the microscope micrometer were suffered to remain in its first state, then, since 5.5 revolutions =5', one revolution would equal 50".454, &c. But, whatever be the value of a revolution, the uses of the moveable wire and the indented slip of brass are the same. A star is observed on the centre of the cross-wires of the telescope. On looking through the microscope, the index, or what serves as one in the slip of brass, occupies a place between two gradua- tions. The wire moved from the index, either to the upper or lower graduation, measures by the revolutions of the screw-head, the distance from the mark of graduation : and, for convenience, each tooth of the indented brass answers (one revolution of the screw being equal to one minute) to one minute : so that, if the wire is moved from the index past two teeth, and the index of the screw-head points to 55, then 2' 55'' are to be added to or sub- tracted * from the degrees and minutes which are read off by the naked eye, or without the aid of the micrometer microscope. In every observation all the six microscopes are to be used for the purpose of diminishing the errors of division,, and the effects of partial expansion. In reading off the angles at the several microscopes, we need onlv to attend to the seconds ; which may be thus explained. Suppose a star to be in the pole and that the telescope is to be directed to it. The whole circle then must be turned round in the direction from B towards C, D, &c. and the end of the telescope containing the object-glass, instead of being directed as it is in the Figure, to a point in the south, between B and C, will be directed to a point between D and A. If, the telescope being directed to the pole, the reading off at the micrometer at A were O 7 0", the Index error, as it is called, would be 0. The readings off at the other microscopes JP, E,*.B, C, D, (were those microscopes placed at exactly equal distances from each * Accordingly, as the distance of the index from the upper or lower graduation is measured. 113 other) would be 60, 120, 180, 240, 300, But these circum- stances are not likely to take place. The index error will probably be of some magnitude : a few seconds, for instance : that is> when the telescope is directed to the pole, the reading off at the microscope A, instead of being 0' 0", may be + 5", or 7"> or S", &c. In like manner, the readings off at F, E, B, C, D, may be, from their not being placed at exactly equal distances, or from inequality of graduation, or from partial ex- pansion, or conjointly from all these causes (for in practice they may all operate) either -60 0' 7", or 60 0' 10", or &c. 120 0' 8", or 120 O' 12", or &c. &c. &c.. Suppose, independently of the dejgrees and minutes, the seconds at the six microscopes to be respectively, +5", +7", +4", +12", +8", +9"; then these are the several index errors : and, if the polar distance of an observed star were read off only at one microscope, the index error belonging to such microscope must be added to, or subtracted from, the distance so read off. Thus, if the microscope 3 were only used, the index error of which is -{- 12", and the north polar distance of /3 Ursa Minoris read off -were 1Q5 4' 46", then, deducting 180 for the position of the microscope, and 12" for the index error, we should have the north polar of /3 UrsaB Minoris = 15 4' 34". But, all the six microscopes being used, it is convenient to consider a mean index error, which will be one-sixth of the several index errors, and, which, in the preceding instance (see 45" 1. 16.) will be -, or ?".5. 6 We have in the preceding illustration, for the sake of sim- plicity, supposed the telescope to be directed to the pole, which, as it has been several times stated, is not marked by any star, but is a point to be assigned by calculation and angular measure- ment. But the illustration will be, in substance, the same if we suppose the telescope directed to a known star., Polaris, for 114 instance. If, by previous Catalogues and Tables, we should know the north polar distance of this star to be 1 4l' 4l".3, the micrometer microscope A marking 1 4l' 48". 5 ; then the index error would be -r-7".2* ? and, in like manner, we should know, by the same star, the index errors for the other micro- scopes, and thence the mean index error. We shall, in another part of the Work, explain the use of the observations made with this instrument, and of the index error, in correcting the catalogues of polar distances. At present we shall be content in shewing, by a kind of exemplifi- cation, that the uses of the instrument do not depend on the accurate positions of the several microscopes. Suppose, the telescope being directed to the pole, the number of seconds indicated by the micrometer microscope A to be 7. Let B indicate b-\-Q3" (b, c, d, &c. denoting degrees and minutes) C c+ 4 D d+ 5 E e+ 9 JP - Let X be the north polar distance of any star, (of Capella, for instance, X being = 44 12' 16"), and let the number of seconds in X be 16, so that, Y being the degrees and minutes,, Jfesy+16"; then, the instrument being directed to Capella, (and, consequently, turned round through an angle X} and the errors of division, expansion, and the uncertainty of the reading off not being considered, the number of seconds in A, will be 23, in B 39, in C 20, in D 21, in E ..... 25, in jF ..... 31, the sum of these is 159, and one sixth is 26". 5; the north polar distance, therefore, of Capella by the instrument, and, by the above method of taking the mean of the seconds, is F+ 26".5 ( = 44 12' 26".5), * Therefore, the equation for the north polar distance is 7"-2, 115 and, consequently, the mean index error F + 26".5 - X, or F + 26".5 - (Y + 16"), or 10".5. This is the index error from one star ; but the process is the same with any other star, since X may be any angle. If the catalogues were exact, and there existed no source of error from inequality of graduation, &c. the same index error would result whatever star were observed. Thus, suppose the number of seconds in X, instead of 16, to be 36, then the number of seconds from the six microscopes instead of being 159 would be 15Q4-6 x 20, and consequently, the mean number would be 26".5 -1- 20 = 46".5, and in this case the index error would be F + 46".5 - (P + 36) = 10". 5. But neither are the catalogues of stars perfect, nor is the in- strument altogether exempt from the errors of graduation, and of partial expansion. It will, therefore, happen in practice, that the index error is different with different stars. If the index error resulting from the observations of twelve stars, should be re- spectively, 10".5, 9".S, 6".8, 13". I, 1J".2, 9".l, 8.4, 13.2, 8.5, 10.2, 7-9, 8-7, 116.9 the sum being 1 16".9, the mean would be = 9 // .74. This is not the place to enter more fully into the special uses of the instrument. We will, however, give a specimen of the method of registering the readings off by the six microscopes. Oct. 15, 1812. Position of the telescope 0. ar. Tl.< In. srm. Out. Names of Stars. Deg.& Min. Microscopes. Mean. N. P. D. A, B, C. D. E. F. )A2 51 53 7 Drac. 38 28 40".2 44".5 46".0 11".2 42".5 41".4 42". 6 38 28 42.6 49 o Lyrae. 51 22 28.1 30.0 33.5 28.0 27.2 29-8 29.4 51 2229.4 9.13 50 47 aAquilae. 81 35 56.5 58.0 1.4 57-2 57.4 58.6 58.2 81 35 58.5 116 The sum of the seconds, belonging to the six microscopes, is, in the first row, equal to 255.8 ; one-sixth of which is 42.6, the mean. The sum in the second row, is 1 76.6 : one-sixth of which (as far as one decimal place) is 29-4 the mean. The sum in the third row is 289. 1 : divide by 6, and the quotient is nearly 48.19 ; but, it is clear, it ought to be 58.19. Now if we look to the number of seconds under C, which are 1.4, it is obvious that if we attended solely to that microscope, the number of minutes instead of being 35, would be 36, or the north polar distance of a Aquihfi would be 81 36' l".4; but, as it is clear, from the number of seconds belonging to the other microscopes, that the mean number of minutes cannot exceed 35, we must, in taking the mean of the seconds, consider 81 36' I 11 A, as 81 35' 6l".4, or we must add 60 to the seconds added together in the usual way, or, which is the more simple way, we must add 10 (=^-60) to one-sixth of the former result; in which case, the mean becomes 58.19, or nearly 58.2. In like manner, we must treat other like cases, should they occur : which, it is plain, can be but seldom. In some cases it may be necessary to add 120 to the sum of the seconds : for instance, if the several seconds were 57.1, 59,5, 1.9, 57.8, 57.8, 57.9, 1.1, 355 3 their sum is 235.3, add 120, and the mean is ^- = 59.2, or, 6 by the former rule, (see 1. 15,) - (235.3) + 20 = 59.2. o At the head of the preceding Table of results, (see p. 1 16,) is written, * Position of the telescope 0.' For the purpose of still farther lessening the errors of division, the telescope can be placed in several positions. When it is at the position 0, the telescope is directed to the pole, and the microscope A, which is the reading microscope, marks-O : and it is at the positions 10, 20, 30, when, the telescope, in each case, being pointed to the pole, the microscope A marks 10, 20, 30, respectively. The mural circle, like the transit instrument, requires three adjustments. 1st, Its axis must be made horizontal. 2dly, Its 117 line of collimation must be made perpendicular to the horizontal axis. 3dly, The line of collimation must be made to move in the plane of the meridian. A simple mechanical contrivance exists for carrying the first of the adjustments into complete effect. When the axis is made horizontal, the line of collimation describes a vertical circle : but it may describe a small vertical circle. To make it necessarily describe a great vertical circle, and a meridional circle, there are no mechanical means. Astronomical ones must be resorted to : and even with those, the two latter corrections are not accom- plished without great difficulty. We may, on this occasion, use (as it was stated in p. 70,) the transit instrument. When a star is on the meridional wire of the transit instrument, so move the mural circle that the star may be on its middle wire. Next, observe by the transit instrument when a star, on, or very near to, the zenith, crosses the meridian : if, at that time, the star is on the middle vertical wire of the telescope of the mural circle, then its line of collimation is rightly adjusted. If the star is on the middle wires of the two telescopes at different times, note their difference and adjust accordingly*. The great difficulties attending the verification of the line of collimation of the mural circle, will always prevent its becoming a good transit instrument. It acts, however, better in this last office than the telescope of the mural quadrant, which slides along the limb of the quadrant, the plane of which cannot be made to be wholly in the plane of the meridian. The mural circle is sufficient, as it is plain from its descrip- tion, to determine, to the extent of 180 degrees, the differences of the declinations of stars that are to the south and the north of the zenith of the observer. There must be two quadrants to effect the same object. Besides this advantage (the advantage of a single instrument) the circle is better balanced, and its six microscopes, which are firmly fixed in a stone wall, together with the power of changing the position of its telescope (see p. 116,) * This adjustment must be conducted by some formula which ex- presses the relation between the difference of the times, and the inclination of the line of collimation to the plane of the meridian. 118 must, when we take the mean results of a great number of obser- vations, do away with, or, at the least, very considerably lessen the errors of division and of partial expansion. But, it may be said, there being no plumb-line to mark the zenith point, the mural circle is defective inasmuch as it does not determine the zenith distances of stars : which distances are necessary to be known, if we would determine the refraction. The direct and special office of the mural circle is to determine the angular meridional distances of stars. If we extend the principle of its uses, and view the image of the pole star by re- flection from a basin of quicksilver, we obtain the angular distance between the star and its image. Such angular distance is twice the elevation of the pole star above the horizon. Hence its - zenith distance becomes known, and the zenith distances of other stars ; the meridional angular distances of which, from the pole star, are determined by the instrument. Since we can make observations, like the preceding, of the pole star both in its superior and inferior passage, we can thence determine (on an assumed law and quantity of refraction) the height of the pole itself above the horizon, which height (see p. 10.) equals the latitude of the place of observation. We cannot with the mural quadrant view the reflected image of the pole star ; nor can we at once, even if we use a plumb-line, determine by it the zenith distances of stars. These distances can only be truly known by knowing the error of colli- mation. The instrument of itself is unable to determine that error, and, in aid of its deficiences, we are obliged to have re- course (see pp. 67, &c.) to a zenith sector. This latter instrument, by double observations of a star near the zenith, one set being made, with the face of the instrument towards the east, the other with the face towards the west, determines the star's true zenith distance (see pp. 63, 67, 68, &c.) y Draconis is the star that has been most frequently observed at Greenwich. If we observe, on any particular day, either with the mural circle or mural quadrant, that star and other stars, we obtain their meridional angular distances, or the differences of their north polar distances. Hence, the zenith distance of y Draconis being determined by the zenith sector, the zenith distances of the above observed stars become known. 119 Thus, suppose by observations in 1812, with the north mural quadrant, that the zenith distances of y Draconis, /BUrsae Minoris, ct Cassiopeae appeared to be, respectively, 2' 20". 5, 23 26' 49".27, 4 l' 4l".l8 ; but, with the zenith sector, the true zenith distance of y Draconis appeared to be 2' 18".5, the true zenith distances of /3 Ursae Minoris, and a Cassiopeze, consequently, were 23 26' 47".27, and 4 l' 39". 18. At the time the instrumental zenith distances are read off, the quadrant is adjusted to a certain position, by making the plumb-line (see the figures of pp. 59, 60.) pass over the two crosses that are on the face of the instrument. It is the office of this plumb-line to keep the quadrant in a given position ; to be so kept, in order to use observations made of stars when we are unable to observe y Draconis. The error of the line of collima- tion is presumed to be the same when the quadrant is adjusted by making the plumb-line pass over the two crosses. But, it is plain, the zenith sector may be used as an auxiliary instrument to the mural circle as well as to the quadrant, and we may determine by their means the latitude of the place of the observation, and the zenith distances of stars. Thus, by the mean of a great number of observations made in 1812, at Green- wich, with the zenith sector, the zenith distance * of y Draconis was found to be 2' 18".5 ...... = 7, see the Figure in the next page. The north polar distance of the same star, found by the mural circle, and reduced to the same period, was equal to 38 29' 3" ..... ,, = Zy+Py = 38 Si' * The distance reduced to January 1812. The meaning of this phrase will be explained in the following Chapters. 120 the co-latitude (see pp. 9, 10, &c.) of Greenwich. Again, for all stars north of the zenith, and between Z and P, Z* = ZP P*, for stars south of the zenith, Z* = P* - ZP, from which formulae, Z % may be found, P 4? being determined by the mural circle. We may use then the zenith sector with the mural circle to deter- mine the error of the line of collimation in the latter, and thence to determine the zenith distances of stars. But, if we observe stars by reflection, we may with the mural circle, and without the aid of another instrument, determine the latitude of the place of observation, and the zenith distances of stars. The peculiar office, however, of the mural circle is to determine the angular distances of those points at which the several stars pass the meridian. These distances are used in correcting the existing catalogues of stars, and in determining, to greater degrees of exactness, their north polar distances. In this use the mural circle need not, like the quadrant, be brought by a plumb-line, or other means, to a given position. That operation is superseded by ascertaining the value of the index error. Thus, if at a certain period the instrumental polar distance of j3 Ursae Minoris appeared to be 15 4' 33".04 - i 19 and by the catalogue 15 4 34.23 121 l".19 would be the index error by that star; the mean index error is the sum of the several index errors divided by their number. If the position of the telescope be changed, or if the same number of microscopes (see p. 1 13,) be not used, the index error will be different : but whatever it is, it stands in lieu of, if we may so express ourselves, a mechanical adjustment of the instrument. But we may use in the same way, and on the same principle, the two fixed mural quadrants. With the north mural quadrant we can observe (supposing Greenwich to be the place of obser- vation) y Draconis and other stars to the north of the zenith. With the south mural quadrant, were its limb an exact quad- rant, we should be unable to observe y Draconis: but (see pp. 59, 60, 64.) the limb being extended a little beyond the limits of an exact quadrant, we are enabled to observe y Draconis : we can also observe with it (for this indeed, is its use) stars to the south of the zenith. By connecting, therefore, the two sets of observations, by means of the intermediate and common star y Draconis, we can, without the plumb-line, determine the meridional angular distances of all stars visible at Greenwich. We may also, as with the mural circle, determine their north polar distances by the aid of catalogues, and the use of an index error*. * It appears from the preceding matter, that neither mural quadrants nor mural circles are perfect instruments. The directions of their lines of collimation cannot be found without a zenith sector. Quadrants and circles with azimuth motions resemble that latter instrument, and are all capable of determining the directions of their lines of collimation, or of making observations independent of the errors of collimation. In prin- ciple then they are much more perfect instruments than fixed quadrants and circles. But large instruments are absolutely necessary in the present state of Astronomical Science, and for its future advancement, and it is difficult to construct large instruments capable of being turned half way round in azimuth on a vertical axis. Yet Ramsden constructed for the Dublin Observatory a circle of eight feet diameter turning round a vertical axis ; and it seems natural to presume that such an instrument must have been defective, since, of late years, its construction has been abandoned, Q 122 We have in the preceding pages given a description of the capital Instruments of an Observatory, and which are used for the making of observations in the meridian. On such obser- abandoned, and fixed mural circles invented. But, theoretically viewed, there seem eminent advantages attached to the former instrument. Within the space of a few minutes it is capable of making a double observation on a star, one with its face towards the east, the other towards the west ; the first before the star is on the meridian, the other after. Both observations must be reduced to the meridian by computation from the intervals of the times at which they were made, and the passage of the star over the meridian : which intervals may be most exactly known from the transit telescope and the Astronomical clock. The vertically of the axis, at each observation, is verified by a plumb-line. It may in practice be difficult to make these observations, but they have a singular and vast advantage in being free of all index error, and in determining, simply and directly, and within a short period of time, the zenith distance of a star. The index error of the mural circle, as it is proposed to be found, is a complex quantity, neither admitting of a brief definition, nor to be found by a single and simple process. Small zenith sectors have an azimuthal motion round a vertical axis. The reversion of the face of the Greenwich zenith sector is obtained by moving the instrument from an eastern to a western wall. This is an operation not easily performed. Mr. Troughton now proposes to construct a zenith sector (or an instrument for like purposes) of twenty-five feet radius, and capable of being turned round a vertical axis. Its range will be small, not exceeding five minutes on each side of the zenith : it is specially designed for observations of 7 Draconis which is distant, less than three minutes, from the zenith of Greenwich. The observations to be made with this instrument will be nearly free of all inequality from refraction, and entirely free from index errors; they will also from the great length and power of the telescope be, it is pro- bable, very exact, and will serve to determine, to a greater degree of exactness than has hitherto been done, the quantities of aberration and nutation. They may also settle the question, now agitated, of the ex- istence and quantity of parallax. The detection of this latter inequality, it may be here stated, has been made by an instrument, revolving, like the instrument just described, round 123 vations Astronomical Science is mainly founded. We must resort to the same source whether we seek for exact data to institute processes in Physical Astronomy, or to confirm their results. Of the other Astronomical instruments there are some from which we derive neither the elements of Astronomical Science, nor the verification of the results of its processes : but which are employed in a practical application of those results. Of such character are, Hadley's Quadrant, the Sextant and Reflecting Circle, instruments of the same class and principle of construc- tion. These are not instruments belonging to an observatory ; but the equatoreal instrument does belong. Its use is to observe phenomena, such as comets, and new planets, when they are out of the meridian. Besides these instruments, there is a repeating circle, portable, and principally useful in deter- mining the latitudes of different stations, and their bearings with respect to each other. An equal altitude and azimuth instrument of which, amongst others, one use is to ascertain the quantity and law of refraction. Some of these instruments will be briefly described in a future Chapter of this Work. From the description of the construction and uses of instru- ments, we will proceed to consider the results of certain obser- vations made with them : and the first observations that claim our attention are those made on the Sun at the time of his passing the meridian. round a fixed axis : or rather the proof of its existence depends on the accuracy of the Dublin circle (see p. 121.) The observations made with the mural circle of Greenwich do not verify such parallax. On this dis- cordance of the two instruments, much controversy has arisen which is not yet settled ; and, whatever be the excellence of the latter instrument, yet it must be allowed that the method of determining its index error involves many debat cable points. CHAP. VI. Sun's Motion Path Ecliptic Obliquity of Ecliptic. BY means of the Astronomical Quadrant, or of the circle, or by any instrument of a like class, we are able to observe the height either of the Sun's upper or his lower limb, and thence, of determining, by measurement and computation, the height of his centre. Let us first examine the observations of the north polar distances (N. P. D.) of the Sun's limbs, N. P. D. Difference in two Days. Jan. 2, 1816, U. L. 112 40' 22".5 S, .... 0L. L. 113 7 37.5 4, . . . . U. L. 112 29 21.7 11' 0".8. Again, N. P. D, Difference in two Days. March 31, 1816, U. L. 85 29' 26".2 April 1, .... L. L. 85 38 22.6 April 2,, . . U. L. 84 43 27.5 .". . 45' 58 /7 .7. - From these observations two inferences may be drawn : the first is, that the Sun, in the interval between January 2, and April 2, has approached the north pole by an angular ascent equal 27 56' 55" (=-- 112 40' 22".5 - 84 43' 27".5). The second is, that his daily portions of ascent are not equal : since, between January 2 and January 4, he ascended through 1 l' 0".8, and between March 31 and April 2, through 45' 58 // .7. So that the Sun's daily meridional ascent, or change of declination, or change of north polar distance (the fact is the same, but under different denominations) in the former period was about one- fourth of what it was in the latter. We shall have like facts, and may make like inferences, in 125 the heights of the Sun's centre observed at Cambridge, in the months of January March and June of the year 1810. Altitudes. 1810, Jan. 1 14 44' 40" 2 14 49 44 3 14 55 15 4, . . 15 1 13 Differences. 5' 4" 5 31 5 58 Therefore, the Sun, during these four days, was ascending in the meridian, but not by equal increases of altitudes, as it appears by the column of differences. Again, the altitudes of the Sun on four successive days in March and June, were Altitudes. Diff. Altitudes. Diff. Mar. 19 ... 37 5' 46" 23' 41" June 20 . . .61 14' 32" 29" 20 ...37 29 27 21.. .61 15 1 23 41 4 21 ...37 53 8 23 39 22.. .61 15 5 21 22 . ..38 16 47 23 .. .61 14 44 During the four days in March, then, the Sun was continually ascending, and by increments of ascent very nearly equal : in June he was still higher^ and on the twenty-second at his greatest alti- tude ; for, on the succeeding day, his altitude was diminished by twenty-one seconds. The increments of altitude, as appears by the column of differences, are, like those in January, unequal. Thus far it appears then ; the phenomenon of the Sun's continually varying altitude cannot be accounted for, by sup- posing the Sun to have an equable motion in the meridian ; ascend- ing for half the year from December 1809 to June 22, 1810. and then descending : let us next consider whether an explanation of other phenomena attending the Sun's transits over the meridian can be obtained, from attributing to the Sun an unequal motion in the direction of the meridian and merely in that direction *. * In proving the Sun's motion in the meridian not uniform, we have supposed, what is not strictly true, the intervals between his successive transits over the meridian to be equal. But the result will be the same, that is, his motion will be found to be unequal, if we correct the sup- position, and allow for the inequalities between successive transits. 126 If the Sun had a motion merely in the meridian, then, since the Earth's rotation is supposed to be equable, the intervals of successive transits over the meridian would always be equal, one with another and, besides, would be equal to the intervals of the transits of a fixed star ; or, of a star having neither a motion in the meridian nor one transversely to it. Now, neither of these conditions takes place; for on Aug. 21, 1810, Hegulus was on the meridian 1 minute 20 seconds before the Sun : on the succeeding day 5 minutes 2 seconds : on the next, or twenty- third, 8 minutes 44 seconds : so that it is plain, during these intervals, the Sun must have shifted away from the meridian, and moved transversely towards the east of it. Hence, to account for the phenomena of the Sun on the meridian, namely, the changes there both in the places and in the times of his transits, two motions must be attributed to the Sun, one in the plane of the meridian, the other transverse to it ; which two motions, (ac- cording to the doctrine of the composition of motion) are equiva- lent to, or may be compounded into, one single oblique motion. From the preceding instance it appears, that the Sun moves to the east of the meridian, and of a fixed star (Regnlus), through an angle which, in time, is equal to 3 minutes 42 seconds : but this angle is not constant : if, for instance, one of the stars of Sagittarius was, with the Sun, on the meridian, January, 2, 1810, the next day the Sun would come later, than the star, to the meridian, by 4 minutes 24 seconds ; on January 4th, later by 8 minutes 48 seconds ; on the 5th, by 13 minutes 12 seconds, &c. Hence, neither is the Sun's motion perpendicular to the meridian equable, nor, as it has appeared, his motion in the meridian. These two may be considered as the two resolved parts of the Sun's oblique motion. 127 The following Table exhibits the Sun's meridian heights on the 22d days of the several months of the year 1810. January. February. March. April. 18 2' 7" 23 55' 19" 38 16' 47" 39 17' 28" May. June. July. August. 41 32' 2?" 61 15' 5" 58 10' 54" 49 44' 19" September. October. November. December. January, 1811 38 16' 23" 26 5 1 7 49" 17 42' 42" 14 19' 42" 17 58' 47" From this Table we may determine, in a general way, the form of the curve in which the Sun may be supposed to move. For, if MM' be taken to represent the whole space from March 1810 to March 1811, and perpendiculars be erected re- spectively equal to the Sun's meridian altitudes on the several intervening days, the curve drawn through their extremities will M M' K be ESWE'. If E be the Sun's place on March 20, e, e 1 on the two successive days, then ME, me, me 1 must be taken re- spectively proportional to 37 29' 27", 37 53' 8", 38 16' 47" (see p. 125.) The intervals Mm, mm, &c. are not exactly equal, since they are the spaces through which the Sun retires each 128 day, from his place on the meridian the preceding day [see Note, p. 125.] : and in the present case they are respectively equal, in degrees, &c. to 54' 33", 54' 3l".5, 54' 3l". The spaces te, t'e, &c., or the increments of the Sun's altitude in the meridian, are respectively equal to 23' 41", 23' 39"; and the motions, in these directions, combined with the transverse motions in the directions Et, et, compound, as has been before remarked (p. 126,) the oblique motions Ee, ee f , &c.. In the Figure ESE 1 ', there are two altitudes nS, r W, one the greatest, the other the least, which for the year 1810, (see p. 125,) would happen on June 22, and December 22d ; and the mean of these two altitudes is \ {(61 15' 5") + (14 19' 42")} =37 5?' 2S"-5, which is, very nearly, the Sun's altitude (ME) on March 21, or Kk his altitude, Sept. 22*. Now when the Sun is at these mean heights ME, K k, he is in the equator. If, therefore, we knew when the Sun was in the equator, we could, by then observing the altitude of his centre determine that of the equator, which altitude (see p. 10.) is the co-latitude of the place of observation. Contrary wise, if, by observations other than those of the Sun, we determine the latitude of the place of observation, we are thence enabled to ascertain when the Sun is in the equator ; which must happen when his zenith distance is equal to the latitude. In pages 10 and 11, &c. we have given some instances of the method of determining the differences of the latitudes of places. But the latitude itself may be found, from the greatest and least altitudes of a circumpolar star above the horizon. Thus, if Hu, Hv denote those altitudes of a star, the parallel of which is vu, * The greatest and least altitudes (nS 9 rW) are supposed to happen on the noons of June, and of Dec. 22 ; which is not exactly true, as it will be hereafter shewn. 129 the latitude = PH = Hv -f- = Hv + Pw = If v -f Htt = H v + Hu latitude consequently, the latitude = -^(Hv -f- /f?/), and the co-latitude equals half the sum of the greatest and least zenith distances. Thus, by observations made at Blackheath, corr d . least zen. dist. Ursa min. Bod. 4 ...... 37 35' 55" corr d . greatest zen. dist ............. .... 39 27 57 |)77 3 52 co-latitude of Observatory .......... 38 31 56 Again, corr d . least zen. dist. o Cephei ............. 15 35' 2l" corr d . greatest zen. dist .................. 61 28 31 |)77 3 52 co-latitude ........ 38 31 56 By these means we should be able to recognise that twice in a year, in March and September, the Sun was in the equator. But, if the latitude were determined accurately, we should find no meridian altitude of the Sun to be exactly equal to the co-lati- tude : for instance, in the former cases, (pp. 125, &c.) the latitude R 130 of the place of observation being supposed, by observations of the pole star, to be 52 12' 36", its co-latitude is, consequently, 37 47' 24". Now, amongst the altitudes stated in p. 125, there is no one exactly equal to 37 4?' 24" : the altitude on March 20th is too small, that on the 21st too large: the reason of this is that, when the Sun was exactly in the equator, he was not on the meridian of the observer's station. There is some place to the east of Cambridge, at which the Sun was on the meridian when in the equator: and this place may easily be determined. We may now pass from the Sun's tabulated places, obtained by daily observations of his meridian altitudes, to the explanation of the changes of places, as originating from, or explicable by, his oblique motion. The line MM 1 (see Fig. p. 127,) is intended to represent the aggregate of the angular distances through which the Sun recedes each day from a fixed star, that was with the Sun on the meridian at e. This aggregate is 360* ; MK = KM ', moreover Kk = M'E' is the height of the equator, and a line EkE' Z containing 360, and extended oil a plane, may be conceived to represent the equator. Reversely, the lines E kE', ES kE' may be conceived to be wound round a sphere, the line EkE', coinciding * This part being intended for general explanation only, t&e pre* cession of the equinoxes is not taken account of. 131 with Qeq, &c. ESkE 1 with eSs, &c., and the points c, e, e' &c. in Fig. p. 127, with the points in Fig. p. 330, denoted by the same letters. Suppose now the Sun to be in the equator at e ; then, by the revolution of the sphere, the point e and the Sun, would be transferred to the meridian at the point Q, and hQ=ME will be the height of the equator: next, let the Sun recede through the space ee ; the point e and the Sun will be on the meridian at f, and fh = me (Fig. p. 127,) will be the meridian altitude : on the succeeding day let the Sun, having stiU farther receded through the space ee' , be at e' ; then his place on the meridian will be f 1 , and his meridian altitude fh = m e 1 (Fig. p. 127.): and similar circumstances will take place till the Sun has receded through the space e S (e S = a quadrant) when his place on the meridian will be at g, and his meridian altitude gh nS (Fig. p. 127,) then the greatest: after this the meridian altitudes will decrease. By supposing therefore the Sun to move in the curve eS, &c. from c towards S, whilst the sphere revolves in the opposite direction, from e towards Q, all the phenomena indicated by observation admit of an adequate explanation. And, as the diurnal phenomena were shewn (p. 8,) equally explicable either by supposing the whole celestial sphere to revolve, the Earth being quiescent, or, the Earth to revolve in a contrary direction, the Heavens being at rest; so, these latter phenomena may be accounted for, either by supposing the Sun to move in an orbit such as e Ss, &c., and the Earth to be at rest, or the Earth to move, but in a reverse direction, in an orbit similar to S whilst the Sun remains at rest. The above explanation does not depend, on the real form ojf the orbit eSs, which may be either circular or elliptical, or of any figure, provided that it lies in the same plane. For, the Sun is continually seen in the direction of a line drawn from him to the Earth ; but, whatever be his place in that line, he will always, by the observer, be transferred to the imaginary concave spherical surface of the Heavens. The imaginary path of the Sun in the Heavens is called the Ecliptic : the points E, E', (fig. p. 127,) of its intersection with the equator, are called the Equinoctial points : they are the nodes of 132 the equator ; the points S, W, those of the greatest and least ele- vations above the horizon, or, the places of the Sun, at his greatest northern and southern declinations, are called the Solstitial points. The points of the intersection of the equator and the ecliptic have been called the Nodes of the former ; which they may be, by likening the equator to the orbit of a revolving body ; for, gene- rally, nodes are defined to be the intersections of the orbit of a planet, or other revolving body, with the plane of the ecliptic. The planes in which the ecliptic and equator lie, are inclined to each other. The angle of their inclination is, for distinction, called the Obliquity 'of the Ecliptic : the angle of the inclination of the planes is the same as the angle made by two tangents, at the point e, to the arcs ee, eq*. (see Fig. p. 130.) If from S a solstitial point, a great circle PS be drawn per- pendicular to the ecliptic, and n S be taken equal to a quadrant, then n is the pole of the ecliptic f. The circle Gg, a tangent to the ecliptic at the solstitial point S, and consequently parallel to the equator (and therefore a parallel of declination) is called a Tropical Circle. A similar one touches the ecliptic at the other solstitial point. The small circle described round P in the circumference of which the pole of the ecliptic is always found, is called a Polar circle : sometimes the Arctic Circle (p. 38) ; and a similar one about the Earth's opposite pole is called the Antarctic circle. A secondary (see p. 8,) to the equator, passing through E, the equinoctial point, is called the Equinoctial Colure: one passing through S, the Solstitial Colure. Astronomers have divided the ecliptic into twelve equal parts called Signs : consequently, the ecliptic containing 360 degrees, each sign contains thirty degrees. Their names and characteristic symbols are, * Trigonometry, p. 128. f Ibid. P. 89. 1. 2. from bottom. This pole is situated in the sign of the Dragon between the stars r and f, but nearer to the latter. 133 Northern. Southern. Aries T Taurus 8 Gemini n Cancer 25 L-eo SI Virgo rrg Libra =a= Scorpio nt Sagittarius $ Capricornus V? Aquarius zz Pisces . . X These signs are situated within an imaginary belt, called the Zodiac, extending eight degrees on each side of the ecliptic. To each of the signSj certain clusters, or groups of stars, called Constellations *, are appropriated. But the signs, astro- nomically, serve merely to denote a certain number of degrees : thus, in the Nautical Almanack, the Sun's longitude for July 1, 1810, is stated to be 3 signs, 8 degrees, 54 minutes, 19 seconds ; which is equivalent to 98 degrees, 54 minutes, 19 seconds. The longitude is also sometimes expressed by means of the symbols of the constellations of the Zodiac. Thus, in Flamstead's catalogue of the fixed %stars, the longitude of 7 Draconis is ex- pressed by : ? 23 4/ 48", which, since Sagittarius, represented by t, is the 9th sign, (the * These groups of stars, or constellations, are by fancy imagined to form the outlines of the figures of animals and instruments, and are de- signated by their names. Thus, one group forms the figure of a Bear, another that of a Lion, a third of a Dragon, a fourth of a Lyre. So there are stars in the tail of the Bear, the head of the Dragon, the heart of the Lion : which are farther distinguished by Greek characters ; the charac- ters, according to their order, denoting the relative magnitudes of the stars. Thus, a Arietis designates the largest star in Aries, fi Draconis, the second star of the Dragon, tj Ursae Majoris, the star the fifth in size of the greater Bear, &c. t The particular stars of a constellation also are usually symbolically represented : thus a $ means the first or principal star in Taurus or the Bull ', \z, one of the inferior stars in Aquarius; /?n, a star of the second magnitude in Virgo ; 7 : , a star of the third magnitude in Libra. 134 first point of which is accordingly distant from that of arcs by 8 x 90, or 240), denotes the longitude of 7 Draconis to be 263 9 43' 48". The term Longitude, which has been just introduced, means an angular distance measured or computed along the ecliptic, and from one of the intersections of the equator and ecliptic : which intersection is called the First Point of Aries. After having passed through the 30 of Aries, the Sun enters Taurus, then Gemini, and, successively, the signs accord- ing to the order in which they were enumerated (p. 133). The motion of the Sun according to this order is said to be direct, or in consequentia ; any motion in the reverse direction is said to be retrograde, or in antecedentia. What longitude is with respect to the ecliptic, right ascension is with respect to the equator. It is angular distance, from the first point of Aries, (see 1. 7,) measured along the equator. And, what declination is relatively to the equator, latitude is to the ecliptic : it is angular distance from the ecliptic, measured by that arc of a secondary to the ecliptic passing through the star, which lies between the star and the ecliptic. Thus, if be the first point of Aries, or denote the intersection of the equator and ecliptic, and St be perpendicular to the part ^ t of a great circle, St, T t are, respectively, the latitude and longitude of S, if T t be part of the ecliptic : or, they are, respectively, the declination and right ascension of were known. If 5 be the Sun, SX part of the ecliptic, But it plainly does not represent the greatest declination, since, if it did, the two adjacent decli- nations would be equal, which they are not : the greatest declina- tion then must have happened sometime between the noons of June 21st, and June 22d, but nearer to the noon of the latter day. It is a quantity somewhat greater than 23 27' 4l", and certainly not differing from it by four seconds. For, assume it to be the greatest declination, then, in fact, we assume the Sun's longitude to be (what it is at the Solstice) 3 signs or 90. Now, this latter assumption cannot err 30' from the truth, since the change in the Sun's longitude for 12 hours is not quite equal to that quantity. Suppose it, however, to be 30', that is, in the Figure referred to, let X be the true place of the solstice, and SX=30', or T S = 89 30', then by Naper's rule*, rad. x sin. St = sin. T X sin. S Y , and rad. x sin. Xy = sin. T X sin. X T ; consequently, eliminating sin. T, there results (since sin. X f = l) * Woodhouse's Trigonometry, p. 146. sin. St .*. log. sin. ^y=10 + log. sin. 23 27'4l"-log. cos. SO', but, 10 + log. sin. 23 27' 41". . . =19-6000260 log. cos. 30' ....... = 9.9999835 9.6000425 .*. Xy = 23 27' 44".5. But since, in the case we have taken, the error in longitude must be less than 30', the real obliquity must be some quantity be- tween 23 27' 4l", and 23 27' 44". And, if the error in longi- tude, instead of being 30', were only 3', the error in declination, instead of being 3".5, would be only 3".5 . - 5 , or .035" *. In the present instance the former error is about 20', and therefore the latter is l".5 nearly, and consequently the obliquity f differs very little from 23 27' 42".5. We have thus, from the greatest observed altitude of the Sun and the latitude of the place of observation, deduced the greatest northern declination of the Sun : which declination is the mea- sure of the obliquity. By a similar process we may observe the least meridional altitude of the Sun, and, if the Sun should not have exactly reached, or should just have past, the point of his greatest depression or declination, we may, as in the former instance, approximate to the time and value of such depression. This extreme southern declination of the Sun is, like the northern, a measure of the obliquity. And the mean obliquity * For the variations in declination near the solstice, are nearly, as the square of the variation in longitude : for, in the former Figure, r x sin. p = sin. y* . sin. I (l = S y*, p = St) .. r . dp . cos. p = dl.sin. V s . cos. I (taking the differentials.) .. dp = . - cos. /= .tan. y .cos. /(since at sols., = V* nearly.) r cos. p r .-. dp = tan. y . sin. (90/)= tan. y.sin. dl= tan. r, since at the solstice /=90 dl nearly, t The obliquity thus determined is the apparent obliquity. 138 for any year, would be half the sum of the two extreme decli- nations computed for that year, or (in which case we do not need to know the latitude) would be half the difference of the greatest und least altitudes of the Sun, or half the difference of the least and greatest zenith distances*. Thus, by observations made in 1807, at Blackheath, Winter solstice, Sun's zenith distance 74 55* 56".02 Summer 28 8 .68 2)46 55 47.34 Mean obliquity for 1807 23 27 53.67 According to received theories, the portions of the ecliptic that lie to the north and south of the equator are exactly similar to each other. The greatest southern declination of the Sun, then, ought to give, for the measure of the obliquity, the same quantity as the greatest northern declination gives. But there is some discordance of observations on this head. According to Dr. Maskelyne, Mr. Pond, Dr. Brinkley, M. Oriani, and M. Arago, the observations of the winter solstice give a less obliquity than observations of the summer solstice. M. Bessel, on the contrary, from his own observations finds the two mea- sures of the obliquity concordant, and labours to shew that the latter observations of Bradley and those of Maskelyne made with the mural quadrant, and corrected for its errors, are of the same character. The anomalous phenomenon (for such it is) of an inequality between the greatest northern and so'uthern declinations of the Sun, may arise from some unknown modification of refraction. The question, certainly, is very intimately connected with the law and quantity of refraction. That source of inequality has not hitherto become the subject of consideration. This, there- fore, would not be the place, did we possess the means, of solving the difficulty that has been stated. We will merely, in addition to what has been said, subjoin the various results of the mean obliquity that different Astronomers, with instruments of different size and construction, have arrived at. * These distances, &c. must be corrected distances if the mean obliquity is to result from them. 139 kStronomers. Instrument. Year. Summer Solstice. Winter Solstice. Bradley. South Mural Quadrant of Greenwich 1755 23 28' 1$".49 23 28' 15".57 *f askelyne * Piazzi The above Quadrant. . . Circle 1795 1798 23 27 55.85 23 27 58.69 23 27 51.46 23 27 50.51 Oriani. Pond. Repeating Circle of 3 feet diameter made by Reichenback .... Mural Circle of Green- wich 1812 1813 23 27 50.77 23 27 50.0 23 27 48.22 23 27 47.34 ' Arago. Repeating Circle of 3 feet diameter by Reichenback. 1813 23 27 50.09 23 27 48.85 Bessel. Circle by Gary of two feet 1814 1815 23 27 47.41 47.48 23 27 47.34 o o 47.75 Brinkley. Circle by Ramsden, 8 feet diametsr 1813 23 27 50.99 23 27 48.14 We must remark on the preceding Table, that they are the largest, and, as they are generally esteemed to be, the best in- struments that give discordant results in the two values of the obliquity. We refer to the mural circle of Greenwich, and the Dublin circle. The circle of M. Bessel, which makes the obser- vations at the summer and winter solstice to accord so closely, is only two feet in diameter. The obliquity of the ecliptic may be determined (see pp. 137, &c.) either from the greatest and least altitudes, or the least and greatest zenith distances. But that is not the sole method. For instance, half the difference between the greatest and least north polar distances of the Sun is the value of the obliquity. In fact, the method to be employed depends on the * Bessel contends that these observations corrected for the errors of the quadrant (errors which the instrument became liable to after J 755) would render the quantities for the obliquity at the summer and winter solstice nearly equal, the seconds in the first being 57".79, in the latter 57''.52. 140 instrument of observation. The Dublin circle, by a double obser- vation, gives the true zenith distance. The repeating circle does the same : so, probably, does M. Bessel's two feet circle. The mural circle of Greenwich also gives the Sun's zenith distance, not, however, after the manner of the preceding instruments, but by the mediation of the zenith sector. By the mural circle, in which neither level nor plumb-line is employed, the Sun's north polar distance is determined. One of the methods, that have been briefly described for determining the obliquity of the ecliptic, consists in deducting from the Sun's greatest altitude, found by computation from the greatest observed meridional altitude, the co-latitude of the place of observation : the latter quantity being determined (see pp. 129, &c.) from the greatest and least altitudes of circumpolar stars. The quantity remaining after the above deduction is the Sun's greatest declination. By a like method, we may, at any time, whether the Sun be on or past the meridian, find his declination. In the first case, the declination is merely the dif- ference between the meridional altitude and co-latitude : In the second, the difference between the meridional altitude increased or diminished by the change of altitude proportional to the time from the passage over the meridian, and the co-latitude. For instance, in the first case, June 21, 1810, Sun's U. L ......... 61 29' 16" L. L ......... 61 46 30 2 Altitude of Sun's centre. ....... 61 15 1 Co-latitude of Cambridge ...... 37 47 24 Sun's declination June 21, at 12 h app*. time 23 27 37 To illustrate the second case, let it be required to find the Sun's declination on June 21, at three o'clock in the afternoon (civil time). Let the meridional altitude of the Sun's centre be found as above On June 22, let it be ......... 6l 15' 5" Altitude on 21st . . . . 6l 15 1 Increase of altitude in 24 hours .004 141 Therefore, the increase in three hours (supposing the increase to A!' be equable) is equal to 4" x = = 0".5, consequently, The alt. of Sun's centre, June 21, 3 o'clock = 61 15' l".5 Co-latitude . . 37 47 24 Declination of Sun, June 21, 3 o'clock . . 