QB Zo? UC-NRLF *+*& AZIMUTH BY GEORGE L. HOSMER ASSOCIATE PROFESSOR OF TOPOGRAPHICAL ENGINEERING MASSACHUSETTS INSTITUTE OF TECHNOLOGY SECOND EDITION REVISED NEW YORK JOHN WILEY & SONS, INC. LONDON: CHAPMAN & HALL, LIMITED 1916 ft* \-1-u Copyright, 1909 and 1916, BY GEORGE L. HOSMER TYI>OGRAPHY AND ELBCTROTYPING BY F. H. GILSON COMPANY BOSTON, MASS. PRHSSWORK BY BERWICK & SMITH CO. NORWOOD, MASS. PREFACE. THE purpose of this volume is to present in compact form certain approximate methods of determining the true bearing of a line, together with the necessary rules and tables arranged in a simple manner so that they will be useful to the practical surveyor. It is a handbook rather than a text-book, hence many subjects have been wholly omitted which are ordinarily included in books on Practical Astronomy but which are not essential in learning to make the observations described in this book. In all of the methods here treated the object sought is to secure sufficient accuracy for the purpose of checking the measured angles of a survey with the least expenditure of time. For this reason many approxima- tions have been made and many refinements omitted which simplify the calculations without introducing serious error into the results, and although such a treatment would scarcely be proper in a text-book the gain in simplicity and convenience would seem to justify its use in a book of this character. The necessity for making astronomical observations for azimuth is confined chiefly to geodetic work, and arises so seldom in general engi- neering practice that many persons engaged in surveying are not familiar with astronomical methods and will not feel confident of obtaining reliable results, and therefore are likely to avoid making use of such observations even when they might be of great practical value. This consideration together with the fact that the rigorous methods of calculating azimuth are rather long and complex, have tended to prevent astronomical obser- vations from being more generally applied to surveying. The author has endeavored to so present the subject that a person who is unfamiliar with astronomy will be able to apply these methods and obtain satisfactory results without taking the time to completely master the theory under- lying the method used. The rules and tables have been put in compact form so that the book may be carried in the field and the results of obser- vations worked up at once if desired. The value of having, from time to time, an independent check on the angular measurements of an exten- sive survey will certainly warrant spending a few minutes' time in making observations and computing the results. The methods here presented are not new but have all appeared in one form or another in works on Navigation, Astronomy, and Surveying. 359274 iv PREFACE Much valuable matter written on this subject, however, is so scattered that it is difficult to find in one small book all that would be needed by the surveyor in making azimuth observations. The author desires to acknowledge his indebtedness to Professor C. F. Allen for the use of the electrotype of Table XVIII taken from his " Field and Office Tables," and to Professor F. E. Turneaure for permis- sion to reprint Table V from Johnson's " Theory and Practice of Survey- ing." Thanks are due to Professor C. B. Breed for valuable suggestions and criticism of the manuscript. G. L. H. BOSTON, MASS., January, 1909. PREFACE TO THE SECOND EDITION. IN the present edition the following changes have been made : (i) A new method is given for finding the azimuth by an ^observation on the pole star at any hour angle when the local time is known. This will be found convenient by those who prefer observations on Polaris to direct solar observations. (2) The tables of the sun's declination have been extended to 1919 and rules have been added for computing the declina- tions for subsequent years. (3) The star maps are new and are more complete than those of the first edition. G. L. H. June, 1916. CONTENTS. ART. PAGE 1. CHECKING THE ANGLES OF A SURVEY i 2. SUN AND STAR OBSERVATIONS i 3. APPARENT MOTIONS OF THE STARS MERIDIAN . 2 4. POLAR DISTANCE DECLINATION 2 5. HOUR ANGLE 3 6. LATITUDE AND ELEVATION OF POLE 3 7. CORRECTING AN ALTITUDE REFRACTION CORRECTION .... 3 8. INDEX CORRECTION 3 9. MAKING SOLAR OBSERVATIONS 4 10. MAKING STAR OBSERVATIONS 4 11. AZIMUTH MARK 5 12. CONVERGENCE OF MERIDIANS 5 METHODS OF OBSERVING 13. AZIMUTH BY AN OBSERVED ALTITUDE OF THE SUN 6 14. COMPUTING THE AZIMUTH 8 15. WHEN TO OBSERVE 10 16. AZIMUTH BY AN ALTITUDE OF A STAR 13 17. OBSERVATIONS FOR LATITUDE 13 18. LATITUDE BY THE SUN AT NOON 13 19. AZIMUTH AND LATITUDE BY OBSERVATION ON 3 CASSIOPEIA AND POLARIS 15 20. FINDING THE STARS 15 21. EXPLANATION OF THE METHOD 15 22. THE TABLES 19 23. MAKING THE OBSERVATIONS 20 24. OBSERVATIONS ON d DRACONIS 23 25. MERIDIAN LINE BY POLARIS AT CULMINATION 25 26. AZIMUTH BY POLARIS WHEN THE TIME is KNOWN 26 27. ACCURATE DETERMINATION OF AZIMUTH BY POLARIS 27 28. DETERMINING THE HOUR ANGLE 28 29. DETERMINING THE AZIMUTH 29 30. COMPUTING THE AZIMUTH 29 31. MERIDIAN BY POLARIS AT ELONGATION 33 32. MERIDIAN BY EQUAL ALTITUDES OF A STAR 35 33. MERIDIAN BY EQUAL ALTITUDES OF THE SUN 37 TABLES 39-73 v AZIMUTH. 1. Checking the Angles of a Survey by Astronomical Azimuths. In the following pages are given several short and convenient methods of determining the azimuth of a line with an engineer's transit. While these methods may be used to determine an azimuth for any purpose which does not require great precision, the formulae and the tables have been specially arranged so that it will be practicable to compute the azimuth in the field for the purpose of checking the angles of a survey. In a preliminary railroad survey, for instance, or in running long traverses by the stadia method, there will ordinarily be no reliable check on the measured angles such as that obtained by closing a circuit or by connect- ing the survey with some line of known azimuth. In such cases the angles can be checked, with all the accuracy required, by determining the azimuth of some line of the survey either by means of a sun observation or by an observation on the pole-star, and comparing the azimuth thus determined with the azimuth computed by means of the measured angles. With convenient tables the azimuth may be computed in the field in a few minutes' time, so that it will be known at once whether the preceding angles are correct. The methods explained in Arts. 13 to 24 inclusive will give results sufficiently accurate for a check of the angles of an ordinary survey, but in these methods extreme accuracy is sacrificed to convenience and rapidity in the computations in order that results may be quickly obtained in the field. In Art. 27 is given a more accurate method which may be used when it is necessary to obtain an azimuth that is correct within a few seconds. 2. Sun and Star Observations. With regard to the relative advantages of sun observations and observations on the pole-star it may be said that observations on the sun are the more convenient of the two, but will not give results of great accuracy; their particular advantage is that they can be made while the survey is in progress and with the loss of but a few minutes' time, and if the azimuth is desired only to about one minute of angle, sun observations will be sufficiently accurate. By observations on the pole-star the azimuth may be determined with great accuracy, and such observations are the ones most commonly employed when the best i 2 AZIMUTH results ?re oougrr.. Sta*- ob^rvuiic/ns, however, have the disadvantage that they must be made at night, in which case the surveyor has to make a special trip to the point of observation, and must carry on his work under various practical difficulties which are not encountered in making sun observations. If a precise azimuth is desired it is necessary to observe on the pole-star, or some other star close to the pole, and also to make auxiliary observations for latitude and hour angle, on which the com- puted azimuth depends. The data for such an observation must be obtained from the Nautical Almanac. If, however, only an approximate result is desired, say within about i minute, the observations may be made on the pole-star by the method given in Arts. 19 to 24, and the azimuth found by Tables VII to XII. This method has the advantages that no Nautical Almanac is required and that there is little calculation except interpolation in the tables. 3. Apparent Motions of the Stars Meridian. If one watches the stars for several hours he will observe that they all appear to move from east to west in circular paths as though they were all attached to the surface of a great sphere, and this sphere were turning about an axis, the earth being at the centre of the sphere. Those stars which are near the equator all move in large circles. As the observer looks farther north he sees that these circles grow smaller, their common centre being an imaginary point called the pole. A vertical plane through the pole cuts out on the sphere a great circle called the meridian of the observer. The line in which this meridian plane cuts a horizontal plane through the observer is called the meridian line. At a distance of about i 10' from the north pole is a bright star which moves around the pole in a very small circle and is known as the pole-star, or Polaris. Since its circle is small its apparent motion is very slow, and consequently it is easy to determine with accuracy its true bearing at any instant and to measure an angle from the star to a reference mark on the ground. The relative position of the fixed stars * is always practically the same, and the pole stays nearly in the same position among the stars year 'after year. We may think of all of the stars in the north as moving in circles around the pole once each day, the size of the circle of any star and the speed of the star's motion depending upon how far that star is from the pole. 4. Polar Distance Declination. The angular distance to a star from the pole is known as its polar distance, an angle which changes slightly from year to year and may be obtained for any date from the * The term fixed star is used to distinguish the very distant stars from the planets; the latter are within the solar system and consequently appear to change their positions rapidly. The fixed stars have but a slight motion, imperceptible to the naked eye. INDEX CORRECTION 3 American Rphemeris and Nautical Almanac.* In case of the sun or a star which is far from the pole it is more convenient to define its position by its angular distance from the equator, i.e., by the declination, which is the complement of the polar distance. 5. Hour Angle. The hour angle of a body is the number of hours, minutes, and seconds that have elapsed since the body was on the obser- ver's meridian. Hence it is simply the angle through which the body has appeared to move since it passed the meridian of the observer. It will be seen then that a star which has an hour angle between oh and 12^ is west of the meridian; if the hour angle is between 12*1 and 24^ the star is east of the meridian. Hours or degrees are simply units of measure- ment of a circumference; in one case the circle is divided into 24 hours, and in the other case into 360 degrees. Hence an hour angle which is expressed in hours, minutes, and seconds of time may be converted into degrees, minutes, and seconds of arc. Since 24^ = 360 we have ih = 15, or, in a more convenient form, i = 4 m and i' = 48. 6. Latitude and Elevation of Pole. A very important principle in astronomy is that the angular altitude of the pole above an observer's horizon equals the latitude of the observer. Hence the latitude may be determined by simply finding, by some means, the altitude of the pole above the horizon. It may also be determined by finding the merid- ian altitude of a point on the equator, which is the complement of the latitude. 7. Correcting an Altitude Refraction Correction. In measuring an altitude of any heavenly body it is necessary to apply a correction to the measured altitude to allow for the bending (refraction) of the rays of light in passing through the earth's atmosphere. This correction is always subtracted from the observed altitude to reduce it to the true altitude, because the ray of light is concave downward, and hence the object appears too high above the horizon. This correction in minutes of angle may be taken from Table I. 8. Index Correction. In measuring altitudes with a transit the plate bubbles should not be relied upon, but after an altitude has been measured the vertical arc should be examined to see if the vernier reads o when the telescope bubble is in the middle of its tube. If it does not read o the reading of the vertical arc must be corrected by an amount equal to this error. If the o line of the vernier is on the same side of the o line of the arc at both readings, the index correction is to be subtracted; if the o line of the vernier passes the o graduation when the telescope is brought down to the horizontal position the correction is to be added. * Published annually (three years in advance) by the Bureau of Equipment, Navy Department, Washington, D. C. 4 AZIMUTH 9. Making Solar Observations. In making observations on the sun with the surveyor's transit it is desirable to have a dark glass placed over the eyepiece to cut down the light so that it will not be too bright for the eye. If no such dark glass accompanies the instrument the observation can be made by throwing the sun's image on a piece of paper held behind the eyepiece. If the objective is focussed on a distant object and the eyepiece tube drawn out, the image of the sun and the cross-hairs can be seen on the paper. The disc can be sharply focussed by moving the paper toward or away from the eyepiece. By this device observations can be made almost as well as by means of the dark glass. 10. Making Star Observations. In making observations at night it is necessary to illuminate the field of view of the telescope in order to make the cross-hairs visible. This should be done in such a manner as to avoid heating either the instrument or the air just in front of the telescope. If the instrument is heated the adjustments will be disturbed; if the air is heated the image will appear very unsteady. With some instruments there is a special reflector placed in a shade tube, fitting to the objective slide, so that when a lantern is held at one side of the telescope the light is reflected into the tube. If no such reflector is at hand a satisfactory result may be obtained by placing a piece of tracing cloth, or oiled paper, in front of the objective, folding the edges back over the tube and fastening it in place by means of a rubber band. A hole about one-half inch in diameter should be cut in the centre of the paper so that light from the star can pass through the central portion of the lens while the outer edge of the lens is covered. The cross-hairs will then be made visible by light diffused by the tracing cloth. For a few minutes just before dark Polaris can be easily seen through tne telescope before it can be seen with the unaided eye, and the cross- hairs will be visible against the sky without artificial illumination. In order to find Polaris under these circumstances it is necessary to know its approximate altitude at the time. The telescope must be focussed on some very distant object and then raised until the vernier indicates the star's altitude. By pointing the telescope about north and then moving the instrument very slowly right and left the star can be found. For the method of finding the altitude of the star see Arts. 19 to 21, and equa- tion [7], p. 29. Observations on Polaris just at dusk can be utilized when making observations by the method of Art. 26. If accurate results are desired in the observations for azimuth the instrument should be firmly set up and allowed to stand for some time before the observations are begun; the observations should be made as quickly as is consistent with careful work, as delay simply allows the instrument more opportunity to change its position and thus introduce CONVERGENCE OF MERIDIANS 5 error. Unless the transit is in perfect adjustment it is well to make two observations, one with the telescope direct and the other with the telescope reversed, and to use the mean result. ii. Azimuth Mark. In finding the azimuth of a transit line by star observations the instrument is set up at one end of the line, and at the other end is placed some sort of azimuth mark on which pointings can be made. If it is inconvenient or impossible to set this mark at the other end of the transit line it may be set in any convenient position, not too near the instrument, and its azimuth from the instrument determined. This observed direction is then connected with the survey by means of an angle measured in the daytime between the line to the azimuth mark and the transit line. In this case the mark should be so arranged that it can be sighted on either in the daytime or at night. For an accurate determination of the azimuth the mark usually consists of a box with a small hole cut in the side toward the observer so that light from a lantern placed inside can shine through the opening. The diameter of the hole should be such as to subtend an angle of about one second (0.3 inch per mile). The light should have the appearance of a small point of light like a star; it should not appear large or blurred. If the line to be sighted over is not one of the lines of the survey but is to be connected with the survey by an angle measured in the daytime, the box should have a target or a stripe painted on it to serve as a mark when sighting on it in the daytime. The centre of the hole and the centre of the target should coincide and should be placed carefully on the line to be sighted over. For the best results the mark should be placed far enough away so that the focus of the telescope does not have to be altered when changing from the star to the mark. \ FIG. i. Convergence of the Meridians. 