mim vvyvvyw; > ' > a^ > > " LIBRARY UNIVERSITY OF CALIFORNIA < "- 1 KT . O e*. Received A, ce$$W*i .\o. Shelf No. > > . ^ * >' > - - t : > ^ - ^ > v ^ >v > , > > > > V^ >' '* ^ Compliments of the U. S, HYDROGRAPHIC OFFICE, COMMANDER J. R. BARTLETT, Q. S. N., Hydrograpfier. Washington, D. C. FINDING THE ERROR OP THE MARINE COMPASS OIST BOA.RD SHIP. BY B. F. GREENE, PROFESSOR OF MATHEMATICS, UNITED STATES NAVY, SUPERINTENDENT OF COMPASSES. OF NAVY DEPARTMENT. WASHINGTON: GOVERNMENT FEINTING OFFICE. 1875. PREFATORY NOTE. So much of tiie Text, comprising nearly all of one Chapter, together with the entire set of Tables, as here given, from the Manual of the Marine Compass, in course of preparation by the undersigned, it has been deemed expedient to print, in advance of its general publication, for the immediate use of the United States Navy. Leaving any statement of particulars, as to the author's intentions, peculiarities of treatment, and acknowledgments, to a more appropriate time, he would merely remark, at present, that his aim has been, so far as "Finding the Error of the Compass" is concerned, to aid the Navi- gator with some additions to his appliances for an easy, expeditious, and reliable determination of this important element. And this is done, from the point of view occupied by him, That, because there is so much unavoidable liability to error and consequent uncertainty in the use of the Compass at sea, there is the greater reason why ice should neglect no available means for reducing these errors on the one hand and insuring greater certainty on the other. If he shall succeed in commending this idea to the approval of Navigators, arid in making their arduous duties enough easier in this respect to induce a more frequent determination of their Compass-Errors, he will feel somewhat compensated for the labor bestowed even upon this one Chapter of his work, not to mention that involved in the preparation of the appended Tables. B. P. GKEENE. BUREAU OF NAVIGATION, NAVY DEPARTMENT, Washington, March, 1875. CONTENTS. 1. NATURE OF THE COMPASS-ERROR. Art. Page. 1. Different Errors of the Compass 1 2. Pointing-Errors of the Compass 1 a) Adjustment- Errors of the Compass. b) Adjustment- Err or 8 of the Azimuth-Circle. c) Error from Defective Sensibility. d) Observation- Errors of the Compass. 3. Reduction-Errors of the Compass 3 a) The Compass-Deviation. b) The Magnetic Variation. 4. Total Error : called simply the Compass-Error 3 Two Different Definitions of the Compass-Error. 2. FINDING THE COMPASS-ERROR. 5. The Basis of Procedure in all' Cases 4 6. Different Methods of Finding the True Azimuth 4 I. BY OBSERVATIONS OF CELESTIAL OBJECTS. A. METHOD OF HORIZON-AZIMUTHS. 7. Fundamental Principles of the Horizon- Azimuth 6 8. Remark : The Amplitude 6 9. Rule : To make the Observation 7 a) Observation of the Sun. &) Observation of the Moon. c) Observation of a Planet or Fixed Star. 10. Remark: Auxiliary Observations 7 11. Rule : To correct the Observed Compass- Azimuth 8 ) For the Sun, a Planet, or Fixed Star. b) For the Moon. 12. Examples of Correcting Observed Azimuths 8- 13. Rule : To find the True Azimuth 8 a) Preparation of the Data. b) Solution by Computation. c) Examples of True Azimuths by Computation. d) Solution by Inspection : Use of Tab. XXIV. e) Remark : On taking Means of Tabular Quantities. f) Examples of True Azimuths by Inspection. 14. Examples of Finding the^Cornpass-Error by Horizon-Azimuths 11 B. METHOD OF TIME-AZIMUTHS. 15. Fundamental Principles of the Time- Azimuth 12 16. The Ship-Time ; its Importance, etc 12 VI CONTENTS. Art. I'ajie. 17. Finding the Compass-Error for a Single Heading of the Ship 13 18. To make a Single Time-Azimuth Observation 13 19. Finding the True Azimuth 14 20. Rule : Preparation of the Data 14 a) The Object being the Sun. &) The Object being the Moon, a Planet, or Fixed Star. Remark on taking out the R. A. 21. Examples in Finding the Hour- Angle 16 22. Rule : Solution by Logarithmic Computation 17 23. Examples of True Azimuths by Computation 17 24. The Tables of Time-Azimuths 18 a) Tab. XXX, or First Part. 6) Tab. XXX. A, or Second Part. c) Tab. XXX. B, or Third Part. d) Intention, etc., u'ith respect to these Tables. 25. Direct and Limiting Values of Time- Azimuths 20 26. Rule : Solution by the Azimuth-Tables 20 27. Examples of True Azimuths by the Azimuth-Tables '21 28. Remark I : On the Two Cases of the Time-Azimuth 22 29. Remark II : On the Conditions introducing the Use of the Symbols oo and -{- oc , 23 a) The Dec. equal to the Lat. 6) The Dec. nearly equal to the Lat. 30. Examples of Finding the Compass-Error by Time-Azimuths 24 31. Finding the Compass-Error f o a Series of Different Headings of the Ship 25 32. To make the Observations of Serial Time-Azimuths 25 Remark I : Position of the Object. Remark IT: Steadying the Ship. Remark III: Heading- Intervals. Remark IF: Care in Sighting and Reading off. 33. Rule : To get the Serial True Azimuths by Computation 26 34. Rule : To get the Serial True Azimuths by the Azimuth-Tables 27 35. Examples of Finding Serial Compass-Errors by Time- Azimuths 28 C. METHOD OF CIRCUMPOLAR AZIMUTHS. 36. The Circumpolar Azimuth a Modified Form of the Time- Azimuth 32 37. Table of Circumpolar Azimuths for Polaris : Description of Tab. XL 32 38. Two Distinct Problems in the Use of Tab. XL 33 39. Rule : To find the True Azimuth of Polaris 33 a) To take out the True Azimuth for any Ship-Time. 6) To find the Ship-Time of the greatest W. or E. E. of Polaris. 40. Examples of Finding the Compass Error by Circumpolar Azimuths 34 D. METHOD OF ALTITUDE-AZIMUTHS. 41. Fundamental principles of the Altitude-Azimuth 35 42. Rule : To make the Observation 35 43. Rule : To find the True Azimuth by Computation 35 44. Rule : To find the True Azimuth by the Azimuth-Tables 36 Remark I : Arguments to Odd Numbers. Remark II: Direct and Limiting Values of Altitude- Azimuths. 45. Examples of Finding the Compass-Error by Altitude-Azimuths 37 CON-TENTS. VII E. METHOD OF TIME- ALTITUDE-AZIMUTHS. Art. Page. 46. Fundamental Characteristics ; Advantages and Defects 38 47. Rale : To find the True Azimuth by Computation 38 48. Examples of Finding the True Azimuth 39 49. Table of Time-Alt. Azimuths : Use of Tab. XLVIII 39 50. Examples of the Use of Tab. XLVIII 40 F. METHOD OF TRANSITION- AZIMUTHS. 51. Fundamental Principles of the Transition-Azimuth 40 52. Remark on the Preparation of the Data 41 53. Rule: To find the True Azimuth 41 ) By Logarithmic Computation. &) By Tabular Inspection. 54. Examples of Finding the Compass-Error by Transition- Azimuths 42 G. DEPENDENCE TO BE PLACED ON THESE METHODS. 55. The Data of a True Azimuth always liable to be in Error 42 56. Estimating Errors of the Azimuth-Data 42 a) The Latitude-Error. 6) The Declination-Error. c) The Hour-Angle Error. d) The Altitude-Error. e) Recapitulation of the Limits of the Data-Errors. 57. Tables of Azimuth-Errors : Auxiliary Tables 44 a) For Errors of Horizon- Azimuths. &) For Errors of Time- Azimuths. c) For Errors of Altitude- Azimuths. d) For Errors of Time-Alt. Azimuths. 58. Finding Partial Azimuth-Errors 45 a) Partial Errors of Horizon- Azimuths. I) Partial Errors of Time- Azimuths. c) Partial Errors of Altitude- Azimuths. d) Partial Errors of Time-Alt. Azimuths. 59. Probable Total Azimuth-Error 47 60. Favorable and Unfavorable Conditions 48 a) Conditions of Horizon-Azimuths. b) Conditions of Time- Azimuths. c) Conditions of Altitude- Azimuths. d) Conditions of Time-Alt. Azimuths. 61. Limits of allowable Azimuth Error 49 62. Dependence on Horizon-Azimuths 51 63. Dependence on Time-Azimuths 52 a) Single Time- Azimuths. b) Serial Time- Azimuths. 64. Dependence on Circumpolar Azimuths 54 65. Dependence on Altitude- Azimuths 54 a) Single Altitude- Azimuths. b) Serial Altitude- Azimuths. 66. Relative Advantages of these Methods 56 a) Horizon-Azimuths. b) Time-Azimuths. c) Circumpolar Azimuths. d) Altitude- Azimuths. VIII CONTENTS. II. BY OBSERVATIONS OF TERRESTRIAL OBJECTS. A. METHOD BY DIRECT BEARINGS. Art. Page- 07. Fundamental Conditions 58 o) First Condition : A Fixed Station. &) Second Condition:' Means of Swinging the Ship. c) Third Condition : A Limited Parallax of Swing. 68. Process of Finding Serial Compass-Errors 60 69. Examples of the Method by Direct Bearings 61 70. Dependence to be placed on Results <>:> 71. Tables of Parallactic Errors : Examples ^3 72. To correct the Observations for Parallactic Errors 6:5 a) Case of Using the True Bearing of Object. &) Case of Using the Magnetic Bearing. c) Remark I: Observing two or more Objects. d) Remark II: Swinging with a Taut Cable. e) Remark III : Using the nearest whole Degree. 73. Examples of Correcting for Parallactic Errors 65 74. Limiting Distance-Ratios 67 75. Different Methods of Finding the True Bearing of Object 67 76. Finding True Bearing by the Geographical Position 67 77. Examples of Finding T. B. .by the Geographical Position 68 78. Finding True Bearing by an Astronomical Bearing : firt a) Rule : To make the Observation. 6) Rule : To make the Computation. 79. Examples of Finding T. B. by an Astronomical Bearing 69 80. Finding True Bearing by an Azimuth of the Vertical Circle 70 a) Rule: To take the Observation. 6) Rule: To find the True Bearing of the Object. 81. Examples of Fuding T. B. by an Azimuth of the Vertical Circle 71 82. Remarks relative to Finding the True Bearing of the Object 71 B. METHOD BY ALLIGNMENTS. 83. Fundamental Idea 72 84. Process of Finding Serial Compass-Errors 72 85. Examples of the Method by Alignments 73 86. Dependence to be placed on Results 74 C. METHOD BY RECIPROCAL BEARINGS. 87. Preliminary Explanat ions 74 88. Two Distinct Cases of Reciprocal Bearings 75 89. Essential Kequisites of the Method under the First Case 75 90. Essential Requisites of the Method under the Second Case 76 a) The Angles measured from the True Meridian. b) The Angles measured from an Arbitrary Line. 91. Preparations for Observations by this Method % 77 92. Process of Conducting Reciprocal Observations 77 93. Reduction of Reciprocal Observations 78 a) Magnetic Observations on Shore. &) Angles Measured from the True Meridian on Shore. c) Angles Measured from any Arbitrary Line. CONTENTS. IX Art. Page. 94. Examples of the Method hy Eeciprocal Bearings , 79 95. Dependence to he placed on Results 82 96. To find the True Meridian 82 97. Remark : To set the Instrument in a True Meridian 83 98. Example of Finding the True Meridian 83 D. RELATIVE ADVANTAGES OF THE SEVERAL METHODS OF FINDING SERIAL COMPASS-ERRORS. 99. Methods hy Celestial Azimuths 84 100. Method hy Direct Bearings 85 101. Method hy Reciprocal Bearings 85 APPENDIX. COMPASS-COMPARISONS. 102. Definition 87 A. COMPASS-COMPARISONS ON BOARD SHIP. 103. Characteristics and Uses of Comparisons 87 104. To make a Compass-Corn parison 87 105. Examples of Compass-Comparisons 88 106. To convert a Given Course hy Standard Compass into an Equivalent Course hy'a Compared Compass and Reciprocally 88 107. Examples of Direct and Reciprocal Conversions - >. 88 108. Rule : To find the Error or Deviation of a Compared Compass 88 109. Examples of Fiodjng the Errors of Compared Compasses . 89 B. COMPASS-COMPARISONS ON SHORE. 110. Comparison on Shore: Detection of Local Magnetism 89 IT INDEX TO THE TABLES. Table. Page. I. Compass-Points and their Equivalents in Degrees 1 II. Conversion of Arc into Time and Time into Arc 2 III. Conversion of Mean Solar into Sidereal Time IV. Conversion of Sidereal into Mean Solar Time 3 V. Length of a Degree in Latitude or Longitude 4 VI. Logarithms of Numbers and Small Arcs 5 VII; Logarithms of Numbers 1 to 1009 24 VIII. Logarithmic Sines, Tangents, and Secants to every Eighth of a Com- pass-Point 26 IX. Logarithmic Sines, Tangents, and Secants to every Minute of Arc and Fourth Minute of Time 27 X. Logarithmic Sines to every Tenth of a Degree 72 XI. Logarithmic Secants to every Tenth of a Degree 74 XII. Logarithmic Tangents to every Tenth of a Degree , 76 XIII. Logarithmic Tangents to every Minute of Time 78 XIV. Lengths of Circular Arcs to every Tenth of a Degree 79 XV. Natural Sines and Cosines to every Tenth of a Degree HO XVI. Natural Tangents to every Tenth of a Degree 82 XVII. Natural Versed Sines to every Tenth of a Degree 84 XVIII. Decimal Parts and their Multiples of a Day 85 XIX. Decimal Equivalents of Common Fractions 86 XX. Proportional Parts 89 XXI. Squares of Numbers from 0.0 to 100.9 90 XXII. Square Roots of Numbers from 0.0 to 100.9 - 92 XXIII. True Rising and Setting 94 XXIV. Horizon- Azimuths 99 XXV. Position-Angles for Horizon-Azimuths 104 XXVI. Limiting-Errors of Horizon-Azimuths 104 XXVII. Correction of the Compass- Azimuth on the Apparent Horizon 105 XXVIII. Error of the Horizon-Azimuth for an Error of 0.2 in the Latitude. .. 105 XXIX. Error of the Horizon-Azimuth for an Error of 0.l in the Declina- tion 105 XXX. Time- Azimuths : Logs A and B 106 XXX. A. Time- Azimuths : Log C 124 XXX. B. Time- Azimuths : Log Tangents X and Y 130 XXXI. Time-Azimuths : Direct and Limiting Values 132 XXXII. Position- Angles of Direct and Limiting Time- Azimuths 134 XXXIII. Altitudes of Direct and Limiting Time- Azimuths ' 136 XXXIV. Error of the Time-Azimuth for an Error of l m in the Hour-Angle 138 XXXV. Error of the Time-Azimuth for an Error of 0.2 in the Latitude 141 XXXVI. Error of the Time-Azimuth for an Error of 0.l in the Declination. . - 141 XXXVII. Limiting-Errors of Time-Azimuths 142 XXXVIII. Limiting-Errors of Time-Azimuths in High Latitudes 142 XXXIX. Limiting-Errors of Serial Time-Azimuths 142 XL. Circumpolar Azimuths : Polaris or the Pole-Star , 143 XLI. Altitude-Azimuths: Part I . 144 XLI. Altitude-Azimuths: Part II . 146 XLI. Altitude-Azimuths: Part III.. 146 XII INDEX TO THE TABLES. Table. Page. XLII. Altitude Azimuths : Direct and Limiting Values 148 XLIII. Position-Angles of Direct and Limiting Altitude-Azimuths , 150 XLIV. Error of the Altitude- Azimuth for an Error of 0.l in the Altitude. .. 150 XLV. Error of the Altitude-Azimuth for an Error of 0.2 in the Latitude.. 151 XLVI. Error of the Altitude- Azimuth for an Error of 0.l in the Declina- tion 152 XL VII. Limiting-Errors of Altitude-Azimuths 152 XLVIII. Time-Alt. Azimuths : Log A 153 XLVIII. Time-Alt. Azimuths : Log B 154 XLIX. Error of the Time-Alt. Azimuth for an Error of 0'.2 in the Hour- Angle ^ 158 L. Error of the Time-Alt. Azimuth for an Error of 3 in the Altitude or Declination , 158 LI. Transition- Azimuths 159 LII. Direct Bearings of a Fixed Object: Limiting Distance 160 LIII. Direct Bearings of a Fixed Object : Parallactic Errors 160 LI V. Products of Arcs Multiplied by the Sines of the Rhumbs 161 LV. Magnetic Elements of the Earth: The Magnetic Variation in Arctic Latitudes 169 LVI. Magnetic Elements of the Earth : The Magnetic Variation in Lati- tudes 70 N to 60 S no LVII. Magnetic Elements of the Earth : The Magnetic Dip 174 LVIII. Magnetic Elements of the Earth : The Horizontal Force 175 LIX. Kight Ascension of the True Sun and Equation of Time 176 LX. Declination of the Sun 180 LXI. Mean Places of T wenty-fi ve Fixed Stars 182 LXII. Meridian-Passages of Twenty-five Fixed Stars 182 LXIII. Reductions of Meridian- Passages of the Fixed Stars 183 LXIV. Reduction of Daily or Hourly Changes in Right Ascension 184 LX V. Reduction of Hourly Changes in the Moon's Right Ascension 186 LXVI. Reduction of the Mean Sun's Right Ascension 188 THE COMPASS-ERROR, AND METHODS OF FINDING BOARD SHIP. 1. NATURE OF THE COMPASS-ERKOR. 1. Different Errors of the Compass. The Errors of the Marine Compass may be considered in two different relations: First, as to the pointing of the compass, which, but for the influence of certain errors, would always be in precise accordance with the Directive Force, whether on board ship or on the land ; and, secondly, as to the reduction to the True Meridian of the corrected compass-pointing, which, although in no proper sense an error of the compass, is treated like one from considerations of practical convenience. All the errors of the compass explained or referred to in the preceding Chapters 1 may therefore be regarded as forming parts of either the Pointing-Errors or Reduction- Errors of this instrument, and they will now be recapitulated under these two heads. . Pointing-Errors or the Compass. The Pointing-Errors include all instrumental and observational errors which affect the accu- racy of the compass-pointing. Under this head there are the following errors : a] Adjustment-Errors of the Compass. These comprise (Chap. I) 1. The divergence of the zero-line from the magnetic axis of the com- pass-card ; 2. The eccentricity of the cap with respect to the centre of the card- circle ; and 3. The eccentricity of the pivot with respect to the centre of the bowl-circle. b) Adjustment-Errors of the Azimuth- Circle*. These are 1. The Direction-Ernpr of the sight-vanes, or divergence of the verti- cal plane of sight from the diameter of the circle (Chap. I) ; and 2. Several possibilities of error from defective construction, which need not be mentioned in this place. (See Chap. I.) c] Error from Defective Sensibility (Chap. T). d) Observation- Errors of the Compass. These comprise i. Equilibrium- Errors of the compass-card and bowl-circle (Chap. V) ; and 1 The references here and elsewhere made, are to different parts of the Manual of the Marine Compass, not yet published, of which this is one Chapter (the sixth, with cer- tain modifications), as explained in the Prefatory Note. FINDING THE COMPASS-ERROR. 2. Personal Errors iu sighting and reading off (Chap. V). The Adjustment-Errors of both the compass and azimnth-circle should be practically unimportant, when these instruments are delivered by the makers; and the. adjustments should be sufficiently stable to re- main unimpaired for a long period, with careful and considerate usage. Unfortunately, even the best intentions of the most reliable maker are liable to be occasionally frustrated ; and hence the only certain way to arrive at satisfactory conclusions concerning these errors is to subject the instruments to careful tests of their actual condition (Chap. I). The error from Defective Sensibility of a well-made liquid compass, having a buoyant card, adjusted to a minimum pressure at the pivot, is commonly inappreciable ; and, under circumstances ordinarily favora- ble, it may in general be relied on to continue in that condition for a long period. But other compasses, whether air or liquid, with heavy cards, and having a pivot-pressure ranging from 1,500 to 5,000 grains, are liable to an appreciable error from this source, proportional, at the least, to the greater pressure at the pivot ; and not only that, but such compasses are even more than proportionally liable to the development of a very serious error as the cap and pivot become sensibly worn or otherwise altered in form (Chap. I). The compasses of the United States Navy, under the inspection-tests to which they are subjected, are required to satisfy the following conditions: The greatest error to which they may be liable must not exceed For Adjustment-Errors of the compass (a) o.i For Adjustment-Errors of the az. circle (I) rto .1 For 'Error from Defective Sensibility (c) d= o .01 or, in the last case, it must be inappreciable. Such a limitation of these errors is equivalent to a probable limit, in the aggregate, 1 of o.i4. These conditions, or something equivalent to them, by which th<* Navy compasses are required to be " practically perfect " when put on board ship, should be required by all Navigators, or at the least for one (standard) compass of their ships. The Observation-Errors, whether depending on the circumstances of the observation or on the personal peculiarities of the observer, are alike impossible of previous estimate. Nor, indeed, can they be esti- mated with much pretension to accuracy even as to their limits after the observation has been made, unless conducted'with a view to such an estimate; and this is rarely practicable or expedient with the Compass- Observations generally required for the purpose of shaping or correct- ing a course. Nothing can therefore be relied on but intelligence and care in making these observations as a remedy for this source of Com- pass-Error. It is otherwise with observations of a serial kind, which are made for the purpose of obtaining a set of Compass-Deviations ; for it is then quite practicable to get a definite estimate of the total Pointing-Error, 'The probable sum of these errors is l/(o.i) a -f.(o .i)* + (o.oi) 8 = o .14. NATURE OF THE COMPASS-ERROR. including the errors of observation, at least as to its limits, both in sign and amount (Chap. IV). 3. Reduction-Errors of the Compass. The Eeductiou- Errors include such as are not, strictly speaking, errors of the compass, but those which are treated as if they were, in the reduction of the cor- rected compass-pointing to the Geographical or True Meridian. There are two of these, as follows : x a] The Compass-Deviation. Ry this is meant the reduction of the cor- rected pointing to the Magnetic Meridian, whatever the place or head- ing of the ship, and whether the ship be upright or in a state of heel (Chap. IV). b) The Magnetic Variation. This is to be understood as the reduc- tion of the Magnetic Meridian at the time and place to the True Meridian (Chap. II). These are sometimes of the same name and sometimes of different names, and it is their resultant or combined effect which constitutes the Keduction-Error in a given case. 4. The Total Error ; called simply the Compass- Error. Each of the errors heretofore mentioned, whether belonging to the first or second class, admits of being measured on the compass- card; and each, according as it has the effect to carry the zero or N point of the card to the eastward or westward, is easterly and marked E, or westerly and marked W. Hence, as in other analogous cases, if all the individual or component errors have the same name, their sum is the total error, and takes the common name of its several compo- nents ; but if a part of the components have one name, and a part the other name, then the difference of the two partial sums is the total error, and takes the name of the greater partial sum. The total or combined effect of the several component errors of a compass will be called the Compass- Error, and will always be under- stood to represent the angular distance east or west from the True Me- ridian to the observed compass-pointing as read off from the compass. It may therefore be defined in two different ways, as follows : a) The Compass-Error is the angle formed at the centre of the com- pass, between the N and S or zero line of the card and the True Merid- ian ; and it is marked J<7 or W, according as the N or zero point of the card falls to the eastward or westward of the true north ; otherwise, b) The Compass-Error is the difference between the compass-bearing and the true bearing of the same object ; and it is marked E or W, according as the compass-bearing falls to the left or right of the true bearing, the observer's eye being supposed at the centre of the compass. FINDING THE COMPASS-ERROR. 2. FINDING THE COMPASS ERROR. There are several different methods of Finding' I lie C'oiiipass- JError, by which will always be meant the total or resultant error, as defined in the preceding articles. It is one of the most important prob 16ms with which the Navigator has to do ; since, how important soever the problems of the Position, no accurate steering of .a given course, from one position to another, can be had without a reliable determina- tion of the Compass-Error on that course. 5. The Basis of Procedure in all Cases. To find the Com- pass-Error in a given case depends on the possibility of knowing the True Azimuth of some celestial or terrestrial object, whenever the Com- pass-Azimuth of the same object can be had by observation. Then The Compass-Error is found by taking the difference between the Compass- Azimuth and the True Azimuth, and is marked E or JF, accord- ing as the Com pass- Azimuth falls to the left or right of the True A/.i- muth (4). The problem will therefore consist of two parts : First, of finding by actual observation the Compass- Azimuth of some object; and, secondly, of finding, by computation or otherwise, the True Azimuth of the same object, if not already known by a previous determination. If the object be a celestial body, in motion, the True Azimuth must be found for the instant of time at which the Compass-Observation is made; but if the object be a terrestrial one, at rest, it is obviously not important, so far as the mere result is concerned, whether the True Azimuth be found at the time it is required, or happens to be already known, provided the observer's position remains unchanged. 6. Different methods of* Finding the True Azi- muth. There are several different methods of finding the True Azi- muth of both celestial and terrestrial objects; and, accordingly, there may be at least as many different methods of finding the Compass- Error. The following methods will be considered : I. By Observations of* Celestial Objects. Method of Horizon-Azimuths. Method of Time- Azimuths. Method of Circumpolar Azimuths. Method of Altitude- Azimuths. Method of Time-Altitude Azimuths. Method of Transition- Azimuths. II. By Observations of Terrestrial Objects. Method of Direct Bearings. Method of Allignments. Method of Reciprocal Bearings. DIFFERENT METHODS OF FINDING IT. Other processes of getting the True Azimuth, incidental for the most part to determinations of the Time and Position, are known to Nautical Astronomers. These, if made available by corresponding Compass- Observations, would supply so many additional means of finding the Compass-Error; but the methods named above include all, probably, that can be usefully employed in a direct manner for this purpose. 1 In practice, the Navigator may be required First, to find the Compass-Error on a particular heading of the ship, for immediate use, upon every change of course; or, Secondly, to find the corresponding Compass-Errors on a series of dif- ferent headings, for subsequent use, with the opportunity chosen at pleasure, either at sea or in port. In the first case, the Compass- Err or 2 is what is required, whether in shaping a course or in working up, in which the direct relation of the compass-course to the true course is always sought ; but, in the second case, the Com pass- Errors are generally required to be reduced to Compass- Deviations? by separating the magnetic variation, and also the Pointing- Error if required (Chap. IV). 'The method of Time- Altitude-Azimuths, included in the foregoing list, requiring both the Altitude and Hour-Angle as data, belongs rather to the class of incidental methods, as it could hardly ever seem to be expedient to use this as a direct method in preference to the Time-Azimuth, although it is sometimes so employed. 2 Still called the " Variation" in most of the books on navigation. 3 The reduction of Compass-Errors to Compass-Deviations by separating the Mag- netic Variation has been fully explained in a previous Chapter. It may, however, be best to repeat so much of it as may suffice to insure that this important operation shall always be performed with certainty. 1. Recall an application of the rule of Art. 4: That, when the Var. and Dev. have the same name their Sum is the Error, which takes the name common to both ; but that, if they have unlike names, their Difference is the Error, which then takes the name of the greater quantity. 2. Conceive the Error and Var. to be laid off on the compass-card, beginning at the N or zero point, and extending East or West, according as they are marked E or W, and measured on the arc of reading. Then the Dev. is always equal to the Difference between the points of reading of the Error and Var. (i. e., equal to their Sum or Difference, according as they have unlike or the same names), and is marked E or W, according as the Var. falls to the left or right of the Error. Always mentally test the result by the first rule. FINDING THE COMPASS-ERROR. I. BY OBSERVATIONS OF CELESTIAL OBJECTS. A. METHOD OF HoRIZON-AfclMUTHs, 7. Fundamental Principles of" the Horizon -Azi- muth. The Metbod of Horizon-Aziinutbs consists in observing tbe compass-bearing of tbe Sun or otber celestial body, wbile its centre is in tbe True Horizon; that is to say, either at the True Rising or True Setting of the body. But since, during an observation, the object can only be referred to the Apparent or Visible Horizon, and since, moreover, it is subject to vertical displacement from refraction, parallax, and dip, it is necessary to adopt some provision by means of which the actual observation in a given case may be made to satisfy the preceding condition. There are two ways of doing this: one by observing the bearing of the object while it is elevated at a certain estimated distance above the Visible Horizon, sufficient to correct the effect of the vertical displacement above mentioned ; the other, by observing the body when its centre is apparently in the Visible Horizon, and applying a suitable correction. The former of these two provisions is that which is more commonly in use, at least by American Navigators; but, apart from the uncer- tainty of the estimated height, it wholly excludes the Moon as one of the objects of observation. The latter, on the other hand, not only admits of a more precise observation, but readily permits the use of the Moon as an object certainly only inferior to the Sun in importance for this purpose. 8. Remark : The Amplitude. The method now being con- sidered is commonly treated in the books on Navigation as an Ampli- tude; that is to say, by using the complement of the Azimuth, reckoning from the E and W points toward the N or S. But the Azimuth, besides being more natural and more convenient in the compass-reading than the Amplitude, has the advantage of being consistent with all the other forms of finding the Compass-Error. 1 'The author may bo deemed a little hardy iti venturing to propose the change indi- cated in the text. But, so far as mere terms are concerned, why may not " Kising- Azimuth" and " Setting-Azimuth" replace the old phraseology of " Rising-Amplitude" and " Setting-Amplitude" without doing serious violence to nautical usage .' With the change proposed, every observation at sea for Compass-Error will be an Azimuth, always referred to the Meridian, and always read forward from the zero-point, rather than, as in the exceptional case of the Amplitude, referred to the prime vertical and read backward from 90. The Table of Horizon- Jzimulhs (XXIV) here given will at least enable the Naviga- tor to test this question from a practical point of view ; but any one, still desirous of preserving the old form, may treat his observation as an Amplitude, by simply using the difference between 90 and the quantity taken from this Table, which is the same a* that taken from a Table of Amplitudes. METHOD OF HORIZON-AZIMUTHS. 9. Rule : To make the Observation. In observing objects Laving appreciable disks, like the Sun and Moon, no preparation is necessary beyond that of being ready at the compass a few minutes in advance, and keeping the sight- vanes pointed in the proper direction. a) Observation of the Sun. At the rising, when the upper. limb ap- pears in the sea-horizon, take the bearing by compass ; and continue to take bearings of the centre, bisecting the disk, and reading off each bearing for an assistant to note, till the lower limb appears. Or, at the setting, when the lower limb touches the horizon, proceed in the same manner till the upper limb disappears. In either case take the mean of the several bearings, which will be the apparent Compass-Azimuth of the Sun's centre. 6) Observation of the Moon. Proceed in the same manner as in observing the Sun, whenever the luminous portion of the disk is suffi- ciently large and bright for the rising to be anticipated in time to get the bearing of the upper limb as it comes upon the horizon. Other- wise, if the brightness be insufficient for a reliable observation in the horizon (as when the Moon is near her conjunction), it may be better to make the observation at a convenient altitude by one of the methods to be described. c) Observation of a Planet or Fixed Star. It is seldom practicable to obtain a reliable observation of a Planet or Fixed Star in the horizon, and especially at rising; so that it is generally better to anticipate the setting, or delay until after the rising, and take the object low in alti- tude by the method of Time- Azimuths. Whenever the attempt is made to obtain a Bising-Azimutb, it will be expedient to prepare for the obser- vation as follows : With the Lat. in, and the Dec. taken from the Naut. Almanac, or from Tab. LX, find the Time, Apparent or Mean, of rising (Tab. XXIII and Introd.} Also take out, from Tab. XXIV, the Horizon-Azimuth. Then, a few minutes before the time of rising, having applied the latest known Compass- Error to the Horizon Azimuth for the particular heading of the ship, and thus obtained the approximate Compass-Azi- muth, place the sight vanes upon the indicated bearing and watch for the Planet or Star to appear; and at the instant it does come distinctly into view, on or extremely near the Visible Horizon, carefully take the bearing. 1 1O. Remark: Ofoier rations auxiliary to Compass- Azimuths. There arc certain auxiliary observations which should always be made at the instant of taking any Azimuth-Observation for Com pass- Error. These are i) The Ship's Heading by the Standard or Azimuth Compass, and by each of the other compasses on board which are suitably mounted for such observations; and 1 In this case, of course, but a single bearing can be taken, and greater care is there- fore requisite. FINDING THE COMPASS-ERROR. 2) If the ship be iron-built, the Angle of Heel by clinometer, and whether to starboard or port (Chap. IV). The labor of making and recording these auxiliary observations is trifling; and they are essential to a complete knowledge of the compass- situation (Chap. IV). 11. Rule: To correct the Observed Azimuth. As already remarked (7), all observed Azimuths in the Apparent Hori/on require certain corrections, in order to convert them into Oompass-Azi muths on the True Horizon. For this correction, enter Tab. XXVII, with the Lat. and Dec., both to the nearest degree, and when taken out apply it to the observed Azimuth in the following manner: a) For the Sun, a Planet, or Fixed Star. At the rising in N Lat. and setting in S Lat., apply the whole correction to the right; or at the set- ting in N Lat. and rising in S Lat., apply it to the left. b) For the Moon. Apply the 7m?/-correction in the contrary manner The result in either case will be the proper Compass-Azimuth, which, by comparison with the True Azimuth, will give the Compass- Error. 12. Examples of Correcting Azimuths ohserved in the Apparent Horizon. Ex. 1. With Lat. 55 N and Dec. 14 N, observed Rising-Azimuth of the Sun was N 47 E. By Tab. XXVII: Obs. Az. N 47.o E Corr. (R.) i .o COMP. Az. N 48 .o E Ex. 2. With Lat. 47 N, Dec. 27 S, observed Setting-Azimuth of Vertus was S 45 -5 W. Ex. 3. With Lat. 65" S and Dec. n S, observed Rising-Azimuth of the Snn was S 56 E. By Tab. XXVII: Obs. Az. S 56.o E Corr. (L.) i .6 COMP. Az. S 57 .6 E Ex. 4. With Lat. 350.6 S, Dec. 20^.2 N, observed Setting-Azimuth of the Moon was N 7o.2 W. Obs. Az. S 45 G -5 W Obs. Az. N 70 .2 W Corr. (L.) o .9 COMP. Az. N 135 .4 W Corr. (| L.) o .3 COMP. Az. S 129 .5 W It will be noticed, in cases where the Lat. and Dec. have contrary names, in which it maybe convenient to take the supplement of tbe Compass-Azimuth for comparison with the True Azimuth, as in Exam- ples 2, 3, 4, 5, and 6 (14), that the result is the same whether the cor- rection bo applied before or after taking the supplement. Of course, when the correction is quite small it may be neglected, as it might be in Ex. i of Art. 14. 13. Rule : To find the True Azimuth. For this, proceed as follows : a) Preparation of Data. Find the Latitude in, and take out the Decli- nation, both to the nearest tenth of a degree. 1 For the latter, observe that 'It will be seen that little use is made by the author of sexagesimal computations, tabular or otherwise, in anything relating to the Azimuth and Compass-Error. In METHOD OF HORIZON-AZIMUTHS. 1. For the Sun, it is sufficient to take it off-hand from Tab. LX. 2. For a Fixed Star, it may be taken at sight from Tab. LXI. 3. For one of the four Planets, it is sufficient to take it off-hand from the Tables of the Nautical Almanac. 4. For the Moon, it will be sufficient to find the Greenwich date to the nearest tenth of an hour, and with this take out the Declination from the Nautical Almanac. b) Solution ~by Computation : Use of Tables X and XI. To the secant of the Lat. add the sine of the Dec.; their sum, rejecting 10 from the index, is the cosine of an angle. This taken at sight from Tab. X to the nearest tenth of a degree, if the Lat. and Dec. have the same name, will be the required True Azimuth less than 90; but, if they have contrary names, the supplement of the angle, or that found by sub- tracting the angle from 180, will be the True Azimuth greater than 90. In either case, mark the Azimuth 1ST or S, according to the Lat., and E or W, according as it is found for a rising or setting of the object observed. c) Examples of True Azimuths by Computation : Ex. 1. With Lat. 42 25' N and Dec. 21. 7 N, what is the True Azimuth at Rising? Lat. 42 .4 N 8600.1317 Dec. 21 .7 N sin 9.5679 T. Az. N 60 .0 E cos 9.6996 Ex. 2. With Lat. 51 30' N and Dec. 16 17' S, what is the True Azimuth at Setting ? Lat. 5 l0 -5 N sec 0.2058 Dec. 1 6 .3 S sin 9.4482 T. Az. N 116 .8 W cos 9.6540 Ex. 3 With Lat. 55 5' S and Dec. i3-9 S 3 what is the True Azimuth at Set- ting ? Lat. 55. i S sec 0.2425 Dec. 13 .9 S sin 9.3806 T. Az. S65'.2\V 0039.6231 Ex. 4. With Lat 16 57' S and Dec. 29 13' N, what is" the True Azimuth at Rising? Lat. i6.9 S sec 0.0194 Dec. 29 .2 N sin 9.6883 T. Az. S 120 .7 E cos 9.7077 fact, at least in his opinion, all refinements, such as the consideration of quantities smaller than minutes of arc and tenths of a minute in time, are a waste of time and pa- tience, being utterly useless in computations of this kind. And, moreover, so far as the finding of any single Compass-Error is concerned, it is always quite sufficient to work to the nearest tenth of the degree in arc, and to the nearest minute in time. The practice of changing sexagesimal quantities of a lower denomination into tenths of a higher is readily done at sight, without thinking about it, after a little practice. Take the series in minutes of arc I' 2' 0.02 0.03 or, o.o o.o 3 o.o5 o.o Again, take the series 24' 25' 26' 2/ o-45 5' o.o8 28' o.io to tenths and hundredths of degrees, o.i to the nearest tenth. 29 3o' o47 o48 o.5o to tenths and hundredths, o 5 o.5 o.5 to the nearest tenth. And similarly for all other cases. To annex one cipher to the minutes, divide by 6, and point off two decimal places, is to convert the sexagesimal into tenths and hun- dredths of degrees. Or, to divide simply by 6 and point off one decimal, is to have an equivalent in tenths. But, as remarked above, the operation becomes as involuntary as the use of the multiplication-table. 10 FINDING THE COMPASS-ERROR. Ex. 5. With Lat. 67 i/N and Dec. 19 28' S, what is the True Azimuth at Rising ? Lat. 6;.3 N sec 0.4135 Dec. 19 .5 S sin 9-5 2 35 T. Az. N 149 .9 E cos 9.9370 Ex. 6. With Lat. 71 55' N and Dec. 12 47' N, what is the True Azimuth at Setting ? Lat. 7i'9 N sec 0.5077 Dec. 12 .8 N sin 9.3455 T. \z. N 44 .5 W cos 9.8532 d) Solution by Inspection : Use of Tab. XXIV. Enter the Table of Horizon- Azimuths with the Lat. and Dec., each to the nearest tabular argument, and take out the corresponding angle ; and this, or its sup- pleinent, according as the Lat. and Dec. have the same name or con- trary names, will be the True Azimuth. Mark as above (6). e) Remark. If either the given Lat. or Dec. be more nearly half-way between two adjacent arguments than equal to either of them, take the Mean of the two numbers corresponding to the two arguments. If both the Lat. and Dec. be more nearly half-way between adjacent argu- ments than equal to them, take the Mean crosswise of two of the four corresponding numbers. The Mean of two numbers is most conveniently found by adding their Half-Difference to the less one. f) Examples of True Azimuths by Inspection (the Data from the Ex- amples [c] ) : Ex. 1. With Lat. 42^4 N and Dec. 2i-5 N, Tab. XXIV gives- True Az. N 6o.2 E Ex. 2. With Lat. 5i.5 N and Dec. i6.3 S, we get TrueAz. N iiG .; W Ex. 3. With Lat. 55.! S and Dec. if. 9 S- True Az. S 65.! W Ex. 4. With Lat. i6.9 S and Dec. 29.2 N True Az. S I2o.5 E Ex. 5. With Lat. 67.3 N and Dec. 190.5 S- True Az. Nfi49.7 E Ex. 6. i2.8 N -With Lat. 7i.9 N and Dec. True Az. N 44. 4 W In Ex. i, on entering Tab. XXIV, we find that the Lat. falls more nearly half-way between 42 and 43 than equal to either, and we there- fore take the Mean of the corresponding Tabular Azimuths, 6o.4 and 59.9, by subtracting their Half-Difference (o.2) from the greater. In Ex. 2, both the Lat. and Dec. are more nearly half-way between 51, 52 and i6.o, i6.5 respectively; and, consequently, we take a crosswise Mean of the four Tabular Azimuths [ /* '' 2 ' 2 > , by adding ( 03 .4, 02 .5 ). the Half-Difference (o.i) of 63.4, 63^2 to the less, or subtracting it from the greater. By a similar mental process, executed at sight, the results in the other Examples are obtained, as well as those in all Examples of this kind. It is thus seen that with no greater care than to take the Means of the Tabular Azimuths, corresponding to the arguments between which the given data fall, the results in these Examples, alike for the higher and lower Latitudes, do not differ more than o.2 from those obtained by computation. And they will seldom differ as much as that. METHOD OF HORIZON-AZIMUTHS. In taking Cross-Means of four numbers we rnay use either pair ; but it will generally be found that one pair, as in this instance, has a smaller difference than the other, and we use that in preference. 14. Examples of Fipcliiig the Compass-Error by Hori- zon-Azimuths. Ex. 1. 1875, June 5 : At sea, in Lat. 1 1 29' N, Long. 30 W, about 6 h io m A. M v the observed bearing of the Sun at rising was N 59 E : Required, the Compass-Error. Greenwich date, June 4 d 2O 1 ' 0's Dec. (Tab. LX) N 22.5 By Tab. XXIV: Latitude n.5 N 's Dec. 22 .5 N Obs. Az. N 59 .o E Corr. (R.) o .1 True Az. N 67. i E Comp. Az. N 59 .1 E COMP. ERROR 8 .o E Ex. 2. 1875, May 30: At sea, in Lat. 25 3' S, Long. 22 W, about 6 h 42 A. M., the observed bearing of the Sun at rising was N 71. 5 E : Required, the Compass- Error. Greenwich date, May 29 d 2o h 's Dec. N 2i.7 By Tab. XXIV: Latitude 25.o S 's Dec. 21 .7 N Obs. Az. S 108 .5 E Corr. (L.) o .3 True Az. S 114.! E Comp. Az. S 1 08 .8 E COMP. ERROR 5 .3 W Ex. 3. 1875, November 27: At sea, in Lat. 40 27' N, Long. 20 7' W, about 4 h 43 P. M., the observed bearing of the San at setting was S 73 W : Required, the Compass-Error. Greenwich date, November 27 d 6 h 's Dec. S 2i.2 By Tab. XXIV: Latitude 4O.5 N > ~ [TrueAz. N ii8.5 W 's Dec. 21 .2 S > Obs. Az. S 73 .o W ? n > Comp. Az. N 107 .7 W Corr. (L.) o .7 S COMP. ERROR . 10 .8 W i Ex. 4. 1875, December 18: At sea, in Lat. 31 35' 'S, Long. 60 13' E, about 7 h o m P. M., the observed bearing of the Sun at setting was N 83 W: Required, the Com- pass-Error. Greenwich date, December i8 (1 3^ 's Dec. S 234 By Tab. XXIV: Latitude 0's Dec. 3i.6 S > True Az. 23 .4 8 S 62. i W Obs.Az. N 83 .4W Corr. (R.) o .4 COMP.ERROR 34 .9 W Ex. 5. 1875, July 1 7 : ^ sea > in La ^- 51 13' N, Long. 33 47' W, about 2 h 7 m A. M., the observed bearing of the Moon at setting was S 62 W : Required, the Compass-Error ; also the Deviation. Ship heading W % S by Standard Corn- Greenwich date, July i6 d i6 h .4 ([ 's Dec. 28.2 S By Tab. XXIV: Latitude 510.2 N ^R.) o .6 c N ^ w m N w COMP.ERROR ............. 21 .3 W Tab. LVL, Mag. Var. 37. 7 W Comp. Dev. 16 .4 E Ex. 0. 1871, August 16: In port, Disco Island, in Lat. 69 14' N, Long. 53 18' W, about 8 h 40 P. M., the observed bearing of the Sun at setting was N 6.8 E : Re- quired, the Compass-Error ; also the Devi- ation. Greenwich date, August i6 d i2 h 's Dec. N 13 40' Latitude 6 9 .2 N 's Dec. 13 .7 N Obs.A z N 6.8E? , Corr. (L.) 2 .4 ) N COMP. ERROR ............. 52 .2 W By Tab. LVI, Mag. Var. 72. 5 W Hence, also, Comp. Dev. 20 .3 E 12 FINDING THE COMPASS-ERROR. By Tab. XXIV : Latitude 48. i N > ^ ^ N Jo60 2 If 'a Dec. 10 .7 S $ Approx. Coinp. Er. 5. 2 E Approx. Coiiip. Az. N 101 .o E or, S 79 .o E A few minutes before the time of rising, the compass is set upon the approximate bearing (S 79.o E), the appearance of the Planet in the horizon watched for, and, as soon as distinctly visible, the bearing carefully taken. In this case it is S 8o.5 E. Hence Obs. Az. N 99. 5 E ) True Az. N io6..2 E Corr. (R.) o .8 > Comp.Az. N 100 .3 E Ex. 1 *. 1875, February 2: Atsea,inLat. 48 5' N, Long. 174 15' W, about 10 P. M., desired to get an observation of Jupiter at rising for Compass-Error on present course of ship, which is North. Preparation : Approx. Gr. date, February 2 d 22 h Ifs Dec. S 10. 7 14' s Mer. Pass. i; 11 $ m M. T. at ship 2_fs Hour- Angle 5 h I2 m at rising (Tab. XXIII} Ship M. T. i2 h 53 m of rising Mag. Var. i5.S E (Tab. LVI) Coiup. Dev. io.3W?Dev. Table for Comp. Er. 5 .2 E > Stand. Comp. COMP. ERROR 5 -9 E Remark. Example 7 is given at length to illustrate the procedure iu such cases, which, although in appearance somewhat complicated, is really very simple and but the work of a few moments. Still, it is quite evident that, by waiting a few minutes for the Planet to rise above the horizon, a bearing then taken and compared with a Time-Azimuth for that instant will be much simpler, and generally preferable, except when the Ship-Time is very uncertain. In such case the Horizon- Azimuth, as above, might be useful. B. METHOD OF TIME-AZIMUTHS. lo. Fundamental Principles of the Time- Azimuth. In a Time-Azimuth, the bearing of the Sun or other heavenly body is observed with the compass whenever practicable, the Ship or Local Time being reliably known within certain limits. The observed bearing is itself the proper Com pass- Azimuth of the body. The True Azimuth is found by a solution of the Triangle of Position (Int.) The Data required are The Hour-Angle, deduced from the Local Time of the observation -(Int.)-, The Declination, taken from the Nautical Almanac or Nautical Tables (Int.}} and The Latitude in, as brought up by the Beckoning or known from pre- vious observation. 16. The Ship-Time. The Ship or Local Time upon which the Hour-Angle depends, is the important element in this problem. There are two distinct means by which it is obtained, whenever required on board ship. The first, which is the ordinary one, consists in the use of^a reliable Comparing- Watch, which is always kept regulated to tbe Ship-Time, either Apparent or Mean, generally to the former ; being set to that time, or the error noted upon it, as often as anew determination is made METHOD OF TIME-AZIMUTHS. 13 by any of the astronomical methods employed for that purpose. Then, whenever required at any intervening moment prior to the next deter- mination, the Ship T. is readily deduced from the W. T., by correcting the latter for the error at the last preceding determination, for that accu- mulated in the rate during the subsequent interval, and for any differ- ence of Longitude in time made good during the same interval, subtract- ing or adding the latter, according as it is W or E of Greenwich. The second means is that of deducing the Ship T. from the Chronome- ter on board, by correcting the Chro. T. for its error on Greenwich T., and then for the place of the ship by subtracting or adding the Longitude in time, according as it is W or E of Greenwich. The Ship T., as thus found, will generally be M. T. ; and if A. T. be required, it will be necessary to take out for the Greenwich date the Equation of Time and apply it to the M. T. For the observations of the Sun, it is more convenient to have A. T. ; but for observations of the Moon, Planets, or Fixed Stars, it is almost a matter of indifference whether A. T. or M. T. be used ; consequently, it is generally expedient to keep one Com paring- Watch at least regulated to ship A. T. 17. Finding- the Compass-Error for a Single Heading of the Ship : Objects which may be employed. In finding the Compass-Error for a single heading of the ship, every recognizable celestial object, which is sufficiently bright to be seen through the sight- vanes of the compass, may be employed in Time-Azimuths with more or less convenience. During the day, the Sun is most commonly resorted to ; next to it, the Moon may often be used with nearly equal facility, whether by daylight or during the night, except when near her conjunc- tion ; and, besides these, the planets Yenus, Mars, Jupiter, and Saturn are frequently available for this purpose. The brighter Fixed Stars may be observed with advantage whenever the more conspicuous objects are in less favorable positions. In general, it will conduce to accuracy to select an object, on a given occasion, relatively low in altitude, not only that, in being seen by direct vision through the sight- vanes rather than by reflection, the Compass- Azimuth is more reliable (Chap. I), but because the condition is more favorable for a reliable True Azimuth when the Data are considerably uncertain. Still, as will be seen in the sequel, Time-Azimuths are really available with sufficient accuracy for a single Compass-Error at sea under a wide range of circumstances. 1 8. To make a Single Time- Azimuth Observation. Take a bearing of the object with the Standard Compass, or preferably a set of two or three bearings, as quickly as possible, bisecting it each time, if it have a sensible disk (Sun or Moon), and noting the times with a watch, whose error on Ship T. is known. Note the heading of the ship with the same compass, and the corre- sponding headings with the other compasses ; also, the Angle of Heel, if the ship be iron-built, with the clinometer (10). 14 FINDING THE COMPASS-ERROR. The observed bearing, or the Mean if several be taken, is the re- quired Compass- Azimuth ; and the noted time, or mean if several be taken, is the corresponding W. T. of observation. 19. Finding* the True Azimuth. This method of finding the True Azimuth consists of two parts : First, the Preparation of the Data, and, secondly, the Solution of the Triangle of Position. The second part may be accomplished by Logarithmic Computation, by Azimuth Tables, or even by a Graphical Construction, and each may have cer- tain advantages and disadvantages as compared with the others ; but the first part must necessarily be the same, and can neither be simpli- fied nor abridged, with reference to one or another of the processes employed for the second part. .30. Rule: Preparation of the Data. In preparing the Data of a Time-Azimuth, we may proceed in the following manner, ac- cording as the object observed is the Sun, or one of the other heavenly bodies : a) The object observed being the Sun. This presents the simplest case. Thus 1. Find the Greemcich date for the Ship-Time of observation to the nearest hour. 2. Take out the Declination of the Sun, with this date, from Tab. LX, at sight, to the nearest tenth of the degree 5 and note the Pol. Dist. and Co-Lat., each to the nearest tenth of the degree. 3. Find the Hour- Angle of the Sun. For this, convert the Watch-Time of observation into ship Ap. Time, by applying the Watch-Error on that time. Or, if the Watch-Error be known only on ship M. Time, apply this error, and in addition the Eq. of Time, taken at sight from Tab. LIX, to the nearest tenth of a minute, according to its sign, adding if -+-, sub- tracting if . Then, if the observation be P. M., the ship A. T. so found is the Hour- Angle West 5 or, if the observation be A. M., the ship A. T. subtracted from i2 h gives the Hour-Angle East. b) The object observed being the Moon, a Planet, or Fixed Star. For these objects there is a little more to be done in getting the Hour- Angle; still, the process of finding this datum is the same for all objects except the Sun, and requires less time to execute than to describe. 1. Find the Greenwich dale, in the usual manner : For the Moon, to the nearest minute ; and for either of the Planets or a Fixed Star, to the nearest tenth of the hour. 2. TaTceoutthe Declination of the object, at sight, to the nearest tenth of the degree; for the Moon and Planets from the Naut. Almanac, and for the Fixed Stars from Tab. LXI ; and note the Pol. Dist. and Co-Lai., each to the nearest tenth of the degree. 3. Find the Hour-Angle of the object. For this, convert the Watch-Time of observation into ship A. T.by applying the Watch-Error on that time. METHOD OF TIME-AZIMUTHS. 15 Xext, to the ship A. T., expressed astronomically, add the Eight Ascen- sion of the True Sun (taken from Tab. LIX and reduced to the Green- wich date by Tab. LXIV), rejecting 24^ from the sum if greater than 24 h , and the result will be the Eight Ascension of the Meridian. Then, the difference between this B. A. and the E. A. of the object (taken from the Naut. Aim. or from Tab. LXI and reduced by Tab. LXIV or LXV), will be the Hour-Angle, which will be W or E according as the latter B. A. is less or greater than the former. If the H. A. thus found be greater than i2 h , subtract it from 24^ and take the remainder as the proper H. A. with the contrary name. Otherwise, if the Watch-Error be known only on ship M. T., convert the Watch-Time of observation into ship M. T. by applying the Watch-Error on that time. Next, to the ship M. T., expressed astronomically, add the Right Ascension of the Mean Sun (taken from Tab. LIX and reduced to the Greenwich date by Tab. LXVI), rejecting 24 h from the sum if greater than 24 b , and the result will be the Eight Ascension of the Meridian. Then the difference between this B. A. and the E. A. of the object will be the Hour- Angle, as explained in the preceding paragraph. 4. Remark. It is always sufficient to take out the Bight Ascensions and to deduce the corresponding Hour- Angles to the nearest tenth of the minute ; and Tables LXIV-LXVI afford the requisite facilities for taking out the proper corrections, at sight, to reduce the Tabular Bight Ascensions 1 to the given Greenwich date. 1 In order to facilitate the taking out of Right Ascensions, to the nearest tenth of the minute, without the necessity of consuming time and patience in the requisite in- terpolations, the several Tables referred to in the text have been constructed. The following explanations of their use may be convenient : 1. To take out tlie E. A. of the True Sun. Enter Tab. LIX with the day of G. date and take out the corresponding R. A. and the Daily Diff. between that R. A. and the next following ; or, enter the Naut. Aim. with the day of date and take out the R. A., noting the adjacent Hourly Diff.; then, in either case^ enter Tab. LXIV with the Diif. and the hour (including tenths, if any) of the G. date, and take out the corresponding correc- tion, which add to the R. A. for the clay, and the result will be the required E. A. True Sun reduced to the date. 2. To take out the R.A. of the Mean Sun. Enter Tab. LIX with the day of G. date, and take out the R. A. of the True Sun for that day, to which apply the corresponding Eq. of Time, according to its sign, adding if -J-, subtracting if ; or, take the R. A. of the Mean Sun directly from the Naut. Aim. for the day of date ; then, in either case, enter Tab. LXVI with the hour (and tenths, if any) of the G. date, and take out the corre- sponding correction, which add to the before-mentioned sum, or R. A. of the Mean Sun for the day of G. date, and the result will be the required JR. A. Mean Sun reduced to the date. 3. To take out the R. A. of the Moon. Enter the Tables of the Naut. Aim. with the day and hour of the G. date and take out .the corresponding R. A., and also note the adja- cent Diff. for i m ; then enter Tab. LXV with that Diff. (found at foot) and the minutes of the G. date and take out the corresponding correction, which add to the R. A. above mentioned, and the result will be the required R. A. of the Moon reduced to the date. 4..To take out the R. A. of a Planet. Enter the Tables of the Naut. Aim. with the day of the G. date and take out the corresponding R. A., and note the adjacent Diff '. for i h with its sign; then enter Tab. LXIV with that Diff. and the /four (and tenths, if any) of the G. date and take out the corresponding correction, which apply to the before-mentioned 16 FINDING THE COMPASS-ERROR. 21. examples in Finding the If our- Angle. Ex. 1. 1875, July 17: Required, the Moon's H. A. iu Long. 33 47' W at i h 37 20* A. M. Ship A. T. d. h. m. h. m. Ship A. T. July 16 13 37 Ship A. T. 13 37.3 Long, iu +215 R. A.T.0 1 744 Green wich date 16 15 53 R. A. Mend. 21 21.3 R. A.d 18 36.4 H. A. 2 44 .6 W Ex. 2. 1875, February 23 : Required, the H. A. of Jupiter in Long. 174 15' W at i h 5 m A. M. Ship A. T. Ship A. T. Feb. 22 13.1 Long, iu -f ii 6 Greenwich date 23 0.7 h. m. Ship A. T. 13 5.0 H. A. T.0 22 25.6 R. A.Merid. ii 30.6 R. A. Jup. 13 59.6 H. A. Jup. 2 29.0 E Ex. 3. 1875, April 25 : Required, the Hour-Angle of the Moon, in Long. 155 20' W at o b 15'" 30* A. M. Ship M. T. d. Ji. m. h. m. Ship M. T. Apr. 24 12 15.5 Ship M. T. 12 15.5 Long, in + 10 21.3 K. A. M.0 2 12.1 Greenwich date 24 22 37 R. A. Mend. 14 27.6 R. A. d 17 44.5 H. A. 3 16.9 E Ex. 4. 1875, October 2 : Required, the H. A. *>f Venus, in Long. 42 W at 5 h 45 m 25 s P. M. Ship A. T. d. Ji. h. m. Ship A. T. Oct. 2 5.8 Ship A. T. 5 45.4 Long, in +2.8 R. A. T. 11233.9 Greenwich date 2 8.6 R. A. Merid. 18 19.3 R. A. Venus 12 45.1 H. A. Venus s ^4-^ W Ex. 5.^1875, April 4: Required, the H. A. of Castor, in Long. 80 39' E at ii h 9 P. M. Ship A. T. a. h. n. m. Ship A. T. April 411.2 Ship A. T. n 9 Long, in 5.4 R. A. T.0 o 53.6 Greenwich date 4 5.8 T R. A. Merid. 12 26 R. A. Star 7 26 6 H. A. Star 4 36 o W Ex. 6. J 875, September 9 : Required, the.H. A. of Fomalhaut, iu Long. 33 55' W at 8 h 20 io s P. M. Ship M. T. d. h h. m. Ship M. T. Sept. 9 8.3 Ship M. T. 8 20.2 Long, in + 2.3 R. A. M.0 n 14.2 Greenwich date 9 10.6 R A Merid ~ R. A. Star 22 50.7 H. A. Star 3 16.3 E R. A. according to the sign of the Diif., and the result will be the required E. A. of the Planet reduced to the date. 5. To take out the R. A. of a Fixed Star. Enter Tab. LXI, or the Naut. Aim., with the year of G. date and takeout at sight the corresponding R. A. to the nearest tenth of the minute. In taking out corrections from Tables LXIV-LXVI, it will be sufficient to enter the Tables with the nearest tabular difference, and to retain the nearest tenth's figure in the decimal part of the correction. i In Ex. i : By Tab. LTX Tab. LXIV In Ex. 2 : By Tab. LIX Tab. LXV In Ex. i : By N. Almanac ' Tab. LXV In Ex. 4 : By N. Almanac, Tab. LXIV R. A. True Sun Red" for i5. 9 R. A. True Su)i [ R. A. True Sun ! Eq. of Time R. A. Mean Sun Red" for 22 h .6 R. A. Mean Sun R. A. Moon Red" for 52 m R. A. Moon R. A. Venus Red" IbrS'-.e 7.'. A. Venus h. in. 7 41.4 at Greenwich noon July 16. + 2.6 Daily Difi'. 4.o. 7 .44.0 tor Greenwich date. h. m. 2 6.5 at Greenwich noon April 24. 4- 1.9 at Greenwich noon. 2 8.4 at Greenwich noon. + 3-7 2 12. i for Greenwich date. h. m. 18 34.4 at Greenwich i5 h .o July 16. + 2.0 Diff. 18 36.4 for Greenwich date. h. m. 12 43.5 at Greenwich noon Oct. 2. I- 1.6 Diff. 1 1 ".4 for i 1 '. 12 45.1 for Greenwich date. METHOD OF TIME-AZIMUTHS. 17 . Rule : Solution by Logarithmic Computation. Using Tables X, XI, XII, and XIII, we may proceed as follows : Take the Half-Difference and Half-Sum of the Pol. Dist. and Co- Lat., also Half the Hour-Angle ; then, to the sine of the y 2 D. add the cosecant of the ^ S. and the cotangent of the y 2 H. A.; the Sum of these three logarithms, rejecting tens from the indices, is the log tangent of the Angle X. Similarly, to the cosine of the y 2 D. add the secant of the y 2 S. and the same cotangent of the y 2 H. A. ; and the Sum of these three loga- rithms, rejecting tens from the indices, is the log tangent of the Angle Y, There will now be two Cases to consider, according as the y z S. of the Pol. Dist. and Co-Lat. is less or greater than 90. First Case : y z S. of Pol. Dist. and Co-Lat. less than 90. Take the Sum or Difference of the Angles X and Y ? according as the Pol. Dist. is greater or less than the Co-Lat., and the result will be the True Azimuth. Second Case : y> S. of Pol. Dist. and Co-Lat. greater 1 than 90. Always take the Difference of the Angles X and Y 5 which subtract from 1 80, and the result will be the True Azimuth. In either Case, mark the True Azimuth N or S according to the Lat., and E or W according to the Hour- Angle. Remark. It may sometimes be convenient to use the Supplement of the True Azimuth by subtracting from 180 and reversing the prefix N or S, in order to make it correspond to the Compass-Azimuth less than o 90 . . Examples of" True Azimuths by Computation. Ex. 1. With Lat. 20 50' N and Dec. 2.7 S, what is the True Az. of the Sun at i h 2i m P. M. ship Ap. Time? 's Hour-Angle i h 2i m W. Pol. dist. Co-Lat. Diff. KDiff. Yz Sum 92. 7 6 9 .2 23 -5 ii .7 80 .9 Yz H. A. oh 4o m .5 T. Az. N i 37 .4 W : sin 9-3070 cos 9.9961 cosec 0.0055 sec 0.8009 cot 0.7488 cot 0.7488 tan 0.0613 tan 1.5458 X 4 9 .o Y 88.4 Ex. 2. With Lat. 40 33' N and Dec. N 2i.8, what is the True Az. of the Sun at 5 h 24 in 43 s A. M. ship A. T. ? 's Hour- Angle 6 h 35 Pol. dist. Co-Lat. Diff. ^ Diff. Yz Sum .3 E. 68.2 Yz H. A. 3 h i 7 .6 T. Az. N 6 7 . 7 E : sin 9.2085 cos 9.9943 cosec 0.0678 sec 0.2856 cot 9-9330 cot 9-9330 tan 9.2093 tan 0.2129 X 9^.2 Y 5 8. 5 Ex. 3. With Lat. 12 if S and Dec. 22.9 S, what is the True Az. of the Sun at 8 h 20 5 s A. M. ship A. T. ? 's H. A. 3 h 39 m -9 E. Pol. dist. Co-Lat. Diff. Yz Sum Yz H. A. i 67. i 77 -7 10 .6 5 -3 72 .4 50 sin 8 9655 cos 9.9981 cosec 0.0208 sec 0.5195 cot 0.2835 cot 0.2835 tan 9.2698 tan 0.8011 X io.s Y 8i.o T. Az. S 7 o.5 E : Ex. 4. With Lat. 6 n' N and Dec. 2 3-3 S, what is the True Az. of the Sun at 5" 10 P. M. ship A. T. ? 0's H. A. 5" io m W. Pol. dist. Co-Lat. Diff. y 2 T)iff. Yz Sum H. A. 83 .8 2 9 -5 14 .7 98 -5 T. Az. N 1 140.7 W: sin 9-4044 cos 9.9855 cosec 0.0048 sec 0.8303 cot 0.0955 cot 0.0955 tan 9.5047 tan 0.91 13 X 170.7 Y 8 3 .o 1 lu taking out the cosecant and secant in this case, use the Supplement of % S. by subtracting from 180 ; otherwise, and more conveniently, use the excess of the y 2 S. over 90, taking out the secant for cosecant and the cosecant for secant. 18 FINDING THE COMPASS-ERROR. Ex. 5. With Lat. 23 24' N and Dec. 234 N, what is the True Az. of the Sun at 6" 10 P. M. ship A. T. ? 's II. A. 6" io m W T . Pol. dist. 66.6 Co-Lat. . 66 .6 Diff. o .o Yz Diff. o .o Yz Sum 66 .6 Yz H. A. 3 1. 5 m T. Az. X 6 7 4 . W : sin oo cosec cot tan oo X o^.o COS 0.0000 sec 0.4010 cot 9 9810 tan o 3820 Y 6 7 o. 4 Ex. 6. WithLat.2i23'NandDec.2i 23' S, what is the True Az. of the Sun at 4 n 30 P. M. ship A. T. ? 's H. A . 4 h 30 W. Pol. dist. m. 4 Co-Lat. 68 .6 Diff. 42 .8 Yz Diff. 21 .4 Yz Sum 90 .o Yz H. A. 2 h 15- T. Az. Kn8.6 W: sin 9.5621 cosec o.oooo cot 0.1751 tan 9.7372 X 28.6 COS sec -}-oo cot tan +00 Y o.o Ex. 7. With Lat, 12 if N and Dec. ii .; N, what is the True Az. of the Sun at 5* 5 P. M. ship A. T. ? 's H. A. 5" 5 m W. Pol. dist. 7 8. 3 Co-Lat. 77 .7 Diff. o .6 Yz Diff. o .3 sin 7.7190 cos o.oooo Yz Sum 78 .o cosec 0.0096 sec 0.6821 Yz H. A. 2^ 32 m .5 cot o.'io53 cot 0.1053 tan 7.8339 tan 0.7874 T. Az. N 8i.i W: X o. 4 Y 80^.7 Ex. 8. With Lat. 72 u' N and Dec. 29 32' N, required the True Az. of a Star whose Hour- Angle is 9 h 15 W. Pol. dist. 6o.s Co-Lat. 17 .8 Diff. 42 , 7 Yz Diff. 21 .3 sin 9.5602 Yz Sum 39 .1 cosec 0.2002 cot tan Yz H. A. 4 h 37 m -5 T. Az.N 360.5 W: cos 9.9693 sec o.noi 9.5757 cot 9.5757 9.3361 tan 9.6551 I2.2 Y 24-3 Ex. 9. With Lat. 30 f N and Dec. 9 if S, what is the True Az. of the Moon when its II. A. iso' 1 13"'. 3 E .' Pol. dist. Co-Lat. Yz Diff. Yz Sum Yz H. A. c 99-3 59 -9 39 -4 19 .7 79 .6 6 m . 6 1 A T. Az. N i74.9 E : sin 9.5278 COS 9-9738 cosec 0.0072 sec 0.7435 cot 1-5384 cot 1.5384 tan 1.0734 tan 2.2557 X 8 5 . 2 Y 890.7 Ex. 1O. With Lat. 12 if N and Dec. 11 42' S, what is the True Az. of the Sun at 6 1 ' 30'" A. M. ship A. T. ? 's II. A. 5'' 30 E. Pol. dist. 101. 7 Co-Lat. Diff. Yz Diff. Yz Sum Yz H. A. 77 .7 24 .o 12.0 89 .7 45 m sin 9-3179 cos 9 9904 cosec o.oooo sec 2 2810 cot 0.0570 cot 0.0570 tan 9 3749 tan 2.3284 X i3-3 Y T. Az. N 103. i E : Ex. 1 1. With Lat. 70 50' N and Dec. 2o.6 1ST, required the True Az. of the Sun at o b 20 A. M. ship A. T. 's H. A. n h 40 E. Pol. dist. 690.4 Co-Lat. 19 .2 Diff. ' 50 .2 Yz Diff. 25 .1 sin 9.6276 cos 9. 9569 Yz Sum 44 .3 cosec 0.1559 sec I 453 Yz H. A. s 1 ' 50 cot 8.6401 cot 8.6401 tan 8.4236 tan 8.7423 T. Az. N4. 7 E: X i.s Y 3.2 Ex. 12. With Lat. i 2' S and Dec. 32. i N, required the True Az. of Castor when its H. A. is 4 h 35 m .8 W. Pol. dist. i22 ? .i Co-Lat. 89 .o Diff. 33 -i Yz Diff- 16 .5 sin 9.4533 cos 9.9817 . Yz Sum 105 .5 cosec 0.0161 sec o 5731 cot 0.1632 cot 0.1632 tan 9.6326 tan 0.7180 Yz H. A. 2'i 17. 9 T. Az. S i2 4 .oW: X 2 3 .2 Y Always, in computing, take secant for cosecant and cosecant for secant of the excess whenever the Half-Sum, as in Ex. 4, is greater than 9o.o. In the use of Table XXX, this distinction is not presented ; the nec- essary changes being provided for in the construction of the Tables. This is one among the other advantages of tabular inspection in avoid- ing the mistakes of a hasty computation. 24. The Tallies of Time-Azimuths. Although, under the preceding Eule, the computation of a True Azimuth by this method is really quite simple, and may be accomplished in a few moments for any METHOD OF TIME-AZIMUTHS. 19 given case, still, it is possible, with the aid of special Tables, to greatly facilitate the determination of this important element. Such is the object of the Tables, in three parts (XXX> XXX. A, and XXX. _B), of this Manual. a) Tab. XXX, or the First Part, provides for taking out Logs A and B with the Lat. and Dec., or with the corresponding Co-Lat. and Pol. Dist., either being used at pleasure as arguments. These argu- ments comprise all Latitudes from o to 80 north or south of the Equa- tor, and all Declinations from o to 35 of the same and contrary name. The arguments are given at intervals of one degree ; but the Table may be entered to tbe nearest half-degree, and the quantities taken at sight, with a sacrifice of precision that, as compared with a computation to the nearest tenth of the degree, is seldom so great as io.2, or 12'; much within the requirements of practice in finding a single Compass- Error. The Table may, however, readily be entered with the Lat. and Dec. to the nearest tenth of the degree, by a simple interpolation, but this is rarely, if ever, required for a single Compass-Error. 1} Tab. XXX. A, or Second Part, provides for taking out Log C, with the Hour-Angle as argument. This argument comprises all Hour- Angles from o 11 to i2 h west or east of the Meridian, and is given through- out at intervals of one-tenth of a wi>ii^(every'six seconds) in time. c) Tab. XXX. B, or Third Part, provides for taking out the Angles X and Y among the arguments, which arejgiven in degrees and tenths. This Table is entered to the nearest tabular quantities with the two sums obtained by adding Log C to Log A and to Log B. d) It was intended in the construction of these Tables First, that they should admit of furnishing an^im mediate or off- hand determination of the True Azimuth, with every probable combi- nation of Data, without a moment of time or patience expended on mere interpolation, and with a degree of precision and certainty to meet every practical requirement, whether in finding single or serial Com- pass-Errors; and, Secondly, that they should be as compact and portable as possib!e, consistently with the realization of the foregoing conditions. 1 1 Other Tables of Time-Azimuths Lave been given under different forms. Such are the following : 1. Sun's True Hearing or Azimuth Tulles, computed for intervals of four minutes, between the Parallels of Latitude 30 and 60 inclusive : By JOHN BURDWOOD, Staff-Com- mander K. N., London, 1866. 2. Tables des Azimuts du Soleil corrcspondant d Vhcure vraie du lord, entre les parallelcs 55 sud et 55 nord : par F. LABROSSE, Ancien Officier de Marine, Paris, 1868. (The explana- tions are given in French and English.) 3. A set of Azimuth-Tables (pp. 58-107), contained in Practical Information on the Deviation of the Compass, for the use of Masters and Mates of Iron Ships : By JOHN THOMAS TOWSON, F.R.G.S., etc., London, 1869. 4. Azimuth and Hour- Angle, for Latitude and Declination ; or Tables for finding Azimuth at sea ly means of the Hour-Angle, in all navigable Latitudes, at every two degrees of Dcclina- 20 FINDING THE COMPASS-ERROR. Direct and Limiting Values of Time-Azimuths : Use of Tab. XXXI. Tab. XXXI gives the Time-Azimuth directly, which, under very favorable conditions (60), may be taken out quite readily and with sufficient accuracy for finding the Compass-Error in numerous special instances ; but the argum'ent-intervals are too large for convenient use in the general case. This Table is useful in showing the limits of the Time-Azimuth ; in furnishing a check against serious error, through a hasty or inconsiderate procedure in getting out the True Azimuth by the ordinary methods ; and it is frequently useful in giving an approximate True Azimuth, at sight, as required in estimating probable errors, etc. 26. Rule : Solution l>y the Azimuth-Tables. The Data having been prepared (20), proceed as follows : Enter Tab. XXX, either with the Lat. and Dec., or with the Co-Lat. and Pol. Dist., to the nearest half-degree, and take out at sight Log A from the left and Log B from the right page ; next, enter Tab. XXX. A, with the Hour- Angle, to the nearest tenth of the minute, and take out Log (7, which add to both Log A and Log B, calling the results Tang X and Tang Y; then, enter Tab. XXX. B , and take out (from the arguments) Ang. X with the Tang X and Ang. Y with the Tang Y ; and, finally, complete the solution, according as it falls under the First or Second Case (Art. 22 and/ootf of Tab. XXX. B). b) Remark. When either the Lat. or Dec. (in Tab. XXX) falls more nearly upon the half-degree than whole degree of the tabular argu- ment, take the Mean of the two adjacent tabular l quantities ; and when both the Lat. and Dec. fall more nearly upon the half than whole degree of the tabular arguments, take the Cross-Mean of the four adjacent quantities. 2 The same rule will be observed in using the Co Lat. and Pol. Dist. as arguments. tion between ihe limits of the Zodiac, etc. : By Major-General R. SHORTREDE, F.R.A.S., London. Also, a Time-Azimuth Diagram : By HUGH GODFRAY, M. A., London, 1858. This graphical method applies to all Latitudes, to all Declinations between 25 N and 25 S, and to all Hour- Angles from 2 h to io b , E or W of the Meridian. 1 The Mean of two quantities is most readily found by subtracting their Half-Differ- ence from the greater, or adding it to the smaller quantity. Most commonly, the Mean of the two adjacent quantities in this Table may be taken at sight. 2 The Mean from each cross-pair of the four adjacent quantities in this Table is the same, or nearly so, and either may be used ; but, in general, it will be seen that the quantities of one cross-pair in each case are very nearly equal, and, accordingly, the Mean of this pair will be taken in preference. METHOD OF TIME-AZIMUTHS. 21 Examples of True Azimuths by the Azimuth- Tables (the data from the Examples of Art. 23). Enter with Ex. 1. Lat. 20.8 N,Dec. 2.; S, and H. A. i' 1 2i m W : Required, the True Az. P. D. Q2 .5 C. L. 69 .o Diff. 23 .5 Yz Sum 80 .7 Log A 9.315 Log B 0.785 Tab. XXX Log C 0.748 Log C 0.748 XXX. A Tan X 0.063 Tan Y i. 533 X 49-2 Y 88^.3 TRUE Az. N i37.5 W XXX. B Ex. 2. Lat. 40.5 N, Dec. 2i.8 N, and H. A. 6 h 35 m -3 E : Required, the True Az. P. D. 68.o C. L. 49 .5 Diff. 1 8 .5 y 2 Sum 58 .7 9.274 9-933 0.280 9-933 9.207 0.213 X 9. i Y 58.5 TRUE Az. N 6j.6 E Ex. 3. Lat. i2.3 S, Dec. 22.9 S, and H. A. 3 h 39 m .9 E : Required, the True Az. P. D. 67.o C. L. 77 .5 Diff. 10 .5 )^ Sum 72 .2 0.514 0.284 9.266 io.5 0.798 sr.o TRUE Az. S 7o.5 E Ex. 4. Lat. 6.2 N, Dec. 230.3 S, and H. A. 5 h 10 W: Required, the True Az. P. D. ii3.5 C. L. 84.0 Diff. 29 .5 Y 2 Sum 98 .7 9.410 0.096 0.803 0.096 9-506 0.899 X i7.8 Y 82.8 TRUE Az. Nii5.o\V Ex. 5. Lat. 23.4 N, Dec. 23. 4 N, and H. A. 6 h io m W : Required, the True Az. P. D. 66-5 C. L. 66 .5 Diff. o .o y 2 Sum 66 .5 oo 0.399 9-98i oo 0.380 X o.o Y 6;.4 TRUE Az. N 67.4 W Ex. 6. Lat. 2i4 N, Dec. 2i-4 S, and H. A. 4 h 30 W : Required, the True Az. P. D. iu.5 C. L. 68 .5 Diff. 43 .o y 2 Sum 90 .o 9.564 0.175 -f- oo 9-739 +<* X 28. 7 Y 90.o TRUE Az. N u8.7\V Ex. 7. Lat. i2.3 N, Dec. ii.7 N, and H. A. 5 11 5 m W : Required, the True Az. P. D. 78.s C. L. 77 .5 Diff. i .o y 2 Sum 78 .o 7-95 0.105 0.682 0.105 8.055 0.787 X o.7 Y 80. 7 TRUE Az. N 8i4 W Ex. 8. Lat. 72.2 N, Dec. 29.$ N, and H. A. 9 h 15 W : Required, the True Az. P. D. 6o. 5 C. L. 18 .o Diff. 42 .5 */2 Sum 39 .2 9.758 0.080 9-576 9-334 9-656 X I2.2 Y 24.3 TRUE Az. N 36-5 W 22 FINDING THE COMPASS-ERROR. Ex. 9. Lat. 30. i N, Dec. 9. 3 S, and H. A. o h i3 m -3 E : Required, the True Az. .P. D. 99.5 C. L. 60 .o Diff. 39 .5 Yi Sum 79 .7 1-537 I-Q73 0.723 1-537 2.260 X 85.2 Y 8 9 .7 TRUE A z. N i74-9E Ex. 1O. Lat. 12^.3 N, Dec. ii.7 S, and H. A. 5 h 30 E^: Required, the True Az. P. D. ioi.5 C. L. 77 .5 Diff. 24 .o Yz Sum 89 .5 9.318 0.057 2.050 0.057 9-375 2.107 X i3. 3 . Y 8 9 .6 TRUE Az. N io2.9 E Ex. 1 1. Lat. 7o.8 N, Doc. 2o.6 N, and II. A. u h 40 E : Required, the True Az. P. D. 6 9 .5 C. L. 19 .o Diff. 50 .5 YL Sum 44 .2 9.786 8.640 8.426 X i.s Y TRUE Az. N 4.6 E O.IOO 8.640 8.740 Ex. 12. Lat. i.o S, Dec. 32. i N, and H. A. 4 h 35 m .8 W : Required, the True Az. P. D. I22.o C. L. 89 .o Diff. 33 .o Yz Sum 105 .5 9-469 0.555 0.163 0.163 9.632 0.718 X 230.2 Y 79-2 TRUE Az. S i24.o W 28. Remark I: Oil the Two Cases of the Time- Azimiith. In obtaining the True Azimutli by tbis method, tbe first thing to be noticed, after getting tbe Angles X and Y, is tbe relation of tbe Half-Sum of Pol. Dist. and Co-Lat. to 90. If tbe Half-Sum be less than 90, tbe Example falls under Case I ; if greater, under Case II. Thus, of tbe several preceding Examples, Nos. i, 2, 3, 5, 7, 8, 9, 10, and ii, fall under Case I; Nos. 4 and 12 under Case II ; while No. 6, having tbe Half-Sum equal to 90. falls ^indifferently into either Case. And the next thing after deciding the Case : If the Example fall under Case I, it is sufficient to observe whether tbe Pol. Dist. be greater or less than the Co-Lat, and to take, correspondingly, the Sum or the Differ- ence of the Angles X and Y as the True Azimuth ; or, if the Example fall under Case II, it is only necessary to take the Difference of the Angles X and Y and use its Supplement (i. e. subtracting the Difference from i8o.o) as the True Azimuth. Tbe simplicity of these precepts in tbe Rule of tbe Time-Azimuth must be evident, if it be kept in mind that they are entirely general $ that is to say, are applicable to the findingjof tbe True Azimuth of any celestial object, whatever its Declination, injany ^Latitude, and at any hour of the day or night. As an aid to the memory, whenever necessary to recall these precepts (given at the foot of Tab. XXX. B, as well as in the text), tbe following consideration may be useful: Observing that, while " Pol. Dist." is the distance of the object from tbe elevated pole, Co-L:vt. is also'tbe Polar Dis- tance of the observer, tbe significance of the fundamental distinction into Two Cases is found in this : that, accord ing]as the Half-Sum orJVteau of METHOD OF TIME-AZIMUTHS. the Polar Distances of the observer and object is less or greater than 90 (that is, less or greater than the Polar Distance of the Equator), the prob- lem falls under Case I or Case II. The subordinate distinction into two varieties of Case I is obviously suggestive $ while the Difference and Sup- plement, or Double Difference, is hardly less so, when reaching beyond the Equator of Case II. 39. Remark II : On the Conditions introducing the Use of the Symbols co and +00. There are two conditions of the Lat. and Dec., which bring into view certain relations of the symbols co and + co , as found in the Azimuth-Tables, with respect to which a brief explanation may be of service to some one, possibly, who may use these Tables. a) The Dec. equal to the Lat. This condition is one very liable to occur, alike with the Dec. of the same and the contrary name. First, when the Dec. has the same name as the Lat. : Then, whether both these Data be entered in Tab. XXX to the w^hole or to the half degree, 1 the Log A is always co ; and as the Log C cannot affect it, 2 the Ang. X for co is always o.o, as in Ex. 5. Secondly, when the Dec. has the contrary name : Then, whether both these Data be entered in Tab. XXX to the whole or half degree, 3 the Log JB is always +00 ; and as the Log C cannot affect it, 4 the Ang. Y for + oo is always 90, as in Ex. 6. &) The Dec. nearly equal to the tat. This is a condition, both when the Dec. has the same and when the contrary name, which is of course more frequently liable to occur than the first. First, when the Dec. has the same name as the Lat. : Then, the smaller limit in Tab. XXX of the Log A is oo , and the adjacent or larger limit (varying from 7,941 to 8.025 J H different parts of the Table) ahvays differs about 0.300 from each of its own outward adjacents. Accordingly, if both the Lat. and Dec. be entered to the half-degree, the Cross-Mean is still to be taken, as in Ex. 7 ; but, if one be entered to the whole and the other to the half degree, the proportion is made to- wards the oo , by always subtracting 0.150 from the adjacent and larger limit f as, for example, if the Lat. were i2.o and Dec. n.5, we should take 7.800 as the Log A. Secondly, when the Dec. has the contrary name : Then, the larger limit of the Log B is +00 , and the adjacent or smaller limit (varying from 2.059 to 1.975 in different parts of the Table) always differs about 0.300 from each of its outward adjacents. Accordingly, if loth the Lat. and Dec. be entered to the half-degree, the Cross-Mean is to be taken, as in Ex. 10 ; but if one be entered to the whole and the other to the 1 Since the l / 2 Diff.'of the Pol. Dist. and Co-Lat. being o.o, the Log sine is . oo . 2 Except in the rare instance of having the Hour-Angle equal to o h .o, when the Ang. X is indeterminate. 3 Since the ^ Sum of the Pol. Dist. and Co-Lat. being 90, the Log secant is + oo . 4 In any instance that can occur in practice. 6 That is, by subtracting the Half Adjacent Diff. from the greater limit, which, although only approximate, always gives a result within about 0^.3. 24 FINDING THE COMPASS-ERROR. half degree, the proportion is made towards the +00 , by always adding 0.150 to the adjacent and smaller limit; 1 as, for example, if the Lat. were i2.o and Dec. n.5, we should take 2.200 as the Log B. Although the preceding explanation of these relations may seem at first view to imply it, there need be really no occasion whatever for em- barrassment in the treatment of these symbols whenever encountered; only remembering that o represents an extremely small quantity or nothing, and that -f oo represents an extremely large quantity or infinity. 3O. Examples of Finding the Compass-Error by a Time-Azimuth. The Examples already given (Arts. 21, 23, 27) fully illustrate the process of getting a True Azimuth by this method, under a considerable variety of circumstances. It will be sufficient to present a few more, merely to exhibit the entire form of procedure in finding the Compass-Error by. a Time-Azimuth. Ex. 1. 1871, August 16: At auclior, Disco Island, in Lat. 69 14' N, Long. 53 18' W, at 4 h 2i m 46 s P. M., Ship A. T., ob- served bearing of O*s centre was N 56 W : Required, the Compass-Error. Greenwich date, August i6 d 8 h 's Dec. 1 3. 7 N : 's H. A. 4* 2i m .8 W Pol. Dist. 763 Co-Lat. 20 .8 0.128 9.792 0.192 9.984 0.192 0.320 X 44-o Y 6 4 .4 True Az. N io8.4 W Comp. Az. N 56 .o W COMP. EK. 52 .4 W Ex. 2. 1875, February 23 : At sea, in Lat. 60 53' N, Long. 174 15' W, at i h 5 m A. M., Ship A. T., observed bearing of Jupiter was S 49 E : Required, the Coin- pass-Error and Deviation. Greenwich date, February 23 d o h .7 ZTsDec. io.7S It's H. A. 2 h 29 m .o E (Ex. 2, Art. 21) Pol. Dist. 100. 7 Co-Lat. 29 .1 9.811 0.472 0.283 X 62. 4 True A/. Coinp. Az. COMP. ER. Mag. Var. COMP. DEV. 0.280 0.472 0.752 Y 8o.o. N I424 E N 131 .o E ii .4 E 19 .3 E (Tdb.LVI) 7 . 9 W Ex. 3. 1875, April 25 : At sea, in Lat. 50 37' N, Long 155 20' W, at o h i5 m 3o 3 A. M., Ship M. T., the observed bearing of d's centre was S 66.5 E: Required, the Compass-Error. Greenwich date, April 24 d 22 b 37 m d 's Dec. S 28. 3 d 's H. A. 3 h i6\9 E (Ex. 3, Art. 21) Pol. Dist. ii8.5 Co-Lat. 39 .5 9.812 0-339 0.151 X 54-7 True Az. Comp. Az. COMP. ER. 0607 Q-339 0.946 Y 830.5 N I38.2 E Nii3-5 E 24.7 E Ex. 4. 1875, July 17 : At sea, in Lat. 52 38' N, Long. 33 47' W, at i h 37 2O 8 A. M., Ship A. T., observed bearing of <[ 's centre was S 66.o W : Required, the Com- pass-Error. Greenwich date, July i6 d I3 h 37'" <[ 's Dec. S 28-3 <['s H. A. 2 1 ' 44 m .9 W (Ex. I, Art. 21) Pol. Dist. ii8 c .5 Co-Lat. 52 .5 9.738 1.029 0.425 0.425 0.163 55-5 M54 Y B8.o True Az. N I43.5 W Comp. Az. N 114 .o W COMP. ER. 29 .5 W ir lhat is, by adding the Half Adjacent Diff. to the smaller limit, which, although only approximately correct, always gives a result within about o.3. METHOD OF TIME-AZIMUTHS. 25 31. Finding 0111 pass-Errors for a Scries of Differ- ent Headings of" the Ship. It is occasionally desirable (Chap. IV) to determine the Compass-Errors for a series of different Headings of the Ship. For this, the Method of Time- Azimuths is peculiarly con- venient and reliable, whether at sea or near the laud. For these observations, the Sun is the object commonly employed. Other objects, such as the Moon, the brighter Planets, and the brighter Stars, might be used, sometimes, with advantage. Generally, the opportunity may be chosen at will ; and the considerations controlling the choice of an object should have reference to the slowness of its change of Declination, the ease and certainty with which it may be sighted, and the convenience with which the auxiliary observations may be made. In all cases, the object should be observed only while it is low in altitude never, at least if it can be avoided, when the H. A. is less than 4 h ; for the precision with which serial Compass-Errors should be had, in order to serve any really useful purpose, requires that those con- ditions be satisfied. To make the Observations of Serial Time-Azi- muths. With a line day and a smooth sea, and with the means of placing the ship upon her different Headings by steaming or sailing, and whenever necessary, near the land, by towing or warping, the requisite observations may be conducted as follows: The ship being steady, take a bearing of the Sun's centre with the Standard Compass, noting the time with a watch, whose error on Ship T., Apparent or Mean, is known within allowable limits (G). Note the Heading of the ship with the same compass; also the corresponding Headings with the Steering-Compasses, and the Angle of Heel with the clinometer (10). After changing the Heading to any desired extent, and again steady- ing, repeat the' observation in all particulars; and so proceed, changing the Heading and repeating the observation, till the desired number shall have been made, completely or partially, round the Compass Circle. Remark Is Position of the Object, The period should be so chosen that the Hour- Angles of the object shall not, in general, be less than 4 h , E or W of the Meridian. In no case should an H. A. be less than 3 h , E or W of the Meridian, except in cases of urgent necessity, as when threatened with thick weather. Remark II : Steadying the Ship. While the observations should be made with the least practicable delay, no flying bearings should be taken ; but the ship should be steadied upon the recognized Heading, not only during the time of making each observation, but for a minute at the least before commencing it. 1 j See Cliap. IF. There must be time not only for the changing induction to be accom- plished, but also for the vibrations of the compass-card to cease, should there be any, if we would aim at getting really satisfactory results. 26 FINDING THE COMPASS-ERROR. Remark III: Heading-Intervals. It is always preferable to make the observations with the Headings upon the regular points by compass, thirty-two, sixteen, or eight, whenever practicable; but, when the circumstances make this difficult or time-consuming, the results will be equally available if obtained on any Headings distributed with approximate equality round the whole circle, or similarly through any part of it, when those on a certain part only are desired or can be had. Remark IT: Care in Sighting and Reading oil". Strict attention should be given to Sighting and Beading off. It should be seen that the Standard Compass is in nice equilibrium. With a well- made and properly-adjusted Compass, and otherwise favorable circum- stances, it should be practicable to sight and read off to the nearest quarter of a degree. 33. Rule: To get the Serial True Azimuths by Com- putation. First, prepare the Data, as follows : a) Find the Greenwich date for the Middle Ship T. of the set of obser- vations, either by applying the Longitude or deducing it from the chro- nometer. ft) With this date take out the Sun's Declination ; also, the Equation of Time, whenever necessary to convert Ship M. T. into Ship A. T. c) Deduce the Co-Lat. and Mid. Pol. Dist. each to the nearest minute for observations in port or near the land, or when the greatest precision is desired ; otherwise, to the nearest tenth of the degree, which is gen- erally sufficient for observations at sea. d) Prepare the Sun's Hour- Angles. For this, convert the W. Times of observation into Ap. Times; then, if the observations be P. M., the Ap. Times are the Sun's Hour- Angles W; if the observations be A. M., the Ap. Times subtracted from i2 h are the Sun's Hour- Angles E. Next, proceed with the computations, as follows : e) Take the Half-Difference and Half-Sum of the Mid. Pol. Dist. and Co-Lat.; then, using Tab. IX, To the sine of the }4 D. add the cosecant of the y S. ; the result, reject- ing 10 from the index, will be the Log A. Similarly, to the cosine of the y?, D. add the secant of the ^ S. 5 the result, rejecting 10 from the index, will be the Log B. /) Now, proceed in tabular form, by ruling several vertical columns, and, dividing each Hour-Angle by 2, place the series of y 2 Hour-Angles, in their order, in Col. I. Take the cotangent of each y 2 H. A. and place it opposite in Col. II. Add Log A to each cotangent and place the result- ing Tangents X in Col. Ill ; also, do the same with Log B, and place the resulting Tangents Y in Col. IV, rejecting 10 from the index in each case. Then, take out for each tangent in Col. Ill the corresponding Angles X, and place them in Col. Y ; also, from the tangents in Col. IV the cor- responding Angles Y, which place in Col. VI. METHOD OF TIME-AZIMUTHS. 27 There will now be two Cases to consider, according as the y 2 S. of the P. D. and C. L. is less or greater than 90. g) First Case: H.S.of P..andC.li.Ies$than!&O. Take the Sum or Difference of the corresponding Angles X and Y, according as the IP. D. is greater or less than the C. L., and place the resulting True. Azimuths in Col. VII ; or, in the h) Second Case : H. S. of P. D. and C. JL. greater than S>0. Always take the Difference of the corresponding Angles X and Y, which subtract from 180, and place the resulting True Azimuths in CoL VII. 1} In either Case, mark the True Azimuth N or S according to the Latitude, and E or W according as the observations are East or West of the Meridian. Remark* It may sometimes be convenient to use the Supple- ments of the True Azimuths (22, e). 34. To get the Serial True Azimuths t>y the Azimuth- Tables. The preceding Kule for computing the True Azimuths of a Series, for precision and facility, leaves little to be desired in an opera- tion that is only occasionally encountered by the Navigator ; neverthe- less, they may be found with the requisite precision and greater facility by the Azimuth -Tables (XXX, XXX. A, XXX. B). The Data having been prepared (33 o-d), proceed as follows : a) Enter Tab. XXX with the Middle Dec. and Lat., or with the Mid. Pol. Dist. and Co-Lat., both to the nearest tenth of the degree, and take out Log A and Log B, proportioning for the tenths of the degree ; otherwise, compute Log A and Log B (33 e), as may be deemed most convenient. 6) Now, proceed in tabular form, by ruling several vertical columns and placing the series of Hour- Angles, in their order, in Col. I. Then, entering Tab. XXX. A, with each H. A., in succession, take out at sight the corresponding Log C, and place it in Col. II, opposite to the H. A. to which it belongs. Next, add Log A to each Log C and place the Sum 1 opposite in Col. Ill; also, do the same with Log B, and place the cor- responding Sums in Col. IV. Finally, enter Tab. XXX. B with each Sum in Col. Ill and take out the corresponding Aug. X, which place in Col. V; also, enter the same Table with each Sum in Col. IV and take out the corresponding Arig. Y, which place in Col. VI. Finally, there will be the two Cases to consider, according as the Half- Sum of Pol. Dist. and Co-Lat. is less or greater than 90. c) First Case : H. S. of Pol. Dist. and Co-Lat. less than 90. Take the Sum or Difference of the corresponding Angles X and Y, according as 1 The Sums in Col. Ill being Tangents X and those in Col. IV being Tangents Y. 28 FINDING THE COMPASS-ERROR. the Pol. Dist. is greater or less than the Co-Lat, and place the resulting True Azimuths in Col. VII. d] Second Case : H. S. of Pol. Dist. and Co-Lat. greater than 90. Always take the Difference of the corresponding Angles X and Y, which subtract from 180, and place the resulting True Azimuths in Col. A T II. e) In either Case, mark the True Azimuths N or S according to the Lat., and E or W according as the observations are East or West of the Meridian. Remark* If it be desired to take the Mid. Dec. and Lat. to the nearest minute, it will be necessary to proceed according to Par. e of the preceding Rule (33), instead of Par. a of the present Kule, to find Log A and Log B; after which, proceed according to the present Rule, observ- ing that it will generally be sufficient to use Log A and Log B with the nearest third decimal figure, rejecting the fourth figure. 35. Examples or Finding Serial Compass^Errorg by the Iflethod of Time-Azimuths. Ex. 1. 1875, May 13: ID New York Bay, Lat. 40 41' N, Long. 74 2' W, about 5 h 45 m P. M., made the following observations of the Suu, on a careful swinging of the Ship, for Compass-Deviations. Watch-Error on Loc. M. T. 47 s fast; mean from comparisons before and after the observations. Mag. Variation -j.g W, from previous determinations. a) Tabular Record of Observations. Watcb Time. Standard Compass. Watch- Time. Standard Compass. Ship's Head. Q's Centre. Ship's Head. 's Centre. h m s 5 18 10 20 30 22 25 10 North NNE NE ENE N 72.0 W 64-5 57-0 5i-5 h m s 5 37 5 41 o 46 30 48 30 South ssw sw wsw N 70. 5 W 79-5 80.0 89-5 27 50 30 20 32 30 34 40 East ESE SE SSE N 49.0 W 4&S p.5 60.5 55 o 57 20 6 o 30 3 10 West WNW NW NNW 88.5 85.5 80.0 73-o b) Preparation of the Data. Mid. date A. T. May 13* 5 h .7 Pol. Dist. 71 32' Long. W -f- 4 .9 Co-Lat. 49 19 Greenwich date 13 10 .6 Diflf. 22 13 Y Z Diff. 1 1 6^ 's Dec. N 18 2i'.3| -f o'.6 ^ Sum 60 25 > Red n for io h .6 -f 6 .4 0's Dec. for date 18 28 N By Tab. IX 9.2848 0.0606 sn cosec cos sec 9.9918 _o-3o67 Log A 9.3454 Log B 0.2985 Eq. of T. 3 m 51*-!- to M. T. Error ol Watch on Loc. A. T. -f 3 METHOD OF TIME-AZIMUTHS. 29 c) Tabular Form of Solution by Azimuth- Tables. Sun's Hour-An- gles West. Table XIX. A. Log A 9.34514. LpgB Table XXX. B. Sun's True Azimuth. Swn's Compass- Azimuth. Compass- Error. Compasb- i)ev. TanX. Tan Y. Ang. X. Ang. T. h m o o o o o 5 21.2 0.074 9.419 0.372 14.7 67.02 N8i.7W N;2.oW 9-7 W i.8W 23.6 069 414 367 14. 55 66.77 81.3 64-5 16.8 8-9 25.1 066 411 364 14-45 66.62 81.1 57-0 24.1 16.2 28.2 060 405 358 14-3 66.32 80.6 5^-5 29.1 21.2 5 3-9 0-055 400 353 14 i 66.12 80.2 49-o 31.2 2 3-3 33-4 051 396 349 14.0 65.92 79-9 48.5 23-5 35-6 046 391 344 13-85 65.67 79-5 52-5 27.0 I9.I 37-7 042 387 340 13 7 6547 79-2 60.5 18.7 10.8 5 40.9 0.036 381 334 13-55 65.17 78-7 70-5 8.2 W 0.3 W 44.1 030 375 328 13-35 64.87 78.2 79-5 1.3 E 9.2 E 49-6 020 365 13-05 64.32 77-4 86.0 8.6 16.5 51.6 016 361 3H 12.95 64.12 77.1 89-5 12.4 20.3 5 58.1 0.004 349 302 12.6 63.52 76.1 88.5 12.4 2-0.3 6 0.4 9-999 344 297 12.45 63.22 75-7 85.5 9.8 17.7 3-6 993 338 291 12 3 62.92 75-2 80.0 4.8 E 12.7 6.2 3 88 333 286 12 17 62.67 74-8 73-o 1.8 W 6.1 E In taking out the Angles X and Y it is well, in a case like this, where considerable precision is desired and can be had, to allow for the in- fluence of the fourth figure (cut off) in Log A and Log B, and write the nearest second decimal, as here done, in the values of those Angles, retaining only the nearest tenth in the values of the Azimuths. Ex . 2. 1866, May 5 : At sea, in Lat. 50 10' N, Long. 13 10' W, about 6 h P. M., made the following observations of the Sun for Compass Errors and Deviations, watch being fast 26 m .o on Ship A. T. : Observations. 1 By Standard Compass. Watch- Tim e. Ship's Head. 's Centre. Ship Mid. A. T. May 5 d 6 h _5 Long.W + I .0 h m 6 15 N 116 E N 64 W Greenwich date 5 7 -5 22 40 72 28 66 75 3 3 S 88 E 6c 72 hT 0's Dec. N 160 16' 1 +o'. 7 5 7 Red* for 7 h . 5 -f C 4 . a 6 44 S 40 E 6 N 60 W Red. Dec. 16 J y 21 .3 56 S 20 W 43 7 o 10 7 IS- N 75 84 W N2 3 W P. D. 730.6 c Log A 9.541 { C. L. 39 .8* LogB 0.242 5 XXX 21 62 21 , JC 42 23 *- j 31 25 o 28 Mag Var. 26 W 38 4 36 1 The Data of this Example are taken from Traite de Deviation et de Regulation des Com- pas : par E. GIQUKL, Professenr d 'hydrographie, Paris, 1868. The observations are said to have been made on board the Ville de Paris f one of the ships of the New York and Havre Line. 30 FINDING THE COMPASS-ERROR. Reductions and Results. Sun's Ho. Ang. West. Tab. XXX.A. Log A 9.541. LogB 0.242. Tab. XXX. B. Sun's True Azimuth. Sun's Com pass- Azimuth. Comp. Error. Comp. Dev. TanX. TanT. Ang.X. Ang. T. h in o o o o 5 49 O.O2I 9.562 0.263 20.05 61.4 N 81.4 W N64 W 17.4 W 8.6 E 56 008 549 250 19-5 60.65 80. i 72 8.1 17.9 6 2 9.996 537 238 19.0 59-95 79.0 75 4.0 22.O 7 987 528 229 18.6 59-45 78.1 72 6.1 19.9 12 977 518 219 18.25 58.85 77.1 67 IO.I 15-9 6 18 9.966 9-507 0.208 17.8 58.2 76.0 60 16.0 10.0 25 953 494 195 17-3 57-45 74-7 5 1 23-7 2.3 E 30 943 484 185 16.95 56.8s 73-8 43 30.8 4.8 W 34 935 476 177 16-7 56.35 38 35- 1 Q I 44 916 457 158 16.0 55-2 71.2 28 43-2 17.2 6 49 9-9 06 9-447 0.148 15.6 S4-6 70.2 23 47.2 '21.2 55 5 436 137 15-3 53-9 69.2 21 48.1 : 22.1 59 8 428 129 15.0 53-4 i 68.4 23 45-4 19.4 7 5 87S 416 117 14.6 52.6 67.2 28 39-2 13.2 12 861 402 103 14.2 51-7 ! 