THE ELEMENTS OF ASTRONOMY FOR SURVEYORS. THIRD EDITION, Thoroughly Revised and greatly Enlarged. In Crown Svo. Pp. i-xiii-f 430. Cloth. Fully Illustrated. A HANDBOOK ON THEODOLITE SURVEYING & LEVELLING. For the use of Students in Land and Mine Surveying. BY PROFESSOR JAMES PARK, F.G.S. CONTENTS. Scope and Object of Surveying. Theodolite. Chains and Steel Bands. Obstacles to Alignment. Meridian and Bearings. Theodolite Traverse. Co-ordinates of a Station. Calculation of Omitted or Connecting Line in a Traverse. Calculation of Areas. Subdivision of Land. Triangulation. Determination of True Meridian, Latitude, and Time. Levelling. Railway Curves. Mine Surveying. INDEX. " A book which should prove as useful to the professional surveyor as to the student. ' ' Nature. In Crown 80. Pp. i-viii + 204. Cloth. With 87 Diagrams. PRACTICAL SURVEYING & FIELD-WORK. Including the Mechanical Forms of Office Calculations, with Examples Completely Worked Out. BY VICTOR G. SALMON, M.A., Government Land and Mine Surveyor, Johannesburg. CONTENTS. Co-ordinate Calculations. Area. Base Measurement. Reduction of Field-book. Various Problems. Adjustment of Instruments. INDEX. In Crown Svo. Fully Illustrated. Cloth. PROBLEMS IN LAND AND MINE SURVEYING. Being 400 Questions and Answers (200 fully worked). Many Examples taken from the Papers set by the Home Office, City and Guilds of London, Ac., at the Surveying Examinations. BY DANIEL DA VIES, M.I.M.E., County Lecturer in Mining, Surveyingj Ac. In Cloth. Pp. i-xi + 179. Fully Illustrated. THE EFFECTS OF ERRORS IN SURVEYING. BY HENRY BRIGGS, M.Sc. CONTENTS. Introduction. Analysis of Error. The Best Shape of Triangles. Propagation of Error in Traversing. Application of the, Methods of determining Average Error to certain Problems in Traversing. Propagation of Error in Minor Triangulation. Summary of Results. APPENDIX. INDEX. " Likely to be of the highest service to surveyors . . . it is a most able treatise " Engineer. FOURTEENTH EDITION, Revised. Enlarged (by 100 pages). Re-set. With Numerous Diagrams. Cloth. A TREATISE ON MINE-SURVEYING. FOP the use of Managers of Mines and Collieries, Students at the Royal School of Mines, &c. BY BENNETT H. BROUGH, Assoc.R.S.M., F.G.S. Revised and Enlarged by HARRY DEAN, M.SC., A.R.S.M. CONTENTS. General Explanations. Measurement of Distances. Chain Sur- veying. Traverse Surveying. Variations of the Magnetic-Needle. Loose-Needle Traversing. Local Variations of the Magnetic-Needle. The German Dial. The Vernier Dial. The Theodolite. Fixed-Needle Traversing. Surface-Surveying with the Theodolite. Plotting the Survey. Plane-Table Surveying. Calculation of Areas. Levelling. Underground and Surface Surveys. Measuring Distances by Telescope. Setting-out. Mine-Surveying Problems. Mine Plans. Applications of the Magnetic-Needle in Mining. Photographic Surveying. APPENDICES.- INDEX. " Its CLEARNESS of STYLE, LUCIDITY of DESCRIPTION, and FULNESS of DETAIL have long ago won for it a place unique in the literature of this branch of mining engineering, and the present edition fully maintains the high standard of its prede- cessors. To the student, and to the mining engineer alike, ITS VALUE is inestimable. The illustrations arc excellent." The Mining Journal. London : CHARLES GRIFFIN & CO., Ltd., Exefer St., Strand, W.C.2. PHILADELPHIA: J. B. LIPPINCOTT COMPANY. THE ELEMENTS OF ASTRONOMY FOR SURVEYORS BY R. W. CHAPMAN, M.A., B.C.E., F.R.A.S., PROFESSOR OF MATHEMATICS AND MECHANICS IN THE I'XIVERSITY OF ADELAIDE. WITH 56 DIAGRAMS. BH1 JSK- /^i LONDON: CHARLES GRIFFIN AND COMPANY, LIMITED. PHILADELPHIA: J. B. LIPPINCOTT COMPANY. 1919. [All Rights Resewed.} PREFACE. ALTHOUGH there are several excellent books on Surveying that deal more or less thoroughly with astronomical obser- vation, it appeared to the writer, as the result of his experience in teaching the subject, that there is a distinct need of an elementary work suitable for the student and for the surveyor who is taking up astronomical observa- tion for the first time. Most of the purely surveying books are content to quote practical formulae for the reduction of the observations, with little or no attempt to expound the principles by which the formulae are derive d. On the other hand, the theoretical works on astronomy in which the mathematical theory is developed are gener- ally too recondite for the beginner, and deal to a large extent with matters of no special interest to the surveyor. The present work is an attempt to provide an elementary exposition, not only of the practical methods of observa- tion and computation, but of the main principles that must be thoroughly understood if the surveyor is to be master b 405417 vi PREFACE. of his profession. Throughout the work the methods of observation are illustrated with numerous fully worked-out actual observations, and a prominent feature of the book is the attention that is given to the effects of observational and instrumental errors of different kinds. A large pro- portion of the examples set for working have been taken from the papers set for candidates at the examinations for Licensed Surveyors in Australia. R. W. C. ADELAIDE, September, 1918. CONTENTS. CHAPTER I. THE SOLUTION OF SPHERICAL TRIANGLES. PAGES A Review of the Principal Formulae of Spherical Trigonometry, . 1-7 CHAPTER II. THE CELESTIAL SPHERE AND ASTRONOMICAL CO-ORDINATES. The Celestial Sphere The Apparent Motion of the Stars Definitions of some Fundamental Terms Astronomical Co- ordinates Altitude and Azimuth Right Ascension and Declination Comparative Advantages of the Two Co- ordinate Systems The Sidereal Day and Sidereal Time Hour Angle Synopsis of Astronomical Terms, . . . 8-20 CHAPTER III. THE EARTH. The Earth as a Globe Terrestrial Latitude and Longitude The Zones of the Earth The Altitude of the Celestial Poles equal to the Latitude of the Place of Observation To Find the Shortest Distance Between Two Places whose Latitudes and Longitudes are given The Earth as an Oblate Spheroid Geographical and Geocentric Latitude Examples, . . 21-34 CHAPTER IV. THE SUN. The Sun's Apparent Motion among the Stars The Earth's Orbit Round the Sun The Equinoxes The Sun's Motion in Right Ascension and Declination The Sun's Semi-Diameter Plotting the Position of the Sun's Centre on the Celestial Sphere The Sun's Apparent Annual Path, .... 35-43 viii CONTENTS. CHAPTER V. TIME. PAGES Sidereal Time Apparent Solar Time Mean Time The Three Systems of Time Measurement Equation of Time Local Mean Time Local Sidereal Time Standard Time To Change Standard Time to Local Mean Time To Reduce an Interval of Mean Time to Sidereal Time, and vice versa To Find Local Sidereal Time at Local Mean Noon, given the Sidereal Time at Mean Noon at Greenwich Given the Local Mean Time at any instant to Determine the Local Sidereal Time Given the Sidereal Time to Find the Corre- sponding Local Mean Time Alternative Methods for Preceding Problems Determination of Time of Transit of a known Star across the Meridian Time of Transit of the First Point of Aries The Use of the Greenwich Time of Transit of the First Point of Aries in Computations of Local Mean and Sidereal Time Nautical Almanac Data with Regard to Time Calculations Examples, ...... 44-70 CHAPTER VI. THE LOCATION OF OBJECTS ON THE CELESTIAL SPHERE. Given the Right Ascension and Declination of a Star, to De- termine its Altitude and Azimuth at any Time Having Observed the Altitude and Azimuth of a Star, and Noted the Time, to Compute its Right Ascension and Declination Having Determined the Altitude aud Azimuth at a given Time, to Find the Approximate Position of the Star at some Short Interval of Time afterwards Examples, . . . 71-79 CHAPTER VII. ASTRONOMICAL AND INSTRUMENTAL CORRECTIONS TO OBSERVATIONS OF ALTITUDE AND AZIMUTH. Parallax Horizontal Parallax Atmospheric Refraction Correc- tions to Observations on Account of Residual Instrumental Errors The Effect of an -Error in Collimation The Elimina- tion of Instrumental Errors by Changing Face The Error made if the Transverse Axis of the Telescope is not Hori- zontal The Use of the Striding Level Allowance for Error of Alidade Level 80-96 CONTENTS. ix CHAPTER VIII. THK DETERMINATION OF TRUE MERIDIAN. PAGES Referring Marks First Method : By Equal Altitudes of a Cir- cumpolar Star. Second Method : By a Circumpolar Star at Elongation Calculation of the Time of Elongation The Effect of an Error in Latitude Star Observations in Day- light. Third Method : By Extra-Meridian Observations on Sun or Star SUD Observations Computation of Sun's De- clination from Nautical Almanac Data The Effects of Errors in Latitude, Declination and Longitude. Fourth Method : By Time Observations upon a Close Circumpolar Star Circum- Elongation Observations for Azimuth Examples, - 97-148 CHAPTER IX. THE DETERMINATION OF LATITUDE. First Method : By Meridian Transits Zenith Pair Observations of Stars Meridian Altitudes at both Lower and Upper Culminations. Second Method : By Circum-Meridian Obser- vations Circum-Meridian Observations of the Sun. Third Method : By Prime Vertical Transits The Effects of Errors in the Time Measurement and in the Setting Out of the Prime Vertical Striding Level Correction. Fourth Method : By the Altitude of the Pole Star at any Time A Rough Method by Noting the Rate at which Altitude of Sun or Star changes near the Prime Vertical A Method by the Measure- ment of the Horizontal Angle between Two Circumpolar Stars at their Greatest Elongations. Table for Reduction of Circum-Meridian Observations - Examples, . . . .149-181 CHAPTER X. THE DETERMINATION OF TIME BY OBSERVATION. First Method : By Meridian Transits The Effect of an Error in the Direction of the Meridian The Effect of an Error in the Horizontality of the Transverse Axis Meridian Transits on both Sides of the Zenith. Second Method : By Extra- Meridian Observations of Sun or Star Arrangement of the Computation Averaging several Observations Observations on both East and West Stars The Effect of Errors in Latitude, Declination and the Measured Altitude. Third CONTENTS. I'AGKSr Method : By Equal Altitudes Error Due; to Inequality in the Altitudes Application of Method to Sun Observations. Fourth Method: Almucantar Method for Time Observations- Sun Dials The Horizontal Dial Prime Vertical Dial Oblique Dials Time of Rising or Setting of a Celestial Body Examples, ". 182-21S CHAPTER XL THE DETERMINATION or LONGITUDE. 1. By Portable Chronometers. 2. By Electric Telegraph or Wire- less Telegraphy Recording and Receiving Signals Com- parison of Chronometers Personal Equation Programme of Operations. 3. By Flash Lamp Signals Longitude by Lunar Observations (a) By Lunar Distances (b) By Lunar Cul- minations (c) By Lunar Occultations, . . . .219-235 CHAPTER XII. THE CONVERGENCE OF MERIDIANS, ..... .236-241 INDEX, . . . .243-247 ASTRONOMY FOR SURVEYORS. CHAPTER I. THE SOLUTION OF SPHERICAL TRIANGLES. IN this chapter the principal formulae of spherical trigonometry, such as will be afterwards applied to calculations on the celestial sphere, are brought together for convenient reference. No attempt will be made to establish the formulae, which are demonstrated in any of the ordinary books on spherical trigonometry, but a brief synopsis will be given of the usual methods for the solution of spherical triangles under different conditions. Great Circles. The line of intersection made with the surface of a sphere by a plane passing through the centre of the sphere is known as a great circle. If this circle passes through two points A and B on the surface of the sphere, then the shortest distance between A and B, measured along the sphere's surface, is that measured along the arc of the great circle joining them. Only one great circle can be drawn to pass through two given points on the surface of a sphere, unless they happen to be at opposite extremities of a diameter, and the length of the shorter arc of this great circle between the two points is the shortest distance between them. Meri- dians of longitude on the earth's surface are great circles. In spherical trigonometry it is always assumed that the arcs representing the sides of the triangles considered are arcs of great circles. 2 ASTRONOMY^ FOR SURVEYORS. Small Cfrcltts:-^Tne line ' of l intersection made with the surface of a sphere by a plane that does not pass through the centre is known as a small circle. The ordinary formulae of spherical trigonometry do not apply to tri- angles having sides that are arcs of small circles. A parallel of latitude on the earth's surface is a small circle. It follows that the shortest distance between two points in the same latitude is not that measured along the parallel of latitude, but is measured along the arc of the great circle joining them. Spherical Triangles. Denoting the angles of a spherical triangle by A, B, and C, and the sides opposite to these angles by a, b, and c respectively, the sides being as usual measured by the angles which they subtend at the centre of the sphere, then we have the following funda- mental relations : (a) The sines of the angles are proportional to the sines of the opposite sides : sin A _ sin B _ sin C sin a sin b sin c (b) One side of a triangle is expressed in terms of the two other sides, and the angle included between them by one of the three formulae : cos a = cos b cos c + sin b sin c cos A\ cos b = cos c cos a + sin c sin a cos B > . (2) cos c = cos a cos b + sin a sin b cos C; (c) From these may be derived another set of six use- ful relationships of which the following two are types : cot a sin b = cot A sin C + cos 6 cos C^ ( O } cot b sin a = cot B sin C -f- cos a cos Cj v Whilst the formulae (2) and (3) are extremely useful in all sorts of investigations into the properties of spherical triangles, they are not adapted to logarithmic computation, THE SOLUTION OF SPHERICAL TRIANGLES. 3 and are consequently not suitable for use in the numerical solution of triangles. For this purpose other formulae are commonly used, derived from these fundamental relation- ships but expressed in a form more suitable for use with logarithms. The Solution of Right- Ar.gled Spherical Triangles. The relationships between the sides and angles of a right- angled spherical triangle are very conveniently summarised by the mnemonic rules due to Napier, the inventor of logarithms, and known as Napier's Rules of Circular Parts. Denoting the right angle by C, Napier defines five "circular" parts (i.e., a, 6, 90 A, 90 c, 90 B), and these are supposed, as in the figure, to be ranged round a circle in the order in which they stand in the triangle. Then, if any one of these five parts is selected and called the middle part, the two parts on each side of it are called the adjacent parts, and the remain- ing two are called trie opposite parts. For instance, if a is chosen as the middle part, 90 B and b are the adjacent parts,, and 90 c and 90 A are the opposite parts. Then Napier's Rules are : Sine of middle part ^product of tangents of adjacent parts. Sine of middle part =product of cosines of opposite parts. Thus sin a =cot B tan b and sin a =sin c sin A. 4 ASTRONOMY FOR SURVEYORS. As an aid to memory, it may be noticed that the vowels in the words sine and middle are the same, so with tangent and adjacent, cosine and opposite. By choosing different parts in turn as the middle parts, we obtain all the possible relationships between the sides and angles, and with a little practice it is easy to choose the particular ones wanted. If we want a relationship between a, 6, and c, for example, 90 c must be taken as the middle part, and we have cos c=cos a cos 6. [f a relationship between a, A, and B is required, take 90 A as the middle part, whence cos A =sin B cos a and so on. There are six cases to consider in the solution of right- angled triangles, and the formulae required, readily obtained from Napier's rules, are as follows : (1) Given the hypotenuse c and an angle A. tan b = tan c cos A, cot B = cos c tan A, sin a --= sin c sin A. (2) Given a side b and the adjacent angle A. tan b tan c = cos A' tan a = tan A sin b, cos B = cos b sin A. (3) Given the two sides a and b. cos c = cos a cos b, cot A= cot a sin b, cot B = cot b sin a . THE SOLUTION OF SPHERICAL TRIANGLES, 5 (4) Given the hypotenuse c and side a, cos c cos o = , cos a tan a cos B = - , tan c sin a sin A= . . sin c (5) Given the two angles A and B, cos c = cot A cot B, cos A cos a = -.- -, sin B cos B cos o = - -. sin A (6) Given a side a and opposite angle A, sin a sin c = - , sin A sin b = tan a cot A, cos A sin B = . cos a The Solution of Oblique Spherical Triangles. (1) Given the three sides, a, b, and c. Let s=l (a +6+ c). Then the angle A may be computed from any one of the following three formulae : A A / sin (s b) . sin (s c) sin - A/ - L -^-T L: -T- 2 sin b . sin c A /sin s . sin (s a) 2 * sin b . sin c /sin (5 6) . sin ( , denoting a pair of ram's horns. The exact nature of this point we shall discuss a little later on, but for the present all 16 ASTRONOMY FOR SURVEYORS. that we want to know is that it is a point whose position 'can always be accurately determined. If we know, then, the angular measure of the arc v D that is to say, the angle which the arc subtends at the centre 0, and also the direction in which it is measured from towards the East that is to say, in the opposite direction to that in which Hour Angle. In Fig. 4 the angle R^P^-^whMir-is-the angle that the plane of the declination circle P B D makes with the plane of the meridian, is known as the hour angle of the star B. If we know the hour angle of a star, and also its polar distance, we can clearly mark the position of the star on the celestial globe, so that these two may be used as another system of co-ordinates. The hour angle of a star is continually changing, but owing to the uniform character of the star's motion, it varies .f p pppafg/nf. raff j If the hour angle is 90 measured towards the East, then the star will take 6 sidereal hours to reach the meridian. Thus a knowledge of the hour, angle at once gives us the time the star will take to reach the meridian, if it be on the East side of it, or the time that has elapsed since the star crossed the meridian, if it be on the Western side. Prime Vertical. The plane through the zenith at right angles to the meridian that is, the vertical plane running CELESTIAL StflERE AND CO-ORDINATES. 19 East and West is Is East and West line, 1 lown as the Prime Vertical. The >yhich is the line of intersection of th the plane of the horizon, is also of the plane of the celestial equator ill be evident from Fig. 2. al Terms. For purposes of reference, es dealt with in this chapter are e. the Prime Vertical w the line of mfefsectidi with the horizon, as j\ Synopsis of Astronomi the principal quantit illustrated in one figu Fig. 5a is drawn for an observer in the Southern Hemi sphere, and Fig. 5b for the Northern Hemisphere. EXAMPLES. 1 . The R.A. of a star being 35 20', what is the local sidereal time when the star is in the meridian ? Ans. 2 hrs. 21 min. 20 sec. 2. If the R.A. of a star is 295 and the sidereal time is 15 hours, is the star to the East or West of the Meridian ? Ans. To the East. 3. What is the declination of a star that rises exactly in the East ? Ans. 0. 4. What is the hour angle of the star in Question 2 ? Ans. 70. 5. The declination of a star is 35 South. Determine its S.P.D. and its N.P.D. Ans. 55 and 125. 6. If the First Point of Aries crosses the meridian exactly two hours, as measured by a sidereal clock, after a certain star, what is the R.A. of the star ? Ans. 330. 7. The declination of the Pole Star is 88 51' North. What is the difference between its greatest and least zenith distances ? Ans. 2 18'. 8. At the time of the year when the R.A. of the sun is zero, determine approximately the time of rising of a star with declination and R.A. 150. Ans. 4 p.m. 9. What is the point whose altitude is 90 and hour angle zero ? Ans. The zenith. 20 ASTRONOMY FOR SURVEYORS A I Fier. 5. Fig. 56. O is the observer. S W N E, the plane of the horizon. Z, the zenith. P, the celestial pole ; P, the polar axis. S P Z N, the plane of the meridjan. K' W Q E, the celestial equator. W Z E, the prime vertical. N, S, W, K, the North, .South, West, and East points. B, any star. Z P B, the hour angle of B. P B D, the declination circle of B. P B, the polar distance of 1>. B D, the declination of B. of> U, the right ascension of B. Z B P, the vertical through B. BF, the altitude of B. B Z, the zenith distance of B. N F, the azimuth of B. 2! CHAPTER III. THE EARTH. The Earth a Globe. That the earth is a globe is no longer a matter for dispute. It has been circumnavigated and .mapped and measured, and no other supposition will fit the facts. We see its round shadow as cast upon the moon during a partial eclipse. We see the planets as great balls of similar dimensions revolving at different distances round the great central sun. The law of gravi- tation explains the form of their orbits and enables their movements to be predicted with the greatest exactness. That our earth is a globe like these, revolving in a similar way around the sun, is the only satisfactory hypothesis that will account for their apparently involved move- ments in the heavens. The whole of the apparent move- ments of the heavenly bodies are readily accounted for on the supposition that the earth is a globe, and no explanation even plausibly satisfactory has been advanced on any other supposition. In the case of some of the planets we can actually observe that they are in rotation in a manner similar to that in which we assume our own earth must rotate to account for the phenomena of night and day and of the diurnal rotation of the stars. In the planet Mars we see the poles or extremities of the axis of rotation surrounded by white caps apparently similar to the great caps of ice and snow that surround the poles of our own earth. Terrestrial Latitude and Longitude. The extremities of the axis of rotation of the earth are called the Poles, and are distinguished as the North and South Poles. 22 ASTRONOMY FOR SURVEYORS. A plane through the earth's centre at right angles to the axis cuts the earth's surface in a circle known as the Equator. Every point on the terrestrial equator is thus equidistant from the North and South Poles. In order to mark the position of a point on the earth's surface, it is necessary to have a system of co-ordinates similar to those we have already discussed in connection with the celestial sphere. Suppose that P (Fig. 6) is a point on the earth's surface, the position of which it is desired to locate. A plane Eig. 0. passing through P and the earth's axis will cut the earth's surface in a great circle N P M S, which is known as a Meridian. Suppose this Meridian cuts the equator at the point M. Then clearly, if we know the position of the point M on the equator, and also the length of the arc P M or the angle which it subtends at the earth's centre, we shall be able to fix the point P. The position of M on the equator is determined by the longitude of P. To measure this, some arbitrary place A must be THE EARTH. 23 selected on the equator as a starting point. The point actually chosen is the point of intersection of the meridian passing through Greenwich, shown as N G A S in the figure, and the equator. The angular measure of the arc A M that is to say, the angle A M is known, as the longitude of P. Thus, all points on the meridian passing through P have the same longitude. All points on the meridian N G A S, passing through Greenwich, have zero longitude. The longitude of other places is reckoned as so many degrees East or West of Greenwich until we come, to 180, which is the longitude of the meridian exactly opposite to the Greenwich meridian. The angle POM, which is the angle between the direction of the vertical at P and the vertical at M, measures what is known as the latitude of P. If we draw a plane through P at right angles to the earth's axis, it will intersect the earth in a small circle L P L' parallel to the equator. Such a circle is called a Parallel of Lati- tude, and all points on the same parallel clearly have the same latitude. Latitude is measured as so many degrees North or South of the Equator. The latitude of the North Pole is 90 N. Thus, if we know the position of the meridian of 'zero longitude, the latitude and longitude of a place are suffi- cient to enable us to mark its position on the globe. The Length of a Degree of Longitude. If the parallel of latitude through P intersects the meridian through Greenwich in B, it is clear that the arc B P will be much smaller than the arc AM. It will have the same angular measurement on a much smaller circle. If P were very near to the North Pole, the arc B P would be very small indeed. Thus two places in the same latitude but differing by, say, ten degrees of longitude, will be very much closer together if they are in a " high " latitude that is to say, a latitude approaching 90 than they will be if both 24 ASTRONOMY FOR SURVEYORS. are on or near the equator. Thus a degree of longitude has its greatest value, when measured in distance along the earth's surface, at the equator, its value becoming less and less as we approach the poles. At the equator a degree of longitude is equivalent to a distance of about 69 miles. A degree of latitude, on the other hand, is always of approximately the same value, about 69 miles, whether it is measured near the poles or near the equator, because it is measured along meridians which are all great circles of the same diameter. The Zones of the Earth. Certain parallels of latitude Arctic Circle Tropic of Cancer. Equator. Tropic of. Capri divide the earth's surface into five belts or divisions, termed zones. These mark in a general way a natural division of the earth's surface according to climate. The parallel of latitude 23 27J' North of the Equator is termed The Tropic of Cancer, and the corresponding parallel South of the Equator is termed The Tropic of Capricorn. As we shall presently see, at all places between THE EAETH. 