23 27 37.5 In the present Chapter, some instances have been given of the uses of the quadrant *, and transit instrument. The Sun has been observed on the meridian, and the attention of the Student directed to the changes both in the place and in the time of the Sun's passage. Twice a year, in March and September, the Sun is in the equator. From the first of these periods he continually, in his passages over the meridian, ascends towards the zenith till about the end of June when he becomes,, with regard to his zenith distance, which is then the least, nearly stationary. From about the end of June to the latter end of September, the Sun's zenith distance, at his passage over the meridian, continually in- creases and with daily increments larger and larger. From his passage cross the equator, in September, the Sun's zenith distances increase till December, but at a diminished rate of in- crease ; so that, towards the end of December, the Sun having reached his greatest zenith distance, becomes, with regard to such zenith distance, nearly stationary, or is at his solstice. The Sun's declination at this latter (our winter's) solstice is equal his declination at the other, the summer solstice, and either decli- nation is the measure of the obliquity of the ecliptic. The above are obvious inferences from the registered obser- vations of the Astronomical quadrant. Like inferences may be made from the observations of the transit instrument and clock. If the Sun and a star are on the meridian together on a certain day, on the following day the star will pass before the Sun : but the interval of time by which it precedes the Sun will not be * The quadrant is here, as in many other places, used as the generic term of all instruments that are used for determining meridional angular distances. 142 constant, whatever be the star, or, which amounts to the sanie thing, whatever be the day of observation. Thus, if on June 20, the Sun and a star are on the meridian together, on June 21, the star will pass the meridian about 4 9 s before the Sun ; but a month after, the time of a like precedence, or acceleration of passage, of another star will not exceed 3 m 5o 8 : a month after, it will be farther reduced to 3 m 41 s ; and, a month after, to 3 m 35 s . Now this motion, to the east of a star, is a motion in right ascension. The Sun, therefore, has a motion in right ascension but not an equable one: he has also (see p. 141,) a motion in declination and not an equable one. We will consider farther, in the next Chapter, the method of estimating the right ascension of an heavenly body. We might also with the Quadrant and Circle make other observations of the Sun than those already mentioned. Thus, by moving the instrument itself round its axis, or (the instrument being steady) by means of a moveable horizontal wire placed in the focus of the eye-glass of the telescope of the instrument, we can measure the Sun's diameter. Now such measurements are found to vary according to the season of the year at which they are made. The inference from this is, that the Sun is, in differ- ent parts of the year, at different distances from the observer. So that, with respect to the Sun, the observations indicate a third inequality in addition to the two already mentioned. But, it is to be remarked, .the observations hitherto re- ferred to of the Sun, whether of his north polar or zenith distance, or of the time of his passage over the meridian, are real observations (in the literal and natural signification of the term), such as faithful instruments ought to give us. They are, indeed, first in importance to the Astronomer, and the foundation of all his theories. But -they soon become subservient to the deduction of another kind of right ascensions and declinations, more abstract in their nature, and independent of the circumstances of individual observations. For instance, the zenith distance of Sirius (the great dog star) might be 68 43' 30" on one day, and 68 43' 22" on the suc- ceeding day. Each distance might be truly given by the instru- ment, but either the one or the other, or each, must be viewed as a modification of the true, distance (which in twenty-four 143 would not be changed) produced by some deranging cause. The Astronomer would contend, in this instance, the cause to be in the atmosphere, which, by bending the ray of light coming from the star, makes each zenith distance less than it would have been had not the light passed through a refracting medium : and he could go on to account for the difference (eight seconds) of the refrac- tion of light from the same star from changes in the weight and temperature of the air. A change, for instance, of ten degrees of Fahrenheit's thermometer, and of 1 inch of the barometer *, would produce a change of eight seconds in the apparent altitude of Sirius : and other variations of the thermometer and barometer would account for the same .fact : in this instance, then, the in- strumental, or apparent zenith distance, is noted and reduced, by correcting it, to a mean zenith distance, or which would be such, did no other cause than that we have mentioned prevent the apparent and mean places of the star from coinciding. Besides the one mentioned, there are, however, several other causes that produce the same effect. But, whatever they are, the observer, in the first instance, must be sure that his instrument is correct, and then must attend to its faithful report of phenomena. The observation is made just as it would be, were the observer placed in the centre of the Earth, at rest, and in an atmosphere that permitted light to pass through in right lines. What other phenomena, observations, so made, are indicative of, or proceed from, it is the business of Astronomical Science to explain. Towards such explanation our present course is now proceed- ing. * In registering an observation the states of the thermometer and barometer are always put down, see pp. 98, 99 CHAP. VII. On the Methods of finding the Right Ascensions of Stars ; from equal Altitudes near the Equinoxes, and from the Obliquity and Declination. Latitudes and Longitudes of Stars. Angles of Position. THE position of a star has been made to depend, as we have seen, on the arcs of two great circles perpendicular to each other. One of these circles is the equator, the other a great circle passing through its pole. The declination of a star is its distance from the equator j and its measure is the arc, of a great circle passing through the pole of the equator and the star, intercepted between the star and equator. The polar distance is the complement* of the decli- nation : these terms are sufficiently significant, and the practical methods of instrumentally measuring, by observation, the quan- tities signified by the terms, admit of an easy explanation. With regard to polar distances ; there is no star in the pole from which we can, by our instruments, at once determine the angular distances of other stars : but (see pp. 120, Sec.) we can always, by observations of circ unipolar stars, determine where the pole is : that is, we are always able, at an assigned time, to state what number of the degrees, minutes and seconds of our instru- ment, the star Polaris, for instance, is distant from the north pole, and, consequently, since we can also by the same instrument observe the angular distances of Polaris and other stars, we can assign their north polar distances. But with regard to the right ascensions of stars, the pro- ceeding is not so natural and obvious. There is no point in the equator permanently marked by a star, or other phenomenon, from which we can take our departure in measuring right ascen- sions ; nor, as in the former case, is there any point assignable by being the middle point between two phenomena. To find, 145 therefore, what we are in quest of, we must not confine our views to the stars and their apparent revolutions. If we look to the Sun, however, we shall find a convenient point, for dating right ascensions from, in the intersection of his path with the equator. This point, in several of its qualities or in the circumstances attending it, is like that other point, the celestial pole, from which polar distances are measured. It is a point neither marked by a star nor capable of being permanently so marked. But though, like the pole, it be variable relatively to the stars, supposing them to be really fixed, yet it can, at any specified time, be assigned : that is, the Astronomer, if he knows his business, is able to tell in how many hours, minutes, seconds, and parts of seconds, after the passage of Sirius (for instance), this point, the intersection of the equator and ecliptic, shaft also pass the meridian. If this can be done, the right ascen- sions of all stars become known from the intervals between their passages over the meridian and that of Sirius. The place of the pole is determined from the zenith distances of a circumpolar star, at its superior and inferior passage over the meridian. The star, at each passage, is at the same distance from the pole. The intersection of the equator and ecliptic, the Vernal Equinox, or as, still more technically, it is called, the First Point of Aries, may be determined from equal meridional altitudes of the Sun, and according to a method which we shall now proceed to describe. In the subjoined Figure, E represents the vernal equinox, k E' 31 jtij{\ the autumnal, ESkE' is the ecliptic, EkE 1 the equator, and T 146 me, m'e', &c. are the several meridional heights corresponding to the intervals of time Mm, Mm, &c. Let eon a day near the time of the equinox be the Sun's place, me his meridional altitude : let b be his place, after an interval of nearly six months, then if a b were equal to me, Ka would equal Mm, since the portion of the curve SbK is supposed to be similar to SeE. But since the ordinates me, me', ab, &c. re- present meridional altitudes only, it will happen that there is no meridional altitude near to K exactly equal to me: a b may be very nearly equal to me, but it will be either a little greater or a little less : suppose it the next less, or that the preceding meri- dional altitude is greater than me: then the Sun's declination at b ( = ab KK) is less than the declination ate ( = meME), but equal to it, at some time between the noon at e and the previous noon : which time must be determined by compu- tation. Let X represent the Sun's right ascension at e, then if Ka were equal to Mm, 180^, or 12 h X would represent his right ascension at b. But Ka being less than Mm, 12 b X represents the Sun's right ascension at some time previous to his being at b, and some greater quantity, 12 h X-\-e, for instance, (e being a small quantity) will represent his right ascension at b. Let also Y represent the star's right ascension. On the day at which the Sun is at e observe the transits of the Sun and star, and, by the clock, note the difference of their transits : represent the difference by d. Let also d 1 represent the difference of the transits of the star and Sun when the latter is at b : then, we have (supposing the star's apparent place not to bave changed) these two equations, Y - X = d, 12 h - A' -f e - Y = d', whence V "~"" _________ 2 r = : ' 8> + <*-* + * 14? Example from the Greenwich Observations of 1816. Transits of Pollux and the Sun. f7 h 36 ra (f.5 I Pollux, March 31, -I <0 41 9-31 I Sun's centre. App*. diff. of transits 6 54 51. 19 f 7 U 33 m 31 S .32 Sept. 12, ] 9 Ml 21 3.7 Pollux, Sun's centre. 3 47 32.37 The parts of the bottom line represent the apparent differ- ences of the transits : but these differences must be corrected (see pp. 103, 104, &c.) on account of the clock's daily rate. Now on March 31, the clock's daily rate (estimated from three stars was -r-8). On Sept. 12, from six stars, 1.45*: and the portions of these, proportional to the differences of the respective transits are + .23 and .226. But, if the clock gains, the difference of transits shewn by it must be greater than the real difference, or than the difference of the right ascensions : and the contrary must take place, if the clock loses. Hence, diminishing the first difference by .23, and increasing the second % .226, d = 6 h 54 m 5Cr\96, d' = 3 47 32.596; * Sept. 1 1 . Sept. 12. Reduction of Wires. Reduction. Rate. U".30 ir. 90 1''.4 2.84 1.44 1.4 20.52 19.10 1.42 48.72 47.20 T.52 10.02 8.54 1.48 32.80 31.32 1.48 8.7 .'. mean rate = = - 1.4,5. 148 whence d + d' = 10 h 42 m 23 s .556, d - d'= 3 7 18.374. We must now see what the altitudes of the Sun were on the noons of March 31, and Sept. 12th. Barometer. Therm. N.P.D. March 31, 1816, 30 s . 1 42 85 29' 26".2 U. L. Sept. 12. 29-95 57 85 34 28.1 U. L. The above are the north polar distances of the Sun's upper limb observed at Greenwich with the mural circle. The first correction to be applied to them is the index error, (see pp. 112, &c.) Then, deducting the co-latitude (ZP, see fig. p. 7.) there remains the zenith distance of the Sun's upper limb : but this distance is too small, by reason of refraction : it requires, therefore, a correction on that account. This is sufficient to explain, in a general way, the following process : March 31, U.L. = 85 29' 26".2 Sept. 12, = 85 34' 28". 1 Index error 4-4 00 2.5 85 29 30.2 co-latitude., 38 3l"21.5 app 1 . Z. D. U. L. 46 58 8.7 refraction 1 3.5 corrected Z. D. . .. . 46 59 12.2 alt. U. L. ..... 43 47.8 Sun's semi-diameter 16 1.12 42 44 46.6 parallax 6.4 true alt. Sun's centre. 42 44 53 85 34 30.6 38 31 21.5 47 3 9-1 1 1.6 47 4 10.7 42 55 49.3 15 56.1 42 39 53.2 6.5 42 39 59.7 The Sun's diameter which is subtracted from the altitude of his upper limb in order to obtain the altitude of the centre, may 149 be found by immediate observation, (see p. 98.) or may be taken from the Nautical Almanack. The correction for parallax will be explained in a subsequent Chapter. We have now me = 42 44' 53" a b = 42 39 59.7 4 53.3 the difference of meridional altitudes, or the difference of the Sun's declinations on the days of March 31, and Sept* 12. Now by observations on September llth, the Sun's altitude was . 43 2' 53".7, and his right ascension, on the same day, by the clock (allowing for its rate), ll h 17 m 28 s . 1. Between the two apparent noons, then, of Sept. llth, and Sept. 12th, the Sun's altitude from 42 52' 53" had decreased to 42 39' 59".7. The decrement then of altitude, which is also- the decrement of declination, was 22' 54'', whilst, in the same interval, the increment of right ascension was 3 m 35 s . 6. Hence, 22' 54" : 3 m 35 8 .6 :: 4* 53". 7 : 46 s .023*. The fourth term 46 s . 023, is the value of e. Hence, (see p. 146,) * Log. 3 m 35*.6 = 2.3336488 log. 4 53.3 = 2.4673121 4,8009609 log. 22 54 3.1379867 1.6629742 = log. 46.022, 150 This determination of the star's right ascension is not quite- exact ; for, in the process by which it was obtained it was assumed, that the star's right ascension was the same on Sept. 12, as on March 31. This assumption, however, is not correct. The apparent right ascension of the star is different at the two periods : or, in other and plainer words, the index of the Astro- nomical Clock would not mark the same time when the Star, on Sept. 12th, was on the meridional wire of the telescope, as it did on March 31, supposing the clock, in the interval, adjusted to sidereal time to have preserved a perfectly equable motion. The difference of the apparent right ascensions of the star at the two periods, is, indeed, but small, not exceeding eight-tenths of a second. If, as it will be in the present instance, the apparent right ascension be greater on Sept. 12th, than on March 31, the second of the equations of p. 146, instead of being 12 h - X + e Y=d', will become 12" - X + e-(Y+y} = d', Y+y, representing the star's right ascension on Sept 12th ; or y representing the increase of right ascension ; consequently, the resulting values of X and Y, will be + e y - ------------ 2 2' Y _ I2 h + d - d' + t y 2 " 2 ' Hence, if we makey = 0".71 (which is nearly its value) we shall have X, or Sun's R. A. = O h S9 m 10 $ .88, F, or Star's R. A. = 7 34 1.85, nearly. The student at present, must be content to take for granted that the value of y is rightly assigned. It is the result of four small corrections due to inequalities not yet explained. It, in truth, happens here as it will repeatedly happen again, that, in conducting an Astronomical process we are obliged to anticipate 151 the results of future demonstrations and to draw on funds not yet established. The preceding value of y is very small, and, as it will be hereafter shewn, it can in no case be much larger. It is merely the difference of the apparent right ascensions of Pollux at the two periods of March 31, and Sept. 12, and it is only a portion (not a proportional portion) of the star's annual increase of right ascension. That there is such an increase may be easily shewn by finding from two observations, made at different periods, the corresponding right ascensions of the star : and, in order to obviate an objection that may be made against the preceding method, inasmuch as it is therein assumed, that the increase of the star's right ascension, during an interval of about six months, is either nothing or a small but undetermined quantity, we shall find the right ascension by a different method. The method consists in finding the Sun's right ascension from two observed or known declinations : one the solstitial declination, or (see p. 136.) the obliquity of the ecliptic, the other an observed declination near the equinox. The star's right ascen- sion will be the Sun's right ascension at the latter observation plus the difference of the times of the meridional transits of the Sun and star. The two periods of observation are March 31, 1816, and March 24,, 1768. For the first of these periods, we have (see p. 149.) me, or altitude of Sun's centre ... =42 44' 53" co-latitude of Greenwich =38 31 21.5 Sun's decimation March 31, 1816 . . 4 13 31.5 Let the obliquity of the ecliptic for that time be assumed equal to 23 27' 50".8. For the latter period we have from Maskelyne's Observations, reduced according to the methods of p. 148. alt. Sun's centre. 40 16' 2" co-latitude 38 3 1 21 .5 Sun's declination, March 21, 1768 1 44 40.5 Let the apparent obliquity at that period be assumed equal to 23 28' 14".*. 152 To find the right ascension, we have, in each case, this formula : rad. X sin. Xtan. St, X or, rad, X sin. R.A. = cotan. obliquity X tan. dec. March 31, 1816. rad = 30 tan. 4 IS 7 3l".5 . . = 8.8685352 cot. 23 27 50.8. . = 10.3624424 (=log. 9 47 59) 9.2309776 March 24, 1768. rad = - 10 tan. 1 44' 40".5 = 8.4837088 cot. 23 28 14.8 = 10.3623042 (=log. 4 1 21) : . . 8.8460130 Hence, (reducing the angular measures into measures of time) March 31, 1816, Sun's R.A = O h 39 m ll s .8 diff. of transits of Sun and star 6 54 50.96 Star's R. A 7 34 2.76 March 24, 1768, Sun's R. A O h l6 m 5 8 .6 diff. of transits (from Maskelyne's Obsv n '.) 7 15 2.46 Star's R. A 7 31 8.06 The difference between these two right ascensions is 2 m 54 8 .7, an increase that has taken place in forty-eight years, and, conse- 153 quently, if the same increase would always happen in every forty- eight years, the mean annual increase would be S 8 .63. But it so happens that, of the inequalities causing the right ascen- sion to vary, one inequality is variable both in degree and direc- tion : it is not the same on March 24, 18 16, and March 24, 1817 : and, in the case before us, it diminishes the right ascension of Pollux by l".22 in the March of 181G, and increases it by l".3 in the March of 1768. The difference of these quantities is 2". 52 : so that, setting aside this variable inequality (which has its cause in the variable action of the Moon on the Earth) the increase of Pollux's right ascension in forty-eight years will be 2 m 54 8 .7 -f 2".52 ; or 2 m 57".22, and the mean annual increase of right ascension, 3 9 .69. This augmentation of right ascension then exists : and it is part of this, (but not, as we said in p. 151, a proportional part) that causes the right ascension of Pollux on March 31, 1816, to be different from its right ascension on Sept. 12, 1816. The first method which has been described for finding the right ascension is due to Flamstead. It is held by practical Astronomers to be a good method. The Sun is observed at equal zenith distances,, and, therefore, any error assigned by the Tables in the quantity of refraction, or any error in the instru- ment, would equally affect each observation. We are tolerably sure of ascertaining (which is the essential part of the method) when the Sun is at equal distances from the zenith. It is less important to know the exact quantities of those zenith distances. We must not hope to obtain, exactly, the right ascension of a star by one observation and process. On this, as on all like occasions, the process must be repeated, and the mean taken of several results ; taken, as the true result, or as the result that is most nearly true. Thus, in the instance adduced, observations (should circumstances permit it) should be made on March 30, and Sept. 13 : on March 29, and Sept. 14, &c. : and like sets of observations should be made on different years. The right ascension of one star being settled, the right ascen- sions of other stars may thence be deduced. Thus, taking the u 154 apparent right ascension of Pollux on March 31, 18 16, to be 7 b 34 m 2 8 .2, let the index of the clock be set to that time when Pollux is on the meridional wire of the transit telescope. The clock, if it goes rightly, will denote the right ascensions of other stars when they are bisected by the meridional wire. Thus, on the above day, Capella passing the meridional wire at 5 h 3 m 5 s . 5 Ald6baran 5 3 21.2 Procyon 7 29 3Q.7 a Arietis 1 56 47.6 such times would be the apparent right ascensions of those stars. a Arietis, the principal star in the constellation of the Ram, passes the meridian, as we see, atl h 56 m 47 s . 6. But the Jirst point of Aries, is, as it has been already mentioned, a term alto- gether technical. It is, if we conceive the ecliptic and equator to be traced out in the Heavens, one of the intersections of those circles, namely, that in which the Sun would be at the time of the vernal equinox. When this point is on the meridian on March 31, 1816, the clock would note O h O m s , if, going regularly, it noted 7 h 34 m 2 s . 2 when Pollux was on the meridian ; 7 h 34 m 2 s .2 being supposed to be the truly computed apparent right ascen- sion of that star. In illustrating the preceding method of finding the right ascensions of stars, we have employed the star Pollux : but, it is plain, there are many other stars that would, equally well, have served that purpose. Dr. Maskelyne found, according to the preceding methods (or at least on their principle) the right ascen- sion of a Aquilae : that was, what he called, his fundamental star, the right ascension of which regulated the right ascensions of other stars. But it may be here noted that, whatever the star and the time of observation, the result of the process merely gives the apparent right ascension of the star at that time : and, conse- quently, the right ascensions of other stars, deduced from that of the fundamental one by means of the differences of their transits, will be merely their apparent right ascensions for the same time. The day after the observation, the right ascensions will, in strict- ne^ss of theory, be different, although imperceptibly so. But, a month after the time of the first observation, the right ascen- 155 sions of the stars will be found to have altered, or, the clock, going rightly, will no longer indicate the original times of their transits. This has been (see pp. 106, &c.) already adverted to; the full explanation of the phenomenon cannot be given till all the inequalities^ that prevent the times of the recurrence of a star to the same horary wire, or to the meridional wire, of a transit telescope, from being entirely regulated by the time of the Earth's rotation. The present purpose of mentioning the circumstance is to shew that a catalogue of the right ascensions of stars made from the observations of two, three, or four years, and even by the best instruments, would, without the aid of theory, or of results obtained during long periods of time, be an imperfect catalogue. The results from the observations of-a few years are quite insuf- ficient. For, although we might by such establish as a fact that both the right ascensions and declinations did not remain the same, but, upon the whole, did either continually increase or diminish, still the mean values of such increases and diminutions would be entirely vitiated, by the operation of certain variable and recurring inequalities. We have already seen an instance of this in the case of Pollux, The interval of forty-eight years is not sufficiently large to give accurately the mean value of the annual increase of the right ascension of that star. If, however, we possessed good observations distant from each other by two hundred years, then, since the utmost effect of the variable ine- quality, of which we have spoken, must be less than 3 s , the mean annual increase of the star's right ascension found as it was in p. 153, cannot be erroneous beyond s . 015. But this mean increase is only one point gained in the formation of the cata- logue of stars. We may know the right ascension of Pollux on March 31, 1816, and its annual change, and still not be able to determine its right ascension on March 31, 1817, or on Sept. 12, 1816. But although the exact determination of these and other points is not within our present reach, still, enough has been done for the elucidation of the general principles of the methods by which the places of the pole and of the first point of Aries, are determined. Both these points are perpetually changing their positions ; but, such is the advanced state of Astronomical Science, they can alwavs be exactly found. 156 We will now proceed to subjects related to the one which has been just discussed. The right ascensions and declinations of the Sun and stars are deduced from observation. Their longitudes and latitudes, not being subjects of immediate observation, are deduced, by computation, and by processes purely mathematical, from right ascensions and declinations. In the case of the Sun, the com- putation is very easy, resting on the solution of a right-angled triangle. One or two examples will be sufficient for the illus- tration of this part of the subject. Thus, let v be part of the ecliptic, and t part of the equator, and let S t be part of a circle of declination : and let the Sun's longitude Nov. 28, 1810, be required, his declination being 21 16' 4", and right ascension I6 h 14 m 58 8 .4, or in space, 243 44' 36". By Naper's rule, r X cos. /> S = cos. T t X cos. St ; /. log., cos. T t or log. cos. 243 44' 36". .. =9.6458083 log. cos. St, or log. cos. 21 16 4 =9.9693672 10-hlog. cos. vS .. 19.6151755 /. 5=245 39' 10" the longitude required ; or = 8 s 5 39' 10". 2dly, Required the Sun's longitude Nov. 29, from his decli- nation = 21 26' 35", and obliquity = 23 27' 4l".S. 157 By Naper, r X sin. st = sin. S X sin. S*p f ; /. log. r + log. sin. 21 26' 35" ...... = 19.5629781 log. sin. 23 27 41.3 ...... = 9.6000276 log. sin. T S ............ = 9.962Q505 .'. longitude = 246 40' 6", or 8" 6 40' 6". 3dly, Required the Sun's longitude Nov. 30, from his J I6 h 23 m 34% and the obliquity of the ecliptic = 23 27' 42".3. By Naper, r X cos. S X tan. ; .'. log. r + log. cos. 23 27' 42".3 = 19-9625237 log tan. I6 h 23 m 34 s . 1 = 10.3492191 log. cotan. T S ........ = 9-6133046 /. longitude ....... , . - = 247 40' 56'', or = 8 s 7 40' 56". The longitude in these examples is computed from the right as- cension and declination, conditions given by observation. But, in the construction of the Nautical Almanack, the reverse operation takes place. The Solar Tables give the Sun's longitude : thence, and from the obliquity of the ecliptic, the right ascension and declination are computed, by trigonometrical operations, similar to the preceding. The longitudes and latitudes of stars (their respective angular distances from the first point of Aries and the ecliptic) are found from their right ascensions and declinations ; by processes, how- ever, less simple than the preceding. Let P, TT be the poles of the equator A Q, and of the ecliptic 158 EC : then since PQ=90, and 7rC = 90, P^r=Ca, but CQ is the measure of the obliquity ; therefore PTT is. Let s be a star, Psn part of a circle of decimation passing through it, irsm part of a circle of latitude. Ps is the star's north polar, TTS is the complement of the star's latitude ; the angle s P TT depends on the star's right ascension, and the angle STrP on the star's longitude. In the present figure, T represents the first point of Aries, and the order of the signs is from Trm, which latter angle equals 90 ZsTrP; consequently, in this case (the star being in the first quadrant), the longitude (L) = 90 - L STrP, = 90 - C (C= ZSTrP). The star's right ascension is Tw, which measures ^ TPn, which angle equals Z TrPs 90 ; consequently, in this case, the right ascension (JR). . = Z ?rPs 90, U= Z.TrPs) = A - 90, and, consequently, A == 90 -{- JR. This is the case in thejirst quadrant, as it is called, or, the above equation is true when the star (s) is situated within the quadrant PT Q, or, when the star's right ascension is less than six hours of sidereal time. The relation of A to the right ascension is different in the other quadrants ; in the 2d quadrant A = 270 JR, and cos. A~ sin. JR in the 3d quadrant A = 270 JR, and cos. A = sin. JR in the 4th quadrant A = JR - 270, and cos. A = - sin. JR in the 1st (as we have seen) A = 90 + -#^ and cos. A = sin. JR In all these cases, cos. A = sin. ^l : which equation, will enable us to lay down a simple and general formula for the value of the star's latitude. In the oblique spherical triangle s TT P, cos. TTS = COS. PTT.COS. Ps -}- sin. PTr.sin. Ps.cos. irPs. Let TTS = A, PTT = /, Ps = 5, then cos. A = cos. /.cos. S sin. /sin. S sin. JR,. 159 Hence, (see Trigonometry, pp. 39, 171.) 1 2 sin.* = cos. I.cos.S sin. /.sin. + . r . . o x90 JR.\ % sin. I. sin. o. sin. f 1 = cos. (/+ J) + 2 sin. I sin. 5 . sin. 3 ( 9 ~ and ' . o A a (/+) . - * , /90 JR\ 2 sin. = 2 sin. 2 sin. I. sin. c . sin. ( 1 . 2 2 V 2 X Let sin. /,sin. $. sin. 2 ( ) = sin.* M \ 2 X then A xT _l_ -Sv - sin.*M, (Trig. p. 31.) = sin. f -f in I . sin. and, logarithmically expressed, sin. = Hence, we have this rule for finding the latitude of a star from its right ascension and north polar distance, and the obliquity of the ecliptic. 1st. Add twice the logarithmic sine of half the difference be- tween the right ascension and 90, to the sum of the logarithmic sines of the obliquity, and the star's polar distance : half this whole sum diminished by 20 will be the logarithmic sine of an auxiliary angle M. 2d. Form two arcs by adding M to the half sum of the obliquity and north polar distance, and by taking M away from that half sum; half the sum of the logarithmic sines of these two latter arcs is the logarithmic sine of half the complement of the star's latitude. 160 EXAMPLE 1. Required the latitude of a Arietis, its mean right ascension (for 1815) being .................... l h 56 m 45".9 its mean north polar distance ........ 67 25 1 .7 and the mean obliquity of the ecliptic*. 23 27 46.3 Reduce J3, to degrees at the rate of twenty-four hours to 360, or of l h to 15 : then 1st, JR = 29 ll'28".5 90^1 = 60 48 31.5 I (90- jR)= 30 24 15.75 ........ log. sin. 9.7042361 19.4084722 N. P. D. = 67 25 1.7 ........ log. sin. 9-9653546 I = 23 27 46.3 . . . ..... log. sin. 9.6000517 2 log. rad. 20 ^.f.u.-r _ ^ ^ ^ 2)18.9738785 M = 17 52 12.3 log. sin. M 9-4869392 2dly, .'. M = 17 52' 12".3 + M= 63 18 36.3 log. sin. 9.9510705 - M = 27 34 1 1.7 log. sin. 9-6654221 2 N.P.D.-hJT 2 2)19.6164926 (log. sin. 40 l' ll /7 .3).. . 9-8082463 Hence, the complement of latitude = 80 2 7 22".6 and the latitude = 9 57 37 . 4. * This is very nearly the mean obliquity for 1815, supposing, accord- ing to Bradley, the mean obliquity for 1750 to be 23 28' 18", and also that the secular diminution of the obliquity is 50". The true value of the mean obliquity is, probably, two seconds greater. 161 EXAMPLE 2. Required the latitude of Pollux in 1815, M = 7 h 33 m 58 8 .7 = 113 29' 40"-5, N.P.D = 61 32 12.4. 1st, M =113 29' 40".5 (M - 90) = 11 44 50.25 log. sin. = 9-3087681 2 18.61/5362 N.P.D. = 61 32' 12".4 log. sin. 9-9440497 / (obliquity) = 23 27 46.3 log. sin. 9.6000517 2 log. rad. - 20 N.P.D + /= 84 59 58.7 2)l8.l6l6376 (log. sin. M) 9.0808188 N.P.D. +/ = 42 29 59.3 M = 6 55' 5".75 2 2dly, M = 6 55' 5".75 N.P.D.+I \-M= 49 25 5.05 .... log. sin. = 9-8805143 N.P.D. + / -~M= 35 34 53.55.- --log. sin. 9-7648191 2 2)19-6453334 (log. sin. 41 39' 50".8) 9-8226667 Hence the complement of star's latitude is . . 83 19' 41'". 6 and the star's latitude 6 40 18.4. 162 EXAMPLE 3. Required the latitude of Spica Virginis (in 1815.) its M = I3 h 15 m 27 S .53 = 198 5l' 52".9-5 N. P. D. .. = 100 11 28.9 1st, JR =198 51' 52".95 M - 90 = 54 25 &( . A7 log. sin.= 9-9103198 2 19-8206396 N. P. D. = 100 1 1 7 28".9 log.' sin. 9-9930933 / = 23 27 46.3 log. sin. 9-6000517 2 log. rad. - 20 N.P.D.-f / =123 39 15.2 2)19-4137846 N P D 4-1 - = 61 49 37.6 (log. sin. M} 9-7068923 .'. M = 30 36' 39". 1 2dly, M = 30 36'"39".1 f ' ' -~ t "' +M = 92 26 16.7 log. sin. = 9-9996068 2 ' : : M=3l 12 58.5 log.sin.= 9.7145557 2)19.7141625 (log. sin. 46 l' 12".4) 9-8570812 Hence the star's distance from the pole of the ecliptic is 92* 2' 24 /; .8, and the star's south latitude is 2 2 ; 24". 8. 163 EXAMPLE 4. Required the latitude of a Aquilae for the year 1815. M =295 26' 1 7". 7 N. P. D.= 81 36 40.6 1st, JR = 295 26' I?''.? - 90 = 102 43 8.8. ..... * log. sin. = 9-9892102 2 2 19.9784204 N. P. D. = 81 36' 40".6 ...... log. sin. 9-9953285 1= 23 2746.3 log. sin. 9-6000517 2 log. rad.= -20 N.P.D. + 1 = 105 426-9 2)19-5738006 N - P -P- +/ = 5.3.13.45 Gog. sin. 37 44* 38 /; .09) 9-7869003 2dly, N.P D +/ - ' '^ -^-^ =37 44 X 38 /; .09 = 90 17 11-5 .......... log. sin. 9-9999946 M=14 47 15.4 ......... -log. sin. 9-4069438 2)19-4069384 ( = log. sin. 30 20 7 42 /7 .25) 9-/034692 Hence, the star's distance from the pole of the ecliptic is 60 41' 24".5, and, consequently, the sta^s latitude is 29 18 7 3o".5 164 EXAMPLE 5. Required the latitude of a Pegasi (in 1815). its M = 343 53' 15".75, aad N.P.D. = 75 47 12.7- 1st, jfl = S43 53' 15".75 /R on =126 58 37.87 .log. sin. = 9-9026692 19.8053384 N. P. D. = 75 47' 12".7 - log. sin. 9-9864981 / = 23 27 46.3 log. sin. 9-6000517 2 log. rad. 20 N.P. D.-f/ = 99 14 59^ 2)19-3918882 (log. sin. M) 9-6959441 N.P.D.-M _ 4g0 S7/ 2g ,, 3 . M = 2g0 46 , u /, 25 j|f=29 46' 14"".25 N.P.D. 4- / 2 N.P.D. -fM=79 23 43.75 log. sin. 9-9925185 -M=19 51 15.25 log. sin. 9.5310040 2)19-5235225 (log. sin. 35 17' 40") 9.7617612 Hence, the distance of the star from the pole of the ecliptic is 70 35' 20" : and, consequently, the star's latitude is 19 24 7 40". 165 EXAMPLE 6. Required the latitude of 7 Draconis in 1750. itsJR =267 42' 7" itsN.P.D = 38 28 16 and obliquity = 23 28 18 M = 267 42' 7" 90 2)177 42 7 JR - 90 = 88 51 3.5 log. sin. = 9-9999126 2 19.9998253 N. P. D. = 38 28' 16" -log. sin. 9.7938741 1= 23 28 18 log. 9-6002054 N.P. D4-/= 61 56 34 2)19-3939048 N.P.D. + J^ 5g (log. sin. M) 96969524 2 M = 29 50 48.4 M= 29 50' 48".4 N.P.D. + 7 2 N.P.D. r = 60 49 5.4 log. sin. = 9.9410524 - M = 1 7 28,6 log. sin. = 8.2928518 2)18.2339042 (log. sin. 7 31' 18".55) 9-1 169521 consequently, the complement of star's latitude = 15 2 37 .1 and star's latitude * 74 57 22.9 166 EXAMPLE 7. Required the latitude of 7 Draconis in 1815, its right ascen- sion being - VV 268 4/ 40 "' 2 its north polar distance 38 29 4.95 and the obliquity of the ecliptic 23 27 52.5 M =268 4' 40".2 3D QQO =89 2 20.1 log, sin. 9-Q999389 2 19.9998778 N. P. D.= 38 29' 4".95 log. sin. 9-7940038 I = 23 27 52.5 log. sin. 9-6000817 2 log. rad. 20 N. P.D. + /= 61 56 57.45 2)19-3939633 N P D 4-7 = 30 58 28.72 (log. sin. M) 9-6969816 M = 29 50 56.43 ' ' ' - + M = 60 49 25..15 log. sin.= 9.9410757 2 N.P.D. -M= 1 7 32.29 log. sin. = 8.2932485 2)18.2343242 (log. sin. 7 Si' 3l' ; .7) 9- 1 7 1621 complement of the star's latitude is 15 3 7 3' 7 .4 and star's latitude 74 56 56.6. If instead of the above value for the obliquity, we had assumed it equal to 23 27 46 .3, the resulting value of the star's latitude would have been 74 56 X 5l". 167 EXAMPLE 8. Required the latitude of Polaris in 1800 : its right ascensiou being ---- j ......... . ............ .'. . 13 b' 15" its north polar distance (S) .......... 1 4o 34".5 and the obliquity of the ecliptic ...... 23 2? 54.8 ^l=J3 o 15" =38 2722.5 .................. 9.7937323 2 19.5874646 3=1 45' 34".5 log. sin. 8.4872189 / ^-23 27 54.8 log. sin. 9-6000929 2 log. r= 20 3+ 7 = 25 13 29-3 2)17.6747764 log. sin. (3 56' 35".7) 8.8373882 ^! r = 1236'44".6 2 M = 3 56 35.7 J I T - -h M = 16 33 20.2 log. sin. = 9.4547617 Jlf= 8 40 8.9 log. sin. 9-1781950 2)18.6329567 (log. sin. 1 1 57' 38".9) 9-3164783 % complement of star's latitude = 1 1 57 7 38 /x .9 /. complement . = 23 55 17-8 /. star's latitude. . - . . = 66 4 42.2 168 We will now proceed the investigate a formula for the star's longitude. The angle sir P (C), as it was mentioned in p. 158, depends on the longitude. In the subjoined Figure _L = 90 C ; consequently, in the 1st quadrant C=90 L; .*. cos. C=sin. L in the 2d C= L - 90, cos. C=sin. L in the 3d C= L 90, cos. C=sin. L in the 4th C = 360 -f 90 - L, cos. C=sin. L. Now, COS. sP - COS. S7T . COS. P V cos. sin. STT . sin. PTT cos. ^ cos. A . cos. / '. sin. L ( = cos. C) == : : 7 sin. A sin. / But, (see Trigonometry, p. 39) 2 sin. 2 in. L, sin consequently, . a /90-f\ _cos. 5 (cos. A cos. I sin. A sin. J) \ Q s sin. A sin. JL and (see Trigonometry, pp. 30, 33.) sm sin. 169 (A -f-/-h^\ i ) sm - \ 2 / sin. A . sin. / and, logarithmically expressed, 90 4- L , . log. sm. o /^++ , - /^_ A |1 20 + log, sm. { - - - ) 4 log. sin. (^ - - --- SJ log. sin. A log. sin. / EXAMPLE 1. To find the longitude of a Arietis for the beginning of the year 1"815. (seep. 160.) A = 80 2 ; 24".56. . . . log. sin. = 99934050 / = 23 27 46.3 log. sin. = 9-600051? S = 67 25 1.7 W) 19.5934567 2)170 55 12.56 |-sum 85 27 36.28 .log. sin. = 9.9986352 sum $ 18 2 34.58 log. sin. - 9.4909872 2 log. rad. = 20 39.4896224 (d) 19.5934567 2)19.8961657 (log. sin. 62 32 7 20 /; .5) 9-9480828 90 + L = 125 4' 41" L = 35 4 41 = 1 s 5 4 4K 170 EXAMPLE 2. Required the longitude of Pollux for 1815. (see p. 161.) A =83 19' 4l".6 log. sin. =9-9970490 1=23 27 46.3 log. sin.=9 6000517 3 = 61 32 12.4 (d) 19.5971007 2)168 19 40.3 sum = 84 9 50.15 log. sin. =9.997743 1 sum - S = 22 37 37.7 - log. sin, = 9-585 1587 2 log. rad. = 20 39.5829018 (d) 19.5971007 2)19.9858011 9.9929005 Now 9.9929005 is the logarithmic sine of 79 40' 6" : it is also the logarithmic sine of 100 19' 6'' the supplement of the former : and this latter angle is the proper angle, since the star (see p. 158, 161,) is situated in the second quadrant. Hence, 90 + L = 200 39' 48", L = 110 39 48 or = 3' 20 39 48. 171 EXAMPLE 3. Required the longitude of Spica Virginis in 1815. /see p. 162.) A = 92 2' 24".8 log. sin. = 9-9997247 /= 23 27 46.3 log. sin. = 9-6000517 S=100 11 28.9 (d) 195997764 2)215 41 40 \ sum =107 50 50 . log. sin.= 9-9785809 sum - I = 7 39 22 log. sin.= 9.1245924 2 log. rad. =20 39-1031733 (d) 19.5997764 2)19-5033969 9.7516984 Now 9-7516984 is the logarithmic sine of 34 22' 14".5, and of the supplement 145 37' 45 /7 .5, and, since the star's M (see p. 163,) is greater than 12 h ^ it must be the latter of these angles that is the true one ; consequently, 90 + L = 291 15' 31" and L = 201 15 31 or = 6" 21 15 31. 172 EXAMPLE 4. Required the longitude of a Aquilae in 1815. A = 60 41' 24".5 ........ log. sin.= 9-940509O 1= 23 27 46.3 ....... ...log. sin. = 9-6000517 i= 81 36 40.6 19.5405607 165 45 51.4 |sum= 82 52 55.7 ...... ---- log. sin. = 9-9966401 \ sum 3 = 1 16 15.1 .......... log. sin. = 8.3459399 $ log. rad. =20 38.3425800 19.5405607 2)18.8020193 9.4010096 Now 9.4010096 is, the logarithmic sine of the arcs 14 34' 56".9 165 25 3.1 374 3456.9 &c. Now, if either of the two first arcs were taken, the star's longi- tude would be less than 9 signs ; whereas, since its right ascen- sion is 19 h 41* 45% it must be greater. Taking, therefore, the third arc, we have 90 + L = 749 9' 53 // .8 and L = 659 9 53.8 = 360 + 299 9 53.8, and thence, rejecting 360, we have the longitude = 9" 29 9' 53".8. 173 EXAMPLE 5. Required the longitude of a Pegasi in 1815. (see p. 164.) A = 70 35' 20" log. sin. = 9-9745846 1= 23 27 46.3 V-A-. log. sin. = 9.6000517 3= 75 47 12.7 (d) 19-5746363 2)169 50 19 \ sum = 84 55 9.5 - log. sin.= 9-9982902 sum = 9 7 56.8 log. sin. = 9-2006233 2 log. rad. = 20 39.1989135 (d) 19-5746363 2)19.6242772" 9.8121386 Now 9.8121386 is the logarithmic sine of the arcs 40 27' 15".5, 180 (40 27' 15".5), 360 + 40 27' 15".5, &c. Assuming the third^ for reasons such as are stated in the last Example, 90 + L = 2 X (400 27' 15".5) = 800 54' 31" and L = 710 54 31 = 360 + 350 54 7 Si"; and rejecting 360, the longitude = 1 1 1 20 54' 3.l"; 174 EXAMPLE 6. Required the longitude of 7 Draconis in 1750. (see p. 165,) A = 15 2' 37".l log. sin. 9.4142288 /= 23 28 18 log. sin. 9-6002054 3= 38 28 16 (d) 19.0144342 2)76 59 11.1 J sum =38 29 35.5 log. sin. 9.7940847.7 \ sum $ = 1 19.5 log sin. 6.5859420 2 log. rad. 20 36.3800267.7 19.0144342 2)17.3655925 8.6827962 Now 8.6827962 is the logarithm of 2 45' 40" and of (180 2 45' 40"), and if, for reasons such as have been alledged, we take the latter arc, we have 90 + L = 354 28' 40" and L = 264 28 40 = 8 s 24 28 40. 175 EXAMPLE 7. Required the longitude of Polaris in 1800. A = 23 55' 17".8 log. sin. = 9.6079754 /= 23 27 54.8 log. sin. = 9.6000930 $= 1 45 34.5 19-2080684 2)49 8 47.1 f sum = 24 34 23.5 log. sin. = 9.6189423 sum 5 = 22 48 49 log. sin. = 9.5885345 2 log. rad. =20 39.2074768 19-2080684 2)19-9994084 (log. sin. 87 53' 8") 9.9997042 = 175 46' 16", and L = 85 46' 16", or 2 s 25 46' 16". The longitudes and latitudes of stars are of some, but not of frequent use, in Astronomy. They are useful in the Theory of the Aberration of Light, and in certain methods founded on the occultations of stars by the Moon. They are also useful in the comparison of catalogues of stars made at different epochs, and afford us, as we shall hereafter see, the most direct mode of find- ing the quantity of the Precession of the Equinoxes *. There are certain angles, technically called Angles of Position, dependent, like the latitudes and longitudes of stars, on their right ascensions and declinations, and the obliquity of the ecliptic, and thence deducible. * Tfyere are Tables of the latitudes and longitudes of stars in Lalande's Astronomy , edition 3. In the Connoissance des Terns of 1788, for the Epoch of 1756 : and in that of 1804, for the Epoch of 1800. 176 Now, the angle of position of a star, or of any point in the Heavens, is the angle formed at the star or point, by the arcs of a circle of declination and of a circle of latitude passing through that star or point. In the subjoined Figure it is the angle TrsP. In the Chapter on the Aberration of Light, we shall see the uses of these angles of position. Our present business is concerning the method of computing them. Let P denote the angle TrsP,, then, (see Trig. p. 141.) sin. P X sin. $ = sin. sirP x sin. /; but (see p. 168,) sin. STrP cos. i/; therefore, to compute P, we have, sin. P X sin. $ = cos. L x sin. /, and, in logarithms, log. sin. P = log. cos. L -f- log. sin. I log. sin. $ ; or we may compute the angle of position thus, sin. P : sin. sPw :: sin. / : sin. A ; but (see p. 156,) sin. sPir = cos. JR; .'. sin. P X sin. A == + cos. M x sin. /, and log. sin. P = log. cos. JR -f log. sin. J log. sin. A . 177 EXAMPLE 1. Required the angle of position of a Arietis for 1815, (see p. 169,) log. cos. 1 s 5 4' 41" = 9.9129496 log. sin. 23 2? 46.3 = 9-600051? 19-5130013 log. shi. O 8 67 25' l".7 = 9-9653546 (log. sin. 20 39 52.3) == 9-5476467 Therefore the angle of position is 20 39' 52".3. EXAMPLE 2. Required the angle of position of y Draconis in 1815, (see p. 166.) In this Example we will use the second formula of com- putation log. cos. 268 2' 40".2 = 8.5255869 log. sin. 23 27 52.5 = 9-6000816 18.1256685 log. sin. 74 56' 56" 9-4144395 (log. sin. 2 56 53.2) 8.71 12290 The angle of position, therefore, is 2 56' 53".S. 178 EXAMPLE 3. Required the angle of position of Polaris in 1800, see pp. 167, 175. log. cos. 13 5'15"...> = 8.86770Q3 log. sin. 23 27 54.8 = 9.6000936 18.4678029 log. sin. 1 45 34.5 = 8.4872199 Gog. 72 59 39.3) 9-9805830 therefore the angle of position is 72 59' 39".3. If a star be situated on the solstitial colure, its right ascen- sion is either 90 or 270 : in each case, cos. JR. = 0, conse- quently, since sin. P. sin. A = cos. JR. sin. /, P = 0. 7 Draconis, as we have seen in p. 166, is very near to the solstitial colure (its longitude = 8 8 24 28' 40"), and its angle of position is less than three degrees. The mean right ascension of a star not continuing the same from year to year, and even the star's latitude and the obliquity of the ecliptic being subject to certain minute changes (secular variations) the angle of position must vary. Lalande's Astronomy, vol. I. p. 488, and the Connoisance des Terns for 1804, contain the angles of positions of several stars, together with their annual variations. The values of those angles thus then become known for several years adjacent to the year for which they are computed. The most simple method of computing the variations is to take the fluxion or differential of some expression involving P, M, S, &c. Thus, we have (see p. 176,) sin. P. sin. A = cos. M .sin. /; /. dP . cos. P sin. A -f d A . sin. P . cos. A = d JR. . sin. JR sin. / + dl . cos. JR. . cos. /. 179 If we neglect, by reason of their smallness, the second and fourth terms, there remains for computing dP, this equation, siu. JR. . sin. / dP cos. P. sin. A EXAMPLE I. It is required to find the annual variation of the angle of position of *y Draconis in 1815 (see pp. 166, 177) sin. M = sin. 268 4' 40" = - .9994 sin. / = sin. 23 27 46 = .398 cos. P = cos. 2 56 53 = .9986 sin. A = sin. 15 3 3 = .2597 and dJR (to be subsequently computed) = 20".7 ; EXAMPLE II. Required the annual variation of the angle of position of a Arietis, the epoch being the year 1815, (see pp. 160, 177,) sin. M = sin. 29 ll' 28" = .487 sin. / = sin. 23 27 46 = .398 cos. P = cos. 20 39 52 = .935 sin. A = cos. 9 57 35 = .985 and dM = 50 /7 .25 * ; Hence, the angle of position for the year 1 800, would be 20 39' 5^. 3 + 10". 5 X 15; that is, 20 42' 29 /7 .8. * The variation of the right ascension will be computed in a subse- quent Chapter. 180 This result does not exactly agree with that which is given at p. 431, of the Connoisance des Terns for 1804. The angle of position for 1800 is there set down at 20 42' 44". Part of the! difference between the two results arises from the obliquity of the ecliptic being assumed of different values in the two processes. M. Chabrol (the computer in the French Almanac) has assumed the value of the mean obliquity equal to 23 27' 58". If we take the secular diminution of the obliquity to be 45".7, then,, in fifteen years (the interval between 1800 and 1815) the diminution would amount to 6".85: consequently, the obliquity in 1815, on the above grounds,, would be 23 27' 58" - 6".85 = 23 P 27' 5l".15*, whereas (see p. 160,) we have assumed it equal to 23 27' 46".3. Now, if the obliquity be lessened, the other quantities, such as the right ascension and north polar distance, remaining the same, the angle of position will also be lessened : and its diminution may be computed from this formula, (see p. 178,) cos. JT . sin. A If we take d/=5", cos. ^l=.873, cos. J=r.917, we have dP = 5" x - X - 917 = 4".3. .935 x .985 This quantity added to 29 42 ; 29".8 (see p. 179,) will make the angle of position equal 29 42' 34". 1. * The mean obliquity for 1813 is stated by Mr. Pond, (Phil. Trans. 1813,) to be 23o 2r 5(fm Therefore if the secular diminution be 45".7, the mean obliquity for 1815 is 23 27' 49".196\ 181 By formulae, in principle like the preceding, may the latitude of a star computed for one value of the obliquity, be changed into the latitude due to another value of the obliquity, sup- posing the obliquity alone to vary and to vary by a small quantity. The latitudes and longitudes of stars, the angles of position, their annual variations, &c. are, as we have already said, mere matters of computation. They are useful, like other Astrono- mical formulae, in the elucidation of theories, in the succinct expressions of results, and in the construction of Tables. The quantities on which they depend, or from which they are derived, are the obliquity of the ecliptic, the right ascensions and declinations of stars. These latter are determined by observa- tions, not, indeed, from single observations, nor from those of one or two years, but from observations made at different and distant epochs and continued through a series of years. The longitude of a star may be computed in a few minutes ; but it requires the observations of fifty years to settle its right ascen- sion. We wish to make one remark more before we quit this subject. The preceding latitudes, longitudes, &c. are intended to be the mean latitudes and longitudes, and are computed from the mean values of the obliquity, and of the right ascensions and declinations. This, for the present, must be taken as a mere statement. We have not hitherto advanced far enough to give a distinct explanation of those mean quantities which are, indeed, (it may be here premised) the fictions of Astronomers : abstract quantities never seen nor observed, but which would be, if our theories be right, were certain obstructive or deranging circum- stances removed. In order to aid the computation of formulae by which the variations of the latitude, longitude, and angle of position of a star may be expressed, we subjoin a few formulae which the attentive Student, by the aid of the annexed references, may easily investigate : 182 P the angle of position, X the latitude of a star, L its longitude, 5 its north polar distance, JR its right ascension, / the obliquity of the ecliptic. Then, 1. tan. JR. = tan. L . cos. / tan. X sec. X sin. / - Trig. p. 117, 2. cos. 3 =sin. L . cos. X sin. J+sin. X cos. / 140 3. tan. L =sin. I cot. $ sec. JR -j- tan. JR. cos. / 157 4. sin. X = cos. $ cos. / sin. $ sin. /. sin. JR, 140 5. cot. P =cos. $. tan. JR, + sin. $ . sec. JR . cot. J. . . . 