65.9 36 29.9 3-9 W Ex. 3. 1875, Nov. 20 : At sea , in Lat. 3 58' N, Long. 14 3' W, about 6 h 45 in A. M., Ship A. T., weather threatening, made the following observations, on a partial swing, to cover probable courses of the Ship, during two or three days : Re'cord of Observations. Watch 2 m io s slow on Ship A. T. Stand. Compass. Ship Mid. A. T. Nov. 19* i8 h .8 Watcb- Time. Corrected A.T. Long, in W Greenwich date + o .9 19 19 .7 Ship's Head. Sun's Centre. h m s h m o O's Dec. S 19^.7 6 38 20 6 40.5 South. S 5i-5 E 44 48 55 50 IO 46.2 51.0 57-3 S 15 E 29 46 49-o 46.0 42-5 Pol. Dist. K>9.7< Log A 9.316 ) Tab. Co-Lat. 86 .0 ( Log B 0.855 ] XXX 59 7 1.2 60 38.0 Mag. Var. 190.3 W (Tab. LVI) Tabular Form by Azimuth-Tables. Sun's H. Angles Tab. 1 XXX.A. Log A Log B 9.310. O.H55. Tab. XXX. E. O's True Azimuth. Compass- Azimuth. Compass Error. Compass- l)cv. TanX. TanT. Ang.X Ang.T. h m 5 i9-5 0.077 9-393 -93 2 13.9 833 N uo C .6 E N i28.5 E I 7 ? 9 W 1.4 E 13-8 088 404 943 [4.2 83.5 110.7 131.0 20.3 i.o W 8.0 099 415 954 14.6 83-7 110.9 134.0 23.1 3.8 W 2-7 no 426 965 [4.9 83.8 1 1 1. 1 137.5 26.4 7.1 458-8 117 433 972 15.2 83-9 111.3 142.0 30-7 11.4 METHOD OF TIME-AZIMUTHS. 31 JEx. 4. 1875, July 16: At sea, in Lat. o 31' S, Long. 20 15' W, about n h 30 P. M., made the following observations of the Moon, on a partial swing of the Ship, for Com- pass-Deviations : Observations and Deduced Data. Standard Compass. R.A.M.0 7' 31 '".S R. A. Moon. Watch- Ship MJMIII Green- Time. Ship's Hed. Moon's Centre. Time. date. R. A. Mer. At G. I2h T.LXV Corr. Reduced R.A. Q h in s o h m h in h m h m in h m II 14 20 ssw Si6.oW II 2O.O 12 41 18 57.8 18 27.5 +1.6 18 29.1 17 o SbyW 17.0 22.7 44 19 0.5 27-5 1.7 29.2 19 30 South 18.5 25.2 46 3- 27-5 1.8 29-3 23 2O SbvE 21.5 29.0 5 6.8 27-5 1.9 29-4 27 o SSE 25-5 32.7 54 10.5 27-5 2.1 29.6 3 SEbyS 29.0 35-7 57 13-5 27-5 2.2 29-7 35 40 SE 33-o 41.4 13 2 19.2 29.8 O.I 29.9 W. Time slow 5. 7 on Ship M. T. Mid. M.T.July 16*11^.5 Long. W + i .3 Mid. Greenwich date 16 12 .8 R. A. Mean Sun 7 h 3 7. 8 Long, in i h 21 W (['s Dec. S 28 if Pol. Dist. 6i-7 ( Log A 9.398 ) Tab. Co-Lat. 89 .5 I Log B 0.588 5 XXX Mag. Var. I 9 .2 W (Tab.LVI) Tabular Form by Azimuth-Tables. Moon's Log A 9 39S Log B 5M8 Table XXX. B. Hour- Angles Table XXX. A. G 's True Azlmutn. Compass Azimuth. Compass Error. Compass- l)ev. West. TanX. TanT. Ang. X. Ang. Y. h m o 28.7 1.203 0.601 1.791 75-9 89.1 S i6.oW 2.8 W I6.4E 31-3 165 563 753 74-7 89.0 14-3 17.0 2-7 16.5 33-7 133 721 73-6 88.9 15-3 18.5 3-2 16.0 37-4 089 485 675 71.9 88.8 16.9 21-5 4.6 14.6 40.9. 048 446 636 70.3 88.7 18.4 25-5 7- 1 12. 1 43-8 1.018 416 606 69.0 88.6 19.6 29.0 9-4 9-8 . 49-3 0.967 365 555 66.6 88.4 21.8 33- II. 2 o.O In this Example for a partial swing, the Moon being taken when her change of Declination was comparatively slow, the results are as relia- ble as they would have been if obtained from observations of the Sun under similar circumstances. The additional labor of the working-out is trifling, as it only involves the six extra columns in the first Table? which are formed easily and rapidly. The Hour- Angles are very small. Such an operation, although performed at an unusual hour of the day, might be the means of anticipating serious embarrassment, espe- cially if threatened with thick weather, and actually deprived of obser- vations for two or three days. 32 FINDING THE COMPASS-ERROR. C. METHOD OF CIRCUMPOLAR AZIMUTHS. 36. The Circumpolar Azimuth a Modified Form of the Time-Azimuth. Circuinpolar Azimuths, as the name implies, are applicable solely to Circumpolar Stars ; that is, to those which never set in the horizon of the observer, but are always found above it. The method is a modified form of the Time- Azimuth, receiving certain special simplifications, which are the more conspicuous and practically useful in proportion as the Polar Distance of the Star is less, or in proportion to its nearness at all times to the Elevated Pole. This will be evident, since, as all the Fixed Stars require sensibly the same time (nearly 24 h ) in making their apparent revolutions about the Earth, or its axis, supposed indefi- nitely produced, it follows that the nearer they are to that axis, and the smaller the circle described by them, the slower their apparent daily motions; so that, when quite near the Pole, they appear almost entirely at rest, in comparison with those Stars which are much farther from it. Polaris, or the "Pole Star," is a notable example of this condition of things to observers in the Northern Hemisphere; for, its Polar Dis- tance being at present less than i^, its greatest change in Azimuth from east to west and from west to east, to an observer so far north as in the Latitude of 60, is less than 5.$ during I2 11 of time, while to an observer at the Equator it is even less than 3.o during the same time. So well is this apparent fixedness of the Pole-Star understood by Navigators, that not a few have regarded it as identical with the North Pole itself, and, in sighting it with their compasses, have deduced their Compass-Errors therefrom as if it were actually at the North Pole. But such an assumption may involve, though not necessarily so, a quarter to nearly half a Point of Error; that is to say. 3.o to 5.5, accord- ing to the time and place of observation within the range of Latitudes indicated. Now the motion, in Azimuth, of a Circumpolar Star, having a small Pol. Dist., is not only very slow, but really quite variable ; being slowest at or near its greatest Elongations, east or west, and quickest when on or near the Meridian, above or below the Pole. Hence, in assuming it to be practicable to get reliable Azimuths of such a Star with the compass, it is desirable to know when to observe, in order to find the Star at its greatest Elongation, which is the most favorable time, or, losing that, to have its True Azimuth at any intermediate time of observation. 37. Tahle or Circumpolar Azimuths : Description of Tah. XL< Tor Polaris. Such is the object of Tab. XL, which is adapted to Polaris, and comprises the True Azimuths of this Star, at intervals of every hour in Sid. Time, and for all Latitudes from the Equator to 60 north. Thus, for any stated Lat., the True Azimuth of the Star is given ' J^ METHOD OF CIRCUMPOLAR AZIMUTHS. First, at its greatest Western and Eastern Elongations; being set down in Cols. Ill and XV, adjacent to the corresponding Sid. Time. As, for example, in Lat. 30, at the greatest Western Elongation, the True Az. is K i.6 W at y 11 9 m -9 Sid. T., while at the greatest Eastern Elongation the True Az. is X i.6 E at iQ 11 i6 m .i Sid. T. Secondly, opposite to the Latitudes in the intermediate columns, IV-XIV, the Sid. Times of which being placed at top and bottom. The Sid. Times at top, commencing with the middle column at i h i3 m , when the Star is on the Meridian above the Pole, increase by hourly intervals, as the Star moves westwardly from the Meridian to its greatest W. E., when the Sid. Times are adjacent to the Latitudes; after which, during its return eastwardly towards the Meridian, the Sid. Times are found at bottom, till at 13* 13 the Star is again on the Meridian below the Pole. Then, as the Star continues its eastwardly movement below the Pole again moving from the Meridian, the Sid. Times are still found at bottom, till it arrives at its greatest E. E., when the Sid. Times are adjacent to the Latitudes; after which, during its return westwardly towards the Meridian, the Sid. Times are found at top till it finally arrives at the point of departure on the Meridian above the Pole, when the Sid. T. is again i h 13. This Table is computed for January i, 1875, but will not be materially at fault for at least ten years. 38. Two Distinct Problems in the Use of Tab. XL. Although the practical object in the use of Tab. XL, so far as the com- pass is concerned, is to get the True Azimuth of the Star, there are two distinct problems to be solved in connection therewith. First,wiih a tolerable Lat. and Ship-Time, having observed a Compass- Azimuth of Polaris and noted the Ship T. thereof, it is required to take out the corresponding True Azimuth of the Star, and thus to obtain the Error of the Compass; or, Secondly, with a very uncertain Lat. and Ship-Time, it is required to find the Ship T. of the Star's greatest Western or Eastern Elongation, whichever may occur during the night, to observe a Compass- Azimuth at that Time, and to take out the corresponding True Azimuth, in order to obtain a reliable Compass-Error, independently of the uncertainties of the Data. 39. Rule: To find the True Azimuth' of Polaris by Tab. XL. Thus, we shall have the following liule: a) To take out the True Azimuth for any Ship- Time. Find the Green- wich date for the Ship T. to the nearest tenth of the hour, in the usual manner ; then, according as the T. is A. or M., take out the K. A. of the True or Mean Sun from Tab. LIX, correcting it for the G. date by Tab. LX1V or LXVI, and add it to the Ship T. (rejecting 24^ if the sum be greater than 24^), and the result will be the Ship Sid. Time. 1 With this Time and the Lat. enter Tab. XL and take out the corresponding True Azimuth, as required. 1 Which is the same thing as the E. A. of the Meridian of the preceding Articles. 34 FINDING THE COMPASS-ERROR. b) To find the Ship- Time of the greatest W. or E. Elongation, and to take out the corresponding True Azimuth. Find the Greenwich date for the supposed Time at Ship. With this date take out from Tab. LIX the R. A. of the True or Mean San, according as the Time is A. or M. Then, enter Tab. XL with the supposed Lat., and subtract the R. A. before men- tioned from the corresponding Sid. T. (increased by 24 1 ' if necessary) of that greatest Elongation, Eastern or Western, which gives a remainder as Ship-Time appropriate to the night, and take out the corresponding True Azimuth. Also, make the Compass-Observation at that Time. c) Mark the Azimuth N according to the Lat. and E or W according as the Star is making its Eastern or Western Elongation. 4O. Examples of Finding Compass-Error by Circum* polar Azimuths. Ex. 1. 1875, March 15 : At sea, in Lat, 50 1 1' N, Long. 45 W, about 9 h 2o in P. M., Ship A. T., the observed bearing of Polaris was N I4.5 E : Required, the Compass- Error. Ship A. T. Mar. is d 9 b -3 Long. W +3-o Greenwich date 15 12 .3 Ship A. T. g h 2o m .o R. A. True 23 41 .7 Ship Sid. T. 9 i -7 True Az. N i.8 W (Tab. XL) Comp Az. N 14 .5 E COMP. ER. 16 .3 W Ex. 2. 1875, June 25: At sea, in Lat. 42 N, Long. 155 E, at n h 50 P. M v Ship M. T., the observed bearing of Polaris was N 27 W: Required, the Compass-Error and Deviation. ShipM. T.June 25** n h .8 Long. E - 10 .3 Greenwich date 25 1.5 ShipM. T., ii h 5o m .o R. A. M. 6 13 .2 Ship Sid. T. 18 3 .2 TrueAz. N i.8 E (Tab. XL) Comp. Az. N 27 .o W COMP. ER. 28 .8 E Mag. Var. 17 .5 E (Tab. LVI) COMP. DEV. 1 1 .3 E Ex. 3 1875, February 19: At sea, in thick weather, blowing a gale for two days, but now clearing to the northward, and Polaris visible, obtained an observa- tion for Compass-Error of N 11 W. Lat. 20 N \ Supposed, with possible Long. 30 W ) error of 5 each. About 7 h P. M., Ship A. T. Ship A. T. Feb. i9 d 7 h Long W -f- 2 Greenwich date 19 9 Ship A. T. R. A. True Ship Sid. T. True Az. N W Comp. Az. N ii .o W COMP. ER. 9 .7 E And this is true within* o.i, even with an error of 10 in Lat. and 5 in Long, of the Ship's place. Ex. 4. 1875, October 17: At sea, in Lat. 52 57' N, Long. 26 19' W, about 7" P. M., ship A. T. : Required, the Ship T. of the next greatest Elongation of Polaris. Ship A. T. Oct. I7 d 7 h .o Long. W -f i .8 Greenwich date 17 8 .8 Sid. T. W. E. 7 h 8 m R. A. True 13 29 Ship A. T. 17 39 Ex. 5. 1875, May 31 : At sea, in Lat. 49 15' N, Long. 43 50' W, about i2 h P. M. approaching the American coast, and anticipating thick weather, desired to get a set of Azimuths for Compass-Deviations, sufficient to cover the probable courses of the ship for the next day. Turning to Tab. LIX, the R. A. True Sun is seen to be 4 h 32 at Greenwich noon ; next, opening at Tab. XL and comparing this R. A. with the Sid. Times of the Elon- METHOD OF ALTITUDE-AZIMUTHS. 35 gatioiis of Polaris, it is seen that the next (Eastern) Elongation will take place at i4 b 47 m Ship A. T., or at 2 h 47 hence. Accordingly, preparations are immediately made to take the observations with the following results. Mid. Greenwich date, May 3i d I5 h .6 ; E. A. True Sun, 4 b 34 m .6. Ship A. T. Ships Head. Compass- Azimuth. Ship Sid. TJ A J2J | V 1 Compass Error. Mag. Var. Compass Deviation. h m 12 30 35 39 45 50 54 WXW WbyN West W by S WSW SW by W N 26.5 E 23.0 20.5 18.5 17.0 18.0 h m j o 17 35 N 1.9 E 40 1.9 44 i-9 5 i-9 55 2,0 59 2.0 24.6 W 21. 1 18.6 16.6 15.0 16.0 36.0 W 36.0 36.0 36.0 ' 36.0 36.0 i ME 14.9 17.4 19.4 21.0 2O.O D. METHOD OF ALTITUDE-AZIMUTHS. 41. Fundamental Principles of the Altitude- Azi- muth. In an Altitude- Azimuth the bearing of the Sun or other heavenly body is observed with the compass, at the same time that its Altitude is measured with a sextant, octant, or other reflecting instru- ment. The True Azimuth is found by a solution of the Triangle of Position. The Data required are The True Altitude, deduced by correcting the Observed Altitude; The Declination, taken from the Nautical Almanac or Naut. Tables; The Latitude in, as brought up by the .Reckoning or known from previous observation. 42. Rule : To make the Observation of an Altitude- Azimuth. Take a bearing of the object with the standard compass, or preferably a set of two or three bearings as quickly as possible; bisecting it each time, if it have a sensible disk (Sun or Moon), and noting the times with a watch ; also, simultaneously, with each bearing, take the Altitude of the body with a sextant or octant. Note the Heading of the ship with the same compass, and the corresponding Headings with the steering-compasses; also, if on board an iron ship, the Angle of Heel, and whether to starboard or port. The observed bearing, or the mean if several be taken, is the required Compass-Azimuth ; and the single Altitude, or the Mean if several be taken, is the observed Altitude. 43. Rule: To find the True Azimuth hy Computa- tion. First, prepare the Data, as follows: a) Find the Greenwich date for the Ship T. of observation, either by applying the Longitude to the Time, or deducing it from the chronome- ter. b) With this date, take out the Declination of the body, and in the case of the Moon the Semi-Diameter and Horizontal Parallax. c) Get the True Altitude of the centre by applying the proper cor- rections to the observed Altitude. 36 FINDING THE COMPASS-ERROR. Next, compute the True Azimuth, as follows : d) Add together the True Altitude, the Lat., and the Pol. Dist., and take the Difference between their Half-Sum and the Pol. Dist. Then, using: Tab. IX or Tables X and XI, add together the secant of the Lat., the secant of the Alt., the cosine of the ^ Sum, and the cosine of the Difference. The Half-Sum of these four logarithms, rejecting tens from the indices, will be the cosine of half the True Azimuth, which take out and double, and the result will be the required True Azimuth. Mark it N or S according to the Lat., and E or W according as the Altitude is increasing or decreasing, or as the object is East or West of the Meridian. 44. Rule: To find the True Azimuth by Aziiiiuth- Tables. The preceding Kule, by Computation, is that usually given. But the process of getting out a True Azimuth may be greatly abridged, with results sufficiently exact for practical purposes, by special Azimuth- Tables. Such is Table XLI, which has been extended and otherwise modified from that originally given by another hand. 1 It is adapted to all Lati- tudes from the Equator to 80 north or south; to all Declinations from o to 30 of the same or contrary name ; and to all Altitudes from o to 82. This Table consists of three parts, to use which, after preparing the Data as before described (43), proceed as follows : With the Lat. and Alt., both to the nearest. degree, enter Tab. XLI, Part I, and take out the corresponding tabular number. . With the Dec. to the nearest degree, and the Diff. of the Lat. and Alt., enter Part If, and take out the corresponding number. Then, with the Sum of these two tabular numbers, enter Part III, and take out the corresponding True Azimuth, which mark in the manner already described in the pre- ceding Rule. Remark I. The arguments of the Table being set down for every even second degree, it will be necessary to take Means of the tab- ular numbers in Parts I and II, adjacent or cross, according as one or both of the given arguments are odd. In general, the Azimuth will be obtained in this manner within a half- degree, or o.5, of that found by computation. But, whenever deemed expedient to take the Data more closely, it is easy to proportion for a more precise result. Remark II : Table of Direct and Limiting Value* of the Altitude-Azimuth. Table XLII will be found useful, in furnishing a convenient check against any serious error in getting out an Altitude- Azimuth, whether by computation or by Table XLI. 1 Azimuth Table ; to facilitate the process of finding the True Bearing of the Sun, etc., by A. .C. JOIIXSON T , R. N., London, 1867. This is contained in a small pamphlet by th#t author under the title How to find the Lvnyilude simultaneously with the Latitude at Noon, etc. There is also an extensive set of Tables for finding the Altitude-Azimuth directly, consisting of 365 pages, 4to, by THOMAS LYNN, London, 1829. METHOD OF ALTITUDE-AZIMUTHS. 37 45. Examples of Altitude- Azimuths. Ex. 1. 1868, August 5 : At sea, in Lat. By Azimuth-Tables: 23 47' N, Long. 110 15' W, at 4 h 6 m P. M., Lat. 27.3Alt. 22.5 (Tub. XLI) 43 Ship A. T., all by account, the observed Diff. 4 .8 Dec. 10 .4 884.6 bearing of the Sim's centre was S 85 W and TrueAz. Sii5.8W Sum 927.6 its True Alt. was 30 26^.4 : Required, the Comp. Az. S 105 .0 W Compass-Error. COMP. ER. 10 .8 E Ship T. 5 d 4 b .i Long, iu -f- 7 .3 Ex. 3. -1871, August 16: At anchor, Gr date 5114 < Disco Island > in Lat - 6 9 *4' N, Long. 53 ! 1 8' W. With the Data of Ex. i in Time- Q'sDec. N 1 6 51' | o'.7 Azimuths (29), and, in addition, the True Red n forii h 4 8 7.7 Altitude of the Sun's centre 21 24' : Re- Red. Dec. 16 43 .3 quired, the Compass-Error. Pol. Dist. 73 17 By Computation : By Computation : Pol. Dist. 76 17' Pol. Dist. 73i7 / Lat. in 69 14 sec 0.4503 Lat. in 23 47 sec 0.0385 True Alt. 21 24 sec 0.0310 True Alt. 30 26 sec 0.0644 Sum 166 55 Sum 127 30 y 2 s lim 83 27^ cos 9.0566 ^ Sum 63 45 cos 9.6457 Diff. 7 io)4 cos 9.9966 Diff. 9 32 cos 9.9940 J9-5345 19.7426 y 2 Az. 54 12 cos 9.7672 y z Az. 41 58 cos 9.8713 TrueAz.NioS 24 W True Az. N 83 56 W By Azimuth-Tables : By Azimuth-Tables : Lat. 69.2 Alt. 2i.4(Tab. XLI} 240 Lat. 24 Alt. 300.0 (Tab. X LI) 51 Diff. 47 .8 Dec. 13.7 668 Diff. 6 Dec. 17.0 772 TrueAz. N 108 .3 W Sum 908 True Az. N 83 .5 W Sum 823 Comp. Az. N 36 .0 W Comp. Az. N 95 .0 W COMP. ER. 52 .3 W COMP. ER. 1 1 .5 E Ex. 4. 1868, December 24: At sea, in Ex. 2. 1871, April 17: At sea, in Lat. Lat. 41 57' N, Long. 20 19' W, with a 27 19' S,Long. 75 52' E, the Sun's bear- Time-Sight of the Moon. ing by compass was N 75 W, at 3 h 51 P. M., Ship A. T., and the True Alt. at the same time was 22 29' : What was the Com- Compass-Bear. S 40 o' E ) At 4h p M True Altitude 37 22 > S hin M.' V pass-Error ? Dec. 7 37 3 ShipT. i7 d 3*1.8 By Computation : Long, in 5 .0 Pol. Dist. 82 23' Gr. date Apr. 16 22 .8 Lat. in 41 57 sec 0.1286 171 .2 True Alt. 37 22 sec 0.0998 O's Dec. N 10 25' | -f o'.9 Sum 161 42 Red n fori h .2 i i ^o X Sum 80 51 cos 9.2015 Red. Dec. 10 24 Diff. i 32 cos 9.9998 Pol. Dist. 100 24 I9-4297 By Computation: # Az ' 5 8 4 6 cos 9.7148 TrueAz.Nii7 32 E Pol. Dist. 100 24' Lat. in 27 19 sec 0.0514 ! By Inspection : . True Alt. 22 29 sec 0.0343 Lat. 42 Alt. 37.o (Tab. XLI) 113 Sum 150 12 Diff. 5 Dec. 7 .5 818 l /2 Sum 75 6 cos 9.4102 TrueAz. N 117 .3 E Sum 931 Diff. 25 18 cos 99562 Comp. Az. N 140.0 E T c\ A *j i COMP. ER. 22 .7 W 1 y-45 zl y 2 Az. 57 21 cos 9.7260 Tr. Az. S 115 42 W 38 FINDING THE COMPASS-ERROR. E. METHOD OF TIME- ALTITUDE-AZIMUTHS. 46. Fundamental Characteristics. The Time-Altitude- Azimuth has for its Data the Hour- Angle, Altitude, aud Declination ; and as these comprise a part of the Data of the Time and Altitude Azimuth, which have already been fully considered when treating the last-named methods, they need no further explanation in this place. The solution of the Triangle of Position for the Azimuth, with these Data, is at first sight more simple than for either the Time or Altitude Azimuth, inasmuch as it requires only the addition of three logarithms directly from the Data, the sum of which being the logarithm of the Azimuth. But, unfortunately, it has the very serious disadvantage of giving an ambiguous result, which, although removable in certain special cases, is not in general freed from the ambiguity, even with con- siderable complexity of cases. Moreover, as a method for finding Serial Compass-Errors, its use is attended with the further disadvantage that, as the H. A. and Alt. are both changeable from one observation to another of the series, there is really more labor required to get out a set of True Azimuths by this method than by either of the preceding. On the whole, therefore, if to what has already been said it be added that the Data for the solution of the Time- Azimuth are generally availa- ble whenever those for the Time- Alt. Azimuth are, there would seem to be little, if any, occasion to resort to the use of the latter method in preference to the former. Nevertheless, the following Kule for its solution is given, together with a few Examples to illustrate the application, so that it may be con- veniently in hand for any use that may be made of it. 47. Rule: To find the True Azimuth by Computa- tion. The Data being given or prepared by the preceding Rules, pro- ceed as follows : a) Using Tab. IX : To the sine of the H. A. add the secant of the Alt. and the cosine of the Dec. ; their Sum, rejecting 10 from the index, is the sine of the Angle Q, which take out. There will now be two general Cases, according as the Pol. Dist. is greater or less than 90. b) Case I : The Pol. Dist. greater than 90. Subtract the Aug. Q from 1 80, and the result will be the True Azimuth, always greater than 90. c) Case II : The Pol. Dist. less than 90. In this case there is gen- erally a doubt, whether the Aug. Q or its Supplement is to be taken as the True Azimuth, with the following exception : Whenever the Alt. is less than the Dec., the Aug. Q will be the True Azimuth, less than 90. ' Otherwise, whenever the Alt. is greater than the Dec. (a very com- mon condition), the result will be in doubt, aud reference must be made with the Data to Tab. XXXI, where it will generally be apparent, at sight, whether the True Azimuth is really less or greater than 90. If METHOD OF TIME-ALTITUDE-AZIMUTHS. 39 less, the Ang. Q is the True Azimuth ; if greater, the Supplement by subtracting from 180 is to be taken as the True Azimuth. In many instances it will be easy to decide during the observation, whether the object is on the side of the Prime Vertical towards the Elevated Pole, or the reverse. In the former condition, the Ang. Q will be taken as the True Azimuth ; in the latter, its Supplement will be taken, greater than 90. cl) In every case, mark the True Azimuth N or S according to the Lat., and E or W according as the H. A. is E or Wof the Meridian. 48. Examples of Finding the True Azimuth. Ex. 1. At sea, in Lat. 30 N, obtained the following Data in an observation of the Sun : H. A. West 4 h o T. Alt. I3.6 Dec. S 20 .o Required, the True Az. By Computation : H. A. 4 h o' Alt. i3.6 20 .0 sin 9.9375 sec 0.0124 cos 9.9730 sin 9.9229 Dec. Ang. Q 56 .9 or, TRUE Az. N 123. i W (Since this falls into Case I.) Ex. 2. At sea, in Lat. 9 N", obtained the following Data in an observation of the Sun : H. A. East 5 b 42 Alt. 7 31' Dec. N 22 o Required, the True Az. H. A. 5* 42 Alt. 7 3 1' Dec. 22.o Ang. Q 68 .8 or, TRUE Az. N 68.8 E (Since this falls into the exception of Case II.) sin 9.9987 sec 0.0037 cos 9.9672 sin 9.9696 Ex. 3. Given the Data in Lat 40. i N for the Son : H. A. West 5 h 6.8 Alt. 22 53' Dec. N 20 .3 Required, the True Az. H. A. 5 h 6 m .8 sin 9.9882 Alt. 22 53' sec 0.0356 Dec. 200 .3 cos 9.9722 Ang. Q 82 .2 sin 9.9960 This is doubtful ; but a reference to Tab. XXXI shows that Ang. Q is to be taken ; or, TRUE Az. N 82.2 W Ex. 4. With the same Lat. and Dec., and H. A. 3 h 56 m .6 Alt. 36 is' Required, the True Az. H. A. 3 b 56 m .6 sin 9.9337 Alt. 36 15' sec 0.0934 Dec. 2o.3 cos 9.9722 Ang. Q 86 .7 sin 9.9993 Here, again, we are in doubt, but, by Tab. XXXI, it is evident that we must use the Supplement of Q ; or, TRUE Az. N 93-3 W 49. Table of Time-Alt. Azimuths. Tab. XLVIII consists of two parts, namely, of a first part, of which the arguments are the Azimuth and Declination, and of a second part, of which the arguments are either the Hour-Angle and Altitude, or the Position-Angle and Lati- tude; that is to say, either pair of arguments at the same time. This Table, although 'having the title of Time-Alt. Azimuths, is not designed exclusively for that object, but to answer several purposes, as follows : 1. To find the Time- Alt. Azimuth, having the H. A., Alt, and Dec. ; 2. To find the Altitude, having the Azimuth, Dec., and H. A. ; 40 FINDING THE COMPASS-ERROR. 3. To find the Hour- Angle, having the Azimuth, Dec., and Alt.; and, 4. To find the Position-Angle, having the Azimuth, Dec., and Lati- tude. Whenever at least two of the Data, one of which for each part of this Table, correspond exactly or very nearly to the tabular arguments, a True Azimuth, or either of the other quantities, with the requisite Data, may be taken out at sight with sufficient accuracy for any ordi- nary requirement ; but, if the Data do not correspond, and interpolations are necessary, it will generally be easier to compute the Azimuth by the preceding Rule. For an off-hand, rough approximation, the Table is convenient for the several purposes for which it is designed. . Examples or the Use of Tali, XLVOI. Ex. 1. With the Data of Ex. i (48), what is the True Azimuth ? With H. A. 4 h o m and Alt. i3.6, we find for Log B 0.050 And with this and Dec. 20 in first part, we find for True Az. i23.o Ex. 2. With Az. 70, Dec. 10, and H. A. 2 h 20, what is the True Alt. f Entering first part, we fiud for Log A 0.020 With this and H. A. in second part, we find for- TrueAlt 53.] Ex. 3. With Az. 151, Dec. 20, and Alt. 60, what is the H. A. ? With Az. and Dec. in first part, we find for Log A 0.287 With this and Alt. in second part, we find for H. A. i h o m Ex. 4. With T. Az. 21, Dec. 17, and Lat. 42, what is the Pos. Angle ? With Az. and Dec. in first part, we find for Log A 0.426 With this and Lat. in second part, we find for Pos. Ang. i6.2 F. TRANSITION- AZIMUTHS. 51. Fundamental Principles. The method of Transition- Azimuths is based on the relative changes of the Hour- Angle and Alti- tude of a celestial object. The H. A. of an object, whatever its Position, changes uniformly ; but the changes of the Alt. are variable, depend- ing on the Position of the object with respect to the observer's Zenith. Accordingly, whenever two or more observations of an Altitude are made and the corresponding Hour-Angles noted, it will be easy to obtain a characteristic or distinguishing relation of these two elements, by deducing the change of Alt. in a unit of the H. A. Interval; and this definite relation, in a given case, will always correspond to a particular Azimuth of the object ; so that whenever we have obtained the former, together with the observer's Latitude, they constitute the requisite Data, by means of which we may readily find the latter. This method is rarely, if ever, resorted to as a direct means of getting) the True Azi- muth 5 but it may be used, often with advantage, in connection with METHOD OF TRANSITION- AZIMUTHS. 41 observations for the Time and Position near the Meridian, when the peculiar Data required for its use are sometimes obtained ; it being only necessary that a Compass-Azimuth of the object be taken as near the middle of the Time-Interval, between the two observations of Altitude, as practicable. With the True Azimuth and Compass- Azimuth of the object known, the Compass-Error is found by the usual comparison. *2. Preliminary Remark oil the Preparation of the Data. The Data being furnished to hand by the direct observation, from which the True Azimuth is merely an incidental determination, there is no necessity to enter into details as to its preparation. It will be sufficient to remark that, the Times of the observed Altitudes being noted with a watch whose error on Ship Ap. Time is known, the Diff. of the corrected W. Times will be the required H. A. Interval, when the Sun is the object observed. Moreover, as the W. Time Interval in these cases should hardly exceed a half-hour, the small Error due to the Difference between Ap. and M. Time (only seldom so much as o 8 .6 in that Interval) may be disregarded, and the W. T. Interval itself used as the H. A. Interval. With this con- dition, tliis method is entirely independent, even with a large uncer- tainty, 1 of the Ship-Time. For all other objects than the Sun, it will not be advisable to use the W. T. Interval for the proper H. A. Interval, as the difference between them, and the consequent Error, might be too large to be admitted ; but, in such cases, the H. A. Interval should be deduced by taking the Diff. of the Hour -Angles 2 found for the two noted W. Times of observation in the usual manner (20, b). 53. Rule : To get the True Azimuth. Having the Data prepared, the True Azimuth by this method may be found either by Logarithmic Computation or by Tabular Inspection. a] By Logarithmic Computation. Using Tab. IX : To the cosecant of the H. A. Interval (or W. Time Interval for the Sun) add the sine of the Alt. Diff. and the secant of the Lat.; their Sum, rejecting tens from their indices, will be the sine of the True Azimuth, which take out, and mark it N or S according as the object is north or south of the Prime Vertical, and E or W according as it is east or west of the Meridian. b) By Tabular Inspection. Divide the Alt. Diff. by the H. A. or W. T. Interval, as the case may be, the seconds of both being reduced to deci- mals of the minute, and the result will be the Change of Alt. in i m . With this, to the nearest tenth of the minute, and the Lat. enter Tab. LI, interpolating for the tenths, and the corresponding tabular number will be the True Azimuth, which mark as prescribed in the preceding Rule. 1 It is of course necessary that the rate of the watch should be sensibly uniform during the Interval. 3 These Hour- Angles would, however, generally be deduced among the Data for the direct use of the observations in question. 42 FINDING THE COMPASS-ERROR. 54. Examples of Finding the Compass-Error by a Transition-Azimuth. Ex. 1. At sea, in Lat. 36 N, with an I observation of the Sun east of the Merid. and south of the P. V., the Alt. Diff. was 5 7'. 5 and W. T. Interval 2o m 24". The Coinpass-Az. at the middle of the Time- Interval being S 2i.5 E, what is the Compass-Error ? By Computation : H. A. Int. 2o m 24* cosec 1.0511 Alt. Diff. 57'.5 sin 8.2236 Latitude 36.o sec 0.0920 True Az. S 13 .4 E sin 9.3667 Comp. Az. S 21 .5 E COMP. ER. 8 .1 E By Inspection : Dividing 57'. 5 by 2o m .4 we get 2'. 8, as the Change of Alt. in i m . Then, enter- ing Table LI, we find 9.5 as the next less Azimuth, with a Tab. Diff. of 4 .8, of which 0.8 part is 3.8 (=4.8 X 0.8) ; and hence, adding to 9.5, we get TRUE Az. S i3.3 E Ex. 2. At sea, in Lat. 51 35' N, an observation of the Sun, W and S, gave the Alt. Diff. 38'.6 for a W. T. Int. of i2 m 43 s . The Compass-Az. being S 3-5 E, what is the Compass-Error ? By Computation : H. A. Int. I2 m 43 8 cosec 1.2561 Alt. Diff. 38'.6 sin 8.0511 Latitude 51 35' sec 0.2066 True A z. S 190 .o W sin 9.5138 Comp. Az. S 3 .5 E COMP. ER. 22 .5 E By Inspection : Dividing 38'.6 by 12. 72 we get 3'.o to the nearest tenth of the minute. Then, entering Table LI, we find TRUE Az. S 18.; W G. DEPENDENCE TO BE PLACED ON THESE METHODS. 55. Possible Errors of the Azimuth-Data. The Data employed in finding the True Azimuth are always liable to be in Error, more or less, according to circumstances. Consequently, the resulting Azimuth, as obtained by either of the preceding methods, will be cor- respondingly uncertain ; and, to the end that'we may be able to form an intelligent opinion of the dependence to be placed on the results, it will be necessary to consider the several relations of the Data, as well as their possible Errors, under the variable conditions of experience. 56. Estimating the Data-Errors; their Ordinary Limits. The Data are uncertain by amounts which are different, not only among themselves but at diiferent times. What these uncertainties are, on a given occasion, can only be a matter of special estimate under the circumstances of the case. The following considerations may serve to fix our ideas on this subject : a] The Latitude-Error. On shore it is not difficult for tbe Navigator to obtain his Latitude within i', by suitable observations to which he is accustomed. At sea, when the Latitude is known from a recent observation, it is commonly reliable within* 2' to 5', at the most ; when found through the Reckoning, after a lapse of i2 h to i8 h from a preceding observation, it is frequently uncertain by 5' to 10', and when obtained, after a lapse of i8 h to 36 h without an observation, the uncertainty may be not less than 15' to 30'. In cases of thick and stormy weather, no reliable estimate can be formed as to the Limit of the Latitude- Err or. DEPENDENCE TO BE PLACED ON THESE METHODS. 43 b) Declination-Error. With a chronometer, which will allow the Greenwich date to be had within 30* or o m .5, the Declination, as taken from the Nautical Almanac, may be depended on within i" for the Sun, the Planets, and Fixed Stars ; 7" to 10" for the Moon. On the other hand, if the G. D. is obtained from the Longitude and Ship-Time, the Declination will be more uncertain. But, even then, when the G. I), is reliable within 2 m , or 30', of Longitude, the Declination may still be depended 011 within 2." to 3" for the Sun, the Planets, and Fixed Stars; 2 5" to 35" for the Moon. These are Limits for the extreme cases of greatest change in the Declination ; but, in proportion as the G. D. is more or less uncertain than the Limits above stated, the Declination will be more or less uncertain. c) The IIour : Angle Error. Since the Hour- Angle depends principally on the Local or Ship Time, it follows that the former cannot be expected to be more accurate than the latter. In reality, for the Sun, the Error of tbe Hour- Angle is identical with that of the Ship Apparent Time, to which it corresponds. For the Moon, the Planets, and Fixed Stars, the Error of the Hour- Angle, composed mainly of the Error in the Ship-Time, is also affected by the small Errors of the Eight Ascensions employed in getting the Hour- Angle (20, fc). But, for practical purposes, the influence of the latter, even in the case of the Moon, is unimportant in comparison to that of the Ship-Time; so that it will be sufficiently exact 1 to regard the Error of the Hour-Angle, for any one of these objects, as that of the Ship-Time to which it corresponds. Now the Local Time, on shore, may, without much difficulty, be found by the Navigator within 4*. At sea, it may be obtained, according to circumstances, by direct Observation, within io 8 to 15*; while, if deduced through the Reckoning, without a very recent observation, it may often be at fault by so much as i m or 2 m , or even more. d) The Altitude-Error. It is not difficult for the Navigator on shore to observe an Altitude within 10", but it is entirely otherwise when he is at sea. Then he is quite fortunate if, under the most favorable cir- cumstances, he can rely on his Altitude from a single observation within i / of arc. Ordinarily, it is doubtful if Altitudes can be relied on, even in the day-time, much within 2' ; while, on a rough sea, or during the night, they may be in Error from 4' to 6'. e) Recapitulation of the Limits of the Data-Errors. In recapitulation, we may take as ordinary Limits of the Errors of the Azimuth-Data, as follows :_, 1 For example, with a Ship T. Error of i m and a G. D. Error of i m , the greatest Error in the R. A. of the San would be o s .2, and the greatest Error in the R. A. of the Moon 3 S ; so that the several Errors and the probable total would stand V (6o 8 ) 3 -f (3 8 ) 2 + (o".2) 3 = V 3609.04 = 60*. i or only o s .i different from that of the Ship T. Error alone. 44 FINDING THE COMPASS-ERROR. On shore Lat. Error, within dLi' Dec. Error, within o'.o5 for all celestial objects H. A. Error, within i 4 s Alt. Error, within i 0^5 At sea Lat. Error, within 3' to 30' Dec. Error, within i' for all objects H. A. Error, within io 8 to i2 m Alt. Error, within 2' to i6' in which the signs indicate, what must always be understood, the uncertainty as to the Error being one of excess (+) or deficiency ( ). 57. Tables of Azimuth-Errors ; Auxiliary Tables. It may be expedient to estimate the effect of supposed or assumed Errors of the Data, sometimes before, but frequently after, computing or other- wise finding a True Azimuth. In either case, this may always be done accurately and quite simply, with the requisite Data, by logarithmic com- putation. But it may also be accomplished without any calculation, and with sufficient accuracy, by mere inspection, with the use of suitable Tables. Such is the object of the several Tables of Azimuth-Errors. They are specified in the following list : a) For Errors of Horizon-Azimuths. Tab. XXVIII for a Lat. Error of 12' or o. 2 . Tab. XXIX for a Dec. Error of 6' or o.i. b) For Errors of Time-Azimuths. Tab. XXX1Y for an H. A. Error of i ni of time or 15' of arc. Tab. XXXV for a Lat. Error of 12' or o.2. Tab. XXXVI for a Dec. Error of 6' or o.i. c) For Errors of Altitude- Azimuths. Tab. XLIV for an Alt. Error of 6' or o.i. Tab. XLV for a Lat. Error of 12' or o.2. Tab. XL VI for a Dec. Error of 6' or o.i. d) For Errors of Time- Alt. Azimuths. Tab. XLIX for an H. A. Error of o m .2 or 12". Tab. L for an Alt. or Dec. Error of 3' or o.o5. 4 In these Tables the arguments are not generally the same as the Data of the Azimuth, but these are wholly or partially replaced by other quantities, such as the Azimuth itself, the Position-Angle (Jwt), the Altitude in a Time-Azimuth, and the Hour-Angle in an Altitude-Azi- muth, etc. Hence the need of auxiliary Tables to furnish these quanti- DEPENDENCE TO BE PLACED ON THESE METHODS. 45 ties for the convenient use of the Tables of Azimuth-Errors. 1 Such are the following : Tab. XXV.- Position- Angles for Horizon- Azimuths. Tab. XXXII. Position-Angles for Time- Azimuths. Tab. XXXIII. Altitudes for Time- Azimuths. Tab. XLIII. Position -Angles for Altitude- Azimuths. With these Tables, and the aid of the two Tables XXXI and XLLI, of Direct and Limiting Azimuths, it will be easy, whenever desired, whether in advance of an observation or afterwards, to estimate the effect of a supposed or assumed Datum-Error upon the Azimuth. If the Data- Errors in a given case are the same as those for which the Tables of Azimuth-Errors are constructed, the Azimuth-Errors may be taken out at sight ; if they are not the same, it will only be necessary to take multiples or submultiples of the Tabular Errors, according as the assumed Data-Errors are larger or smaller than those of the Tables. Ordinarily? the Azimuth-Errors are given to the nearest tenth of the degree, which is sufficiently exact in practice. The result obtained in each special instance will be a Partial Azimuth- Error for the particular Datum-Error considered. S8. To find Partial Azimuth-Errors. In order to illus- trate the use of these Tables in finding Partial Az. Errors, for certain supposed or assumed Errors of the Data, we shall give in succession the Eules for taking out these Errors under each Azimuth-Method, following Avith Examples of application to the several Examples of Azimuths given under that method. a) Partial Errors of Horizon- Azimuths. 1. For the Lat. Error: Enter Tab. XXVIII with the Lat. and True Az., each to the nearest whole degree, and take out the corresponding Error. 2. For the Dec. Error: First enter Tab. XXV with the Lat. and Dec. and take out the Pos. Ang. roughly to the nearest degree ; then, with this, enter Tab. XXIX and take out the corresponding Error. 3. Remark : These Errors may obviously be taken out as well before as after the observation is made. As Examples of application, we may take those of Art. 14, which we shall place in tabular form, including the Azimuth-Data, the Auxiliary Data, the Assumed Data-Errors, and the resulting Partial Az. Errors. 1 The Tables of Errors in all cases might be constructed with the Azimuth-Data solely as arguments; but as each, with the exception of Horizon -Azimuths, would require three arguments, the Tables would need to be considerably extended, and even then be less convenient to use than these. 46 FINDING THE COMPASS-ERROR. Examples of the Partial Errors of Horizon-Azimuths. Az. Data. Aux. Data. Sup. Data-Errors. Par. Az. Errors. No. Prob. of Tot. A/. Ex. Lat. Dec. T. Az. Pos. Aug. Lat. Error. Dec. Error. For Lat. Error. For Dec. Error. Error. 1 11 N 23 N 7 o 79 o 0.2 o 0.1 o O.O 0.1 o 0.1 2 25 S 22 N 114 63 O.I 0.05 O.O O.O O.O 3 40 N 21 S 119 46 0.4 0.2 0.2 0.2 0.3 4 328 23 S 62 55 0.4 0.2 O.2 O.2 0.3 5 51 N 28 S 139 28 O.2 O.I o-3 O.2 0.4 6 69 N 14 N 4 8 16 O.I 0.05 -3 0.2 0.4 7 48 N ii S 106 4i O.2 O.O5 O.I O.O O.I b) Partial Errors of" Time-Azimuths. i. For the H. A. Error : First, enter Tab. XXXII with the H. A., Lat., and Dec., and take out the Pos. Ang. ; then, with the II. A., T. Az., and Pos. Aug., take out the Az. Error from Tab. XXXIV. 2. For the Lat. Error; First, enter Tab. XXXIII with the H. A., Dec., and T. Az., and take out the Alt. ; then, with the T. Az. and Alt., take out the Az. Error from Tab. XXXV. 3. For the Dec. Error : .First, enter Tab. XXXII with the H. A., Lat., and Dec., and take out the Pos. Ang. ; also, take out the Alt. from Tab. XXXIII, if not already found (2)5 then, ^vith the Pos. Ang. and Alt., take out the Az. Error from Tab. XXXVI. 4. Remark: If the T. Az. be unknown, as in the case of finding the Az. Error in advance of an observation, take it out approximately from Tab. XXXI, with the H. A., Lat, and Dec. Examples of the Partial Errors of Time- Azimuths. Azimuth Data. Auxiliary Data. Partial Az. Errors. Prob. No. of Total Ex. H. A. Lat. Dec. Az. Pos. Ang. Alt. For H. A. Err. 1 For Lat. Err. 12' For Dec. Err. tt' Az. Error. h m 6 o o o o 1 I 21 21 N 3 S 137 39 61 0.4 0.3 0. 0.5 2 6 35 41 N 22 N 68 49 8 0.2 O.O O. O.2 3 3 4 12 S 23 S 70 91 33 O.O O.I o. O. 4 5 I0 6 N 238 H5 65 8 0.1 0.0 0. O. 5 6 10 23 N 23 N 67 67 6 O.I O.O 0. O. 6 4 3 21 N 21 S 119 61 12 O.I O.O o. 0. 7 5 5 12 N 12 N 81 80 1 S 0.05 0.05 o. O. * 9 15 72 N 30 N 36 12 14 O.2 O.O O.O 0.2 9 o 13 30 N 9 S 175 4 5 0.4 0.0 0.0 0.4 1O 5 3 12 N 12 S 103 7 6 5 O.I O.O O.I O.I 11 u 40 71 N 21 N 5 2 0-3 O.O 0.0 -3 12 4 3 6 i S 32 N 124 56 22 0.2 O.I O.I -3 c) Partial Errors of Altitude- Azimuths. i. For the Alt. Error : First, enter Tab. XLIII with the Alt., Lat., and Dec., and take out the Pos. Ang. ; then, with the Alt. and Pos. Aug., take out the Az. Error from Tab. XLIV. DEPENDENCE TO BE PLACED ON THESE METHODS. 47 2. For the Lat. Error : First, enter Tab. XLYIII with the T. Az., Dec., and Alt. and take out the H. A.; then, with the Lat. and H. A., take out the Az. Error from Tab. XLV. 3. For the Dec. Error: First, take out the II. A., if not already taken out (2) $ then, with the Lat. and H. A., take out the Az. Error from Tab. XLYI. 4. Remark: If the T. Az. be unknown, as in the case of finding the Az. Error in advance of an observation, take it out approximately from Tab. XLII, with the Alt., Lat., and Dec. Examples of the Partial Errors of Altitude-Azimuths. Azimuth-Data. Auxiliary Data. Partial Az. Errors. Prob. Tolal Az. Ex. Alt. Lat. Dec. Az. Pos. Aug. H. A. For Alt. Err. 6' For Lat. Err. 12' For Dec. Err. 6' Error. o c o h o o o 1 3 24 N i;N 84 70 4-7 0.05 0.1 0.1 0.1 2 22 2 7 S 10 N 116 SS 3.6 O.I O.2 0.2 o-3 3 21 69 N 14 N 1 08 20 1.4 0-3 1.6 0.9 1.9 4 37 42 N 8N 117 45 2.8 O.2 -3 0.2 0.4 d) Partial Errors of Time-Alt. Azimuths. The Partial Errors are taken out in a similar manner from Tables XLIX and L for Time- Alt. Azimuths, and need not be explained nor illustrated by Examples. 59. Total Azimuth-Error. Since the signs of the Data- Errors are uncertain, it is obvious not only that the corresponding Par- tial Azimuth -Errors are uncertain in sign, but that the Total Azimuth- Error is uncertain in both amount and sign. Thus, the Partial Azi- muth-Errors may be all + or additive, for which the Total Azimuth- Error will be equal to their Sum and also additive ; they may be all - or subtractive, for which the Total Azimuth-Error will be equal to their Sum and subtractive; or they may be partly -f and partly , in which case the Total Azimuth -Error will be equal to the Difference between the Sum of the + and the Sum of the Partial Azimuth-Errors, and will take the sign of the greater Sum ; so that it is even quite possible for the -f- Errors to balance the Errors and leave no effective Total Azimuth-Error. Now, when there are several quantities of uncertain sign, whose effect- ive total it is desired to determine, we can only resort to the most prob- able estimate. This may be done, in the present case, in assuming the Errors to have equal weights, by the following Rule : The Probable Total of several Partial Errors is equal to the Square Root of the Sum of their Squares. Thus, Ex. 5 of Horizon-Azimuths gives the two Partial Azimuth- Errors o.3 and o.2. Then, by the rule, Probable Total Az. Error = V(io. 3 ) 2 + or 48 FINDING THE COMPASS-ERROR. And in this manner the several Probable Totals of the last column in each of the Tables are obtained ; that is to say, by squaring each Par- tial Azimuth-Error of an Example, taking the Sum of those squares, and extracting the Square Root of that Sum, taking the result to the nearest tenth of a degree. Of course, the sign of the Probable Total is also uncertain. All that we determine is this : not knowing from the signs of the Partial Errors whether we should take a Sum or Difference of these quantities, we sim- ply get the most probable mean opinion between extreme conclusions. GO. Favorable and Unfavorable Conditions. A glance at the Tables of Azimuth-Errors is sufficient to show how varied are the values of the Azimuth-Error with respect to the same Datum-Error. Thus, referring to Table XXXIV (Errors of the Time-Azimuth for an Error of i m in the Hour-Angle), it is seen that the Azimuth-Error is always comparatively small when the Azimuth itself approaches to o or 1 80, or when the Position-Angle is nearly 90, whether greater or less; that the Azimuth-Error is comparatively large when the Position- Angle approaches o or 180, or when the Azimuth itself is nearly 90, whether greater or less. Also, that, with the same Azimuth and Position-Angle, the Azimuth-Error is small so long as the Hour-Angle differs but little from 6 b ; but increases, first slowly, at length quite rapidly, as the Hour- Angle recedes from 6 h , becoming very large as the Hour- Angle approaches o 11 or i2 h . These illustrations show that there are certain relations between the several parts of the Triangle of Position (Int.) which are better than others for finding the Azimuth ; front which it may be said that the conditions of an Azimuth are favorable or unfavorable, according as it is affected to a less or greater degree by the same Errors of the Data. These conditions of the several Azimuth-Methods may be briefly stated in the following terms: a) Conditions of Horizon- Azimuths. In these First, for an Error of the Latitude, the conditions are most favorable when the Latitude is lowest and the Azimuth itself nearest 90; they are most unfavorable when the Latitude is highest and the Azimuth nearest o or 180. Secondly, for an Error of the Declination, the conditions are most favorable when the Position-Angle is nearest 90 ; they are most unfa- vorable when the Position- Angle is nearest o or 180. Otherwise, it may be said that the conditions are most favorable when the Latitudes are lowest and the Declinations least; they are most unfavorable when the Latitudes are highest and- the Declinations greatest. b) Conditions of Time-Azimuths. In these First, for an Error of the Hour-Angle, the conditions are most favor- able when the Hour-Angle and Position-Angle are both nearest 90, and the Azimuth itself nearest o or i8oj they are most unfavorable when DEPENDENCE TO BE PLACED ON THESE METHODS. 49 the Hour- Angle and Position- Angle are both nearest o or 180, and the Azimuth nearest 90. Secondly, for an Error of the Latitudes, the conditions are most favor- able when the Altitude of the object is least and the Azimuth itself nearest o or 180; they are most unfavorable when the Altitude is greatest and the Azimuth nearest 90. Thirdly ', for an Error of the Declination, the conditions are most favor- able when the Altitude is least and the Position-Angle nearest o or 1 80 ; they are most unfavorable when the Altitude is greatest and the Position- Angle nearest 90. c) Conditions of Altitude-Azimuths. In these First, for an Error of the Altitude, the conditions are most favorable when the Altitude is least and. the Posi tion- Angle nearest 90 ; they are most unfavorable when the Altitude is greatest and the Position- Angle nearest o or 180. Secondly, for an Error of either the Latitude or Declination, the con- ditions are most favorable when the Latitude is lowest and the Hour- Angle nearest 6 h ; they are most unfavorable when the Latitude is highest and the Hour- Angle nearest o h or i2 h . <7) Conditions of" Time-Alt. Azimuths. These conditions need not be particularized, as they are comprised in those stated under the two preceding heads, and may be inferred from a glance at Tables XLIX and L. 61. Limits of Allowable Azimuth-Error. It is com- monly regarded as "sufficient for the requirements of Navigation" if the Helmsman is able to steer within a quarter of the point, or 2.8 of the course set for him. Let this be conceded as a limit for steering ; then, it is also regarded by many Navigators as " sufficiently exact," in view of the Steersman's margin, to take out their True Azimuths, when- ever it can be done by Tables, to the nearest degree ; while it is also fre- quently admitted that little pretension is made, in the observation of their Compass- Azimuths, to get them more closely than to the nearest degree. Now, we have here three distinct sources of Error, each of quite an appreciable amount, which are allowed to occur on every occasion of shaping and steering a ship's course, without mentioning the Error from Defective Sensibility, which, although nothing for some compasses, may be several degrees for others. There are, then, in supposing the com- pass to be practically perfect, as follows : Pointing-Error io.5 in observing the Comp. Az. ; Azimuth-Error o.5 in taking out from Tables ;' Steering- Error =t i.4 in steering the ship's course ; giving a Probable Total Error of i i.6. It is true that these Errors might have opposite signs in some cases, when the total would be reduced to o,4; but they might with equal probability have the same signs, when the total would be increased to 2.4. 50 FINDING THE COMPASS-ERROR. But it must always be remembered that, as already exemplified in the preceding Articles, in consequence of the inevitable Errors of the Data, there is quite likely to be an Error of the True Azimuth, however care- fully it may be computed or taken out from the Tables. This may be quite small under very favorable conditions j but it may also be quite large under different circumstances. Hence, on the simple principle that, if " we cannot save everything, we should strive to save all we can," it would seem to be obvious that it is always advisable to avoid all unnecessary Errors, or to diminish them, at the least, whenever we may. Without in this place entering further on the question of the need to allow so large a margin for Steering-Error, or so large an Error of Observation, it may be remarked, with respect to the two sources of the True Azimuth Error, that it has already been shown to be easy enough, by the methods described, to take out the True Azimuth from the Tables, within o.2 to io.3j while, of the Examples given of Az. Errors due to the Errors of the Data (58, 59), it is seldom that the Prob. Tot. Azimuth-Error from this source, with the actual conditions, need exceed =L o.4. It is evident, therefore, that we should be able, in find- ing any Single Compass-Error, to depend on our True Azimuth, within V(o.3) 2 +(o .4) 2 = o.5 for all ordinary cases of requiring it at sea. On the other hand, in the operation of finding Serial Compass-Errors for Different Headings of the Ship, it is necessary and sufficient that for port or inshore observations our True Azimuth should be depended on within one-tenth of a degree, or dLo.i ; while for observations at sea we must be content to get the best we can ; in general, however, we may expect to get these within three-tenths of the degree, or i o.3. Accordingly, we shall take these as Requirement- Limits in those cases of nautical practice within which True Azimuths are to be employed in finding the Compass-Error $ that is to say : 1. For the Single Compass-Error, the greatest Error of Azimuth not to exceed =t o.5 ; 2. For Serial Compass-Errors, the greatest Error of Azimuth not to exceed rto.i in port, or o.3 at sea. In order to satisfy these requirements in practice, we must be assured, in the first place, that our Data-Errors (56) in any given case are cer- tainly within the limits assumed for them, whatever those may be ; and, secondly, that the conditions of the Azimuth give rise to Partial Azi- muth-Errors, as found by a reference to the proper Tables (57), whose Probable Total is within the assigned Eequirement-Limit. By these means we shall be able to make a reliable estimate of the dependence to be placed on our Compass-Observations, so far at least as finding the True Azimuth is concerned, as well in advance as after- wards ; so that, if the results cannot be depended on within the required limits, we may secure more favorable conditions, either by choosing a different object, or waiting for a favorable change ; or so that, at the least, we may take our succeeding steps with intelligence and certainty. DEPENDENCE TO BE PLACED ON THESE METHODS. 51 We shall next proceed, in further elucidation of this important sub- ject, with the aid of a series of Tables giving the Limiting-Errors of Azimuths for certain assumed limits of the Data-Errors, to show how we may safely estimate, in any given case, the dependence to be placed upon either of the several preceding methods of finding the True Azi- muth. 62. Dependence on Horizon-Azimuths : Use of* Tab. XXVI. Since an Horizon- Azimuth can only be resorted to for a Sin- gle Compass-Error, it is unnecessary to consider the second Bequirement- Limit. If, then, these Azimuths be restricted to the Sun, Moon, Planets, and those Stars whose Declinations never exceed about 30 north or south; and if we assume a Lat. Error within 12', or io .2, and a Dec. Error within =L6', or o.i, Tab. XXVI may be formed directly from Tables XXIV, XXV, XXVIII, and XXIX. This Table consists of four parts, corresponding to the Declinations of 12, 1 8, 24, and 30; the first and second columns of each division con- taining the greatest Par. Az. Errors, to the nearest superior tenth of the degree, for the assumed Lat. and Dec. Errors, answering to the Lati- tudes in the left-hand column, to which they are set opposite. The third column in each division contains the Probable Total Az. Errors ; each of which being found by taking tne Square Boot of the Sum of the Squares of the corresponding Par. Az. Errors. It is thus seen that, with the Data-Errors within the limits here stated, and a Dec. not greater than 12, the Prob. Tot. Error of an Horizon -Azimuth will not exceed o.2 for all Latitudes up to 60 N or S; not exceed o .6 for all Latitudes between 60 and 70; nor exceed i.o even up to 76 of Latitude. But, as the Dec. is greater, the corresponding Az. Errors range at higher limits; so that, for example, with a Dec. of 30, the Az. Error might be o .7 for a Lat. of5S- The small Table at the right is a similar one for all Declinations up to 24, in which the Lat. and Dec. Errors are each assumed to be o.5, or 30'. It shows the Prob. Tot. Error to which one is liable, when careless or indifferent about the Data, either in their preparation, or in taking out from the Azimuth-Tables. It is not an uncommon practice with Navigators to enter the Table of Horizon- Azimuths (or Amplitudes) with the Lat. and Dec. to the nearest whole degree, on the ground of greater ease in taking out the required quantity. If these Data could be supposed precisely correct, then, in using them to the nearest degree, we should involve at the most an Error of o.5 in each; and, accordingly, the small Table, before referred to, would give the corresponding greatest Prob. Tot. Errors to which such a practice would be liable. Thus, we might still depend on our True Azimuths within i.o for all Latitudes up to 50, BT or S. If, on the other hand, as is most certain to be the case, the Data are both in Error, the Lat. especially being sometimes uncertain by 12', 20', or even 30', we may involve the Lat. in Error, by at least a whole degree, in taking it to the nearest degree. Hence, the Par. Az. Errors in the 52 FINDING THE COMPASS-EKKOK. second column of the small Table might be doubled, and the Prob. Tot. Error of the True Az. for a Lat. of 50 might be i-4. 63. Dependence on Time-Azimuth* : Use of the Tables of Limiting-Errors. Regarding Time-Azimuths as available, alike in finding Single and Serial Compass-Errors, we shall consider the True Azimuths with respect to both Requirement-Limits. a) Single Time-Azimuth* within the First Require- ment-Limit. In order to furnish some practical illustrations of the Errors to which a Time- Azimuth may be liable, under different possible circumstances at sea, and thus establish certain Limits of Dependence, three distinct suppositions of Data-Errors will be made, as follows : 1. Ordinary or fair-weather sailing, but with some dependence on the D. R; 2. Cloudy, without regular or reliable observations for Time and Position, with increased dependence on the Reckoning ; 3. Thick weather, but ship kept nearly to her course. Evidently any assumptions of Data-Errors, under these supposed con- ditions, must be mainly arbitrary; but this will be of little consequence, with the use to be made of these TaJbles. Tables XXXVII and XXXVIII are formed in accordance with these suppositions. The former answers to all Latitudes from o to 60 N or S, and for all Declinations from o to 30 of the same or contrary name; while the latter answers to all Latitudes between 60 and 80, and for the same range of Declinations. The greatest Par. Az. Errors are given in each division of the Tables for the H. A. Limits set down in the first column. From this Table the following conclusions may be drawn : First. With the Data-Errors limited according to the first supposition, a Time-Azimuth of any object may be had during the day or night, without being liable to a greater Probable Error than i.o, whatever the Lat. up to 80, even when the least H. A. is no more than i h . As the limiting H. A. is greater, the Azimuth-Error becomes less, until it comes to be less than o.3, for all H. Angles greater than 7 h . Moreover, for all H. Angles in Polar Latitudes, and for all H. Angles greater than 3 h in all other Latitudes, the True Azimuth may be depended on within o'. S . Secondly. With the Data-Errors limited by the second supposition, the Prob. Tot. Azimuth-Error does not exceed io.7, for all Latitudes, pro- vided no H. A. smaller than 3 l1 is admitted. And in this case the greatest Prob. Azimuth-Error is less than io.5 for all H. Angles greater than 8 h . Thirdly. Even with the Data-Errors so large as the limits of the third supposition,' an uncertainty of the Data which should never exist except under severe stress of weather, the True Azimuth may still be depended on within o.S, provided the H. A. of the object is not allowed to be less than 5 1 '. DEPENDENCE TO BE PLACED ON THESE METHODS. 53 In reality, however, all the extreme conditions of Tables XXXVII and XXXVIII very seldom, if ever, concur at the same time ; on the contrary, it most commonly happens that the conditions are much more favorable, even with the same Errors of Data. Consequently, the Azi- muth-Errors for the same Data-Errors will generally be smaller than the maximum or limiting values of this Table. This is seen in the Examples of Time- Azimuths (23, 25), which repre- sent a considerable variety of actual conditions, and whose Errors are given in the Table of Art. 58, for assumed Data-Errors in accordance with the first supposition of Table XXXVII. Table XXXVIII illustrates the convenience with which Time-Azi- muths are adapted to the wants of the Navigator, even in the highest Latitudes. Thus, it is seen that the probable total effect upon the Azi- muth of quite large Data-Errors is not only within the limits of o.5, but is nearly uniform for all Hour- Angles in the entire circle, from o h to i2 h , E or W of the Meridian. b) Serial Azimuths within the Second Requirement- Limit. Unlike the ordinary case, in which the Navigator may desire to find his Compass-Error with every important change of course, and where he is compelled to accept the situation and the best available result, whatever the circumstances, in the case of getting a set of Serial Compass-Errors, he may generally choose his opportunity and thus com- mand the best obtainable conditions. When in port, or otherwise near the land, the ship's Position and Time are always known or may be obtained with considerable precision. Thus, the Navigator may generally find his Time and Latitude, if not otherwise available, the former within 4 8 and the latter within i', by his own observations on shore. At sea, moreover, in all Latitudes from the Equator to 60 north or south, with a proper choice of circumstances, suitable for Serial Azi- muths, the Navigator should be able to get his Time within i io s to is 8 and his Latitude within dL 3', by excluding all dependence on the Reckon- ing. Beyond the Latitude of 60, the Time will be found with less pre- cision, but the Latitude with somewhat greater certainty. Table XXXIX is formed with reference to these supposed circum- stances. The Declination is taken at a limit of 25 N or S, for the Sun and other zodiacal bodies ; and the greatest Prob. Azimuth-Errors are given in each division of the Table for the H. A. Limit, set down in the first column. From this Table, in assuming the Data-Errors to be kept within the limits here given, the following conclusions may be drawn : First, that, while in port or near the land, in all Latitudes, Serial Time- Azimuths may be taken with any Hour- Angle, not less than 2^ h from the Meridian, without being liable to a greater Probable Error than 3', or io.o5. Secondly, that, anywhere at sea, within the parallels of 80 N and 80 S, Serial Time-Azimuths may be taken within the same Limit of the 54 FINDING THE COMPASS-ERROR. Hoar- Angle, under ordinary circumstances, without being liable to a greater Probable Error than 7', or io.i2. Thirdly, that, anywhere at sea, Serial Time- Azimuths may be had within the same Limit of Hour-Angle, without being liable to a greater Probable Error than 16', or rt o.3, even when the Data are considerably uncertain, as, for example, with the Data-Errors assumed in the third division. Keeping in mind, however, that the Azimuth-Errors set down in these Tables correspond to extreme conditions, it will be seen that it is quite within the reach of the Navigator's appliances and of the condi- tions under his control to obtain a series of True Azimuths, by this method, sufficient for a complete circle of different Headings of the ship, in whatever part of the sea likely to be traversed by him, which may in general be depended on within two to three tenths of the degree. Nor need the Navigator confine his observations to the Sun. For the Planets and Fixed Stars may be observed with equal accuracy, so long as the Time and Latitude are known within the above-mentioned Limits, and the observations are made under similar restrictions as to the smallness of the Hour- Angle. 1 The Moon, with the exception of a little additional labor in prepar- ing her Hour- Angles (20, 21), may be used with a facility hardly inferior to that in using the Sun, if taken during the period of greatest Declination. And, at other times, even when her Dec. changes most rapidly, a short or partial series of Azimuths may be had from this object, when, for a period of half an hour, the Error of the Middle Dec. need not exceed 4^'; a condition, that should still give us results which may be depended on within o.3 or o.4, quite sufficient for an imme- diate stress of circumstances which should compel a resort to this object in the absence of one more favorably conditioned. 64. Dependence on Cireumpolar Azimuths. Table XL, while giving direct values of the True Azimuth of Polaris, also furnishes at sight the requisite indications of the Errors due to un- certainties of the Data. And, with respect to these Errors, this Table illustrates, in a striking manner, how nearly complete in its independ- ence, is this method of getting a True Azimuth, of all its Data. Not only is it sensibly free from any dependence on the Dec., but it is practically independent of the Lat. ; while, as to the Hour-Angle, represented by the Sid. Time, the Azimuth varies less than o.i during an hour of Time, when the Star is near one of its greatest Elongations. And, during other periods of the apparent daily revolution of this Star, its changes in Azimuth are still quite small in comparison to the changes in its H. A., or in the Time derived therefrom. 65. Dependence on Altitude-Azimuths: Use of the Tallies of Limiting-Errors. As already evident in what has 1 The principal practical objection to the use of these objects will bo found in the difficulty of sighting them with sufficient distinctness when near the Horizon, and the other attending inconveniences, especially of night-observations. DEPENDENCE TO BE PLACED ON THESE METHODS. 55 preceded, but will be further referred to in a succeeding Article, the computations, as well as the observations of Altitude- Azimuths, are less convenient than those of Time-Azimuths. Especially is this true of Serial Azimuths by this method. We shall also see, from the follow- ing considerations, that their Errors develop more rapidly under certain conditions than those of the latter. a) Single Azimuths within the First Requirement- Limit* In Table XLYII are given the greatest Azimuth-Errors rela- tive to certain assumed Data-Errors, for a series of different Latitudes, under three different limitations of the smallest Hour- Angle, but answer- ing also to the corresponding Supplements, or largest Hour-Angles. They are taken out for a Declination of 30, and therefore cover all Dec- linations of the zodiacal bodies, like the Sun, Moon, etc. Thus, the Data-Errors being within the limits of this Table so long as the Hour- Angle is not less than i h nor greater than n h , an Altitude- Azimuth of any object whose Declination does not exceed 30, in what- ever Latitude from the Equator to 40, may. be had with a certainty of no greater Probable Error than i i.2. Again, so long as the Hour- Angle is not less than 2 h nor greater than io h , an Altitude- Azimuth may be had in any Latitude up to 70 with no greater Probable Error than i i.2, and so on ; as the H. A. Limit is more remote from the Me- ridian, upper and lower, the conditions are improved, and the maximum Probable Error diminished. />) Serial Azimuths within the Second Requirement- Limit. Altitude- Azimuths are occasionally resorted to in serial obser- vations for Compass-Error 5 and it is accordingly advisable to consider the question of their liability to Error, and how far they can be depended on, when employed for this purpose. The subjoined little Table gives the Errors of Altitude- Azimuths under the same conditions as those of Table XLVII, except that the several Data-Errors are reduced to their minimum limits for favorable circum- stances at sea, and the Hour- Angle Limits of approach to the Meridian are made larger. Limiting Errors of Serial Altitude-Azimuths. Par. Azimuth-Error. For Limits of For Latitudes Prob. Total Hour -Angles. N or S. Alt. Lat. Dec. Az. Error Error Error Error. 1' 3' 1' h h o o ( / 3 to 9 o to 30 2 5 2 6 4 8 30 60 2 4 3 6 5 7 60 70 3 3 3 6 5 7 70 30 4 4 6 10 Thus, if the Data-Errors may be safely assumed within i i' for the Altitude, 3' for the Latitude, i' for the Declination, the correspond- ing Partial Azimuth-Errors will not exceed those set down, and the 56 FINDING THE COMPASS-ERROR. Prob. Tot. Error of the Azimuth will not be greater than 6' for all Latitudes from o to 70, provided the Hour- Angles are kept within the limits stated in the first column of the Table. For Latitudes beyond 70, the Total Errors will exceed 6', for an Hour- Angle/ even if pre- cisely 6 h . 66. Relative Advantages of these Methods. We shall now consider very briefly the relative advantages of these methods. a) Horizon- Azimuths. This method, in the well-known form of Ampli- tudes, has long been used by Navigators, generally, as a means of find- ing the Compass-Error. The facility with which the observation is made, and the convenience with which the Table is used, have alike contributed to make this a popular method. Whether in the form of Azimuths, as here given, or in that of Amplitudes, as heretofore used, it has been shown to admit of suffi- ciently precise results under a great variety of circumstances (62); and, with the provision of observing the object in the Apparent Horizon (7), it is equally available in observations of the Moon, and thereby its practi- cal utilities are largely extended. This method is subject, however, to the following disadvantages : 1. It is unavailable, except at certain moments of the day or night. 2 2. It is entirely unavailable in high Latitudes, with respect to bodies which remain above the Horizon, or do not set. 3. Even when available for certain objects in high Latitudes, it is lia- ble to large Errors, and must be used with caution. 1)) Time- Azimuths. Time- Azimuths have the following advantages: 1. In the simplicity and ease of the observation, which even exceeds that of the Horizon-Azimuth, or Amplitude. 2. In the facility and precision with which it may be employed in serial observations of Compass-Error, whether in port or at sea. 3. In often being available when neither of the other methods can be employed, as in the case of an obscured or ill-defined Horizon. 4. In being available whenever any celestial body is distinctly visible at a convenient Altitude for a compass-bearing, whether Sun, Moon, Planet, or Fixed Star. 5. In being available even when the Time and Latitude are uncertain to a very considerable extent, by a suitable choice of object, in order to avoid unfavorable conditions. A prejudice against this method seems to have existed to a consider- able extent, even among intelligent Navigators. Doubtless this has been due in some degree to the supposed greater labor of computation and perplexity of cases. Nevertheless, the computation, even of a ^he consideration of the H. A. Limit in Alt. Az. Errors has been adhered to in the text, not only because it conveniently enters into the estimate of these Errors, but for the purpose of a comparison with the conditions of Time-Azimuths. 2 This objection may not be very serious with those Navigators who depend soldy on morning and evening Amplitudes of the Sun for their Compass-Errors. DEPENDENCE TO BE PLACED ON THESE METHODS. 57 Time- Azimuth, is really quite as short and convenient as that of an Altitude- Azimuth of the same body, more commonly employed. But the simplicity and certainty of this method are still further assured with the use of the Time-Azimuth Tables. By these means the operations of taking out an Azimuth may commonly be done at sight, with suffi- cient accuracy, in ordinary cases of finding the Compass-Error ; while it is always easy to find the Azimuth to the nearest tenth of the degree, whenever required for a series of different Headings of a ship. c) Circumpolar Azimuths. This modification of the Time- Azimuth Method, subject only to the condition that the Star (Polaris) can be conveniently observed through the sight- vanes of the azimuth-circle, must be especially useful whenever the Hour- Angle and Latitude are very uncertain, as they not un frequently are after long-continued thick weather, with entire absence of observations, and the Reckoning even very much at fault. d) Altitude- Azimuths. Altitude- Azimuths have the following disad- vantages: 1. In the necessity of measuring the Altitude of the object, in addi- tion to that of observing the Compass-Azimuth. 2. In being inconvenient for serial observations of Compass-Errors, from the much greater labor of the requisite observations and computa- tions. 3. In being frequently unavailable, from the difficulty of observing an Altitude, in consequence of an obscured or ill-defined Horizon. 4. In being liable to rapidly-increasing Error in the higher Latitudes from the Errors of the Data. In view of these disadvantages, without any very obvious compensa- tion, there would see in to be little occasion for resorting to the observa- tion of an Altitude, merely to find the Compass-Error ; since, in general, whenever an Altitude- Azimuth is available, a Time-Azimuth would be both available and preferable. If, however, the Navigator would follow the commendable practice of taking the Compass-Bearing of the object, whenever practicable, in every observation made for Time and Position, then the Altitude ob- tained for other purposes might be used in an Altitude- Azimuth, for finding the Compass-Error on the same occasion. Even in this case the Time-Azimuth might still be preferred, except with the somewhat rare occurrence that the Navigator has a very uncertain Ship-Time but a good Latitude, in which case the Altitude-Azimuth might give a more reliable result. 58 FINDING THE COMPASS-ERROR. II. OBSERVATIONS OF TERRESTRIAL OBJECTS. A. METHOD BY DIRECT BEARINGS. The Method by Direct Bearings consists in observing the Compass-Bearing of some prominent and well-defined object on the land, whose True Bearing is already known, or may be found, from the station at which the observations are made. The difference between the Compass-Bearing and True Bearing of the object is the Compass Error (5). 67. Fundamental Conditions. There are three requirements in the use of this method, which may be regarded as fundamental condi- tions. a) First Condition : A Fixed Station. The first requirement is & fixed station, from which, as a centre, the True Bearing of the object is determined ; to which the ship may be held either directly by her anchor, or through a mooriug-buoy, and about which she may be swung into every desired Heading round the compass-circle. This requirement is usually fulfilled by selecting a point in a harbor or roadstead, round which, whatever the Heading of the ship, an unob- structed view of the distant object may be had from the compass-posi- tion on board. The selected point, as a centre or station, is sufficiently fixed for a given occasion, by casting the ship's anchor ; but it is some- times permanently marked, by leaving a mooring-buoy strongly held by a mushroom or other form of heavy anchor. ft) Second Condition: ITIeans of Swinging the Ship. The second requirement has reference to the means of swinging the ship upon the desired Headings. These depend somewhat on the cir- cumstances of the location and the resources at command. When the station is located in still or slack water, the ship may be swung by means of warps, using for this purpose kedges and hawsers ; or the ship may be pulled about with great advantage, especially if large, by a steam-tug. When the location is in a stream or tide-way, and when from other circumstances the use of warps maybe inconvenient, advantage may be taken of the changing positions of the ship, as she swings at her anchor to the wind and tide, to obtain a sufficient number of different Headings ; and these may sometimes be increased in number and variety with the aid of a steam-tug. In the first case, it is generally practicable to place the ship with her Head upon the regular points by compass 32, 16, 8, or 4, as may be de- sired $ but, in the second case, it is generally difficult, if not impossible, METHOD BY DIRECT BEARINGS. 59 to do this ; and the only resource is to obtain a sufficient number of dif- ferent Headings round the whole compass-circle, even if they are not upon the regular points. c) Third Condition: A limited Parallax of Swing. The third requirement is that the distance of the object shall be sufficiently great, in comparison with the radius of swing, to cause no appreciable change in the Compass -Bear ing whatever the Position or Heading of the ship round the station. This requirement, in order to be clearly understood, may need a little de- tailed explanation. Eef erring to the fig- ure, let O represent the place of the distant object; S the centre of swing, or station-centre ; C the position of the Standard Compass on board. Then, C S is the radius of siting, equal to the horizontal distance be- tween the compass and the vertical at the mooring-anchor, and supposed to remain constantly the same, or, at any rate, not greater than the distance fixed by a certain length of cable ; O S is the distance of the object, which is sensibly the same as O C in all positions of the ship ; and A S C is the angle of swing, which may have any value from o to 1 80, on each side of the zero-line, O A S B, through the object and centre of sta- tion. Now, unless the distance of the ob- ject be quite considerable in comparison to the radius of swing, it will be found that the direction of the object O, as seen from the compass at C, may be sensibly different from the true direc- tion S O as seen from any point A or B in the zero-line. This possible change of direction, for different positions of the ship, is measured by the Angle O C O', or its equal COS, and is called the Parallax of Swing. It is evidently nothing when the ship is in the zero-line, as when headed towards O or in the opposite direction 5 and it increases the further the ship is swung from that line, becoming a maximum when perpendicular to it, on either side. The Compass-Bearings of the object are accordingly liable to an Error, in consequence of this parallax, which is called the Parallactic Error of Swing, whose magnitude in a given case depends 1. On the Distance-Ratio of the Object; that is, on the distance of the ob- ject divided by the radius of swing ; and 2. On the Angle of Swing; being nothing when this angle is o Or 180, and having equal maxima when the angle is 90 and 270. 60 FINDING THE COMPASS-ERROR. 68. Process of Finding Serial Compass-Errors. The Method by Bearings of a Distant Object is that which is, perhaps, most commonly resorted to for obtaining a series of Compass-Errors round the compass-circle. It may be conducted in the following manner : Everything being ready and the requisite observers at their stations, swing the ship so as to bring her head upon the nearest point by Stand- ard Compass, gently checking her motion, and keeping her steady upon the point; then, take the Bearing of the distant object with the Stand- ard Compass, and note the Heading of the ship with the same compass. Note, also, the Headings with the steering-compasses, and the Angle of Heel with the clinometer. Proceed in the same careful manner to bring the ship's Head upon the next point, and, when duly stopped and steadied there, again take the Bearing of the object and note the Heading of the ship with the Standard Compass, noting also the Heading with the steering-compasses, and the Angle of Heel, as before ; and so on, point after point, till the Bearings shall have been taken of the object, with the corresponding Headings of the ship and the Angles of Heel, for every point of the com- pass round to the point of beginning. The operation, as thus described, may be performed upon each of the thirty-two points, sixteen points, or eight points (preferably in that case upon the octantal points), as It may be deemed expedient at the time. Otherwise, if the circumstances of the swinging are such as to make it inexpedient to attempt obtaining the observations on the regular com- pass-points, proceed rn all respects in the same manner, except that of trying to bring the ship's Head upon any of the regular points. It is, however, desirable in this case to distribute the observations upon Headings, approximately at an equal distance apart, round the compass- circle. In either case, it should be the aim of the observer to get his Bearings of the object with certainty to the nearest quarter of a degree, and his Headings to the nearest eighth of a point or nearest whole degree, accord- ing as he is making his observations upon the compass-points or other- wise. The record of the observations, which should be completely made as the work proceeds, may be kept according to the form in the subjoined Examples. METHOD BY DIRECT BEARINGS, 61 69. Examples of the Method by Direct Bearing's. Ex. 1. 1872 : At , in Lat. . , Long. , Ship , swung for a set of Compass-Errors, by observations of a distant object,. .. Distance of Object, 10 N. miles : Radius of Swing, 200 feet. True Bearing of object, N 41. 2 W. 1 ship's Head. Direct Bearing of Object. Error of Standard By Standard Compass. By Standard Compass. True. Compass. o o o North. N 22.7 W N 41.2 W 18.5 W N by E 26.3 14.9 NNE 29.8 11.4 NE by N 30.8 < 10.4 NE 31.0 ' 10.2 NE bv E 33-8 7-4 ENE 33- < 8.2 E by N 33- ' 8.2 East. 31.8 < 94 EbyS 30.2 * II.O ESE 29.2 I2.O SEbyE 28.2 < I 3 .0 SE 25-5 1 J 5-7 SE by S 24.0 1 17.2 SSB 22.8 1 18.4 SbyE 19.8 1 21.4 South. 18.0 i 23.2 .. S by W *5-3 ' 2 5-9 ssw 15.0 < 26.2 SW by S 13.0 ' 28.2 SW 12.8- < 28.4 SW by W n.6 ' 29.6 wsw IO.O ' 31.2 WbyS 8.8 ' 32-4 / West. 8.4 ' 32.8 Wby N 9.0 < 32.2 WNW 9.8 < 3M NW by W 11.4 * 29.8 NW 12.4 < 28.8 NW by N 14.8 26.4 NNW 18.0 ' 23.2 Nby W 19.6 " 21.6 W 62 . FINDING THE COMPASS-ERROR. Ex. 2. 1872, February 29 : At a station ill New York Bay, Lat. 40.5 N, Long. 74 W, U. S. Ship CANANDAIGUA (wood-built screw-steamer). Swung for a set of Compass- Deviations, by Direct Bearings. Objects observed. Trite Bearings. Distances. Kad. of Swing. Sandy Hook L. H. S 88 .g E 1.2 N. M. 250 feet. Navesink Lights. S 27 .1 E 4.0 " " East Beacon. N 60. .9 E i.i " " The Mag. Var. wa.s 7.6 W at the place and date. Observations. Reductions. Ship's Head. Direct Bearing of Objects. Deviation of the Standard By Standard Compass. By Standard Compass. Magnetic. Compass. o o o North, S 81.5 E S 81.3 E 0.2 E N byE S 85.0 E 81.3 3-7 NNE S 27.7 E S 19.5 E 8.2 .NE byN S 31.2 E 19-5 11.7 NE S 33.2 E 19-5 l 3-7 NEbvE N 84.0 E 81.3 14.7 ENE S 36.8 E 19-5 17-3 E by N S 37-o E 19-5 17-5 East. S 37.2 E S 19.5 E 17-7 E bvS N 51.5 E N 68.5 E 17.0 ES'E N 85.2 E S 81.3 E "3-5 SE by E N 87.0 E 81.3 11.7 ( SB N 88.8 E 81.3 9-9 SE by S S 89.0 E 81-3 7-7 SSE S 86.5 E 81.3 5-2 S by E S 84.0 E 81.3 2.7 E South. S 81.5 E S 81.3 E 0.2 E Sby W S 79.0 E 81.3 2.3 W ssw S 75-5 E 81.3 5.8 SW by S S 13.5 E S 19.5 E 6.0 sw S n.2 -R 19-5 8-3 SW by W \VSW S 8.8 E S 6.8 E 19-5 19-5 10.7 12.7 Wby S S 6.0 E 19-5 13-5 West. S 5.0 E S 19.5 E H-5 WbyN S 67.3 E S 81.3 E 14.0 WNW S 4.713 S 19.5 E 14.8 NW by W S 5.7 E 19-5 13-8 . NW S 8.2 E I9-S "3 NWbyN NNW Nby W S 9.8 E S 75-5 E N 66.7 E !9-5 S 81.3 E N 68.5 E It w 1.8 E METHOD BY DIRECT BEARINGS. 63 70. Dependence to be placed on Results. There are two distinct sources of Error which pertain to the Method by Bearings of a Distant Object First, the uncertainty in the True Bearing of the object, incident to the method employed in finding it; and, Secondly, inaccuracies in the Compass-Bearings of the object, due to the Parallactic Errors of Observation. The first, as will be explained hereafter, should never exceed the tenth of a degree; that is, o.i or =L6'. This Error, therefore, being small and constant, or affecting all the observations alike, may gener- ally be disregarded. The second, on the other hand, is not un frequently a much more seri- ous matter, since it may not only be several times as large as the former, but is attended with the complication of being unequal for the different Headings, and of having contrary signs on opposite sides of the zero- line. Nevertheless, with strict attention to the Distance-Eatio, as will presently be explained, these Errors may be kept within admis- sible limits; and, besides, they may always be estimated and applied as corrections of the observations, whenever it is certain that the swinging is done throughout with a taut cable. 71. Tables .11 and LI SI of Parallactic Errors: Ex- amples. Tables LII and LIII are constructed in order to afford the means of readily estimating these Errors whenever desired. They re- quire no explanation. A few Examples will illustrate their use. Ex. 1. The Distance of the object is 7.4 iiaut. miles, the Radius of Swing 150 feet, and the Angle of Swing 90 ; What is the Distance-Ratio, and what the greatest Paral- lactic Error I From Tab. LII the Dist. Ratio is 300, and from Tab. LIII the greatest Parallactic Error is 12', or o.2. Ex. 2. What is the Distance-Ratio, for which the greatest Parallactic Error shall not exceed 6', or o.i ? From Tab. LIII the Dist. Ratio is between 500 and 600. Ex. 3. The Distance of the object is 8 naut. miles, and the greatest Parallactic Error must not exceed 6' : What is the greatest admissible Radius of Swing ? Am. The Dist. Ratio is 500, and the Radius of Swing must not exceed 100 feet. Ex. 4. The Distance of the object is 2 naut. miles, and the Radius of Swing 200 feet : W T hat is the greatest Parallactic Error ? Ans. i.o. Ex. 5. The Radius of Swing is 150 fest, and the greatest Parallactic Error is not to exceed 6', or o.i : What is the least Distance of Object that will be required ? Ans. 12 N. M. 73. To correct the Observations for Parallactic Errors, It will be the work of but a few moments, and it may some- times be expedient, to correct the Observations of a Distant Object for Parallactic Error ; or, what is the same thing, to reduce them to what they would have been if they had been made at the centre, or on the zero-line, at the station. 64 FINDING THE COMPASS-ERROR. Thus, having deduced the Compass-Errors or Deviations, as the case may be, if it be desired to correct them for Parallax of Swing, proceed in the following manner : a) Rule : Case of Using the True Bearing of* the Object. Change the Compass-Headings into True Headings, by ap- plying the Compass-Errors to the former, to the Bight when these Errors are -/, to the Left when they are W. Next, find the Angles of Swing (68), by comparing the True Headings of the ship in succession with the True Bearing of the object; setting down the Differences, or their Supplements when greater than 90, and marking them R. or Z/., according as the True Heading is Right or Left of the True Bearing of the object. 1 Lastly, enter Tab. LIII with the Distance-Ratio and the Angles of Swing, and take out the corresponding Parallactic Errors ; applying to the Eight or Left of the Compass-Errors, according as the Angles of Swing are L. or JR. of the zero-line (68) or True Bearing of the object from the station-centre. It) Rule : Case of 11 sing the Magnetic Bearing of the Object, If the Magnetic Bearing of the object be used instead of the True Bearing, and the Deviations deduced instead of the Total Errors, the Rule will be the same in every particular as the preceding, except in substituting therein Magnetic for True, and Deviations for Compass- Errors. c) Remark I : Observing two or more Objects. If, in- stead of one object, two or more be employed with different Bearings, True or Magnetic, proceed according to the Rule, only using the proper Bearing for the Compass Error or Deviation considered. d) Remark II : Swinging with a Taut Cable. It must, of course, be understood that these corrections can only be reliably made when the entire swinging is done with a taut cable. e) Remark III: Using the nearest whole Degree. It is sufficient, in these corrections, to use the Bearings and Headings to the nearest whole degree. J The observer's eye being supposed at the centre of the com pass and looking toward the object. METHOD BY DIRECT BEARINGS. 65 73. Examples of Correcting for Parallactic Errors. Ex. 1, The Data and Results of Ex. 2 of Art. 69: The True Bearings being used and Compass-Errors deduced: To correct them for Parallactic Errors. Data and Results of Ex. 2. Correcting: for Paral. Errors. Ship's Head by Standard Compass. True Bear- Ings of Ob- jects. Error of Standard Compass. Ship's Head, True. Angles of Swing. Parallactic Errors. Corrected Compass- Errors. North. S 89 E o 7.4 W N 7 W 82 L o 1.9 5-5 W NbvE " 3-9 W 7 E 84 1.9 2.0 W NNE S 27 E 0.6 E 23 5 0.9 E NE bv N " 4.1 38 65 0.3 4.4 NE " 6.1 78 0.4 6.5 NE by E S 89 E 7- 1 63 28 I.O 8.1 ENE S 27 E 9-7 77 76 0.4 IO.I E by N IO.O 89 64 10.3 East. S 27 E 10.1 E S 80 E 53 L o-3 10.4 E E bv S N6i E 9-4 69 50 R ' 1.5 7-9 ESE S 89 E 5-9 62 27 0.9 5- SE bv E 4.1 52 37 .1 3- SE 2 -3 43 46 4 0.9 E SE by S o.i E 34 55 .6 1.5 W 8SE 2.4 W 25 64 7 4.1 Sby E 4-9 16 73 .8 6-7 South. S 89 E 7-4 W S 7E 81 R 9 9-3 W S by W 9-9 i W 90 9 11.8 ssw 13-4 9 82 9 15-3 SW bv S S 27 E 13.6 20 47 13-9 SW 29 56 0.3 16.2 SW bv W 18.3 38 6 5 0.3 18.6 WSW 20.3 47 74 0.4 20.7 W by S 21. 1 58 85 0.4 21.5 West. S 27 E 22.1 S 68 W 85 R 0.4 22.5 W W by N S 89 E 21.6 80 ii 0.4 22.O WNW S 27 E 22.4 N 90 W 63 22.7 NW bv W " 21.4 78 51 0.3 21.7 NW " 18.9 64 37 O.2 I9.I NW bv N " 17-3 5 1 24 R O.2 NNW S 89 E 13-4 36 53 L 1.6 ii.S N by W N6i E 5.8 W i7 78 1.9 3-9 This Example supposes the Compass-Errors instead of the Deviations to have been found. It illustrates the more complicated case of several objects being observed instead of one. The procedure is entirely the same. /Al*W T T" T5 T) d T f Vtt fm$ A -* XH 66 FINDING THE COMPASS-ERROR. Ex. 2. The Data and Results of Ex. 2 of Art. 69: To correct tbo Deviations for Parallactic Errors. Data and Results tff Ex. 2. Correcting for Para!. Errors. Ship's Head by Standard Compass. Magnetic Bearings of Objects. Deviation of Standard Compass. Ship's Head, Magnetic. Angles of Suing. Paralhutic Errors. Corrected Compass- Deviations. North. S 8i E o 0.2 E North. O' 81 L o 1.9 o 2.1 E N bv E . 81 3-7 N 15 E 84 1.9 5 .6 NNE S IQ E 8.2 3 1 50 o-3 .8-5 NE by N 19 11.7 45 64 -3 12.0 NE 19 J 3-7 59' 78 0.4 I4.I NE by E S Si E 14.7 7i 28 I.O 15-7 ENE 19 J 7-3 85 76 0.4 177 E by N 19 17-5 S 84 E 65 0-3 17.9 East. S 19 E 17-7 S 72 E 53 L -3 1 8.0 E E by S N6 9 E 17.0 62 49 R i-5 15-5 ESE S 81 E J3-5 54 27 0.9 12.6 SE by E 81 11. 7 44 37 .1 10.6 SE 81 9-9 35 46 4 8.5 SE by E 8r 7-7 26 55 .6 6.1 SSE 81 5-2 17 64 7 3-5 Sby E 81 2.7 8 73 .8 0.9 E South. S 81 E 0.2 E South. 81 R 9 1.7 W S bv W 81 2.3 W S 9W 90 9 4.2 ssvv 81 5.8 17 82 9 7-7 SW by S S 19 E 6.0 28 47 -3 6-3 SW 19 8-3 37 56 o-3 8.6 SW by W 19 10.7 45 64 -3 II.O WSW 19 12.7 55 74 0.4 J3- 1 W byS 19 13-5 65 84 0.4 13-9 West. S 19 E 14-5 S 76 W 85 R 0.4 14.9 W WbvN S 81 E 14.0 87 12 0.4 14.4 WNW S 19 E 14.8 N82 W 63 -3 i5-i NW by W 19 13.8 70 51 o-3 14.1 NW J 9 "3 56 37 O.2 n-5 NW bv N *9 9-7 43 24 R 0.2 9-9 NNW S 81 E 5.8 W 28 53 L 1.6 4.2 W Nby W N69 E 1.8 E 9 78 1.9 3-7 E It will be seen that, the Angles of Swing are the same, as they should lu>, in these two Cases of the same Example. The Parallactic Errors are consequently the same. It is quite evident that Ex. 2, Art. 69, should be corrected for Parallactic Errors. METHOD BY DIRECT BEARINGS. 67 74. Limiting Distance- Ratios. From an inspection of Tables LII, LIII, and the Examples of Art. 71, the need of caution in the use of this method will be evident. Assuming that we are content to admit a maximum Parallactic Error of o.i, or 6', the Distance-Ratio will have to be not less than 500; and, with this as a limit, it will be seen that, even with a Eadius of Swing no greater than 100 feet, we shall require the distance of the object to be not less than 8 nautical miles. If, however, in a given case, the Distance-Ratio be much less than 500, and we' can neither find an available object farther off, nor in any prac- ticable manner materially diminish the Radius of Swing, we shall then have to consider First, whether we shall still proceed and correct the observations for Parallactic Errors ; or, Secondly, whether, if this be impracticable (72, ) The Angles Measured from any Arbitrary Line. For this it is only necessary to clamp the graduated limb, without regard to the zero, 1 and during the observations to direct the line of sight to the compass-station on board, when the readings obtained will be True Angles from the Arbitrary Line supposed to pass through the zero of the graduation. Moreover, it will be sufficient, whenever the station on shore is changed during the progress of the observations, to keep the limb clamped and continue the Angles as before. But, since 1 It will be convenient, though not essential, to clamp so that the zero shall fall out- side of the field in which the angles are to be observed. METHOD BY RECIPROCAL BEARINGS. 77 the bearings on shore are taken without regard to any known line of reference, while the position of the ship and that of the compass-station on board are continually changing, it will be necessary to reduce the observations on board, in the manner to be explained, to what they would have been if they had been made at a fixed station and in an unchangeable direction. 91. Preparations lor Observations by this Method. All observations by this method must be directed from the ship ; and, preparatory to undertaking them, proper signals should be arranged for display on board to guide the observer on shore. The following signals should, at the least, be provided for : 1. Prepare to observe! 2. Observe and register ! 3. Observation is satisfactory ! 4. Prepare to repeat the observation ! 