12. Convergence of Meridians. In comparing azimuth observations made at points in different longitudes it will be necessary to allow for the angular convergence of the meridians at the two places. In running 6 AZIMUTH westward, in the northern hemisphere, the meridians are turned farther toward the right, as shown in Fig. i. If an azimuth observation were made at A and a traverse run west- ward to B and another azimuth observation made at point B, then it would be necessary to add to the azimuth observed at B the correction c in order to reduce this azimuth to what it would be if referred to the meridian at A. The difference between this corrected azimuth and the azimuth computed from the angles represents the accumulated error of the angular measurements of the survey. The amount of this correction for convergence of the meridians is shown in Table II. METHODS OF OBSERVING. 13. Azimuth from an Observed Altitude of the Sun. In the following paragraphs it is assumed that the instrument is set up at some regular transit station of a survey and that it can be sighted at another station. The lower motion of the transit should remain clamped during the obser- vation. The telescope is pointed at the sun, and the sun's image "found" in the field and sharply focussed. When a dark glass is used the cross- hairs usually cannot be seen except when they appear against the sun's disc. The telescope should be moved up and down a little so that the cross-hairs and also the stadia hairs can be identified (against the sun's disc) in order to avoid observing on a wrong hair. The observation is made by measuring the altitude of the upper and lower edges of the sun's disc and measuring the horizontal angles to the right and left edges so that the mean of these pairs of observations gives the altitude of the sun's centre and a horizontal angle to the centre corresponding to the same instant. In the first half of the observation the vertical cross-hair is set tangent to the left edge of the sun (by means of the upper plate tangent screw), and the horizontal cross-hair is set tangent to the upper edge of the sun; the vertical arc and the horizontal circle are then read and recorded. The watch time of the observation is also noted. The accuracy of the result may be increased by taking several such pointings and using the mean. In the second half of the observation the vertical cross-hair is set tangent to the right edge and the horizontal cross-hair tangent to the lower edge, thus placing the sun in the opposite quarter of the field from that used before. Both angles and the time are again recorded. The same number of pointings should be made in the second half as in the first. The telescope should then be levelled and the vernier examined to see if there is any index correction to be applied to the readings of the vertical arc. If the plate is set so that the vernier reads azimuths, as is customary in stadia surveying, then these vernier readings AZIMUTH FROM AN OBSERVED ALTITUDE OF THE SUN 7 alone will give the horizontal angle between the sun and the meridian from which the azimuths are being read. If the circle is not set for azimuths the upper clamp should be loosened and the telescope sighted on some line of the survey and the vernier read again, so that this vernier reading, combined with the two readings on the sun ? will give the horizontal angle between the sun and the station sighted. Theoretically it is immaterial whether the observations are made in the exact order given above or not, provided the sun is observed first in one of the quadrants formed by the cross-hairs and then in the opposite quadrant. It will be found, however, that the sun moves so rapidly that it is difficult to set both cross-hairs accurately in position at the same instant, hence the observation will be easier and also more accurate if we select that pair of opposite quadrants in which it will be necessary to make but one setting with the tangent screw, the other setting being made by the motion of the sun itself. This may be done in the following manner. If the observation is to be made in the forenoon set the vertical cross-hair a little in advance of the left edge of the sun. (see the lower FIG. 2a. A,M. Observations. FlG. 2b. P.M. Observations. Diagram showing position of sun's disc a few seconds before the instant of observation for azimuth (northern hemisphere). The arrows show the direction of the sun's motion. Stadia hairs are shown as dotted lines. part of Fig. 2a), and keep the horizontal cross-hair tangent to the upper edge of the sun by means of the vertical tangent screw, following it until the left edge of the sun has moved up to the vertical cross-hair. At this instant stop moving the tangent screw, note the time, and read the angles. For the right and lower edges the position is as shown in the upper part of Fig. 2a, the horizontal cross-hair cutting across the lower portion of AZIMUTH the sun and the vertical cross-hair being kept tangent to the right edge by means of the plate tangent screw. For observations in the afternoon the positions will be as shown in Fig. 2b. If a transit with an inverting eyepiece is being used these positions will of course appear reversed. If the transit is provided with stadia hairs care should be taken not to mistake one of them for the middle cross-hair. In the figure the stadia hairs are represented by dotted lines, the distance between them being i/ioo part of the focal length of the objective. The angular distance between them is therefore about o 34', a little greater than the diameter of the sun. If the transit has a complete vertical circle the telescope should be reversed between the two pointings on the sun to eliminate errors of adjustment. Repeated trials indicate that if the sun's disc is bisected with both cross-hairs at the same instant, the error in the resulting azimuth usually will not exceed i'. For rapid work where great accuracy is not de- manded this method may be used, as it saves time and removes all doubt as to which limb or which cross-hair was observed. 14. Computing the Azimuth from Sun Observations. In order to compute the azimuth we proceed as follows : i.) ALTITUDE. Take the mean of the altitudes of the upper and lower edges of the sun, and subtract from it the refraction correction taken from Table I, thus obtaining the true altitude. The index correction must also be applied. 2.) HORIZONTAL ANGLE. Take the mean of the vernier readings of the horizontal circle for the pointings on the sun. If a pointing has been made on some reference mark, take the difference between this vernier reading and the mean vernier reading for the sun, the result being the horizontal angle between the mark and the sun. 3.) DECLINATION. Take from the Nautical Almanac* (or any solar ephemeris) the declination f of the sun at Greenwich Mean Noon (G.M.N.) for the date of the observation, and also the difference for i hour and its algebraic sign, found in the next column to the right. In order to obtain the declination at the instant of the observation it will be necessary to allow for the change in the declination since the instant of Greenwich Mean Noon. If the watch keeps Standard Time the correction for * It is not necessary to use the large Nautical Almanac for obtaining the sun's declina- tion. Pamphlets containing the declination and the equation of time are issued by the Hydrographic Office (Publication No. 118) and may be obtained from the regular agents. Copies of the solar ephemeris are also published in the form of handbooks for engineers and for the use of navigators. Values of the sun's declination for the years 1916 to 1919 inclusive will be found in Table XV. fThis is given in the Almanac under the heading Apparent Declination. AZIMUTH FROM AN OBSERVED ALTITUDE OF THE SUN 9 change in declination can be made in a very simple manner. At the instant of Greenwich Mean Noon it is 7 A.M. Eastern Time, 6 A.M. Central Time, 5 A.M. Mountain Time, and 4 A.M. Pacific Time. Hence we can obtain the time elapsed since G. M. N. by simply subtrapting 7 A.M., 6 A.M., etc., as the case may be, from the observed watch time, first adding 12^ to the time if it is afternoon. The difference for i hour taken from the Almanac is to be multiplied by this elapsed time expressed in hours and the result added to or subtracted from the declination at G. M. N. (This multiplication may be avoided by the use of Table XIV.) An examination of the declination for the preceding or following dates will show whether it is increasing or decreasing and hence show whether the correction is to be added or subtracted. If the declination is considered positive when the sun is north of the equator and negative when south, then the elapsed time multiplied by the difference for i hour as given in the Almanac is always to be added. For example, suppose the declina- tion is desired for Nov. 10, 1909, at 2 h 30 P.M., Eastern time. The decl. for G. M. N., Nov. 10, = 17 03'.!, the diff. for i hour = 42".$. The time elapsed since G. M. N. = 2*1 30 P.M. + 12^ 7 A.M. = 14^ 30 7*1 = 7^.5. The total change is 42". 5X7^.5 = s'.3- The corrected declination is - 17 03'. i - 5^.3 17 o8'.4. The hourly change never exceeds i minute of angle, so that if the watch is in error by as much as 10 minutes the resulting error in the declination will have a small effect on the azimuth. If the watch keeps local time the watch time of G. M. N. is found by subtracting from 12^ the west longitude of the place expressed in hours, minutes, and seconds. For example, if the observer were in longitude 93 W. his (local) noon would occur 6h i2 m after G. M. N., i.e., 5*1 48 m A.M. by his watch (if correct) is the instant of G. M. N. 4.) AZIMUTH. Compute the azimuth of the sun, from the south point, by the formula log vers Az. = log [sin \ go - (Lat. + Alt.) J +sin Decl.] + log sec Lat. + log sec Alt., [i] in which Az. is the azimuth of the sun's centre from the south point, east or west; Lat. is the latitude of the place either taken from a map to the nearest minute or obtained by observation (see Art. 17); Alt. is the cor- rected altitude of the sun's centre; and Decl. is the declination of the sun at the instant of the observation. The arrangement of the compu- tation is as shown in Examples i to 3, pp. n and 12. The latitude and the altitude are written down, their sum taken, and its complement io AZIMUTH written beneath. From Table III we take the log secants of the latitude and the altitude. These are found by looking up the secant for the next smaller angle in the left portion of the table and adding the proportional parts for the minutes from the proper column at the right. The log secant can thus be written down directly, to the nearest unit in the fourth figure.* The characteristics (o for all the log secants occurring within the limits of this table) have been omitted in Table III. From Table IV we obtain the natural sine of 90 (latitude +altitude)f and also nat- ural sin declination and take their algebraic sum. If the declination is the sine is . From Table V we look up the log of this sum and add it to the two log secants. This sum is the log vers of the azimuth reckoned from the south point. In Table VI are given the log versed sines, the arrangement being exactly as in the preceding tables. If the observa- tion is made in the afternoon the angle from Table VI is the azimuth desired; if in the forenoon the angle must be subtracted from 360 degrees, since azimuths are reckoned from the south point in a clockwise direction. This azimuth, combined with the measured horizontal angle, will give the azimuth of the line desired. If it is desired to compute the azimuth with as great precision as the observations will afford, i.e., to about 5 or io seconds, tables carried to five places should be used in the computation. The formula given above applies to the northern hemisphere, but if the algebraic sign of the declination is changed and the azimuth reckoned from the north point instead of the south it will apply to the southern hemisphere. In the southern hemisphere the positions of the sun shown in Fig. i will obviously be changed. 15. When to Observe. The most favorable times for accurate obser- vations by this method are when the sun is nearly east or west. If the best results are desired the observations should not be made within 2 hours of noon nor when the sun's altitude is much less than io degrees. * The table extends only to 60 degrees, which is sufficient for all ordinary cases, f hould it be necessary to find the log secant of an angle greater than 60 degrees it may be done by taking the natural sine of the complement (Table IV), looking up its log (Table V) and subtracting this log from zero. For angles greater than about 80 degrees, however, this method is not sufficiently accurate. t The sine is employed rather than cos (latitude + altitude) so that aH numbers may be taken from the tables in exactly the same manner. If the sum of the latitude and the altitude exceeds 90 degrees the natural sine of this angle is negative. (See Example 2 .) WHEN TO OBSERVE EXAMPLE i. Latitude 42 21' N. OBSERVATION ON THE SUN FOR AZIMUTH. Instrument at Sta. no. Point sighted Hor. Circle Vert. Ar Station in 238 14' oj 311 48 14 4i 0.1 312 20 15 oo fo 312 27 15 55 ro 312 51 16 08 Mean Horizontal Angle 312 21/5 Obs.Alt. 15 26' 238 14 74 07' .5 Refr. 3.5 True Alt. 15 2 2'. 5 Nov. 28, 1905. Watch (Eastern Time) 8 h 4i m A.M. 8 47 8 h 44 i h 44* Gr. Time COMPUTATION. Lat. 42 21' log sec .1313 Alt. 15 22.5 log sec .0158 Sun's Decl.at G. M. N. = 21 14' .9 Diff.i = - 26" .81 Sum 57 43 -5 o *, 26 " - 8l X i h -73 = - o'.8 Nat. sin .5340 Co 32 i6'-s Nat. sin - .3626 Decl. - 21 15' -7 Decl. at 8" 44 m = - ai 15' -7 Alg. Sum .1714 log 9.2340 log vers 9.3811 Az. of Sun 40 34'.? Sta. in N. of Sun 74 07' .5* 114 42' .2 Az. 245 18' Sta. no to Sta. in, N. 65 18' E. * The plate readings (azimuths) indicate that Sta. in was to the left of the sun and hence North of it. 12 AZIMUTH EXAMPLE 2. Latitude 42 30' N. AZIMUTH OBSERVATION ON SUN. Instrument at Sta. B. Hor. Circle Vert. Arc Sta. 7 ooo' |o 99 19' 56 35' 12 99 54' 56 54' ol 99 4o' 57 49' ol 100 07' 58 02' Mean "99^4? 57 20' Refr. 0.7 57 19' -3 July 15, 1907. Eastern Time 9 h 47 m A.M. 9 Si 9 h 49 Decl. at G. M. N. + 21 40'.? 22" .8 X 2 h .8 = - i'.o Decl. = + 21 39' .7 COMPUTATION. Lat. 42 30' log sec .1324 Alt. 57 19.3 log sec .2677 99 49' -3 sin .1706 Co 9 49' -3 sin -3691 Decl. + 21 39'.? sum .1985 log 9-2978 log vers 9-6979 Az. S. 59 55' E. Sta. 7 north of sun 99 45 S. 159 40' E. Sta. B to Sta. 7 = 200 20' The azimuth of Sta. B to Sta. 7 as calculated from the angles of the survey is 200 EXAMPLE 3. AZIMUTH OBSERVATION AT Q 25, Aug. 6, 1907, 5 h o4 m P.M. in latitude 42 29' .2 N. Mean Alt. = 22 29' .3. Mean plate reading = 92 35' (supposed to be true azimuth). Decl. at G. M. N. + 16 57' .6 40". 7 X io h .i = 6'. 9 Decl. + 16 50'.? COMPUTATION. Lat. 42 29' .2 Alt. 22 29 .3 Sum 64 58' .5 nat. sin .4230 Co 25oi'.s ' nat. sin .2898 Decl. + 16 50' .7 Sum .7128 Hence the azimuths read at Q 25 are 4' too small. log sec .1323 log sec 0343 log 9-853 log vers 0.0196 Azimuth 92 39' Vernier 92 35 Error = 04' LATITUDE BY THE SUN AT NOON 13 If the azimuth calculated by the preceding rule exceeds 70 the azimuth from the north point may be calculated as follows: Take the difference between the Latitude and the Altitude, and subtract it from 90. From the natural sin of this angle subtract the natural sin Declination. The log of the result added to the log secants of the Latitude and Altitude gives the log vers Az. measured from the north. This affords a convenient means of checking azimuths between 70 and 110. It will sometimes be found that the two azimuths will differ i', or even 2', owing to the fact that only four places are used in the logarithms. The mean of the two results will always be more accurate than the result of a single computation. 