25 these parallels at some part of the year the sun shines directly overhead at mid-day. As a consequence, the belt included between these is the hottest portion of the earth's surface, and it is known as the Torrid Zone. The parallel of latitude 66 32J' North of the Equator is called the Arctic Circle, and the corresponding parallel South of the Equator the Antarctic Circle. The belt between the Arctic Circle and the Tropic of Cancer is known as the North Temperate Zone, and that between the Antarctic Circle and the Tropic of Capricorn as the South Temperate Zone. The regions around the two poles bounded by the Arctic and Antarctic circles respectively are termed the Frigid Zones. At all places within the frigid zones the sun is below the horizon at mid-day for some portion of the year. The Altitude of the Celestial Pole is Equal to the Latitude of the Place of Observation. In Fig. 8, let O be the position of the observer and C the earth's centre. Then the direction of the pull of gravity at O is in the direction O C. This, then, will mark the direction of the vertical at O, and the zenith, Z, of the observer will be in C O produced. H R. at right angles to O Z, marks the plane of "this horizon. If C P, the earth's axis, be produced to cut the celestial sphere in P 1; then P x will be the celestial pole. Draw O P 2 parallel to C P x . Then the celestial pole being, as we have seen, at a x distance from the earth that is practically infinite in comparison to the earth's radius, P 2 wjll mark the direction in which the celestial pole is seen by the observer at O. Draw the plane of the equator E C Q at right angles to the earth's axis. Then, from our definition, the latitude of is measured by the angle ECO. 26 ASTRONOMY FOR SURVEYORS. Now the angle ZOP 2 =the angle O C P 1? and the complements of these angles are equal. Therefore, the angle P 2 OR=the angle E C i.e., the altitude of the pole = the latitude of the observer. It follows from this that if the observer travels equal distances North and South from 0, since his latitude will change by equal amounts, the altitude of the celestial pole will also be increased or decreased by equal amounts. As this is actually the case from observation, the fact forms a strong proof of the sphericity of the earth. To find the Shortest Distance, measured along the Earth's Surface, between two Places whose Latitude and Longitude are given, assuming the Earth to be a True Sphere. In Fig. 9, let P and R be two places whose latitudes and longitudes are known. The shortest distance between P and R, measured along the earth's surface, will be the length of the arc of the great circle joining them. THE EARTH. 27 Draw the meridians passing through P and R. Then if we know the latitudes, we know the angular measure of the meridian arc* N P and N R, N being the North Pole. If P is in North latitude, the arc N P is the complement of the latitude. If R is the South latitude, the arc N R is 90+ the latitude. The angle P N R is the difference of the longitudes of P and R if both are measured in the same direction, or Fig. 9. the sum of the longitudes, if one is East and the other West. Thus in the spherical triangle N P R, we know the sides N P and N R and the included angle P N R. Then by the ordinary methods of spherical trigonometry we can compute the angular measurement of the great circle arc P R, and consequently its lineal measurement, if we know the radius of the earth. The radius of the earth is approximately 3,960 miles. 28 ASTRONOMY FOR SURVEYORS. EXAMPLE. Find the shortest distance measured along the earth's surface between Perth (long. 115 50' E., lat. 31 57' 8.) and Brisbane (long. 153 I 7 E., lat. 27 28' S.) 9 assuming that the earth is a sphere of radius 3,960 miles. In this case, both places being in the Southern Hemisphere, it will be preferable to solve the triangle S P R (Fig. 9) rather than N P R. If A denotes the position of Brisbane, B of Perth, and C the South Pole, we shall have in the spherical triangle ABC C A = b = 90 - 27 28' = 62 32' C B = a = 90 - 31 57' = 58 03' C = 153 1' - 115 50' - 37 11' Since we only want to find c, the simplest way to solve this triangle is to divide it into two right-angled triangles by drawing a great circle arc B D to cut C A at right angles. Then we have from the right-angled triangle BDC tan C D = cos C tan a. tan a = tan 58 3', cosC = cos 37 11', tan C D, . . . .-. CD = 51 56' 47", and cos c = cos A D . cos B D 10-2050545 9-9012980 10-1063525 Fig. 9 cos a cos 58 3', cos (b - C D) = cos 10 35' 13", cos CD = cos 51 56' 47", cos c, . c = 32 26' 49". 9-7236026 9-9925435 9-7161461 9-7898616 9-9262845 The circular measure of this angle is -5663. .-. The distance required = -5663 X 3,960 = 2,242-5 miles. The more usual method of solving the triangle A B C, having given the THE EARTH. 29 two sides a, b, and the included angle C, would be to first find the angles A and B by means of the formulae cos %(a + 6) tan (A B) = S ~? ( ff ~ b ) cot sin f (a + b) and then find c from the formula sin C . sin a sin A It' this method is adopted to find c, it must be remembered that when sin c is found there are always two possible solutions, since the sine of an angle the sine of its supplement. Some care is, therefore, necessary in selecting the appropriate value from the two values determined by the tables. EXAMPLES FOE SOLUTION. In all of these examples the earth is to be taken as a sphere of radius 3,960 miles. 1. Find the shortest distance measured along the earth's surface between Mount Gambier (Longitude 140 45' E., Latitude 37 50' S.) and Palmerston .(Longitude 130 50' E., Latitude 12 28' S.). Ana. 1,856-8 miles. 2. Find the shortest distance measured along the earth's surface between Baltimore (Lat. 39 17' N., Long. 76 37' AY.) and Cape Town (Lat. 33 .56' S., Long. 18 26' E.). Ana. 7,893 miles. 3. How far would a place be due South from the equator if the altitude of the S. celestial pole was exactly 20 ? Ans. 1,382-3 miles. 4. Two places are in S. latitude 30, one longitude 115 E., and the other 35 E. Find the difference in the paths of the two ships sailing from one port to the other, one along the parallel of latitude and the other along the arc of the great circle joining the places. Ans. 1,127 miles. 5. What is the declination of a star that passes through the zenith at a place in latitude 35 N. ? Ans. 35 North. 6. A ship sails along the great circle joining two places, each of latitude 45 N., the difference between their longitudes being 2 a. Show that the highest latitude I reached during the passage is given by the formula cot / = cos a. 30 ASTRONOMY FOR SURVEYORS. 7. A ship from latitude 8 25' N. sails south for 600 miles. What latitude is she in ? Ant. l c 35'S. 8. At a place in latitude / North, a star with decimation d rises 60 E, of North. Show that cos I = 2 sin d. The Figure of the Earth. If, as in Fig. 10, F and G are two points on the same meridian, their difference of latitude will be measured by the angle FOG. If we know this angle, and also the length of the arc F G, we Fig. 10. shall then be able to calculate the length of the earth's radius F 0. The difference of latitude between F and G may be determined by astronomical observation, meas- uring the altitude of the celestial pole at each place. The length of the arc F G may be either directly measured or it may be computed by means of a triangulation survey from a measured base . line on some suitable adjacent part of the earth's surface. Determinations of the radius of the earth on these simple principles were made by the Greeks 2,000 years ago. THE EARTH. 31 If the earth were a true sphere, measurements of the radius of the earth made in this way at different parts of its surface would be all the same. But when it became possible to make the necessary observations with suffi- cient precision it was found that such was not the case. When Newton discovered and investigated the results of the law of gravitation in the seventeenth century, he proved that one consequence was that if the earth is a plastic body, revolving on an axis and acted on by its own attraction, it must take the form of a slightly flat- tened sphere with its polar diameter less than its equatorial diameter. Measurements of two arcs made by the Cassinis in France seemed, on the other hand, to indicate that the length of a degree of latitude decreased towards the north, which would imply that the shape of the earth was such that its polar diameter was greater than its equatorial diameter, contrary to Newton's gravitational theory. The French Academy equipped two expeditions in order to settle the problem. One of these measured an arc in the equatorial regions of Peru (1735-1741), and the other an arc in the polar regions of Lapland (1736- 1737). The results showed that a degree of latitude was longer in the polar regions than in parts near the equator, and corroborated Newton's theory. Since then many arcs have been measured in different parts of the world, and the observations have conclusively established the fact that the shape of the earth is not a true sphere, but is very approximately an oblate spheroid, the figure formed by revolving an ellipse about its minor axis. The shape of the earth is thus like that of a sphere slightly flattened at the poles. The amount of flattening is not, however, very great. The length of the earth's polar axis may be taken as 7,900 miles, and its equatorial diameter as 7,927 miles. Thus if a model were made 20 feet in diameter, the polar diameter would be shorter than the equatorial by a trifle over three-quarters of an inch. 32 ASTRONOMY FOR SURVEYORS. More exactly still, it is found that the change in the length of a degree of latitude which takes place as we proceed along a meridian is not the same along all meri- dians. It seems that the equatorial section of the earth is not exactly circular, but is very slightly elliptical. The exact shape would thus appear to be more nearly an ellipsoid. For practical purposes, however, all com- putations in geodetic work are based upon the assumption that the figure of the earth is an oblate spheroid. Geographical and Geocentric Latitude. If in Fig. 11 P represents some point on the meridian N Q S, N and 8 being the North and South Poles, then, making allowance for the fact that the section N Q S E is not a circle but an ellipse, the direction of the horizontal at P will not be at right angles to P O, O being the earth's centre, but will be in the direction of the tangent to the ellipse at P. This is the direction taken by the surface of still water at that point. .Consequently the direction of the vertical there is not O P but G P, where G P is the normal at P that is to say, it is at right angles to the tangent. Thus, if we measure the latitude of P by astronomical THE EARTH. 33 methods, observing the altitude of the celestial pole above the horizon at P, we shall measure the angle P G Q and not the angle P O Q. The angle P G Q thus measures what is called the geographical or geodetic latitude. This is the ordinary latitude that is used for astronomical and geodetic purposes. It is clear, however, that the value of the angle P O Q> if it can be readily determined, might be equally well used in order to fix the position of P on the meridian. This angle measures what is termed the geocentric latitude. The difference between the geocentric and geographical latitude of a place is never very great. There is no difference at all, either at the poles or at the Equator, and the maximum difference is in latitude 45, where it amounts to about 11" 44" of arc. The geocentric latitude cannot be directly observed. It is computed from the geodetic latitude by the formula : =~ When speaking of latitude in this book, it will always be the geodetic latitude that is meant unless otherwise specified. EXAMPLES. 1. At a place in Lat. 42 S. a line is run from a point A on a bearing of 220 for a distance of 2,400 chains to a point B. Assuming the earth a sphere of 3,957 miles radius, find the bearing from Bto A. Ans. 40 15' 12". 2. Given that latitude of London is 51 32' N., latitude of Jerusalem 32 44' N., bearing of Jerusalem from London, 1 10 04'. Find the longitude of Jerusalem, its distance from London, and the bearing of London from Jerusalem. Ans. Longitude, 37 25' 12" E. Distance, 2,278 miles. Bearing of London from Jerusalem, 316 00' 16". 3 34 ASTRONOMY FOR SURVEYORS. 3. The latitude of a Trig. Station A is 33 51' S., and its longitude is 151 12' 42" E. The bearing and distance to another Trig. Station B is 284 08' 44", 105,600 feet. Compute the latitude and longitude of B, and the bearing of B to A, on the assumption that the earth is a sphere with radius 20,890,790 feet. Ans. Longitude, 151 28' 48" E. Bearing, 104 56' 20". 4. Find the great-circle distance in English statute miles from Wellington. N.Z., to Panama, treating the earth as a sphere, and one degree as equal to 69 2 V statute miles. Wellington, . . Lat. 41 17' S., Long. 174 47' E. Panama, . . Lat. 9 00' N., Long. 70 31' W. Ans. 4,528-6 miles. 5. Two places are each in latitude 50 N., and their difference of longitude is 47 36'. Find their distance apart. Ans. 2,090 miles. 35 CHAPTER IV. THE SUN. The Sun's Apparent Motion among the Stars. Like the fixed stars, the sun shares in the apparent general daily rota- tion of the heavens, but unlike them it does not always maintain the same position relative to other objects on the celestial sphere. In addition to its daily circling of the sky, it appears to gradually shift its position with respect to the stars. Neither its declination nor its right ascension remain^constant . Very little consideration will show that its declination must alter during the year, for, if it did not, the sun would always describe the same circle in the heavens. If this were the case, then, like the fixed stars, it would always rise and set at the same points on the horizon, and it would always attain the same altitude when on the meridian. Since it does not do this, it is clear that the declination of the sun must change during the year. That the sun has also a movement in right ascension among the stars is not quite so obvious, but the fact may be readily inferred if we watch the stars that are visible in the East on succeeding mornings just before sunrise or in the West just after sunset. Stars in the East that rise just before the sun, so that in a very short time after rising they are masked by the sun's rays, will pn each succeeding morning be seen for a longer time. Similarly stars in the West, setting just after the sun, will be visible for shorter and shorter periods as we watch them on successive evenings until finally they are lost altogether in the strong sunlight, other stars further East taking their places. Hence we infer that the sun has 36 ASTRONOMY FOR SURVEYORS. a progressive movement among the stars from West to East. The problem of determining the sun's place on the celestial sphere with regard to the fixed stars was a difficult one to early astronomers, because as soon as the sun becomes visible its strong light prevents the stars from being observed at the same time. Some used the moon, and Tycho Brahe used the bright planet Venus in order to get the connection, observing the relative position of the sun and moon or of the sun and Venus when both were visible, and afterwards measuring the position of the moon or Venus with regard to the stars when the sun had set. But as both the moon and Venus also move amongst the stars, the movement that had taken place in the interval had to be allowed for, and the method was thus not particularly simple. The sun's position is nowadays determined by much more accurate methods. The Earth's Orbit round the Sun. All of these move- ments of the sun are apparent only and not real. Just as its apparent daily rotation in the heavens is due to the rotation of the earth on its axis, so the sun's apparent movements in right ascension and declination are really due to the fact that the earth moves in a great orbit round the sun once a year. Actually the earth moves round the sun in a path that is very nearly a huge circle with a radius of about 96 millions of miles. More accurately, the path is described as an ellipse, one focus of the ellipse being occupied by the sun. The curve traced out by the centre of the earth lies in a fixed plane that passes through the centre of the sun. The earth traces out its complete orbit once a year, and all the time it is spinning on its own axis once a day, the direction of the spin on its axis being the same as that in which it moves round the sun. The earth's axis is not at right angles to the plane of its orbit, but it makes with the plane a fixed invariable angle of 66 32J'. That THE SUN. 37 the direction of the earth's axis is constant we know from the fact that the position of the celestial pole amongst the fixed stars shows no appreciable shift throughout the year. Thus, as is illustrated in Fig. 12, the earth moves round the sun, spinning on its axis, which is inclined to the plane of the orbit, and the axis always remains parallel to itself, pointing ever in the same direction amongst the fixed stars, whose distances, it must be remembered, are practically infinitely great even in com- parison with the immense distance of the earth from the sun. When the earth is in the position marked 1, the sun will be shining directly overhead in a place such as a North of the equator. If e is a point on the earth's equator on the same meridian of longitude as a, O being the earth's centre, the angle a O e will be the complement of 66 32J' or 23 27|' that is to say, a will be a point on the Tropic of Cancer. In this position, then, the sun at mid-day will be vertically overhead at all points on the Tropic of Cancer. This statement is not quite accurate, 38 ASTRONOMY FOR SURVEYORS. because the earth does not remain in the one position in its orbit while it makes a complete revolution on its axis ; it is moving forward in its orbit all the time, but as it takes a whole year to go round the sun, its relative movement is not very great in one day. As the earth moves from position 1 to position 2, its axis always remaining parallel to its original direction, it will be seen that the sun will appear to shine directly overhead at points successively nearer and nearer to the equator, until in position 2 the sun's rays fall vertically at the equator. Similarly, as the earth moves on to position 3, the sun's rays will fall vertically at points further and further south of the equator, until at position 3 the sun will appear at mid-day to be overhead at a point on the Tropic of Capricorn. From there on to position 4 the sun will shine vertically at points successively nearer to the equator, until at 4 the sun is once more overhead at the equator. The earth is in the position marked 1 on June 2 2nd, hi that marked 2 on September 22nd, at 3 on December 22nd, and at 4 on March 21st. Thus, if we consider the appearance of the sun to an observer at some point P to the south of the Tropic of Capricorn, on June 22nd the sun will appear to be further from the zenith and lower down in the sky than at any other period of the year. On December 22nd, when the earth is in position 3, the sun at mid-day will be nearer the zenith than at any other time of the year. The orbit of the earth being an ellipse, its distance from the sun is not constant. It is furthest from the sun in the position 1, and nearest to the sun in the position 3. The Equinoxes. On March 21st and September 22nd, the sun, being vertically overhead at the equator, will appear to an observer at any part of the earth to be in THE SUN. 39 the celestial equator. Now, we have seen that when any heavenly body is in the celestial equator its path is bisected by the horizon, so that the time during which it can be seen in the sky is equal to the time during which it is invisible. Thus, when the earth is in either of these positions the days and nights are of equal length all over the world. These points are consequently called the Equinoxes. Motion in Right Ascension and Declination. It thus appears that on March 21st and September 22nd the sun's declination is zero, as it lies on the Celestial Equator. From March 21st to September 22nd it will appear in the sky to the North of the equator, so that its declina- tion will be north with a maximum value of 23 27|' on June 22nd. From September 22nd to March 21st its declination will be south with a similar maximum value on December 22nd. It is also evident that the sun's right ascension changes throughout the year, because as the earth revolves round it the apparent position of the sun among the fixed stars must obviously change. The stars that would be seen by an observer on the earth when in position 1, looking in the direction of the sun, would be seen by an observer at 3 when looking in the direction opposite to that of the sun. Clearly, in the course of the year the sun will trace out a complete circle among the fixed stars. The declination and right ascension of the sun are given in the Nautical Almanac for Greenwich noon on every day of the year. The values at intermediate instants may be found by interpolation. Illustrations of such calculations are given in Chapter VIII. when dealing with sun observations. The Sun's Semi-Diameter. The disc of the sun sub- tends at the eye of an observer an angle of about half a degree. By accurately measuring the angle subtended by diameters taken in different directions, we find that 40 ASTRONOMY FOR SURVEYORS. these are all equal, so that the disc is circular in form. In order to mark the position that the sun occupies on the celestial sphere at any time, we require to determine the position of the centre of the circular disc. But there is no mark at the centre that we can recognise, and so in practice we must observe a point on the edge of the sun and then make an allowance for the distance of this point from the sun's centre. From what we have just seen of the nature of the earth's motion round the sun, it is clear that the sun is not at all times of the year at the same distance from us, and consequently we should not expect its diameter to remain constant. As the earth completes its orbit round the sun in a year and then goes over the same path again, we might anticipate that the variations in the value of the sun's apparent diameter would follow a yearly cycle. This is found to be the case, a slow de- crease taking place from the 31st of December to the of July, and a slow increase during the second half the year. As the semi-diameter is frequently required in reducing sun observations, the values are chronicled for every day in the year in the Nautical Almanac (p. 11 of each month). In the almanac for 1914 the maximum value of the semi- diameter is given on January 3rd as 16' 17-55", and the minimum on July 3rd as 15' 45 -38". To Plot the Position of the Sun's Centre on the Celestial Sphere. Supposing that we know the direction of the true North and South, and also the latitude of the place of observation, we may readily measure the declination of the sun at mid-day. With a telescope pointed in the direction of the meridian we may observe the altitude of the sun's upper or. lower edges (limbs, as they are usually called) at the moment when it crosses the meridian. Making due allowance for the sun's semi-diameter, we shall thus obtain the meridian altitude of the sun's centre. THE SUN. 41 Thus, as in Fig. 13, if P represents the Pole, Z the zenith, we measure either S x N or S 2 S, according as the sun is in a position such as Si or as S 2 . Now, we have previously shown that the altitude of the celestial pole, P N, is equal to the latitude of the place. Thus, if the sun is situated as at S 15 on the same side of the