QB 201 ;-NRLF PRACTICAL ASTRONOMY FOR ENGINEERS PRACTICAL ASTRONOMY FOR ENGINEERS BY FREDERICK HANLEY SEARES Professor of Astronomy in the University of Missouri and Director of the Laws Observatory COLUMBIA, MISSOURI THE E. W. STEPHENS PUBLISHING COMPANY 1909 COPYRIGHT, 1909 BY FREDERICK HANLEY SEARES PREFACE The following pages represent the result of several years' experience in presenting to students of engineering the elements of Practical Astronomy. Although the method and the extent of the discussion have been designed to meet the specialized requirements of such students, it is intended that the work shall also serve as an introduction for those who desire a broader knowl- edge of the subject. The order of treatment and the methods proposed for the solution of the various problems have been tested sufficiently to establish their usefulness; and yet the results are to be regarded as tentative, for they possess neither the completeness nor the consistency which, it is hoped, will characterize a later edition. The volume is incomplete in that it includes no discussion of the principles and methods of the art of numerical calculation a question funda- mental for an appreciation of the spirit of the treatment. Difficulties inherent in this defect may be avoided by a careful examination of an article on numerical calculation which appeared in Popular Astronomy, 1908, pp. 349-367, and in the Engineering Quarterly of the University of Missouri, v. 2, pp. 171-192. The final edition will contain this paper, in a revised form, as a preliminary chapter. The inconsistencies of the work are due largely to the fact that the earlier pages were in print before the later ones were written, and to the further fact that the manuscript was prepared with a haste that permitted no careful interadjustment and balancing of the parts. The main purpose of the volume is an exposition of the principal methods of determining latitude, azimuth, and time. Generally speaking, the limit of precision is that corresponding to the engineer's transit or the sextant. Though the discussion has thus been somewhat narrowly restricted, an attempt has been made to place before the student the means of acquiring correct and complete notions of the fundamental conceptions of the subject. But these can scarcely be attained without some knowledge of the salient facts of Descriptive Astronomy. For those who possess this information, the first chapter will serve as a review; for others, it will afford an orientation sufficient for the purpose in question. Chapter II blocks out in broad lines the solutions of the problems of latitude, azimuth, and time. The observational details of these solutions, with a few exceptions, are presented in Chapter IV, while Chapters V-VII consider in succession the special adaptations of the fundamental formulae employed for the reductions. In each instance the method used in deriving the final equations originates in the principles underlying the subject of numerical calculation. Chapter III is devoted to a theoretical considera- tion of the subject of time. It is not customary to introduce historicat data into texts designed for the use of professional students; but the author has found so much that is 198988 vi PREFACE helpful and stimulating in a consideration of the development of astronomical instruments, methods, and theories that he is disposed to offer an apology for the brevity of the historical sections rather than to attempt a justification of their introduction into a work mainly technical in character. To exclude historical material from scientific instruction is to disregard the most effective means of giving the student a full appreciation of the significance and bearing of scientific results. Brief though they are, it is hoped that these sections may incline the reader toward wider excursions into this most fascinating field. -The numerical solutions for most of the examples have been printed in detail in order better to illustrate both the application of the formulae involved and the operations to be performed by the computer. Care has been taken to secure accuracy in the text as well as in the examples, but a considerable number of errors have already been noted. For these the reader is referred to the list of errata on page 132. The use of the text should be supplemented by a study of the prominent constellations. For this purpose the "Constellation Charts" published by the editor of Popular Astronomy, Northfield, Minnesota, are as serviceable as any, and far less expensive than the average. My acknowledgments are due to Mr. E. S. Haynes and Mr. Harlow Shapley, of the Department of Astronomy of the University of Missouri, for much valuable assistance in preparing the manuscript, in checking the calcu- lations, and in reading the proofs. F. H. SEARES. LAWS OBSERVATORY, UNIVERSITY OF MISSOURI, June, 1909. CONTENTS CHAPTER I INTRODUCTION CELESTIAL SPHERECOORDINATES PAGE. 1 . The results of astronomical investigations '. 1 2. The apparent phenomena of the heavens 4 3. Relation of the apparent phenomena to their interpretation 5 4. Relation of the problems of practical astronomy to the phenomena of the heavens 7 5. Coordinates and coordinate systems , 8 6. Characteristics of the three systems. Changes in the coordinates 10 7. Summary. Method of treating the corrections in practice ._ 15 8. Refraction 16 9. Parallax 18 CHAPTER II FORMULAE OF SPHERICAL TRIGONOMETRY TRANSFORMATION OF COORDINATES GENERAL DISCUSSION OF PROBLEMS 1 0. The fundamental formulae of spherical trigonometry 21 11. Relative positions of the reference circles of the three coordinate systems .... 23 12. Transformation of azimuth and zenith distance into hour angle and declina- tion 25 13. Transformation of hour angle and declination into azimuth and zenith dis- tance 29 14. Transformation of hour angle into right ascension, and vice versa 29 1 5. Transformation of azimuth and altitude into right ascension and declination, and vice versa 31 16. Given the latitude of the place, and the declination and zenith distance of an , object, to find its hour angle, azimuth, and parallactic angle 31 17. Application of transformation formulae to the determination of latitude, azimuth, and time 32 CHAPTER III TIME AND TIME TRANSFORMATION 1 8. The basis of time measurement 36 1 9. Apparent, or true, solar time 36 20. Mean solar time > 36 21 . Sidereal time ' 37 22. The tropical year 38 23. The calendar. 38 24. Given the local time at any point, to find the corresponding local time at any other point 39 25. Given the apparent solar time at any place, to find the corresponding mean solar time, and vice versa 40 26. Relation between the values of a time interval expressed in mean solar and sidereal units 42 27. Relation between mean solar time and the corresponding sidereal time 44 28. The right ascension of the mean sun and its determination 44 29. Given the mean solar time at any instant to find the corresponding sidereal time 47 30. Given the sidereal time at any instant to find the corresponding mean solar time , 48 viii CONTENTS CHAPTER IV INSTRUMENTS AND THEIR USE PAGE. 3 1 . Instruments used by the engineer 50 TIMEPIECES 32. Historical 50 33. Error and rate 51 34. -Comparison of timepieces 52 35. The care of timepieces 58 THE ARTIFICIAL HORIZON 36. Description and use 59 THE VERNIER 37. 'Description and theory 59 38. Uncertainty of the result 60 THE ENGINEER'S TRANSIT 39. Historical 61 40. Influence of imperfections of construction and adjustment 62 4 1 . Summary of the preceding section 71 42. The level 71 43. Precepts for the use of the striding level 72 44. Determination of the value of one division of a level 73 45. The measurement of vertical angles 77 46. The measurement of horizontal angles 80 47. The method of repetitions 81 THE SEXTANT 48. Historical and descriptive 85 49. The principle of the sextant 86 50. Conditions fulfilled by the instrument 87 5 1 . Adjustments of the sextant 88 52. Determination of the index correction 89 53. Determination of eccentricity corrections 90 54. Precepts for the use of the sextant , 91 55. The measurement of altitudes 91 CHAPTER V THE DETERMINATION OF LATITUDE 56. Methods 95 1. MERIDIAN ZENITH DISTANCE 57. Theory 96 58. Procedure 96 2. DIFFERENCE OF MERIDIAN ZENITH DISTANCES TALCOTT'S METHOD 59. Theory 97 60. Procedure 98 3. ClRCUMMERIDIAN ALTITUDES 61. Theory 99 62. Procedure . . 101 CONTENTS ix 4. ZENITH DISTANCE AT ANY HOUB ANGLE 63. Theory PAGE. . . 102 . 103 64. Procedure 5. ALTITUDE OF POLABIS 104 65. Theory 66. Procedure 67. Influence of an error in time .............................................. J CHAPTER VI THE DETERMINATION OF AZIMUTH 68. Methods ................................................... 1. AZIMUTH OF THE SUN 1 AQ 69. Theory ......................................................... 70. Procedure .................................................................. * 2. AZIMUTH OF A CIRCUMPOLAB STAB AT ANY HOUB ANGLE 71. Theory .................................................................... 72. Procedure .................................................................. 3. AZIMUTH FROM AN OBSERVED ZENITH DISTANCE 113 73. Theory 74. Procedure 75. Azimuth of a mark 1 76. Influence of an error in the time H* CHAPTER VII THE DETERMINATION OF TIME 77. Methods 116 1. THE ZENITH DISTANCE METHOD 78. Theory J 79. Procedure 118 2. THE METHOD OF EQUAL ALTITUDES 80. Theory 1 8 1 . Procedure ^ 3. THE MERIDIAN METHOD 82. Theory 1 83 . Procedure 123 4. THE POLABIS VERTICAL CIRCLE METHOD SIMULTANEOUS DETERMINATION OF TIME AND AZIMUTH 84. Theory 126 85. Procedure 129 ERRATA 132 INDEX 133 PRACTICAL ASTRONOMY FOR ENGINEERS CHAPTER I INTRODUCTION CELESTIAL SPHERE COORDINATES. 1. The results of astronomical investigations. The investigations of the astronomer have shown that the universe consists of the sun, its attendant planets, satellites, and planetoids; of comets, meteors, the stars, and the nebulae. The sun, planets, satellites, and planetoids form the solar system, and with these we must perhaps include comets and meteors. The stars and nebulae, considered collectively, constitute the stellar system. The sun is the central and dominating body of the solar system. It is an intensely heated luminous mass, largely if not wholly gaseous in consti- tution. The planets and planetoids, which are relatively cool, revolve about the sun. The satellites revolve about the planets. The paths traced out in the motion of revolution are ellipses, nearly circular in form, which vary slowly in size, form, and position. One focus of each elliptical orbit coin- cides with the center of the body about which the revolution takes place. Thus, in the case of the planets and planetoids, one of the foci of each orbit coincides with the sun, while for the satellites, the coincidence is with the planet to which they belong. In all cases the form of the path is such as would be produced by attractive forces exerted mutually by all members of the solar system and varying in accordance with the Newtonian law of gravitation. In addition to the motion of revolution, the sun, planets, and some of the satellites at least, rotate on their axes with respect to the stars. The planets are eight in number. In order from the sun they are : Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and Neptune. Their distances from the sun range from thirty-six million to nearly three thousand million miles. Their diameters vary from about three thousand to nearly ninety thousand miles. Nevertheless, comparatively speaking, they are small, for their collective mass is but little more than one one-thousandth that of the sun. The planetoids, also known as small planets or asteroids, number six hundred or more, and relatively to the planets, are extremely small bodies so small that they are all telescopic objects and many of them can be seen only with large and powerful instruments. Most of them are of compara- tively recent discovery, and a considerable addition to the number already known is made each year as the result of new discoveries. With but few exceptions their paths lie between the orbits of Mars and Jupiter. The only satellite requiring our attention is the moon. This revolves about the earth with a period of about one month, and rotates on its axis once during each revolution. Although one of the smaller bodies of the solar system it is, on account of its nearness, one of the most striking. l PRACTICAL ASTRONOMY The solar and stellar systems are by no means coordinate parts of the universe. On the contrary, the former, vast as it is, is but an insignificant portion of the latter, for the sun is but a star, not very different on the average from the other stars whose total number is to be counted by hun- dreds of millions; and the space containing the entire solar system, includ- ing sun, planets, satellites, and planetoids, is incredibly small as compared with that occupied by the stellar system. To obtain a more definite notion of the relative size of the two .systems consider the following- illustration : Let the various bodies be represented by small spheres whose diameters and mutual distances exhibit the relative dimensions and distribution through space of the sun, planets, and stars. We shall thus have a rough model of the universe, and to make its dimensions more readily comprehensible let the scale be fixed by assuming that the sphere representing the sun is two feet in diameter. The corresponding diameters of the remaining spheres and their distances from the central body are shown by the following table. OBJECT DIAMETER DISTANCE Sun 2 feet Mercury 0.08 inch 83 feet Venus 0.21 inch 155 feet Earth 0.22 inch 215 feet Mars o. 12 inch 327 feet Jupiter 2.42 inch 1116 feet Saturn 2. 02 inch 2048 feet Uranus 0.97 inch 4118 feet Neptune 0.91 inch 6450 feet Nearest Star Unknown nooo miles Tt will be seen that the distance of the outermost planet from the sun is represented in the model by about a mile and a quarter. On the same scale, the distance of the nearest star, the only one included in the table, is approximately equal to one-half the circumference of the earth. When it is remembered that this object is but one of perhaps two hundred million stars, the vast majority of which are probably at least one hundred times more distant, and further that each of these stars is a sun as our owin sun, the very subordinate position of the solar system becomes strikingly ap- parent. The fact that the sun is similar in size and chemical composition to millions of other stars at once raises the question as to whether they too are not provided with attendant systems of planets and satellites. A de- finite answer is wanting, although analogy suggests that such may well be the case. Bodies no larger than the planets and shining only by reflected light would be quite invisible, even in the most powerful telescopes, when situated at distances comparable with those separating us from the stars. , INTRODUCTION 3 We do know, however, that in many instances two or more stars situated relatively near each other revolve about their common center of gravity thus forming binary or multiple systems. The discovery and study of these systems constitutes one of the most interesting- and important lines of modern astronomical investigation. The distances separating the various members of the solar system are such that the motions of the planets and planetoids with respect to the sun, and of the satellites relative to their primaries, produce rapid changes in their positions as seen from the earth. The stars are also in motion and the velocities involved are very large, amounting occasionally to a hundred miles or more per second of time, but to the observer on the earth, their relative positions remain sensibly unchanged. The distances of these ob- jects are so great that it is only when the utmost refinement of observation is employed and the measures are continued for months and years, that any shift in position can be detected even for those which move most rapidly. With minor exceptions, the configuration of the constellations is the same as it was two thousand years ago when the observations upon which are based the earliest known record of star positions were made. To the casual observer there is not a great deal of difference in the ap- pearance of the stars and the planets. The greater size and luminosity of the former is offset by their greater distance. In ancient times the funda- mental difference between them was not known, and they were distinguished only by the fact that the planets change their positions, while relatively to each other the stars are apparently fixed. In fact the word planet means literally, a moving or wandering star, while what appeared to the early ob- servers as the distinguishing characteristic of the stars is shown by the fre- quent use of the expression fixed stars. The nebulae are to be counted by the hundreds of thousand's. They con- sist of widely extended masses of luminous gas, apparently of simple chemical composition. They are irregularly distributed throughout the heavens, and present the greatest imaginable diversity of form, structure, and brightness. Minute disc like objects, rings, double branched spirals, and voluminous masses of extraordinarily complex structure, some of which resemble closely the delicate high-lying clouds of our own atmosphere, are to be found among them. The brightest are barely visible to the unaided eye, while the faintest tax the powers of the largest modern telescopes. Their distances are of the same order of magnitude as those of the stars, and, indeed, there appears to be an intimate relation connecting these two classes of objects, for there is evidence indicating that the stars have been formed from 1 the nebulae through some evolutionary process the details of which are as yet not fully understood. The preceding paragraphs give the barest outline of the interpretation which astronomers have been led to place upon the phenomena of the heavens. The development of this conception of the structure of the universe forms the major part of the history of astronomy during the last four cen- 4 PRACTICAL ASTRONOMY turies. Many have contributed toward the elaboration of its details, but its more significant features are due to Copernicus, Kepler, and Newton. Although the scheme outlined above is the only theory thus far formu- lated which satisfactorily accounts^for the celestial phenomena in their more intricate relations, there is another conception of the universe, one far earlier in its historical origin, which also accounts for the more striking phenomena. This theory bears the name of the Alexandrian astronomer Ptolemy, and, as its central idea is immediately suggested b}' the most casual examination of the motions of the celestial bodies, we shall now turn to a consideration of these motions and the simple, elementary devices which can be used for their description. 2. The apparent phenomena of the heavens. The observer who goes forth under the star-lit sky finds himself enclosed by a hemispherical vault of blue which meets in the distant horizon the seemingly flat earth upon which he stands. The surface of the vault is strewn with points of light of different brightness, whose number depends upon the transparency of the atmosphere and the brightness of the moon, but is never more than two or three thousand. A fewi hours observation shows that the positions of the points are slowly shifting in a peculiar and definite manner. Those in the east are rising from the horizon while those in the west are setting. Those in the northern heavens describe arcs of circles in a counter-clockwise di- rection about a common central point some distance above the horizon. Their distances from each other remain unchanged. The system moves as a whole. The phenomenon can be described by assuming that each individual point is fixed to a spherical surface which rotates uniformly from' east to west about an axis passing through the eye of the observer and the central point mentioned above. The surface to which the light-points seem at- tached is called the Celestial Sphere. Its radius is indefinitely great. Its period of rotation is one day, and the resulting motion of the celestial bodies is called the Diurnal Motion or Diurnal Rotation. The daylight appearance of the heavens i's not unlike that of the night except that the sun, moon, and occasionally Venus, are the only bodies to be seen in the celestial vault. They too seem to be carried along with the celestial sphere in its rotation, rising in the east, descending toward the west, and disappearing beneath the horizon only to rise again in the east ; but if careful observations be made it wall be seen that these bodies can not be thought of as attached to the surface of the sphere, a fact most easily verified in the case of the moon. Observations upon successive nights show that the position of this object changes with respect to the stars. A con- tinuation of the observations will show that it apparently moves eastward over the surface of the sphere along a great circle at such a rate that an entire circuit is completed in about one month. A similar phenomenon in the case of the sun manifests itself by the fact that the time at which any o-iven star rises does not remain the same, but occurs some four minutes INTRODUCTION 5 earlier for each successive night. A star rising two hours after sunset on a given night will rise approximately l h 56 m after sunset on the following night. The average intervals for succeeding nights will be l h 52 m , l h 48 m , l h 44 m , etc., respectively. That the stars rise earlier on successive nights shows that the motion of the sun over the sphere is toward the east. Its path is a great circle called the Ecliptic. Its motion in one day is approximately one degree, which corresponds to the daily change of four minutes in the time of rising of the stars. This amount varies somewhat, being greatest in January and least in July, but its average is such that a circuit of the sphere is completed in one year. This motion is called the Annual Motion of the Sun. With careful attention it will be found that a few of the star-like points of light, half a dozen more or less, are exceptions to the general rule which rigidly fixes these objects to the surface of the celestial sphere. These are the planets, the wandering stars of the ancients. Their motions with respect to the stars are complex. They have a general progressive motion toward the east, but their paths are looped so that there are frequent changes in di- rection and temporary reversals of motion. Two of them, Mercury and Venus, never depart far from the sun, oscillating from one side to the other in paths which deviate but little from the ecliptic. The paths of the others also lie near the ecliptic, but the planets themselves are not confined to the neighborhood of the sun. The sun, moon, and the planets therefore appear to move over the surface of the celestial sphere with respect to the stars, in paths which lie in or near the ecliptic. The direction of motion is opposite, in general, to that of the diurnal rotation. The various motions proceed quite independently. While the sun, moon, and planets move over the surface of the sphere, the sphere itself rotates on its axis with a uniform angular velocity. These elementary facts are the basis upon which the theory of Ptolemy was developed. It assumes the earth, fixed in position, to be the central body of the universe. It supposes the sun, moon, and planets to revolve about the earth in paths which are either circular or the result of a com- bination of uniform circular motions;; and regards the stars as attached to the surface of a sphere, which, concentric with the earth and enclosing the remaining members of the system, rotates from east to west, completing a revolution in one day. 3. Relation of the apparent phenomena to their interpretation. The re- lation of the apparent phenomena to the conception of Ptolemy is obvious, and their connection with the scheme outlined in Section 1 is not difficult to trace. The celestial sphere rs purely an optical phenomenon and has no real existence. The celestial bodies though differing greatly in distance are all so far from the observer that the eye fails to distinguish any difference in their distances. The blue background upon which they seem projected is due partly to reflection, and party to selective absorption of the light rays by the atmosphere surrounding the earth. As already explained, the stars 6 PRACTICAL ASTRONOMY are so distant that, barring a few exceptional cases, their individual motions produce no sensible variation in their relative positions, and, even for the exceptions, the changes are almost vanishingly small. On the other hand, the sun, planets, and satellites are relatively near, and their motions produce marked changes in their mutual distances and in their positions with respect to the stars. The annual motion of the sun in the ecliptic is but a reflection of the motion of the earth in its elliptical orbit about the sun. The monthly motion of the moon is a consequence of its, revolution about the earth, and the complex motions of the planets are due, partly to their own revolutions about the sun, and partly to the rapidly shifting position of the observer. Finally, the diurnal rotation of the celestial sphere, which at first glance seems to carry with it all the celestial bodies,, is but the result Of the axial rotation of the earth. In so far as the more obvious phenomena of the heavens are concerned there is no contradiction involved in either of the conceptions which have been devised for the description of their relations. That such is the case arises from the fact that we are dealing with a question concerning changes of relative distance and direction. Given two points, A and B, we can de- scribe the fact that their distance apart, and the direction of the line joining them, are changing, in either of two ways. We may think of A as fixed and B moving, or we may think of B as fixed and A in motion. Both methods are correct, and each is capable of giving an accurate description of the change in relative distance and direction. So, in the case of the celestial bodies, we may describe the variation in their distances and directions, either by assuming the earth to be fixed with the remaining bodies in motion, or by choosing another body, the sun, as the fixed member of the system and describing the phenomena in terms of motions referred to it. The former method of procedure is the starting point for the system of Ptolemy, the latter, for that of Copernicus. Both methods are correct, and hence neither can give rise to contradiction so long as the problem remains one of motion. Though two ways lie open before us, both leading to the same goal, the choice of route is by no means a matter of indifference, for one is much more direct than the other. For the discussion of many questions the conception of a fixed earth and rotating heavens affords a simpler method of treatment ; but, when a detailed description of the motions of the planets and satellites is required, the Copernican system is the more useful by far, although the geocentric theory presents no formal contradiction unless we pass beyond the consideration of the phenomena as a case of relative motion, and attempt their explanation as the result of the action of forces and accelerations. If this be done, the conception which makes the earth the central body of the universe comes into open conflict with the fundamental principles of me- chanics. With the heliocentric theory there is no such conflict, and herein lies the essence of the various so-called proofs of the correctness of the Copernican system. The problems of practical astronomy are among those which can 1 more simply treated on the basis of the geocent'ric theory, and we might INTRODUCTION 7 have proceeded to an immediate consideration of our subject from this primitive stand-point but for the importance of emphasizing the character of what we are about to do. For the sake of simplicity, we shall make use of ideas which are not universally applicable throughout the science of as- tronomy. We shall speak of a fixed earth and rotating heavens because it is convenient, and for our present purpose, precise ; but, in so doing, it is im- portant always to bear in mind the more elaborate scheme outlined above, and be ever ready to shift our view-point from the relatively simple, elemen- tary conceptions which form a part of our daily experience, to the more ma- jestic structure whose proportions and dimensions must ever be the delight and wonder of the human mind. 4. Relation of the problems of practical astronomy to the phenomena of the heavens. The problems of practical astronomy with which we are concerned are the determination of latitude, azimuth, time, and longitude. (a) The latitude of a point on the earth may be defined roughly as its angular distance from the equator. It can be shown that this is equal to the complement of the inclination of the rotation axis of the celestial sphere to the direction of the plumb line at the point considered. If the inclination of the axis to the plumb line can be determined, the latitude at once becomes known. (b) The azimuth of a point is the angle included between the vertical plane containing the rotation axis of the celestial sphere and the vertical plane through the object. If the orientation of the vertical plane through the axis of the sphere can be found, the determination of the azimuth of the point becomes but a matter of instrumental manipulation. (c) Time measurement is based upon the diurnal rotation of the earth, which appears to us in reflection as the diurnal rotation of the celestial sphere. The rotation of the celestial sphere can therefore be made the basis of time measurement. To determine the time at any instant, we have only to find the angle through which the sphere has rotated since some specified initial epoch. (d) As will be seen later, the determination of the difference in longi- tude of two points is equivalent to finding the difference of their local times. The solution of the longitude problem therefore involves the application of the methods used for the derivation of time, together with some means; of comparing the local times of the two places. The latter can be accomplished by purely mechanical means, quite independently of any astronomical phenomena, although such phenomena are occasionally used for the pur- pose. In brief, therefore, the solution of these four fundamental problems can be connected directly with certain fundamental celestial phenomena. Both latitude and azimuth depend upon the position of the rotation axis of the celestial sphere, the former, upon its inclination to the direction of the plumb line, the latter, upon the orientation of the vertical plane passing through it ; while the determination of time and longitude involve the posi- tion of the sphere as affected by diurnal rotation. 8 PRACTICAL ASTRONOMY A word more, and we are immediately led to the detailed consideration of our subject: The solution of our problems requires a knowledge of the position of the axis of the celestial sphere and of the orientation of the sphere about that axis. W'e meet at the outset a difficulty in that the sphere and its axis have no objective existence. Since our observations and meas- urements must be upon things which have visible existence, the stars for example, we are forced to an indirect method of procedure. We must make our measurements upon the various celestial bodies and then, from the known location of these objects on the sphere, derive the position of the sphere and its axis. This raises at once the general question of coordinates and coordinate systems to which we now give our attention. 5. Coordinates and Coordinate Systems. Position is a relative term. We cannot specify the position of any object without referring it, either explicitly or implicitly, to some other object whose location is assumed to be known. The designation of the position of a point on the surface of a sphere is most conveniently accomplished by a reference to two great circles that intersect at right angles. For example, the position of a point on the earth is fixed by referring it to the equator and some meridian as that of Greenwich or Washington. The angular distance of the point from the circles of reference are its coordinates in this case, longitude and latitude. Our first step, therefore, in the establishment of coordinate systems for the celestial sphere, is the definition of the points and circles of reference which will form the foundation for the various systems. The Direction of the Plumb Line, or the Direction due to Gravity, produced indefinitely in both directions, pierces the celestial sphere above in the Zenith, and below, in the Nadir. The plane through the point of obser- vation, perpendicular to the direction of the plumb line, is called the Hor- izon Plane. Produced indefinitely in all directions, it cuts the celestial sphere in a great circle called the Horizon. Since the radius of the celestial sphere is indefinitely great as compared with the radius of the earth, a plane through the center of the earth perpendicular to the direction of gravity will also cut the celestial sphere in the horizon. For many pur- poses it is more convenient to consider this plane as the horizon plane. The celestial sphere is pierced by its axis of rotation in two points called the North Celestial Pole and the South Celestial Pole, or more briefly, the North Pole and the South Pole, respectively. It is evident from the relations between the phenomena and their interpretation traced in Section 3 that the axis of the celestial sphere must coincide with the earth's axis of rotation. Great circles through the zenith and nadir are called Vertical Circles. Their planes are perpendicular to the horizon plane. The vertical circle passing through the celestial poles is called the Celestial Meridian, or simply, the Meridian. Its plane coincides with the plane of the terrestrial meridian through the point of observation. DEFINITIONS The vertical circle intersecting- the meridian at an angle of ninety degrees is called the Prime Vertical. The intersections of the meridian and prime vertical with the horizon are the cardinal points, North, East, South, and West. Small circles parallel to the horizon are called Circles of Altitude or Almucanters. Great circles through 'the poles of the celestial sphere are called Hour Circles. The great circle equatorial to the poles of the celestial sphere is called the Celestial Equator. The plane of the celestial equator coincides with the plane of the terrestrial equator. Small circles parallel to the celestial equator are called Circles of Dec- lination. The ecliptic, already defined as the great circle of the celestial sphere fol- lowed by the sun in its annual motion among the stars, is inclined to the celestial equator at an angle of about 23^ degrees. The points of inter- section of the ecliptic and the celestial equator are the Equinoxes, Vernal and Autumnal, respectively. The Vernal Equinox is that point at which the sun in its annual motion passes from the south to the north side of the equator; the Autumnal Equinox, that at which it passes from the north to the south. The points on the ecliptic midway between the equinoxes are called the Solstices, Summer and Winter, respectively. The Summer Solstice lies to the north of the celestial equator, the Winter Solstice, to the south. The coordinate systems most frequently used in astronomy present certain features in common, and a clear understanding- of the underlying principles will greatly aid in acquiring a knowledge of the various systems. At the basis of each system is a Fundamental Great Circle. Great circles perpendicular to this are called Secondary Circles. One of these, called the Principal Secondary, and the fundamental great circle, form the reference circles of the system. The Primary Coordinate is measured along the fundamental great circle from the principal secondary to the secondary passing through the object to which the coordinates refer. The Secondary Coordinate is measured along the secondary passing through the object from the fundamental great circle to the object itself. The fundamental great circle and the principal secondary intersect in two points. The intersection from which the primary coordinate is measured, and the direction of measurement of both coordinates, must be specified. In practical astronomy three systems of coordinates -are required. The details are shown by the following table. The symbol used to designate each coordinate is written after its name in the table. It is sometimes more convenient to use as secondary coordinate the dis- tance of the object from one of the poles of the fundamental great circle. Thus in System I we shall frequently use the distance of an object from 10 PR A C TIC A L A S TR ONOM1 ' COORDINATE SYSTEMS. SYSTEM FUNDAMENTAL GREAT CIRCLE SECONDARY CIRCLES PRINCIPAL SECONDARY COORDINATES PRIMARY SECONDARY I Horizon Vertical 'Meridian Azimuth = A Altitude = h Circles -j- from South toward West -\- from Horizon upward II Celestial Equator Hour Circles Hour Circle coinciding with Meridian Hour Angle t -f- from Meridian toward West Declination = $ -f- from Equator toward North; toward South III u Hour Circle through the Vernal Equinox Right Ascension = a + from Vernal Equinox toward East 11 it the zenith, its Zenith Distance = s, instead of the altitude. Similarly, in Systems II and III we shall occasionally find that an object's distance from the north celestial pole, its North Polar Distance -, is more convenient than declination. Between these alternative coordinates we have the relations : (0 r: = 90 o The details of the various systems are also shown graphically in Fig. 1, which represents an orthogonal projection of the celestial sphere upon the horizon plane. In this projection all vertical circles become straight lines. All circles inclined to the horizon at an angle other than 90 become ellipses. The horizon, and all circles parallel to the horizon plane, remain circles. 6. Characteristics of the Three Systems. Changes in the Coordinates. Coordinates are used both for the location of objects on the sphere by actual observation, and as a means of stating positions predicted on the basis of the laws which describe the motions of the various celestial bodies. The practical astronomer and the engineer have occasion to use them in both ways. It is essential that there be a clear understanding of the relative advantages of the various systems, of the changes which may occur in the different coordinates, and of the relations of the systems to each other. We now proceed to a discussion of the first two of these points. The rela- tions between the systems will be discussed in Chapter II. None of the coordinates defined above is absolutely constant for any of the celestial bodies. The changes which occur arise as the result of: (a) a change in the position of the object, (b) a change in the position of the reference circles, (c) a change in the position of the observer, (d) a bending of the light rays by the atmosphere surrounding the earth. COORDINATES 11 Fig. i. Point Z Circle NESW Point P Line NZS Line WZE Points .V, E, S, W, Ellipse WME Ellipse Point Point Line Arc Arc SI- TV m e V O Z Or POp Angle SZO Line Line Arc Mp Angle Arc Arc Arc rO ZO ZPO pO PO vp Zenith Horizon North Celestial Pole Celestial Meridian Prime Vertical Cardinal Points Celestial Equator Ecliptic Vernal Equinox Any Celestial Object Vertical Circle through O Hour Circle through O Azimuth ot O = A Altitude of 6 = h Zenith Distance of O = z Hour Angle of O = t Declination of O 8 North Polar Distance of O = Right Ascension of O = ,7 and - same as in System II I Coordinates f System I I Coordinates \ System II ~| Coordinates f System III Any or all of these causes may enter to affect the position of an object, with the result that the number of possible variations with which we have to deal is considerable. In some instances, however, the variations are small quantities so small that they can be disregarded in all but the most precise investigations. The small changes which cannot be neglected entirely are regarded as corrections, which, applied to the coordinates corresponding to a given position of the object, reference circles, and observer, give their values for some other position. 12 PRACTICAL ASTRONOMY The bending of the light rays by the earth's atmosphere, a phenomenon known as Refraction, affects all of the coordinates but azimuth. 1 The amount of the refraction, which is always small, depends upon the conditions under which the object is observed. The allowance for its influence is there- fore made by each individual observer. The method of determining its amount will be discussed in Section 8. In the first system, the reference circles are fixed for any given point of observation. The azimuth and altitude of terrestrial objects are therefore constant, unless the point of observation is shifted. For celestial bodies, on the contrary, they are continuously varying. The positions of all such objects are rapidly and constantly changing with respect to the circles of reference, as a result of the diurnal rotation. For the nearer bodies, an additional complexity is introduced by their motions over the sphere and the changing position of the earth in its orbit. It appears, therefore, that azimuth and altitude are of special service in surveying and in geodetic operations, but that their range of advantageous application in connection with celestial bodies is limited, for not only are the azimuth and altitude of a celestial object constantly changing, but, for any given instant, their values are different for all points on the earth. But in spite of this disadvantage, altitude, at least, is of great importance. Its determination in the case of a celestial body affords convenient methods of solving two of the fundamental problems with which we are concerned, viz., latitude and time. Since the fundamental circle in the first system depends only upon the direction of the plumb line, the instrument required for the measurment of altitude is extremely simple, both in construction and use. In consequence, altitude is the most readily determined of all the various coordinates. The observational part of the determination of latitude and time is therefore frequently based upon measures of altitude, the final results being derived from the observed data by a process of. coordinate transformation to be developed in Chapter II. In the third system, the reference circles share in the diurnal rotation. Although not absolutely fixed on the sphere, their motions are so slow that the coordinates of objects, which, like the stars, are sensibly fixed, remain practically constant for considerable intervals of time. Right ascension and declination are therefore convenient for listing or cataloguing the positions of the stars. Catalogues of this sort are not only serviceable for long periods of time, but can also be used at all points on the earth. The latter circum- stance renders right ascension and declination an advantageous means of expressing the positions of bodies not fixed on the sphere. For such objects we have only to replace the single pair of coordinates which suffices for a star, by a series giving the right ascension and declination for equi-distant intervals of time. Such a list of positions is called an Ephemeris. If the time intervals separating the successive epochs for which the coordinates are given be properly chosen, the position can be found for any intermediate J The azimuth of objects near the horizon is also affected by refraction. The magnitude of the change in the coordinate is very small, however. COORDINATES 13 instant by a process of interpolation. The interval selected for the tabula- tion is determined by the rapidity and regularity with which the coordinates change. In the case of the sun, one day intervals are sufficient, but for the moon the positions must be given for each hour. For the more distant planets, whose motions are relatively slow, the interval can be increased to several days. Collections of ephemerides of the sun, moon, and the planets, together with the right ascensions and declinations of the brighter stars, are pub- lished annually by the governments of the more important nations. That issued by our own is prepared in the Nautical Almanac Office at Washington, and bears the title "American Ephemeris and Nautical Almanac." It is necessary to examine the character of the variations produced in the coordinates by the slow motion of the reference circles mentioned above. The mutual attractions of the sun, moon, and the planets produce small changes in the positions of the equator and ecliptic. The motion of the ecliptic is relatively unimportant. That of the equator is best understood by tracing the changes in position of the earth's axis of rotation. As the earth moves in its orbit, the axis does not remain absolutely parallel to a given initial position, but describes a conical surface. The change in the direction of the axis takes place very slowly, about 26000 years being required for it to return to its original position. During this interval the inclination of the equator to the ecliptic never deviates greatly from its mean value of about 23^. Consequently, the celestial pole appears to move over the sphere in a path closely approximating a circle with the pole of the ecliptic as center. The direction of the motion is counter-clockwise, and the radius of the circle equal to the inclination of the equator to the ecliptic. The actual motion of the pole is very complex; but its characteristic features are the progressive circular component already mentioned, and a transverse component which causes it to oscillate or nod back and forth with respect to the pole of the ecliptic. The result is a vibratory motion of the equator about a mean position called the Mean Equator, the mean equator itself slowly revolving about a line perpendicular to the plane of the ecliptic. The motion of the equator combined with that of the ecliptic produces an oscillation of the equinox about a mean position called the Mean Vernal Equinox, which, in turn, has a slow progressive motion toward the west. The resulting changes in the right ascension and declination are divided into two classes, called precession and nutation, respectively. Precession is that part of the change in the coordinates arising from the progressive westward motion of the mean vernal equinox, while Nutation is the result of the oscillatory or periodic motion of the true vernal equinox with respect to the mean equinox. The amount of the precession and nutation depends upon the position of the star. For an object on the equator the maximum value of the preces- sion in right ascension for one year is about forty-five seconds of arc or three seconds of time. For stars near the pole it is much larger, amounting in 14 PRACTICAL ASTRONOMY the case of Polaris, for example, to about 2'7 S . The annual precession in declination is relatively small, and does not exceed 20" for any of the stars. There remains to be considered the effect of the object's own motion and that of the observer. We ha\re already seen how the changes arising from the motion of such objects as the sun, moon, and the planets can be expressed by means of an ephemeris giving the right ascension and declina- tion for equi-distant intervals of time. For the stars the matter is much simpler. Their motions over the sphere are so slight as to be entirely inap- preciable in the vast majority of cases, and for those in which the change cannot be disregarded, it is possible to assume that the motion is uniform and along the arc of a great circle. The change in one year is called the star's Proper Motion. If the right ascension and declination are given for any instant, /, and it is desired to find their values as affected by proper motion for any other instant /', it is only necessary to add to the given coordinates the products of the proper motion in right ascension and declina- tion into the interval/ f expressed in years. The position of a star for a given initial epoch and its proper motion are therefore all that is required for the determination of its position at any other epoch, in so far as the position is dependent upon the star's own motion. The motion of the observer may affect the position of a celestial object in two ways : First, the actual change in his position due to the diurnal and annual motions of the earth causes a change in the coordinates called Parallactic Displacement. Second, the fact that the observer is in motion at the instant of observation may produce an apparent change in the direction in which the object is seen, in the same way that the direction of the wind, as noted from a moving boat or train, appears different from that when the observer is at rest. The change thus produced is called Aberration, and is carefully to be distinguished from the parallactic displacement. Aberration depends only upon the observer's velocity, and not at all upon his position, except as position may determine the direction and magnitude of the motion. Parallactic displacement, on the contrary, depends on the distance over which the observer actually moves. For the nearer bodies the parallactic displacement due to the earth's annual motion is large, and is included with the effect of the object's own motion in the ephemeris which expresses its positions. The variation arising from the rotation of the earth on its axis is far smaller, and can always be treated as a correction. In the case of the stars, the distances are so great that the maximum known parallactic displacement due to the earth's annual motion amounts to only three-quarters of a second of arc. For all but a few, a shift in the position of the earth from one side of its orbit to the other, a distance of more than 180,04)0,000 miles, reveals no measurable change in the coordinates. The displacement due to the earth's rotation is of course altogether inappreciable. Parallactic displacement is usually called Parallax, and, when so spoken of, signifies specifically, the correction which must be applied to the observed 15 coordinates of an object in order to reduce them to what they would be were the object seen from a standard position. For the stars, the standard position is the center of the sun ; for all other bodies, the center of the earth. Aberration is due to the fact that the velocity of the observer is a quantity of appreciable magnitude as compared with the velocity of light. For all stars not lying in the direction of the earth's orbital motion, the telescope must be inclined slightly in advance of the star's real position in order that its rays may pass centrally through both objective and eye-piece of the instrument. The star thus appears displaced in the direction of the ob- server's motion. The amount of the displacement is a maximum when the direction of the motion is at right angles to the direction of the star, and equal to zero when the two directions coincide. The rotation of the earth on its axis produces a similar displacement. The Diurnal Aberration is so minute, however, that it requires consideration only in the most refined observations. The coordinates of the second system possess, to a certain degree, the properties of those of both Systems I and, III. Hour angle, like azimuth and altitude, is a coordinate which varies continuously and rapidly, and is dependent on the position of the observer on the earth. The secondary coordinate, declination, is the same as in System III, and the remarks con- cerning it made above, apply with equal force here. The second system is of prime importance in the solution of the problems of practical astronomy, for it serves as an intermediate step in passing from System I to System III, or vice versa. It is also the basis for the construction of the equatorial mounting for telescopes, the form most commonly used in astronomical inves- tigations. 7. Summary. Method of treating the corrections in practice. It is to be remembered, therefore, that the azimuth and altitude of terrestrial objects are constant for a given point of observation, but change as the observer moves over the surface of the earth. For celestial objects they are not only different for each successive instant, but also, for the same instant, they are different for different points of observation. Right ascension and declination are sensibly the same for all points on the earth, and, in con- sequence, are used in the construction of catalogues and ephemerides. One pair of values serves to fix the position of a star for a long period of time, but for the sun, the moon, and the planets an ephemeris is required. The corrections to which the coordinates are subject are proper motion, precession, nutation, annual aberration, diurnal aberration, parallax, stellar or planetary as the case may be, and refraction. Right ascension and decli- nation are affected by all, but only planetary parallax, refraction, and diurnal aberration arise in practice in connection with azimuth and altitude, and of these three the last is usually negligible. In all cases these three are dependent upon local conditions, and consequently, their calculation and application are left to the observer. Since it is impracticable to include them in catalogue and ephemeris positions of right ascension and declination, there remains to be considered, as affecting- such positions, proper motion. 16 PRACTICAL ASTRONOMT precession, nutation, annual aberration, and stellar parallax. The last is so rarely of significance in practical astronomy that it can be disregarded. As for the others, it is sometimes necessary to know; their collective effect, and sometimes, the influence of the individual variations. It thus happens that we have different kinds of positions or places, known as mean place, true place, and apparent place. The mean place of an object at any instant is its position referred to the mean equator and mean equinox of that instant. The mean place is affected by proper motion and precession. The true place of an object at any instant is its position referred to the true equator and true equinox of that instant, that is, to the instantaneous positions of the actual equator and equinox. The true place is equal to the mean place plus the variation due to the nutation. The apparent place of an object at any instant is equal to the true place at that instant plus the effect of annual aberration. It expresses the location of the object as it would appear to an observer situated at the center of the earth. The positions to be found in star catalogues are mean places, and are referred to the mean equator and equinox for the beginning of some year, for example, 1855.0 or 1900.0. Such catalogues usually contain the data neces- sary for the determination of the precession corrections which must be applied to the coordinates in deriving the mean place for any other epoch. Modern catalogues also contain the value of the proper motion when appreciable. The nutation and annual aberration corrections are found from data given by the various annual ephemerides. The ephemerides themselves contain mean places for several hundred of the brighter stars ; but the engineer is rarely concerned with these, or with the catalogue positions mentioned above, for apparent places are also given for the ephemeris stars, and these are all that he needs. The apparent right ascension and declination are given for each star for every ten days throughout the year. Apparent posi- tions are also given by the ephemeris for the sun, the moon, and the planets, for suitably chosen intervals. Positions for all of these bodies for dates intermediate to the ephemeris epochs can be found by interpolation. With this arrangement, the special calculation of the various corrections necessary for the formation of apparent places is avoided entirely in the discussion of all ordinary observations. The observer must understand the origin and significance of all of the changes which occur in the coordinates, in order to use the ephemeris intelligently ; but he has occasion to calculate specially only those which depend upon the local conditions affecting the observations, viz., diurnal aberration, parallax, and refraction. The first we disregard on account of its minuteness. There remains for the consideration of the engi- neer only refraction and parallax. The following is a brief statement of the methods by which their numerical values can be derived. 8. Refraction. The velocity of light depends upon the density of the medium which it traverses. When a luminous disturbance passes from a medium of one density into that of another, the resulting change in velocity REFRACTION 17 shifts the direction of the wave front, unless the direction of propagation is perpendicular to the surface separating the two media. Stated otherwise, a light ray passing from one medium into another of different density under- goes a change in direction, unless the direction of incidence is normal to the bounding surface. This change in direction is called Refraction. The incident ray, the refracted ray, and the normal to the bounding surface at the point of incidence lie in a plane. When the density of the second medium is greater than that of the first, the ray is bent toward the normal. When the conditions of density are reversed, the direction of bending is away from the normal. The light rays from a celestial object which reach the eye of the observer must penetrate the atmosphere surrounding the earth. They pass from a region of zero density into one whose density gradually increases from the smallest conceivable amount to a maximum which occurs at the surface of the earth. The rays undergo a change in direction as indicated above. The effect is to increase the altitude of all celestial bodies, without sensibly changing their azimuth unless they are very near the horizon. For the case of two media of homogeneous density, the phenomenon of refraction is simple; but here, it is extremely complex and its amount difficult of determination. The course of the ray which reaches the observer is affected not only by its initial direction, but also by the refraction which it suffers at each successive point in its path through the atmosphere. The latter is determined by the density of the different strata, which, in turn, is a function of the altitude. This brings us to the most serious difficulty in the problem, for our knowledge of the constitution of the atmosphere, especially in its upper regions, is imperfect. To proceed, an assumption must be made con- cerning the nature of the relation connecting density and altitude. This, combined with the fundamental principles enunciated above, forms the basis of an elaborate mathematical discussion which results in an expression giving the refraction as a function of the zenith distance of the object, and the temperature of the air and the barometric pressure at the point of observa- tion. This expression is complicated and cumbersome, disadvantages over- come, in a measure, by the reduction of its various parts to tabular form in accordance with a method devised by Bessel. With this arrangement, the determination of the refraction involves the interpolation and combination of a half dozen logarithms, more or less. Various hypotheses concerning the relation between density and altitude have been made, each of which gives rise to a distinct theory of refraction, although the differences between the corresponding numerical results are slight. That generally used is due to Gylden. The tables based upon this theory are known as the Pulkova Refraction Tables, and can be found in the more comprehensive works on spherical and practical astronomy. When the highest precision is desired these tables or. their equivalent must be used, but for many purposes a simpler procedure will suffice. For example, the approximate expression, PRACTICAL ASTRONOMY 983 ^ , r= * J . tans , 460 + / (3) derived empirically from the results given by the theoretical development, 1 can be used for the calculation of" the refraction, r, .when the altitude is not less than 15. In this expression, b is the barometer reading in inches; f, the temperature in degrees Fahrenheit ; z, the observed or apparent zenith distance. The refraction is given in seconds of arc. The error of the result will rarely exceed one second. For rough work the matter can be still further simplified by using mean values for b and /. For r=2<9.5 inches, and t =r 50 Fahr. the coefficient of (3) is 57", whence = 57" tan z' . (4) The values of r given by (4) can be derived from columns three and eight of Table I with either the apparent altitude or the apparent zenith distance as argument. For altitudes greater than 20 and normal atmospheric conditions, the error will seldom exceed a tenth of a minute of arc. 9. Parallax. The parallax of an object is equal to the angle at the object subtended by the line joining the center of the earth and the point Fig. 2 of observation. Thus, in Fig. 2, the circle represents a section of the earth coinciding with the vertical plane through the object. C is the center of the earth, the point of observation, Z the zenith, and B the object. The angles z' and z are the apparent and geocentric zenith distances, respectively. Their difference, which is equal to the angle p, is the parallax of B. form was derived by Comstock, Bulletin of the University of Wisconsin, Science Series, v. i, p. 60. PARALLAX 19 We have the relations z = z'p, (5) (6) where h' and h are the apparent and geocentric altitudes, respectively. The effect of parallax, therefore, is to increase zenith distances and decrease altitudes, just the opposite of that produced by refraction. The parallax depends upon p the radius of the earth, r the distance of the object from the earth's center, and the zenith distance z' or s. From the triangle OCB /sin/ =,0sin2'. The angle / does not exceed a few seconds of arc for any celestial body excepting the moon. For this its maximum value is about 1. We therefore write p=smz'. (7) The coefficient p / r, the value of the parallax when the body is the horizon, is called the Horizontal Parallax. Denoting its value by/ we have /=/ sin^'. (8) The value of p Q varies with the distance of the object. It is tabulated in the American Ephemeris for the sun (p. 285), the moon (page IV of each month), and the planets (pp. 218<-249). For the sun, however, the change in / is so slight that we may use its mean value of 8"8, whence p = 8?8 sin z. (9) The error of this expression never exceeds Q"3. The values of p corre- sponding to (9) can be interpolated from columns four and nine of Table I. For approximate work the solar parallax is conveniently combined with the mean refraction given by (4). The difference of the two corrections can be derived from the fifth and tenth columns of Table I with the apparent altitude or the apparent zenith distance as argument. The preceding discussion assumes that the earth is a sphere. On this basis the parallax in azimuth is zero. Actually, the earth is spheroidal in form, whence it results that the radius, />, and consequently the angle OBC, do not, in general, coincide with the vertical plane through B\, for the plumb line does not point toward the center of the earth, except at the poles and at points on the equator. The actual parallax in zenith distance is therefore slightly different from that given by (9), and in addition, there 20 PRACTICAL ASTRONOMY is a minute component affecting- the azimuth. The influence of the spheroidal form of the earth is so slight, however, that it requires consideration only in the most precise investigations. Finally, it should be remarked that the apparent zenith distance used for the calculation of the parallax is the observed zenith distance freed from refraction; that is, of the two corrections, refraction is to be applied first. The zenith distance thus corrected serves for the calculation of the parallax. For the first system of coordinates, therefore, and the limits of precision here considered, the influence of both refraction and parallax is confined to the coordinate altitude, or its alternative, zenith distance. Hour angle, right ascension, and declination are all affected by both refraction and parallax, but, as these coordinates do not appear as observed quantities in the problems with which we are concerned, the development of the expressions which give the corresponding corrections is omitted. TABLE I. MEAN REFRACTION AND SOLAR PARALLAX Barometer, 29.5 in.; Thermometer, 50 Fahr. h' z' r / r-p h' z' r k / r p i5 75 3- '5 8'.'5 3- '4 40 50 i .'i 6'.'7 I'.O 20 70 2.6 8-3 2-5 50 40 0.8 5-7 0.7 25 65 2.O 8.0 1.9 60 30 0.6 4-4 o-5 30 60 1.6 7-6 i-5 70 20 0.4 3-o 0-3 35 55 i-3 7-2 1.2 80 IO O.2 i-5 0. I 40 50 1. 1 6.7 I .O 90 0.0 o.o o.o The Refraction, ;-, and the Refraction Solar Parallax, r-p, are to be subtracted from h', or added to z'. The Solar Parallax, /, is to be added to //', or subtracted from z'. CHAPTER II FORMULA OF SPHERICAL TRIGONOMETRY TRANSFORMATION OF COORDINATES GENERAL DISCUSSION OF PROBLEMS. 10. The fundamental formulae of spherical trigonometry. Transform- ations of coordinates are of fundamental importance for the solution of most of the problems of spherical and practical astronomy. The relations between the different systems should therefore receive careful attention. The more complicated transformations require the solution of a spherical triangle, and, because of this fact, a brief exposition of the fundamental formulae of spherical trigonometry is introduced at this point. Fig. 3- Let ABC, Fig. 3, be any spherical triangle. Denote its angles by A, B, and C; and its sides by a, b, and c. With the center of the sphere, 0, as origin, construct a set of rectangular coordinate axes, XYZ, such that the XY plane contains the side c, and the ^faxis passes through the vertex B. Let the rec- tangular coordinates of the vertex 7 be x, y, and z. Their values in terms of the parts of the triangle and the radius of the sphere are x = rcosa, y = r sin a cos B, z = r sin a syi B, (10) Construct a second set of axes, XYZ, with the origin at 0, the XY plane coinciding with the side c, and the X axis passing through the vertex A. Let the coordinates of (Preferred to this system be x',y, and z. We then have x = r cos b, y == r sin b cos A, J = r sin b sin A, (n) The second set of rectangular axes can be derived from the first by rotat- ing the first about the Zaxis through the angle c. The coordinates of the first 21 22 PRACTICAL ASTRONOMY system can therefore be expressed in terms of those of the second by means of the relations x = x'cos c y sin c, y = x sin c +y cos^r, (12) z = z'.' Substituting into equations (12) the values of x, y, z, *', /, and z from (10) and (n), and dropping the common factor r, we obtain the desired relations cos a. = cos b cos c + sin b sin c cos A, (13) sin a cos B = cos b sin c sin b cos c cos A, (14) sin as\n B = sin sin A. (15) These equations express relations between five of the six parts of the spherical triangle ABC, and are independent of the rectangular coordinate axes introduced for their derivation. Although the parts of the triangle in Fig. 3 are all less than 90, the method of development and the results are general, and apply to all spherical triangles. These relations are the funda- mental formulae of spherical trigonometry. From them all other spherical trigonometry formulas can be derived. They determine without ambiguity a side and an adjacent angle of a spherical triangle in terms of the two remain- ing sides and the angle included between them, provided the algebraic sign of the sine of the required side, or of the sine or cosine of the required angle, be known. Otherwise there will be two solutions. Equations (i3)-(i5) are conveniently arranged as they stand if addition- subtraction logarithms are to be employed for their calculation. For use with the ordinary logarithmic tables, they should be transformed so as to reduce the addition and subtraction terms in the right members of (13) and (14) to single terms (Num. Cat. pp. 13 and 14). Aside from the case covered by equations (i3)-(i 5), two others occur in connection with the problems of practical astronomy, viz., that in which the given parts are two sides of a spherical triangle, and an angle opposite one of them, to find the third side; and that in which the three sides are given, to find one or more of the angles. The first of these can be solved for those cases which arise in astronomical practice by a simple transformation of (13), the details of which will be considered in connection with the determination of latitude. A solution for the third case can also be found by a rearrange- ment of the terms of (13). Thus, cos a cos b cos c cos A = - . = r . (16) sin b sin c Similar expressions for the angles B and C can be derived by a simple permu- tation of the letters in (16). Equation (16) affords a theoretically accurate solution of the problem; but, practically, the application of expressions of this form is limited on account of the necessity of determining the angles from SPHERICAL TRIGONOMETRY 23 their cosines. For numerical calculation it is important to have formulae such that the angles A, B, and C can be interpolated from their tangents. (Num. CaL pp. 3 and 14). The desired relations can be derived by a transformation of (16), (Chauvenet, Spherical Trigonometry, 12 and 16-18), giving tan , A = -- sin s sin (s-a) in which s = YZ (a + b -\- c). Similar expressions for B and C can be derived by a permutation of the letters of (17). When the three angles of a spherical triangle are to be determined simultaneously, it is advantageous to introduce the auxiliary K, defined by the relation sin (s-a) sin (s-b) sin (s-c) , _, sin s Substituting (18) into (17), we find K ian*4 A = -. -. r. (19) sin (s-a) The expressions tan l / 2 B and tan y 2 C are similar in form. Collecting results, the complete formulae for the calculation of the three angles of a spherical triangle from the three sides are Form s-a, s-b, and s - c, and check by (s-a) + ( s - b) -f (s-c) = s. ,. sin (s-a) sin (s-b) sin (s-c) sn s ,. K K K tan -z A = -^-. - -, tan y z B -^, - , tan % C= -r ;. tan 72 u : , j\, LOU 72 *-- / \> sin (s-a) sin (s-b) sin (s-c) /^ Check: tan % A tan V 2 B tan V 2 C = sin 5 Two solutions are possible. The ambiguity is removed if the quadrant of one of the half-angles of the triangle is known. 11. Relative positions of the reference circles of the three coordinate systems. The transformation of the coordinates of one system into those of another requires a knowledge of the relative positions of the reference circles of the various systems. In the case of Systems I and II the principal secondary circles coincide by definition. The fundamental great circles are inclined to each other at an angle which is constant and equal to the complement of the latitude of the place of observation. The proof of this statement can be derived from Fig. 4, which represents a section through the earth and the celestial sphere in the 24 PRACTICAL ASTRONOMY plane of the meridian of the point of observation, 0. The outer circle repre- sents the celestial meridian, and the inner, the terrestrial meridian of 0, the latter being greatly exaggerated with respect to the former. Z and N are the zenith and the nadir; P and P', the poles of the celestial sphere;/ and/*', the poles of the earth; HH' and EE, the lines of intersection of the planes of horizon and equator, respectively, with the meridian plane,, The plane of the celestial equator coincides with that of the terrestrial equator, which cuts the terrestrial meridian in ee. But, whence Fig. 4- Now, by definition the arc eO measures the latitude, y>, of the point Arc EZ= Arc eO H'E = 90 ( (21) (22} which was to be proved. It thus appears that the second system can be derived from the first by rotating the first about an axis passing through the east and west points, through an angle equal to the co-latitude of the place. It is to be noted, further, that and Arc ZP 90 (p = Co-latitude of 0, Arc HP = <<>. (23) (24) From (21) and (24) it follows that the latitude of any point on the earth is equal to the declination of the zenith of that point. It is also equal to the altitude of the pole as seen from the given point. Systems II and III have the same fundamental great circle, viz., the celestial equator. The principal secondary of the third system does not main- RELATIVE POSITION OF COORDINATE SYSTEMS 25 tain a fixed position with respect to that of the first, but rotates uniformly in a clockwise direction as seen from the north side of the equator. Let Fig. 5 represent an orthogonal projection of the celestial sphere upon the plane of the equator as seen from the North. Pis the north celestial pole; M, the point where the meridian of intersects the celestial equator; and V, the vernal equinox. The arc MBV therefore measures the instantaneous position of the principal secondary of the third system with respect to that of the first. This arc is equal to the hour angle of the vernal equinox, or the right ascension of the observer's meridian. It is called the Sidereal Time = 6. We thus have the following important definition: The. sidereal time at any instant is equal to the hour angle of the vernal equinox at that instant. It is also equal to the right ascension of the observer's meridian at the instant considered. It follows, therefore, that the third system can be derived from the second by rotating the second system about the axis of the celestial sphere through an angle equal to the sidereal time. Finally, the third system can be derived from the first by rotating the first into the position of the second, and thence into the position of the third. Briefly stated, the transformation of coordinates involves the determina- tion of the changes arising in the coordinates as a result of a rotation of the various systems in the manner specified above. It is at once evident that the transformation of azimuth and altitude into hour angle and declination requires a knowledge of the latitude; of hour angle and declination into right ascension and declination, a knowledge of the sidereal time; while, to pass from azimuth and altitude to right ascension and declination, both latitude and sidereal time are required. It is scarcely necessary to add that the reverse transformations demand the same knowledge. 12. Transformation of azimuth and zenith distance into hour angle and declination. The transformation requires the solution of the spherical triangle ZPO, Fig. i, p. 11. The essential part of Fig. I is reproduced in Fig. 6 upon an enlarged scale. An inspection of the notation of p. II shows that the parts of the triangle ZPO can be designated as shown in Fig. 6. 26 Assuming the latitude, , to be known, it is seen that the transformation in question involves the determination of the side TT = 90 d and the adja- cent angle t in terms of the other two sides, 96 tp and z = 90 - h, and the angle 180 A included between them. Equations (i3)-(i5) are directly applicable, and it is only necessary Jbo make the following assignment of parts: A = iSo (25) c = 90 cos A, cos d sin / = sin z sin A. (26) (27) (28) To adapt these formulae for use with the ordinary logarithmic tables, the auxiliary quantities nt and M, defined by m sin M = sin z cos A, m cosM= are introduced (Num. Cal. p. 14). Substituting these relations into (26) and (27) and collecting results, we have for the calculation m sin M = sin z cos A, mcosM= cos 2, cos d sin / = sin 2 sin A, cos d cos/ m cos (