5. Pack the instrument and return on board ! Both observers should be prepared to note the Time of each observa- tion, as a check against mistakes of comparison ; their watches being- compared before and after the set of observations. It is essential that the two observations of each pair be made simultaneously. The instru- ment itself at each station may be sighted as the object to be observed ; but it is better to erect a more conspicuous -and more nicely denned signal directly above each instrument. 1 92. Process of Conducting* Reciprocal Observa- tions. All things being ready, and the observers at their respective stations, bring the ship carefully upon the desired Heading, and, while being duly stopped and steadied, make the first signal ; then, when ready to observe, make the second signal, upon which, at the instant, the observer at each instrument, on board and on shore, will note his Bearing of the other, and the Time by watch. Also, at the same mo- ment, note the Ship's Head by the Standard Compass, and by each of the Steering-Compasses. Note, also, the Angle of Heel. All being completed, and the shore-observation, as posted up, apparently satisfactory, make the third signal. Proceed, by bringing the ship upon a different Heading, making and recording the observations in a similar manner; and so on, round the compass-circle, or until the requisite observations shall have been made for a sufficient number of different Headings. As in other cases, observations for Headings on the regular equidis- tant points by compass are preferable, but they are not essential. 1 It is advisable, as a convenient precaution, for the observer on shore to chalk each observation upon a blackboard, so that it may be read (using a glass if necessary) by the observer on board ; by which means, if there should be any apparent inconsistency, the observation may at once be repeated, and the necessity thereby avoided for again swinging the ship. 78 FINDING THE COMPASS-ERROR. 93. RecluctioBi of Reciprocal Observations. There are three Cases to be considered, according to the nature of the shore-obser- vations. a) Magnetic Observations on Shore. In this Case the Difference between the corresponding Ship and Shore Bearings is the Compass-Deviation, which will be marked E or W 9 according as the observation on board falls to the Left or Right of that on shore. b) Angles Measured from the True Meridian on Shore. Here the Differ- ence between Simultaneous Bearings is the Compass-Error, which is marked E or W, according as the observation on board falls to the Left or Eight of that on shore. c) Angles Measured from any Arbitrary Line on Shore. -First, note the least 1 among the Shore Angles, and regard it as the bearing of the Zero- Line, or line of reference; then, compare with this all the other Angles, setting down the corresponding Differences in a column of Reduction* for Shore Instrument, and marking them R or L, according as the Zero- Line falls to the Right or Left of the Angles compared with it. Next, apply these reductions to the corresponding Bearings from the compass on board to the Right or Left, according as they are marked R or L ; which call the Reduced Bearings by the Standard Compass. There will now be tw r o Cases, according as the observations are made on Equi- distant Headings or at Irregular Intervals. First : Observations on Equidistant Headings. Take the Mean of the Reduced Bearings, whether 32, 16, or 8, and this will be the Magnetic Bearing of the station on shore; then, the Differences between this and the other actual Reduced Bearings will be the Deviations, to be marked E or TT, according as the latter Bearings fall to the Left or Eight of the Mean Bearing. Secondly : Observations at Irregular Intervals. Find the Reduced Bear- iugs by the Standard Compass, as in the previous case; thence, proceed as follows: Compare the Reduced Bearings with that of the zero-line, marking the Differenced or TF, according as the former fall to the Left or Right of the zero-line. With these Differences as Ordinates, and the Headings as Distances, construct a curve, in the manner to be explained hereafter, from which deduce the Mean Difference on a series of Equidistant Headings; set off this Difference according to its name, and draw.a parallel to the Line of Distances; then, the Ordinates to the curve with reference to this new line of distances, for whatever Headings taken off, will be the Compass-Deviations corresponding to those Headings. 1 Any other Shore-Angle may be chosen as that of the zero-line, as may be deemed most convenient. The least makes the Differences or Reductions take the same name; all R orallL. METHOD BY RECIPROCAL BEARINGS. 94. Examples of the JTIethocl by Reciprocal Bear- ings. Ex. 1. 1868, June 18 : At Erie, on Lake Erie, in Lat. 42 N, Long. 80 W, U. S. Ship Michigan (iron-built paddle-wheel steamer). Swung for a set of Compass- Deviations, by Reciprocal Bearings. Observations. Reductions. Time. by Standard Compass. Simultaneous Bearings. Standard Compass on Board. Azimuth* Compass on Shore. h m o o II 13 North. S 60.3 E N 59.0 W 15 NbyE 56.7 59- 27 .NNE 59-o 33 NE by N 48.7 59- 12 8 NE 42.8 59-5 ii NE by E 42.0 59-5 *4 ENE 41.5 60.0 17 E by N 38.0 60.0 12 19 East. S 35-3 E N 60.5 22 E by S 37-2 60.5 25 ES'E 36.5 61.0 33 SE by E 37-2 61.0 45 SE 40.7 60.5 5 1 SE by S 40-5 60.0 54 SSE 49-3 60.0 SbyE 53-7 60.0 I 2 South. S'57-3 E N 60.0 W 6 S by W 64-3 60.0 10 SSW 70.0 60.0 14 SW by S 76.0 60.0 \-[ SW 79-7 59-5 22 SW by-W 83.0 ' 59-o 2 7 ws'w 85.0 58.5 36 W by S 86.7 58.0 i 39 West. S 86.0 E N 57-5 W 5 1 WbyN 857 57-o 54 WNW 83.0 57-0 2 4 NW by W 82.7 56.5 7 NW 78.0 56.5 ii NW by N 75-3 15 NNW 70.0 57-5 19 N by W 67.8 58.0 Deviation of the Standard Compass. 1.3 E 2.3 W 6.0 10.3 16.7 ll'.l 22.0 W 25.2 W 23-3 24-5 23.8 19.8 19-5 10.7 6.3 W 2.7 W 4-3 E 10.0 16.0 20.2 24.0 26.5 28.7 E 28.5 E 28.7 26.0 26.2 21.5 18.- 12.5 9.8 E Explanations. The shore-instrument in this case being an Azimuth-Compass, the Bearings are Magnetic, and the Differences of Simultaneous Bearings are Deviations of the Standard Compass on board. There are certain anomalies in the results, which become more obvious by a Graphical Construction, and at the same time admit of being corrected and the results reduced to mf>re probable values. The Standard Compass was placed on top of pilot-house on hurricane-deck, 28 feet above spar-deck. 80 FINDING THE COMPASS-ERROR. JEx. 2. 1868, June 18: At Erie, on Lake Erie, in Lat. 42 N, Long. 80 W, U. S. Ship Michigan : The Azimuth-Compass used on shore in Ex. i supposed to have been replaced by a Dumb-Compass, and the Angles measured from an Arbitrary Lino. Observations. Reductions. Ship's Head by standard Compass. Simultaneous Bearings. Standard Compass on Board. Dumb-Com- pass on Shore. o o North. S 60.3 E 39-o NbyE 56.7 39-o NNE 53- 39-o NEbyN 39-o NE 42.1 39-5 NEby E 42.0 39-5 ' ENE 4 I -5 40.0 EbyN 38.0 40.0 East. S 35-3 E 40-5 E by S 37- 2 40-5 ESE 36.5 41.0 SE by E 37-2 41.0 SE 40.7 40-5 SEby S 40-5 40.0 SSE 49-3 40.0 S by E 53-7 40.0 South. S 57-3 E 40.0 S by W 64-3 40.0 ssw 70.0 40.0 SVV bv S 76.0 40.0 SW 79-7 39-5 SW by W 83.0 39-o WSW 85.0 38.5 Wby S 86.7 38.0 West. S 86.0 E 37-5 Wby N 857 37- WNW 83.0 37- NW by W 82.7 36.5 NW 78.0 36* NW by N 75-3 37-o NNW 70.0 37-5 Nby W 67.8 38.0 Reductions to Zero-Line of Shore- An- gles. Reduced Bearings by Standard Compass. Deviations of the stand- ard Compass. o 2.5 R S 57-8 E o 0.4 W 2-5 54-2 4.0 2-5 5-5 7-7 2-5 46.2 I2.O 3-o 39-8 I8. 4 3- 39-o 19.2 3-5 38.0 20.2 3-5 34-5 23-7 w 4.0 R S 31.3 E 26.9 W 4.0 33- 2 25.0 4-5 32.0 26.2 4-5 3 2 -7 25-5 4.0 36.7 21.5 3-5 37-o 21.2 3-5 45-8 I2. 4 3-5 50.2 . 8.0 W 3-5 R S 53-8 E 4-4 W 3-5 60.8 2.6 E 3-5 66.5 8-3 3-5 72.5 14-3 3- 76.7 18.5 2-5 80.5 22.3 2.O 83.0 24-8 i-5 85.2 27.0 E 1.0 R S 85.0 E 26.8 E 0-5 85.2 27.0 -5 82.5 24-3 0.0 82.7 24-5 o.o 78.0 19.8 0-5 74.8 16.6 l.O 69.0 10.8 1.5 66.3 8.1 E Mean S 58.2 E Explanations. The Least Angle of the shore-observations is *36.5, which is taken as that of the Zero-Line. Comparing this with all the other Shore-Angles, we get the Reductions in Col. I. These are all marked R, because the direction of the Zero-Line falls to the Right of all the other Angles. Next, these Reductions are applied to the Right of the corresponding Bearings by the Standard Compass, and we get Col. II 5 the Mean of which, or S 58.2 E, is the Magnetic Bearing of the shore-station. Finally, comparing the Reduced Bearings of Col. II with the Mean, we get the Devia- tions in Col. III. METHOD BY RECIPROCAL BEARINGS, 81 Ex. 3. 1868, June 18 : At Erie, on Lake Erie, in Lat. 42 N, Long. 80 W, U. S. Ship Michigan : The ship swung upon Irregular Headings, and the Angles on shore wtill measured from an Arbitrary Line with a Theodolite. Observations. Reductions* Simultaneous Bearings. Ship's Head by standard Compass. Reductions to Zero-Line of Shore-An- gles. Reduced Bearings by Standard Compass. Differences of the Standard Compass. Standard Compass on Board. Theodolite on Shore. o o o o c O N 10 E S 57-3 E 43-2 1.7 R S 55.6 E 14.1 E , 22 53-5 44-8 3-3 50.2 8.7 4 45-7 42.8 444 2.9 E 53 42.4 43-9 2.4 - 40.0 1.5 W 41.2 46.0 4-5 36.7 4.8 88 36.9 45-3 3-8 33- i 8.4 S 73 E S 36.1 E 43-9 2.4 33-7 7-8 56 37-0 44.0 2-5 34-5 7.0 40 4-3 42.6 i.i 39-2 2.3 W 27 47-2 45 - 1 3-6 43-6 2.1 E 10 54-7 44-8 3-3 9-9 S 5 W S 61.0 E 42.8 1-3 59-7 18.2 E 22 69.8 42.8 '3 68.5 27.0 36 75-3 42.5 I.O 74-3 32-8 54 82.1 43-9 2.4 79-7 38.2 65 84.5 43- ! 1.6 82.9 41.4 78 86.3 ^42.2 0.7 85.6 44.1 89 85.9 0.0 85.9 444 N 75 W S 85.7 E 41.7 0.2 85.5 44.0 E 62 82.8 41.7 0.2 82.6 41.1 50 80.2 42-3 0.8 794 37-9 36 76.6 42.4 0.9 75-7 34-2 18 68.5 42.1 0.6 67.9 26.4 4 65.1 45.8 4.3 R 60.8 19.3 E Explanations. Having found the Reduced Bearings of Col. II, as in the preceding Example, we next Compare these with the Angle of the Zero-Line (*4i.s) and get the Differences in Col. III. 11 82 FINDING THE COMPASS-ERROR. 95. Dependence to be placed on Results. With the use of an Azimuth-Compass at the shore- station, there is always consider- able uncertainty relative to the Magnetic Meridian, from which the Bear- ings are supposed to be measured, unless the requisite means are employed to have a strict comparison of the two compasses made on shore. Not only is it necessary to detect the possible influence of Local Magnetism, but to note the actual differences of the shore Com- pass- -Readings on different parts of the compass-card, if we would be at all sure of our results. With these comparisons made, and the ascertained Errors, if any, properly applied as corrections, the shore- observations should be depended on. With the use of a Theodolite or Dumb-Compass, on the other hand, a more satisfactory precision may be attained. It is not difficult for the Navigator, with a reliable Geographical Position and a good Local Time, to determine his True Meridian on shore to the nearest minute 5 and this may be established by Meridian-Marks, and worked from to a cor- responding precision, with a Theodolite reading to minutes in proper, adjustment. Even with a Dumb-Compass, reading only to tenths of degrees, the Bearings should be depended on to the nearest tenth of a degree. There will not be the same degree of precision, if, instead of working from the True Meridian, the Shore- Angles are measured from an arbi- trary line. In this case a certain constant Error will be introduced and merged in the Errors of Compass Adjustment and Observation, in consequence of the necessity of employing a Mean Bearing or Difference as deduced from the results of the observations. The Error from this source will be less or greater according to the greater or less care and nicety observed in conducting the operations. In general, it will rarely be so much as a degree, including the Errors of Adjustment and Obser- vation 5 and, since it affects all the results alike, it may commonly be disregarded. 96. To find the True Ifleridian. There are various modes of procedure for finding the True Meridian. It will, however, be quite sufficient, for the present purpose, to make use of means to which the Navigator is well accustomed, namely : That of obtaining the True Azimuth of the Sun, or other celestial object, at a particular moment of Local Time ; thep, of setting off the Azimuth at the designated mo- ment, upon a Theodolite or Dumb Compass. The whole process may be conducted as follows : a) Compute in advance, for a designated Local Time, the True Azi- muth of the Sun or other celestial body (19); and add or subtract the Semi-Diameter, according as the Azimuth is greater or less than 90. The resulting Azimuth will be that of the inner 1 limb, if the body have a disk (Sun or Moon). b) Having selected a convenient observing-station, set the Theodolite or Dumb-Compass at the chosen point; then turn off' the ascertained 1 That towards the Meridian. METHOD BY RECIPROCAL BEARINGS. 83 True Azimuth from the line of zeros, and carefully clamp the alidade (telescope or sight- vanes as the case may be), and level the instrument for observation. c) As the designated Time approaches for which the True Az. is found, point the line of sight a few minutes in advance to the inner limb of the celestial body, turning the whole instrument upon its spindle, and keeping upon the observed point of contact, by following the motion of the body, till the precise moment of time is reached ; then make fast the lower or spindle clamp. d) Uuclamp the line of sight, turn to zero, and again clamp; it will now be in the True Meridian. Set up a Meridian-Mark, in the direction of the line of sight, fifty to a hundred yards in advance. Uuclamp and turn 1 the line of sight 180; uuclamp, and then set up another Meridian- Mark in the opposite direction. Finally, mark the station in a perma- nent manner on the ground below the centre of the instrument, and it will be ready for use. 97. Remark: To set the Instrument in a True Me- ridian. already established. If the True Meridian be already established, proceed as follows to place the instrument: Set the instrument over the station (d) with a centre-plumb, and level it; then clamp the alidade to zero, and direct the line of sight to one of the permanent Meridian-Marks (d), turning the whole instrument upon the spindle, and, when carefully bisected, make fast the lower or spindle clamp. The instrument will be ready for use. 98. Example of Finding the True Meridian of a Shore-Station. Ex. 1. 1872, February 20: Navy- Yard Dock, Washington, iii Lat. 38 52^4 N, Loii y6 59'.6 W. It is required to find the True Meridian from a True Azimuth of the Sun at 4 b .o P. M. Loc. A. T. : Watch fast on LOG. M. T. i m 14 s . Loc. A. T. 4 h o m 'sDec. S 11 2^9 {54" Eq. of T. -f- 14 i 8 .;] o s .2; Long. +58 Rednforg 1 '.! 8.2 486 Red n for9 h .i 2.5 2.43 Gr. date 20 9 8 Red. Dec. 10 54.7 5 Red. Eq. T. 13 59 .2 3 or, 9 ll .l Pol. Dist. 100 54.7 -f- to A. T. Eq. of T. + o h 13 59 s Loc. M. T. 4 13 59 Pol. Dist. 100 54'. 7 Er. of W. fast + i 14 Co-Lat. 51 7.6 T. byW. 4 15 13 Diff. 49 47.1 y 2 Diff. 24 53.5 sili 9.6242 cos 9.9577 ]/z Sum 76 i .1 cosec 0.0131 sec 0.6168 Yz H. A. 2 1 ' o m cot 0.2386 cot 0.2386 tan 9.8759 tan 0.8131 TrueAz. Nii8io / W X 36 55' Y 81 15' 0's Semi-D. + 16 True Az. N 118 26 W Inner Limb or, S 61 34 W A few minutes before 4 h 15 i 3 8 by Watch, place the theodolite firmly in position at the station chosen for the shore-observations, and level it. is unnecessary with a theodolite whose telescope may be revolved transit- like. 84 FINDING THE COMPASS-ERROR, Next, turn off the Angle 61 34' from o, and clamp the limb, leaving the lower or spindle damp free. .Now, turning the whole instrument upon the spindle, pointing to the Sun, and bringing the vertical wire of the telescope into contact with its inner limb, follow its motion, carefully preserving contact, until the moment 4 h 15 43 e arrives, as shown by watch; then make fast the lower clamp. Unclamp the limb, and turn the telescope back to o, when it will be in the True Meridian. Set up a Meridian -Mark in the direction of the line of sight and mark the centre of the instrument, as shown by the plumb, on the ground below. 1 The instrument is ready for the shore-observations, the lower clamp remaining all the time fixed. Remark. We might proceed in the direct manner, by first making the observation to a noted time by watch, subsequently computing the True Azimuth, and, finally, turning it off on the theodolite. The only objection to this mode of procedure, is the necessity of allowing the instrument to stand and the danger of its disturbance while making the computation. D. RELATIVE ADVANTAGES OP THE SEVEKAL METHODS OF FINDING SERIAL COMPASS- ERRORS. ' * We shall briefly sum up the relative advantages of the principal Methods for finding Serial Compass-Errors. 99. Jlletliocls by Celestial Azimuths. The observations by these methods, and preferably by Time-Azimuths, are certainly the most convenient. They are all made on board ship. The ship may be at her anchorage, or standing off outside, or at sea ; and the different Headings, on which the observations are made, may be had, either by swinging about her anchor, by steaming round, or by tacking under sail. When near the land, these methods are susceptible of considerable accuracy. The True Azimuths may be depended on within i 3' round the entire compass-circle, the Data being reliably known and the condi- tions of favorable results being duly satisfied (60). The Compass- Azi- muths are'taken under the most .favorable circumstances for precision of results. The only drawback to this method, in comparison with the others, while superior in every particular besides, is the supposed labor of ob- taining the series of True Azimuths. And yet, a set of thirty-two Azi- muths may be had, with the aid of the Tables here given, in a half- hour or less. 2 1 If a dumb-compass instead of a theodolite bad been used, we should have turned off instead 6i.6 and clamped the alidade, leaving the spindle free ; otherwise, the de- scription of the text is equalty applicable to this instrument, only substituting " alidade " for "limb," and "sight- vanes" for "telescope." 2 Navigators not (infrequently devote the befit part of a day, sometimes two of them, at a " station " to the laborious and disagreeable duty of " swinging ship " for observa- tions of a distant object, without always obtaining results which they can regard as RELATIVE ADVANTAGES FOR SERIAL COMPASS-ERRORS. 85 At sea, of course none but Celestial Azimuths are available for Com- pass-Error ; and the small labor of getting a set wholly or partially round the compass by the Method of Time-Azimuths, should certainly leave no Navigator of a ship, whenever overtaken by prolonged thick weather, without recent determinations of the Errors on the Compass- Courses likely to be sailed at such a time. OO. method by Direct Bearings. The Method by Direct Bearings has. also the convenience of requiring no observer on shore, but it has the disadvantage of requiring a particular station, com. inonly at a considerable distance from the usual anchorage, and of being attended by a large expenditure of time, labor, and patience in warp- ing the ship about, especially if a large one. It has, moreover, the dis- advantage of being liable not only to the Constant Error, whatever that may be, in getting the True Bearing of the object, but also to in- accuracies in the Compass-Bearings, from the more serious Errors of Parallax ; errors which are always variable and occasionally uncertain, and which are frequently much too large to be admitted, with any re- gard to precision of results. 1 101. Ilethod by Reciprocal Bearings. The use of this method is attended with the disadvantage of requiring observers on shore in addition to those on board. But it admits of being employed at the ordinary anchorage, either by swinging at an anchor, or by steaming about when the water-way permits. Neither Parallax nor any other con- sideration need interfere with entire freedom in the movements of the ship, during the operations of placing her upon different Headings, ex- cept that of keeping the distance between the ship and station on shore within the limits of distinct vision. With the use of an instrument for measuring the Bearings, either from the True Meridian or from any Arbitrary Line at the station on shore, not only is the choice of that station independent of all re- gard to the influence of Local Magnetism, but the Angles admit of " entirely satisfactory ; " which in fact, are quite too frequently utterly unreliable and valueless. Whereas by merely steaming about at their anchorage (not being too near auy other ship), they might get a perfect set of observations of the Sun in two or three hours after sunrise or before sunset ; and another half-hour or so would enable them to reduce their observations and obtain a Table of Compass-Deviations. 1 See the interesting and suggestive description, by the late Dr. Scoresby (pp. 168-182 of Journal of a Voyage to Australia, etc., London, 1859), of his arduous labors for more than a week at Melbourne, Australia, in determining Magnetic Bearings of distant objects and in swinging the noble iron ship Royal Charter. With an admitted Par- allax of o.3 to o4 on some of his Headings; with an apparent uncertainty of perhaps a whole degree in the Magnetic Bearings of his objects ; with probably considerable errors i'n the Compass-Bearings from the manner in which they were taken ; and with an im- mense amount of labor and anxiety on his part, aided by gangs of men, boats, kedges, and warps, he obtained a series, as was to be expected, of not very satisfactory results. And yet, with steam up on one of those days, and with two or three hours devoted to a series of Solar Azimuths, what superior results, and what a saving of labor, worry, and anxiety might have been realized ! 86 FINDING THE COMPASS-ERROR. being measured with greater precision. It is quite possible to have the True Meridian laid down on shore within a minute of arc 5 and the Angles may be measured with equal accuracy, using a theodolite for the purpose. When the True Meridian has not been already laid down, the Angles may be measured from Arbitrary Lines ; but, in this case, the Mean of the Reduced Compass Bearings having to be used as the Magnetic Bear- ing of the assumed zero-line, the results may involve a greater Constant Error. Otherwise, in working from the Magnetic Meridian on shore, this method is not only more restricted in the choice of a station, but is lia- ble, without great care and a troublesome comparison on shore, to consid- erable Errors from Local Magnetism, and differences in Compass- .Read- ings. In conclusion, since one or the other of the two Methods by Celestial Azimuths and Reciprocal Bearings is sufficient for all probable circum- stances, and since either of these Methods may be so conducted as to insure the requisite precision of results, there would seem to be little it any occasion for resort to the less accurate Method by Bearings of a Distant Object. APPENDIX. 87 APPENDIX. COMPASS COMPARISONS. 1O3. Definition. The Comparison of one compass with another is the Difference of their Readings, one being 1 taken as Standard for Azimuths or Bearings of the same object: It is marked E or W, according as the Bearing by the Compared Compass falls to ib& Left or Right of the Bearing by the Standard, the eye being supposed at the centre of the latter's compass card. A distinction exists between the Comparison of Compasses on board and on shore, which will be separately considered. A. COMPASS-COMPARISONS ON BOARD SHIP. 10SI. Characteristics and Uses of Comparisons. From what has already been said, as to the nature of the Compass-Error, it may be inferred that, whenever there are several compasses set up for use in different parts of the same ship, they may generally be expected to show different Readings on the same Heading of the ship, and each to vary for different Headings. This has been fully explained in the pre- ceding Chapters. A Compass-Comparison on board serves several useful purposes, of which the two following may be specially mentioned : First, in converting a Course by Standard into an equivalent Course by Compared Compass; or, vice versa. Secondly, in finding the Error of a Compared Compass, whenever we have its Comparison with the Standard and the Error of the latter. In practice, the ship's Head is the object observed in all Compass- Comparisons on board. For this there are two reasons : First, because the Compass-Readings, and therefore the Differences between them, depend on the direction of the ship's Head ; and. Secondly, because the Lubber-Line, which represents the ship's Head for each compass, is the only object which can be either conveniently or accurately observed with the Steering and other Compasses, not fur- nished with the sight- vanes of an Azimuth-Circle. 1O4. To make a Compass-Comparison. Accordingly, the only necessary provision for Compass-Comparisons on board is, to note the Headings of the ship by the different compasses for which com- parisons are desired, whenever observations for Error are made with the Standard or Azimuth Compass (10). The Rule for the Comparison is comprised in the preceding Defini- tion,. 88 COMPASS-COMPARISONS. 1O5. Examples of Compass-Comparisons. -Ex. 1. Stand. Compass N 47 E Steering-Compass N 41 E COMPARISON 6 E Ex. 2. Stand. Compass S 67 E Steering-Compass S 59 E COMPARISON 8 W Ex. 3. Stand. Compass N W Steering-Compass N N W % .W COMPARISON i % pts. W Ex. 4. Stand. Compass S 11 E Steering-Compass S 2 W COMPARISON 13 AV Ex. 5. Stand. Compass N 89 W .; Steering-Compass S 87 W COMPARISON 4 E Ex. 6. Stand. Compass N E by E Steering-Compass E N E |/j E COMPARISON i ! pts. W 106. To convert a Given Course iUy Standard Com- pass into an Equivalent Course by a Compared Com- pass, and Reciprocally. An important application of the Com- parison of Compasses is found in the conversion of a Standard-Compass Course into an equivalent Coinpared-Compass Course. Thus, it com- monly happens, in practice, that the Sailing-Courses, while given for the Standard Compass, must be set for the Helmsman on the Steering- Compass. This conversion of the given Course is made in the fol- lowing manner: Rule : Suppose the eye at the centre of the Standard Compass > looking along the line of the given Course or Heading ; then apply all East Comparisons to the left hand, and all West Comparisons to the right hand. For Reciprocal Conversions, apply the comparison to the Course by * Compared Compass in the contrary manner, and the result will be the equivalent Course by Standard Compass. 107. Examples of Direct and Reciprocal .-011 ver- sions. Ex. 1. The Course or Heading by Standard is NE, and the Comparison of Steering-Compass is 6 E : Required, the Course by Steering-Compass. Course by Standard N 45 E Comparison of Sceering-Comp. 6 E Course by Steering-Compass N 41 E Ex. 2. The Course by Steering-Com- pass is NNW, and its Comparison with the Standard is 11 W: Required, the Course by Standard. Course by Steering-Compass N 22. 5 W Comparison of Steering-Comp. n .o W Course by Standard N 33 .5 W 1O8. Rule: To find the Error or Deviation of" a Com- pared Compass. Having the Comparison of any compass with the Standard, and the Error or Deviation of the latter, the Error or Devia- tion of the Compared Compass is found as follows: Take the Sum or Difference of the Comparison and Error, or Devia- tion, according as they have the same or different names; and the result will be the Error or Deviation of the Compared Compass, which mark with the name of the greater number. COMPASS-COMPARISONS. 89 1O9, Examples of Finding the Errors of Compared Compasses. Ex. 1. A Comparison of Steering-Com- PHSS is 3/ pt. W with the Standard, and the Error of the latter i % pts. W : Re- quired, the Error of the Steering-Compass. Comparison of Steer. Comp. % pt. W Error of Standard i# W Error of Steering-Comp. 2^ W Ex. 2, Comparison 10 W andDev. of Standard 20 E : Required, the Error of the Compared Compass. Comparison 10 W Dev. of Standard 20 E Dev. of Compared Comp. 10 E Ex. 3. Comparison y.5 E, and Error of Standard i6.2 E : Required, the Error of the Compared Compass. Comparison 7.5 E Error of Standard 16 .2 E Error of Compared Comp. 23 .7 E Ex. 4. Comparison i5,5 E, and Error of Standardise W: Required, the Error of the Compared Compass. Comparison I 5-5 W Error of Standard 9 .o E Error of Compared Comp, 6 .5 W B. COMPASS-COMPAHISONS ON SHORE. 11O. Comparison on Shore: Detection of Local Ulagnetism. It is frequently desirable to carry a compass on shore for [Reciprocal Observations with the Standard on board, and for other purposes. In such a case, it will be necessary to compare the com- passes, if any assurance of accuracy be desired, not only to obtain their Constant Differences but to find the Errors due to Local Magnetism, 1 if, indeed, either exist. For this purpose we may proceed in the following manner : Rule : Take both compasses on shore. Place the Shore-Compass in the selected position, and the Standard Compass at a distance of joo to 200 feet from it. 2 Take the Bearing of each from the other, and note their respective Eeadiugs. Again, place the Standard Compass about the same dis- tance off, and about eight points to one side of the former position, and note the respective Eeadiugs. Finally, place the Standard about the same distance off and about sixteen points from the first position, and note their Eeadings as before. Now compare these Bearings, and it will be shown First, if the Bearings of each pair differ exactly 16 points, or 180, there is neither Deviation from Local Magnetism nor disagreement of Heading from any cause ; Secondly, if the difference between the Eeadings be more or less than 1 80, but sensibly constant, there is probably no Deviation from Local Magnetism, but only a difference of Eeadings in the compasses, which must be noted as a correction to be applied ; and, 1 Arising from detached masses of iron, in sight or concealed from view, such as anchors, guns, warping-posts, chains, etc. ; or from magnetic iron-ore, trap or basaltic rock, etc. 3 It will be convenient to have a tripod for the Standard Compass, to be moved about. The other compass may be set up in its box on a table, block, or post ; upon anything of suitable height and stability. 12 90 COMPASS-COMPARISONS. Thirdly, if the difference of Beadings is not sensibly the same, there is a- Deviation from some local cause ; in which case the position selected for the Compared Compass must be changed, in order to iind, by an- other trial, if possible, a suitable position for a station that shall be free from Local Error. Selecting a second position, make a similar series of observations ; and thus proceed till a satisfactory position be found, or the particular locality abandoned as altogether impracticable. Remark. The necessity for comparing compasses, under the cir- cumstances supposed in this Article, as a preparation for the observa- tion of Serial Compass-Errors, should, we think, very seldom occur in practice. As already indicated, it* is far better, in the Method by Reciprocal Bearings, to employ for the shore-observations such an an- gular instrument as a Theodolite or Dumb-Compass, in which any use of the Magnetic Needle or Card is dispensed with ; not only as being in- trinsically a more reliable instrument, but as relieving from all anxiety and embarrassment relative to the possible influence of Local Mag- netism. Otherwise, it may often be far better to make use of Solar Azimuths. TABLE I. Compass Points and their Equivalents in Degrees and parts thereof. ! Degree Equivalents to the Name of Point. No. nearest No. i Name of Point. Second. Hundreth. Tenth. o / // , j j , North East or West. o o o o.oo o.o South West or East. i I 24 22 I.4O 1.4 i ' i 2 48 45 2.81 2.8 i I 4 13 7 4.22 | 4.2 f i 5 37 30 5-63 5-6 * f 7 i 5 2 7-03 7-o '* 1 f 8 26 15 8.44 8.4 f n_ S 9 50 37 ; 9-84 9-8 1 N. by E. or N. by W. 1 ii 15 o 11.25 11.3 1 S. by W. or S. by E. i 12 39 22 12.66 12.7 | 14 3 45 14.06 14.1 JL 1 15 28 7 15.47 15-5 1 i i 16 52 30 16.88 16.9 1 18 16 52 18.28 18.3 1 m : i 19 41 15 19.69 19.7 1 1 21 5 37 , 21.09 2i. i i N.N.E. or N.N. W. 2 22 30 o 22.50 22.5 2 S.S.W. or S.S.E. i 23 54 22 23.90 23.9 i j. 4 25 '8 45 25.31 ; 25.3 i 1 26 43 7 26.72 : 26.7 1 i 28 7 30 28.13 28.1 1 1 29 31 52 29.53 29.5 i : i 30 56 15 30.94 30.9 1 , ; 32 20 37 32.34 32.3 i N.E.byN.orN.W.byN. 3 33 45 o 33.75 33.8 3 S.W.byS.orS.E.byS. i 35 9 22 35.16 35.2 36 33 45 36-56 36-6 **] i 37 5* 7 37-97 3 8 - i i 39 22 50 39.38 39.4 i t 40 46 52 40.78 ; 40.8 * i ; I 42 ii 15 42.19 42.2 f 1 43 35 37 43-59 j 43- 6 i i N.E. or N. W. 4 45 o o 45-o 45- 4 S.W. or S.E. : i 46 24 22 46.40 ; 46.4 i ' 47 48 45 47-8i \ 47- 8 1 49 13 7 49.22 49.2 1 1 50 37 30 50.63 50.6 i f 52 i 52 i 52-03 52-0 i f 53 26 i5 53-44 53-4 f 1 54 5 37 ; 54. 8 4 54-8 I N.E.byE.orN.W.byW.! 5 56 15 o 56.25 57 39 22 57-66 56-3 57-7 5 i S.W.byW orS.E.byE. ! i 59 3 45 ! 59-06 59-i i f 60 28 7 60.47 60.5 f \ ' * 61 52 30 61.88 61.9 f 63 16 52 63.28 63-3 1 i 1 ' 1 64 41 15 : 64.69 66 5 37 1 66.09 64.7 66.1 1 E.N.E. or W.N.W. 6 67 30 o i 67.50 67.5 6 W.S.W. or E.S.E. i 68 54 22 j 68.90 68.9 i i 70 1 8 45 ! 70.31 | 70.3 i 1 71 43 7 i 71.72 7i-7 1 i 73 7 3 i 73-13 73-i 1 i 74 31 52 ! 74-53 74-5 t f 75 56 15 ' 75-94 75-9 1 1 77 20 37 , 77.34 77-3 1 E. by N. or W. by N. 7 78 45 o | 78.75 78.8 7 W. by S. or E. by S. i 80 9 22 i 80. 1 6 80.2 i 81 33 45 ! 81.56 81.6 i f 82 58 7 i 82.97 83.0 84 22 30 ! 84.38 84-4 | 1 85 46 52 ! 85.78 85.8 | 1 87 II 15 1 87.19 87.2 f i 88 35 37 i 88.59 88.6 1 East or West. 8 90 o o 90.00 90.0 8 West or East. TABLE II. TABLE III. TABLE IV. 3 Mean Solar into Sidereal Time. Sidereal into Mean Solar Time. Solar . HH Hours. Add ' Solar Miu. Add. Solar Sec. Add. Sid. Hours. Subtract, Sid. Miu. Subtract. Sid. * Sec. J iubtract. m s s s m s s 1 s 1 o 9.86 1 0.16 1 o.oo 1 o 9.83 1 0.16 1 | o.oo 2 o 19.71 2 0.33 2 o.oo 2 o 19.66 2 -33 2 1 o.oo 3 o 29.57 3 0.49 3 o.oi 3 o 29.49 3 0.49 3 i o.oi 4 o 39.43 4 0.66 4 o.oi 4 o 39.32 4 0.66 4 o.oi 5 o 49.28 5 0.82 5 o.oi 5 o 49.15 5 0.82 5 O.OI 6 o 59.14 6 0.98 6 O.O2 6 o 58.98 6 0.98 6 0.02 7 9.00 7 1-15 7 ! 0.02 7 8.8 1 7 1-15 7 I 0.02 8 18.85 8 1.31 8 O.O2 8 18.64 8 P-3' 8 0.02 9 28.71 9 1.48 9 0.02 9 28.47 9 1.47 9 ! 0.02 ! 10 38-56 1O 1.64 1O 0.03 10 38-30 10 1.64 1O 0.03 11 4842 11 i-Si 1 1 0.03 11 48.13 11 i. 80 11 0.03 12 58.28 12 1.97 12 0.03 12 57-95 12 1.97 12 < 0.03 13 2 8.13 13 2.13 13 0.04 13 2 7.78 13 2.13 13 ' 0.04 14 2 17.99 14 2.30 14 0.04 14 2 I7.6l 14 2.29 14 0.04 15 2 27.85 15 2.46 15 0.04 15 2 27.44 15 2.46 15 0.04 16 2 37.70 16 : 2.63 16 0.04 16 2 37-27 16 2.62 16 i 0.04 17 2 47.56 17 ! 2.79 17 0.05 17 2 47.10 17 2.78 17 : 0.05 18 2 57.42 18 ! 2.96 18 0.05 18 2 56.93 18 2-95 18 0.05 19 3 7-27 19 ! 3- 12 19 0.05 19 3 6.76 19 3- 11 19 0.05 20 3 17-13 20 3-29 20 0.05 20 3 16.59 20 3.28 2O 0.05 21 3 26.99 21 345 21 0.06 21 3 26.42 21 3-44 21 0.06 22 i 3 36.84 22 : 3-61 22 0.06 22 3 36-25 22 3-60 22 0.06 23 3 46-70 23 3-78 23 0.06 23 3 46.08 23 3-77 23 0.06 24 3 56.56 24 3-94 24 0.07 24 3 55-91 34 3-93 24 0.07 25 4.11 25 0.07 25 4.10 25 0.07 26 4-27 26 0.07 26 4.26 26 0.07 27 4-44 27 0.07 27 4.42 27 O.O7 28 4.60 28 0.08 28 4-59 28 0.08 29 4-76 29 0.08 29 4-75 29 0.08 3O 4-93 30 0.08 30 4.91 30 0.08 31 5-09 31 0.08 31 5.08 31 0.08 32 5-26 32 0.09 32 5-24 32 O.O9 33 5.42 33 0.09 33 5-41 33 O.O9 34 5-59 34 0.09 34 5-57 34 0.09 35 5-75 35 0.10 35 5-73 35 O.IO 36 5-91 36 0.10 36 5-9 36 O.IO 37 6.08 37 O.IO 37 6.06 37 O.IO 38 6.24 38 O.I I 38 6.23 38 O.II 39 6.40 39 O.I I 39 6-39 39 O.II 40 6.57 40 O.II 40 6-55 40 O.II 41 6.74 41 O.I I 41 6.72 41 O.II 42 6.90 42 0.12 42 6.88 42 0.12 43 7.06 43 0.12 43 7.04 43 O.I2 44 7-23 44 0.12 44 7.21 44 0.12 45 7-39 45 0.12 45 7-37 45 0.12 46 7.56 46 0.13 46 7-54 46 O.I3 47 7-72 47 0.13 47 7.70 47 0.13 48 7.89 48 0.13 48 7.86 48 O.I 3 49 8.05 49 0.14 49 8.03 49 0.14 50 8.21 50 0.14 5O 8.19 50 0.14 51 8.38 51 O.I4 51 8.36 51 0.14 52 8-54 52 0.14 52 8.52 52 O.I4 53 8.71 53 0.15 53 8.68 53 0.15 54 8.87 54 0.15 54 8.85 54 O.I5 55 9.04 55 0.15 55 9.01 55 0.15 56 9.20 56 0.15 56 9.17 56 0.15 57 9-36 57 0.16 57 9-34 57 0.16 58 9-53 58 0.16 58 9-5 58 0.16 59 9.69 59 0.16 59 9.67 59 0.16 60 9.86 60 0.16 60 9-83 60 0.16 TABLE V. Length of a Degree in Latitude or Longitude. Lat. Deg. of Long. Deg. of Lat. Lat. Deg. of Long. Deg. of Lat. Stat. Miles. Naut. Miles. Stat. Miles. Naut Miles. o Stat. Miles. Naut. Miles. Stat. Miles Naut. Miles. 69.160 60.000 68.698 59.600 45 48.986 42.498 69.044 59.899 1 2 .150 .119 59-991 964 .698 .600 .699 i .601 46 47 .126 47.251 4L752 40.993 .056 .068 .910 .920 3 .066 .919 .700 ' .602 48 46.362 .222 .080 -931 4 68.992 .855 .702 ' .603 49 45-459 39-439 .092 .941 5 6 68.898 .783 59-773 .673 68.704 .706 59.605 .607 50 51 44.542 43.611 38-643 37.835 69.104 .116 59-951 .962 7 647 555 709 .609 52 42.667 .016 .128 8 .491 .419 .712 .612 53 41.710 36.186 .140 .982 9 3H .265 .715 .615 54 40.740 35-344 I 5 I .992 10 11 68.116 67.898 59-093 58.904 68.719 -723 59.618 .621 55 56 39.758 38.763 34-491 33.628 69.162 60.002 .012 12 659 .697 .728 .625 57 37-756 32-755 .184 .022 13 .400 472 733 .629 58 36.737 31.872 195 .032 14 .120 .229 738 -634 59 35-707 30.979 .206 .041 15 66.820 57.968 . 68.744 59.639 60 34.666 30.076 69.217 60.050. 16 499 .690 -75 .645 61 33-615 29.164 .228 .059 17 18 158 65-797 aj 764 .651 .657 62 63 32-553 31.481 28.242 27.311 .238 .248 077 19 .416 56.751 .771 .663 64 30.399 26.372 .258 .086 20 21 65.015 64.594 56.404 .039 6S :W 7 59.669 .676 65 66 29.308 28.208 25-425 24.471 69.268 277 60.094 .102 22 154 55.657 795 683 67 27.100 23-509 .286 .110 23 63-695 .258 .804 .6 9 I 68 25.983 22.540 294 .117 24 .216 54.843 .813 .699 69 24-857 21.564 .302 .124 25 62.718 54-4" 68.822 59.707 70 23-723 20.582 69.310 60.131 26 27 .201 61.665 53.962 497 .831 .840 715 .723 71 72 22.582 21-435 19-593 18.598 .318 .326 137 143 28 .110 .016 850 73 1 73 20.282 17-597 333 .149 29 60.536 52.518 .860 .740 74 19.122 16.590 339 155 30 59-944 52.005. 68.870 59-749 75 17.956 I5-578 69.345 60.161 31 32 33 58706 .060 51.476 50.931 370 .881 .892 903 .758 .767 776 76 77 78 15-607 14.425 14.561 13-539 12-513 351 357 r- -362 .166 .171 .175 34 57.396 49-794 .914 .786 79 13-238 11.484 .367 .179 35 36 56.715 .016 49-203 48.597 68.925 936 59-796 .806 80 81 12.047 10.452 9.417 69-371 -375 60.183 .186 37 55.30 47-976 947 .816 82 9.656 8-379 -378 .189 38 54.568 340 959 .826 83 8.456 7.338 .381 .192 39 46.690 .971 .836 84 7-253 6.294 .384 .194 40 41 42 53.053 52.271 5M73 46.026 45-348 44-656 68.983 * -995 69.007 59.846 !866 85 86 87 6.048 4.841 3-632 5.248 4.200 3.151 69.387 .389 390 60.196 .198 .199 43 50.659 43-95 .019 .877 88 2.422 2.IOI -391 .200 44 49.830 .231 .031 .888 89 1. 211 1.050 392 .201 TABLE VI. 5 Logarithms of Numbers and Small Arcs. N. Log. N. Log. N. Log. N. Log. N. Log. 2O 3 OI 40 6021 60 7782 80 9031 1 oooo 1 3222 1 6128 1 7853 t 9085 2 \ 3010 2 3424 2 6232 2 7924 2 9138 3 4771 3 | 3617 3 6335 3 7993 3 9191 4 6021 4 L 3802 4 6 435 4 8062 4 9243 5 6 6990 7782 5 j 3979 6 | 415 5 6 6 53 2 6628 5 6 8129 8i95 5 6 9294 9345 7 8451 7 | 43H 7 6721 7 8261 7 9395 8 9031 8 4472 6812 8 8325 8 9445 9 9542 9 4624 9 6902 9 8388 9 9494 10 0000 30 477i 50 6990 7O 8451 90 9542 1 2 3 0414 0792 "39 1 1 1 49H 2 ! 5052 ! 5185 1 2 3 7076 7160 7243 1 2 3 8513 8573 8633 1 2 3 9590 9638 9685 4 1461 4 | 5315 4 7324 4 8692 4 973 1 5 6 1761 2041 5 5441 5563 5 6 7404 7482 5 6 8751 8808 5 6 9777 9823 7 * 9 2304 2553 2788 7 5682 8 5798 9 5911 7 9 7559 7634 7709 7 8 9 8865 8921 8976 7 8 9 9868 9912 995 6 20 3010 4O 6021 6O 7782 80 9031 too oooo N. Log. N. Log. N. Log. N. Log. N. Log. 1 Small Arcs. Small Arcs. s h m s s h TO s II Q 1 II II Q 1 II = 000 50 = o o 50 10 = 00 10 60=00 2O = O O 2O 7O =: O IO 30 . = o o 30 So = o 20 40 = o o 40 90 = o 30 50 = o o 50 loo = 40 TABLE VI Logarithms of Numbers and Small Arcs. No. O 1 2 3 4 5 6 7 8 9 100 oooo 0004 0009 0013 0017 0022 0026 0030 0035 0039 1 2 3 4 5 6 7 8 9 0043 0086 0128 0170 O2I2 0253 0294 0334 0374 0048 0090 OI 33 oi75 0216 02 57 0298 0338 0378 0052 0095 0137 0179 0220 026l 0302 0342 0382 0056 0099 0141 0183 0224 0265 0306 0346 0386 0060 0103 0145 0187 0228 0269 0310 0350 0390 0065 OIO7 0149 0191 0233 0274 03H 0354 0394 0069 OIII 0154 oi95 0237 0278 0318 0358 0398 0073 0116 0158 0199 0241 0282 0322 0362 0402 0077 OI2O Ol62 O2O4 0245 0286 0326 3 66 0406 0082 0124 0166 0208 0249 0290 0330 0370 0410 0449 110 0414 0418 0422 0426 0430 0434 0438 0441 0445 1 2 3 4 5 6 7 8 9 0453 0492 Q53 1 0569 0607 0645 0682 0719 0755 0457 0496 535 0573 0611 0648 0686 0723 0759 0461 0500 0538 0577 0615 0652 0689 0726 0763 0465 0504 0542 0580 0618 0656 0693 730 0766 0469 0508 0546 0584 0622 0660 0697 0734 0770 0473 0512 55 0588 0626 0663 0700 0737 0774 0477 05*5 0554 0592 0630 0667 0704 0741 0777 0481 0519 0558 0596 0633 0671 0708 0745 0781 0484 0523 0561 0599 0637 0674 O7II 0748 0785 0488 0527 0565 0603 0641 0678 0715 0752 0788 I 120 0792 0795 0799 0803 0806 0810 0813 0817 0821 0824 1 2 3 4 5 6 7 8 9 0828 0864 0899 0934 0969 1004 1038 1072 1106 0831 0867 0903 0938 0973 1007 1041 '075 1109 0835 0871 0906 0941 0976 IOII 1045 1079 III3 0839 0874 0910 0945 0980 1014 1048 1082 1116 0842 0878 0913 0948 0983 1017 1052 1086 1119 0846 0881 0917 0952 0986 IO2I 1055 1089 "23 0849 0885 0920 0955 0990 1024 1059 1092 1126 0853 0888 0924 0959 0993 1028 1062 1096 1129 0856 0892 0927 0962 0997 1031 1065 1099 "33 0860 0896 0931 0966 1000 I0 35 1069 1103 1136 130 "39 "43 1146 "49 "53 1156 "59 1163 1166 1169 1 2 3 4 5 6 7 8 9 "73 1206 1239 1271 1303 1335 1367 1399 143 1176 1209 1242 1274 I37 1339 1370 1402 H33 "79 1212 1245 1278 I3IO 1342 1374 1405 H36 1183 1216 1248 1281 1313 1345 1377 1408 1440 1186 1219 1252 1284 1316 1348 1380 1411 1443 1189 1222 1255 1287 I3 J 9 I35i 1383 1415 1446 "93 1225 1258 1290 1323 1355 1386 1418 1449 1196 1229 1261 1294 1126 1358 1389 1421 H52 1199 I2O2 1232 1235 1265 1268 1297 1300 1329 1332 1361 1364 1392 1396 1424 1427 1455 1458 140 1461 1464 1467 1471 H74 H77 1480 1483 1486 1489 1 2 3 4 5 6 7 8 9 1492 1523 1553 1584 1614 1644 1673 1703 1732 1495 1526 1556 1587 1617 1647 1676 1706 1735 I 49 8 1529 1559 1590 l62O 1649 1679 1708 1738 1501 1532 1562 1593 1623 1652 1682 1711 1741 i54 1535 1565 '596 1626 1655 1685 1714 1744 1508 1538 i5 6 9 1599 1629 1658 1688 1717 1746 15" 1541 1572 1602 1632 1661 1691 1720 1749 I5H 1544 1575 1605 1635 1664 1694 1723 1752 1517 1547 1578 1608 1638 1667 1697 1726 1755 1520 1550 1581 1611 1641 1670 1700 1729 1758 150 1761 1764 1767 1770 1772 1775 1778 1781 1784 1787 s h TO s s h m s II Q 1 II II t II 100 = o i 40 1000 = o 16 40 no = o i 50 iioo o 18 20 J2O = O2 O 12OO O 2O O 130 = 2 10 1300 = 21 40 J4O = O 2 2O I4OO O 23 2O 150 = 2 30 1500 = 25 TABLE VI. Logarithms of Numbers and Small Arcs. No. 1 % 3 4 5 6 | 8 9 150 1761 1764 I76 7 1770 1772 1775 1778 1 I78l 1784 1787 1 9 3 4 6 ! * 9 1790 1818 1847 1875 1903 193 i 1959 1987 2014 1793 1821 1850 1878 1906 1934 1962 I 9 8 9 2OI7 1796 1824 1853 1881 1909 1937 i9 6 5 1992 2019 1798 1827 1855 1884 1912 1940 1967 1995 2O22 1801 1830 1858 1886 1915 1942 1970 1998 2025 1804 1833 1861 1889 1917 !945 1973 2OOO 2028 1807 1810 1836 1838 1864 j 1867 1892 | 1895 1920 ! 1923 1948 i 1951 1976 1978 2003 : 2006 2030 2033 1813 1841 1870 1898 1926 1953 1981 2009 2036 1816 1844 1872 1901 1928 1956 1984 201 1 2038 i 160 2041 2044 2047 20 49 2052 2055 2057 2060 2063 2066 2092 2119 2146 2172 2198 2225 22 5 I 227O 2302 1 9 3 4 5 6 7 8 9 2068 2095 2122 2148 2J75 2201 2227 2253 2279 2305 2071 2098 | 2125 2I 5 I 2177 22O4 2230 2256 228l 2074 2101 2I2 7 2154 2180 2206 2232 2258 2284 2076 2103 2130 2156 2l8 3 2209 2235 226l 2287 2079 2106 2i33 2159 2185 2212 2238 2263 2289 2O82 2109 2135 2l62 2188 2214 2240 2266 2292 2084 2III 2I 3 8 2164 2191 2217 2243 2269 2294 2087 2II 4 2I4O 2167 2193 2219 2245 2271 2297 2090 2117 2143 2170 2196 2222 2248 2274 2299 170 2307 2310 2312 2315 2 3!7 2320 2322 2325 2327 ! 1 2 3 4 5 6 7 8 9 2330 2355 2380 2405 2430 2455 2480 2504 2529 2333 235* 2383 2408 2433 2458 2482 2507 2531 2335 2360 2385 2410 2435 , 2460 2485 2509 2533 2338 2363 2 3 88 2413 2438 2463 2487 2 5 I2 2 53 6 2340 2365 2390 2415 2440 2465 2490 25H 2538 2343 2368 2393 2418 2443 2467 2492 2516 2541 2345 2 37 2396 2420 2445 2470 2494 2519 2543 2348 2373 2398 2423 2448 2472 2497 2521 2545 2350 2375 2401 2425 2450 2475 2499 2524 2548 2353 , 2378 | 2403 i 2428 2453 ! 2477 ; 2502 2526 j 2550 ! 180 2553 2555 2558 2560 2562 2565 2567 2570 2572 2574 | 1 2 3 4 5 6 7 8 9 2577 2601 2625 2648 2672 2695 2718 2742 2765 2579 2603 2627 2651 2674 2697 2721 2744 2767 2582 2605 2629 2 653 2676 2700 2723 2746 2769 2584 2608 2632 2655 2679 2702 2725 2749 2772 2586 26lO 2634 2658 2681 2704 2728 2751 2774 2589 2613 2636 2660 2683 2707 2730 2753 2776 2591 2615 2639 2662 2686 2709 2732 2755 2778 2594 2617 2641 2665 2688 2711 2735 2758 2 7 8! 2596 2620 2643 2667 2690 27H 2737 2760 2783 2598 i 2622 2646 i 2669 : 2693 2716 2739 1 2762 ; 2785 j 19O 2788 2790 2792 2794 2797 2799 2801 2804 2806 2808 ! 1 2 3 4 5 6 7 8 9 28lO 2833 2856 2878 29OO 2923 2945 2967 2989 2813 2835 2858 2880 2903 2925 2947 2969 2991 2815 2838 2860 2882 2905 2927 2949 2971 2993 2817 2840 2862 2885 2907 2929 2951 2973 2995 28l9 2842 2865 2887 2909 2931 2953 2975 2997 2822 2844 2867 2889 2911 2934 2956 2978 2999 2824 2847 2869 2891 2914 2936 2958 2980 3OO2 2826 2849 2871 28 9 4 2916 2938 2960 2982 3004 2828 2851 2874 2896 2918 2940 2962 2984 3006 2831 2853 ! 2876 2898 2920 2942 2964 2986 3008 2OO 3OIO 3012 3I5 3017 30I 9 3021 3023 3 2 5 3028 3030 s h m s s h m s II O 1 II II V 1 11 150 = 2 30 1500 = 25 l6o =: O 2 40 I6OO = O 26 40 1 70 = O 2 50 1 7OO = O 28 2O 180 = 03 o 1800 = o 30 o 190 = 3 10 1900 = 31 40 200 = o 3 20 2000 = o 33 20 8 TABLE VI. Logarithms of Numbers and Small Arcs. No. O 1 3012 2 3 4 5 6 7 8 * \ 200 "IT 2 3 4 5 6 7 8 9 3010 3015 3oi7 3019 3021 3023 3025 3028 3030 i 3032 i 3034 354 35 6 375 ; 377 3096 3098 3118 3120 3139 3Hi 3160 i 3162 3181 3183 3201 3204 3036 3058 3079 3101 3122 3143 3164 3185 3206 3038 3060 3081 3 I0 3 3124 3H5 3166 3187 3208 34i 3062 3084 3^5 3126 3H7 3168 3189 3210 3043 3064 3086 3*07 3128 3149 3170 3*9i 3212 $& 3088 3109 3!3 3151 3172 3193 3214 347 3069 3090 3111 3132 3153 3'74 3'95 3216 349 3071 3092 3"3 3U4 3156 3176 3197 3218 3051 ; 373 394 3"5 3*37 3158 ! 3179 3199 3220 ' 21O 3222 3224 3226 3228 323 3233 3235 3237 3239 3241 | 1 2 3 4 5 6 7 8 9 3243 3263 3284 3304 3324 3345 3365 3385 3404 3245 3265 3286 3306 3326 3347' 3367 3387 3406 3426 3247 3267 3288 3308 3328 3349 3369 3389 3408 3249 3270 3290 33io 3330 3351 337i 3391 34io 3251 3272 3292 3312 3332 3353 3373 3393 3412 3253 3274 3294 33H 3334 3355 3375 3395 34H 3255 3276 3296 33i6 3336 3357 3377 3397 34i6 3257 3278 3298 33i8 3339 3359 3379 339| 34i8 3259 3280 3300 3320 334i 336i 338i 3400 3420 3261 3282 | 3302 i 3322 3343 3363 3383 3402 3422 22O 3424 3428 343 3432 3434 343 6 3438 3440 3442 1 2 3 4 5 6 7 8 9 3444 3464 3483 35 2 3522 354i 35 60 3579 3598 3446 3465 3485 3504 3524 3543 3562 358i 3600 3448 3467 3487 35o6 3526 3545 3564 3602 3450 3469 3489 3508 3528 3547 3566 3585 3604 3452 347i 349i 35io 353 3549 3568 3 I 8 Z 3606 3454 3473 3493 3512 3531 3551 3570 3456 3475 3495 35H 3533 3553 3572 359i 3610 3458 3477 349*7 35i6 3535 3555 3574 3593 3612 3460 3479 3499 35i8 3537 3556 3576 3595 3614 3462 348i 3501 3520 3539 3558 3577 3596 3615 230 3617 3619 3621 3623 3625 3627 3629 3630 3632 3 6 34 1 2 3 4 5 6 7 8 9 3636 3655 3 6 74 3692 37" 3729 3747 3766 3784 3638 3657 3675 3694 3713 3731 3749 3768 3786 3640 3659 3 6 77 3696 37H 3733 375i 3788 3642 3660 3 6 79 3698 3716 3735 3753 3789 3 6 44 3662 3681 37oo 37i8 3736 3755 3773 379i 3646 3664 3683 3701 3720 3738 3757 3775 3793 3647 3666 3685 3703 3722 3740 3758 3777 3795 3 6 49 3668 3687 3705 3724 3742 3760 3779 3797 3651 $8 3707 3725 3744 3762 378o 3798 3653 3672 3690 3709 3727 3746 3764 3782 3800 3818 240 3802 3804 3806 3808 3809 3811 3813 3815 38i7 1 2 3 4 5 6 7 8 9 3820 3838 3856 3|74 3892 3909 3927 3945 3962 3822 3840 3858 3876 3893 39" 3929 3946 3964 3824 3842 3860 3877 3895 3913 393 3948 3965 3826 3844 3861 3879 3897 3915 3932 3950 3967 3827 3f45 3863 3881 3*99 3916 3934 3952 3969 3829 3847 3865 3883 3901 39i8 393 6 3953 3971 3831 3849 3867 3885 3902 3920 3938 3955 3972 3833 3851 3869 3886 3904 3922 3939 3957 3974 3f35 3852 3870 3888 3906 3923 3941 3959 3976 .3*36 3854 3872 3890 3908 3925 3943 3960 3978 250 3979 398i 3983 3985 3986 3988 | 3990 | 3992 3993 3995 s h m s s h m s 1. Q 1 II II / // 200 = 3 20 2000 33 2O 210 = o 3 30 2100 = o 35 o 22O = O 3 40 22OO = O 36 40 230 =: O 3 5O 23OO = O 38 2O 240 = 04 o 2400 = o 40 o 250 = o 4 10 2500 = 41 40 TABLE VI. j O Logarithm us of Numbers and Small Ares. 1 No. 1 2 3 4 5 6789 250 3979 398i 3983 3985 3.986 3988 3990 3992 3993 i 3995 1 2 3 4 5 6 7 9 3997 4014 403 * 4048 4065 4082 4099 4116 4133 3998 4016 4033 4050 4067 4084 4101 4118 4000 4017 4035 4052 4069 4086 4103 4120 4136 4002 4019 4036 4053 4071 4087 4104 4121 4138 4004 4021 4038 4055 4072 4089 4106 4123 4140 4005 4023 4040 4057 4074 4091 4108 4125 4141 4007 4024 4041 4059 4076 4093 4109 4126 4H3 4009 4026 4043 4060 4077 4094 4111 4128 4H5 4011 4028 4045 4062 4079 4096 4"3 413 4146 4012 4029 4047 4064 4081 4098 4115 4131 4148 260 4150 4151 4153 4155 4156 4158 4160 4161 4163 4165 1 3 4 5 6 7 8 9 4166 4183 4200 4216 4232 4249 4265 4281 4298 4168 4185 4201 4218 4234 4250 4267 4283 4299 4170 4186 4203 4219 4236 4252 4268 4285 43 l 4171 4188 4205 4221 4237 4254 4270 4286 4302 4173 4190 4206 4223 4239 4255 4272 4288 4304 4175 4191 4208 4224 4241 4257 4273 4289 4306 4176 4193 4209 4226 4242 4259 4275 4291 437 4178 4195 4211 4228 4244 4260 4276 4293 439 4180 4196 4213 4229 4246 4262 4278 4294 4i8r 4198 4214 4231 4247 4263 4280 4296 4312 270 43 Z 4 43i5 4317 4318 4320 4322 4323 4325 4326 4328 1 2 3 4 5 6 7 9 433 4346 4362 4378 4393 4409 4425 4440 445 6 433 i 4347 4363 4379 4395 44" 4426 4442 4458 4333 4349 4365 438i 4396 4412 4428 4444 4459 4335 4366 4382 4398 4414 443 4445 4461 433 6 4352 4368 4384 4400 4415 443 * 4447 4462 4338 4354 4370 4385 4401 4417 4433 4448 4464 4339 4355 437i 4387 4403 4419 4434 445 44 6 5 4341 4357 4373 4389 4404 4420 443 6 445i 4467 4342 4358 4374 4390 4406 4422 4437 4453 4468 4344 4360 4376 4392 4408 4423 4439 4454 4470 280 4472 4473 4475 4476 4478 4479 4481 4482 4484 4486 1 2 3 4 5 6 7 9 4487 4502 4518 4533 45 6 4 4579 4594 4609 4489 4504 4519 4535 4550 4565 4580 4595 4610 4490 4506 4521 4536 4552 4567 4582 4597 4612 4492 457 4522 4538 4553 4568 4583 4598 4613 4493 459 4524 4539 4555 4570 4600 4615 4495 4526 4541 4556 457i 4586 4601 4616 4496 45 12 4527 4542 4558 4573 4588 4603 4618 4498 4513 4529 4544 4559 4574 4589 4604 4619 4499 4515 4530 4545 4576 459i 4606 4621 45 i 4532 4547 45 6 2 4577 4592 4607 4622 290 4624 4625 4627 4628 4630 4631 4633 4634 4636 4637 1 2 3 4 5 6 7 8 9 4639 4654 4669 4683 4698 4713 4728 4742 4757 4640 4655 4670 4685 4700 47H 4729 4744 4758 4642 4657 4672 4686 4701 4716 473 4745 4760 4643 4658 4 6 73 4688 4703 4717 4732 4747 4660 4675 4689 4704 4733 4748 4763 4646 4661 4676 4691 4706 4720 4735 4749 4764 4648 4663 4678 4692 4707 4722 473 6 4751 4765 4649 4664 4679 4694 4709 4723 4738 4752 4767 4651 4666 4681 4695 4710 4725 4739 4754 4768 4652 4667 4682 4697 4711 4726 4755 4770 4784 i 300 477i 4773 4774 4776 4777 1 4778 4780 478i 4783 i s h m s II O 1 II 250 = o 4 10 260 = o 4 20 270 = o 4 30 280 o 4 40 290 = o 4 50 300 =.05 o 8 h in s II Q 1 II 2500 o 41 40 2600 ;= o 43 20 2700 = o 45 o 2800 o 46 40 2900 = 48 20 3000 = o 50 o 10 TABLE VI. Logarithms of Numbers and Small Ares. No. O 1 4773 2 3 4 5 7 9 47S4 300 477i 4774 4776 4777 . 4778 4780 | 4781 4783 1 2 3 4 5 6 7 8 9 4786 4800 4814 4829 4843 4857 4871 4886 4900 4787 i 4789 4802 i 4803 4816 j 4817 4830 ! 4832 4844 i 4846 4859 ! 4 86o 4873 i 4874 4887 4888 4901 4902 4790 4804 4819 4833 4847 4861 4876 4890 4904 479i 4806 4820 4834 4849 4863 4877 4891 4905 4793 4807 4822 4836 4850 4864 4878 4893 4907 4794 4809 4823 4837 4852 4866 4880 4894 4908 4922 4796 4810 4824 4839 4853 4867 4881 4895 4909 4797 4812 4826 4840 4854 4869 4883 4897 4911 4799 4813 4827 , 4842 ! 4856 4870 4884 4898 4912 310 ~T 2 3 4 5 6 y 9 4914 4915 | 4916 4918- 4919 4921 4923 4925 4926 4928 4942 4955 4969 4983 4997 5011 5024 5038 4929 4943 4957 4971 4984 4998- 5012 5026 5039 493 4944 4958 4972 4986 5000 5 OI 3 5027 5041 4932 4946 4960 4973 4987 5001 5i5 5028 5042 4933 4947 4961 4975 4989 5002 5016 5030 543 4935 4949 4962 4976 4990 5004 5 OI 7 503i 545 493 6 4950 4964 4978 4991 5005 5 OI 9 5032 5046 4937 4951 4965 4979 4993 5006 5020 534 547 4939 4953 4967 4980 4994 5008 5022 535 549 4940 4954 4968 4982 4996 5009 5023 537 55 320 1 2 3 4 5 6 7 9 5052 5053 5054 5056 5057 5058 5060 5061 575 5088 5101 5"5 5128 5Hi 5i55 5168 5181 5062 5064 5065 579 5092 5io5 5"9 5132 5H5 5159 5'72 5066 5080 5093 5 I0 7 5120 5134 5H7 5160 5173 5068 5081 5095 5108 5122 5i35 5H8 5161 5i75 5069 5083 5096 5109 5123 5136 5H9 5^3 5 J 76 5070 5084 597 5"i 5!24 5138 5 T 5 l6 4 5*77 5072 5085 599 5112 5126 5 ! 39 5152 5i 6 5 5 r 79 5073 5087 5100 5"3 5 I2 7 5 HO 5153 5i 6 7 5180 ias 5 I0 3 5116 5*3 SMS 5156 5169 5183 5077 5Q9i 5104 5118 513 1 5*44 5157 5 1 / 1 5184 330 5185 5186 5188 5189 5190 5!92 5193 5i94 5190 319? 5210 5223 5236 5249 5262 5275 5288 53i 53H 1 2 3 4 5 6 7 8 9 5198 5211 5224 5237 5250 5263 5276 5289 5302 5200 5213 5226 5239 5252 5265 5278 5290 533 5201 5214 5227 5240 5253 5266 5279 5292 535 5202 5215 5228 5241 5254 5267 5280 5293 5306 5204 5217 5230 5243 5256 5269 5281 5294 5307 5205 5218 5231 5244 5257 5270 5283 5296 5308 5206 5207 5219 5221 5232 5234 5245 5247 5258 5260 5271 5272 5284 : 5285 5297 5298 53io 53" 5209 5222 5235 5248 5261 5274 5287 5299 53 M 5325 5338 535 5363 53g S3 5 * 54oi 54i3 5420 5438 340 5315 53i6 5317 53*9 5320 532i 5322 5324 5326 1 2 3 4 5 6 7 8 9 5328 5340 5353 5366 5378 539i 5403 5416 5428 5329 5342 5354 5367 5379 5392 5405 5417 . 5430 533 5343 5355 5368 538i 5393 5406 54i8 543i 533i 5344 5357 5369 5382 5395 5407 5420 5432 5333 5345 5358 537i 5383 5396 5408 5421 5433 5334 5347 5359 5372 5384 5397 54io 5422 5434 5335 5348 536i 5373 5386 5398 54" 5423 ,5436 5336 5349 5362 5374 5387 5400 5412 5425 5437 5339 5352 5364 5377 5390 5402 54i5 5427 5439 350 5441 5442 5443 5444 5446 5447 5448 5449 545i 5452 8 h m s s h m II 1 II II 1 300 = 05 o 3000 = o 50 310 = 5 10 3100 = 51 320 = o 5 20 3200 = 53 33 = o 5 3 . 33 = 55 340 = o 5 40 3400 = o 56 350 = o 5 50 3500 = o 58 s o 40 20 o 40 20 TABLE VI. 11 Logarithms of Numbers and Small Ares. No. 1 2 .3 4 5 6 7 S 9 350 544i 5442 5443 ! 5444 5446 5447 5448 5449 545i 5452 1 2 3 4 5 6 7 8 9 5453 5465 5478 5490 55 02 55i5 55 2 7 5539 555i 5454 5467 5479 549i 5504 55i6 5528 5540 5552 5456 5468. 548o 5492 5505 5517 5529 554i 5553 5457 5469 548i 5494 55o6 55i8 5530 5542 5555 5458 5470 5483 5495 557 5519 5532 5544 5556 5459 5472 5484 5496 5508 552i 5533 5545 5557 5460 5473 5485 5497 55io 5522 5534 5546 5558 5462 5474 5486 5499 55" 5523 5535 5547 5559 5463 5475 5488 55oo 5512 5524 5536 5549 556i 5464 5477 5489 55 01 55i3 5525 5538 5550 5562 36O 5563 5564 55 6 5 557 5568 5.569 5570 557i 5573 5574 1 2 3 4 5 6 7 8 9 5575 5587 5599 5611 5623 5635 5 6 47 5658 5670 5576 5588 5600 5612 5624 5636 5648 5660 5671 5577 5589 5601 5613 5 62 5 5637 5 6 49 5661 5673 5579 559i 5603 5615 5627 5638 5650 5662 5 6 74 558o 5592 5604 5616 5628 5640 5651 5663 5 6 75 558i 5593 5605 5617 5629 5641 5653 5664 5676 5582 5594 5606 5618 5630 5642 5654 5666 5 6 77 5583 5595 5607 5 6l 9 5631 5643 5655 5667 5678 5585 5597 5609 5621 5 6 3 2 5 6 44 5656 5668 5680 5586 5598 5610 5622 5634 5645 5657 5669 5681 370 5682 5683 5684 5686 ! 5687 5688 5689 5690 5691 5693 1 2 3 4 5 6 7 9 5 6 94 5705 5717 5729 5740 5752 5763 5775 ' 5786 5695 5707 57i8 5730 574i 5753 5765 5776 5788 5696 57o8 5719 573i 5743 5754 5766 5777 5789 5 6 97 5709 572i 5732 5744 5755 5767 5778 5790 5698 57io 5722 5733 5745 5757 5768 578o 579i 57 57" 5723 5735 5746 5758 5769 578i 5792 57oi 5712 5724 5736 5747 5759 5770 5782 5793 5702 57*4 5725 5737 5748 5760 577i 5783 5794 573 5715 5726 5738 5750 576i 5773 5784 5796 5704 57i6 5728 5739 575i 5762 5774 5785 5797 380 5798 5799 5800 5801 5802 5804 5805 5806 5807 5808 1 2 3 4 5 6 7 8 9 5809 5821 5832 5843 5855 5866 5877 5888 5900 5810 5822 5833 5844 5856 5867 5878 5889 590i 5812 5823 5834 5846 5857 5868 5879 5891 5902 5813 5824 5835 5847 5858 5869 5880 5892 5903 58H 5825 5837 5848 5859 5870 5882 5893 5904 5815 5826 5838 5849 5860 5871 5883 5894 5905 5816 5827 5839 5850 5861 5873 5884 5895 5906 5817 5829 5840 5851 5863 5874 5885 5896 5907 5818 5830 5841 5852 5864 5875 5886 5897 5908 5820 5831 5842 5853 5865 5876 5887 5898 59io 39O 59" 5912 5913 59M 5915 59i6 5917 5918 | 5920 5921 1 2 3 4 5 6 7 8 9 5922 5933 5944 5955 5966 5977 5988 5999 6010 5923 5934 5945 5956 5967 5978 5989 6000 60 1 1 5924 5925 5926 5935 593 6 5937 5946 5947 5948 5957 : 5958 i 5959 5968 5969 5970 5979 : 598o 5981 5990 ! 5991 5992 6OOI : 6OO2 6OO3 6012 I 6013 6014 5927 5938 5949 5960 597i 5982 5993 6004 6015 5928 5940 595i 5962 5973 5984 5994 6005 6016 593 5941 5952 5963 5974 5985 5996 6006 6017 593 i 5942 5953 5964 s H 6018 5932 5943 5954 5965 5976 5987 5998 6009 6020 400 6021 6022 6023 6024 6025 6026 6027 6028 6029 6030 /i " 350 = 360 = o -?7o = o 380 = 390 = o 400 = o m x a h m s in n & i it S 50 3500 = 58 20 6 o 3600 =i oo 6 10 37 i i 40 6 20 3800 =i 3 20 6 30 3900 i 5 o 6 40 4000 =i 6 40 12 TABLE VI. Logarithms of Numbers and Small Arcs. No. 1 2 3 4 5 . 6 7 8 9 400 6021 6022 6023 6024 6025 6026 6027 6028 6029 6030 1 2 3 4 A 6 7 8 9 6031 6042 6053 6064 6075 6085 6096 6107 6117 6033 6043 6054 6065 6076 6086 6097 6108 6118 6034 6044 6055 6066 6077 6087 6098 6109 6119 SI 6056 6067 6078 6088 6099 6110 6120 6036 6047 6057 6068 6079 6090 6100 6m 6121 6037 6048 6058 6069 6080 6091 6101 6lI2 6123 6038 6049 6060 6070 6081 6092 6102 6113 6124 6134 6039 6050 6061 6071 6082 6093 6103 6114 6125 6040 6051 6062 6072 6083 6094 6104 6115 6126 6041 6052 6063 6073 6084 6095 6106 6116 6127 SCO 6128 6129 6130 6131 6132 6l 33 6135 6146 6156 6167 6177 6188 6198 6209 6219 6229 6136 "37 1 2 3 4 5 6 7 8 9 6138 6149 6160 6170 6180 6191 6201 6212 6222 6139 6150 6161 6171 6182 6192 6202 6213 6223 6141 6151 6162 6172 6183 6193 6203 6214 6224 6142 6152 6163 6i73 6184 6194 6204 6215 6225 6i43 6i53 6164 6174 6185 6i95 6206 6216 6226 6237 6144 6i54 6165 6i75 6186 6196 6207 6217 6227 6145 6 i55 6166 6176 6187 6197 6208 6218 6228 6147 6 i57 6168 6178 6189 6199 6210 6220 6230 6148 6158 6169 6179 6190 6200 6211 6221 6231 6242 I '2 > 6232 6234 6235 6236 6238 6239 6240 6241 1 2 3 4 5 6 7 8 9 6243 6$ 6274 6284 6294 6304 63H 6 3 2 5 6244 6254 6264 till 6295 6305 ^l 6326 6245 6255 6265 6276 6286 6296 6306 6316 6327 6246 6256 6266 6277 6287 6297 6307 6$ 6247 6268 6278 6288 6298 6308 6318 6329 6248 6258 6269 6279 6289 6299 6309 6320 6 33 6249 6259 6270 6280 6290 6300 6310 6321 6 33J 6250 6260 6271 6281 6291 6301 6311 6322 6332 6251 6261 6272 6282 6292 6302 6312 6323 6333 6252 6262 6273 6283 6293 6303 6313 6324 6 334 i 430 6335 6336 6337 6338 6339 6340 6341 6342 6343 6344 1 tt 3 4 5 6 7 8 9 6345 6355 6 365 6 375 6385 6395 6405 6415 6425 6346 6$ 6$ 6396 6406 6416 6426 6347 6357 6367 6377 6387 6397 6407 6417 6427 6348 6358 6368 6378 6388 6398 6408 6418 6428 6 349 6359 6369 6379 6389 6399 6409 6419 6429 6350 6360 6370 6380 6390 6400 6410 6420 6430 635i ' 6361 6 37i 6381 6391 6401 6411 6421 643 * 6 35 2 6362 6372 6382 6392 6402 6412 6422 6432 6353 6363 6373 6383 6393 6403 6413 6423 6 433 6354 6364 6374 6384 6394 6404 6414 6424 6434 440 6435 6436 6437 6437 6438 6439 6440 6441 6442 6 443 1 2 3 f 6 7 9 6444 6454 6464 6474 6484 6493 6503 6513 6522 6445 6455 6465 6475 6485 6494 6504 6514 6523 6446 6456 6466 6476 6486 6495 6505 6515 6524 6447 6457 6467 6477 6487 6496 6506 6516 6525 6448 6458 6468 6478 6488 6497 6507 6517 6526 6449 6459 6469 6479 6488 6498 6508 6518 6527 6450 6460 6470 6480 6489 6499 6509 6519 6528 6451 6461 6471 6481 6490 6500 6510 6520 6529 6452 6462 6472 6482 6491 6501 6511 6521 6530 6 453 6463 6473 6483 6492 6502 6512 6522 6 53! ; 45O 6532 6533 6534 ! 6535 6536 6537 | 6538 6539 6540 6541 s h m s s h m s II Q 1 II II 1 It 400 = o 6 40 4000 = 6 40 410 = o 6 50 4100 = S 20 420 = 07 o 4200 = 10 o 430 = 7 10 43= IJ 40 440 = o 7 20 4400 13 20 45 = 7 3 45 = *5 c TABLE VI. Logarithms of Numbers and Small Arcs. No. 450 1 g 8 4 5 6 7 8 9 o i a s 4 567 8 6540 9 <>532 6542 6551 6561 6571 6580 6590 6599 6609 6618 6533 6534 6535 6545 6554 6564 6573 6583 6593 6602 6612 6621 6536 6537 6538 6539 6541 6543 6552 6562 6572 6581 6591 6600 6610 6619 6544 6553 6563 6572 6582 6592 6601 6611 6620 6546 6555 6565 6574 6584 6593 6603 6612 6622 6547 6556 6566 6575 6585 6594 6604 6613 6623 6548 6557 6567 6576 6586 6595 6605 6614 6624 6549 6558 6568 6577 6587 6596 6606 6615 6625 6549 6559 6569 6578 6588 6597 6607 6616 6626 6550 6560 6570 6579 6589 6598 6608 6617 6627 460 6628 6629 6629 6630 6631 6632 6633 6634 6635 6636 1 2 3 4 5 7 8 9 6637 6646 6656 6665 6684 6693 6702 6712 6638 6647 6657 6666 6675 6685 6694 6703 6713 6639 6648 6658 6667 6676 6686 6695 6704 6714 6640 6649 6659 6668 6677 6687 6696 6705 6715 6641 6650 6660 6669 6678 6688 6697 6706 6715 6642 6651 6661 6670 6679 6689 6698 6707 6716 6643 6652 6661 6671 6680 6689 6699 6708 6717 6644 6653 6662 6672 6681 6690 6700 6709 6718 6645 6654 6663 6673 6682 6691 6701 6710 6719 6645 6655 6664 6674 6683 6692 6702 6711 6720 470 6721 6722 6723 6724 6725 6726 6727 | 6727 6728 6729 1 2 3 4 5 6 7 9 6730 6739 6/49 6758 6767 6776 6785 6794 6803 6731 6740 6750 6759 6768 6777 6786 6 795 6804 6732 6741 6750 6760 6769 6778 6787 8g 6733 6742 675 1 6761 6770 6779 6788 &i 6734 6743 6752 6761 6771 6780 6789 6798 6807 6735 6744 6753 6762 6772 6781 6790 6799 6808 6736 6745 6754 6763 6772 6782 6791 6800 6809 6737 6746 6755 6764 6773 6783 6792 6801 6810 6738 6747 6756 6765 6774 6783 6792 6802 6811 6739 6748 6757 6766 6775 6784 6793 6802 6812 480 6812 6813 6814 6815 6816 6817 6818 6819 6820 6821 1 2 3 ! 7 i 8 9 6821 6830 6839 6848 6857 6866 6875 6884 6893 6822 6831 6840 6849 6858 6867 6876 6885 6894 6823 6832 6841 6850 68 59 6868 6877 6886 6895 6824 6833 6842 6851 6860 6869 6878 6887 6896 6825 6834 6843 6852 6861 6870 6879 6888 6897 6826 6835 6844 '6853 6862 6871 6880 6889 6898 6827 6836 6845 6854 6863 6872 6881 6890 6898 6828 6837 6846 3g 6873 6882 6890 68 99 6829 6838 6847 6856 6865 6874 6882 6891 6900 6830 6839 6848 6857 6865 6874 6883 6892 6901 49O 6902 6903 6904 6905 6906 6906 6907 6908 6909 " 6918 6927 6936 6944 ^953 6962 6971 6979 6988 6910 1 3 f 6 7 8 9 6911 6920 6928 6937 6946 6955 6964 6972 6981 6912 6921 6929 6938 6947 6956 6964 6973 6982' 6913 6921 6930 6939 6948 6957 6965 6974 6983 6913 6922 6931 6940 6949 6957 6966 6975 6984 6914 6923 6932 6941 6950 6958 6967 6976 6985 6915 6924 6933 6942 6950 6959 6968 6977 6985 6916 6925 6934 6943 695 1 6960 6969 6978 6986 6917 6926 6935 6943 6952 6961 6970 6978 6987 6919 6928 6936 6945 6954 6963 6971 6980 6989 500 6990 6991 6991 6992 6993 6994 6995 6996 6997 6998 s It m s s k m s II 1 II II o 1 II 450 7 30 4500 =115 o 460 = o 7 40 4600 =i 16 40 470 = o 7 50 4700 =i 18 20 480 = 08 o 4800 = i 20 o 490 r= O 8 IO 49OO=i I 21 40 500 = o 8 20 5000 = i 23 20 14 TABLE VI. 1 No. Logarithm* of Number* and Small Arcs. 8 o i i 234 5 6 7 9 5OO 6990 6991 6991 6992 7001 7010 7018 7027 7035 7044 7053 7061 7070 6993 7002 7011 7019 7028 7036 7045 754 7062 i 7071 6994 6995 6 99 6 6997 ! 6998 7006 7015 7023 7032 7041 7049 7058 7066 7075 1 2 3 4 5 6 7 8 9 6998 7007 7016 7024 7033 7042 7050 7059 7067 6999 7008 7017 7025 7034 7042 7051 7059 7008 7000 7009 7017 7026 7035 7043 7052 7060 7069 7003 7011 7020 7029 7037 7046 7054 7063 7071 7004 7012 7021 7029 7038 7047 7055 7064 7072 7004 7013 7022 7030 7039 7048 7056 7065 7<>73 7005 i 7014 : 7023 703 I 7040 7048 7057 7074 510 7076 7077 7077 7078 i 7079 7080 7081 7082 7083 j 7083 1 2 3 4 5 6 7 9 7084 7093 7101 7110 7118 7127 7135 7H3 7152 7085 7094 7102 7110 7119 7127 7136 7144 7153 7086 7094 7103 7111 7120 7128 7137 7H5 7153 7162 7087 7095 7104 7112 7121 7129 7137 7146 7154 7088 7096 7105 7121 7130 7138 7147 7155 7088 7097 7114 7122 7131 7139 7H7 7106 7U5 7123 7132 7140 7148 7157 7090 7099 7107 7116 7124 7132 7141 7H9 7158 7091 7099 7108 ; 7116 7125 7133 7142 ; 7150 7158 7092 7100 7109 7117 7126 7134 7142 7151 5 2O 7160 7161 7163 7163 7164 7165 7166 7167 i 7168 1 3 4 5 6 7 8 9 7168 7177 7185 7193 7202 7210 7218 7226 7235 7194 7202 7211 7219 7227 7235 7170 7178 7187 7195 7203 . 7212 7220 7228 7236 7171 71/9 7188 7196 7204 7212 7221 7229 7237 7172 7180 7188 7197 7205 7213 7221 7230 7238 7173 7181 7189 7197 7206 7214 7222 7230 7239 7173 7182 7190 7198 7207 7215 7223 7231 7239 7174 7183 7191 7199 7207 7216 7224 7232 7240 7175 7183 ; 7192 7200 7208 7216 7225 7233 7241 7176 7184 7192 7201 7209 7217 7226 7234 7242 530 7243 7244 7244 7245 7246 7247 7248 7248 7249 7257 7266 7274 7282 7290 7298 7306 73H 7322 7250 1 2 3 4 5 6 7 8 9 7251 7259 7267 7275 7284 7292 7300 7308 7252 7260 7268 7276 7284 7292 73 01 739 7V7 7253 7261 7269 7277 7285 7293 7301 739 7253 7262 7270 7278 7286 7294 7302 7254 7262 7271 7279 7287 7295 7303 73" 73 19 7255 7263 7271 7279 7288 7296 734 7312 7320 7256 7264 7272 7280 7288 7297 7305 7313 7321 7257 7265 7273 7281 7289 7297 735 7322 7258 7266 7275 7283 7291 7299 737 7323 540 7324 7325 7326 7326 7327 7328 7329 733 7330 7331 1 2 3 4 6 7 8 9 7332 7340 7348 7356 73 6 4 7372 7380 7388 7396 7333 734i 7349 7357 7365 7373 7389 7397 7334 7342 7350 7366 7374 738i 7389 7397 7334 7342 7350 7358 7366 7374 7382 7390 7398 7335 7343 7351 7359 73 6 7 7375 7383 7399 7336 7344 7352 7360 7368 7376 7384 7392 7400 7337 7345 7353 7361 73 6 9 7377 '7385 7393 7400 7338 7346 7354 7362 7370 7377 7385 7393 7401 7338 7340 7354 7362 7386 7394 7402 7339 7347 7355 7363 737i 7379 7387 7395 7403 55O 7404 7404 7405 7406 7407 7408 7408 7409 7410 7411 s h m s s h m s 1. / // H 1 'I 500 = 8 20 5000 =1 23 20 510-0 8 30 5100 = i 25 o 520 = o 8 40 5200 =i 26 40 530 o 8 50 5300 =i 28 20 540 = 09 o 540 =13 550 = o 9 jo 5500 = i 31 40 TABLE VI. 15 No. L.O; 1 ^aritlim* of Numbers and Small Ares. 2 3 4 5 6 ~7 4 o8 7 8 9 55O 7404 7412 74^9 7427 7435 7443 745i 7459 7466 7474 7404 7405 74i3 7421 7429 7437 7444 7452 7460 7468 747 6 7406 7407 7408 7409 7410 74" 1 2 3 4 5 6 7 8 9 7412 7420 7428 743 6 7444 7452 7459 7407 7475 7483 74H 7422 743 7437 7445 7453 7461 7469 7476 7415 7423 743 7438 7446 7454 7462 7469 7477 7415 7423 743 r 7439 7447 7455 7462 747 7478 7416 7424 7432 7440 7448 7455 7463 747i 7479 7417 7425 7433 744i 744 s 745 6 7464 7472 7480 7418 7426 7434 744i 7449 7457 7465 7473 7480 7419 7426 7434 7442 745g 7458 7466 7473 748i 560 7482 7483 7484 7485 7486 7487 7487 7488 7489 1 2 3 4 5 6 7 8 9 7490 7497 755 75*3 7520 7528 753 6 7543 755i 749 7498 7506 75 H 752i 7529 7537 7544 7552 75 60 7491 7492 7499 7500 757 757 7514 !. 7515 7522 7523 753 f 753 7537 , 753 8 7545 754 6 7553 ! 7553 7493 7500 7508 75i6 75 2 4 753i 7539 7547 7554 7494 75 01 7509 7517 7524 7532 7540 7547 7555 7494 7502 75io 7517 7525 7533 7540 7548 7556 7495 7503 75 10 75i8 7526 7534 754i 7549 7556 7496 754 75" 7519 7527 7534 7542 7550 7557 7497 7504 7512 7520 7527 7535 7543 7550 7558 570 v '.-l 2 3 4 5 6 7 * 9 7559 7560 7561 7562 75 6 3 75 6 3 75 6 4 7565 7566 7566 7574 7582 7589 7597 7604 7612 7619 7627 75 6 7 7575 7582 7590 7597 7605 7613 7620 7628 7568 75|S 7s3 759i 7598 7606 7613 7621 7628 7569 7576 7584 759i 7599 7614 7622 7629 7569 7577 7585 7592 7600 7607 7615 7622 7630 7570 7578 7585 7593 7600 7608 7616 7623 763! 757i 7579 7586 7594 7601 7609 7616 7624 7631 7572 7579 7587 7594 7602 7610 7617 7625 7632 7572 758o 7588 7595 7603 7610 7618 7625 7633 7573 758i 7588 7596 7603 7611 7619 7626 7634 580 7634 7635 7636 j 7637 7637 7638 7639 7640 7640 7641 1 2 3 4 5 6 7 8 9 7642 7649 7657 7664 7672 7679 7686 7694 7701 7643 ' 7650 7657 7665 7672 7680 7687 7695 7702 7643 7651 7658 7666 7673 7680 7688 7695 7703 7644 7651 7659 7666 7674 7681 7689 7696 7703 7645 7652 7660 7667 7675 7682 7689 7697 7704 7646 7653 7660 7668 7675 7683 7690 7697 7705 7646 7654 7661 7669 7676 7683 7691 7698 7706 7647 7654 7662 7669 7677 7684 7692 7699 7706 7648 7655 7663 7670 7677 7685 7692 7700 7707 7648 7656 7663 7671 7678 7686 7693 7700 7708 59O 7709 7709 7710 77" 77" 7712 7713 77H 77H 7715 1 2 3 4 5 6 8 9 7716 7723 773i 7738 7745 7752 7760 7767 7774 7717 7724 773i 7739 7746 7753 7760 7768 7775 7717 7725 7732 7739 7747 7754 7761 7768 7776 7718 7725 7733 7740 7747 7755 7762 7769 7776 7719 7726 7733 774i 7748 7755 7763 7770 7777 7720 7727 7734 7742 7749 7756 7763 7771 7778 7720 7728 7735 7742 775 7757 7764 7771 7779 7721 7728 7736 7743 7750 7758 7765 7772. 7779 7722 7729 7736 7744 775i 7758 7766 7773 778o 7722 773 7737 7744 7752 7759 7766 7774 778i 600 7782 7782 7783 7784 7784 7785 7786 7787 7787 7788 s h m s 8 h m $ II O 1 II II o 1 II 550 = o 9 10 5500 = 31 40 560 = o 9 20 5600 = 33 20 570 = o 9 30 5700 = 35 o 580 =r O 9 4O 58OO = 36 4O 590 = o 9 50 5900 = 38 20 600 = o 10 o 6000 = 40 o 16 TABLE VI. Logarithms of* Numbers and Small Arcs. No. o 1 2 j t 5 6 7 8 9 600 7782 7782 7783 7784 7784 7785 7786 7787 7787 77KX : 1 2 3 4 5 6 7 9 7789 7796 7803 7810 7818 7825 7832 7839 7846 7789 7804 7811 7818 7825 7833 78 4 o 7847 7790 7797 7805 7812 7819 7826 7833 7840 7848 7791 7798 7805 7813 7820 7827 7834 7841 7848 7792 7799 7806 7813 7820 7828 7835 7842 7849 7792 7800 7807 7814 7821 7828 7835 7843 75o 7793 7800 7807 7815 7822 7829 7836 7843 7850 7794 7801 7808 7815 7823 7830 7837 7844 7851 7795 7802 7809 7810 7823 7830 7838 7845 7852 7795 7802 78 10 7817 ! 7824 7831 7838 7845 7853 610 7853 7854 7855 7855 7856 7857 7864 7871 7878 7885 7892 7899 7906 79i3 7920 7858 7858 7859 786o i 1 2 3 4 5 6 7 8 9 7860 7868 7875 7882 7889 7896 7903 7910 7917 7861 7868 7875 7882 7889 7897 7904 7911 7918 7862 7869 7876 7883 7890 7897 7904 79 M 7918 7863 7870 7877 7884 7891 7898 7905 7912 7919 7863 7870 7877 7885 7892 7899 7906 7913 7920 7865 7872 7879 7886 7893 7900 7907 79H 7921 7865 7872 7880 7887 7894 7901 7908 7915 7922 7866 7873 7880 7887 7894 7901 7908 7916 7923 7867 i 7874 ! 7881 i 7888 ; 7895 ! 7902 ) 7909 7916 ! 7923 ! 620 7924 7925 7925 7926 7927 7927 7928 7929 7930 7930 1 2 3 4 6 7 i 9 7931 7938 7945 7952 7959 7966 7973 7980 7987 7932 7939 7946 7953 7959 7966 7973 7980 7987 7932 7939 7946 7953 7960 7967 7974 798i 7988 7933 7940 7947 7954 7961 7968 7975 7982 7989 7934 794i 7948 7955 7962 7969 797 S 7982 7989 7934 794i 7948 7955 7962 7969 7976 7983 7990 7935 7942 7949 7956 7963 7970 7977 7984 7991 7936 7943 795 7957 7964 7971 7978 7984 7991 7917 7943 795 7957 7964 7971 7978 7985 7992 7937 ; 7944 i 7951 7958 i 7965 ' 7972 : 7979 7986 ! 7993 63O 7993 7994 7995 7995 7996 7997 7998 7998 7999 | 8000 1 2 3 4 5 6 7 8 9 8000 8007 8014 8021 8028 8035 8041 8048 8055 8001 8008 8015 8022 8028 8035 8042 8049 8056 8002 8009 8015 8022 8029 8036 8043 8050 8056 8002 8009 8016 8023 8030 8037 8043 8050 8057 8003 8010 8017 8024 8030 8037 8044 8051 8058 8004 8011 8017 8024 8031 8038 8045 8052 8058 8004 8011 8018 8025 8032 8039 8045 8052 8059 8005 8012 8019 8026 8033 8039 8046 S32 8006 8013 8020 8026 8033 8040 8047 8054 8060 8006 i 8013 8020 i 8027 8034 ! 8041 1 8048 1 8054 ! 8061 640 8062 8062 | 8063 8064 8065 8065 8066 8067 8067 8068 1 2 3 4 5 6 7 9 9 8069 8075 8082 8089 8096 8102 8109 8116 8122 8069 80$ 8083 8090 8096 8103 8110 8116 8123 8070 8077 8083 8090 8097 8104 8110 8117 8124 8130 8071 8077 8084 8091 8098 8104 8111 8118 8124 8071 8078 8085 8092 8098 8105 8112 8118 8125 8072 8079 8085 8092 8099 8106 8112 8119 8126 8073 8079 8086 8093 8100 8106 8113 8120 8126 8073 8080 8087 8094 8100 8107 8114 8120 8127 8074 8081 8088 8094 8101 8108 8114 8121 8128 8075 8081 8088 i 8095 1 8102 i 8108 8115 8122 i 8128 1 8135 650 8129 8130 8131 8132 8132 8i33 8134 8i34 s h m s s h in II Q 1 II II O 1 600 = o 10 o 6000 = 40 610 = o 10 10 6100 = 41 620 = o 10 20 6200 = 43 630 = o 10 30 6300 = 45 640 = o 10 40 6400 = 46 650 = o 10 50 6500 = 48 8 it 4 2O O 40 20 TABLE VI. 17 Logarithms of Numbers and Small Arcs. No. O 1 2 3 4 567 8 9 650 8129 | 8129 8130 8131 8132 8132 8133 8134 8134 8i35 1 2 3 4 5 6 8 9 8136 8142 8149 8156 8162 8169 8176 8182 8189 8136 8i43 8150 S 8176 8183 8190 8137 8144 8150 8157 8164 8170 8177 8190 8138 8144 8151 8158 8164 8171 8178 8184 8191 8138 8i45 8152 8158 8165 8172 8178 8185 8191 8139 | 8140 8146 i 8146 8152 i 8153 8159 : 8160 8166 | 8166 8172 1 8173 8179 i 8180 8186 8186 8192 8193 8140 8141 8147 8148 8154 8154 8160 8161 8167 : 8168 8174 i 8174 8180 ', SiSi 8187 ! 8188 8193 8194 8142 8148 8155 8162 8168 ! 8175 1 8182 8188 8195 660 8i95 8196 8197 8i97 8198 8199 8199 8200 8201 8201 1 2 3 4 5 6 9 8202 8209 8215 8222 8228 8235 8241 8248 8254 8203 8209 8216 8222 8229 8235 8242 8248 8255 8203 8210 8216 8223 8230 8236 8243 8249 8256 8204 8211 8217 8224 8230 8237 8243 8250 8256 8205 8211 8218 8224 8231 8237 8244 8250 8257 8205 8212 8218 8225 8231 8238 8245 8251 8258 8206 8213 8219 8226 8232 8239 8245 8252 8258 8207 8213 8220 8226 8233 8239 8246 8252 8259 8207 8214 8220 8227 8233 8240 8246 8253 8259 8208 8214 8221 8228 - 8234 8241 8247 8254 8260 67O 8261 8261 8262 8263 8263 8264 8265 8265 8266 8267 1 2 3 4 5 6 9 8267 8274 8280 8287 8293 8299 8306 8312 8319 8268 8269 8274 8275 8281 ' 8281 8287 8288 8294 8294 8300 ' 8301 8307 8307 8313 8314 8319 8320 8269 8276 8282 8289 8295 8301 8308 8314 8321 8270 8276 8283 8289 8296 8302 8308 8315 8321 8270 \ 8271 8277 ' 8278 8283 j 8284 8290 8290 8296 ! 8297 8303 : 8303 8309 8310 8316 8316 8322 8323 8272 8272 8278 I 8279 8285 ; 8285 8291 8292 8298 ; 8298 8304 8305 8310 8311 8317 j 8317 8323 8324 8273 8280 8286 8292 8299 8305 8312 8318 8324 680 8325 8326 8326 8327 8328 8328 8329 8330 8330 8331 337 8344 8350 8356 8363 8 3 6 9 8375 8382 1 2 3 4 5 6 9 8332 8338 8344 835i 8357 8363 8370 8376 8382 8332 8338 8345 8351 8358 8364 8370 8377 8383 8333 8339 8345 8352 8358 8365 8371 8377 8383 8333 8340 8346 8352 8359 8365 8371 8378 8384 8334 8340 8347 8353 8359 8366 8372 8378 8385 8335 8341 8347 8354 8360 8366 8373 8379 8385 8335 8342 8348 8354 8361 8367 8373 8380 8386 8336 8342 8349 8355 8361 8368 8374 8380 8387 8337 8343 8349 8356 8362 8368 8375 8381 8387 690 8388 8389 8390 8390 8391 8392 8392 8393 8394 8394 1 2 3 4 5 6 8 9 8395 8401 8407 8414 8420 8426 8432 8439 8445 8395 8402 8408 8414 8420 8427 8433 8439 8445 8396 8402 8409 8415 8421 8427 8434 8440 8397 8403 8409 8415 8422 8428 8434 8440 8447 8397 8404 8410 8416 8422 8429 8435 8441 8447 8398 8404 8410 8417 8423 8429 8435 8442 8448 8399 8405 8411 8417 8424 8430 8436 8442 8449 8399 8405 8 4 I2 8418 8424 8430 8437 8443 8449 8400 8406 8412 8419 8425 8431 8437 8444 8450 8400 8407 8413 8419 8425 8432 8 43 8 8444 8450 7OO 8451 8452 8452 8453 8453 8454 8455 8455 8456 8457 s h m s II O 1 II 650 = o 10 50 660 = o 1 1 o 670 = 011 10 680 = 011 20 690 = o 1 1 30 700 =011 40 s n 6500 = 6600 = 6700 = 6800 = 6900 - 7000 = k m s / // = 48 20 r 50 o = 5 1 40 = 53 20 = 5 6 40 18 TABLE VI. Logarithms of Numbers and Small Ares. No. 1 2 3 1 5 6 7 8455 8 9 700 8451 8452 8452 8453 i 8453 - 454 : 455 8461 8467 8473 8479 8486 8492 8498 8504 8510 8456 8457 i 1 ? 3 4 5 6 7 8 9 8457 8463 8470 j 8476 8482 8488 8494 8500 8506 8458 8464 8470 8476 8 ^3 8489 8495 8501 8507 8458 8465 8471 8477 8483 8489 8495 8502 8508 8459 8460 8465 8466 8471 8472 8478 8478 8484 ; 8484 8490 ! 8491 8496 ' 8496 8^02 8503 8508 8509 8460 8466 8473 8479 i 8485 8491 8497 8503 8510 8462 8468 8474 8480 8486 8492 8498 8505 8511 8462 8468 8474 8481 8487 8493 8499 8505 8511 8463 8469 8475 8481 8487 8494 ! 8500 8506 8512 710 8513 8513 8514 8514 ; 8515 8 5 ib 8516 8522 8528 8535 8541 8547 8553 8 559 8565 857i 8517 5 r 7 8518 1 2 3 4 5 6 7 * 9 8519 8525 853i 8537 8543 8549 %% 8.,6i 8567 8519 8525 8532 8538 8544 8550 8556 8562 8568 8520 8526 8532 8538 8544 8550 85f 8562 8569 8521 i 8521 8527 8527 8533 : 8533 8539 8 539 8545 : 8545 8551 . 8552 8557 ! 8558 8563 ; 8564 8569 ! 8570 8522 8528 8534 8540 8546 8552 8558 8564 8570 8523 8529 8535 8541 8547 8553 8559 8565 8572 8524 8530 8536 8542 8548 8554 8560 8566 8572 8524 8530 8536 8542 8549 i 8555 ! 8561 | 8567 8573 720 8573 8574 8575 8575 8576 8576 8577 8578 8578 8579 ! 1 2 3 4 5 6 7 8 9 8579 8585 8591 8597 8603 8609 8615 8621 8627 8580 8586 8592 8598 8604 8610 8616 8622 8628 8581 8587 8593 8599 8605 8611 8617 8623 8628 8581 8587 8593 8599 8605 8611 8617 8623 8629 8582 8588 8594 8600 8606 8612 8618 8624 8630 8582 8588 8594 8600 8606 8612 8618 8624 8630 8583 8589 8595 8601 8607 8613 8619 8625 8631 8584 8590 8596 8602 8608 8614 8620 8625 8631 8584 8590 8596 8602 8608 8614 8620 8626 8632 8585 i 8591 8597 86o 3 8609 8615 8621 8627 8633 730 8633 8634 | 8634 8635 8636 8636 8637 8637 8638 8639 1 2 3 '4 5 6 7 8 9 8639 8645 8651 8657 8663 8669 8675 8681 8686 8640 8646 8652 8658 8663 8669 8675 8681 8687 8640 8646 8652 8658 8664 8670 8676 8682 8688 8641 8647 8653 8659 8665 8671 8676 8682 8688 8642 8647 8653 8659 8665 f 8671 8677 8683 8689 8642 8648 8654 8660 8666 8672 8678 8684 8689 8643 8649 8655 8661 8666 8672 8678 8684 8690 8643 8649 8655 8661 8667 8673. 8679 8685 8691 8644 8650 8656 8662 8668 8674 8679 8685 8691 8645 8650 8656 8662 8668 8674 8680 8686 8692 740 8692 8693 8693 8694 86 95 8695 8696 8696 8697 8698 1 2 3 4 5 6 7 8 9 8698 8704 8710 8716 8722 8727 8733 8739 8745 8699 8705 8710 8716 8722 8728 8734 8740 8745 8699 8705 8711 8717 8723 8729 8734 8740 8746 8700 8706 8712 8717 8723 8729 8735 8741 8747 8701 8706 8712 8718 8724 8730 8736 8741 8747 8701 8707 8713 8719 8724 8730 8736 8742 8748 8702 8708 8713 8719 8725 8731 8737 8743 8748 8702 8708 8714 8720 8726 8731 8737 8743 8749 8703 8709 8715 8720 8726 8732 8738 8744 8749 8703 8709 8715 8721 8727 8733 8738 8744 8750 750 8751 875i 8752 8752 8753 8754 8754 8755 8755 8756 s h m s s h m s II 1 II II Q 1 II 700 = 011 40 7000 =i 56 40 710 = o ii 50 7io =i 5 8 20 72O =OI2 O 72OO 2 O O 730 = o 12 10 73 = 2 i 4 740 12 20 7400 = 2 3 20 750 = O 12 30 7500 = 2 5 O TABLE VI. 19 Logarithms of Numbers and Small Arcs. No. 1 2 3 4 5 6 7 8 9 I 750 8751 I 8751 8752 8752 8753 8754 8754 i 8755 8755 8756 1 9 3 i I 5 6 7 i 9 875 6 8757 8762 ! 8763 8768 | 8769 8774 8774 8779 ! 8780 8785 i 8786 8791 i 8792 8/97 i 8797 8802 ! 8803 8758 8763 8769 ; 8775 : 8781 8786 ; 8792 i 8798 8804 8758 8764 8770 8775 ! 8781 8787 8793 8798 8804 8759 8764 8770 8776 8782 8788 8793 8799 8805 8759 8765 877i 8777 8782 8788 8794 8800 8805 8760 8760 8766 8766 8771 8772 8777 8778 8783 8783 8789 i 8789 8794 1 8795 8800 j 8801 8806 8806 8761 8767 8773 8778 8784 8790 8796 8801 8807 8762 8767 8773 8779 8785 8790 8796 8802 8808 76O 8808 8809 8814 8820 8826 8832 8837 8843 8849 8854 8860 8809 8810 8810 8811 8812 8812 8813 8813 1 2 3 4 5 6 7 9 8814 8820 8825 8831 8837 8842 8848 8854 8859 8815 8816 8821 8821 8826 8827 8832 : 8833 8838 i 8838 8843 | 8844 8849 1 8850 8855 8855 8860 8861 8816 8822 8828 8833 8839 8845 8850 8856 8862 8817 8822 8828 8834 8839 8845 8851 8856 8862 8817 8823 8829 8834 8840 8846 8851 8857 8863 8818 8824 8829 8835 8841 8846 8852 8858 8863 8818 8824 8830 8835 8841 8847 8852 8858 8864 8819 8825 8830 8836 8842 8847 8853 8859 8864 770 8865 8865 8866 -| 8867 8867 8868 8868 8869 8869 8870 1 2 3 4 5 6 7 8 9 8871 8876 8882 8887 8893 8899 8904 8910 8915 8871 8877 8882 8888 8894 8899 8905 8910 8916 8872 8877 8883 8889 8894 8900 8905 8911 8916 8872 8878 8883 8889 8895 8900 8906 8911 8917 8873 8878 8884 8890 8895 8901 8906 8912 8918 8873 8879 8885 8890 8896 8901 8907 8913 8918 8874 8880 8885 8891 8896 8902 8908 8913 8919 8874 8880 8886 8891 8897 8903 8908 8914 8919 8875 8881 8886 8892 8898 8903 8909 8914 8920 8876 8881 8887 8892 8898 8904 8909 8915 8920 780 8921 8922 8922 8923 8923 8924 8924 | 8925 8925 8926 1 2 3 4 5 6 7 8 9 8927 8932 8938 8943 8949 8954 8960 8965 8971 8927 & 8944 8949 8955 8960 8966 8971 8928 8933 8939 8944 8950 8955 8961 8966 8972 8928 8934 8939 8945 8950 8 95 6 8961 8967 8972 8929 8934 8940 8945 8951 8956 8962 8967 8973 8929 8935 8940 8946 8951 8957 8963 8968 8974 8930 8935 8941 8946 8952 8958 8963 8969 8974 8930 8936 8942 8947 8953 8958 8964 8969 8975 8931 8937 8942 8948 8953 8959 8964 8970 8975 8932 8937 89.43 8948 8954 8959 8965 8970 8976 790 8976 8977 8977 8978 8978 8979 8980 8980 8981 8981 1 2 3 4 5 6 7 8 9 8982 8987 8993 8998 9004 9009 9015 9020 9025 8982 8988 8993 8999 9004 9010 9015 9021 9026 8983 8988 8994 8999 9005 9010 9016 9021 9027 8983 8989 8994 9000 9005 9011 9016 9022 9027 8984 8989 8995 9000 9006 9011 9017 9022 9028 8985 8990 8995 9001 9006 9012 9017 9023 9028 8985 8991 8996 9002 90Q7 9012 9018 9023 9029 8986 8991 8997 9002 9007 9013 9018 9024 9029 8986 8992 8997 9003 9008 9013 9019 9024 9030 8987 8992 8998 9003 9009 9014 9019 9025 9030 SOO 9031 9031 9032 9033 9033 9034 9034 9035 9035 9036 s h m s s h m s II Q t II II Q 1 It 750 = 12 30 7500 = 2 5 760 O 12 40 76OO = 2 6 40 770 = 12 50 7700 = 2 8 20 780 = 13 7800 = 2 10 790 = 0.13 10 7900 = 2 ii 40 800 = 13 20 8000 = 2 13 20 20 TABLE VI. Logarithms of Numbers and Small Arcs. No. O 123 4 5 9034 6 9034 ? 8 935 90*1 9046 9051 9057 9062 9068 9073 i 9078 9084 9 9036 9041 9047 9052 9057 9063 9068 9074 9079 9084 800 9031 9031 9032 9033 9038 : 9043 ; 9049 i 9054 9060 9065 9070 9076 9081 9033 9035 i 1 2 3 4 5 6 7 8 9 9036 9042 9047 953 9058 9063 9069 9074 9079 9037 i 9042 i 9048 9053 9059 9064 9069 9075 9080 | 9037 9043 i 9048 9054 9059 9064 9070 9075 9081 9038 9044 9049 9055 9060 9066 9071 9076 9082 9039 9044 9050 9055 9061 9066 9071 9077 9082 9040 : 9045 9050 9056 9061 9067 9072 9077 9083 9040 9046 9051 9056 9062 9067 9073 9078 9083 810 9085 9085 9086 9 o86 | 9087 9088 9088, 9089 9089 9090 1 2 3 4 5 6 7 8 9 9090 9096 9101 9106 9112 9117 9122 9128 9i33 9091 9096 9101 9107 9112 9117 9123 9128 9133 9091 9097 9102 9107 9"3 9118 9123 9129 9134 9092 9097 9103 9108 9113 9119 9124 9129 9134 9092 9098 9103 9108 9114 9119 9124 9130 9135 993 9098 9104 9109 9114 9120 9125 9130 9135 9093 9099 9104 9109 9"5 9120 9125 9131 9136 9094 9099 9105 9110 9U5 9121 9126 9131 9137 9094 9100 9105 9111 9116 9121 9126 9132 9i37 9095 9100 9106 9111 9116 9122 9127 9132 9138 820 9138 9139 9139 9140 9140 9141 9141 | 9142 9142 9H3 1 2 3 4 5 6 7 8 9 9H3 9149 9154 9159 9165 9170 9175 9180 9186 9144 9149 9i55 9160 9165 9170 9176 9181 9186 9144 9i5 9155 9160 9166 9171 9176 9181 9187 9145 9150 9156 9161 9166 9171 9177 9182 9187 9146 9i5i 9156 9161 9167 9172 9177 9182 9188 9146 9i5i 9157 9162 9167 9172 9178 9183 9188 9H7 9152 9*57 9162 9168 91.73 9178 9183 9189 9H7 9152 9158 9163 9168 9173 9179 9184 9189 9148 9i53 9158 9163 9169 9 J 74 9179 9185 9190 9148 9153 9159 9164 9169 9175 9180 9185 9190 830 9191 9191 9192 9192 9193 9193 9194 9194 9195 9195 1 2 3 *4 5 6 7 8 9 9196 9201 9206 9212 9217 9222 9227 9232 9238 9197 9202 9207 9212 9217 9223 9228 9233 9238 9197 9202 9207 9213 9218 9223 9228 9233 9239 9198 9203 9208 9213 9218 9224 9229 9234 9239 9198 9203 9209 9214 9219 9224 9229 9235 9240 9199 9204 9209 9214 9219 9225 9230 9235 9240 9199 9204 9210 9215 9220 9225 9230 9236 9241 9200 9205 9210 9215 9221 9226 9231 9236 9241 9200 9205 9211 9216 9221 9226 9231 9237 9242 9201 9206 9211 9216 9222 9227 9232 9237 9242 84O 9243 9243 9244 9244 9245 9245 9246 9246 9247 9247 1 2 3 4 5 6 I 8 9 9248 9253 9258 9263 9269 9274 9279 9284 9289 9248 9254 9259 9264 9269 9274 9279 9284 9290 9249 9254 9259 9264 9270 9275 9280 9285 9290 9250 9255 9260 9265 9270 9275 9280 9285 9291 9250 9255 9260 9265 9271 9276 9281 9286 9291 9251 9256 9261 9266 9271 9276 9281 9287 9292 925 1 9256 9261 9267 9272 9277 9282 9287 9292 9252 9257 9262 9267 9272 9277 9282 9288 9293 9252 9257 9262 9268 9273 9278 9283 9288 9293 9253 9258 9263 9268 92/3 9278 9283 9289 9294 850 9294 9295 9295 9296 9296 9297 9297 9298 9298 9299 s /i r/i s s h m s ,, 1 II H I 'I 800 = 13 20 8000 =2 13 20 810 = o 13 30 8100 = 2 15 o 820 = o 13 40 8200 = 2 16 40 830 = 13 50 8300 = 2 18 20 840 O 14 O 8400 = 2 2O O 850 = 14 10 8500 -2 21 40 TABLE VI. 21 Logarithms of Numbers and Small Ares. No. 1 2 3 4 5 6 7 8 9 S5O 9294 9295 9295 9296 9296 9297 9297 9298 9298 9299 1 2 3 4 5 6 7 8 9 9299 934 939 9315 9320 9325 933 9335 9340 9300 9305 9310 935 9320 9325 933 9335 9340 9300 935 93" 9316 9321 9326 933i 9336 9341 9301 9306 93" 9316 932i 9326 933i 933 6 9341 9301 9306 9312 93'7 9322 9327 9332 9337 9342 9302 937 9312 9317 9322 9327 9332 9337 9342 9302 9307 93 ! 3 93i8 9323 9328 9333 9338 9343 9303 9308 93i3 93i8 9323 9328 9333 9338 9343 933 9308 93H 9319 9324 9329 9334 9339 9344 934 939 93 H 93*9 9324 9329 9334 9339 9344 860 9345 9345 935i 935 6 9361 9366 937i 9376 938i 9386 939i 9346 9347 9347 9348 9348 9349 9349 9354 9359 93 6 4 93 6 9 9374 9379 9384 9389 9394 9350 1 2 3 4 5 6 7 8 9 9350 9355 9360 9365 9370 9375 9380 9385 9390 9351 9356 9361 9366 937i 937 6 938i 9386 9391 9352 9357 9362 9367 9372 9377 9382 9387 9392 935 2 9357 9362 9367 9372 9377 9382 9387 9392 9353 9358 93 6 3 9368 9373 9378 9383 9388 9393 9353 9358 9363 9368 9373 9378 9383 9388 9393 9354 9359 93 6 4 93 6 9 9374 9379 9384 9389 9394 9355 9360 93 6 5 9370 9375 9380 9385 9390 9395 870 9395 9396 939 6 9397 9397 9398 9398 9399 9399 9400 1 2 3 4 5 6 7 8 9 9400 9405 9410 9415 9420 9425 943 9435 9440 9401 9406 9411 9416 942i 9426 943 9435 9440 9401 9406 9411 9416 9421 9426 9431 943 6 9441 9402 9407 9412 9417 9422 9427 943 i 9436 9441 9402 9407 9412 9417 9422 9427 9432 9437 9442 9403 9408 9413 9418 9423 9428 9432 9437 9442 9403 9408 9413 9418 9423 9428 9433 9438 9443 9404 9409 9414 9419 9424 9429 9433 943 s 9443 9404 9409 9414 9419 9424 9429 9434 9439 9444 9405 9410 9415 9420 9425 943 9434 9439 9444 i 880 9445 9445 9446 9446 9447 9447 9448 9448 9449 9449 1 2 3 4 5 6 7 8 9 945 9455 9460 9465 9469 9474 9479 9484 9489 9450 9455 9460 9465 9470 9475 9480 9485 9490 9451 9456 9461 9466 9470 9475 9480 9485 9490 9451 945 6 9461 9466 9471 9476 948i 9486 9490 945 2 9457 9462 9466 9471 9476 9481 9486 9491 9452 9457 9462 9467 9472 9477 9482 9487 9491 9453 9458 9463 9467 9472 9477 9482 9487 9492 9453 9458 9463 9468 9473 9478 9483 9488 9492 9454 9459 9464 9468 9473 9478 9483 94 8 9493 9454 9459 9464 9469 9474 9479 9484 9489 9493 890 9494 9494 9495 9495 9500 9505 95 10 9515 9520 9525 9529 9534 9539 9496 9496 9497 9497 9498 9498 1 2 3 4 5 6 7 8 9 9499 954 9509 9513 95i* 9523 9528 9533 9538 9499 954 959 95M 95*9 9524 9528 9533 9538 9500 955 959 95*4 95^9 9524 9529 9534 9539 95 01 9506 95 10 9515 9520 9525 9530 9535 9540 95oi 9506 95" 9516 952i 9526 953 9535 9540 9502 957 95" 95i6 9521 9526 953i 9536 9540 9502 9507 9512 9517 9522 9526 953 1 9536 954i 9503 9508 9512 9517 9522 9527 9532 9537 9541 9503 9508 9513 95i8 9523 9527 953 2 9537 9542 900 9542 9543 9543 9544 9544 9545 9545 9546 9546 9547 j < ft h m s s h m s H ' II II 1 II 850 = 14 10 8500 2. 21 40 860 - 14 20 8600 = 2 23 20 870 = 14 30 8700 = 2 25 880 = 14 40 8800 = 2 26 40 890 = 14 50 8900 - 2 28 20 900 = 015 o 9000 = 2 30 o 22 TABLE VI. Logarithms of Numbers and Small Arcs. No. 1 2 3 | 4 5 6 7 i 8 9 900 9542 9543 9543 9544 9544 9545 9545 9546 | 9546 9547 | 1 2 3 4 5 6 7 8 9 9547 9552 i 9557 95j2 j 9566 9571 9576 958i 9586 9548 9553 9557 9562 9567 9572 9577 958i 9586 9548 9553 9558 9563 9567 9572 9577 9582 9587 9549 9554 9558 95 6 3 9568 9573 9578 9582 9587 9549 9554 9559 9564 9568 9573 9578 9583 9588 955 9554 9559 95 6 4 95 6 9 9574 9578 9583 9588 9550 9555 9560 95 6 5 9569 9574 9579 9584 9589 955i 9555 I 9560 9565 ' 957 9575 9579 9584 9589 955i 9556 956i 9566 1 9570 1 9575 9580 9585 9589 9594 9552 i 9556 ! 956i 9566 957i 9576 958o i 9585 9590 91O 9590 9591 959i 9592 9592 9593 9593 9594 9595 j 1 2 3 4 5 6 7 8 9 9595 9600 9605 9609 9614 9619 9624 9628 9633 9596 9600 9605 9610 9615 9619 9624 9629 9634 9596 9601 9606 9610 9615 9620 9625 9629 9634 9597 9601 9606 9611 9616 9620 9625 9630 9635 9597 9602 9607 9611 9616 9621 9626 9630 9635 9598 9602 9607 9612 9617 9621 9626 963J 9636 9598 9603 9608 9612 9617 9622 9627 9631 9636 9599 9603 9608 9613 9618 9622 9627 9632 9636 9599 9604 9609 9613 9618 9623 9627 9632 9637 9599 9604 9609 9614 9618 9623 9628 9633 9637 920 9638 9638 9639 9639 9640 9640 9641 9641 9642 9642 ! 1 2 3 4 15 6 7 9 9 6 43 9647 9652 9657 9661 9666 9671 9675 9680 9643 9648 9652 9657 9662 9667 9671 9676 9681 9644 9648 9653 9658 9662 9667 9672 9676 9681 9644 9649 9653 9658 9663 9668 9672 9677 9682 9644 9649 9654 9659 9663 9668 9 6 73 9677 9682 9 6 45 9650 9 6 54 9659 9664 9668 9673 9678 9682 9645 9650 9655 9660 9664 9669 9674 9678 9683 9646 9651 9655 9660 9665 9669 9674 9679 9683 9646 9651 9656 9660 9665 9670 9675 9679 9684 9647 i 9652 9656 ' 9661 ! 9666 ; 9670 ! 9675 ' 9680 ; 9684 930 9685 9685 9686 9686 9687 9687 9688 9688 9689 9 68 9 ! 1 2 3 4 5 6 7 8 9 9690 9694 9699 973 9708 97U 9717 9722 9727 9690 9695 9699 9704 9709 9713 9718 9722 9727 9690 9695 9700 9704 9709 97H 9718 9723 9728 9691 9696 9700 9705 9710 97H 9719 9723 9728 9691 9696 9701 9705 9710 9715 9719 9724 9729 9692 9696 9701 9706 9710 9715 9720 9724 9729 9692 9697 9702 9706 9711 9716 9720 9725 9729 9693 9697 9702 9707 9711 9716 9721 9725 973 9693 9698 9703 9707 9712 9716 9721 9726 9730 9694 t 9698 | 9703 | 9708 ; 9712 i 9717 9722 9726 j 973 * I 940 973i 9732 9732 9733 9733 9734 9734 9735 9735 9735 ! 1 2 3 ! 6 7 9 950 9736 9741 9745 9750 9754 9759 9764 9768 9773 9736 974i 9746 975 9755 9759 9764 9769 9773 9737 974i 9746 975i 9755 9759 9764 9769 9774 9737 9742 9747 975i 975 6 9760 9765 9769 9774 9738 9742 9747 9752 9756 9761 9765 9770 9774 9738 9743 9747 9752 9757 9761 9766 9770 9775 9739 9743 9748 9752 9757 9762 9766 9771 9775 9739 9744 9r4 8 9753 9758 9762 9767 9771 9776 9740 9744 9749 9753 , 9758 9763 9767 9772 9776 9740 9745 i 9749 9754 ! 9758 i 97 6 3 i 9768 9772 9777 9777 9778 9778 9779 9779 9780 9780 9780 9781 978i s h m s s h m s II O 1 II II Q 1 II 900 = 0150 . 9000 = 2 30 910 = 015 10 9100 = 2 31 4 o 920 = 015 20 9200 = 2 33 20 930 = 015 30 93 = 2 35 94O =OI5 40 94OO = 2 36 40 950 = 015 50 9500 = 2 38 20 -L^L. UAJJLJ V X. ^U i N ^IFQ'R T ^ L.ogarit9ims of N limbers ami Small Arcs. fo. o 9777 "9782 9786 979 1 9795 9800 9805 9809 9814 9818 1 2 3 4 5 6 7 9 950 9778 9782 9737 9791 9796 9800 9805 9810 9814 9819 9778 9783 9787 9792 9776 9801 9805 9810 9815 9819 9824 9779 9779 978o 9780 9780 9785 9790 9794 9799 9803 9808 9812 9817 9821 9826 978i 9781 1 9 3 4 & 6 7 8 9 97*3 9788 9792 9797 9801 9806 9810 98i5 9820 9784 9788 9793 9797 9802 9806 9811 9815 9820 9825 9784 9789 9793 9798 9802 9807 9811 9816 9820 9785 9789 9794 9798 9803 9807 9812 9816 9821 9785 9790 9795 9813 9817 9822 9786 9790 9795 9800 9804 9809 -9813 9818 9822 960 9823 9827 9832 9836 9841 9845 9850 9854 9859 9803 0823 9824 9825 9825 9826 9827 1 2 3 4 5 6 7 9 0828 9832 9837 9841 9846 "9850 9855 98.S9 9864 9828 9833 9837 9842 9846 9851 9855 9800 9864 9829 9833 9838 9842 9847 9851 9856 9860 9865 9829 9834 9838 9843 9847 9852 9856 9861 9865 9829 9834 9839 9843 9848 9852 9857 9861 9865 9830 9834 9839 9843 9848 9852 ' 9857 9861 9866 9830 9835 9839 9844 9848 9853 9857 9862 9866 9831 9f35 9840 9844 9849 9853 9858 9862 9867 9831 9836 9840 9845 9849 9854 9858 9863 9867 97O 9868 9868 9869 9869 9870 9870 9870 9871 9871 9872 1 2 3 4 5 6 7 i 9 9872 9877 9881 9886 9890 9895 9899 9903 9908 9873 $1 9886 9890 9895 9899 9904 9908 9873 9878 9882 9886 9891 9895 9900 9904 9909 9874 9878 9882 9887 9891 9896 9900 9905 9909 9874 9878 9883 9887 9892 9896 9901 9905 9910 9874 9879 9883 9888 9892 9897 9901 9906 9910 9875 9879 9884 9888 9893 9897 9902 9906 9910 9875 9880 9884 9889 9893 9898 9902 9907 9911 9876 9880 9885 9889 9903 9907 9911 9876 9881 9885 9890 9894 9899 9903 9907 9912 9SO 9912 9913 9913 9914 9914 9914 9915 9915 9916 9916 1 2 3 4 5 6 7 8 1 9 9917 9921 9926 993 9934 9939 9943 9948 9952 9917 9922 9926 993 9935 9939 9944 9948 9952 9918 9922 9926 9931 9935 9940 9944 9948 9953 9918 9922 9927 993 i 9936 9940 9944 9949 9953 9918 9923 9927 9932 993 6 9941 9945 9949 9954 9919 9923 9928 9932 9937 9941 9945 995 9954 9919 9924 9928 9933 9937 9941 9946 995 9955 9920 9924 9929 9933 9937 9942 9946 9951 9955 9920 9925 9929 9933. 9938 9942 9947 995 i 9955 9921 9925 993 9934 9938 9943 9947 995 2 9956 990 995 6 9957 9957 9958 9958 9959 9959 9959 9960 9960 1 2 3 4 5 6 7 8 9 9961 99^5 9969 9974 9978 9983 9987 9991 9996 9961 9966 9970 9974 9979 9983 9987 9992 9996 9962 9966 9970 9975 9979 9983 9988 9992 9997 9962 9966 9971 9975 9980 9984 9988 9993 9997 9962 9967 9971 9976 9980 . 9984 9989 9993 9997 9963 9967 9972 9976 9980 9985 9989 9993 9998 9963 9968 9972 . 9976 998i 9985 9990 9994 9998 9964 9968 9973 9977 9986 9990 9994 9999 9964 9969 9973 9977 9982 9986 9990 9995 9999 9965 9969 9973 9978 9982 9987 9991 9995 oooo 1000 oooo oooo OOOI OOOI OOO2 OOO2 0003 1 0003 0003 0004 s h m $ il Q I II ^50 = o 15 50 960 = o 16 o 970 = o 16 10 986 = o 16 20 990 = o 16 30 1000 = o 16 40 * /* m s II 1 II 95OO 2 38 2O 9600 = 2 40 O 9700 = 2 41 40 9800 2 43 20 9900 2 45 o 10000 2 46 40 TABLE \ Logarithms of Numbers: 1 to 1OO9. Ifo. 1 I 2 3 Proportional Parts. 4 ." 7 N 9 i 1 2 3 4 5 6 7 8 9 oooo 3010 4771 6021 6990 7782 8451 9031 9542 98 196*294 392490587 685 783 88 1 1 OOOO 0414:0792 1130 1461 1761 2041 2305 2553 2788 30 60 90 I2O 150 I8O2IO24O27O 2 3010 3222 3424:3617 3802 3979 4150 4314 44/2 4624! 18; 35: 53 71 88 io6|i24 1 i42 ! iS9 3 477 1 4014 5052 5185 5315 544i 55t>3 5682 5798 5911 13 25 38 50 63 76 88lOI 113 4 6021 6128 623216335 6435 6532 6628 6721 6812 6902 ; 10 19 29 39! 48 58 68 78 87 flf 6990 7076^7160(7243 7324 7404 7482 , 7559 7634 7709 8 16 24 32 40 48 56 64 72 6 7782 7853 7924 7993 8062 8129 8195 8261 8325 8388 7 13 20 27 33 40 47 54 60 7 8451 513 85/3:8633 8692 8751 ; 8808 8865 i 8921 8976 6 12 18 24 29 35 41 47 53 9 9031 9542 9085 9138 9191- 959O 9638 9685 9243 9294 9345 9731 9777 9823 9395 9445 9868 ^912 9494 9956 I IO 9 15 20 18 25 23 28 36 $2 37 46 ! 41 10 oooo 0043 0086 0128 0170 0212 0253 0294 334 0374 4 8 12 17 21 25 29 33 37 0414 045^ 0492 0531 0569 0607 0645 0682 0719 0755 4 8 II 15 19 21 26 10 34 2 0792 0828 0864 0899 0934 0969 1004 1038 1072 1106 3 7 IO M 17 21 2 4 28 3 1 1 3 "39 1173 1206 1239 1271 M03 1335 1367 1399 143 3 6 10 Ib 19 23 26 29 4 1461 1492 1523 1553 1584 : 1614' 1644 1673! 1703 1732 2 6 9 12 15 18 21 24 27 5 1761 1790 1818 1847 1875 1903 1931 1959 ; 1987 2014 2 6 8 II 17 20 22 25 6 2041 2068 2095 2122 2148 2175 22OI 2227 2253 2279 2 5 8 II 13 16 18 21 24 9 2304 -553 2788 233 2 355 2577 2601 2810 2833 2380 2625 2856 2405 2648 2878 2430 2672 2900 2455 2095 2923 2480 ; 2504 2718 2742 2945 2967 2529 2765 2989 2 2 2 5 5 4 7 7 7 10 9 9 12 12 II 15 14 13 11 16 20 19 18 22 ! 21 ; 20 : 20 3010 3032 3054 3075 3096 3"8 3*39 3160 3181 3201 2 4 6 8 II 13 15 17 19 1 ^222 3243 3263 3284 3304 3324 3345 336"5 3385 3404 2 4 6 8 IO 12 14 16 18 2 3424 3444 3464 3483 1502 3522 354* 356o 3579 3598 2 4 6 8 10 12 14 M I 7 i 3 36l/ 3655 3674 3692 37" 3729 3747 3766 3/84 2 4 6 7 9 II 1 3 15 17 j 4 ^802 ^820 3838 3856 3874 3892 3909 3927 3945 3962 2 4 5 7 9 II 12 14 16 ^ 3997 4014 4031 4048 4065 4082 4099 4116 4133 2 3 5 7 9 10 12 H 15 6 4150 4166 4183 4200 4216 4232 4249 4265 4281 4298 2 3 5 7 8 10 II 13 7 43 H 4330 4346 4362 4378 4393 4409 4425 4440 445 6 2 3 5 6 8 9 II M H ; 8 4472 4487 4502 45i8 14533 4548 45^4 4579 4594 4609 2 3 5 b 8 9 II 12 9 4624 4639 4654 4 bb 9 4683 4698 4713 4728 4742 4757 I 3 4 6 7 9 IO 12 J 3 3O 4771 4786 4800 4814 4829 4843 4857 4871 4886 4900 I 3 4 6 7 9 10 II 13 1 4914 4928 4942 4955 4969 4983 4997 5011 5024 5038 3 4 6 7 8 10 II 12 2 5052 5065 579 5092 5 "9 5132 5H5 5159 5172 3 4 5 7 8 9 II 12 3 5185 5198 52" 5224 5237 5250 5263 5276 5289 5302 3 4 5 6 8 9 IO 12 4 5315 5328 5340 5353 5366 5378 5391 5403 5416 5428 3 4 5 6 8 9 IO II 5 5453 S478 5490 5502 5515 5527 5539 555 1 2 4 5 6 7 9 10 II 6 55 6 3 5575 5587 5599 5611 5623 5635 5647 5658 5670 2 4 5 6 7 8 10 II 1 5682 5798 5694 5809 5705 5821 5832' 5729 5843 5740 5855 5752 5866 5763 5775 5877 5888 5786 5^99 2 2 3 3 5 5 6 6 7 7 8 8 9 9 IO 10 9 59" 5922 5933 5944 5955 5977 5988 5999 6010 2 3 4 5 7 8 9 10 40 6021 6031 6042 6053 6064 6075 6085 6096 6107 6117 2 3 4 5 6 8 9 IO 1 6128 6138 6149 6160 6170 6180 6191 6201 6212 6222 2 s 4 5 6 7 8 9 3 6232 6335 6243 6345 6253 6355 6263 6365 6274 6375 6284 6385 6294 6395 6304 6405 6314 6415 6325 6425 2 2 3 3 4 4 5 5 6 6 7 7 8 9 8 9 4 6 6435 6532 6628 6444 6542 6637 6454 6646 6464 6561 6656 6474 6571 6665 6484 6580 6675 6493 6590 6684 6503 6693 6609 6702 6522 6618 6712 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 7 9 1 6721 6812 6730 6821 6739 6830 St? 6758 6848 6767 68s 7 6776 6866 6785 6875 6794 6884 6803 6893 2 2 3 3 4 4 5 4 5 5 6 6 7 7 8 8 9 6902 6911 6920 6928 6 937 6946 6955 6964 6972 6981 2 3 4 4 5 6 7 8 50 6990 6998 7007 7016 7024 7033 7042 7050 7059 7067 I 2 3 3 4 5 6 7 8 TABLE VII. 25 Logarithms of Numbers: 1 to 1OO9. No. 6990 1 2 3 4 5 6 * 8 9 Proportional Parts. 