zenith as the pole, the difference between the observed altitude S x N and the latitude P N gives the sun's polar distance P S,. If the sun is at S 2 , on the opposite side of thejzenith to the pole, then the arc S 2 N is equal to 1802-^the observed altitude S S 2 . The difference between S 2 N and the latitude P N gives the sun's polar distance as before. The declination of the sun is the complement of its polar distance. Fui. 13. / Having measured the declination of the sun in this way, in order to fix its position on the celestial sphere, it only remains to determine the difference between its right ascension and that of some star whose co-ordinates are known. But we have seen that the difference of right ascension of any two stars is measured by the interval in time between their transits across the meridian, as given by the sidereal clock. If, with the sidereal clock, the times be measured when the first and second limbs of the sun cross the meridian, the mean of the two times will give the instant when the centre crosses the meridian. If, therefore, the time of passage across the meridian of some selected known star is also observed, the interval 42 ASTRONOMY FOR SURVEYORS. between the two times, reduced to degrees, will give the difference between the right ascension of the sun and the star. These observations give us the elements necessary to plot the position of the sun. The Sun's Apparent Annual Path on the Celestial Sphere. In Fig. 14, let A represent the position of the selected fixed reference star as plotted on a globe representing the celestial sphere, P being the Pole, Q R the great circle of the equator, and S N the horizon. Then, if we Fig. 14. set out the angle A P B equal to the observed difference of right ascension and measure off the arc P B equal to the observed polar distance of the sun, the point B will represent the position of the sun's centre on the star globe. When observations similar to those just described are made day after day, and the corresponding positions of the sun plotted on -the globe, those positions are all found to lie on a great circle, which cuts the equator at two opposite points one turn less with respect to the sun than it does with respect to the fixed stars. There are approximately 365J mean solar days in the year, and, therefore, in the same period there are 366 J sidereal days. More exactly, according to Bessel, the year contains 365-24222 solar days, and hence 365-24222 solar days^- 366-24222 sidereal days. Therefore, if m be the measure of any interval in mean time and s the corresponding measure in sidereal time, ra_ 365-24222 *T~ 366 -24222* Thus, if m be given, s can be found, or vice versa. Tables to facilitate the reduction are Driven in the TIME. 53 Nautical Almanac, and less elaborate ones in Chambers' Mathematical Tables. When tables are not used, the simplest way to make the computation is as follows : To convert an interval of mean solar time to sidereal time, add 9-8565 seconds for each mean solar hour. Dividing by 60, this gives us -1642 second to be added for each minute and -0027 second for each second of mean time. Thus, to convert an interval of 6 hrs. 33 min. 17 sec. of solar time into the equivalent interval of sidereal time, we have 6x 9-8565= 59-139 33 x -1642= 5-418 17^x -0027- -046 64-603 seconds = 1 min. 4-6 sec. The addition of this to the given solar time gives us 6 hrs. 34 min. 21-6 sec. as the equivalent sidereal interval. To convert an interval of sidereal time to the equivalent interval of mean solar time, subtract 9-8296 seconds for each sidereal hour. Dividing by 60 we get -1638 second to be subtracted for each sidereal minute, or -0027 second for each second. Thus, to find the interval of solar time equivalent to an interval of 6 hrs. 33 min. 17 sec. of sidereal time, we have 6x 9-8296= 58-978 33 x -1638- 5-405 17 x -0027= -046 64 -42 9 seconds = 1 min. 4-43 sec. Subtracting this from the given interval of sidereal 54 ASTRONOMY FOR SURVEYORS. time gives 6 hrs. 32 min. 12-57 sec. as the equivalent mean time interval. Given the Sidereal Time at Mean Noon at Greenwich on any given Date to find the Local Sidereal Time at Local Mean Noon at any other Place on the Same Date. On page 11 for each month in the Nautical Almanac the Greenwich sidereal times are tabulated for Greenwich mean noon on each day. From these it is necessary, in most work in which the time has to be brought into the calculations, that we should be able to deduce the local sidereal time at local mean noon on the corresponding day at the place of observation. In the succeeding pages it will be convenient to use the following abbreviations : G.M.T. to denote Greenwich mean Time. G.S.T. Greenwich sidereal Time. G.M.N. Greenwich mean noon. L.M.T. Local mean Time. L.S.T. ., Local sidereal Time. L.M.N. ,, Local mean noon. From what we have already done, it will be evident that if we have two clocks, one set to keep sidereal time and the other to keep mean time, the sidereal clock will complete its day in a shorter period than the other, and consequently will be continually gaining. According to the last article, it will gain at the rate of 9-8565 seconds for each solar hour. Now, at a place in West Longitude, noon occurs a certain number of hours after noon at Greenwich, the interval depending upon the longitude. But the tabulated sidereal time at Greenwich noon is the difference between the readings of the sidereal and mean time clocks at that instant. Consequently, by the time it becomes noon at the place in question, the sidereal time will have gained still further on the mean time clock at the rate of 9-8565 TIME 55 seconds for each hour of longitude. Thus the L.S.T. at L.M.N. will be greater than the G.S.T. at G.M.N. by an amount computed at the rate of 9-8565 seconds for each hour of West longitude. Similarly, at a place in East Longitude, noon occurs before the corresponding noon at Greenwich, and in this case L.S.T. at L.M.N. will be less than the G.S.T. at G.M.N. by an amount computed in the same way according to the longitude. EXAMPLE. On October 1st, 1914, the G.S.T. at G.M.N. is given in the Nautical Almanac as 12 hrs. 37 min. 29-99 sec. Determine the L.S.T. at L.M.N. (a) at a place in longitude 57 33' 28" West, (6) at a place in the same longitude East. (a) 57 33' 28" is equivalent to 3 hrs. 50 min. 13-87 sec. 3 x 9-8565 = 29-569 50 x 0-1642 = 8-210 13-87 x -0027 = -037 37-816, say 37-82 sees. Therefore, for a place in West longitude we must add this on to the 12 hrs. 37 min. 29-99 sec., giving 12 hrs. 38 min. 07-81 sec. as the L.S.T. at L.M.N. (6) If the place is in East longitude, we must subtract the 37-82 seconds, giving 12 hrs. 36 min. 52-17 sec. as the L.S.T. at L.M.N. in that case. EXAMPLE. On December 1st, 1914, the G.S.T. at G.M.N. is 16 hrs. 37 min. 59-89 sec. Compute (a) the G.S.T. at G.M.N. on December 2nd, (6) the L.S.T. at a place in longitude 43 35' West at L.M.N. on December 1st. Ans. (a) 16 hrs. 41 min. 56-45 sec. (6) 16 hrs. 38 min. 28-52 sec. Given the Local Mean Time at any Instant, to Determine the Local Sidereal Time. The local mean time gives us the interval measured in solar hours, minutes, and seconds, that has elapsed since local noon. We may readily turn this interval into sidereal hours, and so obtain the number of sidereal hours, minutes, and seconds that have elapsed since noon. But in the preceding paragraph we have seen how the L.S.T. at L.M.N. may be determined on any 56 ASTRONOMY FOR SURVEYORS. given date at a place in any longitude. Consequently we have only to add to this the number of sidereal hours, minutes, and seconds that have since elapsed, to deter- mine the sidereal time at the instant. We. therefore, proceed as follows : 1. From the tabulated G.S.T. of G.M.N. on the date in question, compute the L.S.T. of L.M.N. by allowing for difference in longitude. 2. Turn the given L.M.T. into sidereal hours, minutes, and seconds, and add to the L.S.T. of L.M.N. EXAMPLE. Find the sidereal time at Mount Hamilton (Longitude 121 38' 43-35" West) on October 2nd, 1913, the L.M.T. being 9 hrs. 17 min. 32 sec. p.m. Dividing the longitude by 15, we get the difference in local times between Mount Hamilton and Greenwich to be 8 hrs. 06 min. 34-89 sec. The gain of the sidereal over the mean time clock in this interval, at the rate of 9-8565 seconds per hour, is 1 min. 19-93 sec. From the Nautical Almanac, we get G.S.T. at G.M.N. on October 2nd, 1913, 12 hrs. 42 min. 23-50 sec. Add, . hr. 1 min. 19-93 sec. L.S.T. at L.M.N., . 12 hrs. 43 min. 43-43 sec. But 9 hrs. 17 min. 32 sec. of mean time, when turned into sidereal time, . . 9 hrs. 19 min. 03-59 sec. Therefore, L.S.T. required, . . . 22 hrs. 02 min. 47-02 sec. EXAMPLE. Find the sidereal time at Adelaide (longitude 138 35' 04-5" E.) on October 2nd, 1913, the standard time being 9 hrs. 17 min. 32 sec. p.m. The standard time for South Australia is that of the meridian 142 or 9 hrs. 30 min. E. Difference in local times between Adelaide and Greenwich = 9 hrs. 14 min. 20-3 sec. The gain of the sidereal over the mean time clock in this interval at the rate of 9-8565 seconds per hour is 1 min. 31-06 sec. G.S.T. at G.M.N. on October 2nd, 1913, 12 hrs. 42 min. 23-50 sec. Subtract, hr. 1 min. 31-06 sec. L.S.T. at L.M.N 12 hrs. 40 min. 52-44 sec. TIME. 57 The difference between local time and standard time is 15 min. 39-7 sec. Therefore, the local mean time is . . 9 hrs. 01 min. 52-3 sec. Turning the interval into sidereal time, we get 9 hrs. 03 min. 21-31 sec. Therefore, L.S.T. required, . 21 hrs. 44 min. 13-75 sec. It is to be particularly noticed that the local mean time must always be reckoned from noon when making such calculations. Thus, if the mean time is given as 9 hrs. a.m. on October 2nd, this must be reckoned as 21 hrs. October 1st, or 21 hrs. after noon on October 1st. Given the Sidereal Time at a Place whose Longitude is known, to Determine the corresponding Local Mean Time. If we can find the sidereal time at m^an nnnn 1 then by subtracting this from the given sidereal time we find the number of sidereal hours, minutes, and seconds that have elapsed since noon. Turning this interval of time into mean time will give us the number of mean time hours, minutes, and seconds since noon that is to say, the mean local time required. The rules of procedure are thus : 1. From the tabulated G.S.T. of G.M.N. on the date in question, compute the L.S.T. of L.M.N. by allowing for difference in longitude. 2. Subtract the L.S.T. of L.M.N. from the given sidereal time. Turn the difference into mean solar time, and the result will be the mean time required. EXAMPLE. Given that the sidereal time at Mount Hamilton is 22 hrs. 02 min. 47-02 sec. on October 2nd, 1913, the longitude of the place being 121 38' 43-35" West, find the corresponding local mean time. As in the first example of the preceding section, we obtain L.S.T. at L.M.N., 12 hrs. 43 min. 43-43 sec. Given sidereal time, . . . .22 hrs. 02 min. 47-02 sec. Difference, ... 9 hrs. 19 min. 03-59 sec. 58 ASTRONOMY FOR SURVEYORS. Turning this interval into mean solar time, by the aid of the tables, we get 9 hrs. 17 min. 32 sec. as the L.M.T. required. EXAMPLE. Given that the sidereal time at Adelaide (longitude 138 35' 04-5" E.) is 21 hrs. 44 min. 13-75 sec. on October 2nd, 1913, find the corresponding local mean time. As in the second example of the preceding section, we obtain L.S.T. at L.M.N., 12 hrs. 40 min. 52-44 sec. Given sidereal time 21 hrs. 44 min. 13-75 sec. Difference, . . . 9 hrs. 03 min. 21-31 sec. Turning this interval of sidereal time into mean time, we obtain 9 hrs. 01 min. 52-3 sec. as the L.M.T. required. Alternative Method for Determining the L.S.T., having given the L.M.T. In the preceding methods for computing L.S.T. from L.M.T. or vice versa, it is necessary to first of all compute the L.S.T. of L.M.N., and then to trans- form another interval of time from mean to sidereal or from sidereal to mean. In the methods about to be described the theory is perhaps a little more complex, but there is only one transformation of a time interval necessary, so that the actual computation is a little shorter. From the given L.M.T., allowing for the difference of longitude, we readily compute the corresponding mean time at Greenwich. This gives us the interval in mean time that has elapsed since the last Greenwich noon. Turn this interval into sidereal time, and we get the number of sidereal hours, minutes, and seconds that have elapsed since the mean sun was last on the Green- wich meridian. But from the Nautical Almanac we get the G.S.T. at the last G.M.N. Allowing for the difference in longitude, we can thus obtain the L.S.T. at that instant. And as we have already computed the interval in sidereal time TIME. 59 that has since elapsed, we have only to add this on to the L.S.T. at the preceding G.M.N. in order to get the sidereal time required. We thus get the following rules of procedure : 1. Allowing for the difference of longitude, compute the mean time at Greenwich at the instant in question, and turn the interval of mean time so found into sidereal time. 2. From the Nautical Almanac obtain the G.S.T. at the previous G.M.N. , and allowing for the difference of longitude, determine the corresponding L.S.T. at the same instant. 3. The addition of the results of 1 and 2 gives the L.S.T. required. As illustrations, for purposes of comparison, we will take the same examples as those already worked. EXAMPLE. Find the sidereal time at Mount Hamilton (longitude 121 38' 43-35" West) on October 2nd, 1913, the L.M.T. being 9 hrs. 17 min. 32 sec. p.m. L.M.T. at Mount Hamilton, . . 9 hrs. 17 min. 32 sec. Difference due to Longitude (W.), . 8 hrs. 06 min. 34-89 sec. Corresponding G.M.T., . . .17 hrs. 24 min. 06-89 sec. Turned into sidereal time, this is equivalent to 17 hrs. 26 min. 58-41 sec. From the Nautical Almanac we get G.S.T. at G.M.N. on October 2nd, 1913, 12 hrs. 42 min. 23-50 sec. Difference due to longitude 8 hrs. 06 min. 34-89 sec. .-. L.S.T. at G.M.N., ... 4 hrs. 35 min. 48-61 sec. Interval of sidereal time since elapsed . 17 hrs. 26 min. 58-41 sec. .-. L.S.T. required, .... 22 hrs. 02 min. 47-02 sec. 60 ASTRONOMY FOR SURVEYORS. EXAMPLE. Find the sidereal time at Adelaide (longitude 138 35' 04-5" E.) on October 2nd, 1913, the standard time being 9 hrs. 17 mm. 32 sec. p.m. The standard time for South Australia is that of the meridian 142 or 9 hrs. 30 min. E. Standard time at instant, . . . 9 hrs. 17 min. 32 sec. Subtract difference due to longitude, . 9 hrs. 30 min. sec. Corresponding G.M.T. on October 1st, . 23 hrs. 47 min. 32 sec. Turning the interval into sidereal time we get 23 hrs. 51 min. 26-5 sec. From the Nautical Almanac we find G.S.T. at G.M.N. on October 1st, . . 12 hrs. 38 min. 26-95 see. Difference due to longitude of Adelaide, 9 hrs. 14 min. 20-3 sec. .-. L.S.T. at G.M.N. on October 1st, . 21 hrs. 52 min. 47-25 sec. Interval of sidereal time since elapsed, . 23 hrs. 51 min. 26-5 sec. .-. L.S.T. required, .... 21 hrs. 44 min. 13-75 sec. Alternative Method for Determining the L.M.T., having given the L.S.T. Knowing the longitude of the place, we can compute the sidereal time at Greenwich at the same instant. From the Nautical Almanac, as before, we get the G.S.T. at the previous G.M.N. Subtracting these two results gives us the interval in sidereal time that has elapsed since Greenwich noon. If we turn this interval into mean solar time, we, there- fore, get the interval of mean time that has elapsed since G.M.N. But the L.M.T. corresponding to G.M.N. is readily determined by allowing for the difference of longitude. Adding to this, therefore, the interval of mean time that has since elapsed, we obtain the L.M.T. required. The principal difficulty arises in places with East longitude, where it may happen that the instant under consideration really precedes noon on the same day at Greenwich. This cannot happen with places having West longitude. If this is the case, it will be at once TIME. 61 noticed from the fact that the sidereal time at Greenwich mean noon on the day in question, as found from the Nautical Almanac, will be less than the computed Green- wich sidereal time at the instant. We thus get the following rules for determining the L.M.T., having given the L.S.T. : 1. Allowing for the difference of longitude, compute the G.S.T. at the instant in question. 2. From the Nautical Almanac find the G.S.T. at the previous G.M.N. and then by subtraction the number of sidereal hours that have elapsed since. Turn this interval of sidereal time into mean time. 3. Add this interval of mean time on to the L.M.T. corresponding to G.M.N. , and the result is the L.M.T. required. EXAMPLE. Given that the sidereal time at Mount Hamilton is 22 hrs. 02 min. 47-02 sec. on October 2nd> 1913, the longitude of the place being 121 38' 43-35" West, find the corresponding L.M.T. L.S.T. at Mount Hamilton, . . 22 hrs. 02 min. 47-02 sec. Difference due to longitude (W.), . 8 hrs. 06 min. 34-89 sec Corresponding G.S.T., . . . 30 hrs. 09 min. 21-91 sec G.S.T. at G.M.N., October 2nd, 1913, . 12 hrs. 42 min. 23-50 sec. Interval of sidereal time since G.M.N., . 17 hrs. 26 min. 58-41 sec. Equivalent interval of mean time, . 17 hrs. 24 min. 06-89 sec. L.M.T. corresponding to G.M.N., October 2nd = October 1st, . 15 hrs. 53 min. 25-11 sec. .-. L.M.T. required = October 2nd, . 9 hrs. 17 min. 32 sec. 62 ASTRONOMY FOR SURVEYORS. EXAMPLE. Given that the sidereal time at Adelaide (longitude 138 35' 04-5" E.) is 21 hrs. 44 min. 13-75 sec. on October 2nd, 1913, find the corresponding L.M.T. L.S.T. at Adelaide, . . . 21 hrs. 44 min. 13-75 sec. Difference due to E. longitude, . . 9 hrs. 14 min. 20-30 sec. Corresponding G.S.T., . 12 hrs. 29 min. 53-45 sec. G.S.T. at G.M.N., October 2nd, 1913, . 12 hrs. 42 min. 23-50 sec. Instant precedes G.M.N. by hr. 12 min. 30-05 sec. Equivalent interval of mean time, . hr. 12 min. 28 sec. L.M.T. corresponding to G.M.N., October, 2nd, 9 hrs. 14 min. 20-30 sec. .-. L.M.T. required, . 9 hrs. 01 min. 52-3 sec. In this case, since the instant precedes G.M.N., we must subtract the computed interval of mean time from the L.M.T. corresponding to G.M.N. Comparison of the Preceding Methods. As it is a most important thing that the student should thoroughly grasp the principles involved in the transference of time from one system of time measurement to the other, it is a good exercise for him to master both the first method given and the alternative method in each of the preceding cases. The first method, however, involves less thinking and is more mechanical than the other, so that it is the method generally adopted and the one probably most suited for ordinary computations. Determination of the Local Mean Time of Transit of a Known Star across the Meridian. One very important application of the preceding work is the calculation of the time of transit of a known star across the meridian, or, as it is commonly termed, the time of culmination. The Nautical Almanac supplies us with a table of the right ascensions and declinations of the principal stars in the sky, and it has been shown in Chapter II. that the R.A. of a star, expressed in time, is the sidereal time TIME. 63 at the moment when the star is on the meridian. Thus the problem is simply that of determining the' L.M.T. corresponding to the sidereal time measured by the right ascension of the star. This we may do by one of the methods we have been considering. EXAMPLE. Find the time of culmination of a Tricing. Aust. on the evening of August llth, 1913, at a place in South Australia whose longitude is 139 20' E., the time to be measured in the standard time of the meridian 9 hrs. 30 min. E. G.S.T. of G.M.N., August 17th, . . 9 hrs. 41 min. 02 sec. .. L.S.T. of L.M.N. at place in longitude 139 20' E. computed as in previous work, 9 hrs. 39 min. 30-45 sec. R.A. of a Triang. Aust. = L.S.T. at time of culmination, . . . .16 hrs. 39 min. 31 sec. .. interval of sidereal time elapsed since L.M.N. , 7 hrs. 00 min. 00-55 sec. Equivalent interval of mean time, . . 6 hrs. 58 min. 51-74 sec. This, therefore, would be the L.M.T. at time of culmination. Difference between L.M.T. and time of the standard meridian, hr. 12 min. 40 sec. '. Standard time at culmination, . . 7 hrs. 11 min. 31-7 sec. Time of Transit of the First Point of Aries. In the preceding work we have adopted the usual practice of effecting the change from sidereal to mean or vice versa by means of the column in the Nautical Almanac giving the G.S.T. at G.M.N. But on page 3 of each month there is given another column tabulating for each day in the year the G.M.T. of transit of the First Point of Aries, which may also be used for similar transformation of time . As this instant indicates the beginning of the sidereal day, the column might be appropriately headed, the G.M.T. at sidereal noon. 64 ASTRONOMY FOR SURVEYORS. Given the G.M.T. of Transit of the First Point of Aries, to determine the L.M.T. of Transit at a Place in any other Longitude. The sidereal clock, as we have seen, is always gaining on the clock keeping mean solar time, at the rate of 9-8565 seconds per mean solar hour, or at the rate of 9-8296 seconds for each sidereal hour. Now the G.M.T. of transit of the First Point of Aries is the reading of the mean time clock when the sidereal clock reads hr. It is the difference between the readings of the two clocks at this instant. As the sidereal clock is gaining on the other this difference will get less as the time increases. Now, at a place in West longitude the transit of the First Point of Aries will take place after an interval of time measured in sidereal hours, minutes, and seconds by dividing the longitude by 15. Thus, when this transit occurs the mean time clock will not be so far ahead of the sidereal clock as it was at Greenwich, and the Green- wich reading of the mean time clock will have to be diminished by subtracting 9-8296 seconds for each hour of longitude. This reasoning assumes that, whilst different clocks at various places on the earth's surface will have different readings according to the longitude, the difference between the readings of the sidereal and mean time clocks at any place is the same all over the world at the same instant. This must be so according to the reasoning by which we have established the rules for determining the local mean and sidereal times at a place A, having given those at a place B. For we should alter both the sidereal and mean times at B by the same amount, depending on the difference of longitude between B and A, in order to find the corresponding times at A. Accordingly we get the Nautical Almanac rule for finding from the tables the time of transit of the First Point of Aries at any place. " If the place of observation be not TIME. 65 on the meridian of Greenwich, the mean time must be corrected by the subtraction of 9-8296 sec. for each hour (and proportional parts for the minutes and seconds) of longitude, if the place be to the West of Greenwich ; but by its addition, if to the East/' EXAMPLE. On August 1st, 1914, the G.M.T. of transit of the First Point of Aries is 15 hrs. 20 min. 28-63 sec. Compute the local time of transit on the same day (a) at a place in longitude 57 33' 28" West, (b) at a place in the same longitude East. (a) 57 33' 28" is equivalent in time to 3 hrs. 50 min. 13-87 sec. 3 x 9-8296 - 29-488 50 x -1638 = 8-190 13-87 x -0027 - -037 37-715, say 37-72 seconds. Therefore, for a place in West Longitude, we must subtract this from the 15 hrs. 20 min. 28-63 sec., giving 15 hrs. 19 min. 50-91 sec. as the L.M.T. of transit of the First Point of Aries. (b) For a place in East Longitude we must add the 37-72 seconds, giving 15 hrs. 21 min. 06-35 sec. as the L.M.T. of transit in this case. EXAMPLE. Given that the G.M.T. of transit of the First Point of Aries on August 30th is 13 hrs. 26 min. 27-26 min. Find the G.M.T. of transit on August 31st. Find also the local mean time of transit at a place in longitude 45 W. Ans. 13 hrs. 22 min. 31-35 sec. and 13 hrs. 25 min. 57-77 sec. Given the L.S.T. at any Place and the G.M.T. of Transit of the First Point of Aries on the same day, to determine the L.M.T. The local sidereal time measures the interval in sidereal hours since the transit of the First Point of Aries over that meridian. By turning this, therefore, into mean time hours we get the interval since the transit in mean time hours. But we have just seen how we may calculate the L.M.T. of transit of the First Point of Aries from the information in the Nautical Almanac. The addition of the two results will give us the L.M.T. required. The rule of procedure, therefore, may be expressed : Turn the 5 66 ASTRONOMY FOR SURVEYORS. given sidereal time into mean time and add it on to the computed L.M.T. of transit of the First Point of Aries. As the transit of * may take place at any time of the day, some care is necessary in selecting the right transit, as is illustrated in the following example : EXAMPLE. Given that the L.S.T. at Mount Hamilton is 22 hrs. 02 min. 47-02 sec. on October 2nd, 1913, the longitude of the place being 121 38' 43-35" West, find the corresponding L.M.T. Looking up in the Nautical Almanac the G.M.T. of transit of the First Point of Aries on October 2nd we find it is 11 hrs. 15 min. 45-49 sec. This is very near midnight, and the L.M.T. of transit will not be very different, If we were to add 22 hours on to this it will clearly carry us over into the next day, October 3rd, so that the transit we must select to work from, is that on October 1st. G.M.T. of transit of Y on October 1st, 11 hrs. 19 min. 41-39 sec. Allowance for longitude, to be subtracted, hr. 1 min. 19-71 sec. L.M.T. of transit of on October 1st, 11 hrs. 18 min. 21-68 sec. Mean time equivalent to 22 hrs. 02 min. 47-02 sec. sidereal, 21 hrs. 59 min. 10-32 sec. .. L.M.T. required = October 2nd, . 