9.89061 n cos N 9-75294 sin 8 9-i7i2i tan N 9. 41827* sin 38 58 53 B 0.67364 9 06182* cos N 9-98559 cos 8 sin cos t 9-55I64 log 9-76735 A 9.51018 sin ( Ans - 15. Transformation of azimuth and altitude into right ascension and declination, or vice versa. These transformations are effected by a combina- tion of the results of Sections 12-14. For the direct transformation, deter- mine / and S by (26)-(28) or (29), and then a by (35). For the reverse calculate t by (36), and then A and z by (3i)-(33) or (34). Example 5. What is the right ascension of the object whose coordinates, at the sidereal time I7 h 2i m i6'4, are those given in Example i? The hour angle found in the solution of Example i by equations (26)-(28) is 4 h 47 m 464. This, combined with = I7 h 2i m i64 in accordance with equation (35^, gives for the required right ascension I2 h 33 m 3o;o. Example 6. At a place whose latitude is 38 38' 53", what are the azimuth and zenith dis- tances of an object whose right ascension and declination are 9 h 27 m i42 and 8 31 '47", re- spectively, the sidereal time being 5 h 46 m 56fo? By equation (36;, / = 2o h i9 m 4if8. We have, further, d = 8 31 '47" and = 38 38' 53". These quantities are the same as those appearing in Example 2. The solution by equa- tions (39) gave A 300 10' 29", z = 69 42' 30". 16. Given the latitude of the place, and the declination and zenith distance of an object, to find its hour angle, azimuth, and parallactic angle. We have given three sides of the spherical triangle ZPO, Fig. 6, p. 26* to find the three angles, the parallactic angle being the angle at the object. The parallactic angle is not used in engineering astronomy, although its value is frequently required in practical astronomy proper. Equations (20) are directly applicable for the solution of the problem. Assigning the parts of the triangle as in (30), and, further, writing the angle C = q = parallactic angle, we have for the calculation. 32 PRACTICAL ASTRONOMY a = , b = 90 o, c = go s = y-z (a + b -f 4 Check: (j -a) + (.r-) + ($-*) = j, A-*- M ii ^ a) sin \s-vj sin ^j cj sin s K fC sin (s-a) sin (s-t>) A" sin Check: tan ^ / cot y 2 A tan ^ q = sin Object ^ wcsl } of meridian, Y z t, %A, % q in In engineering astronomy the determination of the hour angle, A is usually all that is required. For this case it is simpler to use equation (17). The formulae are a = z, b = 90 - o, c = 90 - , s=y 2 (a + b + c). Check: (s - a) + (s - b} + (s - c) = s, (38) 3n , v f _ sin (s - b} sin (s - c) tan YZ t -. ; . r sin s sin (s - a) where y 2 t is to be taken in the first or second quadrant according as the ob- ject is west or east of the meridian at the time of observation. For those cases in which the object is more than two and one-half or three hours from the meridian, equation (16) written in the form cos z - sin dsin

