Astron. Of r>*. NAVIGATION THE MACMILLAN COMPANY NEW YORK BOSTON CHICAGO DALLAS ATLANTA SAN FRANCISCO MACMILLAN & CO., LIMITED LONDON BOMBAY CALCUTTA MELBOURNE THE MACMILLAN CO. OF CANADA, LTD. TORONTO NAVIGATION BY HAROLD JACOBY RUTHERFURD PROFESSOR OF ASTRONOMY IN COLUMBIA UNIVERSITY Nefo gorfc THE MACMILLAN COMPANY 1917 All rights reserved V "V * vent. D^t. X COPTKIGHT, 1917, BY THE MACMILLAN COMPANY. Set up and electrotyped. Published October, 1917. Xorfcoooti Press J. S. Gushing Co. Berwick & Smith Co. Norwood, Mass., U.S.A. MACLEAB JACOBY QUARTERMASTER, THIRD CLASS, U. 8. N. ENLISTED FOR THE PERIOD OF THE WAR THIS VOLUME IS OFFERED AS A MARK OF RESPECT BY HIS FATHER 3C754o PREFACE THE present volume was undertaken with certain very definite aims. In the first place, it is intended to be com- plete in itself, so that it should be possible to navigate a ship in any ocean not very near the north or south pole without other books or tabular works, excepting only the nautical almanac for the year in which the voyage is made. To attain this end without unduly extending the size of the volume, certain essential nautical tables have been abridged ; but all are given in sufficiently extended form to permit of actual navigation with their aid; and they are especially suitable for beginners, who can here attain the necessary knowledge with less effort than would be necessary with more bulky volumes. In cases where very extended tables are conven- ient, they are mentioned in the text. In the second place, the author has not assumed that the reader possesses formal mathematical and astronomical knowledge, or desires to possess such knowledge. When- ever methods of navigation require for their demonstration an understanding of spherical trigonometry, or some other branch of formal mathematical science, such demonstrations have been replaced with incomplete or ''outline " demonstra- tions designed for the non-mathematical reader. Practical methods are fully explained ; and an attempt has always been made so to word the explanations that the reader, even the beginner, will understand his problem, and will know what he is doing, and why he does it. The requirements of those who may study without a teacher have received constant and special attention. To meet these requirements the whole subject is presented in vii viii PREFACE a somewhat informal manner; such topics as the use of logarithms, or the principles on which all mathematical tables are constructed these less attractive parts of the subject are not presented in a special chapter, but are de- scribed in a sort of digression, when needed in the discussion of an actual navigational problem. Finally, to further simplify and condense his material, the author has made no attempt to include every method that can possibly be used to navigate a ship, or that ever has been used to navigate a ship ; his purpose has been rather to limit the volume to the methods at present thought best by the most reliable modern authorities. Other books on navigation have been used freely, espe- cially in the preparation of the tables. Among these, that admirable encyclopedia of navigation, known as "Bowditch," published by the Hydrographic Office, United States Navy, and Kelvin's " Tables for Sumner's Method at Sea" have been found of the greatest help. Miss Dorothy W. Block, Instructor of Astronomy in Hunter College, New York, has helped with great energy in the preparation of the tables and the correction of the text. It is hoped that such errors as may now remain in the book are few in number. H. J. COLUMBIA UNIVERSITY, August, 1917. TABLE OF CONTENTS CHAPTER . PAGE I. THE FUNDAMENTAL PROBLEM OF NAVIGATION . . 1 The problem stated. Reasons for the existence of the problem. Definition of "ship's position." Longitude meridians and latitude parallels. Greenwich the initial meridian. Position determined by observation; on the coast and at sea. Dead reckoning. Sextant observa- tions. Chronometer. II. DEAD RECKONING WITHOUT LOGARITHMS ... 7 The two problems. Designation of .ship's course. Latitude difference and departure. The traverse table. Use and construction of tables in general. Arguments and tabular numbers. Relation between departure and longitude difference. Middle latitude. III. DEAD RECKONING WITH LOGARITHMS ... 23 Explanation of number logarithms and their use. Multiplication and division. Trigonometric logarithms. Solution of the two problems. Middle latitude sailing. Mercator sailing. Meridional parts. Great circle sail- ing. The rhumb line. Composite sailing. Parallel sailing. Traverse sailing. IV. THE COMPASS 40 The card, how divided. Degrees and points. Boxing the compass. Lubber line. True course and compass course. Error, variation and deviation. Swinging ship. Azimuth circle and pelorus. The compass formulas. The two deviation tables. Comparative table of points and degrees. V. COASTWISE NAVIGATION 53 The "fix." Bow bearings. Doubling the bearing on the bow. Bow and beam bearings. Distance a-beam. Cross bearings. The danger angle. Danger bearing. Soundings. ix X TABLE OF CONTENTS CHAPTER PAGE VI. THE SEXTANT . .61 Description of the instrument and its use. The vernier. Index error. Three adjustments. The artificial horizon. Correcting the altitude. Dip. Refraction. Parallax. VII. THE NAUTICAL ALMANAC . . . . .75 Specimen pages of it. Greenwich mean time. Decli- nation. Equation of time. Astronomic and civil day. Apparent solar time. Chronometers and the rate card. Right ascension. Solar and sidereal time. VIII. OLDER NAVIGATION METHODS ..... 86 The noon-sight for latitude. Tropic observations and the midnight sun in high latitudes. Preparing for the observation. Setting the cabin clock. Star observa- tion. Ex-meridian observation. The time-sight for longitude. Set of current. Star time-sight. Condensed forms of calculation. IX. NEWER NAVIGATION METHODS . . . . . 108 Errors produced by dead reckoning. Captain Sumner, and the Sumner line. Bearing of the line. The Sumner point. Azimuth tables. Condensed form of calcula- tion. Star observations. Comparison of Sumner navi- gation with time-sight navigation. The Kelvin table. Condensed forms of sun and star observations. Inter- section of two Sumner lines obtained with a special table. Motion of ship between observations. X. A NAVIGATOR'S DAY AT SEA . . . . .141 Voyage planned from New York to Colon. Departure at Sandy Hook lightship. The course to Watlings Island. The variation and deviation applied. Azimuth of the sun observed at sunrise. Bow and beam bearings of Barnegat Light. The patent log and the log book. New course from Barnegat. Morning sight worked as a Sumner line. Another Azimuth observation. Weather thickens at 11 : 30. Ex-meridian sight at 11 : 42, worked as a Sumner line. Afternoon sight worked as a Sumner line. Posi- tion of ship fixed from intersection of the two lines. East- erly current estimated. Compass error again tested. The course set for the night. TABLES . 153 LIST OF ABBREVIATIONS USED IN THE PRESENT VOLUME Alt. for altitude ; App. for apparent ; Arg. diff. for argument difference ; Cf . for compare ; Chron. for chronometer ; Comp'd for computed ; Cos for cosine ; Cot for cotangent ; Csc for cosecant ; C. W. for chronometer minus watch ; Dec. for declination ; Dep. for departure ; Dist. for distance ; D. R. for dead reckoning; Eq. for equation of time ; G. A. T. for Greenwich apparent time; G. M. T. for Greenwich mean time; Hav. for haversine ; H. D. for hourly difference; Int. diff. for interpolation difference ; Lat. for latitude ; Lat. diff. for latitude difference ; Log for logarithm ; Long. for longitude ; Long. diff. for longitude difference ; Mer. lat. diff. for meridional latitude difference ; Obs'd for observed ; p for polar distance ; R. A. for right ascension ; s for half sum ; Sec for secant ; Sin for sine ; T for ship's apparent solar time (or star's hour-angle) ; Tab. diff. for tabular difference ; Tan for tangent. xi NAVIGATION CHAPTER I THE FUNDAMENTAL PROBLEM OF NAVIGATION To find one's way in a ship across the trackless ocean is our problem. Most people would like to know how it is solved ; nor is the solution very difficult to understand when set forth in simple language and without too great wealth of technical detail. We hope the reader will find this to be the case after a study of the following pages. Our fundamental problem can be more fully stated quite easily. It consists in the determination of a ship's location on the earth's surface at any given moment. If this loca- tion can be determined, it becomes a comparatively easy matter to ascertain the direction (north, south, northeast, southeast, etc.) in which the ship must be steered in order to reach her port of destination. For the location of the port of destination on the earth's 'surface is of course also known : and if we know where the ship and her destined port both are, we can easily find the right course for the helmsman. With the fundamental' problem stated in this way, it would almost seem as if there were really no such problem in existence. For when the ship begins her voyage, she is necessarily in a known port. Knowing also the port to which she is to go, we should be able to determine her proper course from the one known port to the other. This course being then steered, no further navigational proceedings would be required. But this reasoning is incorrect, because a ship B 1 2 NAVIGATION does not actually advance across the ocean in exactly the direction in which she is steered. Ocean currents deflect her ; and the action of a strong wind blowing against one of her sides will have a similar effect. Currents and winds cannot be predicted with accuracy : and so it becomes necessary to re-determine the ship's position frequently at sea. This should be done at least once daily if possible; and when it has been done, the mariner can take a new "departure," as he calls it, and lay a new course for his intended port. Thus the effect of ocean currents, etc., can be eliminated, and the voyage made as safely as if they did not exist. Now this determination of the ship's position at sea, and when out of sight of land, is strictly an astronomical problem. It can be solved by means of astronomical ob- servations, and in no other way. But before giving an out- line of how this is done, let us first see what is meant by the words " ship's position at sea." How can we describe a ship's position so that one mariner could tell another where she is located, and thus enable the second mariner to find her? To thus indicate the point on the earth's surface occupied by the ship has a certain similarity with giving the address of a house in a city. Such a city address always consists of two separate statements ; as, for instance, the name of a street and the number of the house. An address cannot be given completely unless two different facts are stated. They need not necessarily be a street name and a street number : we can equally well designate such an address by stating that the house is at the corner of a certain street and a certain avenue. But here also the address is made up of two separate facts. This form of stating an address as the intersection of a certain street and avenue is the form having the closest resemblance to the method of the navigator. If the city avenues are supposed to run north and south, and the streets THE FUNDAMENTAL PROBLEM OF NAVIGATION 3 east and west, as they do in New York (approximately), the analogy with navigation will be almost perfect. For the navigator imagines the earth covered with a net- work consisting of " avenues, " running north and south, and " streets/' running east and west. He calls the " avenues" meridians of longitude, and the "streets" parallels of latitude. Then he designates the position of a ship on the ocean by stating that it is at the intersection of a certain meridian of longitude and parallel of latitude. There are 360 such meridians of longitude : each begins at the terrestrial equator, and runs north and south from there to the north and south poles of the earth. Of the latitude parallels there are ISO. 1 They all run east and west, parallel to the terrestrial equator ; 90 are between the equator and the north pole, and the other 90 between the equator and the south pole. One of the longitude meridians (that passing through Greenwich, England) is chosen arbitrarily as the starting point for counting longitude meridians. To this initial meridian is assigned the number 0, and the other meridians are numbered successively 1, 2, 3, etc. So numbered, the meridians are called "degrees" of longitude; the third one, for instance, being written 3. The meridians may be counted either eastward or westward from Greenwich, a ship on the 20th meridian west of Greenwich, for instance, being in longitude 20 west. The latitude parallels are similarly counted north and south from the equator ; and if the above ship were on the 40th latitude parallel north of the equator, her complete "address," or position at sea, would be long. 20 "W. ; lat. 40 N. Of course a ship would only rarely be located exactly at the intersection of a meridian and parallel. Therefore, the space between any two successive meridians and between any two successive parallels is subdivided into 60 parts, called minutes of arc. Thus the above ship, if halfway 1 Including the equator twice, but excluding the two poles. 4 NAVIGATION between a pair of meridians and also halfway between a pair of parallels, might be in longitude 20 30' west, and in latitude 40 30' north. This would be written long. 2030'W.; lat. 4030'N. Each minute of longitude and latitude is further sub- divided, when extreme accuracy is required, into 60 seconds ; so that if the ship were a little to the north and a little to the west of the above position, she might, for instance, be in long. 20 30' 26" W. ; lat. 40 30' 10" N. These meridians and parallels, or longitude and latitude lines, appear on many maps and charts as straight lines, or at least as lines only slightly curved. But being all lines imagined drawn on the earth, which is almost an exact sphere or round ball, they must really all be circles. Thus, the terrestrial equator is really a big circle, girdling the earth, and divided into 360 equal parts, or. degrees. At each of the division points a meridian starts northward toward the pole. This meridian is also a big circle perpendicular to the equator. The distance along the meridian from the equator to the pole is divided into 90 equal parts or 'degrees, and the whole distance from equator to pole is one quarter of a complete circumference of the earth. The 90 degrees, from equator to pole, thus repre- senting one quarter of a circumference of the earth, a com- plete circumference contains 4 X 90, or 360 degrees, the same as the equator. So the degrees measured along the meridians are equal to the degrees measured along the equator. The former are degrees of latitude, the latter degrees of longitude; and degrees of latitude are equal to degrees of longitude, when the latter are measured along the equator. The length of each degree is then 60 nautical miles. Having thus indicated what is meant by a ship's position in latitude and longitude, we shall next describe in outline how such a position may be determined by observation. If the ship is within sight of a coast-line, there will probably THE FUNDAMENTAL PROBLEM OF NAVIGATION 5 be some lighthouse, or other "aid to navigation," in view, from which the navigator can ascertain where he is. Methods for doing this are described later (p. 53). But when the ship is really at sea, with no land in sight, real deep-sea methods must be employed. These methods, when the weather is clear, always include an observation of the sun or some other heavenly body. When the weather does not permit such observations, the mariner can still find his position approximately by means of "dead reckoning" (abbreviated, D. R.). This process will be described in detail in the next chapter; but we can already state that it consists in a calculation based on his astronomic observation of latest date. Knowing where the ship was the last time he observed the sun, and also know- ing both the direction in which he has steered and the (approximate) speed of the ship, the navigator can calculate (also approximately) the location of the point he has reached. Even when astronomical observations are made, the D. R. calculation is always carried out, because the navi- gator is always anxious to know how nearly correct his D. R. result would have been, if the day had been cloudy. Furthermore, this result also acts as a check on the astronomi- cal work, and tends to increase the navigator's confidence in the correctness of his final result as to the ship's location. The manner in which the ship's position is found from astronomic observations will of course be explained in detail later. It is all done with an instrument called a sextant. This is merely a contrivance with which the navigator can measure how high the sun (or other heavenly body) is in the sky at any moment. The sun is highest in the sky daily at noon, but it is not equally high on different days in the year. Nor is it equally high on the same date in different latitudes. Thus, by measuring with the sextant how high it is on any particular date at noon, as seen from the ship, the navigator learns the terrestrial latitude in which the ship is located. 6 NAVIGATION Similar sextaht observations made at other suitable times during the day, when combined with exact readings taken from an accurate chronometer such as every ocean-going ship carries, will similarly make the ship's longitude known. All this will of course be explained in full detail in later chapters. CHAPTER II DEAD RECKONING WITHOUT LOGARITHMS As we have seen (p. 5), this is a process by means of which the mariner can calculate a ship's position in latitude West Longitude 61 6.0' 59 58 57 5,6 5,5 46 Nf 1 45 44 43 42 41 40 FIG. 1. Dead Reckoning. (Diagram not drawn to scale.) 7 8 NAVIGATION and longitude, without special astronomic observations of any kind. In the accompanying Fig. 1, which represents a portion of a chart of the North Atlantic, a ship's position at noon is shown at the point Y. This point we will call the ship's " initial position," in discussing our present prob- lem. We will suppose that it was correctly obtained by as- tronomic observations, and that these showed the ship at Y to be in lat. 42 11' N. and long. 59 28' W. from Green- wich. Sometime in the afternoon, having traveled a dis- tance estimated from the known speed of the ship as 63 miles, and having "made good" this distance in the direction YP, the ship arrives at P. This point P we will call the ship's "final position" ; and our problem now is to find its latitude and longitude. This problem may be called the first fundamental dead- reckoning problem. The second and remaining fundamental problem is the converse of the first, and may be stated as follows : having given the latitude and longitude of the initial point Y, as occupied by the ship, and also the latitude and longitude of the final point P, it is required to find the dis- tance from Y to P in miles, and also the direction of the line yp. 1 To understand these two problems properly it is next necessary to explain how we may define the words "direc- tion YP." This is done by referring the line YP to the direction of the arrow shown in the figure. This arrow is parallel to the longitude meridians on the chart, and therefore points due north. The angle between the arrow YN and the line YP is marked in the figure, and is called the "ship's course." This angle is really the difference in direction of the two lines YN and YP. The point Y is called the "vertex" of the angle, and all angles are designated 1 We think it advisable to place these two important converse problems together, and to call them both problems of dead reckon- ing, though many writers on navigation confine the phrase " dead reckoning" to the first fundamental problem alone. DEAD RECKONING WITHOUT LOGARITHMS Dead by three letters, the letter belonging to the vertex being placed between the other two; in this case the angle is called either NYP or PYN. Now let us draw a line PQ (fig. 2), from P to NY, and perpendicular to NY. Then the motion of the ship from Y to P will have carried her north of the point Y by a distance equal to YQ, and east of the point Y by a distance equal to QP. Q This is not strictly true, unless the earth's surface, throughout the small area involved in the present problem, can be regarded as a flat surface. Such a flat surface is called in geometry a " plane" surface; and these calculations therefore belong to that Reckoning. part of navigation which is called " plane sailing." Plane- sailing calculations are easy calculations, and they are generally sufficiently accurate for the purposes of the navigator. The ship's course, being thus an angle, must be designated by means of a unit of measure suitable for measuring angles. For this purpose the degrees and minutes already used for longi- tude and latitude (p. 3) are usually employed. Fig. 3 shows that a latitude, for instance, is really an angle, and must there- fore also be measured in de- grees. P is the earth's pole, PQ a meridian, and the latitude of FIG. 3. -Latitude Angle. the observer at IS the angle OCQ, here about 40. So it is clear that the ship's course NYP (figs. 1 and 2) will be measured in degrees. Minutes are not really needed in measuring courses, as they are in measuring latitudes; the nearest whole degree is always accurate enough,, because 10 NAVIGATION it is never possible to steer a ship on her proper course with absolute exactness. In fact, many mariners use a still less precise method of measuring courses by means of "the points of the compass." (See p. 40.) Resuming our two fundamental problems (p. 8), let us now begin with the first one, and proceed to find the lati- tude and longitude of the point P (figs. 1 and 2). To solve this problem, we must not only know the distance YP (63 miles), as traveled by the ship, but also the number of degrees in the course angle NYP. Let us suppose this course angle happens also to be 40. The problem then appears as shown in Fig. 4. We now know the distance YP and the angle QYP. Evidently the next step is to find the distances QFand QP. QY, in our present problem, is called a "latitude difference" and QP is called a "departure." 4. -Dead To find the "latitude difference" and "departure" from the course angle and dis- tance we may either use that branch of mathematics called plane trigonometry, or we may find them from a special navigation table, called a "traverse table." Our Table 1 (beginning p. 154) is such a table. Before x beginning its use it will be well for the reader to note in general that all mathematical tables consist of two sets of numbers. The first set of numbers are called " argu- ments" of the table, and the second set are called "tabular numbers." The main object of the table is to furnish us with the proper tabular number when we know the proper argument. The ordinary multiplication table is a good example of a mathematical table. It is usually written as follows and 1 The beginner may find it advisable, on a first reading of the book, to omit this explanation of mathematical tables, returning later when he finds a reference to it in the text. The dead reckoning problem under discussion is resumed on p. 13. DEAD RECKONING WITHOUT LOGARITHMS 11 it affords a good opportunity of .studying the principles underlying all mathematical tables in a case so simple as to offer no difficulty. MULTIPLICATION TABLE (to illustrate " argument " and " tabular number") 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 2 4 6 8 10 12 14 16 18 20 22 24 3 6 9 12 15 18 21 24 27 30 33 36 4 8 12 16 20 24 28 32 36 40 44 48 6 10 15 20 25 30 35 40 45 50 55 60 6 12 18 24 30 36 42 48* 54 60 66 72 7 14 21 28 35 42 49 56 63 7Q 77 84 8 16 24 32 40 48 56 64 72 80 88 96 9 18 27 36 45 54 63 72 81 90 99 108 10 20 30 40 50 60 70 80 90 100 110 120 11 22 33 44 55 66 77 88 99 110 121 132 12 24 36 48 60 72 84 96 108 120 132 144 In this table the arguments are printed in heavy type and are contained in the left-hand column and the topmost horizontal line. In using the table, these arguments are given in pairs, being always the pair of numbers to be mul- tiplied. In fact, in the case of most tables, the arguments are thus given in pairs, though there are some tables with but a single argument. In the present case one number from the pair of arguments will be found in the left-hand column, the other in the top horizontal line. Thus, if we wish to multiply 6 and 8, these two numbers constitute the pair of arguments. We find the right line (belonging to 6) and column (belonging to 8) , and the tabular number 48 (marked with a *) occurs at the intersection of the 6-line and the 8- column. If the pair of arguments are taken in the order 8x6 instead of 6 X 8, we should use the 8-line and the 6-column, again finding the required product (48) as the tabular number at the intersection. 12 NAVIGATION Sometimes the given arguments cannot be found di- rectly in the table. Thus we might wish to multiply 6| (written 6.5) by 8. Evidently the proper tabular number would be halfway between the 6x8 tabular number (48) and the 7 X 8 tabular number (56). The correct answer would therefore be 52. This process, by wh&Ii the tabular number 52 is obtained, is called " in- terpolation." The example 6J X 8 is an extremely simple one. When less easy ones occur, the interpolation is best made as follows : we ascertain by subtraction how much the tabular number increases while the argument changes from 6 to 7. This increase is here 8, because the tabular number changes from 48 to 56 in the 8-column, while the argument in the left-hand column changes from 6 to 7. This increase of 8 in the tabular number is called a " tabular difference." We now compare the given argument (6.5) with the nearest argument (6) occurring in the left-hand column of arguments, and find an " argument difference" of 0.5 (being 6.5 minus 6). Since this " argument dif- ference" is 0.5, we must evidently take 0.5 X 8 (8 being the tabular difference), and increase the tabular number 48 by 0.5 X 8, or 4. This again brings us to 52. Similar exam- ples are : (1) 5.3 X 4 = 21.2; (2) 7.7 X 8 = 61.6. In example (1) the tabular numbers are 20 and 24; the tabular difference is 4. 0.3 X 4 = 1.2; 20 + 1.2 = 21.2, the answer. Both examples may be verified, of course, by ordi- nary multiplication. When both given arguments contain fractions, as, for instance, 5.3 X 8.4, the resulting " double interpolation" is so complicated as to be of little practical use to the navi- gator. To make this general explanation of mathematical tables complete, it remains to show how they can be used in an inverse manner ; i.e. to find the argument from the tabular DEAD RECKONING WITHOUT LOGARITHMS 13 number. Thus, if we were told that the tabular number is 48, and one argument 8, an inspection of the table would at once show that the other argument must be 6. In this way the table might be used for division as well as multi- plication ; and interpolation would evidently also be possible. Many mathematical tables must frequently be thus used in an inverse manner. Having thus explained the peculiarities of mathematical tables, we return to our dead-reckoning problem and its solution by means of the traverse table (p. 154). Referring to that table we find a column (p. 167), headed 40, the course angle of our present problem. On the left-hand side of the page we find the given distance, 63. Then, opposite the distance 63, and under 40, we find the latitude difference (abbreviated, "Lat.") and the departure (abbreviated, "Dep.") to be: lat. = 48.3, dep. = 40.5. The following are additional examples for practice : Given : dist., 84, course 26 ; Arcs., lat. = 75.5, dep. = 36.8. Given : dist., 28, course 11 ; Arcs., lat. = 27.5, dep. = 5.3. When the course is between 1 and 45 the course angle will be found in Table 1 at the head of the column : but when the course is between 45 and 90, it appears at the foot of the column. In the latter case, the tabular lat. and dep. are to be taken from the columns having "Lat." and "Dep." at the foot instead of the top of the column. Examples follow : Given : dist., 63, course 50 ; Arcs., lat. = 40.5, dep. = 48.3. Given : dist., 84, course 64 ; Arcs., lat. = 36.8, dep. = 75.5. Given : dist., 28, course 52 ; Arcs., lat. = 17.2, dep. =22.1, In addition to the course angles from 1 to 90, three ad- ditional angles are given in parentheses at the top and foot of each column. Thus, with the course angle 30 appear also 150, 210, 330. This simply means that the latitudes 14 NAVIGATION and departures are the same for these four course angles. The accompanying Fig. 5 shows, for instance, that the departures QP and Q'P' are equal for 30 and] 150 courses if the two distances YP and YP' are alike. It will be noticed also that our traverse table always gives distances from 1 to 50 on a left- hand page, and from 50 to 100 on a right-hand page. When distances larger than 100 occur, it is necessary to use the 100, 200, etc., given on the lower part of each page. If, for instance, we require the latitude and departure for a distance 363 miles, course 40, we turn again to .the 40 column, and find (near the bottom of 30 and 150. the page) : For 300 miles, lat. = 229.8, dep. = 192.8 and (in the usual way) for 63 miles, lat. = 48.3, dep. = 40.5 Sums, 363 =278.1 233.3 Consequently, for dist. 363, course 40, lat. =278.1, dep. =233.3. Other examples are : Course 25, dist., 452 ; lat. = 409.6, dep. = 191.0. Course 68, dist., 521 ; lat. = 195.2, dep. = 483.1. Course 226, dist., 384 ; lat. = 266.8, dep. = 276.2. When the given distances or course angles, which are really the "pairs of arguments" (p. 11) of the traverse table, contain fractions, interpolation can be used ; but such close accuracy is seldom, if ever, required in navigation. More extended traverse tables will be found in Bowditch's "American Practical Navigator," published by the Navy Department, Washington. They are also printed separately in Bowditch's "Useful Tables." Both volumes can be purchased at any "navigation shop " where instruments and books suitable for navigators are sold. To complete this explanation of our traverse table, it is still necessary to mention that it also provides, with suf- ficiently close approximation, for the method of measuring DEAD RECKONING WITHOUT LOGARITHMS 15 course angles in " points of the compass" (pp. 10, 41). This method is not now in use in the United States Navy, but it is still largely employed in merchant vessels. It is sufficient to state here that a course of 3 points, for instance, is very nearly equal to a course of 34, and the traverse table column for 34 may properly be used for a 3-point course. Similarly, 31 may be used for 2| points, and the mariner desiring to use points can always find from the traverse table itself just what column to use. A special traverse table for points may also be found in Bowditch's Tables, already mentioned. We have now shown how to find latitude difference and departure by means of the traverse table. But our problem is not yet completely solved. Our ship (p. 8) started from the point 7 in lat. 42 11' N. ; long. 59 28' W. She traveled 63 miles on a 40 course, and the traverse table showed that she thus made good a latitude difference of 48.3 miles and a departure of 40.5 miles. It now remains to ascertain how much the ship changed her latitude in degrees and minutes from 42 11' N. and her longitude in degrees and minutes from 59 28' W. When we have found these last changes, we can learn the latitude and longitude of the point P, which we are required to find. To get the latitude change in degrees and minutes from the latitude difference in miles offers no difficulty. If the miles used are nautical miles (and in navigation they always are nautical miles), each mile of latitude difference corre- sponds to 1' of angular measure (p. 9), and 60 miles corre- spond to 1. Thus our ship must have changed her latitude 48'.3, corresponding to a latitude difference of 48.3 miles. Her initial latitude having been 42 11' N., her final latitude at P will be 42 11' + 48' (if we omit the odd .3) or 42 59' N. The relation between departure and difference of longitude is not quite so simple. Our ship's departure of 40.5 miles might correspond to far more than 40.5 minutes of longitude. In fact, in very high latitudes near the north pole, the longi- tude meridians converge so closely that a person traveling 16 NAVIGATION a few miles might change his longitude very greatly. At the pole itself a man might change his longitude 180 by simply stepping across the pole. So it follows that the longitude difference in minutes is greater than the departure in miles (however, cf. p. 4). The difference between the two increases rapidly as we approach high latitudes though it is nil at the equator; in Table 2 (beginning p. 168) we give this excess of longitude difference over departure for all latitudes under 60, and for all longitude differences up to 100. When the longitude differences are greater than 100J it is necessary to use the numbers given for 100, 200, 300, etc., near the bottom of each page in the table, and to sum tabular num- bers, precisely as we did with the traverse table. It will be noticed that Table 2 gives "tabular numbers" for each degree of latitude in a separate column, and that these various latitudes are called "middle latitudes." Thus the middle latitude and the longitude difference are the pair of arguments (p. 11) for Table 2, and, as we shall see pres- ently, the use of the middle latitude avoids any uncertainty in choosing the correct column for use. In our present problem we have at our disposal (p. 15) two different lat- itudes : the initial latitude at the point Y, 42 11' N., and the final latitude at the point P, 42 59' N. In this case, the two latitudes are so nearly equal that we might use either of them as an argument in Table 2 without material inaccu- racy. In fact, in using Table 2 it is unnecessary to consider minutes of latitude, the nearest degree being sufficient. But often the two latitudes available at this stage of the problem differ by many degrees. In such cases mariners always use the average of the two latitudes, and call it the "middle latitude." In the present case, the middle latitude would be found thus : Initial latitude = 42 11' Final latitude = 42 59' Sum = 85 10' sum = middle latitude = 42 35' DEAD RECKONING WITHOUT LOGARITHMS 17 The nearest even degree to 42 35' is 43, and the prob- lem would therefore be worked with the 43 column of middle, latitude in Table 2. Before completing our problem it is necessary to point out that while Table 2 is intended primarily for changing longitude differences in minutes into departures in miles, it can also be used (as stated at the foot of each page) for the inverse transformation of departures into longitude dif- ferences ; and this, is the transformation we must make in our present problem. It is merely necessary to use the departure (40.5) in the left-hand column, at the head of which are the words "Long. Diff. or Dep.," indicating that either of these two may be used as the argument in that column. Then, in the 43 column of middle latitude, we find (using interpolation) the tabular number 10.8. This means that a longitude difference of 40'. 5 corre- sponds to a departure of 40.5 10.8 miles, or 29.7 miles. But when the table, as in the present case, is used for the inverse transformation, the tabular number 10.8 must, before use, be multiplied by the factor given at the bottom of the column. For the middle latitude 43 this factor is 1 .37 ; and so the right tabular number becomes, in the present case : 10.8 X 1.37 = 14.8; and as the longitude difference is always greater than the departure, it follows that the departure of 40.5 miles gives a longitude difference of : 40.5 + 14.8 = 55'.3 = 55', if we omit the odd tenths. The initial longitude of the ship at the point Y was 59 28' W. As her 40 course has carried her nearer to Green- wich, it follows that her final longitude at the point P is : 59 28' W. - 55' = 58 33' W. We shall now discuss the following similar problem : A ship takes her departure from a point about one mile 18 NAVIGATION east of Navesink Highlands Light, New Jersey, in the initial lat. 40 24' N., initial long. 73 58' W., and travels 1377 miles on a course of 166. What final latitude and longitude does she attain? Entering the traverse table in the column headed 166, which is the same as the 14 column, we find : For dist. 900, lat., 873.2, dep., 217.7 For dist. 400, lat., 388.1, dep., 96.7 For dist. 77, lat., 74.7, dep., 18.6 Sums, 1377, 1336.0, 333.0 To make the large given distance (1377 miles) come within the range of Table 1, it has been necessary to enter the 166 column three times, with the arguments 900, 400, and 77, and then to sum the corresponding tabular numbers. The latitude difference, 1336 miles, is equivalent to 1336', or 22 16', counting, as usual, 60' to 1. Then, since the direction of her course (166) carried the ship to the south of her initial position (cf . Fig. 5, p. 14, and p. 19), we have : Initial lat., 40 24' N. Lat. diff., 22 16' N. Final lat., 18 8' N. Middle lat., 29 16' N. Now turning to Table 2, in the proper column for middle latitude 29 : For dep. 300 tabular number is 37.6 For dep. 33 tabular number is 4.1 Sums 333 41.7 As in the former example, this 41.7 must be multiplied by the factor at the bottom of the column. This factor is 1.14. Multiplying, we have: 41.7 X 1.14 = 47.5. Conse- quently, long. diff. = 333 + 47.5 = 380'.5 = 6 20'.5. Since the direction of her course (166) carried the ship eastward, and therefore nearer to Greenwich, it follows that her final longitude is 73 58' W. - 6 20', or 67 38' W. The final position is therefore : lat. 18 8' N. ; long. 67 38' W. DEAD RECKONING WITHOUT LOGARITHMS 19 The point indicated by this final latitude and longitude is just off tjie entrance to the Mona Passage, between Haiti and Porto Rico ; the given course and distance would there- fore be correct for a voyage from New York to Mona Passage. Additional similar problems are : 1. Initial lat., 40 28' N. ; initial long., 73 50' W. ; course, 119; dist., 2924 miles. This would take the ship from Sandy Hook to St. Vincent, Cape Verde Islands. Ans. Final lat., 16 50' N. ; final long., 25 7' W. 2. Initial lat., 40 W N. ; initial long., 70 0' W. ; course, 75 ; dist., 2606 miles. This would take the ship from Nan- tucket Lightship to Fastnet, the nearest point of the Irish coast. Ans. Final lat., 51 24' N. ; final long., 9 37' W. Before proceeding to our second fundamental problem (p. 8), it will be well to explain briefly two further points of interest. The first of these relates to the method of desig- nating a ship's course. We have hitherto supposed it to be measured in degrees, from the north, around by way of the east, through the south and west, and so back to the north again. This is the best way to count courses, and is the way now in use in the United States Navy. Since a whole circle contains 360, it follows that courses may con- tain any number of degrees from to 360. But there is another quite convenient, although older, way of designating courses, in which a 60 course, for instance, is written N. 60 E., showing that the ship must be steered 60 east of north. In a similar way, a 120 course is written S. 60 E., showing that the helmsman should head her 60 east of south, which would be the same as 30 south of east, or 120 from the north toward the south by way of east. The second further point of interest has to do with the relation between Tables 1 and 2. It is possible to avoid entirely the use of Table 2, and to transform longitude differ- ences into departures, and vice versa, by means of Table 1 20 NAVIGATION alone. It so happens that the relation between these two, for any given middle latitude, as, for instance, 9, is iden- tical with the relation between distance and latitude difference in Table 1 for the course 29. In other words, if we have given a middle latitude and a longitude difference, and wish to find the departure, we : Call the middle latitude a course, and Call the longitude difference a distance ; Then, corresponding to that course and distance, find from Table 1 the tabular latitude difference, and it will be the required departure. The same process can also be reversed, so as to find the longitude difference from the departure. While this method with Table 1 is quite correct, we believe beginners (at least) will find the use of Table 2 advantageous in the solution of these problems, especially when the middle latitude is not very great. Coming now to our second fundamental problem of dead reckoning, let us suppose a ship is required to proceed from the initial lat. 42 11' N. and long. 59 28' W. to a final lat. 42 59' N. and long. 58 33' W. We are to find the course she must steer, and the distance she must run. We have at once the latitude difference of 48', or 48 miles, and the middle latitude 42 35', or nearest whole degree of mid- dle latitude, 43 . The longitude difference is 55' ; and with this we find from Table 2 the correction 14.8 in the 43 column of middle latitude. Remembering that this time we are transforming a longitude difference into departure, and con- sequently do not need to use the factor at the foot of the column, we subtract this correction (14.8) from the longi- tude difference (55') and obtain the departure as 40.2 miles. Next we proceed to Table 1, to find the course and distance corresponding to lat. 48, dep. 40.2. To do this, we must find a place in Table 1 where this particular latitude and departure appear side by side. If this pair of numbers DEAD RECKONING WITHOUT LOGARITHMS 21 cannot be found (exactly) side by side, we must take the pair which come nearest to them : in this case such a pair of numbers is found in the 40 course column, opposite dist. 63. So it appears that the ship must steer on a 40 course a distance of 63 miles, to proceed from the given initial to the given final latitude and longitude. This problem is the direct converse of the one first solved (pp. 15, 17). As a second example, let us now calculate the course and distance from Sandy Hook, lat. 40 28' N. ; long. 73 50' W., to St. Vincent, lat. 16 50' N. ; long. 25 7' W. We have, by subtraction, lat. diff. = 23 38' = 1418' = 1418 miles; long. diff. = 48 43' = 2923'. This 2923' must be turned into a departure, the middle latitude being 28 39', or, to the nearest whole degree, 29. Turning to the column of Table 2 which belongs to 29 of middle latitude, we find the correction for 2923' of longitude difference thus : Tabular number for 900 = 113.0, which being multiplied by 3, gives : Tabular number for 2700 = 339.0 Also, tabular number for 200= 25.1 Tabular number for 23 = 2.9 Sums, tabular number for 2923 = 367.0 This must be subtracted from the longitude difference, and so we get : dep. = 2923 - 367.0 = 2556 miles. We have now to seek a place in Table 1 where lat. 1418 and dep. 2556 appear side by side. No traverse tables are suffi- ciently extended to contain these large numbers, but we . can at once obtain an approximate answer to the problem by dividing both numbers by 100. This reduces them to lat. 14.2, dep. 25.6 ; and the nearest numbers to these which can be found side by side in Table 1 are in the column belong- ing to course 119 and opposite dist. 29. This course (119) is the same as would have been obtained if we had not been 22 NAVIGATION forced to divide our latitude and departure by 100, to bring them within the range of Table 1. But the dist. 29 must now be multiplied by 100, to remove the effect of our former division of latitude and departure by 100. Thus we have the closely approximate information that the course and distance from Sandy Hook to St. Vincent are 119 and 2900 miles. The same problem (p. 19), when taken in its inverse form, starts with the numbers 119 and 2924 miles. In discussing such a problem, many beginners have dif- ficulty in choosing correctly the course number (119) from the four (61, 119, '241, 299) to be found at the foot of the same column of Table 1 . This choice is easily made with the help of our knowledge of elementary geography, or with any rough chart or map. From these, we know that St. Vincent is south and east of Sandy Hook, and the only one of the four possible courses that will carry a ship south and east is course 119. The same course might be written in the other notation (p. 19) S. 61 E., which possibly makes the actual direction to be steered a little easier to under- stand. The above result is approximate only, but higher accuracy is seldom required. When desired, it can be obtained by certain kinds of interpolations (p. 12) ; but these are always unsatisfactory, especially as complete precision can always be easily had by the use of logarithms, as explained in the next chapter. CHAPTER III DEAD RECKONING WITH LOGARITHMS SINCE the publication in 1876 of Kelvin's tables for facilitating Sumner's method, it has been possible to navi- gate in the most approved way without using logarithms or trigonometry. Those who desire to study the subject in this manner may do so by simply omitting those parts of the book in which logarithmic or trigonometric formulas and calculations occur. But this method of study is not recommended, except perhaps for a first reading; for a knowledge of logarithmic processes always affords a most desirable check on the accuracy of the other method, and so makes for safety of the ship and peace of mind of the navigator. Proceeding, then, with the subject of logarithms, we may define them as a mathematical device for facilitating calcula- tions. They are merely numbers ; but they are numbers having this peculiarity : every logarithmic number belongs to some ordinary number (like 1, 2, 3, 27, 800, etc.), and belongs to it alone. Its logarithm belongs to the number as a man's shadow belongs to the man. For our present purpose it is unnecessary to enter into the theory of logarithms ; we shall explain only the methods of using them in practice. Logarithms (abbreviated "log") always consist of two parts, a "whole number" part and a "decimal" part. Thus, 3.30103 is a logarithm, of which the whole number part is 3, and the decimal part .30103. The whole number part may even be zero : thus, 0.30103 is also a logarithm. The decimal part of the logarithm is found from a table of logarithms, such as our Table 3 23 24 NAVIGATION (p. 178) ; but the whole number part is found by an inspec- tion of the number to which the logarithm belongs. We shall hereafter, to save space, always write "log 26" in place of "the logarithm belonging to 26": and, with the help of this abbreviation, we may now write the follow- ing tabular statement, which is fundamental in the matter of logarithms : log 1 = 0.00000, log 1000 = 3.00000, log 10 = 1.00000, log 10000 = 4.00000, log 100 = 2.00000, log 100000 = 5.00000, etc. In other words, for these particular numbers, all "mul- tiples" of 10, the decimal part of the log is zero. For numbers intermediate between 1 and 10, the whole number part of the log is 0, and the decimal part lies between .00000 and .99999. For those between 10 and 100 the whole number part is 1, and the decimal part again lies between .00000 and .99999. The general rule is : the whole number part of a log is one less than the number of figures or "digits" in the number to which the log belongs. Thus, the number 26 has two digits : the whole number part of its log is 1. The number 2678 has four digits : the whole number part of its log is therefore 3. If a number is itself partly decimal, we count only the number of digits to the left of the decimal point for the pur- poses of the present rule. Thus, 26.78 has two digits only ; 2.678 has one ; 267.8 has three, etc. If, on the other hand, a number is wholly decimal, as 0.2678, the whole number part of its logarithm should be "negative," or minus, i.e. less than 0; and it will be one greater than the number of zeros immediately following the decimal point in the number. According to this, the whole number part of log 0.2678 should be 1, because this number has no zeros immediately following the decimal point. But as these negative whole number parts are very inconvenient in actual work, it is customary to increase DEAD RECKONING WITH LOGARITHMS 25 all logs of decimal numbers arbitrarily by 10, which will avoid the negative sign. This arbitrary increase is always corrected again in the further or final procedure, so that it cannot possibly introduce error into the work. In the case of log 0.2678, the arbitrary increase of 10 changes the 1 to + 9 l ; and so 9 would be the whole number part of log 0.2678. Similarly, log 0.002678 would have 7 for its whole number part, because there are two zeros after the decimal point. This would make the whole number part of the log 3, which, being increased by 10, gives + 7. In general, this matter of logs of wholly decimal numbers may be summarized as follows : log 0.1 =9.00000, log 0.0001 =6.00000, log 0.01 =8.00000, log 0.00001 =5.00000, log 0.001 = 7.00000, log 0.000001 = 4.00000, etc. In all these cases the decimal part of the log is zero: and if the number lies, for instance, between 0.1 and 0.01, the whole number part of the log will be 8, and the decimal part will lie between .00000 and .99999. The decimal part in the log of any number is taken from Table 3 without regard to the position of the decimal point in the number itself. The numbers 0.2678, 0.002678, 26.78, 2.678, 267.8, and 2678 all have precisely the same decimal part in their logs, so that such logs will differ in their whole number parts only. We can at once obtain this common decimal part from Table 3 (p. 181), where it is found to be .42781. In looking up this log, we again use (p. 11) a pair of arguments. The argument for the left- hand column consists of the first three digits of 2678 (267) ; and in selecting this argument we disregard any zeros that may immediately follow the decimal point, if the number is wholly decimal, like .002678. The other argument, in the top horizontal line of the tabular page is 8, the right- hand digit of the number 2678. In the horizontal line 1 According to Algebra, 9 is greater than - 1 by 10. 26 NAVIGATION opposite 267, and in the column headed 8, appears 781 ; and these are the last three digits of the required log (.42781). The first two digits (.42) are common to a great many logs, and are therefore only printed in the column headed 0. The first two digits of every log are thus taken from the zero column, regularly from the same horizontal line that contains the last three digits of the log, or from some line above it. Only when there is an asterisk printed in the table with the last three digits do we make an exception, and take the first two digits from the line below the one containing the last three. Thus the decimal part of log 2691 is .42991, but the decimal part of log 2692 is .43008. Having thus found the decimal part of log 2678 to be .42781, and the number 2678 having four digits, the com- plete log 2678 = 3.42781 ; and here the reader should once more note that all tabular logs like .42781 are thus always decimals. The correspond- ing logs for the other numbers given above are : log 267.8 = 2.42781, log 26.78 = 1.42781, log 2.678 = 0.42781, log 0.2678 = 9.42781, log 0.002678 = 7.42781. It is clear that Table 3 gives directly the decimal part of the logs of all numbers containing four digits. If the number contains less than four digits, as 26, we should look it up in the table as if it were 2600. We should find 260 as the argument in the left-hand column (p. 181) ; and in the corresponding line, in the column headed (the fourth digit of 2600), is 41497. This is the decimal part, as usual, and the complete log 26 = 1.41497. If, on the other hand, the number whose log is wanted contains more than four digits, as 26782, it is necessary to DEAD RECKONING WITH LOGARITHMS 27 resort to interpolation (p. 12). The number of digits being here 5, the whole number part of the log is 4 (p. 24). The decimal part of the log is to be found quite without regard to decimal points (p. 25). It may therefore be taken from Table 3 just as if we wanted log 2678.2 instead of 26782. Now the table tells us (p. 181) : decimal part of log 2678 = 42781, decimal part of log 2679 = 42797. The tabular difference (p. 12) of these two decimal parts is 16. As 26782 may, for our present purpose, be regarded as lying & of the way from 2678 to 2679, it follows that the decimal part of log 26782 will lie T 2 ^ of the way from 42781 to 42797. Evidently, we must multiply the tabular differ- ence 16 by ^ (giving 3.2) to find how much larger the decimal part of log 26782 is than the decimal part of log 2678. This 3.2 (or 3, in round numbers) must then be added to 42781 ; and we have, as the result of this interpolation : decimal part of log 26782 = .42784. As we have just found the whole number part to be 4, we have for the complete : log 26782 = 4.42784. This whole process of interpolation may perhaps be more clearly understood if we repeat (p. 10) that all tables furnish tabular numbers corresponding to given arguments. In- terpolation is necessary when the given arguments are not to be found in the argument part of the table, but fall between two of the tabular arguments. Then we obtain by subtraction the difference between the given argument and the nearest smaller argument contained in the table. This difference is the " argument difference" (abbreviated, arg. diff.), and it should be expressed as a decimal fraction of the interval between two successive arguments (cf . T %, above). The tabular difference (tab. diff.) between two successive tabular numbers being also obtained by subtrac- 28 NAVIGATION tion, we have only to multiply the tabular difference by the argument difference to find the "interpolation difference" (int. diff.). This is then added 1 to the proper tabular number (belonging to the above-mentioned nearest argu- ment given in the table) to obtain the tabular number re- quired. The multiplication of the tabular difference by the argu- ment difference is facilitated by certain little auxiliary mul- tiplication tables (called tables of " proportional parts") printed in the margins of many mathematical tables. In the example given above, the tabular difference was 16 ; and Table 3 contains on the proper page (p. 181) a proportional part table headed with this same number 16 ; and it shows that for an argument difference .2, and tabular difference 16, the interpolation difference is 3.2, just as we found above. Other examples of logarithms are : log 427 = 2.63043, log 42765 = 4.63109, log 4276 = 3.63104, log 282374 = 5.45082, log 0.4276 = 9.63104, log 2 = 0.30103, log 0.42765 = 9.63109, log .0027 = 7.43136. The above considerations are preparatory only to the actual use of Table 3 ; and they are not yet quite complete. For it is still necessary to explain the inverse use (p. 12) of the table, or, in other words, the finding of the number to which a given log belongs. Thus, if the given log were 3.42781, we should begin by looking up its decimal part among the logs in the table. Finding it there, we take out the number to which it belongs, 2678. We then put in the decimal point according to the whole number part of the log. This being 3, we know (p. 24) that the number required must contain 4 digits. Therefore : number to which the log 3.42781 belongs = 2678. 1 Except when a glance at the table shows that the tabular num- bers are growing smaller, in which case the interpolation difference must be subtracted. This never occurs in Table 3, but happens fre- quently in Table 4. DEAD RECKONING WITH LOGARITHMS 29 If the given log had been 2.42781, the table would furnish the same number 2678, but the decimal point would be differently located. Because the whole number part of the given log is now 2, we know that the number to which it belongs has three digits, and so : number to which the log 2.42781 belongs = 267.8. When the given log is not to be found in the table exactly, a process of inverse interpolation is, of course, necessary. Thus, if the given log is 4.42784, we look for its decimal part in the table, and find it lies between 42781, which belongs to the number 2678, and 42797, which belongs to the number 2679. The decimal part of the given log being 42784 is greater by 3 than the nearest tabular number 42781. This 3 is there- fore the interpolation difference. The tabular difference is 16, obtained by subtraction between 42781 and 42797. We now divide the interpolation difference by the tabular dif- ference, which gives .2 (^ = 0.2, in round numbers). This .2 is the argument difference, and therefore the complete number belonging to the decimal part of the log (42784) is 26782. The whole number part of the given log being 4, the required number must have 5 digits, and will therefore be 26782. Had the given log been 2.42784, we should have arrived at the number 26782 in just the same way; but we should locate the decimal point differently. The whole number part of the log being now 2, there should be only 3 digits in the number, and we should have : number to which the log 2.42784 belongs = 267.82. Other similar examples are : log = 2.71828, corresponding number = 522.73, log = 4.26323, corresponding number = 18333, log = 9.26323, corresponding number = 0.18333, log = 0.21000, corresponding number = 1.6218. The reader will perceive, from a consideration of these interpolated numbers, that work with logarithms is never 30 NAVIGATION exact, absolutely. This is inherent in the nature of our log tables, which really contain only the decimal parts of the logs carried out to five places of decimals. Further decimals of course exist, but are here omitted, because five places always give sufficient accuracy for navigation calculations. The simplest calculations which are facilitated by loga- rithms are the ordinary arithmetical processes of multi- plication and division. These processes can be turned into addition and subtraction by the use of the following principle : The log of a product is equal to the sum of the logs of the factors. According to this principle, if we wish to multiply a series of factors, we simply add their logs. The sum is then a log and the number to which this log belongs is the product of the series of factors. Suppose, for instance, we wish to multiply the factors 2, 3, and 4. The product should be 24. Proceed- ing with logs, we have from Table 3 : log 2 = 0.30103, log 3 = 0.47712, log 4 = 0.60206, log product = sum = 1.38021, and the number to which the log. 1.38021 belongs is, accord- ing to Table 3, 24.00, the correct product. It is evident that the use of the log table is here of no advantage, because the factors are very small : but when large numbers are to be multiplied the advantage is very great. Taking now a similar simple example of division, let us divide 6 by 3. In division, evidently, we must subtract the log of the divisor from the log of the dividend, to obtain the log of the quotient. We have log 6 = 0.77815, log 3 = 0.47712, log | = difference = 0.30103, DEAD RECKONING WITH LOGARITHMS 31 and the number to which the log 0.30103 belongs is 2.000, the correct quotient. Other examples are: 2.426 X 42.78 X 17.26 = 1791 .3, 6.242 X 87.24 X 62.71 = 34149, 2802 1.6234, ^ = 0.75. 1726 18 24 In the last example, we have log 18 = 1.25527, log 24 = 1.38021. The subtraction would lead to a negative log because 1.38021 is larger than 1.25527. Therefore we arbitrarily increase 1.25527 by 10, giving 11.25527, and then the subtraction gives log quotient = 9.87506, which is the log belonging to the number 0.75, the correct quotient. We come now to the solution of the two fundamental problems of dead reckoning (pp.. 8, 10) by means of logs. For this purpose we must use our Table 4, in connection with Table 3. Table 4 is called a trigonometric log table and the tabular numbers in it are certain logs known as : sine, abbreviated sin, cotangent, abbreviated cot, cosine, abbreviated cos, secant, abbreviated sec, tangent, abbreviated tan, cosecant, abbreviated esc. It is not our purpose to consider the theory of trigonom- etry, but it is necessary for the reader to have some understanding of its practical applica- tions. If we have a triangle QPY (fig. 6), we notice that it is made up of six "parts," the three sides and the three angles. Now it is a fact that if we know any three of these six y parts, we can calculate the other three parts, FIG. 6.- Trigo- provided one of the known parts is a side. Trigonometry is the branch of mathematics which enables us 32 NAVIGATION to do this, and the triangle QPY is the very triangle which occurs in the two problems of dead reckoning. In trigonometry, every angle has belonging to it a sin, cos, etc., just as every number has its log. These sines, etc., can be taken out of Table 4 by means of a pair of argu- ments in the usual way. The two arguments are the number of degrees and the number of minutes in the angle (p. 9). The number of degrees is found in Table 4 at the top or bottom of the page, and the number of minutes in the right-hand or left-hand column. Each page (as, for instance, p. 229) has eight degree numbers, four, 33, (213), (326), and 146 at the top, and four, 123, (303), (236), and 56 at the bottom. The proper sines, etc., for all these degrees appear on the same page (p. 229). When the degree number is at the top or bottom of the left-hand column 33, (213), (303), and 123, the minutes must be taken from the left-hand column. But when the number of degrees is at the top or bottom of the right-hand column 146, (326), (236), and 56, the minutes must come from the right-hand column. And when the number of degrees comes from the top of the page, we must look for the proper sine, etc., in a column having the word sin, etc., at the top. But when the degree number comes from the bottom of the page, the sine, etc., will be taken from a column having the word sin, etc., at the bottom. Thus (p. 229) : sin 33 26' = sin 146 34' = cos 56 34' = cos 123 26' = 9.74113. In this way, sines, tangents, etc., can be taken from Table 4. Examples are : sin 28 32' = 9.67913, cot 117 10' = 9.71028, cos 66 14' = 9.60532, sec 12 40' = 0.01070, tan 128 28' = 0.09991, esc 111 11' = 0.03038. These sines, etc., are really all logs. When the whole num- ber part is 9, it indicates that the log belongs to a number which is wholly decimal (see p. 24), and that the log has been arbitrarily increased by 10. DEAD RECKONING WITH LOGARITHMS 33 Of course these trigonometric tables can also be used in the inverse manner. Thus, to find the angle corresponding to the sin 9.28190, we turn to p. 207, and finding 9.28190 in the sin column, we see that the corresponding angle is either 11 2', 191 2', 168 58', or 348 58'. When the sin, etc., cannot be found in the table exactly, we may always take the nearest one : interpolation is never practically necessary in using the trigonometric tables in navigation. Examples are : sec = 0.17177, angle = 47 40', 227 40', 132 20', or 312 20', tan = 0.17177, angle = 56 3', 236 3', 123 57', or 303 57', sin = 9.17177, angle = 8 32', 188 32', 171 28', or 351 28', cos = 9.17177, angle = 81 28', 261 28', 98 32', or 278 32', esc = 0.17177, angle = 42 20', 222 20', 137 40', or 317 40', cot = 0.17177, angle = 33 57', 213 57', 146 3', or 326 3'. Having thus explained the use of Table 4, we shall now apply it to the two problems of dead reckoning. These problems are : 1. To find latitude difference and departure from course and distance ; 2. To find course and distance from latitude difference and departure. These problems are solved by means of the following formulas, in which the letter C represents the course angle : ,j. f log lat. diff. = log dist. + cos C, \ log dep. = log dist. + sin C. tan C = log dep. - log lat. diff., log dist. = log dep. - sin C. Sometimes it is preferable to find the distance from the latitude difference instead of the departure. We then use the following modification of formula (2) : (2') log dist. = log lat. diff. - cos C. Let us now solve with these formulas our former problem (p. 18), in which a ship traveled 1377 miles on a course of 166. Applying formula (1) above, we have: 34 NAVIGATION log dist. (1377) = 3.13893 log dist. (1377) = 3.13893 cos C (166) = 9.98690 sin C (166) = 9.38368 sum = log lat. diff. = 3.12583 * sum = log dep. = 2.52261 l corresponding lat. diff. = 1336.1 corresponding dep. = 333.1 These corresponding latitude difference and departure agree very closely with the results already found (p. 18) from Table 1. If the departure and latitude difference were given, we could find the course and distance by means of formula (2)- In the present case we have : log dep. (333.1) = 2.52261 log dep. (333.1) = 2.52261 log lat. diff. (1336.1) =3.12583 sin C (166) = 9.38368 by subtraction, tan C = 9.3967S 2 by subtraction, log dist. = 3.13893 3 corresponding C = 166 corresponding dist. = 1377 These numbers, 166 and 1377 miles, are the same numbers with which we began this calculation ; so it is clear that the log method of calculation agrees with the traverse table method. For accuracy the log method is superior. The transformations of departure into longitude differ- ence, and vice versa, are accomplished logarithmically with the following formulas : (3) log long. diff. = log dep. cos middle lat. (4) log dep. = log long. diff. + cos middle lat. Thus the longitude difference corresponding to dep. 333.1 would be calculated by formula (3) as follows : log dep. (333.1) =2.52261 .cos mid. lat. (29 16', p. 18) = 9.94069 by subtraction, log long. diff. = 2.58192 corresponding long. diff. = 381'.9 = 6 21'.9. 1 These numbers have been diminished by 10, to allow for the fact that both cos C and sin C have been arbitrarily increased by 10 (p. 32; cf. also p. 25). 2 This number has been increased by 10, and therefore is in accord with the usual practice of avoiding negative whole numbers in the trigonometric Table 4. 3 This subtraction is correct, if we remember that the 9.38368 is really too large by 10. DEAD RECKONING WITH LOGARITHMS 35 This is in close accord with the result on p. 18, where Table 2 gave 6 20'. 5. The logarithmic method is again the more precise, for it takes account of minutes in the course, which were neglected on p. 18. But either result is accurate enough for practical purposes. Before finally leaving these problems of dead reckoning, we shall explain briefly two additional methods of solving them which differ from the method so far employed. These two additional methods are called "Mercator sailing" and " great circle sailing"; whereas, up to the present, we have been using " middle latitude sailing," so named because the middle latitude appears in the calculations. Mercator sailing is based on a kind of chart first designed by Gerhard Mercator, a sixteenth century geographer. Such charts are still widely used for nautical purposes. In calculations based on them, every parallel of latitude is referred directly to the equator by means of a table of " merid- ional parts." Our Table 5 is such a table, and it gives the meridional part for every degree and minute of latitude from the equator to 60. These meridional parts are really the distances from the equator to the several parallels of latitude, such as they would appear on a Mercator chart drawn to such a scale that I' of longitude at the equator would occupy one linear unit on the chart. Thus the meridional part, for lat. 40 is given in Table 5 as 2607.6. Suppose the scale of the chart at the equator were 1 inch to the degree of longitude. That would be -$ inch to the minute. The dis- tance on the chart from the equator to the 40 parallel of latitude would then be 2607.6 X ^V inches = 43.46 inches. It is needless to say that a chart on such a scale could not show a very large part of the ocean on a single sheet. Calculations by Mercator sailing are of course only made when the distances involved are large and great accuracy is required. It is therefore best to do them by means of logarithms, although it is also possible to obtain Mercator results from the traverse table . In such calculations we do not 36 NAVIGATION use the latitude difference of ordinary middle latitude sailing. In its place appears the " meridional latitude difference" (ab- breviated mer. lat. diff.), defined as the difference between the meridional parts (Table 5) belonging to the two. latitudes (initial and final) involved in the problem. With this defini- tion in mind we may now give the Mercator formulas as follows : (5) log mer. lat. diff. = log long. diff. + cot C. (6) log long. diff. = log mer. lat. diff. + tan C. (7) tan C log long. diff. log mer. lat. diff. Let us now apply these formulas to the problem of pp. 18 and 33, in which a ship starts from the initial lat. 40 24' N. ; long. 73 58' W., and travels 1377 miles on a course, C, of 166. What final latitude and longitude does she at- tain ? The latitude difference is found in the ordinary way (p. 34), there being no special Mercator formula for it, and comes out 1336.1 miles, or 1336M = 22 16'. The final lati- tude (p. 18) is therefore 40 24' - 22 16 r = 18 8'. Then, from Table 5, we have : for initial lat?. 40 24', mer. parts = 2638.9 for final lat. 18 8', mer. parts = 1099.4 by subtraction, 1 mer. lat. diff. = 1539.5 Now, applying formula (6), we have : log mer. lat. diff. (1539.5) (Table 3, p. 179) = 3.18738 tan C (166) (Table 4, p. 209) = 9.39677 by addition, log long. diff. = 2.58415 corresponding long. diff. (Table 3, p. 183) = 383'.8 = 6 24' The final longitude is therefore 73 58' - 6 24' = 67 34' W., whereas we obtained before 67 38' W. (p. 18). Finally, we shall apply the Mercator method to the example of p. 21. It is required to find the course and distance from Sandy Hook, lat. 40 28' N. ; long. 73 50' W. to St. Vincent, lat. 16 50' N. ; long. 25 7' W. 1 If one latitude were in the southern hemisphere and the other in the northern, we should add the meridional parts. DEAD RECKONING WITH LOGARITHMS 37 We have from Table 5 : for initial lat. 40 28', mer. parts = 2644.2 for final lat. 16 50', mer. parts = 1018.1 by subtraction, mer. lat. diff. = 1626.1 The longitude difference is found by subtraction to be 73 50' - 25 7' = 48 43' = 2923'. Now applying formula (7), we have: log long. diff. (2923) (Table 3) = 3.46583 log mer. lat. diff. (1626) (Table 3)= 3.21112 by subtraction, tan C = 0.25471 and therefore (Table 4) C = 119 5'. The distance is found in the ordinary way from the latitude difference (not mer. lat. diff.) by means of formula (20, P- 33. The latitude difference is 40 28 / -16 50' = 23 38' = 1418'. Formula (2') then gives : log lat. diff. (1418') (Table 3) = 3.15168 cos C (119 5') (Table 4) = 9.68671 1 by subtraction, log dist. = 3.46497 l corresponding dist. (Table 3) = 2917 Course 119 5', distance 2917 miles is therefore the solution by Mercator sailing. On p. 22, we obtained 119 and 2900 miles; and on p. 19 we began with 119 and 2924 miles. The agreement is satisfactory. Having thus briefly described Mercator sailing, we come next to "great circle sailing. " This is a method of determin- ing the ship's course toward her port of destination in such a way that the distance to be traveled will be as short as possible. If the earth's surface were flat instead of spherical, the shortest course would be a straight line, as used in plane sailing; but on the sphere the shortest course is a curve called a " great circle." Evidently, on all long voyages, the great circle course is the most advantageous one; that mariners do not more frequently use it is due to a peculiarity of their charts. 1 This log is really too large by 10, so the subtraction is correct. 38 NAVIGATION We cannot here enter into the details of chart ''pro- jections," as the theory of chart making is called. It is sufficient to remark that a straight line drawn on the ordi- nary nautical charts (which follow the Mercator system), between any two ports, will not represent the shortest (or great circle) course between them. On such a chart, the great circle course between the two ports will appear to be longer than the straight line course, although it is really shorter. This accounts for the use of the longer Mercator course by many navigators. Now there is a kind of chart, called a " great circle sailing" chart, on which straight lines between ports really represent shortest (or great circle) courses. One would therefore naturally suppose that mariners would entirely discontinue the use of Mercator charts in favor of great circle charts. But there is a reason for not doing this. On Mercator charts, all terrestrial longitude meridians are represented by parallel vertical straight lines. Conse- quently, if we draw another straight line on the Mercator chart joining two ports, that line will make the same course angle (p. 10) with all the meridians. In this way, a navigator can get from a Mercator chart, by simply drawing a straight line, and quite without calculation, a course angle which will carry him from one port to another. And because the course angle so obtained is the same with respect to all meridians to be crossed by the ship it follows that the voyage can be completed (theoretically at least) from the one port to the other with the great advantage of never changing the course to be steered. On the other hand, the great circle track makes a different angle with every meridian it passes : so that the mariner must make very frequent changes in the course angle to be steered during the progress of a voyage. The simple Mercator track, without change of course, is called a " rhumb line" ; the serious objection to it is that it sometimes leads to greatly (and unnecessarily) lengthened voyages. DEAD RECKONING WITH LOGARITHMS 39 The final conclusion is that Mercator charts, on account of their simplicity, are most convenient for short voyages, or for parts of long voyages when the land is not far away. But for shaping the main part of the course on a very long voyage, great circle sailing charts are to be preferred. At times, in order to avoid very high latitudes, or to round some projecting point of land, navigators must substitute for a single great circle track one " composed" of two or more shorter arcs of great circles. This is called " composite" sailing. Finally, for the sake of completeness, we shall merely mention two other kinds of sailing. " Parallel" sailing, which is simply middle latitude sailing when the latitude difference is zero; and " traverse" sailing, from which the traverse table gets its name. This is also the same thing as middle latitude sailing; but the special word " traverse" is used when the ship changes her course frequently, perhaps even during a single day. It is then possible to sum up the result of all the short courses which together make up the day's run. It is merely necessary to take from the traverse table the latitude difference and departure for each short course separately, and then to add 1 all the values of latitude differ- ence for a "summed latitude difference," and all the values of departure for a " summed departure." With these a " composite course and distance" can be taken from the traverse table, or calculated with logs, and these will repre- sent the motion of the ship, just as if she had steered an unchanged course during the entire day. 1 It is necessary to sum separately latitude differences represent- ing northward motion of the ship and those representing southward motion. The difference of the two sums is what we need to know. The same is true of departures representing eastward and westward motion of the ship. CHAPTER IV THE COMPASS THE ship's course has been defined (p. 8) as the angle between the north and the direction in which the ship is sailing. To ascertain what this angle is, or, in other words, to steer the ship, mariners use the compass. The dial (or "card") of this instrument is divided, like any circle, into 360. In the United States Navy these are numbered in such a way (fig. 7) that appears at the north, 90 at the east, 180 at the south, and 270 at the west. The numbers therefore increase in a " clockwise" direction. There are also compasses in which the numbering begins with at both the north and south points, and increases to 90 at the east and west points.' But the United States Navy system of numbering is to be preferred. In addition to the above division and numbering, the dial is also divided into 32 points (pp. 10, 15), each containing oar\o ^-, or 11|. These points are then further subdivided 32 into quarter points, all of which is shown clearly in Fig. 7. The naming of the points has not been done by chance, but in accordance with a definite rule. The four principal, or "cardinal," points are north, east, south, and west. The remaining points are located by a continued process of halving. Halfway between the cardinal points are the "inter-cardinal" points; and each is named by combining the names of the two cardinal points adjacent to it. Thus northeast (abbreviated N.E.) is halfway between north and east. Again halving and combining names, we get points like E.N.E., S.S.E., etc. Still once more halving completes the tally of 32 points : but a combination of names would now be too complicated. However, since 40 THE COMPASS 41 each of these final points must necessarily be adjacent to a cardinal or inter-cardinal point, they are named by simply increasing the name of such adjacent cardinal or inter- cardinal point. This is accomplished with the word "by." FIG. 7. Compass Card. Thus we find, adjacent to N.E., the points N.E.byE., and N.E. by N. In the light of the above, it is easy to "box" the compass, as seamen say, or to name the 32 points in order. When the point system of division is used, and an accuracy 42 NAVIGATION closer than a single point is required, the compass card is still further subdivided into quarter points. In naming these it is customary, in the United States Navy, to "box" from N. and S. towards E. and W. Thus the space between N.N.E. and N.E. byN. would be divided into four parts thus: N.N.E.JE., NrN.E.JE., N.N.E.fE. But an excep- tion is made to this last rule in the case of quarter points adjacent to a cardinal or inter-cardinal point. These last are always put first in naming the quarter points. Thus, between E. by N. and E., if we always boxed from N. towards E., we should have : E. by N.JE., E. by N.|E., E. by N.fE. But it is customary, because shorter, to name these quarter points E.fN., E.JN., and E.JN. Inside the "bowl" of the compass, and adjacent to the card, a black line is marked on the bowl. This line is in plain view of the steersman, through the glass cover of the compass, and is called the "lubber line." When the ship is headed in such a way that this line comes opposite N.E., for instance, on the card, the ship will be on a N.E. course, which makes an angle of 45 with the north. But would the ship really be traveling on a line making a 45 angle with the geographic meridian, or direction of the north pole of the earth? She would be doing so only if the compass were absolutely correct. This is practically the case with the "gyro-compass," a mechanical contrivance now much used in the navy, but not the case with the ordi-' nary "magnetic" compass. In Chapters II and III, concerning dead reckoning, we have always used the word "course" as if all compasses were absolutely correct. But since they are not correct, it is now necessary to make allowance for their errors. In other words, whenever we use a compass, we must first ascertain the difference between the "true course" and the "compass course." It must not be supposed from this statement that a ship can be steered on two different courses at the same moment. There is really only one direction along which THE COMPASS 43 the ship is moving: but the angle between that direction and the true north may be different from the angle between it and the " compass north." It is the course measured from the true north that must be used in all dead-reckoning calculations, and that always results from such calculations : but for steering the ship by means of a compass the steers- man must be furnished with the course as measured from the compass north. Therefore it is essential for the navigator to know the difference between the two. This difference is called the " error" of the compass. Unfortunately, this error is made up of two parts. The first, called " variation" of the compass, is due to peculiari- ties in the earth's magnetism, and is quite different in dif- ferent places on the earth. It also varies in different years at the same place. But at any one time, all ships in the same part of the ocean will have the same variation. The mariner can always ascertain how great the varia- tion is in his part of the ocean, because it is always marked on his chart. Certain curved lines are drawn on the chart ; and if the ship is located on or near a line marked " varia- tion 10," for instance, it follows that the navigator must on that day allow for 10 of variation. It is also important to take into consideration possible changes in the variation. Sometimes the annual change is marked on the chart; if not, it is important to use a chart of recent date. The second part of the error is called " deviation" and is due to peculiarities in the magnetism always developed in the metallic parts of the ship itself. It is different in dif- ferent ships, even in the same part of the ocean, and is even different in the same ship, when she is headed on different courses. Methods have been invented for " compensating " marine compasses, so as to remove the effects of deviation, and these methods are quite effective. But even when they are used, it is necessary, before beginning a long voyage, to have a " compass adjuster" visit the ship. He will then "swing" the ship on a number of different courses, and 44 NAVIGATION adjust the compass so that it will be as nearly correct as pos- sible. Finally, he will determine, by means of astronomic or other observations, just what the remaining compass devia- tion is on all the various courses, and give the navigator a table of these remaining deviations. This table must be taken into account in "shaping" the ship's course during the voyage. The navigator must also, from time to time, check these tabular deviations while at sea by means of astronomic observations of his own, to take care of possible changes. Such astronomic observations are made with an instru- ment (the " azimuth circle"), which can be attached to the compass, and with which the " compass bearing" of the sun or any other object can be observed. The compass bearing is simply the compass direction of the object, as seen from the ship ; or the compass course on which the ship would be steered, if she were moving directly toward the object. When the. sun is used, its true bearing, measured from the true north, can be taken from astronomic tables which will be explained later; and it is called the sun's " azimuth." A comparison of this true bearing with that measured on the compass with the azimuth circle then makes the compass error known. When it is not convenient to observe the sun, it is possible to substitute observations of a distant well-defined terrestrial ob- ject, whose true bearing can be measured on a chart for com- parison with various compass bearings observed while the ship is being swung. Another method is to set up a compass on shore, away from any iron or steel, and use it to determine the bearing of the distant object. And there is still another method, if the above compass and the ship's compass are inter- visible. For the bearing of each may then be taken from the other, and these should differ by exactly 180. If they do not, the variation from 180 must be due to deviation on board. The "pelorus" is another instrument which may at times replace the azimuth circle. It is located anywhere on the ship, at a convenient point for observation, and not neces- THE COMPASS 45 sarily close to the compass. It has a "dummy card" and a lubber line. The dummy card can be turned until the lubber line indicates the same course as the real compass. Observations of bearings with the pelorus will then obviously be the same as if made on the compass with the azimuth circle. The advantage of the pelorus is that it can be used anywhere on board, while the compass must be kept constantly in the exact place where it was "adjusted" before leaving port. The error thus determined astronomically or otherwise is the sum of the variation and deviation. If we indicate by E the total compass error in that place, at that time, on that ship, and on that course ; by D the deviation similarly described ; by V the variation at that time and in that place ; and if all three are counted from in the usual direction around the compass card, then we have the formula : (1) E = V + D. By counting in the usual direc- tion, we mean counting from the north around to the east, as all courses are counted (p. 19) ; so that a compass error of 10, for instance, would mean that the compass north pointed 10 east of the true north, or had a true bearing of N. 10 E. (p. 19). This is shown in Fig. 8, which also shows the ship's course, counted in the same way. It is clear from the figure that if we now indicate : by C, the ship's compass course, by T, the ship's true course, by E, the compass error, we shall have the formula : (2) .T = FIG. 8. Compass Error. 46 NAVIGATION The simple formulas (1) and (2) enable the navigator to make all necessary compass calculations. The following are examples. Suppose, for instance, that the error E has been deter- mined by observation, and the variation V taken from the chart. Formula (I) then makes it possible to calculate the deviation D. For the formula shows that E is the sum of V and D ; and so D must be the difference of E and 7, or: D=E-V. Thus the deviation D becomes known, as a check on the compass adjuster's work, and, while this value of D is cor- rect only for the particular course on which the ship was headed at the time the observation was made, yet that course is the very one for which it is especially important to have correct information. Again, suppose dead-reckoning calculations show that the ship is to sail on a 40 course. These calculations always furnish the true course (p. 43) so that T = 40. The variation being known from the chart, and the deviation from the adjuster's table, we know from (1) E = V + D. Then from (2) we see that C = T E } which gives the compass course. Let us suppose in the present case, that V was 9, D 1 ; then E = V + D = 9 + 1 = 10 ; and since T = 40, C = T - E = 40 - 10 = 30 ; and the helmsman would be directed to steer a 30 course by com- pass. If, in Fig. 8, the compass north happened to be 10 on the left side of the true north, instead of the right, the error E would be 350, instead of 10 (see also fig. 7, p. 40). This might be made up of a variation V of 349 and a deviation D of 1, as before. If the true course is again to be 40, the compass course would be 40 350, according to the formula C = T E. This subtraction being impossible, we increase the 40 by a complete circumference of 360, which is always permissible, and then have : THE COMPASS 47 C = 360 + 40 - 350 = 50. The ship would be steered on a compass course of 50. An alternative way to take care of errors, variations, and deviations on the left side of the true north is to mark them with the negative or minus sign. Instead of calling V 349, we might call it - 11. This is really the best way, and leads to the same result as before, if we remember that the subtraction of a minus quantity is always equivalent to an addition. In the example just given, calling V 11, instead of 349, we should have : E=V + D=-U + 1 = - 10; and C = T - E = 40 - (- 10) = 50, the same compass course as before. An older way of designating variations, deviations, and errors is to call them east when the compass north points to the right of the true north, and west when it points to the left of the true north. This method leads to the necessity of providing various rules or diagrams with which to make compass calculations. We think the best way to avoid error (and such errors may lose ships and lives) is to use the method here given with its two simple formulas. When some other designation of the error, or some other method of numbering the card, is demanded by a captain, it is always possible to conform to that demand, but also to translate every problem into our method (in imagination at least) as a check against mistake. The following is an example of a compass adjuster's ''devia- tion table," taken from Bowditch's "Navigator" (1916 edition). The deviations are set down in degrees and tenths of a degree, instead of degrees and minutes, for convenience in the further calculations. The ship was swung so that her head bore successively around the horizon, and obser- vations were made at intervals of 15. This is a smaller interval than is usually necessary ; and the deviations in the table are much larger than commonly occur in a modern well-compensated compass. 48 NAVIGATION DEVIATION TABLE BEARING BEARING BEARING BEARING OF SHIP'S DEVIA- OP SHIP'S DEVIA- >F SHIP'S DEVIA- OF SHIP'S DEVIA- HEAD BY TION HEAD BY TION rlEAD BY TION HEAD BY TION COMPASS COMPASS COMPASS COMPASS o o o o o - 15.5 90 - 9.1 180 + 17.9 270 + 9.9 15 - 14.9 105 - 9.0 195 + 23.8 285 + 1.9 30 - 13.3 120 - 7.8 210 + 27.1 300 - 4.2 45 - 11.3 135 - 5.9 225 + 25.6 315 - 10.3 60 - 10.0 150 -2.3 240 + 22.0 330 - 13.6 75 - 9.7 165 + 8.5 255 + 15.9 345 - 16.0 To illustrate the use of this table, let us suppose the ship to be sailing on a compass course of 165, in a part of the ocean where the variation is + 10, or 10 E. Using formula (1) (p. 45), and finding from our table that the deviation D for 165 is + 8. 5, we have the compass error E = V + D = + 10 +8.5 = + 18.5. By formula (2) (p. 45) the true course of the ship is T = C + E= 165 +.18.5 = 183.5. We should use this true course 183. 