TREATISE SURVEYING; " IN WHICH THE THEORY AND PEACTICE AEE FULLY EXPLAINED. PRECEDED BY A SHORT TREATISE ON LOGARITHMS: AND ALSO BY A COMPENDIOUS SYSTEM OF PLANE TRIGONOMETRY. %tyt fojjole f Ilttsiratefr bg $hromotts SAMUEL ALSOP, \ AUTHOR OP A TREATISE ON ALGEBRA, ETC. THIKD EDITION. < , PHILADELPHIA: E. C. & J. BIDDLE, No. 508 MINOR ST. (Between Market and CTiestnut, and Fifth and Sixth Sts.) 1865. PREFACE. THE favor shown to this treatise by the author's colaborers in the educational field having called for another edition of it, he has carefully revised the work, and made such amendments as to him seemed desirable. These are not numerous, but he believes have somewhat improved the work. His aim has been to present the subject, in its practical as well as its theoretical relations, in a manner adapted to the capa- city of every student, by presenting the theory plainly and com- prehensively, and giving definite and precise directions for prac- tice ; and to embrace in the work every thing which an extensive business in land-surveying would be likely to require. How nearly his object has been attained, others must determine : he trusts, however, that the treatise will be found to possess merit sufficient to commend it to the favorable notice of his fellow- teachers. The following brief synopsis of its contents presents the plan and scope of the work. Chapter I. consists of a short explanation of the nature and use of Logarithms. 5 f\ PREFACE. Chapter II. contains the geometrical definitions and con- structions needed in the subsequent part of the work. In Chapter III. is presented a treatise on Plane Trigono- metry, including a great variety of examples illustrative of the solution of triangles. In this chapter will also be found a full description of the Theodolite and Surveyor's Transit, and direc- tions for their use. In Chapter IY. the principles of surveying by the Chain are explained. This method is little employed by practical sur- veyors in this country. Since, however, the measurements require no other instrument than a tape-line, or a cord, or some other means of determining distances, it is of importance to the farmer, who frequently desires to know the contents of par- ticular fields, or of portions of enclosures. The second and third sections of this chapter contain a pretty full treatise on Field Geometry, or the method of performing on the ground, with the chain or measuring line only, those operations which are needed in fixing the positions of points or in locating lines. In Great Britain, Chain Surveying is almost exclusively em- ployed. Chapter Y. is devoted to Compass Surveying. Under this head are included all those methods which require the use of an instrument for determining the beariDgs of lines, whether that instrument be a Compass, a Transit, or a Theodolite. This chapter contains a full account of the methods to be employed in locating lines by means of such instruments. The numerous difficulties with which the surveyor will be likely to meet from obstructions on the ground are stated, and the modes of overcoming them explained. This chapter, with that on Plane Trigonometry, constitutes, in fact, a full treatise on Surveying as practised in this country. In selecting the methods to be employed in overcoming the difficulties both in Compass and in Chain Surveying, care has been taken to adopt such only as may be conveniently employed in the field. Chapter VI. contains the general principles of Triangular PREFACE. 7 Surveying. This is the method employed in extensive geodetic operations. The details of this method are so complex that a volume not a chapter would be required for their development. All that has been attempted is to give some of the more simple principles. Chapter VII. treats of Laying out and Dividing Land. It is believed that many of the demonstrations in this chapter will be found to be much more simple than those usually given, almost all of them having been reduced to the development of a single principle. On a subject of this kind, which has so long occupied the attention of mathematicians, any thing new could hardly be expected. It has been the aim of the author to select the best methods, not to introduce any thing merely because it was new. Chapter IX. contains a treatise on Practical Astronomy, embracing all that is needed for the surveyor's purposes or is practicable with his instruments. Various methods of running meridian lines, and of determining the latitude and the time of day, are fully explained. The concluding chapter (X.) is devoted to the subject of the Variation of the Compass. In it will be found information of great value to the practical surveyor. The tables of varia- tion are in all cases drawn from the most recent and authentic sources. In the preparation of this treatise the author has consulted various well-known English and American mathematical works. To Professor GILLESPIE'S excellent " Treatise on Land Surveying" (D. Appleton & Co., New York,) especially, the author is indebted for very valuable hints, particularly in the directions for prac- tice, the descriptions of the instruments, and various new methods of presenting important points. Some of these are referred to in their places. The typographical peculiarities of this volume, in the headings of articles, &c., were also suggested to the publishers by those of the work of Dr. Gillespie. In each department of the subject treated of in this volume 8 PREFACE. the aim of the author has been to explain clearly the principles involved, and, as a general rule, to give only those methods for practice which he deems the best. By pursuing this course he has kept the volume within moderate limits, and has presented, the subject in such a form as will, he trusts, meet the wants of teachers generally, as well as of very many practical surveyors. The tables appended to this treatise have been prepared with much care. That of Latitudes and Departures will be found to be more concise than those usually given, and, being extended to four decimal places, will enable the calculator to give greatei accuracy to his work. The table of Logarithms of Numbers has been carefully compared with those of Babbage, Hutton, and other standard authors. That of Sines and Tangents waa taken from Hutton, and compared with other seven-decimal tables. Besides these, there is a table of Natural Sines and Cosines to every minute, and one of Chords to every five minutes, of the quadrant. CONTENTS. CHAPTER I. ON THE NATURE AND USE OF LOGARITHMS. SECTION 1. On the Nature of Logarithms. PAQA Definition and Illustration 17 Mode of calculating Logarithms 19 Bases of Logarithms 19 Indices of Logarithms 20 Mantissse of Logarithms 20 Description of the Table of Logarithms 20 To find the Logarithm of a Number from the Table 21 To find the Natural Number corresponding to a given Logarithm 23 SECTION 2. On the Use of Logarithms. Multiplication by Logarithms 25 Division, by Logarithms 26 Involution by Logarithms 27 Evolution by Logarithms 27 On the Use of Arithmetical Complements of Logarithms 28 CHAPTER II. PRACTICAL GEOMETRY SECTION 1. Definitions 31 SBTION 2. Geometrical Properties and Problems ; 36 A. Geometrical Properties 36 B. Geometrical Problems 39 To bisect a given Straight Line 39 To draw a Perpendicular to a Straight Line from a Point in it 40 To let fall a Perpendicular to a Line from a Point without it 40 At a given Point, to make an Angle equal to a given Angle 41 To bisect a given Rectilineal Angle 42 To draw a Straight Line touching a Circle 42 Through a given Point to draw a Parallel to a given Straight Line 42 To inscribe a Circle in a given Triangle 43 To describe a Circle about a given Triangle 43 To find a Third Proportional to two Straight Lines 43 To find a Fourth Proportional to three Straight Lines 43 To find a Mean Proportional between two Straight Lines 44 To divide a Line into two Parts having a given Ratio 44 10 CONTENTS. CHAPTER III. PLANE TKIGONOMETRY. SECTION 1. Definitions. PAGE Measure of Angles 45 Trigonometrical Functions 46 Properties of Sines, Tangents, &c - 47 Geometrical Properties employed in Plane Trigonometry 48 SECTION 2. Drafting or Platting. Mode of drawing Straight Lines 49 Mode of drawing Parallels 49 Mode of drawing Perpendiculars 51 Mode of drawing Circles and Arcs 51 Mode of laying off Angles with a Protractor 52 By a Scale of Chords 52 By a Table of Chords 53 Distances 53 Drawing to a Scale 53 Scales 55 Diagonal Scale 55 Proportional Scale 57 Vernier Scale 57 SECTION 3. Tables of Trigonometrical Functions. Description of the Table of Natural Sines and Cosines 58 Description of the Table of Logarithmic Sines and Tangents 59 Use of Table 60 Table of Chords 63 SECTION 4. On the Numerical Solution of Triangles. Definition 64 The Numerical Solution of Right-Angled Triangles 64 By the Use of the Table of Sines and Tangents 64 By the Application of (47.1.) 66 The Numerical Solution of Oblique- Angled Triangles. The Angles and one Side, or two Sides and an Angle opposite one of them, being given, to find the rest 67 Two Sides and the included Angle being given, to find the rest. Rulel 70 Rule 2 71 The three Sides being given, to find the Angles. Rulel 73 Rule 2 74 SECTION 5. Instruments, and Field Operations. The Chain 76 The Pins 78 Chaining , 78 Recording the Outs 79 Horizontal Measurement 80 Tape Lines 82 Angles 82 The Transit and Theodolite. General Description 83 The Telescope 87 CONTENTS. . 11 PAGE The Object Glass 88 The Eye Piece 88 The Spider Lines 89 The Supports 91 The Vertical Limb 91 The Levels 92 The Levelling Plates 92 The Clamp and Tangent Screws...., 93 The Watch Telescope Verniers , 93 The Reading of the Vernier 95 To Read any Vernier , 96 Retrograde Verniers 96 Reading backwards 98 Double Verniers 98 Adjustments 101 First Adjustment: The Level should be parallel to the Horizontal Plates 102 Second Adjustment : The Axis of the Horizontal Plates should be pa- rallel 102 Third Adjustment: The Line of Collimation must be perpendicular to the Horizontal Axis 102 The Line of Collimation in the Theodolite should be parallel to the Axis of the Cylinders on which the Telescope rests in its Ys 104 Fourth Adjustment: The Horizontal Axis must be parallel to the Horizontal Plates 104 Adjustments of the Vertical Limb 105 First Adjustment : The Level must be parallel to the Line of Colli- mation 105 Second Adjustment : The Zeros of the Vernier and Vertical Limb should coincide when the Telescope is horizontal 106 Measuring Angles 107 Repetition of Angles 108 Verification of Angles 109 Reduction to the Centre 109 Angles of Elevation 110 SECTION 6. Miscellaneous Problems to Illustrate the Rules of Plane Trigono- metry 110 CHAPTER IV. CHAIN SURVEYING. SECTION 1. Definitions. Definition 118 Advantages 118 Area Horizontal , 119 SECTION 2. Field Operations. Ranging out Lines 119 To Interpolate Points in a Line 120 On Level Ground 120 Over a Hill 120 By a Random Line 121 Across a Valley 122 To determine the Point of Intersection of two visual Lines 123 To run, a Line towards an invisible Intersection 123 Perpendiculars. To draw a Perpendicular to a given Line from a Point in it. When the Point is accessible 123 12 CONTENTS. PAGH When the Point is inaccessible 125 To let fall a Perpendicular to a Line from a point without it. When the Point and Line are both accessible 125 When the Point is remote or inaccessible 126 When the Line is inaccessible 126 The Surveyor's Cross 127 To verify the Cross 128 The Optical Square 128 To test the Accuracy of the Square 129 Parallels Through a given Point to draw a Parallel to an accessible Line 130 To draw a Parallel to an inaccessible Line 130 To draw a Parallel to a Line through an inaccessible Point 130 SECTION 3. Obstacles in Running and Measuring Lines. To prolong a Line beyond an Obstacle 131 To measure a line when both ends are accessible 132 When one End is inaccessible 133 When the inaccessible End is the intersection of two Lines 133 When both Ends are inaccessible 134 SECTION 4. Keeping Field Notes , 135 Field Book 135 Test Lines 139 General Directions 139 Platting the Survey 140 SECTION 5. Surveying Fields of Particular Form. Rectangles 141 Parallelograms 141 Triangles. First Method 142 Second Method 142 Trapezoids 144 Trapeziums. First Method 145 Second Method 145 Fields of more than four Sides. First Method 147 Second Method 150 Offsets 151 SECTION 6. Tie Lines. Inaccessible Areas 159 Defects of the Method 159 CHAPTER V. COMPASS SURVEYING. SECTION 1. Definitions and Instruments. The Meridian 160 The Points of the Compass 161 Bearing 161 Reverse Bearing 162 The Magnetic Needle 162 The Magnetic Meridian 163 CONTENTS. 13 PAGS The Magnetic Bearing 163 The Compass... 164 The Sights * 166 The Verniers 166 The Pivot 168 The Divided Circle 168 Adjustments -. 169 Defects of the Compass 169 SECTION 2. Field Operations. Bearings 170 Use of the Vernier 171 The Reverse Bearing 171 Local Attraction 171 To correct for Back Sights 172 By the Vernier 172 To survey a Farm General Directions., 172 Random Line 173 To determine the Bearing by a Station near the Middle of the Line 174 Proof Bearings 174 Angles of Deflection 175 SECTION 3. Obstacles in Compass Surveying. To run a Line making a given Angle with a given Line at a given Point within it 176 To run a Line making a given Angle with a given inaccessible Line at a given Point in that Line 177 From a given Point out of a Line, to run a Line making a given Angle with that Line. If the Line be accessible 177 If the Line be inaccessible 178 If the Point be inaccessible, 178 If the Point and the Line be both inaccessible , 179 To run a Line parallel to a given Line through a given Point. If the Line and the Point be accessible 179 If the Point be inaccessible 179 If the Line be inaccessible 179 If the Line and the Point both be inaccessible 180 Prolongation and Interpolation of Lines 180 To Prolong a Line beyond an Obstruction 181 To Interpolate Points in a Line 182 By a Random Line 182 Measurement of Distances. To determine the Distance between two Points visible from each other.... 183 To determine the Distance on a Line to the inaccessible but visible end 185 To determine the Distance when the end is invisible 186 To determine the Distance to the Intersection of two Lines..-. 186 To determine the Distance between two inaccesible Points 187 Examples illustrative of the preceding Rules 188 SECTION 4. Field Notes 190 SECTION 5. Latitudes and Departures. Definitions 192 The Bearing, Distance, Latitude, and Departure, any two being given, to determine the others 193 To determine the Latitude and Departure by the Traverse Table 194 When the Bearing is given by Minutes 196 14 CONTENTS. PAGE By the Table of Natural Sines and Cosines 197 Test of the Accuracy of the Survey 199 Correction of Latitudes and Departures 200 SECTION 6. Platting the Survey. With the Protractor 202 By a Scale of Chords 203 By a Table of Natural Sines 204 By a Table of Chords 205 By Latitudes and Departures 205 SECTION 7. Problems in Compass Surveying. Given the Bearing of one Side, and the Deflection of the next, to deter- mine its Bearing 208 To determine the Deflection between two Courses 209 To determine the Angle between two Lines 210 To change the Bearings of the Sides of a Survey 211 SECTION 8. Supplying Omissions. The Bearings and Distances of all the Sides except one being given, to determine these 213 All the Bearings and Distances except the Bearing of one Side and the Distance of another being given, to find these 217 All the Bearings and Distances except two Distances being given, to find these 219 All the Bearings and Distances except two Bearings being given, to find these 220 SECTION 9. Content of Land. Given two Sides and the included Angle of a Triangle or Parallelogram, to find the Area 224 The Angles and one Side of a Triangle being given, to find the Area 225 To determine the Area of a Trapezium, three Sides and the two included Angles being given 226 The Bearings and Distances of the Sides of a Tract of Land being given, to find its Area 229 Offsets 235 Inaccessible Areas 238 Compass Surveying by Triangulation 243 CHAPTER VI. TRIANGULAR SURVEYING. Base 247 Reduction to the Level of the Sea 248 Signals 248 Triangulation 248 Base of Verification 250 CHAPTER VIL LAYING OUT AND DIVIDING LAND. SECTION 1. Laying out land. To lay out a given Quantity of Land in the form of a Square 251 To lay out a given Quantity of Land in the form of a Rectangle, one Side being given 251 The Adjacent Sides having a given Ratio 252 CONTENTS. 15 PAGE One Side to exceed another by a given Difference 252 To lay out a given quantity of Land in the form of a Triangle or Paral- lelogram, the Base being given 253 One Side and the Adjacent Angle being given 253 Lemma 254 The Direction of two Adjacent Sides being given, to lay out a given quantity of land. By a Line running a given Course , 255 By a Line running through a given Point 256 Three Adjacent Sides of a Tract being given in Position, to lay out a given quantity of land 259 By a Line parallel to the second Side 259 By a Line running a given Course 262 By a Line througha given Point 267 By the shortest Line 269 To cut off a Plat containing a given Area from a Tract of any number of Sides. By a Division line drawn from one of the Angles 269 By a Line running a given Course 273 To straighten Boundary lines 275 To run a new Line between Tracts of different Values. By a Line running a given Course 280 By a Line through a given Point in the old Line 281 By a Line through a given Point in one of the Adjacent Sides 283 SECTION 2. Division of Land. To divide a Triangle into two Parts having a given Ratio. By a Line through one of the Corners 284 By a Line through a Point in one of the Sides 284 By a Line Parallel to one of the Sides 286 By a Line running a given Course 286 By a Line through a given Point 288 To divide a Trapezoid into two parts having a given Ratio. By a Line cutting the Parallel Sides 290 By a Line Parallel to the Parallel Sides 292 To divide a Trapezium into two parts having a given Ratio. By a Line through a given Point on one Side 294 By a Line through any Point 296 By a Line Parallel to one Side 298 By a Line running a given Course , 301 CHAPTER VIII. MISCELLANEOUS EXAMPLES. Miscellaneous Examples 303 CHAPTER IX. MERIDIANS, LATITUDE, AND TIME. SECTION 1. Meridians. Definition 307 To run a Meridian Line. By equal Altitudes of the Sun 308 By a Meridian Altitude of Polaris 309 To determine the Time Polaris is on the Meridian 310 To run a Meridian by a Meridian Passage observed with a Transit or Theodolite .. 314 16 CONTENTS. PAGE By an Observation of Polaris at its greatest Elongation 314 By Equal Altitudes of a Star 318 SECTION 2. Latitude. To determine the Latitude by a Meridian Altitude of Polaris 319 By a Meridian Altitude of the Sun 319 By an Observation on a Star in the Prime Vertical 320 SECTION 3. To find the Time of Day. By a Meridian Line 322 By an observed Meridian Passage of a Star 322 By an Altitude of the Sun or a Star not in the Meridian 323 CHAPTER X. VARIATION OF THE COMPASS. Secular Change 325 Table of Variations 326 Line of no Variation 326 To determine the Change in Variation by old Lines 327 Diurnal Changes 329 Irregular Changes \ 329 APPENDIX. Demonstration of the Rule for finding the Area of a Triangle when three Sides are given 332 A TREATISE ON SURVEYING, CHAPTER I. ON THE NATURE AND USE OF LOGARITHMS. SECTION I. ON THE NATURE OF LOGARITHMS. 1. Definition. LOGARITHMS are a series of numbers, by the aid of which the operations of multiplication, division, the raising of powers, and the extraction of roots, may, respectively, be performed by addition, subtraction, multi- plication, and division. Such a series may be thus constructed. Above a geometric series, the first term of which is 1, place a corresponding arithmetic series, the first term of which is ; thus : Arithmetical series, 0123456 7 8 Geometrical series, 1 2 4 8 16 32 64 128 256 To determine the product of any two terms of the geometric series, it is evidently only necessary to add the correspond- ing terms of the arithmetic series, and to notice the term of the geometric series agreeing to their sum ; which term is the product required. Thus, to find the product of 4 and 32, we add the corresponding terms, 2 and 5, in the arithmetic series. Their sum, 7, corresponds to 128, the product required. 2. In a table of logarithms, the terms of the geometrical series are called the numbers; the ratio in this series is de- nominated the base of the table ; and the terms of the arith- metical series are called the logarithms of the corresponding 2 17 18 THE NATURE AND USE OF LOGARITHMS. [CHAP. I. terms of the geometric series. The numbers, it will be observed, are the powers of the base, and the logarithms are the indices of those powers. Further to illustrate the use of logarithms, we give the following table : Num. Log. Num. Log. Num. Log. 2 4 8 16 32 1 2 3 4 5 64 128 256 512 1024 6 7 8 9 10 2048 4096 8192 16384 32768 11 12 13 14 15 1. Eequired the quotient of 32768 divided by 2048. The indices or logarithms of these numbers are, respectively, 15 and 11. The difference of these logarithms is 4, which is the logarithm of 16, the quotient required. Hence the difference of the logarithms of two numbers is the logarithm of their quotient. 2. Required the third power of 32. The logarithm of 32 is 5. Multiply this by 3, the index of the power to which 32 is to be raised, and the product, 15, is the index of 32768, the required power. Hence, to involve a number to a given power, we multiply its logarithm by the index of the power to which it is to be raised. 3. Required the fourth root of 4096. The index of this is 12. Divide this index by 4, the degree of the root to be extracted, and the quotient will be 3, which is the logarithm of 8, the root required. Hence, to extract the root of a number, we divide its logarithm by the number expressing the degree of the root to be extra6ted, and the quotient is the logarithm of the root required. 3. The table in Art. 2 contains only the integral powers of 2, that being sufficient for the purpose of illustra- tion; but a complete table contains all the numbers of the natural series, within the limits of the table, together with the indices, or logarithms. The logarithms in such a table will, in most instances, be fractions. Thus, the logarithms corresponding to any of the num- bers between 4 and 8 would be 2 and some fraction ; SEC. I.] THE NATURE OF LOGARITHMS. 19 of any number between 8 and 16, the logarithm would be 3 and a fraction ; and so on. 4. Calculation of Logarithms. Since all numbers are considered as the power of some one base, we will have, if a be the base, and n the number, a x = n. The deter- mination of the logarithm will then consist in solving the above equation so as to find x. This, in general, can only be done by approximation. The details to which it would lead are entirely foreign to the present work. Those who desire to become acquainted with the subject may consult the author's " Treatise on Algebra." 5. Bases. Theoretically, it is of no importance what number is assumed as the base of the system ; but prac- tical convenience suggests that 10, the base of our system of notation, should also be the base of the system of loga- rithms. By the use of this base, it becomes unnecessary to insert in the table of logarithms their integral portions. For, as will be seen hereafter, the figures in the decimal por- tion of the logarithm depend on the figures in the number, while the integral portion of the logarithm depends solely on the position of the decimal point in the number. 6. Assuming, then, 10 for a base, we have the following series : lumbers, 1, 10, 100, 1000, 10000, 100000, 1000000; Logarithms, 01234 5 6. The logarithm of any number between 1 and 10 will be wholly decimal; between 10 and 100, it will be 1 and a decimal ; and so on. If the powers of 10 be continued downwards, we have the powers 1 .1 .01 .001 .0001 .00001, and indices 1 2 3 4 5. The logarithm of any number between .1 and 1 is there- fore 1 + a decimal, of a number between .01 and .1 it is 2 -f a decimal, &c. 20 THE NATURE AND USE OF LOGARITHMS. [CHAP. I. 7. Indices of Logarithms. The integral portion of every logarithm is called the index, the decimal portion being sometimes called the mantissa. From the above series, it is manifest that, if the number is greater than 1, the index is positive, and one less than the number of in- tegral figures. Thus, 246.75 coming between 100 and 1000, its logarithm will be 2 and a decimal. If the num- ber is less than 1, the index will be negative. For ex- ample, the logarithm of .0024675, which comes between .001 and .01, will be 3 + a decimal. 8. Mantissse. The mantissas of logarithms to the base 10 depend solely on the figures of the number, without any regard to the position of the decimal point. Let the logarithm of 31.416 be 1.497151: then, since 314.16 is 10 times 31.416, its logarithm will be 1.497151 + 1 = 2.497151. Similarly, the logarithm of 31416, which is 1000 times 31.416, will be 1.497151 + 3 = 4.497151. Again, .031416 = 31.416 -f- 1000 : its logarithm is there- fore 1.497151 - 3 = 2.497151, in which the sign is understood to belong solely to the index 2, and not to the mantissa. Since, then, the index can be supplied by atten- tion to the position of the decimal point, the mantissse alone are inserted in the body of a table of logarithms. The annexed table will illustrate the above more fully : Number. Logarithm. 64790 4.811508 6479 3.811508 647.9 2.811508 64.79 1.811508 6.479 0.811508 .6479 1.811508 .06479 2.811508 .006479 3.811508. 9. Table of Logarithms. A table of logarithms consists of the series of natural numbers, with their logarithms, or, rather, the mantissas of their logarithms, so arranged that SEC, L] THE NATURE OF LOGARITHMS. 21 one can be readily determined from the other. In the table of logarithms appended to this treatise, the mantissse of the logarithms of all numbers, from 1 to 9999 inclusive, are given. On the first page are found the numbers from 1 to 99, with their logarithms in full. The remaining pages contain only the mantissse of the logarithms. The first column, headed $T, contains the numbers, from 100 to 999 ; and the second, headed 0, the mantissse of their logarithms. Thus, the logarithm of the number 897 is 2.952792; the index being 2, because there are three integral figures in the number. The remaining columns contain the last four figures of the mantissse of the logarithms of numbers of four figures, the first three of which are found in the first column, and the fourth, at the head. Thus, if the number were 8976, the last four figures 3083 of the mantissa of its loga- rithm would be found in the column headed 6 ; the first two, 95, found in the second column, being common to them all. The logarithm of 8976 is, therefore, 3.953083. 10. To denote the point in which the second figure changes, when such change does not take place in the first logarithmic column, the first of the four figures from the change to the end of the line is printed as an index figure ; thus, on page 25 of the tables, we have the lines N. 1 2 3 4 5 6 7 8 9 456 457 458 8965 9916 660865 9060 011 0960 9155 106 1055 9250 201 1150 9346 296 1245 9441 391 1339 9536 486 1434 9631 581 1529 9726 676 1623 9821 771 1718 In such cases the first two figures are found in the next line. The logarithm of 4575 is, therefore, 3.660391. 11. To find the Logarithm of a number from the tables. If the number consists of one or two figures only, its logarithm is found on the first page of the table. If the two figures are both integers, the index is given also ; but, if the one or both figures be decimal, the decimal part only 22 THE NATURE AND USE OF LOGARITHMS. [CHAP. I. of the logarithm should be taken out. Thus, the loga- rithm of 8 is 0.903090 ; of 59 is 1.770852. If the number be wholly or part a decimal, the index must be changed in accordance with the principles laid down in Art. 7. Thus, the index must be one less than the number of figures in the integral part of the natural num- ber. But when the natural number is wholly a decimal the index is negative, and must be one more than the num- ber of ciphers between the first significant figure and tho decimal point. Thus, the logarithm of .8 is 1.903090 ; of .059 is 2.770852. If the number consists of three figures, look for it in the remaining pages of the table, in the column headed N". Opposite to it, in the first column, will be found the deci- mal portion of the logarithm ; the first two figures of the logarithm, being common to all the columns, are printed but once, to save room. Thus, the logarithm of 272 is 2.434569 ; of 529 is 2.723456 ; the index being placed in accordance with the above rule. If the number consists of four figures, the first three must be found as before ; and the fourth, at the top of the table. The last four figures of the logarithm are found opposite to the first three figures of the number, and under the fourth; the first two figures of the logarithm being found in the first logarithmic column. Thus, if the num- ber were 445.8, look for 445 in the column headed IN", and opposite thereto, in the column headed 8, the figures 9140 are found; these affixed to 64, found in the first column, give 649140 for the decimal portion of the logarithm ; and, as there are three integral figures, the index is 2. Hence, the complete logarithm is 2.649140. If there are more than four figures in the number, find the logarithm of the first* four figures as before. Take the difference between this logarithm and the next greater in the table ; multiply this difference by the remaining figures in the number,, and from the product separate as many figures from the right hand as are contained in the mul- SEC. I.] THE NATURE OF LOGARITHMS. 23 tiplier ; then add the remainder to the logarithm first taken out : the sum will he the required logarithm. Let the logarithm of 6475.48 be required. The logarithm of 6475 is .811240 The next greater is 1307 ~67 67 x 48 = 32,16 32 added to 811240 gives .811272 ; and the index being 3, the complete logarithm is 3.811272. Next let the logarithm of .0026579 be required. The logarithm of 2657 is .424392 The next greater 4555 Difference 163 9 146,7 424392 + 147 = .424539, and the index being -3, the com- plete logarithm is 3.424539. NOTE. In this last example, the product is 1467 : the figure stricken off being 7, which is more than 5, 147 is taken instead of 146. EXAMPLES. Required the logarithms of the following numbers : 1. Of 7.5 0.875061 2. Of 876 2.942504 3. Of 93.37 1.970207 4. Of .4725 1.674402 5. Of .869427 1.939233 6. Of .01367 2.135769 7. Of .0645775 2.810081 8. Of .004679 3.670153 9. Of 37196.2 4.570499 10. Of .14638 1.165482 11. Of 6273.69 3.797523 12. Of .037429 2.573208 12. To find the natural number corresponding to a given Logarithm. If four figures only be needed in the answer, seek in the columns of logarithms for the one near- est to the decimal part of the given logarithm : the first three figures of the natural number will be found in the column marked N ; and the fourth, at the top of the column in which the logarithm is found. When the index is positive, the number of integral 24 THE NATURE AND USE OF LOGARITHMS. [CHAP. I. figures will be one greater than the number expressed by the index ; but, if the index is negative, the number will be wholly decimal, and have one less cipher between the decimal point and the first significant figure than the num- ber expressed by the index. Thus, the natural number corresponding to the logarithm 2.860996 is 726.1; and that corresponding to 2.860996 is .07261. If the logarithm be found exactly in the tables, and there be not enough figures in the corresponding number, the deficiency must be supplied by ciphers. Thus, the natural number corresponding to 6.891649 is 7792000. But, if five or six figures be required, find in the table the logarithm next less than the given one, and take out the corresponding number as before; subtract this loga- rithm from the next greater in the table, and also from the given logarithm; annex one or two ciphers to the latter remainder, according as five or six figures are required, and divide the result by the former. The quotient annexed to the figures first taken out will give the figures required, the decimal point being placed as before. Required the number corresponding to 2.649378, to six figures Given logarithm .649378 Next less .649335 cor. num. 4460 Difference 43 Next greater logarithm .649432 Next less .649335 Difference ~~97)4300(44 388 420 388 32 Hence, the number is 446.044. EXAMPLES. Required the natural numbers corresponding to the fol- lowing logarithms. SEC. II.] ON THE USE OF LOGARITHMS. 25 1. 2.46T415 2. 1.396143 3. 2.041637 4. 3.167149 Ans. 293.37 .24897 110.062 .0014694 5. 4.617392 6. 1.947138 7. 2.960014 8. 2.760116 Ans. 41437.3 88.54 . .091204 .057559 SECTION II. ON THE USE OF LOGARITHMS. 13. Multiplication. To multiply numbers by means of logarithms. Add together the logarithms of the factors, and take out the natural number corresponding to the sum. If any of the indices be negative, the figure to be carried from the sum of the decimal portions must be con- sidered positive, and added to the sum of the positive, or subtracted from the sum of the negative indices. Then collect the affirmative indices into one sum, and the nega- tive into another, take the difference between these sums, and prefix thereto the sign of the greater sum. EXAMPLES. Ex. 1. Multiply 47.25 and 397.3. log. 1.674402 47.25 397.3 Product, 18772.5 " 2.599119 4.273521 Ex. 2. Required the product of 764.3, .8175, .04729, and .00125. log. 2.883264 " 1.912488 2.674769 3.096910 764.3 .8175 .04729 .00125 Product, .0369344 2.567431 Ex. 3. Required the product of 87.5 and 6.7. Ans. 586.25. 26 THE NATURE AND USE OF LOGAEITHMS. [CHAP. I. Ex. 4. Required the continued product of .0625, 41.67, .81427, and 2.1463. Ans. 4.5516. Ex. 5. Multiply 67.594, .8739, and 463.92 together. Ans. 27404. Ex. 6. Multiply 46.75, .841, .037654, and .5273 together. Ans. .780633. Ex. 7. Multiply .00314, 16.2587, .32734, .05642, and 1.7638 together. Ans. .001663. 14. Division. To divide numbers by logarithms. Subtract the logarithm of the divisor from that of the dividend : the remainder will be the logarithm of the quotient. If one or both of the indices are negative, subtract the decimal portions of the logarithm as before ; and, if there be one to carry from the last figure, add it to the index of the divisor, if this be positive, but subtract if it be nega- tive ; then conceive the sign of the result to be changed, and if, when so changed, the* two indices have the same sign, add them together ; but, if they have different signs, take their difference and prefix the sign of the greater. EXAMPLES. Ex. 1. Divide 6740 log. 3.828660 by 87 log. 1.939519 Quotient, 77.471 Ex. 2. Divide 86.47 by .0124 Quotient, 6973.4 Ex. 3. Divide .0642 by 87.63 Quotient, .00073263 Ex. 4. Divide .0642 by .008763 1.889141 log. 1.936865 log. 2.093422 14.7765 .012642 3.843443 log. 2.807535 log. 1.942653 4.864882 log. 2.807535 log. 3.942653 Quotient, 7.3263 0.864882 Ex. 5. Divide 407.3 by 27.564. Ans. Ex. 6. Divide .80743 by 63.87. Ans. SEC. II.] ON THE USE OF LOGARITHMS. 27 Ex. 7. Divide 963.7 by .00416. Ans. 231659. Ex. 8. Divide 86.39 by .09427. Ans. 916.41. Ex. 9. Divide .006357 by .0574. Ans. .11075. Ex. 10. Divide 76.342 by .09427. Ans. 809.82. 15. To involve a number to a power. Multiply the logarithm of the number by the index of the power to which it is to be raised. If the index of the logarithm is negative, and there is any thing to be carried from the product of the decimal part by the multiplier, instead of adding this to the pro- duct of the index, subtract it: the difference will be the index of the product, and will always be negative. Ex. 1. Required the fourth power of 5.5. 5.5 log. 0.740363 4 915.065 2.961452. Ex. 2. Required the fifth power of .63. .63 log. 1.799341 5 .099244 2.996705. Ex. 3. Required the fourth power of 7.639. Ans. 3405.24. Ex. 4. Required the third power of .03275. Ans. .00003513. Ex. 5. "What is the fifteenth power of 1.06 ? Ans. 2.3966. Ex. 6. What is the sixth power of .1362 ? Ans. .0000063836. Ex. 7. What is the tenth power of .9637 ? Ans. .69091. 16. To extract a given root of a number. Divide the logarithm of the number by the degree of the root to be extracted : the quotient will be the logarithm of the root. If the index of the logarithm is negative, and does not 28 THE NATURE AND USE OF LOGARITHMS. [CHAP. I. contain the divisor an exact number of times, increase it by so many as are necessary to make it do so, and carry the number so borrowed, as so many tens to the first figure of the decimal. Ex. 1. Extract the fourth root of 56.372. 56.372 log. 4)1.751063 ftesult, 2.7401 .437766 Ex. 2. Extract the fifth root of .000763. .000763 log. 5) 4.882525 Result, .23796 1.376505. Ex. 3. What is the fifth root of .00417 ? Ans. .3342. Ex. 4. Required the fourth root of .419. Ans. .80455. Ex. 5. Required the tenth root of 8764.5. Ans. 2.479. Ex. 6. Required the seventh root of .046375. Ans. .6449. Ex. 7. Required the fifth root of .84392. Ans. .96663. Ex. 8. Required the sixth root of .0043667. Ans. .40429. 17. Arithmetical Complements. When several num- bers are to be added, and others subtracted from the sum, it is often more convenient to perform the operation as though it were a simple case of addition. This may be done by conceiving each subtractive quantity to be taken from a unit of the next higher order than any to be found among the numbers employed ; then add the results with the additive numbers, and deduct from the result as many units of the order mentioned as there were subtractive numbers. The difference between any number and a unit of the next higher order than the highest it contains is called the arithmetical complement of the number. Thus, the arithmetical complement of 8765 is 1235. It is easily ob- tained by taking the first significant figure on the right from ten, and each of the others from nine. This may be done mentally, so that the arithmetical complements need not be written down. Thus, suppose A started out with 375 dollars to collect SEC. II.] ON THE USE OF LOGARITHMS. 29 some bills and to pay sundry debts. From B he received $104, to D he pays $215, to E he pays $75, from F he re- ceives $437, and, finally, pays to Gr $137. How much has he left? 375 . 104 which are added as though they were 375' 104 --215 75 437 137 489 785 925 437 863 3489, deducting 3000 from the final result 3489, because there were three subtractive quantities. The arithmetical complements of logarithms are gene- rally employed where there are more subtractive logarithms than one. To give symmetry to the result, it would be neater to employ them in all cases. To a person who has much facility in calculation, it is most convenient to write down the logarithm as taken from the table, and obtain the arithmetical complement as the work is carried on. Thus, in the example above, the numbers could be written as in the first column ; but in the addition, instead of em- ploying the figures as they appear in the subtractive num- ber, the complement of the first significant figure to ten, and of the others to nine, should be employed. As an example of the use of the arithmetical comple- ments of the logarithms of numbers, let it be required to 27 475 work by logarithms the proportion as : -^=- : : 125 : x. oo 17 Here, as the first term is a fraction, it will have to be in- verted ; and the question will be the same as finding the 55 x 475 x 125 value of 27 x l\ log. u 27 17 55 475 125 1.431364^ which are 1.230449 added as 1.740363 V though 2.676694 they were 2.096910 j written Result, 7114.66 3.852154 A. C. 8.568636 A. C. 8.769551 1.740363 2.676694 2.096910 3.852154T 30 THE NATURE AND USE OP LOGARITHMS. [CHAP. I. deducting 20, because there were two arithmetical comple- ments employed. In the examples wrought out in the subsequent part of this work, the arithmetical complements of the logarithms of the first term of every proportion are employed. CHAPTER II. PRACTICAL GEOMETRY. SECTION I. DEFINITIONS, 18. THE practical surveyor will find a good knowledge of Algebra and of the Elements of Geometry an invaluable aid not only in elucidating the principles of the science, but in enabling him to overcome difficulties with which he will be certain to meet. In fact, so completely is Survey- ing dependent on geometrical principles, that no one can obtain other than a mere practical knowledge of it, without first having mastered them; and he who depends solely on his practical experience will be certain to meet with cases which will call for a kind of knowledge which he does not possess, and which he can obtain only from Geometry. Every student, therefore, who desires to become an in- telligent surveyor, should first study Euclid, or some other treatise on Geometry. He will then have a key which will not only unlock the mysteries contained in the ordinary practice, but which will also open the way to the solution of all the more difficult cases which occur. To those who have taken the course above recommended, the problems solved in the present chapter will be familiar. They are inserted for the benefit of those who may not be thus pre- pared, and also as affording some of the most convenient modes of performing the operations on the ground. 19. Geometry is the science of magnitude and position. 31 32 PRACTICAL GEOMETRY. [CHAP. II. 20. A solid is a magnitude having length, breadth, and thickness. All material bodies are solids, and so are all portions of space, whether they are occupied with material substances or not. Geometry, treating only of dimension and posi- tion, has no reference to the physical properties of matter. 21. The surfaces of solids are superficies. A superficies has, therefore, only length and breadth. 22. The boundaries of superficies, and the intersection of superficies, are lines. Hence, a line has length only. 23. The extremities of lines, and the intersections of lines, are points. A point has, therefore, neither length, breadth, or thickness. 24. A pointy therefore, may be defined as that which has position, but not magnitude. 25. A line is that which has length only. 26. A straight line is one the direction of which does not change. It is the shortest line that can be drawn between two points. 27. A superficies has length and breadth only. 28. A plane superficies, generally called simply a plane, is one with which a straight line may be made to coincide in any direction. 29. A plane rectilineal angle, or sim- ply an angle, is the inclination of two lines which meet each other. (Fig. 1.) A. 30. An angle may be read either by the single letter at SEC. L] DEFINITIONS. 33 the intersection of the lines, or by three letters, of which that at the intersection must always occupy the middle. Thus, (Fig. 1,) the angle between BA and AC may be read simply A or BAG. 31. The magnitude of an angle has no reference to the space included between the lines, nor to their length, but solely to their inclination. 32. Where one straight line stands on another so as to make the adjacent angles equal, each of these angles is called a right angle; and the lines are said to be perpendicular to each other. Thus, (Fig. 2,) if ACD = BCD, each is a right angle, and CD is perpendicular to AB. A c B 33. An angle less than a right angle is called an acute angle. Thus, BCE or ECD (Fig. 2) is an acute angle. 34. An angle greater than a right angle is called an obtuse angle. ACE (Fig. 2) is an obtuse angle. 35. The distance of a point from a straight line is the length of the perpendicular from that point to the line. 36. Parallel straight lines are those of which all points in the one are equidistant from the other. 37. A. figure is an enclosed space. 38. A triangle is a figure bounded by three straight lines. 39. An equilateral triangle is one the three sides of which, are equal. 40. An isosceles triangle is one of which two of the sides are equal. The third side is called the base. 34 PRACTICAL GEOMETRY. [CHAP. II. 4L A scalene triangle has three unequal sides. 42. A right-angled triangle has one of its angles a right angle. 43. The side opposite the right angle is called the hypo* thenuse, and the other sides, the legs. 44. An obtuse-angled triangle has one of its angles obtuse. 45. A quadrilateral figure is bounded by four sides. Fig. 3. 46. A parallelogram (Fig. 3) is a quadrilateral, the opposite sides of which are parallel. 47. A rectangle (Fig. 4) is a parallelogram, the adjacent sides of which are perpendicular to each Fig. 4. other. Thus, ABCD is a rectangle. A A rectangle is read either by naming the letters around it in their order, or by naming two of the sides adjacent to any angle. Thus, the rectangle ABCD is read the rectangle AB.BC. Whenever the rectangle of two lines, such as DE.EF, is spoken of, a rectangular parallelogram, the adjacent sides of which are equal to the lines DE and EF, is meant. 