1 2 3 4 5 6 5 7 6 8 9 5O 6998 7007 7016 7024 7033 7042 7050 7059 7067 3 3 4 7 7 I 6 6 6 6 6 6 6 6 6 5 5 5 5 5 5 8 1 3 4 6 7 9 7076 7160 7243 7324 7404 7482 7559 7634 7709 7084 7168 7251 n-1 TJ 7412 7490 7566 7642 7716 7789 793 7177 7259 7340 7419 7497 7574 7649 7723 7101 7185 7267 7348 7427 7505 7582 7657 7731 7110 7118 7193 7202 7275 | 7284 7356 7364 7435 : 7443 7513; 7520 7589 i 7597 7664 7672 773 s 7745 7126 7210 7*92 7372 745i 7528 7604 7679 7752 7135 7218 7300 7380 7459 7536 7612 7686 7760 7832 7H3 7226 7308 7388 7466 7543 7619 7694 7767 7152 7235 73 l6 7396 7474 755i 7627 7701 7774 i i i i i i i 2 2 2 2 2 2 2 I I 3 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 5 5 5 5 5 5 5 4 4 6 6 6 6 5 5 5 5 5 8 7 7 7 7 7 7 7 7 6 6 6 6 6 6 6 6 6 6 60 7782 7796 7803 7810 7818 7825 7839 7846 i i i i i i i i i I I I I I I I I I I 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 2 4 4 4 4 4 4 4 4 4 4 5 1 2 3 4 5 6 7 8 9 7853 7924 7993 8062 8129 8i95 8261 8325 8388 7860 793i 8000 8069 8136 8202 8267 8331 8395 7868 7938 8007 8075 8142 8209 8274 8338 8401 7875 7945 8014 8082 8149 8215 8280 8344 8407 7882 7952 8021 8089 8156 8222 8287 835i 8414 7889 2i 8096 8162 8228 8293 8357 8420 7896 7966 8035 8102 8169 8235 8299 8363 8426 7903 7973 8041 8109 8176 8241 8306 8370 8432 7910 7980 8048 8116 8182 8248 8312 8376 8439 7917 7987 8055 8122 8189 8254 8319 8382 8445 5 5 5 5 5 5 5 4 4 70 8451 8457 8519 8579 8639 8698 8756 8814 8871 8927 8982 8463 8470 8476 8482 8488 8494 8500 8506 i i i i i i i i I 2 2 2 2 2 2 2 2 2 2 2 3 4 4 5 5 5 5 5 4 4 4 6 5 5 '5 5 5 5 5 5 5 1 3 4 5 6 7 9 8513 8573 8633 8692 8751 8808 8865 8921 8976 8525 8585 8645 8704 8762 8820 8876 8932 8987 8531 859i 8651 8710 8768 8825 8882 8938 8993 8537 8597 8657 8716 8774 8831 8887 8943 8998 8543 8603 866 3 8722 8779 8837 8893 8949 9004 8549 8609 8669 8727 8785 8842 8899 8954 9009 fl 55 8615 8675 8733 8791 8848 8904 8960 9015 8561 8621 8681 8739 8797 8854 8910 8965 9020 8567 8627 8686 8745 8802 8859 8915 8971 9025 I I I I I I I I I 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 3 3 3 3 3 4 4 4 4 4 4 4 4 4 80 9031 9036 9042 9047 9053 9058 9063 9069 9074 9079 i I 2 2 3 3 4 4 5 1 3 4 6 8 9 90 9085 9138 9191 9243 9294 9345 9395 9445 9494 9090 9H3 9196 9248 9299 935 9400 945 9499 9096 9149 9201 9253 934 9355 9405 9455 954 9552 9101 9154 9206 9258 939 9360 9410 9460 959 9106 9212 9263 9365 9415 9112 9165 9217 9269 9320 9370 9420 9469 9117 9170 9222 9274 9325 9375 9425 9474 9523 9122 9175 9227 9279 933 9380 943 9479 9528 9128 9180 9232 9284 9335 9385 9435 9484 9533 9238 9289 9340 9390 9440 9489 9538 9586 i i i o o I I I I I I I 2 2 2 2 2 2 I I I 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 4 4 4 4 9542 9547 9557 9562 9566 9571 9576 958i I 2 3 3 3 3 3 3 3 3 4 3 4 6 7 9 9590 9638 9685 973 i 9777 9823 9868 9912 9956 9595 9643 9689 9736 9782 9827 9872 9917 9961 9600 9647 9694 9741 9786 9832 9877 9921 9605 9652 9699 9745 9791 9836 9881 9926 9969 9609 9657 9703 975 9795 9841 9886 993 9974 9614 9661 9708 9754 9800 9845 9890 9934 9978 9619 9666 9713 9759 9805 9850 9894 9939 9983 9624 9671 9717 9764 9809 9854 9899 9943 9987 9628 9722 9768 9814 9859 9903 9948 9991 9633 9680 9727 9773 9818 9863 9908 9952 9996 o o o o o o I I I I I I I I I I I I I I I I 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 100 oooo 0004 0009 0013 0017 0022 0026 0030 0035 0039 I 1 2 2 3 3 4 4 26 TABLE VIII. Logarithmic Sines, Tangents, and Secants to every Eighth Point of the Compass. Degrees. Points. Sine. Cosecant. Tangent. Cotangent. Secant. Cosine. Points. Degrees. * 0.0 O oo + 00 CO + 00 10.0000 10.0000 8 90.0 1.4 i 8.3899 11.6101 8.3900 1 1. 6100 IO.OOOI 9-9999 I 88.. 6 91.9 i 8.6908 11.3092 8.6913 11.3087 10.0005 9-9995 I 87.2 4.2 I 8.8667 "-I333 8.8678 11.1322 10.0012 9.9988 : 85.8 5.6 1 8.9913 11.0087 8-9934 11.0066 10.0021 9-9979 1 84.4 7.0 I 9.0878 10.9122 9.0911 10.9089 10.0033 9.9967 1 83. 8.4 I 9.1665 10.8335 9.I7I3 10.8287 IO.OO47 9-9953 i 81.6 9.8 t s 9.2329 10.7671 9.2394 10.7606 10.0064 9.9936 i 80.2 11.3 1 9.2902 10.7098 9.2987 10.7013 10.0084 9.9916 7 78.7 12.7 i 9.3406 10.6594 9-3513 10.6487 IO.OIO7 9-9893 3 77.3 14.1 i 9.3856 10.6144 9-3988 I O.60 1 2 10.0132 9.9868 f 75.9 15.5 i 9.4260 10.5740 9.4421 10-5579 10.0160 9.9840 f 74.5 16.9 i 9.4628 10.5372 9.4819 I0.5l8l 10.0191 9.9809 i ! 73.1 18.3 i 9.4965 I0 -5035 9.5190 10.4810 10.0225 9-9775 || 71.7 19.7 I 9-5275 10.4725 9-5537 10.4463 10.0262 9.9738 i 70.3 21.1 3. 6 9-5561 10.4439 9.5863 10.4137 10.0301 9.9699 i 68.9 22.5 2 9.5828 10.4172 9.6172 10.3828 10.0344 9-9656 6 67.5 23.9 i 9.6077 10.3923 9.6467 10-3533 10.0390 9.9610 i 66.1 25.3 i 9.6310 10.3690 9.6748 10.3252 10.0438 9.9562 f 64.7 26.7 I 9.6528 10.3472 9.7019 10.2981 10.0490 9.9510 63 . 3 28.1 1 9- 6 734 10.3266 9.7280 10.2720 10.0546 9-9454 i 61.9 29.5 1 9.6928 10.3072 9.7532 10.2468 10.0604 9.9396 1 60.5 30.9 1 9.7111 10.2889 9-7777 10.2223 10.0666 9-9334 i 4 59.1 32.3 i 9.7284 10.2716 9.8016 10.1984 10.0732 9.9268 i 57.7 33.7 3 9-7447 IO - 2 553 i 9-8249 10.1751 10.0802 9.9198 5 56.3 35.2 i 9.7603 10.2397 9-8477 10.1523 10.0875 9-9125 t | 54.8 36.6 i 9-775 10.2250 9.8702 10.1298 10.0952 9.9048 * 53.4 38. i 9.7890 IO.2IIO 9-8923 10.1077 10.1033 9.8967 i 52.0 39.4 i 9.8024 10.1976 9-9142 10.0858 10.1118 9.8882 i 50.6 40.8 f 9.8150 10.1850 j 9-9358 10.0642 10.1208 9.8792 I 49.2 42.2 1 9.8271 10.1729 9-9573 10.0427 10.1302 9.8698 i 47.8 43.6 1 9.8386 10.1614 9.9787 10.0213 10.1401 9.8599 i 46. 4^ 45.0 4 9-8495 10.1505 10.0000 IO.OOOO 10.1505 9-8495 4 45. Degrees. Points. Cosine. Secant. Cotangent. Tangent. Cosecant. Sine. Points. Degrees. The equivalents in Degrees are to the nearest Tenth. TABLE IX. 27 O = O" O] l,og. Sines, Tangents, and Secants. [ll h 56 m = 179 ' m a Sin. Diff. Cosec. Tan. Diff. Cot. Sec. Cos. m a / O'.l 1" O'.l 1 s o 1 AJI 3 4 5 6 7 8 9 1O 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 4 8 12 16 020 24 28 32 36 040 44 48 52 56 1 4 8 12 16 120 24 28 32 36 140 44 48 52 56 2 4 8 12 16 220 24 28 32 36 240 44 48 52 56 3 4 8 12 16 320 24 28 32 36 34O 44 48 52 56 4 CO 6.4637 7648 9408 7-0658 301 176 125 97 79 67 58 5i 40 41.4 37-8 34-8 32-2 29.9 28.0 26.4 24.8 23-5 22.3 21.2 2O.2 19-3 I8. 5 17.7 17.0 I6. 4 I 5 .8 15.2 14.7 753 440 312 242 198 167 H5 128 114 103 94 87 80 75 70 66 62 59 56 53 50 48 46 44 42 41 3 37 + CO 13-5363 2352 0592 12.9342 CO 6.4637 7648 9408 7.0658 301 176 125 97 79 67 58 8 41.4 37-8 34-8 32.2 29.9 28.0 26.4 24.8 23-5 22.3 21.2 20.2 19-3 I8. 5 17.7 17.0 16.4 I 5 .8 15.2 14.7 753 440 312 242 198 167 ;s 114 103 94 75 70 66 62 59 56 53 5o 48 46 44 42 41 39 38 37 + CO 13-5363 2352 0592 12.9342 10.0000 oooo oooo oooo oooo 10.0000 oooo oooo oooo oooo 60 5956 52 48 5944 40 36 32 28 5924 2O 16 12 8 59 4 59 5856 52 48 5844 40 36 32 28 5824 20 16 12 8 58 4 58 O 5756 52 48 5744 40 36 32 28 5724 20 16 12 8 57 4 57 56 56 52 48 5644 40 36 32 28 5624 20 16 12 8 56 4 56 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 1 6 5 4 3 2 1 O 7.1627 2419 3088 3668 4180 12.8373 7581 6912 6332 5820 7.1627 2419 3088 3668 4180 12.8373 7581 6912 6332 5820 IO.OOOO oooo oooo oooo oooo IO.OOOO oooo oooo oooo oooo 74637 5051 5429 5777 6099 12.5363 4949 457^ 4223 3901 74637 5051 5429 5777 6099 12.5363 4949 4571 4223 3901 IO.OOOO oooo oooo oooo oooo IO.OOOO oooo oooo oooo oooo 7.6398 6678 6942 7190 7425 12.3602 3322 3058 2810 2575 7.6398 6678 6942 7190 7425 12.3602 3322 3058 2810 2575 IO.OOOO oooo oooo oooo oooo IO.OOOO oooo oooo oooo oooo 7.7648 7859 8061 8255 8439 12.2352 2141 1939 1745 1561 ~i2~i'&3 1213 1049 0891 0739 7.7648 7860 8062 8255 8439 12.2352 2140 1938 1745 1561 IO.OOOO oooo oooo oooo oooo IO.OOOO oooo oooo oooo oooo 7.8617 8787 8951 9109 9261 7.8617 8787 8951 9109 9261 12.1383 1213 1049 0891 0739 IO.OOOO oooo oooo oooo oooo IO.OOOO oooo oooo oooo oooo 7.9408 955i 9689 9822 9952 8.0078 0200 03'9 0435 0548 !3 25 38 50 63 Is IOI "3 10 19 29 39 4-Q 68 78 87 8 16 24 32 40 47 55 63 7i i 93 25 49 73 20 $ 17 12.0592 0449 0311 0178 0048 7.9409 $& 9823 9952 *3 25 38 50 63 76 88 IOI "3 10 19 29 39 49 y 78 87 8 16 24 32 40 47 3 71 31 62 93 25 49 73 20 39 58 12.0591 0449 0311 0177 0048 IO.OOOO oooo oooo oooo oooo 10.0000 oooo oooo oooo oooo 11.9922 9800 9681 9565 9452 8.0078 O2OO 0319 435 0548 8.06 58 0765 0870 0972 1072 11.9922 9800 9681 9565 9452 10.0000 oooo oooo oooo oooo 10.0000 oooo oooo oooo oooo 8.0658 0765 0870 0972 1072 11.9342 9235 9130 9028 8928 11.9342 9235 9130 9028 8928 IO.OOOO oooo oooo oooo oooo IO.OOOO oooo oooo oooo oooo 8.1169 1265 1358 145 1539 11.8831 8735 8642 8550 8461 8.II70 1265 1359 145 1540 11.8830 8735 8641 8550 8460 IO.OOOO oooo oooo oooo oooo 10.0000 oooo oooo oooo oooo 8.1627 1713 1797 1880 1961 11.8373 8287 8203 8120 8039 8.1627 1713 1798 1880 1962 "-8373 8287 8202 8120 8038 IO.OOOO oooo oooo OOOI OOOI IO.OOOO oooo oooo 9-9999 9999 8.2041 2119 2196 2271 2346 "7959 7881 7804 7729 7654 8.2041 2I2O 2196 2272 2346 n-7959 7880 7804 7728 7654 IO.OOOI OOOI OOOI OOOI OOOI 9-9999 9999 9999 9999 9999 8.2419 11.7581 8.2419 11.7581 IO.OOOI 9.9999 t i m s Cos. O'.l Sec. Cot. OM I 8 Tan. Cosec. Sin. m s / Diff. Diff. 9O - 6 h O ] [ 5^ 56 m = 89 28 TABLE IX. ! ' i 1 - o h 4 ra j Log. Sines, Tangents, and Secants. [ ll h 52 m = 178 ' ni s Sin. 8.2419 2490 2561 2630 2699 Diff. Cosec. Tan. Diff. Cot. Sec. Cos. m 9 ' O'.l 1 s O'.l I 8 1 2 3 4 5 6 7 8 9 1O 11 12 13 ! 14 i 15 16 i 17 ! 18 L 19 2O 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 ! 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 6O 4 4 8 12 16 420 24 28 32 36 440 44 48 52 56 5 4 8 12 16 5 2O 24 28 32 36 540 44 48 52 56 6 4 8 12 16 6 2O 24 28 32 36 64O 44 4 52 56 7 4 8 12 16 720 24 28 32 36 74O 44 48 52 56 8 7 13 20 27 34 40 47 54 60 6 12 17 23 29 35 % 52 5 10 15 20 26 $ 41 4 6 5 9 H 18 23 28 3 2 37 4i J 12 16 21 25 29 33 37 i ii 15 19 23 27 30 34 17 33 5 . 15 29 43 2 3 6 38 12 2 3 34 10 21 31 9 1 9 28 ~o7 11.7581 75 10 7439 7370 73 i 11.7234 7108 7102 7038 6975 8.2419 2491 2562 2631 2700 7 '3 20 27 34 40 47 54 60 6 12 17 23 29 35 4' 46 52 5 10 *5 20 26 P $ 5 9 H 18 3 32 37 4i 4 8 12 16 21 25 29 33 37 4 8 ii 15 19 23 27 3 34 i/ 33 50 15 29 43 i 38 12 23 34 10 21 3* 9 19 28 11.7581 75 9 743 8 73 6 9 7300 IO.OOOI OOOI OOOI OOOI OOOI 9-9999 9999 9999 9999 9999 56 O 5556 52 48 5544 40 36 32 28 5524 20 16 12 8 55 4 55 O 5456 52 48 5444 5440 36 32 28 5424 O A 16 12 8 54 4 54 5356 52 48 5344 40 36 32 28 5324 20 16 12 8 53 4 53 O 5256 52 48 5244 40 36 32 28 5224 20 16 12 8 52 4 52 6O 59 58 57 56 55 54 53 ! 52 51 50 49 48 47 46 45 44 43 42 41 40 I 39 38 ; 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 i 20 19 18 17 16 15 14 i 13 12 11 10 9 tt 7 5 4 3 2 1 o I 8.2766 2832 2898 2962 3 2 5 8.2767 2833 2899 2963 3026 11.7233 7167 7101 7037 6974 IO.OOOI OOOI OOOI OOOI OOOI 9-9999 9999 9999 9999 9999 8.3088 315 3210 3270 3329 8:3388' 3445 3502 3558 3613 11.6912 6850 6790 6730 6671 8.3089 3 J 5 3211 3271 3330 11.6911 6850 6789 6729 6670 IO.OOOI OOOI OOOI OOOI OOOI 9.9999 9999 9999 9999 9999 11.6612 6555 6498 6442 6387 8-3389 3446 3503 3559 11.6611 6554 6497 6441 6386 IO.OOOI 0001 OOOI OOOI OOOI 9-9999 9999 9999 9999 9999 8.3668 3722 377=; 3828 3880 11.6332 6278 6225 6172 6120 8.3669 3723 3776 3829 3881 11.6331 6277 6224 6171 6119 IO.OOOI OOOI OOOI OOOI OOOI 9-9999 9999 9999 9999 9999 8-3931 3982 4032 4082 4131 11.6069 6018 5968 5918 8-3932 3983 4033 4083 4132 1 1. 6068 6017 59 6 7 5917 5868 IO.OOOI OOOI OOOI OOOI OOOI 9-9999 9999 9999 9999 9999 8.4179 .4227 4275 432-2 4368 11.5821 5773 & 5632 8.4181 4229 4276 4323 4370 11.5819 5771 5724 5 6 77 5 6 3 IO.OOOI 0002 OOO2 OOO2 0002 9-9999 9998 9998 9998 9998 8.4414 4459 454 4549 4593 n.5586 554i 5496 545i 5407 8.4416 4461 4506 455i 4595 ii-5584 5539 5494 5449 5405 IO.OOO2 OOO2 0002 OOO2 OOO2 9.9998 9998 9998 9998 9998 8.4637 4680 4723 4765 4807 8.4848 4890 493 4971 5011 "53 6 3 53 20 5277 5235 5*93 8.4638 4682 4725 4767 4809 11.5362 53i8 5275 5233 5i9i IO.OOO2 OOO2 0002 OOO2 OOO2 9.9998 9998 9998 9998 9998 11.5152 5110 5070 5029 4989 8.4851 4892 4933 4973 5013 11.5149 5108 5067 5027 4987 10.0002 OOO2 OOO2 OO02 0002 9.9998 9998 9998 9998 9998 8.5050 5090 5129 5i 6 7 5206 11.4950 4910 4871 4833 4794 8.5053 5092 5*3' 5170 5208 11.4947 4908 4869 4830 4792 IO.OOO2 OOO2 0002 OOO2 OOO2 9.9998 9998 9998 9998 9998 8.5243 5281 53i8 5355 5392 "4757 4719 4682 8.5246 5283 532i 5358 5394 "4754 4717 4679 4642 4606 10.0002 OOO2 OOO3 0003 0003 9.9998 9998 9997 9997 9997 8.5428 11.4572 8.5431 11.4569 IO.OOO3 9.9997 / m a COB. O'.l Sec. Cot. O'.l ! Tan. Cosec. Sin. m a / Diff. Diff. 91 = 6 U 4 m ] [ 5 h 52 m = 88 TABLE IX. ; 2^ = O" 8 ] Log:. Sines, Tangents, and decants. [ ll 1 48' = 177 O 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2O 21 22 23 24 25 26 27 28 29 3O i 31 i 32 33 | 34 ! 35 36 37 ! 38 ! 39 i 40 1 41 ! 42 1 43 | 44 i 45 i46 |47 48 49 50 51 52 53 54 55 56 57 58 59 6O m s 8 4 8 12 16 8 2O 24 28 32 36 84O 44 48 52 56 9 4 8 12 16 920 24 28 32 36 94O 44 48 52 56 1O O 4 8 12 16 1020 24 28 32 36 1O4O 44 48 52 56 11 4 8 12 16 11 2O 24 28 32 36 1140 44 48 52 56 12 Sin. 8.5428 5464 5500 5535 557i Diff. Cosec. Tan. Di 7 .1 ff. 1 s Cot. Sec. 10.0003 0003 ' 0003 0003 0003 Cos. m s ' O'.l 1 s 4 7 n H 17 21 % 3 1 3 6 10 J 3 16 19 22 26 2 9 6 9 12 21 2 4 2 7 6 8 ii H 17 20 22 25 3 10 J 3 16 18 21 2 3 3 5 7 10 12 15 17 20 22 9 if 27 8 16 24 8 15 22 7 H 21 7 13 19 6 12 19 II-4572 4536 4500 4465 4429 8-5431 5467 553 5538 5573 \ ii H 17 21 24 28 31 6 10 *3 16 19 22 26 29 I 9 12 15 18 21 24 2 7 6 8 n H 17 20 22 25 3 IO 13 16 18 21 23 3 5 7 IO 12 15 17 , 2O 22 i 27 8 16 24 8 15 22 7 14 21 7 13 19 6 12 19 11.4569 4533 4497 4462 4427 9-9997 9997 9997 9997 9997 52 51 56 52 48 5144 40 36 32 28 5124 2O 16 12 8 51 4 51 O 5056 52 48 5044 40 36 32 28 5O24 2O 16 12 8 50 4 50 4956 52 48 4944 40 36 32 28 4924 2O 16 12 8 49 4 49 4856 52 48 4844 40 36 32 28 48 24 2O 16 12 8 48 4 48 6O 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 3 2 1 8.5605 5640 5 6 74 57o8 5742 n-4395 4360 4326 4292 4258 8.5608 5643 5677 57" 5745 11.4392 4357 4323 4289 4255 10.0003 0003 0003 0003 0003 10.0003 0003 0003 0003 0003 9-9997 9997 9997 9997 9997 8.5776 5809 5842 5875 597 11.4224 4191 4158 4125 4093 8-5779 5812 l*& 5 8 78 59" 11.4221 4188 4155 4122 4089 9-9997 9997 9997 9997 9997 8-5939 5972 6003 6 035 6066 11.4061 4028 3997 3965 3934 8-5943 5975 6007 6038 6070 11.4057 4025 3993 3962 393 10.0003 0003 0003 0003 0004 9-9997 9997 9997 9997 9996 9.9996 9996 9996 9996 9996 8.6097 6128 6i59 6189 6220 11.3903 3872 3841 3811 378o 8.6101 6132 6163 6193 6223 8.6254 6283 6313 6 343 6372 11.3899 3868 3837 3807 3777 10.0004 0004 0004 0004 0004 8.6250 6279 6309 6339 6368 "3750 372i 3691 3661 3632 11.3746 3717 3687 3657 3628 10.0004 0004 0004 0004 0004 9-9996 9996 9996 9996 9996 8.6397 6426 6454 6483 6511 11.3603 3574 3546 3517 3489 8.6401 6430 6459 6487 6515 11 -3599 3570 354i 3513 3485 10.0004 0004 0004 0004 0004 9-9996 9996 9996 9996 9996 8-6539 6567 6595 6622 6650 11.3461 3433 3405 3378 335 8.6544 6571 6 9 7 6654 1 1 -345 6 3429 340i 3373 3346 10.0004 0004 0005 0005 0005 9.9996 9996 9995 9995 9995 8.6677 6704 6731 6758 6784 11-3323 3296 3269 3242 3216 8.6682 6709 6736 6762 6789 11-3318 3291 3264 3238 3211 10.0005 0005 0005 0005 0005 9-9995 9995 9995 9995 _9991 9-9995 9995 9995 9995 9995 8.6810 6837 6863 6889 6914 11.3190 3163 3137 3111 3086 8.6815 6842 6868 6894 6920 11.3185 3158 3132 3106 3080 10.0005 0005 0005 0005 0005 8.6940 6965 6991 7016 .7041 11.3060 335 3009 2984 2959 8-6945 697J 6996 7021 7046 "3055 3029 3004 2979 2954 11.2929 2904 2879 2855 2830 10.0005 0005 0005 0005 0006 9-9995 9995 9995 9995 9994 8.7066 7090 7H5 7140 7164 11.2934 2910 2885 2860 2836 8.7071 7096 7121 7H5 7170 10.0006 0006 0006 0006 0006 9-9994 9994 9994 9994 9994 8.7188 11.2812 8.7194 11.2806 10.0006 9-9994 / 1. L m s ! Cos. O'.l 1 s Sec. Cot. O'.l I 8 Tan. Cosec. Sin. 111 S / Diflf. Diff. | 92 a = 6 h 8 m ] [ 5h 4gm _ 8 .y 30 TABLE IX. 3 3 = O h 12 m ] L.O&. Sines, Tangents, and Secants. [ ll h 44 ni = 176 111 a Sin. Diff. Cosec. Tan. Diff. Cot. Sec. Cos. m s O'.l ! 1 s O'.l 1 s 1 2 3 4 5 6 7 8 9 1O 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 40 147 49 49 50 51 52 53 54 ! 55 56 57 58 59 60 12 4 8 12 16 12 2O 24 28 32 36 12 4O 44 48 52 56 13 O 4 8 12 16 13 2O 24 28 32 36 13 4O 44 48 52 56 14 O 4 8 12 16 14 2O 24 28 32 36 1440 44 48 52 56 15 4 8 12 16 1520* 24 28 32 36 15 4O 44 48 52 56 16 8.7188 7212 7236 7260 7283 2 5 7 9 12 16 18 21 2 4 7 9 ii II 18 20 2 4 6 8 ii 15 17 19 2 1 8 10 12 \l 18 2 1 8 10 12 14 15 17 2 6 7 9 ii 13 15 17 6 ' 12 17 6 ii 16 5 ii 16 5 10 15 5 10 5 9 11.2812 2788 2764 2740 2717 8.7194 7218 7242 7266 7290 2 5 7 9 12 16 18 21 2 4 7 9 u i 20 2 1 8 ii 13 15 17 19 2 4 6 8 10 12 14 16 18 2 4 6 8 10 12 15 17 2 4 6 7 9 ii 13 15 17 6 12 17 6 ii ,6 5 ii 16 5 10 15 5 10 14 5 9 14 11.2806 2782 2758 2734 2710 10.0006 9.9994 0006 9994 0006 9994 0006 9994 0006 9994 48 O 4756 52 48 4744 4O 36 32 28 47 24 2O 16 12 8 47 4 47 4656 52 48 4644 40 36 32 28 4624 20 16 12 8 46 4 46 4556 52 48 4544 40 36 32 28 4524 2O 16 12 8 45 4 45 4456 52 48 4444 40 36 32 28 4424 20 16 12 8 44 4 44 60 59 58 57 56 55 54 53 52 51 . 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 i 17 15 14 13 12 11 10 9 8 7 6 5 4 1 2 1 8.7307 733 7354 7377 7400 11.2693 2670 2646 2623 2600 87313 7337 7360 7383 7406 11.2687 2663 2640 2617 2594 10.0006 9.9994 0006 9994 0006 9994 0006 9994 0007 9993 8.7423 7445 7468 749i 7513 11.2577 2555 2532 2509 2487 8-7429 7452 7475 7497 7520 11.2571 2548 2525 2503 2480 10.0007 0007 0007 0007 0007 9-9993 9993 9993 9993 9993 8-7535 7557 7602 7623 11.2465 2443 2420 2398 23/7 8.7542 7565 7587 7609 7631 11.2458 2435 2413 2391 2369 10.0007 0007 0007 0007 0007 9-9993 9993 9993 9993 9993 8.7645 7667 7688 7710 773 l 8.7752 7773 7794 7836 "2355 2333 2312 2290 2269 8.7652 7674 7696 7717 7739 11.2348 2326 2304 2283 2261 10.0007 0007 0007 0008 0008 9-9993 9993 9993 9992 9992 11.2248 2227 2206 2185 2164 8.7760 778i 7802 7823 7844 11.2240 2219 2198 2177 2156 10.0008 0008 0008 0008 0008 9.9992 9992 9992 9992 9992 8.7857 7877 7898 7918 7939 11.2143 2123 2IO2 2082 2061 8.7865 7886 7906 7927 7947 ".2135 2114 2094 2073 2053 10.0008 0008 0008 0008 0008 9.9992 9992 9992 9992 9992 8-7959 7979 7999 8019 8039 II.2O4I 2021 2001 I98l I 9 6l 8.7967 7988 8008 8028 8048 11.2033 2OI2 1992 1972 I95 2 10.0009 0009 0009 0009 0009 9.9991 9991 9991 9991 9991 8.8059 8078 8098 8117 8i37 II.I94I 1922 1863 8.8067 8087 8107 8126 8146 H.I933 1913 1893 1874 185.4 10.0009 0009 0009 0009 0009 9.9991 9991 9991 9991 9991 8.8156 8175 8194 8213 8232 11.1844 1825 1806 1787 1768 8.8165 8185 8204 8223 8242 II.I835 1796 1777 1758 10.0009 0009 0009 OOIO OOIO 9.9991 9991 9991 9990 9990 8.8251 8270 8289 8307 8326 11.1749 173 I7II 1693 1674 8.8261 8280 8299 8317 8336 ".1739 1720 1701 1683 1664 IO.OOIO OOIO OOIO OOIO OOIO 9.9990 9990 9990 9990 9990 8.8345 8363 8381 8400 8418 II.I655 1637 1619 1600 1582 8.8355 8373 8392 8410 8428 II.I645 1627 1608 1590 1572 IO.OOIO OOIO OOIO OOIO OOIO 9.9990 9990 9990 9990 9990 8.8436 11.1564 8.8446 11.1554 IO.OOII 9.9989 ' m a Cos. O'.l 1 s Sec. Cot. O'.l | 1 Tan. Cosec. Sin. m s ' j Diff. Diff. 93 = 6 h 12 m ] [ 5 h 44 m = 86 | TABLE IX. 31 4 r=o h 16 11 ] Log:. Sines, Tangents, and Secants. [ll h 4O U1 = 175 i in 8 Sin. Diff. Cosec. Tan. Diff. Cot. Sec. Cos. m s / 0.1 1 O'.l 1 s O 1 2 3 4 5 6 7 8 9 10 11 12 13 | 14 ! 15 16 17 i 18 19 20 21 22 i 23 | 24 ! 25 26 ! 27 28 29 30 31 32 33 34 35 ! 36 i 37 i 38 39 40 41 42 43 44 45 46 47 48 49 50 151 |52 53 54 55 56 57 58 59 60 16 O 1 8 12 16 ' 1620 24 28 32 36 164O 44 48 52 56 17 4 8 12 16 1720 24 28 32 36 17 4O 44 48 52 56 18 4 8 12 16 1820 24 28 32 36 18 4O 44 48 52 56 19 4 8 12 16 1920 24 28 32 36 1940 44 48 52 56 20 8.8436 8454 8472 8490 8508 "8.8525 8543 8560 8578 8595 8.8613 8630 8647 8665 8682 8.8699 8716 8733 8749 8766 ""8^8783" 8799 8816 8833 8849 2 4 4 9 5 13 7 9 ii 12 H 1 6 2 4 3 9 5 '3 7 9 10 12 H 15 2 4 . 3 8 5 : 12 7 . 8 TO ii' J 3 15 2 4 3 8 1 " 8 9 ii 13 14 2 i 4 3 8 Ii" 8 9 ii 12 H i 4 3 7 4 " 6 7 9 i 10 12 13 11.1564 1546 1528 1510 J49 2 "W5 1457 1440 1422 1405 1 1~ 138 7 1370 1353 J335 _iM 11.1301 1284 1267 1251 J 2 34 11.1217 I2OI 1184 1167 "5i 8.8446 8465 8483 8501 8518 2 4 5 7 9 ii 12 14 16 2 3 5 7 9 10 12 H 15 2 3 7 8 10 ii 13 15 2 3 8 Q II 13 14 2 3 6 8 9 ii 12 H i 3 4 6 7 9 10 12 13 4 9 13 1 I 4 9 13 i 12 4 8 12 8 ii 4 7 ii 11.1554 1535 1517 1499 1482 IO.OOII 001 1 OOII OOII OOII 9.9989 9989 9989 9989 9989 44 4356 52 48 4344 40 36 32 28 4324 2O 16 12 8 43 4 43 O 4256 52 48 4244 40 36 32 28 4224 20 16 12 8 42 4 42 O 4156 52 48 4144 40 36 32 28 4124 20 16 12 8 41 4 41 O 4056 52 48 4044 40 36 32 28 4O24 20 16 12 8 40 4 40 60 59 58 57 56 55 54 53 52 51 50 49 S 48 47 46 45 44 43 42 ! 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 8.8536 8554 8572 8589 8607 8.8624 8642 8659 8676 8694 8.8711 8728 8745 8762 8778 8:8795 8812 8829 8845 8862 11.1464 1446 1428 1411 1393 11.1376 1358 H4i i3 2 4 J3o6 11.1289 1272 1255 1238 1222 IO.OOII OOII OOII OOII OOII IO.OOII 0012 OOI2 OOI2 OOI2 IO.OOI2 OOI2 0012 OOI2 0012 9.9989 9989 9989 9989 __99?9_ 9.9989 9988 9988 9988 9988 9.9988 9988 9988 9988 __?9_8S 9.9988 9987 9987 9987 9987 II.I2O5 1188 1171 H55 1138 IO.OOI2 OOI 3 OOI3 0013 OOI3 8.8865 8882 8898 8914 8930 8.8946 8962 8978 8994 9010 Il - ll 35 1118 IIO2 1086 1070 8.8878 8895 8911 8927 8944 11.1122 1105 I08 9 1073 1056 10.0013 OOI3 OOI3 OOI3 OOI3 9.9987 9987 9987 9987 9987 11.1054 1038 IO22 IOO6 0990 8.8960 8976 8992 9008 9024 11.1040 IO24 1008 0992 0976 IO.OOI3 0014 OOI4 0014 OOI4 9.9987 9986 9986 9986 9986 9.9986 9986 9986 9986 9986 8.9026 9042 9057 9073 9089 11.0974 0958 943 0927 0911 8.9040 9056 9071 9087 9103 11.0960 0944 0929 0913 0897 IO.OOI4 OOI4 OOI4 OOI4 0014 8.9104 9119 9135 9i5 9165 11.0896 0881 0865 0850 0835 8.9118 9 r 34 9150 9165 9180 11.0882 0866 0850 0835 O82O IO.OOI4 0015 OOI5 0015 OOI5 9.9986 9985 9985 9985 9985 8.9181 9196 9211 9226 9241 8.9256 9271 9286 9301 9315 11.0819 0804 0789 0774 759 8.9196 9211 9226 9241 9256 11.0804 0789 0774 0759 0744 IO.OOI5 0015 OOI5 0015 0015 9.9985 9985 9985 9985 9985 11.0744 0729 0714 0699 0685 8.9272 9287 9302 9316 933 i 11.0728 0713 0698 0684 0669 11.0654 0639 . 0624 o6lO 0595 10.0015 OOl6 OOl6 OOl6 OOl6 9.9985 9984 9984 9984 9984 8.9330 9345 9359 9374 _9_388 8.9403 11.0670 0655 0641 0626 0612 8.9346 9361 9376 9390 9405 IO.OOI6 OOl6 00l6 OOl6 OOl6 9.9984 9984 9984 9984 9984 11.0597 8.9420 11.0580 IO.OOI7 9.9983 / m s Cos. O'.l 1 s Sec. Cot. O'.l 1 s Tan. Cosec. Sin. m s / Diff. Diff. 949 = 6 h 16 m ] [ 5 U 4O m = 85 32 TABLE IX. i 5 = O n 2O m ] L,og. Sines, Tangents, and Secants. [ ll h 36 m = 174 3 / 1 2 3 4 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 i54 55 56 57 58 59 60 in s Sin. Diff. Cosec. Tan. Diff. Cot. Sec. Cos. m s 60 59 58 ; 57 56 | 55 54 i 53 52 51 5O 49 48 ; 47 46 45 ; 41 43 42 4B 40 39 38 37 36 35 34 33 32 31 30 29 28 i 27 ; 26 25 24 23 i 22 21 2O 19 18 17 16 15 14 i 13 12 11 10 9 8 7 6 5 4 3 2 1 O 0.1 i 3 4 6 7 8 10 ii 1 3 i 3 4 6 I 10 ii 13 i 3 4 5 I 9 10 12 I 3 4 5 9 10 12 I 3 4 5 6 8 9 % i 2 4 5 6 8 10 ii 1" 4 7 ii 4 7 ii I 10 \ 10 10 9 O'.l i 3 A 6 j 8 10 ii I 3 i 3 4 6 8 10 ii 13 i 3 4 5 Q 10 12 I 3 4 5 8 9 10 12 I 3 4 5 6 8 9 10 12 I 2 4 5 6 8 10 ii 1 s 2O O 4 8 12 16 2020 24 28 32 36 2O 4O 44 48 52 56 21 O 4 8 12 16 2120 24 28 32 36 2140 44 48 52 56 22 4 8 12 16 2220 24 28 32 36 22 4O 44 48 52 56 23 4 8 12 16 2320 24 28 32 36 2340 44 48 52 56 24 O 8.9403 9417 9432 9446 9460 8-9475 9489 953 9517 9531 8-9545 9559 9573 95*7 9601 11.0597 0583 0568 0554 ..J?_54 11.0525 0511 0497 0463 0469 "0455 0441 0427 4i3 0399 8.9420 9434 9449 94^3 9477 8.9492 9506 9520 9534 9549 8.9563 9577 959i 9605 9619 4 7 ii 4 7 ii 10 i 10 I 10 9 11.0580 0566 055 i 0537 523 11.0508 0494 0480 0466 045 * 11.0437 0423 0409 0395 0381 10.0017 0017 0017 0017 0017 9.9983 9983 9983 9983 99?3 9-9983 9983 9983 9983 9982 9.9982 9982 9982 9982 9982 4O O 39 56 52 48 3944 40 36 32 28 3924 20 16 12 8 39 4 39 3856 52 48 3844 40 36 32 28 3824 20 16 12 8 38 4 38 3756 52 48 3744 40 36 32 28 3724 20 16 12 8 37 4 37 3656 52 48 3644 40 36 32 28 3624 20 16 12 8 36 4 36 10.0017 0017 0017 0017 0018 10.0018 0018 0018 0018 0018 8.9614 9628 9642 9655 9669 11.0386 0372 0358 345 O33i 8-9633 9646 9660 9674 9688 11.0367 0354 0340 0326 0312 10.0018 0018 0018 0019 0019 9.9982 9982 9982 9981 9981 8.9682 9696 9709 9723 9736 11.0318 0304 0291 0277 02b 4 8.9701 9715 9729 9742 975 6 11.0299 0285 0271 0258 0244 10.0019 0019 0019 0019 0019 9.9981 9981 998i 9981 9981 8.9750 9763 9776 9789 9803 11.0250 0237 0224 021 1 0197 8.9769 9782 9796 9809 9823 8.9836 9849 9862 9875 9888 11.0231 0218 0204 0191 QI77 11.0164 0151 0138 0125 0112 10.0019 002 o OO2O 0020 OO2O 9.9981 9980 9980 9980 9980 8.9816 9829 9842 9855 9868 11.0184 OI7I 0158 0145 0132 IO.OO2O OO2O 0020 OO2O OO2I 9.9980 9980 9980 9980 9979 8.9881 9894 9907 9919 9932 II.OII9 OIO6 0093 0081 0068 11.0055 0042 0030 0017 OOO4 8.9901 9915 9927 9940 9953 II.OO99 0085 0073 OObO 0047 10.0021 OO2I OO2I 0021 OO2I 9-9979 9979 9979 9979 9979 8.9945 9958 9970 9983 9996 8.9966 9979 9992 9.0005 0017 11.0034 OO2I OOO8 10.9995 9983 IO.OO2I OO2I 0022 OO22 OO22 9-9979 9979 9978 9978 9978 9.0008 0021 0033 0046 0058 10.9992 9979 9967 9954 ___9942. 10.9930 9917 9905 9893 9880 10.9868 9856 9844 9832 9820 10.9808 9.0030 0043 0055 0068 0080 10.9970 9957 9945 9932 9920 IO.OO22 0022 OO22 OO22 0022 9.9978 9978 9978 9978 9978 9.0070 0083 0095 0107 OI2O "9^0132 0144 0156 0168 _pi8o 9.0192 9.0093 0105 0118 0130 _._ OI 43 9-0155 0167 0180 0192 0204 10.9907 9895 9882 9870 9857 10.9845 9833 9820 9808 9796 IO.OO23 OO23 0023 0023 OO23 IO.OO23 OO23 0023 OO24 OO24 9-9977 9977 9977 9977 9977 9-9977 9977 9977 9976 9976 9.0216 10.9784 IO.OO24 9.9976 Sin. / m B Cos. O'.l 1" Sec. Cot. O'.l 1 s Tan. Cosec. ni & Diff. Diff. 95 = 6 h 2O' ] [ 5 h 36 m = 84 . TABLE IX. 33 6 - O u 24 m ] tog:. Sines Tangents, and Secants. [ ll b 32"' = 173 Difif. Diff. I / 111 S Sin. Cosec Tan. Cot. Sec. Cos. I m s / O'.l 1 s O'.l 1* O 24 9.0192 10.9808 9.0216 10.9784 10.0024 9.9976 36 60 1 4 0204 i 3 9796 0228 i 3 9772 0024 9976 35 56 59 2 8 0216 2 6 9784 0240 2 6 9760 0024 9976 52 58 * 12 0228 4 9 9772 0253 4 9 9747 0024 } 9976 48 j 57 ; 4 16 0240 5 97 6 0265 5 9735 0024 9976 35 44 56 5 21 20 9.0252 6 110.9748 9.0277 6 10.9723 IO.OO2CJ 9-9975 40 55 : 6 24 0264 7 973 6 0289 7 9711 0025 ! 9975 36 54 7 28 0276 8 9724 0300 8 9700 0025 9975 32 53 8 32 0287 10 9713 0312 10 9688 0025 9975 28 52 9 36 0299 ii 9701 0324 ii 9676 0025 9975 3524 51 10 2440 9.0311 110.9689 9.0336 10.9664 10.0025 9-9975 20 50 | 11 44 0323 i 3 9677 0348 i 3 9652 0025 9975 16 49 12 48 334 2 6 9666 0360 2 6 9640 0025 9975 12 48 i 13 52 0346 3 9 9654 0371 3 9 9629 0026 9974 8 47 i 14 56 357 5 9643 0383 5 9617 0026 9974 35 4 46 15 25 9.0369 6 10.9631 9-0395 6 10.9605 10.0026 9-9974 35 45 16 0380 7 9620 0407 7 9593 0026 9974 3456 44 i 17 : s 0392 8 9608 0418 8 9582 0026 9974 52 43 ! 18 12 0403 9 9597 0430 9 9570 0026 9974 48 42 ! 19 16 0415 10 9585 0441 10 9559 0026 9974 3444 41 I 2O 25 2O 9.0426 10.9574 9-0453 10.9547 10.0027 9-9973 40 40 21 24 0438 i 4 9562 0464 i 3 953 6 0027 9973 36 39 j 22 28 0449 2 6 955 1 0476 2 6 9524 0027 9973 32 38 i 23 32 0460 3 9 9540 0487 3 9 95 1 3 0027 9973 28 37 ! 24 36 __472 4 9528 0499 5 95 01 0027 9973 3424 36 i 25 1 25 40 9.0483 5 10.9517 9.0510 6 10.9490 10.0027 9-9973 2O 35 i 26 44 0494 7 9506 0521 7 9479 0027 9973 16 34 i 27 48 0505 8 9495 533 8 9467 0028 9972 12 33 28 52 0510 9 9484 544 9 945 6 0028 9972 8 32 29 56 _?.5i7 10 9473 0555 10 9445 0028 9972 34 4 31 30 26 O 9-0539 10.9461 9.0567 10-9433 10.0028 9.9972 34 30 31 4 0550 i 3 945 0578 1 3 9422 0028 9972 3356 29 32 8 0561 2 S 9439 0589 2 ! 5 9411 0028 9972 52 28 ; 33 12 0572 3 8 9428 0600 3 8 9400 0028 9972 48 27 ; 34 16 _B53_ 4 9417 0611 4 i 93 8 9 0029 997i 3344 26 35 26 20 9.0594 5 10.9406 9.0622 ( 10.9378 10.0029 9.9971 40 25 36 24 0605 7 9395 0633 7 9367 0029 9971 36 24 i 37 28 0616 8 9384 0645 8' 9355 0029 9971 32 23 38 32 0626 9 9374 0656 o 9344 0029 9971 28 22 39 36 0637 10 93 6 3 0667 10 9333 0029 9971 3324 21 40 2640 9.0648 10.9352 '9.0678 10.9322 10.0029 9.9971 20 20 i 41 44 0659 i 3 934i 0688 I 3 9312 0030 9970 16 19 42 48 0670 2 5 9330 0699 2 9301 0030 9970 12 18 i 43 52 0680 3 8 9320 0710 3 8 9290 0030 9970 8 17 ! 44 56 0691 4 939 0721 4 9279 0030 9970 33 4 16 45 27 9.0702 5 110.9298 9.0732 5 10.9268 10.0030 9.9970 33 15 46 4 0712 7 9288 743 7 9257 0030 99/0 3256 14 i 47 8 0723 8 9277 754 8 9240 0031 9969 52 13 48 12 0734 9 9266 0764 9 9236 0031 9969 48 12 49 16 0744 10 9256 0775 10 9225 0031 9969 3244 11 5O 2720 9-0755 10.9245 9.0786 10.9214 10.0031 9.9969 40 10 51 24 0765 i 3 9235 0796 i 3 9204 0031 9969 36 9 52 53 28 32 0770 0786 3 8 9224 9214 0807 0818 2 S i 9193 9182 0031 0031 9969 9969! 32 28 8 ; 7 ; 54 36 0797 4 9203 0828 4 9172 0032 9968 3224 6 55 56 57 27 4O 44 48 9.0807 0818 0828 7 10.9193 9182 9172 9.0839 0849 0860 7 10.9161 9J5 1 9140 0.0032 0032 0032 9.9968 9968 9968 20 16 12 5 ! 4 i 3 58 59 52 56 0838 0849 8 9 9162 9I5 1 0871 0881 8 9 9129 9119 0032 0032 9968 9968 8 32 4 2 ! 1 i 6O 28 O 9.0859 10.9141 9.0891 10.9109 0.0032 9.9968 32 O'.l 1 s O'.l 1 s 1 ' ' m s Cos. Sec. Cot. Tan. Cosec. Sin. m s / 1 DiflF. DiflF. 1 J 96 = 6 24> ] [ 5 h 89 m = 83 | 34 TABLE IX. | 7 = O h 28'" ] L,og. Sines, Tangents, and Secants. [ ll h 28> - ] m s 32 3156 59 48 31 34 10 36 32 28 3124 20 16 12 8 31 4 31 3O56 52 48 3044 40 36 32 28 3024 20 16 12 8 30 4 3O O 2956 52 48 2944 40 36 32 28 29 24 2O 16 12 8 29 4 29 2856 52 48 2844 40 36 32 28 2824 20 16 12 8 28 4 28 72 t / m B Sin. Diff. Cosec Tan. Diff, Cot. Sec. Cos. J0'.l 1 O'.l 1 3 5 8 3 8 2 7 2 5 7' 2 5 7 2 5 7 1 2 3 4 5 6 7 9 10 11 12 13 14 15 16 17 18 19 2O 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 28 O 4 12 16 2820 24 28 32 36 2840 44 48 52 56 29 4 8 12 16 2920 24 28 32 36 2940 44 48 52 56 30 4 8 12 16 3020 24 28 32 36 3040 44 48 52 56 31 4 8 12 16 3120 24 28 32 36 314O 44 48 52 56 32 9.0859 0869 0879 0890 0900 9.0910 0920 0930 0940 0951 9.0961 0971 0981 0991 IOOI 9.1011 1 020 1030 1040 1050 i 2 3 4 5 6 8 9 i 2 3 4 6 8 9 i 2 3 4 I I 9 i 2 3 4 3 3 5 7 2 S 7 2 5 7 2 5 7 2 5 7 10.9141 9I3 1 9121 9110 9100 10.9090 9080 9070 9060 9049 10.9039 9029 9019 1078989 8980 8970 8960 8950 9.0891 0902 0912 0923 __933 9.0943 0954 0964 0974 0984 9.0995 1005 1015 1025 _35 9.1045 Mil 1076 1086 i 3 4 6 1 9 i 2 3 4 5 6 i 9 i 2 3 4 I 9 i 2 3 4 I 9 i 2 3 4 7 8 7 i 2 3 4 I I 1 0.9 1 of 9098 9088 9077 oob 7 10.9057 9045 9036 9026 9016 : 1 0.9005 8995 8985 8975 8965 10.8955 8945 8934 8924 8914 10.8904 8894 8884 8875 8865 10.003 003 003 003 003 10.003 3o 0034 0034 0034 10.0034 0034 0034 35 ..._35 10.0035 0035 35 0035 0036 9.9968 9967 9967 9967 9967 9.9967 9967 9966 9966 9906 9.9966 9966 9966 9965 _99j>5 9.9965 9965 9965 9965 9964 60 59 58 ; 57 56 55 54 ! 53 : 52 51 50 49 48 47 46 45 44 i 43 42 i 41 40 39 38 : 37 1 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 j 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 9.1060 1070 1080 1089 1099 10.8940 8930 8920 8911 8901 9.1096 1106 1116 1125 "35 10.0036 0036 0036 0036 0036 9.9964 9964 9964 9964 9964 9.1109 1118 1128 1138 1147 10.8891 8872 8862 8853 9-II45 "55 1165 "75 "85 10.8855 8845 8835 8825 8815 10.0036 0037 0037 0037 0037 9.9964 9963 9963 9963 9963 9-"57 1167 1176 1186 "95 10.8843 8833 8824 8814 8805 9.1194 1204 1214 1223 _i?33 9.1243 1252 1262 1272 1281 10.8806 8796 8786 8777 8767 10.0037 0037 0038 0038 0038 9.9963 9963 9962 9962 9962 9.1205 1 5 1214] 6 1224 j 7 1233 8 1242 i 9 10.8795 8786 8776 8767 8758 10.8757 8748 8738 8728 8719 10.0038 0038 0038 0039 0039 9.9962 9962 9962 9961 9961 9-996T 9961 9961 9960 9960 9.1252 1261 1271 1280 1289 i 2 3 4 7 I i 2 3 4 ! i 10.8748 8739 8729 8720 8711 10.8701 8692 8683 8674 8664 1^8655 8646 8637 8628 8619 9.1291 1300 1310 13*9 1329 10.8709 8700 8690 8681 8671 10.0039 0039 0039 0040 0040 9.1299 1308 1317 1326 _JL33l 9.1345 1354 1363 1372 _J3?i 9.1390 1399 1409 1418 -JW 9.1436 9.i33f 1348 1357 1367 1376 9-1385 1395 1404 Hi3 1423 10.8662 8652 8643 8633 _8624 10.8615 8605 8596 8587 8577 10.0040 0040 0040 0040 0041 10.0041 0041 0041 0041 0041 9.9960 9960 9960 9960 _99S9 9-9959 9959 9959 9959 9959 10.8610 8601 8591 8582 -JS73 10.8564 9-1432 1441 H5 1460 __L4 6 9 9.1478 10.8568 8559 8550 8540 _ 8 53i 10.8522 10.0042 0042 0042 0042 0042 0.0042 9.9958 9958 9958 9958 _9958 9.9958 / m B Cos. O'.l Sec. Cot. O'.l 1 s Tan. Cosec. Sin. m s / Diff. Diff. 97 5 = 6 u 28 m ] [5 h 28 1 "^82 TABLE IX. 35 8 = O h 32 ] JLogr. Sines, Tangents, and Secants. [ ll h 24 m = 171 ' 111 B Sill. Diff. Cosec. Tan. Diff. Cot. Sec. Cos. m s ' j O'.l 1 s O'.l 1 s 1 2 3 4 5 6 7 8 9 1O 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 .33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 32 4 8 12 16 32 20 24 28 32 36 324 45 48 52 56 33 4 8 12 16 33 2O 24 28 32 36 3340 44 48 52 56 34 O 4 8 12 16 3420 24 28 32 36 34 4O 44 48 52 56 35 4 8 12 16 35 2O 24 28 32 36 3540 44 48 52 56 36 9.1436 1445 1453 1462 1471 2 3 4 4 6 j i 2 3 4 6 I 2 3 3 A 5 Q I 2 3 3 4 I 6 7 i 2 2 4 1 6 7 i 2 2 3 4 6 6 7 2 s 7 2 5 7 2 6 2 I 2 4 6 2 4 6 10.8564 8555 8547 8538 _?S29_ 10.8520 8511 8502 *493 8484 9.1478 1487 1496 1505 I5i5 i 2 3 4 5 8 i 2 3 4 5 I I i 2 3 4 4 8 i 2 3 3 4 8 i 2 3 3 4 I 6 7 i 2 2 3 4 6 6 7 2 5 7 2 5 7 2 5 7 2 I 2 4 6 2 I 10.8522 8513 8504 8495 8485 10.8476 8467 8458 8449 8440 10.0042 0043 0043 0043 0043 9.9958 9957 9957 9957 "9957 28 2756 52 48 2744 40 36 ' 32 28 2724 20 16 12 8 27 4 27 2656 52 48 2644 40 36 32 28 2624 20 16 12 8 26 4 26 2556 52 48 2544 40 36 32 28 2524 20 16 12 8 25 4 25 2456 52 48 2444 40 36 32 28 2424 20 16 12 8 24 4 24 O 60 59 58 57 56 55 54 53 52! 51 50 49 48 47 S 44! 43 42! 41 40 39 38! 37 i 36 35 j 34 33 32 31 30 29 28 27 26 25 24 23 22 21 2O i 19] 18 17 16 15 14 13 12 11 10 9 8 7! 6 5 4 3 2 1 9.1480 1489 1498 i57 1516 9-i5 2 5 1533 1542 J55 1 1560 "971568 1577 1586 1594 1603 9.1524 1533 1542 *55i 1560 10.0043 0044 0044 0044 0044 9-9957 9956 995 6 995 6 9956 10.8475 8467 8458 8449 8440 9.1569 1578 1587 i59 6 1605 10.8431 8422 8413 8404 8395 10.0044 0044 0045 0045 0045 9.9956 995 6 9955 9955 9955 10.8432 8423 8414 8406 8397 9.1613 1622 1631 1640 1649 9.1658 1667 1675 1684 1693 10.8387 8378 8369 8360 835i 10.0045 0045 0046 0046 0046 9-9955 9955 9954 9954 9954 9.1612 1620 1629 1637 1646 10.8388 8380 8371 8363 8354 10.8342 8333 8325 8316 8307 10.0046 0046 0046 0047 0047 9-9954 9954 9954 9953 9953 9-1655 1663 1672 1680 1689 10.8345 8337 8328 8320 8311 9.1702 1710 1719 1728 i73 6 10.8298 8290 8281 8272 8264 10.0047 0047 0047 0048 0048 9-9953 9953 9953 9952 9952 9.1697 1705 1714 1722 i73i 10.8303 8295 8286 8278 8269 9-1745 1754 1762 1771 1779 10.8255 8246 8238 8229 8221 10.0048 0048 0048 0049 0049 9.9952 9952 9952 995 i 995i 9-1739 1747 i75 6 1764 1772 10.8261 8253 8244 8236 8228 9.1788 1797 1805 1814 1822 10.8212 8203 8*95 8186 8178 10.0049 0049 0049 0049 0050 9-9951 995 i 995i 995 i 995 9.1781. 1789 1797 1806 1814 10.8219 8211 8203 8i94 8186 9.1831 1839 1848 1856 1864 10.8169 8161 8152 8144 8136 10.0050 0050 0050 0050 0051 9.9950 995 995 995 9949 9.1822 1830 1838 1847 1855 10.8178 8170 8162 8i53 8i45 9-i873 1881 1890 1898 1906 10.8127 8119 8110 8102 8094 10.0051 0051 0051 0051 0052 9-9949 9949 9949 9949 9948 9.1863 1871 1879 1887 1895 10.8137 8129 8121 8113 8105 9-I9I5 1923 1931 1940 1948 10.8085 8077 8069 8060 8052 10.0052 0052 0052 0052 53 9.9948 9948 9948 9948 9947 9.1903 1911 1919 1927 T 935 10.8097 8089 8081 8073 8065 9.1956 1964 1973 1981 1989 10.8044 8036 8027 8019 Sou 10.0053 0053 0053 53 0054 9-9947 9947 9947 9947 9946 9-1943 10.8057 9.1997 10.8003 10.0054 9.9946 / m 8 Cos. O'.l 1 s Sec. Cot. O'.l 1 s Tan. Cosec. Sin. m s / Diff. Diff. 98 = 6 h 32 m ] [5 h 24 m = 81 36 TABLE IX. 9 = O h 36 in ] L,og. Sines, Tangents and Secants. [ll h 2O tn s 1?O / m s Sin. Diff. Cosec. Tan. Diff. Cot. Sec. 0.0054 0054 0054! 0054 0055 Cos. m s / ! O'.l 1 s O'.l 1* 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 ,37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 36 O 4 8 12 16 36 2O 24 28 32 36 36 4O 44 48 52 56 37 4 8 12 16 3720 24 28 32 36 3740 44 48 52 56 38 4 8 12 16 3820 24 28 32 36 3840 44 48 52 56 39 4 8 12 16 3920 24 28 32 36 3940 44 48 52 56 40 9-1943 195 i 1959 1967 1975 i 2 2 3 4 6 7 i 2 2 3 4 6 7 i 2 2 3 4 5 :| 7 2 2 3 4 5 7 i 2 2 3 4 4 ' 6 7 i 2 2 3 4 4 1 7 2 4 6 2 4 6 2 6 2 4 6 2 4 6 . 2 6 0.8057 8049 8041 8033 8025 9.1997 2005 2013 2O22 2030 1 i i 2 ' 2 3 4 6 7 2 2 3 4 6 7 2 2 3 4 5 7 i 2 2 3 4 5 6 7 2 2 3 4 4 i 7 i 2 2 3 4 4 I 7 2 4 6 2 4 6 2 4 6 2 6 2 4 6 2 I 10.8003 7995 7987 7978 7970 9.9946 9946 0946 9946 9945 24 23 56 52 48 2344 40 36 32 28 23 24 2O 16 12 8 23 4 23 O 22 56 52 48 2244 40 36 32 28 2224 20 16 12 8 22 4 22 2156 52 48 2144 40 36 32 28 2124 20 16 12 8 21 4 21 2O 56 52 48 2O 44 40 36 32 28 2O 24 20 1O 12 8 20 4 2O O 60! 59 * 57 56 55 51 53 52 51 50 49 48 47 46 45 1 44! 43 i 42 41 I 40 39 38 37j 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 8 7 6 3 4 3i 9 1 9.1983 1991 1999 2007 201 S 0.8017 8009 8001 7993 7985 9.2038 2046 2054 2062 2O7O 10.7962 7954 7946 7938 793 0.0055 0055 55 0055 0056 9-9945 9945 9945 9945 _9944_ 9-9944 9944 9944 9944 9943 9-9943 9943 9943 9943 9942 9-2022 2030 2038 2046 2054 0.7978 7970 7962 7954 7946 9.2078 2086 2094 2102 2110 10.7922 79H 7906 7898 7890 10.0056 0056 0056 0056 0057 10.0057 0057 0057 0057 0058 9.2O6l 2069 2077 2085 2092 0.7939 793 i 7923 7915 7908 9-2II8 2126 2134 2142 2I5O 10.7882 7874 7866 7858" 7850 9-2IOO 2108 2115 2123 2I 3 I 10.7900 7892 7885 7877 7869 9 2158 2166 2174 2181 2189 10.7842 7834 7826 7819 7811 10.0058 0058 0058 0058 0059 9-9942 9942 9942 9942 9941 9.2138 2146 2153 2161 2169 10.7862 7854 7847 7839 7831 9.2197 2205 2213 2221 2228 10.7803 7795 7787 7779 7772 10.0059 0059 0059 0060 0060 9.9941 9941 9941 9940 _994_ 9.9940 9940 9940 9939 9939 9.2176 2184 2191 2199 2206 10.7824 7816 7809 7801 7794 9.2236 2244 2252 2259 2267 10.7764 775 6 7748 774i 7733 10.0060 0060 0060 0061 0061 9.2214 2221 2229 2236 2243 10.7786 7779 7771 7764 7757 9-2275 2282 2290 2298 2305 10.7725 7718 7710 7702 7695 1 0.006 1 0061 0061 0062 0062 9-9939 9939 9939 9938 9938 9.2251 2258 2266 2273 2280 10.7749 7742 7734 7727 7720 9.2313 2 3 2I 2328 2336 2343 10.7687 7679 7672 7664 7657 10.0062 0062 0063 0063 0063 9-9938 9938 9937 9937 9937 9.2288 2295 2303 2310 2317 10.7712 7705 7697 7690 7683 9-235I 2 359 2366 2374 2381 10.7649 7641 7634 7626 7619 10.0063 0063 0064 0064 0064 9-9937 9937 993 6 9936 993 6 9.9936 993 6 9935 9935 __9935_ 9-9935 9934 9934 9934 9934 9.2324 2332 2339 2346 2353 10.7676 7668 7661 7654 7647 9.2389 2396 2404 2411 2419 10.7611 7604 759 6 7589 758i 10.0064 0064 0065 0065 0065 9.2361 2368 2375 2382 2390 10.7639 7632 7625 7618 7610 9.2426 2434 2441 2448 2456 io-7574 7566 7559 7552 7544 10.0065 0066 0066 0066 0066 9-2397 10.7603 9-2463 10.7537 10.0066 9-9934 m s Cos. O'.l 1 s Sec. Cot. O'.l 1 Tan. Cosec. Sin. m s / Diff. Diff. 99 - 6* 36 ] [ 5 h 20 m = 8O TABLE IX 1O = O h 4O m ] Log. Sines, Tangents, and Secants. [ ll h 16 m = 169- . ' m' s Sin. Diff. Cosec. 10.7603 759^ 75^9 7582 7575 Tan. Di 0.1 ff. 1 s 2 4 6 I 2 4 5 2 4 5 2 4 5 2 3 5 1 s Cot. Sec. Cos. m s ' OM i 2 3 4 4 I 6 i i 2 3 4 4 I 6 2 3 4 4 6 i 2 3 4 4 6 i i 2 3 o 4 5 i i 2 3 3 4 5 1 s o "M 2 A 4 6 7 8 9 1O 11 12 13 14 15 16 17 18 19 20 21 i 22 ! 23 | 24 25 ! 26 | 27 28 29 30 31 32 33 34 35 36 37 38 39 4O i 41 i 42 1 43 44 45 46 47 4 49 50 51 52 53 54 55 56 57 58 59 60 4O O 4 8 12 16 4O2 24 28 32 36 4O4O 44 48 52 56 41 4 8 12 16 4120 24 28 32 36 41 4O 44 48 52 56 42 4 8 12 16 42 2O 24 28 32 36 4240 44 48 52 56 43 O 4 8 12 16 43 2O 24 28 32 36 43 4O 44 48 52 56 44 9.2397 2404 2411 2418 2425 2 4 5 2 5 2 4 5 2 4 5 2 3 5 2 3 5 9.2463 2471 2478 2485 2493 i 2 2 3 4 4 7 i 2 2 3 4 4 7 2 2 3 4 4 6 6 i 2 3 4 4 I 6 i i 2 3 4 4 3 6 i i 2 3 4 4 6 10-7537 7529 7522 75i5 757 10.0066 0067 0067 0067 0067 9-9934 9933 9933 9933 9933 2O O 1956 52 48 1944 40 36 32 28 1924 20 16 ' 12 8 19 4 19 O 1856 52 48 1844 40 36 32 28 1824 20 16 12 8 18 4 18 1756 52 48 1744 40 36 32 28 1724 20 16 12 8 1.7 4 17 O 1656 52 48 1644 40 36 32 28 1624 2O 16 12 8 16 4 16 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 1O 9 8 7 6 5 4 3 2 1 9.2432 2439 2447 2454 2461 10.7568 75 61 7553 7546 __7539 IO-753 2 7525 75i8 75ii 7504 9.2500 2507 2515 2522 2529 10.7500 7493 7485 7478 747i 10.0068 0068 0068 0068 0069 9.9932 993 2 9932 9932 993 i 9.2468 2475 2482 2489 2496 9-2536 2544 255i 2558 _2565 9.2573 2580 2587 2594 2601 10.7464 745 6 7449 7442 7435 10.0069 0069 0069 0069 0070 9-993 i 993 i 993 * 993 i 993 9-2503 2510 2517 2524 253i 10.7497 7490 7483 7476 7469 10.7427 7420 7413 7406 7399 10.0070 0070 0070 0071 0071 9.9930 993 993 9929 9929 9-253 8 2545 255i 2558 2565 10.7462 7455 7449 7442 7435 9.2609 2616 2623 2630 2637 10.7391 7384 7377 7370 _7363 10.7356 7349 7342 7334 7327 10.0071 0071 0071 0072 0072 9.9929 9929 9929 9928 9928 9-2572 2579 2586 2593 2600 10.7428 7421 74H 7407 7400 9.2644 2651 2658 2666 2673 10.0072 0072 0073 0073 0073 9.9928 9928 9927 9927 9927 9.2606 2613 2620 2627 2634 10.7394 7387 7380 7373 7366 9.2680 2687 2694 2701 2708 10.7320 7313 7306 7299 7292 10.0073 0074 0074 0074 0074 9.9927 9926 9926 9926 9926 9.2640 2647 2654 2661 2667 10.7360 7353 7346 7339 7333 9-27I5 2722 2729 2736 2743 10.7285 7278 7271 7264 7257 10.7250 7243 7236 7230 7223 10.0075 0075 0075 0075 0075 9.9925 9925 9925 9925 9925 9.2674 2681 2687 2694 2701 10.7326 73 19 7313 7306 7299 9-2750 2757 2764 2770 2777 10.0076 0076 0076 0076 0077 9.9924 9924 9924 9924 9923 9.2707 2714 2721 2727 2734 10.7293 7286 7279 7273 7266 9.2784 2791 2798 2805 2812 10.7216 7209 7202 7195 7188 10.0077 0077 0077 0078 0078 9.9923 9923 9923 9922 9922 9.2740 2747 2754 2760 2767 10.7260 7253 7246 7240 7233 9.2819 2825 2832 2839 2846 10.7181 7175 7168 7161 7154 10.0078 0078 0079 0079 0079 9.9922 9922 9921 9921 992 T 9-2773 2780 2786 2793 2799 10.7227 7220 7214 7207 7201 9-2853 2859 2866 2873 2880 10.7147 7141 7134 7127 7120 10.0079 0080 0080 0080 0080 9.9921 9920 9920 9920 9920 9.2806 10.7194 9.2887 10.7113 1 0.008 1 9.9919 / m s Cos. OM 1 s Sec. Cot. OM Tan. Cosec. Sin. m s / Diff. Diff. | 1OO = 6 h 4O m ] [ 5h I6 m = 79 38 TABLE IX. 11 = O h 44 m ] L.og. Sines, Tangents, and Secants. [ ll h 12 m = 168 / ni a Sin. Diff. Cosec. Tan. Diff, Cot. Sec. Cos. m s ' o'.i ! i | O'.I 1 s 1 2 3 4 5 6 7 8 9 1O 11 1 12 13 14 15 ! 16 17 18 19 20 21 1 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 i 55 56 57 58 59 60 44 4 8 12 16 44 2O 24 28 32 36 4440 44 48 52 56 45 4 8 12 16 4520 24 28 32 36 4540 44 48 52 56 46 4 8 12 16 4620 24 28 32 36 4640 44 48 52 56 47 4 8 12 16 4720 24 28 32 36 47 4O 44 48 52 56 48 9.2806 2812 2819 2825 2832 i 2 3 3 4 1 i i 2 3 3 4 5 I i i 2 3 3 4 4 i i 2 2 3 4 4 5 5 i i 2 2 3 4 4 5 5 i i 2 2 3 4 4 5 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 10.7194 7188 7181 7175 7168 9.2887 2893 2900 2907 2913 i 2 3 3 4' 5 I i i 2 3 3 4 5 6 i i 2 3 3 4 5 i 2 3 3 4 4 i 2 3 3 4 4 i i 2 3 3 4 4 I 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 10.7113 7107 7100 7093 __7o87 10.7080 7073 7067 7060 7053 10.7047 7040 7033 7027 7020 10.0081 0081 0081 0081 0082 9.9919 9919 9919 9919 9918 16 1556 52 48 1544 40 36 32 28 1524 20 16 12 8 15 4 15 14 56 52 48 1444 40 36 32 28 11 24 20 16 12 8 14 4 14 O 1356 52 48 1344 4O 36 32 28 1324 2O 16 19 8 13 4 13 1256 52 48 1244 40 36 32 28 1224 20 16 12 8 12 4 12 O 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 ! 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 i 7 6 9 4 3 2 1 9.2838 2845 2851 2858 _ 2864 9.2870 2877 2883 2890 2896 10.7162 7155 7M9 7142 7i3 6 10.7130 7123 7117 7110 7104 9.2920 2927 2933 2940 2947 9-2953 2960 2967 2973 2980 10.0082 0082 0082 0083 0083 10.0083 0083 0084 0084 0084 9.9918 9918 9918 9917 9917 9.9917 9917 9916 9916 9916 9.2902 2909 2915 2921 2928 10.7098 7091 7085 7 79 7072 10.7066 7060 7053 7047 7041 10.7035 7028 7022 7016 7010 9.2987 2993 3000 3006 3013 10.7013 7007 7000 6994 6987 10.0084 0085 008 5 0085 0085 9.9916 9915 9915 99!5 _995. 9.9914 9914 9914 9914 9913 9.2934 2940 2947 2 953 2959 9.3020 3026 333 3039 _3S4A 9-3052 3059 3065 3072 3078 10.6980 6974 6967 6961 6954 10.0086 0086 0086 0086 0087 10.0087 0087 0087 0088 0088 9.2965 2972 2978 2984 2990 10.6948 6941 6 935 6928 6922 9-99I3 9913 9913 9912 9912 9.2997 33 3009 3oi5 3021 10.7003 6997 6991 6985 6979 9-3085 3091 3098 3 I0 4 3110 10.6915 6909 6902 6896 6890 10.0088 0088 0089 0089 0089 9.9912 9912 9911 9911 9911 9.3027 334 3040 3046 _3251 9-3058 3064 3070 377 3083 I0 1$ 6960 6954 _ 6 94l 10.6942 6936 6930 6923 6917 9-3 IJ 7 3^3 3130 3'36 __3J42 9-3 H9 3155 3162 3168 3*74 10.6883 6877 6870 6864 6858 10^6851 6845 6838 6832 6826 10.0089 0090 0090 0090 0090 10.0091 0091 0091 0091 0092 9.9911 9910 9910 9910 _99io 9.9909 9909 9909 9909 9908 9.3089 3095 3101 3107 _3ii3_ 9-3 IJ 9 3125 3i3i 3137 3143 10.6911 9.3181 3187 3 J 93 3200 3206 10.6819 6813 6807 6800 6794 10.6788 6781 6775 6769 _M3 10.6756 6750 6744 6738 6731 10.0092 0092 0092 0093 0093 9.9908 9908 9908 9907 9907 10.6881 6875 6869 6863 6857 9.3212 3219 3225 3231 3237 10.0093 0094 0094 0094 0094 9.9907 9906 9906 9906 9906 9-3I49 3155 3161 3167 3173 10.6851 6845 6839 6833 6827 9-3244 325 3256 3262 3269 10.0095 0095 0095 0095 0096 9.9905 9905 9905 9905 9904 9-3I79 10.6821 9-3275 10.6725 10.0096 9.9904 / m B Cos. O'.l I 8 Sec. Cot. o'.i i I- Tan. Cosec. Sin. m B / Diff. Diff. 101 = 6 h 44 m ] C 5 h 12 m = 78 TABLE IX. 39 12 = O" 48 m ] Log. Sines, Tangents, and Secants. [II 11 = 167 / 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 m s Sin. 9-3I79 31S5 3i9i 3*97 3202 9.3208 32H 3220 3226 3 2 3 2 Diff. Cosec. Tan. 9-3275 3281 3287 3293 3300 Diff. Cot. Sec. 10.0096 0096 0096 0097 0097 Cos. 9.9904 9904 9904 9903 9903 m s / 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38: 37 36 35 34 33 32 31 3O 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 ! 6 5 4: 2 1 O O'.l | 1 s O'.l i i 2 2 3 4 4 5 i 2 2 3 4 4 5 i 2 2 3 4 5 5 4 4 5 5 i i 2 2 3 4 4 5 5 i i 2 2 3 3 4 5 5 1 s 48 O 4 8 12 16 4821* 24 28 32 36 4840 44 48 52 56 49 O 4 8 12 16 4920 24 28 32 36 49 4O 44 48 52 56 5O O 4 8 12 16 50 20 24 28 32 36 5O4O 44 48 52 56 51 4 8 12 16 5120 24 28 32 36 5140 44 48 52 56 52 2 2 3 4 4 5 5 i 2 2 3 4 4 5 5 i i 2 2 3 4 4 5 5 i i 2 3 3 4 5 5 2 2 I \ I 2 2 3 3 4 4 5 2 3 5 2 1 2 3 5 i 3 5 i 3 5 ' i 3 4 10.6821 6815 6809 6803 J5798 10.6792 6786 6780 6774 6768 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 3 5 10.6725 6719 6713 6707 6700 12 1156 52 48 1144 40 36 32 28 1124 20 16 12 8 11 4 11 1O56 52 48 1044 40 36 32 28 1024 20 16 12 8 1O 4 1O O 956 52 48 944 40 36 32 28 924 20 16 12 8 9 4 9 O 856 52 48 844 40 36 32 28 824 2O 16 12 8 8 4 8 9-3306 3312 33i8 3324 3330 10.6694 6688 6682 6676 6670 10.0097 0098 0098 0098 0098 9.9903 9902 9902 9902 9962 9-32 }8~ 3 2 44 3250 3255 3261 10.6762 6756 6750 6 745 6 739 9-3336 3343 3349 3355 336i 10.6664 6657 6651 6645 6639 10.0099 0099 0099 0099 OIOO 9.9901 9901 99oi 9901 990Q 9.9900 9900 9899 9899 9899 9.3267 3273 3279 3284 3290 10.6733 6727 6721 6716 6710 9-33 6 7 3373 3379 3385 339i 10.6633 6627 6621 6615 6609 10.0100 OIOO OIOI OIOI OIOI 9.3296 3302 3308 3313 3319 10.6704 6698 6692 6687 6681 9-3397 3403 3409 34i6 3422 10.6603 6597 6591 6584 6578 10.0101 OIO2 OIO2 0102 OIO3 io.oi6y 0103 0103 0104 0104 10.0104 0104 0105 0105 0105 9-9899 9898 9898 9898 9897 9-3325 3331 333 6 3342 3348 9-3353 3359 3365 3370 3376 10.6675 6669 6664 6658 6652 10.6647 6641 6635 6630 6624 9-3428 3434 3440 344 6 .._345.2 9.3458 3464 3469 3475 348i 10.6572 6566 6560 6554 6548 10.6542 6536 6531 6525 6^19 9.9897 9897 9897 9896 __9M 9.9896 9896 9895 9895 9895 9-3382 3387 3393 3399 3404 10.6618 6613 6607 6601 6596 9-3487 3493 3499 3505 35" 10.6513 6507 6501 6495 6489 10.0106 0106 0106 0106 0107 9.9894 9894 9894 9894 9893 9.3-410 34i 6 342i 3427 3432 10.6590 6584 6579 6 573 6568 9-35 i 7 3523 3529 3535 354i 10.6483 6477 6471 6465 6459 10.0107 0107 0108 0108 0108 9-9893 9893 9892 9892 _992 9.9892 9891 9891 9891 9890 9.9890 9890 9890 9889 9889 9.9889 9888 9888 9888 9888 9-3438 3444 3449 3455 3460 9.3466 3471 3477 3482 3488 9-3493 3499 354 35io 3515 10.6562 6556 6551 6545 6540 10.6534 6529 6 523 6518 6512 10.6507 6501 6496 6490 6485 9.3546 3552 3558 3564 _357o_ 9-3576 358i 3587 3593 _3599_ 9.3605 3611 3616 3622 3628 10.6454 6448 6442 6436 6430 10.6424 6419 6413 6407 6401 10.6395 6389 "6384 6378 6372 10.0108 0109 0109 0109 OIIO IO.OIIO OIIO OIIO OIII OH I 10.01 1 1 0112 OII2 0112 OII2 9-3521 10.6479 9.3634 10.6366 10.0113 9.9887 / m s Cos. O'.l I 8 Sec. Cot. O'.l 1 s Tan. Cosec. Sin. m s / Diff. Diff. 102 = 6 h 48 m ] [ 5 h 8'" = 77" : 40 TABLE IX. 1 : 13 ._ O' : 52 m ] Log. Sines, Tangents, and Secants. [II 1 4 m = 166 ' m s Sin. Diff. Cosec. Tan. Diff. Cot. Sec. 10.0113 0113 0113 0114 0114 10.0114 0115 0115 0115 0115 Cos. Ill S 8 O 756 52 48 744 40 36 32 28 724 20 16 12 8 7 4 7 656 52 48 644 40 36 32 28 624 20 16 12 8 6 4 6 556 52 48 544 4O 36 32 28 524 20 16 12 8 5 4 5 456 52 48 444 40 36 32 28 424 20 16 12 8 4 4 OO 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 | 24 i 23 22 21 2O i 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 \ 2 J O O'.l 1 s O'.l 1 s 1 3 4 5 f I 9 1O 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 36 57 58 59 6O 52 4 8 12 16 5220 24 28 32 36 5240 44 48 52 56 53 4 8 12 16 53 2O 24 28 32 36 5340 44 48 52 56 51 4 8 12 16 54 2O 24 28 32 36 5440 44 48 52 56 55 4 8 12 16 5520 24 28 32 36 5540 44 48 52 56 56 9-3521 3526 3532 3537 3543 9-3548 3554 3559 35t>4 357Q 9-3575 35^1 3586 359i 3597 9.3602 3608 3 6 i3 3618 3624 i i i 3 2 \ 4 2 3 3 ; 4 ! 4 i 5 ' i i i 3 2 \ 4 2 3 3 4 4 5 i ! i 1 i 3 2 \ 4 2 3 i 3 4 [ 4 5 i i i 3 2 4 2 3 3 4 4 5 i i i 3 2 4 2 3 3 4 4 5 i i i 3 2 4 2 3 3 : 4 i 4 10.6479 6474 6468 6463 ..-JJ457. 10.6452 6446 6441 6436 6430 9-3634 3639 3645 3651 _.3 6 57_ 9.3662 3668 3674 3680 3685 i i 2 2 3 3 4 5 5 1 2 2 3 3 4 5 5 i i 2 2 3 3 4 4 5 i 2 2 3 3 4 4 5 i i 2 2 3 3 4 4 5 2 2 3 3 4 4 5 3 5 i i 3 4 i 3 4 i 3 4 i 3 4 o7 10.6366 6361 6355 6349 6 343 10.6338 6332 6326 6320 6315 9.9887 9887 9887 9886 9886 9.9886 9885 9885 9885 9885 9.9884 9884 9884 9883 9883 10.6425 6419 6414 6409 6403 9.3691 3697 3702 3708 37H 10.6309 6303 6298 6292 6286 10.01 16 0116 0116 0117 0117 10.6398 6392 6387 6382 6376 9-3719 3725 373i 3736 3742 10.6281 6275 6269 6264 6258 10.0117 0117 0118 0118 0118 9 11| 9882 9882 9882 9.3629 3634 3640 3645 3650 10.6371 6366 6360 6355 6 35 9-3748 3753 3759 3764 3770 10.6252 6247 6241 6236 6230 10.0119 0119 0119 OI2O 0120 9.9881 9881 9881 9880 9880 9-3655 3661 3666 3671 3677 10.6345 6339 6334 6329 6323 10.6318 6313 6308 6302 6297 9.3776 378i 3787 3792 3798 10.6224 6219 6213 6208 6202 IO.OI2O 0120 OI2I OI2I 0121 9.9880 9880 9879 9879 9879 9.3682 3687 3692 3698 3703 9.3709 3713 3719 3724 3729 9.3804 3809 3815 3820 3826 10.6196 6191 6185 6180 6i74 10.0122 OI22 OI22 0123 OI23 9.9878 9878 9878 9877 9877 10.6291 6287 6281 6276 6271 9-3831 3f37 3842 3848 3853 10.6169 6163 6158 6152 6147 10.0123 OI24 0124 0124 OI24 9.9877 9876 9876 9876 9876 9-3734 3739 3745 375 3755 10.6266 6261 6255 6250 6245 9-3859 3864 3f7o 3875 3881 10.6141 6136 6130 6125 6119 IO.OI25 0125 OI25 OI26 0126 9-9875 9875 9875 9874 9874 9.3760 3765 3770 3775 378o 10.6240 6235 6230 6225 6220 9.3886 3892 3897 3903 3908 10.6114 6108 6103 6097 6092 IO.OI26 OI27 0127 0127 0128 9.9874 9873 9873 9873 9872 9.3786 379i 3796 3801 3806 9.3811 3816 3822 3827 3832 10.6214 6209 6204 6199 ._ 6 194 10.6189 6184 6178 6i73 6168 9-39I4 3919 3924 3930 __3935 9-3941 3946 3952 3957 3962 9.3968 10.6086 6081 6076 6070 6065 10.6059 6054 6048 6043 6038 10.0128 0128 0128 0129 OI29 IO.OI29 OI3O 0130 OI3O OI3I 9.9872 9872 9872 9871 _9?_Z]L 9.9871 9870 9870 9870 9869 9.3837 1 10.6163 10.6032 Tan. IO.OI3I 9.9869 ' m B Cos. O'.l 1" Sec. Cot. O'.l Cosec. Sin. m s / Diff. Diff. 1O3 = 6 h 52 m ] [ 5 h 4 m = 76 rn TABLE IX. 41 ; 14 = O h 56 m ] Log:. Sines, Tangents, and Secants. [ ll h O ni -- 165 O 1 3 4 5 6 7 8 9 i 10 11 ; 12 13 11 15 16 ' B7 18 1 19 20 21 22 23 I 24 : : 25 26 27 : 28 29 30 31 ! 32 I 33 34 35 36 ! 37 38 39 40 41 i 42 ! 43 44 45 46 i 47 ! 48 49 50 51 I 52 ! 53 54 55 ! 56 57 58 59 60 m s Sin. 9-3837 3842 3847 3852 3?57 9.^862 3867 3872 JI 9.3887 3892 3897 3902 3907 9.3912 3917 3922 3927 3932 9-3937 3942 3947 395 2 3957 9.3961 3966 397i 3976 398i 9.3986 399i 3996 4001 4005 Di OM ff. i 3 4 i 3 4 3 4 i 3 4 i 2 4 i 2 4 Cosec. Tan. 9.3968 3973 3978 3984 __3?89 9-3995 4000 4005 4011 4016 9.4021 4027 4032 4037 4042 Di OM i i 2 2 3 3 4 4 5 i 2 2 3 3 4 4 5 i i 2 2 3 3 4 4 5 i i 2 2 3 3 4 4 5 i i 2 2 3 3 4 4 5 i i 2 2 3 3 4 4 5 ff. 1 s i 3 4 i 3 4 3 4 3 4 i 3 4 3 4 Cot. 10.6032 6027 6022 6016 6011 10.6005 6000 5995 5989 5984 10.5979 5973 5968 5963 5958 Sec. 10.0131 0131 0132 0132 0132 10.0133 OI 33 oi33 oi33 oi34 10.0134 oi34 OI 35 oi35 OI 35 Cos. m s / 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 4O 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 56 4 8 12 16 56 2t> 24 28 32 36 56 4O 44 48 52 56 37 4 8 12 16 5720 24 28 32 36 374O 44 48 52 56 58 O 4 8 12 16 58 20 24 23 32 36 584O 44 48 52 56 59 O 4 8 12 16 59 20 24 28 32 36 5940 44 48 52 56 60 i 2 2 3 q 4 4 S 2 2 3 3 4 4 5 i i 2 2 3 3 4 4 5 i i 2 2 3 3 4 4 5 i 2 2 3 3 4 4 i i 2 2 3 3 4 4 10.616-^ 6158 6i53 6148 _6j4_3_ 10.6138 6 i33 6128 6123 6118 1 0.6 ii 3 6108 6103 6098 6093 9.9869 9869 9868 9868 9868 15867 9867 9867 9867 9866 9.9866 9866 9865 9865 9865 4 356 52 48 344 40 36 32 28 324 20 16 12 8 3 4 3 256 52 48 244 40 36 32 28 224 2O 16 12 8 2 4 2 156 52 48 144 40 36 32 28 124 2O 16 12 8 1 4 1 O56 52 48 044 40 36 32 28 O24 2O 16 12 8 4 O 10.6088 6083 6078 6073 6068 9.4048 4053 4058 4064 4069 10.5952 5947 5942 5936 593 1 10.0136 0136 0136 0137 0137 9.9864 9864 9864 0863 9863 10.6063 6058 6053 6048 6043 9.4074 4079 4085 4090 4095 10.5926 592i 5915 59io 5905 10.0137 0138 0138 0138 0139 9.9863 9862 9862 9862 9861 10.6039 6034 6029 6024 6019 10.6014 6009 6004 5999 5995 10.5990 5985 5980 5975 __597<> 10.5965 596i 5956 595 i 5946 9.4100 4106 4111 4116 4121 "94127" 4i3 2 4137 4142 4H7 10,5900 5894 5889 5884 _ 5879 10.5873 5868 5863 5858 5853 10.0139 0139 0140 0140 0140 10.0141 0141 0141 0143 0142 9.9861 9861 9860 9860 9860 9.9859 9859 9859 9858 9858 9.9858 9857 9857 9857 9856 9.4010 4015 4020 4025 4030 94035" 4039 4044 4049 4054 94153 4158 4163 4168 _4J73 9.4178 4184 4189 4194 4199 10.5847 5842 5f37 5832 _5_82_7 10.5822 5816 5811 5806 5801 10.0142 0143 0143 0143 0144 10.0144 0144 0145 0145 0145 9.9856 9856 9855 9855 9855 9.4059 4063 4068 4073 _4_78 9.4083 4087 4092 4097 4102 10.5941 5937 5932 5927 5922 9.4204 4209 4214 4220 4225 10.5796 579i 5786 578o .._577i 10.5770 5765 5760 5755 5750 10.0146 0146 0146 0147 0147 9.9854 9854 9854 9853 9853 10.5917 5913 5908 5903 5898 9.4230 4235 4240 4245 4250 10.0147 0148 0148 0148 0149 9-9853 9852 9852 9852 9851 9.4106 4111 4116 4121 4125 10.5894 5889 5884 5879 5875 94255 4260 4265 4270 4275 io.5745 5740 5735 5730 5725 10.0.149 0149 0150 '0150 0150 9.9851 9851 9850 9850 9850 9.4130 10.5870 94281 10.5719 10.0151 9.9849 / m s Cos. OM 1 s Sec. Cot. OM 1 s Tan. Cosec. Sin. m s ' Diff. Diff. 1O4 = 6 h 56'" ] [ 5 h O m = 75 42 TABLE IX. 15 = I O ] Log. Sines, Tangents, and Secants. (. lO h 56 m = 164 / in a Sin. Dill. Cosec. Tan. Diff. Cot. Sec. Cos. 1 m s ' i O'.l I 8 O'.l I 8 O 1 2 3 \ 6 7 8 9 10 11 12 13 14 15 i 1C 17 18 19 20 21 ! 22 i 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 4 8 12 16 020 24 28 32 36 040 44 48 52 56 1 O 4 8 12 16 12O 24 28 32 36 140 44 48 52 56 2 4 8 12 16 220 24 28 32 36 240 44 48 52 56 3 4 8 12 16 320 24 28 32 36 34O 44 48 52 56 4 9.4130 4135 4139 4144 4149 2 2 3 3 4 4 i i i 2 2 3 3 4 4 2 2 3 3 4 4 o i i 2 2 3 3 4 4 o i i 2 2 3 3 4 4 o i i 2 2 3 3 4 4 i 2 4 2 4 i 2 4 i 2 3 2 3 2 3 10.5870 5865 5861 5856 5851 9.4281 4286 4291 4296 43i i i 2 2 3 3 4 4 5 i i 2 2 3 3 4 1 2 2 3 3 4 4 5 i i 2 2 3 3 4 4 5 i i i 2 2 3 3 4 4 i 2 2 3 3 4 4 3 4 i 3 4 i 3 4 i 3 4 2 4 i 2 4 10.5719 57H 5709 5704 _5 6 99 10.5694 5689 5684 5679 __5^74 10.5669 5664 5659 5654 5 6 49 10.0151 0151 0151 0152 0152 9.9849 9849 9849 9848 9848 60 59 56 52 48 5944 40 36 32 28 59 24 20 16 12 8 59 4 59 O 5856 52 48 5844 40 36 32 28 58 24 20 16 12 8 58 4 58 5756 52 48 5744 40 36 32 28 5724 20 16 12 8 57 4 57 O 56 56 52 48 5644 40 36 32 28 56 24 20 16 12 8 56 4 56 O 6O 59 58 ; 57 56 : 55 ! 54 53 52 51 50 49 ! 48 17 46 45 44 43 42 ! 41 40 39 38 ! 37 j 36 3-> 34 33 32 31 30 29 28 i 27 26 | 25 24 23 22 21 JO 19 IS 17 16 15 14 13 12 11 BO 9 8 7 6 5 4 a 2 1 O / 94I53 4158 4163 4168 4172 9-4177 4181 4186 4191 4195 10.5847 5842 5f37 5832 5828 10.5823 58i9 5814 5809 5805 9.4306 43" 43 l6 4321 4326 94331 4336 4341 4346 4351 9-435 6 436i 4366 437i 4376 10.0152 oi53 oi53 oi53 0154 10.0154 OI 54 OI 55 oi55 oi55 9.9848 9847 9847 9847 -Jfc* 6 9.9846 9846 9845 9845 __9_?45 9-9844 9844 9844 9843 9843 9.4200 4205 4209 4214 4219 10.5800 5795 579i 5786 _57?L 10.5777 5772 5768 5763 5758 10.5644 5639 5634 5629 5624 10.0156 0156 0156 oi57 0157 9.4223 4228 4232 4237 4242 9.4246 4251 4255 4260 4264 9.4381 4386 4390 4395 __4400 9.4405 4410 4415 4420 4425 10.5619 5 6l 4 5610 5605 5600 10.0157 0158 0158 0158 oi59 10.0159 0159 0160 0160 0161 9-9843 9842 9842 9842 __9l4i 9.9841 9841 9840 9840 9839 10.5754 5749 5745 5740 5736 10-5595 5590 5585 558o 5575 9.4269 4274 4278 4283 4287 io.573i 5726 5722 5717 _5_73 10.5708 5704 5699 5695 5690 10.5686 5681 5 6 77 5672 5668 9.4430 4435 4440 4445 4449 10.5570 5565 556o 5555 555i 10,0161 0161 0162 0162 0162 9.9839 9839 9838 9838 9838 9.4292 4296 43 I 4305 43 10 9-4454 4459 4464 4469 4474 10.5546 554i 553 6 553i 5526 10.0163 0163 0163 0164 0164 9-9837 9837 9837 9836 9836 9.43H 43 '9 4323 4328 4332 9-4479 4484 4488 4493 4498 10.5521 55i6 55i2 557 5502 10.0164 0165 0165 0165 0166 9.9836 9835 9835 9835 9834 9-4337 434i 4346 * 4350 _^35_5 9-4359 4364 4368 4372 4377 9.4381 4386 4390 4395 4399 10.5633 5 6 59 5654 5650 5 6 45 10.5641 5636 5 6 3 2 5628 .^5 6 13 10.5619 5 6l 4 5610 5605 5601 9-4503 4508 4513 4517 4522 10.5497 5492 5487 5483 5478 10.0166 0167 0167 0167 0168 9.9834 9833 9*33 9833 9832 9-4527 4532 4537 4541 4546 9-45 5 i 455 6 456i 4565 4570 io.5473 5468 5463 5459 _5454 10.5449 5444 5439 5435 5450 10.5425 10.0168 0168 0169 0169 0169 10.0170 0170 0170 0171 0171 9.9832 9832 9831 9831 9831 9.9830 9830 9830 9829 9829 9.9828 9.4403 10-5597 9-4575 10.0172 / m B Cos. O'.l 1" Sec. Cot. O'.l | I 8 Tan. Cosec. Sin. m s Diff. Diff. 1O5 = 7 h O m ] [ 4 h 56" 1 = 74 TABLE IX. 16 = l h 4 M J Log. Sines, Tangents, and Secants. [ 1O' 1 52 m = 163 / 111 S Sin. Diff. Cosec. Tan. Diff. Cot. Sec. Cos. m s 60 59 58 57 56 55 54 53: 52 51 50 49 48 47 46 45 44 43 42 41 40! 39 38! 37 36 35 34! 33 32 i 31 30 29 28 ! 27! 26 25 24 23 22 21 20 19 18: 17: 16 15 14 13 12 11 10 9 8 6 5: 3! 2 1 O'.l | 1 s 0.1 JL- i 2 4 2 4 i 2 4 i 2 4 i 2 3 2 3 17 O 1 2 3 4 5 6 7 8 9 1O 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 4 O 4 8 12 16 420 24 28 32 36 440 44 48 52 56 5 4 8 12 *e 5 2O 24 28 32 36 540 44 48 52 56 6 4 8 12 16 620 24 29 32 36 640 44 48 52 56 7 O 4 8 12 16 720 24 28 32 36 74O 44 48 52 56 8 9.4403 4408 4412 4417 4421 I I 2 1 3 2 2 3 3 4 i 4 o i I 2 i 3 2 2 3 3 4 4 I I 2 i 3 2 2 3 3 i 3 4 ! O I I 2 i ! 3 2 2 3 3 3 4 1 i O I I 2 1 3 2 \ 2 \ 3 ; 3 ! 3 4 oj , ! j 2 I I 4 ! . 10-5597 5592 5588 5583 _5579 iQ-5575 5570 556t> 5562 5557 9-4575 458o 4584 4589 4594 i 2 2 3 3 4 4 i i i 2 2 3 3 4 4 i i i 2 2 3 3 4 4 i i i 2 2 3 3 4 4 o i i 2 2 3 3 4 4 o i i 2 2 3 3 4 4 10.5425 5420 54i6 54" 5406 10.0172 0172 0172 oi73 0173 9.9828 9828 9828 9827 9827 56 55 56 52 48 55 44 40 36 32 28 55 24 2O 16 12 8 55 4 55 5456 52 48 5444 40 36 32 28 5424 20 16 12 8 54 4 54 5356 52 48 5344 40 36 32 28 53 24 2O 16 12 8 53 4 53 52 56 52 48 5244 40 36 32 28 5224 20 16 12 8 52 4 52 9.4425 443 4434 443 s 4443 9-4599 4603 4608 4613 4618 10.5401 5397 5392 5387 5382 10.5378 5373 5368 5363 5359 10.0173 0174 0174 0174 oi75 9.9827 9826 9826 9826 9825 9-4447 445 2 445^ 4400 4465 10-5553 554* 5544 5540 5535 9.4622 4627 4632 4637 4641 10.0175 0176 0176 0176 0177 9.9825 9824 9824 9824 9823 9.4469 4473 4478 4482 4486 io.553i 5527 5522 55i8 55H 9.4646 4651 4655 4660 4665 10-5354 5349 5345 5340 5335 10.0177 0177 0178 0178 0179 9.9823 9823 9822 9822 9821 9.4491 4495 4499 453 4508 10.5509 5505 55i 5497 5492 9.4669 4674 4679 4687 4688 io.533i 5326 532i 5317 5312 10.5307 5303 5298 5293 5289 10.0179 0179 0180 0180 0180 9.9821 9821 9820 9820 9820 9.4512 45i 6 4521 4525 4529 10.5488 5484 5479 5475 547i 9.4693 4697 4702 4707 4711 10.0181 0181 0182 0182 0182 9.9819 9819 9818 9818 9818 9-4533 453^ 4542 454 6 .455 10.5467 5462 5458 5454 5450 9.4716 4721 4725 473 4735 10.5284 52/9 5275 5270 5265 10.0183 0183 0183 0184 0184 9.9817 9817 9817 9816 9816 9-4555 4559 45^3 45 6 7 4572 10.5445 544i 5437 5433 5428 94739 4744 4748 4753 4758 10.5261 5256 5252 5247 5242 10.0185 0185 0185 0180 0186 9-9815 9815 9815 9814 9814 9-457 4580 4584 4588 4593 10.5424 5420 54i6 5412 5407 9.4762 4767 477i 4776 4781 10.5238 5233 5229 5224 5219 10.0180 0187 0187 0188 0188 9.9814 9813 9813 9812 9812 94597 4601 4605 4609 4614 10.5403 5399 5395 539 1 5386 9-4785 4790 4794 4799 4803 10.5215 5210 5206 5?oi 5i97 10.5192 5187 5183 5i78 5174 10.0188 0189 0189 0189 0190 9.9812 9811 9811 9811 9810 9.4618 4622 4626 4630 4634 10.5382 5378 5374 5370 _5366 10.5361 5357 5353 5349 5345 9.4808 4813 4817 4822 4826 10.0190 0191 0191 0191 0192 9.9810 9809 9809 9809 9808 9.4639 4643 4647 4651 4655 9-4831 4835 4840 4844 4849 9-4853 10.5169 5165 5160 5156 5i5i 10.0192 0192 OI 93 0193 0194 9.9808 9808 9807 9807 9806 9.4659 Cos. 10.5341 10.5147 10.0194 9-9806 Toe m s O'.l 1 s Sec. Cot. O'.l Tan. Cosec. Sin. m s / Diff. Diff. , = 7 h 4 m ] [ 4 h 52 m = 73" 44 TABLE IX. 17= l h 8 11 ] L.og. Sines, Tangents, and Secants. [1O 11 48 m = 162 1 2 3 4 5 I 8 9 10 11 12 13 14 15 16 17 18 19 i 20 i 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 ' 57 58 59 60 | m s Sin. Diff. Cosec. Tan. Diff. Cot. Sec. 10.0194 0194 oi95 oi95 0196 10.0196 0196 0197 0197 0198 10.0198 0198 0199 0199 QI99 IO.O2OO 0200 O2OI O2OI 0201 IO.O2O2 O2O2 0203 O2O3 O2O3 Cos. m s 52 O 51 56 52 48 5144 40 36 32 28 5124 2O 16 12 8 51 4 51 5O56 52 48 5044 40 36 32 28 5024 2O 16 12 8 5O 4 50 4956 52 48 4944 40 36 32 28 4924 2O 16 12 8 49 4 49 4856 52 48 4844 40 36 32 28 4824 20 16 12 8 48 4 48 i O'.l 1 O'.l ! 1* 8 4 8 12 16 820 24 28 32 36 840 41 48 52 56 9 4 8 12 16 9 2O 24 28 32 36 94O 44 48 52 56 1O O 4 8 12 16 1020 24 28 32 36 1040 44 48 52 56 11 4 8 12 16 1120 21 28 32 36 1140 44 48 52 56 12 9.4659 4603 4668 4672 4676 o i 2 3 3 4 o i i 2 2 2 3 3 4 o i i 2 2 2 4 o 2 3 3 4 o i 2 2 2 3 3 4 o i 2 2 2 3 3 4. i 2 3 2 3 i 2 3 2 3 I 2 3 i 2 3 110.5341 5337 5332 5328 i ...S3?4 10.5320 53 l6 53i2 5308 534 10:5300 5295 5291 5287 5283 10.5279 5275 5271 5267 __5 26 3 10.5259 5255 5251 5247 _5243_ 10.5239 5235 5231 5227 5223 10.5219 5215 5211 5207 5203 94853 4858 4862 4867 4871 o i I 2 2 3 2 3 3 ; 4 4 o , i I 2 \ 3 2 3 3 4 4 ! I I i 2 1 ; 3 2 ! 2 3 3 4 4 ; O I I 2 1 3 2 2 3 3 4 4 O I I 2 1 3 2 2 3 3 4 4 I I 2 1 3 2 2 3 3 4 4 .10.5147 5H2 5138 5133 5L 2 9 10.5124 5120 5H5 5"i 5106 10.5102 5097 5093 5088 5084 10.5079 5075 5070 5066 5061 IO -5057 553 5048 544 539 9.9806 9806 9805 9805 9804 60 59 58 57 56 .>> 54 53 52! 51 50 49 48 i 47 \ 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 i 31 30 29 28: 27 i 26 25 24 23 22 21 20 19 18 17 16 15! 14 13 12 11 10 9 8 7 6 5 4 3 2 1 O 9.4680 4684 4688 4692 .._46?6_ 9.4700 4705 4709 4713 _47i7_ 9.4721 4725 4729 4733 4737 9.4870 4880 4885 4889 4894 9.4898 4903 4907 4912 _49i 9.4921 4925 493 4934 _4939_ 9-4943 4947 4952 495 6 4961 9.9804 9804 9803 9803 9802 9.9802 9802 9801 9801 9801 9.9800 9800 9799 9799 9799 9.4741 4745 4749 4753 4757 9.9798 9798 9797 9797 9797 9.4761 4765 4769 4773 4777 9.4965 4970 4974 4978 4983 i-535 5030 5026 5022 5017 10.5013 5008 5004 5000 4995 IO.O2O4 0204 O2O5 O2O5 0205 IO.O2O6 O2O6 O2O7 0207 O2O7 9.9796 9796 9795 9795 9795 9.4781 4785 4789 4793 4797 9.4801 4805 4809 4813 4817 9.4987 4992 4996 5000 5005 9-9794 9794 9793 9793 9793 10.5199 5195 5i9i 5187 5183 9.5009 5H 5018 5022 5027 10.4991 4986 4982 4978 4973 IO.O2O8 O2O8 0209 O2O9 O2O9 9.9792 9792 9791 9791 9791 9.4821 4825 4829 4833 4837 10.5179 . 5175 5i7i 5 l6 7 5163 9-503I 5035 5040 544 549 10.4969 4965 4960 495 6 4951 IO.O2IO 0210 O2 1 1 O2II 021 1 9.9790 9790 9789 9789 9789 9.4841 4845 4849 4853 4857 9.4861 4865 4869 4873 4876 9.4880 4884 4888 4892 4896 10.5159 5 J 55 5i5i 5H7 5'43 9-5053 5057 5062 5066 5070 10.4947 4943 4938 4934 493 IO.O2I2 O2I2 0213 0213 O2I3 9.9788 9788 9787 9787 9787 10.5139 5*35 53 5127 5124 9-5075 5079 5083 5088 5092 10.4925 4921 4917 4912 4908 10.4904 4899 4895 10.0214 0214 O2I5 O2I5 0215 I0.02I6 O2l6 0217 0217 0218 9.9786 9786 9785 97f5 9785 10.5120 5116 5"2 5108 5104 9.5096 5101 5105 5109 5"3 9.9784 9784 9783 97f3 9782 9.4900 10.5100 9.5118 10.4882 10.0218 9.9782 r , m & Cos. O'.l 1 s Sec. Cot. O'.l I 8 Tan. Cosec. Sin. m a ' Diff. Diff. ;iO7 3 = 7 h 8 m ] [4 h 48" l = 72 TABLE IX. 45 18= - 1 |J 12'" ] ,og. Sines, Tangents, and Secants. [ 1O 1 44 m = 161 O I 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 m s 12 4 8 12 16 1220 24 28 32 36 124O 44 48 52 56 13 4 8 12 16 13 2O 24 28 32 36 1340 44 48 52 56 14 O 4 8 12 16 1420 21 28 32 36 1440 44 48 52 56 15 O 4 8 12 16 1520 24 28 32 36 1540 44 48 52 56 16 Sin. 94900 ! 49041 4908 4911 49 IS Dif OM F. 1" Cosec. Tan. 9.5118 5122 5120 5Ui 5135 Dif 0.1 F. 1" Cot. Sec. 0.0218 0218 0219 0219 O22O Cos. Ill S , 1 o 2 2 2 3 3 4 o i 2 2 2 3 3 4 2 2 2 3 3 4 o i i 2 2 2 3 3 4 o i 2 2 2 3 3 4 o i i 2 2 Tl O 3 3 i 2 3 2 3 i 2 3 2 3 I 2 3 2 3 10.5100 5096 5092 5089 5085 o i ! i 2 2 3 3 4 4 i i 2 2 3 3 4 4 o i 2 2 3 3 4 4 i i 2 2 3 3 4 4 o i 5 2 2 3 3 4 I i i 2 2 2 3 3 4 i 1 2 3 2 3 I 2 3 2 3 2 3 2 3 T^ 10.4882 4878 4874 4869 4865 9-9782 9782 9781 9781 978o 48 4756 52 48 4744 40 36 32 28 4724 20 16 12 8 47 4 47 4656 52 48 4644 40 36 32 28 46 24 20 16 12 8 46 4 46 45 56 52 48 4544 40 36 32 28 4524 2O 16 12 8 45 4 45 4456 52 48 4444 40 36 32 28 4424 2O 16 12 8 44 4 44 60 59 58 i 57 56 55 54 53 52 51 50 ! 49 | 48 ! 47 46 45 1 44 43 42 41 40 39 38 37 I 36 35 34 33 i 32 31 30 i 29 28 ! 27 26 25 24 23 22 21 20 19 I 18 17 ! 16 15 i 14 ; 13 12 11 10 9 8 7 ! 6 5 4 3 \ 2 1 O 9.4919 49 2 3 4927 493 i 4935 10.5081 577 5073 5069 5065 9-5I39 5H3 5H8 5152 5156 10.4861 4857 4852 4848 4844 0.0220 O22O O22I 0221 O222 9.9780 978o 9779 9779 9778 9-4939 4942 4946 495 4954 10.5061 5058 5054 5050 5046 9.5161 5^5 5 l6 9 5i73 5178 10.4839 4835 4831 4827 4822 IO.O222 O222 0223 O223 0224 9-9778 9778 9777 9777 9776 9.4958 4962 4965 4969 4973 10.5042 5038 535 5031 5027 9.5182 5186 5*90 5J95 5199 10.4818 4814 4810 4805 4801 IO.O224 0225 O225 0225 O226 9.9776 9775 9775 9775 9774 9-4977 4981 4984 4988 4992 10.5023 5019 5016 5012 5008 9-5203 5207 5212 5216 5220 10.4797 4793 4788 4784 478o IO.O226 0227 0227 O227 0228 9-9774 9773 9773 9773 9772 9.4996 5000 5003 5007 5011 10.5004 5000 4997 4993 4989 9.5224 5228 5233 5237 5241 9-5245 5249 5254 5258 5262 10.4776 4772 4767 4763 4759 10.0228 O229 0229 0230 0230 9.9772 9771 977i 9770 9770 9-50I5 5 OI 9 5022 5026 53 10.4985 498i 4978 4974 4970 10-4755 475 i 4746 4742 4738 10.0230 023! 0231 0232 0232 9.9770 9769 9769 9768 9768 9.5 34 537 5041 5045 ' 549 10.4966 49 6 3 4959 4955 495i 9.5266 5270 5275 5279 5283 10.4734 473 4725 4721 4717 0.0233 0233 0233 0234 0234 9.9767 9767 9767 9766 9766 9.5052 5056 5060 5064 5067 10.4948 4944 4940 493 6 4933 9.5287 529 1 5295 53o 534 10.4713 4709 4705 4700 4696 10.0235 0235 0236 0236 0236 9-9765 9765 9764 9764 9764 9-507I 575 507^ 5082 5086 10.4929 4925 4922 4918 49 H 9-5308 5312 53i6 5320 5324 10.4692 4688 4684 4680 4676 IO.O237 0237 0238 0238 0239 9-9763 9763 9762 9762 9761 9.5090 5093 597 5101 5104 10.4910 4907 4903 4899 4896 9-5329 5333 5337 534i 5345 10.4671 4667 4663 4659 4655 10.0239 0239 O24O 0240 024! 9.9761 9761 9760 9760 9759 9.5108 5112 5"5 5"9 5123 10.4892 4888 4885 4881 4877 9-5349 5353 5357 53g 5366 10.4651 4647 4643 4638 4634 IO.O24I O242 0242 O242 0243 9-9759 9758 9758 9758 9757 9.5126 10.4874 9-5370 10.4630 IO.O243 9-9757 / m s Cos. OM 1 s Sec. Cot. O'.l Tan. Cosec. Sin. m s / Diff. Diff. 1O8 = 7 h 12 m ] [ 4 h 44 m = 71 | 46 TABLE IX. 19 = I 16 ] Log. Sines, Tangents, and Secants. [ lO h 4O m - 16O i / m & Sin. Diff. Cosec. Tan. Diff. Cot. Sec. Cos. m s ' ; O'.l I 8 O'.l | 1 O 1 2 3 4 5 6 7 8 9 10 11 12 13 14 I 15 16 17 18 1 19 20 21 22 23 i 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 i 57 58 59 60 16 A 8 12 16 16 2O 24 28 32 36 1640 44 48 52 56 17 4 8 12 16 1720 24 28 32 36 1740 44 48 52 56 18 4 8 12 16 1820 24 28 32 36 1840 44 48 52 56 19 4 8 12 16 1920 24 28 32 36 1940 44 48 52 56 20 9.5126 5130 5134 5137 SHi 9-5 H5 5H8 5152 5i5 6 5159 i i i 2 2 3 3 3 i i i 2 2 3 3 3 o I I I 2 2 3 3 3 i i 2 2 3 3 3 i 2 2 3 3 3 o i i 2 2 3 3 3 2 3 3 2 3 I 2 3 i 2 3 I 2 3 10.4874 4870 4866 4863 4859 9-537 5374 5378 5382 5386 o i i 2 2 2 3 3 4 i i 2 2 2 3 3 4 i i 2 2 2 3 3 4 o i i 2 2 2 3 3 4 o i i 2 2 2 3 3 4 o I I 2 2 2 3 3 4 i 2 3 2 3 i 2 3 I 2 3 i 2 3 2 3 10.4630 4626 4622 4618 4614 10.0243 0244 0244 0245 0245 9-9757 9756 975 6 9755 9755 44 4356 52 48 4344 40 36 32 28 4324 20 16 12 8 43 4 43 42 56 52 48 4244 40 36 32 28 12 24 2O 16 12 8 42 4 42 O 4156 52 48 4144 40 36 32 28 4124 20 16 12 8 41 4 41 4056 52 48 4044 4O 36 32 28 4O24 20 16 12 8 40 4 4O O 60 59 58 57 56 55 54 53 52 51 5O 49 48 : 47 ! 46 45 i 44 i 43 42 i 41 40 39 38 37 36 i 35 34 33 32 31 30 ! 29 28 27 26 i 25 ! 24 i 23 22 21 20 19 18 17 16 15 14 13 12 1 11 1 1O i 8 i 7 I 4 3 2 1 O 10.4855 4852 4848 4844 4841 9-5390 5394 5398 5402 _sw. 9-54" 5415 5419 5423 5427 9-543 r 5435 5439 5443 5447 10.4610 4606 4602 4598 4593 10.0245 0246 0246 0247 0247 9-9755 9754 9754 9753 9753 9-975 2 9752 975 J 975 1 975i 9-5 163 5167 5!7o 5174 5*77 9.5181 5185 5188 5*92 519^ 10.4837 4833 4830 4826 4823 10.4819 4815 4812 4808 4804 10.4589 4585 458i 4577 4573 10.0248 0248 0249 0249 0249 10.4569 4565 45 6 i 4557 4553 10.0250 0250 0251 0251 0252 9-975 975 9749 9749 9748 9-5 199 5203 5206 5210 5213 10.4801 4797 4794 4790 4787 9-545 i 5455 5459 5463 5467 10.4549 4545 454i 4537 4533 10.0252 0253 0253 0253 0254 9.9748 9747 9747 9747 9746 9-52I7 5221 5224 5228 5231 10.4783 4779 4776 4772 4769 9-5471 5475 5479 5483 5487 10.4529 4525 452i 45*7 4513 10.0254 0255 0255 0256 0256 9.9746 9745 9745 9744 9744 9-5235 5239 5242 5246 5249 10.4765 4761 4758 4754 47J5I 9-5491 5495 55oo 5504 55o8 10.4509 4505 4500 4496 4492 10.0257 0257 0257 0258 0258 9-9743 9743 9743 9742 9742 9.5253 5256 5260 5263 5267 10.4747 4744 4740 4737 4733 9-5512 55i6 5520 5524 5528 10.4488 4484 4480 4476 4472 10.0259 0259 0260 0260 0261 9.9741 974i 974 9740 9739 ' 9-5270 5274 5278 5281 5285 10.4730 4726 4722 4719 4715 9-5531 5535 5539 5543 5547 10.4469 44 6 5 4461 4457 4453 10.0261 0261 0262 0262 0263 9-9739 9739 9738 9738 9737 9-5288 5292 5295 5299 5302 10.4712 4708 4705 4701 4698 9-5551 5555 5559 5563 5567 10.4449 4445 4441 4437 4433 10.0263 0264 0264 0265 0265 9-9737 9736 973 6 9735 9735 9-5306 539 5313 53i6 5320 10.4694 4687 4684 4680 9-5571 5575 5579 5583 5587 10.4429 4425 4421 4417 4413 10.0266 0266 0266 0267 0267 9-9734 9734 9734 9733 9733 9-5323 5327 5330 5334 5337 10.4677 4673 4670 4666 4663 10.4659 9-5591 5595 5599 5603 . 5607 9.5611 10.4409 4405 4401 4397 4393 10.0268 0268 0269 0269 0270 9.9732 9732 973 1 9731 973 9-5341 10.4389 10.0270 9-9730 , / m a Cos. O'.l I 8 Sec. Cot. O'.l | I 8 Tan. Cosec. Sin. m s / Diff. Diff. 109 = 7 h 16 m ] [ 4 h 4O ra = 7O 7 j TABLE IX. 47 2O J =a l b 2O' n ] JLog. Sines, Tangents, and Secants. [ lO h 36 m = 159 O 1 2 3 4 5 6 7 8 9 i to 11 12 13 ; 14 15 i 16 17 1 18 19 30 21 22 23 i 24 ! 25 126 27 i 28 ! 29 30 31 32 i 33 i 34 i 35 ! 36 j 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 in s 2O O 4 8 12 16 2O 2O 24 28 32 36 2O 4O 44 48 52 56 21 4 8 12 16 21 2O 24 28 32 36 2140 44 48 52 56 22 O 4 8 12 16 22 2O 24 28 32 36 2240 44 48 52 56 23 4 8 12 16 23 2O 24 28 32 36 2340 44 48 52 56 24 Sin. 9-5341 5344 5347 535 1 5354 9-5358 536i 53 6 5 5368 5372 Diff. Cosec. Tan. Diff. Cot. Sec. ' Cos. m s 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 i 32 31 1 30 29 28 27 26 251 24 23 22 21 20 19 18 17 16 | 15 I 14 13! 12 11 10 9! s| * 6 5! * 3 2 1 0.1 i 1* O'.l o i i 2 2 2 3 3 4 o 2 2 2 3 3 4 o i i 2 2 2 3 3 4 o I 2 2 2 3 3 4 o I 2 2 1 4 o i 2 2 2 3 3 4 1 s o i i i 2 3 3 3 o 2 2 3 3 3 o i i i 2 2 3 3 3 i i i 2 2 3 3 3 o I I 2 2 2 3 3 o I I I 2 2 3 3 i 2 3 i 2 3 2 3 2 3 I 2 3 i 2 3 10.4659 4656 4653 4649 4646 9.5611 5615 5 6l 9 5622 5626 2 3 I 2 3 I 2 3 i 2 3 i 2 3 I 2 3 10.4389 4385 4381 4378 4374 10.0270 i 9.9730 0271 i 9729 0271 ! 9729 0272 i 9728 0272 9728 4O O 3956 52 48 3944 40 36 32 28 3924 2O 16 12 8 39 4 39 O 3856 52 48 3S44 40 36 32 28 3824 20 16 12 8 38 4 38 3756 52 48 3744 40 36 32 28 3724 2O 16 12 8 37 4 37 3656 52 48 3644 40 36 32 28 3624 20 16 12 8 36 4 36 10.4642 4639 4635 4632 4628 9-5 6 3 $$ 5642 5646 10.4370 4366 4362 4358 4354 10.0272 0273 0273 0274 0274 9.9728 9727 9727 9726 9726 9-5375 5379 5382 53g5 53^9 10.4625 4621 4618 4615 4611 9-5 6 5o 5654 5658 5662 5665 9.5669 5673 56/7 5681 5685 9.5689 5693 5696 5700 5704 10.4350 4346 4342 433 s 4335 10.0275 0275 0276 0276 0277 9.9725 9725 9724 9724 9723 9-9723 9722 9722 9722 9721 9.9721 9720 9720 97i9 9719 9-5392 5396 5399 5402 54o6_ 9.5409 5413 54i6 5420 5423 10.4608 4604 4601 4598 4594 10.4591 4587 4584 458o 4577 10.4574 4570 45 6 7 45 6 4 4560 1 0-433 ! 4327 4323 43 9 4315 10.4311 437 434 4300 4296 10.4292 4288 4284 4280 4276 10.0277 0278 0278 0278 __0279_ 10.0279 0280 0280 0281 0281 9.5426 543 5433 5436 5440 9.5708 5712 57i6 5720 5724 10.0282 0282 0283 0283 0284 9.9718 9718 9717 9717 9716 9-5443 5447 5450 5453 5457 10-4557 4553 4550 4547 4543 9-5727 573i 5735 5739 5743 10.4273 4269 4265 4261 4257 10.0284 0285 0285 0286 0286 9.9716 97i5 97i5 97H 97H 9.5460 5463 5467 5470 5474 10.4540 4537 4533 4530 4526 9-5747 5750 5754 5758 5762 10.4253 4250 4246 4242 4238 10.0286 0287 0287 0288 0288 9.9714 9713 9713 9712 9712 9-5477 548o 5484 5487 5490 10.4523 4520 45 i 6 . 4513 45 10 9.5766 5770 5773 578i 10.4234 4230 4227 4223 4219 10.0289 0289 0290 0290 0291 9.9711 9711 9710 9710 9709 9-5494 5497 55oo 5504 557 10.4506 4503 4500 4496 4493 9-5785 5789 5792 5796 5800 10.4215 4211 4208 4204 4200 10.0291 0292 0292 0293 0293 9-9709 9708 9708 9707 9707 9-5510 55H 5517 5520 5523 10.4490 4486 4483 4480 4477 9.5804 5808 5811 5815 5819 10.4196 4192 4189 4185 4181 10.0294 0294 0295 0295 0296 9.9706 9706 9705 9705 9704 9.5527 553 5533 5537 5540 10.4473 4470 4467 4463 4460 9-5823 5827 5830 5834 5838 10.4177 4173 4170 4166 4162 10.0296 0297 0297 0298 0298 9.9704 9703 9703 9702 9702 9-5543 10.4457 9.5842 10.4158 10.0298 9.9702 / m B lo'.i Cos. Di 1 s ff. Sec. Qot. 0.1 1 s Tan. Cosec. Sin. in s / Diff. 11O = 7 h 2O m ] [ 4 h 36 m = 69 48 TABLE IX. 21= I 24 m ] Log. Sines, Tangents, and Secants. [lO h 32"- = 158 in a Sin. Di 0.1 Bf. Cosec. Tan. Diff. Cot. Se::. 10.0298 0299 0299 0300 10.0301 0301 0302 0302 0303 Cos. Ill S 36 35 36 52 48 3544 40 36 32 28 35 21 2O 16 12 8 35 4 35 O 34 56 52 48 3444 10 36 32 28 3424 20 16 12 8 34 4 34 3356 52 48 3344 40 36 32 28 3324 20 16 12 8 33 4 33 32 56 52 48 3244 40 36 32 28 32 24 20 16 12 8 32 I 32 6O 39 58 57 36 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 4.0 39 38 37 36 35 34 33 32 31 i 3O 29' 28 27 26 25 24 23 22 21! 20 19 18 17 16 15 14 13 12 11 | 10 i 9 8 7 ; 6 5 3 1 O I 8 O'.l 1 s 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 |55 56 57 58 59 60 24 4 8 12 16 24 20 24 28 32 36 2440 44 48 52 56 25 4 8 12 16 2520 24 28 32 36 2540 44 48 52 56 26 4 8 12 16 26 2O 24 28 32 36 2640 44 48 52 56 27 4 8 12 16 27 2O 24 28 32 36 27 4O 44 48 52 56 28 9-5543 5547 555 5553 5556 9-556o 5563 5566 5570 5573 i i i 2 2 2 3 3 o i i i 2 2 2 3 3 i i i 2 2 2 3 3 o i i i 2 2 2 3 o i 2 2 2 3 3 o i i i 2 2 2 3 3 i 2 3 i 2 3 2 2 I 2 2 I 2 2 2 2 io.4457 4453 445 4447 4444 9.5842 5846 5849 5857 o i i 2 2 3 3 3 4 o 2 2 3 3 3 4 o i 2 2 3 3 3 4 o i i 2 2 3 3 3 4 o 2 2 3 3 3 4 i i 2 2 3 3 3 4 i 3 i 2 3 i 2 3 I 2 3 i 2 3 i 2 3 10.4158 4154 4I5 1 4M7 4I4J 10.4139 4136 4132 4128 4124 9.9702 9701 9701 9700 9700 9.9699 9699 9698 9698 9697 9.9697 9696 9696 9695 ._9. 6 9_5. 9.9694 9694 9693 9693 9692 10.4440 4437 4434 443 4427 95861 5864 5868 5872 5876 9-5576 5579 5583 5586 9-5592~ 5596 5599 5602 5605 10.4424 4421 4417 4414 44" 10.4408 4404 4401 4398 4395 9-5879 5883 5887 5891 5894 10.4121 4117 4"3 4109 4106 10.0303 0304 0304 0305 35 9-5898 5902 5906 5909 5913 10.4102 4098 4094 4091 4087 10.0306 0306 0307 0307 0308 10.0308 0309 0309 0310 0310 10.0311 0311 0312 0312 0313 9.5609 5612 5615 5618 5621 '5628 5631 5634 5638 10.4391 4388 4382 4379 10.4375 4372 4369 4366 4362 95917 5921 5924 5928 5932 9-5935 5939 5943 5947 5950 10.4083 4079 4076 4072 4068 10.4065 4061 4057 4053 4050 9.9692 9691 9691 9690 9.9689 9689 9688 9688 9687 9.5641 5644 5647 5650 5654 10-4359 4356 4353 4350 4346 9-5954 5958 596i 5965 5969 10.4046 4042 . 4039 4035 4031 10.0313 3 H 03H 0315 0315 9.9687 9686 9686 9685 9685 9-5657 5660 5663 5666 5670 10-4343 4340 4337 4334 433 9-5972 5976 598o 5984 5987 10.4028 4024 4020 4016 4013 10.0316 0316 0317 0317 0318 9.9684 9684 9683 9683 9682 9-5673 5676 5679 5682 5685 10.4327 4324 4321 4315 9-5991 5995 5998 6002 6006 9.6009 6013 6017 6020 6024 6031 6035 6039 6042 10.4009 4005 4002 3998 3994 10.0318 0319 0319 0320 0320 9.9682 9681 9681 9680 9680 9.5689 5692 5695 5698 10.4311 4308 435 4302 4299 10.3991 3987 3983 398o 3976 10.0321 0321 0322 0322 0323 10.0323 0324 0324 0325 0325 9.9679 9679 9678 9678 9677 9-5704 57" 57H 5717 10.4296 4292 4289 4286 4283 10.3972 3969 3965 3958 9-9677 9676 9676 9675 9675 9-5720 5723 5726 5729 5733 10.4280 4277 4274 4271 4267 9.6046 6050 6053 6057 6060 10-3954 3950 3947 3943 3940 10.0326 0326 0327 0327 0328 9-9674 9674 9673 9673 9672 9-5736 10.4264 9.6064 10.3936 10.0328 9.9672 t 111 B Cos. O'.l 1 s Sec. Cot. O'.l 1 s Tan. Cosec. Sin. m s Diff. Diff. 111 = 7 24-] [4*32':=68 | TABLE IX. 22= _ | 2:i> i Log. Sines, Tangents, and Secants. [10" Cos. 9.9672 9671 9671 9670 9670 9.9669 9669 9668 9668 9667 28 n ' = 157 i in s 28 O 4 8 12 16 2N 2O 24 28 32 36 2840 44 48 52 56 29 4 8 12 16 2920 24 28 32 36 2940 44 48 52 56 3O O 4 8 12 16 3020 24 28 32 36 3O4O 44 48 52 56 31 4 8 12 16 3120 24 28 32 36 3140 44 48 52 56 32 Sin. Di 0.1 o i J 2 2 2 3 3 i i 2 2 2 3 3 o 2 2 2 3 3 i i i 2 2 2 3 3 o i i i 2 2 2 3 3 o 2 2 2 3 3 if. 1 s Cosec 10.4264 4261 4258 4255 _4 2 5 2 10.4249 4246 4242 4239 4236 Tan. 9.6064 6068 6071 6075 _._ 6 79 9.6082 6086 6090 6093 6097 Di O'.l o i i 2 2 3 3 3 4 2 2 3 3 3 4 i i 2 2 3 3 3 4 o i 2 2 3 3 3 4 i i 2 2 ^ J 3 3 4 o 2 2 3 3 3 4 ff. 1 s i 2 3 i 2 3 2 3 j 2 3 2 3 i 2 3 Cot. Sec. m s ' O 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 / 9-5736 5739 5742 5745 _ 5748 9-575 1 5754 5758 576i 5764 i 2 2 I 2 2 I 2 2 I 2 2 2 .2 I 2 2 10.3936 3932 3929 3925 _ 3921 10.3918 39H 3910 3907 3903 10.0328 0329 0329 033 __?3359 9.9659 9658 9658 9657 ^965 7 9.9656 9656 9655 9655 _9_654 9.9654 9653 9652 9652 9651 9-58i3 5816 5819 5822 5825 9.5828 5831 5g34 5838 _5?4i 9.5844 5847 5850 5853 5856 9-5859 5862 5865 5868 5871 10.4141 4138 4135 4I3 2 4129 9.6208 6211 6215 6219 6222 10.3792 3789 3785 378i 3778 10.0349 35 035 0351 035i 9.9651 9650 9650 9649 9649 9-5874 5877 5880 5883 5886 10.4126 4123 4120 4117 4114 9.6226 6229 6233 6236 6240 io-3774 3771 3767 3764 3760 10.0352 0352 0353 0353 354 9.9648 9648 9647 9647 9646 9.5889 5892 5895 5898 5901 10.4111 4108 4105 4102 4099 9.6243 6247 6250 6254 6257 iQ-3757 3753 3750 3746 3743 10.0354 355 0355 0356 0357 9.9646 9645 9 6 45 9644 9643 9.5904 5907 59io 5913 59i6 10.4096 4093 4090 4087 4084 9.6261 6264 6268 6271 6275 10.3739 3736 3732 3729 3725 10.0357 0358 0358 0359 359 9.9643 9642 9642 9641 9641 9-59I9 10.4081 9.6279 10.3721 0.0360 9.9640 m s Cos. O'.l 1 s Sec. Cot. O'.l 1 s Tan. Cosec. Sin. m s 1 Diflf. Diff. 112 = 7 h 28 ra ] [ 411 28 m 67 50 TABLE IX. 23 = l h 32"'] Log. Sim-*, Tangents, and Secants. [ lo *! - 156 / m s Sin. Di OM S. 1 Cosec. Tan. Diff. Cot. Sec. Cos. m s 28 2756 52 48 2744 40 36 32 28 27 24 20 16 12 8 27 4 27 26 56 52 48 2644 40 36 32 28 2624 2O 16 12 8 26 4 26 O 2556 52 48 2544 40 36 32 28 25 24 20 16 12 8 25 4 25 O 2456 52 48 2444 40 36 32 28 2424 20 16 12 8 24 4 24 6O 39 58 57 56 55 54 5:i 52 51 50 49 ! 48 47 i 46 \ 45 44 43 ; 42 41 40 39 38 j 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 j 17 ! 16 15 ! 14 i 13 1 12 01 10 9 8 6 5 4 3 2 1 O'.l 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 ! 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 32 4 8 12 16 32 20 24 28 32 36 3240 44 48 52 56 33 4 8 12 16 3 8 2O 24 28 32 36 3340 44 48 52 56 34 4 8 12 16 34 2O 24 28 32 36 3440 44 48 52 56 35 4 8 12 16 3520 24 28 32 36 3540 44 48 52 56 36 O 9-59I9 5922 5925 592 5931 i i i 2 2 2 3 3 i i 2 3 3 i i i 2 2 2 3 3 o I 2 3 3 o i i i 2 2 2 3 3 o I I I 2 2 2 3 3 i 2 2 2 2 I 2 2 I 2 2 2 2 2 2 10.4081 4078 4 75 4072 4069 9.6279 6282 6286 6289 6293 6 i I 2 1 3 : 3 3 1 3 4 j I I 2 2 3 o 3 3 3 4 O I I 2 1 3 2 2 1 3 3 3 ! 4 ; O I I 2 1 3 2 2 3 3 3 4 I I 2 i 3 2 2 3 3 3 ! I I 2 i 3 2 2 3 3 3 4 10.3721 37i8 37H 37" 3707 10.0360 0360 0361 0361 0362 IO.O362 0363 0364 0364 0365 9.9640 9640 9639 9639 9638 9.9638 9637 9636 9636 9635 9-5934 5937 5940 5943 5945 10.4066 4063 4060 4057 4055 9.6296 6300 6303 6307 6310 9.6314 6317 6321 6324 6328 10.3704 3700 3697 3693 3690 9.5948 595i 5954 5957 5960 10.4052 4049 4046 4043 4040 10.3686 3683 3679 3676 3672 10.0365 0366 0366 0367 0367 9-9635 9634 9634 9633 __.9 6 33 9.9632 9632 9631 9631 9630 9.9629 9629 9628 9628 9627 9-59 6 3 5966 5969 5972 _5975 9-59/8 598i 5984 5987 5990 10.4037 4034 4031 4028 4025 9-633I 6334 6^38 6341 6 345 10.3669 3666 3662 3659 3^55 10.0368 0368 0369 0369 03/0 10.4022 4019 4016 4013 4010 9.6348 6352 6355 6359 6362 10.3652 3648 3645 3641 _3 6 J8 10.3634 3631 3627 3624 3620 IO.O37I 0371 0372 0372 0373 9-5992 5995 5998 6001 6004 10.4008 4005 4002 3999 399 6 9.6366 6369 6373 6376 6380 10.0373 0374 0374 375 _175 10.0376 0377 0377 0378 0378 9.9627 9626 9626 9625 9625 9.6007 6010 6013 6016 6019 9.6021 6024 6027 6030 __ 6 33 9.6036 6039 6042 6045 6047 10.3993 3990 3987 3984 398i 9-6383 6386 6390 6393 6397 9.6400 6404 6407 6411 6414 9.6417 6421 6424 6428 6431 10.3617 3614 . 3610 3607 3603 9.9624 9623 9623 9622 9622 10.3979 3976 3973 3970 3967 10.3964 3961 3958 3955 3953 10.3600 3596 3593 3589 3586 10-3583 3579 3576 3572 3569 10.0379 0379 0380 0380 0381 10.0382 0382 0383 0383 0384 9.9621 9621 9620 9620 __? 6l 9 9.9618 9618 9617 9617 9616 9.9616 9615 9615 9614 9613 9.9613 9612 9612 961 1 96 n 9.9610 9610 9609 9608 9608 9.6050 ? 5 ? 6056 6059 6062 9.6065 6068 6070 6073 6076 0.6079 6082 6085 6087 6090 10.3950 3947 3944 394i .3938 10-3935 3932 3930 3927 3924 10.3921 39i8 3915 3913 3910 9-6435 6438 6441 6445 6448 9.6452 6455 6459 6462 _ 6465 9.6469 6472 6476 6479 6482 10.3565 3562 3559 3555 __3552 10.3548 3545 354J 3538 3535 i-353 i 3528 3524 352i 35i8 10.0384 0385 0385 0386 _3 8 7 10.0387 0388 0388 0389 _53?9 10.0390 0390 0391 0392 0392 9.6093 10.3907 9.6486 10.3514 10.0393 9.9607 / m s Cos. O'.l 1* Sec. Cot. O'.l 1 s Tan. Cosec. Sin. m s / Diff. Diff. 113^7 h 32 m ] [4 h 24 m = 66 TABLE IX. 51 24 = l h 36 rn ] Log. Sines, Tangents, and Secants. [ lO h 2O m - 155 ' m s Sin. Diff. Cosec. Tan. Diff. Cot. Sec. Cos. m s i O'.l I 8 O'.l I 8 O 1 2 3 4 U 8 9 i! 10 11 ! 12 ! 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 6 36 O 4 8 12 16 36 2O 24 28 32 36 36 4O 44 48 52 56 37 4 8 12 16 37 2O 24 28 32 36 3740 44 48 52 56 38 4 8 12 16 38 2O 24 28 32 36 3840 44 48 52 56 39 4 8 12 16 3920 24 28 32 36 39 4O 44 48 52 56 4O O 9.6093 6096 6099 6102 6104 o i i i 2 2 2 2 O I I I I 2 2 2 2 I I I 2 2 2 2 O I I I 2 2 2 2 2 2 2 2 O I I I I 2 2 2 2 2 2 \ 2 2 I 2 2 2 2 I 2 2 10.3907 394 3901 3898 3896 9.6486 6489 6493 6496 6499 o i i 2 2 3 3 3 4 i i 2 2 3 3 3 4 o 2 2 3 3 3 4 o i i 2' 2 3 3 3 4 i 2 2 3 3 3 4 i i 2 2 3 3 3 4 i 2 3. I 2 3 i 2 3 i 2 3 I 2 3 i 2 3 10.3514 35" 357 3504 35oi 0.0393 j 9.9607 0393 9607 0394 ! 9606 0394 9606 0395 9605 24 O 2356 52 48 2344 40 36 32 28 2324 20 16 12 8 23 4 23 O 2256 52 48 2244 40 36 32 28 2224 20 16 12 8 22 4 22 2156 52 48 2144 40 36 32 28 2124 2O 16 12 8 21 4 21 2056 52 48 2044 40 36 32 28 2024 20 16 12 8 20 4 2O O 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 9.6107 6110 6113 6116 6119 9.6121 6124 6127 6130 6133 10.3893 3890 3887 3884 3881 9-6503 6506 6510 6516 10.3497 3494 3490 3487 3484 10.3480 3477 3473 3470 3467 0.0396 0396 0397 0397 0398! 9.9604 9604 9603 9603 9602 "9^9602 9601 9601 9600 9599 9-9599 9598 9598 9597 9597 10.3879 3876 3873 3870 3867 9.6520 6523 6527 6530 6533 10.0398 0399 0399 0400 0401 9-6I35 6138 6141 6144 10.3865 3862 3859 9-6537 6540 6543 6547 6550 9-6553 6557 6560 6564 6567 9.6570 6574 6577 6580 6584 10.3463 3460 3457 3453 3450 10.0401 0402 0402 0403 0403 9.6149 6152 6158 6161 9.6163 6166 6169 6172 6174 10.3851 3848 3845 3842 3839 10.3447 3443 3440 3436 3433 10.0404 0405 0405 0406 0406 9.9596 9595 9595 9594 9594 10.3837 3834 3831 3828 10.3823 3820 3814 3812 10.3430 3426 3423 3420 10.0407 0407 0408 0409 0409 9-9593 9593 9592 9591 9591 9.6177 6180 6183 6186 6188 9.6587 6590 6594 6597 6600 10.3413 3406 3403 3400 10.0410 0410 0411 0411 0412 9.9590 9590 9589 9589 9588 9.6191 6194 6i97 6199 6202 10.3809 3806 3803 3801 3798 9.6604 6607 6610 6614 6617 10.3396 3393 339 3386 3383 10.3380 3376 3373 3370 3366 10.0413 0413 0414 0414 0415 9.9587 9587 9586 9586 9585 9.6205 6208 6210 6213 6216 10-3795 3792 3790 3787 3784 10.3781 3779 3776 3773 3770 10.3768 3765 3762 3760 3757 9.6620 6624 6627 6630 6634 10.0416 0416 0417 0417 0418 9-9584 9584 9583 9583 9582 9.6219 6221 6224 6227 6230 9.6232 6235 6238 6240 6243 9.6637 6640 6644 6647 6650 10.3363 3356 3353 _335?_ 10.3346 3343 3340 3336 3333 10.0418 0419 0420 0420 0421 9.9582 958o 9579 9.6654 6657 6660 6664 6667 10.0421 0422 0423 0423 0424 9-9579 9578 9577 9577 9576 9.6246 6249 6251 6254 6257 10-3754 375i 3749 3746 3743 9.6670 6674 6677 6680 6683 10.3330 3326 3323 3320 10.0424 0425 0425 0426 0427 9-9576 9575 9575 9574 9573 9-9573 9.6259 10.3741 9.6687 10.3313 10.0427 L m B Cos. O'.l 1 s Sec. Cot. O'.l Tan. Cosec. Sin. m s ' Diff. Diff. | 114 = 7 h 36 m ] [ 4 h 2O ra = 65 52 TABLE IX. 25 = 1" 4O' n ] JLog. Sines, Tangents, and Secants. [ 1O 1 ' 16'" = 1 51 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 3O 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 1O 9 8 7 6 5 4 3 9 1 m s Sin. Diff. 0.1 1^ Cosec. Tan. 9.6687 6690 6693 6697 6700 Di 0.1 2 2 3 3 3 i i i 2 2 3 3 3 i i i 2 2 3 3 3 i i 2 2 3 3 3 i i i 2 2 3 3 3 I I I 2 2 3 3 3 TvT 0f. 1- i 2 2 I 2 2 2 2 2 2 2 2 I 2 2 Cot. io.33'3 33i 3307 3303 33 Sec. Cos. 9-9573 9572 9572 957i 9570 in s 20 O 1956 52 48 1944 40 36 32 28 1924 20 16 12 8 19 4 19 1856 52 48 1844 40 36 32 28 1824 2O 16 12 8 18 4 18 1756 52 48 1744 40 36 32 28 1724 20 16 12 8 17 4 17 1656 52 48 1644 40 36 32 28 1624 20 16 12 8 16 4 16 O 1 2 3 4 5 6 7 8 9 10 11 12 13 i ri 15 ! 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 6O 40 4 8 12 16 4020 24 28 32 36 4O4O 44 48 52 56 41 4 8 12 16 4120 24 28 32 36 41 4O 44 48 52 56 42 4 8 12 16 4220 24 28 32 36 42 4O 44 48 52 56 43 4 8 12 16 4320 24 28 32 36 4340 44 48 52 56 44 9.6259 6262 6265 6268 6270 i i i 2 2 2 2 O I I I I 2 2 2 2 O I 2 2 2 2 I I I I 2 2 2 2 I I I I 2 2 2 2 O I I I I 2 2 2 2 i 2 2 1 2 2 I 2 2 I 2 2 I 2 2 10.3741 3738 3735 3732 3730 10.0427 0428 0428 0429 0430 9.6273 6276 6278 6281 6284 10.3727 3724 3722 3719 37i6 9.6703 6706 6710 J7I3 6716 10.3297 3294 3290 3287 3284 10.0430 43 * 043 I 0432 433 9-9570 9569 9569 9568 9567 9-9567 9566 9566 9565 9564 9.6286 6289 6292 6295 6297 10.3714 37" 3708 3705 3703 9.6720 6723 6726 6729 6733 10.3280 3277 3274 3271 3267 10.0433 0434 0434 0435 0436 9.6300 6303 6305 6308 6311 10.3700 3 6 97 3 6 95 3692 3689 9.6736 6739 6743 6746 6749 10.3264 3261 3257 3254 3251 10.0436 0437 0437 43 8 0439 9.9564 9563 9563 9562 _ 956 ' 9-6561 9560 9560 9559 9558 9-63 1 3 6316 6319 6321 6324 10.3687 3684 3681 3679 3676 9.6752 6756 6759 6762 6765 10.3248 3244 3241 3238 3235 10.3231 3228 3225 3222 3218 10.0439 0440 0440 0441 0442 9.6327 6329 6332 6335 6 337 10.3673 3671 3668 3665 3663 10.3660 3658 3655 3652 _36 5 o 10.3647 3 6 44 3642 3639 3636 9.6769 6772 6775 6778 6782 10.0442 0443 443 0444 0445 9-9558 9557 9557 955 6 9555 9.6340 6342 6 345 6348 6350 9.6785 6788 6791 6795 6798 9.6801 6811 6814 10.3215 3212 3209 3205 3202 10.0445 0446 0446 0447 0448 9-9555 9554 9554 9553 . 9552 9-6353 6356 6358 6361 6364 10.3199 3196 3192 3189 3186 10.0448 0449 0449 0450 -JH5.L 10.0451 0452 0452 0453 Q454 10.0454 0455 455 0456 Q457 10.0457 0458 0458 0459 6460 9-9552 955i 955 1 955 _?549 9-9549 9548 9548 9547 9546 9.9546 9545 9545 9544 9543 9.6366 6369 6371 6374 6377 10.3634 3631 3629 3626 3623 9.6817 6821 6824 6827 6830 10.3183 3179 3176 3173 3170 9.6379 6382 6385 6387 6390 10.3621 3618 3615 3613 3610 10.3608 3605 3602 3600 3597 9.6834 6837 6840 6843 6846 10.3166 3 l6 3 3160 3157 _3154 10.3150 3H7 3 J 44 3*4 3U7 9.6392 6395 6398 6400 6403 9.6850 6853 6856 6859 6863 9-9543 9542 9542 954i 9540 1$ 6411 6413 6416 9.6418 io.3595 3592 3589 35f7 _35?4 10.3582 9.6866 6869 6872 6875 6879 9.6882 10.3134 3131 3128 3125 3121 10.3118 10.0460 0461 0462 0462 0463 10.0463 9.9540 9539 9538 9538 _953.7 9-9537 Siii. / m s Cos. O'.l I 8 Diff. Sec. Cot. ! Tan. Cosec. m s / Diff. 115 = 7" 40'" ] [ 4 h 16 m = 64 \ TABLE IX. 53 26^ 1 2 3 4 5 6 7 I ! 10 11 12 13 14 15 16 17 18 I 19 20 21 22 123 i 24 j 25 ! 26 1 27 i 28 29 30 i 31 ! 32 i 33 34 j 35 36 37 38 i 39 |40 11 '42 43 11 45 46 47 48 49 50 51 52 53 54 55 56 J57 |58 !59 60 = 1 44 n, j Sill. 9.6418 6421 6424 6426 6429 9.6431 6434 6437 6439 6442 L,o&. Sines, Tangents, and Secants. [ IO" 12 !tl = 153 in s 44 O 4 8 12 16 44 2O 24 28 32 36 44 4O 44 48 52 56 45 4 8 12 16 45 2O 24 28 32 36 4540 44 48 52 56 46 O 4 8 12 16 4620 24 28 32 36 4640 44 48 52 56 47 4 8 12 16 4720 24 28 32 36 47 4O 44 48 52 56 48 Diff. Cosec. Tan. Diff. Cot. Sec. 10.0463 0464 0465 0465 0466 Cos. 9-9537 9536 9535 9535 __.9534 9-9534 9533 9532 9532 953 1 ni s 60 59 58 57 56 55 i 54 53 52 i 51 | 50 49 48 47 ! 46 45 i 44 i 43 i 42 ! 41 40 39 38 i 37 1 36 35 34 33 ! 32 31 30 29 28 27 26 25 24 23 22 21 2O 19 ! 18 17 ! 16 | 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 O'.l o i i 2 2 2 2 O I I I I 2 2 2 2 O T I I 2 2 2 2 I 2 2 2 2 O I I I 2 2 2 2 O I I 2 2 2 2 O'.l I- 2 2 I 2 2 I 2 2 I 2 2 2 2 2 2 O'.l i i 1 2 2 3 3 3 o i i i 2 2 3 3 3 i i i 2 2 3 3 3 o i i i 2 2 3 3 3 o i 2 2 3 3 3 i i i 2 2 3 3 3 1 s 10.3582 3579 3576 3574 ._..357JL 10.3569 35f 3563 356i 3558 9.6882 6885 6888 6891 6895 9.6898 6901 6904 6907 6911 "9.6914 6917 6920 6923 _6917 9.6930 6933 6936 6939 6942 i 2 'J 2 2 I 2 2 2 2 I 2 2 2 2 10.3118 3H5 3112 3109 3105 16 O 1556 52 48 1544 40 36 32 28 1524 20 16 12 8 15 4 15 14 56 52 48 1444 40 36 32 28 1424 2O 16 12 8 14 4 14 1356 52 48 1344 40 36 32 28 1324 20 16 12 8 13 4 13 1256 52 48 1244 40 36 32 28 1224 20 16 12 8 12 4 12 10.3102 399 3096 3093 3089 10.0466 0467 0468 0468 0469 9.6444 6447 6449 6452 _J>454_ 9-6457 6460 6462 6465 6467 !o-355 6 3553 355J 3548 __3_546 iQ-3543 3540 3538 3535 3533 10.3086 3083 3080 377 3Q73 10.3070 3067 3064 3061 3058 10.0470 0470 0471 0471 _547?_ 10.0473 0473 0474 0475 0475 9-953 953 9529 9529 _9S?8 9-9527 9527 9526 9525 9525 9.6470 6472 6475 6477 6480 10.3530 3528 3525 3523 3520 9.6946 6949 6952 6955 6958 10.3054 35! 3048 3045 3042 10.3038 3035 33 2 3029 3026 10.0476 0476 0477 0478 0478 9.9524 9524 9523 9522 9522 9.6483 6485 6488 6490 6493 10.3517 3515 3512 351 3507 9.6962 6965 6968 6971 6974 10.0479 0480 0480 0481 0481 9.9521 9520 9520 95*9 9519 9.95i8 9517 9517 95i6 9515 9.6495 6498 6500 6503 6505 10.3505 3502 35 3497 3495 9.6977 6981 6984 6987 6990 10.3023 3019 3016 3013 3010 10.0482 0483 0483 0484 048-5 9.6508 6510 6513 65*5 6518 10.3492 3490 3487 3485 3482 10.3479 3477 . 3474 3472 3469 9-6993 6996 6999 7003 7006 10.3007 34 3001 2997 2994 10.0485 0486 0487 0487 0488 9-95 i 5 95H 9513 9513 95i2 9.6521 *$*l 6526 6528 6531 9.7009 7012 7015 7018 7022 10.2991 2988 2985 2982 2978 10.0488 0489 0490 0490 0491 9.9512 95" 95io 95io 9509 9-6533 6536 6538 6541 6 543 10.3467 3464 3462 3459 3457 9.7025 7028 7031 7034 7037 10.2975 2972 2969 2966 2963 10.0492 0492 0493 0494 0494 9.9508 9508 957 9506 9506 9.6546 6548 655i 6 553 6556 10.3454 3452 3449 3447 3444 9.7040 743 7047 7050 7053 10.2960 2957 2953 2950 2947 10.0495 0495 0496 0497 0497 9.9505 955 954 953 953 9-6558 6561 6563 6566 6568 10.3442 3439 3437 3434 3432 9.7056 7059 7062 7065 7069 10.2944 2941 2938 2935 2931 10.0498 0499 0499 0500 0501 9.9502 95i 95i 9500 9499 9.6570 10.3430 9.7072 10.2928 10.0501 9-9499 ' j in s Cos. 1 s Sec. Cot. O'.l 1 s Tan. Cosec. Sin. m s / Diff. Diff. 116 = 7 h 44 m ] [ 4 h 12 m = 63 54 TABLE IX. 27= l h 48 :D ] Log. Sines, Tangents, and Secants. [lO h 8 m = 152 i m s Sin. Diff. Cosec. Tan. Diff. Cot. Sec. Cos. Ill S i O'.l 1 s O'.l 1 s i 2 2 I 2 2 I 2 2 2 2 I 2 2 I 2 2 I 8 1 2 3 4 5 6 7 8 9 10 11 i 1~ 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 I 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 48 4 8 12 16 48 20 24 28 32 36 4840 44 48 52 56 49 8 12 16 49 2O 24 28 32 36 49 4O 44 48 52 56 50 4' 8 12 16 5O2O 24 28, 32 36 5040 44 48. 52 56 51 4 8 12 16 51 2O 24 28 32 36 5140 44 48 52 56 52 9.6570 6573 J575 6578 6580 i i i i 2 2 2 2 O 1 I I I 2 2 2 O I I 2 2 2 2 I I I I 2 2 2 2 O I I I I 2 2 2 2 I I I 2 2 2 2 i 2 2 I 2 I I 2 2 I I 2 10.3430 34<7 3425 3422 3420 9.7072 7075 7078 7081 7084 o i i i 2 2 3 3 3 o 2 2 3 3 3 o i i i 2 2 3 3 3 I I 2 2 3 3 3 2 2 ' 3 3 3 o 2 2 3 3 3 10.2928 2925 2922 2919 2916 10.0501 O5O2 O5O2 0503 0504 IO.O5O4 0505 0506 0506 0507 9-9499 9498 9498 9497 9496 12 1156 52 48 1144 40 36 32 28 1124 2O 16 12 8 11 4 11 1056 52 48 1044 40 36 32 28 1024 20 16 12 8 1O 4 10 956 52 48 944 4O 36 32 28 924 20 16 12 8 9 4 9 O 856 52 48 844 40 36 32 28 824 20 16 12 8 8 4 8 00 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 T 8 7 6 5 4 3 2 j 1 9.6583 6585 6588 6590 6593 10.3417 3415 3412 34o 3407 9.7087 7090 7093 7097 7100 10.2913 2910 2907 2903 2900 9.9496 9495 9494 9494 9493 9.9492 9492 9491 9490 9490 9-6595 6598 6600 6603 6605 10.3405 3402 3400 3397 3395 9.7103 7106 7109 7112 7"5 10.2897 2894 2891 2888 2885 IO.O5O8 0508 0509 O5IO 0510 9.6607 6610 6612 6615 6617 10-3393 3390 3388 33f5 ^l 8 ! 10.3380 3378 3375 3373 337i 9.7118 7121 7125 7128 7131 10.2882 2879 2875 2872 2869 IO.O5II O5I2 0512 0513 0514 9.9489 9488 9488 9487 9486 9.6620 6622 6625 6627 6629 9.6632 6634 6637 6639 6642 9-7I34 7137 7140 7143 7146 10.2866 2863 2860 2857 2854 IO.O5I4 0515 S 1 S 0516 0517 9.9486 9485 9485 9484 9483 10.3368 3366 33 6 3 336i 3358 9.7149 7152 7i5 6 7i59 7162 10.281; i 2848 2844 2841 2838 io.o | j 2 2. j O I 1 ' I j 2. i i i I 2 2 2 O I I I I 2 I ! I I 2 2 10.3144 3H2 3MO 3137 Mi 10.3133 & 3126 3124 9-7438 7440 7443 7446 7449 9-745 2 7455 745 8 7461 7464 o i i 2 2 2 3 o i 2 2 2 3 i i i i 2 2 2 3 o I I 2 2 2 3 i i 2 2 2 3 2 2 2 3 2 2 2 2 2 2 I 2 2 I 2 2 i 2 2 10.2562 2560 2557 2554 2551 10.2548 2545 2542 2539 2536 10.0582 0583 0583 0584 0585 10.0585 0586 0587 0587 0588 9.9418 9417 9417 9416 9415 9-94I5 9414 9413 9413 9412 4 356 52 '48 314 4O 36 32 28 324 20 16 12 8 3 4 3 256 52 48 244 40 36 32 28 224 20 16 12 8 2 4 2 156 52 48 144 40 36 32 28 124 20 16 12 8 1 4 1 056 52 48 044 4O 36 32 28 024 20 16 12 8 4 O O 6O 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 4O 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 O 9.6878 6 38 1 638} 6885 633? 