5 in calculating later the ship's position by dead reckoning (p. 10). If the compass variation were everywhere the same, it would be more convenient to have a table of compass errors, instead of a table of deviations ; but because the variation, as given on the chart, varies greatly, the table must be specially made for deviations only. Equally important with the above use of our deviation table is its inverse use. When the navigator has calculated by dead reckoning the course he must steer, that course, as it comes from the calculations, will be a true course (p. 43) ; and it is necessary to turn it into a compass course for the use of the steersman. To do this we must know the deviation ; and we cannot get it directly from the deviation table above, because the use of that table presupposes a knowledge of the compass course, the very thing we are trying to find. The best THE COMPASS 49 way to avoid this difficulty is to imagine the deviation to be non-existent, for the moment, and to make use of the "mag- netic course," defined as the course which would be indi- cated by the compass, if deviation were thus totally absent. Under these circumstances, formula (1) gives E = V, since D = ; and if we designate the magnetic course by M t we may write, in place of formula (2) (p. 45) : (3) M = T - V. Let us suppose a case in which the variation is + 10> and the desired true course of the ship 175. Then the magnetic course, allowing for variation only, will be, by formula (3) : M = T - V = 175 - 10 = 165. This course is not really a compass course, because no account has yet been taken of the deviation. Nor can we yet find the deviation directly from the deviation table, because in that table we must still know the compass course to use as the argument (p. 10), whereas we know as yet only the magnetic course. Therefore navigators should always request the compass adjuster to furnish a " second deviation table," in which the argument is the magnetic course, in- stead of the compass course. Such a second table can al- ways be calculated from the other. We here give one that has been calculated from the table on the preceding page. SECOND DEVIATION TABLE MAG- MAG- MAG- MAG- NETIC NETIC NETIC NETIC BEARING DEVIA- BEARING DEVIA- BEARING DEVIA- BEARING DEVIA- OF SHIP'S TION OF SHIP'S TION OF SHIP'S TION OF SHIP'S TION HEAD HEAD HEAD HEAD o o o o o o - 14.9 90 -9.0 180 + 11.0 270 + 16.5 15 - 13.4 105 -8.4 195 + 16.9 '285 + 4.1 30 - 11.7 120 - 6.9 210 + 21.3 300 - 7.1 45 - 10.4 135 ' -4.8 225 + 24.9 315 - 13.2 60 - 9.8 '150 - 1.4 240 + 26.8 330 - 15.7 75 - 9.3 165 + 5.0 255 + 24.1 345 - 15.5 50 NAVIGATION We also add as an example the calculation of one number in the second table from those given in the first. We shall find the deviation corresponding to the magnetic course 165 ; and we do it by a kind of interpolation (p. 12). From the first table we have the deviation 2.3 for the compass course 150. Since the deviation is the only difference between compass and magnetic courses, it follows that 150 - 2.3, or 147.7 magnetic, corresponds to 150 by com- pass. Similarly, 173.5 magnetic corresponds to 165 by compass. The magnetic course 165 for which we are making the calculation falls between 147. 7 and 173. 5, and exceeds the smaller of the two by 17.3. The whole difference be- tween 147.7 and 173.5 is 25.8. Similarly, the whole dif- ference between the two compass courses involved is 15. Therefore we may write the proportion : 25.8 : 15 = 17.3 : x, where x is the excess over 150 of the compass course corre- sponding to 165 magnetic. Solving this proportion by the ordinary rules of arithmetic, we have : The compass course belonging to 165 magnetic is there- fore 150 + 10.0 = 160.0. The corresponding deviation is 165 - 160.0 = + 5.(V which is therefore the deviation for 165 magnetic, and appears as such in the second table. This entire table can be computed from the first table in an hour. Sometimes the second deviation table gives compass courses instead of deviations. It is then often called a " table of 1 A comparison of formulas (1), (2), and (3) shows that D = M C ; so that the deviation is obtained by subtracting the compass course from the magnetic course. This is also evident from the definition of a magnetic course (p. 49). THE COMPASS 51 steering courses " ; and in the example just calculated it would give the compass or steering course 160 for the mag- netic course 165, instead of giving the deviation -f 5. We shall still further illustrate this important matter by an example, supposed to occur on board a ship for which our two deviation tables hold good. What is the compass course to be given the helmsman at Sandy Hook, on a voyage to St. Vincent? We have already found, from dead-reckoning calculations (p. 22) the course 119. Being the result of a dead-reckon- ing calculation, this is a true course. The track chart of the north Atlantic gives the variation at Sandy Hook as 10 W., or - 10. The true course being 119, we get the magnetic course, allowing for variation only, by formula (3), M = 7 7 -F = 119-(- 10) = 129. The second devia- tion table shows that : for magnetic course 120, the deviation is 6. 9, and for magnetic course 135, the deviation is 4. 8. Magnetic course 129 falls between 120 and 135, so that an interpolation (to be extremely exact) between 6. 9 and 4.8 makes the deviation for magnetic course 129 come out 5.6. Formulas (1) and (2) now give : Error =E = V+D=-W- 5.6 = - 15.6 Compass course = C = T-E = 119 -(- 15.6) = 134.6. To check this, we can now solve the same problem in the inverse way with the first deviation table. For the compass course 134. 6, this table gives the deviation as 5. 9. The variation being 10, we have : E = V + D = -10 - 5.9 = - 15.9 and T = C + E = 134.6 - 15.9 = 118.7, agreeing very closely with the true course 119, with which we started. This shows that the two deviation tables are quite consistent in this case, and also checks the accuracy of the calculation. 52 NAVIGATION We shall close this chapter with the following little table, showing the correspondence between the two methods of dividing the compass card into points, and into degrees. COMPASS POINTS AND DEGREES / o , / / North East 90 South 180 West 270 N. by E. 11 15 E. by S. 101 15 S. by W. 191 15 W. by N. 281 15 N.N.E. 22 30 E.S.E. 112 30 s.s.w. 202 30 W.N.W. 29230 N.E. by N. 3345 S.E. by E. 123 45 S.W. by S. 213 45 N.W. by W. 30345 N.E. 45 S.E. 135 s.w. 225 N.W. 315 N.E. by E. 56 15 S.E. by S. 146 15 S.W. by W. 236 15 N.W. by N. 326 15 E.N.E. 67 30 S.S.E. 15730 W.S.W. 247 30 N.N.W. 33730 E. by N. 7845 S. by E. 16845 W. by S. 25845 N. by W. 34845 J pt. = 2 49' pt. = 5 38' 8 26' 1 pt. = 11 15' CHAPTER V COASTWISE NAVIGATION BEFORE proceeding to a consideration of navigation by means of astronomic observations, as it is practiced on the high seas, we must first explain certain methods by which it is possible to ascertain a ship's position in latitude and longitude while she is in sight of land. Often such methods suffice to complete a long coastwise voyage in safety; they are always important for a last determination of the ship's position before a deep-sea voyage actually begins. Such a last determination is called " taking a departure" (cf. p. 2), and from such point of departure dead-reckoning calcula- tions begin for the first day of the voyage. Any determination or fixing of a ship's position, by astro- nomic observations or otherwise, is often called, for brevity, a "fix." To obtain one while in sight of land it is customary to make observations upon well-known objects ashore, such, for instance, as lighthouses, or other conspicuous objects marked on the chart. It is always possible to ob- serve the bearings of such objects from the ship's deck with the compass, azimuth circle, or pelorus (p. 44). When there is but one such object in sight, it is impossible to secure a fix with ordinary instruments, if the vessel is at anchor. But if she is running, it is merely necessary to take two bearings, and to estimate the distance run by the ship in the interval between the two. Figure 9 will make this matter clear. A lighthouse ashore is at L. SS" is the direction of the ship's course; S her position when the first bearing was observed, and S' her position at the time of the second bearing. SN is the direction of the north. 53 54 NAVIGATION After taking the first bearing, the navigator must calculate the angle S"SL, between the ship's course SS" and the lighthouse direction SL. Thus, if the ship's course angle NSS" (p. 10) was 20, and the bearing NSL was 42, the angle S"SL would be 42 - 20 =22. As the ship proceeds on her course, the angle S"SL will become larger, and a second bearing must be taken at the moment when the ship reaches the point S', where the angle S"SL has become S"S'L. This point S' must be so chosen that the angle S"S'L is just twice the angle S"SL ob- served at S ; or, in this case, 44. This is called " doubling the bear- Ship's Position by Two ing from the bow," and it can easily be accomplished if we con- tinue watching the compass bearing of L as the ship goes ahead, and catch the observation at the right moment. The ship's course not having been changed from 20 (this is important), the right moment will occur when L bears 20 + 44 = 64 by the compass. It can easily be proved by geometry that the distance S'L between the ship at S' and the lighthouse at L will be equal to the distance SS' traveled by the ship in the inter- val between the two observations. * This distance can be estimated quite accurately with an instrument called a "log," or "patent log," which is towed astern of the ship. It is so constructed that it turns as it is pulled through the water, and the number of turns is automatically counted by an attached contrivance on deck. The count is (also auto- matically) turned into miles of distance ; so that the log on deck will indicate how far the ship traveled from S to S'. COASTWISE NAVIGATION 55 As soon as we know the distance S'L and the bearing of the line S'L, we can ''lay down" or "plot" the position of S' on the chart; and this will be a "good fix." To do this, let us indicate by B' the bearing of the line S'L, and then draw on the chart, through the lighthouse L, a pencil line whose bearing from L is E' + 180, or " B' reversed." This can be done with a "course protractor," or with "parallel rulers," instruments to be purchased from any dealer in navigators' supplies. Next we measure or "lay off" on that line the distance S'L, equal to the run SS' as it came from the log. We always know the right "scale" of the chart (or fraction of an inch corresponding to one logged mile) which must be used in laying off the distance S'L; for we know that one mile always corresponds to 1 minute of latitude (p. 15), and the right- and left-hand edges of the chart are always divided into degrees and minutes of latitude. Since the above bearings were observed by compass, it is now important to consider the compass error (p. 43). This will not affect the observations, because it will be the same for both ship's course and lighthouse bearing, so the angles S f/ SL and S"S'L, which are obtained by subtraction, will be the same as if there were no compass error. But when we come to plotting on the chart, the compass bearing B' must be corrected by adding the deviation from the deviation table (pp. 48, 49). The resulting magnetic bear- ing (p. 49) must be used for B f , if the chart has printed on it a compass card (p. 41) showing magnetic bearings. If the printed card shows true bearings only, B' must be corrected for both deviation and variation (p. 43). A specially important case of the foregoing occurs when the two angles S"SL and S"S'L are 45 and 90. The second bearing B' will then put the light just abeam, and the distance by- log, SS', is the distance at which the ship passes the light abeam. This case is called a "bow-and- beam bearing." The navigator sights the light when it bears 45 or 4 points (p. 52) "broad" on the bow, "starboard," 56 NAVIGATION or "port." He then "reads" the log. When he brings the light abeam through the motion of the ship, he reads the log again, and the run in the interval, as taken from the log, is the light's distance abeam. When sailing along the coast, it is particularly important so to shape the ship's course that lights and other promi- nent landmarks will be passed at the right distance abeam. The chart shows what the right distance is : if the navigator shapes a course which makes the distance abeam too small, he may fail to clear rocks or shoals extending seaward ; and if he makes it too large, he may lengthen his voyage unneces- sarily in rounding the light. There are certain pairs of angles (S"SL and S"S'L) which will make known the coming distance abeam long before the ship is dangerously near the light. These angles, S"SL and S"S'L, are called "bearings from the bow" (see p. 54), since they are really measured from the ship's bow instead of the north. If the two bearings from the bow are either of the following pairs : 22 and 34, 32 and 59, 27 and 46, 40 ? and 79, then the logged distance in the interval between the two observations is the distance at which the ship will pass the light abeam if she continues on her present course. This kind of observation will inform the navigator whether his course is safe in ample time to change it if necessary ; and, since in this case no bearings are marked on the chart, no attention need be paid to compass error. When two or more known and conspicuous landmarks are visible from the ship, it is possible to secure a fix by means of "cross-bearings." Observe the bearings of the objects as nearly simultaneously as possible. Allow for compass error in the manner just explained. Calculate for each object a reversed bearing by adding 180 to its observed bearing. Draw on the chart through each object COASTWISE NAVIGATION 57 a pencil line having the proper reversed bearing and these lines will intersect at the point on the chart where the ship is located. Figure 10 illustrates this matter. L, L', L" are lights or landmarks ashore, visible from the ship, and also printed on the chart. The ship is at S. The lines in- tersecting at S repre- sent the reversed bearings of L, L', L", as observed from S. Only two lines are nec- essary; and they should be chosen so that the angle be- tween them is as near FIG. 10. Ship's Position by Cross Bearings. a right angle as possible, if high accuracy is required in the fix. The third object and line merely serve as an additional check or safeguard against error. In addition to the foregoing methods of locating a ship by observations of objects ashore, there is a way to avoid sunken rocks or shoals without actually locating the ship on the chart. It is called the " danger angle," and is shown in Fig. 11. The small circle is supposed drawn on the chart around a rocky shoal K which must be cleared by the ship traveling along the course SS'. To make certain of clearing it safely, the navigator selects two visible objects ashore, and shown on the chart at L and L'. He draws on the chart a large circle passing through L and I/, and just touch- ing the dangerous small circle at T. There is no difficulty in finding the center of the large circle, because it must be somewhere on the line PQ, which is drawn at right angles to the line LL' at its middle point P. A few trials with a 58 NAVIGATION pair of compasses will locate the center. Next, the two lines LT and L'T are drawn. Then the angle LTL' is called the danger angle. Now it is a principle of geometry that if we select other points on the large circle, such as T' and T", the angles FIG. 11. The Danger Angle. LT'L', LT"L', etc., will all be equal, and will contain the same number of degrees as the danger angle LTL'. It fol- lows that if the navigator measures from the deck the angle formed by two lines drawn to the ship from L and -I/, and if he finds it equal to the danger angle LTL', as measured on the chart with a protractor (p. 55), he then knows that the ship is somewhere on the large circle, and is therefore perhaps too near the small dangerous circle. If, on the other hand, the ship is entirely outside the large circle, and therefore surely safe from the dangers of the small circle, COASTWISE NAVIGATION 59 N/l the measured angle at the ship between the objects L and U will always be smaller than the danger angle LTL'. Angles can be measured from the deck by taking compass bearings of L and L' . The difference of the two will be the deck angle, which should be smaller than the danger angle measured on the chart. But the very best way to measure the deck angle is to use the sextant, an angle-measuring instrument to be described later (p. 61). The danger angle can also be used when it is necessary to pass between a sunken danger circle and the shore. The large circle is then drawn through L and L' as before, but in such a way as just to touch the inside of the small circle instead of the outside. To pass inshore of the small circle it is then necessary for the navigator to keep his measured deck angle larger than the danger angle, instead of smaller. Navigators also use at times a means of safety known as the " danger bearing," illustrated in Fig. 12. There is but one charted object in sight ashore at the point L. The ship at S must steer in such a way as to avoid sunken rocks at K. Evidently, she must pass outside the line SQ, of which the bearing from the north is the angle NSQ, which can be meas- ured on the chart. This is the danger bearing, and the ship's course SS', to be safe, must be greater than the danger bearing. In the case shown in the figure, the danger bearing would be very useful long before a fix could be had by means of bearings from the bow or bow-and-beam bearings. Finally, to complete this part of our subject, it is neces- sary to mention " soundings, " which are a method of feel- ing the land, even when it cannot be seen. By means of FIG. 12. The Danger Bearing. 60 NAVIGATION the " lead-line" the mariner can ascertain when he is in shoal water ; and as depths of water are always marked on the chart, he can often get valuable information as to the ship's position. As she runs along her course, he can take a "line of soundings" and upon examining the chart he will often find but a single possible line on the chart where the charted depths correspond with those observed. It follows that the ship's course must have been along that line on the chart ; and at an anxious moment, in a fog, such a check will be a great relief to the navigator. Even in the ocean, far from land, it is possible to take soundings with the "sounding machine" at great depths, and in some parts of the ocean quite accurate locating of the ship will result. Specimens from the ocean floor can also be brought up by attaching some sticky grease to the bottom of the lead, and at times these specimens also give information of value, for the charts always specify the kind of bottom existing in various parts of the ocean. . CHAPTER VI THE SEXTANT WE have twice made reference to this instrument once (p. 5) as a contrivance for ascertaining by observation how high the sun is in the sky, and again (p. 59) in the measure- ment of the danger angle. These two uses of the sextant are not inconsistent, for it is really intended for the measure- ment of any angle (p. 8) formed at the observer's eye by two lines drawn to two distant objects. In the case of the danger angle these two distant objects are landmarks ashore; in the case of the sun they are the " horizon" line (where sea and sky seem to meet), and the sun itself. This height of the sun (or of any star) in the sky is called its " altitude"; and so the altitude is always an angle, to be measured in degrees and minutes. The point directly over- head is the "zenith"; the angle between lines drawn, to horizon and zenith is 90, or a right angle. An altitude of 40, for instance, simply means that the distance from the horizon to the sun is f$ of the total distance from horizon to zenith. Figure 13 will give an idea of the construction of the sex- tant. 1 The essential parts are two small silvered mirrors, M and m; a telescope, EK\ and a circle, A A, engraved with " graduations," by means of which angles may be measured upon it in degrees, minutes, and seconds. The mirror m and the telescope EK are firmly attached to the sextant ; but the mirror M is pivoted in such a way that it 1 Quoted in part from Jacoby's "Astronomy, a Popular Hand- book," Macmillan, 1913; reprinted 1915. 61 62 NAVIGATION can be turned, and the angle through which it is turned measured on the circle by means of the index CB. When the mirror M is turned until it is parallel to the fixed mirror m, the circle " reads" or indicates 0, because the angle be- tween the two mirrors is then 0. In all other positions FIG. 13. The Sextant. of the mirror M the circle measures the angle between the two mirrors. P and Q are sets of colored glasses, which can be interposed temporarily, when the sun's rays are so bril- liant as to be hurtful to the observer's eye. R is a small magnifying glass, pivoted at S, intended to facilitate the examination of the index CB. At C and B are shown the "clamp," by which the index can be fastened to the circle, and the " tangent screw," or " slow-motion screw" which will adjust it delicately, after it has been clamped. / and F are additional telescopes or accessories. The mirror m has an important peculiarity. The silver- ing is scraped away at the back of the mirror from half its THE SEXTANT 63 surface. Thus only one half reflects ; the other half is simply transparent glass. A navigator looking into the telescope at E will therefore look through the mirror m with half his telescope, and with the other half he will look into the mirror. Now it is a fact that half a telescope acts just like a whole one. If a person using an ordinary spy-glass half covers the big end with his hand, he will see the same view he saw with the whole glass. Only, as half the " light-gathering " power is cut off, this view will be fainter, less luminous. Applying this to the sextant telescope, it is clear that the observer will see two things at once : with half the telescope he will see what is visible through the mirror m ; and with the other half he will see what is visible by reflection from the mirror m. If he holds the sextant in such a position that the telescope is horizontal, while the frame of the instrument is vertical, he will see the visible sea horizon with half the telescope through the mirror m. If the other mirror M is then turned to the proper position, it is possible to see the sun in the sky at the same time, with the other half of the telescope, the solar rays having been reflected successively from both mir- rors, M and m. To make this possible, the sextant tele- scope must be aimed at that point of the sea horizon which is directly under the sun. The solar rays will then strike the mirror M first ; be thence reflected to the silvered part of the mirror m; and finally reflected a second time into the telescope. Therefore the observation consists in so turning the movable mirror M, that the sun and horizon can be seen coincidently in the telescope. The angle between the mirrors can then be measured on the circle ; and it is easy to prove by geometry that the angular altitude of the sun will be twice the angle between the two mirrors. Thus it should merely be necessary to double the mirror angle, as indicated by the sextant index, to obtain the solar altitude. But the sextant makers always 64 NAVIGATION save the navigator the trouble of doubling the angle by the simple device of numbering half degrees on the arc A A as if they were whole degrees ; so the angle as it comes from the sextant is already doubled for further use. The mirror m is called the " horizon glass," because the navigator looks through it at the horizon. The other mirror M is the "index glass," because it is attached to the index arm. When the sextant is used for non-astronomical observa- tions, such as the danger angle, the frame is held horizontally, instead of vertically, as in observations of the sun. The telescope is aimed at the left-hand object ashore, and that object is viewed through the horizon glass m. The index glass M is then turned until light from the right-hand object is also brought into the telescope, after successive reflections from the two mirrors M and m. The two objects will then be seen " superposed," and the sextant arc will give the angle between two lines drawn from the observer on board to the two objects ashore. This angle should be smaller than the danger angle to keep the ship safely off-shore of sunken dangers (p. 59). Reading the sextant circle, or ascertaining from it the angle that has been measured, is accomplished by means of a "vernier." This is a short circular arc, engraved with graduations resembling those on the sextant circle, attached to the index CB (fig. 13) just under the little magnifier R. It is so placed that the graduations on the sextant circle and the vernier are close together and can be seen at the same time through the magnifier R. Figure 14 gives an idea of the vernier and a part of the sextant circle near the zero of its graduations. Numbers on both circle and vernier increase toward the left. On the circle, the largest spaces, marked by long lines, are whole degree spaces. Each is usually divided into two halves of 30' each indicated by shorter lines, and these are again subdivided into three small spaces of 10' each. The divisions on the vernier resemble those on the circle, except that the degree spaces THE SEXTANT 65 of the former are here called min- ute spaces, and the 10' spaces of the former are called 10" spaces. The real index of the instru- ment is the zero mark on the vernier, sometimes provided with an engraved "arrow." If this falls exactly on a degree mark of the circle, say the 1 mark, the reading of the instrument is ex- actly 1 0' 0". If it falls exactly on a small line of the circle, say the second to the left of the 1 mark, the reading is exactly 1 20' 0". But if it falls between two of the small lines, say between the 20' and 30' marks to the left of the 1 mark (as shown in the figure), the reading must be 1 20' and a "bit." It is the busi- ness of the vernier to estimate the size of that bit. To do this look along the vernier until you find a line which is exactly op- posite some line on the circle. There will always be such a line : in the figure it is the 6' line of the vernier. Pay no further atten- tion to noting which line on the circle is the one thus " exactly opposite"; it matters not which line it is. But read carefully the number on the vernier belonging to the "exactly opposite" line you have found there. Being on this occasion the 6' line, it follows VA A/1 66 NAVIGATION that the bit is 6' ; and as we found the reading to be 1 20' and a bit, the complete reading is 1 20' + 6' = 1 26'. If the vernier line that happened to be " exactly opposite" was not one of the ten long minute lines, but fell between two of them, it would indicate that the bit was made up of minutes and seconds, instead of being an exact number of minutes. For each space the " exactly opposite" vernier line happens to lie to the left of a long vernier minute line, 10" must be added to the bit. For instance, if in the figure the "exactly opposite" vernier line was the next short one to the left of the 6' long line, the bit would be 6'' 10", and the complete reading 1 26' 10", instead of 1 26'. But seconds are not really required when observing aboard ship, so that it will be sufficient; in using the vernier, to find the number of the long vernier line that comes nearest to being " exactly opposite." It will also be noticed in the figure that the sextant circle has some additional graduations to the right of the mark. These are called "off the arc" graduations, and it is some- times necessary to read a small angle upon them, measuring from the mark to the right instead of the left. This makes it necessary to read the vernier backwards, calling the 0' mark of the vernier 10' and the 10' mark 0'. This backward reading of the vernier offers no particular difficulty, and it is especially useful in determining by ob- servation the "index error" of the sextant. We have seen (p. 62) that when the two sextant mirrors are parallel, the index should read 0' 0". But it is seldom possible to adjust the instrument so that this condition will be satis- fied exactly ; nor would the adjustment remain perfect very long. A better plan is to determine by observation how much the reading differs from 0' 0", when the mirrors are parallel. This difference is the index error, and must be applied as a correction to all angles observed with the instrument. It is easy to make the mirrors parallel : we have merely THE SEXTANT 67 to sight some distant well-defined terrestrial object like the gilt ball on the top of a flagpole (or the sea horizon, if aboard ship at sea), after clamping the index near 0. We shall then see in the telescope two images of the distant object; one by direct vision through the unsilvered part of the hori- zon glass, the other after reflection from both mirrors. By means of the tangent screw, the observer, with his eye at the telescope, can bring these two images together, so that they will appear as a single image. Then the mirrors will be parallel, and the vernier should read 0' 0". If it actually reads 8', for instance, instead of 0' 0", it means that the reading is 8' too large on account of index error ; and every angle measured with that sextant at that time will be 8' too large, and must be corrected by subtracting 8' from it. If, on the other hand, the reading is 8' "off the arc," when it should be 0', the instrument reads &' too small, and any angle measured with it must be corrected by adding 8' to it. For accurate determination of the index error (and it should be checked frequently), navigators prefer to observe the sun, or at night, a star. If a star is used, the process is the same as just described for a flagpole ball. But if the sun is used, a slightly different method is required. The sun, as seen in the telescope, shows a round disk of con- siderable size, and it is not possible to superpose the two images accurately. Therefore it is better to make them just touch, as shown in Fig. 15, when they are said to be "tangent" to each other. This must be done successively in two positions, AB and BA. In other words, after the first "tangency " FlG * 15 -~ Index Error - has been observed, the tangent screw (B, fig. 13) is manipu- lated until the image A passes across B from top to bottom, and gives a new tangency in the second position. Each tangency will give a reading of the vernier. Unless 68 NAVIGATION the sextant is greatly out of adjustment, one of these read- ings will be off the arc, the other on the arc. If there were no index error, the off-arc and on-arc readings would be equal; if they differ, half the difference is the index error. If the off-arc reading is the larger, all altitudes measured with that sextant must be increased by the amount of the index error ; and if the on-arc reading is the larger, all such altitudes must be similarly diminished. The following is an example of an index error determina- tion: On-arc readings, Off-arc readings, 31' 20" 33' 20" 31 40 33 50 30 50 34 Means, 31' 17" 33' 43" The difference is 33' 43" - 31' 17" = 2' 26". Half the difference, or 1' 13", is the index error ; and because readings on the arc are the smaller, all angles read with this instru- ment must be increased by 1' 13", or, for ordinary purposes of navigation, by 1'. In addition to certain " ad justing screws" with which the index error can be reduced when it becomes unduly large, means are provided for three other sextant adjust- ments. These are : 1. To make the index glass perpendicular to the frame of the instrument. 2. To do the same with the horizon glass. 3. To set the telescope parallel to the frame of the instru- ment. These adjustments are always completed by the maker before a sextant is sent out, nor does the navigator usually need to correct them himself. But it is important to know how to test them occasionally. Perpendicularity of the index glass can be examined by looking into the glass very obliquely with the index set near 0. It is then possible to see the inner edge of the sextant circle both by looking at THE SEXTANT 69 it directly, past the edge of the index glass, and also by reflec- tion in the glass itself. The inner edge of the circle should form a continuous line when so examined, if the glass is perpendicular ; but if it is inclined, the line will appear broken, instead of continuous. Secondly, perpendicularity of the horizon glass can be tested at the same time the index error is determined by observing a star or a distant terrestrial point (p. 67). The index glass having been properly adjusted to perpendic- ularity, the two mirrors can never be made parallel by moving the index, unless the horizon glass is also properly perpendicular. Any existing lack of adjustment will there- fore betray itself in the index error determination, because the two images of the star or distant object will not be super- posed in any position of the index. Thirdly, the parallelism of the telescope to the frame of the instrument can usually be best tested with an ordinary pair of " calipers." Having thus described the sextant, its adjustments, and its use from the deck, we have still to explain how it can be used ashore. Sometimes it is necessary for the navigator to make observations ashore, when it is not usually possible to see the horizon line (p. 61). Recourse must then be had to an "artificial horizon," which is simply an iron basin full of mercury covered with a glass roof. The mercury furnishes an almost perfectly horizontal mirror, and the glass roof prevents wind from ruffling the mercury surface, and thus destroying the mirror. FlG . IG. Artificial' Figure 16 explains the principle of the Horizon, artificial horizon. HH is the mercury mirror, S the sun, and X the sextant. The observer aims the sextant telescope at the mercury where he can see a reflection of the sun. He then measures with the instrument the angle between a line 70 NAVIGATION drawn to the sun as seen reflected in the mercury and another line drawn to the actual sun in the sky. It can be shown by geometry that this measured angle will be just twice the real altitude of the sun, such as it would be if observed from the sea horizon. Therefore, in using the artificial horizon, it is merely necessary to divide the sextant angle by 2 to ob- tain the correct altitude of the sun. In observations of this kind two "suns" are seen at the same time in the telescope, just as is the case in index error observations (p. 67) ; whereas in observing from the sea horizon, the telescope shows only one solar image and the horizon line. When there are thus two solar images, they must be brought into tangency, just as we have already explained for index error (p. 67). When there is but one, it must be brought into tangency with the visible sea horizon line. But this altitude is not yet ready to be used in the further calculations for obtaining the position of the ship in latitude and longitude. Further pre- paratory corrections must be applied, in addition to the index error (p. 66), which is always the first correction to receive attention. These pre- paratory corrections are : 1. "Dip" of the sea hori- zon, due to the elevation of the navigator on the ship's deck above the surface of the sea. Its cause is shown in Fig. 17. C is the center of the FIG. 17. Dip of the Horizon. earth, K a point at sea level, and the navigator, elevated a distance OK above the sea. OZ is the direction of the ze- nith (p. 61), OS the direction of the sun, and OH a horizontal line from 0. OT is a line drawn through 0, and just touch- THE SEXTANT 71 ing the sea surface at T f . Evidently OT will be the direc- tion of the sea horizon, where sky and sea seem to meet. Therefore, the altitude of the sun, as measured from the visible sea horizon, will be the angle SOT; whereas the angle we require is the angle SOH, or the altitude of the sun above the true horizontal line OH. Therefore the angle HOT is a correction for dip which must be subtracted from all measured altitudes, and the amount of the correction depends on the height of the navigator's eye above the sea surface. 2. " Refraction" is a bending of the light rays as they come down to us from the sun through the terrestrial atmos- phere. It always makes the sun seem higher in the sky than it really is, giving another subtractive correction for the observed altitude. The bending here involved is due to the passage of the sun's light rays through atmospheric strata of increasing density as the light approaches the earth's surface. 3. "Parallax" is a small correction which must be added to the observed altitude of the sun. In strict theory, all astro- nomic observations are supposed to be made from the earth's center instead of its surface where the ship floats ; and the small parallax correction allows for this minor theoretic point. In the case of star observations this correction is zero. 4. "Semidiameter" is a correction depending on the choice by the navigator of a particular point on the sun's disk (p. 67) for observation. The sun's altitude, as used in the further calculations, should be the altitude of the sun's center ; but it is impossible to locate the center of the disk accurately in the telescope, so the navigator always observes the lowest point of the disk. This is called the " lower limb" of the sun. Beginners sometimes have difficulty in distinguishing the upper from the lower limb in the telescope. The best way to do this is to focus the telescope on some distant 72 NAVIGATION object, and note whether it appears upside-down in the field of view. If so, the telescope is an " inverting" one, and the top of the sun must be observed, as it appears in the telescope, though it will really be the correct (or lower) limb, because of inversion by the telescope. When using the artificial horizon with an inverting telescope, the tangency must be made by bringing the bottom of the mercury image in contact with the top of the other image. The high-pow- ered telescopes supplied with good sextants are usually in- verting telescopes. Evidently the measured altitude, as it comes from the sextant, must be increased by the amount by which the sun's center is higher than the lower limb, and this is the sun's semidiameter. The index correction, together with the above four additional corrections, will fully prepare a meas- ured sextant altitude of the sun for further use in naviga- tional calculations. In the case of a star, which appears in the telescope as a point of light only, without any per- ceptible disk, no semidiameter or parallax corrections are required; and in using the artificial horizon (p. 69), no correction for dip is necessary, either for the sun or a star. It is possible to arrange these various corrections in con- venient tables. Thus, in Table 6 (p. 247), we give a combi- nation of corrections 2 (refraction), 3 (parallax), and 4 (semi- diameter), to be used for observations of the sun's lower limb, and the same combination without the semidiameter and parallax 1 to be used for star observations. It will be noticed that the tabular corrections vary for different values of the observed altitude, which appears in the left-hand col- umn of the table. This variation comes mainly from the refraction part of the combined correction, for the refrac- tion is much greater when the sun or star is observed at a low altitude near the horizon than it is at a high altitude near the zenith. At the foot of the page is given a small supplementary correction depending on the date in the year. 1 Which leaves refraction only. THE SEXTANT 73 This small correction is not important in navigation, but is given here for the sake of completeness. It arises from the semidiameter part of the combined correction, for the an- nual orbit of the earth around the sun is of such a shape that the earth is nearer the sun in January than it is in July, which makes the sun appear bigger in January. And when the sun appears big, the semidiameter will of course be large too. Table 7 gives the dip of the sea horizon, the number in the left-hand column being the height (in feet) of the navigator's eye above sea level. This will be the height of the ship's deck, increased by the height of the man's eye above the deck. Unfortunately, the dip, as given in Table 7, at times varies considerably from the dip as it actually exists at the ship. The 'cause can be seen from Fig. 17 (p. 70), where it will be noticed that the line from the observer at to the sea horizon at T' passes very near the surface of the ocean. It is therefore entirely in the lowest strata of the terrestrial atmosphere, and there quite irregular refractions sometimes occur. These have been known to produce errors in the dip amounting to 10' or 20', and it is principally the existence of these unavoidable errors that makes it unnecessary to read the sextant closer than the nearest minute (p. 66), when observing from the deck. But when observing ashore with the artificial horizon, which has no dip, the navigator may, if he chooses, read seconds, especially if he intends to use in his further calculations the "mean" or average of a considerable number of observations. We shall now give an example of the complete correction of a sextant observation. Suppose the angle read from the sextant was 30 28', the index error (p. 68) 1', addi- tive, height of observer's eye 26 feet. We should then have: observed altitude, lower limb = 30 28' index correction = + 1' correction from Table 6 (p. 247) = + 14' correction from Table 7 (p. 247) = - 5' corrected altitude, for further use = 30 38' 74 NAVIGATION If the altitude had been observed ashore with an arti- ficial horizon, it might have been desirable to retain seconds. The calculation might then have been as follows : observed double altitude (see p. 70), lower limb = 63 0' 20" index correction (p. 68) = -f 1 13 corrected double altitude =63 1 33 resulting altitude =31 30 46 correction from Table 6 (interpolated) = + 14 31 corrected altitude, for further use = 31 45 17 CHAPTER VII THE NAUTICAL ALMANAC BEFORE beginning the further utilization of altitude ob- servations in our navigation calculations, it is necessary to understand the use of the Nautical Almanac. This is an annual publication, issued in two different editions by the Nautical Almanac Office, United States Naval Observatory. Copies can be obtained from the Superintendent of Docu- ments, Washington, D. C., or through any dealer in nautical supplies. Navigators do not need the larger edition, of which the title is " American Ephemeris and Nautical Almanac"; accordingly, all our references are made to the smaller edi- tion for the year 1917. Parts of certain pages from that edition are reprinted in the present volume for convenience of reference, and we shall give a somewhat detailed explana- tion of the almanac page 29 (our p. 76). Let us consider the date Monday, Dec. 17. We find for that date, and for every even hour (0*, 2*, 4*, 6 h , etc.) of " Greenwich Mean Time" (abbreviated G. M. T. 1 ), two tabular numbers (p. 10) called " sun's declination" and "equation of time." To understand these it is necessary to bear in mind that the kind of time in ordinary use is "solar time," as kept by the sun. The "solar day" begins at "noon," called O ft in astronomic navigation, and it continues through twenty-four hours, without any confusing A.M. and P.M. In ordinary life the day begins twelve hours sooner, at midnight, and runs through two twelve-hour periods of A.M. and P.M. to 1 The reader is requested to note carefully this abbreviation, as it will be used very frequently. 75 76 NAVIGATION SUN, DECEMBER, 1917. From Nautical Almanac, p. 29 G. M. T. SUN'S DEC- LINATION EQUATION OF TIME SUN'S DEC- LINATION EQUATION OF TIME SUN'S DEC- LINATION EQUATION OF TIME Monday 17 Tuesday 25 Saturday 29 h / m s / m s / m s - 23 21.3 + 3 56.8 - 23 24.7 -0 1.6 - 23 15.2 - 1 59.7 2 23 21.5 3 54.4 23 24.6 4.1 23 14.9 2 2.1 4 23 21.7 3 51.9 23 24.5 6.5 23 14.6 2 4.6 6 23 21.9 3 49.5 23 24.4 9.0 23 14.3 2 7.0 8 23 22.1 3 47.0 23 24.2 11.5 23 14.0 2 9.4 10 23 22.2 3 44.5 23 24.1 14.0 23 13.7 2 11.9 12 23 22.4 3 42.1 23 24.0 16.5 23 13.4 2 14.3 14 23 22.6 3 39.6 23 23.8 18.9 23 13.1 2 16.7 16 23 22.8 3 37.1 23 23.7 21.4 23 12.8 2 19.1 18 23 22.9 3 34.7 23 23.5 23.9 23 12.5 2 21.5 20 23 23.1 3 32.2 23 23.4 26.4 23 12.2 2 24.0 22 23 23.2 3 29.8 23 23.2 28.8 23 11.9 2 26.4 H. D. 0.1 1.2 0.1 1.2 0.1 1.2 Tuesday 18 Wednesday 26 Sunday 30 - 23 23.4 + 3 27.3 - 23 23.1 - 31.3 - 23 11.6 - 2 28.8 2 23 23.6 3 24.8 23 22.9 33.8 23 11.3 2 31.2 4 23 23.7 3 22.3 23 22.7 36.3 23 11.0 2 33.6 6 23 23.8 3 19.9 23 22.5 38.7 23 10.6 2 36.0 8 23 24.0 3 17.4 23 22.4 41.2 23 10.3 2 38.4 10 23 24.1 3 14.9 23 22.2 43.7 23 10.0 2 40.9 12 23 24.3 3 12.5 23 22.0 46.2 23 9.7 2 43.3 14 23 24.4 3 10.0 23 21.8 48.6 23 9.3 2 45.7 16 23 24.5 3 7.5 23 21.7 51.1 23 9.0 2 48.1 18 23 24.6 3 5.0 23 21.5 53.6 23 8.6 2 50.5 20 23 24.8 3 2.6 23 21.3 56.0 23 8.3 2 52.9 22 23 24.9 3 0.1 23 21.1 58.5 23 7.9 2 55.3 H. D. 0.1 1.2 0.1 1.2 0.2 1.2 Wednesday 19 Thursday 27 Monday 31 - 23 25.0 + 2 57.6 - 23 20.9 - 1 0.9 - 23 7.6 - 2 57.7 2 23 25.1 2 55.1 23 20.7 1 3.4 23 7.2 3 0.1 4 23 25.2 2 52.6 23 20.5 1 5.9 23 6.9 3 2.4 6 23 25.3 2 50.2 23 20.3 1 8.3 23 6.5 3 4.8 8 23 25.4 2 47.7 23 20.1 1 10.8 23 6.1 3 7.2 10 23 25.5 2 45.2 23 19.8 1 13.2 23 5.8 3 9.6 12 23 25.6 2 42.7 23 19.6 1 15.7 23 5.4 3 12.0 14 23 25.7 2 40.2 23 19.4 1 18.1 23 5.0 3 14.4 16 23 25.8 2 37.8 23 19.2 1 20.6 23 4.6 3 16.7 18 23 25.9 2 35.3 23 19.0 1 23.1 23 4.3 3 19.1 20 23 26.0 2 32.8 23 18.7 1 25.5 23 3.9 3 21.5 22 23 26.1 2 30.3 23 18.5 1 28.0 - 23 3.5 - 3 23.9 H. D. 0.0 1.2 0.1 1.2 0.2 1.2 Thursday 20 Friday 28 o 23 26.1 + 2 27.8 - 23 18.3 - 1 30.4 2 23 26.2 2 25.3 23 18.0 1 32.9 4 23 26.3 2 22.8 23 17.8 1 35.3 6 23 26.3 2 20.4 23 17.5 1 37.8 8 23 26.4 2 17.9 23 17.3 1 40.2 SEMIDIAMETER 10 23 26.5 2 15.4 23 17.0 1 42.6 12 23 26.5 2 12.9 23 16.8 1 45.1 14 23 26.6 2 10.4 23 16.5 1 47.5 Dec. 1 16'26 16 18 20 23 26.6 23 26.7 23 26.7 2 7.9 2 5.4 2 2.9 23 16.3 23 16.0 23 15.7 1 50.0 1 52.4 1 54.8 11 21 31 16'28 16'29 16'30 22 - 23 26.8 + 2 0.4 - 23 15.4 - 1 57.3 H. D. 0.0 1.2 0.1 1.2 NOTE. The Equation of Time is to be applied to the G. M. T. in accordance with the sign as given. THE NAUTICAL ALMANAC 77 the following midnight; but this " civil day/' as it is called, does not for the moment concern us. Solar time, as kept by the visible sun, is a very incon- venient kind of time, because there are certain peculiarities in the astronomic motion of the earth which make these solar days of unequal length. They are called " apparent solar days" and the corresponding kind of time is " apparent solar time." To avoid the above inconvenience, an imaginary "mean sun" and a "mean solar day" have been invented. The mean sun conforms as nearly as possible to the average per- formance of the visible sun, and the length of the mean solar day is the average of all the apparent solar days through- out the year. The corresponding kind of time, kept by the mean sun, is "mean solar time" ; and this is the kind of time recorded by all our watches and marine chronometers (p. 6). The difference between these two kinds of solar time varies on different dates, and even at different hours on the same date. It is this difference which is called the "equation of time " and which is one of the tabular numbers in the nautical almanac page 29 (our p. 76). This equation of time is of great importance in navigation, and it is easy to see how page 29 of the almanac may be used to find it. Suppose, for instance, we wish to know what the equation is on Dec. 17, 1917, on board ship, when the ship's chronometer indicates on its face 3 P.M., civil time, or (which is the same thing) 3 h , astronomical time (p. 75). Ship's chronometers are always set to Greenwich mean time, so that 3* by the chronometer signifies that the time at Green- wich was 3\ We then look in the almanac page 29 (our p. 76), and find that the equation was + 3 54*.4 at 2 h , G. M. T., and + 3 m 5P.9 at 4 A , G. M. T. Its value at 3 h must be half- way between these two, or -f 3 m 53M5. This we would call -f 3 m 53*.2, so as to avoid the use of hundredths of seconds, which do not need attention in navigation. And 78 NAVIGATION since the equation is merely the difference between the two kinds of solar time, the + sign means that it must be added to G. M. T., to obtain Greenwich apparent time, in accordance with the "Note" at the foot of the almanac page 29. Consequently, the G. M. T. by chronometer having been 3 A O m s , the Greenwich apparent time at the same in- stant was 3 h O m s + 3 53 S .2 = 3 h 3 53 a .2. It will be noticed that the process we have here used for obtaining the equation from the almanac is merely an inter- polation (see p. 12). Let us, as another example, find the equation for Sunday, Dec. 30, at I0 h 26 W A.M., civil time by chronometer, and we have purposely here retained the civil method of reckoning time to make certain that the reader understands the difference between civil and astro- nomic (or navigation) time. The given time is 10* 26 m A.M., civil time, Dec. 30. But the astronomic Dec. 30 does not begin until noon (p. 75), so that it is not yet Dec. 30 by astronomic reckoning. By that reckoning it is really only 22^ 26 m on Dec. 29. In other words, when the civil time is P.M., as in the first example, the astronomic time is the same as the civil time. But when the civil time is A.M., as in the present example, the astronomic time is found by adding 12* to the civil time, and deducting 1 from the date. These complications emphasize the advantage of the astronomic count, which avoids A.M. and P.M. altogether. We now have from the almanac (p. 76) : equation of time, Dec. 29, 22*, G. M. T. = - 2 m 26.4, equation of time, Dec. 30, 0*, G. M. T. = - 2 m 28.8 ; and the numbers in this example have been purposely so chosen that the above two tabular values of the equation (between which the required value falls) come from different dates in the almanac. This creates no confusion, for these two values of the equation are really consecutive tabular numbers, just as much as if they occurred on a single date. The difference between the two values of the equation is THE NAUTICAL ALMANAC 79 2 a .4; and as this difference corresponds to 2 h in the left= hand (or argument) column, it follows that the difference for l h is here P.2. This is the change of the equation per hour of time; it is called the " hourly difference" (abbre- viated H. D.) and is printed in the almanac at the foot of each daily column. Now we want the equation for Dec. 29, 22* 26 W , by the chronometer. The 26 m must next be changed into a decimal fraction of an hour. 26 m = ff of an hour = 0\43. So the time for which we want the equation becomes Dec. 29, 22 A .43. The H. D. being K2, the change in 0^.43 will be 1 3 .2 X 0.43 = (K5. The almanac shows that at 22 h the equa- tion was 2 m 26 s . 4, and was increasing numerically. There- fore, at 22\43, it was 2 m 26 S .4 + 5 .5 = 2 m 26 S .9. And this number has the minus sign. Therefore, the G. M. T. being Dec. 29, 22 h 26 m , the Greenwich apparent time at the same instant will be Dec. 29, 22 A 26 - 2 m 26 8 .9 = Dec. 29, 22* 23- 33M. Most of these minor interpolation calculations, which are here set forth in great detail for the benefit of the beginner, can be made with sufficient accuracy by a skilled navigator mentally. In the foregoing two examples we have assumed that the chronometer was right, but these instruments practically never run quite correctly. Therefore, before leaving port, navigators always have their chronometers " rated" by a chronometer expert; and when the instrument is returned to the ship just before sailing, a "rate card" (or "rate paper") always comes with it. Let us suppose that in the present example this card stated that the chronometer was slow 8 m 22 8 .5 l on Dec. 20, at noon, and was "losing" 2 P.8 daily. The 8 m 22". 5 would then be the "chronometer error" on Dec. 20 ; and the 1'.8 would be its "daily rate." 1 This number is here purposely chosen much larger than would ever occur in practice. 2 The opposite kind of "rate" is called "gaining." 80 NAVIGATION From Dec. 20, noon, to Dec. 30, 10 h 26 W A.M. is an interval of 9 days 22 hours 26 minutes. This interval must now be reduced to a decimal of a day. 26 m = f f of an hour = 0\43. The interval is therefore 9* 22M3. But 22M3 =4* days = O d .93. Therefore, in days, the interval is 9 tf .93. This transformation of hours and minutes into decimals of a day can be accomplished with less trouble by means of our Table 8 (p. 248). Having a losing rate of 1 s . 8 daily, the chronometer lost 1-.8 X 9.93 = 17 S .9 in the interval of 9.93 days. And as it was already slow 8 m 22 S .5 on Dec. 20, it was slow 8 m 22*.5 +17*.9 = 8 m 40 8 .4 at the time for which the equation is required. Now the equation was required for Dec. 29, 22 h 26 m by the chronometer; and that instrument being slow 8 m 40 S .4, the correct G. M. T. was : Dec. 29, 22^ 26 m + 8 m 40 8 .4 = Dec. 29, 22 h 34 m 40 S .4. Turned into a decimal fraction of an hour, this becomes Dec. 29, 22*.58, instead of 22\43, as we found before, when the chronometer error was omitted from the calculation. The H. D. is l s .2, as before, and the change in 0^.58 = P.2 X 0.58 = 0-.7. Therefore, at 22\5S the equation is 2 m 26 5 .4 + O s .7 = 2 m 27M. This still has the minus sign, so that the correct Greenwich apparent time becomes Dec. 29, 22* 34 40-.4 - 2 m 27M = 22* 32 13'.3. All the above calculations have been carried out here with unnecessary accuracy. There would be no harm if the result were in error by a few tenths of a second ; and it is this cir- cumstance that makes it possible to perform these inter- polations largely mentally. In the foregoing examples no account was taken of the ship's location on the ocean ; yet this location may have an indirect influence on the calculations. To understand this, we must consider for a moment the time-differences which exist between different places on the earth. The sun rises in the east and travels across the sky toward the west ; so that if we consider two places like Greenwich, England, and New York, for instance, the sun, because of this motion from east THE NAUTICAL ALMANAC 81 to west, will pass Greenwich first. Consequently, when it is noon in New York, it has already been noon in Greenwich, and is afternoon there. Greenwich time is therefore always later than New York time. The same is true of any other two places ; there is always a time-difference between them, and the easterly place has the later or " faster" time. The amount of such time-difference of course depends on the relative location of the two places, and the relation is such that 15 of longitude-difference corresponds exactly to l h of time-difference. Thus Sandy Hook, which is in longitude 73 50' west of Greenwich, has a time-difference from Greenwich of 4 A 55 m 20 s . This conversion of longitude into time-difference is best accomplished by means of our Table 9 (p. 249). According to that table : 73 = 4* 52" 0* 50' 3 20 73 50' = 4* 55" 20* The indirect influence of such time-differences upon the use of the almanac is that they may at times, especially when they are large, make the Greenwich date of the ob- servation different from the date on board. Thus a vessel off Manila Bay, in longitude 120 east of Greenwich, would have her local time 8 h (120) later than Greenwich time. If a sextant observation was made on board at 4 P.M., civil time, on a Thursday, the chronometer would indicate 8 A , and it would be 8 A.M. on Thursday, because Greenwich is S h earlier than the ship. This 8 A.M. would really be 20 h of the preceding Wednesday by astronomic time, and so the almanac date used would be one day earlier than the date of the observation. The chronometer will always give the right Greenwich time, but the navigator must be very care- ful to interpolate the almanac numbers on the right date. We have now learned how to ascertain the equation of time from the almanac, and how to use it for transforming G. M. T. into Greenwich apparent time. The contrary transformation, from Greenwich apparent time to G. M. T., 82 NAVIGATION can be made by applying the equation in the opposite way : subtracting when it has the + sign in the almanac, and add- ing when it has the sign. The great importance of these time transformations comes from the fact that sextant observations must necessarily be made upon the visible sun. When they are made for the purpose of calculating the local time on board, this local time will therefore necessarily be local apparent solar time, as kept by the visible sun. At the instant of the observation (p. 6), the chronometer face (corrected for error and rate) tells us the G. M. T. If this is turned into Greenwich ap- parent time by applying the equation, we have only to com- pare the Greenwich and the ship's apparent times to get the time-difference between the ship and Greenwich. This time-difference can then be turned into degrees and minutes, and will be the ship's longitude. Examples of this calcu- lation will be given in detail (p. 99). It is also worth noting here that the time-difference between any two places is precisely the same, quite irrespective of the kind of time in which it is counted. To complete our explanation of the almanac page 29 (our p. 76), it remains to give an example of a calculation of the sun's declination. This is an angle in degrees and minutes, and it is interpolated just like the equation by the aid of its H. D. Thus, for Dec. 29, 22\58 (p. 80) the declination is obtained thus : Dec. 29, 22*, declination = 23 11 '.9 H.D. (O'.l) X A .58 0.1, declination decreasing ; by subtraction, at 22*.58, dec. = 23 11 '.8, and according to the almanac, this declination must be given the minus sign. When the sign should be +, that fact is indicated in the almanac. The use of the declination will be explained later; the accuracy required in the interpo- lation of it is not so great as we have used here, for the nearest minute suffices in practically all navigation work. In addition to the sun's declination, navigators require THE NAUTICAL ALMANAC 83 in their further calculations another number called the sun's "right ascension" (abbreviated, R. A.). This is obtained from pages like the almanac page 3 (reprinted in part below). It is always the R. A. of the "mean sun" that we need, and the almanac gives it for Greenwich mean noon of each day in the year. When needed in our further calcula- tions, it is of course always required for the exact moment when a sextant observation was made. In fact, this state- ment applies also to the equation of time and declination. They must always be interpolated from the almanac for the moment when the navigator actually observed the sun ; and SUN, 1917. From Nautical Almanac, p. 3 DAY OF MONTH RIGHT ASCENSION OF THE MEAN SUN AT GREENWICH MEAN NOON July August September October November December h m s h m s h m s h m s h m s h m s 1 6 35 52.2 8 38 5.5 10 40 18.7 12 38 35.3 14 40 48.4 16 39 5.1 2 6 39 48.8 8 42 2.0 10 44 15.2 12 42 31.8 14 44 45.0 16 43 1.7 3 6 43 45.3 8 45 58.6 10 48 11.8 12 46 28.4 14 48 41.5 16 46 58.2 4 6 47 41.9 8 49 55.1 10 52 8.3 12 50 24.9 14 52 38.1 16 50 54.8 5 6 51 38.4 8 53 51.7 10 56 4.9 12 54 21.5 14 56 34.6 16 54 51.3 6 6 55 35.0 8 57 48.2 11 1.4 12 58 18.0 15 31.2 16 58 47.9 7 6 59 31.6 9 1 44.8 11 3 58.0 13 2 14.6 15 4 27.8 17 2 44.5 8 7 3 28.1 9 5 41.4 11 7 54.5 13 6 11.1 15 8 24.3 17 6 41.0 9 7 7 24.7 9 9 37.9 11 11 51.1 13 10 7.7 15 12 20.9 17 10 37.6 10 7 11 21.2 9 13 34.5 11 15 47.6 13 14 4.2 15 16 17.4 17 14 34.1 11 7 15 17.8 9 17 31.0 11 19 44.2 13 18 0.8 15 20 14.0 17 18 30.7 12 7 19 14.3 9 21 27.6 11 23 40.8 13 21 57.3 15 24 10.5 17 22 27.2 13 7 23 10.9 9 25 24.1 11 27 37.3 13 25 53.9 15 28 7.1 17 26 23.8 14 7 27 7.4 9 29 20.7 11 31 33.9 13 29 50.4 15 32 3.6 17 30 20.4 15 7 31 4.0 9 33 17.2 11 35 30.4 13 33 47.0 15 36 0.2 17 34 16.9 16 7 35 0.6 9 37 13.8 11 39 27.0 13 37 43.6 15 39 56.8 17 38 13.5 17 7 38 57.1 9 41 10.4 11 43 23.5 13 41 40.1 15 43 53.3 17 42 10.0 18 7 42 53.7 9 45 6.9 11 47 20.1 13 45 36.7 15 47 49.9 17 46 6.6 19 7 46 50.2 9 49 3.5 11 51 16.6 13 49 33.2 15 51 46.4 17 50 3.2 20 7 50 46.8 9 53 0.0 11 55 13.2 13 53 29.8 15 55 43.0 17 53 59.7 21 7 54 43.4 9 56 56.6 11 59 9.7 13 57 26.3 15 59 39.5 17 57 56.3 22 7 58 39.9 10 53.1 12 3 6.3 14 1 22.9 16 3 36.1 18 1 52.8 23 8 2 36.5 10 4 49.7 12 7 2.8 14 5 19.4 16 7 32.6 18 5 49.4 24 8 6 33.0 10 8 46.2 12 10 59.4 14 9 16.0 16 11 29.2 18 9 46.0 25 8 10 29.6 10 12 42.8 12 14 55.9 14 13 12.5 16 15 25.8 18 13 42.5 26 8 14 26.1 10 16 39.4 12 18 52.5 14 17 9.1 16 19 22.3 18 17 39.1 27 8 18 22.7 10 20 35.9 12 22 49.0 14 21 5.6 16 23 18.9 18 21 35.6 28 8 22 19.2 10 24 32.4 12 26 45.6 14 25 2.2 16 27 15.4 18 25 32.2 29 8 26 15.8 10 28 29.0 12 30 42.2 14 28 58.8 16 31 12.0 18 29 28.7 30 8 30 12.4 10 32 25.6 12 34 38.7 14 32 55.3 16 35 8.6 18 33 25.3 31 8 34 8.9 10 36 22.1 12 38 35.3 14 36 51.9 16 39 5.1 18 37 21.9 84 NAVIGATION CORRECTION TO BE ADDED TO R. A. M. S. AT G. M. N. FOR TIME PAST NOON From Nautical Almanac, p. 3, Continued TIME Q m 6- 12 IS- 24 30 36 4 2 48 TIME h m s m s m s m s m s m s m s m s m s h 12 1 58.3 1 59.3 2 0.2 2 1.2 2 2.2 2 3.2 2 4.2 2 5.2 2 6.2 12 13 2 8.1 2 9.1 2 10.1 2 1L1 2 12.1 2 13.1 2 14.0 2 15.0 2 16.0 13 14 2 18.0 2 19.0 2 20.0 2 20.9 2 21.9 2 22.9 2 23.9 2 24.9 2 25.9 14 15 2 27.8 2 28.8 2 29.8 2 30.8 2 31.8 2 32.8 2 33.8 2 34.7 2 35.7 15 16 2 37.7 2 38.7 2 39.7 2 40.7 2 41.6 2 42.6 2 43.6 2 44.6 2 45.6 16 17 2 47.6 2 48.5 2 49.5 2 50.5 2 51.5 2 52.5 2 53.5 2 54.5 2 55.4 17 18 2 57.4 2 58.4 2 59.4 3 0.4 3 1.4 3 2.3 3 3.3 3 4.3 3 5.3 18 19 3 7.3 3 8.3 3 9.2 3 10.2 3 11.2 3 12.2 3 13.2 3 14.2 3 15.2 19 20 3 17.1 3 18.1 3 19.1 3 20.1 3 21.1 3 22.1 3 23.0 3 24.0 3 25.0 20 21 3 27.0 3 28.0 3 29.0 3 29.9 3 30.9 3 31.9 3 32.9 3 33.9 3 34.9 21 22 3 36.8 3 37.8 3 38.8 3 39.8 3 40.8 3 41.8 3 42.8 3 43.7 3 44.7 22 23 3 46.7 3 47.7 3 48.7 3 49.7 3 50.6 3 51.6 3 52.6 3 53.6 3 54.6 23 the Greenwich time of this event is of course always taken from the chronometer (duly corrected for error and rate). Thus, if the R. A. of the mean sun is required for Dec. 29, 22* 34 W 40 8 .4, G. M. T. (p. 80), we find from the almanac page 3 (our p. 83) that the R. A. of the mean sun at Green- wich mean noon is 18* 29 m 28'.?. 1 This, according to the sup- plementary table quoted above from page 3, must be increased by a correction for "time past noon." In this case the time past noon is 22* 34 W 40*.4. The tabular correction for 22* 30 m is 3 m 41.8, and for 22* 36 W it is 3 m 42'.8. Ours falls between these two, and an interpolation makes the correction 3 m 42 S .6. Consequently, the R. A. of the mean sun for Dec. 29, 22* 34" 40-.4, G. M. T. is 18* 29 m 28'.7 + 3 W 42'.6 = 18* 33 M 11-.3. It will be noticed that the small supplementary table (quoted above from almanac page 3) only runs from 12* to 24*. The other half of the table, from 0* to 12*, is printed on the opposite page 2 of the almanac. There is also another longer table, printed near the end of the almanac, and there called Table. Ill, from which the supplementary correction can be taken without the necessity of interpolation. It is not absolutely essential that the navigator learn what 1 Right ascensions are always thus measured in hours, minutes, and seconds, like time, and they are counted from 0* to 24*. THE NAUTICAL ALMANAC 85 the words "right ascension" and " decimation" really mean. But for the benefit of those who are curious in such matters we may state that these numbers locate the position of the sun (or of a star) on the sky. The sky is a great globe, called by astronomers the "celestial sphere," and all heavenly bodies are located upon it precisely as points on the earth are there located by their latitudes and longitudes (p. 3). There is a "celestial equator" with two "celestial poles," corresponding accurately to the terrestrial equator and poles. Declination then corresponds exactly to latitude on the earth, and so it measures the distance of a heavenly body from the celestial equator. When the body is north of the celestial equator, the declination is called +. Right ascension similarly corresponds to longitude ; and for the beginning point of right ascensions on the sky there is a "celestial Greenwich," which is called the "vernal equinox." After this brief digression into astronomy, we return to our subject. We have seen (p. 82) that observations of the sun will tell us only apparent solar time, because it is only the visible sun that we can observe. If the observations are made upon a star, the kind of time is different from any so far mentioned. It is called "sidereal time," or star time. It is always possible to change mean solar time into sidereal time, and vice versa, by a simple process of calculation ; but the only change of this kind required in navigation is the transformation of G. M. T. into Greenwich sidereal time. To make this transformation, we have only to take from the almanac, for the given G. M. T., the R. A. of the mean sun, and then to add it to the given G. M. T. Thus, to find the Greenwich sidereal time corresponding to Dec. 29, 22* 34" 40-.4, G. M. T., we have already found (p. 84) that the R. A. of the mean sun = 18*33 m 11-.3 To this must be added the given G. M. T. = 22 34 40.4 Sum = corresponding Greenwich sidereal time = 17 A1 7 m 51*.7 1 The number of hours was here really 41* : but whenever it is larger than 24*, we must drop or reject 24*. CHAPTER VIII OLDER NAVIGATION METHODS WE shall now explain in detail certain standard methods of determining a ship's latitude and longitude by means of sextant observations. An understanding of these methods is essential to a proper comprehension of the newer naviga- tional processes to be described later ; and the older methods are in fact still very widely used at sea, although most re- cent authorities believe they should be rejected in favor of the newer procedure. The simplest of these older processes, and the one most frequently employed, is the determination of the ship's latitude by a noon or " meridian" observation (" noon- sight") of the sun's altitude (p. 61). Now the sun is higher in the sky at noon than it is at any other time during the day; and so it is possible to get the noon-sight by be- ginning to observe the sun with the sextant a few minutes before noon, and continuing the observation as long as the sun's altitude is increasing. The moment it begins to diminish, or the sun to "dip," as sailors say, the observation should be terminated, and the vernier read. The altitude thus observed will be an altitude of the lower limb (p. 71) ; and before it is used further it must be fully corrected for index error ; for refraction parallax and semi- diameter ; and for dip ; all as in the example on p. 73, where the observed altitude was 30 28', and we found the corrected altitude to be 30 38'. Next, the sun's declination must be taken from the al- manac, being interpolated for the Greenwich time of the "86 OLDER NAVIGATION METHODS 87 observation, as in the example on p. 82, where we found the decimation to be - 23 12' on Dec. 29, at 22* 34 W 40'.4, G. M. T. We shall suppose the above altitude 30 28' to have been observed at the Greenwich time stated, so as to make use of the results of our former calculated examples. Nor is there any inconsistency in supposing a noon observa- tion to have been made at 22* 34 W 40*.4. For the noon observation is made when it is noon on board ship, while the 22* 34 m 40-.4 is the G. M. T. at the same moment. The difference is simply the time-difference (p. 80) between Greenwich and the ship. The calculation of the ship's latitude is now made by the following formula : Latitude = 90 + Declination - Altitude. In this formula, the plus sign signifies that the declination must be added; and the minus sign signifies that the altitude must be subtracted. Furthermore, it is most important to remember that if the declination is itself a " minus declina- tion," as in this example, the addition of it according to the formula is really a subtraction. Or, in other words, and in general, whenever a formula calls for an addition, and the number to be added is a minus number, then that number must be subtracted instead of added. And similarly, if the formula calls for a subtraction, and the number to be sub- tracted is a minus number, then that number must be added instead of subtracted. Two minus signs neutralize each other. In the present case we have, omitting seconds : 90 0' declination =-23 12 90 + declination = 66 48 altitude = 30 38 latitude = 36 10 In considering this result it is of interest to inquire where this observation really locates the ship. Now we have not yet stated what the date was, on board, when the observa- 88 NAVIGATION tion was made ; but we have given the G. M. T. as Dec. 29, 22* 34 OT 40*.4. The noon-sight was taken, as a matter of fact, at noon on Dec. 30, or at the moment when the date Dec. 30 commenced by astronomic reckoning. Therefore the ship's time was later than the Greenwich time by about 1* 25"*; or 21 15', allowing 15 to 1* (p. 81) ; and the ship was (approximately) in 21 15' east longitude from Greenwich. This, together with the latitude 36 10', locates the ship in the Mediterranean, south of Greece, and west of Candia. Although we have thus apparently located the ship com- pletely in latitude and longitude from a single noon-sight, it must not be supposed that we have really accomplished this. The noon-sight is only suitable for ascertaining the ship's latitude ; the longitude is determined so inaccurately as to be practically useless. The reason for this is that near noon the sun changes its altitude very slowly, because it is then near the turning-point where its upward morning motion is about to become a downward afternoon motion. For the sun's daily motion in the sky is upward in the morn- ing and downward in the afternoon. Near noon it runs along horizontally, or very nearly so, for several minutes, so that its altitude 'change is insignificant during that time. It follows from this temporary invariability of altitude that we cannot determine the exact moment when noon occurs by observing altitude changes with the sextant. But the latitude determination is not affected; because, for the latitude, we only need to know the noon altitude. And if we happen to measure it a little too soon or too late, on account of the difficulty of fixing the moment of noon, no harm will result, because the altitude very near noon is the same as it is at noon precisely, as we have just seen. It is, in general, practically impossible to determine both latitude and longitude from a single observation. To deter- mine two unknown things, at least two different observations must be made. Nor can any skillful method of planning the observation overcome this fundamental circumstance. OLDER NAVIGATION METHODS 89 Returning now to our latitude formula (p. 87), it is necessary to modify it somewhat in case we happen to be in the tropics, where the sun may pass between the zenith and the celestial pole. Even in temperate latitudes a celestial body may do this, if we happen to observe a star instead of the sun. In such a case, if the ship is in the northern hemisphere, the navigator will observe the sun's altitude toward the north at noon instead of toward the south, as usual. Furthermore, in very high northern latitudes, the " midnight sun," as it is called, can be observed toward the north, and below the celestial pole. This is the minimum altitude during the day, instead of the maximum ; but it is usable for a latitude determination. Such an observation is called a " lower transit" ; and it can often be observed in the case of stars in temperate latitudes. If we now remember to call northerly latitudes and declinations plus, and southerly ones minus, we have the following complete set of formulas for the present problem, including observations in both hemispheres. These formulas are so arranged that we can easily choose the right formula, by having regard to the -f- and signs. But the right formula once chosen, the latitude is calculated without marking declinations with either the -f or sign. lat. 1 and if lat. greater than dec., lat. = 90 + dec. - alt. (1) dec. both + if dec. greater than lat., lat. = dec. + alt. - 90 (2) or both - if lower transit, lat. = 90 + alt. - dec. (3) lat. and dec., 1 lat = 9QO _ a]t _ dec (4) one +, one - j We shall now give some more examples ; and to enable the reader to follow star observations correctly we reprint part of the upper halves of pages 94 and 95 (our pp. 91, 92) of the Nautical Almanac. These contain the right ascensions and declinations (p. 85) of a quantity of bright stars for various dates in the year. These numbers are correct for the moment of " upper transit," which is the moment when these 1 Latitude and declination are abbreviated lat. and dec. 90 NAVIGATION stars attain their maximum altitudes. This event cannot be called a noon-sight in the case of a star ; but it is observable in a manner perfectly similar to a solar noon-sight. These stellar right ascensions and declinations change so slowly that it is unnecessary to use interpolation when taking them from the almanac pages. Proceeding now to our examples, suppose that on shore, at Sandy Hook Light, approximate latitude and longitude 40 28' N., 74 0' W., on Monday, Dec. 17, 1917, at noon, the double altitude of the sun's lower limb was observed with a sextant and artificial horizon, and found to be 51 48'. The index correction required by the sextant was + 4' ; and the G. M. T. by chronometer was 4* 56 m at the moment the observation was made. Find the latitude. We have : Observed double altitude 51 48' (1) Index correction + 4 (2) Adding (1) and (2) gives corrected double altitude 51 52' (3) Halving (3) gives observed altitude 25 56 (4) Correction from Table 6 1 (p. 247) 4- 1 (5) Adding (4) and (5) gives fully corrected altitude 26 10' (6) Now use formula (4) (p. 89) because latitude is + and declination is - . Write 90 (7) Subtracting (6) from (7) gives 90 - corrected altitude . . 63 50 (8) Interpolate declination from almanac (p. 76). This gives declination 23 22 (9) Subtracting (9) from (8) gives for the latitude 40 28 (10) With regard to the foregoing example it is worth remark- ing that if there had been no available chronometer set to Greenwich time, it would still have been possible to calculate the observation. For the known approximate longitude, even if only a dead-reckoning (p. 5) longitude, would be quite accurate enough to make possible the interpolation of the declination from the almanac. And in the present example, the chronometer was only used in getting the declination printed in line (9) above. 1 Dip correction from Table 7 not needed because the artificial horizon was used. OLDER NAVIGATION METHODS 91 APPARENT PLACES OF STARS, 1917 From Nautical Almanac, p. 94 FOR THE UPPER TRANSIT AT GREENWICH No. CONSTELLA- TION NAME RIGHT ASCENSION > "3 - i < I I 1 1 h m s s s s a s a s s s 1 aAndrom. 4 6.3 6.4 7.4 8.4 9.4 10.0 10.3 10.3 10.0 9.6 2 ft Cassiop. 4 44.8 44.4 45.7 47.3 48.7 49.7 50.1 49.9 49.3 48.4 3 PCeti 039 26.5 26.3 27.0 28.0 28.9 29.7 30.0 30.1 29.8 29.5 4 5 Cassiop. 1 20 23.9 22.3 23.5 25.1 26.7 28.1 28.9 29.2 29.0 28.2 5 Urs. Min. 1 29 89.0 22.9 45.5 77.6 112.8 142.4 161.2 166.4 155.3 129.0 6 a Eridani 1 34 39.1 36.8 37.6 38.8 40.3 41.5 42.3 42.4 41.9 41.1 7 a Arietis 2 2 31.0 30.1 30.8 31.7 32.7 33.6 34.3 34.6 34.7 34.5 8 & Eridani 255 8.8 6.8 7.2 7.9 9.0 10.0 10.8 11.3 11.4 11.0 9 a Persei 3 18 25.9 23.9 24.4 25.5 26.8 28.2 29.3 30.2 30.6 30.5 10 Tauri 431 11.7 10.3 10.5 11.0 11.9 12.8 13.7 14.5 15.0 15.2 11 ft Orionis 5 10 35.1 33.7 33.7 34.2 34.7 35.6 36.5 37.3 37.8 38.1 12 a Aurigae 5 10 36.5 34.5 34.6 35.2 36.2 37.5 38.7 39.9 40.7 41.1 13 y Orionis 5 20 43.1 41.7 41.7 42.1 42.8 43.7 44.6 45.4 46.0 46.4 14 e Orionis 5 32 2.4 1.0 1.0 1.3 2.0 2.8 3.7 4.5 5.2 5.5 15 a Orionis 550 43.1 41.8 41.7 42.0 42.7 43.5 44.4 45.3 46.0 46.4 16 Argus 6 22 9.2 6.1 5.5 5.4 6.0 6.9 8.1 9.3 10.2 10.6 17 a Can. Maj. 641 31.6 30.2 30.0 30.1 30.6 31.3 32.2 33.1 33.8 34.3 18 e Can. Maj. 655 24.1 22.6 22.2 22.2 22.6 23.3 24.2 25.2 26.0 26.5 19 a Can. Min. 734 59.7 59.0 58.7 58.8 59.1 59.8 60.5 61.5 62.3 63.0 20 /3 Gemin. 740 17.1 16.3 16.0 16.0 16.4 17.1 18.0 19.0 20.0 20.8 21 Argus 820 51.4 49.0 48.0 47.3 47.2 47.8 48.9 50.4 51.8 52.8 22 A Argus 9 4 58.6 57.9 57.3 56.9 56.8 57.1 57.8 58.9 60.1 61.0 23 ft Argus 9 12 20.6 18.1 16.4 15.1 14.5 14.8 16.0 17.9 20.0 21.7 24 a Hydra 9 23 32.5 32.6 32.2 32.0 32.0 32.3 32.9 33.7 34.7 35.6 25 a Leonis 10 3 59.2 59.7 59.3 59.1 59.0 59.2 59.7 60.5 61.4 62.4 Had it been thus necessary to get the declination without using the chronometer, we should have proceeded as follows : Apparent solar time of noon (p. 75) 0* Approximate longitude = 74 0' W. = (at 15 to the hour) Adding (1) and (2) (p. 81) gives approximate Greenwich apparent time Approx, eq. of time, Dec. 17, at 4 A 56 m (p. 76) Subtracting 1 (4) from (3) gives approximate G. M. T Declination interpolated for G. M. T. in line (5) is 23 1 The equation is additive to G. M. T., according to the note at the foot of p. 76, and therefore to be subtracted from Greenwich apparent time. Oft o w (1) 4 56 W. (2) 4 56 + 4 (3) (4) 4 52 23 22' (5) (6) 92 NAVIGATION APPARENT PLACES OF STARS, 1917 From Nautical Almanac, p. 95 FOR THE UPPER TRANSIT AT GREENWICH DECLINATION "Nn \r . _ \ INO. I ^5 3 1 Ei 1 1 1 I w i SPECIAL NAME MAG.* / / / / / i / , / 1 + 28 38.2 38.1 38.0 38.0 38.0 38.4 38.5 38.5 38.5 Alpheratz 2.2 2 + 58 41.9 41.8 41.7 41.6 41.5 42.0 42.1 42.2 42.2 Caph 2.4 3 - 18 26.5 26.5 26.5 26.4 26.3 26.0 26.1 26.2 26.2 Deneb Kaitos 2.2 4 + 59 48.7 48.7 48.6 48.4 48.3 48.6 48.8 48.9 49.0 Ruchbah 2.8 5 + 88 52.2 52.2 52.1 52.0 51.8 52.0 52.2 52.4 52.5 Polaris 2.1 6 -57 39.7 39.7 39.6 39.4 39.2 39.0 39.2 39.3 39.4 Achernar 0.6 7 + 23 4.5 4.4 4.4 4.3 4.3 4.6 4.7 4.7 4.7 Hamal 2.2 8 -40 38.3 38.3 38.3 38.2 38.1 37.7 37.8 38.0 38.1 Acamar 3.0 9 + 49 34.3 34.3 34.3 34.2 34.1 34.3 34.3 34.4 34.5 1.9 10 + 16 20.7 20.7 20.7 20.7 20.7 20.8 20.8 20.8 20.8 Aldebaran 1.1 11 - 8 17.8 17.8 17.9 17.9 17.8 17.5 17.6 17.7 17.7 Rigel 0.3 12 + 45 55.0 55.1 55.1 55.1 55.0 54.9 54.9 55.0 55.1 Capella 0.2 13 + 6 16.6 16.5 16.5 16.5 16.5 16.7 16.7 16.6 16.6 Bellatrix 1.7 14 - 1 15.2 15.3 15.3 15.3 15.3 15.0 15.1 15.1 15.2 Alnitam 1.8 15 + 7 23.6 23.5 23.5 23.5 23.5 23.7 23.7 23.6 23.6 Betelgeux 1.0-1.4 16 -52 39.0 39.2 39.3 39.3 39.2 38.7 38.7 38.9 39.1 Canopus -0.9 17 - 16 36.1 36.2 36.3 36.3 36.3 35.9 36.0 36.1 36.2 Sirius - 1.6 18 -28 51.5 51.7 51.7 51.8 51.7 51.3 51.4 51.5 51.6 Adhara 1.6 19 + 5 26.3 26.2 26.2 26.2 26.2 26.3 26.2 26.2 26.1 Procyon 0.5 20 + 28 13.6 13.6 13.6 13.7 13.7 13.5 13.5 13.4 13.4 Pollux 1.2 21 -59 14.4 14.6 14.8 14.9 14.9 14.4 14.4 14.5 14.7 1.7 22 - 43 5.7 5.9 6.1 6.2 6.2 5.8 5.8 5.9 6.0 2.2 23 - 69 22.4 22.6 22.8 22.9 23.0 22.5 22.4 22.5 22.7 Miaplacidus 1.8 24 - 8 17.9 18.1 18.1 18.2 18.2 18.0 18.0 18.1 18.2 Alphard 2.2 25 + 12 22.2 22.2 22.2 22.2 22.2 22.2 22.1 22.0 21.9 Regulus 1.3 1 When the number in this column is very small, and especially when it is minus, the star is very bright. It is further to be noted that as we can thus obtain the approximate G. M. T., we really know in advance the approx- imate moment when the observation should be made. So it is unnecessary to get the sextant ready a long time before the observation ; and it is, in fact, better to observe at the proper predetermined approximate moment rather than to wait for the maximum altitude (p. 86). When the ship's position at noon can be predicted with fair approximation, it is thus possible to have the declination and other numbers for calculating the noon-sight also all ready OLDER NAVIGATION METHODS 93 in advance, so that the latitude will be immediately available when the noon altitude has been read from the sextant. We shall now consider the following example : Off St. Paul de Loando, West Africa, approximate latitude 8 55' south, approximate longitude 12 55' east, both predicted in advance by D. R. for noon on Monday, Dec. 31. The altitude of the sun's lower limb is to be measured. Index correction is 5'. Height of eye, 26 ft. To prepare for the observation, we have, as before : Apparent solar time of noon 0* O" 1 (1) Approximate D. R. longitude = 12 55' east = (at 15 to the hour) 52 E. (2) Subtracting (2) from (1) gives approximate Greenwich apparent time, Dec. 30 23 8 (3) Approximate equation of time, Dec. 30, at 23* 8 m (p. 76) - 3 (4) Subtracting (4) from (3), having regard to sign of (4), gives approximate G. M. T 23 11 (5) The navigator will then make the observation when the G. M. T. is 23* ll m , as indicated by the chronometer, duly corrected for error and rate. This would of course also be noon, or the time when the sun attained its maximum altitude for the day. Now the dials of chronometers are always divided into 12 hours, like ordinary watches, although navigators count time through 24 hours, as we have seen (p. 75). The reason is that the dial would be overloaded with numbers if there were 24 hour divisions. Therefore, when we speak of the chronometer indicating 23* ll m , it must be under- stood that the actual chronometer indication, or "chro- nometer face," as it is sometimes called, would really be 11* ll m ; only, the navigator would call it 23* ll m , astronomic time. In this manner civil time still forces its way into navigation, by way of the chronometer face. To make the observation at the prearranged G. M. T. by chronometer it is not desirable to carry that instrument out into the sunlight, where the observer stands. It is much 94 NAVIGATION better for the navigator to use his watch, and to calculate in advance the " watch time" of the observation. To do this, it is merely necessary to compare the watch with the chro- nometer, and thus ascertain how much the watch is slow or fast of the chronometer. This amount is called "chro- nometer minus watch" (abbreviated C. W.) ; and when the watch is fast of the chronometer, C. W. is marked with the minus sign. To obtain the watch time for the observation, we subtract C. W. from the G. M. T. In the present case we will suppose the watch was 47 m fast of the chronometer. Then C. W. = 47 m . To get the watch time for the observa- tion we must subtract 47 m from 23 h ll m . Subtracting a minus number is equivalent to addition ; and so the watch time is 23 h ll m + 47 m = 23 h 58 TO . The observation would be made as nearly as possible 2 m before noon, by the watch. In this connection it also becomes of interest to inquire how the navigator's watch happened to be 47 TO fast of the chronometer. It is customary aboard ship to set the deck and cabin clocks, and all watches, to the ship's local apparent time once a day at least. To do this, we proceed as follows : Take from chronometer the G. M. T., corrected for error and rate (1) Apply to this G. M. T. the eq. of time, giving Green'h app. time (2) Apply to (2) the approximate D. R. longitude, adding it if longi- tude is E., which gives ship's apparent time (3) And set the watch to the time (3). An example of this proceeding can be had from the data on p. 93. Suppose the watch was to be set; and the chro- nometer time was 23 h TO . We should then prepare to set the watch in about 5 m , when the G. M. T. by chronometer would be 23* 5 TO (1) Chronometer error (corrected for rate) say - 2 (2) Corrected G. M. T. by chronometer, (1) + (2) 23 3 (3) Equation of time (p. 93) - 3 (4) Greenwich apparent time, (3) + (4) 23 (5) Approximate longitude (p. 93) 52 E. (6) Ship's apparent time, (5) +(6) 23 52 (7) OLDER NAVIGATION METHODS 95 And the watch would be set to 23 ft 52 m , when the chro- nometer face was 23* 5 m ; or, which is the same thing, the watch would be set at 8 m to 12 when the chronometer in- dicated 5 minutes past 11. Sometimes the navigator wishes the watch to be correct by ship's apparent time at noon, but desires to set it right half an hour sooner, so as to be free at noon to make an observation. In that case he calculates by D. R. what the longitude will be at noon, and proceeds practically in the same way as before. Resuming now the example of p. 93, we are still off St. Paul de Loando, and at 2 m before noon by the watch (p. 94) the altitude of the sun's lower limb was measured. Suppose it was found to be 75 34' (1) The index correction was 5 (2) Adding (1) and (2), with regard to sign of (2), gives corrected altitude 75 29 (3) Correction from Table 6 +16 (4) Correction from Table 7, for 26 ft. height of eye 5 (5) Adding (3), (4), (5) gives corrected altitude 75 40 (6) Formula (2), p. 89, is the proper one, and the inter- polated declination, disregarding sign, is 23 8 (7) Latitude, by formula, is (6) + (7) - 90, or 8 48 (8) The latitude of the ship is therefore 8 48' south, from the above noon-sight observation. The difference of 7' from the approximate latitude (p. 93) might easily be caused by ocean currents. Our next example is a star observation. Position of ship by D. R. March 23, 1917, at 6 a 30"* ship's time is : latitude 40 25' N., longitude 46 52' W., so that she is near the turning point in the southern "lane route" followed by steamships bound from New York to Fastnet in summer. The upper transit (p. 89) of Sirius was observed; and the sextant altitude was 33 7'. Index correction, 7' ; height of eye, 24ft. 96 NAVIGATION The calculation is as follows : Observed altitude of Sirius 33 7' (1) Index correction 7 (2) Adding (1) and (2), having regard to minus sign of (2), gives corrected altitude 33 (3) Correction Tables 6 and 7, combined 6 (4) Adding (3) and (4) gives finally corrected altitude .... 32 54 (5) Use formula (4), p. 89, because latitude is + and decli- nation of Sirius . We have 90 (6) Subtract (5) from (6), giving (90 - altitude) . ... 57 6 (7) Declination of Sirius (p. 92), disregarding sign, is. . . 16 36 (8) Subtract (8) from (7), giving (90- altitude -declina- tion), or the latitude 40 30 (9) Ship's latitude at the moment of observation was therefore 40 30' N. In making such a star observation, it is of course possible to follow the star with the sextant until it begins to dip (p. 86) toward the horizon exactly as we have ex- plained for the sun. But it is preferable to prepare for the observation in advance, and to make it at a definite prede- termined minute by the navigator's watch. To make such preparation, it is necessary to use pages 96 and 97 of the Nautical Almanac, parts of which pages are reprinted here (pp. 97, 98). The almanac page 96 gives for all the bright stars the G. M. T. of upper transit (p. 158) at Greenwich, for the first day of each month. And it will be noticed that the upper transit is here called " meridian transit," which is practically another name for the same thing. Almanac page 97 (our p. 98) then gives a subtractive correction, applicable to the numbers on page 96, to make them correct on days of the month other than the 1 st . Another small correction is still required to make the numbers right in the approximate D. R. longitude of the ship, instead of the longitude of Greenwich, as used on almanac page 96. This correction is subtractive, if the ship is in west longitude, and additive, if she is in east longitude ; and the OLDER NAVIGATION METHODS 97 MERIDIAN TRANSIT OF STARS, 1917 From Nautical Almanac, p. 96 GREENWICH MEAN TIME OP TRANSIT AT GREENWICH CONSTELLA- TION NAME MAG. z 4 H 1 < 1 (H ! 1 1 t Q h m h m h m h m h m h m h m h m h m a Androm. 2.2 5 21 3 19 1 29 23 23 21 25 13 22 11 24 9 22 7 24 /3 Cassiop. 2.4 5 22 3 20 1 30 23 24 21 26 13 22 11 24 9 22 7 24 PCeti 2.2 5 56 3 54 2 4 f 2) ) 23 68 S 22 13 57 11 59 9 57 7 59 8 Cassiop. 2.8 6 37 4 35 2 45 43 22 41 14 38 12 40 10 38 8 40 a Urs. Min. 2.1 6 47 4 45 2 54 52 22 50 14 49 12 51 10 49 8 51 a Eridani 0.6 6 51 4 49 2 59 57 22 55 14 52 12 54 10 52 8 54 a Arietis 2.2 7 19 5 17 3 27 1 25 23 23 15 20 13 22 11 20 9 22 Eridani 3.0 8 12 6 10 4 20 2 18 20 16 12 14 14 12 12 10 14 a Persei 1.9 8 35 6 33 4 43 2 41 43 16 35 14 38 12 36 10 38 a Tauri 1.1 9 47 7 46 5 55 3 54 1 56 17 48 15 50 13 48 11 50 /3 Orionis 0.3 10 27 8 25 6 35 4 33 2 35 18 27 16 29 14 28 12 30 a Aurigse 0.2 10 27 8 25 6 35 4 33 2 35 18 27 16 29 14 28 12 30 y Orionis 1.7 10 37 8 35 6 45 4 43 2 45 18 37 16 39 14 38 12 40 Orionis 1.8 10 48 8 46 6 56 4 54 2 56 18 49 16 51 14 49 12 51 a Orionis 1.0-1.4 11 7 9 5 7 15 5 13 3 15 19 7 17 9 15 7 13 9 a Argus -0.9 11 38 9 36 7 46 5 44 3 46 19 39 17 41 15 39 13 41 a Can. Maj. -1.6 11 57 9 55 8 5 6 3 4 5 19 58 18 15 58 14 eCan. Maj. 1.6 12 11 10 9 8 19 6 17 4 19 20 12 18 14 16 12 14 14 a Can. Min. 0.5 12 51 10 49 8 59 6 57 4 59 20 51 18 53 16 52 14 54 /3 Gemin. 