9 hrs. 17 min. 32 sec. Given the Sidereal Time at Mean Noon at Greenwich to compute the Mean Time at the next Transit of the First Point of Aries. The Nautical Almanac Columns, one giving the sidereal time at mean noon and the other the mean time of transit of the First Point of Aries, may readily be deduced one from the other. Thus, suppose the sidereal time at mean noon is denoted by s. Then at noon s sidereal hours have elapsed since was on the meridian, and, therefore, in 24 s sidereal hours *Y> will again be on the meridian. If we express 24: s sidereal hours in mean solar time, the result will clearly represent the number of mean solar hours that have then elapsed since noon, and TIME. 6t will consequently represent the mean time at the next transit of . Having observed the Altitude and Azimuth of a Star, the Time of Observation being noted, it is required to determine its Right Ascension and Declination. The latitude and longitude of the place of observation are supposed known. Then in the figure, Z being the zenith point, P the pole, and S the star, as before. In the spherical triangle Z S P, Z P is known, being the co-latitude ; Z S, the zenith distance, is also known, and the angle S Z P, which the vertical plane passing through the star makes with the meridian. Thus we know two sides and the included angle, and the triangle may be solved to find S P and the angle SPZ. LOCATION OF OBJECTS ON CELESTIAL SPHEEE. 75 The formulae to be used are those of the preceding problem. *1*#&$&3R sinSP- sin | (ZP+ZS) sin Z . sin Z S sinP ' The angle S P Z, being turned into hours, minutes, and seconds, at the rate of 15 for one hour, measures the sidereal time that will elapse before S comes to the meri- Fig. 17. dian if S is to the East, or the interval of sidereal time since S was on the meridian if it is to the West. But the right ascension of the star is the sidereal time when it is on the meridian. Therefore, to obtain the right ascension of the star, add the time value of the angle S P Z to the local sidereal time at the moment of observation if the star is to the East of the meridian, and subtract it if the star is to the West. The declination of the star is, of course, the complement of the computed polar distance S P. 76 ASTRONOMY FOR SURVEYORS. C. Having computed the Altitude and Azimuth of a Star for a Given Time of Observation, it is required to determine its Approxi- mate Position at some Short Interval of Time afterwards. When a surveyor is preparing for daylight observations of a star, it will be generally necessary for him to take at least two readings of its position. To give him time to- read the verniers and reverse the instrument before taking the second observation, he requires to know the altitude and azimuth of the star at an interval of five or ten minutes after the first reading. The computation for the second position may, of course, be made in precisely the same way as we have already done for the first, in which case several of the logarithms already taken out will be useful for the calculation. But it is rather shorter to make use of the following formulae : If x denotes the slight increase in the hour angle S P Z (to be reckoned negative if the angle is decreasing), y and z the corresponding small increases in the zenith distance Z S, and the azimuth angle P Z S respectively. Then y= sin PS sin PSZ .x. . . (1) cot^SZ^, sinZS The values of P S Z and Z S to be used in the equations being those found in the first calculation. To establish the formulae, let ABC (Fig. 18) be a spherical triangle. Then if b and c remain unchanged, we require to find the small changes y and z in a and B respectively if the angle A is increased by a small amount x. By the ordinary formulae for spherical triangles we have cos a = cos b cos c -f- sin b sin c cos A and cos (a + y) = cos b cos c + sin b sin c cos (A -f- x) LOCATION OF OBJECTS ON CELESTIAL SPHERE. 77 Subtracting gives cos a cos y sin a sin y cos a = sin b sin c (cos A cos a: sin A sin x cos A). Now, if x and y are very small, we can, if they are measured in circular measure, replace sin x and sin y by x and y respectively, and put cos x, cos y each equal to unity. Doing this, we get - y sin a = sin b sin c sin ^4 . #. Putting sin c sin A = sin C sin a, this becomes y = sin b sin C . #, ' which is the first formula given. Since we have here simply the ratio of y to .r, the result will hold good in whatever system of measurement y , and x are expressed, provided they c j are both measured in the same system, both in degrees or both in circular measure. Further, by the law of sines, sin(B+z) sin (A -fa) sin b sin (a+y)' Expanding and substituting as before, we get (sin B + z cos B) (sin a + y cos a) = sin b (sin A -f x cos A) and sin B . sin a = sin b . sin A. .-. substracting, and neglecting the product of two small quantities y and z, z sin a cos B -f y cos a sin B = x sin b cos A. 78 ASTRONOMY FOR SURVEYORS. y Putting x= . , . - sm b sin C /cos A cos B cos C z sin a cos B = y ( - sin B cos a ) = y : ~ - VsmC sm C ... z = y . , which is the second formula, sin a To illustrate the application of the formulae we will extend the scope of the example already worked out in Section A of this Chapter, and compute the position of Achernar 5 sidereal minutes after 7 p.m. From the previous work the angle P S Z = 123 58', P S = 32 19', Z S = 33 36'. sin PS, . . . . . . . 9-72803 sinPSZ, . 9-91874 -44338, . 1-64677 In this example the hour angle of the star is measured to the East, and, therefore, x is negative, and = 5 minutes of time = 1 15' of arc. .-. y = - -44338 x 75' = - 33'. .-. The new altitude is 56 24' + 33' = 56 57'. cot P S Z. 9-82844 sin Z S, 9-74303 1-2173, . ... . 0-08541 and cot P S Z is negative, .-. z = 1-2173 x ( 33') = 40'. .-. The new azimuth is 31 45' 40' = 31 5' East of South. If results are only required to the nearest minute, the above method is quite sufficient, provided the small differences are not much more than 2 degrees of arc. EXAMPLES. 1. Compute to the nearest minute of arc the altitude and azimuth of Sinus (dec. = 16 35' South, R.A. = 6 hrs. 41 min.) at a place in latitude 31 57' South at 12 hrs. sidereal time. Ans. Azimuth = 260 51'. Altitude 17 12'. 2. Compute the* altitude and azimuth of Sirius 10 sidereal minutes later than in I . Ans. Azimuth = 259 38' Altitude = 15 7'. LOCATION OF OBJECTS ON CELESTIAL SPHERE. 79 3. At a place in latitude 28 South at 1 hr. 37 min. sidereal time, th altitude of Canopus is observed as 33 3' and its azimuth as 136 44'. Com- pute the R.A. and dec. of the star. Ans. R.A. = 6 hrs. 21 min. 58 sec. Dec. = 52 38' 48" S. 4. What is the angular distance between the stars A (R.A., 4 hrs. 23 min. 53 sec., Dec., 16 04' 25" N.) and B (R.A., 2 hrs. 54 min. 34 sec., dec., 40 08'03"N.)? Ans. 30 54' 14". 5. Find the angular distance between A (R.A., 19 hrs. 42 min. 11 sec., dec., 8 23' 52" N.) and B (R.A., 22 hrs. 47 min. 41 sec., dec., 30 33' 17" N.). Ana. 59 06' 04". 6. If the N. dec. of a star is 40, show that the number of hours in the sidereal day during which it will be below the horizon of a place which has latitude 30 N. is 8-136. 80 CHAPTER VII. ASTRONOMICAL AND INSTRUMENTAL CORREC- TIONS TO OBSERVATIONS OF ALTITUDE AND AZIMUTH. Parallax. The fixed stars are so distant from us that their directions always appear to be the same, no matter from what point upon the earth's surface they are observed. Even with our most refined instruments no difference can be detected, because their distance is practically infinitely great in comparison with the diameter of the earth. But with the members of our own system, the sun, the moon, and the planets, we are dealing with bodies incomparably nearer to us, and their relative positions amongst the fixed stars of the sky are not precisely the same when viewed from different places. It is, therefore, essential that their registered right ascensions and declinations should be referred to some definite point upon the earth, in order that they may be available to all observers. The point selected is the earth's centre, because, having observed the direction of a planet from any station on the earth's surface, it is an easy matter to deduce its position as it would appear at the earth's centre, and conversely if the position of the star is tabulated as it would be seen from the centre of the earth we may readily find its position as seen from any place on the earth's surface. The selection of the earth's centre as the imaginary place of observation greatly simplifies the com- putations, and consequently most astronomical obser- vations of bodies in our own solar system are reduced ASTRONOMICAL AND INSTRUMENTAL CORRECTIONS. 81 so as to show what the result would be if the observation could have been at the centre of the earth. The registered right ascensions and declinations of the Nautical Almanac are those the different bodies would have if viewed from the earth's centre. The difference between the directions of a heavenly body as seen from the earth's centre and as seen from the place of observation is known as its Parallax. ^ Thus, as in Fig. 19, if S is the sun or planet observed, d ? Fig. 19. A the point of observation, and the earth's centre, the parallax of the body is the angle A S 0, the difference in the directions of A S and OS. If A H x is the direction of the horizontal at A, the altitude of S is the angle S A H 1 . If H 2 is drawn parallel to A Hj, then the difference of the angles S O H 2 and S A Hj = the difference of the 6 82 ASTRONOMY EOR SURVEYORS. angles SOB and SAB which = the angle A S O. Thus, if we call p the parallax, p= angle A S 0= S H 2 - S A Hj. Clearly the angle S H 2 is always greater than the angle S A Hj. If z denotes the zenith distance of S as observed from A, r the earth's radius A, and d the distance S, then, mn / 7) Y From the triangle A O S, - = - . sin z d If the body is observed on the horizon that is to say, if z= 90 the corresponding value of p is called the horizontal parallax. Call this P. Then sin P = *,. d Therefore, sin p = sin P . sin z. Since p and P are very small, except in the case of the moon, whose parallax sometimes exceeds 1, we may substitute the angles for their sines and write p = P sin z. The horizontal parallax of the moon and principal planets is given in the Nautical Almanac for every day in the year, and that of the sun at intervals of 10 days. The parallax for any other altitude is given by the above simple formula. Parallax is greatest when the body is in the horizon, and diminishes with the altitude until it becomes nothing when the body is in the zenith. We see from Fig. 19 that the effect of the parallax upon a celestial object is to make its altitude appear less when observed from A than it would be if seen from O. Consequently, when reducing observations to the earth's centre, we must add the correction for parallax observed to the altitude, or True altitude = observed altitude -f parallax. ASTRONOMICAL AND INSTRUMENTAL CORRECTIONS. 83 Parallax has no effect upon the azimuth of an object in the sky ; the correction is made to altitude only. This statement is strictly correct only when the earth is regarded as a perfect sphere. If 'the spheroidal form of the earth is taken into account there will be parallax in azimuth as well as in altitude. Even then, however, the correction in azimuth is too small to be worth con- sidering except in the case of certain special lunar obser- vations. The horizontal parallax of the sun ranges between 8-65 and 8-95 seconds. At an altitude of 60 its parallax is reduced to half of this. Atmospheric Refraction. When a ray of light passes from one medium into a denser medium as from air into water or from air into A glass, it is bent or refracted towards the normal to the bounding surface. Thus, as //////////////?/;(////////////// in Fig. 20, if a ray of light passes from the medium A to a denser medium B, travers- ing the path P Q R, the re- fracted ray Q R will always make a smaller angle with the normal to the separating surface than the incident ray P Q. The direction of bending is always such that the bent or refracted ray lies in the same plane as that passing through the incident ray P Q and the normal Q N. The law governing the amount of bending is that the ratio between the sines of the angles P Q N and R Q M is constant for these B Fig. 20. 84 ASTRONOMY FOR SURVEYORS. particular media and the value of this ratio is known as the coefficient of refraction. Similarly, when a ray of light from a celestial body reaches the atmosphere surrounding the earth, it is bent slightly out of its original path. If the atmosphere were a uniform homogeneous medium with a definite upper surface it would be comparatively easy to deter- mine the precise amount of bending of the ray. But the density of the atmospheric air diminishes with the height above the earth's surface. Consequently a ray from a star S (Fig. 21), when it reaches the upper limit of the Fig. 21. earth's atmosphere at A, is only very slightly bent, but the amount of bending gradually increases as it passes into the lower and denser layers of air. Its path from A to an observer on the earth's surface at is thus a curve, and the ray ultimately reaches the observer, so that it appears to him to come in the direction of S 1 . Thus, the observer sees the star apparently at S 1 in the celestial sphere, whereas in reality the star is at S. The effect is that the star is apparently raised above its true position, and its apparent altitude is greater than the true altitude ASTRONOMICAL AND INSTRUMENTAL CORRECTIONS, 85 if it could be observed from with no intervening atmo- sphere. The observed altitude of a celestial body must, therefore, be corrected in order to deduce its true altitude, the correction being always subtracted from the observed altitude. The amount of bending of the ray varies somewhat with the pressure and temperature of the air, but it is greatest for stars on the horizon, and gradually decreases to nothing for a star in the zenith. For a body on the horizon the mean value of the correction is 33' that is to say, a star will be just visible on the horizon when it is really 33' below it. Thus the sun, whose dia- meter is about 32', is visible just above the horizon when it is in reality just below it. It will be seen from the figure, since the refracted ray always lies in the plane containing the incident ray S A, and the normal to the spherical bounding surface at A, that S and S' will lie in the same plane as the vertical at 0. This means that refraction produces its effect entirely in altitude, and has no influence upon the apparent azimuth of a heavenly body. Thus no correction in azi- muth is necessary on account of refraction. As we do not know the exact laws which govern the pressure and temperature of the earth's atmosphere at different heights, nor even the distance to which it extends around the earth, no satisfactory computation of the amount of refraction at different altitudes can be made from theoretical considerations alone. By making different assumptions as to the character of the earth's atmosphere various formulae have been derived, but as their demon- stration generally requires mathematics of a rather advanced character, we shall not attempt the problem here. In any case, as we cannot be sure of the correctness of the assumptions that have to be made in order to derive the formula, the values of the constants used have to be obtained and checked from actual observations. There are various ways by which the amount of refraction 86 ASTRONOMY FOR SURVEYORS. at different altitudes may be actually measured, and for practical purposes that formula is selected which best fits the results of such measurements. The formula that has found most favour, and which has been most used by astronomers for this purpose, is that of Bessel, r= A (B Z) M T N cot a, where a = the apparent altitude, r= the amount of refraction in seconds of arc, B, a factor depending on the height of the barometer, t, a factor depending on the reading of the thermometer attached to the barometer, T, a factor depending on the reading of a ther- mometer so exposed as to give the temperature of the external air. A, M, and N are factors depending on the altitude of the celestial body. When suitable values are given to the different factors, this formula can be made to fit in with the results of actual observations on refraction with great precision, and where great accuracy is required this is the formula that is most generally adopted. To use the formula it is, of course, put into the logarithmic form log r = log A+ M (log B -f log t) + N log T + log cot a, and the values of M, N, log A, log B, log t, and log T are obtained from appropriate tables. Such a table is published in Chambers' Mathematical Tables. The constants M and N in the above formula do not differ sensibly from unity if the altitude is considerable. If these are taken each= 1, the formula may be put into a form which makes the application of tables much simpler. For the values of B, t, and T are each unity for certain particular values of the barometric height, and for certain special temperatures of the attached and ASTRONOMICAL AND INSTRUMENTAL CORRECTIONS. 87 unattached thermometers. Consequently for this par- ticular condition of the atmosphere, which we may take as the standard condition, we have r = A cot a. If now we denote by r 1 the amount of the refraction for any other temperature and pressure, we have ^ = A . B t . T cot a, ... r 1 ='BxtxTxr, or refraction = the refraction for altitude a under the standard or mean conditions multiplied by the factors B, t, and T, depending on the height of the barometer and the temperatures recorded by the attached and unattached thermometers. A table of refractions constructed for standard con- ditions of the atmosphere is commonly termed a table of mean refraction. With the aid of such a table and sub- sidiary tables for B, t, and T, we may first of all find the value of the " mean refraction " for the measured altitude, then pick out the values of B, t, and T for the particular conditions of the atmosphere, and the true refraction the mean refraction x B x t x T. This is the method of determining the refraction most commonly adopted for ordinary purposes, and gives accu- rate enough results unless the altitude is very small. The necessary tables are in Chambers' Mathematical Tables. For many purposes, and more especially for high altitudes, it is quite sufficiently accurate to use the value of the refraction as given in the mean refraction table. The refraction is always less than 1' if the altitude is greater than 45, and for zenith distances up to 20 the refraction is practically 1" per 1. Corrections to Observations on Account of Residual Instrumental Errors. It forms no part of the purpose of this book to enter upon a discussion of the construction of the ordinary instruments of the surveyor and the methods 88 ASTRONOMY FOR SURVEYORS. of adjustment. These are matters dealt with in text-books on Surveying. It will be assumed that the reader is acquainted with the construction of the sur- veyor's transit theodolite and with the usual methods of securing its accurate adjustment. But even when the adjustments have been made with great care, there commonly remain certain residual errors which affect the accuracy of the celestial observations, and must be taken into account if the best results are to be obtained. Of these, the two most important are, (1) an error due to the fact that the line of collimation of the telescope is not accurately at right angles to the transverse axis about which the telescope turns, and (2) an error produced if this transverse axis is not absolutely horizontal. We will consider the effect of each of these in turn. The Effect of an Error of Gollimination. Let us suppose that the line of collimation of the telescope, instead of being accurately at right angles to the axis about which the telescope turns, is in error by a small angle c ; that is to say, the telescope makes an angle 90 c on one side and 90 -fc on the other side with the axis. On turning the telescope about the transverse axis, which is adjusted so as to be horizontal, the line of collimation would, if in accurate adjustment, trace out a vertical plane passing through the zenith. But if in error, and the line of collimation is not at right angles to the axis, then, as it is plunged up and down, it will trace out a conical surface and on the celestial sphere it will trace out a circle parallel to a vertical circle through the zenith. Thus, as in Fig. 22, if there were no collimation error the line of collimation of the telescope would trace out the great circle Z S' N, but if in error it will sweep out the parallel small circle L S M. Now, suppose that the star S is observed in such a telescope, and let S S' be an arc of a great circle drawn at right angles to Z N. S S' = N M = c the collimation error. ASTRONOMICAL AND INSTRUMENTAL CORRECTIONS. 89 If we draw the great circle arc Z S, then Z S is the true. zenith distance of the star. But the observed zenith distance is Z S'. Similarly the correct azimuth is measured by the angle H Z S, whereas the azimuth as read on the instrument is H Z S'. In the right-angled triangle S S' Z, S S' being denoted by c, we have cos S Z = cos S' Z cos c. If c is very small, as should be the case if the instrument is in decent adjustment, we may take cos c= 1, and, therefore, practically S' Z = S Z, or no correction will / 1 I s \ ^ 1 I" N M Fig. 22. usually be necessary to the observed zenith distance or altitude. Also, denoting by Z the angle S Z S', the error in azimuth, we have sin c = sin S Z . sin Z, and since c and Z are both small, we may write Z = c . cosec S Z, or the error in azimuth = the collimation error multiplied by the cosecant of the zenith distance. The error in azimuth thus becomes very great if the star is near the zenith, but is= c for a star on the horizon. 90 ASTRONOMY FOR SURVEYORS. The following table shows the way in which the error varies with the altitude of the star : Error in Azimuth corresponding to a Collimation Error c for Various Altitudes of Object. Altitude of Star, 30 60 70 80 85 89 Error in azimuth, c M5c 2c 2-92c 5-76c ll-47c 57-3c The Elimination of Instrumental Errors by Changing Face. Although we have in the preceding paragraph investi- gated the effect of a given collimation error, it is very seldom that the surveyor will need to take this error into account, because in all important work the observa- tions are taken in such a way as to eliminate its effects. This is done by observing each angle twice, with the vertical circle or face alternately to the left and to the right. After the angle has been read once the telescope is reversed in direction by turning about its horizontal axis, and the whole of the upper part of the theodolite is turned through 180 until the first object is again sighted, and the angle is again read with the instrument in this reversed position. The operation is commonly referred to as " changing face/' and should be adopted in all theodolite observations, as it gives a means both effectual and simple of eliminating the chief instrumental errors. An error in collimation will not affect the hori- zontal angle between two objects if both are at the same altitude, but if the altitudes are different, then if the collimation error makes the measured angle a little too great when the vertical circle is facing the left it will make it just as much too small when the vertical circle faces the right, and thus the "mean of the two readings gives the correct result. Now, when measuring the azimuth of a star, we have to sight the telescope to a moving object, and it is not possible, therefore, to exactly repeat the measurement because in ASTRONOMICAL AND INSTRUMENTAL CORRECTIONS. 