48. A square is a rectangle, all the sides of which are equal. 49. A rhombus is an oblique parallelogram, the sides of which are equal. 50. A rhomboid is an oblique parallelogram, the adjacent sides of which are unequal. SEC. I.] DEFINITIONS. 35 51. All quadrilaterals that are not parallelogiams are called trapeziums. 52. A trapezoid is a trapezium, having two of its sides parallel. 53. Figures of any number of sides are called polygons, though this term is generally restricted to those having more than four sides. 54. The diagonal of a figure is a line joining any two opposite angles. 55. The base of any figure is the side on which it may be supposed to stand. Thus, AB (Fig. 5) is the base of ABCD. Fig. 5. 56. The altitude of a figure is the distance of the highest point from the line of the base. CE (Fig. 5) is the altitude of ABCD. 57. The diameter of a circle is a straight line through the centre, terminating in the circumference. 58. The radius of a circle is a straight line drawn from the centre to the circumference. Fig. 6. 59. A segment of a circle is any part cut off by a straight line. Thus, ABCD is a segment. 36 PRACTICAL GEOMETRY. [CHAP. II. 60. A semicircle is a segment cut off by the diameter. ABCandAEB (Fig. 7) are semicircles. 61. A quadrant is a portion of a circle included between two radii at right angles to each other. ADCandBDC (Fig. 7) are quadrants. 62. The angle in a segment is the angle contained between two straight lines drawn from any point in the arc of a seg- ment to the extremities of that arc. Thus, ABD and ACD (Fig. 6) are angles in the segment ABCD. 63. Similar rectilineal figures have their angles equal, and the sides about the equal angles proportionals. 64. Similar segments of a circle are those which contain equal angles. SECTION II. GEOMETRICAL PROPERTIES AND PROBLEMS. A. GEOMETRICAL PROPERTIES. 65. ALL right angles are equal to each other. 66. The angles which one straight line makes with an- other on one side of it are together equal to two right angles. Thus, ACE and ECB (Fig. 2) are together equal to two right angles. (13.1.) SEC. II.] GEOMETRICAL PROPERTIES AND PROBLEMS. 37 67- If a number of straight lines are drawn from a point in another straight line, all the successive angles are together equal to two right angles. Thus, A CD + DCE + ECB (Fig. 2) make two right angles. Fig. 8. A 68. If two straight lines inter- sect each other, the angles verti- cally opposite are equal. Thus, AEC (Fig. 8) = BED, and AED = BEG. (15.1.) 69. Triangles which have two sides and the included angle of one respectively equal to the two sides and the included angle of the other, are equal in all respects. (4.1.) 70. Triangles which have two angles and the interjacent side of one respectively equal to two angles and the inter- jacent side of the other, are equal in all respects. (26.1.) 71. Triangles which, have two angles of the one respec- tively equal to two angles of the other, and which have also the sides opposite to two equal angles equal to each other, are equal in all respects. (26.1.) 72. If a straight line cuts two pa- Fig. 9- rallel lines, the angles similarly situ- ated in respect to these lines, and \ also those alternately situated, will be A -^ B equal to each other (29.1.) Thus, \ (Fig. 9,) EFB = FGD, BFG = DGS, \ G AFE = CGF, and AFG = CGH, c V~ being similarly situated ; and AFE ' \ = DGH, EFB = CGH, AFG = k FGD, and BFG = FGC, being alternately situated. 73. If a straight line cuts two parallel straight lines, the two exterior angles on the same side of the cutting line, and also the two interior angles, are equal to two right 38 PRACTICAL GEOMETRY. [CHAP. II. angles. Thus, (Fig. 9,) EFB and DGH are equal to two right angles, as are also AFE and CGH. So also the pairs of interior angles AFG and FGC, BFG and FGD, are each equal to two right angles. (29.1.) 74. The angles at the base of an isosceles triangle are equal to each other. (5.1.) 75. If one side of a triangle be produced, the exterior angle so formed will be equal to the two angles adjacent to the opposite side, and the three interior angles are equal to two right angles. Thus, (Fig. 10,) ACD = ABC + BAG, and B ABC + BAC + ACB = two right angles. (32.1.) 76. The interior angles of any rectilineal figure are equal to twice as many right angles as the figure has sides, dimi- nished by four right angles. The interior angles of a quadri- lateral are therefore equal to four right angles. (Cor. 1, 32.1.) 77. The opposite sides and angles of a parallelogram are equal to each other. (34.1.) 78. Conversely, any quadrilateral of which the opposite sides or the opposite angles are equal is a parallelogram. 79. Parallelograms having equal bases and altitudes, and also triangles having equal bases and altitudes, are equal to each other. (35-38.1.) 80. A parallelogram is double a triangle having the same base and altitude. (41.1.) 81. The square on the hypothenuse of a right-angled triangle is equal to the sum of the squares of the legs. (47.1.) SEC. II.] GEOMETRICAL PROPERTIES AND PROBLEMS. 39 82. Any figure described on the hypothenuse of a right- angled triangle is equal to the sum of the similar figures similarly described on the sides. (31.6.) Fig. 11. 83, The angle at the centre of a circle is double the angle at the cir- cumference on the same base. Thus, the angle at C (Fig. 11) is double either D or E. (20.3.) 84. Angles in the same segment of a circle are equal. Thus, D and E (Fig. 11) are equal. 85. The angle in a semicircle is a right angle ; the angle in a segment greater than a semicircle is acute ; and that in a segment less than a semicircle is obtuse. 86. The sides about the equal angles of equiangular tri- angles are proportional. (4.6.) B. GEOMETRICAL PROBLEMS. Under this head are given those methods of construction which are applicable to paper drawings. The methods to be used in field operations will be given in a subsequent chapter. Fig. 12. 87. Problem 1. To bisect a given straight line. Let AB (Fig. 12) be the given line. "With the centres A and B, and radius greater than half AB, iB A \ describe arcs cutting in C and D. Join CD cutting AB in E, and the thing is done. (10.1.) 40 PRACTICAL GEOMETRY. [CHAP. II. Problem 2. To draw a perpendicular to a straight line from a given point in it. a. When the point is not near the. end. 88. Let AB (Fig. 13) be the line and the given point. Lay off CD = CE, and with D and E as centres, and any radius greater than DC, describe arcs cutting in F. Draw CF, and the thing is done. (11.1) Fig. 13. b. When the point is near the end of the line. 89. First Method. Take any point D (Fig. 14) not in the line, and with the centre D and radius DC de- scribe the circle ECF, cutting AB in E. Join ED and produce it to F. Then will CF be the perpendicular. For ECF, being an angle in a semi- A ~~ circle, is a right angle. (85.) 90. Second Method. With C (Fig. 15) and any radius describe DEF ; with D and the same radius cross the circle in E ; and with E as a centre, and the same radius, cross it in F. "With E and F as centres, and any radius, describe arcs cutting in G. Then will C Gi- be the perpendicular. Fig. 14. Fig. 15. C B Problem 3. To let fall a perpendicular to a line from a point without it. a. When the point is not nearly opposite the end of the line. SEC. II.] GEOMETRICAL PROPERTIES AND PROBLEMS. 41 Fig. 16. C 91. Let AB (Fig. 16) be the line and C the given point. With the centre C describe an arc cutting AB in D and E. With the centres D and E and any radius describe arcs cut- ting in F. Join CF, and the thing is done. (12.1.) b. When the point is nearly opposite the end of the line. Fig. 17. 92. First Method.With D and E as centres, and radii DC and EC, de- scribe arcs cutting in F : then will CF be the perpendicular. For, the tri- angles CDE and FDE being equal, (8.1,) DGC and FGD will be equal. (4.1.) X' '/ \" \ \ "\.\ *l I 93. Second Method. Let F (Fig. 14) be the point. From F to any point E in the line AB draw FE. On it describe a semicircle cutting AB in C. Join F and C, and FC will be the perpendicular (85.) Problem 4. At a given point in a given straight line to make an angle equal to a given angle. 94. Let BCD (Fig. 18) be the given angle,, and A the given point in AE. With the centre C and any radius de- scribe BD, cutting the sides of the angle in B and D. With A as a centre and the same radius describe EF ; make EF = DB ; draw AF, and the thing is done. PRACTICAL GEOMETRY. [CHAP. II. Problem 5. To bisect a given angle. 95. Let BAG (Fig. 19) be the given angle. With the centre A and any radius describe an arc cutting the sides in B and C. "With the centres B and C, and the same or any other radius, describe arcs cutting in D. Join AD, and the thing is done. (9.1.) Fig. 20. Problem 6. To draw a straight line touching a circle from a given point without it. 96. Let ABC be the given circle, and D the given point. Join D and the centre E. On DE describe a semicircle cut- ting the circumference in B. Join DB, and it will be the tan- gent required. For DBE, being an angle in a semicircle, is a right angle, (31.3 ;) therefore, DB touches the circle, (16.3.) If the point were in the circumference at B. Join EB, and draw BD perpendicular to it. BD will be the tangent. Problem 7. Through a given point to draw a line parallel to a given straight line. 97. First Method. Let A (Fig. 21) p *ig. 21. A be the given point, and BC the given - line. From A to BC let fall a per- pendicular AD; and at any other point E in BC erect a perpendicular B E EF equal to AD. Through A and F draw AF, which will be the parallel required. 98. Second Method. From A (Fig. 22) to D, any point in BC, draw AD. Make DAE = ADC, and AE will be parallel to BC. (2T.1.) Fig. 22. B D SEC. II.] GEOMETRICAL PROPERTIES AND PROBLEMS. 43 99. Third Method. Through A draw ADE, cutting BC in D. Make DE = AD. Through E draw any other line EFG, cutting BC in F. Make FG = EF : then AG will be parallel to BC. (2.6.) Fig. 23. E \ Problem 8. To inscribe a circle in a given triangle. Fig. 24. 100. Let ABC (Fig. 24) be the given triangle. Bisect two of its angles A and B by the lines AD, BD, cutting in D. Then will D be the centre. (4.4.) Problem 9. To describe a circle about a given triangle. Fig. 25. 101. Bisect two of the sides, as AC and AB, (Fig. 25,) by the perpendicu- lars FE and DE, cutting in E. Then will E be the centre of the required circle. Problem 10. To find a third proportional to two straight lines. Fig. 26. 102, Let M and K" (Fig. 26) be the given lines. Draw two lines AB and AC, making any angle at A. Lay off AD = M, and AE and AF each equal to K Join DF, and draw EG parallel to it. AG A will be the third proportional re- quired. (11.6.) Problem 11. To find a fourth proportional to three given straight lines. 44 PRACTICAL GEOMETRY. [CHAP. IL 103. Let M, 1ST, and (Fig. 27) be the three lines. Draw any two lines AB and AC, meeting at A. Lay off AD = M, AE = N, and AF = 0. Join DF, and draw EG pa- A rallel to it : then AG is the fourth M proportional required. (12.6.) N Fig. 27. Problem 12. To find a mean proportional between two straight lines. 104. First Method. Place the lines Fig. 28. AB and BC (Fig. 28) in the same straight line. On AC describe a semicircle cutting the perpendicular through B in D. BD will be the mean proportional required. (13.6.) 105. Second Method. Let AB and Fig. 29. AC (Fig. 29) be the given lines. On AB describe a semicircle cutting the perpendicular at C in D. Join AD. AD is the mean proportional required. (Cor. 8.6.) MakeAE = AD. NOTE. This is a very convenient construction, and is often employed in the Division of Land. C E Fig. 30. Problem 13. To divide a given line into parts having the same ratio as two given numbers M and 106. Let AB (Fig. 30) be the given line. Draw AC making any angle with AB. Lay off AD = M, taken from any scale of equal parts, and DE = N", taken from the same scale. Join BE, and draw DF parallel to it, and the thing is done. (2.6.) CHAPTER III. PLANE TRIGONOMETRY. SECTION I. DEFINITIONS, 107. PLANE TRIGONOMETRY is the science which treats of the relations between the sides and angles of plane tri- angles; which develops the principles by which, when any three of the six parts of a triangle, viz. : the three angles and the three sides, except the three angles, are given, the others may be found'. It likewise treats of the properties of the trigonometrical functions. 108. Measure of Angles. An angle is the inclination between two straight lines: it is measured by the inter- cepted arc of a circle described about the angular point as a centre. In the measurement of angles, it is not the absolute length of the arc that is needed, but the ratio which that length bears to the whole circumference. For the purpose of expressing this ratio readily, the cir- cumference is supposed to be divided into 360 parts, called degrees, each degree into 60 parts, called minutes, and each minute into 60 seconds. Degrees are marked with a cipher over them, minutes with one accent ', and seconds with two ". Thus, 37 degrees, 45 minutes, and 30 seconds, would be written 37 45' 30". When we speak of an arc of 35, we mean an arc which or is -$^. of the circumference. An arc of 180 is half the obU 45 46 PLANE TRIGONOMETRY. [CHAP. III. circumference, one of 90 is a quadrant, and of 45 the half of a quadrant. It is evident that, if several circles be described about the same point, the arcs intercepted between two lines drawn from the centre will bear the same ratio to the cir- cumferences of which they are portions. Thus, if around Fig. 31. the point A (Fig. 31) two circles BCD and EFG be described, cut- ting AK and AH in B, E, C, F, the arc BC will have to the cir- cumference BCD the same ratio as EF has to the circumference EFGr. In the measurement of angles, it is a matter of indif- ference, therefore, what radius is assumed as that of the circle of reference. The radius which is generally adopted is unity. This value of the radius makes it unnecessary to write it down in the formulse. The radius adopted in tfre construction of the Table of Logarithmic Sines and Tangents, to be described hereafter, is 10,000,000,000. Fig. 32. 109. The complement of an arc or angle is what it differs from a quad- rant, or 90. Thus, DB (Fig. 32) is the complement of AB, and MD of AM. 110. The supplement of an arc or angle is what it wants of 180. Thus, BE (Fig. 32) is the supplement of AB, and ME of AM. 111. Trigonometrical Functions. The trigonometri- cal functions are lines having definite geometrical relations to the arc to which they belcng. Those most in use are the sine, the cosine, the tangent, the cotangent, the secant, and the cosecant. SEC. I.] DEFINITIONS. 47 The chord of an arc is the right line joining the extremi- ties of that arc. Thus, EM (Fig. 32) is the chord of the arc EM. The sine of an arc is the line drawn from one extremity of the arc, perpendicular to the diameter through the other extremity. BF (Fig. 32) is the sine of AB or of EB, and BL of BD. NOTE. The sine of an arc is equal to the sine of its supplement. The cosine of an arc is the line intercepted between the foot of the sine and the centre. CF is the cosine of AB or of BE. Since CF = BL, it is manifest that the cosine of an arc is equal to the sine of its complement. The tangent of an arc is a line touching the arc at one extremity and produced till it meets the radius through the other extremity. Thus, AT is the tangent of AB, and DK of DB. The cotangent of an arc is the tangent of its complement. Thus, DK (Fig. 32) is the cotangent of AB. The secant of an arc is the line intercepted between the centre and the extremity of the tangent. Thus, CT (Fig. 32) is the secant of AB. The cosecant of an arc is the secant of the complement of that arc. Thus, CK (Fig. 32) is the cosecant of AB. The sine, cosine, &c. of an arc are also called the sine, cosine, &c. of the angle measured by that arc. Thus, BF and CF (Fig. 32) are the sine and cosine of the angle ACB. NOTE. The tangent, cotangent, secant, or cosecant of an arc is equal to the tangent, cotangent, secant, or cosecant of its supplement. 112. Properties of the Sines, Tangents, &c. of an arc or angle. The sine of 90, the cosine of 0, the tangent of 45, the cotangent of 45, the secant of 0, and the cosecant of 90, are each equal to radius. The square of the sine -f the square of the cosine of 48 PLANE TRIGONOMETRY. [CHAP. III. any arc is equal to the square of radius. (Sin. 2 a + cos. 2 a R a .) This is evident from the right-angled triangle CFB, (Fig. 32.) (4T.1.) The square of the tangent -f the square of radius is equal to the square of the secant. Tan. 2 a -f E 2 = sec. 2 a. (47.1.) Tan. a : R : : K : cotan. a, or tan. a. cot. a R 2 . This is evident from the similarity of the triangles ACT and DKC, (Fig. 32,) which give (4.6) AT : AC : : CD : DK The sine of 30 and the cosine of 60 is each equal to half radius. 113. Geometrical properties most employed in Plane Trigonometry. The angles at the base of an isosceles triangle are equal ; and conversely, if two angles of a triangle are equal, the sides which subtend them are equal. (5 and 6.1.) The external angle of a triangle is equal to the two opposite internal ones. (32.1.) The three interior angles of a triangle are equal to two right angles or 180. (32.1.) Hence, if the sum of two angles be subtracted from 180, the remainder will be the third angle. If one angle be subtracted from 180, the remainder is O ' the sum of the other angles. If one oblique angle of a right-angled triangle be sub- tracted from 90, the remainder is the other angle. The sum of the squares of the legs of a right-angled tri- angle is equal to the square of the hypothenuse. (47.1.) The angle at the centre of a circle Fig. 11. is double the angle at the circum- ference upon the same arc ; or, in other words, the angle at the cir- cumference of a circle is measured by half the arc intercepted by its sides. (20.3.) Thus, the angle ADB is half ACB ; and is, therefore, mea- sured by one-half of the arc AB. The sides about the equal angles of equiangular tri- angles are proportionals. (4.6.) SEC. II.] DRAFTING OR PLATTING. 49 SECTION II. DRAFTING OR PLATTING.* 114. DRAFTING is making a correct drawing of the parts of an object. Platting is drawing the lines of a tract of land so as correctly to represent its boundaries, divisions, and the various circumstances needful to be recorded. It is, in fact, making a map of the tract. It is of great im- portance to a surveyor to be able to make a correct and neat plat of his surveys. The facility of doing so can only be acquired by practice; the student shouid, therefore, be required to make a neat and accurate draft of every pro- blem in Trigonometry he is required to solve, and of every survey he is required to calculate. It is not sufficient that he should draw a figure, as he does in his demonstrations in Geometry, that will serve to demonstrate his principles or afford him a diagram to refer to, but he should be obliged to make all parts in the exact proportion given by the data, so that he can, if needful, determine the length of any line, or the magnitude of any angle, by measurement. 115. Straight lines. Straight lines are generally drawn with a straight-edged ruler. If a very long straight line is needed, a fine silk thread may be stretched between the points that are to be joined, and points pricked in the paper at convenient distances; these may then be joined by a ruler. In drawing straight lines, care should be taken to avoid determining a long line by producing a short one, as any variation from the true direction will become more mani- fest the farther the line is produced. When it is necessary to produce a line, the ruler is fixed with most ease and cer- tainty by putting the points of the compasses into the line to be produced, and bringing the ruler against them. 116. Parallels. Parallels may be drawn as described in * Various hints in this section have been derived from Gillespie's "Laud Surveying." 50 PLANE TRIGONOMETRY. [CHAP. Ill Arts. 97, 98. Practically, however, it is better to draw them by some instrument specially adapted to the purpose. The square and ruler are very convenient instruments for this purpose. The square consists of two arms, which should be made at right angles to each other, to facilitate the erection of perpendi- culars. Let AB (Fig. 33) be the line to which a parallel is to be drawn through C. Adjust one edge of the square to the line AB, and bring a ruler firmly against the other leg; move the square along the ruler un- til the edge coincides with C : this edge will then be parallel to the given line. If a T square be substituted for a simple right angle, it may be held more firmly against the ruler. Instead of a square, a right-angled triangle is frequently used. The legs should be made accurately at right angles, that it may be used for drawing per- pendiculars. Let AB (Fig. 34) be the line, and C the point through which it is required to draw a parallel. Bring one edge of the triangle accurately to the line, and then place a ruler against one of the other sides. Slide the triangle along the ruler until the point C is in the side which before coincided with the line : this side is then parallel to the given line. The parallel rulers which accompany most cases of in- struments are theoretically accurate. They are, however, generally made with so little care that they cannot be de- pended on where correctness is required ; and, even if made true, they are liable to become inaccurate in consequence of wear of the joints. SEC. II.] DRAFTING OR PLATTING. 51 117. Perpendiculars. Perpendiculars may be drawn as directed, (Art. 88, et seq.) A more ready means is to place one leg of the square (Fig. 33) upon the line : the other will then be perpendicular to that line. The triangle is another very convenient instrument for this purpose. Let AB (Fig. 35) be the line to which a perpendicular is to be drawn. Place the hypothenuse of the triangle coincident with AB, and bring the ruler against one of the other sides. Remove the tri- angle and place it with the third side against the ruler, as at D : then the hypothenuse will be perpendicular to AB. This method requires the angle of the triangle to be pre- cisely a right angle. To test F . ^ whether it is so, bring one leg B against a ruler, as at A, (Fig. 36,) and scribe the other leg. Reverse the triangle, and bring the right angle to the same point A, and A again scribe the leg. If the angle is a right angle, the two scribes will exactly coincide. If they do not coincide, the triangle requires rectification. 118. Circles and Arcs. These are generally drawn w^th the compasses, which should have one leg movable, so that a pen or a pencil may be inserted instead of a point. When circles of long radii are required, the beam compasses should be used. These consist of a bar of wood or metal, dressed to a uniform size, and having two slides furnished with points. These slides can be adjusted to any part of the beam, and clamped, by means of screws adapted to the purpose. The point connected with one of the slides is movable, so that a pencil or drawing pen may be substituted. When the beam compasses are not at hand, a strip of drawing paper or pasteboard may be substituted : a pin through one point will serve as a centre; the pencil 52 PLANE TRIGONOMETRY. [CHAP. III. point can be passed through, a hole at the required distance. 119. Angles. Angles may be laid off by a protractor. This is usually a semicircle of metal, the arc of which is divided into degrees. To use it, place it with the centre at the point at which the angle is to be made, and the straight edge coincident with the given line ; then with a fine point prick off the number of degrees required, and join the point thus determined to the centre. The figures on the protractor should begin at each end of the arc, as represented in Fig. 37. 120. By the Scale of Chords. The scale of chords, which is engraved on the ivory scales contained in a box of instruments, may also be used for making angles. For this purpose take from the scale the chord of 60 for a radius. With the point A, at which the angle is to be made, as a centre, and that radius, describe an arc. Take off from the scale the chord of the required number of degrees and lay it on the arc from the given line, join the extremity of the arc thus laid off to the centre, and the thing is done. Thus, if at the point A (Fig. 38) it were required to make an angle BAG of 47. Fig. 38. SEC. II.] DRAFTING OR PLATTING. 53 "With the centre A and radius equal to the chord of 60 describe the arc BC. Then, taking the chord of 47 from the scale, lay it off from B to C. Join AC, and BAG will be the required angle. If an angle of more than 90 is required : first lay off 90, and from the extremity of that arc lay off the remainder. 121. By the Table of Chords. The table of chords (page 97 of the tables) affords a much more accurate means of laying off angles. Take for a radius the distance 10 from any scale of equal parts, to be described hereafter, and describe the arc BC, (Fig. 38.) Then, finding the chord of the required angle by the table, multiply it by 10, and, taking the product from the same scale, lay it off from B to C as before. Join AC, and the thing is done. If the angle is much over 60 it is best to lay off the 60 first. This is done by using the radius as a chord. The remainder can then be laid off from the extremity of the arc of 60 thus determined. 122. Distances. Every line on a draft should be drawn of such a length as correctly to represent the distance of the points connected, in due relation to the other parts of the drawing. In perspective drawing, the parts are deline- ated so as to present to the eye the same relations that those of the natural object do when viewed from a particular point. To produce this effect the figure must be distorted. Right angles are represented as right, obtuse, or acute, ac- cording to the position of the lines ; and the lengths of lines are proportionally increased or diminished according to their position. In drafting, on the contrary, every part must be represented as it is. The angles should be of the same magnitude as they are in reality, and the lines should bear to each other the exact ratio that those which they are intended to represent do. The plat should, in fact, be a miniature representation of the figure. 123. Drawing to a Scale. In order that the due pro- 54 PLANE TRIGONOMETRY. [CHAP. III. portion should exist in the parts of the figure, every line should be made some definite part of the length of that which it is intended to represent. This is called drawing to a scale. The scale to be used depends on the size of the map or draft that is required, and the purposes for which it is to be used. Carpenters often use the scale of an inch to a foot : the lines will then be the twelfth part of their real length. In plats of surveys, or maps of larger tracts of country, a greater diminution is necessary. The scale should, however, in all cases, be adapted to the purpose intended and to the number of objects to be represented. Where the purpose is merely to give a correct representa- tion of the plat, without filling up the details, the main object will be to make the map of a convenient size; but where many details are to be represented the scale should be proportionally larger. Thus, for example, in delineating a harbor where there are few obstructions to navigation, a map on a small scale may be drawn ; but where the rocks and shoals are nume- rous, the scale should be so large that every part may be perfectly distinct. The scales on which the drawing is made should always be mentioned on the map. They may be expressed by naming the lengths which are used as equivalents, thus, " Scale, 10 feet to an inch, 1 mile to an inch, 3 chains to a foot;" or better fractionally, thus, 1 : 100, 1:250, 1 : 10,000, &c. 124. Surveys of Farms. Where the farm is small, 1 chain* to an inch, (1 : 792,) or 2 chains to the inch, (1 : 1584,) may be used ; but if the tract be large, as this would make a plat of a very inconvenient size, a smaller scale must be adopted. When, however, any calculations are to be based on measurements taken from the plat, a smaller scale than 3 chains to the inch (1 : 2376) should not be employed. * The surveyor's chain commonly called Gunter's Chain is 4 poles, or 66 feet, in length, and is divided into one hundred links, each of which is therefore .66 feet, or 7.92 inches in length. SEC. II.] DRAFTING OR PLATTING. 55 125. Scales. Scales are generally made of ivory or box- wood, having a feather-edge, on which the divisions are marked, The distances can then be laid off by placing the ruler on the line, and pricking the paper or marking it with a fine pointed pencil ; or the length of a line may be read off without any difficulty. Boxwood scales, if the wood is clear from knots, are to be preferred to ivory. They are less liable to warp, and suffer less expansion and con- traction from changes in the hygrometric condition of the atmosphere. Paper scales are often employed. These may be pro- cured with divisions to suit almost any purpose, or the sur- veyor may make them himself. Take a piece of drawing- paper, and cut a slip about an inch in width ; draw a line along its middle, and divide it as desired, either into inches or tenths of a foot. The end division should be subdivided into ten parts, and perpendiculars drawn through all the divisions, as represented in the figure, (Fig. 39.) Each of these parts may then represent a chain, ten chains, &c. Fig. 39. Paper scales, being subject to nearly the same expansion and contraction as the paper on which the map is drawn, are, on this account, preferable to those made of wood or ivory. They cannot, however, be divided with the same accuracy. 126. The plane diagonal scale (Fig. 40) consists of eleven Fig. 40. 5 4 : J 5 5 ] A 2 HP p 46\ 1 U i 8 B s ^B * I 6 56 PLANE TRIGONOMETRY. [CHAP. in. lines drawn parallel and equidistant. These are crossed at right angles by lines 1, 2, 3, drawn usually at intervals of half an inch. The first division, on the upper and lower lines, is subdivided into ten equal parts : diagonal lines are then drawn, as in the figure, from each division of the top to the next on the bottom, the first, from A to the first division on the bottom line; the second, from the first on the top to the second on the bottom ; and so on. It is evident that, whatever distance the primary division from A to 1, or 1 to 2, &c. represents, the parts of the line AB will represent tenth parts of that distance. If then it were required to take off the distance of 47 feet on a scale of half an inch to 10 feet, the compasses should be extended from E to F. The diagonal lines serve to subdivide each of the smaller divisions into tenths, thus: The first diagonal, extending from A to the first division on the bottom line and crossing ten equal spaces, will have advanced ^ of one of those divisions at the first intermediate line, ^ at the second, & at the third, and so on. All the other diagonals will advance in the same manner. If then the distance were taken from the line AC along the horizontal line marked 6 to the fourth diagonal, the distance would be .46, the division AB being a unit, or 4.6 if AB were 10. To take off, then, 39.8 feet on a scale of half an inch to 10 feet, the compasses should be ex- tended to the points marked by the arrow heads G- and H : similarly, 46. 7, on the same scale, would extend from one of the arrow heads on the seventh line to the other. In using the diagonal scale the primary divisions should always be made to represent 1, 10, 100, or 1000. When any other scale is required, say 1 : 300, it is better to divide or multiply all the distances and then take off the results. Thus, if 83.7 were required to be taken off on a scale of J inch to 30 feet, first divide 83.7 by 3, giving 27.9, and then take off the quotient on a scale of J inch to 10 feet. The other lines must all be reduced in the same proportion. The above method requires less calculation, and involves SEC. II.] DRAFTING OR PLATTING. 57 less liability to error, than that of determining the value of each division on the reduced scale. 127. Proportional Scale. On most of the rulers fur- nished with cases of instruments there is another set of scales, divided as below, (Fig. 41.) Fig. 41. The figures on the left express the number of divisions to the inch. To lay off 97 feet on a scale of 40 feet to the inch, the compasses would be extended between the arrow- heads on the line 40. Scales of this kind are very con- venient in altering the size of a drawing. Suppose, for example, it is desired to reduce a drawing in the ratio of 5 to 3 : the lengths of the lines should be determined on the scale marked 30, and the same number of. divisions on the scale 50 will give a line of the desired length. 128. Vernier Scale. Make a scale (Fig. 42) with inches divided into tenths, and mark the end of the first inch 0, of the second 100, and so on. From the zero point, back- wards, lay off a space equal to eleven tenths of an inch, and divide it into ten equal parts, numbering the parts backwards, as represented in the figure. This smaller scale Fig. 42. 1 00 2 00 ~T 1 M 1 i M 1 1 1 1 1 1 1 1 1 1 1 Vi fill 88 66 44 22 A. is a vernier. Now, since the ten divisions of the vernier are equal to eleven of the scale, each of the vernier divisions 58 PLANE TRIGONOMETRY. [CHAP. III. is equal to ^ of ^ = J^ of an inch. From the zero point, therefore, to the second division of the vernier is .22 inch, to the third .33, and so on. To measure any line by the scale, take the distance in the compasses, and move them along the scale until you find that they exactly extend from some division on the vernier to a division on the scale. Add the number on the scale to the number on the vernier for the dis- tance required. Thus, suppose the compasses extended from 66 on the vernier to 110 on the scale, the length is 1T6. To lay off a distance by the scale, for example 175, take 55 from 175, and 120 is left : extend the compass from 120 on the scale to 55 on the vernier. To lay off 268 = 180 + 88, extend the compasses from 180 on the scale to 88 on the vernier, as marked by the arrow heads. The vernier scale is equally accurate with the diagonal scale, and much more readily made. SECTION III. TABLES OP TRIGONOMETRICAL FUNCTIONS. 129. Table of Natural Sines and Cosines. THIS table (page 87 of the Tables) contains the sines and cosines to five decimal places for every minute of the quadrant. The table is calculated to the radius 1. As the sine and cosine are always less than radius, the figures are all decimals. In the table the decimal point is omitted. If the sine and cosine is wanted to any other radius, the number taken from the table must be multiplied by that radius. To take out the sine or cosine of an arc from this table, look for the degrees, if less than 45, at the top of the table, and for the minutes at the left ; then, in the column headed properly, and opposite the minutes, will be the function required. If the degrees are 45 or upwards they will be SEC. III.] TRIGONOMETRICAL FUNCTIONS. 59 found at the bottom, and the minutes at the right. The name of the column is at the bottom. Thus, the sine of 32 17', found under 32 and opposite 17', is .53411. The cosine of 53 24', found over 53 and opposite 24' in the right-hand column, is .59622. 130. The table of natural sines and cosines is of but little use in trigonometrical calculations, these being generally performed by logarithms. It is principally employed in determining the latitudes and departures of lines. 131. Table of Logarithmic Sines, Cosines, &c. This table contains the logarithms of the sines, cosines, tangents, and cotangents, to every minute of the semicircle, the radius being 10 000 000 000 and its logarithm 10. The logarithmic sine of 90, cosine of 0, tangent of 45, and cotangent of 45, is each 10. The sine, cosine, tangent, and cotangent, of every arc being equal to the sine, cosine, tangent, and cotangent, of its supple- ment, and also to the cosine, sine, cotangent, and tangent, of its complement, the table is only extended to forty five pages, the degrees from to 44 inclusive being found at the top, those from 45 to 135 at the bottom, and from 136 to 180 at the top. The minutes are contained in the two outer columns, and agree with the degrees at the top and bottom on the same side of the page. The columns headed Diff. 1" contain the difference of the function for a change of 1" in the arc. These differ- ences are calculated by dividing the differences of the suc- cessive numbers in the columns of the functions by 60. By an inspection of these columns of difference it will be seen that, except in the first few pages, they change very slowly. In these, in consequence of the rapid change of the func- tion, the differences vary very much. The difference set down will not, therefore, be accurate, except for about the middle of the minute. The calculations for seconds, there- fore, are not in these cases to be depended on. To obviate this inconvenience, and give to the first few pages a degree 60 PLANE TRIGONOMETRY. [CHAP. III. of accuracy commensurate with that of the rest of the table, the sines and tangents are calculated to every 10 seconds, and these are the same as the cosines and cotangents of arcs within two degrees of 90.* 132. Use of Table. To take out any function from the table, seek the degrees, if less than 45 or more than 135, at the top of the page, and the minutes in the column on the same side of the page as the degrees. Then, in the proper column, (the title being at the top,) and opposite the minutes, will be found the value required. If the degrees are between 45 and 135, seek them at the bottom of the page, the minutes being found, as before, at the same side of the page as the degrees. The titles of the columns are also at the bottom. EXAMPLES. Ex. 1. Required the sine of 37 17'. Ans. 9.782298. Ex. 2. Required the cosine of 127 43'. Ans. 9.786579. Ex. 3. Required the cotangent of 163 29'. Ans. 10.527932. Ex. 4. Required the tangent of 69 11'. Ans. 10.419991. 133. If there are seconds in the arc, take out the function for the degrees and minutes as before. Multiply the num- ber in the difference column by the number of seconds, and add the product to the number first taken out, if the func- tion is increasing, but subtract, if it is decreasing : the result will be the value required. If the arc is less than 90 the sine and tangent are in- creasing, and the cosine and cotangent are decreasing ; but if the arc is greater than 90 the reverse holds true. * The rectangle of the tangent and cotangent of an arc being equal to the square of radius, their logarithms are arithmetical complements (to 20) of each other. Our column of differences serves for both these functions. It is placed between them. SEC. III.] TRIGONOMETRICAL FUNCTIONS. 61 Ex. 1. What is the tangent of 37 42' 25"? The tangent of 37 42' is 9.888116 Diff. V 4.35 25 2175 870 Diff. 25" 108.75 + 109 Tangent 37 42' 25" 9.888225 Ex. 2. What is the cosine of 129 17' 53"? The cosine of 129 17' is 9.801511 Diff. 1" 2.57 53 771 1285 Diff. 53" 136.21 +136 Cosine 129 17' 53" 9.801647 Ex. 3. What is the sine of 63 19' 23"? Ans. 9.951120. Ex. 4. What is the cosine of 57 28' 37"? Ans. 9.730491. Ex. 5. What is the tangent of 143 52' 16"? Ans. 9.863314. Ex. 6. What is the sine of 172 19' 48"? Ans. 9.125375. If the sine or tangent of an arc less than 2 or more than 178, or the cosine or cotangent of an arc between 88 and 92, is required, it should be taken from the first pages of the table. Take out the function to the ten seconds next less than the given arc, multiply one tenth of the difference between the two numbers in the table by the odd seconds, and add or subtract as before. The cotangent of an arc less than 2 may be found by taking out the tangent, and subtracting it from 20.000000 ; so likewise the tangent of an arc between 178 and 180 is found by taking the complement to 20.000000 of its cotangent. 62 PLANE TRIGONOMETRY, [CHAP. III. Ex. 1. Required the sine of 1 27' 36". Sine of 1 27' 30" is 8.405687 & of difference 82.6 6 Difference 6" 495.6 496 Sine of 1 27' 36" 8.406183 Ex. 2. What is the cosine of 88 18' 48"? Ans. 8.468844. Ex. 3. What is the sine of 179 19' 13"? Ans. 8.074198. 134. To find the Arc corresponding to any Trigo- nometric Function. If degrees and minutes only be required, seek, in the pro- per column, the number nearest that given ; and if the title is at the top the degrees are found at the top, and the minutes under the degrees; but if the title is at the bottom the degrees are at the bottom, and the minutes on the same side as the degrees. If seconds are desired, seek for the number corresponding to the minute next less than the true arc, and take the difference between that number and the given one : divide said difference by the number in the difference column, for the seconds. Ex. 1. What is the arc whose sine is 9.427586 ? 9.427586 Sine of 15 31' is 9.427354 7.58)232.00(31" 2274 4.60 The arc is, therefore, 15 31' 31". SEC. III.] TRIGONOMETRICAL FUNCTIONS. 63 Ex. 2. What is the arc whose cotangent is 10.219684? 10.219684 Cotangent of 31 5' is 10.219797 4.76) 113.00 (23.7" 952 1780 1428 3.52 The arc is, therefore, 31 5' 24". Ex. 3. Required the arc the cosine of which is 9.764227. Ans. 54 28' 27". Ex. 4. Required the arc the tangent of which is 10.876429. Ans. 82 25' 44". Ex. 5. What is the arc the cotangent of which is 11.562147? As this corresponds to an arc less than 2, take it from 20.000000: the remainder, 8.437853, is the tangent. The arc is found as follows : 8.437853 1 34' 10" tang. 8.437732 Diff. tol" 76.8 ) 121.0 (1.6" 768 44.20 The angle is, therefore, 1 34' 11.6". Ex. 6. What arc corresponds to the cotangent 8.164375? Ans. 89 9' 48.6". 135. Table of Chords. This table contains the chords of arcs to 90 for every 5 minutes. It is principally used in laying off angles, as explained in Art. 120, and in pro- tracting surveys by the method of Art. 343. 34 PLANE TRIGONOMETRY. [CHAP. III. SECTION IV. ON THE NUMERICAL SOLUTION OF TRIANGLES. 136. Definition. THE solution of a triangle is the deter- mination of the numerical value of certain parts when others are given. To determine a triangle, three inde- pendent parts must be known, viz. : either the three sides, or two sides and an angle, or the angles and one side. The three angles are not of themselves sufficient, since they are not independent, any one of them being equal to the dif- ference between the sum of the others and 180. In the solution of triangles several cases may be distin- guished ; these will be treated of separately. These cases are applicable to all triangles. But as there are special rules for right-angled triangles, which are simpler than the more general ones, they will first be given. A. THE NUMERICAL SOLUTION OF EIGHT- ANGLED TRIANGLES. 137. The following rules contain all that is necessary for solving the different cases of right-angled triangles. 1. The hypothenuse is to either kg as radius is to the sine of the opposite angle. 2. The hypothenuse is to one leg as radius is to the cosine of the adjacent angle. 3. One leg is to the other as radius is to the tangent of the angle adjacent to the former. DEMONSTRATION. Let ABC (Fig. 43) be a Fig. 43. triangle right-angled at B. Take AD any ra- dius, and describe the arc DE; draw EF and DG perpendicular to AB. Then EF will be the sine, AF the cosine, and DG the tangent, of the angle A. Now, from simifar triangles we have 1. AC : CB : : AE : EF : : r 2. AC : AB : ; AE : AF : : r 3. AB : BC : : AD : DG : : r SEC. IV.] NUMERICAL SOLUTION OF TRIANGLES. 65 EXAMPLES. Ex. 1. In the triangle ABC, right-angled at B, there are given the base AB = 57.23 chains, and the angle A 35 27' 25", to find the other sides. Construction. Make AB (Fig. 44)= 57.23, taken *fe- 44. from a scale of equal parts. At the point A make the angle BAG = 35 27'. Erect the perpendicular BC, meeting AC in C, and ABC is the triangle required. Calculation. Rule 3. r : tan. A : : AB : BC. Eule 2. cos. A : r : : AB : AC. For facility of calculation, the proportions are generally written vertically, as below. As rad. log. 10.000000 : tan. A 35 27' 25" 9.852577 : : AB 57.23 ch. 1.757624 : BC 40.76 1.610201 As cos. A 35 27' 25" Ar. Co. 0.089081 : rad. 10.000000 :: AB 57.23 1.757624 : AC 70.26 1.846705 Ex. 2. Given AB = 47.50 chains, and AC = 63.90 chains, to find the angles and side BC. EULE 2. As AC 63.90 Ar. Co. 8.194499 : AB 47.50 1.676694 :: rad. 10.000000 : cos. A 41 58' 57" 9.871193 90 C 48 V 3" 5 66 PLANE TRIGONOMETRY. [CHAP. III. RULE 1. As rad. 10.000000 : sin. A 41 58' 57" 9.825363 :: AC 63.90 1.805501 : CB 42.74 "l.630864 Ex. 3. Given the two legs AB = 59.47 yards, and BO = 48.52 yards, to find the hypothenuse and the angles. Ans. A 39 12' 36", C 50 47' 24", and AC 76.75 yds. Ex. 4. Given the hypothenuse AC = 97.23 chains, the perpendicular BC = 75.87 chains, to find the rest. Ans. A 51 17' 22", C 38 42' 38", AB 60.81 ch. Ex. 5. Given the angle A = 42 19' 24", and the perpen- dicular BC = 25.54 chains, to find the other sides. Ans. AC 37.932 ch., AB 28.045 ch. Ex. 6. Given the angle C = 72 42' 9", and the hypo- thenuse AC = 495 chains, to find the other sides. Ans. AB 472.612 ch., BC 147.18 ch. Ex. 7. In the right-angled triangle ABC we have the base AB = 63.2 perches, and the angle A 42 8' 45", to find the hypothenuse and the perpendicular. Ans. BC 57.20 p., AC 85.24 p. 138, "When two sides are given, the third may be found by (47.1); thus, 1. Given the hypothenuse and one leg, to find the other. Rule. From the square of the hypothenuse subtract the square of the given leg: the square root of the remainder will be the other leg ; or, Multiply the sum of the hypothenuse and given leg by their difference : the square root of this product will be the other leg. This is evident from (47.1) and (cor. 5.2.) 