16. Azimuth by Altitude of a Star. The azimuth of a star can be determined in the same way as the azimuth of the sun, provided the star can be identified and its declination obtained. Since a star has no appreciable diameter its image should be bisected with both cross-hairs. Any of the brighter stars contained in the list given in the Nautical Almanac* can be used for this observation. For accurate results the star's declination should not be greater than about + 20 degrees nor less than about 20 degrees, and at the time of the observation the star should be nearly due east or due west. Since the declination of a fixed star does not change appreciably in 24 hours it will not be necessary to note the time as in a solar observation. 17. Observation for Latitude. In order to obtain the sun's azimuth it is generally necessary to know the latitude of the place within about i minute. In many cases this can be scaled from some reliable map with sufficient accuracy. If no such map is available the latitude must be observed directly, either by the sun's altitude at noon or by the altitude of the pole-star. If we can find the distance of the point of observation north or south of some other point whose latitude is known, the latitude of the instrument may be found by taking 6080 feet equal to i minute of latitude. The latitude can often be found in this way in places which have been surveyed by the United States Public Lands System, since the latitude of some of the points can be ascertained and the distance north or south to other points found from the township and section numbers. 1 8. Latitude by the Sun at Noon. This observation is made by measuring the altitude of the sun at noon, when it is a maxi- mum. The transit should be set up and levelled some time before * For a condensed list see table of Fixed Stars in the Nautical Almanac under the headimg "Mean Places"; for the exact places, right ascension and declination for any date, see table of " Apparent Places." A short list is given on p. 28 of this book. 14 AZIMUTH noon* and the horizontal cross-hair set on the sun's lower edge. As long as the sun continues to rise it should be followed with the vertical motion of the telescope, keeping the cross-hair exactly tangent to the lower edge of the disc. As soon as the sun begins to drop below the cross- hair the motion of the tangent screw should be stopped and the verti- cal arc read. This altitude must be corrected for (i) index error, (2) refraction, (3) semi-diameter of the sun, and (4) the sun's declination. The refrac- tion may be taken from Table I. The sun's semi-diameter may be taken from the Nautical Almanac, but for approximate results may be taken as 16' in March and in September, 16' 15" in December, and 15' 45" in June. If the lower edge of the sun is observed the correction is to be added to the measured altitude; the upper edge could have been observed, in which case the semi-diameter should be subtracted. The declination of the sun is found as described in Art. 14, or it may be found by taking from the Nautical Almanac the declination for Greenwich Apparent Noon, multiplying the difference for i hour by the number of hours in the longitude and adding this to the declination at Greenwich Apparent Noon. The declination must be subtracted from the altitude if the sun is north of the equator (+ ), added if south ( ). The altitude thus found is the complement of the latitude. EXAMPLE. OBSERVATION FOR LATITUDE BY ALTITUDE OF Q AT NOON, Jan. 13, 1905. Longitude 4 h 4S m W. (approx.). Watch time = n h 53 (Eastern Time). First Method. Second Method Ded. at G. M. N. = 21 32' .6 Decl. at Apparent Noon = 21 32' 31" 25" .2 X 4 b -9 = + 2 .1 25" .2 X 4-75 = + 2' oo" Ded. 21 30' .5 Deci. at Local Noon 21 30' 31" Maximum Altit-de O 25 55' Refraction 2' 25 53' Semi-diameter 16 .3 Altitude of centre 26 09' .3 Declination 21 30 .5 Complement of Latitude 47 39' A Latitude 42 20' .2 * It should be remembered that the time of the sun's maximum altitude may differ considerably from noon by the watch. To obtain the Standard Time of this observation, call the time of the observation i2 h , Apparent Time; reduce this Apparent Time to Mean Tune by adding or subtracting the equation of time as given in the Nautical Almanac; then reduce the Mean Time to Standard Time by taking the difference in longitude (expressed in h. m. s.) between the place and the standard meridian and adding it if the place is west of the standard meridian, subtracting if the place is east. EXPLANATION OF METHOD 15 19. Azimuth and Latitude by Observation of 3 Cassiopeiae and Polaris. The method described in the following articles is applicable when only approximate results are desired, say within about i minute of the true values, and when it is desired to obtain the result quickly and without using the Nautical Almanac. An advantage offered by this method is that it is not necessary to know the local time, since this is determined with sufficient accuracy by the observation itself. Tables VII to XII are so arranged that all of the quantities needed in this obser- vation may be found by interpolation, and usually all the tables required for an observation appear at the same opening of the book. 20. Finding the Stars. In order to make this observation it is necessary to be able to identify certain stars near the north pole. The most conspicuous constellation in the northern sky is the Great Dipper, or Great Bear (Ursa Major}. (See Fig. 3.) Polaris, the star on which the azimuth observation chiefly depends, is readily found by reference to the Great Dipper. The two stars forming the side of the dipper bowl which is farthest from the handle are called the pointers because a line through them points very nearly to Polaris, the distance to Polaris being about five times the distance between the pointers. On the opposite side of the pole from the Great Dipper is the constellation Cassiopeia, shaped like a letter W. The star d (delta) Cassiopeiae, which is to be used in this observation, is the one at the bottom of the first stroke of the W, i.e., the lower left-hand star when the W is right side up. The Little Dipper is an inconspicuous constellation; Polaris is at the end of the dipper handle, and two fairly bright stars form the outer side of the dipper bowl. The other stars in this constellation are quite faint. Another star which w r ill be referred to later is d Draconis. When d Cassiopeiae is above the pole d Draconis will be found west (left) of the meridian at about the same altitude as Polaris, these three stars forming a right triangle (nearly), the right-angle being at Polaris. The distance from Polaris to d Draconis is less than the distance from Polaris to d Cassiopeiae. It will be observed that d Draconis, Polaris, and the lower star in the bowl of the Little Dipper form a triangle which is nearly equilateral, d Draconis is not as bright as d Cassiopeiae. There are several faint stars near d Draconis, one of which might possibly be confused with it. This other star (e Draconis) is nearly on a line drawn from d Draconis to d Cassiopeiae, and its distance from d Draconis is about 4 degrees, a little less than the distance between the pointers, s Draconis is not as bright as d Draconis. 21. Explanation of Method. In order to determine the azimuth of Polaris and the latitude of the place it is necessary to find by some means the position of Polaris with respect to the pole at the instant of the obser- vation. This depends upon the hour angle of Polaris at the time of the i6 AZIMUTH ( - aunp) X Great / ^ **C / V jp. ^^f HORIZON Polaris at Upper Culmination. (December.) DIAGRAM SHOWING THE CONSTELLATIONS ABOUT THE NORTH POLE. The arrows show the direction of the apparent motion of the stars. FIG. 3. EXPLANATION OF METHOD 17 observation. In order to determine the coordinates of Polaris, i.e., its distance above or below the pole and its distance east or west of the meridian, we measure the altitude of the star d Cassiopeiae and also the altitude of Polaris. From these altitudes we can calculate the coordinates of Polaris. By referring to Fig. 4 it will be seen that the pole, Polaris, and d Cassiopeiae are all nearly in the same plane (i.e., on the same hour circle), the two stars being on the same side of the pole. The direction of the apparent motion of the stars is shown by the arrow. Hence the relative position of d Cassiopeiae and Polaris as seen by the observer is at once a key to the position of the pole itself. If d Cassiopeiae is directly above Polaris, then Polaris is above the pole and nearly in the meridian; if d Cassiopeiae is below and to the left of Polaris, the latter is below and to the left of the pole. We may think of these two stars, then, as moving around the pole together as though they were two points on an arm pivoted at the pole. \EA8T HORIZON FIG. 4. If we know the latitude of the observer, and the polar distance and the altitude of d Cassiopeiae at any instant, we may calculate the hour angle of this star, i.e., the arc MS in Fig. 4. Hence if Polaris were exactly on the same hour circle with d Cassiopeiae the hour angle of Polaris would be the same as that computed for d Cassiopeiae. In reality Polaris has :8 AZIMUTH a slightly smaller hour angle than d Cassiopeiae, the difference between the two increasing slowly from year to year. This interval is 6 m 583 for the year 1910, ic 111 573 for 1920, and i5 m 138 for 1930. After we observe the altitude of d Cassiopeiae we may wait until this interval of time has elapsed and then make the observation on Polaris, the latter then being in the position it would have occupied at the first observation if the two stars were on the same hour circle. This instant at which Polaris is to be observed we may call for convenience the computed time. When the hour angle of Polaris is known for the instant of the observation the (Polaris) FIG. 5. Coordinates of Polaris. coordinates may be found at once. In Fig. 5 if p is the polar distance of Polaris and / its hour angle, then PM=p cos/ and SM=psmt. [2] PM is the amount (nearly) which Polaris is above or below the pole, hence Lat.* =Alt. of pole =Alt. of Polaris p cos /. [3] * For a more accurate expression for the latitude we should add to the above series the quantity K, from Table XIII. See equation [7]. THE TABLES 19 If cos t is given its proper algebraic sign in the different quadrants this equation holds true for all positions of the star. The quantities PM and SM may be found in Tables VIII and XI. The coordinate p sin t is the angular distance of the star east or west of the meridian. The azimuth of Polaris depends not only upon the distance p sin t but also upon the altitude of the star as seen by the obser- ver; it may be found by the equation Azimuth =p sin t sec altitude, [4] or Azimuth p sin t + p sin / exsec altitude.* [5] The azimuth is computed by taking from Tables IX or XII the azimuth correction (p sin / exsec altitude) and adding it to p sin /. This azimuth is, of course, reckoned from the north point. When d Cassiopeiae is near the meridian, i.e., nearly above or nearly below Polaris, an accurate determination of its hour angle cannot be made. In this case another star, d Draconis (p. 15), can be substituted for d Cassiopeiae and the hour angle of Polaris derived in a similar manner, so that it is almost always possible to use this method. 22. The Tables. All of the quantities needed in this observation may be taken from Tables VII to XII. In Table VII are given the hour angles of d Cassiopeiae for different altitudes and different latitudes. The hour angle for any latitude and any altitude at which an accurate observation can be made may be found by interpolation in this table. If the star is east of the meridian the tabular hour angle should be sub- tracted from 360 degrees to obtain the true hour angle. Since it is necessary to interpolate in this table both for the altitude and the latitude it will be simpler and also more accurate to observe the star when the altitude is a whole degree, preferably an even numbered degree, and thus confine the interpolation to the latitudes. If the observation on Polaris follows the observation on d Cassiopeiae by the interval of time corre- sponding to the date, as given above, then the hour angle of Polaris at the time it was observed is the same as the hour angle taken from Table VII. In Table VIII will be found the values of p sin / and p cos t for different hour angles of Polaris between 30 degrees and 150 degrees, and for the years 1910, 1920, and 1930. If the date falls between those given in the table it will be necessary to interpolate to obtain the coordinates for the date of the observation. With the value of p sin t found in Table VIII and the measured altitude, we take from- Table IX the correction p sin t exsec altitude. This correction, added to p sin t, gives the azimuth desired. * The external secant, or, the secant minus unity. 20 AZIMUTH It will be noticed that in general we do not know the latitude of the place, and therefore we cannot determine the hour angle in a direct manner as was assumed above. If the latitude is not known we may proceed as follows. From the relative position of d Cassiopeise and Polaris we estimate the latitude, remembering that Polaris is i 10' from the pole, and its direction from the pole is nearly the same as the direction of d Cassiopeiae from Polaris. With this approximate latitude and the measured altitude we take from Table VII an approximate hour angle of d Cassiftpeiae. With this approximate hour angle we take from Table VIII the value of p cos /. This is the correction to be applied to the altitude to give the corrected latitude. (Equation [3].) With this new latitude we take from the table a more accurate value of the hour angle. Using this new hour angle we find from Table VIII the value of p sin t. Two approximations will always give p sin / with sufficient accuracy. The azimuth correction from Table IX may be taken out as before. If the latitude as well as the azimuth is desired the above process of approxi- mation may be continued if necessary until the value of p cos t agrees within about i minute with the preceding value. The latitude is then found by equation [3]. It is not really necessary to observe Polaris at the same hour angle as d Cassiopeiae, although this simplifies the calculation slightly. In case the observation is not made at the computed time we must correct the hour angle accordingly before taking out p cos / from Table VIII. The correction is made by converting this difference in the time interval into degrees and adding it to or subtracting it from the hour angle of d Cassio- peiae. Suppose for instance that the observation on Polaris was made 501 208 before the calculated time, then 5 m 208 = 5 m -33 = i-33 (see Art. 5), which must be subtracted from the hour angle of d Cassiopeiae to obtain the hour angle of Polaris, since Polaris had not reached this hour angle at the instant it was observed. 23. Making the Observations. Set up the instrument at one end of the line whose azimuth is to be found, and set a lantern or arrange an azimuth mark at the other end of the line. See if d Cassiopeiae is in a favorable position for an observation. If d Cassiopeiae is near the merid- ian, either above or below the pole, an accurate observation cannot be made on this star. If the altitude of the star and the latitude of the observer are such that an hour angle can be found in Table VII, then a reliable observation can be made. If it is found that d Cassiopeiae can be used, point the telescope at this star, examine the vertical circle to see what the approximate altitude is, and set it so that the vernier reads a whole degree (preferably an even numbered degree) + the refraction correction for this altitude (Table I). If the star is west of the meridian MAKING THE OBSERVATIONS 21 it is moving downward, and the telescope must be set at some altitude below that of the star. If the star is east of the meridian the telescope must bo set at a higher altitude than that of the star. Watch the star and note the time when it crosses the horizontal cross-hair. The star moves so slowly that the observation is not precise, but it is sufficiently exact for this purpose. Next calculate the time of the observation to be made on Polaris by adding to the watch time just noted the interval from the top of Table VII. Set the plate vernier at o degrees and point at the azimuth mark, using the lower clamp. Loosen the upper clamp, point the telescope toward Polaris, and set both cross-hairs on the star. Follow the star's motion, using the vertical tangent screw and the upper plate tangent screw, until the computed time is indicated by the watch, then see that both cross-hairs are bisecting the star. Read the vertical arc and the plate vernier, and determine the index correction to the vertical angle. The latitude and the azimuth are then found from Tables VII to IX as described on pp. 19 and 20. The method of using these tables is illustrated by the following examples. The first two illustrate observations in which Polaris was observed at the calculated instant. In Examples 3 and 4 Polaris was observed before the computed time arrived. EXAMPLE i . OBSERVATION ON POLARIS AND 8 CASSIOPEI.*: FOR LATITUDE AND AZIMUTH. March 14, 1908. Set telescope at altitude 26 02' (26 + the refraction correction for 26); 8 Cassiopeia passed horizontal cross-hair at p h o6 m 10". Interval for 1008 = 6 m io g (Table VII, tcp). Computed time for observation on Polaris = p h 12 20". Set on mark with vernier at ooo'. Bisected Polaris with both cross-hairs at p h 12 20*. Altitude = 41 52'. 