91 the interval of time taken in changing face the position of the star is slightly changed. But it is characteristic of all the more accurate methods of astronomical measure- ment suitable for the surveyor, that reliance is never placed upon one observation, but the methods are so arranged that a series of observations can be made at short intervals, the face of the instrument being alter- nately changed from right to left, so that a mean may be obtained from which instrumental errors are largely eliminated. The Error made if the Transverse Axis of the Telescope is not truly Horizontal. This error, just as that due to collimation with which we have just dealt, may also M Fig. 23. be largely eliminated by the method of changing face. But in this case the elimination is not so perfect, and as it is an easy matter by means of a striding level to actually measure the departure of the axis from the horizontal at each observation, it is frequently desirable to observe the error and allow for it in the computation. If the axis of the telescope is not truly horizontal, the line of collimation, when the telescope is turned about the axis, will not trace out a great circle in the sky passing through the zenith, as it should do, but will trace out a great circle inclined to the vertical. Thus in Fig. 23, if N Z N 1 denotes the great circle that would be traced 27 44' 00" 227 D 43' 45" 11 8 09' 00" Mean angle between star and R.M., 109 34' 52". R.M. to East of Star. CALCULATION. Formula. Sin A = cos declination x sec latitude. log cos dec., 9-7829945 log sec lat., . . 10-0862497 log sin A, ." A from South, Azimuth of Star, . Angle between R.M. and Star, Bearing of R.M., . 9-8692442 47 44' 227 44' 109 34' 52" 1 18 09' 08" THE DETERMINATION OF TRUE MERIDIAN. 109 io^.-l^ ^ *o o (M kl 1 (M 00 ^ CO CO II sk 1 ^ O O Tf rti 50 2 t> bb ii * 33 "cS 1 33 II O 10 O CO O CO 10 O i i Oi O5 B 0000 00 GO 5 O5 Oi r-H ^H O O 0000 0000 X 00 00 D PH PH as 110 ASTRONOMY FOR SURVEYORS. COMPUTATION FOR AZIMUTH. First Pair. Second Pair. Observed altitude, Refraction, . Corrected altitude, 45 50' 37 -5" 55" 45 49' 42 -5" 46 16' 17-5" 54" 46 15' 23-5" Zenith distance z, . . Co-latitude c, . Polar distance p, . , 44 10' 17-5" 55 04' 22" 37 21' 13" 43 44' 36-5" 55 04' 22" 37 21' 13" 2s, . . . '., , - 5, . . ' . ' . . . S C, > . 136 35' 52 -5" 68 17' 56" 13 13' 34" 136 10' 11-5" 68 05' 06" 13 00' 44" S Z, . . . 24 07' 38-5" 24 20' 29-5" L sin (s z), : ; L sin (. s 0000 CO IO O CO CO O CO O O CO CO 1 F-H r-H O O 1 I 03 X C* we find that when x o d A - =o, dx d 2 A cot p d x 2 sin z d 3 A 3 cot p cos z THE DETERMINATION OF TRUE MERIDIAN. HI Therefore, by Taylor's Theorem .in z 2 COt pOOB .(-..) gin . r; sin 2 z provided that A A and z z are expressed in seconds of arc. To get some idea of the relative values of the terms in this series, we find, if the star observed has a polar distance of 30 and the latitude is also 30, then z = 54 44' 09", and if z~ z = 1, the second term works out at 66" and the last term to 0-8". If z z = 2 the values become 264" and 6" respectively. The last term in (9) is equal to coe'poosc o 3 ,, sin p (cos 2 p cos 2 c) and has, therefore, an infinite value if p= c, in which case the star passes through the zenith. This is clearly of no practical importance. The following are the values of the last terms in different latitudes for a star 30 distant from the celestial pole, if z-s = 1:- Latitude. Value of Last Term in (9). 50, 3-5" 40, 1-5" 30, 0-8" 20, 0-4" 10, 0-2" 0, 0" If z z = 2 the preceding values should be multiplied by 8. It follows, therefore, that for the ordinary work of the surveyor the correction involved in the last term of the series is quite negligible for observations extending over 142 ASTRONOMY FOR SURVEYORS. a range of altitude of 2, or 1 on each side of elongation, provided that the star does not pass within 10 of the zenith. At places near the equator the observations may clearly extend over a very much greater range of altitude with the same degree of precision. To determine over what range of time the observations may extend, we find on differentiating the equation cos z = cos c cos p + sin c sin p cos h d z sin c sin p sin h that - -= sin p for a star at elongation. d h sin z This = -I, if p=30. Thus, the rate of change of altitude at elongation does not depend on the latitude, but simply on the polar distance of the star, and for a star distant 30 from the pole we have dh= 2dz. Therefore, if dz= 1, dh= 120' of arc, or 8 minutes of time, the altitude of the star near elongation thus changes by 1 in about 8 minutes. For stars closer to the pole the time taken for the same change of altitude will be greater. Practical Computation. We conclude that for a set of observations extending over a range of altitude of about 2, or 1 on each side of elongation, occupying, in the case of a star with a polar distance of 30, about 16 minutes of time, it is amply sufficient to use the formula _^ = co^(,-^ sinr/ sin z 2 It should be noticed that the error made by the use of this formula in the final reduction of a set of obser- vations will be very much less than the error made in the reduction of the single observation furthest from elonga- tion. We have based the stated limitations upon the error made in the reduction of the single observation, THE DETERMINATION OF TRUE MERIDIAN. 143 so that for a complete set of observations the time occupied may be extended somewhat beyond the limits given above. In low latitudes the observations may extend over a greater range than in high latitudes. In latitude 10, for instance, the observations may extend over half an hour, and formula (10) will still give the average result of the set of readings correct within less than 1". If the range of altitude is too great, or it is desirable to compute A A with the greatest precision possible, then this value must be reduced if z>2 , or increased if 2 ^ OS OS O OS 00 OS 05 OS 6 - 1 4 " I a 1 M ta 2 H? fi 00 OS I> IO lO CO fO g Th O CO CO O OS O ^H PH WOO O (M O (N F 1 r-4 s ; r . Ill ^ cr j| -a p, ! r I i a ! | J o 5S H c^ a 1 a g - '<- I H I 1 IS! i eg o> o C~i & S 1 s . S 1 WO h^ ft w For Approximate Latitude. vv vTh CO i^H t^-CO OCO t^-O i-HC*5 4 tan 1 -2~- * A, -A, tan 32 09' 30", = 6 40' 8-9597747 9-9064310 18-8662057 9-7984562 9-0677495 and - L ~^ Z =38 52' 30" Aj = 45 32' 30" sin P! = sin 37 22', . sin Aj = sin 45 32' 30", cos lat., .-. latitude = 31 45' 20". 9-7831268 9-8535522 9-9295746 TABLE GIVING VALUES OF m FOR REDUCTION OF CIRCUM-MERIDIAN OBSERVATIONS. 2 sin* ~ The values of m are given in seconds of arc. THE DETERMINATION OF LATITUDE. 177 Additional Seconds of Time. Value of t , of Time. 10 20 30 40 SO 0-0 0-1 0-2 0-5 0-9 1-4 I 2-0 2-7 3-5 4-4 5-4 6-6 2 7-8 9-2 10-7 12-3 14-0 15-8 3 17-7 19-7 21-8 24-0 26-4 28-8 4 31-4 34-1 36-9 39-8 42-8 45-9 5 49-1 52-4 55-8 59-4 63-0 66-8 6 70-7 74-7 78-8 83-0 87-3 91-7 7 96-2 100-8 105-6 110-4 115-4 120-5 8 125-7 130-9 136-3 141-8 147-5 153-2 9 159-0 165-0 171-0 177-2 183-5 189-8 10 196-3 202-9 209-6 216-4 223-4 230-4 11 237-5 244-8 252-2 259-6 267-2 274-9 12 282-7 290-6 298-6 306-7 315-0 323-3 13 i 331-7 340-3 349-0 357-7 366-6 375-6 14 384-7 393-9 403-3 412-7 422-2 431-9 15 441-6 451-5 461-5 471-5 481-7 492-0 16 502-5 513-0 523-6 534-3 545-2 556-1 17 567-2 578-4 589-6 601-0 612-5 624-1 18 635-9 647-7 659-6 671-6 683-8 696-0 For intermediate values of t the corresponding values of m may be found by simple interpolation. EXAMPLES. 1. At a place in latitude North, the true zenith distances of a Cephei (declination 61 58' 21-1") is determined as 26 54' 28-3" N. The zenith distance of a Aquike (declination 8 29' 22-7") is found as 26 34' 27-5" S. Find the latitude of the place. Ans. 35 03' 51-5". 2. In latitude 30 S. the times of transit of a star whose declination is 20 S. are observed across the prime vertical. If the direction of the prime vertical is in error by 1, show that the measured interval of time will be too great by about 14 seconds. 3. An observation made in Antarctica on November 19th, 1912, gave the altitude of the sun's centre as 42 07-8', the temperature being 17 F. and the barometer reading 27-2 inches. Correct for refraction and parallax, and compute the latitude of the place, given that the sun's declination is 19 21-6' S. Ans. 67 14-7' S. 12 178 ASTRONOMY FOR SURVEYORS. 4. The declination of the sun being 20 39-9' S., its meridian altitude is observed as 43 17'. The correction for refraction and parallax being 00-9', determine the latitude of the place. Ans. 67 23-8' S. 5. The sun is observed on the prime vertical, morning and afternoon, the times by watch being 7 hrs. 30 min. and 4 hrs. 14 min. The sun's declination is 17 31' 30". Compute the latitude. Ans. 37 17' 30". 6. At a place in S. latitude the interval between the passages of Sirius across the prime vertical is 6 hrs. 9 min. 19 sec. mean time. The mean readings of the bubble on striding level were 10 N. and 14 S., each division being = 20". The declination of the star is 16 35' 33" S. What was the latitude of the place of observation ? Ans. 23 20' 17" S. 7. The hour angle of Aldebaran (dec. 16 20' 15" S.) when on the prime vertical was found to be 4 hrs. 35 min. 19-5 sec. What was the latitude of the place of observation ? Ans. 39 04' 3" S. 8. At a place in the Southern Hemisphere y z Ceti (dec. 2 51' 22" N.) was observed at equal altitudes of 48 02' 20", and the interval in mean solar time between the two occurrences was 16 min. 12 sec. Required the latitude of the place. Ans. 43 50'. 9. Antares crossed the prime vertical at 13 hrs. 52 min. sidereal time. Find the latitude of the place of observation. R.A. of Antares, 16 hrs. 23 min. Dec. 26 13' S. Ans. 31 54' 49" S. 10. The altitudes of a star when it crosses the meridian and prime vertical are respectively 65 and 10 (corrected). Find the star's declination and latitude of place. Ans. Lat., 29 58' 39". Dec., 4 58' 39" S. in S. lat. or N. in N. lat. 11. The altitude of Sirius on the prime vertical is found to read 39 48'. The declination of Sirius is 16 35' 20" S. Find the latitude of the observing station. Allow for refraction. Ans. Lat., 26 30' 1" S. THE DETERMINATION OF LATITUDE. 179 12. At a place in South latitude the altitude of a star was observed at its upper and at its lower culminations, the altitude corrected for refraction at upper culmination being 60 45' 15". and at lower culmination 10 16' 15". Find the latitude of the place of observation and the declination of the star. Ans. Lat., 35 30' 45". Dec. S., 64 45' 30". 13. On the evening of 8th February, 1914, at a place in S. latitude, the magnetic bearing of ft Hydri at its Western elongation was 185 47' 35", and that of Argus a* its Eastern elongation was 137 24' 42". Declination of ft Hydri, .... 77^44'29"S. Argus, .... 63 56' 36" S. Determine the latitude of the place and the magnetic variation. Ana. Latitude, 36 24' 56". Variation, 9 30' 20" E. 14. The altitude of Regulus at 10 hrs. 08 min. sidereal time was 46 52' 32" (fully corrected). From the Nautical Almanac we find : R. A. of Regulus, 10 hrs. 03 min. 17 sec. Declination of Regulus, . . . .12 26' N. What was the correct altitude when on the meridian ? Ana. 46 52' 37-4". 15. On 9th March, 1914, at a place South of Equator in 140 E. longitude the following altitudes of a Virginis (Spica) were observed near its meridian passage and their times taken with a chronometer keeping local mean time : Observed Altitudes. Local Mean Times. 57 40' 36", 2 hrs. 02 min. 18 sec. a.m. 44' 34", ..... 05 min. 54 sec. 48' 40", ..... 10 min. 50 sec. 50' 10", ..... 15 min. 58 sec. ,, 49' 30", 22 min. 10 sec. 46' 40", 27 min. 00 sec. 42' 35". ..... 31 min. 02 sec. The sidereal time at G.M.N., March 8th,,is 23 hrs. 1 min. 22-91 sec. R.A. of Spica = 13 hrs. 20 min. 41-4 sec. Declination of Spica = 10 43' 00" S. Find the latitude of the place. Ans. 42 52' 51". 16. The declination of a star being 40 S., what are the latitudes of the places where its meridian altitude will be 80 ? Ans. 50 or 30 S. 180 ASTRONOMY FOR SURVEYORS. 17. In south latitude two stars are observed on the meridian, one north and the other south of the zenith, the difference of zenith distances being found to be 13' 03-45" N., the declinations of the stars being 45 38' 37-48" S. and 42 44' 04-63" S. respectively. Find the latitude. An*. 44 17' 52-8". 18. A south circumpolar star was observed at equal intervals shortly before and after its elongation, when it was found to change its altitude from 44 35' to 47 35', during an interval of 19 min. 47 sec., by watch keeping correct mean time. Find the polar distance of the star and the latitude of the place of observation. Ans. 37 20' 30". Latitude = 33 27' 58. 19. At 6.10 p.m., local mean time, by watch on 15th September, 1907^ in longitude 151 06' 30" East, the magnetic bearing of r Octantis was 170 37' 30", the bearing of the referring mark being 72 50' 45", and the observed altitude of the star was 34 36'. R.A. of Octantis, . . . .19 hrs. 12 min. 48 sec. Declination of Octantis, . . 89 14' 49" S. Sidereal time at G.M.N., Sept. 15th, 11 hrs. 33 min. 12 sec. Sept. 14th, 11 hrs. 29 min. 15 sec. Find the latitude of the observer and the true bearing of the referring mark. Ans. Latitude = 33 54' 19". 20. On March 6th, 1914, the altitude of Polaris, when corrected for instrumental errors and refraction, is found to be 46 17' 28", the mean time of observation being 7 hrs. 43 min. 35 sec. p.m. and the longitude of the place 37 W. Sidereal time at G.M.N., March 6th, 22 hrs. 53 min. 29-8 sec. R.A. of Polaris, March 6th, . . 1 hr. 27 min. 37-3 sec. N. declination of Polaris, March 6th, 88 51' 8" Find the latitude. Ans. N. 46 3' 35". 21. The observatory at Stockholm is in latitude 59 20' 33" N., and that at the Cape of Good Hope in latitude 33 56' 3-5" S. The declination of Sirius is 16 35' 22" S. Find the altitudes of Sirius when on the meridian at Stockholm and at the Cape of Good Hope respectively. Ans. 14 04' 05" and 72 39' 18-5". 22. The upper transit of a South circumpolar star was observed to occur at 7 hrs. 05 min. 28 sec. p.m. local mean time, and to reach its greatest THE DETERMINATION OF LATITUDE. 181 western elongation at 11 hrs. 44 min. 30 sec. p.m., when its observed azimuth was 33 48'. Find the latitude of the place of observation and the declination of the star. , Ans. Latitude, 31 02' 52" S. Declination, 61 32' 11" S. 23. On March 13th, 1911, at a place South of the Equator, in longitude 9} hours E., at 6 minutes before apparent noon, the altitude of the sun's lower limb was found to be 58 04' 20", at which time clouds prevented further observation. The sun's declination at G.M.N., March 13th, is 3 15' 07-4" S., and on March 12th 3 38' 41-8" S. Find the latitude of the place by reduction to the meridian, the sun's .semi-diameter being 16' 07", its parallax 5", and refraction 37". Ans. 35 02' 28". 24. The altitudes of a star when it crosses the meridian and the prim vertical of a place are a and b. If Hs the latitude of the place, show that cot I = tan a sec a sin b. 25. The meridian altitude of Altair is 51 55' 45", its declination being 8 34' 34" N. and the meridian altitude of 3 Pavonis is 52 54' 32", its North polar distance being 156 36' 18". Find the latitude of the place of obser- vation. Ans. 29 30' 15-5" S. 26. At a place, south of the equator, the meridian zenith distances of the two stars y* Norma and In the triangle EPS S P = p = polar distance of star, B P= 180 1= supplement of latitude, Angle P B S e = error measured by striding level, Angle B P S= x= required error in time of transit. .-. cot S P sin B P = cot e sin x -f- cos B P co^ x. .-. treating x and e as small quantities, x cot p sin / = - - cos I. sin (Z-f p) sin altitude x=e - , ore- sin p cos dec. This formula gives us the hour angle of the star at the moment of observation. Usually e and, therefore, x will be in seconds of arc, and x must then be divided by 15 to determine the error of the observed time of transit in seconds of time. Clearly the transit will be observed either too soon or too late according to the direction of tilt of the transverse axis. If the star transits below the pole, x will be the supple- ment of the angle B P S, and we get sin (I p) sin alt. x= e - , which again = e - sin p cos dec. The error in time in this case increases with the altitude. EXAMPLE. At a place in latitude 30 S. the sidereal time of transit of a star across the meridian is observed to be 12 hrs. 30 min. 17-5 sec., the declination of the star being 58 30' S. The readings of the striding level, one division of which = 13", are : L. R, 6-0 5-0 3-6 . 7-2 9-6 12-2 9-6 4 ) 2-6 0-65 0-65 X 13 --= 8-45". DETERMINATION OF TIME BY OBSERVATION 187 sin 61 30' .-. error m hour angle = 8-4o X ^rs-^7 = 14-21 . sin 31 30 This is equivalent to 0-95 second of time. As the right-hand side of the axis is the higher, and the telescope is directed towards the South, the transit is, therefore, observed too soon by this amount-, and the corrected time of transit across the meridian is 12 hrs. 30 min. 18-45 sec. Meridian Transits on Both Sides of the Zenith. A consider- able improvement may be made in the accuracy of the method by taking observations of the times of transit of two stars, one on each side of the observer's zenith. In Fig. 43, let Z denote the zenith, P the celestial pole, A Z P B the direction of the true meridian, and C Z D the direction of the meridian actually set out, the figure being drawn as though the celestial sphere were viewed Fig. 43. from above. Suppose that the times of transit of two stars are observed, one at S t and the other on the opposite side of the zenith as at S 2 . Then, since both stars move in the same direction, as shown by the arrows, if the observed time of transit of Sj is later than it should be, owing to the faulty determination of the meridian, the time of transit of S 2 will be correspondingly earlier. If the stars are well selected, it may be that the time errors of the two observations are equal and opposite, so that the mean of the two results will give a correct time determination in spite of the error in the setting out of the meridian. This will be the case if the hour angle S x P Z is = the angle S 2 P Z, for then one observation will be just as much too soon as the other one is too late. 188 ASTRONOMY FOR SURVEYORS. The conditions that this may be the case are readily obtained as follows : Let angle B Z D = e= meridian error, and suppose that the hour angle S x P Z = S 2 P Z == h, c = co-latitude P Z. Then, from the triangles S x P Z, S 2 P Z, % sin h sin Z S x sin Z S 2 sin e sin P S x sin P S 2 But, since the error e is small, we may write very approximately P Sj = c Z Sj^ and P S 2 = c + Z S 2 . sin (c - Z SJ _ sin (c + Z S 2 ) sin Z Sj sin Z S 2 sin c . cot Z S l cos c= sin c cot Z S 2 + cos c. cot Z Sj cot Z S 2 = 2 cot c. This, then, is the condition that has to be satisfied by the zenith distance of the two stars if the observations are to be so balanced that by taking the mean of the two we eliminate, or nearly so, the error due to a faulty setting out of the meridian. The following table, based upon the above formula, gives the proper zenith distance of the star on the opposite side of the zenith to the pole, corresponding to different zenith dis- tances of the other observed star, for different latitudes : Zenith Distance of Star Zenith Distance of Star o to the i Opposit Pole. e Side of Zenith Side as Pole. Lat. 10. Lat. 20. Lat. 30. Lat. 40. Lat. 50. Lat. 00. Lat. 70. Lnt. 80. 5 5 09' 5 20' 5 34' 5 51' 6 18' 7 09' 9 34' 85 0' 10 10 39' 11 26' 12 29' 14 30' 16 55' 24 22' 80 0' 20 22 40' 26 21' 32 08' 43 05' 70 0' 30 35 56' 44 53' 60 0' 86 55' . 40 50 0' 65 07' 87 53' . . 50 64 04' 83 39' . 60 77 20' 70 89 21' DETERMINATION OF TIME BY OBSERVATION. 189 The advantage of selecting the two stars in this way may be illustrated by a computed example. Suppose that the place of observation is in latitude 30, and that the polar distance of the 'star observed on the same side of the zenith as the pole is 40, so that its zenith distance is about 20. Suppose, further, that the marked meridian is as much as 1 in error. Computing with these data the spherical triangle S P Z of Fig. 42, it may be shown that the hour angle S P Z is 2 min. 04-8 sec. In other words, the observed transit will take place too soon by thi& amount. Now, according to the table, the star observed on the opposite side of the zenith should have a zenith distance of 32 08'. Suppose it actually has a zenith distance of 32, equivalent to a polar distance of 92. Then, computing in the same way the hour angle of this star when on the faulty meridian, we find that its observed transit will be too late by 2 min. 04 sec. Thus from one observation the chronometer would be set too fast by 2 min. 04 sec., and from the other it would be set too slow by about the same amount, and the mean of the two observations would give the time correct to the nearest second in spite of the fact that the direction of the meridian is 1 in error. If, however, the zenith distances of the two stars are not balanced in the way indicated, the accuracy of the mean result is nothing like so great. If, for example, the two zenith distances were the same, the star observed on the opposite side of the zenith to the pole having a zenith distance of 20, or a polar distance of 80. Then, on computing the spherical triangle, it will be found that the observed transit of this star is too late by 1 min. 24 sec., so that the mean of the two observations is then in error to the extent of about 20 seconds. 190 ASTRONOMY FOR SURVEYORS. Second Method By Extra Meridian Observations of Sun or Star. This is, as a rule, the most convenient and suitable method for the determination of time by the surveyor. It consists in the measurement of the altitude of sun or star when out of the meridian, at the same instant noting the chronometer time. Then, from a knowledge of the latitude of the place and the declination of the body observed we may compute the proper local time at the instant of observation, and so determine the error of the chronometer. The most favourable time for making such an obser- vation will be when the altitude of the celestial body is Fig. 44. changing most rapidly, and this will be the case when it is near the prime vertical. This position has also other advantages, as we shall see in the course of the discussion. As an altitude has to be measured, refraction must be allowed for, and as there is considerable uncertainty about this at low altitudes, the star observed should have an altitude of at least 15. The method involves the solution of the same spherical triangle that we have discussed in connection with extra- meridian observations for azimuth. Thus, in Fig. 44, if DETERMINATION OF TIME BY OBSERVATION. 191 S is the star observed, then in the spherical triangle Z P S we know the three sides : Z P = c = co-latitude, S P = p = polar distance of star, Z S= z= zenith distance, or the complement of the observed altitude. Therefore, we can compute the hour angle S P Z, from which we can find the local sidereal time if we know the R.A. of the star, or this at once gives us the local apparent time in the case of the sun. Let the angle S P Z =h. Then, we have three available formulae adapted to logarithmic computation, any one of which may be used for computing h. They are if s = ^ (z + c -f- p) h /sin (s c) . sin (s p) sin - = y 2 sin c . sin p h /sin s . sin (s z) cos -- = v 2 sin c . sin p h /sin (s c) . sin (s p) tan - = y - ~ : ; : - 2 sin s . sin (s z) The Choice of a Formula. Of the three formulae, that for cos is somewhat the simplest, as we must find s in any case, and we have then only to find s z in addition. With the sine formula we have one more subtraction to make, but there is the advantage that only tables of log sines are used, and there is less risk of mistake in taking out the logarithms. If, however, we are utilising the same observation, as may be done, for the determination of azimuth in addi- tion, then we shall require to compute also the angle S Z P. In this case it is a decided advantage to select 192 ASTRONOMY FOR SURVEYORS. the tangent formula for the computation of both angles, for we shall then need only to look up four logarithms, as the same expressions sin s, sin (s c), sin (s p), and sin (s z) will occur in the tangent formulae for both angles. If, on the other hand, we use the sine or cosine formula for the two angles, it will be necessary to look up six logarithms. Another important point in the selection of a formula is this. The variation in value of the tangent of an angle, as the angle increases from to 90, is very much greater than in the case of a sine or cosine. Consequently a table of tangents will enable us to determine the value of an angle with greater precision than a table of sines or cosines. This is of practical importance when the angle under consideration is near to or 90. Thus there is very little variation in the value of the cosine of an angle up to 2 or 3, and, if we wish to determine the values of such small angles to seconds, a table of cosines is not nearly so good as a table of tangents. Similarly, there is very little variation in the sine of an angle near to 90, and it becomes difficult to compute such angles with precision from a sine table. It follows, therefore, that if h is near or near to 90, the tangent formula is the best one to adopt. Data Necessary for Computation. In addition to the measured altitude, we require a knowledge of the latitude of the place and the declination of the body observed. The declination for a star is taken straight from the Nautical Almanac, but the declination of the sun has to be found by using approximate values for the longitude and local time. If the result obtained shows that the assumed local time is -very much out, the calculation should be repeated by using the corrected value of the local time found from the first computation. Arrangement of the Computation. It is worth some trouble to make a neat form for the computation. A good DETERMINATION OF TIME BY OBSERVATION. 