2. Given the two legs, tojind the hypothenuse. Rule. Add the squares of the two legs, and extract the square root of the sum: the result will be the hypothenuse. SEC. IV.] NUMERICAL SOLUTION OF TRIANGLES. 67 EXAMPLES. Ex. 1. Given the hypothenuse AC = 45 perches, and the leg BC = 29 perches, to find the other leg. Kule 1. AB = >/AC 2 - BC 2 = N/2025 - 841 = ^1184 34.41. or, AB = x/(AC + BC).(AC - BC) = ^74 x 16 = ^1184= 34.41. Ex. 2. The two legs AB and AC are 6 and 8 respectively : what is the hypothenuse ? Ans. 10. Ex. 3. The hypothenuse AC is 47.92 perches, and the leg AB is 29.45 perches : required the length of BC. Ans. 37.8 perches. Ex. 4. The hypothenuse of a right-angled triangle is 49.27 yards, and the base 37.42 yards : required the perpen- dicular. Ans. 32.05. B. THE NUMERICAL SOLUTION OF OBLIQUE-ANGLED TRIANGLES. CASE 1. 139. The angles and one side, or two sides and an angle oppo- site to one of them, being given, to find the rest. EULE. 1. As the sine of the angle opposite the given side is to the sine of the angle opposite the required side, so is the given side to the required side. 2. As the side opposite the given angle is to the other given side, so is the sine of the angle opposite to the former to the sine of the angle opposite the latter. DEMONSTRATION. Both the above rules are combined in the general propo- sition. The sides are to one another as the sines of their opposite angles. Let ABC (Fig. 45) be any triangle. From C let fall / CD perpendicular to AB. Then (Art. 137) AC : CD : : r / : sin. A, and CD : CB : : sin. B : r. Whence (23.5) AC : / CB : : sin. B : sin. A. / 68 PLANE TRIGONOMETRY. [CHAP. III. EXAMPLES. Ex. 1. In the triangle ABC are given AB = 123.5, the angle B = 39 47' 20", and C = 74 52' 10": required the rest. Construction. The angle A = 180 - (B -f- C) = 180 - 114 39' 30" = 65 20' 30". Draw AB (Fig. 45) = 123.5. At the points A and B draw AC, BC, making the angles BAC and ABC equal, respectively, to 65 20' 30" and 39 47' 20" ; then will ABC be the triangle required. As sin. C : sin. B : : AB : AC As sin. C : sin. A :: AB : BC Calculation. 74 52' 10" 39 47' 20" 123.5 81.87 65 20' 30" 116.27 A. C. 0.015322 9.806154 2.091667 1.913143 A. C. 0.015322 9.958474 2.091667 2.065463 Ex. 2. Given the side AB = 327, the side BC the angle A = 32 27', to determine the rest. 238, and Construction. Make AB (Fig. 46) = 327; and at the point A draw AC making the angle A = 32 47'. With the centre B and radius = 238 describe an arc cutting AC in C ; then will ABC be the triangle required. Calculation. EULE 2. Kg ' As BC : AB : : sin. A : sin. C or 238 327 32 47' 48 4' 6" A. C. 7.623423 2.514548 9.733569 9.871540 131 55' 54" SEC. IV.] NUMERICAL SOLUTION OF TRIANGLES. 69 C acute. As sin. C 48 4' 6" A. C. 0.128460 : sin. B 99 8' 54" 9.994441 : : AB 327 2.514548 : AC 433.9T 2.637459 C obtuse. As sin. C 131 55' 54" A. C. 0.128460 : sin. B 15 IT' 6" 9.4209T9 :: AB 2.514548 : AC 115.87 2.063987 NOTE. It will be seen that in the above example the result is uncertain. The sine of an angle being equal to the sine of its supplement, it is impossible, from the sine alone, to determine whether the angle should be taken acute or obtuse. By reference to the construction, (Fig. 46,) we see that whenever the side opposite the given angle is less than the other given side, and greater than the perpendicular BD, the triangle will admit of two forms: ABC, in which the angle opposite to the side AB is acute, and ABC'', in which it is obtuse. If BC were greater than BA, the point G / would fall on the other side of A, and be excluded by the conditions. If it were less than BD, the circle would not meet AC, and the question would be impossible. Ex. 3. Given the side AB 37.25 chains, the side AC = 42.59 chains, and the angle C 57 29' 15", to determine the rest. Ans. BC 32.774 chains, A = 47 53' 52", and B = 74 36' 53". Ex. 4. Given the angle A 29 47' 29", the angle B = 24 15' 17", and the side AB 325 yards, to find the other sides. Ans. AC = 164.93, BC = 199.48. Ex. 5. The side AB of an obtuse-angled triangle is 127.54 yards, the side AC 106.49 yards, and the angle B 52 27' 18", to determine the remaining angles and the side BC. Ans. C = 108 16' 3", A = 19 16' 39", BC = 44.34. Ex. 6. Given AB = 527.63 yards, AC = 398.47 yards, and the angle B 43 29' 11", to determine the rest. Ans. C = 65 40' 44", A = 70 50' 5", BC = 546.93; or, C = 114 19' 16", A = 22 11' 33", BC = 218.71. 70 PLANE TRIGONOMETRY. [CHAP. Ill CASE 2. 140. Two sides and the included angle being given, to determine the rest. EULE 1. Subtract the given angle from 180 : the remainder will be the sum of the remaining angles. Then, As the sum of the given sides is to their difference, so is the tangent of half the sum of the remaining angles to the tangent of half their difference. This half difference added to the half sum will give the angle opposite the greater side, and subtracted from the half sum will give the angle opposite the less side. Then having the angles, the remaining side may be found by Case 1. DEMONSTRATION. The second paragraph of this rule may be enunciated in general terms ; thus, As the sum of two sides of a plane triangl&is to Fig. 47. their difference, so is the tangent of half the sum of the angles opposite those sides to the tangent of half the difference of those angles. Let ABC (Fig. 47) be the triangle of which the side AC is greater than AB. With the centre A and radius AC describe a circle cutting AB pro- duced in E and F. Join EC and CF, and draw FG parallel to BC. Then, because ABC and AFC have the common angle A, AFC -f- ACF = ABC -f ACB. Whence AFC = & (ABC + ACB) ; and, since the half sum of two quantities taken from the greater leaves their half difference, CFG = EFG EFC == ABC EFC = (ABC ACB). Now, since the angle ECF is an angle in a semicircle, it is a right angle. Therefore, if with the centre F and radius FC an arc be described, EC and CG will be the tangents of EFC and CFG, or of the half sum and half dif- ference of ABC and ACB. But (2.6) EB : BF : : EC : CG. Whence AC -f AB : AC AB : : tan. (ABC -f ACB) : tan. (ABC ACB). EXAMPLES. Ex. 1. Given AJB = 527 yards, AC = 493 yards, and the angle A =37 49'. Here C + B = 180 - 37 49' = 142 11', and SEC. IV.] NUMERICAL SOLUTION OP TRIANGLES. 71 As AB + AC 1020 A.C. 6.991400 : AB-AC 34 1.531479 C + B : : tan. - 71 5' 30" 10.465290 2 C-B : tan. - 5 33' 29" 8.988169 2 C 76 38' 59" B 65 32' 1" As sin. C 76 38' 59" A.C. 0.011897 : sin. A 37 49' 9.787557 ::AB 527 2.721811 : BC 332.10 2.521265 Ex. 2. In the triangle ABC are given AB = 1025.57 yaids, BC = 849.53 yards, and the angle B = 65 43' 20", to find the rest. Ans. A = 48 52' 10", C = 65 24' 30", AC = 1028.13. Ex. 3. Two sides of a triangle are 155.96 feet and 217.43 feet, and their included angle 49 19', to find the rest. Ans. Angles, 85 4' 12", 45 36' 48", side, 165.49. RULE 2. 141. As the less of the two given sides is to the greater, so is radius to the tangent of an angle; and as radius is to the tangent of the excess of this angle above 45, so is the tangent of the half sum of the opposite angles to the tangent of their half difference. Having found the half difference, proceed as in Rule 1. NOTE. This rule is rather shorter than the last, where the two sides have been found in a preceding calculation, and thus their logarithms are known. r2 PLANE TRIGONOMETRY. [CHAP. III. DEMONSTRATION. Let ABC (Fig. 48) be any Fig. 48. plane triangle. Draw BD perpendicular to AB, the greater, and equal to BC, the less side. Make BE = BD, and join ED. Then, since BE = BD, the angle BED =B BDE ; and since EBD is a right angle, BDE ss 45. But BED + BDE = 2 BDE = BAD + BDA, and BDE = } (BDA + BAD). But the half sum of any two quantities being taken from the greater will leave the half difference: therefore ADE is the half difference of BDA and BAD. Now, (Rule 3, Art. 137,) BD or BC : BA : : rad. : tan. ADB ; and (demonstration to last rule) AB -j- BD : AB BD : : tan. \ (BDA -f- BAD) : tan. \ (BDA BAD) : : tan. BDE : tan. ADE; but BDE being equal to 45, its tangent = rad. And ADE SB (ADB 45) . . AB + BD : AB BD : : r : tan. (ADB 45) ; but AB + BC : AB BC : : tan. (ACB + BAG) : tan. (ACB BAC) ; whence r : tan. (ADB 45) : : tan. J (ACB -f BAC) : tan. } (ACB BAC). EXAMPLES. Ex. 1. In the course of a calculation I have found the logarithm of AB = 2.596387, that of BC = 2.846392: now, the angle B being 55 49', required the side AC. Calculation. AsAB A. C. 7.403613 : BC 2.846392 : : Kad. 10.000000 : tan. x 60 38' 58" 10.250005 As rad. A. C. 0.000000 : tan. (x - 45) 15 38' 58" 9.447368 ::tan. J(A+C) 62 5' 30" 10.276004 : tan. J (A - C) 27 52' 28" 9.723372 A 89 57' 58" Then, As sin. A 89 57' 58" A. C. 0.000000 : sin. B 55 49' 9.917634 ::BC 2.846392 : AC 580.8 2.764026 SEC. IV.] NUMERICAL SOLUTION OF TRIANGLES. 73 Ex. 2. Given the logarithms of BC and AC 3.964217 and 3.729415 respectively, and the angle C = 63 17' 24", to find AB. Ans. 8317. Ex. 3. Given the logarithms of AB and BC 1.963425 and 2.416347, and the angle B = 129 42', to find AC. Ans. 327.27. CASE 3. 142. Given the three sides, to find the angles. EULB 1. Call the longest side the base, and on it let fall a perpendicular from the opposite angle. Then, as the base is to the sum of the other sides, so is the difference of those sides to the difference of the segments of the base. Half this difference added to half the base will give the greater segment, and subtracted will give the less segment. Having the segments of the base, and the adjacent sides, the angles may be found by Kule 2, Art. 137. DEMONSTRATION. Let ABC (Fig. 49) be the tri- Fig. 49. angle, AB being the longest side : with the centre C and a radius CB, the less of the other sides, describe a circle, cutting AB in E and AC in F and G. Draw CD perpendicular to AB. Then (3.3) DE = DB; therefore AE is the difference of the segments of the base. Also, AG = AC + CB ; and AF = AC CB. Now, (36.3. cor.,) AB . AE = AG . AF; whence (16.6) AB : AG : : AF : AE, or AB : AC -f CB : : AC CB : AD DB. EXAMPLES. Ex. 1. Given the three sides of a triangle, viz. : AB = 467, AC = 413, and BC = 394, to find the angles. 74 PLANE TRIGONOMETRY. [CHAP. III. As AB 467 Ar. Co. 7.330683 : AC + BC 807 2.906874 : : AC - BC 19 1.278754 : AD-DB 32.833 1.516311 i (AD-DB) 16.4165 JAB 233.5 AD 249.9165 BD 217.0835 As AC 413 Ar. Co. 7.384050 : AD 249.9165 2.397794 ::r 10.000000 : cos. A 52 45' 44" 9.781844 As BC 394 Ar. Co. 7.404504 : BD 217.0835 2.336627 ::r 10.000000 : cos. B 56 33' 58" 9.741131 Whence C = 180 - (A+ B) = 70 40' 18". Ex. 2. Given the three sides of a triangle, BC 167, AB 214, and AC 195 yards, respectively, to find the angles. Ans. A = 47 55' 13", B = 60 4' 19, C = 72 0' 28". Ex. 3. Given AB = 51.67, AC = 43.95, and BC = 27.16, to find the angles. Ans. A = 31 42' 42", B = 58 16' 34", C = 90 0' 44". KULE 2. 143. As the rectangle of two sides is to the rectangle of the half sum of the three sides and the excess thereof above the third side, so is the square of radius to the square of the cosine of half the angle contained by the first mentioned sides. SEC. IV.] NUMERICAL SOLUTION OF TRIANGLES. 75 DEMONSTRATION. Let ABC (Fig. 50) Fig. 50. be a triangle, of which AB is greater than AC. Make AD = AC. Join DC, and bisect it by AEF. Draw EH paral- lel and equal to CB. Join HB, and pro- duce it to meet AEF in F. Then, since j EH is equal and parallel to CB, BH is equal and parallel to CE, (33.1.) Therefore F is a right angle. Again: since BH is equal to ED, and the angle EGD = BGH and EDG = GBH, (26.1,) DG = GB and EG = GH. On EH describe a circle, and it will pass through F. Now, 2 AK = 2 AG-f- 2 GK = AC -f AD-f 2 DG -f 2 GK = AC -f AB-f BC ; or AK = (AC + AB -f BC), and AI = AK KI = J (AC + AB + BC) BC. But, (Rule 2, Art. 137,) As AD : AE : : r : cos. DAE (cos. BAC), and AB : AF : : r : cos. % BAC ; whence (23.6) AB . AD : AE . AF : : r 1 : cos. a \ BAC. But (36.3, Cor.) AE . AF = AK . AI = (AC + AB + BC) . $(A.C + AB -f BC) BC; whence AB . AC : (AC + AB + BC) ( (AC -f AB + BC) BC) : : r 1 : cos. a \ BAC. EXAMPLES. Ex. 1. Given AB = 467, AC = 413, and BC = 394, to find the angle C. Here, put s half sum of the sides : we have s = 637 and s AB = 170; whence AC BcI A 41B A.C. 7.384050 J \ BC 394 A.C. 7.404504 , ATf , }s 637 ; s.(5 AB) < * -r> -.^r/% ' \s-AB 170 2.804139 2.230449 ::R 2 20.000000 : cos. 2 J BCA 2)19.823142 J BCA = 35 20 ; 9'' 9.911571 BCA = 70 40' 18". In the above calculation the R* and its logarithm might have been omitted, since we have to deduct 20 in consequence of having taken two arithmetical complements. The sum of the logarithms is divided by 2, to extract the square root, (Art. 16.) 76 PLANE TRIGONOMETRY. [CHAP. III. The rule may be expressed thus : Add together the arithmetical complements of the logarithms of the two sides containing the required angle, the logarithm of the half sum of the three sides, and the logarithm of the excess of the half sum above the side opposite to the required angle : the half sum of these four logarithms will be the logarithmic cosine of half that angle. Ex. 2. Given AE = 167, AC = 214, and EC = 195, to find the angles. Ans. A = 60 4' 22", B = 72 0' 28", C = 47 55' 16". Ex. 3. Given AB = 51.67, AC = 43.95, and BC = 27.16, to find the angles. Ans. A = 31 42' 40", B = 58 16' 28", C = 90 0' 52". SECTION T. INSTRUMENTS AND FIELD OPERATIONS, 144, The Chain. GUNTEE'S CHAIN is the instrument most commonly employed for measuring distances on the ground. For surveying purposes, it is made 66 feet or 4 perches long, and is formed of one hundred links, each of which is therefore .66 feet or 7.92 inches long. The links are generally connected by two or three elliptic rings, to make the chain more flexible. A swivel link should be inserted in the middle, that the chain may turn without twisting. In order to facilitate the counting of the links, every tenth link is marked by a piece of brass, having one, two, three, or four points, according to the number of tens, reckoned from the nearest end of the chain. Sometimes the number of links is stamped on the brass. The middle link is also indicated by a round piece of brass. The advantage of having a chain of this particular length is, that ten square chains make an acre. The calculations SEC.V.] INSTRUMENTS AND FIELD OPERATIONS. 77 are therefore readily reduced to acres by simply shifting the decimal point. There being one hundred links to the chain, all measures are expressed decimally, which renders the calculations much more convenient. Eighty chains make one mile. In railroad surveying, a chain of one hundred feet long is preferred, the dimensions being thus at once given in feet. When the measurements are required to be made with great accuracy, rods of wood or metal, which have been made of precisely the length intended, are used. In the surveys of the American Coast Survey, the unit of length employed is the French metre, equal to the 10000000th part of the quadrant of the meridian. The metre is 39.37079 inches = 3.280899 feet = 1.093633 yards long. It were much to be desired that the metre, or some other unit founded on the magnitude of the earth, or on some other natural length, such as that of a pendulum beating seconds at a given latitude, were universally adopted as the unit. The metre will probably gradually come into general use. To reduce chains and links to feet, express the links decimally and multiply by 66. Thus, 7 chains 57 links = 7.57 chains are equal to 7.57 X 66 = 499.62 feet = 499 feet 7.4 inches. To reduce feet and inches to chains, divide by 66, or by 6 and 11. The inches must first be reduced to a decimal of a rro f*t7 foot, 'thus, 563 feet 8 inches = 563. 67 feet = - ch. = 8.54 chains. Instead of a chain of 66 feet, one of 33 feet, divided into fifty links, is sometimes used. This is really a half chain, and should be so recorded in the notes. The half chain is more convenient when the ground to be measured is uneven. 145. The chain is liable to become incorrect by use; its connecting rings may be pulled open, and thus the chain become too long, or its links may be bent, which will 78 PLANE TRIGONOMETRY. [CHAP. III. shorten the chain. Every surveyor should, therefore, have a carefully measured standard with which to compare his chain frequently. According to the laws of Pennsylvania, such a standard is directed to be marked in every county town, and all surveyors are required to compare their chain therewith every year. If the chain is too long, it may be shortened by tighten- ing the rings ; if it is too short, which it can only become by some of the links having been bent or some rings tightened too much, these should be rectified. It has been found that a distance measured by a perfectly accurate chain is very generally recorded too long; if then the chain is found slightly too long, say from one fourth to one third of an inch, it need not be altered, a distance measured with such a chain being more accurately recorded than if the chain were correct. In using the chain, care should be taken to stretch it always with the same force, or the different parts of the line will not be correctly recorded. Like all other instruments, it should be carefully handled, as it is liable to injury. 146. The Fins. In using the chain, ten pins are necessary to set in the ground to mark the end of each chain measured. These are usually made of iron, and are about a foot or fif- teen inches long, the upper end being formed into a ring, and. the lower sharpened that they may be readily thrust into the ground. Pieces of red and white cloth should be tied to the ring, to distinguish them when measuring through grass or among dead leaves. 147. Chaining. This operation requires two persons. The leader starts with the ten pins in his left hand and the end of the chain in his right; the follower, remain- ing at the starting point and looking at the staff set up to mark the other end of the line, directs the leader to extend the chain precisely in the proper direction. The leader then sticks one pin perpendicularly into the ground at the end of the chain. They then go on until the follower comes to this pin, when he again puts the leader in line, SEC. V.] INSTRUMENTS AND FIELD OPERATIONS. 79 who places a second pin. The follower then takes up the first pin, and the same operation is repeated until the leader has expended all his pins. "When he has stuck his last pin, he calls to the follower, who comes forward, bringing the pins with him. The distance measured viz. : ten chains is then noted. The leader, taking all the pins, again starts, and the operation is repeated as before. When the leader has arrived at the end of the line, the number of pins in possession of the follower shows the number of chains since the last "out," and the number of links from the last pin to the end of the line, the number of odd links. Thus, supposing there were two "outs," and the follower has six pins, the end of the line being 27 links from the last pin, the length would be 26.27 chains. Some surveyors prefer eleven pins. One pin is then stuck at the beginning of the line, and at every "out" a pin is left in the ground by the leader. If the chain-men are both equally careful, they may change duties from time to time. If otherwise, the more intelligent and careful man should act as follower, that being much the more responsible position. 148, Recording the " Outs." As every " out" indicates ten chains, or five chains, if a two-pole chain is used, it is of great importance to have them carefully kept. Various contrivances have been suggested for that purpose. Some chain -men carry a string, in which they tie a knot for every out ; others place in one pocket a number of pebbles, and shift one to another pocket at each out. Either of these methods is sufficient if faithfully followed out. One rule, however, should be faithfully adhered to, viz.: that the memory should never be trusted. The distractions to which the mind is subject in all such operations, necessarily call off* the attention, so that a mere number, which has no associations to call it up, will be very likely to be forgotten. Perhaps the best method of preserving the "outs" is to have nine iron pins and five or six brass ones. The leader takes all the pins and goes on until he has exhausted hia iron pins; he then goes on one chain, and, sticking a 80 PLANE TRIGONOMETRY. [CHAP. HI. brass pin, calls, " Out." The follower then advances, bring- ing the pins. He delivers to the leader the iron pins but retains the brass ones. On arriving at the end of the line, the brass pins in the follower's possession will show the number of "outs" and the iron pins the number of chains since the last "out." Thus, supposing he has six brass and eight iron pins, and that the end of the line is 63 links from the last pin, the distance is 68.63 chains. 149. Horizontal Measurement. In all cases where the object is to determine the area or the position of points on a survey, the measurements must either be made horizon- tally, or, if made up or down a slope, the distance must be reduced according to the inclination. In chaining down a slope, the follower should hold his end of the chain firmly at the pin. The leader should then elevate his end until the chain is horizontal, and then mark the point directly under the end of the chain. This may be done by means of a staff four or five feet long, which should be held vertical, or by dropping a pin held in the hand with the ring downwards, or by a plumb-line. If the ground slopes much, the whole chain cannot be used at once. In such cases the leader should take the end of the half or the quarter, and, elevating it as before, drop his pin or make a mark. The follower then comes forward, and, holding the 50th or 25th link, as the case may be, the leader goes for- ward to the end of another short portion of the chain, which he holds up, as before. A pin is left only at the end of every whole chain. Chaining up a slope is less accurate than chaining down, from the difficulty of holding the end still, under the strain to which the chain is subjected. The follower should always, in such cases, be provided with a staff four or five feet long, and a plumb-line to keep it vertical. If the slope is so steep that the whole chain cannot be used at once, the leader should take (as before) the end of a short portion, say one fourth, and proceed up hill. The follower then elevates his end, holding it firmly against the staff, which is kept vertical by the plumb-line. The leader, having made his mark, noti- SEC. V.] INSTRUMENTS AND FIELD OPERATIONS. 81 fies the follower, who comes forward and holds up the same link that the leader used. He then goes forward as before. 150. When great accuracy is required, the chaining should be made according to the slope of the ground, leaving stakes where there is any change of the slope, and recording the distances to these stakes in the note book. The inclination of the different parts being then taken, the horizontal dis- tance can be calculated. If a transit with a vertical arc is employed, the slope can be obtained at once, and the proper correction may be made at the time. The best way is to have a table prepared for all slopes likely to be met with, and apply the correction on the ground. Instead of deducting from the distance measured, it is best to increase the length on the slope, calling each length so increased a chain : the horizontal distance will then be correctly recorded. Thus, supposing the slope to be 5, in order that the base may be 1 chain the hypothenuse must be 1.0038 : the follower should therefore advance his end of the chain rather less than half a link. If a compass is used, it may be furnished with a tangent scale, to be described hereafter. The following table contains the ratio of the perpen- dicular to the base, the correction of the base for each chain on the slope, and the correction of the slope for each horizontal chain. If the corrections are made as the work proceeds, the last column should be used ; if in the field- notes after the work is done, the third column furnishes the data. 82 PLANE TRIGONOMETRY. [CHAP. Ill, Angle. Slope, perp.: base. Correction of base, in links. Correction of hypoth. in links. Angle. Slope. Correction of base, in links. Correction of hypoth. in links. 3 1 19.1 0.14 +0.14 17 1 3.3 4.37 +4.57 4 1 14.3 0.24 0.24 18 1 3.1 4.89 5.15 5 1 11.4 0.38 0.38 19 1 2.9 6.45 5.76 6 1 9.5 0.55 0.55 20 1 2.7 6.03 6.42 7 1 8.1 0.75 0.75 21 1 2.6 6.64 7.11 8 1 7.1 0.97 0.98 22 1 2.5 7.28 7.85 9 1 6.3 1.23 1.25 23 1 2.4 7.95 8.64 10 1 6.7 1.52 1.54 24 1 2.2 8.65 9.46 11 1 6.1 1.84 1.87 25 1 2.1 9.37 10.34 12 1 4.7 2.19 2.23 26 1 2.1 10.12 11.26 13 1 4.3 2.56 2.63 27 1 2 10.90 12.23 14 1 4.0 2.97 3.06 28 1 1.9 11.71 13.26 15 1 3.7 3.41 3.53 29 1 1.8 12.54 14.34 16 1 3.5 3.87 4.03 30 1 1.7 13.40 15.47 151. Tape-Lines. A tape-line is sometimes used instead of a chain in measuring short distances. It is, however, very little to be depended on. If used at all, the kind that is made with a wire chain should be employed. It is much less liable to be stretched than those made wholly of linen. 152. Chaining being one of the fundamental operations of surveying, whether for trigonometrical purposes or for the calculation of the contents, it has been described minutely. If correct measurements are needful, accurate notes are no less so. The chief points to be attended to in recording the measurements are precision and conciseness. Some of the most approved methods are given in Chap- ter IV. 153. Angles. For surveying purposes horizontal angles alone are needed, since all the parts of the survey are re- duced to a horizontal plane ; but to fix the direction of a point in space not only the horizontal but vertical angles are required. With the aid of these, and the proper linear measures, its position may be fully determined. 154. Horizontal angles are measured by having a plane, properly divided, and capable of being so adjusted as to be SEC. V.] INSTRUMENTS AND FIELD OPERATIONS. ' 83 perfectly horizontal. Movable about the centre of this plane is another plane, or a movable arm, carrying a pair of sights or a telescope, which can be placed so that the line of sight may pass through the object. If then this line be directed to one object, and the position of the two plates or of the arm on the plate be noted by an index properly situated, and then be turned so as to point to another object, the angle through which the plate or the arm has turned will be the horizontal angle contained by two planes drawn from the centre of the instrument to the two objects. 155. Vertical angles are measured by having a pair of sights or a telescope so adjusted as to move on a horizontal axis, the horizontal position of the sights or the telescope being indicated either by a plumb-line or a level. 156. The transit with a vertical arc, or the theodolite, are so arranged as to perform both these offices. As a full understanding of the use of the different parts of these instruments is necessary to their proper management, we shall enter, considerably in detail, into a description of them. THE TRANSIT AND THE THEODOLITE.* 157. General Description. The Transit or the Theo- dolite (Figs. 51 and 52) consists of a circular plate, divided at its circumference into degrees and parts, and so sup- ported that it can be placed in a perfectly horizontal posi- tion. This divided circle is called the limb. An axis exactly perpendicular to this plate, bearing another cir- cular plate, passes through its centre. This plate is so adjusted as to move very nearly in contact with the former without touching it. By this arrangement the upper plate can be turned freely about their common centre. This plate carries a telescope Q, resting on two upright supports KK, upon which it is movable in a vertical plane. The telescope, having thus a horizontal and a vertical motion, * The author is indebted to Professor Gillespie's "Treatise on Land Sur- veying" for many of the features in his mode of presenting the subjects of the Transit and Theodolite, their verniers and their adjustments. PLANE TRIGONOMETRY. [CHAP. III. THE TRANSIT. Fig. 51. SEC.V.] INSTRUMENTS AND FIELD OPERATIONS. THE THEODOLITE. Fig. 52. 85 86 PLANE TRIGONOMETRY. [CHAP. Ill, can readily be pointed to any object. The second described plate has an index of some kind, moving in close proximity to the divided arc, so that the relative position of the plates may be determined. If then the telescope be directed to one object, and afterwards be turned to another, the index will travel over the arc which measures the horizontal angle between the objects. In order to place the plates in a perfectly horizontal posi- tion, levelling screws and levels are required: these, as well as the other parts of the instrument, will be fully described in their proper place. 158. The above description applies to both instruments. The transit, however, is so arranged that the telescope can turn completely over; it can, therefore, be directed back- wards and forwards in the same line. If the same thing is to be done by the theodolite, the telescope must be taken from its supports and have its position reversed. This ope- ration is troublesome, and is, besides, very apt to derange the position of the instrument. For surveying purposes, therefore, the transit is much to be preferred; and when the axis on which the telescope moves is provided with a vertical arc it serves all the pur- poses of a theodolite. The theodolite has a level attached to the telescope. This is not generally found in the transit. 159. The accuracy of these instruments depends on several particulars : 1. By means of the telescope the object can be dis- tinctly seen at distances at which it would be invisible by the unassisted eye. 2. The circle, with its vernier index, enables the observer to record the position of the telescope with the same degree of precision with which it can be pointed. 3. There are arrangements for giviDg slow and regular motion to the parts, so as to place the telescope precisely in the position required. SEC. V.] INSTRUMENTS AND FIELD OPERATIONS. 87 4. There are other arrangements for making the plates of the instrument truly horizontal. 5. Imperfections in the relative position of the different parts of the instruments may be corrected by screws, the heads of some of which are shown in the drawings. However complicated the arrangements for performing these various operations may make the instruments appear, that complication disappears when they are viewed in detail and properly understood. 160, In the figures of these instruments, V is the vernier, covered with a glass plate. In some theodolites the whole divided limb is seen. In others (and in the transit) but a small portion is exposed, it being completely covered by the other plate, except the small portions near the vernier. Transits have generally but one vernier, though in some instruments there are two. The theodolite has generally two, and sometimes three or four. B is the compass box, containing the magnetic needle ET. A, A, are the levels. C and D are screws; the former of which is designed to clamp the lower plate, and the latter to clamp the plates together. T and U are tangent screws, to give slow and regular motion when the plates are clamped: by the former the whole instrument is turned on its axis, and by the latter the upper plate is moved over the other. P, P are the levelling plates; and S, S, S, are three of the four levelling screws. E is the vertical circle, with its vernier F. G is a level attached to the telescope. H is a gcrew to clamp the horizontal axis, (not visible in the figure of the theodolite,) and I a tangent screw, to give it regular motion. 161. The Telescope. A telescope is a combination of lenses so adjusted in a tube as to give a distinct view of a distant object. It consists, essentially, of an object-glass, placed at the far end of the tube, and an eye-piece at the near end. By the principles of optics, the rays of light proceeding from the different points of the object are brought to a 88 PLANE TRIGONOMETRY. [CHAP. III. focus within the tube, (Fig. 53,) there forming an rig. 53. inverted image. Crossing at this focus, they pro- ceed on to the eye-piece, by the lenses of which they are again refracted, and made to issue in parallel pencils, thus giving a distinct magnified image of the object. 162. The Object-glass. Whenever a beam of light passes through a lens, it is not merely re- fracted, but it is likewise separated into the different colored rays of the solar spectrum. This separa- tion of the colored rays, or the chromatic aberration, causes the edges of all bodies viewed with such a glass to be fringed with prismatic colors, instead of being sharply defined. It has been found, how- ever, that the chromatic aberration may be nearly removed, by making a compound lens of flint and crown glass, as represented in Fig. 54, in which A is a concavo- convex lens of flint glass, and B a double convex lens of crown glass, the convexity of one surface being made to agree with the concavity of the other lens. The two are pressed together by a screw in the rim of the brass box which contains them, thus forming a single compound lens. When the surfaces are properly curved, this arrangement is nearly achromatic. The object-glass is placed in a short tube, movable by a pinion attached to the milled head W. (Figs. 51, 52.) By this means it may be moved backwards and forwards, so as to adjust it to dis- B tinct vision. c 163. The Eye-piece. The eye-piece used in the telescopes employed for surveying purposes consists of two plano-convex lenses, fixed in a short tube, the convex surfaces of the lenses being A SEC. V.] INSTRUMENTS AND FIELD OPERATIONS. 89 towards each other. This arrangement is known as " Eamsden's Eye-piece." 164. A telescope with an object-glass and an eye-piece as above described, inverts objects. By the addition of two more lenses the rays may be made to cross each other again, and thus to give a direct image of the object. As these additional lenses absorb a portion of the light passing through them, they diminish the brightness of the image. They may therefore be considered a defect in telescopes intended for the transit or theodolite. A little practice obviates the inconvenience arising from the inversion of the image. The surveyor soon learns to direct his assistant to the right when the image appears to the left of its proper position, and vice versa. 165, The Spider-Lines. The advantage gained by the telescope in producing distinct vision, would add nothing to the precision of the observations, without some means of directing the attention to the precise point which should be observed in the field of view. The whole field forms a circle, in the centre of which the object should appear at the time its position is to be noted. This centre is de- termined by stretching across the field precisely in the focus of the eye-piece a couple of spider-lines or fine wires, at right angles to each other. The former are generally employed. "When they are properly adjusted in the focus they can be distinctly seen, and the point to be observed can be brought exactly to coincide with their intersection. The magnifying power of the eye-piece enables this to be done with the greatest precision. When it has been effected, a line through the centre of the eye-piece and the centre of the object-glass will pass directly through the object. This line is called the line of collimation of the telescope. The spider-lines are attached by gum to the rim of a circular ring of brass placed in the tube of the telescope at the point indicated by the screw-heads a, a, (Figs. 51, 52,) eome of which are invisible in the figure. These screws 90 PLANE TRIGONOMETRY. [CHAP. III. serve to hold the ring in position, as Fi s- 55 - represented in Fig 55, and to adjust - it to its proper position. The eye- piece is made to slip in and out of the tube of the telescope, so that the focus may be brought to coincide exactly with the intersection of cross-wires. The perfect adjustment of the focus may be determined by moving the eye sideways. If this motion causes the wires to change their position on the object, the adjustment is not perfect: it must be made so before taking the observation. 166. Spider-lines are generally used for making the "cross- wires," though platinum wires drawn out very fine are preferable. The wire is drawn to the requisite degree of fineness by stretching a platinum wire in the axis of a cylindrical mould and casting silver around it. The com- pound wire thus formed is then drawn out as fine as possi- ble and the silver removed by nitric acid. By this means Dr. Wollaston succeeded in obtaining wire not more than one thirty thousandth (^fan) of an inch in diameter. As such wire is very difficult to procure, the spider-threads are generally substituted. The operation of placing them in their proper position is thus performed. A piece of stout wire is bent into the form of the letter U, the distance between the legs being greater than the external diameter of the ring. A cobweb is selected having a spider hanging at the end. It is gradually wound round the wire, his weight keeping it stretched : a number of strands are thus obtained extending from leg to leg of the wire : these are fixeji by a little gum. To fix them in their position, the wire is placed so that one of the lines is over notches previously made in the ring. The thread is then fixed in the position with gum or some other tenacious substance. The wire being removed, the line is left stretched across the opening in the proper position. SEC. V.] INSTRUMENTS AND FIELD OPERATIONS. 91 167. The Supports. Attached to one of the horizontal plates, usually the index-plate of the instrument, are two supports, K, K, (Figs. 51, 52,) bearing the horizontal axis L. These supports should be made of precisely the same height, so that when the plate is level the axis may be hori- zontal. In some instruments there is an arrangement for raising or depressing one end of the axis so as to perfect the adjustment. In most cases, however, the adjustment is made perfect by the maker, and, if found not to be so, it "rnust be remedied by removing the support which is too high and filing some off from the bottom. This should always be done by the manufacturer, In the transit the telescope is attached immediately to the axis ; but in the theodolite the axis bears a bar M at right angles to it. This bar carries at its ends two supports, which from their shape are called Y's, in the crotch of which the telescope rests, being confined there by an arch of metal passing over the top. This arch is movable by a joint at one side, and is fastened by a pin at the other. By remov- ing the pin and lifting the arch the telescope is released and may be taken from the support. It rotates freely on its axis when confined by the arch. The telescope, being attached thus to the horizontal axis, admits of being elevated or depressed in a vertical plane so that it may be directed to any object. 168. The Vertical Limb. In the theodolite, the vertical limb E consists of a semicircle of brass graduated on its face and attached to the bar M. This limb moves with the telescope upon the horizontal axis, and thus by means of the index F, serves to determine the angle of elevation of the object. In the transit with a vertical circle, the circle is attached to the end of the axis, as seen at E, the index then being attached to the support K. In some instru- ments, instead of the axis bearing a circle, an arc of from 60 to 90 is attached to the support, and the index is fixed to the axis by an arm which is either permanently fastened to it or is capable of being clamped in any position. 92 PLANE TRIGONOMETRY. [CHAP. IIL 169. The Levels. Attached to the horizontal plate are two levels A and A set at right angles to each other, so as to determine when that plate is horizontal. They consist of glass tubes very slightly curved, the convexity being upward. They are nearly filled with alcohol, leaving a small bubble of air, which by the principles of hydrostatics will always take the highest point. If they are properly adjusted, the plate to which they are attached will, when these bub- bles have been brought to the middle of their run, be level, however it may be turned about its vertical axis. To the telescope of the theodolite and also to that of some transits another level G is fixed. This should be so adjusted that when the line of collimation of the telescope is horizontal the bubble may be in the centre of its run. 170. The Levelling Plates. The four screws S, S, S, and S, called levelling screws, are arranged at intervals of 90 between the two plates P, P, which are called levelling plates or parallel plates. They screw into one plate and press on the other. By tightening one screw and loosening the opposite one at the same time, the upper plate, with the instrument above, may be tilted. To allow this motion, the column connecting them terminates in a ball, which works in a socket in the centre of the lower plate. A joint of this kind, called a ball-and-socket joint, allows movement in all directions. To level the instrument by means of these levelling screws, loosen the clamp, and turn the plates until the telescope is directly over one pair of the screws. Then, taking hold of two opposite screws, move them in contrary directions with an equal and uniform motion, until the bubble in the tube parallel to the line joining these screws is in the middle. Then turn the other screws in like man- ner until the other bubble comes to the middle of its tube. When they are both brought to this position the plates are level if the instrument is in adjustment. In levelling, care should be taken to move both screws equally. If one is moved faster than the other, the instrument will not be firm, or will be cramped. SEC. V.] INSTRUMENTS AND FIELD OPERATIONS. 93 171. The Clamp and Tangent Screws. The former of these are used for binding parts of the instrument firmly together, the latter for giving a slow motion when they are so bound. The clamp C tightens the collar clasping the vertical axis, and thus holds it and the plate attached to it firmly in their places. The other plate, moving on an axis within the former, may, notwithstanding, move freely. "When this clamp is tightened, the collar may be moved slowly round by means of the tangent screw T. In its motion it carries with it the axis and attached plate. The clamp D fastens the two plates together. They may, how- ever, when so clamped, be made to move slightly on each other by means of the tangent screw U. If both clamps are tight, the instrument is firm, and the telescope can only be turned horizontally by one of the tangent screws. If the clamp C is tight and the other loose, the telescope and upper plate will move while the lower remains fixed. If D is tight and C is loose, the two plates are firmly attached to each other; but the whole instrument can be moved horizontally. Attached to the horizontal axis there is likewise a clamp H and tangent screw I, the purposes of which are similar to those described, the clamp fixing the axis, and the screw moving it slowly and steadily. 172. The Watch-Telescope. Connected with the lower part of theodolites of the larger class there is a second tele- scope B,, the object of which is to determine whether the in- strument has changed position during an observation. It is directed to some well defined object, and after all the ob- servations at the station have been made, or more frequently if thought necessary, it should be examined to see whether or not it has changed its position. If it has, the divided arc has changed also. The instrument, therefore, requires readjustment. 