5. Index correction = Q'.S. Horizontal angle, 38 31'. 5. Mark is west of Polaris. 5 Cassiopeia^ is below and west of Polaris. FIRST APPROXIMATION. Approx. Lat. = 42 Alt. 41 52' .5 True Alt. = 26 Index corr. 0.5 From Table VII, hour angle, / = ii2.o 41 52' .o Refr. i.i True Alt. 41 50' .9 From Table VIII, for 112, we find p cos / = 26' .6 p cos i 26.6 Approx. Lat. 42 17'. 5 SECOND APPROXIMATION. From Table VII, using lat. 42 if. 5, t = 112 .7 From Table VIII, for 112 .7, p cos / = 27' .3 and p sin t = 65' .6 Alt. 41 50' .9 From Table IX, az. corr. = 22 .5 p cos / 27.3 Az. = 8&.i = i 28'.! Lat. 42 i8'.a Measured angle =38 31'. 5 Azimuth of mark is N. 39 59' .6 W. 22 AZIMUTH EXAMPLE 2. OBSERVATION ON POLARIS AND 6 CASSIOPEIA. August 18, 1908. Set telescope at altitude 34oi'.5 (34 + refraction corr.). 8 Cassiopeise passed horizontal cross-hair at 9 h o7 m n 8 . Interval for 1908= 6 m io s . Calculated time for observation on Polaris = p h 13 21*. Set on mark with vernier at o oo'. Set on Polaris at g h i3 m 21". Altitude =41 46'. Index correction = o. Horizontal angle = 89 38'. Mark east of Polaris. 3 Cassiopeia? below and east of Polaris. Approx. Lat. 42 True Alt. 34 Alt = 6 , Table VII, hour angle = 92 -9 Refr = \, x True hour angle = 360 92. 9 = 267. i 41 44^9 From Table VIII, for g2.g, p cos / = 3^.6 p cos / = 3.6 Approx. Lat. = 41 48'. 5 Table VII, lat. 41 48'.s, / = 92.6 41 44'. g Table VIII, for 92.6, p cos / = - 3'. 2 - 3 / . 2 and p sin / = 71' = i n' Lat. = 41 48'. i Table IX, az. corr. = 24 -2 Sum = true az. = Measured horizontal angle = Angle from North Bearing of mark I35'.2 8938 91 13' 2 S. 88 46' 8 E. EXAMPLE 3. OBSERVATION ON POLARIS AND d CASSIOPEIA. Feb. n, 1008. Set telescope at altitude 55 01' (5 5+ refraction corr.). d Cassiopeia; passed horizontal cross-hair at 7 h 03 05'. Interval for 1008 = 6 m io 3 . Computed time of observation on Polaris = 7 h o9 m 15". At 7 h 07 05" Polaris was bisected with both cross-hairs. Altitude = 43 05' . Index correction = + 1 ' . Angle between Polaris and mark = 67 1 1 ' Mark is east of Polaris, d Cassiopeiae is above and west of Polaris. Estimated latitude =42 True altitude = 55 From Table VII, / =49 .9 Interval = 0.5 Hour angle of Polaris =49 -4 From Table VIII, p cos / = + 45' .6 Computed time 7 - 09 - 1 5 Observed time 70705 .'. Latitude =42 19' .4 Interval = 2 io = o. S SECOND APPROXIMATION. Corrected hour angle = 50 .4 Interval = .5 Hour angle of Polaris = 49 .9 Table VIII, p sin /= 54' 4 Table IX, corr . = 1 9 .9 Azimuth of Polaris = 74' .3 = i 14' Measured angle =67 n' Direction of mark =N. 65 57' E. OBSERVATIONS ON 8 DRACONIS 23 EXAMPLE 4. OBSERVATION ON POLARIS AND d CASSIOPKI^F.. Jan. 2, 1908. $ Cassiopeia? Alt. 59 1 1 '.5; time, 9 h o7 m so 3 . Interval, 6 m io 8 . Computed time, 9 h 14 oo 3 . Polaris Alt. 43 13'; time, o b n m oo s . Angle from mark to Polaris, 106 49'; mark west of star, d Cassiopeiae is above and W. of Polaris. Assumed latitude =42 Computed time 9 h i4 m oo' True altitude =59 1 1' Observed time 9 n oo Table VII, /= 41. 4 Table VIII, p cos t = 4- 53' .4 Interval = 3 oo .'. Latitude =42 19' (approx.) SECOND APPROXIMATION. Table VII, corrected / =41. 5 Interval, 3 m ^= _ .8_ Hour angle of Polaris =40 .7 Alt. 43 13' ' Table VIII, p sin / = 46'. 3 Refr. i' Table IX, corr. =17.2 ' ~ __ 1 _ 43 12' 6 3 ' .5 P cos t +53-3 Az. of Polaris = i 03'. 5 Lat. 42 iS'.y Horizontal angle = 106 49' Az. of mark S. 72 07'. 5 W. 24. Observations on d Draconis. When d Cassiopeiae has an hour angle of less than 30 degrees or more than 150 degrees the hour angle cannot be accurately determined from the measured altitude, and con- sequently Table VII does not include such hour angles. In this case the altitude of d Draconis may be observed, since this star is nearly always in a favorable position for an observation at times when d Cassiopeiae is in an unfavorable position. The observation is made in just the same way as for d Cassiopeiae except for the time interval between the two observations. The difference in hour angle of d Draconis and Polaris is 6h i4 m 2is for 1910 too long to wait hence we make the obser- vation on Polaris as soon as convenient after the altitude of d Draconis has been measured, and correct the hour angle of d Draconis as previously described for d Cassiopeiae. For example, if the altitude of d Draconis was taken at 8h i5 m P.M. in the year 1910, adding 6h i4 m 2is gives i^h 29111 2 is as the calculated time when Polaris will have the same hour angle as d Draconis. If the observation were made on Polaris at 8h 2o m oos P.M. we must subtract from the true hour angle of Draconis the quantity i4h 29 m 2is 8h 2o m oo 8 = 6h 09 2is = 92 -3, which will give the hour angle of Polaris at the time it was observed. The difference in hour angle between Polaris and d Draconis for different dates is as follows: 1910, 6h 1401 2is = 93.6; 1920, 6h 18 58^ = 94.;; 1930, 6h 23 m 548 = 96. o. The following examples will illustrate the method 6f making the calculations. AZIMUTH EXAMPLE 5. Jan. 10, 1908. Set telescope at Altitude 37 01'. d Draconis passed horizontal hair at s h 57 m to 1 . Interval for 1008 = 6 h 13 26*. Observed Polaris at 5 h 59 40'; altitude = 43 30' ; plate vernier, 275 2 7'. 5; mark, 216 13' .o. d Draconis west of Polaris. Estimated latitude=42 True altitude= 37 Table X, Hour Angle=93-3 Hour Angle of Polaris*=93.3 92 .7= +o.6 Table XI, p cos /= + 1 n :. Latitude=42 18' Table X, Corrected Hour Angle=Q4.i Correction for interval =g^_^_ Hour Angle of Polaris = i^ Observed time s h 57" Interval 6 13 26 Computed time 12 10 36 Observed time 5 59 40 Diff . 6 h io m 56" = 92 .7 Vernier Readings. 27527 / -S 216 13 Angle = 59 1 4' .5 Table XI, p sin *=oi'.7 Table XII, correction= .6_ Azimuth Polaris= 02' .3 Angle to mark= 59 14' -5 Azimuth of mark=N. 59 i6'. W. Alt. 43 30' Refr. i' 43 29' P cost +i ii Lat. 42 1 8' EXAMPLE 6. Jan. 2, 1908. d Draconis; Observed altitude = 27 25'; observed time = 8 h 32 45*. Inter- val = 6 h 13 26*; computed time=i4 h 46 n 1 ; Polaris; altitude = 43 20'; observed time=8 h 34 45'. Angle, mark to Polaris, 107 01'. Mark is west of Polaris. d Dra- conis is west of Polaris. Assumed latitude=42 True altitude= 27 23' Table X, /= 124 JQ Computed time=i4 h 46 ii Observed time= 8 34 45 Interval = 6 n 26 Hour angle of Polaris=i24.o-92.8=3i .2 = 92 .8 Table VIII for 31 .2, p cos /= +60' .*. Latitude=42 19' (approx.) Table X, corrected /=i25.2 Hour angle of Polaris=32.4 Table VIII, p sin t= Table IX, correction= 3 8'.o 14 a Corrected altitude = 43 19 P cos /= 4-i oo' Latitude = 42 19' Azimuth of Polaris= 52'.2 Angle= 107 01' Azimuth of mark= 107 53-2 = S. 72 07' W. * In case this Hour Angle becomes negative it should be subtracted from 360 degrees. Polaris in that case would be east of the meridian. MERIDIAN LINE BY POLARIS AT CULMINATION 2 5 The above examples have been worked out more elaborately than would be required in many cases. Frequently the latitude will be known or may be estimated closely, so that only one approximation is needed. Nearly all of the interpolation may be done mentally. 25. Meridian Line by Polaris at Culmination. When Polaris is near the meridian (i.e. near culmination) it will sometimes be convenient to use the following simple method, which will give the meridian with about the same accuracy as the method of Articles 19-24. We may use in this case either o Cassiopeiae or the star f in the Great Dipper (see Fig. 3). If we determine by means of a surveyor's transit the instant when one of these stars, say d Cassiopeiae, is vertically above or vertically below Polaris, then we have only to wait a certain interval of time, depend- ing upon the date, when Polaris will be in the meridian. If Polaris is sighted at this instant, the telescope may be lowered and the direction of this line marked on the ground, thus giving a meridian line without further calculation The instant when the two stars are in the same vertical plane cannot be determined precisely, but it can be observed with all the accuracy required by setting the cross-hair on Polaris a few minutes before they are in the same vertical plane and then noting the instant when the other star passes the vertical cross -hair. If the interval between the two observations is not more than, say, 5 m , Polaris will change its direction so little that the effect on the observed time of the other star may be neglected. A convenient way to keep this interval small is to set on Polaris, lower the telescope, and wait until the other star appears in the field; then reset on Polaris, lower the telescope, and observe the instant of transit. The intervals which it is necessary to wait before Polaris is in the meridian are given in the following table for the two stars men- TABLE A, Interval be- Interval be- Date. tween 6 Cassi- opeiae and tween f Ursae Majoris and Polaris. Polaris. 1910 7 m -3 6m. 5 1920 ii -5 10 .3 tioned and for the years 1910 and 1920. These intervals are nearly correct when Polaris is either above or below the pole. In high latitudes it will in general be necessary to use the star which is at lower culmination at the time of the observation. In low latitudes it may be more conven- ient to use the star which is at upper culmination. 26 AZIMUTH The precision with which the time must be observed, when using this method, may be judged from the fact that at the instant of culmination the azimuth of Polaris is changing at the rate of about i ' of angle in 2 m of time (in the latitudes of the United States). Hence it will be seen that although Polaris is in the most unfavorable position for a precise azimuth observation, yet even at this time its azimuth may readily be obtained within i' of the true value. 26. Azimuth of Polaris when the Time is Known. Measure the angle between Polaris and the mark, noting the time of pointing on the star. If possible, measure the altitude of the star. (a) Express the watch time as local time by first applying any known error of the watch and then adding 4 m for each degree that the place is east of the standard meridian; or subtract 4 m for each degree if the place is west. If the time is A.M., add i2 h and subtract one day from the date. (b) Interpolate in the following table for the time of upper culmina- tion for the date and year as directed below. TIME OF UPPER CULMINATION OF POLARIS FOR 1915. Jan. i Jan. 15 Feb. i 6 4 6?9 5 Si -6 4 44-5 July i July 15 Aug. i i8 h sii 17 56.3 16 49-7 Proportional Part for Days. d m Feb. 15 3 49-2 Aug. 15 IS SS-o I 3-9 2 7.8 Mar. i 2 54-o Sept. i 14 48.4 3 n.8 Mar. is i 58.8 Sept. 15 13 53-5 4 iS-7 Apr. i o Si-9 Oct. i 12 50.7 5 19-6 Apr. 15 23 52.9 Oct. 15 ii 55-8 6 23.5 7 27.4 May i 22 SO.O Nov. i 10 48.9 8 31-4 May 15 21 55-1 Nov. 15 9 53-8 9 35-3 June i 20 48.5 Dec. i 8 50.8 10 39.2 June 15 19 53-7 Dec. 15 7 55-6 For 1916, add i m s before Mar. i ; subtract i6 on and after Mar. I. For 1917, subtract 0^7. For 1918, add o9. For 1919, add 2s. For 1920, add 4o before Mar. i; add .1 on and after Mar. i. (c) Take the difference between the time computed under (a) and (6) ; add io s for each hour in this interval, and then express the result in degrees and minutes, remembering that i h = 15 and i m = 15'. (d) Enter Table VIII (or IX) with this number of degrees and find p sin t. If the number of degrees exceeds 180, subtract 180 and look opposite the remainder for p sin /. Correct this for altitude by Table IX (or XII), employing preferably the measured altitude; otherwise use the latitude of the place. ACCURATE DETERMINATION OF AZIMUTH 27 (e) The result is the azimuth of Polaris. If the time of upper cul- mination is greater than the local time of the observation, the star is east of the meridian, unless the number of hours exceeds 12, in which case it is west. If the time of culmination is less than the local time, the star is west if the number of hours is less than 12, otherwise it is east. EXAMPLE. On May 8, 1917, in lat. 40 N., long. 71 W., the angle is measured between Polaris and a reference mark, the observation being taken at 7 h 45 P.M., Eastern Time; the watch is i m fast ; altitude of Polaris = 39 06'. Observed time 7 h 45?o Error i.o 7 44.0 May 8 22 22.6 1915 Longitude corr. 16.0 Upper culm. 22 21.9 1917 Local time 8 h oo?o Local time 14 1/3 X io s From Table VIII, using 216 05' 180 = 36 05', for 1917, we find p sin / = 40'. i. From Table IX, for p sin t = 40'.! and altitude 39 05', the correction = n'.s. The azimuth of Polaris is therefore o si'.6, and is west according to (e). 27. Accurate Determination of Azimuth by Observation on Polaris. An accurate determination of azimuth may be made by using a method similar to that of Article 19 except that the determination of the hour angle must be more precise, and the angle between the pole-star and the azimuth mark must be measured with greater accuracy. The determination of the hour angle should be made by observing several altitudes in quick succession, with their corresponding watch readings, on some star which can be identified (called the lime-star) and which is nearly east or west at the time of the observation. Following is a list of bright stars which may be used for time determinations in the northern hemisphere. The position of these stars may be found by consulting the star maps on pp. 62-3. The exact right ascension and declination for any date must of course be taken from the list of Apparent Places given in the Nautical Almanac. In identifying stars by means of the chart the observer should be on the lookout for the planets. These are not fixed in position and hence cannot be shown on the chart. If a very bright star is seen which cannot be found on the chart it is a planet and should not be used for time observations unless it can be positively identified and its exact position for the date determined. 28 AZIMUTH LIST OF FIXED STARS Constellations and Letters. Name. Right Ascension. Declination. a. Arietis Hamal 2h oi m 4- ^ 02' a Tauri Aldebaran 4. 3O ft Orionis Rigel 5IO 8 18 a Orionis Betelgeux . . . c CQ + 7 23 a Can. Maj Sirius 6 41 7 *6 TO 3C a Can. Min Procyon 7 34 + c 27 a Hydrae Alphard. 23 8 ic a Leonis Regulus IO O3 -f- 12 2 C, 3 Leonis Denebola 1 1 4.4. ct Virginis Spica I 3 2O ~ *3 U J ct Bootis Arcturus 14. 1 1 a Cor Bor Alphecca I ^ IO -|- 27 OI 16* * An approximate meridian, found by one of the methods previously given, will usually be sufficiently accurate for this time observation. COMPUTING THE AZIMUTH 31 COMPUTATION OF THE HOUR ANGLE OF REGULUS. nat. sin= .8666 nat. sin= .3018 Latitude 42 21' Declination + 1 2 25' Difference 29 56' Co 60 04' Altitude 1 7 34' log sec. .1313 log sec. .0102 log 9-7510 log vers=9.8934 Diff .= .5648 Right Ascension Polaris = i 25 32.4 Diff. Right Ascension = 8 h 37 m 56.7 = 129 29' / = - 77 2 6' at 7* i5 m 031 = 129 29' Hour Angle of Polaris ) at 7 h I5m 03 s } - S^ 03 HOUR ANGLES OF POLARIS Intervals ist half-set 2nd half-set Hour Angle of Polaris Corresponding Time r= 7 >>223i 7 h 28 m i6* 53 55' 7* 22 m 31 T=7 15 03 7 15 03 55 22 7 28 16 T'-T= 7 ">28' Red. to ^ sidereal ) 13 I3 . 2 Interval = 7 299 52 03' I3 m I5 3 19' 52 3 ' Hour Angle 53 55' 55 22' COMPUTATION OF AZIMUTH. First half-set. Second half-set. log p = i .8503 log sin t=g. 9075 log sec h = .1361 log azimuth = i .8939 azimuth = 78' .32 azimuth = i i8'-32 Angle =67 1 6' Azimuth of line, N. 65 57' .7 E. N. 65 s8'.3 E. Mean azimuth of mark= 245 58' JQ 32 AZIMUTH EXAMPLE 2. AZIMUTH OBSERVATION. Latitude 42 03' Sept. 5, 1906. HORIZONTAL ANGLES FROM MARK TO POLARIS. (Mark East of Star) Tel. Direct Times on Polaris. Vernier 6 h 39 48* o oo' ob" 40 57 6th Rep. 211 45' 30" 41 39 Mean 35 17' 35" 42 29 43 13 44 01 Mean 6 h 42 oi 8 .i= T Tel. Reversed o oo' oo" 6 h 50 15* 6th Rep. 211 32' oo" 51 16 Mean 35 15' 20" 52 52 54 26 55 37 56 55 T= 7 h 52"" 26" (mean) TIME OBSERVATIONS. Mean Altitude of Arcturus = 27 12' (West) Refraction = 2' Reduced Altitude =27 10' COMPUTATION OF HOUR ANGLE OF ARCTURUS. nat. sin = .9247 nat. sin = 4566 Latitude =42 03' Declination = 19 40.4 log sec. log sec. .1292 0263 Difference =22 2 2' .6 Co =67 37' 4 Altitude = 27 10' Diff. = .4681 log 9.6703 log vers=9.8258 Right Ascension Arcturus=i4 k n m 221.6 / = 70 42' at 7 h 52 m 26" Right Ascension Polaris= i 26 21.7 191 15 Diff. Right Ascension= i2 h 45 m oo .9 Hour Angle = 191 15' of Polaris = 261 57' at MERIDIAN BY POLARIS AT ELONGATION 33 HOUR ANGLES OF POLARIS. Intervals ist half-set 2nd half-set T' 6 h 4 2m oi '.I T 7 52 26 6 h 53 m 33 a -5 7 52 26 T r=-i h io m 259 Red. to i Sidereal j 10 Interval= i h io m 37" = - 17 39' 261 57' o h 59 m 02' - 14 46' 261 57' Hour Angle of Polaris Time 244 1 8' 247 ' 6 h 42 m OI s 6 h 53 m 33 8 -5 Altitudes Latitude = 42 03' & 42 03' .o P cos /= 31 .1 27.8 4i3i'.g 41 35'. 2 K (Table XIII) .5 , .5 Altitude 41 31'. 4 41 34'.; COMPUTATION OF THE AZIMUTH. Direct Reversed p cos / = 31'. 