1.2 12 56 10 54 9 4 7 2 5 4 20 57 18 59 16 57 14 59 Argus 1.7 13 36 11 34 9 44 7 42 5 44 21 37 19 39 17 37 15 39 A Argus 2.2 14 20 12 19 10 28 8 27 6 28 22 21 20 23 18 21 16 23 Argus 1.8 14 28 12 26 10 36 8 34 6 36 22 28 20 30 18 28 16 31 a Hydrse 2.2 14 39 12 37 10 47 8 45 6 47 22 40 20 42 18 40 16 42 a Leonis 1.3 15 19 13 17 11 27 9 25 7 27 23 20 21 22 19 20 17 22 amount of it is 10* for every 15 in the ship's longitude. After it has been applied, the result will be the ship's mean solar time of the star's upper transit. As an example, let us take the preparation for the fore- going observation of Sirius, or a Can. Maj. We have : G. M. T. of upper transit, March 1, from almanac page 96 above 8* 5 m (1) Correction for 23d day of month, from almanac page 97 (our p. 98) - 1 27 (2) Correcting (1) with (2), having regard to sign of (2) 6 38 (3) Further correction for longitude 46 52' W., at 10* per 15 of longitude, approximately 1 (4) Subtracting (4) from (3) gives ship's mean solar time of the observation 6 37 (5) 98 NAVIGATION MERIDIAN TRANSIT OF STARS, 1917 From Nautical Almanac, p. 97 CORRECTIONS TO BE APPLIED TO THE MEAN TIME OF TRANSIT ON THE FIRST DAY OF THE MONTH, TO FIND THE MEAN TIME OF TRANSIT ON ANY OTHER DAY OF THE MONTH DAY OP MONTH CORRECTION DAY OP MONTH CORRECTION DAY OF MONTH CORRECTION h m h m h m 1 -0 11 -0 39 21 - 1 19 2 4 12 43 22 1 23 3 8 13 47 23 27 4 12 14 51 24 30 5 16 15 55 25 34 6 -0 20 16 -0 59 26 - 38 7 24 17 1 3 27 42 8 28 18 1 7 28 46 9 31 19 1 11 29 50 10 35 20 1 15 30 54 11 -0 39 21 - 1 19 31 - 1 58 NOTE. If the quantity taken from this Table is greater than the mean time of transit on the first of the month, increase that time by 23 h 56 W and then apply the correction taken from this Table. The actual observation was made at 6 A 30 m , ship's time, as indicated by the navigator's watch. The difference of 7 m between 6* 30 W , and 6* 37 W in line (5) above, is due to the equation of time (p. 77), which is 7 m on March 23. This 7 W , if applied (with its proper sign from the almanac) to line (5) above, will give the ship's apparent time; and we have seen that watches and clocks on board are usually kept set to apparent and not mean ship's time (p. 94). To complete this part of our subject, we have still to con- sider a few additional points of interest. For instance, a star chosen for observation may be one of the planets : Mars, Jupiter, or Saturn. These look like very bright stars in the sextant telescope; and calculations depending on them are similar to those described for stars. The planetary declinations and the G M. T.'s of their upper transits are given in the .almanac, but not on the pages reprinted here. OLDER NAVIGATION METHODS 99 The moon is now so rarely observed that we have not given examples of lunar observations. Sometimes an " ex-meridian" observation of the sun or a star is made at a time very near the upper transit, on a day when the actual transit observation could not be secured because of clouds. There are special tables l for calculating observations of this kind ; but we have not included them here because all such observations can be satisfactorily treated by a new general method to be explained later (p. 108). Having now fully treated the older standard method of determining the ship's latitude, let us next consider the older way of obtaining the longitude. This cannot be done when the sun (or a star) is near its maximum altitude, as already explained (p. 88). The most favorable opportunity occurs when the observed object bears (p. 44) east or west ; but it is not always possible to get the observation on such a bearing. In that case, the longitude observation, often called a " time-sight," must be taken when the sun is near the desired bearing, but always avoiding, if possible, observa- tions at very low altitudes. And if a very low altitude has been observed in an emergency, it can sometimes be checked by a later observation at a better altitude. The principle on which the time-sight depends is simple. Calculations based on the measured altitude make known the ship's mean time at the moment of observation. At the same moment the chronometer face (p. 93), duly cor- rected for error and rate, tells us the G. M. T. The difference between the two times then grves us the longitude (see p. 82). The calculations for this problem are made by means of Table 4 (trigonometric logarithms) and Table 10 (" haver- sines"). These haversines (abbreviated hav.) are really additional trigonometric logarithms; and Table 10 gives in every case not only the haversine itself, which is really 1 Tables 26 and 27 of Bowditch's " Navigator," for. instance. 100 NAVIGATION a logarithm, but also, in the adjoining heavy type col- umns, the number (abbreviated No.) of which the haver- sine is the log. This additional heavy type number is not given throughout the entire table, but only when necessary for working Sumner line calculations (see Chapter IX, p. 108). It is not needed in working time-sights. The argument (p. 10) of the haversine table is a double argument, not to be confounded with the pairs of arguments already explained (p. 11). In the haversine table, the argu- ment is generally given in degrees and minutes, as well as (for convenience) in hours and minutes of time, allowing the usual 15 to each hour, etc. We shall now solve our time-sight problem for the sun; and in doing so shall make use of two angles not hitherto employed: the " polar distance" (abbreviated p), and the "half sum" (abbreviated s). We shall also, for brevity, indicate the ship's apparent solar time by T. Then we have the following formulas : If lat. and dec. are both + or both . . p = 90 dec. (1) If lat. and dec. are one + and one . . . p = 90 + dec. (2) In every case s = f (alt. + lat. + p} (3) If time-sight was made before noon, ship's time, hav. (24*- T) = sec lat. + esc p + cos s + sin (s - alt.) (4) If time-sight was made after noon, ship's time, hav. T = sec lat. + esc p + cos s + sin (s alt.) (5) In using these formulas, we have to choose between (1) and (2), and also between (4) and (5). Formula (3) is always used. No attention need be given to the signs of the declination *or latitude except in choosing between formulas (1) and (2) for calculating p; and in choosing between (4) and (5), we have merely to note whether the time-sight was taken in the forenoon or afternoon by ship's time. We also desire to emphasize especially that these formulas presuppose the latitude to be known. This is merely another application of the principle (p. 88) that "both lati- OLDER NAVIGATION METHODS 101 tude and longitude cannot be determined from a single observation. It follows that in using this method we must first determine the latitude by a noon-sight before we can calculate the time-sight for longitude. If the time-sight was taken in the afternoon, the noon-sight will naturally have preceded it, and the ship's latitude at noon will be known. This noon latitude must then be carried forward to the moment of the afternoon time-sight by D. R. methods (p. 7) ; and the latitude thus obtained must be used for calculating the time-sight. But if the time-sight was a forenoon observation, it cannot be properly calculated until noon, when the latitude will be determined. After that, the latitude can be carried backwards by D. R. to the moment of the forenoon time- sight, and the latter can be calculated. But if the navigator, because of emergency, needs his longitude at once, after taking the forenoon time-sight, he must obtain the latitude by a D. R. calculation based on the last good noon-sight. Most navigators calculate morning time-sights in this way, and then repeat the calculation after the new noon-sight has been obtained. The latter calculation will be preferable to the former, because the further the latitude is carried along by D. R., the less accurate will it be. And any error in the latitude used in the calcula- tion will impress a consequent error on the calculated longi- tude. We shall now work some time-sight examples. On board ship, at sea, Dec. 18, 1917, in the afternoon, D. R. latitude 42 20' N., D. R. longitude 35 16' W., the altitude of sun's lower limb was observed to be 14 19'. The time was taken with the navigator's watch, and was 2 h 29 m 58*. A com- parison of the watch and ship's chronometer gave C. W. = 2 h 27 m 8*. The chronometer correction was 2 m 8* slow of G. M. T. The index correction of the sextant was + 4' ; height of eye, 24 ft. Calculate the ship's longitude. We have first to find, for the moment of the observation, 102 'NAVIGATION values of the declination and equation of time. To do this, we have : Watch time of observation 2 A 29 m 58 (1) C. -W 2 27 8 (2) Adding (1) and (2) gives chronometer time of observation 4 57 6 (3) Chronometer correction, slow 2 8 (4) Adding (3) and (4) gives G. M. T. of observation 4 59 14 (5) For the G. M. T. (5) we interpolate the declina- tion (p. 76), finding - 23 24' (6) and for the same G. M. T. we interpolate the equation of time + 3 m 21 s (7) Now, adding (5) and (7) gives Greenwich ap- parent time of observation 5 h 2 m 35 (8) Next we inspect the formulas (p. 100), choosing (2) be- cause latitude is + and declination , and (5) because the sight was an afternoon one. We now have, from line (6), declination (disregard- ing sign) 23 24' (9) to which, by formula (2), we add 90 (10) giving p 113 24 (11) The observed altitude was 14 19 (12) Index correction +4 (13) Adding (12) and (13) gives corrected altitude 14 23 (14) Correction, Table 6 +12 (15) Correction, Table 7 : - 5 (16) Adding (14), (15), (16) gives finally corrected altitude 14 30 (17) The latitude by D. R. is 42 20 (18) Adding (11), (17), (18) gives 170 14 (19) Halving (19) gives (by formula (3), p. 100) s 85 7 (20) Subtracting (17) from (20) gives (s - alt.) 70 37 (21) Next we apply formula (5), p. 100. We have: sec lat. (18) from Table 4, page 238 0.13121 (22) esc p (11) from Table 4, page 219 0.03727 (23) cos s (20) from Table 4, page 200 8.93007 (24) sin (s - alt.) (21) from Table 4, page 215 9.97466 (25) sum (22) to (25) = hav. T, by formula (5) 9.07321 l (26) 1 This sum has been diminished by 10 arbitrarily (see p. 25), which must always be done when the sum of logs is larger than 10. OLDER NAVIGATION METHODS 103 T, 1 corresponding to (26) from Table 10, page 260, is 2 ft 40 W 59* (27) Greenwich apparent time (8) by watch and chronometer is 5 2 35 (28) Subtract (27) from (28), giving time difference between ship and Greenwich 2 21 36 (29) Turning (29) into degrees with Table 9, page 249, gives 35 24' W. (30) and (30) is the ship's longitude from this time-sight. Upon comparing the D. R. longitude (35 16' W.) with the result of the time-sight (35 24' W.), we find that the ship is 8' west of her D. R. position. This means, of course, that there has been a westerly "set" of current in the interval between the last accurate determination of longitude and the present one. It would be proper for the navigator to calculate from this the amount of westerly drift per hour, and to allow for it in carrying forward his longitude by D. R. from the present time-sight. It is also clear that the northerly or southerly set of the current can be similarly measured and allowed for by comparing the D. R. latitude with the latitude from a noon-sight (cf. p. 95). It is the general custom of navigators to ascribe such differences to ocean currents, never to uncertainty in the astronomic results. Dead reckoning is never allowed any weight as against a sextant observation. The reader will have noticed that the foregoing calculation has been made in great detail, so that a beginner may have no difficulty in understanding it. But a practiced navigator would of course work the calculation in a much more con- densed form, in such a way as to bring the logarithms next to the numbers to which they belong. We shall therefore now repeat the same example in such a condensed form : 1 If the observation had been made before noon, we should have used formula (4) and should here have obtained 24* T, instead of T. This 24* - T would then be subtracted from 24 A , to get T, before continuing the calculation. Thus the form of calculation would contain another line between (27) and (28), in the case of a forenoon observation. 104 NAVIGATION TIME-SIGHT, CONDENSED FORM. SUN Watch time : C. - W. : Chr. time : Chr. corr'n : 29" 58* (1) 2 27 4 57 + 2 G. M. T. : IS'" 4 59 3 2 Eq. of time : G. app. time : 5 8 6 8 14 21 35 (2) (3) (4) (5) (7) (8) Obs'd alt. : Index : Table 6 : Table 7 : Corr'd alt. : 14 19' (12) + 4 (13) + 12 (15) - 5 (16) 14 30 (17) Decl. 18 th , 4 A H. D.: Decl. 4 59" : p: 23 e 23 113 23'.7 0.1 24 (6) 24 (11) Eq. time, 18 th , 4* : + 3" 22.3 H. D.: 1.2 Eq. time, 4 ft 59 TO : +3 21.1 (7) Corr'd alt. Lat., D. R p: sum of 3 : s : s alt. : By chron., : 14 30' (17) . : 42 20 (18) sec lat. : 113 24 (11) esc p: 0.13121 (22) 0.03727 (23) 8.93007 (24) 9.97466 (25) 2)170 14 (19) 85 7 (20) cos s : 70 37 (21) sin (s-alt.): sum of 4 : T = ship's app. time : Greenwich app. time : Longitude : or: 9.07321 (26) = hav. T (or 24* - T) i 2 40" 59* (27) 5 2 35 (8) 2" 21"* 36 s (29) 35 24' W. (30) When the object observed is a star or planet, the choice between formulas (4) and (5), p. 100, is not quite the same as in the case of a solar time-sight. We must use (4) if there is any east in the star's bearing at the moment of observation ; and (5), if there is west in the bearing. The more nearly the star bears due east or west, the more accurate will be the resulting longitude. The use of formulas (1), (2), and (3) is the same as for the sun ; but T, in the case of a star, is no longer the ship's apparent solar time. Instead, it is called 1 See p. 103, footnote. OLDER NAVIGATION METHODS 105 the star's " hour-angle." To get the longitude, we must first (p. 85) calculate the Greenwich sidereal time corre- sponding to the G. M. T. of the observation, as taken from the chronometer, duly corrected for error and rate; and then use the following formulas : (6) Greenwich sid. time right-ascension of star = Greenwich hour-angle. ,_ J West long. = Greenwich hour-angle T, \ East long. = T Greenwich hour-angle. As an example of a star observation we shall take the following : At sea, just before sunrise, Dec. 17, 1917, off Cape Agulhas, latitude by D. R. 35 20' S., longitude by D. R. 20 41' E., the altitude of Sirius was measured, and found to be 40 3'. The star bore west, and the height of eye was 22 ft. Index correction was + 5'. Time by watch, 16 A 29 m 48', or 4* 29 W 48* A.M., civil time, Dec. 18; C. - W., - l h 23 OT 50"; chro- nometer fast of G. M. T. 2 m 28 s . The calculation would proceed thus : Watch time of observation 16* 29 48 (1) C. - W - 1 23 50 (2) Adding (1) and (2), having regard to sign of (2), gives chronometer time of observation 15 5 58 (3) Chronometer correction, fast 2 28 (4) Adding (3) and (4), having regard to sign of (4), gives G. M. T. of observation 15 3 30 (5) Right ascension mean sun, Greenwich mean noon, Dec. 17 (p. 83) 17 42 10 (6) Correction for " time past noon " (see p. 84) .... 2 28 (7) Adding (6) and (7) gives right ascension of mean sun 17 44 38 (8) Adding (5) and (8) (see p. 85) gives Greenwich sidereal time of the observation 8 l 48 8 (9) Right ascension of Sirius, Dec. 17, is (p. 91) .... 6 41 34 (10) Subtracting (10) from (9) gives Greenwich hour- angle (formula (6), above) 2 6 34 (11) 1 This is really 32* ; but 24* is dropped arbitrarily. 106 NAVIGATION Next we calculate T by formula (5), p. 100. We have : Declination of Sirius, Dec. 17 (p. 92) - 16 36' (12) By formula (1), p. 100, subtract (12) from 90, without attention to sign of (12), giving p. . 73 24 (13) The observed altitude was 40 3 (14) The index correction was +5 (15) Table 6 correction - 1 (16) Table 7 correction 5 (17) Adding (14), (15), (16), (17), having regard to signs, gives corrected altitude 40 2 (18) The latitude by D. R. was 35 20 (19) Adding (13), (18), and (19) gives 148 46 (20) Halving (20) gives s 74 23 (21) Subtracting (18) from (21) gives (s - altitude) . . 34 21 (22) Now applying formula (5), page 100, we have : sec latitude (19) from Table 4, page 231 0.08842 (23) esc p (13) from Table 4, page 212 0.01849 (24) cos s (21) from Table 4, page 211 9.43008 (25) sin (s - altitude) (22) from Table 4, page 230 9.75147 (26) Summing (23) to (26) gives hav. T 7 , by form. (5) . . 9.28846 * (27) T 2 corresponding to (27), from Tab. 10, p. 263 is . . 3* 29" 14 (28) Difference between (28) and (11) is the longi- tude by formula (7), page 105 1 22 40 E. (29) Turning (29) into degrees with Table 9, page 249, gives 20 40' E. (30) The D. R. longitude, 20 41' E., was therefore within 1' of the longitude from this time-sight, and this shows that the ship has not been affected by ocean currents since the last observation. It is also interesting to note how near sunrise the observation was made. The twilight must have been quite strong, and the star therefore dim. But star observa- tions can be made best in twilight because the horizon line can then be seen distinctly. 1 This sum has also been diminished by 10 (see footnote, p. 102). 2 Might be 24* - T, if the star bore E. instead of W. (see footnote, p. 103). OLDER NAVIGATION METHODS 107 The foregoing example can of course also be arranged in condensed form, as follows : TIME-SIGHT, CONDENSED FORM. STAR Watch time : C. - W. : Chr. time : Chr. corr'n : G. M. T. : R. A. mean sun : Corr'n, past noon : Greenw'h sid. time : R. A. of Sirius : Greenwich hour-ang. T 7 ., from (27) : Long.: or: R. A. of Sirius : Dec. of Sirius : p: sec lat. : esc. p : cos s: sin (s-alt.) : sum of 4 : 16* 29" 48- (1) Obs'd alt.: 40 3' (14) -1 23 50 (2) Index : + 5 (15) 15 5 58 (3) Table 6: -1 (16) -2 28 (4) Table 7: -5 (17) 15 3 30 (5) Corr'd alt. : 40 2 (18) 17 42 10 (6) Lat. D. R. : 35 20 (19) 2 28 (7) p: 73 24 (13) 8 48 8 (9) sum: 2)148 46 (20) 6 41 34 (10) s : 74 23 (21) : 2 6 34 (11) (s-alt.): 34 21 (22) 3 29 14 (28) 1 22 40 E. (29) 20 40' E. (30) 6* 41" 34' (10) - 16 36' (12) 73 24 (13) 0.08842 (23) 0.01849 (24) 9.43008 (25) 9.75147 (26) 9.28846 (27) = hav. T (or 24* - T) Having now fully explained both the noon-sight and the time-sight, we shall close this chapter with a strong recom- mendation to young navigators to familiarize themselves with the observation of stars. These always furnish a valuable check on sun observations : and at times of danger may save the ship when clouds have obscured the sun for days, and clearing occurs after sunset. It is easy to learn to know the principal stars from Jacoby's " Astronomy/' Chapter III, "How to Know the Stars." See footnote, p. 103. CHAPTER IX NEWER NAVIGATION METHODS THE reader may have noticed in Chapter VIII that there is a very definite difference between the determination of latitude by a noon-sight and longitude by a time-sight : for the latitude is obtained without previous knowledge of the longitude; but to get the longitude, a previous knowledge of the latitude is essential. This is, of course, a decided disadvantage in determining longitude, nor is there any practicable direct way to get the longitude without first knowing the latitude. We have also seen (p. 101) that any existing uncertainty in our knowledge of the latitude will produce an error in the longitude computed from a time-sight. In situations of danger it is important to ascertain how great this longitude error may be. Suppose, for instance, we have calculated a time-sight with a D. R. latitude that we suspect may be as much as 10' too small ; and we wish to know how much our computed longitude may have been thereby put wrong. The obvious way to find out is to recompute the longitude with an assumed latitude 10' larger than the D. R. latitude. The resulting longitude will then show the extreme range of error that must have been produced if the D. R. latitude was 10' too small. A third calculation, with an assumed latitude 10' smaller than the D. R. latitude, will similarly exhibit the extreme possible range of longitude error in the other direction. Thus these two extra calculations will show the limits of longitude error that might be caused by a range of 20' in the possible error of the D. R. latitude. 108 NEWER NAVIGATION METHODS 109 This rather obvious procedure was probably used long ago by more than one intelligent navigator ; but it was first published in 1837 by Thomas H. Sumner, an American merchant captain. He used the method in dramatic cir- cumstances of great danger ; and he brought his ship safely into port. According to his own account, he made three calculations of the longitude, using three assumed latitudes differing by 10', and he of course obtained three different longitudes. He then marked or plotted (p. 55) on his chart the point indicated by the first assumed latitude and its computed longitude. At this point the ship must have been located, if the first assumed latitude had been correct. The other two latitudes, with their computed longitudes, indicated two more points on the chart ; and at one of these points the ship must have been, if either of these additional latitudes was correct. Sumner found that the three points on the chart lay in a straight line; and it became at once evident that whatever latitude he might assume (within reason) he would always get a point on the same straight line, after computing the longitude. In other words, although he did not know his latitude accurately, and so could not compute his longitude accurately, yet he had found a straight line on the chart upon which his ship was surely situated. Such a line can always be found in the way Sumner found it, or in some preferable modern way; and such a line we shall call a " Sumner line," though some writers on naviga- tion prefer to call it a "line of position." On the occasion of laying down his line, Sumner found that it passed directly through Small's Light, near the Irish coast ; and as the line bore E.N.E. on his chart, he simply put the ship on that course, and in less than an hour he "made" Small's Light, actually bearing E.N.E. J E., and, as he says, "close aboard." He had had no observations after passing longitude 21 W., until the morning of Dec. 17, when these historic events occurred. He was off a rocky lee shore, in 110 NAVIGATION the midst of a winter gale, after crossing the Atlantic ; only a seaman can understand the relief he must have felt when that light suddenly appeared off the bow. We have given this account of Summer's experience to impress on the young navigator that he must positively familiarize himself with the Sumner method of navigation. Should we be so fortunate as to have any experienced navi- gator among our readers, we ask him to try the Sumner method once more, in the manner explained below, even if he may have found it troublesome in the past on account of certain difficulties in its application. For the Sumner method is the best method of navigation on all oceans and at all times : even when a noon-sight is available for latitude, it is better to treat it as a Sumner observation, and work out the Sumner line. The principal objection urged against it by certain prac- tical navigators arises from the small scale of existing ocean track charts, on which a distance of 10' is represented by about -J inch. A line like Sumner's, 20' long, would have only a length of J inch on the chart ; and such a little line would not be long enough to show accurately the direction in which it pointed. When near a coast, as in Sumner 's case, this difficulty disappears, because navigators always have (or always should have and use) the large scale charts that can be obtained for coastwise waters. But it is inconvenient for navigators to begin using a method off the coast, on the last day of a voyage, different from the form employed for many days at sea. Therefore, some authorities recommend the construction of a special large scale chart, with its latitude and longitude lines, each time an observation is made throughout the voyage, so that the Sumner line can always be drawn on a sufficiently large scale. It is no wonder that navigators have not generally adopted this somewhat laborious proceeding; and in the method given below we shall utilize the Sumner idea without requiring any lines to be drawn on charts. NEWER NAVIGATION METHODS 111 Another objection to Sumner navigation is that it requires too much calculation ; three longitude calculations for one observation, as Sumner practiced it. This objection is also quite removed now by the use of suitable tables such as we give in the present volume. But before proceeding to explain these tables, we must outline briefly the real principle on which rests the com- plete utilization of the Sumner method on the open sea. There the navigator wants to know the ship's position in both latitude and longitude ; and will not be satisfied with a mere line, with the ship " somewhere on the line." Along the coast such a line might help him to find Small's Light ; but he is not looking for coast lights at sea. And the Sumner method takes care of this matter in the simplest possible way. We have seen (p. 88) that two different observations are always necessary by any method to get both latitude and longitude. But two such observa- tions by the Sumner method give two different lines on the chart : and as the ship must be located on both lines, her actual position must be at their point of intersection. We shall show how the required latitude and longitude of the ship at the point of intersection can be found by a simple calculation, without the drawing of any lines on the chart. Coming now to the modern method of calculating a Sum- ner line, we must first state a general fundamental principle that may be easily verified by geometrical considerations. The true bearing (p. 44) of a Sumner line on a chart is always 90 greater than the true bearing or azimuth (p. 44) of the sun (or star) at the moment of observation. Or, in other words, the Sumner line bears at right angles to the sun at the time of observation. We shall show how the bearing or azimuth of the sun can always be found from suitable " azimuth tables"; but the Sumner line is not completely known from its bearing alone. To locate it properly it is necessary to know in addition the latitude and longitude of some point on the line, which we 112 NAVIGATION will call a "Sumner point." Then, knowing such a point of the line, and the bearing of the line, we may say we know the line completely, and, if necessary, could draw it on a chart. Now to find the required Sumner point. We always have the D. R. position of the ship at the moment of observation ; which we will call the "D. R. point." It is easy to find out if the D. R. point is also a Sumner point. It is merely necessary to calculate what the sun's altitude would be for a ship at the D. R. point, and then compare this calculated altitude with the one actually observed. If the D. R. point was really a Sumner point (which will rarely happen), the two. altitudes will agree ; if not, the amount of disagreement will show how far the D. R. point is distant from the nearest Sumner point. 1 The first step, then, in Sumner navigation, is the calcula- tion of the altitude, supposing the ship to be at the D. R. point at the moment of observation. To do this for a sun observation, we first calculate the Greenwich apparent time (abbreviated G. A. T.) of the observation, just as was done in the case of a time-sight on p. 102. To this G. A. T. we then add the ship's D. R. longitude, if east, or subtract it, if west, to get T (p. 100), the ship's apparent time of the ob- servation. We then use the formulas on p. 113, in which X and Z are " auxiliary angles" required in the calculations, but not otherwise of special interest. These formulas are called the " cosine-haversine " formulas. There are several other sets of formulas with which the same problem can be solved. One set, called the " haversine " formulas, involves the use of haversines only; another, called the " sine-cosine " formulas, solves the problem with sines and cosines. But neither is preferable to the following cosine-haversine set. 1 This method is often called the Marcq Saint Hilaire method ; but it should probably be credited to Lord Kelvin, who published " Tables for Facilitating Sumner's Method at Sea " in 1876. These tables follow the method described above. NEWER NAVIGATION METHODS 113 If observation was made before noon, ship's time, hav. X = cos lat. + cos dec. + hav. (24* - T) , (1 ) If observation was made after noon, ship's time, hav. X = cos lat. + cos dec. + hav. T, (2) lat. dec. = diff. 1 of lat. and dec., if both are + or both , (3) lat. dec. = sum 1 of lat. and dec. if one is + and one , (4) No. hav. Z = No. hav. (lat. - dec.) + No. hav. X, (5) Alt. = 90 - Z. (6) Now we can compare the altitude computed by formula (6) with the observed altitude, fully corrected for index error, etc. The difference between the two altitudes in minutes will be the distance in miles of the nearest Sumner point from the D. R. point, for the minute and nautical mile here correspond, as they do in the case of differences of latitude (p. 15). The bearing of the Sumner point from the D. R. point will be the same as the sun's azimuth if the ob- served altitude is greater than the computed altitude : but if the observed altitude is less than the computed, the bearing of the Sumner point will be 180 greater than the sun's azimuth. The bearing and distance of the Sumner point from the D. R. point once known, it is easy, by means of the traverse table (p. 10), to obtain the latitude and longitude of the Sumner point from the known latitude and longitude of the D. R. point ; or, which is the same thing, from the ship's D. R. latitude and longitude. Before giving examples of these calculations, it remains to show how the sun's bearing or azimuth can be taken from Table 11 (p. 284), called the azimuth table. The pair of arguments (p. 11) for entering this table are: first, in the left-hand column, the declination, which is here used without regard to its sign; and second, in the four topmost hori- 1 In using formulas (3) and (4), pay no attention to + or signs after the right formula is once chosen. The difference between latitude and declination is always taken by subtracting the smaller from the larger ; and the sum by adding them, without regarding their + or signs. Cf. also p. 89. 114 NAVIGATION zontal lines, T (p. 100), the ship's apparent time at the moment of observation. Having found this pair of arguments, we look in the column under T, and in the horizontal line opposite the declination. There we find an "index number." Next we look up the altitude, as computed by formula (6), page 113, in the right-hand column of the azimuth table, and follow along the horizontal line belonging to that altitude, until we reach a number equal (or nearly equal) to the index number. Then we go down the column containing this second appearance of the index number, and find the azi- muth at the bottom of the page. The table gives approxi- mate azimuths only, but the approximation is sufficient for our present purpose. The azimuths at the bottom of the page appear in four horizontal lines, of which the upper two belong to forenoon observations, and the lower two to afternoon observations. All azimuths are counted from the north, through east, south, and west, from to 360, like compass courses in United States Navy practice (p. 41). It is important for the navigator to record, at the time of observation, the word " forenoon" or " afternoon," and also the sun's roughly approximate bearing, to aid in choosing which of the azi- muths at the bottom of the tabular page is the right one. The record showing whether the observation was made in the forenoon or afternoon limits the choice to two of the lines of azimuths; and if there is any doubt remaining between these two, the following rules may clear it up. When latitude is + and declination , azimuth is between 90 and 270; When latitude is -f and declination +, if declination is greater than latitude, azimuth is not between 90 and 270 ; When latitude is and declination , if declination is greater than latitude, azimuth is between 90 and 270 ; When latitude is and declination -f-, azimuth is not between 90 and 270. NEWER NAVIGATION METHODS 115 In other cases, and especially when latitude and declina- tion are nearly equal, the foregoing rules are insufficient, and we must consult Table 12 (p. 290), the " auxiliary azimuth table." This table has latitude and declination for its pair of arguments, the former in the left-hand vertical column, the latter in the topmost horizontal line : and in using the table it is not necessary to pay attention to the + and - signs of latitude and declination. Start with the latitude, and follow its horizontal line to the right until you reach the column having the declination at its head. There you will find an "auxiliary angle," which must be compared with the altitude computed by formula (6), page 113. Then : If the computed altitude is greater than the auxiliary angle, and if latitude is +, azimuth is between 90 and 270 ; If the computed altitude is less than the auxiliary angle, and if latitude is -, azimuth is between 90 and 270 ; If the computed altitude is less than the auxiliary angle, and if latitude is +, azimuth is not between 90 and 270 ; If the computed altitude is greater than the auxiliary angle, and if latitude is , azimuth is not between 90 and 270. It will rarely happen that any of the foregoing rules will be needed, if the navigator will make a careful observation of the sun's azimuth with the azimuth circle or pelorus (p. 44), as soon as possible after the sextant altitude has been observed. The ship's course should also be specially recorded when this observation is made. This proceeding is not merely a convenience to avoid consulting the fore- going rules in using the azimuth table : it is really essential to safe navigation, for a comparison of the observed azi- muth with that derived from the table will make the com- pass error (p. 43) known. The variation is known from the chart ; so that if we observe the compass error, we can allow for the variation, and get the deviation. This can then be compared with the deviation table (p. 48), to see if there has been any change in the compass since leaving port. It is 116 NAVIGATION a great advantage of the Sumner method that the sun's azimuth comes out as a sort of by-product, so that the com- pass can be verified without any additional special calcu- lations. We shall now illustrate all the above considerations by means of examples ; beginning with the observation already treated as a time-sight (p. 101). That observation we shall now work by the Sumner method. From page 101 we take the following : Date of observation, Dec. 18, 1917, in the afternoon; D. R. latitude, 42 20' N. ; D. R. longitude, 35 16' W. ; altitude observed, 14 19' ; time by watch, 2* 29 W 58* ; C. - W., 2* 27 m 8 ; chronometer correction, 2 m 8* slow of G. M. T. ; index correction, + 4' ; height of eye, 24 ft. From the preparatory part of the calculation (p. 102), we also copy the following additional numbers : Declination, line (6), page 102 -23 24' (1) Greenwich apparent time (G. A. T.) of observation, line (8), page 102 5* 2" 35* (2) We have next to calculate, by the formulas on page 113, the altitude corresponding to the D. R. point, for which the latitude and longitude are given above. The longitude is 35 16' W., or, at 15 to the hour (Table 9, p. 249) : D. R. longitude is 2 h 2l m 4* W. (3) Subtracting (3) from (2), according to page 112, gives ship's apparent time of observation, T. . 2 41 31 (4) We are now prepared to apply formulas (1) to (6), page 113. We choose formula (2) for an afternoon obser- vation l ; and write : 1 For a forenoon observation we should choose formula (1), and should therefore need to know 24* - T instead of T. This would make necessary another line in the form of calculation, and it would follow line (4). This new line might be numbered (4') ; and in it would be written 24* - T, obtained by subtracting T (line 4) from 24*. NEWER NAVIGATION METHODS 117 Cos lat., 42 20' N. by D. R. (see Table 4, p. 238) .... 9.86879 (5) Cos dec., 23 24', line (1) (see Table 4, p. 219) 9.96273 (6) Hav. T, 2* 41* 31', line (4) (see Table 10, p. 260) .... 9.07596 (7) Adding (5) to (7) gives hav. X (dropping 20, p. 25) . . 8.90748 (8) Now we choose formula (4), because latitude and declina- tion are + and ; The latitude is, by D. R. . . . 42 20' (9) Adding (1) and (9) according to formula (4) gives (lat. - dec.) 65 44' (10) Now we have, Table 10, page 266, No. hav. of (10) . . 0.29451 (11) No. hav. X, 1 line (8) 0.08082 (12) Adding (11) and (12), according to formula (5), page 113, gives No. hav. Z 0.37533 (13) , And Z, corresponding to (13) is found from Table 10, page 268 75 34' (14) Then, by formula (6) computed altitude =90 - Z (14), or 14 26' (15) This computed altitude (15) must now be compared with the observed altitude, fully corrected. We find : Obs'd alt., fully corrected, line (17), page 102, is 14 30' (16) Difference between (15) and (16), in minutes, is the distance of Sumner point from D. R. point in miles (p. 113). It is 4 miles (17) Next we must find the sun's azimuth from Table 11, page 286. The top argument for entering the table is T, line (4), and it must be found in the " afternoon" lines. The argument for the left-hand column is the declination, line (1). Under T, and opposite declination, we find the tabular index number 5872. 2 Then we find the computed altitude, line (15), in the right-hand column of Table 11, page 286, arid 1 This No. hav. X comes from Table 10, page 258, without looking up the angle X at all. We simply find hav. X in the table, and take the No. hav. X out of the adjoining heavy type column. No inter- polations are needed, the nearest tabular numbers being sufficiently accurate. 2 The index numbers and the azimuth need not be very accurate : it is sufficient to use the nearest tabular arguments, so that inter- polation is not essential. 118 NAVIGATION follow its horizontal line till we again come upon the index number 5872. It lies about halfway between 5703 and 5973. Going down the two columns containing these index numbers, we find in the afternoon azimuth lines two values of the azimuth, 217 and 323. The choice between these two numbers would be very easy, if the observer's record contained even a rough estimate of the sun's bearing at the time of observation. We have purposely not made this avail- able, so as to show how to consult the directions on page 114, and there we find that when the latitude is + and the declination , the azimuth is between 90 and 270. So we finally choose 217 for the sun's azimuth. Since the observed altitude (16) is greater than the com- puted altitude (15), the bearing of the Sumner point from the D. R. point, according to page 113, is the same as the sun's azimuth, or 217. And as we now know the bearing and distance of the Sumner point from the D. R. point, we can find its latitude and longitude by a simple application of the traverse table (p. 154). We have merely to consider the bearing and distance to be a course angle and distance, and imagine a ship to have sailed from the one point to the other. In the present case, the distance is 4 miles (line 17), the course 217 : and Table 1 (p. 164) gives the corresponding latitude 3'. 2, departure 2.4. The longitude difference is obtained from the departure by Table 2 (p. 174) and is, for latitude 42, about 3'.2. Drop- ping odd fractions, the latitude difference and longitude differ- ence both come out 3'. The Sumner point is therefore 3' dis- tant from the D. R. point in both latitude and longitude. And since the bearing 217 indicates on the compass card that the Sumner point is south and west of the D. R. point, it follows that : Lat. of Sumner point = D. R. lat. - 3' = 42 20' N. (line 9) - 3' 42 17' N. (18) Long, of Sumner point = D. R. long. +3' 35 19 W. (19) Azimuth of Sumner line (p. Ill) 307 (20) NEWER NAVIGATION METHODS 119 It is important for the reader to understand that the fore- going calculation is given in extended detail so as to make it easy for the beginner to follow. In condensed form, we should have the following arrangement of the calculation, corresponding to the condensed time-sight form (p. 104). Part of the work here repeated from page 104 has no attached reference numbers in parentheses : the new part of the work has references to the detailed calculation just given. SUMNER LINE, CONDENSED FORM. SUN Obs'd alt. : 14 19' Index: + 4 Table 6 : +12 Table 7: - 5 Corr'dalt.: 14 30' Decl.4*: 23 23'. 7 S. H. D.: 0.1 Decl. 4* 59 : 23 24' S. Eq. time, 4* : + 3 22.3 H. D.: 1.2 Eq. time, 4* 59"* : +3 21.1 Watch time : 2* 29 58 C. - W.: 2 27 8 Chr. time : 4 57 6 Chr. corr'n : + 2 8 G. M. T. 18th : 4 59 14 Eq. of time : -f 3 21 G. app. time : 5 2 35 D. R. long. : 2 21 4W. (3) Ship's app. time, T : 2 41 31 (4) hav. T (or 24^ -T 7 ) : 9.07596 D. R. lat. : 42 20' N. (9) cos lat. : 9.86879 Dec.: 23 24 S. (D cos dec. : 9.96273 sum = hav. X : 8.90748 No. hav. X : 0.08082 (12) No. hav. (lat. Lat. - Dec. : 65 44 (10) - dec.) : 0.29451 (11) Z: 75 34 (14) No. hav. Z 0.37533 (13) Comp'd alt. : 14 26 (15) Obs'd alt. : 14 30 (16) Diff. : 4 (17) Index No. : 5872 Azimuth : 217 Lat. diff . : 3'.2 Dep. : 2.4 Long. diff. : 3'.2 D. R. lat. : 42 20' N. (9) D. R. long. : 35 16' W. (3) Sumner pt. lat. : 42 17 N. (18) Sumner pt. long. : 35 19 W. (19) Azimuth of Sumner line: 307 (20) See footnote, p. 116. 120 NAVIGATION When the object observed is a star (cf. p. 104) or planet, the choice between formulas (1) and (2), page 113, is not quite the same as in the case of a solar observation. We must use formula (1) if the star was on the east side of the sky when observed, which might be called a " forenoon" observa- tion of the star ; and we must use (2) if the star was on the west side of the sky, giving an " afternoon" star observa- tion. The use of the remaining formulas (3) to (6) is the same as for the sun ; but T is now no longer the ship's appar- ent time. Instead, it is the star's hour-angle (p. 104) ; to find it for use in formulas (1) and (2), and in Table 11, we must first calculate (p. 85) the Greenwich sidereal time corresponding to the G. M. T. of the observation, as taken from the chronometer, duly corrected for error and rate ; and then use the following formulas : (7) Greenwich hour-angle = Greenwich sidereal time right ascen- sion of star, . f T = Greenwich hour-angle + D. R. longitude, if east, \ T = Greenwich hour-angle D. R. longitude, if west. As an application of the Sumner method to a star observa- tion, let us take the observation of Sirius, Dec. 17, 1917, off Cape Agulhas, already treated as a time-sight (p. 105). From the preliminary calculations there given, we have : Greenwich hour-angle, line (11), page 105 2 h G m 34 (1) D. R. longitude (p. 105) is 20 41' E., or by Table 9 (p. 249) 1 22 44 E. (2) By formula (8) above, we add (1) and (2), giving T 3 29 18 (3) The star bore west 1 (p. 105) so we choose formula (2) (p. 113), and write : cos lat. (p. 106, line 19), 35 20' S. by D. R. (see Table 4, p. 231) 9.91158 (4) cos dec. (p. 106, line 12), - 16 36' (Tab. 4, p. 212) 9.98151 (5) hav. T, 3 h 29 m 18 s (line 3, above) (see Table 10, p. 263) 9.28872 (6) Adding (4) to (6) gives, by formula (2), page 1 13, hav.Z, 9.18181 2 (7) 1 See p. 116, footnote. 2 Sum diminished by 20 (see footnote, p. 102). NEWER NAVIGATION METHODS 121 Next we choose formula (3), page 113, since latitude and declination are both . We have : By formula (3), lat. - dec. = 35 20' - 16 36' = 18 44' (8) We now use formula (5), page 113. We have: No. hav. 18 44' (8) (see Table 10, p. 254) 0.02649 (9) No. hav. X 1 (7) (see Table 10, p. 261) 0.15194 (10) Adding (9) and (10) gives No. hav. Z 0.17843 (11) And Z, corresponding to (11) is found from Table 10, page 262 49 59' (12) Then, by formula (6), page 113, computed alt. = 90 - Z (12), or 40 1' (13) This computed altitude (13) must be compared with the observed altitude, fully corrected. This was (p. 106, line 18) 40 2' (14) Difference between (13) and (14), in minutes, or dis- tance of Sumner point from D. R. point in miles (p. 113) 1 mile (15) Next we find the star's azimuth from Table 11, page 287. The top argument for entering the table is T, line (3), and it must be found in the " afternoon" lines, since the star bore W. The argument for the left-hand column is the declination, line (5). Under T (p. 287), and opposite declination, we find (approximately) the tabular index num- ber 7550. Then we find the computed altitude, 40 (13), in the right-hand column of the table (p. 289), and follow along its horizontal line until we again reach the index number 7550. The nearest to 7550 is 7544; and under this number, at the foot of the column, we find the two " afternoon" azimuths 260 and 280. These two numbers are so nearly equal that there is un- certainty in choosing between them. Had the observer taken the star's bearing by compass at the time of observa- tion (p. 115), the uncertainty would be removed. But in the absence of this information, we must have recourse to Table 12 (p. 290), the auxiliary azimuth table. Enter- ing this table with the pair of arguments of the present 1 No. hav. here obtained from hav. without finding the angle X (p. 117, footnote). 122 NAVIGATION problem: viz. latitude 35, declination 17, we find the auxiliary angle 31. The computed altitude (13) being 40, is greater than the auxiliary angle, and the latitude is . Therefore, by the instructions (p. 115), the azimuth is not between 90 and 270. We therefore choose 280 as our final azimuth, since 260, the other possible value, is in the prohibited area between 90 and 270. The computed altitude (13) being less than the observed altitude, this observation places the Sumner point 1 mile (15) from the D. R. point, and bearing from it 280, the same as the sun's azimuth (p. 113). The traverse table (p. 156) gives, for distance 1 and course 280, latitude 0.2, departure 1.0. The longitude difference, by Table 2 (p. 172), is 1'.2, for the departure 1 .0. Therefore, since azimuth 280 indicates on the compass card that the Sumner point is W. and N. of the D. R. point, we have : lat. of Sumner point = - 35 20' (4) + 0'.2 = - 35 20' (16) long, of Sumner point = 20 41' E. (2) - 1'.2 = 20 40' E. (17) The bearing of the Sumner line will be 90 greater than the star's azimuth (p. Ill) ; so we have : Bearing of Sumner line = 280 + 90 = 370 ; or, dropping 360 = 10 (18) The foregoing calculation of the Sumner point from a star observation can of course also be put in condensed form. In doing so, we have repeated certain numbers from page 107 without references in parentheses. But numbers taken from the extended calculation just given have their reference numbers attached. This condensed form, like the others previously given, is the form of calculation which would be used in actual navigation. It is most important, in the interest of numeri- cal accuracy, to make all calculations upon forms ; and no numbers should be written on the forms without having an adjoining statement as to the meaning of the numbers. NEWER NAVIGATION METHODS 123 SUMNER LINE, CONDENSED FORM. STAR Watch time : C. - W. : Chr. time : Chr. corr'n : G. M. T. : R. A. mean sun : Corr'n, past noon : Greenw'h sid. time : R. A. of Sirius : Greenw'h hour-angle D. R. long. : T: 16* 29"48 - 1 23 50 15 5 58 - 2 28 15 3 30 17 42 10 2 28 8 48 8 6 41 34 : 2 6 34 1 22 44 E. (2) 3 29 18 (3) Obs'd alt. : Index : Table 6: Table 7 : Corr'd alt. T or (24* - T) 1 : 3 ft 29" 18* (3) hav. Dec. : - 16 36' cos : D. R. lat. : - 35 20 cos : Sum of 3 = hav. X : No. hav. X : Lat. - Dec. : 18 Sum' of 2 = No. hav. Z: Z: Computed alt. = 90 - Z Obs'd alt., corr'd: Diff.: Index No.: 7550 Azimuth : 280 Lat. diff. : 0'.2 Dep. : 1.0 Long. diff. : 1'.2 Sumner pt. lat. : - 35 20' (16) ; long. : 20 40' E. Bearing of Sumner line : 10 (18) 9.28872 9.98151 9.91158 9.18181 0.15194 44' (8); No. hav.: 0.02649 0.17843 49 59' 40 1 40 2 1 40 3' + 5 - 1 - 5 40 2 (6) (5) (4) (7) (10) (9) (ID (12) (13) (14) (15) (17) We have now, in the foregoing examples, illustrated the manner of determining a Sumner line completely by ascer- taining the latitude and longitude of one point on the line (the Sumner point), and the bearing of the line itself at that point. It may be desired to draw the line on the chart, which will always interest the navigator if he is near the coast and has a large-scale chart. To draw it, we merely locate the Sumner point on the chart by its latitude and longi- 1 See footnote, p. 116. 124 NAVIGATION tude, and then draw the line through the point so that it will make with the meridian an angle equal to the bearing which has been computed for the line. The Sumner line should be extended in both directions from the Sumner point, for any convenient distance, in such a way that the point will be near the middle of the line. We can now gain a better understanding as to Sumner navigation by comparing the results obtained in one of the foregoing examples with the corresponding calculation of the same example as a time-sight. Thus from the same ob- servation (pp. 104, 119) As A TIME-SIGHT As A SUMNER OBSERVATION From D. R. latitude 42 20' N. ; From D. R. latitude 42 20' N. ; D. R. longitude 35 16' W., we D. R. longitude 35 16' W., we found the ship's longitude to be found the Sumner point to be 35 24' W. in latitude 42 17' ; longitude 35 19' W. ; and azimuth of Sumner line, 307. Starting with the same observed altitude, and the same D. R. position of the ship, we get quite different results by the two methods of calculation. The time-sight gives us nothing but a longitude; and it will be the correct ship's longitude only if the D. R. latitude was also correct (p. 101). Therefore the time-sight calculation leaves us with both latitude and longitude still affected by possible errors in the D. R. latitude. On the other hand, the Sumner calculation gives us both a latitude and a longitude, but neither belongs to the ship's position. They both belong to the position of the Sumner point, but they are free from the effects of any D. R. errors. They fix the Sumner point only, but they fix it correctly. Furthermore, our knowledge that the ship is somewhere on the Sumner line is also a fact, free from error. So what we learn from the Sumner method is sure ; what we get by the older methods is all really D. R. information in some NEWER NAVIGATION METHODS 125 degree. The Sumner method is independent of D. R., an advantage of which the value cannot be estimated too highly. Furthermore, it can be shown mathematically (cf. p. Ill) that a single observation can never really do more than determine a line on which the ship must be. Even a noon- sight does no more than this ; for in determining the ship's latitude, it really only makes known a horizontal line (the ship's latitude parallel) on the chart. In other words, for a noon-sight the Sumner line is horizontal, or has a bearing of 90. And it will always come out 90, if a noon-sight is worked as a Sumner observation. But the principal purpose of our present comparison of the two methods of calculation is to warn the navigator against falling into the error of imagining the ship to be at the Sumner point. The observation does no more than tell us where the Sumner point is, and that the ship is somewhere on the line ; so far as the observation is concerned, all points on the line are equally likely to be the ship's true position. Therefore it is misleading to call the Sumner point the ship's "most probable position." Were it so, a second observation, made later in the day, would give another "most probable position" of the ship. We should then be naturally led to take as the ship's final location a point midway between the two "most probables," ascribing their divergence to possible errors of observation. But the ship's real position we already know (p. Ill) to be at the intersection of the two Sumner lines resulting from the two observations. And this inter- secting point may be many miles from both "most proba- bles," and from the above-mentioned midpoint between them. Less than two observations cannot fix the ship's position completely; when two have been made, a correct applica- tion of the Sumner method requires that the intersection point of two Sumner lines be determined by calculation. But before explaining the method of doing this, we must describe an excellent alternative way of making Sumner 126 NAVIGATION calculations such as we have given in the above examples. The results are the same results as before, but they are obtained with less work, and quite without logarithms, by means of special tables such as our Table 13 (p. 292), l which we shall call Kelvin's Sumner Line Table. This table has a pair of arguments (p. 11), a and b, a ap- pearing at the heads of the tabular columns, and b in the left-hand column of each page. Corresponding to these two arguments, the table gives two angles, K and Q ; so that whenever a and b are given we can find the corresponding K and Q ; or, if a and K should be given, we can find the corresponding b and Q. In the Sumner problem we obtain, by preparatory calcu- lation (cf. pp. 119, 123), the following data: Declination of sun (or star) ; D. R. latitude ; D. R. longitude ; T, the ship's apparent time of the observation for the sun, or the hour-angle for a star ; and we wish to get the computed altitude and the azimuth. The principle on which Table 13 depends is that the D. R. latitude and longitude being always somewhat uncertain, we can, if we choose, change them by reasonable amounts before beginning our calculations. The Sumner point will then be determined by its distance and bearing from the changed D. R. point, instead of the original D. R. point. By this device the tabular calculation is much facilitated. The use of the table is easy after a little practice, the work being divided into a series of separate operations. In de- scribing these operations we have used small subscript num- bers, to distinguish the several arguments, etc. ; as, for in- stance, in Operation 1 we use ai, bi, KI. 1 These tables were first published by Lord Kelvin in 1876. More extended ones were recently issued by Lieutenant de Aquino, of the Brazilian Navy; and these were reprinted by the Hydro- graphic Office, United States Navy, in 1917. Aquino also improved Kelvin's method of using his table. NEWER NAVIGATION METHODS 127 OPERATION 1, requiring no interpolation. Enter Table 13 with: Arg. ai = declination, taken without regard to + or sign, and cor- rect to the nearest whole degree only ; Arg. 61 = T, if T is between 0* and 6* ; = 12* - T 1 , if T is between 6* and 12*; = T - 12*, if T is between 12* and 18*; = 24* - r, if T is between 18* and 24* ; and before use 61 must be turned into degrees with Table 9 (p. 249). It need be correct to the nearest degree only. This proceeding will make 61 always less than 90. Then take from the table the tabular angle KI, also correct to the nearest degree only. OPERATION 2, requiring simple interpolation. Enter the table a second time with : Arg. 02 = the Ki, obtained in Operation 1. Then, under this a 2 , run down the ^-column until you find the declination (taken without regard to + or - sign) ; so that, in other words, K 2 = declination. Take from the table the angle Q 2 , which stands next to the declination K 2) and also the 6 2 , which is in the left-hand argument column, in the same horizontal line with the declination K 2 in the ^-column. It will rarely be possible to find the declination (which must this time be exact to the nearest minute) in the K-column ; so that a simple interpolation will be necessary in getting Q 2 and 6 2 . An example of this interpolation will be found on page 129 ; and, as we shall see, it is practically the only numerical calculation required in the whole problem. The Kelvin method is very much shorter than it looks. The angle Q 2 is used in choosing the longitude of the " changed D. R. point"; the latitude of that point will be found in Operation 3. To utilize Q 2 for a sun observation, calculate the Greenwich apparent time (G. A. T.) of the 128 NAVIGATION observation, as on page 102, line (8), and turn it into de- grees with Table 9 (page 249). Then : (1) W. long, of changed D. R. point = G. A. T. Q 2 , if, in Oper- ation 1, T was less than 6*; (2) W. long, of changed D. R. point = G. A. T. - (180 - Q 2 ) if, in Operation 1, T was between 6* and 12*; (3) W. long, of changed D. R. point = G. A. T. - (180 + Q 2 ) if, in Operation 1, T was between 12* and 18*; (4) W. long, of changed D. R. point = G. A. T. - (360 - Q 2 ) if, in Operation 1, T was between 18* and 24*. When the subtractions in these formulas cannot be made, the G. A. T. may be increased by 360 ; and when the west longitude comes out greater than 180, subtract it from 360, and call it east longitude. In the case of a star, we must use, in the above formulas, the Greenwich hour-angle, instead of the G. A. T. See page 105, line (11), for the method of obtaining it. OPERATION 3, requiring no interpolation. Enter the table a third time with : Arg. a 3 = KI, again as obtained in Operation 1. (5) Arg. 6 3 = 90 - (6 2 + changed D. R. lat.), if latitude and declination are of opposite signs, one + and one ; (6) Arg. 6 3 = (6 2 + changed D. R. lat.) - 90, if T was between 90 and 270 ; (7) Arg. 6 3 = 90 - (6 2 - changed D. R. lat,), if latitude is less than 62 ; (8) Arg. 6 3 = 90 + (6 2 - changed D. R. lat.), if latitude is greater than 6 2 . In choosing among formulas (5) to (8), give them pre- cedence in order; do not use (7) or (8) if the conditions stated for (5) or (6) are satisfied. And at this point, use your privilege of choosing any reasonable changed D. R. lati- tude for the ship ; and choose one that differs as little as pos- sible from the original D. R. latitude, and that yet makes 6 3 a whole number of degrees. In this way, all further NEWER NAVIGATION METHODS 129 interpolation is avoided. Having once chosen among the formulas, the latitude is used without regard to + or signs. To complete Operation 3, having entered the table with the pair of arguments a 3 and 6 3 , take out the tabular K 3 and Q 3 . Kz is now the computed altitude, to be used (p. 113) in locating the Sumner point from the changed D. R. point; and Q 3 is the sun's true azimuth, which will always come from the table less than 90. If the ship is in the northern hemisphere, this azimuth must be counted from the north point of the horizon if, in Operation 3, we used formulas (6) or (7) ; or from the south point of the horizon, if we used formulas (5) or (8). With the ship in the southern hemi- sphere, interchange the north and south points of the horizon in these directions. And in both hemispheres, the azimuth will of course be counted toward the east or west, according as the observation was a "forenoon" or " afternoon" one (cf. p. 120). We shall now use Table 13 for the example given on page 119 in condensed form. We have (p. 127) : OPERATION 1. 01 = dec. = 23, p. 119, line (1), to the nearest degree; &! ==: T = 2 h 41" 1 31', p. 119, line (4) = 40, to the nearest degree ; and, with a\ and 61 as arguments, Table 13 gives (p. 298) : Kj. = 36, to the nearest degree. OPERATION 2. 02 = Ki = 36. K 2 = 23 24', p. 119, line (1) and, with 02 and K 2 , we must find Q 2 and 6 2 . Running down the column headed a = 36 (p. 302), we find : When K 2 = 23 5', Q 2 = 39 43', b 2 = 29, When K 2 = 23 51', Q 2 = 40 0', b 2 = 30. We wish to interpolate for 7C 2 = 23 24', which is 19' down from 23 5' toward 23 51'. The whole distance from 130 NAVIGATION 23 5' to 23 51' is 46'. Therefore we must interpolate down f of the whole interval from Q 2 = 39 43' to Q 2 = 40 0'. The difference between these two Q 2 's is 17' ; there- fore the final Q 2 , belonging to K 2 = 23 24', is 39 43' + if X 17' = 39 43' + 7' = 39 50'. Similarly, the difference between the two fr 2 's being 60', the final value of 62, for K 2 = 23 24', is 29 + if X 60' = 29 25'. These two little interpolations are practically all the calculation required in the whole problem. To find the longitude of the changed D. R. point from the above Q 2 = 39 50', we take from page 102, line (8), Greenwich apparent time of observation, 5* 2 m 35 which, by Table 9 (p. 249) is, 75 39' We now use formula (1), page 128, because T, in Opera- tion 1, was less than 6*. We get : W. long, of ch'd D. R. pt. = G. A. T. - Q 2 = 75 39' - 39 50' = 35 49' W. OPERATION 3. a, = Ki = 36. The D. R. latitude is + 42 20' (p. 119, line (9)) ; and as the declination is , we choose formula (5), page 128. This, without changing the D. R. latitude, would give 6 3 = 90_(& 2 +D. R.lat.) =90 -(29 25'+ 42 20') = 90-7145'; but by choosing a changed D. R. latitude of 42 35', we shall make 6 3 a whole number of degrees. So we have: 6 3 = 90 - (6 2 + changed D. R. latitude) = 90 - (29 25' + 42 35') = 90 - 72 = 18. Now we enter the table with the arguments a 3 = 36, and 6 3 = 18, and obtain, without interpolation (p. 302) : K 3 = computed altitude = 14 29', Q 3 = sun's true azimuth = 37 22'. This azimuth must be counted from the south point of the horizon, since we used formula (5) in Operation 3 ; and NEWER NAVIGATION METHODS 131 as the observation was an afternoon one, the correct azi- muth will be S. 37 22' W. (cf. p. 19). Counted in the United States Navy way, from the north toward the east, and so around to 360, the azimuth will be 217 22'. On page 119, we found : Computed altitude, 14 26' ; azi- muth, 217. This computed altitude differs by 3' from the value just found by Table 13. The difference is due to our having changed the D. R. point. From the changed D. R. point, in latitude 42 35' N. ; longitude 35 49' W., we now calculate (see Condensed Form, next page) the position of the Sumner point to be : latitude 42 34' N. ; longitude 35 50' W. The former position, as obtained on page 119, was : latitude 42 17' N. ; longitude 35 19' W. These two Sumner point positions should lie on the same Sumner line if the method of Table 13 gives . correct results ; and they will satisfy this test, if the bearing of a line joining them agrees with the azimuth of the Sumner line, which is 217 + 90 = 307. From the two Sumner point positions we have : latitude difference = 17' ; longitude difference = 31'; departure (Table 2, p. 174) = 23.0. The traverse table (p. 164) gives, for latitude 17, departure 23.0, the distance 28, course 307. The agree- ment is perfect, and shows that the same Sumner line passes through both points, though they are 28 miles apart. This test also shows that the calculation may indicate any point on the Sumner line as the Sumner point, if the D. R. position of the ship is uncertain : and so we again call attention to the error of taking the cal- culated Sumner point as the ship's most probable position (cf. p. 125). We now, as usual, repeat the above calculation by Table 13, in condensed form, and including the final determination of the position of the Sumner point from the changed D. R. point. 132 NAVIGATION SUMNER LINE BY TABLE 13, CONDENSED FORM. SUN [The following is taken from page 119.] Decl., 4*: - 23 23'.7 Eq . of time : + 3 m 22.3 H. D. : 0.1 H. D. : 1.2 Decl., 4*59: -23 24 Eq . time : + 3 21.1 Watch time : 2h 29" 58* Obs'd alt. : 14 19' C. - W. : 2 27 8 Index : + 4 Chr. time : 4 57 6 Table 6 : + 12 Chr. corr'n : + 2 8 Table 7 : -5 G. M. T.: 4 59 14 Corr'd alt. : 14 30 Eq. of time : + 3 21 D. R. lat. : 42 20' N. G. app. time : 5 2 35 D. R. long. ; : 35 16' W. D. R. long. : 2 21 4 W. (3) Ship's app. time, T: 2 41 31 (4) [The following is calculated with Table 13.] OPERATION 1 OPERATION 2 at = dec. =23 a* = Ki = 36 61 = T =2* 41"* 31(4) K 2 = dec. = 23 24' = 40 Table 13, Q 2 = 39 50' Table 13, Ki = 36 Table 13, 62 = 29 25' Greenwich app. time = 5& 2 35 = 75 39' By page 128, form. (1), W. long, of changed D. R. pt. = G. A. T. - Q t = 35 49' W. Lat. of changed D. R. pt. = 42 35' N. OPERATION 3 a s = Ki = 36 & 3 = 90 - (62 + changed D. R. lat.) = 18 Table 13, K 3 = comp'd alt. = 14 29' Table 13, Q a = azimuth of sun = 37 22' or, by U. S. Navy = 217 22' Azimuth of Sumner line = 217 22' + 90 = 307 22' Dist. of Sumner pt. from changed D. R. pt. = corr'd obs'd alt. comp'd alt. = 1' or 1 mile Bearing of Sumner pt. from changed D. R. pt. =217, since comp'd alt. is less than obs'd alt. Dist. 1, on course 217, gives lat. diff., 0'.8 ; dep., 0.6 ; long, diff., 0'.8 Lat. of Sumner pt. = lat. of ch'd D. R. pt. - lat. diff. = 42 34' N. Long, of Sumner pt. = long, of ch'd D. R. pt. + long. diff. = 35 50' W. A practised navigator can make the above complete calcu- lation in a few minutes, as there are no logs used ; and any one can easily obtain the necessary practice at sea by simply forming the habit of working his sights both as time-sights and as Sumners. To illustrate the subject further, we now give, in condensed form, the Star Example of p. 123, worked by Table 13. NEWER NAVIGATION METHODS 133 SUMNER LINE BY TABLE 13, CONDENSED FORM. STAR [The following is taken from page 123.] Watch time: 16* 29 48 Obs'dalt.: 40 3' C. - W. : - 1 23 50 Index : + 5 Chr. time : 15 5 58 Table 6 : - 1 Chr. corr'n : - 2 28 Table 7 : - 5 G. M. T. : 15 3 30 Corr'd obs'd alt. : 40 2 R. A. mean sun : 17 42 10 Corr'n, past noon : 2 28 Dec. of Sirius : - 16 36 Greenwich sid. time : 8 48 8 D. R. lat. : -35 20 R. A. of Sirius : 6 41 34 Green, hour-angle : 2 6 34 D. R. long. : 1 22 44 E. T: 3 29 18 [The following is calculated with Table 13.] OPERATION 1 OPERATION 2 ai = dec. =17 02 = Ki = 49 6, = T = 3* 29"* 18 K 2 = dec. = 16 36' = 52 Table 13, Q 2 =51 57' Table 13, Ki = 49 Table 13, b = 25 49' By page 128, form. (1), W. long, of changed D. R. pt. = Green, hour-angle Qz l 339 41' 20 19' E. Lat. of changed D. R. pt. = - 35 49' OPERATION 3 a, = Ki = 49 By form. (8), page 128, b s = 90 + (61 - changed D. R. lat.) = 80 Table 13, Kt = comp'd alt. = 40 15' Table 13, Q = az. of Sirius = N. 81 25' W. or, by U. S. Navy = 278 35' Az. of Sumner line = 368 35', or 8 35' Dist. of Sumner pt. from changed D. R. pt. = corr'd obs'd alt. comp'd alt. = 13' or 13 miles Bearing of Sumner pt. from changed D. R. pt. = 99, since comp'd alt. is greater than obs'd alt. Dist. 13, on course 99, gives lat. diff., 2'.0 ; dep., 12.8 ; long, diff ., 15'.9 Lat. of Sumner pt. = lat. of ch'd D. R. pt. -f lat. diff. 35 51' Long, of Sumner pt. = long, of ch'd D. R. pt. + long. diff. - 20 35' E. To complete this part of our subject, it remains to show how the position of the ship can be found at the intersec- tion of two Sumner lines (pp. Ill, 125) resulting from two different observations. Figure 18 explains the nature of the problem; and it is almost exactly the same figure and 1 #2 being larger than the Greenwich hour-angle, the latter was increased by 360, to make the subtraction possible (p. 128). 134 NAVIGATION problem treated in Chapter V, when we discussed fixing a ship's position by means of " bearings from the bow" (p. 54). The two Sumner lines in Fig. 18 are SL and S'L, passing through the two Sumner points S and S f , whose latitudes and longitudes are known by calculation from the observed altitudes. The bearings or azimuths of the two Sumner lines from the north are the two angles NSL and N'S'L, which are also known from the pre- vious calculations. It is now required to find the latitude and longitude of the intersection point L, where the ship is situated. The similarity of this problem to the former one /^ in Chapter V becomes plain, F,o. lS.-Interaection of Sumner Lines. if we ima S ine a second shi P sailing from one Sumner point to the other, as from S to $', and taking bearings from her bow upon our ship, located at L. These bearings will be the two angles S'SL and S"S'L. If the second of these angles should happen to be just twice as big as the first, the distance S'L between the two ships at the time of the second bearing would be equal (p. 54) to the distance SS f run by the imagined ship between the two observations. This would enable us to fix the position of the imagined ship at ', if L were a lighthouse ashore. But if L is our ship, and S' a Sumner point of known position, the same observations of bow bearings would fix the position of our ship at L. Nor is it necessary (or possible) to measure NEWER NAVIGATION METHODS 135 such imaginary bearings, or read the patent log to get the distance run by an imagined ship. For the distance and bearing of the second Sumner point from the first can be obtained from their known latitudes and longitudes with the traverse table. Thus the line SS' (marked " distance") and the bearing (or course) angle NSS' become known. Furthermore, the "bow bearing" at S is the angle S'SL, and it is equal to the difference NSL NSS'. We have just seen that NSS' is obtained from the traverse table ; and NSL is the calculated azimuth of the Sumner line through S. In a similar way we get the other "bow bearing" S ff S'L. If this were twice the first one, the "required distance" S'L in the figure would be equal to the known distance SS' between the two Sumner points. If not, it can be easily shown mathematically that : (1) Required distance = known distance X a factor, (2) log factor = sin S'SL - sin (S"S'L - S'SL). By these simple formulas the required distance S'L might be found : and as we also know the latitude and longitude of the Sumner point S', and the azimuth or bearing of S'L, the traverse table will make known the latitude and longi- tude of the ship at L. It is to be noted also that as we are at liberty to call either of the Sumner points S', it is desirable to call that one S' which has the larger "bow bearing," so that there will be no difficulty about subtracting S'SL from S"S'L. The factor of formula (2) above can practically always be found in our Table 14, the Sumner Intersection Table, without using logarithms. The pair of arguments of the table are the smaller "bow bearing" and the larger "bow bearing"; the tabular number is the factor of formula (1) above, and will always give the distance of the intersection point from that one of the two Sumner points for which the bow bearing was the larger. And it should not be forgotten that the Sumner line really 136 NAVIGATION extends equally in both directions (p. 124) from the Sumner point, whereas, in Fig. 18, we have extended it mainly in the direction of the intersection point L. Now the cal- culated azimuth of any Sumner line may be changed 180 at will, because the bearings of the two ends of the line from the Sumner point differ by 180, and we may take the bear- ing of the line to be the bearing of either end from the Sumner point in the middle of the line. Figure 18 shows, however, that for the purpose of the present problem we must choose the bearing of that end of the line which is nearest the point of intersection L ; nor does the choice ever offer difficulty, because the known D. R. position of the ship at L, when compared with the known positions of the two Sumner points, will always indicate whether L bears east or west of either Sumner point, and also whether it bears north or south. And the bearing of L once chosen, we can always find either of the two bow bearings by this formula : (3) Bow bearing = bearing of Sumner line minus bearing of the second Sumner point S f from the first point S. In using formula (3) it is allowable to increase the bear- ings of the Sumner lines by 360, when necessary to make the subtractions possible, and if the formula brings out bow bearings larger than 180, subtract them from 360, and proceed as before. It is also always desirable to draw a rough sketch for every intersection problem occurring on shipboard so as to guard against accidental large errors like 90 or 180 in ob- taining the two bow bearings ; and also to make sure that the latitude and longitude of the intersection point L are correctly computed with the traverse table. The foregoing assumes that the ship did not move from the point L between the two sextant observations from which the two Sumner lines were calculated. This will rarely be the case, because it is very desirable that the two observa- tions, if they are both sun observations, be separated by NEWER NAVIGATION METHODS 137 three or four hours, if possible. The condition of an unmov- ing ship will occur only if she is a sailing vessel becalmed, or a steamer at anchor ; or if the two observations are made at nearly the same time upon two different heavenly bodies, such as two stars. High accuracy in the resulting "fix" (p. 53) of the ship will then be attained, if the azimuths of the two stars differ by about 90 at the time of observation. The same favor- able condition will be secured if one of the observations is made upon a star near upper transit (pp. 89, 96), in the twilight just before sunrise or after sunset; and the other observation, at nearly the same time, upon the sun, when it is about 12 or 15 above the horizon. But if the ship has traveled a considerable distance between the two observations, it is necessary to allow for such travel before calculating the intersection point. Suppose she has gone a distance D, upon a course (7, by D. R., between the two observations. Then simply find from Tables 1 and 2 the difference of latitude and longitude corresponding to distance D and course C ; and apply them as corrections to the latitude and longitude of the Sumner point belonging to the first observation. Everything else, including the bearing of the first Sumner line, remaining unchanged, the calculation then proceeds by Table 14, just as if the ship had not moved. The computed intersection point is then the ship's position at the time of the second sextant observation. We shall now work some intersection examples. Suppose we have two Sumner lines, as shown in the rough sketch, Fig. 19, taken on board a ship becalmed. The two sextant observations give : FOR ONE SUMNER POINT, S FOR THE OTHER POINT, S f lat.: 4234 / N. 42 50' N. long.: 3550 / W. 35 36' W. bearing of Sumner line : 307 93 (changed to 273) 1 As found on page 132. 138 NAVIGATION The rough sketch, Fig. 19, having been made, and the two "bow bearings" marked with little circular arcs as shown, we call that one of the two Sumner points S' t which has the larger bow bearing ; and, for the point S f , we change FIG. 19. Bough Sketch of Sumner Intersection. the bearing of the Sumner line from 93 to 180 + 93 = 273, so as to count the bearing for that end of the line which is toward the intersection point L (p. 136). The other bearing, 307, for the point S, is already correctly counted. We now have, from the two Sumner point latitudes and longitudes : latitude difference = 16' ; longitude difference = 14' ; departure (Table 2, p. 174, for middle latitude 43) = 10.2 ; and, for latitude difference = 16, departure = 10.2, we find (Table 1, p. 162), distance = 19, course = 32. The distance between the two Sumner points is therefore 19 miles, and the bearing of S f from S is 32. Now we apply formula (3), page 136, and find : Smaller bow bearing at S = 307 - 32 = 275. Larger bow bearing at S' = 273 - 32 = 241. Being larger than 180, these must be subtracted from 360 (p. 136), giving: Smaller bow bearing = 85; Larger bow bearing = 119. Next we refer to Table 14, and find with the smaller bearing 85, and the larger 119 the factor 1.78 (p. 322). NEWER NAVIGATION METHODS 139 According to formula (1), page 135, we then have: Required distance LS' = distance SS' X factor = 19X1.78 = 33.8 miles. Therefore the position of the ship at L is distant 33.8 miles from S', and she bears 273. With this distance and bearing or course angle, the traverse table (p. 154) gives : latitude = 1.8, departure = 33.8. For the departure 33.8, Table 2 gives, for the middle latitude 43 (p. 174), differ- ence longitude = 46'.2. The bearing 273 showing that the intersection point L is N. and W. of S f , we have : Latitude of ship at L = 42 50' N. + 1'.8 = 42 51'.8 N. Longitude of ship at L = 35 36' W. + 46 / .2 = 36 22' W. As a second example take the following two Sumner lines, as shown in the rough sketch, Fig. 20. The two sextant observations give : FOR ONE SUMNER POINT, S FOR THE OTHER POINT, S' lat. : 14 26' N. 15 30' N. long. : 77 8' W. 76 22'.5 W. bearing of line : 53 135 And suppose the ship, in the interval between the two sextant observations, has traveled a distance D = 31 miles, on course C = 205. We must begin (p. 137) by shifting the first Sumner point S a dis- tance D, on the course C. For this course and distance, we have (Table 1, p. 160) : lat., 28M; dep., 13.1; diff. long, 13'.5 FIG. 2o.-Rou g h (Table 2, p. 168). Sketch of Sumner Therefore, the latitude and longitude of the first Sumner point must be corrected (p. 137) as follows : For the point S, lat. = 14 26' N. - 28M = 13 58' N. long. = 77 8' W. + 13'. 5 = IT 21'.5 W. Bearing (unchanged) = 53. We now have, for the two Sumner points : lat. diff, 92' ; 140 NAVIGATION long, diff., 59' ; dep., 57.0 (p. 169) ; dist., 108 miles (p. 162) ; bearing of S' from S, 32. Now we have, by formula (3), page 136 : Smaller bow bearing at S = 53 - 32 = 21. Larger bow bearing at S' = 135 - 32 = 103. Table 14 (p. 319) gives the factor 0.36 ; so that the ship at L is distant from S' 108 X .36 = 38.9 miles, and bears 135. For this distance and bearing we have (Table 1, p. 166), latitude = 27'. 6; departure = 27.6; and longitude differ- ence (Table 2, p. 168) = 28'.6. Finally, then, at the time of the second sextant observation, the ship at L was in latitude 15 30' N. - 27'.6 = 15 2'.4 N. ; and in longitude 76 22'.5 W. - 28'.6 = 75 54' W. CHAPTER X A NAVIGATOR'S DAY AT SEA THE present chapter contains a number of examples by means of which the reader can gain facility in the use of the methods set forth in the preceding pages. The steam yacht Nav is bound from New York to Colon, and the captain plans to take his departure from the Sandy Hook Lightship, on Dec. 18, 1917, as early as possible in the morning. The first bit of navigation, to be accomplished before the yacht leaves her anchorage in the "Horseshoe," is to ascer- tain by D. R. methods the proper course to steer from Sandy Hook. A glance at the track chart of the north Atlantic shows that she must go by way of Crooked Island Passage, and the Windward Passage between Cuba and Haiti. It is also apparent from the chart that the first land to be sighted among the islands is Watlings Island, and that the proper course should pass to the eastward of it. The position of Sandy Hook Lightship * is lat. 40 28' N. ; long. 73 50' W. Hinchinbroke Rock, at the southern end of Watlings Island, is in lat. 23 57' N. ; long. 74 28' W. But the course should be shaped for a point about 12 miles east of Watlings Island, to be perfectly safe. The position of such a point is (approximately) lat. 23 57' N. ; long. 74 15' W. 2 1 There is an excellent list of latitudes and longitudes in Bow- ditch's "Navigator." 2 The difference between this longitude and that of Hinchinbroke Rock is 13' ; but 13' here corresponds to about 12 miles, on account of Table 2. 141 142 NAVIGATION ABSTRACT OF LOG. Steam Yacht NOD, Dec. 18, 1917 PATENT LOG COMPASS COURSE TRUE COURSE 7 : 02 A.M. Took departure from Sandy Hook Lightship 26.2 S. 188 7:21 Sunrise, observed azimuth 31.0 S. 188 8:00 41.0 S. 188 9:00 57.2 S. 188 9:36 Bow bearing, Barnegat .... 67.0 S. 188 9:42 Altitude and azimuth 69.1 S. 188 9:57 Beam bearing, Barnegat . . . 72.5 S. 188 (fix, lat. 39 45' N. ; long. 73 59' W.) - 10:00 73.4 S. 188 10:07 Changed course 75.3 S.|E. 182 11:00 88.7 S.|E. 182 11:42 Ex-mer. obs'n lat, 39 19'; D. R. long. 73 58' 98.5 S.E. 182 12:00 102.6 S.|E. 182 1 : 00 P.M. 117.7 S.^E. 182 2:00 133.0 S.E. 182 3:00 149.0 S.|E. 182 4:00 163.8 S.E. 182 4:12 Alt. and az., fix, lat. 38 1 1' ; long. 73 54' 166.9 S.iE. 182 5:00 182.0 *-* 2 *"* S.E. 182 6:00 197.2 S.fE. 182| By the method of page 20, the course from Sandy Hook Lightship should be 181, and the distance is 990 miles. These numbers, and all subsequent numbers in the present chapter, should be verified by the reader. The distance being quite large, it is well to check it by the logarithmic method, page 33. The result by this method is: course 181 14', distance 991.7 miles. The chart also shows that this course will carry the yacht very near Barnegat Light, on the coast of New Jersey. The position of this light is lat. 39 46' N. ; long. 74 6' W. The captain decides that it will be well to plan passing this light A NAVIGATOR'S DAY AT SEA 143 at about 5 miles' distance. The position of a point 5 miles east of Barnegat Light is lat. 39 46' N., long. 73 59' W. The course and distance to this point from Sandy Hook Ship are 189 and 42.5 miles. This course is so nearly the same as the course to Watlings Island that the captain decides to steer the 189 course. All this work must be complete before reaching Sandy Hook, for the course from the lightship must be ready for the quartermaster before the lightship is passed. And there is still more preliminary work. For the courses cal- culated above are true courses (p. 43) and the quarter- master must have the compass course, so that he may be able to steer the yacht. The method of calculating the compass course from the true course is given on page 48 ; and in applying it the captain must have his deviation tables at hand. We shall assume that the tables printed on pages 48 and 49 were the ones furnished by the compass adjuster for the present voyage. An examination of the Atlantic track chart shows that in the vicinity of Sandy Hook, the variation, V, is 10 W., or 10. By formula (3) (p. 49), we then have, since the true course T is 189 : Magnetic course = M = T - V = 189 - (- 10) = 199. The second deviation table (p. 49) shows that when the magnetic course (or magnetic bearing of ship's head) is 199, the deviation, D, is + 18. Then, with V = - 10, D = 18, formula (1), page 45, gives : Compass error = E = V + D = - 10 + 18 = + 8. And from formula (2), page 45 : Compass course C=T-E= 189 - 8 = 181 ; and so the yacht must be steered on a 181 compass course for Barnegat. But the quartermaster is to steer by " points " so that the course nearest the 181 course is due south. The captain decides to have the yacht steered due south by 144 NAVIGATION compass, and is prepared to give the quartermaster his orders as soon as Sandy Hook Lightship shall be reached. The foregoing preliminary work having been completed the previous day, the anchor is tripped at the Horseshoe about an hour before daylight on Dec. 18, the weather being fine, sea smooth, and wind light from the northwest. The lightship is reached and passed at 7 : 02 A.M., ship's time, civil reckoning, the ship then taking her departure. At that moment, the patent log is read, and found to register 26.2 miles. The quartermaster gets his orders to steer south; and all the above facts are duly recorded in the log-book. And at every hour thereafter, 8, 9, 10, etc., a similar record must be made in the log-book. The next event is sunrise, which occurs at 7 : 21, very soon after leaving the lightship. The sun's compass bearing can then be very conveniently observed, and will furnish an excellent check on the compass adjuster. This observa- tion was made at 7:21 A.M., ship's time, civil reckoning, corresponding to 19* 21 m , Dec. 17, ship's apparent time, astronomic reckoning; and the sun's bearing or azimuth was 113 by compass. This was entered in the log-book, and at the same time the patent log was read, and found to be 31.0 miles. To check the deviation table, the procedure was then as follows : By patent log the yacht had proceeded from the light- ship a distance of 31.0 26.2 = 4.8 miles, on a compass course of 180, or true course of 188 ; by D. R., she had therefore reached the position lat. 40 23' N. ; long. 73 51' W. The sun's declination, from the almanac, is 23 23', and the (approximate 1 ) T (p. 100) is 19* 21 W . The sun's true azimuth is found from Table 11 to be 121 ; and in using the table for this purpose take the altitude of the sun, for the 1 If there is any chance of this T being much in error, the cap- tain's watch, by which the observation is timed, must be compared with the chronometer. See p. 94. A NAVIGATOR'S DAY AT SEA 145 moment of sunrise, to be 0. The observed compass azi- muth having been 113, formula (2), page 45, gave E = TC = 121 - 113 = + 8. Then from formula (1), page 45, D = E -V = + 8 - (- 10) = + 18. As expected, this deviation agrees with the deviation table, which would not be likely to go wrong so soon after the beginning of a voyage. At 8 A.M. the patent log read 41.0; and at 9 A.M., 57.2. The course was still S. by compass, or 188, true course. At 9 : 24 Barnegat Light was sighted by the lookout, and the mate was ordered to take bow-and-beam bearings (p. 55) upon it. At 9 : 36, the light bore 225 by compass, or 45 from the bow ; patent log, 67.0. At 9* 42 m 28* by his watch the captain took the altitude of the sun's lower limb with the sextant, and found it to be 18 51'. Index correction was + 3', and height of eye, 15 feet. C. W. was 4 A 51 m 50* ; and the chr. correction by the rate card was 4', slow. Patent log, 69.1. At 9 : 45 by the watch, the sun's azimuth was again observed with pelorus, and found to be 137, compass bearing. It was intended to work a Sumner line from the altitude by Kelvin's table; and the pelorus observation was made because the sun's true azimuth always comes out as a by-product, when Kelvin's table is used, and so it is just as well to have an- other check on the deviation table. This is the peculiar advantage of Kelvin's table. Without any additional cal- culations, the compass is always checked up on the very course the ship is steering. This is just what the good navigator wants. The observations could not be worked up at once, be- cause the captain wished to see the result of the mate's bow-and-beam bearings. At 9 : 57 by the watch, Barnegat bore abeam, on the starboard hand, or 270 by compass, the yacht being still on the 180 compass course. Patent log now 72.5. 146 NAVIGATION Between the bow-and-beam bearings the run by log was 72.5 67 = 5.5 miles. Therefore the yacht is now 5.5 miles from Barnegat Light, and the compass bearing of the light is 270. The compass error being -j- 8, the true bear- ing of the light is 278 ; and the bearing of the yacht from the light is the former bearing reversed, or 278 180 = 98, true. From this comes an accurate and complete position of the yacht. Barnegat Light is in lat. 39 46' N. ; long. 74 6' W. The yacht, 5.5 miles away on the bearing 98, must, by traverse table, be in lat. 39 45' N. ; long. 73 59' W. At 10 A.M., the log was 73.4, course 188, true. Now the captain prepared to shape a new course to be followed from the Barnegat bow-and-beam bearing "fix" in the above lat. 39 45' N. ; long. 73 59' W., at 9 : 57. Allowing ten minutes to work up the new course, the captain plans to change course at 10 : 07. At that time the ship, on her course of 188, will be (at 15-knot speed) 2'. 5 S. and practically 0' W. of the Barnegat position. So the course will be changed when the yacht is in lat. 39 42' N. ; long. 73 59' W., at 10 : 07. The course and distance from there to the point 12 miles east of Hinchinbroke Rock are : distance, 945 miles ; course, 181, true, or 173 by compass. Therefore, by the table on page 52, the quartermaster gets the new course S.JE. by compass, at 10 : 07. This corre- sponds to 174 by compass, or 182 true course; and at 10 : 07, when the course was changed, the patent log read 75.3. Now the Sumner line, from the observation at 9* 42 ra 28 s by the watch, was worked by Kelvin's table ; and the result was : Sumner point is in lat. 39 50' N. ; long. 73 56' W. ; bearing of Sumner line 237. It is necessary, as a check, to ascertain whether this Sum- ner line passes through the position obtained for the ship by the Barnegat bearings. Before doing this, the Sumner point must be shifted by the method of page 137, to allow for A NAVIGATOR'S DAY AT SEA 147 the motion of the yacht between 9 : 42, when the sextant observation was made, and 9 : 57, when Barnegat bore abeam. The difference is 15 minutes, and in that time the ship moved south 3.4 miles by the patent log and an in- significant distance west. Therefore the corrected Sumner data are : Sumner point is in lat. 39 46'.6 N. ; long. 73 56' W. ; bearing of Sumner line 237. If everything fits, this Sumner line must pass through the Barnegat "fix" of the yacht in lat. 39 45' N. ; long. 73 59' W., because the yacht must have been somewhere on the line. The traverse table shows that the bearing of a line passing the Sumner point and the yacht's position is 235, differing only 2 from the Sumner line bearing ; so this check is satis- factory. But a better way to check this matter is to deter- mine the yacht's position from the intersection of two lines, one of which is the Sumner line, and the other the beam bear- ing of Barnegat Light. This can be done by the method of page 133. The data of the problem are : Sumner point : lat. 39 46'.6 N. long. 73 56' W. Line bears 237 Barnegat Light : lat. 39 46' N. long. 74 6' W. Line bears 98 We shall call Barnegat Light S' ; and then formula (3), page 136, gives, for the two bow bearings : At Sumner point, S, 237 - 266 = 29. At Barnegat, S', 98 - 266 = 168. For these two bearings, Table 14 gives the factor 0.74, and the yacht is placed 6 miles from Barnegat, on the 98 bear- ing. The bow-and-beam observations gave 5.5 miles, so the check by the Sumner line is excellent. It remains for the captain to utilize the azimuth observa- 148 NAVIGATION tion made at 9 : 45. The bearing of the Simmer line was 237, and therefore the sun's true azimuth was 147. The observed azimuth, by pelorus (p. 145), was 137. The com- pass error was therefore -f- 10. The variation being - 10, the deviation by formula (1), page 45, is D = 10 - ( -10) = + 20. On page 143 we found that the deviation table made this deviation + 18 ; so that the table appears to require a correction of +2. The captain decides not to correct the table for the present, unless later azimuth observations shall confirm it, especially as the sunrise observation showed the adjuster's results to be correct. Azimuth observa- tions made when the sun is high in the sky are not quite as reliable as sunrise ones. Moreover, the observation was made at 9 : 45, whereas the altitude observation, for which the true azimuth was calculated with Kelvin's table, was made at 9 : 42, so that the true azimuth must have been in error by the sun's azimuth change in three minutes. This could have been avoided by giving the mate orders to ob- serve the azimuth at about the same moment when the captain took the altitude. Or, the sun's azimuth change in three minutes might be taken from the azimuth table, and the computed true azimuth duly corrected. At 11 the log read 88.7, and the course was S.JE. by com- pass, or 182, true. At about 11 : 30, the weather showing signs of becoming thick, no preparations were made for a noon-sight by the method of page 86 ; and rather than take the risk of losing his noon observation altogether, the captain took an ex-me- ridian altitude at 11* 42 s by his watch; log was 98.5; the sextant reading 26 55' ; index + 3' ; height of eye 15 ft. ; C. W. was now 4* 51 W .42 S ; and chronometer slow 4*. The observation was worked by Kelvin's table, and gave the Sumner point in lat. 39 20' N. ; long. 73 40' W. ; bearing of Sumner line 86. Figure 21 is a rough sketch of this Sumner line. It is very nearly horizontal ; had the observation been A NAVIGATOR'S DAY AT SEA 149 3920' made at noon precisely, it would have been perfectly hori- zontal. It would now have been possible to move up the Sumner line observed at 9 : 42, and obtain an intersection to fix the position of the yacht. But this did not seem necessary to the cap- tain, because of the beam bearing obtained at Barnegat at 9 : 57, which gave a good fix. And the present Sumner line being so nearly horizontal, it is not necessary to know the longitude very ac- curately to obtain an exact latitude. The longitude by D. R. is sufficient, and it is 73 58' W. The difference between this longitude and that of the Sumner point (73 40') is 18' ; and the ship at L (fig. 21) bears 180 + 86 = 266 from the Sumner point. Table 2 gives the dep. 14.0 for long. diff. 18', in lat. 39. And for course 266, dep. 14.0', we find in Table 1, lat. diff. I'.O, so the yacht's latitude is I' less than that of the Sumner point, and is therefore 39 19'. This happens to be in exact accord with the D. R. latitude, which was also 39 19'. This was perfectly satisfactory, and the captain decided to carry this Sumner line forward for an intersection, in case he should obtain an observation in the afternoon. At 12, the patent log read 102.6, course S. JE., 182 true ; D. R. lat. 39 15' ; long. 73 58' ; distance to Watlings Island 918 miles. Had the yacht been on a course other than almost due south, .it would have been necessary to set the watch and the FIG. 21. Sumner Line from ex-Meridian Observation. 