173. Verniers. As it would be very difficult to divide a circle to the degree of minuteness to which it is desirable to read the angles, or, if it were so divided, since it would 94 PLANE TRIGONOMETRY. [CHAP. III. be impossible for the eye to detect the divisions, some contrivance is necessary to avoid both difficulties. These difficulties will, perhaps, be made more striking by a simple calculation. The circumference of a circle 6 inches in diameter is 18.849 inches. If the circle is divided into degrees there will be -- =19.1 divisions in the space of an inch. If the divisions are quarter degrees there will be 76.4 to the inch ; and if minutes, there would be 1150 divisions to every inch. The first and second could be read ; but the third, though it might by proper mechanical contrivances be made, yet it would be almost, if not en- tirely, impossible to distinguish the cuts so as to read the proper arc. And yet that division is not so minute as is sometimes desirable on a circle of that diameter. The vernier is a simple contrivance to effect this subdivision of space, in a way to be perfectly distinct and easily read. 174. The principle of the vernier will be best understood by a simple example. In the adjoining figure, (Fig. 56,) AB represents a scale with the inch divided into tenths, the figure being on a scale of 3 to 2 or 1J times the natural size. Fig. 56. / \ 10 5 3 2 9 2 CD is another scale having a space equal nine of the divisions on AB divided into ten equal parts. This second scale is the vernier. Kow, since ten spaces of the vernier are equal to nine of the scale, each of the former is equal to nine tenths of one of the latter. If then the on the vernier corresponds to one of the divisions of the scale, the first division of thervernier will fall ^ of a space or ^ of an inch below the next mark on the scale, the next -division SEC. V.] INSTRUMENTS AND FIELD OPERATIONS. 95 and so on. The will fall ifo f an mcn below, the next __ in the figure stands at 28.7 inches. If now the vernier be slid up so that the first division shall correspond to a division on the scale, the will have been raised ^ inch. If the second be made to coincide, the vernier will have been raised ^ of an inch. If it be placed as in Fig. 57, the reading will be 28.74 inches. Fig. 57. s o 2 S The student should make for himself paper scales, di- vided variously, with verniers on other pieces of paper, so that he may become familiar with the manner of reading them. If his scale is to represent degrees, the portion re- presenting the arc might be drawn as a straight line, for the sake of facility in the drawing. It will illustrate the subject as well as if an arc of a circle were used. He should be- come particularly familiar with the one represented by Fig. 60, as it is the division most commonly used in theodolites and transits. 175. The Beading of the Vernier. To determine the reading of the vernier, that is, the denomination of the parts into which it divides the spaces on the scale, obsj^rvejiow many of the spaces on the scale are equal to a number on the vernier which is greater or less by one. The number of spaces on the vernier, so determined, divided into the value of one of the spaces on the scale, will give the denomination required. Thus, in Figs. 56 and 57, ten spaces of the ver- nier correspond with niae on the scale : the reading is therefore to fa of fa = ^ of an inch. If an arc were divided into half-degrees, and thirty spaces on the vernier were equal to twenty nine or to thirty one 36 PLANE TRIGONOMETRY. [CHAP. IIL spaces on the arc, the reading would be to ^ of | = ^ 5 = 1 minute ; or, as it is usually expressed, to minutes. Fig. 60 is an example of this division. \ 176. To read any Vernier. First, determine as above the reading. Then examine the zero point of the vernier. If it coincides with any division of the scale as in Fig. 56, that division gives the true reading, 28.7 inches. But if, as will generally be the case, it does not so coincide, note the division of the scale next preceding the place of the zero, and then look along the vernier until a division thereof is found which is in the same straight line as some division on the scale. This division of the vernier gives the number of parts to be added to the quantity first taken out. Thus, in Fig. 57, the of the vernier is between 8.7 and 8.8, and the fourth division on the vernier is in a line with a division on the scale : the true reading is therefore 28.74 inches. To assist the eye in determining the coincidence of the lines, a magnifying glass, or sometimes a compound micro- scope, is employed. When no line is found exactly to coincide, then there will be some which will appear equally distant on opposite sides. In such cases, take the middle one. 177. Retrograde Verniers. Most verniers to modern instruments are made as above described. In some in- stances, the vernier is made to correspond to a number of spaces on the arc one greater than that into which it is divided. Such verniers require to be read backwards, and are hence called retrograde verniers. Fig. 58 is an ex- ample of one of this kind. It is the form that is generally used in barometers. It is drawn to one and a half times the natural size : the inches are divided into tenths, and eleven spaces on the scale correspond with ten on the vernier. SEC. V.] INSTRUMENTS AND FIELD OPERATIONS. Fig. 58. - 97 1 > 1 1 1 1 | f | 3 2 9 2 8 The value of one division of the vernier is T ^ inch. If therefore on the vernier corresponds to a division on the scale, 1 on the vernier will be ^ of an inch below the next on the scale, 2 will be yfo below ; and so on. If the vernier is raised so that the 1 on the vernier is in line, it is raised ^ inch; if 2 is in line, it is raised jfo; and so on. The reading in Fig. 58 is 29.7 inches, and in Fig. 59, 29.53 inches. Fig. 59. V 5 l ! 1 1 a 2 9 2 8 178. In Fig. 60, the arc is divided by the longer lines into degrees, and by the shorter into half degrees, or 30' spaces. Fig. 60. 98 PLANE TRIGONOMETRY. [CHAP. III. Thirty spaces on the vernier are equal to twenty nine on the arc. The reading is therefore to ^ of 30 minutes = 1 minute. The zero of the vernier stands between 41 30' and 42. On looking along the vernier, it is seen that the fifth and sixth lines coincide about equally well. The ver- nier therefore reads 41 35' 30" 179. Reading backwards. Sometimes it is required to read backwards from the zero point on the limb. When this is done, the numbers on the vernier must be read in reverse, the highest being called zero, and the zero the highest. Fig. 61. Thus, in Fig. 61, the zero of the vernier standing to the right of 360 on the limb, between 1 30' and 2, and the division marked with an arrow-head being in line, the angle is 1 41'. This mode of reading is needful when using the theodolite to take angles of depression, and also when using the transit to trace a line that bends backwards and for- wards, the angle of deflection being then generally taken, and recorded to the right or to the left, as the case may be. 180. Double Verniers. To avoid the inconvenience of reading backwards, a double vernier is frequently made. It consists of two direct verniers having the same zero point, as shown in Fig. 62. SEC. V.] INSTRUMENTS AND FIELD OPERATIONS. Fig. 62. The arc in this figure is divided into degrees, and eleven spaces on the arc are equal to twelve on the vernier : the reading is therefore to 5 minutes. When the figures "on the arc increase to the right, the right-hand vernier is used, and vice versa. The reading on the- figure is 2 45' to the left. 181. Another form of double vernier is shown in Fig. 63. . 63. In the figure, the vernier reads to minutes. When the zero of the vernier is to the left of that on the limb, the figures begin at the zero and increase towards the left to 15' ; they then pass to the right-hand extremity, and again proceed to the left ; that is, they stop at A and commence again at B. The upper figures of each half are the con- tinuation of the lower figures of the other half. The read- ing in Fig. 63 is 1 8' to the left. In Fig. 64 the reading is 3 19' to the right. 100 PLANE TRIGONOMETRY. Fig. 64. [CHAP. IIL Fig. 65. E 182. If the preceding descriptions have been thoroughly understood, the student will have no difficulty in reading the arc on any limb, however it may be divided. He should study the different positions until he can determine the angle with readiness, however the index may be placed. For this purpose, as before remarked, he should make for himself verniers with different scales, so that they can be placed in various positions. The construction of such verniers is very simple. Suppose, for example, it is desired to divide the arc into degrees and subdivide it by the vernier so as to read to 5 minutes : twelve spaces on the vernier must equal eleven on the arc, or one space on the vernier will equal JJ of a space on the arc. Let (Fig. 65) E be the centre and AB a por- tion of the limb, which, for the purpose intended, should not be of less radius than ten or twelve inches, and let CD be the vernier; with some other radius EG, which should be greater than EB, de- scribe an arc GF; take El : EG : : number A of divisions on the vernier : the number that occupies the same space on the arc, H in this case, as 12 to 11. Take from the table of chords the chord of 1 or , as the case may be, and multiply it by the length of EG ; lay off the product on GF, thus determining the points 1, 2, 3, &c., and lay off the same length on IH, determining the points a, 6, c, &c. ; stick a fine needle in the centre E; then, resting the ruler against the needle, bring it so as to coincide with I, and draw the SEC. V.] INSTRUMENTS AND FIEL1) OPERATIONS. 101 division on AB ; then, keeping it pressed against the needle, bring it successively to the other points on GF, and draw the corresponding divisions on AB. The arc will then be divided. In the same way, resting the ruler against the needle, and bringing it successively to the points on IH, the vernier may be divided. The reason of this process is, that since ab 1.2, the degrees of ab will be to the degrees of 1.2 as the radius of GF is to the radius of HI, as 11 to 12. Hence each division of the vernier is $ of one division of the arc. By this means the divisions may be made with facility and accuracy. 183. Adjustments. In order that the theodolite and transit may give correct results when used, it is necessary that the different parts should bear the precise relations to each other that they are intended to have. By the term adjustment is meant the due relation of the parts to each other : when it is said an instrument is in adjustment, it is meant that every part bears to every other precisely its proper relations, so that the instrument is in perfect work- ing order. Before making any observations with a new instrument, it should be carefully examined to verify the adjustment. If the parts are not found to be properly adjusted, they must be rectified. 184. For measuring horizontal angles, the following con- ditions are necessary : 1. The levels should be parallel to the plates, so that when the bubbles are in the middle of their run, the plates shall be horizontal. 2. The axes of the two horizontal plates should be per- fectly parallel and perpendicular to the plane of the plates. 3. The line of collimation should be perpendicular to the horizontal axis. 4. The horizontal axis should be parallel to the plane of the plates, so that when they are horizontal it may be so likewise. 102 " PLANE TRIGONOMETRY. [CHAP. III. 185. First Adjustment. The levels should be parallel to the horizontal plates. Verification. Clamp the two plates together; loosen the clamp C, (Figs. 51, 52 ;) bring the telescope directly over one pair of levelling screws, and level the plates as directed in Art. 170. Turn the plates half round : if the bubbles retain their position, the plane of the levels is perpendicular to the axis on which the lower plate turns. If either of them inclines to one end of its tube, it is out of adjustment, and requires rectification. To rectify the fault, bring the bubble halfway back to the middle by means of the capstan screw attached to one end, and the other half by the levelling screws. Again reverse the position of the plate : if the bubble now remains in the middle, the rectification is complete ; if not, the operation must be repeated. When both levels have been so arranged that the bubbles retain their position in the middle of their run when the plates are turned all round, the adjustment is perfect, and the axis is perpendicular to the plane of the levels. 186. Second Adjustment. The axes of the horizontal plates should be parallel. Verification. Level the plates, as directed in last article. Clamp the lower plate, and loosen the vernier-plate. Turn it half round : if both bubbles still retain their position the axes are parallel. If the plates move freely over each other without binding in any position, they are perpendi- cular to the axes, or, at least, the upper one is so. If any defects be found in either of these particulars, the instrument should be returned to the maker to be rectified. 187. Third Adjustment. The line of collimation of the telescope of the theodolite should be parallel to the common axis of the cylinders on which it rests in its Y '& SEC. V.] INSTRUMENTS AND FIELD OPERATIONS. 1Q3 Verification. Direct the telescope so that the intersec- tion of the wires bisects some well defined point at a dis- tance. Rotate the telescope so as to bring the level to the top. If the intersection still coincides with the object, the adjustment is perfect. If it has changed its posi- tion, bring it half-way back, b} r the screws a, a, and verify again. 188. Fourth. Adjustment. The Une of collimation must be perpendicular to the horizontal axis. Verification for the Transit. Set the transit on a piece of level ground, as at A, (Fig. 66,) and level it carefully. At some distance say four or five chains set a stake B in the ground, with a nail driven in the head, and direct the telescope so that the cross- c Fi s- 66 - wires may bisect exactly on the nail. Clamp the plates, turn the telescope ' over, and place a second stake C precisely in the line of sight. If the adjustment is perfect, the three points B, A, and C will be in a straight line. To determine whether they are so, turn the plate round until the tele- scope points to B ; turn it over, and, if the line of sight passes again through C, the adjustment is perfect. If it does not, set up a stake at E, in the line of sight : then the prolongation of the line BA bisects EAC. Let FG (Fig. 67) be the horizontal axis. Then, if the line of collimation makes the angle FAB acute, when the telescope D is turned over it will make FAC = FAB. The angle CAD is therefore equal to twice the error. Now, if the plate is turned until the line of sight is directed to B, the axis will be in the position F'G'. Turn the telescope over, and the angle EAF'= F'AB ; CAE is therefore equal to four times the error. 104 PLANE TRIGONOMETRY. [CHAP. III. Hence, to rectify the error, the instrument being in the second position, place a stake at H, one fourth of the dis- tance from E to C, (Fig. 67,) and, by means of the screws , E, F, and C. Suppose AC measures 27.56 chains, AD 10 chains, AE 15 chains, AF 20 chains, and BC = 1.57 chains. Then, 27.56 : 10 : : 1.57 : .57, the correction for D. Similarly, Ee = .85, and F/= 1.14 chains. Set off Dd, Ee, and F/, the calculated distances; set stakes at d, e, and /, and range out the line anew. Instead of working out each proportion, it is more concise to divide the deviation by the num- ber of chains in the measured length : this will give the correction for one chain. This correction, being multi- plied by the distance to each stake, will give the correction for that stake. Thus, in the above example, 1.57 = .057, the correction for 1 chain. 10 X .057 = .57, the correction for D ; 15 x .057 = .85, the correction for E ; 20 x .057 = 1.14, the correction for F. 210. Across a valley. When the line runs across a valley, let two points A and B be determined on opposite sides of the valley, from which the intervening ground can be seen. Then let one person take his station at A, and, holding a plumb-line over the stake, let him sight to B : he can then direct his assistant into the proper position, and thus fix as many points as are desirable. NOTE. These operations are all done more accurately and rapidly by means of the transit or theodolite. SEC. II.] FIELD OPEKATIONS. 123 211, To determine the point of intersection of two visual lines. This is most readily done by three persons, two of whom take their stations in the lines, at some distance from the point of intersection, and, looking along their lines respectively, signal the third until he ranges in both lines. A stake may then be driven at the point of inter- section. This operation may readily be performed by two persons. First, let them run out one of the lines, and stretch a cord or the chain across the course of the other. One of them then taking his station in the second line can signal the other to his proper position. 212. To run a line towards an invisible intersection. Through? (Fig. 81) Fig . 81 . run the line AC, in- tersecting the given lines in A and C. Then through any point B in AB set out BD parallel to AC by one of the modes to be pointed out. (See Arts. 227-229.) Divide BD in F, so that BF : FD : : AP : PC; that is, make BF = -- . Then PF will be the required line. AC* B. PERPENDICULARS. Problem 1. To draw a perpendicular to a given line from a given point in it. 213. (a.) When the Point is accessible. This may be done on the ground by the methods described in Arts. 88, 89, and 90, using the chain for a pair of compasses to sweep the circles, or by the following methods : 124 CHAIN SURVEYING. [CHAP. IV. 214. First Method. Let AB Fig. 82. (Fig. 82) be the line and C the point at which the perpendicular is to be erected. First, lay off CD, 60 links; then, fixing one end of the chain at D, sweep an arc of a circle at E, using the whole chain (100 links) for a j- radius. Next, fix one end at C, and, with 80 links for a radius, sweep an arc cutting the former in E. CE will be perpendicular to AB. Any other distances, in the same ratio as the above, will answer. Thus, DC might be 30, CE 40, and DE 50. "With these numbers no circles need be struck. Lay off DC = 30 links; fix the end of the chain at D, and the end of the ninetieth link at C : then, taking the end of the fiftieth link, stretch both parts of the chain equally tight, and set a stake at the point of intersection. These numbers are very convenient when short perpen- diculars are required ; but when the line is run to some dis- tance the greater lengths are preferable. 215. Second Method. Make AC (Fig. 83) a chain. With the whole length of the chain sweep two arcs cutting in D ; range out AD, making DE = AD : then CE will be the per- pendicular required. For, ADC being equilateral, A= 60, and A and ACD = 120; whence DCE and DEC = 60. But DE = DC : therefore DCE = 30, and ACE = 90, Fig. 83. SEC. II.] FIELD OPERATIONS. 125 216. (o.) When the Point is inaccessible. Erect a perpendicular at some other point D (Fig. 84) of the line. Through F, a point in this perpendicular, draw FH parallel to AB, (Art. 227.) Take FE = FD : range out EC, intersecting FH in Q-. Make GH equal FG: then CHI will be the perpendicular required. FE need not be taken equal to DF. If unequal, GH will be determined by the proportion EF : FD : : FG : GH. (c.) If the line is inaccessible, trigonometrical methods must be employed. Problem 2. To let fall a perpendicular to a line from a point without it. (a.) When the point and line are both accessible. 217. The methods in Arts. 91, 92, 93, may be adopted in this case; or in AB (Fig. 85) take any point D, and measure CD. Make DE = DC, and measure CE. FP 2 Then take EF = - , and F ~X~ D r i. ; 2.ED will be the foot of the perpendicular. Describe the semicircle ECA. Then, if CF is perpen- dicular to AB, EC is a mean proportional between AE EC 2 EC 2 and EF, whence EF = = 126 CHAIN SURVEYING. [CHAP. IV. Fig. 86. (b.) If the point is remote or inaccessible. 218. First Method. In AB (Fig. 86) take any convenient points A and D; erect the perpendicular FDE, making FD = DE; range out AE, and EC cutting AB in H, and FH intersecting AE in G: then GBC will be perpen- dicular to AB. For, by construction, the triangles ADE and ADF, as also FDH and EDH, are equal in all respects. Hence, AFG and AEG, having two angles and the included side of one equal to two angles and the included side of the other, are equal in all respects ; therefore AG = AC. Finally, ABC and ABG have two sides and their included angles respectively equal, whence B is a right angle. 219. Second Method. Select any two convenient stations E and F (Fig. 87) from which C may he seen, and range out FC and EC. To these draw the perpendiculars EG and FH cut- ting in I : then CLD will he the perpendicular required. For the perpendiculars to the three sides of a triangle from the opposite angles intersect in the same point. (c.) If the line be inaccessible. 220. From the given point C towards two visible points A and B (Fig. 88) of the given line range out CA and CB, and by one of the preceding methods draw the perpen- dicular EA and BD inter- secting in F : CF will be the perpendicular required. Fig. 88. 221. The preceding methods will apply in all the cases SEC. II. ] FIELD OPERATIONS. 127 enumerated. They are, however, only to be considered as substitutes for the neater and more accurate methods by the use of the theodolite or transit. Measurements such as those directed above, when they are intended to de- termine the direction of an important line, require to be made with scrupulous accuracy ; for every deviation will be magnified as we proceed. An error of two or three inches, which would be a matter of but little importance in a line of a chain long, would cause a deviation of from twelve to twenty feet if the line were prolonged to a mile. In the absence of a transit or theodolite, the following simple instruments, either of which can be constructed by any one having a moderate degree of facility in the use of tools, will enable the surveyor to lay out perpendiculars with readiness and considerable accuracy. 222. The Surveyor's Cross. This consists of a block of wood four or five inches in diameter, with two saw-cuts across its centre precisely at right angles. An auger hole should be made at the bottom of each saw-cut, to afford a larger field of view. The block is fastened to the top of a staff about eight or ten inches long. It should turn freely but firmly on the head of the staff. Instead of saw-cuts, four wires may be set upright at the ex- tremities of perpendicular diameters ; but, as these are likely to be deranged, the other form is better. 223. To erect a perpendicular with the cross, set it up at the point at which the perpendicular is to be drawn, and turn it round till one of the cuts ranges with the given line; then, looking through the other cut, the surveyor can direct his assistant to set a stake in the required perpendicular. If the point is out of the line, take a station as near as the eye can judge to the position of the foot of the per- pendicular, and, having set the cross so that one cut may range with the given line, look through the other, and see how far the line of sight misses the given point. Move the cross that distance and test it again. A few trials will de- termine the proper position. 128 CHAIN SURVEYING. [CHAP. IV. 224. To verify the Accuracy of the Cross. Place it at a given station: range with one of the cuts to a well- defined object, and place a stake in the perpendicular; then turn the cross one-quarter round, and if the stake is in the perpendicular, the cross is correct, but if not, the instru- ment is in error by half the observed deviation. This will be apparent by reference to Fig. 89. If the angle A CD is acute, the stake will be placed to the left of the true position, as at F. By turning the block one-fourth round, the acute angle will be found at BCE, and the stake will be posited at G-, as far to the right as it was before to the left. 225. The Optical Square. The optical square is a much more convenient instrument for drawing perpendiculars than the cross. It consists of a circular box, having a fine vertical slit cut in one side, and directly opposite a circular or oval opening with a vertical line, such as a horsehair stretched across it. The box contains a piece of looking- glass set across it, so as to make an angle of 45 with the line of sight. From the upper half of this glass the sil- vering must be removed. Half-way between the two open- ings mentioned is another, to allow the rays coming from an object in the perpendicular to fall on the mirror and be reflected to the eye. SEC. II. ] FIELD OPERATIONS. 129 Fig. 90. Fig. 90 represents a plan of this instru- ment. ABC is a sec- tion of the box, A the slit at which the eye is placed, B the opening in the line of sight, C the opening for the perpendicular, and DE the looking-glass. The surveyor holds the box in his hand, and, looking at the other end of the line, through the open- ings A and B, directs his assistant, who is seen by reflec- tion through C, to place his rod in such a position that its image shall coincide with the hair across the opening B. HG is then perpendicular to AF. To find the point in which the perpendicular from a dis- tant point will intersect AF, walk along the line, keeping the line of sight AB directed to the end of the line. When the image of a pole standing at the point from which the perpendicular is to be drawn appears at H, the proper posi- tion has been attained. 226. To test the Accuracy of the Square. Erect a perpendicular with it, as above directed. Then sight along the perpendicular, and if the original line appears perpen- dicular, the instrument is correct ; if it does not, the devia- tion will equal twice the error of the instrument. Set a pole in the true perpendicular, which will be found as in Art. 224, and alter the position of the glass until the re- flected image appears in the proper position. One end of the glass should be movable by screws or by little wedges, so as to allow of its position being rectified. 130 CHAIN SURVEYING. [CHAP. IV. C. PAEALLELS. Problem 1. Through a given point to run a parallel to a given accessible line. Fig. 91. 227. This may be done by Arts. 97, 98, or 99, or thus : -4 2 Let AB (Fig. 91) be the line, and \,$ C the point. From C to any point / \ x D in AB, run out the line CD. -g- ^ From E, any point in CD, run a line cutting AB in F. Then make EG a fourth proportional EF.EC -, and GC will be paral- to DE, EF, and EC, or EG = lei to AB. ED Problem 2. To draw a parallel to an inaccessible line, two points of lohich are visible. 228. Let AB (Fig. 92) be the straight line, and C the given point. Run the line CD per- pendicular to AB, by Art. 220 ; and from C set out CE perpen- dicular to CD. It will be the E~~ c parallel required. Problem 3. To draw a parallel to a given line through an inaccessible point. 229. Let AB (Fig. 93) be the given line, and C the given point. From A, towards C, run AC ; and in CA, or CA produced, take any point D. Run DE parallel to AB. Set off BC towards C, in- SEC. III.] OBSTACLES IN RUNNING AND MEASURING LINES. 131 tersecting DE in E. Measure AB and DE. Run through any point in AB the line BFG-, intersecting DE in F. DE BF Make FG = A p ' , and CG will be parallel to AB. AJB For, since FG = ^> we have AB - DE : DE :: BF : FG. AB Whence AB : DE : : BG : FG ; but AB : DE : : BC : EC ; BG : FG : : BC : EC, and CG is parallel to EF, or to AB. SECTION III. OBSTACLES IN RUNNING AND MEASURING LINES.* A. OBSTACLES IN RUNNING LINES. 230. IN ranging out lines by the method described in Art. 204, obstacles are frequently met with which prevent the operation being directly carried on. In such cases some contrivance is necessary in order that the line may be prolonged beyond such obstacle. Various methods have been devised for this purpose. The following are among the most simple : -231. First Method. By per- Fig. 94. pendiculars. Let AB (Fig. 94) be the line, and M the obsta- cle. At two points C and B in AB, set off two equal per- pendiculars CD and BE long enough to pass the obstacle. Through D and E run the line DG ; and at two points F and G beyond the obstacle, set off perpendiculars FH * In Gillespie's "Land Surveying" may be found a still greater variety of methods for these objects. 132 CHAIN SURVEYING. [CHAP. IV. and GI equal to CD. Then HEK will be the prolongation of AB. A B 232. Second Method. By equilateral triangles. Let AB (Fig. 95) be the line, the obstacle being at 0. By sweeping with the chain, describe the equilateral tri- angle BCD. Prolong BD to E sufficiently far to pass the obstacle. Describe the equilateral triangle FEG, and prolong EG till EH = EB. Describe the equilateral triangle HKI, and KH will be the prolongation of AB. 233. Instead of making BEH an equilateral triangle, which would sometimes require the point E to be incon- veniently remote, run BE (Fig. 96) as before. Set out the per- pendicular EG = 1.T32 x BE. Describe the equilateral triangle GFI. Bisect FI in H. Then HG will be the prolongation of BC. B. OBSTACLES IN MEASURING LINES. 234. When, owing to any obstructions, the distance of a line cannot be directly measured, resort should be had to trigonometrical methods. In the absence, however, of the proper instruments, it may be necessary to determine such distances. The following are a few of the many methods that may be employed in such cases : 1. To measure a line when both ends are accessible. 235. Arts. 231, 232, 233, furnish means of determining the distance in this case. By the method Art. 231, BH = SEC. III.] OBSTACLES IN RUNNING AND MEASURING LINES. 133 EF ; and in that of 232, BH = BE. If the method Art. 233 is employed, BO = 2 BE. 2. When one end is inaccessible. 236. First Method. Eun BE (Fig. 97) in any direction, and AD parallel to it. Through any point D in AD, run DE towards C. Measure AD, AB, and BE : AB.BE then BC Fig. 97. 237. Second Method. Set off AC (Fig. 98) in any direction, and CD parallel to AB. Eun DE towards B. Measure AE, AF PT) EC, and CD : then AB = *; CE Fig. 98. B 238. Third Method. Set off AD (Fig. 99) perpendicular to AB, and of any dis- tance. Eun DC perpendicular to DB. OD 2 Measure DC and CA: then CB = - , OA T> orAB Fig. 99. 3. When the point is the intersection of the line with another, and is inaccessible. 134 CHAIN SURVEYING. [CHAP. IV. 239. First Method. Let AB and CD (Fig. 100) be the lines, the distances of which to their intersection are required. Set off DF parallel to BA, and run CFA. Measure CD, CF, CA,andFD. Then BE = BD.DF BD.DC Fig. 100. 240. Second Method. Through H, (Fig. 101,) any point in CD, run two lines AF and BG. Make FH in any ratio to HA, and GH in the same ratio to HB. Draw FGC, cutting CD in C. Measure FC and HC. Then AE = AH.FC HE = Fig. 101. AH.HC FH~ 4. When both ends are inaccessible. 241. Let AB (Fig. 102) be the in- accessible line. From , any con- venient point C, run the lines CA and CB towards A and B, and, by one of the preceding methods, find CA and CB. In CA and CB, or CA and CB produced, take E and D so that CE : CA : : CD : CB. Measure DE. Then CE : CA : : ED : AB. Fig. 102. SEC. IV.] KEEPING FIELD-NOTES. 135 SECTION IV. KEEPING FIELD-NOTES, 242. THE operation next in importance to that of per- forming the measurements accurately is that of recording them neatly, concisely, and luminously. The first is a requisite that cannot be too much insisted on, not only in the first notes, but in all the calculations and records connected with surveying. A rough, careless mode of re- cording observations of any kind generally indicates an equal carelessness in making them. Carelessness in a sur- veyor, on whose accuracy so much depends, is intolerable. Conciseness is also necessary, but it should never be al- lowed to detract from the luminousness of the notes. By this last quality is meant the recording of all the observa- tions in such a mode as to indicate, in the most clear man- ner, the whole configuration of the plat surveyed, and all the circumstances connected with it which it is intended to preserve. The notes should be, in fact, a full record of all the work, so as to indicate fully not only what was done, but what was left undone. 243. First Method. By a sketch. The simplest mode of recording the notes is to draw a sketch of the tract to be surveyed, on which other lines can be inserted as they are measured. On this sketch may be set down the distances to the various points determined. When the tract is large, however, or contains many base- lines, this sketch becomes so complicated as scarcely to be capable of being deciphered after the mind has been with- drawn from that particular work and the configuration of the plat has been in some measure forgotten. 244. Field-Book. Perhaps the best kind of a field- book is one that is long and comparatively narrow, faint- lined at moderate distances. The right-hand page should 136 CHAIN SURVEYING. [CHAP. IV. be ruled from top to bottom with, two lines, about an inch apart, near the middle of the page. The left-hand page maybe ruled in the same manner; but it is better left for remarks, sketches, and subsidiary calculations. In the space between the vertical lines all the distances are to be inserted: offsets, and other measurements con- nected with the main line, may be recorded in the spaces on each side of the column. In recording the measurements the book should be held in the direction in which the work is proceeding. The right-hand side of the column will then coincide with the right-hand side of the line, and vice versa. The notes should commence at the bottom, and all offsets and other lateral distances must be recorded on the side of the columns corresponding to the side of .the line to which they belong. "When marks are left for starting points for other mea- surements, the distance to them should be recorded in the column, and some sign should be made to indicate the purpose for which such distance was recorded. Stations of this kind are called False Stations, and may be desig- nated by the letters F. S. ; by a triangle, A ; or circle, o ; or by surrounding the number by a circle, thus, f567. ) "Whatever plan is adopted should be scrupulously adhered to, changes in the notation being always liable to lead to confusion. A regular station may be designated either by letters, A, B, or by numbers, 1, 2, 3, prefixed by the letter S or by Sta. In the field-notes in the following pages examples of most of these methods will be found. Lines are referred to, either by having them numbered on the notes as Line 1, Line 2, or by the letters or figures which designate the stations at their ends. Thus, a line from Sta. 1 to Sta. 3 would be referred to as the line 1, 3 ; one from Sta. B to Sta. D, as the line BD. This is perhaps the best mode. Some surveyors, however, refer to them by their lengths. Thus, a line 563 links long would be called the line 563. False stations on a line are named by the line and distance. SEC. IV.] KEEPING FIELD-NOTES. 137 Thus, a station on a line AB at 597 links would be called F. S. 597 AB, orC59T)AB, or A, or O 597 AB. It hardly needs remark, yet it is of importance, that unity of system should be adopted. "Whatever method of designating a line or station has been employed in recording it, should be used in referring to it. The spaces on the right and left of the column will serve, in addition to the purposes already mentioned, to contain sketches of adjoining lines and short remarks to elucidate the work. A fence, road, brook, &c. crossing the line measured, should not be sketched as crossing it in a continuous line, as at 365, marginal plan, but should consist of two lines starting at opposite points, as at 742, so that if we were to suppose the lines forming the vertical column to collapse, those representing the fence would be continuous. When the chainmen, after closing the work on one line, begin the next at the closing station, a single horizontal line should be drawn; but if they pass to some other part of the tract, two lines should indicate the end of the line. To indicate the direction in which a line turns, the marks "1 or f may be used, the former indicating that the new line bears to the left, and the latter to the right. Instead of these, the words right and left may be used, or the simple initials E. and L. "Whichever of the means is used, the sign should be on the left hand of the column if the turn is to the left, and vice versd. The following notes will illustrate all these directions! They belong to the tract Fig. 103. 138 CHAIN SURVEYING. [CHAP. IV. Sta. D ^ 2440 ^>~v 2020 ^v li-^ (1395) ^ 1 Sta. A Sta. A 1135 T Sta. C Sta. C 1760 __ - - Q50) Sta. B Sta.B Sl^ 24^ 1445 \x x '"* ^ 1170 '''^\ ,*''' Sta. A N.45E. Sta. C **^ 2425 }_S "*** 1550 ^^s^ t ^^. 1390 /^J^, Brook. ^ 395 ^ Sta. D (1395) in AD ~~~~'-*^.^ 1440 770 ^-'' 425 ** >fS ^*. **"* ("95JP) In EC southerly. Sta.B 1760 ^^, Brook. *- 515 ^r^ 1 Sta. D Beginning at A, the first line measured is the diagonal AB ; the course N. 45 E. is set down at the right. The first point requiring notice is the intersection of the dia- gonals at 1170 links from A. The diagonal is represented by the dotted line crossing the columns, a continuous line being employed to designate a fence or side, and a dotted line a sight-line. At 1445 the fence EF is crossed. The whole length to B is 2492 links. SEC. IV.] KEEPING FIELD-NOTES. 139 Turning to the left along BC, at 950 we come to the fence bearing to the left: 950 is surrounded by a line, thus, (^ 950 >") because it is to be used as a starting-point for another mea- surement. Having arrived at C, 1760 links from B, again turn to the left towards A: the distance CA is 1135 links. AD is next measured. At 1395 the fence EF is found : the point is marked (fT395~) : at 2020 the brook is crossed, and at 2440 links we find the corner D. Turning to the left along DB, at 515 the brook is again crossed. This line is 1760 links long. Passing now to E, f 950 J in BC, along the cross fence, the diagonal AB is passed at 425; at 770 CD is passed; 1440 links brings us to 1395 in AD. Passing to D : along DC, at 395 the brook is crossed ; at 1390 the fence is found; at 1550 we cross the diagonal AB: 2425 brings us to C, which finishes the work. 245. Test-lines. In the above survey more lines have been measured than are absolutely necessary. It is always better to measure too many than too few. If the redundant lines are not needed in the calculation, they serve as tests by which to prove the work. Tor the mere purpose of calcula- tion, one of the diagonals and the line EF might have been omitted : the other lines afford sufficient data for making a plat and calculating the area. An error in one of the others will not prevent the notes from being platted, and hence they do not in any way afford a criterion by which we can judge of the accuracy of the measurements; but when to these are added the length of the other diagonal we have a series of values, all of which must be correct or the map cannot be made. 246. General Directions. When about to survey a tract by this method, the surveyor should first examine the tract carefully and erect poles at the prominent points, corners, and false stations, along the boundary lines. He should stake out all diagonals and subsidiary lines which he may wish to measure, setting a stake at the points in 140 CHAIN SURVEYING. [CHAP. IV. which such lines intersect each other or cross the former lines, in fact, at every point the position of which it may be desirable to fix on the plat. Having made these preparations, he may, if the tract is at all complicated, make an eye-sketch. This will serve to guide him in regard to the best course to take in his measurements. Commencing then at some convenient point of the tract, he should measure carefully the diagonals and sides in suc- cession, passing from one line to such other as will make the least unnecessary walking, and setting down in his note- book the distance to every stake, fence, brook, or other im- portant object met with. When the tract is large, the work may last through several days. In such cases, each day's work should, if possible, be made complete in itself, that it may be platted in the evening. This will prevent the accumulation of errors which might occur from a mismeasurement of one of the earlier lines. 247. Platting the Survey. To plat a survey from the notes, select three sides of a triangle and construct it. Then, on the sides of this construct other triangles, until the whole of the lines are laid down. Measure test-lines to see whether the work is correct. In all cases commence with large triangles, and fill up the details as the work proceeds. SEC. V.] SURVEYING FIELDS OF PARTICULAR FORMS. 141 SECTION V. ON THE METHOD OF SURVEYING FIELDS OF PAR- TICULAR FORMS, 248. Rectangles. MEASURE two adjacent sides: their product will give the area. EXAMPLES. Ex. 1. Let the adjacent sides of a rectangular field be 756 and 1082 links respectively, to plat the field and calcu- late the content. Calculation. Content = 1082 x 756 = 817992 square links = 8 A., OR., 28.7 P. Ex. 2. The adjacent sides of a rectangular tract are 578 and 924 links : required the area. Ans. 5 A., 1R, 14.51 P. Ex. 3. Required the area of a tract the sides of which are 9.75 and 11.47 chains respectively. Ans. 11 A., R., 29 P. 249. Parallelograms. Measure one side and the per- pendicular distance to the opposite side. Their product will be the area. If a plat is required, a diagonal or the distance from one angle to the foot of the perpendicular let fall from the adja- cent angle may be measured. EXAMPLES. Ex. 1. Given one side of a parallelogram 10.37 chains, and the perpendicular distance from the opposite side 7.63 chains, the distance from one end of the first side to the perpendicular thereon from the adjacent angle being 2.75 chains. Required the area and plat. Ans. 7 A., 3 R., 25.97 P. A 42 CHAIN SURVEYING. [CHAP. IV. Ex. 2. Desiring to find the area of a field in the form of a parallelogram, I measured one side 763 links, and the perpendicular from the other end of the adjacent side 647 links, said perpendicular intersecting the first side 137 links from the beginning. Required the content and plat. Ans. 4 A., 3 E., 29.86 P. 250. Triangles. First Method. Measure one side, and the perpendicular thereon from the opposite angle ; noting, if the plat is required, the distance of the foot of the per- pendicular from one end of the base. Multiply the base by the perpendicular, and half the pro- duct will be the area. EXAMPLES. Ex. 1. Required the area and plat of a triangular tract, the base being 7.85 chains and the perpendicular 5. 47 chains, the foot of the perpendicular being 3.25 chains from one end of the base. Calculation. 7.85x5.47 42.9395 . Area = = = 21.46975 chains = 2 A., R., 23.5 P. Ex. 2. Required the area and plat of a triangle, the base being 10.47 chains, and the perpendicular to a point 4.57 chains from the end, being 7.93 chains. Ex. 3. Required the area of a triangle, the base being 1575 links, and the perpendicular 894 links. 251. Second Method. Measure the three sides, and calcu- late by the following rule: From half the sum of the sides take each side severally; mul- tiply the half-sum and the three remainders continually together, and the square root of the product will be the area. SEC. V.] SURVEYING FIELDS OF PARTICULAR, FORMS. 143 DEMONSTRATION. Let ABC (Fig. 104) be Fig. 104. a triangle. Bisect the angles C and A by the lines CDH and AD, cutting each other in D. Then D is the centre of the inscribed circle. Join DB, and draw DE, DP, and DG perpendicular to the three sides. Then will DE = DF = DG, and (47.1) FB = BG, CE = CF, and AE = AG. Bisect the exterior angle KAB by the line AH, cutting CDH in H. Draw HK, HL, and HM perpendicular to CA, AB, and CB. Join HB. Then (26.1) KH = HM, CK = CM, HL = HK, and AL = AK ; also (47.1) BL = BM. Because AK = AL and BM = BL, CK -f CM will be equal to the sum of the sides AB, AC, and BC ; therefore CK or CM = $ (AB -f AC -f BC) = S, if S stand for the sum of the three sides. But CE -f AE + BG = S ; therefore CK = CM = CA + BG, and AK = AL = BG; whence AG == AE = BL = BM, and EK = AB. Now, since CK = CM = J S, we have AK = S AC, EC = S AB, and AE = BM = J S BC. Because the triangles CDE and CKH, as also ADE and HKA, are similar, we have (4.6) and (23.6) Whence, and CE : ED : : CK : KH, AE : ED : : HK : KA, AE . EC : ED a : : CK : KA : : CK a : CK . KA. . EC : ED : : CK : ^/CK . KA, CK . ED == v'CK . KA . AE . EC. Now, ABC = ACD-f- BCD -f ABD S . ED = CK . ED. AC . ED-f BC . ED + % AB . ED Wherefore, ABC = ^/CK . KA. AE . EG. COR. From the above demonstration, it is apparent that the area of a tri- angle is equal to the rectangle of the half-sum of the sides and the radius of the inscribed circle. For another demonstration of this rule, see Appendix. EXAMPLES. Ex. 1. Eequired the area of a triangle, the three sides being 672, 875, and 763 links respectively. NOTE. In cases of this kind the operation will be much facilitated by using logarithms. 