13 27' .84 log p cos / = i .4932 n. i .4446 n. log cos / =9.6372 n. 9.5886 n. log #=1.8560* 1.8560 log sin / =9.9548 n. 9.9646 n. log sec h =0.1257 0.1261 log Azimuth = 1.9365 n. 1-9467 n. Azimuth = 86' .40 88' .45 = i 26' 24" i 28' 27" Angle 35 17 ,35 35 15 20 Azimuth of line = 36 43' 59'' 36 43' 47" Mean, N 36 43' 53". o E. 31. Meridian by Polaris at Elongation. If Polaris is at its extreme east or west elongation, i.e., if it has its greatest east or west bearing, its azimuth can be accurately determined without knowing the exact time at which elongation occurs. The approximate time, near enough to determine when the observation should be begun, may be taken from Table B. The positions of the constellations at the times of elongations may be seen by reference to Fig. 3. * p cos / is computed by adding upward, and the azimuth by adding downward. 34 AZIMUTH TABLE B.* APPROXIMATE LOCAL TIMES OF ELONGATION OF POLARIS. Date. Eastern Elongation. Western Elongation. Date. Eastern Elongation. Western Elongation. * h m h m h m h m Jan. i 12 50P.M. 12 40A.M. July I 12 58 A.M. 12 48 P.M. " 15 ii 55 A.M. II 45 P.M. " 15 II 59P.M. ii 53 A.M. Feb. i 10 48 10 3 8 Aug. i 1 53 10 47 " 15 9 52 9 42 " 15 9 58 9 S 2 Mar. i (i 8 57 8 47 Sept. i 851 8 45 15 8 02 7 52 " i5 7 57 7 5i Apr. i 6 55 6 45 Oct. i 6 54 6 48 15 6 oo 5 5 " 15 5 59 5 53 May i 4 57 4 47 Nov. i 4 52 4 46 " 15 4 02 3 52 " 15 3 57 3 Si June i 2 5 6 2 46 Dec. i 2 54 2 4 8 " 15 2 OI i 5i " 15 i 59 i 53 TABLE C. POLAR DISTANCES OF POLARIS. 1911 10' 07".82 1916 i 08' 3 4".97 1912 09 49.22 1917 i 08 16.45 1913 09 30.64 1918 i 07 57.94 1914 09 12.07 1919 i 07 39.45 1915 08 53-51 1920 i 07 20.98 In order to make this observation set the transit in position a half hour or more before the pole-star reaches its eastern or western elongation. Set the vertical cross-hair on Polaris and follow it, with the plate tangent screw, as long as the star continues to move away from the meridian. When the star is near its greatest elongation it will appear to move verti- cally for a few minutes and then will begin to move back toward the meridian. While the star is in this extreme position it should be carefully * To find the Local Time for any other date than the first or isth, interpolate between the values given in the table; the daily change is about 4 minutes. To convert this Local Time into Standard Time take the diference between the longitude of the place and the longitude of the Standard meridian (expressed in hours, minutes, and seconds) and add this difference to the Local Time if the place is west of the Standard meridian, subtract if the place is east. MERIDIAN BY EQUAL ALTITUDES OF A STAR 35 bisected with the vertical cross-hair and a stake set at some convenient distance from the instrument in line with the cross-hair. In order to eliminate the errors due to poor adjustment of the transit the telescope should be immediately reversed, the star again bisected and another point set in line with the vertical cross-hair. The mean of these two points will give the direction of the star at elongation. The angle between this direction and the direction of the meridian may be cal- culated by the formula sin azimuth = sin polar distance X sec latitude, [8] or, with sufficient accuracy, azimuth (in seconds) = polar distance (in seconds) X sec latitude. [9] This calculated azimuth may be laid off either by means of the transit, using repetitions, or by measuring the distance from the transit to the point that was set, and then calculating the perpendicular offset which will give a point exactly north of the transit. This offset should be laid off with a steel tape. If desired, angles to some mark may be measured instead of setting stakes in line with the cross-hair. At elongation Polaris changes its azimuth so slowly that there will usually be time to set several points or to measure several angles before the star changes its bearing as much as 5 seconds of angle. In latitude 40 N., for example, the azimuth of Polaris 30 minutes before or after elongation is about one minute (of angle) less than it is at elongation. The change in azimuth varies as the square of the time interval, hence in io m either side of elongation the azimuth would vary about 6 to 7 seconds. The polar distance of Polaris may be taken from Table C, or from Table VIII under p sin / and opposite H. A. = 90. If the latitude is not known, the observed altitude of Polaris may be taken as approxi- mately equal to it. The error resulting from this assumption will be about i" of azimuth for each i' error in the latitude. 32. Meridian by Equal Altitudes of a Star. A very simple method of determining the direction of the meridian is by observing on a star at equal altitudes on opposite sides of the meridian. This method is accu- rate and requires no Nautical Almanac or tables of any kind ; it is not convenient, however, as it requires two observations at night separated by several hours' time, but it may prove of value when for some reason the more rapid and convenient methods are not available, as, for instance, in the southern hemisphere where there is no bright star near the pole. In order to determine the true meridian select some star which is not far from the pole and which is on the west side of the meridian in about the position of A, Fig. 6. The hour angle should be such that the star 36 AZIMUTH will reach an equal altitude on the east side of the meridian about 6 h or 8 h later, so that the second half of the observation will occur before daylight. One of the stars in Cassiopeia could be used, for example, the first observation being made when the star has an hour angle of about 135 degrees and the second at an hour angle of 225 degrees, as shown at A and A' in Fig. 6. The star is bisected with both cross-hairs, the horizontal circle is clamped, and the altitude is read and recorded. The telescope is then lowered and a point set in line. Some memorandum or sketch should be made for identifying the star at the second observa- tion. When the star is approaching the same altitude on the east side of HORIZON Evening- Observations. Morning Observations. FIG. 6. Meridian by Equal Altitudes of a Circumpolar (Northern Hemisphere). the meridian the telescope is set at exactly the same altitude as was read at the first observation. The star is then bisected with the vertical cross- hair, and followed until it passes the horizontal cross-hair. After this instant the tangent screw should not be touched. Another point is then set in line with the vertical cross-hair as before. The bisector of the angle between these two points is the meridian line through the instru- ment. If desired, several pairs of altitudes may be observed, to increase the accuracy, as h and h, h' and h' ', Fig. 6. Each pair should be combined independently of the others. If preferred, angles may be measured from some reference mark instead of setting points in line with the star. Instead of taking the altitudes at random it is well to set the telescope so that the vernier reads some whole degree or half degree and then set the vertical cross-hair on the star and follow it with the upper tangent screw MERIDIAN BY EQUAL ALTITUDES OF THE SUN 37 until both cross-hairs bisect the star. If the transit has a full vertical circle, errors in the instrument may be eliminated by taking the evening observations with the instrument direct and the morning observations with the instrument reversed. 33. Meridian by Equal Altitudes of the Sun. The observation just described may also be made on the sun at equal altitudes in the forenoon and afternoon, the difference being that the sun changes its declination during the interval between the observations and hence a correction must be applied in order to obtain the direction of the true meridian. Since the change for a given date is practically the same each year no Nautical Almanac is necessary, but the hourly change in declination for any date may be taken from Table XV. The observation is made as follows: At some time in the forenoon, say between 8h and ioh A.M., set up the transit and set the plate vernier at o degrees. Point the telescope at some azimuth mark and clamp the lower plate. Loosen the upper plate and turn to the sun. Set the vertical arc so that the vernier reads some whole degree or some even 10 minutes (higher than the sun); then set the vertical cross-hair on the left edge of the sun and follow, with the upper tangent screw, until the lower edge of the sun is in contact with the horizontal cross-hair. Note the time and read the altitude and the horizontal angle. In the after- noon, a little before the sun reaches this same altitude, set the vernier on o degrees and point again on the mark. Set the telescope so that the vernier of the vertical arc reads the same altitude as was used for the A.M. observation. When the sun comes into the field of the telescope set the vertical cross-hair on the right edge of the sun and follow it, with the upper plate tangent screw, until the lower edge of the sun is again in contact with the horizontal cross-hair. Note the time and read the horizontal angle. The mean of the two readings of the horizontal angle is approximately the angle between the mark and the south point. Since, however, the sun's declination is not the same at the two observations it will be neces- sary to apply the correction 2 cos Lat. X sin Hour Angle in which d is the hourly change from Table XV multiplied by the num- ber of hours between the observations, and the Hour Angle equals half this number of hours turned into degrees. In the table, the + sign indi- cates that the sun is going north, the sign indicates that it is going south. If the sun is going north the mean of the two angles gives a point west of the true south; if the sun is going south the mean angle is to a point east of south. AZIMUTH EXAMPLE. Latitude 42 18' N. A.M. Observation Reading on mark o oo' Pointings on left and upper edges Altitude 24 58' Horizontal Angle* 357 14' 15" Time 7 h 19 30* AM. % elapsed time= 4 h 26 221 '=66 35' 30" log sin ^=9.9627 log cos L =9.8690 9-8317 log 230" .9= 2. 3634 2-5317 Correction= - 340" .2= - 5* 40" .2 April 19, 1906. P.M. Observation Reading on mark o oo' Pointings on right and upper edges Altitude 24 58' Horizontal Angle* 162 28' oo" Time 4 h 12 15" P.M. Change in declination in = 2 30" .9 Mean Plate Reading 259 51' 08" 1 80 Uncorrected bearing 79 51' 08" Correction 5' 40" Corrected bearing S. 79 45' 28" E. Azimuth= 280 14' 32" * Read "clockwise" in each case. TABLES I. REFRACTION CORRECTION. II. CONVERGENCE OF MERIDIANS. III. LOGARITHMIC SECANTS. IV. NATURAL SINES. V. LOGARITHMS OF NUMBERS. VI. LOGARITHMIC VERSED SINES. VII. HOUR ANGLES OF d CASSIOPEIA. VIII. COORDINATES OF POLARIS. IX. CORRECTION FOR AZIMUTH. X. HOUR ANGLES OF d DRACONIS. XI. COORDINATES OF POLARIS. XII. CORRECTION FOR AZIMUTH. XIII. VALUES OF K FOR COMPUTING THE ALTITUDE. XIV. CORRECTION TO SUN'S DECLINATION. XV. SUN'S DECLINATION. AZIMUTH TABLE I. REFRACTION CORRECTION. (In Minutes.) (True Alt. = Meas'd Alt. - Refr.) Alt. Refr. Alt. Refr. Alt. Refr. 5 9'- 9 13 4'. i 25 '.i 6 8-5 14 3-8 30 7 7 7-4 15 3-6 35 4 8 6.6 16 3-3 40 .2 9 5-9 17 3-1 45 .0 10 5-3 18 3-0 50 0.8 ii 4-9 19 2.8 55 0.7 12 4-5 20 2.6 60 0.6 TABLE II. CONVERGENCE OF THE MERIDIANS. (In Minutes.) Miles (east or west). Lat. i 2 3 4 5 6 7 8 9 10 30 35 '-5 0.6 I'.Q 1.2 i'.S 1.8 2'.0 2.4 2'. 5 3-o 3'-o 3-6 3'- 5 4-2 4 '.o 4-9 4'- 5 5-5 S'-O 6.! 40 45 50 0.7 0.9 I.O i-5 i-7 2.1 2.2 2.6 3-1 2.9 3-5 4.1 3-6 4-3 5-2 4-4 11 5-i 6.1 7-2 5-8 6.9 8-3 6-5 7-8 9-3 w 10.3 TABLES TABLE III. LOGARITHMIC SECANTS. tf Minutes. Proportional Parts. Q o' 10' 2O' 30' 40' 50' i' 2' 3' 4' 5' 6' 7' 8' 9' o .0000 .0000 .0000 .0000 .0000 .0000 O o o o o o o i .0001 .0001 .0001 .0001 .0002 .0002 o o 2 .0003 i-w-y> .0003 .000^ .0004 .ooo .0005 .0005 o o o o o o I 3 4 .0000 .0011 .0011 __. T o .0012 .0013 .001^ .0015 o o o I I I I I 6 .0017 .0024 . OOIo .0025 .OOI( .0027 .0028 .0029 .0031 o I 1 I I I I 7 g .0032 .0034 .0036 .0037 .0039 .0041 I 1 1 I I I 9 .0054 .0056 .0058 !oo6o !oo6 2 .0052 .0064 I I I I I 2 2 IO .0066 .0069 .0071 .0073 .0076 .0078 I I 1 I 2 2 2 ii .0081 .0083 .0086 .0088 .0091 .0093 o I I I 2 2 2 2 12 .0096 .0099 .OIOI .0104 .0107 .0110 o I I 1 2 2 2 3 13 .0113 .0116 .0119 .0122 .0125 .0128 o I I 2 2 2 2 3 14 .0131 .0134 0137 .0141 .0144 .0147 I I 2 2 2 3 3 IS .0151 .0154 .0157 .0161 .016^: .0168 I I 2 2 2 3 3 16 .0172 .0175 .0179 .0183 .0186 .0190 I I 2 2 3 3 3 i7 .0194 .0198 .0202 .0206 .0210 .0214 o I 2 2 2 3 3 4 18 .021? .0222 .O226 .0230 .0235 .0239 I 2 2 3 3 3 4 iQ .0243 .0248 .0252 .0257 .0261 .0266 o I 2 2 3 3 4 4 20 .027O .0275 .O279 .0284 .0289 .0294 I 2 2 3 3 4 4 21 .0298 0303 .0308 0313 .0318 0323 I '2 2 3 3 4 4 5 22 .0328 ' 3 6s 0339 0344 0,76 .0349 .0382 0354 .0387 I I 2 2 2 3 3 4 4 5 23 24 .0300 .0393 0398 .O37I .0404 * J O/ V -' .0410 .0416 OyfCT .0421 rMC*7 I 7 2 2 3 3 4 5 5 25 26 .0427 .0463 433 .0470 ^43? .0476 .0445 .0482 U 4v> x .0488 U 4o/ 0495 I 2 3 3 4 4 5 6 27 .0501 .0508 .0514 .0521 .0527 0534 I 2 3 3 4 4 5 6 28 .0541 0547 .0554 .0561 .0568 0575 7 2 3 3 4 5 6 29 .0582 .0589 .0596 .0603 .0610 .0617 I 2 3 4 4 5 6 6 3 .0625 .0632 0639 .0647 .0654 .0662 I 2 3 4 5 5 6 7 31 .0669 .0677 .0685 .0692 .0700 .0708 I 2 3 4 5 5 6 7 32 .0716 .0724 .0732 .0740 .0748 .0756 1 2 3 4 5 6 6 7 33 .0764 .0772 .0781 .0789 .0797 .0806 I 2 3 4 5 6 7 7 34 .0814 .0823 .0831 .0840 .0849 .0858 I 3 3 4 5 6 7 8 35 .0866 .0875 .0884 .0893 .0902 .0911 I 3 4 5 5 6 7 8 36 .0920 .0930 0939 .0948 .0958 .0967 1 3 4 5 6 7 7 8 37 .0977 .0986 .0996 .1005 .1015 .1025 T 3 4 $ 6 7 8 9 38 1035 .1045 1055 .1065 .1075 .1085 I 3 4 S 6 7 8 9 39 .1095 .1105 .me .1126 .1136 .1147 I 3 4 5 6 7 8 9 40 ."57 .1168 .1179 .1190 .1200 .1211 T 3 4 5 6 7 9 4i .1222 1233 .1244 1255 .1267 .1278 I 3 4 6 7 8 9 42 .1289 .1301 .1312 .1324 .1335 1347 I 3 5 6 7 n 9 43 1359 I37I .1382 .1394 .1406 .I4l8 I 4 5 6 7 8 o I 44 1431 1443 .1455 .1468 .1480 1493 I 4 5 6 7 9 I 45 1505 .1518 .1531 .1543 .1556 .1569 I 3 4 5 6 8 9 7O 2 46 .1582 .1595 .1609 .1622 1635 .1649 I 3 4 5 7 8 9 I 2 47 .1662 .1676 .1689 .1703 .1717 .1731 I 3 4 6 7 8 II 2 48 1745 1759 1773 .1787 .1802 .l8l6 I 3 4 6 7 9 o II 3 49 .1831 .1845 .1860 .1875 .1889 .1904 I 3 4 6 7 9 10 2 3 50 .1919 1934 .1950 .1965 .1980 .1996 2 3 5 6 8 9 II 12 4 Si .2011 .2027 2043 .2059 .2074 .2090 a 3 5 6 8 II 13 4 52 .2107 .2123 .2139 .2156 .2172 .2189 2 3 5 7 8 12 13 5 53 .220-; .2222 .2239 .2256 .2273 .229O 2 3 5 7 Q o 12 14 5 54 .2308 2325 2343 .2360 .2378 . 2396 2 4 5 7 9 II 12 14 6 55 .2414 2432 .2450 .2469 .2487 .2^06 2 4 6 7 9 I 13 15 7 56 .2524 2543 .2^62 .2581 .2600 .2620 2 4 6 8 o II 13 15 7 57 2639 .2658 .2678 .2698 .2718 2738 2 4 6 8 12 14 76 8 58 .27=^8 .2778 .2799 .2819 .2840 .2861 2 4 6 8 TO T2 M 76 9 59 .2882 .2903 .2924 .2945 2967 .2988 2 4 6 I 13 15 17 9 60 .3010 3032 3054 3077 .3099 ..?I22 2 a 7 n T 3 76 18 20 AZIMUTH TABLE IV. NATURAL SINES. Minutes. Proportional parts. & Q o' 10' 20' 30' 40' 50' i' 2' 3' 4' 5' 6' 7' 8' 9' .0000 .0029 .0058 .0087 .0116 .0145 3 6 12 15 20 23 26 i 0175 .0204 0233 .0262 .0291 .0320 3 ft 9 12 15 j 20 23 26 2 0349 .0378 .0407 0436 .0465 .0494 3 ft 9 12 15 ~ 20 2; 26 3 .0523 0552 .0581 .0610 .0640 .0669 3 ft 9 12 15 7 20 23 26 4 .0698 .0727 .0756 .0785 .0814 0843 3 ft 9 12 15 7 20 2 > 26 5 .0872 .0901 .0929 0958 .0987 .1016 3 6 9 12 I5 7 20 23 26 6 .1045 .1074 .1103 II32 . 1161 .1190 3 ft 9 12 14 j 20 2,3 26 7 .1219 .1248 .1276 I30S 1334 1363 3 6 9 12 i_ 7 20 2c 26 8 1392 .1421 .1449 .1478 1507 .1536 3 6 9 I I i- - 20 23 26 9 1564 1593 .1622 .1650 .1679 .1708 3 6 9 11 14 7 20 23 26 10 .1736 1765 1794 .1822 .1851 .1880 3 ft 9 I I i_ 7 2O 23 26 ii . I9 o8 1937 .1965 .1994 .2022 .2051 3 6 9 1 I 14 - 20 23 26 12 .2079 .2108 .2136 .2164 2193 .2221 3 6 9 I I i_ -. 2O 23 26 13 .2250 .2278 .2306 2334 2363 .2391 3 6 8 11 14 j 20 23 25 14 .2419 .2447 2476 .2504 2532 .2560 3 6 8 11 14 - 20 22 25 IS .2588 .2616 .2644 .2672 .2700 .2728 3 6 8 II 14 - 20 22 25 16 .2756 .2784 .2812 .2840 .2868 .2896 3 6 8 11 14 7 20 22 25 i7 .2924 .2952 .2079 .3007 3035 .3062 3 6 8 1 1 14 20 aa 25 18 .3090 .3118 3145 3173 .3201 .3228 3 6 8 11 '4 7 20 22 25 19 -3256 3283 33" 3338 3365 3393 3 5 8 II 14 16 P 22 25 20 .3420 .3448 3475 3502 3529 3557 3 S 8 II M 1 6 19 22 24 21 3584 .3611 3638 3665 .3692 3719 3 5 8 II 14 1 6 19 23 24 22 3746 3773 .3800 .3827 3854 3881 3 5 8 II 12 16 19 21 24 23 3907 3934 .3961 3987 .4014 .4041 3 5 8 II 13 16 19 21 24 24 .4067 .4094 .4120 .4147 4173 .4200 3 5 8 II 33 16 18 21 24 25 .4226 4253 4279 4305 4331 4358 3 5 8 10 13 16 18 21 24 26 4384 .4410 4436 .4462 .4488 4514 3 5 8 10 13 16 18 21 23 27 4540 -4566 4592 .4617 .4643 .4669 3 5 8 IO 13 15 18 21 23 28 4695 4720 .4746 4772 4797 .4823 3 5 8 10 T ^ 15 18 20 23 29 .4848 4874 .4899 .4924 4950 4975 3 5 8 10 13 15 18 20 23 30 .5000 5025 5050 5075 .5100 5125 3 S 8 TO 13 15 18 20 23 31 5150 5175 .5200 5225 5250 5275 5 7 10 12 i5 *7 20 22 32 5299 5324 5348 5373 5398 5422 5 7 10 12 15 17 20 22 33 .5446 5471 5495 5519 5544 5568 5' 7 10 12 15 *7 19 22 34 5592 .5616 5640 5664 .5688 5712 5.' 7 10 12 14 17 19 22 35 5736 .5760 .5783 .5807 5831 .5854 -$ 7 Q 12 14 17 19 21 36 5878 .5901 5925 5948 5972 5995 S 7 9 12 14 16 19 21 37 .6018 .6041 .6065 .6088 .6111 .6134 5 7 9 12 14 16 18 21 38 6157 .6180 .6202 .6225 .6248 .6271 5 7 9 II 14 16 18 20 39 .6293 -6316 6338 636! 