9.6890 6892 6894 6896 6899 9.6901 6903 6905 6908 6910 9.6912" 6914 6917 6919 6921 10.3122 3"9 3"7 3H5 3 "3 10.3110 3108 3106 3 I0 4 3101 10.3099 397 395 3092 3090 9-7467 7470 7473 7476 _7479_ 9.7482 74 8 5 7488 749i __749 9-7497 7500 7503 7506 7509 10.2533 2530 2527 2524 ._?5*L 10.2518 2515 2512 2509 2506 10.2503 2500 2497 2494 2491 10.0589 0590 0590 0591 __!?2_ 10.0592 0593 0594 0594 0595 10.0596 0597 0597 0598 0599 9.9411 9410 9410 9409 9408 9.9408 9407 9406 9406 9405 9.9404 9403 9403 9402 9401 10.3088 3086 3083 3081 379 9-75 12 7515 75i8 752i 7523 10.2488 2485 2482 2479 2477 10.0599 0600 0601 0602 0602 9.9401 9400 9399 939f 9398 9.6923 6926 6928 6930 693 2 9- 6 935 6937 6 939 6941 6943 10.3077 374 3072 3070 3068 9.7526 7529 7532 7535 7538 10.2474 2471 2468 2465 2462 10.0603 0604 0604 0605 0606 9-9397 9396 939 6 9395 _9394 9-9393 9393 9392 9391 9-9390 9389 92 9388 9387 10.3065 3 63 3061 3059 3057 9-7541 7544 7547 755 7553 10.2459 2456 2453 2450 2447 10.0607 0607 0608 0609 0609 9.6946 6948 6950 6952 6 91L 9.6957 6959 6961 6963 6966 9.6968" 6970 6972 6974 6977. 9.6979 6981 6983 6985 6988 9.6990 10.3054 3052 3050 3048 ._3?45_ 10.3043 34i 3039 337 334 9-7556 7559 7562 3 10.2444 2441 2438 2435 2432 1 0.06 10 0611 0612 0612 0613 9-7571 7573 7576 7579 7582 10.2429 2427 2424 2421 2418 10.0614 0615 0615 0616 0617 9.9386 9385 93f5 9384 9383 10.3032 3030 3028 3026 3023 10.3021 3019 3017 3 OI 5 3012 10.3010 9-7585 7588 759 1 7594 7597 10.2415 2412 2409 2406 2403 10.0617 0618 0619 0620 0620 9.9383 9382 938i 9380 9380 9. 7600 7603 7606 7609 7611 "97614" 10.2400 2397 2394 2391 2389 10.2386 10.0621 0622 0623 0623 0624 10.0625 9-9379 9378 9377 9377 9376 9-9375 / i m B Cos. O'.l 1-' Sec. Cot. O'.l 1 s 5T Tan. Cosec. Sin. 1 m s / Diff. Di 119^--7 h 56"' J [4 h 0"\=60 TABLE IX. 57 ' 30 t: 2 h O m ] L-og. Sines, Tangents, and Secants. [ 9 h 56'" = 149 ' m s Sin. Diff. Cosec. Tan. Diff. Cot. Sec. Cos. m s 60 5956 52 48 59 44 40 36 32 28 5924 20 16 12 8 59 4 59 O 5856 52 48 5844 40 36 32 28 58 24 20 16 12 8 58 4 58 5756 52 48 5744 40 36 32 28 5724 20 16 12 8 57 4 57 5656 52 48 5644 40 36 32 28 5624 20 16 12 8 56 4 56 i 0.1 1 s OM 1 s O 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 j 35 36 37 38 39 40 41 42 43 44 45 46 i 47 148 49 50 51 52 53 54 ; 55 i 56 57 58 59 6O 4 8 12 16 020 24 28 32 36 O4O 44 48 52 56 1 4 8 12 - 16 12O 24 28 32 36 140 44 48 52 56 2 O 4 8 12 16 2 2O 24 28 32 36 24O 44 48 52 56 3 4 8 12 16 3 2O 24 28 32 36 340 44 48 52 56 4 j m s ! 9.6990 6992 6994 6996 6998 o o 2 2 O O 2 2 O 2 2 O 2 2 O O 2 2 i i 2 I I 2 I 2 2 2 I I 2 10.3010 3008 3006 3004 3002 9.7614 7617 7620 7623 7626 o J 2 2 3 o I I I I 2 2 2 3 o I ' 2 2 2 3 I I I 2 2 2 3 I 2 2 2 3 o i i i 2 2 2 3 i 2 2 I 2 2 I 2 2 I 2 2 I 2 2 I 2 2 10.2386 2383 2380 2377 2374 10.06215 0625 0626 0627 0628 9-9375 9375 9374 9373 9372 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 O 9.7001 7003 7005 7007 7009 10.2999 2997 2995 2993 2991 9.7629 7632 7635 7638 7641 10.2371 2368 23 6 5 2362 2359 10.2356 2354 235J 2348 2345 10.0628 0629 0630 0631 0631 9-9372 937 1 937" 93 6 9 9369 9.7012 7014 7016 7018 7020 10.2988 2986 2984 2982 2980 9.7644 7646 7649 7652 7655 10.0632 0633 0033 0634 6 35 9.9368 93 6 7 9367 9366 93 6 5 9.7022 7025 7027 7029 7031 10.2978 2975 2973 2971 2969 9.7658 7661 7664 7667 7670 10.2342 2339 233 6 2333 233 10.0636 0636 0637 0638 0639 9.9364 9364 93 6 3 9362 9361 9-7033 7035 7037 7040 7042 9.7044 7046 7048 7050 75JL 9-755 7057 7059 7061 7063 10.2967 2965 2963 2960 2958 9-7673 7675 7678 7681 7684 10.2327 2325 2322 2319 2316 10.0639 0640 0641 0642 0642 9.9361 9360 9359 9358 9358 10.2956 2954 2952 295 2947 9.7687 7690 7693 7696 7699 10.2313 2310 2307 2304 2301 10.0643 0644 0645 0645 0646 9-9357 9356 9355 9355 9354 10.2945 2943 2941 2939 2937 9.7701 7704 7707 7710 7713 10.2299 2296 2293 2290 2287 10.0647 0648 0648 0649 0650 9-9353 935 2 9352 935 i 935 9.7065 7068 7070 7072 7074 10.2935 2932 2930 2928 2926 9.7716 7719 7722 7725 7727 10.2284 2281 2278 2275 ^73 10.2270 2267 2264 2261 2258 10.0651 0651 0652 0653 0654 9-9349 9349 9348 9347 9346 9.7076 7078 7080 7082 7085 10.2924 2922 2920 2918 2915 9-773 7733 7736 7739 7742 10.0654 0655 0656 0657 0657 9-934 6 9345 9344 9343 9343 9.7087 7089 7091 7093 7095 10.2913 2911 2909 2907 2905 9-7745 7748 775 7753 775 6 10.2255 2252 2250 2247 2244 10.0658 659 0660 0660 0661 9.9342 934i 9340 9340 9339 9.7097 ! 7099 o 7102 1 o 7104 I 7106 I 10.2903 2901 2898 2896 2894 9-7759 7762 7765 7768 7771 10.2241 2238 2235 2232 2229 10.0662 0663 0663 0664 0665 9-9338 9337 9337 9336 9335 9.7108 7110 7112 7114 7116 I I I 2 10.2892 2890 2888 2886 2884 9-7773 7776 7779 7782 7785 10.2227 2224 2221 22l8 22 *5 IO.22I2 10.0666 0666 0667 0668 0669 9-9334 9334 9333 9332 9331 9.7118 10.2882 9.7788 10.0669 9-9331 | , Cos. 0.1 1 s Sec. 1 Cot. O'.l 1 s Tan. Cosec. Sin. m s / Diff. Diff. ; 12O M 8 h O' ] [ 3 h 56 m = 59 58 TABLE IX 31 \> I L,og. Sines, Tangents, and Secants. [ 9 h 52 m = 148 / O 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 m & Sin. Diff. Cosec. Tan. Diff. Cot. Sec. Cos. Ill S 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 : 27 26 25 24 ! 2J{ 22 21 20 19 18 17 16 15 14 i 13 12 11 10 9 8 7 i I 3 2 1 o i O'.l I 8 i i 2 I 1 2 I 2 I 2 2 I I 2 O'.l I 8 4 4 8 12 16 420 24 28 32 36 440 44 48 52 56 5 4 8 12 16 520 24 28 32 36 540 44 48 52 56 6 4 8 12 16 620 24 28 32 36 640 44 48 52 56 7 4 8 12 16 7 2O 24 28 32 36 74O 44 48 52 56 8 9.7118 7120 7123 7125 7127 o o I I I I I 2 2 O 2 2 O O I I I I I 2 2 O 1 I I I I 2 2 O 2 2 O 1 , I I I I 2 2 10.2882 2880 2877 2875 2873 9.7788 7791 7793 7796 7799 o i i i i 2 2 2 3 I 2 2 2 3 I I 2 2 2 3 o i i i i 2 2 2 3 I 2 2 2 3 o 2 2 2 3 i 2 2 I 2 2 2 2 2 2 I 2 2 2 2 IO.22I2 22O9 2207 22O4 22OI 10.0669 0670 0671 0672 0672 9-9331 9330 9329 9328 9328 56 55 56 52 48 5544 40 36 32 28 5524 20 16 12 8 55 4 55 5456 52 48 5444 40 36 32 28 54 24 20 16 12 8 54 4 54 53 56 52 48 5341 40 36 32 28 53 24 20 16 12 8 53 4 53 52 56 52 48 5214 40 36 32 28 5224 20 16 12 8 52 4 52 9.7129 7I3 1 7U3 7135 7137 10.2871 2869 2867 2865 2863 10.2861 2859 2856 2854 2852 10.2850 2848 2846 2844 2842 9.7802 7805 7808 7811 7813 10.2198 2195 2192 2189 2187 10.0673 0674 0675 0675 0676 9-9327 9326 9325 9325 93 2 4 9-7I39 7141 7H4 7146 7148 9.7816 7819 7822 7825 7828 IO.2I84 2181 2178 2175 2172 10.0677 0678 0678 0679 0680 9-9323 9322 9522 9321 9320 9-7I5 7152 7154 7156 7158 9-7831 7833 7f36 7839 7842 10.2169 2167 2164 2161 2158 1 0.068 1 0682 0682 0683 0684 9W9 93i8 93^8 9317 93i6 9.7160 7162 7164 7166 7168 10.2840 2838 2836 2834 2832 9-7845 7848 7850 7853 785<> 10.2155 2152 2150 2147 2144 10.0685 0685 0686 0687 0688 9-93I5 9315 93H 9313 9312 9.7171 7173 7175 7177 7179 9.7181 7183 7185 7187 _2!?9 9.7191 7193 7195 7197 7199 9.7201 7203 7205 7208 7210 10.2829 2827 2825 2823 2821 9-7859 7862 7865 7868 7870 10.2141 2138 2135 2132 2130 10.0688 0689 0690 - 0691 0692 9.9312 93" 9310 939 _^3o8 9.9308 93 7 9306 935 935 10.2819 2817 2815 2813 2811 9-7873 7876 7879 7882 7885 10.2127 2124 2121 2118 2115 10.0692 0693 0694 0695 0695 10.2809 2807 2805 2803 2801 10.2799 2797 2795 2792 2790 9.7887 7890 7893 7896 7899 10.2113 2IIO 2107 2104 2IOI 10.0696 0697 0698 0699 0699 9.9304 933 9302 9301 9301 9.7902 7904 7907 7910 7913 10.2098 2096 2093 2O9O 2087 10.0700 0701 0702 0702 0703 9.9300 9299 9298 9298 9297 9.7212 7214 7216 7218 7220 9.7222 7224 7226 7228 7230 10.2788 2786 2784 2782 2780 9.7916 7918 7921 7924 7927 IO.2O84 2082 2079 2076 2073 10.0704 0705 0706 0706 0707 9.9296 9295 9294 9294 9 2 93 10.2778 2776 2774 2772 2770 9-7930 7933 7935 7938 794i 1O.2O7O 2067 2065 2062 2059 10.0708 0709 0709 0710 0711 9.9292 9291 9291 9290 9289 9.7232 7236 7238 7240 10.2768 2766 2764 2762 2760 9.7944 7947 7949 7952 7955 10.2056 2053 2051 2048 2045 10.0712 0713 7i3 0714 0715 9.9288 9287 9287 9286 9285 9.7242 10.2758 9.7958 IO.2O42 10.0716 9.9284 / m & Cos. O'.l I 8 Sec. Cot. O'.l 1" Tan. Cosec. Sin. m s Diff. Diff. i 121 = 8 h 4 m ] . [ 3 h 52'" = 58 TABLE IX. 32 = 2 a 8 m ] L,og. Sines, Tangents, and Secants. [9 h 48 1U = 147 3 ' m & Sin. Diflf. Cosec. Tan. Diff. Cot. Sec. Cos. m s ' O'.l 1 s O'.l o i i i i 2 2 2 3 I I I I 2 2 2 3 O I I I I 2 2 2 3 o I I I I 2 2 2 3 o 2 2 2 3 I I I I 2 2 2 3 1 s 1 2 3 4 5 6 7 8 10 I 11 ! 12 ! 13 14 15 I 16 17 18 19 20 21 22 1 23 24 25 26 27 i 28 29 30 i 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 ,48 49 50 51 52 53 54 55 56 57 58 59 60 8 4 8 12 16 820 24 28 32 36 840 44 48 52 56 9 4 8 12 16 920 24 28 32 36 94O 44 48 52 56 10 4 8 12 16 1O2O 24 28 32 36 1040 44 48 52 56 11 4 8 12 16 1120 24 28 32 36 1140 44 48 52 56 12 9.7242 7244 7246 7248 7250 o o 2 2 O O 2 2 O O 2 O 2 2 O 2 2 O I I I I I 2 2 i 2 I I 2 I I 2 I I 2 2 I I 2 ~1*~ 10.2758 2756 2754 2752 2750 9-7958 7961 7964 7966 7969 2 2 I 2 2 2 2 2 2 . I 2 2 I 2 2 10.2042 2039 2036 2034 2031 10.0716 0717 0717 0718 0719 9.9284 9283 9283 9282 9281 52 5156 52 48 5144 40 36 32 28 5124 20 16 12 8 51 4 51 5056 52 48 5044 40 36 32 28 5024 20 16 12 8 5O 4 50 4956 52 48 4944 40 36 32 28 4924 20 16 12 8 49 4 49 4856 52 48 4844 4O 36 32 28 4824 20 16 12 8 48 4 48 60 59 58 57 56 55 54 53 52 51 5O 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 i 17 16 15 14 13 12 11 10 .9 8 i I 4 3 2 1 i 9.7252 7254 7256 7258 7260 10.2748 2746 2744 2742 2740 9.7972 7975 7978 7980 7983 10.2028 2025 2022 2O2O 2017 10.0720 0721 0721 0722 0723 9.9280 9279 9279 9278 9277 9.7262 7264 7266 7268 7270 10.2738 2736 2734 2732 2730 9.7986 7989 7992 7994 7997 IO.2OI4 201 1 2008 2OO6 2003 10.0724 0725 0725 0726 0727 9.9276 9275 9275 9274 9273 9.7272 7274 7276 7278 7280 10.2728 2726 2724 2722 2720 9.8000 8003 8006 8008 8011 IO.2OOO 1997 1994 1992 1989 10.0728 0728 0729 0730 0731 9.9272 9272 9271 9270 9269 9.7282 7284 7286 7288 7290 10.2718 2716 2714 2712 2710 9.8014 8017 8020 8022 8025 10.1986 1983 1980 I 97 8 1975 10.0732 0732 733 0734 735 9.9268 9268 9267 9266 9265 9.7292 7294 7296 7298 7300 10.2708 2706 2704 2702 2700 9.8028 8031 8034 8036 8039 10.1972 1969 1966 1964 1961 10.0736 0736 0737 0738 0739 9.9264 9264 9263 9262 9261 9.7302 7304 7306 73o8 73 10 10.2698 2696 2694 2692 2690 9.8042 8045 8047 8050 8053 10.1958 *955 1953 195 1947 10.0740 0741 0741 0742 0743 9.9260 9259 9259 9258 9257 9.7312 73H 73 16 73i8 7320 10.2688 2686 2684 2682 2680 9.8056 8059 8obi 8064 8067 10.1944 1941 !939 1936 J 933 10.0744 0745 0745 0746 0747 9.9256 9255 9255 9254 9253 9.7322 7324 7326 7328 7330 10.2678 2676 2674 2672 2670 9.8070 8072 8075 8078 8081 10.1930 1928 1925 1922 1919 10.0748 0749 0749 0750 0751 9.9252 9251 9251 9250 9249 9-733 2 7334 733^ 7338 7340 10.2668 2666 2664 2662 2660 9.8084 8086 8089 8092 8095 10.1916 1914 1911 1908 1905 10.0752 753 0753 0754 0755 9.9248 9247 9247 9246 9245 9-7342 7344 7345 7347 .7349 10.2658 2656 2655 26 5 i 10.2649 2647 2645 2643 2641 9.8097 8100 8103 8106 8109 10.1903 1900 1897 1894 1891 10.0756 0757 0758 0758 0759 9.9244 9243 9242 9242 9241 9-7351 7353 7355 7357 7359 9.8111 8114 8117 8120 8122 10.1889 1886 1883 1880 1878 10.0760 0761 0762 0762 0763 9.9240 9239 9238 9238 9237 9-73 6 i 10.2639 9.8125 10.1875 10.0764 9.6236 . / m a Cos. 0.1 Sec. Cot. OM l a Tan. Cosec. Sin. 111 S / Diff. Diff. 122 = 8 h 8 m J [ 3 h 48 m = 57^ ; 60 TABLE IX. 33 ^2 U 12 s " ] Log:. Sines, Tangents, and Secants. [9 b S S = 146 i m s Sin. Diff. Cosec. Tan. Diff. Cot. Sec. Cos. m s ' O'.l 1 O'.l o i i i i 2 2 2 3 o 2 2 2 3 i i i 2 2 2 3 O 2 2 2 3 o i i i 2 2 2 3 I I I 2 2 2 3 1* 2 3 4 5 6 7 8 9 10 11 12 18 14 15 16 17 18 19 20 21 22 23 24 25 26 27 : 28 29 30 31 32 33 34 35 i 36 I 37 ; 38 ! 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 156 57 58 59 60 12 4 8 12 16 1220 24 28 32 36 1240* 44 48 52 56 13 O 4 8 12 16 1320 24 28 32 36 1340 44 48 52 56 14 O 4 8 12 16 1420 24 28 32 36 1440 44 48 52 56 15 4 8 12 16 15 2O 24 28 32 36 1540 44 48 52 56 16 9-736i 73 6 5 73 6 7 7369 I . o '. i 2 2 2 I I I j 2 I I I I 2 2 O I I : 2 2 j 2 I 1 I J I ! 2 2 2 I O I 2 2 2 1 I 2 2 2 10.2639 2637 2635 2633 2631 9.8125 8128 8131 8i33 8136 , 2 2 I 2 2 2 2 I 2 2 I 2 2 I 2 2 ff. 10.1875 1872 1869 1867 1864 10.0764 0765 0766 0767 0767 9.9236 9235 9234 9233 9233 48 O 4756 52 48 4744 40 36 32 28 4724 20 16 12 8 47 4 47 4656 52 48 1644 40 36 32 28 4624 20 16 12 8 46 4 46 O 4556 52 48 4544 40 36 32 28 4524 20 16 12 8 45 4 45 4456 52 48 4444 40 36 32 28 4424 20 16 12 8 44 4 44 O 60 59 58 ! 57 56 55 54 53 52 51 5O 49 48 47 46 45 11 43 42 i 41 1 40 39 38 i 37 36 35 ! 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 ! 6 3 2 i 1 9-737J 7373 7375 7377 7379 9.7380 7382 7384 7386 7388 10.2629 2627 2625 2623 2621 9.8139 8142 8i45 8147 8150 10.1861 1858 1855 1853 1850 10.1847 1844 1842 1839 1836 10.0768 0769 0770 0771 0771 9.9232 9231 9230 9229 9229 10.2620 2618 2616 2614 2612 9.8153 8156 8158 8161 8164 10.0772 0773 774 0775 0776 9.9228 9227 9226 9225 9224 9-7390 7392 7394 7396 7398 10.2610 2608 2606 2604 2602 9.8167 8169 8172 8i75 8178 10.1833 1831 1828 1825 1822 10.0776 0777 0778 0779 0780 9.9224 9223 9222 9221 9220 9.7400 7402 7404 7406 7407 10.2600 2598 2596 2594 ^2593 10.2591 2589 2587 2585 2583 9.8180 8183 8186 8189 8191 10.1820 1817 1814 1811 1809 10.0781 0781 0782 0783 0784 9.9219 9219 9218 9217 9216 9.7409 74" 7413 7415 7417 9.8194 8i97 8200 8202 8205 I O.I 806 1803 1800 1798 1795 10.0785 0786 0786 0787 0788 9.9215 9214 9214 9213 9212 9.921! 9210 9209 9209 9208 9.7419 742i 7423 7425 7427 10.2581 2579 2577 2575 10.2572 2570 2568 2566 2564 10.2562 2560 2558 2551 2549 2547 io72543~ 2541 2539 2538 2536 9.8208 8211 8213 8216 8219 10.1792 1789 1787 1784 1781 10.0789 0790 0791 0791 0792 9.7428 7430 7432 7434 743 6 9.8222 8224 8227 8230 8233 10.1778 1776 1773 1770 176-7 10.0793 0794 0795 0796 0796 9.9207 9206 9205 9204 9204 9-7438 7440 7442 7444 7445 9-8235 8238 8241 8243 8246 10.1765 1762 1757 1754 10.0797 0798 0799 0800 0801 10.0802 0802 0803 0804 0805 9.9203 9202 9201 9200 9199 9-7447 7449 7451 7453 7455 9.8249 8252 8254 8257 82*60 IO.I75J 1748 1746 1743 1740 9.9198 9198 9197 9196 9-7457 7459 7461 7462 7464 9.8263 8265 8268 8271 8274 10.1737 1735 1732 1729 1726 10.0806 0807 0807 0808 0809 9.9194 9193 9193 9192 9^190 9189 9188 9187 9187 9.9186 9.7466 7468 7470 7472 7474 9.7476 10.2534 2532 2530 2528 2526 10.2524 9.8276 8279 8282 8284 8287 9.8290 10.1724 1721 1718 1716 1713 10.1710 10.0810 0811 0812 0813 0813 10.0814 Cosec. ' m s Cos. O'.l 1 s Sec. Cot. O'.l Di Tan. Sin. m s Diff. 123 = 8 h 12 ] [ 3" 44 m = 56 TABLE IX. 61 34 rrr 2" 16 m s 16 O 4 8 12 16 16 2O 24 28 32 36 16 4O 44 48 52 56 17 4 8 12 16 17 2O 24 28 32 36 1740 44 48 52 56 18 4 8 12 16 182O 24 28 32 36 1840 44 48 52 56 19 4 8 12 16 1920 24 28 32 36 1940 44 48 52 56 20 "] " Sin. 9.7476 7477 7479 7481 7483 9-7485 7487 7489 749i 7492 I-OJf. MlM'S. Diff. Cosec. O'.l 1" | Tang Tan. 9.8290 8293 8295 8298 8301 eiits, an D1.T. O'.lll 8 id Sec; Cot. tints. Sec. 10.0814 0815 0816 0817 0818 [9" Cos. 4O' =* 145 m s i ' 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 / o 2 2 O O I 1 ; I 2 O 2 2 O 2 2 O O I I I 2 2 O 2 2 i i i i 2 I I 2 I I 2 I I 2 I 2 10.2524 2523 2521 25*9 25 r 7 o i i i i 2 2 2 2 O 2 2 2 2 O I 2 2 2 2 I I I 2 2 2 2 O 2. 2 2 2 I I I I 2 2 2 2 i 2 2 I 2 2 I 2 2 I 2 2 I 2 2 I 2 2 10.1710 1707 1705 1702 1699 9.9186 9185 9184 9183 9182 44 O 4356 52 48 4344 40 36 32 28 4324 20 16 12 8 43 4 43 4256 52 48 4244 40 36 32 28 4224 20 16 12 8 42 4 42 4156 52 48 4144 40 36 32 28 4124 20 16 12 8 41 4 41 4O56 52 48 4O44 40 36 32 28 4024 20 16 12 8 40 4 40 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 ; 27 ! 26 i 25 \ 24 23 ! 22 21 20 ; 19 18 17 16 15 i 14 ! 13 12 11 1O 9 8 7 : 6 5 4 3 2 1 10.2515 2513 2511 2509 2508 9-8303 8306 8309 8312 83H 10.1697 1694 1691 1688 1686 10.0819 0819 0820 0821 0822 9.9181 9181 9180 9179 9178 9-7494 7496 7498 7500 7502 10.2506 2504 2502 2500 2498 9-83 1 7 8320 8323 8325 8328 10.1683 1680 1677 1675 1672 10.0823 0824 0825 0825 0826 10.0827 0828 0829 0830 0831 9.9177 9176 9175 9175 9174 9-7504 7505 7507 7509 75" 9-7513 7515 75i7 75i8 7520 10.2496 2495 2493 2491 2 i 8 _9_ 10.2487 2485 2483 2482 2480 9-8331 8333 8336 8339 8342 10.1669 1667 1664 1661 1658 9-9173 9172 9171 9170 9169 9-8344 8347 8350 8352 8355 10.1656 1653 1650 1648 1645 10.0831 0832 o8J 3 0834 0835 9.9169 9168 9i67 9166 9165 9.7522 75 2 4 7526 7528 7529 10.2478 2476 2474 2472 2471 9.8358 8361 8363 8366 8369 10.1642 1639 1637 1634 1631 10.0836 0837 0837 0838 0839 9.9164 9163 9163 9162 9161 9-753 1 7533 7535 7537 7539 10.2469 2467 2465 2463 2461 9.8371 8374 8377 Us 10.1629 1626 1623 1621 1618 10.0840 0841 0842 0843 0844 9.9160 9159 9158 9157 9156 9-7540 7542 7544 7546 7548 10.2460 2458 2456 2454 2452 9 i 3 3 l 8390 8393 8396 10.1615 1612 1610 1607 1604 10.0844 0845 0846 0847 0848 9.9156 9155 9154 9153 9152 9-755 755i 7553 7555 7557 10.2450 2449 2447 2445 2443 9.8 39 8 8401 8404 8406 8409 10.1602 1599 1596 1594 I59i 10.0849 0850 0851 0851 0852 9-9I5 1 915 9149 9149 9148 9-7559 75 6 i 7562 7564 7566 10.2441 2439 2438 2436 2434 9.8412 8415 8417 8420 _8423_ 9.8425 8428 8431 8433 _84J6 9-8439 8442 8444 8447 8450 10.1588 1585 1583 1580 1577 10.0853 0854 0855 0856 0857 9.9147 9146 9H5 9144 9H3 9.7568 7570 757i 7573 7575 10.2432 2430 2429 2427 _?42i 10.2423 2421 2420 2428 2426 io.i575 1572 1569 1567 I5 6 4 10.0858 0858 0859 0860 0861 9.9142 9142 9141 9140 9139 0.9138 9137 9136 9135 9135 9-7577 7579 758o 7582 _7584 9-7586 10.1561 1558 1556 1553 *55Q 10.1548 10.0862 0863 0864 0865 0865 10.2424 i 9-8452 10.0866 9-9 r 34 ni s Cos. O'.l 1 s Sec. Cot. O'.l 1 s Tan. Cosec. Sin. m s | ' Diff. Diff. 124 = h 16 m ] [ 3 h 4O m = 55 62 TABLE IX. 35 J = 2 1 ' 2 m s m -j Sin. Log. Sines, Tangents, sind Secants. L9 Cos. 9-9I34 9133 9132 9I3 1 913 9.9129 9128 9127 9127 9126 ^9125 9124 9123 9122 9I2I 9.9120 9119 9119 9118 _ 9"7 9.9116 9"5 9114 9"3 9112 9.9111 9110 9110 9109 9108 36' r = 1 m s 10 3956 52 48 3911 10 36 32 28 39 21 20 16 12 8 39 1 39 38 56 52 18 3811 10 36 32 28 3821 20 16 12 8 38 1 38 O 3756 52 18 3711 10 36 32 28 3721 20 16 12 8 37 1 37 3656 52 18 3611 10 36 32 28 3621 20 16 12 8 36 1 36 l-l 3 / 60 59 58 57 56 55 51 53 52 51 50 19 18 17 16 15 44 43 42 41 10 39 38 37 36 35 31 33 32 31 30 29 | 28 27 26 25 21 ! 23 22 21 2O 19 18 17 16 15 11 13 12 11 10 9 8 7 6 5 1 3 2 1 Diff. Cosec. ' Tan. Diff. Cot. Sec. O'.l 1* o i i o I I I I I I I O'.l 1 2 2 2 2 J 2 2 I 2 I 2 2 I 2 2 f 1 2 3 1 5 6 7 8 9 10 11 12 13 11 15 16 17 18 19 20 21 22 23 21 25 26 27 28 29 30 31 32 33 31 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 51 55 56 57 58 59 60 2O O 1 8 12 16 20 20 21 28 32 36 2010 11 18 52 56 21 O 4 8 12 16 2120 21 28 32 36 2110 11 18 52 56 22 1 8 12 16 2220 21 28 32 36 22 1O 11 IN 52 56 23 4 8 12 16 2320 24 28 32 36 23 4O 44 48 52 56 24 9.7586 7588 7590 7591 7593 9-7595 7597 7599 7600 7602 977604" 7606 7607 7609 7611 o o 2 2 O 2 2 2 o 2. O 2 O 2 10.2414 2412 2410 2409 2407 10.2405 2403 2401 2400 _239S 10.2396 2394 2393 2391 2389 10.2387 2385 2384 2582 2380 10.237.* 2376 2375 2373 _._ 2 3Z1 10.2369 2368 2366 2364 2362 9.8452 8455 8458 8460 8463 978^66" 8468 8471 8474 8476 9.8479 8482 8484 8487 8490 o i i : i 2 I I I I 2 2 2 I 1 2 \ I I 2 2 2 2 I I I I 2 2 2 2 O I I I I 2 2 2 2 10.1548 1545 1542 1540 1537 i- I 534 1532 1529 1526 _J524 10.1521 1518 1516 1513 1510 10.0866 0867 0868 0869 0870 10.0871 0872 0873 0873 0874 10.0875 o8"6 0877 0878 0879 10.0880 0881 0881 0882 0883 10.0884 0885 0886 0887 0888 10.0889 0890 0890 0891 0892 9-76I3 76i5 7616 7618 702 "> 9-8493 ^495 8498 8501 8503 10.1507 1505 1502 1499 1497 9.7622 7624 7625 7627 7629 9.8506 8509 8511 8514 8517 9.8519 8522 8525 8527 8530 10.1494 1491 1489 1486 1483 10.1481 1478 1475 H73 1470 9-7631 7632 7634 _ll 9.7640 7641 7643 7645 7647 10.2360 2359 2357 2355 2353 9.8533 8535 8538 8541 8543 10.1467 1465 1462 H59 H57 10.0893 0894 0895 0896 0897 10.0898 0899 0899 0900 0901 9.9107 9106 9105 9104 9103 9.7648 7650 7652 7654 7 6 55 10.2352 2350 2348 2346 2345 9.8546 8549 8551 8554 8557 10.1454 H5i 1449 1446 1443 9.9102 9101 9101 9100 9099 9-7657 $? 7662 7664 10.2343 2341 2 339 233* 2336 1$ 8565 8567 8570 10.1441 H38 J 435 M33 1430 10.0902 0903 0904 0905 0906 9.9098 9097 9096 9095 9094 9.7666 7668 7669 7671 7673 10.2334 2332 2331 2329 2327 9.8573 ?sH 8581 8583 10.1427 1425 1422 1419 1417 10.0907 0908 0909 0909 0910 9.9093 9092 9091 9091 9090 9-7675 7676 7678 7680 7682 10.2325 2324 2322 2320 2318 9.8586 8589 8591 8594 8597 10.1414 1411 1409 1406 1403 10.0911 0912 0913 0914 0915 9.9089 9088 9087 9086 9085 9.7683 7685 7687 7689 7690 10.2317 2315 2 3'3 2311 2310 9.8599 8602 8605 8607 8610 10.1401 1398 1395 1393 1390 10.0916 0917 0918 0919 0920 9.9084 9083 9082 9081 9080 9.7692 10.2308 9.8613 10.1387 10.0920 9.9080 ' i m s Cos. O'.l I 8 Sec. Cot. O'.l 1 s Tan. Cosec. Sin. in 3 / Diff. Diff. 125 = 8 h 2O m ] [ 3 U 36 m = 54 TABLE IX. 36 = 2" 21 m s 210 4 8 12 16 2 1 21> 21 23 31 36 2140 11 4 if ** 2-> O 1 8 12 16 2520 21 23 32 36 25 40 44 48 52 56 26 4 8 12 16 26 2O 24 28 32 36 26 40 44 48 52 56 27 4 8 IS 16 27 20 24 28 32 36 274O 44 48 52 56 28 O > ] L,og. Sines, Tangents, and Secants. [ 9 h 32 m = 143 Difif. Sin - i oUT- Cosec. Tan. Di 0.1 i i i i 2 2 2 O I I I 2 2 ff. 1 s Cot. Sec. Cos. in s / O 1 2 3 4 5 6 7 i 9 10 11 12 13 14 15 16 17 18 19 20 21 ; 22 23 i 24 25 26 27 2 29 1 30 31 ! 32 ! 33 34 i35 36 37 38 39 4O 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 6O 9.7692 . 7694 o 7696 i o 7^97 7^99 9-77i 77^3 77 J 4 7700^ 7708 2 o i i o I o I I o I 10.2308 2306 2304 233 2301 10.2299 2297 229J 22)4 2292 9.8613 8615 8618 8621 8623 9.8620 8629 8631 8034 . 86 37 9-8339 8642 8644 8647 8050 9.8,552 8655 8658 8600 865 1 i 2 I I 2 I I I 2 I I 2 I I 2 10.1387 & 1379 1377 10.1374 1371 1369 1365 I3 6 4 10.0920 0921 0922. 0923 0924 10.0925 0926 0927 0928 0929 9.9080 9079 9078 9077 9076 9-9075 9074 9073 9072 97i_ 9.9070 9069 9069 9068 9067 9.9066 9065 9364 9003 9062 9.9061 9060 9059 9358 9057 36 O 35 56 52 48 3544 40 36 32 28 3524 20 16 12 8 35 4 35 O 3456 52 48 3144 40 36 32 28 3424 20 16 12 8 34 4 34 3356 52 48 3344 40 36 32 28 33 24 20 16 12 8 33 4 33 O 3256 52 48 3244 40 36 32 28 3224 20 16 12 8 32 4 32 60 59 58 57 -:> 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 , 27 26 25 24 23 22 21 20 19 18 i 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 O 0.7710 7711 7/13 7715 77^ 9.7718 7723 7722 7723 7711 9-77^7 7728 773 7732 7734 o 2 O o 2 O o 2 O 2 O I0.229J 2289 2287 22S5 2284 10.2282 2280 2278 2277 2275 10.1361 1358 1356 {353 1350 10.1348 1345 1342 1340 1337 10.0930 0931 093 i 0932 09^ 10.0934 935 0936 0937 0938 10.2273 2272 227O 2263 2266 9.866J j 8568 i o 8671' 8674 : 8675 1 !o.i334 1332 1329 1326 1324 10.0939 0940 0941 0942 0943 9-7735 7737 7739 7740 7742 9-7744 774^ 7747 7749 775] 10.2265 2263 2261 2260 2258 10.2256 2254 2253 2251 2249 9.8679 8682 .8684 8687 8589 9.8692 85 95 8697 8700 8703 2 2 2 O 2 2 2 2 O 2 2 2 10.1321 1318 1316 1313 13" 10.1308 1305 1303 1300 1297 10.0944 0944 0945 0946 0947 10.0948 0949 0950 095 i 0952 9.9056 9056 9055 9054 9053 9.9052 9051 9050 9049 _994& 9.9047 9046 9045 9044 9043 9.9042 9041 9041 9040 9039 9-7752 7754 775 6 7758 7759 10.2248 2246 2244 2242 2241 9.8705 8708 8711 8713 8716 10.1295 1292 1289 1287 1284 10.0953 0954 0955 0956 0957 9.7751 77^3 7754 7766 7788 9.7769 7771 7773 7774 7776 10.2239 2237 2236 2234 2232 9.8718 8721 8724 8726 8729 9.8732 8734 8737 8740 8742 10.1282 1279 1276 1274 1271 10.0958 0959 0959 0960 0961 10.223-1 2229 2227 2226 2224 10.1268 1266 1263 1260 1258 10.1255 1253 1250 1247 1245 10.0962 0963 0964 0965 0966 9.9038 9037 9036 9035 934 9.7778 7780 7781 7783 7785 10.2222 . 2220 2219 2217 2215 9-8745 8747 8750 8753 8755 10.0967 0968 0969 0970 0971 9-9033 9032 9031 9030 9029 9.9028 9027 9026 9025 9024 9-7786 7788; 7790! 779i | 7793 ! 2 IO.22I4 2212 2210 22O9 22O7 IO.22O5 9.8758 8761 8763 8766 8769 10.1242 1239 1237 !234 1231 10.0972 0973 0974 0975 0976 9-7795 i 9.8771 10.1229 10.0977 9.9023 / in s Cos. 0.1 1 s r- Bso - Cot. O'.l 1 s Tan. Cossc. Sin. m s / Di Difif. ! 126 = 8 1 ' 24 m ] [ 3 b 32 m = 53 i 64 TABLE IX. 37 D = 2 ' 28 u ] JLogr. Sines, Tangents, smtl Secants. [ 9 1 ' 28 m = 142 ' m s Diff. Sin. O'.l 1 Cosec. Tan. Diif. O'.'l 1 I i 2 i 2 2 ! I I I I 2 I I I 2 2 2 O I I I 2 4 2 I I i I 2 I I 2 2 2 Oil I 2 1 2 2 \ 2 I I 2 2 2 2 Cot. Sec. Cos. S j 9.9323! 9 3.: 1 1 932 >1 9318 9317 9016 9015 9-9314; 9312 9011 9010 9.9337 9008 9307- 9306 9005 ni s ; 3-2 <& . 31 33 5 49 31 11 JIO 32 28 3124 16 12 8 31 4 31 :^ ; >| 48 3O44 40 36 32 28 30 24 20 16 12 8 ' 3O 4 30 2956 52 48 2944 4O 36 32 28 29 24 20 16 12 8 29 4 29 O 28 56 52 48 2844 4O 36 32 28 2W 2 1 20 16 12 8 28 4 28 O 60 39 57 53 51 53 52 51 50 49 48 47 16 45 44 43 42 41 40 39 38 37 36 35 34 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 21 25 26 27 '28 29 30 31 32 33 34 35 36 i 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 28 4 8 12 16 28 2O 21 28 32 36 9-7795 7790 7793 7833 7831 O I I 2 I J ' I I I I i 2 j o i i I i 2 ' O ; O I I j I I j I | I I I 2 O I 2 j O I I I I I I I 2 10.2205 2204 2202 2203 2 I Q 1 9.8771 8774 8770 8779 878-5 10.1229 122 ) 1224 122( 1218 10.0977 0977 0970 9.7803 7805 7800 7808 7810 10.2197 2195 2194 2192 2191 9.8784 8787 8793 8792 8795 9.879; 8333 8833 8805 8338 1O.I2IO 1213 I2IO I2O8 1 20", 10.1203 I2O3 "97 "95 1192 10.0961 0982 0983 0984 0985 2840 44 48 52 56 29 4 8 12 16 29 2O 24 28 32 36 2940 41 48. 52 53 3$ O 4 8 12 16 3O2O 24 28 32 36 3O4O 44 48 52 56 31 O 4 8 12 16 3120 24 28 32 36 3140 44 48 52 56 32 9.7811 7813 7815 7810 7818 9.7820 7821 7823 7825 7820 I0.2I89 2187 2185 2184 2182 10.0980 0987 0988 0989 0990 10.2180 2179 217; 2175 9.88II 8813 8810 8318 8821 9.8.324 8826 8329 8831 8834 10.1189 1187 1184 1182 ii79 10.1170 "74 1169 1166 10.0991 0992 0993 0994 0991 10.0990 0997 0998 0999 IO33 10.1000 IOOI IOO2 1003 103-1- 10.1005 IOOO 1007 1008 1009 9.7828 7830 7831 7833 7835 10.2172 2170 2169 2167 2165 10.2164 2162 2160 2159 2157 10.2150 2154 2152 2151 2149 9.9004 9033 9032 9001 9000 9.9000 8999 8998 8997 8996 9.8995 8994 8993 8992 8991 9.7830 7838 7840 7841 7_^43 9.7844 7840 7848 7849 7851 9.8837 8839 8842 8^45 8847 9.8850 8852 8855 8858 8860 10.1163 1161 1158 "55 "53 10.1150 1148 "45 1 142 1140 9 '7JJ54 7858 7859 10.2147 2146 2144 2142 2141 8868 8871 8873 10.1137 "35 "32 1129 1127 IO.IOIO KM I IOI2 1013 IOI4 9.8990 8989 8988 8987 8986 9.7861 7803 7804 7855 7867 10.2139 2137 2134 9.8876 8879 8881 8884 8886 9.8889 8892 8894 8897 8899 9.8902 8905 8907 8910 8912 10.1124 II2I III9 1116 1114 IO.IOI5 1016 1017 1018 IOIQ 9.8985 8984 8983 8982 8981 9.7869 7871 7872 7874 7870 10.2131 2129 2128 2126 2124 10.2123 2121 2120 2118 2116 IO.IIII 1108 1106 1103 IIOI IO.IO2O IO2I IO22 1023 1024 9.8980 8979 8978 8977 8976 9-7877 7879 7880 7882 7880 10.1098 1095 1093 IO.IO25 1026 IO27 1028 1029 9-8975 8974 8973 8972 8971 9.7885 7887 7889 7890 7892 10.2115 2113 2III 2IIO 2108 9.8915 8918 8920 8923 8925 10.1085 1082 1080 1077 1075 10.1030 1031 IO32 1034 9.8970 8969 8968 8967 8966 9-7893 Cos. 10.2107 9.8928 10.1072 IO.IO35 9.8965 m s O'.l 1" Sec. Cot. O'.l 1 s Diff Tan. CDSCC. Sin. m s ' Diff. 1 2 1 ?' K'> Q I " 1 f 3 ' 28 ! " K 4T V * 9 | L 52 TABLE IX. 65 38 2" 32 >u j Log. Sines, Tangents, and Secants. [ 9 h 24'" = 141 | ' m B Sin. Diff. Coaec. Tan. Dii O'.l f. Cot. Sec. 10.1035 1036 1037 1038 10.1040 1041 1042 1043 1044 Cos. m s ' O'.l 1 s O 1 2 3 4 5 6 8 9 1O 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 i 38 39 40 141 |42 43 44 45 46 47 48 49 5O 51 52 53 54 55 56 57 58 59 60 32 O 4 8 12 16 3220 24 28 32 36 32 4O 44 48' 52 56 33 4 8 12 16 33 2O 24 28 32 36 33 4O 44 48 52 56 34 O 4 8 12 16 3420 24 28 32 36 34 4O 44 48 52 56 35 4 8 12 16 3520 24 28 32 36 3540 44 48 52 56 36 O 9.7893 7895 7897 7898 9.7901 7903 7905 7906 7908 9.7910 7911 7913 79*4 "9^7918" 7919 7921 7922 7924 9.7926 7927 7929 793 9-7934 7935 7937 7938 7940 o 2 O O 2 2 O o 2 O 2 O O I 2 o i i o I I o I I o o I 10.2107 2105 2103 2102 2IOO 10.2099 2097 2095 2094 2092 9.8928 8931 8933 8936 9.8941 8944 8946 8949 8952 o 2 2 2 O I I I I I 2 2 2 O 2 2 O 2 2 2 O 2 2 2 O 2 2 2 i I 2 2 I I 2 I I 2 2 I I 2 10.1072 1069 1067 1064 1061 10.1059 1056 1054 1051 1048 9.8965 8964 8963 8962 8961 9.8960 8959 8958 f 95 ! 8956 9-8955 8954 8953 8952 8951 28 j 27 56 52 48 27 44 to 36 32 28 2724 2O 16 12 8 27 4 27 2656 52 48 2644 40 36 32 28 2624 2O 16 12 8 26 4 26 25 56 52 48 2544 40 36 32 28 2524 20 16 12 8 25 4 25 24 56 52 48 2444 40 36 32 28 2424 2O 16 12 8 24 4 24 111 S 6O 59 58 57 56 55 54 53 52 51 50 \ 49 48 \ 47 i 46 i 45 i 44 43 42 41 40 39 38 37 36 i 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 I 18 17 16 15 14 13 12 I 11 10 9 8 7 6 5 I \ 3 2 1 . 10.2090 2089 2087 2086 2084 10.2082 2081 2079 2078 2076 9-8954 8957 8959 8962 8965 10.1046 1043 1041 1038 10.1033 1030 1028 1025 1022 IO.IO2O IOI7 1015 IOI2 IOIO 10.1007 IOO4 IOO2 0999 10.0994 0991 0989 0986 0984 10.0981 0978 0976 0973 0971 10.1045 1046 1047 1048 1049 9.8967 8970 8972 8975 8978 9.8980 8983 8985 8988 __99_o 9.8993 8996 8998 9001 9003 10.1050 1051 1052 1053 _154 10.1055 1056 1057 1058 _ I0 59_ i o.i 060 1061 1062 1063 1064 9.8950 8949 8948 8947 8945 9.8945 8944 8943 8941 9.8940 8939 8938 8937 8930 IO.2O74 2073 2071 2O7O 2068 IO.2O66 2065 2063 2062 2O6O 9.7941 7943 7945 7946 9-7949 795 i 7953 7954 795 6 IO.2O59 2057 2055 2054 2052 10.2051 2049 2047 2046 2044 9.9006 9009 9011 9014 9016 9.9019 9022 9024 9027 9029 10.1065 1066 1067 1068 1069 10.1070 1071 1072 1073 1074 9-8935 8934 8933 _831 9.8930 8929 8928 8927 8926 9-7957 7959 7960 7962 7964 10.2043 2041 2040 2038 2036 9.9032 9035 937 9040 9042 10.0968 0965 0963 0960 0958 10.1075 1076 1077 1078 1079 9.8925 8924 8923 8922 8921 0.7965 7967 7968 7970 7972 10.2035 2033 2032 2030 2028 9.9045 9047 9050 9053 9055 10.0955 0953 0950 0947 0945 10.1080 1081 1082 1083 1084 9.8920 8919 8918 8917 8916 9.8915 8914 8913 8912 8911 9-7973 7975 7976 7978 7979 IO.2O27 2025 2O24 2022 2O2I 9.9058 9060 9063 9066 9068 10.0942 0940 937 0934 0932 10.1085 1086 1087 1088 1089 9-798i 7982 7984 7986 7987 IO.2OI9 20l8 2Ol6 2OI4 2013 IO.2OII 9.9071 9073 9076 9079 9081 10.0929 0927 0924 0921 0919 10.1090 1091 1092 1093 1094 9.8910 8909 8908 8907 8906 9.7989 9.9084 10.0916 10.1095 9.8905 ' m B Cos. O'.l Sec. Cot. O'.l 1 s Tan. Cosec. Sin. / Diff. Diff. 128 = 8 h 32 m ] [ 3 h 24 m = 51 j 66 TABLE IX. 39 D O 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 ; 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 = 2 h 36] Log. Si nos, Tangents, and Secants. [9 11 2O m = 1 1O m 3 Sin. Diflf. Cosec. Tan. Diflf. Cot. I Sec. | Cos. 1 m s O'.l I 8 O'.l I 8 36 O 4 8 12 16 36 2O 24 28 32 36 3640 44 48 52 56 37 O 4 8 12 16 3720 24 28 32 36 3740 -44 48 52 56 38 4 8 12 16 3820 24 28 32 36 3840 44 48 52 56 39 4 8 12 16 39 2O 24 28 32 36 39 4O 44 48 52 56 40 9.7989 7990 7992 7993 7995 o o I I I I I I o o o o I o o I o o I o o I o I I o I I , I o I I I I I I IO.2OII 2OIO 2008 2OO7 2OO5 9.9084 9086 9089 9091 9094 o I I I I 2 2 2 O 1 I 2 2 2 O O I I I 2 2 O O I I 2 O O I I 2 2 2 O O I 2 2 2 i i 2 I jj i 2 I I 2 I I 2 10.0916 0914 0911 0909 0906 10.1095 1096 1097 1098 1099 9.8905 8904 8903 8902 8901 24 23 56 52 48 2344 40 36 32 28 2324 20 . 16 12 8 23 4 23 O 22 56 52 48 2244 40 36 32 28 2224 20 16 12 8 22 4 22 2156 52 48 2144 40 36 32 28 21 24 06 16 12 8 21 4 21 O 2056 52 48 2044 40 36 32 28 2024 20 16 12 8 20 4 20 60 59 58 57 56 55 54 53 52 1 31 5O 49 48 47 46 45 44 43 42 41 40 39 38 i 37 36 35 34 33 32 31 3O 29 28 1 27 ! 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 9.7997 7998 8000 8001 8003 10.2003 2002 2OOO 1999 1997 9.9097 9099 9102 9104 9107 10.0903 0901 0898 0896 0893 10. 1 100 IIOI IIO2 1103 1104 9.8900 8899 8898 8897 8896 9:8895 8894 8893 8892 8891 9.8004 8006 8007 8009 8010 10.1996 1994 1993 1991 1990 9.9110 9112 9H5 9117 9120 10.0890 0888 0885 0883 0880 10.1105 1106 1107 1 105 1109 9.8012 8014 8015 8017 8018 10.1988 1986 1985 1983 1982 9.9122 9125 9128 9130 9133 10.0878 0875 0872 0870 0867 IO.IIIO IIII 1112 III 3 ."15 9.8890 8889 8888 8887 8885 9.8020 8021 8023 8024 8026 10.1980 1979 1977 I 97 6 J 974 9-9135 9138 9140 9H3 9146 10.0865 0862 0860 0857 0854 IO.III6 III7 1118 1119 II2O q.888 4 " 8883 8882 8881 8880 9.8027 8029 8031 8032 8034 10.1973 1971 1969 1968 1966 9.9148 9151 9153 9150 9158 10.0852 0849 0847 0844 0842 IO.II2I 1 122 II2 3 II2 4 "25 9.8879 8878 8877 8876 8875 9-8035 8037 8038 8040 8041 10.1965 1963 1962 1960 1959 9.9161 9164 9166 9169 9171 10.0839 0836 0834 0831 ' 0829 IO.II26 1127 1128 1129 1130 9.8874 8873 8872 8871 8870 9.8043 8044 8046 8047 8049 10.1957 1956 1954 1953 I95 1 9.9174 9176 9179 9182 9184 10.0826 0824 0821 0818 0816 IO.II3I II 3 2 "33 "34 "35 9.8869 8868 8867 8866 8865 9.8050 8052 8053 8055 8056 10.1950 1948 1947 1945 1944 9.9187 9189 9192 9194 9197 10.0813 0811 0808 0806 0803 10.1136 "37 1138 1140 1141 9.8864 8863 8862 8860 8859 9.8058 8060 8061 8063 8064 10.1942 1940 '939 1937 1936 9.9200 9202 9205 .9207 9210 10.0800 0798 0795 0793 0790 10.1142 "43 "44 "45 1146 9.8858 8857 8856 8855 8854 9.8066 8067 8069 8070 8072 9.8073 8075 8076 8078 8079 10.1934 1933 I93i 1930 1928 9.9212 9215 9218 9220 9223 9.9225 9228 9230 9233 9236 9.9238 10.0788 0785 0782 0780 0777 10.1147 1148 "49 1150 1151 10.1152 "53 "54 "55 1156 9-8853 8852 8851 8850 8849 10.1927 1925 1924 1922 1921 10.0775 0772 0770 0767 0764 9.8848 8847 8846 8845 8844 9.8843 9.8081 10.1919 10.0762 0.1157 / m B Cos. O'.l 1" Sec. Cot. O'.l I 8 Tan. Cosec. Sin. in s / Diflf. Diflf. 129'= 8 h 36 m ] [ 8 h 2O' n = 5O TABLE IX. 67 1O - 2 h 4O m ] L.ogr. Sines, Tangents, and Secants. [ 9 h 16 m = 139 / m s Sin. Diff. Cosec. Tan. Di CK1 S. Cot. Sec. Cos. m s / O'.l I 8 o I o I I I o I I o I I I I 1 s 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2O 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 144 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 / 4O O 4 8 12 16 4O2O 24 - 28 32 36 4040 44 48 52 56 41 4 8 12 16 41 2O 24 28 32 36 4140 44 48 52 56 42 4 8 12 16 4220 24 28 32 36 4240 44 48 52 56 43 4 8 12 16 43 2O 24 28 32 36 4340 44 48 52 56 44 9.8081 8082 8084 8085 8087 o o o I o o o I o o I o 10.1919 1918 1916 1915 1913 9.9238 9241 9243 9246 9248 o o i i i i 2 2 2 O I I I I 2 2 2 O I I 2 2 2 O I I I I 2 2 2 O I I I I 2 2 2 O I I I I 2 2 2 i 2 I 2 I I 2 I 2 2 I 2 10.0762 0759 0757 0754 0752 10.1157 H59 1160 1161 1162 9:8843 8841 8840 8839 8838 "9^8837 8836 8835 8834 8833 20 1956 52 48 1944 40 36 32 28 1924 20 16 12 8 19 4 19 1856 52 48 1844 4O 36 32 28 1824 20 16 12 8 18 4 18 1756 52 48 1744 40 36 32 28 1724 20 16 12 8 17 4 17 1656 52 48 1644 40 36 32 28 1624 20 16 12 8 16 4 16 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 9.8088 8090 8091 8093 8094 10.1912 1910 1909 1907 1906 9.9251 9254 9256 9259 9261 10.0749 0746 0744 0741 Q739 10.0736 0734 0731 0729 0726 10.1163 1164 1165 1166 1167 9.8096 8097 8099 8100 8102 10.1904 1903 1901 1900 1898 9.9264 9266 9269 9271 9274 10.1168 1169 1170 1171 1172 9.8832 8831 8830 8829 8828 9.8103 8105 8106 8108 8109 10.1897 1895 1894 1892 18 9 i 9.9277 9279 9282 9284 9287 10.0723 0721 0718 0716 0713 10.1173 H75 1176 1177 1178 9.8827 8825 8824 8823 8822 9.8111 8112 8114 8115 8117 10.1889 1888 1886 1885 1883 9.9289 9292 9295 9297 9300 10.0711 0708 0705 0703 0700 10.1179 1180 1181 1182 1183 9.8821 8820 8819 8818 8817 9.8118 8120 '8121 8122 8124 10.1882 1880 1879 1878 1876 9.9302 9305 93 7 9310 9312 10.0698 0695 0693 0690 0688 10.1184 1185 1186 1187 1188 9.8816 8815 8814 8813 8812 9.8125 8127 8128 8130 8131 10.1875 1873 1872 1870 1869 9-93I5 93i8 9320 9323 9325 10.0685 0682 0680 0677 0675 10.1190 1191 1192 "93 "94 9.8810 8809 8808 8807 8806 9.8i33 8134 8136 8i37 8i39 10.1867 1866 1864 1863 1861 9-9328 9330 9333 9335 9338 9-9341 9343 9346 9348 935i 10.0672 0670 0667 0665 0662 10.1195 1196 1197 1198 1199 9.8805 8804 8803 8802 8801 9.8140 8142 8143 8i45 8146 I O.I 860 1858 1857 1855 1854 10.0659 0657 0654 0652 0649 10.1200 1201 1203 1204 I2O5 9.8800 8799 8797 8796 8795 9.8794 8793 8792 8791 8790 9.8148 8149 8150 8152 8i53 10.1852 1851 1850 1848 1847 9-9353 9356 9358 9361 9364 10.0647 0644 0642 <*>39 0636 10.1206 1207 1208 1209 I2IO 9.8i55 8156 8158 8159 8161 10.1845 1844 1842 1841 1839 9.9366 9369 937i 9374 9376 10.0634 0631 0629 0626 0624 IO.I2II 1212 1213 1215 1216 9.8789 8788 8787 8785 8784 9.8162 8164 8165 8167 8168 10.1838 1836 1835 1833 1832 9-9379 938i 9384 9387 9389 10.0621 0619 0616 0613 . 0611 10.1217 1218 1219 1220 1221 9.8783 8782 8781 8780 8779 9.8169 10.1831 9.9392 10.0608 10.1222 9.8778 m a Cos. O'.l 1 s Sec. Cot. O'.l 1 s Tan. Cosec. Sin. m s / Diff. Diff. 13O = 8 h 4O m ] [ 3 h 16 m = 49 TABLE IX. 41 = 2 h 44 m ] Log. Sines, Tangents, and Secants. [ 9 h 12 I1! if 138= / in s Sin. Diff. Cosec. Tan. Diff. Cot. Sec. Cos. Ill S 6O 59 58 57 56 55 54 53 52 51 50 49 48 : 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 ; 28 | 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 i O'.l I 8 0.1 1* 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 i 57 1 58 { 59 60 44 4 8 12 16 4420 24 28 32 36 4440 44 48 52 56 45 4 8 12 16 4520 24 28 32 36 4540 44 48 52 56 46 4 8 12 16 4620 24 28 32 36 4640 44 48 52 56 47 4 8 12 16 4720 24 28 32 36 4740 44 48 52 56 48 9.8169 8171 8172 8174 8i75 o o I o o o o o o I o o o o 1 I o I I o I I o I I I I o I 10.1831 1829 1828 1826 1825 9.9392 9394 9397 9399 9402 o o I 2 2 2 O O I I I 2 2 2 O I I 2 2 2 O O I I I 2 2 2 O I I I 2 2 2 O 2 2 2 2 I 2 I I 2 I I 2 I I 2 I I 2 10.0608 0606 0603 0601 0598 IO.I222 I22 3 1224 1225 1227 9.8778 8/77 8776 8775 8773 9.3772 8771 8770 8769 8768 16 O 1556 52 48 1544 40 36 32 28 1524 20 16 12 8 15 4 15 1456 52 48 1444 40 36 32 28 1424 20 16 12 8 4 14 O 1356 52 48 1344 40 36 32 28 1324 20 16 12 8 13 4 13 1256 52 48 1244 40 36 32 28 1224 2O 16 12 8 12 4 12 9.8177 8178 8180 8181 8182 10.1823 1822 1820 1819 1818 9.9404 9407 9409 9412 9415 10.0596 0593 0591 0588 0585 IO.I228 1229 1230 I2 3 I I2 3 2 9.8184 8185 8187 8188 8190 10.1816 1815 1813 1812 1810 9-94I7 9420 9422 9425 9427 10.0583 0580 0578 0575 0573 IO.I233 1234 1235 1237 1238 9.8767 8766 8765 8763 8762 9.8191 8i93 8194 8i95 8i97 10.1809 1807 1806 1805 1803 9.9430 9432 9435 943 s 944 10.0570 0568 0565 0562 0560 10.1239 I24O 1241 1242 1243 9.8761 8760 8759 8758 8757 9.8198 8200 8201 8203 8204 10.1802 1800 1799 1797 1796 9-9443 9445 9448 9450 9453 10-0557 555 0552 0550 0547 IO.I244 1245 1247 1248 1249 9.8756 8755 8753 8752 875i 9.8205 8207 8208 8210 8211 10.1795 *793 1792 1790 1789 9-9455 9458 9460 9463 9466 10.0545 0542 0540 0537 0534 IO.I25O 1251 1252 1253 - 1254 9.8750 8749 8 74 8747 8746 9.8745 8743 8742 8741 8740 9.8213 8214 8216 8217 8218 10.1787 1786 1784 17.83 1782 9.9468 947i 9473 9476 9478 10.0532 0529 0527 0524 0522 10.1255 1257 1258 1259 I26O 9.8220 8221 8223 8224 8225 10.1780 1779 1777 1776 1775 9.9481 9483 9486 9488 9491 10.0519 0517 05H 0512 0509 IO.I26I 1262 1263 1264 1266 9 ff* tg 8734 9.8227 8228 8230 8231 8233 10.1773 1772 1770 1769 1767 9-9494 9496 9499 95oi 954 10.0506 0504 0501 0499 0496 10.0494 0491 0489 0486 0484 10.1267 1268 1269 I27O 1271 IO.I272 1273 1275 1276 1277 9-8733 8732 8731 8730 8729 9.8234 8235 8237 8238 8240 10.1766 1765 1763 1762 1760 9.9506 959 95" 95 H 95i6 9.8728 8727 8725 8724 8723 9.8241 8242 8244 8245 8247 10.1759 1758 1756 1755 1753 9-95 J 9 9522 9524 9527 9529 10.0481 0478 0476 0473 0471 IO.I278 1279 1280 I28l 1282 9.8722 8721 8720 8719 8718 9.8248 8249 8251 8252 8254 10.1752 I75i 1749 1748 1746 9-9532 9534 9537 9539 9542 10.0468 0466 0463 0461 0458 10.1284 1285 1286 1287 1288 9.8716 8715 8714 8713 8712 9-8255 10.1745 9-9544 10.0456 10.1289 9.8711 / m a Cos. O'.l I 8 Sec. Cot. O'.l 1" Tan. Cosec. Sin. m a / Diff. Diflf. 131 = 8 h 44 m ] [8 h ia m = 48 3 TABLE IX. 69 42 = 2 h 48 m ] Log. 8tne$, Tangents, and Secants. [ 9 h 8 -- 137 ! ni s Sin. Diflf. Cosec. Tan. Diff. Cot. Sec. Cos. HI S / O'.l o I 1 o o I o o I I I I I I o o I o o I o o 1 s O'.l I 8 i i 2 ' 2 ' I 2 I 2 I 2 1 I 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 3O 31 32 33 34 ' 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 {51 52 53 154 55 56 57 58 59 60 48 4 8 12 16 48 2O 24 28 32 36 4840 44 48 52 56 49 O 4 8 12 16 4920 24 28 32 36 4940 44 48 52 56 5O O 4 8 12 16 50 20 24 28 32 36 5O4O 44 48 52 56 51 4 8 12 16 5120 24 28 32 36 5140 44 48 52 56 52 O 9-8255 8257 8258 8259 8261 o o I o I o I 10.1745 1743 1742 1741 1739 9-9544 9547 9549 9552 9555 o I I I 2 2 2 O o I I I I 2 2 2 O I I I I 2 2 2 O I I I I 2 2 2 O I I I I 2 2 2 O I I I 2 2 2 10.0456 453 045 J 0448 0445 10.1289 1290 1292 1293 1294 9.8711 8710 8708 8707 8706 12 1156 52 48 1144 40 36 32 28 1124 20 16 12 8 11 4 11 1056 52 48 1044 40 36 32 28 1024 20 16 12 8 10 4 10 956 52 48 944 40 36 32 28 924 20 16 12 8 9 4 9 856 52 48 844 40 36 32 28 824 20 16 12 8 8 4 8 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 9.8262 8264 8265 8266 8268 10.1738 1736 1735 1734 1732 9-9557 9560 9562 95 6 5 _9_567_ 9-9570 9572 9575 9577 9580 10.0443 0440 0438 0435 0433 10.1295 1296 1297 1298 1300 9.8705 8704 8703 8702 8700 9.8269 8270 8272 8273 8275 10.1731 1730 1728 1727 1725 10.0430 0428 0425 0423 0420 10.1301 1302 I 33 i34 1305 9.8699 8698 8697 8696 8695 9.8276 8277 8279 8280 8282 10.1724 1723 1721 1720 1718 9.9582 9585 9588 959 9593 10.0418 0415 0412 0410 0407 10.1306 1308 1309 1310 1311 10.1312 1313 I3H 1316 I3 1 ; 9.8694 8692 8691 9.8283 8284 8286 8287 8289 10.1717 1716 1714 1713 1711 9-9595 9598 9600 9603 9605 10.0405 0402 0400 0397 0395 9.8688 868; 8686 8684 8683 9.8290 8291 8293 8294- 8295 9.8297 8298 8300 8301 8302 10.1710 1709 1707 1706 1705 9.9608 9610 9613 9615 9618 10.0392 0390 0387 0385 _032_ 10.0379 0377 0374 0372 0369 10.1318 I3 J 9 1320 1321 1323 9.8682 8681 8680 8679 8677 10.1703 1702 1700 1699 1698 9.9621 9623 9626 9628 9631 10.1324 1325 1326 1327 1328 9.8676 8675 8674 8673 8672 9.8304 lol 8308 8309 10.1696 1695 1694 1692 1691 9-9633 9636 9638 9641 ' 9 6 43 10.0367 0364 0362 359 0357 10.1329 i33i '33 2 '333 *334 K^SSS 1336 1338 '339 *340 9.8671 8669 8668 8667 8666 9.8311 8312 8313 8315 8316 10.1689 1688 1687 1685 1684 9.9646 9648 9651 9 ^ 9656 10.0354 035 2 0349 0347 0344 9.8665 8664 8662 8661 8660 9-83 1 7 8319 8320 8322 8323 10.1683 1681 1680 1678 1677 9.9659 9661 9664 9666 9669 10.0341 0339 033 6 334 033 1 10.1341 *342 *343 '345 1346 9.8659 8658 8657 8655 8654 9.8324 8326 8327 8328 8330 9-8331 8332 f334 8335 8336 10.1676 1674 1673 1672 1670 9.9671 9674 9676 9679 9681 10.0329 0326 0324 0321 0319 10.1347 1348 1349 !35o J 35 2 9.8653 8652 8651 8650 8648 10.1669 1668 1666 1665 1664 9.9684 9689 9691 9694 10.0316 3 J 4 0311 0309 0306 10-1353 1354 1355 1356 1358 9.8647 8646 8645 8644 8642 9-8338 10.1662 9.9697 10.0303 Jo.1359 9.8641 ' m s Cos. O'.l I 8 Sec. Cot. (X.I 1 s Tan. Cosec. Sin. m s i Diff. Diff. 132 = 8 h 48 m ] [ 311 gm _ 470 | 70 TABLE IX 43 = 2 b 52" ] Log. Sines, Tangents, and Secants. [ 9 h 4 = 136 i m a Sin. Diff. Cosec. Tan. Diff. Cot. Sec. Cos. m B ' O'.l 1" o I I o I o I I o I I o I I I I O'.l 1" 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 J28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 52 4 8 12 16 5220 24 28 32 36 5240 44 48 52 56 53 4 8 12 16 5320 24 28 32 36 5340 44 48 52 56 54 4 8 12 16 5420 24 28 32 36 5440 44 48 52 56 55 4 8 12 16 5520 24 28 32 36 5540 44 48 52 56 56 O 9-8338 8339 8341 8342 8343 o o o o o o o o o o o o o o 10.1662 1661 1659 1658 1657 10.1655 1654 1653 1651 1650 9.9697 9699 9702 9704 9707 o o i i i i 2 2 2 O O I I I I 2 2 2 O I I I I 2 2 2 O O I I I I 2 2 2 O I I I I 2 2 2 O I I I 2 2 2 I i 2 2 I I 2 I I 2 I 2 I I 2 ~l-~ 10.0303 0301 0298 0296 0293 10.0291 0288 0286 0283 0281 10.1359 1360 1361 1362 1363 9.8641 8640 8639 8638 8637 8 756 52 48 744 40 36 32 28 724 20 16 12 8 7 4 7 656 52 48 644 40 36 32 28 624 20 16 12 8 6 4 6 556 52 48 5 21 40 36 32 28 524 20 16 12 8 5 4 5 456 52 48 444 40 36 32 28 424 2O 16 12 8 4 4 4 O 60 59 I 5N i 57 56 55 54 53 52 51 50 49 48 ! 47 46 45 44| 43 42 i 41 40 39 38 37 ! 36 ; 35 34 33 32 31 3O 29 28 27 26 25 24 23 22 21 20 19 18 i 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 9-8345 8346 8347 8349 8350 9.9709 9712 97U 9717 97*9 10.1365 1366 1367 1368 1369 9-8635 8634 8633 8632 8631 9.f35i 8353 8354 f355 8357 10.1649 1647 1646 1645 1643 9.9722 9724 9727 9729 9732 10.0278 0276 0273 0271 0268 10.1371 1372 1373 1374 1375 9.8629 8628 8627 8626 8625 9-8358 8359 8361 8362 8363 9-8365 8366 8367 8369 8370 10.1642 1641 1639 1638 1637 10.1635 J 634 1633 1631 1630 9-9735 9737 9740 9742 9745 10.0265 0263 0260 0258 0255 10.1376 1378 1379 1380 1381 10.1382 1384 1385 1386 1387 9.8624 8622 8621 8620 8619 9-9747 975 9752 9755 9757 10.0253 0250 0248 0245 0243 9.8618 8616 8615 8614 8613 9-837I 8373 8374 8375 8377 10.1629 1627 1626 1625 1623 9.9760 9762 97 6 5 9767 9770 10.0240 0238 0235 0233 0230 10.1388 1390 i39i 1392 1393 9.8612 8610 8607 9-8378 8379 8381 8382 8383 10.1622 1621 1619 1618 1617 9-9773 9775 9778 9780 9783 10.0227 0225 O222 0220 0217 10.1394 1396 1397 1398 1399 9.8606 8604 8603 8602 8601 9-8385 8386 8387 8389 8390 10.1615 1614 1613 1611 1610 9-9785 9788 9790 9793 9795 IO.O2I5 O2 12 0210 0207 O2O5 10.1400 1402 i43 1404 1405 9.8600 8598 8597 8596 8595 9-839I 8393 8394 8395 8397 10.1609 1607 1606 1605 1603 9.9798 9800 9803 9805 9808 9.9810 9813 9816 9818 9821 IO.O2O2 O2OO 0197 0195 0192 10.1406 1408 1409 1410 1411 9-8594 8592 8591 8590 8589 "9:8588 8586 8585 8584 8583 9-8398 8399 8401 8402 8403 10.1602 1601 1599 1598 1597 10.0190 0187 0184 0182 0179 10.1412 1414 1415 1416 1417 9.8405 8406 8407 8409 8410 10.1595 1594 1593 1591 !59<> 10.1589 1588 1586 1585 1584 9.9823 9826 9828 9831 9833 IO.OI77 0174 0172 0169 0167 10.1418 1420 1421 1422 1423 9-8582, 8580 8579 8578 8577 9.8411 8412 8414 8415 8416 9.9836 9838 9841 9843 9846 IO.OI64 Ol62 0150 0157 0154 10.1425 1426 1427 1428 1429 9-8575 8574 8573 8572 8571 9.8418 10.1582 9.9848 IO.OI52 10.1431 9-8569 / m B Cos. O'.l I 8 Sec. Cot. O'.l Tan. Cosec. Sin. m B / Diff. Diff. 133 = 8 h 52 m ] [ 3 h 4 = 46 TABLE IX. 71 44 .-- 2 Q 56 m ] Log. Sines, Tangents, and Secants. [ 9 h O ra = 135 / m s Sin. Diff. Cosec. Tan. Diff. Cot Sec. Cos. m s ' O'.l 1* 0.1 1 s O 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 56 4 8 12 16 5620 24 28 32 36 5640 44 48 52 56 57 4 8 12 16 5720 24 28 32 36 5740 44 48 52 56 58 4 8 12 16 5820 24 28 32 36 58 4O 44 48 52 56 59 4 8 12 16 5920 24 28 32 36 5940 44 48 52 56 6O O 9.8418 8419 8420 8422 8423 o o o o o I I I o o I I o o o I I o o o I I I o o I I I I o o I I I I o o I o I o I I o I . o I 10.1582 1581 1580 1578 _L577 10.1576 1574 1573 1572 i57i 9.9848 9851 9853 9856 9858 o 2 2 2 O O I I I 2 2 2 O I I 2 2 2 O O I I 2 2 2 O O I I I I 2 2 2 O I I 2 2 2 i 2 I I 2 I 2 I I 2 I I 2 I 2 10.0152 0149 0147 0144 0142 10.1431 H32 H33 H34 1436 9.8569 8568 8567 8566 8564 9-8563 8562 8<6i 8560 8558 4 356 52 48 344 40 36 32 28 324 20 16 12 8 3 4 3 256 52 48 244 40 36 32 28 224 20 16 12 8 2 4 2 156 52 48 144 40 36 32 28 124 20 16 12 8 1 4 1 O56 52 48 044 4O 36 32 28 024 20 16 12 8 4 60 j 59 58 57 ; 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 O 9.8424 8426 8427 8428 8429 9.9861 9864 9866 9869 9871 10.0139 0136 0134 0131 0129 10.1437 1438 H39 1440 1442 9.84^1 8432 f433 8435 8436 10.1569 1568 1567 1565 1564 10.1563 1561 1560 1559 1558 9.9874 9876 9879 9881 9884 10.0126 0124 0121 OII9 0116 10.1443 1444 1445 1447 1448 9.8557 8556 8555 8553 8552 9.8437 8439 8440 8441 8442 9.9886 9889 9891 9894 9896 10.0114 oin 0109 0106 0104 10.1449 1450 1452 J 453 H54 9-855I 8550 8548 8547 8546 9.8444 8445 8446 8448 8449 10.1556 1555 !554 1552 i55i 9.9899 9901 9904 9907 9909 IO.OIOI 0099 0096 0093 0091 10.1455 1456 H58 H59 1460 9-8545 8544 8542 8541 8540 9.8450 8451 8453 8454 8455 10.1550 1549 1547 1546 1545 9.9912 9914 9917 9919 9922 10.0088 0086 0083 0081 0078 10.1461 1463 1464 1465 1466 9-8539 8537 8536 8535 8534 9-8457 8458 8459 8460 8462 io-i543 1542 i54i 1540 1538 9.9924 9927 9929 9932 9934 10.0076 0073 0071 0068 0066 10.1468 1469 1470 1471 H73 9-8532 8531 8530 8529 8527 9.8463 8464 8466 8467 8468 io.i537 1536 *534 1533 1532 9-9937 9939 9942 9944 9947 10.0063 0061 0058 0056 0053 10.1474 1475 1476 1478 H79 9.8526 8525 8524 8522 8521 9.8469 8471 8472 8473 8475 10-1531 1529 1528 1527 1525 10.1524 1523 1522 1520 1519 9-9949 9952 9955 9957 9960 10.0051 0048 0045 0043 0040 10.1480 1481 1483 1484 1485 9-8520 8519 8517 8516 8515 9.8476 8477 8478 8480 8481 9.9962 99 6 5 9967 9970 9972 10.0038 0035 0033 0030 0028 10.1486 1488 1489 1490 1491 9.8514 8512 8511 8510 8509 9.8482 8483 8485 8486 8487 10.1518 1517 1515 I5U 1513 9-9975 9977 9980 9982 9985 10.0025 0023 OO2O 00l8 0015 10.1493 1494 H95 1496 1498 9.8507 8506 8505 8504 8502 9.8489 8490 8491 8492 8494 10.1511 1510 1509 1508 1506 9.9987 9990 9992 9995 9997 IO.OOI3 OOIO 0008 0005 0003 10.1499 1500 1501 1503 1504 9.8501 8500 8499 8497 8496 9.8495 10.1505 10.0000 IO.OOOO 10.1505 9.8495 / m B Cos. O'.l 1 s Sec. Cot. O'.l 1 s Tan. Cosec. Sin. m s / Diff. Diff. 134 = 8 b 56 m ] [ 3 h O m - 45 72 TABLE X. JLog. Sines to every Tenth of a Degree. 1 o.o 0' 6 0.2 12' 0.3 18' O.4 24' 0.5 3O' 0.6 36 0.7 42' O.8 48' 0.9 54 60 egrees. 5 Q CO 7.2419 7.5429 7.7190 7-8439 7.9408 8.0200 8.0870 8.1450 8.1961 8.2419 89 i 8.2419 8.2832 8.3210 8.3558 8.3880 8.4179 4459 4723 497i 5206 5428 88 2 5428 5640 5842 6035 6220 6397 6567 6731 6889 7041 7188 87 3 7188 7330 7468 7602 773 i 7857 7979 8098 8213 8326 8436 86 4 8436 1 8543 8647 8749 8849 8946 9042 9i35 9226 9315 9403 85 5 8.9403 8.9489 8-9573 8.9655 8.9736 8.9816 8.9894 8.9970 9.0046 9.0120 9.0192 84 6 9.0192 9.0264 9-0334 9.0403 9.0472 9-0539 9.0605 9.0670 0734 0797 0859 83 7 0859 0920 0981 1040 1099 "57 1214 1271 1326 1381 1436 82 8 1436 1489 1542 1594 1646 1697 1747 1797 1847 1895 *943 81 9 1943 1991 2038 2085 2131 2176 2221 2266 2310 2353 2397 80 10 11 9.2397 2806 9.2439 2845 9.2482 2883 9.2524 2921 9-2565 2959 9.2606 2997 9.2647 334 9.2687 3070 9.2727 3107 9.2767 3*43 9.2806 79 78 12 13 14 3179 3521 3837 3214 3554 3867 3249 3284 3586 3618 3897 3927 3650 3957 3353 3682 3986 3387 3713 4015 3745 4044 3455 3775 4073 3488 3806 4102 3837 413 77 76 75 15 16 17 9.4130 4403 4659 9-4I58 443 4684 9.4186 4456 4709 9.4214 4482 4733 9.4242 45o8 4757 9.4269 4533 478i 9.4296 4559 4805 9-4323 4584 4829 9-4350 4609 4853 9-4377 4t>34 4876 9-4403 4659 4900 74 73 72 , 18 4900 4923 4946 4969 4992 5015 537 5060 5082 5104 5126 71 19 5126 5H8 5170 5192 5213 5235 5256 5278 5299 5320 5341 70 20 9.5341 9-536i 9.5382 9.5402 9-5423 9-5443 9-5463 9.5484 9-5504 9-5523 9-5543 69 21 22 5543 5736 5563 5754 5583 5773 5602 5792 5621 5810 5641 5828 5660 5847 5679 5865 5883 I 5901 5736 68 j 67 23 5919 5937 5954 5972 5990 6007 6024 6042 6059 6076 6093 66 24 6093 6110 6127 6144 6161 6177 6194 6210 6227 6243 6259 65 25 9.6259 9.6276 9.6292 9.6308 9-6324 9-6340 9-6356 9-6371 9-6387 9.6403 9.6418 64 26 6418 6434 6449 | 6465 6480 6495 6510 6526 6541 6556 6570 63 i 27 6570 6585 6600 6615 6629 6644 6659 6673 6687 6702 6716 62 28 6716 673 6744 6759 6/73 6787 6801 6814 6828 6842 6856 61 29 6856 6869 6883 6896 6910 6923 6937 6950 6963 6977 6990 6O 30 9.6990 9-7003 9.7016 9.7029 9.7042 9-7055 9.7068 9.7080 9.7093 9.7106 9.7118 59 31 7118 7131 7144 7156 7168 7181 7193 7205 7218 7230 7242 58 32 7242 7254 7266 7278 7290 7302 73 H 7326 7338 7349 736i 57 33 7361 7373 7384 7396 7407 7419 743 7442 7453 7464 7476 56 j 34 7476 7487 7498 759 7520 7531 7542 7553 7564 7575 7586 55 ! 35 9-7586 9-7597 9.7607 9.7618 9.7629 9.7640 9-7650 9.7661 9.7671 9.7682 9.7692 54 36 37 7692 7795 7703 7805 7713 7815 7723 7825 7734 7835 7744 7844 7754 7854 7764 7864 7774 7874. 7785 788 4 7795 7893 53 52 38 7903 7913 7922 7932 794i 795 i 7960 7970 7979 7989 51 39 7989 7998 8007 8017 8026 8035 8044 8053 8063 8072 8081 50 40 11 9.8081 8169 9.8090 8178 9.8099 9.8108 19.8117 8187 8191; ! 8204 9.8125 8213 9.8134 8221 9.8143 8230 9.8152 8238 9.8161 8247 9.8169 8255 49 48 42 43 8338 8264 8346 8272 8354 8280 8289 8362 8370 8297 8378 8305 8386 8313 8394 8322 8402 8330 8410 S$ 47 46 44 8418 8426 8433 8441 8449 8457 8464 8472 8480 8487 8495 45 03 c/i 60 54' 48' 42' 36 30 24' 18' 12' 6 6 12' 18' 24' 30 36 42 48' 54' 60' Q 1 oo 8.2419 7.2419 8-2833 7.5429 8.3211 7.7190 7.8439 8.3881 7.9409 8.4181 8.O2OO 4461 8.0870 4/25 8.1450 4973 8.1962 5208 8.2419 89 88 2 543 l 5643 5845 6038 6223 6401 6571 6736 6894 7046 7*94 87 3 4 7 IQ4 8446 7337 8554 7609 8762 7739 8862 7865 8960 7988 9056 8107 915 8223 9241 8336 8446 9420 86 85 5 8.9420 8.9506 8.9591 8.9674 8.9756 8.9836 8.9915 8.9992 9.0068 9.0143 9.0216 84 6 9.0216 9.0289 9.0360 9.0430 9.0499 9-0507 9.0633 9.0699 0764 0828 0891 83 7 0891 0954 1015 1076 "35 1194 1252 1310 1367 1423 1478 82 8 1478 '533 1587 1640 1693 1745 1797 1848 1898 1948 1997 81 9 1997 2040 2094 2142 2189 2230 2282 2328 2374 2419 2463 80 10 9.2463 9.2507 9-255I 9.2594 9.2637 9.2680 9.2722 9-2764 9.2805 9.2846 9.2887 79 11 2887 2927 2967 3006 3040 3085 3123 3162 3200 3237 3275 78 12 3275 3312 3349 3385 3422 3458 3493 3529 3564 3599 3634 77 13 3634 3668 3702 3736 3770 3804 3837 3870 3903 3935 3968 76 14 3968 4000 4032 4064 4095 4127 4158 4189 4220 4250 4281 75 15 9.4281 9-43 ii 9-4341 9-4371 9.4400 9.4430 9-4459 9.4488 9-45 1 7 9.4546 9-4575 74 16 4575 4603 4632 4660 4688 4716 4744 4771 4799 4826 4853 73 17 4853 4880 4907 4934 4961 4987 5 OI 4 5040 5066 5092 5118 72 18 5118 5H3 5 l6 9 5195 5220 5245 5270 5295 5320 5345 5370 71 19 5370 5394 5419 5443 5467 55i6 5539 5563 5587 5611 70 20 9.5611 5842 9-5634 5864 9-5658 5887 9-5681 5909 9.5704 593 2 5954 9-5750 5976 9-5773 5998 9.5796 6020 9.5819 6042 9.5842 6064 69 68 22 6064 6086 6108 6129 6151 6172 6194 6215 6236 6257 6279 67 23 6279 6300 6321 6341 6362 6383 6404 6424 6445 6465 6486 66 24 6480 6506 6 527 6547 6567 6587 6607 6027 6647 6667 6687 65 25 9.6687 9.6706 9.6726 9.6746 9.6765 9.6785 9.6804 9.6824 9.6843 9.6863 9.6882 64 26 6882 6901 6920 6939 6958 6977 6996 7015 7034 7053 7072 63 27 7072 7090 7109 7128 7146 7165 7183 7202 7220 7238 7257 62 28 7257 7275 7293 73" 733 7348 7366 7384 7402 7420 7438 61 29 7438 7455 7473 7491 7509 7526 7544 7562 7579 7597 7614 6O 30 9.7614 9.7632 9.7649 9.7667 9.7684 9.7701 9.7719 9-7736 9-7753 9.77-1 9.7788 59 31 7788 7805 7822 7856 7873 7890 7907 7924 7958 58 32 33 7958 8125 7975 8142 7992 8158 8i75 8025 8191 8042 8208 8059 8224 8241 8092 8257 8109 8274 8125 8290 57 56 34 8290 8306 8323 8339 8355 8371 8388 8404 8420 8436 8452 55 35 36 9.8452 8613 9.8468 8629 9.8484 8644 9.8501 8660 9-8517 8676 9.8533 8092 9.8549 8708 9.8565 8724 9.8581 8740 9.8597 8755 9.8613 8771 54 53 37 8771 8787 8803 8818 8834 8850 8865 8881 8897 8912 8928 52 31 8928 8944 8959 8975 8990 9006 9022 9037 9053 9068 9084 51 39 9084 9099 9"5 9130 9146 9161 9176 9192 9207 9223 9238 5O 40 9.9238 9.9254 9.9269 9.9284 9.9300 9-93 1 5 9-933 9.9346 9.9361 9.9376 9.9392 49 41 9392 9407 9422 9438 9453 9468 9483 9499 95 14 9529 9544 48 42 9544 956o 9575 9590 9605 9621 9636 9651 9666 9681 9697 47 43 44 9697 9848 9712 9864 9727 9879 9742 9894 9757 9909 9772 9924 9788 9939 9803 9955 9818 9970 9833 9985 9848 10.0000 46 45 03 03 S 60 54' 48' 42' 36' 30' 24' 18' 12' 6' O' ! 1 .0 0.9 0.8 0.7 0.6 05 04 0.3 0.2 01 o.o 00 IS Logarithmic Cotangents. TABLE XII. < < Logarithmic Tangents to ei'ery Teuth of a Degree. Degrees. o.o O' 0.l 6' 0.2 12' 0.3 IN 0.4 24' D .5 3O' 0.6 36 0.7 42' 0.8 48' 0.9 54 1.O 6O' Degrees, j 45 IO.OOOO 10.0015 10.0030 10.0045 1 0.006 1 10.0076 10.0091 10.0106 IO.OI2I 10.0136 10.0152 44 46 0152 0167 0182 0197 O2I2 0227 0243 0258 0273 0288 0303 43 4? 0303 0319 0334 0349 0364 0379 0395 0410 0425 0440 0456 42 4S 0456 0471 0486 0501 0517 0532 0547 0562 0578 0593 0608 41 49 0608 0624 0639 0654 0670 0685 0700 0716 0731 0746 0762 40 50 10.0762 10.0777 10.0793 10.0808 10.0824 10.0839 10.0854 10.0870 10.0885 10.0901 10.0916 39 51 0916 0932 0947 0963 0978 0994 IOIO 1025 IO4I 1056 1072 38 52 1072 1088 1103 1119 "35 1150 1166 1182 "97 1213 1229 37 53 1229 1245 1260 . 1276 1292 1308 1324 1340 1356 i37i 1387 36 54 1387 1403 1419 H35 H5 i 1467 1483 1499 1516 1532 1548 35 55 10.1548 10.1564 10.1580 10.1596 10.1612 10.1629 10.1645 10.1661 10.1677 10.1694 10.1710 34 56 1710 1726 1743 1759 1776 1792 1809 1825 1842 1858 1875 33 57 1875 1891 1908 1925 1941 1958 1975 1992. 2008 2025 2042 32 58 2042 2059 2076 2093 2IIO 2127 2144 2161 2178 2195 2212 31 59 2212 2229 2247 2264 228l 2299 2316 2333 235 J 2368 2386 30 60 10.2386 10.2403 10.2421 10.2438 10.2456 10.2474 10.2491 10.2509 10.2527 10.2545 10.2562 29 61 2562 2580 2598 2616 2634 2652 2670 2689 2707 2725 2 743 28 62 2 743 2762 2780 2798 2817 2835 2854 2872 2891 2910 2928 27 63 2928 2947 2966 2985 3004 3023 3042 3601 3080 399 3118 26 64 3Il8 3*37 3i57 3176 3196 3215 3 2 35 3 2 54 3274 3294 3313 25 65 10.3313 10-3333 IO -3353 10-3373 10-3393 10.3413 10-3433 io-3453 10.3473 10.3494 10.3514 24 66 35H 3535 3555 3576 3596 3617 3638 3659 3 6 79 3700 3721 23 67 3721 3743 3764 3785 3806 3828 3849 3871 3892 39 J 4 393 6 22 68 3936 3958 398o 4002 4024 4046 4068 4091 4"3 4136 4158 21 69 4 I58 4181 4204 4227 4250 4273 4296 43 J 9 4342 4366 4389 20 7O 10.4389 10.4413 10.4437 10.4461 10.4484 10.4509 10-4533 10-4557 10.4581 10.4606 10.4630 19 71 4630 4655 4680 4705 473 4755 478o 4805 4831 4857 4882 18 72 4882 4908 4934 4960 4986 5013 539 5066 5093 5120 5H7 17 73 5'47 5174 5201 5229 5256 5284 5312 5340 5368 5397 5425 16 74 5425 5454 5483 55 12 5541 557o 5600 5629 5659 5689 5719 15 75 10.5719 10-5750 10.5780 10.5811 10.5842 10.5873 10.5905 10.5936 10.5968 10.6000 10.6032 14 76 6032 6065 6097 6130 6163 6196 6230 6264 6298 6332 6366 13 | 77 6366 6401 6436 6471 6507 6542 6578 6615 6651 6688 6725 12 78 6725 6763 6800 6838 6877 6915 6 954 6994 7033 7073 7"3 11 79 7H3 7i54 7195 7236 7278 7320 7363 7406 7449 7493 7537 1.0 80 io.7537 10.7581 10.7626 10.7672 10.7718 10.7764 10.7811 10.7858 10.7906 10.7954 10.8003 9 I 81 8003 8052 8102 8152 8203 8255 8307 8360 8413 8467 8522 8 ! 82 8522 8577 8633 8690 8748 8806 8865 8924 8985 9046 9109 7 83 9109 9172 9236 9301 93 6 7 9433 95 i 957 9640 9711 9784 6 84 9784 9857 993 2 11.0008 1 1 .0085 11.0164 11.0244 11.0326 11.0409 11.0494 11.0580 5 85 11.0580 11.0669 11.0759 11.0850 11.0944 11.1040 11.1138 11.1238111.1341 11.1446 "1554 4 86 1554 1664 1777 1893 2OI2 2135 2261 2391 ! 2525 2663 2806 3 87 2806 2954 3106 3264 3429 3599 3777 3962 j 4155 4357 45 6 9 2 1 88 45 6 9 4792 5027 5275 5539 5819 6119 6441 i 6789 11.7167 758i 1 I 89 7581 8038 8550 9130 9800 12.0591 12.1561 12.2810 12.4571 12.7581 -f- CO i 6O' 54' 48' 42' 36 30 24 18' 12' 6 O' 02 $ & 0) 1.0 0.9 0.8 0.7 0.6 0.5 O.4 O.3 0.2 0.l o.o I V Q P Logarithmic Cotangeuts. 78 TABLE XIII. Log. Tangents and Cotangents in Time. o u Diff. l b Diff. 2 h Diff. 3 h Diff. 4 h Diff. 5 h Diff. m 6 h O m .l l^h O ra .l 8 h O m .l 9 h O ra .l 10 O m .l ll h 6 m .i m oo 9.4281 9.7614 IO.OOOO 10.2386 10.5719 eo 1 7.6398 301 4356 7 7658 4 0038 4 2429 4 5796 8 59 2 7.9409 176 443 14 7701 9 0076 8 2474 i 9 5873 16 58 i 3 8.1170 125 453 21 7745 13 0114 ii 2518 f 13 5952 25 57 4 8.2419 97 4575 28 7788 17 0152 15 2562 18 6032 33 56 5 8.3389 ! 79 9.4646 35 9-7831 21 10.0190 19 10.2607 22 10.6114 4 1 55 6 4181 67 y- 47l6 42 7873 26 0228 23 2652 27 6196 49 54 7 4851 58 4785 49 7916 30 0265 1 27 2698 ' 31 6281 57 53 8 5431 51 4853 5 6 7958 34 33 i 3 2743 36 6366 66 52 9 5943 46 4921 6 3 8000 39 0341 34 2789 40 6454 74 51 10 8.6401 4 1 94987 9.8042 10.0379 10.2835 10.6542 50 11 6815 38 5053 6 8084 4 0418 4 2882 5 6633 10 49 12 7194 35 5Il8 13 8125 .8 0456 8 2928 9 6725 20 48 i 13 7542 32 5l82 19 8167 12 0494 ii 2975 13 6819 29 47 14 7865 30 5245 2 5 8208 16 15 3023 19 6915 39 46 ! ; 1$ 8.8165 28 9.5308 31 9.8249 20 10.0570 19 10.3070 24 10.7013 49 45 16 8446 26 5370 38 8290 2 5 0608 23 3Il8 28 7H3 59 44 17 8711 2 5 543 * 44 8331 29 0647 27 3 l66 33 7216 69 43 18 8960 24 549 ! 50 8371 33 0685 30 3215 38 7320 78 42 19 9196 22 5551 56 8412 37 0723 34 3264 43 7427 88 41 20 8.9420 21 9.5611 9.8452 10.0762 10.3313 10-7537 40 21 9633 20 5669 6 8493 4 0800 4 3363 5 7649 13 39 22 9836 19 5/27 ii 8533 8 0839 8 3413 10 7764 26 38 23 9.0030 1 9 5785 17 8573 I2 0878 12 3463 15 7882 39 37 24 0216 18 5842 22 8613 16 0916 16 35H 20 8003 52 36 25 9-0395 17 9.5898 28 9-8652 20 10.0955 19 10.3565 25 10.8127 65 35 > 26 0567 16 5954 34 8692 24 0994 23 3617 31 8255 78 34 27 0732 16 6009 39 8732 28 1033 27 3669 36 8387 91 33 28 0891 15 6064 45 8771 32 1072 31 3721 8522 104 32 29 1045 6118 50 88n 36 mi 35 3774 46 8662 1 1 7 31 30 9.1194 9-6172 9.8850 10.1150 10.3828 10.8806 14 30 31 1338 J 3 6226 5 8889 4 1189 4 3882 6 8955 15 29 32 1478 26 6279 10 8928 8 1229 8 3936 n 9109 15 28 33 1613 39 6331 16 8967 12 1268 12 17 9268 16 27 34 '745 52 6383 21 9006 16 1308 16 4046 22 9433 16 26 35 9.1873 65 9-6435 26 9.9045 19 10.1348 20 10.4102 28 10.9605 17 25 36 1997 78 6486 3 1 9084. 2 3 1387 24 4158 34 9784 18 24 37 2118 6537 36 9122 27 1427 28 4215 39 9970 19 23 38 2236 104 6587 42 9161 3 1 1467 32 4273 45 11.0164 19 22 1 39 2351 117 6637 47 9200 35 1507 36 433 i 50 0367 20 21 40 9.2463 9.6687 9-9238 10.1548 10.4389 11.0580 21 20 41 2573 10 6736 5 9277 4 1588 4 4449 6 0804 22 19 42 2680 20 6785 10 9315 8 1629 8 459 12 1040 24 18 i 43 2784 29 6834 14 9353 n 1669 12 4569 19 1289 2 5 17 44 2887 39 6882 19 9392 15 1710 16 4630 25 1554 26 16 45 9-2987 49 9.6930 24 19 10.1751 20 10.4692 31 11-1835 28 15 46 3085 59 6977 29 9468 2 3 1792 25 4755 37 2135 3 14 ^47 3181 7025 34 9506 27 1833 29 4818 43 2458 3 2 13 48 3275 78 7072 38 9544 3 1875 33 4882 50 2806 35 12 49 3367 88 7118 43 9582 34 1916 37 4947 56 3185 38 11 50 9-3458 9-7165 9.9621 10.1958 10.5013 n-3599 4 1 10 i 51 3546 8 7211 5 9659 4 2000 4 5079 7 4057 46 9 52 3 6 34 16 7257 9 9697 8 2O42 8 5H7 14 4569 5J 8 53 3719 25 732 13 9735 n 2084 13 5215 21 5 I 5 58 7 54 3804 33 7348 18 9773 15 2127 17 5284 28 5819 67 6 55 9-3886 9-7393 22 9.9810 19 I0.2I69 21 10-5354 35 11.6611 79 5 56 3968 49 7438 27 9848 23 2212 25 5425 43 7581 97 4 57 4048 57 7482 31 9886 27 2255 3 5497 5 8830 '25 3 58 4127 66 7526 36 9924 30 2299 5570 57 12.0591 176 2 59 4204 74 757i 40 9962 34 2342 38 5644 64 3602 301 1 60 9.4281 9.7614 10.0000 10.2386 10.5719 + 00 ll h Diff io h Diff 9 h Diff 8 h Diff 7 h Diff 6 h Diff. m 5 h O. m l 4 h O m .l 3" O m .l 2 h O' u .l l h 0.l O h O m .l 1X1 jPp R Read "Tangents " in upper line and "Cotangents" in lower line of Hour Arguments alike at top and bottom, taking the minutes in left or right column according as the hours are taken at top or bottom. TABLE XIV. 79 Degrees. ' 1 00 LengtBis of Circular Arcs. 6 O.2 12' 03 18' 0.4 24' 0.5 30' 0.6 36' 0.7 42' 0.8 48' 0.9 54' O o.oooo 0.0017 0.0035 0.0052 0.0070 0.0087 0.0105 0.0122 0.0140 0.0157 1 0175 0192 0209 0227 0244 0262 0280 0297 0317 0332 2 0349 0367 0384 0401 0419 0436 0454 0471 0489 0506 3 0524 0541 559 0576 0594 0611 0628 0646 0664 0681 4 0698 0716 0733 0750 0768 0785 0803 0820 0838 0855 5 0.0873 0.0890 0.0908 0.0925 0.0943 0.0960 0.0978 0.0995 O.IOI2 0.1030 6 1047 1065 1082 1099 1117 H34 1152 1169 Il87 1204 7 1222 1239 1257 1274 1292 1309 1327 1344 I 3 6l 1379 8 1396 1414 1431 1449 1466 1484 1501 I 5 l8 1536 9 1571 1588 1606 1623 1641 1658 1676 1693 I7IO 1728 10 0.1745 0.1763! 0.1780 0.1798 0.1815 0.1833 0.1850 0.1867 0.1885 0.1902 11 1920 1937 1955 | 1972 1990 2007 2025 2042 2059 2077 12 2094 2112 2129 ! 2147 2164 2182 2199 22l6 2234 2251 13 2269 2286 2304 i 2321 2339 2356 2374 f 2391 2409 2426 14 2443 2461 2478 j 2496 2513 253! 2548 2566 2583 26OI 15 0.26l8 0.2635 0.2653 j 0.2670 0.2688 0.2705 0.2723 0.2740 0.2758 0.2775 16 2793 2810 2827 | 2845 2862 2880 i 2897 2915 I 2932 2950 17 2967 2985 3002 1 3019 3037 3054 3072 3089 ! 3107 3124 18 3H2 3159 i 3177 3194 3212 3229 3246 | 3264 3281 3299 19 3316 3334 i 335 ' 33 68 33 g 6 3403 3421 | 3438 3456 3473 20 0-3491 0.3508 0.3526 0-3543 0.3561 0.3578 0.3596 0.3613 0.3630 0.3648 21 3665 3683 37oo 37i8 3735 3752 3770 3787 3805 3822 22 3840 3857 3875 3892 3910 3927 3945 3962 3979 3997 23 4014 4032 4049 4067 4084 4102 4119 4136 4154 4171 24 4189 4206 4224 4241 4259 4276 4294 43 11 4328 4346 25 0.4363 0.4381 0.4398 0.4416 0.4433 0.4451 0.4468 0.4486 0-4503 0.4520 26 4538 j 4555 4573 4590 4608 4625 4643 4660 4678 4695 27 4712 i 473 4747 4765 4782 4800 4817 4835 4852 4869 28 4887 ! 4904 4922 4939 4957 4974 4992 5009 5027 5044 29 5061 5079 5096 5"4 5*49 5166 5184 5201 5219 30 0.5236 0.5253 0.5271 0.5288 0.5306 0.5323 0.5341 0.5358 0.5376 0-5393 31 5411 5428 5445 5463 548o 5498 5515 5533 5550 5568 32 5585 5603 5620 5638 5655 5673 5690 5707 5725 5743 33 576o 5777 5795 5812 5830 5847 5865 5882 5899 5917 34 5934 5952 5969 5986 6004 6021 6039 6056 6074 6091 35 0.6109 0.6126 0.6144 0.6161 0.6179 0.6196 0.6213 0.6231 0.6248 0.6266 36 6283 6301 6318 6336 6353 6370 6388 6405 6423 6440 37 6458 6475 6493 6510 6528 6545 6563 6580 6597 6615 38 6632 6650 6667 6685 6702 6720 6737 6754 6772 6789 39 6807 6824 6842 6859 6877 6894 6912 6920 6946 6964 40 0.6981 0.6999 0.7016 0.7034 0.7051 0.7069 0.7086 0.7103 0.7121 0.7138 41 7156 7173 7191 7208 7226 7243 7261 7278 7296 7313 42 733 7348 7365 7383 7400 7418 7435 7453 7470 7487 43 755 7522 7540 7557 7575 7592 7610 7627 7645 7662 44 7679 7697 77H 7732 7749 7767 7784 7802 7819 7836 45 0.7854 0.7871 0.7889 0.7906 0.7924 0.7941 0-7959 0.7976 0.7994 0.8011 46 8029 8046 8063 8081 8098 8116 8133 8151 8168 8186 47 8203 8220 8238 8255 8273 8290 8308 8325 8343 8360 48 8378 8395 8413 8430 8448 8465 8482 8500 8517 8535 49 8552 8570 8587 8604 8622 8639 8657 8674 8692 8709 50 0.8727 0.8744 0.8762 0-8779 0.8797 0.8814 0.8831 0.8849 0.8866 0.8884 51 8901 8919 8936 8953 8971 8988 9006 9023 9041 9058 52 9076 9093 ; 9111 9128 9146 9163 9181 9198 9215 9233 53 9250 9268 9285 933 9320 9338 9355 9372 9390 9407 54 9425 * 9442 9460 9477 9495 9512 9530 9547 9564 9582 55 0.9599 0.9617 0.9634 0.9652 0.9669 0.9687 0.9704 0.9721 0.9739 0.9756 56 9774 9791 9809 9826 9844 9861 9879 9896 9914 993 i 57 9948 9966 9983 1. 000 1 1.0018 1.0036 1-0053 1.0071 i. 0088 1.0105 58 1.0123 1.0140 1.0158 0175 0193 02 10 0228 0245 0263 0280 59 0297 0315 0332 0350 0367 0385 0402 0420 0437 0455 1 I 80 TABLE XV. Natural Suit's. Degrees. o.o 6' 0.2 12' 0.3 18'' 0.4 24' 0.5 30' 0.6 36' 0.7 42' 0=.8 48' 0^.9 1 .0 54' 60' Degrees. o.oooo 0.0017 0.0035 0.0052 0.0070 0.0087 0.0105 0.0122 0.0140 0.0157 0.0175 89 1 0175 0192 0209 0227 0244 0262 0279 0297 03H 0332 0349 88 2 0349 0366 0384 0401 0419 0436 0454 0471 0488 0506 0523 87 3 0523 0541 0558 0576 0593 0610 0628 0645 0663 0680 0698 86 4 0698 0715 0732 075 0767 0785 0802 0819 0837 0854 0872 85 5 0.0872 0.0889 0.0906 0.0924 0.0941 0.0958 0.0976 0.0993 O.IOII 0.1028 0.1045 84 6 1045 1063 1080 1097 1115 1132 1149 1167 1184 1201 1219 83 7 1219 1236 1253 1271 1288 1305 1323 1340 1357 1374 1392 82 8 1392 1409 1426 1444 1461 1478 1495 1513 1530 1547 1564 81 9 1582 1599 1616 1633 1650 1668 1685 1702 1719 i/3 6 80 10 0.1736 o.i754 0.1771 0.1788 0.1805 0.1822 0.1840 0.1857 0.1874 0.1891 0.1908 79 11 1908 1925 1942 1959 1977 1994 2OII 2028 2045 2O62 2079 78 12 2079 2096 2113 2130 2147 2164 2181 2198 2215 2233 2250 77 i 13 2250 2267 2284 2300 2317 2334 2351 2368 2385 2 4 02 2419 76 14 2419 2436 2453 2470 2487 2504 2521 2538 2554 2571 2588 75 15 16 0.2588 2756 0.2605 2773 0.2622 2790 0.2639 2807 0.2656 2823 0.2672 2840 0.2689 2857 0.2706 2874 0.2723 2890 0.2740 2907 0.2756 2924 74 73 17 2924 2940 2957 2974 2990 3007 3024 3040 3057 374 3090 72 18 3090 3107 3123 3156 3173 3190 3 206 3223 3239 3256 71 ! 19 3256 3272 3289 335 3322 3338 3355 3371. 3387 3404 3420 7O 2O 0.3420 0-3437 0-3453 0.3469 0.3486 0.3502 0.3518 0-3535 0.3551 o-35 6 7 0.3584 69 21 22 3584 3746 3600 3762 3616 3778 3 6 33 3795 3 6 49 3811 3665 3827 3681 3843 3697 3859 37H 3875 3730 3891 3746 3907 68 67 23 3907 3923 3939 3955 397i 3987 4003 4019 4035 4051 4067 66 24 4067 4083 4099 4H5 4I3 1 4H7 4163 4179 4195 4210 4226 65 25 0.4226 0.4242 0.4258 0.4274 0.4289 0.4305 0.4321 0-4337 0.4352 0.4368 0.4384 64 26 4384 4399 4415 443 * 4446 4462 4478 4493 459 45 2 4 4540 63 27 4540 4555 4571 4586 4602 4617 4633 4648 4664 4679 4695 62 28 29 4848 4710 4863 4726 4879 4741 4894 4756 4909 4772 4924 4787 4939 4802 4955 4818 4970 4833 4985 5000 61 60 30 0.5000 0.5015 0.5030 0.5045 0.5060 0.5075 0.5090 0.5105 0.5120 0.5135 0.5150 59 31 5150 5165 5180 5*95 5210 5225 5240 5255 5270 5284 5299 58 32 5299 53*4 5329 5344 5358 5373 5388 5402 5417 5432 5446 57 33 34 5446 5592 5606 5476 5621 5490 5 6 35 5505 5519 5664 5534 5678 5548 5693 5563 5707 5577 572i 5592 573 6 56 55 I 35 36 0.5736 5878 o.575o 5892 0.5764 5906 0-5779 5920 0-5793 5934 0.5807 0.5821 5962 0-5835 597 6 0.5850 5990 0.5864 6004 0.5878 6018 54 53 37 6018 6032 6046 6060 6074 6088 6101 6115 6129 6143 6157 52 38 6157 6170 6184 6198 6211 6225 6239 6252 6266 6280 6293 51 39 6293 6307 6320 6334 6347 6361 6374 6388 6401 6414 6428 50 40 0.6428 0.6441 0.6455 0.6468 0.6481 0.6494 0.6508 0.6521 0.6534 0.6547 0.6561 49 41 6561 6 574 6587 6600 6613 6626 6639 6652 6665 6678 6691 48 ; 42 6691 6704 6717 6730 6743 6756 6769 6782 6794 6807 6820 47 ! 43 6820 6833 6845 6858 6871 6884 6896 6909 6921 6934 6947 46 44 6947 6959 6972 6984 6997 7009 7022 7034 7046 7059 7071 45 : i 60' 54' 48' 42' 36' 30' 24' 18' 12' 6 0' 1 1 ro 0.9 0.8 O.7 D .6 0.5 04 03 0.2 01 o.o 60 5 ! Natural Cosines. TABLE XV. 81 1 Natural Sim**. co 0) 0.0 O 0.l 6 O a .2 12 0.3 18' 0.4 24' 0.5 3O' 0.6 36' 0.7 42 0.8 48' 0.9 54 1.0 60 egrees. 1 Q i 45 0.7071 0.7083 0.7096 0.7108 0.7120 0.7133 0.7145 0.7157 0.7169 0.7181 0.7193 44 46 7i93 7206 7218 7230 7242 7 2 54 7266 7278 7290 7302 73H 43 47 73 H 73 2 5 7337 7349 736i 7373 7385 7396 7408 7420 743 i 42 48 743 r 7443 7455 7466 74/8 7490 75 01 7513 7524 7536 7547 41 49 7547 7559 7570 758i 7593 7604 ' 7615 7627 7638 7649 7660 40 50 0.7650 0.7672 0.7683 0.7694 0.7705 0.7716 0.7727 0.7738 0.7749 0.7760 0.7771 39 51 777i 7782 7793 7804 781 S 7826 7837 7848 7859 7869 7880 38 *i 7880 7891 7902 7912 7923 7934 7944 7955 7965 7976 7986 37 53 7986 7997 8007 8018 8028 8039 8049 8059 8070 8080 8090 36 54 8090 8100 8m 8121 8131 8141 8151 8161 8171 8181 8192 35 55 0.8192 0.8202 0.8211 0.8221 0.8231 0.8241 0.8251 0.8261 0.8271 0.8281 0.8290 34 50 57 8293 83^7 8300 8396 8310 8406 8320 8415 8329 8425 8339 8434 8348 8443 8358 8453 8368 8462 8377 8471 8387 8480 33 32 5 59 8480 8572 8490 8581 8499 8590 8508 8599 8517 8607 8526 8616 8536 8625 8545 8634 8554 8643 8563 8652 8572 8660 31 1 30 69 o.865o 0.8669 0.8678 0.8686 0.8695 0.8704 0.8712 0.8721 0.8729 0.8738 0.8746 29 61 8746 8755 8763 8771 8780 8788 8796 8805 8813 8821 8829 28 0i 8829 8838 8846 8854 8862 8870 8878 8886 8894 8902 8910 27 63 64 8910 8988 8918 8996 8926 9003 8934 9011 8942 9018 8949 9026 8957 9033 8965 9041 8973 9048 8980 9056 8988 9063 26 25 65 0.9063 0.9070 0.9078 0.908-5 0.9092 0.9100 0.9107 0.9114 0.9121 0.9128 o.9i35 24 66 9135 9143 9i5 9157 9164 9171 9178 9184 9191 9198 9205 23 i 67 9205 9212 9219 9225 9232 9239 9245 9252 9259 9265 9272 22 ! 68 9272 9278 9285 9291 9298 934 93 11 93 J 7 9323 933 9336 21 ! 69 9336 9342 9348 9354 9361 93 6 7 9373 9379 9385 939i 9397 20 70 0-9397 0.9403 0.9409 0.9415 ' 0.9421 0.9426 0.9432 0.9438 0.9444 0.9449 0-9455 19 71 9455 9461 9466 9472 9478 9483 9489 9494 9500 9505 95" 18 72 73 74 95" 9563 9613 95i6 9568 9617 952i 9573 9622 9527 9578 9627 9532 9583 9632 9537 9588 9636 954 2 9593 9641 954f 9598 9646 9553 9603 9650 9558 9608 9655 9563 9613 9 6 59 17 16 15 75 0.9659 0.9664 0.9668 0.9673 0.9677 0.9681 0.9686 0.9690 0.9694 0.9699 0.9703 14 76 973 9707 9711 97i5 9720 9724 9728 9732 973 6 9740 9744 13 77 9744 9748 975i 9755 9759 97 6 3 9767 9770 9774 9778 9781 12 78 9781 9785 9789 9792 9796 9799 9803 9806 9810 9813 9816 11 79 9816 9820 9823 9826 9829 9833 9836 9839 9842 9845 9848 1O SO 0.9848 0.985 1 0.9854 0.9857 0.9860 0.9863 0.9866 0.9869 0.9871 0.9874 0.9877 9 81 9877 9880 9882 9885 9888 9890 9893 9895 9898 9900 9903 8 82 9903 995 9907 9910 9912 9914 9917 9919 992i 9923 9925 7 83 9925 9928 993 993 2 9934 993 6 9938 9940 9942 9943 9945 6 i 84 9945 9947 9949 995 i 9952 9954 9956 9957 9959 9960 9962 5 85 0.9962 0.9963 0.9965 0.9966 0.9968 0.9969 0.9971 0.9972 0.9973 0.9974 0.9976 4 86 9976 9977 9978 9979 9980 998i 9982 9983 9984 9985 9986 3 87 9986 9987 9988 9989 9990 9990 9991 9992 9993 9993 9994 2 88 9994 9995 9995 9996 9996 9997 9997 9997 9998 9998 9998 1 89 9998 9999 9999 9999 9999 I.OOOO I.OOOO I.OOOO I.OOOO i .0000 I.OOOO O 1 60' 54 48' 12- 36' 3O' 24' 18' 12 6 0' i a i 1.O O.9 0.8 0.7 0.6 0.5 0.4 0.3 02 0^.1 O.O 1 Natural Cosines. 82 TABLE XVI. Natural Tangents. Degrees. o.o 6' O.2 12' O.3 18' 0.4 24' 0.5 30' .6 36 0.7 42' 0.8 48' 0.9 54' 1 .0 6O Degrees. | 0.0000 0.0017 0.0035 0.0052 0.0070 0.0087 0.0105 O.OI22 0.0140 0.0157 0.0175 89 1 0175 0192 0209 0227 0244 0262 0279 0297 0314 0332 349 88 2 0349 0367 0384 0402 0419 0437. 0454 0472 0489 0507 0524 87 3 0524 0542 0559 0577 0594 0612 0629 0647 0664 0682 0699 86 ; 4 0699 0717 0734 0752 0769 0787 0805 0822 0840 0857 0875 85 5 0.0875 0.0892 0.0910 0.0928 0.0945 0.0963 0.0981 0.09 9 8 0.1016 0.1033 0.1051 84 6 1051 1069 1086 1104 1 122 "39 "57 "75 1192 1210 1228 83 7 1228 1246 1263 1281 1299 1317 1334 1352 1370 1388 1405 82 8 1405 1423 1441 H59 M77 1495 1512 1530 1548 1566 1584 81 9 1584 1602 1620 1638 1655 1673 1691 1709 1727 1745 1763 80 10 0.1763 0.1781 0.1799 0.1817 0.1835 0.1853 0.1871 0.1890 0.1908 0.1926 0.1944 79 11 1944 1962 1980 1998 2Ol6 2035 2053 2071 2089 2107 2126 78 12 2126 2144 2162 2180 2199 2217 2235 2254 2272 2290 2309 77 ! 13 2309 2327 2345 2364 2382 2401 2419 2438 2456 2475 2493 76 14 2493 2512 2530 2549 2568 2586 2605 2623 2642 266l 2679 75 15 16 0.2679 2867 0.2698 2886 0.2717 2905 0.2736 2924 0.2754 2943 0.2773 2962 0.2792 2981 0.2811 3000 0,2830 3019 0.2849 3038 0.2867 357 74 73 17 3057 3076 3096 3"5 3134 3153 3172 3*91 3211 3230 3249 72 18 3249 3269 3288 337 3327 3346 3365 3385 3404 3424 3443 71 19 3443 3463 3482 3502 3522 3541 356i 358i 3600 3620 3640 7O 2O 0.3640 0-3659 0.3679 0.3699 0.3719 0-3739 0-3759 0-3779 0-3799 0.3819 0-3839 69 2 1 3839 3859 3879 3899 3919 3939 3959 3979 4000 4O2O 4040 68 22 4040 4061 4081 4101 4122 4142 4163 4183 4204 4224 4245 67 23 4245 4265 4286 437 4327 4348 4369 4390 44" 443 * 4452 66 24 4452 4473 4494 4515 4536 4557 4578 4599 4621 4642 4663 65 i 25 0.4663 0.4684 0.4706 0.4727 0.4748 0.4770 0.4791 0.4813 0.4834 0.4856 0.4877 64 26 4877 4899 4921 4942 4964 4986 5008 5029 505 1 573 595 63 27 595 5117 5J39 5161 5^4 5206 5228 5250 5272 5295 5317 62 28 53 * 7 5340 ! 5362 5384 540.7 5430 5452 5475 5498 5520 5543 61 29 5543 5566 5589 5612 5635 5658 5681 5704 5727 5750 5774 60 30 31 0-5774 6009 0-5797 6032 0.5820 6056 ios2 0.5867 6104 0.5890 6128 0.5914 6152 0.5938 6176 0.5961 6200 0.5985 6224 0.6009 6249 59 58 32 33 34 6249 6494 6745 6273 6519 6771 6297 6544 6796 6322 6569 6822 6346 6594 6847 6371 6619 6873 6395 6644 6899 6420 6669 6924 6445 6694 6950 6469 6720 6976 6494 6745 7002 57 56 55 35 0.7002 0.7028 0.7054 0.7080 0.7107 0-7133 0.7159 0.7186 0.7212 0.7239 0.7265 54 i 36 7265 7292 7319 7346 7373 7400 7427 7454 748i 7508 7536 53 37 7536 7813 7563 7841 7590 7869 7618 7898 7646 7926 7673 7954 7701 7983 7729 8012 7757 8040 7785 8069 7813 8098 52 51 j 39 8098 8127 8156 8185 8214 8243 8273 8302 8332 8361 8391 5O 40 0.8391 0.8421 0.8451 0.8481 0.8511 0.8541 0.8571 0.8601 0.8632 0.8662 0.8693 49 41 8693 8724 8754 8785 8816 8847 8878 8910 8941 8972 9004 48 |42 9004 9036 9067 9099 9131 9163 9195 9228 9260 9293 9325 47 43 9325 9358 9391 9424 9457 9490 9556 9590 9623 9657 46 i 44 9657 9691 9725 9759 9793 9827 9861 9896 9930 9965 I.OOOO 45 i 6O' 54' 48' 42' 36' 30' 24' 18' 12' 6' i s !- O.9 0.8 0.7 0.6 0.5 0.4 0.3 O.2 01 O c .O I 3 Natural Cotangents. TABLE XVI. 83 Natural Tangents. Degrees. o.o 6' O.2 12' 0.3 18' 0.4 24' 0.5 3O' 0.6 36 0.7 42' 0.8 48' 0.9 54' 60' Degrees. 45 1. 0000 1-0035 1.0070 1.0105 1.0141 1.0176 I.O2I2 1.0247 1.0283 1.0319 1-0355 44 46 0355 0392 0428 0464 0501 0538 575 0612 0649 0686* 0724 43 j 47 0724 0761 0799 0837 0875 0913 095 1 0990 1028 1067 1106 42 IN 1106 H4S 1184 1224 1263 1303 1343 1383 1423 1463- 1504 41 49 1504 1544 1585 1626 1667 1708 1750 1792 1833 1875 1918 40 50 1.1918 1.1960 I.2OO2 1.2045 1.2088 1.2131 I.2I7 4 1.2218 1.2261 1-2305 1.2349 39 51 2349 2393 2437 2482 2527 2572 2617 2662 2708 2753 2799 3S 52 2799 2846 2892 2938 2985 3032 3079 3127 3175 3222 3270 37 53 3270 33*9 33 6 7 3465 3514 3564 3613 3663 3713 3764 36 54 37 6 4 3865. 3916 3968 4019 4071 4124 4176 4229 4281 35 55 1.4281 1-4335 1.4388 1.4442 1.4496 1-4550 1.4605 1.4660 . I.47I5 1.4770 1.4826 34 56 4826 4882 4938 499.4 5051 5108 5l66i 5224 5282 5340 5399 33 57 5399 545S 55'7 5577 5 6 37 5697 5757 5818 5880 5941 6003 32 58 6003 6066 6128 6191 6255 6319 6383 6447) 6512 6577 6643 31 59 6643 6709 6775 6842 6909 6977 7045 7182 7251 7321 30 60 I -732i| I-739 1 1.7461 I-753 2 1.7603 I-7675 1-7747 1.7820 1-7893 1.7966 1.8040 29 61 8040' 81115 8190 8265 8341 8418 8495 8572! 8650 8728 8807 28 62 8807 8887 8967 9047 9128 9210 9292 9375 9458 95421 9626 27 63 9626 9711 9797 9883 9970 2.0057 2.0145 2.0233 2.0323 2.0413) 2.0503 26 64 2.0503 2.0594 2.0686 2.0778 2.0872 0965 1060 "55 1251 1348 1445 25 65 2.1445 2.1543 2.1642 2.1742 2.1842 2.1943 2.2045 2.2148 2.2251 2.2355 2.2460 24 66 2460 2566 2673 2781 2889 2998 3109 3220 3332 3445 3559 23 67 3559 3673 3789 39o6 4023 4142 4262 4383 4504 4627 22 68 475i 4876 5002 5 I2 9 5257 5386 5517 5649 5782 59i6 6051 21 69 6051 6187 6325 6464 6605 6746 6889 7034 7179 7326 7475 20 7O 2-7475 2.7625 2.7776 2.7929 2.8083 2.8239 2.8397 2.8556 2.8716 2.8878 2.9042 19 71 9042 9208 9375 9544 97H 9887 3.0061 3.0237 3-0415 3.0595 3.0777 18 72 73 3.0777 2709 3.0961 2914 3.1146 3122 3-1334 3332 3-1524 3544 3.1716 3759 1910 3977 2106 2305 4420 2506 4646 2709 4874 17 16 74 4874 5105 5339 5576 5816 6059 6305 6554 6806 7062 15 75 3-7321 3-7583 3.7848 3.8118 3-839I 3.8667 3.8947 3.9232 3-9520 3-9812 4.0108 14 76 4.0108 4.0408 4-0713 4.1022 4.1335 4-1653 4.1976 4-2303 4-2635 4.2972 3315 13 77 3315 3662 4015 4373 4737 5107 5483 5864 6252 6646 7046 12 78 7046 7453 7867 8288 8716 9152 9594 5.0045 5-0504 5.0970 5.1446 11 79 5.1446 5"!929 5.2422 5.2924 5-3435 5-3955 5-4486 5026 5578 6140 6713 1O 80 81 82 83 5-67I3 6.3138 7-"54 8.1443 5-7297 6-3859 7.2066 8.2636 5-7894 6.4596 7.3002 8.3863 5.8502 6-5350 7.3962 8.5126 5-9i2 4 6.6122 7-4947 8.6427 I'? 758 6.6912 7.5958 8.7769 6.0405 6.7720 7.6996 8.9152 6.1066 6.8548 7.8062 9.0579 6.1742 6-9395 7-9158 9.2052 6.2432 7.0264 8.0285 9-3572 6.3138 7-II54 8.1443 9.5144 9 7' 6 84 9.5144 9.6768 9.8448 10.0187 10.1988 10.3854 10.5789 10.7797 10.9882 11.2048 11.4301 & 85 86 11.4301 14.3007 11.6645 14.6685 11.9087 I5-0557 12.1632 15.4638 12.4289 15.8945 12.7062 16.3499 12.9962 16.8319 13.2996 I7.343 2 13.6174 17.8863 I3-9507 18.4645 14.3007 19.0811 4 3 !87 88 89 19.0811 28.6363 57.2900 19.7403 30.1446 63.6567 ^0.4465 31.8205 71.6151 21.2049 33.6935 81.8470 22.0217 35.8006 954895 22.9038 38.1885 i 14.5886 23-8593 40.9174 24.8978 44.0661 190.9842 26.0307 47-7395 286.4777 27.2715 52.0807 572.9572 28.6363 57.2900 00 2 1 Q> 60 54' 48' 42' 36' 30' 24' 18' 12' 6' 0' S 1 1*0 0.9 0.8 0.7 06 0.5 O.4 O.3 0.2 0*1 o.o 1 Natural Cotangents. 84 TABLE XVII. Natural Versed Sines. i o.o 01 .2 0.3 0.4 0^.5 0.6 0.7 0.8 0.9 O O' 6' 12' IS' 24 30 36' 42' 48' 54' Q . o.oooo 0.0000 o.oooo 0.0000 o.oooo 0.0000 O.OOOI O.OOOI O.OOOI O.OOOI 1 OOO2 OOO2 OOO2 0003 0003 0003 1 0004 0004 0005 0015 : 2 0006 0007 0007 0008 0009 0010 0010 OOII 0012 0003 3 0014 0015 0016 0017 0018 0019 OO2O OO2I OO22 0013 4 0024 0026 0027 0028 0029 0031 0032 0034 35 37 5 0.0038 0.0040 0.0041 0.0043 0.0044 0.0046 0.0048 0.0049 O.OO5I 0.0053 6 0055 0057 0058 0060 0062 0064 0066 0068 0070 0072 7 0075 0077 0079 0081 0083 0086 0088 0090 0093 0095 8 0097 OIOO OIO2 0105 0107 OIIO OII2 0115 0118 0120 9 0123 0126 0129 0131 0134 0137 0140 0143 0146 0149 10 0.0152 0.0155 0.0158 0.0161 0.0164 0.0167 O.OI7I 0.0174 0.0177 O.OlSo 11 0184 0187 OIOX) 0194 0197 0201 O2O4 0208 021 1 O2I5 12 0219 O222 0226 0230 0233 0237 024! 0245 0249 0252 13 0256 0260 0264 0268 0272 0276 0280 0285 0289 0293 I 14 0297 0301 0306 0310 3 H 0319 0323 0327 0332 0336 15 0.0341 0.0345 0.0350 0.0354 0-0359 0.0364 0.0368 0.0373 0.0378 0.0383 I 16 0387 0392 0397 0402 0407 0412 0417 0422 0427 0432 17 0437 0442 0447 0452 0458 0463 0468 0473 0479 0484! 18 0489 0495 0500 0506 0511 0517 0522 0528 534 0539 1 19 0545 0551 0556 0562 0568 0574 0579 0585 0591 0597 | 2O 0.0603 0.0609 0.0615 0.0621 0.0627 0.0633 0.0639 0.0646 0.0652 0.0658 i 21 0664 0670 0677 0683 0689 0696 0702 0709 0715 0722 ! 22 0728 735 0741 0748 0755 0761 0768 0775 0788 i 23 0795 0802 0809 0816 0822 0829 0836 0843 0850 0857, 24 0865 0872 0879 0886 0893 0900 9 08 0915 0922 0930 ; 25 0.0937 0.0944 0.0952 0.0959 0.0967 0.0974 0.0982 0.0989 0.0997 0.1004 26 IOI2 IO2O 1027 1035 1043 1051 1058 1066 1074 1082 27 IOOX) 1098 1106 1114 1 122 1130 II 3 8 1146 H54 1162 28 II7I 1179 1187 1195 I2O4 1212 1220 1229 1237 1245 ' 29 1262 1271 1279 1288 1296 1305 1314 1322 1331 3O 0.1340 0.1348 0.1357 0.1366 - I 375 0.1384 o. 393 0.1401 O.I4IO 0.1419 31 1428 H37 1446 H55 1464 H74 1492 I5O1 1510 32 33 1520 1613 1529 1623 1538 1632 1547 1642 i|57 1652 1566 1661 575 671 Io8o 1594 1690 1604 1700 34 1710 1719 1729 1739 J749 1759 769 1779 1789 1798 35 o.i 808 O.l8l9 0.1829 0.1839 0.1849 0.1859 0.1869 0.1879 0.1889 0.1900 36 1910 1920 1930 1941 195 * 1961 1972 1982 1993 2003 37 2014 2O24 2035 2045 2056 2066 2077 2088 2098 2109 38 2120 2131 2141 2152 2163 2174 2185 2196 2207 2218 39 2229 2240 2251 2262 2273 2284 2295 2306 2317 2328 40 0.2340 0.2351 0.2362 0-2373 0.2385 0.2396 0.2407 0.2419 0.2430 0.2441 41 2453 2464 2476 2487 2499 2510 2522 2534 2545 2557 42 2569 2580 2592 2604 2615 2627 2639 2651 2663 2075 43 2686 2698 2710 2722 2734 2746 2758 2770 2782 2794 44 2807 2819 2831 2843 2855 2867 2880 2892 2904 2917 j I 45 O.2929 0.2941 0.2954 0.2966 0.2978 0.2991 0.3003 0.3016 0.3028 0.3041 46 353 3066 379 3091 3 I0 4 3116 3129 3142 3155 3167 47 48 49 3 I80 3309 3439 3193 3322 3453 3206 3335 3466 3218 3348 3479 3231 3492 3244 3374 3506 32^7 337 3270 3400 3532 3283 3413 3545 3296 ; 3426 3559 50 0.3572 0.3586 0-3599 0.3612 0.3626 0.3639 0.3653 0.3666 0.3680 0-3693 51 3707 3720 3734 3748 376i 3775 3789 3802 3816 3830 52 3843 3857 3871 3885 3899 3912 3926 3940 3954 3968 53 3982 3996 4010 4024 4038 4052 4066 4080 4094 4108 | 54 4122 4I5 4165 4179 4193 4207 4221 4236 4250 55 0.4264 0.4279 0.4293 0.4307 04322 0.4336 0-435 0.4365 0-4379 0-4394 56 57 4408 4554 4423 4568 4437 4583 4452 4598 4466 4612 4627 4495 4642 45 10 4656 4524 4671 4686 | 58 4701 4716 473 4745 476o 4775 4790 4805 4820 4835 59 4850 4865 4880 4895 4910 4925 4940 4955 497 49% i TABLE XVIII. 85 Decimal Parts and their Multiples of a Day. Hrs. Dec'l Parts nf Multipliers. Hrs. Dec'l Parts of Multipliers. vJl Day. 2 3 4 5 6 7 8 9 ! Day. j 3 4 5 6 7 8 9 0.0 o.ooo o.oo o.oo 0.00 0.00 o.oo o.oo o.oo o.oo ! 12.O 0.500 I.OO 1.50 2.OO 2.50 3.00 3-5o 4.00 4-5 2 008 02 03 03 04 05! 06 07 08 2 508 02 53 3 54 5 56 07 58 4 017 03 05 07 08 IO 12 13 15 4 517 3 55 07 58 10 62 13 65, 6 025 05 08 IO 13 15 18 20 j 23 6 525 05 5* 10 63 15 68 20 73 8 033 07J 10 13 17 20 23 27 30 8 533 07 60 13 67 20 73 27 1.0 0.042 0.080.13 0.17 0.21 0.25 0.29 0.330.38 13.O 0.542 i. 08 1.63 2.17 2.71 3-25 3-79 4-33 4.88 2 050 Io i *5 20 25 3 35 4 45 2 550 10 6 5 20 75 3 85 40 95 4 058 12 18 23 2 9 35 4 1 47 53 | 4 558 12 68 23 79 35 47 5-03 6 067 13 20 27 33 40 47 53 60 6 567 '3 7o| 27 83 40 97 53 10 8 075 15 23 30 37 45 53 60 68 8 575 15 73 3 87 45 4-03 60 18 2.0 0.083 0.170.25 0-33 0.42 0.50 0.58 0.670.75'! 14.0 0-583 1.17 1.75 2-33 2.92 3-5 4.08 4.67 5-25 2 092 18 28 37 46 55 64 73 83 2 592 18 78 37 96 55 H 73 33 4 IOO 20 3 40 50 60 70 80 90 4 600 20 80 40 3.00 60 20 80 40 6 108 22 '33 43 65 76 87! 98 6 608 22 83 43 04 65 26 87 48 8 117 23 35 47 58 70 82 93 J -5 8 617 23 85 47 08 70 32 93 55 3.0 0.125 0.25 0.38 0.50 0.63 0-75 0.88 1.001.13 15.O 0.625 1.2 5 1.88 2.50 3-13 3-75 4-38 5.00 5.63 2 27 40 53 67 80 93 07 20 2 633 27 90 53 17 80 43 07 70 4 142 28 43 57 71 85 99 13 28 4 642 28 93 57 21 85 49 13 78 6 150 3 45 60 75 90 1.05 201 35 6 650 3 95 60 25 90 55 20 85 8 158 32 48 63 79 95 ii 27 43 8 658 32 98 63 29 95 61 27 93 4.0 0.167 o-33 0.50 0.67 0.83 I.OO 1.17 1.33 1.50 16.0 0.667 i-33 2.00 2.67 3-33 4.00 4.67 5-33 6.00 2 175 35 53 70 87 05 23 40| 58 2 675 35 03 37 05 73 40 08 4 37 55 73 91 10 28] 47! 65 4 683 37 05 73 41 10 78 47 15 6 192 38 77 95 15 34 53 73 6 692 38 08 77 45 15 84 53 23 8 200 40 6 O 80 l-OO 20 40 60 80 8 700 40 10 80 50 20 90 00 3 5.0 0.208 0.42 0.63 0.83 1.04 1.251.461.67 I.881J17.0 0.708 1.42 2.13 2.83 3-54 4-25 4.96 5.67 6.38 2 217 43 65 87 08 30 52 73 95 2 717 43 15 87 I 8 30 5.02 73 45 4 225 45 68 90 12 35 58 80 2.03! 4 725 45 18 90 62 35 08 80 53 6 233 47 70 93 17 40 6 3 87 10 6 733 47 20 93 67 40 13 87 00 8 242 48 73 97 21 45 69 93 i8|| 8 742 48 23 97 7i 45 19 93 68 6.0 0.250 0.50 0-75 I.OO 1.25 1.50 i-75 2.00 2.25 18.0 0.750 1.50 2.25 3-oo 3-75 4-50 5-25 6.00 6-75 2 258 5 2 78 3 29 55 81 07 331 2 758 S 2 28 03 79 55 3 1 07 83 4 26 7 53 80 07 33 60 87 13 40 4 767 53 30 07 83 60 37 90 6 275 55 83 10 37 65 93 20 48 6 775 55 33 10 87 65 43 20 98 8 283 57 8 5 13 42 70 98 27 55 8 783 57 35 13 92 70 48 27 7-05 ro 0.292 0.58 0.88 1.17 1.46 i-75 2.04 2-33 2.63 190 0.792 1.58 2.38 3-'7 3-96 4-75 5-54 6-33 7-13 2 300 60 90 20 50 80 10 40 70 2 800 60 40 20 4.00 80 60 40 20 4 3 08 62 93 23 54 85 16 47 78 4 808 62 43 2 3 04 85 66 47 28 6 63 95 27 58 90 22 53 85 6 817 63 45 27 08 90 7 2 53 35 8 325 65 98 30 62 95 28 60 93 8 825 65 48 30 12 95 78 00 43 8.0 0-333 0.67 I.OO i-33 1.67 2.00 2-33 2.67 3-oo 20.0 0-833 1.67 2.50 3-33 4.17 5.00 5-83 6.67 7-50 2 342 68 03 37 71 05 39 73 08 2 842 68 53 37 21 05 89 73 58 4 350 7o 5 40 75 10 45 80 15 i 4 850 70 55 40 25 io| 95 80 65! 6 358 72 08 43 79 15 87 23 6 858 72 58 43 2 9 15 6.01 87 73 8 367 73 IO 47 83 20 57 93 3 8 867 73 60 47 33 20 07 93 80 9.0 0.375 -75 1.13 1.50 1.87 2.25 2-63 3.00 3.38 21.0 0.875 i-75 2.63 3-50 4-37 5-25 6.13 7.00 7.88 2 383 77 I ; 53 92 30 68 07 45 2 883 77 65 53 42 3 18 07 95 4 6 392 400 78 80 20 57 60 96 2.OO 35 40 80 13 20 8 4 6 892 900 i 68 70 46 50 35 4 24 3 13 20 8.03 10 8 408 82 2 3 63 04 45 86 27 68 8 908 82 73 63 54 45 36 27 18 10.0 0.417 0.83 1.25 1.67 2.08 2.50 2.92 3-33 3-75 22.0 0.917 1.83 2.75 3-67 4.58 5-50 6.42 7-33 8.25 2 425 85 28 70 12 55 98 40 83 2 925 85 78 70 62 55 48 40 33 4 433 87 3 73 17 60 3-3 47 90 4 933 87 80 73 67 60 53 47 40 6 44 2 88 33 77 21 65 09 53 98 6 942 88 f 3 V 7 1 65 59 53 48 8 45 90 35 80 25 70 15 60 4-05 8 95 90 85 80 75 70 65 60 55 110 0.458 0.92 1.38 1.83 2.2 9 2-75 3-21 3-67 4-13 23.0 0.958 1.92 2.88 3-83 4-79 5-75 6.71 7-67 8.63 2 467 93 40 87 33 80 2 7 73 20 2 967 93 90 87 83 80 77 73 70 4 475 95 43 90 37 85 33 80 28 4 975 95 93 90 87 85 80 78 6 483 97 45 93 42 90 38 87 35 6 983 97 95 93 92 90 88 87 85 8 492 98 47 97 46 95 44 93 43! 8 992 98 97 97 96 95 94 93 93 12.O 0.500 I.OO 1.50 2.OO 2.50 3-oo 3-5 4.004.50 24.O I.OOO 2.OO 3.00 4.00 5.00 Ob 7.00 S.oo 9.00! 1 86 TABLE XIX. 1 Decimal Equivalents to two places of Common Fractions. Numerator of Fraction. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 D. 10 00 10 20 3 40 5 60 70 80 90 " 10 11 oo 09 18 27 36 45 54 64 73 82 11 12 00 08 17 25 33 42 5 58 67 75 83 91 12 13 00 08 15 23 31 38 46 54 61 69 77 85 92 13 14 oo 07 H 21 29 36 43 50 57 64 7i 78 86 93 14 15 00 07 '3 20 27 33 40 47 53 60 67 73 80 87 93 15 16 oo 06 12 19 25 3 1 37 44 50 56 62 69 75 Si 87 94 16 17 00 06 12 18 23 35 41 47 53 59 *5 7i 76 82 88 94 17 18 oo 06 II 17 22 28 33 39 44 5 56 61 67 7 2 78 83 89 94 18 19 oo 05 IO 16 21 26 32 37 42 47 53 58 63 68 74 79 84 89 95 19 2O 00 05 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 S o S 90 95 2O 21 oo 05 09 14 IQ 24 29 33 38 43 48 52 57 62 67 7i 76 81 86 90 95 21 22 00 04 09 14 18 23 27 3 2 36 4i 45 5 54 59 64 68 73 77 82 86 9i 22 23 oo 04 09 *3 *7 22 26 3 35 39 43 48 5 2 56 61 65 70 74 78 83 87 23 24 00 04 08 12 17 21 25 29 33 37 42 46 So 54 58 62 67 7i 75 79 83 24 25 oo 04 08 12 16 20 24 28 3 2 36 40 44 48 52 56 60 64 68 72 76 80 25 26 00 04 08 II 15 19 23 27 3 1 35 38 43 46 50 54 58 61 65 69 73 77 26 27 00 04 07 II 15 18 22 26 3 33 37 4i 44 48 52 56 59 63 67 70 74 27 28 oo 04 07 II 14 18 21 25 29 32 36 39 43 46 50 54 57 61 64 68 7i 28 29 00 3 07 10 14 !7 21 24 28 3 1 34 38 4i 45 48 52 55 59 62 65 69 29 3O 31 oo 00 03 3 3 IO 10 13 !3 ;i 20 19 23 2 3 27 26 30 33 3 2 37 35 40 39 43 42 47 45 5 48 53 52 57 55 60 58 63 61 67 64 30 31 32 00 03 06 09 12 16 19 22 2 5 28 3* 34 37 4i 44 47 50 53 56 59 62 32 33 oo 03 06 09 12 15 18 21 24 27 3<> 33 36 39 42 45 48 5i 54 5 f 61 33 34 oo 3 06 09 12 15 18 21 2 3 26 29 32 35 38 4i 44 47 50 53 56 59 34 35 oo 03 06 09 II 14 !7 20 23 26 29 3 1 34 37 40 43 46 49 5 1 54 57 35 36 00 3 06 08 II 14 17 19 22 2 5 28 3 1 33 36 39 42 44 47 5 53 56 36 37 00 03 05 08 II 13 16 19 22 24 27 30 3 2 35 38 40 43 46 49 5 1 54 37 38 oo 03 05 08 IO J 3 16 18 21 24 26 2 9 32 34 37 39 42 45 47 50 53 38 39 oo 3 05 08 10 13 15 18 2O 2 3 26 28 3i 33 36 38 4i 43 46 49 5 1 39 4O oo 02 5 07 IO 12 15 17 20 22 25 27 30 32 35 37 40 42 45 47 50 40 42 oo 02 5 07 09 12 14 17 19 21 24 26 29 3i 33 36 38 40 43 45 48 42 44 00 02 04 06 09 II 14 16 18 2O 2 3 2 5 27 29 32 34 36 39 4i 43 45 44 46 oo 02 04 06 09 II 13 15 17 20 22 24 26 28 30 33 35 37 39 4i 43 46 48 oo 02 04 06 08 IO 12 15 17 19 21 23 25 27 29 3 33 35 37 39 42 48 1 50 00 02 04 06 08 10 12 14 16 18 2O 22 24 26 28 3 32 34 36 38 40 50 I 52 oo 02 04 06 08 IO II 13 15 * 7 19 21 23 25 27 20 3 1 33 35 36 38 52 ! 54 00 02 04 06 07 09 II *3 15 17 18 2O 22 24 26 2o 30 3i 33 35 37 54 56 00 02 04 05 07 09 II 12 H 16 18 20 21 23 25 27 29 3 32 34 3 6 56 58 oo 02 3 05 07 09 10 12 14 15 17 19 21 22 24 26 27 29 3 1 33 34 58 60 00 02 03 5 07 08 IO 12 '3 15 17 18 2O 22 23 25 27 28 3 3 2 33 60 62 oo 02 03 05 06 08 10 II 13 14 16 18 19 21 23 24 26 27 29 3i 32 62 64 oo 02 3 05 06 08 09 II 12 i.: 16 17 18 20 22 23 25 26 28 3 3 1 64 66 oo OI 3 5 06 08 09 II 12 A 15 17 18 2O 21 23 24 26 27 3 66 68 00 01 3 04 06 07 09 10 12 2 15 16 18 19 21 22 23 25 26 i 28 29 68 7O oo 01 3 04 06 07 09 IO II 2 14 16 17 JO 2O 21 23 24 26 27 29 7O 72 00 01 3 04 06 07 08 10 II 2 H 15 17 18 19 21 22 24 25 ! 26 28 72 74 00 01 3 04 05 07 08 09 II 2 *3 15 16 18 19 20 22 23 24 26 27 74 76 oo OI 3 04 i 07 08 09 10 2 13 H 16 17 18 2O 21 22 24 25 26 76 78 00 OI 03 04 Of 06 08 09 IO II *3 14 15 7 18 19 2O 22 23 24 26 78 80 oo 01 02 04 05 06 07 09 10 II 12 14 15 16 17 19 20 21 22 24 25 8O 82 oo OI 02 04 ! 06 07 08 IO II 12 *3 15 16 17 18 19 21 22 23 24 N2 84 oo 01 02 04 l 06 07 08 09 II 12 !3 14 15 17 18 19 20 21 23 24 84 86 00 01 02 3 05 06 07 08 09 IO 12 !3 H 15 16 17 19 20 21 22 23 86 88 oo 01 02 03 04 06 07 08 09 10 II 12 H 15 16 '7 18 19 2O 22 23 88 9O oo 01 02 3 04 06 07 08 09 10 II 12 13 '4 15 17 18 19 20 21 22 90 92 oo 01 02 3 ot 05 06 08 09 IO 11 12 *3 '4 15 16 17 18 20 21 22 92 94 00 01 02 3 Qi Of 06 07 08 09 II 12 13 H 15 16 17 18 19 20 21 94 96 00 OI 02 03 Ot Of 06 07 08 09 10 II 12 J 3 15 16 17 18 19 20 21 96 98 oo OI 02 03 Ot Of 06 07 08 oc 10 II 12 13 H 15 16 17 18 19 20 98 100 00 01 02(03 ot 05 06 07 08 09 10 II 12 13 H 15 16 17 18 19 2O 100 TABLE XIX. 87 Decimal Equivalents to two places of Common Fractions. Numerator of Fraction. D. 1 1 i D. 20 21 22 23 21 25 26 2728 29 30 31 32 33 31 35 36 37 3 39 10 20 20 21 9S 21 ! 22 9S i 22 i 23 87 91 95 1 23 21 83 87 92 96 21 25 83 84 88 92 96 25 i 26 77 81 8S 88 92 96 26 | 27 74 78 81 8S 89 9} 95 27 2 7S 78 82 85 89 93 96 1 28 29 6} 72 79 83 83 93 93 97 29 30 67 70 73 77 83 8} 87 90 93 97 i 30 31 64 68 74 77 Si 84 87 90 94 97 31 32 62 65 69 7 2 7S 78 Si 84 87 9i 94 97 [ ' 32 33 or 64 67 70 79 82 SS 88 91 94197 33 31 S) 62 65 68 7 1 73 79 82 85 88 91 94 97 31 35 57 63 61 65 69 7i 74 77 83 83 85 89 91 94 97 35 36 5*1 61 64 67 6? 72 1 75 78 81 83 86 89 92 94 97 36 37 S4 57 59 62 6S 68 7o 73 76 78 Si 84 86 89 92 95 97 37 3 S3 SS 58 60 63 65 68 71 74 76 79 81 84 87 89 92 95 97 38 39 5 1 54 5^ 59 61 64 67 69 72 74 77 79 82 8 5 87 90 92 95! 97 39 10 S 3 S2 SS S7 60 62 6S 67 70 72 7S 77 80 82! 85 87 90 | 92 i 95 97 10 U 4> 52 5S 57 59 62 64 67 69 74 76 78 j 81 86 88 [ 90 93 9S 12 11 45 4^ 5^ SS 57 59 61 64 65 68 70! 73 7S 77 83 82 84 So 89 91 11 16 43 43 52 54 5 5 59 61 63 6S 67(69 72 74 76 78. So 83 8S 87 16 18 42 44 45 48 53 52 54 5 5 58 60 62 65 67 69 73 75 77 79 Si 83 18 50 40 42 44 45 48 50 S 2 54 S6 58 60 62 64 65 68 70 72 74 76 78 80 50 5^ 33 43 42 44 43 48 50 S4 55 S8 60 61 63 6S 67 69 7i 73 75 77 52 51 37 39 41 43 44 45 48 S o 52 54 56 S7 59 61 63 6S 67 69 7 72 74 51 56 30 37 39 |4i 43 4S 46J48 So 52 S4 SS 57 59 61 62 64 65 68 56 5& 34 38 i 4 3 i 4 l 43 45 47 48 53 55 57 59 60 62 64 65 67 69 58 69 33 35 37 38 43 42 43 45 47 48 5 S2 S3 55 57 S8 60 62 63 65 6/ 60 ! 62 32 34 35137 39 40 42i 44 4) 47 48 50 52 53 55 56 S8 60 61 63 64 62 I 61 31 33 34 33 37 39 41 42 44 4S 47 48! 50 52 S3 55 56 58 59 61 62 61 66 33 32 33 35 38 31 4i 42 44 4S 47 48 So 52 S3 SS 56 ss S9 61 66 63 29 3 32 34 35 37 38(40 4i 43 44 46 47 48 5 5 1 53 54 56 57 59 68 70 29 3 3 3i 33 34 3 5 37 39 40 4 1 43 44 46 47 48 5o 5i 53 54 56 57 70 1 72 28 29 31 32 33 35 37 39 40 42 43 44 46 47 49 5 5i 53 S4 56 72 71 27 28 33 3 1 32 34 35 36 38 39 40 42 43 45 46 47 49 Si S3 54 71 76 23 28 29 30 3 2 33 34 35 37 38 39 41 42 43 4S 46 47 49 5o Si 53 76 78 23 27 28 29 3 1 3 2 33 35 36 37 38 40 41 42 44 45 46 47 49 78j 80 25 23 27 29 30 31 32 34 35 36 37 39 40 41 42 44 45 46 47 49 So 80; 82 24 23 27 28 29 3 32 33 34 35 37 38 39 40 43 44 45 46 48 49 82 81 24 2 5 2t> 27 29 30 32 33 34 36 37 38 39 40 42 43 44 45 46 48 81 86 23 24 25 27 28 29 3 3i 32 34 35 3 37 38 39 41 42 43 44 4S 46 86 88 23 24 2 5 26 27 28 23 3 1 32 33 34 35 36 37 39 40 41 42 43 44 45 88 90 22 23 24 25 27 28 29 30 31 32 33 34 36 37 38 39 40 41 42 43 44 90 92 22 23 24 25 23 27 28 29 30 3 1 33 34 35 36 37 38 39 40 42 43 92 91 21 22 23 24 25 27 28 29 3 31 32 33 34 35 36 37 38 39 40 41 42 91 93 21 22 23 24 25 26 2 7 28 29 30 3 1 32 33 34 35 36 37 38 39 41 42 96 98 20 21 22 23 24 25 23 27 29 3 32 33 34 35 36 37 38 39 40 98 l.OO 2O 21 22 2 3 24 25 26 27 28 29 3 3i 32 33 34 35 36 37 38 39 40 100 . 1 5 10 15 20 25 3} 33 40 45 59 55 60 65 70 75 80 85 90 95 103 D. 125 OI 04 08 12 16 20 24 28 32 36 40 44)48 52 56 60 64 68 72 76 So 125! J176 OI 1 03 06 08 ii 14 17 20 23 2S 28 34 37 40 43 46 48 Si 54 57 1761 3O1 00 j 02 03 5 07 08 10 12 13 15 17 18 20 22 23 2 5 26 28 30 32 33 301 D. 100, 105 110113 120 123 (30 135 140 143 150 155 160 163 170 175 209 225 250275 300 D. 125 So 84 88 i 92 96 I 125 176 57| 60 63165 68 71 74 77 80 82 8S 88 91 94 97 176 3O1J 33 35 36 38 40 4i 43 45j46 48 50 52 53 55 56 58 66 74 83 9i 301 88 TABLE XIX. Decimal Equivalents to two places of Common Fractions. Numerator of Fraction. 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 42 95 97 42 44 93 95 98 44 46 8? 90 9i 93 96 98 46 48 83 85 87 89 92 94 96 98 48 . 50 80 82 84 86 88 90 92 94 96 98 50 52 77 79 81 83 85 86 88 9 92 94 96 98 52 54 74 76 78 8b 81 83 85 87 89 9i 93 94 96 98 54 56 7i 73 75 77 78 80 82 84 86 87 89 9i 93 95 96 98 56 58 69 7i 72 74 76 78 79 81 83 84 86 88 90 9i 93 95 97 98 58 60 67 68 70 72 73 75 77 78 80 82 83 85 87 88 90 Q2 93 95 97 98 6O 62 64 66 68 69 7i 73 74 76 77 79 Si 82 84 85 87 89 90 92 94 95 97 62 64 62 64 66 67 69 70 72 73 75 77 78 80 81 83 84 86 87 89 9i 92 94 64 66 61 62 64 65 67 68 70 7i 73 74 7 6 77 79 80 82 83 85 86 88 89 9i 66 68 59 63 62 63 65 66 68 69 7i 72 73 75 76 78 79 81 82 84 85 87 88 68 70 57 58 60 61 63 64 66 67 69 70 7i 73 74 76 77 78 80 81 83 84 86 70 72 56 57 58 60 61 62 64 65 67 68 69 7i 72 74 75 76 78 79 81 82 83 72 74 54 55 57 58 59 61 62 63 65 66 68 69 70 72 73 74 76 77 78 80 81 74 76 53 54 55 56 58 59 60 62 63 64 65 67 68 70 7i 72 74 75 76 78 79 76 78 5i 53 54 55 56 58 59 60 61 63 64 65 67 68 69 70 72 73 74 76 77 78 80 5 51 5 2 54 55 56 57 59 60 61 62 64 65 66 67 69 70 7i 72 74 75 80 82 49 So 51 5 2 54 55 56 57 59 60 61 62 63 65 66 67 68 69 7 72 73 82 i 84 48 49 5 51 5 2 53 55 56 57 58 59 61 62 63 64 65 67 68 69 70 7 1 84 86 46 48 4-Q 5 51 52 53 55 56 57 58 59 60 61 63 64 65 66 67 68 70 86; 88 45 46 48 49 50 5i 52 53 55 S^ 57 58 59 60 61 62 64 65 66 67 68 88! 