150 NAVIGATION cabin clock to ship's apparent time. In fact, some naviga- tors set their watches to ship's apparent time before every observation (p. 94) : at 1, log read 117.7, misty, at 2, log read 133.0, misty, at 3, log read 149.0 misty, at 4, log read 163.8, clearing. At 4 s I2 m IS 8 by the watch, the weather having cleared, the altitude of the sun was found to be 4 38' ; index + 4' ; eye 15 ft. ; C. W. 4* 51 m 50* ; chronometer slow 4 s ; log 166.9. Sun's azimuth, observed by the mate at the same time, came out 224 by compass. This observation was worked for a Sumner line by the Kelvin table, and gave : Position of Sumner point lat. 38 6' N. ; long. 73 49' W. ; bearing of line 145 ; azimuth of sun 235. The Sumner line obtained at 11* 42 m 0' was brought up to the time of the present observation by D. R. (p. 137), giving : position of 1 1 : 42 Sumner point, after moving it, lat. 38 12' N.; long. 73 43' W.; bearing of the line 86. Both lines were then sketched, as shown in Fig. 22. The point S is the FIG. 22. Rough Sketch of Sumner (moved) Sumner point from the 11:42 observation, S' that from the 4:12 observation. The intersection point L is the position of the ship at 4 : 12, and it came out (p. 134) : lat. 38 11' N. ; long. 73 43' W. The position brought up by D. R. from 11 :42 was : lat. 38 11' ; long. 74 1' ; so that there has been an easterly set of the current, amounting to 7' of longitude in 4J hours. The sun's true azimuth at 4 : 12 was 235, from the Kelvin table ; and the pelorus observation gave 224. The compass error was therefore A NAVIGATOR'S DAY AT SEA 151 -fll. The variation being -- 10, the deviation must beZ) = 11 - ( - 10=) + 21. The deviation table made this deviation + 18, so that table seems to require a correc- tion of +3. The pelorus observation of 9 : 45 gave a correc- tion of + 2 for the deviation table ; and as this is now apparently confirmed, the captain decides to examine the chart again, before finally shaping course for the night, to see if the yacht has not perhaps moved into a region where the variation is different from the Sandy Hook variation so far used. At 5 the log read 182.0, course was still 182 true. The captain now prepared to shape the course for the night, and to change his course, if necessary, at 6 : 00. His first step was to obtain the D. R. position at 6 : 00, starting from the observed position at 4 : 12. This gave position at 6 : 00, by D. R. : lat. 37 41' ; long. 73 55'. The easterly current l of about 2' per hour set the yacht farther east about 3' between 4 : 12 and 6 : 00. Therefore he took the D. R. position at 6 : 00 to be lat. 37 41' ; long. 73 52'. The posi- tion of the point of destination, 12 miles east of Watlings Island, is still : lat. 23 57' ; long. 74 15'. The true course and distance to that point from the yacht's 6 : 00 position is therefore, by traverse table : course 181 J ; dist. 824 miles. A further examination of the track chart shows that the variation, which was 10 at Sandy Hook, is now 8. The compass error, from the last pelorus observation, was + 11. Consequently, by the pelorus observation, the compass course for the night should be 181J 11 = 170?, or S.fE. (see the Table on p. 52). Furthermore, the variation being now 8 and the error + 11 makes the deviation D = E - V = + 11 - (- 8) = + 19. The com- pass adjuster's deviation of -f 18 is therefore vindicated, and the compass course S.JE. can be set for the night. At 6 the log read 197.2, course S.fE., or 182* true. 1 Doubtless the Gulf Stream, 152 NAVIGATION In conclusion, the captain of the Nav hopes he has been able to make his imagined proceedings clear enough to help the young navigator in planning his own first day's work at sea. May it be the first of many happy and successful days. And let him not forget, when attempting to verify the various calculations and problems of the Nav, that every observation in this book has been prepared by calculation, and none is the result of actual sextant observing. Should inconsistencies or errors be found by any young navigator, it is hoped that he will make them known so that they may be corrected, in case the Nav shall be required to make another voyage in a second edition. LIST OF TABLES 1. Traverse Table; explained on pages 10 and 19; and its use in the Sumner method on pages 113, 135 154 2. Conversion of longitude difference and departure; ex- plained on page 16 168 3. Number logarithms ; explained on page 23 178 4. Trigonometric logarithms ; explained on page 31 196 5. Meridional parts ; explained on page 35 241 6. Sextant Correction Table ; explained on page 72 247 7. Dip correction ; explained on page 73 247 8. Conversion of hours and minutes into decimals of a day ; explained on page 80 248 9. Conversion of degrees and minutes of longitude and hours and minutes of time 249 10. Haversines ; explained on page 99 250 11. Azimuth Table ; explained on page 113 284 12. Auxiliary Azimuth Table ; explained on page 115 290 13. Kelvin's Sumner Line Table ; explained on page 126 292 14. Sumner Intersection Table; .explained on page 135 318 PUBLISHERS' NOTE Table 3, Number Logarithms, has been reprinted from "The Macmillan Logarithmic and Trigonometric Tables," New York, 1917. 153 154 Table 1. Traverse Table 1 2 i Pt. 3 4 5 f Pt. 6 7 (179, 181, (178, 182 (177, 183 (176, 184 (175, 185 (174, 186 (173, 187, DlST 359) 358) 357) 356) 355) 354) 353) Lat. Dep. Lat. Dep Lat. Dep Lat. Dep Lat. Dep Lat. Dep Lat. Dep. 1 1.0 0.0 1.0 0.0 1.0 0.1 1.0 0.1 1.0 0.1 1.0 0.1 1.0 0.1 2 2.0 0.0 2.0 0.1 2.0 0.1 2.0 0.1 2.0 0.2 2.0 0.2 2.0 0.2 3 3.0 0.1 3.0 0.1 3.0 0.2 3.0 0.2 3.0 0.3 3.0 0.3 3.0 0.4 4 4.0 0.1 4.0 0.1 4.0 0.2 4.0 0.3 4.0 0.3 4.0 0.4 4.0 0.5 5 5.0 0.1 5.0 0.2 5.0 0.3 5.0 0.3 5.0 0.4 5.0 0.5 5.0 0.6 6 6.0 0.1 6.0 0.2 6.0 0.3 6.0 0.4 6.0 0.5 6.0 0.6 6.0 0.7 7 7.0 0.1 7.0 0.2 7.0 0.4 7.0 0.5 7.0 0.6 7.0 0.7 6.9 0.9 8 8.0 0.1 8.0 0.3 8.0 0.4 8.0 0.6 8.0 0.7 8.0 0.8 7.9 1.0 9 9.0 0.2 9.0 0.3 9.0 0.5 9.0 0.6 9.0 0.8 9.0 0.9 8.9 1.1 10 10.0 0.2 10.0 0.3 10.0 0.5 10.0 0.7 10.0 0.9 9.9 1.0 9.9 1.2 11 11.0 0.2 11.0 0.4 11.0 0.6 11.0 0.8 11.0 1.0 10.9 1.1 10.9 1.3 12 12.0 0.2 12.0 0.4 12.0 0.6 12.0 0.8 12.0 1.0 11.9 1.3 11.9 1.5 13 13.0 0.2 13.0 0.5 13.0 0.7 13.0 0.9 13.0 1.1 12.9 1.4 12.9 1.6 14 14.0 0.2 14.0 0.5 14.0 0.7 14.0 1.0 13.9 1.2 13.9 1.5 13.9 1.7 15 15.0 0.3 15.0 0.5 15.0 0.8 15.0 1.0 14.9 1.3 14.9 1.6 14.9 1.8 16 16.0 0.3 16.0 0.6 16.0 0.8 16.0 1.1 15.9 1.4 15.9 1.7 15.9 1.9 17 17.0 0.3 17.0 0.6 17.0 0.9 17.0 1.2 16.9 1.5 16.9 1.8 16.9 2.1 18 18.0 0.3 18.0 0.6 18.0 0.9 18.0 1.3 17.9 1.6 17.9 1.9 17.9 2.2 19 19.0 0.3 19.0 0.7 19.0 1.0 19.0 1.3 18.9 1.7 18.9 2.0 18.9 2.3 20 20.0 0.3 20.0 0.7 20.0 1.0 20.0 1.4 19.9 1.7 19.9 2.1 19.9 2.4 21 21.0 0.4 21.0 0.7 21.0 1.1 20.9 1.5 20.9 1.8 20.9 2.2 20.8 2.6 22 22.0 0.4 22.0 0.8 22.0 1.2 21.9 1.5 21.9 1.9 21.9 2.3 21.8 2.7 23 23.0 0.4 23.0 0.8 23.0 1.2 22.9 1.6 22.9 2.0 22.9 2.4 22.8 2.8 24 24.0 0.4 24.0 0.8 24.0 1.3 23.9 1.7 23.9 2.1 23.9 2.5 23.8 2.9 25 25.0 0.4 25.0 0.9 25.0 1.3 24.9 1.7 24.9 2.2 24.9 2.6 24.8 3.0 26 26.0 0.5 26.0 0.9 26.0 1.4 25.9 1.8 25.9 2.3 25.9 2.7 25.8 3.2 27 27.0 0.5 27.0 0.9 27.0 1.4 26.9 1.9 26.9 2.4 26.9 2.8 26.8 3.3 28 28.0 0.5 28.0 1.0 28.0 1.5 27.9 2.0 27.9 2.4 27.8 2.9 27.8 3.4 29 29.0 0.5 29.0 1.0 29.0 1 5 28.9 2.0 28.9 2.5 28.8 3.0 28.8 3.5 30 30.0 0.5 30.0 1.0 30.0 i!e 29.9 2.1 29.9 2.6 29.8 3.1 29.8 3.7 31 31.0 0'.5 31.0 1 1 31.0 1 6 30.9 ?,?. 30.9 9 7 30.8 32 30.8 38 32 32.0 0.6 32.0 1 i 32.0 1 7 31.9 9 9 31.9 > 8 31.8 3 3 31.8 3 9 33 33.0 0.6 33.0 1.2 33.0 1.7 32.9 2.3 32.9 2.9 32.8 3.4 32.8 4.0 34 34.0 0.6 34.0 1.2 34.0 1.8 33.9 2.4 33.9 3.0 33.8 3.6 33.7 4.1 35 35.0 0.6 35.0 1.2 35.0 1.8 34.9 2.4 34.9 3.1 34.8 3.7 34.7 4.3 36 36.0 0.6 36.0 1.3 36.0 1.9 35.9 2.5 35.9 3.1 35.8 3.8 35.7 4.4 37 37.0 0.6 37.0 1.3 36.9 1.9 36.9 2.6 36.9 3.2 36.8 3.9 36.7 4.5 38 38.0 0.7 38.0 1.3 37.9 2.0 37.9 2.7 37.9 3.3 37.8 4.0 37.7 4.6 39 39.0 0.7 39.0 1.4 38.9 2.0 38.9 2.7 38.9 3.4 38.8 4.1 38.7 4.8 40 40.0 0.7 40.0 .4 39.9 2.1 39.9 2.8 39.8 3.5 39.8 4.2 39.7 4.9 41 41.0 0.7 41.0 .4 40.9 2.1 40.9 2.9 40.8 3.6 40.8 4.3 40.7 5.0 42 42.0 0.7 42.0 c .0 41.9 2.2 41.9 2.9 41.8 3.7 41.8 4.4 41.7 5.1 43 43.0 0.8 43.0 .5 42.9 2.3 42.9 3.0 42.8 3.7 42.8 4.5 42.7 5.2 44 44.0 0.8 44.0 .5 43.9 2.3 43.9 3.1 43.8 3.8 43.8 4.6 43.7 5.4 45 45.0 0.8 45.0 .6 44.9 2.4 44.9 3.1 44.8 3.9 44.8 4.7 44.7 5.5 46 46.0 0.8 46.0 .6 45.9 2.4 45.9 3.2 45.8 4.0 45.7 4.8 45.7 5.6 47 47.0 0.8 47.0 .6 46.9 2.5 46.9 3.3 46.8 4.1 46.7 4.9 46.6 5.7 48 48.0 0.8 48.0 .7 47.9 2.5 47.9 3.3 47.8 4.2 47.7 5.0 47.6 5.8 49 49.0 0.9 49.0 .7 48.9 2.6 48.9 3.4 48.8 4.3 48.7 5.1 48.6 6.0 50 50.0 0.9 50.0 .7 49.9 2.6 49.9 3.5 49.8 4.4 49.7 5.2 49.6 6.1 100 00.0 1.7 99.9 3.5 99.9 5.2 99.8 7.0 99.6 8.7 99.5 10.5 99.3 12.2 200 200.0 3.5 199.9 7.0 199.7 10.5 199.5 14.0 199.2 17.4 198.9 20.9 198.5 24.4 300 300.0 5.2 299.8 10.5 299.6 15.7 299.3 20.9 298.9 26.1 298.4 31.4 297.8 36.6 400 399.9 7.0 399.8 13.9 399.4 20.9 399.0 27.9 398.5 34.9 397.8 41.8 397.0 48.7 500 499.9 8.8 499.7 17.4 499.3 26.2 498.8 34.8 498.1 43.6 497.3 52.3 496.3 61.0 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. (91, 269 (92, 268, (93, 267, (94, 266, (95 265, (96, 264, (97, 263, 271) 272) 273) 274) 275) 276) 277) 89 88 7|Pt.87 86 85 7iPt.84 83 Table 1. Traverse Table 1 2 1 Pt. 3 4 5 \ Pt. 6 7 DlST (179, 181 359) (178, 182, 358) (177, 183, 357) (176, 184, 356) (175, 185, 355) (174, 186, 354) (173, 187, 353) Lat. Dep Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 51 51.0 0.9 51.0 1.8 50.9 2.7 50.9 3.6 50.8 4.4 50.7 5.3 50.6 6.2 52 52.0 0.9 52.0 1.8 51.9 2.7 51.9 3.6 51.8 4.5 51.7 5.4 51.6 6.3 53 53.0 0.9 53.0 1.8 52.9 2.8 52.9 3.7 52.8 4.6 52.7 5.5 52.6 6.5 54 54.0 0.9 54.0 1.9 53.9 2.8 53.9 3.8 53.8 4.7 53.7 5.6 53.6 6.6 55 55.0 1.0 55.0 1.9 54.9 2.9 54.9 3.8 54.8 4.8 54.7 5.7 54.6 6.7 56 56.0 1.0 56.0 2.0 55.9 2.9 55.9 3.9 55.8 4.9 55.7 5.9 55.6 6.8 57 57.0 1.0 57.0 2.0 56.9 3.0 56.9 4.0 56.8 5.0 56.7 6.0 56.6 6.9 58 58.0 1.0 58.0 2.0 57.9 3.0 57.9 4.0 57.8 5.1 57.7 6.1 57.6 7.1 59 59.0 1.0 59.0 2.1 58.9 3.1 58.9 4.1 58.8 5.1 58.7 6.2 58.6 7.2 60 60.0 1.0 60.0 2.1 59.9 3.1 59.9 4.2 59.8 5.2 59.7 6.3 59.6 7.3 61 61.0 1.1 61.0 2.1 60.9 3.2 60.9 4.3 60.8 5.3 60.7 6.4 60.5 7.4 62 62.0 1.1 62.0 2.2 61.9 3.2 61.8 4.3 61.8 5.4 61.7 6.5 61.5 7.6 63 63.0 1.1 63.0 2.2 62.9 3.3 62.8 4.4 62.8 5.5 62.7 6.6 62.5 7.7 64 64.0 1.1 64.0 2.2 63.9 3.3 63.8 4.5 63.8 5.6 63.6 6.7 63.5 7.8 65 65.0 1.1 65.0 2.3 64.9 3.4 64.8 4.5 64.8 5.7 64.6 6.8 64.5 7.9 66 66.0 1.2 66.0 2.3 65.9 3.5 65.8 4.6 65.7 5.8 65.6 6.9 65.5 8:0 67 67.0 1.2 67.0 2.3 66.9 3.5 66.8 4.7 66.7 5.8 66.6 7.0 66.5 8.2 68 68.0 1.2 68.0 2.4 67.9 3.6 67.8 4.7 67.7 5.9 67.6 7.1 67.5 8.3 69 69.0 1.2 69.0 2.4 68.9 3.6 68.8 4.8 68.7 6.0 68.6 7.2 68.5 8.4 70 70.0 1.2 70.0 2.4 69.9 3.7 69.8 4.9 69.7 6.1 69.6 7.3 69.5 8.5 71 71.0 1.2 71.0 2.5 70.9 3.7 70.8 5.0 70.7 6.2 70.6 7.4 70.5 &7 72 72.0 1.3 72.0 2.5 71.9 3.8 71.8 5.0 71.7 6.3 71.6 7.5 71.5 8.8 73 73.0 1.3 73.0 2.5 72.9 3.8 72.8 5.1 72.7 6.4 72.6 7.6 72.5 8.9 74 74.0 1.3 74.0 2.6 73.9 3.9 73.8 5.2 73.7 6.4 73.6 7.7 73.4 9.0 75 75.0 1.3 75.0 2.6 74.9 3.9 74.8 5.2 74.7 6.5 74.6 7.8 74.4 9.1 76 76.0 1.3 76.0 2.7 75.9 4.0 75.8 5.3 75.7 6.6 75.6 7.9 75.4 9.3 77 77.0 1.3 77.0 2.7 76.9 4.0 76.8 5.4 76.7 6.7 76.6 8.0 76.4 9.4 78 78.0 1.4 78.0 2.7 77.9 4.1 77.8 5.4 77.7 6.8 77.6 8.2 77.4 9.5 79 79.0 1.4 79.0 2.8 78.9 4.1 78.8 5.5 78.7 6.9 78.6 8.3 78.4 9.6 80 80.0 1.4 80.0 2.8 79.9 4.2 79.8 5.6 79.7 7.0 79.6 8.4 79.4 9.7 81 81.0 1.4 81.0 2.8 80.9 4.2 80.8 5.7 80.7 7.1 80.6 8.5 80.4 9.9 82 82.0 1.4 82.0 2.9 81.9 4.3 81.8 5.7 81.7 7.1 81.6 8.6 81.4 10.0 83 83.0 1.4 82.9 2.9 82.9 4.3 82.8 5.8 82.7 7.2 82.5 8.7 82.4 10.1 84 84.0 1.5 83.9 2.9 83.9 4.4 83.8 5.9 83.7 7.3 83.5 8.8 83.4 10.2 85 85.0 1.5 84.9 3.0 84.9 4.4 84.8 5.9 84.7 7.4 84.5 8.9 84.4 10.4 86 86.0 1.5 85.9 3.0 85.9 4.5 85.8 6.0 85.7 7.5 85.5 9.0 85.4 10.5 87 87.0 1.5 86.9 3.0 86.9 4.6 86.8 6.1 86.7 7.6 86.5 9.1 86.4 10.6 88 88.0 1.5 87.9 3.1 87.9 4.6 87.8 6.1 87.7 7.7 87.5 9.2 87.3 10.7 89 89.0 1.6 88.9 3.1 88.9 4.7 88.8 6.2 88.7 7.8 88.5 9.3 88.3 10.8 90 90.0 1.6 89.9 3.1 89.9 4.7 89.8 6.3 89.7 7.8 89.5 9.4 89.3 11.0 " 91 91.0 1.6 90.9 3.2 90.9 4.8 90.8 6.3 90.7 7.9 90.5 9.5 90.3 11.1 92 92.0 1.6 91.9 3.2 91.9 4.8 91.8 6.4 91.6 8.0 91.5 9.6 91.3 11.2 93 93.0 1.6 92.9 3.2 92.9 4.9 92.8 6.5 92.6 8.1 92.5 9.7 92.3 11.3 94 94.0 1.6 93.9 3.3 93.9 4.9 93.8 6.6 93.6 8.2 93.5 9.8 93.3 11.5 95 95.0 1.7 94.9 3.3 94.9 5.0 94.8 6.6 94.6 8.3 94.5 9.9 94.3 11.6 96 96.0 1.7 95.9 3.4 95.9 5.0 95.8 6.7 95.6 8.4 95.5 10.0 95.3 11.7 97 97.0 1.7 96.9 3.4 96.9 5.1 96.8 6.8 96.6 8.5 96.5 10.1 96.3 11.8 98 98.0 1.7 97.9 3.4 97.9 5.1 97.8 6.8 97.6 8.5 97.5 10.2 97.3 11.9 99 99.0 1.7 98.9 3.5 98.9 5.2 98.8 6.9 98.6 8.6 98.5 10.3 98.3 12.1 100 100.0 1.7 99.9 3.5 99.9 5.2 99.8 7.0 99.6 8.7 99.5 10.5 99.3 12.2 600 599.9 10.5 599.6 20.9 599.2 31.4 598.6 41.9 597.7 52.3 596.7 62.7 595.5 73.1 700 699.8 12.2 699.5 24.4 699.0 36.6 698.2 48.8 697.2 61.0 696.1 73.2 694.9 85.3 800 799.8 14.0 799.5 27.9 798.9 41.9 798.0 55.8 796.9 69.7 795.6 83.6 794.1 97.5 900 899.7 15.7 899.3 31.4 898.6 47.1 897.6 62.8 896.4 78.4 895.0 94.1 893.3 109.6 Dep. LaT Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. (91, 269, (92, 268 (93, 267, (94, 266, (95, 265, (96, 264, (97, 263, 271) 272) 273) 274) 275) 276) 277) 89 88 7f Pt. 87 86 85 7 Pt. 84 83 156 Table 1. Traverse Table f Pt. 8 9 10 1 Pt. 11 12 13 1 1 Pt. 14 DlST. (172, 188, 352) (171, 189, 351) (170, 190, 350) (169, 191, 349) (168, 192, 348) (167, 193, 347) (166, 194, 346) Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 1 1.0 0.1 1.0 0.2 1.0 0.2 1.0 0.2 1.0 0.2 1.0 0.2 1.0 0.2 2 2.0 0.3 2.0 0.3 2.0 0.3 2.0 0.4 2.0 0.4 1.9 0.4 1.9 0.5 3 3.0 0.4 3.0 0.5 3.0 0.5 2.9 0.6 .2.9 0.6 2.9 0.7 2.9 0.7 4 4.0 0.6 4.0 0.6 3.9 0.7 3.9 0.8 3.9 0.8 3.9 0.9 3.9 1.0 5 5.0 0.7 4.9 0.8 4.9 0.9 4.9 1.0 4.9 .0 4.9 1.1 4.9 1.2 6 5.9 0.8 5.9 0.9 5.9 .0 5.9 1.1 5.9 2 5.8 1.3 5.8 1.5 7 6.9 .0 6.9 1.1 6.9 .2 6.9 1.3 6.8 '.5 6.8 1.6 6.8 1.7 8 7.9 .1 7.9 1.3 7.9 .4 7.9 1.5 7.8 .7 7.8 1.8 7.8 1.9 9 8.9 .3 8.9 1.4 8.9 .6 8.8 1.7 8.8 .9 8.8 2.0 8.7 2.2 10 9.9 .4 9.9 1.6 9.8 .7 9.8 1.9 9.8 2.1 9.7 2.2 9.7 2.4 11 10.9 .5 10.9 1.7 10.8 1.9 10.8 2.1 10.8 2.3 10.7 2.5 10.7 2.7 12 11.9 .7 11.9 1.9 11.8 2.1 11.8 2.3 11.7 2.5 11.7 2.7 11.6 2.9 13 12.9 .8 12.8 2.0 12.8 2.3 12.8 2.5 12.7 2.7 12.7 2.9 12.6 3.1 14 13.9 1.9 13.8 2.2 13.8 2.4 13.7 2.7 13.7 2.9 13.6 3J 13.6 3.4 15 14.9 2.1 14.8 2.3 14.8 2.6 14.7 2.9 14.7 3.1 14.6 3.4 14.6 3.6 16 15.8 2.2 15.8 2.5 15.8 2.8 15.7 3.1 15.7 3.3 15.6 3.6 15.5 3.9 17 16.8 2.4 16.8 2.7 16.7 3.0 16.7 3.2 16.6 3.5 16.6 3.8 16.5 4.1 18 17.8 2.5 17.8 2.9 17.7 3.1 17.7 3.4 17.6 3.7 17.5 4.0 17.5 4.4 19 18.8 2.6 18.8 3.0 18.7 3.3 18.7 3.6 18.6 4.0 18.5 4.3 18.4 4.6 20 19.8 2.8 19.8 3.1 19.7 3.5 19.6 3.8 19.6 4.2 19.5 4.5 19.4 4.8 21 20.8 2.9 20.7 3.3 20.7 3.6 20.6 4.0 20.5 4.4 20.5 4.7 20.4 5.1 22 21.8 3.1 21.7 3.4 21.7 3.8 21.6 4.2 21.5 4.6 21.4 4.9 21.3 5.3 23 22.8 3.2 22.7 3.6 22.7 4.0 22.6 4.4 22.5 4.8 22.4 5.2 22.3 5.6 24 23.8 3.3 23.7 3.8 23.6 4.2 23.6 4.6 23.5 5.0 23.4 5.4 23.3 5.8 25 24.8 3.5 24.7 3.9 24.6 4.3 24.5 4.8 24.5 5.2 24.4 5.6 24.3 6.0 26 25.7 3.6 25.7 4.1 25.6 4.5 25.5 5.0 25.4 5.4 25.3 5.8 25.2 6.3 27 26,7 3.8 26.7 4.2 26.6 4.7 26.5 5.2 26.4 5.6 26.3 6.1 26.2 6.5 28 27.7 3.9 27.7 4.4 27.6 4.9 27.5 5.3 27.4 5.8 27.3 6.3 27.2 6.8 29 28.7 4.0 28.6 4.5 28.6 5.0 28.5 5.5 28.4 6.0 28.3 6.5 28.1 7.0 30 29.7 4.2 29.6 4.7 29.5 5.2 29.4 5.7 29.3 6.2 29.2 6.7 29.1 7.3 31 30.7 4.3 30.6 4.8 30.5 5.4 30.4 5.9 30.3 6.4 30.2 7.0 30.1 7.5 32 31.7 4.5 31.6 5.0 31.5 5.6 31.4 6.1 31.3 6.7 31.2 7.2 31.0 7.7 33 32.7 4.6 32.6 5.2 32.5 5.7 32.4 6.3 32.3 6.9 32.2 7.4 32.0 8.0 34 33.7 4.7 33.6 5.3 33.5 5.9 33.4 6.5 33.3 7.1 33.1 7.6 33.0 8.2 35 34.7 4.9 34.6 5.5 34.5 6.1 34.4 6.7 34.2 7.3 34.1 7.9 34.0 8.5 36 35.6 5.0 35.6 5.6 35.5 6.3 35.3 6.9 35.2 7.5 35.1 8.1 34.9 8.7 37 36.6 5.1 36.5 5.8 36.4 6.4 36.3 7.1 36.2 7.7 36.1 8.3 35.9 9.0 38 37.6 5.3 37.5 5.9 37.4 6.6 37.3 7.3 37.2 7.9 37.0 8.5 36.9 9.2 39 38.6 5.4 38.5 6.1 38.4 6.8 38.3 7.4 38.1 8.1 38.0 8.8 37.8 9.4 40 39.6 5.6 39.5 6.3 39.4 6.9 39.3 7.6 39.1 8.3 39.0 9.0 38.8 9.7 41 40.6 5.7 40.5 6.4 40.4 7.1 40.2 7.8 40.1 8.5 39.9 9.2 39.8 9.9 42 41.6 5.8 41.5 6.6 41.4 7.3 41.2 8.0 41.1 8.7 40.9 9.4 40.8 10.2 43 42.6 6.0 42.5 6.7 42.3 7.5 42.2 8.2 42.1 8.9 41.9 9.7 41.7 10.4 44 43.6 6.1 43.5 6.9 43.3 7.6 43.2 8.4 43.0 9.1 42.9 9.9 42.7 10.6 45 44.6 6.3 44.4 7.0 44.3 7.8 44.2 8.6 44.0 9.4 43.8 10.1 43.7 10.9 46 45.6 6.4 45.4 7.2 45.3 8.0 45.2 8.8 45.0 9.6 44.8 10.3 44.6 11.1 47 46.5 6.5 46.4 7.4 46.3 8.2 46.1 9.0 46.0 9.8 45.8 10.6 45.6 11.4 48 47.5 6.7 47.4 7.5 47.3 8.3 47.1 9.2 47.0 10.0 46.8 10.8 46.6 11.6 49 48.5 6.8 48.4 7.7 48.3 8.5 48.1 9.3 47.9 10.2 47.7 11.0 47.5 11.9 50 49.5 7.0 49.4 7.8 49.2 8.7 49.1 9.5 48.9 10.4 48.7 11.2 48.5 12.1 100 99.0 13.9 98.8 15.6 98.5 17.4 98.2 19.1 97.8 20.8 97.4 22.5 97.0 24.2 200 198.1 27.8 197.5 31.3 197.0 34.7 196.3 38.2 195.6 41.6 194.9 45.0 194.1 48.4 300 297.1 41.8 296.3 46.9 295.4 52.1 294.5 57.2 293.4 62.4 292.3 67.5 291.1 72.6 400 396.1 55.7 395.1 62.6 393.9 69.5 392.6 76.3 391.3 83.1 389.8 90.0 388.1 96.7 500 495.1 69.6 493.8 78.2 492.4 86.8 490.8 95.4 489.1 104.0 487.2 112.4 485.1 121.0 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. (98, 262, (99, 261, (100, 260, (101, 259, (102, 258, (103, 257, (104, 256, 278) 279) 280) 281) 282) 283) 284) 7i Pt. 82 81 80 7 Pt. 79 78 77 6 f Pt. 76 The 1-Pt. or 11 Courses are : N. by E., N. by W., S. by E., S. by W. Table 1. Traverse Table f Pt. 8 9 10 1 Pt. 11 12 13 UPt. 14 (172, 188, (171, 189. (170, 190, (169, 191, (168, 192, (167, 193, (166, 194, DlST. 352) 351) 350) 349) 348) 347) 346) Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 51 50.5 7.1 50.4 8.0 50.2 8.9 50.1 9.7 49.9 10.6 49.7 11.5 49.5 12.3 52 51.5 7.2 51.4 8.1 51.2 9.0 51.0 9.9 50.9 10.8 50.7 11.7 50.5 12.6 53 52.5 7.4 52.3 8.3 52.2 9.2 52.0 10.1 51.8 11.0 51.6 11.9 51.4 12.8 54 53.5 7.5 53.3 8.4 53.2 9.4 53.0 10.3 52.8 11.2 52.6 12.1 52.4 13.1 55 54.5 7.7 54.3 8.6 54.2 9.6 54.0 10.5 53.8 11.4 53.6 12.4 53.4 13.3 56 55.5 7.8 55.3 8.8 55.1 9.7 55.0 10.7 54.8 11.6 54.6 12.6 54.3 13.5 57 56.4 7.9 56.3 8.9 56.1 9.9 56.0 10.9 55.8 11.9 55.5 12.8 55.3 13.8 :,s 57.4 8.1 57.3 9.1 57.1 10.1 56.9 11.1 56.7 12.1 56.5 13.0 56.3 14.0 59 58.4 8.2 58.3 9.2 58.1 10.2 57.9 11.3 57.7 12.3 57.5 13.3 57.2 14.3 60 59.4 8.4 59.3 9.4 59.1 10.4 58.9 11.4 58.7 12.5 58.5 13.5 58.2 14.5 61 60.4 8.5 60.2 9.5 60.1 10.6 59.9 11.6 59.7 12.7 59.4 13.7 59.2 14.8 62 61.4 8.6 61.2 9.7 61.1 10.8 60.9 11.8 60.6 12.9 60.4 13.9 60.2 15.0 63 62.4 8.8 62.2 9.9 62.0 10.9 61.8 12.0 61.6 13.1 61.4 14.2 61.1 15.2 64 63.4 8.9 63.2 10.0 63.0 11.1 62.8 12.2 62.6 13.3 62.4 14.4 62.1 15.5 65 64.4 9.0 64.2 10.2 64.0 11.3 63.8 12.4 63.6 13.5 63.3 14.6 63.1 15.7 66 65.4 9.2 65.2 10.3 65.0 11.5 64.8 12.6 64.6 13.7 64.3 14.8 64.0 16.0 67 66.3 9.3 66.2 10.5 66.0 11.6 65.8 12.8 65.5 13.9 65.3 15.1 65.0 16.2 68 67.3 9.5 67.2 10.6 67.0 11.8 66.8 13.0 66.5 14.1 66.3 15.3 66.0 16.5 69 68.3 9.6 68.2 10.8 68.0 12.0 67.7 13.2 67.5 14.3 67.2 15.5 67.0 16.7 70 69.3 9.7 69.1 11.0 68.9 12.2 68.7 13.4 68.5 14.6 68.2 15.7 67.9 16.9 71 70.3 9.9 70.1 11.1 69.9 12.3 69.7 13.5 69.4 14.8 69.2 16.0 68.9 17.2 72 71.3 10.0 71.1 11.3 70.9 12.5 70.7 13.7 70.4 15.0 70.2 16.2 69.9 17.4 73 72.3 10.2 72.1 11.4 71.9 12.7 71.7 13.9 71.4 15.2 71.1 16.4 70.8 17.7 74 73.3 10.3 73.1 11.6 72.9 12.8 72.6 14.1 72.4 15.4 72.1 16.6 71.8 17.9 75 74.3 10.4 74.1 11.7 73.9 13.0 73.6 14.3 73.4 15.6 73.1 16.9 72.8 18.1 76 75.3 10.6 75.1 11.9 74.8 13.2 74.6 14.5 74.3 15.8 74.1 17.1 73.7 18.4 77 76.3 10.7 76.1 12.0 75.8 13.4 75.6 14.7 75.3 16.0 75.0 17.3 74.7 18.6 78 77.2 10.9 77.0 12.2 76.8 13.5 76.6 14.9 76.3 16.2 76.0 17.5 75.7 18.9 79 78.2 11.0 78.0 12.4 77.8 13.7 77.5 15.1 77.3 16.4 77.0 17.8 76.7 19.1 80 79.2 11.1 79.0 12.5 78.8 13.9 78.5 15.3 78.3 16.6 77.9 18.0 77.6 19.4 81 80.2 11.3 80.0 12.7 79.8 14.1 79.5 15.5 79.2 16.8 78.9 18.2 78.6 19.6 82 81.2 11.4 81.0 12.8 80.8 14.2 80.5 15.6 80.2 17.0 79.9 18.4 79.6 19.8 83 82.2 11.6 82.0 13.0 81.7 14.4 81.5 15.8 81.2 17.3 80.9 18.7 80.5 20.1 84 83.2 11.7 83.0 13.1 82.7 14.6 82.5 16.0 82.2 17.5 81.8 18.9 81.5 20.3 85 84.2 11.8 S4.0 13.3 83.7 14.8 83.4 16.2 83.1 17.7 82.8 19.1 82.5 20.6 86 85.2 12.0 84.9 13.5 84.7 14.9 84.4 16.4 84.1 17.9 83.8 19.3 83.4 20.8 87 86.2 12.1 85.9 13.6 85.7 15.1 85.4 16.6 85.1 18.1 84.8 19.6 84.4 21.0 88 87.1 12.2 86.9 13.8 86.7 15.3 86.4 16.8 86.1 18.3 85.7 19.8 85.4 21.3 89 88.1 12.4 87.9 13.9 87.6 15.5 87.4 17.0 87.1 18.5 86.7 20.0 86.4 21.5 90 89.1 12.5 88.9 14.1 88.6 15.6 88.3 17.2 88.0 18.7 87.7 20.2 87.3 21.8 91 90.1 12.7 89.9 14.2 89.6 15.8 89.3 17.4 89.0 18.9 88.7 20.5 88.3 22.0 92 91.1 12.8 90.9 14.4 90.6 16.0 90.3 17.6 90.0 19.1 89.6 20.7 89.3 22.3 93 92.1 12.9 91.9 14.5 91.6 16.1 91.3 17.7 91.0 19.3 90.6 20.9 90.2 22.5 94 93.1 13.1 92.8 14.7 92.6 16.3 92.3 17.9 91.9 19.5 91.6 21.1 91.2 22.7 95 94.1 13.2 93.8 14.9 93.6 16.5 93.3 18.1 92.9 19.8 92.6 21.4 92.2 23.0 96 95.1 13.4 94.8 15.0 94.5 16.7 94.2 18.3 93.9 20.0 93.5 21.6 93.1 23.2 97 96.1 13.5 95.8 15.2 95.5 16.8 95.2 18.5 94.9 20.2 94.5 21.8 94.1 23.5 98 97.0 13.6 96.8 15.3 96.5 17.0 96.2 18.7 95.9 20.4 95.5 22.0 95.1 23.7 99 98.0 13.8 97.8 15.5 97.5 17.2 97.2 18.9 96.8 20.6 96.5 22.3 96.1 24.0 100 99.0 13.9 98.8 15.6 98.5 17.4 98.2 19.1 97.8 20.8 97.4 22.5 97.0 24.2 BOO 594.2 83.5 592.6 93.8 590.9 104.2 589.0 114.5 586.9 124.7 584.6 135.0 582.2 145.1 700 693.3 97.4 691.3 109.4 689.5 121.5 687.1 133.6 684.7 145.5 682.1 157.5 679.2 169.3 son 792.3 111.4 790.2 125.1 787.9 139.0 785.2 152.6 782.5 166.3 779.4 180.0 776.2 193.6 900 891.3 125.2 888.8 140.8 886.3 156.3 883.3 171.7 880.2 187.1 876.8 202.4 873.2 217.7 Dep. Lat. Dep. Lat. Dep. Lat. Dep Lat. Dep. Lat. Dep. Lat. Dep. Lat. (98, 262, (99, 261, (100, 260, (101, 259, (102, 258, (103, 257, (104, 256, 278) 279) 280) 281) 282) 283) 284) 7-1 Pt. 82 81 80 7 Pt. 79 78 77 6f Pt. 76 The 7-Pt. or 79 Courses are : E. by N., W. by N., E. by S., W. by S. 158 Table 1. Traverse Table 15 16 UPt.l7 18 19 If Pt. 20 (165, 195, (164, 196, (163, 197, (162, 198, (161, 199, (160, 200, DlST. 345) 344) 343) 342) 341) 340) Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 1 1.0 0.3 1.0 0.3 1.0 0.3 1.0 0.3 0.9 0.3 0.9 0.3 2 1.9 0.5 1.9 0.6 1.9 0.6 1.9 0.6 1.9 0.7 1.9 0.7 3 2.9 0.8 2.9 0.8 2.9 0.9 2.9 0.9 2.8 1.0 2.8 1.0 4 3.9 1.0 3.8 1.1 3.8 1.2 3.8 1.2 3.8 1.3 3.8 1.4 5 4.8 1.3 4.8 1.4 4.8 1.5 4.8 1.5 4.7 1.6 4.7 1.7 6 5.8 1.6 5.8 1.7 5.7 1.8 5.7 1.9 5.7 2.0 5.6 2.1 7 6.8 1.8 6.7 1.9 6.7 2.0 6.7 2.2 6.6 2.3 6.6 2.4 8 7.7 2.1 7.7 2.2 7.7 2.3 . 7.6 2.5 7.6 2.6 7.5 2.7 9 8.7 2.3 8.7 2.5 8.6 2.6 8.6 2.8 8.5 2.9 8.5 3.1 10 9.7 2.6 9.6 2.8 9.6 2.9 9.5 3.1 9.5 3.3 9.4 3.4 11 10.6 2.8 10.6 3.0 10.5 3.2 10.5 3.4 10.4 3.6 10.3 3.8 12 11.6 3.1 11.5 3.3 11.5 3.5 11.4 3.7 11.3 3.9 11.3 4.1 13 12.6 3.4 12.5 3.6 12.4 3.8 12.4 4.0 12.3 4.2 12.2 4.4 14 13.5 3.6 13.5 3.9 13.4 4.1 13.3 4.3 13.2 4.6 13.2 4.8 15 14.5 3.9 14.4 4.1 14.3 4.4 14.3 4.6 14.2 4.9 14.1 5.1 16 15.5 4.1 15.4 4.4 15.3 4.7 15.2 4.9 15.1 5.2 15.0 5.5 17 16.4 4.4 16.3 4.7 16.3 5.0 16.2 5.3 16.1 5.5 16.0 5.8 18 17.4 4.7 17.3 5.0 17.2 5.3 17.1 5.6 17.0 5.9 16.9 6.2 19 18.4 4.9 18.3 5.2 18.2 5.6 18.1 5.9 18.0 6.2 17.9 6.5 20 19.3 5.2 19.2 5.5 19.1 5.8 19.0 6.2 18.9 6.5 18.8 6.8 21 20.3 5.4 20.2 5.8 20.1 6.1 20.0 6.5 19.9 6.8 19.7 7.2 22 21.3 5.7 21.1 6.1 21.0 6.4 20.9 6.8 20.8 7.2 20.7 7.5 23 22.2 6.0 22.1 6.3 22.0 6.7 21.9 7.1 21.7 7.5 21.6 7.9 24 23.2 6.2 23.1 6.6 23.0 7.0 22.8 7.4 22.7 7.8 22.6 8.2 25 24.1 6.5 24.0 6.9 23.9 7.3 23.8 7.7 23.6 8.1 23.5 8.6 26 25.1 6.7 25.0 7.2 24.9 7.6 24.7 8.0 24.6 8.5 24.4 8.9 27 26.1 7.0 26.0 7.4 25.8 7.9 25.7 8.3 25.5 8.8 25.4 9.2 28 27.0 7.2 26.9 7.7 26.8 8.2 26.6 8.7 26.5 9.1 26.3 9.6 29 28.0 7.5 27.9 8.0 27.7 8.5 27.6 9.0 27.4 9.4 27.3 9.9 30 29.0 7.8 28.8 8.3 28.7 8.8 28.5 9.3 28.4 9.8 28.2 10.3 31 29.9 8.0 29.8 8.5 29.6 9.1 29.5 9.6 29.3 10.1 29.1 10.6 32 30.9 8.3 30.8 8.8 30.6 9.4 30.4 9.9 30.3 10.4 30.1 10.9 33 31.9 8.5 31.7 9.1 31.6 9.6 31.4 10.2 31.2 10.7 31.0 11.3 34 32.8 8.8 32.7 9.4 32.5 9.9 32.3 10.5 32.1 11.1 31.9 11.6 35 33.8 9.1 33.6 9.6 33.5 10.2 33.3 10.8 33.1 11.4 32.9 12.0 36 34.8 9.3 34.6 9.9 34.4 10.5 34.2 11.1 34.0 11.7 33.8 12.3 37 35.7 9.6 35.6 10.2 35.4 10.8 35.2 11.4 35.0 12.0 34.8 12.7 38 36.7 9.8 36.5 10.5 36.3 11.1 36.1 11.7 35.9 12.4 35.7 13.0 39 37.7 10.1 37.5 10.7 37.3 11.4 37.1 12.1 36.9 12.7 36.6 13.3 40 38.6 10.4 38.5 11.0 38.3 11.7 38.0 12.4 37.8 13.0 37.6 13.7 41 39.6 10.6 39.4 11.3 39.2 12.0 39.0 12.7 38.8 13.3 38.5 14.0 42 40.6 10.9 40.4 11.6 40.2 12.3 39.9 13.0 39.7 13.7 39.5 14.4 43 41 5 11.1 41.3 11.9 41.1 12.6 40.9 13.3 40.7 14.0 40.4 147 44 ' 42 5 11.4 42.3 12.1 42.1 12.9 41.8 13.6 41.6 14.3 41.3 150 45 43.5 11.6 43.3 12.4 43.0 13.2 42.8 13.9 42.5 14.7 42.3 15.4 46 44.4 11.9 44.2 12.7 44.0 13.4 43.7 14.2 43.5 15.0 43.2 15.7 47 45.4 12.2 45.2 13.0 44.9 13.7 44.7 14.5 44.4 15.3 44.2 16.1 48 46.4 12.4 46.1 13.2 45.9 14.0 45.7 14.8 45.4 15.6 45.1 16.4 49 47.3 12.7 47.1 13.5 46.9 14.3 46.6 15.1 46.3 16.0 46.0 16.8 50 48.3 12.9 48.1 13.8 47.8 14.6 47.6 15.5 47.3 16.3 47.0 17.1 100 96.6 25.9 96.1 27.6 95.6 29.2 95.1 30.9 94.6 32.6 94.0 34.2 200 193.2 51.8 192.3 55.1 191.3 58.5 190.2 61.8 189.1 65.1 187.9 68.4 300 289.8 77.6 288.4 82.7 286.9 87.7 285.3 92.7 283.7 97.7 281.9 102.6 400 386.3 103.5 384.5 110.2 382.5 117.0 380.4 123.6 378.2 130.2 375.9 136.8 500 483.0 129.4 480.6 137.8 478.1 146.2 475.5 154.5 472.8 162.8 469.9 171.0 Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. (105, 255, (106, 254, (107, 253, (108, 252, (109, 251, (110, 250, 285) 286) 287) 288) 289) 290) 75 74 6| Pt. 73 72 71 6J Pt. 70 Table 1. Traverse Table 159 15 16 1 Pt. 17 18 19 l|Pt.20 (165, 195, (164, 196, (163, 197, (162, 198, (161, 199, (160, 200, DlST. 345) 344) 343) 342) 341) 340) Lat. Dep. Lat. Dep, Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 51 49.3 13.2 49.0 14.1 48.8 14.9 48.5 15.8 48.2 16.6 47.9 17.4 52 50.2 13.5 50.0 14.3 49.7 15.2 49.5 16.1 49.2 16.9 48.9 17.8 53 51.2 13.7 50.9 14.6 50.7 15.5 50.4 16.4 50.1 17.3 49.8 18.1 54 52.2 14.0 51.9 14.9 51.6 15.8 51.4 16.7 51.1 17.6 50.7 18.5 55 53.1 14.2 52.9 15.2 52.6 16.1 52.3 17.0 52.0 17.9 51.7 18.8 56 54.1 14.5 53.8 15.4 53.6 16.4 53.3 17.3 52.9 18.2 52.6 19.2 57 55.1 14.8 54.8 15.7 54.5 16.7 54.2 17.6 53.9 18.6 53.6 19.5 58 56.0 15.0 55.8 16.0 55.5 17.0 55.2 17.9 54.8 18.9 54.5 19.8 59 57.0 15.3 56.7 16.3 56.4 17.2 56.1 18.2 55.8 19.2 55.4 20.2 60 58.0 15.5 57.7 16.5 57.4 17.5 57.1 18.5 56.7 19.5 56.4 20.5 61 58.9 15.8 58.6 16.8 58.3 17.8 58.0 18.9 57.7 19.9 57.3 20.9 62 59.9 16.0 59.6 17.1 59.3 18.1 59.0 19.2 5S.6 20.2 58.3 21.2 63 60.9 16.3 60.6 17.4 60.2 18.4 59.9 19.5 59.6 20.5 59.2 21.5 64 61.8 16.6 61.5 17.6 61.2 18.7 60.9 19.8 60.5 20.8 60.1 21.9 65 62.8 16.8 62.5 17.9 62.2 19.0 61.8 20.1 61.5 21.2 61.1 22.2 66 63.8 17.1 63.4 18.2 63.1 19.3 62.8 20.4 62.4 21.5 62.0 22.6 67 64.7 17.3 64.4 18.5 64.1 19.6 63.7 20.7 63.3 21.8 63.0 22.9 68 65.7 17.6 65.4 18.7 65.0 19.9 64.7 21.0 64.3 22.1 63.9 23.3 69 66.6 17.9 66.3 19.0 66.0 20.2 65.6 21.3 65.2 22.5 64.8 23.6 70 67.6 18.1 67.3 19.3 66.9 20.5 66.6 21.6 66.2 22.8 65.8 23.9 71 68.6 18.4 68.2 19.6 67.9 20.8 67.5 21.9 67.1 23.1 66.7 24.3 72 69.5 18.6 69.2 19.8 68.9 21.1 68.5 22.2 68.1 23.4 67.7 24.6 73 70.5 18.9 70.2 20.1 69.8 21.3 69.4 22.6 69.0 23.8 68.6 25.0 74 71.5 19.2 71.1 20.4 70.8 21.6 70.4 22.9 70.0 24.1 69.5 25.3 75 72.4 19.4 72.1 20.7 71.7 21.9 71.3 23.2 70.9 24.4 70.5 25.7 76 73.4 19.7 73.1 20.9 72.7 22.2 72.3 23.5 71.9 24.7 71.4 26.0 77 74.4 19.9 74.0 21.2 73.6 22.5 73.2 23.8 72.8 25.1 72.4 26.3 78 75.3 20.2 75.0 21.5 74.6 22.8 74.2 24.1 73.8 25.4 73.3 26.7 79 76.3 20.4 75.9 21.8 75.5 23.1 75.1 24.4 74.7 25.7 74.2 27.0 80 77.3 20.7 76.9 22.1 76.5 23.4 76.1 24.7 75.6 26.0 75.2 27.4 81 78.2 21.0 77.9 22.3 77.5 23.7 77.0 25.0 76.6 26.4 76.1 27.7 82 79.2 21.2 78.8 22.6 78.4 24.0 78.0 25.3 77.5 26.7 77.1 28.0 83 80.2 21.5 79.8 22.9 79.4 24.3 78.9 25.6 78.5 27.0 78.0 28.4 84 81.1 21.7 80.7 23.2 80.3 24.6 79.9 26.0 79.4 27.3 78.9 28.7 85 82.1 22.0 81.7 23.4 81.3 24.9 80.8 26.3 80.4 27.7 79.9 29.1 86 83.1 22.3 82.7 23.7 82.2 25.1 81.8 26.6 81.3 28.0 80.8 29.4 87 84.0 22.5 83.6 24.0 83.2 25.4 82.7 26.9 82.3 28.3 81.8 29.8 88 85.0 22.8 84.6 24.3 84.2 25.7 83.7 27.2 83.2 28.7 82.7 30.1 80 86.0 23.0 85.6 24.5 85.1 26.0 84.6 27.5 84.2 29.0 83.6 30.4 90 86.9 23.3 86.5 24.8 86.1 26.3 85.6 27.8 85.1 29.3 84.6 30.8 01 87.9 23.6 87.5 25.1 87.0 26.6 86.5 28.1 86.0 29.6 85.5 31.1 92 88.9 23.8 88.4 25.4 88.0 26.9 87.5 28.4 87.0 30.0 86.5 31.5 93 89.8 24.1 89.4 25.6 88.9 27.2 88.4 28.7 87.9 30.3 87.4 31.8 94 90.8 24.3 90.4 25.9 89.9 27.5 89.4 29.0 88.9 30.6 88.3 32.1 95 91.8 24.6 91.3 26.2 90.8 27.8 90.4 29.4 89.8 30.9 89.3 32.5 96 92.7 24.8 92.3 26.5 91.8 28.1 91.3 29.7 90.8 31.3 90.2 32.8 97 93.7 25.1 93.2 26.7 92.8 28.4 92.3 30.0 91.7 31.6 91.2 33.2 98 94.7 25.4 94.2 27.0 93.7 28.7 93.2 30.3 92.7 31.9 92.1 33.5 99 95.6 25.6 95.2 27.3 94.7 28.9 94.2 30.6 93.6 32.2 93.0 33.9 100 96.6 25.9 96.1 27.6 95.6 29.2 95.1 30.9 94.6 32.6 94.0 34.2 600 579.5 155.3 576.8 165.4 573.8 175.4 570.6 185.4 567.3 195.3 563.8 205.2 700 676.1 181.1 672.8 193.0 669.4 204.6 665.8 216.3 661.9 227.9 657.9 239.4 800 772.7 207.0 769.0 220.5 765.0 233.9 760.8 247.3 756.5 260.4 751.8 273.6 900 869.2 232.9 865.0 248.0 860.6 263.1 855.9 278.1 850.9 292.9 845.7 307.8 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. (105, 255, (106, 254, (107, 253, (108, 252, (109, 251, (110, 250, 285) 286) 287) 288) 289) 290) 75 74 6| Pt, 73 72 71 70 160 Table 1. Traverse Table 21 22 2 Pt. 23 24 2|Pt.25 26 DlST. (159, 201, (158, 202, (157, 203, (156, 204, (155, 205, (154, 206, 339) 338) 337) 336) 335) 334) Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 1 0.9 0.4 0.9 0.4 0.9 0.4 0.9 0.4 0.9 0.4 0.9 0.4 2 1.9 0.7 1.9 0.7 1.8 0.8 1.8 0.8 1.8 0.8 1.8 0.9 3 2.8 1.1 2.8 1.1 2.8 1.2 2.7 1.2 2.7 1.3 2.7 1.3 4 3.7 1.4 3.7 1.5 3.7 1.6 3.7 1.6 3.6 1.7 3.6 1.8 5 4.7 1.8 4.6 1.9 4.6 2.0 4.6 2.0 4.5 2.1 4.5 2.2 6 5.6 2.2 5.6 2.2 5.5 2.3 5.5 2.4 5.4 2.5 5.4 2.6 7 6.5 2.5 6.5 2.6 6.4 2.7 6.4 2.8 6.3 3.0 6.3 3.1 8 7.5 2.9 7.4 3.0 7.4 3.1 7.3 3.3 7.3 3.4 7.2 3.5 9 8.4 3.2 8.3 3.4 8.3 3.5 8.2 3.7 8.2 3.8 8.1 3.9 10 9.3 3.6 9.3 3.7 9.2 3.9 9.1 4.1 9.1 4.2 9.0 4.4 11 10.3 3.9 10.2 4.1 10.1 4.3 10.0 4.5 10.0 4.6 9.9 4.8 12 11.2 4.3 11.1 4.5 11.0 4.7 11.0 4.9 10.9 5.1 10.8 5.3 13 12.1 4.7 12.1 4.9 12.0 5.1 11.9 5.3 11.8 5.5 11.7 5.7 14 13.1 5.0 13.0 5.2 12.9 5.5 12.8 5.7 12.7 5.9 12.6 6.1 15 14.0 5.4 13.9 5.6 13.8 5.9 13.7 6.1 13.6 6.3 13.5 6.6 16 14.9 5.7 14.8 6.0 14.7 6.3 14.6 6.5 14.5 6.8 14.4 7.0 17 15.9 6.1 15.8 6.4 15.6 6.6 15.5 6.9 15.4 7.2 15.3 7.5 18 16.8 6.5 16.7 6.7 16.6 7.0 16.4 7.3 16.3 7.6 16.2 7.9 19 17.7 6.8 17.6 7.1 17.5 7.4 17.4 7.7 17.2 8.0 17.1 8.3 20 18.7 7.2 18.5 7.5 18.4 7.8 18.3 8.1 18.1 8.5 18.0 8.8 21 19.6 7.5 19.5 7.9 19.3 8.2 19.2 8.5 19.0 8.9 18.9 9.2 22 20.5 7.9 20.4 8.2 20.3 8.6 20.1 8.9 19.9 9.3 19.8 9.6 23 21.5 8.2 21.3 8.6 21.2 9.0 21.0 9.4 20.8 9.7 20.7 10.1 24 22.4 8.6 22.3 9.0 22.1 9.4 21.9 9.8 21.8 10.1 21.6 10.5 25 23.3 9.0 23.2 9.4 23.0 9.8 22.8 10.2 22.7 10.6 22.5 11.0 26 24.3 9.3 24.1 9.7 23.9 10.2 23.8 10.6 23.6 11.0 23.4 11.4 27 25.2 9.7 25.0 10.1 24.9 10.5 24.7 11.0 24.5 11.4 24.3 11.8 28 26.1 10.0 26.0 10.5 25.8 10.9 25.6 11.4 25.4 11.8 25.2 12.3 29 27.1 10.4 26.9 10.9 26.7 11.3 26.5 11.8 26.3 12.3 26.1 12.7 30 28.0 10.8 27.8 11.2 27.6 11.7 27.4 12.2 27.2 12.7 27.0 13.2 31 28.9 11.1 28.7 11.6 28.5 12.1 28.3 12.6 28.1 13.1 27.9 13.6 32 29.9 11.5 29.7 12.0 29.5 12.5 29.2 13.0 29.0 13.5 28.8 14.0 33 30.8 11.8 30.6 12.4 30.4 12.9 3o!i 13.4 29.9 13.9 29.7 14.5 34 31.7 12.2 31.5 12.7 31.3 13.3 31.1 13.8 30.8 14.4 30.6 14.9 35 32.7 12.5 32.5 13.1 32.2 13.7 32.0 14.2 31.7 14.8 31.5 15.3 36 33.6 12.9 33.4 13.5 33.1 14.1 32.9 14.6 32.6 15.2 32.4 15.8 37 34.5 13.3 34.3 13.9 34.1 14.5 33.8 15.0 33.5 15.6 33.3 16.2 38 35.5 13.6 35.2 14.2 35.0 14.8 34.7 15.5 34.4 16.1 34.2 16.7 39 36.4 14.0 36.2 14.6 35.9 15.2 35.6 15.9 35.3 16.5 35.1 17.1 40 37.3 14.3 37.1 15.0 36.8 15.6 36.5 16.3 36.3 16.9 36.0 17.5 41 38.3 14.7 38.0 15.4 37.7 16.0 37.5 16.7 37.2 17.3 36.9 18.0 42 39.2 15.1 38.9 15.7 38.7 16.4 38.4 17.1 38.1 17.7 37.7 18.4 43 40.1 15.4 39.9 16.1 39.6 16.8 39.3 17.5 39.0 18.2 38.6 18.8 44 41.1 15.8 40.8 16.5 40.5 17.2 40.2 17.9 39.9 18.6 39.5 19.3 45 42.0 16.1 41.7 16.9 41.4 17.6 41.1 18.3 40.8 19.0 40.4 19.7 46 42.9 16.5 42.7 17.2 42.3 18.0 42.0 18.7 41.7 19.4 41.3 20.2 47 43.9 16.8 43.6 17.6 43.3 18.4 42.9 19.1 42.6 19.9 42.2 20.6 48 44.8 17.2 44.5 18.0 44.2 18.8 43.9 19.5 43.5 20.3 43.1 21.0 49 45.7 17.6 45.4 18.4 45.1 19.1 44.8 19.9 44.4 20.7 44.0 21.5 50 46.7 17.9 46.4 18.7 46.0 19.5 45.7 20.3 45.3 21.1 44.9 21.9 100 93.4 35.8 92.7 37.5 92.1 39.1 91.4 40.7 90.6 42.3 89.9 43.8 200 186.7 71.7 185.4 74.9 184.1 78.1 182.7 81.3 181.3 84.5 179.8 87.7 300 280.1 107.5 278.2 112.4 276.2 117.2 274.1 122.0 271.9 126.8 269.6 131.5 400 373.4 143.4 370.9 149.8 368.2 156.3 365.4 162.7 362.5 169.0 359.5 175.4 500 466.8 179.2 463.6 187.3 460.2 195.4 456.8 203.4 453.1 211.3 449.4 219.2 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. (111, 249, (112, 248, (113. 247, (114, 246, (115, 245, (116, 244, 291) 292) 293) 294) 295) 296) 69 6 Pt. 68 67 66 5f Pt. 65 64 The 2-Pt. or 23 Courses are : N.N.E., N.N.W., S.S.E., S.S.W. Table 1. Traverse Table 161 21 22 2 Pt. 23 24 2iPt. 25 26 (159, 201, (158, 202, (157, 203, (156, 204, (155, 205, (154, 206, DlST. 339) 338) 337) 336) 335) 334) Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 51 47.6 18.3 47.3 19.1 46.9 19.9 46.6 20.7 46.2 21.6 45.8 22.4 52 48.5 18.6 48.2 19.5 47.9 20.3 47.5 21.2 47.1 22.0 46.7 22.8 53 49.5 19.0 49.1 19.9 48.8 20.7 48.4 21.6 48.0 22.4 47.6 23.2 54 50.4 19.4 50.1 20.2 49.7 21.1 49.3 22.0 48.9 22.8 48.5 23.7 55 51.3 19.7 51.0 20.6 50.6 21.5 50.2 22.4 49.8 23.2 49.4 24.1 56 52.3 20.1 51.9 21.0 51.5 21.9 51.2 22.8 50.8 23.7 50.3 24.5 57 53.2 20.4 52.8 21.4 52.5 22.3 52.1 23.2 51.7 24.1 51.2 25.0 58 54.1 20.8 53.8 21.7 53.4 22.7 53.0 23.6 52.6 24.5 52.1 25.4 59 55.1 21.1 54.7 22.1 54.3 23.1 53.9 24.0 53.5 24.9 53.0 25.9 60 56.0 21.5 55.6 22.5 55.2 23.4 54.8 24.4 54.4 25.4 53.9 26.3 61 56.9 21.9 56.6 22.9 56.2 23.8 55.7 24.8 55.3 25.8 54.8 26.7 62 57.9 22.2 57.5 23.2 57.1 24.2 56.6 25.2 56.2 26.2 55.7 27.2 63 58.8 22.6 58.4 23.6 58.0 24.6 57.6 25.6 57.1 26.6 56.6 27.6 64 59.7 22.9 59.3 24.0 58.9 25.0 58.5 26.0 58.0 27.0 57.5 28.1 65 60.7 23.3 60.3 24.3 59.8 25.4 59.4 26.4 58.9 27.5 58.4 28.5 66 61.6 23.7 61.2 24.7 60.8 25.8 60.3 26.8 59.8 27.9 59.3 28.9 67 62.5 24.0 62.1 25.1 61.7 26.2 61.2 27.3 60.7 28.3 60.2 29.4 68 63.5 24.4 63.0 25.5 62.6 26.6 62.1 27.7 61.6 28.7 61.1 29.8 69 64.4 24.7 64.0 25.8 63.5 27.0 63.0 28.1 62.5 29.2 62.0 30.2 70 65.4 25.1 64.9 26.2 64.4 27.4 63.9 28.5 63.4 29.6 62.9 30.7 71 66.3 25.4 65.8 26.6 65.4 27.7 64.9 28.9 64.3 30.0 63.8 31.1 72 67.2 25.8 66.8 27.0 66.3 28.1 65.8 29.3 65.3 30.4 64.7 31.6 73 68.2 26.2 67.7 27.3 67.2 28.5 66.7 29.7 66.2 30.9 65.6 32.0 74 69.1 26.5 68.6 27.7 68.1 28.9 67.6 30.1 67.1 31.3 66.5 32.4 75 70.0 26.9 69.5 28.1 69.0 29.3 68.5 30.5 68.0 31.7 67.4 32.9 76 71.0 27.2 70.5 28.5 70.0 29.7 69.4 30.9 68.9 32.1 68.3 33.3 77 71.9 27.6 71.4 28.8 70.9 30.1 70.3 31.3 69.8 32.5 69.2 33.8 78 72.8 28.0 72.3 29.2 71.8 30.5 71.3 31.7 70.7 33.0 70.1 34.2 79 73.0 28.3 73.2 29.6 72.7 30.9 72.2 32.1 71.6 33.4 71.0 34.6 80 74.7 28.7 74.2 30.0 73.6 31.3 73.1 32.5 72.5 33.8 71.9 35.1 81 75.6 29.0 75.1 30.3 74.6 31.6 74.0 32.9 73.4 34.2 72.8 35.5 82 76.6 29.4 76.0 30.7 75.5 32.0 74.9 33.4 74.3 34.7 73.7 35.9 83 77.5 29.7 77.0 31.1 76.4 32.4 75.8 33.8 75.2 35.1 74.6 36.4 84 78.4 30.1 77.9 31.5 77.3 32.8 76.7 34.2 76.1 35.5 75.5 36.8 85 79.4 30.5 78.8 31.8 78.2 33.2 77.7 34.6 77.0 35.9 76.4 37.3 86 80.3 30.8 79.7 32.2 79.2 33.6 78.6 35.0 77.9 36.3 77.3 37.7 87 81.2 31.2 80.7 32.6 80.1 34.0 79.5 35.4 78.8 36.8 78.2 38.1 88 82.2 31.5 81.6 33.0 81.0 34.4 80.4 35.8 79.8 37.2 79.1 38.6 89 83.1 31.9 82.5 33.3 81.9 34.8 81.3 36.2 80.7 37.6 80.0 39.0 90 84.0 32.3 83.4 33.7 82.8 35.2 82.2 36.6 81.6 38.0 80.9 39.5 91 85.0 32.6 84.4 34.1 83.8 35.6 83.1 37.0 82.5 38.5 81.8 39.9 92 85.9 33.0 85.3 34.5 84.7 35.9 84.0 37.4 83.4 38.9 82.7 40.3 93 86.8 33.3 86.2 34.8 85.6 36.3 85.0 37.8 84.3 39.3 83.6 40.8 94 87.8 33.7 87.2 35.2 86.5 36.7 85.9 38.2 85.2 39.7 84.5 41.2 95 88.7 34.0 88.1 35.6 87.4 37.1 86.8 38.6 86.1 40.1 85.4 41.6 96 89.6 34.4 89.0 36.0 88.4 37.5 87.7 39.0 87.0 40.6 86.3 42.1 97 90.6 34.8 89.9 36.3 89.3 37.9 88.6 39.5 87.9 41.0 87.2 42.5 98 91.5 35.1 90.9 36.7 90.2 38.3 89.5 39.9 88.8 41.4 88.1 43.0 99 92.4 35.5 91.8 37.1 91.1 38.7 90.4 40.3 89.7 41.8 89.0 43.4 100 93.4 35.8 92.7 37.5 92.1 39.1 91.4 40.7 90.6 42.3 89.9 43.8 600 560.1 215.0 556.3 224.8 552.3 234.4 548.1 244.0 543.8 253.6 539.3 263.0 700 653.6 250.8 649.1 262.2 644.3 273.5 639.5 284.7 634.5 295.8 629.2 306.8 800 746.9 286.7 741.8 299.7 736.4 312.6 730.8 325.4 725.1 338.1 719.1 350.6 900 840.3 322.5 834.5 IK 7.1 828.3 351.7 822.1 366.0 815.6 380.3 808.9 394.5 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. (111, 249, (112, 248, (113, 247, (114, 246, (115, 245, (116, 244, 291) 292) 293) 294) 295) 296) 69 6 Pt. 68 67 66 5f Pt. 65 64 The 6-Pt. or 68 Courses are: E.N.E., W.N.W., E.S.E., W.S.W. 162 Table 1. Traverse Table 27 1\ Pt. 28 29 30 2f Pt. 31 32 DlST. (153, 207, 333) (152, 208, 332) (151, 209, 331) (150, 210, 330) <149, 211, 329) (148, 212, 328) Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 1 0.9 0.5 0.9 0.5 0.9 0.5 0.9 0.5 0.9 0.5 0.8 0.5 2 1.8 0.9 1.8 0.9 1.7 1.0 1.7 1.0 1.7 1.0 1.7 1.1 3 2.7 1.4 2.6 1.4 2.6 1.5 2.6 1.5 2.6 1.5 2.5 1.6 4 3.6 1.8 3.5 1.9 3.5 1.9 3.5 2.0 3.4 2.1 3.4 2.1 5 4.5 2.3 4.4 2.3 4.4 2.4 4.3 2.5 4.3 2.6 4.2 2.6 6 5.3 2.7 5.3 2.8 5.2 2.9 5.2 3.0 5.1 3.1 5.1 3.2 7 6.2 3.2 6.2 3.3 6.1 3.4 6.1 3.5 6.0 3.6 5.9 3.7 8 7.1 3.6 7.1 3.8 7.0 3.9 6.9 4.0 6.9 4.1 6.8 4.2 9 8.0 4.1 7.9 4.2 7.9 4.4 7.8 4.5 Y.7 4.6 7.6 4.8 10 8.9 4.5 8.8 4.7 8.7 4.8 8.7 5.0 8.6 5.2 8.5 5.3 11 9.8 5.0 9.7 5.2 9.6 5.3 9.5 5.5 9.4 5.7 9.3 5.8 12 10.7 5.4 10.6 5.6 10.5 5.8 10.4 6.0 10.3 6.2 10.2 6.4 13 11.6 5.9 11.5 6.1 11.4 6.3 11.3 6.5 11.1 6.7 11.0 6.9 14 12.5 6.4 12.4 6.6 12.2 6.8 12.1 7.0 12.0 7.2 11.9 7.4 15 13.4 6.8 13.2 7.0 13.1 7.3 13.0 7.5 12.9 7.7 12.7 7.9 16 14.3 7.3 14.1 7.5 14.0 7.8 13.9 8.0 13.7 8.2 13.6 8.5 17 15.1 7.7 15.0 8.0 14.9 8.2 14.7 8.5 14.6 8.8 14.4 9.0 18 16.0 8.2 15.9 8.5 15.7 8.7 15.6 9.0 15.4 9.3 15.3 9.5 19 16.9 8.6 16.8 8.9 16.6 9.2 16.5 9.5 16.3 9.8 16.1 10.1 20 17.8 9.1 17.7 9.4 17.5 9.7 17.3 10.0 17.1 10.3 17.0 10.6 21 18.7 9.5 18.5 9.9 18.4 10.2 18.2 10.5 18.0 10.8 17.8 11.1 22 19.6 10.0 19.4 10.3 19.2 10.7 19.1 11.0 18.9 11.3 18.7 11.7 23 20.5 10.4 20.3 10.8 20.1 11.2 19.9 11.5 19.7 11.8 19.5 12.2 24 21.4 10.9 21.2 11.3 21.0 11.6 20.8 12.0 20.6 12.4 20.4 12.7 25 22.3 11.3 22.1 11.7 21.9 12.1 21.7 12.5 21.4 12.9 21.2 13.2 26 23.2 11.8 23.0 12.2 22 7 12.6 22.5 13.0 22.3 13.4 22.0 13.8 27 24.1 12.3 23.8 12.7 23^6 13.1 23.4 13.5 23.1 13.9 22.9 14.3 28 24.9 12.7 24.7 13.1 24.5 13.6 24.2 14.0 24.0 14.4 23.7 14.8 29 25.8 13.2 25.6 13.6 25.4 14.1 25.1 14.5 24.9 14.9 24.6 15.4 30 26.7 13.6 26.5 14.1 26.2 14.5 26.0 15.0 25.7 15.5 25.4 15.9 31 27.6 14.1 27.4 14.6 27.1 15.0 26.8 15.5 26.6 16.0 26.3 16.4 32 28.5 14.5 28.3 15.0 28.0 15.5 27.7 16.0 27.4 16.5 27.1 17.0 33 29.4 15.0 29.1 15.5 28.9 16.0 28.6 16.5 28.3 17.0 28.0 17.5 34 30.3 15.4 30.0 16.0 29.7 16.5 29.4 17.0 29.1 17.5 28.8 18.0 35 31.2 15.9 30.9 16.4 30.6 17.0 30.3 17.5 30.0 18.0 29.7 18.5 36 32.1 16.3 31.8 16.9 31.5 17.5 31.2 18.0 30.9 18.5 30.5 19.1 37 33.0 16.8 32.7 17.4 32.4 17.9 32.0 18.5 31.7 19.1 31.4 19.6 38 33.9 17.3 33.6 17.8 33.2 18.4 32.9 19.0 32.6 19.6 32.2 20.1 39 34.7 17.7 34.4 18.3 34.1 18.9 33.8 19.5 33.4 20.1 33.1 20.7 40 35.6 18.2 35.3 18.8 35.0 19.4 34.6 20.0 34.3 20.6 33.9 21.2 41 36.5 18.6 36.2 19.2 35.9 19.9 35.5 20.5 35.1 21.1 34.8 21.7 42 37.4 19.1 37.1 19.7 36.7 20.4 36.4 21.0 36.0 21.6 35.6 22.3 43 38.3 19.5 38.0 20.2 37.6 20.8 37.2 21.5 36.9 22.1 36.5 22.8 44 39.2 20.0 38.8 20.7 38.5 21.3 38.1 22.0 37.7 22.7 37.3 23.3 45 40.1 20.4 39.7 21.1 39.4 21.8 39.0 22.5 38.6 23.2 38.2 23.8 46 41.0 20.9 40.6 21.6 40.2 22.3 39.8 23.0 39.4 23.7 39.0 24.4 47 41.9 21.3 41.5 22.1 41.1 22.8 40.7 23.5 40.3 24.2 39.9 24.9 48 42.8 21.8 42.4 22.5 42.0 23.3 41.6 24.0 41.1 24.7 40.7 25.4 49 43.7 22.2 43.3 23.0 42.9 23.8 42.4 24.5 42.0 25.2 41.6 26.0 50 44.6 22.7 44.1 23.5 43.7 24.2 43.3 25.0 42.9 25.8 42.4 26.5 100 89.1 45.4 88.3 46.9 87.5 48.5 86.6 50.0 85.7 51.5 84.8 53.0 200 178.2 90.8 176.6 93.9 174.9 97.0 173.2 100.0 171.4 103.0 169.6 106.0 300 267.3 136.2 264.9 140.8 262.4 145.4 259.8 150.0 257.1 154.5 254.4 159.0 400 356.4 181.6 353.1 187.8 349.8 193.9 346.4 200.0 342.9 206.0 339.2 211.9 500 445.5 227.0 441.5 234.7 437.3 242.4 433.0 250.0 428.6 257.5 424.0 265.0 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. (117, 243, (118, 242, (119, 241, (120, 240, (121, 239, (122, 238, 297) 298) 299) 300) 301) 302) 63 5|Pt. 62 61 60 5|Pt. 59 58 Table 1. Traverse Table 163 27 2 Pt. 28 29 30 2f Pt. 31 32 (153, 207, (152, 208, (151, 209, (150, 210, (149, 211, (148, 212, DlST. 333) 332) 331) 330) 329) 328) Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 51 45.4 23.2 45.0 23.9 44.6 24.7 44.2 25.5 43.7 26.3 43.3 27.0 52 46.3 23.6 45.9 24.4 45.5 25.2 45.0 26.0 44.6 26.8 44.1 27.6 53 47.2 24.1 46.8 24.9 46.4 25.7 45.9 26.5 45.4 27.3 44.9 28.1 54 48.1 24.5 47.7 25.4 47.2 26.2 46.8 27.0 46.3 27.8 45.8 28.6 55 49.0 25.0 48.6 25.8 48.1 26.7 47.6 27.5 47.1 28.3 46.6 29.1 56 49.9 25.4 49.4 26.3 49.0 27.1 48.5 28.0 48.0 28.8 47.5 29.7 57 50.8 25.9 50.3 26.8 49.9 27.6 49.4 28.5 48.9 29.4 48.3 30.2 58 51.7 26.3 51.2 27.2 50.7 28.1 50.2 29.0 49.7 29.9 49.2 30.7 59 52.6 26.8 52.1 27.7 51.6 28.6 51.1 29.5 50.6 30.4 50.0 31.3 60 53.5 27.2 53.0 28.2 52.5 29.1 52.0 30.0 51.4 30.9 50.9 31.8 61 54.4 27.7 53.9 28.6 53.4 29.6 52.8 30.5 52.3 31.4 51.7 32.3 62 55.2 28.1 54.7 29.1 54.2 30.1 53.7 31.0 53.1 31.9 52.6 32.9 63 56.1 28.6 55.6 29.6 55.1 30.5 54.6 31.5 54.0 32.4 53.4 33.4 64 57.0 29.1 56.5 30.0 56.0 31.0 55.4 32.0 54.9 33.0 54.3 33.9 65 57.9 29.5 57.4 30.5 56.9 31.5 56.3 32.5 55.7 33.5 55.1 34.4 66 58.8 30.0 58.3 31.0 57.7 32.0 57.2 "33.0 56.6 34.0 56.0 35.0 67 59.7 30.4 59.2 31.5 58.6 32.5 58.0 33.5 57.4 34.5 56.8 35.5 68 60.6 30.9 60.0 31.9 59.5 33.0 58.9 34.0 58.3 35.0 57.7 36.0 69 61.5 31.3 60.9 32.4 60.3 33.5 59.8 34.5 59.1 35.5 58.5 36.6 70 62.4 31.8 61.8 32.9 61.2 33.9 60.6 35.0 60.0 36.1 59.4 37.1 71 63.3 32.2 62.7 33.3 62.1 34.4 61.5 35.5 60.9 36.6 60.2 37.6 72 64.2 32.7 63.6 33.8 63.0 34.9 62.4 36.0 61.7 37.1 61.1 38.2 73 65.0 33.1 64.5 34.3 63.8 35.4 63.2 36.5 62.6 37.6 61.9 38.7 74 65.9 33.6 65.3 34.7 64.7 35.9 64.1 37.0 63.4 38.1 62.8 39.2 75 66.8 34.0 66.2 35.2 65.6 36.4 65.0 37.5 64.3 38.6 63.6 39.7 76 67.7 34.5 67.1 35.7 66.5 36.8 65.8 38.0 65.1 39.1 64.5 40.3 77 68.6 35.0 68.0 36.1 67.3 37.3 66.7 38.5 66.0 39.7 65.3 40.8 78 69.5 35.4 68.9 36.6 68.2 37.8 67.5 39.0 66.9 40.2 66.1 41.3 79 70.4 35.9 69.8 37.1 69.1 38.3 68.4 39.5 67.7 40.7 67.0 41.9 80 71.3 36.3 70.6 37.6 70.0 38.8 69.3 40.0 68.6 41.2 67.8 42.4 81 72.2 36.8 71.5 38.0 70.8 39.3 70.1 40.5 69.4 41.7 68.7 42.9 82 73.1 37.2 72.4 38.5 71.7 39.8 71.0 41.0 70.3 42.2 69.5 43.5 83 74.0 37.7 73.3 39.0 72.6 40.2 71.9 41.5 71.1 42.7 70.4 44.0 84 74.8 38.1 74.2 39.4 73.5 40.7 72.7 42.0 72.0 43.3 71.2 44.5 85 75.7 38.6 75.1 39.9 74.3 41.2 73.6 42.5 72.9 43.8 72.1 45.0 86 76.6 39.0 75.9 40.4 75.2 41.7 74.5 43.0 73.7 44.3 72.9 45.6 87 77.5 39.5 76.8 40.8 76.1 42.2 75.3 43.5 74.6 44.8 73.8 46.1 88 78.4 40.0 77.7 41.3 77.0 42.7 76.2 44.0 75.4 45.3 74.6 46.6 89 79.3 40.4 78.6 41.8 77.8 43.1 77.1 44.5 76.3 45.8 75.5 47.2 90 80.2 40.9 79.5 42.3 78.7 43.6 77.9 45.0 77.1 46.4 76.3 47.7 91 81.1 41.3 80.3 42.7 79.6 44.1 78.8 45.5 78.0 46.9 77.2 48.2 92 82.0 41.8 81.2 43.2 80.5 44.6 79.7 46.0 78.9 47.4 78.0 48.8 93 82.9 42.2 82.1 43.7 81.3 45.1 80.5 46.5 79.7 47.9 78.9 49.3 94 83.8 42.7 83.0 44.1 82.2 45.6 81.4 47.0 80.6 48.4 79.7 49.8 95 84.6 43.1 83.9 44.6 83.1 46.1 82.3 47.5 81.4 48.9 80.6 50.3 96 85.5 43.6 84.8 45.1 84.0 46.5 83.1 48.0 82.3 49.4 81.4 50.9 97 86.4 44.0 85.6 45.5 84.8 47.0 84.0 48.5 83.1 50.0 82.3 51.4 98 87.3 44.5 86.5 46.0 85.7 47.5 84.9 49.0 84.0 50.5 83.1 51.9 99 88.2 44.9 87.4 46.5 86.6 48.0 85.7 49.5 84.9 51.0 84.0 52.5 100 89.1 45.4 88.3 46.9 87.5 48.5 86.6 50.0 85.7 51.5 84.8 53.0 600 534.6 272.4 529.8 281.7 524.8 290.9 519.6 300.0 514.3 309.0 508.8 318.0 700 623.7 317.8 618.0 328.6 612.2 339.4 606.1 350.0 600.1 360.4 593.6 371.0 800 712.9 363.2 706.3 375.6 699.7 387.9 692.8 400.0 685.8 412.0 678.4 423.9 900 801.9 408.5 794.5 422.5 787.0 436.3 779.3 450.0 771.4 463.4 763.2 476.8 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat Dep. Lat. (117, 243, (118, 242, (119, 241, (120, 240 (121, 239, (122, 238, 297) 298) 299) 300) 301) 302) 63 5^ Pt. 62 61 60 5| Pt. 59 58 164 Table 1. Traverse Table 33 3 Pt. 34 35 36 31 Pt. 37 38 (147, 213, (146, 214, (145, 215, (144, 216, (143, 217, (142, 218, DlST. 327) 326) 325) 324) 323) 322) Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 1 0.8 0.5 0.8 0.6 0.8 0.6 0.8 0.6 0.8 0.6 0.8 0.6 2 1.7 1.1 1.7 1.1 1.6 1.1 1.6 1.2 1.6 1.2 1.6 1.2 3 2.5 1.6 2.5 1.7 2.5 1.7 2.4 1.8 2.4 1.8 2.4 1.8 4 3.4 2.2 3.3 2.2 3.3 2.3 3.2 2.4 3.2 2.4 3.2 2.5 5 4.2 2.7 4.1 2.8 4.1 2.9 4.0 2.9 4.0 3.0 3.9 3.1 6 5.0 3.3 5.0 3.4 4.9 3.4 4.9 3.5 4.8 3.6 4.7 3.7 7 5.9 3.8 5.8 3.9 5.7 4.0 5.7 4.1 5.6 4.2 5.5 4.3 8 6.7 4.4 6.6 4.5 6.6 4.6 6.5 4.7 6.4 4.8 6.3 4.9 9 75 4.9 7.5 5.0 7.4 5.2 7.3 5.3 7.2 5.4 7 1 5.5 10 8.4 5.4 8.3 5.6 8.2 5.7 8.1 5.9 8.0 6.0 7.9 6.2 11 9.2 6.0 9.1 6.2 9.0 6.3 8.9 6.5 8.8 6.6 8.7 6.8 12 10.1 6.5 9.9 6.7 9.8 6.9 9.7 7.1 9.6 7.2 9.5 7.4 13 10.9 7.1 10.8 7.3 10.6 7.5 10.5 7.6 10.4 7.8 10.2 8.0 14 11.7 7.6 11.6 7.8 11.5 8.0 11.3 8.2 11.2 8.4 11.0 8.6 15 12.6 8.2 12.4 8.4 12.3 8.6 12.1 8.8 12.0 9.0 11.8 9.2 16 134 8.7 13.3 8.9 13.1 9.2 12.9 9.4 12.8 9.6 12.6 99 17 14.3 9.3 14.1 9.5 13.9 9.8 13.8 10.0 13.6 10.2 13.4 10.5 18 15.1 9.8 14.9 10.1 14.7 10.3 14.6 10.6 14.4 10.8 14.2 11.1 19 15.9 10.3 15.8 10.6 15.6 10.9 15.4 11.2 15.2 11.4 15.0 11.7 20 16.8 10.9 16.6 11.2 16.4 11.5 16.2 11.8 16.0 12.0 15.8 12.3 21 17.6 11.4 17.4 11.7 17.2 12.0 17.0 12.3 16.8 12.6 16.5 12.9 22 18.5 12.0 18.2 12.3 18.0 12.6 17.8 12.9 17.6 13.2 17.3 13.5 23 19.3 12.5 19.1 12.9 18.8 13.2 18.6 13.5 18.4 13.8 18.1 14.2 24 20.1 13.1 19.9 13.4 19.7 13.8 19.4 14.1 19.2 14.4 18.9 14.8 25 21.0 13.6 20.7 14.0 20.5 14.3 20.2 14.7 20.0 15.0 19.7 15.4 26 21.8 14.2 21.6 14.5 21.3 14.9 21.0 15.3 20.8 15.6 20.5 16.0 27 22.6 14.7 22.4 15.1 22.1 15.5 21.8 15.9 21.6 16.2 21.3 16.6 28 23.5 15.2 23.2 15.7 22.9 16.1 22.7 16.5 22.4 16.9 22.1 17.2 29 24.3 15.8 24.0 16.2 23.8 16.6 23.5 17.0 23.2 17.5 22.9 17.9 30 25.2 16.3 24.9 16.8 24.6 17.2 24.3 17.6 24.0 18.1 23.6 18.5 31 26.0 16.9 25.7 17.3 25.4 17.8 25.1 18.2 24.8 18.7 24.4 19.1 32 26.8 17.4 26.5 17.9 26.2 18.4 25.9 18.8 25.6 19.3 25.2 19.7 33 27.7 18.0 27.4 18.5 27.0 18.9 26.7 19.4 26.4 19.9 26.0 20.3 34 28.5 18.5 28.2 19.0 27.9 19.5 27.5 20.0 27.2 20.5 26.8 20.9 35 29.4 19.1 29.0 19.6 28.7 20.1 28.3 20.6 28.0 21.1 27.6 21.5 36 30.2 19.6 29.8 20.1 29.5 20.6 29.1 21.2 28.8 21.7 28.4 22.2 37 31.0 20.2 30.7 20.7 30.3 21.2 29.9 21.7 29.5 22.3 29.2 22.8 38 31.9 20.7 31.5 21.2 31.1 21.8 30.7 22.3 30.3 22.9 29.9 23.4 39 32.7 21.2 32.3 21.8 31.9 22.4 31.6 22.9 31.1 23.5 30.7 24.0 40 33.5 21.8 33.2 22.4 32.8 22.9 32.4 23.5 31.9 24.1 31.5 24.6 41 34.4 22.3 34.0 22.9 33.6 23.5 33.2 24.1 32.7 24.7 32.3 25.2 42 35.2 22.9 34.8 23.5 34.4 24.1 34.0 24.7 33.5 25.3 33.1 25.9 43 36.1 23.4 35.6 24.0 35.2 24.7 34.8 25.3 34.3 25.9 33.9 26.5 44 36.9 24.0 36.5 24.6 36.0 25.2 35.6 25.9 35.1 26.5 34.7 27.1 45 37.7 24.5 37.3 25.2 36.9 25.8 36.4 26.5 35.9 27.1 35.5 27.7 46 38.6 25.1 38.1 25.7 37.7 26.4 37.2 27.0 36.7 27.7 36.2 28.3 47 39.4 25.6 39.0 26.3 38.5 27.0 38.0 27.6 37.5 28.3 37.0 28.9 48 40.3 26.1 39.8 26.8 39.3 27.5 38.8 28.2 38.3 28.9 37.8 29.6 49 41.1 26.7 40.6 27.4 40.1 28.1 39.6 28.8 39.1 29.5 38.6 30.2 50 41.9 27.2 41.5 28.0 41.0 28.7 40.5 29.4 39.9 30.1 39.4 30.8 100 83.9 54.5 82.9 55.9 81.9 57.4 80.9 58.8 79.9 60.2 78.8 61.6 200 167.7 108.9 165.8 111.8 163.8 114.7 161.8 117.6 159.7 120.4 157.6 123.1 300 251.6 163.4 248.7 167.8 245.7 172.1 242.7 176.3 239.6 180.5 236.4 184.7 400 335.5 217.8 331.6 223.7 327.7 229.4 323.6 235.1 319.4 240.7 315.2 246.3 500 419.3 272.3 414.5 279.6 409.6 286.8 404.5 293.9 399.3 300.9 394.0 307.8 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. (123, 237, (124, 236, (125, 235, (126, 234, (127, 233, (128, 232, 303) 304) 305) 306) 307) 308) 57 5 Pt. 56 55 54 4f Pt, 53 52 The 3-Pt. or 34 Courses are : N.E. by N., N.W. by N., S.E. by S., S.W. by S. Table 1. Traverse Table 165 33 3 Pt, 34 35 36 3iPt.37 38 DlST. (147, 213, 327) !(146, 214, 326) (145, 215, 325) (144, 216, 324) (143, 217, 323) (142, 218, 322) Lat. Dep. iLat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 51 42.8 27.8 42.3 28.5 41.8 29.3 41.3 30.0 40.7 30.7 40.2 31.4 52 43.6 28.3 43.1 29.1 42.6 29.8 42.1 30.6 41.5 31.3 41.0 32.0 53 44.4 28.9 43.9 29.6 43.4 30.4 42.9 31.2 42.3 31.9 41.8 32.6 54 45.3 29.4 44.8 30.2 44.2 31.0 43.7 31.7 43.1 32.5 42.6 33.2 55 46.1 30.0 45.6 30.8 45.1 31.5 44.5 32.3 43.9 33.1 43.3 33.9 56 47.0 30.5 46.4 31.3 45.9 32.1 45.3 32.9 44.7 33.7 44.1 34.5 57 47.8 31.0 47.3 31.9 46.7 32.7 46.1 33.5 45.5 34.3 44.9 35.1 58 48.6 31.6 48.1 32.4 47.5 33.3 46.9 34.1 46.3 34.9 45.7 35.7 59 49.5 32.1 48.9 33.0 48.3 33.8 47.7 34.7 47.1 35.5 46.5 36.3 60 50.3 32.7 49.7 33.6 49.1 34.4 48.5 35.3 47.9 36.1 47.3 36.9 61 51.2 33.2 50.6 34.1 50.0 35.0 49.4 35.9 48.7 36.7 48.1 37.6 62 52.0 33.8 51.4 34.7 50.8 35.6 50.2 36.4 49.5 37.3 48.9 38.2 63 52.8 34.3 52.2 35.2 51.6 36.1 51.0 37.0 50.3 37.9 49.6 38.8 64 53.7 34.9 53.1 35.8 52.4 36.7 51.8 37.6 51.1 38.5 50.4 39.4 65 54.5 35.4 53.9 36.3 53.2 37.3 52.6 38.2 51.9 39.1 51.2 40.0 66 55.4 35.9 54.7 36.9 54.1 37.9 53.4 38.8 52.7 39.7 52.0 40.6 67 56.2 36.5 55.5 37.5 54.9 38.4 54.2 39.4 53.5 40.3 52.8 41.2 68 57.0 37.0 56.4 38.0 55.7 39.0 55.0 40.0 54.3 40.9 53.6 41.9 69 57.9 37.6 57.2 38.6 56.5 39.6 55.8 40.6 55.1 41.5 54.4 42.5 70 58.7 38.1 58.0 39.1 57.3 40.2 56.6 41.1 55.9 42.1 55.2 43.1 71 59.5 38.7 58.9 39.7 58.2 40.7 57.4 41.7 56.7 42.7 55.9 43.7 72 60.4 39.2 59.7 40.3 59.0 41.3 58.2 42.3 57.5 43.3 56.7 44.3 73 61.2 39.8 60.5 40.8 59.8 41.9 59.1 42.9 58.3 43.9 57.5 44.9 74 62.1 40.3 61.3 41.4 60.6 42.4 59.9 43.5 59.1 44.5 58.3 45.6 75 62.9 40.8 62.2 41.9 61.4 43.0 60.7 44.1 59.9 45.1 59.1 46.2 76 63.7 41.4 63.0 42.5 62.3 43.6 61.5 44.7 60.7 45.7 59.9 46.8 77 64.6 41.9 63.8 43.1 63.1 44.2 62.3 45.3 61.5 46.3 60.7 47.4 78 65.4 42.5 64.7 43.6 63.9 44.7 63.1 45.8 62.3 46.9 61.5 48.0 79 66.3 43.0 65.5 44.2 64.7 45.3 63.9 46.4 63.1 47.5 62.3 48.6 80 67.1 43.6 66.3 44.7 65.5 45.9 64.7 47.0 63.9 48.1 63.0 49.3 81 67.9 44.1 67.2 45.3 66.4 46.5 65.5 47.6 64.7 48.7 63.8 49.9 82 68.8 44.7 68.0 45.9 67.2 47.0 66.3 48.2 65.5 49.3 64.6 50.5 83 69.6 45.2 68.8 46.4 68.0 47.6 67.1 48.8 66.3 50.0 65.4 51.1 84 70.4 45.7 69.6 47.0 68.8 48.2 68.0 49.4 67.1 50.6 66.2 51.7 85 71.3 46.3 70.5 47.5 69.6 48.8 68.8 50.0 67.9 51.2 67.0 52.3 86 72.1 46.8 71.3 48.1 70.4 49.3 69.6 50.5 68.7 51.8 67.8 52.9 87 73.0 47.4 72.1 48.6 71.3 49.9 70.4 51.1 69.5 52.4 68.6 53.6 88 73.8 47.9 73.0 49.2 72.1 50.5 71.2 51.7 70.3 53.0 69.3 54.2 89 74.6 48.5 73.8 49.8 72.9 51.0 72.0 52.3 71.1 53.6 70.1 54.8 90 75.5 49.0 74.6 50.3 73.7 51.6 72.8 52.9 71.9 54.2 70.9 55.4 91 76.3 49.6 75.4 50.9 74.5 52.2 73.6 53.5 72.7 54.8 71.7 56.0 92 77.2 50.1 76.3 51.4 75.4 52.8 74.4 54.1 73.5 55.4 72.5 56.6 93 78.0 50.7 77.1 52.0 76.2 53.3 75.2 54.7 74.3 56.0 73.3 57.3 94 78.8 51.2 77.9 52.6 77.0 53.9 76.0 55.3 75.1 56.6 74.1 57.9 95 79.7 51.7 78.8 53.1 77.8 54.5 76.9 55.8 75.9 57.2 74.9 58.5 96 80.5 52.3 79.6 53.7 78.6 55.1 77.7 56.4 76.7 57.8 75.6 59.1 97 81.4 52.8 80.4 54.2 79.5 55.6 78.5 57.0 77.5 58.4 76.4 59.7 98 82.2 53.4 81.2 54.8 80.3 56.2 79.3 57.6 78.3 59.0 77.2 60.3 99 83.0 53.9 82.1 55.4 81.1 56.8 80.1 58.2 79.1 59.6 78.0 61.0 100 83.9 54.5 82.9 55.9 81.9 57.4 80.9 58.8 79.9 60.2 78.8 61.6 600 503.2 326.8 497.4 335.5 491.5 344.1 485.4 352.7 479.2 361.1 472.8 369.4 700 587.0 381.3 580.3 391.4 573.5 401.5 566.2 411.4 559.0 421.3 551.6 430.8 800 671.0 435.7 663.3 447.4 655.4 458.8 647.3 470.2 638.9 481.5 630.4 492.5 900 754.8 490.1 746.1 503.2 737.2 516.2 728.1 528.9 718.6 541.7 709.1 554.0 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. (123, 237, (124, 236, (125, 235, (126, 234, (127, 233, (128, 232, 303) 304) 305) 306) 307) 308) 57 5 Pt. 56 55 54 4f Pt, 53 52 The 5-Pt. or 56 Courses are : N.E. by E., S.E. by E., N.W. by W., S.W. by W. 166 Table 1. Traverse Table 3J Pt. 39 40 41 3f Pt. 42 43 44 4 Pt. 45 DlST (141, 219, 321) (140, 220, 320) (139, 221 319) (138, 222, 318) (137, 223, 317) (136, 224, 316) (135, 225, 315) Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 1 0.8 0.6 0.8 0.6 0.8 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 2 1.6 1.3 1.5 1.3 1.5 1.3 1.5 1.3 1.5 1.4 1.4 1.4 1.4 1.4 3 2 'i 1.9 2.3 1.9 2.3 2.0 2.2 2.0 2.2 2.0 2.2 2.1 2.1 2.1 4 3.1 2.5 3.1 2.6 3.0 2.6 3.0 2.7 2.9 2.7 2.9 2.8 2.8 2.8 5 3.9 3.1 3.8 3.2 3.8 3.3 3.7 3.3 3.7 3.4 3.6 3.5 3.5 3.5 6 4.7 3.8 4.6 3.9 4.5 3.9 4.5 4.0 4.4 4.1 4.3 4.2 4.2 4.2 7 5.4 4.4 5.4 4.5 5.3 4.6 5.2 4.7 5.1 4.8 5.0 4.9 4.9 4.9 8 6.2 5.0 6.1 5.1 6.0 5.2 5.9 5.4 5.9 5.5 5.8 5.6 5.7 5.7 9 7.0 5.7 6.9 5.8 6.8 5.9 6.7 6.0 6.6 6.1 6.5 6.3 6.4 6.4 10 7.8 6.3 7.7 6.4 7.5 6.6 7.4 6.7 7 3 6.8 7.2 6.9 7.1 7.1 11 8.5 6.9 8.4 7.1 8.3 7.2 8.2 7.4 8.0 7.5 7.9 7.6 7.8 7.8 12 9.3 7.6 9.2 7.7 9.1 7.9 8.9 8.0 8.8 8.2 8.6 8.3 8.5 8.5 13 10.1 8.2 10.0 8.4 9.8 8.5 9.7 8.7 9.5 8.9 9.4 9.0 9.2 9.2 14 10.9 8.8 10.7 9.0 10.6 9.2 10.4 9.4 10.2 9.5 10.1 9.7 9.9 9.9 15 11.7 9.4 11.5 9.6 11.3 9.8 11.1 10.0 11.0 10.2 10.8 10.4 10.6 10.6 16 12.4 10.1 12.3 10.3 12.1 10.5 11.9 10.7 11.7 10.9 11.5 11.1 11.3 11.3 17 13.2 10.7 13.0 10.9 12.8 11.2 12.6 11.4 12.4 11.6 12 2 11.8 12.0 12.0 18 14.0 11.3 13.8 11.6 13.6 11.8 13.4 12.0 13.2 12.3 12.9 12.5 12.7 12.7 19 14.8 12.0 14.6 12.2 14.3 12.5 14.1 12.7 13.9 13.0 13.7 13.2 13.4 13.4 20 15.5 12.6 15.3 12.9 15.1 13.1 14.9 13.4 14.6 13.6 14.4 13.9 14.1 14.1 21 16.3 13.2 16.1 13.5 15.8 13.8 15.6 14.1 15.4 14.3 15.1 14.6 14.8 14.8 22 17.1 13.8 16.9 14.1 16.6 14.4 16.3 14.7 16.1 15.0 15.8 15.3 15.6 15.6 23 17.9 14.5 17.6 14.8 17.4 15.1 17.1 15.4 16.8 15.7 16.5 16.0 16.3 16.3 24 18.7 15.1 18.4 15.4 18.1 15.7 17.8 16.1 17.6 16.4 17.3 16.7 17.0 17.0 25 19.4 15.7 19.2 16.1 18.9 16.4 18.6 16.7 18.3 17.0 18.0 17.4 17.7 17.7 26 20.2 16.4 19.9 16.7 19.6 17.1 19.3 17.4 19.0 17.7 18.7 18.1 18.4 18.4 27 21.0 17.0 20.7 17.4 20.4 17.7 20.1 18.1 19.7 18.4 19.4 18.8 19.1 19.1 28 21.8 17.6 21.4 18.0 21.1 18.4 20.8 18.7 20.5 19.1 20.1 19.5 19.8 19.8 29 22.5 18.3 22.2 18.6 21.9 19.0 21.6 19.4 21.2 19.8 20.9 20.1 20.5 20.5 30 23.3 18.9 23.0 19.3 22.6 19.7 22.3 20.1 21.9 20.5 21.6 20.8 21.2 21.2 31 24.1 19.5 23.7 19.9 23.4 20.3 23.0 20.7 22.7 21.1 22.3 21.5 21.9 21.9 32 24.9 20.1 24.5 20.6 24.2 21.0 23.8 21.4 23.4 21.8 23.0 22.2 22.6 22.6 33 25.6 20.8 25.3 21.2 24.9 21.6 24.5 22.1 24.1 22.5 23.7 22.9 23.3 23.3 34 26.4 21.4 26.0 21.9 25.7 22.3 25.3 22.8 24.9 23.2 24.5 23.6 24.0 24.0 35 27.2 22.0 26.8 22.5 26.4 23.0 26.0 23.4 25.6 23.9 25.2 24.3 24.7 24.7 36 28.0 22.7 27.6 23.1 27.2 23.6 26.8 24.1 26.3 24.6 25.9 25.0 25.5 25.5 37 28.8 23.3 28.3 23.8 27.9 24.3 27.5 24.8 27.1 25.2 26.6 25.7 26.2 26.2 38 29.5 23.9 29.1 24.4 28.7 24.9 28.2 25.4 27.8 25.9 27.3 26.4 26.9 26.9 39 30.3 24.5 29.9 25.1 29.4 25.6 29.0 26.1 28.5 26.6 28.1 27.1 27.6 27.6 40 31.1 25.2 30.6 25.7 30.2 26.2 29.7 26.8 29.3 27.3 28.8 27.8 28.3 28.3 41 31.9 25.8 31.4 26.4 30.9 26.9 30.5 27.4 30.0 28.0 29.5 28.5 29.0 29.0 42 32.6 26.4 32.2 27.0 31.7 27.6 31.2 28.1 30.7 28.6 30.2 29.2 29.7 29.7 43 33.4 27.1 32.9 27.6 32.5 28.2 32.0 28.8 31.4 29.3 30.9 29.9 30.4 30.4 44 34.2 27.7 33.7 28.3 33.2 28.9 32.7 29.4 32.2 30.0 31.7 30.6 31.1 31.1 45 35.0 28.3 34.5 28.9 34.0 29.5 33.4 30.1 32.9 30.7 32.4 31.3 31.8 31.8 46 35.7 28.9 35.2 29.6 34.7 30.2 34.2 30.8 33.6 31.4 33.1 32.0 32.5 32.5 47 36.5 29.6 36.0 30.2 35.5 30.8 34.9 31.4 34.4 32.1 33.8 32.6 33.2 33.2 48 37.3 30.2 36.8 30.9 36.2 31.5 35.7 32.1 35.1 32.7 34.5 33.3 33.9 33.9 49 38.1 30.8 37.5 31.5 37.0 32.1 36.4 32.8 35.8 33.4 35.2 34.0 34.6 34.6 50 38.9 31.5 38.3 32.1 37.7 32.8 37.2 33.5 36.6 34.1 36.0 34.7 35.4 35.4 100 77.7 62.9 76.6 64.3 75.5 65.6 74.3 66.9 73.1 68.2 71.9 69.5 70.7 70.7 200 55.4 125.9 53.2 28.6 50.9 131.2 48.6 133.8 46.3 136.4 143.9 138.9 141.4 141.4 300 233.1 188.8 29.8 92.8 226.4 196.8 22.9 200.7 19.4 204.6 215.8 208.4 212.1 212.1 400 310.9 251.7 06.4 257.1 301.9 262.4 97.3 267.7 92.6 272.8 287.7 277.9 282.8 282.8 500 388.6 314.7 83.0 321.4 377.3 328.0 71.6 334.6 365.7 341.0 359.7 347.3 353.5 353.5 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. (129, 231, (130, 230, (131, 229, (132, 228, (133, 227, (134, 226, (135, 225, 309) 310) 311 312) 313) 314) 315) \\ Pt. 51 50 49 4i Pt. 48 47 46 4 Pt. 45 The 4-Pt. or 45 Courses are : N.E., N.W., S.E., S.W. Table 1. Traverse Table 3^ Pt. 39 40 41 3f Pt.42 43 44 4 Pt. 45 (141, 219, (140, 220, (139, 221, (138, 222, (137, 223, (136, 224, (135, 225, DlST. 321) 320) 319) 318) 317) 316) 315) Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 51 39.6 32.1 39.1 32.8 38.5 33.5 37.9 34.1 37.3 34.8 36.7 35.4 36.1 36.1 52 40.4 32.7 39.8 33.4 39.2 34.1 38.6 34.8 38.0 35.5 37.4 36.1 36.8 36.8 53 41.2 33.4 40.6 34.1 40.0 34.8 39.4 35.5 38.8 36.1 38.1 36.8 37.5 37.5 54 42.0 34.0 41.4 34.7 40.8 35.4 40.1 36.1 39.5 36.8 38.8 37.5 38.2 38.2 55 42.7 34.6 42.1 35.4 41.5 36.1 40.9 36.8 40.2 37.5 39.6 38.2 38.9 38.9 56 43.5 35.2 42.9 36.0 42.3 36.7 41.6 37.5 41.0 38.2 40.3 38.9 39.6 39.6 57 44.3 35.9 43.7 36.6 43.0 37.4 42.4 38.1 41.7 38.9 41.0 39.6 40.3 40.3 58 45.1 36.5 44.4 37.3 43.8 38.1 43.1 38.8 42.4 39.6 41.7 40.3 41.0 41.0 59 45.9 37.1 45.2 37.9 44.5 38.7 43.8 39.5 43.1 40.2 42.4 41.0 41.7 41.7 60 46.6 37.8 46.0 38.6 45.3 39.4 44.6 40.1 43.9 40.9 43.2 41.7 42.4 42.4 61 47.4 38.4 46.7 39.2 46.0 40.0 45.3 40.8 44.6 41.6 43.9 42.4 43.1 43.1 62 48.2 39.0 47.5 39.9 46.8 40.7 46.1 41.5 45.3 42.3 44.6 43.1 43.8 43.8 63 49.0 39.6 48.3 40.5 47.5 41.3 46.8 42.2 46.1 43.0 45.3 43.8 44.5 44.5 64 49.7 40.3 49.0 41.1 48.3 42.0 47.6 42.8 46.8 43.6 46.0 44.5 45.3 45.3 65 50.5 40.9 49.8 41.8 49.1 42.6 48.3 43.5 47.5 44.3 46.8 45.2 46.0 46.0 66 51.3 41.5 50.6 42.4 49.8 43.3 49.0 44.2 48.3 45.0 47.5 45.8 46.7 46.7 67 52.1 42.2 51.3 43.1 50.6 44.0 49.8 44.8 49.0 45.7 48.2 46.5 47.4 47.4 68 52.8 42.8 52.1 43.7 51.3 44.6 50.5 45.5 49.7 46.4 48.9 47.2 48.1 48.1 69 53.6 43.4 52.9 44.4 52.1 45.3 51.3 46.2 50.5 47.1 49.6 47.9 48.8 48.8 70 54.4 44.1 53.6 45.0 52.8 45.9 52.0 46.8 51.2 47.7 50.4 48.6 49.5 49.5 71 55.2 44.7 54.4 45.6 53.6 46.6 52.8 47.5 51.9 48.4 51.1 49.3 50.2 50.2 72 56.0 45.3 55.2 46.3 54.3 47.2 53.5 48.2 52.7 49.1 51.8 50.0 50.9 50.9 73 56.7 45.9 55.9 46.9 55.1 47.9 54.2 48.8 53.4 49.8 52.5 50.7 51.6 51.6 74 57.5 46.6 56.7 47.6 55.8 48.5 55.0 49.5 54.1 50.5 53.2 51.4 52.3 52.3 75 58.3 47.2 57.5 48.2 56.6 49.2 55.7 50.2 54.9 51.1 54.0 52.1 53.0 53.0 76 59.1 47.8 58.2 48.9 57.4 49.9 56.5 50.9 55.6 51.8 54.7 52.8 53.7 53.7 77 59.8 48.5 59.0 49.5 58.1 50.5 57.2 51.5 56.3 52.5 55.4 53.5 54.4 54.4 78 60.6 49.1 59.8 50.1 58.9 51.2 58.0 52.2 57.0 53.2 56.1 54.2 55.2 55.2 79 61.4 49.7 60.5 50.8 59.6 51.8 58.7 52.9 57.8 53.9 56.8 54.9 55.9 55.9 80 62.2 50.3 61.3 51.4 60.4 52.5 59.5 53.5 58.5 54.6 57.5 55.6 56.6 56.6 81 62.9 51.0 62.0 52.1 61.1 53.1 60.2 54.2 59.2 55.2 58.3 56.3 57.3 57.3 82 63.7 51.6 62.8 52.7 61.9 53.8 60.9 54.9 60.0 55.9 59.0 57.0 58.0 58.0 83 64.5 52.2 63.6 53.4 62.6 54.5 61.7 55.5 60.7 56.6 59.7 57.7 58.7 58.7 84 65.3 52.9 64.3 54.0 63.4 55.1 62.4 56.2 61.4 57.3 60.4 58.4 59.4 59.4 85 66.1 53.5 65.1 54.6 64.2 55.8 63.2 56.9 62.2 58.0 61.1 59.0 60.1 60.1 86 66.8 54.1 65.9 55.3 64.9 56.4 63.9 57.5 62.9 58.7 61.9 59.7 60.8 60.8 87 67.6 54.8 66.6 55.9 65.7 57.1 64.7 58.2 63.6 59.3 62.6 60.4 61.5 61.5 88 68.4 55.4 67.4 56.6 66.4 57.7 65.4 58.9 64.4 60.0 63.3 61.1 62.2 62.2 89 69.2 56.0 68.2 57.2 67.2 58.4 66.1 59.6 65.1 60.7 64.0 61.8 62.9 62.9 90 69.9 56.6 68.9 57.9 67.9 59.0 66.9 60.2 65.8 61.4 64.7 62.5 63.6 63.6 91 70.7 57.3 69.7 58.5 68.7 59.7 67.6 60.9 66.6 62.1 65.5 63.2 64.3 64.3 92 71.5 57.9 70.5 59.1 69.4 60.4 68.4 61.6 67.3 62.7 66.2 63.9 65.1 65.1 93 72.3 58.5 71.2 59.8 70.2 61.0 69.1 62.2 68.0 63.4 66.9 64.6 65.8 65.8 94 73.1 59.2 72.0 60.4 70.9 61.7 69.9 62.9 68.7 64.1 67.6 65.3 66.5 66.5 95 73.8 59.8 72.8 61.1 71.7 62.3 70.6 63.6 69.5 64.8 68.3 66.0 67.2 67.2 96 74.6 60.4 73.5 61.7 72.5 63.0 71.3 64.2 70.2 65.5 69.1 66.7 67.9 67.9 97 75.4 61.0 74.3 62.4 73.2 63.6 72.1 64.9 70.9 66.2 69.8 67.4 68.6 68.6 98 76.2 61.7 75.1 63.0 74.0 64.3 72.8 65.6 71.7 66.8 70.5 68.1 69.3 69.3 99 76.9 62.3 75.8 63.6 74.7 64.9 73.6 66.2 72.4 67.5 71.2 68.8 70.0 70.0 100 77.7 62.9 76.6 64.3 75.5 65.6 74.3 66.9 73.1 68.2 71.9 69.5 70.7 70.7 600 466.3 377.6 459.6 385.7 452.8 393.6 445.9 401.5 438.8 409.2 431.6 416.8 424.3 424.3 700 543.9 440.6 536.3 450.0 528.3 459.2 520.2 468.4 511.9 477.4 503.5 486.3 495.0 495.0 800 621.8 503.5 613.0 514.2 603.9 524.8 594.6 535.3 585.1 545.6 575.4 555.8 565.7 565.7 900 699.3 566.3 689.5 578.5 679.2 590.3 668.8 602.2 658.2 613.8 647.3 625.2 636.3 636.3 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. (129, 231, (130, 230, (131, 229, (132, 228, (133, 227, (134, 226, (135, 225, 309) 310) 311) 312) 313) 314) 315) 4| Pt. 51 50 49 4J Pt. 48 47 46 4 Pt. 45 The 4-Pt. or 45 Courses are : N.E., N.W., S.E., S.W. 168 Table 2 To CHANGE LONG. DIFF. INTO DEP., SUBTRACT TABULAR NUMBER FROM LONG. DIFF. LONG. DIPF. OR MIDDLE LATITUDE DEP. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 0.0 0.0 ~bTo ~o!b~ ~ocT ~ob~ ~00 ~ob~ ~QA "oo" ~ob~ ~ooT 0.0 0.0 ~ob~ 2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.1 0.1 3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.1 0.1 0.1 o.i 4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.1 0.1 0.1 0.1 0.1 5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.1 0.1 0.1 0.1 0.1 0.2 6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.1 0.1 0.1 0.1 0.2 0.2 0.2 7 0.0 0.0 0.0 0.0 0.0 0.0 0. 0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.2 8 0.0 0.0 0.0 0.0 0.0 0.0 0. 0.1 0. 0.1 0.1 0.2 0.2 0.2 0.3 9 0.0 0.0 0.0 0.0 0.0 0.0 0. 0. 0.1 0.1 0.2 0.2 0.2 0.3 0.3 10 0.0 0.0 0.0 0.0 0.0 0.1 0. 0. 0.1 0.2 0.2 0.2 0.3 0.3 0.3 11 0.0 0.0 0.0 0.0 0.0 0.1 0. 0. 0. 0.2 0.2 0.2 0.3 0.3 0.4 12 0.0 0.0 0.0 0.0 0.0 0.1 0. 0. 0. 0.2 0.2 0.3 0.3 0.4 0.4 13 0.0 0.0 0.0 0.0 0.0 0.1 0. 0. 0.2 0.2 0.2 0.3 0.3 0.4 0.4 14 0.0 0.0 0.0 0.0 0.1 0.1 0. 0. 0.2 0.2 0.3 0.3 0.4 0.4 0.5 15 0.0 0.0 0.0 0.0 0. 0.1 0. 0. 0.2 0.2 0.3 0.3 0.4 0.4 0.5 16 0.0 0.0 0.0 0.0 0. 0.1 0. 0.2 0.2 0.2 0.3 0.3 0.4 0.5 0.5 17 0.0 0.0 0.0 0.0 0. 0. 0. 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.6 18 0.0 0.0 0.0 0.0 0. 0. 0. 0.2 0.2 0.3 0.3 0.4 0.5 0.5 0.6 19 0.0 0.0 0.0 0.0 0. 0. 0. 0.2 0.2 0.3 0.3 0.4 0.5 0.6 0.6 20 0.0 0.0 0.0 0.0 0. 0. 0. 0.2 0.2 0.3 0.4 0.4 0.5 0.6 0.7 21 0.0 0.0 0.0 0. 0.1 0. 0.2 0.2 0.3 0.3 0.4 0.5 0.5 0.6 0.7 22 0.0 0.0 0.0 0. 0.1 0. 0.2 0.2 0.3 0.3 0.4 0.5 0.6 0.7 0.7 23 0.0 0.0 0.0 0. 0.1 0. 0.2 0.2 0.3 0.3 0.4 0.5 0.6 0.7 0.8 24 0.0 0.0 0.0 0. 0.1 0. 0.2 0.2 0.3 0.4 0.4 0.5 0.6 0.7 0.8 25 0.0 0.0 0.0 0. 0. 0. 0.2 0.2 0.3 0.4 0.5 0.5 0.6 0.7 0.9 26 0.0 0.0 0.0 0. 0. 0. 0.2 0.3 0.3 0.4 0.5 0.6 0.7 0.8 0.9 27 0.0 0.0 0.0 0. 0. 0. 0.2 0.3 0.3 0.4 0.5 0.6 0.7 0.8 0.9 28 0.0 0.0 0.0 0.1 0. 0.2 0.2 0.3 0.3 0.4 0.5 0.6 0.7 0.8 .0 29 0.0 0.0 0.0 0.1 0. 0.2 0.2 0.3 0.4 0.4 0.5 0.6 0.7 0.9 .0 30 0.0 0.0 0.0 0.1 0.1 0.2 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 .0 31 0.0 0.0 0.0 0.1 0.1 0.2 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 .1 32 0.0 0.0 0.0 0.1 0.1 0.2 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1.0 .1 33 0.0 0.0 0.0 0.1 0.1 0.2 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1.0 .1 34 0.0 0.0 0.0 0.1 0.1 0.2 0.3 0.3 0.4 0.5 0.6 0.7 0.9 1.0 .2 35 0.0 0.0 0.0 0.1 0.1 0.2 0.3 0.3 0.4 0.5 0.6 0.8 0.9 1.0 .2 36 0.0 0.0 0.0 0.1 0.1 0.2 0.3 0.4 0.4 0.5 0.7 0.8 0.9 1.1 .2 37 0.0 0.0 0. 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.1 .3 38 0.0 0.0 0. 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 .0 1.1 1.3 39 0.0 0.0 0. 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.9 .0 1.2 1.3 40 0.0 0.0 0. 0.1 0.2 0.2 0.3 0.4 0.5 0.6 0.7 0.9 .0 1.2 1.4 41 0.0 0.0 0. 0.1 0.2 0.2 0.3 0.4 0.5 0.6 0.8 0.9 .1 .2 1.4 42 0.0 0.0 0. 0. 0.2 0.2 0.3 0.4 0.5 0.6 0.8 0.9 .1 .2 1.4 43 0.0 0.0 0. 0. 0.2 0.2 0.3 0.4 0.5 0.7 0.8 0.9 .1 .3 1.5 44 0.0 0.0 0. 0. 0.2 0.2 0.3 0.4 0.5 0.7 0.8 1.0 .1 .3 1.5 45 0.0 0.0 0. 0. 0.2 0.2 0.3 0.4 0.6 0.7 0.8 1.0 .2 .3 .5 46 0.0 0.0 0.1 0. 0.2 0.3 0.3 0.4 0.6 0.7 0.8 1.0 .2 .4 .6 47 0.0 0.0 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.9 1.0 .2 1.4 .6 48 0.0 0.0 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.9 1.0 .2 1.4 .6 49 0.0 0.0 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.9 1.1 .3 1.5 .7 50 0.0 0.0 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.8 0.9 1.1 .3 1.5 .7 100 0.0 0.1 0.1 0.2 0.4 0.5 0.7 1.0 1.2 1.5 1.8 2.2 2.6 3.0 3.4 200 0.0 0.1 0.3 0.5 0.8 1.1 1.5 1.9 2.5 3.0 3.7 4.4 5.1 5.9 6.8 300 0.0 0.2 0.4 0.7 1.1 1.6 2.2 2.9 3.7 4.6 5.5 6.6 7.7 8.9 10.2 400 0.1 0.2 0.6 1.0 1.5 2.2 3.0 3.9 4.9 6.1 7.4 8.7 10.2 11.9 13.7 500 0.1 0.3 0.7 1.2 1.9 2.7 3.7 4.9 6.2 7.6 9.2 10.9 12.8 14.9 17.0 1.00 1.00 1.00 1.00 1.00 1.01 1.01 1.01 1.01 1.02 1.02 1.02 1.03 1.03 1.04 FACTOR To CHANGE DEP. INTO LONG. DIFF., MULTIPLY TABULAR NUMBER BY FACTOR AT FOOT OF COLUMN, AND ADD PRODUCT TO DEP. Table 2 To CHANGE LONG. DIFF. INTO DEP. SUBTRACT TABULAR NUMBER FROM LONG. DIFF. LONG. DIFF. OR DEP. MIDDLE LATITXTDE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 51 0.0 0.0 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.8 0.9 1.1 1.3 1.5 1.7 52 0.0 0.0 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.8 1.0 1.1 1.3 1.5 1.8 53 0.0 0.0 0.1 0.1 0.2 0.3 0.4 0.5 0.7 0.8 1.0 1.2 1.4 1.6 1.8 54 0.0 0.0 0.1 0.1 0.2 0.3 0.4 0.5 0.7 0.8 1.0 1.2 1.4 1.6 1.8 55 0.0 0.0 0.1 0.1 0.2 0.3 0.4 0.5 0.7 0.8 1.0 1.2 1.4 1.6 1.9 56 0.0 0.0 0.1 0.1 0.2 0.3 0.4 0.5 0:7 0.9 1.0 1.2 1.4 1.7 1.9 57 0.0 0.0 0.1 0.1 0.2 0.3 0.4 0.6 0.7 0.9 1.0 1.2 1.5 1.7 1.9 58 0.0 0.0 0.1 0.1 0.2 0.3 0.4 0.6 0.7 0.9 1.1 1.3 1.5 1.7 2.0 59 0.0 0.0 0.1 0.1 ).2 0.3 0.4 0.6 0.7 0.9 1.1 1.3 1.5 1.8 2.0 60 0.0 0.0 0.1 0.1 0.2 0.3 0.4 0.6 0.7 0.9 1.1 1.3 1.5 1.8 2.0 61 0.0 0.0 0.1 0.1 0.2 0.3 0.5 0.6 0.8 0.9 1.1 1.3 1.6 1.8 2.1 62 0.0 0.0 0.1 0.2 0.2 0.3 0.5 0.6 0.8 0.9 1.1 1.4 1.6 1.8 2.1 63 0.0 0.0 0.1 0.2 0.2 0.3 0.5 0.6 0.8 1.0 1.2 1.4 1.6 1.9 2.1 64 0.0 0.0 0.1 0.2 0.2 0.4 0.5 0.6 0.8 1.0 1.2 1.4 1.6 1.9 2.2 65 0.0 0.0 0.1 0.2 0.2 0.4 0.5 0.6 0.8 1.0 1.2 1.4 1.7 1.9 2.2 66 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.8 1.0 1.2 1.4 1.7 2.0 2.2 67 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.7 0.8 1.0 1.2 1.5 1.7 2.0 2.3 68 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.7 0.8 1.0 1.2 1.5 1.7 2.0 2.3 69 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.7 0.8 .0 1.3 1.5 1.8 2.0 2.4 70 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.7 0.9 .1 1.3 1.5 1.8 2.1 2.4 71 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.7 0.9 .1 1.3 1.6 1.8 2.1 2.4 72 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.7 0.9 .1 1.3 1.6 1.8 2.1 2.5 73 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.7 0.9 .1 1.3 1.6 1.9 2.2 2.5 74 0.0 0.0 0.1 0.2 0.3 0.4 0.6 0.7 0.9 .1 1.4 1.6 1.9 2.2 2.5 75 0.0 0.0 0.1 0.2 0.3 0.4 0.6 0.7 0.9 .1 1.4 1.6 1.9 2.2 2.6 76 0.0 0.0 0.1 0.2 0.3 0.4 0.6 0.7 0.9 .2 1.4 1.7 1.9 2.3 2.6 77 0.0 0.0 0.1 0.2 0.3 0.4 0.6 0.7 0.9 .2 1.4 1.7 2.0 2.3 2.6 78 0.0 0.0 0.1 0.2 0.3 0.4 0.6 0.8 .0 .2 1.4 1.7 2.0 2.3 2.7 79 0.0 0.0 0.1 0.2 0.3 0.4 0.6 0.8 .0 .2 1.5 1.7 2.0 2.3 2.7 80 0.0 0.0 0.1 0.2 0.3 0.4 0.6 0.8 .0 .2 1.5 1.7 2.1 2.4 2.7 81 0.0 0.0 0.1 0.2 0.3 0.4 0.6 0.8 .0 .2 1.5 1.8 2.1 2.4 2.8 82 0.0 0.0 0.1 0.2 0.3 0.4 0.6 0.8 .0 .2 1.5 1.8 2.1 2.4 2.8 83 0.0 0.1 0.1 0.2 0.3 0.5 0.6 0.8 .0 .3 1.5 1.8 2.1 2.5 2.8 84 0.0 0.1 0.1 0.2 0.3 0.5 0.6 0.8 .0 .3 1.5 1.8 2.2 2.5 2.9 85 0.0 0.1 0.1 0.2 0.3 0.5 0.6 0.8 .0 1.3 1.6 1.9 2.2 2.5 2.9 86 0.0 0.1 0.1 0.2 0.3 0.5 0.6 0.8 .1 1.3 1.6 1.9 2.2 2.6 2.9 87 0.0 0.1 0.1 0.2 0.3 0.5 0.6 0.8 1.3 1.6 1.9 2.2 2.6 3.0 88 0.0 0.1 0.1 0.2 0.3 0.5 0.7 0.9 1.3 1.6 1.9 2.3 2.6 3.0 89 0.0 0.1 0.1 0.2 0.3 0.5 0.7 0.9 1.4 1.6 1.9 2.3 2.6 3.0 90 0.0 0.1 0.1 0.2 0.3 0.5 0.7 0.9 | 1.4 1.7 2.0 2.3 2.7 3.1 91 0.0 0.1 0.1 0.2 0.3 0.5 0.7 0.9 .4 "l.7 2.0 2.3 2.7 3.1 92 0.0 0.1 0.1 0.2 0.4 0.5 0.7 0.9 < .4 1.7 2.0 2.4 2.7 3.1 93 0.0 0.1 0.1 0.2 0.4 0.5 0.7 0.9 .4 1.7 2.0 2.4 2.8 3.2 94 0.0 0.1 0.1 0.2 0.4 0.5 0.7 0.9 '.2 .4 1.7 2.1 2.4 2.8 3.2 95 0.0 0.1 0.1 0.2 0.4 0.5 0.7 0.9 .2 .4 1.7 2.1 2.4 2.8 3.2 96 0.0 0.1 0.1 0.2 0.4 0.5 0.7 0.9 .2 .5 1.8 2.1 2.5 2.9 3.3 97 0.0 0.1 0.1 0.2 0.4 0.5 0.7 0.9 .2 .5 1.8 2.1 2.5 2.9 3.3 98 0.0 0.1 0.1 0.2 0.4 0.5 0.7 1.0 .2 .5 1.8 2.1 2.5 2.9 3.3 99 0.0 0.1 0.1 0.2 0.4 0.5 0.7 1.0 .2 .5 1.8 2.2 2.5 2.9 3.4 100 0.0 0.1 0.1 0.2 0.4 0.5 0.7 1.0 .2 .5 1.8 2.2 2.6 3.0 3.4 600 0.1 0.4 0.8 1.4 2.3 3.3 4.5 5.8 7.4 9.1 10.0 13.1 15.4 17.8 20.5 700 0.2 0.5 1.0 1.8 2.8 3.9 5.1 6.7 8.7 10.5 12.9 15.3 17.9 20.8 23.9 800 0.2 0.5 1.1 2.0 3.1 4.4 5.9 7.7 9.8 12.1 14.8 17.5 20.6 23.8 27.3 900 0.3 0.7 1.4 2.4 3.6 5.0 6.7 8.7 11.2 13.7 16.7 19.8 23.2 26.8 30.8 LOO LOO LOO LOO LOO L01 L01 1.01 1.01 1.02 1.02 1.02 1.03 1.03 1.04 FACTOR To CHANGE DEP. INTO LONG. DIFF. MULTIPLY TABULAR NUMBER BY FACTOR AT FOOT OF COLUMN AND ADD PRODUCT TO DEP. 170 Table To CHANGE LONG. DIFF. INTO DEP., SUBTRACT TABULAR NUMBER FROM LONG. DIFF. LONG. DIFF. OR MIDDLE LATITUDE DEP. 16 17 18 19 20 21 22 23 24 25 26 27 28 1 0.0 0.0 0.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 2 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0.2 0.2 3 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3 0.4 4 0.2 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3 0.4 0.4 0.4 0.5 5 0.2 0.2 0.2 0.3 0.3 0.3 0.4 0.4 0.4 0.5 0.5 0.5 0.6 6 0.2 0.3 0.3 0.3 0.4 0.4 0.4 0.5 0.5 0.6 0.6 0.7 0.7 7 0.3 0.3 0.3 0.4 0.4 0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 8 0.3 0.3 0.4 0.4 0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.9 0.9 9 0.3 0.4 0.4 0.5 0.5 0.6 0.7 0.7 0.8 0.8 0.9 1.0 1.1 10 0.4 0.4 0.5 0.5 0.6 0.7 0.7 0.8 0.9 0.9 1.0 1.1 1.2 11 0.4 0.5 0.5 0.6 0.7 0.7 0.8 0.9 1.0 1.0 1.1 1.2 1.3 12 0.5 0.5 0.6 0.7 0.7 0.8 0.9 1.0 1.0 1.1 1.2 1.3 1.4 13 0.5 0.6 0.6 0.7 0.8 0.9 0.9 1.0 1.1 1.2 1.3 1.4 1.5 14 05 0.6 0.7 0.8 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1 5 1 6 15 0.6 0.7 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.8 16 06 0.7 0.8 0.9 1.0 1 i 1 2 1.3 1.4 1.5 1 6 1 7 1 9 17 0.7 0.7 0.8 0.9 1.0 1.1 1.2 1.4 1.5 1.6 1.7 1.9 2.0 18 0.7 0.8 0.9 .0 1.1 1.2 1.3 1.4 1:6 1.7 1.8 2.0 2.1 19 0.7 0.8 0.9 .0 1.1 .3 1.4 1.5 1.6 1.8 1.9 2.1 2.2 20 0.8 0.9 .0 .1 .2 .3 1.5 1.6 1.7 1.9 2.0 2.2 2.3 21 0.8 0.9 .0 .1 .3 .4 1.5 1.7 1.8 2.0 2.1 2.3 2.5 22 09 .0 .1 .2 .3 .5 1.6 1.7 1.9 2.1 2.2 24 9 6 23 0.9 .0 .1 .3 .4 .5 . 1.7 1.8 2.0 2.2 2.3 2.5 2.7 24 0.9 .0 .2 .3 .4 .6 1.7 1.9 2.1 2.2 2.4 2.6 2.8 25 1.0 .1 .2 .4 .5 .7 1.8 2.0 2.2 2.3 2.5 2.7 2.9 26 1.0 .1 .3 .4 .6 .7 1.9 2.1 2.2 2.4 2.6 2.8 3.0 27 .0 .2 .3 .5 .6 1.8 2.0 2.1 2.3 2.5 2.7 2.9 3.2 28 .1 .2 .4 .5 .7 1.9 2.0 2.2 2.4 2.6 2.8 3.1 3.3 29 .1 .3 .4 .6 1.7 1.9 2.1 2.3 2.5 2.7 2.9 3.2 3.4 30 .75 .3 .5 .6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.3 3 5 31 .2 .4 .5 .7 1.9 2.1 2.3 2.5 2.7 2.9 3.1 3.4 3.6 32 .2 .4 .6 1.7 1.9 2.1 2.3 2.5 2.8 3.0 3.2 3.5 3.7 33 .3 .4 .6 1.8 2.0 2.2 2.4 2.6 2.9 3.1 3.3 3.6 3.9 34 .3 .5 .7 1.9 2.1 2.3 2.5 2.7 2.9 3.2 3.4 3.7 4.0 35 .4 .5 .7 1.9 2.1 2.3 2.5 2.8 3.0 3.3 3.5 3.8 4.1 36 .4 .6 .8 2.0 2.2 2.4 2.6 2.9 3.1 3.4 3.6 3.9 4.2 37 4 .6 1.8 2.0 2.2 2.5 2.7 2.9 3.2 3.5 3.7 4.0 43 38 5 .7 1.9 2.1 2.3 2.5 2.8 3.0 3.3 3.6 3.8 4.1 44 39 5 1.7 1.9 2.1 2.4 2.6 2.8 3.1 3.4 3.7 3.9 4.3 4 6 40 5 1.7 2.0 2.2 2.4 2.7 2.9 3.2 3.5 3.7 4.0 4.4 47 41 .6 1.8 2.0 2.2 2.5 2.7 3.0 3.3 3.5 3.8 4.1 4.5 4.8 42 .6 1.8 2.1 2.3 2.5 2.8 3.1 3.3 3.6 3.9 4.3 4.6 4.9 43 .7 1.9 2.1 2.3 2.6 2.9 3.1 3.4 3.7 4.0 4.4 4.7 5.0 44 .7 1.9 2.2 2.4 2.7 2.9 3.2 3.5 3.8 4.1 4.5 4.8 5.2 45 .7 2.0 2.2 2.5 2.7 3.0 3.3 3.6 3.9 4.2 4.6 4.9 5.3 46 .8 2.0 2.3 2.5 2.8 3.1 3.3 3.7 4.0 4.3 4.7 5.0 5.4 47 .8 2.1 2.3 2.6 2.8 3.1 3.4 3.7 4.1 4.4 4.8 5.1 5.5 48 .9 2.1 2.3 2.6 2.9 3.2 3.5 3.8 4.1 4.5 4.9 5.2 5.6 49 , .9 2.1 2.4 2.7 3.0 3.3 3.6 3.9 4.2 4.6 5.0 5.3 5.7 50 1.9 2.2 2.4 2.7 3.0 3.3 3.6 4.0 4.3 4.7 5.1 5.4 5.9 100 3.9 4.4 4.9 5.4 6.0 6.6 7.3 7.9 8.6 9.4 10.1 10.9 11.7 200 7.7 8.7 9.8 10.9 12.1 13.3 14.6 15.9 17.3 18.7 20.2 21.8 23.4 300 11.6 13.1 14.7 16.3 18.1 19.9 21.8 23.8 25.9 28.1 30.4 32.7 35.1 400 15.5 17.5 19.6 21.8 24.1 26.6 29.1 31.8 34.6 37.5 40.5 43.6 46.9 500 19.4 21.9 24.5 27.2 30.1 33.2 36.4 39.8 43.2 46.9 50.6 54.5 58.5 1.04 1.05 1.05 1.06 1.06 1.07 1.08 1.09 1.09 1.10 1.11 1.12 1.13 FACTOR To CHANGE DEP. INTO LONG. DIFF., MULTIPLY TABULAR NUMBER BY FACTOR AT FOOT OF COLUMN AND ADD PRODUCT TO DEP. Table 2 171 To CHANGE LONG. DIFF. INTO DEP. SUBTRACT TABULAR NUMBER FROM LONG. DIFF. LONG. DIFF. MIDDLE LATITUDE OR DEP. 16 17 18 19 20 21 22 23 24 25 26 27 28 51 2.0 2.2 2.5 2.8 3.1 3.4 3.7 4.1 4.4 4.8 5.2 5.6 6.0 52 2 2 3 2.5 2.8 3.1 35 3.8 4.1 4.5 4.9 5.3 5.7 6 1 53 2.1 2.3 2.6 2.9 3.2 3.5 3.9 4.2 4.6 5.0 5.4 5.8 6.2 54 2.1 2.4 2.6 2.9 3.3 3.6 3.9 4.3 4.7 5.1 5.5 5.9 6.3 55 2.1 2.4 2.7 3.0 3.3 3.7 4.0 4.4 4.8 5.2 5.6 6.0 6.4 56 2? 2.4 2.7 3.1 3.4 3 7 4.1 4.5 4.8 5.2 5.7 6.1 66 57 2 2 2.5 28 3.1 34 3 8 4.2 4.5 49 5.3 5.8 6.2 67 58 2.2 2.5 2.8 3.2 3.5 3.9 4.2 4.6 5.0 5.4 5.9 6.3 6.8 59 2.3 2.6 2.9 3.2 3.6 3.9 4.3 4.7 5.1 5.5 6.0 6.4 6.9 60 2.3 2.6 2.9 3.3 3.6 4.0 4.4 4.8 5.2 5.6 6.1 6.5 7.0 61 2.4 2.7 3.0 3.3 3.7 4.1 4.4 4.8 5.3 5.7 6.2 6.6 7.1 62 2.4 2.7 3.0 3.4 3.7 4.1 4.5 4.9 5.4 5.8 6.3 6.8 7.3 63 2.4 2.8 3.1 3.4 3.8 4.2 4.6 5.0 5.4 5.9 6.4 6.9 7.4 64 2.5 2.8 3.1 3.5 3.9 4.3 4.7 5.1 5.5 6.0 6.5 7.0 7.5 65 2.5 2.8 3.2 3.5 3.9 4.3 4.7 5.2 5.6 6.1 6.6 7.1 7.6 66 2.6 2.9 3.2 3.6 4.0 4.4 4.8 5.2 5.7 6.2 6.7 7.2 7.7 67 2.6 2.9 3.3 3.7 4.0 4.5 4.9 5.3 5.8 6.3 6.8 7.3 7.8 68 2.6 3.0 3.3 3.7 4.1 4.5 5.0 5.4 5.9 6.4 6.9 7.4 8.0 69 2.7 3.0 3.4 3.8 4.2 4.6 5.0 5.5 6.0 6.5 7.0 7.5 8.1 70 2.7 3.1 3.4 3.8 4.2 4.6 5.1 5.6 6.1 6.6 7.1 7.6 8.2 71 2.8 3.1 3.5 3.9 4.3 4.7 5.2 5.6 6.1 6.7 7.2 7.7 8.3 72 2.8 3.1 3.5 3.9 4.3 4.8 5.2 5.7 6.2 6.7 7.3 7.8 8.4 73 2.8 3.2 3.6 4.0 4.4 4.8 5.3 5.8 6.3 6.8 7.4 8.0 8.5 74 2.9 3.2 3.6 4.0 4.5 4.9 5.4 5.9 6.4 6.9 7.5 8.1 8.7 75 2.9 3.3 3.7 4.1 4.5 5.0 5.5 6.0 6.5 7.0 7.6 8.2 8.8 76 2.9 3.3 3.7 4.1 4.6 5.0 5.5 6.0 6.6 7.1 7.7 8.3 8.9 77 3.0 3.4 3.8 4.2 4.6 5.1 5.6 6.1 6.7 7.2 7.8 8.4 9.0 78 3.0 3.4 3.8 4.2 4.7 5.2 5.7 6.2 6.7 7.3 7.9 8.5- 9.1 79 3.1 3.5 3.9 4.3 4.8 5.2 5.8 6.3 6.8 7.4 8.0 8.6 9.2 80 3.1 3.5 3.9 4.4 4.8 5.3 5.8 6.4 6.9 7.5 8.1 8.7 9.4 81 3.1 3.5 4.0 4.4 4.9 5.4 5.9 6.4 7.0 7.6 8.2 8.8 9.5 82 3.2 3.6 4.0 4.5 4.9 5.4 6.0 6.5 7.1 7.7 8.3 8.9 9.6 83 3.2 3.6 4.1 4.5 5.0 5.5 6.0 6.6 7.2 7.8 8.4 9.0 9.7 84 3.3 3.7 4.1 4.6 5.1 5.6 6.1 6.7 7.3 7.9 8.5 9.2 9.8 85 3.3 3.7 4.2 4.6 5.1 5.6 6.2 6.8 7.3 8.0 8.6 9.3 9.9 86 3.3 3.8 4.2 4.7 5.2 5.7 6.3 6.8 7.4 8.1 8.7 9.4 10.1 87 3.4 3.8 4.3 4.7 5.2 5.8 6.3 6.9 7.5 8.2 8.8 9.5 10.2 88 3.4 3.8 4.3 4.8 5.3 5.8 6.4 7.0 7.6 8.2 8.9 9.6 10.3 89 3.4 3.9 4.4 4.8 5.4 5.9 6.5 7.1 7.7 8.3 9.0 9.7 10.4 90 3.5 3.9 4.4 4.9 5.4 6.0 6.6 7.2 7.8 8.4 9.1 9.8 10.5 91 3.5 4.0 4.5 5.0 5.5 6.0 6.6 7.2 7.9 8.5 9.2 9.9 10.7 92 3.6 4.0 4.5 5.0 5.5 6.1 6.7 7.3 8.0 8.6 9.3 10.0 10.8 93 3.6 4.1 4.6 5.1 5.6 6.2 6.8 7.4 8.0 8.7 9.4 10.1 10.9 94 3.6 4.1 4.6 5.1 5.7 6.2 6.8 7.5 8.1 8.8 9.5 10.2 11.0 95 3.7 4.2 4.6 5.2 5.7 6.3 6.9 7.6 8.2 8.9 9.6 10.4 11.1 96 3.7 4.2 4.7 5.2 5.8 6.4 7.0 7.6 8.3 9.0 9.7 10.5 11.2 97 3.8 4.2 4.7 5.3 5.8 6.4 7.1. 7.7 8.4 9.1 9.8 10.6 11.4 98 3.8 4.3 4.8 5.3 5.9 6.5 7.1 7.8 8.5 9.2 9.9 10.7 11.5 99 3.8 4.3 4.8 5.4 6.0 6.6 7.2 7.9 8.6 9.3 10.0 10.8 11.6 100 3.9 4.4 4.9 5.4 6.0 6.6 7.3 7.9 8.6 9.4 10.1 10.9 11.7 600 23.2 26.2 29.4 32.7 36.2 39.9 43.7 47.7 51.9 56.2 60.7 65.4 70.2 700 27.2 30.6 34.2 38.1 42.1 46.4 50.9 55.7 60.5 65.5 70.8 76.3 82.0 800 31.0 35.0 39.2 43.5 48.2 53.1 58.2 63.6 69.2 74.9 80.9 87.1 93.7 900 35.0 39.4 44.1 49.1 54.3 59.7 65.5 71.7 77.9 84.4 91.1 98.1 105.5 1.04 1.05 1.05 1.06 1.06 1.07 1.08 1.09 1.10 1.10 1.11 1.12 1.13 FACTOB To CHANGE DEP. INTO LONG. DIFF. MULTIPLY T ABULAR NUMBER BY FACTOR AT FOOT OF COLUMN, AND ADD PRODUCT TO DEP. 172 Table 2 To CHANGE LONG. DIFF. INTO DEP., SUBTRACT TABULAR NUMBER FROM LONG. DIFF. LONG. DIFF. OR MIDDLE LATITUDE DEP. 29 30 31 32 33 34 35 36 37 38 39 40 1 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 2 0.3 0.3 0.3 0.3 0.3 0.3 0.4 0.4 0.4 0.4 0.4 0.5 3 0.4 0.4 0.4 0.5 0.5 0.5 0.5 0.6 0.6 0.6 0.7 0.7 4 0.5 0.5 0.6 0.6 0.6 0.7 0.7 0.8 0.8 0.8 0.9 0.9 5 0.6 0.7 0.7 0.8 0.8 0.9 0.9 1.0 1.0 1.1 1.1 1.2 6 0.8 0.8 0.9 0.9 1.0 1.0 1.1 1.1 1.2 1.3 1.3 1.4 7 0.9 0.9 1.0 1.1 1.1 1.2 1.3 1.3 1.4 1.5 1.6 1.6 8 .0 .1 1.1 1.2 1.3 1.4 1.4 1.5 1.6 1.7 1.8 1.9 9 .1 .2 1.3 1.4 1.5 1.5 1.6 1.7 1.8 1.9 2.0 2.1 10 .3 .3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 11 .4 .5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.5 2.6 12 .5 .6 1.7 1.8 1.9 2.1 2.2 2.3 2.4 2.5 2.7 2.8 13 .6 .7 1.9 2.0 2.1 2.2 2.4 2.5 2.6 2.8 2.9 3.0 14 .8 .9 2.0 2.1 2.3 2.4 2.5 2.7 2.8 3.0 3.1 3.3 15 .9 2.0 2.1 2.3 2.4 2.6 2.7 2.9 3.0 3.2 3.3 3.5 16 2.0 2.1 2.3 2.4 2.6 2.7 2.9 3.1 3.2 3.4 3.6 3.7 17 2.1 2.3 2.4 2.6 2.7 2.9 3.1 3.2 3.4 3.6 3.8 4.0 18 2.3 2.4 2.6 2.7 2.9 3.1 3.3 3.4 3.6 3.8 4.0 4.2 19 2.4 2.5 2.7 2.9 3.1 3.2 3.4 3.6 3.8 4.0 4.2 4.4 20 2.5 2.7 2.9 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.5 4.7 21 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.5 4.7 4.9 22 2.8 2.9 3.1 3.3 3.5 3.8 4.0 4.2 4.4 4.7 4.9 5.1 23 2.9 3.1 3.3 3.5 3.7 3,9 4.2 4.4 4.6 4.9 5.1 5.4 24 3.0 3.2 3.4 3.6 3.9 4.1 4.3 4.6 4.8 5.1 5.3 5.6 25 3.1 3.3 3.6 3.8 4.0 4.3 4.5 4.8 5.0 5.3 5.6 5.8 26 3.3 3.5 3.7 4.0 4.2 4.4 4.7 5.0 5.2 5.5 5.8 6.1 27 3.4 3.6 3.9 4.1 4.4 4.6 4.9 5.2 5.4 5.7 6.0 6.3 28 3.5 3.8 4.0 4.3 4.5 4.8 5.1 5.3 5.6 5.9 6.2 6.6 29 3.6 3.9 4.1 4.4 4.7 5.0 5.2 5.5 5.8 6.1 6.5 6.8 30 3.8 4.0 4.3 4.6 4.8 5.1 5.4 5.7 6.0 6.4 6.7 7.0 31 3.9 4.2 4.4 4.7 5.0 5.3 5.6 5.9 6.2 6.6 6.9 7.3 32 4.0 4.3 4.6 4.9 5.2 5.5 5.8 6.1 6.4 6.8 7.1 7.5 33 4.1 4.4 4.7 5.0 5.3 5.6 6.0 6.3 6.6 7.0 7.4 7.7 34 4.3 4.6 4.9 5.2 5.5 5.8 6.1 6.5 6.8 7.2 7.6 8.0 35 4.4 4.7 5.0 5.3 5.6 6.0 6.3 6.7 7.0 7.4 7.8 8.2 36 4.5 4.8 5.1 5.5 5.8 6.2 6.5 6.9 7.2 7.6 8.0 8.4 37 4.6 5.0 5.3 5.6 6.0 6.3 6.7 7.1 7.5 7.8 8.2 8.7 38 4.8 5.1 5.4 5.8 6.1 6.5 6.9 7.3 7.7 8.1 8.5 8.9 39 4.9 5.2 5.6 5.9 6.3 6.7 7.1 7.4 7.9 8.3 8.7 9.1 40 5.0 5.4 5.7 6.1 6.5 6.8 7.2 7.6 8.1 8.5 8.9 9.4 41 5.1 5.5 5.9 6.2 6.6 7.0 7.4 7.8 8.3 8.7 9.1 9.6 42 5.3 5.6 6.0 6.4 6.8 7.2 7.6 8.0 8.5 8.9 9.4 9.8 43 5.4 5.8 6.1 6.5 6.9 7.4 7.8 8.2 8.7 9.1 9.6 10.1 44 5.5 5.9 6.3 6.7 7.1 7.5 8.0 8.4 8.9 9.3 9.8 10.3 45 5.6 6.0 6.4 6.8 7.3 7.7 8.1 8.6 9.1 9.5 10.0 10.5 46 5.8 6.2 6.6 7.0 7.4 7.9 8.3 8.8 9.3 9.8 10.3 10.8 47 5.9 6.3 6.7 7.1 7.6 8.0 8.5 9.0 9.5 10.0 10.5 11.0 48 6.0 6.4 6.9 7.3 7.7 8.2 8.7 9.2 9.7 10.2 10.7 11.2 49 6.1 6.6 7.0 7.4 7.9 8.4 8.9 9.4 9.9 10.4 10.9 11.5 50 6.3 6.7 7.1 7.6 8.1 8.5 9.0 9.5 10.1 10.6 11.1 11.7 100 12.5 13.4 14.3 15.2 16.1 17.1 18.1 19.1 20.1 21.2 22.3 23.4 200 25.1 26.8 28.6 30.4 32.3 34.2 36.2 38.2 40.3 42.4 44.6 46.8 300 37.6 40.2 42.9 45.6 48.4 51.3 54.3 57.3 60.4 63.6 66.9 70.2 400 50.2 53.6 57.1 60.8 64.5 68.4 72.3 76.4 80.6 84.8 89.1 93.6 500 62.7 67.0 71.4 76.0 80.7 85.5 90.4 95.5 100.7 106.0 111.4 117.0 1.14 1.15 1.17 1.18 1.19 1.21 1.22 1.24 1.25 1.27 1.29 1.31 FACTOK To CHANGE DEP. INTO LONG. DIFF., MULTIPLY TABULAR NUMBER BY FACTOR AT FOOT OF COLUMN, AND ADD PRODUCT TO DEP. Table 2 To CHANGE LONG. DIFF. INTO DEP. SUBTRACT TABULAR NUMBER FROM LONG. DIFF. LONG. DIFF. MIDDLE LATITUDE OR DEP. 29 30 31 32 33 34 35 36 37 38 39 40 51 6.4 6.8 7.3 7.7 8.2 8.7 9.2 9.7 10.3 10.8 11.4 11.9 52 6.5 7.0 7.4 7.9 8.4 8.9 9.4 9.9 10.5 11.0 11.6 12.2 53 6.6 7.1 7.6 8.1 8.6 9.1 9.6 10.1 10.7 11.2 11.8 12.4 54 6.8 7.2 7.7 8.2 8.7 9.2 9.8 10.3 10.9 11.4 12.0 12.6 55 6.9 7.4 7.9 8.4 8.9 9.4 9.9 10.5 11.1 11.7 12.3 12.9 56 7.0 7.5 8.0 8.5 9.0 9.6 10.1 10.7 11.3 11.9 12.5 13.1 57 7.1 7.6 8.1 8.7 9.2 9.7 10.3 10.9 11.5 12.1 12.7 13.3 58 7.3 7.8 8.3 8.8 9.4 9.9 10.5 11.1 11.7 12.3 12.9 13.6 59 7.4 7.9 8.4 9.0 9.5 10.1 10.7 11.3 11.9 12.5 13.1 13.8 60 7.5 8.0 8.6 9.1 9.7 10.3 10.9 11.5 12.1 12.7 13.4 14.0 61 7.6 8.2 8.7 9.3 9.8 10.4 11.0 11.6 12.3 12.9 13.6 14.3 62 7.8 8.3 8.9 9.4 10.0 10.6 11.2 11.8 12.5 13.1 13.8 14.5 63 7.9 8.4 9.0 9.6 10.2 10.8 11.4 12.0 12.7 13.4 14.0 14.7 64 8.0 8.6 9.1 9.7 10.3 10.9 11.6 12.2 12.9 13.6 14.3 15.0 65 8.1 8.7 9.3 9.9 10.5 11.1 11.8 12.4 13.1 13.8 14.5 15.2 66 8.3 8.8 9.4 10.0 10.6 11.3 11.9 12.6 13.3 14.0 14.7 15.4 67 8.4 9.0 9.6 10.2 10.8 11.5 12.1 ' 12.8 13.5 14.2 14.9 15.7 68 8.5 9.1 9.7 10.3 11.0 11.6 12.3 13.0 13.7 14.4 15.2 15.9 69 8.7 9.2 9.9 10.5 11.1 11.8 12.5 13.2 13.9 14.6 15.4 16.1 70 8.8 9.4 10.0 10.6 11.3 12.0 12.7 13.4 14.1 14.8 15.6 16.4 71 8.9 9.5 10.1 10.8 11.5 12.1 12.8 13.6 14.3 15.1 15.8 16.6 72 9.0 9.6 10.3 10.9 11.6 12.3 13.0 13.8 14.5 15.3 16.0 16.8 73 9.2 9.8 10.4 11.1 11.8 12.5 13.2 13.9 14.7 15.5 16.3 17.1 74 9.3 9.9 10.6 11.2 11.9 12.7 13.4 14.1 14.9 15.7 16.5 17.3 75 9.4 10.0 10.7 11.4 12.1 12.8 13.6 14.3 15.1 15.9 16.7 17.5 76 9.5 10.2 10.9 11.5 12.3 13.0 13.7 14.5 15.3 16.1 16.9 17.8 77 9.7 10.3 11.0 11.7 12.4 13.2 13.9 14.7 15.5 16.3 17.2 18.0 78 98 10.5 11.1 11.9 12.6 13.3 14.1 14.9 15.7 16.5 174 182 79 9.9 10.6 11.3 12.0 12.7 13.5 14.3 15.1 15.9 16.7 17.6 18.5 80 10.0 10.7 11.4 12.2 12.9 13.7 14.5 15.3 16.1 17.0 17.8 18.7 81 10.2 10.9 11.6 12.3 13.1 13.8 14.6 15.5 16.3 17.2 18.1 19.0 82 10.3 11.0 11.7 12.5 13.2 14.0 14.8 15.7 16.5 17.4 18.3 19.2 83 10.4 11.1 11.9 12.6 13.4 14.2 15.0 15.9 16.7 17.6 18.5 19.4 84 10.5 11.3 12.0 12.8 13.6 14.4 15.2 16.0 16.9 17.8 18.7 19.7 85 10.7 11.4 12.1 12.9 13.7 14.5 15.4 16.2 17.1 18.0 18.9 19.9 86 10.8 11.5 12.3 13.1 13.9 14.7 15.6 16.4 17.3 18.2 19.2 20.1 87 10.9 11.7 12.4 13.2 14.0 14.9 15.7 16.6 17.5 18.4 19.4 20.4 88 11.0 11.8 12.6 13.4 14.2 15.0 15.9 16.8 17.7 18.7 19.6 20.6 89 11.2 11.9 12.7 13.5 14.4 15.2 16.1 17.0 17.9 18.9 19.8 20.8 90 11.3 12.1 12.9 13.7 14.5 15.4 16.3 17.2 18.1 19.1 20.1 21.1 91 11.4 12.2 13.0 13.8 14.7 15.6 16.5 17.4 18.3 19.3 20.3 21.3 92 11.5 12.3 13.1 14.0 14.8 15.7 16.6 17.6 18.5 19.5 20.5 21.5 93 11.7 12.5 13.3 14.1 15.0 15.9 16.8 17.8 18.7 19.7 20.7 21.8 94 11.8 12.6 13.4 14.3 15.2 16.1 17.0 18.0 18.9 19.9 20.9 22.0 95 11.9 12.7 13.6 14.4 15.3 16.2 17.2 18.1 19.1 20.1 21.2 22.2 96 12.0 12.9 13.7 14.6 15.5 16.4 17.4 18.3 19.3 20.4 21.4 22.5 97 12.2 13.0 13.9 14.7 15.6 16.6 17.5 18.5 19.5 20.6 21.6 22.7 98 12.3 13.1 14.0 14.9 15.8 16.8 17.7 18.7 19.7 20.8 21.8 22.9 99 12.4 13.3 14.1 15.0 16.0 16.9 17.9 18.9 19.9 21.0 22.1 23.2 100 12.5 13.4 14.3 15.2 16.1 17.1 18.1 19.1 20.1 21.2 22.3 23.4 600 75.2 80.4 85.7 91.2 96.8 102.6 108.5 114.6 120.8 127.2 133.7 140.4 700 87.8 93.9 99.9 106.4 113.0 119.7 126.5 133.8 141.0 148.4 156.1 163.7 800 100.3 107.2 114.2 121.6 129.0 136.7 144.6 152.7 161.1 169.6 178.2 187.0 900 113.0 120.7 128.6 136.8 145.2 153.9 162.8 171.9 181.4 190.9 200.7 210.5 1.14 1.15 1.17 1.18 1.19 1.21 1.22 1.24 1.25 1.27 1.29 1.31 ! FACTOR To CHANGE DEP. INTO LONG. DIFF. MULTIPLY TABULAR NUMBER BY FACTOR AT FOOT OF COLUMN AND ADD PRODUCT TO DEP. 174 Table 2 To CHANGE LONG. DIFP. INTO DEP., SUBTRACT TABULAR NUMBER FROM LONG. DIFF. LONG. DIPF. OR MIDDLE LATITUDE DEP. 41 42 43 44 45 46 47 48 49 50 51 1 0.2 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.4 0.4 2 0.5 0.5 0.5 0.6 0.6 0.6 0.6 0.7 0.7 0.7 0.7 3 0.7 0.8 0.8 0.8 0.9 0.9 1.0 1.0 1.0 1.1 1.1 4 1.0 1.0 1.1 1.1 1.2 1.2 1.3 1.3 1.4 1.4 15 5 1.2 1.3 1.3 1.4 1.5 1.5 1.6 1.7 1.7 1.8 1.9 6 1.5 1.5 1.6 1.7 1.8 1.8 1.9 2.0 2.1 2.1 2.2 7 1.7 1.8 1.9 2.0 2.1 2.1 2.2 2.3 2.4 2.5 2.6 8 2.0 2.1 2.1 2.2 2.3 2.4 2.5 2.6 2.8 2.9 3.0 9 2.2 2.3 2.4 2.5 2.6 2.7 2.9 3.0 3.1 3.2 3.3 10 2.5 2.6 2.7 2.8 2.9 3.1 3.2 3.3 3.4 3.6 3.7 11 2.7 2.8 3.0 3.1 3.2 3.4 3.5 3.6 3.8 3.9 4.1 12 2.9 3.1 3.2 3.4 3.5 3.7 3.8 4.0 4.1 4.3 4.4 13 3.2 3.3 3.5 3.6 3.8 4.0 4.1 4.3 4.5 4.6 4.8 14 3.4 3.6 3.8 3.9 4.1 4.3 4.5 4.6 4.8 5.0 5.2 15 3.7 3.9 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 16 3.9 4.1 4.3 4.5 4.7 4.9 5.1 5.3 5.5 5.7 5.9 17 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.1 6.3 18 4.4 4.6 4.8 5.1 5.3 5.5 5.7 6.0 6.2 6.4 6.7 19 4.7 4.9 5.1 5.3 5.6 5.8 6.0 6.3 6.5 6.8 7.0 20 4.9 5.1 5.4 5.6 5.9 6.1 6.4 6.6 6.9 7.1 7.4 21 5.2 5.4 5.6 5.9 6.2 6.4 6.7 6.9 7.2 7.5 7.8 22 5.4 5.7 5.9 6.2 6.4 6.7 7.0 7.3 7.6 7.9 8.2 23 5.6 5.9 6.2 6.5 6.7 7.0 7.3 7.6 7.9 8.2 8.5 24 5.9 6.2 6.4 6.7 7.0 7.3 7.6 7.9 8.3 8.6 8.9 25 6.1 6.4 6.7 7.0 7.3 7.6 8.0 8.3 8.6 8.9 9.3 26 6.4 6.7 7.0 7.3 7.6 7.9 8.3 8.6 8.9 9.3 9.6 27 6.6 6.9 7.3 7.6 7.9 8.2 8.6 8.9 9.3 9.6 10.0 28 6.9 7.2 7.5 7.9 8.2 8.5 8.9 9.3 9.6 10.0 10.4 29 7.1 7.4 7.8 8.1 8.5 8.9 9.2 9.6 10.0 10.4 10.7 30 7.4 7.7 8.1 8.4 8.8 9.2 9.5 9.9 10.3 10.7 11.1 31 . 