6383 .6406 4 7 9 II 13 16 18 20 40 .6428 6450 .6472 .6494 6517 6539 4 7 9 I I *3 15 18 20 4i .6561 6583 .6604 .6626 .6648 .6670 4 7 9 I I 13 15 17 2O 42 .6691 6713 6734 .6756 .6777 .6799 4 6 9 II 13 15 i7 19 43 .6820 .6841 .6862 .6884 .6905 .6926 4 6 8 II 13 15 17 19 44 .6947 .6967 .6988 .7009 .7030 .7050 4 6 8 10 12 14 i? 19 i o' 10' 20' 30' 40' So' i' 2' 3' 4' 5' 6' 7' 8' 9' Minutes. Proportional parts. TABLES TABLE IV. (Continued) . 43 Minutes. Proportional parts. bib Q o' 10' 20' 30' 40' 50' i' a' 3' 4' 5' 6' f 8' 9' 45 .7071 .7092 .7112 7133 7i53 7i73 4 6 8 JO la M 16 18 46 .7193 .7214 7234 7254 .7274 7294 4 6 8 JO 12 *4 JO 18 47 7314 7333 7353 7373 7392 .7412 4 6 8 JO la 14 1 6 18 48 7431 7451 .7470 .7490 7509 .7528 4 6 8 10 12 14 IS i7 49 7547 .7566 7585 .7604 .7623 .7642 4 6 8 II 13 IS i? SO .7660 .7679 .7698 .7716 7735 7753 4 6 7 9 II 13 15 17 51 .7771 .7790 .7808 .7826 .7844 .7862 4 5 7 II 13 14 16 52 .7880 .7898 .7916 7934 7951 .7969 4 s 7 1J 12 14 16 53 .7986 .8004 .8021 8039 .8056 .8073 3 s 7 9 JO 12 14 IS 54 .8090 .8107 .8124 .8141 .8158 8i75 3 S 7 8 10 12 14 15 55 .8192 .8208 .8225 .8241 .8258 .8274 3 s 7 8 10 T2 13 15 56 .8290 .8307 8323 8339 8355 8371 3 5 6 8 IO II *3 14 57 .8387 8403 .8418 8434 8450 8465 3 5 6 8 9 11 12 14 58 .8480 .8496 8511 .8526 .8542 8557 3 5 6 8 9 n ia 14 59 .8572 8587 .8601 .8616 8631 .8646 4 6 7 9 10 la 13 60 .8660 .8675 .8689 .8704 .8718 .8732 3 4 6 7 9 TO 12 13 61 .8746 .8760 .8774 .8788 .8802 .8816 3 4 6 7 8 10 II 13 62 .8829 8843 .8857 .8870 .8884 .8897 3 4 5 7 8 II 12 63 .8910 .8923 8936 .8949 .8962 8975 3 4 5 6 8 9 IO 13 64 .8988 .9001 .9013 .9026 .9038 .9051 3 4 5 6 8 9 10 II 65 .9063 9075 .9088 .9100 .9112 .9124 4 5 6 ' 7 S 10 II 66 9135 .9147 9159 .9171 .9182 .9194 3 5 6 7 ' 8 . 9 10 67 .9205 .9216 .9228 9239 .9250 .9261 3 4 6 7 8 9 IO 68 .9272 .9283 9293 9304 9315 9325 3 4 S 6 8 9 10 69 9336 9346 9356 9367 9377 9387 3 4 5 6 7 8 9 70 9397 .9407 .9417 .9426 9436 .9446 3 4 5 6 7 8 9 7 1 9455 9465 9474 9483 9492 .9502 3 4 5 6 7 7 8 72 9511 .9520 .9528 9537 9546 9555 3 4 4 5 6 . 7 8 73 9563 9572 9580 9588 9596 .9605 3 4 S 6 7 7 74 .9613 .9621 .9628 .9636 .9644 .9652 3 4 S 5 6 7 75 .9659 .9667 .9674 .9681 .9689 .9696 3 4 4 S & 7 76 9703 .9710 .9717 .9724 9730 9737 3 3 4 5 5 6 77 9744 9750 9757 9763 .9769 9775 3 3 4 4 5 6 78 .978i .9787 9793 9799 9805 .9811 3 3 4 .. S 5 79 .9816 .9822 .9827 9833 0838 9843 3 3 4 4 5 80 .9848 9853 9858 .9863 .9868 .9872 2 3 3 4 4 81 82 9877 . 99O3 .9881 . 99O7 .9886 . 99H .9890 . 9914 9894 .9918 .9899 . 9922 2 3 3 3 4 83 9925 .9929 9932 9936 9939 .9942 2 2 2 3 3 84 9945 .9948 9951 9954 9957 9959 I 2 2 2 3 85 86 .9962 .9964 rv"i*7& .9967 0080 .9969 0081 .9971 Qng } 9974 008 <; I I a 2 2 87 997^ .9986 . 997 .9988 . yyou .9989 yy 01 .9990 yy o o .9992 yyj 9993 I I i I I 88 9994 9995 .9996 9997 9997 .9008 O o I I 89 .9998 9999 9999 .0000 .0000 .0000 o o O o O bib Q o' 10' 20' 30' 40' So' x' a' 3' 4' 5' 6' -7? 8' 9' Minutes. Proportional parts. 44 AZIMUTH TABLE V. LOGARITHMS OF NUMBERS. 8 Proportional parts . o I 2 _ 4 5 6 M 8 g I 2 3 4 5 6 7 8 9 I0 .0000 0043 .0086 .0128 .0170 .0212 0253 .0294 0334 0374 ' 4 8 12 21 2 5 20 33 37 II -0414 0453 .0492 0531 .0569 .0607 .0645 .0682 .0719 0755 4 8 II 5 10 23 2( 30 34 12 -0792 .0828 .0864 .0899 0934 .0969 .1004 .1038 .1072 . 1106 3 7 IO 4 17 21 2 4 28 3J 13 -"39 1173 .1206 .1239 .1271 1303 .1335 1367 1399 .1430 3 ft 10 3 1 6 10 2 3 2ft 20 14 . 1461 .1492 1523 1553 .1584 .1614 .1644 1673 1703 1732 3 ( 9 2 15 18 21 24 27 15 .1761 .1790 .1818 .1847 1875 .1903 1931 1959 .1987 .2014 3 6 8 ii 14 17 20 22 2 5 16 -2041 .2068 .2095 . 2122 .2148 2175 .2201 .2227 2253 .2279 3 5 8 ii 13 1 6 1 8 21 24 17 -2304 18 2553 2330 2577 . 2601 .2380 .2625 2405 .2648 .2430 .2672 2455 .2695 .2480 .27l8 2504 .2742 2529 2765 5 5 7 7 10 9 12 15 14 17 2O 10 22 21 19 .2788 .2810 2833 .2856 .2878 .2900 .2923 2945 .2967 .2989 4 7 9 I 1 13 1 6 1 8 2O 20 -3010 3032 3054 3075 3096 3118 3139 .3160 .3181 3201 4 6 8 I I 13 ,- ,- 10 21 -3222 3243 3263 .3284 3304 3324 3345 3365 3385 3404 4 6 8 10 12 M ift 1 8 22 .3424 3444 3464 3483 3502 3522 3541 3560 3579 3598 4 ft 8 IO 12 '4 i 5 T 7 23 .3617 3636 .3655 3674 .3692 37" 3729 3747 .3766 3784 4 6 7 II 13 15 17 24 .3802 .3820 3838 .3856 3874 3892 3909 3927 3945 3962 4 5 7 9 II 12 '4 16 25 -3979 3997 .4014 .4031 .4048 -4065 .4082 4099 .4116 4133 3 5 7 () 10 12 14 15 26.4150 .4166 4183 .4200 .4216 4232 .4249 4265 .4281 .4298 3 5 7 8 10 II 13 15 27 -43 J 4 4330 4346 .4362 4378 4393 .4409 4425 .4440 4456 3 3 ft 8 9 II 13 14 28 -4472 .4487 .4502 .4518 4533 4548 4564 4579 4594 .4609 3 5 ft 8 9 II 12 '4 29 .4624 4639 .4654 .4669 4683 .4698 4713 .4728 4742 4757 3 4 6 7 9 IO 12 13 30 -4771 .4786 .4800 .4814 .4829 4843 4857 .4871 .4886 .4900 3 4 6 7 9 IO II 13 31 -49I4 .4928 .4942 4955 4969 4983 4997 .5011 .5024 5038 3 4 6 7 8 10 II 12 32 -5051 5065 5079 .5092 5105 5"9 5132 5145 5159 5172 3 4 5 7 8 9 II 12 33 -5185 .5198 .5211 5224 5237 5250 5263 5276 .5289 5302 3 4 j ft 8 9 IO 12 34 -5315 .5328 5340 5353 5366 5378 5391 5403 .5416 5428 3 4 5 ft 8 9 10 II 35 -5441 5453 5465 5478 5490 5502 5514 .5527 5539 5551 2 4 5 ft 7 10 36 -5563 5575 .5587 5599 5611 5623 5635 5647 .5670 3 4 5 ft 7 8 !O TI 37 . 5682 5694 5705 5717 5729 5740 5752 5763 5775 .5786 3 5 ft 7 8 IO 38 5798 .5809 .5821 5832 5843 .5866 5877 .5888 5899 3 5 ft 7 8 9 10 39 -59" 5922 5933 5944 5955 .5966 5977 5988 5999 .6010 3 4 5 7 8 c !O 40 .6021 6031 .6042 6053 .6064 6075 .6085 .6096 .6107 .6117 5 4 5 6 8 ( :o 41 .6128 -6138 .6149 .6160 .6170 .6180 .6191 .6201 .6212 .6222 3 4 5 6 7 8 9 42 .6232 .6243 6253 .6263 6274 .6284 .6294 6304 6314 6325 3 4 5 6 7 8 9 43 -6335 6345 6355 6365 6375 -6385 6395 .6405 6415 .6425 3 4 5 6 7 8 9 44 -6435 6444 6454 .6464 .6474 .6484 6493 6503 6513 6522 3 4 5 6 7 8 9 45 -6532 6542 6551 -6561 6571 6580 .67 8261 8267 8274 8280 8287 8293 8299 8306 8312 8319 i '-. 3 4 6 ,S 8325 8331 8338 8344 8351 8357 8363 837 8376 8382 i - j 4 4 6 6c> 8388 8395 8401 8407 8414 8420 8426 8432 8439 8445 i 2 3 4 4 6 7 8451 8457 8463 8470 8476 8482 8488 8494 8500 .8506 i 2 3 4 4 5 6 71 8513 8519 852=; 8531 8537 8543 8549 8555 8561 .8567 i 2 3 4 4 5 5 7 2 8573 3?79 8585 8591 8597 8603 8609 8615 8621 .8627 i 2 j 4 4 5 S 73 8633 8639 8645 8651 8657 8663 8669 8675 8681 .8686 i 2 4 4 5 5 74 8692 8698 8704 8710 8716 8722 8727 8733 8739 8745 i 2 3 4 4 5 5 75 8751 8756 8762 8768 8774 8779 8785 8791 8797 .8802 i 2 . 3 4 5 S 6 8808 8814 8820 8825 8831 8837 8842 8848 8854 8859 i 2 3 4 5 5 7 8865 8871 8876 8882 8887 8893 8899 8904 8910 8915 T 2 3 4 4 5 S 8921 8927 8932 8938 8943 8949 8954 8960 8965 .8971 I 2 3 4 4 5 o 8976 8982 8987 8993 8998 9004 9009 9015 9020 .9025 I 2 3 4 4 S o 9031 9036 9042 947 9053 9058 9063 9069 9074 .9079 I j 3 4 4 5 Si 9085 9090 9006 9101 9106 9112 9117 9122 9128 9i33 I 2 3 4 4 S 82 9138 9M3 9149 9154 9159 9165 9170 9i75 9180 .9186 I 2 3 4 4 5 S 9191 9196 9201 9206 9212 9217 9222 9227 9232 9238 1 2 3 4 4 s 84 9 2 43 9248 9253 9258 9263 9269 9274 9279 9284 .9289 I 2 3 4 4 5 85 9294 9299 9304 9309 9315 9320 9325 9330 9335 934 I 2 3 4 4 5 6 9345 9350 9355 9360 9365 9370 9375 9380 9385 939 I 2 3 4 4 5 87 9395 .9400 9405 9410 9415 9420 9425 943 9435 .9440 O 2 3 3 4 4 s 9445 9450 9455 9460 9465 9469 9474 9479 9484 .9489 2 3 3 4 4 80 9494 9499 954 959 9513 95i8 9523 9528 9533 .9538 o 2 2 3 3 4 4 00 9542 9547 9552 9557 9562 9566 957i 9576 958i 9586 o 2 2 3 3 4 4 01 9590 9595 9600 9605 9609 9614 9619 9624 9628 9633 2 2 3 3 4 4 02 9638 9643 9647 9652 9657 9661 9666 9671 9675 .9680 o 2 2 3 3 4 4 3 9685 .9689 9694 9699 9703 9708 9713 9717 9722 .9727 2 2 3 3 4 4 04 9731 9736 9741 9745 9750 9754 9759 9763 9768 9773 o 2 2 3 3 4 4 05 9777 .9782 9786 9791 9795 9800 9805 9809 9814 .9818 o 2 2 3 3 4 4 X> 9823 .9827 9832 9836 9841 9845 9850 9854 9859 9863 2 2 3 3 4 4 07 9868 .9872 9877 9881 9886 9890 9894 9899 9903 .9908 2 2 3 3 4 4 >8 9912 .9917 9921 9926 9930 9934 9939 9943 9948 9952 2 2 3 3 4 4 00 9956 .9961 99<55 9969 9974 9978 9983 9987 9991 9996 2 2 3 3 3 4 46 AZIMUTH TABLE VI. LOGARITHMIC VERSED SINES. 1 Minutes. Proportional parts. j o' 10' 2O' 30' 40' 50' i a' 3' 4 5 t> 7' ' 9' 10 8.1816 8.1959 8.2IOC 8. =239 8.2375 8.2510 14 27 41 55 69 82 96 i 10 123 II .2642 .2772 .2900 .3027 3151 3274 12 25 38 50 63 75 88 100 113 12 3395 3514 .3632 .3748 3863 3976 12 23 |35 46 58 69 81 92 104 13 .4087 .4198 .4414 .4520 .4625 11 21 32 43 53 64 75 85 96 14 .4728 .4830 493 5031 513 .5228 10 20 30 40 5 60 70 80 89 IS 5324 .5420 5514 5607 .5700 579 " 19 28 37 46 56 65 74 84 16 5881 5971 .6055 .6147 6234 .6319 \- 2t> 35 44 52 61 70 79 17 .6404 .6488 .6572 .6654 .6736 .6817 8 16 25 33 49 57 66 74 18 .6897 .6976 7055 7133 .7210 .7287 8 IS 23 131 39 46 | 54 62 | 70 19 .7362 .7438 751 7586 7659 7732 7 IS 22 30 37 44 52 59 66 20 7804 7875 794^ .8016 .8086 8155 7 14 21 28 35 42 49 56 63 21 .8223 .8291 -8358 8425 .8491 855- 7 i3 20 27 33 40 47 53 60 22 .8622 .8687 875 8815 .887^ .894 6 13 19 25 32 38 44 5i 57 23 9003 .9065 .9127 .9188 .9248 .9308 6 12 18 24 30 36 43 49 55 24 .9368 .9427 .9486 9544 .9602 .0660 12 17 23 29 35 41 47 5-2 25 8.9717 8.9774 8.9830 8.9886 8.9942 8-9997 6 IZ i7 22 28 34 39 45 50 26 27 9-0052 0374 9.0107 9.016 . 0426 . 0475 9.0215 0530 9.0268 .0582 19.0321 .0633 5 5 II IO 16 16 21 21 27 26 32 38 36 43 41 48 47 28 .0684 0734 .0785 0834 .0884 0933 5 10 15 2O 25 30 35 40 45 29 .0982 .1031 .1079 .1128 "75 .1223 5 IO 14 TO 24 29 34 38 43 30 .1270 1317 .1364 .1410 1457 1503 5 '4 19 23 28 32 37 42 31 .1548 1594 .1684 .1728 1773 4 9 13 18 22 27 3i 36 40 32 .1817 .1861 .1905 .1948 .1991 .2034 4 13 i - 22 26 30 35 39 33 .2077 .2120 .2162 .2204 .2246 .2288 4 8 13 17 21 25 29 34 38 34 2329 2370 .2411 2452 2493 2533 4 8 12 1 6 20 24 28 33 37 35 2573 .2613 2653 .2692 .2732 .2771 4 8 12 16 20 24 28 32 36 36 .2810 .2849 .2887 .2926] .2964 .3002 4 8 II 15 19 23 27 3 1 34 37 .3040 3077 3115 .3152 -3189 .3226 4 7 II 15 19 22 26 30 ] 33 38 3263 3300 3336 3372 -3409 3444 4 7 II 14 18 22 25 29 33 39 3480 3516 3551 3586 .3622 .3657 4 7 II 14 18 21 25 28 32 40 .3691 3726 .3760 3795 3829 3863 3 7 10 14 17 21 24 27 3 i .3897 3931 -3964 3998 .4031 .4064 3 7 10 '3 i? 20 23 27 30 42 .4097 .4130 .4162 4195 .4227 .4260 3 7 IO 16 20 23 26 29 43 .4292 4324 4356 .4387 .4419 4450 3 6 10 13 16 19 22 25 29 44 .4482 4513 4544 4575 .4606 .4637 3 6 9 12 15 19 22 25 28 45 .4667 .4698 4728 4758 4788 .4818 3 6 9 12 is 18 21 24 27 46 .4848 .4878 .4907 4937 .4966 4995 3 6 9 12 18 21 24 26 47 .5024 5053 5082 .5140 -5i68 3 6 9 I I 14 17 2O 23 26 48 5197 5225 5253 ;& 5309 5337 3 6 8 11 14 17 20 22 25 49 5365 5393 .5420 5448 5475 5502 3 5 8 I I 14 16 19 22 25 50 5529 5556 .5583 5610 5637 5663 3 5 8 II 13 16 19 21 24 Si 52 5690 5847 -5716 .5873 5743 5899 .5769 5924 5795 5950 5821 5975 3 3 5 5 8 8 IO 10 13 13 16 15 18 18 21 21 24 23 53 .6001 .6026 6051 .6076 .6101 .6126 3 5 8 IO 13 IS 18 20 23 54 .6151 .6176 .6201 .6225 6250 .6274 2 5 7 o 12 ' IS J 7 20 22 55 56 .6298 .6442 6323 .6466 6347 .6490 6371 6513 .6395 6537 .6419 .6560 5 5 7 7 IO D 12 12 14 li 19 19 22 21 57 6584 .6607 .6630 6653 .66-76 .6699 5 7 o II 14 16 18 21 58 .6722 .6744 .6767 .6790 .6812 6835 5 7 II 14 16 18 20 59 6857 .6879 .6902 .6924 .6946 .6968 4 7 II 13 IS 18 20 TABLES TABLE VI. (Continued). 47 bb Q Minutes. Proportional parts. o' 10' 20' 30' 40' So' l' 2 f 3' 4' 5' 6' 7' 8' 9' 60 9.6990 9.7012 9-7033 9-7055 9.7077 9.7098 4 6 9 ii 13 15 17 20 61 .7120 .7141 .7162 .7184 7205 .7226 4 8 ii 13 15 17 19 62 .7247 .7268 7289 7310 7331 7351 4 6 8 10 12 15 17 19 63 7372 7393 7413 7434 7454 7474 4 6 8 10 12 14 16 18 64 7494 7515 7535 7555 7575 7595 4 6 8 IO 12 14 16 18 65 7615 7634 .7654 .7674 7693 7713 4 6 8 IO 12 14 16 18 66 7732 7752 .7771 .7791 .7810 .7829 4 6 8 10 12 '14 15 i7 67 .7848 .7867 .7886 7905 .7924 7943 4 6 8 9 II 13 15 i? 68 .7962 .7980 7999 .8017 8036 -8054 4 6 7 9 II 13 15 i? 69 .8073 .8091 .8110 .8128 .8146 .8164 4 s 7 9 II 13 IS 16 70 .8182 .8200 .8218 .8236 8254 .8272 4 5 7 9 II 13 14 16 ?i .8289 .8307 8325 -8342 .8360 8377 4 5 7 9 II 12 T 4 16 72 8395 .8412 .8429 8447 .8464 .8481 3 5 7 9 IO 12 14 15 73 .8498 8515 8532 8549 .8566 -8583 3 5 7 8 10 12 14 15 74 .8600 .8616 8633 .8650 .8666 .8683 3 5 7 8 ID 12 13 15 75 .8699 .8716 .8732 .8748 8765 .8781 3 5 7 8 10 II 13 15 76 .8797 8813 .8829 .8845 .8861 8877 3 S 6 8 10 II 13 14 77 .8909 8925 .8941 8956 .8972 3 5 (> 8 9 II 13 14 78 .8988 .9003 .9019 9034 .9050 9065 3 5 6 8 9 II 12 14 79 .9081 r 9096 .9111 .9126 .9141 9157 3 5 6 8 9 II 12 14 80 .9172 .9187 .9202 .9217 .9232 .9246 3 4 6 7 9 IO 12 13 81 .9261 .9276 .9291 9305 9320 9335 3 4 7 9 10 12 13 82 9349 9364 9378 9393 .9407 9421 3 4 6 7 9 10 12 T 3 83 9436 945 .9464 9478 .9492 95o6 3 4 6 7 9 10. II 13 84 9521 9535 9548 9562 9576 9590 3 4 6 7 8 10 II 13 85 .9604 .9618 .9631 9645 9659 .9672 3 4 5 7 8 10 II 12 86 .9686 9699 9713 .9726 .9740 9753 3 4 5 7 8 9 II 12 8? 9767 .9780 9793 .9806 .9819 9833 3 4 5 7 8 9 II 12 88 .9846 .9859 .9872 9885 .9898 .9911 3 4 5 6 8 9 10 12 89 9.9924 9.9936 9-9949 9.9962 9-9975 9.9987 3 4 5 6 8 9 IO II 90 0.0000 0.0013 0.0025 0.0038 0.0050 0.0063 3 4 5 6 8 9 IO II 9i .0075 .0088 .0100 .0112 0125 0137 2 4 5 6 7 9 10 II 92 .0149 .0161 0173 -0185 .0197 .0210 2 4 5 6 7 9 10 II 93 .0222 .0234 .0245 0257 .0269 .0281 2 4 5 6 7 8 9 II 94 .0293 0305 .0316 .0328 .0340 0351 2 3 S 6 7 8 9 10 95 0363 0374 .0386 0397 .0409 .O42O 2 3 5 6 7 8 9 10 96 .0432 0443 0454 .0466 0477 .0488 2 3 4 6 7 8 9 10 97 .0499 .0511 .0522 533 0544 0555 2 3 4 6 7 8 9 IO 98 .0566 0577 .0588 0599 .0610 .O62O 2 3 4 5 6 8 9 10 99 .0631 .0642 0653 .0663 .0674 .0685 2 3 4 5 6 8 9 10 100 0695 .0706 .0717 .0727 .0738 .0748 2 3 4 5 6 7 8 10 101 .0758 .0769 .0779 .0790 .0800 .O8l0 2 3 4 5 6 7 8 9 IO2 .0820 .0831 .0841 .0851 .0861 .0871 2 3 4 5 6 7 8 9 103 .0881 .0891 .0901 .0911 .0921 .0931 2 3 4 5 6 7 8 9 104 .0941 .0951 .0961 .0970 .0980 .0990 2 3 4 5 6 7 8 9 IOS .1000 .1009 .1019 .1029 .1038 .1048 2 3 4 5 6 7 8 9 106 1057 . 1067 . 1076 .1086 .1095 HO5 2 3 4 5 6 7 8 9 107 .1114 .1123 1133 .1142 .1151 . 1160 2 3 4 5 6 6 7 8 108 .1169 .1179 .1188 .1197 .1206 .1215 2 3 4 5 5 6 7 8 109 .1224 1233 . 1242 .1251 .1260 .1269 2 3 4 5 5 6 7 8 AZIMUTH TABLE VII. HOUR ANGLES OF d CASSIOPEIA. Intervals: 1910, 6 m s8 s ; 1920, io m s7 s ; 1930, i5 m i3 s . Latitudes. Alt. 16 18 20 22 24 26 28 30 32 34 6 io6.o 109. 9 113. 8 n8.o 122. 5 127. 3 132- 7 138. 7 145. 9 I55- 2 8 101.8 105-5 109.3 ii3-3 "7-5 122. O 126. 9 132.2 138.3 145-5 10 97-7 IOI.2 104.9 108.8 112. 8 I.I7.0 121.5 126.4 131-8 137-9 12 93-6 97-1 100.7 104.4 108.2 112. 2 116.4 121. 125.9 14 89.6 93-o 96-5 100. I 103.8 IO7.6 in. 6 115.9 i 120.4 125-3 16 85.6 89-0 92-4 95-9 99-5 103.2 107.0 III.O ii5-3 II9-8 18 81.6 85-0 88.4 91.8 95-3 98.9 IO2.6 106.4 110.4 II4.7 20 77-6 81.0 84.4 87-7 91. 94-7 98.2 101.9 105.8 109-8 22 73-6 77-o 80.4 83-7 87- 90-5 94-o 97-6 101.3 105 r 24 69.6 73-o 76.4 79-7 83- 86.5 89-9 93-3 06.9 IOJ.O 26 65-6 69.0 72.4 75-8 79- 82.4 85-8 89-2 92.6 96.2 28 61.4 65.0 68.4 71.8 75- 78.4 81.7 85-1 88.4 91.9 30 60.9 64.4 67-8 74-5 77-7 81.0 84-3 87.7 32 52-9 56.7 60.3 63.8 67 '. 70-5 73-8 77.0 80.3 83.6 34 48-4 52-4 56-1 59-7 63- 66.5 69.8 73-0 76-3 79-5 36 43-7 47-9 51-8 55-5 59- 62.5 65-8 69.1 72-3 75-5 38 38.7 43-2 47-4 5i-3 54-9 58.5 61.8 65.1 08.3 71-5 40 33-2 38-3 42.8 46.9 50-7 54-3 57-8 61.1 64-3 67-5 42 27.0 32-9 37-8 42.2 46-3 50-1 53-6 57-o 60.3 63-5 44 26.7 32-4 37-3 41.7 45-7 49-4 53-o 56-3 59-6 46 26.3 32-0 30-9 41.2 48.8 52-2 55-5 48 26. o 31 6 36.3 40 6 44- 5 48.1 51 * 5 50 25-6 31.2 35-8 40.0 43-8 47-3 25. 2 so. 6 35- 2 39- 3 AT.. T 54 o w * ^ 24-8 30.1 34-7 *tO' * 38-7 Latitudes. Alt. 36 38 40 42 44 46 48 50 52 54 8 10 154 -9 145.2 154.6 144.8 154. 4 j-70 H 154- o 16 130.0 124.8 130-3 136^5 144.0 153-7 18 119.2 124.2 129.7 136.0 143.6 153-4 20 114.0 118.6 123.6 129.2 135-5 143-1 153-0 22 109.1 II3-4 n8.o 123.0 128.6 135-0 142.6 152-7 24 104.4 108.4 112. 7 117-3 122.3 127.9 134-4 142.1 152-2 26 99-8 103-7 107.7 112. 116.6 121. 6 127.2 133-7 I4L5 151-8 28 95-4 99-1 102.9 106.9 III. 2 115-8 120.9 126.5 133-0 140.9 30 M.I 94.6 98.3 102. I 106.1 110.4 115-0 I2O.O 125-7 132-3 32 86.9 90-3 93-8 97-4 IOI.2 105.2 109.5 II4.I 119.2 124.9 34 82.8 86.1 89-4 92-9 96.5 100.3 104.3 I08.5 113-1 118.2 36 78-7 81.9 85-2 88.5 92.0 95-5 99-3 103-3 107-5 112. I 38 74-7 77-8 81.0 84-3 87.6 91.0 94-5 98.2 102.2 I06.4 40 70.7 73-8 76.9 80. i 83.3 86.5 89-9 93-4 97.1 IOI.O 42 66.7 69.8 72-9 75-9 79-o 82.2 85-4 88.7 92.2 95-8 62.7 65-8 68.8 71.9 74-9 77-9 81.0 84-3 87-5 90.9 46 58.7 61.8 64-8 67-8 70.8 73-8 76.8 79-8 82.9 86.1 48 54-7 57-8 60.9 63-8 66.8 69-7 72.6 75-5 78-4 81.4 50 50.7 53-8 56-9 59-9 62.8 65.6 68.4 71.2 74.1 76.9 52 46-5 49-8 52-9 55-9 58.8 61.6 64-3 67-1 69.8 72-5 54 42-3 45-7 48.9 51-9 54-8 57-6 60.3 63.0 65-6 68.2 56 37-9 4i-5 44-8 47-9 50.8 53-6 56-3 58-9 6l. S 63-9 58 33-3 37-2 40.6 43-8 46.8 49.6 52-3 54-9 57-4 59-8 60 28.4 32-6 36-3 39-7 42.8 45-6 48-3 50-9 53-3 55-7 NOTE. If the star is east of the meridian subtract this hour angle from 360. TABLES TABLE VIII. COORDINATES OF POLARIS. 