will usually give satisfactory results. In any case, (39) affords a valuable con- trol upon the value of / given by (38). The numerator of the right member of (39) is readily calculated by means of addition-subtraction logarithms. 17. Application of transformation formulae to the determination of latitude, azimuth, and time. It was shown in Section 4 that the solution of the fundamental problems of practical astronomy requires the determination of the position of the axis of the celestial sphere and the orientation of the sphere as affected by the diurnal rotation. In practice this is accomplished indirectly by observing the positions of various celestial bodies with respect to the horizon, the observed data being combined with the known position of the bodies on the sphere for the determination of the position of the sphere itself. The means for effecting the coordinate transformation hereby implied are to be found in the formulae of Sections 12-16. Although the most advantageous determination of latitude, azimuth, and time requires a modification of these formulae, it is, nevertheless, easy to see that the solution of the various problems is within our grasp, and that the TRA NSFORMA TIONS GENERA L DISC US SI ON 33 Example 7. For a place whose latitude is 3856'5i", find the hour angle, azimuth, and parallactic angle of an object east of the meridian whose declination and zenith distance are 8 16' 14" and 54 16' 12", respectively. Equations (37) are used for the solution, which is given below in the column on the left. If only the hour angle were required, equations (38) or (39) would be used. As an illustra- tion of the' application of these formulae, the problem is also solved on this assumption. The first ten lines of the computation for (38), being the same as that for (37), are omitted. The remainder of the calculation for (38) occupies the upper part of the right-hand column. The solution by (39) is in the lower part of this column. The object is rather too near the merid- ian for the satisfactory use of equation (39), although it happens that the resulting value of the hour angle agrees well with that from (37) and (38). tan 8 b c 25 S s - a s-b s - c sin (s - ) sin (5 - b) sin (5 - c) cosec 5 tan y z t cot } A tan % q i co\.y z A t&n%g K cosec 5 y*t 8 16' 14" 38 56 51 54 16 12 98 16 14 5i 3 9 203 35 35 101 47 48 Ck. 47 3i 36 3 3i 34 50 44 39 9.86782 8.78890 9.88892 0.00927 8-55491 sin (.?-) sin (s-c) cosec (s-a~) cosec s tan* % t tan t 8.78890 9.88892 0.13218 0.00927 8.81927 9.40964,, 165 35' 46' 33i ii 32 9.40964,, 0.48856* 9-38854* 9.28674* 9.28673,, Ck. 165' 162 1 66 33i 324 332 35 o 15 i 30 46" 36 20 sin d sin y cos 8 COS (ft cos z sin d sin ,_- A B cos z sin d sin cos d cos