193 arrangement reduces the work, and is an aid to accuracy. The following, for instance, is the method adopted in the printed forms of the Queensland Survey Department : p = 5934 / 48 // log sin 9-9356770 c= 76 05' log sin 9-9870611 z= 66 34' 19" 19-9227381 2 ) 202 14'0r' subtract from 20- 9 n _ 41 32' 15-5" lo S~ -- : = 0-0772619 s ~ P ~ sin p sin c 8-c = 25 02' 03-5- i ogsin 9-8215856 log sin = 9-6265032 2 ) 19^5253507 .-. i h= 35 22' 48" log sin 9-7626753 Where the same observation is to be utilised for both time and azimuth, a neat device is to proceed as follows : log sin (s c) = say 9-949960 ' log sin (sp) = 9-046045 Iogsin (s z) 9-875721 28-871726 subtract log sin s 9-945558 2 ) 18-926168 9-463084 From this we have simply to subtract log sin (s z) 7 ry and log sin (s p) in order to get tan and tan - f respectively. 9-463084 9-463084 log sin (sz) = 9-875721 9-046045 log tan- 9-587363 log tan - 10-417039 Having Computed the Hour Angle to Find the Time of the Observation. In the case of a star the angle S P Z, turned into time by dividing by 15, measures the interval 13 194 ASTRONOMY- FOR SURVEYORS. of sidereal time after or before the time of culmination, according as the star is observed on the West or East of the meridian. But the R.A. of the star is equal to the sidereal time at the instant of culmination. Therefore, the sidereal time at the moment of observation is obtained by adding (or subtracting) the value of h to the R.A. of the star. This may be turned into mean time in the way already discussed. Thus, if the R.A. of the star is 7 hrs. 30 min., and the angle h is 35, the star being observed in the West, then the local sidereal time at the moment of observation is 7 hrs. 30 min. + 2 hrs. 20 min. = 9 hrs. 50 min. If the sun has been observed, the value of the angle h at once gives us the interval of solar time before or after the meridian transit of the sun that is to say, it gives us the local apparent time. To convert this into mean time the equation of time must be determined at that par- ticular instant. To do this we first find the corresponding Greenwich apparent time, by allowing for the difference of longitude, and then take the equation of time from page 1 of the Nautical Almanac, allowing for the hourly variation. Suppose, for example, that the angle k, for a sun observation, is 48 20', the observation being made at a place in longitude 60 W. on May 23rd in the afternoon. We have, therefore, Local apparent time, . . .3 hrs. 13 min. 20 sec. Longitude, . . . . .4 hrs. min. sec. Greenwich apparent time, May 23rd, 7 hrs. 13 min. 20 sec. We have then to find the equation of time at this instant. The Nautical Almanac gives for this date, 1914, the equation of time at apparent noon, Greenwich, as 3 min. 30-40 sec. The variation in one hour is given as 0-191 second, the equation decreasing on successive days. The Almanac states that the equation of time is to be subtracted from apparent time. Hence, at the given instant, Equation of time = 3 min. 30-40 sec. 7-222 x 0-191 sec. = 3 min. 29-02 sec. Therefore, the required mean time is 3 hrs. 13 min. 20 sec. 3 min. 29-02 sec. = 3 hrs. 09 min. 50-98 sec. DETERMINATION OF TIME BY OBSERVATION. 195 Averaging Several Observations of the Same Star. In practice it is usual to take at least two, and commonly four, observations in as quick succession as possible, half being taken with F.L. and half with F.R. The computa- tion is then made as though one observation only had been taken, the mean of the altitudes being assumed to be the true altitude at the mean of the noted chrono- meter times. The object of this procedure is to eliminate instrumental errors, but this is done at the expense of introducing another error due to the fact that the assumption made is not mathematically exact. The investigation of the magnitude of the error thus introduced into the work is too complex for insertion here, but it may be stated that the surveyor is quite safe in thus averaging altitude observations extending over a range of 2 in altitude under ordinary conditions. The error thus made in an extra- meridian time determination is then generally only a small fraction of a second of time, its exact magnitude depending upon the latitude of the observer, the declina- tion, and hour angle of the heavenly body. It is least when the hour angle is nearly 90. Observations on Both East and West Stars. It is a great improvement in accuracy to take one set of observations upon a star in the east and another corresponding set, under as similar conditions as possible, upon a star in the West. The averaging of two such sets of observations tends to eliminate certain classes of errors, and this should always be done where the highest accuracy is sought. If, for example, the refraction assumed is too great, the corrected altitude will be too low, and the computed time will be too early for a star in the east, while it will be correspondingly too late for a star in the west. If the two errors are about equal, as will be the case if the E. and W. stars make about the same horizontal angle with the meridian, and are observed at about the same 196 ASTRONOMY FOR SURVEYORS. altitude, then the average of the two sets of results will be correct. Similarly, the effects of any systematic error in the measurement of altitude are eliminated by pairing sets of observations in this way. The same applies to extra meridian observations for azimuth. EXAMPLE OF EXTRA MERIDIAN OBSERVATION ON SUN FOR TIME. Forenoon Observations. Place Survey Office, Adelaide. Thermometer 56. Longitude 9 hrs. 14 min. 20 sec. E. Barometer 30-49 inches. Latitude 34 55' 38" S. Date -15th July, 1914. Value of 1 division of bubble 10". Standard Meridian 9 hrs. 30 min. E. Chronometer keeping approximately standard time. OBSERVATIONS. Vertical Angles. Level. Observed Limb. Face. Chronometer Time. A. B. Mean. E. 0. H. M. S. L L 20 08' 50" 20 09' 10" 20 09' 00" 10 10 9 30 23 U R 20 49' 10" 20 49' 00 '' 20 49' 05" 11 9 9 31 29 L R 20 27' 20" 20 27' 00" 20 27' 10" 11-5 8-5 9 32 40 U L 21 07' 20" 21 07' 50" j 21 07' 35" : 10 1 10 9 33 44 Means, . ' 20 38' 12" 10-6 9-4 9 32 04 Computation for sun's declination at assumed approximate time of observation. Approximate standard time of observation, 14/7/14, . Difference for standard meridian, Corresponding G.M.T., . Declination : 14th July, 1914 (G.M.N.), . Difference for 12 hrs. 02 min. 04 sec., 21 hrs. 32 min. 04 sec. 9 hrs. 30 min. 00 sec. 12 hrs. 02 min. 04 sec. 21 47' 03-3" 04' 28-6" Declination at instant of observation (North), 21 42' 34-7" Sun's South Polar Distance, . . 111 42' 34-7" DETERMINATION OF TIME BY OBSERVATION. 197 h I sin (s c) sin (s p) Formula Tan - = / ' * L. 'Y sin s sm (a z) CALCULATION. Mean of observed altitudes, . . . 20 38' 12" Level correction, ..... 6" 20 38' 06" Refraction and parallax, .... 2' 21" Corrected altitude, . . . .20 35' 45" Zenith distance = z, .... 69 24' 15" Co-latitude = c, 55 04' 22" Sun's polar distance = p, . . . 111 42' 35" 2s, 236 11' 12" *, 118 05' 36" s-c, 63 01' 14" s - p, 6 23' 01" s-z, 48 41' 21" log sin (s-c), 9-949960 log sin (s - p), ... . 9-046045 logcosecs, .... . 10-054442 log cosec (.9 - z), 10-124279 log tan"*, 19-174726 log tan | = tan 21 08' 28", . - 9-587363 h, 42 16' 56" h (in time), ...... 2 hrs. 49 min. 08 sec. Local apparent time = 24 hrs. h, . .21 hrs. 10 min. 52 sec. Longitude, , . . . . . . 9 hrs. 14 min. 20 sec. Greenwich apparent time, . . . 11 hrs. 56 min. 32 sec. 198 ASTRONOMY FOR SURVEYORS Equation time at G.A.N., ... 5 min. 33 sec. Correction for 11 hrs. 56 min. 32 sec., . 3 sec. Equation time instant observation, . 5 min. 36 sec. L.A.T., . .... 21 hrs. 10 min. 52 sec. L.M.T., . ...... . .. . 21 hrs. 16 min. 28 sec. Diff. Standard Merid., . 15 min. 40 sec. Local Standard time, . . . . 21 hrs. 32 min. 08 sec. Chronometer time, . . .-." . . 21 hrs. 32 min. 04 sec. Error of Chronometer, . . * 04 sec. slow EXAMPLE FOR REDUCTION. With the same instrument as that used in the preceding observation a similar set of four sun observations was taken on the afternoon of July 21st, 1914, at the same place. The mean altitude obtained was 23 53' 36", the average alidade level readings were E. 10-5, 0, 9-5. The mean of the chronometer times was 2 hrs. 52 min. 52-5 sec. From the Nautical Almanac Declination of sun, at G.M.N., July 20th, 1914, 20 47' 18-2" N. Variation in one hour at noon on the 20th, . 27-60" 21st, . 28-47" Equation of Time, G.A.N., July 20th (to be added to apparent time), . . . . . 6 min. 05-99 sec. Variation in one hour, . ? . . . 0-165 sec. Longitude, standard time, and latitude are given in the preceding case. The chronometer being supposed to keep standard time, determine its error. Ans. 02-1 sec. slow. The Effect of an Error in Latitude. It is important that we should know to what degree of precision the latitude must be known in order that the time may be determined. This may be readily investigated in a manner similar to that adopted with corresponding problems previously. From the spherical triangle S Z P of Fig. 44, cos z = cos c cos p-\- sin c sin p cos h. If c is too large by a small amount y, then, for the same measured zenith distance z, h will be too small by an amount x, and we shall have cos z = cos (c + y) cos p-\- sin (c + y) sin p cos (h x). DETERMINATION OF TIME BY OBSERVATION. 199 Subtracting these two equations, and treating x and h as small quantities, we readily get cos c cos h sin p -}- sin c cos p x=y cot Z sin c sin c sin p sin h where Z denotes the azimuth angle S Z P. This shows that x will be very large compared with y, if Z is nearly equal to 0, or if c is nearly 0. That is to say, a small error in the latitude will produce a very large error in the time if the body is observed near to the meridian, or if the observation is made in high latitudes near to either terrestrial pole. On the other hand, if Z is 90 i.e., if the observation is made on the prime vertical x is 0, and an error in latitude makes no difference. In this case the angle S Z P is a right- angled triangle, and we can get a relation between p, z, and h that does not involve c at all, so that a knowledge of the latitude is unnecessary. If the observation is made near to the prime vertical, therefore, an error in latitude will produce very little effect on the time determination. The following table, based upon the above formula, gives the error in time corresponding to an error of 1' in the latitude for different azimuth angles : ERROR IN TIME CORRESPONDING TO 1' ERROR IN LATITUDE.* Azimuth of Observed Body. Latitude of Place. 0. 30. 40. 50. 60. Seconds. Seconds. Seconds. * Seconds. Seconds. 45 4-0 4-6 5-2 6-2 8-0 60 2-3 2-6 3-0 3-5 4-5 80 0-7 0-8 0-9 1-1 1-4 90 0-0 0-0 0-0 0-0 0-0 * If the word Declination be substituted for latitude, the same table will give the error in time due to an error of 1' in the Declination, the first column representing, not the azimuth, but the angle Z S P. 200 ASTRONOMY FOR SURVEYORS. This all points to the desirableness of making the observation as near to the prime vertical as possible. The Effect of an Error in the Measured Altitude. By a method similar to that adopted in the last paragraph it may be readily shown, if x is the error in the hour angle corresponding to an error y in the observed altitude, that x = y cosec Z cosec c x clearly becomes very great if either Z or c are small, and it has its least value when Z and c are each 90. Thus, again, an error of observation has the least effect when the observation is made on a celestial body near the prime vertical, and the most favourable place for making the observation is at the equator. TABLE SHOWING ERROR IN TIME DETERMINATION OWING TO AN ERROR OF 1' IN THE MEASURED ALTITUDE, WITH DIFFERENT AZIMUTHS OF THE OBSERVED BODY. Latitude of Place. Azimuth of Observed Body. 0. 30. 40. 50. 60. Seconds. Seconds. Seconds Seconds. Seconds. 45 5-6 6-4 7-3 8-7 11-3 60 4-6 5-3 6-0 7-1 9-2 80 4-1 4-7 5-3 6-3 8-1 90 4-0 4-6 5-2 6-2 8-0 1 This table deserves a little careful consideration, as it shows the degree of precision with which altitudes must be measured if the time is to be determined within one second. Under the most favourable possible conditions an error of J minute of arc will cause an error of one second in the time, and it may produce an error of two seconds or even more. EXAMPLE. In the extra -meridian observation for time set out at length in paragraph just preceding show that an error of 1' in the measured altitude will produce an error of 7 seconds in the time. DETERMINATION OF TIME BY OBSERVATION. 201 The Effect of an Error in the Declination of the Sun caused by a Defective Knowledge of Longitude or Local Time. With star observations the Nautical Almanac gives us the declination of the star with all the precision that is re- quired, but with sun observations the surveyor has first of all to compute the declination. To do this he requires to know both his longitude and the approximate local mean time. From the formula cos z = cos c cos p + sin c sin p cos h it appears that the relation between an error in p and an error in h will be of precisely the same nature as the relation between an error in c and an error in h. So that if x denotes the error in the hour angle corresponding to an error y in the declination cot Z S P x= - . y. sin p Thus the table already given, showing the error in time caused by 1' error in latitude, also gives the error in time caused by 1' error in declination, provided that the first column is taken as representing the angle Z S P instead of the azimuth. We have already seen that the maximum rate of varia- tion of the declination of the sun is a little less than 1' per hour. So that to get the declination of the sun to the nearest minute it is sufficient to know the time to the nearest hour. But one hour of time corresponds to 15 of longitude, so that it is seldom that the surveyor will not know his longitude sufficiently well for this purpose. It will be seen from the table that, in order to deter- mine the time to the nearest second, it will be necessary to know the declination within only about one-fifth of a minute of arc under almost the worst conditions of obser- vation considered in the table. For this it will be usually sufficient to know the local time within a quarter of an hour. 202 ASTRONOMY FOR SURVEYORS. If the local time is not known with sufficient accuracy, its value must be assumed for the purpose of finding the approximate declination. This is then used in a preliminary calculation made to determine the time. The calculation is then made over again, using the approxi- mate local time so found in order to get a more accurate value of the sun's declination, which in turn is used in the computation to obtain a more accurate determination of the local mean time. Third Method By Equal Altitudes. If a star be observed at the same altitude on opposite sides of the meridian , the two observations must clearly be made at equal intervals of time before and after the star's meridian Fig. 45. transit. Thus, in Fig. 45, if the star be observed in the two positions, Sj and S 2 , so that the zenith distances- Z Sj and Z S 2 are equal, then, if P is the celestial pole, the two hour angles Z P S x and Z P S 2 must be equal. It follows that the mean of these two observed times is the time of the star's meridian transit. But the local sidereal time at the instant of the star's meridian transit is determined by the star's R.A., which is given by the Nautical Almanac. This local sidereal time may be reduced to mean time, and a comparison of this with the average of the two observed chronometer tinier determines the error of the chronometer. DETERMINATION OF TIME BY OBSERVATION. 203 With stars the method is capable of giving very accurate results, and it has the great advantage that no knowledge is required of latitude, declination, or even azimuth, and errors of graduation of the instrument have no effect upon the result. But to the surveyor it has the obvious drawback that a considerable interval of time must elapse between the observations. As the accuracy of the determination depends upon the altitude being the same at the two observations, the star should have an altitude of something more than 45, in order to get rid of the uncertainties of refraction near the horizon. EXAMPLE. On September 1st, 1914, jj Crucis was observed East of the meridian at 10 hrs. 42 min. 30-5 sec. by a chronometer keeping sidereal time. It was again at the same altitude West of the meridian at 14 hrs. 51 min. 20-7 sec. Find the error of the clock. East 10 hrs. 42 min. 30-5 sec. West. 14 hrs. 51 min. 20-7 sec. 2 ) 25 hrs. 33 min. 51-2 sec. Meridian transit by chronometer, . 12 hrs. 46 min. 55-6 sec. R.A. of star, 12 hrs. 42 min. 41 sec. Chronometer correction, . . . 4 min. 14-6 sec. As the chronometer is too fast, the correction is to be subtracted from the chronometer reading. If, as is more usual, the chronometer keeps local mean time, the sidereal time at the meridian transit of the star must be reduced to local mean time in order to compare with the chronometer time. This cannot be done without a knowledge of the longitude. EXAMPLE. At a place in longitude 8 hrs. 35 min. 27 sec. East, on the evening of September 1st, 1914, the star a Pavonis is observed East of the meridian at 7 hrs. 9 min. 20-5 sec., with a watch keeping local mean time. It is again observed at the same altitude to the West of the meridian at 9 min. 30-2 sec. after midnight. Find the error of the watch, having given G.S.T. at G.M.N., September 1st, 1914, 10 hrs. 39 min. 13-38 sec. R.A. of a Pavonis, .... 20 hrs. 18 min. 57-4 sec. Ans. 8-1 seconds slow. 204 ASTRONOMY FOR SURVEYORS. It is desirable, in order to make the determination as precise as possible, that a series of observations should be made upon the star on each side of the meridian, instead of one observation only. A few times should be taken when the star is on the East of the meridian at altitudes differing by 20 or 30 minutes of arc. A corre- sponding series of times should then be taken when the star is on the West of the meridian at the same altitudes. Since all that we want to ensure is that the altitude is the same at corresponding observations East and West of the meridian, there is no particular object in reversing the face of the instrument. The whole set of observations may be taken with the one face. The Error due to a Slight Inequality in the Altitudes of two Corresponding Observations. If in Fig. 45 Z S = zenith dis- tance of the first observation = z, Z P = co-latitude = c P Sj = polar distance = p h = hour angle Z P Sj_ Z = angle S x Z P = azimuth of star cos z = cos c cos p -f- sin c sin p cos h t . ( 1) Suppose now that at the second observation the zenith distance, instead of being z, is z + y, being in error by a small amount y. Then the hour angle Z P S 2 will be in error by a corresponding amount x, so that instead of being h, it will be h + x. Then, from the spherical triangle Z P S 2 , cos (z+ y) = cos c cos p+ sin c sin p . cos (h+ x). (2) Subtracting (2) from (1), treating x and y as small quantities, we get y . sin z = x sin c sin p sin h . sin z sin p But sin h sin Z ' *=-- -V- sin c sin Z DETERMINATION OF TIME BY OBSERVATION. 205 We see thus that the error x in the hour angle, corre- sponding to an error y in the second altitude, will be least when Z = 90, and will be greater the smaller the value of Z, We draw, therefore, the practical conclusion that the ob- servations are best made on stars near the prime vertical. If the declination of a star is slightly less than the latitude, it will cross the prime vertical near the zenith and the interval between the times of transit will be small. This, therefore, is a convenient observation to make, and the conditions are favourable to accuracy. The Determination of Time by Equal Altitudes of the Sun. The above method is an extremely simple one as applied to the stars, because the -declination of a star remains constant during the period over which the observations extend. But in the case of the sun the declination changes so rapidly that it cannot be considered as constant, and the theory becomes complicated by the fact that allowance must be made for the alteration of declination in the interval between the observations. Referring again to Fig. 45, if p denotes the polar distance of the sun when it is on the meridian, then at the first sight, when the sun is at S l5 the polar distance will be py, and at the second sight, when the sun is at S 2 , the polar distance will be p=f y. The -f or sign is to be taken in the first of these expressions according as the sun is approaching or leaving the elevated pole. If p were constant, we should have cos z = cos p cos c + sin p sin c cos h. But if at the first observation, S 1? the polar distance is p+ y, the hour angle will be h + x, and we have cos z = cos (p + y) cos c -f sin (p + y) sin c cos (h -f x). Subtracting these two equations, and treating x and y as small quantities, we get = y sin p cos cy cos p sin c cos h + x sin h sin c sin p. x = y (cot c cosec h cot p cot h). 206 ASTRONOMY FOR SURVEYORS. Under these conditions the first observation will be made when the sun is at an hour angle h -fa before apparent noon, where x is given by the preceding expression, and it may be positive or negative according as cot c cosec h is < or > cot p cot h. Similarly the second observation will be made with the sun at an hour angle h x after apparent noon, and it may be shown in the same way as before that the value of x is given in this case also by the same mathematical expression. The mean of these two observed times will therefore be when the sun is at an hour angle x before apparent noon. When the sun is leaving the elevated pole, instead of approaching it, the mean of the two observed times will be when the sun is at an hour angle x after apparent noon. Thus, the true time of transit i.e., the time of apparent noon is given by Mean of observed times yV y (cot c cosec h cot p cot h). y is the alteration in the sun's declination in half the time interval between the two observations. h is half the time interval between the two observations reduced to angular measure. The + sign is to be taken if the sun is leaving the elevated pole, and the -- sign when it is approaching the elevated pole. Just as with star observations, it is necessary, in order to obtain the best results, that a series, say four or six, of observations should be taken to the sun in the forenoon and a corresponding set in the afternoon, the sights in each case being taken alternately to the upper and lower limbs. EXAMPLE. At Adelaide, longitude 9 hrs. 14 min. 20 sec. E., latitude 34 55' 38" S., on July 21st, 1914, equal altitude observations of the sun DETEKMINATION OF TIME BY OBSERVATION. 207 were taken in the forenoon and afternoon. The means of the noted times were 9 hrs. 35 min. 03 sec. a.m. and 2 hrs. 37 min. 15 sec, p.m. by a watch keeping mean time. 12 hrs. 00 min-. 00 sec. subtract 9 hrs. 35 min. 03 sec. 2 hrs. 24 min. 57 sec. add 2 hrs. 37 min. 15 sec. 2 ) 5 hrs. 02 min. 12 sec. = time between observations. 2 hrs. 31 min. 06 sec. .'. h=3T 46' 30". subtract from 2 hrs. 37 min. 15 sec. hr. 6 min. 09 sec. = time by watch at apparent noon. c = 55 04' 22" Declination at G.A.N., July 21st, . . 20 36' 02-5" Correction for longitude, .... 2' 41 -5" Declination at L.A.N., . . . .20 38' 44' .'. p, 110 38' 44' cot c . cosec h . . . = 1-140 cot p cot h . . . . = -486 cot c cosec h cot p cot h . . = 1-626 Change in declination in 2 hrs. 31 min. 06 sec. = 71-69", and sun is approaching elevated pole, 1-626 x 71-69 .