144 CHAIN SURVEYING. [CHAP. IV. 672 + 8T5 + 763 2310 = - = 1155 = half-sum of sides. 2i L J sum = 1155 log. 3.062582 J sum 672 = 483 log. 2.683947 \ sum 875 = 280 log. 2.447158 \ sum 763 = 392 log. 2.593286 2)10.786973 Area, 247449 square links, 5.393486 = 2 A., 1 K., 35.9 P. Ex. 2. Eequired the area of a triangular tract, the sides of which are 17.25 chains, 16.43 chains, and 14.65 chains respectively. Ans. 11 A., R., 14.4 P. Ex. 3. Given the three sides, 19.58 chains, 16.92 chains, and 12.76 chains, of a triangular field : required the area. Ans. 10 A., 2 R., 27 P. 252. Trapezoids. Measure the parallel sides and the per- pendicular distance between them. If a plat is desired, a diagonal, or the Fi s- 105 - distance AE, (Fig. 105,) may be mea- sured. Multiply the sum of the parallel sides by A half the perpendicular : the product is the area. DEMONSTRATION. ABCD == ABD + BCD = $ AB . DE + $ DC . DE == (AB-f DC). DE. EXAMPLES. Ex. 1. Given AB = 7.75 chains, DC = 5.47 chains, and DE = 4.43 chains, to calculate the content and plat the map, AC being 7.00 chains. Ans. Area, 2 A., 3 R., 28.5 P. Ex. 2. Given the parallel sides of a trapezoid, 16.25 chains and 14.23 chains, respectively: the perpendicular from the end of the shorter side being 12.76 chains, and the distance SEC. V.] SURVEYING FIELDS OF PARTICULAR FORMS. 145 from the foot of said perpendicular to the adjacent end of the longer side 1.37 chains. Required the area and plat. Ans. 19 A., 1 E., 31.4 P. '253. Trapeziums. First Method. Measure a diagonal, and the perpendiculars thereon, from the opposite angle. The area of a trapezium is equal to the rectangle of the diagonal and half the sum of the perpendiculars from the opposite angles. This is evident from the triangles of which the trapezium is composed. EXAMPLES. Ex. 1. To plat and calculate the area of a trapezium, the diagonal being 15.63 chains, and the perpendiculars thereto from the opposite angles being 8.97 and 6.43 chains, and meeting the diagonal at the distances of 4.65 and 13.23 chains. Ans. Area, 12 A., R., 5.6 P. Ex. 2. Given (Fig. 106) AC = 19.68 Fig . i 06 . chains, AE = 7.84 chains, AP = 16.23 chains, ED = 10.42 chains, and FB = 8.73 chains, to plat the figure and find the area. Ans. 18 A., 3 R., 14.98 P. Ex. 3. Required the area of a trape- zium, the diagonal being 17.63 chains, and the perpen- diculars 6.47 and 12.51 chains respectively. Ans. 16 A., 2 R., 36.94 P. 254. Second Method. Measure one side, and the perpen diculars thereon from the extremities of the opposite side, with the distances of the feet of these perpendiculars from one end of the base. 10 146 CHAIN SURVEYING. [CHAP. IV. Fig, 107. C EXAMPLES. Ex. 1. Let ABCD (Fig. 107) be a trapezium, of which the fol- lowing dimensions are given, viz. : AE = 3.27 chains, AF = 10.17 chains, AB = 17.62 chains, ED = 7.29 chains, and FC = 13.19 chains. Required to plat it, and calculate the area. Lay off the distances AE, AF, and AB ; then erect the perpendiculars ED and FC, and draw AD, DC, and CB. The trapezium is divided into two triangles and the trapezoid, the areas of which may be found by the pre- ceding rules. Thus, 2AED= AE.ED = 23.8383 2 EFCD = EF.(ED +FC)= 141.3120 2 CFB = CF. FB = 98.2655 whence ABCD = J of 263.4158 = 131.7079 chains = 13 A., E., 27.3 P. If either of the angles A or B were obtuse, the perpen- dicular would fall outside the base, and the area of the corresponding triangle should be subtracted. Ex. 2. Plat and calculate the area of a trapezium from the following field-notes : OB 1143 perp. 936 917 perp. 825 415 A Ans. 7 A., E., 30.3 P. Ex. 3. Calculate the area from the following field- notes : Ans. 6 A., 2 E., 2 P. perp. 892 1365 967 Stat. B. perp. 568 376 A SEC. V.] SURVEYING FIELDS OF PARTICULAR FORMS. 147 Fields of more than four sides, bounded by straight lines. 255. First Method. Divide the tract into triangles and trapeziums, and calculate the areas by some of the pre- ceding rules. In applying this method, as many of the measurements as practicable should be made on the ground ; the field then being platted with care, the other distances may be measured on the map. When it is intended to depend on the map for the distances, every part of the plat should be laid down with scrupulous ac- curacy, on a scale of not less than three chains to the inch. Ex. 1. To draw the map and calculate from the follow^ ing field-notes the area of the pentagonal field ABODE : oD 1 690 1 570 510 C 1 E. 350 280 OA N.15E. C oC 1 770 4 a 915 1 510 250 B z 585 Brook. ! 360 Brook. H 365 AD A E. of AD o E The construction is plain. Calculation. Twice trapezium ACDE = AD x (Ea + 60) = 6.90 x 8.60 = 59.34; twice triangle ABC = AC x Be = 7.70 x 2.50 = 19.25; Fig. 108. = 39.295 ch. = 3 A., 3 R, 28.72 P. Ex. 2. Map the plat, and calculate the content from the following field- notes : Fig. 109. 148 CHAIN SURVEYING. [CHAP. IV. 0D 520 288 80 E G120 206 F G 440 D230 150 00 Lof CA c 550 B180 410 135 130 G A East. Construction. Commencing at A, (Fig. 109,) draw the line AC east 5.50 chains, marking the points a and b at 1.35 and 4.10 chains respectively : at a and b erect the perpendiculars aG 1.30 and B l.SO chains. From C to G draw CG, which should be 4.40 chains long. At c, 1.50 chains from C, draw cD perpendicular to CG and equal to 2.30 chains. With the centre G and radius 1.20 chains, describe a circle, and from D draw the line DF 5.20 chains long, touching the circle at e, which should be 2.06 chains from F. At d, 2.88 chains from F, draw the perpendicular dE .80 chains: then will A B C D E F G be the corners of the tract. Calculation. 2 ABCG = AC (Ga + B6) = 5.50 x 3.10 = 1T.05; 2 GCD = GC . cD = 4.40 x 2.30 = 10.12; 2 GDEF = FD (Ge + <2E) = 5.20 x 2.00 = 10.40. Therefore area 3 E., 20.56 P. 37.5T , . - chains 18.785 chains = 1 A., Ex. 3. Required the plans and areas of the adjoining fields, of which the following are the field-notes, the two fields to be platted on one map. SEC.V.] SURVEYING FIELDS OF PARTICULAR FORMS. 149 0(4) 970 (3) 772 830 395 284 (5) 0(6) KE. (2) 395 320 100 0(1) 715 (6) K 10 E. Area 10 A., 2 E., 18.576 P. o 7 1150 675 0(8) 432 (11) (8) 565 0(9) 1285 1000 960 0(7) 155 (10) L. of (7,5) (4) 562 ISi 390 282 0(5) 313 (10) R. of (4) Area 12 A., 3 R., 18.1 P. Ex. 4. Required the plan and areas of the adjoining fields from the following field-notes, tracing thereon the course of the brooks. 0(7) 1051 Brook + (6.7) 680 -v^^ 648 540 (1) ~ N ~~^-^ 510 N^^^ Brook. 365 Brook + (1.5) (6) 380 130 0(5) r 0(5) 1255 853 T65 (1) (4) 500 440 0(3) r 0(3) 1150 & Brook + (2.3) 850 VN ' N -\-V- 490 -v_. Brook. (2) 482 200 ^^s 0(1) (11) 620 0(10) 1080 730 0(8) 465 (9) KE. (6) 665 0(11) 1395 1095 748 325 270 55 0(7) Brook + (8.11) /635 (8) \ Junction ^-N-^ of Brooks. Brook + (7.8; R. of (7.5) Area 14 A., 3R., 28.24 P. Area 15 A., 2 R., 7 P. NOTE. In the above field-notes the marginal references, such as " Brook -j 6.7," means to the point in which the brook crosses the line (6.7.) 150 CHAIN SURVEYING. [CHAP. IV. 256. Second Method. Instead of running diagonals, it may sometimes be more convenient to run one or more lines through the tract and take the perpendiculars to the several angles, as in the following example. Let the field be of the form ABCDEF, (Fig. 110.) Run the line AC, and take the perpendiculars /F, eE, B, and dD. The field will thus be divided into triangles and trape- zoids, the area of which may be calculated by the preceding rules. Thus, let the field-notes of the preceding tract be as follows : 0C 1185 D420 840 760 200 B E280 590 F220 250 0A East. Dist. Perp. Int. Dist. Sum of Perp. Double Areas. 2 AFf 2 /FE 2eEDd 2DdC Left-hand areas. Eight " " 250 590 840 1185 220 280 420 250 340 250 345 220 500 700 420 55000 170000 175000 144900 1185 x 200 544900 237000 2) 781900 39.0950 ch. = 3 A., 3 E., 25.5 P. The calculation being performed thus: In the first column are placed the distances to the feet of the left-hand perpendiculars. In the second the perpendiculars them- selves. The numbers in the third column are found by subtracting each number in column 1 from the succeeding number in the same column. The numbers in column 3 SEC.V.] SURVEYING FIELDS OF PARTICULAR FORMS. 151 therefore represent the distances A/, fe, ed, and dC. The numbers in the fourth column are found by adding each number in column 2 to the succeeding number in the same column ; they therefore are the sums of the adjacent perpendiculars. Those in the fifth column are found by multiplying the corresponding numbers in columns 3 and 4. They therefore are the double areas of the several trapezoids and triangles. Ex. 2. Required to calculate the content and make plats from the following field-notes : 1 oG 312T 2590 476 F H3T5 2145 20TO 642 E 1400 1920 1485 523 D 840 5160 K600 790 200 465 B A E. F 4025 3617 792 G 3254 826 H E594 2846 D435 2137 1548 3191 C729' 1026 429 623 K B237 175 0A K 15 E. Area 25 A., 1 R, 5 P. Area 38 A., 3 R, 17.5 P. 257. Offsets. In what precedes, the sides have been sup- posed to be right lines. This, is ordinarily the case except when the tract bounds on a stream. It then, if the stream is not navigable, generally takes in half the bed. Lands bounding on tide-water go to low-water mark. In all such cases the area and configuration of the boundary are most readily determined by offsets. A base is run near the crooked boundary, and perpen- dicular offsets are taken to the line, whether it be the middle of the stream or low- water mark. If the positions of these offsets are judiciously chosen, so that the part of the boun- dary intercepted between any two consecutive ones is nearly straight, the correct area may be calculated precisely as in last article. In the field-notes the distances are written in the column and the offsets on the right or left hand, accord- ing as they are to the right or left of the line run. 152 CHAIN SUEVEYING. [CHAP. IV. Thus, supposing it were required to find the area contained between the line AB and the stream, (Fig. Ill,) the following being the field-notes. Fig. 111. B 25 865 70 725 165 580 165 475 100 355 115 195 90 75 40 A K10E. The calculation would be as below, the same formula being used as in last article. Dist. Offs. Int. Dist. Sum of Offs. Double Areas. 40 75 90 75 130 9750 195 115 120 205 24600 355 100 160 215 34400 475 165 120 265 31800 580 165 105 330 34650 725 70 145 235 34075 865 25 140 95 13300 2) 182575 Area = 3 E., 26 P. 9.12875 ch. SEC.V.] SURVEYING FIELDS OP PARTICULAR FORMS. 153 Ex. 1. Eequired the area and plan from the following notes : \ A 4830 2040 F r ^ eo E 1471 930 485 D / f s' D 5000 3585 G / E355 F 2175 1428 D 95 140 60 r r A 3000 Q r C665 55 270 396 310 340 50 D 4175 3335 1929 1408 1015 610 A on creek-bank. 60 130 B 85 55 D 1072 750 390 C G 4241 F r r F 826 420 E r 55 55 K56JE. 1 c 1350 (2160) 75 100 60 B on A.D. Fig. 112 is a plat of this tract. 154 CHAIN SURVEYING. [CHAI Calculation. To find AGF, Art. 251. AG 3000 F& 4241 FA 4830 2)12071 i sum 6035.5 3.780713 i s-AG 3035.5 3.482230 i s-FG 1794.5 3.253943 i 5 -AF 1205.5 3.081167 2)13.598053 AGF = 6295435 6.799026 To find AFD. AF 4830 AD 4175 FD 2175 2)11180 i sum 5590 3.747412 i 5-AF 760 2.880814 i s-AD 1415 3.150756 i s FD 3415 3.533391 2)13.312373 AFD = 4530917 6.656186 SEC. V.] SURVEYING FIELDS OF PARTICULAR FORMS. To find BCD. 155 BC 1350 BD 2015 CD 1072 i sum 2)4437 3.346059 2218.5 1 s-BC 868.5 2.938770 i s-BD 203.5 2.308564 i 5 -CD BCD = 1146.5 670475 3.059374 2 ) 11.652767 5.826383 To find DEF. DE 1471 EF 826 J 1 DF sum s -DE 2175 3.349472 2.883661 2)4472 2236 765 I s-EF 1410 3.149219 I 5-DF 61 1.785330 2)11.167682 DEF 383567 5.583841 156 CHAIN SURVEYING. [CHAP. IV. Base. Dist. Offsets. Inter. Dist. Sum of Offsets. Double Areas. AB 610 1015 1408 1929 2160 50 340 310 396 270 55 610 405 393 521 231 390 650 706 666 325 237900 263250 277458 346986 75075 BO 1350 110 148500 CD 390 750 1072 55 85 130 60 390 360 322 140 215 190 54600 77400 61180 DE 485 930 1471 60 140 95 60 485 445 541 200 235 155 97000 104575 83855 EF 420 826 60 100 75 420 406 160 175 67200 71050 2 ) 1966029 Area of part cut off by bases, 983014.5 AGF 6295435 AFD 4530917 BCD 670475 * DEF 383567 128 A., 2 R., 21.5 P. 12863408.5 links. SEC. V.] SURVEYING FIELDS OF PARTICULAR FORMS. 157 The field-notes of a meadow, bounding on a river and divided into four fields, are as follows, the measurements being to low-water mark. Required the map and the content of the whole: 55 72 97 75 D cq CO 1054 896 739 480 C C 97 739 ; 75 48 -$ I > Sta. 3 Sta. 3 Limestone on 1450 bank of run. 1030 ^>> Sta. 2 Sta. 2 a limestone. 1344 (752) N.5910'E. Sta. 1 a limestone. Sta. 1 1396 585 <^o Sta. 5 \ Sta. 5 a mai 1740 corn 63 1414 Phil 35 1237 87 1016 rt 45 824 -tfr*x^ 50 652 P*^^ 551 462 75 295 75 Sta. 4 192 COMPASS SURVEYING. [CHAP. V. Fig. 141 is a plat of this tract. Fig. 141. SECTION Y. LATITUDES AND DEPARTURES, DEFINITIONS. 330. THE difference of latitude or, as it is concisely called, the latitude of a line is the distance one end is farther north or south than the other. It is reckoned north or south according as the bearing is northerly or southerly. 331. The difference of longitude or the departure of a line is the distance one end is farther east or west than the other, and is reckoned east or west as the bearing is easterly or westerly. 332. "Where the course is directly north or south, the latitude is equal to the distance, and the departure is zero ; but where the bearing is east or west, the latitude is zero, SEC. V.] LATITUDES AND DEPARTURES. 193 and the departure is equal to the distance. In all other cases the latitude and departure will each be less 'than the distance, the latter being the hypothenuse of a right-angled triangle, of which the others are the legs, and the angle adjacent to the latitude the bearing. Thus, Fi 142 AB (Fig. 142) being the line, AC is the N latitude north, and CB the departure east. Strictly speaking, the triangle is a right- angled spherical triangle ; but the deviation from a plane is so small as to be abso- lutely unappreciable except in lines of great length. No notice is, therefore, taken of the rotundity of the earth in "Land Surveying." 333. The latitude, departure, and distance being the sides of a right-angled triangle, of which the bearing is one of the acute angles, any two of these may be found if the others are known. 1. Given the bearing and distance, to find latitude and departure. As radius : cosine of bearing : : distance : latitude ; and as radius : sine of bearing : : distance : departure. 2. Given the latitude and departure, to find the bearing and distance. As latitude : departure : : radius : tangent of bearing. As cosine of bearing : radius : : latitude : distance. 3. Given the bearing and departure, to find the distance and latitude. As sine of bearing : radius : : departure : distance. As radius : cotangent of bearing : : departure : latitude. 4. Given the bearing and latitude, to find the distance and departure. As cosine of bearing : radius : : latitude : distance. As radius : tangent of bearing : : latitude : departure. 13 194 COMPASS SURVEYING. [CHAP. V 5. Given the distance and latitude, to find the bearing and departure. As distance : latitude : : radius : cosine of bearing. As radius : sine of bearing : : distance : departure. 6. Given the distance and departure, to find the bearing and latitude. As distance : departure : : radius : sine of bearing. As radius : cosine of bearing : : distance : latitude. EXAMPLES. Ex. 1. Giving the bearing and distance of a line 1ST. 56 TV. 37.56 chains, to find the latitude and departure. Ans. Lat. 20.87 K; Dep. 31.23 W. Ex. 2. Given the difference of latitude 36.17 K, and the distance 52.95, to find the bearing and departure, east. Ans. Bearing = K 46 55' E.; Dep. = 38.67. Ex. 3. Given the difference of latitude 19.25 K, and the departure 26.45 W,, to find the bearing and distance. Ans. Bearing = K 53 57' W. ; dist. = 32.71. Ex. 4. Given the bearing S. 33| W., and the departure 18.33 chains, to find the distance and difference of latitude. Ans. Dist. = 33.21 ch.; Lat. = 27.69 S. 334. Traverse Table. The traverse table contains the latitudes and departures for every quarter degree of the quadrant to all distances up to ten. From these, the lati- tude and departure, corresponding to any bearing and dis- tance, may readily be found by the following rule : If the distance be not greater than ten. Seek the degrees at the top or bottom of the table according as their number is less or greater than 45, and in the columns marked Lati- tude and Departure, opposite to the distance, will be found the latitude and departure. If the degrees are found at the bottom of the table, the name of the column is there like- wise. For all degrees less than forty five, the left-hand SEC. V.] LATITUDES AND DEPARTURES. 195 column is the latitude, but the departure, for those greater than 45. If the distance be more than ten, and consist of whole tens. Take out the number from the table as before, and remove the decimal point as many places to the right as there are ciphers at the right of the distance in the table. If the distance is not composed simply of tens. Take from the table the latitude and departure corresponding to every figure, removing the decimal point as many places to the right or to the left as the digit is removed to the left or the right of the unit's place, and take the sum of the results. EXAMPLES. Ex. 1. Required the latitude and departure of a line bearing 1ST. 37J E. 8 chains. Opposite to 8 chains, under the degrees 37J, are found, Lat. 6.3680, Dep. 4.8424. The latitude and departure required are, therefore, 6.37 K, 4.84 E. If the distance had been 80 chains, the latitude and de- parture would have been 63.68 K, 48.42 E. Ex. 2. Required the latitude and departure of a line run- ning S. 63J E. 75 chains. 70 ch. Lat. 31.234 D e p. 62.645 5 2.231 4.475 33.465 67.120 Hence the result is Lat. 33.46 S. ; Dep. 67.12 E. Ex. 3. Required the latitude and departure of a line run- ning K 35f W. 58.65 chains. 50 ch. Lat. 40.579 Dep. 29.212 8 " 6.493 4.674 .6 487 351 .05 41^ 29 Lat. 47.600 K Dep. 34.266 W. 196 COMPASS SURVEYING. [CHAP. V. Ex. 4. What are the latitude and departure of a line bear- ing S. 63J W. 27.49 chains? Ans. Lat. 12.27 S.; Dep. 24.60 W. Ex. 5. What are the latitude and departure of a line IN". 55f E. 27 chains ? Ans. Lat. 15.20 ET.; Dep. 22.32 E. Ex. 6. "What are the latitude and departure of a line bear- ing K 84f E. 123.56 chains? Ans. Lat. 11.31 ST. ; Dep. 123.04 E. Ex. 7. What are the latitude and departure, the bearing and distance being S. 24f W. 97.56 chains ? Ans. Lat. 88.60 S. ; Dep. 40.84 W. 335. When the bearing is given to minutes. Take out the numbers in the table for the quarter degrees between which the minutes fall. Then say, As 15 minutes is to the excess of the given number of minutes above the less of the two quarters, so is the dif- ference of the numbers in the table to a fourth term, which must be subtracted from the number corresponding to the less of the two quarters if the quantity is a latitude, but added if it is a departure. Thus, supposing the line were N. 41 18' E. 43.27 chains. Take the difference between the latitude for 41 J and that for 41 J-% and say, As 15' is to the difference between 41 J and 41 18', or 3', so is the difference between the latitudes to the correction for 3'. This correction subtracted from the latitude for 41 J will give the latitude required. Do the same with the departure, except that the correc- tion found as above must be added to the departure for 41 J. In the example, we have for the distance 40 in the column for 41J the Lat. 30.074 Dep. 26.374 41J 29.958 26.505 Differences .116 .131 Then, As 15' : 3' : : .116 : .023, correction of latitude ; and, As 15' : 3' : : .131 : .026, correction of departure. SEC. V.] LATITUDES AND DEPARTURES. 197 The corrected latitude and departure for 41 18', distance 40 chains, are Lat. 30.051., Dep. 26.400. In like manner, the latitudes and departures for each of the remaining figures may be calculated, being as below : For 40 ch. Lat. 30.051 Dep. 26.400 3 " 2.254 1.980 .2 150 132 .07 53 46 32.508 K 28.558 E. There will rarely be any calculation necessary for the decimal figures of the distance, as the variation caused by a quarter of a degree will seldom change more than a unit any of the figures that need be retained. Ex. 1. The bearing and distance being 1ST. 76 42' E. 39.76 chains, to find the difference of latitude and departure. Ans. Lat. 9.147 K ; Dep. 38.694 E. Ex. 2. Given the bearing and distance S. 37 9' E. 63.45 chains, to find the difference of latitude and departure. Ans. Lat. 50.573 S. ; Dep. 38.317 E. Ex. 3. Required the difference of latitude and departure of a line running S. 29 17' E. 123.75 chains. Ans. Lat, 107.937 S. ; Dep. 60.529 E. 336. By Table of Natural Sines and Cosines. The differ- ence of latitude and departure, when the bearing is given to minutes, is more readily found from the table of natural sines and cosines than from the traverse table. The dif- ference of latitude and departure are the cosine and the sine of the bearing to a radius equal to the distance. Therefore, to find the difference of latitude and departure of a line, take out the natural cosine and sine of the bear- ing, and multiply them by the distance. Ex. 1. Required the difference of latitude and departure of a line bearing N. 41 18' E. 43.27 chains. COMPASS SURVEYING. [CHAP.V. 41 18' Cosine .75126 Sine 66000 Dist. Diff. Lat. Dep. 40 ch. 30.0504 26.4000 3 2.2538 1.9800 .2 1503 1320 .07 526 462 Lat. 32.5071 N. Dep. 28.5582 E. The result by this method may be depended on to the third decimal figure, unless the distance is several hundred chains, and then it will rarely affect the second decimal figure. Ex. 2. Eequired the latitude and departure of a line K 29 38' E. 26.47 chains. 29 38' Cosine .86921 Sine.49445 20 ch. 17.3842 9.8890 6 5.2153 2.9667 .4 .3477 1978 .07 608 346 Lat, 23.0080 K Dep. 13.0881 E. The calculation need not, in general, be carried beyond the third decimal place. In the above example the work would then stand thus : 29 38' Cosine .86921 Sine.49445 20 ch. 17.384 9.889 6 " 5.215 2.967 .4 348 198 .07 61 34 Lat. 23.008 ST. Dep. 13.088 E. Ex. 3. Eequired the latitude and departure of a line bear- ing S. 56 V E. 63.48 chains. Ans. Lat, 35.39 S. ; Dep. 52.70 E. Ex. 4. Eequired the latitude and departure of a line bear- ing K 52 49' W. 136.75 chains. Ans. Lat. 82.65 K; Dep. 108.95 W. SEC. V.] LATITUDES AND DEPARTURES. 199 Ex. 5. Given the bearing and distance S. 23 47' W. 13.62 chains, to find the latitude and departure. Ans. Lat. 12.46 S.; Dep. 5.49 W. 337. Test of the Accuracy of the Survey. When the surveyor has gone round a tract, and has come back to the point from which he started, it is self-evident that he has travelled as far in a southerly direction as he has in a northerly, and as far easterly as westerly. His whole northing must equal his whole southing, and his whole easting equal his whole westing. If then the north latitudes are placed in one column and the south lati- tudes in another, the sum of the numbers in these columns will be equal, provided the bearings and distances are correct. So also the columns of departures will balance each other. Owing to the unavoidable errors in taking the measure- ments, and also to the fact that the bearings are generally taken to quarter degrees, this exact balancing rarely occurs in practice. When the sums are nearly equal, we may attribute the error to the want of precision in the instru- ments ; but, if the error is considerable, a new survey should be made. * It not unfrequently happens that the mistake has been made on a single side. This can often be detected by taking the errors of latitude and departure, and calculating or estimating the bearing of a line which should produce such an error by a mismeasurement of its length or a mis- take in its bearing. A little ingenuity will then frequently enable the surveyor to' judge of the probable position of the error, and thus obviate the necessity of a complete resurvey of the tract. It is laid down as a rule by some good surveyors tha,t an error of one link for every five chains in the whole distance is the most that is allowable. When the transit or theodo- lite is used, a much closer limit should be drawn. One link for ten or fifteen chains is quite enough, unless the ground is very difficult. Every surveyor will, however, 200 COMPASS SURVEYING. [CHAP. V. form a rule for himself, dependent on his experience of the precision to which he usually obtains. A young surveyor should set a high standard of excellence, as he will find this to be a very good method of making himself accurate. If he begins by being satisfied with poor results, the chances are that he will never attain to a high rank in his profession. 338. Correction of Latitudes and Departures. When the northings and southings, or the eastings and westings, do not balance, the error should be distributed among the sides before making any calculations dependent upon them. The usual mode of distributing the error is to apply to each line a portion proportioned to its length. Rule a table, and head the columns as in the adjoining example. Take the latitudes and departures of the several sides, and place them in their proper columns. Take the difference between the sum of the northings and that of the southings. The result is the error in lati- tude, and should be marked with the name of the less sum. Do the same with the eastings and westings : the result is the error in departure, of the same name as the less sum. Divide the error of latitude by the sum of the distances : the quotient is the correction for 1 chain. Multiply the correction for 1 chain by the number of chains in the several sides : the products will be the correc- tions for those sides, which may be set down in a column prepared for the purpose, or at once applied to the latitude. Operate the same way with the error in departure, to obtain the corrections of departure of the several sides. The corrections are of the same name as the errors. The corrections above found are to be applied by adding them when of the same name, but subtracting if of different names. If one side of a tract is hilly, or otherwise difficult to measure, a larger share of the error should be attributed to that side. When a change of bearing of a long side will lessen the SEC. V.] LATITUDES AND DEPARTURES. 201 error, this change should be made, especially if the survey was made with a compass. The corrections may be made in the original columns by using red ink. New columns are, however, to be preferred. Ex. 1. Given the bearing and distances as follows, to find the corrected latitudes and departures. 1 K43JW. 28.43 2 ff. 29} E. 30.55 3 S. 80 E. 28.74 4 East. 40.00 5 S. 10J E. 23.70 6 S. 64 W. 25.18 7 K63fW. 20.82 8 S. 57JW. 31.65 Bearings. Dirt. N. S. E. W. Cor. N. Cor. W. N. S. E. W. 1 N.43^W. 28.43 20.62 19.57 .01 20.62 19.58 2 N.29%E. 30.55 26.52 15.16 .02 26.52 15.14 3 S. 80 E. 28.74 40.00 4.99 28.30 TbT .02 ^02~ 4.99 28.28 4 East. 40.00 .01 39.98 5 S.10^ E. 23.70 23.32 4.22 .01 23.32 4.21 6 S. 64 W. 25.18 11.04 22.63 .01 11.04 22.64 7 N.63%W. 20.82 9.21 18.67 .01 9.21 18.68 8 S.57%W. 31.65 17.01 26.69 .02 17.01 26.71 229.07 56.35 56.36 87.68 87.56 .01 .12 56.36 56.36 87.61 87.61 56.33 87.56 Er.N. .01 .12Er.W. Ex. 2. Correct the latitudes and departures from the fol- lowing notes: 1. S. 49 W. 12.93 ch.; 2. S. 88 W. 13.68 ch. ; 3. K 25J W. 14.09 ch. ; 4. K 43J E. 14.70 ch. ; 5. K 12J W. 17.95 ch. ; 6. ST. 88J E. 17.68 ch. ; 7. S. 36J E. 35.80 ch.; 8. S. 77J W. 16.15 ch. Ans. 1. S. 8.48, W. 9.76; 2. S.. 48, W. 13.67; 3. K 12.73, W. 6.01; 4. K 10.70, E. 10.07; 5. K 17.51, W. 3.88 ; 6. K 38, E. 17.69; 7. S. 28.79, E. 21.30; 8. S. 3.57, W. 15.74. ?,03 COMPASS SURVEYING. [CHAP. V. SECTION VI. PLATTING THE SURVEY,* 339. With the Protractor. First Method. DRAW a line !N"S, on any convenient part of the paper, to represent the meridian. Place the protractor with its straight edge to this line, and its arc turned to the right if the bearing be easterly, but to the left if it be westerly, and with a fine point mark off the number of degrees. Draw a straight line from the centre to this point, and on it lay off Fi s- the distance. The point 2 (Fig. 143) will thus be determined. Through 2 draw a line parallel to N" S. Place the protractor with its centre at 2 and its straight side coincident with the me- ridian, and prick off the degrees in the bearing of the second side. Join this point to 2, and on the line thus determined lay off 2.3 equal to the second side. Through 3 draw another meridian ; and so proceed until all the bearings and distances have been laid down. "When the last line has been platted, it should end at the starting point: if it does not, either the notes are incorrect or an error has been made in the platting The proper position of the protractor after the first may be determined without drawing meridians, by placing the centre at the point and turning the protractor until the number of degrees in the bearing of the last line coin- cides with that line. Its position is then parallel to the former one, and the bearing of the next line may be pricked off. This method is the one commonly employed. It has, however, the disadvantage of accumulating errors, since any mistake in laying down the bearing of one line will alter * Various hints in this section have been derived from Gillespie's Surveying." Land SEC. VI.] PLATTING THE SURVEY. 203 both the direction and position of -every subsequent line on the plat. The figure is the plat from the following field-notes : 1. K 27 J E. 7.T5; 2. S. 60J E. 10.80; 3. S. 8 E. 9.50; 4. S. 47J E. 9.37; 5. S. 54J W. 8.42; 6. K 37J "W. 23.69. 340. Second Method. Draw a number of parallel lines to represent meridians. They may be equidistant or not. The. faint lines on ruled paper will answer very well. Select any convenient point for Fig. 144 a place of beginning, and draw the line AB (Fig. 144) for the first side. Place the protractor so that its centre shall be on one of the me- ridians, and turn it until the num- ber of degrees in the next side coincides with the same meridian, as at C : slip it down the line, maintaining the coincidence of the centre and degree mark with the meridian, until the straight side passes through the point Draw a line along this side. It will be the direction of the required line, on which lay off" the given distance. So continue until all the sides have been platted. The figure will close, if the work is properly done. This method is quite as accurate as the last, and admits of very rapid execution. / \ \ \ 341. By a Scale of Chords. With a radius equal to the chord of 60 describe a circle near the middle of the paper. Through its centre (Fig. 145) draw a line RTS to represent the meridian. Lay off from the north and south points the different bearings, 5 marking them 1, 2, &c. Through A, any convenient point, draw AB parallel to 0.1, and on it lay off AB equal to the length of the first side Fig. 145. 204 COMPASS SURVEYING. [CHAP. Y. taken from any convenient scale. Through B draw BC parallel to 0.2: on it lay off BC equal, to the second side. Through C draw CD parallel to 0.3; and so proceed till all the lines have been platted. With an accurate scale of chords of a good size, this method is probably preferable to either of the others. The scale on the rule sold with cases of instruments, however, is so small that no great precision can be obtained by its use. It is still, however, preferable to the other methods if the protractor in similar cases of instruments is employed. 342. By a Table of Natural Sines. The sine of any arc is equal to half the chord of twice that arc, or to the chord of twice the number of degrees on a circle of half the radius. "We may therefore use a table of natural sines to lay off angles. Its use in protracting a survey is ex- plained below. (Fig. 146) the paper to 5 on a This scale Fig. 146. N Describe a circle about the centre of with a radius equal scale of equal parts, should be taken as large as con- venient. Through its centre A draw JSTS to represent the me- ridian, and cross the circle at the points marked 60, with the centres N" and S, and radius equal to that of the circle: also draw EW perpendicular to !N"S. The points marked 30 may be obtained by crossing the circle with the compasses opened to the radius and one leg at E and W. A skeleton protractor is thus formed, having the E"orth, South, East, and West points, as well as the 30 and 60 points, accurately laid down. Commencing with the first bearing, which in the figure is N". 27J E., divide it by 2, and from the table of natural sines take out the sine of the quotient 13 45'. It is found to be 2.3769, the decimal point being removed 1 place to the right. Take this distance 2.38 from the scale of equal parts, and lay it off from N" to 1. SEC. VI.] PLATTING THE SURVEY. 205 The second bearing is S. 60J E. The half of J is 15' : the sine of this is 0.0436. Lay off .04 from 60 to 2. The third bearing is S. 8 E. : the sine of 4 is 0.6976. Lay off .70 from S. towards E. : the point 3 is thus determined. The fourth is S. 47J E., which exceeds 30 by 17J : the half of 17J is 8 45', of which the sine is 1.5212. 1.52 laid off from 30 towards E. determines the point 4. An accurate protractor is thus formed on the paper, con- taining all the bearings in the field-notes. The subsequent work will be as in last article. 343. By a Table of Chords. Instead of a table of natural sines, a table of chords, when it can be procured, is more convenient. Prepare a circle, as in last article, with the E"., S., E., W., and the 30 and 60 points, the radius being 10, taken from a scale of equal parts. Take from the table the chord of the number of degrees, or of its excess above 30 or 60, and lay it off from the proper point, as directed in last article: an accurate pro- tractor is thus formed on the paper, and the work proceeds as before. The object in determining the 30 and 60 points is to avoid the necessity of laying off long distances. "When the compasses are much stretched, the points strike the paper very obliquely, and are apt to sink in so as to make the dis- tance laid off slightly too short. This method is preferable to any of those which precede it : it is only to be excelled by the one next given. 344. By Latitudes and Departures. "Where the latitudes have been calculated and balanced, they afford the most convenient and accurate means of platting the survey. Rule five columns, heading them Sta., N"., S., E., "W", Commencing at any convenient station, place the latitude and departure of the side beginning at this station oppo- site the next station in the table, and in their appropriate columns. When the latitude set down is of the same name 206 COMPASS SURVEYING. [CHAP. V. as that of the next side, add them together, and place the result in the proper column of latitudes opposite the next side. But if they be of different names, take their differ- ence, and place it in the column of the same name as the greater. Proceed in the same way with this result and the next latitude, and so continue till all the latitudes have been used. The results will be the latitude of the stations opposite which they are placed, all counted from the point at which we commenced. Proceed in the same manner with the departures. Thus, if it were required to plat the survey of which the field- notes are given Ex. 1, Art. 338, we have the latitudes and departures, as in the following table. (See the example re- ferred to): Sta. N. s. E. w. 1 20.62 19.58 2 26.52 15.14 3 4.99 28.28 4 .01 39.98 5 23.32 4.21 6 11.04 22.64 7 9.21 18.68 8 17.01 26.71 Preparing a table as above directed, and beginning at the fourth station, the total latitudes .and departures will be as below : Sta. N. s. E. w. 1 42.15 23.84 2 21.53 43.42 3 4.99 28.28 4 00 0.00 5 .01 39.98 6 23.31 44.19 7 34.35 21.55 8 25.14 2.87 ;. VI.] PLATTING THE SURVEY. 207 The latitude of the fourth side is .01 K This is put in the column headed north, opposite the fifth station. The next latitude being south, take the difference 23.31 ; place it in the south: add 23.31 and 11.04, both being south, and we have 34.35 S. Subtract from this 9.21 K leaves 25.14 S. This, added to 1T.01 S., gives 42.15 S. Subtract 20.62 K leaves 21.53 S.; 21.53 S. from 26.52 K, the next latitude, leaves 4.99 K Finally, 4.99 K and 4.99 S. cancel, leaving for the latitude of the fourth station. In the same man- ner we find the total departures. As the latitude and departure of the station with which we begin are zero, the work proves itself. It is usual to begin with the first side. The table having been prepared as above, draw on any convenient part of the paper a meridian line, !N"S, (Fig. 147,) and take any point E for the starting point. From this Fig. 147. N point, lay off the several total latitudes contained in the table above or 'below the point as the latitude is north or south, and number them according to the station to which they are op- posite in the table. Through these points draw perpendiculars to the me- ridian, and make them equal to the several total de- partures, laying the distance to the right hand if the departure be east, but to the left if it be west. The cor- 208 COMPASS SURVEYING. [CHAP. V. ners will thus be determined. "When these are joined, the plat will be completed. SECTION VII. PROBLEMS IN COMPASS SURVEYING, 345. Problem 1. GIVEN the bearing of one side, and the deflection of the next, to determine its bearing. If the given bearing is northeasterly or southwesterly, add the deflection if it is to the right hand. If the sum exceeds 90, take its supplement, and change north to south, or south to north. If the deflection is to the left hand, subtract it from the bearing ; but if it is greater than the bearing from which it is to be subtracted, take the difference, and change east to west, or west to east. "When the given bearing is northwesterly or southeasterly, add the left-hand and subtract the right-hand defections, ap- plying the same rules as above. EXAMPLES. Ex. 1. Given AB (Fig. 148) K". 37 E., Kg.i. and the deflection of the next side 43 c ' * D 15' to the right. BD = K 37 E. w _ DBC= 43 15' "Whence BC is K 80 15' E. A g Ex. 2. Given AB K 37 E., and the deflection of BC' 43 15' to the left. BD = K 37 E. DBC' = 43 15' Whence BC' is N. 6 15' W. SEC. VII.] PROBLEMS IN COMPASS SURVEYING. 209 Ex. 3. Given the bearing of AB, K 39 W., and BC de- flects to the left 75 26': required the bearing of BC. Ans. S. 65 34' W. Ex. 4. Given the bearing of a line S. 63 29' E., and the deflection of the next 29 17' to the right : required its bearing. Ans. S. 34 12' E. Ex. 5. The bearing of one line being S. 34 12' E., and the deflection of the next 75 32' to the right: required its bearing. Ans. S. 4120'W. 346. Problem 2. To determine the angle of deflection between two courses. 1. If the lines run between the same points of the com- pass, take the difference of their bearings. 2. If they run between points directly opposite, subtract the difference of the bearings from 180. 3. If they run from the same point towards different points, add the bearings. 4. If they run from different points towards the same point, take the sum of the bearings from 180. EXAMPLES. Ex. 1. AB (Fig. 149) runs S. 56 W., and BC S. 25 W. : required the de- flection. w 56 25 Deflection 31 to the left. 14 210 COMPASS SURVEYING. [CHAP. V. Ex. 2. Given AB (Fig. 150) K 46 W., and BC S. 79 E. : required the de- D flection. 1ST. 46 W. w S. 79 E. 33 180 Eig.150. AB BC ABC DBC 147 = deflection to the right. Ex. 3. Given AB (Fig. 151) K 39 E., and BC K 63 W., to find the de- c flection. AB N. 39 E. BC K 63 W. Fig. 151. DBC 102 = deflection to the left. Ex.4. Given AB (Fig. 152)8. 82 E., and BC BT. 67 E., to find the de- flection. AB S. 82 E. BC K 67 E. Fig. 152. DBC 149 180 31 = deflection to the left. Ex. 5. The bearing of a line is K 46 30' E., and that of ,the next S. 63 29' "W. : required the deflection. Ans. 163 1' to the left. Ex. 6. What is the deflection in passing from a course S. 63 W. to one K 29 W.? Ans. 88 to the right. Ex. 7. "What is the deflection in passing from a course N. 82JW.to one K 29J W.? Ans. 53J to the right. 347, Angle between lines. If the angle between two SEC. VII.] PROBLEMS IN COMPASS SURVEYING. 211 lines is required, reverse the first bearing, and apply the above rules. EXAMPLES. Ex. 1. Given AB K 87 E., and BC S. 25 W., to find the angle ABC. Ans. ABC = 62. Ex. 2. Given AB~S. 63 E., and BC K 56 E. : required the angle ABC. Ans. ABC = 119. Ex. 3. Given CD K 15 W., and DE 1ST. 56 W.: required the angle CDE. Ans. CDE = 139. Problem 3. To change the bearings of the sides of a survey. 348. It is frequently useful to change the bearings of a survey so as to determine what they would be if one side were made a meridian. This change is made on the sup- position that the whole plat is turned around without alter- ing the relative positions of the sides. Every bearing will thus be altered by the same angle. The following rules take in all the possible cases. The reason of these rules will be made apparent by drawing a figure to represent any particular case. 1. Deduct the bearing of the side that is to be made a meridian from all those bearings that are between the same points as it is, and also from those that are between points directly opposite to them. If it is greater than any of those bearings, take the difference, and change west to east, or east to west. 2. Add the bearing of the side that is to be made a meridian to those bearings that are neither between the same points as it is, nor between points directly opposite. If either of the sums exceeds 90, take the supplement, and change south to north, or north to south. EXAMPLES. Ex. 1. The bearings of a tract of land are, 1. K 57 E.; 212 COMPASS SURVEYING. [CHAP. V. 2. K 89 E.; 3. S. 49J E.; 4. South; 5. S. 27f "W.; 6. S. 53| W.; 7. F. 89 W.; 8. F. 37 "W.; 9. 1ST. 43 E. to the place of beginning. Required to change the bearings, so that the ninth side may be a meridian. 1. F. 57 E. 2. F. 89 E. 3. S. 49J E. F. 43 E. F. 43 E. F. 43 E. N. 14 E. F. 46 E. 92J 180 F. 87J E. 4. S. W. 5. S. 27} W. 6. S. 53J W. K 43 E. K 43 E. 1ST. 43 E. S. 43 E. S. 15i E. S. 10J W. 7. K 89 W. 8. F. 37 W. 9. Forth. K 43 E. K 43 E. 132 F. 80 W. 180 S. 48 W. Ex. 2. Change the bearings in the following notes, so that the second side may be a meridian : 1. IN". 43 25' W. ; 2. K 29 48 r E. ; 3. S. 80 E. ; 4. K 89 55 r E. ; 5. S. 10 13' E.; 6. K6355'W.; 7. S.6345'W.; 8. N.5735'W. Ans. 1. K 73 13' W.; 2. North; 3. K 70 12' E.; 4. K 60 7' E. ; 5. S. 40 1' E.; 6. S. 86 17' W.; 7. 5. 33 57' W. ; 8. K 87 23' W. Ex. 3. Change the bearings in the following notes, so that the fourth side may be a meridian : 1. S. 63 E. ; 2. 8. 47 E.; 3. S. 59J W.; 4. K 841^.; 5. K 12 W.; 6. K 17J E., and 7. S. 29| W. Ans. S. 21J W.; 2. S. 37i "W".; 3. K36i"W.; 4. Forth; 5. K 72J E.; 6. S. 78 E.; 7. F. 65 j W. SEC. VIII. ] SUPPLYING OMISSIONS. 213 SECTION VIII. SUPPPLYING OMISSIONS, 349. WHEN any two of the dimensions have been omit- ted to be taken, or have become obliterated from the field- notes, these may be supplied. This should never lead the surveyor to neglect to take every bearing and every dis- tance. It is far better to use almost any means, however indirect, to obtain all the bearings and distances indepen- dently of one another than to determine any one from the rest. If one side is determined from the others, all the errors committed in the measurements are accumulated on that side, and thus the means of proving the work by the balancing of the latitudes and departures is lost. The various problems in Section % will enable the young sur- veyor to solve almost every case of difficulty that will be likely to occur in making his measurements. Should any difficulty arise to which none of the methods there de- veloped are applicable, a knowledge of the principles of Trigonometry will afford him the means of overcoming it. CASE 1. 350. The bearings and distances of all the sides except one, being given, to determine these. Determine the latitudes and departures of those sides of which the bearings and distances are given. Take the difference between the sums of the northings and southings, and also between the sums of the eastings and westings : the remainders will be the latitude and departure of the side the bearing and distance of which are unknown. With this latitude and departure calculate the bearing and distance by Art. 333. This principle will enable us to determine a side when it cannot be directly measured. Thus, run a series of courses and distances, so as to join the two points to be connected. 