. Again, the elimination of z from (32) and (33) gives an expression for A as a function for , d, and t = 6 - a. Let it be assumed that tp and 6 are known. The azimuth of a star of known right ascension and declination can therefore be calculated. The calculated azimuth applied to the observed difference in azimuth of star and mark gives the azimuth of the mark. Finally, equations (38) and (39) express the hour angle, t, as a function of #, ^, and d. If the zenith distance of a star of known right ascension and dec- lination be measured in a place of known latitude, the hour angle can be cal- culated. Equation (35), in the form 6 = t + a, then gives the sidereal time of observation. The solutions thus outlined require, for the determination of latitude, a knowledge of the time; for the determination of time, a knowledge of the lat- itude; and, for azimuth, both time and latitude. For the first two, time and latitude, it might appear that the methods proposed are fallacious. If each is required for the determination of the other, how can either ever be determined? The explanation is to be found in the fact that the formulae can be arranged in such a way that an approximate value for either of these quantities suffices for the determination of a relatively precise value of the other. Thus, a mere guess as to the time will lead to a relatively accurate value of the latitude, which, in turn, can be used for the determination of a more precise value of the time. The process can be repeated as many times as may be necessary to se- cure the desired degree of precision. The principle involved in the procedure thus outlined is called the Method of Successive Approximations. In numerical investigations it is of great importance. The method amounts, practically, to replacing a single complex process by a series, consisting of repetitions of some relatively simple operation. Ordinarily, the success of the method depends upon the number of repetitions or approximations which must be made in order to arrive at the desired result. If the convergence is rapid, so that one or two approximations suffice, the saving in time and labor as compared with the direct solution is frequently very great. Indeed, in some instances, the method of successive approximations is the only method of pro- cedure, the direct solution being impossible as a result of the complexity of the relation connecting the various quantities involved. GENERAL DISCUSSION 35 The general method of procedure for the solution of the problems of latitude, time, and azimuth has been outlined. There remains the formulation of the details. But, before proceeding to a detailed development, we must consider the subject of time in its theoretical aspects the different kinds of time, their definition and their relations. Chapter III will be devoted to this question. We must also consider the various astronomical instruments that find application in engineering astronomy their characteristics and the con- ditions under which they are employed, since the nature of the data obtained through their use will influence the arrangement of the solutions. Chapter IV is therefore devoted to a discussion of various astronomical instruments. In arranging the details of the methods for the determination of latitude, time, and azimuth, it is to be remembered that the various problems are not merely to be solved, but they are to be solved with a definite degree of precision, and with a minimum expenditure of labor. This requirement renders the question one of some complexity, for the precision required may vary within wide limits. For many purposes approximate results will suffice, and it is then desirable to sacrifice accuracy and thus reduce the labor involved. On the other hand, in astronomical work of the highest precision, no means should be overlooked which can in any way contribute toward an elimination or reduction of the errors of observation and calculation. The problems with which we have to deal therefore present themselves under the most diverse conditions, and, if an intelligent arrangement of the methods is to be accomplished, one must constantly bear in mind the results which will be established in the two following chapters, as well as those already obtained in the discussion of the principles of numerical calculation. CHAPTER III TIME AND TIME TRANSFORMATION 18. The basis of time measurement The rotation of the earth is the basis for the measurement of time. Svmce motion is relative, we must specify the object to which the rotation is referred. By referring to different objects, it is obvious that we may have several different kinds of time. Actually, the rotation of the earth is referred to three different things: the apparent, or true, sun, a fictitious object called the mean sun, and the vernal equinox. In practice, how-' ever, we turn the matter about and take the apparent diurnal rotations of these objects with reference to the meridian of the observer, considered to be fixed, as the basis of time measurement. We have, accordingly, three kinds of time: Apparent, or True, Solar Time, Mean Solar Time, and Sidereal Time. 19. Apparent, or True, Solar Time=A.S.T. The apparent, or true, solar time at any instant is equal to the hour angle of the apparent, or true, sun at that instant. The interval between two successive transits of the apparent, or true, sun across the same meridian is called an Apparent, or True, Solar Day=A. S. D. The instant of transit of the apparent sun is called Apparent Noon A. N. In astronomical practice the apparent solar day begins at apparent noon. It is subdivided into 24 hours, which are counted continuously from o to 24. The earth revolves about the sun in an elliptical orbit, the sun itself occupying one of the foci of the ellipse. The earth's motion is such that the radius vector connecting it with the sun sweeps over equal areas in equal times. Since the distance of the earth from the sun varies, it follows that the angular velocity of the earth in its orbit is variable. Hence, the angular motion of the sun along the ecliptic, which is but a reflection of the earth's orbital motion, is also variable. The projection into the equator of the motion along the ecliptic is like- wise variable, not only because the ecliptical motion is variable, but also on account of the fact that the angle of projection changes, being o degrees at the solstices, and about 23^ degrees at the equinoxes. Apparent solar time is not, therefore, a uniformly varying quantity, nor are apparent solar days of the same length. The adoption of such a time system for the regulation of the affairs of everyday life would bring with it many inconveniences, the first of which would be the impossibility of constructing a timepiece capable of following accurately the irregular variations of apparent solar time. On this account there has been devised a uniformly varying time, based upon the motion of a fictitious body called the mean sun. 20. Mean Solar Time=M. S. T. The mean sun is an imaginary body supposed to move with a constant angular velocity eastward along the equator, such that it completes a circuit of the sphere in the same time as the apparent, or true, sun. Further, the mean sun is so chosen that its right ascension differs as little as possible, on the average, from that of the true sun. 36 DEFINITIONS 37 The Mean Solar Time at any instant is equal to the hour angle of the mean sun at that instant. The interval between two successive transits of the mean sun across the same meridian is called a Mean Solar Day M. S. D. The instant of transit of the mean sun is called Mean Noon=M. N. Mean solar time is a uniformly varying quantity and all mean solar days are of the same length. Mean solar time is the time indicated by watches and clocks, generally, throughout the civilized world, and the mean solar day is the standard unit for the measurement of time. In astronomical practice the mean solar day begins at mean noon. It is subdivided into 24 hours which are numbered continuously from o to 24. The astronomical date therefore changes at noon. But since a change of date during the daylight hours would be inconvenient and confusing for the affairs of every- day life, the Calendar Date, or Civil Date, is supposed to change 1 2 hours before the transit of the mean sun, i.e. at the midnight preceding the astronomical change of date. Further, in most countries, the hours of the civil mean solar day are not numbered continuously from o to 24, but from o to 12, ana then again from o to 12, the letters A. M. or P. M. being affixed to the time in order to avoid ambiguity. For example the civil date 1907, Oct. 8, io h A. M., is equivalent to the astronomical date, 1907, Oct. 7, 22 h . The astronomical day Oct. 8 did not begin until the mean sun was on the meridian on Oct. 8 of the calendar. From the manner of definition, it is evident that at any instant the mean solar time for different places not on the same meridian is different. If each place were to attempt to regulate its affairs in accordance with its own local mean solar time, confusion would arise, especially in connection with railway traffic. To avoid this difficulty all points within certain limits of longitude use the time of the same meridian. The meridians selected for this purpose are all an exact multiple of 15 degrees from the meridian of Greenwich, with the result that all timepieces referred to them indicate at any instant the same number of minutes and seconds, and differ among themselves, and from the local mean solar time of the meridian of Greenwich, by an exact number of hours. The system thus defined is called Standard Time. Although, theoretically, all points within 7^2 degrees of longitude of a standard meridian use the local mean solar time of that meridian, actually, the boundaries separating adjacent regions whose standard times differ by one hour are quite irregular. The standard meridians for the United States are 75, 90, 105, and 120 degrees west of Greenwich. The corresponding standard times are Eastern, Central, Mountain, and Pacific. These are slow as compared with Greenwich mean solar time by 5, 6, 7, and 8 hours, respectively. 21. Sidereal Time. The sidereal time at any instant is equal to the hour angle of the true vernal equinox at that instant. (See p. 25.) The interval between two successive transits of the true vernal equinox across the same meridian is called a Sidereal Day S. D. 38 PRACTICAL ASTRONOMY The instant of transit of the true vernal equinox is called Sidereal Noon^S. N. Since the precessional and nutational motions of the true equinox are not uniform, sidereal time is not, strictly speaking, a uniformly varying quantity, but practically it may be considered .as such, for the variations in the motion of the equinox take place so slowly that, for the purposes of observational astron- omy, all sidereal days are of the same length. The importance of sidereal time in the transformation of the coordinates of the second system into those of the third, and vice versa, has already been shown in Sections n and 14. It also plays an important role in the determina- tion of time generally, for sidereal time is more easily determined than either apparent or mean solar time. The usual order of procedure in time determination is as follows : Every observatory possesses at least one sidereal timepiece whose error is determined by observations on stars. The true sidereal time thus obtained is transformed into mean solar time by calculation, and used for the correction of the mean solar timepieces of the observatory. Certain observatories, in particular the United States Naval Observatory at Washington, and the Lick Observatory at Mt. Hamilton in California, send out daily over the wires of the various tele- graph companies, series of time signals which indicate accurately the instant of mean noon. These signals reach every part of the country, and serve for the regulation of watches and clocks generally. 22. The Tropical Year. Several different kinds of years are employed in astronomy. The most important are the tropical and the Julian. The Tropical Year is the interval between two successive passages of the mean sun through the mean vernal equinox. Its length is 365.2422 M. S. D. During this interval the mean sun makes one circuit of the celestial sphere from equinox to equinox again, in a direction opposite to that of the rotation of the sphere itself, whence it follows that during a tropical year the equinox must complete 366.2422 revolutions with respect to the observer's meridian. We therefore have the important relation: One Tropical 63^=365.2422 M. S. 0.^366.2422 S. D. (40) In accordance with a suggestion due to Bessel, the tropical year begins at the instant when the mean right ascension of the mean sun plus the constant part of the annual aberration is equal to 280 or i8 h 4O m . The symbol for this instant is formed by affixing a decimal point and a zero to the corresponding year number; thus for 1909, the beginning of the tropical year is indicated by 1909.0. This epoch is independent of the position of the observer on the earth and does not, in general, coincide with the beginning of the calendar year, although the difference between the two never exceeds a fractional part of a day. 23. The Calendar. For chronological purposes the use of a year involving fractional parts of a day would be inconvenient. That actually used has its origin in a decree promulgated by Julius Caesar in 45 B. C. which ordered that THE CALENDAR 39 the calendar year should consist of 365 days for three years in succession, these to be followed by a fourth of 366 days. The extra day of the fourth year was introduced by counting twice the sixth day before the calends of March in the Roman system. In consequence such years were long distinguished by the designation bissextile, although they are now called Leap Years. The years of 365 days are Common Years. With this arrangement the average length of the calendar year was 365^ days. This period is called a Julian Year, and the calendar based upon it, the Julian Calendar. The difference between the Julian and the tropical years is about n m . In order to avoid the gradual displacement of the calendar dates with respect to the seasons resulting from the accumulation of this difference, a slight mod- ification in the method of counting leap years was introduced in 1582 by Pope Gregory XIII. The accumulated difference amounts approximately to three days in 400 years, and, as the Julian year is longer than the tropical, the Julian calendar falls behind the seasons by this amount. Gregory therefore ordered that the century years, all of which are leap years under the Julian rule, should not be counted as such unless the year numbers are exactly divisible by 400. At the same time it was ordered that 10 days should be dropped from the calendar in order to bring the date of the passage of the sun through the vernal equinox back to the 2ist of March, where it was at the time of the Council of Nice in 325 A. D. The Julian system thus modified is called the Gregorian Calendar. The revised rule for the determination of leap years is as follows : All years whose numbers are exactly divisible by four are leap years, excepting the century years. These are leap years only when exactly divisible by four hundred. All other years are common years. The average length of the Gregorian calendar year differs from that of the tropical year by only 0.0003 day or 26 s . In the modern system the extra day in leap years appears as the 2Qth of February. The Gregorian calendar was soon adopted by all Roman Catholic countries and by England in 1752. Russia and Greece and other countries under the dominion of the Eastern or Greek Church, still use the Julian Calendar, which, at present, differs from the Gregorian by 13 days. 24. Given the local time at any point, to find the corresponding local time at any other point. From the definitions of apparent solar, mean solar, and sidereal time, it follows that at any instant the difference between two local times is equal to the angular distance between the celestial meridians to 'which the times are referred. But this is equal to the angular distance between the geographical meridians of the two places, i.e. their difference of longitude. Let Te. = the time of the eastern place, TV, = the time of the western place, L = longitude difference of the two places, We then have the relations: T e = T w + L (41) 40 PRACTICAL ASTRONOMY Equations (41) are true whether the times be apparent solar, mean solar, or sidereal. Example 8. Given, Columbia mean solar time i2 h i4 m 16541, find the corresponding Greenwich mean solar time. T w = i& I4 m 16541 L = 6 9 18.33 T e = 18 23 34.74 Ans. Example 9. Given, Greenwich mean solar time 1907, Oct. 6 3 h i4 m 21', find the corre- sponding Washington mean solar time. T e 1907, Oct. 6 3 h I4 m 2i 8 L = 5 8 16 TV = 1907, Oct. 522 65 Ans. Example 10. Given, central standard time 1907, Oct. 12 6 h i8 h o" A.M., find the cor- responding Greenwich mean solar time, astronomical and civil. TV 1907, Oct. 12 6 h i8 m o 8 A.M. Oct. ii 18 18 o astronomical L = 600 7e = 1907, Oct. 12 o 18 o astronomical \ Oct. 12 o 18 o P.M. civil / Example 11. Given, central standard time 1907, Oct. n o h 3 m 16518 P.M. find the corresponding Columbia mean solar lime, civil and astronomical. Te = 1907, Oct. ii o h 3 m 16518 P.M. L = 9 i8-33 TV = I907 Oct. 1 1 ii 53 57.85 A.M. civil j ^^ Oct. 10 23 53 57.85 astronomical / 25. Given the apparent solar time at any place, to find the corre- sponding mean solar time, and vice versa. From equation (36), t = 6 , and the definitions of mean solar and apparent solar time, we find M. S. T. = R. A. of M. S., A. S. T. = R. A. of A. S. whence M. S. T. A. S. T. = R. A. of A. S. R. A. of M. S. The difference E = M. S. T. A. S. T. (42) is called the Equation of Time. The equation of time varies irregularly throughout the year, its maximum absolute value being about i6 m . It is some- times positive, and sometimes negative, since the right ascension of the apparent sun is sometimes smaller and sometimes greater than that of the mean sun. The right ascension of the apparent sun is calculated from the known orbital motion of the earth. The right ascension of the mean sun is known from its manner TIME TRANSFORMATION 41 of definition. This data suffices for .the calculation of E, whose values are tabu- lated in the various astronomical ephemerides. In the American Ephemeris they are given for instants of Greenwich apparent noon on page I for each month, and for Greenwich mean noon, on page II. The former page is used when apparent time is converted into the corresponding mean solar time, and the latter when apparent solar time is to be found from a given mean solar time. The algebraic sign of E is not given in the American Ephemeris, but the column containing its values is headed by a precept which indicates whether it is to be added to or subtracted from the given time. Values of E for times other than Greenwich apparent noon and Greenwich mean noon must be obtained by inter- polation. This operation is facilitated by the use of the hourly change in E printed in the columns headed "Difference for I Hour," which immediately fol- low those containing the equation of time. If the time to be converted refers to a meridian other than that of Greenwich, the corresponding Greenwich time must be calculated before the interpolation is made. Note that for each date the difference of the right ascension of the apparent, or true, sun in column two of page II, and the right ascension of the mean sun in the last column of the same page, is equal to the corresponding value of E, in accordance with the definition. Example 12. Given, Greenwich apparent solar time 1907, Oct. 15 2 h 6 m 12506, find the corresponding Greenwich mean solar time. E for Gr. A. N. 1907, Oct. 15 13 56575 (Eph. p. 164) Change in E during 2 h 6 m 12" -(- 1.20 E (to be subtracted from A. S. T.) 13 57.95 Gr. A. S. T. 1907, Oct. 15 26 12.06 Gr. M. S. T. 1907, Oct. 15 i 52 14.11 Ans. Example 13. Given, Greenwich mean solar time 1907, Oct. 15 i h 52*" 14511, find the corresponding Greenwich apparant solar time. E for Gr. M. N. 1907, Oct. 15 13 56588 (Eph. p. 165) Change in E during i h 52"" 14" -(- 1.07 E (to be added to M. S. T.) 13 57.95 Gr. M. S.T. 1907, Oct. 15 i 52 14.11 Gr. A. S. T. 1907, Oct. 15 2 6 12.06 Ans. Example 14. Given, central standard time 1907, Oct. 20 n h i8 m 1252 A.M., find the corresponding Columbia apparent solar time. C. S. T. 1907, Oct. 20 n h i8 m 1212 A.M. L 9 18.3 Columbia M. S. T. 1907, Oct. 19 23 8 53.9 astronomical Gr. M. S. T. 1907, Oct, 20 5 18 12.2 E for Gr. M. N. 1907, Oct. 20 14 58.63 (Eph. p. 165) Change in E during 5 h i8 m 12" -(- 2.39 E (to be added to Columbia M. S. T.) 15 i.o Columbia A. S. T. 1907, Oct. 19 23 23 54.9 Ans. 42 PRACTICAL ASTRONOMY Example 15. Given Columbia apparent solar time 1907, Oct. 19 23^ 23 5419, find the corresponding central standard time. Columbia A. S. T. 1907, Oct. 19 23 h 23'" 5419 L 69 18.3 Gr. A. S. T. 1907, Oct. 20 5 33 13.2 E for Gr. A. N. 1907, Ot't. 20 14 58.52 Change in during 5 h 33 m i3 B _|_ 2.50 E (to be sub. from Columbia A. S. T.) 15 i.o Columbia M. S. T. 1907, Oct. 19 23 8 53.9 L 9 18.3 C. s - T. 1907, Oct. 20 ii 18 12.2 A.M. Ans. 26. Relation between the values of a time interval expressed in mean solar and sidereal units. Equation (40) is the fundamental relation connecting the units of mean solar and sidereal time. If we let /s = the value of any interval /in mean solar units, 7m = the value of /in sidereal units, we find from (40) / s _ / m + (43) /S (44) Writing HI = r^r- II = 365.2422 366.2422 (43) and (44) become / s = / m + in/ m / m = / s H/ s (46) Assuming /m = 24 h we find from (45) 24 h o m o'ooo M. S. = 24 h 3 m 56:555 Sid. Similarly, by supposing / s = 24 h we obtain from (46) 24 h o m o?ooo Sid. = 23 h 56 m 4^091 M. S. Hence Gain of on M. S. T. in I M. S. D. = IIl24 h = 2365555 Gain of 6 on M. S. T. in i S. D. = 1124 = 235.909 ' 47 > RELATION OF SIDEREAL AND SOLAR INTERVALS 43 and further Gain of on M. S. T. in I M. S. hour = IIIi = 958565 Gainof0onM.S.T.ini S. hour = Hi =9.8296 For many purposes these expressions may be replaced by the following approx- imate relations: IIl24 h 4 m (i 1/70), Error = 0*016 1124 = 4 (i 1/60), Error = 0.081 IIIi =IO(I 1/70), Error = 0.0006 Hi =10 (i 1/60), Error 0.0037 Equations (45) and (46) may be used for the conversion of the value of a time interval expressed in mean solar units into its corresponding value in sidereal units, and vice versa. The calculations are most conveniently made by Tables II and III printed at the end of the American Ephemeris. Table II contains the numerical values of II/s, while Table III gives those of III/m, the arguments being the values of /s and 7m, respectively. It will be observed that the first factors of II/s and III/m indicate the table, and the second the argu- ment which is to be used for the interpolation. In case tables are not available the conversion can be based upon equations (47) or (48), or more simply, upon (49), provided the highest precision is not required. Example 16. Given the mean solar interval i6 h i8 m 2i!2o, find the equivalent sidereal interval. By Eq. (45) 7 m = i6 h *l8 m 21520 Ill/m = 2 40.72 (Eph. Table III) /s = 16 21 1.92 Ans. The calculation of III/ m by the third of (49) is as follows: / m = i6'.'3o6 io s / m = i63fo6 i/7oXio'/ m = 2.33 Ill/m = 160.73 = 2 m 4573 The value thus found differs onlyo'oi from that derived from Table III of the Ephemeris. Example 17. Given the sidereal interval 2o h 28 m 42517, find the equivalent mean solar interval. By Eq. (46) 7 S = 2O h 28 42517 II/s = 3 21.29 ( E P h - T ble II) /m = 20 25 2O.88 Ann. The calculation of II/ S by the last of (49) is as follows: /s = 20^478 io 8 / s = 204578 i/6oXio 9 /s = 3.41 II/s = 201.37 = 3 m 21537 44 PRACTICAL ASTRONOMY 27. Relation between mean solar time and the corresponding side- real time. In Section 14 it was shown that the relation connecting the hour angle of an object with the sidereal time is where a represents the right ascension of the object. Applying this equation to the mean sun, we find M = e R (50) in which R represents the right ascension of the mean sun, and M its hour angle. The latter, however, is equal by definition to the mean solar time. Equa- tion (50) therefore expresses a relation between mean solar time and the cor- responding sidereal time, which can be made the basis for the conversion of the one into the other. The transformation requires a knowledge of R, the right ascension of the mean sun, at the instant to which the given time refers. We now turn our attention to a consideration of the methods which are avail- able for the determination of this quantity. 28. The right ascension of the mean sun and its determination. It is shown in works on theoretical astronomy that the right ascension of the mean sun at any instant of Greenwich mean time is given by the expression ^ G = i8 h 3 8 m 45*836 + (236:555 X 365.25)' + of 0000093 ' a (5 1 ) + nutation in right ascension, in which t is reckoned in Julian years from the epoch 1900, Jan. o d o h Gr. M. T. It thus appears that the increase in the right ascension of the mean sun is not strictly proportional to the increase in the time. This, in connection with equation (50), shows that sidereal time is not a uniformly varying quantity, a fact already indicated in Section 21. The nutation in right ascension oscillates between limits which are approximately-}- I s and I s with a period of about 19 years. Its change in one day is therefore very small, and, as the same is true of the term involving t 2 in (51), it follows that the increase in the right ascension of the mean sun in one mean solar day is sensibly 236. S 555. From equation (50) it is seen that the gain of sidereal on mean solar time during any interval is equal to the increase in R during that interval; and, indeed, we have exact numerical agree- ment between the change in the latter for one mean solar day, as given by equa- tion (51), and the gain of the former during the same period as shown by the first of (47). From this it follows that the methods given in Section 26, including Tables II and III of the Ephemeris and the approximate relations (49), can equally well be applied to the determination of the increase in R, provided only that the interval for which the cKange is to be calculated is small enough to render the variations in the last two terms of (51) negligible. RIGHT ASCENSION OF MEAN SUN 45 To facilitate the solution of problems in which R is required, its precise numerical values are tabulated in the various astronomical ephemerides for every day in the year. In the American Ephetnieris they are given for the instant of Greenwich mean noon, and are to be found in the last column of page II for each month. If these tabular values be represented by Ro, and if J?L represent the right ascension of the mean sun at the instant of mean noon for a point whose longitude west of Greenwich is L, it follows from the preceding paragraph that RL = Ro + IIIZ, (52) for L is equal to the time interval separating mean noon of the place from the preceding Greenwich mean noon. Further, the value of R at any mean time, M, at a point whose longitude west of Greenwich is L is given by R = XL + II W, (53) or R = Ro + IIIZ + HIM. (54) Equations (52) and (53), or their equivalent, (54), suffice for the determi- nation of R at any instant at any place when the value of Ro for the preceding mean noon is known. For a given place the term lllL is a constant. Its value can be calculated once for all, and can then be added mentally to the value of Ro as the latter is taken from the Ephemeris. The quantity HIM may be derived from Table III of the Ephemeris with M as argument. If an Ephemeris is not available the values of R can still be found; approxi- mately at least, by the use of Tables II-IV, page 46. The first of these contains the values of Ro computed from (51) for the date Jan. o for each of the years 1907-1920. Denoting these by Roo and neglecting the variations in the last two terms of (51) we have for Greenwich mean noon of any other date Ro = Roo -f- IIIZ? (55) where D indicates the number of mean solar days that have elapsed since the preceding Jan. o. Substituting (55) into (54). R = R 00 + IIIZ + lll(D + M). (56) The value of D may be obtained from Table III by adding the day of the month to the tabular number standing opposite the name of the month in question. M is conveniently expressed in decimals of a day by means of Table IV. The value thus found is to be combined with D. If the precise value of III, viz., 236. S 555, be used, the uncertainty in R derived from (56) will be only that arising from the neglect of the variation in the last terms of (51). If care be taken to count D from the nearest Jan. o the error will never exceed o. 8 3 or 46 PRACTICAL ASTRONOMY TABLE II RIGHT ASCENSION OF THE MEAN SUN FOR THE EPOCH JAN. o d o h GR. M. T. Year ROO * Year R oo 1907 i8 h 36 0547 1914 i8 h 37 1 " 13562 1908 35 3-04 1915 36 16.64 1909 38 2.28 1916 35 19.63 1910 37 5-09 1917 38 1907 1911 36 7-99 1918 37 21.85 1912 35 10.98 1919 36 24.51 1913 38 10.57 1920 35 28.05 Date D Common Year Leap Year Jan. o o o Feb. o 31 31 Mar. o 59 60 Apr. o 90 9i May o 1 20 121 June o 151 152 July o isn l82l Aug. o 2I2\ 153 J 213) 153 J Sept. o 243 \ 122 / 244 \ 122 j Oct. o 2731 9 2 / 2741 9 2 / Nov. o 304 1 3051 Dec. o 334 1 335 j PRECEPT: Add the day of the month to the tabular value corresponding to the given month. The use of the nega- tive values gives the day num- ber from Vnz following Jan. o. Decimals Decimals Decimals Hour of a Min. of a Min. of a Day Day Day I 0.042 i O.OOI 10 0.007 2 0.083 2 I 20 0.014 3 0.125 3 2 30 O.02I 4 0.167 4 3 40 0.02S 5 0.208 5 3 50 0.035 6 0.250 6 4 60 O.O42 7 0.292 7 5 8 0-333 8 6 9 0-375 9 6 10 0.417 10 0.007 ii 0.458 12 0.500 PRECEPT: When the given hour is greater than 12, drop i2 h from the argument and add o d 5oo to the result given by the table. Thus, for I7 h 28 m enter the table with the argument 5 h 28 m , giving o d 228, whence i7 h 28 m = o d 228 -j- o d 50o = o d 728. TRANSFORMATION OF MEAN SOLAR INTO SIDEREAL TIME 47 0*4. This requires that for D> i83 d the negative value of Table III be employed, together with the value of Roo for ihe following Jan. o. If a somewhat greater uncertainty is permissible, the result may be more expeditiously found by using 4 m (i 1/70) for III. If D be reckoned from the nearest Jan. o as above, the corresponding error will not exceed 3 s . Example 18. Find the right ascension of the mean sun for the epoch 1907, June 16 8 h 2i m 14500 Columbia M. S. T. By Equation (54) Xo = 5 h 34 m 25510 (Eph. p. 93) L = 6 h 9 m 18133 11IL = i 0.67 (Eph. Table III) Af=8 21 14.00 IIU/= i 22.34 (Eph. Table III) R = 5 36 48.11 Ans. By Equation (56) (D + M) = i67d3 4 8 (Tables III and IV; R 00 (1907) = i8 h 36 m 055 (Table II) 4 m (D + M*) = 669T392 IIIZ, = i 0.7 (^ + ^) = 91-563 lll(D + M) = io 59 49.7 ^=5 3 6 5 1 Example 19. Find the right ascension of the mean sun for the epoch 1909, Sept. 21 I9 h 2 6m 248 Columbia M. S. T. = 101 d + o d 8io (Tables III and IV) J ff o (1910) = i8 h 37 5*1 (Table II) = 100^190 III/, i 0.7 4 m (D+M) = 4001760 III(Z? -f M) = 6 35 2.1 i/7oX4 m (>+Af) = 5.725 R= 12 34 Ans. The precise value given by (54) is I2 h 3 m 5f2i. 29. Given the mean solar time at any instant to find the corre- sponding sidereal time. From equation (50) we find d = M+R (57) Introducing the value of R from (53) we have 6 = M+i. + lUM, (58) where L = R + HIZ. (59) Equations (59) and (58) solve the problem. Equation (58) may be interpreted as follows: RL is the right ascension: of the mean sun at the preceding mean noon for a place in longitude L west of Greenwich. It is therefore also equal to the hour angle of the vernal equinox at that instant, i.e. to the sidereal time of the preceding mean noon at the place considered. Now M is the mean time interval since preceding mean noon, and by (45) Af-f-IIIM is the equivalent sidereal interval. The right member of (58) therefore expresses the sum of the sidereal time of the preceding mean noon and the number of sidereal hours, minutes, and seconds that have elapsed 48 PRACTICAL ASTRONOMY since noon. In other words it is the sidereal time corresponding to the mean time, M } as indicated by the equation. In case the Ephemeris is not at hand, R may be obtained from (56) and substituted into (57) for the determination of 6. The uncertainty in the sidereal time thus found will be the same as th#t of R derived from (56). Oftentimes a rough approximation for 6 is all that is required. In such cases the following, designed for use at the meridian of Columbia, is useful : = i8 h 3717 -f M + 4 m (i 1/70) (D + M). (60) The first term in the right member of this formula is the average value of Roo plus the constant term IIIL, which for Columbia may be taken equal to i m . The expression can be adapted for use at any other meridian by introducing the appropriate value of IIIL. The maximum error in the value of 6 derived from (60) is i. m 7. Example 20. Given Columbia mean solar time i6 h 27 m 32517 on 1909, Nov. 16, find the corresponding sidereal time. By equations (58) and (59) M ' = i6 h 27 m 32517 Ro = 15 39 39-98 IIIL = i 0.67 IIIM= 2 42.23 S = 8 10 55.05 Ans. By equations (56) and (57) D + M = 45 d + o d 686 M = i6 h 27 3252 = 44 d 3H -ffoo= 18 37 5.1 4 m (Z> + ^7) = 177^256 I1IZ, = i 0.7 i/70X 4 m (Z>-f^/) = - 2.532 lll(D + M)= 2 54 43.4 B = 8 10 55 Ans. By equation (60) M= 16 27.5 4 m (i 1/70) (Z> + M) = 2 54.7 = 8 10. 5 Ans. 30. Given the sidereal time at any instant to find the corresponding mean solar time. We make use of equation (50), viz. Substituting as in Section 29 we have M= 6 XLHIM or d R L (61) 49 Multiplying equations (45) and (46), member by member, and dropping the common factor 7m 7s we find (i +III)(i -II)- i Combining this with (61) we find M = 6 RL 11(0 XL), (62) where, as before, R L = R -f III7 (63) Equations (63) and (62) solve the problem. Equation (62) is susceptible of an interpretation similar to that given (58) in the preceding section. Since 6 is the given sidereal time, and R\. the sidereal time of the preceding mean noon, 6 T^L is the sidereal interval that has elapsed since noon. To find the equivalent mean time interval we must, in accordance with equation (46), subtract from 6 RL the quantity 1 1(0 RL). The right member of (62) therefore expresses the number of mean solar hours, minutes, and seconds that have elapsed since the preceding mean noon, i.e. the mean solar time corresponding to the given 6. Example 21. Given, 1908, May 12, Columbia sidereal time i h 7 m 19*27, find the corre- sponding central standard time. By equations (62) and (63) S = i h 7 m 19128 ^? L = 3 20 25.46 9 #L=2i 46 53.82 11(0 7? L )= 3 34-io (Eph. Table II) M = 21 43 19.72 Z, = 9 18.33 C. S. T. = 9 52 38.05 A.M. May 13. Ans. CHAPTER IV INSTRUMENTS AND THEIR USE 31. Instruments used by the engineer. The instruments employed by the engineer for the determination of latitude, time and azimuth are the watch or chronometer, the artificial horizon, arid the engineer's transit or the sextant. The following pages give a brief account of the theory of these instruments and a statement of the methods to be followed in using them. The use of both the engineer's transit and the sextant presupposes an under- standing of the vernier. In consequence, the construction and theory of this at- tachment is treated separately before the discussion of the transit and sextant is undertaken. TIMEPIECES 32. Historical. Contrivances for the measurement of time have been used since the beginning of civilization, but it was not until the end of the sixteenth century that they reached the degree of perfection which made them of service in astronomical observations. The pendulum seems first to have been used as a means of governing the motion of a clock by Biirgi of the observatory of Landgrave William IV at Cassel about 1580, though it is not certain that the principle employed was that involved in the modern method of regulation. How- ever this may be, the method now used was certainly suggested by Galileo about 1637; but Galileo was then near the end of his life, blind and enfeebled, and it was not until some years later that his idea found material realization in a clock constructed by his son Vincenzio. It remained for Huygens, however, the Dutch physicist and astronomer, to rediscover the principle, and in 1657 give it an appli- cation that attracted general attention. Some sixty years later Harrison and Graham devised methods of pendulum compensation for changes of temperature, which, with important modifications in the escapement mechanism introduced by Graham in 1713, made the clock an instrument of precision. Since then its devel- opment in design and construction has kept pace with that of other forms of astronomical apparatus. The pendulum clock must be mounted in a fixed position. It can not be transported from place to place, and it does not, therefore, fulfill all the requirements that may be demanded of a timepiece. By the beginning of the eighteenth century the need of accurate portable timepieces had become pressing, not so much for the work of the astronomer as for that of the navigator. The most difficult thing in finding the position of a ship is the determination of longi- tude. At that time no method was known capable of giving this with anything more than the roughest approximation, although the question had been attacked by the most capable minds of the two centuries immediately preceding. The matter was of such importance that the governments of Spain, France, and the Netherlands established large money prizes for a successful solution, and in 1714 that of Great Britain offered a reward of 20,000 for a method which would give the longitude of a ship within half a degree. With an accurate portable timepiece, 50 TIMEPIECES 51 which conlcl be set to indicate the time of some standard meridian before begin- ning a voyage, the solution would have been simple. Notwithstanding the stimu- lus of reward no solution was forthcoming for many years. In 1735 Harrison succeeded in constructing a chronometer which was compensated for changes of temperature; and about 1760 one of his instruments was sent on a trial voyage to Jamaica. Upon return its variation was found to be such as to bring the values of the longitudes based on its readings within the permissible limit of error. The ideal timepiece, so far as uniformity is concerned, would be a body moving under the action of no forces, but in practice this can not be realized. The modern timepiece of precision is a close approximation to something equivalent, but falls short of the ideal. Thus far it has been impossible completely to nullify the effect of certain influences which affect the uniformity of motion. Changes in temperature, variations in barometric pressure, and the gradual thickening of the oil lubricating the mechanism produce irregularities, even when the skill of the designer and clockmaker is exercised to its utmost. No timepiece is perfect. We can say only that some are better than others. Further, it is impossible to set a timepiece with such exactness that it does not differ from the true time by a quantity greater than the uncertainty with which the latter can be determined. Thus it happens that a timepiece seldom if ever indicates the true time; and, in general, no attempt is made to remove the error. The timepiece is started under conditions as favorable as possible, and set to indicate approximately the true time. It is then left to run as it will, the astronomer, in the meantime, directing his attention to a precise determination of the amount and the rate of change of the error. These being known, the true time at any instant is easily found. 33. Error and rate. The error, or correction, of a timepiece is the quantity which added algebraically to the indicated time gives the true lime. The error of a timepiece which is slow is therefore positive. If the timepiece is fast the algebraic sign of its correction is negative. The error of a mean solar timepiece is denoted by the symbol JT; of a sidereal timepiece, by Jt). To designate the timepiece to which the correction refers subscripts may be added. Thus the error of a Fauth sidereal clock may be indicated by J6 f ; of a Negus mean time chronometer, by JT^. Sometimes it is convenient to use the number of the timepiece as subscript. If 6' be the indicated sidereal time at a given instant, and JO the cor- responding error of the timepiece, the true time of the instant will be e = d'-\-Jd'. (64) The analogous formula for a mean solar timepiece is T= T + JT. (65) The daily rate, or simply the rate, of a timepiece is the change in the error during one day. 52 PRACTICAL ASTRONOMY If the error of a timepiece increases algebraically, the rate is positive; if it decreases, the rate is negative. The symbols od and oT with appropriate sub- scripts are used for the designation of the rates of sidereal and mean solar time- pieces, respectively. The hourly rate,/.^. the change during one hour, is some- times more conveniently employed than the daily rate. It is convenient, but in no wise important, that the rate of a timepiece should be small. On the other hand, it is of the utmost consequence that the rate should be constant; for the reliability of the instrument depends wholly upon the degree to which this condition is fulfilled. Generally it is impossible to determine by observation the error at the instant for which the true time is required. We must therefore be able to calculate its value for the instant in question from values previously observed. If the rate is constant this can be done with precision; otherwise, the result will be affected by an uncertainty which will be the greater, the longer is the interval separating the epochs of the observed and the calculated errors. If J# and JO' be values of the observed error for the epochs t and /', the daily rate will be given by =^ < 66 > in which t' /must be expressed in days and fractions of a day. The rate having thus been found, the error for any other epoch, /", may be calculated by the formula Jd" = J6' + dd(t" t') (67) Example 22. The error of a sidereal clock was -f- 5 27161 on 1909, Feb. 3, at 6'.'4 sidereal time, and + 5 m 33!io on 1909, Feb. 11, at 5'.'2; find the daily rate, and the correction on Feb. 14 at 7^6 sidereal time. We have J0 = + 5 m 27*61, J0' --= + 5 m 33! 10, and t' t = i i d 5^2 3 d 61'4 7 d 22V8 = 7^95. Equation (66) then gives 80 = + 5H9/7-9S = 4 0569, which is the required value of the rate. To find the error for Feb. 14, 7^6, we have t" t 1 i4 d 7>?6 i i d 5'.'2 = 3 d 2l'4 = 3 the point where the rotation axis intersects the plane of the circle; O, the zero of the graduations; and V^ and F 2 the zeros of the verniers. The distance aC = e is the eccentricity of the circle. The perpendicular distance of a from the line joining V l and V ., is the eccentricity of the verniers. The reading of l / l is the angle OCF : , and of F 2 , OCF 2 . Denote these by R^ and R 2 , respect- ively. The angles through which the instrument must be rotated in order that the zeros of the verniers may move from O to the positions indicated, are l ^=A l and OaV '. 