49 4 1910 1920 1930 ^ ^ I9IO 1920 1930 < E P P t p P E w P * P P P P E sin t COS / sin t cos t sin t cos t sin t COS/ sin / cost sin / cost 30 32 35-2 37-3 61.0 33-7 35-7 58.3 57-1 32.1 34-1 55-6 54-5 150 148 60 62 61.0 62.2 35-2 33- 1 58-3 59-5 33-7 31-6 55-6 1 56.7 32.1 30.2 120 118 34 39-4 58.4 37-7 55-8 35-9 53-3 146 64 63-3 3-9 60.5 29-5 57-8 28.2 116 36 41.4 57-o 39-6 54-5 37-8 52.0 144 66 64.4 28.6 61-5 27.4 58.7 26.1 114 38 43-4 55-5 53- i 39-6 50.6 142 68 65-3 26.4 62.4 25-2 59-6 24.1 112 40 45-3 54-o 43-3 51-6 41-3 49-2 140 70 66.2 24.1 63-3 23-0 60.4 22.0 no 4 2 47-i 52.4 45-1 50.0 43-o 47-7 138 72 67.0 21.8 64.1 20.8 61.1 19.9 108 44 48.9 50-7 46.8 48.4 44-6 46.2 136 74 67.7 19.4 64.7 18.6 61.8 17.7 106 46 50.7 48.9 48.4 46.8 46.2 44.6 134 76 68.3 17.0 65-3 16.3 62.3 15-5 104 48 52-4 47.1 50.0 45-i 47-7 43-o 132 68.9 14.6 14.0 62.9 13-4 102 50 54-o 45-3 51-6 43-3 49-2 41-3 130 80 69.4 12.2 66.1 11.7 63-3 II. 2 100 52 55-5 43-4 53-i 41-5 50.6 39-6 128 8a 69-8 9 .8 66.7 9-4 63.6 8.9 98 54 57-o 41.4 54-5 39- ^ 52.0 37-8 126 84 70.1 7-4 67-0 7.0 63-9 6. 7 96 56 58-4 39-4 55-8 37-7 53-3 35-9 124 86 70.3 4-9 67.2 4-7 64.1 4-5 94 58 59-7 37-3 57-i 35-7 54-5 34-1 122 88 70.4 2-5 67-3 2.4 64.2 2.2 92 60 61.0 35-2 58.3 33-7 55-6 32- i 120 90 70.4 0.0 67.4 o.o 64-3 O.O 90 TABLE IX. CORRECTION FOR AZIMUTH. p sin t. Proportional parts. Alt. 30' 40' So' 60 70' i' ~ 2' 3' 4' 5' 6' 7' 8' 9' 15 I* . i i'-4 ; '.8 2' . i 2'. 5 / .'i .'j .'2 .'2 .'2 '3 .'3 18 i -S 2.1 2.6 3.1 3-6 .2 . 2 3 3 4 4 5 21 1 .X 2.8 3-6 4-3 5-o .2 3 4 4 5 .6 .6 24 9 .8 3-8 4-7 5-7 6.6 3 4 5 .6 7 .8 9 27 3 7 4-9 6.1 7-3 8.6 4 5 .6 7 -9 X .0 i.i 30 4 .6 6.2 7-7 9-3 10.8 3 5 .6 .8 9 .1 .2 4 31 5 .0 6-7 8-3 10. 11.7 3 5 7 .8 .0 . 2 3 5 32 's 4 7-2 9.0 10.8 12.5 4 -5 7 9 .1 3 4 .6 33 i .8 7-7 9.6 "5 13-5 4 .6 .8 .0 .2 3 5 7 34 6 4 8.2 j 0. 1 12.4 14.4 4 4 ( o . 2 6 35 6.6 8.8 II. 13-2 15-5 4 7 9 .1 3 5 .S .0 36 7 . I 9-4 i 1.8 14.2 16.5 -5 7 9 . 2 4 7 .1 37 7 .6 10. I i 2.6 I5-I 17-6 3 5 .8 .0 3 -5 .8 .0 3 38 8.1 8.6 10.8 TT C 13-5 14- 3 16.1 17. 2 18.8 2O. I 3 7 5 .6 .8 r] .1 3 .6 9 . i 4 .6 39 40 9.2 11 ' 3 12.2 15-3 18.3 21.4 " O 3 .6 V Q .2 5 .8 . i 3 4 7 9.8 III i6. 3 i** ^ 19-5 22.8 3 7 .c 3 .6 .0 3 .6 .8 9 42 43 10. 4 II. 14.7 i / o 8. 4 22.0 25-7 4 -7 . i 5 .8 .2 4 .6 -0 3- J 3-3 44 ii 7 15.6 19-5 23-4 27-3 4 .8 .2 .6 .0 3 7 3 . I 3-5 45 12 4 16.6 20.7 24.9 29.0 4 .8 .2 7 . i 5 2.9 3 3 3-7 46 13 .2 17.6 2 2.0 26. 4 30.8 4 9 3 .8 .2 .6 3- I 3 5 3-9 47 14 .0 18.7 2 3-3 28.0 32.6 5 9 -4 9 3 .8 3-3 3 7 4.2 48 14.8 19.8 24-7 29.7 34-6 5 .0 5 .0 5 3.0 3-5 4 4-4 49 15.7 21. O 26.2 31-5 36-7 5 .0 .6 .6 3- 1 3-7 4 . i 4-7 5 16.7 22.2 27-8 33-3 38-9 .6 .1 7 . 2 .8 3-3 3-9 4 4 5-o T7 .7 23-6 29-5 35-3 41.2 .6 . 2 .8 4 9 3-5 4.1 4 7 5-3 52 18 -7 25.0 ', 1.2 37-5 43-7 .6 .2 Q 5 3-J 3-7 4-4 5 5-6 53 19.8 26.5 33-1 39-7 46-3 7 3 .c .6 3-3 4.0 4.6 5 3 6.0 54 21 .0 28.1 2 5. 1 42.1 49.1 -7 4 . i .8 3-5 4-2 4-9 5 .6 6-3 55 22 3 29.7 37.2 44.6 52.0 7 5 .2 3-0 3-7 4-5 5-2 5 Q 6.7 5 AZIMUTH TABLE X. HOUR ANGLES OF d DRACONIS. Intervals: 1910, 6 h i4 m 2i s ; 1920, 6 h i8 m 58 s ; 1930, 6 h 2 Latitudes. Alt. 16 18 20 22 24 26 28 30 32 34 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 H 4 .i 108.3 102.7 97-3 92.0 86.7 81.5 76.3- 71.0 65-6 60.0 54-2 48.! 4i-5 34-1 25-0 1 19 .8 "3-7 107.9 102.3 96.9 91.6 86.3 81.1 75-8 70-5 65.2 59-6 53-9 47-8 41.2 33-8 24-8 1 26. i II9-5 "3-3 107-5 101.9 96.5 91.1 85-9 80.6 75-4 70.1 64-7 59-2 53-5 47-4 40.9 33-5 24.6 133 -0 125.7 119.1 112.9 107.1 101.5 96.0 90.7 85-4 80.2 74-9 69.6 64-3 58-8 53-1 47.0 40-5 33-2 24.4 140 .6 132.6 125-4 118.7 112.5 106.7 IOI.O 95-6 00.2 84.9 79-7 74-5 69-2 63-8 58-3 52.6 46.6 40.2 32-9 24.1 i5i-o 140.7 132.3 125.0 118.3 112. 1 106.2 100.6 95-i 89.7 84.4 79-2 74.0 68.7 63-4 57-9 52.2 46.2 39-8 32-6 23-9 "iSO.'s" 140.4 132.0 124.6 117.9 in. 6 105.8 100. I 94-6 89.2 83-9 78.7 73-5 68.2 62.9 57-4 51-8 45-8 39-4 32-3 23-6 150.5 140.1 131-6 124.2 "7-5 III. 2 105.3 99-6 94.1 88.7 83-4 78.2 72.9 67.7 62.4 56-9 5i-3 45-4 39-o 3i-9 139.8 I3I-3 123-8 117.1 no. 8 104.8 09.1 93-6 88.2 82.9 77-6 72-4 66.9 61.8 56.4 50.8 44-9 38.6 139-5 130.9 123.4 116.6 110.3 104.3 98.6 93-o 87.6 82.3 77.0 71-8 66.6 61.3 55-9 50-3 44.4 38.1 Latitudes. Alt. 36 38 40 42 44 46 48 50 52 54 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 I 60 139 -2 I30-5 123.0 116.2 109.8 103.8 98.0 92.4 87.0 81.7 76.5 71.2 66.0 60.7 55-3 49-7 43-9 37-7 i38-'8" 130.1 122.5 "5-7 109.3 103.2 97-4 91.8 86.4 81.1 75-8 70.6 65-3 60.0 54-7 49.1 43-3 37-2 138.5 129.7 122. I H5-I 108.7 IO2.6 96.8 91.2 85.7 80.4 75-i 69-9 64.7 59-4 54-o 48-5 42.8 36-6 138.! 129.3 121. 6 114.6 108.1 102.0 96.2 9-5 8 5 -I 79.6 74-4 69-2 63-9 58.7 53-3 47-8 42.1 36.0 137-7 128.8 121. II4.0 107-5 IOI-3 95-5 89.8 84-3 78.9 73-6 68.4 63-2 57-9 52-6 47-1 41.4 35-4 137-3 128.3 120.5 "3-4 106.8 100.6 94-7 89-0 83-5 78.1 72.8 67.6 62.3 57-i 51.8 46.3 40-7 136.8 127.8 119.9 112.7 106.1 99-9 93-9 88.2 82.7 77-2 71.9 66.7 61.4 56.2 50.9 45-5 136.4 127.2 119.2 112. 105.3 99.1 93-1 87-3 81.7 76-3 71.0 65-7 60.5 55-2 49-9 135-8 126.6 118.5 III. 2 104-5 98.2 92.1 86. 3 80.7 75-3 69-9 64.6 59-4 54-i 135-3 125.9 117.8 110.4 103.6 97-2 91.1 85-3 79-6 74-i 68.7 63-4 58-2 NOTE. If the star is east of the meridian subtract this hour angle from 360, TABLES TABLE XI. COORDINATES OF POLARIS. I9io 1920 1930 < 1910 < 1920 1930 ^ ffi p P P P P P K I ij P # P P P P ffi HH sin t cos / sint cos / sin / c ost sin t COS* sin * < sin t cost o o.< > 70.4 0.0 67.4 o.o 54-3 180 5 18.2 68.0 17.4 65.1 16.6 62.1 165 I I. J 70.4 1.2 67.4 i.i 54.2 79 6 19.4 67.7 18.6 64.7 17.7 61.8 164 2 2.. > 70.4 2.4 67.4 2. 2 54.2 [78 7 20.6 67.4 19.7 64.4 18.8 61.5 163 3 3- 4 4-( J 70.3 > 70.2 3-5 4-7 7.3 67.2 3-4 4-5 54.2 54.1 77 76 8 21.8 9 22.9 67.0 66.6 20.8 21.9 64.1 63-7 19.9 20.9 61.1 60.8 162 161 S 6. 70.1 5-9 67.1 5-6 54.0 75 '- o 24.1 66.2 23-0 63-3 22.0 60.4 160 6 ?.< 70.0 7.0 67.0 6.7 53.9 74 : >i 25.2 65.8; 24.1 62.9 23.0 60.0 159 7 8.< ) 69.9 8.2 66.9 7-8 53.8 73 - 2 26.4 65.3 25.2 62.4 24.1 59-6 158 8 9.* 5 69.7 9-4 66.7 8.9 53.6 72 : 3 27.5 64.8 26.3 62.0 25-1 59-1 i57 9 ii. c > 69.5 10.5 66.5 10. I 53.5 [71 '. 4 28.6 64.4 27.4 61.5 26.1 58.7 156 IO 12. i 69-3 11.7 66.3 II. 2 53.3 170 : '5 29.8 63.8 28.5 61.0 27.2 58.2 155 II 13.-: t 69.1 12.8 66.2 12-3 53.1 169 . >6 30.9 63-3 29-5 60.5 28.2 57-8 154 12 I4-C > 68.9 14.0 65-9 13-4 52.9 [68 27 32.0 62.8 30.6 60.0 29.2 57-3 153 13 is-* 5 68.6 65-7 14.4 52.6 [67 28 33-1 62.2 31.6 59-5 30-2 56.7 152 14 17. c > 68.3 i6'. 3 65-4 15-5 62.3 r66 29 34.2 61.6 32-7 58-9 3L2 56.2 151 15 18.2 68.0 17.4 65.1 16.6 62.! 165 ?o 35-2 61.0 33-7 58.3 32.1 55-6 150 TABLE XII. CORRECTIONS FOR AZIMUTH. p sin t. Proportional parts. Alt. 10' 20' 30' 40' i' 2' 3' 4' 5' 6' 7 8' 9' 18 5 I.O 2.1 . i . i .2 2 3 3 4 '3 4 '3 5 21 7 1.4 2.1 2.8 .1 .1 .2 3 4 4 5 . 6 .6 24 9 1.9 2.8 3-8 .1 .2 3 4 5 .6 7 . 8 9 27 1.2 2.4 3-7 4.9 .1 .2 4 5 .6 7 9 I.O i.i 3 .5 3- 1 4.6 6.2 .2 3 5 .6 .8 9 i.i 2 1.4 .7 g 3-3 i 6 5-o 6-7 . 2 3 5 7 IV .8 I.O 1. 1 1.2 I 1 3 f A 1:1 3 2 33 9 3- 3-8 5-8 7-7 .2 4 5 .6 / .8 I.O I. 2 * o 1-3 *+ 5 34 . i 4.1 6^2 8.2 .2 4 .6 .8 I.O 1.2 1.4 .6 1.9 g .2 4-4 6.6 8.8 .2 4 7 9 . 9 . i 1-3 1-5 8 2.0 3 37 5 5-o 7.6 10. 1 3 5 5 !s .O 3 i-5 1.8 .0 2-3 38 7 5-4 8,1 10.8 3 5 .8 .1 3 1.6 1.9 i 2-4 39 9 5-7 8.6 n-5 3 .6 9 .1 4 i-7 2.0 3 2.6 40 3- 1 6.1 9.2 12.2 3 .6 9 . 2 5 1.8 2. I 4 2.7 3-3 6-5 6,, 9-8 13-0 T , g 3 7 .0 3 .6 2.0 2-3 .6 .8 2.9 42 43 3-5 3-7 9 7-3 II. O 13.0 14.7 3 4 7 .1 ! 3 .8 2.2 2.6 9 3' ^ 3-3 44 3-9 7-8 11.7 15-6 4 .8 .2 .6 .0 2-3 2.7 3- 1 3-5 45 4.1 8-3 12.4 16.6 4 .8 .2 .7 . i 2-5 2.9 3- 3 3-7 46 4-4 8.8 13-2 17-6 4 9 3 .8 .2 2.6 3-1 3- 5 3-9 47 4-7 9-3 14.0 18.7 5 9 4 Q .3 2.8 3-3 3- 7 4.2 48 4-9 9-9 14.8 19.8 5 s c .0 5 3-o 3-5 4.0 4-4 49 5-2 10.5 iS-7 21.0 5 .6 . i .6 3- 1 3-7 4- 1 4-7 5 5-6 ii. i 16.7 22.2 .6 . 7 ,2 .8 3-3 3-9 4- 4 5.0 Si 5-9 ii. 8 17.7 23-6 .6 .8 4 5 3-5 4-i 4- 7 5-3 52 6.2 12.5 18.7 25.0 .6 9 . $ 3- I 3-7 4.4 5- o 5-6 53 6.6 13.2 19.8 26.5 7 3 .0 .6 3-3 4.0 4-6 5- 3 6.0 54 7.0 14.0 21.0 28.1 7 4 . i .8 3-5 4.2 4-9 5-6 6-3 55 7.4 14.9 22.3 29-7 .7 5 2 3 o 3-7 4-5 5-2 5- 9 6.7 AZIMUTH TABLE XIII. VALUES OF FOR COMPUTING THE ALTITUDE. J/6 p" 1 sin i' sin 2 t tan h for/ = i 10' A' TABLE XIII (A). Latitudes . Equation of time. Hour angle 30 40 50 April 15, o m May 15, 3111.8 May i , 311.0 June , -2111.5 June 15, o m July , +3 m .5 15 30 45 60 75 90 105 120 .0 .0 .1 .2 3 4 4 4 3 .0 .0 .2 3 5 .6 .6 .6 4 .0 . i .2 4 7 .8 9 .8 .6 July 26 + 611.3 Aug. 15, +4 m .4 Sept. 15, - 4 m.8 Oct. 15, -1411.1 Nov. 15, -15111.3 Dec. 24, c m Jan. 15, + Qm.2 Feb. 12, +1411.4 Mar. 15, +911.1 Aug. , + 6m.i Sept. i,om Oct. , -IQH.2 Nov. 3, -1611.3 Dec. , io m .9 Jan. , +311.2 Feb. , +13:11.6 Mar. , +1211.5 April i, +411.0 135 .2 .3 4 150 . i . i . 2 ' 165 .0 .0 .1 180 .0 .0 .O For an increase of i' in p the term increases about 3 %. TABLE XIII (B), Sines of Azimuth and Hour Angle. 1 2 3 4 5 6 7 8 9 10 .086 80 10 .240 .281 318 352 384 4i3 .440 .466 .490 513 534 70 20 534 554 574 592 .609 .626 .642 657 .672 .686 609 60 30 .609 .712 724 736 .748 759 .769 779 .789 799 .808 50 40 .808. .817 .826 834 .842 .849 857 .864 .871 .878 .884 40 50 .884 .891 .897 .902 .908 913 .919 924 .928 933 .938 30 60 .938 942 .946 950 954 957 .961 .964 .967 .970 973 20 70 973 .976 .978 .981 983 985 .987 .989 990 992 993 10 80 993 995 .996 997 .998 998 999 999 .000 .000 .000 10 9 8 7 6 5 4 3 2 1 Cosines of Declination and Altitude. TABLES 53 TABLE XIV. FOR FINDING THE CORRECTION TO THE SUN'S DECLINATION. (In minutes and tenths.) 8^ . Difference for i hour. Proportional parts. <*-" C IP 10" 3" 4" 5" 6" 7" 8" 9" 2O 30" 40" So" o .0 .1 .1 .2 .2 .0 .0 .0 .0 .0 . o .0 .0 .0 .0 .0 1 .1 3 4 5 .6 .0 .0 .0 .1 .1 . i . I .1 I .2 3 5 7 .8 .0 .0 _, . i . I .2 i .2 3 3 4 5 .6 .6 7 9 .8 .0 .2 I.O .0 .0 .0 .0 .1 . I . i .2 .1 .2 .2 .2 .2 .2 . 2 .2 3 2 3 7 .0 3 i-7 .0 .1 .2 .2 .2 3 3 J 4 .8 . i 5 1.9 .0 .1 . 2 . 2 3 3 3 4 .8 3 7 2. I .0 .1 .2 3 3 3 4 I 5 9 4 .8 2-3 .1 .1 .2 3 3 4 4 3 5 I.O 5 .0 2-5 . I 3 3 4 4 5 5 .1 .6 . 2 2-7 . I 3 3 4 4 5 ^ . 6 . 2 .8 3 2.9 .1 3 4 4 5 . r 2 .6 3 9 5 3- 1 .1 3 3 4 4 5 .6 4 .7 3 .0 7 3-3 . I 3 3 4 5 5 .6 7 4 . i .8 3-5 .1 3 4 4 -5 .6 .6 .8 5 3 3-o 3-8 .2 3 4 5 5 .6 7 i .8 .6 4 3-2 4.0 . 2 3 4 5 .6 .6 7 5" .8 . 7 3- 3 4 2 2 3 3 c .6 .8 9 .8 ^6 3-5 4-4 .2 3 4 4 J 5 .6 7 I 9 .8 .8 3-7 4.6 . 2 3 4 5 .6 .6 7 '.8 ji I O 9 9 3.8 4.8 . 2 .3 4 6 M 8 O 6 I.O .0 3-Q 4.0 5-o .2 3 4 5 .6 7 .8 9 i I.O .1 4-2 5-2 .2 3 -4 5 .6 7 .8 9 j^ 1. 1 . 2 3-3 4-3 5-4 .2 3 4 5 7 .8 9 .0 * I.I 3 3-4 4-5 5-6 .2 3 5 .6 7 .8 9 .0 7 1.2 3 3-5 4-7 5-8 .2 4 5 .6 7 .8 9 i I. 2 4 5 .6 3-6 3-8 3-9 4.8 5-0 5-2 6.0 6-3 6-5 .2 3 3 4 4 4 5 5 5 .6 .6 7 'a 9 9 9 .0 .0 .0 8 1-3 7 4.0 5-3 6.7 3 4 5 7 .8 9 .1 1.4 .8 4.1 5-5 6.9 3 4 .6 7 .8 .0 . i 1.4 .8 4-3 5-7 7- I 3 4 .6 7 9 .0 . i .3 * i-5 9 4-4 5-8 7-3 3 4 .6 7 9 .0 . 2 3 9 i-5 3-o 4-5 6.0 7-5 3 5 .6 .8 9 . i . 2 4 J i. 5 3- z 4.6 6.2 7-7 3 5 .6 .8 9 .1 .2 4 * 1.6 3-2 4.8 6.3 7-9 3 5 .6 .8 .0 . i 3 .4 * 1.6 3-3 4.9 6-5 8.1 3 5 .7 .8 .0 .1 3 5 IO i-7 3-3 5-o 6.7 8-3 3 5 7 .8 .0 .2 3 -5 i. 7 3-4 6.8 8-5 3 5 7 9 .0 .2 4 .5 i 1.8 3-5 5-3 7.0 8.8 4 5 7 9 .1 .2 4 .6 * 1.8 3-6 5-4 7-2 9.0 4 5 7 9 . I 3 4 .6 ii 1.8 3-7 5-5 7-3 9-2 4 .6 7 9 . I 3 .5 7 4 1.9 3-8 5-6 7-5 9-4 4 .6 .8 9 .1 3 5 7 1.9 3-8 5-8 7-7 9-6 4 .6 .8 I.O . 2 .3 .5 .7 * 2.0 3-9 5-9 7-8 9.8 4 .6 .8 i .0 .2 4 .6 .8 12 2.O 4.0 6.0 8.0 IO.O .2 4 .6 .8 i .0 1.2 1-4 1.6 1.8 54 AZIMUTH TABLES OF THE SUN'S DECLINATION. The following tables contain the sun's declination for the instant of Greenwich Mean Noon for each day in the year 1916-9 inclusive. To find the declination for any other instant of time than Greenwich Mean Noon increase or decrease the tabular declination by an amount equal to the "difference for one hour" multiplied by the number of hours elapsed since Greenwich Mean Noon. If the watch used is regulated to standard time, the Greenwich Mean Time is found at once by adding 5 h for Eastern Time, 6 h for Central Time, etc. If the watch keeps local time, the Greenwich Time is obtained by adding the west longi- tude of the meridian for which the watch is regulated, expressed in hours, minutes and seconds. The work of multiplying the " difference for one hour "by the number of hours since Greenwich Noon may be avoided by employing Table XIV on page 53. The hours since Green- wich Noon will be found in the left-hand column, for intervals of a quar- ter of an hour. The " difference for one hour" is given at the top for intervals of ten seconds. The proportional parts for seconds of declina- tion at the right are employed exactly as in the other tables similarly arranged. The correction to the declination is in minutes and tenths of minutes. TABLES OF THE SUN'S DECLINATION 55 The tables for 1916-9 may be used to find the sun's declination for the years 1920-3 by applying the following simple rule : To find the declination for any day in the year 1920 compute the declination for the corresponding day in the year 1916 but for one hour later in the day than the actual Greenwich Mean Time. To find the declination for any day in the year 1921 employ the table for the year 1917; time, one hour later than the given time. To find the declination for any day in the year 1922 employ the table for the year 1918; time, one hour later than the given time. To find the declination for any day in the year 1923 employ the table for the year 1919; time, one hour later than the given time. EXAMPLE : Find the sun's declination at 5 P.M., Eastern Time, May 19, 1920. From the table on PP- 56-7 the declination for May 19, 1916, at Greenwich Mean Noon is N. 19 45' -5 and the " diff. for one hour " is 32". 2. According to the rule we must compute the declination for 6 P.M. Eastern Time. 6 h P.M. Eastern Time + 5 h = " h P-M., G. M. T. May 19, 1916, at G. M. N. decl. = N. 19 45'. 5 32 ".2 X n h (Table XIV) = $. Declination at s h P.M. May 19, 1920 = N. 19 5i'-4 For a method of calculating the declination for any date by employing the ephemeris for the preceding years see p. 69. 22 1 Declination. Ei^for "? .,,0.0.0 ,0,0.0.0.0 ,0.0.0.,., ,.,,. n ,.>, 0.0000 ; + + ; +' 1 | SSSS2 SSSSc? >;?;? c7SS ^ : Dedination. Diftta ^ V****.** sjsi *45M**ilfri j3!j 6S^ S % 2 ! Declination. DH-for Ov |s^| : -M4rid3do6Moo Sl!^5vd^o6dd do odd m : o ^ ^ : 1 Declination. Wta o toioioioio ^Jioioirsio ioioir>ioio 10*^101010 10*0101010 +* * " " + cO.to.O ..-.COO ^tocOO ..toco. Mcog^ ^^ * February. 1 1 Q t^, M rf t^- . .t^.toO>M Ot^MWQ t^ O cs M oO co r^OO t^ rf O co*O O to ?00 t^ O * O OO ooo oo t- : : in C/2 January. Declination. Lte ^n^^SS? 55555^5^^ ^53S5 ^Sd : + + + + + + + OOtOtO. COCOCNMO tOTj-COWM OtO^fcOfN QtOTfN*-! 00 COCO t-t-0000 OOOvONOvO O M H M M o o co co co rf P CO CO O i* * ^ 1 ** p H 1 1 10 10 10 10 10 10 10 10 10 1 1 1 OO OOOO OO 00 10 10 10 10 10 1 1 I .0 OOI>-IOCOO OOOcoOOO io 01 OvO co O t- - ro ^^ ^o t-co co Ov MO^J-WO Tt O M .N M (N CO CO CO 1 COCOOVM CO CO Tf OVO O co CO CO co CO co 1 1 M >- qv qvo & ta _g ^OOiOio^ ^co.c,. co M O 10 TJ- M M 5 't CO l-l i P 5? i IOJ\[ H N CO -00 Ov O H J itBQ 1 s C '5 Q VO CT-OO t-^vo in ->j- rr) pi M o ONOO t>-vo ingrown OOMpim ^J- iTjvO f^OO I i + + + + +1 8^8 otro S M M 8 S 55 t^. rn d> ^ 0> -i-od oi inoo' d oi 4- invd ti tivd in 4 oi oo' in PlPiOiPlP! PI010?0101 01N010IPI P?P?P?0?0? P?0?P? P?C? 55 1 Declination. |or O - O cOvO O 33S Pi H o ON 00 oo tivd m 4 -*- ro PI ON ONOO ti vd m 4 ro oi w ^ N . co W CO M 21s. in oi O H CO ID M -^- IT) m mvO + + + + + + t^. m 6 vd Pi oo' rood ro ti. M in & pi in tC. ON M pi ro w S c3 w w S VO VO f~ t^ t> t^OO OOOOOO O\ONOvO>ON OOOOO 55 1 2 * c .2 I ON vd vo" vd in in in 4 4 4 en ro pi 01 oi H M O O d>d< + + + + + + + o?^-S ro Jn < 5CT ft S'**S*S- '5*?S# ff! : I 55 55 : M 2' 3 M c .2 rt C 1 Q ON ONOO vq in ro M ^M^^RRMM^M ^Eroo* fs txvO VO vo vo m Ovo co OH m ot oo m nr^coovo t^M ^-co M ^OJincoM -^-OJOCOM mo pi 4- ^.oo" 6 M M M M CTOO vd ro M r^ 4 vd moiMOO m * m N M m--roPii-i m-^roi-iO JOXBQ H.cn^m vO^OOONO HWro^-^vO^COONO -wy^^ WM & <25 ^- \o to in -t- r^ N M O oo t^\o ^ m M ooo vO ^- w M M cp iovo oo O w co n vo ^ro OO t^,vO ^ ro WHO O>OO vOiO^fOw wdwwro TJ-VO t^OO ON O | s"S ---. i+ + H 6 c ^ A 4^' " 3, ^- *A o 406 S"3, S !? A S>vd S 2-vS"^ ^ ^ do? S*5 ^ 1 Declinatic ------- ~ I i- o 1 s r __ J?^5 iii** ?mmmm SrsSa . 6 % lination. ssffits asffsj s!sa?ff iaajrs, SJE-K,* are** : as C/3 Cfi ' CO i *2 u-iu-jirjiou-) iou-iminirj iAu">mu"v> tniDinmio mmminio mmiorj-ri- ^- i i i i i i i 2 O :lination * t^ O co t-^ O covo OH -<}-t^oc)- M p S ON O M N ro ** mvo t^.co o O M N ro ^- mvo txt^oooOHts c*m^ mvo ^O i i i i i i i *3 I aa? wajt ? S si S wsa* ft!i ^ aMa* s 1 r""" "'"""' ""'""" """"" """ m uo w M N co * in vo t^oo oo M(Nco- (N HO ONOO tx \O m ^J- CO N M o' M pi ro 4 invo' f^-oo \ , S " 4 I + + + + ^ p M vo t>. * rwo p) ro o ro pj r^oo moo vo M M + 1 1 t>. qv t>. M q vq t-. moo tv ; i * 55 : .2 * p M >> .. rovq O ro vo O rpvo oo M -^-vO oo O ro in*o oo O - in m pi t-* ooo -^-oo O^vOwroNoo OOvo ooo M ro * mvq t>oo o O M H a i . C 3 O oo' vo 4 M oo' in pi oo' 4 o' vd M vo' O "i c> pi inoo in o P) " O M ro in O P -*inMpi-* mopiro-* 00000 OMMMM M 1 55 p K ^ 00 vq <* M q>vo root^ -^ot^roov mi-irxCioo OO t^. ti t^. t^. vo vo' vo' vo' in in in 4 4 ro ro ro pi pi M rooo rooo ro oo pi t^. M in 1 c >fifi***jas'**ftaft as*** su!?a : 1 Z & ' oo . ^Kftss^^^sssissaa ?