157 6. cot. P =cos. X sec. L cot, I sin. X tan. L 157 7. cos. JR. sin. S == cos. L .cos. X TWg. p. 141 8. sin. P sin. 3 == sin. J, cos. L 141 9. sin. P cos. X = sin. Z". cos. Jil 141 10. sin. JR = sin. L . cos. P + cos. L sin. P. sin. X. 11. cos. / cos. JR= cos. L. cos. P sin. L sin. P sin. L. CHAP. VIII. Comparison of the Catalogues of Stars made for different Epochs. The annual Increments of the Longitudes of all Stars nearly the same. The Precession of the Equinoxes. Comparison of the Latitudes of Stars computed for different Epochs. The Latitudes of Stars subject only to slight. Variations. Com- parison of the North Polar Distances and Right Ascensions of Stars. Suggested Formula of their Variations. Conse- quences respecting the Length of the Year, fyc. that follow from the fact of the Precession. IN the preceding Chapter, the terms Mean and Apparent Right Ascension, Mean and Apparent North Polar Distance, &c. have frequently occurred, without a formal definition of their meanings. Indeed, a definition is not easily given : for, in order to its being intelligible, it ought to enumerate the several cir- cumstances that make a star's mean place to differ from its apparent : which enumeration depends on what is to follow. The mean place of a star differs from its apparent place at a given epoch, not for one cause only, but for several. The mean place of a star at one epoch, differs from its mean place at another epoch, almost solely, from one cause : with the expla- nation of this latter point our course of explanation will begin. We will first shew that the place of a star is different in the year 1815 from what it was in 1760. The place of a star depends on its distance from the first point of Aries and from the pole of the equator : or (for so also may its position be determined) from the first point of Aries and the pole of ecliptic. We may determine then whether a star's place is changed or not, by comparing together its registered right ascensions and polar distances for two different epochs : or, by comparing together its longitudes and latitudes computed 184 (see pp. 160, &c.) from those right ascensions and polar distances. We will begin with the latter comparison, although it may seem to be more simple to compare together right ascensions and polar distances, which, indeed, may be considered as objects of im- mediate observation. - ^ -*. In order to deduce the variations of the longitudes and lati- tudes of some of the principal stars, we will compare our results (see pp. 160, &c.) with Delambre's Catalogue of Longitudes and Latitudes inserted in the Connoisance des Terns for the year 1756. The following Tables contain this comparison (see pp. 160, 161, &c.) LONGITUDES. Stars. 1815. 1756. Diff. of Long, in 59 Years. Mean Annua Increase. a Arietis. 1 s 5 4/41" 1 s 4 15' 3" 49' 38" 50".47 Pollux. 3 20 39 48 3 19 50 55 48 53 49.7 Spica Virginis. 6 21 15 31 6 20 26 20 49 11 50.1 a Aquila? 9 29 9 53.8 9 28 20 6* 49 57.8 50.8 a Pegasi. 11 20 54 31 11 20 5 19 49 12 50.1 Here it appears, notwithstanding the different positions of these stars with regard either to the pole of the equator or ecliptic, that their longitudes are increased by nearly the same * The star's longitude would be 9* 28 20' I'M, the elements of the computation being & = 294 43' 7".36 N. P.D = 81 45 27.92 and obliquity = 23 28 14-.86. 185 quantity. If we divide the numbers in the fourth column by 59 1, [the results will be (as they are expressed in the fifth column) the [mean annual increases of the longitudes. By the mere comparison then of the longitudes of stars at different epochs, we arrive at the important fact of the nearly \equal increases of those longitudes at the rate of about 50" mnually. We may account for it (or assign a probable reason for the fact) either by supposing the whole sphere, on which the I stars seem placed, to be slowly turned (in addition to its diurnal I rotation round the prolonged axis of the Earth) round an axis passing through the poles of the ecliptic, or by supposing the intersection of the equator and ecliptic, the first point of Aries as it technically is called, to have retrograded. This retrogradation (the fact in its relation to Astronomical calculations is the same in either supposition) is technically called the Precession of the Equinoxes*. Its mean value estimated from the five preceding stars is = 50".23. o But the Precession is an Astronomical element of too much importance to be estimated from a, few pbservations, or (we should say, if we did not know the past state of science) from observations not distant from each other by more than sixty years. If the observations of Hipparchus, who lived one hundred and twenty years before the Christian a?ra were as accurate as the observations now made, or as the observations made in Flamstead's time, we should be able thence to determine the mean quantity of the precession with the greatest precision. But the antient observations are very little to be relied on. t Since the number of years is 59 (= 601), we may easily com- pute the annual increase from the whole increase : thus, if the latter be 1 23' 34", ri' 23" 34"' 1 the former is / 1 23 34 \ ) 1 23, &c.V 1 24 58 59 A A 186 That they are inaccurate, we have evidence from the very state- ments that have come down to us from Hipparchus and Ptolemy' The longitude of Spica Virginis (see p. 171,) in 1815 was 6 s 21 15' 3 l"; therefore, since the longitude of the autumnal equinox is 6 s , it may be said, that the latter precedes the star by 21 15' 3l". Hipparchus (according to Ptolemy) says that, in his time, the star preceded the autumnal equinox by 6 instead of 8*, which it did, according to the observations of Timocharis, made in the year 2Q5 before our aera. Now, M. Delambre very justly observes, first, that these round numbers of 6 and 8 degrees throw great doubts on the precision of the observations ; secondly that the quantity 2 of the precession, at the rate 'of 50" annually would give an interval of time equal to 144 years instead of 160 or 170 years that intervened between the two observations: so that, it is probable, the observations made or computed were inaccurate to the amount of a quarter of a degree. Now such an error diffused over even as great an interval as 1800 years would still be of moment : it would amount to 0".833, and alto- gether vitiate the investigation. On this account it is better to compute the precession by comparing together observations that are now making, with observations made about the year 1750 by those distinguished Astronomers Bradley and Lacaille. And this M. Delambre has done ; by comparing a great number of his own observations with those of Mayer and of the two last-mentioned Astronomers, he finds the mean quantity of the precession to be 50".l. M. Lalande, in his Astronomy, has computed the precession, by comparing the longitude of Spica Virginis as assigned by Hipparchus with the longitude of the same star computed in 1750. Thus, 128. A.C. Longitude of Spica Virginis. . =5 8 24 0' 1750. A. D .=6 2021 difference of longitudes .....=: 26 2 1 Ptolemy lived in the year 137 of our sera. 187 Therefore the mean annual precession = 26 21' 1878 = 50" SO'" = 5CT.5. By a number of like comparisons, the same author finds the secular precession, that is, the amount of the accumulated pre- cessions for 100 years, to be 1 23' 34". The mean annual pre- cession corresponding to this is 50 // .34 ; and the sum of such annual precessions amounts to 1 in 7H- years. If we suppose the precession to be 50".l, then, in (360\ = j years, the fast point of Arfes will have retro- 5O. IS graded through an entire circle. The quantity 50''.1, which is the mean value of the precession, is obtained from the differences of the longitudes of a great many stars (three or four hundred for instance) computed at dif- ferent epochs. This mean quantity may not agree with the mean quantity derived from the observations of a single star, however many, or accurately made, those observations may be. It will not be the case with Pollux, the second star in the preceding Table. The difference, however, between the mean quantities of the precession, as they result from 300 stars or a single star, is, in all cases, very small. Still the difference, which is proved to exist, points out to some peculiarity in the single star. It cannot be, like most of the other stars, entirely fixed, but must have, what is called, (or what we are obliged to call from default of a know- ledge of its cause) a proper motion. For this reason, namely, that the mean longitude of a star is not altered solely from the regression of the intersection of the equator and ecliptic, or by the precession, Astronomers employ the term of Annual Varia- tion, comprehending under it the effect both of precession and of annual proper motion. This subject will be more fully treated of in a subsequent Chapter. The comparison of the longitudes of stars computed for the two epochs of 1750 and 1815 establishes, as we have seen, the important fact of the prece'ssion of the equinoxes. Let us now compare the latitudes. 188 LATITUDES. Stars. 1815. 1756. Diff. in 59 Years. a Arietis 9 57'S7".4N 9 57' 32" ' + 5".4 Pollux . 6 40 18.4 N 6 40 3 + 15.4 Spica Virginis. 2 2 24.8 S 226 + 18.8 a Aquilae .... 29 18 35.5 N 29 18 44* - 8.5 a Pegasi 19 24 40. N 19 24 44 - 4 It appears from this Table that the changes of latitudes are very small ; in no case amounting, annually, to 0".4. The Astronomical fact then is, a minute annual change of latitude with a considerable annual change of longitude. With regard to the cause of the former change, we may conjecture that it arises either partly from the precession of the equinoxes and partly from other causes, or, that is altogether independent of the precession. We shall consider this matter again ; at present we have not suf- ficient means to remove our uncertainty. In the mean time we will examine (what are indeed the foundations of the preceding Tables of longitudes and latitudes) the right ascensions and north polar distances of stars observed at the two different epochs of 1756 and 1815. To the former stars we shall add some others for the farther elucidation of the subject. * The latitude computed, from the star's JR =294 43' 1". 3d its N. P. D.= 81 45 27.92 and obliquity = 23 28 14.86 would be 29 1 8' 3Q".2. 189 NORTH POLAR DISTANCES. Stars. 1756. 1815. Variation in 59 Years. 76 10 25.85 67 42 12.86 74 14.89 75 50 40.6 6725 1.69 73 52 19.46 / " 19 45.25 -17 11.17 - 7 55.43 20.08 -17.4 - 8 a Arietis Aldebaran(aTauri) rj Geminorum 67 26 52 6727 1.18 + 9.18 f 0.15 Pollux (J3 Gemin.) 61 24 26.8 61 32 12.34 + 7 45.54 + 7.9 3 Ursae majoris ... 31 36 32.75 31 56 17-95 + 19 45.2 +20.09 Spica Virginis (a) 7 Draconis........ 99 52 46.76 38 2820.294 11954 15.535 81 45 27.92 76 6 11.51 100 11 28.9 38 29 5.05 119 53 35.28 81 36 40.54 75 47 12.84 + 18 42.14 -f 44.756 40.255 8 47.38 18 58.67 +19 -f 0.7'5 - 0,68 - 8.9 -19.3 S Sagittarii a Aquilae a Pegasi RIGHT . ASCENSIONS. Stars. 1756. 1815. Variation in 59 Years. Annual Variation. y Pegasi . . 10 27.511 55 46.95 45 19.44 46.09 a Arietis 28 22 7.201 29 11 27.3 49 20.1 50.1 Aldebaran 65 29 12.999 66 19 42.9 50 29.9 51.34 rj Geminorum . . . Pollux 90 2 14.924 112 35 16.989 90 55 39.44 1 1 3 29 39.6 53 24.51 54 22.61 54.3 55.3 8 Ursae majoris . . Spica Virginis . . . 7 Draconis 180 48 23.344 198 534.856 267 44 11.386 181 33 8.22 198 51 52.95 268 4 40.2 44 44.87 46 18.1 20 28.81 45.5 47.09 20.82 S Sagittarii ... . . a Aquilae 271 2038.827 294 43 7.36 272 16 42.9 295 26 17.7 56 4.06 43 10.34 57 43.9 a Pegasi 343 9 20.86 843 53 15.15 43 54.29 44.65 190 We will first examine the Table of North Polar Distances. The first star, 7 Pegasi, is subjected to the greatest diminution in north polar distance, a Arietis suffers less, and Aldebaran still less. The north polar distance of the fourth star (rj Geminorum) is augmented, but by a very small quantity. The north polar distance of Pollux is augmented by a greater quantity, and 5 Ursa majoris by the greatest (20".09). The north polar distance of 7 Draconis is very slightly augmented. Those of the remaining stars are diminished, and the last star (a Pegasi) suffers a diminution of north polar distance nearly equal to that of 7 Pegasi. Now the slightest inspection of the Table will shew us that these variations of north polar distances, whether we regard their quantities or their directions, are entirely independent of the north polar distances themselves. We must look, therefore, elsewhere for a clue to lead us to the detection of the law (if any such should exist) that regulates these variations of polar distance. If we look to the second Table we shall easily perceive a connexion between the above-mentioned variations and the right ascensions. For instance, y Pegasi which has the smallest right ascen- sion, suffers the greatest diminution in north polar distance. The change in the north polar distance of r] Geminorum is very small and positive, and its right ascension is a little more than 90. It is, therefore, immediately suggested to us that, if its right ascension had been exactly 90, its polar distance would have been unaltered. Again, of the following stars, the north polar distance of $ Ursae majoris is augmented by the largest quan- tity (by a quantity equal the diminution of the north polar distance of 7 Pegasi) and its right ascension a little exceeds 180. We have next 7 Draconis ; its right ascension is a little less than three quadrants, and the variation of its north polar distance small and additive; then, 5 Sagittarii, its right ascension is a little greater than three quadrants whilst the variation of its north polar distance is nearly equal to that of the former star but subtractive. The right ascension of a Pegasi is about seventeen degrees less than 360, and the variation of its north polar distance is of the same sign with that of 7 Pegasi, but somewhat less, 191 To those who are acquainted, in the slightest degree, with the properties of Trigonometrical lines, it will be obvious that the above variations are analogous to the variations of the cosine of an arc that passes through all its degrees of magnitude from to 360. Suppose, then, we should conjecture the variation in north polar distance to be comprised under this formula C . cos. JR, in order to examine the truth of the conjecture, we have (taking the mean of the two cosines), cos. 33' 6" = .9999 cos. 28 46 47 = .8764 cos. 65 54 27 = .4028 cos. 90 28 57 = .0083. Equate - C.cos. 33' 6", and - 20".08, and We have 20" .08 = ~9999 = 20 * 82 ' nearly ' and, accordingly, -20".082X .8764 = - 17".5, nearly, 20".082X .4082 = - 8.1 - 20".082 X -.0083 = 0".l6, which results agree, very nearly, with those of the last column of of the Table, p. 189. And if we were to make a like experi- ment of the truth of the formula C . cos. JR, with the remain- ing stars, we should find a like near* agreement between its results and the numbers of the above-mentioned last column. * We cannot expect an exact agreement. In order to try whether the conjectural formula (C.cos, JR) were true, we took the cosine of the mean of the two arcs and multiplied it into i ft part of the whole variation of north polar distance between the years 1756 and 1815. But this mode of proceeding was adopted merely, as we said, for the purpose of pro- curing a test of the truth of the formula. If C . cos. -#l be a true formula 20".08 x cos. 29 1 1' 27 ''.3 is the annual variation of a Arietis for the year 1815 : which will differ (a little indeed) from *- 7 ** ' 1? X cos. (6> 42' 12".86 -f- 67 25' l".o9). We have now then, from the mere examination of the regis- tered right ascensions and declinations of stars and their com- puted latitudes and longitudes, established, or rendered probable the existence of, three important facts. The first is, the nearly equal increases of the longitudes of all stars at the rate of about fifty seconds of space annually ; the second is, the very small annual changes of their latitudes, or the nearly permanent position of the pole of the ecliptic : the third is, the annual variations of the declinations of stars regulated, both in their directions and quantities, by the right ascensions. The variations of the right ascensions of stars (see Table, p. 18Q,) we have not examined for the purpose of finding out their law, which is not so obvious, nor so easy to be detected, as that of the variations in declination. We have, however, found out sufficient to enable us to make further conjecture concerning the cause (if there should be only one) which produces those variations that have resulted from the preceding investigations.. For instance, let TT be the pole of the ecliptic L, P the pole of the equator v Q. Let the retrogradation of the inter- section ( ) of the equator and ecliptic be expressed by T T ' - then Q the solstitial point, 90 distant from y* * must also regress to some point q, and the position of the solstitial colure, from 19S ,that of TrPLQ, will become that of vrplq. Now, it has appeared (see p. 179,) that the obliquity of the ecliptic is subject to very slight variation. It may be supposed nearly constant during small portions of time (during portions of a year, for instance,) in which case, P may be supposed to have regressed to p through a circular arc Pp, the radius of which is ?rP. Let cr be a star : P<7 is its polar distance when the pole is at P, per its polar distance when the pole is transferred to p: and per P & will be its change of polar distance arising from precession. The right ascension of the star, when the pole is at P, will be wQ, , and, when the pole is transferred to p, stq<^'. The increase, therefore, of right ascension arising from precession, will be st or st -f- tqv 4- v'v ~ wQ r f , i ( y v being perpendicular to v), or (since tqv = wQ) st + T ' We have then, on the supposition that the motion of the pole of the equator is rightly represented by the preceding scheme, a mode of computing the variation of right ascension. If its variation and that of the polar distance, computed according to the same scheme, should accord with the results of observations, there would arise a presumption that the scheme was true, or that it adequately represented the nature and law of the change of the intersection of the equator and ecliptic. We may add farther that, according to the above method of representing the change of the position of the equator, the lati- tudes of stars will remain unaltered : a consequence (see p. 188.) which accords, very nearly, with the results of observation. In a future Chapter we shall enter more fully into this subject. It is sufficient for our present purpose to have shewn, that the mere examination and comparison of registered obser- vations is sufficient to suggest the laws of the variations of the polar distances and right ascensions of stars and of schemes and BB 194 formulae for representing and computing them. Whether the; laws and formulae so suggested be true or not must be decided by the test of observations. We can do nothing else than ti whether or not the results of the formulae of computations accoi with those of observations. Science furnishes us no surer clu< than this to guide our researches. And Astronomy, exacted as il may now seem, is merely a system built up by like trials ai processes. Before we proceed to this verification, we wish to deduce a few results that necessarily follow from the Astronomical fact of the precession of the equinoxes. The intersection of the equator and ecliptic happens when the altitude of the Sun's centre is equal to the co-latitude of the place. The instant of time, therefore, of the intersection, or of the Sun's being in the equator is that, at which the above equality takes place. Now, with fixed instruments, it is the Sun's meridional altitude only which is observed. It may happen, indeed, but it is very unlikely to happen, that the Sun's meridional altitude shall be equal to the co-latitude of the place. The instant of time then of the Sun's being in the equator, must, in almost every case, be determined by computation : by observing one altitude less than the co-latitude, and, on the succeeding noon, an altitude greater than the co-latitude, and then by computing the time between the two successive noons, at which the Sun was at that intermediate altitude which is equal the co-latitude of the place. For instance, by Observations at Greenwich in March 1770, Z. D. Sun's Centre. Mean Time. March 20. ..... 51 29' 59".5 12 h 7 m 3GP.5 March 21 51 6 18.5.... 12 7 18.5 23 41 18 But latitude of Greenwich = 51 28' 38".5 Zenith distance of the Sun on 20th. .=51 29 59 . 5 1 21 195 Hence, (since the interval of time, corresponding to the change of 23' 4l" in the Sun's zenith distance, is 24 h 18 s = 86382 s ), we have 23'4l"( = 85260'") : 86382 s :: l' 2l" : l h 22 3'.2; consequently, at 12 h 7 ra 3&.5 + l h 22 m 3 S .2, or, at 13 h 29 m 39 8 .7 mean solar time, the Sun's zenith distance was equal 51 28' 38".5 ; or, in other words, the Sun was then in the equator. Find, by a like process, the time of the Sun's being in the equator in 1771, and the interval of the times is the length of an equinoctial year. Or, by finding the time of the Sun's being in the equator at some other epoch, in 1820 for instance, we may find the interval of time due to fifty equinoctial years, and thence the mean value of one equinoctial year. For instance, Sun's Zenith Distance. Mean Time. March 20, 1820. .51 32 7 52".5 12 h 7 m 37 8 .8 March 21 51 9 11.5 12 7 19.5 23 41 00 18.3 Sun's zenith distance 51 32' 52".5 Equator's zenith distance 51 28 38.5 O 4 14 Hence, 85260'" : 86381'.7 :: 4' 14" : 4 h 17 m 18". The Sun therefore entered the equinox on March 20, 1820, at l6 h 24 m 55 s . 8, and, consequently, the interval of time, between the equinoxes of 1770 and 1820, equals 50 years -f l6 h 24 m 55 9 .8 - I3 h 29 m 37M : now, out of the fifty years, twelve are Bissextiles, or contain 366 days, consequently, the above interval equals 50 X 365 d + 12 d 2 h 55* 18 9 .7, and one-fiftieth of this sum, or the mean value of one year equals 365 d 5 h 49 m 6*.374. 196 This is an easy consequence when the interval between two equinoxes is already known. But if, independently of previous results, we sought, by direct processes, the length of the year, we could more simply effect it, thus : Sun's Zenith Distance. Mean Solar Time. March 21, 1820 51 9' 1 1".5 12 h 7 m 19 S .5 March 21, 1770 51 6 18.5... 12 7 18.2 02 53 00 1.3 We must then enquire at what time on March 21, 1770, (civil time) the Sun's zenith distance was 51 9' ll".5 : because, the interval between two equal zenith distances observed in 1770 and 1820, and each equal to the latitude of the place, must equal the interval between any other two equal zenith distances observed, respectively, in 1770 and 1820, and towards the same equinox. The enquiry, then, is reduced to the finding of the time due to a decrease of 2' 53" in zenith distance. Now, (see p. 194,) the Sun's zenith distance on March 20, 1770, at 12 h 7 m S6 8 .5 was 5 1 ' 29' 59". 5 ; therefore the decrease in zenith distance in 24 h 18 s . 3 was 23' 4l". Hence, as before, 23' 41" : 863S1 S .7 :: 2' 53" : 2 h 55 m 14 S .94; consequently, the interval of time between two equal zenith distances of the Sun, near the same equinoctial point> in 1770 and 1820, is 365 d x 50 + 12 J + 2 h 55 m 16 s . 24, which is, nearly, the same result as was obtained before in p. 195. Instead of equal zenith distances of the Sun's centre, we may use (which will be a more simple operation) equal zenith distances of his upper limb, or of his lower limb : and, since the equal zenith distances may be apy where assumed (provided they are referred to the same equinoctial point) we may find, by pro- cesses like to the preceding, the length of the year from observed equal distances of the Sun's centre, or of one of his limbs, at or near the solstices. The following is an instance of the determination of the length of the year from observations of the altitudes of the Sun's upper limb at Paris. 197 Altitude of Sun's Upper Limb. March 20, 1672. . 41 43' 0" March 20, 1716. .41 27 10 41 27' 10" March 21, 1716. . 41 51 15 50 23 50 15' 50" But 7 r. x 24 h = 15 h 56 m 39 s , nearly. 23 50" Hence, since in this interval of forty-four years there were ten bissextile years, the whole interval between the equal altitudes of the Sun's upper limb is 44 X 365 d 4- 10 d 15 h 56 m 39 s , and one forty-fourth part, or the mean length of one of the years is 365 d 5 h 49 m s . 88. This value is different from the preceding one of p. 195 : and, if we were to select and operate on the observations of other epochs, the resulting value of the length of a mean year would, most probably, differ from both of the preceding values. The difference is too considerable to be attributed to the errors of observation. There exists, as it will be shewn in the solar theory, a real difference, which arises from the motion of the Earth in an ellipse, the ellipse itself' being moveable. Since then the Sun, after quitting the equinoctial, or the solstitial point, does not continue to return to the same points in intervals of time exactly equal, the lengths of all real equinoctial years or tropical years, as they sometimes are called, cannot be equal. The preceding value then of the length of a year, (see pp. 195, 197,) whether it be the fiftieth or forty-fourth part of the time absolved between similar positions of the Sun, ought not, if we would preserve the analogies of language, to be called the length of a mean solar year. It may be called, as, indeed, it usually is, the mean length of an apparent solar year. The length of a mean solar year, if we would derive it solely from observation, must be obtained from the comparison of observa- - 198 lions, distant from each other by an interval of time equal to that in which the apogee of the Earth's orbit progresses through 360*. But, as it will be shewn hereafter, there are other means of computing the length of a mean solar year, than those which are founded on the comparison of observations separated from each other by so long an interval. M. Delambre estimates the length of the year at 365 d 5 h 48 m 51 8 .6 (=365.226396593684). A sidereal year is the time elapsed from the Sun's quitting a particular star to his next return to the same star ; or the in- terval between two successive periods, at each of which the difference of longitude between the Sun and a star was the same quantity. This year must differ from the equinoctial year, by reason of the precession : it must be greater than the latter year by the time taken up by the Sun in describing 50".l. The length of a sidereal year, then, can be determined by no method more obvious or more correct, than that of adding, to the com- puted length of Jthe equinoctial year, the time of describing 50".l : or, -which rests on the same principle, than that of finding the time of describing 360 from the previously ascertained time of describing 360-50".l. Thus, T denoting the length of the equinoctial year, -...," 360 -oOM the length of the sidereal year : which therefore equals 365 d .2236396593684 x __!^6000_ 1295949.9 or 365 d 6 h 9 m 11 8 .5, the difference, therefore, of the length of a sidereal and equinoc- tial year is 20 m 19 s .9- " The progression of the apogee is not the sole cause of the inequality of solar years (see vol. II. pp, 453, &c.) 199 But as it appears by p. 198, the length of a sidereal year may be immediately determined from observations. Thus, April 1, 1669, at O h 3 m 47 s , longitude of Procyon Sun's longitude = 3 9 8 59' 36", In April 2, 1745, at ll h I0 m 45 s , there was the same differ- ence of longitudes. The interval of the two epochs is 76 years (18 of which were bissextiles) -f l d 1 l h 6 m 58 8 , or 27759 d 1 l h 6 m 58 s , But since an exact number of sidereal years had elapsed, which number can be no other than 76, we have the mean length of one year equal to In the Chapters on the Solar Theory and the Calendar, two other kinds of years the Anomalistic and the Julian will be con- sidered, and their lengths found. Indeed, in now treating of the equinoctial and sidereal years, we have anticipated what will be the subject of future enquiries. But the quantity of the pre- cession of the equinoxes being found, the digression towards the subject of the equinoctial ^ears was natural and of little dif- ficulty. But there are subjects of primary importance that now claim our attention. If the Astronomical doctrine ' that there are fixed stars,' be true, it must be shewn why the apparent places of such stars do not remain the same : why the apparent place of a star is sometimes not the same on two successive days ; in different months of the same year ; on the same days of different years. We must, in fact, go into that investigation, the entrance to which was merely pointed out in pp. 44, &c. The objects of investigation are the causes that render unequal the places of stars at different temperatures, in different seasons, in different years, and in different positions of the observer. The know- ledge of these causes, and the determination of their quantities and laws, constitute, what is called, the Theory of Corrections, 200 by which an observed, or apparent place of a star is reduced to a mean place ; and by which, if we illustrate it by an instance, an observation of a Aquilae made at Palermo on April 1, 1800, may be compared with an observation of the same star made at Greenwich on June 15, 1819. CHAP. IX. THEORY OF CORRECTIONS. On the ^Corrections to be made to the observed or Apparent Right Ascensions and Declinations of Stars in order to reduce them to the Mean. Refraction. Aberration. Preces- sion. Inequality of Precession. Nutation. Parallax. THE inequalities that cause the apparent places of stars to differ, sometimes from themselves and always from their mean places, are not discernible without the aid of instruments. They are exceedingly minute, and their existence does not affect that explanation of the general phenomena of the Heavens, (see pp. 7, 8, &c.) which is founded on the rotation of the Earth and the permanence of the places of certain stars on the apparently concave surface of the Heavens. To the fact of the mere rotation of the Earth round its axis, which is sufficient for the general explanation of phenomena, we must add another, quite essential to their minute and particular explanation, which is the uniformity of the Earth's rotation. This latter is a fundamental principle admitting neither great nor slight modifications, and proved to be true by its being the basis of a large class of Astronomical calculations, and by the agree- ment of their results with observations. We cannot add to this, as fundamental principles of the same kind and invariability, the parallelism of the Earth's axis of rotation and the permanence of the places of the fixed stars. Both these latter principles are very nearly, but not exactly, in- variable. They require certain modifications which it will be part of the business of the present Chapter to explain. cc 202 The fixed stars are so called from preserving (what nearly takes place) the same distances from each other. If their mutual distances were strictly invariable, still it would not follow that their places were invariable: for, as we have seen, (see pp. 46, 152, &c.) the place of a fixed star is referred to two points, one, the pole of the equator, the other, the intersection of the equator and the ecliptic ; which two points are not fixed. The inequalities that arise from the motions of the pole and the equinoctial point affect a star's place, and continue to do so by the same kind of effect. That is, if the north polar distance of a star be diminished from March to June, it will be still farther diminished in the next October, and continue to be diminished during succeeding portions of time. There is no periodical variation nor recurrence of effect, except after exceedingly long periods. But there are inequalities, if we may so express our- selves, of a simpler kind, that affect, and irregularly, the star's polar distance : on one day diminishing that distance by a certain quantity : on the next, perhaps, by a larger quantity, and on the third day, perhaps, by a quantity less than that of the first diminution. The inequality we allude to is refraction; depending, at a given altitude of the observed star, on the weight, tempera- ture, and probably, moisture of the amosphere, and, consequently, to be computed by means of the barometer, thermometer, and hygrometer. We will begin with this inequality : first explaining, in a general way, its cause, and then establishing its existence as a phenomenon, by means of one or two simple observations. The atmosphere which surrounds the Earth is to be con- sidered as a medium of variable density, decreasing as the distance from the Earth's surface increases. A ray of light then from a star, in its progress through the atmosphere to the eye of the observer, continually passes from a rarer to a denser medium, and, consequently, according to optical laws, is continually de- flected. The deflections take place towards a perpendicular to the medium at the point of the light's impact, and, consequently, the variation of the density of the atmosphere being gradual, the 203 path of the star's light through the atmosphere will be curvilinear and, if we may so express it, convex towards the star. The star, then, will seem to be in the direction of a tangent to that last portion of its curvilinear path which enters the eye of the observer : and, consequently, the star will appear to be elevated above its true place. This deviation must take place in a plane perpendicular to the atmosphere, and passing through the star and the spectator. At each point of the light's progress, the medium to the right and left, in the direction of a perpendicular to the above plane, being supposed, for small distances, to be the same, no lateral devi- ation can take place. Hence the refraction takes place, and entirely, in the plane of a vertical circle. The vertical plane becomes the plane of the meridian when the star is to the north or south of the spectator. Hence, in observations made with a mural quadrant or circle, the whole effect of refraction takes place in declination, whilst the right ascension continues unaltered. The observations, therefore, made with a transit instrument are independent of refraction, so that, if its middle wire be meridional, the time of a star's transit will be the same at whatever point of the wire it passes (see p. 83.). This is a brief explanation of the inequality of refraction from a consideration of its cause. We will now shew, by a simple instance or two, its existence as a fact or phenomenon. If the angular distance of Polaris and 7 Andromeda (both being below the pole) be observed at Cambridge, it will be found to be about* 46 4 1 7 : but if the same stars be observed when y Andromeda passes the meridian to the south of the zenith, its distance from Polaris will be greater than before, by about eight minutes, and be very nearly 46 50'. If similar observations be made at Paris of the same stars, then, in the first case, their distance will be about 46 22', and, in the latter, about 46 45'. * The variations in the state of the air prevent the distance from always being the same. 204 We have then the angular distance of two stars (which ought, were their light not impeded, to remain unaltered) represented by four different instrumental angles. But the inequalities in these angles are perfectly explicable on the principles that have been laid down. For instance, by the observation at Paris, the distance of the two stars, below the pole, was 46 22', whereas it was 46 4l' at Cambridge. But the latitude of Paris being 48 50' 13".3, and the north polar distance of y Andromedae 48 33' 12", the latter star, when on the meridian below the pole, would not be much more than 1?' above the horizon ; whereas, in a similar position at Cambridge, the latitude of which is 52 13' 24" the star's elevation would be more than 3 40 ; . Now, at the former elevation, the light from y Andromedae would suffer much greater refraction than at the latter, and, consequently, the apparent distance between it and Polaris would be less at Cambridge than at Paris, although Polaris itself would be a little more elevated by refraction at the latter than at the former place. The other cases admit of a like explanation ; the distance between the stars when they are above the pole must be greater than when below, because y Andromedae, in the first case, is not more than 10 from the zenith when its light will not suffer much refraction ; whereas, in the latter, it is very near the horizon when its light will be refracted as much as it can be. It is easy to find other instances of like nature : for example, in the Greenwich Observations of Dec. 8, 1815, we have Barometer. Thermo In. meter. Out. Zenith Distance. 29.88 29.94 29.88 29.96 27 29 27 28 24 21 23 25 Polaris 36 50' 25" 4O 10 44.4 53 35 37.8 23 25 17.9 Polaris, S. P ft Ursae minoris, S. P. /3 Ursae minoris 205 Now (see p. 129,) half the sum of the greatest and least zenith distances of a circumpolar star, is equal to the co-latitude of the place of observation. Hence the co-latitude of Green- wich from the above two observations of Polaris is \ (77 l' 9".4), or 38 30' 34".7, from the observations of /3 Ursae minoris, (77 0' 55".7), or 38 30' 27".85 : the co-latitude, then, which should be an unalterable quantity, is represented, by reason of some inequality, by two different quanti- ties. But of this circumstance, as of the former, the explanation (at least the general explanation) easily follows from the cause that has been propounded. Each star, in both its positions, will be apparently elevated by the deflection of its rays ; its two zenith distances, therefore, will be less than they ought to be, and, con- sequently, half their sum, which is to represent the co-latitude, will be less than its true value. Again, the defect of this half sum from the true value of the co-latitude is greater in /3 Ursae minoris than in Polaris ; because the former star in its greatest distance from the zenith is distant from it nearly 54, and there- fore the course of its light is much more bent than that of the light from Polaris. But it is not only that different circumpolar stars give, ac- cording to the preceding method, different values for the latitude, but the same star will, on two different days, give different values. Thus, by the Greenwich Observations of 1812. Barometer. Thermc In. >meter. Out. Zenith Distance, Oct. 14. 28.82 29.10 50 47 47 45 a Cephei. . . . a Cephei, S. P. 10 19' 15".6 66 41 7.9 Oct. 16. 29.50 50 47 a Cephei .... 10 19 17.2 42 39 a Cephei, S. P. 66 41 3.6 Half the sum of the zenith distances on Oct. 14, is 38 30' ll".7 _ on Oct. 16, is 38 30 10.4 206 Two results not only differing from each other, but from those that were given in p. 205. But here, as before, the differ- ences in the results may be probably accounted for,, if we admit the preceding principles of explanation. In the first place, a Cephei at its lowest altitude will be much nearer to the horizon than /3 Ursae minoris, and, therefore, will be more elevated (more than proportionally elevated). In the second place, the observations of a Cephei, below the pole, on the sixteenth arid fourteenth, were made under different circumstances. These different circumstances are the weight and temperature of the air. On the latter day the barometer was four-tenths of an inch higher, and the thermometer six degrees lower than on the former day. For both reasons then (for each instrument shewed the air to be denser) the star would be more elevated, and its zenith distance more diminished, on the 16th than on the 14th; which is the fact shewn by the observations. If the above were a solitary instance, no great reliance ought to be placed on the preceding reasonings ; for, the errors of one or two seconds of space may be the errors of the instrument, or of the observer. But the Greenwich Observations contain numerous similar instances, all tending to the same conclusion and to exclude the supposition of instrumental or accidental error. This inequality of refraction then causes the north polar distances, and the zenith distances of stars to be apparently un- equal on contiguous days. The law of the inequality is only that which can be inferred from experiments and observations on the state of the atmosphere. The knowledge of the theory of refraction, and of the ex- pression of its laws by formulae, enables us to divest observations in altitude of one kind of inequality. Two north polar distances or two zenith distances, then, of the same star on two following days, or on days distant from each by short intervals, if unequal by the instrument, ought when corrected for refraction, to appear equal. In so small a portion of time, as that of two or three days, the effect of other inequalities would not be sensible by the instrument. Again, two observations of the zenith distances of the same star, made at the interval of two, three, or four months, and 201 corrected for refraction, if then unequal, would be so from some other cause or causes. If we exclude, by our supposition, error from the instrument and observation, and suppose the correction for refraction properly made, two such observations as those that have been just mentioned would be unequal from precession. For, if this latter inequality causes, as it does in certain stars, an annual variation of 20" in their north polar distances, it would, supposing its operation uniform, cause a variation of 10" in half a year, and of ,5 /; in three months. These two inequalities, then, of refraction and precession being known and their effects rescinded, if the zenith distances of the same star, observed at the interval of three, or of six months, should still be unequal, it would be necessary to investigate the source of the inequality. By such steps as have been described, that is, by cor- recting observations for all known and ascertained causes, and then by comparing the observations so corrected, Bradley found the north polar distances and right ascensions of stars to be different at different parts of the same year, but to be the same again in like periods of different years. If the north polar distance of a star were increased in March, it would be diminished in September. In June the inequality would affect the star's right ascension, and if it then increased the right ascension, it would in January, diminish it. This inequality is now succinctly designated by the term Aberration. Bradley discovered its cause, (and the discovery is altogether a wonderful one) in the combination of the motion of light with the motion of the Earth in her orbit. His explanation is founded mainly on the fact which Roemer had established from observations of the eclipses of Jupiter's satellites, namely, that light is not instantaneously transmitted but successively communicated and propagated, or that some portion of time is necessary, (no matter how small that portion,) for the light to pass along so short a space as that, for instance, of the tube of a telescope. The method of detecting a fourth or fifth inequality, would resemble the method alfeady used. If observations of y Draconis made during March 18 J 5, and corrected for the inequalities of refraction, precession, and aberration, differed from the corrected 208 observations of the same star made during July 1820, that is, if its north polar distance and right ascension were represented in the two periods by different quantities, there would be some new cause or causes of inequality to be detected. Such would be the fact. Observations, like those described, would not agree by reason of an inequality called Nutation. We will briefly explain its cause. We have hitherto (see pp. 185, &c.) viewed the precession of the equinoxes as an Astronomical fact, established by the comparison of the longitudes of stars at different epochs. This effect may be conceived to take place, from the intersection of the ecliptic and equator being endowed with a regressive motion, describing, thereby, gradually and equably the whole arc of the precession. Now the cause of this regression is the action of the Sun and Moon on the bulging equatoreal parts of the terrestrial, spheroid. The actions of these two luminaries vary, that of the first from its declination, that of the latter from the inclination of its orbit to the ecliptic. From the variableness, then, of these two actions there will arise two inequalities, one called the Solar Inequality of Precession, the other the Nutatioti *. When the preceding inequalities are known, a star's apparent place may be divested of their effects and reduced to its mean place. But another correction is still wanting in the case of the Sun and planets. An Observer at the Cape of Good Hope, and an Observer at Greenwich, would, on the same day, refer a planet to different parts of the Heavens. But observations, made at the above-mentioned places, ought to be true to all the world. Each observer then applies, in addition to the five enu- merated corrections, a sixth correction, and reduces his observa- tions to the centre of the Earth. This last correction is called * In the Tables published by Dr. Maskelyne in the Greenwich Obser- vations, the Nutation is separated into two equations : one called the Equation of the Equinoxes, the other deviation in right ascension, and deviation in north polar distance. 209 Parallax : it cannot be said to arise from any inequality : but is used for the sake of simplicity and the convenience of Astrono- mical computations* In simple cases the effect of parallax may be easily shewn. A spectator at A would see a star s, in the direction Asn. Another spectator, situated in the point at which Cs cuts the Earth's surface, or situated at C the Earth's centre, would see the star s in the direction Csm. The difference of the star's places, seen from A and C, is measured by the angle AsC called the Parallax. This is the parallax arising from the situation of the observer on the surface of the Earth. But there is a parallax called an Annual Parallax ; which is the difference of the places of a star seen respectively from two opposite points of a diameter of the Earth's orbit. We have now enumerated, and briefly described, the causes of those corrections by which the apparent places of stars at one epoch may be reduced to their mean places, at the same, or at any other, epoch. The first, refraction, is independent of the time of the day and year, and varies with the state of the atmo- D D 210 sphere and the, altitude of the observed body. A knowledge of its laws enables us to translate the angular distance, shewn by the Astronomical instrument, into another distance, such as the instrument would shew, did light pass through a perfectly per- vious metlium. The next inequality is that of precession, which changes the place of a star by changing, relatively to the Heavens, the place of the observer. It depends on time, inasmuch as, if it augments the north polar distance of a star in May 18 16, it will still farther augment it in December, and still farther after an increased lapse of time *. The former inequality affects only the declinations of stars^ but this alters both their declinations and right ascensions. The third inequality _, Aberration, depends not on the year, but on the time of the year. If it diminishes the right ascension of a star in May 1, 1816, by a certain quantity, it will equally diminish it in May 1817, and at similar times of succeeding years. In the month of October it will augment the right ascension : and, if in these months of May and October,, its diminishing and augmenting effects on the star's right ascension are the greatest, in the intermediate months of August and January/ its greatest effects will be on the star's declination. The term Nutation was originally meant to be significant, agreeably to its import, of a like motion in the pole of the equator, produced by the variable action of the Moon on that protuberant shell of matter by which the Earth is made greater than a sphere. The variableness of the action depends on the Moon's distance from the equator : which depends,, for its mean quantity, on the inclination of the lunar orbit to the ecliptic : which again, as it will hereafter appear, depends on the longitude of the node of the Moon's orbit. The inequality, then, of nutation will be the same, when the longitude of the node is : it will vary, whilst the place of the node continues (which is the fact) to regress, and will have experienced all its vicissitudes of augmentation, max- imum, diminution and mean state, during a period of regression ; which period is about eighteen years and an half. * We exclude from this statement all extraordinary cases of exception. It is proposed to consider each of these inequalities in a separate Chapter, and to obtain (except which we do nothing) their mean values, and the formulas of the laws of their variations. These being obtained, Tables may be constructed which will con- t veniently exhibit, at any given epoch, the respective quantities of the several inequalities affecting any particular star. The numbers in the Tables are, technically, called Corrections, and, when they are applied, with their proper signs, to the apparent places of stars, the latter are said to be reduced to their mean places. The apparent places are what the observer sees and what his instru- ments shew to him. The mean places can never be the objects of observation; but are, as it has tyeen already said, abstract quantities, the results of computations, the last conclusions which Astronomical Science, in its progress towards perfection, has arrived at. CHAP. X. REFRACTION. Refraction. General Explanation of its Effects. Computation of its Effects on the Supposition of the Earth's Surface being plane, and the Lamina of the Atmosphere parallel to it. Error produced by that Supposition at 80 Zenith Distance. Tycho Brake's, and Bradley's Methods of determining the Refraction. Method of determining the Refraction by Obser- vations of Circumpolar Stars. Different Formula of Re- fraction. Dependence of the Value of the Latitude of the Observatory on the Mean Refraction at 45 of Zenith Distance.