7.6 8.0 8.3 8.7 9.1 9.5 9.9 10.3 10.7 11.1 11.5 32 7.8 8.2 8.6 9.0 9.4 9.8 10.2 10.6 11.0 11.4 11.9 33 8.1 8.5 8.9 9.3 9.7 10.1 10.5 10.9 11.4 11.8 12.2 34 8.3 8.7 9.1 9.5 10.0 10.4 10.8 11.2 11.7 12.1 12.6 35 8.6 9.0 9.4 9.8 10.3 10.7 11.1 11.6 12.0 12.5 13.0 36 8.8 9.2 9.7 10.1 10.5 11.0 11.4 11.9 12.4 12.9 13.3 37 9.1 9.5 9.9 10.4 10.8 11.3 11.8 12.2 12.7 13.2 13.7 38 9 3 9.8 10.2 10.7 11.1 11.6 12.1 12.6 13.1 13.6 14 1 39 9 6 10 10.5 10.9 11.4 11.9 12.4 12.9 13.4 13.9 145 40 98 10.3 107 11.2 11.7 12.2 12.7 13.2 13.8 14.3 148 41 10.1 10.5 11.0 11.5 12.0 12.5 13.0 13.6 14.1 14.6 15.2 42 10.3 10.8 11.3 11.8 12.3 12.8 13.4 13.9 14.4 15.0 15.6 43 10.5 11.0 11.6 12.1 12.6 13.1 13.7 14.2 14.8 15.4 15.9 44 10.8 11.3 11.8 12.3 12.9 13.4 14.0 14.6 15.1 15.7 16.3 45 11.0 11.6 12.1 12.6 13.2 13.7 14.3 14.9 15.5 16.1 16.7 46 11.3 11.8 12.4 12.9 13.5 14.0 14.6 15.2 15.8 16.4 17.1 47 11.5 12.1 12.6 13.2 13.8 14.4 14.9 15.6 16.2 16.8 17.4 48 11.8 12.3 12.9 13.5 14.1 14.7 15.3 15.9 16.5 17.1 17.8 49 12.0 12.6 13.2 13.8 14.4 15.0 15.6 16.2 16.9 17.5 18.2 50 12.3 12.8 13.4 14.0 14.6 15.3 15.9 16.5 17.2 17.9 18.5 100 24.5 25.7 26.9 28.1 29.3 30.5 31.8 33.1 34.4 35.7 37.1 200 49.1 51.4 53.7 56.1 58.6 61.1 63.6 66.2 68.8 71.4 74.1 300 73.6 77.1 80.6 84.2 87.9 91.6 95.4 99.3 103..2 107.2 111.2 400 98.1 102.7 107.4 112.3 117.2 122.1 127.2 132.3 137.6 142.9 148.3 500 122.7 128.4 134.3 140.3 146.5 152.7 159.0 165.4 172.0 178.6 185.3 1.33 1.35 1.37 1.39 1.41 1.44 1.47 1.50 1.52 1.56 1.59 FACTOR To CHANGE DEP. INTO LONG. DIFF., MULTIPLY TABULAR NUMBER BY FACTOR AT FOOT OF COLUMN, AND ADD PRODUCT TO DEP. Table 2 175 To CHANGE LONG. DIFF. INTO DEP. SUBTRACT TABULAE NUMBER FROM LONG. DIFF. LONG. DIFF OR MIDDLE LATITUDE DEP. 41 42 43 44 45 46 47 48 49 50 51 51 12.5 13.1 13.7 14.3 14.9 15.6 16.2 16.9 17.5 18.2 18.9 52 12.8 13.4 14.0 14.6 15.2 15.9 16.5 17.2 17.9 18.6 19.3 53 13.0 13.6 14.2 14.9 15.5 16.2 16.9 17.5 18.2 18.9 19.6 54 13.2 13.9 14.5 15.2 15.8 16.5 17.2 17.9 18.6 19.3 20.0 55 13.5 14.1 14.8 15.4 16.1 16.8 17.5 18.2 18.9 19.6 20.4 56 13.7 14.4 15.0 15.7 16.4 17.1 17.8 18.5 19.3 20.0 20.8 57 14.0 14.6 15.3 16.0 16.7 17.4 18.1 18.9 19.6 20.4 21.1 58 14.2 14.9 15.6 16.3 17.0 17.7 18.4 19.2 19.9 20.7 21.5 59 14.5 15.2 15.9 16.6 17.3 18.0 18.8 19.5 20.3 21.1 21.9 60 14.7 15.4 16.1 16.8 17.6 18.3 19.1 19.9 20.6 21.4 22.2 61 15.0 15.7 16.4 17.1 17.9 18.6 19.4 20.2 21.0 21.8 22.6 62 15.2 15.9 16.7 17.4 18.2 18.9 19.7 20.5 21.3 22.1 23.0 63 15.5 16.2 16.9 17.7 18.5 19.2 20.0 20.8 21.7 22.5 23.4 64 15.7 16.4 17.2 18.0 18.7 19.5 20.4 21.2 22.0 22.9 23.7 65 15.9 16.7 17.5 18.2 19.0 19.8 20.7 21.5 22.4 23.2 24.1 66 16.2 17.0 17.7 18.5 19.3 20.2 21.0 21.8 22.7 23.6 24.5 67 16.4 17.2 18.0 18.8 19.6 20.5 21.3 22.2 23.0 23.9 24.8 68 16.7 17.5 18.3 19.1 19.9 20.8 21.6 22.5 23.4 24.3 25.2 69 16.9 17.7 18.5 19.4 20.2 21.1 21.9 22.8 23.7 24.6 25.6 70 17.2 18.0 18.8 19.6 20.5 21.4 22.3 23.2 24.1 25.0 25.9 71 17.4 18.2 19.1 19.9 20.8 21.7 22.6 23.5 24.4 25.4 26.3 72 17.7 18.5 19.3 20.2 21.1 22.0 22.9 23.8 24.8 25.7 26.7 73 17.9 18.8 19.6 20.5 21.4 22.3 23.2 24.2 25.1 26.1 27.1 74 18.2 19.0 19.9 20.8 21.7 22.6 23.5 24.5 25.5 26.4 27.4 75 18.4 19.3 20.1 21.0 22.0 22.9 23.9 24.8 25.8 26.8 27.8 76 18.6 19.5 20.4 21.3 22.3 23.2 24.2 25.1 26.1 27.1 28.2 77 18.9 19.8 20.7 21.6 22.6 23.5 24.5 25.5 26.5 27.5 28.5 78 19.1 20.0 21.0 21.9 22.8 23.8 24.8 25.8 26.8 27.9 28.9 79 19.4 20.3 21.2 22.2 23.1 24.1 25.1 26.1 27.2 28.2 29.3 80 19.6 20.5 21.5 22.5 23.4 24.4 25.4 26.5 27.5 28.6 29.7 81 19.9 20.8 21.8 22.7 23.7 24.7 25.8 26.8 27.9 28.9 30.0 82 20.1 21.1 22.0 23.0 24.0 25.0 26.1 27.1 28.2 29.3 30.4 83 20.4 21.3 22.3 23.3 24.3 25.3 26.4 27.5 28.5 29.6 30.8 84 20.6 21.6 22.6 23.6 24.6 25.6 26.7 27.8 28.9 30.0 31.1 85 20.8 21.8 22.8 23.9 24.9 26.0 27.0 28.1 29.2 30.4 31.5 86 21.1 22.1 23.1 24.1 25.2 26.3 27.3 28.5 29.6 30.7 31.9 87 21.3 22.3 23.4 24 .4 25.5 26.6 27.7 28.8 29.9 31.1 32.2 88 21.6 22.6 23.6 24.7 25.8 26.9 28.0 29.1 30.3 31.4 32.6 89 21.8 22.9 23.9 25.0 26.1 27.2 28.3 29.4 30.6 31.8 33.0 90 22.1 23.1 24.2 25.3 26.4 27.5 28.6 29.8 31.0 32.1 33.4 91 22.3 23.4 24.4 25.5 26.7 27.8 28.9 30.1 31.3 32.5 33.7 92 22.6 23.6 24.7 25.8 26.9 28.1 29.3 30.4 31.6 32.9 34.1 93 22.8 23.9 25.0 26.1 27.2 28.4 29.6 30.8 32.0 33.2 34.5 94 23.1 24.1 25.3 26.4 27.5 28.7 29.9 31.1 32.3 33.6 34.8 95 23.3 24.4 25.5 26.7 27.8 29.0 30.2 31.4 32.7 33.9 35.2 96 23.5 24.7 25.8 26.9 28.1 29.3 30.5 31.8 33.0 34.3 35.6 97 23.8 24.9 26.1 27.2 28.4 29.6 30.8 32.1 33.4 34.6 36.0 98 24.0 25.2 26.3 27.5 28.7 29.9 31.2 32.4 33.7 35.0 36.3 99 24.3 25.4 26.6 27.8 29.0 30.2 31.5 32.8 34.1 35.4 36.7 100 24.5 25.7 26.9 28.1 29.3 30.5 31.8 33.1 34.4 35.7 37.1 600 147.2 154.1 161.2 168.4 175.7 183.2 190.8 198.5 206.4 214.3 222.4 700 171.7 179.8 188.1 196.5 205.0 213.7 222.6 231.6 240.8 250.0 259.4 800 196.1 205.4 214.9 224.6 234.3 244.2 254.4 264.7 275.2 285.8 296.5 900 220.8 231.2 241.8 252.7 263.7 274.8 286.2 297.8 309.7 321.5 333.7 1.33 1.35 1.37 1.39 1.41 1.44 1.47 1.50 1.52 1.56 1.59 FACTOR To CHANGE DEP. INTO LONG. DIFF. MULTIPLY TABULAR NUMBER BY FACTOR AT FOOT OF COLUMN AND ADD PRODUCT TO DEP. 176 Table 2 To CHANGE LONG. DIFF. INTO DEP., SUBTRACT TABULAR NUMBER FROM LONG. DIFF. LONG. DIFP. OH MIDDLE LATITUDE DEP. 52 53 54 55 56 57 58 59 60 1 0.4 0.4 0.4 0.4 0.4 0.5 0.5 0.5 0.5 2 0.8 0.8 0.8 0.9 0.9 0.9 0.9 1.0 1.0 3 1.2 1.2 1.2 1.3 1.3 1.4 1.4 1.5 1.5 4 1.5 1.6 1.6 1.7 1.8 1.8 1.9 1.9 2.0 5 1.9 2.0 2.1 2.1 2.2 2.3 2.4 2.4 2.5 6 2.3 2.4 2.5 2.6 2.6 2.7 2.8 2.9 3.0 7 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 8 3.1 3.2 3.3 3.4 3.5 3.6 3.8 3.9 4.0 9 3.5 3.6 3.7 3.8 4.0 4.1 4.2 4.4 4.5 10 3.8 4.0 4.1 4.3 4.4 4.6 4.7 4.8 5.0 11 4.2 4.4 4.5 4.7 4.8 5.0 5.2 5.3 5.5 12 4.6 4.8 4.9 5.1 5.3 5.5 5.6 5.8 6.0 13 5.0 5.2 5.4 5.5 5.7 5.9 6.1 6.3 6.5 14 5.4 5.6 5.8 6.0 6.2 6.4 6.6 6.8 7.0 15 5.8 6.0 6.2 6.4 6.6 6.8 7.1 7.3 7.5 16 6.1 6.4 6.6 6.8 7.1 7.3 7.5 7.8 8.0 17 6.5 6.8 7.0 7.2 7.5 7.7 8.0 8.2 8.5 18 6.9 7.2 7.4 7.7 7.9 8.2 8.5 8.7 9.0 19 7.3 7.6 7.8 8.1 8.4 8.7 8.9 9.2 9.5 20 7.7 8.0 8.2 8.5 8.8 9.1 9.4 9.7 10.0 21 8.1 8.4 8.7 9.0 9.3 9.6 9.9 10.2 10.5 22 8.5 8.8 9.1 9.4 9.7 10.0 10.3 10.7 11.0 23 8.8 9.2 9.5 9.8 10.1 10.5 10.8 11.2 11.5 24 9.2 9.6 9.9 10.2 10.6 10.9 11.3 11.6 12.0 25 9.6 10.0 10.3 10.7 11.0 11.4 11.8 12.1 12.5 26 10.0 10.4 10.7 11.1 11.5 11.8 12.2 12.6 13.0 27 10.4 10.8 11.1 11.5 11.9 12.3 12.7 13.1 13.5 28 10.8 11.1 11.5 11.9 12.3 12.8 13.2 13.6 14.0 29 11.1 11.5 12.0 12.4 12.8 13.2 13.6 14.1 14.5 30 11.5 11.9 12.4 12.8 13.2 13.7 14.1 14.5 15.0 31 11.9 12.3 12.8 13.2 13.7 14.1 14.6 15.0 15.5 32 12.3 12.7 13.2 13.6 14.1 14.6 15.0 15.5 16.0 33 12.7 13.1 13.6 14.1 14.5 15.0 15.5 16.0 16.5 34 13.1 13.5 14.0 14.5 15.0 15.5 16.0 16.5 17.0 35 13.5 13.9 14.4 14.9 15.4 15.9 16.5 17.0 17.5 36 13.8 14.3 14.8 15.4 15.9 16.4 16.9 17.5 18.0 37 14.2 14.7 15.3 15.8 16.3 16.8 17.4 17.9 18.5 38 14.6 15.1 15.7 16.2 16.8 17.3 17.9 18.4 19.0 39 15.0 15.5 16.1 16.6 17.2 17.8 18.3 18.9 19.5 40 15.4 15.9 16.5 17.1 17.6 18.2 18.8 19.4 20.0 41 15.8 16.3 16.9 17.5 18.1 18.7 19.3 19.9 20.5 42 16.1 16.7 17.3 17.9 18.5 19.1 19.7 20.4 21.0 43 16.5 17.1 17.7 18.3 19.0 19.6 20.2 20.9 21.5 44 16.9 17.5 18.1 18.8 19.4 20.0 20.7 21.3 22.0 45 17.3 17.9 18.5 19.2 19.8 20.5 21.2 21.8 22.5 46 17.7 18.3 19.0 19.6 20.3 20.9 21.6 22.3 23.0 47 18.1 18.7 19.4 20.0 20.7 21.4 22.1 22.8 23.5 48 18.4 19.1 19.8 20.5 21.2 21.9 22.6 23.3 24.0 49 18.8 19.5 20.2 20.9 21.6 22.3 23.0 23.8 24.5 50 19.2 19.9 20.6 21.3 22.0 22.8 23.5 24.2 25.0 100 38.4 39.8 41.2 42.6 44.1 45.5 47.0 48.5 50.0 200 76.9 79.6 82.4 85.3 88.2 91.1 94.0 97.0 100.0 300 115.3 119.5 123.7 127.9 132.2 136.6 141.0 145.5 150.0 400 153.7 159.3 164.9 170.6 176.3 182.2 188.1 194.0 200.0 500 192.2 199.1 206.1 213.2 220.4 227.7 235.0 242.5 250.0 1.62 1.66 1.70 1.74 1.79 1.84 1.89 1.94 2.00 FACTOR To CHANGE DEP. INTO LONG. DIFF., MULTIPLY TABULAR NUMBER BY FACTOR AT FOOT OF COLUMN AND ADD PRODUCT TO DEP. Table 2 177 To CHANGE LONG. DIFF. INTO DEP. SUBTRACT TABULAR NUMBER FROM LONG. DIFF. LONG. DIFF. MIDDLE LATITUDE OR DEP. 52 53 54 55 56 57 58 59 60 51 19.6 20.3 21.0 21.7 22.5 23.2 24.0 24.7 25.5 52 20.0 20.7 21.4 22.2 22.9 23.7 24.4 25.2 26.0 53 20.4 21.1 21.8 22.6 23.4 24.1 24.9 25.7 26.5 54 20.8 21.5 22.3 23.0 23.8 24.6 25.4 26.2 27.0 55 21.1 21.9 22.7 23.5 24.2 25.0 25.9 26.7 27.5 56 21.5 22.3 23.1 23.9 24.7 25.5 26.3 27.2 28.0 57 21.9 22.7 23.5 24.3 25.1 26.0 26.8 27.6 28.5 58 22.3 23.1 23.9 24.7 25.6 26.4 27.3 28.1 29.0 59 22.7 23.5 24.3 25.2 26.0 26.9 27.7 28.6 29.5 60 23.1 23.9 24.7 25.6 26.4 27.3 28.2 29.1 30.0 61 23.4 24.3 25.1 26.0 26.9 27.8 28.7 29.6 30.5 62 23.8 24.7 25.6 26.4 27.3 28.2 29.1 30.1 31.0 63 24.2 25.1 26.0 26.9 27.8 28.7 29.6 30.6 31.5 64 24.6 25.5 26.4 27.3 28.2 29.1 30.1 31.0 32.0 65 25.0 25.9 26.8 27.7 28.7 29.6 30.6 31.5 32.5 66 25.4 26.3 27.2 28.1 29.1 30.1 31.0 32.0 33.0 67 25.8 26.7 27.6 28.6 29.5 30.5 31.5 32.5 33.5 68 26.1 27.1 28.0 29.0 30.0 31.0 32.0 33.0 34.0 69 26.5 27.5 28.4 29.4 30.4 31.4 32.4 33.5 34.5 70 26.9 27.9 28.9 29.8 30.9 31.9 32.9 33.9 35.0 71 27.3 28.3 29.3 30.3 31.3 32.3 33.4 34.4 35.5 72 27.7 28.7 29.7 30.7 31.7 32.8 33.8 34.9 36.0 73 28.1 29.1 30.1 31.1 32.2 33.2 34.3 35.4 36.5 74 28.4 29.5 30.5 31.6 32.6 33.7 34.8 35.9 37.0 75 28.8 29.9 30.9 32.0 33.1 34.2 35.3 36.4 37.5 76 29.2 30.3 31.3 32.4 33.5 34.6 35.7 36.9 38.0 77 29.6 30.7 31.7 32.8 33.9 35.1 36.2 37.3 38.5 78 30.0 31.1 32.2 33.3 34.4 35.5 36.7 37.8 39.0 79 30.4 31.5 32.6 33.7 34.8 36.0 37.1 38.3 39.5 80 30.7 31.9 33.0 34.1 35.3 36.4 37.6 38.8 40.0 81 31.1 32.3 33.4 34.5 35.7 36.9 38.1 39.3 40.5 82 31.5 32.7 33.8 35.0 36.1 37.3 38.5 39.8 41.0 83 31.9 33.0 34.2 35.4 36.6 37.8 39.0 40.3 41.5 84 32.3 33.4 34.6 35.8 37.0 38.3 39.5 40.7 42.0 85 32.7 33.8 35.0 36.2 37.5 38.7 40.0 41.2 42.5 86 33.1 34.2 35.5 36.7 37.9 39.2 40.4 41.7 43.0 87 33.4 34.6 35.9 37.1 38.4 39.6 40.9 42.2 43.5 88 33.8 35.0 36.3 37.5 38.8 40.1 41.4 42.7 44.0 89 34.2 35.4 36.7 38.0 39.2 40.5 41.8 43.2 44.5 90 34.6 35.8 37.1 38.4 39.7 41.0 42.3 43.6 45.0 91 35.0 36.2 37.5 38.8 40.1 41.4 42.8 44.1 45.5 92 35.4 36.6 37.9 39.2 40.6 41.9 43.2 44.6 46.0 93 35.7 37.0 38.3 39.7 41.0 42.3 43.7 45.1 46.5 94 36.1 37.4 38.7 40.1 41.4 42.8 44.2 45.6 47.0 95 36.5 37.8 39.2 40.5 41.9 43.3 44.7 46.1 47.5 96 36.9 38.2 39.6 40.9 42.3 43.7 45.1 46.6 48.0 97 37.3 38.6 40.0 41.4 42.8 44.2 45.6 47.0 48.5 98 37.7 39.0 40.4 41.8 43.2 44.6 46.1 47.5 49.0 99 38.0 39.4 40.8 42.2 43.6 45.1 46.5 48.0 49.5 100 38.4 39.8 41.2 42.6 44.1 45.5 47.0 48.5 50.0 600 230.6 238.9 247.3 255.9 264.5 273.2 282.0 291.0 300.0 700 269.2 279.7 288.6 298.5 308.6 318.7 329.0 339.6 350.0 800 307.5 319.5 329.8 341.2 352.6 364.3 376.1 388.0 400.0 900 346.0 358.3 371.1 383.8 396.8 409.9 423.2 436.6 450.0 1.63 1.66 1.70 1.74 1.79 1.84 1.89 1.94 2.00 FACTOR To CHANGE DEP. INTO LONG. DIFF. MULTIPLY TABULAR NUMBER BY FACTOR AT FOOT OF COLUMN AND ADD PRODUCT TO DEP. 178 Table 3. Number Logarithms 1 2 3 4 5 6 7 8 9 Prop, Pts, 100 00000 043 087 130 173 217 260 303 346 389 01 432 475 518 561 604 647 689 732 775 817 44 43 42 02 860 903 945 988 *030 *072 *115 *157 *199 *242 1 4.4 4.3 4.2 03 01284 326 368 410 452 494 536 578 620 662 2 8.8 8.6 8.4 04 703 745 787 828 870 912 953 995 *036 *078 3 13.2 12.9 12.6 05 02119 160 202 243 284 325 366 407 449 490 4 17.6 17.2 16.8 06 531 572 612 653 694 735 776 816 857 898 5 22.0 21.5 21.0 6 26.4 25.8 25.2 07 938 979 *019 *060 *100 *141 *181 *222 *262 *302 7 30.8 30.1 29.4 08 03342 383 423 463 503 543 583 623 663 703 8 35.2 34.4 33.6 09 743 782 822 862 902 941 981 *021 *060 *100 9 39.6 38.7 37.8 110 04139 179 218 258 297 336 376 415 454 493 11 532 571 610 650 689 727 766 805 844 883 41 40 39 12 922 961 999 *038 *077 *115 *154 *192 *231 *269 1 4.1 4.0 3.9 13 05308 346 385 423 461 500 538 576 614 652 2 8.2 8.0 7.8 14 15 16 690 06070 446 729 108 483 767 145 521 805 183 558 843 221 595 881 258 633 918 296 670 956 333 707 994 371 744 *032 408 781 3 4 5 6 12.3 16.4 20.5 24.6 12.0 16.0 20.0 24.0 11.7 15.6 19.5 23.4 17 819 856 893 930 967 *004 *041 *078 *115 *151 7 28.7 28.0 27.3 18 07188 225 262 298 335 372 408 445 482 518 8 32.8 32.0 31.2 19 555 591 628 664 700 737 773 809 846 882 9 36.9 36.0 35.1 120 918 954 990 *027 *063 *099 *135 *171 *207 *243 21 08279 314 350 386 422 458 493 529 565 600 38 37 36 22 636 672 707 743 778 814 849 884 920 955 1 3.8 3.7 3.6 23 991 *026 *061 *096 *132 *167 *202 *237 *272 *307 2 7.6 7.4 7.2 24 09342 377 412 447 482 517 552 587 621 656 3 11.4 11.1 Mo 10.8 25 26 691 10037 726 072 760 106 795 140 830 175 864 209 899 243 934 278 968 312 *003 346 4 5 6 15.2 19.0 22.8 .8 18.5 22.2 14.4 18.0 21.6 27 380 415 449 483 517 551 585 619 653 687 7 26.6 25.9 25.2 28 721 755 789 823 857 890 924 958 992 *025 8 30.4 29.6 28.8 29 11059 093 126 160 193 227 261 294 327 361 9 34.2 33.3 32.4 130 394 428 461 494 528 561 594 628 661 694 31 727 760 793 826 860 893 926 959 992 *024 35 34 33 32 12057 090 123 156 189 222 254 287 320 352 1 3.5 3.4 3.3 33 385 418 450 483 516 548 581 613 646 678 2 7.0 6.8 6.6 34 35 36 710 13033 354 743 066 386 775 098 418 808 130 450 840 162 481 872 194 513 905 226 545 937 258 577 969 290 609 *001 322 640 3 4 5 6 10.5 14.0 17.5 21.0 10.2 13.6 17.0 20.4 9.9 13.2 16.5 19.8 37 672 704 735 767 799 830 862 893 925 956 7 24.5 23.8 23.1 38 988 *019 *051 *082 *114 *145 *176 *208 *239 *270 8 28.0 27.2 26.4 39 14301 333 364 395 426 457 489 520 551 582 9 31.5 30.6 29.7 140 613 644 675 706 737 768 799 829 860 891 41 922 953 983 *014 *045 *076 *106 *137 *168 *198 32 31 30 42 15229 259 290 320 351 381 412 442 473 503 1 3.2 3.1 3.0 43 534 564 594 625 655 685 715 746 776 806 2 6.4 6.2 6.0 44 836 866 897 927 957 987 *017 *047 *077 *107 3 19 K 9.3 12.4 9.0 45 46 16 137 435 167 465 197 495 227 524 256 554 286 584 316 613 346 643 376 673 406 702 5 6 &40O 16.0 19.2 15^5 18.6 15!o 18.0 47 732 761 791 820 850 879 909 938 967 997 7 22.4 21.7 21.0 48 17026 056 085 114 143 173 202 231 260 289 8 25.6 24.8 24.0 49 319 348 377 406 435 464 493 522 551 580 9 28.8 27.9 27.0 150 609 638 667 696 725 754 782 811 840 869 1 2 3 4 5 6 7 8 9 Prop, Pts, Table 3. Number Logarithms 179 1 2 3 4 5 6 7 8 9 Prop, Pts, 150 17609 638 667 696 725 754 782 811 840 869 51 898 926 955 984 *013 *041 *070 *099 *127 *156 52 18 184 213 241 270 298 327 355 384 412 441 53 469 498 526 554 583 611 639 667 696 724 54 752 780 808 837 865 893 921 949 977 *005 55 19033 061 089 117 145 173 201 229 257 285 56 312 340 368 396 424 451 479 507 535 562 57 590 618 645 673 700 728 756 783 811 838 58 866 893 921 948 976 *003 *030 *058 085 *112 59 20140 167 194 222 249 276 303 330 358 385 160 412 439 466 493 520 548 575 602 629 656 61 683 710 737 763 790 817 844 871 898 925 29 28 27 62 952 978 *005 *032 *059 *085 *112 *139 *165 *192 1 2.9 2.8 2.7 63 21219 245 272 299 325 352 378 405 431 458 2 5.8 5.6 5.4 64 65 66 484 748 22011 511 775 037 537 801 063 564 827 089 590 854 115 617 880 141 643 906 167 669 932 194 696 958 220 722 985 246 3 4 5 6 8.7 11.6 14.5 17.4 8.4 11.2 14.0 16.8 8.1 10.8 13.5 16.2 67 272 298 324 350 376 401 427 453 479 505 7 20.3 19.6 18.9 68 531 557 583 608 634 660 686 712 737 763 8 23.2 22.4 21.6 69 789 814 840 866 891 917 943 968 994 *019 9 26.1 25.2 24.3 170 23045 070 096 121 147 172 198 223 249 274 71 300 325 350 376 401 426 452 477 502 528 26 25 24 72 553 578 603 629 654 679 704 729 754 779 1 2.6 2.5 2.4 73 805 830 855 880 905 930 955 980 *005 *030 2 5.2 5.0 4.8 74 24055 080 105 130 155 180 204 229 254 279 3 7.8 7.5 7.2 75 304 329 353 378 403 428 452 477 502 527 4 10.4 10.0 9.6 76 551 576 601 625 650 674 699 724 748 773 5 13.0 12.5 12.0 (5 15.6 15.0 14.4 77 797 822 846 871 895 920 944 969 993 *018 7 18.2 17.5 16.8 78 25042 066 091 115 139 164 188 212 237 261 8 20.8 20.0 19.2 79 285 310 334 358 382 406 431 455 479 503 9 23.4 22.5 21.6 180 527 551 575 600 624 648 672 696 720 744 81 768 792 816 840 864 888 912 935 959 983 23 22 21 82 26007 031 055 079 102 126 150 174 198 221 1 Q O 1 83 245 269 293 316 340 364 387 411 435 458 2 9*0 4.6 &* 4.4 4 4.2 84 482 505 529 553 576 600 623 647 670 694 3 6.9 6.6 6.3 85 717 741 764 788 811 834 858 881 905 928 4 9.2 8.8 8.4 86 951 975 *021 *045 *068 *091 *114 *138 *161 5 11.5 11.0 10.5 6 13.8 13.2 12.6 87 27184 207 231 254 277 300 323 346 370 393 7 16.1 15.4 14.7 88 416 439 462 485 508 531 554 577 600 623 8 18.4 17.6 16.8 89 646 669 692 715 738 761 784 807 8:30 852 9 20.7 19.8 18.9 190 875 898 921 944 967 989 *012 *035 *058 *081 91 28103 126 149 171 194 217 240 262 285 307 92 330 353 375 398 421 443 466 488 511 533 93 556 578 601 623 646 668 691 713 735 758 94 780 803 825 847 870 892 914 937 959 981 95 29003 026 048 070 092 115 137 159 181 203 96 226 248 270 292 314 336 358 380 403 425 97 447 469 491 513 535 557 579 601 623 645 98 667 688 710 732 754 776 798 820 842 863 99 885 907 929 951 973 994 *016 *038 *060 *081 200 30103 125 146 168 190 211 233 255 276 298 1 2 3 4 5 6 7 8 9 Prop, Pts, 180 Table 3. Number Logarithms 1 2 3 4 5 6 7 8 9 Prop, Pts, 200 30103 125 146 168 190 211 233 255 276 298 01 320 341 363 384 406 428 449 471 492 514 02 535 557 578 600 621 643 664 685 707 728 03 750 771 792 814 835 856 878 899 920 942 04 963 984 *006 *027 *048 *069 *091 *112 *133 *154 05 31175 197 218 239 260 281 302 323 345 366 06 387 408 429 450 471 492 513 534 555 576 07 597 618 639 660 681 702 723 744 765 785 08 806 827 848 869 890 911 931 952 973 994 09 32015 035 056 077 098 118 139 160 181 201 210 222 243 263 284 305 325 346 366 387 408 Aft ft 1 ft/\ 11 428 449 469 490 510 531 552 572 593 613 22 21 20 12 634 654 675 695 715 736 756 777 797 818 1 2.2 2.1 2.0 13 838 858 879 899 919 940 960 980 *001 *021 2 4.4 4.2 4.0 3 6.6 6.3 6.0 14 33041 062 082 102 122 143 163 183 203 224 4 8.8 8.4 8.0 15 244 264 284 304 325 345 365 385 405 425 5 11.0 10.5 10.0 16 445 465 486 506 526 546 566 586 606 626 6 13.2 12.6 12.0 17 18 19 646 846 34044 666 866 064 686 885 084 706 905 104 726 925 124 746 945 143 766 965 163 786 985 183 806 *005 203 826 *025 223 7 8 9 15.4 17.6 19.8 14.7 16.8 18.9 14.0 16.0 18.0 220 242 262 282 301 321 341 361 380 400 420 21 439 459 479 498 518 537 557 577 596 616 22 635 655 674 694 713 733 753 772 792 811 23 830 850 869 889 908 928 947 967 986 *005 24 35025 044 064 083 102 122 141 160 180 199 25 218 238 257 276 295 315 334 353 372 392 26 411 430 449 468 488 507 526 545 564 583 27 603 622 641 660 679 698 717 736 755 774 28 793 813 832 851 870 889 908 927 946 965 29 984 *003 *021 *040 *059 *078 *097 *116 *135 *154 230 36173 192 211 229 248 267 286 305 324 342 31 361 380 399 418 436 455 474 493 511 530 19 18 17 32 549 568 586 605 624 642 661 680 698 717 1 1.9 1.8 1.7 33 736 754 773 791 810 829 847 866 884 903 2 3.8 3.6 3.4 3 5.7 5.4 5.1 34 922 940 959 977 996 *014 *033 *051 *070 *088 4 y.g 7.2 6.8 35 37107 125 144 162 181 199 218 2:36 254 273 5 95 9.0 8*5 36 291 310 328 346 365 383 401 420 438 457 6 11.4 10.8 10.2 37 475 493 511 530 548 566 585 603 621 639 7 13.3 12.6 11.9 38 658 676 694 712 731 749 767 785 803 822 8 15.2 14.4 13.6 39 840 858 876 894 912 931 949 967 985 *003 9 17.1 16.2 15.3 240 38021 039 057 075 093 112 130 148 166 184 41 202 220 238 256 274 292 310 328 346 364 42 382 399 417 435 453 471 489 507 525 543 43 561 578 596 614 632 650 668 686 703 721 44 739 757 775 792 810 828 846 863 881 899 45 917 934 952 970 987 *005 *023 *041 *058 *076 46 39094 111 129 146 164 182 199 217 235 252 47 270 287 305 322 340 358 375 393 410 428 48 445 463 480 498 515 533 550 568 585 602 49 620 637 655 672 690 707 724 742 759 777 250 794 811 829 846 863 881 898 915 933 950 1 2 3 4 5 6 7 8 9 Prop. Pts, Table 3. Number Logarithms 181 1 2 3 4 5 6 7 8 9 Prop, Pts, 250 39794 811 829 846 863 881 898 915 933 950 51 967 985 *002 *019 *037 *054 *071 *088 *106 *123 52 40140 157 175 192 209 226 243 261 278 295 53 312 329 346 364 381 398 415 432 449 466 54 483 500 518 535 552 569 586 603 620 637 55 654 671 688 705 722 739 756 773 790 807 56 824 841 858 875 892 909 926 943 960 976 57 993 *010 *027 #044 *061 *078 *095 *111 *128 *145 58 41162 179 196 212 229 246 263 280 296 313 59 330 347 3(33 380 397 414 430 447 464 481 260 497 514 531 547 564 581 597 614 631 647 61 664 681 697 714 731 747 764 780 797 814 18 17 16 62 830 847 863 880 896 913 929 946 963 979 1 1.8 I/ r 1.6 63 996 *012 *029 *045 *062 *078 *095 *111 *127 *144 2 3.6 3.^ t 3.2 64 65 66 42160 325 488 177 341 504 193 357 521 210 374 537 226 390 553 243 406 570 259 423 586 275 439 602 292 455 619 308 472 635 3 5.4 5J 4 7.2 6.* 5 9.0 8.^ (3- 10.8 10.5 L 4.8 J 6.4 > 8.0 J 9.6 67 651 667 684 700 716 732 749 765 781 797 7 12.6 ll.< ) 11.2 68 813 830 846 8(32 878 894 911 927 943 959 8 14.4 13.( 3 12.8 69 975 991 *008 *024 *040 *056 *072 *088 *104 *120 9 16.2 15.: J 14.4 270 43136 152 169 185 201 217 233 249 265 281 71 297 313 329 345 361 377 393 409 425 441 72 457 473 489 505 521 537 553 569 584 600 73 616 632 648 664 680 696 712 727 743 759 74 775 791 807 823 838 854 870 886 902 917 75 933 949 965 981 996 *012 *028 *044 *059 *075 76 44091 107 122 138 154 170 185 201 217 232 77 248 264 279 295 311 326 342 358 373 389 78 404 420 436 451 467 483 498 514 529 545 79 560 576 592 607 623 638 654 669 685 700 280 716 731 747 762 778 793 809 824 840 855 81 871 886 902 917 932 948 963 979 994 *010 15 14 82 45025 040 056 071 086 102 117 133 148 163 11 PC 1 -I 83 179 194 209 225 240 255 271 286 301 317 1.0 2 3.0 2.S 84 332 347 362 378 393 408 423 439 454 469 3 4.5 4.2 85 484 500 515 530 545 561 576 591 606 621 4 6.0 5.6 86 637 652 667 682 697 712 728 743 758 773 5 7.5 7.0 6 9.0 8.4 87 788 803 818 834 849 864 879 894 909 924 7 10.5 9.8 88 939 954 969 984 *000 *015 *030 *045 *060 *075 8 12.0 1 1.2 89 46090 105 120 135 150 165 180 195 210 225 9 13.5 ] 2.6 290 240 255 270 285 300 315 330 345 359 374 91 389 404 419 434 449 464 479 494 509 523 92 538 553 568 583 598 613 627 642 657 672 93 687 702 716 731 746 761 776 790 805 820 94 835 850 864 879 894 909 923 938 953 967 95 982 997 *012 *026 *041 *056 *070 *085 *100 #114 96 47129 144 159 173 188 202 217 232 246 261 97 276 290 305 319 334 349 363 378 392 407 98 422 436 451 465 480 494 509 524 538 553 99 567 582 596 611 625 640 654 669 683 698 300 712 727 741 756 770 784 799 813 828 842 1 2 3 4 5 6 7 8 9 Prop, Pts, 182 Table 3. Number Logarithms 1 2 3 4 5 6 7 8 9 Prop, Pts, 300 47712 727 741 756 770 784 799 813 828 842 01 857 871 885 900 914 929 943 958 972 986 02 48001 015 029 044 058 073 087 101 116 130 03 144 159 173 187 202 216 230 244 259 273 04 287 302 316 330 344 359 373 387 401 416 05 430 444 458 473 487 501 515 530 544 558 06 572 586 601 615 629 643 657 671 686 700 07 714 728 742 756 770 785 799 813 827 841 08 855 869 883 897 911 926 940 954 968 982 09 996 *010 *024 *038 *052 066 *080 *094 *108 *122 310 49136 150 164 178 192 206 220 234 248 262 11 276 290 304 318 332 346 360 374 388 402 15 14 12 415 429 443 457 471 485 499 513 527 541 1 1.5 1.4 13 554 568 582 596 610 624 638 651 665 679 2 3.0 2.8 3 4.5 4.2 14 693 707 721 734 748 762 776 790 803 817 4 6.0 5.6 15 831 845 859 872 886 900 914 927 941 955 5 7.5 7.0 16 969 982 996 *010 *024 *037 *051 *065 *079 *092 6 9X) 8^4 17 50106 120 133 147 161 174 188 202 215 229 7 10.5 9.8 18 243 256 270 284 297 311 325 338 352 365 8 12.0 11.2 19 379 393 406 420 433 447 461 474 488 501 9 13.5 12.6 320 515 529 542 556 569 583 596 610 623 637 21 651 664 678 691 705 718 732 745 ITKQ 772 22 786 799 813 826 840 853 866 880 893 907 23 920 934 947 961 974 987 *001 *014 *028 *041 24 51 055 068 081 095 108 121 135 148 162 175 25 188 202 215 228 242 255 268 282 295 308 26 322 335 348 362 375 388 402 415 428 441 27 455 468 481 495 508 521 534 548 561 574 28 587 601 614 627 640 654 667 680 693 706 29 720 733 746 759 772 786 799 812 825 838 330 851 865 878 891 904 917 930 943 957 970 31 983 996 *009 *022 *035 *048 *061 *075 *088 *101 13 12 32 52114 127 140 153 166 179 192 205 218 231 1 1.3 1.2 33 244 257 270 284 297 310 323 336 349 362 2 2.6 2.4 34 375 388 401 414 427 440 453 466 479 492 3 3.9 4 5.2 3.6 4.8 35 504 517 530 543 556 569 582 595 608 621 5 6.5 fi 36 634 647 660 673 686 699 711 724 737 750 6 7^8 \j,\j 7.2 37 763 776 789 802 815 827 840 853 866 879 7 9.1 8.4 38 892 905 917 930 943 956 969 982 994 *007 8 10.4 9.6 39 53020 033 046 058 071 084 097 110 122 135 9 11.7 10.8 340 148 161 173 186 199 212 224 237 250 263 41 275 288 301 314 326 339 352 364 377 390 42 403 415 428 441 453 466 479 491 504 517 43 529 542 555 567 580 593 605 618 631 643 44 656 668 681 694 706 719 732 744 757 769 45 782 794 807 820 832 845 857 870 882 895 46 908 920 933 945 958 970 983 995 *008 *020 47 54033 045 058 070 083 095 108 120 133 145 48 158 170 183 195 208 220 233 245 258 270 49 283 295 307 320 332 345 357 370 382 394 . 350 407 419 432 444 456 469 481 494 506 518 1 2 3 4 5 6 7 8 9 Prop, Pts, Table 3. Number Logarithms 183 1 2 3 4 5 6 7 8 9 Prop, Pts, 350 54407 419 432 444 456 469 481 494 506 518 51 531 543 555 568 580 593 605 617 630 642 52 (i54 667 679 691 704 716 728 741 753 765 53 777 790 802 814 827 839 851 864 876 888 54 900 913 925 937 949 962 974 986 998 *011 55 55023 035 047 060 072 084 096 108 121 133 56 145 157 169 182 194 206 218 230 242 255 57 267 279 291 303 315 328 340 352 364 376 58 388 400 413 425 437 449 461 473 485 497 59 509 522 534 546 558 570 582 594 606 618 360 630 642 654 666 678 691 703 715 727 739 61 751 763 775 787 799 811 823 835 847 859 13 12 62 871 883 895 907 919 931 943 955 967 979 1 1.3 1.2 63 991 *003 *015 *027 *038 *050 *062 *074 *086 *098 2 2.6 2.4 64 56110 122 134 146 158 170 182 194 205 217 3 3.9 3.6 65 229 241 253 265 277 289 301 312 324 336 4 5.2 4.8 66 348 360 372 384 396 407 419 431 443 455 5 6 6.5 7.8 6.0 7.2 67 467 478 490 502 514 526 538 549 561 573 7 9.1 8.4 68 585 597 608 620 632 644 656 667 679 691 8 10.4 9.6 69 703 714 726 738 750 761 773 785 797 808 9 11.7 10.8 370 820 832 844 855 867 879 891 902 914 926 71 937 949 961 972 984 996 *008 *019 *031 *043 72 57054 066 078 089 101 113 124 136 148 159 73 171 183 194 206 217 229 241 252 264 276 74 287 299 310 322 334 345 357 368 380 392 75 403 415 426 438 449 461 473 484 496 507 76 519 530 542 553 565 576 588 600 611 623 77 634 646 657 669 680 692 703 715 726 738 78 749 761 772 784 795 807 818 830 841 852 79 864 875 887 898 910 921 933 944 955 967 380 978 990 *001 *013 *024 *035 *047 *058 *070 *081 81 58092 104 115 127 138 149 161 172 184 195 11 10 82 206 213 229 240 252 263 274 286 297 309 1 I "I 1 83 320 331 343 354 365 377 388 399 410 422 m 2 J..X 2.2 J.-VJ 2.0 84 433 444 456 467 478 490 501 512 524 535 3 3.3 3.0 85 546 557 569 580 591 602 614 625 636 647 4 4.4 4.0 86 659 670 681 692 704 715 726 737 749 760 5 5.5 5.0 6 6.6 6.0 87 771 782 794 805 816 827 838 850 861 872 7 7.7 7.0 88 883 894 906 917 928 939 950 961 973 984 8 8.8 8.0 89 995 *006 *017 *028 *040 *051 *062 *073 *084 *095 9 9.9 9.0 390 59106 118 129 140 151 162 173 184 195 207 91 218 229 240 251 262 273 284 295 306 318 92 329 340 351 362 373 384 395 406 417 428 93 439 450 461 472 483 494 506 517 528 539 94 550 561 572 583 594 605 616 627 638 649 95 660 671 682 693 704 715 726 737 748 759 96 770 780 791 802 813 824 835 846 857 868 97 879 890 901 912 923 934 945 956 966 977 98 988 999 *010 *021 *032 *04.". *054 *065 *076 *086 99 60097 108 119 130 141 152 163 173 184 195 400 206 217 228 239 249 260 271 282 293 304 1 2 3 4 5 6 7 8 9 Prop, Pts, 184 Table 3. Number Logarithms 1 2 3 4 5 6 7 8 9 Prop. Pts. 400 60206 217 228 23!) 249 260 271 282 293 304 01 314 325 336 347 358 369 379 390 401 412 02 423 433 444 455 4(5(5 477 487 498 509 520 03 531 541 552 563 574 584 595 606 617 627 04 638 649 660 670 681 692 703 713 724 735 05 746 756 767 778 788 799 810 821 831 842 06 853 863 874 885 895 906 917 927 938 949 07 959 970 981 991 *002 *013 *023 *034 *045 *055 06 61 066 077 087 0!)8 109 119 130 140 151 162 09 172 183 194 204 215 225 236 247 257 268 410 278 289 300 310 321 331 342 352 363 374 11 384 395 405 416 426 437 448 458 4(59 479 12 490 500 511 521 532 542 553 563 574 584 13 595 606 616 627 637 648 658 669 679 690 14 700 711 721 731 742 752 '763 773 784 794 15 805 815 826 836 847 857 868 878 888 899 16 909 920 930 941 951 962 972 982 993 *003 17 62014 024 034 045 055 066 076 086 097 107 18 118 128 138 149 159 170 180 190 201 211 19 221 232 242 252 263 273 284 294 304 315 420 325 335 346 356 366 377 387 397 408 418 21 428 439 449 459 469 480 490 500 511 521 11 10 9 22 531 542 552 562 572 583 593 603 613 624 1 1.1 1.0 0.9 23 634 644 655 (565 675 685 696 706 716 726 2 2.2 2.0 l!8 24 737 747 757 767 778 788 798 808 818 829 3 3.3 3.0 2.7 25 839 849 859 870 880 890 900 910 921 931 4 4.4 4.0 3.6 26 941 951 961 972 982 992 *002 *012 *022 *033 5 5.5 5.0 4.5 6 6.6 6.0 5.4 27 63043 053 063 073 083 094 104 114 124 134 7 7.7 7.0 6.3 28 144 155 1(55 175 185 195 205 215 225 23(5 8 8.8 8.0 7.2 29 246 256 266 276 286 296 306 317 327 337 9 9.9 9.0 8.1 430 347 357 367 377 387 397 407 417 428 438 31 448 458 468 478 488 498 508 518 528 538 32 548 558 568 579 589 599 609 619 629 639 33 649 659 669 679 689 699 709 719 729 739 34 749 759 769 779 789 799 809 819 829 839 35 849 859 869 879 889 899 909 919 929 939 36 949 959 969 979 988 998 *008 *018 *028 *038 37 64048 058 068 078 088 098 108 118 128 137 38 147 157 167 177 187 197 207 217 227 237 39 246 256 266 276 286 296 306 316 326 335 440 345 355 365 375 385 395 404 414 424 434 41 444 454 464 473 483 493 503 513 523 532 42 542 552 562 572 582 591 601 611 621 631 43 640 650 660 670 680 689 699 709 719 729 44 738 748 758 768 777 787 797 807 816 826 45 836 846 856 866 875 885 895 904 914 924 46 933 943 953 963 972 982 992 *002 *011 *021 47 65031 040 050 060 070 079 089 099 108 118 48 128 137 147 157 167 176 186 196 205 215 49 225 234 244 254 263 273 283 292 302 312 450 321 331 341 350 360 369 379 389 398 408 1 2 3 4 5 6 7 8 9 Prop. Pts. Table 3. Number Logarithms 185 1 2 3 4 5 6 7 8 9 Prop, Pts, 450 65 321 331 341 350 360 369 379 389 398 408 51 418 427 437 447 456 466 475 485 495 504 52 514 523 533 543 552 562 571 581 591 600 53 610 619 629 639 648 658 667 677 686 696 54 706 715 725 734 744 753 763 772 782 792 55 801 811 820 830 839 849 858 868 877 887 56 896 906 916 925 935 944 954 963 973 982 57 992 *001 *011 *020 *030 *039 *049 *058 *068 *077 58 66087 096 106 115 124 134 143 153 162 172 59 181 191 200 210 219 229 238 247 257 266 460 276 285 295 304 314 323 332 342 351 361 61 370 380 389 398 408 417 427 436 445 455 82 464 474 483 492 502 511 521 530 539 549 63 558 567 577 586 596 605 614 624 633 642 64 652 661 671 680 689 699 708 717 727 736 65 745 755 764 773 783 792 801 811 820 829 66 839 848 857 867 876 885 894 904 913 922 67 932 941 950 960 969 978 987 997 *006 *015 68 67025 034 043 052 062 071 080 089 099 108 69 117 127 136 145 154 164 173 182 191 201 470 210 219 228 237 247 256 265 274 284 293 71 302 311 321 330 339 348 357 367 376 385 10 9 8 72 394 403 413 422 431 440 449 459 468 477 1 1.0 0.9 0.8 73 486 495 504 514 523 532 541 550 560 569 2 2.0 1.8 1.6 74 75 76 578 669 761 587 679 770 596 688 779 605 697 788 614 706 797 624 715 806 633 724 815 642 733 825 651 742 834 660 752 843 3 3.0 2.7 2.4 4 4.0 3.6 3.2 5 5.0 4.5 4.0 6 6.0 5.4 4.8 77 852 861 870 879 888 897 906 916 925 934 7 7.0 6.3 5.6 78 943 952 961 970 979 988 997 *006 *015 *024 8 8.0 7.2 6.4 79 68034 043 052 061 070 079 088 097 106 115 9 9.0 8.1 7.2 480 124 133 142 151 160 169 178 187 196 205 81 215 224 233 242 251 260 269 278 287 296 82 305 314 323 332 341 350 359 368 377 386 83 395 404 413 422 431 440 449 458 467 476 84 485 494 502 511 520 529 538 547 556 565 85 574 583 592 601 610 619 628 637 646 655 86 664 673 681 690 699 708 717 726 735 744 87 753 762 771 780 789 797 806 815 824 833 88 842 851 860 869 878 886 895 904 913 922 89 931 940 949 958 9(J6 975 984 993 *002 *011 490 69020 028 037 046 055 064 073 082 090 099 91 108 117 126 135 144 152 161 170 179 188 92 197 205 214 223 232 241 249 258 267 276 93 285 294 302 311 320 329 338 346 355 364 94 373 381 390 399 408 417 425 434 443 452 95 461 469 478 487 496 504 513 522 531 539 96 548 557 566 574 583 592 601 609 618 627 97 636 644 653 662 671 679 688 697 705 714 98 723 732 740 749 758 767 775 784 793 801 99 810 819 827 836 845 854 862 871 880 888 500 897 906 914 923 932 940 949 958 966 975 1 i 2 3 4 5 6 7 8 9 Prop, Pts, 186 Table 3. Number Logarithms 1 2 3 4 5 6 7 8 9 Prop, Pts, 500 69 897 906 914 923 932 940 949 958 966 975 01 984 992 *001 *010 *018 *027 *036 *044 *053 *062 02 70070 079 088 096 105 114 122 131 140 148 03 157 165 174 183 191 200 209 217 226 234 04 243 252 260 269 278 286 295 303 312 321 05 329 338 346 355 364 372 381 389 398 406 06 415 424 432 441 449 458 467 475 484 492 07 501 509 518 526 535 544 552 561 569 578 08 586 595 603 612 621 629 638 646 655 663 09 672 680 689 697 706 714 723 731 740 749 510 757 766 774 783 791 800 808 817 825 834 11 842 851 859 868 876 885 893 902 910 919 12 927 935 944 952 961 9(59 978 986 995 *003 13 71012 020 029 037 046 054 063 071 079 088 14 096 105 113 122 130 139 147 155 164 172 15 181 189 198 206 214 223 231 240 248 257 16 265 273 282 290 299 307 315 324 332 341 17 349 357 366 374 383 391 399 408 416 425 18 433 441 450 458 466 475 483 492 500 508 19 517 525 533 542 550 559 567 575 584 592 520 600 609 617 625 634 642 650 659 667 675 21 684 692 700 709 717 725 734 742 750 759 987 22 767 775 784 792 800 809 817 825 834 842 1 0.9 0.8 0.7 23 850 858 867 875 883 892 900 908 917 925 2 1.8 1.6 1.4 24 933 941 950 958 966 975 983 991 999 *008 3 2.7 2.4 2.1 25 72016 024 032 041 049 057 066 074 082 090 4 3.6 3.2 2.8 26 099 107 115 123 132 140 148 156 165 173 5 4.5 4.0 3.5 6 5.4 4.8 4.2 27 181 189 198 206 214 222 230 239 247 255 7 6.3 5.6 4.9 28 263 272 280 288 296 304 313 321 329 337 8 7.2 6.4 5.6 29 346 354 362 370 378 387 395 403 411 419 9 8.1 7.2 6.3 530 428 436 444 452 460 469 477 485 493 501 31 509 518 526 534 542 550 558 567 575 583 32 591 599 607 616 624 632 640 648 656 665 33 673 681 689 697 705 713 722 730 738 746 34 754 762 770 779 787 795 803 811 819 827 35 835 843 852 860 868 876 884 892 900 908 36 916 925 933 941 949 957 965 973 981 989 37 997 *006 *014 *022 *030 *038 *046 *054 *062 *070 38 73078 086 094 102 111 119 127 135 143 151 39 159 167 175 183 191 199 207 215 223 231 540 239 247 255 263 272 280 288 296 304 312 41 320 328 336 344 352 360 368 376 384 392 42 400 408 416 424 432 440 448 456 464 472 43 480 488 496 504 512 520 528 536 544 552 44 560 568 576 584 592 600 608 616 624 632 45 640 648 656 664 672 679 687 695 703 711 46 719 727 735 743 751 759 767 775 783 791 47 799 807 815 823 830 838 846 854 862 870 48 878 886 894 902 910 918 926 933 941 949 49 957 965 973 981 989 997 *005 *013 *020 *028 550 74036 044 052 060 068 076 084 092 099 107 1 2 3 4 5 6 7 8 9 Prop, Pts, Table 3. Number Logarithms 187 1 2 3 4 5 6 7 8 9 Prop. Pts, 550 74036 044 052 060 068 076 084 092 099 107 51 115 123 131 139 147 155 162 170 178 186 52 194 202 210 218 225 233 241 249 257 265 53 273 280 288 296 304 312 320 327 335 343 54 351 359 367 374 382 390 398 406 414 421 55 429 437 445 453 461 468 476 484 492 500 56 507 515 523 531 539 547 554 562 570 578 57 586 593 601 609 617 624 632 640 648 656 58 663 671 679 687 695 702 710 718 726 733 59 741 749 757 764 772 780 788 796 803 811 560 819 827 834 842 850 858 865 873 881 889 I 61 896 904 912 920 927 935 943 950 958 966 62 974 981 989 997 *005 *012 *020 *028 *035 *043 63 75051 059 066 074 082 089 097 105 113 120 64 128 136 143 151 159 166 174 182 189 197 65 205 213 220 228 236 243 251 259 266 274 66 282 289 297 305 312 320 328 335 343 351 67 358 366 374 381 389 397 404 412 420 427 68 435 442 450 458 465 473 481 488 496 504 69 511 519 526 534 542 549 557 565 572 580 570 587 595 603 610 618 626 633 641 648 656 71 664 671 679 686 694 702 709 717 724 732 8 7 72 740 747 755 762 770 778 785 793 800 808 1 0.8 0.7 73 815 823 831 838 846 853 861 868 876 884 2 1.6 1.4 74 891 899 906 914 921 929 937 944 952 959 3 2.4 2.1 75 967 974 982 989 997 *005 *012 *020 *027 *035 4 3.2 2.8 76 76042 050 057 065 072 080 087 095 103 110 5 4.0 3.5 6 4.8 4.2 77 118 125 133 140 148 155 163 170 178 185 7 5.6 4.9 78 193 200 208 215 223 230 238 245 253 260 8 6.4 5.6 79 268 275 283 290 298 305 313 320 328 335 9 7.2 6.3 580 343 350 358 365 373 380 388 395 403 410 81 418 425 433 440 448 455 462 470 477 485 82 492 500 507 515 522 530 537 545 552 559 83 567 574 582 589 597 604 612 619 626 634 84 641 649 656 664 671 678 686 693 701 708 85 716 723 730 738 745 753 760 768 775 782 86 790 797 805 812 819 827 834 842 849 856 87 864 871 879 886 893 901 908 916 923 930 88 938 945 953 960 967 975 982 989 997 *004 89 77012 019 026 034 041 048 056 063 070 078 590 085 093 100 107 115 122 129 137 144 151 91 159 166 173 181 188 195 203 210 217 225 92 232 240 247 254 262 269 276 283 291 298 93 305 313 320 327 335 342 349 357 364 371 94 379 386 393 401 408 415 422 430 437 444 95 452 459 466 474 481 488 495 503 510 517 96 525 532 539 546 554 561 568 576 583 590 97 597 605 612 619 627 634 641 648 656 663 98 670 677 685 692 699 706 714 721 728 735 99 743 750 757 764 772 779 786 793 801 808 600 815 822 830 837 844 851 859 866 873 880 1 2 3 4 5 6 7 8 9 Prop, Pts, 188 Table 3. Number Logarithms 1 2 3 4 5 6 7 8 9 Prop, Pts, 600 77815 822 8:30 837 844 851 859 866 873 880 01 887 895 902 909 916 924 931 938 945 952 02 960 967 974 981 988 996 *003 *010 *017 *025 03 78032 039 046 053 061 068 075 082 089 097 04 104 111 118 125 132 140 147 154 161 168 05 176 183 190 197 204 211 219 226 233 240 06 247 254 262 269 276 283 290 297 305 312 07 319 326 333 340 347 355 362 369 376 383 08 390 398 405 412 419 426 433 440 447 455 09 462 469 476 483 490 497 504 512 519 526 610 533 540 547 554 561 569 576 583 590 597 11 604 611 618 625 633 640 647 654 661 668 12 675 682 689 696 704 711 718 725 732 739 13 746 753 760 767 774 781 789 796 803 810 14 817 824 831 838 845 852 859 866 873 880 15 888 895 902 909 916 923 930 937 944 951 16 958 965 972 979 986 993 *000 *007 *014 *021 17 79029 036 043 050 057 064 071 078 085 092 18 099 106 113 120 127 134 141 148 155 162 19 169 176 183 190 197 204 211 218 225 232 620 239 246 253 260 267 274 281 288 295 302 21 309 316 323 330 337 344 351 358 365 372 876 22 379 386 393 400 407 414 421 428 435 442 1 0.8 0.7 0.6 23 449 456 463 470 477 484 491 498 505 511 2 1.6 1.4 1.2 24 25 26 518 588 657 525 595 664 532 602 671 539 609 678 546 616 685 553 623 692 560 630 699 567 637 706 574 644 713 581 650 720 3 2.4 2.1 1.8 4 3.2 2.8 2.4 5 4.0 3.5 3.0 6 4.8 4.2 3.6 27 727 734 741 748 754 761 768 775 782 789 7 5.6 4.9 4.2 28 796 803 810 817 824 831 837 844 851 858 8 6.4 5.6 4.8 29 865 872 879 886 893 900 906 913 920 927 9 7.2 6.3 5.4 630 934 941 948 955 962 969 975 982 989 996 31 80003 010 017 024 030 037 044 051 058 065 32 072 079 085 092 099 106 113 120 127 134 33 140 147 154 161 168 175 182 188 195 202 34 209 216 223 229 236 243 250 257 264 271 35 277 284 291 298 305 312 318 325 332 339 36 346 353 359 366 373 380 387 393 400 407 37 414 421 428 434 441 448 455 462 468 475 38 482 489 496 502 509 516 523 530 536 543 39 550 557 564 570 577 584 591 598 604 611 640 618 625 632 638 645 652 659 665 672 679 41 686 693 699 706 713 720 726 733 740 747 42 754 760 767 774 781 787 794 801 808 814 43 821 828 835 841 848 855 862 868 875 882 44 889 895 902 909 916 922 929 936 943 949 45 956 963 969 976 983 990 99(5 *003 *010 017 46 81023 030 037 043 050 057 064 070 077 084 47 090 097 104 111 117 124 131 137 144 151 48 158 164 171 178 184 191 198 204 211 218 49 224 231 238 245 251 258 265 271 278 285 650 291 298 305 311 318 325 331 338 345 351 1 2 3 4 5 6 7 8 9 Prop. Pts, Table 3. Number Logarithms 189 1 2 3 4 5 6 7 8 9 Prop, Pts. 650 81291 298 305 311 318 325 331 338 345 351 51 358 365 371 378 385 391. 398 405 411 418 52 425 431 438 445 451 458 465 471 478 485 53 491 498 605 511 518 525 531 538 544 551 54 558 564 571 578 584 591 598 604 611 617 55 624 631 637 (544 651 657 6(54 671 677 684 56 690 697 704 710 717 723 730 737 743 750 57 757 763 770 776 783 790 796 803 809 816 58 823 829 836 842 849 856 862 869 875 882 59 889 895 902 908 915 921 928 935 941 948 660 954 961 968 974 981 987 994 *000 *007 *014 61 82020 027 033 040 046 053 060 066 073 079 62 086 092 099 105 112 119 125 132 138 145 63 151 158 164 171 178 184 191 197 204 210 64 217 223 230 236 243 249 256 263 269 276 65 282 289 295 302 308 315 321 328 334 341 66 347 354 360 367 373 380 387 393 400 406 67 413 419 426 432 439 445 452 458 465 471 68 478 484 491 497 504 510 517 523 530 536 69 543 549 556 562 569 575 582 588 595 601 670 607 614 620 627 633 640 646 653 659 666 71 672 679 685 692 698 705 711 718 724 730 7 6 72 737 743 750 756 763 769 776 782 789 795 1 0.7 0.6 73 802 808 814 821 827 834 840 847 853 860 2 1.4 1.2 74 866 872 879 885 892 898 905 911 918 924 3 2.1 1.8 75 930 937 943 950 956 963 969 975 982 988 4 2.8 2.4 76 995 *001 *008 *014 *020 *027 *033 *040 *046 *052 5 3.5 3.0 6 4.2 3.6 77 83059 065 072 078 085 091 097 104 110 117 7 4.9 4.2 78 123 129 136 142 149 155 161 168 174 181 8 5.6 4.8 79 187 193 200 206 213 219 225 232 238 245 9 6.3 5.4 680 251 257 264 270 276 283 289 296 302 308 81 315 321 327 334 340 347 353 359 366 372 82 378 385 391 398 404 410 417 423 429 436 83 442 448 455 461 467 474 480 487 493 499 84 506 512 518 525 531 537 544 550 556 563 85 569 575 582 588 594 601 607 613 620 626 86 632 639 645 651 658 664 670 677 683 689 87 696 702 708 715 721 727 734 740 746 753 88 759 705 771 778 784 790 797 803 809 816 89 822 828 835 841 847 853 860 866 872 879 690 885 891 897 904 910 916 923 929 935 942 91 948 954 960 967 973 979 985 992 998 *004 92 84011 017 023 029 036 042 048 055 061 067 93 073 080 086 092 098 105 111 117 123 130 94 136 142 148 155 161 167 173 180 186 192 95 198 205 211 217 223 230 236 242 248 255 96 261 267 273 280 286 292 298 305 311 317 97 323 330 336 342 348 354 361 367 373 379 98 386 392 398 404 410 417 423 429 435 442 99 448 454 460 466 473 479 485 491 497 504 700 510 516 522 528 535 541 547 553 559 566 1 2 3 4 5 6 7 8 9 Prop, Pts, 190 Table 3. Number Logarithms 1 2 3 4 5 6 7 8 9 Prop, Pts. 700 84510 516 522 528 535 541 547 553 559 566 01 572 578 584 590 597 603 609 615 621 628 02 634 640 646 652 658 665 671 677 683 689 03 696 702 708 714 720 726 733 739 745 751 04 757 763 770 776 782 788 794 800 807 813 05 819 825 831 837 844 850 856 862 868 874 06 880 887 893 899 905 911 917 924 930 936 07 942 948 954 960 967 973 979 985 991 997 08 85003 009 016 022 028 034 040 046 052 058 09 065 071 077 083 089 095 101 107 114 120 710 126 132 138 144 150 156 163 169 175 181 11 187 193 199 205 211 217 224 230 236 242 12 248 254 260 266 272 278 285 291 297 303 13 309 315 321 327 333 339 345 352 358 364 14 370 376 382 388 394 400 406 412 418 425 15 431 437 443 449 455 461 467 473 479 485 16 491 497 503 509 516 522 528 534 540 546 17 552 558 564 570 576 582 588 594 600 606 18 612 618 625 631 637 643 649 655 661 667 19 673 679 685 691 697 703 709 715 721 727 720 733 739 745 751 757 763 769 775 781 788 21 794 800 806 812 818 824 830 836 842 848 765 22 854 860 866 872 878 884 890 896 902 908 1 0.7 0.6 0.5 23 914 920 926 932 938 944 950 956 962 968 2 1.4 1.2 1.0 24 25 26 974 86034 094 980 040 100 986 046 106 992 052 112 998 058 118 *004 064 124 *010 070 130 *016 076 136 *022 082 141 *028 088 147 3 2.1 1.8 1.5 4 2.8 2.4 2.0 5 3.5 3.0 2.5 6 4.2 3.6 3.0 27 153 159 165 171 177 183 189 195 201 207 7 4.9 4.2 3.5 28 213 219 225 231 237 243 249 255 261 267 8 5.6 4.8 4.0 29 273 279 285 291 297 303 308 314 320 326 9 6.3 5.4 4.5 730 332 338 344 350 356 362 3(58 374 380 386 31 392 398 404 410 415 421 427 433 439 445 32 451 457 463 469 475 481 487 493 499 504 33 510 516 522 528 534 540 546 552 558 564 34 570 576 581 587 593 599 605 611 617 623 35 629 635 641 646 652 658 664 670 676 682 36 688 694 700 705 711 717 723 729 735 741 37 747 753 759 764 770 776 782 788 794 800 38 806 812 817 823 829 835 841 847 853 859 39 864 870 876 882 888 894 900 906 911 917 740 923 929 935 941 947 953 958 964 970 976 41 982 988 994 999 *005 *011 *017 *023 *029 *035 42 87040 046 052 058 064 070 075 081 087 093 43 099 105 111 116 122 128 134 140 146 151 44 157 163 169 175 181 186 192 198 204 210 45 216 221 227 233 239 245 251 256 262 268 46 274 280 286 291 297 303 309 315 320 326 47 332 338 344 349 355 361 367 373 379 384 48 390 396 402 408 413 419 425 431 437 442 49 448 454 460 466 471 477 483 489 495 500 750 506 512 518 523 529 535 541 547 552 558 1 2 3 4 5 6 7 8 9 Prop, Pts, Table 3. Number Logarithms 191 1 2 3 4 5 6 7 8 9 Prop, Pts, 750 87506 512 518 523 529 535 541 547 552 558 51 564 570 576 581 587 593 599 604 610 616 52 622 628 633 639 645 651 656 662 6(58 674 53 <>7'. 685 691 697 703 708 714 720 726 731 54 737 743 749 754 760 766 772 777 783 789 55 795 800 806 812 818 823 829 835 841 846 56 852 858 864 869 875 881 887 892 898 904 57 910 915 921 927 933 938 944 950 955 961 58 967 973 978 984 990 996 *001 *007 *013 *018 59 88024 030 036 041 047 053 058 064 070 076 760 081 087 093 098 104 110 116 121 127 133 61 138 144 150 156 161 167 173 178 184 190 62 195 201 207 213 218 224 230 235 241 247 63 252 258 264 270 275 281 287 292 298 304 64 309 315 321 326 332 338 343 349 355 360 65 366 372 377 383 389 395 400 406 412 417 66 423 429 434 440 446 451 457 463 468 474 67 480 485 491 497 502 508 513 519 525 530 68 536 542 547 553 559 564 570 576 581 587 69 593 598 604 610 615 621 627 632 638 643 770 649 655 660 666 672 677 683 689 694 700 71 705 711 717 722 728 734 739 745 750 756 6 5 72 762 767 773 779 784 790 795 801 807 812 1 0.6 0.5 73 818 824 829 835 840 846 852 857 863 868 2 1.2 1.0 74 874 880 885 891 897 902 908 913 919 925 3 1.8 1.5 75 930 936 941 947 953 958 9(54 969 975 981 4 2.4 2.0 76 986 992 997 *003 *009 *014 *020 *025 *031 *037 5 3.0 2.5 6 3.6 3.0 77 89042 048 053 059 064 070 076 081 087 092 7 4.2 3.5 78 098 104 109 115 120 126 131 137 143 148 8 4.8 4.0 79 154 159 165 170 176 182 187 193 198 204 9 5.4 4.5 780 209 215 221 226 232 237 243 248 254 260 81 265 271 276 282 287 293 298 304 310 315 82 321 326 332 337 343 348 354 360 365 371 83 376 382 387 393 398 404 409 415 421 426 84 432 437 443 448 454 459 465 470 476 481 85 487 492 498 504 509 515 520 526 531 537 86 542 548 553 559 564 570 575 581 586 592 87 597 603 609 614 (520 625 631 636 642 647 88 653 658 664 669 675 680 686 691 697 702 89 708 713 719 724 730 735 741 746 752 757 790 763 768 774 779 785 790 796 801 807 812 91 818 823 829 834 840 845 851 856 862 867 92 873 878 883 889 894 900 905 911 916 922 93 927 933 938 944 949 955 960 966 971 977 &4 982 988 993 998 *004 *009 *015 *020 *026 *031 95 90037 042 048 053 059 064 069 075 080 086 96 091 097 102 108 113 119 124 129 135 140 97 146 151 157 162 168 173 179 184 189 195 98 200 206 211 217 222 227 233 238 244 249 99 255 260 266 271 276 282 287 293 298 304 800 309 314 320 325 331 336 342 347 352 358 1 2 3 4 5 6 7 8 9 Prop, Pta, 192 Table 3. Number Logarithms 1 2 3 4 5 6 7 8 9 Prop, Pts, 800 90309 314 320 325 331 336 342 347 352 358 6 5 1 0.6 0.5 2 1.2 1.0 3 1.8 1.5 4 2.4 2.0 5 3.0 2.5 6 3.6 3.0 7 4.2 3.5 8 4.8 4.0 9 5.4 4.5 01 02 03 04 05 06 07 08 09 363 417 472 526 580 634 687 741 795 369 423 477 531 585 639 693 747 800 374 428 482 536 590 644 698 752 806 380 434 488 542 596 650 703 757 811 385 439 493 547 601 655 709 763 816 390 445 499 553 607 660 714 768 822 396 450 504 558 612 666 720 773 827 401 455 509 563 617 671 725 779 832 407 461 515 569 623 677 730 784 838 412 466 520 574 628 682 736 789 843 810 849 854 859 865 870 875 881 886 891 897 11 12 13 14 15 16 17 18 19 902 956 91009 062 116 169 222 275 328 907 961 014 068 121 174 228 281 334 913 966 020 073 126 180 233 286 339 918 972 025 078 132 185 238 291 344 924 977 030 084 137 190 243 297 350 929 982 036 089 142 196 249 302 355 934 988 041 094 148 201 254 307 360 940 993 046 100 153 206 259 312 365 945 998 052 105 158 212 265 318 371 950 *004 057 110 164 217 270 323 376 820 381 387 392 397 403 408 413 418 424 429 21 22 23 24 25 26 27 28 29 434 487 540 593 645 698 751 803 855 440 492 545 598 651 703 756 808 861 445 498 551 603 656 709 761 814 866 450 503 556 609 661 714 766 819 871 455 508 561 614 666 719 772 824 876 461 514 566 619 672 724 777 829 882 466 519 572 624 677 730 782 834 887 471 524 577 630 682 735 787 840 892 477 529 582 635 687 740 793 845 897 482 535 587 640 693 745 798 850 903 830 908 913 918 924 929 934 939 944 950 955 31 32 33 34 35 36 37 38 39 960 92012 065 117 169 221 273 324 376 965 018 070 122 174 226 278 330 381 971 023 075 127 179 231 283 335 387 976 028 080 132 184 236 288 340 392 981 033 085 137 189 241 293 345 397 986 038 091 143 195 247 298 350 402 991 044 096 148 200 252 304 355 407 997 049 101 153 205 257 309 361 412 *002 054 106 158 210 262 314 366 418 *007 059 111 163 215 267 319 371 423 840 428 433 438 443 449 454 459 464 469 474 41 42 43 44 45 46 47 48 49 480 531 583 634 686 737 788 840 891 485 536 588 639 691 742 793 845 896 490 542 593 645 696 747 799 850 901 495 547 598 650 701 752 804 855 906 500 552 603 655 706 758 809 860 911 505 557 609 660 711 763 814 865 916 511 562 614 665 716 768 819 870 921 516 567 619 670 722 773 824 875 927 521 572 624 675 727 778 829 881 932 526 578 629 681 732 783 834 886 937 850 942 947 952 957 962 967 973 978 983 988 1 g 3 4 5 6 7 8 9 Prop, Pts, Table 3. Number Logarithms 193 1 2 3 4 5 6 7 8 9 Prop, Pts, 850 92 942 947 952 957 962 967 973 978 983 988 51 993 998 *003 *008 *013 *018 *024 *029 *034 039 52 93044 049 054 059 064 069 075 080 085 090 53 095 100 105 110 115- 120 125 131 136 141 54 146 151 156 161 166 171 176 181 186 192 55 197 202 207 212 217 222 227 232 237 242 56 247 252 258 263 268 273 278 283 288 293 57 298 303 308 313 318 323 328 334 339 344 58 349 354 359 364 369 374 379 384 389 394 59 399 404 409 414 420 425 430 435 440 445 860 4,50 455 460 465 470 475 480 485 490 495 61 500 505 510 515 520 526 531 536 541 546 (52 551 556 561 566 571 576 581 586 591 596 63 601 606 611 616 621 626 631 636 641 646 64 651 656 661 666 671 676 682 687 692 697 65 702 707 712 717 722 727 732 737 742 747 66 752 757 762 767 772 777 782 787 792 797 67 802 807 812 817 822 827 832 837 842 847 68 852 857 862 867 872 877 882 887 892 897 69 902 907 912 917 922 927 932 937 942 947 870 952 957 962 967 972 977 982 987 992 997 71 94002 007 012 017 022 027 032 037 042 047 654 72 052 057 062 067 072 077 082 086 091 096 1 0.6 0.5 0.4 73 101 106 111 116 121 126 131 136 141 146 2 1.2 1.0 0.8 74 75 76 151 201 250 156 206 255 161 211 260 166 216 265 171 221 270 176 226 275 181 231 280 186 236 285 191 240 290 196 245 295 3 1.8 1.5 1.2 4 2.4 2.0 1.6 5 3.0 2.5 2.0 6 3.6 3.0 2.4 77 300 305 310 315 320 325 330 335 340 345 7 4.2 3.5 2.8 78 349 354 359 364 369 374 379 384 389 394 8 4.8 4.0 3,2 79 395) 404 409 414 419 424 429 433 438 443 9 5.4 4.5 3.6 880 448 453 458 463 468 473 478 483 488 493 81 498 503 507 512 517 522 527 532 537 542 82 547 552 557 562 567 571 576 581 586 591 83 596 601 606 611 616 621 626 630 635 640 84 645 650 655 660 665 670 675 680 685 689 85 694 699 704 709 714 719 724 729 734 738 86 743 748 753 758 763 768 773 778 783 787 87 792 797 802 807 812 817 822 827 832 836 88 841 846 851 856 861 866 871 876 880 885 89 890 895 900 905 910 915 919 924 929 934 890 9o9 944 949 954 959 963 968 973 978 983 91 988 993 998 *002 *007 *012 *017 *022 027 *032 92 95 Otfi 041 046 051 056 061 066 071 075 080 93 085 090 095 100 105 109 114 119 124 129 94 134 139 143 148 153 158 163 168 173 177 95 182 187 192 197 202 207 211 216 221 226 96 231 236 240 245 250 255 260 265 270 274 97 279 284 289 294 299 303 308 313 318 323 98 328 332 337 342 347 352 357 361 366 371 99 376 381 386 390 395 400 405 410 415 419 900 424 429 434 439 444 448 453 458 463 468 1 2 3 4 5 6 7 8 9 Prop, Pts, 194 Table 3. Number Logarithms 1 2 3 4 5 6 7 8 9 Prop, Pts= 900 95424 429 434 439 444 448 453 458 463 468 01 472 477 482 487 492 497 501 506 511 516 02 521 525 530 535 540 545 550 554 559 564 03 569 574 578 583 588 593 598 602 607 612 04 617 622 626 631 636 641 646 650 655 660 05 665 670 674 679 684 689 694 698 703 708 06 713 718 722 727 732 737 742 746 751 756 07 761 766 770 775 780 785 789 794 799 804 08 809 813 818 823 828 832 837 842 847 852 09 856 861 866 871 875 880 885 890 895 899 910 904 909 914 918 923 928 933 938 942 947 11 952 957 961 966 971 976 980 985 990 995 12 999 004 *009 *014 *019 *023 *028 *033 *038 *042 13 96.047 052 057 061 066 071 076 080 085 090 14 095 099 104 109 114 118 123 128 133 137 15 142 147 152 156 161 166 171 175 180 185 16 190 194 199 204 209 213 218 223 227 232 17 237 242 246 251 256 261 265 270 275 280 18 284 289 294 298 303 308 313 317 322 327 19 332 336 341 346 350 355 360 365 369 374 920 379 384 388 393 398 402 407 412 417 421 21 426 431 4a5 440 445 450 454 459 464 468 5 4 22 473 478 483 487 492 497 501 506 511 515 1 0.5 0.4 23 520 525 530 534 539 544 548 553 558 562 2 1.0 0.8 24 567 572 577 581 586 591 595 600 605 609 3 1.5 1.2 25 614 619 624 628 633 638 642 647 652 656 4 2.0 1.6 26 661 666 670 675 680 685 689 694 699 703 5 2.5 2.0 6 3.0 2.4 27 708 713 717 722 727 731 736 741 745 750 7 3.5 2.8 28 755 759 764 769 774 778 783 788 792 797 8 4.0 3.2 29 802 806 811 816 820 825 830 834 839 844 9 4.5 3.6 930 848 853 858 862 867 872 876 881 886 890 31 895 900 904 909 914 918 923 928 932 937 32 942 946 951 956 960 965 970 974 979 984 33 988 993 997 *002 *007 *011 *016 *021 *025 *030 34 97035 039 044 049 053 058 063 067 072 077 35 081 086 090 095 100 104 109 114 118 123 36 128 132 137 142 146 151 155 160 165 169 37 174 179 183 188 192 197 202 206 211 216 38 220 225 230 234 239 243 248 253 257 262 39 267 271 276 280 285 290 294 299 304 308 940 313 317 322 327 331 336 340 345 350 354 41 359 3T4 368 373 377 382 387 391 396 400 42 405 410 414 419 424 428 433 437 442 447 43 451 .456 460 465 470 474 479 483 488 493 44 497 502 506 511 516 520 525 529 534 539 45 543 548 552 557 562 566 571 575 580 585 46 589 594 598 603 607 612 617 621 626 630 47 635 640 644 649 653 658 663 667 672 676 48 681 685 690 695 699 704 708 713 717 722 49 727 731 736 740 745 749 754 759 763 768 950 772 777 782 786 791 795 800 804 809 813 1 2 3 4 5 6 7 8 9 Prop, Pts, Table 3. Number Logarithms 195 1 2 3 4 5 6 7 8 9 Prop, Pts, 950 97772 777 782 786 791 795 800 804 809 813 51 818 823 827 832 836 841 845 850 855 859 52 864 868 873 877 882 886 891 896 900 905 53 909 914 918 923 928 932 937 941 946 950 54 955 959 964 968 973 978 982 987 991 996 55 98000 005 009 014 019 023 028 032 037 041 56 046 050 055 059 064 068 073 078 082 087 57 091 096 100 105 109 114 118 123 127 132 58 137 141 146 150 155 159 164 168 173 177 59 182 186 191 195 200 204 209 214 218 223 960 227 232 236 241 245 250 254 259 263 268 61 272 277 281 286 290 295 299 304 303 313 62 318 322 327 331 336 340 345 349 354 358 63 363 3G7 372 376 381 385 390 394 399 403 64 408 412 417 421 426 430 435 439 444 448 65 453 457 462 466 471 475 480 484 489 493 66 498 502 507 511 516 520 525 529 534 538 67 543 547 552 556 561 565 570 574 579 583 68 588 592 597 601 605 610 614 619 623 628 69 632 (>37 641 646 650 655 659 664 668 673 970 677 682 686 691 695 700 704 709 713 717 71 722 726 731 735 740 744 749 753 758 762 5 4 72 767 771 776 780 784 789 793 798 802 807 1 0.5 0.4 73 811 816 820 825 829 834 838 843 847 851 2 1.0 0.8 74 856 860 865 869 874 878 883 887 892 896 3 1.5 1.2 75 900 905 909 914 918 923 927 932 936 941 4 2.Q 1.6 76 945 949 954 958 963 967 972 976 981 985 5 2.5 2.0 6 3.0 2.4 77 989 994 998 *003 *007 *012 *016 *021 *025 *029 7 3.5 2.8 78 99034 038 043 047 052 056 061 065 069 074 8 4.0 3.2 79 078 083 087 092 096 100 105 109 114 118 9 4.5 3.6 980 123 127 131 136 140 145 149 154 158 162 81 167 171 176 180 185 189 193 198 202 207 82 211 216 220 224 229 233 238 242 247 251 83 255 260 264 269 273 277 282 286 291 295 84 300 304 308 313 317 322 326 330 335 339 85 344 348 352 357 361 366 370 374 379 383 86 388 392 396 401 405 410 414 419 423 427 87 432 436 441 445 449 454 458 463 467 471 88 476 480 484 489 493 498 502 506 511 515 89 520 524 528 533 537 542 546 550 555 559 990 564 568 572 577 581 585 590 594 599 603 91 607 612 616 621 625 629 634 638 642 647 92 651 656 660 664 669 673 677 082 686 691 93 695 699 704 708 712 717 721 726 730 734 94 739 743 747 752 756 760 765 769 774 778 95 782 787 791 795 800 804 808 813 817 822 96 826 830 835 839 843 848 852 856 861 865 97 870 874 878 883 887 891 896 900 904 909 98 913 917 922 926 930 935 939 944 948 952 99 957 961 965 970 974 978 983 987 991 996 1000 00000 004 009 013 017 022 026 030 035 039 1 2 3 4 5 6 7 8 9 Prop, Pts, 196 Table 4. Trigonometric Logarithms (180) (359) 179 ' Sin Cos Tan Cot Sec Csc 0.00 000 0.00 000 60 1 6.46 373 .00 000 6.46 373 3.53 627 .00 000 3.53 627 59 2 6.76 476 .00 000 6.76 476 3.23 524 .00 000 .23 524 58 3 6.94 085 .00 000 6.94 085 3.05 915 .00 000 .05 915 57 4 7.06 579 .00 000 7.06 579 2.93 421 .00 000 2.93 421 56 5 7.16 270 0.00 000 7.16 270 2.83 730 0.00 000 2.83 730 55 6 .24 188 .00 000 .24 188 .75 812 .00 000 .75 812 54 7 .30 882 .00 000 .30 882 .69 118 .00 000 .69 118 53 8 .36 682 .00 000 .36 682 .63 318 .00 000 .63 318 52 9 .41 797 .00 000 .41 797 .58 203 .00 000 .58 203 51 10 7.46 373 0.00 000 7.46 373 2.53 627 0.00 000 2.53 627 50 11 .50 512 .00 000 .50 512 .49 488 .00 000 .49 488 49 12 .54 291 .00 000 .54 291 .45 709 .00 000 .45 709 48 13 .57 767 .00 000 .57 767 .42 233 .00 000 .42 233 47 14 .60 985 .00 000 .60 986 .39 014 .00 000 .39 015 46 15 7.63 982 0.00 000 7.63 982 2.36 018 0.00 000 2.36 018 45 16 .66 784 .00 000 .66 785 .33 215 .00 000 .33 216 44 17 .69417 9.99 999 .69 418 .30 582 .00 001 .30583 43 18 .71 900 .99 999 .71 900 .28 100 .00 001 .28 100 42 19 .74 248 .99 999 .74 248 .25 752 .00 001 .25 752 41 20 7.76 475 9.99 999 7.76 476 2.23 524 0.00 001 2.23 525 40 21 .78 594 .99 999 .78 595 .21 405 .00 001 .21 406 39 22 .80 615 .99 999 .80 615 .19 385 .00 001 .19 385 38 23 .82 545 .99 999 .82 546 .17454 .00 001 .17 455 37 24 .84 393 .99 999 .84 394 .15 606 .00 001 .15 607 36 25 7.86 166 9.99 999 7.86 167 2.13 833 0.00 001 2.13 834 35 26 .87 870 .99 999 .87 871 .12 129 .00 001 .12 130 34 27 .89 509 .99 999 .89 510 .10 490 .00 001 .10491 33 28 .91 088 .99 999 .91 089 .08911 .00 001 .08 912 32 29 .92 612 .99 998 .92 613 .07 387 .00 002 .07 388 31 30 7.94 084 9.99 998 7.94 086 2.05 914 0.00 002 2.05 916 30 31 .95 508 .99 998 .95 510 .04 490 .00 002 .04 492 29 32 .96 887 .99 998 .96 889 .03 111 .00 002 .03 113 28 33 .98 223 .99 998 .98 225 .01 775 .00 002 .01 777 27 34 .99 520 .99 998 .99 522 .00 478 .00 002 .00 480 26 35 8.00 779 9.99 998 8.00 781 1.99219 0.00 002 1.99 221 25 36 .02 002 .99 998 .02 004 .97 996 .00 002 .97 998 24 37 .03 192 .99 997 .03 194 .96 806 .00 003 .96 808 23 38 .04 350 .99 997 . .04 353 .95 647 .00 003 .95 650 22 39 .05 478 .99 997 .05 481 .94 519 .00 003 .94 522 21 40 8.06 578 9.99 997 8.06 581 1.93419 0.00 003 1.93422 20 41 .07 650 .99 997 .07 653 .92 347 .00 003 .92 350 19 42 .08 696 .99 997 .08 700 .91 300 .00 003 .91 304 18 43 .09 718 .99 997 .09 722 .90 278 .00 003 .90 282 17 44 .10717 .99 996 .10 720 .89 280 .00 004 .89 283 16 45 8.11 693 9.99 996 8.11 696 1.88304 0.00 004 1.88307 15 46 .12 647 .99 996 .12651 .87 349 .00 004 .87 353 14 47 .13 581 .99 996 .13 585 .86 415 .00 004 .86 419 13 48 .14 495 .99 996 .14 500 .85 500 .00 004 .85 505 12 49 .15391 .99 996 .15 395 .84605 .00 004 .84 609 11 50 8.16268 9.99 995 8.16 273 1.83727 0.00 005 1.83 732 10 51 .17 128 .99 995 .17 133 .82 867 .00 005 .82 872 9 52 .17971 .99 995 .17 976 .82 024 .00 005 .82 029 8 53 .18 798 .99 995 .18804 .81 196 .00 005 .81 202 7 54 .19610 .99 995 .19616 .80 384 .00 005 .80 390 6 55 8.20 407 9.99 994 8.20413 1.79 587 0.00 006 1.79 593 5 56 .21 189 .99 994 .21 195 .78 805 .00 006 .78811 4 57 .21 958 .99 994 .21 964 .78 036 .00 006 .78 042 3 58 .22 713 .99 994 .22 720 .77 280 .00 006 .77 287 2 59 .23 456 .99 994 .23 462 .76 538 .00 006 .76 544 1 60 8.24 186 9.99 993 8.24 192 1.75808 0.00 007 1.75 814 Cos Sin Cot Tan Csc Sec 1* 90 (270) (269) 89 Table 4. Trigonometric Logarithms 197 1 (181) (358) 178 ' Sin Cos Tan Cot Sec Csc 8.24 186 9.99 993 8.24 192 1.75 808 0.00 007 1.75 814 60 1 .24 903 .99 993 .24 910 .75 090 .00 007 .75 097 59 2 .25 609 .99 993 .25 616 .74384 .00 007 .74 391 58 3 .26 304 .99 993 .26 312 .73 688 .00 007 .73 696 57 4 .26 988 .99 992 .26 996 .73 004 .00 008 .73 012 56 5 8.27 661 9.99 992 8.27 669 1.72 331 0.00 008 1.72 339 55 6 .28 324 .99 992 .28 332 .71 668 .00 008 .71 676 54 7 .28 977 .99 992 .28 986 .71 014 .00 008 .71 023 53 8 .29 621 .99 992 .29 629 .70 371 .00 008 .70 379 52 9 .30 255 .99 991 .30 263 .69 737 .00 009 .69 745 51 10 8.30 879 9.99 991 8.30 888 1.69 112 0.00 009 1.69 121 50 11 .31 495 .99 991 .31 505 .68 495 .00 009 .68 505 49 12 .32 103 .99 990 .32 112 .67 888 .00 010 .67 897 48 13 .32 702 .99 990 .32711 .67 289 .00 010 .67 298 47 14 .33 292 .99 990 .33 302 .66 698 .00 010 .66 708 46 15 8.33 875 9.99 990 8.33 886 1.66 114 0.00 010 1.66 125 45 16 .34 450 .99 989 .34 461 .65 539 00.011 65550 44 17 .35 018 .99 989 .35 029 .64 971 .00011 .64 982 43 18 .35 578 .99 989 .35 590 .64 410 .00011 .64 422 42 19 .36 131 .99 989 .36 143 .63 857 .00011 .63 869 41 20 8.36 678 9.99 988 8.36 689 1.63311 0.00 012 1.63 322 40 21 .37 217 .99 988 .37 229 .62 771 .00 012 .62 783 39 22 .37 750 .99 988 .37 762 .62 238 .00 012 .62 250 38 23 .38 276 .99 987 .38 289 .61 711 .00 013 .61 724 37 24 .38 796 .99 987 .38 809 .61 191 .00 013 .61 204 36 25 8.39 310 9.99 987 8.39 323 1.60677 0.00 013 1.60 690 35 26 .39 818 .99 986 .39 832 .60 168 .00 014 .60 182 34 27 .40 320 .99 986 .40 334 .59 666 .00 014 .59 680 33 28 .40 816 .99 986 .40 830 .59 170 .00 014 .59 184 32 29 .41 307 .99 985 .41 321 .58 679 .00 015 .58 693 31 30 8.41 792 9.99 985 8.41 807 1.58 193 0.00 015 1.58208 30 31 .42 272 .99 985 .42 287 .57713 .00 015 .57 728 29 32 .42 746 .99 984 .42 762 .57 238 .00 016 .57 254 28 33 .43 216 .99984 .43 232 .56 768 .00 016 .56 784 27 34 .43 680 .99984 .43 696 .56 304 .00 016 .56 320 26 35 8.44 139 9.99 983 8.44 156 1.55 844 0.00 017 1.55 861 25 36 .44 594 .99 983 .44611 .55 389 .00 017 .55 406 24 37 .45 044 .99 983 .45 061 .54 939 .00 017 .54 956 23 38 .45 489 .99 982 .45 507 .54 493 .00 018 .54 511 22 39 .45 930 .99 982 .45 948 .54 052 .00 018 .54 070 21 40 8.46 366 9.99 982 8.46 385 1.53615 0.00 018 1.53 634 20 41 .46 799 .99 981 .46 817 .53 183 .00 019 .53 201 19 42 .47 226 .99 981 .47 245 .52755 .00 019 .52 774 18 43 .47 650 .99 981 .47 669 .52 331 .00 019 .52 350 17 44 .48 069 .99 980 .48 089 .51 911 .00 020 .51 931 16 45 8.48 485 9.99 980 8.48 505 1.51 495 0.00 020 1.51 515 15 46 .48 896 .99 979 .48 917 .51 083 .00 021 .51 104 14 47 .49 304 .99 979 .49 325 .50 675 .00 021 .50 696 13 48 .49 708 .99 979 .49 729 .50 271 .00 021 .50 292 12 49 .50 108 .99 978 .50 130 .49 870 .00 022 .49 892 11 50 8.50 504 9.99 978 8.50 527 1.49 473 0.00 022 1.49 496 10 51 .50 897 .99 977 .50 920 .49 080 .00 023 .49 103 9 52 .51 287 .99 977 .51 310 .48 690 .00 023 .48 713 8 53 .51 673 .99 977 .51 696 .48 304 .00 023 .48 327 7 54 .52 055 .99 976 .52 079 .47 921 .00 024 .47 945 6 55 8.52 434 9.99 976 8.52 459 1.47 541 0.00 024 1.47 566 5 56 .52 810 .99 975 .52 835 .47 165 .00 025 .47 190 4 57 .53 183 .99 975 .53 208 .46 792 .00 025 .46 817 3 58 .53 552 .99 974 .53 578 .46 422 .00 026 .46 448 2 59 .53 919 .99 974 .53 945 .46 055 .00 026 .46 081 1 60 8.54 282 9.99 974 8.54 308 1.45692 0.00 026 1.45718 Cos Sin Cot Tan Csc Sec ' 91 (271) (268) 88 198 Table 4. Trigonometric Logarithms 2 (182) (357) 177 C ' Sin Cos Tan Cot Sec Csc 8.54 282 9.99 974 8.54 308 1.45 692 0.00 026 1.45 718 60 1 .54 642 .99 973 .54 669 .45 331 .00 027 .45 358 59 2 .54 999 .99 973 .55 027 .44 973 .00 027 .45 001 58 3 .55 354 .99 972 .55 382 .44 618 .00 028 .44 646 57 4 .55 705 .99 972 .55 734 .44 266 .00 028 .44 295 56 5 8.56 054 9.99 971 8.56 083 1.43 917 0.00 029 1.43 946 55 6 .56 400 .99 971 .56 429 .43 571 .00 029 .43 600 54 7 .56 743 .99 970 .56 773 .43 227 .00 030 .43 257 53 8 .57 084 .99 970 .57 114 .42 886 .00 030 .42 916 52 9 .57 421 .99 969 .57 452 .42 548 .00 031 .42 579 51 10 8.57 757 9.99 969 8.57 788 1.42212 0.00 031 1.42 243 50 11 .58 089 .99 968 .58 121 .41 879 .00 032 .41 911 49 12 .58 419 .99 968 .58 451 .41 549 .00 032 .41 581 48 13 .58 747 .99 967 .58 779 .41 221 .00 033 .41 253 47 14 .59 072 .99 967 .59 105 .40895 .00 033 .40 928 46 15 8.59 395 9.99 967 8.59 428 1.40572 0.00 033 1.40 605 45 16 .59 715 .99 966 .59 749 .40 251 .00 034 .40 285 44 17 .60 033 .99 966 .60 068 .39 932 .00 034 .39 967 43 18 .60 349 .99 965 .60 384 .39 616 .00 035 .39 651 42 19 .60 662 .99 964 .60 698 .39 302 .00 036 .39 338 41 20 8.60 973 9.99 964 8.61 009 1.38991 0.00 036 1.39 027 40 21 .61 282 .99 963 .61 319 .38 681 .00 037 .38 718 39 22 .61 589 .99 963 .61 626 .38 374 .00 037 .38411 38 23 .61 894 .99 962 .61 931 .38 069 .00 038 .38 106 37 24 .62 196 .99 962 .62 234 .37 766 .00 038 .37 804 36 25 8.62 497 9.99 961 8.62 535 1.37465 0.00 039 1.37503 35 26 .62 795 .99 961 .62 834 .37 166 .00 039 .37 205 34 27 .63 091 .99 960 .63 131 .36 869 .00 940 .36 909 33 28 .63 385 .99 960 .63 426 .36 574 .00 040 .36 615 32 29 .63 678 .99 959 .63 718 .36 282 .00 041 .36 322 31 30 8.63 968 9.99 959 8.64 009 1.35 991 0.00 041 1.36032 30 31 .64 256 .99 958 .64 298 .35 702 .00 042 .35 744 29 32 .64 543 .99 958 .64 585 .35415 .00 042 .35 457 28 33 .64 827 .99 957 .64 870 .35 130 .00 043 .35 173 27 34 .65 110 .99 956 .65 154 .34846 .00 044 .34 890 26 35 8.65 391 9.99 956 8.65 435 1.34 565 0.00 044 1.34 609 25 36 .65 670 .99 955 .65 715 .34 285 .00 045 .34 330 24 37 .65 947 .99 955 .65 993 .34 007 .00 045 .34 053 23 38 .66 223 .99 954 .66 269 .33 731 .00 046 .33 777 22 39 .66 497 .99 954 .66 543 .33 457 .00 046 .33 503 21 40 8.66 769 9.99 953 8.66 816 1.33 184 0.00 047 1.33231 20 41 .67 039 .99 952 .67 087 .32 913 .00 048 .32 961 19 42 .67 308 .99 952 .67 356 .32 644 .00 048 .32 692 18 43 .67 575 .99 951 .67 624 .32 376 .00 049 .32 425 17 44 .67 841 .99 951 .67 890 .32 110 .00 049 .32 159 16 45 8.68 104 9.99 950 8.68 154 1.31 846 0.00 050 1.31 896 15 46 .68 367 .99 949 .68417 .31 583 .00 051 .31 633 14 47 .68 627 .99 949 .68 678 .31 322 .00 051 .31 373 13 48 .68 886 .99 948 .68 938 .31 062 .00 052 .31 114 12 49 .69 144 .99 948 .69 196 .30 804 .00 052 .30 856 11 50 8.69 400 9.99 947 8.69 453 1.30 547 0.00 053 1.30 600 10 51 .69 654 .99 946 .69 708 .30 292 .00 054 .30 346 9 52 .69 907 .99 946 .69 962 .30 038 .00 054 .30 093 8 53 .70 159 .99 945 .70 214 .29 786 .00 055 .29 841 7 54 .70 409 .99 944 .70 465 .29 535 .00 056 .29 591 6 55 8.70 658 9.99 944 8.70 714 1.29 286 0.00 056 1.29 342 5 56 .70 905 .99 943 .70 962 .29 038 .00 057 .29 095 4 57 .71 151 .99 942 .71 208 .28 792 .00 058 .28849 3 58 .71 395 .99 942 .71 453 .28 547 .00 058 .28 605 2 59 .71 638 .99 941 .71 697 .28 303 .00 059 .28 362 1 60 8.71 880 9.99 940 8.71 940 1.28060 0.00 060 1.28 120 Cos Sin Cot 1 Tan Csc Sec ' 92 (272) (267) 87 Table 4. Trigonometric Logarithms 199 3 (183) (356) 176 ' Sin Cos Tan Cot Sec Csc 8.71 880 9.99 940 8.71 940 1.28 060 0.00 060 1.28 120 60 1 .72 120 .99 940 .72 181 .27 819 .00 060 .27 880 59 2 .72 359 .99 939 .72 420 .27 580 .00 061 .27 641 58 3 .72 597 .99 938 .72 659 .27 341 .00 062 .27 403 57 4 .72 834 .99 938 .72 896 .27 104 .00 062 .27 166 56 5 8.73 069 9.99 937 8.73 132 1.26 868 0.00 063 1.26931 55 6 .73 303 .99 936 .73 366 .26 634 .00 064 .26 697 54 7 .73 535 .99 936 .73 600 .26 400 .00 064 .26 465 53 8 .73 767 .99 935 .73832 .26 168 .00 065 .26 233 52 9 .73 997 .99 934 .74 063 .25 937 .00 066 .26 003 51 10 8.74 226 9.99 934 8.74 292 1.25 708 0.00 066 1.25774 50 11 .74 454 .99 933 .74 521 .25 479 .00 067 .25 546 49 12 .74 680 .99 932 .74 748 .25 252 .00 068 .25 320 48 13 .74 906 .99 932 .74 974 .25 026 .00 068 .25 094 47 14 .75 130 .99 931 .75 199 .24 801 .00 069 .24 870 46 15 8.75 353 9.99 930 8.75 423 1.24577 0.00 070 1.24 647 45 16 .75 575 .99 929 .75 645 .24 355 .00 071 .24 425 44 17 .75 795 .99 929 .75 867 .24 133 .00 071 .24 205 43 18 .76 015 .99 928 .76 087 .23 913 .00 072 .23 985 42 19 .76 234 .99 927 .76 306 .23 694 .00 073 .23 766 41 20 8.76 451 9.99 926 8.76 525 1.23 475 0.00 074 1.23 549 40 21 .76 667 .99 926 .76 742 .23 258 .00 074 .23 333 39 22 .76 883 .99 925 .76 958 .23 042 .00 075 .23 117 38 23 .77 097 .99 924 .77 173 .22 827 .00 076 .22 903 37 24 .77 310 .99 923 .77 387 .22 613 .00 077 .22 690 36 25 8.77 522 9.99 923 8.77 600 1.22 400 0.00 077 1.22 478 35 26 .77 733 .99 922 .77811 .22 189 .00 078 .22 267 34 27 .77 943 .99 921 .78 022 .21 978 .00 079 .22 057 33 28 .78 152 .99 920 .78 232 .21 768 .00 080 .21 848 32 29 .78 360 .99 920 .78 441 .21 559 .00 080 .21 640 31 30 8.78 568 9.99 919 8.78 649 1.21 351 0.00 081 1.21 432 30 31 .78 774 .99 918 .78855 .21 145 .00 082 .21 226 29 32 .78 979 .99 917 .79 061 .20 939 .00 083 .21 021 28 33 .79 183 .99 917 .79 266 .20 734 .00 083 .20 817 27 34 .79 386 .99 916 .79 470 .20 530 .00084 .20 614 26 35 8.79 588 9.99 915 8.79 673 1.20 327 0.00 085 1.20412 25 36 .79 789 .99 914 .79 875 .20 125 .00 086 .20211 24 37 .79 990 .99 913 .80 076 .19324 .00 087 .20 010 23 38 .80 189 .99 913 .80 277 .19 723 .00 087 .19811 22 39 .80 388 .99 912 .80 476 .19 524 .00 088 .19612 21 40 8.80 585 9.99911 8.80 674 1.19 326 0.00 089 1.19415 20 41 .80 782 .99 910 .80 872 .19 128 .00 090 .19218 19 42 .80 978 .99 909 .81 068 .18 932 .00 091 .19 022 18 43 .81 173 .99 909 .81 264 .18 736 .00 091 .18 827 17 44 .81 367 .99 908 .81 459 .18541 .00 092 .18 633 16 45 8.81 560 9.99 907 8.81 653 1.18347 0.00 093 1.18440 15 46 .81 752 .99 906 .81 846 .18 154 .00 094 .18 248 14 47 .81 944 .99 905 .82 038 .17 962 .00 095 .18 056 13 48 .82 134 .99 904 .82 230 .17 770 .00 096 .17 866 12 49 .82 324 .99 904 .82 420 .17 580 .00 096 .17676 11 50 8.82 513 9.99 903 8.82 610 1.17 390 0.00 097 1.17 487 10 51 .82 701 .99 902 .82 799 .17 201 .00 098 .17 299 9 52 .82 888 .99 901 .82 987 .17013 .00 099 .17112 8 53 .83 075 .99 900 .83 175 .16 825 .00 100 .16 925 7 54 .83 261 .99 899 .83361 .16 639 .00 101 .16 739 6 55 8.83 446 9.99 898 8.83 547 1.16453 0.00 102 1.16554 5 56 .83 630 .99 898 .83 732 .16 268 .00 102 .16 370 4 57 .83 813 .99 897 .83916 .16084 .00 103 .16 187 3 58 .83 996 .99 896 .84 100 .15 900 .00 104 .16 004 2 59 .84177 .99 895 .84282 .15718 .00 105 .15 823 1 60 8.84 358 9.99 894 8.84 464 1.15536 0.00 106 1.15 642 Cos Sin Cot Tan Csc Sec ' 93 (273) (266) 86 C 200 Table 4:. Trigonometric Logarithms 4 (184) (355) 175 ' Sin Cos Tan Cot Sec Csc 8.84 358 9.99 894 8.84 464 1.15 536 0.00 106 1.15 642 60 1 .84 539 .99 893 .84646 .15 354 .00 107 .15461 59 2 .84718 .99 892 .84826 .15 174 .00 108 .15 282 58 3 .84 897 .99 891 .85006 .14 994 .00 109 .15 103 57 4 .85 075 .99 891 .85 185 .14 815 .00 109 .14 925 56 5 8.85 252 9.99 890 8.85 363 1.14 637 0.00 110 1.14 748 55 6 .85 429 .99 889 .85 540 .14 460 .00 111 .14 571 54 7 .85 605 .99 888 .85 717 .14 283 .00 112 .14 395 53 8 .85 780 .99 887 .85 893 .14 107 .00 113 .14 220 52 9 .85 955 .99 886 .86 069 .13 931 .00 114 .14 045 51 10 8.86 128 9.99 885 8.86 243 1.13 757 0.00 115 1.13 872 50 11 .86 301 .99884 .86 417 .13 583 .00 116 .13 699 49 12 .86 474 .99 883 .86 591 .13409 .00 117 .13 526 48 13 .86 645 .99 882 .86 763 .13 237 .00 118 .13 355 47 14 .86 816 .99 881 .86 935 .13 065 .00 119 .13 184 46 15 8.86 987 9.99 880 8.87 106 1.12 894 0.00 120 1.13013 45 16 .87 156 .99 879 .87 277 .12 723 .00 121 .12844 '44 17 .87 325 .99 879 .87 447 .12 553 .00 121 .12 675 43 18 .87 494 .99 878 .87 616 .12384 .00 122 .12506 42 19 .87 661 .99 877 .87 785 .12215 .00 123 .12 339 41 20 8.87 829 9.99 876 8.87 953 1.12 047 0.00 124 1.12 171 40 21 .87 995 .99 875 .88 120 .11 880 .00 125 .12 005 39 22 .88 161 .99 874 .88 287 .11713 00 126 .11 839 38 23 .88 326 .99 873 .88 453 .11 547 .00 127 .11 674 37 24 .88 490 .99 872 .88 618 .11 382 .00 128 .11 510 36 25 8.88 654 9.99 871 8.88 783 1.11 217 0.00 129 1.11 346 35 26 .88 817 .99 870 .88 948 .11 052 .00 130 .11 183 34 27 .88 980 .99 869 .89 111 .10 889 .00131 .11 020 33 28 .89 142 .99 868 .89 274 .10 726 .00 132 .10 858 32 29 .89 304 .99 867 .89 437 .10563 .00 133 .10 696 31 30 8.89 464 9.99 866 8.89 598 1.10402 0.00 134 1.10536 30 31 .89 625 .99 865 .89 760 .10 240 .00 135 .10 375 29 32 .89784 .99 864 .89 920 .10 080 .00 136 .10216 28 33 .89 943 .99 863 .90 080 .09 920 .00 137 .10 057 27 34 .90 102 .99 862 .90 240 .09 760 .00 138 .09 898 26 35 8.90 260 9.99 861 8.90 399 1.09 601 0.00 139 1.09 740 25 36 .90417 .99 860 .90 557 .09 443 .00 140 .09 583 24 37 .90 574 .99 859 .90 715 .09 285 .00 141 .09 426 23 38 .90 730 .99 858 .90 872 .09 128 .00 142 .09 270 22 39 .90 885 .99 857 .91 029 .08 971 .00 143 .09 115 21 40 8.91 040 9.99 856 8.91 185 1.08815 0.00 144 1.08 960 20 41 .91 195 .99 855 .91 340 .08 660 .00 145 .08 805 19 42 .91 349 .99 854 .91 495 .08 505 .00 146 .08 651 18 43 .91 502 .99 853 .91 650 .08 350 .00 147 .08 498 17 44 .91 655 .99 852 .91 803 .08 197 .00 148 .08 345 16 45 8.91 807 9.99 851 8.91 957 1.08043 0.00 149 1.08 193 15 46 .91 959 .99 850 .92 110 .07 890 .00 150 .08 041 14 47 .92 110 .99 848 .92 262 .07 738 .00 152 .07 890 13 48 .92 261 .99 847 .92 414 .07 586 .00 153 .07 739 12 49 .92411 .99846 .92 565 .07 435 .00 154 .07 589 11 50 8.92 561 9.99 845 8.92 716 1.07 284 0.00 155 1.07 439 10 51 .92 710 .99 844 .92 866 .07 134 .00 156 .07 290 9 52 .92 859 .99 843 .93 016 .06984 .00 157 .07 141 8 53 .93 007 .99 842 .93 165 .06 835 .00 158 .06 993 7 54 .93 154 .99 841 .93 313 .06 687 .00 159 .06 846 6 55 8.93 301 9.99 840 8.93 462 1.06 538 0.00 160 1.06 699 5 56 .93 448 .99 839 .93 609 .06 391 .00 161 .06 552 4 57 .93 594 .99 838 .93 756 .06 244 .00 162 .06 406 3 58 .93 740 .99 837 .93 903 .06 097 .00 163 .06 260 2 59 .93 885 .99 836 .94 049 .05 951 .00 164 .06 115 1 60 8.94 030 9.99 834 8.94 195 1.05 805 0.00 166 1.05 970 Cos Sin Cot Tan Csc Sec ' 94 (274) (265) 85 Table 4. Trigonometric Logarithms 201 5 (185) (354) 174 ' Sin Cos Tan Cot Sec Csc 8.94 030 9.99 834 8.94 195 l.UO &U> 0.00 166 1.05 970 60 1 .94 174 .99 833 .94 340 .05 660 .00 167 .05 826 59 2 .94 317 .99 832 .94 485 .05 515 .00 168 .05 683 58 3 .94 461 .99 831 .94 630 .05 370 .00169 .05 539 57 4 .94 603 .99 830 .94 773 .05227 .00 170 .05 397 56 5 8.94 746 9.99 829 8.94 917 1.05083 0.00 171 1.05 254 55 6 .94 887 .99 828 .95 060 .04 940 .00 172 .05 113 54 7 .95 029 .99 827 .95 202 .04798 .00 173 .04 971 53 8 .95 170 .99 825 .95344 .04 656 .00 175 .04 830 52 9 .95 310 .99 824 .95 486 .04 514 .00 176 .04 690 51 10 8.95 450 9.99 823 8.95 627 1.04 373 0.00 177 1.04 550 50 11 .95 589 .99 822 .95 767 .04233 .00 178 .04411 49 12 .95 728 .99 821 .95 908 .04 092 .00 179 .04 272 48 13 .95 867 .99 820 .96 047 .03 953 .00 180 .04 133 47 14 .96 005 .99 819 .96 187 .03 813 .00181 .03 995 46 15 8.96 143 9.99 817 8.96 325 1.03 675 0.00 183 1.03 857 45 16 .96 280 .99 816 .96 464 .03 536 .00 184 .03 720 44 17 .96 417 .99 815 .96 602 .03 398 .00 185 .03 583 43 18 .96 553 .99 814 .96 739 .03 261 .00 186 .03 447 42 19 .96 689 .99 813 .96 877 .03 123 .00 187 .03311 41 20 8.96 825 9.99 812 8.97 013 1.02987 0.00 188 1.03 175 40 21 .96 960 .99 810 .97 150 .02 850 .00 190 .03 040 39 22 .97 095 .99 809 .97 285 .02 715 .00 191 .02 905 38 23 .97 229 .99 808 .97 421 .02 579 .00 192 .02 771 37 24 .97 363 .99 807 .97 556 .02 444 .00 193 .02 637 36 25 8.97 496 9.99 806 8.97 691 1.02309 0.00 194 1.02 504 35 26 .97 629 .99 804 .97 825 .02 175 .00 196 .02 371 34 27 .97 762 .99 803 .97 959 .02 041 .00 197 .02 238 33 28 .97 894 .99 802 .98 092 .01 908 .00 198 .02 106 32 29 .98 026 .99 801 .98 225 .01 775 .00 199 .01 974 31 30 8.98 157 9.99 800 8.98 358 1.01 642 0.00 200 1.01 843 30 31 .98 288 .99 798 .98 490 .01 510 .00 202 .01 712 29 32 .98419 .99 797 .98 622 .01 378 .00 203 .01 581 28 33 .98549 .99 796 .98 753 .01 247 .00 204 .01 451 27 34 .98 679 .99 795 .98884 .01 116 .00 205 .01 321 26 35 8.98 808 9.99 793 8.99 015 1.00985 0.00 207 1.01 192 25 36 .98 937 .99 792 .99 145 .00 855 .00 208 .01 063 24 37 .99066 .99 791 .99 275 .00 725 .00 209 .00934 23 38 .99 194 .99 790 .99 405 .00 595 .00 210 .00 806 22 39 .99 322 .99 788 .99 534 .00 466 .00 212 .00678 21 40 8.99 450 9.99 787 8.99 662 1.00338 0.00 213 1.00550 20 41 .99 577 .99 786 .99 791 .00 209 .00 214 .00 423 19 42 .99 704 .99 785 .99 919 .00 081 .00215 .00 296 18 43 .99 830 .99 783 9.00 046 0.99 954 .00217 .00 170 17 44 .99 956 .99 782 .00 174 .99 826 .00 218 .00 044 16 45 9.00 082 9.99 781 9.00 301 0.99 699 0.00 219 0.99 918 15 46 .00207 .99 780 .00 427 .99 573 .00 220 .99 793 14 47 .00 332 .99 778 .00 553 .99 447 .00 222 .99 668 13 48 .00 456 .99 777 .00 679 .99 321 .00 223 .99 544 12 49 .00 581 .99 776 .00 805 .99 195 .00 224 .99419 11 50 9.00 704 9.99 775 9.00 930 0.99 070 0.00 225 0.99 296 10 51 .00828 .99 773 .01 055 .98 945 .00 227 .99 172 9 52 .00 951 .99 772 .01 179 .98 821 .00 228 .99 049 8 53 .01 074 .99 771 .01 303 .98 697 .00 229 .98 926 7 54 .01 196 .99 769 .01 427 .98 573 .00231 .98 804 6 55 9.01 318 9.99 768 9.01 550 0.98 450 0.00 232 0.98 682 5 56 .01 440 .99 767 .01 673 .98 327 .00 233 .98 560 4 57 .01 561 .99 765 .01 796 .98 204 .00 235 .98 439 3 58 .01 682 .99764 .01 918 .98 082 .00 236 .98318 2 59 .01 803 .99 763 .02040 .97 960 .00 237 .98 197 1 60 9.01 923 9.99 761 9.02 162 0.97 838 0.00 239 0.98 077 Cos Sin Cot Tan Csc Sec ' 95 (275) (264) 84 202 Table 4. Trigonometric Logarithms 6 (186) (353) 173 C ' Sin Cos Tan Cot Sec Csc 9.01 923 9.99 761 9.02 162 0.97 838 0.00 239 0.98 077 60 1 .02 043 .99 760 .02 283 .97 717 00240 .97 957 59 2 .02 163 .99 759 .02 404 .97 596 .00 241 .97 837 58 3 .02 283 .99 757 .02 525 .97 475 .00 243 .97 717 57 4 .02 402 .99 756 .02 645 .97 355 .00 244 .97 598 56 5 9.02 520 9.99 755 9.02 766 0.97 234 0.00 245 0.97 480 55 6 .02 639 .99 753 .02 885 .97 115 .00 247 .97 361 54 7 .02 757 .99 752 .03 005 .96 995 .00 248 .97 243 53 8 .02 874 .99 751 .03 124 .96 876 .00 249 .97 126 52 9 .02 992 .99 749 .03 242 .96 758 .00 251 .97 008 51 10 9.03 109 9.99 748 9.03 361 0.96 639 0.00 252 0.96 891 50 11 .03 226 .99 747 .03 479 .96 521 .00 253 .96 774 49 12 .03 342 .99 745 .03 597 .96 403 .00 255 .96 658 48 13 .03 458 .99 744 .03 714 .96 286 .00 256 .96 542 47 14 .03 574 .99 742 .03 832 .96 168 .00 258 .96 426 46 15 9.03 690 9.99 741 9.03 948 0.96 052 0.00 259 0.96 310 45 16 .03 805 .99 740 .04 065 .95 935 .00 260 .96 195 44 17 .03 920 .99 738 .04 181 .95 819 .00 262 .96 080 43 18 .04 034 .99 737 .04 297 .95 703 .00 263 .95 966 42 19 .04 149 .99 736 .04 413 .95 587 .00 264 .95 851 41 20 9.04 262 9.99 734 9.04 528 0.95 472 0.00 266 0.95 738 40 21 .04 376 .99 733 .04 643 .95 357 .00 267 .95 624 39 22 .04 490 .99 731 .04 758 .95 242 .00 269 .95 510 38 23 .04 603 .99 730 .04 873 .95 127 .00 270 .95 397 37 24 .04 715 .99 728 .04 987 .95 013 .00 272 .95 285 36 25 9.04 828 9.99 727 9.05 101 0.94 899 0.00 273 0.95 172 35 26 .04 940 .99 726 .05 214 .94 786 .00 274 .95 060 34 27 .05 052 .99 724 .05 328 .94 672 .00 276 .94 948 33 28 .05 164 .99 723 .05 441 . 94559 .00 277 .94 836 32 29 .05 275 .99 721 .05 553 .94 447 .00 279 .94 725 31 30 9.05 386 9.99 720 9.05 666 0.94 334 0.00 280 0.94 614 30 31 .05 497 .99 718 .05 778 .94 222 .00 282 .94 503 29 32 .05 607 .99 717 .05 890 .94 110 .00 283 .94 393 28 33 .05 717 .99 716 .06 002 .93 998 .00 284 .94 283 27 34 .05 827 .99 714 .06 113 .93 887 .00 286 .94 173 26 35 9.05 937 9.99 713 9.06 224 0.93 776 0.00 287 0.94 063 25 36 .06 046 .99 711 .06 335 .93 665 .00 289 .93 954 24 37 .06 155 .99 710 .06 445 .93555 .00 290 .93 845 23 38 .06 264 .99 708 .06 556 .93 444 .00 292 .93 736 22 39 .06 372 .99 707 .06 666 .93 334 .00 293 .93 628 21 40 9.06 481 9.99 705 9.06 775 0.93 225 0.00 295 0.93 519 20 41 .06 589 .99 704 .06 885 .93 115 .00 296 .93411 19 42 .06 696 .99 702 .06 994 .93 006 .00 298 .93 304 18 43 .06 804 .99 701 .07 103 .92 897 .00 299 .93 196 17 44 .06911 .99 699 .07211 .92 789 .00 301 .93 089 16 45 9.07 018 9.99 698 9.07 320 0.92 680 0.00 302 0.92 982 15 46 .07 124 .99 696 .07 428 .92 572 .00 304 .92 876 14 47 .07 231 .99 695 .07 536 .92 464 .00 305 .92 769 13 48 .07 337 .99 693 .07 643 .92 357 00.307 .92 663 12 49 .07 442 .99 692 .07 751 .92 249 .00 308 .92 558 11 50 9.07 548 9.99 690 9.07 858 0.92 142 0.00 310 0.92 452 10 51 .07 653 .99 689 .07 964 .92 036 .00311 .92 347 9 52 .07 758 .99 687 .08 071 .91 929 .00 313 .92 242 8 53 .07 863 .99 686 .08 177 .91 823 .00 314 .92 137 7 54 .07 968 .99 684 .08 283 .91 717 .00316 .92 032 6 55 9.08 072 9.99 683 9.08 389 0.91 611 0.00317 0.91 928 5 56 .08 176 .99 681 .08 495 .91 505 .00 319 .91 824 4 57 .08 280 .99 680 .08 600 .91 400 .00 320 .91 720 3 58 .08 383 .99 678 .08 705 .91 295 .00 322 .91 617 2 59 .08 486 .99 677 .08 810 .91 190 .00 323 .91 514 1 60 9.08 589 9.99 675 9.08 914 0.91 086 0.00 325 0.91411 Cos Sin Cot Tan Csc Sec ' (276) (263) 83 Table 4. Trigonometric Logarithms 203 7 (187) (352) 172 C / Sin Cos Tan Cot Sec Csc 9.08 589 9.99 675 9.08 914 0.91 086 0.00 325 0.91 411 60 1 .08 692 .99 674 .09 019 .90 981 .00326 .91 308 59 2 .08 795 .99 672 .09 123 .90 877 .00328 .91 205 58 3 .08 897 .99 670 .09 227 .90 773 .00 330 .91 103 57 4 .08 999 .99 669 .09 330 .90 670 .00331 .91 001 56 5 9.09 101 9.99 667 9.09 434 0.90 566 0.00 333 0.90 899 55 6 .09 202 .99 666 .09 537 .90 463 .00334 .90 798 54 7 .09 304 .99 664 .09 640 .90 360 .00 336 .90 696 53 8 .09 405 .99 663 .09 742 .90 258 .00 337 .90 595 52 9 .09 506 .99 661 .09845 .90 155 .00339 .90 494 51 10 9.09 606 9.99 659 9.09 947 0.90 053 0.00 34 1 0.90 394 50 11 .09 707 .99 658 .10 049 .89 951 .00 342 .90 293 49 12 .09 807 .99 656 .10 150 .89 850 .00344 .90 193 48 13 .09 907 .99 655 .10 252 .89 748 .00 345 .90 093 47 14 .10 006 .99 653 .10 353 .89647 .00347 .89 994 46 15 9.10 106 9.99 651 9.10 454 0.89 546 0.00 349 0.89 894 45 16 .10 205 .99 650 .10 555 .89 445 .00 350 .89 795 44 17 .10304 .99648 .10 656 .89 344 .00352 .89 696 43 18 .10 402 .99 647 .10 756 .89 244 .00 353 .89 598 42 19 .10 501 .99645 .10 856 .89 144 .00 355 .89 499 41 20 '9. 10 599 9.99 643 9.10 956 0.89 044 0.00 357 0.89 401 40 21 .10 697 .99642 .11 056 .88 944 .00 358 .89 303 39 22 .10 795 .99640 .11 155 .88845 .00 360 .89 205 38 23 .10 893 .99 638 .11 254 .88 746 .00 362 .89 107 37 24 .10 990 .99 637 .11 353 .88647 .00 363 .89 010 36 25 9.11 087 9.99 635 9.11 452 0.88 548 0.00 365 0.88 913 35 26 .11 184 .99 633 .11551 .88 449 .00 367 .88 816 34 27 .11 281 .99 632 .11649 .88 351 .00 368 .88 719 33 28 .11 377 .99 630 .11 747 .88 253 .00 370 .88 623 32 29 .11 474 .99 629 .11 845 .88 155 .00 371 .88 526 31 30 9.11 570 9.99 627 9.11 943 0.88 057 0.00 373 0.88 430 30 31 .11 666 .99 625 .12 040 .87 960 .00375 .88 334 29 32 .1.1 761 .99 624 .12 138 .87 862 .00 376 .88 239 28 33 .11 857 .99 622 .12 235 .87 765 .00 378 .88 143 27 34 .11 952 .99 620 .12 332 .87 668 .00 380 .88048 26 35 9.12 047 9.99 618 9.12 428 0.87 572 0.00 382 0.87 953 25 36 .12 142 .99 617 .12 525 .87 475 .00383 .87 858 24 37 .12 236 .99 615 .12 621 .87 379 .00 385 .87 764 23 38 .12331 .99 613 .12717 .87 283 .00 387 .87 669 22 39 .12 425 .99 612 .12813 .87 187 .00388 .87 575 21 40 9.12519 9.99 610 9.12 909 0.87 091 0.00 390 0.87 481 20 41 .12612 .99 608 .13004 .86 996 .00 392 .87 388 19 42 .12 706 .99 607 .13 099 .86 901 .00 393 .87 294 18 43 .12 799 .99 605 .13 194 .86 806 .00 395 .87 201 17 44 .12 892 .99 603 .13 289 .86711 .00 397 .87 108 16 45 9.12 985 9.99 601 9.13 384 0.86 616 0.00 399 0.87 015 15 46 .13 078 .99 600 .13 478 .86 522 .00 400 .86 922 14 47 .13 171 .99 598 .13 573 .86 427 .00402 .86 829 13 48 .13 263 .99 596 .13 667 .86 333 .00 404 .86 737 12 49 .13 355 .99 595 .13 761 .86 239 .00405 .86 645 11 50 9.13 447 9.99 593 9.13 854 0.86 146 0.00 407 0.86 553 10 51 .13 539 .99 591 .13 948 .86 052 .00 409 .86 461 9 52 .13 630 .99589 .14041 .85 959 .00411 .86 370 8 53 .13 722 .99 588 .14 134 .85 866 .00412 .86 278 7 54 .13 813 .99 586 .14 227 .85 773 .00 414 .86 187 6 55 9.13 904 9.99 584 9.14 320 0.85 680 0.00416 0.86 096 5 56 .13 994 .99 582 .14412 .85 588 .00 418 .86 006 4 57 .14 085 .99 581 .14 504 .85 496 .00 419 .85 915 3 58 .14 175 .99 579 .14 597 .85 403 .00 421 .85825 2 59 .14 266 .99 577 .14 688 .85 312 .00423 .85 734 1 60 9.14 356 9.99 575 9.14 780 0.85 220 0.00 425 0.85 644 Cos Sin Cot Tan Csc Sec ' 97 (277) (262) 82 { 204 Table 4. Trigonometric Logarithms 8 (188) (351) 171 C Sin Cos Tan Cot Sec Csc 9.14 356 9.99 575 9.14 780 0.85 220 0.00 425 0.85 644 60 1 .14445 .99 574 .14872 .85 128 .00 426 .85 555 59 2 .14 535 .99 572 .14 963 .65 037 .00 428 .85 465 58 3 .14 624 .99 570 .15 054 .84 946 .00 430 .85 376 57 4 .14 714 .99 568 .15 145 .84 855 .00 432 .85 286 56 5 9.14 803 9.99 566 9.15 236 0.84 764 0.00 434 0.85 197 55 6 .14 891 .99 565 .15327 .84 673 .00 435 .85 109 54 7 .14 980 .99 563 .15417 .84 583 .00 437 .85 020 53 8 .15069 .99 561 .15 508 .84492 .00 439 .84931 52 9 .15 157 .99 559 .15 598 .84 402 .00 441 .84 843 51 10 9.15 245 9.99 557 9.15 688 0.84 312 0.00 443 0.84 755 50 11 .15 333 .99 556 .15 777 .84 223 .00 444 .84 667 49 12 .15421 .99 554 .15 867 .84 133 .00 446 .84579 48 13 .15508 .99 552 .15 956 .84 044 .00 448 .84 492 47 14 .15 596 .99 550 .16 046 .83 954 .00 450 .84404 46 15 9.15 683 9.99 548 9.16 135 0.83 865 0.00 452 0.84 317 45 16 .15 770 .99 546 .16 224 .83 776 .00 454 .84230 44 17 .15857 .99 545 .16312 .83 688 .00 455 .84 143 43 18 .15944 .99 543 .16401 .83 599 .00 457 .84056 42 19 .16 030 .99 541 .16 489 .83511 .00 459 .83 970 41 20 9.16 116 9.99 539 9.16 577 0.83 423 0.00 461 0.83 884 40 21 .16 203 .99 537 .16 665 .83 335 .00 463 .83 797 39 22 .16289 .99 535 .16 753 .83 247 .00 465 .83 711 38 23 .16 374 .99 533 .16 841 .83 159 .00 467 .83 626 37 24 .16460 .99 532 .16 928 .83 072 .00 468 .83 540 36 25 9.16 545 9.99 530 9.17016 0.82 984 0.00 470 0.83 455 35 26 .16631 .99 528 .17 103 .82 897 .00 472 .83 369 34 27 .16716 .99 526 .17 190 .82 810 .00 474 .83 284 33 28 .16801 .99524 .17277 .82 723 .00 476 .83199 32 29 .16 886 .99 522 .17363 .82 637 .00 478 .83 114 31 30 9.16 970 9.99 520 9.17450 0.82 550 0.00 480 0.83 030 30 31 .17 055 .99 518 .17 536 .82 464 .00 482 .82 945 29 32 .17 139 .99 517 17622 .82 378 .00 483 .82 861 28 33 .17 223 .99 515 .17 708 .82 292 .00 485 .82 777 27 34 .17 307 .99 513 .17 794 .82 206 .00 487 .82 693 26 35 9.17391 9.99511 9.17 880 0.82 120 0.00 489 0.82 609 25 36 .17474 .99 509 .17 965 .82 035 .00 491 .82 526 24 37 .17 558 .99 507 .18051 .81 949 .00 493 .82 442 23 38 .17 641 .99 505 .18 136 .81 864 .00 495 .82 359 22 39 .17 724 .99 503 .18221 .81 779 .00 497 .82 276 21 40 9.17 807 9.99 501 9.18306 0.81 694 0.00 499 0.82 193 20 41 .17 890 .99 499 .18391 .81 609 .00 501 .82 110 19 42 .17 973 .99 497 .18475 .81 525 .00 503 .82 027 18 43 .18 055 .99 495 .18 560 .81 440 .00 505 .81 945 17 44 .18 137 .99 494 .18 644 .81 356 .00 506 .81 863 16 45 9.18 220 9.99 492 9.18 728 0.81 272 0.00 508 0.81 780 15 46' .18 302 .99 490 .18812 .81 188 .00 510 .81 698 14 47 .18 383 .99 488 .18896 .81 104 .00 512 .81 617 13 48 .18465 .99 486 .18979 .81 021 .00 514 .81 535 12 49 .18 547 .99484 .19 063 .80 937 .00 516 .81 453 11 50 9.18 628 9.99 482 9.19 146 0.80 854 0.00 518 0.81 372 10 51 .18 709 .99 480 .19 229 .80 771 .00 520 .81 291 9 52 .18 790 .99 478 .19312 .80 688 .00 522 .81 210 8 53 .18871 .99 476 .19 395 .80 605 .00 524 .81 129 7 54 .18 952 .99 474 .19478 .80 522 .00 526 .81 048 6 55 9.19 033 9.99 472 9.19 561 0.80 439 0.00 528 0.80 967 5 56 .19 113 .99 470 .19 643 .80 357 .00 530 .80 887 4 57 .19 193 .99 468 .19 725 .80 275 .00 532 .80 807 3 58 .19 273 .99 466 .19 807 .80 193 .00 534 .80 727 2 59 .19 353 .99 464 .19 889 .80 111 .00 536 .80 647 1 60 9.19433 9.99 462 9.19971 0.80 029 0.00 538 0.80 567 Cos Sin Cot Tan Csc Sec / 98 (278) (261) 81 c Table 4. Trigonometric Logarithms 205 9 (189) (350) 170 C ' Sin Cos Tan Cot Sec Csc 9.19 433 9.99 462 9.19971 0.80 029 0.00 538 0.80 567 60 1 .19513 .99 460 .20 053 .79 947 .00 540 .80 487 59 2 .19 592 .99 458 .20 134 .79 866 .00542 .80 408 58 3 .19 672 .99 456 .20 216 .79 784 .00544 .80 328 57 4 .19 751 .99 454 .20 297 .79 703 .00546 .80 249 56 5 9.19 830 9.99 452 9.20 378 0.79 622 0.00 548 0.80 170 55 6 .19 909 .99 450 .20 459 .79 541 .00 550 .80 091 54 7 .19 988 .99 448 .20 540 .79 460 .00 552 .80 012 53 8 .20 067 .99 446 .20 621 .79 379 .00 554 .79 933 52 9 .20 145 .99 444 .20 701 .79 299 .00 556 .79 855 51 10 9.20 223 9.99 442 9.20 782 0.79 218 0.00 558 0.79 777 50 11 .20 302 .99 440 .20 862 .79 138 .00 560 .79 698 49 12 .20 380 .99 438 .20 942 .79 058 .00562 .79 620 48 13 .20 458 .99 436 .21 022 .78 978 .00 564 .79 542 47 14 .20 535 .99 434 .21 102 .78 898 .00 566 .79 465 46 15 9.20 613 9.99 432 9.21 182 0.78 818 0.00 568 0.79 387 45 16 .20 691 .99 429 .21 261 .78 739 .00 571 .79 309 44 17 .20 768 .99 427 .21 341 .78 659 .00573 .79 232 43 18 .20845 .99 425 .21 420 .78 580 .00 575 .79 155 42 19 .20 922 .99 423 .21 499 .78 501 .00 577 .79 078 41 20 9.20 999 9.99 421 9.21 578 0.78 422 0.00 579 0.79 001 40 21 .21 076 .99419 .21 657 .78 343 .00 581 .78 924 39 22 .21 153 .99 417 .21 736 .78 264 .00 583 .78847 38 23 .21 229 .99 415 .21 814 .78 186 .00 585 .78 771 37 24 .21 306 .99 413 .21 893 .78 107 .00 587 .78 694 36 25 9.21 382 9.99411 9.21 971 0.78 029 0.00 589 0.78 618 35 26 .21 458 .99 409 .22 049 .77 951 .00 591 .78 542 34 27 .21 534 .99 407 .22 127 .77 873 .00593 .78 466 33 28 .21 610 .99 404 .22 205 .77 795 .00 596 .78 390 32 29 .21 685 .99 402 .22 283 .77 717 .00 598 .78 315 31 30 9.21 761 9.99 400 9.22 361 0.77 639 0.00 600 0.78 239 30 31 .21 836 .99 398 .22 438 .77 562 .00 602 .78 164 29 32 .21 912 .99 396 .22 516 .77 484 .00 604 .78 088 28 33 .21 987 .99 394 .22 593 .77 407 .00 606 .78 013 27 34 .22 062 .99 392 .22 670 .77 330 .00 608 .77 938 26 35 9.22 137 9.99 390 9.22 747 0.77 253 0.00 610 0.77 863 25 36 .22211 .99 388 .22 824 .77 176 .00 612 .77 789 24 37 .22 286 .99 385 .22 901 .77 099 .00 615 .77 714 23 38 .22 361 .99 383 .22 977 .77 023 .00617 .77 639 22 39 .22 435 .99 381 .23 054 .76 946 .00619 .77 565 21 40 9.22 509 9.99 379 9.23 130 0.76 870 0.00 621 0.77 491 20 41 .22 583 .99 377 .23 206 .76 794 .00 623 .77417 19 42 .22 657 .99 375 .23 283 .76 717 .00 625 .77 343 18 43 .22 731 .99 372 .23 359 .76 641 .00 628 .77 269 17 44 .22 805 .99 370 .23 435 .76 565 .00 630 .77 195 16 45 9.22 878 9.99 368 9.23 510 0.76 490 0.00 632 0.77 122 15 46 .22 952 .99 366 .23 586 .76 414 .00 634 .77 048 14 47 .23 025 .99364 .23 661 .76 339 .00636 .76 975 13 48 .23 098 .99 362 .23 737 .76 263 .00 638 .76 902 12 49 .23 171 .99 359 .23 812 .76 188 .00641 .76 829 11 50 9.23 244 9.99 357 9.23 887 0.76 113 0.00 643 0.76 756 10 51 .23 317 .99 355 .23 962 .76 038 .00645 .76 683 9 52 .23 390 .99 353 .24 037 .75 963 .00647 .76 610 8 53 .23 462 .99 351 .24 112 .75 888 .00 649 .76 538 7 54 .23 535 .99 348 .24 186 .75 814 .00652 .76 465 6 55 9.23 607 9.99 346 9.24 261 0.75 739 0.00 654 0.76 393 5 56 .23 679 .99 344 .24 335 .75 665 .00 656 .76 321 4 57 .23 752 .99 342 .24 410 .75 590 .00 658 .76 248 3 58 .23 823 .99 340 .24484 .75 516 .00 660 .76 177 2 59 .23 895 .99 337 .24 558 .75 442 .00663 .76 105 1 60 9.23 967 9.99 335 9.24 632 0.75 368 0.00 665 0.76 033 Cos Sin Cot Tan Csc Sec 99 (279) (260) 80 C 206 Table 4:. Trigonometric Logarithms 10 (190) (349) 169 C ' Sin Cos Tan Cot Sec Csc 9.23 967 9.99 335 9.24 632 0.75 368 0.00 665 0.76 033 60 1 .24 039 .99 333 .24 706 .75 294 .00 667 .75 961 59 2 .24 110 .99 331 .24 779 .75 221 .00 669 .75 890 58 3 .24 181 .99 328 .24 853 .75 147 .00 672 .75 819 57 4 .24 253 .99 326 .24 926 .75 074 .00 674 .75 747 56 5 9.24 324 9.99 324 9.25 000 0.75 000 0.00 676 0.75 676 55 6 .24 395 .99 322 .25 073 .74 927 .00 678 .75 605 54 7 .24 466 .99 319 .25 146 .74 854 .00 681 .75 534 53 8 .24 536 .99317 .25 219 .74 781 .00 683 .75 464 52 9 .24 607 .99 315 .25 292 .74 708 .00 685 .75 393 51 10 9.24 677 9.99 313 9.25 365 0.74 635 0.00 687 0.75 323 50 11 .24 748 .99 310 .25 437 .74 563 .00 690 .75 252 49 12 .24 818 .99 308 .25 510 .74 490 .00 692 .75 182 48 13 .24 888 .99 306 .25 582 .74 418 .00 694 .75 112 47 14 .24 958 .99 304 .25 655 .74 345 .00 696 .75 042 46 15 9.25 028 9.99 301 9.25 727 0.74 273 0.00 699 0.74 972 45 16 .25 098 .99 299 .25 799 .74 201 .00 701 .74 902 44 17 .25 168 .99 297 .25 871 .74 129 .00 703 .74 832 43 18 .25 237 .99 294 .25 943 .74 057 .00 706 .74763 42 19 .25 307 .99 292 .26 015 .73 985 .00 708 .74 693 41 20 9.25 376 9.99 290 9.26 086 0.73 914 0.00 710 0.74 624 40 21 .25 445 .99 288 .26 158 .73 842 .00 712 .74 555 39 22 .25 514 .99 285 .26 229 .73 771 .00 715 .74 486 38 23 .25 583 .99 283 .26 301 .73 699 .00 717 .74 417 37 24 .25 652 .99 281 .26 372 .73 628 .00 719 .74 348 36 25 9.25 721 9.99 278 9.26 443 0.73 557 0.00 722 0.74 279 35 26 .25 790 .99 276 .26 514 .73 486 .00 724 .74 210 34 27 .25 858 .99 274 .26 585 .73415 .00 726 .74 142 33 28 .25 927 .99 271 .26 655 .73 345 .00 729 .74 073 32 29 .25 995 .99 269 .26 726 .73 274 .00 731 .74 005 31 30 9.26 063 9.99 267 9.26 797 0.73 203 0.00 733 0.73 937 30 31 .26 131 .99 264 .26 867 .73 133 .00 736 .73 869 29 32 .26 199 .99 262 .26 937 .73 063 .00 738 .73 801 28 33 .26 267 .99 260 .27 008 .72 992 .00 740 .73 733 27 34 .26 335 .99 257 .27 078 .72 922 .00 743 .73 665 26 35 9.26 403 9.99 255 9.27 148 0.72 852 0.00 745 0.73 597 25 36 .26 470 .99 252 .27 218 .72 782 .00 748 .73 530 24 37 .26 538 .99 250 .27 288 .72 712 .00 750 .73 462 23 38 .26 605 .99 248 .27 357 .72643 .00 752 .73 395 22 39 .26 672 .99 245 .27 427 .72 573 .00 755 .73 328 21 40 9.26 739 9.99 243 9.27 496 0.72 504 0.00 757 0.73 261 20 41 .26 806 .99 241 .27 566 .72 434 .00 759 .73 194 19 42 .26 873 .99 238 .27 635 .72 365 .00 762 .73 127 18 43 .26 940 .99 236 .27 704 .72 296 .00 764 .73 060 17 44 .27 007 .99 233 .27 773 .72 227 .00 767 .72 993 16 45 9.27 073 9.99 231 9.27 842 0.72 158 0.00 769 0.72 927 15 46 .27 140 .99 229 .27911 .72 089 .00 771 .72 860 14 47 .27 206 .99 226 .27 980 .72 020 .00 774 .72 794 13 48 .27 273 .99 224 .28 049 .71 951 .00 776 .72 727 12 49 .27 339 .99 221 .28 117 .71 883 .00 779 .72 661 11 50 9.27 405 9.99 219 9.28 186 0.71 814 0.00 781 0.72 595 10 51 .27 471 .99 217 .28 254 .71 746 .00 783 .72 529 9 52 .27 537 .99 214 .28 323 .71 677 .00 786 .72 463 8 53 .27 602 .99 212 .28 391 .71 609 .00 788 .72 398 7 54 .27 668 .99 209 .28 459 .71 541 .00 791 .72 332 6 55 9.27 734 9.99 207 9.28 527 0.71 473 0.00 793 0.72 266 5 56 .27 799 .99 204 .28 595 .71 405 .00 796 .72 201 4 57 .27 864 .99 202 .28 662 .71 338 .00 798 .72 136 3 58 .27 930 .99 200 .28 730 .71 270 .00 800 .72 070 2 59 .27 995 .99 197 .28 798 .71 202 .00 803 .72 005 1 60 9.28 060 9.99 195 9.28 865 0.71 135 0.00 805 0.71 940 Cos Sin Cot Tan Csc Sec ' 100 (280) (259) 79 Table 4. Trigonometric Logarithms 20< 11 (191) (348) 168 C ' Sin Cos Tan Cot Sec Csc 9.28 060 9.99 195 9.28 865 0.71 135 0.00 805 0.71 940 60 1 .28 125 .99 192 .28 933 .71 067 .00 808 .71 875 59 2 .28 190 .99 190 .29 000 .71 000 .00 810 .71 810 58 3 .28 254 .99 187 .29 067 .70 933 .00 813 .71 746 57 4 .28 319 .99 185 .29 134 .70 866 .00 815 .71 681 56 5 9.28 384 9.99 182 9.29 201 0.70 799 0.00 818 0.71 616 55 6 .28 448 .99 180 .29 268 .70 732 .00 820 .71 552 54 7 .28 512 .99 177 .29 335 .70 665 .00 823 .71 488 53 8 .28 577 .99 175 .29 402 .70 598 .00 825 .71 423 52 9 .28 641 .99 172 .29 468 .70 532 .00 828 .71 359 51 10 9.28 705 9.99 170 9.29 535 0.70 465 0.00 830 0.71 295 50 11 .28 769 .99 167 .29 601 .70 399 .00 833 .71 231 49 12 .28 833 .99 165 .29 668 .70 332 .00835 .71 167 48 13 .28 896 .99 162 .29 734 .70 266 .00 838 .71 104 47 14 .28 960 .99 160 .29 800 .70 200 .00840 .71 040 46 15 9.29 024 9.99 157 9.29 866 0.70 134 0.00 843 0.70 976 45 16 .29 087 .99 155 .29 932 .70 068 .00845 .70 913 44 17 .29 150 .99 152 .29 998 .70 002 .00 848 .70 850 43 18 .29 214 .99 150 .30 064 .69 936 .00 850 .70 786 42 19 .29 277 .99 147 .30 130 .69 870 .00853 .70 723 41 20 9.29 340 9.99 145 9.30 195 0.69 805 0.00 855 0.70 660 40 21 .29 403 .99 142 .30 261 .69 739 .00858 .70 597 39 22 .29 466 .99 140 .30 326 .69 674 .00 860 .70 534 38 23 .29 