-. time of apparent noon = 6 09 - =-= seconds 15 = 6' 09"- 7-6" = 6' 01 -4". But, from the Nautical Almanac, the equation of time to be added to apparent time at L.A.N. is 6' 08-3", which is, therefore, the true time of apparent noon. Thus the watch is 7 seconds slow. Fourth Method -Almucantar Method for Time Observations. In 1884 Mr. S. C. Chandler, at the Harvard College Observatory, U.S.A., devised a form of instrument in which the telescope was fixed at a constant angle with the vertical, so that the line of sight traced out a hori- zontal circle on the celestial sphere, and observations for the determination of latitude and other purposes were made by noting the times of transit of stars across the fixed horizontal circle. The instrument was named an 208 ASTRONOMY FOR SURVEYORS. " almucantar," and it proved to be capable of very- remarkable work. The same principle may be readily applied with an ordinary theodolite, and experience has shown that extremely accurate determinations of time are possible in this way.* Any horizontal circle may be used for the observations, but the most convenient is the one that passes through the pole of the observer. This has been named the " co- latitude circle/' its zenith distance being everywhere equal to the co-latitude. The formulse for reduction then become very simple. The method consists in observing the times of transit of a series of East and West stars, some- where near the prime vertical, across the horizontal Fig. 45a. wire of a telescope that is set to an altitude equal to that of the pole. Allowance must be made for refraction, and, therefore, the telescope is actually set so that its altitude as read off on the vertical circle is equal to the latitude of the place plus refraction. In Fig. 45a, Z denotes the zenith, P the celestial pole, A and B the North and South points, P S Q the co-latitude circle. Let S denote the position of a star, somewhere near the prime vertical, as it crosses the co-latitude circle. * See paper by W. E. Cooke, " On a New and Accurate Method of deter- mining Time, Latitude, and Azimuth with a Theodolite " Monthly Noticee,. Royal Astronomical Society, January, 1903. DETERMINATION OF TIME BY OBSERVATION. 209 Let Z P = c = co-latitude. P S = p= star's polar distance, measured, of course, along the great circle arc P N S and not along the small circle P S Q. Angle S P Z = h = hour angle of star. Angle S Z P = Z = azimuth of star measured from elevated pole. Then, since Z S = c, Z S P is an isosceles triangle, and, if Z N be drawn perpendicular to the great circle arc joining S and P, it will divide S Z P into two equal right- angled triangles. From the triangle Z N P cos N P Z = tan P N cot Z P P P cos h tan - . cot c = tan . tan I . (1) ' 2i if I is the latitude of the place. To determine the azimuth at which a star will cross the co-latitude circle, from the same triangle cos Z P = cot N Z P cot N P Z. Z cos c=cot h . cot , 2 or cot f sin I . tan h. . . (2) 2i Formula ( 1 ) enables the time of transit to be com- puted, and formula (2) gives the azimuth if required. If an observation on one star in the East is balanced by a corresponding observation on a star in the West of somewhere about the same declination, then the mean of the two time observations will give a correct result even if the co-latitude circle is considerably out. If, for instance, the co-latitude circle is set out too low, the observed time of transit in the East will be too soon, but that in the West will be too late, and if there is not much 14 210 ASTRONOMY FOB SURVEYORS. difference in the declinations of the stars the time of transit will be just as much too soon in the one case as it is too late in the other. Thus by averaging the two results any small error in the setting out of the co-latitude circle is practically eliminated, and it is not necessary, therefore, in order to apply the method that the latitude of the place should be known with precision. An approxi- mate latitude will suffice. For precisely the same reasons as have been investi- gated when dealing with extra-meridian observations for time, slight errors in latitude, declination, and altitude will have least effect upon the result when the stars observed are near the prime vertical. The stars should be selected from a zone of about 20 on each side of the prime vertical. EXAMPLE. On May 3rd, 1903, in Lai. 31 56' 45" S., the transit of /? Orionis was observed in the West across the co-latitude circle at 8 hrs. 55 min . 1 -5 sec. by a watch keeping sidereal time. The transit oj a Virginia icas similarly observed in the East at 9 hrs. 20 min. 23-4 sec. Determine the error of the watch. B Orionis. a Virginis. Declination, . . 8 19' 2-7" S. 10 39' 30-1" S. p, . . ..'".. 8140'57-3" 7920'29-9" \ ]>, . - . . 40 50' 28-6" 39 40' 15" log tan |, . . 9-9367323 9-9187412 log tan/, -. . 9*7948752 9-7948752 log cos h, . . . 9-7316075 9-7136164 /*, . . . . 57 22' 58" 58 51 '31" I) in time, . . 3 hrs. 49 min. 32 sec. 3 hrs. 55 min. 26 sec. II. A. of star, . . 5 hrs. 09 min. 52-6 sec. 13 hrs. 20 min. 07-5 sec. Computed time, . 8 hrs. 59 min. 24-6 sec. 9 hrs. 24 min. 41 -5 sec. Observed time, . 8 hrs. 55 min. 01-5 sec. 9 hrs. 20 min. 23-4 sec. Error of watch (slow), 4 min. 23-1 sec. 4 min. 18-1 sec. Mean determination of watch error. 4 min. 20-6 sec. slow. Adjustment of Telescope during Observation. It is the most essential thing for accurate work, in observations DETERMINATION OF TIME BY OBSERVATION. 211 of this kind, that the telescope should throughout make exactly the same angle with the horizontal. It is not of such importance that the -altitude should be exactly equal to the latitude, ; a,s r it is that the altitude should remain the same throughout the observations. Now, no matter how carefully a transit theodolite is adjusted, the bubble attached to the vertical circle will not remain precisely in the centre of its run as the telescope is turned from star to star. It is, therefore, essential to accurate work that this bubble should be adjusted to the centre of its run just before the star crosses the horizontal wire in each case. This must be done, of course, by the ad- justing screw on every transit theodolite that moves both telescope and vertical circle together without affecting the altitude reading. After .the reading on the vertical circle has been set for the first star so that the altitude is equal to the latitude plus refraction, the altitude screw which would alter this reading must on no accdunt be touched. But at each observation the horizontal line of the vertical circle must be adjusted without altering the reading of the vernier. To get the most accurate results observations must be made upon a number of stars, at least six in the East and six in the West, and the mean of all the determinations is taken. The East and West stars should be selected so that the angles in azimuth that one set make to the East are as nearly as possible equal to the angles that the other set make to the West. Sun Dials. Whilst the sun dial does not provide the surveyor with a means of determining local time with anything like the precision obtainable by the methods that have been described, it enables the time to be fixed quite sufficiently near for the regulation of watches and clocks for ordinary 212 ASTRONOMY FOR SURVEYORS. purposes, and the instrument may be read just as easily as a clock. It is especially useful in the remote parts of sparsely populated countries where no other means of checking the clock times are available. When a sun dial is illuminated by the direct light of the sun the shadow of a straight line or sharp straight edge is thrown upon a plane containing a graduated circle so marked that the apparent solar time is indicated by the reading at the place where the shadow intersects the circle. The plane containing the graduated circle may be either horizontal, vertical, or inclined. The straight edge, the shadow of which is thrown upon the circle, is always set up so as to be parallel to the earth's axis. It is called the stile, or gnomon of the dial. When the gradu- ated circle or " plane of the dial " is horizontal we have what is known as a horizontal dial, and as this is the most common form we will consider it first. The Horizontal Dial. In Fig. 46, let M B L A represent the plane of the dial, which we may suppose to be ex- tended indefinitely so that M B L A is the circle in which it intersects the celestial sphere. C P is the direction of the gnomon, which again we may suppose to be produced to intersect the celestial sphere in the celestial pole P. B P A is the plane of the meridian. If now S denotes the position of the sun, the line of intersection of the shadow of the gnomon C P with the plane of the dial will be the line of intersection of the plane containing C P and S with the plane M B L A. MPL represents in the figure the plane passing through S and C P, and M C L is the line of intersection of this plane with the plane of the dial, or C L is the direction of the shadow of the gnomon. Neglecting the slight alteration in the declination of the sun during the hours of daylight, S will describe a circle uniformly on the celestial sphere about P as centre. The angle S P B is the hour angle of the sun, DETERMINATION OF TIME BY OBSERVATION. 213 decreasing or increasing uniformly with the time according as the observation is made in the morning or in the after- noon. Then in the right-angled triangle L P A A P = I = latitude of place. Angle A P L = h = hour angle of sun. A L = x = required division along the dial corresponding to hour angle h. sin 1= cot h tan x, or tan x= sin / tan h. Thus, to graduate the dial for the hourly intervals before and after noon, we must put h= 15, 30, 45, etc., in succession and compute the corresponding values of x, knowing, of course, the value of /. Thus, if the latitude of the place is 30, the first hourly division on each side of noon will be marked out at an angle with C A given by log tan x = log sin 30 -f log tan 15, from which x= 7 38'. The next hourly division, indicating either 10 a.m. or 2 p.m. will make an angle with C A given by log tan x = log sin 30 + log tan 30, from which x 16 6', and so on. The reading of the shadow of the gnomon gives the 214 ASTRONOMY FOR SURVEYORS. local apparent time which must be corrected by the equa- tion of time, as given by the Nautical Almanac, in order to obtain the mean time. A table of corrections may easily be drawn out for different times of the year. The Prime Vertical Dial. In this case the plane of the dial lies in the prime vertical. In Fig. 47 let A L B M be the plane of the dial, which we will again suppose is continued on indefinitely, so as to cut the celestial sphere. C P, the direction of the stile or gnomon, is again parallel to the earth's axis, but this time P will be the celestial pole below the visible horizon. APB is the plane of the meridian. Then if, as in the previous case, S denotes the position of the sun on the celestial sphere, the apparent movement of S is to describe a circle on the celestial sphere with P as centre, and the hour angle of S is the angle SPA. The shadow of P C thrown by S upon the plane of the dial will be C M, the line of intersection of the plane passing through S and P C with the plane of the dial. In the right-angled spherical triangle P B M P B = 90 - / = co-latitude. Angle B P M = h = hour angle of sun. B M = x = required division along the dial correspond- ing to the hour angle h. cos I = cot h tan x or tan x = cos I tan ^, and by this formula the dial may be graduated in a similar manner to the horizontal dial. Fig. 47. DETERMINATION OF TIME BY OBSERVATION. 215 Oblique Eials. If the plane of the dial is inclined to the horizontal the dial is said to be " oblique/' There is one case that is particularly simple, and has given rise to some of the simplest sun dial constructions. This is the case in which the plane of the dial is tilted so as to be perpendicular to the stile, so that it coincides with the plane of the celestial equator. With this arrangement the shadow of the stile on the dial moves round uniformly with the revolution of the sun and the hour divisions on the dial are consequently uniformly spaced. Fig. 48. Time of Rising or Setting of a Celestial Body. This is not of much value for the determination of time, because of the uncertainty of refraction on the horizon. In Fig. 48, if A S B be the plane of the horizon, Z the zenith, P the celestial pole, and S the body, which is exactly on the celestial horizon, then the spherical triangle P S A is right-angled at A, and cos SPA=cot SP tan PA. cos (hour angle S P Z) = tan dec. tan lat. 216 ASTRONOMY FOR SURVEYORS. From this the hour angle of the body at rising or setting may be computed, and this will determine the apparent solar time in the case of the sun or the sidereal time if a star is observed. We have here neglected the effect of refraction, which, amounting as it does to about 36' on the horizon, will cause stars to be just visible when they are really 36' below the horizon. To find the azimuth of the body, we have cos S P = cos S A cos P A, sin dec. or cos S A = cos lat.' EXAMPLES. 1. At a place in lat. 35 S., the bearing of a wall is 1 10. Find the apparent time at the equinox when it casts no shadow. Ans. 3 hrs. 50 min. 24-5 sec. p.m. 2. Find the true bearing and apparent time of sunrise in lat. 32 S. when the sun's declination is 20 S. (Take the sun's centre and neglect refraction and parallax.) Ans. Bearing, 113 47' 05". Time, 5 hrs. 07 min. 25 sec. 3. Rigel was observed East of the meridian on the horizontal wire of a theodolite at 7 hrs. 05 min. 20 sec. p.m. by a watch which is supposed to keep West Australian standard time (120th meridian). It was also observed at the same altitude West to cross the horizontal wire at 1 hr. 25 min. 30 sec. a.m. Neglecting the rate of the watch, find its error. Date of first observation, . . . January 5th, 1908. Longitude of locality, .... 1 15 50' 26" E. Sidereal time at G.M.N., January 5th, . 18 hrs. 54 min. 45-83 sec. Sidereal time at G.M.N., January 6th, . 18 hrs. 58 min. 42-39 sec. R.A. of Rigel, 5 hrs. 10 min. 07-29 sec. Ans. 16 min. 9-8 sec. slow. 4. On July 16th, 1910, in latitude 33 15' 13" S. and longitude 10 hrs. 04 min. 50 sec. E., the observed altitude of the sun's centre was 31 54' 45" bearing 10 35' 15" magnetic, the referring mark bearing 86 54' 15" magnetic, time by watch being 10 hrs. 48 min. DETERMINATION OF TIME BY OBSERVATION. 217 The sun's declination at noon on July 15th at Greenwich was 21 38' 18" N., and the mean hourly difference 23-05" decreasing. The equation of time to be added to apparent time is 5 min. 46-18 sec., and the hourly increase 0-25 sec. Find the true bearing of the referring mark, the magnetic variation, and the error of the watch. Ans. Bearing, 98 34' 46". Variation, 11 40' 31" E. Watch error, 3' 04-3" fast. 5. At a place 40 51' 20" S., 140 20' 30" E., at 9 hrs. 10 min. 20 sec. a.m. by a watch on 2nd September, 1910, the sun's preceding limb was found by compass bearing to be 58 14' 20", and the observed altitude of the upper limb 27 11' 15". Declination at G.M.N., September 1st, 8 31' 00-7" N. ; hourly variation, 54-24". Declination at G.M.N., September 2nd, 8 09' 14-4" N. ; hourly variation, 54-58". Sun's semi-diameter, G.M.N., September 1st, 15' 52-61". September 2nd, 15' 52-84". Equation of time (to be added to apparent time), G.A.N., September 1st, 9-04 sec. Equation of time (to be subtracted from apparent time), G.A.N., Sep- tember 2nd, 9-66 sec. What was the declination of the compass and the correct mean time of observation ? Ans. Declination, 9 21' 15" West. Mean time, 9 hrs. 07 min. 57 sec. 6. At a place in latitude 32 S. a vertical rod 6 feet high casts a shadow 15 feet long in a direction bearing 75 12'. What is the apparent time and the approximate time of year ? Ans. 5 hrs. 5 min. p.m. December. 7. If the time be found by a single altitude, show that a small error in the latitude will have no effect on the time when the body is in the prime vertical. 8. At 5 p.m. by watch on September 8th at a place in latitude 31 57' 08-4" S., longitude 7 hrs. 43 min. E., the observed altitude of the sun's centre (corrected for instrumental errors) was 29 58' 25-2". Sun's declina- tion at G.A.N., September 8th = 5 45' 55-9" N., variation in one hour 56-40". 218 ASTRONOMY FOR SURVEYORS. Equation of time to be subtracted from apparent time = 2 min. 18 sec. Find the sun's true bearing and the error of the watch on West Australian standard time (120th meridian). AIM. Bearing, 299 49' 06-32". 9. On January 3rd, 1914, at a place latitude 30 15' S., longitude 148 E., the following sun observation was taken : Observed Altitude. Alidade Approximate Local Mean A,, W I A f r -,m T? \r Bubble. Time by Watch. roui B.M. |O 38 07' 15" E. 0. 37 8 hrs. 6 min. a.m. 112 14' 40" O, 39 18' 37" 2 8 8 hrs. 10 min. a.m. 114 51' 20" Magnetic bearing of R.M , 200 10' 20". Bubble divisions on Alidade = 20". Required : Magnetic Variation and Error of Watch . Data from Nautical Almanac : Sun's Declination. Hourly Variation. Jan. 3rd, G.M.N., 22 53' 02-4" S., . . . 14-08" Jan. 4th, G.M.N., 22 47' 11-0" S., . . . 15-21" Equation of time (to be added to apparent time). Jan. 3rd, G.M.N., 4 min. 23-81 sec., . . 1-162" Jan. 4th, G.M.N., 4 min. 51-51 sec., . . 1-145" Ans. Magnetic variation = 9 44 11" E. Error of watch = 6 min. 42 sec. slow. 219 CHAPTER XI. DETERMINATION OF LONGITUDE. THE difference of longitude between any two places on the earth's surface, as we have already seen, is measured by the difference between either their local sidereal times or their local mean times at the same instant. The problem, then, of the determination of the difference in longitude between A and B amounts to that of the determination of the difference in the local times at A and B. By the methods we have considered in the last chapter we may by astronomical observation determine the local time at A at some instant, and a means must be found of determining what is the local time at B at the same instant, if we are to ascertain the difference of longitude. The problem presented is usually that of the deter- mination of the difference of longitude between two places rather than the fixing of the absolute longitude of a place as measured from the now universal standard meridian, that of Greenwich. Usually we seek to find the difference in longitude between a point on a survey and some fixed observatory in the country or some other point on the survey, the longitude of which has been previously determined. In all cases the local time at some instant must be determined at the place whose longitude is required by one of the astronomical methods of the last chapter. The corresponding local time at the reference station 220 ASTRONOMY FOR SURVEYORS. is then in modern practice usually found by one of three ways : (a) By portable chronometers. (b) By electric telegraph or wireless telegraphy. (c) By flash-light signals. (a) By Portable Chronometers. Since the time when chronometers that will retain a fairly uniform rate have been generally available, this has been the general method for the determination of longitude at sea. Every ship carries a chronometer, which keeps either Greenwich time or the local time at some known port, and from an astronomical observation the Captain is thus able to ascertain the difference between his local time and that of the chronometer. The method is very simple and con- venient, but wireless telegraphy, which is capable of much greater precision, may perhaps largely supersede it in the near future. To obtain accurate results it is essential that the chronometer should keep a constant rate, and the conditions on board a ship are much more favourable for this than is usually the case when chronometers are carried about from place to place on land. So that for land work the box chronometers used at sea are com- monly replaced by chronometer watches which are more easily carried and are found to be more satisfactory. Suppose now that it is required to determine the difference in longitude between A and B. The watch or chronometer must first be regulated at station A. Its error on the local time at that place must be deter- mined and its "rate" i.e., the amount that it gains or loses in 24 hours must be found. On the assumption that the rate remains constant this will enable the local time at A to be found from a reading of the chronometer at any time afterwards. If then the chronometer be transported to B and an astronomical observation be made there for the determination of local time, it will DETERMINATION OF LONGITUDE. 221 be possible to find from the chronometer the local time at A at the same instant. EXAMPLE. At A, September 8th, 1914, the chronometer at 8 p.m. was found to be 2 min. 6-5 sec. fast, and it was gaining at the rate of 2-58 sec. in 24 chronometer hours. At B, September 9th, 1914, from an astronomical observation which gave the local time as 9 hrs. 12 min. 35 sec. p.m., the reading of the chrono- meter was 9 hrs. 12 min. 30-6 sec. What is the difference of longitude ? The interval of time, as measured on the chronometer, between the two readings is 25 hrs. 10 min. 24-1 sec. = 1-049 days. Therefore, in this interval the chronometer has gained 1-049 X 2-58 sec. = 2-7 sec. Thus, at B the chronometer was fast by 2 min. 9-2 sec., and the local time at A was 9 hrs. 10 min. 21 -4 sec., corresponding to the local time of 9 hrs. 12 min. 35 sec. at B. Thus, the time at B is in advance of that at A by 2 min. 13-6 sec., or B is to the East of A by 33' 24". The accuracy of the method is affected by the fact that the rates of chronometers are not perfectly constant, and particularly by the fact that the rate whilst being carried is not the same as when at rest. The best way to minimise the error is to use several chronometers, from each of which a longitude determination is obtained, and the average of the results is taken. If possible, after the observations have been made at B, the chrono- meters should be carried back again to A and another comparison made with the local time there. This method is now never used by surveyors except where telegraphic communication is not available. (b) By Electric Telegraph or Wireless Telegraphy. If two- places are connected by electric telegraph the difference of longitude may be obtained with great accuracy. Suppose that A and B are two stations so connected,. A being to the east of B, so that the local time at A i& in advance of that at B. Then if an operator at A taps a telegraphic key that 222 ASTRONOMY FOR SURVEYORS. produces a corresponding tap in a telegraphic key at B, the two taps will be very nearly simultaneous, but not quite. A certain slight interval of time, a fraction of a second, will be required to transit the electric current from A to B and to produce the motion of the recording instruments. But whether the signal be transmitted from A to B or in the reverse direction from B to A, the time taken in transmission will be the same. If now the operators at A and B note the exact instant of each tap on chronometers keeping local time, either mean solar or sidereal, the difference in the times would at once give the difference in longitude if the taps were absolutely simultaneous. But, actually, when the message is sent from A to B, owing to the time taken in transmission, the tap at B will be a little later than it should, and the result obtained for the difference in longitude will be correspondingly too small. And similarly when the message is sent from B to A, the tap at A will be made later than should be the case if the transmission were instantaneous, and A being to the east of B, the difference of time will now appear too great. Thus by averaging the results of sending messages in opposite directions a correct value is obtained for the difference in longitude, and the error due to the time of transmission is completely eliminated. With signals sent by wireless telegraphy the velocity of the electric wave is so great that practically there is no measurable difference in the results obtained, whether the signals are sent from A to B or from B to A. For the most refined determinations the signals as received are automatically recorded on a chronograph, but very good work can be done by noting the times of signals with a chronometer if proper methods are adopted . DETERMINATION OF LONGITUDE. 