- Corrections of Refraction due to the Thermometer and Barometer. Instances and Uses of the Formula and Tables of Refraction. Explanation of certain Phenomena arising from Refraction. IN the preceding Chapter, we have explained, in a general way, how stars become apparently elevated by the deflections of their rays of light in their passages through the atmosphere. The general effect is the same, whether the atmosphere be of an uniform or of a variable density. Suppose s to be a star, and that a ray of light sp falls on Mp m> the boundary between a denser and a rarer medium, the former being beneath Mpm. Let Pp be a perpendicular to this bounding surface ; then, the ray instead of pursuing the course spx, is deflected at the pointy, 213 into the direction/? a. Let Nn be a second boundary similar to the former : then the ray, after the first refraction at p, instead of pursuing the course pay, is deflected, at a, into that of a b. Again, the ray is, a third time, deflected at b from the course a b z into be. The eye of the spectator, supposed to be at c, sees the star in the direction of cb. The inclination of the lines c b and p s is the whole refraction, or deflection, which the ray has undergone. In what has preceded, the medium contained between Mpm and Qcq has been parcelled out into different strata. But cir- cumstances, similar to those that have been described, would take place, if the medium had been distributed into a greater number of strata. The deflections would have been more, but all the same way : and, if we suppose the parcelling out of the whole medium between Mpm and Qcq into minute portions to be indefinitely continued, the course of the ray pabc will become curvilinear, of which pa, a b, be, &c. are the elements. We may, therefore, thus represent the course of the ray. Let 5 be the star, c the spectator, and suppose a plane perpen- dicular to the Earth's surface to pass through 5 and c; then (see p. 203.) the media at the several points, on both sides of the plane, being supposed to be the same, the refraction will take place entirely in the plane j that is, entirely in the plane of a 214 vertical circle. The refraction also taking place, at every point of the light's course, on the supposition that it is made through a medium of a continually varying density, the path of the ray of light will be such as spa be is-, and, cs', being a tangent to such curve at its extreme element at c, will be apparently the star's direction. Or, / will be the apparent place of the star s, and the angle scs 1 will be the whole refraction. Let A = the angle sch, equal to the star's elevation above the horizon, then, if r be the refraction due to that elevation, the star's apparent elevation = A -\- r ; and, if = 90 A, Z being the zenith distance, the star's apparent zenith distance If the star be in the zenith, r = ; since the light, in its descent, cuts the tangent plane of each succeeding stratum of the atmosphere perpendicularly; consequently, there is no reason why it should be deflected towards one part rather than towards any other. In the zenith then there is no refraction, in the horizon, the greatest. In intermediate points, the refraction is of some mean values, but not proportional to the angular distance of those points from the horizon. The question we have now to consider is, since the refraction varies with the star's elevation, that is, since it is greater (other things being equal) the less the zenith distance, what function of the zenith distance, or, what terms involving the sine or tangent of such zenith distance, will repre- sent the law of refraction. This is one part of the enquiry ; the other part respects the actual quantity of refraction at some certain zenith distance, at 45, for instance, and at some state of the air, held to be its mean state. We have already in a previous Chapter, (see pp. 203, &c.) established the general fact of refraction, and the facts of the in- crease of refraction with the increase of zenith distance, and of the small quantities of refraction at zenith distances less than 90. On this latter fact as a condition, and the constant ratio existing between the sines of incidence and refraction, we will found a simple formula * of atmospheric refraction. ' This is principally taken from Dr. Brinkley's Investigations, (see Irish Transactio?it>, for 1814.) 215 Let O be the Earth's centre, OC its radius, a b a boundary between two media (such as were spoken of in p. 213,) sa a ray of light refracted at a into the direction Ca. Let Oa = + $, $ very small relatively to a, z. ZCa = 0, / = angle of incidence . R == angle of refraction, r = refraction measured by s a s , and sin. I = w.sin. R ; then sin. J = w.sin. OaC = m sin. OCa. OC - Oa =. m sin. ^ T 1 J , nearly, But sin. I = sin. (.R + r) = sin. 1^ cos. r -j- cos. U .sin. r = sin. .R + r sin. l" . cos. R, since, r being very small, sin. r = r sin. 1 ;/ , and cos. r= 1, nearly. Substitute now for sin. /, sin. R, cos. R, and we have 216 ro.sin. Z. ( 1 -- ^ = sin. Z. (l -- ) V / V as + r. sin. 1" . \/[l - sin. 2 Z (l - jj)*] ; but 1 - sin. 2 Z . (l - ~ a )* = 1 - sin. 2 Z.(l-^y, nearly, * = cos. 9 Hence, the square root = cos. Z x fl + - tan. 2 Zj , nearly, and (m 1)A J .sin. Z V d/ r = sin. l ;/ .cos. Z. (l +- tan.* \ a (m T . .tan. Z X ( I tan. 2 Z ) , nearly, sin. 1 \ a / sm. 1* sin. 1 .a or, since tan . tan. JS + tan. 3 Z = tan. Z sec.* , m 1 _, m 1 5 r = : - . tan. Z -- : - . - . tan. Z . sec.* Z. sm. 1 sin. 1 a If the terms (see 1. 2,) had been farther expanded, terms involving tan. 5 Z, &c. would have been introduced into the preceding formula, which would then have been of this from r = A tan. Z + B tan. 3 Z + C.tan. 5 Z + &c. In the former expression, if a be made infinite, the second term vanishes, and other terms, had they been introduced, would 217 also have vanished, since they would have involved the power* of - . In this case then a m - I r = -jf . tan. Z ; sin. 1 and this is the expression for the refraction, supposing the Earth's surface to be a plane, and light to be transmitted through a stratum of uniformly dense air parallel to the Earth's surface. But, if the Earth were a plane, and light were transmitted through a number of parallel strata of increasing densities, the refraction would be the same, as if the light, with its first angle of incidence, impinged immediately on the last stratum of air, or on that which is nearest to the Earth's surface. The refraction in that case would be represented by m - 1 77 . tan. Z. sin. 1 The other terms of the series, therefore, arise, from the spherical form of the Earth, and from the supposition of concentric laminae of the atmosphere. Let us estimate the value of the second term, namely, of m 1 o . JT . - . tan. Z < sec. Z, sin. l" a when Z = 80. Let $, the height of an uniform atmosphere of the same density as at the Earth's surface, = 5.095 miles, a, the Earth's radius,, = 3979. m (the ratio of the sines of incidence and refraction, the baro- meter being =29.6, and the thermometer = 50) = 1.0002803 ; ) then, since - = .00128, nearly, a E 218 we have *^_HJ . ^ tan. Z . sec. 2 Z = 13".92, nearly, sin. 1 a In computing then the refraction, on the supposition of the Earth being a plane, we fall, at 80 zenith distance, into an error of about 14", the first term of the refraction being 5 7 27".9- At 45, when tan. = 1, the first term, namely, and the second. Tl . - tan. Z sec. 2 Z = 0".148. sin. 1 a Hence the mean refraction (barometer = 29.6 inches, and thermometer = 50) is equal to v i 57 // .817 - 0".148 = 57".67, nearly. At distances from the zenith less than 45, the second term will bear a still less proportion to the first term : so that, we may safely conclude, for all zenith distances less than 45, the refraction will vary nearly as the tangent of the zenith distance and its mean quantity will be expounded by the term Tf . tan. Z, equal to 5 7". 82 . tan. Z. sin. 1 vn 1 In determining the value of the coefficient : 77 , no refer- sm. 1 ence has been made to astronomical refractions. The value o m was assumed equal to 1 .0002803, which value was taken from certain direct experiments on the refractive power of air. We shall, however, see that the observations of circumpolar star * Log. .0002803 = 6.4476251 log. tan. 80 ' '.;.''.''. :. ''.'".; . ..=10.7536812 2 log. sec. 80 .v/.. ; . =21.5206596 log. .00128 = 7.1072100 arithmetical complement of log. sin. 1" = 5.3144251 41.1436010 tak away 40 and 1,143601 = log. 13.919. 219 will enable us to compute the coefficient jj , and also the sin. 1 coefficients of other terms, supposing the refraction to be repre- sented by A . tan. Z + B . tan. 3 Z + C tan. 5 Z + &c. We will now briefly describe the methods by which Tycho Brah6 and Bradley determined, astronomically, the quantities of refraction. Let H denote the latitude of the Observatory, / the obliquity of the ecliptic, S the polar distance of a circumpolar star, Z, Z' its two apparent meridional zenith distances, S the Sun's apparent summer solstitial zenith distance, S 1 , his winter; if p, p', r, r 1 , be the quantities of refraction due, respectively, to the last apparent distances, then (see pp. 129, 140, 145, &c.) H - I = S + r, X^? H + I = S' + r', 180-2H= Z + p + Z' + p', adding these three equations together, we have 180 = S + S f + Z + Z' + r + r' + p + p'. If the refraction varied as the tangent of the zenith distance, or (see p. 217,) could adequately be expressed by .A. tan. Z, the first term of the series, we should have, by substituting in the preceding equation, 180 = S + S' + Z + Z'+ A (tan. S + tan. S' + tan. Z + tan. Z\ from which equation, A would immediately become known, since S, S', &c. are known from observations. But the first term, A tan. Z, of the formula of refraction will not represent the refraction with sufficient exactness when the observed star is far from the zenith, 220 The Sun, for instance, at the winter solstice, if Greenwich' be the place of observation, will be distant from the zenith by 51 28' 39". 5 + 23 2?' 50", or nearly by 75. At such a distance, in order to represent the refraction with sufficient exact- ness, we must take account, at least, of the second term of the formula. If B. tan. 3 Z represent that second term, the preceding equation of p. 209, 1. 24, will be augmented by B (tan. 3 S + tan. 3 S' + tan. 3 Z + tan. 3 '), in which case, if we had observations of only one circumpolar star (the pole star, for instance,) we should have one equation involving two indeterminate quantities A and B. We cannot, therefore, by the preceding method, and with the aid of only one circumpolar star, determine the formula of refraction, if we suppose it to consist of two terms. If we compute B from the formula of p. 216, supposing m to be known, by direct ex- periments on the refractive power of air, it will be equal to 0".073, nearly. Since A 180 - (S+S' + Z+Z') - B(tan. 3 S + tan. 3 S'-ftan. 3 Z + tan. 3 Z 1 } tan. S + tan. S' + tan. Z -f- tan Z' if we attribute to B certain small values, such as 0".05, 0".1, &c. we may deduce, from the above expression, corresponding values of A. But, with these changes in the values of A and B, the latitude will also vary : for (see p. 219.) 2//= 180 - [Z + Z'+A (tan. Z + tan. Z 1 ) + (tan. 3 Z + tan. 3 ')] j consequently, 2 d H = dA (tan. +tan. ') - dB (tan. 3 Z + tan. 3 Z'). From such expressions, and from Bradley's Observations (observations determining the values of Z, Z 1 , S, S') M. Delambre has formed a small Table exhibiting the alterations which will take place in the resulting values of the latitude, from the differences of values assigned to the coefficient of the principal term of the formula of refraction. This Table is subjoined. 221 B. ^. H. dA. dH. Refraction at 45. 0".05 56". 92 51 28'39".65 f 0".33 56".87 .10 57.36 39.32 0".44 0.32 57.26 .15 57.81 39.00 0.45 0.33 57.66 .20 58.25 38.67 0.44 0.33 58.05 .25 58.70 38.34 0.45 0.33 58.45 .30 59.14 38.01 0.44 0.33 58.74 We may perceive in this Table a circumstance worthy of notice ; which is that, if we diminish by a certain quantity (0".4 in the present instance) the refraction at 45, we increase, by nearly as much (by 0".33), the latitude of the place, and vice versa ; when we speak, therefore, of the latitude of a place, we must be supposed to speak of it as computed by a certain Table of refractions, or according to a certain theory of refrac- tions. The latitude of the Greenwich Observatory computed by the French Tables of refractions, in which the mean refraction, at 45 zenith distance, is 57".5, is 51 28' 38"; by Bradley's Re- fractions, in which the mean refraction, at 45, is 56".9, it is 5128 / 39 // . Nor is there, as we shall hereafter see, any in- fallible method of determining the latitude of a place, the problem of refractions being, to a certain extent, an indetermi- nate one. The preceding method of determining the refraction from observations of the Sun at the solstices is due to Tycho Brahe. It was imitated, and made better, by Bradley, whose method we will now explain. Instead of observing the Sun near the solstices, Bradley observed the Sun near the equinoxes. For instance, he observed the Sun (see fig. p. 222.) at e and b, when it had equal zenith distances ; or rather (as the principle has been explained in pages 146, 147, &c.) he observed a zenith distance about the end of March ; and then, by observing, on two successive days in September, two other 222 zenith distances, one less, the other greater than the original one, S M M' he was able to -compute at what time between the two latter days the Sun, could it have been observed on the meridian, would have had the same zenith distance as at the first observation in March. By observing also the difference .of transits between the Sun and a star, Bradley was enabled to compute the arc compre- hended between e and b (supposing those to be the positions of the Sun when he was at equal distances from the zenith) : when am then became 180, the Sun was in the equator : and by com- puting (see pp. 149, &c.) the changes of zenith distance propor- tional to the changes of right ascension, the illustrious Astro- nomer, of whom we are speaking, was enabled to compute the Sun's zenith distance when he was in the equator ; or, those two equal zenith distances which were distant from each other by twelve hours of right ascension. Now these observed zenith distances were less than the true, by reason of refraction. Let S repre- sent the zenith distance, and r its refraction, then the true zenith distance is Or | - O "| I and S, according to Bradley, was 51 2?' 28"*. The half sum of the observed zenith distances of Polaris, above and below the pole, was found t equal to 38 30' 35". * This result was the mean of several observations in 1746 and 1747. f The mean apparent zenith distance of the pole was obtained by a multitude of observations, of the pole star above and below the pole, made between 1750 and 1752, and reduced by being corrected for Precession, Aberration, and Nutation, lo January 175 J, Old Stile, (see Bradley *s Observations, p. .0.) 223 This, if there were no refraction (r') would be the value of ZP (see fig. p. 7.) the co-latitude ; the other value 51 27' 28", ought, were there no refraction, to be the latitude, or the value of the arc ZE : their sum 89 58' 3", is less than 90, by l' 57" ; which expresses the sum of the equatoreal and polar refractions. Now, as before, (see p. 218,) if the refraction varied as the tangent of the zenith distance, we should have r = A. tan. 51 27' 28", / = A. tan. 38 30 35, and, adding the equations together, l' 57" ( = r + r') = A (tan. 51 27' 28" + tan. 38 30' 35"). Hence, since tan. 51 27' 28" = 1.2553, nearly, and tan. 38 30 35 = .7956 2.0509 we have A = -^ = 57".045, 2.0509 and accordingly, the equatoreal refraction = 5?".045 X 1.2553 = 7l".6l, the polar refraction. . . = 57 /; .045 X .7956= 45".39. Hence, the apparent zenith distance of the equator being (see 1. 3,) 51 27' 28" and the refraction 1 11.6 51 28 39.6 is the latitude, and 57".045 is the mean refraction at the zenith distance of 45. But these results depend, as it is clear, on the refraction being represented, with sufficient exactness, by the term 57".045 tan. Z, at the apparent zenith distances of 51 27' 28", and 38 30' 35" ; or, which amounts to the same thing, on the smallness of the coefficient (B) of the second term of the fol- lowing formula, refraction = A tan. Z + B . tan. 3 Z. Suppose B to equal (/'.I, then the sum of the terms in- volving tan. 3 51 27' 28", and tan. 3 38 30' 35", (which terms are neglected in Bradley 's Computation) will be 0".2482, and (see p. 223,) A will equal to 57"-l67, and the latitude will equal to 57". 167 X tan. 51 27' 28" 0". 1 . tan. 8 51 27' 28", that is to 51 28' 39".56. The quantity l' 57" is the sum of the equatoreal and polar refractions ; the exactness of which depends, on the accuracy of the observations of the zenith distances of the Sun and of the pole star. If we suppose an error of l" in each observation, the sum of the refractions may remain the same, or be 1 1 7" 2". But (see p. 223,), 117" 2" _ 117" l" 2.0509 " 2.0509 ~~ 1.0254 = 57 7/ .045 0".975 = 58".02, or 56".07, in the first of which cases the resulting value of the latitude will be greater than 51 28' 39".5, and, in the second, less. When we suppose, in the preceding method, the refraction to be represented by the single term, 57".045 tan. Z, + we determine the refractions for all zenith distances that are less than the latitude of Greenwich. But how shall the refraction be determined when the zenith distances are greater than that latitude ? The above term, it is clear, will not apply to all zenith distances : for it fails when Z = 90. Bradley determined 225 the quantities of refraction, at zenith distances greater than the latitude of his Observatory, by means of circumpolar stars. An instance will best illustrate his method : Zenith Distance of a Cassiopeae above the pole = 4 23' 20" refraction = 57".04 x tan. 4 23' 20". =0 4.5 true zenith distance 4 23 24.5 but co-latitude 38 31 39.5 north polar distance 34 8 15 zenith distance below the pole .72 39 54.5 observed zenith distance 72 36 55.5 refraction at last zenith distance 2 59 By these means, the quantity of refraction, at the zenith distance of 72 36' 55", was determined : but the term 57".045 x tan. 72 36' 55", gives a larger quantity. That single term, therefore, does not represent the law of refraction at zenith distances equal to or greater than 72 36' 55". If we assume a formula of two terms to represent the refraction, we have from the above observations, 2' 59" = 57".045 . tan. 72 36' 55" - jB.tan. 3 72 36' 55", and if from such equation we determine JB, we determine it (see p. 220,) on an assumed value of the latitude, the errors in the determination of which may exceed B. But, if we assume the refraction, to be represented by a for- mula of two terms with indeterminate coefficients, and suppose the latitude also to be undetermined, we must have, at least, the observations of three circumpolar stars to furnish us with three equations to determine the three above-mentioned unknown quantities : or, if we suppose that a formula of three terms will represent, more correctly, the refraction, there will be need of four circumpolar stars. In the second Volume of the Systeme du Base Metrique, &c. we have an instance of the determination of the latitude and refraction, by assuming the latter to be repre- sented by the following empirical formula, refraction = A tan. Z + B tan. 3 Z -f C . tan. 6 Z. FF 226 The four circumpolar stars,, were Polaris, ft Ursae minoris, a Draconis, and Ursae majoris, and the place of observation was Mountjoy. Now, see p. 219, if H be the latitude, Z, Z 1 the least and greatest zenith distances of a circumpolar star, p, p the refractions corresponding to Z, Z , 180 2H= Z 4- Z 1 + p + /'. Now, with the pole star, Z + Z' = 97 14' 2l".47 with ft Ursse minoris = 97 13 58 . 17 with a Draconis = 97 12 58 .6 with Ursae majoris = 97 9 31.65 computing, therefore, p + p ', from the formula, A .(tan. Z+ tan. Z') + 5 (tan. 3 Z + tan. 3 Z'} + C. (tan. 5 + tan. 5 2') we obtain four values of 180 QH, or four equations involving four indeterminate quantities, namely, H, A 3 ~B y C. The four equations are 180 2 H = 97 14' 2l".47 + 2.27634 A - 2.984405 + 3.9740 C = 97 13 58.17 + 2.67976 A - 8.470445 + 33.31284 C =97 12 58.6 + 3.75994^ -36.50251 B + 400.06998 C = 97 9 31 .65 + 7. 86330^4 - 439-269635 -f- 25581.807 C. The resulting values of the coefficients are A = 61". 1766 B = 0".2648 C = 0".002485. Substitute these values in the four preceding equations, and those equations become 180 - QH = 97 !6 / 39 // .95 180 - 2 fif = 97 16 39.95 180 - 2# = 97 16 39.95 180 - 2# = 97 16 39.94. Taking, then, the mean, 180 - 2 H = 97 16' 39".9475, and H = 41 21 40.02625. 227 The formula of refraction is r = 6l".1766 tan. Z - 0".2648 tan. 3 Z + 0".002485 tan. 5 Z, and making Z = 45, r = 60".914285, instead of 57" .045, which it would be, nearly, according to Bradley. In the above instance, we have, as M. Delambre justly observes, a formula of refraction derived (with regard to the numerical values of its coefficients) entirely from observations, and satisfying eight observed zenith distances. The formula, however, gives a mean refraction at 45 much greater than Bradley's formula gives. If we were to correct the apparent zenith distance of the pole (rather the half sum of the apparent zenith distances of the pole star above and below the pole) found by Bradley' s method (see p. 223,) by the preceding formula, we should have true co-latitude =38 30' 35"+6l".176 x tan. 38 30' 35" - .26485 X tan. 3 38 3C/35" + &c. = 38 31' 23".5, nearly, and the latitude would = 51 28 7 36",5, a quantity less, by three seconds, than the latitude found by Bradley's formula, in which A (the coefficient) is 57". We have already seen a similar instance (p. 221.). If we increase the value of the coefficient (A) of the first term of the formula of refraction, or increase the mean refraction at 45 of zenith distance, we diminish the resulting value of the latitude. In the preceding instance of observations made at Mountjoy, the latitude of which is (see p. 226,) about 41 2 1 7 40", we diminish the latitude as much as we increase A, and vice versa. And of this circumstance (which is worthy of attention in the theory of re- fractions) M. Delambre furnishes us with an additional confirma- tion. The true, or actual, refraction is reduced to the mean, on the score of temperature, (we shall soon more fully explain this part 228 - 350 + 1 of the subject) by multiplying the former by , in Bradley ' Theory, t denoting the number of degrees in Fahrenheit's scale above zero. By this multiplier for temperature, the observations made at Mountjoy were reduced, and the formula of p. 227, obtained. But Astronomers are not all agreed upon the cor- rectness of the above multiplier. The French use a different one : according to them, the correcting multiplier is more 450 -f- t nearly - - If the actual observations then are reduced by 500 this last fraction, or by any other not the same as Bradley's, the resulting coefficients of the formula of refraction will be dif- ferent. M. Delambre informs us that he did reduce the observa- tions by Mayer's Tables, and the formula of refraction became r = 63".302 tan. Z-0".34396 tan. 3 Z+0".0033923 tan. 5 Z, very different from the former one of p. 227, but, apparently, equally well adapted to the observed zenith distances. In this case, however, the latitude was diminished by 2", that is, nearly, by the difference between 63".302, and 6l".176. In deducing the coefficients of the formula of refraction,' and the latitude, from the eight observed zenith distances of four cir- cumpolar stars, the formula was made to consist of three terms. If a fourth term, D tan. 7 Z, had been introduced, the problem could not have been resolved : since there would have been five unknown quantities, (A 9 JB, C, D and H) and but four equations. But, if we assume A to be of some value between 57" and 6l", we may, from equations, similar to those of p. 226, and resulting from the observations, deduce the latitude and the coefficients of the second, third, and fourth terms. But, in each assumption, that will happen, which has before been noted to happen. As A is assumed of greater value H, the latitude j will, and by equal degrees, result of less. Thus, Values of A Corresponding Values of H. 57". 13 ............ 41 21' 44". 1 58 .............. 41 21 43 . <2 .59 .............. 41 21 42 . 12 60 . .41 21 41 .2. 229 The inference drawn, by M. Delambre, from this and other instances, is, that the construction q/ a Table of Refractions by observations is a truly indeterminate problem. In determining the polar and equatoreal refractions from the sum of refractions, Bradley assumed the refraction to vary as the tangent of the zenith distance. But, he afterwards expressed the law of its variation, more correctly, by the formula, r = 57". tan. (Z 3r). We are ignorant of the means by which he arrived at this formula : whether they were empirical or theoretical. But the formula is compact and elegant, and not difficult of application. It gives the mean refraction at 45 of zenith distance equal to 57" . tan. (45 3r). Suppose, in order to approximate to the value., that, at first, Sr is neglected : then r = 51" tan. 45 = 57" ; .*. r at 45 = 57" tan. (45 2' 5l") = 56".9, nearly. Again, and similarly, in order to find the mean refractions at 60 of zenith distance, 1 st , r = 51" . tan. 60 = 5l" X 1.732 = l' 3S".7 ; /. 3r = 4' 56". 1, and secondly, (r) = 57" . tan. 59 55' 3".9 = l' 38".4. Again, and similarly, if the altitudes were 27 39' 17", 62 13' 6", First, Zenith distance = 62 20' 43". r=57" . tan. 62 20' 43"==57" x 1.908 = l' 48".75, Sr= 5' 26".25 = 5' 26", nearly. i Secondly, (r) = 57" . tan. 62 15' 17" = l' 48".3, Again, when zenith distance = 27 46' 54". First, r = 57" . tan. 27 46' 54" = 30", nearly, 3 r = l' 30". Secondly, (r) = 57" ! tan. 27 4o 24" = 29".99- 230 By these means, that is, either by Bradley's formula or by Brinkley's, or by that of the French mathematicians, the mean re- fraction may be deduced. The results will not agree : accord- ing to Bradley, the mean refraction at 45 = 56". 9 according to the French = 57 . 5 * according to Brinkley = 57-72. But the true or actual refraction differs from the mean, if the temperature and weight of the air are not the same, when the observation is made, as they are supposed to be in the mean state of the air. Such mean state is denoted or represented by fifty degrees of Fahrenheit, and 29.60 inches of the common barometer. If the temperature be at its mean state, but the air less dense than at its mean state, or the height of the barometer be less than 29-6, if it be, for instance, 29-35, then the actual refraction is less than the mean, and in order to reduce it to the latter state we must multiply the former by ; and, generally, 29.35 if h be the height of the barometer, by j . If the barometer stand at its mean height, but the temperature be greater or less than 50, the actual refraction will be less or greater than the mean, and the correction, by which the former is to be reduced to the 350 + 1 latter, is had by multiplying the former by , according to Bradley, and by according to the French. If, there- oUO fore, (which will almost always happen) neither the barometer nor thermometer be at their mean states, 29.6 350 -f- the a c tual refraction x x .= mean refraction, n 400 * The latitudes of Observatories determined from observations of cir- cumpolar stars will vary according to the Tables of refractions by which the observations are reduced. Thus, Latitude of Dublin . . . Latitude of Greenwich, By French Tables. 53 23' 13". 5 51 28 38 Bradley's. 53<> 23' 14".2 51 528 39.5 231 and, if, according to Bradley, the mean refraction (r) be denoted by 56".9 X tan. (Z 3r), we have the actual refraction (r) = g ^ X x 56".9 . tan. (Z 3 r), or, if we use the French corrections, and take one of M. De- lambre's formulae, 500 h (60". tan. Z 0". 14207 tan. 3 Z) Ji C60".t s 29.6 1-0". 450 + 1 29.6 I 0".0045053 tan. 5 Z+ &c. J We must now examine the grounds on which the preceding corrections have been made. With regard to the first correction, that of the barometer, it is founded on this assumption (which is confirmed, very nearly, by experiment) of the refraction increasing and diminishing, and proportionally, with the increased and diminished densities of the air. Of which latter, the greater and less heights of the baro- meter are the indications and measures. Hence, if dr, dh, re- present the corresponding variations of the refraction, and of the height of the column of mercury in the barometer, h + dh s , dh^ r + dr=rx - - = r x (l + -) . The other correction, that for the thermometer, is obtained on principles less simple and sure. The temperature increasing increases the volume of air, which varies inversely as the density, and the greater the density, the greater the refraction. What is required to be known then, is, the relation between the increases of temperature, and of the volume of air : or, in a more scientific form, how much will the volume of air be increased by an increase of 1 of temperature ? Let m be an indeterminate coefficient : then if the volume of air at the mean temperature be V, it may be re- presented by V X (1 -h m X 1), when the thermometer indicates an increase of 1 of temperature, and TX (1 + mk) when k is the increase of temperature : if r be the refraction in the first case, r' in the second, we have r' t ' V I r '' V.(\ + mk) ~ l+rnk"' 232 and in order to determine w, we must have recourse to experi- ment. Now in order to determine m from actual observations, we must get rid of r, and use another equation similar to the former : let it be r" = m X f ' . in which the same star is observed and at the same place : for then, we shall have m by the common process of elimination. Thus, by the first and second equations, r = /(I + m X /c) = r"(l + m x f) ; r" m = in which equation, r" , r , k, /, are known from actual obser- vation. To determine m, with correctness, select stars having low altitudes, and compare those altitudes observed under the cir- cumstances of great differences of temperature. M. Delambre has selected such altitudes from Lemonnier's Histoire Celeste, p. 32 : in which r 1 = 10' 40", A; = 54, r"= Q' 20", r = - 4. 5; 80" /. m = - = .002415, nearly. 33120 Now 50 is the temperature at which mean refraction is held to take place. The multiplier, therefore, for reducing the true or actual refraction to the mean is 1 +.002415 x(f 50), or, .87924 + .002415*, 879240 + 2415* 364 + * ' ' whlch nearly equals to 414 > a not differing much from Bradley's (see p. 230,) which is - 40O 233 If we examine the preceding method, it will be found liable to considerable uncertainty. In order to procure large differ- ences of refraction, those stars were, selected which are at con- siderable distances from the zenith. Now, of such stars the re- fractions are very irregular : by which it is to be understood, that the refractions are not always the same, whilst those circum- stances, that are supposed to cause refraction, do remain the same. As a proof too of the uncertainty of the method, there are considerable differences of opinion respecting the value of m. We subjoin its values according to different authors. Value of m. Bradley ................... 002444 Lemonnier. ................. 002415 Mayer. .. ................. .002012 Lacaille ................... 001644 Bonne .................... 001777 Laplace .................. 002186 The uncertainty respecting the correction of refraction for difference of temperature, is rather an embarrassing circumstance, when minute inequalities are to be detected, or when a question arises concerning the exact mean places of stars*. * In the preceding instance the correction for temperature was astro- nomically determined. But it has been determined, independently of the observations of stars, and by direct experiments. Thus, a column of air called J , at 32 of Fahrenheit, becomes 1.375, at 212. If the expansion be held to be equable, at a temperature t, the column will be equal to 1 + - x (* - 32), or 1 + ,002083 x (t - 32); 1 oU .'. at 50 (when mean refraction is held to happen) it will equal to 1.0375, nearly. But the refraction, if it vary as the density, will vary inversely as the volume of the same column of air : hence, the true refraction equals the mean x 933343 + 2 Q83f ' which latter fraction is nearly equal to - (see Irish Trans. 1815, Dr. Brinkley.) GG 234 We will take an instance to elucidate the preceding state- ment. Let the observed star be Procyon and its zenith distance (Greenwich being the place of observation) 45 46' 53", and suppose, at the time of observation, Fahrenheit's Thermometer to be 70. The mean refraction by Bradley 's formula, [56".9 tan. (Z- 3r)] equals 58".44 : and if we use Bradley's Correction for Tem- perature (see p. 230,) we have the actual refraction equal to 58^44 X 4Q = 58".44X = 5o".66, nearly. 350 -f 70 42 If we use the French Correction, then the actual refraction is equal to 58 ". 44 x 50 = 58".44 x = 56".2, nearly. 450 + 70 104 The true zenith distance then of Procyon by Bradley's Cor- rection would be 45 47' 48".66 and by the French Tables k 45 47 49-2 so that, under the circumstances of the observation, (the material circumstance being the height of the thermometer) there would be a difference in the north polar distance of the star of 0".54. We have taken the temperature at 70 which is not enormous in the month of July *, about the hour of noon, when Procyon would pass the meridian. Suppose, now_, the same star to be observed, half a year after, in January, when it would pass the meridian about midnight, and that the thermometer is at 30 : in 400 this case the true refraction, by Bradley = 58".44 X = 6l".5, 500 by the French. =58".44 X = 6o".87. * In July the Sun's right ascension is from 7 h to 8 h and Procyon's right ascension being about 7 h 30 m , the star passes the meridian about noon-tide. In January the Sun's right ascension is from 19 h to 20 h , and consequently, Procyon will be on the meridian about midnight. 235 -* ^v In this case, the contrary to what happened in the former takes place. The correction for reducing the apparent zenith dis- i tance to the true is larger by Bradley 's than by the French Tables. If, therefore, the star's place were deduced by observations like the preceding, there would be uncertainty to the amount, in each case, of more than half a second respecting the star's mean polar C i* distance*. &*$&$ t If we had used the French mean refraction at 45, instead of tf ^ *;" the English, the mean refraction for Procyon instead of 58".44 t *~ 0- would have been 59"4. * ft, JJ It appears, from what has preceded, that there is, at present,*- ^ Q \ considerable doubt respecting that correction of refraction, which-f JJ j is due to a variation of temperature. With regard to the cor- '<. rection due to an increased or diminished density of the air, as fi indicated by the height of the mercury in the barometer, there is, amongst Astronomers, no difference of opinion. If h be the J height of the barometer, the true refraction is less or greater than the mean refraction (which is held to take place when h is 29.6 inches) as h is less or greater than 29.6 inchest, and in that pro- portion ; the principle is, the variation of the refraction as the density of the air. The states of the barometer and thermometer must be noted down at the time of each observation. But, Astronomers hold it needless to consult the hygrometer. According to M. Laplace, Gay Lussac, and Biot, refraction is not influenced by the relative moisture of the atmosphere. * This illustration is taken from Dr. Brinkley's Paper on Parallax (Irish Trans. 1815), in which he shews the effect of the uncertainty of the correction for temperature on the index error of the mural circle. f A very small correction must be applied to the height of the baro- meter when the temperature is other than 50 its mean state. If the tem- perature be above 50, part of the height of the mercury in the barometer is owing to the expansion of the mercury : that part, therefore, must be subducted. If the expansion be .0001 inch for one degree of Fahrenheit, for 50 degrees, it will be( 50) x ,0001, hence the correcting frac- tion (see p. 233.) for the barometer, instead of being ,. , will be ' 236 There remains, however, much to be done on the subject of refraction. We may, amongst other guesses, conjecture that, from some defect in its theory, arises the difference in the values of the obliquity of the ecliptic, as they result from observations oi the winter and summer solstices. Maskelyne, with Bradley's refractions, made the obliquity from the winter, less by 8" than from the summer solstice. Delambre makes it less by 4 /; The formula of refraction may be so altered, as to make the two values of the obliquity to agree. But, then, the altered formula, applied to the observations of circumpolar stars, would produce anomalous results ; it would, for instance, produce different values of the co-latitude. There are other methods, than those already mentioned, for determining the quantities of refraction. The method of the Abb6 Lacaille is ingenious and founded on good principles. It does not happen to every Astronomer to be able to practise a similar method. Lacaille observed certain stars at Paris, and the same stars at the Cape of Good Hope, and from such obser- vations deduced the quantities of refraction corresponding to the different altitudes of the same stars at the respective places of observation. In order to illustrate his method, let Z be the 237 zenith of the Paris Observatory, Z' that of the Cape, and let the true zenith distances of a star S be esteemed to be ZAS, Z'BS, respectively, then ZAS = ACS + CSA Z'BS BCS + CSB .'. ZAS + ZBS = ACB + BSA = ACB, if the angle of parallax (see p. 42,) BSA be neglected. If then, Z, Z' be the apparent or observed zenith distances of a star, r, r', the corresponding refractions, we have ACB = Z + Z 1 + r + /: A second star, the zenith distances of which are V, V, and the refractions p, p', will give a similar equation, viz. ACB =V+ V + P + p', and, a third star will give a third similar equation, a fourth star, a fourth equation, and so on : if we suppose the refraction to vary as the zenith distance (to equal A tan. Z\ we have, by equating the two first equations, A (tan. Z + tan. Z 1 - tan. V - tan. V 1 } = V + V - Z - 2', from which equation, Z, Z', F, F', the observed zenith distances, being known, J. may be determined, and thence the angle ACB. But if instead of r = A tan. , which imperfectly expresses the law of refraction, we assume ^ r = A tan. Z + B . tan. 3 , we must have three values of ACB t to determine A and B, and ACB: and if we add to the last formula a third term C tan. 5 Z, we must, in order to deduce A, B, C, and ACB have four values of ACB 9 or the zenith distances of four stars observed both at the Cape of Good Hope, and at Paris. According to the results of Lacaille obtained by the pre- ceding method, the mean refraction at 45 is much greater than 56".9, which is Bradley's value : the coefficient A would equal 66". The preceding method of Lacaille is one that cannot often be resorted to. We will explain another method for determining 238 the refraction which does not require the observer to change the place of his observations, but which will admit of his changing it. Let M be the star's apparent place, raised, by refraction, above m its true place. Suppose, by previous observations, to be known, PZ, the co-latitude = 90 - H Pm, the star's north polar distance. . = $ and, by immediate observations, ZM, the star's zenith distance = Z y ZPm, the hour angle = P, then (see Trigonometry, p. 13Q>) cos. (-f-7')( = cos. Zm) cos. P. cos. H . sin. $ + sin. H.cos. S, from which formula, Z -f r becomes known, and thence r = (Z + r) - Z. This is, what may be called, the bare scientific process, which, however, in practice, becomes invested with circumstances that require great attention. For instance, the observations of the zenith distances may be made at a place, that is not the scite of an Observatory, and the latitude of which may be uncertain to the amount of two or three seconds. Indeed, if we are un- certain about the quantities of refraction due to the zenith dis- tances of stars, we must be uncertain with respect to the co- latitude, which (see p. 129,) is half the sum of the greatest and 239 least zenith distances of a circumpolar star. Let p, p 1 , be the apparent zenith distances of such a star, p, p, the corresponding refractions, then 90 - H = (p+p' + p + p'}', but d H representing the variation of the latitude from refraction, /. d H = \ (p + p'). Hence, the error in determining the latitude, is half the sum of the refractions due to the two zenith distances of the circum- polar star used in determining it. If Polaris be that circum- polar star, -j (p 4- p 1 ) = p, nearly. In order then to determine the error of the quantity of refraction resulting from the formula of p. 238,, take its differential and substitute p instead of dH ; if this be done, dr . sin.(-f-r) jo(cos. H.cos. 5 cos. P.sin. H.sin.S) whence dr may be computed. In this deduction $ has been supposed constant, or not subject to error. But, if the process of p. 237, be viewed as an original one, in which the observer (and Lacaille was so cir- cumstanced, nearly,) had to determine not only the quantities of refraction, but the latitude of his Observatory and the declinations of his stars, it is plain that the resulting values of the refraction would be erroneous, from the errors of the latter quantities. Now, the apparent polar distance, or, $ + dS = ~(p p f ) the real polar distance, or S r: \ (p p 1 + p p')-, .: di = i(p'-p), which, if Polaris be the circumpolar star, is a very small quan- tity. In the above method, then, the refraction, an unknown quantity, is to be determined from quantities which themselves involve the refraction : a kind of dilemma, in which the Astro- nomer repeatedly finds himself, but which the same kind of 240 artifice, almost always, enables him to loose himself from. He first neglects the indeterminate quantity where it appears in its in- volved state and finds an approximate value. This first value is then substituted, in every part of the original equation, and a second value is obtained, which second value serves, as a stepping stone, to ascend to a value nearer to the truth. The third, fourth, &c. values (although it is scarcely ever necessary to proceed so far) may be either taken as the true values, or may be made alike subservient to truer values. Thus, in the instance before us, the quantity of refraction is to be computed from the formula of p. 238, the values of H and $ being those which result from observation. The resulting value of the refraction will then serve to correct H and 5, and, their corrected values being sub- stituted in the formula, a second value of the refraction is to be deduced, with which H and 3 are again to be corrected, &c. &c. In the preceding method we must use a clock to determine the hour angle P ; but there is an instrument, called an Altitude and Azimuth Instrument, which will enable us to determine the refraction without the aid of a clock. Now this instrument determines, at once, both the altitude and azimuth of the star : the latter truly, the former as it is made greater by refraction. If we use the former figure and symbols, and make, besides, A to represent the azimuth, we have sin. PmZ (sin. _B) = sin. A . . ' . : sin. o thence B becomes known. Again, by Naper's Analogies, (Trig. p. 169,) Z m /% -f- rx j , cos. tan. = tan. ( ) = tan. ? (flOP- H+i) EXAMPLE. Latitude of place of observation ...... 51 3l' 0" Star's observed altitude ............ 18 13 5 ................. azimuth. ..... 74 53 30 . IS. P. D. . 66 32 * 24* B found. sin. 74 53' 30" = 9*9847229 cos. 51 31 = 97939907 19.7787136 sin. 66 32 0. ... 9-9625076 9.8162060 = log. sin. 40 54' 56" = B. Again, (90 H+) = 52 30' 30" tan. = 10.1151503 (4 + B). . . . = 57 54 13 cos. = 9.7253768 (/4-JB), .. .= 15 59 17. .arith. comp. cos. =10.0193759 29.8599030 Rejecting 20, we have log. tan. = 9.8599030, Z + r thence = 35 54' 53".5 Z + r = 71 49 47 but Z = 71 46 55 therefore, r, the refraction = 2 52 In the preceding instance the zenith distance is about 72; up to that distance, and beyond it by about ten degrees, the formula? and their deduced Tables, satisfy, (to borrow a French mode of expression) the observations. That is, the half sum of the greatest and least corrected zenith distances of a circumpolar star, is, very nearly, the same quantity, whether the greatest zenith distance of the star, be forty or eighty degrees. If we go beyond eighty, the refractions are irregular. Dr. Brinkley has shewn those of Capella * to be so. Some * The co-latitude of the Dublin Observatory being 36 36' 46".7, and the north polar distance of Capella being greater than 44, the greatest zenith of that star exceeds 80. n 11 Tables (the French, for instance,) represent the refractions neai to the horizon, more nearly than others : but, hitherto, there has been invented no formula that restricts the irregularity of re- fraction that begins to take place about 80 of zenith distance Laplace's formula does not extend to distances beyond 74. At 82 30', the formulae of Bradley and Simpson are erroneous, to the amount of 8". But, if we advert to the results which M. Delambre has given us of observations of stars near the horizon, it is hopeless to expect to reduce all refractions under one law. Those bor- dering on ninety degrees of zenith distance seem freed from al restraint. They disagree amongst themselves, and are, in this way, irregular ; namely, they are not the same, when other circumstances, the altitude, and the heights of the barometer and thermometer are the same. It is certain, then, that the theory of refraction is imperfect : not solely because it does not restrict all its cases within the same law, but because it has no tests of, or means of measuring certain circumstances, on which, at great zenith distances, the refraction must depend. This is a perplexity, from which mathematical skill alone can never extricate us. If the theory, however, be imperfect, the results of its for- mulae, or its Tables, are easy of application ; and we now subjoin one or two specimens of those Tables and instances of their uses. The specimens and the instances are both taken from the Volumes of the Greenwich Observations. From the Table I. of mean refractions, computed to every ten minutes of zenith distance, Zenith Distance. Refraction. Diff. for 10'. 77 10' 4' 6". 1.5 3".fi5 77 CO 4 9.40 3 .31 77 30 4 12.71 243 From TABLE II. Apparent Zen. Dist. Log. Refraction. Differences. 28 50' 1.49665 304 29 1 .49969 &c. &, 47 1.78533 256 47 10 1.78789 Sec. Stc. 77 20 2.39690 77 30 2. 40262 572 From TABLE III. Height of the Barometer in English Inches. Thermora. 29.6 29.7 29.8 29.9 54 9.99568 9-99715 9, Sic. 9, &c. 55 0.99460 0.99607 &c. " 56 0.99353 0.99500 - 57 0.99247 0.99394 9-99685 58 0.99140 0.99287 &c. $9 0.99034 0.99181 60 0.98Q28 0.99075 244 Extracts from the Greenwich Observations. June, 1812. Barometer. Thermc Out. meler. In. Star. Zenith Distance. 14 29.71 57 60 Antares. 77 24' 34".7 16 29.60 54 57 Antares. 77 24' 33".9 22 29.76 56 54 a Arietis. 28 54' 10".2 1816- Barometer. Thermom. N. P. D. Sept. 12. 29.95 57 0'sU.L. 8534 / 28".l Suppose it were required, in the first place, to find the mean refraction of Antares on June 14, by Table I. This Table gives the refraction for 77 20', and 77 30' : but the zenith distance of Antares is between these two zenith distances : it must, there- fore, be found by interpolation, or by proportion, just as we find the logarithmic sine of an arc expressed in degrees, minutes and seconds, from Tables not extended beyond minutes. Thus, see Table I, diff. for l' = .331 .'. for 4 = 1.324 030 . 165 4 . 22 4 34 1.51, nearly. But refraction for 77 20' 0" = 4' 9".4 .'. for 77 24' 34" ... =4' 10".9, nearly. Hence the zenith distance of Antares on the 14th, corrected for mean refraction, is 77 28' 45".6. But it is scarcely ever necessary to correct an apparent zenith 245 distance by the mean refraction. It is the true refraction that must be added to the observed zenith distance : and that refraction must be computed by Tables II. and III. The Rule is : Take from Table 17, the logarithm (A) cor- responding to the apparent zenith distance, and add. it to a loga- rithm (_B) of Table III, answering to the proposed heights of the barometer and thermometer. The sum (rejecting 10) is the loga- rithm of the true refraction. It will be necessary in this, as in the former, computation, to deduce, by proportion, logarithms intermediate to those ex- pressed in the Tables. To compute the refraction on the 14th. By Table II, diff. for l'. = 57.2 .\ for 4. . < . . 228.S 30" 28.6 4.7 4.5 4 34.7. 261.9 but logarithm for 77 20' 0" is 2.39690 ; .'. logarithm for 77 24' 34".7 is 2.39951.9 U); next, log. barometer 29.7, thermometer 57. 9-99394 thermometer 60 9.99075 2)19.98469 999234.5 correction for .01 of barometer 14 9-99248,5 (B) 2.39951.9 (A) (log. of 246.6) 2.39200.4 Hence the true refraction is 4' 6".6 and since the apparent zenith distance is . .77 24 34 . 7 the true zenith distance is. . . . 77 28 41 .3 246 Again, in order to compute the refraction on the J6th r log. barometer 29.6, thermometer 54 . . . . = 9-99568 thermometer 57 . . . . = 9-99247 '2)19.98815 9.99407.5 (A will be a little less than in the former instance) 2.39950 (A} (log. of 247.5) : 2.39357-5 Hence the true refraction is .............. 4' 7".5 and since the apparent zenith distance is . . 77 24 33.9 the true zenith distance is ...... f . f 77 28 41 .4 In these two instances the mean of the two thermometers has been taken to represent the temperature. In the next instance (that of June 22,) we will compute the refraction from the In thermometer. From Table II, diff. for I 1 o" /, for 4 r o" is 30.4 40 is 121.6 10 5 06 02 0.1 4 10.2 126.76 but log, for 28 50' O' 7 ! is 1.49665; : *'. log. for 28 54' 10".2 is 1.49791-76 (J). Again, log. barometer 29.7, thermometer 54 9-99715 (B) correction for .06 , 87 (log. of 31 32') 1.49593.76 Hence, the true refraction is 0' 3 1 ".3 and since the apparent zenith distance is 28 54 10.2 the true zenith distance is. . . t , 28 54 41,5. 247 In these three instances, the apparent zenith distances, observed (see pp. 65, 66, &c.) by a mural quadrant, are expressed. In the next instance we must deduce the zenith distance, the north polar distance of the Sun's upper limb being observed by the mural circle, (see pp. 1 10, &c.) North polar distance, Sun's upper limb = 85 34' 28". 