529 .99 137 .30 391 .69 609 .00 863 .70 471 37 24 .29 591 .99 135 .30 457 .69 543 .00865 .70 409 36 25 9.29 654 9.99 132 9.30 522 0.69 478 0.00 868 0.70 346 35 26 .29 716 .99 130 .30 587 .69 413 .00 870 .70 284 34 27 .29 779 .99 127 .30 652 .69 348 .00 873 .70 221 33 28 .29 841 .99 124 .30 717 .69 283 .00 876 .70 159 32 29 .29 903 .99 122 .30 782 .69 218 .00 878 .70 097 31 30 9.29 966 9.99 119 9.30 846 0.69 154 0.00 881 0.70 034 30 31 .30 028 .99 117 .30911 .69 089 .00 883 .69 972 29 32 .30 090 .99 114 .30 975 .69 025 .00 886 .69 910 28 33 .30 151 .99 112 .31 040 .68 960 .00 888 .69849 27 34 .30 213 .99 109 .31 104 .68 896 .00 891 .69 787 26 35 9.30 275 9.99 106 9.31 168 0.68 832 0.00 894 0.69 725 25 36 .30 336 .99 104 .31 233 .68 767 .00 896 .69 664 24 37 .30 398 .99 101 .31 297 .68 703 .00 899 .69 602 23 38 .30 459 .99 099 .31 361 .68 639 .00 901 .69 541 22 39 .30 521 .99 096 .31 425 .68 575 .00904 .69 479 21 40 9.30 582 9.99 093 9.31 489 0.68511 0.00 907 0.69 418 20 41 .30 643 .99 091 .31 552 .68 448 .00 909 .69 357 19 42 .30 704 .99 088 .31 616 .68384 .00 912 .69 296 18 43 .30 765 .99 086 .31 679 .68 321 .00 914 .69 235 17 44 .30 826 .99 083 .31 743 .68 257 .00917 .69 174 16 45 9.30 887 9.99 080 9.31 806 0.68 194 0.00 920 0.69 113 15 46 .30 947 .99 078 .31 870 .68 130 .00 922 .69 053 14 47 .31 008 .99 075 .31 933 .68 067 .00 925 .68 992 13 48 .31 068 .99 072 .31 996 .68 004 .00 928 .68 932 12 49 .31 129 .99 070 .32 059 .67 941 .00930 .68 871 11 50 9.31 189 9.99 067 9.32 122 0.67 878 0.00 933 0.68811 10 51 .31 250 .99064 .32 185 .67 815 .00 936 .68 750 9 52 .31 310 .99 062 .32 248 .67 752 .00 938 .68 690 8 53 .31 370 .99 059 .32311 .67 689 .00 941 .68 630 7 54 .31 430 .99 056 .32 373 .67 627 .00 944 .68 570 6 55 9.31 490 9.99 054 9.32 436 0.67 564 0.00 946 0.68 510 5 56 .31 549 .99 051 .32 498 .67 502 .00949 .68 451 4 57 .31 609 .99 048 .32 561 .67 439 .00 952 .68 391 3 58 .31 669 .99 046 .32 623 .67 377 .00 954 .68 331 2 59 .31 728 .99 043 .32 685 .67 315 .00957 .68 272 1 60 9.31 788 9.99 040 9.32 747 0.67 253 0.00 960 0.68 212 Cos Sin Cot 'fan Csc Sec ' 101 (281) (258) 78 208 Table 4. Trigonometric Logarithms 12 (192) (347) 167 c ' Sin Cos Tan Cot Sec | Csc 9.31 788 9.99 040 9.32 747 0.67 253 0.00 960 0.68212 60 1 .31 847 .99 038 .32 810 .67 190 .00 962 .68 153 59 2 .31 907 .99 035 .32 872 .67 128 .00 965 .68 093 58 3 .31 966 .99 032 .32 933 .67 067 .00 968 .68 034 57 4 .32 025 .99 030 .32 995 .67 005 .00 970 .67 975 56 5 9.32 084 9.99 027 9.33 057 0.66 943 0.00 973 0.67 916 55 6 .32 143 .99 024 .33 119 .66 881 .00 976 .67 857 54 7 .32 202 . .99 022 .33 180 .66 820 .00 978 .67 798 53 8 .32 261 .99 019 .33 242 .66 758 .00 981 .67 739 52 9 .32 319 .99 016 .33 303 .66 697 .00 984 .67 681 51 10 9.32 378 9.99 013 9.33 365 0.66 635 0.00 987 0.67 622 50 11 .32 437 .99011 .33 426 .66 574 .00 989 .67 563 49 12 .32 495 .99 008 .33 487 .66 513 .00 992 .67 505 48 13 .32 553 .99 005 .33 548 .66 452 .00 995 .67 447 47 14 .32 612 .99 002 .33 609 .66 391 .00 998 .67 388 46 15 9.32 670 9.99000 9.33 670 0.66 330 0.01 000 0.67 330 45 16 .32 728 .98 997 .33 731 .66 269 .01 003 .67 272 44 17 .32 786 .98 994 .33 792 .66 208 .01 006 .67 214 43 18 .32 844 .98 991 .33 853 .66 147 .01 009 .67 156 42 19 .32 902 .98 989 .33 913 .66 087 .01 Oil .67 098 41 20 9.32 960 9.98 986 9.33 974 0.66 026 0.01 014 0.67 040 40 21 .33 018 .98 983 .34 034 .65 966 .01 017 .66 982 39 22 .33 075 .98 980 .34 095 .65 905 .01 020 .66 925 38 23 .33 133 .98 978 .34 155 .65 845 .01 022 .66 867 37 24 .33 190 .98 975 .34 215 .65 785 .01 025 .66 810 36 25 9.33 248 9.98 972 9.34 276 0.65 724 0.01 028 0.66 752 35 26 .33 305 .98 969 .34 336 .65 664 .01 031 .66 695 34 27 .33 362 .98 967 .34 396 .65 604 .01 033 .66 638 33 28 .33 420 .98 964 .34 456 .65 544 .01 036 .66 580 32 29 .33 477 .98 961 .34 516 .65 484 .01 039 .66 523 31 30 9.33 534 9.98 958 9.34 576 0.65 424 0.01 042 0.66 466 30 31 .33 591 .98 955 .34 635 .65 365 .01 045 .66 409 29 32 .33 647 .98 953 .34 695 .65 305 .01 047 .66 353 28 33 .33 704 .98 950 .34 755 .65 245 .01 050 .66 296 27 34 .33 761 .98 947 .34 814 .65 186 .01 053 .66 239 26 35 9.33 818 9.98 944 9.34 874 0.65 126 0.01 056 0.66 182 25 36 .33 874 .98 941 .34 933 .65 067 .01 059 .66 126 24 37 .33 931 .98 938 .34 992 .65 008 .01 062 .66 069 23 38 .33 987 .98 936 .35 051 .64 949 .01 064 .66 013 22 39 .34 043 .98 933 .35 111 .64 889 .01 067 .65 957 21 40 9.34 100 9.98 930 9.35 170 0.64 830 0.01 070 0.65 900 20 41 .34 156 .98 927 .35 229 .64 771 .01 073 .65 844 19 42 .34 212 .98 924 .35 288 .64 712 .01 076 .65 788 18 43 .34 268 .98 921 .35 347 .64 653 .01 079 .65 732 17 44 .34 324 ,98 919 .35 405 .64 595 .01 081 .65 676 16 45 9.34 380 9.98 916 9.35 464 0.64 536 0.01 084 0.65 620 15 46 .34 436 .98 913 .35 523 .64 477 .01 087 .65 564 14 47 .34 491 .98 910 .35 581 .64 419 .01 090 .65 509 13 48 .34 547 .98 907 .35 640 .64 360 .01 093 .65 453 12 49 .34 602 .98 904 .35 698 .64 302 .01 096 .65 398 11 50 9.34 658 9.98 901 9.35 757 0.64 243 0.01 099 0.65 342 10 51 .34 713 .98 898 .35 815 .64 185 .01 102 .65 287 9 52 .34 769 .98 896 .35 873 .64 127 .01 104 .65 231 8 53 .34 824 .98 893 .35 931 .64 069 .01 107 .65 176 7 54 .34 879 .98 890 .35 989 .64011 .01 110 .65 121 6 55 9.34 934 9.98 887 9.36 047 0.63 953 0.01 113 0.65 066 5 56 .34 989 .98 884 .36 105 .63 895 .01 116 .65011 4 57 .35 044 .98 881 .36 163 .63 837 .01 119 .64 956 3 58 .35 099 .98 878 .36 221 .63 779 .01 122 .64 901 2 59 .35 154 .98 875 .36 279 .63 721 .01 125 .64 846 1 60 9.35 209 9.98 872 9.36 336 0.63 664 0.01 128 0.64 791 Cos Sin 102 (282) (257) 77 Table 4. Trigonometric Logarithms 209 13 (193) (346) 166 C ' Sin Cos Tan Cot Sec Csc 9.35 209 9.98 872 9.36 336 0.63 664 0.01 128 0.64 791 60 1 .35 263 .98 869 .36 394 .63 606 .01 131 .64 737 59 2 .35 318 .98 867 .36 452 .63 548 .01 133 .64 682 58 3 .35 373 .98864 .36 509 .63 491 .01 136 .64 627 57 4 .35 427 .98 861 .36 566 .63 434 .01 139 .64 573 56 5 9.35 481 9.98 858 9.36 624 0.63 376 0.01 142 0.64 519 55 6 .35 536 .98 855 .36 681 .63 319 .01 145 .64464 54 7 .35 590 .98 852 .36 738 .63 262 .01 148 .64 410 53 8 .35 644 .98849 .36 795 .63 205 .01 151 .64 356 52 9 .35 698 .98846 .36 852 .63 148 .01 154 .64302 51 10 9.35 752 9.98 843 9.36 909 0.63 091 0.01 157 0.64 248 50 11 .35 806 .98840 .36 966 .63 034 .01 160 .64 194 49 12 .35 860 .98 837 .37 023 .62 977 .01 163 .64 140 48 13 .35 914 .98 834 .37 080 .62 920 .01 166 .64 086 47 14 .35 968 .98 831 .37 137 .62 863 .01 169 .64 032 46 15 9.36 022 9.98 828 9.37 193 0.62 807 0.01 172 0.63 978 45 16 .36 075 .98 825 .37 250 .62 750 .01 175 .63 925 44 17 .36 129 .98 822 .37 306 .62 694 .01 178 .63 871 43 18 .36 182 .98 819 .37 363 .62 637 .01 181 .63 818 42 19 .36 236 .98 816 .37 419 .62 581 .01 184 .63 764 41 20 9.36 289 9.98 813 9.37 476 0.62 524 0.01 187 0.63 711 40 21 .36 342 .98 810 .37 532 .62 468 .01 190 .63 658 39 22 .36 395 .98 807 .37 588 .62 412 .01 193 .63 605 38 23 .36 449 .98 804 .37 644 .62 356 .01 196 .63 551 37 24 .36 502 .98 801 .37 700 .62 300 .01 199 .63 498 36 25 9.36 555 9.98 798 9.37 756 0.62 244 0.01 202 0.63 445 35 26 .36 608 .98 795 .37 812 .62 188 .01 205 .63 392 34 27 .36 660 .98 792 .37 868 .62 132 .01 208 .63 340 33 28 .36 713 .98 789 .37 924 .62 076 .01 211 .63 287 32 29 .36 766 .98 786 .37 980 .62 020 .01 214 .63 234 31 30 9.36 819 9.98 783 9.38 035 0.61 965 0.01 217 0.63 181 30 31 .36 871 .98 780 .38 091 .61 909 .01 220 .63 129 29 32 .36 924 .98 777 .38 147 .61 853 .01 223 .63 076 28 33 .36 976 .98 774 .38 202 .61 798 .01 226 .63 024 27 34 .37 028 .98 771 .38 257 .61 743 .01 229 .62 972 26 35 9.37 081 9.98 768 9.38 313 0.61 687 0.01 232 0.62 919 25 36 .37 133 .98 765 .38 368 .61 632 .01 235 .62 867 24 37 .37 185 .98 762 .38 423 .61 577 .01 238 .62 815 23 38 .37 237 .98 759 .38 479 .61 521 .01 241 .62 763 22 39 .37 289 .98 756 .38 534 .61 466 .01 244 .62711 21 40 9.37 341 9.98 753 9.38 589 0.61 411 0.01 247 0.62 659 20 41 .37 393 .98 750 .38644 .61 356 .01 250 .62 607 19 42 .37445 .98 746 .38 699 .61 301 .01 254 .62 555 18 43 .37 497 .98 743 .38 754 .61 246 .01 257 .62 503 17 44 .37 549 .98 740 .38 808 .61 192 .01 260 .62 451 16 45 9.37 600 9.98 737 9.38 863 0.61 137 0.01 263 0.62 400 15 46 .37 652 .98 734 .38 918 .61 082 .01 266 .62 348 14 47 .37 703 .98 731 .38 972 .61 028 .01 269 .62 297 13 48 .37 755 .98 728 .39 027 .60 973 .01 272 .62 245 12 49 .37 806 .98 725 .39 082 .60 918 .01 275 .62 194 11 50 9.37 858 9.98 722 9.39 136 0.60 864 0.01 278 0.62 142 10 51 .37 909 .98 719 .39 190 .60 810 .01 281 .62 091 9 52 .37 960 .98 715 .39 245 .60 755 .01285 .62040 8 53 .38011 .98 712 .39 299 .60 701 .01 288 .61 989 7 54 .38 062 .98 709 .39 353 .60 647 .01 291 .61 938 6 55 9.38 113 9.98 706 9.39 407 0.60 593 0.01 294 0.61 887 5 56 .38 164 .98 703 .39 461 .60 539 .01 297 .61 836 4 57 .38 215 .98700 .39 515 .60 485 .01 300 .61785 3 58 .38266 .98 697 .39 569 .60 431 .01 303 .61 734 2 59 .38 317 .98 694 .39 623 .60 377 .01 306 .61 683 1 60 9.38 368 9.98 690 9.39 677 0.60 323 0.01 310 0.61 632 Cos Sin Cot Tan Csc Sec ' 103 (283) (256) 76 210 Table 4. Trigonometric Logarithms 14 (194) (345) 165 C ' Sin Cos Tan Cot Sec Csc 9.38 368 9.98 690 9.39 677 0.60 323 0.01 310 0.61 632 60 1 .38418 .98 687 .39 731 .60 269 .01 313 .61 582 59 2 .38 469 .98 684 .39 785 .60 215 .0,1 316 .61 531 58 3 .38 519 .98 681 .39 838 .60 162 .01 319 .61 481 57 4 .38 570 .98 678 .39 892 .60 108 .01 322 .61 430 56 5 9.38 620 9.98 675 9.39 945 0.60 055 0.01 325 0.61 380 55 6 .38 670 .98 671 .39 999 .60 001 .01 329 .61 330 54 7 .38 721 .98 668 .40 052 .59 948 .01 332 .61 279 53 8 .38 771 .98 665 .40 106 .59 894 .01 335 .61 229 52 9 .38 821 .98 662 .40 159 .59 841 .01 338 .61 179 51 10 9.38 871 9.98 659 9.40 212 0.59 788 0.01 341 0.61 129 50 11 .38 921 .98 656 .40 266 .59 734 .01 344 .61 079 49 12 .38 971 .98 652 .40 319 .59 681 .01 348 .61 029 48 13 .39 021 .98 649 .40 372 .59 628 .01 351 .60 979 47 14 .39 071 .98 646 .40 425 .59 575 .01 354 .60 929 46 15 9.39 121 9.98 643 9.40 478 0.59 522 0.01 357 0.60 879 45 16 .39 170 .98 640 .40 531 .59 469 .01 360 .60 830 44 17 .39 220 .98 636 .40 584 .59 416 .01 364 .60 780 43 18 .39 270 .98 633 .40 636 .59364 .01 367 .60 730 42 19 .39 319 .98 630 .40 689 .59 311 .01 370 .60 681 41 20 9.39 369 9.98 627 9.40 742 0.59 258 0.01 373 0.60 631 40 21 .39 418 .98 623 .40 795 .59 205 .01 377 .60 582 39 22 .39 467 .98 620 .40 847 .59 153 .01 380 .60 533 38 23 .39 517 .98 617 .40 900 .59 100 .01 383 .60 483 37 24 .39 566 .98 614 .40 952 .59 048 .01 386 .60 434 36 25 9.39 615 9.98 610 9.41 005 0.58995 0.01 390 0.60 385 35 26 .39 664 .98 607 .41 057 .58 943 .01 393 .60 336 34 27 .39 713 .98 604 .41 109 .58 891 .01 396 .60 287 33 28 .39 762 .98 601 .41 161 .58 839 .01 399 .60 238 32 29 .39811 .98 597 .41 214 .58 786 .01 403 .60 189 31 30 9.39 860 9.98 594 9.41 266 0.58 734 0.01 406 0.60 140 30 31 .39 909 .98 591 .41 318 .58 682 .01 409 .60 091 29 32 .39 958 .98 588 .41 370 .58 630 .01 412 .60 042 28 33 .40 006 .98 584 .41 422 .58 578 .01 416 .59 994 27 34 .40 055 .98 581 .41 474 .58 526 .01 419 .59 945 26 35 9.40 103 9.98 578 9.41 526 0.58 474 0.01 422 0.59 897 25 36 .40 152 .98 574 .41 578 .58 422 .01 426 .59 848 24 37 .40 200 .98 571 .41 629 .58 371 .01 429 .59 800 23 38 .40 249 .98 568 .41 681 .58 319 .01 432 .59 751 22 39 .40 297 .98 565 .41 733 .58 267 .01 435 .59 703 21 40 9.40 346 9.98 561 9.41 784 0.58 216 0.01 439 0.59 654 20 41 .40 394 .98 558 .41 836 .58 164 .01 442 .59 606 19 42 .40 442 .98 555 .41 887 .58 113 .01 445 .59 558 18 43 .40 490 .98 551 .41 939 .58 061 .01 449 .59 510 17 44 .40 538 .98 548 .41 990 .58 010 .01 452 .59 462 16 45 9.40 586 9.98 545 9.42 041 0.57 959 0.01 455 0.59 414 15 46 .40 634 .98 541 .42 093 .57 907 .01 459 .59 366 14 47 .40 682 .98 538 .42 144 .57 856 .01 462 .59 318 13 48 .40 730 .98 535 .42 195 .57 805 .01465 .59 270 12 49 .40 778 .98 531 .42 246 .57 754 .01 469 .59 222 11 50 9.40 825 9.98 528 9.42 297 0.57 703 0.01 472 0.59 175 10 51 .40 873 .98 525 .42 348 .57 652 .01 475 .59 127 9 52 .40 921 .98 521 .42 399 .57 601 .01 479 .59 079 8 53 .40 968 .98518 .42 450 .57 550 .01 482 .59 032 7 54 .41 016 .98515 .42 501 .57 499 .01 485 .58984 6 55 9.41 063 9.98511 9.42 552 0.57 448 0.01 489 0.58 937 5 56 .41 111 .98 508 .42 603 .57 397 .01 492 .58 889 4 57 .41 158 .98 505 .42 653 .57 347 .01 495 .58 842 3 58 .41 205 .98 501 .42 704 .57 296 .01 499 .58 795 2 59 .41 252 .98 498 .42 755 .57 245 .01 502 .58 748 1 60 9.41 300 9.98 494 9.42 805 0.57 195 0.01 506 0.58 700 Cos Sin Cot Tan Csc Sec ' 104 (284) (255) 75 Table 4. Trigonometric Logarithms 211 15 (195) (344) 164 ' Sin Cos Tan Cot Sec Csc 9.41 300 9.98 494 9.42 805 0.57 195 0.01 506 0.58 700 60 1 .41 347 .98 491 .42 856 .57 144 .01 509 .58 653 59 2 .41 394 .98 488 .42 906 .57 094 .01 512 .58 606 58 3 .41 441 .98484 .42 957 .57 043 .01 516 .58 559 57 4 .41 488 .98 481 .43 007 .56 993 .01 519 .58 512 56 5 9.41 535 9.98 477 9.43 057 0.56 943 0.01 523 0.58 465 55 6 .41 582 .98 474 .43 108 .56 892 .01 526 .58418 54 7 .41 628 .98 471 .43 158 .56842 .01 529 .58 372 53 8 .41 675 .98 467 .43 208 .56 792 .01 533 .58 325 52 9 .41 722 .98464 .43 258 .56 742 .01 536 .58 278 51 10 9.41 768 9.98 460 9.43 308 0.56 692 0.01 540 0.58 232 50 11 .41 815 .98 457 .43 358 .56 642 .01 543 .58 185 49 12 .41 861 .98 453 .43 408 .56 592 .01 547 .58 139 48 13 .41 908 .98 450 .43 458 .56 542 .01 550 .58 092 47 14 .41 954 .98447 .43 508 .56 492 .01 553 .58 046 46 15 9.42 001 9.98 443 9.43 558 0.56 442 0.01 557 0.57 999 45 16 .42047 .98440 .43 607 .56 393 .01 560 .57 953 44 17 .42 093 .98 436 .43 657 .56 343 .01 564 .57 907 43 18 .42 140 .98 433 .43 707 .56 293 .01 567 .57 860 42 19 .42 186 .98 429 .43 756 .56 244 .01 571 .57 814 41 20 9.42 232 9.98 426 9.43 806 0.56 194 0.01 574 0.57 768 40 21 .42 278 .98 422 .43 855 .56 145 .01 578 .57 722 39 22 .42 324 .98419 .43 905 .56 095 .01 581 .57 676 38 23 .42 370 .98 415 .43 954 .56046 .01 585 .57 630 37 24 .42 416 .98412 .44 004 .55 996 .01 588 .57 584 36 25 9.42 461 9.98 409 9.44 053 0.55 947 0.01 591 0.57 539 35 26 .42 507 .98 405 .44 102 .55 898 .01 595 .57 493 34 27 .42 553 .98 402 .44 151 .55849 .01 598 .57 447 33 28 .42 599 .98 398 .44201 .55 799 .01 602 .57 401 32 29 .42 644 .98 395 .44250 .55 750 .01 605 .57 356 31 30 9.42 690 9.98 391 9.44 299 0.55 701 0.01 609 0.57 310 30 31 .42 735 .98 388 .44348 .55 652 .01 612 .57 265 29 32 .42 781 .98384 .44397 .55 603 .01 616 .57 219 28 33 .42 826 .98 381 .44446 .55 554 .01 619 .57 174 27 34 .42872 .98 377 .44495 .55 505 .01 623 .57 128 26 35 9.42 917 9.98 373 9.44 544 0.55 456 0.01 627 0.57 083 25 36 .42 962 .98 370 .44 592 .55 408 .01 630 .57 038 24 37 .43 008 .98 366 .44641 .55 359 .01 634 .56 992 23 38 .43 053 .98 363 .44 690 .55 310 .01 637 .56 947 22 39 .43 098 .98 359 .44738 .55 262 .01 641 .56 902 21 40 9.43 143 9.98 356 9.44 787 0.55 213 0.01 644 0.56 857 20 41 .43 188 .98 352 .44836 .55 164 .01 648 .56 812 19 42 .43 233 .98 349 .44884 .55 116 .01 651 .56 767 18 43 .43 278 .98 345 .44933 .55 067 .01 655 .56 722 17 44 .43 323 .98 342 .44981 .55 019 .01 658 .56 677 16 45 9.43 367 9.98 338 9.45 029 0.54 971 0.01 662 0.56 633 15 46 .43 412 .98 334 .45 078 .54 922 .01 666 .56 588 14 47 .43 457 .98 331 .45 126 .54 874 .01 669 .56 543 13 48 .43 502 .98 327 .45 174 .54826 .01 673 .56 498 12 49 .43546 .98 324 .45 222 .54778 .01 676 .56 454 11 50 9.43 591 9.98 320 9.45 271 0.54 729 0.01 680 0.56 409 10 51 .43 635 .98 317 .45 319 .54 681 .01 683 .56 365 9 52 .43 680 .98 313 .45 367 .54 633 .01 687 .56 320 8 53 .43 724 .98 309 .45 415 .54 585 .01 691 .56 276 7 54 .43 769 .98 306 .45 463 .54 537 .01 694 .56 231 6 55 9.43 813 9.98 302 9.45511 0.54 489 0.01 698 0.56 187 5 56 .43 857 .98 299 .45 559 .54441 .01 701 .56 143 4 57 .43 901 .98 295 .45 606 .54 394 .01 705 .56 099 3 58 .43 946 .98 291 .45 654 .54 346 .01 709 .56 054 2 59 .43 990 .98 288 .45 702 .54298 .01 712 .56 010 1 60 9.44 034 9.98 284 9.45 750 0.54 250 0.01 716 0.55 966 Cos Sin Cot Tan Csc Sec ' 105 (285) (254) 74 212 Table 4. Trigonometric Logarithms 16 (196) (343) 163 ' Sin Cos Tan Cot Sec Csc 9.44 034 9.98 284 9.45 750 0.54 250 0.01 716 0.55 966 60 1 .44 078 .98 281 .45 797 .54 203 .01 719 .55 922 59 2 .44 122 .98 277 .45 845 .54 155 .01 723 .55 878 58 3 .44 166 .98 273 .45 892 .54 108 .01 727 .55 834 57 4 .44 210 .98 270 .45 940 .54 060 .01 730 .55 790 56 5 9.44 253 9.98 266 9.45 987 0.54 013 0.01 734 0.55 747 55 6 .44 297 .98 262 .46 035 .53 965 .01 738 .55 703 54 7 .44 341 .98 259 .46 082 .53 918 .01 741 .55 659 53 8 .44 385 .98 255 .46 130 .53 870 .01 745 .55 615 52 9 .44 428 .98 251 .46 177 .53 823 .01 749 .55 572 51 10 9.44 472 9.98 248 9.46 224 0.53 776 0.01 752 0.55 528 50 11 .44 516 .98 244 .46 271 .53 729 .01 756 .55484 49 12 .44 559 .98 240 .46 319 .53 681 .01 760 .55 441 48 13 .44 602 .98 237 .46 366 .53 634 .01 763 .55 398 47 14 .44 646 .98 233 .46 413 .53 587 .01 767 .55 354 46 15 9.44 689 9.98 229 9.46 460 0.53 540 0.01 771 0.55311 45 16 .44 733 .98 226 .46 507 .53 493 .01 774 .55 267 44 17 .44 776 .98 222 .46 554 .53 446 .01 778 .55 224 43 18 .44 819 .98218 .46 601 .53 399 .01 782 .55 181 42 19 .44 862 .98 215 .46 648 .53 352 .01 785 .55 138 41 20 9.44 905 9.98211 9.46 694 0.53 306 0.01 789 0.55 095 40 21 .44 948 .98 207 .46 741 .53 259 .01 793 .55 052 39 22 .44 992 .98 204 .46 788 .53 212 .01 796 .55 008 38 23 .45 035 .98 200 .46 835 .53 165 .01 800 .54 965 37 24 .45 077 .98 196 .46 881 .53 119 .01 804 .54 923 36 25 9.45 120 9.98 192 9.46 928 0.53 072 0.01 808 0.54 880 35 26 .45 163 .98 189 .46 975 .53 025 .01 811 .54 837 34 27 .45 206 .98 185 .47 021 .52 979 .01 815 .54 794 33 28 .45 249 .98 181 .47 068 .52 932 .01 819 .54 751 32 29 .45 292 .98 177 .47 114 .52 886 .01 823 .54 708 31 30 9.45 334 9.98 174 9.47 160 0.52 840 0.01 826 0.54 666 30 31 .45 377 .98 170 .47 207 .52 793 .01 830 .54 623 29 32 .45 419 .98 166 .47 253 .52 747 .01 834 .54 581 28 33 .45 462 .98 162 .47 299 .52 701 .01 838 .54 538 27 34 .45 504 .98 159 .47 346 .52 654 .01 841 .54 496 26 35 9.45 547 9.98 155 9.47 392 0.52 608 0.01 845 0.54 453 25 36 .45 589 .98 151 .47 438 .52 562 .01 849 .54411 24 37 .45 632 .98 147 .47 484 .52 516 .01 853 .54 368 23 38 .45 674 .98 144 .47 530 .52 470 .01 856 .54 326 22 39 .45 716 .98 140 .47 576 .52 424 .01 860 .54 284 21 40 9.45 758 9.98 136 9.47 622 0.52 378 0.01 864 0.54 242 20 41 .45 801 .98 132 .47 668 .52 332 .01 868 .54 199 19 42 .45 843 .98 129 .47 714 .52 286 .01 871 .54 157 18 43 .45 885 .98 125 .47 760 .52 240 .01 875 .54 115 17 44 .45 927 .98 121 .47 806 .52 194 .01 879 .54 073 16 45 9.45 969 9.98 117 9.47 852 0.52 148 0.01 883 0.54 031 15 46 .46011 .98 113 .47 897 .52 103 .01 887 .53 989 14 47 .46 053 .98 110 .47 943 .52 057 .01 890 .53 947 13 48 .46 095 .98 106 .47 989 .52011 .01 894 .53 905 12 49 .46 136 .98 102 .48 035 .51 965 .01 898 .53 864 11 50 9.46 178 9.98 098 9.48 080 0.51 920 0.01 902 0.53 822 10 51 .46 220 .98 094 .48 126 .51 874 .01 906 .53 780 9 52 .46 262 .98 090 .48 171 .51 829 .01 910 .53 738 8 53 .46 303 .98 087 .48 217 .51 783 .01 913 .53 697 7 54 .46 345 .98 083 .48 262 .51 738 .01 917 .53 655 6 55 9.46 386 9.98 079 9.48 307 0.51 693 0.01 921 0.53 614 5 56 .46 428 .98 075 .48 353 .51 647 .01 925 .53 572 4 57 .46 469 .98 071 .48 398 .51 602 .01 929 .53 531 3 58 .46511 .98 067 .48 443 .51 557 .01 933 .53 489 2 59 .46 552 .98 063 .48 489 .51 511 .01 937 .53 448 1 60 9.46 594 9.98 060 9.48 534 0.51 466 0.01 940 0.53 406 Cos Sin Cot Tan Csc Sec ' 106 (286) (253) 73 Table 4. Trigonometric Logarithms 213 17 (197) (342) 162 C ' Sin Cos Tan Cot Sec Csc 9.46 594 9.98 060 9.48 534 0.51 466 0.01 940 0.53 406 60 1 .46 635 .98 056 .48 579 .51 421 .01 944 .53 365 59 2 .46 676 .98 052 .48 624 .51 376 .01 948 .53 324 58 3 .46 717 .98 048 .48 669 .51 331 .01 952 .53 283 57 4 .46 758 .98044 .48 714 .51 286 .01 956 .53 242 56 5 9.46 800 9.98 040 9.48 759 0.51 241 0.01 960 0.53 200 55 6 .46841 .98 036 .48 804 .51 196 .01 964 .53 159 54 7 .46 882 .98 032 .48849 .51 151 .01 968 .53 118 53 8 .46 923 .98 029 .48 894 .51 106 .01 971 .53 077 52 9 .46964 .98 025 .48 939 .51 061 .01 975 .53 036 51 10 9.47 005 9.98 021 9.48 984 0.51 016 0.01 979 0.52 995 50 11 .47045 .98 017 .49 029 .50 971 .01 983 .52 955 49 12 .47 086 .98 013 .49 073 .50 927 .01 987 .52 914 48 13 .47 127 ' .98009 .49 118 .50 882 .01 991 .52 873 47 14 .47 168 .98 005 .49 163 .50 837 .01 995 .52 832 46 15 9.47 209 9.98 001 9.49 207 0.50 793 0.01 999 0.52 791 45 16 .47 249 .97 997 .49 252 .50 748 .02 003 .52 751 44 17 .47 290 .97 993 .49 296 .50 704 .02 007 .52 710 43 18 .47 330 .97 989 .49 341 .50 659 .02011 .52 670 42 19 .47 371 .97 986 .49 385 .50 615 .02 014 .52 629 41 20 9.47411 9.97 982 9.49 430 0.50 570 0.02 018 0.52 589 40 21 .47 452 .97 978 .49 474 .50 526 .02 022 .52 548 39 22 .47 492 .97 974 .49 519 .50 481 .02 026 .52 508 38 23 .47 533 .97 970 .49 563 .50 437 .02 030 .52 467 37 24 .47 573 .97 966 .49 607 .50 393 .02 034 .52 427 36 25 9.47 613 9.97 962 9.49 652 0.50 348 0.02 038 0.52 387 35 26 .47 654 .97 958 .49 696 .50 304 .02042 .52 346 34 27 .47 694 .97 954 .49 740 .50 260 .02 046 .52 306 33 28 .47 734 .97 950 .49784 .50 216 .02 050 .52 266 32 29 .47 774 .97 946 .49 828 .50 172 .02 054 .52 226 31 30 9.47 814 9.97 942 9.49 872 0.50 128 0.02 058 0.52 186 30 31 .47 854 .97 938 .49 916 .50084 .02 062 .52 146 29 32 .47 894 .97 934 .49 960 .50 040 .02 066 .52 106 28 33 .47 934 .97 930 .50 004 .49 996 .02 070 .52 066 27 34 .47 974 .97 926 .50 048 .49 952 .02 074 .52 026 26 35 9.48 014 9.97 922 9.50 092 0.49 908 0.02 078 0.51 986 25 36 .48 054 .97 918 .50 136 .49 864 .02 082 .51 946 24 37 .48 094 .97 914 .50 180 .49 820 .02 086 .51 906 23 38 .48 133 .97 910 .50 223 .49 777 .02 090 .51 867 22 39 .48 173 .97 906 .50 267 .49 733 .02 094 .51 827 21 40 9.48 213 9.97 902 9.50311 0.49 689 0.02 098 0.51 787 20 41 .48 252 .97 898 .50 355 .49 645 .02 102 .51 748 19 42 .48 292 .97 894 .50 398 .49 602 .02 106 .51 708 18 43 .48 332 .97 890 .50 442 .49 558 .02 110 .51 668 17 44 .48 371 .97 886 .50 485 .49 515 .02 114 .51 629 16 45 9.48411 9.97 882 9.50 529 0.49 471 0.02 118 0.51 589 15 46 .48 450 .97 878 .50 572 .49 428 .02 122 .51 550 14 47 .48 490 .97 874 .50 616 .49384 .02 126 .51 510 13 48 .48 529 .97 870 .50 659 .49 341 .02 130 .51 471 12 49 .48 568 .97 866 .50 703 .49 297 .02 134 .51 432 11 50 9.48 607 9.97 861 9.50 746 0.49 254 0.02 139 0.51 393 10 51 .48 647 .97 857 .50 789 .49 211 .02 143 .51 353 9 52 .48 686 .97853 .50833 .49 167 .02 147 .51 314 8 53 .48 725 .97849 .50 876 .49 124 .02 151 .51 275 7 54 .48 764 .97845 .50 919 .49 081 .02 155 .51 236 6 55 9.48 803 9.97 841 9.50 962 0.49 038 0.02 159 0.51 197 5 56 .48842 .97 837 .51 005 .48 995 .02 163 .51 158 4 57 .48 881 .97 833 .51048 .48 952 .02 167 .51 119 3 58 .48 920 .97 829 .51 092 .48 908 .02 171 .51 080 2 59 .48 959 .97 825 .51 135 .48 865 .02 175 .51 041 1 60 9.48 998 9.97 821 9.51 178 0.48 822 0.02 179 0.51 002 Cos Sin Cot Tan Csc Sec ' 107 (287) (252) 72 214 Table 4. Trigonometric Logarithms 18 (198) (341) 161 ' Sin Cos Tan Cot Sec Csc 9.48 998 9.97 821 9.51 178 0.48 822 0.02 179 0.51 002 60 1 .49 037 .97 817 .51 221 .48 779 .02 183 .50 963 59 2 .49 076 .97 812 .51 264 .48 736 .02 188 .50 924 58 3 .49 115 .97 808 .51 306 .48 694 .02 192 .50 885 57 4 .49 153 .97 804 .51 349 .48 651 .02 196 .50 847 56 5 9.49 192 9.97 800 9.51 392 0.48 608 0.02 200 0.50 808 55 6 .49 231 .97 796 .51 435 .48 565 .02 204 .50 769 54 7 .49 269 .97 792 .51 478 .48 522 .02 208 .50 731 53 8 .49 308 .97 788 .51 520 .48 480 .02 212 .50 692 52 9 .49 347 .97784 .51 563 .48 437 .02 216 .50 653 51 10 9.49 385 9.97 779 9.51 606 0.48 394 0.02 221 0.50 615 50 11 .49 424 .97 775 .51 648 .48 352 .02 225 .50 576 49 12 .49 462 .97 771 .51 691 .48 309 .02 229 .50 538 48 13 .49 500 .97 767 .51 734 .48 266 .02 233 .50 500 47 14 .49 539 .97 763 .51 776 .48 224 .02 237 .50 461 46 15 9.49 577 9.97 759 9.51 819 0.48 181 0.02 241 0.50 423 45 16 .49 615 .97 754 .51 861 .48 139 .02 246 .50 385 44 17 .49 654 .97 750 .51 903 .48 097 .02 250 .50 346 43 18 .49 692 .97 746 .51 946 .48 054 .02 254 .50 308 42 19 ,49 730 .97 742 .51 988 .48 012 .02 258 .50 270 41 20 9.49 768 9.97 738 9.52 031 0.47 969 0.02 262 0.50 232 40 21 .49 806 .97 734 .52 073 .47 927 .02 266 .50 194 39 22 .49844 .97 729 .52 115 .47 885 .02 271 .50 156 38 23 .49 882 .97 725 .52 157 .47843 .02 275 .50 118 37 24 .49 920 .97 721 .52 200 .47 800 .02 279 .50 080 36 25 9.49 958 9.97 717 9.52 242 0.47 758 0.02 283 0.50 042 35 26 .49 996 .97 713 .52284 .47 716 .02 287 .50 004 34 27 .50 034 .97 708 .52 326 .47 674 .02 292 .49 966 33 28 .50 072 .97 704 .52 368 .47 632 .02 296 .49 928 32 29 .50 110 .97 700 .52 410 .47 590 .02 300 .49 890 31 30 9.50 148 9.97 696 9.52 452 0.47 548 0.02 304 0.49 852 30 31 .50 185 .97 691 .52 494 .47 506 .02 309 .49 815 29 32 .50 223 .97 687 .52 536 .47 464 .02 313 .49 777 28 33 .50 261 .97 683 .52 578 .47 422 .02 317 .49 739 27 34 .50 298 .97 679 .52 620 .47 380 .02 321 .49 702 26 35 9.50 336 9.97 674 9.52 661 0.47 339 0.02 326 0.49 664 25 36 .50 374 .97 670 .52 703 .47 297 .02 330 .49 626 24 37 .50411 .97 666 .52 745 .47255 .02 334 .49 5,89 23 38 .50 449 .97 662 .52 787 .47 213 .02 338 .49 551 22 39 .50 486 .97 657 .52 829 .47 171 .02 343 .49 514 21 40 9.50 523 9.97 653 9.52 870 0.47 130 0.02 347 0.49 477 20 41 .50 561 .97 649 .52912 .47 088 .02 351 .49 439 19 42 .50 598 .97645 .52 953 .47 047 .02 355 .49 402 18 43 .50 635 .97 640 .52 995 .47 005 .02 360 .49 365 17 44 .50 673 .97 636 .53 037 .46 963 .02 364 .49 327 16 45 9.50 710 9.97 632 9.53 078 0.46 922 0.02 368 0.49 290 15 46 .50 747 .97 628 .53 120 .46 880 .02 372 .49 253 14 47 .50 784 .97 623 .53 161 .46 839 .02 377 .49 216 13 48 .50 821 .97 619 .53 202 .46 798 .02 381 .49 179 12 49 .50 858 .97 615 .53 244 .46 756 .02 385 .49 142 11 50 9.50 896 9.97 610 9.53 285 0.46 715 0.02 390 0.49 104 10 51 .50 933 .97 606 .53 327 .46 673 .02 394 .49 067 9 52 .50 970 .97 602 .53 368 .46 632 .02 398 .49 030 8 53 .51 007 .97 597 .53 409 .46 591 .02 403 .48 993 7 54 .51 043 .97 593 .53 450 .46 550 .02 407 .48 957 6 55 9.51 080 9.97 589 9.53 492 0.46 508 0.02411 0.48 920 5 56 .51 117 .97 584 .53 533 .46 467 .02 416 .48 883 4 57 .51 154 .97 580 .53 574 .46 426 .02 420 .48 846 3 58 .51 191 .97 576 .53 615 .46 385 .02 424 .48 809 2 59 .51 227 .97 571 .53 656 .46 344 .02 429 .48 773 1 60 9.51 264 9.97 567 9.53 697 0.46 303 0.02 433 0.48 736 Cos Sin Cot Tan Csc Sec ' 108 (288) (251) 71 C Table 4. Trigonometric Logarithms 215 19 (199) (340) 160 ' Sin Cos Tan Cot Sec Csc 9.51 264 9.97 567 9.53 697 0.46 303 0.02 433 0.48 736 60 1 .51 301 .97 563 .53 738 .46 262 .02 437 .48 699 59 2 .51 338 .97 558 .53 779 .46 221 .02442 .48 662 58 3 .51 374 .97 554 .53 820 .46 180 .02 446 .48 626 57 4 .51411 .97 550 .53 861 .46 139 .02 450 .48 589 56 5 9.51 447 9.97 545 9.53 902 0.46 098 0.02 455 0.48 553 55 6 .51 484 .97 541 .53 943 .46 057 .02 459 .48 516 54 7 .51 520 .97 536 .53 984 .46 016 .02 464 .48 480 53 8 .51 557 .97 532 .54 025 .45 975 .02 468 .48 443 52 9 .51 593 .97 528 .54 065 .45 935 .02 472 .48 407 51 10 9.51 629 9.97 523 9.54 106 0.45 894 0.02 477 0.48 371 50 11 .51 666 .97 519 .54 147 .45 853 .02 481 .48 334 49 12 .51 702 .97 515 .54187 .45 813 .02 485 .48 298 48 13 .51 738 .97 510 .54 228 .45 772 .02 490 .48 262 47 14 .51 774 .97 506 .54 269 .45 731 .02 494 .48 226 46 15 9.51 811 9.97 501 9.54 309 0.45 691 0.02 499 0.48 189 45 16 .51 847 .97 497 .54350 .45 650 .02 503 .48 153 44 17 .51 883 .97 492 .54 390 .45 610 .02 508 .48 117 43 18 .51 919 .97 488 .54 431 .45 569 .02 512 .48 081 42 19 .51 955 .97484 .54 471 .45 529 .02 516 .48 045 41 20 9.51 991 9.97 479 9.54 512 0.45 488 0.02 521 0.48 009 40 21 .52 027 .97 475 .54 552 .45 448 .02 525 .47 973 39 22 .52 063 .97 470 .54 593 .45 407 .02 530 .47 937 38 23 .52 099 .97 466 .54 633 .45 367 .02 534 .47 901 37 24 .52 135 .97 461 .54 673 .45 327 .02 539 .47 865 36 25 9.52 171 9.97 457 9.54 714 0.45 286 0.02 543 0.47 829 35 26 .52 207 .97 453 .54 754 .45 246 .02 547 .47 793 34 27 .52 242 .97448 .54 794 .45 206 .02 552 .47 758 33 28 .52 278 .97 444 .54 835 .45 165 .02 556 .47 722 32 29 .52 314 .97 439 .54 875 .45 125 .02 561 .47 686 31 30 9.52 350 9.97 435 9.54 915 0.45 085 0.02 565 0.47 650 30 31 .52 385 .97 430 .54 955 .45 045 .02 570 .47 615 29 32 .52 421 .97 426 .54 995 .45 005 .02 574 .47 579 28 33 .52 456 .97 421 .55 035 .44965 .02 579 .47 544 27 34 .52 492 .97 417 .55 075 .44 925 .02 583 .47 508 26 35 9.52 527 9.97 412 9.55 115 0.44 885 0.02 588 0.47 473 25 36 .52 563 .97 408 .55 155 .44845 .02 592 .47 437 24 37 .52 598 .97 403 .55 195 .44 805 .02 597 .47 402 23 38 .52 634 .97 399 .55 235 .44765 .02 601 .47 366 22 39 .52 669 .97 394 .55 275 .44725 .02 606 .47 331 21 40 9.52 705 9.97 390 9.55 315 0.44 685 0.02 610 0.47 295 20 41 .52 740 .97 385 .55 355 .44645 .02 615 .47 260 19 42 .52 775 .97 381 .55 395 .44605 .02 619 .47 225 18 43 .52811 .97 376 .55 434 .44566 .02 624 .47 189 17 44 .52846 .97 372 .55 474 .44526 .02 628 .47 154 16 45 9.52 881 9.97 367 9.55 514 0.44 486 0.02 633 0.47 119 15 46 .52 916 .97 363 .55554 .44446 .02 637 .47084 14 47 .52 951 .97 358 .55 593 .44407 .02642 .47 049 13 48 .52 986 .97 353 .55 633 .44367 .02647 .47 014 12 49 .53 021 .97 349 .55 673 .44327 .02 651 .46 979 11 50 9.53 056 9.97 344 9.55 712 0.44 288 0.02 656 0.46 944 10 51 .53 092 .97 340 .55 752 .44248 .02 660 .46 908 9 52 .53 126 .97 335 .55 791 .44209 .02 665 .46 874 8 53 .53 161 .97 331 .55 831 .44169 .02 669 .46 839 7 54 .53 196 .97 326 .55 870 .44130 .02 674 .46804 6 55 9.53 231 9.97 322 9.55 910 0.44 090 0.02 678 0.46 769 5 56 .53 266 .97 317 .55 949 .44051 .02683 .46 734 4 57 .53 301 .97 312 .55 989 .44011 .02 688 .46 699 3 58 .53 336 .97 308 .56 028 .43 972 .02 692 .46664 2 59 .53 370 .97 303 .56 067 .43 933 .02 697 .46 630 1 60 9.53 405 9.97 299 9.56 107 9.43 893 0.02 701 0.46 595 Cos Sin Cot Tan Csc Sec / 109 (289) (250) 70 216 Table 4. Trigonometric Logarithms 20 (200) (339) 159 ' Sin Cos Tan Cot Sec Csc 9.53 405 9.97 299 9.56 107 0.43 893 0.02 701 0.46 595 60 1 .53 440 .97 294 .56 146 .43 854 .02 706 .46 560 59 2 .53 475 .97 289 .56 185 .43 815 .02711 .46 525 58 3 .53 509 .97 285 .56 224 .43 776 .02 715 .46 491 57 4 .53 544 .97 280 .56 264 .43 736 .02 720 .46 456 56 5 9.53 578 9.97 276 9.56 303 0.43 697 0.02 724 0.46 422 55 6 .53 613 .97 271 .56 342 .43 658 .02 729 .46 387 54 7 .53 647 .97 266 .56 381 .43 619 .02 734 .46 353 53 8 .53 682 .97 262 .56 420 .43 580 .02 738 .46 318 52 9 .53 716 .97 257 .56 459 .43 541 .02 743 .46 284 51 10 9.53 751 9.97 252 9.56 498 0.43 502 0.02 748 0.46 249 50 11 .53 785 .97 248 .56 537 .43 463 .02 752 .46 215 49 12 .53 819 .97 243 .56 576 .43 424 .02 757 .46 181 48 13 .53 854 .97 238 .56 615 .43 385 .02 762 .46 146 47 14 .53 888 .97 234 .56 654 .43 346 .02 766 .46 112 46 15 9.53 922 9.97 229 9.56 693 0.43 307 0.02 771 0.46 078 45 16 .53 957 .97 224 .56 732 .43 268 .02 776 .46 043 44 17 .53 991 .97 220 .56 771 .43 229 .02 780 .46 009 43 18 .54 025 .97 215 .56 810 .43 190 .02 785 .45 975 42 19 .54 059 .97 210 .56 849 .43 151 .02 790 .45 941 41 20 9.54 093 9.97 206 9.56 887 0.43 113 0.02 794 0.45 907 40 21 .54 127 .97 201 .56 926 .43 074 .02 799 .45 873 39 22 .54 161 .97 196 .56 965 .43 035 .02 804 .45 839 38 23 .54 195 .97 192 .57 004 .42 996 .02 808 .45 805 37 24 .54 229 .97 187 .57 042 .42 958 .02 813 .45 771 36 25 9.54 263 9.97 182 9.57 081 0.42 919 0.02 818 0.45 737 35 26 .54 297 .97 178 .57 120 .42 880 .02 822 .45 703 34 27 .54 331 .97 173 .57 158 .42842 .02 827 .45 669 33 28 .54 365 .97 168 .57 197 .42 803 .02 832 .45 635 32 29 .54 399 .97 163 .57 235 .42 765 .02 837 .45 601 31 30 9.54 433 9.97 159 9.57 274 0.42 726 0.02 841 0.45 567 30 31 .54 466 .97 154 .57 312 .42 688 .02846 .45 534 29 32 .54 500 .97 149 .57 351 .42 649 .02 851 .45 500 28 33 .54 534 .97 145 .57 389 .42 611 .02 855 .45 466 27 34 .54 567 .97 140 .57 428 .42 572 .02 860 .45 433 26 35 9.54 601 9.97 135 9.57 466 0.42 534 0.02 865 0.45 399 25 36 .54 635 .97 130 .57 504 .42 496 .02 870 .45 365 24 37 .54 668 .97 126 .57 543 .42 457 .02 874 .45 332 23 38 .54 702 .97 121 .57 581 .42419 .02 879 .45 298 22 39 .54 735 .97 116 .57 619 .42 381 .02 884 .45 265 21 40 9.54 769 9.97 111 9.57 658 0.42 342 0.02 889 0.45 231 20 41 .54 802 .97 107 .57 696 .42 304 .02 893 .45 198 19 42 .54 836 .97 102 .57 734 .42 266 .02 898 .45 164 18 43 .54 869 .97 097 ' .57 772 .42 228 .02 903 .45 131 17 44 .54 903 .97 092 .57 810 .42 190 .02 908 .45 097 16 45 9.54 936 9.97 087 9.57 849 0.42 151 0.02 913 0.45 064 15 46 .54 969 .97 083 .57 887 .42 113 .02 917 .45 031 14 47 .55 003 .97 078 .57 925 .42 075 .02 922 .44 997 13 48 .55 036 .97 073 .57 963 .42 037 .02 927 .44 964 12 49 .55 069 .97 068 .58 001 .41 999 .02 932 .44 931 11 50 9.55 102 9.97 063 9.58 039 0.41 961 0.02 937 0.44 898 10 51 .55 136 .97 059 .58 077 .41 923 .02 941 .44864 9 52 .55 169 .97 054 .58 115 .41 885 .02 946 .44 831 8 53 .55 202 .97 049 .58 153 .41 847 .02 951 .44 798 7 54 .55 235 .97 044 .58 191 .41 809 .02 956 .44 765 6 55 9.55 268 9.97 039 9.58 229 0.41 771 0.02 961 0.44 732 5 56 .55 301 .97 035 .58 267 .41 733 .02 965 .44 699 4 57 .55 334 .97 030 .58 304 .41 696 .02 970 .44 666 3 58 .55 367 .97 025 .58 342 .41 658 .02 975 .44633 2 59 .55 400 .97 020 .58 380 .41 620 .02 980 .44 600 1 60 9.55 433 9.97 015 9.58418 0.41 582 0.02 985 0.44 567 Cos Sin Cot Tan Csc Sec ' 110 (290) (249) 69 C Table 4. Trigonometric Logarithms 217 21 (201) (338) 158 C / Sin Cos Tan Cot Sec Csc 9.55 433 9.97 015 9.58418 0.41 582 0.02 985 0.44 567 60 1 .55 466 .97 010 .58 455 .41 545 .02 990 .44 534 59 2 .55 499 .97 005 .58 493 .41 507 .02 995 .44 501 58 3 .55 532 .97 001 .58 531 .41 469 .02 999 .44 468 57 4 .55 564 .96 996 .58 569 .41 431 .03 004 .44 436 56 5 9.55 597 9.96 991 9.58 606 0.41 394 0.03 009 0.44 403 55 6 .55 630 .96 986 .58644 .41 356 .03 014 .44 370 54 7 .55 663 .96 981 .58 681 .41 319 .03 019 .44 337 53 8 .55 695 .96 976 .58 719 .41 281 .03 024 .44 305 52 9 .55 728 .96 971 .58 757 .41 243 .03 029 .44 272 51 10 9.55 761 9.96 966 9.58 794 0.41 206 0.03 034 0.44 239 50 11 .55 793 .96 962 .58 832 .41 168 .03 038 .44 207 49 12 .55 826 .96 957 .58 869 .41 131 .03 043 .44174 48 13 .55 858 .96 952 .58 907 .41 093 .03048 .44142 47 14 .55 891 .96 947 .58944 .41 056 .03 053 .44109 46 15 9.55 923 9.96 942 9.58 981 0.41 019 0.03 058 0.44 077 45 16 .55 956 .96 937 .59 019 .40 981 .03 063 . .44044 44 17 .55 988 .96 932 .59 056 .40 944 .03 068 .44 012 43 18 .56 021 .96 927 .59 094 .40 906 .03 073 .43 979 42 19 .56 053 .96 922 .59 131 .40 869 .03 078 .43 947 41 20 9.56 085 9.96 917 9.59 168 0.40 832 0.03 083 0.43 915 40 21 .56 118 .96 912 .59 205 .40 795 .03 088 .43 882 39 22 .56 150 .96 907 .59 243 .40 757 .03 093 .43 850 38 23 .56 182 .96 903 .59 280 .40 720 .03 097 .43 818 37 24 .56215 .96 898 .59 317 .40 683 .03 102 .43 785 36 25 9.56 247 9.96 893 9.59 354 0.40 646 0.03 107 0.43 753 35 26 .56 279 .96 888 .59 391 .40 609 .03 112 .43 721 34 27 .56311 .96 883 .59 429 .40 571 .03 117 .43 689 33 28 .56 343 .96 878 .59 466 .40 534 .03 122 .43 657 32 29 .56 375 .96 873 .59 503 .40 497 .03 127 .43 625 31 30 9.56 408 9.96 868 9.59 540 0.40 460 0.03 132 0.43 592 30 31 .56440 .96 863 .59 577 .40 423 .03 137 .43 560 29 32 .56 472 .96 858 .59 614 .40 386 .03 142 .43 528 28 33 .56 504 .96853 .59 651 .40 349 .03 147 .43 496 27 34 .56 536 .96848 .59 688 .40 312 .03 152 .43 464 26 35 9.56 568 9.96 843 9.59 725 0.40 275 0.03 157 0.43 432 25 36 .56 599 .96 838 .59 762 .40 238 .03 162 .43 401 24 37 .56 631 .96 833 .59 799 .40 201 .03 167 .43 369 23 38 .56 663 .96 828 .59 835 .40 165 .03 172 .43 337 22 39 .56 695 .96 823 .59 872 .40 128 .03 177 .43 305 21 40 9.56 727 9.96 818 9.59 909 0.40 091 0.03 182 0.43 273 20 41 .56 759 .96 813 .59 946 .40 054 .03 187 .43 241 19 42 .56 790 .96 808 .59 983 .40017 .03 192 .43 210 18 43 .56 822 .96 803 .60 019 .39 981 .03 197 .43 178 17 44 .56854 .96 798 .60 056 .39 944 .03 202 .43 146 16 45 9.56 886 9.96 793 9.60 093 0.39 907 0.03 207 0.43 114 15 46 .56 917 .96 788 .60 130 .39 870 .03 212 .43 083 14 47 .56 949 .96 783 .60 166 .39 834 .03 217 .43 051 13 48 .56 980 .96 778 .60 203 .39 797 .03 222 .43 020 12 49 .57 012 .96 772 .60 240 .39 760 .03 228 .42 988 11 50 9.57 044 9.96 767 9.60 276 0.39 724 0.03 233 0.42 956 10 51 .57 075 .96 762 .60 313 .39 687 .03 238 .42 925 9 52 .57 107 .96 757 .60 349 .39 651 .03 243 .42 893 8 53 .57 138 .96 752 .60 386 .39 614 .03 248 .42 862 7 54 .57 169 .96 747 .60 422 .39 578 .03 253 .42 831 6 55 9.57 201 9.96 742 9.60 459 0.39 541 0.03 258 0.42 799 5 56 .57 232 .96 737 .60 495 .39 505 .03263 .42 768 4 57 .57 264 .96732 .60 532 .39 468 .03 268 .42 736 3 58 .57 295 .96 727 .60 568 .39 432 .03 273 .42 705 2 59 .57 326 .96 722 .60 605 .39 395 .03 278 .42 674 1 60 9.57 358 9.96 717 9.60 641 0.39 359 0.03 283 0.42 642 Cos Sin Cot Tan Csc Sec ' 111 (291) (248) 68 218 Table 4. Trigonometric Logarithms 22 (202) (337) 157 ' Sin Cos Tan Cot Sec Csc 9.57 358 9.96 717 9.60 641 0.39 359 0.03 283 0.42 642 60 1 .57 389 .96711 .60 677 .39 323 .03 289 .42 611 59 2 .57 420 .96 706 .60 714 .39 286 .03 294 .42 580 58 3 .57 451 .96 701 .60 750 .39 250 .03 299 .42 549 57 4 .57 482 .96 696 .60 786 .39 214 .03 304 .42 518 56 5 9.57 514 9.96 691 9.60 823 0.39 177 0.03 309 0.42 486 55 6 .57 545 .96 686 .60 859 .39 141 .03 314 .42 455 54 7 .57 576 .96 681 .60 895 .39 105 .03 319 .42 424 53 8 .57 607 .96 676 .60 931 .39 069 .03 324 .42 393 52 9 .57 638 .96 670 .60 967 .39 033 .03 330 .42 362 51 10 9.57 669 9.96 665 9.61 004 9.38 996 0.03 335 0.42 331 50 11 .57 700 .96 660 .61 040 .38 960 .03 340 .42 300 49 12 .57 731 .96 655 .61 076 .38 924 .03 345 .42 269 48 13 .57 762 .96 650 .61 112 .38 888 .03 350 .42 238 47 14 .57 793 .96 645 .61 148 .38 852 .03 355 .42 207 46 15 9.57 824 9.96 640 9.61 184 0.38 816 0.03 360 0.42 176 45 16 .57 855 .96 634 .61 220 .38 780 .03 366 .42 145 44 17 .57 885 .96 629 .61 256 .38 744 .03 371 .42 115 43 18 .57 916 .96 624 .61 292 .38 708 .03 376 .42084 42 19 .57 947 .96 619 .61 328 .38 672 .03 381 .42 053 41 20 9.57 978 9.96 614 9.61 364 0.38 636 0.03 386 0.42 022 40 21 .58 008 .96 608 .61 400 .38 600 .03 392 .41 992 39 22 .58 039 .96 603 .61 436 .38 564 .03 397 .41 961 38 23 .58 070 .96 598 .61 472 .38 528 .03 402 .41 930 37 24 .58 101 .96 593 .61 508 .38 492 .03 407 .41 899 36 25 9.58 131 9.96 588 9.61 544 0.38 456 0.03 412 0.41 869 35 26 .58 162 .96 582 .61 579 .38 421 .03 418 .41 838 34 27 .58 192 .96 577 .61 615 .38 385 .03 423 .41 808 33 28 .58 223 .96 572 .61 651 .38 349 .03 428 .41 777 32 29 .58 253 .96 567 .61 687 .38313 .03 433 .41 747 31 30 9.58 284 9.96 562 9.61 722 0.38 278 0.03 438 0.41 716 30 31 .58 314 .96 556 .61 758 .38 242 .03 444 .41 686 29 32 .58 345 .96 551 .61 794 .38 206 .03 449 .41 655 28 33 .58 375 .96 546 .61 830 .38 170 .03 454 .41 625 27 34 .58 406 .96 541 .61 865 .38 135 .03 459 .41 594 26 35 9.58 436 9.96 535 9.61 901 0.38 099 0.03 465 0.41 564 25 36 .58 467 .96 530 .61 936 .38 064 .03 470 .41 533 24 37 .58 497 .96 525 .61 972 .38 028 .03 475 .41 503 23 38 .58 527 .96 520 .62 008 .37 992 .03 480 .41 473 22 39 .58 557 .96 514 .62 043 .37 957 .03 486 .41 443 21 40 9.58 588 9.96 509 9.62 079 0.37 921 0.03 491 0.41 412 20 41 .58 618 .96 504 .62 114 .37 886 .03 496 .41 382 19 42 .58648 .96 498 .62 150 .37 850 .03 502 .41 352 18 43 .58 678 .96 493 .62 185 .37 815 .03 507 .41 322 17 44 .58 709 .96 488 .62 221 .37 779 .03 512 .41 291 16 45 9.58 739 9.96 483 9.62 256 0.37 744 0.03 517 0.41 261 15 46 .58 769 .96 477 .62 292 .37 708 .03 523 .41 231 14 47 .58 799 .96 472 .62 327 .37 673 .03 528 .41 201 13 48 .58 829 .96 467 .62 362 .37 638 .03 533 .41 171 12 49 .58 859 .96 461 .62 398 .37 602 .03 539 .41 141 11 50 9.58 889 9.96 456 9.62 433 0.37 567 0.03 544 0.41 111 10 51 .58 919 .96 451 .62 468 .37 532 .03 549 .41 081 9 52 .58 949 .96 445 .62 504 .37 496 .03 555 .41 051 8 53 .58 979 .96 440 .62 539 .37 461 .03 560 .41 021 7 54 .59 009 .96 435 .62 574 .37 426 .03 565 .40 991 6 55 9.59 039 9.96 429 9.62 609 0.37 391 0.03 571 0.40 961 5 56 .59 069 .96 424 .62 645 .37 355 .03 576 .40 931 4 57 .59 098 .96 419 .62 680 .37 320 .03 581 .40 902 3 58 .59 128 .96 413 .62 715 .37 285 .03 587 .40 872 2 59 .59 158 .96 408 .62 750 .37 250 .03 592 .40842 1 60 9.59 188 9.96 403 9.62 785 0.37 215 0.03 597 0.40 812 Cos Sin Cot Tan Csc Sec ' 112 (292) (247) 67 Table 4. Trigonometric Logarithms 219 23 (203) (336) 156 ' Sin Cos Tan Cot Sec Csc 9.59 188 9.96 403 9.62 785 0.37 215 0.03 597 0.40 812 60 1 .59 218 .96 397 .62 820 .37 180 .03 603 .40 782 59 2 .59 247 .96 392 .62 855 .37 145 .03 608 .40 753 58 3 .59 277 .96 387 .62 890 .37 110 .03 613 .40 723 57 4 .59 307 .96 381 .62 926 .37 074 .03 619 .40 693 56 5 9.59 336 9.96 376 9.62 961 0.37 039 0.03 624 0.40 664 55 6 .59 366 .96 370 .62 996 .37 004 .03 630 .40 634 54 7 .59 396 .96 365 .63 031 .36 969 .03 635 .40 604 53 8 .59 425 .96 360 .63 066 .36 934 .03 640 .40 575 52 9 .59 455 .96 354 .63 101 .36 899 .03 646 .40 545 51 10 9.59 484 9.96 349 9.63 135 0.36 865 0.03 651 0.40 516 50 11 .59 514 .96 343 .63 170 .36 830 .03 657 .40 486 49 12 .59 543 .96 338 .63 205 .36 795 .03 662 .40 457 48 13 .59 573 .96 333 .63 240 .36 760 .03 667 .40 427 47 14 .59 602 .96 327 .63 275 .36 725 .03 673 .40 398 46 15 9.59 632 9.96 322 9.63 310 0.36 690 0.03 678 0.40 368 45 16 .59 661 .96 316 .63 345 .36 655 .03 684 .40 339 44 17 .59 690 .96311 .63 379 .36 621 .03 689 .40 310 43 18 .59 720 .96 305 .63 414 .36 586 .03 695 .40 280 42 19 .59 749 .96 300 .63 449 .36 551 .03 700 .40 251 41 20 9.59 778 9.96 294 9.63 484 0.36 516 0.03 706 0.40 222 40 21 .59 808 .96 289 .63 519 .36 481 .03 711 .40 192 39 22 .59 837 .96284 .63 553 .36 447 .03 716 .40 163 38 23 .59 866 .96 278 .63 588 .36 412 .03 722 .40 134 37 24 .59 895 .96 273 .63 623 .36 377 .03 727 .40 105 36 25 9.59 924 9.96 267 9.63 657 0.36 343 0.03 733 0.40 076 35 26 .59 954 .96 262 .63 692 .36 308 .03 738 .40 046 34 27 .59 983 .96 256 .63 726 .36 274 .03 744 .40 017 33 28 .60 012 .96 251 .63 761 .36 239 .03 749 .39 988 32 29 .60041 .96 245 .63 796 .36 204 .03 755 .39 959 31 30 9.60 070 9.96 240 9.63 830 0.36 170 0.03 760 .39 930 30 31 .60 099 .96 234 .63 865 .36 135 .03 766 .39 901 29 32 .60 128 .96 229 .63 899 .36 101 .03 771 .39 872 28 33 .60 157 .96 223 .63 934 .36 066 .03 777 .39843 27 34 .60 186 .96 218 .63 968 .36 032 .03 782 .39 814 26 35 9.60 215 9.96 212 9.64 003 0.35 997 0.03 788 0.39 785 25 36 .60244 .96 207 .64 037 .35 963 .03 793 .39 756 24 37 .60 273 .96 201 .64072 .35 928 .03 799 .39 727 23 38 .60 302 .96 196 .64 106 .35 894 .03 804 .39 698 22 39 .60 331 .96 190 .64 140 .35 860 .03 810 .39 669 21 40 9.60 359 9.96 185 9.64 175 0.35 825 0.03 815 0.39 641 20 41 .60 388 .96 179 .64 209 .35 791 .03 821 .39 612 19 42 .60 417 .96 174 .64243 .35 757 .03 826 .39 583 18 43 .60446 .96 168 .64278 .35 722 .03 832 .39 554 17 44 .60 474 .96 162 .64312 .35 688 .03 838 .39 526 16 45 9.60 503 9.96 157 9.64 346 0.35 654 0.03 843 0.39 497 15 46 .60 532 .96 151 .64 381 .35 619 .03849 .39 468 14 47 .60 561 .96 146 .64 415 .35 585 .03 854 .39 439 13 48 .60 589 .96 140 .64449 .35 551 .03 860 .39411 12 49 .60 618 .96 135 .64483 .35 517 .03 865 .39 382 11 50 9.60 646 9.96 129 9.64517 0.35 483 0.03 871 0.39 354 10 51 .60 675 .96 123 .64 552 .35448 .03 877 .39 325 9 52 .60 704 .96 118 .64586 .35 414 .03 882 .39 296 8 53 .60 732 .96112 .64620 .35 380 .03 888 .39 268 7 54 .60 761 .96 107 .64654 .35 346 .03 893 .39 239 6 55 9.60 789 9.96 101 9.64 688 0.35 312 0.03 899 0.39 211 5 56 .60 818 .96 095 .64 722 .35 278 .03 905 .39 182 4 57 .60846 .96 090 .64 756 .35244 .03 910 .39 154 3 58 .60 875 .96084 .64 790 .35 210 .03 916 .39 125 2 59 .60 903 .96 079 .64 824 .35 176 .03 921 ,39 097 1 60 9.60 931 9.96 073 9.64 858 0.35 142 0.03 927 0.39 069 Cos Sin Cot Tan Csc Sec ' 113 (293) (246) 66 C 220 Table 4. Trigonometric Logarithms 24 (204) (335) 155' ' Sin Cos Tan Cot Sec Csc 9.60 931 9.96 073 9.64 858 0.35 142 0.03 927 0.39 069 60 1 .60 960 .96 067 .64 892 .35 108 .03 933 .39 040 59 2 .60 988 .96 062 .64926 .35 074 .03 938 .39 012 58 3 .61 016 .96 056 .64 960 .35 040 .03 944 .38 984 57 4 .61 045 .96 050 .64 994 .35 006 .03 950 .38 955 56 5 9.61 073 9.96 045 9.65 028 0.34 972 0.03 955 0.38 927 55 6 .61 101 .96 039 .65 062 .34 938 .03 961 .38 899 54 7 .61 129 .96 034 .65 096 .34 904 .03 966 .38 871 53 8 .61 158 .96 028 .65 130 .34 870 .03 972 .38 842 52 9 .61 186 .96 022 .65 164 .34 836 .03 978 .38 814 51 10 9.61 214 9.96 017 9.65 197 0.34 803 0.03 983 0.38 786 50 11 .61 242 .96 Oil .65 231 .34 769 .03 989 .38 758 49 12 .61 270 .96 005 .65 265 .34 735 .03 995 .38 730 48 13 .61 298 .96 000 .65 299 .34 701 .04 000 .38 702 47 14 .61 326 .95 994 .65 333 .34 667 .04 006 .38 674 46 15 9.61 354 9.95 988 9.65 366 0.34 634 0.04 012 0.38 646 45 16 .61 382 .95 982 .65 400 .34 600 .04 018 .38618 44 17 .61 411 .95 977 .65 434 .34 566 .04 023 .38 589 43 18 .61 438 .95 971 .65 467 .34 533 .04 029 .38 562 42 19 .61 466 .95 965 .65 501 .34 499 .04 035 .38 534 41 20 9.61 494 9.95 960 9.65 535 0.34 465 0.04 040 0.38 506 40 21 .61 522 .95 954 .65 568 .34 432 .04 046 .38 478 39 22 .61 550 .95 948 .65 602 .34 398 .04 052 .38 450 38 23 .61 578 .95 942 .65 636 .34 364 .04 058 .38 422 37 24 .61 606 .95 937 .65 669 .34 331 .04 063 .38 394 36 25 9.61 634 9.95 931 9.65 703 0.34 297 0.04 069 0.38 366 35 26 .61 662 .95 925 .65 736 .34 264 .04 075 .38 338 34 27 .61 689 .95 920 .65 770 .34 230 .04 080 .38311 33 28 .61 717 .95 914 .65 803 .34 197 .04 086 .38 283 32 29 .61 745 .95 908 .65 837 .34 163 .04 092 .38 255 31 30 9.61 773 9.95 902 9.65 870 0.34 130 0.04 098 0.38 227 30 31 .61 800 .95 897 .65 904 .34 096 .04 103 .38 200 29 32 .61 828 .95 891 .65 937 .34 063 .04 109 .38 172 28 33 .61 856 .95 885 .65 971 .34 029 .04 115 .38 144 27 34 .61 883 .95 879 .66 004 .33 996 .04 121 .38 117 26 35 9.61 911 9.95 873 9.66 038 0.33 962 0.04 127 0.38 089 25 36 .61 939 .95 868 .66 071 .33 929 .04 132 .38 061 24 37 .61 966 .95 862 .66 104 .33 896 .04 138 .38 034 23 38 .61 994 .95 856 .66 138 .33 862 .04 144 .38 006 22 39 .62 021 .95 850 .66 171 .33 829 .04 150 .37 979 21 40 9.62 049 9.95 844 9.66 204 0.33 796 0.04 156 0.37 951 20 41 .62 076 .95 839 .66 238 .33 762 .04 161 .37 924 19 42 .62 104 .95 833 .66 271 .33 729 .04 167 .37 896 18 43 .62 131 .95 827 .66 304 .33 696 .04 173 .37 869 17 44 .62 159 .95 821 .66 337 .33 663 .04 179 .37841 16 45 9.62 186 9.95 815 9.66 371 0.33 629 0.04 185 0.37 814 15 46 .62 214 .95 810 .66 404 .33 596 .04 190 .37 786 14 47 .62 241 .95 804 .66 437 .33 563 .04 196 .37 759 13 48 .62 268 .95 798 .66 470 .33 530 .04 202 .37 732 12 49 .62 296 .95 792 .66 503 .33 497 .04 208 .37 704 11 50 9.62 323 9.95 786 9.66 537 0.33 463 0.04 214 0.37 677 10 51 .62 350 .95 780 .66 570 .33 430 .04 220 .37 650 9 52 .62 377 .95 775 .66 603 .33 397 .04 225 .37 623 8 53 .62 405 .95 769 .66 636 .33 364 .04 231 .37 595 7 54 .62 432 .95 763 .66 669 .33 331 .04 237 .37 568 6 55 9.62 459 9.95 757 9.66 702 0.33 298 0.04 243 0.37 541 5 56 .62 486 .95 751 .66 735 .33 265 .04 249 .37 514 4 57 .62 513 .95 745 .66 768 .33 232 .04 255 .37 487 3 58 .62 541 .95 739 .66 801 .33 199 .04 261 .37 459 2 59 .62 568 .95 733 .66 834 .33 166 .04 267 .37 432 1 60 9.62 595 9.95 728 9.66 867 0.33 133 0.04 272 0.37 405 Cos Sin Cot Tan Csc Sec ' 114 (294) (245) 65 C Table 4. Trigonometric Logarithms 221 25 (205) (334) 154 C ' Sin Cos Tan Cot Sec Csc 9.62 595 9.95 728 9.66 867 0.33 133 0.04 272 0.37 405 60 1 .62 622 .95 722 .66 900 .33 100 .04 278 .37 378 59 2 .62 649 .95 716 .66 933 .33 067 .04284 .37 351 58 3 .62 676 .95 710 .66 966 .33 034 .04 290 .37 324 57 4 .62 703 .95 704 .66999 .33 001 .04 296 .37 297 56 5 9.62 730 9.95 698 9.67 032 0.32 968 0.04 302 0.37 270 55 6 .62 757 .95 692 .67 065 .32 935 .04308 .37 243 54 7 .62784 .95 686 .67 098 .32 902 .04 314 .37 216 53 8 .62811 .95 680 .67 131 .32 869 .04 320 .37 189 52 9 .62838 .95 674 .67 163 .32 837 .04 326 .37 162 51 10 9.62 865 9.95 668 9.67 196 0.32 804 0.04 332 0.37 135 50 11 .62 892 .95 663 .67 229 .32 771 .04337 .37 108 49 12 .62 918 .95 657 .67 262 .32 738 .04 343 .37 082 48 13 .62 945 .95 651 .67 295 .32 705 .04 349 .37 055 47 14 .62 972 .95645 .67 327 .32 673 .04355 .37 028 46 15 9.62 999 9.95 639 9.67 360 0.32 640 0.04 361 0.37 001 45 16 .63 026 .95 633 .67 393 .32 607 .04 367 .36 974 44 17 .63 052 .95 627 .67 426 .32 574 .04 373 .36 948 43 18 .63 079 .95 621 .67 458 .32 542 .04 379 .36 921 42 19 .63 106 .95 615 .67 491 .32 509 .04385 .36 894 41 20 9.63 133 9.95 609 9.67 524 0.32 476 0.04 391 0.36 867 40 21 .63 159 .95 603 .67 556 .32 444 .04 397 .36841 39 22 .63 186 .95 597 .67 589 .32411 .04 403 .36 814 38 23 .63 213 .95 591 .67 622 .32 378 .04409 .36 787 37 24 .63 239 .95 585 .67 654 .32 346 .04 415 .36 761 36 25 9.63 266 9.95 579 9.67 687 0.32 313 0.04 421 0.36 734 35 26 .63 292 .95 573 .67 719 .32 281 .04 427 .36 708 34 27 .63 319 .95 567 .67 752 .32 248 .04 433 .36 681 33 28 .63 345 .95 561 .67 785 .32 215 .04 439 .36 655 32 29 .63 372 .95 555 .67 817 .32 183 .04 445 .36 628 31 30 9.63 398 9.95 549 9.67 850 0.32 150 0.04 451 0.36 602 30 31 .63 425 .95 543 .67 882 .32 118 .04 457 .36 575 29 32 .63 451 .95 537 .67 915 .32 085 .04 463 .36 549 28 33 .63 478 .95 531 .67 947 .32 053 .04 469 .36 522 27 34 .63 504 .95 525 .67 980 .32 020 .04 475 .36 496 26 35 9.63 531 9.95 519 9.68012 0.31 988 0.04 481 0.36 469 25 36 .63 557 .95 513 .68 044 .31 956 .04 487 .36 443 24 37 .63583 .95 507 .68 077 .31 923 .04 493 .36 417 23 38 .63 610 .95 500 .68 109 .31 891 .04 500 .36 390 22 39 .63 636 .95 494 .68 142 .31 858 .04506 .36 364 21 40 9.63 662 9.95 488 9.68 174 0.31 826 0.04 512 0.36 338 20 41 .63 689 .95 482 .68 206 .31 794 .04 518 . 36311 19 42 .63 715 .95 476 .68 239 .31 761 .04 524 .36 285 18 43 .63 741 .95 470 .68 271 .31 729 .04 530 .36 259 17 44 .63 767 .95 464 .68 303 .31 697 .04536 .36 233 16 45 9.63 794 9.95 458 9.68 336 0.31 664 0.04 542 0.36 206 15 46 .63 820 .95 452 .68 368 .31 632 .04 548 .36 180 14 47 .63846 .95 446 .68 400 .31 600 .04554 .36 154 13 48 .63 872 .95 440 .68 432 .31 568 .04 560 .36 128 12 49 .63 898 .95 434 .68 465 .31 535 .04 566 .36 102 11 50 9.63 924 9.95 427 9.68 497 0.31 503 0.04 573 0.36 076 10 51 .63 950 .95 421 .68 529 .31 471 .04 579 .36 050 9 52 .63 976 .95 415 .68 561 .31 439 .04 585 .36 024 8 53 .64 002 .95 409 .68 593 .31 407 .04 591 .35 998 7 54 .64 028 .95 403 .68 626 .31 374 .04597 .35 972 6 55 9.64 054 9.95 397 9.68 658 0.31 342 0.04 603 0.35 946 5 56 .64080 .95 391 .68 690 .31 310 .04 609 .35 920 4 57 .64 106 .95 384 .68 722 .31 278 .04 616 .35 894 3 58 .64132 .95 378 .68 754 .31 246 .04 622 .35 868 2 59 .64158 .95 372 .68 786 .31 214 .04 628 .35 842 1 60 9.64 184 9.95 366 9.68 818 0.31 182 0.04 634 0.35 816 Cos Sin Cot Tan Csc Sec ' 115 (295) (244) 64' 222 Table 4. Trigonometric Logarithms 26 (206) (333) 153 C ' Sin Cos Tan Cot Sec Csc 9.64 184 9.95 366 9.68818 0.31 182 0.04 634 0.35 816 60 1 .64 210 .95 360 .68 850 .31 150 .04 640 .35 790 59 2 .64 236 .95 354 .68 882 .31 118 .04 646 .35 764 58 3 .64 262 .95 348 .68 914 .31 086 .04 652 .35 738 57 4 .64 288 .95 341 .68 946 .31 054 .04 659 .35 712 56 5 9.64 313 9.95 335 9.68 978 0.31 022 0.04 665 0.35 687 55 6 .64339 .95 329 .69 010 .30990 .04 671 .35 661 54 7 .64 365 .95 323 .69 042 .30 958 .04677 .35 635 53 8 .64 391 .95 317 .69 074 .30 926 .04 683 .35 609 52 9 .64 417 .95 310 .69 106 .30 894 .04 690 .35 583 51 10 9.64 442 9.95 304 9.69 138 0.30 862 0.04 696 0.35 558 50 11 .64 468 .95 298 .69 170 .30 830 .04 702 .35 532 49 12 .64 494 .95 292 .69 202 .30 798 .04 708 .35 506 48 13 .64 519 .95 286 .69 234 .30 766 .04 714 .35 481 47 14 .64 545 .95 279 .69 266 .30 734 .04 721 .35 455 46 15 9.64 571 9.95 273 9.69 298 0.30 702 0.04 727 0.35 429 45 16 .64 596 .95 267 .69 329 .30 671 .04 733 .35 404 44 17 .64 622 .95 261 .69 361 .30 639 .04 739 .35 378 43 18 .64 647 .95 254 .69 393 .30 607 .04 746 .35353 42 19 .64 673 .95 248 .69 425 .30 575 .04 752 .35 327 41 20 9.64 698 9.95 242 9.69 457 0.30 543 0.04 758 0.35 302 40 21 .64 724 .95 236 .69 488 .30 512 .04 764 .35 276 39 22 .64 749 .95 229 .69 520 .30 480 .04 771 .35 251 38 23 .64 775 .95 223 .69 552 .30 448 .04 777 .35 225 37 24 .64 800 .95 217 .69 584 .30 416 .04 783 .35 200 36 25 9.64 826 9.95211 9.69 615 0.30 385 0.04 789 0.35 174 35 26 .64 851 .95 204 .69 647 .30 353 .04 796 .35 149 34 27 .64877 .95 198 .69 679 .30 321 .04 802 .35 123 33 28 .64 902 .95 192 .69 710 .30 290 .04808 .35 098 32 29 .64 927 .95 185 .69 742 .30 258 .04 815 .35 073 31 30 9.64 953 9.95 179 9.69 774 0.30 226 0.04 821 0.35 047 30 31 .64 978 .95 173 .69 805 .30 195 .04 827 .35 022 29 32 .65 003 .95 167 .69 837 .30 163 .04 833 .34 997 28 33 .65 029 .95 160 .69 868 .30 132 .04 840 .34 971 27 34 .65 054 .95 154 .69 900 .30 100 .04846 .34 946 26 35 9.65 079 9.95 148 9.69 932 0.30 068 0.04 852 0.34 921 25 36 .65 104 .95 141 .69 963 .30 037 .04 859 .34 896 24 37 .65 130 .95 135 .69 995 .30 005 .04 865 .34 870 23 38 .65 155 .95 129 .70 026 .29 974 .04 871 .34 845 22 39 .65 180 .95 122 .70 058 .29 942 .04 878 .34 820 21 40 9.65 205 9.95 116 9.70 089 0.29911 0.04 884 0.34 795 20 41 .65 230 .95 110 .70 121 .29 879 .04 890 .34 770 19 42 .65 255 .95 103 .70 152 .29848 .04 897 .34 745 18 43 .65 281 .95 097 .70 184 .29 816 .04 903 .34 719 17 44 .65 306 .95 090 .70 215 .29 785 .04 910 .34 694 16 45 9.65 331 9.95 084 9.70 247 0.29 753 0.04 916 0.34 669 15 46 .65 356 .95 078 .70 278 .29 722 .04 922 .34 644 14 47 .65 381 .95 071 .70 309 .29 691 .04 929 .34 619 13 48 .65 406 .95 065 .70 341 .29 659 .04 935 .34 594 12 49 .65 431 .95 059 .70 372 .29 628 .04 941 .34 569 11 50 9.65 456 9.95 052 9.70 404 0.29 596 0.04 948 0.34 544 10 51 .65 481 .95 046 .70 435 .29 565 .04 954 .34 519 9 52 .65 506 .95 039 .70 466 .29 534 .04 961 .34 494 8 53 .65 531 .95 033 .70 498 .29 502 .04 967 .34 469 7 54 .65 556 .95 027 .70 529 .29 471 .04 973 .34 444 6 55 9.65 580 9.95 020 9.70 560 0.29 440 0.04 980 0.34 420 5 56 .65 605 .95 014 .70 592 .29 408 .04 986 .34 395 4 57 .65 630 .95 007 .70 623 .29 377 .04 993 .34 370 3 58 .65 655 .95 001 .70 654 .29 346 .04 999 .34 345 2 59 .65 680 .94 995 .70 685 .29 315 .05 005 .34 320 1 60 9.65 705 9.94 988 9.70 717 0.29 283 0.05 012 0.34 295 Cos Sin Cot Tan Csc Sec ' 116 (296) (243) 63 C Table 4. Trigonometric Logarithms 223 27 (207) (332) 152 Sin Cos Tan Cot Sec Csc 9.65 705 9.94 988 9.70717 0.29 283 0.05 012 0.34 295 60 1 .65 729 .94 982 .70 748 .29 252 .05 018 .34 271 59 2 .65 754 .94 975 .70 779 .29 221 .05 025 .34 246 58 3 .65 779 .94 969 .70 810 .29 190 .05 031 .34 221 57 4 .65 804 .94 962 .70841 .29 159 .05 038 .34 196 56 5 9.65 828 9.94 956 9.70 873 0.29 127 0.05 044 0.34 172 55 6 .65 853 .94 949 .70 904 .29 096 .05 051 .34 147 54 7 .65 878 .94 943 .70 935 .29 065 .05 057 .34 122 53 8 .65 902 .94 936 .70 966 .29 034 .05 064 .34 098 52 9 .65 927 .94 930 .70 997 .29 003 .05 070 .34 073 51 10 9.65 952 9.94 923 9.71 028 0,28 972 0.05 077 0.34 048 50 11 .65 976 .94 917 .71 059 .28 941 .05 083 .34 024 49 12 .66 001 .94911 .71 090 .28 910 .05 089 .33 999 48 13 .66 025 .94 904 .71 121 .28 879 .05 096 .33 975 47 14 .66 050 .94 898 .71 153 .28847 .05 102 .33 950 46 15 9.66 075 9.94 891 9.71 184 0.28 816 0.05 109 0.33 925 45 16 .66 099 .94 885 .71 215 .28 785 .05 115 .33 901 44 17 .66 124 .94 878 .71 246 .28 754 .05 122 .33 876 43 18 .66 148 .94 871 .71 277 .28 723 .05 129 .33 852 42 19 .66 173 .94 865 .71 308 .28 692 .05 135 .33 827 41 20 9.66 197 9.94 858 9.71 339 0.28 661 0.05 142 0.33 803 40 21 .66 221 .94 852 .71 370 .28 630 .05 148 .33 779 39 22 .66 246 .94845 .71 401 .28 599 .05 155 .33 754 38 23 .66 270 .94 839 .71 431 .28 569 .05 161 .33 730 37 24 .66 295 .94 832 .71 462 .28 538 .05 168 .33 705 36 25 9.66 319 9.94 826 9.71 493 0.28 507 0.05 174 0.33 681 35 26 .66 343 .94 819 .71 524 .28 476 .05 181 .33 657 34 27 .66 368 .94 813 .71 555 .28 445 .05 187 .33 632 33 28 .66 392 .94 806 .71 586 .28 414 .05 194 .33 608 32 29 .66 416 .94 799 .71 617 .28 383 .05 201 .33 584 31 30 9.66 441 9.94 793 9.71 648 0.28 352 0.05 207 0.33 559 30 31 .66 465 .94 786 .71 679 .28 321 .05 214 .33 535 29 32 .66 489 .94 780 .71 709 .28 291 .05 220 .33511 28 33 .66 513 .94 773 .71 740 .28 260 .05 227 .33 487 27 34 .66 537 .94 767 .71 771 .28 229 .05 233 .33 463 26 35 9.66 562 9.94 760 9.71 802 0.28 198 0.05 240 0.33 438 25 36 .66 586 .94 753 .71 833 .28 167 .05 247 .33 414 24 37 .66 610 .94 747 .71 863 .28 137 .05 253 .33 390 23 38 .66 634 .94 740 .71 894 .28 106 .05 260 .33 366 22 39 .66 658 .94 734 .71 925 .28 075 .05 266 .33 342 21 40 9.66 682 9.94 727 9.71 955 0.28 045 0.05 273 0.33 318 20 41 .66 706 .94 720 .71 986 .28 014 .05 280 .33 294 19 42 .66 731 .94 714 .72 017 .27 983 .05 286 .33 269 18 43 .66 755 .94 707 .72 048 .27 952 .05 293 .33 245 17 44 .66 779 .94 700 .72 078 .27 922 .05 300 .33 221 16 45 9.66 803 9.94 694 9.72 109 0.27 891 0.05 306 0.33 197 15 46 .66 827 .94 687 .72 140 .27 860 .05 313 .33 173 14 47 .66 851 .94 680 .72 170 .27 830 .05 320 .33 149 13 48 .66 875 .94 674 .72 201 .27 799 .05 326 .33 125 12 49 .66 899 .94667 .72 231 .27 769 .05 333 .33 101 11 50 9.66 922 9.94 660 9.72 262 0.27 738 0.05 340 0.33 078 10 51 .66 946 .94 654 .72 293 .27 707 .05 346 .33 054 9 52 .66 970 .94647 .72 323 .27 677 .05 353 .33 030 8 53 .66 994 .94 640 .72 354 .27646 .05 360 .33 006 7 54 .67 018 .94 634 .72384 .27 616 .05 366 .32 982 6 55 9.67 042 9.94 627 9.72 415 0.27 585 0.05 373 0.32 958 5 56 .67 066 .94 620 .72445 .27 555 .05 380 .32 934 4 57 .67 090 .94 614 .72 476 .27 524 .05 386 .32 910 3 58 .67 113 .94 607 .72 506 .27 494 .05 393 .32 887 2 59 .67 137 .94 600 .72 537 .27 463 .05400 .32 863 1 60 9.67 161 9.94 593 9.72 567 0.27 433 0.05 407 0.32 839 Cos Sin Cot Tan Csc Sec ' 117 (297) (242) 62 224 Table 4. Trigonometric Logarithms 28 (208) (331) 151 C ' Sin Cos Tan Cot Sec Csc 9.67 161 9.94 593 9.72 567 0.27 433 0.05 407 0.32 839 60 1 .67 185 .94 587 .72 598 .27 402 .05413 .32 815 59 2 .67 208 .94 580 .72 628 .27 372 .05 420 .32 792 58 3 .67 232 .94 573 .72 659 .27 341 .05 427 .32 768 57 4 .67 256 .94 567 .72 689 .27311 .05 433 .32 744 56 5 9.67 280 9.94 560 9.72 720 0.27 280 0.05 440 0.32 720 55 6 .67 303 .94 553 .72 750 .27 250 .05 447 .32 697 54 7 .67 327 .94 546 .72 780 .27 220 .05 454 .32 673 53 8 .67 350 .94 540 .72811 .27 189 .05 460 .32 650 52 9 .67 374 .94 533 .72841 .27 159 .05 467 .32 626 51 10 9.67 398 9.94 526 9.72 872 0.27 128 0.05 474 0.32 602 50 11 .67 421 .94 519 .72 902 .27 098 .05 481 .32 579 49 12 .67 445 .94 513 .72 932 .27 068 .05 487 .32 555 48 13 .67 468 .94 506 .72 963 .27 037 .05 494 .32 532 47 14 .67 492 .94 499 .72 993 .27 007 .05 501 .32 508 46 15 9.67 515 9.94 492 9.73 023 0.26 977 0.05 508 0.32 485 45 16 .67 539 .94 485 .73 054 .26 946 .05 515 .32 461 44 17 .67 562 .94 479 .73084 .26 916 .05 521 .32 438 43 18 .67 586 .94 472 .73 114 .26 886 .05 528 .32 414 42 19 .67 609 .94 465 .73 144 .26 856 .05 535 .32 391 41 20 9.67 633 9.94 458 9.73 175 0.26 825 0.05 542 0.32 367 40 21 .67 656 .94 451 .73 205 .26 795 .05 549 .32 344 39 22 .67 680 .94 445 .73 235 .26 765 .05 555 .32 320 38 23 .67 703 .94 438 .73 265 .26 735 .05 562 .32 297 37 24 .67 726 .94 431 .73 295 .26 705 .05 569 .32 274 36 25 9.67 750 9.94 424 9.73 326 0.26 674 0.05 576 0.32 250 35 26 .67 773 .94 417 .73 356 .26 644 .05 583 .32 227 34 27 .67 796 .94 410 .73 386 .26 614 .05 590 .32 204 33 28 .67 820 .94 404 .73 416 .26 584 .05 596 .32 180 32 29 .67 843 .94 397 .73 446 .26 554 .05 603 .32 157 31 30 9.67 866 9.94 390 9.73 476 0.26 524 0.05 610 0.32 134 30 31 .67 890 .94 383 .73 507 .26 493 .05 617 .32 110 29 32 .67 913 .94 376 .73 537 .26 463 .05 624 .32 087 28 33 .67 936 .94 369 .73 567 .26 433 .05 631 .32 064 27 34 .67 959 .94 362 .73 597 .26 403 .05 638 .32 041 26 35 9.67 982 9.94 355 9.73 627 0.26 373 0.05 645 0.32 018 25 36 .68 006 .94 349 .73 657 .26 343 .05 651 .31 994 24 37 .68 029 .94 342 .73 687 .26 313 .05 658 .31 971 23 38 .68 052 .94 335 .73 717 .26 283 .05 665 .31 948 22 39 .68 075 .94 328 .73 747 .26 253 .05 672 .31 925 21 40 9.68 098 9.94 321 9.73 777 0.26 223 0.05 679 0.31 902 20 41 .68 121 .94 314 .73 807 .26 193 .05 686 .31 879 19 42 .68 144 .94 307 .73 837 .26 163 .05 693 .31 856 18 43 .68 167 .94 300 .73 867 .26 133 .05 700 .31 833 17 44 .68 190 .94 293 .73 897 .26 103 .05 707 .31 810 16 45 9.68 213 9.94 286 9.73 927 0.26 073 0.05 714 0.31 787 15 46 .68 237 .94 279 .73 957 .26 043 .05 721 .31 763 14 47 .68 260 .94 273 .73 987 .26 013 .05 727 .31 740 13 48 .68 283 .94 266 .74 017 .25 983 .05 734 .31 717 12 49 .68 305 .94 259 .74 047 .25 953 .05 741 .31 695 11 50 9.68 328 9.94 252 9.74 077 0.25 923 0.05 748 0.31 672 10 51 .68 351 .94 245 .74 107 .25 893 .05 755 .31 649 9 52 .68 374 .94 238 .74 137 .25 863 .05 762 .31 626 8 53 .68 397 .94 231 .74 166 .25 834 .05 769 .31 603 7 54 .68 420 .94 224 .74 196 .25 804 .05 776 .31 580 6 55 9.68 443 9.94 217 9.74 226 0.25 774 0.05 783 0.31 557 5 56 .68 466 .94 210 .74 256 .25 744 .05 790 .31 534 4 57 .68 489 .94 203 .74 286 .25 714 .05 797 .31 511 3 58 .68512 .94 196 .74 316 .25 684 .05 804 .31 488 2 59 .68 534 .94 189 .74 345 .25 655 .05811 .31 466 1 60 9.68 557 9.94 182 9.74 375 0.25 625 0.05 818 0.31 443 Cos Sin Cot Tan Csc Sec ' 118 (298) (241) 61' Table 4. Trigonometric Logarithms 225 29 (209) (330) 150 ' Sin Cos Tan Cot Sec Csc 9.68 557 9.94 182 9.74 375 0.25 625 0.05 818 0.31 443 60 1 .68 580 .94 175 .74 405 .25 595 .05 825 .31 420 59 2 .68 603 .94 168 .74 435 .25 565 .05 832 .31 397 58 3 .68 625 .94 161 .74 465 .25535 .05 839 .31 375 57 4 .68 648 .94 154 .74 494 .25 506 .05846 .31 352 56 5 9.68 671 9.94 147 9.74 524 0.25 476 0.05 853 0.31 329 55 6 .68 694 .94 140 .74 554 .25 446 .05 860 .31 306 54 7 .68 716 .94 133 .74 583 .25417 .05 867 .31284 53 8 .68 739 .94 126 .74 613 .25 387 .05 874 .31 261 52 9 .68 762 .94 119 .74643 .25 357 .05 881 .31 238 51 10 9.68 784 9.94 112 9.74 673 0.25 327 0.05 888 0.31 216 50 11 .68 807 .94 105 .74 702 .25 298 .05 895 .31 193 49 12 .68 829 .94 098 .74 732 .25 268 .05 902 .31 171 48 13 .68 852 .94 090 .74 762 .25 238 .05 910 .31 148 47 14 .68 875 .94 083 .74 791 .25 209 .05 917 .31 125 46 15 9.68 897 9.94 076 9.74 821 0.25 179 0.05 924 0.31 103 45 16 .68 920 .94 069 .74 851 .25 149 .05 931 .31 080 44 17 .68 942 .94 062 .74 880 .25 120 .05 938 .31 058 43 18 .68 965 .94 055 .74 910 .25 090 .05 945 .31 035 42 19 .68 987 .94 048 .74 939 .25 061 .05 952 .31 013 41 20 9.69 010 9.94 041 9.74 969 0.25 031 0.05 959 0.30 990 40 21 .69 032 .94 034 .74 998 .25 002 .05 966 .30 968 39 22 .69 055 .94 027 .75 028 .24 972 .05 973 .30 945 38 23 .69 077 .94 020 .75 058 .24 942 .05 980 .30 923 37 24 .69 100 .94 012 .75 087 .24 913 .05 988 .30 900 36 25 9.69 122 9.94 005 9.75 117 0.24 883 0.05 995 0.30 878 35 26 .69 144 .93 998 .75 146 .24 854 .06 002 .30 856 34 27 .69 167 .93 991 .75 176 .24 824 .06 009 .30 833 33 28 .69 189 .93 984 .75 205 .24 795 .06 016 .30 811 32 29 .69 212 .93 977 .75 235 .24 765 .06 023 .30 788 31 30 9.69 234 9.93 970 9.75 264 0.24 736 0.06 030 0.30 766 30 31 .69 256 .93 963 .75 294 .24 706 .06 037 .30 744 29 32 .69 279 .93 955 .75 323 .24 677 .06 045 .30 721 28 33 .69 301 .93 948 .75 353 .24647 .06 052 .30 699 27 34 .69 323 .93 941 .75 382 .24 618 .06059 .30 677 26 35 9.69 345 9.93 934 9.75411 0.24 589 0.06 066 0.30 655 25 36 .69 368 .93 927 .75 441 .24 559 .06 073 .30 632 24 37 .69 390 .93 920 .75 470 .24 530 .06 080 .301610 23 38 .69 412 .93 912 .75 500 .24 500 .06 088 .30 588 22 39 .69 434 .93 905 .75 529 .24 471 .06 095 .30 566 21 40 9.69 456 9.93 898 9.75 558 0.24 442 0.06 102 0.30 544 20 41 .69 479 .93 891 .75 588 .24 412 .06 109 .30 521 19 42 .69 501 .93884 .75 617 .24 383 .06 116 .30 499 18 43 .69 523 .93 876 .75647 .24 353 .06 124 .30 477 17 44 .69 545 .93 869 .75 676 .24 324 .06 131 .30 455 16 45 9.69 567 9.93 862 9.75 705 0.24 295 0.06 138 0.30 433 15 46 .69 589 .93855 .75 735 .24 265 .06 145 .30411 14 47 .69611 .93847 .75764 .24 236 .06 153 .30 389 13 48 .69 633 .93840 .75 793 .24 207 .06 160 .30 367 12 49 .69 655 .93833 .75 822 .24 178 .06167 .30 345 11 50 9.69 677 9.93 826 9.75 852 0.24 148 0.06 174 0.30 323 10 51 .69 699 .93 819 .75 881 .24 119 .06 181 .30 301 9 52 .69 721 .93811 .75 910 .24 090 .06 189 .30 279 8 53 .69 743 .93804 .75 939 .24 061 .06 196 .30 257 7 54 .69 765 .93 797 .75 969 .24 031 .06 203 .30 235 6 55 9.69 787 9.93 789 9.75 998 0.24 002 0.06211 0.30213 5 56 .69 809 .93 782 .76 027 .23 973 .06 218 .30 191 4 57 .69 831 .93 775 .76 056 .23 944 .06 225 .30 169 3 58 .69 853 .93 768 .76 086 .23 914 .06 232 .30 147 2 59 .69 875 .93 760 .76 115 .23 885 .06 240 .30 125 1 60 9.69 897 9.93 753 9.76 144 0.23 856 0.06 247 0.30 103 Cos Sin Cot Tan Csc Sec ' 119 (299) (240) 60 226 Table 4. Trigonometric Logarithms 30 (210) (329) 149 ' Sin Cos Tan Cot Sec Csc 9.69 897 9.93 753 9.76 144 0.23 856 0.06 247 0.30 103 60 1 .69 919 .93 746 .76 173 .23 827 .06 254 .30 081 59 2 .69 941 .93 738 .76 202 .23 798 .06 262 .30 059 58 3 .69 963 .93 731 .76 231 .23 769 .06 269 .30 037 57 4 .69 984 .93 724 .76 261 .23 739 .06 276 .30 016 56 5 9.70 006 9.93 717 9.76 290 0.23 710 0.06 283 0.29 994 55 6 .70 028 .93 709 .76319 .23 681 .06 291 .29 972 54 7 .70 050 .93 702 .76 348 .23 652 .06 298 .29 950 53 8 .70 072 .93 695 .76 377 .23 623 .06 305 .29 928 52 9 .70 093 .93 687 .76 406 .23 594 .06 313 .29 907 51 10 9.70 115 9.93 680 9.76 435 0.23 565 0.06 320 0.29 885 50 11 .70 137 .93 673 .76 464 .23 536 .06 327 .29 863 49 12 .70 159 .93 665 .76 493 .23 507 .06 335 .29841 48 13 .70 180 .93 658 .76 522 .23 478 .06 342 .29 820 45 14 .70 202 .93 650 .76 551 .23 449 .06 350 .29 798 46 15 9.70 224 9.93 643 9.76 580 0.23 420 0.06 357 0.29 776 45 16 .70 245 .93 636 .76 609 .23 391 .06 364 .29 755 44 17 .70 267 .93 628 .76 639 .23 361 .06 372 .29 733 43 18 .70 288 .93 621 .76 668 .23 332 .06 379 .29 712 42 19 .70 310 .93 614 .76 697 .23 303 .06 386 .29 690 41 20 9.70 332 9.93 606 9.76 725 0.23 275 0.06 394 0.29 668 40 21 .70 353 .93 599 .76 754 .23 246 .06 401 .29 647 39 22 .70 375 .93 591 .76 783 .23 217 .06 409 .29 625 38 23 .70 396 .93 584 .76 812 .23 188 .06 416 .29 604 37 24 .70 418 .93 577 .76 841 .23 159 .06 423 .29 582 36 25 9.70 439 9.93 569 9.76 870 0.23 130 0.06 431 0.29 561 35 26 .70 461 .93 562 .76 899 .23 101 .06 438 .29 539 34 27 .70 482 .93 554 .76 928 .23 072 .06 446 .29 518 33 28 .70 504 .93 547 .76 957 .23 043 .06 453 .29 496 32 29 .70 525 .93 539 .76 986 .23 014 .06 461 .29 475 31 30 9.70 547 9.93 532 9.77 015 0.22 985 0.06 468 0.29 453 30 31 .70 568 .93 525 .77 044 .22 956 .06 475 .29 432 29 32 .70 590 .93 517 .77 073 .22 927 .06 483 .29 410 28 33 .70611 .93 510 .77 101 .22 899 .06 490 .29 389 27 34 .70 633 .93 502 .77 130 .22 870 .06 498 .29 367 26 35 9.70 654 9.93 495 9.77 159 0.22 841 0.06 505 0.29 346 25 36 .70 675 .93 487 .77 188 .22 812 .06 513 .29 325 24 37 .70 697 .93 480 .77 217 .22 783 .06 520 .29 303 23 38 .70 718 .93 472 .77 246 .22 754 .06 528 .29 282 22 39 .70 739 .93 465 .77 274 .22 726 .06 535 .29 261 21 40 9.70 761 9.93 457 9.77 303 0.22 697 0.06 543 0.29 239 20 41 .70 782 .93 450 .77 332 .22 668 .06 550 .29 218 19 42 .70 803 .93 442 .77 361 .22 639 .06 558 .29 197 18 43 .70 824 .93 435 .77 390 .22 610 .06 565 .29 176 17- 44 .70 846 .93 427 .77 418 .22 582 .06 573 .29 154 16 45 9.70 867 9.93 420 9.77 447 0.22 553 0.06 580 0.29 133 15 46 .70 888 .93 412 .77 476 .22 524 .06 588 .29 112 14 47 .70 909 .93 405 .77 505 .22 495 .06 595 .29 091 13 48 .70 931 .93 397 .77 533 .22 467 .06 603 .29 069 12 49 .70 952 .93 390 .77 562 .22 438 .06 610 .29 048 11 50 9.70 973 9.93 382 9.77 591 0.22 409 0.06 618 0.29 027 10 51 .70 994 .93 375 .77 619 .22 381 .06 625 .29 006 9 52 .71 015 .93 367 .77 648 .22 352 .06 633 .28 985 8 53 .71 036 .93 360 .77 677 .22 323 .06 640 .28 964 7 54 .71 058 .93 352 .77 706 .22 294 .06 648 .28 942 6 55 9.71 079 9.93 344 9.77 734 0.22 266 0.06 656 0.28 921 5 56 .71 100 .93 337 .77 763 .22 237 .06 663 .28 900 4 57 .71 121 .93 329 .77 791 .22 209 .06 671 .28 879 3 58 .71 142 .93 322 .77 820 .22 180 .06 678 .28 858 2 59 .71 163 .93 314 .77 849 .22 151 .06 686 .28837 1 60 9.71 184 9.93 307 9.77 877 0.22 123 0.06 693 0.28816 Cos Sin Cot Tan Csc Sec ' 120 (300) (239) 59 Table 4. Trigonometric Logarithms 227 31 (211) (328) 148 C ' Sin Cos Tan Cot Sec Csc 9.71 184 9.93 307 9.77 877 0.22 123 0.06 693 0.28 816 60 1 .71 205 .93 299 .77 906 .22 094 .06 701 .28 795 59 2 .71 226 .93 291 .77 935 .22 065 .06 709 .28 774 58 3 .71 247 .93284 .77 963 .22 037 .06 716 .28 753 57 4 .71 268 .93 276 .77 992 .22 008 .06 724 .28 732 56 5 9.71 289 9.93 269 9.78 020 0.21 980 0.06 731 0.28711 55 6 .71 310 .93 261 .78 049 .21 951 .06 739 .28 690 54 7 .71 331 .93 253 .78 077 .21 -923 .06 747 .28 669 53 8 .71 352 .93 246 .78 106 .21 894 .06 754 .28 648 52 9 .71 373 .93 238 .78 135 .21 865 .06 762 .28 627 51 10 9.71 393 9.93 230 9.78 163 0.21 837 0.06 770 0.28 607 50 11 .71 414 .93 223 .78 192 '.21 808 .06 777 .28 586 49 12 .71 435 .93 215 .78 220 .21 780 .06 785 .28 565 48 13 .71 456 .93 207 .78 249 .21 751 .06 793 .28 544 47 14 .71 477 .93 200 .78 277 .21 723 .06 800 .28 523 46 15 9.71 498 9.93 192 9.78 306 0.21 694 0.06 808 0.28 502 45 16 .71 519 .93184 .78 334 .21 666 .06 816 .28 481 44 17 .71 539 .93 177 .78 363 .21 637 .06 823 .28 461 43 18 .71 560 .93 169 .78 391 .21 609 .06 831 .28440 42 19 .71 581 .93 161 .78 419 .21 581 .06 839 .28 419 41 20 9.71 602 9.93 154 .78 448 0.21 552 0.06 846 0.28 398 40 21 .71 622 .93 146 .78 476 .21 524 .06 854 .28 378 39 22 .71 643 .93 138 .78 505 .21 495 .06 862 .28 357 38 23 .71 664 .93 131 .78 533 .21 467 .06 869 .28 336 37 24 .71 685 .93 123 .78 562 .21 438 .06 877 .28 315 36 25 9.71 705 9.93 115 9.78 590 0.21 410 0.06 885 0.28 295 35 26 .71 726 .93 108 .78 618 .21 382 .06 892 .28 274 34 27 .71 747 .93 100 .78647 .21 353 .06 900 .28 253 33 28 .71 767 .93 092 .78 675 .21 325 .06 908 .28 233 32 29 .71 788 .93084 .78 704 .21 296 .06 916 .28 212 31 30 9.71 809 9.93 077 9.78 732 0.21 268 0.06 923 0.28 191 30 31 .71 829 .93 069 .78 760 .21 240 .06 931 .28 171 29 32 .71 850 .93 061 .78 789 .21 211 .06 939 .28 150 28 33 .71 870 .93 053 .78 817 .21 183 .06 947 .28 130 27 34 .71 891 .93046 .78845 .21 155 .06 954 .28 109 26 35 9.71 911 9.93 038 9.78 874 0.21 126 0.06 962 0.28 089 25 36 .71 932 .93 030 .78 902 .21 098 .06 970 .28 068 24 37 .71 952 .93 022 .78 930 .21 070 .06 978 .28 048 23 38 .71 973 .93 014 .78 959 .21041 .06 986 .28 027 22 39 .71 994 .93 007 .78 987 .21 013 .06 993 .28 006 21 40 9.72 014 9.92 999 9.79 015 0.20 985 0.07 001 0.27 986 20 41 .72 034 .92 991 .79 043 .20 957 .07 009 .27 966 19 42 .72 055 .92 983 .79 072 .20 928 .07 017 .27 945 18 43 .72 075 .92 976 .79 100 .20900 .07 024 .27 925 17 44 .72 096 .92 968 .79 128 .20 872 .07 032 .27 904 16 45 9.72 116 9.92 960 9.79 156 0.20844 0.07 040 0.27 884 15 46 .72 137 .92 952 .79 185 .20 815 .07 048 .27 863 14 47 .72 157 .92 944 .79 213 .20 787 .07 056 .27843 13 48 .72 177 .92 936 .79 241 .20 759 .07 064 .27 823 12 49 .72 198 .92 929 .79 269 .20 731 .07 071 .27 802 11 50 9.72 218 9.92 921 9.79 297 0.20 703 0.07 079 0.27 782 10 51 .72 238 .92 913 .79 326 .20 674 .07 087 .27 762 9 52 .72 259 .92 905 .79 354 .20646 .07 095 .27 741 8 53 .72 279 .92 897 .79 382 .20 618 .07 103 .27 721 7 54 .72 299 .92 889 .79 410 .20 590 .07 111 .27 701 6 55 9.72 320 9.92 881 9.79 438 0.20 562 0.07 119 0.27 680 5 56 .72 340 .92 874 .79 466 .20 534 .07 126 .27 660 4 57 .72 360 .92 866 .79 495 .20 505 .07 134 .27 640 3 58 .72 381 .92 858 .79 523 .20 477 .07 142 .27 619 2 59 .72 401 .92 850 .79 551 .20 449 .07 150 .27 599 1 60 9.72 421 9.92 842 9.79 579 0.20 421 0.07 158 0.27 579 Cos Sin Cot Tan Csc Sec ' 121 (301) (238) 58 228 Table 4. Trigonometric Logarithms 32 (212) (327) 147< / Sin Cos Tan Cot Sec Csc 9.72 421 9.92 842 9.79 579 0.20 421 0.07 158 0.27 579 60 1 .72 441 .92 834 .79 607 .20 393 .07 166 .27 559 59 2 .72 461 .92 826 .79 635 .20 365 .07 174 .27 539 58 3 .72 482 .92 818 .79 663 .20 337 .07 182 .27 518 57 4 .72 502 .92 810 .79 691 .20 309 .07 190 .27 498 56 5 9.72 522 9.92 803 9.79 719 0.20 281 0.07 197 0.27 478 55 6 .72 542 .92 795 .79 747 .20 253 .07 205 .27 458 54 7 .72 562 .92 787 .79 776 .20 224 .07 213 .27 438 53 8 .72 582 .92 779 .79 804 .20 196 .07 221 .27 418 52 9 .72 602 .92 771 .79 832 .20 168 .07 229 .27 398 51 10 9.72 622 9.92 763 9.79 860 0.20 140 0.07 237 0.27 378 50 11 .72 643 .92 755 .79 888 .20 112 .07 245 .27 357 49 12 .72 663 .92 747 .79 916 .20 084 .07 253 .27 337 48 13 .72 683 .92 739 .79 944 .20 056 .07 261 .27 317 47 14 .72 703 .92 731 .79 972 .20 028 .07 269 .27 297 46 15 9.72 723 9.92 723 9.80 000 0.20 000 0.07 277 0.27 277 45 16 .72 743 .92 715 .80 028 .19 972 .07285 .27 257 44 17 .72 763 .92 707 .80 056 .19 944 .07 293 .27 237 43 18 72 783 .92 699 .80 084 .19916 .07 301 .27 217 42 19 .72803 .92 691 .80 112 .19 888 .07 309 .27 197 41 20 9.72 823 9.92 683 9.80 140 0.19 860 0.07 317 0.27 177 40 21 .72 843 .92 675 .80 168 .19 832 .07 325 .27 157 39 22 .72 863 .92 667 .80 195 .19 805 .07 333 .27 137 38 23 .72 883 .92 659 .80 223 .19 777 .07 341 .27 117 37 24 .72 902 .92 651 .80 251 .19 749 .07 349 .27 098 36 25 9.72 922 9.92 643 9.80 279 0.19 721 0.07 357 0.27 078 35 26 .72 942 .92 635 .80 307 .19 693 .07 365 .27 058 34 27 .72 962 .92 627 .80 335 .19 665 .07 373 .27 038 33 28 .72 982 .92 619 .80 363 .19 637 .07 381 .27 018 32 29 .73 002 .92611 .80 391 .19 609 .07 389 .26 998 31 30 9.73 022 0.92 603 9.80 419 0.19 581 0.07 397 0.26 978 30 31 .73 041 .92 595 .80 447 .19 553 .07 405 .26 959 29 32 73 061 .92 587 .80 474 .19 526 .07413 .26 939 28 33 .73 081 .92 579 .80 502 .19 498 .07 421 .26 919 27 34 .73 101 .92 571 .80 530 .19 470 .07 429 .26 899 26 35 9.73 121 9.92 563 9.80 558 0.19 442 0.07 437 0.26 879 25 36 .73 140 .92 555 .80 586 .19414 .07 445 .26 860 24 37 .73 160 .92 546 .80 614 .19 386 .07 454 .26840 23 38 .73 180 .92 538 .80 642 .19 358 .07 462 .26 820 22 39 .73 200 .92 530 .80 669 .19 331 .07 470 .26 800 21 40 9.73 219 9.92 522 9.80 697 0.19 303 0.07 478 0.26 781 20 41 .73 239 .92 514 .80 725 .19 275 .07 486 .26 761 19 42 .73 259 .92 506 .80 753 .19 247 .07 494 .26 741 18 43 .73 278 .92 498 .80 781 .19219 .07 502 .26 722 17 44 .73 298 .92 490 .80 808 .19 192 .07 510 .26 702 16 45 9.73 318 9.92 482 9.80 836 0.19 164 0.07 518 0.26 682 15 46 .73 337 .92 473 .80 864 .19 136 .07 527 .26 663 14 47 .73 357 .92 465 .80 892 .19 108 .07 535 .26 643 13 48 .73 377 .92 457 .80 919 .19 081 .07 543 .26 623 12 49 .73 396 .92 449 .80 947 .19 053 .07 551 .26 604 11 50 9.73 416 9.92 441 9.80 975 0.19 025 0.07 559 0.26 584 10 51 .73 435 .92 433 .81 003 .18 997 .07 567 .26 565 9 52 .73 455 .92 425 .81 030 .18 970 .07 575 .26 545 8 53 .73 474 .92 416 .81 058 .18 942 .07 584 .26 526 7 54 .73 494 .92 408 .81 086 .18914 .07 592 .26 506 6 55 9.73 513 9.92 400 9.81 113 0.18 887 0.07 600 0.26 487 5 56 .73 533 .92 392 .81 141 .18 859 .07 608 .26 467 4 57 .73 552 .92 384 .81 169 .18831 .07 616 .26 448 3 58 .73 572 .92 376 .81 196 .18 804 .07 624 .26 428 2 59 .73 591 .92 367 .81 224 .18776 .07 633 .26 409 1 60 9.73611 9.92 359 9.81 252 0.18 748 0.07 641 0.26 389 Cos Sin Cot Tan Csc Sec ' 122 (302) (237) 57 C Table 4. Trigonometric Logarithms 229 33 (213) (326) 146 C ' Sin Cos Tan Cot Sec Csc 9.73611 9.92 359 9.81 252 0.18748 0.07 641 0.26 389 60 1 .73 630 .92 351 .81 279 .18721 .07 649 .26 370 59 2 .73 650 .92 343 .81 307 .18 693 .07 657 .26 350 58 3 .73 669 .92 335 .81 335 .18 665 .07 665 .26 331 57 4 .73 689 .92 326 .81 362 .18 638 .07 674 .26311 56 5 9.73 708 9.92 318 9.81 390 0.18 610 0.07 682 0.26 292 55 6 .73 727 .92 310 .81 418 .18 582 .07 690 .26 273 54 7 .73 747 .92 302 .81 445 .18 555 .07 698 .26 253 53 8 .73 766 .92 293 .81 473 .18527 .07 707 .26 234 52 9 .73 785 .92 285 .81 500 .18 500 .07 715 .26 215 51 10 9.73 805 9.92 277 9.81 528 0.18 472 0.07 723 0.26 195" 50 11 .73 824 .92 269 .81 556 .18 444 .07 731 .26 176 49 12 .73 843 .92 260 .81 583 .18417 .07 740 .26 157 48 13 .73 863 .92 252 .81 611 .18 389 .07 748 .26 137 47 14 .73 882 .92 244 .81 638 .18362 .07 756 .26 118 46 15 9.73 901 9.92 235 9.81 666 0.18 334 0.07 765 0.26 099 45 16 .73 921 .92 227 .81 693 .18 307 .07 773 .26 079 44 17 .73 940 .92 219 .81 721 .18 279 .07 781 .26 060 43 18 .73 959 .92211 .81 748 .18252 .07 789 .26 041 42 19 .73 978 .92 202 .81 776 .18 224 .07 798 .26 022 41 20 9.73 997 9.92 194 9.81 803 0.18 197 0.07 806 0.26 003 40 21 .74 017 .92 186 .81 831 .18 169 .07 814 .25 983 39 22 .74 036 .92 177 .81 858 .18 142 .07 823 .25 964 38 23 .74 055 .92 169 .81 886 .18 114 .07 831 .25 945 37 24 .74 074 .92 161 .81 913 .18 087 .07 839 .25 926 36 25 9.74 093 9.92 152 9.81 941 0.18 059 0.07 848 0.25 907 35 26 .74 113 .92 144 .81 968 .18032 .07 856 .25 887 34 27 .74 132 .92 136 .81 996 .18 004 .07 864 .25 868 33 28 .74 151 .92 127 .82 023 .17 977 .07 873 .25849 . 32 29 ,74 170 .92 119 .82 051 .17 949 .07 881 .25 830 31 30 9.74 189 9.92 111 9.82 078 0.17922 0.07 889 0.25 811 30 31 .74 208 .92 102 .82 106 .17 894 .07 898 .25 792 29 32 .74 227 .92 094 .82 133 .17 867 .07 906 .25 773 28 33 .74 246 .92 086 .82 161 .17 839 .07 914 .25 754 27 34 .74 265 .92 077 .82 188 .17812 .07 923 .25 735 26 35 9.74 284' 9.92 069 9.82 215 0.17 785 0.07 931 0.25 716 25 36 .74 303 .92 060 .82 243 .17 757 .07 940 .25 697 24 37 .74 322 .92 052 .82 270 .17 730 .07 948 .25 678 23 38 .74 341 .92 044 .82 298 .17 702 .07 956 ,25 659 ' 22 39 .74 360 .92 035 .82 325 .17 675 .07 965 .25 640 21 40 9.74 379 9.92 027 9.82 352 0.17 648 0.07 973 0.25 621 20 41 .74 398 .92 018 .82 380 .17 620 .07 982 .25 602 19 42 .74 417 .92 010 .82 407 .17 593 .07 990 .25 583 18 43 .74 436 .92 002 .82 435 .17 565 .07 998 .25 564 17 44 .74 455 .91 993 .82 462 .17 538 .08 007 .25 545 16 45 9.74 474 9.91 985 9.82 489 0.17511 0.08 015 0.25 526 15 46 .74 493 .91 976 .82 517 .17483 .08 024 .25 507 14 47 .74 512 .91 968 .82 544 .17 456 .08 032 .25 488 13 48 .74 531 .91 959 .82 571 .17 429 .08 041 .25 469 12 49 .74 549 .91 951 .82 599 .17401 .08 049 .25 451 11 50 9.74 568 9.91 942 9.82 626 0.17 374 0.08 058 0.25 432 10 51 .74 587 .91 934 .82 653 .17 347 .08 066 .25 413 9 52 .74 606 .91 925 .82 681 .17319 .08 075 .25 394 8 53 .74 625 .91 917 .82 708 .17 292 .08 083 .25 375 7 54 .74644 .91 908 .82 735 .17 265 .08 092 .25 356 6 55 9.74 662 9.91 900 9.82 762 0.17 238 0.08 100 0.25 338 5 56 .74 681 .91 891 .82 790 .17210 .08 109 .25 319 4 57 .74 700 .91 883 .82 817 .17 183 .08 117 .25 300 3 58 .74 719 .91 874 .82844 .17 156 .08 126 .25 281 2 59 .74 737 .91 866 .82 871 .17 129 .08 134 .25 263 1 60 9.74 756 9.91 857 9.82 899 0.17 101 0.08 143 0.25 244 Cos Sin Cot Tan Csc Sec ' 123 (303) (236) 56 C 230 Table 4. Trigonometric Logarithms 34 (214) (325) 145 ' Sin Cos Tan Cot Sec Csc 9.74 756 9.91 857 9.82 899 0.17 101 0.08 143 0.25 244 60 1 .74 775 .91 849 .82 926 .17 074 .08 151 .25 225 59 2 .74 794 .91 840 .82 953 .17 047 .08 160 .25 206 58 3 .74 812 .91 832 .82 980 .17020 .08 168 .25 188 57 4 .74 831 .91 823 .83 008 .16 992 .08 177 .25 169 56 5 9.74 850 9.91 815 9.83 035 0.16 965 0.08 185 0.25 150 55 6 .74 868 .91 806 .83 062 .16 938 .08 194 .25 132 54 7 .74 887 .91 798 .83 089 .16911 .08 202 .25 113 53 8 .74 906 .91 789 .83 117 .16 883 .08211 .25 094 52 9 .74 924 .91 781 .83 144 .16 856 .08 219 .25 076 51 10 9.74 943 9.91 772 9.83 171 0.16 829 0.08 228 0.25 057 50 11 .74 961 .91 763 .83 198 .16 802 .08 237 .25 039 49 12 .74 980 .91 755 .83 225 .16 775 .08 245 .25 020 48 13 .74 999 .91 746 .83 252 .16 748 .08 254 .25 001 47 14 .75 017 .91 738 .83 280 16 720 .08 262 .24 983 46 15 9.75 036 9.91 729 9.83 307 0.16 693 0.08 271 0.24 964 45 16 .75 054 .91 720 .83 334 .16 666 .08 280 .24 946 44 17 .75 073 .91 712 .83 361 .16 639 .08 288 .24 927 43 18 .75 091 .91 703 .83 388 .16612 .08 297 .24 909 42 19 .75 110 .91 695 .83 415 .16 585 .08 305 .24 890 41 20 9.75 128 9.91 686 9.83 442 0.16 558 0.08 314 0.24 872 40 21 .75 147 .91 677 .83 470 .16 530 .08 323 .24 853 39 22 .75 165 .91 669 .83 497 .16 503 .08 331 .24 835 38 23 .75 184 .91 660 .83 524 .16476 .08 340 .24 816 37 24 .75 202 .91 651 .83 551 .16449 .08 349 .24 798 36 25 9.75 221 9.91 643 9.83 578 0.16422 0.08 357 0.24 779 35 26 .75 239 .91 634 .83 605 .16 395 .08 366 .24 761 34 27 .75 258 .91 625 .83 632 .16 368 .08 375 .24 742 33 28 .75 276 .91 617 .83 659 .16341 .08 383 .24 724 32 29 .75 294 .91 608 .83 686 .16314 .08 392 .24 706 31 30 9.75 313 9.91 599 9.83 713 0.16 287 0.08 401 0.24 687 30 31 .75 331 .91 591 .83 740 .16 260 .08 409 .24 669 29 32 .75 350 .91 582 .83 768 .16 232 .08 418 .24 650 28 33 .75 368 .91 573 .83 795 .16 205 .08 427 .24 632 27 34 .75 386 .91 565 .83 822 .16 178 .08 435 .24 614 26 35 9.75 405 9.91 556 9.83 849 0.16 151 0.08 444 0.24 595 25 36 .75 423 .91 547 .83 876 .16 124 .08 453 .24 577 24 37 .75 441 .91 538 .83 903 .16 097 .08 462 .24 559 23 38 .75 459 .91 530 .83 930 .16 070 .08 470 .24 541 22 39 .75 478 .91 521 .83 957 .16 043 .08 479 .24 522 21 40 9.75 496 9.91 512 9.83 984 0.16016 Q.08 488 0.24 504 20 41 .75 514 .91 504 .84011 .15 989 .08 496 .24 486 19 42 .75 533 .91 495 .84 038 .15 962 .08 505 .24 467 18 43 .75 551 .91 486 .84 065 .15 935 .08 514 .24 449 17 44 .75 569 .91 477 .84 092 .15 908 .08 523 .24 431 16 45 9.75 587 9.91 469 9.84 119 0.15 881 0.08 531 0.24 413 15 46 .75 605 .91 460 .84 146 .15 854 .08 540 .24 395 14 47 .75 624 .91 451 .84 173 .15 827 .08 549 .24 376 13 48 .75 642 .91 442 .84200 .15 800 .08 558 .24 358 12 49 .75 660 .91 433 .84227 .15 773 .08 567 .24 340 11 50 9.75 678 9.91 425 9.84 254 0.15 746 0.08 575 0.24 322 10 51 .75 696 .91 416 .84 280 .15 720 .08 584 .24 304 9 52 .75 714 .91 407 .84307 .15 693 .08 593 .24 286 8 53 .75 733 .91 398 .84334 .15 666 .08 602 .24 267 7 54 .75 751 .91 389 .84361 .15 639 .08611 .24 249 6 55 9.75 769 9.91 381 9.84 388 0.15 612 0.08 619 0.24 231 5 56 .75 787 .91 372 .84415 .15 585 .08 628 .24 213 4 57 .75 805 .91 363 .84442 .15 558 .08 637 .24 195 3 58 .75 823 .91 354 .84 469 .15531 .08 646 .24 177 2 59 .75 841 .91 345 .84 496 .15 504 .08 655 .24 159 1 60 9.75 859 9.91 336 9.84 523 0.15 477 0.08 664 0.24 141 Cos Sin Cot Tan Csc Sec ' 124 (304) (235) 55 Table 4. Trigonometric Logarithms 231 35 (215) (324) 144 ' Sin Cos Tan Cot Sec Csc 9.75 859 9.91 336 9.84 523 0.15477 0.08 664 0.24 141 60 1 .75 877 .91 328 .84550 .15 450 .08 672 .24 123 59 2 .75 895 .91 319 .84576 .15 424 .08 681 .24 105 58 3 .75 913 .91 310 .84603 .15 397 .08 690 .24 087 57 4 .75 931 .91 301 .84630 .15 370 .08 699 .24 069 56 5 9.75 949 9.91 292 9.84 657 0.15 343 0.08 708 0.24 051 55 6 .75 967 .91 283 .84684 .15316 .08 717 .24 033 54 7 .75 985 .91 274 .84711 .15 289 .08 726 .24 015 53 8 .76 003 .91 266 .84738 .15 262 .08 734 .23 997 52 9 .76 021 .91 257 .84764 .15 236 .08 743 .23 979 51 10 9.76 039 9.91 248 9.84 791 0.15 209 0.08 752 0.23 961 50 11 .76 057 .91 239 .84818 .15 182 .08 761 .23 943 49 12 .76 075 .91 230 .84845 .15 155 .08 770 .23 925 48 13 .76 093 .91 221 .84872 .15 128 .08 779 .23 907 47 14 .76 111 .91 212 .84899 .15 101 .08 788 .23 889 46 15 9.76 129 9.91 203 9.84 925 0.15 075 0.08 797 0.23 871 45 16 .76 146 .91 194 .84952 .15 048 .08 806 .23 854 44 17 .76164 .91 185 .84979 .15 021 .08 815 .23 836 43 18 .76 182 .91 176 .85 006 .14 994 .08 824 .23 818 42 19 .76 200 .91 167 .85 033 .14 967 .08 833 .23 800 41 20 9.76 218 9.91 158 9.85 059 0.14 941 0.08 842 0.23 782 40 21 .76 236 .91 149 .85 086 .14914 .08 851 .23764 39 22 .76 253 .91 141 .85 113 .14 887 .08 859 .23 747 38 23 .76 271 .91 132 .85 140 .14 860 .08 868 .23 729 37 24 .76 289 .91 123 .85 166 .14 834 .08 877 .23711 36 25 9.76 307 9.91 114 9.85 193 0.14 807 0.08 886 0.23 693 35 26 .76 324 .91 105 .85 220 .14 780 .08 895 .23 676 34 27 .76 342 .91 096 .85 247 .14 753 .08 904 .23 658 33 28 .76 360 .91 087 .85 273 .14 727 .08 913 .23 640 32 29 .76 378 .91 078 .85 300 .14 700 .08 922 .23 622 31 30 9.76 395 9.91 069 9.85 327 0.14 673 0.08 931 0.23 605 30 31 .76 413 .91 060 .85 354 .14 646 .08 940 .23 587 29 32 .76 431 .91 051 .85 380 .14 620 .08 949 .23 569 28 33 .76 448 .91 042 .85 407 .14 593 .08 958 .23 552 27 34 .76 466 .91 033 .85 434 .14 566 .08 967 .23 534 26 35 9.76 484 9.91 023 9.85 460 0.14 540 0.08 977 0.23 516 25 36 .76 501 .91 014 .85 487 .14513 .08 986 .23 499 24 37 .76 519 .91 005 .85 514 .14 486 .08 995 .23 481 23 38 .76 537 .90 996 .85 540 .14 460 .09 004 .23 463 22 39 .76 554 .90 987 .85 567 .14433 .09 013 .23 446 21 40 9.76 572 9.90 978 9.85 594 0.14406 0.09 022 0.23 428 20 41 .76 590 .90 969 .85 620 .14 380 .09 031 .23 410 19 42 .76 607 .90 960 .85 647 .14 353 .09 040 .23 393 18 43 .76 625 .90 951 .85674 .14 326 .09 049 .23 375 17 44 .76 642 .90 942 .85 700 .14 300 .09 058 .23 358 16 45 9.76 660 9.90 933 9.85 727 0.14 273 0.09 067 0.23 340 15 46 .76 677 .90 924 .85 754 .14 246 .09 076 .23 323 14 47 .76 695 .90 915 .85 780 .14 220 .09 085 .23 305 13 48 .76 712 .90 906 .85 807 .14 193 .09 094 .23 288 12 49 .76 730 .90 896 .85 834 .14 166 .09 104 .23 270 11 50 9.76 747 9.90 887 9.85 860 0.14 140 0.09 113 0.23 253 10 51 .76 765 .90878 .85 887 .14 113 .09 122 .23 235 9 52 .76 782 .90 869 .85913 .14 087 .09 131 .23 218 8 53 .76 800 .90 860 .85940 .14 060 .09 140 .23 200 7 54 .76 817 .90 851 .85 967 .14 033 .09 149 .23 183 6 55 9.76 835 9.90 842 9.85 993 0.14 007 0.09 158 0.23 165 5 56 .76 852 .90 832 .86 020 .13 980 .09 168 .23 148 4 57 .76 870 .90 823 .86 046 .13 954 .09 177 .23 130 3 58 .76 887 .90 814 .86 073 .13 927 .09 186 .23 113 2 59 .76 904 .90 805 .86 100 .13 900 .09 195 .23 096 1 60 9.76 922 9.90 796 9.86 126 0.13 874 0.09 204 0.23 078 Cos Sin Cot Tan Csc Sec ' 125 (305) (234) 54 232 Table 4. Trigonometric Logarithms 36 (216) (323) 143 ' Sin Cos Tan Cot Sec Csc 9.76 922 9.90 796 9.86 126 0.13 874 0.09 204 0.23 078 60 1 .76 939 .90 787 .86 153 .13 847 .09 213 .23 061 59 2 .76 957 .90 777 .86 179 .13821 .09 223 .23 043 58 3 .76 974 .90 768 .86 206 .13 794 .09 232 .23 026 57 4 .76 991 .90 759 .86 232 .13 768 .09 241 .23 009 56 5 9.77 009 9.90 750 9.86 259 0.13 741 0.09 250 0.22 991 55 6 .77 026 .90 741 .86 285 .13715 .09 259 .22 974 54 7 .77 043 .90 731 .86 312 .13 688 .09 269 .22 957 53 8 .77 061 .90 722 .86 338 .13 662 .09 278 .22 939 52 9 .77 078 .90 713 .86 365 .13 635 .09 287 .22 922 51 10 9.77 095 9.90 704 9.86 392 0.13 608 0.09 296 0.22 905 50 11 .77 112 .90 694 .86 418 .13 582 .09 306 .22 888 49 12 .77 130 .90 685 .86 445 .13 555 .09 315 .22 870 48 13 .77 147 .90 676 .86 471 .13 529 .09 324 .22 853 47 14 .77 164 .90 667 .86 498 .13 502 .09 333 .22 836 46 15 9.77 181 9.90 657 9.86 524 0.13 476 0.09 343 0.22 819 45 16 .77 199 .90 648 .86 551 .13 449 .09 352 .22 801 44 17 .77 216 .90 639 .86 577 .13 423 .09 361 .22 784 43 18 .77 233 .90 630 .86 603 .13 397 .09 370 .22 767 42 19 .77 250 .90 620 .86 630 .13 370 .09 380 .22 750 41 20 9.77 268 9.90611 9.86 656 0.13 344 0.09 389 0.22 732 40 21 .77 285 .90 602 .86 683 .13317 .09 398 .22 715 39 22 .77 302 .90 592 .86 709 .13 291 .09 408 .22 698 38 23 .77 319 .90 583 .86 736 .13 264 .09 417 .22 681 37 24 .77 336 .90 574 .86 762 .13238 .09 426 .22 664 36 25 9.77 353 9.90 565 9.86 789 0.13211 0.09 435 0.22 647 35 26 .77 370 .90 555 .86 815 .13 185 .09 445 .22 630 34 27 .77 387 .90 546 .86842 .13 158 .09 454 .22 613 33 28 .77 405 .90 537 .86 868 .13 132 .09 463 .22 595 32 29 .77 422 .90 527 .86 894 13 106 .09 473 .22 578 31 30 9.77 439 9.90 518 9.86 921 0.13 079 0.09 482 0.22 561 30 31 .77 456 .90 509 .86 947 .13 053 .09 491 .22 544 29 32 .77 473 .90 499 .86 974 .13 026 .09 501 .22 527 28 33 .77 490 .90 490 .87 000 .13 000 .09 510 .22 510 27 34 .77 507 .90 480 .87 027 .12 973 .09 520 .22 493 26 35 9.77 524 9.90 471 9.87 053 0.12 947 0.09 529 0.22 476 25 36 .77 541 .90 462 .87 079 .12921 .09 538 .22 459 24 37 .77 558 .90 452 .87 106 .12 894 .09 548 .22 442 23 38 .77 575 .90 443 .87 132 .12 868 .09 557 .22 425 22 39 .77 592 .90 434 .87 158 .12 842 .09 566 .22 408 21 40 9.77 609 9.90 424 9.87 185 0.12815 0.09 576 0.22 391 20 41 .77 626 .90415 .87211 .12 789 .09 585 .22 374 19 42 .77 643 .90 405 .87 238 .12 762 .09 595 .22 357 18 43 .77 660 .90 396 .87 264 .12 736 .09 604 .22 340 17 44 .77 677 .90386 .87 290 .12710 .09 614 .22 323 16 45 9.77 694 9.90 377 9.87317 0.12 683 0.09 623 0.22 306 15 46 .77711 .90 368 .87 343 .12 657 .09 632 .22 289 14 47 .77-728 .90 358 .87 369 .12 631 .09 642 .22 272 13 48 .77744 .90 349 .87 396 .12 604 .09 651 .22 256 12 49 .77.761 .90 339 .87 422 .12 578 .09 661 .22 239 11 50 9.77 778 9.90 330 9.87 448 0.12 552 0.09 670 0.22 222 10 51 .77795 .90 320 .87 475 .12 525 .09 680 .22 205 9 52 .77.812 .90311 .87 501 .12 499 .09 689 .22 188 8 53 .77 829 .90 301 .87 527 .12 473 .09 699 .22 171 7 54 .77846 .90 292 .87 554 .12 446 .09 708 .22 154 6 55 9.77 862 9.90 282 9.87 580 0.12 420 0.09 718 0.22 138 5 56 .77879 .90 273 .87 606 .12 394 .09 727 .22 121 4 57 .77 896 .90 263 .87 633 .12 367 .09 737 .22 104 3 58 .77 913 .90 254 .87 659 .12 341 .09 746 .22 087 2 59 .77 930 .90 244 .87 685 .12315 .09 756 .22 070 1 60 9.77 946 9.90 235 9.87711 0.12 289 0.09 765 0.22 054 Cos Sin Cot Tan Csc Sec ' 126 (306) (233) 53 Table 4. Trigonometric Logarithms 233 37 (217) (322) 142 C ' Sin Cos Tan Cot Sec Csc 9.77 946 9.90 235 9.87711 0.12 289 0.09 765 0.22 054 60 1 .77 963 .90 225 .87 738 .12 262 .09 775 .22 037 59 2 .77 980 .90 216 .87 764 .12236 .09 784 .22 020 58 3 .77 997 .90 206 .87 790 .12210 .09 794 .22 003 57 4 .78 013 .90 197 .87 817 .12 183 .09 803 .21 987 56 5 9.78 030 9.90 187 9.87 843 0.12 157 0.09 813 0.21 970 55 6 .78 047 .90 178 .87 869 .12 131 .09 822 .21 953 54 7 .78 063 .90 168 .87 895 .12 105 .09 832 .21 937 53 8 .78 080 .90 159 .87 922 .12 078 .09 841 .21 920 52 9 .78 097 .90 149 .87 948 .12 052 .09 851 .21 903 51 10 9.78113 9.90 139 9.87 974 0.12 026 0.09 861 0.21 887 50 11 .78 130 .90 130 .88 000 .12 000 .09 870 .21 870 49 12 .78 147 .90 120 .88 027 .11 973 .09 880 .21 853 48 13 .78 163 .90 111 .88 053 .11 947 .09 889 .21 837 47 14 .78 180 .90 101 .88 079 .11 921 .09 899 .21 820 46 15 9.78 197 9.90 091 9.88 105 0.11 895 0.09 909 0.21 803 45 16 .78 213 .90 082 .88 131 .11 869 .09 918 .21 787 44 17 .78 230 .90 072 .88 158 .11 842 .09 928 .21 770 43 18 .78 246 .90 063 .88184 .11 816 .09 937 .21 754 42 19 .78 263 .90 053 .88 210 .11 790 .09 947 .21 737 41 20 9.78 280 9.90 043 9.88 236 0.11 764 0.09 957 0.21 720 40 21 .78 296 .90 034 .88 262 .11 738 .09 966 .21 704 39 22 .78313 .90 024 .88 289 .11 711 .09 976 .21 687 38 23 .78 329 .90 014 .88 315 .11 685 .09 986 .21 671 37 24 .78 346 .90 005 .88 341 .11 659 .09 995 .21 654 36 25 9.78 362 9.89 995 9.88 367 0.11 633 0.10 005 0.21 638 35 26 .78 379 .89 985 .88 393 .11 607 .10015 .21 621 34 27 .78 395 .89 976 .88 420 .11 580 .10 024 .21 605 33 28 .78412 .89 966 .88 446 .11 554 .10034 .21 588 32 29 .78 428 .89 956 .88 472 .11 528 .10 044 .21 572 31 30 9.78 445 9.89 947 9.88 498 0.11 502 0.10 053 0.21 555 30 31 .78 461 .89 937 .88 524 .11 476 .10 063 .21 539 29 32 .78 478 .89 927 .88 550 .11450 .10 073 .21 522 28 33 .78 494 .89 918 .88 577 .11 423 .10 082 .21 506 27 34 .78 510 .89 908 .88 603 .11397 .10 092 .21 490 26 35 9.78 527 9.89 898 9.88 629 0.11 371 0.10 102 0.21 473 25 36 .78 543 .89 888 .88 655 .11 345 .10 112 .21 457 24 37 .78 560 .89 879 .88 681 .11 319 .10 121 .21 440 23 38 .78 576 .89 869 .88 707 .11 293 .10 131 .21 424 22 39 .78 592 .89 859 .88733 .11 267 .10 141 .21 408 21 40 9.78 609 9.89 849 