214 COMPASS SURVEYING. [CHAP. V. These, with the unknown side, form, a closed tract, the sides of which are all known except one. It will likewise enable us to determine the course and distance of a straight road between two points already connected by a crooked one. In both these cases it is best, where the nature of the ground will admit of it, to run the courses at right angles to each other, as in Fig. 153, in which AB is the distance to be determined. Run AC any direction, CD perpendicular to AB, DE to CD, EF to DE, FG to EF, and, finally, GB per- pendicular to FG through B. Then, assuming AC as a meridian, AC + DE + FG will be the latitude of AB and CD + EF + GB the departure. From these calculate the distance AB and the bearing BAC. This angle applied to the true bearing of AC will give that of AB. EXAMPLES. Ex. 1. The bearings and distances of the sides of a tract of land being as follows, it is desired to find the bearing and distance of the third side, viz. : 1. K 56J "W. 15.35 chains; 2. K 9 W. 19.51 ch.; 3. Unknown; 4. S. 39| E. 13.35 ch.; 5. K 82J E. 12.65 ch.; 6. S. 6j "W. 12.18 ch.; 7. S. 52J W. 20.95 ch. SEC.VIIL] SUPPLYING OMISSIONS. 215 Sta. Bearing. Distance. N. s. E. w. 1 ET. 56i W. 15.35 8.53 12.76 2 N. 9 W. 19.51 19.2T 3.05 3 4 S. 39| E. 13.35 10.26 8.54 5 K 82J E. 12.65 1.65 12.54 6 S. 6 W. 12.18 12.10 1.43 7 S. 52J W. 20.95 12.75 16.62 r-~ 29.45 35.11 29.45 21.08 33.86 21.08 5.66 N. 12.78 E. Diff. Lat. Departure, Bearing, 5.66 12.78 K 66 7' E. log. 0.752816 log. 1.106531 tang. 10.353715 Bearing, Diff. Lat. Distance, 66 T 13.98 cos. 9.607322 log. 0.752816 1.145494 Ex. 2. One side AB of a tract of land running through a swamp, it was impossible to take the bearing and distance directly. I therefore took the following bearings and dis- tances on the fast land, viz. : AC, K 47 W. 16.55 chains ; CD, K 19 5' E. 11.48 ch. ; DE, K 11 5' W. 15.53 ch. ; EF, K 23 E. 9.72 ch., and FB, K 75 12' E. 14.00 chains. Required the bearing and distance of AB. 216 COMPASS SURVEYING. [CHAP. V. Sta. Bearing. Distance. N. s. E. w. A 1ST. 47 W. 16.55 ( 11.29 12.10 C K195'E. 11.48 10.85 3.75 D N.H 5'W. 15.53 15.24 2.99 E 1ST. 23 E. 9.72 8.95 3.80 F 1ST. 75 12' E. 14.00 3.58 13.54 B (49.91) (6.00) 49.91 21.09 15.09 Diff. Lat. Departure, Bearing AB, Bearing, Diff. Lat. Distance, 49.91 6.00 6 51' E. 6 51' 50.27 15.09 6.00 log. 1.698188 log. 0.778151 tang. 9.077963 cos. 9.996889 1.698188 1.701299 NOTE. In calculations of this kind, it is sufficiently accurate to confine the operations to two decimal places, unless the number of sides is large. In Ex. 2, had the work been extended to the third decimal place, it would not have made more than 15" difference in the bearing and 1 link in the distance. Ex. 3. Given the bearings and distances as follows, viz. : 1. S. 29f E. 3.19; 2. S.37iW. 5.86; 3. S.39JE. 11.29; 4. N. 53 E. 19.32; 5. Unknown; 6. S. 60 W.7.12; 7. S. 29JE. 2.18; 8.S. 60| W.8.12; to find the bearing and dis- tance of the fifth side. Ans. N". 31 5' W. 16.26 ch. Ex. 4. Required the bearing and distance of the third side from the following notes : 1. K 46 40' W. 18.41 chains; 2. K 54 J E. 13.45 chains; 3. Unknown; 4. S. 74 55' E. 17.58 chains; 5. S. 47 50' E. 15.86 chains ; 6. 5. 47 25' W. 16.36 chains; 7. S. 62 35' W. 14.69 chains. Ans. 3d side, K 5 26' W. 12.67 ch. Ex. 5. It being impossible to take the bearing and dis- tance of one side AB of a tract of land directly, in con- SEC. VIII.] SUPPLYING OMISSIONS. 217 sequence of a marsh grown up with thick bushes, I took bearings and distances on the fast land as below, viz.: AC S. 49J W. 9.30 chains ; CD S. 32J E. 10.25 chains ; DE S. 5J W. 6.75 chains ; and EB K 79f E. 8.10 chains. Ke- quired the bearing and distance of the side AB. Ans. S. 16 12' E. 20.82 ch. Ex. 6. The bearings and distances taken along the middle of a road which it is desired to straighten are as below, 1. S. 27 30' E. 12.65 chains; 2. S. 10J E. 23.45 chains; 3. S. 14 W. 124.33 chains; 4. S. 67 E. 82.43 chains; 5. S. 17 E. 96.35 chains. Required the bearing and distance of a new road that shall connect the extremities. Ans. S. 16 44' E. 291.63 ch. CASE 2. 351. The bearings and distances of the sides of a tract of land being given, except two, one of which has the bearing given, and the other the distance and the points between which it runs, to determine the unknown bearing and distance. RULE. Change the bearings so that the side whose bearing only is given, may be a meridian. Take out the latitudes and departures according to these changed bearings. Take the difference of the eastings and westings: this difference will be the departure of the side not made a meridian. "With this departure and the given distance, calculate by Art. 333 the changed bearing and difference of latitude, and place the latter in the column of latitude. From the changed bearing the true bearing may readily be found. Take the difference between the northings and south- ings. This difference is the difference of latitude of the side made a meridian, and is equal to the distance. NOTE. In general, there will be no difficulty in determining whether the changed bearing found should be north or south. In some cases, however, either will render the true bearing conformable to the points given. In this case the question is ambiguous, and can only be determined from the other data, except when the true bearing is nearly known. 218 COMPASS SURVEYING. [CHAP. Y. EXAMPLES. Ex. 1. Given the courses and distances as below, to find the unknown bearing and distance. Sta. Bearing. Changed Bearing. Dist. N. s. E. w. 1 K 56J- "W. S. 57f W. 15.35 8.19 12.98 2 K 9 W. K 75 W. 19.51 5.05 18.85 3 K 66 E. North. (14.00) 4 S. 39f E. K 74J E. 13.35 3.62 12.85 5 N. .E. 12.65 (12.12) (3.62) 6 S. 6f W. S. 59J E. 12.18 6.23 10.47 7 S. 52i w. S. 131 E. 20.95 20.37 4.89 34.79 34.79 31.83 31.83 Dist., fifth side, Dep. Ch. bear. 12.65 3.62 le^ 66 A. C. 8.897909 0.558709 sin. 9.456618 K 82 38' E., bearing of fifth side. Ch. bear., fifth side, 16 38 ; cos. 9.981436 Dist. 1.102091 Diff. Lat. . 12.12 "1.083527 Dist, third side, 14.00 ch. Ex. 2. Given 1. K 47 W. 16.55 chains; 2. N. 19 5' W. 11.48 chains; 3. K - W. 15.53 chains ; 4. K 23 E. 9.72 chains; 5. !N". 75J E. 14 chains; 6. S. 7 E., unknown; to determine the bearing of the third and the distance of the sixth side. Ans. 3d side, K 28J W. ; 6th, 48.67 ch. SEC. VIII.] SUPPLYING OMISSIONS. 219 CASE 3. 352. The bearings and distances of the sides of a tract of land being given, except the distances of two sides, to determine these. RULE. Change the bearings so that one of the sides the dis- tance of which is unknown may be a meridian. Take out the latitudes and departures with these changed bearings. The difference of the eastings and westings will be the de- parture of the side not made a meridian. With this de- parture and the changed bearing, find the distance and difference of latitude. Place the latter in its proper place in the table. Take the difference between the northings and southings: this difference will be the difference of latitude of the side made a meridian, and will be equal to the distance. EXAMPLES. Given as follow,!. K 56J W. 15.35 chains ; 2. K 9 W., unknown ; 3. K 66 E. 14.00 chains ; 4. S. 39f E. 13.35 chains; 5. K 82} E., unknown ; 6. S. 6}"W. 12.18 chains; 7. S. 52J W. 20.95 chains; to find the distances of the second and fifth sides. Sta. Bearing. Changed Bearing. Dist. N. S. E. w. 1 K56JW. K47JW. 15.35 10.42 11.27 2 K 9 W. North. (19.54) (19.54) 3 N. 66 E. K 75 E. 14.00 3.62 13.52 4 S. 39| E. S. 30f E. 13.35 11.47 6.83 5 K82fE. S. 88J- E. .39 (12.64) 6 S. 6f W. S. 15f W. 12.18 11.72 3.31 7 S.52JW. S. 61J W. 20.95 10.00 18.41 33.58 33.58 32.99 32.99 220 COMPASS SURVEYING. [CHAP. V. Ch. bear., fifth side, 88 15' A. C. sin. 0.000203 Dep. 12.64 1.10174T Disk " 12.65 1.101950 Ch. bear. cos. 8.484848 Bist. 1.101950 Diff. Lat. 0.39 S. 1.596798 Ex. 2. Given 1. S. 29| E. 3.19 chains; 2. S. 37J W. 5.86 chains; 3. S. 39J E., unknown; 4. K 53 E. 19.32 chains; 5. K 31 5' W., unknown; 6. S. 60f W. 7.12 chains; 7. S.29|E.2.18 chains; 8. S. 60J W. 8.12 chains; to find the distances of the third and fifth sides. Ans. 3d side, 11.28 chains; 5th, 16.26 chains. CASE 4. 353. The bearings and distances of all the sides of a tract of land being known except the bearings of two sides, to determine these. EULE. Take out the differences of latitude and the departures of the sides whose bearings and distances are known. The differences of the northings and southings will be the dif- ference of latitude, and that of the eastings and westings the departure, of a line which, with the known sides of the survey, will form a closed figure, and may therefore be called the closing line. With this closing line and the distances of the two other sides form a triangle. Calculate two angles of this triangle. These angles applied to the bearing of the closing line will give the bearings required. SEC. VIII.] SUPPLYING OMISSIONS. 221 EXAMPLES. Ex. 1. Given AB (Fig. 154) K 56 J W. 15.35 chains; BC K 9 W. 19.51 chains ; CD K E. 14 chains ; DE S. 39f E. 13.35; EF F. 82J E. 12.65 chains; FG S. W. 12.18 chains; GA S. 52J W. 20.95 chains; to find the bearings of the third and sixth sides. Bearing. Disk N. S. E. w. AB N. 56J W. 15.35 8.53 12.76 BC K 9 W. 19.51 19.2T 3.05 Ce S. 39| E. 13.35 10.26 8.54 ef K 82J E. 12.65 1.65 12.54 GA S. 52J W. 20.95 12.75 16.62 29.45 23.01 23.01 21.08 32.43 21.08 6.44 11.35 Diff. Lat. Dep. Tang, closing line, Cos. bear. Diff. Lat. Dist. closing line, FG /G /F 6.44 11.35 S.6026'E. 60 26' 13.05 A. C. 9.191114 1.054996 10.246110 A.C. 0.306769 0.808886 1.115655 A. C. 8.884388 " " 8.853872 1.292588 0.871281 JF/G F/G 26 41' 53 22' 2)19.902129 cos. 9.951064 222 COMPASS SURVEYING. [CHAP. V. FG 12.18 A. C. 8.914353 /F 14.00 1.146128 sin. F/G 53 22' 9.904429 sin./GF 67 17' 9.964910 60 26' Bear, of /G S. 651'W. " GF 180 - (53 22' + 60 26') = 66 12'; therefore, K 66 12' E. is the bearing of CD. Ex. 2. Given 1. S. 29f E. 3.19 chains; 2. S. 37J W. 5.86 chains ; 3. S. E. 11.29 chains ; 4. BT. 53 E. 19.32 chains; 5. K W. 16.26 chains; 6. S.60fW. 7.12 chains; 7. S. 29J E. 2.18'chains; 8. S. 60| W. 8.12 chains; to find the bearing of the third and fifth sides. Ans. 3d side, S. 39 8'E.; 5th, K 31 W. 354. The first three of the preceding rules are so simple as hardly to need any explanation. The principle of the last will be seen from the following illustration. The figure being protracted from the field-notes in Ex 1, Case 4, these are, as will be seen, the same as Ex. 1 in the other cases. Let ABCDEFG (Fig. 154) be the plat of the tract, the bearings of CD and FG being supposed unknown. If Ce and ef be drawn parallel to the sides DE and EF, and /G be joined, then will ABCe/G form a closed figure, the bearings and dis- tances of all the sides except /G being known. The course and dis- tance of this side, which is the closing line, are found as directed in the rule. Join /F and eE. Then /F is equal and parallel to eE and therefore to CD. The sides of the triangle /FG are therefore the closing line, the side FG, and the line /F equal and parallel to the side CD. In/FG find the angles /and G: these applied to the bearing of /G will give the bearings of /F or CD and of FG. SEC. IX.] CONTENT OF LAND. 223 This method might have been employed in Cases 2 and 3. Those given in the rules are, however, more concise, and are therefore to be preferred. 355. Though the methods illustrated above will serve to supply omissions in all cases where not more than two of the dimensions are unknown, yet it will not be amiss again to impress on the young practitioner the necessity, in all cases in which it is practicable, of determining each side independently of every other. The rules for supplying omissions should only be used in cases where one or more of the data have been accidentally omitted, or have become defaced on the notes. However accurate the field-work may be, there is always a liability to error, and if one side is determined by the rest no means are left of detecting any error. When a side cannot be measured directly, the best way is to determine it by some of the trigonometrical methods, taking the angles and base-lines with great care. In this way a degree of accuracy may be obtained equal to that of the sides measured directly. The latitudes and de- partures may then be balanced as usual. SECTION IX. CONTENT OF LAND. 356. FROM the bearings and distances of the sides of a tract of land, or from the angles arid the lengths of the sides, the area may be found, however numerous the sides may be. This may be done by Problem 4, which is entirely general, it being applicable whatever the number of sides may be, provided they are straight lines. As, however, there are other more concise methods applicable to triangles and quadrilaterals, those are first given. If one or more of the boundaries is irregular, instead of multiplying the number of sides by taking the bearings of 224 COMPASS SURVEYING. [CHAP. V. all the sinuosities of tlie boundary, it is better to run one or more base lines and take offsets, as directed in chain sur- veying. The content within the base lines is then to be calculated, and the area cut off by the base lines, being found by the method Art. 256, is to be added to or sub- tracted from the former area, according as the boundary is without or within the base. As has been already remarked, (Art. 257,) when the tract bounds on a brook or rivulet, the middle of the stream is the boundary, unless otherwise declared in the deed. Lands bordering on tide water go to low-water mark. "When* the stream, though not tide water, is large, the area is generally limited by the low-water mark, or by the regular banks of the stream. If the farm bounds on a public road, the boundary is, except in special cases, the middle of the road, and the measures are to be taken accordingly. 357. Problem 1. Given two sides and the included angle of a triangle or parallelogram, to determine the area. Say, As radius is to the sine of the included angle, so is the rectangle of the given sides to double the area of the tri- angle, or to the area of the parallelogram. DEMONSTRATION. We have, (Fig. 155,) by Art. 137, Asrad. : sin. A : : AC : CD : : AB.AC : AB.CP CD, (Cor. 1.6) ; but AB . CD = 2 ABC. EXAMPLES. Ex. 1. Given AB = 12.36 chains, BC = 14.36 chains, and ABC = 47 35', to determine the area of the triangle. As rad. A.C. 0.000000 : sin. B 47 35' 9.868209 . f AB 12.36 ch. 1.092018 : I BC 14.36 1.157154 : 2 ABC 2)131.033 2.117381 65.5165 ch. = 6 A., 2 K., 8.26 P. SEC. IX.] CONTENT OF LAND. 225 Ex. 2. Given AB K 37 14' W. 1T.25 chains, and BC K 74 29' W. 10.8T chains, to determine the area of the triangle ABC. Ans. 5 A., 2 R,, 28 P. Ex. 3. Given AB = 23.56 chains, AC = 16.42 chains, and the angle A 126 47'. Required the area of the triangle. Ans. 15 A., 1 R,, 38.7 P. 358. Problem 2. The angles and one side of a triangle being given, to determine the area. Say, As the rectangle of radius and sine of the angle op- posite the given side is to the rectangle of the sines of the other angles, so is the square of the given side to double the area. DEMONSTRATION. We have (Fig. 155) r : sin. A : : AC : CD (Art. 137), and Bin. B : sin. C : : AC : AB (Art. 139). (23.6). r . sin. B : sin. A. sin. C : : AC 2 : AB . CD, or 2 ABC. EXAMPLES. Ex. 1. Given AB = 21.62 chains, and the angle A= 47 56' and B = 76 15', to find the area. frad. A.C. 0.000000 8 (sin. C 55 49' 0.082366 fsin. A 47 56' 9.870618 I sin. B 76 15' 9.987372 JAB 21.62 ch. 1.334856 1 : t AB 21.62 1.334856 : 2 ABC 2)407.444 2.610068 Area = 203.722 ch. = 20 A., 1 R., 19.5 P. Ex. 2. Given AB 17.63 chains, and the angle A = 63 52' and B 73 47', to find the area. Ans. 19 A., 3 R., 22 P. Ex. 3. Given one side 15.65 chains, and the adjacent angles 63 17' and 59 12', to determine the area of the triangle. Ans. 11 A., R., 22 P. 15 226 COMPASS SURVEYING. [CHAP. V. 359. Problem 3. To determine the area of a trapezium, three sides and the two included angles being given. KULE. 1. Consider two adjacent sides and their contained angle as the sides and included angle of a triangle, and find its double area by Prob. 1. 2. In like manner, find the double area of a triangle of which the two other adjacent sides and their contained angle are two sides and the included angle. 3. Take the difference between the sum of the given angles and 180, and consider the two opposite given sides and this difference as two sides and the included angle of a triangle, and find its double area. 4. If the sum of the given angles is greater than 180, add this third area to the sum of the others ; but if the sum of the given angles is less than 180, subtract the third area from the sum of the others: the result will be double the area of the trapezium. DEMONSTRATION. Let ABCD (Figs. 156, 157) be the trapezium, of which AB, BC, and CD, and the angles B and C, are given. Join BD, and draw DE and CG perpendiculai\to AB, and CF perpendicular to ED. Then will DC5\ = 180 oo (B -f C.) Also, draw AH parallel to \G CB, and join DH. Then will 2 ABD == AB . DE = AB (EF db DF) = AB.EFAB.DF = 2 ABC 2 CDH. Whence 2 ABCD = 2 BDC + 2 ADB = 2 BCD -f 2 ABC 2 CDH: the plus sign being used (Fig. 157) when the sum of the angles is greater than 180. Fig. 156. SEC. IX.] CONTENT OF LAND. 227 EXAMPLES. Ex. 1. Given AB = 6.95 chains, BC = 8.37 chains, CD = 5.43 chains, ABC = 85 17', and BCD = 54 12', to find the area of- the trapezium. As r 0.000000 : sin. B 85 17' 9.998527 f AB 6.95 0.841985 1 : I BC 8.37 0.922725 : 2ABC 57.975 As r : sin. 180 - (B + C) 40 31' AB 6.95 CD 5.43 2 CDH 25.031 1.763237 0.000000 9.812692 0.841985 0.743800 1.398477 As r : sin. C (BC : [CD : 2 BCD 0.000000 9.909055 0.922725 0.734800 1.566580 34.903ch. = 3 A., IE., 38.45R Ex. 2. Given AB S. 27 E. 12.47 chains, BC K. 66 E. 11.43, and CD K 8 "W. 9.16 chains, to find the area of the trapezium. Ans. 14 A., R, 1.56 P. Ex. 3. Given AB S. 45 W. 8.63 chains, BC S. 86 30' E. 9.27 chains, and CD K 34 E. 11.23 chains, to find the area of the trapezium. Ans. 6 A., 2 R., 9 R 228 COMPASS SURVEYING. [CHAP. V 360, The above rule is a particular example of a more general problem, which may be enunciated thus : Let A, B, C, D, &c. be the sides of any polygon., and let the angle contained between the directions of any two sides, as B and D, be designated [BD]. Then, leaving out any side, we shall have the double area equal to the sum of the products of all the other pairs into the sine of their included angle. Thus, if the figure were a pentagon, we should have 2 the area = BC sin. [BC] + BD sin. [BD] + BE sin. [BE] + CD sin. [CD] + CE sin. [CE] + DE sin. [DE]. Observing that any product must be taken negative, if the angle is turned in a contrary direction from the general convexity of the figure with reference to the side A. Thus, in Fig. 156, we have 2 ABCD = AB . BC sin. [AB . BC] + BC . CD sin. [BC . CD] - AB . CD sin. [AB . CD], the lines BA and CD meeting so as to make the angle [AB . CD] present its convexity in the opposite direction from that of the figure. But, in Fig. 157, we have 2 ABCD = AB . BC sin. [AB.BC] + BC.CD sin. [BC.CD] + AB . CD sin. [AB.CD]. In the pentagon (Fig. 158) we shall have 2 Area = B.C.sin.[B.C.] + B.D.sin. [B.D.] + B.E.sin.[B.E.]-f C.D.sin. [C.D.]+C.E.sin.[C.E.] + D.E.sin. [D.E]. In Fig. 159 we have 2 Area = B.C. sin. [B.C.]+ B.D.sin. [B.D.]- B.E.sin. [B.E.]+ C.D.sin. [C.D.] + C.E.sin. fC.E.l + D.E.sin. [D.E]. Fig. 158. A Fig. 159. SEC. IX.] CONTENT OP LAND. 229 361, Problem 4. The bearings and distances of the boun- daries of a tract of land being given, to determine its area by means of the latitudes and departures of the sides. Let ABCDEFG- (Fig. 160) Fig. ieo. be the plat of a tract, and let N MS be a meridian anywhere a-- on the map. Through the corners draw the perpendicu- lars Aa, B6, &c. Then, it is evi- dent that ABCDEFG = AagGc + DdeE Aa&B - CcdD - Ee/F. Now, these various figures being trapezoids, their areas aL will be found by multiplying their perpendiculars by the g half-sums of their parallel sides. The perpendiculars are the differences of latitude of the sides of the tract. The sums of their parallel sides may be found as follows : The position of the line ETS being arbitrary, the sum Aa + B6, corresponding to the first side AB, may be taken at pleasure. Now, if from Aa + B6 we take AA, the whole departure of the two sides AB and BC, we have B6 4- Cc, the sum of the parallel sides of B6cC. Similarly, if to B -f Cc we add iD, the departure of the two sides BC and CD, we have Cc + Dd; and so on. The whole may be arranged in a tabular form, as below, Sides. N. S. E. W. E. D. D. W. D. D. Multipliers. N. Areas. 8. Areas. AB Bfc AA; Afc + Go Aa+B6,E. 2 Aa6B BC fO Bp Afc + B^> B6 + Cc, E. 2B6cC CD Cq qD gD Bp Cc + Dd, E. 2 CcdD DE Dl ffi qD + lE Dd+Ee, E. 2DefeE EF ~Em mF ZE + mF Ee+F/, E. 2Ee/F FG nG Fn mF Fra P/+ Gsr, E. 2 FfgG QA oA Go Fn + Go G0 + Aa.E. ZQgaA. in which the first column contains the sides, and the next four the differences of latitude and the departures; the 230 COMPASS SURVEYING. [CHAP. V. fifth and sixth columns contain the whole departures of two consecutive sides. These may be called the double departures, and the columns headed, accordingly, E.D.D. and "W.D.D. These double departures are found thus: The first, AA; 4- Go, is the sum of the departures of GA and AB, and is placed in the column of west double departures, because both departures are westerly ; the second, AA; -f B>, is the sum of those of AB and BC, and is west; the third is Dq Bp, and is east, because D is east of B ; the fourth, Dq + E, is east ; and so on. The eighth column contains the sums of the parallel sides. These may be called the multipliers. They are found by the following process. Assuming the first, Aa + B6, at pleasure, designate it either east or west. In the figure, the line MS being to the west of AB, the multiplier is east. The double de- parture AA; + Dp = Ah being west, subtract it from Aa -f B6, and we have Db + Cc. To B6 + Cc add the next double departure, qD pE = D, and we have Cc + Dd ; qD + ZE added to Cc + Dd gives Dd + fte; IE + mF added to Dd + Ee gives Ee -f F/ ; mF Fn added to Ee + F/ gives F/ + Gg ; and, lastly, F/i + Go taken from F/ + Gg leaves Gg + Aa. The areas are arranged in the last two columns, which are headed north areas and south areas for distinction. These areas are placed in the above table in the columns of the same name as the difference of latitudes of the sides to which they belong. Had the line NS been drawn so Ei s- 1 ^J- as to intersect the plat, some of the areas would have been to the west of it, and some of the multipliers might have been west. Fig. 161 is an example of this. In this case, we have * 2 ABCDEFG = 2 AafrB + 2 DbcC -f 2 CcdD 2Ddr + 2reE 2Ee/F + 2 F/^G + 2 Ggs - 2 saA = 2 AabD + 2 DbcC + 2 CcdD - 2 (Ddr ; s - reE) - 2 Ee/F + 2 F/#G + 2 (Ggs - saA.) SEC. IX.] CONTENT OF LAND. 231 But 2(Ddr - reE) = Dd . dr Ee . er = Dd . de - Dd . cr - Ee.de + Ee.dr; and since Dd : dr : : Ee : er, Dd.er = Ee dr. 2 (Ddr - reE) = Dd . de - Ee . de = (Dd - Ee) de. Whence 2 ABCDEFG = (Aa + Bb) ab + (Bb + Cc) be + (Cc + Dd) cd - (Dd - Ee) de - (Ee -f /F) e/ + (/F + + (Gg Aa) ag. The following table exhibits the whole. Sides. N. ~I S. E. W. E. D. D. W. D. D. Multipliers. r N. Areas. S. Areas. AB pB jpB+Go B6 + Aa, W. 2AafcB BC B? ?c pB + qC B&-J-CC, W. 2B6Cc CD Di Gi Gi qG Cc + Dd, W. 2Cafl> DE Et J)t Ci + Vt Dd Ee, W. 2(Ddr Eer) EF ~FOT Em mV Dt + m Ee+F/, E. 2(Ee/F) Gn ~Ao" Fn "oT" Fw Fra F/+G<7,E. 2F/flrG GA F + Go G^r Aa, E. 2(Gps Aas) Here the first multiplier is west, the meridian being to the east of the line AB. The subsequent multipliers are found as follow: (Bb + Aa) -f (^B + #0) = Bb + Cc; (Bb + Cc) - (Ci - qC) = Cc + Dd; (Cc + Dd) - (Of + D*) = Dd - Ee ; (D* + Fw) - (Dd - Ee) = (Ee + F/), which must be marked east, not only from its position on the figure, but also from the fact that the east double departure is greater than the west multiplier, which is taken from it ; (Ee + F/) 4- (Fw Fn) = F/+ G#; and (F/ -f G#) (Fn -f Go) = Gg Aa. The areas are arranged so that the additive quantities may be in the column of south areas and the subtractive in that of north areas. From the above investigation the following rule is de- rived : EULE. Kule a table as in the adjoining examples. Find the cor- rected latitudes and departures by Art. 338. Then, if the departures of the first and last sides are of the same name, add them together, and place their sum opposite the first side in the column of double departures of that name ; but 232 COMPASS SURVEYING. [CHAP. V. if they are of different names, take their difference and place it in the column of the same name as the greater. Proceed in the same way with the departures of the first and second sides, placing the result opposite the second side ; and so on. Assume any number for a multiplier for the first side, marking it E. for east or W. for west, as may be preferred. Then, if this multiplier and the double departure corre- sponding to the second side are of the same name, add them together, and place the sum with that name in the column of multipliers, for a multiplier for that side ; but, if the multiplier and double departure be of different names, take their difference and mark it with the name of the greater, for the next multiplier. Proceed in the same manner with the multiplier thus determined and the third double departure, to find the multiplier for the third side. So continue until all the multipliers have been found. Multiply the difference of latitude of each side by the corresponding multiplier, for the area corresponding to that side. If the multiplier be east, place the product in the column of areas which is of the same name as the dif- ference of latitude ; but, if the multiplier be west, place the product in the column of the opposite name. Sum the north and the south areas. Half the difference of the sums will be the area of the tract. NOTE. In working any area, the columns of double departures should balance. The first multiplier is generally assumed zero. One multiplication is thus avoided. When this is done, the last multiplier will be equal to the first double departure, but of a different name. EXAMPLES. Ex. 1. Given the bearings and distances as follow, to find the area: 1. K 56J W. 15.35 ch. ; 2. N. 9 W. 19.51 ch. ; 3. K 66 E. 14.01 ch. ; 4. S. 39J E. 13.35 ch. ; 5. K 82i E. 12.65 ch. ; 6. S. 6f W. 12.18 ch. ; 7. S. 52J W. 20.95 ch. ; to find the area. IX.] CONTENT OF LAND. 233 I CO OS CD | CD xO o rH si CD CO 1C OJ O ^O 5 02 CO CO CO XO xO -0 xO OS iO ^W. 30.00 24.12 li~.oT 17.84 47.96 .OOE. N.56%E. 21.60 11.92 .17 .17E. 2.0264 3 ~T~ ~5~ N. 26% E. 13.50 12.06 6.08 24.09 24.26E. 292.5756 S. 84% E. 10.72 .98 10.68 16.76 41.02E. 40.1996 S.51^E. 14.71 9.16 11.51 22.19 63.21E. 579.0036 6 S. 11% E. | 8.26 8.09 1.68 13.19 76.40E. 618.0760 7 s. 4514 w. 42.41 29.87 30.12 28.44 47.96E. 1432.5652 48.10 48.10 47.96 47.96 76.40 76.40 294.6020 2669.8444 Area of offsets calculated as in Ex. 1, Art. 257. = 128.592265 128 A., 2 R., 14.76 P. Ex. 2. Given the field-notes as below of a meadow bounding on a small brook, to calculate the area: of (2) 1132 55 1054 1 72 896 I 97 739 75 480 On brook. ^T A 63 1414 35 1237 87 1016 45 824 1 50 652 i 551 452 75 295 75 (4) 1396 Ans. 34 A., 3 K., 0.6 P. Ex. 3. Required the area of the meadow bordering on a mill-race, of which the boundaries are contained in the fol- lowing field-notes, the angles given being the deflections from the last course : (2) 11.28 (D s. 53i __x Jx^^ 785 90 750 70 ^o27 ) < & 500 585 82 650 37 400 350 10 400 300 30 185 25 225 40 150 25 20 15 15 57 27 70 Sta. A S.6637W. Sta.C f5622' Sta.E |-7029'. Area, 24 A., 3 K., 20 P. SEC. IX.] CONTENT OF LAND. 364. Compass Surveying by Triangulation. 243 When the tract is bounded by straight lines, the area may be found by determining the position of each of the angular points with reference to one or more base lines properly chosen. To do this, measure a base from the ends of which all the corners of the tract can be seen, and take their angles of position. There will thus be a system of triangles formed, giving data for calcu- lating the content of the tract. Thus, if ABODE (Fig. 165) re- present a field, measure a base FG, and from F and G take the E bearings, or the angles of posi- tion, of A, B, C, D, and E. Cal- culate FA, FB, FC, FD, FE, and thence the areas of the tri- angles FAB, FBC, FCD, FDE, and FEA. * Then, ABODE = FBC + FCD + FDE - FEA - FAB. EXAMPLE. To determine the area of a field ABODE, I mea- sured a base line FG of 12.25 chains, and at F and G I took the angles of position, as follow: GFA = 63 15', GFB = 27 33', GFC = 35 35', GFD = 58 25', GFE - 92 10', FGA = 26 5', FGB = 58 30', FGC = 97 12', FGD = 72 28', and FGE = 37 32'. Fig. 165 is a plat of this tract, on a scale of 1 inch to 10 chains. As sin. FAG : sin. FGA :: FG : FA Calculation. t 1. To find FA, 90 40' 26 5' 12.25 .000029 9.643135 1.088136 0.731300 244 COMPASS SURVEYING. [CHAP. V. To find FB. As sin. FBa 93 57' .001033 : sin. BaF 58 30' 9.930766 : : Fa 1.088136 : FB 1.019935 % find FC. As sin. FCa 47 13' 0.134347 : sin.FaC 97 12' 9.996562 : : Fa 1.088136 : FC 1.219045 To find FD. As sin. FDa 49 7' 0.121453 : sin. FaD 72 28' 9.979340 : : FG 1.088136 : FD 1.188929 To find FE. As sin. FEa 50 18' 0.113848 : sin. FaE 37 32' 9.784776 :: Fa 1.088136 FE 0.986760 To find 2 FAB. sin. AFB 35 42' 9.766072 FA 0.731300 FB 1.019935 2 FAB 32.9084 1.517307 To find 2 FBC. sinBFC 8 2' 9.145349 BF 1.019935 FC 1.219045 2 FBC 24.2286 1.384329 SEC. IX.] CONTENT OF LAND. 245 sin. CFD CF FD 2 FCD sin. DFE DF FE 2 FDE sin. AFE FE FA 2FEA To find 2 FCD. 22 50' 99.2805 To find 2 FDE. 33 45' 83.2585 To find 2 FEA. 28 55' 25.2633 2FBC 2 FCD 2 FDE 2 FAB 2FAE 7 A., IE., 28.76 P. 32.9084 25.2633 9.588890 1.219045 1.188929 1.996864 9.744739 1.188929 0.986760 1.920428 9.684430 0.986760 0.731300 1.402490 24.2286 99.2805 83.2585 206.7676 58.1717 2)148.5959 74.29795 sq.ch. 365, If no two points can be found from which all the corners can be seen, several points may be taken, and these all being connected by a system of triangles with a single measured base, or with several if suitable ground for mea- suring them can be found, the area may then be calculated. 246 COMPASS SURVEYING. [CHAP. V. Thus, (Fig. 166,) if ABCDEFG represent a tract, and H, I, and K, three points such that, from H, B, C, D, and E, can be seen. From I, all the corners can be seen, and from E! we can see A, H G, F, and E. If the angles of position of the corners relatively to the base lines HI and HK be taken, the distances IA, IB, 1C, ID, &c. may be found, and thence the areas of IAB, IBC, ICD, &c. Consequently, ABCDEFG = ICD + IDE + IEF + IFG IGA IAB - IBC becomes known. 366. The same principle may be applied to surveying a farm by means of one or more base lines within the tract. If such lines be run, and the corners be connected by triangles with given stations in these bases, the tract may be platted and the area calculated. In all cases of the application of this method, care should be taken to have the triangles as nearly equilateral as possi- ble. If any of the angles are very acute or very obtuse, the amount of error from any mistake in measuring the base or in taking the angles is much increased. CHAPTER VI. TRIANGULAR SURVEYING. 367. THE method pursued in the last few articles of' Chap. Y. constitutes what is called triangular surveying. It consists in connecting prominent points with one or more base lines by means of a system of triangles, the sides of these forming bases for other systems until the whole tract is covered. The positions of these points having thus been accurately determined, the minuter configurations may be filled up by means of secondary triangles, or by any of the other methods of surveying of which we have already treated. 368, Base. In triangular surveying there is generally but a single base measured as a foundation for the work. This therefore requires to be measured with extreme care, since an error will vitiate the whole work. The precautions to insure extreme accuracy are such as almost to preclude the possibility of an error. Delambre, in speaking of a measurement of this kind in France, says, "To give some idea of the precision of the methods employed, it is sufficient to relate the following occurrence during the measurement of the base near Perpignan: One day, a violent wind seemed every moment about to derange our rules, by slipping them on their supports. After having struggled a long time against these difficulties, we finally abandoned the work. Three days after, on a calm day, we recommenced the work of that whole day, and we only found a fourth of a line [one-twelfth of a French inch] dif 247 248 TRIANGULAR SURVEYING. [CHAP. VI. ference between two measurements, with the one of which' we were entirely satisfied, but of which the other was esteemed so doubtful that we considered it necessary to perform the whole work anew." 369. Reduction to the Level of the Sea. The base should if possible be measured on level ground. A smooth beach, if one can be found of sufficient length, affords one of the best locations. The work then requires no further reduction. If the ground is considerably elevated, the length must be reduced to what it would have been if the same arc of a great circle had been measured on the sea- 'level. This will be shorter than the measured arc in the ratio of the radius of the circle of which the measured arc forms part to that of the earth. Thus, suppose the arc was on ground elevated 300 feet, and a base of 5000 yards had been measured: then say, As 3956 miles + 300 feet : 3956 miles : : 5000 yards : the length required. The radius used should be that belonging to the latitude in which the work was performed, it being different in dif- ferent latitudes in consequence of the oblateness of the earth. 370. Signals. The base having been measured, the next object is to place signals on prominent points over the coun- try. Any prominent object may be selected for this pur- pose. A tree on a hill, provided it stands so that its trunk is visible projected against the sky, the spire of a church or any other object so elevated as to be seen from a great distance, may be employed. It is in general best, however, to employ signals constructed expressly for the purpose. Perhaps one of the best is a tall mast with a flag floating from the top. The flag waving in the wind can frequently be seen when a still object would not attract the attention. The observation must, however, be taken to the centre of the mast, and not to the flag. The Drummond light, reflected in the proper direction by a parabolic mirror, is the best of all. Such a signal may be seen at the distance of sixty miles. 371. Triangulation. The signals having been placed, SEC. IX.] TRIANGULAR SURVEYING. 249 their relative position should then be determined by de- termining their angles of position with each other. In this triangulation it is very important to have all the triangles as nearly equilateral as possible. It is not always possible to obtain triangles so "well conditioned" as would be de- sirable. The rule should, however, be strictly observed never to employ a triangle with a very acute angle opposite to the known side, as a very small error in 4 one of the adjacent angles will then produce a very sensible error in the calculated distance. For example, suppose the base AB were 500 yards long and the adjacent angles were A = 88 39' 15" and B = 88 17' 45"; the third angle C would then be 3 3'. The calculated distance of AC would be 9394.6 yards : an error of 5" in one of the observed angles would cause an error in this result of half a yard, a quantity utterly in- admissible in operations of this nature. The base generally being short, compared to the sides of the tri- angles which it is desirable to employ, these should be gradually enlarged, without allowing any of them to become " ill conditioned." The mode in which this is done may be seen from Fig. 167, in which AB is the base, on which two triangles ABC and ABD, both well conditioned, are founded. The line CD joining their vertices, becomes the base for two other triangles DCE and DCF, by means of which the line EF may be found. The angles at all the points of the triangle should be measured. The sum of these should be 180. If it differs but little, a few seconds, from this, the error should be dis- tributed among the angles, giving one-third to each. If, however, a greater number of observations have been made at some stations than at others, they should have a pro portionally less share of the error. Thus, suppose the recorded angle A is the mean of 5 observations, B the mean 250 TRIANGULAR SURVEYING. [CHAP. VI. of 8, and C of 2 : $ = of the error should be applied to A, & to B, and ^ to C. 372. Base of Verification. In order to prove the cor- rectness of the observations and calculations, some part of the work as distant as possible from the base should be con- nected with another carefully measured base, the coinci- dence of the measured and calculated distance of which will prove the whole work. In a system of triangulation carried over the whole of France, a distance of more than 600 miles, the base of verification was found to be by calculation 38406.54 feet long, and by measurement 3840T.5 The difference being only .96 feet, which was the total error arising from observations on more than sixty triangles. In the United States Coast Survey, carried on under the supervision of Prof. A. D. Bache, still greater accuracy has been obtained. CHAPTER VII. - LAYING OUT AND DIVIDING LAND, SECTION I. LAYING OUT LAND, Problem 1. To lay out a given area in the form of a square. 373. REDUCE the given area to square perches or square chains, and extract the square root. This root will be the length of one side. Erect perpendiculars at the ends equal to the base, and the thing is done. The side of a square acre is 316.23 links = 12.65 poles = 69.57 yards. Problem 2. To lay out a given area in the form of a rect- angle, one side being given. 374. Reduce the area to a denomination of the same name as the side. Divide the former by the latter, and the quotient will be the length of the other side. EXAMPLES. Ex. 1. Lay out 10 acres in a rectangular form, one side being 12 chains long. Required the other side. Ans. 8.33 chains. Ex. 2. "What must be the length of one side of a rect- angle, the area being 15 acres and one side 37.95 perches ? Ans. 63.24 perches. 251 252 LAYING OUT AND DIVIDING LAND. [CHAP. VII. Problem 3. To lay out a given area in a rectangular form, the adjacent sides to have a given ratio. 375. Divide the given area expressed in square chains or square perches by the product of the numbers expressing the ratio. The square root of the quotient multiplied by these numbers respectively will give the length of the sides. DEMONSTRATION. If mx and nx represent the sides, and A the area, then will mnx* = A. Whence x = I . EXAMPLES. Ex. 1. Required to lay out an acre in a rectangular form, so that the length shall be to the breadth as 3 to 2. What must be the length of the sides ? Ans. 3.873 chains and 2.582 chains. Ex. 2. It is desired to lay out a field of 10 acres in a rect- angular form, so that the sides may be in the ratio of 4 to 5. "What are these sides ? Ans. 8.944 chains and 11.18 chains. Problem 4. To lay out a given area in a rectangular form, one side to exceed the other by a given difference. 376. To the given area add the square of half the given difference of the sides. To the square root of the result add said half difference for the greater side, and subtract it for the less. CONSTRUCTION. Make AE (Fig. 168) equal to the given difference of the sides. Erect the perpendicu- lar EG equal to the square root of the given area. Bisect AE in F, and make FB = FG : then will AB be the greater side, and BE the less. For (29.6) AB . BE = EG*. The rule may be proved thus : FB a &= FG" = GE a -f- EF a = area -|- square of half the difference of the sides. Then, AB = AF -f FB, BC = FB FE. SEC. I.] LAYING OUT LAND. 253 EXAMPLES. Ex. 1. It is desired to lay out 10 acres in the form of a rectangle, the length to exceed the breadth by 2 chains. Ans. Length, 11.05 chains; breadth, 9.05 chains. Ex. 2. Required to lay out 17 A., 3 R., 16 P. in a rect- angular form, so that one side may exceed the other by 50 perches. Ans. Length 84, and breadth 34 perches. Problem 5. To lay out a given area in the form of a tri- angle or parallelogram, the base being given. 377. Divide the area of the parallelogram, or twice the area of the triangle, by the base. At any point of the base erect a perpendicular equal to the quotient. The summit will be the vertex of the triangle, or the end of a side of the parallelogram. If through the summit of the perpendicular a parallel to the base be drawn, any point in that parallel may be taken for the vertex of the triangle. Problem 6. To lay out a given area in the form of a tri- angle or parallelogram, one side and the adjacent angle being given. 378. As the rectangle of a given side and sine of the given angle is to twice the area of the triangle or the area of the parallelogram, so is radius to the other side adjacent to that angle. DEMONSTRATION. By Art. 357 we have (Fig. 169) r : sin. A : : AB . AC : 2 ABC, or (1.6) r . AB : sin. A . AB : : AB . AC : 2 ABC ; whence sin. A . AB : 2 ABC : : r . AB : AB . AC : : r : AC. % EXAMPLES. Ex. 1. Required to lay out 43 A., 2 R. in the form of a parallelogram, one side AB being 54 chains, and the adja- cent angle BAG 63. 254 LAYING OUT AND DIVIDING LAND. [CHAP. VII. /AB 54 A. C. 8.267606 As Ai5 - sin - A \ s in. A 63 0.050119 : ABCD 435 ch. 2.638489 :: r 10.000000 : AC 9.04 ch. 1.956214 Ex. 2. Required to lay out 3.5 acres in the form of a tri- angle, one side being 11.25 chains, and the adjacent angle 73 25'. Ans. AC 6.49 chains. Ex. 3. Given AB K 85 W. 16.37 chains, BDS. 32 J W., to determine its length so that the parallelogram ABCD may contain 16 acres. Ans. BD = 10.99 chains. Ex. 4. The bearings of two adjacent sides of a tract of land being K 85 E. and S. 23 E., required to lay off 10 acres by a line running from a point in one side 14.37 chains from the angle and falling on the other side. Ans. Distance, 14.63 chains. 379. Lemma. If ABC (Fig. 170) be any triangle, and CD a line in any direction from the vertex cut- ting AB in D, and if AF be taken a mean proportional between AB A and AD, and FE be drawn parallel to DC, the triangle AFE = ABC. DEMONSTRATION. Since AD : AF : : AF : AB, we have (Cor. 2, 20.6) AD : AB : : ADC : AFE ; but (1.6) AD : AB : : ADC : ABC, therefore ABC = AFE. The above lemma will be found very useful in the con- structions required in dividing land. The mean proportional AF may be found by describing a semicircle on AD, erecting a perpendicular BG, and making AF = AG ; or, if the point A is, remote, we may draw BH parallel to AC, meeting CD in H ; draw HI per- pendicular to CD, cutting the semicircle on CD in I ; make SEC. L] LAYING OUT LAND. 255 CK = CI, and draw KF parallel to CA. Then, since BH and FK are parallel to AC, the line AD is divided similarly to CD (10.6) ; but CK is a mean proportional between CH and CD, therefore AF is a mean proportional between AB and AD. 380. Problem 7. Two adjacent sides of a tract of land being given in direction, to lay off a given area by a line running a given course. Fig. 171. CONSTRUCTION. Take AD (Fig. 171) any convenient length. Erect the per- 2 Area pendicular AE = A _ . Draw the AD parallel EF cutting AF in F. Run FG the given course. Take AB a mean pro- A DBG portional between AD and AG or = -v/AD . AG. Then BC parallel to GF will be the division line. For, by construction, ADF = the given area, and, by lem- ma, ABC = ADF. AB may be calculated by the following rule : As the rectangle of the sines of the angles adjacent to the required side is to the rectangle of radius and the sine of the angle opposite to that side, so is twice the area to be cu^ off to the square of that side. The truth of this rule is evident from Art. 358. EXAMPLES. Ex. 1. Given AB S. 63 E. and AC K 47 15' E., to lay off 7 acres by a line BC running due north. Required the distance on the first side. 256 LAYING OUT AND DIVIDING LAND. [CHAP. VII. Here the angles are A = 69 45', B = 63, and C = 47 15'. Whence As {sin. A sin. B frad. 1 sin. C 2 ABC AB 2 AB 69 45' 63 47 15' 140 chains 11.09 Ar. Co. 0.027709 " 0.050119 10.000000 9.865887 2.146128 2)2.089843 1.044921. Ex. 2. Given the bearings of two adjacent sides, taken at the same station, K". 57 15' "W. and 1ST. 45 30' E., to deter- mine the distance from the angular point of a station on the first side from which a line running 1ST. 77 E. will cut off 5 acres. Ans. 8.648 chains. Ex. 3. Given AB S. 57 E. and AC S. 5 16' W., to lay off 12 acres by a line running N". 75 E. Required the dis- tance on the first side. Ans. 18.50 chains. 381. Problem 8. The directions of two adjacent sides of a tract of land being given, to lay off a given area by a line running through a given point. Fi s- CONSTRUCTION. Divide the given area by the perpendicular distance from P to AC, (Fig. 172.) Lay off ?