1 (pp. 1 12, &c.) index error +2.5 85 34 30. 6 co-latitude.. .38 31 21.5 apparent zenith distance Sun's upper limb 47 3 9-1 Computation for refraction, From Table II, diff. for I 7 0" is 25.6 /. for 3 76.8 09 3 . 96 S 9 80.76 but log. for 47 0' 0" is 1.78533 /. for 47 3' 9" is 1.78613.76 (A). But, from Table II, log. barometer 29.9, thermometer 57, is 9-99685 correction for .05 72 ( B) (log. of 60.75) 1.78360.76 If, therefore, we add this refraction ( = l' 0".75) to the zenith distance of the Sun's upper limb, and add the Sun's semi- diameter, we shall haVe the zenith distance of the Sun's centre. Apparent zenith distance of the Sun's upper limb 47 3' 9"- 1 refraction 1 .75 Sun's semi-diameter 15 56.1 zenith distance of the Sun's centre 47 20 5.95 (see p. 209,) parallax 6.5 true zenith distance of the Sun's centre 47 19 59-45 and Irue altitude ; . . : . 4<2 4O 0.55 248 The explanation of the theory of refraction, the deduction of its formula, the construction of Tables and their application, are the main objects of the present Chapter. There is begun in it, what will be continued, a series of investigations of those cor- rections by which the star's apparent place may be reduced to its mean place. The first in this series, the correction for refrac- tion, is a correction for an inequality unlike, in its nature, to all other inequalities. It can never, even during short intervals in the same day, be presumed to be the same. It varies, every hour, with the temperature, and requires the unceasing attention of the observer to his thermometer and barometer. But although what has been principally aimed at is, the divesting of instrumental zenith distances of the errors of refrac- tion, yet the principle, or the ascertained effects, of that ine- quality may be applied (as to collateral objects) to the ex- planation of certain ordinary phenomena. Such are the elliptical forms of the orbs of the Sun and of the full Moon when near to the horizon ; or their then curtate vertical diameters. The appearance of the Sun above the horizon, previously to the com- puted time of its rising, &c. The first phenomenon arises from the rapid variation of the refraction when the observed body is near to the horizon. For instance, the upper limb of the Sun in the horizon is elevated by refraction, but the lower limb is much more than proportionally elevated. Let zenith distance of the Sun's upper limb be 90 if the refraction be ." 28 29 apparent zenith distance of the Sun's upper limb 89 31 31 Suppose the Sun's diameter to be 32,' ; then, zenith distance of the Sun's lower limb is 90 32' 0" but the refraction . , 32 46 .'. apparent zenith distance of the Sun's lower limb 89 59 14 subtract 89 31 31 Sun's apparent diameter 27 43 249 The vertical diameter is, therefore, in this case, shortened 4 ; 17" by the effect of refraction, whilst the horizontal diameter s scarcely at all affected. The upper and lower boundaries of the Sun's disk, in the preceding case, will be nearly elliptical : for, conceive a vertical circle to pass through the Sun contiguous to that which passes through his centre. That part of the vertical circle, which is in- tercepted between the Sun's horizontal diameter and either the upper or lower boundary of his disk, is nearly parallel to the rertical semi-diameter. It may, then, be conceived as an ordinate of the boundary curve. It would have been, were there no refrac- ion, an ordinate of a circle (the Sun's orb being circular). It is ess than this latter ordinate in the same proportion, nearly, as the curtate vertical semi-diameter of the Sun is less than his horizontal semi-diameter : and the above is the property of an ordinate to an ellipse. The Sun's disk, or a star, may also appear above the horizon when it is, in fact, or, astronomically, below it. For instance, the star v, the course of the light of which is vwc, will be seen in the direction cv. On like principles, it is possible to see both the Sun and the Moon above the horizon at the time of a central I I 250 eclipse, For suppose, at such a conjuncture, the Sun to be just above the horizon ; the Moon, being diametrically opposite, must, indeed, be beneath the horizon, but may be so little beneath, as, by refraction, to appear above. This phenomenon is recorded to have happened at Paris on July 19, 1750. The next correction is due to an inequality called Aberration It is difficult to prescribe the natural order of the inequalities, and, perhaps, we have already departed from it, in not first treating o Precession. In fact, if the historical were the natural order, we have already done so. The latter inequality was known to th< antients, and its quantity, not very exactly indeed, assigned whereas, it was not until the time of Tycho Brahe am Dominic Cassini that the effects of refraction on observations were computed and allowed for. The researches of preceding Astronomers did not extend beyond some speculations concerning its cause. The historical order (the order of their successive discoveries) of the inequalities, is, Precession, Refraction, Aberration, Nutation. As for the inequality of Parallax (we are now speaking of those inequalities that affect the fixed stars) we are doubtful what place we ought to assign it. One hundred and fifty years ago. Flam- stead thought he had discovered it, whereas its existence is now doubted of at Greenwich. It cannot, therefore, be even now said to be discovered. For the historical place of a discovery must be dated from the time at which it is, beyond controversy, esta- blished, and not from that at which it may have been either vaguely surmised, or erroneously affirmed, to exist. But, dismissing this enquiry, we have no difficulty .in assigning a place to parallax in a scientific arrangement. It will be immediately after aberration: because, the formulae of the latter inequality, become, with a very slight alteration, the formula? of parallax. We shall not, indeed, use those formulae in correcting observations, because the effects of parallax on the right ascensions and declinations of stars, if any, are, certainly, very inconsiderable. It is necessary, however, to be possessed of its appropriate formulae ; to know, in fact, the laws of its variation, that, should the comparison of reduced observations present us any anomalies, we may be able to ascertain whether, and to what degree, such anomalies are attributable to parallax. 251 According, then, to the plan of the present Treatise, parallax will be treated of immediately after aberration ; Next, Precession, 4he Inequality of Precession, Nutation. The north polar distances and right ascensions of stars, corrected for these inequalities (which with refraction are,, at present, the only accredited inequa- lities) become, or are to be held as, the mean north polar distances and mean right ascensions. Should these mean quantities, com- puted for two, or more, epochs, and then compared by being re- duced to the same, or a common epoch, be found to differ, the causes of the differences would become subjects of enquiry : and till such causes are detected, might be designated by the title of Proper Motions. The formal propositions of a scientific Treatise have many advantages, but are not exempt from this objection : namely, that the Student is too suddenly carried into the middle of the subject, and too abruptly introduced into a system. He finds himself, with little preparation, amongst arrangements that are the results of many ,trials, many failures, and much thought. -This evil will be felt in the following subject. The principle on which aberration depends is not an obvious one. Its effects do not admit of easy proof or familiar illustration. They cannot be exhibited separately, but are mixed up with embarrassing circumstances. But, in truth, it does not happen in this, otherwise than it does in other subjects. Jf the Student would thoroughly understand the doctrine of aber- jration, he must look to the history of its rise and first promulgation. Its propositions and precepts he must view, not as the first and natural suggestions which arose in the mind of its author, but as ideas carefully methodised and arranged for imparting instruc- tion in the most convenient and concise form. CHAP. XI. ABERRATION. Jts Principle. Illustration of it. Roemers Discovery of the Progressive Motion of Light. The general Effect of Aber- ration is the Apparent Translation of a Stars Place toward the Path of the Earth's Motion. The partial Effects oj Aberration on the Right Ascension and Declination of a Star on its Latitude and Longitude. The Effects of Aberration on a Star situated on the Solstitial Colure at the Seasons of the Equinoxes and Solstices. Formula for the Aberration, in Right Ascension; in Declination; in Latitude; in Lon- gitude. Application and Use of such Formula. SUPPOSE a" to be the place of a star, and the eye of the observer, who is at rest, to be at c, then, (if there were no refraction) the star would be seen in the direction c t ; and this would be the case, whether the light were instantaneously transmitted from a to the eye at c, or gradually descended to it in the line y aberration. In the other Figure (Fig. 2.) and on a like sup- position of the tube and star, the star's latitude would be aug- mented by aberration. The plane passing through ae and c represents the first point of Aries, the longitude of T is T T ; and the lon- gitude of the Sun, which, by a spectator on the Earth's surface, is referred to , is T O = T T+90 . We have now obtained what may be called a general expres- sion for the aberration : an expression for the aberration which takes place in the circle v T, and which, except in particular cases, does not affect, with its whole quantity, the observations of right ascension and declination. The resolved parts, therefore, of the general effect of aberration become the proper objects of enquiry : and, with the view of investigating, most conveniently, such resolved parts, we shall first determine those positions of the point T (see the Figure of p. 267,) in which the resolved parts, the aberrations in right ascension and declination, &c. are nothing. * In Fig. p. 252, sin. cte : sin. tee (= sin. tec, nearly,) :: ce : t e, or sin. 5/0- : sin. tec :: vel. : velocity of light. General Construction for the Point T, when any resolved part of the Aberration is equal to nothing. Let, as before, cr be the star, TT the pole of the ecliptic CTL, P the pole of the equator, JEQ, and . In order to 270 compute D Q Z and d Q Z, we have, since the spherical triangle D 1 J.lJ '\i-: ..- ^.y'jL .\i>"i Draw o-'w perpendicularly to P'.<> Z =95 33 55 but * long. = 6 26 19 6 26 19 /. N = 32 26 58 .-.- = 89 7 36 Secondly, M m log. r 10 log. 20".25 1.30642 * 1.30642 log. sin. 24 4' 44" 9-61063. . log. cos. 24 4 ; 44" 9-96046 1091705 2 1.26688 (a) log. sin. d Z 9-95362 log. sin. 75 54' 4" 9.98671 (log. M ) 0.96343 log. cos. 5 3S 55 9-99795 19-98466 (b) M = 9". 19 (log. i) 1.28222 hence m = J9 // -15. 276 Hence, (see p. 87 O A = M.cos. (0 -JV) = M.cos. (0 - 32 26' 58"), and fl = msin.(0 - w)= w. sin. (0 +89 7 36), from which two expressions, the aberrations in right ascensioi and north polar distance may be determined for every day in year. The first of the tw : o preceding expressions involves the cosine of the difference of two arcs : the second the sine of the sum of two arcs, but negatively expressed. If we wish to express (and it is convenient they should be so expressed) both aberrations by the positive sines of the sums of two arcs, we must transform the preceding formulae after the following manner : M cos. (032 26' 58") = M cos. (0+57 33' 2" - 90) = M sin. (0 +57 33 2) - m sin. ( /!).== m . sin. (0+89 7' 36") = -w.sin. (0 +6 s + 89 ?' 36" ff) = m.sin. (0+8 s 29 l' 36"), and if we express the above two formulae logarithmically, we have log. .4 = log. sin. (0 + 1 s 27 33' 2") -f- .96343, log. a = log. sin. (0+8 29 7 44) + 1 .28222, and, if we wish to express the aberration in right ascension in time,, we must subtract from 1.28222 (log. m) the log. 15- (=1.17609) in which case, 0.10613 will be the logarithm of the maximum. 277 EXAMPLE II. a Arietis. For the latitude, longitude, angle of position of this star in 1815, see pp. 160, &c. T rj A n CT Z -4 Z 10 4- 10 4- log. tan. 20 39' 52" 19.57653 cot 20.42346 log. sin. 9 57 37 9.23796 9.23796 (tan. d Z) 10.33857 (cot. A Z) 1 1.18550 d Z = 65 21' 50" A Z = 3 43' 56" /. D Z = 24 38 10 /. a Z =93 43 56 *'s long. =35 4 41 35 4 41 N = 59 42 51 -w = 58 39 15 Again, M m log. r 10 log. 20.25 ..... 1.30642 1.30642 log. sin. 9 57' 37" 9.54764 log. cos. 20 39' 52" 9.971 12 10.85406 21.27754 (a) log. sin. 65 21' 50" 9.95855 log. sin. 67 25' l" 9-96535 (log. M) 89551 log. cos. 3 43 56 9-99908 M = 7".86 19-96443 (6) (log. w) 1.31311 (, 278 EXAMPLE III. Polaris, (see pp. 167, 175, 178.) d Z 4 Z 10-f 10 + log. tan. 72 59' 39". . < . 20.51450 log. cot. = 19.48549 log. sin. 66 4 42 9.96099 9.96099 (log. tan. d o zy 10.55351 (log. cot. A Z) 9-52450 d Z = 74 22' 50" A Z= 71 30' 2" .'. D Z = 15 37 10 .-.fl Z=l6l 30 2 but *'s long. = 85 46 16 85 46 16 /. N. =101 23 26 7z= 75 43 46 Again, M m log. r 10 log. 20.25 1.30642 1.30642 log. sin. 72 59' 39" 9-98058 log. cos, 72 59' 39" 9.46608 11.28700 20.77250 (a) log. sin. 74 22' 50" 9-98460 log sin. 1 45 ; 34" 8.48722 (log. M) 1.30240 log. cos. 71 30 2 9.50146 M = 20 // .06 17.98868 () (log. m) 2.78382 (a m = 607".9 = 10' 7^.9. Hence, A =20".06 . cos. ( Q - N) = 20 /x .06 . cos. ( O - 3 s 1 1 23 / ^26 // ) a= - 607".9 . sin. ( O -/z)= - 607 /; .( sin. ( +2 15 43 46), or, as before, (see pp. 276, 277,) A = 20".o6.sin. (0 -f 11 s 18 36' 34"), a = 607".9 sin. ( G + 8 15 43 46). 279 Ex AMPLE IV. TI Ursa, majoris. Suppose the latitude, longitude, and angle of position of this star (to be computed by the formulae of pp. 159, 168., 175) to be, in the year 1725 (the time of Bradley 's Observations) as follow : latitude 54 23' 53" longitude 173 3 15 angle of position (P) 38 37 26 theN.P. D. is 39 18 5 dZ AJL 10+ 10 + log. tan. 38* 3?' 26". 19-90253. .cot. 38 37' 26" 20.09746 log. sin. 54 23 53. . . 9-91013 9-91013 (log. tan. d Z) 9.99240 (log. cot. A Z) 10.18733 d Z = 1 s 14 29' 55" A Z = I 9 3 0' 33" .'. D Z= 4 14 29 55 a Z = 1 26 59 27 but *'s long. =a 5 23 315 5 23 315 AT=10 7 33 10 n = 3 26 3 48 M m log. r 10 log. 20.25 1.30642 1.30642 log. sin. 38 37' 26" 9-79532 log. cos. 38 37' 26" 9-89279 11.10174 21.19921 (a) log. sin. 44 29 25 9-84565 log. sin. 56 59 27 9.92354 (log. M} 1.25609 log. sin. 39 18 5 9.80167 M = 18".03 19-72521 (6) (log. m) 1.47400(0 6 m = 29". 78. Hence, A = 18".03.cos. (O 10 s 7 33' 10 r/ ), or = 18".03 .sin. (0 + 4 22 26 50), a 29".78 .sin. (0 3 26 3 48), or =s 29 // .78 . sin. (0 + 23 56 12). 280 EXAMPLE V. y Draconis. (See the latitude, longitude, angle of position of this star for 1815, at pp. 166, 174, 177.) d Z J Z 10+ 10 + log. tan. 2 56' 53" . . 18.71 180 . . log cot. 2 56' 53" 21.28820 log. sin. 74 56 51 . . 9-98484 9-98484 (log. tan. d Z) 8.72696 (log. cot. A Q Z) 11.30336 d Z = s 3 3' 9" 4 Z = s 2 50' 50" .'. D Z =3339 fl Z = 87 9 10 but #'s long. = 8 25 14 36 8 25 14 36 JV = 11 28 17 45 n = 5 28 5 26 M m log. r ,10 log. 20.25 1.30642. 1.30642 log. sin. 2 56 7 53" 8.71122 log. cos. 2 56' 53" 9-99942 10.01764 21.30584 (a) log. sin. 3 3' 9" 8.72633 log. cos. 2 50 ; 50" 9-99946 (log. M) ........ 1.29131 sin P 38 29 5 9-79400 M = 19".557 19-79346 (log. m) 1.51238 m = 32".53. Hence, A = 19".557 . cos. (- 11 s 28 17 ; 45"), or = 19"-557 .sin.(0 -f 3 1 42 15), fl =-.32 // .53.sin. (0 - 5 28 5 26), or = 32".53 . sin. ( -f 1 .54 34). 281 EXAMPLE VI. a Aquarii. The Latitude &c. of this Star for the year 1800 are as follow : latitude s 10 4l' 34", longitude 11 35 45, (P) angle of position .... 20 17 48, N. P. D. . 91 16 58. d Z J Z 10-f 10 + log. tan. 20 1?' 48 // , . 19.56802. . log. cot. 20 17' 48" 20.43198 log. sin. 10 41 34 . . 9.26777. ."T^"' 'f 9-26777 (log. tan. d Z} 10.30025 (log. cot. A Z) 11.16421 d Z = 2 s 3 23' 37" A Z = s 3 55' 50" .'. D Z = 26 36 23 a Q Z = 3 3 55 50 but #'s long. =11 33 45 1 1 33 45 .'. N=ll 27 10 8 n = 7 26 37 55 M m log. r 10 log. 20".25 1.30642 1.30642 log. sin. 20 1?' 48" 9-54018 log. cos. 20 17' 48" 9-97216 10.84660 21.27858 (a) log. sin. 63 23' 37" 9-95137 log. cos. 3 55' 50'' 9-99897 (log. M} 0.89523 log. cos. 1 16 58 9-99989 19-99886 (ft M= 7".S56 (log. m) 1 .27972 (a- ft m = 19".04. Hence, A = 7 /; .856 . cos. ( O 1 1 s 27 10'' 8"), or, = 7 /; .856 , sin. ( O + 3 2 49 52), a == - 19 /7 .04 . sin. (O - 7 26 37 55), or, = 19 // -04. sin. (O + 10 3 22 5). NN 282 From these expressions * A and a may be deduced, for any assigned time, by one operation. The assigned time gives O , the Sun's longitude : and we may deduce A and , either by multiplying the coefficients, that express the maxima of aberra- tion, into the natural sines of the sums of the arcs, or by a loga- rithmic process : for instance, suppose the aberrations of y Draconis were required for December 3 ; By the Nautical Almanack, . . = 8 9 10 34' arc to be added (see p. 280.) =3 1 42 (negl g . the seconds) 11 12 16 natural sine of II 8 12 16' = - .3045; .'. A = \9". 55 x - .3045 = 5".95, a quantity, with its affixed sign, to be added to the mean north polar distance in order to obtain the apparent north polar distance. -v **"* "'"- o*v Again, to find, by the logarithmic process, the aberration of ij Ursae majoris on the same day, Q =8 8 10 34' arc to be added (see p. 279.) =4 22 27 13 3 1 .... log. sin. =9.73630 log. M = 1.25609 (10-Hog.) A 10.99239 .*. A = 9"82 to be added to the mean north polar distance in order to obtain the apparent north polar distance, * These values of A, a, are to be added, as it has been already remarked, to the mean north polar distances and mean right ascensions, in order to obtain the apparent. If we wish for expressions still additive, and for the reverse operation, we must increase, or diminish, according to the case, the arcs, which are the arguments, by 6 s : thus in the last case, A' = 7".856 . sin. (0 -H 9 s 2 4Q/ 52"), a' = 19".04 . sin. (0 + 4 3 22 5) A' y and a', being quantities to be added to the apparent in order to obtain the mean north polar distances and right ascension. 283 With equal facility and brevity may the aberrations of other stars be deduced for any assigned period : but still more conve- niently by means of a Table : the columns of which should contain log. M, log. m, N 9 n, or, the sines of the sums of arcs being used, quantities (N ; , n') analogous to N, n. We will shew a specimen of such a Table by means of the results we have already obtained. Stars. Log. M. A?" Log. m. n' 96343 1 s 27 33' 1 28222 8 s 29 7' .89551 1 17 1.31311 7 28 39 Polaris . 1 30240 11 18 36 2.78382 8 15 44 rj Ursae majoris . y Draconis. . . . a Aquarii 1.25609 1.29131 .89523 4 22 27 3 1 42 3 2 50 1.47400 1.51238 1.27972 2 3 56 1 54 10 3 22 With such a Table the two rules for rinding the aberration in north polar distance and right ascension would be log. A = log. sin. (0 + N r ) + log. M, log. a log. sin. (<>.+ n 7 ) + log. m. In computing M, N', &c. we have used different periods : that of 1815, for y Pegasi ; of 1725 for rj Ursae majoris, &c. In a Table properly constructed, the numbers ought to be com- puted for the same epoch. Still, the present Table will give, if we exclude Polaris, results nearly true : for, although the angles of position and the longitudes of stars, or, to go farther, the right ascensions and declinations of stars and the obliquity of the ecliptic are continually varying, yet they may considerably vary without much affecting the values of the aberrations in right as- cension and north polar distance. Thus, of the last star in the above Tables, a change of 1 in its right ascension will not pro- duce a change exceeding 0".34 in its aberration in right ascension. A special Table then of aberrations will last 50 or 60 years, for 284 stars that have not great declinations. A Table for Polaris will require to be renewed every eight or ten years. The resulting values of A and a are to be added to the mean north polar dis- tances, and mean right ascensions, respectively, in order to obtain the apparent north polar distances and right ascensions. In the Volume of the Greenwich Observations for 1812, &c. there is inserted, at p. 250, a Table for ninety-six stars, similar to the preceding one. It is founded, however, upon thejirst of the formula which have been investigated for expressing the aber- rations in north polar distance : upon this A = M.cos. (O - N), OC BS 7 '.i.r j II ! Icc(]8..; ... . . . and since the Table is to be used for reducing apparent north polar distances to mean, it gives results with signs different from those that belong to the Table of p. 283. The two Tables, however, are essentially the same *. In the precept, (see p. 282,) for using the Table we are directed to take G , the Sun's longitude, from the Nautical Almanack. This part of the rule, however, stands in need of some modification: for, if we look (see pp. 269, &c.) to the inves- gation of the formulae, it is, clearly, a condition of such investi- gation that the Sun's longitude should be that, which it ought to be, at the time of the star's passage over the meridian f. The Sun's longitude, therefore, taken from the Nautical Almanack is not truly expressed, except (which is a particular case) the Sun * The Tables may be deduced, the one from the other. For, in the Tables for the aberrations in north polar distance, the sum of the respective numbers (such as J\ T/ , N~) always equals 9 s or 21 s . Thus, if the numbers under the column JV ', in a Table so constructed, should be for the stars, a Cassiopeae, a Ceti, a Persei, a Coronae Bor. a Herculis, respectively, c 8 s 29 22' The numbers in a I A 1 ft * Table constructed like the Greenwich Table I, \ 9 25 34 wouldbe I 5 J 56 V. 5 24 24- t The observations of right ascension and north polar distance are supposed to be made on the meridian. 285 and star are on the meridian together : for instance, the right ascension of tj Ursae majoris is 13 h 40 m 7 s , and on June 20, 1812, the Sun's right ascension was = 5 h 55 m 14 s ; consequently, the star was on the meridian about 7 h 45 m after noon, at which latter time, the Sun's longitude, by the Nautical Almanack, was 2 s 28 54' 32" : this longitude, therefore, must be increased (if 59' be the Sun's increase of longitude in twenty-four hours) by yh 4501 X 59', or, by about 19' 24". This quantity, then, in 24 the above instance, and like proportional quantities, in other instances, must, in forming the arguments ( + N f ), &c. be added, as corrections to the Sun's longitude. A Table, in the Greenwich Observations, immediately following that we have already noticed, contains, for each star, the correction due to it for every tenth day of the year. The labour of an Astronomer, in reducing his observations, is so great, that the construction of convenient Tables is a matter of considerable importance. The Tables, which we have de- scribed, hold a middle place between special and general Tables. Special Tables express, in numbers,, the aberrations of certain stars for every tenth day, or for every ten degrees of the Sun's longitude. Such Tables are the most convenient and the most sure in practice. They have, over other Tables^ that kind of advantage which Taylor's Logarithms have over Sherwin's. But they are inconvenient from the largeness of their Volume *. General Tables of aberration are^ indeed, small in size, but cannot be used without considerable computation. Besides the labour of using them there are the chances (which are excluded from Special Tables) of mistakes. From the right ascension, and declination of the star and the day we may deduce from these Tables the star's aberration ; but not without six or seven small processes of computation. * M. Zach has, in his Tabula? Speciales Aberrationis et Nutationis : Gotha, 1806, given the aberrations of six hundred zodiacal stars. These are contained in two thick Volumes. Their learned Author remarks that like Tables, for Piazzi's Catalogue of nine thousand stars, would require fourteen octavo Volumes of five hundred pages each. 286 By the Table which has been described and constructed, we arrive, with little risk of a mistake, at the result after twice using the Trigonometrical and Logarithmic Tables : once to take out a logarithmic sine : again, to take out the number coresponding to a resulting logarithm. But, perhaps, it will be better to shew the convenience of the Table by one or two illustrations. EXAMPLE I. Required the aberration in right ascension and north polar distance of Polaris* on July 23, 1800. N = 11 o c 18 27' 36 15 19 3, sin. = 9.97554 log. M = 1-30240 (10 + log. ^)=1 1.27794 A = 18".96 0=4' = 8 a 27' 15 44 12 16 11 sin. = 9.44515 log. m = 2.78382 (10+ log. m) 12.22897 .-. m = 169".42 = 2' 49".42. * It was mentioned in page 2.83, that Tables of Aberration will serve, during fifty years, for stars, the polar distances of which are not very small. But Polaris, the north polar distance of which is less than 2, requires to have a new Table of aberration constructed for it every ten years, ft Ursae minoris is in a like predicament. We subjoin what, ac- cording to M. Zach, would be tbe numbers and logarithms of maxima for Polaris for the years 1790, 1800, 1810, 1820. Log. M 1790 1.3035 1800 1.3034 1810 1.3033 ] 820 1 .3032 N' ll 5 19 1' 11 18 39 11 18 9 11 17 33 Log. m in time. 1.5945 1.6081 1.6215 1.6361 ri 8 s 16 18' 8 15 46 8 15 14 8 14 35 287 EXAMPLE II. Required the aberration in north polar distance of rj Ursze majoris on Feb. 20th : and its aberration in right ascension on Dec. 3, 1726. Feb. 20th, O = 11 s 1 & 20".2.5 _ tan. a = : sin. d v cos. Let tan . N =- = tan. ( 4- 37) ; cos. / pp 298 _ cos. G cos. I / cos. JR cos. ( O x sin. ( O -f sin. cr 300 But, since ZaT is a right-angled spherical triangle, by Naper's rule, we have 1 x sin. ZT sin. aT X sin. ZcrT; k " - sin. ZT __ cos. Z "cos. star's latitude 'cos. star's latitude Hence A: is a maximum (ri) when cos. is the greatest, that is when either =0, or 180 : in other words, when the Earth, or Sun, is in syzygy with the star : hence the maximum, or n cos. star's latitude and U n. cos. @Z (10). We might have avoided this direct process and deduced the aberrations in latitude and longitude from those in right ascension and declination. In its technical enunciation, the enquiry woulc have been to find the errors in latitude and longitude from the given errors in right ascension and declination : which error might have been found by two ways : either after Cotes's manner as Cagnoli has done, or by deducing the values of dL, d\ from some of the formulae given in p. 182. It is plain, in investigating the formulae of aberration, that we might have pursued a method the reverse of that which has been now described : that is, the first steps of investigation might have been directed to the finding out (and they are, in fact, the most easily found) the aberrations in longitude and latitude : thence we might have proceeded, by a route strictly mathematical, and without any clue furnished by the nature of the enquiry, to the aberrations in north polar distance and right ascension. Such, generally, has been the course of investigation. Clairaut, Thomas Simpson, Cagnoli, and Suanberg have followed it. The last-mentioned Author, in his Treatise*, has derived his for- mulae, from the differentials or the fluxions of equations 1 3 2, of * Exposition des Operationsfaites en Lapponie, pour la determination cr*un arc du meridien, &c. Stockholm, 1805. 301 p. 182. He goes, however, farther than the generality of authors*, and adds, to his formulae, certain minute corrections which are due to the eccentricity of the Earth's orbit. But, as it has been already remarked, there is a great variety in the ways of deducing the formulae of aberration. The formulae of aberration in right ascension and north polar distance, being the most important, have been investigated by the most direct and shortest methods. Of such investigations, the curve of the star's aberration, described during a year, was not a condition. It was not enquired, since it was not essential to enquire, whether the curve were circular or elliptical. The laws of the several aber- rations (which laws are expressed by their appropriate formulae) are indeed connected with the form of the curve, inasmuch as two results derived from a common source are connected. If the one were varied the other would : and, in consequence of this sort of connexion, it is easy to see that, from one established or proved, the other might be deduced as a Corollary or conse- quence. From the nature, or law, then, of the curve apparently described, during a year, by a star, in consequence of the prin- ciple of aberration of light t the respective formulae expressing the aberration in its several directions may be supposed to be derived. And, in fact, the original proposition in the present theory was ' that the apparent path described by a star, in con- sequence of aberration, was a circle the plane of which was parallel to the ecliptic.' * Delambre has done the same thing (see his Astronomy, pp. 110, &c.) We have not entered into these investigations, of which, perhaps, the chief use is the shewing that the corrections sought are so small, that they may safely be neglected. If IT be the longitude of the perigee the term to be added to the aberration in right ascension, (see p. 297-) is -0".34 : 5- (cos. I . cos. JR. . cos. 7r 4- sin. JR sin. TT). sin* o The aberration on north polar distance will be 20".25 (cos. M sin. cos. I. sin. JR cos. ) + 20",25 . sin. / cos. cos. S - 0".34 sin. I cos. TT cos. S -f tf',34 {sin. 3 (cos. I sin. M cos. TT - cos. M sin. *)}. 302 This certainly was a happy beginning of a beautiful Theory. But the origin of the formulae was thrown a little farther back than when they were made to commence from the effects of aber- ration on the latitudes and longitudes of stars. These latter quantities are not, except in particular cases, objects of observa- tion. But the circle of aberration, is still less so; it is a mere fiction whatever be the star in the Heavens we select for our observation. It cannot be called a phenomenon, because it would be only such were the circumstances of observation dif- ferent from w^hat they really are. The curve would be a circle, if we saw the aberrations in directions parallel to the plane of the ecliptic. And this point we will now proceed to establish. By p. 267, the aberration always takes place in a plane passing through and P = B . sin. Ps. The Earth's orbit being, nearly,' circular, ET is, nearly, per- pendicular to SEP, and PT (SE being extremely small rela- tively to SP) is, nearly, a quadrant. Now (see p. 268,) the aberration varies as the sine of s T, and, as we have just seen, the parallax varies as the sine of sP; sP will become sT after E has described 90 : consequently, the formula which ex- pressed the variation of the aberration, three months previously, will now express that of the parallax : or, since the aberrations in opposite points of the Earth's orbit are equal, although in dif- ferent directions, the formula for the aberration, three months hence, will be the formula for the parallax at the present time. 313 We will now consider which of these two formulae ought to be taken, so that the signs may be the same in each. The star . parallax = 0".78, the sine of 8 s O e 44' being negative. -Again, August 1, Q = 4 8 8 59' 5 53 9 9 52. .. log. sin. = 9.9935 1.9549 (log. .88) 19.9484 /. parallax = <)",88. Again, Nov. 11, =r 7 s 18 69' 5 53 12 19 52 log. sin. = 9.5312 T.9549 (log. .306) 19.4861 *. parallax = O x/ .306. 315 Generally, if log. A = log. sin. ( + N') + log. M 9 log.P(inN.P.D.) = log. sin. (O + 90 + 2V ') + log. M~ 1.3064, \" being the semi-annual parallax. We might, then, were it worth the while, from a Table of Aberrations, (see p. 283,) form a Table of Parallaxes : for instance, Stars. Number for N.P. D. Log. maximum for N. P. D. Number for Right Ascen. Log. maximum for Right Ascen. a Lyrae . . 2 s 24 42' T.9452 2 8 23 5' 0.1066 a Cygni. . 5 53 L9549 I 23 50 0.1334 ft Aurigae. 7 17 1.5652 9 3 28 0.1497 a Aquilae. 5 13 9 T.7175 2 6 34 T.9982 From this Table, and by the above formula, we may imme- diately compute the parallaxes of the above stars, for any day in the year. With a facility like that which the preceding transformations admit of, and on the same principle, we may transform any other formulae of aberration into formulae of parallax : for instance, let us take Delambre's formulae, (see p. 301.) A(in N. P. D.) = 20".25. cos. S(cos. JR. sin. O - cos. I. sin. ^3lcos. O ) -f- 20".25 sin. I. cos. ; therefore, adding by the rule (see p. 3 1 3,) 3 s to , par x .(inN.P.D.)=l".cos.5(cos. Isin.JRsin. -f cos.^Rcos. 0) l". sin. /. sin. O Again, 20 /7 .25 1 1 cos. M. cos. O + sin. M sin. 0); Aberr Q . in JEl= sin. .*. par*, (in Jt) = - s (cos. /, cos. JR . sin. O - sin. JR cos. ). sin. 4 316 Again, L and X representing, respectively, the longitude and latitude of a star, 20". 25 the aberration in longitude = cos. (L G ) ; cos. X l" .'. the parallax in longitude = sin. (L G ). Again., Aberration in latitude = 20".25 . sin. X . sin. (L G ) ; .*. parallax in latitude = \" . sin. X . cos. (L G ) Hence, when the aberrations in longitude and latitude are the greatest, the parallaxes in longitude and latitude are the least : and conversely. Since the variations of the parallax and aberration are ex- pressed by the same formulae, the curves (should any question arise concerning them) of aberration and parallax are similar. That is, (see p. 303.) the curve apparently described by a star, in consequence of parallax, is an ellipse, of which the minor axis is 2". sin. X, 2" representing the major axis, or, generally, if 2vr should represent the major axis, STT sin. X would represent the minor. These ellipses, like those that represent the aberration, are easily traced out. M. Lalande in his Astronomy, vol. Ill, has traced out the ellipses of parallaxes of Sirius and Arcturus : now the latitude of Sirius (1805) =0 S 39 33' 40" of Arcturus 30 52 1? longitude of- Sirius 3 1 1 - 23 of Arcturus 6 21 30 0. Hence, (see 1. 7,) paral. in lat. of Sirius = \" . sin. (39 33' 40") x cos. (3 s 1 1 23' - G ) .'. the parallax is a maximum either when G = 3 s 11 23' or when G = 9 11 23 the two maxima of parallax happens then, about July 3, and Januarys, and are, respectively, equal to 0".6367, -f-0 A '.6367: 317 draw, therefore, as line parallel to the ecliptic equal to 2", and, through its middle point, draw, on each side of it, a line equal to .636? which will be the semi-minor axis. The whole line will be the minor axis ; at the extremity farthest from the ecliptic the star will appear to be on January 3,' at the extremity nearest to the ecliptic on July 3. In order to find when Sirius will appear to be at the ex- tremity of the major axis, we must make in the expression for the parallax in longitude, namely, in - sec. X sin. (3 s 1 1 23' - ), 3 s 11 23' = 3 s ; whence = 11 23': the time, corresponding to this longitude, is April 1 : at the in- terval of half a year, the star is at the other extremity of the axis major. With regard to Arcturus, the expressions for his parallaxes in latitude and longitude are, respectively, l" . sin. X . cos. (6 8 21 30' - ), and l" . sec. X . sin. (6 21 30 0); consequently, he is at the extremities of his minor axis, either when = 6 s 21 30', or =21 30' : that is, on October 15th, and April llth ; and he is at the extremities of the major axis of the ellipse of parallax, either when =3 8 21 30', or =9* 21 30', that is, on July 14, and January 12. The minor axis of the ellipse is 2". sin. 30 52' 17", or 2 X .513 = l".026. It was in the observations of the pole star that Flamstead thought he discovered the existence of parallax. Now (see p. 278,) the aberration in N. P. D. = 20".06 sin. ( -f 1 1 s 1 8 17') ; 2C/' 06 /. (see p. 313,) the parallax = ' sin.( + 14 18 17) = 0".99.sin (O -f 2 18 17). The parallax, therefore, is nothing when = 3 s 11 43', and very small, when O is nearly of the above value ; that is, about the middle of summer. In winter the same thing will take place ; that is, the parallax in declination, (supposing the star to 318 have au annual parallax) will be extremely small. Now, Flamstead, from his observations, found the declination of the pole star to be less in summer than in winter by about 40 ' ; 4 or, which is the same thing, he found the diameter of the small circle, described by Polaris round the pole,, to be larger in sum- mer than in winter by about l' 20". But this phenomenon could not, as we have shewn, arise from parallax : still it was a phenomenon : in other words, the observations of Flamstead were just*: there was such a difference as he noted, but it arose not from parallax but aberration : which it is easy to shew : thus, by the expression in the preceding page, when O = 3 8 11 43', the aberration in N. P. D.= 20".06 . sin. 15" = 20".06, and, at the opposite point of the Earth's orbit, = 20".06. Hence, the north polar distance was greater, or the declination less, in the former period than in the latter by 20".06-f-20" .06, or 40".12, which agrees exactly with Flamstead's Observations, but overturns his inferences. The preceding part of this Chapter relates to the fixed stars. Should these, or any of them, have an annual parallax, their apparent places in right ascension and north polar distance will, by reason of such parallax, differ from their mean. The correc- tions, for reducing the one to the other are furnished by the preceding formula: which formulas (see p. 315,) are, with regard to the analytical law of their construction, the same as the formulee of aberration. We could easily, then, correct the * < The observations of Mr. Flamstead of the different distances of the pole star at different times of the year, which were through mistake looked upon by some as a proof of the annual parallax of it, seem to have been made with much greater care than those of Dr. Hook. For though they do not all exactly correspond with each other, yet from the whole Mr. Flamstead concluded that the star was 35", 40", or 45" nearer the pole in December than hi May or July; and according to my hypothesis it ought to appear 40" nearer in December than in June/ Bradley, Phil. Trans. No. 106, p. 66l. 319 right ascension and north polar distance of a star on account of parallax, if a certain annual parallax were assigned to it; or, from observing the maxima of parallax in north polar distance, we could (as in the case of aberration, see p. 2Q2,) assign the radius of that circle of parallax which a star situated in the pole of the ecliptic would apparently describe. There is no difficulty, indeed, in correcting an observed distance on an assumed quan- tity of parallax, nor, should any differences in the places of stars, not accounted for on established theories, be observed, any dif- ficulty in determining, whether such differences can be imputed to parallax. The real difficulty is to make observations that can be relied on to the fractions of a second of space : since the question is, whether there can be shewn in observations parts of a second of space not accounted for on known theories, and not attributable to the errors of observation. The subject has, at various times, occupied the attention of Astronomers. Before the discovery of the aberration of light, the main object, in the search after parallax, was the establish- ment of the Copernican System * as far as that could be effected by the proof of the Earth's motion. This was Hook's object, Flamstead's, and Bradley ? s. The first asserted the existence ot parallax by relying on his own faulty observations : the second by faulty inferences. From good observations Bradley shewed the errors of Hook's observations, and of Flamstead's reasonings ; he made it evident that the latter Astronomer, in his search after parallax, had stumbled on the "effects of aberration, which he mistook for those of the former inequality. Bradley himself thought f that the stars had no sensible parallax, and that he * ' To furnish the learned with an Experimentum Crucis to determine between the Tychonic and Copernican Systems/ Hooke's Treatise en- titled, An attempt to prove the Motion of the Earth by Observations. f- * I am of opinion, that if it were 1", I should have perceived it, in the great number of observations that I made especially of y Draconis : which agreeing with the hypothesis (without allowing any thing for parallax) nearly as when the Sun was in conjunction with, as in opposition to this star, it seems very probable that the parallax of it is not so great as one single second; and consequently, that it is above 400000 times farther from us than the Sun.' Phil. Trans, for Dec. 1/28. p. 66*0. 320 must have discovered such an inequality in 7 Draconis and rj Ursae majoris, had the parallax in either of these stars amounted to l". It is right, however, to observe that this remark of Bradley is not much to be relied on, since it was made at a time when the stars were subject to an inequality, of the existence of which he was then ignorant. The question of parallax has, of late years, been revived by Dr. Brinkley* who thinks that he has found parallax in a Aquilae, a Lyra, a Cygni. In consequence of this opinion, fixed tele- scopes have been directed, at the Greenwich Observatory, towards certain stars, with this special object in view : namely, that each telescope should take into its field of view, at least, two stars differing from each other in right ascension. One of these telescopes is directed towards a Cygni, of which the north polar distance is about 45 22' and right ascension 20 h 35 m . But the north polar polar distance of /3 Auriga? is about 45 5'. There- fore, since the difference of their declinations does not exceed 18', the telescope can be so placed that each star, when it passes the meridian, shall be in the field of view : one passing to the north of a middle wire, the other to the south. Now the right ascen- sion of /3 Aurigae is about 5 h 46 m , and, therefore, it will pass the meridian about 14 h 40 m before a Cygni. The effects of parallax will in certain seasons be to decrease the north polar distances of both stars : in other seasons to increase them : and in others to increase the north polar distance of one whilst it decreases that of the other : and conversely. These combined effects are shewn by means of a micrometer which measures every day (every day on which an observation can be made) the difference of the declinations of the two stars. We ought rather to have said, that the combined effects of parallax (should there be any) may be extricated from the differences of declination which, by means of the micrometer attached to the telescope, are instrumentally shewn. For, besides the substantial difference of the mean declinations, the apparent distance of the two stars arises from the different effects of aberration, nutation, &c. on the two stars. These latter effects being accounted for, or the * Irish Transactions, vol. XII. Trans. Royal Society, 1818. 321 observations being corrected, if there should appear to be any other difference than that of the mean declination, the causes of such difference would become matter of enquiry. In order to ascertain whether parallax is one, or a sole cause, we must compare the unaccounted for differences with the mathema- tical calculations of the combined effects of parallax. This is easily effected : Thus, computing, on the principles and according to the methods of pp. 274, &c. the numbers due to the parallaxes in north polar distance of a Cygni and /3 Aurigae, and the maxima, we have the number for a Cygni = 5 s 53', maximum = 0".913 for /3 Auriga* = 17 17, maximum =0".367 ; consequently, the combined effect of parallax in north polar distance on these two stars is 0".913 X sin. ( O + 5 8 53') 0".367 . sin. ( O +0 9 17 I?')- For instance, if we wish to find the combined effect on July 1 , October 1, and Oct. 11, we have or Cygni, July 1. f3 Auriga, July 1. = 3 s 9 23' .............. 3 s 9 23' 5 53 .............. 17 17 8 10 16 log. sin. = 9.9737 3 26 40 log. sin. = 9.9549 log. 913 = 9-9549 log. 367=9.5652 log. (-.848) 9.9286 (log. .327) 9-5157 Hence, the combined effect = -O".848 0". 32 7= l // .75. Again, a Cygni, Oct. 1. /3 Aurigae, Oct. I. = 6" 8 8'. 6 s 8 8' 5 53 17 17 11 9 1 log.sin.=9.5540 6 25 25 log. sin. = 9-6326 log. 91 3=9.9549 log. 3678 =9-5652 (log. .318) 9.5089 log. (-.157) 9.1978 /. combined effect = 0".3J8 + 0".157 = 0".l6l. ss Again, a Cygni, Oct. 11. ft Aurigae, Oct. 11. O = 6" 18 0' 6 s 18 0' 5 53 17 17 11 18 53 log. sin. = 9.2851 7 5 17 log. sin. =9-76l6 log. 913 = 9.9549 log. .3678 = 9.5652 (log. -.173) 9-2400 (-.212) 9.3268 .*. combined effect = 0".173 + 0".212 = // .039. But we will now dismiss this class of parallaxes, the discus- sion of which is not suited to an Elementary Treatise, and turn our attention to the parallaxes of the planets, which can not only be proved to exist, but which are, in Practical Astronomy, usec for determining the distances of those planets that are affected by them. We have already (see p. 209.) entered on this subject. If s be a planet, C the centre of the Earth, Z the zenith of the spec- tator placed at A, then s is seen from A in the direction Asm, and from C in the direction Csm. If the latter be held, or be de- fined, to be the true or Astronomical direction, then A s n is the apparent. As far as parallax is concerned, m is the planet's true 323 place, n its apparent, and the angular distance of these places, or the angle nsm( = AsC) is the diurnal parallax. By Plane Trigonometry, n A AC rj AC - VA sin. CsA = -77- x sin. CAs = -77- . sin. ZA s ; Cs Ls but ZAs is the star's apparent zenith distance^ consequently, if CA, Cs be held to be constant, sin. CsA, or sin. parallax oc sin. apparent zenith distance. Hence, the parallax is greatest when the zenith distance = 90, or when the planet appears in the horizon. Let P be the greatest parallax, p the parallax at any other zenith distance, then sin. P : sin. p :: 1 : sin. apparent zenith distance ; .*. sin. p = sin. P . sin. zenith distance = sin. P . sin. (D -f- p), D being the angle ZCs. We may thus 'approximate to the value of p in terms of P and D, sin. p =. sin. P sin. D cos. |> + sin. P cos. D sin. p ; .*. tan. p = sin. P sin. jD + sin. P cos. D. tan.jj. Instead of tan. p in the last term, substitute its value as expressed by the equation just obtained ; then an.p= sin. P.sin.D-f-sin. : *P.sin.Dcos.D + sin. a P.cos/*Z)tan.j)' Repeat the operation, and tan. p = sin. P . sin. D + sin. 2 P . sin . D . cos. D -f sin. 3 P. cos. 2 D sin. D + sin. 4 P cos. 3 D . sin. D -f- sin. 4 P cos. 4 JD tan. p : the law of the terms is evident ; but, for almost every case that can occur, the summation of the three first terms will be suf- ficient, P not exceeding 1. If, instead of tan. p, we wish to have an expression for p, we may easily obtain such by means of the expression p = tan. p y tan. 3 p, which is the approximate expression for the arc in terms of the tangent, when the arc is small. The second term of this expres- sion ( Y tan. 3 p) will produce from the equation 324 tan. p = sin. D.sin. P + sin. D. cos. I). sin.* P + sin. D.cos.* D. sin. 3 P sin 3 D sin 3 P a term such as ' '- ; if, therefore, we do not con- tinue the series beyond terms involving sin. 3 P, we have p = sin. D . sin. P + | sin. 2 D sin. 2 P -f sin. 3 P (sin. D cos. 2 D - ^ ) ; sin. 3 D 3 ^ sin. 3 D but sin. D cos.* D = sin. D sin. 3 D 3 3 4 sin. 3 D 3 "' sin. 3 D (see Trigonometry, p. 47.) = Hence, ;> = sin. D . sin. P -f^ sin. 2D . sin. 2 P -f j sin. 3D . sin. 3 P, which, as it may be fairly conjectured, is only part of a series obeying the same law. In an assigned instance, the resulting arithmetical value of p would be in terms of the radius. It is more convenient to have such value expressed in seconds of angular space. Now, p being small, p : arc (=r jp) :: sin. l" : l"; ""' 1 "--'- arc(= ^ ) = ^ ; .'. expressed in seconds, ,=>" ^ .- in .p | sin ' 2jP .cin.P 1 s ' m ' 3D , 3 j n p sin. l" 2 . sin. l" ' 3 . sin. l" * There are, besides what we have given., other series and ex- pressions for computing the parallax, from D the zenith distance, and P the horizontal parallax. D can always be determined (very nearly at least) by observation, but hitherto no method has been given of determining P. In short, although we know what parallax is, can symbolically express it, and can compute 325 the parallax at a given zenith distance from the horizontal, yet we are at present without a practical method of determining it. We will now turn our attention to that point. Let A and B be two places on the Earth's surface situated in the same meridian ; and suppose, by the methods described /* in pp. 12, 129, their latitudes to be determined. When the planet S is on the meridian, let its zenith distances ZAS (z), Z'BS (*'), be, respectively, observed at A and B ; then, since ACB the difference or sum of the latitudes (in the diagram the sum) is known, we have L JSB = 360 (180 2 + 180 - z 1 + ACB) = z + z' - ACB ; hence the angle ASB, (sometimes called the parallax, being the angle which a chord AB subtends at S,) is known : call this angle Ay and the angles CSB, CSA, p', p, respectively. A is not the angle (see the former Figure) which we are seek- ing : it is, either the angle CSB (p) or the angle CSA (p). Now , CB CA CB sin. p ==sm. 2.-^xT, and sin. p ~ sin. z .777; = sm z ~7rz : C3 Co L/o 326 sin. z f hence, sin. p' , or, sin. (A p) =: sin.jp. , or, sin. z 1 sin. A . cos. p cos. A . sm. p = sin. p . : r sin. z whence, dividing by sin. A . sin. p, and transposing, sin. z' cot. p = cot. A -\ : : . sin. z . sin. A This formula may be thus adapted to logarithmic computa- tion : / sin. z' \ cot. p = cot. A I 1 -f I ; V sin. z . cos. A/ make - = (tan.0) 9 ; /. cot. p = cot. A . (sec. Of-, sin. z . cos. A and consequently, log. cot. p = log. cot. A -r- 2 log. sec. 20 ; being determined from log. tan. = | (30 -f- log- sin. z log. sin. z log. cos. A). From this formula, p may be computed ; but since, in point of fact, the parallax of all heavenly bodies that are observed is very small, a much simpler formula, and accurate enough for computation, may be exhibited : Thus, A, p, p, being very small, are nearly equal their sines \ instead of sin. (Ap) = sin. p. ~~ , we may write sin. z sin. z' A p = p . ; ; whence sin. z A sin. z sin. z + sin. z' ' A . sin. z or = If we wish to express the horizontal parallax, since sin. p = sin. P . sin. z, or p = P . sin. z, P= -. _ sin. z + sin. z and, if we restore the value of A, making / ACB = L + L f -z' - (LL') p = sin. 2;-f-sin. z' As an example to this formula, we may take the observations of Lacaille, at the Cape of Good Hope, and of Wargentin, at Stockholm : 1751, Oct. 6. At the Cape, zen. dist. (z) of r= 10;059 rad. = 60.1. rad. . 