9.88 759 0.11 241 0.10 151 0.21 391 20 41 .78 625 .89840 .88 786 .11 214 .10 160 .21 375 19 42 .78 642 .89 830 .88812 .11 188 .10 170 .21 358 18 43 .78 658 .89 820 .88 838 .11 162 .10 180 .21 342 17 44 .78 674 .89 810 .88864 .11 136 .10 190 .21 326 16 45 9.78 691 9.89 801 9.88 890 0.11 110 0.10 199 0.21 309 15 46 .78 707 .89 791 .88 916 .11 084 .10 209 .21 293 14 47 .78 723 .89 781 .88 942 .11058 .10219 .21 277 13 48 .78 739 .89 771 .88 968 .11 032 .10229 .21 261 12 49 .78 756 .89 761 .88 994 .11 006 .10 239 .21 244 11 50 9.78 772 9.89 752 9.89 020 0.10980 0.10248 0.21 228 10 51 .78 788 .89 742 .89 046 .10954 .10 258 .21 212 9 52 .78 805 .89 732 .89 073 .10927 .10 268 .21 195 8 53 .78 821 .89 722 .89 099 .10901 .10 278 .21 179 7 54 .78 837 .89 712 .89 125 .10 875 .10 288 .21 163 6 55 9.78 853 9.89 702 9.89 151 0.10849 0.10 298 0.21 147 5 56 .78 869 .89 693 .89 177 .10 823 .10 307 .21 131 4 57 .78 886 .89 683 .89 203 .10 797 .10317 .21 114 3 58 .78 902 .89 673 .89 229 .10 771 .10327 .21 098 2 59 .78918 .89 663 .89 255 .10 745 .10337 .21 082 1 60 9.78 934 9.89 653 9.89 281 0.10719 0.10 347 0.21 066 Cos Sin Cot Tan Csc Sec ' 127 (307) (232) 52 234 Table 4. Trigonometric Logarithms 38 (218) (321) 141 C ' Sin Cos Tan Cot Sec Csc 9.78 934 9.89 653 9.89 281 0.10719 0.10347 0.21 066 60 1 .78 950 .89 643 .89 307 .10 693 .10357 .21 050 59 2 .78 967 .89 633 .89333 .10 667 .10367 .21 033 58 3 .78 983 .89 624 .89 359 .10 641 .10376 .21 017 57 4 .78 999 .89 614 .89 385 .10615 .10 386 .21 001 56 5 9.79 015 9.89 604 9.89411 0.10589 0.10 396 0.20 985 55 6 .79 031 .89 594 .89 437 .10 563 .10 406 .20 969 54 7 .79 047 .89 584 .89 463 .10 537 .10416 .20 953 53 8 .79 063 .89 574 .89 489 .10511 .10426 .20 937 52 9 .79 079 .89 564 .89 515 .10 485 .10436 .20 921 51 10 9.79 095 9.89 554 9.89 541 0.10459 0.10446 0.20 905 50 11 .79 111 .89 544 .89 567 .10 433 .10 456 .20 889 49 12 .79 128 .89 534 .89 593 .10 407 .10 466 .20 872 48 13 .79 144 .89 524 .89 619 .10381 .10476 .20 856 47 14 .79 160 .89 514 .89 645 .10 355 .10 486 .20840 46 15 9.79 176 9.89 504 9.89 671 0.10 329 0.10 496 0.20 824 45 16 .79 192 .89 495 .89 697 .10303 .10505 .20 808 44 17 .79 208 .89 485 .89 723 .10277 .10515 .20 792 43 18 .79 224 .89 475 .89 749 .10251 .10 525 .20 776 42 19 .79 240 .89 465 .89 775 .10 225 .10535 .20 760 41 20 9.79 256 9.89 455 9.89 801 0.10 199 0.10 545 0.20 744 40 21 .79 272 .89 445 .89 827 .10 173 .10 555 .20 728 39 22 .79 288 .89 435 .89 853 .10 147 .10 565 .20 712 38 23 .79 304 .89 425 .89 879 .10 121 .10 575 .20 696 37 24 .79 319 .89 415 .89 905 .10095 .10 585 .20 681 36 25 9.79 335 9.89 405 9.89 931 0.10 069 0.10595 0.20 665 35 26 .79 351 .89 395 .89 957 .10 043 .10 605 .20 649 34 27 .79 367 .89 385 .89 983 .10017 .10615 .20 633 33 28 .79 383 .89 375 .90 009 .09 991 .10 625 .20 617 32 29 .79 399 .89 364 .90 035 .09 965 .10 636 .20 601 31 30 9.79415 9.89 354 9.90 061 0.09 939 0.10646 0.20 585 30 31 .79 431 .89 344 .90 086 .09 914 .10 656 .20 569 29 32 .79 447 .89 334 .90 112 .09 888 .10 666 .20 553 28 33 .79 463 .89 324 .90 138 .09 862 .10 676 .20 537 27 34 .79 478 .89 314 .90 164 .09 836 .10 686 .20 522 26 35 9.79 494 9.89 304 9.90 190 0.09 810 0.10 696 0.20 506 25 36 .79 510 .89 294 .90 216 .09 784 .10 706 .20 490 24 37 .79 526 .89 284 .90 242 .09 758 .10716 .20 474 23 38 .79 542 .89 274 .90 268 .09 732 .10 726 .20 458 22 39 .79 558 .89 264 .90 294 .09 706 .10 736 .20 442 21 40 9.79 573 9.89 254 9.90 320 0.09 680 O'lO 746 0.20 427 20 41 .79 589 .89 244 .90 346 .09 654 .10 756 .20411 19 42 .79 605 .89 233 .90 371 .09 629 .10 767 .20 395 18 43 .79 621 .89 223 .90 397 .09 603 .10 777 .20 379 17 44 .79 636 .89 213 .90 423 .09 577 .10 787 .20 364 16 45 9.79 652 9.89 203 9.90 449 0.09 551 0.10 797 0.20 348 15 46 .79 668 .89 193 .90 475 .09 525 .10 807 .20 332 14 47 .79 684 .89 183 .90 501 .09 499 .10817 .20 316 13 48 .79 699 .89 173 .90 527 .09 473 .10 827 .20 301 12 49 .79 715 .89 162 .90 553 .09 447 .10 838 .20 285 11 50 9.79 731 9.89 152 9.90 578 0.09 422 0.10 848 0.20 269 10 51 .79 746 .89 142 .90 604 .09 396 .10 858 .20 254 9 52 .79 762 .89 132 .90 630 .09 370 .10 868 .20 238 8 53 .79 778 .89 122 .90 656 .09 344 .10 878 .20 222 7 54 .79 793 .89112 .90 682 .09 318 .10 888 .20 207 6 55 9.79 809 9.89 101 9.90 708 0.09 292 0.10 899 0.20 191 5 56 .79 825 .89 091 .90 734 .09 266 .10 909 .20 175 4 57 .79840 .89 081 .90 759 .09 241 .10919 .20 160 3 58 .79 856 .89 071 .90 785 .09 215 .10 929 .20 144 2 59 .79 872 .89 060 .90811 .09 189 .10 940 .20 128 1 60 9.79 887 9.89 050 9.90 837 0.09 163 0.10 950 0.20 113 Cos Sin Cot Tan Csc Sec ' 128 (308) (231) 51 C Table 4. Trigonometric Logarithms 235 39 (219) (320) 140 C ' Sin Cos Tan Cot Sec Csc 9.79 887 9.89 050 9.90 837 0.09 163 0.10950 0.20 113 60 1 .79 903 .89 040 .90 863 .09 137 .10 960 .20 097 59 2 .79 918 .89 030 .90 889 .09 111 .10 970 .20 082 58 3 .79 934 .89 020 .90 914 .09 086 .10 980 .20 066 57 4 .79 950 .89 009 .90 940 .09060 .10991 .20 050 56 5 9.79 965 9.88 999 9.90 966 0.09 034 0.11 001 0.20 035 55 6 .79 981 .88 989 .90 992 .09 008 .11 Oil .20 019 54 7 .79 996 .88 978 .91 018 .08 982 .11 022 .20004 53 8 .80012 .88 968 .91 043 .08 957 .11 032 .19 988 52 9 .80027 .88 958 .91 069 .08 931 .11 042 .19 973 51 10 9.80 043 9.88 948 9.91 095 0.08 905 0.11 052 0.19 957 50 11 .80 058 .88 937 .91 121 .08 879 .11 063 .19 942 49 12 .80 074 .88 927 .91 147 .08 853 .11 073 .19 926 48 13 .80 089 .88 917 .91 172 .08 828 .11 083 .19911 47 14 .80 105 .88 906 .91 198 .08 802 .11 094 .19 895 46 15 9.80 120 9.88 896 9.91 224 0.08 776 0.11 104 0.19 880 45 16 .80 136 .88 886 .91 250 .08 750 .11 114 .19 864 44 17 .80 151 .88 875 .91 276 .08 724 .11 125 .19 849 43 18 .80 166 .88 865 .91 301 .08 699 .11 135 .19 834 42 19 .80 182 .88855 .91 327 .08 673 .11 145 .19818 41 20 9.80 197 9.88 844 9.91 353 0.08 647 0.11 156 0.19 803 40 21 .80213 .88 834 .91 379 .08 621 .11 166 .19 787 39 22 .80 228 .88 824 .91 404 .08 596 .11 176 .19 772 38 23 .80 244 .88 813 .91 430 .08 570 .11 187 .19 756 37 24 .80 259 .88 803 .91456 .08 544 .11 197 .19 741 36 25 9.80 274 9.88 793 9.91 482 0.08 518 0.11 207 0.19 726 35 26 .80 290 .88 782 .91 507 .08 493 .11218 .19710 34 27 .80 305 .88 772 .91 533 .08 467 .11 228 .19 695 33 28 .80 320 .88 761 .91 559 .08 441 .11 239 .19 680 32 29 .80 336 .88 751 .91 585 .08 415 .11 249 .19 664 31 30 9.80 351 9.88 741 9.91 610 0.08 390 0.11 259 0.19 649 30 31 .80 366 .88730 .91 636 .08364 .11 270 .19 634 29 32 .80 382 .88 720 .91 662 .08 338 .11 280 .19618 28 33 .80 397 .88 709 .91 688 .08 312 .11 291 .19 603 27 34 .80 412 .88 699 .91 713 .08 287 .11 301 .19 588 26 35 9.80 428 9.88 688 9.91 739 0.08 261 0.11312 0.19 572 25 36 .80 443 .88 678 .91 765 .08 235 .11 322 .19 557 24 37 .80 458 .88 668 .91 791 .08 209 .11 332 .19 542 23 38 .80 473 .88 657 .91 816 .08 184 .11 343 .19 527 22 39 .80 489 .88 647 .91 842 .08 158 .11 353 .19511 21 40 9.80 504 9.88 636 9.91 868 0.08 132 0.11 364 0.19496 20 41 .80 519 .88 626 .91 893 .08 107 .11 374 .19 481 19 42 .80 534 .88 615 .91 919 .08 081 .11 385 .19 466 18 43 .80 550 .88605 .91 945 .08 055 .11 395 .19 450 17 44 .80 565 .88 594 .91 971 .08 029 .11 406 .19435 16 45 9.80 580 9.88 584 9.91 996 0.08 004 0.11416 0.19 420 15 46 .80 595 .88 573 .92 022 .07 978 .11 427 .19 405 14 47 .80 610 .88 563 .92 048 .07 952 .11437 .19 390 13 48 .80 625 .88 552 .92 073 .07 927 .11 448 .19 375 12 49 .80641 .88542 .92 099 .07 901 .11458 .19 359 11 50 9.80 656 9.88 531 9.92 125 0.07 875 0.11 469 0.19 344 10 51 .80 671 .88 521 .92 150 .07 850 .11 479 .19329 9 52 .80 686 .88 510 .92 176 .07 824 .11 490 .19314 8 53 .80 701 .88 499 .92 202 .07 798 .11 501 .19 299 7 54 .80716 .88 489 .92 227 .07 773 .11 511 .19284 6 55 9.80 731 9.88 478 9.92 253 0.07 747 0.11 522 0.19 269 5 56 .80746 .88 468 .92 279 .07 721 .11 532 .19 254 4 57 .80 762 .88 457 .92 304 .07 696 .11 543 .19 238 3 58 .80777 .88 447 .92 330 .07 670 .11 553 .19 223 2 59 .80 792 .88 436 .92 356 .07644 .11 564 .19 208 1 60 9.80 807 9.88 425 9.92 381 0.07 619 0.11 575 0.19 193 Cos Sin Cot Tan Csc Sec ' 129 (309) (230) 50 236 Table 4. Trigonometric Logarithms 40 (220) (319) 139 C ' Sin Cos Tan Cot Sec Csc 9.80 807 9.88 425 9.92 381 0.07 619 0.11 575 0.19 193 60 1 .80 822 .88 415 .92 407 .07 593 .11 585 .19 178 59 2 .80 837 .88 404 .92 433 .07 567 .11 596 .19 163 58 3 .80 852 .88 394 .92 458 .07 542 .11 606 .19 148 57 4 .80 867 .88 383 .92 484 .07 516 .11 617 .19 133 56 5 9.80 882 9.88 372 9.92 510 0.07 490 0.11 628 0.19 118 55 6 .80 897 .88 362 .92 535 .07 465 .11 638 .19 103 54 7 .80 912 .88 351 .92 561 .07 439 .11 649 .19 088 53 8 .80 927 .88 340 .92 587 .07 413 .11 660 .19 073 52 9 .80 942 .88 330 .92 612 .07 388 .11 670 .19 058 51 10 9.80 957 9.88319 9.92 638 0.07 362 0.11 681 0.19043 50 11 .80 972 .88 308 .92 663 .07 337 .11 692 .19 028 49 12 .80 987 .88 298 .92 689 .07311 .11 702 .19013 48 13 .81 002 .88 287 .92 715 .07 285 .11713 .18 998 47 14 .81 017 .88 276 .92 740 07 260 .11 724 .18983 46 15 9.81 032 9.88 266 9.92 766 0.07 234 0.11 734 0.18 968 45 16 .81 047 .88 255 .92 792 .07 208 .11 745 .18 953 44 17 .81 061 .88 244 .92 817 .07 183 .11 756 .18 939 43 18 .81 076 .88 234 .92 843 .07 157 .11 766 .18 924 42 19 .81 091 .88 223 .92 868 07 132 .11 777 .18 909 41 20 9.81 106 9.88 212 9.92 894 0.07 106 0.11 788 0.18 894 40 21 .81 121 .88 201 .92 920 .07 080 .11 799 .18 879 39 22 .81 136 .88 191 .92 945 .07 055 .11 809 .18 864 38 23 .81 151 .88 180 .92 971 .07 029 .11 820 .18849 37 24 .81 166 .88 169 .92 996 .07 004 .11 831 .18 834 36 25 9.81 180 9.88 158 9.93 022 0.06 978 0.11 842 0.18 820 35 26 .81 195 .88 148 .93 048 .06 952 .11 852 .18 805 34 27 .81 210 .88 137 .93 073 .06 927 .11 863 .18 790 33 28 .81 225 .88 126 .93 099 .06 901 .11 874 .18 775 32 29 .81 240 .88 115 .93 124 .06 876 .11 885 .18 760 31 30 9.81 254 9.88 105 9.93 150 0.06 850 0.11 895 0.18 746 30 31 .81 269 .88 094 .93 175 .06 825 .11 906 .18731 29 32 .81 284 .88 083 .93 201 .06 799 .11917 .18716 28 33 .81 299 .88 072 .93 227 .06 773 .11 928 .18701 27 34 .81 314 .88 061 .93 252 .06 748 .11 939 .18686 26 35 9.81 328 9.88 051 9.93 278 0.06 722 0.11 949 0.18 672 25 36 .81 343 .88 040 .93 303 .06 697 .11 960 .18657 24 37 .81 358 .88 029 .93 329 .06 671 .11 971 .18642 23 38 .81 372 .88 018 .93 354 .06 646 .11 982 .18628 22 39 .81 387 .88 007 .93 380 .06 620 .11 993 .18613 21 40 9.81 402 9.87 996 9.93 406 0.06 594 0.12 004 0.18598 20 41 .81 417 .87 985 .93 431 .06 569 .12015 .18 583 19 42 .81 431 .87 975 .93 457 .06 543 .12 025 .18 569 18 43 .81 446 .87 964 .92 482 .06 518 .12 036 .18 554 17 44 .81 461 .87 953 .93 508 .06 492 .12 047 .18539 16 45 9.81 475 9.87 942 9.93 533 0.06 467 0.12 058 0.18 525 15 46 .81 490 .87 931 .93 559 .06 441 .12 069 .18510 14 47 .81 505 .87 920 .93584 .06 416 .12 080 .18495 13 48 .81 519 .87 909 .93 610 .06 390 .12091 .18481 12 49 .81 534 .87 898 .93 636 .06 364 .12 102 .18466 11 50 9.81 549 9.87 887 9.93 661 0.06 339 0.12113 0.18451 10 51 .81 563 .87 877 .93 687 .06 313 .12 123 .18437 9 52 .81 578 .87 866 .93 712 .06 288 .12 134 .18422 8 53 .81 592 .87 855 .93 738 .06 262 .12 145 .18 408 7 54 .81 607 .87844 .93 763 .06 237 .12 156 .18393 6 55 9.81 622 9.87 833 9.93 789 0.06211 0.12 167 0.18378 5 56 .81 636 .87 822 .93 814 .06 186 .12 178 .18 364 4 57 .81 651 .87811 .93 840 .06 160 .12 189 ..18349 3 58 .81 665 .87 800 .93 865 .06 135 .12 200 .18335 2 59 .81 680 .87 789 .93 891 .06 109 .12211 .18320 1 60 9.81 694 9.87 778 9.93 916 0.06 084 0.12 222 .18 306 Cos Sin Cot | Tan Csc Sec ' 130 (310) (229) 49 < Table 4. Trigonometric Logarithms 237 41 (221) (318) 138 ' Sin Cos Tan Cot Sec Csc 9.81 694 9.87 778 9.93 916 0.06 084 0.12222 0.18 306 60 1 .81 709 .87 767 .93 942 .06 058 .12 233 .18291 59 2 .81 723 .87 756 .93 967 .06 033 .12 244 .18 277 58 3 .81 738 .87 745 .93 993 .06007 .12255 .18 262 57 4 .81 752 .87 734 .94 018 .05982 .12 266 .18 248 56 5 9.81 767 9.87 723 9.94 044 0.05 956 0.12 277 0.18233 55 6 .81 781 .87 712 .94 069 .05 931 .12 288 .18219 54 7 .81 796 .87 701 .94095 .05 905 .12 299 .18 204 53 8 .81 810 .87 690 .94 120 .05 880 .12310 .18 190 52 9 .81 825 .87 679 .94 146 .05 854 .12321 .18 175 51 10 9.81 839 9.87 668 9.94 171 0.05 829 0.12 332 0.18 161 50 11 .81854 .87 657 .94 197 .05 803 .12 343 .18 146 49 12 .81 868 .87 646 .94 222 .05 778 .12 354 .18 132 48 13 .81 882 .87 635 .94 248 .05 752 .12 365 .18118 47 14 .81 897 .87 624 .94 273 .05 727 .12376 .18 103 46 15 9.81 911 9.87 613 9.94 299 0.05 701 0.12 387 0.18 089 45 16 .81 926 .87 601 .94324 .05 676 .12 399 .18074 44 17 .81 940 .87 590 .94 350 .05 650 .12410 48 060 43 18 .81 955 .87 579 .94 375 .05 625 .12421 .18 045 42 19 .81 969 .87 568 .94 401 .05 599 .12 432 .18031 41 20 9.81 983 9.87 557 9.94 426 0.05 574 0.12443 0.18017 40 21 .81 998 .87 546 .94 452 .05 548 .12 454 .18 002 39 22 .82 012 .87 535 .94477 .05 523 .12 465 .17 988 38 23 .82 026 .87 524 .94 503 .05 497 .12 476 .17 974 37 24 .82041 .87 513 .94 528 .05 472 .12 487 .17 959 36 25 9.82 055 9.87 501 9.94 554 0.05 446 0.12 499 0.17 945 35 26 .82 069 .87 490 .94579 .05 421 .12510 .17931 34 27 .82084 .87 479 .94604 .05 396 .12521 .17916 33 28 .82 098 .87 468 .94 630 .05 370 .12 532 .17 902 32 29 .82 112 .87 457 .94655 .05 345 .12 543 .17 888 31 30 9.82 126 9.87 446 9.94 681 0.05 319 0.12 554 0.17 874 30 31 .82 141 .87 434 .94706 .05 294 .12 566 .17 859 29 32 .82 155 .87 423 .94 732 .05 268 .12 577 .17845 28 33 .82 169 .87 412 .94 757 .05 243 .12 588 .17831 27 34 .82184 .87 401 .94 783 05 217 .12 599 .17 816 26 35 9.82 198 9.87 390 9.94 808 0.05 192 0.12610 0.17 802 25 36 .82 212 .87 378 .94834 .05 166 .12 622 .17 788 24 37 .82 226 .87 367 .94 859 .05 141 .12 633 .17 774 23 38 .82 240 .87 356 .94884 .05 116 .12644 .17 760 22 39 .82 255 .87 345 .94 910 .05 090 .12 655 .17 745 21 40 9.82 269 9.87 334 9.94 935 0.05 065 0.12 666 0.17 731 20 41 .82 283 .87 322 .94 961 .05 039 .12 678 .17717 19 42 .82 297 .87311 .94 986 .05 014 .12 689 .17 703 18 43 .82311 .87 300 .95 012 .04988 .12 700 .17 689 17 44 .82 326 .87 288 .95 037 .04963 .12712 .17 674 16 45 9.82 340 9.87 277 9.95 062 0.04 938 0.12 723 0.17 660 15 46 .82 354 .87 266 .95 088 .04912 .12 734 .17 646 14 47 .82 368 .87 255 .95 113 .04887 .12 745 .17 632 13 48 82 382 .87 243 .95 139 .04861 .12 757 .17618 12 49 .82 396 .87 232 .95 164 .04836 .12 768 .17 604 11 50 9.82 410 9.87 221 9.95 190 0.04 810 0.12 779 0.17 590 10 51 .82 424 .87 209 .95 215 .04785 .12 791 .17 576 9 52 .82 439 .87 198 .95 240 .04760 .12 802 .17561 8 53 .82 453 .87 187 .95 266 .04734 .12813 .17 547 7 54 .82 467 .87 175 .95 291 .04709 .12 825 .17 533 6 55 9.82 481 9.87 164 9.95 317 0.04 683 0.12 836 0.17519 5 56 .82 495 .87 153 .95 342 .04658 .12847 .17 505 4 57 .82 509 .87 141 .95 368 .04632 .12 859 .17491 3 58 .82 523 .87 130 .95 393 .04607 .12 870 .17 477 2 59 .82 537 .87 119 .95418 .04582 .12 881 .17 463 1 60 9.82 551 9.87 107 9.95 444 0.04 556 0.12 893 0.17449 Cos Sin Cot Tan Csc Sec ' 131 (311) (228) 48 238 Table 4. Trigonometric Logarithms 42 (222) (317) 137 C ' Sin Cos Tan Cot Sec Csc 9.82 551 9.87 107 9.95 444 0.04 556 0.12 893 0.17 449 60 1 .82 565 .87 096 .95 469 .04531 .12 904 .17 435 59 2 .82 579 .87 085 .95 495 .04 505 .12915 .17421 58 3 .82 593 .87 073 .95 520 .04 480 .12 927 .17 407 57 4 .82 607 .87 062 .95 545 .04 455 .12 938 .17 393 56 5 9.82 621 9.87 050 9.95 571 0.04 429 0.12 950 0.17 379 55 6 .82 635 .87 039 .95 596 .04 404 .12 961 .17 365 54 7 .82 649 .87 028 .95 622 .04 378 .12 972 .17351 53 8 .82 663 .87 016 .95 647 .04 353 .12 984 .17 337 52 9 .82 677 .87 005 .95 672 .04 328 .12 995 .17 323 51 10 9.82 691 9.86 993 9.95 698 0.04 302 0.13 007 0.17 309 50 11 .82 705 .86 982 .95 723 .04 277 .13018 .17 295 49 12 .82 719 .86 970 .95 748 .04 252 .13 030 .17 281 48 13 .82 733 .86 959 .95 774 .04 226 .13 041 .17 267 47 14 .82 747 .86 947 .95 799 .04 201 .13 053 .17 253 46 15 9.82 761 9.86 936 9.95 825 0.04 175 0.13 064 0.17 239 45 16 .82 775 .86 924 .95 850 .04 150 .13 076 .17 225 44 17 .82 788 .86 913 .95 875 .04125 .13 087 .17212 43 18 .82 802 .86 902 .95 901 .04 099 .13 098 .17 198 42 19 .82 816 .86 890 .95 926 .04 074 .13 110 ' .17184 41 20 9.82 830 9.86 879 9.95 952 0.04 048 0.13 121 0.17 170 40 21 .82844 .86 867 .95 977 .04 023 .13 133 .17 156 39 22 .82 858 .86 855 .96 002 .03 998 .13 145 .17 142 38 23 .82 872 .86844 .96 028 .03 972 .13 156 .17 128 37 24 .82 885 .86 832 .96 053 .03 947 .13 168 .17 115 36 25 9.82 899 9.86 821 9.96 078 0.03 922 0.13 179 0.17 101 35 26 .82 913 .86 809 .96 104 .03 896 .13 191 .17 087 34 27 .82 927 .86 798 .96 129 .03 871 .13 202 .17 073 33 28 .82 941 .86 786 .96 155 .03 845 .13214 .17 059 32 29 .82 955 .86 775 .96 180 .03 820 .13 225 .17 045 31 30 9.82 968 9.86 763 9.96 205 0.03 795 0.13 237 0.17 032 30 31 .82 982 .86 752 .96 231 .03 769 .13 248 .17018 29 32 .82 996 .86 740 .96 256 .03 744 .13 260 .17 004 28 33 .83 010 .86 728 .96 281 .03 719 .13 272 .16 990 27 34 .83 023 .86 717 .96 307 .03 693 .13 283 .16 977 26 35 9.83 037 9.86 705 9.96 332 0.03 668 0.13 295 0.16 963 25 36 .83 051 .86 694 .96 357 .03 643 .13 306 .16 949 24 37 .83 065 .86 682 .96 383 .03 617 .13318 .16 935 23 38 .83 078 .86 670 .96 408 .03 592 .13 330 .16 922 22 39 .83 092 .86 659 .96 433 .03 567 .13 341 .16 908 21 40 9.83 106 9.86 647 9.96 459 0.03 541 0.13 353 0.16 894 20 41 .83 120 .86 635 .96 484 .03 516 .13 365 .16 880 19 42 .83 133 .86 624 .96 510 .03 490 .13 376 .16 867 18 43 .83 147 .86 612 .96 535 .03 465 .13 388 .16 853 17 44 .83 161 .86 600 .96 560 .03 440 .13 400 .16 839 16 45 9.83 174 9.86 589 9.96 586 0.03 414 0.13411 0.16 826 15 46 .83 188 .86 577 .96611 .03 389 .13 423 .16812 14 47 .83 202 .86 565 .96 636 .03 364 .13435 .16 798 13 48 .83 215 .86 554 .96 662 .03 338 .13 446 .16 785 12 49 .83 229 .86 542 .96 687 .03 313 .13 458 .16771 11 50 9.83 242 9.86 530 9.96 712 0.03 288 0.13 470 0.16 758 10 51 .83 256 .86 518 .96 738 .03 262 .13 482 .16 744 9 52 .83 270 .86 507 .96 763 .03 237 .13 493 .16 730 8 53 .83 283 .86 495 .96 788 .03 212 .13 505 .16717 7 54 .83 297 .86 483 .96 814 .03 186 .13517 .16 703 6 55 9.83 310 9.86 472 9.96 839 0.03 161 0.13 528 0.16 690 5 56 .83 324 .86 460 .96 864 .03 136 .13 540 .16 676 4 57 .83 338 .86 448 .96 890 .03 110 .13 552 .16 662 3 58 .83 351 .86 436 .96 915 .03 085 .13 564 .16 649 2 59 .83 365 .86 425 .96 940 .03 060 .13 575 .16 635 1 60 9.83 378 9.86 413 9.96 966 0.03 034 0.13 587 0.16 622 Cos Sin Cot Tan Csc Sec ' 132 (312) (227) 47 Table 4. Trigonometric Logarithms 239 43 (223) (316) 136 ' Sin Cos Tan Cot Sec Csc 9.83 378 9.86413 9.96 966 0.03 034 0.13 587 0.16622 60 1 .83 392 .86 401 .96 991 .03 009 .13 599 .16 608 59 2 .83 405 .86 389 .97 016 .02 984 .13611 .16 595 58 3 .83419 .86 377 .97 042 .02 958 .13 623 .16 581 57 4 .83 432 .86 366 .97 067 .02 933 .13 634 .16 568 56 5 9.83 446 9.86 354 9.97 092 0.02 908 0.13 646 0.16 554 55 6 .83459 .86 342 .97 118 .02 882 .13 658 .16541 54 7 .83 473 .86 330 .97 143 .02 857 .13 670 .16 527 53 8 .83 486 .86 318 .97 168 .02 832 .13 682 .16514 52 9 .83500 .86 306 .97 193 .02 807 .13 694 .16 500 51 10 9.83 513 9.86 295 9.97 219 0.02 781 0.13 705 0.16 487 50 11 .83 527 .86 283 .97 244 .02 756 .13717 .16 473 49 12 .83 540 .86 271 .97 269 .02 731 .13 729 .16 460 48 13 .83 554 .86 259 .97 295 .02 705 .13 741 .16 446 47 14 .83567 .86 247 .97 320 .02 680 .13 753 .16 433 46 15 9.83 581 9.86 235 9.97 345 0.02 655 0.13 765 0.16419 45 16 .83594 .86 223 .97 371 .02 629 .13 777 .16 406 44 17 .83 608 .86211 .97 396 .02 604 .13 789 .16 392 43 18 .83 621 .86 200 .97 421 .02 579 .13 800 .16 379 42 19 .83 634 .86 188 .97 447 .02 553 .13812 .16 366 41 20 9.83 648 9.86 176 9.97 472 0.02 528 0.13 824 0.16 352 40 21 .83 661 .86 164 .97 497 .02 503 .13 836 .16 339 39 22 .83 674 .86 152 .97 523 .02 477 .13848 .16326 38 23 .83 688 .86 140 .97 548 .02 452 .13 860 .16312 37 24 .83 701 .86 128 .97 573 .02427 .13 872 .16 299 36 25 9.83 715 9.86 116 9.97 598 0.02 402 0.13 884 0.16 285 35 26 .83 728 .86 104 .97 624 .02 376 .13 896 .16 272 34 27 .83741 .86 092 .97 649 .02 351 .13 908 .16 259 33 28 .83 755 .86 080 .97 674 .02 326 .13 920 .16 245 32 29 .83 768 .86 068 .97 700 .02 300 .13 932 .16 232 31 30 9.83 781 9.86 056 9.97 725 0.02 275 0.13 944 0.16219 30 31 .83 795 .86 044 .97 750 .02 250 .13 956 .16 205 29 32 .83 808 .86 032 .97 776 .02 224 .13 968 .16 192 28 33 .83 821 .86 020 .97 801 .02 199 .13 980 .16 179 27 34 .83834 .86 008 .97 826 .02 174 .13 992 .16 166 26 35 9.83 848 9.85 996 9.97 851 0.02 149 0.14 004 0.16 152 25 36 .83 861 .85 984 .97 877 .02 123 .14 016 .16 139 24 37 .83 874 .85 972 .97 902 .02 098 .14 028 .16 126 23 38 .83 887 .85 960 .97 927 .02 073 .14 040 .16 113 22 39 .83 901 .85948 .97 953 .02 047 .14 052 .16 099 21 40 9.83 914 9.85 936 9.97 978 0.02 022 0.14 064 0.16 086 20 41 .83 927 .85 924 .98 003 .01 997 .14 076 .16 073 19 42 .83 940 .85 912 .98 029 .01 971 .14 088 .16 060 18 43 .83 954 .85 900 .98 054 .01 946 .14 100 .16 046 17 44 .83 967 .85 888 .98 079 .01 921 .14 112 .16 033 16 45 9.83 980 9.85 876 9.98 104 0.01 896 0.14 124 0.16 020 15 46 .83 993 .85864 .98 130 .01 870 .14 136 .16 007 14 47 .84006 .85851 .98 155 .01845 .14 149 .15 994 13 48 .84020 .85 839 .98 180 .01 820 .14 161 .15 980 12 49 .84033 .85 827 .98 206 .01 794 .14 173 .15 967 11 50 9.84046 9.85 815 9.98 231 0.01 769 0.14 185 0.15 954 10 51 .84059 .85 803 .98 256 .01 744 .14 197 .15 941 9 52 .84072 .85 791 .98 281 .01 719 .14 209 .15 928 8 53 .84085 .85 779 .98 307 .01 693 .14 221 .15915 7 54 .84098 .85 766 .98 332 .01 668 .14 234 .15 902 6 55 9.84112 9.85 754 9.98 357 0.01 643 0.14 246 0.15 888 5 56 .84125 .85742 .98 383 .01 617 .14 258 .15 875 4 57 .84 138 .85 730 .98 408 .01 592 .14 270 .15 862 3 58 .84151 .85 718 .98 433 .01 567 .14 282 .15849 2 59 .84 164 .85 706 .98 458 .01 542 .14 294 .15 836 1 60 9.84 177 9.85 693 9.98 484 0.01 516 0.14 307 0.15 823 Cos Sin Cot Tan Csc Sec ' 133 (313) (226) 46 C 240 Table 4. Trigonometric Logarithms 44 (224) (315) 135= Sin Cos Tan Cot Sec Csc 9.84 177 9.85 693 9.98 484 0.01 516 0.14307 0.15 823 60 1 .84 190 .85 681 .98 509 .01 491 .14319 .15810 59 2 .84203 .85 669 .98 534 .01 466 .14331 .15 797 58 3 .84216 .85 657 .98 560 .01 440 .14 343 .15 784 57 4 .84229 .85 645 .98 585 .01 415 .14 355 .15 771 56 5 9.84 242 9.85 632 9.98610 0.01 390 0.14 368 0.15 758 55 6 .84 255 .85 620 .98 635 .01 365 .14 380 .15 745 54 7 .84 269 .85 608 .98 661 .01 339 .14 392 .15 731 53 8 .84282 .85 596 .98 686 .01 314 .14404 .15718 52 9 .84295 .85 583 .98 711 .01 289 .14417 .15 705 51 10 9.84 308 9.85 571 9.98 737 0.01 263 0.14429 0.15 692 50 11 .84 321 .85 559 .98 762 .01 238 .14 441 .15 679 49 12 .84334 .85 547 .98 787 .01 213 .14 453 .15 666 48 13 .84347 .85 534 .98 812 .01 188 .14 466 .15 653 47 14 .84 360 .85 522 .98 838 .01 162 .14 478 .15 640 46 15 9.84 373 9.85 510 9.98 863 0.01 137 0.14490 0.15 627 45 16 .84385 .85 497 .98 888 .01 112 .14 503 .15 615 44 17 .84 398 .85 485 .98 913 .01 087 .14515 .15 602 43 18 .84411 .85 473 .98 939 .01 061 .14 527 .15 589 42 19 .84 424 .85 460 .98 964 .01 036 .14 540 .15 576 41 20 9.84 437 9.85 448 9.98 989 0.01 Oil 0.14 552 0.15 563 40 21 .84 450 .85 436 .99 015 .00 985 .14 564 .15 550 39 22 .84 463 .85 423 .99 040 .00 960 .14 577 .15 537 38 23 .84476 .85411 .99 065 .00 935 .14 589 .15 524 37 24 .84 489 .85399 .99 090 .00 910 .14 601 .15511 36 25 9.84 502 9.85 386 9.99 116 0.00 884 0.14 614 0.15 498 35 26 .84 515 .85 374 .99 141 .00 859 .14 626 .15485 34 27 .84528 .85 361 .99 166 .00 834 .14 639 .15472 33 28 .84 540 .85 349 .99 191 .00 809 .14 651 .15 460 32 29 .84553 .85 337 .99 217 .00783 .14 663 .15447 31 30 9.84 566 9.85 324 9.99 242 0.00 758 0.14 676 0.15 434 30 31 .84579 .85 312 .99 267 .00 733 .14 688 .15421 29 32 .84592 .85 299 .99 293 .00 707 .14 701 .15 408 28 33 .84 605 .85 287 .99 318 .00682 .14713 .15 395 27 34 .84 618 .85 274 .99 343 .00657 .14 726 .15 382 26 35 9.84 630 9.85 262 9.99 368 0.00 632 0.14 738 0.15 370 25 36 .84643 .85 250 .99 394 .00606 .14 750 .15 357 24 37 .84 656 .85237 .99 419 .00581 .14 763 .15 344 23 38 .84669 .85 225 .99 444 .00 556 .14 775 .15331 22 39 .84682 .85 212 .99 469 .00 531 .14 788 .15318 21 40 9.84 694 9.85 200 9.99 495 0.00 505 0.14 800 0.15306 20 41 .84 707 .85 187 .99 520 .00 480 .14813 .15 293 19 42 .84 720 .85 175 .99 545 .00455 .14 825 .15 280 18 43 .84733 .85 162 .99 570 .00430 .14 838 .15 267 17 44 .84745 .85 150 .99 596 .00 404 .14 850 .15 255 16 45 9.84 758 9.85 137 9.99 621 0.00 379 0.14 863 0.15 242 15 46 .84771 .85 125 .99 646 .00354 .14 875 .15 229 14 47 .84784 .85 112 .99 672 .00 328 .14 888 .15216 13 48 .84 796 .85100 .99 697 .00303 .14 900 .15 204 12 49 .84809 .85087 .99 722 .00278 .14 913 .15 191 11 50 9.84 822 9.85 074 9.99 747 0.00 253 0.14 926 0.15 178 10 51 .84835 .85 062 .99 773 .00 227 .14 938 .15 165 9 52 .84 847 .85049 .99 798 .00202 .14951 .15 153 8 53 .84860 .85 037 .99 823 .00 177 .14 963 .15 140 7 54 .84873 .85 024 .99848 .00 152 .14 976 .15 127 6 55 9.84 885 9.85012 9.99 874 0.00 126 0.14 988 0.15 115 5 56 .84898 .84999 .99 899 .00 101 .15 001 .15 102 4 57 .84911 .84 986 .99 924 .00 076 .15014 .15 089 3 58 .84923 .84 974 .99 949 .00 051 .15 026 .15 077 2 59 .84 936 .84 961 .99 975 .00 025 .15 039 .15 064 1 60 9.84 949 9.84 949 0.00 000 0.00 000 0.15 051 0.15 051 Cos Sin Cot Tan Csc Sec ' 134 (314) (225) 45 Table 5. Meridional Parts 241 1 2 3 4 5 6 7 9 0.0 59.6 119.2 178.9 238.6 298.3 358.2 418.2 478.3 538.6 1 1.0 60.6 20.2 79.9 39.6 99.3 59.2 19.2 79.3 39.6 1 2 2.0 61.6 21.2 80.8 40.6 300.3 60.2 20.2 80.3 40.6 2 3 3.0 62.6 22.2 81.8 41.6 01.3 61.2 21.2 81.3 41.6 3 4 4.0 63.6 23.2 82.8 42.5 02.3 62.2 22.2 82.3 42.6 4 5 5.0 64.6 124.2 183.8 243.5 303.3 363.2 423.2 483.3 543.6 5 6 6.0 65.6 25.2 84.8 44.5 04.3 64.2 24.2 84.3 44.6 6 7 7.0 66.5 26.2 85.8 45.5 05.3 65.2 25.2 85.3 45.6 7 8 7.9 67.5 27.2 86.8 46.5 06.3 66.2 26.2 86.3 46.6 8 9 8.9 68.5 28.2 87.8 47.5 07.3 67.2 27.2 87.3 47.6 9 10 9.9 69.5 129.1 188.8 248.5 308.3 368.2 428.2 488.3 548.6 10 11 10.9 70.5 30.1 89.8 49.5 09.3 69.2 29.2 89.3 49.6 11 12 11.9 71.5 31.1 90.8 50.5 10.3 70.2 30.2 90.4 50.6 12 13 12.9 72.5 32.1 91.8 51.5 11.3 71.2 31.2 91.4 51.7 13 14 13.9 73.5 33.1 92.8 52.5 12.3 72.2 32.2 92.4 52.7 14 15 14.9 74.5 134.1 193.8 253.5 313.3 373.2 433.2 493.4 553.7 15 16 15.9 75.5 35.1 94.8 54.5 14.3 74.2 34.2 94.4 54.7 16 17 16.9 76.5 36.1 95.8 55.5 15.3 75.2 35.2 95.4 55.7 17 18 17.9 77.5 37.1 96.8 56.5 16.3 76.2 36.2 96.4 56.7 18 19 18.9 78.5 38.1 97.8 57.5 17.3 77.2 37.2 97.4 57.7 19 20 19.9 79.5 139.1 198.8 258.5 318.3 378.2 438.2 498.4 558.7 20 21 20.9 80.5 40.1 99.7 59.5 19.3 79.2 39.2 99.4 59.7 21 22 21.9 81.5 41.1 200.7 60.5 20.3 80.2 40.2 500.4 60.7 22 23 22.8 82.4 42.1 01.7 61.5 21.3 81.2 41.2 01.4 61.7 23 24 23.8 83.4 43.1 02.7 62.5 22.3 82.2 42.2 02.4 62.7 24 25 24.8 84.4 144.1 203.7 263.5 323.3 383.2 443.2 503.4 563.7 25 26 25.8 85.4 45.1 04.7 64.5 24.3 84.2 44.2 04.4 64.7 26 27 26.8 86.4 46.0 05.7 65.5 25.3 85.2 45.2 05.4 65.7 27 28 27.8 87.4 47.0 06.7 66.5 26.3 86.2 46.2 06.4 66.8 28 29 28.8 88.4 48.0 07.7 67.4 27.3 87.2 47.2 07.4 67.8 29 30 29.8 89.4 149.0 208.7 268.4 328.3 388.2 448.2 508.4 568.8 30 31 30.8 90.4 50.0 09.7 69.4 29.3 89.2 49.2 09.4 69.8 31 32 31.8 91.4 51.0 10.7 70.4 30.3 90.2 50.2 10.4 70.8 32 33 32.8 92.4 52.0 11.7 71.4 31.3 91.2 51.2 11.4 71.8 33 34 33.8 93.4 53.0 12.7 72.4 32.3 92.2 52.2 12.4 72.8 34 35 34.8 94.4 154.0 213.7 273.4 333.3 393.2 453.2 513.4 573.8 35 36 35.8 95.4 55.0 14.7 74.4 34.3 94.2 54.3 14.5 74.8 36 37 36.7 96.4 56.0 15.7 75.4 35.3 95.2 55.3 15.5 75.8 37 38 37.7 97.3 57.0 16.7 76.4 36.2 96.2 56.3 16.5 76.8 38 39 38.7 98.3 58.0 17.7 77.4 37.2 97.2 57.3 17.5 77.8 39 40 39.7 99.3 159.0 218.7 278.4 338.2 398.2 458.3 518.5 578.8 40 41 40.7 100.3 60.0 19.7 79.4 39.2 99.2 59.3 19.5 79.9 41 42 41.7 01.3 61.0 20.6 80.4 40.2 400.2 60.3 20.5 80.9 42 43 42.7 02.3 62.0 21.6 81.4 41.2 01.2 61.3 21.5 81.9 43 44 43.7 03.3 63.0 22.6 82.4 42.2 02.2 62.3 22.5 82.9 44 45 44.7 104.3 164.0 223.6 283.4 343.2 403.2 463.3 523.5 583.9 45 46 45.7 05.3 65.0 24.6 84.4 44.2 04.2 64.3 24.5 84.9 46 47 46.7 06.3 66.0 25.6 85.4 45.2 05.2 65.3 25.5 85.9 47 48 47.7 07.3 67.0 26.6 86.4 46.2 06.2 66.3 26.5 86.9 48 49 48.7 08.3 68.0 27.6 87.4 47.2 07.2 67.3 27.5 87.9 49 50 49.7 109.3 168.9 228.6 288.4 348.2 408.2 468.3 528.5 588.9 50 51 50.7 10.3 69.9 29.6 89.4 49.2 09.2 69.3 29.5 89.9 51 52 51.6 11.3 70.9 30.6 90.4 50.2 10.2 70.3 30.5 90.9 52 53 52.6 12.3 71.9 31.6 91.4 51.2 11.2 71.3 31.5 91.9 53 54 53.6 13.2 72.9 32.6 92.4 52.2 12.2 72.3 32.5 93.0 54 55 54.6 114.2 173.9 233.6 293.4 353.2 413.2 473.3 533.5 594.0 55 56 55.6 15.2 74.9 34.6 94.4 54.2 14.2 74.3 34.6 95.0 56 57 56.6 16.2 75.9 35.6 95.4 55.2 15.2 75.3 35.6 96.0 57 58 57.6 17.2 76.9 36.6 96.3 56.2 16.2 76.3 36.6 97.0 58 59 58.6 18.2 77.9 37.6 97.3 57.2 17.2 77.3 37.6 98.0 59 60 59.6 119.2 178.9 238.6 298.3 358.2 418.2 478.3 538.6 599.0 60 ' 1 2 3 4 5 6 7 8 9 ' 242 Table 5. Meridional Parts ' 10 11 12 13 14 15 16 17 18 19 ' 599.0 659.6 720.5 781.5 842.8 904.4 966.3 1028.5 1091.0 1153.9 1 600.0 60.6 21.5 82.5 43.9 05.4 67.3 29.5 92.0 54.9 1 2 01.0 61.7 22.5 83.6 44.9 06.5 68.3 30.5 93.1 56.0 2 3 02.0 62.7 23.5 84.6 45.9 07.5 69.4 31.6 94.1 57.0 ! 3 4 03.0 63.7 24.5 85.6 46.9 08.5 70.4 32.6 95.2 58.1 4 5 604.1 664.7 725.5 786.6 847.9 909.6 971.4 1033.7 1096.2 1159.1 5 6 05.1 65.7 26.6 87.6 49.0 10.6 72.5 34.7 97.3 60.2 6 7 06.1 66.7 27.6 88.7 50.0 11.6 73.5 35.7 98.3 61.2 7 8 07.1 67.7 28.6 89.7 51.0 12.6 74.6 36.8 99.4 62.3 8 9 08.1 68.7 29.6 90.7 52.0 13.7 75.6 37.8 1100.4 63.3 9 10 609.1 669.8 730.6 791.7 853.1 914.7 976.6 1038.9 1101.4 1164.4 10 11 10.1 70.8 31.6 92.7 54.1 15.7 77.7 39.9 02.5 65.4 11 12 11.1 71.8 32.7 93.8 55.1 16.8 78.7 40.9 03.5 66.5 12 13 12.1 72.8 33.7 94.8 56.1 17.8 79.7 42.0 04.6 67.5 13 14 13.1 73.8 34.7 95.8 57.2 18.8 80.8 43.0 05.6 68.6 14 15 614.1 674.8 735.7 796.8 858.2 919.8 981.8 1044.1 1106.7 1169.7 15 16 15.2 75.8 36.7 97.8 59.2 20.9 82.8 45.1 07.7 70.7 16 17 16.2 76.8 37.7 98.9 60.2 21.9 83.9 46.1 08.8 71.8 17 18 17.2 77.9 38.8 99.9 61.3 22.9 84.9 47.2 09.8 72.8 18 19 18.2 78.9 39.8 800.9 62.3 24.0 85.9 48.2 10.9 73.9 19 20 619.2 679.9 740.8 801.9 863.3 925.0 987.0 1049.3 1111.9 1174.9 20 21 20.2 80.9 41.8 02.9 64.3 26.0 88.0 50.3 13.0 76.0 21 22 21.2 81.9 42.8 04.0 65.4 27.1 89.0 51.3 14.0 77.0 22 23 22.2 82.9 43.8 05.0 66.4 28.1 90.1 52.4 15.0 78.1 23 24 23.2 83.9 44.9 06.0 67.4 29.1 91.1 53.4 16.1 79.1 24 25 624.2 684.9 745.9 807.0 868.5 930.1 992.1 1054.5 1117.1 1180.2 25 26 25.3 86.0 46.9 08.1 69.5 31.2 93.2 55.5 18.2 81.2 26 27 26.3 87.0 47.9 09.1 70.5 32.2 94.2 56.6 19.2 82.3 27 28 27.3 88.0 48.9 10.1 71.5 33.2 95.3 57.6 20.3 83.3 28 29 28.3 89.0 49.9 11.1 72.6 34.3 96.3 58.6 21.3 84.4 29 30 629.3 690.0 751.0 812.1 873.6 935.3 997.3 1059.7 1122.4 1185.5 30 31 30.3 91.0 52.0 13.2 74.6 36.3 98.4 60.7 23.4 86.5 31 32 31.3 92.0 53.0 14.2 75.6 37.4 99.4 61.8 24.5 87.6 32 33 32.3 93. 54.0 15.2 76.7 38.4 1000.4 62.8 25.5 88.6 33 34 33.3 94. 55.0 16.2 77.7 39.4 01.5 63.9 26.6 89.7 34 35 634.3 695. 756.0 817.3 878.7 940.5 1002.5 1064.9 1127.6 1190.7 35 36 35.4 96. 57.1 18.3 79.7 41.5 03.6 65.9 28.7 91.8 36 37 36.4 97. 58.1 19.3 80.8 42.5 04.6 67.0 29.7 92.8 37 38 37.4 98.1 59.1 20.3 81.8 43.6 05.6 68.0 30.8 93.9 38 39 38.4 99.1 60.1 21.3 82.8 44.6 06.7 69.1 31.8 95.0 39 40 639.4 700.2 761.1 822.4 883.8 945.6 1007.7 1070.1 1132.9 1196.0 40 41 40.4 01.2 62.2 23.4 84.9 46.7 08.7 71.2 33.9 97.1 41 42 41.4 02.2 63.2 24.4 85.9 47.7 09.8 72.2 35.0 98.1 42 43 42.4 03.2 64.2 25.4 86.9 48.7 10.8 73.2 36.0 99.2 43 44 43.4 04.2 65.2 26.5 88.0 49.7 11.8 74.3 37.1 1200.2 44 45 644.5 705.2 766.2 827.5 889.0 950.8 1012.9 1075.3 1138.1 1201.3 45 46 45.5 06.2 67.3 28.5 90.0 51.8 13.9 76.4 39.2 02.3 46 47 46.5 07.3 68.3 29.5 91.0 52.8 15.0 77.4 40.2 03.4 47 48 47.5 08.3 69.3 30.5 92.1 53.9 16.0 78.5 41.3 04.5 48 49 48.5 09.3 70.3 31.6 93.1 54.9 17.0 79.5 42.3 05.5 49 50 649.5 710.3 771.3 832.6 894.1 955.9 1018.1 1080.5 1143.4 1206.6 50 51 50.5 11.3 72.3 33.6 95.2 57.0 19.1 81.6 44.4 07.6 51 52 51.5 12.3 73.4 34.6 96.2 58.0 20.2 82.6 45.5 08.7 52 53 52.5 13.4 74.4 35.7 97.2 59.0 21.2 83.7 46.5 09.7 53 54 53.6 14.4 75.4 36.7 98.2 60.1 22.2 84.7 47.6 10.8 54 55 654.6 715.4 776.4 837.7 899.3 961.1 1023.3 1085.8 1148.6 1211.8 55 56 55.6 16.4 77.4 38.7 900.3 62.1 24.3 86.8 49.7 12.9 56 57 56.6 17.4 78.5 39.8 01.3 63.2 25.3 87.9 50.7 14.0 57 58 57.6 18.4 79.5 40.8 02.3 64.2 26.4 88.9 51.8 15.0 58 59 58.6 19.4 80.5 41.8 03.4 65.2 27.4 89.9 52.8 16.1 59 60 659.6 720.5 781.5 842.8 904.4 966.3 1028.5 1091.0 1153.9 1217.1 60 ' 10 11 12 13 14 15 16 17 18 19 ' Table 5. Meridional Parts 243 1 20 21 22 23 24 25 26 27 28 29 ' 1217.1 1280.8 1344.9 1409.5 1474.5 1540.1 1606.2 1672.9 1740.2 1808.1 1 18.2 81.9 46.0 10.6 75.6 41.2 07.3 74.0 41.3 09.2 1 2 19.3 82.9 47.1 11.6 76.7 42.3 08.4 75.1 42.4 10.4 2 3 20.3 84.0 48.1 12.7 77.8 43.4 09.5 76.2 43.6 11.5 3 4 21.4 85.1 49.2 13.8 78.9 44.5 10.6 77.4 44.7 12.6 4 5 1222.4 1286.1 1350.3 1414.9 1480.0 1545.6 1611.7 1678.5 1745.8 1813.8 5 6 23.5 87.2 51.4 16.0 81.1 46.7 12.9 79.6 46.9 14.9 6 7 24.5 88.3 52.4 17.1 82.2 47.8 14.0 80.7 48.1 16.1 7 8 25.6 89.3 53.5 18.1 83.3 48.9 15.1 81.8 49.2 17.2 8 9 26.7 90.4 54.6 19.2 84.3 50.0 16.2 82.9 50.3 18.3 9 10 1227.7 1291.5 1355.7 1420.3 1485.4 1551.1 1617.3 1684.1 1751.5 1819.5 10 11 28.8 92.5 56.7 21.4 86.5 52.2 18.4 85.2 52.6 20.6 11 12 29.8 93.6 57.8 22.5 87.6 53.3 19.5 86.3 53.7 21.8 12 13 30.9 94.7 58.9 23.5 88.7 54.4 20.6 87.4 54.8 22.9 13 14 32.0 95.7 59.9 24.6 89.8 55.5 21.7 88.5 56.0 24.0 14 15 1233.0 1296.8 1361.0 1425.7 1490.9 1556.6 1622.8 1689.7 1757.1 1825.2 15 16 34.1 97.9 62.1 26.8 92.0 57.7 23.9 90.8 58.2 26.3 16 17 35.1 98.9 63.2 27.9 93.1 58.8 25.0 91.9 59.4 27.5 17 18 36.2 1300.0 64.2 29.0 94.2 59.9 26.2 93.0 60.5 28.6 18 19 37.3 01.1 65.3 30.0 95.2 61.0 27.3 94.1 61.6 29.7 19 20 1238.3 1302.1 1366.4 1431.1 1496.3 1562.1 1628.4 1695.3 1762.7 1830.9 20 21 39.4 03.2 67.5 32.2 97.4 63.2 29.5 96.4 63.9 32.0 21 22 40.4 04.3 68.5 33.3 98.5 64.3 30.6 97.5 65.0 33.2 22 23 41.5 05.3 69.6 34.4 99.6 65.4 31.7 98.6 66.1 34.3 23 24 42.6 06.4 70.7 35.4 1500.7 66.5 32.8 99.7 67.3 35.4 24 25 1243.6 1307.5 1371.8 1436.5 1501.8 1567.6 1633.9 1700.9 1768.4 1836.6 25 26 44.7 08.5 72.8 37.6 02.9 68.7 35.0 02.0 69.5 37.7 26 27 45.7 09.6 73.9 38.7 04.0 69.8 36.1 03.1 70.7 38.9 27 28 46.8 10.7 75.0 39.8 05.1 70.9 37.3 04.2 71.8 40.0 28 29 47.9 11.7 76.1 40.9 06.2 72.0 38.4 05.3 72.9 41.2 29 30 1248.9 1312.8 1377.1 1442.0 1507.3 1573.1 1639.5 1706.5 1774.1 1842.3 30 31 50.0 13.9 78.2 43.0 08.4 74.2 40.6 07.6 75.2 43.4 31 32 51.0 14.9 79.3 44.1 09.4 75.3 41.7 08.7 76.3 44.6 32 33 52.1 16.0 80.4 45.2 10.5 76.4 42.8 09.8 77.4 45.7 33 34 53.2 17.1 81.5 46.3 11.6 77.5 43.9 10.9 78.6 46.9 34 35 1254.2 1318.2 1382.5 1447.4 1512.7 1578.6 1645.0 1712.1 1779.7 1848.0 35 36 55.3 19.2 83.6 48.5 13.8 79.7 46.2 13.2 80.8 49.2 36 37 56.4 20.3 84.7 49.5 14.9 80.8 47.3 14.3 82.0 50.3 37 38 57.4 21.4 85.8 50.6 16.0 81.9 48.4 15.4 83.1 51.4 38 39 58.5 22.4 86.8 51.7 17.1 83.0 49.5 16.6 84.2 52.6 39 40 1259.5 1323.5 1387.9 1452.8 1518.2 1584.1 1650.6 1717.7 1785.4 1853.7 40 41 60.6 24.6 89.0 53.9 19.3 85.2 51.7 18.8 86.5 54.9 41 42 61.7 25.6 90.1 55.0 20.4 86.3 52.8 19.9 87.6 56.0 42 43 62.7 26.7 91.1 56.1 21.5 87.4 53.9 21.1 88.8 57.2 43 44 63.8 27.8 92.2 57.1 22.6 88.5 55.1 22.2 89.9 58.3 44 45 1264.9 1328.9 1393.3 1458.2 1523.7 1589.6 1656.2 1723.3 1791.1 1859.5 45 46 65.9 29.9 94.4 59.3 24.8 90.7 57.3 24.4 92.2 60.6 46 47 67.0 31.0 95.5 60.4 25.9 91.8 58.4 25.5 93.3 61.8 47 48 68.0 32.1 96.5 61.5 27.0 92.9 59.5 26.7 94.5 62.9 48 49 69.1 33.1 97.6 62.6 28.0 94.1 60.6 27.8 95.6 64.0 49 50 1270.2 1334.2 1398.7 1463.7 1529.1 1595.2 1661.7 1728.9 1796.7 1865.2 50 51 71.2 35.3 99.8 64.8 30.2 96.3 62.9 30.0 97.9 66.3 51 52 72.3 36.3 1400.9 65.8 31.3 97.4 64.0 31.2 99.0 67.5 52 53 73.4 37.4 01.9 66.9 32.4 98.5 65.1 32.3 1800.1 68.6 53 54 74.4 38.5 03.0 68.0 33.5 99.6 66.2 33.4 01.3 69.8 54 55 1275.5 1339.6 1404.1 1469.1 1534.6 1600.7 1667.3 1734.5 1802.4 1870.9 55 56 76.6 40.6 05.2 70.2 35.7 01.8 68.4 35.7 03.5 72.1 56 57 77.6 41.7 06.2 71.3 36.8 02.9 69.5 36.8 04.7 73.2 57 58 78.7 42.8 07.3 72.4 37.9 04.0 70.7 37.9 05.8 74.4 58 59 79.7 43.8 08.4 73.5 39.0 05.1 71.8 39.1 07.0 75.5 59 60 1280.8 1344.9 1409.5 1474.5 1540.1 1606.2 1672.9 1740.2 1808.1 1876.7 60 ' 20 21 22 23 24 25 26 27 28 29 ' 244 Table 5. Meridional Parts ' 30 31 32 33 34 35 36 37 38 39 ; 1876.7 1946.0 2016.0 2086.8 2158.4 2230.9 2304.2 2378.5 2453.8 2530.2 o 1 77.8 47.1 17.2 88.0 59.6 32.1 05.5 79.8 55.1 31.5 1 2 79.0 48.3 18.3 89.2 60.8 33.3 06.7 81.0 56.4 32.8 2 3 80.1 49.4 19.5 90.3 62.0 34.5 07.9 82.3 57.6 34.0 3 4 81.3 50.6 20.7 91.5 63.2 35.7 09.2 83.5 58.9 35.3 4 5 1882.4 1951.8 2021.9 2092.7 2164.4 2236.9 2310.4 2384.8 2460.2 2536.6 5 6 83.6 52.9 23.0 93.9 65.6 38.2 11.6 86.0 61.4 37.9 6 7 84.7 54.1 24.2 95.1 66.8 39.4 12.9 87.3 62.7 39.2 7 8 85.9 55.3 25.4 96.3 68.0 40.6 14.1 88.5 64.0 40.5 8 9 87.0 56.4 26.6 97.5 69.2 41.8 15.3 89.8 65.2 41.7 9 10 1888.2 1957.6 2027.7 2098.7 2170.4 2243.0 2316.5 2391.0 2466.5 2543.0 10 11 89.3 58.7 28.9 99.8 71.6 44.2 17.8 92.3 67.8 44.3 11 12 90.5 59.9 30.1 2101.0 72.8 45.5 19.0 93.5 69.0 45.6 12 13 91.6 61.1 31.3 02.2 74.0 46.7 20.3 94.8 70.3 46.9 13 14 92.8 62.2 32.4 03.4 75.2 47.9 21.5 96.0 71.6 48.2 14 15 1893.9 1963.4 2033.6 2104.6 2176.4 2249.1 2322.7 2397.3 2472.8 2549.5 15 16 95.1 64.6 34.8 05.8 77.6 50.3 24.0 98.5 74.1 50.7 16 17 96.2 65.7 36.0 07.0 78.8 51.6 25.2 99.8 75.4 52.0 17 18 97.4 66.9 37.1 08.2 80.0 52.8 26.4 2401.0 76.6 53.3 18 19 98.5 68.1 38.3 09.4 81.2 54.0 27.7 02.3 77.9 54.6 19 20 1899.7 1969.2 2039.5 2110.6 2182.5 2255.2 2328.9 2403.5 2479.2 2555.9 20 21 1900.8 70.4 40.7 11.8 83.7 56.4 30.1 04.8 80.4 57.2 21 22 02.0 71.5 41.8 12.9 84.9 57.7 31.4 06.0 81.7 58.5 22 23 03.1 72.7 43.0 14.1 86.1 58.9 32.6 07.3 83.0 59.8 23 24 04.3 73.9 44.2 15.3 87.3 60.1 33.8 08.5 84.3 61.0 24 25 1905.5 1975.0 2045.4 2116.5 2188.5 2261 3 2335.1 2409.8 2485.5 2562.3 25 26 06.6 76.2 46.6 17.7 89.7 62.5 36.3 11.1 86.8 63.6 26 27 07.8 77.4 47.7 18.9 90.9 63.8 37.6 12.3 88.1 64.9 27 28 08.9 78.5 48.9 20.1 92.1 65.0 38.8 13.6 89.3 66.2 28 29 10.1 79.7 50.1 21.3 93.3 66.2 40.0 14.8 90.6 67.5 29 30 1911.2 1980.9 2051.3 2122.5 2194.5 2267.4 2341.3 2416.1 2491.9 2568.8 30 31 12.4 82.0 52.5 23.7 95.7 68.7 42.5 17.3 93.2 70.1 31 32 13.5 83.2 53.6 24.9 96.9 69.9 43.7 18.6 94.4 71.4 32 33 14.7 84.4 54.8 26.1 98.1 71.1 45.0 19.8 95.7 72.7 33 34 15.8 85.5 56.0 27.3 99.4 72.3 46.2 21.1 97.0 73.9 34 35 1917.0 1986.7 2057.2 2128.5 2200.6 2273.5 2347.5 2422.3 2498.3 2575.2 35 36 18.2 87.9 58.4 29.6 01.8 74.8 48.7 23.6 99.5 76.5 36 37 19.3 89.1 59.5 30.8 03.0 76.0 49.9 24.9 2500.8 77.8 37 38 20.5 90.2 60.7 32.0 04.2 77.2 51.2 26.1 02.1 79.1 38 39 21.6 91.4 61.9 33.2 05.4 78.4 52.4 27.4 03.4 80.4 39 40 1922.8 1992.6 2063.1 2134.4 2206.6 2279.7 2353.7 2428.6 2504.6 2581.7 40 41 23.9 93.7 64.3 35.6 07.8 80.9 54.9 29.9 05.9 83.0 41 42 25.1 94.9 65.5 36.8 09.0 82.1 56.1 31.2 07.2 84.3 42 43 26.3 96.1 66.6 38.0 10.2 83.3 57.4 32.4 08.5 85.6 43 44 27.4 97.2 67.8 39.2 11.5 84.6 58.6 33.7 09.7 86.9 44 45 1928.6 1998.4 2069.0 2140.4 2212.7 2285.8 2359.9 2434.9 2511.0 2588.2 45 46 29.7 99.6 70.2 41.6 13.9 87.0 61.1 36.2 12.3 89.5 46 47 30.9 2000.7 71.4 42.8 15.1 88.3 62.4 37.4 13.6 90.8 47 48 32.0 01.9 72.6 44.0 16.3 89.5 63.6 38.7 14.8 92.1 48 49 33.2 03.1 73.7 45.2 17.5 90.7 64.8 40.0 16.1 93.4 49 50 1934.4 2004.3 2074.9 2146.4 2218.7 2291.9 2366.1 2441.2 2517.4 2594.7 50 51 35.5 05.4 76.1 47.6 19.9 93.2 67.3 42.5 18.7 96.0 51 52 36.7 06.6 77.3 48.8 21.1 94.4 68.6 43.7 20.0 97.3 52 53 37.8 07.8 78.5 50.0 22.4 95.6 69.8 45.0 21.2 98.5 53 54 39.0 08.9 79.7 51.2 23.6 96.9 71.1 46.3 22.5 99.8 54 55 1940.2 2010.1 2080.8 2152.4 2224.8 2298.1 2372.3 2447.5 2523.8 2601.1 55 56 41.3 11.3 82.0 53.6 26.0 99.3 73.6 48.8 25.1 02.4 56 57 42.5 12.5 83.2 54.8 27.2 2300.5 74.8 50.1 26.4 03.7 57 58 43.6 13.6 84.4 56.0 28.4 01.8 76.1 51.3 27.6 05.0 58 59 44.8 14.8 85.6 57.2 29.6 03.0 77.3 52.6 28.9 06.3 59 60 1946.0 2016.0 2086.8 2158.4 2230.9 2304.2 2378.5 2453.8 2530.2 2607.6 60 ' 30 31 32 33 34 35 36 37 38 39 ' Table 5. Meridional Parts 245 ' 40 41 42 43 44 45 46 47 48 49 ' 2607.6 2686.2 2766.0 2847.1 2929.5 3013.4 3098.7 3185.6 3274.1 3364.4 1 08.9 87.6 67.4 48.5 30.9 14.8 3100.1 87.1 75.6 65.9 1 2 10.2 88.9 68.7 49.9 32.3 16.2 01.6 88.5 77.1 67.4 2 3 11.5 90.2 70.1 51.2 33.7 17.6 03.0 90.0 78.6 69.0 3 4 12.8 91.5 71.4 52.6 35.1 19.0 04.4 91.4 80.1 70.5 4 5 2614.1 2692.8 2772.8 2853.9 2936.5 3020.4 3105.9 3192.9 3281.6 3372.0 5 6 15.4 94.2 74.1 55.3 37.9 21.8 07.3 94.4 83.1 73.5 6 7 16.8 95.5 75.4 56.7 39.3 23.3 08.8 95.8 84.6 75.1 7 8 18.1 96.8 76.8 58.0 40.6 24.7 10.2 97.3 86.1 76.6 8 9 19.4 98.1 78.1 59.4 42.0 26.1 11.6 98.8 87.6 78.1 9 10 2620.7 2699.5 2779.5 2860.8 2943.4 3027.5 3113.1 3200.2 3289.0 3379.6 10 11 22.0 2700.8 80.8 62.1 44.8 28.9 14.5 01.7 90.5 81.2 11 12 23.3 02.1 82.2 63.5 46.2 30.3 16.0 03.2 92.0 82.7 12 13 24.6 03.4 83.5 64.9 47.6 31.7 17.4 04.6 93.5 84.2 13 14 25.9 04.8 84.8 66.2 49.0 33.2 18.8 06.1 95.0 85.7 14 15 2627.2 2706.1 2786.2 2867.6 2950.4 3034,6 3120.3 3207.6 3296.5 3387.3 15 16 28.5 07.4 87.5 69.0 51.8 36.0 21.7 09.0 98.0 88.8 16 17 29.8 08.7 88.9 70.3 53.2 37.4 23.2 10.5 99.5 90.3 17 18 31.1 10.1 90.2 71.7 54.5 38.8 24.6 12.0 3301.0 91.8 18 19 32.4 11.4 91.6 73.1 55.9 40.2 26.0 13.4 02.5 93.4 19 20 2633.7 2712.7 2792.9 2874.4 2957.3 3041.7 3127.5 3214.9 3304.0 3394.9 20 21 35.0 14.0 94.3 75.8 58.7 43.1 28.9 16.4 05.5 96.4 21 22 36.3 15.4 95.6 77.2 60.1 44.5 30.4 17.9 07.0 98.0 22 23 37.6 16.7 97.0 78.6 61.5 45.9 31.8 19.3 08.5 99.5 23 24 38.9 18.0 98.3 79.9 62.9 47.3 33.3 20.8 10.0 3401.0 24 25 2640.2 2719.3 2799.7 2881.3 2964.3 3048.7 3134.7 3222.3 3311.5 3402.6 25 26 41.6 20.7 2801.0 82.7 65.7 50.2 36.2 23.7 13.0 04.1 26 27 42.9 22.0 02.4 84.0 67.1 51.6 37.6 25.2 14.5 05.6 27 28 44.2 23.3 03.7 85.4 68.5 53.0 39.0 26.7 16.0 07.2 28 29 45.5 24.7 05.1 86.8 69.9 54.4 40.5 28.2 17.5 08.7 29 30 2646.8 2726.0 2806.4 2888.2 2971.3 3055.9 3141.9 3229.6 3319.0 3410.2 30 31 48.1 27.3 07.8 89.5 72.7 57.3 43.4 31.1 20.5 11.8 31 32 49.4 28.6 09.1 90.9 74.1 58.7 44.8 32.6 22.1 13.3 32 33 50.7 30.0 10.5 92.3 75.5 60.1 46.3 34.1 23.6 14.8 33 34 52.0 31.3 11.8 93.7 76.9 61.5 47.7 35.6 25.1 16.4 34 35 2653.3 2732.6 2813.2 2895.0 2978.3 3063.0 3149.2 3237.0 3326.6 3417.9 35 36 54.7 34.0 14.5 96.4 79.7 64.4 50.6 38.5 28.1 19.5 36 37 56.0 35.3 15.9 97.8 81.1 65.8 52.1 40.0 29.6 21.0 37 38 57.3 36.6 17.2 99.2 82.5 67.2 53.5 41.5 31.1 22.5 38 39 58.6 38.0 18.6 2900.5 83.9 68.7 55.0 42.9 32.6 24.1 39 40 2659.9 2739.3 2820.0 2901.9 2985.3 3070.1 3156.4 3244.4 3334.1 3425.6 40 41 61.2 40.6 21.3 03.3 86.7 71.5 57.9 45.9 35.6 27.2 41 42 62.5 42.0 22.7 04.7 88.1 72.9 59.4 47.4 37.1 28.7 42 43 63.9 43.3 24.0 06.1 89.5 74.4 60.8 48.9 38.6 30.2 43 44 65.2 44.6 25.4 07.4 90.9 75.8 62.3 50.3 40.2 31.8 44 45 2666.5 2746.0 2826.7 2908.8 2992.3 3077.2 3163.7 3251.8 3341.7 3433.3 45 46 67.8 47.3 28.1 10.2 93.7 78.7 65.2 53.3 43.2 34.9 46 47 69.1 48.6 29.4 11.6 95.1 80.1 66.6 54.8 44.7 36.4 47 48 70.4 50.0 30.8 13.0 96.5 81.5 68.1 56.3 46.2 38.0 48 49 71.7 51.3 32.2 14.3 97.9 82.9 69.5 57.8 47.7 39.5 49 50 2673.1 2752.7 2833.5 2915.7 2999.3 3084.4 3171.0 3259.3 3349.2 3441.0 50 51 74.4 54.0 34.9 17.1 3000.7 85.8 72.5 60.7 50.8 42.6 51 52 75.7 55.3 36.2 18.5 02.1 87.2 73.9 62.2 52.3 44.1 52 53 77.0 56.7 37.6 19.9 03.5 88.7 75.4 63.7 53.8 45.7 53 54 78.3 58.0 39.0 21.2 04.9 90.1 76.8 65.2 55.3 47.2 54 55 2679.6 2759.3 2840.3 2922.6 3006.3 3091.5 3178.3 3266.7 3356.8 3448.8 55 56 81.0 60.7 41.7 24.0 07.7 93.0 79.7 68.2 58.3 50.3 56 57 82.3 62.0 43.0 25.4 09.2 94.4 81.2 69.7 59.9 51.9 57 58 83.6 63.4 44.4 26.8 10.6 95.8 82.7 71.1 61.4 53.4 58 59 84.9 64.7 45.8 28.2 12.0 97.3 84.1 72.6 62.9 55.0 59 60 2686.2 2766.0 2847.1 2929.5 3013.4 3098.7 3185.6 3274.1 3364.4 3456.5 60 ' 40 41 42 43 44 45 46 47 48 49 / 246 Table 5. Meridional Parts / 50 51 52 53 54 55 56 57 58 59 / 3456.5 3550.6 3646.7 3745.1 3845.7 3948.8 4054.5 4163.0 4274.4 4389.1 1 58.1 52.2 48.4 46.7 47.4 50.5 56.3 64.8 76.3 91.0 1 2 59.6 53.8 50.0 48.4 49.1 52.3 58.1 66.6 78.2 92.9 2 3 61.2 55.4 51.6 50.0 50.8 54.0 59.8 68.5 80.1 94.9 3 4 62.7 56.9 53.2 51.7 52.5 55.7 61.6 70.3 82.0 96.8 4 5 464.3 558.5 3654.8 3753.4 3854.2 3957.5 4063.4 4172.1 4283.9 4398.8 5 6 65.9 60.1 56.5 55.0 55.9 59.2 65.2 74.0 85.7 4400.7 6 ' 7 67.4 61.7 58.1 56.7 57.6 61.0 67.0 75.8 87.6 02.6 7 8 69.0 63.3 59.7 58.3 59.3 62.7 68.8 77.7 89.5 04.6 8 9 70.5 64.9 61.3 60.0 61.0 64.5 70.6 79.5 91.4 06.5 9 10 3472.1 566.5 3663.0 3761.7 3862.7 3966.2 4072.4 4181.3 4293.3 4408.5 10 11 73.6 68.1 64.6 63.3 64.4 68.0 74.2 83.2 95.2 10.4 11 12 75.2 69.7 66.2 65.0 66.1 69.7 76.0 85.0 97.1 12.4 12 13 76.7 71.3 67.9 66.7 67.8 71.5 77.7 86.9 99.0 14.3 13 14 78.3 72.8 69.5 68.3 69.5 73.2 79.5 88.7 4300.9 16.3 14 15 3479.9 3574.4 3671.1 3770.0 3871.2 3975.0 4081.3 4190.6 4302.8 4418.2 15 16 81.4 76.0 72.7 71.7 72.9 76.7 83.1 92.4 04.7 20.2 16 17 83.0 77.6 74.4 73.3 74.6 78.5 84.9 94.2 06.6 22.1 17 18 84.5 79.2 76.0 75.0 76.3 80.2 86.7 96.1 08.5 24.1 18 19 86.1 80.8 77.6 76.7 78.1 82.0 88.5 97.9 10.4 26.1 19 20 3487.7 3582.4 3679.3 3778.3 3879.8 3983.7 4090.3 4199.8 4312.3 4428.0 20 21 89.2 84.0 80.9 80.0 81.5 85.5 92.1 4201.6 14.2 30.0 21 22 90.8 85.6 82.5 81.7 83.2 87.2 93.9 03.5 16.1 31.9 22 23 92.4 87.2 84.2 83.3 84.9 89.0 95.7 05.3 18.0 33.9 23 24 93.9 88.8 85.8 85.0 86.6 90.7 97.5 07.2 19.9 35.8 24 25 3495.5 3590.4 3687.4 3786.7 3888.3 3992.5 4099.3 4209.0 4321.8 4437.8 25 26 97.1 92.0 89.1 88.4 90.0 94.3 4101.1 10.9 23.7 39.8 26 27 98.6 93.6 90.7 90.0 91.8 96.0 02.9 12.8 25.6 41.7 27 28 3500.2 95.2 92.3 91.7 93.5 97.8 04.8 14.6 27.5 43.7 28 29 01.8 96.8 94.0 93.4 95.2 99.5 06.6 16.5 29.4 45.7 29 30 3503.3 3598.4 3695.6 3795.1 3896.9 4001.3 4108.4 4218.3 4331.3 4447.6 30 31 04.9 3600.0 97.3 96.8 98.6 03.1 10.2 20.2 33.2 49.6 31 32 06.5 01.6 98.9 98.4 3900.4 04.8 12.0 22.0 35.2 51.6 32 33 08.0 03.2 3700.5 3800.1 02.1 06.6 13.8 23.9 37.1 53.5 33 34 09.6 04.8 02.2 01.8 03.8 08.3 15.6 25.8 39.0 55.5 34 35 3511.2 3606.4 3703.8 3803.5 3905.5 4010.1 4117.4 4227.6 4340.9 4457.5 35 36 12.7 08.0 05.5 05.1 07.2 11.9 19.2 29.5 42.8 59.4 36 37 14.3 09.6 07.1 06.8 09.0 13.6 21.0 31.3 44.7 61.4 37 38 15.9 11.2 08.7 08.5 10.7 15.4 22.9 33.2 46.6 63.4 38 39 17.5 12.8 10.4 10.2 12.4 17.2 24.7 35.1 48.6 65.4 39 40 3519.0 3614.5 3712.0 3811.9 3914.1 4018.9 4126.5 4236.9 4350.5 4467. 40 41 20.6 16.1 13.7 13.6 15.9 20.7 28.3 38.8 52.4 69. 41 42 22.2 17.7 15.3 15.2 17.6 22.5 30.1 40.7 54.3 71. 42 43 23.7 19.3 17.0 17.0 19.3 24.3 31.9 42.5 56.2 73. 43 44 25.3 20.9 18.6 18.6 21.0 26.0 33.8 44.4 58.2 75. 44 45 3526.9 3622.5 3720.3 3820.3 3922.8 4027.8 4135.6 4246.3 4360.1 4477. 45 46 28.5 24.1 21.9 22.0 24.5 29.6 37.4 48.1 62.0 79. 46 47 30.1 25.7 23.6 23.7 26.2 31.4 39.2 50.0 63.9 81. 47 48 31.6 27.3 25.2 25.4 28.0 33.1 41.0 51.9 65.9 83. 48 49 33.2 29.0 26.9 27.1 29.7 34.9 42.9 53.8 67.8 85. 49 50 3534.8 3630.6 3728.5 3828.7 3931.4 4036.7 4144.7 4255.6 4369.7 4487. 50 51 36.4 32.2 30.2 30.4 33.2 38.5 46.5 57.5 71.7 89. 51 52 37.9 33.8 31.8 32.1 34.9 40.2 48.3 59.4 73.6 91. 52 53 39.5 35.4 33.5 33.8 36.6 42.0 50.2 61.3 75.5 93. 53 54 41.1 37.0 35.1 35.5 38.4 43.8 52.0 63.1 77.4 95. 54 55 3542.7 3638.6 3736.8 3837.2 3940.1 4045.6 4153.8 4265.0 4379.4 4497. 55 56 44.3 40.3 38.4 38.9 41.8 47.4 55.7 66.9 81.3 99. 56 57 45.9 41.9 40.1 40.6 43.6 49.1 57.5 68.8 83.2 4501.1 57 58 47.4 43.5 41.7 42.3 45.3 50.9 59.3 70.7 85.2 03.1 58 59 49.0 45.1 43.4 45.0 47.0 52.7 61.1 72.5 87.1 05.1 59 60 3550.6 3646.7 3745.1 3845.7 3948.8 4054.5 4163.0 4274.4 4389.1 4507.1 60 ' 50 51 52 53 54 55 56 57 58 59 / Table 6 Table 7 247 Combined Correction for Observed Sextant Altitudes Correction for Dip of Sea Horizon (Sun or Star) OBSERVED ALTITUDE CORRECTION For Sun (to be added to observed alti- tude) For Star (to be subtracted from observed altitude) 5 6' 14" 9' 55" 6 7 41 8 28 7 8 45 7 24 8 9 35 6 34 9 10 16 5 53 10 10 50 5 19 11 11 17 4 51 12 11 41 4 27 13 12 2 4 7 14 12 19 3 49 15 12 34 3 34 20 13 29 2 39 25 14 3 2 5 30 14 26 1 41 35 14 44 1 23 40 14 57 1 10 45 15 8 58 50 15 17 49 55 15 25 40 60 15 31 34 65 15 37 27 70 15 42 21 75 15 47 16 80 15 52 10 85 15 55 5 HEIGHT op OBSERVER'S EYE ABOVE SEA LEVEL (feet) DIP CORREC- TION (to be subtracted from observed altitude) 4 1' 58" 6 2 24 8 2 46 10 3 06 12 3 24 14 3 40 16 3 55 18 4 9 20 4 23 22 4 36 24 4 48 26 5 28 5 11 30 5 22 35 5 48 40 6 12 45 6 36 50 6 56 55 7 16 60 7 35 70 8 12 85 9 2 100 9 48 Small supplementary correction, for Sun only. Jan. to March \ , , in ,, and Oct. toDec./ addl April to Sept., subtract 10". The dip correction is not required when the artificial horizon is used. 248 Table 8 To Change Hours and Minutes into Decimals of a Day HOURS EXPRESSED AS DECIMAL PARTS OF A DAY HOURS DECIMAL 1 .0416 2 .0833 3 .1250 4 .1666 5 .2083 6 .2500 7 .2916 8 .3333 9 .3750 10 .4166 11 .4583 12 .5000 13 .5416 14 .5833 15 .6249 16 .6666 17 .7083 18 .7500 19 .7916 20 .8333 21 .8749 22 .9166 23 .9583 24 1.0000 MINUTES EXPRESSED AS DECIMAL PARTS OF A DAY MINUTES DECIMAL MINUTES DECIMAL 1 .0006 31 .0215 2 .0013 32 .0222 3 .0020 33 .0229 4 .0027 34 .0236 5 .0034 35 .0243 6 .0041 36 .0250 7 .0048 37 .0256 8 .0055 38 .0263 9 .0062 39 .0270 10 .0069 40 .0277 11 .0076 41 .0284 12 .0083 42 .0291 13 .0090 43 .0298 14 .0097 44 .0305 15 .0104 45 .0312 16 .0111 46 .0319 17 .0118 47 .0326 18 .0125 48 .0333 19 .0131 49 .0340 20 .0138 50 .0347 21 .0145 51 .0354 22 .0152 52 .0361 23 .0159 53 .0368 24 .0166 54 .0375 25 .0173 55 .0381 26 .0180 56 .0388 27 .0187 57 .0395 28 .0194 58 .0402 29 .0201 59 .0409 30 .0208 60 .0416 Table 9 249 To Interchange Degrees and Minutes of Longitude and Hours, Minutes, and Seconds of Time. Part 1 1* 2* 3* 4h 6* 6* 7* 8* 9* 10* 11* Qm 15 30 45 60 75 90 105 120 135 150 165 4 1 16 31 46 61 76 91 106 121 136 151 166 8 2 17 32 47 62 77 92 107 122 137 152 167 12 3 18 33 48 63 78 93 108 123 138 153 168 16 4 19 34 49 64 79 94 109 124 139 154 169 20 5 20 35 50 65 80 95 110 125 140 155 170 24 6 21 36 51 66 81 96 111 126 141 156 171 28 7 22 37 52 67 82 97 112 127 142 157 172 32 8 23 38 53 68 83 98 113 128 143 158 173 36 9 24 39 54 69 84 99 114 129 144 159 174 40 10 25 40 55 70 85 100 115 130 145 160 175 44 11 26 41 56 71 86 101 116 131 146 161 176 48 12 27 42 57 72 87 102 117 132 147 162 177 52 13 28 43 58 73 88 103 118 133 148 163 178 :><; 14 29 44 59 74 89 104 119 134 149 164 179 12* 13* I4h 15* 16* 17* 18* 19* 20* 21* 22* 23* O m 180 195 210 225 240 255 270 285 300 315 330 345 4 181 196 211 226 241 256 271 286 301 316 331 346 8 182 197 212 227 242 257 272 287 302 317 332 347 12 183 198 213 228 243 258 273 288 303 318 333 348 16 184 199 214 229 244 259 274 289 304 319 334 349 20 185 200 215 230 245 260 275 290 305 320 335 350 24 186 201 216 231 246 261 276 291 306 321 336 351 28 187 202 217 232 247 262 277 292 307 322 337 352 32 188 203 218 233 248 263 278 293 308 323 338 353 36 189 204 219 234 249 264 279 294 309 324 339 354 40 190 205 220 235 250 265 280 295 310 325 340 355 44 191 206 221 236 251 266 281 296 311 326 341 356 48 192 207 222 237 252 267 282 297 312 327 342 357 52 193 208 223 238 253 268 283 298 313 328 343 358 56 194 209 224 239 254 269 284 299 314 329 344 359 Part 2 EXPLANATION OF TABLE 9 1. To change degrees of longitude into hours and minutes of time: Find the number of degrees in Part 1. The required hours will then be found at the head of the column containing the degrees, and the required min- utes at the left-hand end of the line containing the degrees. Examples: 113 = 7* 32 m ; 294 = 19*36. 2. To change minutes of longitude into minutes and seconds of time : Find the minutes of longitude in Part 2. The required minutes and seconds of time will again be found at the head of the column and the left-hand end of the line. Examples : 43' = 2 m 52' 28' = l m 52*. 3. 1 and 2 can be combined by addition. Examples : 113 43' = lh 34 m 52*. 294 28' = 19h 37m 52 *. 4. To change hours and minutes of time into degrees and minutes of longitude : Find the number of hours at the head of one of the columns of Part 1 ; then run down the column until you reach a line having at its left-hand end a number of minutes equal to (or just smaller than) the given number of minutes of time. Where that line and column meet you will find the required degrees of longitude. Examples: lh 32 m = 113; 19* 36 m =294. 5. To change minutes and seconds of time into minutes of longitude : Find the number of minutes of time at the head of one of the columns of Part 2 ; then run down the column until you reach a line having at its left-hand end a number of seconds equal (or nearly equal) to the given number of seconds of time. Where that line and column meet you will find the minutes of longitude. Examples : 2 m 52* = 43' ; l m 52' = 28'. 6. 4 and 5 can be combined by addition : Examples: 734 m 52' = 113 43'; 19/'37 m 52 =294 28'. Q,n l m 2 m 3 m 0' 15' 30' 45' 4 1 16 31 46 8 2 17 32 47 12 3 18 33 48 16 4 19 34 49 20 5 20 35 50 24 6 21 36 51 28 7 22 37 52 32 8 23 38 53 36 9 24 39 54 40 10 25 40 55 44 11 26 41 56 48 12 27 42 57 52 13 28 43 58 56 14 29 44 59 250 Table 10. Haversine Table s ' 0* O m O h 4 m. 10 Oh s m 2 O h jgm 30 Hav. No. Hav. No. Hav. No. Hay. No. 0.00000 5.88168 0.00008 6.48371 0.00030 6.83584 0.00069 4 1 2.32539 .00000 .89604 .00008 .49092 .00031 .84065 .00069 8 2 .92745 .00000 .91016 .00008 .49807 .00031 .84543 .00070 12 3 3.27963 .00000 .92406 .00008 .50516 .00032 .85019 .00071 16 4 .52951 .00000 .93774 .00009 .51219 .00033 .85492 .00072 20 5 3.72333 0.00000 5.95121 0.00009 6.51916 0.00033 6.85963 0.00072 24 6 .88169 .00000 .96447 .00009 .52608 .00034 .86431 .00073 28 7 4.01559 .00000 .97753 .00010 .53295 .00034 .86897 .00074 32 8 .13157 .00000 .99040 .00010 .53976 .00035 .87360 .00075 36 9 .23388 .00000 6.00308 .00010 .54652 .00035 .87821 .00076 40 10 4.32539 0.00000 6.01557 0.00010 6.55323 0.00036 6.88279 0.00076 44 11 .40818 .00000 .02789 .00011 .55988 .00036 .88735 .00077 48 12 .48375 .00000 .04004 .00011 .56649 .00037 .89188 .00078 52 13 .55328 .00000 .05202 .00011 .57304 .00037 .89639 .00079 56 14 .61765 .00000 .06384 .00012 .57955 .00038 .90088 .00080 s ' O h l m O h 5 m 1 O h 9 m 2 Qh I 3 m 30 15 4.67757 0.00000 6.07550 0.00012 6.58600 0.00039 6.90535 0.00080 4 16 .73363 .00001 .08700 .00012 .59241 .00039 .90979 .00081 5 17 .78629 .00001 .09836 .00013 .59878 .00040 .91421 .00082 12 18 .83594 .00001 .10956 .00013 .60509 .00040 .91860 .00083 ^S 19 .88290 .00001 .12063 .00013 .61136 .00041 .92298 .00084 20 20 4.92745 0.00001 6.13155 0.00014 6.61759 0.00041 6.92733 0.00085 24 21 .96983 .00001 .14234 .00014 .62377 .00042 .93166 .00085 25 22 5.01024 .00001 .15300 .00014 .62991 .00043 .93597 .00086 32 23 .04885 .00001 .16353 .00015 .63600 .00043 .94026 .00087 3 24 .08581 .00001 .17393 .00015 .64205 .00044 .94453 .00088 4(9 25 5.12127 0.00001 6.18421 0.00015 6.64806 0.00044 6.94877 0.00089 44 26 .15534 .00001 .19437 .00016 .65403 .00045 .95300 .00090 45 27 .18812 .00002 .20441 .00016 .65996 .00046 .95720 .00091 52 28 .21971 .00002 .21433 .00016 .66585 .00046 .96139 .00091 5 29 .25019 .00002 .22415 .00017 .67170 .00047 .96555 .00092 s ' Qhgm QO Qh Qm jo QK io m 2 Qh Ijrm 30 30 5.27963 0.00002 6.23385 0.00017 6.67751 0.00048 6.96970 0.00093 4 31 .30811 .00002 .24345 .00018 .68328 .00048 .97382 .00094 8 32 .33569 .00002 .25294 .00018 .68901 .00049 .97793 .00095 ^2 33 .36242 .00002 .26233 .00018 .69470 .00050 .98201 .00096 .70 34 .38835 .00002 .27162 .00019 .70036 .00050 .98608 .00097 20 35 5.41352 0.00003 6.28081 0.00019 6.70598 0.00051 6.99013 0.00098 24 36 .43799 .00003 .28991 .00019 .71157 .00051 .99416 .00099 28 37 .46179 .00003 .29891 .00020 .71712 .00052 .99817 .00100 32 38 .48496 .00003 .30781 .00020 .72263 .00053 7.00216 .00101 35 39 .50752 .00003 .31663 .00021 .72811 .00053 .00613 .00101 40 40 5.52951 0.00003 6.32536 0.00021 6.73355 0.00054 7.01009 0.00102 44 41 .55095 .00004 .33400 .00022 .73896 .00055 .01403 .00103 48 42 .57189 .00004 .34256 .00022 .74434 .00056 .01795 .00104 52 43 .59232 .00004 .35103 .00022 .74969 .00056 .02185 .00105 5 44 .61229 .00004 .35943 .00023 .75500 .00057 .02573 .00106 s Qh 3 m QO Qh 7 m 10 Qh Hm 2 0^ 15 m 3 45 5.63181 0.00004 6.36774 0.00023 6.76028 0.00058 7.02960 0.00107 4 46 .65090 .00004 .37597 .00024 .76552 .00058 .03345 .00108 8 47 .66958 .00005 .38412 .00024 .77074 .00059 .03729 .00109 .72 48 .68787 .00005 .39220 .00025 .77592 .00060 .04110 .00110 16 49 .70578 .00005 .40021 .00025 .78108 .00060 .04490 .00111 20 50 5.72332 0.00005 6.40814 0.00026 6.78620 0.00061 7.04869 0.00112 24 51 .74052 .00006 .41600 .00026 .79129 .00062 .05245 .00113 28 52 .75739 .00006 .42379 .00027 .79630 .00063 .05620 .00114 32 53 .77394 .00006 .43151 .00027 .80139 .00063 .05994 .00115 36 54 .79017 .00006 .43916 .00027 .80640 .00064 .06366 .00116 40 55 5.80611 0.00006 6.44675 0.00028 6.81137 0.00065 7.06736 0.00117 44 56 .82176 .00007 .45427 .00028 .81632 .00066 .07105 .00118 48 57 .83713 .00007 .46172 .00029 .82124 .00066 .07472 .00119 52 58 .85224 .00007 .46911 .00029 .82614 .00067 .07837 .00120 56 59 .86709 .00007 .47644 .00030 .83100 .00068 .08201 .00121 60 5.88168 0.00008 6.48371 0.00030 6.83584 0.00069 7.08564 0.00122 Table 10. Haversine Table 251 s / O h iem 4 Oh 20 m 5 Oh 24 m 6 Qh 28 m 7 Hay. No. Hav. No. Hav. No. Hav. No. 7.08564 0.00122 7.27936 0.00190 7.43760 0.00274 7.57135 0.00373 4 1 .08925 .00123 .28225 .00192 .44001 .00275 .57341 .00374 8 2 .09284 .00124 .28513 .00193 .44241 .00277 .57547 .00376 t* 3 .09642 .00125 .28800 .00194 .44480 .00278 .57752 .00378 16 4 .09999 .00126 .29086 .00195 .44719 .00280 .57957 .00380 20 5 7.10354 0.00127 7.29371 0.00197 7.44957 0.00282 7.58162 0.00382 24 6 .10708 .00128 .29655 .00198 .45194 .00283 .58366 .00383 28 7 .11060 .00129 .29938 .00199 .45431 .00285 .58569 .00385 32 8 .11411 .00130 .30220 .00201 .45667 .00286 .58772 .00387 36 9 .11760 .00131 .30502 .00202 .45903 .00288 .58974 .00389 40 10 7.12108 0.00132 7.30782 0.00203 7.46138 0.00289 7.59176 0.00391 44 11 .12455 .00133 .31062 .00204 .46372 .00291 .59378 .00392 48 12 .12800 .00134 .31340 .00206 .46605 .00292 .59579 .00394 52 13 .13144 .00135 .31618 .00207 .46838 .00294 .59779 .00396 56 14 .13486 .00136 .31895 .00208 .47071 .00296 .59979 .00398 s ' Qh 1? 40 Qh 21 m 5 O h 25 m 6 0*29 m 7 15 7.13827 0.00137 7.32171 0.00210 7.47302 0.00297 7.60179 0.00400 4 16 .14167 .00139 .32446 .00211 .47533 .00299 .60378 .00402 5 17 .14506 .00140 .32720 .00212 .47764 .00300 .60577 .00403 12 18 .14843 .00141 .32994 .00214 .47994 .00302 .60775 .00405 ^5 19 .15179 .00142 .33266 .00215 .48223 .00304 .60973 .00407 20 20 7.15513 0.00143 7.33538 0.00216 7.48452 0.00305 7.61170 0.00409 24 21 .15846 .00144 .33809 .00218 .48680 .00307 .61367 .00411 25 22 .16178 .00145 .34079 .00219 .48907 .00308 .61564 .00413 32 23 .16509 .00146 .34348 .00221 .49134 .00310 .61760 .00415 35 24 .16839 .00147 .34616 .00222 .49360 .00312 .61955 .00416 40 25 7.17167 0.00148 7.34884 0.00223 7.49586 0.00313 7.62151 0.00418 44 26 .17494 .00150 .35150 .00225 .49811 .00315 .62345 .00420 4ti) OO t->^5 Ad .64932 .04460 .68368 .04827 .71667 .05208 .74839 .05603 55 24 .64990 .04466 .68424 .04833 .71721 .05214 .74890 .05609 40 25 8.65049 0.04472 8.68480 0.04839 8.71774 0.05221 8.74942 0.05616 44 26 .65107 .04478 .68536 .04846 .71828 .05227 .74994 .05623 48 27 .65165 .04484 .68592 .04852 .71882 .05234 .75046 .05629 52 28 .65224 .04490 .68648 .04858 .71936 .05240 .75097 .05636 55 29 .65282 .04496 .68704 .04864 .71989 .05247 .75149 .05643 s lh ss m 24 lh 42 25 lh 46 m 26 lh 5Q m 27 30 8.65340 0.04502 8.68760 0.04871 8.72043 0.05253 8.75201 0.05649 4 31 .65398 .04508 .68815 .04877 .72097 .05260 .75252 .05656 8 32 .65456 .04514 .68871 .04883 .72150 .05266 .75304 .05663 ^2 33 .65514 .04520 .68927 .04890 .72204 .05273 .75355 .05670 16 34 .65572 .04526 .68983 .04896 .72257 .05279 .75407 .05676 20 35 8.65630 0.04532 8.69038 0.04902 8.72311 0.05286 8.75458 0.05683 24 36 .65688 .04538 .69094 .04908 .72364 .05292 .75510 .05690 28 37 .65746 .04544 .69149 .04915 .72418 .05299 .75561 .05697 32 38 .65804 .04550 .69205 .04921 .72471 .05305 .75613 .05703 36 39 .65862 .04556 .69260 .04927 .72525 .05312 .75664 .05710 40 40 8.65920 0.04562 8.69316 0.04934 8.72578 0.05318 8.75715 0.05717 44 41 .65978 .04569 .69371 .04940 .72631 .05325 .75767 .05724 48 42 .66035 .04575 .69427 .04946 .72684 .05331 .75818 .05730 52 43 .66093 .04581 .69482 .04952 .72738 .05338 .75869 .05737 56 44 .66151 .04587 .69537 .04959 .72791 .05345 .75920 .05744 s ' lh 39 m 24 lh 43 25 lh ^m 26 lh 51 m 27 45 8.66208 0.04593 8.69593 0.04965 8.72844 0.05351 8.75972 O.C5751 4 46 .66266 .04599 .69648 .04971 .72897 .05358 .76023 .05757 5 47 .66323 .04605 .69703 .04978 .72950 .05364 .76074 .05764 12 48 .66381 .04611 .69758 .04984 .73003 .05371 .76125 .05771 iff 49 .66438 .04617 .69814 .04990 .73056 .05377 .76176 .05778 20 50 8.66496 0.04623 8.69869 0.04997 8.73109 0.05384 8.76227 O.C5785 24 51 .66553 .04629 .69924 .05003 .73162 .05390 .76278 .05791 28 52 .66610 .04636 .69979 .05009 .73215 .05397 .76329 .05798 32 53 .66668 .04642 .70034 .05016 .73268 .05404 .76380 .05805 3S 54 .66725 .04648 .70089 .05022 .73321 .05410 .76431 .05812 40 55 8.66782 0.04654 8.70144 0.05028 8.73374 0.05417 8.76481 0.05819 44 56 .66839 .04660 .70198 .05035 .73426 .05423 .76532 .05825 48 57 .66896 .04666 .70253 .05041 .73479 .05430 .76583 .05832 52 58 .66953 .04672 .70308 .05048 .73532 .05436 .76634 .05839 56 59 .67010 .04678 .70363 .05054 .73584 .05443 .76684 .05846 (90 60 8.67067 0.04685 8.70418 005060 8.73637 0.05450 8.76735 0.05853 Table 10. Haversine Table 257 s ' 1* 52 28 lh S6 m 29 2 h O m 30 2h jrn. 31 Hay. No. Hav. No. Hav. No. Hav. No. 8.76735 0.05853 8.79720 0.06269 8.82599 0.06699 8.85380 0.07142 4 1 .76786 .05859 .79769 .06276 .82646 .06706 .85425 .07149 8 2 .76836 .05866 .79818 .06283 .82694 .06713 .85471 .07157 12 3 .76887 .05873 .79866 .06290 .82741 .06721 .85516 .07164 16 4 .76938 .05880 .79915 .06297 .82788 .06728 .85562 .07172 20 5 8.76988 0.05887 8.79964 0.06304 8.82835 0.06735 8.85607 0.07179 24 6 .77039 .05894 .80013 .06311 .82882 .06742 .85653 .07187 28 7 .77089 .05901 .80061 .06318 .82929 .06750 .85698 .07194 32 8 .77139 .05907 .80110 .06326 .82976 .06757 .85743 .07202 36 9 .77190 .05914 .80158 .06333 .83023 .06764 .85789 .07209 40 10 8.77240 0.05921 8.80207 0.06340 8.83069 0.06772 8.85834 0.07217 44 11 .77291 .05928 .80256 .06347 .83116 .06779 .85879 .07224 48 12 .77341 .05935 .80304 .06354 .83163 .06786 .85925 .07232 52 13 .77391 .05942 .80353 .06361 .83210 .06794 .85970 .07239 56 14 .77441 .05949 .80401 .06368 .83257 .06801 .86015 .07247 s ' 1* 53 m 28 lh 57 29 2h I'* 30 9h Qm 31 15 8.77492 0.05955 8.80449 0.06375 8.83303 0.06808 8.86060 0.07254 4 16 .77542 .05962 .80498 .06382 .83350 .06816 .86105 .07262 5 17 .77592 .05969 .80546 .06389 .83397 .06823 .86151 .07270 12 18 .77642 .05976 .80595 .06397 .83444 .06830 .86196 .07277 10 19 .77692 .05983 .80643 .06404 .83490 .06838 .86241 .07285 20 20 8,77742 0.05990 8.80691 0.06411 8.83537 0.06845 8.86286 0.07292 24 21 .77792 .05997 .80739 .06418 .83583 .06852 .86331 .07300 2S 22 .77842 .06004 .80788 .06425 .83630 .06860 .86376 .07307 32 23 .77892 .06011 .80836 .06432 .83676 .06867 .86421 .07315 30 24 .77942 .06018 .80884 .06439 .83723 .06874 .86466 .07322 40 25 8.77992 0.06024 8.80932 0.06446 8.83769 0.06882 8.86511 0.07330 44 26 .78042 .06031 .80980 .06454 .83816 .06889 .86556 .07338 45 27 .78092 .06038 .81028 .06461 .83862 .06896 .86600 .07345 2 28 .78142 .06045 .81076 .06468 .83909 .06904 .86645 .07353 50 29 .78191 .06052 .81124 .06475 .83955 .06911 .86690 .07360 s ' 1* 54 m 28 lh ggm 29 2 h 2" 1 30 gh Qm 31 30 8.78241 0.06059 8.81172 0.06482 8.84002 0.06919 8.86735 0.07368 4 31 .78291 .06066 .81220 .06489 .84048 .06926 .86780 .07376 8 32 .78341 .06073 .81268 .06497 .84094 .06933 .86825 .07383 12 33 .78390 .06080 .81316 .06504 .84140 .06941 .86869 .07391 10 34 .78440 .06087 .81364 .06511 .84187 .06948 .86914 .07398 20 35 8.78490 0.06094 8.81412 0.06518 8.84233 0.06956 8.86959 0.07406 24 36 .78539 .06101 .81460 .06525 .84279 .06963 .87003 .07414 28 37 .78589 .06108 .81508 .06532 .84325 .06970 .87048 .07421 32 38 .78638 .06115 .81555 .06540 .84371 .06978 .87093 .07429 S 39 .78688 .06122 .81603 .06547 .84417 .06985 .87137 .07437 40 40 8.78737 0.06129 8.81651 0.06554 8.84464 0.06993 8.87182 0.07444 44 41 .78787 .06136 .81699 .06561 .84510 .07000 .87226 .07452 48 42 .78836 .06143 .81746 .06568 .84556 .07007 .87271 .07459 52 43 .78885 .06150 .81794 .06576 .84602 .07015 .87315 .07467 50 44 .78935 .06157 .81841 .06583 .S404S 07022 .87360 .07475 s ' lh 55 m 28 lh 5gm 29 2 h 3 m 30 2 h 7m 31 45 8.78984 0.06164 8.81889 0.06590 8.84694 0.07030 8.87404 0.07482 4 46 .79033 .06171 .81937 .06597 .84740 .07037 .87448 .07490 5 47 .79082 .06178 .81984 .06605 .84785 .07045 .87493 .07498 12 48 .79132 .06185 .82032 .06612 .84831 .07052 .87537 .07505 10 49 .79181 .06192 .82079 .06619 .84877 .07059 .87582 .07513 20 50 8.79230 0.06199 8.82126 0.06626 8.84923 0.07067 8.87626 0.07521 24 51 .79279 .06206 .82174 .06633 .84969 .07074 .87670 .07528 28 52 .79328 .06213 .82221 .06641 .85015 .07082 .87714 .07536 32 53 .79377 .06220 .82269 .06648 .85060 .07089 .87759 .07544 30 54 .79426 .06227 .82316 .06655 .851t)6 .07097 .87803 .07551 40 55 8.79475 0.06234 8.82363 0.06662 8.85152 0.07104 8.87847 0.07559 44 56 .79524 .06241 .82410 .06670 .85197 .07112 .87891 .07567 48 57 .79573 .06248 .82458 .06677 .85243 .07119 .87935 .07574 52 58 .79622 .06255 .82505 .066S4 .85289 .07127 .87980 .07582 50 59 .79671 .06262 .82552 .06691 .85334 .07134 .88024 .07590 00 60 8.79720 0.06269 8.82599 0.06699 8.85380 0.07142 8.88068 0.07598 258 Table 10. Haversine Table s ' 2 h 8 32 2 h 12 m 33 2 h 16 34 2^ 20 m 35 Hav. No. Hav. No. Hav. No. Hav. No. 8.88068 0.07598 8.90668 0.08066 8.93187 0.08548 8.95628 0.09042 4 1 .88112 .07605 .90711 .08074 .93228 .08556 .95668 .09051 8 2 .88156 .07613 .90754 .08082 .93270 .08564 .95709 .09059 12 3 .88200 .07621 .90796 .08090 .93311 .08573 .95749 .09067 16 4 .88244 .07628 .90839 .08098 .93352 .08581 .95789 .09076 20 5 8.88288 0.07636 8.90881 0.08106 8.93393 0.08589 8.95828 0.09084 24 6 .88332 .07644 .90924 .08114 .93435 .08597 .95868 .09093 28 7 .88375 .07652 .90966 .08122 .93476 .08605 .95908 .09101 32 8 .88419 .07659 .91009 .08130 .93517 .08613 .95948 .09109 36 9 .88463 .07667 .91051 .08138 .93558 .08621 .95988 .09118 40 10 8.88507 0.07675 8.91094 0.08146 8.93599 0.08630 8.96028 0.09126 44 11 .88551 .07683 .91136 .08154 .93640 .08638 .96068 .09134 48 12 .88595 .07690 .9117& .08162 .93681 .08646 .96108 .09143 52 13 .88638 .07698 .91221 .08170 .93722 .08654 .96148 .09151 56 14 .88682 .07706 .91263 .08178 .93764 .08662 .96187 .09160 s ' 2 h 9 m 32 2 h IS" 1 33 2 h 17 m 34 2 h 21 m 35 15 8.88726 0.07714 8.91306 0.08186 8.93805 0.08671 8.96227 0.09168 4 16 .88769 .07721 .91348 .08194 .93846 .08679 .96267 .09176 5 17 .88813 .07729 .91390 .08202 .93886 .08687 .96307 .09185 12 18 .88857 .07737 .91432 .08210 .93927 .08695 .96346 .09193 iff 19 .88900 .07745 .91475 .08218 .93968 .08703 .96386 .09202 20 20 8.88944 0.07752 8.91517 0.08226 8.94009 0.08711 8.96426 0.09210 24 21 .88988 .07760 .91559 .08234 .94050 .08720 .96465 .09218 25 22 .89031 .07768 .91601 .08242 .94091 .08728 .96505 .09227 32 23 .89075 .07776 .91643 .08250 .94132 .08736 .96545 .09235 30 24 .89118 .07784 .91685 .08258 .94173 .08744 .96584 .09244 40 25 8.89162 0.07791 8.91728 0.08266 8.94213 0.08753 8.96624 0.09252 44 26 .89205 .07799 .91770 .08274 .94254 .08761 .96663 .09260 48 27 .89248 .07807 .91812 .08282 .94295 .08769 .96703 .09269 52 28 .89292 .07815 .91854 .08290 .94336 .08777 .96742 .09277 50 29 .89335 .07823 .91896 .08298 .94376 .08785 .96782 .09286 s ' %h iQm 32 2 h 14 m 33 & 18 m 34 2* 22 m 35 30 8.89379 0.07830 8.91938 0.08306 8.94417 0.08794 8.96821 0.09294 4 31 .89422 .07838 .91980 .08314 .94458 .08802 .96861 .09303 8 32 .89465 .07846 .92022 .08322 .94498 .08810 .96900 .09311 12 33 .89509 .07854 .92064 .08330 .94539 .08818 .96940 .09320 10 34 .89552 .07862 .92105 .08338 .94580 .08827 .96979 .09328 20 35 8.89595 0.07870 8.92147 0.08346 8.94620 0.08835 8.97018 0.09337 24 36 .89638 .07877 .92189 .08354 .94661 .08843 .97058 .09345 28 37 .89681 .07885 .92231 .08362 .94701 .08851 .97097 .09353 32 38 .89725 .07893 .92273 .08370 .94742 .08860 .97136 .09362 30 39 .89768 .07901 .92315 .08378 .94782 .08868 .97176 .09370 40 40 8.89811 0.07909 8.92356 0.08386 8.94823 0.08876 8.97215 0.09379 44 41 .89854 .07917 .92398 .08394 .94863 .08885 .97254 .09387 48 42 .89897 .07924 .92440 .08402 .94904 .08893 .97294 .09396 52 43 .89940 .07932 .92482 .08410 .94944 .08901 .97333 .09404 50 44 .89983 .07940 .92523 .08418 .94985 .08909 .97372 .09413 s ' 2^ ll m 32 %h IQm 33 2 h 19 m 34 2 h 23 35 45 8.90026 0.07948 8.92565 0.08427 8.95025 0.08918 8.97411 0.09421 4 46 .90069 .07956 .92607 .08435 .95065 .08926 .97450 .09430 5 47 .90112 .07964 .92648 .08443 .95106 .08934 .97489 .09438 12 48 .90155 .07972 .92690 .08451 .95146 .08943 .97529 .09447 16 49 .90198 .07980 .92731 .08459 .95186 .08951 .97568 .09455 20 50 8.90241 0.07987 8.92773 0.08467 8.95227 0.08959 8.97607 0.09464 24 51 .90284 .07995 .92814 .08475 .95267 .08967 .97646 .09472 28 52 .90326 .08003 .92856 .08483 .95307 .08976 .97685 .09481 32 53 .90369 .08011 .92897 .08491 .95347 .08984 .97724 .09489 30 54 .90412 .08019 .92939 .08499 .95388 .08992 .97763 .09498 40 55 8.90455 0.08027 8.92980 0.08508 8.95428 0.09001 8.97802 0.09506 44 56 .90498 .08035 .93022 .08516 .95468 .09009 .97841 .09515 48 57 .90540 .08043 .93063 .08524 .95508 .09017 .97880 .09524 52 58 .90583 .08051 .93104 .08532 .95548 .09026 .97919 .09532 50 59 .90626 .08059 .93146 .08540 .95588 .09034 .97958 .09541 00 60 8.90668 0.08066 8.93187 0.08548 8.95628 0.09042 8.97997 0.09549 Table 10. Haversine Table 259 s ' 2* 24 m 36 2* 28 37 2* 32 38 2* 36 m 39 Hav. No. Hav. No. Hav. No. Hav. No. 8.97997 0.09549 9.00295 0.10068 9.02528 0.10599 9.04699 0.11143 4 1 .98035 .09558 .00333 .10077 .02565 .10608 .04735 .11152 8 2 .98074 .09566 .00371 .10086 .02602 .10617 .04770 .11161 12 3 .98113 .09575 .00408 .10095 .02638 .10626 .04806 .11170 16 4 .98152 .09583 .00446 .10103 .02675 .10635 .04842 .11179 20 5 8.98191 0.09592 9.00484 0.10112 9.02712 0.10644 9.04877 0.11189 24 6 .98229 .09601 .00522 .10121 .02748 .10653 .04913 .11198 28 7 .98268 .09609 .00559 .10130 .02785 .10662 .04948 .11207 32 8 .98307 .09618 .00597 .10138 .02821 .10671 .04984 .11216 36 9 .98346 .09626 .00634 .10147 .02858 .10680 .05019 .11225 40 10 8.98384 0.09635 9.00672 0.10156 9.02894 0.10689 9.05055 0.11234 44 11 .98423 .09643 .00710 .10165 .02931 .10698 .05090 .11244 48 12 .98462 .09652 .00747 .10174 .02967 .10707 .05126 .11253 52 13 .98500 .09661 .00785 .10182 .03004 .10716 .05161 .11262 56 14 .98539 .09669 .00822 .10191 .03040 .10725 .05197 .11271 s ' 36 2 h 2 gm 37 2 h 33 m 38 2 h 37 m 39 15 8.98578 0.09678 9.00860 0.10200 9,03077 0.10734 9.05232 0.11280 4 16 .98616 .09686 .00897 .10209 .03113 .10743 .05268 .11290 5 17 .98655 .09695 .00935 .10218 .03150 .10752 .05303 .11299 12 18 .98693 .09704 .00972 .10226 .03186 .10761 .05339 .11308 .70 19 .98732 .09712 .01009 .10235 .03222 .10770 .05374 .11317 20 20 8.98770 0.09721 9.01047 0.10244 9.03259 0.10779 9.05409 0.11326 24 21 .98809 .09729 .01084 .10253 .03295 .10788 .05445 .11336 25 22 .98847 .09738 .01122 .10262 .03331 .10797 .05480 .11345 32 23 .98886 .09747 .01159 .10270 .03368 .10806 .05515 .11354 36 24 .98924 .09755 .01196 .10279 .03404 .10815 .05551 .11363 40 25 8.98963 0.09764 9.01234 0.10288 9.03440 0.10824 9.05586 0.11373 44 26 .99001 .09773 .01271 .10297 .03476 .10833 .05621 .11382 45 27 .99039 .09781 .01308 .10306 .03513 .10842 .05656 .11391 52 28 .99078 .09790 .01345 .10315 .03549 .10851 .05692 .11400 56 29 .99116 .09799 .01383 .10323 .03585 .10861 .05727 .11410 s ' 36 2h SQ 37 2* 34 m 38 2* 38 39 30 8.99154 0.09807 9.01420 0.10332 9.03621 0.10870 9.05762 0.11419 4 31 .99193 .09816 .01457 .10341 .03657 .10879 .05797 .11428 8 32 .99231 .09824 .01494 .10350 .03694 .10888 .05832 .11437 12 33 .99269 .09833 .01531 .10359 .03730 .10897 .05867 .11447 10 34 .99307 .09842 .01569 .10368 .03766 .10906 .05903 .11456 20 35 8.99346 0.09850 9.01606 0.10377 9.03802 0.10915 9.05938 0.11465 24 36 .99384 .09859 .01643 .10386 .03838 .10924 .05973 .11474 28 37 .99422 .09868 .01680 .10394 .03874 .10933 .06008 .11484 32 38 .99460 .09876 .01717 .10403 .03910 .10942 .06043 .11493 30 39 .99498 .09885 .01754 .10412 .03946 .10951 .06078 .11502 40 40 8.99536 0.09894 9.01791 0.10421 9.03982 0.10960 9.06113 0.11511 44 41 .99575 .09903 .01828 .10430 .04018 .10969 .06148 .11521 48 42 .99613 .09911 .01865 .10439 .04054 .10978 .06183 .11530 52 43 .99651 .09920 .01902 .10448 .04090 .10988 .06218 .11539 50 44 .99689 .09929 .01939 .10457 .04126 .10997 .06253 .11549 s %h 27 m 36 2h si m 37 2 h 35 m 38 2 h 39 39 45 8.99727 0.09937 9.01976 0.10466 9.04162 0.11006 9.06288 0.11558 4 46 .99765 .09946 .02013 .10474 .04198 .11015 .06323 .11567 8 47 .99803 .09955 .02050 .10483 .04234 .11024 .06358 .11577 12 48 .99841 .09963 .02087 .10492 .04270 .11033 .06393 .11586 16 49 .99879 .09972 .02124 .10501 .04306 .11042 .06428 .11595 20 50 8.99917 0.09981 9.02161 0.10510 9.04341 0.11051 9.06462 0.11604 24 51 .99955 .09990 .02197 .10519 .04377 .11060 .06497 .11614 28 52 .99993 .09998 .02234 .10528 .04413 .11070 .06532 .11623 32 53 9.00031 .10007 .02271 .10537 .04449 .11079 .06567 .11632 30 54 .00068 .10016 .02308 .10546 .04485 .11088 .06602 .11642 40 55 9.00106 0.10025 9.02345 0.10555 9.04520 0.11097 9.06637 0.11651 44 56 .00144 .10033 .02381 .10564 .04556 .11106 .06671 .11660 48 57 .00182 .10042 .02418 .10573 .04592 .11115 .06706 .11670 52 58 .00220 .10051 .02455 .10582 .04628 .11124 .06741 .11679 50 59 .00258 .10059 .02492 .10591 .04663 .11134 .06776 .11688 (10 60 9.00295 0.10068 9.02528 0.10599 9.04699 0.11143 9.06810 0.11698 260 Table 10. Haversine Table s ' 2 h 40 m 40 2 h 44 m 41 2 h 48 m 42 & 52 43 Hav. No. Hav. No. Hav. No. Hav. No. 9.06810 0.11698 9.08865 0.12265 9.10866 0.12843 9.12815 0.13432 4 1 .06845 .11707 .08899 .12274 .10899 .12852 .12847 .13442 8 2 .06880 .11716 .08933 .12284 .10932 .12862 .12879 .13452 12 3 .06914 .11726 .08966 .12293 .10965 .12872 .12911 .13462 16 4 .06949 .11735 .09000 .12303 .10997 .12882 .12943 .13472 20 5 9.06984 0.11745 9.09034 0.12312 9.11030 0.12891 9.12975 0.13482 24 6 .07018 .11754 .09068 .12322 .11063 .12901 .13007 .13492 28 7 .07053 .11763 .09101 .12331 .11096 .12911 .13039 .13502 32 8 .07088 .11773 .09135 .12341 .11129 .12921 .13071 .13512 36 9 .07122 .11782 .09169 .12351 .11161 .12930 .13103 .13522 40 10 3.07157 0.11791 9.09202 0.12360 9.11194 0.12940 9.13135 0.13532 44 11 .07191 .11801 .09236 .12370 .11227 .12950 .13167 .13542 48 12 .07226 .11810 .09269 .12379 .11260 .12960 .13199 .13552 52 13 .07260 .11820 .09303 .12389 .11292 .12970 .13231 .13562 56 14 .07295 .11829 .09337 .12398 .11325 .12979 .13263 .13571 s ' 2^ 41 40 %h 4.5 41 o 2 h 49 m 42 2 h 53 43 15 9.07329 0.11838 9.09370 0.12408 9.11358 0.12989 9.13295 0.13581 4 16 .07364 .11848 .09404 .12418 .11391 .12999 .13326 .13591 5 17 .07398 .11857 .09437 .12427 .11423 .13009 .13358 .13601 12 18 .07433 .11867 .09471 .12437 .11456 .13018 .13390 .13611 ^6 19 .07467 .11876 .09504 .12446 .11489 .13028 .13422 .13621 20 20 9.07501 0.11885 9.09538 0.12456 9.11521 0.13038 9.13454 0.13631 24 21 .07536 .11895 .09571 .12466 .11554 .13048 .13486 .13641 2S 22 .07570 .11904 .09605 .12475 .11586 .13058 .13517 .13651 32 23 .07605 .11914 .09638 .12485 .11619 .13067 .13549 .13661 36 24 .07639 .11923 .09672 .12494 .11652 .13077 .13581 .13671 40 25 9.07673 0.11933 9.09705 0.12504 9.11684 0.13087 9.13613 0.13681 44 26 .07708 .11942 .09739 .12514 .11717 .13097 .13644 .13691 4n 41 2 h 51 m 42 & 55 43 45 9.08357 0.12122 9.10371 0.12697 9.12332 0.13284 9.14245 0.13882 4 46 .08391 .12131 .10404 .12707 .12365 .13294 .14276 .13892 5 47 .08425 .12141 .10437 .12717 ,12397 .13304 .14307 .13902 12 48 .08459 .12150 .10470 .12726 .12429 .13314 .14339 .13912 16 49 .08492 .12160 .10503 .12736 .12461 .13323 .14370 .13922 20 50 9.08526 0.12169 9.10536 0.12746 9.12494 0.13333 9.14402 0.13932 24 51 .08560 .12179 .10569 .12755 .12526 .13343 .14433 .13942 2S 52 .08594 .12188 .10602 .12765 .12558 .13353 .14465 .13952 32 - 53 .08628 .12198 .10635 .12775 .12590 .13363 .14496 .13962 36 54 .08662 .12207 .10668 .12784 .12622 .13373 .14527 .13972 40 55 9.08696 0.12217 9.10701 0.12794 9.12655 0.13383 9.14559 0.13983 44 56 .08730 .12226 .10734 .12804 .12687 .13393 .14590 .13993 48 57 .08764 .12236 .10767 .12814 .12719 .13403 .14621 .14003 52 58 .08797 .12245 .10800 .12823 .12751 .13412 .14653 .14013 -50 59 .08831 .12255 .10833 .12833 .12783 .13422 .14684 .14023 60 60 9.08865 0.12265 9.10866 0.12843 9.12815 0.13432 9.14715 0.14033 Table 10. Haversine Table 261 s ' 2* 56 m 44 3 h O m 45 3 h 4 m 46 3 h 8 m 47 Hav. No. Hav. No. Hav. No. Hav. No. 9.14715 0.14033 9.16568 0.14645 9.18376 0.15267 9.20140 0.15900 4 .14746 .14043 .16598 .14655 .18405 .15278 .20169 .15911 8 .14778 .14053 .16629 .14665 .18435 .15288 .20198 .15921 12 .14809 .14063 .16659 .14676 .18465 .15298 .20227 .15932 16 .14840 .14073 .16690 .14686 .18495 .15309 .20256 .15943 20 9.14871 0.14084 9.16720 0.14696 9.18524 0.15319 9.20285 0.15953 24 .14902 .14094 .16751 .14706 .18554 .15330 .20314 .15964 28 7 .14934 .14104 .16781 .14717 .18584 .15340 .20343 .15975 32 8 .14965 .14114 .16812 .14727 .18613 .15351 .20372 .15985 36 9 .14996 .14124 .16842 .14737 .18643 .15361 .20401 .15996 40 10 9.15027 0.14134 9.16872 0.14748 9.18673 0.15372 9.20430 0.16007 44 11 .15058 .14144 .16903 .14758 .18702 .15382 .20459 .16017 48 12 .15089 .14154 .16933 .14768 .18732 .15393 .20488 .16028 52 13 .15120 .14165 .16963 .14779 .18762 .15403 .20517 .16039 56 14 .15152 .14175 .16994 .14789 .18791 .15414 .20546 .16049 s ' 2 h 57 m 44 3 h jm 45 &g* 46 3 h 9 m 47 15 9.15183 0.14185 9.17024 0.14799 9.18821 0.15424 9.20574 0.16060 4 '16 .15214 .14195 .17054 .14810 .18850 .15435 .20603 .16071 8 17 .15245 .14205 .17085 .14820 .18880 .15445 .20632 .16081 J 18 .15276 .14215 .17115 .14830 .18909 .15456 .20661 .16092 16 19 .15307 .14226 .17145 .14841 .18939 .15466 .20690 .16103 20 20 9.15338 0.14236 9.17175 0.14851 9.18968 0.15477 9.20719 0.16113 24 21 .15369 .14246 .17206 .14861 .18998 .15487 .20748 .16124 25 22 .15400 .14256 .17236 .14872 .19027 .15498 .20776 .16135 32 23 .15431 .14266 .17266 .14882 .19057 .15509 .20805 .16146 36 24 .15462 .14276 .17296 .14892 .19086 .15519 .20834 .16156 40 25 9.15493 0.14287 9.17327 0.14903 9.19116 0.15530 9.20863 0.16167 44 26 .15524 .14297 .17357 .14913 .19145 .15540 .20891 .16178 4 18 .49849 .31513 .50956 .32326 .52042 .33145 .53109 .33969 /0 19 .49867 .31526 .50974 .32340 .52060 .33159 .53126 .33983 20 20 9.49886 0.31540 9.50992 0.32353 9.52078 0.33173 9.53144 0.33997 24 21 .49904 .31553 .51010 .32367 .52096 .33186 .53162 .34011 2S 22 .49923 .31567 .51029 .32381 .52114 .33200 .53179 .34024 32 23 .49942 .31580 .51047 .32394 .52132 .33214 .53197 .34038 35 24 .49960 .31594 .51065 .32408 .52150 .33227 .53214 .34052 40 25 9.49979 0.31607 9.51083 0.32422 9.52168 0.33241 9.53232 0.34066 44 26 .49997 .31621 .51102 .32435 .52185 .33255 .53249 .34080 45 27 .50016 .31634 .51120 .32449 .52203 .33269 .53267 .34093 .52 28 .50034 .31648 .51138 .32462 .52221 .33282 .53285 .34107 5(7 29 .50053 .31661 .51156 .32476 .52239 .33296 .53302 .34121 s ' 4h 34 m 68 4* 38 m 69 4h 4% 70 4* 46 71 30 9.50072 0.31675 9.51174 0.32490 9.52257 0.33310 9.53320 0.34135 4 31 .50090 .31688 .51193 .32503 .52275 .33323 .53337 .34149 8 32 .50109 .31702 .51211 .32517 .52293 .33337 .53355 .34162 J2 33 .50127 .31716 .51229 .32531 .52311 .33351 .53372 .34176 16 34 .50146 .31729 .51247 .32544 .52328 .33365 .53390 .34190 20 35 9.50164 0.31742 9.51265 0.32558 9.52346 0.33378 9.53407 0.34204 24 36 .50183 .31756 .51284 .32571 .52364 .33392 .53425 .34218 25 37 .50201 .31770 .51302 .32585 .52382 .33406 .53442 .34231 32 38 .50220 .31783 .51320 .32599 .52400 .33419 .53460 .34245 36 39 .50238 .31797 .51338 .32612 .52418 .33433 .53477 .34259 40 40 9.50257 0.31810 9.51356 0.32626 9.52436 0.33447 9.53495 0.34273 44 41 .50275 .31824 .51374 .32640 .52453 .33461 .53512 .34287 48 42 .50294 .31837 .51393 .32653 .52471 .33474 .53530 .34300 52 43 .50312 .31851 .51411 .32667 .52489 .33488 .53547 .34314 56 44 .50331 .31865 .51429 .32681 .52507 .33502 .53565 .34328 s ' 4* 35 m 68 4* 39 m 69 4h ^^m 70 o 4 h 47 m 71 45 9.50349 0.31878 9.51447 0.32694 9.52525 0.33515 9.53582 0.34342 4 46 .50368 .31892 .51465 .32708 .52542 .33529 .53600 .34356 2 47 .50386 .31905 .51483 .32721 .52560 .33543 .53617 .34369 12 48 .50405 .31919 .51501 .32735 .52578 .33557 .53635 .34383 /0 49 .50423 .31932 .51519 .32749 .52596 .33570 .53652 .34397 20 50 9.50442 0.31946 9.51538 0.32762 9.52613 0.33584 9.53670 0.34411 24 51 .50460 .31959 .51556 .32776 .52631 .33598 .53687 .34425 28 52 .50478 .31973 .51574 .32790 .52649 .33612 .53704 .34439 32 53 .50497 .31987 .51592 .32803 .52667 .33625 .53722 .34452 36 54 .50515 .32000 .51610 .32817 .52684 .33639 .53739 .34466 40 55 9.50534 0.32014 9.51628 0.32831 9.52702 0.33653 9.53757 0.34480 44 56 .50552 .32027 .51646 .32844 .52720 .33667 .53774 .34494 48 57 .50570 .32041 .51664 .32858 .52738 .33680 .53792 .34508 52 58 .50589 .32054 .51682 .32872 .52755 .33694 .53809 .34521 56 59 .50607 .32068 .51700 .32885 .52773 .33708 .53826 .34535 60 60 9.50626 0.32082 9.51718 0.32899 9.52791 0.33722 9.53844 0.34549 268 Table 10. Haversine Table s ' 4 h 48 m 72 4* 52 m 73 ^h 5 ffm 74 5h 50 9.54707 0.35242 9.55725 0.36078 9.56725 0.36919 9.57706 0.37763 24 51 .54724 .35256 .55742 .36092 .56741 .36933 .57723 .37777 28 52 .54741 .35270 .55758 .36106 .56758 .36947 .57739 .37791 32 53 .54758 .35284 .55775 .36120 .56774 .36961 .57755 .37805 36? 54 .54775 .35298 .55792 .36134 .56791 .36975 .57771 .37819 40 55 9.54792 0.35312 9.55809 0.36148 9.56807 0.36989 9.57787 0.37833 44 56 .54809 .35326 .55826 .36162 .56824 .37003 .57804 .37847 48 57 .54826 .35340 .55842 .36176 .56840 .37017 .57820 .37862 52 58 .54843 .35354 .55859 .36190 .56856 37031 .57836 .37876 56? 59 .54860 .35368 .55876 .36204 .56873 .37045 .57852 .37890 6?6> 60 9.54878 0.35381 9.55893 0.36218 9.56889 0.37059 9.57868 0.37904 Table 10. Haversine Table 269 s ' 5* 4 m 76 5 h gm 77 5 h i2 m 78 5 h 16 m 79 Hav. No. Hav. No. Hav. No. Hav. No. 9.57868 0.37904 9.58830 0.38752 9.59774 0.39604 9.60702 0.40460 4 1 .57885 .37918 .58846 .38767 .59790 .39619 .60717 .40474 8 2 .57901 .37932 .58862 .38781 .59806 .39633 .60733 .40488 12 3 .57917 .37946 .58878 .38795 .59821 .39647 .60748 .40502 16 4 .57933 .37960 .58893 .38809 .59837 .39661 .60763 .40517 20 5 9.57949 0.37974 9.58909 0.38823 9.59852 0.39676 9.60779 0.40531 24 6 .57965 .37989 .58925 .38837 .59868 .39690 .60794 .40545 28 7 .57981 .38003 .58941 .38852 .50883 .39704 .60809 .40560 32 8 .57998 .38017 .58957 .38866 .59899 .39718 .60825 .40574 36 9 .58014 .38031 .58973 .38880 .59915 .39732 .60840 .40588 40 10 9.58030 0.38045 9.58989 0.38894 9.59930 0.39746 9.60855 0.40602 44 11 .58046 .38059 .59004 .38908 .59946 .39761 .60870 .40617 48 12 .58062 .38073 .59020 .38923 .59961 .39775 .60886 .40631 52 13 .58078 .38087 .59036 .38937 .59977 .39789 .60901 .40645 56 14 .58094 .38102 .59052 .38951 .59992 .39803 .60916 .40660 s ' 5 h 5 m 76 6 h 9 m 77 5h IS" 1 78 6 h 17 m 79 15 9.58110 0.38116 9.59068 0.38965 9.60008 0.39818 9.60931 0.40674 4 16 .58126 .38130 .59083 .38979 .60023 .39832 .60947 .40688 S 17 .58143 .38144 .59099 .38994 .60039 .39846 .60962 .40702 12 18 .58159 .38158 .59115 .39008 .60054 .39861 .60977 .40717 16 19 .58175 .38172 .59131 .39022 .60070 .39875 .60992 .40731 20 20 9.58191 0.38186 9.59147 0.39036 9.60085 0.39889 9.61008 0.40745 24 21 .58207 .38200 .59162 .39050 .60101 .39903 .61023 .40760 2S 22 .58223 .38215 .59178 .39064 .60116 .39918 .61038 .40774 32 23 .58239 .38229 .59194 .39079 .60132 .39932 .61053 .40788 35 24 .58255 .38243 .59210 .39093 .60147 .39946 .61069 .40802 40 25 9.58271 0.38257 9.59225 0.39107 9.60163 0.39960 9.61084 0.40817 44 26 .58287 .38271 .59241 .39121 .60178 .39975 .61099 .40831 45 27 .58303 .38285 .59257 .39135 .60194 .39989 .61114 .40845 52 28 .58319 .38299 .59273 .39150 .60209 .40003 .61129 .40860 55 29 .58335 .38314 .59289 .39164 .60225 .40017 .61145 .40874 s ' 5 h &n 76 5 h 10 m 77 5h 14 78 5 h is 79 30 9.58351 0.38328 9.59304 0.39178 9.60240 0.40032 9.61160 10.40888 4 31 .58367 .38342 .59320 .39192 .60256 .40046 .61175 .40903 8 32 .58383 .38356 .59336 .39206 .60271 .40060 .61190 .40917 12 33 .58399 .38370 .59351 .39221 .60287 .40074 .61205 .40931 16 34 .58415 .38384 .59367 .39235 .60302 .40089 .61221 .40945 20 35 9.58431 0.38398 9.59383 0.39249 9.60318 0.40103 9.61236 0.40960 24 36 .58447 .38413 .59399 .39263 .60333 .40117 .61251 .40974 28 37 .58463 .38427 .59414 .39277 .60348 .40131 .61266 .40988 32 38 .58479 .38441 .59430 .39292 .60364 .40146 .61281 .41003 35 39 .58495 .38455 .59446 .39306 .60379 .40160 .61296 .41017 40 40 9.58511 0.38469 9.59461 0.39320 9.60395 0.40174 9.61312 0.41031 44 41 .58527 .38483 .59477 .39334 .60410 .40188 .61327 .41046 48 42 .58543 .38498 .59493 .39348 .60426 .40203 .61342 .41060 52 43 .58559 .38512 .59508 .39363 .60441 .40217 .61357 .41074 56 44 .58575 .38526 .59524 .39377 .60456 .40231 .61372 .41089 s ' 5 h 7" 76 5 h nm 77 5 h 15 >n ?8 o 6 h igm 79 45 9.58591 0.38540 9.59540 0.39391 9.60472 0.40245 9.61387 0.41103 4 46 .58607 .38554 .59556 .39405 .60487 .40260 .61402 .41117 5 47 .58623 .38568 .59571 .39420 .60502 .40274 .61417 .41131 12 48 .58639 .38582 .59587 .39434 .60518 .40288 .61433 .41146 16 49 .58655 .38597 .59602 .39448 .60533 .40303 .61448 .41160 20 50 9.58671 0.38611 9.59618 0.39462 9.60549 0.40317 9.61463 0.41174 24 51 .58687 .38625 .59634 .39476 .60564 .40331 .61478 .41189 28 52 .58703 .38639 .59649 .39491 .60579 .40345 .61493 .41203 32 53 .58719 .38653 .59665 .39505 .60595 .40360 .61508 .41217 36 54 .58735 .38667 .59681 .39519 .60610 .40374 .61523 .41232 40 55 9.58750 0.38682 9.59696 0.39533 9.60625 0.40388 9.61538 0.41246 44 56 .58766 .OOOJD .59712 .39548 .60641 .40402 .61553 .41260 48 57 .58782 .38710 .59728 .39562 .60656 .40417 .61568 .41275 52 58 .58798 .38724 .59743 .39576 .60671 .40431 .61583 .41289 56 59 .58814 .38738 .59759 .39590 .60687 .40445 .61598 .41303 60 60 9.58830 0.38752 9.59774 0.39604 9.60702 0.40460 9.61614 0.41318 270 Table 10. Haversine Table s ' 5* 20 m 80 5h 24 m 81 gh 28 m 82 oh 32 m 83 Hav. No. Hav. No. Hav. No. Hav. No. 9.61614 0.41318 9.62509 0.42178 9.63389 0.43041 9.64253 0.43907 4 1 .61629 .41332 .62524 .42193 .63403 .43056 .64267 .43921 8 2 .61644 .41346 .62538 .42207 .63418 .43070 .64281 .43935 12 3 .61659 .41361 .62553 .42221 .63432 .43085 .64296 .43950 16 4 .61674 .41375 .62568 .42236 .63447 .43099 .64310 .43964 20 5 9.61689 0.41389 9.62583 0.42250 9.63461 0.43113 0.64324 0.43979 24 6 .61704 .41404 .62598 .42264 .63476 .43128 .64339 .43993 28 7 .61719 .41418 .62612 .42279 .63490 .43142 .64353 .44008 32 8 .61734 .41432 .62627 .42293 .63505 .43157 .64367 .44022 36 9 .61749 .41447 .62642 .42308 .63519 .43171 .64381 .44036 40 10 9.61764 0.41461 9.62657 0.42322 9.63534 0.43185 9.64396 0.44051 44 11 .61779 .41475 .62671 .42336 .63548 .43200 .64410 .44065 48 12 .61794 .41490 .62686 .42351 .63563 .43214 .64424 .44080 52 13 .61809 .41504 .62701 .42365 .63577 .43229 .64438 .44094 56 14 .61824 .41518 .62716 .42379 .63592 .43243 .64452 .44109 s ' gh 21 m 80 5h 25 81 gh 29 m 82 gh 33 m 83 15 9.61839 0.41533 9.62730 0.42394 9.63606 0.43257 9.64467 0.44123 4 16 .61854 .41547 .62745 .42408 .63621 .43272 .64481 .44138 5 17 .61869 .41561 .62760 .42423 .63635 .43286 .64495 .44152 12 18 .61884 .41576 .62774 .42437 .63649 .43301 .64509 .44166 15 19 .61899 .41590 .62789 .42451 .63664 .43315 .64523 .44181 20 9.61914 0.41604 9.62804 0.42466 9.63678 0.43330 9.64538 0.44195 24 21 .61929 .41619 .62819 .42480 .63693 .43344 .64552 .44210 ^o 22 .61944 .41633 .62833 .42494 .63707 .43358 .64566 .44224 32 23 .61959 .41647 .62848 .42509 .63722 .43373 .64580 .44239 35 24 .61974 .41662 .62863 .42523 .63736 .43387 .64594 .44253 40 25 9.61989 0.41676 9.62877 0.42538 9.63751 0.43402 9.64609 0.44268 44 26 .62003 .41690 .62892 .42552 .63765 .43416 .64623 .44282 45 27 .62018 .41705 .62907 .42566 .63779 .43430 .64637 .44296 52 28 .62033 .41719 .62921 .42581 .63794 .43445 .64651 .44311 55 29 .62048 .41733 .62936 .42595 .63808 .43459 .64665 .44325 s ' 5 h 2%m 80 5h 25 m 81 gh so m 82 gh 34 m 83 30 9.62063 0.41748 9.62951 0.42610 9.63823 0.43474 9.64679 0.44340 4 31 .62078 .41762 .62965 .42624 .63837 .43488 .64694 .44354 8 32 .62093 .41776 .62980 .42638 .63851 .43503 .64708 .44369 ^2 33 .62108 .41791 .62995 .42653 .63866 .43517 .64722 .44383 ^5 34 .62123 .41805 .63009 .42667 .63880 .43531 .64736 .44398 20 35 9.62138 0.41819 9.63024 0.42681 9.63895 0.43546 9.64750 0.44412 #4 36 .62153 .41834 .63039 .42696 .63909 .43560 .64764 .44427 28 37 .62168 .41848 .63063 .42710 .63923 .43575 .64778 .44441 32 38 .62182 .41862 .63068 .42725 .63938 .43589 .64793 .44455 35 39 .62197 .41877 .63082 .42739 .63952 .43603 .64807 .44470 40 40 9.62212 0.41891 9.63097 0.42753 9.63966 0.43618 9.64821 0.44484 44 41 .62227 .41905 .63112 .42768 .63981 .43632 .64835 .44499 48 42 .62242 .41920 .63126 .42782 .63995 .43647 .64849 .44513 52 43 .62257 .41934 .63141 .42797 .64010 .43661 .64863 .44528 55 44 .62272 .41949 .63156 .42811 .64024 .43676 .64877 .44542 s ' 5 h 23 m 80 gh 27 m 81 gh 31 m 82 gh 35 m 83 45 9.62287 0.41963 9.63170 0.42825 9.64038 0.43690 9.64891 0.44557 4 46 .62301 .41977 .63185 .42840 .64053 .43704 .64905 .44571 5 47 .62316 .41992 .63199 .42854 .64067 .43719 .64919 .44586 .72 48 .62331 .42006 .63214 .42869 .64081 .43733 .64934 .44600 16 49 .62346 .42020 .63228 .42883 .64096 .43748 .64948 .44614 20 50 9.62361 0.42035 9.63243 0.42897 9.64110 0.43762 9.64962 0.44629 24 51 .62376 .42049 .63258 .42912 .64124 .43777 .64976 .44643 28 52 .62390 .42063 .63272 .42926 .64139 .43791 .64990 .44658 32 53 .62405 .42078 .63287 .42941 .64153 .43805 .65004 .44672 35 54 .62420 .42092 .63301 .42955 .64167 .43820 .65018 .44687 40 55 9.62435 0.42106 9.63316 0.42969 9.64181 0.43834 9.65032 0.44701 44 56 .62450 .42121 .63330 .42984 .64196 .43849 .65046 .44716 48 57 .62464 .42135 .63345 .42998 .64210 .43863 .65060 .44730 52 58 .62479 .42150 .63360 .43013 .64224 .43878 .65074 .44745 55 59 .62494 .42164 .63374 .43027 .64239 .43892 .65088 .44759 60 60 9.62509 0.42178 9.63389 0.43041 9.64253 0.43907 9.65102 0.44774 Table 10. Haversine Table 271 s ' 5* 36 m 84 5* 40 m 85 5* 44 m 86 5 h 48 m 87 Hav. No. Hav. No. Hav. No. Hav. No. 9.65102 0.44774 9.65937 0.45642 9.66757 0.46512 9.67562 0.47383 4 l .65116 .44788 .65950 .45657 .66770 .46527 .67576 .47398 8 2 .65130 .44803 .65964 .45671 .66784 .46541 .67589 .47412 12 3 .65144 .44817 .65978 .45686 .66797 .46556 .67602 .47427 16 4 .65158 .44831 .65992 .45700 .66811 .46570 .67616 .47441 20 5 9.65172 0.44846 9.66006 0.45715 9.66824 0.46585 9.67629 0.47456 24 6 .65186 .44860 .66019 .45729 .66838 .46599 .67642 .47470 28 7 .65200 .44875 .66033 .45744 .66851 .46614 .67656 .47485 32 8 .65214 .44889 .66047 .45758 .66865 .46628 .67669 .47499 36 9 .65228 .44904 .66061 .45773 .66878 .46643 .67682 .47514 40 10 9.65242 0.44918 9.66074 0.45787 9.66892 0.46657 9.67695 0.47528 44 11 .65256 .44933 .66088 .45802 .66905 .46672 .67709 .47543 48 12 .65270 .44947 .66102 .45816 .66919 .46686 .67722 .47558 52 13 .65284 .44962 .66116 .45831 .66932 .46701 .67735 .47572 56 14 .65298 .44976 .66129 .45845 .66946 .46715 .67748 .47587 s ' oh 37 m 84 5 h 41 m 85 5^ 45 m 86 5 h 49 m 87 15 9.65312 0.44991 9.66143 0.45860 9.66959 0.46730 9.67762 0.47601 4 16 .65326 .45005 .66157 .45874 .66973 .46744 .67775 .47616 5 17 .65340 .45020 .66170 .45889 .66986 .46759 .67788 .47630 12 18 .65354 .45034 .66184 .45903 .67000 .46773 .67801 .47645 76 19 .65368 .45048 .66198 .45918 .67013 .46788 .67815 .47659 20 20 9.65382 0.45063 9.66212 0.45932 9.67027 0.46802 9.67828 0.47674 24 21 .65396 .45077 .66225 .45947 .67040 .46817 .67841 .47688 2S 22 .65410 .45092 .66239 .45961 .67054 .46831 .67854 .47703 32 23 .65424 .45106 .66253 .45976 .67067 .46846 .67868 .47717 56 24 .65438 .45121 .66266 .45990 .67081 .46860 .67881 .47732 40 25 9.65452 0.45135 9.66280 0.46005 9.67094 0.46875 9.67894 0.47746 44 26 .65466 .45150 .66294 .46019 .67108 .46890 .67907 .47761 45 27 .65480 .45164 .66307 .46034 .67121 .46904 .67920 .47775 52 28 .65493 .45179 .66321 .46048 .67134 .46919 .67934 .47790 55 29 .65507 .45193 .66335 .46063 .67148 .46933 .67947 .47805 s ' 5* 38 m 84 5 h 42 85 5^ 46 86 5* 50 87 30 9.65521 0.45208 9.66348 0.46077 9.67161 0.46948 9.67960 0.47819 4 31 .65535 .45222 .66362 .46092 .67175 .46962 .67973 .47834 8 32 .65549 .45237 .66376 .46106 .67188 .46977 .67986 .47848 72 33 .65563 .45251 .66389 .46121 .67202 .46991 .68000 .47863 76 34 .65577 .45266 .66403 .46135 .67215 .47006 .68013 .47877 20 35 9.65591 0.45280 9.66417 0.46150 9.67228 0.47020 9.68026 0.47892 24 36 .65605 .45295 .66430 .46164 .67242 .47035 .68039 .47906 28 37 .65619 .45309 .66444 .46179 .67255 .47049 .68052 .47921 32 38 .65632 .45324 .66458 .46193 .67269 .47064 .68066 .47935 36 39 .65646 .45338 .66471 .46208 .67282 .47078 .68079 .47950 40 40 9.65660 0.45353 9.66485 0.46222 9.67295- 0.47093 9.68092 0.47964 44 41 .65674 .45367 .66499 .46237 .67309 .47107 .68105 .47979 48 42 .65688 .45381 .66512 .46251 .67322 .47122 .68118 .47993 52 43 .65702 .45396 .66526 .46266 .67336 .47136 .68131 .48008 56 44 .65716 .45410 .66539 .46280 .67349 .47151 .68144 .48022 6' ' 5* 39 m 84 5* 43 85 5 h 47 m 86 5 h S1 m 87 45 9.65729 0.45425 9.66553 0.46295 9.67362 0.47165 9.68158 0.48037 4 46 .65743 .45439 .66567 .46309 .67376 .47180 .68171 .48052 S 47 .65757 .45454 .66580 .46324 .67389 .47194 .68184 .48066 /2 48 .65771 .45468 .66594 .46338 .67402 .47209 .68197 .48081 16 49 .65785 .45483 .66607 .46353 .67416 .47223 .68210 .48095 20 50 9.65799 0.45497 9.66621 0.46367 9.67429 0.47238 9.68223 0.48110 24 51 .65812 .45512 .66635 .46382 .67443 .47252 .68236 .48124 28 52 .65826 .45526 .66648 .46396 .67456 .47267 .68249 .48139 32 53 .65840 .45541 .66662 .46411 .67469 .47282 .68263 .48153 36 54 .65854 .45555 .66675 .46425 .67483 .47296 .68276 .48168 40 55 9.65868 0.45570 9.66689 0.46440 9.67496 0.47311 9.68289 0.48182 44 56 .65881 .45584 .66702 .46454 .67509 .47325 .68302 .48197 48 57 .65895 .45599 .66716 .46469 .67522 .47340 .68315 .48211 52 58 .65909 .45613 .66730 .46483 .67536 .47354 .68328 .48226 56 59 .65923 .45628 .66743 .46498 .67549 .47369 .68341 .48241 60 60 9.65937 0.45642 9.66757 0.46512 9.67562 0.47383 9.68354 0.48255 272 Table 10. Haversine Table s ' 5 h 52 m 88 5h 56 m 89 6h Q m 6^ 4 Hav. No. Hav. No. S Hav. Hav. 9.68354 0.48255 9.69132 0.49127 9.69897 9.70648 4 1 .68367 .48269 .69145 .49142 4 .69910 .70661 8 2 .68380 .48284 .69158 .49156 8 .69922 .70673 12 3 .68393 .48299 .69171 .49171 12 .69935 .70686 16 4 .68407 .48313 .69184 .49186 16 .69948 .70698 20 5 9.68420 0.48328 9.69197 0.49200 20 9.69960 9.70710 24 6 .68433 .48342 .69209 .49215 24 .69973 .70723 28 7 .68446 .48357 .69222 .49229 28 .69985 .70735 32 8 .68459 .48371 .69235 .49244 32 .69998 .70748 36 9 .68472 .48386 .69248 .49258 36 .70011 .70760 40 10 9.68485 0.48400 9.69261 0.49273 40 9.70023 9.70772 44 11 .68498 .48415 .69274 .49287 44 .70036 .70785 45 12 .68511 .48429 .69286 .49302 48 .70048 .70797 52 13 .68524 .48444 .69299 .49316 52 .70061 .70809 56 14 .68537 .48459 .69312 .49331 56 .70074 .70822 s ' 5* 53 88 5h 57 m 89 | s 6 h l m 6 h 5 m 15 9.68550 0.48473 9.69325 0.49346 > o3 9.7008'6 9.70834 4 16 .68563 .48488 .69338 .49360 A 4 .70099 .70847 5 17 .68576 .48502 .69350 .49375 6 8 .70111 .70859 12 18 .68589 .48517 .69363 .49389 & 12 .70124 .70871 .76 19 .68602 .48531 .69376 .49404 S 16 .70136 .70884 20 9.68615 0.48546 9.69389 0.49418 20 9.70149 9.70896 24 21 .68628 .48560 .69402 .49433 3 24 .70161 .70908 S 22 .68641 .48575 .69414 .49447 A\ 28 .70174 .70921 32 23 .68654 .48589 .69427 .49462 ! 