2 =A. 2 , respectively. A t and A 2 are therefore to be regarded 64 PRACTICAL ASTRONOMY as the angles which determine the positions of the verniers with respect to O for the pointing in question. The relations connecting A 1 and A,, with the vernier readings, R : and R 2 , are where E , E 1} and E 2 are the corrections for eccentricity for the points O, and V z . The mean of (71) and (72) is R,) + E. + ^(E,-E I ). (73) For any other pointing of the telescope, we have the analogous equation + + y 2 (E 2 '-E I '). ( 74 ) It is easily shown that E 2 E l and E.,' / are of the order of ee'/r 2 , where e' is the eccentricity of the verniers and r the radius of the circle. The last terms of (73) and (74) are entirely insensible in a well constructed instrument. The difference of (73) and (74) is therefore #(A,'+A,') - %(A t +A.) = #(*/+*,') - X(R t +R,). (75) The left member of (75) is the angular distance through which the instru- ment is rotated in passing from the first position to the second, and the equation shows that this angle is equal to the difference in the means of the vernier read- ings for the final and initial positions. The eccentricity is therefore eliminated by combining the means of the readings of both verniers. It can be shown that the eccentricity will also be eliminated by combining the means of any number of verniers, greater than two, uniformly distributed about the circle. In practice it is sufficient to use the degrees indicated by the first vernier with the means of the minutes and seconds of the two readings. Nos. 4 7. Horizontal Angles: In the measurement of horizontal angles an error of adjustment in No. 7 has no influence. To investigate the effect of residual errors in Nos. 4 6, let i= inclination of the vertical axis to the true vertical, 90 /^inclination of the horizontal axis to the vertical axis, &r=inclination of the horizontal axis to the horizon plane, 90 + ^^inclination of the line of sight to the horizontal axis. The quantities b and c are the errors in level and collimation, respectively. Then, in Fig. 8, which represents a projection of the celestial sphere on the plane of the horizon, let Z be the zenith, Z' the intersection with the celestial sphere of the vertical axis produced, O an object whose zenith distance is Z Q , and A the intersection of the horizontal axis produced with the celestial sphere when O is seen at the intersection of the threads. The sides of the triangles ZAZ' UNIVERSITY Of INSTRUMENTAL ERRORS 65 and ZAO have the values indicated in the figure. Finally, let k, K and / be the directions of ZA, ZO, and Z'A, respectively, referred to ZP. Applying equations (13) and (15) to triangle ZAZ', we find sin b = sin/ cos i -{- cosy' sin /cos /, cos b sin k = cosy sin /. (76) (77) In a carefully adjusted instrument i, j, and b are very small, and we may neglect their squares as insensible. Equations (76) and (77) thus reduce to b =j -(-/cos/, (78) (79) Equation (13) applied to triangle ZAO gives sin c = sin b cos -f- cos b sin z cos (K k). (80) Since c and 90 K -{- k are also very small, equation (80) may be written c = b cos s -f- (90 K + k) sin ^ or 90 = b cot z a -\- c cosec z . (81) Were there no errors of adjustment, the direction of A referred to P would be AT 90. The direction given by the instrument, determined by the angle through which it must be rotated to bring A from coincidence with ZP to its actual position, is /. Since the verniers maintain a fixed position with re- spect to A, the difference K 90 / represents" the effect of the residual errors 66 PRACTICAL ASTRONOMY on the horizontal circle readings. But by (79) l=k, sensibly, whence it follows that the amount of the error is given by (81). If, therefore, R be the actual horizontal circle reading, and R^, the value for a perfectly adjusted instrument. we have R o = R + ffcot z + c cosec * , C. R. (82) in which the value of b is given by equation (78). Assuming that equation (82) refers to that position of the instrument for which the vertical circle is on the right as the observer stands facing the eyepiece (C. R.), we find by a precisely similar investigation for circle left (C. L.), i = R = R l b 1 cotz c cosec z a , C. L. where R l is the circle reading less 180, and b^, the inclination of the horizontal axis to the plane of the horizon for C. L. The mean of equations (82) and (84) is , (85) or, substituting the values of b and b 1 from (78) and (83) (86) It therefore appears that the mean of the readings of the horizontal circle taken C. R. and C. L. for settings on any object is free from the influence of j, c, and the component of i in the direction of the line of sight, viz., i sin /.. More- over, for objects near the horizon the effect of /cos/, the component of i par- allel to the horizontal axis, is small, for it appears in (86) multiplied by cot ^ ft . If the instrument be provided with a striding level, the values of b and /?, may be determined by observation. Their substitution into (85) will then give the horizontal circle reading completely freed from i, j, and c- The readings may also be freed from the influence of b by combining the results of a setting on O with those obtained by pointing on the image of O seen reflected in a dish of mercury, both observations being made in the same position of the instrument, either C. R. or C. L. The reflected image, 0', will be on the vertical circle through O, and as far below the horizon as O is above. Since the horizontal axis is not truly horizontal, it will be necessary to rotate the instru- ment slightly about the vertical axis in turning from O down to '. A will thus move a small amount to a new position A'. To investigate the effect of the errors for a pointing on O' we must therefore consider the triangle A'ZO' in place of AZO in Fig. 8. The sides of A'ZO' are ZA' = ZA = gtf b, A'0' = AO = 9Q+c, and ZO' = 180 S Q . The angle at Z is K k' where k' is the direction of ZA' referred to ZP. We then find, simi- larly to equation (80), INSTRUMENTAL ERRORS 67 sin c = sin $cos z -j- cos sin z cos (A" /'), whence A' 90 k' b cot s -\- c cosec , and, finally, if /?' be the horizontal circle reading for the setting on O' R =R' b cot z + c cosec z . C. R. (87) Equations (82) and (87) both refer to C. R. Their mean is R o = % (R + /?') + c cosec s , C. R. (88) By the same method we find from the reflected observation, C. L., R = A?/ -f- ^ cot z c cosec z , C. L. (89) in which R t ' is the circle reading less 180 for C. L. This equation combined with (84) gives R o = y,(R t + R t ') c cosec z , C. L. (90) Equations (88) and (90) show that the mean of the horizontal circle read- ings for direct and reflected observations of an object in the same position of the instrument is free from the influence of any adjustment error in level. Finally the combination of (88) and (90) gives R =y 4 (R + R'+R>-\-R:\ (91) in other words the mean of the readings, direct and reflected, for both positions of the instrument, is free not only from b, but from the collimation error as well. Vertical Circle Readings: To investigate the influence of /, / and c upon the readings of the vertical circle, consider again Fig. 8. The true zenith dis- tance of is Z0=z t} ; that given by the vertical circle readings is equal to the angle Z'AO. From the triangle ZAO we find cos ~ sin $sin c -\- cos b cos c cos (ZAO}. The squares and products of the errors of adjustment are ordinarily quite insen- sible, whence we find with all necessary precision. ^ o r= Angle ZAO. Denoting the instrumental zenith distance Z'AO by r, we find = angle ZAZ ', and from triangle ZAZ' cos $sin (z a s] sin i sin /, G8 PRACTICAL ASTROXOMY or, since b, z #, and i are very small, z = s + i sin /, C. R. (92) A similar investigation gives for the reversed position of the instrument * = *, + , 'sin/, C.L. (93) in which z^ is the instrumental zenith distance for C. L. The angles z and z v are not read directly from the circles. The ordinary engineer's transit reads altitudes, but if there is any deviation from the condi- tion expressed in No. 7, the readings will not be the true altitudes, for they will include the effect of the index error. If r and r l be the vertical circle readings for C. R. and C. L., respectively, and / the reading 1 when the line of sight is hor- izontal, we have s =gor +/, C. R. (94) ,=90 r t /, C. L. (95) Substituting (94) and (95) into (92) and (93) s o = 9 o r + /+ /sin /, C. R. (96) z 90 r t / + / si n /, C. L. (97) The mean of (96) and (97) is z = 90 */z (r + r t ) + i sin /. (98) For an instrument whose vertical circle is graduated continuously from o to 360 it is easily shown that the equation corresponding to (98) is *, = ^ (^. 7',-) + i si n /, (99) in which z\ and v 2 are the circle readings, the subscripts being assigned so that It therefore appears that the vertical circle readings are not sensibly affected by /, c, or the component of i parallel to the horizontal axis. The com- ponent of i in the direction of the line of sight, viz., /sin/ enters with i'ts full value, and (98) and (99) show that it cannot be eliminated even when readings taken C. R. and C. L. are combined. The formation of the mean for the two positions of the instrument does eliminate the index error, however, i.e. the residual error of adjustment in No. 7. To free the results from z'sin/ we may combine observations direct and reflected, using the mercurial horizon. Considering the triangle A'ZO' previously defined, we find for the reflected observation cos (180 ) = sin^sin^ + cos b cos c cos (ZA'O'} INSTRUMENTAL ERRORS 69 whence, neglecting products and squares of the errors of adjustment, the true zenith distance of 0' is 180 z = angle ZA' 0'. Denoting the instrumental zenith distance of O' , which is the angle Z'A'O', by 180 s' we find Angle ZA'Z' = (i8o z } (180 z') = z' z . In the triangle ZA'Z' the sides are ZA' = 90 b, Z'A'-go _/, and ZZ' = t, and denoting the angle ZZ'A' by /'we find cos $ sin (#' ') = sin /sin /', or with sufficient approximation z z' z'sin/', C..R. (100) Now if r' be the vertical circle reading for C. R., reflected, and / the cir- cle reading when the line of sight is horizontal, we shall have, similarly to (94), z' == 90 r' I, C. R. This substituted into (100) gives = 90 r' / /sin/' C. R. (102) Since /' differs from / by a quantity of the order of the errors, the difference between i sin / and i sin /' will be insensible, so that when equations (96) and (102) are combined to form the mean we have simply *, = 90 % (r + r'). C. R. ( 103) Similar considerations for observations direct and reflected, C. L., give o r= 90 l A(r I + r/). C. L. (104) In other words, the formation of the means of the vertical circle readings for observations direct and reflected in the same position of the instrument elimi- nates not only the component of i in the direction of the line of sight, but the index correction as well. The influence of z'cos/, j and c is insensible. So far as the errors here considered are concerned, observations direct and reflected in a single position of the instrument are sufficient. Nevertheless it is desirable that measures be made both C. R. and C. L. for in this way different parts of the vertical circle are used, thus partially neutralizing errors of graduation. For an instrument with a vertical circle graduated continuously from o to 360, it is easily shown as before that in (103) and (104) the sum of the circle readings must be replaced by their difference taken in such a way that it is less than 180. 70 PRACTICAL ASTRONOMY The preceding discussion assumes that the adjustments of the instrument remain unchanged throughout the observations. If this is not so, the elimination of the errors will, in general, be incomplete. It is not always convenient to make use of the artificial horizon, and it is therefore desirable to be able to apply a method of elimination which does not depend upon this accessory. * It is easily shown that if the instrument be rclerelled before observing in the reversed position, the mean of the readings C. R. and C. L., both for the horizontal and the vertical circle, will be free from the errors in all of the ad- justment under Nos. 4 7, within quantities of the order of the products and squares of the errors. The same will be true, even though the plate bubbles are not accurately centered during the direct observations, provided, after re- versal, they be brought to the same position in the tubes that they occupied before. That such will be the case follows from a consideration of Fig. 8. The reversal and relevelling is equivalent to rotating the triangle ZAZ' about Zthrough the angle i8o-|-2r, its dimensions remaining unchanged. A thus assumes a new position A l} distant from O by 90-)-^, and Z' a position Z/. The triangle ZA^O leads to an equation differing from (84) only in that b i is replaced by b. The mean of the new equation and (82) is simply R.= X(R + R^ (105) where R and R l are the horizontal circle readings, the latter having been reduced by 1 80. The result is therefore free from both b and c. Again, from triangle ZA^Z^, we find for circle left analogously to (97), z = C) R* -^I D=- . (120) n The method of repetitions derives its advantage from the fact that the circle is not read for the intermediate settings on A and B. Not only is the observer thus spared considerable labor, but, what is of more importance, the errors which necessarily would affect the readings do not enter into the result. Consequently, that part of the resultant error of observation arising from the intermediate settings is due solely to the imperfect setting of the cross threads on the object. For instruments such as the engineer's transit, in which the uncertainty accompanying the reading of the angle is large as compared with that of the pointing on the object, the precision of the result given by (120) will be considerably greater than that of the mean of n separate measurements of the angle D, each of which requires two readings of the circle. But for instruments in which the accuracy of the readings is comparable with that of the pointings, as is the case with the modern theodolite provided with read- ing microscopes, the method of repetitions is not to be recommended. Although there is even here a theoretical advantage, it is offset by the fact that the peculiar observing program required for the application of the method presupposes the stability of the instrument for a relatively long interval, and hence affords an unusual opportunity for small variations in position to affect the precision of the measures. Moreover, experience has shown that there are small systematic errors dependent upon the direction of measurement, i.e. upon whether the initial setting is made on A or on B; and, although these may be eliminated in part by combining series measured in opposite direc- tions, it is not certain that the compensation is of the completeness requisite for observations of the highest precision. With the engineer's transit, how- ever, the method of repetitions may be used with advantage. Since rotation takes place on both the upper and the lower motions, any non-parallelism of the vertical axes will affect the readings; and the observing program must be arranged to eliminate this along with the other instru- mental errors. For any given setting the deviation of the axis from parallel- ism,/, unites with the inclination of the lower axes to the true vertical, i' , and determines the value of z, the inclination of the upper axis to the vertical, for the setting in question. For different settings i will be different, for a rotation of the instrument on the lower motion causes the upper axis to describe a cone whose apex angle is 2p and whose axis is inclined to the true vertical by i'. But no matter what the magnitude of i may be, within certain limits easily including allvalues arising in practice, it may be eliminated by forming the mean of direct and reflected readings made in the same position METHOD OF REPETITIONS 83 of the instrument, provided that i is the same in direction and magnitude for both settings. This follows from the discussion on pages 66 and 67 whose result is expressed by equation (88). Hence, if after a series of n repetitions observed C.R. direct, n further repetitions be made C.R. reflected, such that the vernier readings for the corresponding settings in the two series are approximately the same, the instrumental errors i' and /will be eliminated. Equation (88) shows thaty, the deviation of the upper vertical axis from per- pendicularity with the horizontal axis, will also be eliminated. To remove the influence of the collimation, c, the entire process must be repeated C.L.; and to neutralize the systematic error dependent upon the direction of meas- urement, the direct and reflected series should be measured in opposite direc- tions. We thus have the following observing program, in which A' and B' denote the reflected images of A and B, respectively: Level on the lower motion. rSet on A and read the horizontal circle. Direct -j Turn from A to B on the upper motion n times. I Read the horizontal circle for last setting on B. j-C.R. i Set on B' and read the horizontal circle. Reflected < Turn from B' to A' on the upper motion n times. ' Read the horizontal circle for last setting on A'. Repeat for C.L. The circle reading for the first setting on B' must be the same, approximately at least, as that for the last setting on B, The mean of the values of D calculated from the four series is the required azimuth difference of A and B. Uusually one of the objects, say A, will be near the horizon, in which case reflected settings on A' will be impossible. A must then be substituted for A' in the above program. The error due to i will not be eliminated from these settings; but, owing to the presence of the factor cot , it may be disregarded. When the artificial horizon is not used the program must be modified. Were i' zero, /would constantly be equal to/, although the direction of the deflection would change with a rotation of the instrument on the lower motion. If a series of n repetitions C.R. be made under these circumstances,, equation (82) shows that each setting will be affected by an error of the form /cot + /> cos /cot z -f- c cosec,sr . The first and last terms of this expression will have the same values for all' pointings on the same object. Equations (82), (84), and (86) show that they may be eliminated by combining with a similar series made C.L. The values of the second term will be different for each setting owing to the change in /, but their sum will be zero if the values of / are uniformly distributed throughout 360, or any multiple of 360. In order that this may be the case, approxi- mately at least, it is only necessary that n be the integer most nearly equaling, kT)6o /D, where the k is any integer, in practice usually i or 2. It is also easily seen that, if after any arbitrary number of settings the instrument be reversed about the lozver motion and the series repeated in. the 84 PRACTICAL ASTRONOMY reverse order, the sum of the errors involving/ will be zero, provided that the circle readings for corresponding settings C.R. and C.L. are the same, or ap- proximately so. The reversal of the instrument on the lower motion changes the direction of the deflection/ by 180. The values of / for corresponding settings C.R. and C.L. will therefore differ by 180, and the errors will be oppo- site in sign and will cancel when the mean of the two series is formed. The reversal also eliminates the influence of j and c as indicated in the preceding paragraph. The above assumes that the deflection of the lower axis, /', is zero. If this is not the case, each setting will be affected by an additional error of the form i' cos /' cot z , in which /' is constant so long as i' remains unchanged in direction. If i' be the result of a non-adjustment of the plate bubbles, the error which it produces may be eliminated from the mean of two series, one C.R. and one C.L., by relevelling after reversal. (See page 70.) This will change the direction of i' by 180. Consequently, the values of /' for C.R. and C.L. will differ by 180, and the errors for the two positions will neutralize each other when the mean is formed. The consideration of these results leads to the following arrangement of the observing program. Level on the lower motion. Set on A and read horizontal circle. ^ Turn from A to B on upper motion times. \ C.R. Read horizontal circle for last setting on B. Reverse on lower motion and relevel. Set on B and read horizontal circle. \ Turn from B to A on upper motion n times. I C.L. Read horizontal circle for last setting on A. J The circle reading for the first setting on B, C.L. should be the same, approximately at least, as that for the last setting on B, C.R. The mean of the values of D calculated from the two series is the required azimuth difference of A and B. With this arrangement the instrumental errors i',p,j, and c will be com- pletely eliminated, whether the settings are distributed through 360 or not, provided only that the instrumental errors remain constant during the obser- vations. Practically, it is desirable that the value of n should be such that nD equals 360, or a multiple of 360, at least approximately; but when D is small this may unduly prolong the observations. The maximum number of repeti- tions which can be made advantageously depends upon the stability of the instrument and must be determined by experience. If the instrument is provided with a striding level, the influence of z', /, and / may be taken into account by measuring the inclination of the horizontal axis for each setting and applying a correction to R l and R a of the form b cot z , in which b denotes the sum of all the observed inclinations for settings on A and B respectively. When one of the objects, say B, is a star, the time of each setting on B must be noted. The calculated value of D will then correspond sensibly to the mean of the times, provided the observing program be not too long. THE SEXTANT 85 Example 34. On 1909, April 9, the following observations of the difference in azimuth of Polaris and a mark were made by the method of repetitions with a Buff & Buff engineer's transit. The recorded times are those of a Fauth sidereal clock whose error was -}-6 m 36 B . After four repetitions C.R., the instrument was reversed on the lower motion, relevelled, and the series repeated in the reverse order. Since the azimuth difference is approximately 174, 720 must be added to the readings on the star before combining them with those on the mark. The results for the two halves are derived separately, although the means for the set are also given. Hor. Circle Object F Ver. A Ver. B Circle Means Mark 179 59'- 5 359 59'- 5 R 179 59' 30" Polaris 9 h 27 3" 353 32 R Polaris 3 48 R Polaris 33 36 R Polaris 35 49 i 54 13 5 334 13-5 R 874 13 3 4) 7 43 4)694 i4 o 9 3i 56 = 9 h 3S m 338 .. ?-M = 173 33 30 Polaris 9 39 12 154 i3-o 334 13-5 L 874 13 15 Polaris 42 28 L Polaris 44 9 L Polaris 46 20 L Mark 179 51.0 359 5i-o L 179 5i o 4) 12 9 4)694 22 15 9 43 2 = 9* 49 m 3S 8 S" M = 173 35 34 Final Means = 9 44 5 S M = 173 34 32 THE SEXTANT 48. Historical and descriptive. The instruments typified by the engi- neer's transit may be used for the measurement of horizontal or vertical angles only. Simultaneously with the development of the altazimuth principle there was gradually evolved a contrivance adapted for the measurement of angles lying in any plane. Beginning with the astrolabe of the ancients, the applica- tion of various ideas gave in succession the Jacob's staff, or cross-staff, which dates apparently from the middle of the fourteenth century, the back-staff, or Davis quadrant; the sextant of Tycho, which was also used by the Arabs in the tenth century; the octants of Hooke and Fouchy, in which a mirror was used for the first time; and, finally, the reflecting octant whose principle was due to Newton, although the construction was first carried out by John Hadley about 1731. The instrument of Hadley has been improved in design, but no essential modification has been made in its principle. In its modern form it is known as the reflecting sextant, or more generally, simply as the sextant. With the exception of the astrolabe and the large fixed sextants of Tycho, the various forms mentioned are characterized by the fact that they may be held in the hand during observations, small oscillations and variations in the position of the instrument offering no serious difficulty in the execution of the measures. These instruments have therefore played an important part in the 86 PRACTICAL ASTRONOMY practice of navigation, and to-day the sextant is the only instrument which can advantageously be employed in the observations necessary for the deter- mination of a ship's position. In addition, its compactness and lightness, and the precision of the results that may be obtained with it render it one of the most convenient and valuable instruments at our command. The modern sextant consists of a light, flat, metal frame supporting a graduated arc, usually 70 in length; a movable index arm; two small mirrors perpendicular to the plane of the arc; and a small telescope. The index arm is pivoted at the center of the arc and has rigidly attached to it one of the mirrors, the index glass, whose reflecting surface contains the rotation axis of the arm and the attached mirror. The position of the index glass corre- sponding to any setting may be read from the graduated arc by means of a vernier. The second mirror, the horizon glass, is firmly attached to the frame of the sextant in a manner such that when the vernier reads zero the two mirrors are parallel. Only that half of the horizon glass adjacent to the frame is silvered. The telescope, whose line of sight is parallel to the frame, is directed toward the horizon glass, and with it a distant object may be seen through the unsilvered portion. When the frame is brought into coincidence with the plane determined by the object, the eye, of the observer, and any other object, a reflected image of the second object may be seen in the field of the telescope, simultaneously with the first, by giving the index arm a certain definite position depending upon the angular distance separat- ing the objects. If the position of the arm is such that the rays of the second object reflected by the index glass to the horizon glass, and then from the silvered portion of the latter, enter the telescope parallel to the rays that pass from the first object through the unsilvered portion of the horizon glass, the two images will be seen in coincidence. This being the case, the relative incli- nation of the mirrors as shown below, will be one-half the angular distance separating the objects; and, since the construction is such that the inclination may be read from the graduated arc, it becomes possible to find the angular distance between the objects. The use of the instrument is simplified by graduating the arc so that the vernier reading is twice the inclination of the mirrors, and hence, directly, the angular distance of the objects. With the usual form of the instrument the maximum angle that can be measured is therefore about 140. The two mirrors and the telescope are provided with adjusting screws, which may be used to bring them accurately into the posi- tions presupposed by the theory of the instrument. In addition, the tele- scope may be moved perpendicularly back and forth with respect to the frame thus permitting an equalization of the intensity of the direct and reflected images by varying the ratio of the reflected and transmitted light that enters the telescope. Adjustable shade glasses adapt the instrument for observa- tions on the sun. 49. The principle of the sextant. In Fig. 10 let 0V represent the graduated arc; /and//, the index glass and the horizon glass, respectively; and IV, the index arm, pivoted at the center of the arc and provided with a THE SEXTANT 87 vernier at V. When V coincides with 0, the mirrors are parallel. The posi- tion indicated in the figure is such that the two objects S t and S 2 are seen in coincidence, for the rays from 5 r pass through the unsilvered portion of H and enter the telescope in the direction HE, while those from S 2 falling on / are reflected to //"and thence in the direction HE. The two beams therefore enter the telescope parallel. It is to be shown that the inclination of 7 to H is one-half the angular distanced separating the objects. IN and HN are the normals to the mirrors, and by the fundamental laws of reflection they bisect the angles S,IH and IHE, respectively. In the triangle IHE. 2+ y 2 A. But in the triangle IHN a = b + M. Therefore, M=y 2 A. But M, being the angle between the normals to the mirrors, measures their inclination, and is equal to the angle subtended by the arc 0F, whence A = 20V. (121} But since the arc is graduated so that the reading is twice the angle subtended by OF the angular distance between the two objects is given directly by the scale. 50. Conditions fulfilled by the instrument. The following conditions, among others, must be fulfilled by the perfectly adjusted sextant. 88 PRACTICAL ASTRONOMY . 1. The index glass must be perpendicular to the plane of the arc. 2. The horizon glass must be perpendicular to the plane of the arc. 3. The axis of the telescope must be parallel to the plane of the arc. 4. The vernier must read zero when the mirrors are parallel. 5. The center of rotation of th index arm must coincide with the center of the graduated arc. Since the positions of the mirrors and the telescope are liable to derange- ment, methods must be available for adjusting the instrument as perfectly as possible. This is the more important inasmuch as it is impossible to eliminate from the measures the influence of any residual errors in the adjustments. Although elimination is impossible, it should be remarked that the errors arising in connection with Nos. 4 and 5, at least, may be determined by the methods given in Sections 52 and 53, and applied as corrections to the read- ings obtained with the instrument. Conditions Nos. 1-4 are within the control of the observer. No. 5 must be satisfied as perfectly as possible by the manufacturer. 51. Adjustments of the sextant. No. i. Index glass. To test the perpendicularity of the index glass, place the sextant in a horizontal position, unscrew the telescope and stand it on the arc just in front of the surface of the index glass produced. If then the eye be placed close to the mirror, the observer will see the reflected image of the upright telescope alongside the telescope itself. By carefully moving the index arm, the telescope and its image may be brought nearly into coincidence. If the two are parallel, the index glass is in adjustment. The telescope should be rotated about its axis in order to be sure that it is perpendicular to the plane of the arc. If the adjustment is imperfect, correction must be made by the screws at the base of the mirror. Some instruments are not provided with the necessary screws, and in such cases the adjustment had best be entrusted to an instrument maker. The test can also be made by looking into the index glass as before, and noting whether the arc and its reflected image lie in the same place. If not, the position of the mirror must be changed until such is the case. No. 2. The horizon glass. The adjustment of the horizon glass may be tested by directing the telescope toward a distant, sharply defined object, preferably a star, and bringing the index arm near the zero of the scale. Two images of the object will then be seen in the field of view one formed by the rays transmitted by the horizon glass, the other, by those reflected into the telescope by the mirrors. The reflected image should pass through the direct image as the index arm is moved back and forth by the slow motion. If it does not, the horizon glass is not perpendicular to the plane of the arc, and must be adjusted until the direct and reflected images of the same object can be made accurately coincident. No. 3. The telescope. The parallelism of the telescope to the frame may be tested by bringing the images of two objects about 120 apart into coin- cidence at the edge of the field nearest the frame. Then, without changing INDEX CORRECTION 89 the reading, shift the images to the opposite side of the field. If they remain in coincidence, the telescope is in adjustment. If not, its position must be varied by means of the adjusting screws of the supporting collar until the test is satisfactory. No. 4. Index adjustment. If the fourth condition is not fulfilled, an index error will be introduced into the angles read from the scale. To test the adjustment, bring the direct and reflected images of the same distant object into the coincidence as in the adjustment of the horizon glass. The corres- ponding scale reading is called the zero reading = ^ . If R is zero, the adjustment is correct. If not, set the index at o, and bring the images into coincidence by means of the proper adjusting screws attached to the horizon glass. // is better, however, to disregard this adjustment and correct the readings by the amount of the index error. It can be shown that the errors affecting the readings as a result of an imperfect adjustment of the index glass, the horizon glass, and the telescope are of the order of the squares of the residual errors of adjustment. If care be exercised in making the adjustments, the resulting errors will be negligible as compared with the uncertainty in the readings arising from other sources. 52. Determination of the index correction. Make a series of zero readings on a distant, sharply defined object, a star if possible. If the zero of the vernier falls to the right of the zero of the scale, do not use negative readings, but consider the last degree graduation preceding the zero of the scale as 359, and read in the direction of increasing graduations. The zero reading is what the instrument actually reads when it should read zero. The index correction, /,. is the quantity which must be added algebraically to the scale readings to obtain the true reading. We therefore have 7=0 *, (122) /= 3 6o R . (123) The latter expression is to be used for the determination of / when the zero of the vernier falls to the right of that of the scale for coincidence of the direct and reflected images of the same object. When observations are to be made on the sun, the index correction should be determined from measures on this object. Since it is impossible, on account of their size, to bring the solar images accurately into coincidence, we deter- mine the zero reading as follows: Make the two images externally tangent, the reflected being above the direct, and read the vernier. Let R I represent the mean of a series of such readings. Then make an equal number of settings for tangency with the reflected image below. Call the mean of the corres- ponding readings R 2 . The mean of R t and R 2 will then be the value of the zero reading, and we shall have 7=o #(*,+*,), (124) 7=360 j*(* x + /y. (125) 90 PRACTICAL ASTRONOMY The readings thus obtained will also give the value of S, the sun's semi- diameter. Since the center of the reflected image moves over a distance of four semi-diameters in shifting from the first position to the second, we have S=MR-R^. (126) Owing to the brilliancy of the solar image, its diameter appears larger than it really is a phenomenon known as irradiation. Should the value of 5" be required for the reduction of observations on the sun (see Section 55), the value calculated from equation (126) should be used rather than that derived from the Ephemeris, in order that the influence of irradiation may be eliminated. 53. Determination of eccentricity corrections. Any defect in the fifth condition introduces an eccentricity error into the readings. Since, with the usual forjn of the instrument there is but a single vernier, this cannot be eliminated. Each sextant must be investigated specially for the determination of the eccentricity errors affecting the readings for different parts of the scale. These may be found by measuring a series of known angles of different mag- nitudes. The mean result for each angle, A, gives by (71) an equation of the form A = R + I + E -E< (127) where R is the sextant reading for coincidence of the two objects whose angular distance is A; /, the index correction; and E and E, the eccentricity corrections for those graduations of the scale which coincide with vernier graduations for the readings R Q and R, respectively. The readings of the coinciding graduations when the vernier reads R and R may be denoted by R ' and R', respectively. E E is the correction which must be applied to the sextant reading, freed from index correction, in order to obtain the true value of the angle. Denoting its value by e, (127) may be written e=A (R + I). (128) Having determined e from (128) for a considerable number of angles dis- tributed as uniformly as possible over the scale, the results may be plotted as ordinates with the corresponding values of R' as abscissas. From the plot a table may be constructed giving the values of for equidistant values of R', from which the value of e for any other reading, R, can then be derived. Care should be taken always to enter the table with the R' corresponding to the given R as argument. It should be noted that the usefulness of the table depends upon / remaining sensibly constant, for if the index correction changes by any considerable amount, R ' may change sufficiently to render the tabular values of no longer applicable. The chief difficulty in investigating the eccentricity of a sextant consists in securing a suitable series of known angles. A simple method is to measure with a good theodolite the angles between a series of distant objects, nearly MEASUREMENT OF ALTITUDES 91 in the horizon, care being taken to tilt the instrument so that in turning from one object to the next no rotation about the horizontal axis is necessary. 54. Precepts for the use of the sextant. The following points should carefully be noted in using the sextant: Focus the telescope accurately. The image of a star should be a sharply defined point; that of the sun must show the limb clearly defined and free from all blurring. For solar observa- tions, use, whenever possible, shade glasses attached to the eyepiece rather than those in front of the mirrors; and reduce the intensity of the images as much as is consistent with clear definition. If the use of the mirror shade glasses cannot be avoided, select those which will make the direct and reflected images of the same color, and reverse them through 180 at the middle of the observing program to eliminate the effect of any non-parallelism of their surfaces. If a roof is used to protect the surface of the mercury from wind, it also should be reversed at the middle of the program. In all cases make the direct and reflected images of the same intensity by regulating the distance of the telescope from the frame. Make the adjustments in the order in which they are given above, and always test them before beginning obser- vations. The index correction should be determined both before and after each series of settings. Make all coincidences and contacts in the center of the field. Finally, the instrument should be handled with great care, for a slight shock may disturb the adjustment of the mirrors and change the value of the index correction. 55. The measurement of altitudes. Although the sextant may be used for the measurement of angles lying in any plane, it finds its widest application in practical astronomy in the determination of the altitude of a celestial body. At sea the observations are made by bringing the reflected image of the body into contact with the image of the distant horizon seen directly through the unsilvered portion of the horizon glass. To obtain the true reading the plane of the arc must be vertical. Practically, the matter is accomplished by rotating the instrument back and forth slightly about the axis of the tele- scope, which causes the reflected image to oscillate along a circular arc in the field. The index is to be set so that the arc is tangent to the image of the horizon. The corresponding reading corrected for index correction, dip of horizon, and refraction is the required altitude. The correction for dip is necessary, since, owing to the elevation of the observer, the visible horizon lies below the astronomical horizon. The square root of the altitude of the observer above the level of the sea, expressed in feet, will be the numerical value of the correction in minutes of arc. The observations are not suscept- ible of high precision, and the correction for eccentricity may be disregarded as relatively unimportant. For observation's on land the artificial horizon must be used. The meas- urement of the angular distance between the object and its mercury image gives the value of the double altitude of the object. Some practice is required in order to be able to bring the object and its mercury image into 92 coincidence quickly and accurately. In case the object is a star, care must be taken that the images coinciding are really those of the object and its reflec- tion in the mercury. The following is the simplest method of procedure: Stand in a position such that the mercury image is clearly visible in the center of the horizon, and direct the telescope toward the object. By bring- ing the index near zero the reflected image will appear in the field. The telescope is then turned slowly downward toward the mercury, the index being moved forward along the arc at the same time at a rate such that the reflected image of the object remains constantly in the field. If the plane of the sextant is kept vertical, and if the observer is careful to stand so that the mercury reflection can be seen, its image seen directly through the unsilvered portion of the horizon glass will come into the field when, the telescope has been sufficiently lowered. Both images should then be visible. The varying altitude of the object will cause them to change their relative positions. The index is set so that the images are approaching and clamped. When they become coincident the time is noted and the vernier read. The instant of coincidence is best determined by giving the instrument a slight oscillatory motion about the axis of the telescope and noting the time when the reflected image in its motion back and forth across the field passes through the direct image. To obtain an accurate value of the altitude, a series of such settings should be taken in quick succession, the time and the vernier reading being noted for each. It is not necessary to use the method described above for bringing the images into the field for any of the settings but the first; for if, after reading, the index be left clamped and the telescope be directed toward the mercury image, the plane of the arc being held vertical, the reflected image will also be in the field. If it is not at once seen, a slight rotation about the axis of the telescope will bring it into view, unless too long an interval has elapsed. Measures for altitude may also be made by setting the zero of the vernier accurately on one of the scale divisions so that the images are near each other and approaching a coincidence. The time of coincidence and the vernier reading are noted. The index is then moved 20' so that the images will again be approaching coincidence. The time and the reading are noted as before and the process is repeated until a sufficient number of measures has been secured. The consistency of the measures should always be tested, as in the case of the engineer's transit (see page 79) by calculating the rate of change of the readings per minute of time. If however, the observations have been made by noting the times of coincidence for equidistant readings of the vernier, the constancy of the time intervals between the successive settings will be a sufficient test. If R denote the mean of the vernier readings, the apparent double altitude of the object will be given by (129) MEASUREMENT OF ALTITUDES 93 in which /is the index correction, and e the correction for eccentricity. The true altitude corresponding to the mean of the observed times is found from where the refraction, r, may be derived from Table I, page 20, or if more accurate results are required, by equation (3), page 18. If the zenith distance is desired instead of the altitude, we calculate z' from z' = goh', (130) and z from z = z' + r. (131) For measures on the sun coincidences are not observed, but, instead, the instants when the images are externally tangent. To eliminate the influence of semidiameter, the same number of contacts should be observed for both images approaching and images receding. If for any reason this cannot be done, a correction for semidiameter must be applied. Let a = number of settings for images approaching, w r = number of settings for images receding, n = total number of settings, 5 = the semidiameter of the sun calculated by equation (126). We shall then have for solar observations ?/.' p-u^a n r g. . , /Upper sign, altitude decreasing.) ( ^ n ' \ Lower sign, altitude increasing. / in which h' is the apparent altitude of the sun's center corresponding to the mean of the observed times; and the term involving S, the correction for semidiameter. The true altitude and zenith distance are then given by (133) z z' + rp, (134) The solar parallax,/, may be obtained from columns four an-d eight of Table I, page 20. For approximate results r p may be taken from the fifth and tenth columns of this same table. Example 85. On 1909, April 10, the following sextant observations of the altitude of the sun were made at the Laws Observatory near the time of meridian transit. The error of the timepiece was A0 f = + 6 m 37 s . The observations will be reduced later for the determina- tion of latitude. 94 PRACTICAL ASTRONOMY Readings on Sun for Index Correction F Reading Limb /? R* o h 59 H 1 22 s 117 18' 50" Lower 3 1 ' 30" 359 28' 0" I i 28 118 23 10 Upper 3i 40 28 20 2 29 118 25 50 Upper 32 28 o " 4 9 117 24 o Lower 32 10 28 10 5 19 117 24 30 Lower , r 5 359 28 8 6 53 118 28 40 Upper 8 27 118 29 o Upper ro Reading = 359 59 59 12 8 117 24 IO Lower Index Corr. = -j- i /?! J? 2 = i 3 42 Semidiameter = 15 56 CHAPTER V THE DETERMINATION OF LATITUDE 56. Methods. On page 34 it was shown that if the zenith distance or altitude of a star of known right ascension and declination be measured at a known time, the latitude of the place of observation can be determined by means of equation (31). The preceding chapter indicates the methods that may be employed for the measurement of the zenith distance. It is the pur- pose of the present chapter to determine the most advantageous method of using the fundamental equation and to develop the formulae necessary for the practical solution of the problem. To establish a criterion for the use of equation (31), it is to be noted that the resultant error of observation in