333 Hence the mean distance of the Moon is about 60 radii of the Earth. Here, the greatest and least distances are., respectively, 63. 6619 ,63. 6619 - TT- rad. eH, and rad. B, or 0.99567 1.13714 63.94145 X rad. , and 55.98725 rad. *. The general use of parallax is, then, to determine the distances of heavenly bodies : but the special object for which it has been here introduced, is the reduction or correction., which must be made, by means of it, to the observed place of a body ; to pre- pare, for instance, an observed altitude of the Moon, for the deducing its declination. Now since, by the principle of the re- duction, we imagine a spectator to be in the centre of the Earth, it is plain, from the inspection of the Figure, p. 322, that the planet seen from the surface, must be lower, that is, nearer to the hori- zon than seen from the centre. But, this last is assumed to be the true place, or, it is made the place in Astronomical computa- tions : and, accordingly, a body seen from the surface must be said to be below its true place, or to be depressed by parallax. This depression takes place in a plane passing through, the centre of the Earth, the spectator, and the observed heavenly body ; it takes place, therefore, like refraction, in the plane of a vertical circle. Now, the meridian is a vertical circle ; the declination of an heavenly body then, as determined by its me- ridian altitude (see p. 151,) w'ill be affected by the whole quantity of parallax ; but its right ascension, as determined by the time of transit over the meridian, will not be at all affected. We will now subjoin two instances, in the first of which the Sun's declination is deduced from his observed zenith distance : in the second the Moon's declination from her observed altitude : both observations are corrected for refraction and parallax : in * It is plain from the above instances, that it is shorter to compute by the French than by the English expression : for, in the former, we may immediately divide the numerator (63. 66 19) by the denominator ; which we cannot do in the latter. 334 the former instance, then, the refraction is added to the zenith distance, the parallax subtracted : in the latter the refraction is taken away from the altitude, the parallax added. EXAMPLE I. Altitude of Sun's upper limb. 62 30' 30".5 error of collimation 34 . 5 62 29 56 apparent zenith distance .......... 27 30 4 sin. .4617 refraction ..................... 29 27 30 33 (8" X .4617) Parallax ......... 4 27 30 29 semi-diameter of the Sun ......... 15 46 27 46 15 latitude of the place of observation . . 48 50 14 declination of the Sun 21 3 59 EXAMPLE II. Altitude of Moon's upper limb 51 1 1' 24'' refraction r 45 ri; 51 10 39 (55' 24" x .6246) Parallax. . , 34 36.2 51 45 15.2 semi-diameter . . . . 15 8.8 altitude of Moon's center 51 30 6.4 co-latitude of Greenwich 33 3] 21 .5 decimation of the Moon 12 58 44.9 In this case, the horizontal parallax for Greenwich is taken = 55' 24"; and the multiplier .6246 is the natural sine of 335 38 $9' 18", which is the zenith distance 38 49' 2l" diminished by 10' 3", the value of the vertical angle (see p. 330.) 55' 24" represents the horizontal parallax for Greenwich, being the parallax on a spheroid at the latitude 51 28' 40", de- duced from, what is called, the Equatoreal parallax ; which is the difference of the Moon's place in the Heavens seen from the equator and the Earth's centre : the Moon being in the horizon i of the spectator. But this equatoreal parallax is deduced from 1 the equatoreal parallax at the mean distance of the Moon*, which according to Mayer, is 57' ll".4. There is, therefore, the equatoreal parallax at the mean distance ; the horizontal equa- toreal at any distance ; the horizontal for any latitude, and the common parallax for any altitude : and, in observations of the ' Moon and in calculations from them, all these circumstances must be attended to. The quantity of parallax has been computed (see p. 327,) by means of observations made in the meridian. It may also be I computed, as refraction was (p. 238,) by observations out of the plane of the meridian ; for, in these latter, parallax, which causes a variation in the right ascension, may be computed from such variation. For instance, let M be a planet in its true place, m in its apparent place, Mm lying in a vertical circle ZM m (see p. 238.). Now, m being the place instead of M, the time from the passage over the meridian, will be represented by the angle * The Moon's greatest parallax is 6l' 32"; her least 53' 52". 336 ZPm, instead of the angle ZPM : the change, therefore, in the time, or in the apparent right ascension of the planet, caused by parallax, is represented by the angle FPv; and this change may be thus estimated: if M were a fixed star, Mm would be nothing, and there would be no parallax affecting the time, 01 the right ascension : two fixed stars then, near to each other, that crossed the vertical wire of a telescope in the plane of th< meridian, after an interval of t seconds, would also cross th< vertical wire of the telescope in a plane, not that of the meridian, after the same interval t. But if, instead of one of the fixed stars, we take a planet having parallax, then if the above-men- tioned interval were t seconds on the meridian (where parallax does not affect the right ascension,) it could not be t seconds out of the meridian, but, as the figure shews, something more ; for instance, t-\-e seconds. Now e is reckoned, or known, by means of a chronometer ; and thence, a horizontal parallax (P) may be computed from this formula 15 X eX cos. dec. cos. lat. x sin. hour angle ' which may be thus proved : Vv = Mn . sec. VM = M m. sin. ZMP . sec. VM = P . sin. ZM . sin. ZMP . sec. VM = P. sin. ZP . sin. ZPM . sec. VM, (for sin. ZM. sin. ZMP = sin. ZP . sin. ZPM Trig. p. 155.) Hence, sin. 2P.sin. ZPM. sec. VM' or, since 360 : 24 h :: Vv : e ; and since sin. ZP = cos. latitude, I 1 sin. ZPM=sin. hour angle (h\ sec. VM = P = cos. VM cos. dec. 15 .e.cos. dec. cos. lat. X sin. hour angle This expression applies to the case when the planet and star are observed, firstly, on the meridian, and afterwards when they have passed it : if they are observed before they are on the me- 337 ridian, then a similar expression would obtain for a line V 1 V analogous to Vv ; and we should have P cos. lat. sin. ti - VV = cos. dec. Hence, if the difference e belongs to two observations of the star and planet, the one made to the east, the other to the west of the meridian, we have T , . T7 ., , P cos. lat. sin. h P cos. lat. sin. A' Vv+V'v, or ex 15 = -\ , cos. dec. cos. dec. and accordingly, e X 15 cos, dec. P = cos. lat. x (sin. A-j-sin. K} \A 1 K. \S rtr\c* /!.,< (Trig. p. 31.) cos. lat. x (sin. A+sin. e X 15 x cos. / . ft*t-A\ s 2 cos. lat. f sin. J cos. f In the preceding investigation it has been supposed, that e arises solely from parallax : but since, during the observations, the planet will have moved either from, or towards the star, the noted difference of time, or excess above t seconds., will be com- pounded of the effect of parallax, and of the time due to the planet's motion, during the interval of the observations. EXAMPLE. Aug. 15, 1719. Paris. By the observations of M. Maraldi at 9 h I8 m , Mars passed the vertical wire 10 m 17 s after a small star in Aquarius; and, seven hours being elapsed, 10 m J s after. But in this interval (seven hours) Mars had approached the star by fourteen seconds ; that is, had there been no parallax, the former difference of passage, which was 10 17 s , would have been reduced to 10 m 17 s 14 s , or, 10 m 3 s : but, by the second observation, the difference of passage is only 10 m 1 s , consequently, the effect of parallax is (10 m 3 s )-(10 m 1 s ), or 2 8 : and this is the value to be substituted for e in the preceding expression : and since, by observations at the time, it appeared that U U 338 Declination = 15 0' 0" log. cos. . . .9-9849438 h = 56 39 (log. 15 1.1760913) h' = 49 15 h + h' = 52 57 log, sin. . ..9-9020628 h- h' = 3 42 log. cos.. . .9-9990938 latitude of Paris = 48 50 12 log. cos 9-8183630 We have, from the logarithmic formula of p. 337, log. P = log. 15 + 20 + log. cos. 15 - (log. cos. 48 50' 12" -f log. sin. 52 57' + log. cos. 3 42') = 1.4415155 ; .-. P, the horizontal parallax of Mars, is 27 7/ .638 (See Mem. de VAcad. 1722 ; and Lalande's Jstron. torn. II. p. 356). Some additions to the preceding investigations will be subse- quently given in the Chapters on the ' Occultations of fixed Stars,' and ' the Transit of Venus J CHAP. XIII. ON PRECESSION. Formula for computing the Precession in North Polar Distance and Right Ascension. Uses of the Formula in correcting Observations. W E have already shewn in pp. 189> &c. by a mere compa- rison of the catalogues of stars formed for different epochs, that the north polar distances and right ascensions of stars are conti- nually varying ; the latter, generally increasing with the time : the former, however, decreasing, if the stars be in either the first or fourth quadrant of right ascension, but increasing, if the stars be in the second or third. It was also shewn in the pages just referred to, that the above phenomena, of the changes in the north polar distances and right ascensions, could be accounted for, by attributing to the pole of the equator a slow circular motion, and contrary to the order of the signs, round the pole of the ecliptic. This circular motion of the pole produces a corresponding change in the equator. It makes to vary, the intersection of the equator and ecliptic, and produces, in that point of intersection, a retrograde motion denominated the Precession of the Equinoxes, The philosophers, who observed the phenomena of precession and of the changes in the positions of stars, conjectured that such phenomena could be accounted for, by attributing a motion to the pole of the equator round that of the ecliptic. Physical Astro- nomy has shewn this conjecture to be true. We may, therefore, recur to the diagram of p. 192, in which the path of the pole of the Earth is represented by a small circle described round the pole of the ecliptic, and, by means of it, compute the formulas of the changes of the north polar distance and right ascensions of stars. 340 If cr then be a star situated in the second quadrant of right ascension, the polar distance, by the translation of the pole from P to p, is changed from Per to pa- ; and the variation of north polar distance is pa- Pa- ~pr, nearly, but pr = Pp. cos. Ppr = Pp . sin. qps = Z PTrp.sin. TrP.sin. qps\ now, the angle P ?rp, is the angle described in a given time (during the translation of P to p} by P round TT : its measure is equal to the arc that represents the retrogradation of the point T , or, in other words, the precession T ' Suppose Pp to represent the motion of P during a year, then (see p. 187.) T T ' = 50".l : again, qps is equal to (according to the position of the star in the present diagram) #'s M - 3 s : and TT P measures the obliquity (I) of the ecliptic, hence, per Po- = 50". 1 sin. J.sin.(#'s Jl 3 f ) = - 50". 1 . sin. /.cos. #'s jJl. 341 According to the construction of the present diagram, the star is in the second quadrant of right ascension : consequently, cos. JR. is negative, and pa- P 270, d$ is negative ; when > 90 and <270, positive. When JR = 90, or = 270^, that is, when the star is in the solstitial colure, cos. jR=0, and there is no pre- cession in north polar distance. When JR = 0, or = 180, cos. JR = + 1, consequently, the precession in north polar distance of stars situated in the equinoctial colure is the greatest, y Pegasi is nearly so situated, and its precession in north polar distance, nearly equals to 50". 1 sin. 23 27' 50" = - 19".9 : all which conclusions agree with those of Chapter VIII. By a like way we may arrive at the precession in right ascen- sion. Thus, the equation (4) of p. 182, is sin. X = cos. $ . cos. / sin. 3 . sin. / . sin. JR ; .*. = dS (sin. 3 cos. I + cos. $ sin. I sin. JR) dJR . sin. 5 sin. / cos. JR k 342 In this equation, instead of d$, write its value dL. sin. I. cos. JR : after which substitution, each side of the equation is divisible by sin. /.cos. ^l; accordingly, dJR, sin. $ = dL (sin. $ .cos. /-fcos.$. sin. /.sin. JR), and dJR=z50".l (cos. I -f cot. 3 sin. /. sin. J&\ which is the expression for the precession in right ascension during a year*. The precession in north polar distance depends, as the ex- pression for it shews, on the right ascension and not on the decli- nation. The precession in right ascension depends both on the star's right ascension and its declination. The first term in the preceding formula is 50". 1 .cos./, which involves neither the right ascension nor the declination. It is independent, therefore, of the star's place, or, in other words (and as it is commonly stated) it is expressive of that part of the precession in right ascension which is common to all stars. Let a star be situated in the equator, then $ = 90, and cot. < = 0, consequently, dJR = 50".l .cos. / = 50".l X .917S=45".95. The precession, therefore, of an equatoreal star in right ascen- sion is expressed by that part of the precession which is said to be common to all the stars, and which is also the same as the retrogradation of the first point of Aries in right ascension. * We may easily obtain the same expression from the diagram of p. 340. Thus, the right ascension VQw becomes, by the effect of pre- cession, T'qs-, and T'g = T'v+^-M* = fv -f yQw-f-fs. The variation in right ascension, therefore, is E=rT. cos. /= 50.l. cos. J^^ sin. .sin. Ppr.cot. Po- = . sin. TT P .sin. Ppr . cot. P 12 h , the sin. JR, becomes negative, and consequently, the second term of the above formula is negative. If it should exceed the first term, the precession in right ascension would be negative : and this happens with one (/3 Ursae minoris) of the forty-five principal stars inserted in the Nautical Almanack. Its annual precession in right ascension is nearly (y.267 *. This circumstance (that of a negative precession in right ascension) will not take place with any of the thirty-six principal stars formerly inserted in Dr. Maskelyne's Catalogue : for, amongst the last twenty stars, the right ascensions of which are * According to the subjoined computation, log. cot. S, or log. cot. 15 6' 24" ......... 10.5690 log. sin. JR, or log. sin. 42 35 ......... 9.8303 log. sin. 7, or log. sin. 23 27 50 ......... 9.6000 cos. I = .9172; .-. precession = 50". 1 (.9 172 --9985) = 50". 1 x .0813 = 4.''.07, nearly, and in time = 5 .267, nearly. XX 346 greater than 12 h , the star that has the least north polar distance, is a Lyrae, the co-tangent of which distance (51 24') is <1: consequently, since sin. JR cannot, in its greatest negative value, exceed 1, and since cos. 23 27' 50" is > sin. 23 2?' 50", cos. 23 27' 50" + sin. 23 2?' 50" . sin. M . cot. 51 24', must be positive. In Wollaston's Catalogue of circumpolar stars there are abundant instances of stars, the annual precessions of which in right ascension are negative. Since the precessions in right ascension of some stars are positive, of others negative, there must be some stars so situated as not, during short periods, at least, to be affected in their right ascensions by the precession. The places of such stars must depend on the equation, cos. / -f sin. /. sin. JR cot. $ = 0, which equation, in other terms, is sin. JR -j- tan. $ cot. 1=0. But the equation (5) of p. 182, is cot. P = cos. $ tan. JR + in. $ . sec- JR . cot. /, cot. P. cos. M . or = sin. JR + tan. o . cot. 1 ; cos. o therefore cot. P must =0, or P the angle of position must = 90. The sixth equation of p. 182, is cot. P = cos. X sec. L cot. I sin. X . tan. L ; therefore if P = 90, cot. X = tan. I . sin. L. Hence, by assuming certain values of the longitude, we may determine the corresponding latitudes : for instance, let L = 10, then log. sin. 10 =19-23967 log. tan. 23 28'. . . . = 9.63761 28.87728 /. cot. X = 8.87728, and X = 85 4l'. 347 In like manner we may form other corresponding values, and arrange them thus Dng s . 0, 10 0', 20 0', 30 0', 50 0', 70 0', 90 0', at s . 90, 85 41, 81 34, 79 44, 71 36, 67 47, 66 32. We will now give one or two Examples of the formulae of precession. EXAMPLE I. Required the annual precession in JR, of 7 Pegasi (Algeriib). supposing its right ascension to be O h 2 m 56".79, and its north polar distance to = 75 55' 44", 50".l . cos. I, computed. log. r. 10 log. 50".l = 1.69983 log. cos. I = 9.96251 11. 66234 = log. 45.95. 50". 1 . sin. J. sin. JR. cot. S, computed. -log. r 3 -30 log. 50".l. ; 1.69983 log. sin. 23 28' 9.60012 log. sin. 2 m 56 8 8.10716 log. cot. 76 55' 44" , . . , &.39906 2.80617 = log. 0.640 Hence the annual precession in right ascension is equal to 45".95 + 0".0640 = 46".014, and, in time, = 3 s . 067, nearly. 348 EXAMPLE II. Required the annual precession in north polar distance of the same star. - log.r* 20 log.50".l 1.6998 log. sin. 23 28' 9.6001 log. cos. 2 m 56" 9.9999 1.2998 = log. 19".94. EXAMPLE III. The right ascension of a Serpentis being, in 1800,, = 15 h 34 m 25*.2, and its north polar distance = 82 56* 9".2, it is required to find its precessions in right ascension and north polar distance. 50". 1 . sin. I . sin. JR . cot. $, computed. - log. r s . . .-30 log. 50".l 1.6998 log. sin. 15 h 34 m 25 8 9.9057 log. sin. 23 28' 0" 9.6001 log. cot. 82 56 9 9.0933 .2989 = log. l".99. But, since JR > 12 b , this part of the precession must be taken negatively, and written l".99. Hence, since the common part of the precession (see p. 342,) is 45".98, we have the annual precession of a Serpentis in JR = 45".98 - l".99 = 43".99, 50".l . sin. I. cos. jH, computed. log. r 2 = 20 log.50".l = 1.6998 log. sin. 23 28' = 9.6001 log. cos. 15 b 34 m = 9.7733 1.0732 = log. ll".8S 349 and since the ^R is > 12 h , the precession in north polar distance is negative and = 11 ".83. By such easy computations, may the annual precessions be found, and as easily may the precessions for parts of a year be found. In fact, if t be the number of days elapsed since the beginning of the year, the precession must be equal to the annual precession multiplied into the fraction - ; for instance, EXAMPLE IV. Let it be required to find the precession in north polar distance of a Arietis on May 22, 1812. -log. r a ...... -20 log. 50". 1 ...... 1.6998 log. sin 2328 / 0" 9-6001 log. cos. 29 8 56 9-9412 1.2411 = log. 17".42. Again, log. 142 ... .2.1522 (142 days from Jan. 1, to May 22.) 3.3933 log. 365 ........ 2.5622 .8311 = log. 6".778. Hence, the annual precession is 17"*42, and the precession up to May 22 = 6".78, nearly, on the supposition of an equable generation of precession. The uses of the formula? of precession are like those of aber- ration ; they enable us to correct observations : to reduce north polar or zenith distances, observed at different times to the same time. For, since the pole of the Earth is, within certain and narrow limits, continually pointing to different parts of the Heavens, the distances of the pole from the stars must be con- tinually changing. The distance, therefore, of the zenith of a place from any particular star is continually varying ; for the distance of the zenith from the pole must remain the same, whilst the Earth preserves its axis of rotation. If, therefore, we had 350 to determine the difference of the latitudes of Greenwich and Blenheim, from two observed zenith distances of the same star, we should be unable to determine the difference, except, amongst other conditions, we knew that of the times at which the observations were respectively made. For instance, if the star y Draconis were, on the 1st of January 1800, 19' 23" south of the zenith of Blenheim, it would the next year be 19' 23".? south : the succeeding year 19' 24".4 south. The difference, therefore, of the latitudes of Greenwich And Blenheim, determined by adding the mean zenith distance of y Draconis at Blenheim on March 1, 1800, to the mean zenith distance of the same star at Greenwich on April 30, 1801, would be altogether an erroneous determination/ In order to procure a right one, we must reduce, by other corrections^ as well as by that of precession, the zenith distance of y Draconis observed at Greenwich on April 30, 1801, to that which was its zenith distance on March 1, 1800; or the zenith distance of the same star observed at Blenheim on March 1, 1800, to what would be its distance on April 30, 1801 : or, both the zenith distances must be reduced to those zenith distances which would be, or were, the true zenith distances at some common epoch; either, for instance, the 1 st of January 1798, or the 1 st of October 1803. As far then as the preceding matter of the Treatise informs us concerning the influence of the inequalities, that make the apparent place of a star different from its mean place, we must, in order to use observations like the preceding, and for the pur- poses specified, know the states of the barometer and thermo- meter at each place of observation, that we may thence determine the respective quantities of refraction. We must also know the days of the months, in order to determine the difference of the effects of aberration, which inequality, with a given star, is inde- pendent of latitude and of every condition save that of the Sun's longitude : and, in the third place, we must know the year and the number of days elapsed from its beginning, in order to know how much, past a given epoch, the zenith of the place has altered with respect to the star, by reason of the pole's motion. In the ensuing Chapters, we shall see the necessity of correcting the star's place for other reasons than those already stated. 351 From the formulae that have been given, Tables have been constructed. In the Greenwich Observations for 1812, two Tables of M. Zach's are inserted, by which the precession in north polar distance of any star for any day in the year may be found. The first, from the star's right ascension, which is the argument, gives the annual precession in north polar distance : the second Table gives a decimal number, corresponding to the day of the month (on which it is required to find the precession) by which the annual precession is to be multiplied. Thus, the number of seconds in the first Table belonging to the argument 29 8 7 (which is the right ascension of a Arietis) is 17".47. The decimal number corresponding to May 22, is .386 : there- fore, the precession of a Arietis from January I , to May 22, is 17".47 x .386 equal to 6".7 (see p. 349.) In order to shew the usefulness of the formulae of precession in right ascension, we will take, as an instance, the method of regulating Astronomical Clocks. In order to know (see pp. 103, &c.) whether a clock be too fast or too slow, we observe the hour, minute, and second noted by it, when a known star is, or is computed to be, on the meridional wire of the Transit Telescope. If the clock were neither too fast nor too slow, it would, at that instant, denote the star's apparent right ascension. In order to ascertain this circumstance, we must compute the star's apparent right ascension. The first step in such computation, is to take the star's mean right ascen- sion from a catalogue of stars constructed for a certain epoch ; the next step is to add the increase of right ascension that has accrued, in the interval between the above epoch and the time of observation. This increase, in other words, is (leaving out of consideration any proper or peculiar motion which the star may have) the star's precession in right ascension. So that, if we would make the comparison of the clock's time with time com- puted from Astronomical elements, we must, in the second step of our computation, be able to assign the star's precession in right ascension. For instance, suppose a Arietis to be on the meridional wire of the transit telescope on May 20th, 1822, when the sidereal 352 clock indicates l h 58 m 0', and that we are obliged to use a catalogue of stars computed for the epoch of Jan. 1, 1819 " then, from such catalogue, we have, firstly, mean right ascension, January 1, 1819 l h 56 59 9 .36 secondly, from the same catalogue, and by the for- i mula of p. 344, the annual precession = 3".34;/"0 10.02 therefore for three years, precesssion J precession to May 20th, (=:3 8 .34 x .381) 1.27 mean right ascen. of a Arietis on May 20, 1822. ..15? 10.65 If we stopped at these corrections, the clock would appear to be too fast by 48 s . 93 : but, as we have shewn, in the Chapter on Aberration, the star, by the effect of that inequality, will appear sobner on the meridional wire than it otherwise would do, and by i s .205. From that cause, therefore, the apparent right ascension will be greater than the mean : in computing, there- fore, the former from the latter, we must subtract 1 3 .205 from the latter; and, accordingly, since Mean right ascension, on May 20, 1822, is = l h 57 m 10 s .65 Aberration =0 1 .205 Apparent right ascension 1 57 9.445 We have not, at present, theory and formulae, to continue farther the process of corrections, and to compute, to a greater degree of exactness, the star's apparent right ascension. If the last result were true and final, it would make the clock too fast by 50*.555. But it will be soon our business, to explain the existence of two other inequalities, and to assign their quantities and laws. It will, then, appear that, in the instance before us, the apparent right ascension of a Arietis must be increased by lunar nutation (by s . 584) and diminished by Solar nutation. What are the causes of these two inequalities, and the laws to which they are subjected, we will proceed to explain in the following Chapter. CHAP. XIV. ON SOLAR AND LUNAR NUTATION. The Origin of the Nutations in the Inequable Generation of the Precession. Formula of the Lunar Nutation in Right As- cension and North Polar Distance; made similar to the For- mula of Aberration and Parallax. Formula of the Solar similar to those of the Lunar Nutation. History of the Discovery of Nutation. * THE two inequalities that give the title to the present Chapter, are intimately connected with that of the preceding. For the purpose of pointing out that connexion, we must look to the physical causes of these inequalities ; and, in the inequable action of the cause of precession, we shall be able to trace the origin of the Lunar and Solar Nutations. The actions of the Sun and Moon on the excess of the Earth, which is an oblate spheroid, above the inscribed sphere, produce the retrogradation of the equinoctial points, or, as it is technically called, the Precession of the Equinoxes. The material circumstance in the production of this phenomenon, is that excess of matter, just spoken of. The other circumstances, scarcely less material, and, indeed, essential to the phenomenon, are the inclination of the Sun's orbit to the equator, and the inclination of the Moon's orbit to that of the Sun's, and, consequently, to the Earth's equator. If the Sun and Moon were constantly in he plane of the equator, there would, notwithstanding the Earth's spheroidical form, be no precession. When either luminary is in the equator, its action, in producing precession, is nothing. Twice a jear, therefore, namely, at the two equinoxes, the Sun's force in causing precession is nothing, and twice a year, at the solstices, it is the greatest. It must, therefore, be of some mean value, in the intermediate times. The retrogradation, therefore, of the equinoctial points, inasmuch as it arises from the Y Y 354 Sun, cannot be equable, since the cause producing it is, on no two successive days of the year, exactly the same. There arises, therefore, an inequality of precession. In consequence of such inequality, the precession in right ascension of a Arietis (taking one of the instances of the last Chapter, see p. 352,) on May 20th, will not bear that proportion to the annual precession (3 8 .34) which the number of days, between January 1, and May 20, bears to three hundred and sixty-five days ; and, gene- rally, the precession for fifty days, whether it be in right ascen- sion or in north polar distance, will not be necessarily equal to - x p, p representing the precession. The exact portion of the annual precession (in right ascension or in north polar distance) to which it is equal, or the correction necessary to be made to the mean portion, will depend on the season of the year to which the fifty days belong. The precession, therefore, after being used as a correction, itself requires to be corrected. This, however, is easily effected by altering the number by which (see p. 349,) it is necessary to multiply the annual precession, in order to obtain its proportional part. Thus, of the star a Serpentis, the annual precession in right ascension of which is 2 S .Q35, the mean proportional precession on 120 April 30, would be x 2*.Q35 = .328 x 2 S .Q35, and .328 would be the multiplier : but this is too large, the actual precession gene- rated from January 1st to April 30th, being less than the pro- portional part of the mean. It may be made duly less then, by merely lessening the multiplier. .328 : in the present instance, it would be reduced to .30, which number, and like numbers, in like instances, are furnished by proper Tables (see Wollaston's Fasci- culus, Appendix, p. 42). This, however, it is to be noted, is not the sole method for correcting the precession. The inequable retrogradation of the equinoctial points, or the inequality of precession, is not the sole effect produced by the unequal action of the Sun on the Earth's excess of matter above an inscribed sphere. The obliquity of the ecliptic, which, were the precession uniform, would not be affected by the cause pro- 355 ducing precession, is subject to a semi-annual equation : since, as in the inequality of precession, the force causing a change in the obliquity arrives, twice in a year, at its maximum. These two effects, one of an inequality of precession, the other of an oscillation of the plane of the equator, constitute, what technically is called, the Solar Nutation. There is also, as it may be conjectured from the arguments just alledged, a Lunar Nutation. The precession of the equinoxes is produced by the joint action of the Sun and Moon. As the Sun not being in the equator, causes that part of the precession, which is due to his action, to be inequably generated, so the Moon, continually altering her declination, is continually causing precession with an unequal force. But the period of the ine- qualities of its action, from their evanescent state to a state of maximum, is different from the period of the inequality of the Sun's action. It is no semi-annual period. The lunar period depends, however, on principles the same as those that regulate the solar. When the Moon's orbit, which is continually chang- ing its position, returns, and not, at the end of an interval, to the same position which it had at the beginning, the interval *so cir- cumstanced is the period required. Now this is regulated by the motion of the Moon's nodes. The Moon's orbit is inclined to the ecliptic, and its nodes retrograde in the ecliptic. The period of this retrogradation is about eighteen years and seven months. At the beginning, suppose the Moon's node to have been in the node of the equator and ecliptic, then, at the end of eighteen years and seven months, the same node will have described 360 contrary to the order of the signs, and returned to thejirst point of Aries, and, during this retrogradation of the node, the lunar orbit will have occupied every position which it can occupy relative to the equator. The inequality of the Moon's action, then, in causing precession, will have passed through all its vicissitudes. But, as in the former case, this is not the sole effect of the inequality of the Moon's force. The plane of the equator will be made to oscillate ; so that, according to the longitude of the node of the Moon's orbit, it will be necessary to correct the mean obliquity on account of the lunar nutation. 356 We have seen in pp. 192, 193, that the phenomena of pre- cession can be accounted for, by supposing the pole of the equa- tor to describe uniformly a small circle round the pole of the ecliptic in a period of 25869 years. But these new phenomena of precession render some modification necessary in the preceding hypothesis. By reason of the solar nutation, the pole of the equator will oscillate, during half a year, about its mean place in the above-mentioned small circle, and the retrogradation of the pole will not be uniform. There will be a like oscillation and a like inequability of precession from the lunar nutation, but during a longer period. From both causes then, the north polar distances and the right ascensions of stars will be changed. Their precessions in north polar distance and right ascension computed according to the methods of pp. 344, &c. will not be the true precessions. In order to make the former the true precessions, we must correct them both for solar and for lunar nutation. We have, in the preceding pages, described the causes of the lunar and solar nutations. But the lunar nutation, which is, by far, the most considerable, was not sought for and found out from a previous persuasion, or belief of the existence of its cause. Bradley, soon after the discovery of the aberration of light, noticed it as a phenomenon, and then assigned its cause, and the laws of its variation. But the solar nutation has never appeared to Astronomers as a phenomenon. It could scarcely be expected to be noticed as such, since its maximum is less than half a second. Its existence and quantity are derived from Physical Astronomy ; and, on such authority, it is introduced as a correc- tion of Astronomical observations. We will now proceed to the deduction of the formulae of the lunar nutation. Similar formulae will express the laws of the solar nutation : the formula;, considered as variable expressions, will differ only in their^coefficients. One set of formula? belong to the inequality of the Moon's action on the Earth's excess above an inscribed sphere ; the other to the inequality of the Sun's action on the same excess. In deriving these formula, we must begin with borrowing certain results established by Physical Astronomy. It has been 357 proved, in confirmation of Bradley's conjectures, that the pheno- mena of nutation are explicable on the hypothesis of the pole of the Earth, describing, round its mean place (that place which, see p. 337, it would hold in the small circle described round the pole of the ecliptic, were there no inequality of precession) an ellipse, in a period equal to the revolution of the Moon's nodes. The major axis of this ellipse is situated in the solstitial colure and equal to 19 /7 .296 ; it bears that proportion to the minor axis (such are the results of theory) which the cosine of the obliquity bears to the cosine of twice the obliquity : consequently, the minor axis will be 14". 364*. Let CdA represent such an ellipse, P being the mean place of the pole, TT the pole of the ecliptic. CDAO is a circle TT described with the centre P and radius CP. <^ L is the ecliptic, w the equator, IT PL the solstitial colure. In order to de- termine the true place of the pole, take the angle APO "equal to * These are M. Zach's numbers. Bradley's are 18", 16". Maske- lyne's 19".l, and Laplace's J9".l6\ See Mecanique Celeste, Liv. V. p. 351. 358 the retrogradation of the Moon's ascending node from : draw Ot perpendicular to PA, and the point in the ellipse, through which Oi passes, is the true place of the pole. This construc- tion being admitted, the nutations in right ascensions and north polar distance may, Pp being very small, be thus easily com- puted. Nutation in North Polar Distance. The nutation in N. P. D.=P 'tfs = Y v ts sin. ers = Y Y cos. (= f T' cos. r T'V) is, as in the case of precession, common to all stars. In order to reduce farther the above expression, we have pP(T = APp + AP(r = (in the present figure) APp + M 90, , sin. APp and YT =Ll = Pp . : ; sin. PTT .'. - T'v - ts = Pp. sin. APp. co-tan. I - Pp . sin. (^Pp-f^l - 90) . cot. N. P. D. = Pp . sin. APp . cot. 1+ Pp . cos. (APp + JR.) . cot. 3, S representing the north polar distance. But these forms are not convenient for computation. In order to render them convenient, we must, from the properties of the ellipse, deduce the values of Pp, and of the tangent of APp, and then substitute such values in the above expressions : thus, 359 Pp _ sec. APp _ cos. APO _ cos. (12 s SI) PO "" sec. APO ' cos. APp cos. ^Pp = - '- - , a designating the longitude of the Moon's as- cos. APp cending node. Again, tan. APp pi Pd Pd tan. APO " Oi Z PD = '* PO' T> 7 T> i .-. tan. ^Pp = . tan. ^PO = .tan. (12 s a) Now substitute, and there will result The Nutation in North Polar Distance = - . ' - (sin. jtPp.cos. JR,-r-cos.^P.sin. JR) cos. APp = PO (tan. APp . cos. M cos. SI + cos. SI . sin. JR) = P rf cos. JR sin. SI + PO . cos. SI . sin. JR .7 ;/ .i82 cos. M sin. SI +9 /; .648 . cos. a sin. M, which is the difference, as far as nutation is concerned, between the mean and apparent north polar distance. The apparent north polar distance, therefore, must be had by adding the pre- ceding quantity, with its sign changed, to the mean. Nutation in right ascension = Pd. sin. a cot. / + PO . cos. a -cos. JR cot. S + P d . sin. a sin. JR cot. $, which, as far as nutation is concerned, is the difference of the mean and apparent right ascensions : and, consequently, the above expression must be subtracted from the mean, in order to obtain the apparent right ascension ; or, which is the same, must be added after a negative sign has been prefixed ; in which case, we have, substituting for PO, Pd their numericaf values, The Nutation in Right Ascension = 7". 182. sin. a cot. / 9".648 . cos. a cos. JR . cot. S 7 ;/ .182 . sin. a sin. JR cot. S. 360 Of the expressions for the nutations in north polar distance and right ascension, each admits of a maximum value : in order to find that value of Si which gives the nutation in north polar distance a maximum, we have = 7". 182 . cos. JR cos. Si -f9".648 sin. Si sin. JR ; 7". 182 b a which is the value of tan. Si , when the nutation is a hiaximum. Let A" be the corresponding value of Si , M the maximum, then (see p. 359, 1. 16, &c.) M = b . cos. JR sin. X a . sin. JR cos. X = a . sin. JR ( co-tan. JR sin. A"-j- cos. X ) \ a / = a sin. JR (sin. X . tan. X + cos. X) sin. JR = a . T? . cos. A We may now express the nutation at any time (what takes place with any given longitude of the Moon's node) in terms of the maximum, and of the corresponding value of the longitude of the node : thus, The Nutation in North Polar Distance = b cos. JR . sin. Si a . sin. JR cos. 3 = a . sin. JR ( cot. JR sin. Si +cos. SI ) \ a / = 3f .cos. X (tan. A sin. SI + cos. SI) = M (sin. X sin. Si + cos. X cos. 1) = M.cos. (SI - A) = M.cos. (SI 4- 15 s A 15 s ) = M. sin. (SI + 15 s AT), or (should M, when its arithmetical value is deduced from the expression of 1. 13, be negative) = iUT.shi. (SI +21' A)*. * Since X cannot exceed 12 s , we are sure, by using 15 s and 21', of having 15' A, and 21 g A, expressed by a positive arc. If the result- ing arc exceeds 12 we may cast out 12* : for sin. (12'+^) = sin. A. 361 Hence, as in the case of aberration (see pp. 274, &c.) we can always find the nutation by adding, to, the longitude of the Moon's ascending node, an arc equal to 15 s X, or = 2 1 8 X, the value of which arc will depend on the star's right ascension. In the same way we may reduce the expression for the nuta- tion in right ascension. Thus, the nutation in right ascension, = b. sin. Si cot. I a . cos. Si cos. JR cot. $ b . sin. Si sin. JR cot. . In order then to obtain that value (F) of Si which shall make the nutation a maximum, we have b cos. Si . (cot. / + sin. JR cot. ) = a . sin. SI . cos. JR cot. 8 ; therefore, writing Y instead of Si , b cot. I + sin. ^R.cot. $ a ' cos. JR. cot. 8 Hence the maximum (TO) of nutation in right ascension, = b . sin. F. (cot. J-f sin. JR cot. 3) a . cos. F cos. ^R cot. 5 a . cos. JR cot. $ cos. F By means of this expression we may, as in the case of the nutation in north polar distance, (see p. 360.) express the nuta- tion in right ascension in terms of m, and of the corresponding value of Si ' thus, the nutation in right ascension, (ri) =: b. sin. Si (cot. J+ sin. JR cot. 5) a . cos. SI cos. JR cot. 5 = a . cos. JR cot. 8 . tan. Fsin. Si a. cos. JR cot. 8 cos. Si * /sin. F sin. Si + cos. F cos. SI \ = a . cos. JR cot. o ( =r I V cos. F / = m . cos. (SI F) = m sin. (Si + 15 s F), or (should the value of m y when arithmetically deduced, be nega- tive) = m. sin. (a + 1 s - F). Hence* as in the former case, (see p. 360.) and in the case zz of aberrations, the nutation may be expressed by the sine of a positive arc. We may, then, thus symbolically express the formulae of the Nutation. Formula of the Nutation in North Polar Distance. 7 f/ 182 tan. X = cot. M= 0".744 cot. JR. ...... (l) 9.648 in logarithms, log. tan. Z = 9.87182 + log. cot. JR 10; next, M== - 9.648 . = - 9".648 . sin. M sec. X (2 cos. X in logarithms, log. M= .984437 + log. sin. JR + log. sec. Jf 20, and thirdly, (nutation) N = M sin. (SI + 15 s - X) (3) in logarithms, log. IV = log. .M+log. sin. (SI -f 15 s X) 10. Formula of the Nutation in Right Ascension. /cot. / + sin. JR . cot. $\ tan. F= 0".744 ( 3- ) (4) V cos. JR cot. d / 9 /7 .648 . cos. M . cot. S m = -* (5) cos. F in logarithms, log. m= .984437-f log. cos. JR + log. cot. $ - log. cos. F 10, and thirdly, n = m . sin. (Si + 15 s - F) (6) in logarithms, log. w=log. m + log. sin. (SI + 15 s F) 10. It now remains to assign, in particular instances, the peculiar values of the arcs 15 s - X, 15 s - F, or 2 1* - X, 21 s - F, which are to be added to SI . 363 EXAMPLE I. 7 Pegasi, (1800.) X computed. Y computed. - 10 .... - 10 - 40= 40 = 44' ll".85. . cot. = 11. 890852 log. .744= 9.871820 (see p. 362.) 9.871820 cot. / . . = 10.362458 log. tan. (89 ; 38") . . 11.762672 sec. M. . = 10.000036 but tan. Xis negative; /.X = 90 59', nearly tan. S . . = 10.601938 and + 15 s - X = 11 s 29 l' ( = log. 6.858) .836252 Again, 20 = 20 log. .744 = 9.871808 tan. M = 8.109147 (log. .0095) 3.981955 But, 6.858 + .0095 = 6.868, and 6.868 = tan. 81 43', nearly, .-. F = 2 S 21 43'. M computed, see p. 362. 20 -20 log. 9-648 984437 log. sin. M 8.109111 log. sec. X 11.765443 (log. M) t 858985 m computed. 30 -30 984431 cos. M. . 9.999963 cot. S. . . 9.398061 sec. F. . 10.819448 (log. m). . 1.201903 Hence, N = M. sin. (SI + 11 s 29 l'), and log. N = .8589 + log. sin. (Si + 11 s 29 l'). Again, w = m.sin. (a + 15 s 2 8 21 43') = wa.sin.'($l -f 21 2 21 43) = m.sin. (SI +18 8 17) = m. sin. (SI +608 17), and log. w = L202 + log. sin. (a + 6 s 8 170 364 EXAMPLE II. a Arietis (1815.) X computed. Y computed. 10= -10 40 =-40 m=29 II 7 28". . cot. = 10.252838 log. .744= 9-871820 9.871820 cot. I. . = 10.3624; (tan. 53 6' 44") ...... 10.124658 sec. M- . = 10.05898^ but tan. X is negative; /. X= 126 53 7 16" tan. S. . = 10.38199 = 4 s 6 53' 16" (log. 4.7342) .675256 i .-. 15 8 - X = 10 s 23 7 7 , nearly. Again, - 10 = 10 log. .744 = 9.871820 tan. M = 9.747161 (log. .41588) * 9.618981 Now, 4.7342 4- .4158 = 5.15008, and 5.15008 = tan. 79 O 7 4l"; .-. F = 2 s 19 I 7 , nearly, M computed. ra computed, see p. 362. -20= 20 30 =-30 .984437 .984437 sin. M = 9.688174 cos. M 9-941012 sec. X = 10.221662 cot. 8 9.618008 (log. M) .894273 sec. F 10.719845 ) 1.263302 Hence, N= M. sin. (SI + 10 s 23 e 7 7 ), and log. N = .8943 + log. sin. (a + 10 s 23 7'), Again, w= -m.sin. (SI -f 15 s -2 s 19 I 7 ) = m.s'm. (SI + 21 s 2 s 19 l') = m.sin. (a + 18 s 10 59 7 ) = m.sin. (& +6 8 10 59 7 ), and log. n = 1.2633 + log. sin. (SI + 6 s 10 59'). 365 EXAMPLE III. Polaris, (1800). X computed. Y computed. -10 =-10 -40= 40 jR=13 5' 15". . cot. = 10.633762 log. .744= 9.871820 9.871820 cot. /. . = 10.362458 tan. (72 39' 35") 10.505582 sec. ^t..= 10.011422 but tan. X is negative; .'. X= 107 20' 25" tan. S . . = 8.486050 = 3 s 1 7 20' 25" (log. .05392) 2.73 1 750 /. 15 s - X= II 8 12 39' 35". Again, - 10 = -10 log. 744 = 9.871820 tan. M = 9.366237 (log. .1730) 9.238057 Now .1730 4- .05392 = .2269, and .2269 = tan. 12 47' ; /. Y = 12 47'. M computed. m computed. -20= 20 -30= 30 .98443 98443 sin, M = 9.35481 cos. M. . 9-98857 sec. X = 10.52588 cot.S. . 11.51261 (log. M) .86512 sec. Y= 10.01091 (log. m) 2.49652 Hence, N = M. sin. (SI + 11 s 12 39' 35"), and log. N = .8651 + log. sin. (SI + 1 I s 12 39 7 35"). Again, n = - m . sin. (Si + 15 s - 12 47') = m. sin. (SI +21 12 47) = m.sin. (Sl+20 8 17 13') = 77z.sin.(a + 8 17 13), and log. n = 2.4965 + log. sin. (Si + 8* 17 13'). 366 EXAMPLE IV. a Aquarii, (1800), see pp. 281, &c. .X computed. Y computed. - 10= 10 -40= -40 JR = 328 59! 26". . cot.= 10.21906 log. .744= 9-87182 9.87182 cot. /. . =10.36246 tan. (50 57' 6") 10.09088 sec. M . = 10.06751 now, since cot. JR is negative tan. S. . . = 11.65989 tan. X (see p. 362.) is positive (log. 91.556) 1.96168 .'. X = 1 s 20 57', nearly. -Again, - 20= 20 log. .744 9-87182 tan. M 9-78037 ( = log. .4489) 7.65219 Now 91.556 +.4489 = 92.005, and 92.005 = tan. 89 22' ; but tan. Y is negative ; .-. Y = 6 s - (2 8 29 22') = 3 38. M computed. m computed. -20= -20 30= 30 .98443 98443 sin. M. . . 9-71342 cos. M 9-93248 sec. X. . . . 10.20067 "cot. $ 8.34010 (log. M). . .89852 sec. F 11.95665 (log. m) 1.21366 Hence, N=M.s\n.(Sl + 15 s -I 8 20 57'), = M.sin. (SI + 1 s 9 3') and log. N= .8985 + log. sin. (ft + l 8 9 3') -Again, n = m.sin. (Si + 21 s 3 38) = m.sin. (SI -f 17 s 29 22') = m. sin. (SI + 5 29 22), and log. n = 1.2136+ log. sin; (Si + 5 8 29 22'). 367 The Solar Nutation arises from like causes as the Lunar, and admits of similar formulae. As an ellipse, made the locus of the true place of the pole, served to exhibit the effects of the lunar nutation, so an ellipse, of different, and much smaller dimensions may be made to represent the path which the true pole of the equator would, by reason of the Sun's inequality of force in causing precession, describe about the mean place of the pole. Thus, in the present Figure, the ellipse A dC will serve to re- present the locus of the pole, when ^P = .435, Pd = .3Q9, and APO, instead of being = Si, is equal to 2 , or twice the Sun's longitude, according to the order of the signs; the equations, therefore, for the solar nutation in north polar distance, and right ascension., analogous to those of p. 362, will be The Solar Nutation in North Polar Distance = - .399 cos. JR . sin. 20+ .435 sin. M . cos. 2 , The Solar Nutation in Right Ascension = .399 sin. 2 cot. I .435 cos. 2O cos. M . cot. S .399 sin. 2 sin. M cot. S, which equations admit of transformations precisely similar to 368 those which the equations of the lunar nutation (see pp. 360, &c.) were made to undergo. Hence, we have these formulae similar to those of p. 362. for north polar distance, SQQ tan. X'= - cot. JR, and .435 log. tan. X 1 = 9-96248 + log. cot. M - 10, M' = - .435 . sin. M . sec. X 1 and log. M' = .63849 + log. sin. M + log. sec. X' -20, N 1 = M'. sin. (20 + 15 s - X'), or (see p. S6l.) = M' sin. (2 O + 21 s - X'), log. N' = log. M' + log. sin. (2 O + 15 s - X 1 } 10, for right ascension, 3QQ tan. Y' = '- (cot. I sec. M . tan. $ -f tan. M) ; .435 Q17 /. tan. Y' = tan. Y x , 744 m' = .435 cos. M cot. S sec. F', and log. m 1 = 9-63849 + log. cos. JR + log. cot. -f log. sec. Y' 30, w'= ro'.sin. (2 + 15 s - F'), log. w'= log. wi' + log. sin. (2 O + 15 s Y') - 10. Hence it appears, that we may very easily deduce the solar nutations, if we have computed the lunar, since in the processes of computation, there are several parts nearly the same. Thus, if we take the last instance, that of a Aquarii, (see p. 366.) we have - 10= 10 tan. M 10.21906 log. tan. Y = 11.96381 9-96248 .09166 (log. tan. X'} 10.18154 log. tan. Y 1 = 12.05547 .-. X 1 = 56 38' 28", 15"- X 1 = 13 s 3 22'; but 12.05547 = log. tan. 89 28 7 , nearly, 369 and since tan Y' is negative, Y 1 = 3 s 32', 15' Y 1 = 11 29 28. M f computed. m computed. 20= -20 30= 30 log. ratio of axes. . .63849 63849 sin. M 9.71342 cos. M 9.93248 sec. X' 10.25973 cot. 8.34010 (log. M 1 } 61 164 sec. Y' 12.03113 (log. m'} 94220 Hence, JV' = M' sin. (2O + 13 s 3 22') = M 1 sin. (80-4-13 22), and log. N' = .6116 +log. sin. (2 + 1 s 3 22') 10, n 1 = m . sin. (2 + 15 8 3 s 32') = m' sin. (2 + 5 s 29 28'), and log. n = .942 -f- log. sin. (2 +5 S 29 28') 10*. * The expressions for the solar nutation are thus made similar to the expressions for the lunar: but they require a separate investigation. There is not the same ratio between the axes of the ellipse that belongs to the solar nutation, as between the axes belonging to the ellipse of the lunar nutation. M. Zach, however, (seep. 120. Tabulce Speciales Aberrationis et Nutationis) and M. Delambre commenting on him (see Connoissance des Terns of 1810. p. 463.) derive the solar from the lunar corrections by merely multiplying the latter by a constant quantity: which is" no just operation. 3 A 370 The lunar and solar nutations are now then expressed by for- mulae similar to those by which the aberrations of stars (see pp. 283, &c.) have been expressed; and, we might form Tables like that of which a specimen has been given in p. 283. Thus, LUNAR NUTATION. SOLAR NUTATJON. North Polar Dist. Right Ascen. North Polar Dist. Right Ascen. Number. y Pegasi. log. max" 1 . 11 s 29 l' .8589 6" 8 17' 1.202 II 8 29 12' 9.6026 6" 6 43' 9-968 a Arietis, 10 s 23 I 1 .8943 6 s '10 59' 1 .2633 10 s 28 39' 9.61050 6 s 8 55' .00733 Polaris. 11 s 12 40' .8651 8 s 17 13' 2.4965 11 s 15 13' 9.603 8 s 1421 / 9.1561 a Aquarii. I s 9 3' .8985 5 8 29 22' 1.2136 1 s 3 22' 9.6116 5 s 29 28' 9-942 . By means of a Table like the preceding, and of a Table like the one of p. 283, we may easily compute the corrections which it is necessary to apply to the mean north polar distance and mean right ascension of a star, in order to deduce the apparent north polar distance and apparent right ascension. There will be indeed another correction, besides those just mentioned, to be taken account of, namely, the correction for precession. Sup- pose, for instance, it were required to express the apparent north polar distance of a Aquarii for some time in the year 1800 : its mean north polar distance (see p. 281.) is 91 16' 58": its mean right ascension 328 52' 26" ; .'