90 44- 45 47 48 49 5o 51 5 2 53 54 S^ 57 58 59 60 61 62 63 64 66 67 90; 92 43 44 46 47 48 4 2 5o 51 52 53 54 55 56 58 59 60 61 62 63 64 65 92 1 94 42 44 45 46 47 48 49 5 5i 52 53 54 55 56 57 58 60 61 62 63 64 94 | 96 42 43 44 45 46 47 48 49 50 5 1 52 53 54 55 56 57 58 CO 60 61 62 96 j 98 4i 42 43 44 45 46 47 48 49 50 5 1 52 53 54 55 56 57 58 59 60 61 98 100 40 4i 42 43 44 45 46 47 48 49 50 5i 52 53 54 55 S^ 57 58 59 60 100 D. 63 61 62 iM > oo j. 61 63 69 70 n 72 73 74 75 76 77 78 79 80 D. 62 97 98 62! 04 9+ 95 97 9? 64) 66 9i 92 94 95 97 98 66 68 83 93 9i 93 94 9 5 97 98 681 70 85 87 89 93 91 93 94 96 97 9 3 70 72 83 85 86 83 89 90 92 93 94 96 97 99 72 74 81 82 84 85 86 88 89 9 2 93 95 96 97 99 74 i 76 79 SD 81 83 84 85 87 88 89 9i 92 93 95 96 97 99 76! 78 77 78 79 81 82 83 85 86 87 88 90 9i 92 93 95 96 97 99 78 80 75 7 6 77 79 80 81 82 84 85 86 87 89 90 9 1 92 94 95 96 97 99 80 82 73 74 75 77 78 79 80 82 83 84 85 87 88 89 90 9i 93 94 95 96 98 82 84 7i 72 74 75 76 77 78 80 81 82 83 84 86 87 88 89 90 92 93 94 95 84 86 70 7i 72 73 74 75 77 78 79 80 81 83 84 85 86 87 88 89 9i 92 93 86! ! 88 68 69 70 72 73 74 75 76 77 78 80 81 82 83 84 85 86 87 89 90 9i 88 90 67 68 69 70 7i 72 73 74 76 77 78 79 80 81 82 83 84 86 87 88 89 9O : 92 65 65 67 63 69 7i 72 73 74 75 76 77 78 79 80 81 83 84 85 86 8 7 92 91 64 65 66 67 68 69 70 7 1 72 73 74 76 77 78 79 80 81 82 83 84 85 94 96 62 63 65 66 67 68 69 70 7i 72 73 74 75 76 77 78 79 80 81 82 83 96 93 61 62 63 64 65 66 67 68 69 70 7i 73 73 74 75 76 77 78 80 81 82 98 199 60 61 62 1L JH 65 66 Jli 68 69 TO 7i Zi Zi 74 21 21 2L 21 79 80 1OO D 1*9 Si "sT ii 81 83 "8? 81 83"[~89 91) 91 92 93 91 95 9 97 98 99 100 D. 82 98 99 82 81 95 96 9 8 99 84 86 93 94 95 96 98 99 86 88 9i 92 93 94 95 97 98 99 88 90 89 90 9i 92 93 94 96 97 98 99 90 92 87 88 89 90 91 92 93 95 96 97 98 99 92 91 85 86 87 88 89 90 91 92 94 95 96 97 98 99 94 96 83 84 85 85 87 88 89 92 93 94 95 96 97 98 99 96 94 82 83 84 85 86 87 88 89 90 9i 92 93 94 95 96 97 98 99 98 100 80 81 82 83 84 85 85 87 88 89 90 9i 92 93 94 95 96 97 98 99 100 TABLE XX. 89 Proportional Parts. I Tenths and their Multiples. 1 Tenths and their Multiples. 1 01 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 j O.I 0.2 0.3 0.4 0.5 0.6 O.7 0.8 0.9 1 O.I 0.2 0-3 0.4 o-5 0.6 0.7 0.8 0.9 56 5-6 1 1.2 16.8 22.4 28.0 33-6 39-2 44.8 504 2 O.2 0.4 o!6 0.8 I.O 1.2 1.4 1.6 1.8 57 5-7 II.4 17.1 22.8 28.5 34-2 39-9 45-6 5*'3 ! 3 -3 0,6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 58 5-8 11.6 17.4 23.2 29.0 34-8 40.6 46.4 52.2; 4 0.4 0.8 1.2 1.6 2.O 2.4 2.8 3-2 3-6 59 5-9 11.8 17.7 23-6 29-5 354 41-3 47-2 53.1 ! 5 I.O !-5 2.0 2-5 3-5 4.0 4-5 60 6.0 12.0 18.0 24.0 30.0 36.0 42.0 48.0 54-0 6 0.6 1.2 1.8 2.4 3- 3-6 4.2 4.8 54 61 6.1 12.2 18.3 24.4 30-5 36.6 42.7 48.8 54-9; 7 0.7 1.4 2.1 2.8 3-5 4.2 4.9 5.6 6-3 62 6.2 12.4 18.6 24.8 31.0 37-2 434 49.6 55-8 j 0.8 1.6 2.4 3-2 4.0 4.8 5.6 6.4 7-2 63 6-3 12.6 18.9 25.2 37-8 44.1 54 56.7 9 0.9 1.8 2.7 3.6 4-5 54 6.3 7.2 8.1 64 6.4 12.8 19.2 25.6 32.0 384 44.8 51.2 57.6 1O I.O 2.O 3- 4.0 6.0 7.0 8.0 9.0 65 6-5 13.0 19-5 26.0 32.5 39-0 45-5 52.0 58.5 11 i.i 2.2 3-3 44 5-5 6.6 7-7 8.8 9-9 66 6.6 13.2 19-8 26.4 33- 39.6 46.2 52.8 594 i.2 2.4 3.6 4.8 6.0 7.2 8.4 9.6 10.8 67 6.7 13-4 20.1 26.8 33-5 40.2 46.9 53-6 60.3 13 J .-5 2.6 3-9 5-2 6.5 7.8 9.1 10.4 11.7 68 6.8 13.6 20.4 27.2 34-o 40.8 47.6 544 61.2 14 1.4 2.8 4.2 5-6 7.0 8.4 9.8 1 1.2 12.6 69 6.9 13.8 20-7 2 7 .6 34-5 41.4 48.3 55-2 62.1 15 i-5 3- 4-5 6.0 7-5 9.0 10.5 12.0 13-5 70 7.0 14.0 21.0 28.0 35-o 42.0 49.0 56.0 63.0 16 1.6 3-2 4.8 6.4 8.0 9.6 11.2 12.8 14.4 71 7-1 14.2 21-3 28. 4 35-5 42.6 49-7 56.8 63-9 17 i-7 34 5- 1 6.8 8-5 IO.2 II.9 I 3-6j 15-3 72 7-2 14.4 21.6 28.8 36.0 43-2 54 57.6 64.8 18 1.8 3-6 54 7.2 9.0 10.8 12.6 14.4 16.2 73 7-3 14.4 21.9 29.2 36.5 43-8 5 1 - 1 584 65-7! 19 1.9 3-8 5-7 7-6 9-5 11.4 13.3 15.2 17.1 74 .74 14.8 22.2 29.6 37-o 444 51.8 59-2 66.6 20 2.0 4.0 6.0 8.0 IO.O 12.0 14.0 1 6.0 1 8.0 75 7-5 15.0 22-5 30.0 37-5 45> o 52-5 60.0 67-5 21 2.1 4.2 6-3 8.4 10.5 12.6 14.7 16.8! 18.9 76 7.6 15.2 22.8 304 38.0 45-6 53-2 60.8 68.4 22 2.2 44 6.6 8.8 II.O 13.2 154 17.61 19.8 77 7-7 154 23.1 30.8 38.5 46.2 53-9 61.6 69-3 |23 2-3 4.6 6.9 9.2 "5 I 3 .8 16.1 18.4 20.7 78 7.8 15-6 234 31.2 39-o 46.8 54.6 62.4 70.2 24 2.4 4.8 7.2 9.6 I2.O 144 16.8 19.2121.6 79 7-9 15-8 23.7 3L6 39-5 474 55-3 63.2 71.1 25 2-5 5- 7-5 IO.O 12.5 17-5 2O.O 22.5 80 8.0 16.0 24.0 32.0 40.0 48.0 56.0 64.0 72.0 26 2.6 5-2 7-8 10.4 13.0 15.6 18.2 20.8 234 81 8.1 16.2 24-3 324 40.5 48.6 56.7 64.8 72.9 27 2.7 54 8.1 10.8 16.2 18.9 21.6 24.3 82 8.2 16.4 2 4 .6 32-8 41.0 49.2 574 65.6 73-8 28 2.8 5.6 8.4 II.2 I 4 .0 i6.8j 19.6 22.4 25.2 83 8-3 16.6 24.9 33-2 4i.5 49.8 58.1 66.4 74-7 29 2.9 5-8 8.7 n.6 14-5 17.4)20.3 23.2 26.1 84 8.4 16.8 25-2 33-6 42.0 504 58.8 67.2 75-6 3O 3-0 6.0 9.0 I2.O 15.0 iS.Oj 21.0 24.0 27.0 85 8.5 17.0 25-5 34-0 42-5 51.0 59-5 68.0 76.5 i 31 3- 1 6.2 9-3 12.4 18.6 21.7 24.8 27.9 86 8.6 17.2 25.8 344 43- 51.6 60.2 68.8 774 32 3-2 6.4 9.6 12.8 16.0 19.2 22.4 25.6 28.8 87 8.7 17.4 26.1 34-8 43-5 52.2 60.9 69.6 78.3 33 3-3 6.6 9.9 13.2 16.5 19.8 23.1 26.4 29.7 88 8.8 17.6 26.4 35-2 44.0 52.8 61.6 70.4 79.2 34 34 6.8 10.2 13.6 17.0 20.4 23.8 27.2 30.6 89 8-9 17.8 26. 7 35-6 44-5 534 62.3 71.2 80. i 35 3-5 7.0 10.5 14.0 17-5 2 1. d 24.5 28.0 3L5 90 9.0 18.0 27.0 36-0 45- 54o 63.0 72.0 81.0 36 3-6 7-2 10.8 14.4 18.0 21.6! 25.2 28.8 324 91 9.1 18.2 27.3 364 45-5 54-6 63-7 72.8 81.9 37 3-7 74 n. i 14.8 18.5 22.2! 25.9 29.6 33-3 92 9.2 18.4 27.6 36.8 46.0 55-2 64.4 73-6 82.8 ; |38 3-8 7-6 11.4 15.2 19.0 22.8 26.6 304 34-2 93 9-3 18.6! 27.9 37-2 46.5 55-8 65.1 744 837 39 40 3-9 4.0 7-8 8.0 11.7 12.0 15.6 16.0 19-5 20.0 234 24.0 27-3 28.0 31.2 32.0 35-i 36.0 94 95 94 9-5 18.8 19.0 28.2 28.5 37-6 38.0 47.0 47-5 564 65.8 66.5 75-2 76.0 84.6 85.5 41 4.1 8.2 12-3 16.4 20-5 2 4 .6 28.7 32.8 36.9 96 9.6 19.2 28.8 384 48.0 57-6 67.2 76.8 86.4 42 4-2 8.4 12.6 16.8 21.0 25.2 29.4 33.6 37-8 97 9-7 19.4 29.1 38.8 48.5 58.2 67.9 77-6 87.3 43 4-3 8.6 12.9 17.2 21.5 25.8 30.1 344 38.7 98 9,8 19.6 29.4 39-2 49.0 58.8 68.6 78.4 88.2 44 44 8.8 13.2 17.6 22.O 26.4 30-8 35-2 39-6 99 9-9 19.8 29.7 39-6 49-5 594 69-3 79-2 89.1 45 4-5 9.0 13-5 1 8.0 22.5 27.0 36.0 40.5 100 IO.O 20.0 3O.O 40.0 50.0 60.0 70.0 80.0 90.0 46 4.6 9-2 13-8 18.4 23.0 27.6 32.2 36.8 41.4 101 10. 1 2O.2 30.3 40.4 50.5 60.6 70.7 80.8 90.9 !47 4-7 94 I4.I 18.8 23-5 282 32-9 37-6 42-3 102 IO.2 20-4 30.6 40.8 51.0 61.2 71.4 81.6 91.8: ! 48 4.8 9.6 144 19.2 24.0 28.8 33-6 384 43-2 103 10.3 20.6 30-9 41.2 61.8 72.1 82.4 92.7 1 49 4-9 9.8 14.7 19.6 24.5 29.4 34-3 39-2 44.1 104 10.4 20.8 31.2 41.6 52.0 62.4 72.8 83.2 93-6 50 IO.O 15-0 20.0 25.0 3O.O 40.0 45-0 105 10.5 21.0 31-5 42.0 52.5 63.0 73-5 84.0 94-5 51 52 53 5-2 5-3 10.2 10^6 15-3 15.6 15-9 20.4 20.8 21.2 25^ 26.0 26.5 30.6 31.2 31.8 35-7 364 37-i 40.8 41.6 424 45-9 46.8 47-7 1O6 107 108 10.6 10.7 10.8 21.2 21. 4 21.6 31-8 32.1 32-4 42.4 42.8 43-2 53-o 53-5 54-0 63.6 64.2 64.8 74-2 74.9 75.6 84.8 85.6 86.4 97-2 54 54 10.8 16.2 21.6 27.0 324 37-8 43-2 48.6 1O9 10.9 21.8 32.7 43-6 54-5 654 76.3 87.2 98.1 55 5-5 II.O 16.5 22.0 27-5 33- 38.5 44.0 49.5 110 II.O 22.d 33.0 44.0 55-0 66.0 77.0* 88.0 99.0, TABLE XXL Squai 's of Nnmbcrs increasing by Tenths from O.O to 10O.9. No. O.O 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 09 Diff. 0.01 | o.o 0.0 0.0 O.I O.2 o-3 0.4 o-5 0.6 0.8 O.OI 1 I.O 1.2 1.4 1-7 2.0 2-3 2.6 2.9 3-2 3.6 03 2 4.0 4.4 4.8 5-3 5 .8 6-3 6.8 7-3 7-8 8.4 5 3 9.0 9 .6 10.2 10.9 1 1.6 12.3 13.0 '3-7 14.4 15.2 07 4 16.0 1 6.8 I 7 .6 18.5 19.4 20.3 21.2 22.1 23.0 24.0 09 i 5 25.0 26.0 27.0 28.1 29.2 30.3 3M 32.5 33-6 34-8 O.I I i 6 36.0 37-2 38.4 39-7 41.0 42.3 43-6 44.9 46.2 47-6 13 1 V 49.0 504 51.8 53-3 54-8 56.3 57-8 59-3 60.8 62.4 15 8 64.0 65.6 67.2 68.9 70.6 72-3 74-o 75-7 77-4 79-2 17 9 81.0 82.8 84.6 86.5 88.4 90-3 92.2 94.1 96.0 98.0 19 10 1 00.0 IO2.O 104.0 1 06. i 108.2 110.3 112.4 "4-5 116.6 118.8 0.21 11 12 121.0 144.0 123.2 146.4 125.4 148.8 127.7 !5i-3 130.0 153-8 132.3 156.3 134.6 158.8 136.9 161.3 igt 141.6 166.4 23 25 13 169.0 I7I.6 174.2 176.9 179.6 182.3 185.0 187.7 - 190.4 193.2 2 7 14 196.0 198.8 2OI.6 204.5 207.4 210.3 213.2 216.1 219.0 222.0 29 : 15 225.0 228.0 231.0 234-1 237-2 240.3 243-4 246.5 249.6 252.8 0.31 16 256.0 259.2 262.4 265.7 269.0 272.3 275.6 278.9 282.2 285.6 33 ' 17 289.0 292.4 295.8 299-3 302.8 306-3 309.8 3I3-3 316.8 320.4 35 ! 18 19 324.0 361.0 327.6 364-8 33 i - 2 368.6 334-9 372.5 338.6 376.4 342-3 380.3 346.0 384-2 349-7 388.1 353-4 392.0 357-2 396.0 37 : 39 ; 20 400.0 404.0 408.0 412.1 416.2 420.3 424.4 428.5 432.6 436.8 0.41 21 441.0 445-2 449-4 453-7 458.o 462.3 466.6 470.9 475-2 479.6 43 22 484.0 488.4 492.8 497-3 501.8 506.3 510.8 5I5-3 519.8 524.4 45 ' 23 529.0 533- 6 538.2 542.9 547-6 552.3 557-0 561-7 566.4 57 1 - 2 47 24 576.0 580.8 585-6 590.5 595-4 600.3 605.2 610.1 615.0 620.0 49 i 25 26 625.0 676.0 630.0 681.2 635- 686.4 640.1 691.7 645.2 697.0 650.3 702.3 6554 707.6 660.5 712.9 665.6 718.2 670.8 723.6 0.51 ; 53 27 28 29 729.0 784.0 841.0 734-4 789.6 8 4 t>.8 739-8 795-2 852.6 745-3 800.9 858.5 750.8 806.6 864.4 756.3. 812.3 870.3 761.8 818.0 876.2 767-3 Ilil 772.8 829.4 888.0 778.4 835-2 894.0 55 57 59 ; 30 900.0 906.0 912.0 918.1 924.2 930.3 93 6 -4 942.5 948.6 954-8 0.61 31 961.0 967.2 973-4 979-7 986.0 992.3 998.6 1004.9 ion. 2 1017.6 63 32 1024.0 1030.4 1036.8 1043-3 1049.8 1056.3 1062.8 1069.3 1075-8 1082.4 65 33 1089.0 1095.6 IIO2.2 1108.9 1115.6 1122.3 1129.0 XI 35-7 1142.4 1149.2 67 i 34 1156.0 1162.8 1169.6 1176.5 1183.4 1190.3 1197.2 1204.1 I2II.O 1218.0 6 9 : 35 1225.0 1232.0 1239.0 1246.1 1253.2 1260.3 1267.4 1274-5 I28I.6 1288.8 0.71 36 1296.0 1303.2 I3I0.4 I3I7.7 | 1325.0 1332.3 I339.6 1346.9 1354-2 1361.6 73 i 37 1369.0 1376.4 I383-8 I39I-3 '398.8 1406.3 1413.8 1421.3 1428.8 1436.4 75 : 38 1444.0 1451.6 1459.2 1466.9 1 1474.6 1482.3 1490.0 1497-7 I505-4 1513-2 77 39 1521.0 1528.8 1536.6 1544-5 1 1552.4 1560.3 1568.2 1576.1 1584.0 1592.0 79 1 I 40 1 600.0 1 608.0 1616.0 1624.1 1632.2 1640.3 1 1648.4 1656.5 I66 4 .6 1672.8 0.8 1 41 1681.0 1689.2 1697.4 1705-7 1714.0 1722.3 i 1730.6 1738-9 1747.2 !755-6 83 42 43 1764.0 1849.0 1772.4 1857.6 1780.8 1866.2 1789-3 1874.9 1797.8 1883.6 1806.3 1892.3 1814.8 1901.0 1823.3 1831.8 1918.4 1840.4 1927.2 85 87 44 1936.0 1944.8 1953-6 1962.5 1971.4 1980.3 1989.2 1998.1 2OO7.O 2016.0 89 ! 45 2025.0 2034.0 2043.0 2052.1 2061.2 2070.3 2079.4 2088.5 2097.6 2106.8 0.91 46 2116.0 2I25J.2 2134-4 2H3.7 2153.0 2162.3 2171.6 2180.9 2I9O.2 2199.6 93 47 2209.0 2218.4 2227.8 2237-3 2246.8 2256.3 2265.8 2275-3 2284.8 2294.4 95 48 2304.0 23I3.6 2323.2 2332-9 2342.6 2352-3 2362.0 237I-7 2381.4 2391.2 97 49 2401.0 2410.8 2420.6 2430-5 2440.4 2450-3 2460.2 2470.1 2480.0 2490.0 99 50 2500.0 2510.0 2520.0 2530.1 2540.2 2550-3 2560.4 2570-5 2580.6 2590.8 I.OI . I TABLE XXI. 91 Squares of Numbers increasing by Tenths from O.O to 1OO.9. No. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Diff. 0.01 50 2500.0 2510.0 2520.0 2530-1 2540.2 2550.3 2560.4 2570.5 2580.6 2590.8 I.OI 51 2601.0 2611.2 2621.4 2631.7 2642.0 2652.3 2662.6 2672.9 2683.2 2693.6 3 2704.0 2714.4 2724.8 2735-3 2745-8 2756.3 2766.8 2777-3 2787.8 2798.4 05 53 2809.0 2819.6 2830.2 2840.9 2851.6 2862.3 2873.0 2883.7 2894.4 2905.2 07 54 2916.0 2926.8 2937.6 2948.5 2959.4 2970-3 2981.2 2992.1 3003.0 3014.0 09 55 3025.0 3036.0 3047.0 3058.1 3069.2 3080.3 3091.4 3002.5 3013.6 3124.8 i. ii 56 3136.0 3H7-2 3158.4 3169.7 3181.0 3192-3 3203-6 3214.9 3226.2 3237.6 13 57 3249.0 3260.4 3271.8 3283-3 3294.8 3306.3 3317.8 3329-3 3340-8 33524 '5 1 58 3364-0 3375-6 3387-2 3398.9 3410.6 3422.3 3434-0 3445-7 3457-4 3469.2 17 59 3481.0 3492-8 3504.6 3528.4 3540-3 3552-2 3564-1 3576.0 3588.0 6O 3600.0 3612.0 3624.0 3636-1 3648.2 3660.3 3672-4 3684.5 3696.6 3708.8 I.2I 61 3721.0 3733-2 3745-4 3757-7 3770.0 3782.3 3794-6 3806.9 3819-2 3831-6 23 62 3844.0 3856-4 3868.8 3881.3 3893.8 3906.3 3918.8 3931-3 3943-8 3956.4 25 63 3969.0 3981.6 3994-2 4006.9 4019.6 4032-3 4045.0 4057-7 4070.4 4083.2 27 64 4096.0 4108.8 4121.6 4134.5 4147.4 4160.3 4I73-2 4186.1 4199.0 4212.0 29 65 4225.0 4238.0 4251.0 4264.1 4277-2 4290.3 4303-4 43*6-5 4329.6 4342.8 !. 3I 66 4356.0 4369.2 4382.4 4395-7 4409.0 4422.3 4435-6 4448.9 4462.2 4475-6 33 67 4489.0 4502.4 4515-8 4529-3 4542.8 4556.3 4569.8 4583-3 4596.8 4610.4 35 68 4624.0 4637-6 4651.2 4664.9 4678.6 4692.3 4706.0 47I9.7 4733-4 4747-2 37 69 4761.0 4774.8 4788.6 4802.5 4816.4 4830.3 4844.2 4858.1 4872.0 4886.0 39 70 4900.0 4914.0 4928.0 4942.1 4956.2 4970.3 4984.4 4998.5 5012.6 5026.8 1.41 71 5041.0 5055-2 5069.4 5083.7 5098.0 5"2.3 5126.6 5140.9 5I55-2 5169-6 43 72 5184.0 5198.4 5212.8 5227-3 5241.8 5270.8 5285-3 5299.8 45 73 5329-0 5343-6 5358.2 5372-9 5387.6 5402.3 54I7-0 5431-7 5446.4 5461.2 47 74 5476.0 5490.8 5505-6 5520.5 5535-4 5550.3 5565-2 5580.1 5595-0 5610.0 49 75 5625.0 5640.0 5655-0 5670.1 5685-2 5700.3 57154 5730.5 5745-6 5760.8 1.51 76 5776.0 5791-2 5806.4 5821.7 5837-0 5852-3 5867.6 5882.9 5898.2 59I3-6 53 77 5929.0 5944-4 5959-8 5975-3 5990.8 6006.3 6021.8 6037-3 6052.8 6068.4 55 78 6084.0 ' 6099.6 6115.2 6130-9 6146.6 6162.3 6178.0 6193-7 6209.4 6225.2 57 79 6241.0] 6256.8 6272.6 6288.5 6304.4 6320.3 6336-2 6352-1 6368.0 6384.0 59 80 6400.0 6416.0 6432.0 6448.1 6464.2 6480.3 6496.4 6512.5 6528.6 6544-8 1.61 81 6561.0 6577.2 6593-4 6609.7 6626.0 6642.3 6658.6 6674.9 6691.2 6707.6 63 82 6724.0 6740.4 6756.8 6773-3 6789.8 6806.3 6822.8 6839-3 6*855.8 6872.4 .65 83 6889.0 6905.6 6922.2 6938-9 6955-6 6972.3 6989.0 7005.7 7022.4 7039.2 67 ; 84 7056.0 7072.8 7089.6 7106.5 7123.4 7H0.3 7I57.2 7174.1 7191.0 7208.0 6 9 85 7225.0 7242.0 7259.0 7276.1 7293.2 73 i o-3 73274 7344-5 7361.6 7378.8 1.71 86 7390.0 74I3- 2 7430.4 7447-7 7465.0 7482.3 7499.6 75 i 6.9 7534-2 7551-6 73 87 7569.0 7586.4 7603.8 7621.3 7638.8 7656-3 7673.8 7691-3 7708.8 7726.4 75 88 7 744.0 | 7761.6 7779-2 7796.9 7814.6 7832.3 7850.0 7807.7 7885.4 7903.2 V 89 7921.0 I 7938.8 7956.6 7974-5 7992.4 8010.3 8028.2 8046.1 8064.0 8082.0 79 90 8100.0 8118.0 8136.0 8154-1 8172.2 8190.3 8208.4 8226.5 8244.6 8262.8 1.81 91 92 93 8281.0 8464.0 8649.0 8299.2 8482.4 8667.6 8317-4 8500.8 8686.2 8335-7 8519-3 8704.9 8354.0 8537-8 8723.6 8372-3 8390.6 8556-3 8574.8 8742.3 8761.0 8408.9 8427.2 8593-3! 8611.8 8779.7 8798.4 8445-6 8630.4 8817.2 II 87 94 8836.0 8854.8 8873.6 8892.5 8911.4 8930-3 8949.2 896 S.i 8987.0 9006.0 89 95 9025.0 9044.0 9063.0 9082.1 9ior.2 9120.3 91394 9158.5 9177.6 9196.8 1.91 96 9216.0 9235-2 9254.4) 9273.7 9293.0 93 i 2.3 933 i -6 9350-9 i 9370.2 9389-6 93 97 9409.0 9428.4 9447.8! 9467.3 9486.8 9506.3 9525.8 9545-3 9564-8 9584.4 95 98 9604.0 9623.6 9643.2 ! 9662.9 9682.6 9702.3! 9722.0 9741.7 9701.4 9781.2 97 99 9801.0 9820.8 9840.6 1 9860.5 9880.4 9900.3 i 9920.2 9940.1 9960.0 9980.0 1.99 i r 100 IOOOO.O IOO2O.O 10040.0 j 10060. i 10080.2 10100.3 10120.4 10140.5 10160.6 10180.8 2.01 92 TABLE XXII. Square Roots of Numbers increasing by Tenths from O.O to 1OO.9. No. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Diff. 0.01 0.000 0.316 0.447 0.548 0.632 0.707 0.775 0.837 0.894 0.949 1 2 1. 000 414 4.049 449 1.095 483 1.140 549 1.225 581 1.265 612 1.304 643 1.342 673 I.378. 703 4-15 3 732 761 789 817 871 897 924 95 975 2.69 4 2.000 2.025 2.049 2.074 2.098 2.I2I 2.145 2.168 2.191 2.214 2-37 5 2.236 2.258 2.280 2.302 2.324 2.345 2.366 2.387 2.408 2.429 2.14 6 449 470 490 5 10 53 550 569 588 608 627 1.97 7 9 646 828 3.000 665 846 3.017 683 864 3-033 702 88 1 3-050 720 898 3.066 739 915 3.082 757 3.098 775 95 3-iH 966 3-130 811 983 3.146 1.83 1.72 1.63 10 3.162 3.178 3- J 94 3.210 3-225 3-240 3-256 3.271 3.286 3-302 i-55 11 317 332 347 362 376 391 406 421 435 45 48 13 464 606 479 619 493 6 33 647 521 661 536 674 688 564 701 578 715 592 728 42 3 6 14 742 755 768 782 795 808 821 834 847 860 32 15 3-873 3.886 3-899 3.912 3-924 3-937 3-950 3-962 3-975 3.987 1.27 16 4.000 4.012 4-025 4-037 4.050 4.062 4.074 4.087 4.099 4.111 23 17 123 135 159 171 183 195 207 219 231 20 18 243 254 266 278 290 301 3*3 324 336 348 16 19 359 370 382 393 405 416 427 438 450 461 1 3 20 4-472 4-483 4-494 4.506 4-5I7 4-528 4-539 4-550 4.561 4-572 1. 10 21 583 593 604 615 626 637 658 669 680 08 22 690 701 712 . 722 733 743 754 764 775 785 06 23 796 806 817 827 837 848 858 868 879 889 3 24 899 909 919 93 940 950 960 970 980 990 OI 25 5.000 S-oio 5.020 5-030 5.040 5-050 5.060 5-070 5-079 5-089 0.99 26 099 109 119 128 138 148 I 58 167 177 187 97 27 28 196 292 206 301 215 310 225 320 235 329 244 339 348 263 357 273 367 282 376 95 94 29 385 394 44 422 441 450 459 468 9 2 30 31 5 1S 5.486 577 5-495 5-505 S-SH 604 5-523 612 5-532 621 630 5-550 639 5-559 648 0.91 32 657 666 675 683 692 701 710 718 727 736 88 33 745 753 762 771 779 788 797 805 814 822 87 34 831 840 848 857 865 874 882 891 899 908 85 35 36 5.916 6.000 5-925 6.008 5-933 6.017 5-941 6.025 5-95 6.033 5.958 6.042 5-967 6.050 5-975 6.058 Ifel 5-992 6.075 0.85 83 37 083 091 099 107 116 124 132 140 '148 156 82 38 164 173 181 189 197 205 213 221 229 237 81 39 245 253 261 269 277 285 293 3 OI 309 80 40 6-325 6-332 6.340 6-348 6.356 6.364 6-372 6.380 6.387 6-395 0.79 41 403 411 419 427 434 442 450 458 465 473 78 42 43 44 481 557 633 488 496 III 504 580 656 512 588 663 519 595 671 67^ 535 611 686 542 . 618 693 550 626 701 77 76 1 75 | 45 46 6.708 782 6.716 790 6.723 797 6.731 804 6.738 812 6-745 819 6-753 826 6.760 834 6.768 841 6 ill 0.74 73 47 856 863 870 877 885 892 899 907 914 921 73 48 928 935 943 95 957 964 971 979 986 993 72 49 7.000 7.007 7.014 7.021 7.029 7.036 7-043 7.050 7-057 7.064 7i 50 7.071 7-078 7.085 7.092 7.099 7.106 7-"3 7.120 7.127 7-134 0.70 TABLE XXII. Square Roots of Numbers increasing: by Tentlts front O.O to 1OO.9. No. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 O.9 Diff. 0.01 . 50 7.071 7.078 7.085 7.092 7.099 7.106 7-113 7.120 7.127 7.134 0.70 51 141 148 155 162 169 176 183 190 197 204 70 52 211 218 225 232 239 246 253 259 266 273 69 53 280 287 294 301 308 314 321 328 335 342 68 54 348 355 362 369 376 382 389 396 403 409 68 55 7.416 7-423 7-43 7-436 7-443 7-45 7-457 7463 7.470 7.477 0.67 56 483 490 497 53 510 517 523 530 537 543 67 57 55 556 563 570 576 583 589 596 603 609 66 ! 58 616 622 629 6 35 642 649 655 662 668 675 65 59 681 688 694 701 707 7H 720 727 733 74 6 5 i 6O 7.746 7-752 7-759 7-765 7.772 7.778 7.785 7.791 7-797 7.804 0.64 61 810 817 823 829 836 , 842 849 855 86 1 868 6 4 i 62 874 880 887 893 899 906 912 918 925 93 i 63 63 64 937 8.000 944* 8.006 950 8.012 956 8.019 962 8.025 969 8.031 975 8-037 981 8.044 987 8.050 994 8.056 63 62 i 65 8.062 8.068 8.075 8.081 8.087 8.093 8.099 8.106 8.II2 8.118 0.62 ; 66 124 130 136 142 149 '55 161 167 173 179 61 i 67 185 191 198 204 210 216 222 228 234 240 61 1 68 246 252 258 264 270 276 283 289 295 301 60 69 307 313 319 325 331 337 343 349 355 60 7D 8.367 8-373 8-379 8.385 8.390 8.396 8.402 8.408 8.414 8.420 O.6O ; 71 426 432 438 444 45 456 462 468 473 479 59 > 72 485 491 497 53 509 515 .521 526 532 538 59 ; 73 544 55 556 562 567 573 579 585 597 58- 74 602 608 614 620 626 631 637 643. 649 654 58 ; 75 8.660 8.666 -8.672 8.678 8.683 8.689 8.695 8.701 8.706 8.712 0.58 76 718 724 729 735 74* 746 752 758 764 769 57 77 775 781 786 792 798 803 809 815 820 826 57 78 o _ 837 843 849 854 860 866 871 877 883 56 79 888 894 899 905 911 916 922 927 933 939 56 i 89 8.944 8.950 8-955 8.961 8.967 8.972 8.978 8.983 8.989 8.994 0.56 81 9.000 9.006 9.011 9.017 9.022 9.028 9-033 9-039 9-044 9.050 56; 82 055 061 066 072 077 083 088 094 099 I0 5 55 83 no 116 121 127 132 138 143 149 154 1 60 55 84 165 171 I 7 6 182 187 192 198 203 209 214 54 85 9.220 9-225 9.230 9.236 9.241 9.247 9.252 9- 2 57 9.263 9.268 -54 : 88 274 279 284 290 295 301 306 311 322 54 87 327 338 343 349 354 359 365 37o 375 53 88 381 386 391 397 402 407 418 423 429 53 : 89 434 439 445 45 455 460 466 47 1 476 482 53 90 9.487 9-492 9-497 9-53 9.508 9.513 9.518 9-524 9-529 9-534 o-53 91 539 545 55 555 560 566 576 581 586 52 : 92 592 597 602 607 612 618 623 628 6 33 638 52 93 644 649 654 659 664 670 675 680 685 690 52 1 94 695 701 706 711 716 721 726 73 i 737 742 51 95 9-747 9-752 9-757 9.762 9.767 9.772 9.778 9.783 9-788 9-793 0.51 96 798 803 808 813 818 823 829 834 839 844 5 1 97 849 854 859 864 869 874 879 884 889 894 5 1 98 899 905 910 920 925 930 935 940 945 99 950 955 960 965 970 975 980 985 990 995 50 ; 100 IO.OOO 10.005 IO.OIO 10.015 IO.O2O 10.025 10.030 10.035 10.040 10.045 0.50 94 TABLE XXIII. True Rising* and Setting*. Declination of same name as the Latitude. 6 o 6 o 5 58 6 2 5 55 6 5 5 53 6 7 5 5 1 6 9 5 48 6 12 5 46 6 14 3O 32 o 58 2 55 5 53 7 50 IO 47 13 45 15 32 34 o 57 3 55 52 8 49 ii 46 14 44 16 34 : 36 o o 57 3 54 6 5 1 9 48 12 45 15 42 18 36 ! 38 o 57 3 54 6 9 47 13 44 16 19 38 40 6 o 6 o 5 57 6 3 5 53 6 7 5 50 6 10 5 47 6 13 5 43 6 17 5 40 6 20 4O 41 o 57 3 53 7 5 10 46 14 43 17 39 21 41 42 o 56 4 53 7 49 ii 46 14 42 18 38 22 42 ! 43 o o 56 4 53 7 49 ii 45 15 41 19 38 22 43 44 o o 56 4 52 8 48 12 45 15 41 19 37 2 3 44 45 6 o 6 o 5 56 6 4 5 S 2 6 8 548 6 12 5 44 6 16 5 40 6 20 5 36 6 24 45 46 o 56 4 5 2 8 48 12 43 17 21 35 2 5 46 47 o 56 4 51 9 47 13 43 17 38 22 34 26 47 48 o 56 4 51 9 47 42 18 38 22 33 27 48 49 o 55 5 9 46 4 42 18 37 23 32 28 49 50 6 o 6 o 5 55 6 5 5 5 6 10 5 46 6 14 5 4 1 6 19 536 6 24 5 3 1 629 5O 51 o 55 5 5 IO 45 15 40 20 35 25 30 3 51 52 o 55 5 . 10 45 15 39 21 34 26 29 31 52 53 o 55 5 49 ii 44 16 39 21 33 27 28 3 2 53 i 54 o o 55 5 49 ii 44 16 38 22 32 28 27 33 51 i 55 6 o 6 o 5 54 6 6 5 49 6 ii 5 43 6 17 5 37 623 5 3 1 6 29 5 25 635 55 56 o 54 6 48 12 42 18 36 24 3<> 30 24 36 56 i 57 o o 54 6 48 12 41 19 35 25 29 3i 2 3 37 57 ! : 58 o 54 6 47 IT 41 19 34 26 28 3 2 21 39 58 59 o o 53 7 47 13 40 20 33 27 27 33 20 40 59 60 6 o 6 o 5 53 6 7 5 46 6 14 5 39 6 21 5 32 6 28 5 25 635 5 18 6 42 60 61 o o 53 46 14 38 22 29 24 36 16 44 61 62 o 5 2 8 45 15 37 2 3 30 3 22 38 14 46 62 63 o o 5 2 8 44 16 3 6 24 28 3 2 2O 40 12 48 63 64 o o 52 8 44 16 35 25 27 33 19 4 1 IO 50 64 65 6 o 6 o 5 5 1 6 9 5 43 6 17 5 34 6 26 5 26 634 5 18 6 42 5 8 652 65 66 o o 5 1 9 42 18 33 27 24 36 IS 45 5 55 66 67 o 51 9 41 19 32 28 22 38 12 48 3 57 67 i 68 o 5 10 40 20 3 30 20 40 IO 5 7 o 68 69 o IO 39 21 29 3 1 18 42 7 53 4 56 4 69 ! 70 6 o 6 o 5 49 6 ii 5 38 6 22 5 27 633 5 16 6 44 5 4 6 56 4 53 7 7 70 71 o o 48 12 37 23 25 35 13 47 i 59 49 ii 71 72 o 48 12 35 2 5 23 37 10 50 4 58 7 2 45 15 72 73 o o 47 13 34 26 21 39 7 53 53 7 40 20 73 74 o o 46 14 32 28 18 42 4 56 49 ii 34 26 74 75 6 o 6 o 5 45 6 15 5 30 6 30 5 15 645 c o 7 o 4 44 7 16 4 28 7 3 2 75 76 o 44 16 28 3 2 ii 49 4 55 5 38 22 20 40 76 77 o o 43 17 25 35 7 53 49 ii 31 29 ii 49 77 78 o 19 22 38 3 57 43 17 2 3 37 i o 59 78 179 39 21 19 4 57 7 3 3 6 24 13 47 3 49 8 u 79 r Set. Ris. Set. Ris. Set. Ris. Set. Ris. Set. Ris. Set. Ris. Set. Ris. 6 0) | 1 2 3 4 5 6 1 | Declination of contrary name. 1 TABLE XXIII. 95 1 True Rising- aiicl Setting-. i o> Declination of same name as the Latitude. . 6 7 8 9 10 11 12 '! 3 Ris. Set. Ris. Set. Ris. Set. Ris. Set. Ris. Set. Ris. Set. Ris. Set. nS i O hm hm h m \ h m h m h m h m h m h m /* m h m h m h m h m 6060 6 06 o 6 o 6 o 6060 6 o 6 o 6 o 6 o 6 o 6 o 4 5 58 \ 2 5 58 i 2 5 58 2 5 57 3 5 57 3 S 57 3 5 57 3 4 8 57 3 5 6 1 4 55 5 55 54 6 54 6 53 7 8 12 55' 5 54 6 53 7 5 2 8 9 9 10 12 16 53! 7 52; 8 9 50 10 48 12 47 13 46 H 16 1 2O 551 ! 6 9 5 5; 6 10 S 48 6 12 5 47 6 13 5 45 6 15 S 44 6 16 5 42 6 18 20 1 22 50 10 49 n 47 13 45 VS 44 16 42 18 40 20 22 ! 24 49 u 47 13 46 14 44 16 42 18 40 20 38! 22 24 26 48 12 46; 14 44 16 42 18 40 20 38 22 3 6 | 24 26 28 47 J 3 45 15 43 17 4 1 19 38 22 36 24 34 26 28 ! 30 5 46 6 14 5 44' 6 16 5 4i 6 19 5 39 6 21 S 37 62 S 5 34 6 26 5 32 | 6 28 30 ; 32 45 15 42 1 18 40 1 20 37 23 35 25 32] 28 29 3i 32 34 44 i 16 4i 19 38 22 35 25 33 27 3 3 27| 33 34 ! 36 42 18 40 20 37 23 34 26 31 29 28 32 24 30 36 i > 38 41 19 38! 22 35 25 3 2 28 2t> 32 25 35 22 38 38 40 5 40 i 6 20 5 3 6 i 6 24 5 33 6 27 5 29 631 5 26 6 34 5 22 6 38 5 19 6 4i 40 i 41 39 i 21 35; 25 3 2 28 28 32 25 35 21 39 17 43 43 i 42 22 35 25 31 29 27 33 23 37 20 i 40 1 6 4.4 42 i 43 3 s 22 34 26 301 30 26 54 22 38 18 42 14 1 4J 43 44 37 2 3 33 27 29 31 25 35 21 39 17 43 13 47 44 j 45 536 624 5 32 6 28 ^28 632 5 24 6 36 5 19 641 S IS 6 45 5 n 649 45 46 35 2S 3i 29 27 33 22 38 18 42 14 46 9 46 ! |47 34 26 3 30 25 35 21 39 16 44 12 48 7 53 47 ! 33 27 29 31 24 3 19 IS 45 10 5 5 55 48 j 49 28 28 32 23 37 18 42 13 47 8 52 3! 57 49 50 5 3i 6 29 526 6 34 5 21 6 39 5 i 6 6 44 5 ii 6 49 5 6 6 54 5 i 659 50 ' 51 3 3 2,S 35 20 40 IS 4S 10 So A 4 59 7 i 51 i 52 29 31 24 19 41 13 47 8 52 2 58 57 52 ! 53 28 32 22 38 17 43 ii 49 6 54 oj 7 o 54 6 53 54 27 33 21 39 15 45 10 50 4 56 4 58! 2 52! 8 54 ! 55 525 63S S 20 6 40 5 ^4 6 46 S 8 652 5 2 6 58 4 s6 7 4 4 49 7 ii 55 i 56 24 3 5 18 42 12 48 6 54 4 59 7 i 53 7 47 13 56 57 23 37 16 44 10 50 4 57 So 10 44 16 57 58 21 39 IS 45 8 52 i 59 54 6 48 12 19 58 j 59 20 40 13 47 6 54 4 59 7 i 5 2 8 44 16 37 23 59 ; 60 518 642 S ii 6 49 5 4 6 56 4 56 7 4 4 49 7 u 4 41 7 19 4 34 7 26 6O 61 16 44 9 i 59 54 6 4'o H 38 22 3 30 61 62 14 4 b 7 53 4 59 7 i 9 43 17 34 26 26 34 62 63 12 48 4 5 6 S6 4 48 12 39 21 3 3 21 39 63 64 10 50 2 58 53 7 44 16 35 25 26 34 17 43 64 j 65 5 8 652 C O 7 o 4 5 7 10 4 41 7 19 4 3 1 7 29 4 21 7 39 4 12 7 48 65 66 5 55 4$6 4 46 14 37 2 3 27 33 16 44 6 54 66 6? 3 57 53 7 43 17 33 27 22 38 ii 49 o 8 o 67 68 o 7 o 49 u 39 21 28 3 2 16 44 5 55 3 53 7 68 ! 69 45<> 4 45 15 34 26 23 37 II 49 3 58 8 2 46 H 69 70 453 7 7 4 4i 7 19 4 29 7 31 4 17 743 4 4 756 3 5 8 9 3 37 823 70 71 49 ii 3& 24 23 37 10 50 3 57 8 3 43 17 28 32 71 72 45 15 31 29 17 43 3 57 49 ii 33! 27 17 43 72 73 40 20 2 5 35 10 So 3 SS 8 S 39 21 22 38 4 so 73 74 34 26 19 3 57 46 H 28 32 9 51 2 49 9 ii 74 75 428 732 4 u 7 49 3 54 8 6 3 35 8 2<; 3 15 8 45 2 54 9 6 2 30 9 3 75 76 20 40 2 58 43 17 22 38 o O O 35 25 6 54 76 77 II 49 3 5 1 8 9 3 3 71 53 2 41 19 ii 49 i 32 10 28 77 78 I 59 39 21 14 46 247 9 13 16 44 i 35 10 25 12 78 79 349 811 23 37 2 55 9 5 22 38 i 40 10 20 OJI2 79 Set. Ris. Set. Ris. Set. Ris. Set. Ris. Set. Ris. Set. Ris. Set. Ris. $ o> O 9 6 7 8 9 10 11 12 M Declination of contrary name. 3 TABLE XXIII. True Rising and Setting-. Declination of same name as the Latitude. 0) B " 12 13 14 15 16 17 18 'O t j*^ I Ris. Set. Ris Set Ris. Set. Ris. Set. Ris. Set. Ris. Set. Ris. Set. u i o TwT h ni h in h m h m k m // ni h m h m h >n h in k m h m \ h in 6 o 6 o 6 o 6 o 6 oi 6 o 6 o 6 o 6 o 6 o 6 o 6 o 6 06 o O 2 558 2 558 2 5 58 2 5 58 2 558 2 558 2 5 57 f 3 2 4 57 3 56 4 5 6 4 56 4 55 5 55 5 55i 5 4 6 55 5 54 6 54 6 54 6 53 7 53 7 52 8 6 8 53 7 53 7 52 8 9 9 50 10 50 10 8 10 5 5 1 6 9 5 5 1 6 9 5 50; 6 10 5 49 6 ii 548 6 12 548 6 12 5 47 6 13 10 12 50 10 49 ii 4 8 j 12 47 IT 46 *4 45 15 44 16 12 14 48 12 47 13 46; 14 45 15 44 16 43 17 41 1 19 14 16 46 H 45 15 44 16 42 18 41 19 40 20 39 21 1G 18 44 16 43 41 j 19 40 20 39 21 37 2 3 36 24 18 20 542 618 5 4 1 6 19 5 39J 6 21 538 6 22 5 36 | 6 24 5 34 6 26 5 33; 6 27 20 24 40 38 20 22 39 36 21 24 37 i 23 34! 26 35 33 25 27 33 27 3 1 29 32 29 28 3 1 30 i 30 27' 33 22 21 26 36 24 34 26 32 28 30 3 28 32 26 54 24' 3 6 26 28 34 26 32 28 30 30 27 33 25 35 2 3 37 20 40 28 J 30 532 628 5 29 631 5 27 6 33 5 24 6 36 5 22 6 38 5 19 6 41 5 17 6 43 30 : 32 29 3 1 27 33 24 3 6 21 j ,Q 19 41 16 44 '3 47 32 , 34 27 33 24 36 21 ; 39 18 42 15 45 12 48 9 34 36 24 3 o 21 39 18 42 15 45 12 48 9 51 Si 55 36 38 22 38 18 42 15 45 12 48 8 52 5 55 1 59 38 40 519 641 5 15 645 5 12 6 48 5 8 6 52 5 4 6 56 5 i 6 59 4 57 7 3 40 41 142 II 43 44 H 12 4* 48 10 8 50 52 6 4 54 56 2 58 07 o 458 56 7 2 4 54 5 2 6 8 41 42 '43 H 46 10 50 6 54 2 58 458 ' 2 54 6 49 ii 43 i 44 13 47 8 52 4 56 070 5 6 4 5 1 9 47 13 44 '45 5" 649 5 7 653 5 2 6 58 4 58 ! 7 2 4 53 | 7 7 4 49 7 11 4 44 7 16 45 46 9 51 5 55 0)70 56 4 5 1 9 46 14 41 19 4G 47 7 53 3 57 4 5 8 t 2 53 7 48 1 12 43 17 38 22 47 ,48 5 55 i 59 56 4 5 1 9 46 H 19 35 25 48 ;49 3 57 4 58 7 2 53 7 48 12 43 17 38 22 32 28 49 50 5 i 659 456 7 4 4 51 7 9 4 46 7 *4 4 40 7 20 4 35 7 25 4 29 7 3 1 50 51 459 7 * 54 6 48 12 43 17 37 23 29 25 35 51 52 57 5 1 9 46 14 40 20 34 26 28 32 22 38 52 53 54 6 49 ii 43 17 37 23 3 1 29 24 36 18 42 53 54 52 8 46 J 4 40 20 33 27 2 7 33 20 40 J 4 46 54 55 449 711 4 43 7 17 4 37 7 23 .4 3 7 3 4 23 7 37 4 16 7 44 4 9 7 5 1 55 56 47 40 20 33 27 26 34 19 12 48 5 55 56 57 44 16 37 23 3 3 22 38 15 45 8 52 o 8 o 57 58 41 19 33 27 26 34 18 42 ii 49 3 57 3 55 5 58 i 59 37 23 3 30 22 38 14 46 6 54 358 8 2 49 ii 59 60 434 726 4 26 7 34 4 18 7 42 4 9 7 51 4 i 7 59 3 52 8 8 3 43 8 17 60 61 3 30 22 38 47 4 56 3 55 8 5 46 14 36 24 61 62 26 34 17 43 8 52 3 59 8 i 49 ii 40 20 291 3 1 62 63 21 39 12 48 3 57 53 7 43 17 33 27 22 38 63 64 17 43 7 53 3 57 8 3 47 13 3 6 24 25 35 13 47 64 65 4 I2 748 4 i 7 59 3 5 1 8 9 3 40 8 20 328 832 3 16 8 44 3 3 857 65 66 6 54 3 54 8 6 44 16 32 28 20 40 7 53 2 53 9 7 66 67 o 8 o 48 12 36 24 2 3 37 10 5 2 56 9 4 40 20 67 68 353 7 19 28 32 14 46 2 59 9 i 43 17 26 34 6 ; 69 46 14 32 28 18 42 3 57 46 14 29 3 1 9 5' 69 70 337 823 3 23 837 3 7 853 2 50 9 10 2 32 9 28 2 12 948 i 47 10 13 7O 71 28 32 12 48 2 54 9 6 36 24 15 45 I 5 10 10 43 71 72 17 43 2 59 9 i 40 20 18 42 I 5 2 10 8 I9| 41 O O 12 O 72 73 4 56 44 16 21 39 i 55 'o 5 21 39 O O 12 O 73 74 249 911 25 35 I 58 10 2 23 37 12 74 ; Set. Ris. Set. Ris. Set. Ris. Set. Ris. Set. Ris. Set. Ris. Set. Ris. 1 0) 0) ? 12 ' 13 14 15 16 17 18 3 5 "S 01 Declination of contrary name. Li TABLE XXIII. 97 True Rising: and Setting:. Declination of same name as the Latitude. . .. o3 o T3 3 18 19 3 20 21 22 23 24 5 ! ^4 H \ * Ris. I Set. Ris. Set. Ris. Set. Ris. Set. Ris. Set. Ris. Set. Ris. Set. nt O h m \ h m h m // nt h in h m h m h m h m h lit h m h m h m h m o 6060 6 o 6 o 6 o 6 o 6 o 6 o 6 o 6 o 6 0160 6060 O 557J 3 S 57 3 5 57 3 5 57 3 5 57 3 5 57 3 5 5 6 i 4 2 4 55 5 54 6 54 6 54 6 54 6 53 7 53 7 4 6 S2 8 S 2 8 Si 9 Si 9 5 10 5 10 49 ii 6 8 50 IO 49 ii 48 12 48 12 47 13 46 14 46 H 8 IO 547 613 5 4^ 6 14 S 4S 6 IS 5 44 6 16 5 44 6 16 5 43 6 17 5 42 6 18 10 12 44 16 43 17 42 18 19 40 20 39 21 38 22 12 14 41 19 40 20 39 21 38 22 37 23 36 2 4 35 2 5 14 16 39 21 37 2 3 36 2 4 35 25 33 27 32 28 29 16 18 36 24 34 26 33 27 2 9 3 30 28 3 2 27 33 18 2O 533 627 5 3i 6 29 5 30 6 30 528 6 32 5 26 6 34 5 2 4 6 36 S 23 637 20 22 3 30 28 32 26 34 24 36 22 38 20 1 40 19 22 24 27 33 25 35 23 37 21 39 19 i6| 44 46 24 26 24 36 21 39 19 17 43 IS 4S 12 4 8 IO So 26 28 20 40 18 42 15 45 13 47 IO 5 8 1 52 5 55 28 J 30 5 1 7 1 6 43 5 i4 6 46 5 n 6 49 S 9 6 51 S 6 6 54 S 3 6 57 5 o 7 o 30 31 15 45 12 48 9 Si 7 53 4 $6 i 59 4 S8 2 31 32 13 47 10 50 7 53 4 2 58 4 59 7 i 55 5 32 33 ii 49 8 5 2 5 55 2 58 4 59 7 i 56 4 53 7 33 34 91 5i 6 54 3 57 O 7 o 57 3 53 7 5 IO 34 35 5 7 653 5 4 656 5 i 6 59 4 58 7 2 4 S4 7 6 4 51 7 9 4 47 7 IS 35 36 51 55 2 58 4 59 7 i SS S 5 2 8 48 12 44 16 36 37 31 57 O 7 o 4 S3 7 49 ii 45 15 42 18 37 38 i ! 59 4 58 2 54 6 So 10 46 H 43 17 39 21 38 i 39 459' 7 i 55 5 9 48 12 44 16 40 2O 35 25 39 40 457J7 3 4 S3 7 7 4 49 7 ii 4 4S 7 15 4 4i 7 19 4 37 7 23 4 32 7 28 40 41 54 6 5 IO 46 14 42 18 S8 22 33 27 29 41 42 521 8 48 12 43! 17 39 21 SS 2S 3 3 2 5 35 42 43 49! ii 45 15 19 36 24 29 27 33 22 43 .] 44 47! 13 42 1 8 38 22 33 27 28 32 2 3 37 IS 42 44 45 444! 716 4 39 7 21 4 35 7 2$ 4 3 7 30 4 2 5 7 35 4 26 7 40 4 14 7 46 45 46 41 19 36 24 31 29 26 34 21 39 16 44 IO So 46 47 38 22 33 27 28 32 2 3 37 17 43 12 48 6 54 47 48 35 2 5 30 30 2 5 3S 19 13 47 7 SS i 59 48 49 3 2 28 27 33 21 39 15 45 9 51 3 57 3 57 49 50 429 731 4 23 7 37 4 17 7 43 4 ii 7 49 4 5 7 SS 3 S8 8 2 3 5 2 8 8 5O 51 52 25 22 35 38 19 15 45 13 9 47 7 2 53 58 3 5S 8 o S $ 7 12 47 13 19 51 52 53 18 42 ii 49 4 c6 3 57 8 3 5 IO 43 17 35 2 5 53 54 14 46 7 53 o 8 o 52 8 4S IS 37 23 29 54 55 4 9 75i 4 2 758 3 55 8 5 3 47 8 13 3 39 8 21 3 3i 8 29 3 22 8 38 55 56 5 55 3 57 8 3 49 ii 19 33 27 24 36 IS 45 56 57 o 8 o 5 2 8 44 16 35 25 26 34 17 43 53 57 58 3 55 S 46 14 38 22 28 3 2 19 9 2 58 9 2 58 59 49 ii 40 20 31 29 21 39 ii 49 9 o 49 ii 59 J60 343 817 3 34 8 26 3 24 8 36 3 13 8 47 3 2 8 58 2 5i 9 9 238 9 22 60 ! 61 24 26 34 16 44 5 55 2 53 9 7 40 20 26 34 61 62 29 31 18 42 7 53 2 55 9 5 42 18 28 3 2 13 47 62 63 22 38 IO 50 2 58 9 2 44 16 3 3 14 46 i 10 4 63 64 13 47 i 59 47 13 3 2 28 16 44 i S8 IO 2 36 24 64 65 3 3 857 2 50 9 10 2 3S 9 2 S 2 18 9 42 2 O IO O i 38 IO 22 I 9 10 51 65 66 253 9 7 37 23 21 39 2 S8 i 39 21 10 50 o o 12 66 i 67 40 20 23 37 4 56 i 41 10 19 ii 49 12 67 68 26 34 6 54 1 43 10 17 13 47 o o 12 O 68 69 9 5 1 i 45 10 15 14 46 12 69 0) Set. Ris. Set. Ris. Set. Ris. Set. Ris. Set. Ris. Set. Ris. Set. Ris. I 1 M 18 19 2O 21 22 23 24 nt | Declination of contrary name. | 98 TABLE XXIII. 1 True Rising; and Setting. Declination of same name as the Latitude. O Q) T3 'O 3 24 25 26 27 28 29 30 2 M IH 8 Ris. Set. Ris. Set. Ris. Set. Ris. Set. Ris. Set. Ris. Set. Ris. Set. 8 o /( m h m h m h m h m h m k m h m h m h m h m h m h m h m j 6 o 6 o 6 o 6 o 6 o 6 o 6 o 6 o 6 o 6 o 6 o 6 o 6 o 6 o 2 556 4 556 4 5 56 4 556 4 556 4 556 4 5 55 5 2 4 53 7' 53 7 5 2 8 5 2 8 9 9 9 4 6 49 ii 49 ii 48 12 48 12 47 47 '3 46 1*4 6 8 46 14 45 15 44 16 44 16 43 17 42 18 4i 19 8 j 10 542 6 18 ) 4^ 6 19 5 40 6 20 5 39 6 21 538 6 22 5 38 6 22 5 37 6 23 10 12 38 22 37 23 36 24 35 25 34 26 33 27 32 28 12 14 35 25 33 27 3 2 28 29 30 30 28 32 27 33 14 16 29 29 3 1 28 32 26 34 25 35 2 3 37 22 38 16 18 27 33 25 35 24 36 22 38 20 40 18 42 17 43 18 20 523 637 5 21 6 39 5 i9 6 41 5 I 7 643 5 15 6 45 5 i3 6 47 5 6 49 20 22 19 41 17 43 15 45 12 48 IO 5 8 52 6 54 22 24 14 46 12 48 10 50 8 52 5 55 3 57 o 7 o 24 26 10 50 7 53 5 55 2 58 o 7 o 4 57 7 3 4 55 5 26 28 5 j 55 3 57 o 7 o 4 57 7 3 4 54 6 9 49 ii 28 30 5070 458 7 2 4 55 7 5 4 52 7 8 448 7 12 4 45 7 15 4 42 7 18 30 31 458| 2 55 5 5 2 8 49 ii 46 42 18 39 21 31 32 55 5 52 8 49 ii 46 14 42 $i 39 21 35 25 32 33 53 7 49 ii 46 H 43 17 39 21 36 24 32 28 33 34 10 47 13 43 17 40 20 36 24 32 28 28 32 34 35 447 7 J 3 4 44 7 16 4 40 7 20 436 7 24 4 33 7 27 4 29 7 31 4 25 7 35 35 1 36 44 16 41 19 37 23 33 27 29 31 25 35 21 39 36 37 42 18 38 22 34 26 3 30 26 34 21 39 17 43 37 38 39 21 35 25' 30 30 26 34 22 38 17 43 13 47 38 39 35 25 29 27 33 23 37 18 42 13 47 9 5 1 39 40 432 7 28 428 7 32 4 23 7 37 4 19 7 41 4 14 746 4 9 7 5 1 4 4 756 40 41 29 3 1 24 36 20 40 15 45 10 50 5 o 5S o 8 o 41 42 25 35 21 39 16 44 ii 49 5 55 o 8 o 3 55 5 42 43 22 38 17 43 12 48 7 53 i 59 3 56 4 50 10 43 44 iS 42 13 47 8 52 2 58 3 56 8 4 9 44 16 44 45 414 7 46 4 9 T 5 1 4 3 7 57 3 57 8 3 3 52 8 8 3 45 8 15 3 39 8 21 45 46 10 5 55 3 59 8 i 7 46 14 40 20 33 27 46 47 6 54 8 o 54 6 48 12 19 34 26 27 33 47 48 i 59 3 55 5 49 ii 42 18 35 25 28 32 20 40 48 49 357 8 3 10 43 17 36 24 29 3 1 22 38 13 47 49 50 352 8 8 3 45 8 15 338 8 22 3 30 8 30 3 23 837 3 15 845 3 6 854 50 51 47 13 39 21 32 28 24 36 16 44 7 53 2 58 9 2 51 52 19 33 27 25 35 17 43 8 52 2 59 9 i 49 ii 52 53 35 25 27 33 19 10 50 9 9 40 20 53 54 29 20 40 ii 49 2 58 2 52 8 4 1 19 30 3 54 55 322 838 3 13 847 3 3 857 2 53 9 7 2 42 9 18 2 3 I 9 29 2 18 9 42 55 56 15 45 55 2 55 9 5 44 16 32 28 19 41 5 55 56 57 7 53 2 56 9 4 45 15 33 27 20 40 6 54 i 49 IO II 57 58 258 9 2 47 13 35 25 22 38 7 53 i 50 IO IO 3 30 58 59 49 ii 36 24 23 37 8 52 1 51 10 9 3 1 29 4 56 59 60 238 9 22 2 25 9 35 2 9 9 5 1 1 52 10 8 1 32 10 28 i 5 10 55 12 6O 61 26 34 II 49 1 53 10 7 33 27 I 6 54 12 61 62 J 3 47 i 55 10 5 34 26 6 54 12 O 62 63 1 S^ 10 4 35 25 7 53 12 63 64 36 24 52 12 O 64 65 i 9 10 51 O O ! I2 O 65 i 66 o o 12 66 ! 67 67 1 68 1 68 69 69 Set. Ris. Set. Ris. Set. Ris. Set. Ris. Set. Ris. Set. Ris. Set. Ris. d CD d +J 24 25 26 27 28 29 3O 1 I Declination of contrary name. rt TABLE XXIV. 99 Horizon-Azimuths. 1 Declination of same name as the Latitude. | % rt 0?0 o 0.5 1.0 15 2.0 o 2.5 o 3.0 o 3.5 o 4.0 o 4.5 50 55 60 1 o o o o o o o o , o O 90.0 89.5 89.0 88.5 88.0 87.5 87.0 86.5 86.0 85.5 85.0 84.5 84.0 I0 - O.O 9-5 9.0 8.5 8.0 7-5 7.0 6.5 5-9 54 4.9 44 3-9 10 O.O 9-5 9.0 8.5 7-9 74 O.Q 6.4 5-8 5-3 4.8 4-3 3-8 15 i 2O 0.0 9-5 8.9 8.4 7-9 7-3 O.o 6-3 5-7 5-2 4-7 4.2 3-6 20 I 25 o.o 9-5 8.9 8.4 7.8 7.2 6.7 6.2 5.6 4-5 4.0 34 25 1 SO 90.0 89.4 88.8 88.3 87.7 87.1 86.6 86.0 854 84.8 84.2 837 83-1 30 32 o.o 94 8.8 8.2 7.6 7.1 6-5 5-9 5-3 4-7 4.1 3-5 3- 32 34 0.0 94 8.8 8.2 7.6 7.0 6.4 5.8 5.2 4.6 4.0 34 2.8 34 36 o.o 94 8.8 8.2 7-5 6.9 6-3 5-7 44 3-9 3-2 2.6 36 1 38 o.o 94 8.7 8.1 7-5 6.8 6.2 5.6 4-9 4-3 3-7 2.4 38 40 90.0 89-3 88.7 88.0 87.4 86.7 86.1 854 84.8 84.1 83.5 82.8 "82.2 40 ! 42 o.o 9-3 8-7 8.0 7-'3 6.6 6.0 5-3 4.6 3-9 3-3 2.6 2.0 42 44 o.o 9-3 8.6 7-9 7.2 6-5 5.8 44 3-7 2.4 1.7 44 > 46 0.0 9-3 8.6 7-8 6.4 5-7 5.0 4.2 3-5 2.S 2.1 1.4 46 48 o.o 9-3 8-5 7-8 7.0 6-3 5-5 4.8 4.0 3-3 2-5 1.8 I.O 48 1 50 90.0 89.2 88.5 87.7 86.9 86.1 85-3 84.6 83.8 83.0 82.2 81.4 80.7 50 51 o.o 9.2 8-4 7.6 6.8 6.0 5- 2 44 3-6 2.8 2.0 1.2 0.5 51 52 0.0 9.2 8.4 7.6 6.7 5-9 4-3 3-5 2.7 1.9 I.O 52 ! 53 o.o 9-2 8.4 7-5 6.7 5-8 5- 4.2 3-3 2-5 o.S 0.0 53 54 0.0 9.1 8-3 7-5 6.6 5-7 4-9 4.0 3-2 2-3 J -5 0.6 79-8 54 ! 55 90.0 89.1 88.3 87.4 86.5 85.6 84.8 83.9 83.0 82.1 81.3 80.4 79-5 55 ! 56 o.o 9.1 8.2 7-3 6.4 5-5 4.6 3-7 2.8 1.9 I.O O.I 9.2 56 j 57 0.0 9.1 8:2 7-3 6-3 54 4-5 3-6 2.6 0.8 79-9 .8.9 57 58 o.o 9.1 8.1 7.2 6.2 5-3 4-3 34 24 '5 -5 9.6 8.6 58 59 0.0 9.0 8.1 6.1 4.2 3.2 2.2 1.2 o-3 9-3 8-3 59 60 90.0 89.0 88.0 87.0 86.0 85.0 84.0 83.0 82.0 81.0 80.0 79.0 7/-9 60 61' o.o 9.0 7-9 6.9 5-9 4.8 3-8 2.8 i-7 0.7 79-7 O U 7-5 61 62 0.0 8.9 7-9 6.8 5-7 4-7 3-6 2-5 0.4 9-3 8.2 62 63 o.o 8-9 7-8 6-7 5-5 4.5 34 2-3 1.2 O.I 8.9 7.8 6.6 63 64 0.0 8.9 7-7 6.6 54 4-3 3- 1 2.0 0.8 79.7 8-5 74 6.1 64 65.0 90.0 88.8 87.6 86.5 85.2 84.1 82.9 81.7 80.5 79.3 78.1 76.9 75.6 65.0 55 o.o 8.8 7-6 6.4 S- 2 4.0 2.8 1.5 7-9 6.6 54 5.5 60 0.0 8.8 7-5 6.3 3-9 2.6 1.4 O.I o Q 7.6 6.4 6.O 65 o.o 8.8 7-5 6.2 5- 3-7 2-5 1.2 79-9 8.7 74 6.1 1*8 6.5 7.O 0.0 8.7 74 6.2 4-9 3-6 2-3 I.O 9-7 84 5-8 4-5 7.0 67.5 90.0 88.7 874 86.1 84.8 83.5 82.1 80.8 79-5 78.2 76.8 75-5 74-1 67.5 8.0 0.0 8-7 7-3 6.0 4-7 3-3 2.O 0.6 9-3 7.9 6-5 5-2 3-8 8.O 8.5 0.0 8.6 7-3 5-9 4.6 3-2 1.8 0.4 9.0 7.6 6.2 4.8 34 8.5 9.0 o.o 8.6 7-2 5-8 4-5 3-0 1.6 O.2 8.8 74 5-9 4-5 9.0 9.5 o.o 8.6 5-7 4-3 2.8 1.4 0.0 8-5 5.6 4.1 2^6 9.5 70.0 90.0 88.5 87-1 85.6 84.2 82.7 81.2 79.7 78.2 76.7 75-2 73-7 72.2 70.0 O.5 0.0 8.5 7.0 5-5 4.0 2-5 I.O 9.5 7-9 6.4 4.9 3-3 1.8 0.5 1.0 o.o 8-5 6-9 54 3-8 2-3 0.7 9.2 7.6 6.1 4-5 2.9 1.3 1.0 1.5 0.0 8.4 6.8 5-3 3-7 2.1 0.5 8.9 7-3 5-7 4.1 2.4 0.8 1.5 2.0 o.o 8.4 6.8 3-5 1-9 0.2 8.6 5-3 3-6 1.9 O.2 2.0 72.5 90.0 88.3 86.7 85.0 83.3 8l. 7 80.0 78.3 76.6 74-9 73- r 71.4 6 9 .7 72.5 3.O o.o 8-3 6.6 4-9 3- 1 1.4 79-7 8.0 6.2 4-5 2.6 0.9 Q. I 3.0 3.5 0.0 8.2 6.5 4.8 2.9 1.2 94 7.6 5-8 4.0 2.1 8.4 3.5 40 o.o 8.2 6.4 4.6 2.7 0. 9 9.1 7-2 54 3-5 1.6 697 7-7 4.0 4.5 o.o 8.1 6.3 44 2-5 0.6 8.7 6.8 4-9 2.9 I.O 9.0 7.0 4.5 75.0 90.0 88.1 86.2 84.2 82.3 80.3 78.3 76.4 744 72-3 70.3 68.3 66.2 75.O 5.5 0.0 8.0 6.1 4.0 2.0 0.0 7-9 5-9 3-8 1.7 69.6 7-5 5-3 5.5 6.0 o.o 7-9 6.0 3-8 1.7 79.6 7-5 54 3-2 i.i 8.9 6.7 44 6.0 6.5 o.o 7-9 5-8 3-6 1.4 7.0 4.8 2.6 0.4 8.1 5.8 34 6.5 7.O 0.0 7.8 5.6 34 I.I 8.8 6.5 4.2 1.9 69.6 7-2 4.8 2-3 7.O With Declination of contrary name enter the Table as above, but subtract the I tabular azimuth from 180. 0. 100 TABLE XXIV. Horizon-Azimuths. 'O 'O 3 5 ft I o 6.0 65 70 75 So 8.5 90 9.5 o 10.0 105 110 11.5 12.0 1 o o o o o o o o o o 84.0 83.5 83.0 82.5 82.0 81.5 Si.o 80.5 80.0 79.5 79.0 78.5 78.0 10 3-9 3-4 2.9 2.4 1.9 1.4 0.9 0.3 79-9 9-3 8.8 8.3 7-8 10 15 3-8 3-3 2.8 2.2 1.2 0.7 0.2 9.6 9.1 8.6 8.1 7-5 15 20 3-6 3- 1 2.6 2.0 1.5 0.9 0.4 79-9 9-3 8.8 8-3 7-7 7.2 20 25 3-4 2.9 2.3 1-7 1.2 0.6 O.I 9-5 8.9 8-4 7-8 7-2 6-7 25 30 83.1 82.5 81.9 8l. 3 80.7 80.2 79.6 79.0 78.5 77.9 77-3 76.7 76.1 30 32 3- 2-3 1.7 1.2 0.5 o.o 9.4 8.8 8.2 7.6 7.0 6.4 5.8 32 34 2.8 2.2 I.c I.O 79-7 Q.2 8-5 7-9 7-3 6.7 6.1 5-5 34 36 2.6 2.0 '3 0.7 O.I 9-5 O.Q 8.2 7-6 7.0 6.4 5-7 5- 1 36 38 2.4 1.8 i.i 0.5 79-8 9.2 8.6 7-9 7-3 6.6 6.0 5-3 4-7 38 40 82.2' 81.5 80.9 80.2 79-5 78.9 78.3 77-6 76.9 76.2 75-6 74-9 74-3 40 42 2.0 1.2 0.6 79-9 Q.2 8-5 7-9 7.2 6.5 5-8 5-2 4-4 3-8 42 44 1.7 0.9 o> 3 9-5 o.Q 8.1 7-5 6.7 6.0 5-3 4-7 3-9 3-2 44 46 1.4 0.6 79-9 9.2 8-5 7-7 7.0 6.2 5-5 4-8 4.1 3-3 2.6 46 48 I.O -3 9-5 8.8 8.0 7-2 6-5 5-7 5-o 4.2 3-4 2.7 1.9 48 50 80.7 79-9 79.1 78.3 77-5 76-7 75-9 75.1 74-3 73-5 72.7 71.9 71.1 50 51 9.6 8.8 8.0 7.2 6.4 5-6 4.8 4.0 3-2 2-3 i-S 0.7 51 52 -3 9.4 8.6 7.8 6.9 6.1 5-3 4.4 3-6 2.8 1.9 i.i O. I 52 53 o.o 9.2 8-3 7-5 6.6 5-8 4-9 4.1 3-2 2.4 0.6 69.8 53 54 79.8 8.9 8.0 7.2 6-3 5-4 4.6 3-7 2.8 1-9 i.i O.2 9-3 54 55 79-5 78.6 77-7 76.9 76.0 75-i 74.2 73-3 72.4 71.5 70.6 69.7 68.8 55 56 9.2 8-3 7-4 6.5 5-6 4-7 3-8 2.8 1.9 I.O O.I 9-1 8.2 56 57 8-9 8.0 6.1 5-2 4.2 3-3 2.3 1.4 0.4 69.5 8-5 7-6 57 58 8.6 7-7 ? 5-7 4.8 3.8 2.8 1.8 0.9 69.9 8.9 7-9 6.9 58 59 8.3 7-3 6-3 5-3 4-3 3-3 2-3 l -3 -3 9-3 8.3 7-2 6.2 59 6O 77-9 76.9 75-9 74-9 73-8 72.8 71.8 70.7 69.7 68.6 67.6 66.5 65.4 6O 61 7-5 6.5 5-4 4.4 3-3 2.2 1.2 O.I 9.0 7-9 6.9 5-7 4.6 61 62 6.1 4.9 3-9 2.7 1.6 0.6 69.4 8-3 7-i 6.1 4.8 3-7 62 63 6.6 5.6 4.4 3-3 2.1 I.O 69.9 8-7 7.5 6-3 5-2 3-9 2.8 63 64 6.1 5.0 3.8 2.7 l 'S 0.3 9.1 7-9 6.7 5-4 4-3 2.9 1-7 64 65. 75.6 74-5 73-2 72.0 70.7 69.5 68.3 67.0 65.8 64.4 63.2 61.8 60.5 65.0 5.5 5-4 4.2 2.9 1.7 0.4 9.1 7.8 6.5 5-3 3-9 2.6 1.3 59-9 5.5 6.0 5- 1 3-8 2.6 1.3 0.0 8.7 7-4 6.1 4-7 3-4 2.0 0.7 9-3 6.0 6.5 4.8 3-5 2.2 0.9 69.6 8.2 6.9 5.6 4.2 2.8 1.4 o.o 8.6 6.5 7.0 4.5 3-2 1.8 9.1 7.8 6.4 3-6 2.2 0.8 59.3 7-9 7.0 67.5 74.1 72.8 71.4 70.1 68.7 67.3 65.9 64.5 63.0 6l.6 60. i 58.6 57-1 67.5 80 3-8 2.4 I.O 69.6 8.2 6.8 5-3 3-9 2.4 0. 9 59-4 7.8 6-3 8.0 8.5 34 2.0 0.6 9-i 7-7 6.2 4-7 3-2 1.7 O.2 8.6 7.0 5-4 8.5 9.0 3-9 1.6 O.I 8.6 7.2 5-6 4.1 2.6 I.O 59-4 7.8 6.2 4-5 9.0 9.5 2.6 i.i 69.6 8.1 6.6 3-5 1.9 -3 8.6 7.0 5.3 3-6 9.5 70.0 72.2 70.7 69.1 67.6 66.0 64.4 62.8 61.2 59-5 57.8 56.1 54.3 52.6 70.0 0.5 1.8 O.2 8.6 7.0 5-4 3-7 2.1 0.4 8.7 6.9 5- 1 3-3 1.5 O.5 ! 10 '3 69.7 8.0 6.4 4-7 3- i-3 59-5 7-8 6.0 4.1 2.2 o-3 1.0 1.5 0.8 9.1 7-4 5-7 4.0 2.2 0.5 8.6 6.8 5' 3- I.I 49.1 1.5 2.0 O.2 8. 5 6.8 3-2 1.4 59-6 7-7 5-8 3-9 1.9 49 .8 7-7 2.0 72.5 69-7 67.9 66.1 64-3 62.4 60.5 58.6 56.7 54-7 52-7 50.6 48.5 46.3 72.5 3.0 9.1 7.2 5-4 3-5 1.6 59-6 7.6 5.6 3-5 1.4 49.2 7.0 4-7 3.O 3.5 8. 4 6-5 4.6 2.6 0.7 8.6 6.6 4-5 2-3 O.I 7.8 5-4 3.0 3.5 4.0 7-7 5-7 3-8 1.7 59-7 7-5 5-4 3-2 0.9 48.6 6.2 3-7 i.i 4.0 4.5 7.0 4-9 2.9 0.7 8.6 6.4 4.2 1.8 49-5 7.0 4-4 1.8 38.9 4.5 75.0 5.5 66.2 5-3 64.0 3- 1 61.9 0.9 'I! 57-5 6.2 55-2 3-8 52.8 '3 5-4 48.8 47-9 6.1 45-2 3-3 42.5 o-3 39-6 7-2 36-5 3-8 75.0 ! 3.5 6.O 4-4 2.1 59-8 7-4 4.9 2-3 49-7 7.0 4.1 i.i 37-9 4-5 0.7 6.O 6.5 3-4 I.O 8.6 6.0 3-4 0.7 7-9 5- 1.9 38.7 5-2 1.3 27.0 6.5 | 7.0 2-3 59.8 7.2 4-5 1.8 48.9 5-9 2.8 39-5 5-9 2.O 27.6 22.4 7.0 i With Declination of contrary name enter the Table as above, but subtract the | tabular azimuth from 180.0. TABLE XXIV. 101 Horizon-Azimuths. 1 Declination of same name as the Latitude. | ; 1 3 12.O 12.5 o 13.O o 13.5 14.0 14.5 1.5.0 15.5 16.0 16.5 170 175 18.0 1 a o" o o o o o o o o o 78.0 77-5 77.0 76.5 76.0 75-5 75.0 74-5 74.0 73-5 73.O 72.5 72.0 O 10 77.8 7-3 6.8 6.3 5.8 5-3 4-7 4.2 3-7 3-2 2.7 2.2 10 15 7-5 7-1 6.5 6.0 5-5 44 3-9 34 2.9 2-3 1.8 1.3 15 2O 7-2 6.7 6.2 5.6 4-5 4.0 3-5 2.9 2.4 1.9 1.3 0.8 20 1 25 6.7 6.2 5.6 4-5 3-9 34 2.9 2-3 1.7 1.2 0.6 O.I 25 3O 76, 75-5 75- 744 73-8 73-2 72.6 72.0 71.4 70.8 70.3 69.7 69.1 30 32 5-8 5-2 4-7 4.0 34 2.8 2.2 1.6 l.O 0.4 6 9 .8 9-2 8.6 32 34 5-5 4.9 4-3 3-6 3- 2.4 1.8 1.2 -5 0.0 9-3 8.7 8.1 34 36 5- 1 4-5 3-9 3-2 2.6 2.O 1.3 0.7 o.o 69.5 8.8 8.2 7-5 36 3 4-7 4.0 34 2.8 2.1 J -5 0.8 O.2 69.5 8.9 8.2 7.6 6.9 38 4O 74-3 73-6 72.9 72.2 71.6 70.9 70-3 69.6 68.9 68.2 67.6 66.9 66.2 40 41 4.0 3-3 2-7 2.O *-3 0.6 0.0 9.2 8.6 7-9 7.2 6.5 5-8 41 42 3-8 3- 1 2.4 1.7 1.0 0-3 69.6 8. 9 8.2 7-5 6.8 6.1 54 42 43 3-5 2.8 2.1 1.4 0.7 o.o 9.3 8.6 7-8 7-1 6.4 5-7 5- 43 44 3-2 2-5 1.8 I.I 0.4 69.6 8.9 8.2 74 6.7 6.0 5-3 4.6 44 ! 45 72.9 72.2 7i.5 70.7 70.0 69.3 68.5 67.8 67.0 66.3 65.6 64.8 64.1 45 46 2.6 1.8 i.i 0.4 69.6 8-9 8.1 74 6.6 5-9 C.I 4-3 3-6 46 47 2-3 1 5 0.7 0.0 9.2 8-5 7-7 6.9 6.2 54 4.6 3-8 3- 1 47 48 1.9 i.i 69.6 8.8 8.0 7-2 6.4 5-7 4.9 4.1 3-3 2-5 48 49 f-S 0.7 69.9 9-2 8.4 7.6 6.8 5-9 4-3 3-5 2.7 1.9 49 5O 71.1 70.3 69.5 68.7 67.9 67-1 66.3 654 64.6 63.8 63.0 62.1 61-3 50 51 0.7 69.9 9.1 8.2 74 6.5 5-7 4.9 4.0 3-2 2.4 1-5 0.6 51 52 0.3 94 8.6 7.7 6.9 6.0 4-3 34 2.5 0.8 59-9 52 53 69.8 8.9 8.1 7.2 6-3 54 4-5 3- 6 2.7 1.8 1.0 o.o 9.1 53 54 9-3 8.4 7-5 6.6 5-7 4.8 3-9 2.9 2.0 i.i 0.2 59-2 8-3 54 55 68.8 67.8 66.9 66.0 65.1 64.1 63-2 62.2 6l. 3 60.3 594 584 574 55 56 57 8.2 7.6 7-2 6.6 6-3 5-6 5-3 4.6 4.4 3-6 34 2.6 24 1.6 1.4 0.6 0.5 s ll 8-5 7-5 n 6.4 54 56 57 58 6.9 5-9 4.9 3-9 2.8 1.8 0.8 59-7 87 7 .6 6.5 54 4-3 58 59 6.2 5-2 4.1 2.0 0.9 59-8 8-7 77 6. 5 54 4-3 59 60 654 64.4 63-3 62.2 61.1 59-9 58.8 57-7 56.6 554 54-2 53.1 51.8 69 61 4.6 3-5 2.4 1.2 O.I 8.9 7-8 6-5 54 4.2 2.9 1.7 0.4 61 62 3-7 2-5 1.4 O.2 59-0 7-8 6.6 5-3 4.1 2.8 0.2 48.8 62 63 2.8 o-3 59-o 7-8 6.5 5-3 3-9 2.6 1.3 49-9 48.5 7- 1 63 64 i-7 0.4 7.8 6.5 5-2 3-8 2.4 l.O 49.6 8.2 6. 7 5-2 64 65.O 60.5 59-2 57-8 56.5 55-i 53-7 52.2 50.8 49.3 47-8 46.2 44.6 43-o 65.0 5.5 59-9 8-5 7-1 5-7 4-3 2.9 1.4 49-9 8.4 6.8 5-2 3-5 1.8 5.5 6.O 9.3 7-8 6.4 5-0 3-5 2.0 0.5 8.9 7-3 5-7 4.1 2-3 0.6 6.0 6.5 8.6 7.1 5-7 4.2 2-7 I.I 49-5 7-9 6.2 4.6 2.9 i.i 39-2 6.5 7.0 7-9 6.4 4.9 3-3 1.8 0.2 8-5 6.8 5- 1 34 !.6 39-7 7-7 7.0 167.5 57-1 55-6 54 524 50.8 49-2 474 45-7 43-9 42.1 40.2 38.2 36.1 67.5 8.O 6.3 4-7 3.1 1.4 49.8 8.1 6-3 4-5 2.6 0.7 38.7 6.6 44 8.0 ' 8.5 54 3.8 2.1 0.4 8-7 6.9 5.1 3-2 1.2 39-2 7- 1 4.9 2-5 8.5 9O 4-5 2.8 I.I 49-3 7-5 5-7 3-8 1.8 39-7 7.6 54 0.4 9.0 9.5 3-6 1.8 o.o 8.2 6-3 44 2.4 0-3 5-8 3-5 0.9 28.1 9.5 7O.O 52.6 50.7 48.9 47.0 45-o 43- 40.8 38.6 36.3 33-9 3 r -3 28.5 254 70.O O.5 1.5 49.6 7.6 5.6 3.6 1.4 39-2 6.8 4-3 28.9 5-7 2.2 0.5 l.O o-3 8-3. 6-3 4.2 2.0 39-7 74 4.8 2.1 29-3 6.1 2.5 I8. 3 1.0 1.5 49.1 7-o 4-9 2.6 0-3 7-9 54 2.6 29-7 6.5 2.9 18.6 '3- 1 1.5 2.O 7-7 5.6 3-3 0.9 38.5 5-9 O.I 6. 9 3-2 18.9 13-3 o.o 2.0 72.5 46.3 44.0 41.6 39.1 3 6 4 33-6 30.6 27.3 2 3 .6 19.1 13-5 o.o 72.5 3.O 4-7 2-3 39-7 7.0 4.1 27.8 3-9 194 13-7 o.o 3.O 3.5 3- 0.4 7-7 4-7 1.6 28^2 44 19.7 13-9 0.0 3.5 4.0 i.i 38.3 5-3 2.1 28.6 4-7 O.2 14.1 OiO 4.0 4.5 38-9 5-9 2.7 29.1 5.1 o-5 14-5 0.0 4.5 i "With Declination of contrary name enter the Table as above, but subtract the tabular azimuth from 180.0. 102 TABLE XXIV. Horizon- Azimuths. 1 Declination of same name as the Latitude. 6 1 o 18.0 o 18.5 19.0 19.5 20.0 20.5 o 21.0 o 21.5 o 22.0 22.5 23.0 23.5 24.O I) 1 o o o 72.0 71-5 71.0 7-5 70.0 69.5 69.0 68.5 68.0 67.5 67.0 66.5 66.0 O 10 1.2 0.7 O.2 69.7 9.2 8.7 8.2 7-7 7- 1 6.6 6.1 5.6 10 15 1.3 0.8 69.8 9-3 8.7 8.2 7-7 7.2 6.7 6.1 5.6 5- 1 15 20 0.8 0.3 69J 9.2 8.6 8.1 7 .6 7.0 6.5 6.0 54 4.9 4-3 20 25 O.I 69.5 8. 9 8. 4 7.8 7-3 6. 7 6.1 5.6 5- 4-5 3-9 3-3 25 30 69.1 68.5 67.9 67-3 66.7 66.2 65-6 65.0 64.4 64.8 63-2 62.6 62.0 30 32 8.6 8.0 74 6.8 6.2 5.6 5.O 44 3-8 3-2 2.6 2.O i-3 32 34 8.1 7-5 6.9 6.2 5.6 5- 44 3-8 3- 1 2 -5 1.9 1-3 0.6 34 36 7-5 6.9 6.3 5-6 4-3 3-7 1.8 i.i 0.5 59-8 36 38 6.9 6.2 5-6 4.9 4-3 3- 6 3- 2-3 1.6 0.9 -3 59-6 8-9 38 4O 66.2 65.6 64.9 64.2 63-5 62.8 62.1 61.4 60.7 60.0 59-3 58.7 57-9 4O 41 42 5.8 54 5-2 4-7 4-5 4.0 3-8 3-3 & 2-3 1.9 1-7 1.2 0.9 0.4 O.2 59-7 59-5 9.0 8^ 8.2 7.6 74 6.8 41 42 43 5- 4-3 3-6 2.8 2.1 1.4 0.7 59-9 9.2 8.4 7-7 7.0 6.2 43 44 4.6 3-8 3- 1 2-3 1.6 0.9 O.2 94 8.6 7.8 6.4 5.6 44 45 64.1 63.3 62.6 61.8 61.1 60.3 59-6 58.8 58.0 57-2 56.5 55-7 54-9 45 46 3.6 2.8 2.1 J -3 o> 5 59-7 9-0- 8.2 74 6.6 5-8 5 >o 4.2 46 47 3- 1 2-3 I -S 0.7 59-9 9.1 8-3 7-5 6.7 5-9 4-3 34 47 48 2-5 P-7 0.9 O.I 9-3 8.4 7.6 6.8 6.0 4-3 3-5 2.6 48 49 1.9 i.i o-3 594 8.6 7-7 6.9 6.0 5-2 4-3 3-5 2.6 J -7 49 50 61.3 60.4 59-6 58.7 57-9 57- 56.1 55-2 544 53-5 52.6 5!-7 50.8 50 51 0.6 59-7 8.9 8.0 6.2 5-3 44 3-5 2.6 1.6 0.7 49.8 51 52 59-9 9.0 8.1 7-2 3 5-3 44 3-5 2-5 1.6 0.6 49-7 8.7 52 53 9-i 8.2 7-3 6.3 54 44 34 2-5 0-5 49-5 8.6 7-5 53 54 8-3 7-3 6.4 54 44 34 2.4 1.4 0.4 494 8-3 74 6.2 54 55 574 56.4 554 544 534 524 51.3 50-3 49-2 48.1 47.1 46.0 44-8 55 56 6.4 54 44 3-3 2-3 1.2 0.2 49.0 7-9 6.8 5-7 4.6 3-3 56 57 54 44 3-3 2.2 i.i O.O 48.9 7-7 6-5 54 4.2 3.0 i-7 57 58 4-3 2.1 0.9 49-8 48.6 7-5 6.2 3-8 2.5 1.2 39-9 58 59 2.0 0.8 49.6 8.4 7.2 5-9 4.6 3-3 2.0 0.7 39-3 7.8 59 60.0 51.8 50.6 494 4 8.1 46.8 45-5 44-2 42.8 41.4 4 O.I 38.6 3M 35-6 60.0 0.5 i.i 49.9 8.6 7-3 6.0 4.6 3-3 1.9 0.4 39-o 7-5 5-9 4-3 O5 l.O 0.4 7.8 6.5 5- 1 3-7 2-3 0.9 394 7-9 4-7 3.0 10 1.5 49.6 8'3 7.0 5-6 4.2 2.7 39-8 6-7 3-3 1.5 1.5 i 2.0 8.8 7-5 6.1 4-7 3-2 O.2 8.7 7- 1 54 3-7 1.9 o.o 2.0 62.5 48.0 46.6 45- * 43-7 42.2 40.6 29.0 374 35-8 34.0 32.2 30-3 28.3 62.5 3.0 5-7 4.1 2.6 39-5 7.8 6.1 44 2-5 0.6 28.6 6.4 3.O 3.5 6.2 4-7 3- 1 1.5 39-9 8-3 6.5 4-7 2.9 0.9 28.9 6.6 4-3 3.5 4.0 5' 2 3-6 2.0 o3 8-7 7.0 e i 3-3 29.2 7.0 4-5 1.9 4.0 4.5 4.1 2-5 0.8 39-1 74 5-5 3'-6 1.6 29-5 7.2 4.8 2.2 19.1 4.5 65.0 43.0 41-3 39-6 37-8 36.0 34-o 32.0 29.8 27-5 25.1 22.4 194 15.6 65.0 5.5 1.8 0.0 8.2 6.4 44 2.4 0.2 7.8 5-3 2.7 19.6 15-9 n. i 5.5 60 0.6 38.7 6.8 4.9 2.7 0.6 28.2 5-6 2.9 19.8 16.2 II.4 o.o 6.O 65 39-2 7-3 5-3 3- 2 0.9 28.6 6.0 3-2 o.o 16.3 1 1.6 O.O 6.5 7.0 7-7 5-7 3.6 28.9 6-3 3-5 0.2 16.5 11.7 0.0 7.0 67.5 36.1 34-o 31.7 29-3 26.6 23-8 20.5 I6. 7 11.8 0.0 67.5 8.O 44 2.1 29.7 7.0 4.1 0.8 17.0 II-9 o.o 8.O 8.5 2-5 o.o 74 44 i.i 17.2 I2.I O.O 8.5 9.0 0.4 27.7 4-7 17-3 12.3 O.O 9.O 9.5 28.1 1.6 17.6 12.4 0.0 9.5 70.0 254 21.9 17.8 12.6 o.o 70.0 0.5 2.2 18.1 12.8 0.0 0.5 ! 1.0 I8. 3 12.9 0.0 10 i 1.5 !3-! o.o 1.5 2.0 0.0 2O "With Declination of contrary name enter the Table as above, but subtract the tabular azimuth from 180 .0. TABLE XXIV. 103 Horizou- Azimuths. 0) Declination of same name as the Latitude. | 1 24.O 245 25.0 255 26.O 26.5 27.0 27.5 o 28.O 28.5 29.0 29.5 30.0 2 H) o o o O o o o O o o 66.0 65.5 65.0 64.5 64.0 63'5 63.0 62.5 62.0 61.5 6 i.o 60.5 6O.O 4 5-9 5-4 4-9 4-4 3-9 3-4 2-9 2.4 1.9 1.4 0.9 0.4 59-9 4 8 5-7 5-2 4-7 4.2 3-7 3-2 2-7 2.2 1.7 1.2 0.7 O.2 9-7 8 12 5-4 4-9 4.4 3-9 3-4 2.9 2.4 1.9 1.3 0.8 0.3 59-8 9-3 12 16 4-4 3-9 3-4 2.9 2.4 1.8 I -3 0.8 O.2 59-7 9-2 8-7 16 20 64-3 63.8 63-3 62.7 62.2 .61.7 61.1 60.6 60.0 59-5 58.9 58.4 57-9 20 22 4.0 3-4 2.9 2-3 1.8 1.2 0.7 O.I 59-6 9.0 8.5 7-9 7-4 22 24 3-6 3- 2.4 1.9 1.3 o.S O.2 59-6 9-i 8-5 8.0 7-4 6.8 24 26 3- 1 i 2.5 1.9 1.4 0.8 0.3 59-7 9.1 8.5 7-9 7-4 6.8 6.2 26 28 2.6 2.O 1.4 0.8 0.2 59-7 9.1 8.5 7-9 7-3 6-7 6.1 5-5 28 30 62.0 6l. 4 60.8 60.2 59-6 59.0 58.4 57.8 57-2 56.6 56.0 55-3 54-7 30 31 i.j I.I -5 59-9 9.2 o O 8.0 7-4 6.8 6.2 5-5 4-9 4-3 31 32 1.3 0.7 O.I 9-5 8.9 8-3 7.6 7.0 6.4 5.8 4-5 3-9 32 33 I.O 0.4 59-8 9.1 8.5 7-9 7-2 6.6 6.0 5-3 4-7 4.0 3-4 33 34 0.6 O.O 9-4 8.7 8.1 7-4 6.8 6.2 5-5 4-9 4.2 3-6 2.9 34 35 60.2 59-6 58.9 58.3 57-7 57-0 56.4 55-7 55-o 54-4 53-7 53-i ! 52-4 35 36 59.8 9.2 8.5 7-9 7.2 6.5 5-9 S- 2 4-5 3-9 3.2 2.5 1.8 33 37 9.4 8.7 8.1 7-4 6.7 6.0 5-4 4-7 4.0 3-3 2.6 1.9 1.2 37 ' 38 . 8.9 8.3 7.6 6.9 6.2 5-5 4.8 4.1 3-4 2. 7 j 2.0 '3 0.6 33 39 8.4 7-8 6.4 5-7 4-3 3-5 2.8 2.1 1.4 0.7 | o.o 39 40 57-9 57-2 56.5 55-8 55- 1 54-4 53-7 52-9 52.2 51.5 | 50.7 50.0 ! 49.3 40 41 7-4 6.7 5-9 5-2 4-5 3-8 3- 2.3 1.5 0.8 o.o 49.3 8.5 41 42 6.8 6.1 5-3 4-6 3-9 2.3 1.6 0.8 O.I 49-3 8-5 7-7 42 43 44 6.2 5-6 5-5 4.8 4-7 4.0 3-9 3-2 3-2 2-5 2.4 1.6 0.9 0.8 o.o O.I 49-3 49-3 8.4 8.5 7.6 7-7 6.9 6.8 6.0 43 44 i 45 54-9 54-1 53-3 52.5 51.7 5-9 50.1 49-2 48.4 47-5 46.7 45-9 45 - 45 1 46 4.2 3-4 2-5 0.9 o.o 49.2 8-3 7-5 6.6 5-7 4.9 4.0 46 47 3-4 2.6 1-7 0.9 0.0 49.1 8-3 7-4 6.5 5-6 4-7 3.8 ; 2.9 47 48 2.6 1.7 0.8 o.o 49.1 8.2 7-3 6.4 5-4 4-5 3-6 2.6 ; 1-7 48 49 J -7 0.8 49.9 49.0 8.1 7-2 6.2 5-3 4-3 3-3 2.4 1.4 ! 0.4 49 50 50.8 49.8 48.9 48.0 47.0 46.1 45- i 44.1 43-i 42-1 41.1 4O.O i 38.9 50 51 49.8 1 8.8 7.8 6.8 5-9 4-9 - 3-8 2.8 1.8 0.7 39-6 38.5 ! 7-4 51 52 8.7 7.7 6.7 5-6 4.6 3.6 2-5 1.4 o-3 39-2 8.0 6.9 5.7 52 53 7-5 6.5 5-4 4-3 3-3 2.2 I.O 39-9 38.7 7-5 6-3 5.i ; 3-8 53 54 6.2 5- 1 . 4.0 2.9 1.8 0.6 39-4 8.2 7.0 5-7 4-4 54 55.0 41.8 43-7 ! 42.5 41.4 40.2 38.9 37-7 36-4 35-i 33-7 32-3 30.9 29.3 55.0 5.5 4.1 2.9 1.7 0.5 39-3 8.0 6-7 5-4 4.0 2.6 i.i 29.6 8.0 5.5 6.0 3-3 2.1 0.9 8-4 7.1 5-7 4-3 2-9 1.4 29.9 8-3 6.6 6.0 65 2.5 1.2 o.o 3 1: 7 7-4 6.1 4.6 3-2 O.I 8-5 6.8 5- 6.5 7.0 1.7 0.4 39-1 7-8 6-4 5- 3-5 2.O o-5 28.8 7- 1 5-3 3-4 7.0 57.5 3.0 40.8 39-9 39-5 8-5 38.1 36.8 5-7 35-3 4.2 33-8 2.6 32-3 3-7 29.4 29.1 7.6 27.4 5-8 3 23.6 21.5 19-3 57.5 8.0 85 8.9 7-5 6!o 4-5 3- 1.4 29.7 7-9 6.1 4.0 1.9 19.6 16.9 8.5 9.0 7.8 6.4 4-9 3-3 o.o 8.2 6-3 4-3 2.1 19.7 17.0 13.8 9.O 9.5 6.7 5.2 3-6 2.O -3 28.5 6.6 4-5 2-3 19.9 17.2 14,1 -9.9 9.5 600 35-6 34-0 32.3 30.6 28.8 26.8 24.8 2.6 20. r 17.4 14.2 IO.O o.o 60.0 0.5 4-3 2.6 0.9 29.0 7.1 5 2.8 0.4 17.6 14.2 IO.I o.o 0.5 1.0 1.2 29-3 7-4 5-3 o-5 17.7 14-5 10.2 0.0 1.0 1.5 1.5 29.7 7-7 5.6 3-3 pj 18.0 14.6 10.3 0.0 1.5 2.O o.o 8.0 5.8 3-5 I.O 18.1 14.8 10.4 o.o 2.0 62.5 28.3 26.1 23-8 21.2 18.3 14.9 10.5 0.0 62.5 3.O 6-4 4.0 1.4 I8. 5 IS- 1 10.6 o.o 3.O 3.5 4-3 18.7 15.2 10.7 o.o 3.5 4.0 1.9 18.9 10.8 o.o 4.0 4.5 19.1 15.6 I I.O o.o . 4.5 With Declination of contrary name enter the Table as above, but subtract the tabular azimuth from 180.0. 104 TABLE XXV. Position- Angles for Horizon-Azimuths. Lat. Declination of same or contrary name. O 5 10 12 14 16 18 20 22 24 26 28 3O Lat. o o o o o o o o o 90.0 90.0 90.0 90.0 90.0 90.0 90.0 90.0 9c .0 9 3.O 90.0 90.0 90.0 2 88.0 88.0 88.0 88.0 87.9 87.9 87.9 87. -) 87.8 8 7 .8 87.8 87.7 87.7 2 4 6.0 6.0 6.0 5-9 5-9 5-8 5.8 5- j 5 7 5.6 5-6 5-5 5-4 4 6 4.0 4.0 3-9 3-9 3-8 3-8 3-7 3-6 3-5 3-4 3-3 3-2 6 8 2.0 2.0 9 1.8 1.8 1.6 i. 5 i 4 1.2 i.i 0.9 0.8 8 10 8o.O 80.0 79-9 79.8 79-7 79.6 79-5 79-4 79.2 79.0 78.9 78.6 78.4 10 12 78.0 77-9 7-8 7-7 7.6 7-5 7-4 7.2 7.0 6.8 6.6 6.4 6.1 12 14 6.0 5-9 i .0 5-7 5-6 5-4 5-3 5- i 4-9 4.6 4-4 4.1 3-8 14 16 4.0 3-9 3-8 3-6 3-5 3-3 3- 1 3- 2.7 2.4 2.1 1.8 1.4 16 18 2.O 1.9 ] 7 1.6 1.4 1.2 I.O 0.8 c 5 0.2 69.9 69-5 69.1 18 20 7O.O 69.9 69.7 69.5 694 69.2 68.9 68.7 68.4 68.0 67.6 67.2 66.7 20 22 68.0 7-9 7-7 7-5 7-3 7.1 6.8 6.5 6.2 5-8 54 4.9 4.4 22 24 6.0 5-9 i .6 5-4 5-2 5'0 4.7 4.4 4.0 3-6 3- 1 2.6 2.0 24 26 4.0 3-9 3-6 3-4 3- 1 2.9 2-5 2.2 .8 1.3 0.8 O.2 59-6 26 28 2.0 1.9 ] 5 i.i 0.8 0.4 O.O 59.6 59-1 58.5 57-9 7.2 28 30 60.0 59-9 59-5 59-2 59-0 58.6 58.3 57.8 57-4 56.8 56.2 55-5 54-7 3O ' 32 58.0 7-9 7-5 7.2 6.9 6.5 6.1 5- 7 5 * 4-5 3-9 2.2 32 j 34 6.0 5- ^ 5-4 4.8 4.4 4.0 3-5 2.9 2.2 i-S 0.7 49-7 34 36 4.0 3- S 3-4 3- 1 2.7 2-3 1.8 ^ 0.7 49.9 49.2 48-3 7.2 36 38 2.0 i. S 3 I.O 0.6 0.2 49-7 49. i 48.4 7.6 6.8 5-8 4-7 38 40 5O.O 49-8 49-3 48.9 48.5 48.0 47-5 46.8 46.1 45-3 44-3 43-3 42.1 40 42 48.0 7.8 7.2 6.8 6.4 5-9 5-3 4.6 3-8 2.9 1.9 0.7 39-4 42 44 6.0 5-8 c I 4-7 4-3 3-7 3- 1 2-3 i 5 0.5 39-4 38.1 6.7 44 46 4.0 3-8 >-I 2.7 2.1 0.9 0.0 39-i 38.0 6.8 5-4 3-8 46 48 2.0 i. 7 ] .O 0.6 0.0 39-3 38.6 37.7 6.7 5-5 4.2 2.7 0.9 48 : 50 52 40.0 38.0 39- 7- 7 7 38.9 6.9 38.5 | 37-9 6-3 5-7 4-9 36.3 4.0 35-4 3- 34.3 1.8 33- 0.4 31-5 28.8 29.8 6.8 27.8 4-5 50 52 54 6.0 5- 7 4.8 4.2 3-5 2.7 i-7 0.6 29.2 27.7 5-8 3-6 0.9 54 56 4.0 3-7 2.7 2.O 0.4 29-3 28. i 6.6 4.8 2.7 O.I 17.0 56 58 2.0 1.6 0.6 29.9 29.1 28.1 6.9 5-5 3-8 1.8 19.4 16.1 11.7 58 60 3O.O 29.6 28.4 27.7 26.8 25-7 24.4 22.9 20.9 1 8.6 15.5 II.2 o.o 60 62 28.0 7.6 6-3 5-5 4-5 3-3 1.8 20.0 17.8 14.9 10.8 0.0 62 61 6.0 5-6 4.1 3-2 2.1 0.7 19.1 17.0 14.2 IO.2 o.o 64 66 4.0 3-5 1.9 0.9 197 18.1 16.1 I 3 .6 9.9 0.0 66 68 2.0 5 19.7 1 8.6 I7.I 15-3 12.8 9.4 o.o 68 70 20.0 19.4 17.4 16.1 14.4 12.1 8.8 O.O 70 72 1 8.0 17.3 15.0 i 3-5 IJ.4 8-3 ao 72 74 16.0 *5- 2 12.6 10.7 .| 7.9 0.0 74 76 14.0 '3- I 9.8 7-3 ! 0.0 76 78 12.0 10.9 6-7 o.o 78 80 I O.O 8. 7 o.o 80 TABLE XXVI. Limiting Errors of Horizon-Azimuths. Dec. 12. Dec. 18. Dec. 24. Dec. 30. Dec. 24. : Q> Partial Az. Partial Az. Partial Az. Partial Az. & Partial Az. *0 Error. Prob. T^it .1 I Error. Prob. rr.*i Error. Prob. Tiki 41 1 Error. Prob. T ttiil 2 Error. Prob T*\< 11 1 tj Lat. Dec. rotai Az. Lat. IUI.ll Dec. Az. Lat. Dec. loiai Az. Lat. Dec. 1 M II I Az. Lat. Dec. rouu Az. : rt Error 1 12'. Error Error. W Error !2'. ___ Error Error Error 12'. Error ] irror. Error 12'. Error Error rt Error n3 0.5 Error Error. 1 . o o O O.O to.i 0.1 0.0 0.1 0.1 0.0 0.1 0.1 0.0 0.1 0.1 O o.o 0.5 0.5 30 0.0 O.I O.I 0.1 O.I 0.2 0.1 O.I O.2 0.1 O.I 0.2 IO 0.1 0.5 40 0.1 0.2 0.2 O.I O.2 O.2 O.2 O.2 0.3 0.2 O.I 0.2 2O 0.2 0.5 o.'6 50 O.I 0.2 0.2 0.2 O.2 0.3 0.2 0.2 0.3 -3 O.2 0.4 3O 0.2 0.6 0.7! 60 O.I 0.2 0.2 0.3 0.3 0.4 0-5 0.4 0.7 *o.6 *o. 3 0.7 4O 0.3 0.8 0.9 62 O.2 o-3 0.4 0.4 0-3 0.5 0.7 0.4 0.8 ti.o to.6 1.2 50 0.5 0.9 I.O 64 O.2 0.4 O-5 0.4 0.7 I.I 0.7 '3 60 1.3 '7 2.1 66 '3 o-3 0.4 0.7 0-5 0.9 68 0-3 0.3 o. ** 0.6 1.3 70 0.4 0.4 0.6 72 0.5 0.4 0.7 74 76 0.6 1 0.5 0.8 , 0.6 0.8 I.O * For Lat. 55. t For Lat. 58. TABLE XXVIL 105 Correction of the observed Compass-Azimuth as taken on the Apparent Horizon. Declination of same or contrary name. Lat. 5 10 12 14 16 18 20 82 24 26 28 30 Lat. o o o o o O 0.0 0.0 O.O o.o O.O 0.0 0.0 o.o o.o 0.0 0.0 0.0 0.0 5 I I I I I I I I I I I I I 5 10 I I I I I I I I I I I I I 10 15 2 2 2 2 2 2 2 2 2 2 2 2 2 15 20 2 2 2 2 2 2 3 3 3 3 3 3 3 20 24 28 O. 3 3 0-3 0-3 0-3 0-3 0.3 0-3 o-3 o-3 0-3 0-3 0.4 4 0.4 4 24 28 32 36 S 5 5 * 1 | 5 5 1 I 32 36 38 5 5 5 5 6 6 6 6 6 6 6 7 7 38 40 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.7 0.7 0.7 0.7 4O 42 6 6 6 6 6 7 7 7 7 8 8 8 42 44 6 6 7 7 7 7 7 7 8 8 8 9 9 44 46 7 7 7 7 7 8 8 8 8 9 9 9 I.O 46 48 7 8 8 8 8 8 8 9 9 I.O I.O I.O i 48 50 0.8 0.8 0.8 0.8 0.9 0.9 0.9 0.9 I.O i.i i.i i.i 1.3 50 52 S 9 9 9 9 I.O I.O I.O i 2 2 3 5 52 54 9 9 I.O I.O I.O i i i 2 3 4 8 54 56 I.O I.O i i i 2 2 2 3 6 8 2.2 56 58 i i 2 2 2 3 3 4 5 7 9 2-3 3-2 58 60 i. 2 1.2 1.3 1.3 1.3 1.4 i-S 1.6 1.7 2.0 2.4 3-4 00 6O 62 S 3 4 4 4 6 7 8 2.1 ij 3-5 00 62 64 4 4 5 S 6 8 9 2.2 6 3-7 00 64 66 5 5 7 7 9 2.0 2-3 S 3-8 00 66 68 b 7 9 2.0 2.2 4 9 4.0 00 68 70 1.8 1.9 2.1 2-3 2.6 3- 1 4-3 oo 70 72 2.0 2.1 5 8 '3-3 4.6 00 72 74 2. 5 3-5 4.8 oo 74 76 6 3.0 8 00 76 78 3- I 6 5-7 00 78 80 3-8 44 CO 8O TABLE XXVII L TABLE XXIX. Error of the Horizon- Azimuth for an Error of 12', or O.2, in the Latitude. Error of the Horizon- Azimuth for an Error Azimuth. of 6', or 0.l, in the 90 80 70 60 50 40 30 20 15 10 5 Lat. Declination. o o 10 0.0 0.0 0.0 0.0 0.0 O 0.0 o 0.0 I o O.O I 0.0 I O.O 2 0.0 0.0 oo o 10 Position- Angle. Azimuth. Error. 20 o o o o I I I 2 7 4 8 oo 15 30 o I I 2 3 4 6 M oo 20 o 9O o o 35 o I I 2 2 4 5 8 6 00 25 8O 40 0.0 0.0 0. 0. O.I 0.2 0-3 0.5 0.6 I.O 1.9 00 40 7O 45 o 2 2 3 6 8 i 2-3 00 45 60 5O o o 2 3 4 7 9 3 8 00 50 50 55 2 2 3 r 8 i.i 6 3-3 oo 55 60 o 2 3 4 6 9 3 2.0 4.0 00 60 4O O.I 62 0.0 0. 0. 0.2 0-3 0-5 0.6 I.O 1.4 2.1 4-3 00 62 3O 2 64 o 2 3 5 7 i 5 3 7 00 64 25 2 66 o 2 3 4 5 8 2 7 00 66 2O 68 2 3 4 6 9 4 8 8 7 oo 68 7O o 2 3 5 7 I.O 5 2.0 3.1 6.3 00 70 18 0.3 72 0.0 o. 0.2 o.S 0.8 i.i 2-3 3.5 7.0 00 72 16 4 74 76 78 o 2 2 3 3 4 5 9 I.O i 2 4 6 9 2.2 6 6 9 4-5 5-3 ' 8.0 9.2 0.7 00 oo 00 74 76 78 14 12 1O 4 I 8O 2 4 b 9 3 2.0 3- 1 4.2 6.4 i 3-o 00 8O 8 0.7 Lat. o a I O o i S ee 1 o i o I o o 1 Lat. 6 4 2 I.O 4 2.9 Azimuth. oo 106 TABLE XXX. Time-Azimuths : Log A. Latitude. Dec. Polar Dist. 1 2 3 4 5 6 7 8> 9 10 35 9-499 9.488 9.476 9.465 9453 9.440 9-427 9413 9-399 9-385 9-370 55 34 485 474 462 45 437 424 411 397 382 367 56 33 472 460 447 435 422 408 394 379 364 348 33 * 57 32 458 445 432 419 406 391 376 345 328 310 58 31 443 43 417 403 389 374 358 342 325 308 289 59 3O 9.428 9415 9.401 9-386 9-371 9.356 9-340 9-323 9-305 9.286 9.267 60 29 413 399 384 369 354 337 320 302 284 264 243 61 28 27 397 380 382 365 367 349 333 335 316 3i8 298 300 279 281 259 262 238 241 216 219 193 62 63 26 363 347 331 3H 296 277 257 236 214 190 165 64 25 9.346 9.329 9.312 9.294 9-275 9-255 9-234 9.211 9.188 9.163 9.136 65 24' 328 310 292 273 253 232 209 1 86 161 134 105 66 23 39 290 271 251 230 207 184 159 132 103 072 67 22 289 269 249 228 206 182 '57 130 IOI 070 036 68 21 268 248 226 204 1 80 155 128 099 068 034 8.998 69 20 9.246 9.225 9.202 9.179 9.153 9.126 9.097 9.066 9.032 8.996 8-955 7O 19 224 20 1 177 152 125 096 064 030 8-994 953 909 71 18 200 176 150 123 094 063 029 8.992 952 907 857 72 17 175 149 122 093 06 1 027 8. 99 o 95 905 855 798 73 16 I 4 8 121 091 060 026 8.989 948 903 853 796 730 74 ! 15 9 .II9 9.090 9-059 9.025 8.988 8.947 8.902 8.852 8-795 8.728 8.650 75 14 089 058 8.986 946 901 850 793 727 649 552 76 13 057 022 8.985 944 899 849 792 725 647 551 427 77 12 022 8.984 943 898 848 791 724 646 549 425 250 78 11 8.984 943 897 847 789 723 645 548 424 249 7.948 79 10 8.942 8.897 8.846 8.789 8.722 8.643 8-547 8.423 8.247 7-947 oo 80 9 896 845 788 721 643 546 422 246 7.946 oo 7-947 81 8 8 45 787 721 642 545 421 245 7-945 00 7.946 8.247 82 7 787 720 641 545 420 244 7-944 00 7-945 8.246 423 83 6 719 641 544 419 244 7-943 00 7-944 8.245 422 547 84 5 8.640 8-543 8.419 8.243 7.942 00 7-943 8.244 8.421 8.546 8.643 85 4 543 418 243 7-94 1 00 7.942 8.244 420 545 643 722 86 \ 3 418 242 7.941 oc 7.941 8.243 419 545 642 721 789 87 ! 2 242 7.941 00 7.941 8.242 419 544 641 721 788 846 88 1 7.941 00 7.941 8.242 418 543 641 720 787 845 897 89 oc 7.941 8.242 8.418 8-543 8.640 8.719 8.787 8.845 8.896 8.942 90 1 7-941 8.242 418 543 640 719 786 844 895 941 983 91 2 8.242 418 543 640 7^9 786 844 895 982 9.020 92 3 418 543 640 719 786 844 895 941 982 9.020 055 93 4 543 640 719 786 844 895 940 982 9.019 054 087 94 5 8.640 8.719 8.786 8.844 8.895 8.940 8.982 9.019 9-054 9.086 9.116 95 1 6 719 786 844 895 940 982 9.019 054 086 116 1 144 96 7 787 844 895 941 982 9.019 054 086 116 144 1 170 97 8 895 941 982 9.019 054 086 116 144 170 | 194 98 9 896 941 982 9.020 054 086 116 144 170 194 | 218 99 10 11 8.942 984 8.983 9.021 9.020 9-55 087 9.087 117 9.116 144 9.144 170 9.170 194 9.194 218 9.218 240 9.240 261 100 101 12 9.022 056 088 117 H5 170 218 240 261 281 1O2 -13 57 088 118 145 171 '95 218 240 201 281 300 103 14 089 119 146 172 196 219 241 261 28l 300 104 -15 9.119 9.147 9-173 9.197 9.220 9.241 9.262 9.282 9.301 9.318 9-336 105 16 148 173 197 220 242 263 282 301 336 353 106 17 198 221 243 263 283 302 320 337 353 369 107 18 200 222 244 264 284 33 320 337 354 385 1O$ 19 224 245 265 285 303 321 338 354 370 3 400 1O9 20 9.246 9.267 9.286 9-305 9.322 9-339 9-355 9-371 9.386 9.401 9415 110 21 268 287 306 323 34 356 372 387 401 416 429 111 22 289 307 325 357 373 388 402 416 430 443 112 23 -24 39 328 326 344 343 360 359 375 374 390 389 405 403 418 417 432 43 i 444 444 457 469 113 114 90 89 88 87 86 85 84 83 82 81 80 Dec. Polar Dist. tfo-Latitude. TABLE XXX. 107 Time- Azimuths: Log B. Latitude. Dec. Polar Dist. 1 2 3 4 5 6 7 8 9 10' 35 0.501 0.491 0.480 0.470 0.460 0.451 0.442 o-433 0.424 0.415 0.407 o 55 34 5H 54 493 482 472 462 453 444 434 425 56 33 528 506 495 484 474 464 455 445 43 6 427 57 32 542 531 5*9 508 497 486 476 466 45 6 447 437 58 31 557 545 .533 499 488 478 467 458 448 59 30 0.572 0-559 0-547 0-534 0-523 0.512 0.500 0.490 0.479 0.469 0459 60 29 587 574 561 548 536 524 513 502 491 481 470 61 28 603 589 576 563 55 "2 526 5*5 503 493 482 62 27 620 605 578 564 539 528 516 55 494 63 26 637 621 607 593 579 566 553 529 517 506 64 25 24 0.654 672 0.638 656 0.623 640 0.608 624 0-594 610 0.580 595 0.567 582 168 0.542 556 o-530 543 0.518 53 1 65 i 68 23 691 674 657 641 626 611 597 583 570 556 544 67 22 711 693 675 659 643 627 612 598 584 570 557 68 21 732 7i3 694 677 660 644 628 613 599 585 69 20 0-754 0-733 0.714 0.695 0.678 0.661 0.645 0.629 0.614 0.600 0.585 70 19 776 755 735 715 697 679 662 646 630 615 600 71 18 800 778 75 6 73 6 716 697 680 663 646 631 615 72 17 825 802 779 757 737 698 680 663 647 631 73 16 852 827 803 758 737 718 699 68 1 664 647 74 15 0.881 0-853 0.827 0.803 0.780 o-759 0.738 0.718 0.699 0.681 0.664 75 14 911 88 1 854 828 804 759 739 719 700 682 76 13 943 912 882 855 829 805 782 760 739 719 700 77 12 978 944 912 883 855 829 805 782 760 739 719 78 11 1.016 979 945 883 856 830 805 782 760 739 79 1O 1.058 1. 117 0.979 0-945 0.913 0.884 0.856 0.830 0.806 0.782 0.760 80 9 104 059 1.018 980 946 914 884 856 830 806 782 81 155 105 059 1.018 981 946 914 884 856 830 806 82 7 213 156 105 059 1.018 981 946 914 884 856 830 83 6 281 214 156 105 060 1.018 981 946 914 884 856 84 5 1,360 1.281 1.214 1.156 1.105 i. 060 1.018 0.981 0.946 0.914 0.884 85 4 457 360 281 214 156 105 060 1.018 981 946 913 86 3 582- 360 281 214 156 I0 5 059 1.018 980 945 87 2 758 582 457 360 281 214 156 105 059 1.018 980 88 1 2.059 758 582 457 360 281 214 156 105 059 1.017 89 + 00 2.059 1-758 1.582 1-457 1.360 1.281 1.213 I-I55 1.104 1.058 9O 2.059 + 00 2.059 758 582 457 . 359 280 213 155 103 91 2 1.758 2.059 + 2.059 757 581 456 359 279 212 154 92 3 582 1.758 2.059 -j- 00 2.058 757 455 358 279 211 93 4 457 582 J-757 2.058 + 00 2.058 756 580 455 357 2 7 8 94 - 5 1.360 1.457 1.58! 1-757 2.058 -f CO 2.057 1.756 1-579 1-454 1-357 95 6 281 359 45 6 581 1.756 2.057 + 00 2.056 755 578 453 96 7 213 280 359 455 580 2.056 + 00 2.055 754 577 97 8 155 213 279 358 455 579 1-755 2.055 + 00 2.054 753 98 9 104 155 212 279 357 454 578 J-754 2.054 -j-oo 2-053 99 10 1.058 1.103 I.I54 I.2I1 1.278 1-357 1-453 1-577 1-753 2-053 + 00 100 11 016 057 103 153 211 277 355 452 576 I-75 1 2.052 1O1 12 0.978 016 057 I O2 *5 2 209 276 354 575 i.75o 1O2 -13 943 0.978 015 056 IO1 208 2 75 353 449 573 103 14 911 942 0.977 OI 4 054 099 150 207 273 35 i 448 104 15 0.881 0.910 0.941 0-975 I.OI2 I -53 1.098 1.148 1.205 1.272 i-35o 105 -16 852 879 909 94 0.974 on 052 097 147 204 270 1O6 17 825 . 851 878 907 939 o.973 010 050 095 145 202 107 -18 800 824 850 877 906 937 0.971 008 048 93 J 43 108 19 776 799 823 848 875 904 93 6 0.970 006 047 091 1O9 ! -20 0.754 0.775 0.798 0.821 0.847 0.874 0.903 0-934 0.968 1.004 1.045 110 21 732 752 774 796 820 845 872 901 Q32 0.966 002 111 22 711 731 772 794 818 843 870 899 930 0.964 112 23 691 710 729 749 770 793 816 841 868 897 9 28 113 24 672 690 708 727 747 768 791 814 839 866 895 114 90 89 88 87 86 85 84 83 82 81 80 Polar Co-Latitude. Dist. 108 TABLE XXX. Time-Azimuths: Log A. i Latitude. Dec. Polar Dist. 10 11 12 13 14 15 16 17 18 19 2O o 35 9.370 9-354 9-337 9-320 9-302 9.282 9.262 9.241 9.218 9.194 9.168 55 34 351 334 298 279 259 237 214 190 164 136 56 i i 33 331 295 276 255 234 211 1 86 1 60 132 102 57 32 310 292 273 252 230 207 183 157 129 098 065 58 31 289 270 249 227 204 1 80 153 125 : 095 062 026 59 3O 9.267 9.246 9.224 9.201 9.176 9.150 9-122 9.092 9.058 9.023 8.983 60 29 243 221 198 173 147 119 088 055 019 8.980 936 61 28 219 195 171 144 116 085 052 016 8.976 93 2 883 62 27 193 1 68 141 "3 082 049 013 8-973 929 880 823 63 2S 165 ; 139 no 080 046 OIO 8.970 926 876 820 755 64 25 9.136 9.108 9.077 9.043 9.007 8.967 8.923 8-873 8.817 8.752 8.674 65 24 105 075 041 005 8.965 920 8 7 I 814 749 671 576 66 i 23 072 ; 039 002 8.962 918 868 Nil 746 668 573 449 67 22 036 ooo 8.960 915 805 809 743 665 i 570 446 8.272 68 21 8.998 8.957 913 863 806 741 663 567 444 269 7-969 69 20 8.955 i 8.911 8.861 8.804 8.738 8.660 8.565 8.441 8.266 7.967 00 70 19 909 859 802 736 658 562 439 264 7-964 oo 7.966 71 ! 17 857 i 800 798 ! 73 2 734 654 656 558 560 434 436 259 7-959 7.961 00 oc 7.961 7.964 8.264 8.266 441 72 73 ! IS 73 652 556 432 257 7-957 oo 7.959 8.261 439 565 74 15 8.650 8.554 8.430 8.255 7-955 00 7.957 8.259 8.436 8.562 8.660 75 14 552 428 253 7-953 00 7-955 8.257 434 560 658 738 76 13 427 252 7-95 i 00 7-953 8-255 432 558 736 804 77 12 250 7.950 oo 7-95 i 8-253 43 556 654 734 802 86 1 78 11 7.948 00 7-95 8.252 428 554 652 732 800 859 911 79 1O 00 7.948 8.250 8.427 8-552 8.650 8.730 8.798 8.857 8.909 8-955 80 9 7-947 8.249 425 55 1 649 728 796 855 907 953 996 81 8 8.247 424 549 647 727 795 853 905 952 994 9.032 82 i 7 6 423 547 548 645 646 724 725 792 793 850 852 902 903 948 950 990 992 9.029 9.030 064 066 097 83 84 ~ 8.643 8.723 8.791 8.849 8.901 8.947 8.989 9.027 9.063 9.096 9.126 85 4 722 - 789 848 899 946 988 9.026 061 094 I2 5 '53 86 3 789 847 898 944 986 9.025 060 093 123 179 87 2 846 897 943 985 9.023 059 092 122 150 177 202 88 1 897 943 984 9.022 058 090 121 149 176 2OI 225 89 8.942 8.984 9.022 9-057 9.089 9.119 9.148 9-175 9.200 9.224 9.246 90 1 983 9.021 056 088 119 147 173 199 222 245 267 91 i 2 9.020 055 088 118 146 173 197 221 244 ! 265 286 92 - 3 4 oil 087 117 117 171 '72 196 197 220 22O 242 243 263 264 284 285 303 305 322 93 j 94 5 9.116 9.144 9.171 9-195 9.219 9.241 9.263 9.283 9-303 9.321 9-339 95 6 144 170 195 218 241 262 282 302 3 20 338 355 96 - * 170 195 218 240 261 282 3 OI 320 337 354 97 8 194 218 240 261 281 3 OI 319 337 354 370 386 98 - 9 218 240 261 281 300 319 336 353 370 385 401 99 10 9.240 9.261 9.281 9.300 9.318 9.336 9-353 9-369 9-385 9.400 9-4I5 100 11 261 281 3 3*8 335 352 369 384 400 414 428 101 -12 281 300 3*8 335 352 368 384 399 414 428 441 1O2 13 300 3i8 335 352 368 384 399 413 428 441 454 103 14 3*8 335 352 368 384 399 413 427 441 454 466 104 | -15 9-336 9.352 9.368 9-384 9-399 9-4I3 9.427 9.440 9-454 9.466 9.478 1O5 16 353 369 384 399 413 427 440 453 466 478 490 106 17 369 384 399 413 427 440 453 466 478 490 502 107 18 385 399 4*4 427 441 453 466 478 490 502 108 19 400 414 428 441 454 466 478 490 502 5*3 524 1O9 20 9415 9.428 9.442 9-454 9.466 9-479 9.490 9-502 9.5I3 9-524 9-534 110 21 429 442 455 467 479 491 502 513 524 534 544 111 22 23 24 443 $ 481 468 480 492 479 492 504 491 503 502 525 524 535 524 535 545 535 545 555 544 554 564 564 574 112 j 113 114 80 79 78 77 D 76 75 74 73 72 71 70 i Polar Co-Latitude. Dist. TABLE XXX. 109 Time-Azimuths: Log 15. Latitude. TX^ _ Polar Dec. 10 11 12 13 14 15 16 17 18 19 20 Dist. o 35 0.407 0.398 0.390 0-383 0-375 0.367 0.360 o-353 0.346 0-339 0.332 55 34 4*7 408 400 392 384 376 369 361 354 347 340 56 33 427 418 410 401 393 385 378 370 362 355 348 57 32 437 429 420 411 403 394 387 363 356 58 31 448 439 43 421 412 44 396 388 379 372 3 6 4 59 3O 0-459 0.450 0.440 0.431 0.422 0.413 0.405 0-397 0.388 0.380 0.372 60 29 470 461 45 * 442 432 423 414 406 397 389 381 61 28 482 472 462 452 442 433 424 415 406 398 390 62 27 494 483 473 463 453 443 434 425 416 407 399 63 26 506 495 484 474 463 454 444 435 425 416 408 64 25 0.518 0.507 0.496 0.485 0-474 0.464 0-454 0-445 0435 0.426 0.417 65 24 53 1 5 J 9 508 496 486 475 465 455 445 43 6 426 66 23 544 520 508 497 486 476 4 6 5 455 446 436 67 22 557 545 532 520 59 498 487 476 466 45 6 446 68 21 558 545 533 521 510 498 487 476 466 45 6 69 20 0.585 0.572 0-559 0.546 0.533 0.522 0.510 0.498 0.487 0.476 0.466 70 19 600 586 572 559 546 534 522 510 498 487 476 71 18 615 60 1 586 573 559 547 534 522 5 IQ 498 487 72 17 616 60 1 573 560 547 534 522 51 498 73 i 16 647 631 616 601 587 573 560 547 534 522 74 15 0.664 0.648 0.632 0.616 0.60 1 0.587 0-573 0.560 0.546 0-534 0.522 75 14 682 665 648 632 616 60 1 587 573 559 546 534 76 13 700 682 665 648 632 616 601 573 559 546 77 12 719 700 682 665 648 632 616 60 1 587 559 78 11 739 719 700 682 665 648 631 616 601 586 572 79 1O 0.760 0-739 0.719 0.700 0.682 0.664 0.647 0.631 0.615 0.600 0.585 80 9 782 760 739 719 700 681 664 647 630 615 599 81 8 806 782 760 739 719 699 68 1 663 646 630 614 82 7 830 805 782 760 739 718 699 680 663 646 629 83 6 856 830 805 782 759 738 718 698 680 662 645 84 5 0.884 0.856 0.829 0.805 0.781 0.759 0-737 0.717 0.698 0.679 0.661 85 4 3 913 883 913 855 883 829 855 804 828 I g 737 757 716 736 697 715 678 695 86 87 2 980 945 912 882 854 827 803 779 756 735 88 ! 1 1.017 979 944 912 882 853 827 802 778 755 733 89 O 1.058 1.016 0.978 0-943 0.911 0.881 0.852 0.826 0.800 0.776 0-754 90 1 103 057 1.016 978 942 910 879 851 824 799 775 91 2 154 103 056 1.015 977 94i 909 878 850 823 798 92 3 211 153 102 056 1.014 975 940 907 877 848 821 93 4 2 7 8 211 152 101 054 I.OI2 974 939 906 875 847 94 5 1-357 1.277 1.209 1.151 1.099 1-053 I.OII 0-973 0-937 0.904 0.874 95 - 6 453 355 276 208 150 9 8 052 I.OIO 971 936 903 96 - 7 577 452 354 275 207 148 097 050 1.008 970 934 97 8 753 576 353 273 205 095 048 i. 006 968 98 9 2-053 575 449 35 i 272 204 145 093 047 1.004 99 10 + oc 2.052 i-75 1-573 1.448 1-350 1.270 1.202 I-I43 1.091 1.045 1OO 11 2.052 + 00 2.050 748 572 446 348 268 200 141 089 101 12 i-75 2.050 -f- 00 2.049 747 570 444 34 6 266 198 139 102 13 573 1.748 2.049 + oo 2.047 745 568 442 344 264 196 1O3 14 448 572 1-747 2.047 + 00 2.045 743 566 440 342 262 104 15 16 270 1.446 348 1-570 444 1-745 568 2-045 1-743 + 00 2.043 2.043 -4- 00 1.741 2.041 1.564 739 1.438 5 61 1.340 435 1O5 106 17 202 268 346 442 566 1.741 2.041 -f oo 2.039 736 559 107 18 H3 200 266 344 440 564 1-739 2.039 -f- oo 2.036 734 108 19 091 141 198 264 342 438 561 i-73 6 2.036 -f 00 2.033 1O9 -20 1.045 1.089 I-I39 1.196 1.262 1.340 1-435 1-559 1-734 2.033 4- oo 110 21 002 043 087 137 194 259 337 433 556 1-731 2.031 111 22 0.964 ooo 040 085 J 35 191 257 335 43 554 1.728 112 -23 928 0.961 0.998 038 082 132 189 254 33 2 427 55 1 113 24 895 925 959 0-995 035 080 129 186 251 329 424 114 Dec 80 79 78 77 76 75 74 73 72 71 7O Polar Diat Co-Latitude. XJlOl* 110 TABLE XXX. Time-Azimuths: JLogr A. Latitude. Dec. Polar Dist. 20 21 22 23 24 25 26 27 28 29 30 35 9.168 9.140 9.110 9.077 9.042 9.003 8-959 8.911 8.855 8.790 8.714 55 34 136 106 073 038 8.998 8-955 906 850 786 709 614 56 33 1 02 069 034 8-994 95 i 902 846 781 704 610 487 57 32 065 030 8.990 947 898 842 777 7oo 60; 483 39 58 31 026 8.987 943 894 838 773 696 60 1 478 34 006 59 30 8.983 8-939 8.890 8.834 8.769 8.692 8-597 8-474 8.300 8.001 00 6O 29 936 886 830 765 688 593 470 296 7-997 OO 8.001 61 28 883 827 762 684 589 466 292 7-993 oo 7-997 300 62 27 823 758 681 586 462 288 7-989 00 7-993 8.296 474 63 : 26 755 677 582 459 285 7-985 oo 7.989 8.292 470 597 61 25 8.674 8-579 8.456 8.281 7.982 00 7-985 8.288 8.466 8-593 8.692 65 24 576 452 278 7-978 00 7.982 8.285 462 589 688 769 66 23 449 275 7-975 oo 7.978 8.281 459 586 684 765 834 67 22 272 7-972 00 7-975 8.278 456 582 681 762 830 890 68 21 7.969 oo 7-972 8.275 452 579 677 758 827 886 939 69 20 00 7.969 8.272 8-449 8.576 8.674 8-755 8.823 8.883 8.936 8.983 70 19 7.966 8.269 446 573 671 752 820 880 932 980 9.023 7.1 18 8.266 444 57 668 749 817 876 929 976 9.019 059 72 17 441 567 665 746 814 873 926 973 9.016 55 092 73 16 565 663 743 811 871 923 9/0 9.013 052 088 122 74 i 15 8.660 8.741 9.809 8.868 8.920 8.967 9.010 9.049 9-085 9.119 9.150 75 ! 14 738 806 865 918 965 9.007 046 082 116 147 I 7 6 76 13 804 863 915 962 9.005 44 080 113 144 173 201 77 ! 12 861 913 960 9.002 041 077 no 141 171 198 224 78 I 11 911 957 9.000 039 075 108 139 1 68 195 221 246 79 j 10 8.955 8.998 9.036 9.072 9.105 9.136 9.165 9-193 9.219 9-243 9.267 80 i 9 996 9.034 070 103 134 163 190 216 241 264 286 81 8 9.032 068 101 132 161 188 214 238 262 284 35 82 7 066 099 130 186 212 236 259 281 3 02 323 83 6 097 128 157 184 209 234 257 279 300 320 340 84 5 9.126 9-155 9.182 9.207 9.232 9-255 9.277 9.298 9.318 9-337 9-356 85 4 153 1 80 206 230 253 275 296 316 335 354 372 86 3 179 204 228 251 273 294 3i4 333 35 1 369 387 87 2 202 226 249 271 292 3 I2 33i 349 367 384 401 88 1 22 5 248 269 290 310 329 348 365 382 399 415 89 9-246 9.268 9.289 9-309 9-328 9.346 9-363 9.380 9-397 9-4I3 9428 90 1 267 287 307 326 344 362 379 395 411 426 441 91 2 286 306 325 343 360 377 393 409 424 439 453 92 3 4 305 322 323 340 357 359 374 376 392 406 407 421 423 436 437 45 464 466 477 93 94 - 5 9-339 9-373 9-389 9-405 9.420 9-434 9.448 9.462 9.476 9.489 95 6 355 372 388 403 418 433 447 461 474 487 500 96 7 37i 387 402 417 432 446 459 473 485 498 5 IQ 97 9 386 401 416 431 445 458 47i 484 497 509 521 98 9 401 415 430 444 457 470 483 495 508 519 531 99 10 9-4I5 9.429 9-443 9-456 9.469 9.482 9.494 9.506 9.518 9-530 9-541 100 11 428 442 455 468 481 493 505 5*7 528 540 550 1O1 12 441 455 467 480 492 54 527 538 549 560 102 13 454 467 479 492 53 526 537 548 559 569 1O3 14 466 479 491 503 5H 526 537 547 558 568 578 104 15 9.478 9.491 9.502 9.SH 9.525 9.536 9.546 9-557 9.567 9-577 9-587 105 16 490 502 513 524 535 546 556 566 576 586 595 1O6 17 502 513 524 535 545 555 565 575 585 594 604 1O7 -18 513 524 534 545 555 565 575 584 594 603 612 1O8 19 524 534 544 555 564 574 584 593 602 611 620 10?) i 20 9-534 9-544 9-554 9-564 9-574 9.583 9-593 9.602 9.610 9.619 9.628 110 21 544 554 564 574 583 592 60 1 610 619 627 635 111 22 554 564 574 583 592 60 1 610, 618 627 635 643 112 -23 564 574 583 592 601 609 618 626 634 642 650 113 24 574 583 592 601 609 618 626 634 642 650 658 114 Dec. 70 69 68 67 66 65 64 63 62 61 60 Polar Co-Latitude. Dist. TABLE XXX. Ill Time-Azimuths : Log B. Latitude. Polar Dec 20 21 i 22 23 24 25 26 27 28 29 30 Dist. o 35 0-332 | j 0.325 0.319 0.312 '0.306 0.299 0.293 0.287 0.281 0-275 0.269 55 34 34 333 ' 3 26 319 3*3 306 300 294 288 282 276 56 33 348 34i 333 326 320 313 37 300 294 288 282 57 32 356 349 i 34i 334 327 320 3H 307 301 294 288 58 31 364 357 349 342 335 328 321 3H 308 301 295 59 30 0.372 0.365 0.357 1 0.350 0.342 0-335 0.328 0.321 0.314 0.308 0.301 60 29 381 373 365 358 350 343 335 328 321 314 308 61 39 373 366 358 350 343 336 328 321 3H 62 27 399 390 382 374 366 358 350 343 336 328 321 63 26 408 399 390 382 374 366 358 350 343 335 328 64 25 0.417 0.408 ! 0.399 0.391 0.382 0-374 0.366 0.358 0.350 0-342 0-335 65 24 426 417 408 399 391 382 374 366 358 350 342 66 1 23 436 426 ! 417 408 399 39 1 382 374 366 358 350 67 22 446 436 426 417 408 399 390 382 374 365 357 68 21 45 6 446 43 6 426 417 408 399 390 382 373 365 69 1 20 0.466 0.456 0.446 0.436 0.426 0.417 0.407 0-399 0.390 0.381 0.372 70 19 476 466 j 456 i 445 435 426 416 407 398 389 380 71 18 487 476 \ 466 i 455 445 435 425 416 406 397 388 72 498 487 476 465 455 445 435 425 415 406 396 73 16 498 i 487 476 4 6 5 454 444 434 424 414 405 74 15 0.522 0.509 i 0.498 0.486 0-475 0.464 0-454 0-443 0-433 0.423 0.413 75 : 14 534 521 : 509 j 497 486 474 464 453 442 432 422 76 13 546 533 i 520 5 8 497 485 474 463 452 441 77 12 559 545 532 520 508 496 484 473 462 451 440 78 11 572 558 544 532 519 57 495 483 472 461 450 79 10 0.585 o.57i 0-557 0-544 0.531 0.518 0.506 0.494 0.482 0.471 0459 8O 9 599 585 556 543 5^0 517 505 492 481 469 81 8 614 599 584 569 542 529 516 53 491 479 82 7 629 613 598 583 568 554 541 527 515 502 490 83 6 6 45 628 612 597 581 567 553 539 526 513 500 84 5 0.661 0.644 0.627 o.6n 0-595 0.580 0.566 0-552 0.538 0.524 0.511 85 4 678 660 643 626 609 594 579 564 55 536 523 86 3 695 677 659 641 624 608 593 577 548 535 - 87 2 694 675 657 640 623 607 576 561 547 88 ! 1 733 712 693 674 656 638 621 605 589 574 559 89 0-754 0.732 0.711 0.691 0.672 0.654 0.637 0.620 0.603 0.587 0.572 90 1 1 775 752 73i 710 690 671 653 635 618 60 1 585 91 - 2 798 774 75 i 729 708 688 669 651 633 616 599 92 3 821 796 772 749 727 706 686 667 649 631 614 93 4 847 820 794 770 747 725 704 684 665 646 629 94 5 0.874 0.845 0.818 0-793 0.768 0-745 0.723 0.702 0.682 0.663 0.644 95 6 903 872 843 816 791 766 743 721 700 680 660 96 7 934 901 870 841 814 789 764 719 698 677 97 8 968 932 899 868 839 812 786 762 738 716 695 98 9 1.004 966 93 897 866 837 810 784 759 736 7H 99 1O 1.045 1.002 0.964 0.928 0.895 0.864 0-835 0.807 0.781 0-757 0-733 100 11 089 043 1. 000 9 6i 925 892 861 832 805 779 101 12 139 08 7 040 998 959 923 890 859 829 802 776 102 -13 196 137 085 1.038 995 957 920 887 856 827 799 103 14 262 194 135 082 1-035 993 954 918 884 853 824 1O4 -15 1.340 1.259 1.191 1.132 1.080 I -33 0.990 0.951 0.915 0.881 0.850 105 -16 435 337 257 189 129 077 1.030 987 948 912 878 106 17 559 433 335 254 1 86 126 074 1.027 984 945 908 107 1 734 55 i 430 332 25 1 183 123 071 1.024 981 942 108 19 2.033 554 427 329 248 180 120 068 1.020 977 109 -20 + 00 2.031 1.728 I -55 I 1.424 1.326 1.245 I.I77 1.117 1.064 1.017 110 21 2.031 + 00 2.028 725 548 421 323 242 173 114 06 1 111 22 1.728 2.028 + 00 2.025 722 544 418 319 238 170 no 112 23 551 1-725 2.025 + 00 2.O22 719 54i 414 316 235 166 113 24 424 548 1.722 2.022 + 00 2.018 538 411 3 I2 231 114 Dec. 70 69 68 67 66 65 64 63 62 61 60 Polar Co-Latitude. Dist. 112 TABLE XXX. Time-Azimuths : Log A. Latitude. Dec. 35 3 * Polar Dist. 30 31 3i> 33 34 37 38 39 a 40 35 8.714 8.619 8.497 8.3-3 8 - 02 5 CO 8.030 8-334 S.SM 8.641 : 8.740 o &S 34 614 492 3'8 8.020 co 8.025 329 57 635 7^ Si 7 56 33 487 3H 8.015 00 8.O2O 3 2 3 502 .629 729 Si i i S8i 57 32 39 8.010 00 8.015 318 497 624 724 805 8?5 936 5N 31 8.006 00 8.010 3!3 492 619 719 800 870 93 984 59 30 oo 8.006 8.308 8.487 8.614 8.714 8.795 8.865 8.925 8-979 9.027 60 29 8.001 34 483 610 709 790 860 920 974 9.022 066 61 * 300 478 605 704 786 855 9i5 969 9.017 06 1 IOI 2 27 474 601 700 781 850 911 9 5 4 9.012 056 096 133 63 26 597 696 777 846 906 959 9.007 051 091 128 162 64 25 8.692 8-773 8.842 8.902 8-955 9.003 9.046 9.086 9.123 9.158 9.190 65 24 769 838 898 95i 999 042 082 119 153 185 2I 5 66 23 834 894 947 994 9-038 077 114 148 180 211 239 67 22 890 943 990 9.034 073 IIO 144 176 206 234 261 6$ 21 939 987 9.030 069 IOJ 140 172 202 230 257 282 69 20 8.983 9.026 9.066 9.102 9.136 9.168 9.198 9.226 9-253 9.278 9.302 7O 19 9.023 062 098 132 164 194 222 248 274 2 9 8 321 71 IS 059 095 129 1 60 190 218 244 270 294 317 339 72 17 092 I2 5 157 186 214 241 265 290 3 J 3 335 356 73 16 122 153 183 211 237 262 286 309 33i 352 372 74 15 9.150 9.180 9.207 9-234 9-259 9.282 9-35 9.327 9-348 9.368 9-387 75 : 14 I 7 6 204 230 255 2 79 302 323 344 364 384 402 76 13 201 227 252 2 7 6 298 320 34i 361 380 399 416 77 12 224 249 273 295 317 337 357 376 395 413 43 78 11 246 270 292 3U 334 354 373 39i 409 427 443 79 10 9.267 9.289 9.310 9.331 9-35 J 9-370 9.388 9.406 j 9.423 9.440 9-456 80 9 286 3 oS 328 348 367 385 403 420 436 452 468 81 35 325 345 364 382 400 417 433 449 465 480 82 7 323 342 361 379 397 414 430 446 461 476 49 i 83 6 34 358 376 394 411 427 443 458 473 4 83 502 84 5 9-356 9-374 9-39 i 9.408 9-424 9.440 9-455 9.470 9485 9-499 9-5I3 85 4 372 389 406 422 437 453 467 482 496 5io 523 86 3 387 403 419 435 45 465 479 493 507 520 533 87 2 401 417 432 448 462 476 490 504 517 530 542 88 1 415 430 445 460 474 488 5oi 5H 52/ 540 552 89 9.428 9-443 9-457 9-472 9-485 9-499 9-512 9-524 9-537 9-549 9.561 90 1 441 455 469 483 496 509 522 534 547 558 570 91 2 453 467 481 494 507 520 532 544 556 567 579 92 3 465 479 492 55 518 530 542 554 565 576 587 93 4 477 490 53 5i6 528 54 551 563 574 585 595 94 5 9.488 9.501 9-5I4 9.526 9.538 9-549 9.560 9-572 9.582 9-593 9.603 95 g 499 512 524 536 547 558 569 580 59i 60 1 611 96 7 5 10 522 534 545 557 567 578 589 599 609 619 97 8 ft .521 531 532 542 544 553 5 6 4 566 575 576 585 587 595 605 607 615 6i7 625 627 634 98 99 ! 10 9-54* 9-552 9.562 9-573 9-583 9-593 9.603 9.613 9.623 9.632 9.641 100 i 11 55 1 56i 57i 582 592 602 611 621 630 639 648 1O1 12 560 570 580 590 600 610 619 628 637 646 655 1O2 -13 5 6 9 579 589 599 608 617 627 636 644 653 662 10* 14 578 588 597 607 616 625 6 3 4 643 651 660 668 104 -15 9-587 9-596 9.606 9.615 9-624 9-633 9.641 9.650 9.658 9.667 9-675 105 16 595 605 614 ' 623 631 640 648 657 665 673 68 1 106 17 604 613 622 630 639 647 655 664 671 6So 688 107 18 612 62! 629 638 646 654 662 670 678 686 694 1O8 19 620 628 637 645 653 66 1 669 677 685 692 700 109 20 9.628 9.636 9.644 9.652 9.660 9668 9.676 9.684 9.691 9.698 9.706 110 21 635 644 652 6 59 667 675 683 690 697 704 712 111 22 643 651 659 666 674 682 689 696 703 710 717 112 -23 650 658 666 673 68 1 688 695 702 709 716 723 113 24 ! 658 665 673 680 687 694 701 708 715 722 728 114 60 59 5 57 56 55 54 53 52 51 50 Polar Dsc. ! Co-Latitude. Dist. TABLE XXX. 113 Time- Azimuths: Log B. Latitude. Polar Dec. 30 31 32 33 34 35 36 37 38 39 40 Dist. 1 35 0.269 0.264 0.258 | 0.252 i 0.247 0.241 0.236 i 0.231 0.226 O.22O 0.215 55 : 34 276 270 264 258 ! 252 247 241 236 231 225 220 56 33 282 276 270 264 258 252 247 241 236 I 2^0 225 57 32 288 282 276 270 264 258 252 247 241 2^5 230 58 i 31 295 288 282 276 270 264 258 252 246 240 235 59 30 0.301 0.295 0.288 0.282 0.276 0.269 0.263 0-257 0.251 0.246 0.240 60 29 308 301 294 288 282 275 269 263 257 25 1 245 61 28 3H 308 301 294 288 281 275 269 262 256 250 62 27 321 314 307 294 287 281 274 268 261 2 55 63 26 328 321 3H 307 300 293 287 280 273 267 261 64 25 0.335 0.328 0.320 0-313 0.306 0.299 0.293 0.286 O.279 0.273 0.266 65 24 342 335 327 320 313 306 299 292 285 278 272 66 23 350 342 334 327 319 312 305 298 291 284 67 22 357 349 341 334 326 318 3" 34 297 290 283 68 21 365 356 348 I 34i 333 325 318 310 33 296 289 69 23 0.372 0.364 -35 6 0.348 0.340 0.332 0.324 0.316 0.309 0.302 0.294 70 19 380 372 3 6 3 355 347 339 331 323 315 308 300 71 18 388 380 37i 362 354 346 338 330 322 314 306 72 17 396 388 379 37o 361 353 345 336 328 320 312 73 16 405 396 387 377 3 6 9 360 352 343 335 327 319 74 15 0.413 0.404 Q-394 0-385 0.376 0.367 0-359 0-350 0.342 0.333 -3 2 5 75 14 422 412 403 393 384 375 366 357 349 34 332 76 13 43^ 421 411 401 392 f3 373 364 356 347 338 77 12 440 43 420 410 400 381 372 3 6 3 354 345 78 ; 11 45 439 429 418 408 399 389 379 370 361 352 79 10 0-459 0.448 0.438 0.427 0.417 0.407 0-397 0.387 0.378 0.368 0-359 80 9 469 458 447 436 426 415 405 395 385 376 366 81 8 479 468 456 445 435 424 413 403 393 383 374 82 7 490 478 466 455 444 433 422 411 401 391 83 6 500 488 476 464 453 442 420 409 399 389 84 ! 5 0.511 0-499 0.486 -474 0.462 0-451 0.440 0.428 0.418 0.407 0-397 85 ! 4 523 497 484 472 460 449 437 426 405 86 3 535 52i 508 495 482 470 458 447 435 424 413 87 2 547 533 519 566 493 480 468 45 6 444 433 421 88 1 559 545 531 5 r 7 504 491 478 466 453 442 43 89 : O 0.572 0-557 0-543 0.528 0.515 0.501 0.488 0.476 0.463 0.451 0-439 90 I 585 570 555 540 526 5 12 499 486 473 460 448 91 2 599 583 568 552 538 524 496 483 470 458 92 3 614 597 581 565 55 535 521 507 493 480 467 93 4 629 611 594 578 563 547 533 518 54 491 477 94 5 0.644 0.626 0.609 0.592 0.576 0.560 0-545 0.530 0.515 0.501 0.488 95 6 660 642 624 606 589 573- 557 542 527 512 498 96 7 677 658 639 621 603 587 570 554 539 524 59 97 8 695 675 655 636 618 60 1 584 567 536 520 98 9 7H 692 672 652 633 615 598 580 564 548 53 2 99 1O 0-733 0.711 0.690 0.669 0.649 0.630 0.612 0-594 0-577 0.560 o-544 100 11 754 73 708 687 666 646 627 609 59i '573 557 101 12 776 727 705 683 663 6 43 624 605 570 102 13 799 773 748 724 702 680 6 59 639 620 602 584 103 14 824 796 770 745 721 698 677 ! 656 636 617 598 104 15 0.850 0.820 0-793 0.766 0.741 0.718 0.695 ! 0.673 0.652 0.632 0.613 105 16 878 847 817 789 753 738 691 669 648 628 106 17 908 875 -843 814 786 759 734 710 687 665 644 1O7 -18 942 905 871 840 810 782 756 73 706 683 66 1 108 19 977 938 902 868 836 806 778 752 726 702 679 109 2O 1.017 0-974 o-935 0.898 0.864 0.832 0.802 0.774 0.748 0.722 0.698 110 21 22 06 1 x -uo 1.013 057 970 I.OIO 966 894 927 860 890 828 856 7^8 824 770 794 III 718 739 111 112 23 166 106 53 i. 006 962 9 2 3 886 852 820 789 yf 76l 113 24 231 162 102 049 I.OO2 958 918 881 847 815 785 114 60 59 58 57 56 55 54 53 52 51 5O Polar Co-Latitude. Dist. 114 TABLE XXX. 1 Time- Azimuths : Log A. Latitude. Dec. 46 | Polar Dist. 4O 41 42 43 44 45 47 48 i 49 50 o 35 8.740 8.822 8.892 8-953 9.007 9.056 9.101 9.141 9.179 9-215 9.248 o 55 34 817 886 947 9.001 050 094 J35 173 208 241 273 56 33 88 1 941 995 044 088 129 167 202 235 266 295 57 32 936 989 9.038 082 123 1 60 195 228 289 58 31 984 9-032 076 117 154 189 222 253 282 310 337 59 30 9.027 9.071 9.111 9.149 9.184 9.216 9.247 9.276 9-304 9-330 9-355 60 29 066 106 '43 178 211 241 270 298 324 349 373 61 28 IOI 138 173 205 2 3 6 264 292 318 343 367 390 62 27 133 167 200 230 259 286 3 I2 337 384 406 63 26 162 195 225 254 28l 307 332 355 378 400 421 64 25 9.190 9.220 9.249 9.276 9.302 9-326 9-35 9-373 9-395 9.416 9436 65 24 215 244 271 297 3 2I 345 367 389 410 43 45 66 23 239 266 2 9 2 316 340 362 384 405 425 444 463 67 22 261 287 335 357 379 399 4*9 439 457 476 68 21 282 307 330 352 374 394 414 434 452 470 488 69 20 19 9.302 321 9-325 343 9.348 364 9 385 9-390 405 9.409 424 9-429 442 9-447 460 9-465 478 9483 495 9.50 5 11 70 71 18 339 360 380 400 419 437 455 473 490 506 522 72 17 356 376 396 415 433 45 r 485 5 O1 5 J 7 S3 2 73 16 372 39i 410 429 446 463 480 496 512 527 542 74 15 9-387 9.406 9424 9.442 9-459 9.476 9.492 9-507 9-523 9.538 9-552 75 14 402 420 438 455 487 503 518 533 547 562 76 13 416 434 451 467 483 498 528 543 557 77 12 43 447 463 479 494 59 524 538 553 566 580 78 11 443 459 475 490 505 520 534 548 562 575 588 79 10 9-456 9471 9487 9.502 9.516 9-53 9-544 9-558 9-571 9-584 9-597 80 9 468 483 498 512 526 540 554 567 580 592 605 81 8 480 494 509 523 536 550 563 576 588 600 613 82 7 491 505 533 546 559 572 584 596 608 620 83 6 502 529 543 555 568 580 593 604 616 628 84 5 9-5I3 9.526 9-539 9-552 9-565 9-577 9-589 9.601 9.612 9.624 9-635 85 4 523 536 549 573 585 597 609 620 631 642 86 3 533 545 558 570 582 594 605 616 627 638 649 87 2 1 542 552 555 564 567 576 579 587 590 598 602 610 620 624 631 635 642 645 652 2 88 89 9.561 9-573 9.584 9-595 9.606 9-617 9.628 9-638 9-649 9.659 9.669 90 1 570 581 593 603 614 625 635 645 655 665 675 91 2 579 60 1 611 622 632 642 652 662 672 68 1 92 - 3 587 598 609 619 629 639 649 659 668 678 687 93 4 595 606 616 626 636 646 656 665 675 684 693 94 5 9.603 9.614 9.624 9-634 9-643 9.653 9.662 9.672 9.681 9.690 9.699 95 6 611 621 631 641 650 660 669 678 687 696 74 96 7 619 629 638 648 657 666 675 684 693 701 710 97 8 627 636 645 6 55 664 673 681 690 699 707 98 9 634 643 652 661 670 679 687 696 704 713 721 99 10 9.641 9.650 9.659 9.668 9.676 9-685 9-693 9.702 9.710 9.718 9.726 1OO 11 648 657 666 674 683 691 699 707 715 723 731 101 12 655 664 672 681 689 697 705 713 721 729 736 102 13 662 670 679 687 695 703 711 718 726 734 103 14 668 677 685 693 701 708 716 724 731 739 746 1O4 -15 9-675 9-683 9.691 9.699 9.706 9-7H 9.722 9.729 9-736 9-744 9.751 105 16 68 1 689 697 705 712 720 727 734 741 749 756 1O6 17 688 695 703 710 718 725 732 739 746 753 760 107 18 694 701 709 716 723 73 737 744 751 765 108 19 700 707 7H 721 729 736 742 . 749 756 763 769 109 20 9.706 9-713 9.720 9727 9-734 9-741 9-747 9-754 9.761 9.767 9-774 110 21 712 718 725 732 739 746 752 765 772 778 111 22 717 724 73 i 738 744 75 1 - 757 764 770 776 782 112 23 723 736 743 749 756 762 768 774 780 787 113 24 728 735 748 754 760 767 773 779 785 791 114 5O 49 48 47 46 45 44 43 42 41 40 Polar Co-Latitude. Dist. TABLE XXX. 115 Time- Azimuths: Log 12. Latitude. Dec. Polar Dist. 40 41 42 43 44 45 46 47 48 49 50 35 0.215 O.2IO 0.205 0.200 0-195 0.190 o.i 86 0.181 0.176 0.171 0.167 o 55 34 220 215 210 205 199 194 190 185 1 80 '75 170 56 33 225 219 214 209 204 199 189 184 179 174 57 32 230 224 219 214 208 203 198 193 1 88 183 178 58 31 235 229 224 218 213 208 202 197 192 187 181 59 3O 0.240 0.234 0.228 0.223 0.217 0.212 0.207 0.201 0.196 0.191 0.185 60 29 245 239 233 228 222 216 211 205 200 195 189 61 28 250 244 238 232 227 221 215 210 2O4 199 193 62 27 255 249 243 237 2 3 I 225 22O 214 208 203 197 63 26 261 254 248 242 236 2 3 224 218 212 207 20 1 64 25 0.266 0.260 0-253 0.247 0.241 0.235 0.229 O.223 0.217 0.21 1 0.205 65 24 272 265 259 252 2 4 6 240 233 227 221 215 209 66 23 277 270 264 257 25 I 244 238 232 226 220 213 67 22 283 276 269 263 2 5 6 249 243 2 3 6 230 224 218 68 21 289 282 275 268 26l 254 248 2 4 I 235 228 222 69 2O 0.294 0.287 0.280 0.273 0.266 0-259 0-253 0.246 0-239 0-233 0.226 70 19 300 293 286 279 271 264 258 251 244 237 231 71 18 306. 299 291 284 277 27O 263 2 5 6 249 2 4 2 235 72 17 312 305 297 290 282 275 268 261 254 247 240 73 1 16 3*9 3" 303 296 288 280 273 266 259 251 245 74 i 15 0-325 0.317 0.309 0.301 0.294 0-286 0.278 O.27I 0-264 0.256 0.249 75 14 332 323 315 307 299 292 284 2 7 6 269 26l 254 76 13 338 330 322 3'3 305 297 289 282 274 266 259 77 12 345 336 328 3" 33 295 287 279 271 264 78 11 352 343 334 326 39 3 OI 293 28 5 277 269 79 10 -359 0-35 0-341 0-332 0.324 0.315 0.307 0.298 0.290 0.282 0.274 80 9 366 357 348 339 330 321 313 34 2 9 6 28 7 279 81 8 374 364 355 345 336 327 3*9 3 IO 301 293 285 82 7 381 371 362 352 343 334 325 316 37 299 290 83 6 389 379 369 359 350 340 322 313 34 296 84 5 0-397 0.386 0.376 0.366 o-357 0-347 0-338 0.328 0.319 0.310 0.301 85 4 405 394 384 374 3 6 4 354 344 335 325 316 307 86 3 4i3 402 392 381 37i 361 35 i 341 33 2 322 87 2 421 410 399 389 378 368 358 348 338 328 319 88 1 43 419 407 397 386 375 365 355 345 335 325 89 O 0-439 0.427 0.416 0.405 0-394 0-383 0.372 0.362 0.351 0.341 0.331 90 . | 448 436 424 413 402 390 380 369 358 348 338 91 2 458 445 433 421 410 398 387 376 3 6 5 355 344 92 3 467 455 442 43 418 406 395 384 373 362 93 4 477 464 451 439 427 415 403 380 369 358 94 5 0.488 0-474 0.461 0.448 0.436 0.423 0.411 0-399 0.388 0.376 0-365 95 6 498 484 47i 457 445 432 420 396 384 372 96 7 509 495 481 467 454 44i 428 416 404 392 380 97 8 520 506 491 477 464 45 437 424 412 400 387 98 9 532 517 502 488 474 460 446 433 420 408 395 99 10 0-544 0.529 0-513 0-499 0.484 0.470 0.456 0.442 0.429 0.416 0.403 100 11 557 54i 525 5^0 495 480 466 452 438 425 412 101 12 570 553 537 521 506 491 476 462 447 434 421 102 13 584 566 549 533 517 502 486 472 457 443 429 103 14 598 580 562 545 629 513 497 482 467 453 438 104 15 0.613 0-594 0.576 0.558 0.541 0-525 0.508 0-493 0.477 0.463 0.448 1O5 16 628 609 590 554 537 520 54 488 473 458 106 18 19 644 66 1 679 624 640 657 604 620 636 615 567 581 596 549 563 577 532 545 558 515 527 540 499 5*0 522 483 494 505 468 478 489 107 108 109 -20 21 0.698 718 0.675 693 0.653 670 0.631 648 0.611 626 'oo6 0.572 586 0-553 567 0-535 548 0.517 530 0.500 5*2 110 111 22 739 689 665 643 .621 01 581 561 543 524 112 23 24 761 785 734 756 729 684 703 660 679 638 655 616 633 595 611 575 590 556 570 537 113 1 114 ! 5O 49 48 47 46 45 44 43 42 41 40 Dec Polar Co-Latitude. Dist. 116 TABLE XXX. Time- Azimuths: Log A. Latitude. Dec. 8 > 6O Polar Dist. 50 51 52 53 54 55 56- 57" 59 o 9.248 9.280 9-309 9-337 9-364 9-39 9-415 9-439 9.462 9.484 9.506 55 34 273 302 33 357 383 407 431 454 476 498 518 56 33 295 323 35 376 400 424 447 468 490 5 10 530 57 32 343 369 393 417 439 461 482 503 523 542 58 ] 31 337 362 386 410 432 454 475 495 534 553 59 30 9-355 9.380 9403 9.425 9-447 9.468 9.488 9-508 9-527 9-545 9-564 60 29 373 396 419 440 461 481 5i 520 538 556 574 61 28 390 412 434 454 474 494 513 531 549 566 583 62 27 406 427 448 468 487 506 524 542 559 576 593 63 26 421 442 462 481 499 518 535 552 569 586 602 64 25 9436 9.456 9-475 9-493 9-5U 9-529 9-546 9.562 9-579 9-595 9.611 65 24 45 469 487 505 523 540 556 572 588 604 619 66 i 23 463 481 499 533 55 566 582 597 612 627 67 22 476 493 5 11 527 544 560 575 591 606 620 635 68 ! 21 488 505 522 538 554 569 585 600 614 628 643 69 20 9.500 9.516 9-532 9-548 9-564 9-579 9-594 9.608 9.622 9.636 9.650 7O 19 5" 527 543 558 573 588 602 616 630 644 657 71 18 522 537 553 567 582 596 610 624 638 651 664 72 17 532 547 562 577 605 618 632 645 658 6/1 73 16 542 557 57i 586 599 613 626 639 652 665 677 74 15 9-552 9.566 9.580 9-594 9.608 9.621 9-634 9.646 0.659 9.671 9-683 75 14 562 575 589 603 616 628 641 653 666 678 689 76 13 12 584 593 606 618 623 63 1 636 643 648 655 660 667 672 679 684 690 695 701 77 78 11 588 601 614 626 638 650 662 673 696 707 79 10 9-597 9.609 9.621 9-633 9.645 9-657 9.668 9.680 9.691 9.702 9-7I3 80 9 605 617 629 641 652 664 675 686 697 707 718 81 8 613 624 636 648 659 670 68 1 692 702 713 723 82 : 7 620 632 643 654 665 676 687 698 708 718 728 83 i 6 628 639 650 661 672 682 693 703 713 724 733 84 | 5 9.635 9.646 9.657 9.668 9.678 9.688 9.699 9-709 9.719 9.729 9-738 85 4 642 653 663 674 684 694 74 7H 724 734 743 86 3 649 659 670 680 690 700 710 720 729 739 748 87 2 656 666 676 686 696 706 7*5 725 734 743 753 88 1 662 672 682 692 702 711 721 730 739 757 89 ! 9.669 9.679 9.688 9.698 9.707 9.717 9.726 9-735 9-744 9-753 9.762 90 1 675 685 694 703 713 722 731 74 748 757 766 91 ! 2 681 691 700 79 718 727 736 744 753 762 770 92 3 687 696 705 723 732 749 758 766 774 93 4 693 702 711 720 728 737 745 754 762 770 778 94 5 9.699 9.708 9.716 9-725 9-733 9.742 9-750 9-758 9.766 9-774 9.782 95 6 704 722 73 738 747 755 763 771 778 786 96 7 710 7i8 727 735 743 75i 759 767 775 ^ 790 97 g 715 724 732 740 748 756 763 771 779 786 794 98 ! 