cos<5 sin 2 YT. t. (150) To express Z explicitly we may replace the left member of (150) by its expan- sion by Taylor's theorem. Since Z is small the convergence will be rapid. Introducing at the same time A = cos

, both of which are required for the com- putation. Finally, calculate the latitude for each observation by means of (155) or (156). The declination to be used is that corresponding to the instant of observation. The final result may also be obtained by applying the mean of all the values of Am and of Bn to the mean of all the zenith distances in accordance with equations (155) or (156). This method, however, gives no indication as to the consistency of the observations, and it is better to reduce the results separately, or, at least, to reduce separately the means of not more than two or three consecutive measures. The method of circummeridian altitudes may advantageously be com- bined with that of Talcott. When this is done there will be given a series of values of ^ (z\ ' N ) derived from observations made near the meridian. Each of these must be reduced to the meridian by adding to (142) the term y-2 (Z s - Z N ), in which Z s and Z N are to be calculated by (154). Example 37. The reduction of the circummeridian altitudes given in Ex. 35, p. 93, is as follows: To eliminate the semidiameter the means are formed for the ist and 2nd, 3rd and 4th, 5th and 6th, and the 7th and 8th observations. These results are in the first and sixth lines of the calculation below. The eccentricity corrections are unknown. The index cor- rection found in Ex. 35 is -f- i". In Ex. 36, p. 96, the clock time of transit was found to be i h 8 m 5 s . / (sidereal) t (solar) m jh o m 2f* 7 40 7 39 115" 4 4 ' 19" 4 6 45 44" jh fcm I I 6 59 59 8" I h lO m + 2 + 2 18" 13 10" n o R ii7 c ' Si' o" "7 54' 55" 117 56' 35" II7 56' 35" h' 58 55 30 58 57 28 58 58 18 58 58 18 z' 3i 4 30 31 2 3i 3i i 42 31 I 42 r-p +30 +30 +30 -I -3 3 +7 54 22 +7 54 25 +7 54 27 + 7 54 3i Am 2 52 i 6 12 15 9 38 56 30 38 56 20 38 56 27 38 56 28 From the 3rd column COS > 9- 8908 P 38 57' COS(J 9- 9959 3 +7 54 cosec z^ > - 2877 z o 3 1 2 log A o. 1744 The mean of the four values of

N] I 9.8034 The calculated

38 57.5 Ant. 104 PRACTICAL ASTRONOMY The C.S.T. is converted into the corresponding Columbia by (41) and (58). a and 8 are from p. 321 from the Ephemeris. The value of t shows that Polaris was east of the mer- idian at the time of the observation, whence cos A and sin (y N) are negative. Since cos (a> N} is positive,

+ sin TT cos

2// 2 , COS7T=I - ^7T 2 , with errors which are still smaller, thus obtaining H 7rcos/-i-^tan^(7r 2 //'). (165) Neglecting terms involving ;r 3 , H* 7T Z COS 2 /, and substituting H' into (165) we have H TT cos / + J^TT 2 tan y? sin 2 /. (166) Finally, by (163) LATITUDE FROM POLARIS 105 in which

is larger ^ 323 34-0 than the true value byo'. I. The application of equations (169) to the data of Ex. 38 gives & = 38 57'.5, which agrees exactly with the result obtained by the formulae of Section 64. Example 4O. Find the latitude by means of Table IV of the Ephemeris from the data of Exs. 38 and 39. The hour angle is to be calculated as before. Its value in both cases is greater than i2 h . Consequently, H is to be interpolated from Table IV, Ephemeris^. 595, with 2 produced by a small change in 6 is by (136) dy = tan A cos with a given degree of precision. If dd is expressed in seconds of time, the factor 15 must be introduced into the right members of the various equations in order that dtp may be expressed in seconds of arc. It is evident that, aside from the dependence of d

1.1849 sin t 8.5334 = i5" log 300 2.4771 Example 43. How accurately must the time be known in order that the altitude of Polaris given in Ex. 33, p. Si, may yield a value of the latitude uncertain by not more than o'. i ? By (173) and the data in Ex. 39, p. 105, we find t 3 2 3 34'- o d y o'.i sin/ -595 0.02 sin/ 0.012 de = 8'. 33= 33 s Atts. CHAPTER VI THE DETERMINATION OF AZIMUTH 68. Methods. The azimuth of a terrestrial mark may be found by ob- serving the difference in azimuth of. the mark and a celestial object and applying to this difference the calculated azimuth of the object corresponding to the in- stant of observation. The methods to be employed for the observational part of the process have been discussed in detail in Chapter IV. We have now to examine the means by which the azimuth of the celestial body may be com- puted. A rigorous and general method of procedure leading to the fundamental equation cosd sin / tan A= = 5 . (174) sin o cos (p cos o sin (p cos / was outlined on page 34. Before proceeding to the adaptation of this equation to the purposes of calculation it is desirable to investigate the conditions under which it may most advantageously be employed. The calculated azimuth will depend upon the right ascension and declination of the star, the time, and the latitude of the place of observation. The first two quantities may be assumed to be known with precision, but the last are likely to be affected by relatively large uncertainties. To determine the influence of these upon the calculated azimuth, and thus derive a precept for the choice of objects to be observed, we differen- tiate (33), A, z, and t being considered variable, and substitute for dz its value from page 95. Writing at the same time dt =. dd we find after simplification. dA = cot z sin A dtp -f- (sin z sin (p + cos z cos ip cos A) cosec z dd. If in Fig. 6 we denote the angle at O by q, the expression in parenthesis re- duces by the second of the fundamental formulae of spherical trigonometry to cos d cos ^, whence dA = cot 2 s\nAd

will have the least influence when t and q are as near 90 or 270 as possible. These conditions cannot both be fulfilled at the same time. But for circumpolar stars observed near elonga- tion the magnitude of cosg and cott in (176) will be such that errors in z and tp will have only an insignificant influence on the calculated azimuth. The consideration of the preceding results indicates that we shall need adaptations of the fundamental azimuth equations designed for the calculation of 1. The azimuth of the sun. 2. The azimuth of a circumpolar star at any hour angle. 3. Azimuth from an observed zenith distance. I. AZIMUTH OF THE SUN 69. Theory. The first four equations of (34) are the equivalent of (32) and (33) from which the fundamental equation (174) was derived. By their combination we find the following group which for the purposes of calculation replaces (174). tan d ., tanN= - , cos t cos N ta.nA= -. -- =77 tan /. (178) sin (

sin / tanA = - , (179) i tan TT tan

, k = tan TT tan

must be used, and that the value of the coefficient TT appearing in (183) must correspond to the date of observation. In case a number of azimuth determinations are to be made at a given station, the corresponding local value of log sec