. its precession, proportional to a time t elapsed from the beginning of the year, is (see p. 341.) 50". 1 x t sin. (23. 2?' 50") X cos. (328 52' 26") = 1 7".07 1. 371 Hence, the apparent north polar distance =91 16' 58" ~ 17".07* -f 7".856 .sin. (O -f-3 8 2 49' 52") (aberration, p. 281.) + 7".915 . sin. (Si + 1 9 3' 0). .. .(lunar nutation, p. 370.) 4- 0".4ti8.sin.(2Q + 1 3 22 0).... (solar nutation, p. 370.) and, in a specific instance, when the values of the longitudes of the Sun and of the Moon's ascending node would be assigned, the resulting value of the apparent north polar distance would agree with the observed and instrumental distance cleared of the effects of refraction. On the footing of mere theory, we ought to add to the pre- ceding terms (see 11. 2, &c.) that express the several corrections, a similar term (see p. 313, &c.) on account of parallax. But its co- efficient is, at present, either insensible or unknown. We do not, therefore, correct for parallax : but we must correct on account of the 'star's proper motion: the quantity of which correction, resting on no theory, is determined solely by observation. We have assigned the formula for determining the apparent north polar distance of a Aquarii for some time (t) in the year 1800. But, as it has been explained, the same formula (except- ing its first and fourth term) will serve to express the north polar distance of a Aquarii for any time in any other year ; provided such year be not too remote from the epoch for which the numbers and maxima (see p. 366.) have been computed. Thus, the numbers and maxima have been computed for a Arietis, supposing the epoch to be 1815 : but, the same numbers and maxima will serve to compute, with no practical error, the aberrations and nuta- tions of that star for any time during 1822. The like will happen with other stars ; for instance, suppose it were necessary to express and compute the apparent right ascension of a Arietis on May 20th, 1822 : then we have, from the Catalogue of 1819- Star's mean right ascension, Jan. I, 1819 l h 56 m 59*.36 three years precession ( = 3. x 3 8 .34) , . . 10 . 02 proportional precession to May 22 ( = 3 .34 x .3801) 1 . 27 and from the Nautical Almanack and Tables, 372 O = 1 s 28 50' 39" SI = 10 20 20 also see p. 283, No. for aberration = 7 28 39 p. 370, for lunar nutation = -6 10 59 p. 370, for solar = 6 8-55 consequently, see pp. 283, &c. the argument for aberration is 9 9 27 '29'.5, nearly, for lunar nutation 17 1 19 for solar nutation 10 6 36 whilst, by the same pages, the maxima (expressed in time) are, respectively, & (20'.564), ' (18.335), & (1 9 .017). Hence on May 20th, 1822, the apparent right ascension of a Arietis is l h 57 m 10 S .65 ^ = l h 57 m 10'.65 + ^ 5 (20.564) sin. ( 9 27 29'.5) .....= - 1.21 + i (18.335) sin. (17 1 19) = +0.58 + 5 ( 1.017) sin. (10 6 36) = -0.05 1 57 9-97 The apparent right ascension, therefore, of a Arietis will be very nearly l h 57 m 1.0". Of the four corrections that have been applied, in the pre- ceding instance (and of like corrections with other stars) three are dependent on the time elapsed from the beginning of the year ; namely, the proportional part of the precession, the aberration, and the solar nutation. If these corrections, therefore, be com- puted for every day of a certain year and their results taken, such results will serve for every day on succeeding years, and, without material error, will be right results during a century. Such is, nearly, the practice of Astronomers. They compute to every tenth day of the year, and insert in Tables, the results of preces- sion, and aberration. Thus, in the preceding instance, we have 373 the precession, in right ascension = I 8 . 27 the inequality of preces. n in JR, or solar nut at. in JR. = . 05 the aberration in right ascension = 1.21 + 0.01 and this result is called, (see Table X, in the first Volume of the Greenwich Observations) the Correction oj the Star's Right Ascension in time. The fourth correction, that of the lunar nutation, depends on the longitude of the Moon's ascending node, and conse- quently will not, like the other corrections, be the same on the corresponding days of successive years. It is computed from a separate Table, of which the argument is the longitude of the Moon's node. The deduction of formulae for correcting, on account of the lunar and solar nutations, the apparent north polar distances and right ascensions of stars, ought to be considered as the essential business of the present Chapter ; which, therefore, might here be closed. But, before this is done, we wish to make a short digression towards certain collateral objects : some of little moment, or merely curious : others distinguished solely by their celebrity in Astronomical history. In aberration, we pointed out its origin and cause, and then, with such means as we were using, deduced its formulae. Nothing was then borrowed from a foreign or an unexplained theory. But it has been otherwise in the subject of nutation. Some general idea, indeed, was given of its cause, but no formulae deduced from such explanation. The means of deducing them were borrowed from Physical Astronomy and taken on trust. And, in order to obtain the most convenient means of computing such formulae, we supposed (which indeed is one of the results of Physical Astronomy on this subject) the locus of the pole to be an ellipse. But, it is to be observed, this is only one way of viewing the subject : it is neither the essential nor the only way. All the computations might have conducted, and their results arrived at, without an ellipse to represent either the solar or lunar nutation. The inequality of the lunar force in causing precession 374 produces an equation of precession, and an equation of the ob- liquity. The inequality of the solar force does the same thing. Let the lunar equation of precession, or d^ = 18 .03584 sin. Si the equation of obliquity, or dl = 9 -6 . cos. St , and from these two equations, by the means either of spherical triangles, (as Cagnoli has done pp. 439, &c. of his Trigonometry} or by taking the differentials of some of the equations of page 182, (as Suanberg has done, pp. 108, &c. of his Exposition des Ope- rations, &c.) the corresponding variations of the right ascension and north polar distance, or, technically, the nutations in right ascension and north polar distance, may be deduced. In like manner, if we represent the inequality or equation of precession arising from the Sun by dy '= - l". 002 . sin. 2 O , and the equation of obliquity, by d I = 0".435 . cos. 2 O 5 we may deduce, by the method just described, the corresponding variations in the right ascensions and north polar distances of stars, which, technically, will be the solar nutations in right as- censions and north polar distance, or which, as it is sometimes said, arise from the solar inequality of precession. Instead of this, which is, perhaps, the most direct method, we have followed Bradley's. This latter is usually adopted in Astronomical Treatises,, and, certainly, possesses the merit of being clear and intelligible. But it is apt to cause the student to form erroneous conceptions : to make him view as a fact, or phe- nomenon, what is merely a mathematical fiction. If we could trace out in the Heavens the path of the pole of the equator it would not be an ellipse. It would be such a curve were there no inequality of the Sun's force, and were not the mean place of the pole itself in motion along a circular arc. But this latter motion takes place, and, besides, the path of the true pole, by reason of the solar nutation, would, were other causes abstracted, itself be elliptical. The path, therefore, described by the true pole, by virtue of three motions, or in consequence of precession, and (for such 375 they are) its two inequalities, is some epicycloidal curve *, of fto very easy description. Instead of deducing the nutations in north polar distance and right ascension from the nutation of the obliquity, and the nuta- tion of longitude, (see pp, 374, 357.) we have deduced them from the assumption of the locus of the pole being an ellipse. From formulae so deduced we may derive, as consequences, the nuta- tions of obliquity and longitude. Thus, N(m N. P. D.) = 9".64.sin. M .cos. a -7". 18 cos. M. sin. SI . Now the change produced, by nutation, in the north polar distance of a star situated on the solstitial colure, will be equal to the change of obliquity from the same cause. Let the right ascension, therefore, = 90, in which case sin. JR, = 1, cos. JR, = 0, and the nutation of obliquity = 9"'64 . cos. a . Hence, the nutation of the obliquity is the greatest when SI, either = 0, or = 180, that is, either when the Moon's node is in and cos. M sin. a = sin. (M+ a) - |- sin. (M- a). Hence, the above formula for the nutation in north polai distance, becomes N= l".233 . sin. (M + a ) + 8".415 sin. OR a ), 379 in this expression, substitute, instead of JR, 180-t-jil, and the resulting expression will be one for the nutation of a star having a right ascension opposite to the former, that is, N' =s l".233.sin. (m-fSl) 8".415 sin.(.l- ft), an expression equal in quantity to the former, but in a different direction (see Phil. Trans. No. 435, pp. 12, 13, also p. 294. of this Treatise). By a similar transformation, the expression for the nutation in right ascension becomes n = - 8".415 . cot. S . cos. (A - SI ) l".233 . cot. . cos. (JR + SI ) - 16".544 sin. & , and under these two latter forms, which are Lambert's,, the nuta- tions in north polar distance and right ascension are usually expressed (see Delambre, Chap. xxx". torn. 3. Connoissance des Terns. 1810. p. 463. Cagnoli's Trigonometry, p. 440. Vince's Astronomy, Vol. II. p. 132.) With regard to the Astronomical uses of the theory of nuta- tion and of its formula, the same may be said, both for explana- tion and illustration, as has been already said on the subjects of precession and aberration. The aberration will be nearly the same, on the same days of different years : so will be the solar nutation. The lunar nutation will, almost certainly, be different. If, therefore, to take our old instance, we would determine, the difference of the latitudes of the Observatories of Greenwich and Blenheim, from two recorded instrumental observations of *y Draconis, we must know, besides the zenith distances deter- mined by the instruments, the states of the thermometer and barometer at the instants of the two observations in order thence to determine the refractions ; secondly, the interval of years between the two observations in order to determine the difference of the precessions in north polar distance ; thirdly, the days of the year in order to determine the respective proportional changes of precession ; the inequalities of precession (or the solar nuta- tions in north polar distance) and the aberrations ; and, lastly, we must know the particular years and days, in order to know the respective positions of the Moon's ascending node and thence to determine the lunar nutations. 380 But it is not on the occasions of determining the latitudes, which are rare occasions, that the knowledge of the nutation, and ' of the other corrections is chiefly necessary. Such knowledge is * required in the daily business of an Observatory. The first operation is to observe the star and to register the observation : the second is to clear the observation of its inequalities, or to reduce it. For instance, in order to determine the error of the line of collimation of the brass quadrant of Greenwich, there were observed with it forty-six zenith distances of y Draconis, at different times of the year, from Feb. 21, 1811, to Dec. 29, 1811. Each of these distances was an apparent distance, and each different, the one from the other. Reduced to the same eppch, which was the beginning of the year, each would express the mean zenith distance, and (were the observations exactly made, and the theories by which they are reduced, correct) by the same quantity. This is not the case, if it were, one observation would do as well as forty-six. But the accuracy which cannot be hoped for from one observation is attained by taking the mean of many. In order to effect this, each zenith distance must, as it has been said, be reduced, or corrected for aberration, solar nutation, lunar nutation, and precession. The following is a specimen of registering the observations and their reductions : Observed Zenith Distances of y Draconis reduced to the beginning of the Year. Observed Z. D. iberration. Solar Mutation. Lunar Mutation. Secession. Sum of Equations. Mean Z. | Jan. 1, m Feb. 21. 2' 7" + 17".42 -f-0".30 -9". 54 + 0".09 + 8" .27 2' 15"J 22, 2 6.8 17.56 0.31 9.54 0.1 8.43 15 .! 26, 2 4.8 18.08 0.36 -9.54 0.1 9-01 13 .i! &c. &c. : &c. &c. &c. &c. &c. J Aug. 2, 2 36 - 12.70 0.0 -9-41 0.40 -21.79 9! 14.( 6, 2 37 -13.66 0.02 -9.41 0.41 - 22.66 14.: 381 The corrections in the several columns are, in practice, taken from appropriate Tables : but they may, amongst other methods of computation, be computed by those which have been deduced in this Treatise : for instance, we may compute the numbers under the third column from 19".55 sin. (0 -f 3 8 1 42' 15"), 5th, from 0".966 sin. (SI + 3 1 30'), 6th, from 50". 1 x -^- . sin. (23 280 . cos. 268', 365 when, instead of O and Si , the respective values of the longi- tudes of the Sun and of the Moon's node are substituted on Feb. 21, 22, &c. and Aug. 2, 6,, &c. The preceding expressions for computing the aberration, &c. are adapted (see p. 380.) to reduce the mean to the apparent polar distance. Now, 7 Draconis is north of the zenith of the Observatory of Greenwich. The resulting numerical values, therefore, of the aberration, &c. if additive of the north polar distance, are subtractive of the zenith distance ; and conversely : but, the above Table reduces the apparent zenith distance to the mean ; consequently, the numbers in that Table will have the same sign as the numbers computed by the preceding formulae. The numbers 'expressing the aberration, &c. being either taken from Tables, or computed, are added together with their proper signs. The results are inserted in the seventh column. The numbers in the seventh column added to those in the second (which express the observed zenith distances) give the mean zenith distance on the beginning of the year 1811. For instance, the sum of the equations in the first horizontal line is -f- 8". 27 ; the observed zenith distance, in the second column, is 2' f ; conse- quently, the mean zenith distance is 2 7 15". 27 ; which number, as well as all similarly obtained numbers, is inserted in the eighth column. The numbers in the eighth column added together and divided by their number (46 in the Table of which we have given a specimen) give the mean value of the mean zenith distances *. * In the above specimen and its explanation, we may perceive the use of the theories of the inequalities in deducing certain results. The result, 382 The corrections, the theories of which have been investigated, are precession, aberration, the lunar and solar nutations. By these the instrumental and apparent zenith distances of 7 Draconis observed in March, April, &c. of any particular year may be reduced to the mean zenith distance at the beginning of the same year ; and, consequently, the mean zenith distances of the same star on the beginnings of different years may be compared toge- ther. A like reduction and comparison may be made of the right ascension of a star. With regard to the mean zenith distances and mean right ascensions of stars at the beginnings of different years, if these differ, they ought to differ, supposing all the corrections to have been accurately assigned, solely from the effect of pre- cession. If the differences should not be accounted for from that effect, a new source of inequality would be indicated, if the effects of precession will account for the differences in the mean zenith distances of some stars, but not of others, there would, in that case, arise an indication of some peculiarity affecting these latter stars. But instead of describing, in general terms, the result, in the adduced instance, is the. mean zenith distance of 7 Draconis on Jan. 1, 1811. We may be required to go a step farther, and to shew the use of the results obtained in the above Table. Those results were obtained for the purpose of thence deducing the error of the line of colli- mation of the brass mural quadrant of Greenwich. But, see p. 67, such results alone are not sufficient. They must be compared with other results obtained by the Zenith sector (see p. 6*8.) Thus, in the Volume of Observations, a few pages after those we have quoted, there is a Table of the zenith distances of 7 Draconis observed with the zenith sector, partly with its face towards the east, and partly towards the west. All these observed zenith distances are reduced, precisely as the preceding ones have been, to the beginning of 1811. The mean of such reduced zenith distances is (see p. 6*8.) the true mean zenith distance of 7 Draconis. The difference of this last mean and the mean obtained by the brass mural quadrant is the error of the line of collimation. We may infer,. and not wrongly, from the preceding instance (which is one of many similar ones) that the business of an Observatory does not admit of being very leisurely and not laboriously conducted. The method of finding the error of the line of collimation is, usually described in four or five lines. The actual finding of it requires, as we have seen, the observations of fifty days, many arithmetical computations and the use of extensive Tables. 383 / method of detecting inequalities, it will be better to exemplify it. And, as a first instance, we will describe the method which Bradley followed in extricating, from certain observed differences in the declinations of stars firstly, the inequality of aberration, and, secondly, that of nutation. This being done we will shew, on like principles, that there is still some change in the places of stars to be explained, or at least to make account of, even when they are reduced to the same epoch by the corrections that are due to the inequalities of precession, aberration, lunar and solar nutation. These investigations will be carried on in the next Chapter, in which, we will also briefly advert to the methods which were first resorted to, for representing the lunar and solar nutations. CHAP. XV. On the Means by which Bradley separated Nutation from the Inequalities of Precession and Aberration. On the successive Corrections applied to the Apparent Place of a Star. On the Secular Diminution of the Obliquity. IN treating of the several inequalities of precession, aberra- tion, and nutation, it is necessary, in order to avoid being per- plexed by the mere words of a theory, to recur to the simple facts of observation. Now, the observations of Bradley were on the Declinations of Stars, or, what amounts to the same thing at a given place, on their zenith distances*; and, the phenomena of his observations, were changes in the observed zenith distances of the same stars ; happening sometimes, at different parts of the same year, and at other times, at corresponding seasons of dif- ferent years. The star y Draconis, passing the meridian very near the zenith of Bradley's Observatory, and being/ consequently, little affected by refraction, was the chief star of his observations. This star (see pp. 288, &c.) in March passed more to the south of the zenith by about 39" than it did in September : that is, whatever was its mean place, the difference of its two zenith distances, or of its declinations, was, in half a year, observed to be about 3Q 1 '. Other stars, also, changed their declinations. The changes of declination of a small star in Camelopardalus (the 35th of Hevelius), with an opposite right ascension to that of 7 Draconis, were observed at the same times as those of the latter star : and, it was Bradley's argument, that, if these phe- * Bradley's observations were made with a Zenith Sector, adapted to measure the small portions, or minutes of declination, and of zenith distances, near the zenith. 385 nomena (changes of declination) arose from a real nutation of the Earth's axis, the pole must have moved as much towards y Draconis, as from the star in Camelopardalus ; but (see p. 295.) this not being the case, the hypothesis of a nutation of the Earth's axis would not account for the observed phenomenon : more strictly speaking, it would not completely account for it, for, in fact, some part of the observed changes of declination was due to the effect of nutation, Bradley, as we have seen, (p. 295.) solved the above pheno- mena by the theory of aberration. Now. if such theory, with the known one of precession, would account for all observed changes of zenith distances, or, of north polar distances, then, there could be no changes but what arose from precession and aberration. Hence, since (p. 276.) the aberration is the same, at the same season of the year, the distance of y Draconis, in September 1728, ought to have differed from his distance, in September 1727, only by the annual precession in north polar distances the distance, in September 1729, from the distance, in September 1727, by twice the annual precession in north polar distance ; and so on. Such, however, was not the observed fact. In 1728, after the effect of precession had been allowed for, y Draconis was nearer the north, by about 0".8 than in 1727. In 1729, nearer than in 1727, by l".5. In 1730, by 4". 5. In 1731, by nearly 8". Here then was a new pheno*- menon, a change of north polar distance, indicating an inequality not yet discovered. Bradley observed other stars besides y Draconis; amongst others, the small star above-mentioned (p. 295.) of Camelo- pardalus : and, it is not a little worthy of notice, this same star, which, in the case of the former inequality, (that of aberration) directed him to reject the hypothesis of a nutation of the Earth's axis, here determined him to adopt it. For, within the same periods, the changes in north polar distance of y Draconis and of the star in Camelopardalus, were equal and in contrary direc- tions : that is, whilst the former, through the years 1728, 1729, 1730, 1731, was approaching the zenith, and consequently, the pole, the latter was, by equal steps, receding from the zenith, and consequently from the pole. These phenomena then of the 3c 386 changes in north polar distances, which (not like those of aberra- tion that take place in different parts of the same year, and recur in the corresponding parts of different years) were observed, through a term of years, could adequately be explained by sup- posing, during that term, a nutation in the Earth's axis, towards 7 Draconis, and, from the small star in Camelopardalus. After 1731, Bradley observed contrary effects to happen; that is, 7 Draconis receded from the zenith and north pole, and the star in Camelopardalus, by equal steps, approached those points; and this access and recess continued till 1741, (a period of more than nine years) ; after which, the former star again began to approach the zenith, and the latter to recede from it. These phenomena, then, that took place between 1731 and 1741, could be adequately explained by supposing, during that term, a nuta- tion in the Earth's axis, from 7 Draconis and towards the small star in Camelopardalus. The mere hypothesis of a nutation, or vibratory motion in the Earth's axis, would have found little reception amongst men of science, if no arguments had been adduced to render such nutation probable : that is, if some physical cause, likely to pro- duce it, had not been suggested. Previously, however, to the suggestion of the real and immediate physical cause, Bradley enquired, whether this seeming nutation of the Earth's axis was connected with any concomitant circumstance, or phenomenon : such circumstance he found to be the position of the nodes of; the Moon's orbit. The star 7 Draconis was (after the effects of precession had been allowed for) most remote from the pole, when the Moon's node was in Aries, and least, when in Libra : and, after a com- plete revolution of the Moon's nodes^ the distances of all the 1 observed stars, at the end, differed from the distances at the beginning, by the effect of precession only. Hence, the phe- nomenon of a nutation, and the longitude of the Moon's node were connected. But, the inclination of the Moon's orbit varies with the longitude of the node : the former is greatest, when the latter is equal to nothing ; and least, when the latter is six signs. Hence, the nutation and inclination were connected together.- 387 But, the Moon's action, on the bulging equatoreal parts of the Earth, is greater the more distant the Moon is from the equator ; and her mean action greater, the greater the inclination of her orbit. Hence, the phenomenon of the nutation was connected, with the variable action of the Moon in causing precession ; and this last connexion made nutation the effect, and the variation of the Moon's action the cause. And this was the physical cause which seemed to Bradley to afford an adequate solution of the phenomenon he observed : and subsequent researches have con- firmed the sagacity of his conjectures. The real distance of any star (y Draconis for instance) from the north pole of the equator, is changed continually and con- stantly, by the effect of precession only. The variations in that distance from aberration and nutation are periodical, and recur, the former in the space of a year, the latter in the time of a revolution of the Moon's nodes. Hence, although, in any phe- nomenon of a change in the north polar distance of a star, the effects of several causes may be blended together and com- pounded ; yet the method is plain, by which we may disengage and separate them. For instance, since the revolution of the Moon's nodes is completed in about eighteen years, and since the -aberration and the solar inequality are the same, at the same time of the year, the north polar distance of y Draconis in 1745, ought to differ from its north polar distance in 1727, almost solely by the effect of precession : that is, since the latter north polar distance was 38 28' 10",2, and the precession 0".8, the north polar distance in 1745 ought to have been 38 27' 56". And such difference was, by Bradley's observations, (see Phil. Trans. No. 485, p. 27.) found very nearly to exist. Again, between September 6, 1728, and September 6, 1730, the aberration and solar inequality being the same, the respective north polar distances of y Draconis at those periods ought to differ, by twice the annual precession in north polar, distance, and by the effect of nutation : and hence the effect of nutation in an interval of two years, between two known positions of the Moon's ascending node, would be known. Again, between September 6, 1728, and March 6, 1729, the 38$ solar inequality being the same, the respective north polar distances of y Draconis ought to differ from each other by the precession in north polar distance due to half a year, by the nutation for the same time, and (see p. 122.) nearly by the sum of the greatest aberrations in north polar distance ; and the whole difference would consist almost entirely of aberration, since the precession and nutation together would aot amount to a second. Again, the Moon's ascending node being, March 28, 1727, in Aries, and July 17, 1736, in Libra; the respective north polar distance of y Draconis would differ by the precession due to nine years three months, by the solar inequality of precession, by aberration, and by the sum of the two maximum effects of nutation. But, between March 28, 1727, and March 28, J736, (since then the solar inequality and the aberration would be the same) the north polar distances would differ by the effect of pre- cession, (a known quantity) and, nearly, by the sum of the two maximum effects in nutation* Hence, it would be easy to disengage, and numerically exhibit, (what is a material element), the maximum effect of nutation. By examining various and numerous observations and by dis- criminating those that happened at particular conjunctures, Bradley found abundant confirmation of the truth of his two theories, aberration and nutation. During a period of more than twenty years, he accounted for the phenomena of observation, that is, the changes in the declinations of various stars, by making those changes or variations consist of three parts ; the first due to precession ; the second to aberration ; and, the third to nutation ; the quantities and laws of the two latter, being assigned on the principles and by the formulae of his theories, We cannot sufficiently admire the patience, the sagacity, and the genius of this Astronomer, who, from a previously unobserved variation not amounting to more than forty seconds, extricated, and reduced to form and regularity, two curious and beautiful theories. The following Table exhibits the coincidence of his theories with observations. (See Phil. Trans. No, 485, p. 27,) 389 7 Draconis. South of 38 25'. Precessioiit Aberration. Nutation. Mean Distance. 1727 Sept. 3 70".5 -0".4 + I9".2 -8".9 80".4 1728 Mar. 18 108.7 -0.8 -19 -8.6 80.3 Sept. 6 70.2 1.2 + 19.3 8.1 80.2 1729 Mar. 6 108.3 -1.6 - 19.3 ~7.4 80.0 Sept. 8 69.4 2,1 + 19.3 -6.9 80.2 1730 Sept. 8 68.0 2.9 + 19.3 3.4 80.5 1731 Sept. 8 66.0 3.8 4-19-3 l.O 80.5 1732 Sept. 6 64.3 4.6 + 19.3 + 2.0 81.0 1733 Aug. 29 60.8 5.4 + 19.0 + 4.8 79.2 17^4 Aug. 1 1 62.3 -6.2 + 16.9 + 6.9 79.9 1735 Sept. 10 60.6 7.1 + 19.3 + 7.9 80.1 1736 Sept. 9 59.3 8.0 + 19.3 +9-0 79.6 1737 Sept. 6 60.8 8.8 . + 19.3 + 8.5 79.8 1738 Sept. 13 62 -9.6 + 19.3 + 7.0 78.7 1739 Sept. 2 66.6 10.5 + 19.2 + 4.7 80.0 1740 Sept. 5 70.8 -11.3 + 19.3 + 1.9 80.7 1741 Sept. 2 75.4 12.1 + 19.2 1.1 81.4 1742 Sept. 5 76.7 12.9 + 19.3 -4.0 79.1 1743 Sept. 2 81.6 -13,7 + 19.1 -6.4 80.6 1745 Sept. 5 86.3 -15.4 + 19.2 -8.9 81.2 1746 Sept. 17 66.5 16.2 + 19.2 8.7 80.8 1747 Sept. 2 86.1 -17.0 + 19.2 -7.6 80.7 A brief explanation will suffice for this Table. The appa- rent place of a star is deduced from the mean, by applying to the latter the several corrections : or, the mean is deduced from the apparent, by applying the same corrections with contrary signs. If therefore y Draconis were, at the beginning of any period, . certain number of seconds, south of the zenith, or south of 390 any particular division in the zenith sector; it would, at the end of the period, be really farther from the zenith by preces- sion ; really farther or nearer, by nutation ; and apparently nearer or farther by aberration. By the mean distance of the star from the division 38 25' of the zenith sector (see last column in preceding Table), Bradley means the distance on March 27, 1727, such as would have been the distance, had there been neither nutation, nor aberration. But, in that year, the nutation, (the node of the Moon's orbit being in Aries} was the greatest. Hence, in September 1727, (see the first horizontal row of the preceding Table) the observed or apparent distance of *y Draconis would differ from the mean, by the effect of precession (j x .8) in half a year, by the maximum effect of aberration, and by nearly the greatest effect of nutation. The apparent distance then of the star being 70 r/ .5, the mean (according to Bradley) would be 70". 5 0".4 -f 19".2 - 8".9 = 80".4. Again^ reversing the process. If SO" were the mean distance, then, on March 6, 1729, the star would appear by aberration farther distant about I9 r/ .3 : would really be more distant by the effect of two years' precession in north polar distance (2 X .8 ;) and would really be more distant than it would be if the Moon's orbit were at its mean inclination (the 1 being either in ffi or in V?) by the effect of nutation (7". 4). The apparent* distance therefore would be 80" + l".6 + 19".3 + 7".4 = 108".3. The mean distances deduced according to the preceding ex- planation, by means of corrections,, from Bradley 's two theories of aberration and nutation,, and from the known effect of pre- cession, ought, if the theories be true, to be invariably the same : and their very near equality (see last column in Table, p. 389.) establishes, almost beyond a doubt, the truth of those theories. * There is some violation of the propriety of language in calling that apparent, which depends on real causes, viz. the changes of the place of the pole from precession and nutation. In strictness, apparent should have been confined to aberration, refraction. 391 The division in the zenith sector, from which, as a fixed point, Bradley measured the distances of y Draconis, (the north polar distance of which he calls 38 25 ; ) is not the division cor- responding to the zenith of the Observatory at Wansted. If it had been, the apparent north polar distances of y Draconis on Sept. 3, 1727, and on March 6, 172Q, would have been, re- spectively, 38 26' 10".5, and 38 26' 48".3. Having now explained the method by which Bradley detected, in the small differences of the declinations of certain fixed stars, the existence of several inequalities, we will briefly state, after what manner, he first thought that the effects of nutation could be represented. Bradley supposed the path described by the true pole round its mean place, by reason of the inequality of the Moon's force in causing precession, to be a circle. Thus, in the subjoined Figure, let TT be the pole of the ecliptic, P the mean place of the pole of the equator, and let DOAB be a circle described round P as a centre and with a radius PA ( = Q".6). Moreover, A is to be the true place of the pole, when the ascending node of the Moon's orbit is y (the first point of Aries). The other positions of the true pole are to be determined by supposing it to move equably along the circle AOD, &c. in a direction contrary to the order of the signs, and to describe the circle, in a period equal to that of the retrogradation of the Moon's nodes. The Moon's node then being at any distance from , take the angle APO equal to that distance, and O is the true place of the pole. Such point being assumed to be the true place, the changes in the north polar distances and in right ascensions of stars are to be computed, exactly as they were when p, a point in the ellipse, was assumed to be the pole's place (see p. 358.) This Memoir of Bradley 's is inserted in No. 485, of the Philosophical Transactions. Towards the end of it, its Author suggests that the effects of nutation would be more truly repre- sented, by supposing the locus of the pole to be an ellipse, instead of a circle, the transverse and conjugate axes, AC and DB, being nearly 18" and 16" respectively. Not, however, perfectly satisfied of the justness of this last suggestion, Bradley wished it to be tried by theory ; and, such trial, as we have seen in the last Chapter, has, since Bradley's time, been made. In the inequalities of precession, aberration, and the lunar nutation, observation has preceded theory. These inequalities were first detected as phenomena, and then their physical causes assigned. It has not been so with the solar nutation, which was never, (such is its minuteness), distinctly perceived as a phe- nomenon. It was first conjectured to exist from analogy. The inequality of the Moon's force in generating precession being found to cause a lunar nutation, the inequality of the Sun's force it was presumed, would also cause a solar nutation resembling the lunar. Its law and quantity have, accordingly, been com- puted, and the numerical results applied as corrections of obser- vations . In the solar, as in the lunar nutation, the true pole describes, round the mean place of the pole, an ellipse, the semi-axis major of which ellipse, in the case of the solar nutation, is less than half a second. The corrections, therefore, of the north polar distances and right ascensions of stars, in consequence of this deviation of the pole, are very small. And this last circumstance makes it of little consequence, whether, in computing the above 393 errors in the places of stars, we consider the pole to be erratic in an ellipse or in a circle * which Dr. Maskelyne f and other writers consider as its locus. We will again mention, for the sake of preventing any false conceptions on this subject, that the two ellipses, as the respective curves of deviation of the true pole, in consequence of the inequa- lities of the lunar and solar force in causing precession, are merely mathematical schemes and contrivances for the convenient com- putation of the changes produced in the places of stars. The changes to be computed are very small : which is the reason * The solar inequality has been thus represented : TT, P, are, respec- tively, the poles of the ecliptic and the equator. By virtue of the preces- sion, P will describe, and contrary to the order of the signs, the arcs of a small circle Ppk. After a lapse of time, suppose p to be the mtan place of the pole : the true place will be nearer to *-, or farther from TT, or to the right or left of jo, according to the position of the Sun. In order to determine its place, describe round p as a centre, and with a radius = 0".435, avs, and take, according to the order of the signs, the angle apv equal to twice the Sun's longitude : then, v is the true place of the pole, the pole being at b (P b = radius of the small circle) when the Sun was at V. f Explanation and use of the Tables inserted in the 1st Volume of the Greenwich Observations. 3D 394 why we may separately compute the effects of precession and of the two nutations, combine them and obtain a result scarcely different from the true result ; the true result being that which would be obtained by placing the pole in that curve which would be described, by the combination of its three movements ; one circular round the pole of the ecliptic, and representing the mean effect of the luni-solar precession : the second elliptical by reason of the inequality of the lunar precession : the third also elliptical and caused by the inequality of the solar precession. In the next Chapter we will consider whether the theories of the preceding inequalities completely explain the differences ol the declinations and of the right ascensions of stars, either com- puted or observed, at different epochs. CHAP. XVI. On the proper Motions of Stars : the Means of discovering such; their ambiguous Nature ; arising from the Means used for determining the Precession. Instances of the Methods used for folding the proper Motions of a Star in Right Ascension and Declination. IN order to compute the right ascensions and declinations of stars, there are necessary, firstly, a catalogue of their mean right ascensions and mean declinations at a certain epoch ; and, secondly, Tables, computed by the aid of theory and observation, for supplying the amount of those differences which will be found to exist between the above-mentioned right ascensions and declinations, and their values at a different epoch, whether such values be the mean or the true values. If the comparison between the right ascensions and declina- tions be a comparison between their mean values at different epochs, then certain inequalities will be rescinded and have none effect in producing differences of the mean values : for instance, aberration and the two nutations (or the two sources of ine- qualities of precession) will be in such predicament. The differences of the mean values will depend solely (if there be no other inequalities than those treated of in the pre- ceding part of the Treatise) on Precession. Thus, if in July 9, 1821, the north polar distance of a Lyrae be observed, such north polar distance, is an apparent distance, or true distance. When corrected for aberration, for the solar and lunar nutation, and for that part of the annual precession which is proportional to the interval between January 1, and July 9, it will express the mean north polar distance of a Lyrae for January 1, 1821. And it ought, were the preceding enumeration of the sources of ine- quality complete, to differ from the mean value of 1815 (sup- posing that to be the epoch of the existing or standard catalogue) 396 by the sum of six annual precessions. And the like is to be said of the polar distances and right ascensions of other stars. Now the fact is that, after the completion of the above process, the differences of the polar distances and right ascen- sions of stars are not found to be exactly accounted for by the quantity and law of precession : in some stars the differences are greater than what they ought to be by the effect of precession, in others less. And this fact being ascertained, our attention is drawn towards the mode by which the quantity of precession is ascertained. The precession, or the retrogradation of the intersection of the equator and ecliptic on the ecliptic, is not a phenomenon of immediate observation. It requires, in all cases, some slight computation ; which computation may be made either from the changes it produces on the right ascensions of stars, or from the changes in north polar distances, or from the differences of the longitudes of stars computed, for different epochs *, and from the respective values of the right ascensions and polar distances of stars belonging to those epochs (see Chapter VIII). Now, whichever be the method used, the mean quantity of the pre- cession is that which results from a great number of stars, three or four hundred, for instance ; and even if there were any undetected inequality equally affecting, however, all the stars, yet, since the effect of such inequality would be blended with that of pre- cession, the quantity of the precession (or what is so deemed) obtained from all the stars, ought to agree with the mean pre- cession deduced from numerous observations of any one star. But, if we suppose any peculiar movements to belong to any one, or to more stars, such peculiar movements would affect the quantity of precession determined by the preceding method, and vitiate it. Reversely, if the mean quantity of the precession deduced from the comparison of three hundred stars should differ from the quantity resulting from the comparison of fifty longi- tudes of the star Arcturus, for instance, it would infallibly follow * See pp. 184, &c. of this Work. M. Zach computed the longitudes of thirty-five principal stars, in order to determine the precession. 39T as a consequence, either that Arcturus was subjected to some motion to which all other stars were not, or that some or all of those stars were subject to motions from which Arcturus was exempt. Such motions, not generally affecting all stars, are called by Astronomers, Proper Motions, and are to be assigned to stars not from any theory but solely by observation. It is clear, however, that if there exists no other method of detecting these proper motions than what has been just described, that they can never be entirely disengaged from the effect of pre- cession and exhibited separately ; since they themselves enter into the composition of precession. All that can be done is to deter- mine how much the annual changes of the mean right ascension and mean north polar distance of each star differ from its mean precession in right ascension and mean precession in north polar distance; understanding, thereby, those values which are com- puted, by the formulae of pp. 340, 341, and from a quantity (50".l) held to be the mean quantity of the precession. These annual changes, which are compounded of the pre- cession and certain proper motions, are technically denominated the Annual Variations in right ascension and north polar distance, and are inserted as such in the catalogues of stars. The annual precessions in north polar distance and right ascen- sion are then subjected to a certain law (see pp. 1QO, &c. 340, 341.) but the annual variations are altogether irregular, never differing, however, from the former, except by minute quantities. We have seen, then, that the precession, as its results from Astronomical methods, is not the actual retrogradation of the in- tersection of the equator and ecliptic on the ecliptic. We will now consider another point. Is the retrogradation (supposing it capable of being determined) produced solely by the influence of the Sun and Moon on the excess of the Earth above a sphere ? Is it, in fact, a luni-solar precession ? This is a question which we must go out of the precincts of Plane Astronomy to find an answer to. 398 We have already seen, (Chap. XV.) that the obliquity of thet ediptic, besides its periodical variations (see pp. 375, 376.) is subject to an inequality of a very long period * and called a Secular Inequality. The effect of this inequality is the diminu- tion of the mean inclination at the rate of 45".7 in a century. It is caused by the action of the planets on the Sun : the effect of which action is to draw the Sun out of the plane of the curve in which he is moving ; so that, unlike the periodical changes of obliquity, which arise from the oscillations of the equator, or the nutations of the Earth's axis, the secular diminution of the obliquity arises from the displacement of the ecliptic itself. But this is not the sole effect of the action of the planets ; for, besides the changes of obliquity, the intersection of the equator and ecliptic is made to move, not by a retrograde, but by a direct motion, or according to the order of the signs along the equator. And, estimated in that direction, its annual amount (a quantity too small to be determined by observation) is O".20174: in the direction of the ecliptic its quantity is 0".18505 ( = .20174 . cos. 23 28'). Now this inequality is under the predicament described in p. 396 : it equally affects the longitudes of all stars : and, con- sequently, in determining the precession from the differences of the longitudes of stars at different epochs, (see p. 186, &c.) we determine not the luni-solar precession, but the luni-solar pre- cession diminished by this quantity 0". 18505. If, therefore, 50".l be the precession determined by observation, the preces- sion due solely to the action of the Sun and Moon is 50".1 -f // .18505. = 50".28505. In like manner the precession in right ascension, determined by observation, is less than the luni-solar precession in right ascension, by the quantity 0".20174, and, consequently, the actual change of right ascension is (see p. 344.) 50 /7 .28505 (cos. I + sin. J . sin. M . cot. S) 0".20174. * Physical Astronomy, Chap. XXII. 399 This is the expression for the precession in right ascension, on the supposition that the obliquity of the ecliptic remains the same : but, if that should be variable, it would be necessary to add the term - dl.cos. JR. cot. S to the preceding : so that, if d L represent the luni-solar pre- cession, the change in right ascension equals to dL(cos. I + sm - ! sin. J& cot. 5) -0".20174 dl.cos. M. cot. $*, * It may be right to explain the grounds and the method of deducing this and similar expressions, especially, since the subject, as it is found in some Authors, is not free from ambiguity. The equations from which the variations in right ascension, north polar distance, longitude and latitude may be readily deduced, are the first four equations of p. 182 : to wit, (1.) tan. JR = tan. L cos. I tan. \ sec. L . sin. 7. (2.) sin. (90 S) = sin. \ cos. I -f- sin. L . cos. X . sin. J. (3.) tan. L = tan. M cos. I + tan. (90 - S) . sec. .1 . sin. /. (4.) sin. \ = sin. (90 3) cos. Z sin. M . cos. (90 S) . sin. J. Which equations, by a slight transformation, are made similar, for a pur- pose which will be soon explained. If in the equation (1) we make L and JR to vary, \ and I remaining constant, we have, as in p. 341, dM = dL (cos. I -f-cot. S .sin. /.sin. A); but, if we suppose J to vary, then there will be introduced a term such as d I . cos. M . cot. S : so that (a) d JR = dL (cos. I + cot. S . sin. / sin. M) d I . cos. M . cot. 3. In like manner, from the equation (2), we have, \ and J being constant, (see p. 341.) dS= dL.cos.J&.sin. I; but if we make I to vary, we have (b) d$ = dL . cos. M, sin. I - d/.sin. M. Now 400 which is Laplace's expression, (see Mecanique Celeste, torn. II. p. 350.) Now the operations of reduction by which these differential expres- sions have been deduced from the equations (1) and (2), will, when applied to the equations (3) and (4), (which see p. 399. are similar), produce similar differential equations ; and accordingly, (c) dL = dJ& (cos. I tan. A . sin. 7. sin. L) d J. cos. L tan. \, (d) d\ =. dJR, . cos. L . sin. / dl . sin. /. We must now consider the conditions or circumstances under which these variations take place, and the several values which d J, the change of obliquity, will, according to those circumstances, possess. In the first place we may observe, that, the expressions (a), (b) were obtained by supposing the latitude (\) to be constant. The star, there- fore, being fixed, the ecliptic must be supposed not to change its position. But I the obliquity is variable : such variation, therefore, must be sup- posed to arise from a change in the position of the equator ; and the pre- ceding expression (a), by means of its last term, will express the change of right ascension due to that variation, of whatever kind the variation be, whether it be periodical or secular : provided its quantity be very small. The expression, therefore, will represent those variations of the right ascension and declination which arise from the changes in the obliquity produced by the solar and lunar nutations ; for, these changes are pro- duced by an oscillation of the equator : but they will not serve (for the reasons alledged in 1. 13, &c.) to represent the variations produced by that change of the obliquity which a change in the position of the plane of the ecliptic gives rise to. The expressions will also serve to represent any variations of right ascension and north polar distance, produced by a secular change of the -obliquity, provided such secular change arise from a change in the plane of the equator. Now it is such a change which Laplace and other writers must have had in their minds, when they represented the variation in right ascension, by the expression (a), of p. 399- For, such expression is used in determining, from an assigned mean right ascension of a star at a certain epoch, its mean right ascension at another epoch. In such kind of calculation (used, for instance, in determining the proper motion of a star) all periodical in- equalities, such as the nutations of the obliquity, are rescinded or ac- counted 401 The right ascension of a star will be affected, as we have seen, by that Progression of the first point of Aries which is caused by counted for ; and, as we have already explained, no secular changes of obliquity can influence the right ascension, except such as arise, not from a change in the position of the plane of the ecliptic, but from a change in the position of the plane of the equator. But we must, in this case, have recourse to Physical Astronomy. The secular equation, which we are in search of, is so small that observa- tions are unable to indicate it. Let I denote the mean obliquity of the ecliptic at the beginning of the year 1750, which ecliptic, for distinction's sake, is called the Fixed Ecliptic ; let t denote the number of years reck- oned from 1750: then, by the results of Physical Astronomy, (see* Laplace, torn. III. p. 158.) 