- AD equal to the quotient. Draw PE parallel to AB. Make DF perpendicular to AD and equal to AE. Lay off FC = DE. Then CPB will be the division line. DEMONSTRATION. Complete the parallelogram ADHI. By construction, APD is half the required area ; and, therefore, AIHD con- tains the required area. Now, because the triangles PIB, HPK, and CDK are similar, and their homo- logous sides IP, DC, and HP are equal to the three sides DF, DC, and CF of the right-angled triangle DCF, we shall have (31.6) HPK = PBI+ CDK. To SEC. I.] LAYING OUT LAND. 257 these equals add AIPKD, and we have AIHD = ABC ; whence ABC contains the required area. If the directions of AB and AC and the position of the point P be given by bearings, AC maybe calculated as follows: In API find PI; also find the perpendicular PL. Then AD = area -*- PL. Then in DFC we have DF = PI and FC = DE to find DC, which added to AD will give AC. If FC be laid off on both sides, another point C' will be determined, through which the line may run. EXAMPLES. Ex. 1. Given the bearings of AB K". 34 W., and of AC West, to lay off 18 acres by a line running through a point P bearing from A K 41 W. 18.85 chains. To find PL As sin. I 56 A. C. 0.081426 : sin. PAI 7 9.085894 :: AP 18.85 1.275311 : PI 2.77 0.442631 To find PL and AD. As rad. A. C. 0.000000 : sin. PAL 49 9.877780 : : PA 18.85 1.275311 : PL 1.153091 Given area, 180 ch. 2.255273 AD 12.65 1.102182; whence ED = AD - PI =- 12.65 - 2.77 = 9.88. To find DC. FC + ED = 12.65 1.102182 EC - ED = 7.11 0.851870 2)1.954052 DC = 9.485 .977026; therefore AC = AD + DC = 12.65 + 9.485 = 22.135 ch. Ex. 2. Given the angle BAC = 85, to lay off 6 acres by a line through a spring the perpendicular distances 17 258 LAYING OUT AND DIVIDING LAND. [CHAP. VII. of which from AB and AC are 3.25 chains and 7.92 chains respectively. Kequired AC. Ans. AC == 10.40 chains. Ex. 3. A has sold B 3J acres, to be laid off in a corner of a field, by a line through a tree bearing North 5.64 chains from the angular point. Now, the bearings of the sides being N. 46 15' E. and K 42 W., it is required to find the distance to the division line, measured on the first side. Ans. 11.58 ch. 382. If the point P were exterior to the angle, the con- struction and calculation would be perfectly analogous to the preceding. The following is an example : Given the angle A = 60, (Fig. 173,) EAP = 90, and AP = 23.42 chains, to cut off 14 A. by a line running through P. Make AD = ^| = 5.98. Draw PE parallel to AB. Erect the perpendicular DF = AE, and make FC = ED. Then CB will be the divi- sion-line. For, as before, AIHD = the given area; but PEH = PBI + CKD ; .-. HIBK = CKD, and AIHD = ABC. r : tan. 30: : AP (23.42) : AE = DF = 13.52; whence 4 CF = DE = AE + AD = 19.50, and DC = ^/CF 2 -FD 2 = ^33.02x5.98 = 14.05 ; AC = 5.98 + 14.05 = 20.03 chains. Problem 9. Three adjacent sides of a tract of land being given in position, to lay off a given area in a quadrilateral form by i line running from the first side to the third. SEC. L] LAYING OUT LAND. 299 CASE 1. 383. The division line to be parallel to the second side. Conceive the lines CB and DA (Figs. 174, 175) to be pro- duced till they meet, and cal- culate the area of ABE. Add this to the given area if the sum of the angles A and B is greater than 180, as in Fig. 174 ; but if the sum be less, as in Fig. 175, subtract ] ABCD from ABE: the re- mainder will be the area of ECD. Fig. 174. Fig. 175. Then say, As EAB is to ECD, so is AB 2 to CD 2 . is ABvCDtoAD. And, as sin. E is to sine of B, so The following is a neat construction : Produce (0tB and GA to meet in E. Erect AF perpen- dicular to AB, and equal to double the area divided by AB. Draw FG parallel to AB, meeting AE in G. Then the tri- angle ABG will contain the required area. Take ED a mean proportional between EA and EG, or let ED = >/EA.EG. Through D draw the division line CD : ABCD will contain the required area. For (lemma) ECD = EBG ; whence ABCD = ABG. The calculation is more concisely made by the following rule : As the rectangle of the sines of the angles A and B is to the rectangle of radius and the sine of E, so is twice the given area to the difference between AB 2 and CD 2 . This difference, added to AB 2 when the sum of the angles A and B is greater than 180, but subtracted when the sum is less, will give CD 2 . Then, As sine of E is to the sine of B, so is the difference between CD and AB to the distance AD. 260 LAYING OUT AND DIVIDING LAND. DEMONSTRATION. ECD : EBA : : CD a : AB ; Whence, by division, ABCD : EBA : : CD* e* AB* : AB" ; consequently, and But (Art. 380) [CHAP. VII. 2 ABCD : 2 EBA : : CD 3 >AB a : AB, 2 EBA: AB a :: 2 ABCD :CD a fiNAB". sin. A. sin. B : rad. sin. E : : 2 EBA : AB a ; whence sin. A. sin. B : rad. sin. E : : 2 ABCD : CD a *a AB a . EXAMPLES. Ex. 1. Given 1. K 62 15' E. ; 2. 1ST. 19 12' W. 7.92 chains ; 3. S. 87 W., to cut off 5 acres by a line parallel to the second side. Required the length of the division line, and the distance on the first side. First Method. To find ABE, (Art.358.) f rad. A. C. 0.000000 I sin. E 24 45' " " 0.378139 ( sin. A 98 33' 9.995146 \ sin. B 106 12' 9.982404 JAB 7.92 0.898725 1 AB 0.898725 2 ABE 142.278 2.153139 2 ABCD 100 2 ECD 242.278 As 2 ABE 142.278 A. C. 7.846861 : 2 ECD 242.278 2.384314 :: AB 3 J7.92 17.92 0.898725 0.898725 : CD 2 2)2.028625 CD 10.335 1.014312 As sin. E 24 45' A. C. 0.378139 : sin. B 106 12' 9.982404 :: CD-AB 2.415 0.382917 : AD 5.539 0.743460 Sic. I.] LAYING OUT LAND. 261 As {sin. sin. Second Method. 98 33' 106 12' 100 ch. 44.08T 62.7264 A. C. 0.004854 0.017596 10.000000 9.621861 2.000000 1.644311 A B {rad. sin. E : : 2 ABCD : CD 2 -AB 2 AB 2 Whence CD = as before. Ex. 2. Given GA North, AB 1ST. 89 E. 7.86 chains, and BO 8. 1 30' W., to cut off 10 acres by a line parallel to AB. Eequired the distance of the division line from A. Ans. 13.00 ch. CASE 2. 385. By a line running a given course. Construct, as in last case, ABG to contain the given area. Draw BL according to the given course. Take ED a mean proportional B " AWL SEC. I.] LAYING OUT LAND. 263 between EL and EG : CD p Fig. us. parallel to BL will be the division line. For, by the lemma, ECD = EBG; whence ABCD = ABG, the required area. A*^ W UT \D G *"^J * ::; *'"A The calculation may be performed by the finding .ATC and the area of ABE ; whence ECD becomes known. The dis- tance ED may then be found by Art. 380 ; or, Conceive Wn to be drawn parallel to CD, making EWn = EAB. Then say, As the rectangle of the sines of the angles C and D is to the rectangle of the sines of A and B, so is the square of AB to the square of Wn. And, As the rectangle of the sines of C and D is to the rectangle of radius and sine of E, so is twice the given area to a fourth term. If the sum of the angles A and B is greater than 180, add these fourth terms together ; but, if the sum of A and B is less than 180, subtract the second fourth term from the first : the result will be the square of the division line CD. Then, As sine of C is to sine of B, so is AB to a fourth term ; take the difference between this fourth term and CD, and say, As sine of E is to the sine of C, so is this dif- ference to AD. DEMONSTRATION. Since EnW = EAB, EW is a mean proportional between E A and EL. Whence riW is a mean proportional between AP and BL ; there- fore AP . BL = nW a . Now, by similar triangles, we have sin. L (sin. D) : sin. A : : AB : BL, and sin. P (sin. C) : sin. B : : AB : AP. Whence (23.6) sin. C . sin. D : sin. A . sin. B : : AB 9 : AP . BL = nW; and, by demonstration to last case, sin. C . sin. D : rad. sin. E : : 2 wWDC : CD a cnW a . Draw AMN parallel to BC. Then, in the triangle ABM, we have sin. M (sin. C) : sin. BAM (sin. B) : : AB : BM ; and, in AND, we have sin. NAD (sin. E) : sin. N (sin. C) : : DN (CD *cBM) : AD. 264 LAYING OUT AND DIVIDING LAND. [CHAP. VII. EXAMPLES. Ex. 1. Given 1. ST. 62 15' E. ; 2. N. 19 12' W. 7.92 chains ; 3. S. 87 W., to cut off 5 acres by a line perpen- dicular to the first side. Required the length of the divi- sion line, and its distance from the end of the first side. First Method. As sin. E 24 45' Ar. Co. 0.378139 : sin. B 106 12' 9.9S2404 :: AB 7.92 0.898725 : EA 18.166 1.259268 AB 0.898725 sin. A 98 33' 9.995146 2 ABE 142.278 2.153139 2ABCD 100 2ECD 242.278 Then, (Art. 380,) r sin. E 24 45' Ar. Co. 0.378139 As 1 - -n ( sin. D 90 " " 0.000000 ( rad. 10.000000 ' I sin. 65 15' 9.958154 : : 2 ECD 242.278 2.384314 : ED 2 2)2.720607 ED 22.93 1.360303 AE 18.17 AD 4.76 As sin. C 65 15' Ar. Co. 0.041846 : sin. E 24 45' 9.621861 :: ED 1.360303 : CD 10.57 1.024010 SEC. L] LAYING OUT LAND. 265 Second Method. f sin. C 65 15' Ar. Co. 0.041846 ( sin. D 90 " 0.000000 r sin. A { sin. B 98 33' 106 12' 9.995146 9.982404 rAB 7.92 chains 0.898725 : nW 2 65.5913 0.898725 1.816846 f sin. C Ar. Co. 0.041846 \ sin. D 0.000000 f rad. \ sin. E 24 45' 10.000000 9.621861 :: 2ABCD 100 chains 46.1006 2.000000 1.663707 CD! - 65.5913 = 10.57. -/111.6919 = As sin. C 65 15' Ar. Co. 0.041846 : sin. B 106 12' 9.982404 :: AB : BM 7.92 8.375 0.898725 0.922975 CD 10.57 DN ^ 2.195 As sin. E 24 45' Ar. Co. 0.378139 : sin. C 65 15' 9.958154 : AD 2.195 4.76 0.341435 0.677728 266 LAYING OUT AND DIVIDING LAND. [CHAP. VII. It will be seen from the above that the first method is in this case the shorter. It has the advantage, also, of first giving the value of AD, which of itself is sufficient to de- termine the position of the division line. In the second method, if AG and BH are nearly parallel, the calculation for CD and DN should be carried to the third decimal figure. The construction given for this and the preceding case admits of easy application on the ground. Run the lines CB and GA to their point of intersection ; lay out the perpendicular AF ; run FG parallel to AB and BL parallel to the division line. Measure EL and EG, and make ED = v/EL . EG. Ex. 2. The bearings of three adjacent sides of a tract of land are 1. K 26 47' W. ; 2. K 63 13' E. 12.72 chains ; 3. S. 8 17' E., to cut off 7 acres by a line running due east. The distance on the first side and the length of the division line are required. Ans. Distance, 3.37 ; division line, 11.11. Ex. 3. The bearings of three adjacent sides of a tract of land being 1. K 78 17' E; 2. N. 5 13' E. 15.62 chains; and 3. K 63 43' W., it is desired to cut off 10 acres by a line making equal angles with the first and third sides. What is the bearing of the division line, and its distance from the end of the first side ? Ans. Bearing, K 7 17' E. ; distance on first side, 6.316. If the first and third sides are nearly parallel, the area of ABL may be calculated. This taken from ABCD, or added to it, according as BL falls within or without the tract, will give the area of BLDC, which may be parted off as directed in Art. 384. SEC. 1.] LAYING OUT LAND. 267 CASE 3. 386. By a line through a given point. Produce CB and DA Fig. 179. (Fig. 179) to meet in E, and calculate the area EAB. Thence ECD is found. Proceed as in Art. 381. Thus, calculate or measure the perpendicular FOT) PL Lay off EF = -p . Draw PK parallel to BE, meeting AE in K. Erect the perpendicular FG = EK or BP, and make GD = FK. Then will the division line pass through D. Calculation. Determine AE. Then ED = EF + ft O QO CO QO rH CD O P o O -fH tO ri CO CO rH fc** CO 00 ^ . rH CD QO * 51 g CO b- rH rH o CO QO w rH CD rH rH 5 O O s CO CD b- (N 02 QO rH rH O & b- AD : EF sss AD : : AB : AE. DEMONSTRATION. GBC : GAD :: BC a : AD; .-. (17.5) ABCD : GAD :: BC a AD' : AD*. Similarly, GEF : GAD :: EF a : AD 9 .-. (17.5) AEFD: GAD : : FE a AD*: AD*; whence ABCD : AEFD : : BC 3 AD a FE a AD a ; or, m + n : m : : BC a AD a : FE a AD a : consequently (m -f- n) FE a m AD a n AD a = m BC a m AD a ; m BC a + n AD ft or, (m + n) FE a = m BC a + n AD2, and FE a = . in + n Again : Draw AKL parallel to DC. Then BL : EK : : AB : AE ; . or, BC AD : FE AD : : AB : AE. SEC. II.] DIVISION OF LAND. 293 Second Method. The distance AE may be calculated thus : Find GA and GD; thence GO and GB are known: then GC : GD : : GA : GH; whence HB and HI are known, and therefore GE = */ GI.GB is known. EXAMPLES. Ex. 1. Given AB S. 14 W. 4.39 ch., BC E. 9.10 ch., CD K 14 20' W. 4.40 chains, and DA W. 6.95 chains, to divide the trapezoid into two parts AEFD and BEFC in the ratio of 2 to 3, by a line EF parallel to the sides BC and DA. Kequired the distance AE on the first side. m . BC 2 + n . AD 2 165.62 + 144.9075 = m + n 5 whence EF = v' 62.1055 = 7.88. And BC - AD (2.15) : EF - AD (.93) : : AB (4.39) : AE = 1.90. Ex. 2. Given AB S. 87 15' E. 6.47 chains, BC N. 23 30' E. 10.32 chains, CD S. 64 45' W. 9.30 chains, and DA S. 23 30' W. 5.55 chains, to determine the distance AE of a point E, situated in AB, such that EF parallel to AD may divide the trapezoid into two parts AEFD and EBCF having the ratio of 4 to 5. Ans. AE = 3.36 chains. 294 LAYING OUT AND DIVIDING LAND. [CHAP.VIL Problem 3. To divide a trapezium into two parts having a given ratio. CASE 1. 405. The division line to run through a given point in one of the sides. Let ABCD (Fig. 194) represent the trapezium and P the given point ; and let m : n represent the given ratio. CONSTRUCTION. Determine I, as in Art. 404. Join PI, and draw G CF parallel to it : then will PF be the division line. For if CH and CI be joined, CHD = ABCD ; and, since HCI : ICD : : m : n, HCI and ICD will be equal to the two parts into which the quadrilateral is to be divided. But, since PI is parallel to CF, we have GC : GP : : GF : GI; .-. (15.6) GPF = GCI, andPFDC = CID. Calculation. In GAB find GA and GB. Then GC : GB : : G A : GH; whence HD and HI become known ; and GP : GC : : GI : GF. Finally, AF = GF - GA. EXAMPLES. Ex. 1. Given AB K 25| E. 4.65 chains, BC K 77 E. 6.30 chains, CD South 7.30 chains, and DA K 78J W. 8.35 chains, to divide the trapezium into two equal parts by a line EF running through a point P in BC distant 2.50 chains from B. AF is required. SBO. II.] DIVISION OF LAND. 295 Calculation. To find GA and GB. As sin. G 24 45' A. C. 0.378139 : sin. GBA 51 15' 9.892030 : : AB 4.65 0.667453 : AG 8.662 0.937622 AD 8.35 GD 17.012 As sin. a 24 45' A. C. 0.378139 : sin. GAB 104 9.986904 :: AB 0.667453 : BG 10.777 1.032496 BC GO To find GH. As GO 17.077 A. C. 8.767588 : GB 10.777 1.032496 : : GA 8.662 0.937622 : GH 5.466 0.737706 HI = J (GD - GH) = 5.773 and GI = GH + HI = 11.239, To find GF and AF. AsGP : GC :: GI : GF AG AF 5.794. Ex. 2. Given AB K 27| W. 19.55 chains, BC East 18.92 chains, CD S. 1J E. 10.49 chains, and DA S. 56 W. 12.25 chains, to find BF, so that a line run from a point 13.277 A. C. 8.876900 17.077 1.232412 11.239 1.050727 14.456 1.160039 8.662 296 LAYING OUT AND DIVIDING LAND. [CHAP. VII P in AD 6 chains from A may divide the trapezium into two parts ABFP and PFCD having the ratio of 5 to 4. Ans. BF = 9.00 ch. CASE 2. 406. The division line to run through any point. Let ABCD (Fig. 195) Fig. 195. "be the given trapezium and P the given point. Determine I, as in the G ^---~ , ^ y - __ M last two articles, and bi- sect GI in K Through P draw 0PM parallel to GD, meeting GB in 0. }/ Join EX), and draw CL N parallel to it. Through L draw LM parallel to GB. Make ~LN perpendicular to AD and equal to OP. With the centre N" and radius equal to PM, describe an arc cutting AD in F. Then FPE will be the division line. DEMONSTRATION. As was proven, Art. 381, GFE = GOML = 2 GOL = 2 GCK = GCI : whence ABEF = ABCI. But CI divides the trapezium into two parts having the given ratio ; therefore, EF does so likewise. Calculation. Find GB, GA, GH, and GI. Then in OBP find OB and OP: thus GO is known. And because GO : GO : : GK : GL, GL is known ; but PM = GL OP. Hence, in LNF we have LN and NF to find LF, EXAMPLES. Ex. 1. Given AB K 25| E. 4.65 chains, BC N". 77 E. 6.30 chains, GD South 7.30 chains, and DA N. 78J W. 8.35 chains, to part off two-fifths of the tract next to AB by a line through a spring S. 54| E. 2.95 chains from the second corner. The distance AF is required. SEC. II.] DIVISION OF LAND. Calculation. 297 As in Ex. 1, last case : GB = 10.77T, GA = 8.662, GO = 17.077, GD = 17.012, GH = 5.466, GI = (GH + f HD) = 10.084, and GK = 5.042. As sin. BOP : sin. BPO :: BP : OB GB GO As sin. BOP : sin. OBP :: BP : OP As GO : GO :: GK : GL To find OB and OP. 24 45' A. C. 0.378139 9.600700 0.469822 24 45' 131 45' 5.257 To find GL. 7.967 17.077 5.042 10.807 0.448661 A. C. 0.378139 9.872772 0.469822 0.720733 9.098705 1.232412 0.702603 1.033720 KF = GL - OP Whence LF = ^ETF 2 -LK 3 whence AF = GL + LF GA 5.55. 1.779; 3.924. Ex. 2. Given AB K 27J W. 19.55 chains, BC East 18.92 chains, CD S. 1} E. 10.49 chains, and DA S. 56 W. 12.25 chains, to divide the quadrilateral into two parts ABEF and FECD in the ratio of 5 to 4, by a line EF through a spring P, which bears from B S. 70| E. 11.52 chains. The distance AF is required. Ans. AF = 5.01 ch. 298 LAYING OUT AND DIVIDING LAND. [CHAP. VII. CASE 3. 407. The division line to be parallel to one side. Fig. 196. Let ABCD (Fig. 196) re- present the trapezium which is to be divided into two parts having the ratio of m to n by a line parallel to CD. CONSTRUCTION. Deter- mine H and I, as in the pre- ceding articles. Take GF a mean proportional between GI and GD : then EF, parallel to CD, will be the division line. For, as was demonstrated, (Art. 404,) ABCD = HCD, and CHI : CID :: m : n. But (lemma) GCI = GEF ; ICD = EFDC, and HCI = ABEF: whence ABEF : FECD : : m : n. If the division line Mg.W. is to be parallel to the shorter side AB (Fig. 197.) Draw CK paral- lel to AB, and take GF a mean proportional between GI and GK; or, join BD, and draw CH' parallel to it. AI' : I'H' : : m : n, and take GF a mean proportional between GA and GF. Then will EF, parallel to AB, be the division line. H A I F K Divide AH' in I', so that SEC. II.]. DIVISION OF LAND. 299 Calculation. First Method. Find, as in the preceding articles, GH and GI. Then GF = / GI . GK = 11.765 1.070585 8.662 AF = 3.103 To find EF. As GA 8.662 A. C. 9.062378 : AB 4.65 0.667453 : : GF 11.765 1.070585 : EF 6.316 1.800416 Second Method. C sin. E As sin. E 128 45' A. C. 0.107970 sin. F 76 0.013096 sin. C 77 9.988724 sin. D 78 15' 9.990803 CD 7.30 0.863323 CD 0.863323 f 67.18 1.827239 2 134.36 AB 2 64.8675 5)199.2275 EF = V 39.8455 = 6.312. To find AF. As sin. G 24 45' A. C. 0.378139 : sin. E 128 45' 9.892030 : : FE - AB 1.662 0.220631 : AF 3.096 0.490800 SEC. II.]- DIVISION OP LAND. 301 Ex. 2. Given the bearings and distances as in Ex. 1, to divide the trapezium into two parts AFED and FECB, having the ratio of 3 to 2, by a line EF parallel to BC. AF and EF are required. Ans. AF = 1.60 chains ; EF = 7.66 chains. Ex. 3. Given as in Ex. 1, to divide the trapezium into two parts ABEF and FECD, in the ratio of 2 to 3, by a line EF parallel to CD. AF and EF are required. Ans. AF = 4.44 chains ; EF = 5.62 chains. CASE 4. 408. The division line to run any direction. Fig. 198. Kv D Let ABCD (Fig. 198) be the trapezium to be divided into two parts ABEF and FECD, in the ratio of m to ft, by a line EF running any course. The construction of this case is the same as that of the last, CK being drawn so as to be of the same course as EF. Calculation. Conceive xy and vw to be drawn so as to make Gxy = GAB, and Gvw = GCD : then will vwyx be equal to ABCD. It will also be divided by EF into two parts having the ratio of m to n. Find xy 2 and viv 3 by the proportions sin. E . sin. F : sin. A . sin. B : : AB 2 : xy\ and sin. E . sin. F : sin. C . sin. D : : CD 3 : viv 2 , the truth of which has been proven in the demonstration to rule for Art. 407. Then (Art. 404) EF* = >* + *&. m + n Draw AOP parallel to BC, meeting BE" and EF in O and P. 302 LAYING OUT AND DIVIDING LAND. [CHAP. VII. Then sin. BOA (sin. E) : sin. BAG (sin. B) : : AB : BO, and sin. PAF (sin. G) : sin. P (sin. E) : : PF (EF BO) : AF. The calculation may otherwise be made by finding GH and GI, as in Arts. 406, 407, and also GK. Then GF = I ; AWD DEPARTURES. jATXTTTDXSS D. iDeg. *Deg. I Deg. IDeg. D. 1 2 3 4 5 6 7 8 9 10 Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 1 2 3 4 5 4 7 8 9 1O 1 2 3 4 5 6 7 8 9 1O I.OOOO 2.0000 3.0000 4.0000 5.0000 5-9999 6.9999 7-9999 8-9999 9.9999 .0044 .0087 .0131 .0175 .0218 .0262 .0305 .0349 0393 .0436 I.OOOO 1.9999 2.9999 3.9998 4.9998 5.9998 6.9997 7-9997 8.9997 9.9996 .0087 .0175 .0262 349 .0436 .0524 .0011 .0698 .0785 .0873 9999 1.9998 2.9997 3-9997 4.9996 5-9995 6.9994 7-9993 8.9992 9.9991 .0131 .0262 0393 .0524 .0654 .0785 .0916 .1047 .1178 .1309 .9998 1.9997 2.9995 3-9994 4.9992 5.9991 6.9989 7.9988 8.9986 9.9985 .0175 349 .0524 .0698 .0873 .1047 .1222 .1396 .I5 7 I I 745 89f Deg. 89 * Deg. 89 1 Deg. 89 Deg. It Deg. U Deg. If Deg. 2 Deg. .9998 1.9995 2.9993 3.9990 4.9988 5.9986 6.9983 7.9981 8-9979 9.9976 .0218 .0436 .0654 .0873 .1091 .1309 .1527 1745 .1963 .2181 9997 1-9993 2.9990 3.9986 4.9983 5-9979 6.9976 7-9973 8.9969 9.9966 .0262 .0524 .0785 .1047 .1309 .1571 1832 .2094 .2356 .2618 9995 1.9991 2.9986 3.9981 4-9977 5.9972 6.9967 7.9963 8.9958 9-9953 .0305 .0611 .0916 .1222 .1527 .1832 .2138 .2443 .2748 354 9994 1.9988 2.9982 3.9976 4.9970 5.9963 6.9957 7-995 1 8-9945 9-9939 .0349 .0698 .1047 .1396 1745 .2094 .2443 .2792 .3141 349 1 2 3 4 5 6 7 8 9 10 88f Deg. 88J Deg. 88i Deg. 88 Deg. 2t Deg. 2* Deg. 2f Deg. 3 Deg. 1 2 3 4 5 6 7 8 9 10 .9992 1.9985 2.9977 3.9969 4.9961 5-9954 6.9946 7-9938 8.9931 9-99*3 0393 .0785 .1178 .1570 .1963 .2356 .2748 .3140 3533 .3926 .9990 1.9981 2.9971 3.9962 4.9952 5-9943 6.9933 7.9924 8.9914 9.9905 .0436 .0872 .1308 1745 .2l8l .2617 353 349 .3926 .4362 .9988 1.9977 2.9965 3-9954 4.9942 5-993 1 6.9919 7.9908 8.9896 9.9885 .0480 .0960 1439 .1919 2399 .2879 3358 .3838 .4318 4798 .9986 1.9973 2.9959 3-9945 4-993 * 5.9918 6.9904 7.9890 8.9877 9.9863 .0523 .1047 .1570 .2093 .2617 :$: .4187 .4710 .5234 1 2 3 4 5 6 7 8 9 10 871 Deg. 87 * Deg. 87i Deg. 87 Deg. 3t Deg. 3 Deg. 3| Deg. 4 Deg. 1 2 3 4 5 6 7 8 9 1O 9984 1.9968 2.9952 3.9936 4.9920 5.9904 6.9887 7.9871 8.9855 9-9839 .0567 .1134 ' I7 2o .2268 .2835 3402 .3968 4535 .5102 .5669 .9981 1.9963 2.9944 3-99^5 4.9907 5.9888 6.9869 7.9851 8.9832 9.9813 .0610 .1221 .1831 .2442 .3052 .3663 4*73 .4884 5494 .6105 Lat. 9979 J-9957 2.9936 3-99 J 4 4.9893 5.9872 6.9850 7.9829 8.9807 9.9786 .0654 .1308 .1962 .2616 .3270 .3924 4578 .5232 .5886 .6540 .9976 1.9951 2.9927 3.9903 4.9878 6.9829 7.9805 8.9781 9.9756 .0698 1395 .2093 .2790 .3488 .4185 .4883 i s8 5 .6278 .6976 1 2 3 4 5 6 7 8 9 10 D. Dep. Lat. Dep. Dep. Lat. Dep. Lat. D. 86f Deg. 86* Deg. 861 Deg. 86 Deg. 22 LATITUDES AND DEPARTURES. D. 41 Deg. 4i Deg. 4| Deg. 5 Deg. D. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 1 2 3 4 5 6 7 8 9 10 9973 1.9945 2.9918 3.9890 4.9863 5.9835 6.9808 7.9780 8-9753 9.9725 .0741 .1482 .2223 .2964 3705 4447 .5188 .5929 .6670 .7411 .9969 1.9938 2.9908 3-9^77 4.9846 5-9 8l 5 6.9784 7-9753 8.9723 9.9692 .0785 .1569 2354 3138 39*3 .4708 5492 .6277 .7061 .7846 .9966 I-993 1 2.9897 3.9867 4.9828 5-9794 6.9700 7.9725 8.9691 9.9657 .0828 .1656 .2484 33 12 .4140 .4968 5797 .6625 7453 .8281 .9962 1.9924 2.9886 3.9848 4.9810 5.9772 6.9734 7.9696 8.9658 9.9619 .0872 1743 .2615 .3486 4358 5229 .6101 .6972 .7844 .8716 1 2 \ 3 4 5 6 7 8 9 10 85| Deg. 85 Deg. 5 Deg. 85i Deg. 5f Deg. 85 Deg. 5J Deg. 6 Deg. 1 2 3 4 5 6 7 8 9 10 .9958 1.9916 2.9874 3.9832 4.9790 5.9748 6.9706 7.9664 8.9622 9.9580 .0915 .1830 2 I1 5 .3660 4575 .5490 .6405 .7320 .8235 .9150 9954 1.9908 2.9862 3.9816 4.9770 5-97^4 6.9678 7.9632 8.9586 9.9540 .0958 .1917 .2875 3834 .4792 5751- .6709 .7668 .8626 9585 .9950 1.9899 2.9849 3-9799 4.9748 5.9698 6.9648 7-9597 8-9547 9-9497 .1002 .2004 .3006 .4008 .5009 .6011 .7013 .8015 .90x7 1.0019 9945 1.9890 2.9836 3-978i 4.9726 5-9671 6.9617 7.9562 8.9507 9-9452 .1045 .2091 .3136 .4181 .5226 .6272 :l%l .9408 1-0453 1 2 3 4 5 6 7 8 9 10 841 Deg. 84J Deg. 841 Deg. 6f Deg. 84 Deg. 61 Deg. 6J Deg. 7 Deg. 1 2 3 4 5 6 7 8 9 10 -994 1 1.9881 2.9822 3.9762 4-9703 5-9 6 43 6.9584 7.9524 8.9465 9.9406 .1089 .2177 .3266 4355 5443 .6532 .7621 .8709 .9798 1.0887 .9936 1.9871 2.9807 3-9743 4.9679 5.9614 6.9550 7.9486 8.9421 9-9357 .1132 .2264 3396 4528 .5660 .6792 .7924 .9056 1.0188 1.1320 993 1 1.9861 2.9792 3-9723 4-9 6 53 5-9584 6.9515 7-9445 8.9376 9.9307 "75 2351 .3526 .4701 .5877 .7052 .8228 943 1.0578 I-I754 .9925 1.9851 2.9776 3.9702 4.9627 S* 55 i 6.9478 7.9404 8.9329 9.9255 .1219 2 ! 3 Z 3656 4875 .6093 .7312 8531 .9750 1.0968 1.2187 1 2 3 4 5 6 7 8 9 10 I 2 3 4 5 6 7 8 9 10 83f Deg. 83 J Deg. 83 1 Deg. 83 Deg. 71 Deg. 7J Deg. 71 Deg. 8 Deg. .9920 1.9840 2.9760 3.9680 4.9600 5.9520 6.9440 7.9360 8.9280 9.9200 .1262 .2524 .3786 .5048 .6310 .7572 .8834 1.0096 1.1358 1.2620 .9914 1.9829 2.9743 3.9658 4.9572 5-9487 6.9401 7.9316 8.9230 9.9144 1305 .2611 .3916 .5221 .6526 .7832 .9137 1.0442 1.1747 I-3053 Lat. .9909 1.9817 2.9726 3-9635 4-9543 5-9452 6.9361 7.9269 8.9178 9.9087 1349 .2697 .4046 5394 6 743 .8091 .9440 1.0788 1.2137 i*34 8 5 993 1.9805 2.9708 3.9611 4-95I3 5.9416 6.9319 7.9221 8.9124 9.9027 .1392 .2783 4 r 75 5567 .6959 8350 .9742 1.1134 1.2526 i-39 J 7 Lat. 1 2 3 4 5 6 7 8 9 10 D. D. Dep. Lat. Dep. Dep. Lat. Dep. 82| Deg. 82 Deg. 821 Deg. 82 Deg. LATITUDES AND DEPARTURES. D. 8i Deg. 8i Deg. 8f Deg. 9 Deg. D. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 1 2 3 4 5 6 7 8 9 1O .9897 1 -9793 2.9690 3.9586 4.9483 5-9379 6.9276 7.9172 8.9069 9.8965 1435 .2870 435j 574 7*75 .8610 1.0044 1.1479 1.2914 1.4349 .9890 1.9780 2.9670 3-95 61 4-945 1 5-934 1 6.9231 7.9121 8.9011 9.8902 .1478 .2956 4434 .5912 .7390 .8869 1.0347 1.1825 i-333 1.4781 .9884 1.9767 2.9651 3-9534 4.9418 5-9302 6.9185 7.9069 8.8953 9.8836 .1521 .3042 .4564 .6085 .7606 .9127 1.0649 1.2170 1.3691) 1.5212 .9877 1-9754 2.9631 3.9508 4-9384 5.9261 6.9138 7.9015 8.8892 9.8769 .1564 .3129 .4693 .6257 .7822 .9386 1.0950 1.2515 1.4079 I-5643 1 2 ' 3 I 4 5 6 7 8 9 10 81f Deg. 81 J Deg. 8H Deg. 81 Deg. 9i Deg. 9i Deg. 9f Deg. 10 Deg. 1 2 3 4 5 6 7 8 9 10 .9870 1.9740 2.9610 3.9480 4-935 5.9220 6.9090 7.8960 8.8830 9.8700 .1607 3215 .4822 .6430 .8037 .9645 1.1252 1.2859 1.4467 1.6074 .9863 1.9726 2.9589 3-9451 4-93H 5-9*77 6.9040 7.8903 8.8766 9.8629 .1650 33 01 ! 495 i | .6602 .8252 .9903 I-I553 1.3204 1.4854 1.6505 .9856 1.9711 2.9567 3.9422 4.9278 5-9I33 6.8989 7.8844 8.8700 9.8556 .1693 3387 .5080 .6774 .8467 1.0161 1.1854 1-3548 1.5241 1-6935 .9848 1.9696 2.9544 3-9392 4.9240 5.9088 6.8937 7.8785 8-8633 9.8481 .1736 3473 .5209 .6946 .8682 1.0419 1.2155 1.3892 1.5628 1.7365 1 2 3 4 5 6 7 8 9 10 80f Deg. 80* Deg. 80i Deg. 80 Deg. 10i Deg. 10* Deg. 10| Deg. 11 Deg. 1 2 3 4 5 6 7 8 9 10 .9840 1.9681 2.9521 3.9362 4.9202 5.9042 6.8883 7.8723 8.8564 9.8404 .1779 3559 .5338 .7118 .8897 1.0677 1.2456 1-4^35 1.6015 1.7794 9833 1.9665 2.9498 3.9330 4.9163 5.8995 6.8828 7.8660 8.8493 9.8325 .1822 3 6 45 .5467 .7289 .9112 1.0934 1.2756 1-4579 1.6401 1.8224 .9825 1.9649 2-9474 3.9298 4.9123 5-8947 6.8772 7.8596 8.8421 9-8245 .1865 373 5596 .7461 9326 1.1191 i-357 1.4922 1.6787 1.8652 .9816 i-9 6 33 2-9449 3.9265 4.9081 5.8898 6.8714 7-853 8.8346 9.8163 .1908 .3816 .5724 7632 954 1.1449 1-3357 1.5265 1.7173 1.9081 1 2 3 4 5 6 7 8 9 1O ~T 2 3 4 5 6 7 8 9 10 D. 79f Deg. 79 * Deg. 79i Deg. 79 Deg. lit Deg. Hi Deg. Hi Deg. 12 Deg. .9808 1.9616 2.9424 3-9 2 3i 4.9039 5.8847 6.8655 7.8463 8.8271 9.8079 .1951 .3902 5*53 .7804 9755 1.1705 1.3656 1.5607 1.7558 1.9509 9799 1.9598 2.9398 3-9!97 4.8996 5-8795 6.8595 7.8394 8.8193 9.7992 .1994 .3987 .5981 7975 .9968 1.1962 1.3956 1-5949 1-7943 1.9937 .9790 1.9581 2.9371 3.9162 4.8952 5-8743 6-8533 7-8324 8.8114 9.7905 .2036 .4073 .6109 .8146 1.0182 1.2219 1.4255 1.6291 1.8328 2.0364 .9781 1-95*3 2-9344 3.9126 4.8907 5.8689 6.8470 7.8252 8.8033 9-78i5 .2079 4158 .6237 .8316 1.0396 1.2475 1-4554 1.6633 1.8712 2.0791 Lat. 1 i 2 3 4 5 6 7 j 8 9 ; 10 Dep. Lat. Dep. Lat. Dep. Lat. Dep. D. 78f Deg. 78i Deg. 78i Deg. 78 Deg. LATITUDES AND DEPARTURES. D. 12i Deg. 12* Deg. 12| Deg. 13 Deg. D. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 1 2 3 4 5 6 7 8 9 10 .9772 1-9545 2.9317 3.9089 4.8862 5.8634 6.8406 7.8178 8.7951 9.7723 .2122 .4244 .6365 .8 4 8 7 1.0609 I.273I I. 4 852 1.6974 1.9096 2.I2I8 .9761 1.9526 2.9289 3.9052 4.8815 5.8578 6.8341 7.8104 8.7867 9.7630 .2164 4329 6493 .8658 1.0822 1.2986 1.5151 i-73 J 5 1.9480 2.1644 9753 1.9507 2.9260 3.9014 4.8767 5.8521 6.8274 7.8027 8.7781 9-7534 .2207 44 * 4 .6621 .8828 1.1035 1.3242 1.5449 1.7656 1.9863 2.2070 9744 1.9487 2.9231 3-8975 4.8719 5.8462 6.8206 7.7950 8.7693 9-7437 .2250 4499 .6749 .8998 1.1248 J-3497 r -5747 1.7996 2.0246 2.2495 1 2 3 4 5 6 7 8 9 1O 2 3 4 5 6 7 8 9 - 10 77f Deg. 77* Deg. 77i Deg. 77 Deg. ~T 2 3 4 5 6 7 8 9 10 13i Deg. 13* Deg. 13 f Deg. 14 Deg. 9734 1.9468 2.9201 3-8935 4.8669 5.8403 6.8137 7.7870 8.7604 9-7338 .2292 4584 .6876 .9168 1.1460 1-3752 1.6044 1.8336 2.0628 2.2920 9724 1.9447 2.9171 3-8895 4.8618 5-8342 6.8066 7.7790 8-75*3 9.7237 * 2 H 4 .4669 .7003 9338 1.1672 1.4007 1.6341 1.8676 2.IOIO 2-3345 9713 1.9427 2.9140 3-8854 4-8567 5.8281 6.7994 7.7707 8.7421 9-7I34 .2377 4754 7I3 1 .9507 1.1884 1.4261 1.6638 1.9015 2.1392 2.3769 .9703 1.9406 2.9109 3.8812 4-8515 5.8218 6.7921 7.7624 8.7327 9.7030 .2419 .4838 .7258 .9677 1.2096 1.4515 I-6935 1-9354 2.1773 2.4192 ~T 2 3 4 5 6 7 8 9 10 76| Deg. 76* Deg. 76i Deg. 76 Deg. 14i Deg. 14* Deg. 14f Deg. 15 Deg. .9692 1.9385 2.9077 3.8769 4.8462 5.8154 6.7846 7.7538 8.7231 9.6923 .2462 .4923 7385 .9846 1.2308 1.4769 I.723I 1.9692 2.2154 2.4615 .9681 1.9363 ! 2.9044 3.8726 4.8407 5.8089 6.7770 7-7452 8.7133 9.6815 .2504 .5008 .7511 1.0015 1.2519 1.5023 1.7527 2.0030 2.2534 2.5038 .9670 1-9341 2.9011 3.8682 4-8352 5.8023 6.7693 7.7364 8.7034 9.6705 .2546 .5092 .7638 1.0184 1.2730 1.5276 1.7822 2.0368 2.2914 2.5460 9659 1.9319 2.8978 3.8637 4.8296 5-7956 6.7615 7.7274 8.6933 9-6593 .2588 5*76 7765 1-0353 1.2941 1.5529 1.8117 2.0706 2.3294 2.5882 1 2 3 4 5 6 7 8 9 10 ~T 2 3 4 5 6 7 8 9 10 75f Deg. 75* Deg. 75i Deg. 75 Deg. 16 Deg. 15i Deg. 15* Deg. 15f Deg. .9648 1.9296 2.8944 3-8591 4.8239 5-7887 6-7535 7.7183 8.6831 9.6479 .2630 .5261 .7891 I.052I I.3I52 1.5782 I.84I2 2.1042 2.3673 2.630^ .9636 1.9273 2.8909 3-8545 4.8182 5.7818 6.7454 7.7090 8-6727 9-6363 .2672 5345 .8017 1.0690 1.3362 1.6034 1.8707 2.1379 2.4051 2.6724 .9625 1.9249 2.8874 3-8498 4.8123 5-7747 6.7372 7.6996 8.6621 9.6246 .2714 .5429 .8147 1.0858 1-3572 1.6286 1.9001 2.1715 2.4430 2.7144 .9613 1.9225 2.8838 3-8450 4.8063 7.6901 8-6514 9.6120 .2756 5513 .8269 1.1025 1.3782 1-6538 1.9295 2.2051 2.4807 2.7564 1 2 3 4 5 6 7 8 9 10 D. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. D. 74f Deg. 74* Deg. 74i Deg. 74 Deg. LATITUDES AND DEPARTURES. D. 161 Deg. 16* Deg. 16f Deg. 17 Deg. D. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 1 2 3 4 5 6 7 8 9 10 ~T 2 3 4 5 6 7 8 9 10 .9600 1.9201 2.8801 3.8402 4.8002 5.7603 6.7203 7.6804 8.6404 9.6005 .2798 5597 8395 1.1193 I-399 1 1.6790 1.9588 2.2386 2.5185 2.7983 .9588 1.9176 2.8765 3-8353 4.7941 5-75*9 6.7117 7.6706 8.6294 9.5882 .2840 .5680 .8520 1.1361 1.4201 1.7041 1.9881 2.2721 2.5561 2.8402 .9576 1.9151 2.8727 3-8303 4.7879 5-7454 6.7030 7.6606 8.6181 9-5757 .2882 5764 .8646 1.1528 1.4410 1.7292 2.0174 2.3056 2.5938 2.8820 95 6 3 1.9126 2.8689 3.8252 4.7815 5.7378 6.6941 7.6504 8.6067 9.5630 .2924 5847 .8771 1.1695 1.4619 '754* 2.0466 2.3390 2.6313 2.9237 1 2 3 4 5 6 7 8 9 10 73f Deg. 73 Deg. 73* Deg. 73 Deg. 17* Deg. 17 Deg. 17f Deg. 18 Deg. .9550 1.9100 2.8651 3.8201 4-7751 5-73 01 6.6851 7.6402 8.5952 9.5502 .2965 ^8896 1.1862 1.4827 1.7792 2.0758 2.3723 2.6689 2.9654 9537 1.9074 2.8612 3.8149 4.7686 H**3 6.6700 7.629} 8-5835 9.5372 .3007 .6014 .9021 1.2028 I-5035 1.8042 2.1049 2.4056 2.7064 3.0071 95*4 1.9048 2.8572 3.8096 4.7620 5-7I44 6.6668 7.6192 8.5716 9.5240 349 .6097 .9146 1.2195 i-5*43 1.8292 2.134! 2.4389 2.7438 3.0486 95" 1.9021 2.8532 3.8042 4-7553 5.7063 6.6574 7.6085 8-5595 9.5106 .3090 .6180 .9271 1.2361 1.5451 1.8541 2.1631 2.4721 2.7812 3.0902 1 2 3 4 5 6 7 8 9 1O 72f Deg. 72* Deg. 72* Deg. 72 Deg. 18* Deg. 18J Deg. 18f Deg. 19 Deg. 1 2 3 4 5 6 7 8 9 10 9497 1.8994 2.8491 3.7988 4.7485 5.6982 6.6479 7-5976 8-5473 9.4970 .6263 9395 1.2527 1.5658 1.8790 2.1921 *-553 2.8185 3.1316 -9483 1.8966 2.8450 3-7933 4.7416 5.6899 6.6383 7.5866 8.5349 9.4832 3173 .6346 .9519 1.2692 1.5865 1.9038 2.221 1 2.5384 2.8557 3.1730 .9469 1-8939 2.8408 3-7877 4-7347 5.6816 6.6285 7-5754 8.5224 9-4693 .3214 .6429 .9643 1.2858 1.6072 1.9286 2.2501 2.5715 2.8930 3.2144 9455 1.8910 2.8366 3.7821 4.7276 5- 6 73 T 6.6186 7.5641 8.5097 9-455* .3256 .6511 .9767 1.3023 1.6278 1-9534 2.2790 2.6045 2.9301 3-*557 1 2 3 4 5 6 7 8 9 10 71| Deg. 71i Deg. 71* Deg. 71 Deg. 191 Deg. 19 j- Deg. 19| Deg. 20 Deg. 1 2 3 4 5 6 7 8 9 1O .9441 1.8882 2.8323 3-7764 4.7204 5.6645 6.6086 7-55*7 8.4968 9.4409 3*97 .6594 .9891 1.3188 1.6485 1.9781 2.3078 2.6375 2.9672 3.2969 .9426 1.8853 2.8279 3.7706 4-713* 5.6558 6.5985 7.5411 8.4838 9.4264 3338 .6676 I.OOI4 'IP 1.6690 2.0028 2.3366 2.6705 3.0043 3-338I .9412 1.8824 2.8235 3.7647 4.7059 5.6471 7.5294 8.4706 9.4118 3379 .6758 1.0138 'SS 1 ? 1.6896 2.0275 2.3654 2.7033 3.0413 3-379* 9397 1.8794 2.8191 3-7588 4.6985 5.6382 6.5778 7-5J75 8-457* 9-3969 %& 1.0261 1.3681 1.7101 2.0521 2.3941 2.7362 3.0782 3.4202 1 2 3 4 5 6 7 8 9 10 D. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. D. 70f Deg. 70* Deg. 70* Deg. 70 Deg. LATITUDES AITO DEPARTURES. D. 20* Deg. 20* Deg. 20f Deg. 21 Deg. D. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 1 2 3 4 5 6 7 8 9 1O .9382 1.8764 2.8146 3-75^8 4.6910 5.6291 6.5673 7-555 8-4437 9.3819 .3461 .6922! 1.0384] 1.3845 1.7306 2.0767 2.4228 2.7689 3-"5i 3.4612 93 6 7 1-8733 2.8100 3.7467 4.6834 5.6200 6.5567 7-4934 8.4300 9.3667 .3502 .7004 ! 1.0506 1.4008 1.7510 2.IOI2 2.4515 2.8017 1- I 5*9 3.5021 9351 1.8703 2.8054 3-745 4-6757 5.6108 6-5459 7.4811 8.4162 9-35H 3543 .7086 1.0629 1.4172 1-77*5 2.1257 2.4800 2.8343 3.1886 3-5429 933 6 1.8672 2.8007 3-7343 4.6679 5.6015 6-5351 7.4686 8.4022 9-335 8 3584 .7167 1.0751 J-4335 1.7918 2.1502 2.5086 2.8669 3.2253 3-5837 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 i 9 10 69| Deg. 69 J Deg. 21 J Deg. 69i Deg. 69 Deg. 21 1 Deg. 21f Deg. 22 Deg. 1 2 3 4 5 6 7 8 9 10 .9320 1.8640 2.7960 3.7280 4.6600 5-592 6.5241 7-456i 8.3881 9.3201 .3624 .7249 1 1.0873 1.4498 1.8122 2.1746 2-537 1 2.8995 3.2619] 3.6244 934 i. 8608 2.7913 3.7217 4.6521 5.5825 6.5129 7-4433 8.3738 9.3042 .3665 733 1.0995 1.4660 1.8325 2.1990 2-5655 2.9320; 3.2985 3.6650 .9288 1.8576 2.7864 3-7I5 2 4.6440 5-5729 6.5017 7.4305 9.2881 .3706 .7411 1.1117 1.4822 1.8528 2.2233 2-5939 1.9645 3-335 3.7056 .9272 1.8544 2.7816 3.7087 4-6359 5-5631 6.4903 7-4*75 8-3447 9.2718 3746 .7492 1.1238 1.4984 1.8730 2.2476 2.6222 2.9969 3-37I5 3.7461 68f Deg. 68 J Deg. 68i Deg. 68 Deg. . ~T| 2 3 4 5 6 7 8 9 1O 22i Deg. 22 Deg. 22f Deg. 23 Deg. 1 2 3 4 5 6 7 8 9 1O .9255 1.8511 2.7766 3.7022 4.6277 8i 7.4043 8.3299 9-2554 .3786 7573 1.1359 1.5146 1.8932 2.2719 2.6505 3.0292 3.4078 3-7865 .9239 1.8478 2.7716 3- 6 955 4.6194 5-5433 6.4672 7.3910 8.3149 9.2388 .3827 7654 1.1481 i-537 I-9J34 2.2961 2.6788 3.0615 3.4442 3.8268 .9222 1.8444 2.7666 3.6888 4.6110 5-5332 6-4554 7.3776 8.2998 9.2220 .3867 7734 I.IOOI 1.5468 I-933 6 2.3203 2.7070 3.0937 3.4804 3.8671 .9205 1.8410 2.7615 3.6820 4.6025 5-523 6-4435 7.3640 8.2845 9.2050 397 .7815 1.1722 1.5629 J-9537 2.3444 2-735 1 3 ' 12 ^ 3.5166 3.9073 671 Deg. 23i Deg. 67 i Deg. 67i Deg. 67 Deg. 23 Deg. 23| Deg. 24 Deg. 1 2 3 4 5 6 7 8 9 10 .9188 1.8376 2.7564 3.6752 4-594 5-5 I2 7 6-43 T 5 7-3503 8.2691 9.1879 3947 .7895 1.1842 1.5790 1-9737 2.3685 2.7632 3-158 3-55^7 3-9474 .9171 1.8341 2.7512 3.6682 4-5853 5.5024 6.4194 7-3365 8.2535 9.1706 3987 7975 1.1962 1.5950 1.9937 2.3925 2.7912 3.1900 3-5887 3-9875 '% I5 l 1.8306 2.7459 3.6612 4-5766 5-49*9 6.4072 7-3225 8.2378 1 9-J53 1 .4027 .8055 1.2082 1.6110 2.0137 2.4165 2.8192 3.2220 3.6247 4.0275 9*35 1.8271 2.7406 3-6542 4-5677 5'48i3 6.3948 7.3084 8.2219 9-'355 .4067 8i35 I.22O2 1.6269 2.0337 2.4404 2.8472 3-2539 3.6606 4.0674 1 2 3 4 5 6 7 8 i 9 10 D. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. D. 66f Deg. 66J Deg. 66i Deg. 66 Deg. 10 _ . - -. LATITUDES AND DEPARTURES. i D - 24i Deg. 24 Deg. 24f Deg. 25 Deg. D. 1 2 3 4 5 6 7 8 9 10 Lat. Dep. Lat. Dep. Lat. Dep. Lat. .9063 1.8126 2.7189 3.6252 4'53!5 5-4378 6.3442 7-2505 8.1568 9.0631 Dep. 1 2 3 4 5 6 7 8 9 10 .9118 1.8235 2-7353 3.6470 4.5588 5.4706 6.3823 7.2941 8.2059 9.1176 .4107 .8214 1.2322 1.6429 2.0536 2.4643 2.8750 3.2858 3.6965 4.1072 .9100 1.8199 2.7299 3.6398 4.5498 5-4598 6.3697 7.2797 8.1897 9.0996 4 I 47| .82945 1.2441 1.6588 2.0735 2.4882 2.9029 3-3'75 3-7322 4.1469 .9081 1.8163 2.7244 3.6326 4.5407 5.4489 6.3570 7.2651 8-1733 9.0814 .4187 8373 1.2560 1.6746 2.0933 2.5120 2.9306 3-3493 3.7679 4.1866 .4226 .8452 1.2679 1.6905 2.1131 2-5357 2.9583 3.3809 3.8036 4.2262 65f Deg. 65 \ Deg. 25 Deg. 65i Deg. 25f Deg. 65 Deg. 25i Deg. 26 Deg. 1 2 3 4 5 6 7 8 9 10 .9045 1.8089 2.7134 3.61/8 4.5223 5.4267 6.3312 7.2356 8.1401 9.0446 .4266 .8531 1.2797 1.7063 2.1328 2-5594 2.9860 3-4^5 3-839I 4.2657 .9026 1.8052 2.7078 3.6103 4.5129 5-4I55 6.3181 7.2207 8.1233 9.0259 435 .8610 1.2915 1.7220 2.1526 2.5831 3.0136 3-4441 3.8746 4.3051 .9007 1.8014 2.7021 3.6028 4-535 5.4042 6.3049 7.2056 8.1063 9.0070 4344 .8689 I-3033 1.7378 2.1722 2.6067 3.0411 3-475 6 3.9100 4-3445 .8988 1.7976 2.6964 3-5952 4.4940 5.3928 6.2916 7.1904 8.0891 8.9879 .4384 .8767 I-3J5 1 1-7535 2.1919 2.6302 3.0686 3.5070 3-9453 4-3837 1 2 3 4 5 6 7. 8 9 10 ~T 2 3 4 5 6 7 8 9 1O 64f Deg. 64 Deg. 64i Deg. 64 Deg. 26i Deg. 26 Deg. 26f Deg. 27 Deg. .8969 1.7937 2.6906 3-5^75 4.4844 5.3812 6.2781 7.1750 8.0719 8.9687 4423 .8846 1.3269 1.7692 2.2114 2.6537 3.0960 3-5383 3.9806 4.4229 .8949 1.7899 2.6848 3-5797 4-4747 5.3696 6.2645 7-1595 8.0544 8-9493 .4462 .8924 1.3386 1.7848 2.2310 2.6772 3>I f 3 t 3.5696 4.0158 4.4620 .8930 1.7860 2.6789 3-5719 4.4649 5-3579 6.2509 7.1438 8.0368 8.9298 .4501 .9002 I-3503 1.8004 2.2505 2.7006 3- I 57 3.6008 4.0509 4.5010 .8910 1.7820 2.6730 3.5640 4.4550 5.3460 6.2370 7.1281 8.0191 8.9101 4540 .9080 1.3620 i. 8160 2.2700 2.7239 3-1779 3.6319 4.0859 4-5399 1 2 3 4 5 6 7 8 9 10 63| Deg. 63 i Deg. 63i Deg. 63 Deg. 27i Deg. 27 Deg. 271 Deg. 28 Deg. 1 2 3 4 5 6 7 8 9 10 .8890 1.7780 2.6671 3-556i 4.4451 5-3341 6.2231 7.1121 8.0012 8.8902 4579 9 J 57 1.3736 1.8315 2.2894 2.7472 3-25' 3.6630 4.1209 4-5787 .8870 'i! 40 2.6610 3.5480 4-43 5 1 5.3221 6.2091 7.0961 8.8701 .4617 9235 1.3852 1.8470 2.3087 2.7705 3.2322 3.6940 4-1557 4.6175 .8850 1.7700 2.6550 3.5400 4.4249 5-399 6.1949 7.0799 7.9649 8.8499 .4656 .9312 1.3968 1.8625 2.3281 2.7937 3-2593 3.7249 4.1905 4.6561 .8829 1.7659 2.6488 3-53i8 4.4147 5.2977 6.1806 7.0636 7.9465 8.8295 .4695 .9389 1.4084 1.8779 2-3474 2.8168 3.2863 3-7558 4.2252 4.6947 1 2 3 4 5 6 7 8 9 10 D. D. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 62f Deg. 62 } Deg. 62i Deg. 62 Deg. 11 LATITUDES AIT3 D DEPART CJKZSS. D. 28i Peg. 28J Deg. 28f Deg. 29 Deg. D. 1 2 3 4 5 6 7 8 9 10 Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 1 2 3 4 5 6 7 8 9 1O .8809 1.7618 2.6427 3-5236 4.4045 l^l 3 6.1662 7.0471 7.9280 8.8089 4733 .9466 1.4200 '8933 2.3666 2.8399 3-3132 3.7866 4.2599 4.7332 .8788 1 '1 S 1 6 2.6365 3-5I53 4.3941 5-2729 6.1517 7.0305 7.9094 8.7882 .4772 9543 I-43I5 1.9086 2.3858 2.8630 3.3401 3- 8 i73 4.2944 4.7716 .8767 1-7535 2.6302 3.5069 4-3836 5.2604 6.1371 7.0138 7.8905 8.7673 .4810 .9620 1.4430 1.9240 2.4049 2.8859 3.3669 3.8479 4-3289 4.8099 .8746 1.7492 2.6239 3-4985 4-3731 5-2477 6.1223 6.9970 7.8716 8.7462 .4848 .9696 1.4544 1.9392 2.4240 2.9089 3-3937 3-8785 4-3633 4.8481 ~~T 2 3 i 4 5 6 7 8 9 10 61f Deg. 61 Deg. 61 i Deg. 61 Deg. 30 Deg. 29J Deg. 29* Deg. 29| Deg. .8725 1.7450 2.6175 3.4900 4.3625 5- 2 35o 6.1075 6.9800 7.8525 8.7250 .4886 .9772 1.4659 1-9545 2.4431 2.9317 3.4203 3.9090 4.3976 4.8862 .8704 1.7407 2.6lH 3.4814 4.3518 5.2221 6.0925 6.9628 7-8332 8.7036 .4924 .9848 J-4773 1.9697 2.4621 2-9545 3-4470 3-9394 4.4318 4.9242 .8682 1.7364 2.6046 3-4728 4.3410 5.2092 6.0774 6.9456 7.8138 8.6820 .4962 .9924 1.4886 1.9849 2.4811 2.9773 3-4735 3-9697 4.4659 4.9622 .8660 1.7321 2.5981 3.4641 4-33 01 5.1962 6.0622 6.9282 EBS .5000 I.OOOO 1.5000 2.OOOO 2.5000 3.0000 3.5000 4.0000 4.5000 5.0000 1 2 3 4 5 6 7 8 9 10 60f Deg. 60 Deg. 60i Deg. 60 Deg. 30 i Deg. 30 Deg. 30| Deg. 31 Deg. 1 2 3 4 5 6 7 8 9 10 .8638 1.7277 2.5915 3-4553 4.3192 5.1830 6.0468 6.9107 7-7745 8.6384 .5038 1.0075 *-5"3 2.0151 2.5189 3.0226 3.5264 4.0302 4.5340 5.0377 .8616 1.7233 2.5849 3-4465 4.3081 5.1698 6.0314 6.8930 7-7547 8.6163 575 1.0151 1.5226 2.0302 2-5377 3.0452 3-5528 4.0603 4-5678 5-0754 .8594 1.7188 2.5782 3-4376 4.2970 5.1564 6.0158 6.8753 7-7347 8.5941 5"3 1.0226 1-5339 2.0452 2-5565 3.0678 3-579 1 4.0903 4.6016 5.1129 .8572 i7*43 2.5715 3.4287 4.2858 5-143 6.OOO2 6.8573 7-7145 8-5717 .5150 1.0301 1-5451 2.0602 2.5752 3.0902 3.6053 4.1203 4-6353 5^504 1 2 3 4 5 6 7 8 9 10 59| Deg. 59* Deg. 3H Deg. 59i Deg. 59 Deg. 3H Deg. 31f Deg. 32 Deg. 1 2 3 4 5 6 7 8 9 10 .8549 1.7098 2.5647 3.4196 4.2746 5-i 2 95 5.9844 6.8393 7.6942 8.5491 .5188 1-0375 i-55 6 3 2.0751 2-5939 3.1126 3- 6 3i4 4.1502 4.6690 5-1877 .8526 1-7053 2-5579 3.4106 4.2632 5.1158 5.9685 6.8211 7.6738 8.5264 .5225 1.0450 1-5675 2.0900 2.6125 3-I350 3-6575 4.1800 4.7025 5.2250 .8504 1.7007 2.5511 3.4014 4.2518 5.1021 5-9525 6.8028 7.6532 8.5035 .5262 1.0524 1.5786 2.1049 2.6311 3-'573 3-6835 4.2097 4-7359 5.2621 I .8480 1.6961 2.5441 3.3922 4.2402 5-0883 5-9363 6.7844 7-6324 8.4805 .5299 1.0598 1.5898 *"1 19 1 2.6496 3-J795 3.7094 4.2394 4-7693 5.2992 1 2 3 4 5 6 7 8 9 10 D. D. Dep. Laf. Dep. Lat. Dep. Lat. Dep. Lat. 58f Deg. 58* Deg. 58 1 Deg. 58 Deg. 12 LATITUDES AND DEPARTURES. D. 1 2 3 4 5 6 7 8 9 10 321 Deg. 32* Deg. 32f Deg. 33 Deg. D. 1 2 3 4 5 6 7 8 9 10 Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. .8457 1.6915 2.5372 3.3829 4.2286 5.0744 5.9201 6.7658 7.6116 8-4573 .5336 1.0672 1.6008 2.1345 2.6681 3.2017 3-7353 4.2689 4.8025 5.33 6 i .8434 1.6868 2.5302 3-3736 4.2170 5.0603 5-937 6.7471 7'595 8.4339 5373 1.0746 1.6119 2.1492 2.6865 3.2238 3-76ii 4.2984 4-8357 5.3730 .8410 1.6821 2.5231 3.3642 4.2052 5.0462 5-8873 6.7283 7.5694 8.4104 .5410 1.0819 1.6229 2.1639 2.7049 3-2458 3.7868 4-3278 4.8688 5.4097 -8387 1.6773 2.5160 3-3547 4-1934 5.0320 5.8707 6.7094 7.5480 8.3867 .5446 1.0893 1.6339 2.1786 2.7232 3.2678 3.8125 4-3571 4.9018 5-4464 ~T 2 3 4 5 6 7 8 9 10 57i Deg. 57* Deg. 571 Deg. 57 Deg. 331 Deg. 33* Deg. 33f Deg. 34 Deg. 8363 1.6726 2.5089 3-345 1 4.1814 5.0177 5.8540 6.6903 7.5266 8.3629 5483 1.0966 1.6449 2.1932 2.7415 3.2898 3.8381 4.3863 4.9346 5.4829 .8339 1.6678 2.5017 3-3355 4.1694 5-33 5-837* 6.6711 7-55 8-3389 55i9 1.1039 1.6558 2.2077 2.7597 3.3116 3.8636 4-4I55 4-9674 5-5*94 8315 1.6629 2.4944 3-3259 4-1573 4.9888 5.8203 6.6518 7.4832 8.3147 5556 I.IIII 1.6667 2.2223 2.7779 3-3334 3.8890 4.4446 5.0001 5-5557 .8290 1.6581 2.4871 3.3162 4.1452 4.9742 5-8033 6.6323 7.4613 8.2904 5592 1.1184 1.6776 2.2368 2.7960 3-3552 3.9144 4-4735 5.0327 5-59I9 1 2 3 4 5 6 7 8 9 10 56| Deg. 56* Deg. 561 Deg. 56 Deg. 341 Deg. 34* Deg. 34| Deg. 35 Deg. 1 2 3 4 5 6 7 8 9 1O .8266 1.6532 2.4798 3.3064 4.1329 4-9595 5.7861 6.6127 7-4393 8.2659 .5628 1.1256 1.6884 2.2512 2.8140 3.3768 3.9396 4.5024 5.0652 5.6280 .8241 1.6483 2.4724 3.2965 4.1206 4.9448 5.7689 6-593 7.4171 8.2413 .5664 1.1328 1.6992 2.2656 2.8320 3-3984 3.9648 4-53*2 5.0977 5.6641 .8216 1-6433 2.4649 3.2866 4.1082 4.9299 5-75I5 6.5732 7.3948 8.2165 .5700 1.1400 1.7100 2.2800 2.8500 3.4200 3.9900 4.5600 5.1300 5.7000 .8192 1.6383 2 -4575 3.2766 4.0958 4.9149 5-7341 6.5532 7-3724 8.1915 .5736 1.1472 1.7207 2.2943 2.8679 3-44I5 4.0150 4.5886 5.1622 5-7358 1 2 3 4 5 6 7 8 9 10 55| Deg. 55* Deg. 551 Deg. 351 Deg. 55 Deg. 351 Deg. 353 Deg. 36 Deg. 1 2 3 4 5 6 7 8 9 10 .8166 1-6333 2 ' 4 1^ 3.2666 4.0832 4.8998 5-7165 6-533 1 7.3498 8.1664 5771 i- I 543 i-73 J 4 2.3086 2.8857 3.4629 4.0400 4.6172 5-'943 5-77I5 .8141 1.6282 2.4423 3-^565 4.0706 4.8847 5.6988 6.5129 7.3270 8.1412 .5807 1.1614 1.7421 2.3228 2.9035 3.4842 4.0649 4.6456 5.2263 5.8070 .8116 1.6231 2-4347 3-2463 4.0579 4.8694 5.6810 6.4926 7.3042 8.1157 .5842 1.1685 1.7527 2.3370 2.9212 3-5055 4.0897 4.6740 5.2582 5.8425 .8090 1.6180 2.4271 3.2361 4.0451 4.8541 5.6631 6.4721 7.2812 8.0902 5878 1.1756 1.7634 2.3511 2.9389 3.5267 4.1145 4.7023 5.2901 5-8779 1 2 3 4 5 6 7 8 9 10 D. Dep. Lat. Dep. Lat. Dep. Lat. 1 Dep. Lat. D. 54i Deg. 54* Deg. i 541 Deg. 54 Deg. LATITUDES AND DEPARTURES. D. 1 2 3 4 5 6 7 8 i 9 10 ~~T 2 3 4 5 6 7 8 9 10 361: Deg. 361 Deg. 36f Deg. 37 Deg. D. 1 2 3 4 5 6 7 S 9 10 Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. .8064 1.6129 2.4193 3.2258 4.0322 4-8387 5-645I 6.4516 7.2580 8.0644 59'3 1.1826 1-7739 2.3652 2-95 6 5 3-5479 4.1392 4-73 5 5.3218 5-9J3 1 .8039 1.6077 2.4116 3-2154 4.0193 4.8231 5.6270 6.4309 7-2347 8.0386 5948 1.1896 1.7845 2-3793 2-9741 3-5689 4.1638 4.7586 5-3534 5.9482 .8013 1.6025 2.4038 3-2050 4.0063 4.8075 5.6088 6.4100 7.2113 8.0125 5983 1.1966- 1.7950 2-3933 2.9916 3-5899 4.1883 4.7866 5-3849 5-9832 .7986 '5973 2.3959 3-1945 3.9932 4.7918 5-5904 6.3891 7.1877 7.9864 .6018 1.2036 1.8054 2.473 3.0091 3.6109 4.2127 4.8145 5-4 l6 3 6.0181 53f Deg. 53^ Deg. 53 i Deg. 53 Deg. ~T 2 3 4 5 6 7 8 9 10 37i Deg. 371 Deg. 371 Deg. 38 Deg. .7960 1.5920 2.3880 3.1840 3.9800 4.7760 5-57* 6.3680 7.1640 7.9600 6053, i. 2106 1.8159 2.4212 3.0265 3-6318 4.2371 4.8424 5-4476 6.0529 7934 1.5867 2.3801 3>I Hi 3.9668 4.7601 5-5535 6.3468 7.1402 7-9335 .6088 1.2175 1.8263 2.4350 3-0438 3.6526 4.2613 4.8701 5-4789 6.0876 .7907 1.5814 2.3721 3.1628 3-9534 4.7441 5-5348 6-3255 7.1162 7.9069 .6122 1.2244 1.8367 2.4489 3.0611 3-6733 4.2855 4-8977 5.5100 6.1222 .7880 1.5760 2.3640 3.1520 3.9401 4.7281 5.5161 6.3041 7.0921 7.8801 .6157 1-2313 1.8470 2.4626 3.0783 3.6940 4.3096 4-9253 I' 54 ^ 6.1566 52| Deg. 521 Deg. 52i Deg. 52 Deg. 39 Deg. 1 2 3 4 5 6 7 8 9 1O 38i Deg. 381 Deg. 38f Deg. 1 , 2 3 4 5 6 7 8 9 10 7853 1.5706 2.3560 3 ' 14 il 3.9266 4.7119 5-4972 6.2825 7.0679 7-8532 .6191 1.2382 1.8573 2.4764 3-0955 3-7I46 4-3337 4.9528 5.5718 6.1909 .7826 1.5652 2.3478 3.1304 3.9130 4.6956 5-4783 6.2609 7.0435 7.8261 .6225 1.2450 1.8675 2.4901 3.1126 3-735 1 4-3576 4.9801 5.6026 6.2251 7799 1.5598 2.3397 3-"95 3.8994 4.6793 5-4592 6.2391 7.0190 7-7988 .6259 1.2518 1.8 77 8 2.5037 3.1296 3-7555 4-3815 5-oo74 5-6333 6.2592 .7771 1-5543 2-33H 3.1086 3-8857 4.6629 5.4400 6.2172 6-9943 7-77I5 .6293 1.2586 1.8880 2 -5'73 3.1466 3-7759 4.4052 5.0346 5.6639 6.2932 51| Deg. 39i Deg. 511 Deg. 5H Deg. 51 Deg. 391 Deg. 39f Deg. 40 Deg. 1 2 3 4 5 6 7 8 9 1O 7744 1.5488 2.3232 3.0976 3.8720 4.6464 5.4207 *- 19 : 51 6.9695 7-7439 .6327 1.2654 1.8981 2.5308 3- l6 35 3.7962 4.4289 5.0616 5-6943 6.3271 .7716 1-5432 2.3149 3.0865 3.8581 4.6297 5.4014 6.1730 6.9440 7.7162 Dep. .6361 1.2722 1.9082 2-5443 3.1804 3.8165 4.4525 5.0886 5-7247 6.3608 .7688 i-5"377 2.3065 3-0754 3.8442 4.6131 5-38i9 6.1507 6.9196 7.6884 6394 1.2789 1.9183 .2-5578 3.1972 3.8366 4.4761 5-"55 5-755 6-3944 .7660 1.5321 2.2981 3.0642 3.8302 4-5963 5-3623 6.1284 6.8944 7.6604 .6428 1.2856 1.9284 2.5712 3.2139 3-8567 4-4995 5-i4 2 3 5-7851 6.4279 1 2 3 4 5 6 7 8 9 10 D. Dep. Lat. Lat. Dep. Lat. Dep. Lat. D 50| Deg. 501 Deg. 50J Deg. 50 Deg. LATITUDES AND DEPARTURES. D. 40i Deg. 40 Deg. 40f Deg. 41 Deg. D. 1 2 3 4 5 6 7 8 9 10 Lat. Dep. Lat. Dep. Lat. - Dep. Lat. Dep. 1 2 1 3 4 5 6 7 8 9 10 .7632 1.5265 2.2897 3.0529 3.8162 4-5794 5.3426 6.1059 6.8691 7.6323 .6461 1.2922 1.9384 2.5845 3.2306 3.8767 4.5229 5.1690 5.8151 6.4612 .7604 1.5208 2.2812 3.0416 3.8020 4.5624 5.3228 6.0832 6.8437 7.6041 .6494 1.2989 1.9483 2.5978 3-2472 3.8967 4.5461 5-I956 5.8450 6.4945 .7576 1.5151 2.2727 3.0303 3-7878 4-5454 5-3030 6.0605 6.8181 7-5756 .6528 1-3055 1-9583 2.6110 3.2638 3.9166 4-5693 5.2221 5.8748 6.5276 7547 1.5094 2.2641 3.0188 3-7735 4-5283 5.2830 6.0377 6.7924 7-5471 .6561 1.3121 1.9682 2.6242 3-2803 3-9364 4.5924 5.2485 5-9045 6.5606 ~T ! 