directly from the table. The values of log G in Table IX are based upon y> = 40, and it = i 10'. the latter of which is the mean north polar distance for 1910.0. The maximum absolute errors in the azimuth resulting from the use of this table for various latitudes are Latitude 30 35 40 45 50 Error in A o'.24 o'. 12 o'.oo o'. 15 o'.38 The values of log G sec in the third column of Table IX refer to the lati- tude of the Laws Observatory, which is 38 57'. 72. Procedure. Interpolate a and 3 for the instant of observation from the list of apparent places of circumpolar stars, Ephemeris, pp. 312-323. Cal- culate 71 = 90 , / = a, where 6 is the true sidereal time of observation for which the azimuth difference of the star and the mark has been measured. Then : For a precise azimuth, calculate A from (180). The value of logG may be taken from Kept. Supt. U. S. Coast and Geodetic Survey, 1897-8, pp. 399-407, or from some similar table, with the argument log h cos t; or it may be calculated by means of (182). For an approximate azimuth from Polaris, interpolate log G, or log G sec ^>, from table IX with t as argument. Then calculate A from A i8o TrCsec^sinA (184) If TT be expressed in minutes of arc, the last term of (184) will also be given in minutes of arc. Example 43. Determine the azimuth of the mark from the data given in Ex. 34, p. 85. The latitude of the place of observation is 38 56' 52". Equations (180) are used for the calculation, the results for the two positions of the instrument being reduced separately. The azimuth of the mark is found by subtracting the difference S M, taken from p. 85, from the calculated azimuth of Polaris. The difference of the two values of M is not to be taken as an indication of the precision of the result, as these quantities are affected by instrumental errors whose influence is not eliminated until the mean is formed. 113 a i 7T 1 < 38 log// 25 m 19* 10' 46" S^ 52 0.10918 8.31362 9.90756 8.42280 8.22118 C. R. C. L. 9 h 38 32 s 9 h 49 m 38 8 / 8 13 13 8 24 19 t 123 18' 15" 126 4 ' 45" cost 9-739 6 4n 9-77005 n h cos t = A. 7.96082,1 7 ^IZSn B = log G 9.99605 9-99577 sin / 9.92209 9.90752 tan A 8. 34094 8.32609,, A 178 44 ' 38" 178 47' 10' S M 173 33 30 173 35 34 M 5 ii 8 5 ii 3 6 Mean 5 ii' 22" 3- AZIMUTH FROM AN OBSERVED ZENITH DISTANCE 73. Theory. Equation (26) rewritten in the form cos A = sin S cos z sin sn z costp A . Substituting these results into (187) and (188) and writing cosd = x, we find dA= Trsec ^> c dA= sec^pcot/dfe. ( 1 9 2 ) The first of these can also be derived from (184) by differentiating and writing G= I. Example 45. The altitude of the sun and the difference of its azimuth and that of a mark were measured with an engineer's transit at the Laws Observatory on 1909, April 27. The results were 7\v = 4 h i m n?o, P.M., J7\v i m 44?5 (referred to C.S.T.), h' = 33 19'. 6, 6" j1/=8i 2417. Find the azimuth of the mark, calculating the azimuth of the sun both by method i and method 3. The computation of the solar azimuth by (177) and (178) is in the first column; that for (186), in the second. In the latter instance the time is required only with such precision as as may be necessary for the interpolation of declination from the Ephemeris for the instant of observation. C.S.T. 3 h 59 2655 h> 33 19-6 Col. M.S. T. 3 50 8.2 r p 1.3 E 2 24.7 a = z 56 41.7 ; = Col. A.S.T. 3 52 32.9 b -n: 76 8.9 t 58 8'. 2 c = go (p 51 3.1 d +13 5 1 -* s 9i 5 6 -9 tan (5 9.39196 sin (s c) 9.81604 cos t 9.72255 sin (s a) 9.76132 tan TV 9.66941 cosec s 0.00025 N 25 2'. 2 cosec (s ) 0.56498 (p 38 56.9 co\.%A 0.07130 tp N 13 54.7 A 80 38:2 Ck. cosyV 9-957I5 -S M 81 24.7 tan t 0.20651 M 359 13.5 cosec ((p TV) 0.61902 tan A 0.78268 A 80 38'.! CHAPTER VII THE DETERMINATION OF TIME 77. Methods. The determination of time means, practically, finding the error of a timepiece. .To accomplish this the true time 6 or T is calculated from observations on a star or the sun and compared with the clock time at which the observations were made. The required error is given by M=e-e\ (193) or jr=r-r, (i 94 ) according as the timepiece is sidereal or mean solar, 6' and T' being the clock values of the time of observation. The fundamental equation for the determination of time is 6 = a + t. (195) Applied to any celestial object this equation gives the sidereal time, from which the mean solar or apparent solar time may be derived by the transform- ation processes of Chapter III. For the sun, however, the hour angle / is directly the apparent solar time, and, in case of observations on this object, the mean solar time may be found from (42) written in the form T=t + E. (196) When the timepiece is solar the use of (196) is simpler than that of (195). Since a and E may be regarded as known, the problem is reduced to the determination of the hour angle of the object for the instant of observation. As indicated on page 34 this may be accomplished by measuring the zenith distance of the object at a place of known latitude and using equation (38) or (39)- The problem can also be solved by determining the clock time 6 ' of the instant for which the hour angle of the object is zero. For this case the fundamental equation reduces to (197) and 4d = a 6 '. (198) In outlining the methods that may be employed for the determination of 6 ' it will be assumed that the object is a star and that the timepiece used is sidereal. The modifications necessary for the removal of these limitations will be considered in connection with the discussion of the details presented in the following sections. 116 METHODS 117 To determine ' we may note the time 0, when a star has a certain zenith distance, or altitude, east of the meridian, and, again, the time 2 when it has the same zenith distance west of the meridian. Since the celestial sphere rotates uniformly, we shall have 0.'=X(0>+0,). (199) The method is known as that of equal altitudes. The clock time of meridian transit, ', may also be determined by noting the instant of passage of an object across the vertical thread of a transit instrument mounted so that the line of sight of the telescope lies in the plane of the meridian. This is the meridian method of time determination. Finally, d ' may be found by observing the transit of an object across the vertical thread of an instrument nearly in the plane of the meridian. The application of a small correction to the observed time depending upon the displacement of the instrument from the meridian gives the clock time for which ^ = o. In practice the deviation of the instrument is such that the line of sight lies in the plane of the vertical circle passing through Polaris at a definite instant. The process is accordingly known as the Polaris vertical circle method of time determination. It is of special interest on account of the fact that it is readily adapted to a simultaneous determination of time and azimuth. There are other methods of determining the true time, but those outlined afford a sufficient variety to meet the conditions arising in practice. We there- fore proceed to a detailed consideration of 1. The zenith distance method. 2. The method of equal zenith distances or altitudes. 3. The meridian method. 4. The Polaris vertical circle method. I. THE ZENITH DISTANCE METHOD 78. Theory. The formulae necessary for the calculation of / from d, cosec t tan 3 cot /) dd, (203) in which /is one-half the interval between the two observations expressed in solar units, d the declination for apparent noon, and dd the change in d during the interval t. Both the observed times will be too late by the quantity dt. Hence, for solar observations made with a sidereal timepiece, AQ = a #(0 f + e m ) + dt. (204) If the timepiece is solar, we have from (196) and (202), since / = for the instant of meridian transit, AT=E- % (T, + T.) + dt. (205) It is sometimes convenient to combine afternoon observations with others made on the following morning. In this case the mean of the observed times corrected for the change in declination is the clock time of lower culmination. The quantity t in (203) is one-half the interval between the observations expressed in solar units as before; but d must be interpolated from the Ephemeris for the instant of the sun's lower transit, and the resulting value of dt must be added to the clock times of observation. The expressions for the clock correction are J0 = i2 h + #(0, + 6,} dt, (206) AT= 12" + E-% (T, + T,) -dt, (207) in which the values of a and E refer to the instant of lower culmination. 81. Procedure. The object observed should be near the prime vertical. When three or four hours east of the meridian note the time of transit across the horizontal thread of the transit for a definite reading of the vertical circle, most conveniently an exact degree or half degree. Change the reading by 10' or 20' and note the time of transit as before. Repeat a number of times, always changing the reading by the same amount. After meridian passage observe the times of transit over the horizontal thread for the same readings of the vertical circle as before, but in the reverse order. If the sextant is used, note the times of contact of the direct and reflected images for the same series of equidistant readings of the vernier before and after meridian passage. Denote the means of the two series of times by 0, and a , or T f and T,, according as the timepiece is sidereal or mean solar. For a star the error of the clock will be given by (201) or (202). For the sun, calculate dt by (203), and the clock error by (204) or (205) in case the observations are made in the morning and after- 120 PRACTICAL ASTRONOMY noon of the same day, or by (206) and (207) when they are secured in the afternoon and on the following morning. Care must be taken not to disturb the instrumental adjustments between the two sets of measures. If these remain unchanged no correction need be applied for index error, eccentricity, refraction, parallax or semidiameter. This fact taken in connection with the simplicity of the reductions constitutes the chief advantage of the method. It is subject, however, to the serious objection that an interval of several hours must elapse before the observing program can be completed, with the danger that clouds may interfere with the second series of measures. When the engineer's transit is used for the observations, all the measures should be made in the same position of the verticle circle, and the angles should all be set from the same vernier. As in the case of the zenith distance method of time determination, an approximate knowledge of the time is necessary when the object observed is the sun. If the clock correction is quite unknown, this may be derived from the observations themselves as before. It is only necessary to interpolate the sun's right ascension, or the equation of time, as may be required, on the assumption that the clock error is zero. This approximate result will lead to an approximation for the error of the timepiece with which the calculation may be repeated for the determination of the final value. 3. THE MERIDIAN METHOD 82. Theory. The meridian method of time determination requires a transit instrument mounted so that, when perfectly adjusted, the line of sight lies constantly in the plane of the meridian, whatever the elevation of the telescope. In order that this may be the case, the horizontal axis must coincide with the intersection of the planes of the prime vertical and the horizon, and the line of sight must be perpendicular to the horizontal axis. The instant of a star's transit across the vertical thread will then be the same as that of its meridian passage. Denoting the clock time of this instant by 6 ' the error of the timepiece will be given by J0 = _ '. (208) In general, however, the conditions of perfect adjustment will not be satisfied. The horizontal axis will not lie exactly in the plane of the prime vertical, nor will it be truly horizontal. When produced it will cut the celestial sphere in a point A, Fig. 8, page 65, whose azimuth referred to the east point and whose altitude we may denote by a and <, respectively. Further, the line of sight will not be exactly perpendicular to the horizontal axis, but will form with it an angle 90 -\-c. The quantities a, , and c are known as the azimuth, level, and collimation constants, respectively. In general, therefore, the star will not be on the meridian at the instant of its transit across the vertical thread, but will have a small hour angle t whose value will depend upon the magnitude of the instrumental constants a, b, and c and the position of the star. To obtain the clock time of meridian transit we must subtract t from the clock time of observation, 0', whence THE MERIDIAN METHOD 121 d ' = d' t, (20g) and by (208) Jd = a 6' + t. (210) The values of a, b, and c can always be found. Consequently Ad can be determined by (210) when t has been expressed as a function of the instru- mental constants. For this purpose we make use of equations (82), (89), and (33). The last two terms of (82) express the influence of the level and colli- mation constants, b and c, upon the reading of the horizontal circle of the engi- neer's transit for C. R., or, what amounts to the same thing, the amount by which the azimuth difference of the point A and the object O, when on the vertical thread, exceeds 90. The last two terms of (89) express the corres- ponding quantity for C. L. These results may be applied directly to the meridian transit to determine the azimuth of the star at the instant of its transit across the vertical thread. For, denoting this azimuth by A s , and assuming that a, the azimuth of the point A referred to the east point, is measured positive toward the south, we have at once A s = a-\-& cot zc cosecz, (211) in which the upper sign refers to C. R., and the lower to C. L. In the present case, however, the positions of the instrument are less ambiguously expressed by circle west (C. W.) and circle east (C. E.), respectively. We may now use (33) to determine the hour angle of the star when its azimuth is equal to A s . Replacing A in (33) by A s and writing A s and / instead of their sines, which we may do since both are very small angles, we find t cos d = A s sin z (2 12) whence by (211) /cos d = a sin z -f-^cos^ c. ( 2I 3) Equations (211) and (212) become indeterminate for = o, on account of the presence of A, but the conditions of the problem show that there can be no such discontinuity in the expression which gives t as a function of #, , and c. Equation (213) is therefore valid for 2 = 0, and becomes inapplicable only for stars very near the pole. Since the star is near the meridian at the instant of observation, z in (213) may be replaced by the meridian zenith distance d. Writing A = sin (

L'E -p t^ w which determines the collimation constant. Should there be more than one pair of stars of equal declination, (217) may be applied to each. The mean of the resulting values of c will then be accepted as the final value. Next, consider two stars observed in the same position of the instrument whose declinations differ as widely as possible. One of these should be a northern star, a circumpolar preferably, the other, a southern star. Writing Ad" = A6'cC (218) we find for these objects from (215) THE MERIDIAN METHOD 123 Ad = Ad" s + aA & whence Inasmuch as there is danger of a change in the azimuth constant during the reversal, a should be determined by (219) for both positions of the instrument. The value of Ad is then to be calculated by Ad = Ad"-{-aA. (220) The mean of all such values is the final value of the clock correction. The chief advantage of the meridian method of time determination is to be found in the fact that the results do not depend upon a reading of the circles. Since the uncertainty of an observed transit is considerably less than that of an angle measured with a graduated circle, the precision is relatively high. It is the standard method of determining time in observatories. When carried out with a large and stable instrument mounted permanently in the plane of the meridian, with the inclusion of certain refinements not con- sidered in the preceding sections, it affords results not surpassed by those of any other method, either in precision or in the amount of labor involved in the reductions. 83. Procedure. To place the instrument in the meridian we may make use of a distant object of known azimuth. Set off the value of the azimuth on the horizontal circle and bring the object on the vertical thread by rotating on the lower motion. Having clamped the lower motion, rotate on the upper motion until the reading is zero. The line of sight will then be approximately in the plane of the meridian. In case no object of known azimuth is available, Polaris may be used in- stead. In this case the star is brought on the vertical thread at an instant for which its azimuth has previously been calculated by (184). With the ex- ception that the setting must be made at a definite instant, the procedure is the same as that for a distant terrestrial object. The determination of the azimuth of Polaris requires a knowledge of the approximate time, but (191) shows that if 6 be known within two or three minutes, the azimuth will not be in error by more than one or two minutes of arc, which is sufficiently accurate. In case the clock correction is entirely unknown, an approximation may be derived as follows: Set on Polaris and clamp in azimuth. Then rotate the telescope on the horizontal axis and observe the transit across the vertical thread of a southern star of small zenith distance. Denoting the sidereal clock time of transit by 6', the approximate error of the timepiece will be given by A6 = a 6'. (221) 124 PRACTICAL ASTRONOMY Since the azimuth of Polaris differs but little from 180, the line of sight will not deviate greatly from the plane of the meridian, especially when directed toward points near the zenith. If the zenith distance of the time star is not more than 25 or 30 the error in 6 will not, ordinarily, exceed two or three minutes, and this, as stated above, is sufficient for the calculation of the azimuth of Polaris with the precision necessa'ry for the orientation of the instrument. The program will include the observation of four or five stars in each posi- tion of the instrument, reversal being made at the middle of the series. Each group should contain one northern star to be used for the determination of the azimuth constant. The remaining objects should be southern stars culminating preferably between the zenith and the equator. In order that there may be sufficient data for the determination of the collimation constant, care should be taken to observe at least one pair of stars, one C. W., the other, C. E., whose declinations are equal or nearly so. For an instrument whose vertical circle reads altitudes, the settings which will give the telescope the proper elevation to bring the stars into the field at the time of culmination are to be calculated by. Setting = 90 (p <5), (222) in which the upper sign refers to northern stars. The star list with the setting for each object should be prepared in ad- vance. This having been done, the instrument is to be levelled and adjusted in azimuth. Three or four minutes before the transit of the first star, which will occur at the clock time a J0, set the vertical circle at the proper read- ing, and as the star comes into the field adjust in altitude until it moves along the horizontal thread. Note the time of its transit across the vertical thread to the nearest tenth of a second. After one-half the stars have been observed in this manner, reverse the instrument about the vertical axis through 180 and proceed with the observation of the remaining stars. Observations with the striding level for the determination of b should be made at frequent intervals throughout the observing program. Level read- ings increasing toward the east should be recorded as positive; toward the west, as negative. If a striding level is not available, the plate levels, especially the transverse level, should be very carefully adjusted before beginning the observations and the bubbles should be kept centered during the measures. The reduction is begun by collecting the right ascension, the declination, and the transit factors for each star. The coordinates are to be interpolated for the instant of observation from the list of apparent places in the Ephemeris. The transit factors may be computed by (214), or better still, they may be in- terpolated from the transit factor tables. (See page 122.) If the inclination of the horizontal axis has been measured, the values of b are to be computed by (113). The value of Ad' is then to be calculated for each star by (216). Then select two stars of equal or nearly equal declination, one observed C. W., the other C. E., and calculate c by (217). Compute as many such values of c as there are pairs of stars of equal declination, and form the mean of all. With the mean value of c calculate Jd" for each star by (218). Then determine a for THE MERIDIAN METHOD 125 each position of the instrument by (219), using for this purpose the stars of extreme northern and southern declination. Finally calculate Aft for each object by (220). The mean of all such values of Jd is the final value of the clock correction corresponding to the mean of the observed clock times of transit. In case the rate of the timepiece is large, each observed 6' should be cor- rected for rate before forming the values of M\ the corrections being applied in such a way that each 6' becomes what it would have been had all the observa- tions been made at the same instant. The epoch to which the values of 6' are reduced is usually the exact hour or half-hour nearest the middle of the series. Example 47. On 1909, May 19, Wed. P. M., the error of the Fauth sidereal clock of the Laws Observatory was determined by the meridian method, the instrument used being a Buff & Buff engineer's transit. The error of the clock was known to be approximately -f- 7 m o 8 . The azimuth of Polaris ca'culated by (184) for the clock time n h 5i m o 8 was 179 26'$. Vernier A of the horizontal circle was set at this value, and at the clock time indicated Polaris was brought to the inter- section of the threads by means of the lower motion. After clamping, the upper motion was released and vernier A was made to read o. The instrument having thus been placed in the meridian, the transits of four stars were observed. The reversal was then made by changing the reading of vernier A from o to 180, after which four more stars were observed. The plate levels were carefully adjusted at the beginning, and the bubbles were kept centered throughout the observations. The first of the tables gives the observing program and the data of observation. The various columns contain, respectively, the number, name, magnitude or brightness, and the apparent right ascension and declination of the stars; the setting of the vertical circle, the the observed clock time of transit, and the position of the circle. The settings were obtained by adding the colatitude 51 3' to the values of the declination. For northern stars this sum must be subtracted from 180. The second table contains the reduction and the value of the clock correction derived from each star. The values of J#' are obtained by subtracting each 0' from the correspond- ing a in accordance with (216). The third and fourth columns contain the values of the transit factors interpolated from the tables of the Laws Observatory. None of the pairs of stars observed are suitable for the determination of the collimation by (217). To avoid this difficulty, approximate values of the azimuth constant are derived by (219) from stars i and 4, and 6 and 8, J#" being replaced by J&' for this purpose. The results are = -)- 252 and rt e = + 454. These values are uncertain owing to the fact that the influence of the collimation has been neglected in deriving them, but they are sufficiently accurate for a determination of c by (2i6a), provided we use for this calculation stars whose declinations differ as little as pos- sible. Substituting the numerical values of , A, and C into (2i6a) for stars 3 and 5, and 2 and 8 we find A9 = + 7 m 4?4 + i -04^ J# = + 7 m 4?4 + i -o.<5c j# = + 7 4.6 i.ooc j0 = -f 7 5.0 i.ooc These two sets of equations give for c, -\- 0510 and -)- - 2 7> respectively. The mean, -f-fi9, is accepted as the value of the collimation constant. Multiplying this by the value of C for each star gives the corrections for collimation contained in the fifth column. The combina- tion of these with the value of J$' gives the quantities in the column headed J#". It should be noted that the algebraic sign of the collimation correction changes with the reversal of the instrument. The azimuth constant is now redetermined for each position of the circle, using for this purpose the value of J#" for stars i and 4, and 6 and 8. The results are w = + 2138 and e =r-^4;i6. From these we find the values of the azimuth corrections aA, which, added to the values of J#" in accordance with (220), give J#, the clock correction for each star. 126 PRACTICAL ASTRONOMY The last column contains the weight assigned to each result in forming the mean value of the clock correction. The mean J0 for the southern stars is the same for each position of the instrument, which shows that the influence of the collimation has been satisfactorily eliminated. It should be noted, as a control upon the calculation of the azimuth constant, that the values of AO for each pair of azimuth stars must agree within one unit of the last place of decimals. In the present case the.agreement is exact for both pairs. No. Star Mag. a d Setting r Circle i e Corvi 3-2 I2 h 5x127:3 22 7' 28 56' S n h 58 m 258 W 2 Y Corvi 2.7 II 8.2 17 2 34 i S 12 4 5-7 W 3 <5* Corvi 3-1 25 10. i 16 i 35 2 S 18 7.6 W 4 x Draconis 3-8 29 39.0 + 70 17 58 40 N 22 32.0 W 5 r- Virginis 2.9 37 3-9 o 57 50 6 S 3 2- 1 E 6 32* Camelop. 5-2 48 36.1 + 83 54 45 3 N 4 I 2 E 7 e Virginis 3-i 12 57 39-8 + n 27 62 30 S 50 37-i E 8 Virginis 4.6 13 5 15-2 5 3 46 o S 58 13-3 E No. AO' A C cC JT aA AO Wt. i + 7i!5 + o-94 -|- i. 08 + 052 + 7 m i! 7 + 252 + 7 m 3 . 9 i 2 25 + 0.87 1.05 + 0.2 2.7 + 2.1 4-8 i 3 2-5 + 0.85 i .04 + O.2 2.7 + 2.0 4-7 i 4 7.0 - 1-54 2.96 + 0.6 7.6 3-7 3-9 5 1.8 -(-0.64 I .00 O.2 1.6 + 2.7 4-3 i 6 34-i -6.65 9.41 1.8 32-3 27.7 4-6 o 7 2.7 + 0-47 i .02 0.2 2-5 + 2.0 4-5 i 8 1.9 + 0.70 + I .00 O.2 1-7 + 2.9 4.6 i At 0'= 4. THE POLARIS VERTICAL CIRCLE METHOD SIMULTANEOUS DETERMINATION OF TIME AND AZIMUTH 84. Theory. In the method now to be discussed the transits of stars are observed across the vertical circle passing through Polaris, the instrument being adjusted with reference to the plane of this circle by bringing Polaris on the vertical thread immediately before each transit. Since the azimuth of Polaris is always a small angle, that of each time star at the instant of its observation will also be small. The conditions do not therefore differ essen- tially from those in the meridian method, and the clock correction may be calculated by (215) as before. The only question to be considered is whether the approximations introduced in deriving this equation are justifiable in view of the fact that the value of a in the vertical circle method may amount to i or 2, while with the meridian method it need not exceed i' or 2'. It can be shown that, when it is a question of hundredths of a second of time in the THE VERTICAL CIRCLE METHOD 127 final result, (215) is insufficient; but for those cases in which an uncertainty of one or two tenths of a second is permissible, the approximation is ample. In the meridian method both a and c are determined from the observations themselves. Here we determine c as before, but a is to be calculated from the known position of Polaris* The azimuth constant will nearly equal the azimuth of Polaris measured from the north point positive toward the east at the instant of setting, but not exactly, owing to the presence of the instru- mental constants b and c. If a represent the azimuth of Polaris defined as above, we have by (82) and (89) a = a + b cot z c cosec z , z being the zenith distance of Polaris. Since b and c are very small, z may be replaced by 90 ^, whence tany? ^rsec sin 4i (226) which may be used for the calculation of a . This leaves in (225) only two unknowns, J# and c, and the observation of any two time stars therefore affords the data necessary for a complete solution of the problem. For the sake of precision one of these should be observed C. W., the other, C. E. To determine c write Jd' = a d' + a A + bB'. (227) We then find from (225) jo=je' w +cCv, 40 = 40', cC n whence Jtf',-J0' c r' \ r> ( 22 ) I* E I W W There is here no necessity for an equality in declination of the two stars as in the case of the meridian method, for the influence of the azimuth is in this 128 PRACTICAL ASTRONOMY case included in J0'. Having found c from (228) we calculate Ad from (225) written in the form M=A6'cC. (229) The factor A in (227) is the s^me as that in (215), but it must be more accurately known than in the meridian method, on account of the magnitude of a . The quantities B' and C are easily reduced by (214) to B' = sec y, C = E -\- tan y, (230) in which E = secd tan o. (23 1 ) The values of E may be taken from Table X with d as argument, whence C may be found by the simple addition of tan^>. For any given latitude C' itself may be tabulated with d as argument. The third column of Table X contains such a series of values for the latitude of the Laws Observatory, viz., 38 57'. The vertical circle method is easily adapted to a simultaneous determi- nation of time and azimuth. If the horizontal circle be read at the instant of setting on Polaris, and if in addition, readings be taken on a mark, the azimuth of the mark will be given at once; for the azimuth of Polaris is calculated in the course of the reduction of the observations for time, and the horizontal circle readings give the azimuth difference of the star and the mark. Since a is measured from the north point positive toward the east, the azimuth of the mark measured in the conventional manner will be A m = MS-\-a 1 80 (232) in which 5 and M are the means of the horizontal circle readings on the star and the mark, respectively; and a , the mean of the calculated azimuths of Polaris. The vertical circle method of time determination, like that of the meridian method, is not dependent upon the reading of graduated circles, and in conse- quence, yields results of a relatively high degree of precision. It possesses the further advantage that no preliminary adjustment in the plane of the meridian is necessary. It is especially valuable for use with unstable instru- ments, for the constancy of the quantities a, b, and c is assumed for only a very short interval, much less than in the meridian method. It is necessary that the azimuth and level constants remain unchanged only during the interval separating the setting on Polaris and the transit of the time star immediately following, and this need not exceed two or three minutes. More- over, each set of two time stars is complete in itself and gives a complete determination of the error of the timepiece. The instrument used should be carefully constructed, however, for any irregularity in the form of the pivots is likely to produce serious errors in the results. THE VERTICAL CIRCLE METHOD 129 85. Procedure. The instrument is carefully levelled, and three or four minutes before the transit of a southern star across the vertical circle through Polaris, the telescope is turned to the north, and Polaris itself is brought to the intersection of the vertical and horizontal threads. The instrument is clamped in azimuth and the sidereal time of setting, , is noted. The tele- scope is then rotated about the horizontal axis until its position is such that the southern or time star will pass through the field of view. The transit of the time star is observed, and the entire process is then repeated for a second time star, with the instrument in the reversed position. The data thus obtained constitute a set and permit a determination of the error of the time- piece. If a simultaneous determination of time and azimuth is required, the program for a set will be Set on the mark and read the H. C. Set on Polaris, note the time, and read the H. C. \ C. W. Observe the transit of the time star. Set on Polaris, note the time, and read the H. C. 1 Observe the transit of the time star. L C. E. Set on the mark and read the H. C. in which C. VV. and C. E. are to be interpreted as meaning that if the instru- ment be turned from the mark to the north by rotating about the vertical axis, the vertical circle will then be west or east, respectively. The plate levels should be carefully watched, and if there is any evidence of creeping, the in- strument should be relevelled. The observing list with the settings for the time stars should be prepared in advance. It is also desirable, in order to save time in observing and to avoid errors in the identification of the stars, to calculate in advance the approxim- ate times of transit. Disregarding the errors in level and collimation we have from (225) 0' = -f d)secd, C' = tan(/> -\- E, Ad' = a d' J ra A+bszc

to O h 0.0075 o. 1167 24 h I 0.0073 0.1165 23 2 0.0065 0.1156 22 3 0.0053 0.1145 21 4 0.0037 o. 1129 2O 5 0.0019 O. IIII 19 6 o oooo 0.1092 18 7 9.9981 o. 1072 17 8 9.9963 0.1055 16 9 9.9948 o 1039 15 10 9.9936 o. 1028 H ii 9.9928 O. IO2O 13 12 9.9926 o. 1018 12 TABLE X d E C' + 30 0.58 i-39 + 25 0.64 i-45 -f- 20 0.70 '5i + 15 0.77 1.58 + 10 0.84 1.65 + 5 0.92 1.72 o I .00 i. Si 5 1.09 i .90 10 1.19 2.00 15 1.30 2. II 20 i-43 2.24 25 i-57 2.38 3 i-73 2-54 Example 48. On 1909, May 19, immediately after securing the meridian observations given in Ex. 47, a simultaneous determination of time and azimuth was made by the Polaris vertical circle method, the instrument used being the same as that employed for the meridian observations. The stars observed were Object Mag. R. A. Polaris 2.2 i h 25 a Virginis i . i 13 20 Virginis 3.6 13 30 Dec. Setting 34; + 88 49' 3" 24.9 IO 41 40 22 4-4 o 8 50 55 During the observations the vertical circle of Polaris was so nearly in coincidence with the meridian that no special calculation of the instant of transit of the time stars across this circle was necessary. The record of the observations is as follows: Horizontal Circle Object Mark Polaris a Virginis Polaris Virginis Mark Fauth Clk. Ver. A 7 46'.o 179 56-5 Ver. B ( 187 46 '.o 359 56.5 Circle W W W E E E 13 10 13 16 22 20 10.5 2 57-2 359 187 58.5 45-5 179 58 5 7 45 -5 We have n 7o'.95, log n = 1.8510. For the calculation of the azimuth of Polaris we use the approximate clock correction A0 = + 7 m o", whence a d0 = I h i8 m 34 . The combination of this with Q in accordance with (237) gives t . 132 PRACTICAL ASTRO NOMT The azimuth a whose logarithm is given in the fourth line is expressed in minutes of arc. Since the correction a A must be expressed in seconds of time the logarithm of 4, viz., 0.6020, is also included when log a and the two logarithms immediately following it are added to form log a A. The final value of the clock correction is in satisfactory agreement with that found in Ex. 47. a Virg-fnis, C.W. Virginis, C. E. sin /, G &QC