7 = 23 28' 18" -f- t* x 0".000009842. Hence, JR, L, /, &c. being supposed functions of the time, or dJR, dL,dI, in the formula of p. 399. standing for we have dJR. dL_ dl . -dr d *> 7r dt > in df > ~ = 2* x 0".000009842 = t X 0".000019684, and accordingly, the formula of p. 399. 1. 27. for expressing that in- crement of the right ascension, which is to be added to the mean right ascension of 1750, jn order to obtain the mean right ascension at any other epoch distant from 1750 by t years, will be d JR. * 50".239055 (cos. 7+sin. I. sin. JR. . cot. 8)- 0".20l6*33 1 ~ /; .0000 19684 . cos. JR . cot. S x *, which is a formula like that which M. Zach has given at p. 12, of his Supplement aux Nouvelles Tables d' Aberration. If we substitute the value of dl, just obtained, in the expression for the variation in north polar distance, we have d* = - t x 50".239055 sin. I - 0".0000 1 96844 x t. In these expressions, it is to be observed, that allowance is made for all * According to Laplace this coefficient equals 50".2875. 3E 402 the action of the planets. Hence, if we estimate the precession from observations, made on the observed differences of right all the inequalities that affect the precession in right ascension and north polar distance : for instance, the whole effect of the lum-solar precession in right ascension is diminished by 0".20l633 (the effect of the planets in causing the equinoctial point to progress) and by 0".0000196844 cos. J& cot. S, which is the variation in the right ascension produced by a change of obliquity : which last correction it is scarcely ever necessary to make account of. Having now spoken of the change produced, in the right ascensions and declinations of stars, by the obliquity varying from- a change in the posi- tion of the plane of the equator, we will consider what effect on the positions of stars will be produced by the obliquity varying from a change in the ecliptic itself. A mere oscillation of the plane of the ecliptic, round an axis passing through the two equinoctial points, will affect neither the declinations nor the right ascensions of stars : but the latter quantities will be affected, if, as is the case (see Chap. xxii. Vol. II. of Astronomy} the change of obliquity is accompanied by a progression of the equinoctial points : both inequalities, indeed, arise from the same cause. The declination, how- ever, will remain constant ; and, accordingly, in deducing the equations (c), (d), of p. 399, we supposed 3 to remain constant. Those equations, there- fore, will represent the variations of the longitudes and latitudes of stars due to any change of the obliquity, provided such change arise from the displacement of the ecliptic itself. But there are no periodical changes of the ecliptic : the sole change to which it is subject is a secular variation produced by the action of the planets : the annual value of which is about 0".5 (see Vol. II. Astronomy, 'p. 451.) Accordingly, the expres- sions (c), (d), become dL=0".20l633 (cos. I - tan. A sin. I. sin. L) - 0".5t cos. L tan. A, and d\= *(0".20l633 cos. L sin. 7+ 0".5 . sin. L) the former of which expressions represents solely the change produced in the longitude of a star by the action of the planets on the plane of the Earth's orbit, and makes no account of the effect of precession. In deducing the above values of dL and d\ we have supposed the annual variation of the inclination produced by a change of the ecliptic to be 403 Ascensions at different epochs, we determine a quantity with which the above-mentioned progression is blended. But the be 0".5 : its value, unlike the other of which we have deduced (see p. 401. 1. 19.) may be obtained from the comparison of observations : but (see As- tronomy, Vol. II. Chap, xxii.) it may be also derived from theory, which, besides the term involving t and the coefficient 0".5, furnishes another term involving t* : thus, according to Laplace, (Mec. Celest. torn. III. p. 153.) /' representing the true ecliptic, /'= 23 28' 18"-f x 0".52114 - t 2 x 0".000002723, consequently, ^- = - // .52114 - t x 0".00000545, at and, accordingly, the former expressions of dL and d\, in order to be more correct, ought to be increased by the term t X 0".00000545 : but the practical correctness thence ensuing, is, as it is plain, of very little moment. In the preceding investigations, account has been made solely of the variations to which the mean inclination is subject : whether such mean inclination be the inclination of the plane of the true ecliptic, or of the Jixed ecliptic of 1750. But the true ecliptic, besides its secular diminu- tion, is subject to periodical variations ; one the solar, the other the lunar nutation. In order then to represent the true obliquity at any time of the year, let /', determined by the above equations of 1. p. be the mean value of the true ecliptic at the beginning of the year : let n be the number of days elapsed from the beginning, then the trite value of the true ecliptic is I' - rc x 0".52114 + 0".435 cos. 20 -f 9".6"4 cos. SI. oOo*xo We have already quoted from Laplace the values of the obliquity, &c. we subjoin, from the same Author, the values of some other quantities connected with this subject of enquiry, the precession ty) on the fixed ecliptic=50".2876*- 0".00012I7945'% (x//) on the true ecliptic=50".09915+0".0001 22148**. Hence making f = 1, , ^ - \j,' = 0".18848, nearly, which was, in the year 1750, the progression of the equinoctial points in longitude 404 declination of a star is not affected by such progression ; conse- quently, the precession determined from the annual precession in north polar distance, will be different from that which is deter- mined from the annual precession in right ascension. The differ- ence, perhaps, is too minute to be detected by observation ; but, if the results of Physical Astronomy be relied on, it exists as really as the precession itself. Thus, if the precession in north polar distance, of a star situated in the equinoctial colure, should be 2(/.04, then the pre- 20".04 cession in longitude = : = 50 .324, sin. I and the precession in M would = 20".04 . cot. / = 46". 162. But the precession in right ascension obtained by computing the right ascensions of the equinoctial point, at two different epochs, would be 46". 162 0".202 = 45".Q6. Having now ascertained the causes which affect the preces- sion, we will explain, by means of an instance, the method of detecting the proper motions of stars. longitude occasioned by the displacement of the ecliptic. The progression, therefore, of the equinoctial points in right ascension = 0".18848xsec. 23 28' 18" = // .20415, and, for a time t, = 0".20415. The access of the equinoctial point in the direction of latitude towards the south pole of the ecliptic of 1750 = t x 0".1884S x tan. 23 28' 18'' = 0".081 1, nearly. The expression of p. 403. is for the mean precession : but, as we have seen in Chap. XIV. there is an inequality arising from the unequal actions of the Sun and Moon : if, therefore, \^ ' deduced from p. 403. 1. 31. be the precession from 1750 to the beginning of a year distant from 1 750 by the time , and if n be the number of days elapsed from the beginning of this latter year, we have, at the end of these number of days, the true retrogradation of the equinoctial point, or the true preces- sion equal to *' + 36^25 X P " 1 "* Sin * 2 ~~ l6 "' 544 sin ' &> p being the annual precession in the proposed year, 405 It is required to determine, from the observations of 1755, and 1802, the proper motion of Arcturus *. The proper motion in right ascension will be the difference between the mean right ascension in 1755 increased by the pre- cession in right ascension due to the interval of forty-seven years, and the mean right ascension of 1802. Now, as it has been explained in pp. 343, &c. the annual precession, whether in right ascension or north polar distance, depending on the star's right ascension and north polar distance, must be different according to the epoch for which it is com- puted. Its values, therefore, in 1755 and 1802 will be different, although in a small degree. Suppose the precession of Arcturus to be that which would result from the mean value of its right ascension and north polar distance : then, since according to Bradley in 1755, its right ascension = 7 s 1 7' 25". 155 and according to Maskelyne in 1802 =7 1 39 27.6 its mean right ascension for the middle time =7 1 23 26 .378 Again, north polar distance in 1755 =69 3l' 54" -in 1802. . . 69 46 49.8 its mean north polar distance for middle time. . =69 39 21.9 We must now find the precession in right ascension from the formula of p. 399- but, previously, we must determine the value of the luni-solar precession to be used in that formula. In 1750 its value was 50".239055 t the prop 1 , part of its secular equat n . for 28.5 years. .0 .006693 .*. the value for the mean time of 1778.5 ......... .50.245748 * This instance is taken from the Supplement aux Nowcelles Tables $ Aberration et de Nutation. By the Baron de Zach, Marseilles, 1813. f According to M. Zach (Supplement aux Nouvelles Tables d' Aber- ration, p. 13.) the year 1750 being the epoch, the luni-solar precession = 50",239055 0".00023485 1. 406 which is the value of dL to be used in the present instance (see p. 400.) log. r . . 10 log. 50".245748. 1.7010994 log. cos. obliquity.. 9.9625002 1 .6635996 ( = log. 46".08924). Again, -log. r 3 30 log. 50".245748 1.7010994 log. sin. obliquity 9.6001570 log. sin. M . . . 9.7167296 log. cot. $ 9.5691196 0.5871056 = (log. 3".864l) 42.22463 But this value (42". 22463) is (see p. 398.) the value of the luni-solar precession in right ascension. In order, then, to find that value, which the observations give, we must diminish it by the progression of the equinoctial points : consequently, such value must equal to 42".22463 - 0".20172 = 42".02291 *. This last quantity, then, is the value of the precession in right ascension deduced from those values of the luni-solar pre- cession, and of the right ascension and north polar distance of Arcturus corresponding to an epoch, which is the mean of 2756 and 1802. Hence, the mean right ascension of Arcturus, computed from such precession and its mean right ascension in 1755, is equal to 7 s 1 7' 25". 155 ( + 47 X 42".02291 =) 32 55 . 07677 7 1 40 20.23177 but the observed right ascen n . of * in 1802= 7 1 39 27 . 6 the unaccounted for differ 6 .*. in 47 years is 52.63177. ' The epoch being 1750, the pregression caused by the action of the planets is 0".20l68 -f- 0".0000012 1. 407 This difference, for want of an explanatory theory, or from ignorance of its cause, is attributed to the star's proper motion: and its annual proper motion, thence computed, is =1 // .1196. Now, by reason of this proper motion, the computed right ascension (see p. 406^ 1. 36.) is greater than the observed right ascension. In order to make the two right ascen- sions agree, therefore, the proper motion must be applied, with a negative sign, or must be made to diminish the precession, or must be thus written 1 // .1196. The annual precession being then (see p. 406.) 42".02291, the annual variation (which is the term given in catalogues -and in the Nautical Almanack to the sum of the precession and of proper motion) is 42".02291 - l".1196 =40 // .90331 = ,in time, 2 8 .727. In order to compute the proper motion of the same star in north polar distance, we have (see pp. 348.) - log. r 2 -20 log. 50".245748 1.7010994 log. sin. obliquity 9-6001570 log. cos. right ascension .... 9.9312726 1.2325290 ( = log. I7".08l62). The mean north polar distance of Arcturus then, computed from such precession and its north polar distance in 1755, is equal to 69 31' 54" -f 17".08l62 X 47 13 22.83614 69 45 16.83614 but the observed north polar dist. in 1802. . 69 46 49 .3 the accounted for diff. .*. in 47 years is. . . .0 1 32.96386 The proper annual motion, therefore, in north polar distance l' 32".96386 is equal to = 1.97795. And as this proper motion makes the computed north polar distance .less than the observed, 408 it must be added to the precession in north polar distance, and written l".97795. The precession therefore, being 17".08l62 and the proper motion 1 -97795 the annual variation in north polar distance 19.05957 * In like manner the proper motions of other stars are to be determined : and Dr. Maskelyne computed, in the first Volume of the Greenwich Observations, the proper motions of Sirius, Castor, Procyon, Pollux, Regulus and a Aquilae. The list has subsequently been much increased (see Greenwich Observations Vol. I. tab. 9.) It is plain, from what has been said, and from the preceding computation, that the proper motions of the stars are determi- nable by no formula. They are ascertained solely by obser- vation. We are ignorant of their causes and laws. We cannot even presume that the proper motions, determined by the com- parison of observations made at different epochs, were the same in the preceding, or will be the same in future periods. All that can be said is, that, if such a presumption were made, the error consequent on it will be very small, inasmuch as the proper motions themselves, as far as they have been hitherto ascertained, are very small. But, notwithstanding our ignorance of the causes of these proper motions, still it is essential to know their quantities, since they affect observations precisely, as any other inequality does, and lessen or augment the right ascensions and declinations of stars. Their effects, therefore, are now, as we have said, regularly combined with the results from precession, and then inserted in catalogues. It is the excellence of modern instruments and observations that causes the proper motions of stars to be known. They were formerly blended with the effects of other inequalities, and not dis- tinguishable ; principally for this reason, that their quantities were far less than the probable errors of observations. They are not even now easily made out : for, as it appears by the instance of p. 405, &c. they are not determined by single observstions, or by R0 jf Dr. Maskelyne's explanation and Use of the Tables, p. 4. makes the proper motion of Arcturus in right ascension = l".395, and in north polar distance = + 2".01. 409 the comparison of observations made during short intervals, but by the comparison of observations made in former times with present observations. Now, as Dr. Maskelyne remarks (p. 9. Explanation and Use of the Greenwich Tables) ; ' we are in want of good antient observations/ But, even if we did possess ober- vations of the latter character, the question concerning prbper motions, would be one of considerable difficulty ; or, rather, the nature of these motions is, as it must always be, ambiguous. The great Astronomer, whose words we have just quoted, says a little farther on ' the other stars of the Table do not appear to have any proper motions/ It should rather have been said that they do not appear to have proper motions according to the method used for determining them. For it is easy to feign a case in which a star should have a proper motion, which should not be indicated by the method used in detecting it. For instance, since the precession, as determined by the differences of the longitudes of stars at different epochs, or of their right as- censions, is the m^an of the precessions due to the^ several stars diminished or augmented by the proper motions of those stars, such mean may be exactly equal to the precession plus or minus the star's proper motion : in which case, the star would appear to have no proper motion. If we would state the case symbolically, let P be the precession, a, /3, 7, 5, &c. (either positive or nega- tive) the proper motions in longitude, of the stars which are used in determining the precession, then the precession, determined according to the method we have described, is P + -f P + /3-f P + 7 + &c. m m being the number of stars ; let p be the proper motion of the star the proper motion of which is sought, then it equals mP + a + )3 + 7 + &c - m _a -f ft +7 +_&c. which may become equal to by a variety of ways, th variously adjusting the proportional values of a, /3, 7, ^ .'3 F that is, by &c. 410 There will be an error or uncertainty of a like nature what- ever the star be, the proper motion of which should be required : that is, the result of its proper motion will be an ambiguous result, whether the star be or be not one of those stars that are used in determining the precession. If the former be its predi- cament, we then, in deducing the star's proper motion, are arguing (which is a common case in Astronomy) in a vicious circle: since the quantity to be determined is already implicated in the quantities from which it is to be determined. Dr. Maskelyne mentions only a few stars as having proper motions. But M. Bessel, by examining 2959 stars of Piazzi's Catalogue, finds 425 that have an annual motion not less than 0".2. As there is no law dependent on the places of these stars regulating their proper motions, so there is no connexion sub- sisting between the magnitudes of stars and the quantities of such motions. The only circumstance worthy of note seems to be that, amongst the stars apparently endowed with considerable proper motions, there are many double stars, a Cassiopeae and a Geminorum are two instances, the proper motion of the first being, in JR = l // .85, in N. P. D. = 0".47, of the latter being, in M = .58, in N. P. D. = . 64. But the stars with the largest proper motions are 40 D of Eridanus and 6l * of Cygnus : that of the former, in north polar distance, being 4", of the latter 3". 3. So that, according to M. Zach's method of illustrating the subject, if we were to determine the latitude of Greenwich by means both of one star and the other, and the two determinations should exactly agree in 1821, then, in 1822, they would differ by ?", if in the process of correcting the observations, we made no account of, or were ignorant of, these proper motions. The preceding discussions relate to very minute changes in the positions of stars : of which minute changes there are two kinds : one of the points or planes from which a star's place is measured : (as for instance, the changes of intersection of the equator and ecliptic and the plane of the ecliptic from the action of the planets :) the other of the position of the star from some * 6*1 Cygni in Flamstead's Catalogue. 411 unexplained motion of the star itself. The former change, like the other ciianges from precession and nutation, leaves to the star unimpaired its character of being fixed : the latter takes away from the propriety of that denomination. But the estimation of these minute changes, whether they be those of the observer, or of the star itself, is, as it must have appeared, a matter of considerable nicety. We must compare present observations made with large instruments with tolerably good catalogues. It is needless to say that, the better the cata- logue the more exact will be our operations : but, it is to be noted that, beyond Bradley's time, there are no catalogues of sufficient exactness for determining quantities so small as those of the proper motions of stars. We are thus deprived of the means of remedying, the almost inevitable errors of particular observations, by the comparison of observations distant from each other by very large intervals of time : and, in truth, the fixing of the laws and quantities of those minute variations, which have just been the subjects of discussion, is a point reserved for times to come. It is a matter not to be questioned that our present in- struments, and our present means of forming catalogues of stars (which at the epochs of their construction are the most faithful registers of the mean places of stars) are better now than they were seventy years ago. We speak of the catalogues made, for particular epochs, by Flamstead, Bradley, Mayer, &c. : but, in fact, in a modern Observatory the business of making, or of improving, the cata- logue of stars is an operation continually going on. We will, for a short time, turn our attention to that point, and first notice the practice of the Greenwich Observatory. Observations of north polar distances are now made at Greenwich by means of a Mural Circle, of which a short description has been given in pages 10Q., &c- The accuracy of the divisions of the instrument is examined by other means than Astronomical. It is, in fact, presumed to be a perfect instru- ment : an imperfect instrument, used with a perfect catalogue, (admitting, for an instant, the possibility of the latter circum- stance) would necessarily tend to make the catalogue erroneous. 412 The first operation with the mural circle is, (the telescope occupying that position in which the star was seen bisected by the centre of the cross-wires) to read off the six microscopes, and, see pp. 112, 113, to take the mean of these as the instru- mental polar distance of the star. This distance is next to be corrected for refraction, precession, aberration, and nutation, and reduced to the first day of the year on which the observation was made (see Chapters IX, to XV.) Such reduced distance is the mean distance, and it ought, supposing the observation to be truly made, to agree, allowing for the index error, with the tabulated mean polar distance : or, the mean polar distance of the catalogue. If it does not agree, the error is held to be in the catalogue. Thus, suppose Polaris to have been observed on June II, 1812, both above and below the pole, and the reduced north polar distance above the pole to be 1 4l' 55". 98 , . . . below , 358 18 31 . 79 then, were it not for the index error, the sum of these ought to equal 360 : if the sum differs, it differs by twice the index error. In the present instance, then, since the sum is 360 0' 27".77, the index error is 13".885 : consequently, in order to obtain the true north polar distance above and below the pole, we must diminish the former by 13".885, and increase the latter by the same quantity. Reversely, the observed distance minus or plus the true distance must give the index error. Now, if the north polar distance of the catalogue were the true north polar distance, the observed instrumental north polar distance, minus or plus the north polar distance of the catalogue, ought, if the latter were correct, to give the same index error. If it does not, the error is the error of the catalogue. Thus, in the instance before us, the north polar distance being . , . 1 4l' 55".98 and the index error 13 .885 the true north polar distance . . . c ... 1 41 42 .095 but, if the north polar distance of the catalogue were 1 4l' 4l' / .30 the difference between it and 1 4-1 ' 55".98, would be equal 413 to 14".68, instead of 13".885 : the difference, therefore, of these two differences,, that is, 14".68 13".885 = .795, must be the error of the catalogued north polar distance. The principle of illustration used in this instance extends to all other like instances. Consider, therefore, the mean north polar distances of the catalogue to be the true mean distances, subtract them from the observed,, and the results are the index errors : their mean is the mean index error. Subtract this index error from the observed north polar distance of a circumpolar star above the pole, and add it to the north polar distance of the same star below the pole : the sum, as it has been shewn, ought to equal 360, if the catalogued be the correct distances : if not, the error is that of the catalogue, which, from the defect from 360, become known. The thing will be made more clear by examples. Observations made with the mural circle from June II, to June 18, 1812, the position of the telescope being O, (seep. 115.) No. of Obser. Reduced Observa 8 . by Instruments. N. P. D. by Catalogue. Difference. Diff. X No. of Observations. Polaris. . . . 2 1 41' 55".Q8 41 // .30 14".68 29". 36 S P. ..... 7 358 18 31.79 18.70 13 .09 91.63 /3 Ursae Min. 6 15 4 46.88 34.23 12.65 75.90 a Cassiopeae. 1 34 29 54.20 42 . 48 11 .72 11.72 y Draconis. . 3 38 29 15.88 2.85 13.03 39.09 fj Ursae maj. 6 39 44 52.09 39.61 12.48 74.88 Capella .... 1 44 12 37.76 25.03 12.73 12.73 a Cor. Bor. 5 62 38 55 42.96 .12.04 60.20 Arcturus. . 5 69 50 11.81 0. 11 .81 59.05 /3 Leonis. . 3 74 21 49-18 37.11 12.07 36.21 Regulus . . 3 77 7 19.26 5.29 13.97 41.91 aSerpentis. 3 82 58 38.52 27.50 11 .02 33.06 Total .... 45 sum 565 . 74 565". 74 ' = 12".57 equation for north polar distance subtractive, 4D 414 In the above process, that operation is made with eleven stars which was (see p. 412.) illustrated by means of one. The result is now a mean result. If we went no farther than we are allowed to go by these observations made, during seven days, and on eleven stars, we should have a quantity 12".57 re- presenting the index error, and which it is necessary to subtract from the observed distances, in order to obtain distances which would be the true distances, were the catalogues correct. The test of that correctness is to be found, as we have already shewn, in the sum of the two polar distances of a circumpolar star. Taking Polaris, then, as such a star, we have Observed N. P. D. Equation to N. P. D. Corrected N. P. D. above pole 1 4l' 55".Q8. ... - 12".57. ... 1 4l' 43".41 below pole 358 18 SI .79. ... - 12 . 57. .. .358 18 19 . 22 360 2 . 63 The catalogue, therefore, (see p. 413.) cannot be right; the mean subtractive equation of north polar distance ought to be greater than 12".57 by f (2".63), or l".315 : but, in order to obtain a greater index error, or subtractive equation of north polar distance, we must lessen the mean north polar dis- tances of the catalogue : consequently, the correction of the catalogue would be l".315: and, that being made, the sum of the two north polar distances would be, as it ought to be, exactly 360. But, in a matter of such astronomical importance as the correction of a catalogue, it would be unsafe to trust to the observations of a few stars, made during a short period. If the instrument were, with regard to its divisions, a perfect one, it would still not be exempt from the effects of partial expansion. To annul these effects and those of the inequalities of graduation, (for in degree, at least, such must be supposed to exist) it is necessary to multiply observations and (we are speaking of the Greenwich mural circle) to change the position of the telescope, and to observe the same stars (see p. 413.) when it shall occupy the positions 0, 30, &c. Thus, as a specimen of the operations when the two former positions are employed : 415 1812. Posi- tion. Equation to N. P. D. Above Pole uflcorrected N. P. D. Corrected. Below Pole uncorrected N.P.D. Corrected. ne 11 to 18. -12".57 1 41 55.98 / // 1 41 43.41 358 1831.79 ' // 358 18 19.22 19 to 20. -10.08 051.90 41.82 29.26 19.18 to July 1. 30 -23.57 1 42 5.39 041.82 41.57 18.00 Reductions, similar to those indicated in the above schedule, are made of observations for the year 1812. If, in order to obtain greater accuracy, we employ the observations of 1813, we must reduce the mean of the latter to the mean of the former, by adding, to the north polar distance of Polaris, the quantity + 19 /; .45, which expresses its annual precession in north polar distance. In the pages of the Volume of the Greenwich Observations (for 18 15) from which the preceding extracts have been made, the observations of 1812 are combined with those of 1813 : and the mean of the N. P. D's of Polaris above the pole = 1 4 1' 2 1".50 below =358 18 38.32 359 59 59-82 The mean result, then, of the observations of two years differs from that (see p. 414, 1. 13, &c.) of the observations of seven days : and, since the sum of the two north polar distances is less than 360 by 0". 1 8, the correction of the catalogue becomes additive and equal to 0".09. By these means a new catalogue is made, which is, by like operations, again to be reformed. The above method will, generally speaking, render more correct the mean polar distances of stars : it may, indeed, by its peculiar nature, render the mean distances of some less correct than they were before the process of correction, For, in finding the index error, a great variety of stars are used, some of which are considerably distant from the zenith of Greenwich. Now, a previous operation in finding the index error is the reduction of the several observed distances ; the accuracy, therefore, of the 416 index error so to be found, depends on the reductions of all the stars being operations of like certainty. One of these reduc- tions is the correction for refraction : which, as Dr. Brinkley has observed in his Memoir (Irish Transactions, 1815.) on the paral- laxes of stars, is a correction of considerable uncertainty (see pp. 233, &c.) If the correction used for refraction should be wrong, the index error cannot be right. The like may be said of the other corrections. In short the index error, and the con- sequent correction of the catalogue must, in degree at least, partake of that uncertainty to which the reductions of any of the stars used in finding the index error are liable. This method, then, of correcting the mean declination of stars requires, as Dr. Brinkley notes, great attention in all enquiries concerning (what indeed modern Astronomy is now conversant about) minute changes in the places of stars. But the excellent Astronomer, whom we have just quoted, uses a different instrument and method for determining the mean places of stars. The method may indeed be said to be essen- tially different, since it has no .concern with the index error, which, as we have seen, plays so great a part in the uses of the mural circle. We will now speak briefly of the description of the Circle of the Observatory of Trinity College, Dublin, and more fully of its uses and application. The Circle planned and partly executed by Ramsden, is eight feet in diameter. What is peculiar to it, being so large an instrument, is its capability of being turned round a vertical axis : so that the same face of the instrument may be turned both to the east and west. In this principle of its construction it resembles a zenith sector, and those small quadrants and declination circles that are furnished with azimuthal motions (see pp. 65, &c. also Phil. Trans. 1806, pp. 406, See.): and its maker intended that it should derive from that principle, the same advantage which the smaller instruments possess, namely, that of determining the true zenith distance of a star, inde- pendently of the line of collimation (see pp. 67, &c.) A plumb-line is used in the present instrument, not for determining the zenith point on the instrument, but for adjusting 417 the vertical axis. The divisions of the limb are read off by means of three microscopes : one at the bottom, opposite to the lowest part of the circle : the other two respectively opposite the left and right extremities of the horizontal diameter. This short description is sufficient for our purpose : a fuller descrip- tion has been given by Dr. Brinkley in the Irish Transactions for 1815. It was the original intention of the maker of the instrument that meridional observations should be made with it. And the instrument can readily be placed in the plane of the meridian : but, in that case, only one observation of the same star can be made with it on the same day. Such (see pp. 67, &c.) is to be reckoned only half an observation : we must wait twenty-four hours at least before we reverse the instrument and complete the observation. If the weather should be unfavourable we may be obliged to wait several days. But, even in the interval of one day, the temperature may have altered and affected the instrument. To prevent this evil, or to obviate the objection that may be founded on its supposed existence, Dr. Brinkley observes the star with the face of the instrument to the east, once or twice before it reaches the meridian, and then, as-often, with-the face of the instrument to the west, after the star has passed the meridian. Thus the two essential parts of an observation are made within the space of ten or twelve minutes. But the ob- servations thus made are, in a certain sense, imperfect ones, since they are not observations of meridional zenith distances. They- may, however, by the aid of calculation, be made to become, or be reduced to, such observations. The main con- dition necessary to be known is the time of the observation, or, rather, the interval of time between the observation of the zenith distance and the star's transit over the meridian. This is easily had in an Observatory. What else remains is a matter altogether of calculation, which, as on like occasions, will furnish us with a formula and rule of solution. We will now direct our attention to the formula, which is to express the difference of the meridional zenith distance of a star, and of its zenith distance observed very near to the 3 G 418 meridian, in terms of the interval between the times of obser- vation and the star's transit, and of certain given quantities. Let L denote the latitude of the place of observation, D the star's polar distance, z, z' f two zenith distances, h, h 1 , the corresponding hour-angles ; then, if we form two spherical triangles ZP s, ZP s 1 , Z being the zenith, P the pole, and 5, s' two positions of the star, we have (see Trigonometry, Chap. IX.) cos. z sin. L . cos. D cos. h = - : ~ , cos. Li . sin. JJ f cos. z' sin. L . cos. D cos. L . sin. D ., cos. J cos. z consequently, cos. h cos. h = : - . cos. L . sin. JD or (see Trigonometry, p. 33.) h'+h ti h z' + z z'z 1 sin. . sin. = sin. . sin. 22 22 cos. L . sin. D ' Let one observation (that to which h } z belong) be made on the meridian, then, since h = 0, U . z'+z . z'-z sin. = sin. . sin. x 2 2 2 cos L . sin. D. ' Now z'. z are nearly equal : let $ denote their difference, then z' + z a 2 2* z'-z 3 2 2' .'. sin. fi . d k ' ^ Q = COS * ^ * S ^ n< ^ ' Sm ** """ but, S being very small, 419 sin. - =s -. sin. l", nearly, / 5 \ 3 sin. f 2 + j 1 = sin. 2 4- - sin. 1 .cos. 2; f ^ 1 A' .'. -sin. \" x < sin. 2+ - sin. l". cos.sj = cos. L. sin. D.sin. 2 - , . o h' cos. L . sin. D and - as sin. . -5 O o sin. l" (sin. 2 -^ sin. l". cos. zj . e h' cos. Z/.sin. Z) f $ ; ) = sin. - . - Tl : . < 1 sin. 1 . cot. Z(, nearly. 2 sin. l" . sin. z I 2 If no great accuracy be required we may reject the second term, in which case, we have 5 . 9 h' cos. L . sin. D 2 " 2 * sin. l". sin. z Substitute this value in the preceding expression, and we i shall have a second approximate value of - , in the expression Sill ~ ^ 5 . A 7 cos. I/ . sin. D ' 2 /cos.L . sin. Z>v - = sin . . ff : : TT/ I : I - cot.*; 2 2 sin. 1 . sin. z sin. 1 V sin. z s which, since z = L 90 D, is Delambre's expression, and from which the correction ^ or the reduction to the meridian, may be computed. Dr. Brinkley, however, very rightly prefers another formula (or rather a transformation of the above formula) in which the arc and its powers, should be involved instead of the powers of its sine. Thus, since if z/ ., slJ^ 3 tl /* . If 1 /K\ . M sin. =s - sin. 1 - I - 1 . sin. 1 , nearly, 420 writing, therefore, in the above formula, cosec, z instead of , and \5h' instead of ti , in order to convert h', expressed sin. z in parts of space, into time, J = !!5lj! . 15 . cos. L . sin. D . cosec. z X (h'f 2 . 15 4 . cos. L . sin. D . cosec. z X (h'} 4 24 , n j . 15 4 . (cos. L . sin. D . cosec, zf. cot. z x (h') 4 , 8 from which $ may be computed. We may, however, for the purposes of computation, express $ more commodiously. Let the first term (C), the first correction, = A sin. D . cosec. z . (h'f then the 2d term, orcor.(C') = . \&.A sin. D. cosec. z.(A'y 12 and the third term, or correction (C") = - C 2 . . cot. z. Hence, since A = . 1 5 2 . cos. L, we have log. A = log. sin. l"-f 2 log. 15 -f- log. cos. L + ar. com. 2 20, log. C = log. -4 4- log. sin. D + log. cosec. z 20 -f- 2 log. A', log. C'= log. -^ -f 2 log. sin. l"-f 2 log. 15 + ar. com. 12 + log. sin. D -H log. cosec. 2 30 -h 4 log. hf, log. C" =: 2 log. C -f log. sin. l" -H ar. com. 2 + log. cot. z 20. When the observations are made at the same place, the log. cos. L, which is a given quantity, may be added to the other constant quantities : for instance, the latitude of the Dublin Observatory being 53 23' 13".5, its log. cos. = 9-77552, which being combined with the logarithms of the three first terms of the expression for log. C, the result is 16,51225. Hence 421 * log. C = 6.51225 + log. sin. D + log. cosec. z 20 + 2 log. ti, log. C' = 7.15638 + log. sin. D + log. cosec. z 20 + 4log.#, log.C" = 4.38454 + 2 log. C + log. cot. z - 10. These are the formulae of computation for any star, the lati- tude of the place being equal to that of the Observatory of Trinity College, Dublin. But, as it is necessary to make a great number of observations of the same star, it is convenient to possess peculiar formulae of computation adapted to the several stars. For instance, if the star should be Arcturus, the sum of the second and third terms of log. C is a constant quantity (=20.23344); the third term of log. C" is also a constant quantity (= 10.1831) : and accordingly, the three formulae for Arcturus, observed at the Observatory of Trinity College, Dublin, become tlog. C = 6.74569 + 2 log. ti, log. C 1 =3.7.3898 + 4 log. ti, log. C" = 8.05902 + 4 log. ti. Above the pole z' > z, and z' z = & ; .*. z = z 1 3. Hence Constant Number in log. C computed. Constant Number in log. O computed. * Log.cos.lat 9.77554 log. A 6.51225 log. sin. 1" 4.68557 2 log. sin. 1" 9.37114 2 log. 15 2.35218 2 log. 15 2.35218 Arith. comp.2 9.69896 ar.comp. 12 8.92081 (log. A) 26.51225 7-15638 ViuniurmLu-^-.. I. Constant Number in log. C" computed. log. sin. 1" = 4.68557 log. 2 = .30103 4.38454 t Log. sin. N. P. D. (=69 52' 46") 9.97265 log. cosec. z (=33 16) ., 10.26079 20.23344 .23344 6.51225 7.15638 6.7456"9 7-38982 2 3.49138 4.38454 1831 8.05902 (see p. 418,) in order to reduce the observations, we have this formula, meridional zen. dist. sr observed zen. dist. (C C' C"), and below the pole meridional zen. dist. = observed zen. dist. + C C + C". The following instance of the star Arcturus observed, May 12, 1820, at the Dublin Observatory, contains the appli- cation of the preceding formulae, Latitude of the Observatory 53 23' 13".46 mean N. P. D. of Arcturus for 1820 69 52 31 . 89 mean M 21 % 1 5151. 6 place of Moon's node 1 1 s 29 26 Time by Clock. Left Micros. Bottom Microscopes. Right. Micros. Mean of the three Microscopes. Refrac- tion. I3 h 56 m 28 s 49".7 33 19' 50". 5 E 4".3 33 19' 54".83 37".82 14 28 31.7 33 17 32. 6 E 47.1 17 37.13 37.77 14 9 51 X 50.6 33 14 54. 5W 45.0 14 50.03 37.74 14 14 52 38.0 33 16 41. OW 31.7 16 36.90 37.77 Barometer 29.67. Thermometer Int. 52.5. Thermometer Ext. 48. 423 sage by clock) time of obser n . 135628 14 h 7 m 3 9 .3 14 28 14 h 7 ro 3 S .S 14 9 51 14 h 7 m 3 8 .3 14 1452 values of h' . . . 010 35.3 6 35.3 2 47.7 7 48.7 values of A' in seconds 635.3 395.3 167.7 468.7 logarithms of h! 2.80298 2 logarithms of h 1 5.60596 constant quantity 6.74569 log. C 2.35165 2.59693 5.19386 6.74569 1 .93955 2.22453 4.44906 6.74569 1.19475 2.67089 5.34178 6.74569 2.08747 122.31 C . . . . . . 224 72 87.008 15.66 4 log. A' .... 1.21192 7.3898 log C 7 .... 8.60172 C 1 .... 0399 0.68356 8.05902 8.74258 0.055 122.31 2' 2".25 37.77 4 log h' .... 1.21192 . constant quantity 8 05902 log. C" . . 9.27094 87".008 37.77 15 7/ .66 37.74 C" 0.186 C r . 003QQ C 224 72 (C-C 7 -C").. 3' 44 /7 .49 refraction .... 37.82 3 6.67 meanofSmic*. 33 19 54.83 E 49.23 17 37.3 E 22.08 50.03 1 24.54 16 36.90 33 1648.16E 48.07 1648.07 12.11 12.36 15 12.36 W 33 16 48.1 IE 33 15 12.23 W 33 15 12.23 W. , January, 1820. 33 16 0.17 aber. prec n .nut. . . 13.53 33 15 46.64 mean zen. dist, 424 The above is the whole of the process necessary for reducing each of the four observed zenith distances to the meridional zenith distance. In the left hand column, since the interval (h'} between the time of observation and the transit was 10 m 35 s , all the three corrections C, C', C", were computed : but in the second and third columns, when the values of h' are only 6 m 35% 2 m 47 s , the * The value of h' is made the difference between the times of obser- vation by the clock and of the star's transit by the clock. The most ready way of determining it, is to observe the star's transit by the transit instrument, and to note its time by the clock. The difference of that time and of the time of observation by the same clock is the value of A'. If it be not convenient to observe the star's passage, we must compute its JR and thence, and from the error and rate of the clock, compute h'. The special object of the example in the text is the illustration of the method of finding the meridional zenith distance of a star by means of zenith distances observed before and after the star's transit. But the example may be made to serve another purpose : it is a kind of practical proof that the duties of an Observatory, are laborious duties. The com- putation, as it stands in the text, is a long one ; yet the whole of it is not given : for instance, the computations of the four refractions and of the inequalities of aberration, nutation, and precession are omitted. Again, we have considered only one star: but, if ten or more stars be observed, they will all require reductions similar to the preceding reduc- tion. Observations, then, by means of a circle, such as we have been speaking of, and so used, are considerably more operose than those made by a mural circle or quadrant. The above method of deducing zenith distance is peculiar to the Observatory of Trinity College, Dublin. It renders the duties more laborious than when the meridional zenith distance is observed by means of mural quadrants or circles. The other parts of the daily business in an Observatory, the observations and computations of right ascensions, occwltations, eclipses of satellites, are nearly the same at Greenwich, Paris, and Dublin. Bradley's theories, and instruments like Bird's, make one Observer quite unequal to the proper discharge of the duties of an Ob- servatory. We 425 values of d , C" (they are to the values first obtained nearly as the squares of the times) are too small to be taken account of. In the fourth column C' is too inconsiderable to be computed. The catalogues of mean right ascensions stand, also, in need of continual corrections : and such corrections are effected on grounds not altogether unlike those that have been used in cor- recting north polar distances. We have endeavoured to draw the attention of the reader to the respective constructions and uses of the two instruments of Greenwich and Dublin for measuring north polar distances and zenith distances. The former, as we have seen, does not determine the north polar distance of a star, except by the intervention of several other stars, used for de- termining the index error. The latter is capable of determining the zenith distance of a star, if such star should be a solitary one in the heavens. It is capable of determining, within so short an interval as fifteen minutes, the zenith distance of a star : and, on that account, is admirably adapted to note (should there be any such) the peculiar motions of a star. On the mere footing of theory, no instrument is better adapted for dis- covering (should it be capable of being discovered) parallax. But various objections are made to it. Istly, The great mass of metal that forms the frame of the instrument, and revolves with it, and likely, from its derangements from heat, &c. to derange the instrument : 2dly, theunfixednessofits microscropes, and their position : 3dly, the uncertainty of the permanence of position of the plumb-line, by which, at each ob- servation, the instrument is adjusted. These objections are certainly deserving of attention, and ought, (as much as they can be,) to be ex- amined by the observations which the instrument itself furnishes, in the same way as the recorded observations by the mural circle are perhaps sufficient to determine its precision in settling the mean place of any proposed star. In the mural circle the derangements from unequal temperature can arise only from the circle itself being affected. Its six microscopes are fixed : the position of its telescope may be varied. These are its great excellencies. The accuracy of its division (of which, however, the Astronomer Royal has given the fullest testimony) is without the present question, which regards not an individual instrument, but a class of instruments and the principles of their construction. With 3 H 426 The method of finding the right ascensions of stars has been explained in Chap. VII. of this Treatise. It depends on the finding the time of the Sun's entering the equator. The same process determines the right ascensions of the Sun and of the stars employed in that process. If by that, and like processes, we obtain, for a certain epoch, a correct catalogue of the mean right ascensions of stars, we are able, by a knowledge of the several inequalities to which the places of stars are subject, to determine their right ascensions for any other epoch, and thence to regulate the Astronomical Clock. We could thence determine the right ascension of the Sun. If, therefore, by any means, other than those of the right ascensions of stars, we are able to determine, at the latter epoch, the Sun's right ascension, such determination, compared with the former, would be a test of its accuracy ; and, consequently, of the computed right ascensions of stars. Now we possess such means of determining the Sun's right ascension in a knowledge of his declination, which can be observed, and of the obliquity of the ecliptic which can be com- puted. An instance has been given of this method in pages 151, &c. The Sun's right ascension, then, is when he is near to the equinoxes, to be computed from the clock ; in other words, from the right ascensions of stars j and from his observed north polar distances. The differences of the two results, then, would (sup- posing the latter computations to be exact) be the errors of the catalogue. The following Table will exemplify the method. With regard to the mural circle, it ought farther to be noted, that, although it uses no plumb-line, and cannot be reversed as the Dublin circle can, it is not destitute of the- means of determining the zenith point. Such means are found in an artificial horizon (a basin of oil or of quicksilver) ; and then certain stars are observed both by reflection and by immediate vision. This operation is now practised at the Greenwich Observatory. 427 Observations of the SUN about the EQUINOXES. 812. Sun's R. A. by Stars. Sun's R. A. by Dec. 1817- Sun's R. A. by Stars. un's R. A. by Dec. Diff. Sum. ug. 8, 138 11 41.1 11 50.0 // + 8.9 May 5, / // 42 641.8 6 34.2 - 7'.'6 + l".3 13, 142 55 56.4 55 49.6 - 6.8 Ap r . 30, 37 19 53.2 7.8 + 14.6 + 7.8 21, 15023 50.7 24 3.2 + 12.5 22, 29 4731.0 7 45.0 + 14.0 +26.5 26, 155 5.5 59 57.5 - 8.0 16 24 12 52.8 3 6.8 + 14.0 + 6.0 ept. 2, l6l 22 40.9 22 43.1 + 2.2 10 184135.2 41 48.5 + 13.3 + 15.5 4, 163 11 21.0 11 19-6 1.4 8 16 51 45.6 52 6.7 + 21. + 19-7 5, 164 527.9 5 28.2 + 0.3 7 15 57 0.0 57 8.3 + 8. + 8.6 12, 1702332.1 23 23.3 - 8.8 Mar. 31 9 34 40. 34 45.3 + 4. - 4.3 14, 172 11 17.8 11 11.0 - 6.8 30 8 40 10. 40 22.0 4-12. + 5.2 20, 1773433.1 34 33.2 + 0.1 23 2 18 54. 19 4.8 + 10. + 10.7 23, 1801619.9 16 23.4 + 3.5 20 35935 10. 35 23.4 + 12. + 16.1 25 182 436.7 4 25.6 -11.1 18 357 45 53. 45 59.8 + 6 - 4.4 >ct.!5, 200 20 15.7 19 57.0 -18.7 Feb. 26 339 17 14. 17 31.4 + 17. 1.5 26, 2104447.0 44 40.2 - 6.8 l6 32943 58- 44 19.3 +20. + 14.8 27, 211 4237.5 42 30.0 - 7- 15 328 45 43. 46 3.3 + 19. + 12.1 Jov.3, 218 3224.3 32 7-4 -16. 8 3215236. 52 45.5 + 9- - 7.8 7, 22231 3.4 30 52.0 11. 3185249 53 1.5 + 11 + 0.4 10, 225 32 14.7 32 2.5 -12. 31449 56 50 17.3 + 21 + 9-0 11, 22633 1.8 32 37.0 -24. Jan. 3 313 48 53 49 12.0 + 18 - 6.3 The sum of the positive results in the last column is 153".7 : of the negative 24".3, their difference, therefore, is 129 /7 .4. There are thirty-eight comparisons, and Jo which, in time, is Cf.%8, is the correction of the catalogue : or, is the common quantity by which the mean right ascensions of the catalogue are to be increased. By a repetition of such processes and the use of improved instruments, the catalogues of the right ascensions will continue to be improved. We have now given, not enough indeed for all purposes, but sufficiently for the plan of the Treatise, the theory ofthejixed stars. It, we may say, necessarily precedes that of wandering stars or planets. The right ascensions and declinations of the 428 fixed stars are used in determining those of the planets. They have many things in common. Both are subject to the inequa- lities of refraction, aberration, precession and nutation. But, without going far into the circumstances of distinction between the fixed stars and the planets, it is obvious that there must be peculiarities belonging to the latter from their relative proximity to the observer, and their continual change of place. The former circumstance renders them subject to parallax, and the latter (to mention one instance) modifies the quantity of aberration : for, in the time of the transmission of light from the planet to the Earth, the former has changed its place. But these are only slight circumstances of distinction. The planets' distances, velocities, the forms of their orbits must be investigated : subjects of enquiry to which there is nothing like in the preceding discussions, and depending on principles not yet laid down. Our attention will be directed to these points in the succeeding part of this Volume : and as, amongst the planets, the Earth claims the chief consideration, its theory shall be first discussed. 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