9 721 729 737 745 753 760 768 775 783 790 798 99 -10 9.726 9-734 9.742 9.750 9-757 9-765 9.772 9.780 9.787 9-794 9.801 100 1 11 73 i 739 747 754 762 769 776 784 791 798 805 101 12 736 744 75 i 759 766 773 780 787 795 802 809 1O2 13 74 i 749 756 763 77 777 784 791 798 805 812 1O3 ' -14 746 753 760 768 775 782 788 795 802 809 816 104 ; 15 9-751 9-765 9.772 9-779 9.786 9-792 9-799 9.806 9.812 9.819 1O5 16 756 763 769 776 783 790 796 803 809 816 822 106 17 760 767 774 780 787 794 800 807 813 819 825 107 -18 765 771 778 785 79 i 797 804 810 816 823 829 1O8 : 19 769 776 782 789 795 801 807 814 820 826 832 1O9 2O 9-774 9.780 9.786 9-793 9-799 9.805 9.811 9.817 9-823 9.829 9.835 1 10 21 778 784 791 797 803 809 815 821 827 832 838 Mi i 22 782 789 795 801 807 812 818 824 830 836 841 112 ! -23 787 793 804 810 816 822 828 833 839 844 113 -24 791 797 802 808 814 820 825 831 836 842 847 114 4O 39 38 37 36 35 34 33 32 31 3O ! Polar | Co-Latitude. Dist. TABLE XXX. 117 Time-Azimuths: L>og !J. Latitude. Polar Dsc. 50 51 52 53 54 55 56 57 58 59 6O Dist. 35 0.167 0.162 0.157 0.153 i 0.148 0.144 0.139 i 0.135 0.131 0.126 O.I22 o 55 170 1 66 161 156 ! 152 H7 142 ! 138 133 129 I2 5 56 33 174 169 164 1 60 155 150 145 141 136 132 127 57 32 178 173 1 68 163 158 153 149 H4 139 135 I 3 58 31 181 176 171 1 66 161 157 152 147 142 137 133 59 30 0,185 0.180 0.175 0.170 0.165 o.i 60 0.155 0.150 j 0.145 0.140 0.135 60 29 28 199 193. 184 188 179 182 173 177 1 68 172 I 3 167 158 161 '53 156 148 H3 146 138 141 61 62 27 197 191 186 181 170 165 159 154 149 1 44 63 26 201 195 190 184 179 173 1 68 163 157 152 *47 64 25 0.205 0.199 0.194 o.i 88 0.182 0.177 0.171 0.166 o.i 60 0.155 0.150 65 24 209 203 198 192 1 86 1 80 175 169 164 158 153 66 23 213 207 201 196 1 190 184 178 173 167 161 156 67 22 218 212 2O5 200 194 187 182 176 170 164 68 21 222 216 210 203 197 191 185 179 173 1 68 162 69 20 0.226 0.220 O.2I4 0.207 0.201 0.195 0.189 0.183 0.177 0.171 0.165 70 19 231 224 218 211 205 199 193 186 1 80 174 1 68 71 18 235 229 222 215 209 203 196 190 184 177 171 72 17 240 233 226 220 213 207 200 194 187 181 '75 73 16 245 2 3 I 224 217 210 2O4 197 191 184 178 74 15 0.249 0.242 0- 2 35 0.228 0.221 O.2I4 0.208 0.201 0.194 0.188 0.181 75 14 254 247 240 2 3 2 225 218 212 2O5 198 191 185 76 13 259 252 244 237 2 3 223 216 2O9 j 2O2 195 188 77 12 264 2 5 6 249 2 4 I 234 227 220 213 205 198 192 78 11 269 26l 253 246 239 231 22 4 216 209 202 195 79 10 0.274 0.266 0.258 0.251 0.243 0.235 0.228 0.220 0.213 0.206 0.199 80 9 279 271 263 255 2 4 8 240 232 225 217 210 202 81 8 285 2 7 6 268 260 252 244 237 229 221 214 2O6 82 7 29O 282 2 73 265 257 249 241 233 225 218 210 83 6 296 287 2 7 8 270 262 253 245 237 230 222 214 84 5 0.301 0.292 0.284 0.275 0.267 0.258 0.250 0.242 0.234 0.226 0.218 85 4 3 307 313 298 304 289 295 280 286 272 277 268 2 55 260 2 4 6 251 2 3 8 242 230 234 222 226 86 87 2 1 319 325 39 315 3 00 306 2 9 I 297 282 287 273 278 264 269 260 252 2 3 8 243 230 234 88 89 Q-33 1 0.321 0.312 0.302 0.293 0.283 0.274 0.265 0.256 0.247 0.239 90 I 1 338 328 318 308 2 9 8 289 279 270 26l 252 243 91 '' 2 344 334 324 34 294 285 275 266 2 57 247 92 3 351 33 320 3 IO 300 29O 28l 271 26l 252 93 4 358 347 337 326 316 306 296 286 2 7 6 266 257 94 5 0-365 o-354 0-343 0.332 0.322 0.312 0.301 0.291 0.28l 0.271 0.262 95 6 372 361 35o 339 3 28 318 307 297 287 2 7 6 267 96 i 7 380 368 357 346 335 324 313 302 292 282 272 97 8 387 376 364 353 330 319 3 08 298 287 277 98 9 395 383 37 1 360 348 336 325 3H 303 293 282 99 10 0.403 0.391 0-379 0.367 0-355 0-343 0.33 2 0.320 0.309 0.298 0.287 10O 11 412 399 386 374 362 350 338 327 34 293 101 12 421 407 394 382 369 357 345 333 3 2I 3 IO 299 1O2 -13 429 416 402 390 377 364 352 340 328 3 I6 305 103 14 438 425 411 398 384 372 359 347 334 322 310 104 -15 0.448 0-434 0.420 0.406 0.392 0.379 0.366 0-354 0.341 0.329 0-317 105 16 458 443 429 414 401 387 374 361 348 335 323 106 ! 17 468 453 438 423 409 395 382 368 355 342 329 107 i 18 478 463 447 433 418 404 390 376 362 349 336 108 19 489 473 457 442 427 412 398 384 370 356 343 1O9 20 0.500 0.484 0.468 0.452 0.436 0.421 0.406 0.392 0.378 0.364 0-35 110 21 512 495 478 462 446 43 * 415 400 386 372 357 111 22 524 507 489 473 456 440 425 409 394 380 3 6 5 112 23 537 519 484 467 45 434 418 403 388 373 113 24 550 53i 513 495 478 461 444 428 412 396 114 4O 39 38 37 36 35 34 33 32 31 30 Polar Co-Latitude. Dist. 118 TABLE XXX. Time-Azimuths: JLog A. Latitude. _ Polar 60 61 62 63 64 65 66 67 68 69 70 Dist. 35 9.506 9-527 9-547 9-567 9.586 9.605 9-623 9.641 9-659 9-677 9-694 55 34 518 538 558 577 596 614 632 650 667 684 701 56 33 530 569 587 606 623 641 658 675 691 707 57 32 542 56i 579 597 615 632 649 666 682 698 58 31 553 589 606 624 640 657 673 689 705 720 59 30 9-564 9.581 9-598 9.615 9.632 9.648 9-665 9.680 9.696 9.711 9.726 60 29 574 59i 607 624 640 656 672 687 702 717 73 2 61 28 583 600 616 632 648 664 679 694 709 723 737 62 27 593 609 625 640 656 671 686 700 729 743 63 26 602 618 633 648 663 678 692 706 721 734 ! 748 64 25 9.611 9.626 9.641 9.656 9.670 9-685 9-699 9.712 9.726 9.740 j 9.753 65 24 619 634 649 663 677 691 705 718 732 745 758 66 23 22 627 635 642 649 656 663 670 677 684 690 697 703 711 717. 724 73 737 742 750 755 i 67 68 21 643 656 670 683 696 709 722 735 747 760 772 69 2O 9.650 9-663 9-677 9.690 9.702 9715 9.728 9.740 9-752 9-765 9-776 7O 19 18 657 664 670 677 689 696 702 708 714 721 726 733 738 745 75 III 769 773 781 785 71 72 17 16 677 683 689 695 701 708 713 720 725 737 743 748 760 766 778 782 789 793 73 74 15 9.683 9.695 9.707 9.719 9-73 9.742 9-753 9.764 9-775 9.786 9-797 75 14 689 701 713 724 735 747 769 779 790 801 76 13 695 707 729 740 762 773 783 794 804 77 12 701 713 724 734 745 756 767 777 787 798 808 78 11 707 718 729 739 750 760 781 79 i 80 1 811 79 10 9-7I3 9-723 9-734 9-744 9-755 9765 9-775 9.785 9-795 9.805 9.815 80 9 718 728 739 749 759 769 779 789 799 809 . 818 81 8 723 733 744 754 764 774 783 793 803 812 821 82 7 728 738 748 758 768 778 787 797 806 815 825 83 6 733 743 753 763 772 782 791 800 810 819 828 84 5 9-738 9.748 9758 9-767 9.776 9.786 9-795 9.804 9-813 9.822 9-831 85 4 743 753 762 771 780 789 798 807 816 825 834 86 3 748 757 766 775 793 802 811 819 828 837 87 2 753 761 771 779 788 797 806 814 823 831 840 88 1 757 766 775 783 792 801 809 818 826 834 842 89 9.762 9.770 9-779 9.787 9.796 9.804 9.813 9.821 9.829 9.837 9-845 90 1 766 774 783 791 799 808 816 824 832 840 848 91 1 2 770 778 787 795 803 811 819 827 835 843 851 92 * 774 782 790 799 807 815 822 830 838 846 853 93 I 4 778 786 794 802 810 818 826 833 841 848 856 94 5 9.782 9.790 9.798 9.806 9.813 9.821 9.829 9.836 9.844 9.851 9-858 95 [ __ ** 786 794 802 809 817 824 832 839 846 854 861 96 7 790 798 805 813 820 827 835 842 849 856 863 97 8 794 801 809 816 823 830 838 845 852 859 866 98 9 798 805 812 819 826 833 841 848 854 861 868 99 1O 9.801 9.809 9-815 9.823 9.830 9.836 9-843 9.850 9.857 9.864 9.871 1OO 11 805 812 819 826 833 839 846 853 8f)0 866 873 1O1 12 809 815 822 829 836 842 849 856 862 869 875 102 -13 14 812 816 819 822 825 829 832 835 839 842 845 848 852 854 858 861 865 867 871 873 877 880 103 104 -15 9.819 9.825 9.832 9.838 9-845 9.851 9-857 9-863 9.869 9.876 9.882 105 16 822 828 835 841 847 854 860 866 872 878 884 106 17 825 832 838 844 850 856 862 868 874 880 886 1O7 18 829 835 841 847 853 859 865 871 877 882 888 1O8 19 832 838 844 850 856 862 867 873 879 885 890 109 20 9.835 9.841 9.847 9.853 9-858 9.864 9.870 9-875 9.881 9.887 9.892 no 21 838 844 850 855 86 1 867 872 878 883 889 894 in 22 841 847 852 858 864 869 875 880 886 891 896 112 23 844 850 855 86 1 866 872 877 882 888 893 898 113 24 847 853 858 864 869 874 879 885 890 895 900 114 30 29 28 27 26 25 24 23 22 21 20 i Dec. Polar Dist. i Co-Latitude. TABLE XXX. 119 Time- Azimuths: Log B. Latitude. Polar Dec. 6O 61 62 63 64 65 66 67 68 69 70 Dist. o 35 0.122 0.118 0.113 0.109 0.105 O.IOI 0.097 0.092 0.088 0.084 0.080 55 34 125 120 116 in 107 103 099 094 090 086 082 56 33 127 123 118 114 109 105 101 096 092 088 083 57 32 130 125 121 116 TI2 107 103 098 094 090 085 58 31 33 128 I2 3 119 114 110 105 100 096 091 087 59 30 0.135 0.131 0.126 O.I2I O.II7 O.II2 0.107 0.103 0.098 0.093 0.089 60 29 138 *33 129 124 119 114 109 105 100 095 091 61 28 141 136 131 126 121 116 112 107 102 097 092 62 27 144 139 J 34 129 124 119 114 109 IO4 099 094 63 26 H7 142 136 '3 1 126 121 116 in 106 101 096 64 25 0.150 0.144 0.139 0.134 0.129 0.124 0.118 0.113 0.108 0.103 0.098 65 24 153 H7 142 136 131 126 121 115 no 105 IOO 66 23 I 5 6 !5 H5 139 134 128 123 118 112 107 102 67 22 159 153 147 142- I 3 6 131 125 1 20 114 109 I0 4 68 21 156 15 H5 139 '33 128 122 117 in 106 69 20 0.165 0.159 - I 53 0.147 0.142 0.136 0.130 0.125 0.119 0.113 0.108 70 19 1 68 162 156 150 144 138 133 127 121 115 no 71 18 171 165 J 59 *53 H7 141 135 129 123 118 112 72 17 *75 168 162 156 ^o 144 138 132 126 120 114 73 16 178 171 165 159 153 146 140 134 128 122 116 74 15 0.181 o.i75 o.i 68 0.162 0-155 0.149 0.143 0.137 O.I3I O.I24 0.118 75 14 185 178 171 165 I 5 8 !5 2 146 139 133 127 120 76 13 iSS 181 ' 175 168 161 155 148 142 135 I2 9 123 77 12 192 185 178 171 164 158 151 144 138 131 125 78 11 195 1 88 181 174 167 161 154 H7 140 134 127 79 10 0.199 0.191 0.185 0.177 0.170 0.163 0.157 0.150 0.143 0.136 0.129 8O 9 202 195 1 88 181 174 1 66 159 153 I 4 6 I 3 9 I 3 2 81 8 206 199 191 184 177 170 162 155 148 I 4 I '34 82 7 2IO 202 J 95 187 1 80 J 73 165 151 144 137 83 6 214 206 198 191 183 176 1 68 161 154 146 139 84 5 0.2 1 8 O.2IO 0.202 0.194 0.187 o.i79 0.171 0.164 0.156 0.149 0.142 85 4 222 214 206 198 190 182 174 167 159 152 144 86 3 226 218 2IO 201 193 185 178 170 154 147 87 2 230 222 2I 3 205 197 189 181 173 165 157 149 88 1 234 226 217 209 2OI 192 184 i'/6 168 1 60 152 89 O 0.239 0.230 0.221 O.2I3 0.204 0.196 0.188 0.179 0.171 0.163 0-155 90 1 243 234 225 217 208 199 191 183 174 1 66 158 91 2 247 239 229 221 212 203 194 - 1 86 177 169 1 60 92 3 252 243 234 225 216 207 198 189 181 172 l6 3 93 4 257 247 2 3 8 22 9 220 211 202 193 184 175 1 66 94 - 5 0.262 0.252 O.242 0.233 O.224 0.214 O.2O5 0.196 0.187 0.178 0.169 95 6 267 257 247 237 228 218 20 9 200 190 181 172 96 7 272 262 252 242 232 222 213 203 194 185 *75 97 ! 8 277 267 2 5 6 246 2 3 6 226 217 207 197 188 179 98 9 282 272 26l 251 2 4 I 2 3 I 221 211 201 191 182 99 i 10 0.287 0.277 0.266 0.256 0.245 0-235 0.225 O.2I5 0.205 o.i95 0.185 100 11 293 282 271 26l 250 239 229 219 209 199 189 101 ! 12 299 287 2 7 6 266 2 55 244 233 223 213 202 192 102 13 35 293 282 271 260 249 238 227 217 206 196 103 14 310 299 28 7 2 7 6 265 253 242 231 221 210 199 104 15 0.317 0.305 0.293 0.28l 0.270 0.258 0.247 0.236 0.225 0.214 0.203 1O5 16 323 3" 299 287 275 263 252 240 229 218 207 106 17 3 2 9 3i7 35 292 280 269 257 245 234 222 211" 107 -18 336 323 3 11 298 286 274 262 250 2 3 8 227 215 108 19 343 330 3i7 34 292 279 26 7 255 243 231 219 109 -20 o-35o 0-337 0.324 0.310 0.298 0.285 0.272 0.260 0.248 0.235 0.224 110 ! 21 357 344 33 3*7 34 291 2 7 8 265 253 240 228 Ill 22 365 351 337 323 3 IO 297 283 270 258 245 232 112 23 24 373 381 358 366 344 35 i 330 337 316 323 33 39 289 295 2 7 6 282 263 268 250 255 237 242 113 114 Ti*r 30 29 28 27 26 25 24 23 22 21 2O Polar uec. Co-Latitude. Dist. 120 TABLE XXX. Time-Azimuths: Log A. Latitude. _ Polar 70 71 72 73 D 74 75 76 77 78 79 8O Dist. 35 9.694 9.711 9.727 9-743 9.760 9-775 9.791 9.807 9.822 9.838 9.853 55 34 701 717 733 749 765 780 796 811 826 841 856 56 33 32 707 723 729' 739 745 770 775 785 790 800 804 815 819 830 833 844 848 859 862 57 58 31 720 735 750 765 780 794 808 823 837 851 865 59 30 9.726 9.741 9-755 9.770 9.784 9.798 9.812 9.826 9.840 9-854 9.867 60 29 28 732 737 746 752 761 766 775 779 789 793 802 806 816 820 830 833 843 846 & 870 873 61 62 26 743 748 III 770 775 784 788 810 814 824 827 837 840 849 852 862 865 875 878 63 64 25 9-753 9.766 9.779 9.792 9.805 9.818 9.830 9-843 9.855 9.868 9.880 65 24 771 796 809 821 834 846 858 870 882 66 23 22 21 i 772 jg 784 792 796 800 804 808 813 816 820 825 828 831 837 840 843 849 852 855 861 866 873 875 878 885 887 889 67 68 69 20 9.776 9.788 9.800 9.812 9-823 9.835 9.846 9-857 9.869 9.880 9.891 70 19 781 792 804 815 827 838 849 860 872 882 893 71 18 785 796 807 819 830 841 852 863 874 884 895 72 17 789 800 Bn 822 833 844 854 865 876 886 897 73 16 793 804 815 826 836 847 857 868 878 888 899 74 15 9-797 9.808 9.818 9.829 9-839 9.850 9.860 9.870 9.880 9.890 9.900 75 14 801 8n 822 832 842 852 862 872 882 892 902 76 13 12 804 808 815 818 828 835 838 845 848 855 857 865 867 875 877 884 887 896 904 906 77 78 11 811 821 831 841 850 860 870 879 889 898 907 79 10 9.815 9.825 9-834 9.844 9-853 9.863 9.872 9.881 9.891 9.900 9.909 8O '9 818 828 837 846 856 865 874 883 893 902 911 81 8 7 821 825 831 834 840 843 849 852 858 861 867 870 877 879 885 887 gg 903 95 912 914 82 83 6 828 837 846 855 863 872 881 889 898 907 915 84 5 4 9.831 834 *f 9 ifi **& 9.866 868 9-874 877 9 885 9.891 893 9.900 902 9.908 910 9.917 918 85 86 3 2 837 840 ss 856 862 86 5 871 873 879 881 887 889 895 897 903 905 911 913 920 921 87 88 1 842 851 859 867 875 883 891 899 907 923 89 o 9-845 9.853 9.861 9.86 9 9.877 9.885 9-893 9.901 9.908 9.916 9.924 90 1 848 856 864 872 879 887 895 902 910 918 925 91 1 3 $1 X 866 868 $ 88 1 883 889 891 897 898 904 906 912 913 919 920 926 928 92 93 ! 4 856 863 871 878 885 893 900 907 915 922 929 94 1 5 9.858 9.866 9-873 9 .88o 9.887 9.895 9.902 9.909 9.916 9-923 9.93 95 ' 6 86 1 868 875 882 889 897 94 911 918 925 93 2 96 ! 7 863 871 878 884 891 898 905 912 919 926 933 97 i 8 866 873 880 886 893 900 907 914 920 927 934 98 9 868 875 882 888 895 902 909 915 922 928 935 99 ! 10 9.871 9.877 9.884 9.890 9.897 9.904 9.910 9.917 9-923 9-93 9.936 100 ! 11 873 879 886 892 899 905 912 918 925 93 i 937 101 1 12 875 882 888 894 901 907 9i3 920 926 932 938 102 13 fo 77 884 890 896 903 909 915 921 927 933 939 103 14 880 886 892 898 904 910 916 922 929 935 940 104 -15 9.882 9.888 9.894 9.900 9.906 9.912 9.918 9.924 9-930 9.942 105 16 884 890 896 902 908 914 919 925 931 937 943 106 i 1"^ 886 892 898 904 909 915 921 927 932 938 944 107 18 888 894 900 95 911 917 922 928 934 939 945 108 -19 890 896 901 907 913 918 924 929 935 940 946 109 20 9.892 9.898 9.903 9.909 9.914 9.920 9-925 9-93 1 9-93 6 9.941 9-947 110 i 21 894 900 905 910 916 921 927 932 937 942 948 111 22 896 902 907 912 917 923 928 933 938 943 949 112 23 898 903 909 914 919 924 929 934 939 945 95 113 24 900 905 910 915 921 926 93 1 93 6 941 946 95 i 114 20 19 ! 18 17 16 15 14 13 12 11 10 n r i Polar Co-Latitude. Dist. TABLE XXX. 121 Time-Azimuths: Log B. Latitude. Polar Dec. 70 3 71 72 73 74 75 76 77 78 79 80 Dist. 35 0.080 0.076 0.072 0.068 0.064 0.060 0.056 0.052 0.048 0.044 0.040 O | 55 34 082 078 073 069 065 06 1 057 053 049 044 040 56 33 32 083 085 079 081 075 076 071 072 066 068 062 064 058 059 054 055 050 045 046 041 042 57 58 31 087 082 078 074 069 065 060 056 052 047 043 59 30 0.089 0.084 0.079 0.075 0.071 0.066 0.062 0.057 -53 0.048 0.044 6t> 29 091 086 08 1 076 072 067 063 058 054 049 45 61 28 092 088 083 078 073 069 064 060 055 050 046 62 094 089 085 080 075 070 065 06 1 056 051 047 63 26 096 091 086 08 1 076 072 067 062 057 052 048 64 , 25 0.098 0.093 0.088 0.083 0.078 0.073 0.068 0.063 0.058 0-053 0.048 65 i 24 IOO 095 090 085 079 074 069 064 059 054 049 66 i 23 102 97 091 086 08 1 076 071 066 060 055 050 67 ! 22 104 098 093 088 083 077 072 067 062 056 051 68 21 1 06 IOO 095 090 084 079 073 068 063 057 052 69 2O o.i 08 O.I O2 0.097 0.091 0.086 0.080 0.075 0.069 0.064 0.059 0-053 70 19 no 104 099 93 087 082 076 071 065 060 54 71 112 1 06 IOO 95 089 083 078 072 066 061 055 73 i 17 114 1 08 1 02 096 091 085 079 073 068 062 056 73 16 116 no 104 098 092 087 08 1 075 069 063 057 74 15 0.118 O.I 12 o.i 06 O.IOO 0.094 0.088 0.082 0.076 0.070 0.064 0.058 75 ! 14 120 114 1 08 1 02 096 090 084 078 071 065 059 76 13 I2 3 116 110 104 097 091 085 079 073 067 060 77 : 12 125 118 112 106 099 093 087 080 074 068 062 78 i 11 127 121 114 108 101 095 088 082 075 069 063 79 | 10 0.129 0.123 0.116 O.IIO 0.103 0.096 0.090 0.083 0.077 0.070 0.064 8O 9 132 I2 5 118 112 i5 098 091 085 078 072 065 Nl 127 120 114 107 IOO 093 086 080 073 066 82 i 7 137 129 122 116 109 102 095 088 08 1 074 067 83 6 139 132 125 118 in I0 3 096 089 082 075 068 84 ; 5 0.142 0.134 O.I27 0.120 0.113 0.105 0.098 0.091 0.084 0.077 0.070 85 4 144 137 129 122 115 107 IOO 093 085 078 071 86 3 147 139 I 3 2 124 117 IO9 102 094 087 080 072 87 2 149 142 *34 126 119 III I0 3 096 088 08 1 074 88 ; 1 152 144 136 128 121 113 105 098 090 082 075 89 0-155 0.147 0.139 0.131 0.123 O.II5 0.107 0.099 0.092 0.084 0.076 90 I 158 149 141 133 125 117 IO9 101 093 085 077 91 ! 2 1 60 144 135 127 119 III 103 095 087 079 92 3 163 155 146 138 I 3 121 113 105 097 089 080 93 ! 4 166 158 149 140 I 3 2 I2 3 "5 107 098 090 082 94 5 0.169 o.i 60 0.152 0.143 0.134 O.I26 O.II7 0.109 O.IOO 0.092 0.083 95 - 6 172 163 154 145 137 128 119 in 102 093 085 96 - 7 175 166 157 T48 139 I 3 121 113 104 095 086 97 ! 8 179 169 1 60 151 142 123 1 06 097 088 98 ; 9 182 172 163 154 144 135 126 117 108 098 089 99 10 0.185 0.176 0.166 0.156 0.147 0.137 0.128 0.119 0.109 O.IOO 0.091 100 11 189 179 169 159 150 140 I 3 121 in 1 02 . 093 101 12 192 182 172 162 M3 123 113 104 094 102 13 196 185 175 165 155 13S 125 116 106 096 103 14 199 189 178 168 158 148 128 118 1 08 098 104 15 0.203 0.192 0,182 0.171 0.161 0.150 0.140 0.130 O.I 2O O.IIO O.IOO 105 ! 16 207 196 185 174 164 153 143 I 3 2 122 112 101 1O6 -17 211 200 189 178 167 156 146 135 I2 4 114 103 107 18 215 2O4 192 181 170 148 137 127 116 I0 5 108 -19 219 208 196 185 173 162 I 5 I 140 129 118 107 109 -20 0.224 O.2I2 0.200 0.188 0.177 0.165 0.154 0.143 O.I3I O.I2O 0.109 110 21 228 216 204 192 1 80 169 H5 134 122 in 111 -22 232 220 208 196 184 172 1 60 I 4 8 I 3 6 "5 "3 112 23 237 224 212 200 187 175 163 139 127 115 113 24 242 229 216 203 191 179 166 154 142 J 3 118 114 1 i Dec. 20 19 18 17 16 15 14 13 12 11 10 Polar Diat I Co-Latitude. JLJl<* 122 TABLE XXX. Time-Azimuths: JLog: A. Declination. T.afr Co- JJcll. 24 -25 26 D 27 28 29 30 -31 -32 33 34 Lat. o 9.328 9-346 9-363 9-38o 9-397 9-4I3 9.428 9-443 9-457 9.471 9-485 o 9O 1 344 362 379 395 411 426 441 455 469 483 496 89 2 360 377 393 409 424 439 453 467 481 494 57 88 3 375 392 407 423 437 45 2 465 479 492 55 5 l8 87 4 390 406 421 436 45 464 477 490 503 516 528 86 5 9.404 9.420 9-434 9.448 9.462 9.476 9.489 9.501 9-5H 9.526 9.538 85 ; 6 418 433 447 461 474 487 500 512 524 536 547 84 i 7 43i 446 459 472 485 498 5io 522 534 545 556 83 8 444 458 47i 484 497 509 52i 532 544 555 566 82 I 9 457 470 483 495 508 519 53i 542 553 564 575 81 10 9.469 9.482 9.494 9.506 9.5i8 9-529 9-541 9-552 9.562 9-573 9-584 wo 11 481 493 505 517 528 539 550 56i 57i 582 592 79 ' 1*2 492 54 516 527 538 549 560 570 580 590 j doo 7 13 503 5'5 526 537 548 558 569 589 599 i 608 77 ! 14 5H 525 536 547 557 568 578 588 597 607 616 76 15 9o 2 5 9-53 6 9.546 9-557 9-567 9-577 9.586 9-596 9.605 9.615 i 9.624 75 16 535 546 556 566 576 586 595 604 613 623 631 74 17 545 555 5<>5 575 585 594 603 612 621 630 6 39 73 IN 555 5&5 575 584 594 603 612 620 629 638 646 7:2 19 5 6 4 574 584 593 602 611 620 628 637 645 653 71 20 9-574 9-583 9-593 9.601 9.610 9.619 9.628 9.636 9.644 9.652 9.660 70 21 5*3 592 602 610 618 627 635 643 652 659 667 69 22 592 601 610 618 626 635 643 651 659 666 674. 6 23 631 609 618 626 634 642 650 658 666 673 68 1 67 : 24 609 618 626 634 642 650 658 665 673 680 687 66 25 9.618 9.626 9-634 9.642 9.650 9-657 9.665 9.672 9.680 9.687 9.694 T 65 26 626 634 642 649 657 664 672 679 686 693 700 61 27 634 642 649 657 664 672 679 686 693 700 706 63 28 642 650 657 664 672 679 686 692 699 706 712 62 29 650 657 664 672 679 686 692 699 706 712 718 61 30 9.658 9.665 9.672 9.679 9.686 9.692' 9.699 9.706 9.712 9.718 9.724 60 31 65 5 672 679 6^6 692 699 705 712 718 724 73 59 32 673 679 686 693 699 706 712 7i8 724 73 736 5 33 6So 687 693 700 706 712 718 724 73 736 742 57 34 687 694 700 706 712 718 724 730 736 742 748 56 j 35 9.694 9.701 9.707 9-713 9.719 9.725 9-73 1 9-736 9.742 9.748 9-753 55 1 36 701 708 713 719 725 73i 737 742 748 753 759 54 37 708 7H 720 726 73i' 737 743 748 753 759 764 53 3 715 721 727 732 738 743 749 754 759 764 769 52 39 722 727 733 738 744 749 754 760 765 770 775 51 40 9.728 9-734 9-739 9-745 9-750 9-755 9.760 9-765 9.771 9-775 9.780 50 41 735 740 746 7Si 756 761 766 771 776 781 785 49 42 74i 747 752 757 762 767 772 776 781 786 790 48 43 748 753 758 763 768 772 777 782 786 791 795 47 44 754 759 704 769 773 778 783 787 792 796 800 46 45 9.760 9-765 9.770 9-775 9-779 9.784 9.788 9.792 9-797 9.801 9.805 4,5 46 767 . 77i 776 780 785 789 793 798 802 806 810 44 47 773 777 782 786 790 795 799 803 807 811 8i5 43 48 779 783 788 792 796 800 804 808 812 816 820 42 49 785 789 793 797 801 805 809 813 817 821 825 41 50 9.791 9-795 9-799 9-803 9.807 9.811 9.815 9.818 9.822 9.826 9.830 4O 51 797 801 804 808 812 816 820 823 827 831 8.S4 39 52 802 806 810 814 818 821 825 829 832 836 839 38 53 808 812 816 819 823 826 830 834 837 840 844 37 54 814 818 821 825 828 832 835 838 842 845 848 36 55 9.820 9-823 9.827 9.830 9.833 9-837 9.840 9-843 9.847 9.850 9-853 35 56 825 829 832 835 839 842 845 8 4 g 851 854 858 34 57 831 834 837 841 844 847 850 853 856 859 862 33 58 836 840 843 846 849 852 855 858 861 864 866 32 59 842 845 848 851 854 857 860 863 865 868 871 31 114 115 116 117 118 119 120 121 122 123 124 1 JLat. Co- ! Lat. Polar Distance. TABLE XXX. 123 Time-Azimuths: Log B. Declination. T ** Co- JjK. 24 25 26 27 -28 29 30 31 32 33 34 Lat. o 0.672 0.654 0-637 0.620 0.603 0.587 0.572 0-557 0.542 0.528 0-515 90 1 690 671 653 635 618 60 1 585 57o 555 540 526 89 2 708 688 669 651 633 616 599 583 568 552 538 88 3 4 727 747 706 725 686 704 667 684 649 665 631 646 628 597 611 581 594 578 550 563 87 86 5 0.768 0-745 0.723 0.702 0.682 0.663 0.644 0.626 0.609 0.592 0.576 85 5 791 766 743 721 700 680 660 642 624 606 589 84 7 814 788 764 719 698 677 658 6 39 621 603 83 839 812 786 762 738 716 695 675 655 636 618 82 9 866 837 810 784 759 736 692 672 652 633 81 1O 0.895 0.864 0-835 0.807 0.781 0-757 0-733 0.711 0.690 0.669 0.649 8O 11 925 892 861 832 805 779 754 730 708 687 666 79 12 959 923 890 859 829 776 727 705 683 78 13 995 956 920 887 856 827 799 773 748 724 702 77 14 1-035 993. 954 918 884 853 824 796 770 745 721 76 15 1.080 I -33 0.990 0.951 0.915 0.881 0.850 0.820 0-793 0.766 0.741 75 16 129 077 1.030 987 948 912 878 847 817 789 763 74 17 1 86 127 074 1.027 984 945 909 875 843 814 786 73 18 25 l 183 124 071 1.024 981 941 95 871 840 810 72 19 329 248 180 1 20 068 1.020 977 938 902 868 836 71 23 1.424 1.326 1-245 1.177 1.117 1.064 1.017 0.974 0-934 0.898 0.864 70 21 548 421 323 242 J73 114 06 1 1.013 970 931 894 69 22 722 544 418 319 238 170 no 057 I.OIO 966 927 68 23 2.022 719 414 316 235 166 1 06 053 i. 006 962 67 24 + 00 2.018 715 538 411 3 I2 231 162 102 049 I.OOI 66 25 2.018 + 00 2.015 1.712 1-534 1.407 1.308 1.227 I.I58 1.098 1.045 65 26 I-7I5 2.015 + 00 2.01 1 708 53 403 34 223 154 094 .64 27 538 1.712 2.OII + 00 2.007 704 526 399 300 219 150 63 2 411 534 1.708 2.007 + 00 2.003 700 522 395 296 214 62 29 312 407 530 1.704 2.003 -h 999 696 390 291 61 SO I.23I 1.308 1.403 1.526 1.700 1.999 + 00 1.994 1.691 I-5I3 1.386 60 31 162 227 34 399 522 696 J-995 + 00 99 686 508 59 33 102 158 223 300 395 5i7 691 1.990 + 00 985 682 58 33 049 098 154 219 296 390 513 686 1.985 + 00 980 57 34 002 45 094 150 214 291 386 508 682 1.980 + 00 56 35 0.958 0-997 I.04I 1.089 I-I45 I.2IO 1.286 1.381 I-503 1-677 1-975 55 36 954 0-993 036 085 140 205 281 376 ' 498 671 54 37 88 1 914 949 0.988 031 080 135 200 276 371 493 53 33 847 877 909 944 0.983 O26 75 130 195 271 365 52 33 815 842 872 904 939 0.978 021 070 125 189 265 51 4D 0.785 0.810 0.838 0.867 0.899 0-934 0.973 1.016 1.064 1.119 1.183 50 41 756 780 805 833 862 894 929 0.968 Oil 059 114 49 42 729 75 i 775 800 828 857 88 9 924 0.962 005 053 48 ! 43 44 703 679 724 698 746 719 770 795 764 822 789 8 5 I 816 846 877 0.956 912 0.999 950 47 46 45 0.655 0.674 0.693 0.714 0-735 0-759 0.784 0.811 0.840 0.871 0.906 45 43 47 633 611 650 627 668 645 688 663 708 682 73 702 753 724 778 747 805 772 8 865 827 44 43 4 590 605 622 639 657 676 696 718 765 792 42 49 570 58 4 600 616 633 ' 651 670 690 711 734 759 41 50 0.564 0-579 0-594 0.610 0.627 0.645 0.663 0.683 0.705 0.727 40 51 531 544 558 573 588 604 620 638 657 877 698 39 52 513 525 538 552 566 581 597 614 631 650 670 38 53 54 495 477 .507 489 519 532 546 526 560 539 575 553 590 568 607 583 624 600 643 617 37 36 55 0.460 0.471 0.482 0.494 0.506 0.519 0-532 0.546 0.561 0.576 0-593 35 56 444 454 465 476 487 499 5 12 525 539 553 569 34 57 428 437 448 458 469 480 492 505 518 53 2 546 33 58 412 421 43 i 441 462 473 485 497 5 IQ 524 32 59 396 45 414 424 434 444 455 466 477 490 502 31 Lat. 114 115 116 117 118 119 120 121 122 123 124 Co- Lat. Polar Distance. 124 TABLE XXX. A. 0* Time- Azimuths : Log C. m O a .O 0~.l 0' n .2 O in .3 O m .4 O m .5 O m .6 O n '.7 O m .8 O m .9 l m .O m O -f 00 3.661 3-360 3.184 3-059 2.962 2.883 2.816 2.758 2.707 2.661 59 1 2,66 1 2.620 2.582 2-547 2.515 485 457 431 406 382 360 58 2 360 339 3i9 299 281 263 246 230 214 199 184 57 3 184 170 156 H3 130 117 105 093 08 1 070 059 56 4 059 048 038 028 018 008 1.998 1.989 1.980 1.971 1.962 55 5 6 1.962 883 1-954 876 !-945 869 1-937 62 1.929 855 1.921 848 1.913 842 1.905 835 1.898 829 1.890 822 1.883 816 54 53 7 816 810 804 798 792 7 86 780 775 769 764 758 52 8 9 758 707 753 .702 747 697 742 693 688 732 683 727 679 722 674 717 670 712 666 707 66 1 51 50 10 i.65i 1-657 1.653 1.648 1.644 1.640 1.636 1.632 1.628 1.624 1.620 49 11 620 616 612 608 604 600 597 593 589 586 582 48 12 582 578 575 571 568 564 561 557 554 547 47 13 547 544 541 537 534 53 1 528 524 r ig 46 11 5*5 512 509 506 503 500 497 494 491 488 485 45 15 1.485 1.482 1-479 1.476 1-474 1.471 1.468 1.465 1.462 1.460 1-457 44 16 457 454 452 449 446 444 441 438 436 433 43 ! 17 43* 428 425 423 420 418 415 413 411 408 406 42 19 4 o5 382 403 380 401 378 399 375 396 373 394 369 389 366 387 364 384 362 382 360 41 40 , 2D 1.360 1-358 1-356 J-353 1.351 1-349 1-347 1-345 1-343 I-34I 1-339 39 21 339 337 335 333 330 328 326 324 322 320 38 1 *j 3 ! 8 316 315 313 3" 39 307 35 303 301 299 37 23 299 297 295 293 292 290 288 286 284 282 281 36 21 281 279 277 275 273 272 270 268 266 265 263 35 2.5 1.263 1.261 1.259 1.258 1.256 1.254 1-253 1.251 1.249 1.247 1.246 34 ! 23 246 244 242 241 239 237 236 234 233 231 229 33 27 229 228 226 225 223 221 220 218 217 215 214 32 23 214 212 2IO 209 207 206 204 203 20 1 200 198 31 23 19$ 197 195 194 192 191 189 188 186 I8 5 183 30 , 33 1.183 I.I82 1.181 1.179 1.178 I.I76 I.I75 I - I 73 1.172 I.I7I 1.169 29 31 169 168 166 165 164 162 161 159 158 J 55 28 i 1 3.2 155 154 153 151 150 149 H7 146 H5 H3 142 27 33 142 141 139 138 '37 134 133 132 130 129 26 ! 34 129 128 126 125 124 I2 3 121 1 20 119 118 116 25 35 I.II6 1.115 1.114 1.113 i. in I.IIO I.I09 1.108 1.106 1.105 1.104 24 33 104 103 102 100 099 9 8 097 096 094 093 092 23 37 092 091 090 089 087 086 085 084 083 082 080 22 38 080 079 7 8 077 076 75 074 072 071 070 069 21 | 33 069 068 067 066 065 064 062 061 060 059 058 20 40 1.058 1.057 1.056 1-055 1.054 1-053 1.052 1.050 1.049 1.048 1.047 19 41 047 046 045 044 043 4 2 041 040 039 038 37 18 : 42 037 036 035 34 33 3 2 3 I 030 029 027 026 17 43 026 025 O24 023 022 021 020 019 018 017 016 16 ; 41 016 OI4 013 OI2 on Oil OIO 009 008 007 15 45 1.007 i. 006 1.005 1.004 1.003 i. 002 I.OOI I.OOO 0.999 0.998 0-997 14 : 43 0.997 0.996 0-995 0.994 0-993 0.992 0.991 0.990 989 989 988 13! 47 ; 48 988 978 987 977 986 977 985 976 984 975 983 974 982 973 981 972 980 971 979 970 978 969 12 11 49 969 968 968 9 5 7 966 965 964 963 962 961 961 10 | 50 0.961 0.960 0.959 0.958 0-957 0.956 0-955 0.954 0-954 0-953 0.952 9 i 51 952 95 * 950 949 948 948 947 946 945 944 943 8 ; j 52 943 942 942 941 940 939 938 937 937 936 935 7 53 935 934 933 933 932 93 929 928 928 927 6 ! 54 927 926 925 924 924 923 922 921 920 920 919 5 55 0.919 0.918 0.917 0.916 0.916 0.915 0.914 0.913 0.912 0.912 0.911 4 56 911 910 909 909 908 907 906 905 905 904 93 3 i 57 903 902 902 901 900 899 899 898 895 2 i 58 895 895 894 893 892 892 890 889 888 888 1 59 888 887 886 886 885 884 884 883 882 88 1 881 I m 1-0 0-.9 O m .8 O m .7 O m .6 O m .5 O m .4 0-3 O m .2 O m .l O m .O m With Hour- Angles greater than 6 h read from bottom, and subtract tabular [ 1 l h Log from 10.000. TABLE XXX. A. 125 l h Time-Azimuths: Log . m O".O O m .l O m .2 O n .3 0'.4 O ;n .5 O ra .6 :n .7 O !n .8 O m .9 1-0 'm ! O 0.88 1 0.880 0.879 0.878 0.878 0.877 0.876 0.875 0.875 0.874 0-873 59 ! i 873 873 872 871 870 870 869 868 868 85 7 855 58 ! 2 866 865 865 864 863 85 3 862 861 86 1 860 859 57 3 859 858 858 857 856 856 855 854 854 853 852 53 : 4 852 852 851 850 849 849 848 847 847 846 845 55 5 0.845 0.845 0.844 0.843 0.843 0.842 0.841 0.841 0.840 0.839 0.839 > i ! 6 839 838 837 837 836 835 835 834 833 833 832 53 i 7 832 831 831 830 829 829 828 827 827 826 826 52 8 826 825 824 824 823 822 822 821 820 820 819 51 i 9 819 818 818 817 817 816 815 815 814 813 813 50 1O 0.813 0.812 0.811 0.811 0.810 0.810 0.809 0.808 0.808 0.807 0.806 49 ! 11 806 806 805 805 804 803 803 802 802 831 800 48 i 12 800 800 799 798 798 797 797 796 795 795 794 47 : 794 794 793 792 792 791 791 790 789 789 788 46 14 788 7 83 787 786 786 785 785 784 783 783 782 45 j 15 0.782 0.782 0.781 0.780 0.780 0.779 0.779 0.778 0.778 0.777 0.776 44 i 16 776 775 775 775 774 773 773 772 772 771 771 43 ! 771 770 769 769 758 768 767 767 766 765 765 42 1 8 7^5 764 764 763 753 762 762 761 760 700 759 41 j 19 759 759 758 758 757 756 756 755 755 754 754 40 2O 0-754 0-753 0-753 0.752 0.751 0.751 0.750 0.750 0.749 0-749 0.748 39 21 748 748 747 747 746 745 745 744 744 743 743 38 i 22 743 742 742 741 740 739 739 738 738 737 37 23 737 737 736 736 735 735 734 734 733 733 732 36 24 732 732 73 i 730 730 729 729 728 728 727 727 35 25 0.727 0.726 0.726 0.725 0.725 0.724 0.724 0.723 0.723 0.722 0.722 34 26 722 721 721 720 720 719 718 718 717 717 716 33 27 716 716 715 7H 714 713 713 712 712 711 32 : 2 711 711 710 710 709 709 708 7 oS 707 707 706 31 ! 29 706 7 o5 705 705 7^4 704 703 703 702 702 701 30 30 0.701 0.701 0.700 0.700 0.699 0.699 0.698 0.698 0.697 0.697 0.696 29 31 696 696 695 695 694 694 693 693 692 28 | ! 32 692 691 690 690 689 689 688 688 687 687 27 33 687 635 685 685 685 684 684 683 683 682 682 26 34 682 68 1 68 1 6Si 680 680 679 679 678 678 677 25 35 0.677 0.677 0.676 0.676 0.675 0.675 0.674 0.6 "4 0.673 0.673 0.673 24 36 673 672 672 671 671 670 670 669 669 668 668 23 37 668 667 667 65 7 666 666 665 665 664 664 663 22 38 663 663 662 662 661 661 661 660 660 659 659 21 39 659 658 658 657 657 656 656 656 655 655 654 20 40 0.654 0.654 0-653 0.653 0.652 0.652 0.652 0.651 0.651 0.650 0.650 19 41 650 649 649 648 648 648 647 647 646 646 645 18 42 645 645 644 644 644 643 643 642 642 641 641 17 43 641 641 640 640 639 639 638 638 638 637 637 16 ! 44 637 636 630 635 635 634 634 634 633 633 632 15 45 0.632 0.632 0.631 0.631 0.631 0.630 0.630 0.629 0.629 0.628 0.628 14 ; ! 46 628 628 627 627 626 626 626 625 625 624 624 13 ! 47 624 623 623 1 623 622 622 621 621 620 620 620 12 ! 48 620 619 619 i 618 618 618 617 617 616 616 615 11 49 615 6i5 615 614 614 613 613 613 612 612 611 1O ! 5*> 0.611 0.611 0.611 0.610 0.6 10 0.609 0.609 0.609 0.608 0.608 0.607 9 51 607 607 606 606 606 605 605 604 604 604 603 8 52 603 603 602 602 602 60 1 60 1 600 600 600 599 7 53 599 599 598 598 598 597 597 596 596 596 595 6 54 595 595 594 594 594 593 593 592 592 592 59 1 5 J55 0.591 0.591 0.590 0.590 0.590 0.589 0.589 0.589 0.588 0.587 0.587 4 56 587 587 587 586 586 585 585 585 584 584 583 3 57 583 583 583 582 582 582 581 ,581 580 580 580 2 58 580 579 579 578 578 578 577 577 577 576 5/6 1 59 576 575 575 575 574 574 573 573 573 572 572 m 1.0 0-.9 O m .8 0^.7 O :n .6 0-.5 O m .4 O m .3 O m .2 O m .l O in .O m "With Hour- Angles greater than 6 b read from bottom, and subtract tabular L * ** l Log from 14X000. 126 TABLE XXX. A. 9h Time-Azimuths: Log: C. t m o.o O-.l 0' u .2 0".3 0,4 0.5 0.6 O m .7 0-8 09 1>0 m 0.572 0.572 0.571 0571 0.570 0.570 0.570 0.569 0.569 0.569 0.568 59 1 5 58 568 567 567 567 566 566 566 565 5 6 5 5$4 58 2 504 564 564 5^3 5^3 563 562 562 561 561 561 57 3 55o 560 560 559 559 558 558 558 j 557 557 56 4 557 557 ! 55$ 55$ 556 555 555 554 554 554 553 55 5 o-553 o-553 0.553 0.552 0-552 0.552 0.551 0.551 0.550 0-55 0.550 54 6 550 549 549 549 548 548 548 547 547 54$ 54$ 53 7 54$ 54^ 545 545 545 544 544 544 543 543 543 52 8 543 542 542 542 541 541 540 540 540 539 539 51 9 539 539 ; 538 538 538 537 537 536 53$ 53$ 535 5O 19 -535 o-535 o-535 0-534 0-534 0-534 o-533 0-533 0.533 0.532 | 0.532 49 11 532 532 53' 531 530 530 530 529 529 529 ; 528 48 Id 528 528 528 527 527 527 526 526 5 25 525 525 47 13 525 525 ; 524 524 5 2 4 523 523 523 522 522 : 521 48 14 521 521 521 520 520 520 5^9 5'9 519 45 15 0.518 0.518 0.517 0.517 0.517 0.516 0.516 0.516 0.515 0.515 i 0.515 44 1 16 515 i 514 514 514 513 513 513 512 512 5 12 5 11 43 17 5" 5" 5 11 5* 5 r o 510 59 59 59 508 i 508 42 13 508 508 . 5^7 507 507 506 506 506 505 505 : 505 41 19 505 504 504 504 533 503 503 502 j 502 502 \ 501 40 1 2D 0.501 0.501 i 0.501 0.500 0.500 0.500 0-499 0.499 0.499 0.498 ! 0.498 39 21 498 498 ' 497 497 497 496 496 49$ 495 495 495 38 2J 495 494 494 494 493 493 493 492 492 492 ! 49 37 23 491 491 i 491 493 493 490 493 489 489 489 483 36 24 488 488 l 488 487 487 487 486 486 . 485 485 485 35 25 0.485 0485 0.484 0.484 0.484 0.483 0.483 0.483 0.482 0.482 1 0.482 34 23 482 481 481 481 481 480 480 480 479 479 479 33 27 479 478 478 478 477 477 477 476 47$ 47$ 475 32 2 475 475 475 475 474 474 474 473 4/3 473 472 31 29 472 472 472 47i 47 1 47 470 470 470 469 30 30 0.469 0.469 0.469 0.468 0.468 0.468 0.467 0.467 0.467 0.466 0.466 29 31 ' 466 466 4<>5 465 465 465 464 464 464 4$3 4$3 28 32 463 463 462 462 462 461 461 461 461 460 460 27 33 460 460 459 459 459 458 458 458 458 457 457 26 34 457 457 45$ 45 $ 456 455 455 455 454 454 454 25 35 0-454 0-454 0-453 0-453 0-453 0.452 0.452 0.452 0-45 l 0.451 0.451 24 36 451 45 45 45 449 449 449 448 448 448 23 37 448 448 447 447 447 446 446 446 445 445 445 22 33 445 445 444 444 444 443 443 443 442 442 442 21 39 442 442 441 441 441 44 440 440 440 439 439 2O 40 o-439 0-439 0.438 0.438 0.438 0-437 0-437 0.437 0-437 0,436 0.436 19 41 436 43$ 435 435 435 435 434 434 434 433 433 18 ! 42 433 433 I 432 432 432 432 43 l 43 43 17 43 43 43 1 430 429 429 429 4~28 428 428 428 427 16 41 427 427 427 426 426 426 426 425 425 425 424 15 45 0.424 0.424 0.424 0.424 0.423 0.423 0.423 0.422 0.422 0.422 0.422 14 46 422 421 421 421 420 420 420 420 419 419 419 13 47 419 418 418 418 418 417 417 417 416 416 416 12 48 416 416 415 415 415 414 414 414 414 413 413 11 49 413 413 412 412 412 412 411 411 411 410 410 1O 50 0.410 0.410 0.410 0.409 0.409 0.409 0.409 0.408 0.408 0.408 0.407 9 51 407 407 407 406 406 406 405 45 405 405 8 52 405 404 404 404 403 403 403 403 402 402 402 7 53 402 402 401 401 401 400 400 400 400 399 399 6 54 399 399 399 398 398 398 397 397 397 397 39$ 5 55 0.396 0.395 0.396 0.396 0-395 o-395 o.395 0-394 0-394 0-394 0-394 i 56 394 393 393 393 393 392 392 392 39i 391 3 io 3 57 39* 39i 39 393 390 390 389 389 389 388 388 i > 58 388 388 388 387 387 387 387 3 85 386 386 385 1 59 385 385 385 385 384 384 384 384 383 383 383 m 1-0 O n .9 O m .8 O m .7 O ra .6 0-5 O m .4 O m .3 O m .2 O m .l O'".O m With Hour- Angles greater than 6 h read from bottom, and subtract tabular 9 h Log from 10.000. TABLE XXX. A. 127 3 h Time- Azimuths: Log C. m 0'".0 O'M O m .2 0.3 O m .4 O m .5 O m .6 0.7 0.8 0.9 l m .O m O 0-383 0-383 0.382 0.382 0.382 0.381 0.381 0.381 0.381 0.380 0.380 59 1 380 380 380 379 379 379 379 378 378 378 377 58 377 377 377 377 376 376 376 376 375 375 375 57 3 375 375 374 374 374 373 373 373 373 372 372 56 * 372 372 372 37i 37i 37i 37i 370 37o 37 370 55 5 0.370 ! 0.369 0.369 0.369 0.368 0.368 0.368 0.368 0.367 0.367 0.367 54 6 367 367 366 366 366 366 365 365 365 365 364 53 f 3 6 4 | 3 6 4 3 6 4 364 363 363 363 362 362 362 362 52 362 ; 361 361 361 361 360 360 360 360 359 359 51 9 359 | 359 359 358 358 358 358 357 357 357 357 50 10 0-357 i 0.356 0-356 0-356 0-356 0-355 0-355 0-355 0-354 0-354 o-354 49 11 354 ! 354 353 353 353 353 352 352 352 352 351 48 12 35i 351 35i 35i 350 350 350 350 349 349 349 47 13 349 349 348 348 348 348 347 347 347 347 346 46 14 346 | 346 34 6 346 345 345 345 345 344 344 344 45 15 0.344 0.344 o-343 o-343 o-343 o-343 0.342 0.342 0.342 0.342 0.341 44 16 34i 34i 34i 34i 340 340 340 340 339 339 339 43 ! 17 339 339 33 f 338 338 338 337 337 337 337 336 42 ; IS 33 6 : 33 6 336 336 335 335 335 335 334 334 334 41 19 334 ! 334 333 333 333 333 33 2 332 332 332 33i 40 i 20 0-331 0.331 o-33i 0-331 0-330 0-330 0-33 0-33 0.329 0.329 . 0.329 39 j 91 329 1 329 328 328 328 328 327 327 327 327 326 38 1 1 22 326 ! 326 326 326 3*5 325 325 325 324 324 324 37 23 324 324 323 323 323 323 322 322 322 322 322 36 24 322 ; 321 321 321 3 21 320 320 320 320 319 3*9 35 i 25 0.319 0.319 0.319 0.318 0.318 0.318 0.318 0.317 0.317 o-3i7 0-317 34 26 317 316 316 316 316 315 315 315 315 3H 3*4 33 27 3H 3H 3H 3H 3i3 3 J 3 313 313 312 312 312 32 1 28 312 312 3ii 3" 3" 3H 310 310 310 310 39 31 29 309 39 39 309 308 308 308 308 308 307 307 3O 3O 0-30.7 0.307 0.307 0.306 0.306 0.306 0.306 0-305 0.305 0-305 0-305 29 31 305 304 304 34 34 33 303 303 303 303 302 28 i 32 302 302 32 302 301 301 301 301 300 300 3 27 33 300 300 299 299 299 299 298 298 298 298 298 26 34 298 297 297 297 297 296 296 296 296 295 295 25 35 0.295 0.295 0.295 0.294 0.294 0.294 0.294 0.294 0.293 0.293 0.293 24 36 293 293 292 292 292 292 291 291 291 290 23 37 290 290 290 290 290 289 289 289 289 288 288 22 38 288 288 288 287 287 287 287 287 286 286 286 21 39 286 286 285 285 285 285 284 284 284 284 284 20 4O 0.284 0.283 0.283 0.283 0.283 0.282 0.282 0.282 0.282 0.281 0.281 19 41 281 281 281 281 286 280 280 280 279 279 279 18 42 279 279 278 278 278 278 278 277 277 277 277 17 43 277 276 276 276 276 275 275 275 275 275 274 16 44 274 274 274 274 273 273 273 273 272 272 272 15 45 0.272 0.272 0.272 0.271 0.271 0.271 0.271 0.270 0.270 0.270 0.270 14 46 270 270 269 269 269 269 268 268 268 268 267 13 47 267 267 267 267 267 266 266 266 266 265 265 12 48 49 265 263 265 263 265 263 265 262 264 262 264 262 264 262, 264 261 263 261 4 261 4 261 11 10 50 51 0.261 258 0.261 258 0.260 258 0.260 258 0.260 258 0.260 257 0.259 257 0.259 257 0.259 257 0.259 256 0.258 256 9 8 52 256 256 256 256 255 255 255 255 254 254 254 7 53 254 254 254 253 253 253 253 252 252 252 252 6 54 252 252 251 25 i 251 251 250 250 250 250 250 5 55 0.250 0.249 0.249 0.249 0.249 0.248 0.248 0.248 0.248 0.248 0.247 4 56 247 247 247 247 246 246 246 246 246 245 245 3 57 245 245 245 244 244 244 244 244 243 243 243 2 58 243 243 243 242 242 242 242 241 241 241 241 1 59 241 241 240 240 240 240 239 239 239 .239 239 m l m .O O m .9 O ra .8 O ra -7 O m .6 O m .5 O m .4 O m .3 O m .2 O ra .l O m .O m With Hour- Angles greater than 6 11 read from bottom, and subtract tabular 8 h | Log from 10.000. 128 TABLE XXX. A. 4'.] Time-Azimuths: Log . m 0'".0 0-1 0".2 .3 O :u 4 0.5 O".6 O n .7 O !1 '.8 O : ".9 ; 1"'.O 111 0.239 0.238 0.238 0.238 0.238 0.237 0.237 ! 0.237 0.237 o. 37 0.236 59 1 236 236 236 236 236 235 235 ! 235 235 ^4 234 5S 2 234 234 234 -.U 233 233 233 2 33 ! 232 j ^2 232 57 3 232 232 232 231 i 231 231 231 231 ; 230 \ 30 230 .53 4 230 230 229 229 229 229 229 228 j 228 j .8 i 228 5.5 5 0.228 0.227 0.227 0.227 0.227 0.227 0.226 0.226 0.226 0.226 0.226 51 6 226 225 225 225 225 224 224 224 224 224 223 53 7 223 223 223 223 223 222 222 222 222 222 222 52 8 222 221 221 221 220 220 22O 220 22O 219 210 51 9 219 2I 9 2I 9 218 218 218 218 218 2I 7 217 217 50 10 0.217 O.2I7 0.217 0.2 1 6 0.216 0.2 1 6 0.216 0.215 O.2I5 O.2I5 O.2I5 49 11 215 215 2I 4 214 214 214 214 213 213 2I 3 ; 2I 3 48 12 2I 3 212 212 212 212 212 211 211 211 211 211 47 13 211 210 2IO 2IO 2IO 209 209 209 209 209 ; 208 46 14 208 208 208 208 208 207 207 207 207 207 206 45 15 O.2O6 0.206 O.2O6 0.206 O.2O5 0.205 O.2O5 O.2O5 O.2O5 O.2O4 ! 0.2O4 44 16 204 204 20 4 204 203 203 203 203 203 202 202 43 17 202 202 202 2O I 201 2OI 201 201 200 200 200 42 18 200 2OO 2OO 199 199 199 199 I 99 I 9 8 198 198 11 19 I 9 8 198 I 9 7 197 197 197 197 I 9 6 I 9 6 196 196 40 20 0.195 O.I95 0.195 0.195 0.195 0.195 0.195 0.194 0.194 0.194 0.194 39 21 194 I 94 193 193 193 192 192 192 192 192 38 22 I 9 2 191 191 191 191 191 190 190 I 9 190 ' 190 37 23 189 189 I8 9 l8 9 189 1 88 1 88 1 88 1 88 I8 7 36 < 24 187 I8 7 I8 7 I8 7 l8 7 1 86 1 86 1 86 1 86 1 86 I8 5 35 25 0.185 0.185 0.185 0.185 0.185 0.184 0.184 0.184 0.184 0.184 0.183 34 26 183 !&3 I8 3 183 183 182 182 182 182 181 181 33 27 181 181 181 181 180 1 80 1 80 1 80 180 179 179 32 ! 28 1 7J 179 179 179 178 178 178 178 178 177 177 31 29 177 177 177 177 176 176 176 176 176 175 '75 30 30. 0.175 0.175 0.175 o.i74 0.174 0.174 0.174 0.174 0-173 0,173 - I 73 29 31 173 173 '73 172 172 172 172 172 171 171 171 28 ; 32 171 171 170 170 170 170 170 169 169 169 27 33 169 ^69 169 1 68 1 68 1 68 1 68 168 167 167 167 26 34 167 167 167 1 66 1 66 166 166 1 66 165 165 165 2,5 35 0.165 0.165 0.164 0.164 0.164 0.164 0.164 0.163 0.163 0.163 0.163 24 36 163 163 162 162 162 162 162 161 161 161 161 23 37 161 161 1 60 1 60 1 60 1 60 1 60 !59 '59 159 159 22 38 '59 159 158 158 158 158 158 '57 '57 157 21 39 i57 '57 156 156 156 156 156 155 '55 '55 20 40 0-155 0.155 0.154 0.154 0.154 0.154 0.154 0-153 o-!53 0-153 0-153 19 41 153 153 J52 '52 152 152 152 151 151 151 151 18 42 151 '5 1 '5 I S 150 149 149 149 149 17 43 149 149 148 [48 148 148 148 H7 H7 147 16 44 H7 '47 146 146 146 146 146 *45 '45 '45 '45 15 45 0.145 0.145 0.144 0.144 0.144 0.144 0.144 0.143 0.143 0.143 0.143 14 46 143 143 142 142 142 142 142 141 141 141 141 13 47 141 141 140 140 140 140 140 *39 139 i39 139. 12 48 139 139 '38 138 138 138 138 37 137 137 137 11 49 137 136 136. 136 136 136 135 135 1O 50 0.135 o.i35 0.134 0.134 0.134 0.134 0.134 0.133 0*133 0.133 0.133 9 51 133 i33 132 132 132 132 132 131 131 I 3 I 131 52 131 130 130 130 130 130 129 129 129 129 7 53 129 129 128 128 128 128 128 127 127 127 12J (i 54 127 127 126 126 126 126 126 I2 5 125 125 125 5 55 0.125 0.125 0.124 0.124 0.124 0.124 0.124 0.123 0.123 0.123 0.123 1 56 123 123 122 122 122 122 122 122 121 121 121 3 57 121 121 121 I2O 1 2O 1 2O 1 2O 1 2O 119 119 1 19 9 58 119 119 119 118 118 118 118 118 117 117 117 1 59 117 117 116 116 116 116 116 "5 H5 m l' r -.O O m .9 0'".8 O m .7 0"'.6 O m .5 O m .4 O'".3 O m .2 O m .l O in .O m With Hour- Angles greater than 6 h read from bottom, and subtract tabular r Log from 10.000. TABLE XXX. A. 129 5 h Time- Azimuths: Log C. m O m .O O m .l O'".2 0'.3 0'.4 0'.5 0'.6 0'".7 0\8 0'.9 1"'.0 m ; O 0.115 0.115 0.115 0.114 0.114 0.114 0.114 0.114 0.113 0.113 0.113 59 1 113 113 113 112 112 112 112 112 in in III 58 ! * in III III III 110 IIO IIO IIO IIO 109 109 57 3 109 109 109 109 108 1 08 1 08 108 108 107 107 56 4 107 107 107 107 106 1 06 106 1 06 1 06 105 105 55 5 0.105 0.105 0.105 0.105 0.104 O.IO4 0.104 0.104 0.104 0.103 0.103 54 6 103 103 103 103 103 102 102 102 1 02 1 02 IOI 53 7 101 IOI IOI IOI IOI IOO IOO IOO IOO IOO 099 52 099 099 099 099 099 098 098 098 098 098 097 51 9 097 097 097 097 097 096 9 6 096 096 096 096 50 1O 0.096 0.095 0.095 0.095 0.095 0.095 0.094 0.094 0.094 0.094 0.094 49 ! 11 094 093 93 093 93 093 092 092 092 092 092 48 13 092 091 091 091 091 091 090 090 090 090 090 47 13 090 090 089 089 089 089 089 088 088 088 088 46 14 088 088 087 087 087 087 087 086 086 086 086 45 i 15 0.086 0.086 0.085 0.085 0.085 0.085 0.085 0.084 0.084 0.084 0.084 44 16 084 084 084 083 083 08 3 083 08 3 082 082 082 43 17 082 082 082 081 08 1 081 081 08 1 080 080 080 42 18 080 080 ' 080 079 079 079 079 079 078 078 078 41 19 078 078 078 078 077 077 077 077 077 076 076 40 20 0.076 0.076 0.076 0.076 0.075 ' 0.075 0.075 0.075 0.075 0.074 0.074 39 21 074 074 074 074 073 73 073 073 073 073 072 38 22 072 072 072 072 072 071 071 071 071 071 070 37 23 070 070 070 070 070 069 069 069 069 069 069 36 24 069 068 068 068 068 068 067 06 7 067 067 067 35 25 0.067 0.066 0.066 0.066 0.066 0.066 0.065 0.065 0.065 0.065 0.065 34 26 065 064 064 064 064 064 064 06 3 063 063 063 33 27 063 063 062 062 062 062 062 061 06 1 061 061 32 | 2 061 06 1 060 060 060 060 060 059 059 59 059 31 29 059 059 59 058 058 058 058 058 057 057 057 3O 30 0.057 0.057 0.057 0.056 0.056 0.056 0.056 0.056 o-055 0.055 0.055 29 31 055 055 55 055 054 054 054 054 054 53 053 28 32 053 053 53 053 052 052 052 052 052 051 OS 1 27 33 051 5 J 5 ! OS 1 OS 1 050 050 05,0 050 050 049 26 34 049 049 049 049 049 048 048 048 048 048 047 25 35 0.047 0.047 0.047 0.047 0.047 0.047 0.046 0.046 0.046 0.046 0.046 24 36 046 045 045 045 045 045 044 044 044 044 044 23 37 044 043 043 043 043 043 043 042 042 042 042 22 as 042 042 041 041 041 041 041 040 040 040 040 21 39 040 040 039 039 039 39 039 039 038 038 038 2O 40 0.038 0.038 0.038 0.037 0.037 0.037 0.037 0.037 0.036 0.036 0.036 19 41 036 036 036 035 035 035 035 035 035 034 034 18 i 42 034 034 034 034 33 033 033 033 033 032 032 17 43 032 032 032 032 031 031 031 031 031 031 030 16 44 030 030 030 030 030 029 029 029 029 029 028 15 45 0.028 0.028 0.028 0.028 0.028 0.027 0.027 0.027 0.027 0.027 0.027 14 I 46 027 026 026 026 026 026 025 025 025 025 025 13 1 47 025 024 024 024 024 024 024 023 023 023 023 12 48 023 023 022 022 022 022 022 021 021 O2I' 021 11 49 02 1 021 020 O2O 020 O2O O2O O2O OI9 OI9 OI9 1O 1 50 0.019 O.OI9 O.OI9 o.o 1 8 0.018 0.018 0.018* O.OlS 0.017 0.017 0.017 9 51 017 017 017 016 016 016 016 016 016 015 015 8 I 52 i5 OI 5 I5 oi5 014 014 014 014 014 OI 3 OI 3 7 | 53 013 013 OI 3 013 013 OI2 OI2 012 012 012 Oil 6 j 54 on on on on Oil 010 OIO OIO OIO OIO 009 5 55 0.009 0.009 0.009 0.009 0.009 O.OO9 O.OO8 0.008 0.008 0.008 0.008 4 56 008 007 007 007 007 007 006 006 OO6 006 006 3 1 57 006 005 005 005 005 005 OO5 004 OO4 OO4 004 2 i 58 004 004 003 003 003 003 00 3 OO2 002 002 OO2 * 59 002 002 002 OOI OOI OOI OOI OOI OOO OOO OOO O m l m .O O m .9 O m .8 O m .7 O m .6 O m .5 O m .4 O m .3 O m .2 O m .l O in .O m With Hour- Angles greater than 6 h read from bottom, and subtract tabular 1 " Log from 10.000. 130 TABLE XXX. B. - Time-Azimutlts : I.ou Tangents X and Y. *1 00 O'l 0.2 O.3 O.4 O.5 0.6 3 .7 0.8 O .9 -'S 'Sb'S $" |a| o 00 7.242 7-543 7-719 7-844 7.941 8.020 8.087 8.145 8.196 o ' o 1 8.242 8.283 8.321 8.356 8.388 8.418 446 472 497 521 i 2 543 564 585 604 622 640 657 674 689 705 2 3 4 719 845 734 855 Itl 761 876 774 886 786 896 799 906 811 822 924 834 933 3 1 5 8.942 8.951 8-959 8.967 8.976 8.984 8.991 8.999 9.007 9.014 5 6 9.022 9.029 9.036 9-043 9.050 9-057 9.063 9.070 076 083 6 7 089 095 102 108 114 119 I2 5 137 142 7 8 148 153 159 164 169 174 180 185 190 195 8 9 200 i 205 209 214 219 224 228 233 237 242 9 1O 9.246 i 9.251 9-255 9-259 9.264 9.268 9.272 9.276 9.280 9.285 10 11 289 293 297 301 305 308 3^2 316 320 324 11 12 327 33i 335 339 342 349 353 356 360 12 13 363 367 370 374 377 380 384 387 390 394 13 14 397 400 403 406 410 4i3 416 419 422 425 14 16 9.428 457 9-431 460 9-434 463 9-437 466 9.440 469 9-443 472 9.446 474 9.449 477 9-452 480 9-455 483 15 16 17 485 488 491 493 496 499 54 507 59 17 18 512 5H 517 519 522 525 527 530 S3 2 535 18 19 537 539 542 544 547 549 552 554 556 559 19 20 9.561 9-563 9.566 9.568 9-570 9-573 9-575 9-577 9.580 9-582 2O 22 609 589 611 6?] 593 615 595 617 598 619 600 621 602 624 604 626 21 22 23 628 630 632 634 636 638 640 642 644 647 23 24 649 651 653 655 657 659 661 663 665 667 24 25 9.669 9.671 9-673 9-675 9-677 9.678 9.680 9.682 9.684 9.686 25 26 688 690 692 694 696 698 700 702 703 705 26 27 28 707 726 3 711 729 713 73 1 733 716 735 718 737 720 738 722 740 724 742 27 28 29 744 746 747 749 753 754 756 758 760 29 30 9.761 9-763 9-765 9-767 9.768 9.770 9-772 9-774 9-775 9-777 30 31 779 780 782 784 786 787 789 791 794 31 32 797 799 80 1 803 804 806 808 809 811 32 33 gj-2 814 816 817 819 821 822 824 826 827 33 34 829 831 832 834 836 837 839 840 842 844 34 35 9-845 9.847 9.848 9.850 9-852 9-853 9-855 9.856 9.858 9.860 35 36 861 863 864 866 868 869 871 872 874 876 36 37 877 879 880 882 883 885 887 888 890 891 37 38 893 894 896 897 899 901 902 94 905 907 38 39 908 910 911 913 9*5 916 918 919 921 922 39 40 9.924 9-925 9-927 9.928 9-93 9-931 9-933 9-935 9-936 9-938 40 41 939 941 942 944 945 947 948 95 951 953 41 42 954 956 957 959 961 962 964 965 967 968 42 43 970 971 973 974 976 977 979 980 982 983 43 44 985 986 988 989 991 992 994 995 997 998 44 45 0.000 O.OO2 0.003 0.005 0.006 0.008 0.009 o.on O.OI2 0.014 45 FIRST CASE : Half sum of Polar Distance and Co-latitude less than 90. The j sum or difference of the Angles X and Y, according as the Polar Distance is greater or less than the Co-latitude, is the Azimuth. SECOND CASE: Half sum of Polar Distance and Co-latitude greater than 90. The difference of the Angles X and Y subtracted from 180 is always the Azimuth. TABLE XXX. B. 131 Time-Azimuths : Log Tangents X and Y. M . N . f! O C .O 01 0.2 O.3 0.4 0.5 O.6 0.7 0.8 0.9 }l o 45 o.ooo 0.002 0.003 0.005 0.006 0.008 0.009 o.on 0.012 0.014 45 OI 5 017 018 020 02 1 023 024 026 027 029 46 47 030 032 033 035 036 038 039 041 043 044 47 48 046 047 049 050 052 053 055 056 058 059 48 49 06 1 062 064 065 067 069 070 072 073 075 49 50 0.076 0.078 0.079 0.081 0.082 0.084 0.085 0.087 0.089 0.090 50 51 092 093 095 096 098 099 IOI 103 104 106 51 52 107 IO9 no 112 "3 U5 117 118 120 121 52 53 123 124 126 128 129 132 134 136 137 53 54 139 140 142 144 147 148 150 152 153 54 55 0.155 0.156 0.158 O.I 6O 0.161 0.163 0.164 0.166 o.i 68 0.169 55 56 171 173 174 I 7 6 178 179 181 183 184 1 86 56 57 I8 7 I8 9 191 192 194 I9 6 197 199 201 203 57 58 204 2O6 208 209 211 213 214 216 218 220 58 59 221 223 225 226 228 230 232 233 235 237 59 60 0.239 0.240 0.242 0.244 0.246 0.247 0.249 0.251 0-253 0.254 60 61 256 258 260 262 263 265 267 269 271 272 61 62 274 2 7 6 278 280 282 284 285 287 289 291 62 63 293 295 297 2 9 8 3 00 302 34 306 308 310 63 | 64 312 3*4 318 3 20 322 323 325 327 329 64 65 0.331 0-333 0.335 0-337 0-339 0.341 0-343 0.345 0-347 0-349 65 66 351 353 356 358 3 60 362 364 366 368 370 66 67 372 376 379 38l 383 385 387 389 391 67 68 394 39 6 398 400 402 405 407 409 411 414 68 69 416 418 420 423 425 427 43 432 434 437 69 70 0-439 0.441 0.444 0.446 0.448 0.451 0-453 0.456 0.458 0.461 7O 71 463 465 468 470 473 475 478 481 483 486 71 72 488 491 493 496 499 504 507 509 512 72 73 74 515 543 545 520 548 523 526 554 58 557 531 560 534 503 III 540 569 73 74 75 0.572 o-575 0.578 0.581 0.584 0.587 0.590 0-594 0-597 0.600 75 76 603 606 610 613 616 620 623 626 630 633 76 77 637 640 644 647 651 654 658 66 1 665 669 77 78 673 676 680 684 688 692 695 699 703 707 78 79 711 7i5 720 724 728 732 736 745 749 79 80 0-754 0.758 0.763 0.767 0.772 0.776 0.781 0.786 0.791 0.795 8O 81 800 805 810 815 820 826 831 836 841 847 81 82 852 858 863 869 875 881 886 892 898 905 82 83 911 917 924 930 937 943 95 957 964 971 83 84 978 986 993 1. 00 1 1.009 1.016 1.024 1.041 1.049 84 85 1.058 1.067 1.076 1.085 1.094 1.104 1.114 1.124 I-I34 I-I45 85 86 87 III 166 295 178 326 201 343 214 360 226 378 239 396 253 266 43 6 86 87 88 457 479 53 528 554 582 612 644 679 717 88 89 758 804 855 980 2.059 2.156 2.281 2-457 2.758 89 j 90 -f 00 9O The Azimuth is marked N or S according to the Latitude, and E or W according to the Hour- Angle. The Position- Angle is found by reversing the process of the preceding Rule ; that is to say, by operating with the difference instead of the sum, and the sum instead of the difference, in the two Cases. 132 TABLE XXXI. Time-Azimuths: Direct and Limiting Values. Lat. Hour-Angle West or East of the Meridian. 11 12 On Hori- zon. O h l h 2" 3 h 4 h 5 h 6 h 7" 8 h 9 h 10 Az. H.A. o 10 20 30 40 ! 5O 60 70 o 10 ! 20 30 40 50 60 70 80 O 10 20 30 40 50 60 ! 70 80 Polar Distance 30, or Declination 60 of same name as the Latitude. 0.0 o.o o.o 0.0 0.0 0.0 Indet. 180.0 180.0 8 i 9.6 H-3 14-3 20.1 34-7 140.6 158.3 o 16.1 17.8 20.6 25-1 33- 76.9 H3-9 137-8 o 22.2 24.1 27.0 3L7 39-0 51.0 7-3 95-8 119.2 o 26.6 28.2 30.7 34.7 40.7 49.9 63.4 82.0 102.5 o 29.1 30.2 32.1 35-2 3 2i 46.6 5 6 -4 70.1 87-3 30.0 30.4 31.6 33-7 37-o 41.9 49.1 59-4 73-3 29.1 28.9 29.4 30.7 32.9 3 6 4 41.6 49.2 60. i o 25.8 25.7 26.3 27.7 30.1 33-7 39-2 47-5 o 20.7 20.9 21.7 23.1 25.6 29.4 35-3 o 14.6 14.5 14.9 15.7 17.2 19.6 23.4 7-5 74 7.6 7-9 8.6 9-8 11.7 o.o 0.0 O.O O.O 0.0 o.o o.o o 30.0 28.4 22.8 O.O h in 6 o 7" 836 ] 12 O Polar Distance 40 D , or Declination 50 of same name. o.o o.o o.o o.o o.o Iiidet. 180.0 180.0 1 80.0 o 12.2 14.4 18.1 25.2 41.6 84.2 132.9 152.6 160.8 22.8 26.0 Si 1% 107.1 129.1 142.3 30-7 46.2 57-i 72.4 9L3 1 1 O.O 124.7 36.0 38.5 42.4 47-9 79-4 94.1 108.3 o 39-0 40.6 43-i 47-o 52-3 59.6 69.0 80.4 92.8 o 4O.O 40.4 41,7 44.1 47.6 5 2.6 59-2 67.8 78.3 39-o 38.4 3 8.6 39-7 41.8 45-o 49-7 56.0 64.4 33-8 34-o 35-i 37-o 40.1 44.6 Si-i 270 27-3 28.4 30-3 33-4 38.0 o 18.8 1 8.8 19-3 20.4 22.3 25-3 o 9-7 9.6 9.8 IO.2 II. I 12.6 o.o 0.0 O.O O.O o.o o.o o 40.0 38.9 354 27.8 o.o h m 6 o 649 i 743 850 I2~0 Polar Distance 50, or Declination 4O of same name. 0.0 0.0 o.o 0.0 Indet. 1 80.0 1 80.0 180.0 180.0 I7.I 21. 4 29-5 46.7 8 5 .2 127.8 148.2 157-4 162.2 30.8 36.5 45-4 103.9 123.5 136.5 144.8 40.1 45-2 52-3 62.2 75-1 90.2 105-3 II8.I 127.9 45-9 49-5 54-5 61.2 69.6 79-8 90.9 101.9 1 1 1.8 49o 5i.o K |j 87.4 96.4 o 5O.O 5-4 51-7 54-o 57-3 61.7 67.2 74-o 81.7 49.0 48.0 47-8 48.4 50.0 52.6 56.3 61.2 67.5 o 42.1 41-6 41.9 43-2 454 48.8 53-6 32.8 33-2 344 36.6 40.0 22^6 22.6 23.2 24.4 26.6 o 1 1.6 II.4 1 1.6 12.2 13-3 o O.O O.O o.o o.o o.o o 50.0 49-2 46.8 42.1 32-9 O.O h m 6 o 634 711 756 859 12 Polar Distance 60, or Declination 30 of same name. 10 20 30 40 5O 60 70 j 80 10 20 30 40 50 60 70 80 o 0.0 0.0 0.0 Ind't. iSo.O 1 80.0 ISO.O 1 80.0 1 80.0 24.1 3 2.8 5O.6 86.2 124.6 144.9 154-7 1 60.0 163.1 40.9 50.1 63.8 82.4 102.9 120.3 132.7 140.9 146.4 50.8 57.8 67.0 78.3 91.0 103.6 114.6 123.4 130.1 o 60.9 66.8 73-9 82.1 90.8 99-5 107-5 114.4 59-i 64.8 69.0 74.1 79-8 86.2 92.7 99.1 , 60.0 60.4 61.5 63-4 66.1 69.6 73-9 78.8 84.3 59-i 57-6 56.8 56.9 57-8 59-5 62.0 65.5 69.8 49.1 48.6 49.0 50.2 52.4 55-6 o 37-8 38.1 394 41.6 o 25.8 25-7 26.3 27-7 o T 3-! 12.9 13.2 13-8 o o.o O.O 0.0 o.o o 6O.O 59-5 57-9 54-7 49-3 38.9 o.o /' in 6 o 623 649 i 7i8 756 854 12 Polar Distance 70, or Declination 20 of same name. o 0.0 0.0 Indet. iSo.O 180.0 iSo.O iSo.O iSo.O iSo.O 354 53-6 874 123.0 142.9 152.9 .158.4 161.7 163.8 o 53-9 & I0 3-3 119.1 130.7 138.6 144.0 147.6 62.7 71.6 81.9 93- ' 104.0 II3-5 121.3 127.4 131.8 6 h 72.6 78.8 85-7 92.8 99-8 106.2 111.7 116.4 o 69.4 72.0 75-3 79.1 834 87.9 92-5 97.0 IOI.2 o 70.0 70.3 71.1 72.5 744 76.8 79-7 82.9 86.4 o 69.4 67.3 66.0 65.3 65-3 65.9 67.2 6 9 .2 71.8 o 55-3 54-5 54-6 55-5 57-3 o 41.7 41.9 43-o o 28.2 28.1 28.6 o J 4'3 I4.I 14-3 o o.o O.O O.O 7O.O 69.7 68.6 66.7 63-5 57.9 46.8 o.o h in 6 * l l\ 6 30 ; 649 711 j 743 836 12 O The Azimuths in smaller figures answer to actual depressions of the object below the True Horizon, and are used for differences with the adjacent Azimuths. TABLE XXXI. 133 Time-Azimuths: Direct and Limiting Values. Lat. Hour-Angle West or East of the Meridian. On Hori- zon. 01 1" 2 h 3 h 4 h 5 h 6" 7 h 8" 9 h 10 h ll h 12 h Az. H.A. Pol. Dist. 80, or Declination 10 of same name as the Latitude. O 1O i 20 30 ! 4O 50 60 70 80 o O 10 20 30 4O 50 6O 70 8O o 10 20 3O 40 50 6O 1 70 80 o 10 20 30 ! 40 i 50 60 70 o 10 20 30 40 50 60 O 10 20 30 40 j 50 o 0.0 Indet. 180.0 l8o.O l8o.O l8o.O l8o.O l8o.O l8o.O 8 122.5 141.9 152.0 157-6 160.9 163.0 164.3 o 70.6 87.3 104.6 "9-3 130.1 137-7 142.9 146.4 148.7 o 76.0 85.9 96.2 105.9 114-3 I2I.2 126.6 I30-5 J 33-3 785 84-3 90.4 96.4 IO2.2 !07-3 in. 7 II5-3 118.1 o 79-7 82.4 85.4 88.6 91.9 95- 98.0 100.7 103.1 80.0 80.2 80.6 81.3 82.3 83-5 85.0 86.6 88.3 o 79-7 77.2 75-3 73-7 72.7 72.1 72.1 7 2.6 73-5 o 58.9 58.5 58.9 o 49-3 44.2 o 29.8 29-5 o 15-0 14.8 o o.o 0.0 o ft in 80.0 6 o 79.8 6 7 79-3 6 1 S 78.4 i 6 23 i 76.9 6 34 i 74.3 649 j 69.7 711 ! 59-5 7 56 I O.o ; 12 o Pol. Dist. 90, or Declination 0. Indet. 180.0 1 8o.O 180.0 180.0 180.0 180.0 180.0 180.0 o 9O.O 123.0 I4I.9 I5I.8 1574 160.7 162.8 I6 4 .I 164.8 o 90.0 106.7 120.6 130.9 138.0 143.0 146.3 148.4 149.6 o 90.0 99-9 108.9 116.6 122.7 127-5 130.9 133.2 134.6 90.0 95-7 IOI.2 I06.I IIO.4 "3-9 116.6 118.5 119.6 90.0 92.7 95.2 97.6 99-8 101.6 103.1 104.1 104.8 90.0 90.0 90.0 90.0 90.0 90.0 90.0 90.0 90.0 o 90.0 87.3 84.8 82.4 80.2 78.4 76.9 75-9 75.2 9O.O 90.0 9O.O 9O.O 9O.O 9O.O 9O.O 9O.O 9O.O h m \ 6 6 6 o 6 o 6 o 6 o 6 o 6 o 6 o Pol. Dist. 100, or Declination 10 of contrary name. 180.0 180.0 180.0 1 80.0 180.0 180.0 1 80.0 180.0 180.0 124.3 142.8 152-5 157.8 161.1 163.1 164.4 165.0 165.2 109.4 123.0 132.7 139-5 144.1 147.2 149.2 150.2 *5-5 o 104.0 112.7 I2O.O 125.6 129.8 132.8 134-7 135-7 135-8 101.5 106.7 111.3 114.9 117.8 119.8 121. 1 I2I-5 121. 1 o 100.3 102.8 104.8 106.3 107.3 107.8 107.9 107.5 IOO.O 99.8 99-4 98.7 97-7 96.5 o 100.3 - o IOO.O 100.2 IOO-7 101.6 103.1 105.7 110.3 120.5 180.0 k m 6 o 553 545 537 526 5" 449 4 4 Pol. Dist. 110, or Declination 20 of contrary name. 180.0 180.0 180.0 180.0 180.0 180.0 1 80.0 180.0 o 144.6 153-8 159.0 162.0 164.0 165.1 165.7 165.9 o I26.O 135-5 I4I.9 146.2 I49.I 150.9 I5I.8 I5L9 o "7-3 124.2 129.5 133-4 136.1 137.6 138-3 138.1 1 12.8 117.2 120.6 123.1 124.7 125.5 125.4 o 110.6 112.7 114.0 114.7 114.7 114.1 IIO.O 109.7 108.9 I0 7-5 1 10.6 1 1 0.0 110.3 111.4 II3-3 116.5 1 22. 1 133-2 I So.O h m 6 o 545 530 5" 449 417 324 o o ! Pol. Dist. 120, or Declination 30 of contrary name. i8oo 1 80.0 180.0 180.0 180.0 1 80.0 180.0 '$ 160.6 '63.5 165.2 166.3 166.9 167.0 I39-I 145.2 149.2 I5I.8 1534 154.2 154-3 129.2 134.4 138.0 140.4 141.7 142.2 141.9 123.7 I27.I 129.5 130.9 1314 i 3 i.i 120.9 122.4 123.2 123.1 122.3 120.0 119.6 118.5 120.9 120.0 120.5 1 22. 1 125.3 130.7 I4I.I l8o.O h m 6 o 1 537 5" i 442 | 4 4 3 6 o o Pol. Dist. 130, or Declination 40 of contrary name. h TO~j 6 o ! 526 449 4 4 3 * o o 180.0 180.0 180.0 180.0 180.0 180.0 162.9 165.4 167.0 167.9 168.4 168.6 149.2 152.9 155-3 156-7 157-4 157-4 139.9 143-3 145.6 146.8 147.2 146.8 134-1 136-5 137-9 138.4 138.1 o 131.0 132.0 132.2 131.6 130.0 129.6 o 131.0 o I3O.O 130.8 133-2 *37-9 147.9 180.0 134 TABLE XXXII. Position-Angles for Direct and Limiting Time- Azimuths. Lat. Hour- Angle West or East of the Meridian. Lat. O h l h 2 h 3 h 4 h 5 h 6 h 7 h 8 h 9 h BO 11" 12 h o 10 20 30 40 50 60 70 80 o O 10 20 3O 40 50 60 70 8O 10 20 3O 40 50 60 7O 80 o 10 20 30 40 50 60 1 70 80 o 10 20 30 40 50 60 70 80 PoL Dist. 3O, or Declination 60 of same name as the Latitude. o 10 2O 30 40 50 GO 7O i 80 o 10 20 30 40 5O 60 70 8O i8o!o 180.0 180.0 180.0 180.0 180.0 Indet. O.O 0.0 162.8 160.9 158.4 1547 148.2 132.9 83-5 25-7 7-4 146.3 142.9 138-7 132-7 123-5 107.1 76.9 38.7 13-5 I3 2? 126.6 121. 2 II4.6 105.3 91-3 70-3 42.9 17-6 n66 111.7 106.2 99-5 80.9 79-4 63-4 42.6 19.8 103.1 98.0 92.5 86.2 78.6 69.0 5 6 -4 40.0 20.3 90.0 85.0 79-7 73-9 67.2 59-2 49.1 36.1 19.4 76.9 71.1 67.2 62.0 56.3 49-7 41.6 31.2 17-5 o 59-o 54-6 50.2 45-4 40.1 33-7 25-6 14-8 41.7 38.1 34-4 30-3 25.6 19.6 n.6 28.2 25-7 23.2 20.4 17.2 *3-3 7-9 14.3 12.9 11.6 IO.2 8.6 6.7 4.0 o O.O 0.0 O.O O.O 0.0 O.O O.O Pol. Dist. 40, or Declination 50 of same name. i8oo 180.0 180.0 180.0 180.0 Indet. O.O O.O 0.0 o 160.7 157-6 152.9 144.9 127.8 84.2 34-7 14.2 5-i 143.0 137-7 130.7 120.3 103.9 78.4 48.0 24.4 9-5 o 127.4 I2I.2 "3-5 103.6 90.2 72.4 51.0 30.0 12.8 o "3-9 107.3 99.8 90.8 79-8 66.1 49-9 32.1 14.9 o 101.6 95-o 87.9 79-9 70-5 59-6 46.6 31-7 15.6 o 90.0 83.5 76.9 69.6 61.7 52.6 41.9 29-5 15-3 o 78.4 72.1 65-9 59-5 52-6 45- 36-4 26.2 14.1 o 54-5 48.9 43-2 37-o 30.1 22.0 12. 1 378 2 22.1 17.0 9.6 25.8 22.6 19-3 I5 i u.6 6.6 *3-i II.4 9 .8 8.0 5-9 3-4 o 0.0 0.0 O.O 0.0 0.0 O.O Pol. Dist. 50, or Declination 40 of same name. I 10 20 3O 40 ! 50 60 7O 80 i8oo 180.0 180.0 180.0 Indet 0.0 O.O 0.0 O.O o 157-4 152.0 142.9 124.6 85.2 41.6 20.1 9-9 4.0 138 130.1 119.1 102.9 80.2 54-6 33-o 17.9 7-5 o 122.8 II4-3 104.0 9I.O 75-i 57-i 39-o 23-2 10.3 o 110.4 IO2.2 92.8 82.1 69.6 55-7 40.7 25.9 I2.I 99.8 91.9 834 $ 1 26.5 13.0 o 9O.O 82.3 74-4 66.1 57-3 47.6 37-o 25-4 13.0 o 80.2 72.7 65.3 57.8 50.0 41.8 32-9 23.0 12.1 o 55-3 48.6 41.9 35-o 27.7 19.6 10.5 32-8 27-3 21.6 8.4 o 22.6 18.8 14.9 10.6 5-8 o it 6 9-6 7.6 5-4 3-o O.O O.O O.O O.O O.O Pol. Dist. 60, or Declination 30 of same name. 10 20 3O 40 50 OO 7O 80 \ o O 10 20 30 40 50 6O 70 80 rt o 1 80.0 180.0 1 80.0 Indet. 0.0 O.O O.O 0.0 0.0 151.8 141.9 123.0 86.2 46.7 25.2 14-3 7-8 3-3 130.9 "9-3 103-3 82.4 59-6 39-8 25.1 14.4 6.4 n66 105.9 93-i 78.3 62.2 46.2 3i-7 19.2 8.8 o 106.1 96.4 85.7 73-9 61.2 48.0 34-7 22.1 10.5 o 7*1 69.0 58.3 46.9 35-2 23-2 11.4 o 9O.O 8l. 3 72.5 634 54-0 44.1 33-7 22.8 "5 8,! 4 73-7 65.3 56.9 48.4 39-7 30.7 21. 1 10.8 o 49.1 41-5 34-o 26.3 18.2 9-5 o 27.0 20.7 14-5 7.6 o 18.8 14-5 10. 1 5-4 o 9-7 7-4 5-2 2.7 o O.O O.O 0.0 0.0 Pol. Dist. 70, or Declination 20 of same name. i8oo 1 80.0 Indet. 0.0 O.O O.O 0.0 0.0 0.0 141.9 122.5 87.4 50.6 33 tj 3-o 120.7 104.6 84.8 63.8 45-4 31-3 20.6 12.3 5.7 io89 96.1 81.9 67.0 III 27.0 1 6.8 7-9 o IOI.2 90.4 78.8 66.8 54-5 42.4 30-7 19.8 9-5 o 95-2 85-4 S3 54-i 43-i 32.1 21.2 10.5 o 90.0 80.6 6 7 ;.-5 Si 31.6 21.2 10.6 8 4 8 753 66.0 56.8 47-7 38.6 29.4 19.9 10. 1 42.1 33-8 25-7 17-5 8. 9 20 7 14.1 7-2 I4 6 9-8 5-i o 7-5 8 O.O O.O O.O Position- Angles in smaller figures correspond to depressions of the object below the True Horizon. I TABLE XXXII. 135 Position-Angles for Direct and Limiting Time-Azimuths. Lat. Hour- Angle West o r Eas 6 h b of the Meridian. T >* ' O h l h 2 h 3 h 4 h -1 7 h 8 h 9 h 10'' ll b 12 h j-iat. o O 10 20 30 4O 50 60 70 80 o 10 20 30 40 50 60 70 80 10 20 30 40 50 60 7O 80 o 10 20 30 40 50 60 70 10 20 30 40 50 60 10 20 30 40 5O Pol. Dist. 80, or Declination 10 of same name as the Latitude. o ; 1O 20 ! 3O 40 i 50 60 70 ! 80 ! i8o?o Indet. O.O O.O O.O O.O O.O O.O O.O 123.0 88.7 53-6 3 2 -9 21.5 14.4 9-5 5-8 2.7 o 106.7 87.3 67.4 50.1 36.5 26.0 17.8 II. I 5-3 99-8 85-9 71.6 57.8 45-2 33-9 24.1 '5-3 7-4 o 95-7 84-3 72.6 60.9 49-5 38.5 28.2 18.3 8.9 o 92.7 82.4 72.0 61.5 51.0 40.6 30.2 19.9 9-9 o 90.0 80.2 70.3 60.4 50-4 40.4 3-4 20.3 IO.2 o 87.3 77-2 673 57-6 48.0 38-4 28.9 194 9-7 25.8 17.1 8.7 140 7-i o 9-9 5 .0 2.6 0.0 O.O Pol. Dist. 90, or Declination 0. o O 10 | 20 30 i 40 50 60 ,' 70 8O o 10 20 3O 40 50 60 70 80 i Indet. 0.0 O.O 0.0 O.O 0.0 O.O O.O 0.0 90.0 55-7 35-4 24.1 17.1 12.2 8. 5 5-4 2.6 o 90.0 7O.6 54-0 40.9 30.8 22.7 16.1 10.3 5-o o 90.0 76.0 62.8 50.8 40.1 30.6 22.2 14.4 7-i o 90.0 78.5 67.2 56.3 45-9 36.0 26.5 17-5 8.7 o 90.0 79-7 69.4 59-1 49.0 39-o 29.1 19.4 9-7 o 9O.O 80.0 70.0 6o.O 5O.O 4O.O 30.0 2O.O 1 0.0 90 o 79-7 69.4 59- 1 49.0 39-o 29.1 19.4 9-7 i | Pol. Dist. 100, or Declination 10 of contrary name. o O.O O.O 0.0 O.O 0.0 O.O O.O O.O O.O o 57-o 37-i 26.1 19-3 14.6 10.9 7.8 5 J o 73-3 57-o 44-5 34-8 27.1 20.7 I5-* 9-9 5-o 80?! 67.2 55-7 45- 6 36.7 28.6 21. 1 14.0 7-i 8*3 m 5 2.8 43-4 34-5 25.8 17.2 8.7 87-3 77-2 67-3 57-6 47-9 38.5 28 9 19-3 90.0 80. i 70-3 60.4 50-4 40.4 o_ 92.7 Pol. Dist. 110, or Declination 20 of contrary name. o 0.0 O.O O.O O.O 0.0 O.O O.O 0.0 o 38.1 27-5 21.0 I6. 5 I 3 .0 10. 1 7-5 5- 1 59-4 47.2 38-1 30.8 24.7 19.4 14.6 9-9 71.1 60.0 5-4 42.0 34-4 27.4 20.7 14 i 78.8 68.8 59-4 50-5 42.0 33-8 2 5-7 84.8 75-3 66.0 56.8 47-8 38-6 90.0 80.6 71.1 61.5 95-2 o ! 10 ! 20 i 30 i 40 ! 50 6O 70 ! o i o 10 ! 20 ! 30 I 40 1 50 6O Pol. Dist. 120, or Declination 30 of contrary name. o O.O 0.0 0.0 O.O O.O O.O 0.0 2& 22.1 I7. 9 14.7 i2.i 9-7 7-4 o 49.1 40-5 %.l S3 14-5 o 63.4 54-4 46.6 39-6 33-2 27.0 20.9 o 73.9 65.1 56.9 49.1 41.6 34-o 82. 4 73-5 65-3 56.9 48.4 o 90.0 81.3 72.5 o 97.6 Pol. Dist. 130, or Declination 40 of contrary name. o 0.0 O.O 0.0 0.0 0.0 0.0 o 22.6 18.9 1 6.0 13-7 n.6 9.6 o 41.9 35-9 3;9 26.6 22.6 18.8 57-2 50.2 43-9 38.2 32-8 27.4 o 69.6 62.2 48.6 41.9 80.2 72.7 65-3 578 o 90.0 82.3 o 99.8 o 10 20 30 40 5O 136 TABLE XXXIII Altitudes for Direct and Limiting Time-Azimuths. Lat. ! 10 20 30 40 5O 60 70 8O 10 20 30 40 ' 50 60 i 70 80 10 20 ! 3O 40 50 6O 70 80 o 10 20 I 30 140 50 60 i 70 80 o 10 20 30 40 50 6O 70 80 Hour-Angle West or East of the Meridian. Lat. 6 b 7 h 8 h 9 h 10 h ll h 8-2 PoL Dist. 30, or Declination 60 of same name as the Latitude. o O 10 20 30 40 ! 50 60 70 80 O 10 20 30 40 50 i 6O 70 ! 8O 30.0 40.0 50.0 60.0 70.0 80.0 90.0 80.0 70.0 28.9 38.7 48.6 58.4 67.9 76.9 82.3 78.2 69-5 25-5 35-0 44-6 53-9 62.7 70.4 75-i m 20.7 30.0 38.9 47.5 55.8 62.9 68.0 69.1 66.0 14.4 23.4 32.1 40.5 48.4 55-5 6 4 !i 637 ,? 24.7 33-o 40.9 48.3 54-6 59-1 61.1 0.0 8-7 17-3 25-6 33-8 41.6 48.6 54-5 58.5 7-3 1.2 1 0.0 I8. 7 27.2 35-5 43-3 50.3 56.1 o 5-5 3-3 12.4 21.4 30.2 38.7 46.7 54-0 o i-7 16^6 2 5 .8 34-9 43-8 52.3 o 6.6 3-2 I3-I 22.7 33 5i-i o 8.9 0.9 10.8 20.7 30.6 40.4 50.2 o IO.O O.O IO.O 2O.O 30.0 4O.O 5O.O Pol. Dist. 40 3 , or Declination 50 of same name. o 40.0 50.0 60.0 70.0 80.0 90.0 80.0 70.0 60.0 o 3 8.2 48.1 57-7 67.0 SI 76.8 68.7 59-6 33-8 42.9 51.8 59-9 66.8 70.8 70.4 65.5 58.3 o 27.1 35-5 43-6 5-9 57-2 61.5 62.9 61.1 56.4 i8.8 26.7 34-3 414 47-6 52.5 55-5 56.1 54-1 o 9-7 17.2 24.6 31.8 38.3 43-9 48-3 51.0 51.6 o O.O 7.7 15.2 22.5 29-5 35-9 41-5 46.1 49.0 o 9-7 1.8 6.0 13.8 21.4 28.7 35-5 41-5 46.5 5-9 14-3 22.2 30.2 37-6 44-3 o. 7 8.1 17.1 25.8 34-3 42.6 5-8 3-9 13-4 22.7 32.0 41.2 8.6 0.9 10.8 20.7 30.5 40.3 IO.O 0.0 IO.O 2O.O 30.0 4O.O Pol. Dist. 50, or Declination 40 of same name. I 10 20 30 40 50! 60 70 8O 50.0 60.0 70.0 80.0 90.0 80.0 70.0 60.0 50.0 47-7 57-2 66.2 74-2 78.5 75.6 67-9 59-o 49-6 o 41.5 49.9 m 67.1 66.7 62.7 56.2 48.4 32.8 40.2 46.8 52.2 55-9 57-2 5 2l 46.6 o 22.5 29-3 35-4 40.8 45 - 47-6 48.4 47.3 44-4 o 11.4 17.9 24.0 29-5 34-4 38.3 41.0 42.2 41.9 o 0.0 6-4 12.7 I8. 7 24-4 29-5 33-8 3M 39-3 11.4 4.8 2.1 8.8 15.2 21.4 27-3 32-4 36.8 o 8.0 o.7 6. 9 14-3 21.4 28.2 34-5 o O.I 8.2 16.6 24.8 32.6 o 5-8 3-7 '3-i 22.2 31.2 o 8.8 0.6 10.6 20.5 3-4 o IO.O O.O IO.O 2O.O 30.0 Pol. Dist. 60, or Declination 30 of same name. o; 10 20 30 40 50 6O 70 80 o 10 20 30 40 50 6O 70 80 o 60.0 70.0 80.0 70.0 60.0 50.0 40.0 56.8 65.6 73-2 77.0 74.2 67.0 58.3 49.1 39-6 48^6 55.6 61.1 64.1 63.6 59-9 53-9 46.6 38.5 37^3 43-6 48.3 5i-3 52-2 50-9 47-7 42.8 36.8 o 25.6 3-9 $ 40.8 41.4 40-5 38.2 34-6 o 12.9 17.9 22.4 26.3 29.5 31.8 33-o 33-i 32-1 o O.O 5-o 9-9 14-5 1 8.8 22.5 25-7 28.0 29-5 o 12.9 7.8 2-5 3- 8.6 13-8 18.7 23-2 20.9 o 7.0 0.6 5-9 12.4 18.8 24.6 o 0.8 7-2 I5-I 22.5 6.0 3-o 12.4 21.4 9.0 0.5 10.3 20. 2 o IO.O O.O IO.O 20.0 Pol. Dist. 70, or Declination 20 of same name. 70.0 80.0 90.0 80.0 70.0 60.0 50.0 40.0 30.0 6s.-, 72.4 75-9 73-2 66.2 57-7 48.6 39-2 29.6 54-5 l?i 61.8 61.1 57-5 5i-7 $1 28.6 41^6 45.6 47-8 48-3 46.8 43-6 3*9 33-3 26.9 28.0 31-5 33-9 35-3 35-4 34-3 1:1 24-7 o 14.1 17.4 20.2 22.4 24.0 24-7 24.7 23-9 22.3 o 0.0 3-3 6.7 9.9 12.7 15-2 17.2 18.7 19.7 14.1 10.4 6.4 2.2 2.0 6.1 10. 1 13.8 17.1 o 8.1 2.4 3-5 9.2 14.8 o 1.8 ,ij o 6.2 2.5 "3 o 9.2 0.4 10.3 o IO.O 0.0 IO.O The Altitudes in smaller figures are depressions below the True Horizon ; that is, negative Altitudes. TABLE XXXIII. 137 Altitudes for Direct and Limiting Time-Azimuths. ^ Lat. Hour- Angle West or East of the Meridian. Lat. O h l h 2 h 3 h 4h 5 h 6 h 7 h 8* 9 h lO h ll h 12 h Pol. Dist. 80, or Declination 10 of same name as the Latitude. o O 10 20 30 40 50 60 70 80 c 10 2O 30 40 50 60 70 80 o 10 20 30 50 6O 70 o 80.0 90.0 80.0 70.0 60.0 50.0 40.0 30.0 20.0 72.1 75-2 Hi 57-2 48.1 38.8 29.2 19.6 58-5 60.5 594 55- 6 49.9 42.9 35-2 27.1 18.6 o 44.1 45.7 45 - 6 43-6 40.2 35-5 29.9 23-7 17.0 o 29-5 31.0 31.5 3-9 29-3 26.7 234 19.4 14.8 14.8 16.3 17.4 17.9 17.9 '7-3 16.1 H-5 12.4 O.O 1.8 3-3 4.9 6.4 7.6 8.6 9-9 9.8 14.8 12.8 10.4 7-7 4-9 1.8 -3 4.2 7-3 o 5-5 O.O 4.9 5-8 3-o 7- 1 2.0 o 9-4 0.2 IO.O 0.0 Pol. Dist. 90, or Declination 0. o 10 20 30 40 50 60 7O 80 o 90.0 80.0 70.0 60.0 5 40.0 30.0 20.0 10.0 75-o 72.0 65.2 56.8 47-7 384 28.9 19-3 9-7 , 60.0 58.5 54-5 48.6 41.6 33-8 25-7 17.2 8.7 45- 44.1 41.6 37-7 32.8 27.0 20.7 14.0 7.0 o 30.0 29-5 28.0 25.7 22.5 18.7 14-5 9-9 5-i o 15.0 14.8 14.1 12.9 11.5 9.6 7-5 5-2 2.7 o O.O O.O 0.0 O.O 0.0 O.O 0.0 0.0 O.O 15-0 14.8 14.1 12.9 "5 9.6 7-5 5-2 2.7 Pol. Dist. 100, or Declination 10 of contrary name. o 10 20 30 4O 50 I 60 7O 80 o 10 20 30 40 50 60 70 O 10 20 30 40 50 60 o O 10 20 30 40 50 80.0 70.0 60.0 50.0 40.0 30.0 2O.O 10.0 O.O o 72.0 65.0 56.5 47-5 38.1 28.6 18.9 9-3 O.I 58^5 54-1 47-9 40.7 32.8 24-5 1 6.0 74 1.2 O 44.1 41.0 36.5 3I-I 2J.9 18.3 II.4 4-3 2.9 29.5 27.0 23-7 19.8 154 10.5 54 -3 4.9 14.8 12.8 10.4 p 2-3 I.I 4.4 0.0 w 3-4 5-o 6.4 7.6 o 14.8 o O 10 20 30 40 5O 60 70 80 Pol. Dist. 110, or Declination 20 of contrary name. 70.0 60.0 50.0 40.0 30.0 2O.O 1 0.0 O.O o 65.2 56.5 474 38.0 28.4 18.8 9.1 O.2 O 54-5 47-9 40.4 32.2 23.8 i5-i 6.4 2.4 9. 41.6 3^5 30-5 23-9 16.8 94 2.O 5-4 280 23.7 18.9 13.6 8.1 2.4 3-4 14.1 10.4 6-5 2.4 i. 9 6.1 0.0 3-9 6.7 14.1 O 10 20 30 40 50 60 70 Pol. Dist. 120, or Declination 30 of contrary name. 10 20 30 40 50 60 o 10 20 30 40 50 , o oo.o 50.0 40.0 30.0 20.0 IO.O O.O 568 474 38.0 28.3 18.6 8.8 0.5 o 48.6 40.7 32.2 23-5 14.6 5.8 3-2 o 37-8 3i-i 23-9 16.2 8.5 0.9 7-3 o 25.6 19.8 I 3 .6 7-i 0.4 6.0 o 12.9 7.7 2.3 3-2 8.3 o 0.0 5-i 9-9 o 129 Pol. Dist. 130, or Declination 40 of contrary name. 50.0 40.0 30.0 2O.O IO.O 0.0 47-7 38.1 28.3 18.7 8.8 0.6 41-5 32.8 23.8 14.6 5-5 3-6 o 32.8 25.0 1 6.8 8.6 . 0.5 8.3 o 22.5 'fc 0.4 6. 9 11.4 4.8 1.9 8.2 0.0 6.4 11.4 138 TABLE XXXIV. Error of the Time-Azimuth for an Error of 1 in the Hour-Angle. Hour-Angle l h or ll h . Az. Position- Angle. Az. 90 85 80 75 70 65 3 60 5-5 50 45 40 30 20= 10 0=> o O 5 10 15 20 25 30 3.) 40 45 M ! 60 | 70 ! SO 9O O 7 o o o o O 7 I I 2 2. 3 3 3 4 4 4 5 5 5 0' i 2 3 3 4 I 6 7 8 9 9 10 10 o' I 3 4 5 6 7 9 10 ii 12 13 14 15 J 5 o' 2 3 5 7 8 10 ii 13 H '5 17 19 20 20 o' 2 6 8 10 12 14 16 17 19 21 23 24 25 0' 3 5 7 10 12 H 17 19 2O 22 25 3 29 O 7 i 9 ii H 17 ^9 21 23 26 29 31 33 33 0' 10 13 16 19 21 24 26 28 32 35 37 37 o 7 4 7 ii 14 17 20 8 29 31 p 40 41 o' i 12 15 19 22 26 28 31 34 39 42 44 44 0' 4 9 13 17 21 25 29 32 35 39 43 47 49 50 0' 5 9 H 19 23 27 31 ? 42 47 5i 54 5,5 o' 5 10 15 20 11 33 37 40 44 49 ? 57 O 7 5 IO 15 20 25 29 33 37 4i 44 5 55 i 10 175 17O , 165 160 155 150 j 145 140 135 130 12O 110 100 90 Az. 90 95 100 105! 110 115 120 125 130 1 135 140 150 160 170 180 Az. Position- Angle. 1 Az. Hour-Angle 2 h or 10 h . Position- Angle. Az. 90 85 80 75 7O 65 0' I 2 3 4 6 I 9 10 ii 12 12 y 60 O 7 3 4 5 6 7 9 IO ii 12 '3 14 15 15 55 50 45 40 30 20 10 0? 5 10 15 20 25 30 35 40 4> 50 60 70 80 90 0' o o o o o o' o 2 2 2 2 3 3 3 0' o I I 2 2 3 3 3 4 4 5 5 5 5 o' I 2 3 3 4 4 I 6 7 8 0' 2 3 . 3 4 7 7 8 9 10 10 IO o' i 3 4 6 7 9 10 ii 12 *3 !l 17 *7 0' 2 3 5 7 8 10 ii 12 14 15 19 19 o 7 2 I 7 9 ii 12 H 15 16 18 20 21 21 0' 2 4 6 8 IO 12 !3 15 16 18 20 22 23 23 140 o' 2 5 7 9 ii 13 15 17 18 20 22 24 26 2_6_ 150 0' 3 5 7 IO 12 H 16 18 20 22 2 4 26 28 28 160 o' 3 8 IO 12 15 17 19 21 s 28 29 29 o' i IO 13 15 17 19 21 2 3 26 28 29 >o 1*0 175 i 170 165 160 155 150 145 140 135 13O 120 i a > j 1OO i 90 i Az. 90 95 1OO 105 110 115 120 125 130 135 170 J 180 Az. Position- Angle. Hour- Angle 3 U or 9 h . Az. Position- Angle. Az. | 9 170 165 160 155 150 140 145 135 13O 120 110 1OO 9O i Az. 90 95 100 105 110 1 15 j 120 125- 130 135 14O 150 160J17O 180 Az l Position- Angle. TABLE XXXIV. 139 Error of the Time-Azimuth for an Error of l m in the Hour-Angle. Hour- Angle 4 h or 8 h . Az. Position- Angle. Az. 90 85 80 75 70 65 60 55 50 45 40 30 20 10 o 5 10 15 20 25 30 35 4O 45 50 60 70 90 Az. o 1 o o o o o o o o o o o' o o I 2 o' o o I I I I 2 2 2 2 3 3 3 3 o' I I I 2 2 3 3 3 3 4 4 4 5 o' I I I 2 2 3 3 4 4 4 6 6 6 0' I I 2 2 3 4 4 5 5 6 6 7 7 7 o' i i 2 3 4 4 6 7 8 9 o' 2 3 3 4 I 6 7 8 9 9 10 IO o' I 2 3 4 I 6 8 9 10 10 ii ii o' i 2 3 4 i 9 9 ii ii 12 12 o' 2 3 4 6 8 9 9 10 ii 12 '3 J 3 0' i 3 4 5 6 7 9 10 ii ii 13 14 15 15 o' i 3 4 6 9 10 ii 12 H 15 IO 16 o' I 3 I I 10 ii 12 13 11 17 17 o' 2 3 7 9 10 ii 12 13 16 17 J 7 180 175 170 165 160 155 150 145 14O 135 130 12O 110 100 90 90 95 100 105 11O 115 12O 125 130 135 140 150 16O 17O 180 Az. Position- Angle. Hour-Angle 5 h or 7 h . Az. T 5 10 15 20 25 30 35 40 45 50 60 70 90 Position- Angle. Az. 90 85 80 75 70 65 60 55 50 45 4O 30 2

40 45 50 60 70 80 i 90 O.O o o o.o o o o o.o o o o o 0.0 o I I 2 0.2 2 3 3 4 0.4 4 5 5 g o 0.0 I t 3 3 0.4 6 6 7 0.8 9 9 I.O o 0.0 I 3 4 5 0.6 7 9 I.O i.i 3 4 5 5 O.O 2 3 5 7 0.8 I.O i 3 4 i-5 I 9 2.0 o 0.0 2 4 6 8 I.O 2 I 7 1.9 2.1 3 4 4 O.O 2 S 7 I.O 1.2 4 6 8 2.0 2.2 5 8 9 o O.O I 9 i.i 1.4 6 9 2.1 3 2.5 3-i 2 3 o O.O 3 6 I.O 3 1.6 8 2.1 4 6 2.8 3-2 I 7 o 0.0 4 7 I.O 4 i-7 2.0 9 3-i 4.0 o 0.0 \ I.I 5 1.9 2.2 3- 1 fj 4-1 3 4 o O.O 4 9 i-3 7 2.1 3-2 5 3-8 4-3 7 9 5-o o 0.0 5 9 1.4 8 2-3 7 3-i 4.1 7 5-i 3 4 o 0.0 5 I.O 5 9 2.4 8 3 6 4.0 4-3 9 5 i 6 o 0.0 5 I.O 5 2.O 2.4 9 3-3 7 4.1 4-4 5-o 4 6 7 180 175 170 165 160 155 150 145 140 135 13O 120 110 1OO 90 Az. 90 95 100 105 110 115 120 125 130 135 140 150 160 170 180 Az. Position- Angle. Hour-Angle O h 20 m or ll h 40 m . Az. Position- Angle. Az. 90 85 80 75 70 65 60 55 50 45 40 30 20 10 O 3 O 5 10 15 20 25 30 35 40 43 50 60 7O 80 90 0.0 o o o o.o o o o.o o o 90 0.0 o 0. 2 2 O.2 2 2 2 3 O.O o I I 2 0.2 2 3 3 4 0.4 4 5 5 f 0.0 I I 2 3 0-3 4 4 5 5 0.6 6 7 7 7 o 0.0 I 2 3 3 0.4 I 6 7 0.8 9 9 I.O o 0.0 I 2 3 4 -5 6 8 9 0.9 i.i i 2 2 0.0 I 2 4 5 0.6 8 9 I.O i.i 2 3 4 4 O.O I 3 4 6 0.7 8 9 2 i-3 4 6 6 o O.O 2 I 0.8 9 2 3 '1 8 8 o o.o 2 4 5 7 0.9 I.O 2 3 4 1.6 8 9 2.0 o o O.O 2 4 6 8 0.9 i.i 3 i i-7 9 2.1 2 2 o 0.0 2 | 9 i.i 2 i 8 1.9 2.2 3 4 150 o O.O 2 S 7 9 i.i 3 5 7 9 2.1 3 S 7 7 o O.O 2 5 7 I.O 1.2 4 6 8 2.0 2.2 4 I 8 o o.o 3 5 7 I.O 1.2 4 6 8 2.0 2.2 I 9 180 175 170 165 160 155 15O 145 140 135 130 12O 110 , 1OO 90 Az. i 95 100 105 j 110=> 115 120 125 130 135 140? 160 170 180 Az. Position- Angle. Hour-Angle O h 30 ra or ll h 30'". j Az. Position- Angle. Az. 90 85 80 75 70 65 60 55 50 45 40 30 20 10 5 1O 15 20 25 30 35 40 45 50 60 70 80 90 0.0 o 0.0 o o o o.o o o 0.0 o o I O.I I I I I O.I I 2 2 2 o O.O I I I O.I 2 2 2 2 o-3 3 3 3 3 o 0.0 I I 2 0.2 2 3 3 3 0.4 4 5 5 5 o 0.0 I I 2 2 0-3 3 4 4 5 1 6 6 7 0.0 I I 2 3 0.3 4 5 I 0.6 1 8 8 0.0 I 2 2 3 0.4 i 7 1 9 9 I.O 0.6 I 2 3 4 -5 I I 0.8 9 I.O I o o.o I 2 3 4 0.5 I 9 0.9 I.O 2 2 2 o O.O I 2 3 5 0.6 8 9 I.O I.O 2 3 3 4 o 0.0 I 3 4 5 0.6 I 9 I.O 3 4 4 5 0.0 I 3 6 0.7 8 9 I.O 2 i-3 4 6 6 7 0.0 2 i 0.8 9 I.O 2 3 '1 8 7 8 O.O 2 3 1 0.8 9 i.i 2 3 1.4 6 8 9 9 o 0.0 2 3 5 7 0.8 I.O i 2 4 1 9 9 ISO 175 170 165 i 160 155 15O 145 140 135 13O 12O 110 100 9O u*. 90 93 100 105 110 115 l*O 125 130 135^ 140 150 160 170 18O Az. Position-Angle. TABLE XXXV. 141 Error of Time-Azimuth for an Error of 12 , or O.2, in Latitude. Altitude. A * Az. JU. O 10 20 30 40 i 50 60 7O 75 8O 82 84 86 88 90 o o o' o' o' o' o' 0' o' o' o 0.0 0.0 O.O o O.O o O.O o . 0.0 bidet. ISO 2 o o o o o i i i O I I I 2 00 178 4 o o o I i i 2 I I I I 2 4 00 176 6 o 1 2 2 3 I I 2 2 3 6 00 174 8 o I 2 3 5 I 2 2 3 4 8 00 172 1O o 2 3 4 6 O.I O.2 o-3 0.3 0.5 I.O 00 170 12 14 o o o 2 2 2 I 3 4 4 5 I 2 2 2 3 3 3 4 5 6 7 2 4 00 OO 168 166 16 o 2 3 4 6 9 2 3 4 5 8 6 00 164 18 o 2 3 4 6 10 2 3 4 6 9 8 00 162 2O o 2 2 3 5 7 ii '3 0.4 0.5 0.7 I.O 2.0 00 160 ! 25 o 2 3 4 6 8 14 3 5 6 8 2 4 00 155 3O o 2 7 n 10 16 4 6 7 I.O 4 9 OO 150 35 o 3 O 4 6 8 12 19 4 6 8 i 6 3-3 00 145 1 4O 3 5 7 9 '3 21 5 7 9 2 8 7 00 140 45 2. 3 5 10 15 23 0.5 0.8 I.O 1.4 2.0 4.1 00 135 5O o 2. 3 8 ii 16 25 6 9 i 5 2 4 00 130 55 o 2. 4 6 8 12 !7 27 6 9 2 6 3 7 00 125 60 o 2. 4 6 9 12 18 29 6 I.O 2 7 5- 00 12O 65 2. 4 6 9 13 19 3 7 3 7 6 2 00 115 70 o 2. 4 7 10 '3 19 0.7 i. 1.3 1.8 2-7 54 00 110 75 2, 4 7 10 H 20 32 7 4 8 8 5 00 1O5 80 o 2. 4 7 10 H 20 32 7 4 9 8 6 00 1OO 85. o 2. 4 7 10 H 21 33 7 4 9 8 7 00 95 90 2 4 7 10 H 21 33 7 4 9 9 7 00 9O i TABLE XXXVI. Error of Time-Azimuth for an Error of 6', or O.l, in Declination. Pos. Altitude. Pos. Ang. 10 2O 30 4O 50 6O 70 75 80 82 84 86" 88> 90 Ang. o 0' o' o' o' o' o' o' o' O.O 0.0 o O.O 0.0 0.0 o 0.0 Indet. 180 2 o o o i O o I I 00 178 4 o o I I I i o o I I I 2 00 176 6 I I I I I 2 I I I 2 3 00 174 8 I I I I 2 2 I I I I 2 4 OO 172 10 I I I 2. 2 3 O.I O.I O.I 0.2 -3 0.5 00 170 i 12 I I 2 2. 3 4 I I 2 2 3 6 00 168 14 2 2 2 2 2 3 4 I I 2 2 3 7 00 166 16 2, 2 2 2 2 3 3 5 I 2 2 3 4 8 00 164 18 2, 2 2 2 2 3 4 5 I 2 2 3 4 9 00 162 20 2. 2 2 2 3 3 4 6 O.I 0.2 0.2 0-3 0-5 I.O 00 16O 25 2, 3 3 3 3 4 5 7 2 2 3 4 6 2 00 155 30 3 3 3 4 4 5 6 9 2 3 4 5 7 4 00 150 35 3 4 4 4 5 7 10 2 3 4 5 8 6 00 145 40 4 4 4 5 -5 6 8 ii 2 4 5 6 9 8 00 140 45 4 4 5 5 6 7 9 12 0.3 0.4 0-5 0.7 I.O 2.0 00 135 5O 55 5 5 5 5 5 5 6 6 8 9 10 '3 14 3 3 4 5 I I i 2 2 4 00 00 130 125 60 5 6 6 7 8 10 3 5 6 8 2 00 120 65 5 6 6 6 7 9 ii 16 3 5 7 9 3 6 00 115 70 6 6 6 7 7 9 ii 17 0.4 0.5 0.7 0.9 1.3 2.7 00 110 75 6 6 6 7 8 9 12 17 4 6 7 9 4 8 00 105 80 6 6 6 7 8 9 12 17 4 6 7 9 4 8 OO 1OO 85 -6 6. 6 7 8 9 12 18 4 6 7 I.O 4 9 00 95 9O 6 6 6 7 8 9 12 18 4 6 7 o 4 9 00 9O 142 TABLE XXXVII. Limiting* Errors of Time-Azimuths. Hour- Angle. First Supposition. Second Supposition. Third Supposition. Partial Az. Error. Prob. Total Az. Error. Partial Az. Error. rrOD. Partial Az. Error. Prob. Total Az. Error. 1.9 | I.O 0.8 0.8 ; 0.8 0.8 i 0.7 0.7 0.7 0.7 H.A. Error l m . Lat. Error 12'. o | o's 0.4 O.2 O.I O.I O.I O.O o.o o.o 0.0 Dec. Error 3'. H. A. Lat. Error Error 2. IS'. n Total AZ. Y5? p Error. H.A. j Lat. Error Error 3-. 30'. Dec. i Error 3. h m 1 2 3 4 5 O 6 7 O 8 9 i 1O O i 11 12 o 0.6 0.4 o-3 o-3 o-3 o-3 0-3 0.2 0.2 O.I O.I O.I O.I O.O 0.0 0.0 o.o o.o 0.0 1.0 0.7 0.6 0-5 0.4 0.4 0.4 o-3 o-3 0-3 o 0.9 0.7 o-5 0-5 o-5 0-5 0-5 0-5 0.5 o-5 0.8 o-5 0.4 0-3 0.2 O.I O.I O.I 0.0 o.o o.o o 0.2 0.2 O.I O.I O.I O.I O.O e.o o.o 0.0 0.0 0.0 o 1.4 I.I 0.8 0.7 0.6 0.6 0.6 0.6 0-5 o-5 o-5 o 1.9 I.I : 0.9 i 0.8 ; 0.7 0.7 0.7: 0.7 0.7 0.7 0.7 0.7 i o 1.9 0.9 0.6 0.4 o-3 O.2 0.2 O.I O.I 0.0 0.0 o 0.1 0.2 O.I O.I O.I O.I O.O o.o 0.0 o.o ! o.o 0.0 TABLE XXXVIII. Limiting: Errors of Ti sae-Azimuths in High Latitudes. Hour- Angle. First Supposition. Second Supposition. Third Supposition. Partial Az. Error. Prob. Total Az. Error. Partial Az. Error. Prob. Total Az. Error. Partial Az. Error. Prob. Total Az. Error. 1-. 12'. 3'. 2-. 18'. 3'. 3*. 30'. 0'. h m 010 O3O 1 2 O 3 4 5 6 7 8 9 1O O 11 12 O o 0.4 0.4 0.4 0.4 o-3 o-3 0-3 0-3 o-3 o-3 o-3 0.0 0.0 0.0 0.1 0.2 O.2 0.2 O.I O.I O.I O.I 0.0 O.O o.o o.o o o.o 0.0 o.o 0.0 0.0 0.0 0.0 o.o 0.0 0.0 0.0 0.0 o.o 0.0 0.0 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 o-3 0-3 0-3 0.7 0.7 0.7 0.7 0.6 0-5 0-5 o-5 o-5 0-5 0-5 0.0 o.o 0.1 0.2 0.2 0.2 O.2 0.2 0.2 O.I O.I O.I 0.0 0.0 O.O o.o 0.0 0.0 0.0 0.0 0.0 0.0 o.o 0.0 0.0 0.0 0.0 o.o o.o 0.0 o 0.7 0.7 0.7 0.8 0.7 0.6 0.6 0.6 0.6 o-S 0-5 o-5 o-5 o 1.1 I.I I.I I.I 0.9 0.8 0.8 0.8 0.8 0.7 0.7 0.7 0.7 0.7 0.7 o 0.0 0.1 0.2 0.4 0.4 0.4 0.3 O.2 O.2 O.I 0.0 O.O O.O o o.o o.o o.o 0.1 O.I O.I O.I O.I O.I O.I O.I O.I o.o 0.0 o.o 1.1 I.I I.I 1.2 I.O 0.9 0.9 0.9 0.8 0.8 0.8 0.7 0.7 0.7 0.7 TABLE XXXIX. Limiting Errors of Serial Time-Azimuths. Hour- Angle. Observations in Port. At Sea: In all Latitudes. At Sea: In all Latitudes. Partial Az. Error. Prob. Total Az. Error. Partial Az. Error. Prob. Total Az. Error. Partial Az. Error. Prob. Total Az. Error. H.A. Error 4*. Lat. Error 1'. Dec. Error 0'.5. H.A. Error 10-. Lat. Error 3'. Dec. Error 1'. H.A. Error 30. Lat. Error 0'. Dec. Error 1'. /' m 6 5 4 3 230 0.9 0.9 I.O 1.3 0.4 0.6 0.8 1.2 1.6 0.5 0.6 0.8 I.O 1.1 1.2 1.4 1.9 2.4 2.2 2.2 2.6 3-3 4.0 1.2 4-9 1.1 i.i 1.2 1.6 1.9 2.7 2.8 3-7 / 6.7 7.8 1 0.0 I2.I 2.5 34 4.6 7.2 9-7 1.1 i.i 1.2 1.6 1.9 7.; 7 .6 9.1 15.6 TABLE XL. 143 4 i rr ii 111 polar Azimuths: Polaris, or the Pole-Star. Western Elongations. Sid.T. TUT 1 Eastern Elongations. Lat. Greatest Elongation West- Star moving fro n Meridian. Sid. Time of Azimuths- Mer'n above Pole. Star moving towards Merid an. Sid. Time of Azimuths- Greatest Elongation East- Lat. Sid. T. Az. h m 613 5 13 413 313 213 113 013 2313 2213 21132013 Az. Sid. T. o h m o o o o o o o o h m O 7 13- 1.4 1.3 1.2 I.O 0.7 0-3 0.0 o.3 0.7 I.O 1.2 1.3 1.4 19 13.0 O 2 12.8 4 3 2 O 7 3 3 7 O 2 3 4 13.2 2 4 12.6 4 3 2 o 7 3 o 3 7 2 3 4 13-4 4 6 12.4 4 3 2 o 7 4 o 4 7 o 2 3 4 13.6 6 8 12.2 4 3 2 o 7 4 o 4 7 2 3 4 13.8 8 ! 10 7 12.0 1.4 1.3 1.2 I.O 0.7 0.4 o.o 0.4 0.7 I.O 1.2 1.3 1.4 19 14.0 10 12 11.8 4 3 2 o 7 4 4 7 2 3 4 14.2 12 14 11.6 4 4 2 o 7 4 o 4 7 o 2 4 4 14.4 14 16 11.4 4 4 3 o 7 4 o 4 7 3 4 4 14.6 16 18 11.2 4 4 3 o 7 4 o 4 7 o 3 4 4 14.8 18 20 7 i i.o 1.5 1.4 1.3 I.O 0.7 0.4 0.0 0.4 0.7 I.O 1.3 1.4 1.5 19 15.0 20 21 10.9 5 4 3 7 4 o 4 7 3 4 5 21 22 10.8 5 4 3 o 7 4 o 4 7 o 3 4 5 15.2 22 23 10.7 5 4 3 o 7 4 o 4 7 o 3 4 5 '5-3 23 24 10.6 5 4 3 I 7 4 o 4 7 3 4 5 15-4 24 25 7 IO -5 1.5 1.4 1.3 i. 0.8 0.4 o.o 0.4 0.8 I. J -3 1.4 1.5 19 15-5 25 26 10.4 5 5 3 8 4 o 4 8 3 5 5 15.6 26 27 IO.2 5 5 3 8 4 o 4 8 3 5 5 15.8 27 28 IO.I 5 3 j 8 4 o 4 8 3 5 5 15-9 28 29 IO.O 6 5 4 8 4 4 8 4 5 6 16.0 29 30 7 9-9 1.6 1.5 1.4 i. 0.8 04 o.o 0.4 0.8 I. 1.4 1.5 1.6 19 16.1 30 31 9-7 6 5 4 8 4 4 8 4 5 6 16.3 31 32 9.6 6 6 4 8 4 4 8 I 4 6 6 16.4 32 33 9-5 6 6 4 2 8 4 o 4 8 2 4 6 6 16.5 33 34 9-3 6 6 4 2 8 4 4 8 2 4 6 6 16.7 34 35 7 9-2 1-7 1.6 1.4 1.2 0.8 0.4 o.o 0.4 0.8 1.2 1.4 1.6 l.y 19 16.8 35 36 Q. I 7 6 5 2 8 4 4 8 2 5 6 7 16.9 36 37 o.Q 7 6 5 2 8 4 o 4 8 2 5 6 7 17.1 37 38 8.8 7 7 5 2 9 4 o 4 9 2 5 7 7 17.2 38 39 8.6 8 7 5 2 9 5 o 5 9 2 5 7 8 17.4 39 40 7 8.4 1.8 1.7 '5 1.3 0.9 0.5 o.o 0.9 1.3 1.5 1.7 1.8 19 17.6 4O 41 8-3 8 7 6 3 9 5 5 9 3 6 7 8 17.7 41 42 8.1 8 8 6 3 9 5 o 5 9 3 6 8 8 17.9 42 43 7-9 9 8 6 3 9 5 o 5 9 3 6 8 9 18.1 43 44 7.8 9 8 6 3 I.O 5 5 I.O 3 6 8 9 18.2 44 45 7 7-6 1.9 1.9 1-7 1.4 I.O o-5 0.0 I.O 1.4 1.7 1.9 1.9 19 18.4 45 46 7-4 9 9 7 4 5 o 5 4 7 9 9 1 8.6 46 47 7.2 2.0 9 4 o 5 o 5 o 4 7 9 2.O 18.8 47 48 7.0 o 2.0 8 4 o 5 o 5 4 8 2.0 O 19.0 48 49 6.8 I 8 5 5 o 5 5 8 O I 19.2 49 50 7 6.5 2.1 2.0 1.8 i.i 0.5 o.o I.I 1.8 2.0 2.1 1 9 J9-S 50 51 6-3 2 I 9 5 i 6 6 I 5 9 I 2 19.7 51 52 6.0 2 I 9 6 i 6 6 I 6 9 I 2 20.0 52 53 5.8 3 2 2.0 6 i 6 o 6 I 6 2.0 2 3 20.2 53 54 5-5 3 2 6 2 6 6 2 6 2 3 20.5 54 55 7 5-2 2.4 2-3 2.1 1.7 1.2 0.6 0.0 0.6 1.2 1-7 2.1 2 -3 2.4 19 20.8 55 56 4-9 4 4 I 7 2 6 o 6 2 I 4 4 21. 1 56 57 4.6 4 2 8 3 6 o 6 3 8 2 4 5 21.4 57 58 4-3 6 5 2 8 3 7 7 3 8 2 6 21.7 58 59 4.0 6 6 3 9 3 7 7 3 9 3 6 6 22.O 59 6O 7 3-6 2.7 2.6 2.4 2.0 1.4 0.7 0.0 0.7 1.4 2.0 2.4 2.6 2.7 19 22.4 60 Sid. T. Az. 111 913 1013 1113 1213 h m 1313 h m 1413 1513 1613 1713 1813 Az. Sid. T. Lat. Greatest Elongation West- Sid- Time of Azimuths- Star moving towards Meridian, Sid.T. Me?'n below the Sid. Time of Azimuths. Star nwing from Meridian. Greatest Elongation East. Lat Pole- Subtract R. A. True Sun (Tab. LIX) from Sid. T. to get Ship A. T., or subtract R. A. Mean Sun (by applying E. T. to R. A. True Sun) to get Ship M. T. 144 TABLE XLI. Altitude- Azimuths : Part 1. Alt. Latitude. Alt. i D 2 4 6 8= 10 12 14 16 iw 20 22 24 26 28 3O o i 2 3 5 6 8 ii 13 16 20 23 27 3* O i 2 o i 2 3 5 6 8 ii 13 16 20 23 27 2 4 o o I 2 3 4 7 9 n 17 2O 24 28 3 2 4 6 I I 2 2 3 4 6 8 10 12 15 18 21 24 28 3 2 6 8 2 2 3 3 4 5 7 9 n '3 16 19 22 25 29 33 8 10 3 3 4 4 5 6 8 10 12 H *7 20 23 26 3 34 10 12 5 5 5 6 7 8 10 ii 13 16 18 21 24 28 32 36 12 14 6 6 7 8 9 10 ii 13 15 17 20 23 26 30 34 38 14 16 8 8 9 10 n 12 13 15 17 19 22 25 28 3 2 36 40 16 18 ii n ii 12 13 H 16 17 19 22 24 27 30 34 38 42 18 20 13 13 14 15 16 17 18 20 22 24 27 30 33 37 41 45 20 22 24 16 20 16 20 17 20 21 19 22 2O 2 3 21 24 2 11 27 3 30 33 36 39 39 43 43 47 48 22 21 26 23 23 23 24 25 26 28 3 32 37 4 43 46 5 54 26 28 27 27 27 28 29 30 3 2 33 35 38 4i 44 47 50 54 58 28 30 3 1 3 i 3 1 3 2 33 34 36 38 40 42 45 48 5 1 54 58 62 30 i 32 36 36 36 37 38 39 41 42 44 47 49 52 59 6 3 67 32 34 42 43 44 4 6 47 49 52 54 57 60 64 68 72 34 36 46 46 46 47 48 49 51 59 62 66 69 73 77 36 38 52 52 5 2 53 54 55 57 58 60 63 65 68 7i 75 79 83 38 | 40 58 58 S 8 59 60 61 63 64 66 69 7i 74 77 81 85 89 40 42 64 64 6 5 66 67 68 69 7i 73 75 78 81 84 88 92 96 42 44 7 1 7 2 73 74 75 76 78 80 82 85 88 91 95 99 I0 3 41 i 46 79 79 80 80 81 82 84 86 88 90 93 96 99 102 1 06 no 46 48 87 8 7 88 88 8 9 90 92 94 96 98 IOI 104 106 IIO 114 118 48 , 50 96 96 96 97 98 99 101 102 104 107 109 112 "5 119 123 127 50 ' 52 105 105 106 106 107 109 IIO 112 114 116 119 122 125 128 132 136 52 54 "5 "5 116 116 117 119 120 122 124 126 129 I 3 2 135 138 142 146 54 56 126 126 127 127 128 129 I 3 I 133 135 137 140 H3 146 149 '53 157 56 58 138 138 138 139 140 141 H3 144 146 149 151 154 157 161 165 169 58 60 62 n: 55 iS l6 5 III '54 167 'I 5 169 157 171 159 161 175 164 178 I6 7 181 170 174 187 178 191 182 J 95 60 62 64 179 179 180 181 182 184 I8 S 187 190 192 195 199 202 206 2IO 61 66 '95 196 196 197 199 200 202 204 206 209 212 215 218 222 226 66 68 213 213 214 214 215 217 218 220 222 224 227 2 3 233 236 2 4 2 44 68 70 233 233 233 234 235 236 238 239 241 244 246 249 253 256 260 26 4 70 72 255 256 257 25 260 26l 263 266 268 271 275 278 282 286 72 74 280 280 280 281 282 283 285 286 288 291 293 2 9 6 299 303 37 3" 71 76 308 308 39 39 310 3" 315 317 319 322 324 328 33 i 335 339 76 78 34i 34i 342 343 344 346 347 349 352 354 357 361 364 368 372 78 1 80 380 380 381 382 383 384 385 387 389 39i 394 397 400 404 408 411 NO 82 428 428 429 429 43 43 i 433 435 437 439 442 445 448 45i 455 459 82 84 490 491 491 491 492 494 495 497 499 5 O1 54 507 51 5*3 5i7 52i 84 86 578 578 579 579 580 581 583 585 587 589 592 595 598 601 605 609 86 88 729 729 729 73 731 732 733 735 737 739 742 745 748 752 756 759 88 Entering Part I with the Lat. and Alt., and Part II with the Dec. and Diflf. of the Lat. and Alt., take out the corresponding numbers. Then, with the Sum of these two numbers, the corresponding Azimuth is found in Part HI. TABLE XLI. 145 Altitude-Azifiimtlis: Part I. Alt. Latitude. Alt. . j 3O 32 34 36 38 40 42 44 46 48 50 52 54 56 58 6O O 31 I 36 4 1 46 5 2 58 64 7I 79 87 06 I0 5 "5 126 138 150 O 2 3 1 3 6 4 1 46 52 58 64 71 79 87 96 105 "5 126 138 2 4 6 32 3 2 3^ 37 42 46 47 52 53 58 59 a 72 73 80 80 88 88 96 1 06 106 116 116 127 127 139 139 152 4 6 8 33 38 43 48 54 60 67 74 81 89 98 107 117 128 140 153 8 1O 34 39 44 49 55 61 68 75 82 90 99 109 119 129 141 i54 1O 12 36 46 57 63 69 76 , 84 92 101 no 120 '3 1 143 155 12 14 38 42 47 S 2 58 64 78 86 94 102 112 122 133 144 157 14 16 4 44 49 54 60 66 73 80 88 96 IO4 114 124 135 146 159 16 18 4 2 47 52 57 63 69 75 82 90 98 107 116 126 137 149 161 18 20 45 49 54 59 65 7i 78' 85 93 101 109 119 129 140 151 164 20 22 48 5 2 57 62 68 74 81 88 96 104 112 122 I 3 2 H3 154 167 22 24 51 55 60 66 71 77 84 9 1 99 107 "5 125 '35 146 158 170 24 26 54 59 64 69 75 81 88 95 102 HO 119 128 138 149 161 '74 26 28 6 3 68 73 79 85 9 2 99 1 06 II 4 123 I 3 2 142 153 165 178 28 30 62 67 72 77 83 89 96 103 no 118 127 I 3 6 147 157 169 182 30 32 67 72 77 82 88 94 100 107 115 123 132 141 151 162 174 186 3 ' 34 72 76 81 87 92 98 I0 5 112 120 128 137 146 156 167 179 191 34 36 77 82 87 9 2 98 104 HO 117 J 25 133 142 151 161 172 184 196 36 38 83 88 93 98 104 no 116 I2 3 139 148 157 167 178 190 202 38 40 89 94 99 104 IIO 116 122 I2 9 : 37 H5 154 I6 3 173 184 196 208 40 42 96 100 I0 5 no 116 122 129 I 3 6 144 161 170 180 191 202 215 42 44 103 107 112 117 123 I2 9 I 3 6 J43 '59 1 68 177 187 198 209 222 44 46 no "5 120 J 25 J 37 H3 158 1 66 176 1841 194 205 217 2 3 46 48 118 123 128 139 152 '59 166 174 184 192 202 213 225 238 48 50 127 132 J 37 I 4 2 148 154 161 168 i;5 183 192 201 211 222 234 246 50 52 136 141 146 I 5 I 157 163 170 177 184 192 201 211 221 231 243 255 52 54 146 151 156 161 167 173 180 187 194 202 211 221 2 3 I 2 4 I 253 266 54 56 58 '57 169 162 174 167 179 172 184 178 190 184 196 191 202 198 209 205 217 2I 3 225 222 233 2 3 I 243 242 253 264 264 275 277 288 56 58 i 60 182 186 191 196 202 208 215 222 230 238 2 4 6 2 5 6 266 277 288 301 6O 62 J 95 200 205 210 216 222 229 2 3 6 243 251 260 26 9 280 291 302 315 62 64 2IO 215 220 22 5 2 3 I 237 244 251 258 266 275 284 295 3 06 317 330 64 66 226 231 2 3 6 2 4 I 247 253 260 267 274 283 291 3 OI 323 334 66 68 244 249 254 259 265 271 277 285 292 300 309 319 329 341 68 70 264 269 274 279 285 291 297 305 312 320 329 338 349 7O 72 286 291 296 301 37 313 327 334 342 351 360 72 74 3" 321 326 332 338 344 352 359 367 376 74 76 339 344 349 354 300 366 372 3 80 387 395 76 78 372 377 382 387 393 399 405 413 420 78 80 411 416 421 426 432 438 445 452 80 82 459 464 469 474 480 486 493 82 84 52i 526 53 1 536 542 548 84 86 609 614 619 624 630 86 88 759 764 769 774 88 Mark the Azimuth N or S according to the Latitude, and E or W according as the object observed is East or West of the Meridian. When the Lat. exceeds 60 use Alt. for Lat. and Lat. for Alt. 1 146 TABLE XLI. Altitude- Azimuths: Part II. Diff. Declination of same name as the Latitude. Diff. Alt. hM Jl Alt. and i and L:it. 2 i 6 8 10 12 14 3 16 18 20 22 24 26 28 30 Lat. o O 849 842 834 ! 825 817 808 799 789 779 769 759 748 736 724 712 699 2 849 842 834 825^- 817 808 799 7*9 779 769 758 748: 73 6 724 712 699 2 4 849 8 4 I 833 825 - 816 807 798 789 779 768 758 747 735 698 4 6 848 840 i 832 824 815 806 797: 788 778 767 757 74 6 734 722 710 697 6 8 847 839 831 : 823 814 805 796; 786 776 766 755 744 732 720 708 6 95 8 10 846 l 838 1 830 : 822 813 804 796 ! 785 775 764 753 742 73 718 705 692 1O 12 845 837 829! 820 j 8n 802 793 783 773 762 75i 740 728 715 ! 702 689 12 14 843 ; 835 1 827 818 i 809 800 791 781 770 760 748 737 725 712! 699 685 14 16 841 833! 824 816! 807 798 788; 778 767 757 745 734 72i 708 695 68? 16 18 838 ; 830; 822; 813! 804 795 785 775 764 753 742 73o! 7i7 704 691 677 18 20 836 828! 819 810! 801 792 782 : 771 t 760 749 738 726 713 699 686; 672 2O 22 833 825 816:807! 798 788 778 767 75 6 745 733 721 708 694 680 665 22 24 830 821 812 ! 803! 794 784 774 763 752 740 728 715 702 688 673 i 658 24 26 826 818 808 ; 7991 790 780 769 758 747 735 722 709 695 68 1 666: 650 26 28 822 814 804 : 795 j 785 775 764' 753 7 4i 729 716 702 688 673 658 641 28 30 818 ; 809 Soo 790 780 769 758 747 735 '723 709 695 680 665 649 ; 631 30 32 814 i 804 | 795 785 i 775 764 752 741 729 716 702 687 672 656 639 620 32 34 809 , 799 790 779 ; 769 758 746i 734 721 708 693 678 662 6 45 627 608 34 36 803 j 7941 784 773! 762 739 j 726 i ^ 699 684 668 651 634 595 36 38 797 788 778 767) 755 744 731 718 704 689 674 657 640 621 60 1 ! 579 38 40 791 781 771 760 748 736 723 709 695 679 663 646 627 607 585: 562 40 42 44 785 778 774 767 764 753 75 6 744 740 73 i 727 7H 704 684 673 668 656 6 5 I 638 a 613 597 574 568! 548! 542 520 42 44 46 770 759 747 735 722 708 693 677 660 642 62 3 602 579 554 526 494 46 48 762 75 738 725 711 697 68 1 665 646 628 60 7 584 558 532 500 464 48 ! 50 753 741 729 715 700 685 669 651 632 611 ,88 563 535 55 469 427 5O '52 744 732 718 704 688 672 655 636 615 593 568 540 59 474 43 2 i 52 54 734 721 707 691 675 658 6 39 619 597 572 545 478 437 3861 321 54 56 723 709 694 678 66 1 642 622 600 576 548 518 482 441 390 326! 236 56 58 711 696 681 664 645 625 603 579 552 521 486 445 395 33 240 87 58 : 60 698 j 683 666 648; 628 606 582 555 525 489 449 399 334 244 92 60 62 685 668 650 631 609 585 558 528 493 452 402 338 248 96 62 64 670 652 633 611 587 561 496 455 405 341 252 99 64 66 654 635 613 589 563 533 499! 458 A ,08 344 2 55 103 66 ! 68 636 615 565 535 460 411 347 258 106 68 i 70 616 593 567 537 503 462 413 350 260 109 70 i 72 594 ! 568 539 505 464 415 352 263 in 72 74 569 540 506 466 417 354 265 113 74 76 541 ! 507 467 418 355 267 76 78 508 468 419 357 268 117 78 80 469 420 358 269, 118 80 > 82 421 358 270 119 82 84 359 271 1 20 84 j 86 2.71 121 86 | 88 121 88 | Altitude- Azimuths : Part III. 1 a ^ a rf a ti a M a si a 2 Kj a M . * i 10 4 -4 4 6 7 20 20 2 2 2 3 3 4 4 S ^ 6 8 40 O 2 2 2 2 3 3 4 5 5 6 7 8 9 2 4O 0.2 0.2 0.2 0.2 6.2 0.2 o-3 0.4 0-5 o-S 0.6 0.6 0.7 0.9 920 ! 20 2 2 2 2 2 3 3 4 6 6 7 8 I.O 40 1 2 2 2 2 3 3 4 S 6 6 7 S I.O 2 1O 1 40 *>o 2 2 3 3 3 4 (> T 7 8 9 i i I.O 2 i 4 20 1 12 0-3 6 7 9 i 2 3 ^ 7 2.1 56 56 4 4 4 S 6 8 I.O 2 4 5 7 2.0 11 4 48 5 6 6 7 I.O i 4 6 7 2.0 12 4O 6 6 6 7 8 9 2 4 7 8 2.1 20 J 32 0.7 0.7 0.8 0.8 0.9 i.i 1.4 J. 7 2.1 2-3 1128 24 I.O 1.0 I.O i. i 1.2 S 9 2-3 2.8 36 i 16 4 5 5 6 9 2.2 2.9 3-4 44 8 2.9 2.9 3-3 3-7 4-4 5-7 52 O oo 00 00 12 TABLE XL VII. Limiting Errors of Altitude-Azimuths. Least Hour-Angle 1 . Least Hour-Angle 2 il . Least Hour- Angle 3 h . Partial Az. Error. Partial Az. Error. Partial Az. Error. Lat. Prob. Prob. Prob. Total Total Total 1 Alt Error Lat. Dec. Error Error Az. Error. Alt. Lat. Error Error Dec. Error Az. Error. Alt. Error Lat. Error Dec. Error Az. Error. 6'. 12'. 3'. 6'- 12'. 3'. 6'. 12'. 3'. o 0.4 0.8 0.2 0.9 0.2 0.4 o 0.1 o 0.5 o 0.1 0.2 0.1 0.3 10 0.4 0.8 0.2 0.9 0.2 0.4 O.I o-5 O.I 0.2 O.I 0.3 20 0.4 0.8 0.2 I.O O.2 0.4 O.I O.I 0.2 O.I 30 0.9 0.3 I.I 0.3 0.4 0.2 0.6 O.I 0.3 O.I 0.4 40 -5 i.o 0.3 1.2 0.3 0.5 0.2 0.6 0.2 O-3 O.I 0.4 50 0.6 1.2 0.3 i-7 0.4 0.6 0.2 0.8 0.2 0.3 0.2 -5 60 0.8 1.6 0.4 1.9 0.4 0.7 O.2 0.8 *3 0.4 O.2 0.6 70 i.i 2.3 0.6 2.6 0.6 i.o O. "1 1.2 0.4 0.6 0.2 0.8 80 2.8 4-5 1-2 5-5 1-3 2.O 0.6 2-5 I.O 1.2 0-4 TABLE XLVIII. 153 Time-Alt. Azimuths : Log. A. Declination. Diff. A r* for A *T JLZ. 5 10 15 2O 25 30 35 40 45 50 IvJ-i J*.Z. , i i o o + 00 + 00 + 00 + oo + 00 + 00 + 00 + 00 JL 00 + 00 + 00 180 1 1.756 I-75I 1-743 I-73 1 1-715 1.695 1.671 1.6 4 2 1.607 1.566 179 457 455 45 442 43 414 394 3/0 3 41 306 265 30.1 _ 178 3 279 274 266 254 238 218 194 165 130 089 17.6 177 4 156 154 149 141 129 113 093 069 040 005 0.964 12.5 176 5 i. 060 1.058 1.053 1.045 1-033 1.017 0.097 0-973 0.944 0.909 0.868 9.6 175 6 0.981 0.979 i 0.974 0.966 0-954 0.938 c >i8 894 865 8 3 769 7-9 174 7 914 912 907 899 887 871 i 5 1 827 798 7 63 722 6.7 173 8 856 854 849 841 829 813 793 769 740 705 664 5- 8 172 9 806 804 799 791 779 763 743 719 690 655 614 5- 171 10 0.760 0-759 o-753 0-745 0-733 0.717 0.697 0.673 0.644 0.6 09 0.568 4.6 17O 11 719 717 712 704 692 676 656 632 603 5 68 527 4.1 169 682 680 675 667 655 639 619 595 5 66 53 1 490 37 168 13 648 646 641 633 621 605 585 5 61 532 497 456 3-4 167 14 616 614 609 60 1 589 573 553 529 500 465 424 3.2 166 15 0.587 0.585 0.580 0.572 0.560 0-544 0.524 0.500 0.471 0.436 0.395 2.9 165 16 560 558 553 545 533 517 497 473 444 409 368 2.6 164 17 534 S3 2 527 519 507 491 t tfi 447 418 383 342 163 18 5 10 508 53 495 483 467 447 423 394 359 3*8 2.4 162 19 488 486 481 473 461 445 425 401 372 337 296 2.2 161 2.2 2O 0.466 0.464 0-459 0.451 0-439 0.423 0.403 0-379 0-350 0-315 0.274 160 21 446 444 439 43 i 419 403 383 359 33 295 254 " 159 22 427 425 420 412 400 384 364 340 3" 276 235 1-9 158 23 408 406 401 393 51 365 345 321 292 257 216 1.9 157 24 391 389 384 376 364 348 328 304 275 240 199 *-7 156 25 o-374 0.372 0.367 0-359 0.347 0.331 0.311 0.287 0.258 0.223 0.182 r -7 1.6 155 26 358 356 3 Si 343 31 315 295 271 242 207 1 66 154 27 343 336 328 316 300 280 256 227 192 151 i5 153 28 328 326 321 313 301 285 265 241 212 177 136 i>5 152 29 3H 312 307 299 287 271 251 227 198 163 122 1.4 151 30 0.301 0.299 0.294 0.286 0.274 0.258 0.238 0.214 0.185 0.150 0.109 i-3 15O 31 288 286 281 273 261 245 225 20 1 172 137 9 6 " 3 149 32 276 274 269 261 249 233 213 i 89 1 60 125 084 1.2 148 33 264 262 257 249 237 221 20 1 177 148 "3 072 ' 147 34 253 251 246 238 226 210 190 166 137 102 06 1 LI 146 I.I 35 0.242 0.240 0-235 0.227 0.215 0.199 0.179 O.I 55 0.126 0.091 0.050 145 36 231 229 224 216 204 1 88 1 68 144 I 15 080 039 * 144 37 221 219 214 206 194 178 158 i 34 I0 5 070 029 I.O 143 38 211 209 204 i 96 184 168 148 124 095 060 OI9 I.O 142 39 201 199 194 i 86 174 158 138 114 085 050 009 I.O 141 40 0.192 0.190 0.185 0.177 0.165 0.149 0.129 O.I 0=5 0.076 0.041 0.000 0.9 _ o.