223 Recording and Receiving Signals. A set of signals usually consists of a series of taps made at intervals of 10 seconds by a sidereal chronometer, the set extending over from 3 to 5 minutes. Each set is ushered in by a warning rattle of the key. The exact time of each tap is recorded at the receiving station by an observer who is counting out the ticks, which represent half seconds, on a chronometer keeping mean time. If the tap occurs between 1-5 and 2-0 seconds, the observer judges whether the time is 1-6, 1-7, 1-8, or 1-9. It is a very important aid to accuracy that the 10 second signals should be sent by means of a sidereal chronometer and recorded by a mean time chronometer. If the chronometer at the sending and receiving ends kept the same kind of time, the taps would always occur at the same decimal of a second, and the recorder, after the first two or three taps, would probably become pre- judiced in favour of some particular value of the decimal which he would retain throughout the set. But if one chronometer keeps sidereal and the other mean time, the tick of the sidereal chronometer Avill coincide with that of the mean time chronometer every three minutes, and in the interval between the coincidences the deci- mals of a second recorded at the receiving station will range from -1 to -9, so that the judgment of the recorder is not likely to be prejudiced in the same way as it would be if both instruments kept the same kind of time. Comparison of Chronometers. If two chronometers keep- ing the same kind of time, both beating half seconds, are to be compared, it will generally happen that the ticks of the one do not exactly coincide with the ticks of the other, but differ by some fraction of a half second that must be estimated by ear. It is difficult and re- quires considerable practice to make this estimate nearer than the fifth of a second. But it is possible to compare a sidereal .and a mean time chronometer with much 224 ASTRONOMY FOR SURVEYORS. greater accuracy, because at intervals of about three minutes the ticks of the two exactly coincide, and, if the comparison be made at the moment of coincidence, there is no difference of a fraction of a beat for the ear to estimate. Thus the difference in the readings of the two chronometers at this particular instant may be obtained exactly. The only error will be that which arises from judging the beats to be in coincidence when they are really separated by a small fraction. But it is found that a difference between the beats as small as 0-02 second is sufficient to enable the practised ear to detect the departure from exact synchronism and con- sequently the comparison may be made with an error not exceeding this quantity. The error of the sidereal chronometer is first obtained by astronomical observation, in the manner described in the previous chapter. Then to determine the error of the mean time chronometer a comparison is made at one of the moments when the beats coincide. List- ening to the beats of the two chronometers the observer judges when a coincidence is about to occur. He then begins to count^ the beats of one chronometer while he watches the face of the other. When he no longer per- ceives any difference in the beats, he notes the corre- sponding half seconds of the two instruments. The observed instant on the sidereal chronometer is then reduced to mean time, after allowing for the error of the chronometer, and the difference between the result and the recorded instant on the mean time chronometer gives its error. Personal Equation. It is found that different men, when performing such operations as sending or record- ing signals, will differ appreciably in their work. One man, when pressing down a telegraphic key at the instant the chronometer ticks, will consistently do so a little too late. Another will invariably press the key a small DETERMINATION OF LONGITUDE. 225 fraction of a second too soon. Similarly when recording the time signals one observer will consistently make a larger error than the other. II is found that the more practised and experienced the observers are, the more regular and consistent are the errors made in this way, and that this personal error or " personal equation/' as it is commonly called, remains fairly constant for long periods of time. Consequently its effects may be largely eliminated, in the average of a considerable number of observations, if the personal equations of the observers be determined both before and after the obser- vations are made. In this case the relative personal equation is required between two observers. It may be most simply obtained by the observers setting up their instruments near to one another at the same station. They then send sets of signals to one another, just as they would do in ordinary field work, in order to determine their difference of longitude. This should be done under conditions as nearly as possible the same as those obtaining at the actual work in the field. The result obtained, which should of course be zero, is the relative personal equation that must be applied in the reduction of the field obser- vations. It is advisable to observe the personal equation in this way for two or three evenings shortly preceding and following the field trip. When a large number of observations is being made probably the best way of eliminating the error due to personal equation is to exchange the observers at the enda of the telegraph line when half the total number of signals have been transmitted. When A sends and B receives, the time recorded at the receiving station should exactly coincide with the time of sending. Usually it does not, owing to the existence of this personal equation, and the time actually recorded by B may be either before or after the chronometer tick that A is transmitting. 15 226 ASTRONOMY FOR SURVEYORS. If the time recorded is always after the chronometer tick, the error will be fairly consistent so long as A is sending and B receiving. If B is at a station to the east of A, the effect of this error will be to make the difference of longitude greater than it really is, but if B is at a station to the west of A the same error will make the difference of longitude appear less than it should be. Thus if the observers change places when half the observations are over, personal equation is eliminated in the mean of the whole set and there is no necessity to make a special determination of it. Programme of Operations. Observations are made on several evenings. Professor W. E. Cooke, who was responsible for the introduction of the almucantar method of time observation in Western Australia i thus summarises the operations for any one evening : Observations. (a) Compare sidereal and mean time chronometers. (b) Take first half of almucantar observations, using sidereal chronometer. (c) Take chronometers to telegraph station and ex- change signals sending from sidereal and receiving by mean time. (d) Complete almucantar observations. (e) Compare the two chronometers. Computations. (/) From the almucantar observations determine the error of the sidereal chronometer at some definite sidereal hour, also its rate. (g) Apply the rate so as to obtain the error at time (a) ; reduce sidereal time (a) to mean, and hence determine error of mean time chronometer at time (a). (h) Do the same for time (e). (i) From (g) and (h) determine the errors of each chronometer at time (c). DETERMINATION OF LONGITUDE. 227 (j) Apply these errors to the average of the signals, also apply the correction for personal equation. Sub- tract the results from the similar results at the other station, and thus the difference of longitude will be obtained. When a determination of difference of longitude is made telegraphically between fixed observatories, the precision of the method is increased by sending the signals from a clock, the pendulum of which automatically completes an electric circuit when at the bottom of its stroke. The record at the other station is then taken on a chronograph, from which the instant can be read off to the hundredth part of a second. Such equipment is, however, not usually available for field work. (c) By Flash-Light Signals. When two stations are visible one from the other, flash light signals may be sent from one at ten second intervals as determined by the tick of a sidereal chronometer and recorded at the other by means of a chronometer keeping mean time, just as with electric telegraph signals. Or the signals may be sent from an intermediate station that is visible from both. The observers at each station must of course have obtained their local time by proper observation, and the difference between their local times at the instant of the signal gives at once the difference of longitude. The signal may be made by the flash of a heliotrope by day or the eclipse of a bright light at night. The following examples gives the results of obser- vations made in this way in Western Australia to de- termine the difference of longitude between the Perth Observatory and Mount Maxwell, about 17 miles away to the east. The signals were made by means of an acetylene lamp placed in a box, the light shining through a hole over which a photographic snap-shutter was fixed. The shutter was released at the proper second and the time of the flash noted as it was seen through a theodolite 228 ASTRONOMY FOR SURVEYORS. at the other station. The example is taken from the Western Australian Handbook for Surveyors : DIFFERENCE OF LONGITUDE. 1909. Mount Maxwell j Observatory to to Observatory. Mount Maxwell. Mean Result. Nov. 6th, Nov. 7th, Nov. 8th, . : .. . Nov. 9th, Nov. 13th, 7-96' I 7 8-64' 7-87' l'S-56' 7-82' 1' 8-54' 7-81' I 7 8-61' 7-93' I' 8-53' i' s-so" I' 8-21" rs-18" 1'8-21" rs-23" Mean, .... Personal equation, l'8-25" + 0'0-06" Difference of time, 1'8-31" Longitude by Lunar Observations. The methods for the determination of longitude that have just been described are those nowadays most usually adopted, but before the invention of the electric telegraph and the perfection of chronometers the only methods available over long distances depended upon observations of the moon. The moon changes its position among the fixed stars much more rapidly than any other celestial body, its relative movement amounting to over 13 in 24 hours, or roughly it moves over a distance equal to its own diameter in one hour. Consequently it is possible to use it as a clock, and, by measuring its position with regard to surrounding stars, we may determine at any instant, with the aid of the tables of the moon's motion given in the Nautical Almanac, the corresponding time at Greenwich. It was chiefly in order that " the moon's motion might be systematically observed for the purpose of providing navigators with accurate tables, which could be used for the determination of longitude, that the Greenwich observatory was originally founded. Lunar observa- DETERMINATION OF LONGITUDE. 229 tions, however, generally entail rather laborious com- putation, and the results, with the exception of those obtained by the method of lunar occupations, are not comparable in accuracy with the determinations made by the simpler methods previously given. Consequently such methods are now rarely used on land, and we shall merely describe the general principles involved. There are three principal methods of making observa- tions upon the moon for longitude. They are : (a) By Lunar Distances. (b) By Lunar Culminations. (c) By Lunar Occupations. (a) By Lunar Distances. The angular distance between the bright limb of the moon and some bright star in its vicinity is measured by means of the sextant, and at the same instant the altitudes of both moon and star are observed. This is best done by three observers, one for each measurement, but if there is only one ob- server, he takes first the altitudes, then the lunar distance, and then the altitudes once more, noting the time of each observation. From these he readily deduces the proper altitudes at the moment when the lunar distance was measured. By adding or subtracting to the observed distance the apparent semi-diameter of the moon, according as the bright limb of the moon is toward or from the star, the apparent distance between the star and the moon's centre is found. The moon's semi-diameter is given on page 3 of each month in the Nautical Almanac, for noon and midnight of each day. From this apparent distance, allowing for refraction and parallax, and knowing the approximate latitude of the place, the observations enable the distance to be computed as it would be observed from the centre of the earth, or the true distance as it is commonly termed. But if we know the true 230 ASTRONOMY FOR SURVEYORS. distance the corresponding time at Greenwich may be found from the information given in the Nautical Almanac. And the local time of the observation is readily found from the observed altitude of either moon or star. The longitude is found, of course, as the differ- ence between the local time and the corresponding Greenwich time. In fig. 49 let S and M denote the apparent positions of the star and the moon's centre respectively, Z being the Zenith. Parallax and Refraction will affect them in the vertical planes Z S and Z M. Now refraction causes a body to appear at a higher altitude than it really has, whilst a body when viewed from the earth's centre will have a greater altitude than when seen from the earth's surface. Thus to allow for refrac- tion we have to decrease the observed altitude, and to allow for parallax we must increase it. Now in the case of the moon parallax is greater than refrac- tion, the contrary being true for a star or planet. Thus the " true " position of S, as observed from the Fijr 49. earth's centre, is at S l5 below S, and the true position of M is at M 1? above M. In the triangle Z S M, the three sides have been directly determined by observation, and, therefore, the angle Z may be computed by the ordinary rules of spherical trigonometry. Then in the triangle Z S l M x , Z S x , and Z M 1? are known, and also the included angle Z, consequently the true 'distance Mj Sj may be computed. The Nautical Almanac used to give a table of true lunar distances, for every third hour of Greenwich mean time, from selected suitable bright stars. But these tables have lately been discontinued as it was decided DETERMINATION OF LONGITUDE. 231 that they were no longer of sufficient use to warrant their retention. The method is not capable of any degree of precision, about 5 seconds of time representing the accuracy attainable, and, now that the tables of lunar distances are no longer published, involves a lot of computation. The measurements cannot be made by a theodolite, the sextant being essential, and the method can only be classed as a rough one under the best circumstances. (b) By Lunar Culminations. As the moon moves right round the earth in a lunar month of about 28 days, its right ascension must change by 360 in that period, or at an average of about 13 in 24 hours. Thus in one hour its right ascension will alter on the average by something over 30 minutes of arc or two minutes of time. Now the right ascension of the moon may be most easily measured by observing the difference in time between its transit across the meridian and that of some known star. If the local time at the place of observation is also known, this determines the right ascension of the moon at a given instant of local time. But the Nautical Almanac gives the right ascension of the moon for every hour of Greenwich time throughout the year, and, by interpolation between the values in the tables, the Greenwich time corresponding to the measured right ascension may be found. Then the difference between the local time of observation and the corresponding Greenwich time as thus determined gives the longitude required. The computations are thus simple, and the method is the easiest of all the lunar methods for finding longitude. The observations are facilitated by the tables of moon- culminating stars given in the Nautical Almanac on p. 412 and succeeding pages. In these tables for each day in the year there are tabulated one or two stars, known as moon-culminating stars, that do not differ much from 232 ASTRONOMY FOR SURVEYORS. the moon in either right ascension or declination, and are consequently suitable for meridian transit observa- tions in comparison with the moon. For if the declina- tion of the observed star does not differ much from that of the moon, any error in the setting out of the meridian will affect the times of both transits to the same extent, and in the difference between the two times of transit, which is what is sought, the error will be eliminated. The times of meridian transit are unaffected by parallax and refraction which introduce complications in other lunar methods. A disadvantage is that for a considerable part of the month transits occur at very inconvenient times. The method in any case is not capable of great accuracy. An error of one second in the measurement of the time of transit of the moon's limb will cause an error of about 30 seconds of time in the longitude. Thus a good observation will only determine the longi- tude within about 10 seconds of time, and only by the average of a number of careful observations will it be possible to determine the longitude by this method within 5 seconds of time, corresponding to IJ minutes of arc, or to a distance of over one mile near the equator. EXAMPLE. At a place in approximate longitude 9 hrs. 06 min. E. the times of transit across the meridian of the moon's bright limb and of the star y Aquarii icere recorded by means of a chronometer keeping local mean time on the evening of September 30th, 1914. Observed time of transit of Moon I.*, . 9 hrs. 14 min. 22-8 sec. a Aquarii, . 9 hrs. 52 min. 30-2 sec. Determine the longitude of the place. Difference in times of transit, . . 38 min. 07-4 sec. Equivalent interval of sidereal time, . 38 min. 13-66 sec. R. A. of r Aquarii, . . . . 22 hrs. 26 min. 09-87 sec. R.A. of Moon I., 21 hrs. 47 min. ott-21 sec. * The Roman numerals I. and II. are used in the Nautical Almanac to indicate the moon's preceding and following limbs respectively. DETERMINATION OF LONGITUDE. 233 Allowing for the approximate longitude, the transit takes place at about 8 minutes after Greenwich noon on September 30th. From the Nautical Almanac we obtain Time of Meridian Sidereal Time of Passage at Semi-diameter Greenwich. Passing Meridian. Sept. 30th, . . 9hrs. 32-1 min. (upper) 63-92 seconds Sept. 29th, . . 21 hrs. 10-1 min. (lower) 65-02 Thus, the sidereal time for the semi-diameter to pass the meridian is given by 63-92 + - .^ = 33 hrs. 32 mm. 21 hrs. 10 mm. .. R.A. of moon's centre at instant of observation = 21 hrs. 49 min. 00-98 sec. Again, from the Nautical Almanac, R.A. of moon. at Greenwich, hr. = 21 hrs. 48 min. 44-20 sec. 1 hr. --= 21 hrs. 50 min. 41-41 sec. Therefore, by interpolation, the Greenwich mean time corresponding to the R.A. of 21 hrs. 49 min. 00-98 sec. is hr. 08 min. 35-4 sec. But the observed local time of the observation is 9 hrs. 14 min. 22-8 sec. Therefore, the longitude is 9 hrs. 05 min. 47-4 sec. East. (c) By Lunar Occultations. In the course of its monthly revolution round the earth the moon covers or " occults " in turn a number of the fixed stars. As the moon ap- parently moves from West to East among the stars, the stars in its track first disappear under the Eastern limb and afterwards reappear on the other side. The covering of a star in this way by the moon is known as an " occupation," the disappearance of the star behind the Eastern limb of the moon being known as the " immersion/' and its reappearance as the " emersion/' The method by lunar occultations consists in observing the local time of immersion or emersion, or both, at the occupation of a known star. At such moments the apparent right ascension of the star is the same as that of the Eastern or Western limb of the moon, and, after making proper allowance for refraction, parallax, and semi- diameter, the true right ascension of the moon may be 234 ASTRONOMY FOR SURVEYORS. determined at the instant, and hence, from the tables in the Nautical Almanac, the corresponding Greenwich time may be found. The method is capable of much greater accuracy than any other method by lunar observations. The two methods previously described, even under the most favourable conditions, can give but roughly approxi- mate results. But from several observations of lunar occultations a longitude may be determined within less than one second of time. Unfortunately, however, the prediction of the circumstances of an occultation and the complete computation of the observations involve principles that are rather complex for an elementary work. Partly on this account, and partly because suit- able observations can only be made at any one place some three or four times in a month as a rule, the method is not one used to any extent by surveyors, and no further elaboration of the method will in consequence be attempted here. Relative Accuracy of Different Methods. Major Close, in his Text Book of Topographical Surveying, gives the fol- lowing table showing the terminal error in longitude which might be expected after a march of 300 miles in a hilly tropical country. Method. Probable Error in Longitude. Triangulation, . . . .100 yards to J mile. Telegraph, \ -to \ ; mile. Chronometers, . . .1 mile. Occultation, . . . . \ mile. Moon culminations, . . .1 mile. Lunar distance, . . . .10 miles. The probable errors "are here stated as distances measured parallel to the equator, but, as the actual measurements of longitude are made in time, and as the distance measured along the earth's surface correspond- ing to a given difference of time gets less and less as we DETERMINATION OF LONGITUDE. 235 proceed further from the equator, it follows that the probable errors in distance would be considerably less than those chronicled at places remote from the equator. Where a triangulation can be carried on to directly connect the two places whose difference of longitude is- required, the determination may be made with the greatest precision possible. The telegraphic method comes next in order of accuracy, and is nowadays the method most commonly used. In order to get anything like the same accuracy by the method of lunar occultations, the observa- tions would have to extend over several months, and the tabulated values for the right ascension of the moon given in the Nautical Almanac would have to be corrected from observations made at some fixed observatory. 236 CHAPTER XII. THE CONVERGENCE OF MERIDIANS. THE line of sight of the telescope of a theodolite in ac- curate adjustment, as the telescope is turned about its horizontal axis, traces out a vertical plane. This, if we regard the earth as spherical, we may consider to be a plane passing through the centre of the earth. There- fore, the straight line that is set out by a theodolite is in reality always the arc of a great circle on the earth's surface. Now, unless it happens to coincide with the equator or with a meridian of longitude, any great circle will cut different meridians at different angles. In other words, its bearing will vary from point to point. Thus as we proceed along a straight line set out by a theodolite on the earth's surface, the bearing of the line will not remain constant but will gradually alter. A line the bearing of which was everywhere the same would not be a straight line. A parallel of latitude for instance is such a line, but if the telescope of a theodolite is set out truly East and West at any place its direction would not mark out the parallel of latitude, which is a small circle, but a great circle that would ultimately intersect the equator. This alteration in the bearing of a straight line is an important matter in surveys of any magnitude, as in latitudes in the neighbourhood of 60 it amounts to con- siderably over a minute of arc in a line one mile long, and in higher latitudes the alteration is still greater. In fig. 50, let N and S denote the North and South terrestrial poles, E L M Q is the equator, and A and B THE CONVERGENCE OF MERIDIANS. 237 any two points between which the great circle arc A B has been set out. Let N A M S and N B L S be the meridians through A and B. Then the bearing of the line B A at B is the angle NBA, and the bearing of the same line at A is 180 -NAB. The difference between the bearings of the line A B at the points A and B is known as the convergence of the meridians between A and B. If A B is plotted as a straight line on a plane, then the meridians through A and B will not be drawn as parallel lines, but as lines making an angle with one another equal to the convergence. Denote the convergence by c. Then c = 180 - N A B - N B A. Let /= latitude of A and V = latitude of B. NA=90-/, NB=90 /'. Denote the difference of longitude between A andp* by m, so that m = angle B N A. 238 ASTRONOMY FOR SURVEYORS. Then in the spherical triangle NBA, having given two sides and the included angle, tan \ rNBA+NAB) _ cos \ (NB-NA) cot \rn cos I (NB+NAJT .-. cot } (180- NB A-N AB) cos \ (I I') cot I m or, inverting r tan i COS i cos \ C' i (i 8 o .- I - r)' UtlJQoaU-f, . .. sin I sin i L r \d+l') L 3Dec'61BP Vlt/OS ! v o \ D Rh.C u uu t n. 1QC1 NOV 19 bl \ A 1 /JO\/f* i 3nC*63TL REC'D LD OCTl2'63-2pjyi LD 21A-50m-12,'60 (BG221slO)476B General Library University of California Berkeley ,'6968 LIBRARY USE RETURN TO DESK FROM WHICH BORROWED LOAN DEPT. THIS BOOK IS DUE BEFORE CLOSING TIME ON LAST DATE STAMPED BELOW FEB FEB26'65-2P LD 62A-50m-2,'64 (E3494slO)9412A General Library University of California Berkeley