**-r~-z S -2 c an si ssasjgftffw jissa* 3SV9S, SE-5-JS- S, 3 CfJ * * ^ b " H 3 j4$^5|4^|ldaa^ SSSS5B iraft: : : 1 j '58*fr?aR?S3 * o> rooo r in OinroPtO inrowom ro SOOsOOOOOOOOOOr>- t^ P VI JOXEQ M P4 ro < in vo t^oo OO M N ro * in vo r^oo O - pi ro <* in vo t-oo O O H M JgJ I 3 " 1 1 1 1 11+ + + 6 1 1 wwsj a *?a SB* *& NO NO vd NO 10 rr> M ONVO O1 ON P cn cn 10 ii f. 5 ; 00 00 f^vO NO 10 T|- 01 N Pino OVOO 00 t-^VO 10 * ^ 10 10 -f ON rp 01 M NO ON OO 1OOO ONNO O w ON O1 OJ) "H O ONOO vd looiw'oo'io pioNiOMrV oSod P) r^ -." O1 PJ P) M O ONOO t-xNO 1O 1 1 1 o ^-*q PJ. HNONO oivo 1OOO M -4-vd OO ON O H H I si | ONONOOO OOMMH n cn cn 01 1 n j: T 10 T^ " i " T 10 _^ PJ * rvOO qOOOoO NOPJOOOICX ONHMOt^ 010101NP1 HHOOON ON IOIO.OIO1O lOlOlOlO^J- Tt- II 1 01 rx O M M 00 -j-00 ON ON t>. PJTt-OPlO MO11OHO1 10 1 cn cn h ^ ,0.0 tO^l ^OO ON OOO NO 010O Olrx O>MP)fOM MOOONOO1 1 II ONNO PI P) NO O Tj-00 PJ 10 *P)OOOvO -^-PlONrN.-* H ONNO O1 O 1-. * O t~. * o'r-.-4-ONpJ NOONpivdo D. = <* S I ^cn cn S S " ^ C - ONNO 01O tv.Tt-Mr-.Tj- ONOP10001 O^ONOM ~r~. riod CJ> o" o" "' P) P) 01 ij- ^(- 10 IONO r^. t^-oo' od ON 1 | 1 i NOMNOOlO ONOIt^MlOOO ? & 10 to 10 in 10 10 io lo Io 1 1 1 1 j --"= saswaoass 8^s? J??9S?5 ^8%^1 o 1 55 * ^ 000 ^ in 2 3 ON O w PI 01 -\D . ^ 1 1 1 1 R 1 1 1 Jii Declination. rOCOrOC,N WWNNW WWHMH HHHHO O O O O ON ON ON ONOO OO OO muojv M C* CO ^ 10 -vO t^OO ONQ MNrO^iOvO (^OO O O HP, 0,^,0 NO ^OOONO M JOXEQ CJ <2 ** "H o" ONOO rx vd n * ro d H o" d>oo' t^. vd in -4- m ci H o" H rn -4- invd rxob I a* + + + + +11 N H vo t*s ^~ txvo Pi ro o ^ N r^oo moo VOHH t^O^t^MQ vot^ moo t^ c vo in tr> d t^ -4- d vd pi t>. ci vd O rovd d* H tr> mo vd vd vd vd in ro M o^vd ro . Declinatk K Z '' fr <2 * taJ 5 c o ^ ' mvo O co vo O rovo oo H -^vo oo O en mvo oo o H m <* mvo t^oo ON O H M m'o "8 ? *S SI m*? iT ^"m S "8 5- m o" S 3- O M 1 c? ^^- m t? S cT M ^3- % in in invo vo vo vo t^ tv f> tvoo ooooooo>OvO>O>OOOOO OMMHH H 1 55 X II ^ oo vo -* M q*\q <^Oi>> "l"*?^^*? *9 t ^ ! . T 00 ^T 00 co oo N tx M m - p 1 Mmow-rt- OfOiOM OM^ON mwroiOM roirji-trom wcomMco 1 Q z * : w te'l 3 M c iiiiHmn^Jmii^n^n & CO ^12 X N hi i 1 te'J 3 H - OO O H ON IOOO O t-- r^\o t^vo OJ tx ON O OO m O -^ mvo -^- w tx W * i d coM^-r^-c^ ONco-^-mvo\o t ovo\o lo-^-^-w^ oodvo^-tN oi*^ 1 ^* i i o 1 oc : : 3 H fe t^oo OH N rr> * mvq tx t^oo oo oo ON ooo vo_ * 3 lination. TJ- dv ^-od d inoo" d ci -4- invd vdvdm ^-mHONt*. ^-Hts.rod* ^ON rood w in O in m^- * mWMHO inrj-rooiH Om-^-Nn OmroiNO mrooOm m 1 CO J JO N XBQ M W CO -^- vT) vO txOO O^O MWm^-iOVO txOO (>O MnmTj-irj'O t^OO O^ O w December. c .2 i 1 s " q q <* co ^ >o^ c, H qoo t. IOCOPJ qoo vq ? -o M ? vo oo H 1 1 1 1 1 1 + + + 'M d ooo' \6 4 M oo 10 M voHvoo'4 i^. d pi 4 to vd vd vo* vd 10 COM d>\d m o CA3 OJ November. Q* a _o G 10 1 1 III II H d d>co ^d loroHtooio wd>iot-it^ food N ^ M 1006 M 4^o oo <> 6 w H i C/3 C/3 ' October. 21 Q - o c ro coco"o6odl>. l^r^i-xi^.tC.idvdvdvdio ioio444 rororoNPi MMO'OO> O> 1 1 1 1 1 1 1 p \d ONN 10 o pi toco d rn6 ^""^fj," 1 CD 0! September. 3 M "4 4 4 10 10 xovd vdvdvd vdt^.t^.i~.ti. t^. t^.oo odod oooo'cdcooo odoo'oo'oo'od '. 1010101010 1010101010 ioiotnioio 1010101010 10^0 iomio 1010101010 1 1 1 1 1 1 1 xo Declination XS-ftK SS?2 H94 ***** 0-HOco.n^oo, . ^ ^^ C/3 : 3 M 3 Declination. *for tx. t^.OO CJ>O O^WWCO "ooo H ro in l>-od d N i ro ro ro ro ro g M M g oo' t^. \d oot-^tvt^r^vOvOvOvOiO lotn-^- 1 ^-^- ^rorocop) CJNHHM OOOOOOO *3 1 , Declination. 1 1 1 1 1 1 1 jo XBQ M N CO ^ IO vO t^-OO O>O MNfOTf-iOVO t^OO ONO tHNrnTfiOvO t^OO O> O * 6 4 AZIMUTH PLATES 66 AZIMUTH COKSTELLATIOXS ABOUT THE NOBTH POLE MISCELLANEOUS RULES AND TABLES CORRECTING THE WATCH. To find the approximate time, for correcting the sun's declination: 1. Add the log sin azimuth (Table XIII B) to the log cos altitude and from the sum subtract the log cos declination. The result is the log sin hour angle. Convert this angle into hours, minutes, and seconds. If it is forenoon, subtract this hour angle from i2h to obtain thfe solar time. 2. Convert the time thus found into Mean Time by adding or sub- tracting the Equation of Time (Table XIII A). 3. Convert this Mean Time into Standard Time by taking the differ- ence in longitude between the place and the standard meridian, express- ing it as hours, minutes, and seconds, and adding it to the mean time if the place is west of the Standard Meridian, subtracting if it is east. The difference between this and the average watch reading is the error of the watch.* This problem might be applied as follows. If the surveyor is far from a place where the time can be obtained he may work up his azimuth observation as previously explained and then compute the time by this method. If it is discovered that the watch is largely in error the azimuth should be recomputed, using the corrected declination of the sun. If the time is thus computed with each azimuth observation the error of the watch may be known approximately at all times. The only doubtful case in the above solution is when the sun is about 6 n east or west of the meridian. The sine will be the same for the hour angle or its supplement. To remove this ambiguity compute the altitude of the sun when the hour angle is 6h, which is done by adding the log sin latitude to the log sin declination. The result is the log sin altitude at the instant when the sun is 6h from the meridian. If the observed altitude is less than this computed altitude the hour angle is greater than 90 and vice versa. * If it is desired to compute the sun's hour angle without first computing the azimuth this may be done by means of equation [6]. 68 CORRECTING THE WATCH 69 EXAMPLE. From example 2, p. 12, we have Declination = + 21 40', Altitude = 57 20', Azimuth = S. 59 55' E., Eastern Time p h 49, A.M. The longitude is approxi- mately 71 01' W. Date, July 15. log sin azimuth = .938 Standard Meridian 75 log cos altitude = .732 Local Meridian 71 or' i . 670 Difference = 3 59' log cos declination = .969 = 15 56 s , log sin hour angle = .701 hour angle = 30 . 2 = 2 h OI m Time = 9 h 59'" Equation = +6 Mean Time = io h os m Longitude correction 16 Eastern Time 9 h 49 m showing that the watch was correct. In example 3, p. 12, the declination is + 16 50' .7 and the latitude is 42 29' .2. log sin declination = .462 log sin latitude = .830 log sin altitude = .292 Altitude at 6 h =n.3- Hence the hour angle of the sun at the time of the observation was less than 6 h , since the observed altitude is greater than n.3. To obtain the sun's declination on any day at G. M. N. from an Almanac of the pre- vious year. Take out the declination for G. M. N. of the same date and compute its value for a time 6 h earlier. This may be done conveniently by taking the " diff. for one hour " in seconds, moving the decimal point one place to the left, and calling the result minutes. This correction is to be added if the declination is numerically decreasing; subtracted if increasing. The resulting declination is generally correct within a small fraction of one minute. The declination may be made more accurate by applying a further correction as follows: Take two tenths of the " diff. for one hour," convert it into minutes, and apply this correction the opposite way from the preceding correction. Notice that this correc- tion is one thirtieth of the first correction. On a leap year, after March i, one day must be added to the given date before entering the almanac. For example, March 2, 1912, would correspond to March 3, 1911, when applying this rule. EXAMPLES, i. It is desired to know the declination of the sun at G. M. N. Jan. 3, 1912, only a 1911 almanac being available. For Jan. 3, 1911 the declination is S. 22 54' 34".4 = S. 22 54'. 57. The diff. for one hour = i3".75- The first correction = i'.375. The declination is numerically decreasing, so the correction is added, giving S. 22 55'.94- To make the second correction we subtract 0.2 X 13.75 * 60 =o'.os, giving S. 22 55'. &g. 2. To find the declination for June 10, 1912 from the almanac for 1911. June 10, 1912, corresponds to June n, 1911. The declination for June n, 1911 is N. 23oi'.92| the difference for i h is n".36. The declination for June 10, 1912 is therefore N. 23 oi'.92 i'.i4 = N. 23 oo'-78. Making the second correction reduces it to N. 23 oo'.82. 7 AZIMUTH TABLE XVII. CORRECTIONS FOR REDUCING SLOPE MEASUREMENTS TO HORIZONTAL. 1 ft. ft. ft. F set. 1 100 200 300 IO 2O 30 40 50 60 70 80 90 o 30' .00 .01 .01 40' . OI .01 .02 .01 .01 .01 50 .01 .02 03 .01 .01 .01 .01 .01 ioo' .02 03 05 .01 .01 .01 .01 .01 .01 IO X .02 P4 .06 . OI . OI .01 . OI .01 .02 .02 20' 03 05 .08 .OI .OI .01 .01 .02 .02 .02 .02 30' .03 .07 . IO .OI .OI .01 .02 .02 .02 .03 .03 40' .04 .08 13 v .01 .01 .02 .02 03 03 03 .04 50 05 . IO 15 .01 .01 .02 .02 03 03 .04 .04 05 200' .06 . 12 .18 .01 .01 .02 02 03 .04 .04 05 05 10' .07 .14 .21 .01 .01 .02 03 .04 .04 05 .06 .06 2o' .08 17 25 .01 .02 .02 03 .04 05 .06 07 .07 30' .10 .29 .01 .02 03 .04 05 .06 .07 .08 .09 40' .11 .22 32 .01 .02 03 .04 05 .06 .08 .09 . IO 50 .12 .24 37 .01 .02 .04 05 .06 .07 .09 .10 .11 3oo' .14 27 .41 .01 03 .04 05 .07 .08 .10 .11 .12 10' 15 3 1 .46 .02 03 05 .06 .08 .09 .11 .12 .14 20' 17 34 .02 03 05 .07 .08 .10 .12 14 15 30' .19 37 56 .02 .04 .06 07 .09 . II '3 15 17 40; .20 .41 .61 .02 .04 .06 .08 .10 .12 .14 .16 .18 .22 45 .67 .02 .04 07 .09 .11 13 .16 .18 .20 4oo' .24 49 73 .02 05 07 . IO .12 15 17 19 .22 10' .26 53 79 03 05 .08 .11 13 .16 .19 .21 .24 20' .29 57 .86 03 06 .09 .11 .14 17 .20 23 .26 30' 31 .62 92 03 .06 .09 .12 IS .18 .22 25 .28 40' 33 .66 99 03 .07 .10 13 17 .20 23 27 30 36 7i 1.07 .04 .07 . ii .14 .18 .21 25 .28 32 5 oo' 38 .76 .14 .04 .08 . ii IS .19 23 .27 30 34 10' .41 .81 .22 .04 .08 .12 .16 . 2O 24 .28 33 37 20' 43 .87 30 .04 .09 13 17 .22 .26 30 35 39 3 r .46 .92 .38 05 .09 .14 .18 23 .28 32 37 .41 49 .08 47 05 .10 IS .20 24 .29 34 39 44 50' 52 1.04 55 05 . IO .16 .21 .26 31 36 .41 47 6oo' 55 .10 .64 05 . II .16 .22 .27 33 38 44 49 10' 58 .16 74 05 .12 17 23 .29 35 .41 .46 52 20' .61 .22 83 .06 .12 .18 24 31 37 43 49 55 30' .64 .29 93 .06 13 .19 .26 32 39 45 58 40' .68 35 03 .07 .14 .20 27 34 .41 47 54 .61 50' .42 13 .07 .14 . 21 .28 36 43 50 57 .64 J EXAMPLE. Required the horizontal distance for a slope distance of 272.46 ft., vertical angle of 4 16'. For 4 10' the correction is 0.53 + 0.19 = 072 ft. Inter- polating between 4 10' and 4 20' for distance 300 we find 0.04 as the increase for 6'. Hence correction = 0.76, and horizontal distance = 271.70 ft. MISCELLANEOUS TABLES TABLE XVII (Continued). M I ft. ft. ft. Feet. i 100 200 300 IO 2O 30 40 So; 60 70 80 QO 7oo' 75 1.49 2.24 .07 IS .22 30 37 45 52 .60 .67 05' .76 53 2.29 .08 15 23 31 38 .46 53 .61 .69 10' 78 56 2.34 .08 .16 23 31 39 47 55 .62 70 15' .80 .60 2.40 .08 ,16 .24 32 .40 .48 56 .64 .72 20' .82 .64 2-45 .08 .16 25 33 .41 49 57 65 74 25' .84 .67 2.51 .08 17 25 33 .42 50 59 67 75 30' .86 71 2-57 .09 17 .26 34 43 Si .60 .68 77 35' .88 75 2.63 .09 .18 .26 35 44 53 .61 70 79 4o' .89 79 2.68 .09 .18 27 36 45 54 63 72 .80 45' .91 1.83 2.74 .09 .18 27 37 .46 55 .64 73 .82 So' 93 1.87 2.80 .09 .19 .28 37 47 56 .65 75 .84 55' 95 1.91 2.86 .10 .19 .29 38 .48 57 .67 .76 .86 8oo' 97 i-95 2.92 .10 .19 29 39 49 58 .68 78 .88 05' 99 1.99 2.98 .10 .20 30 .40 50 .60 .70 .80 .89 10' I.OI 2.03 3-04 .10 .20 30 .41 51 .61 .71 .81 .91 15' 1.04 .07 .10 .21 .41 52 .62 72 83 93 20' i. 06 .11 3-17 .11 .21 32 .42 53 63 74 84 95 25' i. 08 15 3-23 .11 .22 32 43 34 65 75 .86 97 3o' .10 .20 3-29 .11 .22 33 44 55 .66 77 .88 99 35' .12 .24 3.36 .11 .22 34 45 56 .67 78 .90 I.OI 40' .14 .28 3-43 .11 23 34 .46 57 .69 .80 .91 1.03 45 / .16 2-33 3-49 .12 23 47 58 .70 .81 93 1.05 .19 2-37 3-56 .12 .24 .36 47 59 83 95 1.07 55' .21 2.42 3-63 .12 .24 36 .48 .60 73 85 97 1.09 9 oo' 23 2.46 3.69 .12 25 37 49 .62 74 .86 .98 i. ii 05' 25 2.51 3-76 13 25 38 50 63 75 .88 1. 00 1.13 10' .28 2-55 3-83 13 .26 38 51 .64 77 .89 1.02 i.iS 15^ 30 2.60 3-90 13 .26 39 52 65 .78 .91 1.04 1.17 32 2-65 3-97 13 .26 .40 53 .66 79 93 1. 06 1.19 25' 35 2.70 4.04 13 .27 .40 54 67 .81 94 1.08 1. 21 30' 37 2.74 4.11 .14 27 .41 55 .69 .82 .96 .10 1.23 35' .40 2.79 4.19 .14 .28 .42 56 .70 .84 .98 .12 1.26 40' .42 2.84 4.26 .14 .28 43 57 7 1 85 99 .14 1.28 45' 44 2.89 4-33 .14 29 43 58 .72 87 I.OI .16 1.30 50' 47 2.94 4.41 15 .29 44 59 73 .88 1.03 .18 1-32 55' 49 2-99 4.48 15 30 45 .60 75 90 1.05 .20 1-34 10 oo' 1-52 3-04 4.56 15 30 .46 .61 .76 .91 1.06 1.22 1-37 To reduce slope measurements to horizontal when the angle is greater than 10, add the log vers. of the angle (Table VI) to the log slope distance (Table V). The sum is the log correction. Subtract the correction from the slope distance . To reduce a slope measurement to horizontal when the difference in elevation of the ends of the tape is given: Divide the square of the height by twice the slope length and subtract the result from the slope length. EXAMPLE. Height, 8.1 feet, slope length, 85 feet. (8.1)2^-170 = 0.38. Horizontal distance = 84.62 feet. If the height is large compared with the distance make a second computation, dividing the square of the height by the sum of the slope length and the approximate horizontal distance. Use this correction instead of the first. EXAMPLE. Height, 20 feet, slope length, loofeet. (2o) 2 -;- 200 = 2.00 feet. Approxi- mate horizontal distance = 98.0 feet. (2o) 2 -H 198.0 = 2.02. Horizontal distance = 97.98 feet. 72 AZIMUTH TABLE XVIII. INCHES IN DECIMALS OF A FOOT. In. X 3 3 4 5 6 7 8 9 10 ii iln. Feet .0833 .I66 7 .2500 3333 .4167 .5000 .5833 .6667 .7500 8333 .9167 A .0026 .0859 1693 .2526 3359 4193 .5026 5859 6693 .7526 8359 9193 A A .0052 .0885 .1719 2552 .3385 .4219 5052 5885 .6719 7552 8385 .9219 A A .0078 .0911 1745 2578 34" 4245 .5078 59" 6745 7578 .8411 9245 & 1 .0104 .0938 .1771 .2604 3438 .4271 .5104 5938 .6771 .7604 8438 9271 * & .0130 .0964 .1797 .2630 3464 .4297 5130 5964 .6797 .7630 .8464 9297 32 A .0156 .0990 .1823 .2656 349 4323 5156 5990 .6823 .7656 .8490 9323 i 3 s 3 ^ .0182 .1016 .1849 .2682 35i6 4349 .5182 .6016 .6849 .7682 -8516 9349 35 i .0208 .1042 1875 .2708 3542 4375 .5208 .6042 .6875 .7708 8542 9375 A .0234 .1068 . 1901 2734 3568 .4401 .5=34 .6068 .6901 7734 8568 .9401 & A .0260 .1094 .1927 .2760 3594 4427 .5260 .6094 .6927 7760 8594 9427 i 5 3 U .0286 .1120 1953 .2786 3620 4453 .5286 .6120 6953 7786 8620 9453 & i 0313 .1146 .1979 .2813 3646 4479 5313 .6146 .6979 7813 8646 9479 1 if 0339 .1172 .2005 .2839 3672 45<>5 5339 .6172 .7005 7839 8672 9505 i3 A 0365 .1198 .2031 .2865 3698 4531 5365 .6198 7031 7865 8698 9531 /B ?,:} .0391 .1224 .2057 .2891 3724 4557 5391 .6224 7Q57 7891 8724 9557 M i .0417 .1250 .2083 .2917 3750 4583 5417 .6250 .7083 7917 8750- 9583 i U 0443 .1276 .2109 2943 3776 4609 5443 .6276 .7109 7943 8776 9609 a T 9 * .0469 .I3O2 2135 .2969 3802 4635 5469 .6302 7135 7969 8802 9635 T s U 495 .1328 .2l6l .2995 3828 4661 5495 .6328 7161 7995 8828 9661 * i .0521 1354 .2188 .3021 3854 4688 5521 6354 .7188 8021 8854 9688 i 3^ .0547 .1380 .2214 347 3880 4714 5547 .6380 .7214 8047 8880 97H & fi .0573 .1406 .2240 .3073 3906 4740 5573 .6406 7240 8073 8906 9740 tt H 0599 .1432 .2266 3099 3932 4766 5599 .6432 .7266 8099 8932 9766 ff i .0625 .1458 .2292 3125 3958 4792 5625 .6458 7292 8125 8958 9792 i H .0651 .1484 .2318 3151 3984 4818 5651 .6484 73i8 8151 8984 9818 5 13 .0677 .1510 2344 .3177 4010 4844 5677 .6510 7344 8i77 9010 9844 ii H .0703 1536 .2370 .3203 4036 4870 5703 .6536 7370 8203 9036 9870 fi i .0729 .1563 .2396 .3229 4063 4896 5729 .6563 739 s 8229 9063 9896 1 5 755 .1589 .2422 3255 .4089 4922 5755 .6589 7422 8255 9089 9922 H .0781 .1615 .2448 .3281 4"5 4948 .578i .6615 7448 8281 95 9948 15 H .0807 . 1641 .2474 3307 4141 4974 5807 .6641 7474 8307 9141 9974 & In. I a 3 4 5 6 7 8 9 10 ii 'In. MISCELLANEOUS TABLES^ : 73 TABLE XIX. CONSTANTS. Number. Logarithm. Ratio of circumference to diameter Base of hyperbolic logarithms 3- HIS9 2 71828 0.49715 o 4.3420 Modulus of common system of logs o. 43420 o. 6377810 Length of seconds pendulum at N. Y. (inches) 30. 1017 I. CTQ22O Acceleration due to gravity at N. Y Cubic inches in i U. S. gallon 3 2 -i5949 231. 1-50731 2. 36361 Cubic feet in i U. S. gallon o. 1337 9- 1261310 U. S. gallons in i cubic foot 7 480? o 87303 Pounds of water in i cubic foot 62 t; i. 7QC88 Pounds of water in i U. S. gallon . ... 8. 3SS o. 02102; Pounds per square inch due to i atmos.. . . Pounds per square inch due to i foot head of water Feet of head for pressure of i pound per square inch 14.7 0-434 2 3O4 1.16732 9- 63749-io o 36248 Inches in i centimeter O 3Q37 9CQCI 7IO Centimeters in i inch 2 ^4OO o 40483 Feet in i meter 3 2808 o sisoS Meters in i foot i o 304.8 9 4840210 Miles in r kilometer : . . . o 62137 o 7033^10 Kilometers in i mile Square inches in i square centimeter.. . . Square centimeters in i square inch. . . . Square feet in i square meter Square meters in i square foot Cubic feet in i cubic meter 1.60935 0.155 6.4520 10. 764 0.09290 3C. 31^6 o. 20665 9. 19033-10 o. 80969 1.03197 8.96802-10 I. ^4797 Pounds (av.) in i kilogram 2. 2046 o. 34333 Kilograms in i pound (av.) Ft.-lbs. in i kilogram-meter 0-453 6 7 2 3 308 9. 65667-10 o. 8^032 Natural sine of o oo' 01" = .00000485; log Natural sine of o 01' = .00029089; log Natural sine of o i' Natural sine of o oo' 01" 4-68557 6.46373 o. 03 ft. 100 ft. 0.3 inch i mile /rf) UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW Books not returned on time are subject to a fine of 50c per volume after the third day overdue, increasing to $1.00 per volume after the sixth day. Books not in demand may be renewed if application is made before expiration of loan period. RPR 18 1918 5 1321 .'928 5 Mil i^j APR 2 1931 50m-7.'lG YB 27106 UNIVERSITY OF CALIFORNIA LIBRARY