- 140 42 175 173 i 68 160 148 132 112 o 88 059 024 9.983 0.05 0.8^ 138 44 158 156 i 51 H3 i 31 115 095 071 042 007 966 136 46 143 141 136 i 28 116 IOO 080 056 027 9.992 951 o-7S 134 48 129 127 122 114 102 086 066 042 013 97 8 937 0.70 132 0.65 I 50 0.116 0.114 0.109 O.IOI 0.089 0.073 0.053 0.029 0.000 9.965 9.924 13O 52 104 102 O 97 089 077 06 1 041 017 9.988 953 912 128 54 092 5 077 06 5 049 029 005 976 941 126 56 082 075 067 55 039 OI9 9-9 95 966 9 3i 0.50 124 58 072 070 065 057 045 029 OO9 9 85 956 921 0.50 122 60 0.063 0.06 1 0.056 0.048 0.036 O.O2O O.OOO 9-976 9-947 9.9 12 9.871 ' 45 12O 65 043 041 036 028 016 000 9.980 956 927 8 92 851 0.40 115 70 027 025 O2O 012 ooo 9.984 964 940 ii 876 835 0.32 110 75 013 008 000 9.988 972 952 928 99 864 823 0.24 105 80 007 005 000 9.992 980 964 944 920 891 856 815 O.I 100 90 o.ooo 9-998 9-993 9.985 9-973 9-957 9-937 9.913 9.884 9.849 9.808 0.07 90 1 0.04 o.xo 0.16 0.24 0.40 0.48 0.58 0.70 0.82 Az. I For Az. Az. Difference for 0.l of Declination, 154 TABLE XLVIII. Time Alt Azimuths : Log B. Hour-Angle or Position- Altitude or Latitude. Hour-Angle or Position- Angle. Angle. Arc. Time. O 10 15 20 25 30 32 34 34 i 3* |0 4O Time- Arc. o A m k m + 00 _|_ oo _f_ oc _(- 00 _|_ 00 + 00 _|_ 00 -j_ oo + oo + oc -j- 00 12 180 1 4 1-758 1.7 51 1-743 1.7 3 1 1.715 1.695 1.686 1.676 1.666 1.654 1.642 11 56 179 2 8 457 450 442 43 414 394 385 375 365 353 341 11 52 178 3 O 12 281 274 266 254 238 218 209 199 i 9 177 165 11 48 177 4 16 156 149 141 129 "3 093 084 074 064 052 040 11 44 176 5 020 i. 060 1-053 1.045 I -33 1.017 0-997 0.9 SS 0.978 0.968 0.956 0.944 11 40 175 6 24 0.981 0-974 0.966 o.954 0.9 ^8 918 909 899 8 89 8 77 865 11 36 174 7 O 28 914 9 37 899 887 8 7i 851 8 42 832 822 810 798 11 32 173 8 O 32 856 8 49 841 829 813 793 7 84 774 764 752 740 11 28 172 9 36 806 799 791 779 763 743 734 724 7H 702 690 11 24 171 10 040 0.760 o-753 0-745 o-733 0.717 0.697 0.688 0.678 0.668 0.656 0.644 11 2O 170 11 044 719 712 704 692 676 656 647 637 627 6 15 603 11 16 169 12 O 48 682 675 667 655 639 619 610 600 590 578 566 11 12 168 13 52 648 641 633 621 605 585 576 566 556 544 532 11 8 167 14 56 616 609 601 589 573 553 544 534 524 512 500 11 4 166 15 0.587 0.580 0-572 0.560 0-544 0.524 0.515 0-505 0-495 0.483 0.471 11 O 165 16 4 560 553 545 533 5 17 497 488 478 468 45 6 444 10 56 164 | 17 8 534 527 5 10 57 491 462 452 442 43 418 10 5-2 163 18 12 5*0 53 495 483 467 447 438 428 418 406 394 1O 48 162 19 16 488 481 473 461 445 425 416 406 396 384 372 10 44 161 29 20 0.466 o.459 0-45 * 0-439 0.423 0.403 0-394 0.384 0-374 0.362 0-350 10 40 16O 21 24 446 439 43 1 419 403 383 374 364 354 342 330 10 36 159 22 28 427 420 412 400 384 364 355 345 335 323 3" 1O 32 158 23 32 408 401 393 3 Si 365 345 336 326 316 34 292 1O 28 157 24 36 39 * 384 3 76 364 3 48 328 3 19 39 299 287 275 10 21 156 25 ! 26 40 44 o-374 358 0.367 35 i o-359 343 0-347 33 i o.33i 0.311 295 0.302 286 0.292 276 0.282 266 0.270 254 0.258 242 10 20 10 16 155 154 27 48 343 336 328 316 300 280 271 261 25 1 239 227 10 12 i 153 28 52 328 321 313 301 285 265 256 246 236 224 212 1O 8 152 29 56 3*4 307 299 287 271 25 1 242 23? 222 210 I 9 8 10 4 151 30 2 0.701 0.294 0.286 0.274 0.258 0.238 0.229 0.219 0.209 O.I 97 0.185 1O O 15t> 31 2 4 288 281 273 261 245 225 216 206 I 96 I 84 172 9 56 149 32 2 8 2 7 6 269 261 249 233 213 204 194 I 84 172 1 60 9 52 148 ; 33 2 12 264 257 249 237 221 20 1 i 92 182 172 1 60 148 9 48 147 34 2 16 253 246 238 226 2IO 190 i 81 171 161 149 137 9 44 146 35 2 20 O.242 0-235 0.227 0.215 0.199 0.179 0.170 o.i 60 0.150 O.I 38 0.126 9 4O 145 36 2 24 231 224 216 204 1 88 1 68 '59 149 139 127 H5 9 36 144 37 2 28 221 214 206 194 178 158 149 139 129 117 105 9 32 143 ! 38 2 32 211 204 i 96 184 1 68 148 139 129 119 107 095 9 28 142 39 2 36 2O I 194 i 86 174 158 138 129 119 109 097 08 5 9 24 141 40 240 O.I92 0.185 0.177 0.165 0.149 0.129 0.120 O.IIO O.I 00 0.088 0.076 9 2O 140 42 2 48 175 168 160 148 132 112 103 093 083 071 059 9 12 138 44 2 56 158 i 5i 143 '3 1 "5 095 O 86 076 066 054 042 9 4 136 46 3 4 136 i 28 116 IOO 080 071 06 1 05 I 039 027 8 56 134 48 3 12 129 122 114 1 02 086 066 057 047 037 025 013 8 48 132 5O 3 20 0.116 O.IO9 O.IOI 0.089 0.073 0-53 0.044 0.034 0.024 0.012 O.OOO 8 40 130 52 3 28 104 O 97 089 077 061 041 032 O2 2 012 OOO 9.988 8 32 128 54 3 36 092 85 077 065 049 029 O2O 010 OOO 9.988 976 8 24 126 56 3 44 082 75 067 55 039 019 010 ooo 9.990 978 966 8 16 124 58 3 52 072 06 5 057 045 029 O09 OOO 9.990 i 980 9 68 95 6 8 8 122 60 4 0.063 0.056 0.048 0.036 0.020 0.000 9.991 9.981 9.971 9-959 9-947 8 12O 65 4 20 043 036 028 016 OOO 9.980 971 961 951 939 927 7 40 115 70 4 40 027 O2O 012 ooo 9-9 84 964 955 945 935 923 911 7 2O no 75 5 015 008 000 9.988 972 952 943 933 923 911 899 7 O 105 80 5 20 007 OOO 9.992 980 964 944 935 925 915 903 891 6 40 1OO 90 6 o.ooo 9.993 9-985 9-973 9-957 9-937 9.928 9.918 9.908 9.896 9.884 6 O 90 0.07 0.16 024 0.32 0.40 1 045 0.50 o 50 0.60 0.60 Differences for 0.l of Altitude or Latitude. Differences for Side- Arguments are the same as for Azimuths under Log A. TABLE XLVIII. 155 Time-Alt. Azimuths: L,og B. Hour-Angle Altitude or Latitude. Hour-Angle or Position or Position- Angle. Angle. Arc. Time. 40 D 42 44 46 48 50 51 52 53 54 55 Time. Arc. o h m h m o -f- oo 4 oo 4 oo 4 oo 4 oo 4 oo 4 4 oo 4 oo 4 oo 4 oo 12 180 1 4 1.642 1.629 1.615 i. 600 I. 583 1.566 i-557 1-547 1-537 1.527 1.517 11 56 179 2 8 341 328 3H 299 282 265 256 246 236 226 216 11 52 178 3 12 165 152 138 123 1 06 089 080 070 060 050 040 11 48 177 4 O 16 040 027 013 0.998 0.981 0.964 0-955 0-945 0-935 0.925 0.915 11 44 176 5 20 0.944 0.931 o. }i7 o. 902 0.885 0.868 0.859 0.849 0.839 0.829 0.819 11 40 175 6 O 24 865 852 i 3?8 ^23 806 789 780 770 760 740 11 36 174 7 O 28 798 7*5 771 756 739 722 703 693 683 673 11 32 173 8 32 740 727 713 < 598 58i 664 655 6 45 635 625 615 11 28 172 9 36 690 677 663 648 631 614 605 595 585 575 565 11 24 171 10 040 0.644 0.631 0.617 0.602 0.585 0.568 0-559 0-549 0-539 0.529 0.519 11 20 170 11 O 44 603 590 576 561 544 527 518 508 498 488 478 11 16 169 12 048 566 553 539 524 507 490 481 < tfi 46! i I5i 441 11 12 168 13 O 52 532 t ;?9 505 490 473 45 6 447 437 427 i 407 11 8 167 14 56 500 487 473 458 441 424 415 405 395 385 375 11 4 166 15 O 0.471 0.458 0.444 0.429 0.412 o-395 0.386 0.376 0.366 0.356 0.346 11 165 16 4 444 43i 417 402 385 368 359 349 339 329 319 10 56 164 17 8 418 405 39i 376 3 6 9 342 333 323 ,'3 303 293 10 52 163 18 12 394 381 367 352 335 309 299 289 279 269 1048 162 19 16 372 359 345 330 513 1 9 6 287 277 267 257 247 1044 161 2O 20 0.350 0.337 0-323 0.308 0.291 0.274 0.265 0-255 0.245 0-235 0.225 1040 16O 21 24 33 317 33 288 271 254 245 235 225 215 205 1O 36 159 22 28 3 11 298 284 269 252 235 226 216 206 196 186 10 32 158 23 32 292 279 265 250 2 33 216 207 197 187 177 167 10 28 157 24 36 275 262 248 233 216 199 190 80 170 1 60 150 10 24 156 25 40 0.258 0.245 0.231 0.216 0.199 0.182 0.173 0.163 0.153 0-143 0.133 1O 2O 155 26 44 242 229 215 200 183 166 147 J 37 127 117 10 16 154 27 48 227 214 200 I8 5 68 I5 1 142 132 122 112 102 10 12 153 28 52 212 i 99 185 170 ] 53 136 127 117 107 C ,97 08 7 10 8 152 29 56 198 i 85 171 I 5 6 149 122 "3 103 093 C 83 073 10 4 151 30 2 0.185 0.172 0.158 0.143 0.126 0.109 O.I 00 0.090 0.080 0.070 0.060 10 150 31 2 4 172 159 i 45 130 "3 096 087 077 067 057 047 9 56 149 32 2 8 1 60 147 133 n8 101 084 075 065 055 045 35 9 52 148 33 2 12 148 i 35 121 1 06 089 072 063 53 043 033 023 9 48 147 34 2 16 137 124 no 095 078 06 1 052 042 032 O22 012 944 146 35 2 2O 0.126 0.113 0.099 0.084 0.067 0.050 0.041 0.031 0.021 O.OII O.OOO 9 40 145 36 2 24 "5 102 088 073 056 039 036 ' 020 OIO 000 9.990 9 36 144 37 2 28 105 C 92 078 063 046 029 O2O OIO OOO 9.990 980 9 32 143 38 2 32 095 c 82 068 053 036 OI9 OIO ooo 9-9 9 980 970 9 28 142 39 2 36 085 072 058 043 026 00 9 000 9.990 9 80 970 960 9 24 141 40 2 40 0.076 0.063 0.049 0.034 0.017 0.000 9.991 9.98! 9.971 9.961 9-951 9 20 14O 42 248 059 046 3 2 017 ooo 9.983 974 9 6 4 954 944 934 9 12 138 44 2 56 042 029 OI 5 000 9-C 83 966 957 947 937 927 917 9 4 136 46 3 4 027 014 000 9-985 968 951 942 932 922 9 12 8 56 134 48 3 12 013 000 9.986 971 954 937 928 918 908 S 9 8 888 8 48 132 50 3 20 o.ooo 9.987 9-973 9.9S8 9.941 9.924 9-9I5 9-9 5 9.8 95 9.885 9-875 8 40 130 52 3 28 9.988 975 961 946 939 912 903 g 93 8 8} 873 863 8 32 128 54 3 36 976 963 949 934 917 900 891 881 871 861 851 8 24 126 56 3 44 966 953 939 924 907 890 88 1 871 861 851 841 8 16 124 58 3 52 956 943 929 914 897 880 871 86 1 851 841 831 8 8 122 60 65 4 4 20 9.947 927 9-934 J9-92Q 19.905 r4 900! 88q 9.888 868 9.871 851 9.862 842 9.852 832 9.842 822 9.832 812 9.822 802 8 7 40 120 115 70 440 911 98 8 84 8 69 852 835 826 816 806 9 6 786 7 20 110 75 5 899 886 872 8^7 840 823 814 804 7 Q4 . 4 774 7 105 80 5 20 891 878 864 849 832 815 806 796 7 S6 776 766 640 100 90 6 9.884 9.871 9.857 9.842 9.825 9.808 799 9-789 9-779 9-7 6 9 9-759 6 90 0.65 0.70 o-75 0.85 0.85 1 0.9 1 i.o I.O I.O I.O Differences for 0-1 of Altitude or Latitude. Differences for Side- Arguments are the same as for Azimuths under Log A. 156 TABLE XL VIII. Time- A It . Azimuths : Log B. Hour-Angle or Position- Altitude or Latitude. Hour-Angle or Position- Angle. Angle. Ar;. Time. 55 56 57 58 59 60 61 62 63 64 65 Time. Arc. o h in h m + 00 -\- 00 + 00 -f 00 -}- oo 4- oo + 00 + 00 _i_ oo _|_ oo 4- 00 12 180 1 2 4 8 1.517 216 1.506 205 1.494:1.482 193! 181 1.470 169 1-457 156 1.444 1.430 1.415 143 1291 H4 1.400 099 1.384 083 11 56 11 52 179 ! 178 i 3 O 12 040 029 01 7 j 005 O.Q 93 0.980 0.967 0-953 0.938 0.923 0.907 11 48 177 4 16 0.915 0.904 0.892 !o.88o 8 bS 855 842 828. 813 798 782 11 44 176 5 20 0.819 0.808 0.796 0.784 0.772 0-759 0.746 0.732 0.717 0.702 0.686 11 40 175 6 24 740 729 717 705 693 680 667 6 53 638 623 607 11 36 174 7 28 673 662 650 638 626 613 600 586J 5 71 55 6 54 11 32 173 i 8 32 615 604 592 580 568 555 542 528 513 498 482 11 28 172 9 36 565 554 1 542 53 518 505 492 478 463 448 43 2 11 24 171 1O 040 0.519 0.508 10.496 '0.484 0.472 0-459 0.446 0.432 0.4I7J0.402 0.386 li 20 170 11 O 44 478 467 455: 443| 43 1 418 405 39i 376 361 345 11 16 169 I j_^ O 48 441 43 418 406 394 381 368 354 i 339 324 38 11 12 168 13 O 52 407 396 384 372 360 347 334 320! 305 290 274 11 8 167 14 56 375 3 6 4 352 340 328 302 2881 273 258 242 11 4 166 15 o 0.346 0-335 0.323 0.311 0.299 0.286 0.273 0.259 0.244 10.229 0.213 11 165 16 319 308 296 284 272 259 246 232 1 217 202 j 1 86 10 56 164 17 8 293 282 270 258 246 233 220 206 191 176 160 1O 52 163 I 18 12 269 258 246 234 222 209 196 182 167 152 136 1O 48 164 I 19 16 247 236 224 212 2OO 187 174 160 ' 145 i 30 114 10 44 161 ! 20 20 0.225 0.214 0.202 0.190 0.178 0.165 0.152 0.138 0.123 - I0 8 0.092 1O 4O 169 21 24 205 194! l82j I /o I 5 8 145 132 ii8j 103 088 ! 072 10 36 159 22 28 186 175 l63J 151 139 126 113 099 084 1 069 i 053 1O 34 158 23 32 167 156 144 132 1 2O 107 094 080 1 065 050 034 1O 28 157 24 36 150 139 127 "5 I0 3 090 077 063 ! 048 i 033 i 017 10 24 a .* i 25 4O 0.133 O.I22 O.IIO 0.098 0.086 0.073 0.060 0.046 0.031 o.o 1 6 j o.ooo 1O 2O 155 ! 26 44 117 1 06 094 082 070 057 044 030 015 ooo 9.984 14> 16 ! 154 27 48 1 02 091 079 ! 067 055 042 029 015 ooo 985 969 1O 12 i 153 28 52 087 076 064 i 052 040 027 014 i ooo 9.985 9.970 954 10 8 i 152 29 56 073 062 050 038 O26 013 ooo 9.986 971 956 940 1O 4 151 30 2 O 0.060 0.049 0.037 0.025 O.OI3 0.000 9-987 9-973 9.958 9-943 9-927 10 150 31 2 4 047 036 024; OI2 OOO 9.987 974 960 945 930 914 9 56 149 32 2 8 035 024 OI2 ! OOO 9.988 975 962 948 933 918 902 9 52 148 j |33 2 12 023 012 ooo 19.988 976 963 95o| 93 6 921 906 890 9 4 147 | 34 2 16 012 OOO 9.989 977 965 952 939 925 909 895 879 9 44 146 35 2 20 O.OOO 9.990 9.978 9.966. 9.954 9.941 9.928 9.914 9.899 9.884 9.868 9 40 145 36 2 24 9.990 979 967 955 943 93 917 903 888 873 857 9 36 144 37 2 28 980 969 957 945 933 920 878 863 847 9 32 143 38 2 32 970 959 947 935 923 910 7 883 868 853 837 9 28 142 39 2 36 960 949 937 925 913 900 7 873 858 843 827 9 24 141 40 240 9-95 ! 9.940 9.928 9.916 9-904 9.891 9.878 9-864 9.849 34 9.818 9 20 140 42 2 48 934 923 9 ii 899 887 874 86 1 847 832 8i7 80 1 9 12 138 44 2 56 917 9 06 8 94 882 870 857 844 830 815 800 784 9 4 136 46 3 4 902 891 879 867 855 842 829 815 800 785 8 56 134 48 3 12 888 877 865 853 841 828 815 801 786 8 48 132 5O 3 20 9.875 9.864 9-852 9.840 9.828 9.815 9.802 9788 8 4O 130 52 3 28 863 852 840 828 816 803 790 8 32 128 \ 54 3 36 851 840 828 816 804 791 8 24 126 56 3 44 841 830 818 806 7Q4 8 16 124 58 3 52 831 820 808 796 8 8 122 60 4 9.822 9.811 9-799 8 120 65 4 20 802 791 7 4O 115 70 440 7 20 110 75 5 7 105 ! 80 5 20 6 4O 100 | 90 6 6 90 ! 1 x 2 1.3 1 1.3 1 1.4 1 1.5 i-S 1.6 Differences for 0.l of Altitude or Latitude. Differences for Side- Arguments are the same as those for Azimuths under Log A. TABLE XLVIII. 157 Time-Alt. Azimuths: Log B. Hour-Angle or Position- Altitude or Latitude. Hour-Angle or Position- Angle. Angle. Arc. Time. 65 66 67 68 69 70 71 72 73 74 75 Time. Arc. \ o h m h m + 00 + 00 + oo + 00 + 00 + 00 + 00 + oo + 00 + 00 + 05 12 18O 1 4 1.384 1.367 I -35 1.331 1.312 1.292 1.270 1.2 48 1.224 1.198 1.171 11 56 179 2 8 083 066 049 030 01 1 0.991 0.979 0.947 0.923 0.897 0.870 11 52 178 3 4 12 O 16 0.907 782 0.890 765 0.873 748 0.854 0.835 729 710 815 690 1^ 771 636 747 622 721 596 694 569 11 48 11 44 177 176 5 20 0.686 0.669 0.652 0.633 0.614 0-594 0.572 -5 5 0.526 0.500 0-473 11 40 175 6 O 24 607 590 573 5 54 5 35 493 47i 447 421 394 11 36 174 7 O 28 540 523 506 4 7 4 68 448 426 404 380 3 54 327 11 32 173 8 32 482 465 448 429 410 390 34 6 322 2 ?6 269 11 28 172 9 36 432 415 398 379 360 34 3 i8 296 272 2 4 6 219 11 24 171 10 O 4O 0.386 0.369 0.352 0-333 -3i4 0.294 0.272 0.250 0.226 O.2OO 0.173 11 20 170 11 044 345 328 3 II 292 1 273 253 231 209 185 159 132 11 16 169 12 O 48 308 291 274 2 55 236 216 194 172 148 122 095 11 12 168 13 O 52 274 257 240 221 202 182 1 60 138 114 088 061 11 8 167 14 56 242 225 208 189 1 70 150 128 1 06 082 056 029 11 4 166 15 O 0.213 0.196 0.179 0.160 0.141 O.I2I 0.099 0.077 0-053 O.O27 O.OOO 11 165 16 4 1 86 169 i 52 I 33 114 094 072 050 026 OOO 9-973 10 56 164 17 8 1 60 126 107 088 068 046 024 000 9-974 947 10 52 163 18 12 136 119 102 083 064 044 022 ooo 9.976 95 923 10 48 162 19 16 114 097 080 O6 1 042 022 000 9.978 954 928 901 10 44 161 2t> 20 0.092 0.075 0.058 0.039 :o.o2o O.OOO 9.978 9-95 6 9-932 9-9 36 9.879 10 40 160 21 21 072 055 O }8 019 ooo 9.980 958 936 912 886 859 1.0 36 159 22 28 053 036 019 ooo 9.981 961 949 i7 893 867 840 10 32 158 23 32 034 017 ooo 9.981 962 942 920 98 874 8 iB 821 10 28 157 24 36 017 000 9.983 964 945 925 93 881 857 831 804 10 24 156 | 25 4O 0.000 9-983 9.966 9.947 9.928 9.908 9.886 9.864 9.840 9.814 9787 1O 2O 155 26 44 984 967 9 5 931 ! 912 892 870 8 48 824 7 58 10 16 154 27 48 969 952 9 916 i 8 97 877 855 833 809 7 S} 10 12 153 28 52 954 937 920 9 91 c> O 82 862 840 818 794 10 8 152 29 56 940 923 906 8 87 868 848 826 804 1O 4 151 30 2 9.927 9.910 9.893 9.874 9-855 9.835 9.813 9.791 10 150 ! 31 2 4 914 897 880 861 842 822 800 9 56 149 32 2 8 902 885 868 849 830 810 788 9 52 148 33 2 12 890 873 856 837 818 798 9 48 147 34 2 16 879 862 845 826 807 787 9 44 146 35 2 2O 9.868 9.851 9-834 9.815 9.796 9 4O 145 36 2 24 857 840 823 804 9 36 144 37 2 28 847 830 *3 794 9 32 143 38 2 32 837 820 8 03 9 28 142 39 2 36 827 810 793 9 24 141 40 2 40 9.818 9.801 9 20 140 42 2 48 801 9 12 138 44 2 56 9 4 136 46 3 4 8 56 134 48 3 12 8 48 132 50 3 2O 8 40 130 52 3 28 8 32 128 54 3 36 8 24 126 56 3 44 8 16 124 58 3 52 8 8 122 60 4 8 120 65 4 20 7 40 115 70 440 7 2O 110 75 5 7 105 80 5 20 6 40 1OO 90 6 6 O 90 1.7 1 1.7 1.9 1.9 2.0 1 2.2 j 2.2 2.4 2.6 2.7 Differences for 0.l of Altitude or Latitude. Differences for Side- Arguments are the same as those for Azimuths under Log A. 158 TABLE XLIX. Error of Time- Alt. Azimuth for an Error of O m .2, or 12% in Hour-Angle. Hour- Angle. Az. i Az. 6 5 20 4 40 4 O ! 3 20 2 40 2 1 40 1 20 100 40 20 1 o O.O o' o' o' o' o' o' O.O o 0.0 c O.O O.O o 0.0 Indet, 180 IO O o i I I i o o I I 00 170 20 o I I I I 2 o I I I 2 oo 16O 30 o I i 2 2 3 I I I 2 3 00 15O 40 o 1 2 2 3 4 I 1 2 2 5 00 140 50 O.O I 2 3 4 6 O.I 0.2 0.2 0. 3 0.7 oo 13O 60 o 2 3 4 6 9 2 2 3 5 I.O 00 12O 65 o 2 4 5 8 ii 2 3 4 6 2 00 115 70 3 5 7 10 14 3 4 5 8 6 00 no 75 o 2 4 7 9 13 19 4 5 7 i.i 2.1 00 1O5 8O 0.0 3 6 IO I5 20 29 0.6 0.8 i.i 1.6 3-2 00 100 81 o 3 7 ii 16 23 33 7 9 2 8 6 00 99 82 4 8 12 18 25 37 8 I.O 3 2.0 4.1 00 98 83 4 9 H 21 29 42 9 i 5 3 6 00 97 84 o 5 10 17 24 34 49 I.O 3 8 7 5-4 oc 96 85 O.O 6 I3 20 29 4i 60 1.2 1.6 2.1 3- 2 6.s 95 86 o 8 It) 2S 51 75 5 2.O 7 4.0 94 87 O IO 21 33 48 69 99 2.0 6 3.6 93 j 88 89 o 15 o 30 is 99 72 144 103 205 149 297 6J 3-9 92 91 9O Iiidet. 00 00 oo 00 00 00 00 90 6 6 40 7 20 8 8 40 9 20 10 1020 1040 11 1120 1140 12 Az Az. Hour-Angle. TABLE L. Error of Time- Alt. Azimuth for an Error of 3' in Altitude or Declination. Altitude or Declination. Az Az. 10 2O 30 40 5O 60 65 7O 75 80 85 90 o o' o' o' o' o' o' o' O.O o 0.0 o 0.0 O.O o 0.0 Indet. O | 180 IO o o i i i i O I I 00 17O 2O I i I i 2 o I I I 2 oo 16O 3O o I i 2 2 3 I I I 2 3 00 15O 40 I 2 2 3 4 I I 2 2 5 00 14O 5O I 2 3 4 6 O.I 0.2 0.2 0-3 0.7 oo 130 60 2 3 4 6 9 2 2 3 I.O 00 12O 65 o 2 4 5 8 ii 2 3 4 6 2 00 115 7O 75 2 3 4 5 7 7 9 10 13 H 19 3 4 4 5 5 7 8 i.i 6 2.1 00 00 no 105 80 3 6 10 I5 20 29 0.6 0.8 i.i 1.6 3-2 oo 100 | 81 82 o 3 4 2 ii 12 16 18 23 25 33 37 I 9 I.O 2 3 8 2.O 6 4.1 00 oc 99 98 ; 83 4 9 14 21 29 42 9 i 3 6 00 97 ! 84 o 5 10 17 24 34 49 I.O 3 8 7 5-4 00 96 - 85 6 13 20 29 4 1 60 1.2 1.6 2.1 3-2 6.5 95 ! 86 87 o o 8 IO 1 6 21 25 .36 '48 69 75 99 5 2.0 2.0 6 3! 4.0 94 1 93 88 89 '5 30 32 63 99 72 144 103 205 149 297 6.? 3-9 92 91 90 Indet 00 oo 00 00 00 00 90 TABLE LI. 159 Transition- Azimuths. T _4 Change of Altituds in One Minute of Time. Li at. 01 2 3 4' 5 6 7' 8' 9 10 11 12' 13 14' 15'' o o o o o o o o o o o o o o o O o.o 3-8 7-7 "5 15-5 19-5 23.6 27.8 32.2 3 6 -9 41.8 47-2 53- 1 60. 1 69.0 90.0 1 4 o ! 3.8 7.7 II.6 15-5 19.5 23.61 27.9 32.3 1 37.0 41.9 47-3 53-3 60.3 69.3 o 3-9 7-7 II. 7 15.6 19.7 23.8 28.1 32-6} 37-3 42.3 47-8 53-9 61.1 70.5 12 o 3.9! 7.8 | 1 1.8 15.8 19.9 24.1 28.5 33-o | 37-8 43-o 48.6 54-9 62.4 72.6 16 o 4.0 1 8.0 I2.O 16.1 20.3 24.6; 29.0 j 33.7 38.6 43-9 49-7 5 6 -3 64.4! 7 6-I 20 o.o | 4.1 8.2 12.3 16.5 20.8 25.2 i 29.8! 34.6 39.7 45-2 5i-3 58-3 67-3 83.3 22 o 4.1 8.3 12.4 16.7 21. 1 2 5-5i 3-2: 35- 1 40.3 46.0 52-3 59-6 69.2 24 o 4.2 8.4 12.6 17.0 21.4 26.0 30.7 35.7 41.0 46.9 534 61.1 71.6 26 o 4.2 : 8.5 12.8 *7-3 21.8 26.4! 31.3 1 36.4 41.9 47-9 54-7 62.9 74.6 28 o | 4.3 8.7 I3-I 17.6 22.2 26.9 31.9 37-2 42.8 49.0 56.1 65.0 79.0 3O o.o 4.4 8.8 r 3-3 17.9 22.6 27.5 32.6 38.0 43-8 5-3 57-9 67.5 31 o 4.5 8.9 13-5 18.1 22.9 2 7-8i 33-o 38-5 444 51.0 58.8 68.9 32 o 4.5 9.0 13-6 18.3 23.1 28.1 3341 39- 45- 51.8 59-8 70.6 33 o 4.6 9.1 13.8 18.5 234 28-5 33-8 39-5 45-7 52.6 61.0 72.5 34 o 4.6 9.2 14.0 18.8 23-7 28.8 1 34.3 40.0 46.4 53-5 62.2 74.8 1 | 35 0.0 4-7 ! 9.4 14.1 19.0 24.0 29.2 34.7 40.6 47.1 54-5 63-5 77.6 36 4.7 ! 9.5 H-3 19.2 24-3 29.6; 35.2 41.2 47-9 55-5 65.0 81.4 i 37 o 4.8 j 9.6] 14.5 19-5 24.7 30.0 1 35.7 41.9 48.7 56.6 66.7 | j 38 o 4.8 9.7 14.7 19.8 25.0 3-5 i 3 6 -3 42.6 i 49.6 57-8 68.5 1 39 o 4.9 j 9.9 14.9 20. i 254 31.0; 36.9; 43.3 50.5 59-i 70.7 1 . 4O 41 O.O 5-o 5-i IO.O 10.2 i5-i J 54 20.4 20.7 25-8 26.2 3i-5 i 37-5 32.0 ! 38.2 44.1 45-o 51.6 52.6 60.5 62.0 3 42 o 5-2 10.3 15.6 21.2 26.6 32.6 38.9 45.9 53-8 63.8 80.7 43 o 5.2 10.5 15-9 21.4 27.1 33- i 39-6 46.8 55-i 65-7 44 o 5-3 I io-7 16.1 21.8 27.6 33-8 40.4 47-8 56.5 67.9 45 o.o 5.4 IO-9 16.4 22.1 28.1 344 41.3 49:0 .58.0 70.5 46 o 5.5 ii. i 16.7 22.6 28.7 35- 1 42.2 50.2 59-7 73-7 47 o 5.6 11.3 17.0 23.0 29-3 35-9 43-2 514 77-8 48 49 o 5-7 o i 5.8 "5 1.1.7 17.4 17.7 23-5 24.0 29.9 30.5 36.7 37-6 44.21 52.9 45-3 544 63^7 66.1 85.1 50 o.o 6.0 12.0 18.1 24-5 31.2 38.5 46.5 56.1 69.0 51 o 6.1 12.2 18.5 25.1 32.0 39-5 47-9 57-9 72.4 52 o 6.2 12.5 18.9 25-7 3 2.8 40-5 49-3 60.0 77.0 53 o 6.4 12.8 19.4 26.3 33-6 41.6 50.8 62.4 85.6 54 o 6.5 !3-i 19.9 27.0 34-5 42.9 52.5 65.1 55 0.0 6.7 134 20.4 277 35-5 44.2 544 68.4 56 o 6.8 13.8 20.9 28.5 36.6 45-7 56.6 72.5 57 o 7-o 14.2 21.5 ' 2 9-3 37-7 47-3 59-o 78.3 58 7.2 14.6 22.2 30.2 39-o 49.0 61.7 59 74 15.0 22.8 3 1.2 40-3 5-9 65.0 60 o.o 7-7 *5-5 2 3 .6 32.2 41.8 53-i 69.0 61 o 7-9 16.0 24.4 334 434 55.6 74-3 62 8.2 16.5 25.2 ! 34-6 45-2 584 83-7 63 8.4 17.1 26.1 36.0 47.2 61.8 64 o 8.7 17.7 27.1 37-5 49-5 65.8 65 o.o 9.1 18.4 28.2 39-i 52.1 71.2 66 9.4 19-1 29.4 41.0 55- 79-5 67 o 9.8 19.9 30.8 ! 43- 68 o 10.3 2O.8 3 2 -3 ' 454 62^8 69 10.7 21.8 33-9 48.1 68.5 7O o.o II. 2 22.9 35-8 ! 51-2 77.1 71 1 1.8 24.2 37-9 55- 72 12.5 25.6 40-3 59-6 73 o 13.2 27.1 43-2 65.8 74 14.0 28.9 46.5 I I ! | The Azimuths of this Table correspond to the stated changes in the Altitude of a Heavenly Body in One Minute of Time. 160 TABLE Lll. Direct Bearings of a Fixed Object : Limiting Distance of the Object. 1 i Si 2 a Distance-Ratio, or Distance of the Object divided by the Radius of owing. Sj? ! If, ll ' K 30 40 50 75 100 150 200 250 300 .{.*!> 400 50O 600 700 SOU 90O 1OOO K Ftet. \.M .V. -V. .v. jr. V V .V. X A'. .V. A. M. .V. M. A .!/. A". M. A'. M. A. ./. A .)/ V. M. \. \J .V. M. \. M. ,,.,. 1 50 0.2 0-3 0.4 0.6 0.8 1.2 1.6 2.1 2.S 2.9 3-3 4 5 6 7 7 I 8 50 75 0.4 0.6 0.9 1.2 1.8 2-S 3-i 3-7 4-3 4.9 6 9 i 10 ii 12 75 1 1OO O-5 0.7 0.8 1.2 1.6 2.S 3-3 4.1 4.9 S-7 6.6 8 10 ii IT JC 16 100 125 0.6 0.8 I.O 1.5 2.0 4.1 S-i 6.1 7.2 8.2 10 12 14 16 18 20 125 150 0.7 I.O .2 1.8 2-5 3-7 4.9 6.2 7-4 8.6 9.9 12 '5 17 2O 22 25 15O 175 0.9 1.2 4 2.2 2.9 4-1 S-7 7.2 8.6 IO.I ii. 5 14 17 20 23 26 29 175 200 I.O 1.3 .6 2.4 3-3 4-9 6.6 8.2 9-9 II -5 13.1 16 2O 23 2b 30 33 2OO ' 225 i.i '5 .8 2.8 3-7 S-S 7-4 9.2 ii. I 12.9 14.7 18 22 26 30 33 225 250 1.2 1.6 2.O 3-1 4.1 6.2 8.2 10.3 12.3 14.4 16.4 21 2 5 29 33 38 42 250 275 1.4 1.8 2-3 3-4 4-5 6.8 9.0 1 1-3! 13-5 15.8 1 8.0 23 27 32 36 45 275 300 *$ 2.0 2.S 3-7 4.9 7-4 9.9 12.3 ! 14-8 17.2 19.7 2S 30 34 39 44 49 30O 325 1.6 2.1 2.7 4.0 .5-3 8.0 10.7 13.3 1 16.0 18.7 21.4 27 32 37 43 48 325 350 375 400 i!8 2.0 2-3 2. 9 3-3 4-3 4.6 4.9 5.8 6.2 6.6 8.6 9.2 9-9 11.5 12.3 14.4 '5-4 16.4 17.2 18.5 19.7 20. i 21.6 23.0 23.0 24.6 26.3 29 31 33 34 37 40 40 43 47 46 49 53 52 1 67 350 375 4OO The Distances are expressed in Nautical Miles of 6,086 feet. 1 TABLE LIII. Direct Bearings of a Fixed Object: Parallactic Errors of the Bearings. 3.S Distance-Ratio, or Distance of the Object divided by the Radius of Swing. *, o A |l 30 40 50 75 100 150 200 250 300 350 400 500 600 7OO 800 900 1000 I s o o o o o o '0 | O.O O.O O.O O.O O.O O.O 0.0 O.O 0.0 0.0 O.O O.O O.O 0.0 O.O O.O O.O O; 10 3 2 2 I I I I O o O o 10 2O 7 S 4 3 2 I I I I O o o O o 20 i 30 I.O 7 6 4 3 2 I I I I I O o O 30 i 40 2 9 7 5 4 2 2 I I I I I I o o o 1O 50 i-S i.i 0.9 0.6 0.4 0-3 0.2 0.2 O.I O.I 0. O.I o. o. O.I O.I 0.0 50 6O 7 2 I.O 7 s 3 2 2 2 I I I o oo 70 8 3 i 7 4 3 2 2 2 I I I 70 9 4 2 8 6 4 3 2 2 2 I I I 80 ! 90 1.9 1.4 1.2 0.8 0.6 0.4 0.3 0.2 O.2 0.2 O.2 O.I 0. 0. O.I 0. O.I 90 100 1.9 1.4 1.2 0.8 0.6 0.4 0.3 O.2 O.2 0.2 O.I O.I o. o. 0.1 0.. O.I 100 110 8 3 I 7 s 4 3 2 2 2 I I I I 110 120 7 2 7 5 3 2 2 2 I I I I 120 13O 5 I 0.9 6 4 3 2 2 I I ' I I 130 i 140 1.2 O.q 0-7 0.5 0.4 0.2 0.2 O.I O.I O.I O.I O.I O.I O.I O.Q. 0.0 0-0 140 15O 7 6 4 3 2 I I I I I I o o o o 150 I 16O 0.7 S 4 2 I I I I O o o o o 1 92 54 49 69 IO 1 2 3 o o 13 15 ii 15 78 83 99 15.06 62 11 89 20 3 2 3 4 17 19 89 13 79 77 99 40 4 21.5 0.0 4.19 8.23 11.94 15.20 17.87 ,9.86 21.09 21.50 21.5 6 o 21 27 12.00 28 96 96 19 60 6 7 23 30 06 35 18.04 20.05 2 9 70 7 8 25 34 II 42 13 H 38 80 8 9 o 27 38 17 49 21 23 48 90 9 22.O 1 o.o 4.29 31 8.42 46 12.22 28 I5-56 63 l8.29 37 20.33 42 "if 22.00 IO 22.O 1 2 o 33 50 33 70 46 5i 78 20 2 3 o 35 53 39 77 54 60 87 3 3 4 37 57 45 84 62 69 97 40 4 22.5 0.0 4-39 8.61 12.50 I 5-9 I 18.71 20.79 22.07 22.50 22.5 6 o 4 1 65 56 98 79 88 17 60 6 7 o 43 69 61 16.05 Q 97 26 70 7 8 45 72 67 12 96 21.06 36 80 8 ! 9 47 76 72 19 19.04 15 46 90 9 23.0 0.0 4-49 8.80 12.78 16.26 19.12 21.25 22 -56 23.00 23.0 1 o 51 84 83 34 21 34 66 10 1 2 53 88 89 29 43 76 20 2 3 4 II 92 95 95 13.00 48 55 46 e 95 3 40 3 4 23.5 0.0 4-58 8-99 13.06 16.62 19.54 21.71 23-05 23.50 23.5 6 o 00 9-03 ii 69 62 80 15 60 6 7 o 62 07 17 76 71 90 25 70 7 8 o 64 ii 22 83 79 99 35 80 8 9 66 15 28 90 7 22.08 45 90 9 24.0 0.0 4.68 9.18 '3-33 16.97 19.95 22.17 23-54 24.00 24.0 TABLE LIV. 165 Products of Arcs multiplied by the Sines of the Rhumbs. Arcs. Sines of the Rhumbs. Arcs. t i 0.000 0.195 J0.383 O.556 0.707 0.831 0.924 g O.981 l.OOO 240 o o o.o 4.68 9-18 13.33 o 16.97 o o 19.95 22.17 2354 o 24.00 24.0 1 o 70 22 39 17.04 20.03 26 64 IO 1 2 o 72 26 44 II 12 36 74 20 2 3 o 74 30 50 18 20 45 83 3 3 4 o 76 34 56 25 29 54 93 40 4 24.5 o.o 4.78 9.38 13.61 17.32 20.37 22.63 24-03 24.50 24.5 6 o 80 41 66 39 45 72 '3 60 6 7 o 82 45 72 46 53 82 2 3 70 7 8 o 84 49 78 54 62 9 1 33 80 8 9 o 86 53 83 61 70 23.00 42 90 9 25.0 o.o 4.88 9-57 13.89 17.68 20.78 23.09 24.52 25.00 25.0 1 o 90 60 94 75 87 19 62 10 1 2 o 92 64 14.00 82 95 28 7 2 20 2 3 93 68 05 89 21-03 37 81 3 3 4 i o 95 72 ii 96 ii 46 9i 40 4 25.5 o.o 4-97 9.76 14.16 18.03 21. 2O 23.56 25.01 25-50 25.5 6 o 99 79 22 IO 28 65 ii 60 6 7 o 5.01 83 28 17 37 74 21 70 7 8 o 3 87 33 25 45 83 30 80 8 9 5 9 1 39 32 53 93 4 90 9 26.0 0.0 5.07 9-95 14.44 18.39 21.62 24.02 25.50 26.00 26.O 1 o 09 99 5 46 70 ii 60 10 1 2 o ii 53 78 20 7 20 2 3 I 3 06 61 60 87 3 80 3 3 4 o 15 IO 67 67 95 39 89 40 4 265 o.o 5.17 10.14 14.72 18.74 22.03 j 24.48 25-99 26.50 26.5 6 19 18 78 81 12 57 26.09 60 6 7 o 21 22 83 88 20 67 19 70 7 8 23 25 89 95 28 76 2 9 80 8 9 25 29 94 19.02 36 85 38 90 9 27.0 o.o 5.27 IO -33 15.00 19.09 22.45 24-95 26.48 27.00 27.O 1 o 29 37 06 16 53 25-04 58 IO 1 2 o 31 ii 23 62 13 68 20 2 3 33 45 17 3 1 7 22 78 3 3 4 o 35 48 22 38 78 31 88 40 4 27.5 o.o 5.36 10.52 15.28 19-45 22.86 2540 26.97 27.50 27.5 6 38 56 33 52 95 50 27.07 60 6 7 o 40 60 39 59 23-03 59 17 70 7 8 o 42 | 64 44 66 ir 68 27 80 8 9 o 44 j 68 50 73 20 77 37 90 9 28.0 o.o i 5.46 10.72 15-55 19.80 23.28 25-87 27.47 28.00 28.0 1 2 o o 48 75 79 61 67 87 94 36 45 96 26.05 II 10 20 I 2 3 o 52 83 72 2O.OI 53 15 76 3 3 4 o 54 87 78 08 61 .24 86 40 4 28.5 o.o 5' 56 > 10.91 15-83 20.15 23.69 26.33 27.95 28.50 28.5 6 58 94 89 22 78 42 28.05 60 6 7 60 98 94 29 86 5 2 15 70 7 8 o 62 11.02 16.00 37 94 61 25 80 8 9 o 64 06 5 44 24.03 70 35 90 9 29.O o.o 5-66 II. 10 i6.n 20.51 24.11 26.79 28.44 29.00 29.0 1 o 68 13 17 58 19 88 54 IO 1 2 o 70 17 22 65 28 98 64 20 2 3 o 72 21 28 72 36 27.07 74 3 3 4 o 74 25 33 79 44 16 84 40 4 29.5 o.o 5-75 11.29 16,39 20.86 24.52 27.25 28.93 29.50 29.5 6 77 33 44 93 61 35 29.03 60 6 1 7 o 79 36 50 21.00 69 44 70 7 8 9 o o 81 83 40 44 If 7 11 23 33 80 90 8 9 30 o.o 5.85 11.48 1 6.66 21.22 24-94 27.72 29-43 30.00 30.0 166 TABLE LIV. Products of Arcs multiplied by the Sines of the Rhumbs. Arcs. Sines of the Rhumbs. Arcs. o.ooo Si = 0.195 83 = 0.383 S 3 = 0.556 0.707 S 5 = 0.831 0924 S 7 -_ 0.981 1.000 30.0 o.o 5 8 5 11.48 i666 21.22 2494 o 27.72 29-43 30.00 30.0 1 '87 52 72 29 25-03 Si 53 10 1 2 o 89 56 78 36 ii 90 62 20 2 3 4 o 91 93 59 63 83 89 43 50 19 28 99 28.09 g 30 40 3 4 30.5 o.o 5-95 11.67 26.94 21-57 25.36 28.18 29.92 30-50 30.5 6 97 71 17.00 64 44 27 30.01 60 6 7 o 99 75 05 7i 52 36 ii 70 7 8 6.01 79 ii 78 61 46 21 80 8 9 o 03 83 17 85 69 55 31 90 9 31.0 o.o 6.05 11.86 17.22 21.92 25.78 28.64 30.41 31.00 31.O 1 o 07 90 28 99 86 73 51 10 1 2 o 09 94 33 22.06 94 82 61 20 2 3 10 98 .39 13 26.02 91 70 30 3 4 12 12.02 44 20 10 29.01 80 40 4 31.5 0.0 6.14 12.05 17.50 22.27 26.19 29.10 30.90 3 I -5 31.5 6 o 16 09 55 35 27 19 31.00 60 6 7 18 13 61 42 35 29 10 70 7 8 o 20 17 66 49 44 38 19 80 8 9 o 22 21 72 56 47 29 90 9 32.O 0.0 6.24 12.24 17.78 22.63 26.61 29.57 31-39 32.00 32.0 1 o 26 28 83 70 69 66 49 10 1 2 28 3 2 89 77 77 75 59 20 2 3 3 36 94 84 85 84 68 30 3 4 32 40 18.00 94 94 78 40 4 32.5 0.0 6.34 12.44 18.05 22.98 27.02 30.03 31.88 32-50 32.5 6 o 36 47 ii 23-05 10 12 98 60 6 7 38 17 13 19 22 32.08 7 7 8 o 40 55 22 20 27 3 1 17 80 8 9 42 59 28 27 35 40 27 90 9 33.0 0.0 6.44 12.63 18.33 23-34 27.44 30-49 32.37 33-0 33.0 1 46 67 39 4 i 5 2 58 47 10 1 2 o 48 7 44* 4 8 60 67 56 20 2 3 5 74 5 55 68 76 66 3 3 4 52 78 56 62 77 86 76 40 4 33.5 0.0 6-53 12.82 18.61 23.69 27.85 30.95 32.86 33-50 335 6 o 55 86 66 76 93 21.04 95 60 6 7 o 57 90 72 83 28.02 13 33-5 70 7 8 o 59 93 78 90 10 23 15 80 8 9 o 61 97 83 97 19 32 25 90 9 34.0 0.0 6.63 13.01 18.89 24.04 28.27 3M 1 33-35 34.00 340 1 o 65 5 94 12 35 51 45 10 1 2 67 09 19.00 19 43 60 20 2 3 4 o 69 11 05 ii 26 33 g 69 78 74 30 40 3 4 31.5 o.o 6-73 13.20 19.16 24.40 28.68 3I-87 33.84 34-50 31.5 6 o 75 24 22 47 77 97 94 60 6 7 8 o 77 79 32 2 7 33 I? 85 93 32.06 15 34.03 13 g 7 8 o 81 35 39 68 29.01 24 23 90 9 35.0 o.o 6.83 13-39 19.44 24.75 29.10 32-34 34-33 35-00 35.0 1 o 85 43 5 82 18 43 43 10 1 2 3 o 87 89 47 51 55 61 89 96 26 35 g 20 3 2 3 4 o 54 67 25-03 43 7 72 40 4 35.5 0.0 6.92 I3-58 19.72 25.10 29.51 32.80 34-82 35-50 35.5 6 o 94 62 78 18 60 89 92 60 6 7 o 96 66 83 2 5 68 98 35-02 70 7 8 o 98 70 89 32 77 33-o8 12 80 8 9 o 7.00 74 94 39 g 5 17 21 90 9 36.O 0.0 7.02 13-78 20.00 25.46 29-93 33-26 35.31 36.00 36.0 TABLE LIV. 16' Products of Arcs multiplied by the Sines of the Rhumbs. Arcs. Sines of the Rhumbs. Arcs. ! o.ooo 0.195 82 = 0.383 0.556 0.707 S.,= 0.831 0.924 0.981 1.000 36.0 o o.o o 7.02 I3 7 8 20.00 2546 2993 3326 35-31 36.00 36.O . 1 o 04 8l 05 53 30.01 35 41 10 1 2 o 06 85 II 60 10 44 51 20 2 3 08 89 16 67 18 54 61 3 3 4 o 10 | 93 22 74 26 63 70 40 4 36.5 0.0 7.12 13-97 20.28 25.81 30-35 33-72 35.80 36.50 36.5 6 o 14 14.00 33 88 43 81 90 60 6 7 o 16 04 39 95 9 1 36.00 7 7 8 o 18 08 44 26.02 60 34.00 10 80 8 9 o 20 12 50 09 68 09 19 90 9 37.0 o.o 7.22 I4.I6 20-55 26.16 30.76 34-18 36.29 37-00 37.0 1 o 24 20 61 23 85 28 39 10 1 2 26 23 66 3 93 37 49 20 2 3 o 28 27 72 38 31.01 46 3 3 4 o 3 31 78 45 ib 55 69 40 4 37.5 o.o 7.31 14-35 20.83 26.52 31.18 34-64 36.78 37-50 37.5 6 33 39 89 59 26 74 88 60 6 7 o 35 42 94 66 34 83 98 70 7 8 37 46 21.00 73 43 92 37.08 80 8 9 o 39 5 5 80 35- 01 17 90 9 38.0 0.0 7.41 14-54 21. II 26.87 31-59 35- 11 37-27 38.00 38.O 1 o 43 58 16 94 67 20 37 10 1 2 45 62 22 27.01 76 29 47 20 2 3 o 47 66 28 08 84 38 57 30 3 4 o 49 70 33 15 92 47 66 40 4 38.5 0.0 7-51 H-73 21-39 27.22 32.01 35-57 37-76 38.50 38.5 6 53 77 44 3 09 86 60 6 7 o 55 81 5 37 17 75 96 70 7 8 6 57 85 55 44 26 84 38.06 80 8 9 o 59 89 61 34 94 16 90 9 39.0 o.o 7-6! 14.92 21.67 27.58 32.42 36-03 38.26 39.00 39.0 1 o 63 96 72 65 51 J3 35 10 1 2 65 15.00 78 72 59 22 45 20 2 3 o 67 04 83 79 68 31 55 3 3 4 69 08 89 86 76 40 65 40 4 39.5 o.o 7.71 15-11 21.94 27-93 32.84 36.49 38.74 39-5 39.5 6 o 73 15 22.00 28.00 92 84 60 6 7 o 75 19 5 07 33- 01 68 94 70 7 8 77 23 ii 15 09 77 39-04 80 8 9 o 79 27 17 22 17 86 *4 90 9 4O.O o.o 7.80 I5-3I 22.22 28.29 33-26 36.95 39.24 40.00 4O.O 1 2 o o 82 84 * 27 33 36 43 34 42 37-04 H 33 43 10 20 1 2 3 4 o 86 88 42 46 39 44 50 57 . 59 23 33 i 30 40 3 4 40.5 0.0 7.90 I5-50 22.50 28.64 33-68 37-42 39-73 40.50 40.5 6 92 54 55 71 76 51 82 60 6 7 8 o o 94 96 if 61 67 78 85 84 93 60 92 40.02 7 8 9 o 98 65 72 92 34-oi 78 12 90 9 41.0 0.0 8.00 15.69 22.78 28.99 34-09 37-88 40.22 41.00 41.O 1 o 02 72 83 29.07 17 97 31 10 1 2 o 04 76 89 14 2 5 38.06 41 20 2 3 06 80 95 21 34 16 51 3 3 4 o 08 84 23.00 28 42 25 61 40 4 41.5 o.o 8.10 15.88 23.06 29,35 34-51 38.34 40.71 41.50 41.5 i 6 o 12 92 ii 42 59 43 80 60 6 ' 7 o 13 95 j 16 49 67 52 90 70 7 8 o 15 99 22 56 75 62 41.00 80 8 9 o 17 16.03 28 63 83 7 1 10 90 9 42.O o.o 8.19 16.07 23-33 29.70 34-92 38.81 41.19 42.00 42.0 168 TABLE L1V. Products of Arcs multiplied by the Sines of the Rhumbs. Arcs. Sines of the Rhumbs. Arcs. S = o.ooo 81 = 0.195 0.383 0.556 0.7O7 0.831 0.924 t m O.981 1.000 o 42.0 o o.o 8% o 16.07 2333 2970 3492 3 88i o 41.19 o 42.00 42.O 1 o 21 II 39 77 35-0 90 29 10 1 2 23 15 44 84 08 99 39 20 2 3 o 25 19 5 9' 16 39-oS 49 3 3 4 o 27 23 56 98 25 17 59 40 4 42.5 6 o.o 8.29 31 16.26 3 23.61 66 30-05 12 35-34 42 39.26 36 41.69 78 42.50 60 42.5 6 7 33 72 19 5 45 88 70 7 8 o 35 38 77 26 58 54 98 80 8 9 o 37 42 83 24 67 64 42.08 90 9 43.O o.o 8-39 16.46 23.89 30.41 35-75 39-73 42.18 43-0 43.0 1 o 49 94 48 83 82 28 10 1 2 o 43 53 24.00 55 92 91 37 20 2 3 o 45 57 05 62 36.00 40.00 47 30 3 4 o 47 pi ii 6 9 08 10 57 40 4 43.5 6 0.0 o 8-49 I6 ^ 24.17 22 30.76 36- i 7 2 5 40.19 28 42.67 77 43-50 60 43.5 6 7 53 72 28 90 33 37 86 70 7 8 o 55 76 33 97 42 47 96 80 8 9 o 57 80 39 31.04 50 56 43-o6 9 9 44.0 o.o 8.58 16.84 24-45 31.11 36.58 40.65 43.16 44-00 44.O 1 60 87 5 18 66 74 25 10 1 2 o 62 91 55 26 75 83 35 20 2 3 o 64 95 61 33 83 93 45 30 3 4 66 99 67 40 92 41.02 55 40 4 44.5 0.0 8.68 17-03 24.72 31-47 37-00 41.11 43-65 44-50 44.5 6 o 70 07 77 54 08 20 74 60 6 7 o 72 10 83 pi 16 3 84 70 7 8 74 4 89 68 25 39 94 So 8 9 o 76 18 95 75 33 48 44.04 90 9 45.O o.o 8.78 17.22 25.00 31.82 37-41 41-57 44.14 45.00 45.O 1 80 26 06 89 5 67 24 10 1 2 3 o 82 84 29 33 ii 17 96 32-03 58 66 c 33 43 20 3 2 3 4 86 37 22 ii 75 95 53 40 4 45.5 0.0 8.88 17.41 25.28 32.18 37.83 42.04 44-63 45-50 45.5 6 o 90 45 33 25 73 60 6 7 o 92 94 49 53 39 45 32 39 99 38.08 22 32 83 93 g 7 8 9 95 56 50 46 16 41 45.02 90 9 46.0 0.0 8.97 17.60 25-56 32-53 38.25 42.50 45-12 46.00 46.0 1 o 99 64 Si 60 33 59 22 IO 1 2 9.01 68 66 67 68 3 1 20 2 3 o 3 72 72 74 49 78 41 3 3 4 o 05 75 78 81 57 87 5' 40 4 46.5 o.o 9.07 17.79 25-84 32.88 38.66 42.96 45.61 46.50 46.5 6 09 83 89 95 74 43-05 7i 60 6 I o v o ii 87 9i 94 26.00 33-02 10 82 4 24 81 g 7 8 9 15 95 06 17 99 33 46.01 90 9 47.0 0.0 9.17 17.99 26.11 33-24 39.08 43-42 46.10 47.00 47.0 1 19 18.02 16 3 1 16 51 20 10 ] 2 o 21 06 22 38 24 60 29 20 2 3 o 23 10 28 45 33 70 39 3 3 4 25 14 33 52 79 49 40 4 47.5 0.0 9.27 18.18 26.39 33-59 39-49 43-88 46.59 47.50 47.5 6 29 22 44 66 57 97 69 60 6 7 o 3 1 25 5 73 66 44.07 / 70 7 8 o 3 2 29 55 80 74 16 88 80 8 9 o 34 33 ll 87 82 25 98 90 9 48.0 o.o 9-36 18.37 26.66 33-94 39-90 44-34 47.08 48.00 48.O TABLE LV. 169 Magnetic Element* of the Earth: The Magnetic Variation. IBf ARCTIC LATITUDES. T * Longitude West of Greenwich. Lat. Juat. O 5 1O 15 20 25 30 35 40 45 60 N, ! 65 I 7O I 75 | SO 60N 65 70 75 80 60 N. 65 70 75 SO 22 W. 27 28 o 25 W. 28 3 3i 32 30 w. 33 it 37 35 W. 3 39 40 4i 39 W. 43 44 45 45 43 W. 46 48 49 50 46 W. 5 53 54 55 49 W. it ii 51 W. | 52 w. 61 67 69 70 60 N. 65 70 75 SO 60 N. 65 70 75 SO 60 N. 65 70 75 SO 45 50 55 60 65 7O 75 80 85 90 52 W. 61 67 69 70 54 W. 64 72 75 77 B w - & 8 4 53 W. O7 91 52 W. 67 81 90 98 49 W. 66 82 96 105 43 W. 64 83 103 114 32 w. 60 84 112 I2 3 20 W. 5 85 124 132 4W. 25 85 135 141 90 95 100 105 11O 115 12O 125 13O 135 o 4 w. Si 135 I 4 I o loE. 10 E. 80 W. 150 150 o 19 E. 70 E. I 7 2\V. 1 60 o 24 E. & 156 E. I 75 W. o 27 E. ii 130 I72E. 29 E. 43 65 no 1 60 3?E. 45 63 95 140 33E. 45 60 88 128 33 E. 44 i no 33 E. 42 54 60 N. 65 70 75 SO 135 140 145 150 155 160 165 170 175 ISO 33 E. 4 2 54 ^s 3*E. 40 1 62 8 4 3?*. \ S^ 70 29 E. 35 42 So 59 27 E. 33 39 45 54 o 25 E. 30 35 40 49 o 23 E. 27 32 37 45 o 21 E. 24 29 35 42 i8E. 21 26 32 40 o i6E. 19 23 30 38 60 N. 65 70 75 SO Lftt Longitude East of Greenwich. Lat. 5 1O 15 2O 25 3O 35 40 45 60 N. 65 70 75 SO 60 N. 65 70 75 SO 60 N. 65 70 75 SO 60 N. 65 70 75 SO 22 W. 24 26 27 28 I9W. 20 22 22 23 i5W. 18 18 19 12 W. 14 J 3 13 '3 r- 9 10 10 6W. 5 6 2W. 2 2 2 i 2 2 2 2 4 E. 1 6 7 o 6E. 8 9 9 10 60 N. 65 70 75 80 60 N. 65 70 75 80 45 50 55 60 65 7O 75 8O 85 9O o 6E. 8 9 9 10 8E. 10 12 13 13 o 10 E. 12 H 15 15 i?E. 11 18 18 12 E. 16 18 21 22 12 E. 16 18 24 25 13 E. 10 18 1 o 12 E. 16 18 25 3i o ii E. H 17 24 33 o 10 E. 13 15 22 32 9O 95 10O 105 110 115 12O 125 130 135 o loE. '3 15 22 32 7E. 9 13 20 31 SB. 7 ii 18 30 SB. 5 9 16 2 9 o 3E. 7 15 28 o 2W. 6E. H 27 o 3W. i 5 E. 13 27 o sw. 2 4 E. 13 27 o 6W. ?* : 12 27 o 5 W. U 12 27 60 N. 65 70 75 SO 60 N. 65 70 75 SO 135 140 145 15O= 155 160 165 170 175 ISO o 5 W. U 12 27 o 4 W. i 4 E. 13 27 2W. IE. 6 14 28 o o 2E. 8 16 29 ffc 10 18 30 \ E - 12 2O 31 KF EJ. 10 i"S 22 3 2 o 10 E. 13 17 24 34 ;i E - 20 I i6E. 19 23 30 38 170 TABLE LVI. i magnetic Elements of the Earth : The magnetic Variation. IN LATITUDES FRO 11 70 X. TO 6O<> S. Longitude West of Greenwich. Lat. Lat. o 5 10 15 20 25 30 35 40 45 70 N. 65 257W. 23-7 3 0?2W. 28.0 35-oW. 32.8 395W. 38.0 44^2 W. 42.6 48?6W. 46.6 50-5 575W. 54-5 58.5^ 67.5 W. 61.0 7ON. 65 i 60 22.0 254 30.0 34-8 394 43-3 46.5 49.0 5-3 52-6 6O 55 20.6 234 27.0 31.3 43-2 44-3 44-8 55 50 19-3 21.8 244 27-5 3o!8 33-5 35-5 37-o 37-8 37-6 50 45 N. i8.2W. 20.3 W. 22.6 W. 24.9 W. 2 7 . 4 W. 29.8 W, 31-3 w- 32. oW. 31.9 W. 3 i.o\V. 45 N. 40 35 \u 19.1 18.2 21.0 19.8 23.0 21.4 24.7 23.8 26.4 24.0 27-3 24.2 27-5 23.8 27.0 j 25.5 23.0 21.4 40 35 30 16.4 17.6 1 8.9 20.2 21.2 21.7 21.5 20.8 19-5 17.0 30 25 16.4 17.4 18.4 194 2O.2 20.3 19.9 18.8 16.5 13.8 25 i 2O N. i6.6W. I7-5W. 18.2 W. 19.0 W. 19-5 W. 19.5 W. i8.7\V. 16.5 W. i4.o\V. io.7\V. 20 N. 15 17.1 17-7 18.3 I9.O 194 19.2 17-3 15.0 1 1.6 8.2 15 10 17.6 18.2 18.6 19.2 19-5 18.7 16.4 9.8 6.0 10 5 18.4 18.8 19.3 19-5 19.4 18.2 15.8 12.5 8.8 5- 5 19.8 20.5 20.7 20.3 19-5 18.9 15-3 12.0 8-3 44 O 58. 10 21.8 W. 24.0 22.4 W. 24.2 22.5 W. 23.8 2I.6W. 22.O i9-9\V. 19.6 I 7 . 5 W. 17.0 15-oW. H-3 II.O 8.0 W. 74 4.0 W. 3-6 58. ! 10 15 25-8 25.6 24.2 21.6 18.8 16.2 13-5 10.4 6.6 3- 15 20 27.0 25.8 234 20.7 1 8.0 15-3 12.5 9.2 54 2.2 20 25 26.9 25.0 22.3 19.6 1 6.8 14.1 1 1.2 7-7 4-3 1.2 25 308. 26.0 W. 23-7 w. 21. 1 W. 18.4 W. I5.6W. i2.7\V. 9.6 W. 6.4 W. 3.0 W. O.O 308. 35 24.8 22.4 19.8 17.0 H-3 II.2 8.2 5- 1.6 i. 9 E. 35 4O 234 21.0 18.4 15.7 12.8 9 .8 6.6 34 O.O 3-5 40 45 21.9 19-5 16.8 14.2 II. 2 8.0 5- 2.0 E. 5-3 45 50 20.3 17-7 15.2 12.4 9-5 6-3 3- 0.3 E. 3-7 7.0 5O 558. 18.5 W. I5-9W. 13.3 W. 10.5 W. 74 W. 4.2 W. 0.8 W. 2.4 E. 5.6 E. 8.6 E. 55 S. 60 16.8 14.0 11.4 8.3 5-2 2.8 1.2 E. 44 7-3 10.3 6O Longitude West of Greenwich. Lat. Lat. 45 5O 55 60 65 70 75 80 85 90 70 N. 67^5 w. 7i5W 75oW. 78?o W. 8i?oW. 82?oW. 83.0 W. 84 W. 85^0 W. 8 4 ?o\V. 70 N. 65 61.0 64*5 66.5 67.0 67.5 66.5 64-5 60.0 50.0 22.5 65 60 52.6 53-8 54.0 53-3 44.0 42-5 32.0 2O.O 3-5 60 55 44-8 44.8 43-8 42.0 38^0 33- 27.0 1 8.0 8.0 3-0 E. 55 50 37-6 36-3 34-2 31.0 27.0 22.4 16.5 I O.O 2.2 6.0 50 45 N. 3 i.oW. 29.3 W. 26.8 W. 23.7 W. 20.3 W. iS.oW. 10.0 W. 4.5 W. i.oE. 7.0 E. 45 N. 40 25-5 23-5 20.9 17-5 134 94 5-3 I.O 3-3 7-3 4O 35 21.4 19.0 15-7 12.3 9.0 5-8 2-3 i.i E. 44 74 35 30 17.0 14.0 n. i 8.5 5-7 2-7 0.0 2.6 5- 7-3 30 25 13.8 10.9 8.4 5-8 1.6 E. 3-5. 54 7.2 25 20 N. 10.7 W. 7.0 W. 5.2 W. 2.7\V. o.iW. 1.4 E. 2.8 E. 4.4 E. 5.9 E. 7-3 K. 2O N. | 15 8.2 5-2 2.4 0.2 E. 1.4 E. 2-5 3-8 5-2 6.5 74 15 10 6.0 3- o.i E. 1.4 2-5 3-7 4-9 6-3 7.2 7-7 10 5 5- 1.8 0.8 2.1 3-3 4.6 6.0 7-i 7-8 8-3 5 O 44 I.O 1.3 2.7 4.0 5.6 7.0 8.0 8.6 8.7 O 58. 4.0 W. 0.6 W. 1.7 E. 3-3 E. 5.0 E. 6.5 E. 8.0 E. 9.0 E. 9.4 E. 9.5 E. 58. 10 3-6 0.2 2.1 4.2 5.8 7-7 9.0 IO.I 10.5 10.5 1O 15 3- 0.2 E. 2.6 5.0 7.0 9.0 10.2 1 1.2 "5 11.5 15 20 2.2 I.O 3-7 6 3 8-5 10.3 u.6 12.5 12.8 12.6 20 25 1.2 2.3 7^8 1 0.0 1 1. 8 12.2 I 4 .2 14-3 13.8 25 308. O.O 3-5 E. 6.5 E. 9.4 E. ii.6E. 13.8 E. 15.2 E. 15.8 E. 15.8 E. 15.1 E. 308. 35 1.9 E. S- 2 8-3 II.O J3-5 I 5-5 1 6.8 17.4 17.2 16.4 35 40 3-5 6.8 10.0 12.8 154 i7-3 18.5 19.1 19.0 18.2 4O 45 5-3 8.4 1 1.6 14.7 19.0 20.4 21.2 21. 1 20.4 45 50 7.0 I O.O 13-3 16.1 18.3 20.5 22.1 23.2 23.2 22.7 5O 558. 8.6 E. ii.8E. 15.0 E. I7-5E. 20.0 E. 21.8 E. 24.0 E. 2t;.oE. 25.5 E. 25-3 E. 558. 60 10.3 13-5 16.5 19.2 21.6 24-3 26.O 27.1 27.7 27.8 6O TABLE LVI. 171 Magnetic Elements of the Earth : The Magnetic Variation. IN LATITUDES FROM 7O X. TO 60 S. Longitude West of Greenwich. Lat. Lat. 90 95 I OO 105 1100 115 120 125 130 135 70 N. 84.0 W. 8o.oW. o 70.0 E. 68.0 E. 67.0 E. 65.o E. 63.0 E. 60.0 E. 58.0 E. 54.0 E. 7ON. 65 22.5 7-5 E. 23.0 31.5 38.0 42.6 45 - 45-o 44-0 42.0 65 60 3-5 IO.O 19.0 24.0 27.2 29-5 31.0 33-o 33-o 32.6 60 55 3.0 E. II.O 17.0 2I.O 23.1 25.0 26.3 27.2 27-5 27.4 55 5O 6.0 10.8 15-3 l8. 4 21.6 22.0 22.8 23-4 23-7 23-6 50 45 N. 7.0 E. 10.4 E. 13.4 E. i6.oE. I7.8E. 19.3 E. 20.0 E. 20.4 E. 20.5 E. 20.4 E. 45 N. 4O 7-3 IO.O 12.0 14.0 15-4 16.4 17.0 17-5 17.7 17.7 40 35 7-4 9-3 II.O 12.5 13-5 14.3 14.9 15.2 15-4 15-4 35 30 7-3 8.9 IO.O II.O 11.7 12.2 12.5 12.7 12.8 12.9 30 25 7-2 8-4 9-1 9.8 10.3 IO.6 10.7 10.8 10.8 10.8 25 20 N. 7-3 E 8.1 E. 8.5 E. 8.8 E. 9.0 E. 9.0 E. 9.0 E. 9.0 E. 8.9 E. 8.9 E. 20 N. 15 7-4 8.0 8-3 8.1 7-9 7.6 7-3 7.1 7.0 7.0 15 1O 7-7 8.1 7-9 7-3 6.9 6.5 6.2 6.0 5.8 5.8 10 5 8.3 8-3 7-7 7.0 6.4 6.0 5-5 5-0 4.8 4.8 5 8-7 8.6 8.0 6-3 5-7 4-5 4.0 4.0 58. 9-5 E. 9.2 E. 8.4 E. 7.5 E. 6.6 E. 6.0 E. 5-4E. 5.0 E. 4.5 E. 4.4 E. 58. 10 10.5 IO.O 9.1 8.1 7.2 6.6 6.0 5-7 5-5 5-5 10 15 10.9 9.9 9.0 8.0 7-4 7.0 6.6 6.4 6-3 15 20 12.6 11.8 10.8 9.9 8.9 8.2 7-8 7-5 7-3 7.2 20 25 13-8 12.9 11.7 10.7 9.8 9.0 8.6 . 8-3 8.2 8.1 25 SOS, 15.1 E. 14.0 E. 13.0 E. II.8E. 10.7 E. 9.8 E. 9-3 E. 9.0 E. 8.7 E. 8.7E. 308. 35 16.4 15-4 14.1 13.0 11.9 10.9 IO.I 9-7 9-5 9.4 35 40 18.2 17.0 15.6 14.4 13.2 12.2 "3 10.6 10.3 IO.I 40 45 20.4 17.8 16.2 14.6 I 3 .6 12.7 I2.O "5 1 1.2 45 50 22.7 21.6 20.2 18.6 17.0 15-3 14.2 13-5 13-0 12.6 50 558. 25.3 E. 24.5 E. 23.0 E. 2I.4E. 19.8 E. 18.2 E. I6.6E. 15-5 E. 14.8 E. 14.4 E. 558. 60 27.8 27-5 26.7 25.2 23-6 22.0 20.4 19.1 18.1 17-5 60 Longitude West of Greenwich. Lat. Lat. 135 140 145 150 155 160 165 170' 175 ISO 70 N. 54.0 E. 50.0 E. 46.5 E. 42.0 E. 39.0 E. 35 -E. 29.3 E. 25.8 E. 23.0 E. 70 N. 65 42.0 40.0 37-0 34-5 32.5 30.0 26^6 / 23-9 2I.O 18.5 65 60 32-6 32.0 30-7 28.7 26.7 24.8 22.9 20.7 l8. 3 16.0 60 55 27.4 26.8 26.0 24.8 23-5 22.0 20.6 18.7 1 6.8 14.7 55 50 23.6 23.2 22.6 21.9 21.0 19.9 18.5 17.0 15.6 14.0 5O 45 N. 20.4 E. 2O.O E. 19.8 E. 19.3 E. i8. 7 E. iS.oE. i6.oE. 15.8 E. 14.7 E. 13-5 E. 45 N. 40 17.7 17.6 17-3 17.0 1 6.6 1 6.0 15-4 14.7 14.0 13.0 40 35 15-4 15-3 15.0 14.9 14.7 14.4 14-3 13-9 13-3 12.4 35 30 12.9 12.9 13.0 13.0 13.0 13.0 13.0 12.9 12.6 12.0 30 25 10.8 10.9 II.O 11. i "5 11.7 1 1. 8 11.9 u.6 25 20 N. 8.9 E. 8.9 E. 8.9 E. 9.0 E. 9-3-E. 9.7 E. lo.iE. 10.6 E. 10.8 E. 10.8 E. 20 N. 15 7.0 7.1 7-3 7-5 8.0 8.4 9.0 9-5 9.8 IO.O 15 1O 5.8 5-9 6.t 6.4 6.8 7-3 8.0 8.7 9.1 9-5 10 5 4.8 4.8 5 55 8.0 6.0 3-8 4 0.7 W. 2.4 4.1 5-7 6.5 6.0 55 50 7-i 5-4 3-3 .0 0.8 2-5 4-2 5.8 6.5 6.1 50 45 N. 6.7E. 5.0 E. 3.0 E. .oE. o.7W. 2.2 W. 3-9 W. 5.6 W. 6.4 W. 6.0 W. 45 N. 40 6-3 4.8 3- .2 o-3 i-7 3-3 4-8 5-7 5-5 40 35 5-2 4-3 3-i 4 0.2 1.2 2-5 3-9 4-5 4-4 35 30 3-6 3-4 2.7 7 0.6 E. 0.6 i-7 2.6 3- 2.9 30 25 2.8 2.9 2-5 2.0 I.O o.i E. 0.8 1.6 1.8 1.6 25 20 N. 2-3 E. 2.5 E. 2.3 E. 2.2 E. 1.5 E. 0.6 E. o.o o.7\V. 0.9 W. 0.5 W. 20 N. 15 2.O 2.2 2.4 2.2 i-7 1.2 0.6 E. o.o o.o 0.5 E. 15 10 1.6 1.9 2.1 2.O 1.6 1.4 i-3 .iE. i.i E. i-4 10 5 1.4 1.6 i.i i-7 i-7 1.6 1.4 4 1.5 1.9 5 O i.i i-3 1.4 1.4 1.4 1.4 i-5 7 1.8 2-3 O 5S. 0.5 E. 0.8 E. i.oE. i.iE. i.iE. 1.2 E. 1.3 E. .6E. 1.8 E. 2.7 E. 53. 1O 0.2 W. O.I 0.2 -3 0.4 -5 I.O 4 i-9 3- 1O 15 2.1 i.7\V. i. 4 W. i.iW. 0.8 W. 0.5 W. 0.0 0.8 i-7 3- 1 15 20 4-5 4.0 3-4 2.8 2.2 1.5 0.8 W. o.o 1.4 3-2 2O 25 7-7 7.0 6.0 4.9 3-8 2.6 i-7 0.6 W. 0.9 3-3 25 3OS. n.5\V. 10.6 W. 9.3 W. 7.7 w. 6.0 W. 4.3 W. 2.5 W. i.i W. 0.5 E. 3-3 E. 3OS 35 16.0 14.6, 13.0 II.O 9.0 6.5 4-3 1.8 o.o 3- 2 35 40 20.5 19.0 16.8 14.4 1 1.6 9.0 6.0 3- 0.3 W. 3- 40 45 25-7 24.0 21.4 18.4 15-3 11.9 8-3 4.8 I.O 2.8 45 50 31.2 29-5 27.0 23.0 19.0 15.0 10.7 6.5 2.O 2.5 50 558. 38.3 w. 35-8 W. 33-oW. 29.0 W. 24.3 W. 19.8 W. 14.7 W. 9.2 W. 3.8 W. 1.8 E. 55 S. 6O 45-o 43- 2 41.0 36.0 31.0 25.0 2O.O 13-8 7.0 o.o 60 Longitude East of Green wich. Lat. Lat. 135 140 145 150 155 160 165 170> 175 18O 70 N. 3-5 E. o 4-5 E. 6.0 E. 8?oE. 10.0 E. o 12. 5 E. K.oE. o 17.5 E. 20.0 E. o 23.0 E. 70 N. 65 2.7\V. I.4VV. '5 2-5 4.6 7-3 9.8 12.6 15.8 18.5 65 6O 5-2 3-6 i.8W. O.2 2.8 5-3 7-8 10.3 13.2 16.0 60 55 6.0 4-6 2.5 0.4 W. 2.O 4.4 7.0 9-7 12.3 14.7 55 5O 6.1 4.8 2.8 0.7 1.9 4-3 7.0 9-6 12.0 14.0 50 45 N. 6.0 W. 4-7^. 2.8 W. 0.5 W. 2.0 E. 4-7E. 7-3 E. 9.8 E. H.8E. 13-5 E. 45 N. 4O 5-5 4-4 2.2 o.o 2-5 5-o 7.6 10.0 1 1.6 13.0 40 35 4.4 3-4 !-5 0.5 E. 3-i 5-5 7-8 IO.O 114 12.4 35 3O 2.9 2.0 0.4 1.4 3-8 6.0 8.1 IO.O n. i 12.0 30 25 1.6 0.8 0.5 E. 2.2 4-3 6-4 8.2 IO.O II.O 1 1.6 25 2O N. 0.5 W. 0.5 E. 1.5 E. 3.1 E. 5.0 E. 6.8 E. 8.4 E. 9-7 E. 10.5 E. 10.8 E. 2O N. 15 0.5 E. 1.2 2-5 3-9 5-6 7-i 8.4 9.4 9.9 IO.O 15 1O 1.4 2.2 3-2 4-7 6.0 74 8.4 9.2 9-5 9-5 10 5 1.9 2. 9 4.0 5-2 6.4 7-7 8.4 9.0 9-3 9.0 5 O 2-3 3-4 4.6 5-7 6.8 8.0 8.4 8-7 9.0 8-7 ] 5S. 2.7 E. 3.8 E. 5.0 E. 6.iE. 7.2 E. 8.2 E. 8.6 E. 9.0 E. 9.1 E. 8.8 E. 58. 1O 3-o 4.2 5 i 6.6 7.6 8.4 9.0 9-3 9-3 9.2 10 15 3- 1 4-5 5.8 7-i 8.2 9.0 9-5 9-7 9-7 9.6 15 2O 3-2 4.8 6.3 7-7 8.8 9-5 fO.O ! IO-3 10.4 IO.2 20 25 3-3 5-2 7.0 8-3 9-5 10.3 ii.o J 11.3 11.4 II. I 25 22 s - 3-3 E. 5.6 E. 7.5 E. 9 .iE. 10.3 E. 11.3 E. 12.0 E. 12.5 E. 12.6 E. 12.3 E. 30 S. 35 3-2 6.0 8.1 1 0.0 "3 12.4 13.2 *3-7 13-7 l 3-4 35 4O 3> o 6-3 8.8 10.7 12.3 13.6 14.4 14.9 15.0 14.6 40 45 2.8 6.6 9-7 11.7 J 3-5 14.8 15.7 16.2 16.3 1 6.0 45 &O 2.5 7.0 10.3 12.7 14.7 16.2 17.2 17-7 17.8 17-5 50 5$ ft 1.8 E. 6.8 E. ii.oE. 14.0 E. 16.3 E. 18.1 E. 14.1 E. 19.6 E. 18.4 E. i9.oE. 558. 6O 0.0 6.0 11.4 15.6 18.5 20.7 22.0 22.4 22.0 21.2 60 174 TABLE LVII. Magnetic Elements of the Earth: The Magnetic Dip. Longitude West of Greenwich. i Lat. Lat. o O 10 |*0 I 3O 40 si o o 60 70 80 90 10 o 110 1*0 130 o 140 o o 1 o ' o 150 160J170 ISO o o o o i o o o i n o o r o o . , 75N. +8 1 +82 +83 +83+85 +86 +86 +87 +89 +89 +89 +87 +87 +86 +85 +83 : +82 +82 ! +82 75N.J 70 79 81 81 82! 84 85 85, 86 88 89 89 87 86 84 83 Si 80 80 i 79 7O 65 75 76 77 78 i 79 81 83! 85 87 87 86 85 85 83 81 79 77 76 75 65 !60 72 73 74 76 78 80 i 82 84 86 86 85 83 82 79 77 75 73 72! 7i 60 55 70 72 73 74 76 78 So: 82 83 83 82 80 78 76 74 72 70 68 66 55 |50N. +67 +69 +71 +73 >75 +77 j+78 +79 +80 +79 +78 +76 +74 +72 +69 +57 +65 +63 i +6 1 5ON. 145 64 66 68 7i 73 75 I 7 6 77 77 76 74 72 70 68 65 63 60 58 56 45 J4O 60 62 65 oa 70 72 72; 72 72 71 69 67 65 62 60 57 55 52 51 40 35 56 58 61 65 67 68 63 ; 68 67 65 64 62 59 57 55 52 50 48 47 35 30 54 57 61 63 64! 64! 63 62 60 58 56 54 52 48 47 45 43 30 ! 25 N. +46 +5 +54 i+57+58 +59+58 +57+j6 +53 +5 1 +49 +48 +47 +46 +44 +43 +42 i +39 25N. : 2O 38 43 48 52 54 55 ! 53 52 49 45 43 42 42 42 41 40 38 35 i 30 2O 1 5 30 37: 42 47; 49 49 47! 44 40 38 36 34 33 33 32 3 2 3 27 22 15 10 20 29 30 40 42 41 : 40 i 37 33 28 26 24 24 2 3 23 21 19 14 10 5 IO 20 29 33 35 34 33 3 27 22 17 H 13 '3 13 13 II + 9+5 5 +I0 j +20 +26 :+ 2 8 +27 +25 +20 +16 +10 + 5 + 3 + 2 + 2 + 2 + 2 + I -1-4 58. - 8+ 2\ 12 19 21 23 IS! 12 + 82+ 2 -3-5 - 7 - 7 - 7 7 - 7 10 '4 58. 10 18 -6 + 4 ii 14! 13 9 + 2-3-9 12 14 16 16 16 16 17 20 2 3 10 15 28 16-5 +2+5+4 0-7 12 17 20 22 24 24 24 26 27 3 32 15 20 34 26 j 15 - 8 - 4 - 5 -io ; 15 i 20 25 28 3 3 2 33 33 34 35 37 39 20 25 S. -38 32-24 -i8 ! -i4 -14 -18 -23J-28 -32 -35 -38 -40 -41 -41 : -41 -42 -44 -46 258. 30 42 37j 3i 27 23! 23 : ' 26! 30 34 38 44 46 47 48 1 49 50 5 1 53 30 35 46 41! 36 33 3 3 33' 3 6 4 44 47 5 51 52 53 54 55 5 6 S 8 35 !4O 5 45 40 38 i 36 36 39 , 42 ] 46 50 53 55 56 57 58 59! 60 61 62 40 145 53 50 45 43! 42 42 45j 48| 51 55 58 60 61 62 63 63! 64 65 66 45 508. -55 -52 -49 -47 j-47 -48 -50 -52 -55 -59 -62 -3 4 -65 -66 -67 -67! -68 -69 -70 5OS. 55 57 55 53 52! 52 53 55l 57 60 64 67 68 69 70 71 71 72 73 73 55 6O 59 57 56 55 56 57 59 1 62 65 68 70 72 73 74 75 75 76 76 76 60 Longitude East of Greenwich. Lat. Lat. c 10 *o :n 10 50 60 70 80 100 110 1*0 130 145 150 160 170 180 J75N. +8? +8 1 +8 1 +80 +sS O j O +8oj+8i +85 +81 +8? +81 +81 +81 +81 +8? +8^ +8? ~& +82 75N. 7O 79 79 78 77 77 77| 78 79 79 80 80 80 80 So 80 79 79 79 79 70 65 75 75 74 74 74 75 76 76 76 77 77 77 77 77 76 76 75 74 75 65 60 72 72 71 71 71 7 1 72 72 72 73 73 73 73 73 72 72 70 70 60 55 70 69 68 68 68 68 69 70 70 70 69 68 66 65 66 55 50 N. +67 +66 +65 +64 +64 +64 +64 +65 +65 fb5 +66 +66 +66 +65 +64 +63 +6 1 +60 +5 1 50 N. 45 64 62 61 60 60 60 60 60 60 60 60 61 61 60 59 57 55 55 56 45 40 60 57 55 54 53 53 53 53 54 55 55 56 56 55 53 49 49 5* 40 35 56 53 5 48 47 47 47 48 49 50 So 50 5 49 48 46 44 45 47 35 3O 51 47 43 41 40 39 39 40 40 41 42 43 43 42 42 40 40 43 30 25 N. +46 +40 +35 +31 +30 +29 +29 +30 +31 +33 +34 +35 +35 +35 +35 +33 +33 +35 +39 25N. 2O 38 30 2 3 20 19 19 20 20 21 22 23 25 27 28 28 24 24 2 5 30 20 15 3 20 14 10 10 IO IO 10 IO II 12 15 17 18 19 15 15 22 15 10 20 10 + 4 o O + I + 3 + 5 + 8 9 9 8 9 14 IO 5 IO - 6 -IO -10 -10 -10 -10 -10 - 9 - 7 - 5 - 3 - i o o o + 5 5 O O - 9 -16 -20 -22 -22 -21 -2O -20 -18 -16 -14 -12 -n -IO -10 -10 - 9 - 4 58. - 8 19 25 30 30 3 30 3 29 27 24 22 20 20 20 21 21 19 14 58. IO 18 28 33 37 39 40 40 39 38 36 33 3 2 3 1 3 3 1 3 2 31 28 2 3 10 15 27 34 40 44 47 48 48 47 46 44 43 42 4 1 40 40 4 38 35 32 15 20 34 40 46 50 53 54 53 52 52 5 1 5 1 5 1 50 49 47 46 44 42 39 2O 25 S 30 35 -38 42 46 -44 49 5 2 -5o -54 57 -57 59 63 -57 61 64 t 64 t 65 65 -55 59 65 -55 59 65 -55 59 6 5 1 -53 H -53 57 *3 1 -50 -46 53 258. 30 35 40 5 54 58 61 63 65 66 67 67 68 69 70 70 69 68 67 66 64 62 40 45 53 56 60 62 65 67 68 69 70 71 72 74 75 74 73 72 70 68 66 45 508. 55 6O -55 57 59 -f 60 62 -6 1 62 64 -63 it -66 11 -68 69 70 -69 72 72 73 -72 74 75 -73 76 77 1 -76 79 82 1o 7 83 - 3 -76 79 82 18 Si -74 -72 75 77 -70 8 508. 55 60 TABLE LVIII. 175 Magnetic Elements of the Earth : The Horizontal Force. Longitude West of Greenwich. Lat. Lat. o o 10 20 o 30 40 50 60 o 70 80 90 o 100 HO 1*0 130 140 150 160 o 170 o ISO 75 N. 0.6 O. ^ 0.5 0.4 0.4 0-3 O.I 0.0 o.o o.o O.O o.o O.I 0.2 -3 0-3 0.4 0.4 0.4 75 N. 70 7 6 6 5 4 3 2 o o o o 3 4 5 5 6 6 7 70 65 8 7 7 6 5 4 3 I o I 3 4 6 6 7 8 8 9 65 60 9 8 7 6 6 4 3 3 3 4 5 6 7 8 9 I.O I.O I.O 60 55 9 9 8 8 7 6 6 5 4 5 6 7 8 9 I.O I.O i 2 2 55 i50N. I.O I.O 0.9 0.9 0.8 0.7 0.7 0.6 0.6 0.7 0.8 0.9 I.O i.i i.i 1.2 '3 1.3 1.4 50 N. 145 2 i I.O 9 9 8 8 8 9 I.O i.i i.i 2 2 3 4 4 4 4 45 40 35 3 4 2 3 i 2 4" I.O 2 I.O i I.O i I.O 2 I.O 2 i.i 3 2 4 6 3 5 3 5 J 4 6 I I 6 6 I 6 6 40 35 25 N. 1.6 1-5 1.4 1.4 1.6 1.8 1.9 1.9 1.9 1.8 1.8 '7 25 N. 2O 7 '6 6 5 'e 7 8 9 2.O 2.0 2.O 9 9 8 7 7 7 7 20 15 8 7 7 6 6 7 8 9 2.O I I I 2.0 9 8 8 8 8 8 15 10 9 8 8 7 7 8 9 2.0 I 2 2 I I 2.O 9 889 9 10 5 9 8 8 8 8 9 2.0 o I 2 2 I I O 9 9 9 9 9 5 1.6 1.9 1.9 1.9 1.9 1.9 2.O 2.0 2.1 2.2 2.1 2.1 2.1 2.0 2.Q 1.9 1.9 i 2.0 2.O O 58. 8 8 8 9 9 O I I I I O Q 2.0 2.O O 58. 10 7 7 7 8 8 9 1.9 O I I O O i G O O O O 10 15 6 6 6 7 7 8 ' 8 1.9 O O O O O O O O ! 15 2O 4 5 6 6 6 7 8 9 1.9 O oj o O 20 258. 14 1.4 T -5 I -5 1.6 1.6 J -7 1.8 1.9 2.0 2.O 2.0 1.9 1.9 1 9 j 1.9 i 1.9 1.9 1.9 258. 3O 3 4 4 5 5 6 7 8 9 1.9 1.9 1.9 9 8 8 8 7 30 35 3 3 4 5 S 6 7 8 8 8 8 7 7 7 6 35 40 45 3 3 4 4* 5 5 6 6 6 7 8 8 8 5 I 7 6 6 6 5 5 40 45 508. 1.4 5 1.6 1.6 1.6 7 1.6 1.6 . 1.4 1.4 1.4 J -3 J3 I3 1.2 508. 55 3 4 4 5 5 6 6 5 5 5 4 3 3 2 2 2 i i 55 60 3 4 4 5 5 5 5 5 4 3 2 i o 0.9 0.9 0.8 0.8 60 Longitude East of Greenwich. Lat. Lat. o 20 O O 30 40 o 50 60 o 70 80 o 90 100 110 120 130 140 o 150 o 160 o 170 o ISO 75 N. 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 o-S 0-5 0-5 0.5 0-5 0-5 0:5 0.4 75 N. 70 7 7 7 8 8 8 8 8 7 7 6 5 5 6 6 6 8 8 7 70 65 8 8 8 8 8 8 8 9 9 8 8 8 8 8 8 9 9 9 9 65 6O 9 9 9 9 9 9 I.O I.O I.O I.O I.O I.O I.O I.O I.O I.O I.O I.O I.O 60 55 9 I.O I.O i.i i.i i.i 2 2 i i i i i i i 2 2 2 2 55 5ON I.O i.i T T i ? 1.2 i ? I 3 I "\ i "3 i ^ i ^ 1 1 T 1 1.2 i 3 I "* 1-4. 5ON. 45 2 2 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 T* 4 4 45 40 35 3 4 3 4 4 5 ^ 5 7 6 7 6 7 6 7 6 7 7 6 6 7 6 7 6 i 40 35 30 5 S 6 7 8 8 9 9 2.0 2.0 9 9 8 8 7 7 6 6 6 30 25 N. 1.6 i-7 1.7 1.8 1.9 1.9 2.0 2.1 2.2 2.1 2.0 2.0 1.9 1.9 1.8 1.8 i-7 1.7 1.7 25 N. 20 7 8 9 9 9 2.O I 3 2 I I 2.O 2.0 9 8 8 7 7 20 15 8 9 9 9 1.9 O I 2 3 2 I I 2.O 9 9 8 8 15 1O 9 9 8 8 9 1.9 I 2 2 2 I I 2.O 9 9 9 10 5 9 9 8 7 7 8 9 O I I 2 2 2 I I 2.0 2.0 9 5 O 1.9 1.8 1.7 1.6 1.6 1 7 1.8 1.9 2.O 2.1 2.1 2.1 2.2 2.2 2.1 2.1 2.0 2.0 2.0 O 58. 8 7 6 5 5 6 7 8 1-9 I I 2 2 I I I 58. 10 7 6 5 5 5 5 6 7 8 1-9 I I I I I I O 1O 15 6 - 6 7 8 T rt j j I o Q o 15 20 4 4 3 3 3 3 4 5 6 7 1 1.8 1.9 o 20 258. 1.4 I> 3 I -3 1 '3 x -3 x -3 *'3 1.4 I -5 j e 1.6 I -7 1.8 1.8 1.8 1.8 1.8 1.8 1.9 25 S. 30 35 3 3 3 2 3 2 2 2 I 2 I 2 I 3 2 4 2 4 3 5 3 6 4 6' 4 7 4 7 4 7 5 7 5 I I 30 35 4O 3 2 2 I I I I I 2 2 3 3 3 3 3 4 4 5 4O 45 3 2 I I O O I I I i i 2 2 2 2 3 3 45 508. 1-3 1.2 I.I I. I.O I.O I.O I.O I.O I.O I.O i.o I.O I.O I.O I.O I.I i.i 1.2 508. 55 3 2 2 0.9 0.9 0.9 0.8 0.8 0.7 0.7 0.8 0.8 0.9 0.9 55 6O 3 3 2 I o o 9 8 8 7 6 5 5 5 5 7 8 8 6O 176 TABLE LIX. Right Ascension of the True Sun and Equation of Time. 1875: At Greenwich Mean Noon. Da January. February. March. April. May. June. ay. 0.R.A. E.T. ,R.A.| E.T. 0.R.A. E.T. 0', R. A. E. T. 0-.R.A. E.T. 0,11. A. E.T. h m m 1, in m, h in in h m in h 'in in // m in 1 18 46.7 - 3-8 20 59.0 -13-8 22 48.1 12.6 o 41.8 4.0 2 33- 4- 3- 4 357 2 5 1 - 1 4.2 21 3-0 14.0 51.9 12.4 454 3-7 36-9 3-J 39.8 2.4 : 3 55-5 4-7 7.1 1 14.1 55.6 12.2 49.0 3.4 40.7 3-2 43-9 2.2 j 4 59.9 5- 1 II. I 14.2 59-3 I2.O 52.7 3-i 44-5 3-3 48.0 2.0 5 19 4-3 5-6 15-2 14-3 23 3-o II.7 56.3 2.8 48.4 34 52-2 9 6 8-7 6.0 19.2 H-3 6.8 II.5 i o.o 2.5 52.2 3-5 56.3 .7 7 I 3- 1 6.5 23.2 14.4 10.5 "3 3-6 , 2.2 56.1 3- 6 5 0.4 .5 ^ 19 17.4 -6.9 21 27.2 14.4 23 14-2 II.O i 7-3 2.9 3 - + 3-6 5 4-5 + .3 9 21.8 7-3 31.2 14.5 17.9 10.8 ii.o 1.7 3-9 3-7 8.7 .1 1O 26.2 7-7 35- 1 14-5 21.5 10.5 14.6 1.4 7-7 3-8 12.8 0.9 11 3-5 8.2 14-5 25.2 10.3 18.3 I.I ".7 3-8 16.9 0.7 13 39-2 8.5 8-9 43-o 46.9 14.5 14-5 28.9 32-6 IO.O 9-7 22.O 0.9 25.7 0.6 15.6 19-5 !i 21. 1 25.2 o-5 14 43-5 9-3 50.8 14.4 36-2 94 29-3 : 0.3 234 3-9 294 O.I 15 19 47.8 9-7 21 54-7 14.4 23 39-9 9.2 i 33.0 o.i 3 274 -f 3-9 5 33-5 O.I 16 52.1 IO.O 58.6 14-3 43-5 8.9 36.7 -f- 0.2 31.3 3-8 37-7 0.3 17 56.3 10.3 22 2.5 14-3 47.2 8.6 40.4 0.4 35-3 3-8 41.8 -5 18 20 0.6 10.7 6. 4 14.2 50.8 8.3 44.1 0.6 39-2 3-8 46.0 0.7 19 4-9 1 1.0 IO.2 14.1 54-5 8.0 47.9 0.9 43-2 3-8 50.1 0.9 20 9.1 "3 14.0 14.0 58.1 7-7 51.6 .1 47.2 3-7 54-3 1.2 21 M-5 17.9 13-9 o 1.8 74 55-3 i -3 51.2 3-7 58.5 1.4 22 20 17.5 11.8 22 21-7 13.8 o 5.4! 7.1 i 59-o 4- -5 3 55-2 -f 3-6 6 2.6 1.6 23 21.7 I2.I 25-5 13.6 9.0 1 6.8 2 2.8 .7 59-2 3-5 6.8 1.8 24 25.9 12.3 29-3 13-5 12.7 6.5 6.5 .9 4 3-2 3-5 10.9 2.0 25 3 O.I 12.6 33-i 13-3 16.3 6.2 10.3 2.1 7-3 34 15.1 2.2 26 34-3 12.8 36.8 19.9 5-8 I4.I ! 2.2 "3 3-3 19.2 2.4 27 13.0 40.6 13.0 23-6 5-5 I 7 .8 2.4 154 3-2 234 2.7 28 42.6 13.2 444 12.8 27.2 5-2 21.6 2.6 19.4 3- 27.6 2. 9 29 30 20 46.7 50.8 54-9 -134 13.5 o 30.8 1! -4-9 4.6 4-3 2 25.4 4- 2.7 29.2 2.9 4 23.5 4- 2.9 27.6 2.8 31.6 2.7 6 31.7 35.9 3-3 Da July. August. September. October. XoTember. December. ay. 0'.R.A. E. T. 0.R.A. E.T. 0' S R.A. E.T. '.R.A. E.T. 8 R.A. E.T. 0',R.A. E.T. h m m h m m h m in h m m h m m h m m 1 6 40.0 3-5 844-8 6.1 10 40.9 4- 0.0 12 29.0 + 10-3 14 25.2 +16.3 1 6 28.9 2 44.1 3-6 48.7 6.0 44-5 0.4 3 2.6 10.6 29.1 16,3 33-2 10.5 3 48.2 3-8 5 2.6 6.0 48.2 0.7 36.2 10.9 33-0 I6. 3 37-5 IO.I 4 52.4 4.0 56.5 CQ 51.8 1.0 39-9 1 1.2 37-0 16.3 41.9 9-7 5 56.5 4.2 9 -3 5.8 554 i*3 43-5 II-5 40.9 16.3 46.2 9-3 6 7 0.6 44 4.2 5-7 59-o 47-2 u.8 44.9 16,3 50.6 8.9 7 4.7 4-5 8.0 5.6 II 2.6 2.0 50.8 I2.I 48.9 16.2 55-0 8.4 8 7 8.8 4-7 9 u.8 54 II 6.2 + 2.4 12 54.4 4-12.4 14 52.9 4-16.1 16 59-3 4- 8.0, 9 12.9 4-8 15.6 5-3 9.8 2.7 58.2 12.6 5 6 -9 16.1 i7 3-7 7.6 10 17.0 5.0 19.4 5-2 134 3' 13 1.8 12.9 15 i.o 16.0 8.1 7-1 11 21. 1 S- 2 23-2 5- 17.0 34 5-5 13.2 5- 15-9 12.5 6.6 12 25.2 5-3 27.0 4-9 2O.6 3-7 9.2 134 9.1 15-7 16.9 6.2 13 29.2 54 30.8 4-7 24.2 4.1 12.9 13.7 13.2 l$.6 21.4 5-7 14 33-3 5-5 34-5 4-5 27.8 44 1 6.6 13.9 17.2 15-5 25.8 5-2 15 7 37-3 -5-6 9 38-3 4-3 II 31.4 + 4-8 13 20.3 4-14.1 15 21.4 +15,3 17 30.2 + 4-7 16 41.4 5-7 42.0 4.1 35-o 5- 1 24.0 14.4 25-5 I 5 <1 34-6 4-3 17 454 5-8 45-7 3-9 38.6 5-5 27.8 '4-5 29.6 14.9 3-* 18 49-5 5-9 49-5 3-7 42.2 5-8 3L5 14.8 33-8 14.7 43-5 3-3' 19 53-5 6.0 53-2 3-5 45-7 6.2 35-3 14.9 37-9 14.5 47-9 2.8 20 57-5 6.0 5 6 -9 3-3 49-3 6-5 39-0 I5* 1 42.1 14-3 524 2.3! 21 8 0.5 6.1 10 0.6 3-0 52-9 6.9 42.8 15-3 46.3 14.0 56.8 1.8 22 8 5-5 6.2 10 4.3 2.8 ii 56.5 + 7-2 13 46.6 + 154 15 50-5 +13-8 18 1.2 4- 1.3 23 9-5 6.2 8.0 2-5 12 O.I 7.6 504 15.6 54-7 i3-5 5-7 0.8 24 25 134 17.4 6.2 6.2 11.7 15.4 2-3 2.0 3-7 7-3 7-9 8-3 $ i5-7 15.8 58-9 16 3.2 13.2 12.9 IO.I 14.6 d* 26 21.3 6.2 19.0 1.7 10.9 8.6 14 1.9 15-9 74 12.6 19.0 0.7! 27 25-3 6.2 22.7 1-4 14-5 9.0 5-7 1 6.0 7 12.3 23-5 1.2 | 28 29.2 6.2 26.4 1.2 18.1 9-3 9.6 16.1 16.0 11.9 27.9 1.7 29 8 33-i 6.2 10 30.0 0.9 12 21-7 4- 9.6 14 "34 4-16.2 1 6 20.3 4-11.6 18 32-3 2.2 30 37- 1 6.2 33.6 0.5 254 9-9 17.4 16.2 24.6 1 1.2 36.8 2-7 31 40.9 6.1 37-3 0.3 21.2 16.3 41.2 3-2 TABLE LlX. 177 ISig'Sat Ascension of the True 1111 and Equation of Time. 1876: At Greenwich Mean Noon. Hiiv January. February. March. April. M.iy. June. uaj. ' S R,A. E. T. 0'sR.A. E.T. 0' S R.A. E. T. S R.A. E.T. 0' S R. A. E.T. S R.A. E.T. h 'in | tn h m tn h in m h m in, 7l Trt m h m in 1 18 45.6 -3-6 20 58.0 -I 3 .8 22 51.0 12.5 44-5 -3-8 2 35-9 + 3-1 4 38.8 + 2.4 2 50.0 4.1 21 2.1 13-9 54-7 12.2 48.2 3-5 39-8 3-2 43-0 2.2 3 54-4 4.6 6.1 14.0 58.4 12.6 51.8 3-2 43-6 3-3 47-1 2.1 4 58.8 i 5.0 10.2 14.2 2 3 2.2 11.8 55-5 2.9 474 34 51.2 1.9 i 5 19 3-2 5-5 14.2 14.2 5-9 1 1.6 59-1 2.6 51-3 3-5 55-3 i-7 6 7.6 5-9 18.2 14-3 9.6 M-3 I 2.8 2-3 55-i 3-6 594 i-5 7 12.0 6.4 22.2 14.4 13-3 11. i 6.4 2.0 59- 3-6 5 3-5 1.4 8 19 16.4 _ 6.8 21 26.2 14.4 23 17- 10.8 I IO.I 1-7 3 2.9 + 3-7 5 7-7 + 1.2! 9 20.7 7.2 30.2 H-5 20.7 10.6 13-7 1.4 6.8 3-8 11.8 I.O 10 25.1 7.6 34-2 14-5 24.3 10.3 17.4 1.2 10.7 3-8 15-9 0.8 11 29.4 8.0 38.1 14-5 28.0 IO.I 21. 1 0.9 14.6 3-8 20. i 0.6 12 33-8 8.4 42.0 H-5 3 I -7 9.8 24.8 0.7 18.5 3-9 24.2 0.41 13 38.1 8.8 46.0 14-5 35-3 9-5 28.5 0.4 22.5 3-9 28.4 0.2 14 42.4 9.2 49.9 14.4 39-o 9.2 3 2.1 O.I 26.4 3-9 32.5 o.o 15 19 46.7 9-5 21 53-8 14.4 23 42.6 - 8.9 i 35-8 + o.i 3 34 H-3-9 5 3 6 -7 0.2 16 51.0 9-9 57-7 H-3 46-3 8.6 39-5 o-3 34.3 3-8 40.8 0.4 17 55-3 IO.2 22 1-5 H-3 49-9 8-3 43-2 0.6 38.3 3-8 45-o 0.7 IS 59-6 10.6 5-4 14.2 53-6 8.1 47.0 0.8 42.3 3-8 49.2 0.9 19 20 3.8 10.9 9-3 14.1 57-2 7-7 5-7 I.O 46.2 3-8 53-3 20 8.1 II. 2 14.0 o 0.9 7-5 54-4 1.2 50.2 3-7 57-5 *-3 21 12.3 "5 16.9 13-9 4-5 7- 1 58.1 i-5 54-3 3-6 6 1.6 i-5 22 20 16.5 11.7 22 20.8 -13.8 o 8.1 6.8 2 1.9 + 1.6 3 58.3 + 3-5 6 5.8 j 1.8; 23 20.7 I2.O 24.6 13-7 11.8 6.5 5.6 j 1.8 4 2.3 3-5 10.0 2.O 24 24.9 12-3 28.4 13-5 iS-4 6.2 9.4 2.O 6-3 34 14.1 2.2 25 29.1 12-5 32.2 13-4 19.1 5-9 13.2 2.2 10.3 3-3 18.3 2.4 26 33-3 12.7 35-9 13.2 22.7 5-6 16.9 2.4 14.4 3-2 22.4 2.6 27 374 12.9 39-7 13.0 26.3 5-3 20.7 2-5 18.5 3-o 26.6 2.8 28 41.6 I 3 .2 43-5 12.8 30.0 5-o 24-5 2.7 22.5 2.9 30-7 3-o 29 20 45.7 !3-3 22 47.2 -12.6 o 33-6 4-7 2 28.3 + 2.8 4 26.6 + 2.8 6 34-9 3-2 i 30 49.8 13-5 37-3 4.4 3 2.1 3-o 3-7 2-7 39-0 1 34 31 53-9 13-7 40.9 4.1 34-8 2-5 I Day 1 July. August. September. October. \ovember. December. 0' 3 R.A. E. T. S ? ,R,A. E. T. 0' S R.A. E.T. 0' s R. A. E.T. 0'sR.A. E.T. 0'sR.A. E.T. i j h m m h tn in h m m, h m tn h m m h m 1 m \ 1 6 43.1 -3-6 8 47.8 6.0 10 43-7 + 0. 3 12 31-7 + 10.5 14 28.1 + 16.3 16 32.2 +10.6 2 47-3 3-8 5J-7 6.0 47-3 0.6 354 10.8 32.1 I6. 3 36.5 10.2 3 5i-4 4.0 55-5 5-9 50-9 0.9 39-o n. i 36.0 I6. 3 4 0.8 9.8 4 55-5 4.2 59-4 5-8 54-5 M 42.6 11.4 40.0 I6. 3 45-2 94 5 59-6 ! 4-3 9 3-2 5-7 58.1 1.6 46.3 11.7 43-9 I6. 3 49-5 O.O 6 7 3-7 4-5 7.0 5.6 H 1.7 1.9 49-9 12.0 47-9 16.2 53-9 8.5 7 7.8 4.6 10.9 5-5 5-3 2 -3 53-6 I2. 3 5i-9 16.2 58.3 8.1 8 7 1 1-9 -4-8 9 H-7 5-3 ii 9.0 + 2.6 12 57.2 + 12.6 IA 56.0 + 16.1 17 2.7 -f 7-7 9 16.0 5-o 18.5 S- 2 12.5 3-o 13 0.9 12.9 15 o.o 16.0 7-i 7.2 1O 20.1 5-i 22.3 5-i 16.1 3-3 4.6 13.1 4.0 15-9 "5 6.7 11 24.2 S- 2 26.1 4.9 19.7 3-7 8-3 134 8.1 15.8 15-9 6.3 12 28.3 5-4 29.8 4-7 23-3 4.0 I2.O 13.6 12.2 15.6 20.3 5.8 13 32.3 5-5 33-6 4-5 26.9 4.4 15-7 13-9 I6. 3 15.5 24.7 5-3 14 36.4 5-6 37-4 ' 4-4 3^-5 4-7 19.4 14.1 20.4 15-3 29.1 4-9 15 7 40-4 5-7 9 4i-i 4.2 H 34.1 + -5-I 13 23.1 +14.3 15 24-5 +15-2 17 33- 6 -h 44 ' 16 44-5 5-8 44-8 3-9 37-7 5-4 26.9 14-5 28.6 15.0 38.0 3-9 17 48.5 5-9 48.6 3-8 4i-3 5.8 3O.6 14.7 3 2.8 14.8 42.4 34 18 52.5 6.0 52.3 3-5 44.9 6.1 344 14.9 36.9 14.6 46.9 2.9 19 56.5 6.0 56.0 3-3 48.5 6.4 38.1 JS- 1 4I.I 14-3 5'-3 2.4 20 8 0.5 6.1 59-7 3- 1 52-1 6.8 41.9 15.2 45-3 14.1 55-8 1.9 21 4-5 6.1 10 3.4 2.8 55-7 7-i 45-7 154 49-5 13.8 18 0.2 1.4 22 8 8.5 6.2 10 7.1 2.6 n 59-3 + 7-5 13 49-5 -H5-5 15 53-7 +13-6 1 8 4.6 + 0.9 23 12.5 6.2 10.8 2-3 12 2.9 53-3 i5-7 57-9 13-3 9.1 0.4 24 16.4 6.2 14-5 2.1 6. 4 8.2 57-i 15.8 l6 2.2 13.0 *3-5 o.i 25 20.4 6.2 18.2 1.8 10. 1 8-5 14 i.o i5-9 6. 4 12.7 18.0 0.6 26 24-3 6.2 21.8 i-5 I 3 .6 8.9 4.8- 16.0 10.7 12.3 22.4 i.i 27 28.2 6.2 25-5 1.2 17-3 9.2 8-7 16.1 15.0 I2.O 26.8 1.6 28 32-2 6.2 29.1 0.9 20-9 9-5 12.5 16.2 19.2 II-7 31-3 2.1 29 8 36.1 6.2 10 32.8 0.6 12 24.5 + 9-9 14 16.4 +16.2 16 23.5 + II.3 18 35-7 - 2.6 30 40.0 6.1 3 6 -4 0.3 28.1 10.2 20.3 16.3 27.8 II.O 40.1 3-i 31 43-9 6.1 40.0 o.o 24.2 16.3 44-5 ! 3-5 ! 178 TABLE LIX. Right Ascension of the True Sun and Equation of Time. 1877: At Greenwich Mean Noon. Dav January. February. March. April. May. June. ItU) . '>B.A. E.T. 0'.B.A. E.T. 0',B.A. E.T. 0' s B. A. E.T. ' S B.A. E.T. 0'sR.A. E.T. h m m h in at h m m h m m h in m h in in \ I 1 8 49.0 4.0 21 I.I 13-9 22 50.1 12.5 o 43-7 ~ 3-9 2 35-o 4- 3- 1 4 37-9 + 24 i 2 534 4-5 5-i 14.0 53-8 .12-3 47-3 3-6 38.8 3-2 4i-9 2-3 3 57.8 4.9 9-2 14.1 57-5 I2.I 5-9 3-2 42.7 3-3 46.1 2.1 4 19 2.2 54 13.2 14.2 23 i-3 1 1.0 54-6 3- 46.5 34 50.2 1.0 5 6. 5 5.8 17-3 14-3 C.O 3 n.6 58.2 2.7 54 3-5 54-3 1.8 6 10-9 6-3 21.3 14.4 8.7 11.4 i 1.9 2.4 54-2 3-6 584 i.6l 7 15-3 6.7 25-3 14.4 12.4 II. 2 5-5 2.1 58.1 3-6 5 2.5 M- 8 19 19.7 7.1 21 29.3 -14-5 23 16.1 10-9 i 9.2 1.8 3 2.0 + 3-7 5 6.7 + 1.2 9 24.0 7-5 33-2 14-5 19.8 10.7 12.9 i-5 5-9 3-7 10.8 I.O 10 28.4 7-9 37-2 14-5 23-5 IO.4 16.5 1.2 9.8 3-8 14.9 o.S 11 32.7 8-3 41.1 14-5 27.1 IO.I 20.2 I.O 13-7 3-8 19.1 0.6; 12 37-i 8.7 45 ' 14-5 30.8 9.9 23-9 0-7' 17.6 3-8 23.2 0.4 13 41.4 9.1 49.0 14-5 34-5 9.6 2 7 .6 o-5 21.5 3-9 27.4 0.2 14 45-7 9-5 52-9 14.4 38-1 9-3 31-3 0.2 254 3-9 3i'S 0.0 15 19 50.0 -9.8 21 56.8 14.4 23 4i-8 9.0 i 35-o -f- O.O 3 294 + 3-9 5 35-7 O.2 16 54-3 10.2 22 0.6 14-3 454 8-7 38.6 0.3 334 3-8 39-8 0.4 17 58.6 10.5 4-5 14.2 49.1 8.4 42.4 o-5 37-3 3-8 44.0 0.6! 18 20 2.8 10.8 8.4 14.2 52-7 8.1 46.1 0.7 41-3 3-8 48.1 0.8 19 7.0 ii. i 12.2 14.1 5 6 4 7.8 49.8 I.O 45-3 37 52-3 I.O 20 n-3 11.4 16.1 14.0 o o.o 7-5 53-5 1.2 49-3 3-7 56.5 1-3 21 15-5 11.7 19.9 13.8 3-7 7-3 57-3 1.4 53-3 3-6 6 0.6 22 20 19.7 12.0 22 23.7 13-7 o 7-3 -6.9 2 I.O 4- 1.6 3 57-3 + 3-6 6 4.8 1-7; 23 23-9 12.2 27-5 13-6 10.9 6.6 4-7 1.8 4 i-3 3-5 8.9 24 28.1 12.5 31-3 134 14.6 6-3 8.5 2.0 54 34 13.1 2.1 25 3 2 -3 12.7 35- 13.2 18.2 6.0 12.3 2.1 94 3-3 17.2 2-3 : 26 364 12.9 38.8 I3-I 21.8 5-7 16.0 2-3 134 3- 2 21.4 2.6 27 40.6 I3-I 42.6 12.9 25-5 54 19.8 2-5 17-5 3.1 25.6 2.8- 28 44-7 13-3 46.3 12.7 29.1 23.6 2.6 21.5 3-o 29.7 3- 29 20 48.8 13-5 o 32.7 -4-8 2 27.4 + 2.8 4 25.6 4- 2.9 6 33-9 3-2 30 52-9 13.6 364 4-5 31.2 2.9 29.7 2.7 38.0 34 31 57-o I 3 .8 40.0 4.2 33-8 2.6 Dav July. August. September. October. November. December. uy. 0' 8 B.A. E.T. 0'sR.A. E.T. 0'sB.A. E.T. 0' S B.A. E.T. 0'.,R.A. E.T. 0'.,B.A. E.T. h m m h m m h m m h m in h m in h m m 1 6 42.1 3-5 8- 46.8 6.0 10 42.8 + 0.2 12 30.8 + 10-5 14 27.2 4-16-3 16 31.1 + 10.7: 2 46.2 3-7 50-7 6.0 46.4 -5 34-5 10.7 3I-I I6. 3 354 10.3! 3 54 3-9 54-6 C.Q 50.0 0.9 38.1 II. I 35-i I6. 3 39-8 9-9: 4 54-5 4-i 584 5 53-7 1.2 41.7 11.4 39-o I6. 3 44.1 9-5! 5 58.6 4-3 9 2.3 5-7 57-3 i-5 454 II.7 43-o I6. 3 48.5 Q I 6 7 2.7 44 6.1 5-6 ii 0.9 1.8 49.0 12.0 47.0 16.2 52.9 o.o 7 6.8 4.6 9-9 5-5 4-5 2.2 52.7 12.2 51.0 16.2 57-2 8.2 8 7 10.9 -4.8 9 13-8 54 ii 8.1 + 2-5 12 56.4 + 12-5 H 55-o +16.1 17 1.6 4- 7.8! 9 15.0 4.9 17.6 S- 2 11.7 2.9 13 o-o 12.8 59- 16.0 6.0 7-3 1O 19.1 21.4 I 5*3 3.2 3-7 13.0 15 3-i 15-9 10.4 6.9 11 23.2 5-2 25.1 4-9 18.9 3.6 74 13-3 15.8 14.8 6.4 12 27-3 5-3 28.9 4.8 22.5 3-9 ii. i 13-5 1 1. 2 15-7 19.2 5-9 13 54 32-7 4.6 26.1 4.2 14-8 1 13-8 15-3 15-5 23.6 5-5 14 354 5.6 36.5 44 29.7 4.6 18.5 14.0 194 154 28.1 5.0 15 7 39-5 5.7 9 40.2 - 4.2 ii 33-2 4- 5.0 13 22.3 +14.2 15 23-5 4-15.2 17 32.5 + 4-5 16 43-5 5-8 43-9 4.0 36.8 5-3 26.0 14.4 27.6 15.0 3 6 -9 4.0 17 47-5 5-9 47-7 3-8 40.4 29.7 14.7 3 1.8 14.8 41.4 3-5 18 5-9 3-6 44-o 6.0 33-5 14.8 35-9 14.6 45-8 3- 19 55- 6 6.0 55 i . 34 47.6 6.4 37-2 15-0 40.1 14.4 50.2 2-5 20 59.6 6.1 58.8 51.2 6.7 41.0 15-2 44-3 14.1 54-7 2.0 21 8 3-5 6.1 10 2.5 2.9 54-8 44-8 15-3 48.5 13-9 59-1 i-5 22 8 7-5 6.2 10 6.2 - 2.6 n 584 + 74 13 48.6 -H5-5 15 52-7 + 13.6 18 3-5 + I.I 23 6.2 9.9 2.4 12 2.O 7.8 524 15.6 5 5 -9 134 8.0 o-5 24 jCC 6.2 13.6 2.1 5-6 8.1 56.2 15.8 16 i.i 13*1 12.4 O.I 25 19.4 6.2 17.2 1.8 9.2 8.5 14 o.o i5-9 54 12.8 16.9 - 0.5 i 26 234 6.2 20.9 1.6 12.8 8.8 3-9 ib.o 9.6 12.5 21.3 I.O 27 28 27-3 31.2 6.2 6.2 24.6 28.2 I.O 16.4 20.0 9.1 9-5 16.1 16.1 13-9 18.2 I2.I u.8 25-7 30.2 14 1-9 29 6.2 10 31.9 0.7 12 23.6 + 9.8 14 15-5 +16.2 16 22.5 +"4 18 34.6 - 2.4 30 39- 6.1 35-5 0.4 27.2 IO.I 19.4 1 6.2 26.8 ii. i 39-o 2.Q 31 42.9 6.1 39-2 O.I 23-3 16.3 43-5 34 TABLE LIX. 179 Right Ascension of tlae True Sun and Equation of Time. 1878 : At Greenwich Mean Noon. j AAV January. February. March. April. May. June. VRJi 0' S R.A. E. T. sB.A. E.T. 0'.,B.A. E.T. 0'sB.A. E.T. 0'sB.A. E.T. S B.A. E.T. h in 771 h m in k m m h in m h m m h m in 1 18 47.9 3-9 21 O.I -.-13.9 22 49-2 12.5 o 42.8 - 3-9 2 34-1 + 3-0 4 36.9 + 2.4 2 52-3 4-3 4.2 14.0 52-9 12.3 46.4 3-6 37-9 3-i' 41.0 2-3! 3 56.7 4.8 8.2 14.1 56.6 la.i 50.0 3-3 41.8 3-2 45-1 2.1 i 4 19 i.i 5-3 12.3 14.2 23 04 11.9 53-7 3- 45-6 3-3 49-2 2.o! 9 5-5 5-7 I6. 3 14-3 4.1 11.7 57-3 2-7 494 34 53-3 1.8! 6 9-9 6.2 20.3 14.4 7-8 M-5 I I.O 2-5 53-3 3-5 574 1.6 * 14-3 6.6 24-3 14.4 "5 II. 2 4-7 2.2 57-2 5 1.6 1.4 8 19 18.6 - 7-o 21 28.3 -14-5 23 15.2 II.O i 8.3 i-9 3 i.o + 3-7 5 5-7 + 1.2 9 23.0 7.4 32.3 14.5 I8..9 10.7 12.0 1.6 4.9 3-7 9-8 I.I 10 27-3 7-8 36.2 14-5 22.6 10-5 I 5 .6 i-3 8.8 3-8 14.0 0.8 11 31-7 8.2 40.2 H-5 26.2 IO.2 19-3 I.O 12.7 3-8 18.1 0.7 12 36.0 8.6 44.1 H-5 29.9 9-9 23.0 0.8 16.7 3-8 22.2 0.5 13 40.3 9.0 48.0 14.5 33-6 9-7 26.7 o-5 2O.6 3-8 26.4 ! 0.3 14 44.6 94 51-9 14.4 37-2 9-3 34 o-3 24.5 3-9 30.5 ! o.i 15 19 48.9 9-7 21 55.8 14.4 23 40.9 9.1 i 34-i o.o 3 28.5 + 3-8 5 34-7 0.2 16 53.2 10.1 59-7 14-3 44.6 8.8 37-8 4- 0.2 324 3-8 38.9 ! 0.4 17 57-5 ! i4 22 3.5 14.2 48.2 8.5 41.4 0-5 364 3-8 43- 0.6 18 20 i.8j 10.7 74 14.2 51.8 8.2 45-2 0.7 40.4 3-8 47.2 0.8 i 19 6.0 II.O "3 14.1 55-5 7-9 48.9 0.9 44-3 3-8 5!-3 I.O 2O 10.2 n-3 15.1 14.0 59-i 7.6 52.6 I.I 48-3 3-7 55-5 1.2 21 H-5 11.6 18.9 I 3 .8 2.8 7-3 56.3 1.4 52.3 3-6 59-6 1-4 22 20 18.7 11.9 22 22.7 13-7 o 6.4 7-o 2 O.I + 1.6 3 5 6 -3 + 3-6 6 3.8 i-7i 23 22.9 12. 1 26.5 13-6 10.0 6.7 3-8 1.8 4 0.4 3-5 8.0 1.9! 24 27.1 12.4 30-3 134 13-7 6.4 7.6 1.9 44 34 I2.I 2.1 J25 31.2 12.6 34-i 13-3 17-3 6.1 n-3 2.1 8.4 3-3 I6. 3 2-3 1 26 35-4 12.8 37-9 !3-i 2I.O 5.8 2-3 12.5 3-2 20.4 2 -5 27 39-5 13.0 41.7 12.9 24.6 5-5 18.9 2-5 16.5 3-i 24.6 2.7 28 43-7 13.2 454 I2. 7 28.2 5-i 22.7 2.6 20.6 3- 28.7 2.9 1 29 20 47.8 134 o 31.8 -4-8 2 26.5 + 2.8 4 24.6 -1- 2.9 6 32.9 3-i 30 5i-9 13.6 35-5 4-5 30-3 2.9 28.7 2.7 37-o 3-3: 31 56.0 13.7 39-i 4.2 32-8 2.6 Day. July. August. September. October. November. December. 0'sB. 4. E.T. 0' S B.A. E.T. 0' S R.A. E.T. 0'sB.A. E.T. 0'sB.A. E.T. S B.A. E.T. ! h in in h m in h in in h m m h m 771 h m m 1 6 41.1 - 3-5 845-9 6.1 10 42.0 + O.I 12 30.0 + 10.3 14 26.3 + 16.3 16 30.1 + 10.7 2 45-3 3-7 49.8 6.0 45 - 6 0.4 33-6 10.7 30.2 I6. 3 344 10.4 3 494 3-9 53-7 6.0 49.2 0.7 37-3 10.9 34-i I6. 3 38.7 IO.O 4 53-6 4.1 57-5 5*9 52-8 I.I 40.9 "3 38-1 16-3 43-i 9.6 5 57-7 4-3 9 i-4 5-8 564 1.4 44-5 n.6 42.0 I6. 3 474 9.2 6 7 1.8 44 5-2 5-7 II O.I 1.7 48.2 11.9 46.1 16.2 51.8 8.8 7 5-9 4.6 9.1 5.6 3-6 2.1 51.8 12.2 50.0 16.2 56.2 8-3 S 7 10.0 -4-8 9 12.9 54 ii 7-3 + 2. 4 12 55-5 + 124 14 54.0 +16.1 17 0.6 + 7-9 9 14.1 4-9 16.7 5-3 10.8 2.8 59-2 12.7 58.1 16.0 4.9 74 jio 18.2 5-i 20.5 5-2 14.4 3-i 13 2.8 I 3 .0 15 2.1 15.9 9-3 7.0 11 22.3 5-2 24-3 5.0 18.1 34 6.5 13.2 6.2 15.8 13-7 6-5 12 26.3 5-3 28.0 4.8 21.6 3-8 10.2 J 3-5 10.2 15-7 18.1 6.1 13 34 5-5 31.8 4-7 25.2 4.2 13-9 13-7 14-3 15.6 22.6 5-6 14 344 5.6 35-6 4-5 28.8 4-5 I 7 .6 14.0 18.4 154 27.0 5-i 15 738.5 $-7 9 39-3 - 4-3 ii 32.4 + 4-9 I 3 21.4 +14.2 15 22.5 +15-3 17 314 + 4-6 16 42-5 s-s 43-i 4.1 36.0 5- 2 25.1 14.4 26.6 !5-i 35-8 4.2 17 46.6 5-9 46.8 3-9 39-6 5.6 28.8 14.6 30.8 14.9 40.3 3-6 18 50.6 5-9 50-5 3-7 43-2 5-9 32.6 14.8 34-9 14.7 44-7 3-2 19 54.6 6.0 54-2 34 46.7 36.3 15.0 39-1 14.4 49.1 2-7 2O 58.6 6.1 58.0 3-2 54 6^6 4O.I i5-i 43-3 14.2 53-6 2.2 21 8 2.6 6.1 10 1.7 3- 53-9 7.0 43-9 iS-3 474 14.0 58.0 x -7 22 8 6.6 6.2 10 5-3 2.7 ii 57-5 + 7-3 13 47-7 + 154 15 5i-7 + I3-7 18 2.4 4- 1.2 23 10.6 6.2 9.1 2-5 12 I.I 7-7 51-5 15.6 55-9 134 6.9 0.7 i 24 14-5 6.2 12.7 2.2 4-7 8.0 55-3 J 5-7 16 o.i 1 Z' 1 11.3 0.2 25 18.5 6-3 16.4 1.9 8-3 8.4 59-i 15.8 44 12.8 15-8 ! 26 22.4 6.2 20.1 i-7 11.9 8.7 14 3.0 8.6 12.5 2O.2 0.8 1 27 26.4 6.3 237 1.4 15.6 9.0 6.8 16.0 12.9 12.2 24.7 T -3 ; 28 30.3 6.2 27.4 i.i 19.4 94 10.7 16.1 17.2 1 1.8 29.1 1.8 29 8 34.2 6.2 10 31.0 0.8 II 22.8 + 97 H 14-5 + 16.2 16 21.5 +11.5 18 33-5 2.7 3O 38-1 6.2 34-7 0.5 26.4 IO.O 18.4 16.2 25.8 II. I 38.0 -2.8 31 42.0 6.1 38.3 0.2 22.3 16.3 424 3-3 180 TABLE LX. I Declination of the Sun. 1875 : At Greenwich Mean Noon. :Day. Jan. | Feb. Mar. Apr. May. June. July. Aug. Sept. Oct. XOT. Dec. o o o o o o o o O ' O <~> !n i 1 23.0 S. 17.1 S. 7.6 S. 4-5 N - i5.oX. 22.0 X. 23. i X. i8.iX. 8.4 X. 3.18. 144 S. 21.88. 2 3 22.9 22.8 16.8 16.6 7.2 4-9 5-3 '5-3 15.6 22.2 22.3 23.1 23.0 17.8 17.6 8.0 3.5 7-6 i 3-9 14.7 15.0 22.O 22.1 4 22.7 16.3 6.5 5.6 i5-9 22.4 22.9 17-3 7-3 i 4-3 154 22.2 5 22.6 16.0 6.1 6.0 16.2 22-5 22.8 17.0 6.9 4.7 15-7 22.4 6 22-5 15-6 5-7 6.4 16.5 22.6 22.7 16.8 6.5 5.1 16.0 22-5 7 22.4 15-3 5-3 6.8 1 6.8 22.7 22.6 16.5 6.1 5-5 16.3 22.6 8 22.3 S. 15.08. 4.98. 7.2 X. 17.1 X. 22.8 X. 22.5 X. 16.2 X. 5.8 X. 5-88. 16.68. 22.78. 9 22.1 14.7 4-5 7-5 17-3 22. 9 22.4 15-9 54 6.2 16.8 22.8 i 1O 22.0 14.4 4-i 7-9 17.6 23.0 22.3 15.6 6.6 17.1 22.9 i 11 21.8 14.1 3-8 8-3 17.9 23.1 22.1 15-3 4.6 7.0 17.4 23.0 i 12 21.7 13-7 34 8.6 18.1 23.2 22.0 15.0 4.2 74 17.7 2 3 .I 13 21.5 134 9.0 18.4 23.2 21.9 14.7 3-9 7-7 17.9 232 i -*- 1 * 21.3 2.6 94 18.6 23-3 21.7 14.4 3-5 8.1 18.2 23.2 15 21. 1 S. 12.7 S. 2.28. 9 . 7 x. iS.SX. 23-3 N. 21.6 X. 14. i X. 3-iX. 8.58. 18.5 S. 23.3 S. 16 21.0 12.4 1.8 10. 1 19.1 234 21.4 13.8 2-7 8.8 18.7 23-3 17 20.8 12.0 i-4 10.4 19-3 234 21.2 13-5 2-3 9.2 19.0 234 18 20.6 II.7 I.O 10.8 19-5 234 21. 1 13.2 1.9 9-6 19.2 23-4 19 20.4 "3 0.6 n. i 19.7 23.4 2O.9 12.8 9-9 19-5 234 1 20 20. 1 II.O 0.28. H-5 20.0 23-5 20.7 12.5 1.2 10.3 19.7 23-4 i 21 19.9 10.6 0.2 X. 1 1.8 2O.2 23-5 20.5 12.2 0.8 10.7 19.9 23-5 ! 22 19.78. 10.2 S. o.6X. 12.2 X. 20.4 X. i 23.5 X. 20.3 X. H-9X. 0.4 X. 11.08. 20.1 S. 23-5 8. 1 23 19.5 9-9 I.O I2. 5 20.6 23-4 20.1 11.5 o.olS. 11.4 20.3 23-5 24 19.2 9-5 1.4 12.8 20.7 234 19-9 11.2 0.4 11.7 20-5 234 i 25 19.0 9.1 1.8 13.! 2O-9 23-4 197 10.8 0.8 I2.I 20.7 234 26 18.7 8.8 2.2 *3-5 21. 1 23-4 19-5 10.5 1.2 I2. 4 20.9 234 27 18.5 8.4 2.6 13.8 21.3 23-3 19-3 10. 1 1.6 12.8 21. 1 23-3 28 18.2 8.0 2.9 14.1 21.4 23-3 I9.O 9.8 2.0 13-1 21. 3 23-3 29 18.0 S. 3-3 N. 14.4 X. 21.6 X. 23-3 N. I8.8X. 9.4 X. 2.48. 1348. 21.58. 23.28. 3O 17.7 3-7 14.7 21.8 23.2 18.6 9.1 2.7 13.8 21.6 23-2 31 17.4 4.1 21.9 18.3 8.7 14.1 23.1 1876 : At Greenwich Mean Noon. 1 o 23.0 S. 17.28. 7.3S. 4.8 X. o 15-3 N. 22. I X. 23. i X. I7-9N. 8.iX. 3.48. i 4 .6 S. 21.98. j 2 23.0 16.9 7.0 5-2 15.6 22.3 23.0 17.6 7-7 3-8 15.0 22.1 3 22.9 16.6 6.6 5.6 15-9 22.4 22.9 17.4 7-3 4.2 15.3 22.2 4 22.8 16.3 6.2 5-9 1O.2 22.5 22.8 17.1 7.0 4.6 15.6 22.3 5 22.7 16.0 5-8 6-3 16.4 22.6 22.7 16.8 6.6 5- 15-9 22-5 6 22.5 '5-7 54 6.7 l6. 7 22.7 22.6 1 6.6 6.2 54 16.2 22.6 7 22.4 154 5- 17.0 22.8 22.5 16.3 5-9 5-7 16.5 22.7 8 22.3 S. 15.18. 4.68. 74 N. 17.3^. 22.9 X. 22.4 X. i6.oX. 5-5 N. 6.1 S. 16.88. 22.8 S. 9 22.2 14.8 4.2 7-8 17.5 23.0 22.3 15-7 5.1 6-5 17.1 22.9 10 22.0 14-5 3-8 8.2 17.8 23.1 22.2 154 4-7 6.9 17-3 23.0 11 21.9 14.1 3-5 8.6 18.0 23.1 22.0 IS- 1 4-3 7-3 17.6 23.1 12 21.7 13.8 3- 1 8.9 18.3 23.2 21.9 14.8 4.0 7-6 17.9 2 3 .I 13 21-5 13-5 2.7 9-3 18.5 23-3 21.8 H-5 3-6 8.0 18.2 23-2 14 21.4 2-3 9-6 . 18.8 23-3 21.6 14.2 3-2 8.4 18.4 23-3 15 21.28. 12.8 S. 1.98. i o.o X. 19.0 X. 23-3 N. 21.5 X. 13.9 X. 2.8 X. 8.8 S. 18.78. 23-3 s. 16 21.0 12.4 10.3 19.2 234 21.3 13.6 2.4 9-i 18.9 234 17 20.8 I2.I j I.I 10.7 19-5 23-4 21. 1 13.2 2.0 9-5 19.2 234 18 2O.6 11.8 1 0.7 II.O 19.7 23-4 20-9 12.9 1.6 9-9 19.4 23.4 19 20.4 11.4 0.38. 11.4 19.9 234 20.8' 12.6 1.2 10.2 19.6 23-4 20 2O.2 II.O o.iX. 11.7 20. i 23-5 2O.6 12.3 0.9 10.6 19.9 23-5 21 20.0 10.7 0.5 12. 1 20.3 23-5 20.4 11.9 o-5 10.9 20.1 23-5 22 19.88. 10.3 S. 0.9 X. I2.4X. 20.5 X. 23.5 X. 20.2 X. n.6X. o.iX. 11.38. 20.3 S. 23.5 S. 23 !9-5 10.0' 1.3 12.7 20.7 234 20.O 1 1- 3 0.38. ii.o 20.5 23-4 . 1 24 *9-3 9-6 1.7 20.9 234 19.8 10.9 0.7 I 12.0 20.7 234 25 19.0 9.2 2.1 134 21. 1 23-4 19-5 10.6 I.I il2. 3 20.9 23-4 26 18.8 8.8 2-5 13.7 .'21.2 234 19-3 10.2 1.5 12.7 21. 1 23-4 !27 18.5 8.^ 2.8 14.0 i 21-4 23-3 I9.I 9-9 1.9 13.0 21.3 23-3 28 18.3 8.1 3-2 144 21.6 23-3 18.9 9-5 2-3 13-3 21.4 23-3 29 18.08. 7-78. 3-6 X. 14.7 X. 2I.7X. 23.2 X. i8.6X. 9.2 X. 2.6J3. I3-7S. 21.68. 23.2 S. 30 17.8 4.0 15.0 1 21.9 23.2 18.4 8.8 3- 14.0 21.8 23.1 31 17-5 i 4.4 | 22.0 18.1 8.4 14-3 23.1 TABLE LX. 181 Declination of the Sun. 1877: At Greenwich. Mean Noon. Day. Jan. Feb. Mar. Apr. May. June. July. ] Aug. Sept. Oct. Nov. Dec. 1 23.0 S. 17.08. MS. 4 . 7 N. o 1 o I5.2X. I22.IX. o o 23. i N. I7-9N. 8.2 X. o 3-3S. 14.6 S. 2i.9S. 9 22.9 16.7 7.0 5-i 15-5 22.2 23.0 17.7 7.8 3-7 14.9 22.O 3 22.8 16.4 6. 7 I 5 .8 22.4 23.0 17.4 7-4 4.1 15.2 22.2 4 22.7 16.1 6-3 5-8 16.1 22.5 22.9 17.2 7-i 4-5 15-5 22.3 5 22.6 15.8 5-9 6.2 16.4 22.6 22.8. 16.9 6-7 4-9 22.4 6 22-5 15.5 5-5 6.6 16.6 22-7 22.7 16.6 6-3 5-3 16.1 22.6 7 22-3 15.2 5-i 7-0 16.9 22.8 22.6 16.3 5-9 5.6 16.4 22.7 8 22.2 S. 14.9 8. 4.78. 74N- 17.2 N. 22.9 N. 22. 5 X. i6.iX. 5-6X. 6.08. 16.78. 22.8 S. 9 22.1 H-5 4-3 7-7 17-5 23.0 22.3 15.8 5-2 6.4 17.0 22.9 to 21.9 14.2 3-9 8.1 17.7 23.1 22.2 15-5 4.8 6.8 17-3 23.0 11 21.7 13-9 3-5 8-5 18.0 23.1 22.1 15.2 4-4 7-2 17-5 23.1 12 21.6 13.6 3-2 8.8 18.2 23.2 21-9 14.9 4.0 7-5 17.8 23.1 1 13 21.4 13.2 2.8 9.2 18.5 23.2 21.8 14.6 3-7 Z' 9 18.1 23-2 14 21.2 12.9 2.4 9.6 18.7 23.3 21.6 H-3 3-3 8-3 18.3 23-3 15 21. 1 S. 12.58. 2.08. 9.9 N. 19.0 N. 23.3 IS T . 21.5 N. i4.oX. 2.9 X. 8.78. 18.6 S. 23.3S. 16 20-9 12.2 1.6 10.3 19.2 23-4 21.3 13-6 2-5 9.0 18.9 234 17 20.7 11.8 1.2 10.6 19.4 23.4 21.2 13-3 2.1 9-4 19.1 234 IS 20.5 n-5 o.S II. 19.6 23.4 21.0 13.0 i-7 9.8 19-3 23-4 19 20-3 ii. i 0.48. "3 19.9 23.4 20.8 12.7 i-3 10. 1 19.6 234 20 2O.O 10.8 o.o 11.7 20.1 23-5 20.6 12.3 I.O 10.5 19.8 23-5 21 19.8 10.4 0.4 N. 12.0 20. 3 23-5 20.4 12.0 0.6 10.8 2O.O 23-5 22 19.6 S. 10.0 S. 0.8 N. I2.3X. 20.5 N. 23.5 N. 20.2 N. ii. 7N. 0.2X. 11.28. 20.2 S. 23-5 s. 23 19-3 9-7 1.2 12.7 20.7 23-4 2O.O n-3 0.28. n-5 2O.4 23-4 21 19.1 o, 3 1.6 13.0 20.8 23-4 19.8 II.O 0.6 11.9 20.6 234 25 18.9 8.9 2.0 13-3 21.0 23-4 19.6 10.7 1.0 12.2 20.8 23-4 26 18.6 8.6 i 2.4 13.6 21.2 23-4 194 10.3 1.4 12.6 21.0 23-4 27 18.4 8.2 2.8 14.0 21.4 ,23.3 19.2 10.0 1.8 I2. 9 21.2 23-3 28 iS.i 7.8 3-i 14-3 21-5 23.3 l8. 9 9.6 2.2 13-3 21.4 23-3 29 17.88. 3.5 N. 14.6 N. 21. 7 N. 23.2 N. i8.;X. 9.2 N. 2.68. 13.68. 21.68. 23.2 S. 30 17-5' 3-9 14-9 21.8 23.2 18.6 8.9 2.9 13-9 21.7 23.2 31 17-3 4-3 22.0 18.2 8.5 | 14.2 ! 23.1 1878 At Greenwich Mean Noon. 1 23.0 S. 17.1 S. 7-5 S- 4 .6X. i5.iN. 22. i N. o 23. iN. iS.oXJ 8.2 X. o 3.28. o 14.5 S. 21.88. 2 22.9 16.8 7-i 5-o 15-4 22.2 23.0 17.8 7-9 3-6 14.8 22.0 3 22.8 16.5 6.8 5-4 15.7 22.3 23.0 17-5 7-5 4.0 15-1 22.1 4 22.7 16.2 6-4 5.8 16.0 22.5 22.9 17.2 7-i 4-4 15-4 22.3 5 22.6 i5-9 6.0 6.1 16.3 22.6 22.8 17.0 - 6.8 4.8 15-7 22.4 6 22.5 15-6 5.6 6.5 16.6 22-7 22.7 16.7 6.4 5-2 16.0 22.5 7 22.4 i$-3 5-2 6.9 16.9 22.8 22.6 16.4 6.0 5.6 16.3 22.6 22.2S. 14.9 S. 4.88. 7-3 N. 17.1 N. 22.9 N. 22.5 N. i6.iX. 5 . 7 x. 5.98. 16.68. 22.88. 9 22.1 14.6 4.4 7.6 17.4 23.0 22.4 15.8 5-3 6-3 16.9 22.9 10 21.9 H-3 4.0 8.0 17-7 23.0 22.2 15-5 4-9 6.7 17.2 22.9 11 21.8 14.0 3-6 8.4 17.9 23.1 22.1 15-3 4-5 7-i 17-5 23-0' 12 21.6 13.6 3-3 8.7 18.2 23.2 22.O 15.0 4.1 7-5 23.1 13 21.5 J3-3 2.9 9.1 18.4 23.2 21.8 14.6 3-8 7-8 18.0 23.2 14 21.3 13.0 2-5 9-5 18.7 23-3 21.7 H-3 3-4 8.2 18.3 23.2 15 21.18. 12.6 S. 2.IS. 9.8 N. 18.9 N. 23.3 N. 21.5 N. 14. i X. 3.0 X. 8.68. 18.5 S. 23-3 s. 16 20.9 12.3 1-7 10.2 19.1 23-4 21.4 13-7 2.6 8.9 18.8 23.3 17 20.7 11.9 i-3 I0. 5 19.4 23-4 21.2 13-4 2.2 9-3 19.0 23-4 18 20.5 n.6 0.9 10.9 19.6 23-4 2I.O I3- 1 1.8 9-7 19-3 23.4 19 20.3 11.2 o-5 II. 2 I9. 23-4 20.8 12.8 1.4 IO.O 19-5 234 i 20 20.1 10.9 o.i S. 11.6 2O.O 23-5 20.7 12.4 I.O 10.4 19.7 23-4 21 19.9 10.5 0.3 N 11.9 20.2 | 23.5 20.5 I2.I 0.7 10.8 2O.O 23.5 22 19.68. 10.1 S. o.7N 12.2 X. 20.4 N. 23.5 N. 20.3 N. ii.SX. 0.3 X. ii. 18. 20.2 S. 23.5 s. 23 194 9.8 i.i 12.6 : 20.6 23-4 20. 1 11.4 0.18. 11.5 20.4 23-4 24 I 9 .2 9.4 i-5 12.9 20.8 1 23.4 19.9 ii. i 0.5 i 1.8 20.6 23-4 25 18.9 9.0 1.9 13.2 21.0 i23.4 19.7 10.7 0.9 12.2 20.8 23-4 26 I8. 7 8.7 2-3 13-6 21.2 ^23.4 19.4 '10.4 1-3 12-5 21.0 23-4 27 l8. 4 8-3 2-7 13-9 21.3 :2 3 . 3 19.2 IO.O 1.7 12.8 21.2 23-3 28 18.2 7-9 3-i I 4 .2 21-5 23.3 19.0 9-7 2.1 13.2 21.4 23-3 29 17.98. 34* 14.5 N 2 1. 6 N. 23.2 N. 18.7 N. 9.3 X. 2.SS. '13.58. 21.58. 23.2 8. 3O 17.6 3-8 14.8 21.8 i 23.2 18.5 9.0 2.Q 13.8 21.7 23.2 31 J 7-3 4.2 21.9 18.3 8.6 14.2 23.1 182 TABLE LXI. Mean Places of Twenty-five Fixed Stars. 1875 : January 1 , Greenwich. Common Name. ! Right Ascen- sion. Annual Difif. Decli- nation. Annual Diff. Name of Con- stellation. Systematic Name of Star. h m m o o Alpheratz, 2 o 1.93 0.05 28.40 N -{-0.005 Androm'eda, a Andromed e. Polaris (P. Star), 2 I 13.00 35 88.64 N. + 005 The Little Bear, a Ursse Minoris. Acher'nar, I i 33-5 04 57.878. 005 Erid'anus, a Eridani, Hamel, 2 2 0.13' 06 22.87 N + 005 The Ram, a Arietis. Aldebar'an, I 4 28.75 06 16.26 N + 002 The Bull, a Tauri. Capella, 5 746 0.07 45.87 N -j-O.OOI The Charioteer, a Aurigce. Rigel, 5 8.53 5 8.35 s. 001 Ori'on, ft Orionis. Betelguese, Var 5 48.41 5 7.38 N 4- ooo Ori'on, a Orionis, Cano'pus, 6 21.18 02 52.638. -j- ooo The Ship Argo, a Argus. Sir'ius, 6 39.64 4 16.55 S. -j- ooi The Great Dog, a Canis Majoris. Castor, 7 26.62 0.06 32.I6N O.OO2 The Twins, a Geminorum. Pro'cyon, 7 3 2 -76 5 5.54 N 002 The Little Dog, a Canis Minoris. Pollux, 7 37-66 06 28.33 N OO2 The Twins, ft Geminorum. Alpharcl, 9 21.45 05 8.12 S. + 004 The Hydra, a Hydrce. Reg'ulus, 10 1.71 05 12.58 N 005 The Lion, a Leonis. a 1 Crucis, 12 19.65 0.05 62.43 S. +0.005 The Cross, a 1 Crucis. Spica, 13 18.61 5 10.51 S. -f- 005 The Virgin, a Virginis. ft Centauri, I 3 55- 02 07 59-77 S. -r 005 The Centaur, ft Centauri. Arctu'rus, 14 9.96 5 19.83 N 005 The Bear- Watcher, a Bootis. a 2 Centauri, 14 31.14 07 60.31 S. -{- 004 The Centaur, a 2 Centauri. Anta'res, 16 21.75 0.06 26.158. -{-0.002 The Scorpion, a Scorpii. Vega, Altair, 1 18 32.71 19 44.68 03 5 38.67 N 8.54 N 4- ooi -j- 003 The Harp, The Eagle, a Lyrre. a Aquilce. Fo'malhaut, !|22 50.74 5 30.288. 005 The Southern Fish, a Piscis Australis. Mar'kab, 2 22 58.53 05 14-53 N . + 005 Peg'asus, a Pegasi. The DifF. in R. A. ia always additive. The DifF. in Dec. is additive or subtractive according as the sign is + or . TABLE LXII. Meridian-Passages of Twenty-five Fixed Stars. 1875 : At Greenwich Mean Noon on the First Day of the Month. Name of Star. Jan. Feb. Mar. Apr. May. June. July. Aug. Sept. Oct. Nov. Dec. h m h m h m h m h m h m h m h m k m h m h m h in Alpheratz, 5 i5 3 3 I 14 2320 21 29 19 26 17 22 15 17 1321 ii 33 937 733 Polaris, 626 414 225 031 22 40 2037 1833 1628 1432 1244 10 48 844 | Achernar, 646 434 245 05 1 23 o 2057 1853 1648 1452 13 4 ii 8 9 4 i Hamel, 7i3 5 i 312 i 18 2327 21 24 19 20 1715 15 19 I33i "35 93i i Aldebaran, 942 730 541 347 I 5 6 2353 2M9 1944 1748 16 o 14 4 12 O Capella, 10 21 8 9 6 20 426 235 3 2 2228 20 23 1827 1639 1443 1239 Rigel, 1022 8 10 621 427 236 033 22 29 2024 1828 1640 1444 I2 4 Betelguese, II 2 850 7 i 5 7 316 I 13 23 9 21 4 19 8 16 20 1524 13 2O Canopus, H34 922 733 539 348 145 2341 21 36 1940 1752 1556 1352 Sirius, "53 941 752 558 4 7 2 4 24 o 2155 1959 18 ii 1615 14 II Castor, 12 40 1028 839 645 454 251 047 22 42 2046 1858 I 7 2 1458 Procyon, 12 46 1034 845 651 5 o 257 053 2248 2052 19 4 17 8 15 4 Pollux, 1251 1039 850 656 5 5 3 2 058 2253 2057 19 9 1713 15 9 Alphard, H35 1223 1034 840 649 446 242 037 2241 2053 i857 i653 Regulus, 1515 13 3 ii 14 9 20 729 526 322 I 17 23 21 21 33 1937 1733 a 1 Crucis, 1733 15 21 1332 ii 38 947 744 540 335 139 235i 21 55 I95i Spica, 1832 16 20 1431 1237 1046 843 639 434 2 3 8 o 50 2254 20 50 ft Centauri, 19 8 1656 15 7 13 J 3 II 22 919 715 5 10 3 14 i 26 2330 21 26 Arcturus, 1.923 17 II 15 22 | 13 28 H37 934 730 525 329 i 4 1 2345 2I 4 I a' 2 Centauri, 1944 1732 1543 13*49 1158 955 751 546 35 2 2 o 6 22 2 Antares, 2135 1923 1734 1540 1349 ii 46 942 737 54i 353 i 57 2353 Vega, 2346 21 34 1945 1751 16 o 1357 ii 53 948 752 6 4 4 8 2 4! Altair, 058 22 46 2057 19 3 1712 15 9 13 5 II O 9 -4 716 5 2 J 316 ! Fomalhaut, 4 4 1 5 2 3 22 9 20 18 1815 16 ii 14 6 12 IO IO 22 8 26 6 22 : Markab, 412 2 II 22 17 20 26 1823 16 19 H 14 12 18 1030 834 630 TABLE LXIII. 183 Reductions of Meridian Passages of the Fixed Stars. 1875 : At Greenwich Mean Noon on each day of the month. Day. Jan. Feb. Mar. Apr. May. June. July. Aug. Sept. Oct. Nov. Dec. 1 O h O m O h O m o h o m O h O m O h O m O h o m O h O m Q\i Q m O h O m O h O m Q!I o m O h O m 2 4 4 4 4 4 4 4 4 4 4 4 4 3 9 8 7 7 8 8 8 8 7 7 8 9 4 13 12 II ii II 12 12 12 II II 12 !3 5 18 16 15 14 15 16 16 15 14 14 16 17 2.2 20 19 18 19 21 21 19 18 18 20 22 7 26 24 22 22 23 25 25 23 22 22 2 4 26 8 o 31 o 28 26 o 25 o 27 o 29 o 29 o 27 o 25 o 25 o 28 o 30 9 35 32 3 29 31 33 33 31 29 29 32 35 10 39 36 33 33 35 37 37 35 32 33 36 39 11 44 40 37 36 39 4i 4i 38 36 36 40 ^44 12 48 44 4i 40 43 45 45 42 40 40 44 48 13 52 48 44 44 46 49 49 46 43 44 48 52 11 57 52 48 47 50 54 53 5o 47 48 52. 57 15 i i o 56 o 52 o 5i o 54 o 58 o 57 o 53 o 50 o 5i o 56 i i 16 5 I O 55 55 58 I 2 i i 57 54 55 I 6 17 10 3 59 59 I 2 6 5 i i 58 59 4 10 18 H 7 i 3 I 2 6 10 9 5 i i I 2 9 15 19 18 ii 6 6 10 15 !3 8 5 6 13 19 2O 22 15 10 10 H 9 17 12 8 10 17 23 21 27 19 H 13 18 23 21 16 12 14 21 28 22 1 31 i 23 i 17 i 17 I 22 i 27 I 2j i 19 i 16 i 18 I 25 i 32 23 35 26 21 21 26 3i 29 23 19 21 29 37 24 39 30 25 25 30 35 33 27 23 25 34 4i 25 43 34 28 28 34 39 37 3i 26 29 38 46 26 48 38 32 32 38 43 4i 34 30 33 42 50 27 52 42 35 36 42 48 45 38 34 37 46 55 28 56 45 39 40 46 52 49 42 37 4i 5i 59 29 2 i 49 * 43 i 44 i 50 i 56 i 53 i 45 i 41 i 44 i 55 2 3 3O 4 46 47 55 2 57 48 44 48 59 8 31 8 50 59 2 I 52 52 12 The Reduction is to be subtracted from the Time in Table LXII. The result -will be Apparent Time ; and this, although adapted to 1875, will be within 2 m for many years. 184 TABLE LXIV. Reduction of Daily or Hourly changes in Right Ascension. Daily Change. Decimal Parts of the Hour. Hourly Change. O h .O 0".l 0".2 O h .3 O h .4 0".5 !l .6 0' 7 0".8 0".9 l h .O m m m in in m m m in m m m i' o.o O.O 0.00 o.oo o.oo o.oo o.oo o.oo o.oo 0.00 o.oo o.oo o.oo 1 o oo 00 00 00 00 00 00 00 oo 00 25 2 oo oo oo 00 oo 00 01 OI OI 01 50 3 Q 00 00 00 oo OI OI OI 01 01 OI 75 4 Q 00 00 oo -01 01 01 01 01 01 02 l.OO 0.5 0.0 o.oo o.oo O.OI O.OI O.OI O.OI O.OI 0.02 0.02 O.O2 1.25 6 o oo oo 01 01 01 OI 02 02 02 02 50 7 Q 00 01 01 ol 02 02 02 02 3 3 75 8 o OO OI OI OI 02 02 02 3 03 3 2.OO 9 o OO OI OI 01 02 02 3 3 03 04 25 1.0 0.0 O.OO O OI O.OI 0.02 O O2 O.O2 0.03 0.03 0.04 o 04 250 1 o 00 01 01 02 02 : 03 03 04 04 5 75 2 o oo 01 01 02 02 03 03 04 04 05 3.OO 3 o OI OI 02 02 3 03 04 04 5 05 25 j 4 o 01 OI 02 02 03 03 04 05 05 06 50 1.5 o.o O.OI O.OI 002 O.O2 0.03 0.04 0.04 0.05 0.06 0.06 3.75 6 o OI OI 02 3 3 04 05 05 06 07 4.OO 7 o ! or OI 02 3 03 04 05 06 06 07 25 ; 8 o 01 01 02 03 04 04 05 06 07 07 50 9 o 01 02 02 03 04 5 06 06 07 08 75 | 2.0 O.O O.OI o 02 O.O2 0.03 0.04 0.05 0.06 0.07 0.07 0.08 5 OO 1 01 02 03 03 04 05 06 07 08 09 25 2 O OI 02 3 04 05 05 06 07 08 09 50 3 Q 01 02 03 04 05 06 07 08 09 10 75 4 o OI 02 03 04 05 06 ; 07 08 09 IO 6.OO 2.5 o.o ! o.oi 0.02 0.03 0.04 0.05 o 06 0.07 0.08 0.09 O.IO 6.25 6 O OI 02 03 04 05 06 08 09 IO II 50 7 O OI ! O2 03 04 06 07 08 09 10 II 75 | 8 01 \ 02 03 05 06 7 08 09 10 12 7.OO 9 01 02 04 05 06 07 09 IO II 12 25 3.0 O.OI O.O2 o 04 005 0.06 0.07 0.09 10 II 0. 12 7.5O 1 o 01 03 04 05 06 08 09 10 12 13 75 ! 2 o OI 3 04 05 07 08 09 II 12 '3 .OO 3 o 01 03 04 05 07 08 IO II 12 14 25 4 o 01 03 04 06 07 08 10 II 13 14 50 3.5 0.0 O.OI 0.03 0.04 0.06 0.07 0.09 O.IO 0.12 0.13 0.15 8.75 6 01 03 04 06 07 09 10 12 13 15 9.OO j 7 o 02 3 05 06 08 09 II 12 14 15 25 8 o 02 3 05 06 08 09 II 13 14 16- 50 9 02 03 05 06 08 IO II 13 J 5 16 75 4.0 00 0.02 0.03 0.05 007 0.08 O.IO 0.12 013 0.15 0.17 10.00 1* o 02 3 05 07 09 10 12 14 15 17 25 2 02 3 05 07 09 10 12 14 16 17 5O 3 o 02 04 05 07 09 II 13 14 16 18 75 ; 4 o 02 04 05 07 09 II 13 15 16 18 11.00 4.5 o.o O.O2 0.04 0.06 0.08 0.09 0. II 0.13 0.15 0.17 0.19 1125 6 02 04 06 08 09 II 13 15 17 *9 50 ; 7 o 02 04 06 08 10 12 14 16 18 20 75 8 o 02 04 '06 08 10 12 14 16 18 20 12.00 9 02 04 06 08 10 12 14 16 18 20 25 5.0 0.0 O.O2 o 04 0.06 0.08 10 0.12 0.15 0.17 0.19 O.2I 12.50 1 o 02 04 06 08 II J 3 15 17 19 21 75 i 2 o 02 04 06 09 II I 3 15 17 19 22 13.OO 3 02 04 07 09 II '3 15 18 20 22 25 4 o 02 04 07 09 II 13 16 18 20 22 50 5.5 0.0 0.02 0.05 0.07 0.09 0. II 0.14 0.16 0.18 0.21 0.23 13.75 6 o 02 5 07 09 12 14 16 19 21 23 14.OO 7 o 02 5 07 09 12 14 17 19 21 24 25 8 02 5 07 IO 12 14 17 !9 22 24 5O 9 02 5 07 10 12 14 17 20 22 25 75 6.0 0.0 002 005 0.07 O.IO 0.13 0.15 0.17 0.20 0.22 0.25 15.00 TABLE LXIV. 185 Reduction of Daily or Hourly Changes in Right Ascension. Daily , Change. Hours of the Day. Hourly Change. 2*0 4".0 6 b .O 8 h .O 10 b O 12 h .O 14 h .O 16 b .O 18M> 20 h .O 22 h .O m in m m m, m m 771 m m m m s o.o O.OO o.oo 0.00 O.OO 0.00 0.00 0.00 0.00 O.OO 0.00 O.OO o.oo 1 01 01 O2 03 04 05 06 06 07 08 09 25 02 3 05 07 08 IO 12 13 15 16 18 50 3 02 5 07 IO 12 15 17 20 22 25 27 75 4 03 07 IO '3 17 20 23 27 30 33 37 1.00 0.5 0.04 0.08 0.12 0.17 O.2I 0.25 0.29 -33 0.38 0.42 0.46 1.25 6 05 10 15 20 25 30 35 40 45 50 55 5O 7 06 12 17 23 29 35 41 46 52 S 8 64 75 8 07 13 20 27 33 40 46 53 60 67 73 2.00 9 07 15 22 3 37 45 5 2 60 67 75 82. 25 1.0 0.08 0.17 0.25 o-33 0.42 0.50 0.58 0.67 0.75 ' 0.83 0.92 2.5O 1 09 18 27 37 46 55 64 73 82 92 I.OI 75 2 10 20 3 40 60 70 80 90 I.OO IO 3.00 3 II 22 3 2 43 54 65 76 86 97 08 19 25 4 12 2 3 35 47 58 70 81 93 1.05 17 28 50 1.5 0.12 0.25 0.37 0.50 0.62 0.75 0.87 I.OO 1. 12 1.25 1.37 3.75 6 13 27 40 53 67 80 93 07 20 33 47 4.OO 7 H 28 42 57 7 1 85 99 13 27 42 56 25 $ 30 45 60 75 90 1.05 20 35 50 65 50 9 16 S 2 47 63 79 95 ii 26 42 58 74 75 2.0 0.17 o-33 0.50 0.67 0.83 I.OO 1.17 i-33 1.50 1.67 1.83 5.OO 1 17 35 52 70 87 05 22 40 57 75 92 25 2 18 37 55 73 92 10 28 47 65 83 2.02 5O 3 19 38 58 96 15 34 92 II 75 4 20 40 60 80 I.OO 20 40 60 80 2.00 20 6.00 2.5 0.21 0.42 0.62 0.83 1.04 1.25 1.46 1.66 1.87 2.08 2.29 6.25 6 22 43 65 87 08 30 73 95 17 38 50 7 8 22 23 45 70 90 93 12 17 35 40 1 80 87 2.02 IO 25 33 47 57 75 7.OO 9 24 48' 72 97 21 45 69 94 17 42 66 25 3.0 0.25 0.50 o-75 I.OO 1.25 1.50 i-75 2.00 2.2S 2.50 2-75 7.5O 1 26 5 2 77 03 29 55 81 06 32 84 75 2 27 53 80 07 33 60 86 13 40 67 93 8.00 3 27 55 82 10 37 65 92 20 47 75 3.02 25 4 28 57 85 13 42 70 98 27 55 83 12 50 3.5 O.29 0.58 0.87 1. 17 1.46 1.75 2.04 2-33 2.62 2.92 3 .2I 8.75 6 30 60 9 20 80 IO 40 70 3.00 3 9.00 7 3 1 62 92 23 54 85 16 4 6 77 08 'IQ 25 8 3 2 63 95 27 58 90 21 53 85 17 48 50 9 32 65 97 30 62 95 27 60 92 25 57 75 4.0 0.33 0.67 I.OO i-33 1.67 2.00 2-33 2.67 3.00 3-33 3-67 I O.OO 1 34 68 02 37 71 05 39 73 07 42 76 25 2 35 70 05 40 75 10 45 80 15 50 85 5O 3 36 72 07 43 79 15 51 86 22 58 94 75 4 37 73 IO 47 83 20 56 93 30 67 4-03 11.00 4.5 0-37 0-75 1. 12 1.50 1.87 2.25 2.62 3.00 3-37 3-75 4.12 11.25 6 38 77 15 53 92 30 68 07 45 83 22 50 7 39 40 11 17 20 11 96 2.00 35 40 74 80 20 g 92 4.00 3 1 40 75 12.00 9 82 22 63 04 45 86 26 67 08 49 25 5.0 0.42 0.83 1.25 1.67 2.08 2.50 2.91 3-33 3-75 4.17 4.58 12.5O 1 42 85 27 70 12 55 97 40 82 25 67 75 2 43 87 30 73 17 60 3-03 47 90 33 Z 7 13.00 3 44 88 32 21 65 09 53 97 42 86 25 4 45 90 35 80 25 70 15 60 4-05 5 95 50 5.5 6 0.46 47 0.92 93 40 1.83 87 2.29 33 2 85- 3.21 26 3.66 73 4.12 20 4.58 67 5-04 13 13.75 14.00 7 47 95 42 90 37 85 32 80 27 75 22 25 8 48 97 45 . 93 42 90 38 87 35 83 3 2 50 9 49 98 47 97 46 95 44 93 42 92 41 75 6.0 0.50 I.OO 1.50 2.OO 2.50 3.00 3-5 4.00 . 4-50 5.00 5-50 15.00 186 TABLE LXV. Reduction of Hourly Changes in the Moon's Right Ascension. Parts Change in One Hour. Parts Hour. 1 1U .O l m .l l ra .2 1^.3 l m .4 l m .5 l m .6 1-.7 1-8 I 111 . 9 20 of the Hour. m m m . m m m m M m tn in, m m o O.OO O.OO O.OO O.OO O.OO O.OO O.OO 0.00 O.OO 0.00 O.OO 1 02 02 02 02 02 02 03 3 03 03 3 1 2 03 04 04 4 05 05 5 06 06 06 07 2 3 5 05 06 06 07 07 08 08 09 09 10 3 4 07 07 08 9 9 10 ii ii 12 13 13 4 5 0.08 0.09 0.10 O.I I O.I2 O.I2 0.13 0.14 0.15 0.16 0.17 5 6 10 II 12 13 14 15 16 17 18 19 20 6 7 12 13 14 15 16 17 19 20 21 22 2 3 7 8 13 15 16 17 19 20 21 23 24 25 27 8 9 15 16 18 19 21 22 24 25- 27 28 3 9 10 O.I7 0.18 "O.2O O.22 0.23 O.25 0.27 0.28 0.30 0.32 -33 10 11 18 20 22 24 26 27 29 3 1 33 35 37 11 12 13 20 22 22 26 26 28 , 28 30 30 32 32 35 34 37 36 39 38 40 43 12 13 14 23 26 28 30 33 35 37 40 42 44 47 14 15 0.25 0.27 0.30 0.32 -35 o-37 0.40 0.42 0-45 0.47 0.50 15 16 27 29 32 35 37 l 40 43 45 48 5i 53 16 17 28 31 34 37 40 i 42 45 48 5 1 54 57 17 18 30 33 36 39 42 45 48 5 1 54 57 60 18 19 32 35 38 44 47" 51 54 57 60 63 19 20 0-33 0-37 0.40 -43 0.47 0.50 -53 -57 0.60 0.63 0.67 2O 21 35 38 42 45 49 . 52 56 59 63 66 70 21 22 37 40 44 48 51 55 59 62 66 7 73 22 23 24 38 40 42 44 46 48 50 52 54 56 57 60 61 64 a 69 72 77 80- 23 24 25 0.42 0.46 0.50 -54 0.58 0.62 0.67 0.71 0-75 0.79 0.83 25 26 43 48 5 2 56 61 65 69 74 78 82 8? 26 27 45 49 54 58 63 67 72 76 81 85 90 27 28 47 51 56 61 66 70 75 79 84 89 93 28 29 48 53 58 63 68 72 77 82 87 92 97 29 30 0.50 -55 0.60 0.65 0.70 o-75 0.80 0.85 0.90 -95 I.OO 3O 31 52 57 62 67 72 77 83 88 93 98 3 31 32 33 53 55 8 64 66 69 75 77 80 82 85 88 93 96 99 I.OI 04 07 IO 32 33 34 57 62 68 74 79 85 91 96 1.02 08 34 35 0.58 0.64 0.70 0.76 0.82 0.87 -93 0.99 1.05 i. ii 1.17 35 36 60 66 72 78 84 9 96 1.02 08 '4 20 36 37 62 68 74 80 86 92 99 5 ii 17 2 3 37 38 63 70 76 82 89 95 I.OI 08 H 20 27 38 39 65 78 84 97 04 10 17 23 3 39 40 0.67 o-73 0.80 0.87 0-93 I.OO 1.07 1.13 1.20 1.27 r -33 40 41 68 75 82 89 96 02 09 16 23 3 37 41 42 70 77 84 98 05 12 I 9 26 33 40 42 43 72 86 93 I 00 07 15 22 29 3 6 43 43 44 73 81 88 95 03 10 17 25 32 39 47 44 45 0-75 0.82 0.90 0.97 1.05 1. 12 1.20 1.27 i-35 1.42 1.50 45 46 77 84 92 I.OO 07 15 2 3 30 38 46 53 46 47 78 86 94 02 IO 17 2 5 33 41 49 57 47 48 80 88 96 04 12 20 28 44 5 2 60 48 49 82 90 98 06 14 22 '3 1 39 47 55 63 49 50 083 0.92 I.OO 1. 08 I.I7 1.25 i-33 1.42 1.50 1.58 !.6 7 5O 51 85 93 02 . 10 19 27 36 44 53 61 70 51 52 53 87 88 95 97 4 06 13 15 21 24 30 3 2 39 47 5 56 11 73 77 52 53 ! 54 90 99 1. 08 17 26 35 44 53 62 71 80 54 ! 55 0.92 I.OI I.IO I.I9 1.28 i-37 1.47 1.56 1.65 i-74, 1.83 55 i 56 93 3 12 21 31 40 49 59 68 87 56 57 95 14 2 3 33 42 52 61 71 80 9 57 58 97 06 16 26 45 55 64 74 84 93 58 59 98 08 18 28 38 47 57 67 77 87 97 59 60 I.OO I.IO 1.20 I. 3 1.40 1.50 i. 60 1.70 1.80 1.90 2.00 6O i I'.OOil'.lO J.20 1*.30 1--.40 1 -.50 1*.60 1 8 .70 1".80 1".9O 2*.00 L Change in One Minute. i TABLE LXV. 187 Reduction of Hourly Changes in the Moon's Right Ascension^ Parts **. -4-1-* ^ Change in One Hour. Parts f . r. _ ot tne Hour. 2 m .O 2 in .l 2 m .2 2 m .3 2 m -4 2 ra .5 2 m .6 2 m .7 2 m .8 2 m .9 3 m .O pi tne Hour. in m in m m m m m m m m in m O 0.00 0.00 O.OO 0.00 0.00 O.OO 0.00 0.00 O.OO O.OO O.OO O 1 03 03 -04 04 4 04 04 04 05 05 5 1 2 07 07 | 07 08 08 08 09 09 09 IO 10 2 3 10 10 II II 12 12 13 13 14 14 15 3 4 13 H 15 15 16 17 17 18 19 19 20 4 5 0.17 0.17 0.18 0.19 O.2O 0.21 0.22 0.22 0.23 0.24 0.25 5 6 20 21 22 23 2 4 25 26 27 28 29 30 6 7 23 24 26 27 28 29 30 31 33 34 35 7 8 27 28 29 3 1 32 33 35 36 37 39 40 8 9 30 31 33 34 36 37 39 40 42 43 45 9 10 0-33 0-35 0-37 0.38 0.40 0.42 0-43 o-45 0.47 0.48 0.50 10 11 37 38 40 42 44 46 48 49 51 53 55 11 12 40 42 44 46 48 50 52 54 56 58 66 12 13 43 45 48 50 52 54 56 58 61 63 65 13 i 14 47 49 54 56 58 61 63 65 68 70 14 15 0.50 0.52 o-55 0-57 0.60 0.62 0.65 0.67 0.70 0.72 ' 0-75 15 16 53 56 59 61 64 67 69 72 75 77 86 16 17 57 59 62 65 68 74 76 79 82 85 17 18 19 60 63 6 d 66 70 69 73 72 76 75 79 78 82 81 85 84 89 87 92 90 95 18 19 2O 0.67 0.70 o-73 0.77 0.80 0.83 0.87 0.90 0.93 0.97 I.OO 20 21 70 73 77 80 84 87 91 94 98 I.OI 5 21 22 73 77 Si 84 88 92 95 99 1.03 06 IO 22 23 77 80 84 88 92 96 I.OO 1.03 07 ii 15 23 24 80 84 88 92 96 I.OO 04 08 12 16 20 24 25 0.83 0.87 0.92 0.96 I.OO 1.04 1.08 1. 12 I.I7 1. 21 1.25 .25 26 87 91 95 I.OO 04 08 13 17 21 26 3 26 27 90 94 99 03 08 12 17 21 26 3 35 27 28 93 98 1.03 07 12 17 21 26 3 1 35 40 28 29 97 I.OI 06 ii 16 21 26 30 35 40 45 29 3O I.OO 1.05 I.IO 1.15 i. 20 1.25 I. 3 '35 1.40 1-45 1.50 ' 30 31 03 08 14 19 24 29 34 39 45 31 32 07 12 17 2 3 28 33 39 44 49 55 60 32 33 IO I 5 21 26 3 2 37 43 48 54 59 65 33 34 1 3 19 25 3 36 42 47 53 59 64 70 34 35 1.17 1.22 1.28 i-34 1.40 1.46 1.52 I -57 1.63 1.69 J -75 35 36 37 20 2 3 26 29 5 38 42 44 48 50 54 I 62 66 68 73 74 79 80 85 36 37 38 27 33 39 46 52 58 65 71 77 84 90 38 39 3 3 6 43 49 56 62 69 75 82 88 95 39 40 i-33 1.40 1.47 i-53 1.60 1.67 i-73 i. 80 1.87 i-93 2.OO 40 41 37 43 5 57 64 71 84 98 5 41 42 40 47 54 61 68 75 82 89 96 2-03 IO 42 43 43 50 58 65 72 79 86 93 2.OI 08 15 43 44 47 54 61 69 76 .83 91 98 05 13 20 44 45 1.50 1.57 1.65 1.72 i.8o 1.87 i-95 2.02 2.IO 2.17 2.25 45 46 53 61 69 76 84 92 99 07 15 22 3 46 47 57 64 72 80 88 96 2.04 II 19 27 35 47 I 48 60 68 76 84 92 2.OO 08 16 24 3 2 40 48 ! 49 63 71 80 88 96 04 12 20 29 37 45 49 50 1.67 r -75 1.83 1.92 2.00 2.08 2.17 2-25 2-33 2.42 2.50 50 51 70 78 95 04 12 21 29 ' 38 46 55 51 52 73 82 99 08 17 25 34 43 5i 60 52 ! 53 77 85 94 2.03 12 21 3 38 47 56 65 53 j 54 80 89 98 07 16 25 34 43 52 61 70 54 i 55 1.83 1.92 2.02 2. II 2.20 2.29 2.38 2.47 2-57 2.66 2-75 55 i 56 87 96 05 15 24 33 43 5 2 61 7 1 80 56 I 57 90 99 09 18 28 37 47 56 66 75 85 57 58 93 2.03 13 22 3 2 42 51 61 71 80 90 58 59 97 06 16 26 36 46 56 65 75 85 95 59 60 2.OO 2.IO 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 60 2 s .OO 2 s . 10 2 s . 20 2 s . 3O 2 S .4O 2 s . 5O 2 8 .60 2 s . 70 2 s . 80 2 8 .9O 3 S .OO | Change in One Minute. 188 TABLE LXVI. Reduction of the Mean Sun's Right Ascension. Tenths of the Hour. Time. Arc. | O h .O O b .l O h .2 O h .3 O h .4 O b .5 00 o h .7 O h .8 O h .9 ft m 0.00 in O.O2 m 0.03 m 0.05 m 0.07 m O.o8 m O.IO m O.I I m 0.13 m 0.15 o O 1 16 18 20 21 23 25 26 28 29 31 ir > 2 33 34 36 38 39 41 43 44 46 47 30 3 49 5i 53 54 56 57 59 61 62 64 45 4 66 67 69 7i 72 74 75 77 79 80 60 5 0.82 0.84 0.85 0.87 0.89 0.90 0.92 0.94 0-95 0.97 75 6 99 1. 00 1.02 1.03 1.05 1.07 i. 08 1. 10 1. 12 "3 90 r i-'5 17 18 20 22 23 25 26 28 30 1O5 31 33 35 36 38 40 41 43 44 46 12O 9 48 49 5i 53 54 56 58 59 61 63 135 1O 1.64 1.66 1.68 1.69 1.71 1.72 1.74 1.76 1.77 1.79 150 11 81 82 84 86 87 89 90 92 94 95 165 12 97 99 2.OO 2.O2 2.04 2.05 2.07 2.09 2.10 2.12 10 13 2.13 2.15 17 18 20 22 23 25 27' 28 195 14 30 32 33 35 37 38 39 41 43 45 210 15 2.46 2.48 2.50 2.51 2-53 2-55 2.56 2.58 2-59 2.61 225 16 63 64 66 68 69 7i 73 74 76 88 240 17 79 81 83 84 86 87 89 9i 92 94 255 18 96 97 99 3.01 3.02 3-04 3-05 3-07 3-9 3.10 27O 19 3.12 3-'4 3-15 17 19 20 .22 24 25 27 285 20 , 3.28 3-3 3-32 3-33 3-35 3-37 3-38 3-40 342 3-43 300 21 45 47 48 5o 52 53 55 56 58 60 315 22 61 63 65 66 68 70 7i 73 74 76 330 23 78 79 81 83 84 86 88 89 9 1 93 345 o.o 1.5 3.0 4.5 6.0 7.5 9.0 10. 5 12.0 13.5 Tenths of the Hour in Arc. 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