H> 37. ii, 37, 17, 37, 17. 37, 18, 39, last, 40, 2, Ex. ii, " 41, 2, Ex. 13, 42, 4 and 5, Sec. 26, 60, last, 64, eq. (72), 70, 17. 73, 75, prec. eq. (117), 81, 4, Ex. 33, 85, 96, prec. eq. (141), 98, ERRATA Many nebulae show continuous spectra, indicating that they may not be wholly gaseous in constitution. for cos z co

b, and h itself is affected by an uncertainty, an error of observation if you will, the result- ant error of observation will exceed the error affecting h approximately in the ratio of a to 6 2 . With the errors of calculation the case is different. Here there is almost al- ways a certain multiplication of error, so that the accumu- lated error of calculation is usually in excess of the uncertain- ties attached to the individual numbers which enter into the computation; and, generally speaking, the longer the calcula- tion, the greater will be the accumulation or multiplication. Again, the solution of a given problem is frequently cap- able of expression in a variety of ways. Analytically con- sidered, these may be identical, but viewed from the stand- point of practical applications, they may present the greatest diversity. For example, the two expressions y 2 sin 2 1/2 x (3) y = 1 cos x (4) are theoretically equivalent, but when used for the calcula- tion of values of y corresponding to given values of x, they are by no means identical, especially for values of x near 0. As an illustration, consider the final errors resulting from an r-place calculation of (3) and (4) for the determination F. H. Scares ' 5 of y corresponding to x = 2 2', where we may think of x as the result of an observation whose uncertainty is ex- pressed by the appended quantity 2'. An appropriate in- vestigation shows that their maximum values are E [ v] = 0.00002 + 0.003 X 10" r , (3a) E [j] 0.00002 4- 3 X 10~ r . (4a) The first terms in the right members are the resultant errors of observation. The last are the accumulated errors of calculation. So far as the precision to be obtained with a specified number of decimals is concerned, the advantage is obviously in favor of equation (3). "With these facts before us we are in a position to appre- ciate better the qualifications required of the computer. His aim must be so to arrange the calculation that the errors in the data and the errors of calculation will produce the minimum possible effect upon the final result, and, at the same time, to derive this result with the least possible ex- penditure of time and labor. It is evident that his task is one of some complexity. The conditions to be satisfied are, to a certain extent, contradictory. For example, the accumu- lated error of calculation can be reduced to any desired limit by sufficiently increasing the number of decimal places em- ployed ; but any such increase carries with it a notable increase in the labor of calculation. On the other hand, a reduction of labor can often be brought about by a modifi- cation of the formula to be calculated, but this in turn may involve a sacrifice of precision. The adjustment of these variable factors to each other and to the requirements just expressed demands a nice bal- ancing of detail, which becomes only the more difficult when it is considered that the problems presenting themselves for solution, and the conditions under which they arise, are the most diverse imaginable. It is obvious that the computer has to deal with questions whose answers are not to be discovered through the exercise of whatever skill he may possess in the manipulation of figures, however important this accomplish- ment may be for the technical performance of his labors. They can be found only in a detailed knowledge of what has been but outlined in the preceding paragraphs. In addi- tion, the computer must ever be upon the alert \\ith a dis- criminating judgment, if his work is to be consistent in its The Art of Numerical Calculation details, and economical of the time and energy required for its execution. II Leaving now the consideration of the subject in its general aspects, we proceed to a discussion of matters of more imme- diate practical significance. (a) General Arrangement and Procedure. Computations are most conveniently made upon cross section paper whose squares measure one-fifth or one-sixth of an inch on the side. Before any figures are entered, the symbols for the quantities to be combined should be written in a vertical column at the left of the sheet, care being taken to bring together, as nearly as may be, those symbols or arguments whose numerical values are to be combined. Even though the same quantity enter into the calculation at several points, write its argument but once. A very little practice will make it possible to add or subtract numbers which are separated by several intervening quantities. The numbers are to be written in a vertical column immediately to the right of the column of arguments. If the same calculation is to be per- formed for a number of similar sets of data, the work should appear in parallel vertical columns, that for each set occupying a column by itself. In such a case do not complete the first column before beginning the others, but work across the page, inserting all the numbers corresponding to any given argu- ment before proceeding to the others. If, however, several trigonometric functions of the same angle are required, all should be interpolated with a single opening of the table, even though their symbols occupy widely separated positions in the column of arguments. Further, in computing for similar sets of data, do not enter arguments for quantities which are constant for all the sets, but write the values of such constants on the lower edge of a card or slip of paper. This can be held above the numbers with which the constants are to be united and moved along from column to column as the additions or sub- tractions are performed. The beginner will proceed with the greatest security by writing the arguments fully and complete ly, although the experienced computer is able to abbreviate the work by omitting some of the arguments and per- forming the corresponding operations mentally. Thus, it is possible to form the sum of two logarithms, enter the table, and interpolate the corresponding number without writing P. H. Scares 7 down the result of the addition. The argument for the sum can therefore be omitted, but such abbreviations are to Joe introduced gradually, and only after some skill has been acquired. Whenever it becomes necessary to abbreviate a number, say to r places, by dropping the higher decimals, increase the digit of the rth place by one unit when the neglected quan- tity exceeds one-half a unit of this place. If the decimals neg- lected are less than half a unit of the ;-th place, they are to be dropped without change in that place. When the neglected part is exactly a half unit of the rth place, it is a good rule to increase the digit of the rth place by one unit, in case that digit is odd ; otherwise, drop the higher places without change in the rth place. Errors arising from the abbreviation will thus tend to neutralize each other in the long run. (b) Aids to the Computer. Machines for addition, multiplication etc. Their operation is so simple that they require no special treatment in this place. Their construction is such that there is no accumulated error of calculation, unless the quantities involved are abbre- viated by dropping higher decimal places. The sliderule. In effect, this instrument is a graphical table of logarithms. In its usual form, the accumulated error of calculation generally amounts to a few units of the fourth place of decimals. It is, therefore, in nowise a substitute for tables of logarithms of 5, 6, and 7, or even 4 places. It is extremely convenient for certain classes of computation, but many experienced computers maintain that properly con- structed tables of logarithms give more satisfactory results. In any case, its continued use results in a strain upon the eyes far greater than that accompanying the use of well printed tables. Multiplication tables. The best are the Rechentafeln oi Crelle, published by Reimer of Berlin. These tables give directly the exact products of numbers of three figures or less; and can be used for the determination of products of numbers of any magnitude. They can also be used for divi- sion, most conveniently, when the divisor is of three figures or less. Their only objection is their bulk. They should be in the hands of every computer. Logarithmic-trigonometric tables. Those most generally useful are of five places of decimals, although' 3, 4, 6, and 8 The Art of Numerical Calculation 7-place tables. are also frequently required, and should be within reach of all who have to deal with astronomical or geodetic calculations. In purchasing tables, care should be exercised, for many are badly arranged and unfit for the purpose for which they are intended. With the exception of 3-place tables, those not giving the differences of the adjacent logarithms, at least for the tables of trigonometric functions, should be avoided. The same is true of those not containing auxiliary tables of proportional parts. The tabulation of the logarithms of the trigonometric functions to six places of decimals for every minute of arc, only, is likewise a bad arrangement. It is also important that the tables for the sine and cosine should not be separated from those of the tangent and cotangent. And, finally, it is desirable to select tables containing addition-subtraction logarithms. There are numerous other points of minor importance, but a more detailed discussion can be replaced by the following list of satisfactory tables. The list does not pretend to be complete. Four-place tables : Slichter, Macmillan ; Bremiker, Weidmannsche Buchhandlung, Berlin. Five-place tables : Becker, Tauchnitz, Leipzig; Gauss, Strien, Halle; Albrecht, Stankiewicz, Berlin; Newcomb, H. Holt & Co., New York ; Hussey, Allyn & Bacon, Boston ; Bremiker, Weidmannsche Buchhandlung, Berlin. Six-place tables: Bremiker, edited by Albrecht, Nicolaische Verlags-Buchhandlung, Berlin. Seven-place tables : Vega, edited by Bremiker, Weidmannsche Buchhandlung. This is the best 7-place table. Bruhns, Tauchnitz, Leipzig. In Bremiker's 4 and 5-place tables, the arguments for the trigonometric functions are expressed in decimals of a degree, the intervals being 0.l and 0.01, respectively, for the body of the tables. It is assumed that the student is familiar with the funda- mental principles underlying the construction and use of the ordinary logarithmic-trigonometric tables. The details of their usage can therefore be dismissed with the following precepts: F. H. Scares 9 (1) Do not use negative characteristics. When such occur, increase them by ten, and operate as though a minus ten were written after the logarithm. When two such logarithms are added, the sum will have an appended minus twenty, which should be reduced to minus ten, dropping at the same time ten units from the characteristic. (2) In case the number corresponding to a given logarithm is negative indicate that fact by writing a subscript n after the logarithm. In combining a number of logarithms, affix a subscript n to the result when the number of w-logarithms is odd. If this is even, the resulting logarithm needs no subscript. (3) Derive the logarithm of the secant and cosecant from those ot the cosine and sine, respectively, by subtracting the latter from zero. This is most easily accomplished by sub- tracting each digit of the logarithm from 9, proceeding from left to right, until the last is reached, which is to be subtracted from 10. (4) Interpolate all the functions required for any given angle with a single opening of the table. (5) In the formation of powers of numbers, care must be exercised when the power is fractional, and the number less than unity. After the logarithm of the number has been mul- tiplied by the power, p, the appended characteristic, which is normally 10, will be lOp. This must be reduced to 10 by adding 10(1 p) to the characteristic proper and subtracting the same quantity from the characteristic appended to the result of the multiplication. Addition-subtraction logarithms. The purpose of these tables is to determine the logarithm of a + b when the logarithms of a and h are given. The following illustrates the principle underlying their construction and use: Let 4 = log N , B ='log (AT + 1), where N represents any number. The addition-subtraction logarithmic table contains the values of B tabulated with the argument A. Now suppose whence A = log b log a , and 10 The Art of Numerical Calculation B - log (-4 1) = log (a + b) - log a, or log (a + b) log a + B. This is the fundamental equation for addition. The procedure is as follows: Form A = log b log a. Interpolate B with A as argument. Then, log (a + b) = loga + B. Again, let b whence A = log (a fo) log b, and a B = log -7- log a log />. These are the fundamental equations for subtraction. The procedure is as follows : Form B = log a log b. Interpolate A with B as argument. Then, log (a b) = log b -j- A. The arrangement of the tables assumed a>h for the solution of both the addition and the subtraction problem. The appli- cation of these tables involves the performance of one subtrac- tion, one addition, and one interpolation. The use of the ordinary logarithmic table for the derivation of the same result involves three interpolations and one addition or sub- traction. Further, as an illustration, in the 5-place tables of Gauss, the ordinary table covers 18 pages while the addition- subtraction table covers but 12, of which only 4V& are neces- sary for the addition problem. Finally, it can be shown that the uncertainty of a result derived from the addition-subtrac- tion table is less than that accompanying the use of the ordinary table. From every standpoint, therefore, whether that of the number of operations to be performed, the number of pages to be thumbed, or the accuracy of the final result, the advantage is in favor of the addition-subtraction table. Tables of squares, cubes, etc. Special tables. Barlow's Tables, F. H. Scares H containing the squares, cubes, square roots, cube roots, and reciprocals of all integers up to 10,000, is one of the most convenient. The roots are given to seven places of decimals, and the reciprocals partly to nine and partly to ten places. Some of the 5-place logarithmic tables, such as those ot Gauss and Albrecht, also contains tables of squares. Any table of squares or cubes can be used inversely for the derivation of square and cube roots. In addition to the various aids mentioned there are innumer- able special tables designed for the solution of special problems. Almost any problem which has to be solved repeatedly for different sets of data can be simplified through the use of specially constructed tables. In this connection there is abun- dant opportunity for the exercise of ingenuity and skill on the part of the computer. (c) Resultant Error of Observation, Accumulated Error of Calculation. Number of Decimal Places. The estimation of the effect on the final result due to un- certainties in the data, in other words, the evaluation of the resultant error of observation, is most conveniently made by means of the relation obtained by differentiating the formulae to be solved with respect to the final result and the quantities whose values are given. The substitution of the uncertainties in the data for the corresponding differentials in this express- ion leads to a knowledge of the numerical value of the differ- ential of the final result, which may be taken as the resultant error of observation. The determination of the maximum possible value of the accumulated error of calculation for any given set of formulas is a more or less complicated process. Since, however, the accumulated error seldom, if ever, reaches its maximum, a knowledge of its average magnitude is of more practical im- portance. This varies with the character of the equations to be solved, and its exact evaluation presents some difficulty. But for formulas containing no critical features, such as ab- normally large multipliers or small divisors, or differences defined by two relatively large and nearly equal quantities, or angles to be interpolated from sines or cosines, the following, based upon the theory of probabilities, gives an approximate expression of the average uncertainty in the logarithm of a result : tfi. =0.4X lO'Vn 12 The Art of Numerical Calculation where r is the .number of decimal places employed, and n the number of quantities involved in the calculation. If the final result is a number, AT, its approximate average uncertainty will be given by or, if an angle, by Uj, = 0.4 X 10~ r 206265" \/ n. (6) (7) The following table shows the results given by (5), (6), and (7) for tables of 3 to 7 places, n being equal to unity. To obtain the accumulated error of calculation for any given case, it is only necessary to multiply the proper tabular value by the square root of the number of quantities entering into the calculation. 1 No. of Decimals = r n= 1 *7 L ] Ux \ U 3 4 5 6 7 0.4 X 10~ 3 4 0.4 X 10 0.4 X 1(T 0.4 X 10 . 0.4 X 10~' 10~ 3 N 1C" N 10-AT 10-" jv 10-' .v 1/4 8" 0.8 0.08 0.008 Experience shows that these results are in close agreement with the average values actually occurring in practice. The determination of the number of decimal places to be used in any given calculation is a matter of great importance. If the number chosen is too small, the precision of the data will be sacrificed. If too large, much unnecessary labor will be expended just how much, is suggested by the fact that the relative amounts of time required to execute a calculation with 4. 5, 6, and 7 places of decimals are approximately ex- pressed by the numbers 1, 2, 3, and 5, respectively, i. e., five times as much labor is required to complete a given calcula- tion with 7-place logarithms as would be required if only 4-place tables were used. In practice, the number actually to be employed is usually determined by the accuracy of the given data. If this is to be used to its full advantage, a sufficient number of decimals must be employed to make the accumulated error of calculation small as compared with the resultant error of observation. Having determined the amount of the latter, the above table affords such indica- necessarily small as compared with the resultLt er or of ob ervation showing that the use of 6-place tables woud in- volve a needless amount of labor. The second value, on the other hand indicates that 5-place tables will entail he m Li- mum of labor consistent with the precision desired gain, required the number of decimals necessary for a calculation involving 25 logarithms, in which the resultant error of observation is 0.0001, the result itself being a num- 3er whose approximate value is 100. The expression for the accumulated error ot calculation is Uy[ 10 r >< 100 X 5 = 5 X 10 2 ~ r - To make this small as compared with the resultant error of observation, r must be taken equal to 7. For formulae free from the critical features mentioned in the second paragraph of this section, the resultant error of observation will usually be of the order of the uncertainties in the data. The choice of the number of decimals is then very simple. If the data, consists of numbers, it is only nec- essary to choose a number of decimals greater by one than the number of significant figures in the given quantities. If, on the other hand, the data consists of angles, a glance at the tabular values of 7 A affords the necessary information. The above suggestions by no means cover all the cases which may arise in practice, but they give an indication as to the general method of procedure. (d) The Adaptation of Formulae. The following general suggestions indicate the more im- nortant points to be borne in mind. m Whenever possible, transform equations containing ^fFWpnces of terms into expressions containing only Quotients. Thus, the equations 14 The Art of Numerical Calculation sin 5 = cos z sin < sin z cos cos .A, cos 8 cos t = cos z cos + sin z sin cos A, (8) cos 5 sin t = sin z sin /I, defining 8 and t in terms of z, , and A can be reduced to the form sin 5 777 sin ( M ) , cos 5 cos t = m cos( M), (9) cos 5 sin t = sin z sin A, by introducing the auxiliaries m and Af defined b} r m sin M = sin z cos A , (10) 777 COS M = COS Z . The solution of (10) and (9) thus replaces the solution of (8). Although the number of equations involved in (9) and (10) is five as against three in (8), the calculation is usually simpler in arrangement and control. (2) Whenever possible, calculated angles should be deter- mined from the tangent. This does not mean that the formulae are necessarily to be arranged so as to express explicitly the tangent of a required angle. As an illustration, M in (10) will be determined from its tangent, derived by subtracting log 723 cos M from log m sin M, although the equations do not express the tangent of M explicitly. The same is true of the determination of t from the last two of (9). It is possible to replace equations (9) and (10) by a single group of three equa- tions giving directly the tangents of M, t, and 8; but this is not to be recommended, for there is no saving in labor, and the use of (9) and (10) as they stand introduces symmetry into the arrangement of the work, and simplifies the determin- ation of the quadrants and the control of the calculation. (3) Avoid formulas expressing a quantity as the difference of two relatively large and nearly equal numbers. The com- parison of equations (3) and (4) made in a previous section illustrates the disadvantage connected with expressions of this type. (4) When it is necessary to determine a quantity differing but little from a second quantity whose value is known, arrange the formulae in such a way as to express the difference of the two. The calculated difference applied to the known quantity then gives the desired result. Developments in series are frequently useful in this connection. Thus, the geocentric F. H. Scares 15 latitude, <', is given by the relation tan ' = ( I e 2 ) tan , (11) in which <' is the astronomical latitude, and e, the eccentricity of a meridional section of the Earth. The numerical value of the latter is approximately 0.08, whence it follows that <' differs but little from <. In accordance with the above men- tioned principle, (11) is replaced for the purpose of numerical calculation by the following" equivalent expression ' -- 690".65 sin 2 -f- l".16 sin 4< . . . (12) in which the neglected terms are insensible. The first term in the right member of (12) calculated with 5-place and the sec- ond, with 3-plac.e logarithms, gives the same precision as 7- place tables used in connection with (11). (5) Calculations can frequently be much simplified by the use of approximate formulae. It is obviously permissible to neglect those terms in an equation whose numerical values are small as compared with the resultant error of observation. One of the most common methods of introducing such simpli- fications. consists in the substitution of the first terms of the developments in series of the sine, cosine, and the tangent of small angles for the trigonometric functions themselves. Since * 3 -i sin x = x 6 + ... ' .- (13) tan x x + *' -f- ... O it follows that the errors resulting from the substitution men- tioned will be of the order ot x s / 6 , x 2 />, and x a h, respectively. The evaluation of the errors in any given case is readily accomplished by means of the approximate relations A 2 (-Y in degrees) 2 X 1' i (14) A- 3 = (x in degrees) 3 X 1" Thus, for x = 15', the substitution of x for sin * introduces the error 3 i" 384 ' i /i \ _ [ I 6\-T./ 16 The Art of Numerical Calculation For the same value of x, the substitution of 1 for cos x pro- duces the error 1 / 1 V 1' 2Y4 ) xl/== "32 = 2 " ' approximately . It will be noted that the use of the suggestion in (4) will frequently make possible substitutions of this character, for an arrangement of the formulae expressing the difference between the required quantity and a nearly equal known quantity often- times introduces the trigonometric functions of small angles. As an illustration, the equation sin h = cos TT sin + sin TT cos cos t (15) expresses the relation between the north polar distance of a star, TT, its hour angle, t, its altitude h, and the latitude of the place, <. The latitude can be calculated .when the remain- ing quantities are known. One of the most convenient methods results from an application of (15) to the star Polaris. Since the latitude equals the altitude of the celestial pole above the horizon, and since this latter differs at most by 1 12' from the altitude of Polaris, it is desirable to express (15) as a function of H = < h. The resulting equation in H is sin H =. sin IT cos t -j- tan < (cos H cos IT ) . (16) Since H ^ TT = 1 12', (16) can be replaced by the approxim- ate relation 7f2 H = TT cos t -\- -^ tan sin 2 t , (17) in which terms of the order of ?r 3 or higher have been neglected. The maximum error of (17) is V or 2". The appearance of < in the last term of (17) seems to require a knowledge of the quantity for whose determination the equation has been designed, viz., the latitude. But, since the coefficient of this term is very small, a rough approximation for , which can be obtained in a variety of ways, is all that is required. (e) Control and Checking of Computations. The acquirement of accuracy in the performance of numerical calculations is largely a matter of intelligent practice. The mistakes of the beginner are due in part to an inability to concentrate his attention upon the large number of details involved in even a relatively short calculation. With experi- ence, the more frequently occurring operations are reduced to a more or less mechanical process. This reduces the strain F. H. Scares 17 upon the attention, and thereby lessens the liability to error. But this liability is never entirely removed, and the most skill- ful computer occasionally makes mistakes. ' It is therefore essential that all calculations should be controlled. The importance of this cannot be too strongly emphasized, especi- ally for the beginner. The methods of checking are numerous. Generally each type of problem requires its own special method of treatment. One of the most satisfactory methods of control is afforded by the derivation of a result from two independent sets of formulas. There are many cases, however, in which the application of such a test is impossible. Another is the so-called method of differences, which can be applied when the same set of formulae is to be calculated for several uniformly varying sets of data. In such cases the final results, and in- deed, the numerical values of any given quantity in the calcu- lation, present a systematic variation as one passes successively from one set of data to another. The test is applied b} r form- ing the successive differences of the numerical values of the final results or of such quantities as it is deemed necessary to control. A numerical error in one or more of the separate calculations reveals itself through an irregularity in the varia- tion of the differences. The control is a searching one and is capable of bringing to light even the neglected decimals form- ing a part of the accumulated error of calculation. The arithmetic mean of a series of numbers can be checked by forming the differences between the mean and the individual numbers, regard being paid to the algebraic sign. If the calcu- lated mean is correct, the sum of the positive differences will equal that of the negative differences. Other methods of checking will readily suggest themselves to the computer after a moderate amount of experience, but whatever the method, he must constantly be on his guard not to place too much confidence in a mere agreement of numerical results, for no process of checking is absolute. An agreement in numerical results signifies at most a certain probability that a calculation has been correctly performed. This proba- bility may be large or small according to the nature of the test, but it never becomes equal to certainty. For example, the method of differences affords an invaluable control in so far as isolated errors are concerned, but it is quite incapable of discovering systematic errors affecting similarly all of the separate calculations. It is therefore desirable that several different tests should be applied. For those rare cases in 18 The Art of Numerical Calculation which all ordinary processes of checking become impossible, the calculation should, be repeated by a second computer, working quite independently of the first ; and failing this, by the original computer himself. But this last method should be adopted only as a final resort, and then only after a con- siderable interval of time has elapsed, for it is a well-known fact that an error once committed is very likely to recur in the repeated calculation. The preceding remarks relate particularly to the detection of errors already committed. Something should also be said as to the methods of preventing errors. The most important things are care, attention, and deliberation. The systematic arrangement of the work explained under (a) contributes much. The method of combining numbers is of importance. When tw numbers are to be added or subtracted, the combination should be made from left to right. With a little practice this method affords an increase in both accuracy and rapidity as compared with the usual process. With the beginner each result should be verified as the calculation progresses, always by an independent method, if possible. Thus, additions, in the case of two numbers, can be performed both from left to right and from right to left; for more than two numbers, by adding from the top downward and from the bottom up; subtractions can be controlled by adding the difference to the subtrahend ; interpolations of trigonometric functions, by a comparison of the difference of the logarithms of the sine and cosine with the logarithm of the tangent. Finally, the com- puter should be on the watch for such impossible results as values of a sine or cosine greater than unity, and the occurr- ence of negative values for essentially positive numbers. Under no circumstances should these details be neglected unless the computer has acquired a very considerable skill, for it is a matter of experience that the location and correction of errors in a completed calculation consumes far more time than is required for the execution of the work with that de- liberation which assures some degree of accuracy in the final result. In addition, the consciousness of having exercised all possible care gives a feeling of security which, in the attempt to secure freedom from errors, is a psychological factor of no small importance. Laws Observatory, Columbia, Mo.