32 .70187 .70933 35 24 .68667 .48604 .69440 .49476 ^ 36 .70199 .70945 40 25 9.68680 0.48618 9.69453 0.49491 S fa 40 9.70212 9.70958 44 26 .68693 .48633 .69465 .49506 3 44 .70224 .70970 45 27 .68706 .48648 .69478 .49520 5* 48 .70237 .70982 52 28 .68719 .48662 .69491 .49535 -g 52 .70249 .70995 55 29 .68732 .48677 .69504 .49549 9 20 56 .70262 .71007 s ' 5^ 54 m 88 5* 58 89 >> o> 8 s Qh %rn 6 h Qm 30 9.68745 0.48691 9.69516 0.49564 ,8-" +3 rt-l 9.70274 9.71019 4 31 .68758 .48706 .69529 .49578 a| 4 .70287 .71032 8 32 .68771 .48720 .69542 .49593 8 .70299 .71044 ^2 33 .68784 .48735 .69555 .49607 ^o a} o 3 12 .70312 .71056 .76 34 .68797 .48749 .69567 .49622 5-2 16 .70324 .71068 20 35 9.68810 0.48764 9.69580 0.49636 is 20 . 9.70337 9.71081 24 36 .68823 .48778 .69593 .49651 2 24 .70349 .71093 28 37 .68836 .48793 .69605 .49665 "* 03 28 .70362 .71105 S# 38 .68849 .48807 .69618 .49680 R 32 .70374 .71118 56 39 .68862 .48822 .69631 .49695 3 36 .70387 .71130 40 40 9.68875 0.48837 9.69644 0.49709 1 40 9.70399 9.71142 44 41 .68887 .48851 .69656 .49724 44 .70412 .71154 48 42 .68900 .48866 .69669 .49738 o te 48 .70424 .71167 52 43 .68913 .48880 .69682 .49753 c\ 52 .70437 .71179 56 44 .68926 .48895 .69694 .49767 $ ~ 56 .70449 .71191 s ' 5 h 55 m 88 5^ 59 m 89 i s Qh &* 6 h 7m 45 9.68939 0.48909 9.69707 0.49782 i 9.70462 9.71203 4 46 .68952 .48924 .69720 .49796 i 4 .70474 .71216 5 47 .68965 .48938 .69732 .49811 o +7 8 .70487 .71228 ^2 48 .68978 .48953 .69745 .49825 f-+ 12 .70499 .71240 16 49 .68991 .48967 .69758 .49840 16 .70512 .71252 #0 50 9.69004 0.48982 9.69770 0.49855 20 9.70524 9.71265 24 51 .69017 .48997 .69783 .49869 24 .70537 .71277 2* SO" 1 0* 34 m 6 h 38 9.71751 9.72471 9.73177 9.73872 9.74554 9.75225 9.75884 9.76531 4 .71763 .72482 .73189 .73883 .74566 .75236 .75895 .76542 8 .71775 .72494 .73201 .73895 .74577 .75247 .75906 .76553 12 .71787 .72506 .73212 .73906 .74588 .75258 .75917 .76563 16 .71800 .72518 .73224 .73918 .74600 .75269 .75927 .76574 20 9.71812 9.72530 9.73236 9.73929 9.74611 9.75280 9.75938 9.76585 24 .71824 .72542 .73247 .73941 .74622 .75291 .75949 .76595 28 .71836 .72554 .73259 .73952 .74633 .75303 .75960 .76606 32 .71848 .72565 .73271 .73964 .74645 .75314 .75971 .76617 36 .71860 .72577 .73282 .73975 .74656 .75325 .75982 .76627 40 9.71872 9.72589 9.73294 9.73987 9.74667 9.75336 9.75993 9.76638 44 .71884 .72601 .73306 .73998 .74678 .75347 .76004 .76649 48 .71896 .72613 .73317 .74009 .74690 .75358 .76014 .76659 52 .71908 .72625 .73329 .74021 .74701 .75369 .76025 .76670 66 .71920 .72637 .73341 .74032 .74712 .75380 .76036 .76681 8 0* ll m 6* 15 m Qh igm. Qh 23 6 h 27 m 6 h 31 m Qh S5 m 0* 39 9.71932 9.72648 9.73352 9.74044 9.74723 9.75391 9.76047 9.76691 4 .71944 .72660 .73364 .74055 .74734 .75402 .76058 .76702 8 .71956 .72672 .73375 .74067 .74746 .75413 .76069 .76713 12 .71968 .72684 .73387 .74078 .74757 .75424 .76079 .76723 16 .71980 .72696 .73399 .74089 .74768 .75435 .76090 .76734 20 9.71992 9.72708 9.73410 9.74101 9.74779 9.75446 9.76101 9.76745 24 .72004 .72719 .73422 .74112 .74791 .75457 .76112 .76755 28 .72016 .72731 .73433 .74124 .74802 .75468 .76123 .76766 32 .72028 .72743 .73445 .74135 .74813 .75479 .76134 .76777 36 .72040 .72755 .73457 .74146 .74824 .75490 .76144 .76787 40 9.72052 9.72767 9.73468 9.74158 9.74835 9.75501 9.76155 9.76798 44 .72064 .72778 .73480 .74169 .74846 .75512 .76166 .76808 48 .72070 .72790 .73491 .74181 .74858 .75523 .76177 .76819 52 .72088 .72802 .73503 .74192 .74869 .75534 .76188 .76830 56 .72100 .72814 .73515 .74203 .74880 .75545 .76198 .76840 60 9.72112 9.72825 9.73526 9.74215 9.74891 9.75556 9.76209 9.76851 274 Table 10. Haversine Table s 6h 40 m 6 h 44 m Qh 48 6^ 52 Qh 5Qm 7 h Q m 7 h 4 7h 8 Hav. Hav. Hav. Hav. Hav. Hav. Hav. Hav. 9.76851 9.77481 9.78101 9.78709 9.79306 9.79893 9.80470 9.81036 4 .76861 .77492 .78111 .78719 .79316 .79903 .80479 .81045 8 .76872 .77502 .78121 .78729 .79326 .79913 .80489 .81054 12 .76883 .77512 .78131 .78739 .79336 .79922 .80498 .81064 16 .76893 .77523 .78141 .78749 .79346 .79932 .80508 .81073 20 9.76904 9.77533 9.78152 9.78759 9.79356 9.79942 9.80517 9.81082 24 .76914 .77544 .78162 .78769 .79366 .79951 .80527 .81092 28 .76925 .77554 .78172 .78779 .79376 .79961 .80536 .81101 32 .76936 .77564 .78182 .78789 .79385 .79971 .80546 .81110 36 .76946 .77575 .78192 .78799 .79395 .79980 .80555 .81120 40 9.76957 9.77585 9.78203 9.78809 9.79405 9.79990 9.80565 9.88129 44 .76967 .77596 .78213 .78819 .79415 .80000 .80574 .81138 48 .76978 .77606 .78223 .78829 .79425 .80009 .80584 .81148 52 .76988 .77616 .78233 .78839 .79434 .80019 .80593 .81157 56 .76999 .77627 .78243 .78849 .79444 .80029 .80603 .81166 s Qh ^im Q h 5 m 6 h 49 6* 53" 1 Qh gym yh im 7 h 5 7 h 9 9.77009 9.77637 9.78254 9.78859 9.79454 9.80038 9,80612 9.81176 4 .77020 .77647 .78264 .78869 .79464 .80048 .80622 .81185 8 .77031 .77658 .78274 .78879 .79474 .80058 .80631 .81194 12 .77041 .77668 .78284 .78889 .79484 .80067 .80641 .81204 16 .77052 .77679 .78294 .78899 .79493 .80077 .80650 .81213 20 9.77062 9.77689 9.78305 9.78909 9.79503 9.30087 9.80660 9.81222 24 .77073 .77699 .78315 .78919 .79513 .80096 .80669 .81231 28 .77083 .77710 .78325 .78929 .79523 .80106 .80678 .81241 32 .77094 .77720 .78335 .78939 .79533 .80116 .80688 .81250 36 .77104 .77730 .78345 .78949 .79542 .80125 .80697 .81259 40 9.77115 9.77741 9.78355 9.78959 9.79552 9.80135 9.80707 9.81269 44 .77125 .77751 .78365 .78969 .79562 .80144 .80716 .81278 48 .77136 .77761 .78376 .78979 .79572 .80154 .80726 .81287 62 .77146 .77772 .78386 .78989 .79582 .80164 .80735 .81296 66 .77157 .77782 .78396 .78999 .79591 .80173 .80745 .81306 s 6h 42 6^ 46 6 h 50 6>> 54 Qh QS 7* 2 7 h 6 7 h 10 9.77167 9.77792 9.78406 9.79009 9.79601 9.80183 9.80754 9.81315 4 .77178 .77803 .78416 .79019 .79611 .80192 .80763 .81324 8 .77188 .77813 .78426 .79029 .79621 .80202 .80773 .81333 12 .77199 .77823 .78436 .79039 .79631 .80212 .80782 .81343 16 . .77209 .77834 .78447 .79049 .79640 .80221 .80792 .81352 20 9.77220 9.77844 9.78457 9.79059 9.79650 9.80231 9.80801 9.81361 24 .77230 .77854 .78467 .79069 .79660 .80240 .80811 .81370 28 .77241 .77864 .78477 .79079 .79670 .80250 .80820 .81380 32 .77251 .77875 .78487 .79089 .79679 .80260 .80829 .81389 36 .77262 .77885 .78497 .79099 .79689 .80269 .80839 .81398 40 9.77272 9.77895 9.78507 9.79108 9.79699 9.80279 9.80848 9.81407 44 .77283 .77906 .78517 .79118 .79709 .80288 .80858 .81417 48 .77293 .77916 .78528 .79128 .79718 .80298 .80867 .81426 52 .77304 .77926 .78538 .79138 .79728 .80307 .80876 .81435 56 .77314 .77936 .78548 .79148 .79738 .80317 .80886 .81444 s 6h 43 m 6h 47 m Qh Qim 6 h 55 m Qh 59 m 7 h S ?h jrn 7 h Um 9.77325 9.77947 9.78558 9.79158 9.79748 9.80327 9.80895 9.81454 4 .77335 .77957 .78568 .79168 .79757 .80336 .80905 .81463 8 .77346 .77967 .78578 .79178 .79767 .80346 .80914 .81472 12 .77356 .77978 .78588 .79188 .79777 .80355 .80923 .81481 16 .77366 .77988 .78598 .79198 .79787 .80365 .80933 .81490 20 9.77377 9.77998 9.78608 9.79208 9.79796 9.80374 9.80942 9.81500 24 .77387 .78008 .78618 .79217 .79806 .80384 .80952 .81509 28 .77398 .78019 .78628 .79227 .79816 .80393 .80961 .81518 32 .77408 .78029 .78638 .79237 .79825 .80403 .80970 .81527 36 .77419 .78039 .78649 .79247 .79835 .80413 .80980 .81536 40 9.77429 9.78049 9.78659 9.79257 9.79845 9.80422 9.80989 9.81546 44 .77440 .78060 .78669 .79267 .79855 .80432 .80998 .81555 48 .77450 .78070 .78679 .79277 .79864 .80441 .81008 .81564 52 .77460 .78080 .78689 .79287 .79874 .80451 .81017 .81573 56 .77471 .78090 .78699 .79297 .79884 .80460 .81026 .81582 60 9.77481 9.78101 9.78709 9.79306 9.79893 9.80470 9.81036 9.81592 Table 10. Haversine Table 275 s 7 h 12 m 7 h 16 7 h 20 m 7 h 24 m 7 h 28 7 h 82 7h 36 7 h 40 m HOT. Hav. Hav. Hav. Hav. Hav. Hav. Hav. 9.81592 9.82137 9.82673 9.83199 9.83715 9.84221 9.84718 9.85206 4 .81601 .82146 .82682 .83207 .83723 .84230 .84726 .85214 8 .81610 .82155 .82691 .83216 .83732 .84238 .84735 .85222 12 .81619 .82164 .82699 .83225 .83740 .84246 .84743 .85230 16 .81628 .82173 .82708 .83233 .83749 .84255 .84751 .85238 20 9.81637 9.82182 9.82717 9.83242 9.83757 9.84263 9.84759 9.85246 24 .81647 .82191 .82726 .83251 .83766 .84271 .84767 .85254 28 .81656 .82200 .82735 .83259 .83774 .84280 .84776 .85262 32 .81665 .82209 .82744 .83268 .83783 .84288 .84784 .85270 36 .81674 .82218 .82752 .83277 .83791 .84296 .84792 .85278 40 9.81683 9.82227 9.82761 9.83285 9.83800 9.84305 9.84800 9.85286 44 .81692 .82236 .82770 .83294 .83808 .84313 .84808 .85294 48 .81701 .82245 .82779 .83303 .83817 .84321 .84817 .85302 62 .81711 .82254 .82788 .83311 .83825 .84330 .84825 .85310 56 '.81720 .S2263 .82796 .83320 .83834 .84338 .84833 .85318 s 7* is m 7 h 17 m ?h 21 m 7* 25 m 7 h 29 m 7 h 33 m ?h 3?m 7 h ^im 9.81729 9.82272 9.82805 9.83329 9.83842 9.84346 9.84841 9.85326 4 .81738 .82281 .82814 .83337 .83851 .84355 .84849 .85334 8 .81747 .82290 .82823 .83346 .83859 .84363 .84857 .85342 12 .81756 .82299 .82832 .83355 .83868 .84371 .84866 .85350 16 .81765 .82308 .82840 .83363 .83876 .84380 .84874 .85358 20 9.81775 9.82317 9.82849 9.83372 9.83885 9.84388 9.84882 9.85366 24 .81784 .82326 .82858 .83380 .83893 .84396 .84890 .85374 28 .81793 .82335 .82867 .83389 .83902 .84405 .84898 .85382 32 .81802 .82344 .82876 .83398 .83910 .84413 .84906 .85390 36 .81811 .82353 .82884 .83406 .83919 .84421 .84914 .85398 40 9.81820 9.82362 9.82893 9.83415 9.83927 9.84430 9.84923 9.85406 44 .81829 .82371 .82902 .83424 .83935 .84438 .84931 .85414 48 .81838 .82380 .82911 .83432 .83944 .84446 .84939 .85422 52 .81847 .82388 .82920 .83441 .83952 .84454 .84947 .85430 56 .81857 .82397 .82928 .83449 .83961 .84463 .84955 .85438 s 7 h 14 m 7 h 18 m ?h 22 m 7 h 26 7* 30 m 7 h 34 m 7* 38 m 7* 42 m 9.81866 9.82406 9.82937 9.83458 9.83969 9.84471 9.84963 9.85446 4 .81875 .82415 .82946 .83467 .83978 .84479 .84971 .85454 8 .81884 .82424 .82955 .83475 .83986 .84488 .84979 .85462 12 .81893 .82433 .82963 .83484 .83995 .84496 .84988 .85470 16 .81902 .82442 .82972 .83492 .84003 .84504 .84996 .85478 20 9.81911 9.82451 9.82981 9.83501 9.84011 9.84512 9.85004 9.85486 24 .81920 .82460 .82990 .83510 .84020 .84521 .85012 .85494 28 .81929 .82469 .82998 .83518 .84028 .84529 .85020 .85502 32 .81938 .82478 .83007 .83527 .84037 .84537 .85028 .85510 36 .81947 .82487 .83016 .83535 .84045 .84545 .85036 .85518 40 9.81956 9.82495 9.83025 9.83544 9.84054 9.84554 9.85044 9.85526 44 .81965 .82504 .83033 .83552 .84062 .84562 .85052 .85534 48 .81975 .82513 .83042 .83561 .84070 .84570 .85061 .85542 52 .81984 .82522 .83051 .83570 .84079 .84578 .85069 .85550 66 .81993 .82531 .83059 .83578 .84087 .84587 .85077 .85557 8 ?h js m 7 h 19 m 7/ 23 nt 7 h 27 m 7 h 31 m 7 h 35 m 7 h ggm ?h 43 9.82002 9.82540 9.83068 9.83587 9.84096 ( .).S4o,).J 9.85085 9.85565 4 .82011 .82549 .83077 .83595 .84104 .84603 .85093 .85573 8 .82020 .82558 .83086 .83604 .84112 .84611 .85101 .85581 12 .82029 .82567 .83094 .83612 .84121 .84620 .85109 .85589 16 .82038 .82575 .83103 .83621 .84129 .84628 .85117 .85597 20 9.82047 9.82584 9.83112 9.83630 9.84138 9.84636 9.85125 9.85605 24 .82056 .82593 .83120 .83638 .84146 .84644 .85133 .85613 28 .82065 .82602 .83129 .83647 .84154 .84653 .85141 .85621 32 .82074 .82611 .83138 .83655 .84163 .84661 .85149 .85629 36 .82083 .82620 .83147 .83664 .84171 .84669 .85158 .85637 40 9.82092 9.82629 9.83155 9.83672 9.84179 9.84677 9.85166 9.85645 44 .82101 .82638 .83164 .83681 .84188 .84685 .85174 .85653 48 .82110 .82646 .83173 .83689 .84196 .84694 .85182 .85660 52 .82119 .82655 .83181 .83698 .84205 .84702 .85190 .85668 56 .82128 .82664 .83190 .83706 .84213 .84710 .85198 .85676 60 9.82137 9.82673 9.83199 9.83715 9.84221 9.84718 9.85206 9.85684 276 Table 10. Haversine Table s 7 h ^m 7 h 48 7 h ggm 7 h 5 Qm gh Q m 8* 4 m 8h gm 8h 12 Hav. Hav. Hav. Hav. Hav. Hav. Hav. Hav. 9.85684 9.86153 9.86613 9.87064 9.87506 9.87939 9.88364 9.88780 4 .85692 .86161 .86621 .87072 .87513 .87947 .88371 .88787 8 .85700 .86169 .86628 .87079 .87521 .87954 .88378 .88793 12 .85708 .86176 .86636 .87086 .87528 .87961 .88385 .88800 16 .85716 .86184 .86643 .87094 .87535 .87968 .88392 .88807 . 20 9.85724 9.86192 9.86651 9.87101 9.87543 9.87975 9.88399 9.88814 24 .85731 .86200 .86659 .87109 .87550 .87982 .88406 .88821 28 .85739 .86207 .86666 .87116 .87557 .87989 .88413 .88828 32 .85747 .86215 .86674 .87124 .87564 .87996 .88420 .88835 36 .85755 .86223 .86681 .87131 .87572 .88004 .88427 .88841 40 9.85763 9.86230 9.86689 9.87138 9.87579 9.88011 9.88434 9.88848 44 .85771 .86238 .86696 .87146 .87586 .88018 .88441 .88855 48 .85779 .86246 .86704 .87153 .87593 .88025 .88448 .88862 52 .85787 .86254 .86712 .87161 .87601 .88032 .88455 .88869 56 .85794 .86261 .86719 .87168 .87608 .88039 .88.462 .88876 s 7 h 45 m 7 h 49 7 h 53 7 h 57 m gh im 8 h gm gh Qm 8h 13 m 9.85802 9.86269 9.86727 9.87175 9.87615 9.88046 9.88469 9.88882 4 .85810 .86277 .86734 .87183 .87623 .88053 .88476 .88889 8 .85818 .86284 .86742 .87190 .87630 .88061 .88483 .88896 12 .85826 .86292 .86749 .87198 .87637 .88068 .88490 .88903 16 .85834 .86300 .86757 .87205 .87644 .88075 .88496 .88910 20 9.85841 9.86307 9.86764 9.87212 9.87652 9.88082 9.88503 9.88916 24 .85849 .86315 .86772 .87220 .87659 .88089 .88510 .88923 .85857 .86323 .86780 .87227 .87666 .88096 .88517 .88930 32 .85865 .86331 .86787 .87235 .87673 .88103 .88524 .88937 36 .85873 .86338 .86795 .87242 .87680 .88110 .88531 .88944 40 9.85881 9.86346 9.86802 9.87249 9.87688 9.88117 9.88528 9.88950 44 .85888 .86354 .86810 .87257 .87695 .88124 .88545 .88957 48 .85896 .86361 .86817 .87264 .87702 .88131 .88552 .88964 52 .85904 .86369 .86825 .87271 .87709 .88139 .88559 .88971 56 .85912 .86377 .86832 .87279 .87717 .88146 .88566 .88978 s 7 h 46 m 7 h 50 m 7 h 54 m 7 h 58 m 8 h 2 8 h 6 8h io m 8^ 14 m 9.85920 9.86384 9.86840 9.87286 9.87724 9.88153 9.88573 9.88984 4 .85928 .86392 .86847 .87294 .87731 .88160 .88580 .88991 8 .85935 .86400 .86855 .87301 .87738 .88167 .88587 .88998 12 .85943 .86407 .86862 .87308 .87745 .88174 .88594 .89005 16 .85951 .86415 .86870 .87316 .87753 .88181 .88600 .89012 20 9.85959 9.86423 9.86877 9.87323 9.87760 9.88188 9.88607 9.89018 24 .85967 .86430 .86885 .87330 .87767 .88195 .88614 .89025 28 .85974 .86438 .86892 .87338 .87774 .88202 .88621 .89032 32 .85982 .86446 .86900 .87345 .87782 .88209 .88628 .89039 36 .85990 .86453 .86907 .87352 .87789 .88216 .88635 .89045 40 9.85998 9.86461 9.86915 9.87360 9.87796 9.88223 9.88642 9.89052 44 .86006 .86468 .86922 .87367 .87803 .88230 .88649 .89059 48 .86013 .86476 .86930 .87374 .87810 .88237 .88656 .89066 52 .86021 .86484 .86937 .87382 .87818 .88244 .88663 .89072 56 .86029 .86491 .86945 .87389 .87825 .88252 .88670 .89079 s 7 h 47 m 7 h 51 m 7 h 55 m 7 h 59 m gh grn 8 h 7 m 8 h ll m 8 h 15 9.86037 9.86499 9.86952 9.87396 9.87832 9.88259 9.88677 9.89086 4 .86045 .86507 .86960 .87404 .87839 .88266 .88683 .89093 8 .86052 .86514 .86967 .87411 .87846 .88273 .88690 .89099 12 .86060 .86522 .86975 .87418 .87853 .88280 .88697 .89106 16 .86068 .86529 .86982 .87426 .87861 .88287 .88704 .89113 20 9.86076 9.86537 9.86990 9.87433 9.87868 9.88294 9.88711 9.89120 24 .86083 .86545 .86997 .87440 .87875 .88301 .88718 .89126 28 .86091 .86552 .87004 .87448 .87882 .88308 .88725 .89133 32 .86099 .86560 .87012 .87455 .87889 .88315 .88732 .89140 36 .86107 .86568 .87019 .87462 .87896 .88322 .88739 .89147 40 9.86114 9.86575 9.87027 9.87470 9.87904 9.88329 9.88745 9.89153 44 .86122 .86583 .87034 .87477 .87911 .88336 .88752 .89160 48 .86130 .86590 .87042 .87484 .87918 .88343 .88759 .89167 52 .86138 .86598 .87049 .87492 .87925 .88350 .88766 .89174 66 .86145 .86606 .87057 .87499 .87932 .88357 .88773 .89180 60 9.86153 9.86613 9.87064 9.87506 9.87939 9.88364 9.88780 9.89187 Table 10. Haversine Table 277 g 8 h i$m 8* 20" 8 h 24 m 8* 28 8 h 32 m 8 h 36 m 8 h 40 8^ 44 m Hay. Hav. Hav. Hay. Hay. Hav. Hav. Hav. KS91S7 9.89586 9.89976 9.90358 9.90732 9.91098 9.91455 9.91805 4 .89194 .89592 .89983 .90365 .90738 .91104 .91461 .91810 8 .89200 .89599 .89989 .90371 .90744 .91110 .91467 .91816 12 .89207 .89606 .89995 .90377 .90751 .91116 .91473 .91822 16 .89214 .89612 .90002 .90383 .90757 .91122 .91479 .91828 20 9.89221 9.89619 9.90008 9.90390 9.90763 9.91128 9.91485 9.91833 24 .89227 .89625 .90015 .90396 .90769 .91134 .91490 .91839 28 .89234 .89632 .90021 .90402 .90775 .91140 .91496 .91845 32 .89241 .89638 .90028 .90409 .90781 .91146 .91502 .91851 36 .89247 .89645 .90034 .90415 .90787 .91152 .91508 .91856 40 9.89254 9.89651 9.90040 9.90421 9.90794 9.91158 9.91514 9.91862 44 .89261 .89658 .90047 .90428 .90800 .91164 .91520 .91868 48 .89267 .89665 .90053 .90434 .90806 .91170 .91526 .91874 52 .89274 .89671 .90060 .90440 .90812 .91176 .91532 .91879 56 .89281 .89678 .90066 .90446 .90818 .91182 .91537 .91885 s 8 h 1? g h 21 m 8 h 25 m 8 h 29 m 8 h Sgm 8 h 8Jm 8 h ^im 8 h ^ 5 m 9.89287 9.896S4 9.90072 9.90452 9.90824 9.91188 9.91543 9.91891 4 .89294 .89691 .90079 .90459 .90830 .91194 .91549 .91896 8 .89301 .89697 .90085 .90465 .90836 .91200 .91555 .91902 12 .89308 .89704 .90092 .90471 .90843 .91206 .91561 .91908 16 .89314 .89710 .90098 .90478 .90849 .91212 .91567 .91914 20 9.89321 9.89717 9.90104 9.90484 9.90855 9.91218 9.91573 9.91919 24 .89328 .89723 .90111 .90490 .90861 .91224 .91578 .91925 28 .89334 .89730 .90117 .90496 .90867 .91230 .91584 .91931 32 .89341 .89736 .90124 .90503 .90873 .91236 .91590 .91936 36 .89348 .89743 .90130 .90509 .90879 .91242 .91596 .91942 40 9.89354 9.89749 9.90136 9.90515 9.90885 9.91248 9.91602 9.91948 44 .89361 .89756 .90143 .90521 .90892 .91254 .91608 .91954 48 .89368 .89763 .90149 .90527 .90898 .91260 .91613 .91959 62 .89374 .89769 .90156 .90534 .90904 .91265 .91619 .91965 56 .89381 .89776 .90162 .90540 .90910 .91271 .91625 .91971 s 8 h i 8 m 8 h 22 gh 26 8* 30 m 8 h SJm 8 h 3S 8 h j^2 m 8^ 46 9.89387 9.89782 9.90168 9.90546 9.90916 9.91277 9.91631 9.91976 4 .89394 .89789 .90175 .90552 .90922 .91283 .91637 .91982 8 .89400 .89795 .90181 .90559 .90928 .91289 .91643 .91988 12 .89407 .89802 .90187 .90565 .90934 .91295 .91648 .91993 16 .89414 .89808 .90194 .90571 .90940 .91301 .91654 .91999 20 9.89421 9.89815 9.90200 9.90577 9.90946 9.91307 9.91660 9.92005 24 .89427 .89821 .90206 .90584 .90952 .91313 .91666 .92010 28 .89434 .89828 .90213 .90590 .90958 .91319 .91672 .92016 32 .89441 .89834 .90219 .90596 .90965 .91325 .91677 .92022 36 .89447 .89840 .90225 .90602 .90971 .91331 .91683 .92027 40 9.89454 9.89847 9.90232 9.90608 9.90977 9.91337 9.91689 9.92033 44 .89460 .89853 .90238 .90615 .90983 .91343 .91695 .92039 48 .89467 .89860 .90244 .90621 .90989 .91349 .91701 .92044 62 .89474 .89866 .90251 .90627 .90995 .91355 .91706 .92050 56 .89480 .89873 .90257 .90633 .91001 .91361 .91712 .92056 8 gh ICfm g h 23 m 8 h 27 m 8>> 31 m 8 h S5 m 8* 39 m 8* 43 8* 47 m 9.89487 9.89879 9.90264 9.90639 9.91007 9.91367 9.91718 9.92061 4 .89493 .89886 .90270 .90646 .91013 .91372 .91724 .92067 8 .89500 .89n Qh 36 m Qh SQm Qh 43 m 9* 47 m 9h 51 9.94802 9.95069 9.95328 9.95579 9.95824 9.96061 9.96291 9.96514 4 .94806 .95073 .95332 .95584 .95828 .96065 .96295 .96517 8 .94811 .95077 .95336 .95588 .95832 .96069 .96299 .96521 12 .94815 .95082 .95340 .95592 .95836 .96073 .96302 .96525 16 .94820 .95086 .95345 .95596 .95840 .96077 .96306 .96528 20 9.94824 9.95090 9.95349 9.95600 9.95844 9.96081 9.96310 9.96532 24 .94829 .95095 .95353 .95604 .95848 .96084 .96314 .96536 28 .94833 .95099 .95357 .95608 .95852 .96088 .96317 .96539 32 .94838 .95104 .95362 .95613 .95856 .96092 .96321 .96543 36 .94842 .95108 .95366 .95617 .95860 .96096 .96325 .96547 40 9.94847 9.95112 9.95370 9.95621 9.95864 9.96100 9.96329 9.96550 44 .94851 .95117 .95374 .95625 .95868 .96104 .96332 .96554 48 .94856 .95121 .95379 .95629 .95872 .96108 .96336 .96557 52 .94860 .95125 .95383 .95633 .95876 .96112 .96340 .96561 56 .94865 .95130 .95387 .95637 .95880 .96115 .96344 .96565 60 9.94869 9.95134 9.95391 9.95641 9.95884 9.96119 9.96347 9.96568 280 Table 10. Haversine Table s Qh 5% gh ,5gm 10* O m 10* 4 m 10 h 8 m 10 h 12 m IQh 16 10* 20 m Hav. Hav. Hav. Hav. Hav. Hav. Hav. Hav. 9.96568 9.96782 9.96989 9.97188 9.97381 9.97566 9.97745 9.97916 4 .96572 .96786 .96992 .97192 .97384 .97569 .97748 .97919 8 .96576 .96789 .96996 .97195 .97387 .97572 .97751 .97922 12 .96579 .96793 .96999 .97198 .97390 .97575 .97754 .97925 16 .96583 .96796 .97002 .97201 .97393 .97578 .97756 .97927 20 9.96586 9.96800 9.97006 9.97205 9.97397 9.97581 9.97759 9.97930 24 .96590 .96803 .97009 .97208 .97400 .97584 .97762 .97933 28 .96594 .96807 .97012 .97211 .97403 .97587 .97765 .97936 32 .96597 .96810 .97016 .97214 .97406 .97591 .97768 .97939 36 .96601 .96814 .97019 .97218 .97409 .97594 .97771 .97941 40 9.96604 9.96817 9.97022 9.97221 9.97412 9.97597 9.97774 9.97944 44 .96608 .96821 .97026 .97224 .97415 .97600 .97777 .97947 48 .96612 .96824 .97029 .97227 .97418 .97603 .97780 .97950 52 .96615 .96827 .97033 .97231 .97422 .97606 .97783 .97953 56 .96619 .96831 .97036 .97234 .97425 .97609 .97785 .97955 s Qh ggm Qh Q^m 10 h l m 10 h 5 m 10* 9 m 10* 13 m 10* 17 m 10* 21 m 9.96622 9.96834 9.97039 9.97237 9.97428 9.97612 9.97788 9.97958 4 .96626 .96837 .97043 .97240 .97431 .97615 .97791 .97961 8 .96630 .96841 .97046 .97244 .97434 .97618 .97794 .97964 12 .96633 .96845 .97049 .97247 .97437 .97621 .97797 .97966 16 .96637 .96848 .97052 .97250 .97440 .97624 .97800 .97969 20 9.96640 9.96852 9.97056 9.97253 9.97443 9.97627 9.97803 9.97972 24 .96644 .96855 .97059 .97257 .97447 .97630 .97806 .97975 28 .96648 .96859 .97063 .97260 .97450 .97633 .97808 .97977 32 .96651 .96862 .97066 .97263 .97453 .97636 .97811 .97980 36 .96655 .96866 .97069 .97266 .97456 .97639 .97814 .97983 40 9.96658 9.96869 9.97073 9.97269 9.97459 9.97642 9.97817 9.97986 44 .96662 .96873 .97076 .97273 .97462 .97645 .97820 .97988 48 .96665 .96876 .97079 .97276 .97465 .97647 .97823 .97991 62 .96669 .96879 .97083 .97279 .97468 .97650 .97826 .97994 56 .96673 .96883 .97086 .97282 .97471 .97653 .97829 .97997 s Qh Qjm Qh Q 8 m 10* 2 10* 6 10* 10 m 10 h 14 m 10* 18 m 10* 22 m 9.96676 9.96886 9.97089 9.97285 9.97474 9.97656 9.97831 9.97999 4 .96680 .96890 .97093 .97289 .94478 .97659 .97834 .98002 8 .96683 .96894 .97096 .96292 .97481 .97662 .97837 .98005 12 .96687 .96897 .97099 .97295 .97484 .97665 .97840 .98008 16 .96690 .96900 .97103 .97298 .97487 .97668 .97843 .98010 20 9.96694 9.96904 9.97106 9.97301 9.97490 9.97671 9.97846 9.98013 24 .96697 .96907 .97109 .97305 .97493 .97674 .97849 .98016 28 .96701 .96910 .97113 .97308 .97496 .97677 .97851 .98019 32 .96705 .96914 .97116 .97311 .97499 .97680 .97854 .98021 36 .96708 .96917 .97119 .97314 .97502 .97683 .97857 .98024 40 9.96712 9.96921 9.97123 9.97317 9.97505 9.97686 9.97860 9.98027 44 .96715 .96924 .97126 .97321 .97508 .97689 .97863 .98030 48 .96719 .96928 .97129 .97324 .97511 .97692 .97866 .98032 52 .96722 .96931 .97132 .97327 .97514 .97695 .97868 .98035 56 .96726 .96934 .97136 .97330 .97518 .97698 .97871 .98038 s gh 55 m Qh 59 m 10 h 3 m 10* 7 m 10 h ll m 10 h 15 m 10* 19 m 10* 23 m 9.96729 9.96938 9.97139 9.97333 9.97521 9.97701 9.97874 9.98040 4 .96733 .96941 .97142 .97337 .97524 .97704 .97877 .98043 8 .96736 .96945 .97146 .97340 .97527 .97707 .97880 .98046 12 .96740 .96948 .97149 .97343 .97530 .97710 .97883 .98049 16 .96743 .96951 .97152 .97346 .97533 .97713 .97885 .98051 20 9.96747 9.96955 9.97156 9.97349 9.97536 9.97716 9.97888 9.98054 24 .96750 .96958 .97159 .97352 .97539 .97718 .97891 .98057 28 .96754 .96962 .97162 .97356 .97542 .97721 .97894 .98059 32 .96758 .96965 .97165 .97359 .97545 .97724 .97897 .98062 36 .98761 .96968 .97169 .97362 .97548 .97727 .97899 .98065 40 9.96765 9.96972 9.97172 9.97365 9.97551 9.97730 9.97902 9.98067 44 .96768 .96975 .97175 .97368 .97554 .97733 .97905 .98070 48 .96772 .96979 .97179 .97371 .97557 .97736 .97908 .98073 52 .96775 .96982 .97182 .97375 .97560 .97739 .97911 .98076 56 .96779 .96985 .97185 .97378 .97563 .97742 .97914 .98078 60 9.96782 9.96989 9.97188 9.97381 9.97566 9.97745 9.97916 9.98081 Table 10. Haversiiie Table 281 $ 10* 24 m 10* 28 m 10*32 10*36 m 10* 40 m 10* 44 m 10* 48 m 10* 52 Hav. Hav. Hav. Hav. Hav. Hav. Hav. Hav. 9.98081 9.98239 9.98389 9.98533 9.98670 9.98801 9.98924 9.99041 4 .98084 .98241 .98392 .98536 .98673 .98803 .98926 .99043 8 .98086 .98244 .98394 .98538 .98675 .98805 .98928 .99044 12 .98089 .98246 .98397 .98540 .98677 .98807 .98930 .99046 16 .98092 .98249 .98399 .98543 .98679 .98809 \98932 .99048 20 9.98094 9.98251 9.98402 9.98545 9.98681 9.98811 9.98934 9.99050 24 .98097 .98254 .98404 .98547 .98684 .98813 .98936 .99052 28 .98100 .98256 .98406 .98550 .98686 .98815 .98938 .99054 32 .98102 .98259 .98409 .98552 .98688 .98817 .98940 .99056 36 .98105 .98262 .98411 .98554 .98690 .98819 .98942 .99058 40 9.98108 9.98364 9.98414 9.98557 9.98692 9.98822 9.98944 9.99059 44 .98110 .98267 .98416 .98559 .98695 .98824 .98946 .99061 48 .98113 .98269 .98419 .98561 .98697 .98826 .98948 .99063 s* .98116 .98272 .98421 .98564 .98699 .98828 .98950 .99065 56 .98118 .98274 .98424 .98566 .98701 .98830 .98952 .99067 8 10 h 25 m 10 h 29 m 10* 33 m 10* 37 m 10* 41 " l 10* 45 m 10* 49 m 10* 53 m 9.98121 9.98277 9.98426 9.98568 9.98703 9.98832 9.98954 9.99069 4 .98124 .98279 .98428 .98570 .98706 .98834 .98956 .99071 8 .98126 .98282 .98431 .98573 .98708 .98836 .98958 .99072 12 .98129 .98285 .98433 .98575 .98710 .98838 .98960 .99074 16 .98132 .98287 .98436 .98577 .98712 .98840 .98962 .99076 20 9.98134 9.98290 9.98438 9.98580 9.98714 9.98842 9.98964 9.99078 24 .98137 .98292 .98440 .98582 .98717 .98845 .98966 .99080 28 .98139 .98295 .98443 .98584 .98719 .98847 .98968 .99082 32 .98142 .98297 .98445 .98587 .98721 .98849 .98970 .99084 36 .98145 .98300 .98448 .98589 .98723 .98851 .98971 .99085 40 9.98147 9.98302 9.98450 9.98591 9.98725 9.98853 9.98973 9.99087 44 .98150 .98305 .98453 .98593 .98728 .98855 .98975 .99089 48 .98153 .98307 .98455 .98596 .98730 .98857 .98977 .99091 52 .98155 .98310 .98457 .98598 .98732 .98859 .98979 .99093 56 .98158 .98312 .98460 .98600 .98734 .98861 .98981 .99095 s 10* 26 m 10* 30 m 10* 34 m 10* 38 10* 42 m 10* 46 m 10* 50 m 10* 54 m 9.98161 9.98315 9.98462 9.98603 9.98736 9.98863 9.98983 9.99096 4 .98163 .98317 .98465 .98605 .98738 .98865 .98985 .99098 8 .98166 .98320 .98467 .98607 .98741 .98867 .98987 .99100 12 .98168 .98322 .98469 .98609 .98743 .98869 .98989 .99102 16 .98171 .98325 .98472 .98612 .98745 .98871 .98991 .99104 20 9.98174 9.98327 9.98474 9.98614 9.98747 9.98873 9.98993 9.99106 24 .98176 .98330 .98476 .98616 .98749 .98875 .98995 .99107 28 .98179 .98332 .98479 .98619 .98751 .98877 .98997 .99109 32 .98182 .98335 .98481 .98621 .98754 .98880 .98999 .99111 36 .98184 .98337 .98484 .98623 .98756 .98882 .99001 .99113 40 9.98187 9.98340 9.98486 9.98625 9.98758 9.98884 9.99003 9.99115 44 .98189 .98342 .98488 .98628 .98760 .98886 .99004 .99116 48 .98192 .98345 .98491 .98630 .98762 .98888 .99006 .99118 52 .98195 .98347 .98493 .98632 .98764 .98890 .99008 .99120 56 .98197 .98350 .98496 .98634 .98766 .98892 .99010 .99122 s 10*27 m 10*31 10* 35 m 10* 39 m 10* 43 m 10* 47 m 10*51 m 10* 55 m 9.98200 9.98352 9.98498 9.98637 9.98769 9.98894 9.99012 9.99124 4 .98202 .98355 .98500 .98639 .98771 .98896 .99014 .99126 8 .98205 .98357 .98503 .98641 .98773 .98898 .99016 .99127 12 .98208 .98360 .98505 .98643 .98775 .98900 .99018 .99129 16 .98210 .98362 .98507 .98646 .98777 .98902 .99020 .99131 20 9.98213 9.98365 9.98510 9.98648 9.98779 9.98904 9.99022 9.99133 24 .98215 .98367 .98512 .98650 .98781 .98906 .99024 .99135 28 .98218 .98370 .98514 .98652 .98784 .98908 .99026 .99136 32 .98221 .98372 .98517 .98655 .98786 .98910 .99027 .99138 36 .98223 .98375 .98519 .98657 .98788 .98912 .99029 .99140 40 9.98226 9.98377 9.98521 9.98659 9.98790 9.98914 9.99031 9.99142 44 .98228 .98379 .98524 .98661 .98792 .98916 .99033 .99143 48 .98231 .98382 .98526 .98664 .98794 .98918 .99035 .99145 62 .98233 .98384 .98529 .98666 .98796 .98920 .99037 .99147 56 .98236 .98387 .98531 .98668 .98798 .98922 .99039 .99149 60 9.98239 9.98389 9.98533 9.98670 9.98801 9.98924 9.99041 9.99151 282 Table 10. Haversine Table s 10* 56 Ilh0 m llh jrn llh #m 11* 12 m llh Iftn Uh 20 m llh 24 m Hav. Hav. Hav. Hav. Hav. Hav. Hav. Hav. 9.99151 9.99254 9.99350 9.99440 9.99523 9.99599 9.99669 9.99732 4 .99152 .99255 .99352 .99441 .99524 .99600 .99670 .99733 8 .99154 .99257 .99353 .99443 .99526 .99602 .99671 .99734 12 .99156 .99259 .99355 .99444 .99527 .99603 .99672 .99735 16 .99158 .99260 .99356 .99446 .99528 .99604 .99673 .99736 20 9.99159 9.99262 9.99358 9.99447 9.99529 9.99605 9.99674 9.99737 24 .99161 .99264 .99359 .99448 .99531 .99606 .99675 .99738 28 .99163 .99265 .99361 .99450 .99532 .99608 .99677 .99739 32 .99165 .99267 .99362 .99451 .99533 .99609 .99678 .99740 36 .99166 .99269 .99364 .99453 .99535 .99610 .99679 .99741 40 9.99168 9.99270 9.99366 9.99454 9.99536 9.99611 9.99680 9.99742 44 .99170 .99272 .99367 .99456 .99537 .99612 .99681 .99743 48 .99172 .99274 .99369 .99457 .99539 .99614 .99682 .99744 52 .99173 .99275 .99370 .99458 .99540 .99615 .99683 .99745 56 .99175 .99277 .99372 .99460 .99541 .99616 .99684 .99746 s IQh Qfm llh im 11* 5 m 11^ 9^ Hh igm ll h 17 m Ilh21 m Ilh25 m 9.99177 9.99278 9.99373 9.99461 9.99543 9.99617 9.99685 9.99747 4 .99179 .99280 .99375 .99463 .99544 .99618 .99686 .99748 8 .99180 .99282 .99376 .99464 .99545 .99620 .99687 .99748 12 .99182 .99283 .99378 .99465 .99546 .99621 .99688 .99749 16 .99184 .99285 .99379 .99467 .99548 .99622 .99690 .99750 20 9.99186 9.99287 9.99381 9.99468 9.99549 9.99623 9.99691 9.99751 24 .99187 .99288 .99382 .99470 .99550 .99624 .99692 .99752 28 .99189 .99290 .99384 .99471 .99552 .99626 .99693 .99753 32 .99191 .99291 .99385 .99472 .99553 .99627 .99694 .99754 36 .99193 .99293 .99387 .99474 .99554 .99628 .99695 .99755 40 9.99194 9.99295 9.99388 9.99475 9.99555 9.99629 9.99696 9.99756 44 .99196 .99296 .99390 .99477 .99557 .99630 .99697 .99757 48 .99198 .99298 .99391 .99478 .99558 .99631 .99698 .99758 52 .99200 .99300 .99393 .99479 .99559 .99633 .99699 .99759 56 .99201 .99301 .99394 .99481 .99561 .99634 .99700 .99760 s IQh 58 ll h 2 Hh Qm nh i m llh Uf* nh I 8 m llh 22 m Uh 26 9.99203 9.99303 9.99396 9.99482 9.99562 9.99635 9.99701 9.99761 4 .99205 .99304 .99397 .99484 .99563 .99636 .99702 .99762 8 .99206 .99306 .99399 .99485 .99564 .99637 .99703 .99763 12 .99208 .99308 .99400 .99486 .99566 .99638 .99704 .99764 16 .99210 .99309 .99402 .99488 .99567 .99639 .99705 .99765 20 9.99212 9.99311 9.99403 9.99489 9.99568 9.99641 9.99706 9.99766 24 .99213 .99312 .99405 .99490 .99569 .99642 .99707 .99766 28 ' .99215 .99314 .99406 .99492 .99571 .99643 .99708 .99767 32 .99217 .99316 .99408 .99493 .99572 .99644 .99710 .99768 36 .99218 .99317 .99409 .99495 .99573 .99645 .99711 .99769 40 9.99220 9.99319 9.99411 9.99496 9.99575 9.99646 9.99712 9.99770 44 .99222 .99320 .99412 .99497 .99576 .99648 .99713 .99771 48 .99223 .99322 .99414 .99499 .99577 .99649 .99714 .99772 52 .99225 .99324 .99415 .99500 .99578 .99650 .99715 .99773 56 .99227 .99325 .99417 .99501 .99580 .99651 .99716 .99774 s 10h 59 m 11^ sm Hh 7 llh jjm Hh IQm U h igm llh 23 m nh 27 m 9.99229 9.99327 9.99418 9.99503 9.99581 9.99652 9.99717 9.99774 4 .99230 .99328 .99420 .99504 .99582 .99653 .99718 .99775 8 .99232 .99330 .99421 .99505 .99583 .99654 .99719 .99776 12 .99234 .99331 .99422 .99507 .99584 .99655 .99720 .99777 16 .99235 .99333 .99424 .99508 .99586 .99657 .99721 .99778 20 9.99237 9.99335 9.99425 9.99510 9.99587 9.99658 9.99722 9.99779 24 .99239 .99336 .99427 .99511 .99588 .99659 .99723 .99780 28 .99240 .99338 .99429 .99512 .99589 .99660 .99724 .99781 32 .99242 .99339 .99430 .99514 .99591 .99661 .99725 .99782 36 .99244 .99341 .99431 .99515 .99592 .99662 .99726 .99783 40 9.99245 9.99342 9.99433 9.99516 9.99593 9.99663 9.99727 9.99784 44 .99247 .99344 .99434 .99518 .99594 .99664 .99728 .99785 48 .99249 .99345 .99436 .99519 .99596 .99666 .99729 .99786 52 .99250 .99347 .99437 .99520 .99597 .99667 .99730 .99786 56 .99252 .99349 .99438 .99522 .99598 .99668 .99731 .99787 60 9.99254 9.99350 9.99440 9.99523 9.99599 9.99669 9.99732 9.99788 Table 10. Haversine Table 283 s 11* 28 ll h 32 11* 36 11* 40 m 11* U m H h 48 ll*52 m 11*56 Hav. Hav. Hav. Hav. Hav. Hav. Hav. Hav. 9.99788 9.99S3S 9.99881 9.99917 9.99947 9.99970 9.99987 9.99997 4 .99789 .99839 .99882 .99918 .99948 .99971 .99987 .99997 8 .99790 .99839 .99882 .99918 .99948 .99971 .99987 .99997 12 .99791 .99840 .99883 .99919 .99948 .99971 .99987 .99997 16 .99792 .99841 .99884 .99919 .99949 .99972 .99988 .99997 20 9.99793 9.99842 9.99884 9.99920 9.99949 9.99972 9.99988 9.99997 24 .99793 .99842 .99885 .99921 .99950 .99972 .99988 .99997 28 .99794 .99843 .99885 .99921 .99950 .99973 .99988 .99997 32 .99795 .99844 .99886 .99922 .99951 .99973 .99988 .99998 36 .99796 .99845 .99887 .99922 .99951 .99973 .99989 .99998 40 9.99797 9.99845 9.99887 9.99923 9.99951 9.99973 9.99989 9.99998 44 .99798 .99846 .99888 .99923 .99952 .99974 .99989 .99998 48 .99799 .99847 .99889 .99924 .99952 .99974 .99989 .99998 52 .99800 .99848 .99889 .99924 .99953 .99974 .99989 .99998 56 .99800 .99848 .99890 .99925 .99953 .99975 .99990 .99998 s llh 29 m 11* 33 m 11* 37 m 11* 41 m 11* 45 m 11* 49 m 11* 53 11* 57 9.99801 9.99849 9.99891 9.99925 9.99953 9.99975 9.99990 9.99998 4 .99802 .99850 .99891 .99926 .99954 .99975 .99990 .99998 8 .99803 .99851 .99892 .99926 .99954 .99976 .99990 .99998 12 .99804 .99851 .99893 .99927 .99954 .99976 .99990 .99998 16 .99805 .99852 .99893 .99927 .99955 .99976 .99991 .99998 20 9.99805 9.99853 9.99894 9.99928 9.99955 9.99976 9.99991 9.99999 24 .99806 .99854 .99894 .99928 .99956 .99977 .99991 .99999 28 .99807 .99854 .99895 .99929 .99956 .99977 .99991 .99999 32 .99808 .99855 .99896 .99929 .99957 .99977 .99991 .99999 36 .99809 .99856 .99896 .99930 .99957 .99978 .99992 .99999 40 9.99810 9.99857 9.99897 9.99931 9.99958 9.99978 9.99992 9.99999 44 .99811 .99857 .99897 .99931 .99958 .99978 .99992 .99999 48 .99811 ,99858 .99898 .99932 .99958 .99978 .99992 .99999 52 .99812 .99859 .99899 .99932 .99959 .99979 .99992 .99999 66 .99813 .99859 .99899 .99933 .99959 .99979 .99992 .99999 8 llh SO llh $4 11* 38 m 11* 42 11* 46 m 11*50 11* 54 m 11* 58 m 9.99814 9.99860 9.99900 9.99933 9.99959 9.99979 9.99993 9.99999 4 .99815 .99861 .99901 .99934 .99960 .99980 .99993 .99999 8 .99815 .99862 .99901 .99934 .99960 .99980 .99993 .99999 12 .99816 .99862 .99902 .99935 .99961 .99980 .99993 .99999 16 .99817 .99863 .99902 .99935 .99961 .99980 .99993 .99999 20 9.99818 9.99864 9.99903 9.99935 9.99961 9.99981 9.99993 9.99999 24 .99819 .99864 .99904 .99936 .99962 .99981 .99994 .99999 28 .99820 .99865 .99904 .99936 .99962 .99981 .99994 .00000 32 .99820 .99866 .99905 .99937 .99963 .99981 .99994 .00000 36 .99821 .99867 .99905 .99937 .99963 .99982 .99994 .00000 40 9.99822 9.99867 9.99906 9.99938 9.99963 9.99982 9.99994 0.00000 44 .99823 .99868 .99906 .99938 .99964 .99982 .99994 .00000 48 .99824 .99869 .99907 .99939 .99964 .99983 .99994 .00000 52 .99824 .99869 .99908 .99939 .99964 .99983 .99995 .00000 56 .99825 .99870 .99908 .99940 .99965 .99983 .99995 .00000 s llh sim nhsgm 11*39 11* 43 m 11* 47 m 11* 51 m 11* 55 11* 5y n 9.99826 9.99871 9.99909 9.99940 9.99965 9.99983 9.99995 0.00000 4 .99827 .99871 .99909 .99941 .99965 .99983 .99995 .00000 8 .99828 .99872 .99910 .99941 .99966 .99984 .99995 .00000 12 .99828 .99873 .99911 .99942 .99966 .99984 .99995 .00000 16 .99829 .99874 .99911 .99942 .99966 .99984 .99995 .00000 20 9.99830 9.99874 9.99912 9.99943 9.99967 9.99984 9.99996 0.00000 24 .99831 .99875 .99912 .99943 .99967 .99985 .99996 .00000 28 .99832 .99876 .99913 .99943 .99968 .99985 .99996 .00000 32 .99832 .99876 .99913 .99944 .99968 .99985 .99996 .00000 36 .99833 .99877 .99914 .99944 .99968 .99985 .99996 .00000 40 9.99834 9.99878 9.99915 9.99945 9.99969 9.99986 9.99996 0.00000 44 .99835 .99878 .99915 .99945 .99969 .99986 .99996 .00000 48 .99836 .99879 .99916 .99946 .99969 .99986 .99996 .00000 52 .99836 .99880 .99916 .99946 .99970 .99986 .99996 .00000 56 .99837 .99880 .99917 .99947 .99970 .99987 .99997 .00000 60 9.99838 9.99881 9.99917 9.99947 9.99970 9.99987 9.99997 0.00000 284 Table 11. Azimuth T, THE SHIP'S APPARENT TIME FOR A SUN OBSERVATION, OR THE HOUR-ANGLE FOR A STAR OBSERVATION USE USE THESE 12 h 0m 12 h s m 12*16 12 h 24 m 12*32 12 h 40 m 12* 48 m 12* 56 m THESE IN > ^ i \ FORE- 24 28 52 23 44 23 36 23 28 23 20 23 12 23 4 FORE- NOON NOON USE USE THESE Oh Q m Qh gm O h 16 m 0*24 QhS2 m 0*40 0*48 0*56 THESE IN > < IN AFTER- 12 n 52 11 44 11 36 11 28 11 20 11 12 11 4 AFTER- NOON NOON 349 698 1045 1392 1737 2079 2419 Oo 2 349 697 1045 1391 1736 2078 2417 2 4 348 696 1042 1389 1732 2074 2413 4 6 347 694 1040 1384 1726 2067 2406 6 8 346 691 1035 1378 1720 2059 2395 8 10 344 687 1029 1371 1710 2047 2382 10 12 341 682 1022 1361 1698 2033 2367 12 14 339 677 1015 1351 1685 2018 2347 14 16 336 671 1005 1338 1669 1998 2326 16 18 332 663 994 1323 1651 1977 2301 18 20 328 656 982 1308 1632 1954 2274 20 22 324 647 969 1290 1610 1928 2244 22 24 319 637 955 1272 1586 1900 2210 24 26 314 627 940 1251 1561 1868 2174 26 28 308 616 923 1228 1533 1835 2136 28 30 302 604 905 1205 1504 1801 2095 30 32 296 592 886 1180 1472 1763 2051 32 34 289 578 867 1153 1440 1724 2005 34 36 282 564 846 1126 1405 1682 1957 36 01 38 275 550 824 1096 1369 1639 1906 38 X o 40 267 534 801 1066 1330 1592 1853 40 03 W g 42 259 518 777 1034 1290 1545 1798 42 Q p 44 251 502 752 1001 1249 1496 1740 44 H 1 46 242 485 726 967 1206 1444 1681 46 i m 48 234 467 699 931 1162 1391 1619 48 FORE- 22 66 22 48 22 40 22 32 22 24 22 16 22 8 < IN FORE- NOON NOON USE USE THESE Jh jrn 1*12 l*20 m 1*28 1*36 l*44 m 1*52 THESE IN ^ < IN AFTER- 10 56 10 48 10 40 10 32 10 24 10 16 10 8 AFTER- NOON NOON 2756 3090 3421 3746 4067 4383 4695 2 2755 3088 3418 3744 4065 4381 4692 2 4 2750 3082 3412 3737 4058 4373 4684 4 6 2742 3073 3402 3726 4044 4360 4669 6 8 2730 3060 3387 3710 4028 4341 4649 8 10 2714 3043 3368 3689 4005 4317 4624 10 12 2696 3023 3346 3664 3978 4287 4592 12 14 2674 2998 3319 3635 3947 4253 4555 14 16 2650 2970 3288 3600 3910 4214 4513 16 18 2622 2939 3253 3563 3868 4170 4465 18 20 2590 2903 3214 3521 3821 4119 4412 20 22 2556 2865 3171 3473 3771 4064 4353 22 24 2519 2823 3125 3422 3715 4005 4288 24 26 2477 2777 3074 3367 3656 3941 4220 26 28 2434 2729 3020 3308 3591 3871 4145 28 30 2387 2676 2962 3244 3522 3796 4065 30 32 2337 2620 2900 3177 3449 3718 3981 32 34 2285 2563 2835 3106 3372 3634 3892 34 36 2230 2500 2767 3031 3291 3546 3798 36 00 38 2172 2435 2696 2952 3205 3454 3699 38 K O 40 2112 2367 2620 2869 3116 3358 3596 40 H 42 2048 2297 2542 2784 3023 3258 3489 42 Q 44 1983 2223 2460 2695 2925 3154 3377 44 i 46 1914 2147 2375 2602 2826 3045 3261 46 a 48 1845 2067 2289 2507 2721 2934 3142 48 <3 Q 50 1772 1986 2199 2408 2614 2818 3018 50 52 1697 1902 2106 2306 2504 2699 2891 52 54 1620 1818 2010 2202 2391 2577 2759 54 56 1541 1728 1913 2094 2275 2451 2625 56 58 1461 1638 1812 1986 2155 2324 2488 58 60 1378 1545 1710 1874 2033 2192 2347 60 62 1294 1451 1606 1759 1909 2058 2204 62 64 1209 1355 1499 1643 1783 1922 2058 64 66 1121 1257 1391 1524 1654 1783 1909 66 68 1032 1158 1281 1404 1524 1643 1759 68 70 943 1057 1169 1281 1391 1499 1606 70 72 852 955 1057 1158 1257 1355 1451 72 74 760 852 943 1032 1121 1209 1294 74 76 667 748 827 906 984 1060 1136 76 78 573 643 711 779 846 911 976 78 80 479 537 594 651 706 761 815 80 82 384 430 476 521 566 610 653 82 84 288 323 358 392 425 458 491 84 86 192 216 239 261 290 306 328 86 88 96 108 119 131 142 153 164 88 USE USE THESE 16 18 20 22 24 26 28 THESE IN > ^ IN FORE- 164 162 160 158 156 154 152 FORE- NOON NOON USE USE THESE 196 198 200 202 204 206 208 THESE IN > ^ IN AFTER- 344 342 340 338 336 334 332 AFTER- NOON NOON TRUE BEARING OR AZIMUTH 286 Table 11. Azimuth T, THE SHIP'S APPARENT TIME FOR A SUN OBSERVATION, OR THE HOUR-ANGLE FOR A STAR OBSERVATION USE USE THESE 14* O m 14* 8 Up 16 14*S4 m 14*32 14 h 40 m 14*48" 14*56 THESE IN > < IN FORE- 22 21 52 21 44 21 36 21 28 21 20 21 12 21 4 FORE- NOON NOON USE USE THESE 2 h O m 2 h s 2*16 2*24 2*32 2*40 2*48 2*56 THESE IN ^ ^ IN AFTER- 10 9 52 9 44 9 36 9 28 9 20 9 12 9 4 AFTER- NOON NOON 5000 5299 5593 5878 6156 6428 6691 6947 Qo 2 4997 5297 5589 5875 6153 6424 6688 6942 2 4 4987 5286 5578 5864 6142 6412 6676 6929 4 6 4973 5270 5562 5845 6124 6393 6655 6908 6 8 4951 5248 5538 5821 6096 6365 6627 6879 8 10 4923 5219 5507 5789 6063 6330 6591 6841 10 12 4891 5183 5470 5749 6022 6288 6545 6795 12 14 4852 5141 5426 5703 5973 6237 6492 6741 14 16 4806 5094 5375 5650 5919 6179 6433 6677 16 18 4755 5040 5319 5590 5856 6113 6363 6607 18 20 4699 4979 5255 5524 5785 6040 6288 6528 20 22 4635 4914 5184 5450 5709 5960 6204 6440 22 24 4567 4841 5109 5370 5624 5872 6112 6346 24 26 4493 4763 5025 5283 5534 5777 6015 6243 26 28 4415 4678 4938 5190 5437 5675 5907 6134 28 30 4330 4588 4843 5091 5332 5567 5794 6016 30 32 4240 4493 4742 4984 5222 5451 5674 5891 32 34' 4145 4393 4635 4873 5104 5328 5547 5758 34 36 4044 4287 4524 4755 4980 5200 5414 5619 36 w 38 3941 4176 4407 4633 4852 5065 5272 5474 38 * o 40 3830 4060 4284 4503 4717 4923 5126 5321 40 1 1 42 3715 3939 4156 4368 4575 4776 4973 5162 42 Q I 44 3596 3812 4023 4229 4429 4624 4812 4997 44 E 3 46 3473 3681 3885 4083 4277 4465 4648 4825 46 1 48 3346 3546 3742 3932 4120 4301 4477 4648 48 3 Q 50 3214 3406 3594 3779 3958 4131 4301 4465 50 52 3078 3263 3443 3619 3790 3958 4120 4277 52 54 2939 3115 3287 3454 3619 3779 3932 4083 54 56 2796 2963 3127 3287 3443 3594 3742 3885 56 58 2650 2808 2963 3115 3263 3406 3546 3681 58 60 2500 2650 2796 2939 3078 3214 3346 3473 60 62 2347 2488 2625 2760 2891 3018 3142 3261 62 64 2192 2324 2451 2577 2699 2818 2934 3045 64 66 2033 2155 2275 2391 2504 2614 2721 2826 66 68 1874 1986 2094 2202 2306 2408 2507 2602 68 70 1710 1812 1913 2010 2106 2199 2289 2375 70 72 1545 1638 1728 1817 1902 1986 2067 2147 72 74 1378 1461 1541 1620 1697 1720 1845 1914 74 76 1210 1282 1353 1422 1489 1555 1619 1681 76 78 1040 1102 1162 1222 1280 1337 1391 1444 78 80 868 920 971 1021 1069 1116 1162 1206 80 82 696 738 778 818 857 895 931 967 82 84 523 554 585 615 644 672 699 726 84 86 349 370 390 411 429 448 467 485 86 88 175 185 195 205 215 224 234 242 88 USE USE THESE 30 32 34 36 38 40 42 44 THESE IN ^ ^ IN FORE- 150 148 146 144 142 140 138 136 FORE- NOON NOON USE USE THESE 210 212 214 216 218 220 222 224 THESE IN ^ < IN AFTER- 330 328 326 324 322 320 318 316 AFTER- NOON NOON TRUE BEARING OR AZIMUTH Table 11. Azimuth 287 T, THE SHIP'S APPARENT TIME FOR A SUN OBSERVATION, OR THE HOUR-ANGLE FOR A STAR OBSERVATION USE USE THESE 15* 4 m 15* 12 15* 20 15* 28 m 15*36 15* 44 15*52 THESE IN ^ < IN FORE- 20 56 20 48 20 40 20 32 20 24 20 16 20 8 FORE- NOON NOON USE USE THESE 3 h 4 m 3*12 3*20 3*28 3*36 3*44 m 3*52 THESE IN ^ < IN AFTER- 8 56 8 48 8 40 8 32 8 24 8 16 8 8 AFTER- NOON NOON 7193 7432 7661 7879 8091 8290 8480 2 7190 7427 7656 7875 8085 8285 8476 2 4 7176 7413 7642 7861 8071 8269 8461 4 6 7153 7391 7619 7836 8046 8245 8433 6 8 7124 7358 7586 7803 8011 8210 8399 8 10 7084 7318 7544 7761 7968 8164 8352 10 12 7036 7269 7494 7707 7914 8110 8284 12 14 6979 7211 7433 7645 7850 8044 8228 14 16 6915 7144 7364 7575 7776 7969 8153 16 18 6841 7068 7286 7494 7695 7884 8065 18 20 6759 6984 7197 7404 7603 7791 7969 20 22 6670 6890 7103 7307 7501 7686 7863 22 24 6572 6789 6998 7199 7391 7573 7749 24 26 6466 6679 6885 7082 7271 7450 7623 26 28 6352 6561 6764 6958 7144 7319 7489 28 30 6230 6436 6634 6825 7006 7180 7345 30 32 6101 6302 6497 6683 6861 7031 7191 32 34 5964 6160 6351 6533 6707 6873 7031 34 36 5820 6012 6197 6375 6545 6707 6861 36 1C 38 5669 5856 6037 6210 6375 6533 6683 38 fc 40 5511 5693 5868 6037 6197 6351 6497 40 p 42 5346 5522 5693 5856 6012 6160 6302 42 1 < fc 44 5175 5346 5511 5669 5820 5964 6101 44 ^ 1 16 4997 5162 5321 5474 5619 5758 5891 46 H D w 48 4812 4973 5126 5272 5414 5547 5674 48 < Q 50 4624 4776 4923 5065 5200 5328 5451 50 52 4429 4575 4717 4852 4980 5104 5222 52 54 4229 4368 4503 4633 4755 4873 4984 54 56 4023 4156 4284 4407 4524 4635 4742 56 58 3812 3939 4060 4176 4287 4393 4493 58 60 3596 3715 3830 3941 4044 4145 4240 60 62 3378 3489 3596 3700 3798 3892 3981 62 64 3154 3258 3358 3454 3546 3634 3718 64 66 2925 3023 3116 3205 3291 3372 3449 66 68 2695 2784 2869 2952 3031 3106 3177 68 70 2460 2542 2620 2696 2767 2835 2900 70 72 2223 2297 2367 2435 2500 2563 2620 72 74 1983 2048 2112 2172 2230 2285 2337 74 76 1740 1798 1853 1906 1957 2005 2051 76 78 1496 1545 1592 1639 1682 1724 1763 78 80 1249 1290 1330 1369 1405 1440 1472 80 82 1001 1034 1066 1096 1126 1153 1180 82 84 752 777 801 824 846 867 886 84 86 502 518 534 550 564 578 592 86 88 251 259 267 275 282 289 296 88 USE USE THESE 46 48 50 52 54 56 58 THESE IN > ^'" Jjq- FORE- 134 132 130 128 126 124 122 FORE- NOON NOON USE USE THESE 226 228 230 232 234 236 238 THESE IN > AFTER- 314 312 310 308 306 304 302 < IN AFTER- NOON NOON TRUE BEARING OR AZIMUTH 288 Table 11. Azimuth T, THE SHIP'S APPARENT TIME FOR A SUN OBSERVATION, OR THE HOUR-ANGLE FOR A STAR OBSERVATION USE USE THESE I Oh Q m 16* 8" 16*16 16* 24 m 16*32 16*40 16*48 16*36 THESE IN ^ FORE- 20 19 52 19 44 19 36 19 28 19 20 19 12 19 4 FORE- NOON NOON USE USE THESE IN >- 4* o m 4* 8^ 4* 16 4 h 24 m 4*32 m 4 h 40" 4*48 4*36 THESE -^ ||^ AFTER- 8 7 52 7 44 7 36 7 28 7 20 7 12 7 4 AFTER- NOON NOON 8660 8828 8989 9135 9272 9397 9510 9612 2 8656 8824 8982 9131 9266 9391 9506 9607 2 4 8640 8808 8966 9114 9249 9374 9486 9590 4 6 8612 8780 8939 9084 9221 9346 9458 9561 6 8 8576 8744 8900 9046 9181 9305 9419 9519 8 10 8529 8696 8851 8997 9131 9253 9367 9466 10 12 8470 8636 8792 8935 9069 9191 9303 9401 12 14 8403 8567 8722 8863 8997 9118 9228 3326 14 16 8326 8487 8640 8782 8913 9033 9143 9241 16 18 8235 8397 8549 8688 8818 8937 9044 9143 18 20 8137 8296 8447 8584 8714 8831 8937 9033 20 22 8030 8187 8333 8470 8596 8714 8818 8913 22 24 7912 8067 8212 8347 8470 8584 8688 8782 24 26 7784 7936 8078 8212 8333 8447 8549 8640 26 28 7647 7796 7936 8067 8187 8296 8397 8487 28 30 7501 7647 7784 7912 8030 8137 8235 8326 30 32 7345 7489 7623 7749 7863 7969 8065 8153 32 34 7180 7319 7450 7573 7686 7791 7884 7969 34 36 7006 7144 7271 7391 7501 7603 7695 7776 36 38 6825 6958 7082 7199 7307 7404 7494 7575 38 iz; 40 6634 6764 6885 6998 7103 7197 7286 7364 40 I 42 6436 6561 6679 6789 6890 6984 7068 7144 42 Q -<1 fe 44 6230 6352 6466 6572 6670 6759 6841 6915 44 ^ M 46 6016 6134 6243 6346 6440 6528 6607 6677 46 | H 48 5794 5907 6015 6212 6204 6288 6363 6433 48 < Q 50 5567 5675 5777 5872 5960 6040 6113 6179 50 52 5332 5437 5534 5624 5709 5785 5856 5919 52 54 5091 5190 5283 5370 5450 5524 5590 5650 54 56 4843 4938 5025 5109 5184 5255 5319 5375 56 58 4588 4678 4763 4841 4914 4979 5040 5094 58 60 4330 4415 4493 4567 4635 4699 4755 4806 60 62 4065 4145 4220 4288. 4353 4412 4465 4513 62 64 3^96 3871 3941 4005 4064 4119 4170 4214 64 66 3522 3591 3656 3715 3771 3821 3868 3910 66 68 3244 3308 3367 3422 3473 3521 3563 3600 68 70 2962 3020 3074 3125 3171 3214 3253 3288 70 72 2676 2729 2777 2823 2865 2903 2939 2970 72 74 2387 2434 2477 2519 2556 2590 2622 2650 74 76 2095 2136 2174 2210 2244 2274 2301 2326 76 78 1801 1835 1868 1900 1928 1954 1977 1998 78 80 1504 1533 1561 1586 1610 1632 1651 1669 80 82 1205 1228 1251 1272 1290 1308 1323 1338 82 84 905 923 940 955 969 982 994 1005 84 86 604 616 627 637 647 656 663 671 86 88 302 308 314 319 324 328 332 336 88 USE USE THESE 60 62 64 66 68 70 72 74 THESE IN > < IN FORE- 120 118 116 114 112 110 108 106 FORE- NOON NOON USE USE THESE 240 242 244 246 248 250 252 254 THESE IN > ^ IN AFTER- 300 298 296 294 292 290 288 286 AFTER- NOON NOON TRUE BEARING OR AZIMUTH Table 11. Azimuth 289 T, THE SHIP'S APPARENT TIME FOR A SUN OBSERVATION, OR THE HOUR-ANGLE FOR A STAR OBSERVATION USE USE THESE 17>> ^ 17*12" 17*20" 7*28" 7*36" 7*44 m 17*52" 8* 0" THESE IN ^ FORE- 1856 18 48 18 40 8 32 8 24 8 16 18 8 8 < IN FORE- NOON NOON USE USE THESE 5* 4 m 5*12" 5*20" 5*28" 5*36" 5*44 m 5*52" 6* 0" THESE IN > AFTER- 6 56 6 48 6 40 6 32 6 24 6 16 6 8 6 < IN AFTER- NOON NOON 9703 9781 9849 9904 9945 9974 9993 10000 2 9696 9774 9842 9897 9940 9970 9988 9993 2 4 9679 9757 9824 9879 9922 9951 9970 9974 4 6 9649 9727 9795 9849 9891 9922 9940 9945 6 8 9610 9687 9752 9806 9849 9879 9897 9904 8 10 9557 9634 9699 9752 9795 9824 9842 9849 10 12 9491 9568 9634 9687 9727 9757 9774 9781 12 14 9414 9491 9557 9610 9649 9679 9696 9703 14 16 9326 9401 9466 9519 9561 9590 9607 9612 16 18 9228 9303 9367 9419 9458 9486 9506 9510 18 20 9118 9191 9253 9305 9346 9374 9391 9397 20 22 8997 9069 9131 9181 9221 9249 9266 9272 22 24 8863 8935 8997 9046 9084 9114 9131 9135 24 26 8722 8792 8851 8900 8939 8966 8982 8989 26 28 8567 8636 8696 8744 8780 8808 8824 8828 28 30 8403 8470 8529 8576 8612 8640 8656 8660 30 32 8228 8284 8352 8399 8433 8461 8476 8480 32 34 8044 8110 8164 S210 8245 8269 8285 8290 34 36 7850 7914 7968 8011 8046 8071 8085 8091 36 g 38 7645 7707 7761 7803 7836 7861 7875 7879 38 o 40 7433 7494 7544 7586 7619 7642 7656 7661 40 to 42 7211 7269 7318 7358 7391 7413 7427 7432 42 8 i 44 6979 7036 7084 7124 7153 7176 7190 7193 44 | 1 46 6741 6795 6841 6879 6908 6929 6942 6947 46 5 I 48 6492 6545 6591 6627 6655 6676 6688 6691 48 <5 P 50 6237 6288 6330 6365 6393 6412 6424 6428 50 52 5973 6022 6063 6096 6124 6142 6153 6156 52 54 5703 5749 5789 5821 5845 5864 5875 5878 54 56 5426 5470 5507 5538 5562 5578 5589 5593 56 58 5141 5183 5219 5248 5270 5286 5297 5299 58 60 4852 4891 4923 4951 4973 4987 4997 5000 60 62 4555 4592 4624 4649 4669 4684 4692 4695 62 64 4253 4287 4317 4341 4360 4373 4381 4383 64 66 3947 3978 4005 4028 4044 4058 4065 4067- 66 68 3635 3664 3689 3710 3726 3737 3744 3746 68 70 3319 3346 3368 3387 3402 3412 3418 3421 70 72 2998 3023 3043 3060 3073 3082 3088 3090 72 74 2674 2696 2714 2730 2742 2750 2755 2756 74 76 2347 2367 2382 2395 2406 2413 2417 2419 76 78 2018 2033 2047 2059 2067 2074 2078 2079 78 80 1685 1698 1710 1720 1726 1732 1736 1737 80 82 1351 1361 1371 1378 1384 1389 1391 1392 82 84 1015 1022 1029 1035 1040 1042 1045 1045 84 86 677 682 687 691 694 696 697 698 86 88 339 341 344 346 347 348 349 349 88 USE USE THESE 76 78 80 82 84 86 88 90 THESE IN > < IN FORE- 104 102 100 98 96 94 92 90 FORE- NOON NOON USE USE THESE 256 258 260 262 264 266 268 270 THESE IN > < IN AFTER- 284 282 280 278 276 274 272 270 AFTER- NOON NOON TRUE BEARING OR AZIMUTH 290 Table 12. Auxiliary Azimuth Table LATITUDE DECLINATIONS 2 4 6 8 10 12 14 16 18 20 22 24 2 90 4 30 90 6 20 42 90 8 15 30 49 90 10 12 24 37 53 90 12 10 20 30 42 57 90 14 8 17 26 35 46 59 90 16 7 15 22 30 39 49 61 90 18 . 6 13 20 27 34 42 52 63 90 20 6 12 18 24 31 37 45 54 65 90 22 5 11 16 22 28 34 40 47 56 66 90 24 5 10 15 20 25 31 36 43 49 57 67 90 26 5 9 14 19 23 28 34 39 45 51 59 68 28 4 9 13 17 22 26 31 36 41 47 53 60 30 4 8 12 16 20 25 29 33 38 43 49 54 32 4 8 11 15 19 23 27 31 36 40 45 50 34 4 7 11 14 18 22 26 30 34 38 42 47 36 3 7 10 14 17 21 24 28 32 36 40 44 38 3 7 10 13 16 20 23 27 30 34 37 41 40 3 6 9 12 16 19 22 25 29 32 36 39 42 3 6 9 12 15 18 21 24 28 31 34 37 44 3 6 9 12 14 17 20 23 26 30 33 36 46 3 6 8 11 14 17 20 23 25 28 31 34 48 3 5 8 11 14 16 19 22 25 27 30 33 50 3 5 8 10 13 16 18 21 24 27 29 32 52 3 5 8 10 13 15 18 20 23 26 28 31 54 2 5 7 10 12 15 17 20 22 25 28 30 56 2 5 7 10 12 15 17 19 22 24 27 29 58 2 5 7 9 12 14 17 19 21 24 26 29 60 2 5 7 9 12 14 16 19 21 23 26 28 Table 12. Completed LATITUDE DECLINATIONS 26 28 30 32 34 36 38 40 42 44 46 48 50 26 90 28 69 90 30 61 70 90 32 56 62 71 90 34 52 57 63 71 90 36 48 53 58 64 72 90 38 45 50 54 59 65 73 90 40 43 47 51 56 60 66 73 90 42 41 45 48 53 57 61 67 74 90 44 39 43 46 50 54 58 62 68 74 90 46 38 41 44 47 51 55 59 63 68 75 90 48 36 39 42 45 49 52 56 60 64 69 75 90 50 35 38 41 44 47 50 53 57 61 65 70 76 90 52 34 37 39 42 45 48 51 55 58 62 66 71 76 54 33 35 38 41 44 47 50 53 56 59 63 67 71 56 32 34 37 40 42 45 48 51 54 57 60 64 68 58 31 33 36 39 41 44 47 49 52 55 58 61 65 60 30 33 35 38 40 43 45 48 51 53 56 59 62 292 Table 13. Kelvin's Simmer Line Table a = a = 1 a = 2 a = 3 a = 4 a = 5 a = 6 K Q K Q K Q K Q K Q K Q K Q 1 2 3 4 5 6 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 59 3 3 3 3 3 3 59 259 4 4 4 4 4 59 1 359 1 359 1 5 5 5 5 5 1 459 1 459 1 458 1 6 6 6 6 1 6 1 559 1 559 2 558 2 7 7 7 7 1 59 1 659 2 658 2 658 3 8 8 8 1 8 1 759 2 759 2 758 3 757 4 9 9 9 1 9 1 859 2 859 3 858 4 857 5 10 10 o- 10 1 10 2 959 3 959 4 958 5 957 6 11 11 11 1 11 2 1059 3 1058 4 1057 6 1056 7 12 12 12 1 12 3 11 59 4 11 58 5 11 57 7 11 56 8 13 13 13 2 13 3 1259 5 1258 6 1257 8 1256 9 14 14 14 2 59 4 1359 5 1358 7 1357 9 1355 11 15 15 15 2 1459 4 1459 6 1458 8 1456 10 1455 13 16 16 16 2 1559 5 1559 7 1558 10 1556 12 1555 14 17 17 17 3 1659 5 1659 8 1657 11 1656 14 1654 16 18 18 18 3 1759 6 1758 9 1757 12 1756 15 1754 18 19 19 19 3 1859 7 1858 10 1857 14 1855 17 1854 21 20 20 20 4 1959 8 1958 11 1957 15 1955 19 1953 23 21 21 21 4 2059 9 2058 13 2057 17 2055 21 2053 25 22 22 22 5 21 59 9 2158 14 21 57 19 2155 23 2152 28 23 23 23 5 2259 10 2258 15 2256 21 2254 26 2252 31 24 24 24 6 2359 11 2358 17 2356 23 2354 28 2352 34 25 25 25 6 2459 12 2458 18 2456 25 2454 31 2451 37 26 26 26 7 2559 13 2558 20 2556 27 2554 34 2551 40 27 27 27 7 2659 15 2658 22 2656 29 2653 37 2650 44 28 28 28 8 2759 16 2757 24 2756 32 2753 40 2750 47 29 29 29 9 2859 17 2857 26 2855 34 2853 43 2850 51 30 30 30 9 2959 19 2957 28 2955 37 2952 46 2949 55 31 31 31 10 3059 20 3057 30 3055 40 3052 50 3049 59 32 32 32 11 3159 21 31 57 32 31 55 43 31 52 53 3148 7 4 33 33 33 12 3259 23 3257 34 3255 46 3252 57 3248 9 34 34 34 12 3359 25 3357 37 3354 49 3351 6 1 3347 14 35 35 35 13 3459 26 3457 40 3454 53 3451 6 3447 19 36 36 36 14 3558 28 3557 42 3554 56 3551 10 3546 24 37 37 37 15 3658 30 3656 45 3654 5 3650 15 3646 30 38 38 38 16 3758 32 3756 48 3753 4 3750 20 3745 36 39 39 39 17 3858 34 3856 51 3853 8 3849 25 3845 42 40 40 40 18 3958 37 3956 55 3953 13 3949 31 3944 49 41 41 41 19 4058 39 4056 58 4053 18 4049 37 4044 56 42 42 42 21 4158 41 41 56 4 2 41 52 23 4148 43 4143 8 3 43 43 43 22 4258 44 4256 6 4252 28 4248 49 4242 11 44 44 59 23 4358 47 4355 10 4352 33 4347 56 4342 19 45 45 4459 25 4458 50 4455 14 4452 39 4447 7 3 4441 27 Table 13. Kelvin's Sumner Line Table 293 b a = a = 1 a = 2 a = 3 a = 4 a = 5 a = 6 K Q K Q K Q K Q K Q K Q K Q 45 45 4459 125 4458 250 4455 414 4452 539 4447 7' 3 4441 827 46 46 4559 26 4558 53 4555 19 4551 45 4546 11 4541 36 47 47 4659 28 4658 56 4655 24 4651 51 4646 19 4640 46 48 48 4759 30 4758 59 4755 29 4751 58 4745 27 4739 56 49 49 4859 31 4858 3 3 4855 34 4850 6 5 4845 36 4838 9. 6 50 50 4959 33 4958 7 4954 40 4950 13 4944 45 4938 17 51 51 5059 35 5057 11 5054 46 5050 21 5044 55 5037 29 52 52 5159 37 51 57 15 51 54 52 51 49 29 51 43 8 5 5136 41 53 53 5259 40 5257 19 5254 59 5249 38 5243 16 5235 54 54 54 5359 42 5357 24 5354 5 6 5349 47 5342 28 5334 10 8 55 55 5459 45 5457 29 5453 13 5448 57 5441 40 5433 23 56 56 5559 47 5557 34 5553 21 5548 7 8 5541 53 5532 39 57 57 5659 50 5657 40 5653 30 5647 19 5640 9 7 5631 55 58 58 5759 53 5757 46 5752 39 5747 31 5739 22 5730 11 13 59 59 5859 56 5857 53 5852 49 5846 44 5838 38 5829 32 60 60 5959 2 5956 4 5952 59 5946 58 5937 56 5928 52 61 61 6059 4 6056 7 6052 610 6045 813 6036 10 14 6026 12 14 62 62 61 59 8 6156 15 61 51 22 6144 28 6135 33 61 25 37 63 63 6259 12 6256 24 6251 35 6244 45 6234 54 6223 13 2 64 64 6359 17 6356 33 6350 49 6343 9 4 6333 11 17 6322 29 65 65 6459 22 6456 43 6450 7 4 6442 24 6432 42 6420 58 66 66 6559 27 6555 54 6549 20 6541 45 6531 12 8 65 18 1429 67 67 6659 33 6655 5 6 6649 38 6640 10 9 6629 37 66 16 15 3 68 68 6759 40 6755 19 6748 58 6739 34 6728 13 9 67 14 40 69 69 6859 47 6855 34 6848 8 19 6838 11 2 6826 43 6812 1621 70 70 6959 55 6954 50 6947 43 6937 33 6925 1421 69 9 17 5 71 71 7058 3 4 7054 6 7 7046 9 9 7036 12 7 7023 15 2 70 6 54 72 72 7158 14 71 54 27 7146 38 7135 45 71 20 48 71 3 1847 73 73 7258 25 7253 49 7245 10 10 7233 1327 72 18 1640 72 1946 74 74 7358 37 7353 7 13 7344 46 7331 14 14 73 15 1737 56 2052 75 75 7458 51 7452 41 7443 11 27 7429 15 7 7412 1841 7352 22 6 76 76 7558 4 8 7552 8 13 7541 12 13 7527 16 7 75 9 1953 7447 2329 77 77 7658 26 7651 50 7640 13 7 7625 17 16 76 5 21 15 7542 25 3 78 78 7758 48 7750 932 7738 14 9 7722 1835 77 1 2249 7636 2649 79 79 7857 514 7849 1022 7836 1522 7818 20 8 56 2438 7729 2851 80 80 7957 44 7948 1122 7934 1648 7914 21 56 7850 2644 7821 31 11 81 81 8057 622 8047 1235 8031 1831 80 9 24 5 7943 29 13 7912 3354 82 82 8156 7 9 81 45 14 5 8128 2038 81 4 2641 8034 32 9 80 1 37 4 83 83 8256 8 9 8243 1559 8223 23 16 57 2951 81 24 3540 47 4047 84 84 8355 929 8341 1828 8318 2638 8248 3347 82 12 3956 81 31 45 9 85 85 8454 11 20 8437 2150 8410 31 1 8336 3844 56 45 7 82 12 5020 86 86 8553 14 3 8532 2636 85 3655 8421 45 4 8336 51 26 48 5626 87 87 8650 1827 8624 3343 45 45 2 85 53 11 84 10 59 7 83 18 6332 88 88 8746 2634 87 10 45 1 8624 5620 32 6329 37 6815 41 71 38 89 89 8835 45 46 6327 50 71 35 53 7559 54 7843 55 8034 90 90 89 90 88 90 87 90 86 90 85 90 84 90 294 Table 13. Kelvin's Simmer Line Table b a = 7 a = 8 a = 9 a = 10 a = 11 a = 12 a = 13 K Q K Q K Q K Q K Q K Q K Q 7 8 9 10 11 12 13 1 1 59 59 59 59 59 58 2 59 159 159 158 158 157 1 57 3 259 1 258 1 258 1 257 1 257 1 256 1 255 1 4 358 1 358 1 357 1 356 1 356 2 355 2 354 2 5 458 2 457 2 456 2 455 2 454 3 453 3 452 3 6 557 2 556 3 556 3 555 3 553 4 552 4 551 4 7 657 3 656 4 655 4 654 4 652 5 651 5 649 6 8 756 4 755 5 754 5 753 6 751 6 749 7 748 8 9 856 5 855 6 853 7 852 7 850 8 848 9 846 10 10 955 6 954 7 953 8 951 9 949 10 947 11 944 12 11 1055 8 1053 9 1052 10 1050 11 1048 12 1045 13 1043 14 12 11 55 9 1153 11 11 51 12 1149 13 1147 14 1144 16 11 41 17 13 1254 11 1252 13 1250 14 1248 15 1245 17 1243 19 1240 20 14 1354 13 1352 15 1349 16 1347 18 1344 20 1341 22 1338 23 15 1453 15 1451 17 1449 19 1446 21 1443 23 1440 25 1436 26 16 1553 17 1550 19 1548 21 1545 24 1542 26 1538 28 1535 30 17 1652 19 1650 22 1647 24 1644 27 1641 29 1637 32 1633 34 18 1752 21 1749 24 1746 27 1743 30 1739 33 1736 36 1731 39 19 1851 24 1848 27 1845 30 1842 34 1838 37 1834 40 1830 43 20 1951 27 1948 30 1945 34 1941 38 1937 41 1933 45 1928 48 21 2050 30 2047 34 2044 38 2040 42 2036 46 2031 50 2026 53 22 2150 33 2146 37 2143 42 2139 46 21 35 51 2130 55 2124 59 23 2249 36 2246 41 2242 46 2238 51 2233 56 2228 13 2223 14 5 24 2349 39 2345 45 2341 50 2337 56 2332 12 1 2327 6 2321 11 25 2448 43 2444 49 2440 55 2436 11 1 2431 6 2425 12 2419 17 26 2548 47 2544 53 2539 10 2535 6 2529 12 2523 18 2517 24 27 2647 51 2643 58 2638 5 2633 12 2628 18 2622 25 2615 31 28 2746 55 2742 9 3 2738 10 2732 18 2727 25 2720 32 27 13 39 29 2846 59 2841 8 2837 16 2831 24 2825 32 2818 40 28 11 47 30 2945 8 4 2941 13 2936 22 2930 31 2924 39 2917 48 29 9 56 31 3045 9 3040 19 3035 28 3029 38 3022 47 3015 56 30 7 15 5 32 3144 14 31 39 25 3134 35 3127 45 3121 55 31 13 14 4 31 5 14 33 3243 20 3238 31 3233 42 3226 52 3219 13 3 3211 13 32 3 24 34 3343 26 3337 37 3332 49 3325 12 3318 12 3310 23 33 1 34 35 3442 32 3437 44 3430 57 3424 9 3416 21 34 8 33 59 44 36 3541 38 3536 51 3529 11 5 3522 18 3514 31 35 6 43 3456 55 37 3641 45 3635 59 3628 13 3621 27 3613 41 36 4 54 3554 16 7 38 3740 52 3734 10 7 3727 22 3719 37 3711 51 37 2 15 6 3652 20 39 3839 59 3833 15 3826 31 3818 47 38 9 14 2 38 18 3749 33 40 3939 9 6 3932 24 3925 41 3916 58 39 7 14 57 31 3847 46 41 4038 14 4031 33 4023 51 4015 13 9 40 5 27 3955 44 3944 17 42 4137 23 4130 43 4122 12 2 41 13 21 41 3 40 4053 58 4041 15 43 4236 32 4229 53 4221 13 4212 33 42 1 53 41 51 1612 41 39 31 44 4335 41 4328 11 3 43 19 25 4310 46 59 15 7 4248 28 4236 48 45 4434 51 4427 14 4418 38 44 8 14 4357 22 4346 44 4333 18 5 Table 13. Kelvin's Sumner Line Table 295 b a = 7 a=8 a = 9 a = 10 a = 11 a = 12 a = 13 K Q K Q K Q K Q K Q K Q K Q 45 4434 951 4427 11 14 4418 1238 44 8 14 4357 1522 4346 1644 4333 18 5 46 4534 10 1 4526 26 4516 51 45 6 15 4455 38 4443 17 1 4430 23 47 4633 12 4624 39 46 15 13 5 46 4 30 4553 55 4540 19 4527 42 48 4732 24 4723 52 47 13 19 47 2 46 4651 16 12 4638 37 4624 19 2 49 4831 36 4822 12 6 48 12 34 48 15 3 4748 30 4735 57 4720 23 50 4930 49 4920 20 4910 51 58 20 4846 50 4832 1818 48 17 45 51 5029 11 2 50 19 35 50 8 14 8 4956 39 4943 17 10 4929 40 49 13 20 9 52 51 27 17 51 18 52 51 6 26 5054 59 5040 31 5026 19 3 50 9 33 53 5226 32 52 16 13 9 52 4 45 51 52 1620 51 37 54 51 22 27 51 5 59 54 5325 48 5314 27 53 2 15 5 5249 42 5234 1818 52 19 53 52 1 21 27 55 5424 12 5 54 13 46 54 26 5347 17 5 5331 43 53 15 2020 57 56 56 5522 23 55 11 14 6 58 49 5444 30 5428 19 10 54 11 49 5353 2226 57 5621 42 56 9 28 5556 1613 5541 56 5525 39 55 7 21 19 5448 58 58 5719 13 3 57 7 51 5653 39 5638 1824 5621 20 9 56 3 51 5543 2333 59 5818 25 58 5 1516 5751 17 6 5735 54 5717 41 59 2226 5638 24 9 60 59 16 48 59 3 42 5848 35 5832 1926 58 13 21 15 5754 23 2 5733 47 61 6014 1413 60 1 1610 5945 18 6 5928 59 59 9 51 5849 41 5827 2528 62 61 12 39 58 40 6042 39 6024 2035 60 5 2230 5944 2422 5921 26 11 63 62 10 15 8 6156 17 12 61 39 19 14 6120 21 14 61 23 11 6038 25 5 6015 57 64 63 8 39 6253 47 6236 52 62 16 55 55 55 6132 52 61 8 2746 65 64 6 1612 6350 1824 6332 2033 63 12 2239 6250 2442 6226 2642 62 1 2839 66 65 4 48 6447 19 4 6428 21 17 64 7 2326 6344 2533 6320 2736 53 2935 67 66 1 1727 6543 47 6524 22 4 65 2 2417 6438 2627 6413 2833 6345 3035 68 58 18 9 6639 2034 66 19 55 56 25 12 6532 2726 65 5 29 34 6437 31 39 69 6755 55 6735 21 25 6714 2351 6650 26 12 6625 2829 57 3040 6528 3247 70 6852 1945 6831 2220 68 9 2451 6744 2716 6717 2937 6648 31 52 6618 34 1 71 6948 2040 6926 2321 69 3 2556 6837 2826 68 9 3050 6739 33 8 67 7 3520 72 7044 21 40 7021 2427 57 27 8 6929 2943 69 32 10 6829 3431 55 3646 73 7139 2247 71 16 2540 7050 2827 7021 31 6 50 33 37 69 18 36 1 6843 38 18 74 7234 24 1 72 10 27 1 7142 2953 71 12 3236 7040 35 12 70 6 3738 6930 3957 75 7329 2523 73 3 2830 7234 31 28 72 2 3416 7128 3655 53 3924 7015 41 44 76 7423 2655 55 30 9 7324 33 13 51 36 5 72 16 3847 7138 41 18 59 4340 77 7516 2838 7446 32 74 14 35 9 7339 38 5 73 2 4050 7223 4323 71 42 4545 78 76 8 3034 7537 34 4 75 2 3718 7426 40 18 47 43 4 73 6 4538 7223 48 79 59 3246 7626 3623 49 3942 75 11 4244 7430 4532 47 48 5 73 2 5026 80 7749 3516 7713 3859 7635 4222 54 4526 7511 4813 7426 5045 39 53 3 81 7837 38 8 59 41 56 77 18 4521 7635 4825 49 51 10 75 2 5339 74 14 5553 82 7923 41 25 7842 45 17 59 4842 77 13 5143 7626 5424 37 5647 46 5855 83 80 7 45 13 7923 49 4 7837 5225 49 5521 59 5755 76 8 60 10 75 16 62 10 84 47 4936 80 1 5322 7912 5635 7821 5920 7729 6144 36 6349 42 6538 85 8124 5438 35 5812 43 61 11 50 6342 56 6551 77 1 6742 76 5 6919 86 57 6024 81 4 6336 80 9 66 14 79 14 6825 7818 70 16 22 71 50 25 73 11 87 8223 6655 28 6935 31 7143 34 7328 36 7456 38 76 10 40 77 14 88 43 74 8 45 76 3 47 7734 48 7848 49 7949 50 8041 51 81 24 89 56 81 55 56 8255 57 8343 57 8421 57 8452 58 8518 58 8541 90 83 90 82 90 81 90 80 90 79 90 78 90 77 90 296 Table 13. Kelvin's Sumner Line Table b a = 14 a = 15 a = 16 a = 17 a = 18 a= 19 a = 20 K Q K Q K Q K Q K Q K Q K Q 1 14 15 16 17 18 19 20 58 58 58 57 57 57 56 2 156 156 1 55 155 ; 1 54 154 1 53 1 J 255 ] 254 253 252 251 250 2 249 2 4 353 t 352 2 351 2 349 t 348 t 347 3 346 3 5 451 3 450 t 448 3 447 i 445 ^ 444 ^ 442 4 ( 549 4 548 c t. 546 i c 544 i . 542 ( 540 6 538 6 f 647 6 646 r 644 *" 642 *j 639 8 637 t 635 8 8 746 8 744 C 741 c 739 9 736 10 734 10 731 11 c 844 10 841 11 839 11 836 12 833 13 830 13 827 14 10 942 12 939 13 937 14 934 15 930 16 927 16 923 17 11 1040 15 1037 16 1034 17 1031 18 1027 19 1024 20 1020 21 12 1138 18 1135 19 1132 20 1128 21 11 24 23 11 20 2^ 11 16 25 13 1236 21 1233 22 1229 24 1225 25 1221 27 12 17 28 1212 29 14 1335 25 1331 26 1327 28 1323 29 1318 31 13 13 32 13 8 34 15 1433 28 1429 30 1424 32 1420 34 14 15 36 14 10 37 14 5 39 16 1531 32 1526 35 1522 37 15 17 39 15 12 41 15 6 42 15 1 44 17 1629 37 1624 39 16 19 42 16 14 44 16 9 46 16 3 48 57 50 18 1727 41 1722 44 17 17 47 1711 49 17 6 52 59 54 1653 56 19 1825 46 1820 49 18 14 52 18 8 55 18 2 58 1756 20 1 1749 21 3 20 1923 52 19 17 55 19 12 58 19 5 18 1 59 19 4 1852 8 1845 10 21 2021 57 20 15 16 1 20 9 17 4 20 2 8 1956 11 1948 15 1941 18 22 21 19 15 3 21 13 7 21 6 11 59 15 2052 19 2045 22 2037 26 23 22 17 g 2210 14 22 4 18 21 56 22 21 49 27 21 41 30 21 32 34 24 2315 16 23 8 21 23 1 26 2253 30 2245 35 2237 39 2228 43 25 24 13 23 24 6 28 58 34 2350 38 2342 43 2333 48 2324 53 26 25 10 30 25 3 36 2455 42 2447 47 2438 52 2429 58 2420 22 3 27 26 8 38 26 1 44 2552 50 2544 56 2535 20 2 2525 21 8 25 15 13 28 27 6 46 58 53 2650 59 2641 19 6 2631 12 2621 18 2611 24 29 28 4 55 2755 17 2 2747 18 9 2737 16 2727 23 2717 29 27 6 36 30 29 1 16 4 853 12 2844 19 2834 27 2824 34 28 13 41 28 2 48 31 59 13 950 22 2941 30 2930 38 2920 46 29 9 53 57 23 1 32 3057 23 047 32 3037 41 3027 50 3016 58 30 4 22 6 2952 14 33 31 54 33 1 44 43 1 34 53 3123 20 2 31 12 21 11 31 19 3047 28 34 3252 44 242 55 231 19 5 3220 15 32 8 24 55 33 3142 42 35 3349 56 339 8 7 328 18 3 16 28 3 4 38 3251 48 3237 57 36 3446 7 8 436 20 424 31 4 12 42 59 53 3346 23 3 3332 24 13 37 544 20 533 33 521 45 5 8 57 455 22 8 441 19 3426 30 38 641 33 629 47 617 20 6 4 21 12 550 24 536 36 521 48 39 738 47 726 9 1 713 15 7 28 646 41 631 54 6 15 25 6 40 835 8 2 823 17 8 10 31 56 45 741 59 726 24 12 7 9 25 41 932 17 9 19 33 9 6 48 852 22 3 836 23 18 820 32 8 3 45 42 029 33 16 50 2 21 6 947 22 931 37 9 15 52 57 26 6 43 1 26 50 1 12 20 7 58 25 042 41 026 57 9 25 13 951 27 44 223 9 7 2 9 26 154 44 138 23 2 1 21 24 18 1 3 35 045 50 45 3 19 25 3 5 45 249 22 4 233 23 2 16 41 57 58 139 2714 Table 13. Kelvin's Sumner Line Table 297 b a = 14 a = 15 a = 16 a = 17 a = 18 a = 19 a = 20 K K Q K Q K Q K Q K Q K Q 45 43 19 1925 43 5 2045 4249 22 4 4233 2323 42 16 2441 41 57 2558 41 39 27 14 46 44 16 45 44 1 21 6 4345 26 4328 45 43 10 25 4 4251 2G12L 4232 39 47 45 12 20 5 57 27 4440 48 4423 24 9 44 4 28 4345 47 4325 28 5 48 46 8 26 4553 49 4535 23 12 4518 33 58 54 44 39 27 14 44 18 33 49 47 4 48 4648 2213 4630 37 4612 59 4552 2621 4532 42 4510 29 1 50 48 21 12 4744 38 4725 24 3 47 6 2526 4646 49 4625 28 11 46 2 31 5] 56 37 4839 23 4 4820 30 48 55 4739 2718 47 18 41 54 30 3 5S 4952 22 3 4934 31 4915 59 54 2625 4832 49 48 10 29 13 4746 36 53 5048 30 5029 24 50 9 2529 4948 56 4925 2822 49 2 47 4838 31 10 54 5143 59 51 24 30 51 3 26 5041 2729 50 18 56 54 3022 4929 46 55 5238 2330 5218 25 2 57 34 51 34 28 4 51 10 2932 5046 59 5020 3224 56 5333 24 2 5312 36 5250 27 9 5227 40 52 2 3010 51 37 31 37 51 10 33 3 57 5428 36 54 6 2612 5343 46 5320 29 18 54 49 5228 32 18 52 45 58 5522 25 12 55 49 5436 2825 54 12 59 5346 31 31 53 18 33 1 50 3429 59 56 16 50 53 2729 5529 29 6 55 4 3042 5437 32 15 54 8 46 5339 35 15 60 5710 2630 5646 28 11 5621 50 55 31 27 5527 33 1 58 3433 5428 36 3 61 58 4 2713 5739 56 57 13 3036 5646 32 14 56 17 50 5547 3523 55 16 54 62 57 59 5831 2943 58 4 3125 5736 33 4 57 7 3441 5636 36 16 56 4 3747 63 5950 2847 5923 3033 55 3217 5826 57 56 3535 5724 37 11 51 3843 64 6042 2938 6015 3126 5946 3311 59 16 3453 5844 3633 58 11 38 9 5738 3942 65 61 34 3032 61 6 3223 6036 34 9 60 5 3553 5932 3733 58 39 10 5824 4044 66 6225 31 31 56 3323 6125 3511 53 3656 6019 3837 5944 40 15 59 9 41 49 67 63 16 3233 6246 3427 62 14 36 17 61 41 38 3 61 6 3945 6030 41 23 53 4258 68 64 7 33 39 6335 3535 63 2 3726 6228 39 13 52 4056 61 15 4235 6036 4411 69 56 3450 6423 3647 49 3840 63 14 4028 6237 42 12 58 4351 61 19 4527 70 6545 36 5 6511 38 4 6436 3958 59 4148 6321 4332 6241 4511 62 1 4647 71 6633 3727 58 3927 6521 41 22 6443 43 12 64 4 4456 6323 4636 41 4811 72 6720 3854 6644 4056 66 6 4252 6526 4442 45 4626 64 4 48 6 6321 4940 73 68 7 4027 6729 4230 49 4427 66 8 46 17 6526 48 1 43 4940 59 51 14 74 52 42 8 68 12 44 11 6731 46 8 49 4758 66 6 4942 6521 51 19 6436 5252 75 6936 4356 55 4559 6812 4756 6729 4945 44 5128 58 53 4 65 11 5435 76 7018 4552 6936 4755 52 4951 68 7 51 39 6720 5320 6633 5455 45 5623 77 59 4757 7015 4959 6930 51 53 43 5339 55 55 18 67 7 5651 66 18 5817 78 7138 50 11 53 52 12 70 6 54 3 69 18 5547 6829 5723 39 5853 48 60 16 79 7216 5235 71 28 5433 40 5621 50 58 2 69 5935 68 9 61 67 17 6220 80 51 55 9 72 2 57 3 71 12 5848 7021 6024 29 61 53 37 63 14 44 6430 81 7324 5754 34 5943 42 61 23 50 6254 56 64 18 69 3 6534 68 9 6645 82 55 6050 73 3 6233 72 9 64 7 71 16 6531 7021 6649 27 68 31 69 5 83 7423 6357 29 6532 34 6659 39 68 16 44 6926 48 7031 51 71 29 84 48 67 15 52 6841 56 6959 72 71 7 71 3 7210 70 7 73 7 69 9 7359 85 75 9 7044 7412 71 59 73 15 73 6 18 74 5 20 7459 23 7548 25 7632 86 27 7422 29 7525 31 7620 33 H 9 35 7753 36 7833 37 79 9 87 41 78 9 43 7857 44 7939 45 80 17 46 8051 46 81 22 47 81 49 88 52 82 2 52 8235 53 83 4 53 8329 54 8352 54 8413 54 8431 89 58 86 58 8616 58 8631 58 8644 58 8655 58 87 6 59 87 15 90 76 90 75 90 74 90 73 90 72 90 71 90 70 90 298 Table 13. Kelvin's Sumner Line Table b a = 21 a = 22 a = 23 a = 24 a = 25 a = 26 a = 27 K Q K Q K Q K Q K Q K Q K Q 21 22 23 24 25 26 27 1 56 56 55 55 54 54 53 2 152 1 1 51 1 1 51 1 1 50 1 1 49 1 148 1 1 47 1 3 248 2 247 2 246 2 244 2 243 2 242 2 240 2 4 344 3 342 3 341 3 339 3 338 3 336 3 334 3 5 440 4 438 5 436 5 434 5 432 5 430 5 427 5 6 536 6 534 7 531 7 529 7 526 7 523 7 521 8 7 632 9 629 9 626 9 624 10 620 10 617 10 614 11 8 728 11 725 12 722 12 7 18 13 715 13 711 13 7 7 14 9 824 14 820 15 817 15 813 16 8 9 16 8 5 17 8 1 17 10 920 18 916 18 912 19 9 8 20 9 3 20 59 21 54 21 11 1016 22 1011 22 10 7 23 10 2 24 57 24 953 25 947 26 12 11 12 26 11 7 26 11 2 27 57 28 1052 29 1046 30 1041 31 13 12 7 30 12 2 31 57 32 11 52 33 11 46 34 11 40 35 11 34 36 14 13 3 35 58 36 1252 38 1246 39 1240 40 1234 41 1227 42 15 59 40 1353 42 1347 43 1341 45 1334 46 1327 47 1320 49 16 1455 46 1448 48 1442 49 1435 51 1428 53 1421 54 14 13 56 17 1550 52 1544 54 1537 56 1529 58 1522 26 1514 27 1 15 6 28 3 18 1646 59 1639 23 1 1632 24 3 1624 25 5 1616 7 16 8 9 59 11 19 1742 22 6 1734 8 1726 11 17 18 13 1710 15 17 1 17 1652 19 20 1837 13 1829 16 1821 19 1812 21 18 3 23 54 26 1745 28 21 1933 21 1924 24 19 16 27 19 7 30 57 32 1847 35 1837 37 22 2028 29 2019 33 20 10 36 20 1 39 1951 42 1940 45 1930 47 23 2124 38 21 14 42 21 5 45 55 49 2044 52 2033 55 2022 58 24 22 19 47 22 9 52 59 55 21 49 59 21 38 27 3 21 26 28 6 21 15 29 9 25 23 14 57 23 4 24 2 2254 25 6 2243 26 10 2231 14 22 19 17 22 7 21 26 24 9 23 8 59 12 2348 17 2337 21 2325 25 2312 29 23 33 27 25 4 19 2454 23 2442 28 2430 33 2418 37 24 5 42 52 46 28 59 30 2548 35 2536 40 2524 46 2511 50 57 55 2444 59 29 2654 42 2643 48 2630 53 26 17 59 26 4 28 4 2550 29 9 2536 30 13 30 2749 54 2737 25 1 2724 26 7 2711 2713 57 18 2642 23 2627 28 31 2844 24 7 2832 14 28 18 21 28 4 27 2750 33 2734 38 2719 44 32 2939 21 2926 28 29 12 35 57 42 2842 48 2826 54 2811 31 33 3034 36 3020 43 30 5 51 2950 58 2935 29 4 2918 3011 29 2 17 34 31 28 51 31 14 59 59 27 7 3043 28 14 3027 21 3010 28 53 35 35 3223 25 7 32 8 2615 3152 24 3136 31 31 19 39 31 2 46 3044 53 36 33 17 23 33 1 32 3245 41 3229 49 3211 58 53 31 5 3135 32 12 37 3411 40 55 50 3338 59 3321 29 8 33 3 3017 3245 25 3226 32 38 35 5 58 3448 27 9 3431 28 18 3413 28 55 37 3336 45 3316 53 39 59 26 17 3542 28 3524 38 35 5 49 3447 58 3427 32 7 34 6 3315 40 3653 37 3635 48 3617 59 57 3010 3538 3120 3518 29 56 38 41 3746 58 3728 28 10 37 9 2921 3649 32 3629 43 36 8 52 3546 34 1 42 3840 27 19 3821 32 38 1 44 3741 56 3720 32 7 58 3317 3636 26 43 3933 42 39 13 55 53 30 8 3832 31 20 3811 31 3748 42 3725 52 44 4026 28 5 40 6 29 19 3945 33 3923 45 39 1 57 3838 34 8 3814 35 19 45 41 19 30 58 44 4037 59 4014 3212 51 3324 3928 36 39 3 47 Table 13. Kelvin's Sunnier Line Table 299 b a = 31 a = 23 a = 23 a = 24 a = 25 a = 26 a = 27 K Q K Q K Q K Q K Q K Q K Q 45 41 19 2830 4058 2944 4037 3059 40 14 32 12 3951 3324 3928 3436 39 3 3547 46 4211 55 4150 3011 41 28 31 26 41 5 39 4041 52 40 17 35 4 52 36 16 47 43 4 2922 4242 39 4219 54 55 33 8 4131 3422 41 6 34 4040 46 48 56 50 4333 31 8 4310 3223 4245 38 4220 52 55 36 5 41 28 3717 49 4448 3020 4424 38 44 54 4335 3410 43 9 3524 4243 37 42 15 50 50 4540 51 4515 32 9 50 3326 4425 43 58 57 4331 3711 43 2 3824 51 4631 31 23 46 6 42 4540 34 45 14 3517 4447 3632 4418 46 49 39 52 4722 57 56 33 17 4630 35 46 3 53 4535 37 8 45 5 3823 4436 37 53 48 13 3232 4746 53 47 19 35 12 51 3630 4622 46 52 39 1 4522 4015 54 49 3 33 9 4836 3430 48 8 50 4739 37 9 47 9 3826 4639 41 46 7 55 55 53 48 4925 35 10 56 3630 4827 49 56 39 7 4725 4023 52 4137 56 5043 3428 5014 51 4944 37 12 49 14 3832 4842 49 48 10 41 6 4737 4220 57 5132 35 11 51 2 3634 5032 56 50 1 39 16 4928 4034 55 51 4821 43 5 58 5221 55 50 3719 51 19 3842 47 40 2 5014 41 21 4940 4238 49 5 52 59 53 9 3642 5238 38 7 52 6 3930 5133 50 59 42 9 5024 4326 48 4441 60 57 3731 5325 56 52 4020 52 18 4141 5143 43 51 7 44 17 5030 4532 61 5444 3822 5411 3948 5337 41 12 53 2 4234 5226 53 49 45 10 51 12 4625 62 5531 3916 57 4043 5422 42 7 46 4329 53 9 4448 5231 46 6 53 4721 63 5617 40 13 5542 41 40 55 6 43 5 5429 4427 51 4546 53 13 47 3 5233 4818 64 57 3 41 13 5627 4240 49 44 5 5511 4527 5433 4646 53 48 3 53 13 4918 65 48 4215 5711 4343 5632 45 8 53 4630 5513 4749 5433 49 5 51 5020 66 5832 4321 54 4449 5714 46 13 5634 4735 53 4854 5512 5010 5429 5124 67 5915 4430 5836 4558 55 4722 5714 4844 5632 50 2 50 51 18 55 6 5231 68 57 4542 59 17 47 10 5835 4834 53 4955 57 10 51 13 5627 52 28 42 5341 69 6039 4658 57 4826 5915 4950 5831 51 10 47 5227 57 3 5342 56 17 5453 70 61 19 4818 6036 4945 53 51 8 59 8 5228 5823 5344 38 5458 51 56 8 71 58 4942 61 14 51 8 6030 5231 44 5349 58 55 5 58 12 56 17 5724 5725 72 6236 51 10 51 5235 61 6 5357 6019 55 14 5932 5628 44 5739 56 5846 73 63 13 5242 6227 54 7 41 5527 53 5642 60 5 5755 5916 59 4 5826 60 9 74 49 54 19 63 2 5542 6214 57 6125 5814 36 5925 46 6032 55 6135 75 6423 56 1 35 5721 46 5838 56 5950 61 6 6058 60 15 62 3 5923 63 4 76 56 5747 64 7 59 5 6316 6019 6226 61 29 34 6235 42 6337 50 6436 77 6527 5938 37 6053 45 62 5 54 63 12 62 1 6415 61 8 65 14 6015 6611 78 57 61 34 65 5 6246 6413 6354 6320 6458 26 6558 32 6655 38 6748 79 6625 6334 32 6443 38 6548 44 6648 50 6745 55 6838 61 6928 80 50 6540 56 6644 65 2 6745 64 7 6842 63 12 6935 62 16 7024 20 71 11 81 6714 6750 66 19 6850 23 6946 28 7039 32 71 27 35 72 13 39 7256 82 36 70 4 40 71 43 7151 47 7239 50 7323 53 74 4 56 7443 83 55 7223 58 73 13 66 1 7359 65 3 7442 64 6 7521 63 8 7558 62 10 7633 84 6812 7446 6714 7530 16 7610 18 7647 20 7722 22 7754 23 7824 85 26 7712 28 7750 29 7824 31 7855 32 7925 33 7952 35 8018 86 38 7942 39 8012 40 8040 41 81 6 42 8130 43 8152 44 8212 87 48 82 14 48 8237 49 8258 49 8318 50 8336 50 8353 51 84 8 88 55 8448 55 85 4 55 8518 55 8531 56 8543 56 8555 56 86 5 89 59 8724 59 8732 59 8739 59 8745 59 8751 59 8757 59 88 2 90 69 90 68 90 67 90 66 90 65 90 64 90 63 90 300 Table 13. Kelvin's Sumner Line Table b a = 28 a = 29 a = 30 a = 31 3 a = 32 a = 33 a = 34 K Q K Q K Q K Q K Q K Q K Q 28 29 30 31 32 33 34 1 53 52 52 51 51 50 50 2 1 46 1 145 1 144 1 143 1 142 1 141 1 1 39 1 3 2 39 2 237 2 236 2 234 2 233 2 231 2 229 2 4 332 3 330 4 328 4 326 4 323 4 321 4 3 19 4 5 425 5 422 6 420 6 4 17 6 4 14 6 4 12 6 4 9 6 6 5 18 8 5 15 8 5 12 8 5 8 8 5 5 8 5 2 9 58 9 7 6 11 11 6 7 11 6 4 11 6 11 56 11 52 12 548 12 8 7 4 14 59 14 55 15 51 15 647 15 642 15 638 16 9 56 18 752 18 747 19 743 19 737 19 732 19 727 20 10 849 22 844 22 839 23 834 23 828 24 822 .24 8 17 25 11 942 27 936 27 931 28 925 28 9 19 29 9 12 29 9 6 30 12 1035 32 1029 32 1022 33 10 16 34 10 9 34 10 2 35 56 35 13 11 27 37 11 21 38 11 14 39 11 7 40 11 40 52 41 1045 41 14 1220 43 12 13 44 12 6 45 58 46 50 47 1142 48 11 34 48 15 13 13 50 13 5 51 57 52 1249 53 1241 54 1232 55 1223 56 16 14 5 57 57 58 1349 59 1340 32 1 1331 33 2 1322 34 3 13 13 35 4 17 58 29 4 1449 30 6 1440 31 7 1431 9 1421 10 14 12 11 14 2 12 18 1550 12 1541 14 1531 16 1522 17 15 12 18 15 1 20 51 21 19 1642 21 1633 23 1623 25 1612 26 16 2 27 51 29 1540 30 20 1735 30 1724 32 17 14 34 17 3 36 52 37 1640 39 1628 40 21 1827 40 18 16 42 18 5 44 53 46 1742 48 1729 49 17 17 51 22 19 19 50 19 8 52 56 55 1844 57 1831 59 18 19 35 18 6 36 2 23 20 11 30 1 59 31 3 1947 32 6 1934 33 8 1921 34 10 19 8 12 54 14 24 21 3 12 2050 15 2037 18 2024 20 20 11 22 57 24 1942 26 25 55 24 2141 27 21 28 30 21 14 33 21 35 2046 37 2030 39 26 2246 37 2232 40 22 19 43 22 4 46 49 48 21 34 51 21 18 53 27 2338 50 2323 53 23 9 57 54 34 2238 35 2 2223 36 5 22 6 37 8 28 2429 31 3 24 14 32 7 59 33 11 2344 14 2327 17 23 11 20 54 23 29 2521 18 25 5 22 2449 26 2433 29 24 16 33 59 36 2342 38 30 2612 33 56 37 2539 41 2523 45 25 5 49 2447 52 2429 55 31 27 3 49 2647 53 2629 58 2612 35 2 54 36 6 2535 37 9 25 16 3812 32 54 32 5 2737 3310 27 19 3415 27 1 19 2642 23 2623 27 26 3 30 33 2845 22 2827 28 28 9 33 50 37 2730 41 2711 45 50 49 34 29 35 40 29 17 46 58 51 2839 56 28 18 37 58 38 4 2737 39 8 35 3026 59 30 7 34 5 2947 3511 2927 3616 29 6 20 2845 24 2824 28 36 31 16 33 19 56 25 3036 31 30 15 36 54 41 2932 45 2910 49 37 32 6 39 3146 46 3125 52 31 3 57 3041 38 2 3019 39 7 56 40 11 38 56 34 1 3235 35 7 32 13 3614 51 3720 31 28 25 31 5 30 3042 34 39 3346 23 3324 30 33 1 37 3239 43 32 15 48 51 53 3127 57 40 3435 46 34 13 53 49 37 3326 38 7 33 2 39 12 3237 40 17 32 12 41 22 41 3524 35 10 35 1 36 18 3437 25 34 13 31 48 37 3323 43 57 47 42 3613 35 49 43 3525 51 35 57 3434 40 4 34 8 41 9 3342 42 14 43 37 2 36 1 3637 37 10 3612 3817 46 3924 3520 31 53 36 3426 41 44 50 28 3725 37 59 45 3633 52 36 6 59 3538 42 5 35 10 43 10 45 3838 57 38 12 38 5 3746 3914 3719 4021 51 4128 3622 34 53 39 Table 13. Kelvin's Sumner Line Table 301 b a = 28 a = 29 a = 30 a = 31 a = 32 a = 33 a = 34 K Q K Q K Q K Q K Q K Q K Q 45 3838 3657 38 12 38 5 3746 39 14 37 19 4021 3651 41 28 3622 4234 3553 43 39 46 3926 3726 59 35 3832 44 38 4 52 3736 58 37 6 43 4 3636 44 9 47 40 13 56 3946 39 6 39 18 40 15 49 41 23 3820 4230 50 36 37 19 41 48 41 3828 4032 38 40 4 47 3934 55 39 4 43 2 3833 44 9 38 2 45 14 49 47 39 1 41 18 4011 49 4121 40 19 4229 48 36 39 16 43 44 48 50 4234 36 42 4 46 41 34 56 41 3 43 4 4031 4411 59 45 18 3926 4623 51 4320 40 12 49 41 22 42 18 4232 46 40 41 14 48 4041 54 40 7 59 52 44 5 49 4334 42 43 2 4310 4229 44 18 56 4526 41 22 4632 48 4737 53 50 41 28 44 18 39 46 49 43 12 57 4238 46 5 42 3 47 11 41 28 48 16 54 4535 42 8 45 2 43 19 4429 4429 54 4538 43 19 45 44 51 42 7 56 55 46 19 50 46 44 1 4511 4511 4436 4620 44 4727 4324 4833 46 4937 56 47 3 4334 4629 45 53 55 45 17 47 4 40 4810 44 3 49 16 4325 5020 57 46 44 19 47 11 4530 4635 4640 58 49 4520 55 42 50 1 44 3 51 5 58 4829 45 6 53 46 17 47 16 4727 4638 4835 59 4942 4520 47 40 51 59 49 11 55 4834 47 6 56 48 16 47 17 4924 4638 5030 58 51 35 45 17 52 38 60 52 4646 49 14 57 4836 49 6 56 50 14 4716 5120 4635 5224 53 5327 61 5033 4738 54 4850 49 15 59 4834 51 6 53 52 12 47 11 53 15 4628 54 18 62 51 13 4833 5033 4944 53 5053 49 11 52 4829 53 5 46 54 8 47 3 55 10 63 53 4930 51 12 5041 5030 51 49 48 56 49 5 54 4821 55 3 37 56 4 64 5231 5030 49 5140 51 7 5248 5024 5353 40 57 55 59 4810 59 65 53 9 51 31 5226 5241 43 5348 59 5453 5014 5556 4928 5657 42 5756 66 46 5235 53 2 5344 52 18 5450 5133 5555 47 5656 50 1 5757 49 14 5855 67 5422 5341 37 5449 52 5555 52 6 5658 51 19 5759 32 5858 44 5955 68 57 5450 5411 5557 5325 57 1 38 58 3 50 59 4 51 2 60 1 5014 6057 69 5531 56 1 44 57 7 57 58 10 53 9 59 11 5221 6010 32 61 6 43 62 1 70 56 4 57 15 55 16 5819 5428 5921 39 6021 50 61 18 52 1 62 13 51 10 63 7 71 36 5831 47 5934 58 6035 54 8 6133 5318 6229 28 6322 37 64 14 72 57 7 5950 5617 6052 5527 61 51 36 6247 45 6341 54 6433 52 3 65 23 73 36 61 12 46 62 12 55 63 9 55 3 64 3 54 11 6456 53 19 6546 27 6634 74 58 4 6236 5713 6334 5621 6429 29 6521 36 66 12 43 67 50 6746 75 31 64 3 39 6458 46 6551 53 6642 55 6730 54 6 6816 53 12 69 76 57 6532 58 4 6625 5710 6716 5616 68 4 22 6850 28 6934 33 70 16 77 5921 67 4 27 6754 33 6843 38 6929 43 7012 48 7054 53 71 33 78 44 6839 49 6926 54 7012 58 7055 56 3 7136 55 7 72 15 54 11 7252 79 60 5 7016 59 9 71 5813 7143 5717 7223 21 73 1 25 7338 28 74 12 80 24 7155 28 7236 31 73 16 35 7353 38 7428 41 75 2 44 7534 81 42 7336 45 74 14 48 7450 51 7524 53 7557 56 7627 58 7657 82 58 7520 60 7554 59 3 7627 58 5 7657 57 7 7727 56 9 7754 55 11 78 21 83 61 12 77 6 14 7736 16 78 5 18 7832 19 7858 21 7922 22 7946 84 25 7853 26 79 19 28 7944 29 80 8 30 8030 31 8051 32 81 12 85 36 8042 36 81 4 38 8125 38 8145 39 82 4 40 822-1 41 82 38 86 44 8232 45 8250 46 83 7 46 8323 47 8338 47 8352 48 84 6 87 51 8423 51 8436 52 8449 52 85 1 53 8513 53 8523 53 85 34 88 56 86 15 56 8624 56 8632 56 8640 57 8648 57 8655 57 87 2 89 59 88 7 59 88 12 59 8816 59 8820 59 8824 59 8827 59 8831 90 62 90 61 90 60 90 59 90 58 90 57 90 56 90 302 Table 13. Kelvin's Simmer Line Table b a = 35 a = 36 a = 37 a = 38 a = 39 a = 40 a = 41 K Q K Q K Q K Q K Q K Q K Q o f 35 36 37 38 39 40 41 1 49 49 48 47 47 46 45 2 1 38 1 137 1 136 1 135 1 133 1 132 131 1 3 227 2 226 2 224 2 222 ey 220 f 218 f 216 2 4 317 4 3 14 4 3 12 4 3 9 4 3 7 4 3 4 4 3 1 4 5 4 6 6 4 3 6 59 6 56 6 53 6 50 6 46 6 6 55 9 51 g 447 9 444 9 440 9 436 9 431 9 7 544 12 539 12 535 12 531 12 526 13 521 13 5 17 13 8 633 16 628 16 623 16 618 16 613 17 6 7 17 6 2 17 9 722 20 716 20 7 11 20 7 5 21 59 21 53 21 47 21 10 811 25 8 5 25 58 25 52 26 745 26 739 26 732 26 11 9 30 53 30 846 31 839 31 832 31 824 31 8 17 32 12 48 36 941 36 934 37 926 37 918 37 9 10 37 9 2 38 13 1037 42 1029 43 1021 43 1013 43 10 4 44 55 44 47 44 14 1126 49 11 17 50 11 8 50 59 50 50 51 1041 51 1031 51 15 12 14 56 12 5 57 56 57 11 46 58 11 36 59 11 26 59 11 16 59 16 13 3 36 4 53 37 5 1243 38 5 1233 39 6 1222 40 7 12 11 41 7 12 42 7 17 51 13 1341 13 1330 14 13 19 15 13 8 16 56 16 45 16 18 1440 22 1429 22 14 17 23 14 6 24 54 25 1341 25 1329 26 19 1528 31 1516 32 15 4 33 52 34 1440 35 1426 35 14 13 36 20 16 16 41 16 4 43 51 44 1538 44 1525 45 15 11 46 57 46 21 17 4 52 51 54 1638 55 1624 55 16 10 56 56 57 1541 57 22 52 37 4 1738 38 5 1725 39 6 17 10 40 7 55 41 8 1641 42 9 1625 43 9 23 1840 16 1825 17 18 11 18 56 19 1740 20 1725 21 17 9 22 24 1928 28 1912 30 57 31 1842 32 1825 33 18 9 34 53 35 25 20 15 41 59 43 19 43 45 1927 46 19 10 47 53 48 1836 48 26 21 3 55 2046 57 20 29 59 20 13 41 55 42 1 1937 43 2 19 19 44 3 27 50 3810 2133 39 12 21 15 40 13 58 15 2040 16 2021 17 20 2 18 28 2237 25 22 19 27 22 1 29 2143 30 21 24 32 21 5 33 45 33 29 2324 41 23 5 43 47 45 2228 46 22 8 48 48 49 2128 49 30 2411 57 51 40 23 32 41 2 2312 42 3 52 43 5 2231 44 6 2210 45 6 31 57 3915 2437 17 24 17 19 57 21 2336 22 23 14 23 52 24 32 2544 33 2523 35 25 2 37 2441 39 2419 41 57 42 2334 43 33 2630 52 26 9 54 47 56 2525 58 25 2 44 2440 45 1 24 16 46 2 34 2716 40 11 54 41 14 2632 42 16 26 9 43 18 45 20 2522 21 58 22 35 28 2 31 2739 34 27 16 37 52 39 2628 40 26 4 41 2539 42 36 47 52 2824 56 28 58 2735 44 2711 45 2 46 46 3 2620 47 3 37 2932 41 14 29 8 42 18 44 4320 2818 22 53 24 2727 25 27 1 25 38 3017 37 52 41 2927 43 29 1 45 2835 47 28 8 48 41 48 39 31 2 42 1 3036 43 4 3010 44 7 44 45 9 2917 46 11 49 47 12 2821 48 12 40 46 26 31 20 29 53 32 3026 34 58 35 2930 36 29 1 37 41 3230 51 32 3 55 31 36 58 31 8 46 3039 47 1 3010 48 2 41 49 2 42 33 14 43 18 46 4421 32 18 4524 49 26 31 20 28 50 28 3020 28 43 58 45 3329 49 33 52 3230 53 32 55 31.30 55 59 55 44 3441 44 14 34 12 45 17 42 4620 33 11 4722 40 4823 32 9 4924 3137 5023 45 3524 43 54 46 3423 49 52 51 3320 52 48 53 32 15 52 Table 13. Kelvin's Simmer Line Table 303 b a =35 a = 36 a = 37 a = 38 a = 39 a = 40 a = 41 K Q K Q K Q K Q K Q K Q K Q 45 3524 4443 3454 4546 3423 4649 3352 4751 3320 4852 3248 4953 32 15 5052 46 36 6 4514 3535 46 17 35 4 4720 3432 4822 59 4923 3326 5023 53 5122 47 48 45 3616 49 44 51 3512 53 3438 54 34 4 54 3330 53 48 3730 4618 57 4721 3624 4824 51 4925 3517 5026 42 5126 34 7 5225 49 3811 52 3738 55 37 4 57 3630 59 55 59 35 19 59 43 57 50 52 4727 38 18 4830 43 4932 37 8 5033 3632 5133 56 5233 3519 5331 51 3932 48 3 57 49 6 3822 50 8 46 51 9 37 9 52 9 3632 53 8 55 54 6 52 4012 41 3936 43 39 45 3823 46 46 45 37 8 44 3630 42 53 52 49 19 40 15 5022 38 5123 39 5224 3822 5323 43 5421 37 4 55 18 54 41 31 59 53 51 2 4015 52 3 36 53 3 57 54 2 3818 59 38 56 55 42 9 5041 4130 43 52 43 40 12 43 3932 41 52 5539 38 11 5635 56 47 51 23 42 7 5225 41 28 5325 47 5424 40 7 5522 3926 56 19 44 57 15 57 4324 52 7 43 53 9 42 3 54 8 4122 55 7 41 56 4 59 57 1 3916 56 58 44 53 43 19 54 38 53 56 51 41 14 48 4031 44 48 5838 59 36 5340 54 5440 43 12 5539 4229 5636 46 5733 41 3 5828 4019 5921 60 4511 5428 4429 5528 46 5626 43 2 5723 42 18 5819 34 5913 49 60 6 61 46 5518 45 2 56 17 44 19 5715 34 58 11 49 59 6 42 4 59 41 18 51 62 1620 56 10 35 57 8 51 58 5 44 5 59 4320 54 34 6047 47 6138 63 53 57 3 46 7 58 4522 56 36 50 50 6043 43 3 61 35 42 15 6225 64 4725 57 39 54 52 5949 45 6 6042 4419 6134 31 6225 43 63 14 65 56 5853 47 9 5949 4622 6043 35 61 35 47 6226 58 6316 43 9 64 4 66 4827 5951 39 6046 51 61 39 46 3 6230 45 14 6320 4425 64 8 35 55 67 56 6050 48 8 61 44 47 19 6236 30 6326 40 64 15 51 65 2 44 6548 68 4925 61 51 36 6244 46 6334 56 6423 46 6 6511 4516 57 24 6641 69 53 6254 49 3 6345 4813 6434 4722 6522 31 66 8 40 6653 48 6736 70 5020 6358 29 6448 38 6535 46 6622 55 67 6 46 3 6750 45 10 6831 71 46 65 4 54 6552 49 2 6638 4810 6723 47 17 68 6 25 6848 32 6928 72 51 10 66 11 50 18 6658 26 6742 33 6825 39 69 7 46 6947 52 7026 73 34 6720 41 68 5 48 6848 54 6929 48 70 9 47 6 7047 4612 7125 74 57 6831 51 3 69 14 50 9 6955 49 15 7034 20 71 12 26 7149 30 7225 75 5218 6943 24 7024 29 71 3 34 7140 39 7216 44 7251 48 7325 76 38 7056 43 7135 48 72 12 52 7248 57 7322 48 1 7355 47 5 7427 77 57 72 11 52 1 7248 51 6 7323 50 9 7356 49 13 7429 17 75 20 7530 78 53 15 7328 18 74 2 22 7435 25 75 6 29 7536 32 76 5 35 7633 79 31 7445 34 75 17 37 7547 40 7617 43 7645 46 77 11 48 7737 80 46 76 4 49 7633 51 77 1 54 7728 56 7754 58 7818 8 7842 81 54 7724 53 2 7751 52 4 78 16 51 6 7841 8 79 4 4910 7926 12 7948 82 13 7846 14 79 9 16 7932 17 7954 19 SO 15 20 8035 22 8054 83 24 80 8 25 8029 26 8049 27 81 8 29 81 27 29 8144 31 82 1 84 33 81 31 34 8149 35 82 6 36 8223 37 8239 37 8254 39 83 9 85 41 8254 42 83 10 43 8324 43 8338 44 8352 44 84 4 45 8417 86 48 84 19 49 8431 49 8443 49 8454 50 85 5 50 85 15 50 8525 87 53 8544 54 8553 54 86 2 54 8610 54 86 18 54 8626 54 8633 88 57 87 9 57 8715 57 8721 57 8726 57 8732 57 8737 57 8742 89 59 8834 59 8837 59 8840 59 8843 59 8846 59 8848 59 8851 90 55 90 54 90 53 90 52 90 1 90 9 90 304 Table 13. Kelvin's Sumner Line Table b a = 42 a = 43 a = 44 a = 45 a = 46 a = 47 a = 48 K Q K Q K Q K Q K Q K Q K Q 42 43 44 45 46 47 48 1 45 44 43 42 42 41 40 2 129 1 1 28 1 1 26 1 1 25 1 1 23 1 1 22 1 1 20 1