2 3 , 4 5 6 7 8 9 10 49| Deg. 49 J Deg. 41 J Deg. ~ 49i Deg. 49 Deg. 1 2 3 4 5 6 7 8 9 10 4H Deg. 41f Deg. 42 Deg. .7518 i-S>37 2-^555 3.0074 S-759 2 4.5110 5.2629 6.0147 6.7666 7.5184 6 593 1.3187 1.9780 2.6374 3.2967 3-956I 4.6154 5.2748 5-9341 6-5935 .7490 1.4979 2.2469 2.9958 3-7448 4-4937 5.2427 5.9916 6.7406 7.4896 .6626 1.3252 1.9879 2.6505 3-3I3 1 3-9757 4.6383 5 - 3 ^ 5.9636 6.6262 .7461 1.4921 2.2382 2.9842 3-73 3 4.4763 5.2224 5.9685 6 '7'45 7.4606 .6659 1.3318 1.9976 2.6635 3-3294 3-9953 4.6612 5-327I 5-9929 6.6588 743 1 1.4863 2.2294 2.9726 3-7I57 4.4589 5.2020 5-9452 6.6883 743*4 .6691 I-3383 2.0074 2.6765 3-3457 4.0148 4.6839 5-353 6.0222 6.6913" ~T 2 3 4 5 6 7 8 9 10 48f Deg. 48 J Deg. 48 i Deg. 48 Deg. 42| Deg. 42 Deg. 42| Deg. 43 Deg. .7402 1.4804 2.2207 2.9609 3.7011 4.4413 5-1815 5-9217 6.6620 7.4022 .6724 1-3447 2.0171 2.6895 3.3618 4.0342 4.7066 5-3789 6.0513 6.7237 7373 1.4746 2. 2Il8 2.9491 3.6864 4-4237 5.1609 5.8982 6-6355 7.3728 .6756 1.3512 2.0268 2.7024 3-378o 4-535 4.7291 5-4047 6.0803 6-7559 , ,$!! 2.2030 2.9373 3.6716 4.4059 5'4| 5.8746 6.6089 7-3432 .6788 i'3576 2.0364 2.7152 3-3940 4.0728 4.7516 5-4304 6.1092 6.7880 7|i.4 1.4627 2.1941 2.9254 3.6568 4.3881 5-"95 5-8508 6.5822 7-3 J 35 .6820 1.3640 2.0460 2.7280 3.4100 4.0920 4.7740 5.4560 6.1380 6.8200 1 2 3 4 5 6 7 8 9 1O 1 2 3 4 5 6 2 9 10 471 Deg. 47i Deg. 47i Deg. 47 Deg. 44 Deg. 43 J Deg. 43 J Deg. 43f Deg. .7284 1.4567 2.1851 2-9135 3.6419 4.3702 5.0986 5.8270 6-5553 7.2837 .6852 I-3704 2.0555 2.7407 3-4259 4.11 ii 4.7963 5-4^5 6.1666 6.8518 7254 1.4507 2.1761 2.901.5 3.6269 4.3522 5.0776 5.8030 6.5284 7-2537 .6884 1.3767 2.0651 2-7534 3.4418 4.1301 4.8185 5.5068 6.1952 6-8835 Lat. .7224 1.4447 2.1671 2.8895 3.6118 4-3342 5-0565 5-7789 6.5013 7.2236 .6915 1.3830 2.0745 2.7661 3-4576 4.1491 4.8406 5-5321 6.2236 6.9151 .7193 1-4387 2.1580 2.8774 3-5967 4.3160 5-0354 5-7547 6.4741 7-1934 6947 1-3893 2.0840 2.7786 3-4733 4.1680 4.8626 5-5573 6.2519 6.9466 1 2 3 4 5 6 7 8 9 10 1 D. Dep. Lat. Dep. Dep. Lat. Dep. Lat. D. 46f Deg. 46* Deg. 46J Deg. 46 Deg. 15 3 3 AND DEPARTURES. jATITUDE, D. 44J Deg. 44J Deg. 44f Deg. 45 Deg. D. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 1 .7163 .6978 7133 .7009 .7102 .7040 .7071 .7071 1 2 1.4326 1.3956 1.4265 1.4018 1.4204 1.4080 1.4142 1.4142 2 1 3 4 2.1489 2.8652 2.0934 2.7912 2.1398 2.8530 2.1027 2.8036 2.1306 2.8407 2.II20 2.8161 2.1213 2.8284 2.1213 2.8284 3 4 5 3-58I5 3.4890 3.5663 3-545 3-559 3.5201 3-5355 3-5355 5 6 4.2978 4.1867 4.2795 4.2055 4.2611 4.2241 4.2426 4.2426 6 7 5.0141 4.8845 4.9928 4.9064 4-97!3 4.9281 4-9497 4-9497 7 8 9 5-7304 6.4467 5-5823 6.2801 5.7060 6.4193 5.6073 6.3082 5.6815 6.3917 5.6321 6.3361 5.6569 6.3640 5.6569 6.3640 1 9 1O 7.1630 6.9779 7.1325 7.0091 7.1019 7.0401 7.0711 7.0711 10 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. D. D. 45f Deg. 45 * Deg. 451 Deg. 45 Deg. TABLE OF USEFUL NUMBERS. Logarithms. Ratio of circumference to diameter n = 3.1415916536 0.4971499 Area of circle to radius i = " " Surface of sphere to diameter i = " " Area of circle to diameter i = 7853981634 1.8950899 Base of Napierian Logarithms = 2.7182818285 4342945 Modulus of common " = 4342944819 1.6377843 Equatorial radius of the earth, in feet = 20923599.98 7.3206364 Polar " = 20853657.16 7.3191823 Length of seconds pendulum, in London, in inches = 39.13929. " " " Paris = 39.1285. " " " New York =39.1012. U. S. standard gallon contains 231 c. in., or 58372.175 grains = 8.338882 Ibs. avoir- dupois of water at 39.8 Fahr. U. S. standard-bushel contains 2150.42 c. in., or 77.627413 Ibs. av. of water at 39.8 Fahr. British imperial gallon contains 277.274 c. in., = 1.2003 w i ne gallons of 231 c. in. French metre = 39.37079 in. = 3.28089917 feet. " toise = 6.39459252 feet. u are = 100 sq. metres = 1076.4299 sq. ft. " hectare = 100 ares = 2.471143 acres = 107642.9936 sq. ft. " litre = I cubic decimeter = 61.02705 c. in. =.26418637 gallons of 231 c. in. " hectolitre = 100 litres = 26.418637 gallons. 1 pound avoirdupois = 7000 grs. = 1.215277 pounds Troy. I " Troy = 5760 grs. = .822857 pounds avoir. i gramme = 15.442 grains. i kilogramme = 1000 grammes = 15442 grs. = 2.20607 tt>s. avoir. Tropical year = 365 d. 5 h. 45 m. 47.588 sec. 1G TABLE OF THE LOGARITHMS OF NUMBERS, FROM 1 TO 10,000. ir A TABLE OS THE LOGARITHMS OF NUMBERS FKOM 1 TO 10,000. N. Log. N. Log. N. Ug. N. Log. 1 o.oooooo 26 1.414973 51 1.707570 76 1.880814 2 3 0.301030 0.477121 0.602060 27 28 29 1.431364 1.447158 1.462398 52 53 54 1.716003 1.724276 1.732394 17 78 79 1.886491 1.892095 1.897627 5 0.698970 30 1.477121 55 1.740363 80 1.903090 6 0.778151 31 1.491362 56 1.748188 81 1.908485 7 0.845098 32 1.505150 57 1-755875 82 1.913814 8 0.903090 33 1.518514 58 1.763428 83 1.919078 9 o-954 2 43 34 i-53 I 479 59 1.770852 84 1.924279 10 1. 000000 35 1.544068 60 1.778151 85 1.929419 11 1-041393 36 i-55 6 33 61 1-78533 86 1.934498 12 1.079181 37 1.568202 62 1.792392 87 13 IJ 3943 38 1.579784 63 I-79934 1 88 1.944483 14 15 .146128 .176091 39 40 1.591065 1.602060 64 65 i. 806180 1.812913 89 90 1.949390 I-954H3 16 .204120 41 1.612784 66 1.819544 91 1.959041 17 18 .230449 .255273 42 43 1.623249 1.633468 67 68 1.826075 1.832509 92 93 1.963788 1.968483 19 .278754 44 I-643453 69 1.838849 94 1.973128 20 .301030 45 1.653213 70 1.845098 95 1.977724 21 .322219 46 1.662758 71 1.851258 96 1.982271 22 .342423 47 1.672098 72 1.857332 97 1.986772 23 .361728 48 1.681241 73 i 863323 98 1.991226 24 .380211 49 1.690196 74 i 869232 99 1.995635 25 .397940 50 1.698970 75 1.875061 100 2.OOOOOO 19 N. 100. LOGARITHMS. Log. 000. N. 1 2 3 4 5 6 7 8 9 100 oooooo 0434 0868 1301 1734 2166 2598 3029 3461 3891 1 101 ^321 4751 5181 5609 6038 6466 6894 7321 7748 8174 102 103 600 012837 9026 3 2 59 945i 3680 9876 4100 3oo 4521 0724 4940 '147 5360 '570 5779 J 993 6197 8 4'5 6616 104 7033 745 1 7868 8284 8700 9116 9532 9947 3 6i 775 105 021189 1603 2016 2428 2841 3252 3664 475 4486 4896 106 107 5306 9384 57J5 9789 6125 i 9 5 6533 6oo 6942 '004 7350 '408 7757 >8l2 8164 221 6 8571 2619 8978 3 02I 108 109 033424 7426 3826 7825 4227 8223 4628 8620 5029 9017 543 9414 5830 9811 6230 207 6629 %02 7028 110 111 041393 5323 1787 57H 2182 6105 2576 6 n 4 2 5 2969 6885 3362 7275 3755 7664 4148 8 12 3 4540 8442 4932 8830 112 9218 9606 9993 0380 0766 J 53 J 53 8 2309 2694 113 053078 3463 3846 4230 4613 4996 5378 5760 6142 6524 114 6905 7286 7666 8046 8426 8805 9185 95 6 3 9942 0320 115 060698 1075 1452 1829 2206 2582 2958 3333 3709 4083 116 4458 4832 5206 558o 5953 6326 6699 7071 7443 7815 117 8186 8557 8928 9298 9668 0038 0407 0776 145 '514 118 071882 2250 2617 2985 335 2 3718 4085 445 * 4816 5182 119 5547 5912 6276 6040 7004 7368 7731 8094 8457 8819 120 079181 9543 9904 266 0626 0987 '347 1707 2067 2426 121 082785 3503 3861 4219 4576 4934 5291 5647 6004 122 6360 6716 7071 7426 8136 8490 8^45 9198 955 2 123 995 0258 6n 0963 ? V5 '667 2018 2 7 2I 3 o 7 i 124 093422 3772 4122 4820 5169 55 l8 5866 6215 6562 125 6910 7257 7604 795 1 8298 8644 8990 9335 9681 0026 126 100371 0715 1059 1403 1747 2091 2434 2777 3119 3462 127 3804 4146 4487 4828 5 l6 9 5510 5851 6191 6531 68 7 i 128 7210 7549 7888 8227 8565 8903 9241 9579 9916 0253 129 110590 0926 1263 '599 1934 2270 2605 2940 3^75 3609 130 "3943 4277 4611 4944 5^78 5611 5943 6276 6608 6940 131 7271 7603 7934 8265 8926 9256 9586 99*5 0245 132 120574 0903 1231 1560 _ O O I OOO 2216 2 544 2871 3198 35 2 5 133 385* 4178 454 4830 5156 5481 5806 6131 6456 6 7 8i 134 7105 7429 7753 8076 8399 8722 9045 9368 9690 OI2 135 130334 0655 0977 1298 1619 1939 2260 2580 2900 3219 136 3539 3858 4177 4496 4814 5i33 545 1 5769 6086 6403 137 6721 737 7354 7671 7987 8303 8618 934 9249 9564 138 9879 i94 5o8 822 1136 '450 *7^3 2076 2389 2 7 02 139 143015 33 2 7 3 6 39 3951 4263 4574 4885 5196 5507 5818 140 146128 6438 6748 7058 7367 7676 7985 8294 8603 8911 141 9219 9527 9835 I42 449 o 75 6 1370 '676 1982 142 152288 2594 2900 3205 3510 3815 4120 4424 4728 5032 143 5336 5640 5943 6246 6549 6852 7154 7457 7759 8o6l 144 8362 8664 8965 9266 9567 9868 i68 0469 0769 '068 145 161368 1667 1967 2266 2564 2863 3161 3460 3758 4055 146 4353 4650 4947 5244 5541 5838 6134 6430 6726 7 O22 147 73*7 7613 7908 8203 8497 8792 9086 9380 9 6 74 9968 148 170262 0555 0848 1141 1434 1726 2019 2311 2603 2895 149 3186 3478 3769 4060 435 1 4641 4932 5222 55 12 5802 150 176091 6381 6670 6959 7248 753 6 7825 8113 8401 8689 151 8977 9264 9552 9839 I26 0413 0699 0986 12 7 2 1 55 8 152 181844 2129 2415 2700 2985 3270 3555 3839 4 I2 3 4407 153 4691 4975 5 2 59 5542 5^5 6108 6391 6674 6956 7239 154 7521 7803 8084 8366 8647 8928 9209 9490 977i 0051 155 190332 0612 0892 1171 1451 1730 2010 2289 2567 2846 156 3403 3681 3959 4 2 37 45H 4792 5069 5346 5623 157 158 5900 8657 8932 6453 9206 6729 9481 7005 9755 7281 7556 0303 7832 577 8io 7 0850 8382 '124 159 201397 1670 1943 2216 2488 2761 333 335 3577 3848 N. 1 2 3 4 5 6 7 8 9 20 N. 160. LOGARITHMS. Log. 204. ; N. O i 2 3 4 5 6 7 8 9 | 160 204120 439 1 4663 4934 5204 5475 574 6 6016 6286 6556 i 161 6826 7096 7365 7634 7904 8441 8710 8979 9*47 162 9515 9783 0051 0586 853 1388 '654 '921 163 212188 2454 2720 2986 3*5* 35i8 3783 4049 43 J 4 4579 164 4844 5109 5373 5638 5902 6166 6430 6694 6957 7221 165 7484 7747 8010 8*73 8536 8798 9060 93*3 9585 9846 j 166 220108 0370 0631 0892 "53 1414 1675 1936 2196 2456 167 2716 2976 3*3 6 349 6 3755 4015 4*74 4533 479* 5051 168 539 55 6 8 5826 6084 6342 6600 6858 7"5 737* 7630 169 7887 8144 8400 8657 8913 9170 9426 9682 9938 o, 93 170 230449 0704 0960 1215 1470 1724 1979 2234 2488 2742 171 2996 3250 354 3757 4011 4264 45*7 4770 5*3 5276 172 55*8 6033 6285 6537 6789 7041 7292 7 Q 544 7795 173 8046 8*97 8548 8799 9049 9*99 955 9800 0300 174 240549 0799 1048 i*97 I54 6 1795 2044 2293 2541 2790 175 3038 3*86 3534 3782 4030 4*77 45*5 477* 5019 5266 176 55*3 5759 6006 6252 6499 6745 6991 7*37 7482 77*8 177 7973 8219 8464 8709 8954 9198 9443 9687 993* 0176 178 179 250420 *853 0664 3096 0908 3338 1151 1395 3822 1638 4064 1881 4306 2125 4548 2368 4790 2610 5031 180 255273 55H 5755 ~5996 6237 6477 6718 6958 7198 7439 181 7679 7918 8158 8398 8637 8877 9116 9355 9594 9833 182 260071 0310 0548 0787 1025 1263 1501 1739 1976 2214 183 2451 2688 2925 3162 3399 3636 3873 4109 434 6 4582 184 4818 554 5290 55*5 5761 599 6 6232 6467 6702 6937 185 7172 7406 7641 7875 8110 8344 8578 8812 9046 9*79 186 95i3 9746 9980 *i3 44^ 6 7 9 9I2 ! >44 , '377 ! 6o9 187 271842 2074 2306 *538 2770 3001 3*33 34 6 4 3696 39*7 188 189 f 8 6462 6692 4620 0921 4850 7151 5081 7380 53" 7609 554* 7838 577* 8067 6002 8296 6232 85*5 190 278754 8982 9211 9439 9667 9895 I23 35' 57 8 8o6 191 281033 1261 1488 1715 1942 2169 2396 2622 2849 375 192 193 33 01 5557 35*7 5782 3753 6007 3979 6232 4205 6456 443 1 6681 4656 6905 4882 7130 5107 7354 533* 7578 194 7802 8026 8249 8473 8696 8920 9'43 9366 9589 9812 195 290035 0257 0480 0702 0925 "47 1369 1591 1813 2034 196 2256 2478 2699 2920 33 6 3 3584 3804 4025 4246 197 rr 4687 4907 5127 5347 55 6 7 5787 6007 6226 6446 198 6665 6884 7104 73*3 754* 7761 7979 8198 8416 8635 199 8853 9071 9289 9507 97*5 9943 i6i o 37 8 595 8i3 200 301030 i*47 1464 1681 1898 2114 2331 *547 2764 2980 201 3196 3628 3844 4059 4*75 449 1 4706 49* i 5136 202 535 1 5566 5781 5996 6211 6425 6639 6854 7068 7282 203 7496 7710 79*4 8i37 8351 8564 8778 8991 9204 9417 204 9630 9843 0056 268 4 8i 0693 0906 n8 1330 '54* 205 3"754 1966 2177 2389 2600 2812 3023 3*34 3445 3656 206 3867 4078 4289 4499 4710 4920 5*3 534 5551 5760 207 597 6180 6390 6599 6809 7018 7227 743 6 7646 7854 208 8063 8272 8481 8689 8898 9106 93*4 95** 9730 9938 209 320146 0354 0562 0769 0977 1184 I39 1 1598 1805 2OI2 210 322219 2426 T6 33 2839 3046 ~3*5* 3458 3665 3871 4077 211 4282 4488 4694 4899 5105 53 10 5516 57*i 5926 6131 212 6336 6541 6 745 6950 7*55 7359 75 6 3 7767 797* 8176 213 8380 8583 8787 8991 9194 9398 9601 9805 oo8 2II 214 330414 0617 0819 1022 1225 1427 1630 1832 2034 2236 215 2438 2640 2842 344 3246 3447 3 6 49 3850 4051 i 2 ! 3 216 4454 4 6 55 4856 557 5*57 5458 5^58 5859 6059 6260 217 T*3~ 6460 6660 6860 7060 7260 7459 7659 7858 8058 8*57 218 8456 8656 8855 9054 9*53 945 i 9650 9849 o 047 2 4 6 219 340444 0642 0841 1039 i*37 H35 1632 1830 2028 2225 ! N. 1 2 3 * 5 6 7 8 9 21 W. 220. XiOaARITHZKES. Log. 342. N. 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N. 1 2 3 4 5 6 7 8 9 760 880814 0871 0928 0985 1042 1099 1156 1213 1271 1328 i 761 1385 1442 1499 1556 1613 1670 1727 1784 1841 1898 762 1955 2OI2 2069 2126 2183 2240 2297 2354 2411 2468 763 2525 2 5 8l 2638 2695 2752 2809 2866 2923 2980 337 764 393 3*5 3207 3264 3321 3377 3434 3491 3548 3605 i 765 3661 3718 3775 3832 3888 3945 4002 4059 4" 5 4172 766 4229 4285 4342 4399 4455 4512 4569 4625 4682 4739 767 4795 4852 4909 4965 5022 5078 5 J 35 5192 5248 535 768 5361 5418 5474 5587 5 6 44 5757 5813 5870 769 5926 5983 6039 6096 6152 6209 6265 6321 6378 6 434 770 886491 6547 6604 6660 6716 6773 6829 6885 6942 6998 771 754 7111 7167 7223 7280 733 6 7392 7449 755 7561 772 7617 7674 773 7786 7842 7898 7955 8011 8067 8123 773 8179 8236 8292 8348 8404 8460 8516 8573 8629 8685 774 8741 8797 8853 8909 8965 9021 9077 9*34 9190 9246 775 9302 9358 9414 9470 9526 9582 9638 9694 9750 9806 776 9862 9918 9974 o 3 o o86 I97 253 39 0365 777 890421 0477 533 0589 0645 0700 0756 0812 0868 0924 778 0980 i35 1091 "47 1203 1259 13*4 1370 1426 1482 779 1537 1593 1649 1705 1760 1816 1872 1928 1983 2039 780 892095 2150 2206 2262 2317 2 373 2429 2484 2540 2595 781 2651 2707 2762 2818 2873 2929 2985 3040 3096 3151 782 3207 3262 3318 3373 34*9 3484 354 3595 3706 783 3762 3817 3873 3928 3984 439 4094 4150 4205 4261 784 4316 437 1 4427 4482 4538 4593 4648 4704 4759 4814 785 4870 4925 4980 5036 509 1 5146 5201 5257 5312 5367 786 5423 5478 5533 5588 5644 5699 5754 5809 5920 787 5975 6030 6085 6140 6195 6251 6306 6361 6416 6471 788 6526 6581 6636 6692 6747 6802 6857 6912 6967 7022 789 7077 7132 7187 7242 7297 7352 7407 7462 75 J 7 7572 790 897627 7682 7737 7792 7847 7902 7957 8012 8067 8122 791 8176 8231 8286 8396 8451 8506 8561 8615 8670 792 8725 8780 8835 8890 8944 8999 9054 9109 9164 9218 793 9*73 9328 9383 9437 9492 9547 9602 9656 97" 9766 794 9821 9875 993 9985 39 94 I49- 0258 0312 795 900367 0422 0476 0531 0586 0640 0695 0749 0804 0859 796 0913 0968 1022 1077 1131 1186 1240 1295 1349 1404 797 1458 1513 1567 1622 1676 J73 1 1785 1840 1894 1948 1 798 2003 2057 2112 2166 2221 2275 2329 2384 2438 2492 799 2 547 2601 2655 2710 2764 2818 2873 2927 2981 33 6 800 903090 3'44 3199 3*53 337 3361 3416 347 3524 3578 801 3 6 33 3687 374 1 3795 3849 394 3958 4012 4066 4120 802 4 J 74 4229 4283 4337 439 1 4445 4499 4553 4607 4661 803 4716 477 4824 4878 4932 4986 5040 594 5148 5202 804 5256 53 10 53 6 4 5418 5472 5526 558 5 6 34 5688 5742 805 5796 5850 594 5958 6012 6066 6119 6173 6227 6281 806 6 335 6389 6 443 6497 6551 6604 6658 6712 6766 6820 807 6874 6927 6981 735 7089 7143 7196 7250 734 7358 808 74" 7465 75*9 7573 7626 7680 7734 7787 7841 7895 809 7949 8002 8056 8110 8163 8217 8270 8324 8378 8431 810 908485 8539 8592 8646 8699 8753 8807 8860 8914 8967 811 9021 9074 9128 9181 9 2 35 9289 9342 9396 9449 953 812 9556 9610 9663 9716 977 9823 9877 993 9984 37 813 910091 0144 0197 0251 0304 0358 0411 0464 o5i8 0571 814 0624 0678 0731 0784 0838 0891 0944 0998 1051 1104 815 1158 I2II 1264 1317 1371 1424 1477 1530 1584 1637 816 1690 1743 1797 1850 1903 1956 2009 2063 2116 2169 817 2222 2275 2328 2381 2435 2488 2541 2594 2647 2700 ! 818 a 753 2806 2859 2913 2966 3019 3072 3125 3178 3231 1 819 3284 3337 339 3443 3496 3549 3602 3 6 55 3708 3761 1 N. 1 2 3 4 5 6 7 8 9 ai N. 820. LOGARITHMS. Log. 913. N. O 1 2 3 4 5 6 7 8 9 820 913814 3867 3920 3973 4026 4079 4 X 3 2 4184 4237 4290 821 822 4343 4872 4396 4925 4449 4977 4502 5030 4555 5083 4608 5i3 6 4660 5189 47i3 5*4! 4766 5294 4819 5347 823 5400 5453 555 5558 5611 5664 57i6 5769 5822 5875 824 5927 5980 6033 6085 6138 6191 6243 6296 6349 6401 825 6454 6507 6559 6612 6664 6717 6770 6822 6875 6927 826 6980 7033 7085 7138 7190 7*43 7295 7348 7400 7453 827 7506 7611 7663 7716 7768 7820 7873 7925 7978 828 829 8030 8555 8083 8607 8i35 8659 8188 8712 8240 8764 8293 8816 ii*; 8397 8921 8450 8973 8502 9026 830 919078 9130 9183 9*35 9287 9340 9392 9444 9496 9549 831 9601 9 6 53 9706 9758 9810 9862 9914 9967 oi9 o7i 832 920123 0176 0228 0280 0332 0384 0436 0489 0541 0593 833 0645 0697 0749 0801 0853 0906 0958 1010 1062 1114 834 1166 1218 1270 1322 1374 1426 1478 153 1582 1634 835 1686 1738 1790 1842 1894 1946 i 99 8 2050 2102 2154 836 2206 2258 2310 2362 2414 2466 2518 2570 2622 2674 837 2725 2777 2826 2881 ^933 2985 337 3089 3140 3192 838 3 2 44 3296 3348 3399 345i 3503 3555 3607 3658 3710 839 3762 3814 3865 3917 39 6 9 4021 4072 4124 4176 4228 840 924279 433 1 4383 4434 4486 4538 4589 4641 4693 4744 841 4796 4848 4899 495 * 5003 554 5106 5'57 5209 5261 842 si" 53 6 4 54M 5467 5518 557 5621 5673 5725 5776 843 5828 5879 593 1 5982 6034 6085 6137 6188 6240 6291 844 6342 6394 6445 6497 6548 6600 66 5 z 6702 6 754 6805 845 68 S7 6908 6959 7011 7062 7114 7165 7216 7268 7319 846 7370 7422 7473 75*4 7576 7627 7678 773 77 8l 7832 847 7883 7935 7986 8037 8088 8140 8191 8242 8293 8345 848 8 39 6 8447 8498 8549 8601 8652 8703 8754 8805 8857 849 8908 8959 9010 9061 9112 9163 9215 9266 93*7 9368 850 851 929419 993 9470 9981 9521 0032 9572 0083 9623 i34 9674 0185 9725 0236 9776 0287 9827 0338 ? 8 8 9 852 930440 0491 0542 0592 0643 0694 0745 0796 0847 0898 853 0949 1000 1051 1 102 1153 1204 1254 i35 1356 1407 854 1458 1509 1560 1610 1661 1712 1763 1814 1865 1915 855 1966 2017 2068 2118 2169 222O 2271 2322 2372 2423 856 2474 2524 *575 2626 2677 2727 2778 2829 2879 2930 857 2981 3031 3082 3 r 33 3183 3*34 3^85 3335 3386 3437 858 3487 3538 3589 3 6 39 3690 374 379i 3841 3892 3943 859 3993 4044 4094 4H5 4*95 4246 4296 4347 4397 4448 860 934498 4549 4599 4650 4700 475 * 4801 4852 4902 4953 861 5003 554 5104 5i54 5^05 5255 5306 5356 5406 5457 862 5507 5558 5608 5658 5709 F 5 : 9 5809 5860 5910 5960 863 6011 6061 6m 6162 6212 6262 6313 6363 6413 6463 864 6514 6564 6614 6665 6715 6765 6815 6865 6916 6966 865 7016 7066 7117 7167 7217 7267 7317 7367 7418 7468 866 7518 7568 7618 7668 7718 7769 7819 7869 7919 7969 867 8019 8069 8119 8169 8219 8269 8320 8370 8420 8470 868 8520 8570 8620 8670 8720 8770 8820 8870 8920 8970 869 9020 9070 9120 9170 9220 9270 9320 9369 9419 9469 870 9395*9 9569 9619 9669 9719 9769 9819 9869 9918 9968 871 940018 0068 0118 0168 0218 0267 0317 0367 0417 0467 872 0516 0566 0616 0666 0716 0765 0815 0865 0915 0964 873 1014 1064 1114 1163 1213 1263 I3'3 1362 1412 1462 874 1511 1561 1611 1660 1710 1760 1809 1859 1909 1958 875 2008 2058 2107 2157 2207 2256 2306 2 255 2405 2455 876 2504 2 554 2603 2653 2702 2752 2801 2851 2901 2950 877 3000 3049 3099 3148 3^8 3*47 3297 3346 3396 3445 878 3495 3544 3593 3 6 43 3692 3742 3791 3841 3890 3939 879 3989 4038 4088 4137 4186 4236 4285 4335 4384 4433 N. 1 2 3 4 5 6 7 8 9 32 N. 880. LOCfr/ 5. Log. 944. LRITIX1VES N. O 1 2 3 4 5 6 7 8 9 880 944483 453* 4581 4631 4680 47*9 4779 4828 4877 49*7 881 4976 5025 574 5124 5*73 5222 5*7* 5321 537 54 J 9 882 54 6 9 5518 55 6 7 5616 5665 5715 5764 5813 5862 5912 883 59^1 6010 6059 6108 6157 6207 6256 6305 6 354 6403 884 6452 6501 655 1 6600 6649 6698 6747 6796 6845 6894 885 6 943 6992 7041 7090 7140 7189 7238 7287 7336 7385 886 7434 7483 753* / J 7630 7679 77*8 7777 7826 7875 887 7924 7973 8022 8070 8119 8168 8217 8266 8315 8364 888 8413 8462 8511 8560 8609 8657 8706 8755 8804 8853 889 8902 8951 8999 9048 9097 9146 9'95 9*44 9292 890 949390 9439 9488 9536 9585 9 o 6 34 9683 9731 9780 9829 891 9878 9926 9975 024 0073 I 7 "219 0267 0316 892 0414 0462 0511 0560 0608 0657 0706 0754 0803 893 95 o8 S i 0900 0949 0997 1046 1095 1143 1192 1240 1289 894 1338 1386 H35 1483 153* 1580 1629 1677 1726 1775 895 1823 1872 1920 1969 2017 2066 2114 2163 221 1 2260 896 2308 2356 2405 *453 2502 *55 2599 2647 2696 2744 897 2792 2841 2889 2938 2986 334 3083 3i3i 3180 3**8 898 ! 899 3276 3760 33*5 3808 3373 3856 3905 3470 3953 3518 4001 3566 4049 36i5 4098 3663 4146 37" 4194 900 954*43 4*9 i 4339 4387 4435 4484 453* 4580 4628 4677 901 47*5 4773 4821 4869 4918 4966 5014 5062 5IIO 5158 902 5207 5*55 5303 5351 5399 5447 5495 5543 559* 5640 903 5688 573 6 5784 583* 5880 59*8 5976 6024 6072 6120 904 6168 6216 6265 6313 6361 6409 6 457 6505 6 553 6601 905 6649 6697 6745 6793 6840 6888 6936 6984 7032 7080 906 7128 7176 7224 7272 7320 7368 7416 7464 7512 7559 907 7607 7 6 55 7703 7799 7847 7894 794* 799 8038 908 8086 8134 8181 8229 8277 8373 8421 8468 8516 909 8564 8612 8659 8707 8755 8803 8850 8898 8946 8994 910 959041 9089 9137 9185 9232 9280 9328 9375 94*3 9471 911 9518 ? 5 9614 9661 9 fl 757 9804 985* 9900 9947 912 9995 0090 i 3 8 i85 28o 3*8 o 37 6 4*3 913 960471 0518 0566 0613 0661 0709 0756 0804 0851 0899 914 0946 0994 1041 1089 1136 1184 1231 i*79 1326 1374 915 1421 1469 1516 1563 1611 1658 1706 1753 i8or 1848 916 1895 J 943 1990 2038 2085 2132 2180 2227 2275 2322 917 2369 2417 2464 2511 *559 2606 2653 2701 2748 2795 918 919 2843 3316 2890 33 6 3 3410 2985 3457 33* 3504 379 355* 3126 3599 I 3221 3 6 93 3268 374i 920 963788 3835 3882 39*9 3977 4024 4071 4118 4165 4212 921 4260 437 4354 4401 4448 4495 454* 459 4637 4684 922 473 * 4778 4825 4872 4919 4966 5013 5061 5108 5J55 923 5202 5*49 5296 5343 539 5437 5484 5531 5578 5625 924 5672 5719 5766 5813 5860 5907 5954 6001 6048 6095 925 6142 6189 6236 6283 6329 6376 6423 6470 6517 6564 926 6611 6658 6705 6752 6799 6845 6892 6939 6986 733 927 7080 7127 7*73 7220 7267 7314 7361 7408 7454 7501 928 7548 7595 7642 7688 7735 7782 7829 7875 7922 7969 929 8016 8062 8109 8156 8203 8249 8296 8343 8390 8436 930 968483 853 8576 8623 8670 8716 "87^63 8810 8856 8903 931 8950 8996 9043 9090 9136 9183 9229 9276 93*3 93 6 9 932 9416 9463 9509 9556 9602 9 Q 6 49 9695 974* 9789 9835 933 9882 9928 9975 02I o68 i6i 2O7 *54 "300 934 970347 0393 0440 0486 0533 0579 0626 0672 0719 0765 935 0812 0858 0904 0951 0997 1044 1090 "37 1183 1229 936 1276 1322 1369 1415 1461 1508 1554 1601 1647 1693 937 1740 1786 1832 l8 79 1925 1971 2018 2064 2110 2157 938 2203 2249 2295 2342 2388 2434 2481 2527 *573 2619 939 2666 2712 2758 2804 2851 2897 2943 2989 335 3082 1 N. O 1 2 3 4 5 6 7 8 9 33 1ST. 940. IiOaARITHlYES. Log. 973. N. 1 2 3 4 5 6 7 8 9 940 941 973128 359 "jjj* 3220 3682 3266 3728 3313 3774 3359 3820 345 3866 345i 3913 3497 3959 3543 i 4005 i 942 4051 4097 4*43 4189 4 2 35 4281 43 2 7 4374 4420 4466 943 45i 2 4558 4604 4650 4696 4742 4788 4834 4880 4926 944 4972 5018 5064 5110 5156 5202 5248 5294 534 5386 945 543 2 5478 55*4 5570 5616 5662 5707 5753 5799 5845 946 5891 5937 5983 6029 6075 6121 6167 6212 6258 6304 947 6350 6396 6442 6488 6 533 6579 6625 6671 6717 6763 948 6808 6854 6900 6946 6992 737 7083 7129 7175 7220 949 7266 7312 7358 743 7449 7495 754 1 7586 7632 7678 950 977724 7769 7815 7861 7906 7952 7998 8043 8089 8i35 951 8181 8226 8272 8317 8363 8409 8454 8500 8546 8591 952 8637 8683 8728 8774 8819 8865 8911 8956 9002 9047 953 9093 9138 9184 9230 9275 9321 9366 9412 9457 9503 954 9548 9594 9 6 39 9685 973 9776 9821 9867 9912 9958 955 980003 0049 0094 0140 0185 0231 0276 0322 0367 0412 956 0458 0503 0549 0594 0640 0685 0730 0776 0821 0867 957 0912 0957 1003 1048 1093 1139 1184 1229 1275 1320 958 1366 l ft l 1456 1501 1547 1592 1637 1683 1728 J 773 959 1819 1864 1909 '954 2000 2045 2090 2135 2181 2226 ~960" 982271 2316 2362 2407 2452 2497 2 543 2588 2633 2678 961 2723 2769 2814 2859 2904 2949 2994 3040 3085 3130 962 3*75 3220 3265 3310 3356 3401 3446 349i 353 6 358i 963 3626 3671 3716 3762 3807 3852 3897 394 2 3987 4032 964 4077 4122 4167 4212 4^57 4302 4347 4392 4437 4482 965 4527 4572 4617 4662 4707 4752 4797 4842 4887 493 2 966 4977 5022 5067 5112 5'57 5202 5*47 5292 5337 5382 967 5426 547i 55i 6 55 6 i 5606 5651 5696 574 1 5786 583 968 5875 5920 59 6 5 6010 6055 6100 6144 6189 6234 6279 969 6324 6369 6413 6458 6503 6548 6593 6637 6682 6727 970 986772 6817 6861 6906 6951 6996 7040 7085 7130 7175 971 7219 7264 739 7353 7398 7443 7488 753 2 7577 7622 972 7666 77" 7756 7800 7845 7890 7934 7979 8024 8068 973 8"3 8157 8202 8247 8291 8336 8381 8425 8470 85H 974 8559 8604 8648 8693 8737 8782 8826 8871 8916 8960 975 9005 9049 9094 9138 9183 9227 9272 9316 9361 9405 976 977 9450 9895 9494 9939 9539 9983 9583 028 9628 072 9672 U7 9717 i6i 9761 0206 9806 0250 9850 *94 978 990339 0383 0428 0472 0516 0561 0605 0650 0694 0738 979 0783 0827 0871 0916 0960 1004 1049 1093 "37 1182 980 991226 1270 1315 1359 1403 1448 1492 I53 6 1580 1625 981 1669 1713 1758 1802 1846 1890 1935 1979 2023 2067 982 2III 2156 2200 2244 2288 2 333 2377 2421 2465 2509 983 *554 2598 2642 2686 2730 2774 2819 2863 2907 2951 984 2995 339 3083 3127 3172 3216 3260 334 3348 339 2 985 343 6 3480 3524 3568 3 6l 3 3657 3701 3745 3789 3833 986 3877 3921 39 6 5 4009 4053 4097 4141 4185 4229 4273 987 43 r 7 4361 445 4449 4493 4537 458i 4625 4669 4713 988 4757 4801 4845 4889 4933 4977 5021 5 65 5108 5152 989 5 J 9 6 5240 5284 53^8 5372 5416 5460 554 5547 5591 990 995635 5 6 79 5723 57 6 7 5811 5854 5898 5942 5986 6030 991 6074 6117 6161 6205 6249 6293 6 337 6380 6424 6468 992 6512 6 555 6599 6643 6687 6731 6774 6818 6862 6906 993 6949 6 993 737 7080 7124 7168 7212 7255 7299 7343 994 7386 743 7474 7517 7561 7605 7648 7692 7736 7779 995 7823 7867 7910 7954 7998 8041 8085 8129 8172 8216 996 8259 8303 8347 8390 8434 8477 8521 8564 8608 8652 997 8695 8739 8782 8826 8869 8913 8956 9000 9043 9087 998 9131 9*74 9218 9261 9305 9348 9392 9435 9479 9522 999 95 6 5 9609 9652 9696 9739 9783 9826 9870 9913 9957 ! N 1 2 3 4: 5 ~6~ 7 8 9 TABLE OF LOGARITHMIC SINES TANGENTS. 35 OCU [C 179 LRXTH1MU M. Sec. Sine. Tang. M. Sec. Sine. Tang. ~60 10 7.463725 7.463727 50 I 10 20 30 5-685575 5.986605 6.162696 5.685575 6.162696 50 40 30 10 20 30 70904 77966 _ 84915 70906 77968 84917 50 40 30 40 50 .287635 .384545 .287635 384545 20 10 40 50 QI7<4 735*4*7 7-598490 20 10 1 10 4637*6 .530673 .463726 530673 50 59 11 10 7.505118 11649 05120 11651 50 49 20 .588665 .588665 40 20 18083 18085 40 30 .619817 .639817 30 30 24423 24426 30 40 685575 685575 20 40 30672 30675 20 50 .726968 .726968 10 50 36832 36835 10 2 .764756 764756 ~58 12 42906 42909 48 10 .799518 .799518 50 10 48897 48899 50 20 .831703 .831703 40 20 54806 54808 40 30 .861666 .861666 30 30 60635 60638 30 40 .889695 .889695 20 40 66387 66390 20 50 .916024 .916024 10 50 72065 72068 10 3 .940847 .940847 57 13 77668 77671 47 10 .964328 .964329 50 10 83201 83204 50 20 6.986605 6.986605 40 20 88664 88667 40 30 7.007794 7.077794 30 30 9459 94062 30 40 50 27998 47303 27998 4733 20 10 40 50 7-599388 7.604652 7-599391 7.604655 20 10 4 65786 6 57 86 56 14 09853 09857 46 10 7.08351? 7.083515 50 10 H993 14996 50 20 7.100548 7.100548 40 20 20072 20076 40 30 16938 16939 30 30 25091 25097 30 40 3*733 3*733 20 40 ^co5 30060 20 50 47973 47973 10 50 34962 34968 10 5 1*696 62696 55 15 Kill % C ?20 45 10 76936 76937 50 10 44615 446*9 50 20 7.190725 7.190725 40 20 49361 49366 40 30 7.204089 7.204089 30 30 54056 54061 30 40 17054 17054 20 40 58701 58706 20 50 29643 29643 10 50 63297 63301 10 6 41877 41878 ~54 16 67845 67849 44 10 53776 53777 50 10 7*345 72350 50 20 30 65358 76639 6 I1 59 76640 40 30 20 30 76799 81208 76804 8l2I1 40 30 40 50 87635 7.298358 87635 7.298359 20 10 40 60 85573 89894 8557 89900 20 10 7 7.308824 7.308825 53 17 94173 94179 43 10 19043 19044 50 10 7.698410 7.698416 50 20 29027 29028 40 20 7.702606 7.7026l2 40 30 38787 38788 30 30 06762 06768 30 40 48332 48333 20 40 10879 10885 20 50 57672 57673 10 50 14957 14962 10 8 66816 66817 52 18 18997 19003 42 10 75770 7577* 50 10 22999 23005 50 20 84544 84546 40 20 26966 26972 40 30 7-393H5 7-393H 6 30 30 30896 30902 30 40 7.401578 7-4 OI 579 20 40 3479 1 34797 20 50 09850 09852 10 50 38651 38658 10 9 17968 17970 51 19 4*477 42484 41 10 25937 25939 50 10 46270 46277 50 20 33762 33764 40 20 50031 50037 40 30 41449 41451 30 30 53758 53765 30 40 49002 49004 20 40 57454 57462 20 50 56426 56428 10 50 61119 61127 10 10 7.463725 7.463727 50 20 7.764754 7.764761 4O Cosine. Cotang. Sec M. Cosine. Cotang. s^cT M. 90 89 36 SINES AND TANGENTS. 179 M. Sec. Sine. Tang. M. Sec. Sine. Tang. 20 7.764754 7.764761 40 30 7.940842 7.940858 30 10 68365 50 10 43*48 43265 50 20 71932 71940 40 20 45641 45 6 57 40 30 75477 75485 30 30 48020 48037 30 40 78994 79002 20 40 50387 50404 20 50 82482 82490 10 50 52741 5*758 10 21 85943 85951 39 31 55082 55100 29 10 89376 893H 50 10 574*0 574*8 50 20 30 92782 96162 92790 96170 40 30 20 30 597*7 62031 59745 62049 40 30 40 50 7-799515 7-80^843 7-7995*4 7.802852 20 10 40 50 66602 66621 20 10 22 06146 06155 ~38 32 68870 68889 28 10 20 09423 12677 09472 12686 50 40 10 20 71126 73370 7**45 73389 50 40 30 J 59O5 15915 30 30 75603 75622 30 40 19111 19120 20 40 77824 77844 20 50 22292 22302 10 50 80034 80054 10 23 *545* 25460 37 33 82233 82253 27 10 28586 28596 50 10 84421 84441 50 20 31700 31710 40 20 86598 86618 40 30 34791 348oi 30 30 88764 88785 30 40 37860 37870 20 40 90919 90940 20 50 40907 40918 10 50 93064 93085 10 "24 10 43934 4 6 939 43944 46950 50 ~36 ~34 10 95198 97322 95219 97343 50 26 20 499*4 49935 40 20 7-999435 7.999456 40 30 52888 52900 30 30 8.001538 8.001560 30 40 55833 55844 20 40 03631 03653 20 50 58757 58769 10 50 05714 05736 10 25 61662 61674 35 35 07787 07809 25 10 20 64548 67414 64560 67426 50 40 10 20 09850 11903 09872 11926 50 40 30 70262 70274 30 30 13947 13970 30 40 73092 73104 20 40 15981 16004 20 50 75902 759*5 10 50 18005 18029 10 ~26 78695 78708 in ~36 20021 20044 24 10 81470 81483 50 10 22027 22051 50 20 84228 84240 40 20 24023 24047 40 30 86968 86981 30 30 260II 26035 30 40 89690 89704 20 40 27989 28014 20 50 92396 92410 10 50 29959 29984 10 27 95085 95099 33 37 3I9I9 3*945 23 10 7-897758 7.897771 50 10 337I 33897 50 20 7.900414 7.900428 40 20 35840 40 30 40 50 03054 05678 08287 03068 05692 08301 30 20 10 30 40 50 37749 39675 41592 37775 39701 41618 30 20 10 28 10879 10894 ~32 38 435 01 435*7 22~ 10 13457 13471 50 10 45401 454*8 50 20 16019 16034 40 20 47*94 40 30 18566 18581 30 30 49178 49*05 30 40 21098 21113 20 40 5*054 51081 20 50 23616 23631 10 50 52922 5*949 10 29 26119 26134 31 39 54781 54809 /?! 10 28608 2862! 50 10 56633 56661 50 20 30 31082 33543 31098 33559 40 30 20 30 58477 60314 58506 60342 40 30 40 50 35989 38422 36006 38439 20 10 40 50 62142 63963 62171 63992 20 10 30 7.940842 7.940858 30 40 8.065776 8.065806 20 Cosine. Cotang. Sec. IT Cosine. Cotang. SecT M. 90 89 24 37 LOGARITHMIC 179 M. Sec. Sine. Tang. M. Sec. Sine. Tang. 40 8.065776 8.065806 20 50 8.162681 8.162727 10 10 67582 67612 50 10 64126 64172 50 20 69380 69410 40 20 65566 65613 40 30 71171 71201 30 30 67002 67049 30 40 7*955 72985 20 40 68433 68480 20 50 74731 74761 10 50 69859 69906 10 41 76500 76531 19 51 71280 71328 9 10 78261 78293 50 10 72697 72745 50 20 80016 80047 40 20 74109 74158 40 30 81764 81795 30 30 755*7 75566 30 40 83504 8 353 6 20 40 76920 76969 20 50 '85238 85270 10 50 78319 78368 10 ~4Z 86965 86997 ~18 ~52 79713 79763 8 10 88684 88717 50 10 81102 81152 50 20 90398 90430 40 20 82488 82538 40 30 40 50 92104 93804 95497 92137 93 8 37 9553 30 20 10 30 40 50 83868 85245 86617 83919 85296 86668 30 20 10 43 10 20 8.098863 8.100537 97217 8.098897 8.100571 50 40 17 53 10 20 87985 89348 90707 88036 89400 90760 50 40 7 30 02204 02239 30 30 92062 92115 30 40 03864 03899 20 40 934*3 93466 20 50 05519 5554 10 50 94760 94813 10 44 10 07167 08809 07202 08845 50 ^16 54 10 96102 97440 96156 97494 50 6 20 10444 10481 40 20 8.198774 8.198829 40 30 12074 IZIIO 30 30 8.200104 8.200159 30 40 13697 13734 20 40 01430 01485 20 50 '53*5 15352 10 50 02752 02808 10 45 16926 16963 15 55 04070 04126 5 10 20 18532 20131 18569 20169 50 40 10 20 05384 06694 05440 0.6750 50 40 30 21725 21763 30 30 08000 08057 30 40 23313 23351 20 40 09302 09359 20 50 24895 24933 10 50 10601 10658 10 ~46 8.126471 8.126510 ~14 56 11895 JI 953 4 10 28042 28081 50 10 13185 J 3 2 43 50 20 29606 29646 40 20 14472 14530 40 30 31166 31206 30 30 J5755 15814 30 40 50 32720 34268 32760 34308 20 10 40 50 17034 18309 17093 18369 20 10 47 35810 35851 13 57 8.219581 8.219641 3 10 37348 37389 50 10 20849 20909 50 20 38879 38921 40 20 22113 22174 40 30 40406 40447 30 30 23374 23434 30 40 41927 41969 20 40 24631 24692 20 50 43443 43485 10 50 25884 25945 10 ~48 10 44953 46458 44996 46501 50 ~12 58 10 27133 28380 27195 28442 50 ~2~ 20 47959 48001 40 20 29622 29685 40 30 49453 49497 30 30 30861 30924 30 40 50943 50987 20 40 32096 32160 20 50 52428 5 2 472 10 50 33328 33392 10 49 53907 53952 11 59 34557 34621 1 10 553^ 55426 50 10 3572 35846 50 20 56852 56896 40 20 37003 37068 40 30 58316 58361 30 30 38221 38286 30 40 50 59776 61231 59821 61276 20 10 40 50 39436 40647 395oi 4 7i3 20 10 50 8.162681 8.162727 10 60 8.241855 8.241921 Cosine. Cotang. Sec. M. Cosine. Cotang. Sel" M. 90 89 88 1 SINES AND TANaENTS. 178 M. Sec. Sine. Tang. M. Sec. Sine. Tang. O 8.241855 8.241921 60 10 8.308794 8.308884 50 10 3060 3126 50 10 8.309827 8.309917 50 20 4261 43*8 40 20 8.310857 8.310948 40 30 5459 55*6 30 30 1885 1976 30 40 6654 6721 20 40 2910 3002 20 50 7845 7913 10 50 3933 4025 10 1 8.249033 8.249101 59 11 4954 5046 49 10 8.250218 8.250287 50 10 597* 6065 50 20 1400 1469 40 20 6987 7081 40 30 2578 2648 30 30 8001 8095 30 40 3753 38*3 20 40 8.319012 8.319106 20 50 49*5 4996 10 50 8.320021 8.320115 10 2 6094 6165 58 12 1027 1122 48 10 7260 733 1 50 10 2031 2127 50 20 8423 8494 40 20 3033 3129 40 30 8.259582 8.259654 30 30 4032 4128 30 40 8.260739 8.260811 20 40 5029 5126 20 50 1892 1965 10 50 6024 6121 10 3 3042 3 JI 5 57 13 7016 7114 47 10 4190 4263 50 10 8007 8105 50 20 5334 5408 40 20 8995 8.329093 40 30 40 6 475 7613 ps 30 20 30 40 8.329980 8.330964 8.330080 1064 30 20 50 8749 8824 10 50 1945 2045 10 * 10 8.269881 8.271010 8.269956 8.271086 50 ~56 14 10 2924 3901 3025 4002 50 46 20 2137 2213 40 20 4876 4977 40 30 3260 3337 30 30 5848 595 30 40 4381 4458 20 40 6819 6921 20 50 5499 5576 10 50 7787 7890 10 5 6614 6691 55 15 8753 8856 45 10 77*6 7804 50 10 8.339717 8.339821 50 20 8835 8.278913 40 20 8.340679 8.340783 40 30 8.279941 8.280020 30 30 1638 1743 30 40 8.281045 1124 20 40 2596 2701 20 50 "45 2225 10 50 3 6 57 10 6 3*43 33*3 ~54 16 45 4 4610 44~ 10 4339 4419 50 10 545 6 5562 50 20 543 * 40 20 6405 6512 40 30 40 6521 7608 6602 7689 30 20 30 40 735* 8*97 7459 8405 30 20 50 8692 8774 10 50 8.349240 8.349348 10 7 8.289773 8.289856 53 17 8.350181 8.350289 43 10 8.290852 8.290935 50 10 1119 1229 50 20 1928 2O I 2 40 20 2056 2166 40 30 3002 3086 30 30 2991 3101 30 40 4073 4157 20 40 39*4 435 20 50 5141 5226 10 50 4855 4966 10 8 6207 6292 ~52 18 5783 5f95 42 10 7270 7355 50 10 6710 6823 50 20 8330 8416 40 20 7635 7748 40 30 8.299388 8.299474 30 30 8558 8671 30 i 40 8.300443 8.300530 20 40 8-359479 8-359593 20 50 1496 1583 10 50 8.360398 8.360512 10 9 2546 2633 51 19 1315 1430 41 10 3594 3682 50 10 2230 *345 50 20 4639 47*7 40 20 3H3 3*59 40 30 40 5681 6721 5770 6811 30 20 30 40 4054 4964 4171 5080 30 20 50 7759 7849 10 50 5871 5988 10 10 8.308794 8.308884 50 20 8.366777 8.366894 40 Cosine. Cotang. j Sec. M. Cosine. Cotang. SecT M. 91 88 39 1 LOGARITHMIC 178 M. Sec. Sine. Tang. M. Sec. Sine. Tang. 20 8.366777 8.366894 40 ~30 8.417919 8.418068 30 10 7681 7799 50 10 8722 8872 50 20 8582 8701 40 20 8.419524 8.419674 40 30 8.369482 8.369601 30 30 8.420324 8.420475 30 40 8.370380 8.370500 20 40 II2 3 1274 20 50 1277 1397 10 50 1921 2072 10 21 2171 2291 39 31 2717 2869 29 10 3063 3184 50 10 35" 3664 50 20 3954 4076 40 20 4304 4458 40 30 4 8 43 4965 30 30 5096 5250 30 40 573 5853 20 40 5886 6040 20 50 6615 6738 10 50 6675 6830 10 22 7499 7622 38 32 7462 7618 28 10 8380 8504 50 10 8248 8404 50 20 8.379260 8.379385 40 20 9032 9189 40 30 8.380138 8.380263 30 30 8.429815 8-4*9973 30 40 1015 1140 20 40 8.430597 8-43755 20 50 1889 2015 10 50 1377 i53 6 10 23 2762 2889 37 33 2156 2 3 J 5 27 10 3 6 33 3760 50 10 2 933 393 50 20 4502 4630 40 20 379 3870 40 30 5370 5498 30 30 4484 4645 30 40 6236 6364 20 40 5257 5419 20 50 7100 7229 10 50 6029 6191 10 24 7962 8092 36 ~3l 6800 6962 26 10 8823 8953 50 10 7569 773 2 50 20 8.389682 8.389812 40 20 8337 8500 40 30 40 8.390539 1395 8.390670 1526 30 20 30 40 9103 8.439868 8.439267 8.440033 30 20 50 2249 2381 10 50 8.440632 0797 10 25 3101 3*34 35 35 1394 1560 25 10 395i 4085 50 10 2155 2322 50 20 4800 4934 40 20 2915 3082 40 30 5 6 47 5782 30 30 3 6 74 3841 30 40 6493 6628 20 40 443 1 4599 20 50 7337 7472 10 50 5186 5355 10 ~26 8179 8315 34 36 594i 6110 24 10 9020 9156 50 10 6694 6864 50 20 8.399859 8.399996 40 20 7446 7616 40 30 8.400696 8.400834 30 30 8196 8367 30 40 i53 2 1670 20 40 8946 9117 20 50 2366 2505 10 50 8.449694 8.449866 10 27 3!99 3338 33 37 8.450440 8.450613 23 10 4030 4170 50 10 1186 J 359 50 20 4859 5000 40 20 1930 2104 40 30 5687 5828 30 30 2672 2847 30 40 6513 66 55 20 40 34*4 3589 20 50 7338 7480 10 50 4'54 433 10 28 8161 8304 32 38 4893 5070 22 10 8983 9126 50 10 5 6 3 X 5808 50 20 8.409803 8.409946 40 20 6368 6 545 40 30 8.410621 8.410765 30 30 7103 7281 30 40 H38 1583 20 40 7837 8016 20 50 2254 2 399 10 50 8570 8749 10 29 3068 3^13 31 39 8.459301 8.459481 21 10 3880 4026 50 10 8.460032 8.460212 50 20 4691 4837 40 20 0761 0942 40 30 55 5 6 47 30 30 1489 1670 30 40 6308 6456 20 40 2215 2398 20 50 7114 7262 10 50 2941 3 I2 4 10 30 8.417919 8.418068 30 40 8.463665 8.463849 20 Cosine. Cotang. Sec. M. Cosine. Cotang. Sec. M. 91 88 40 1 SINES AND TANGENTS. 178 M. Sec. Sine. Ts&g. M. Sec. Sine. Tang. 4O 8.463665 8.463849 20 50 8.505045 8.505267 10 10 20 4388 5110 4572 50 40 10 20 5702 6358 I 9 o 5 6582 50 40 30 5830 6016 30 30 7014 7238 30 40 6 55 6736 20 40 7668 7893 20 50 7268 7455 10 50 8321 8547 10 41 7985 8172 19 51 8974 9200 9 10 8701 8889 50 10 8.509625 8.509852 50 20 8.469416 8.469604 40 20 8.510275 8.510503 40 30 8.470129 8.470318 30 30 0925 II 53 30 40 0841 1031 20 40 1573 1802 20 50 1553 J 743 10 50 2221 2451 10 42 2263 2454 18 52 2867 3098 8 10 2971 3 l6 3 50 10 3513 3744 50 20 3 6 79 3871 40 20 4 J 57 4389 40 30 4386 4579 30 30 4801 534 30 40 50 5795 5^5 599 20 10 40 50 m 5677 6319 20 10 43 6498 6693 17 53 6726 6961 7 10 7200 739 6 50 10 7366 7602 50 20 7901 8097 40 20 8005 8241 40 30 8601 8798 30 30 8643 8880 30 40 9299 8-479497 20 40 9280 8.519517 20 50 8.479997 8.480195 10 50 8.519916 8.520154 10 44 8.480693 0892 16 ~54 8.520551 0790 ~6~ 10 1388 1588 50 10 1186 50 20 2082 2283 40 20 1819 2059 40 30 2775 2976 30 30 2451 2692 30 40 34 6 7 3669 20 40 3083 33*4 20 50 4158 4360 10 50 3713 395 6 10 45 4848 5050 15 55 4343 4586 5 10 5536 574 50 10 4972 50 20 6224 6428 40 20 5599 5844 40 30 6910 7115 30 30 6226 6472 30 40 794 6.76 6906 45 16 8468 6.02 4969 74 3499 6.76 6501 44 17 8829 6.01 49*5 395 6.75 6095 43 18 9190 6.01 4880 4310 6.75 5690 42 19 6.00 4836 4715 6.74 5*85 41 20 9.519911 6.00 479* 5"9 6.74 4881 40 21 9.520271 5-99 9.974748 9-5455*4 6.73 10.454476 39 22 23 0631 0990 5-99 5.98 473 4659 59*8 6331 6.73 6.72 3669 38 37 24 1349 5.58 4614 6735 6.72 3265 36 25 1707 5-97 457 7138 6.71 2862 35 26 2066 5-96 45*5 7540 6.71 2460 34 27 2424 5.96 4481 7943 6.70 2057 33 28 2781 5-95 443 6 8345 6.70 l6 55 32 29 3138 5-95 439' 74 8747 6.69 i*53 31 30 3495 5-94 4347 75 9149 6.69 0851 30 31 9.523852 5-94 9.974302 9550 6.68 0450 29 32 4208 5-93 4*57 9.549951 6.68 10.450049 28 33 4564 5-93 4212 9-5535* 6.67 10.449648 27 34 4920 5-9* 4167 0752 6.67 9248 26 35 5*75 5.91 4122 1152 6.66 8848 25 36 5630 5-9 1 4077 1552 6.66 8448 24 37 5984 5.90 4032 1952 6.65 8048 23 38 6 339 5-9 3987 6.65 7649 22 39 6693 5.89 394* 2750 6.65 7*5 21 40 7046 5-89 3897 3H9 6.64 6851 20 41 9.527400 5.88 9.973852 9-553548 6.64 10.446452 19 42 ' 7753 5.88 3807 394 6 6.63 6054 18 43 8105 5-87 3761 75 4344 6.63 5656 17 44 8458 5-87 3716 .76 474 1 6.62 5*59 16 45 8810 5.86 3671 5'39 6.62 4861 15 46 9161 5.86 3625 553 6 6.61 4464 14 47 95*3 5-85 5933 6.61 4067 13 48 9.529864 5-85 353^5 6329 6.60 3671 12 -49 9.530215 5.84 6725 6.60 3*75 11 50 0565 5-84 3444 7121 6.59 2879 10 51 9-539 I 5 5.83 9-973398 9-5575I7 6.59 10.442483 9 52 1265 5.82 335* 79 X 3 6.59 2087 8 53 1614 5.82 337 8308 6.58 1692 7 54 1963 5.81 3261 8702 6.58 1298 6 55 2312 5.81 3**5 9097 6.57 0903 5 56 2661 5.80 3169 949 * 6.57 0509 4 57 3009 5.80 3 I2 4 9-559885 6.56 10.440115 3 58 3357 5-79 3078 .76 9.560279 6.56 10.439721 2 59 374 5-79 3032 77 0673 6.55 93*7 1 60 9.534052 9.972986 9.561066 10.438934 Cosine. Diff. 1" Sine. Diff.l" Cotang. Diff. 1" Tang. M. 109 70 til 20 XiOaARITHlKXC 159 M. Sine. Diff. 1" Cosine. Diff.l" Tang. Diff 1" Cotang. 9.534052 5-78 9.972986 77 9.561066 6.55 10.438934 60 1 1 4399 2940 H59 6.54 8541 59 2 4745 5-77 2894 1851 6.54 8149 58 3 5092 5-77 2848 2244 6-53 7756 57 4 5438 2802 2636 6. 53 7364 56 5 5783 76 *755 3028 6. 53 6972 55 6 6129 5-75 2709 3419 6.52 6581 54 7 6474 5-74 2663 3811 6.52 6189 53 8 6818 5-74 2617 4202 6.51 5798 52 9 7163 5-73 2570 459* 6.51 5408 51 10 7507 5-73 2524 4983 6.50 5017 50 11 9-53785I 5-7* 9.972478 77 9-565373 6.50 10.434627 49 12 13 8194 8538 5-7* *43* *385 .78 5763 6l 53 6.49 6.49 4*37 3847 48 47 14 8880 5.71 2338 6542 6.49 3458 46 15 9223 5.70 2291 6932 6.48 3068 45 16 9565 5-7 2245 7320 6.48 2680 44 17 9.539907 5-69 2198 7709 6.47 2291 43 18 9.540249 5-69 2151 8098 6.47 1902 42 19 0590 5.68 2105 8486 6.46 1514 41 20 0931 5.68 2058 8873 6.46 1127 40 21 22 9.541272 1613 5-67 9.972011 1964 9261 9.569648 6-45 6.45 0739 10.430352 39 38 23 1953 5^66 1917 9-570035 6.45 10.429965 37 24 2293 5.66 1870 0422 6.44 9578 36 25 2632 5.65 1823 0809 6.44 9191 35 26 2971 5.65 1776 78 1195 6.43 8805 34 27 3310 1729 79 1581 6.43 8419 33 28 3 6 49 5-4 1682 1967 6.42 8033 32 29 3987 1635 2352 6.42 7648 31 30 43*5 5-63 1588 2738 6.42 7262 30 31 9.544663 5.62 9.971540 9.573123 6.41 10.426877 29 32 5000 5.62 H93 357 6.41 6 493 28 33 34 5338 5674 5.61 5.61 1446 1398 389* 4*76 6.40 6.40 6108 57*4 27 26 35 6011 5.60 I35i 4660 6-39 534 25 36 6347 5.60 1303 544 6-39 4956 24 37 6683 5-59 1256 54*7 6-39 4573 23 38 39 40 7010 7354 7689 5-59 5-5* 1208 1161 1113 79 5810 6193 6576 6.38 6.38 6.37 4190 3807 34*4 22 21 20 ^41 9.548024 5-57 9.971066 .80 9.576958 6-37 10.423042 19 42 8359 5-57 1018 7341 6.36 2659 18 43 8693 5.56 0970 77*3 6.36 2277 17 44 9027 5-56 0922 8104 6.36 1896 16 45 9360 5-55 0874 8486 6-35 I5H 15 46 9-549693 5-55 0827 8867 6-35 "33 14 47 9.550026 5-54 0779 9248 6.34 0752 13 48 359 5-54 0731 9-5796*9 6-34 10.420371 12 49 0692 5-53 0683 9.580009 6-34 10.419991 11 50 102^ 5-53 0635 0389 6-33 9611 10 61 9-55*356 5-5* 9.970586 9.580769 6-33 10.419231 9 52 1687 5.52 0538 1149 6.32 8851 8 53 2018 5-5* 0490 15*8 6.32 8472 7 54 *349 0442 1907 6.32 8093 6 55 2680 5-5i 0394 .80 2286 6.31 7714 5 56 3010 5-5 345 .81 2665 6.31 7335 4 57 58 334i 3670 5-50 5-49 0297 0249 343 34** 6.30 6.30 6957 6578 3 2 59 4000 5-49 O2OO .81 3800 6.29 6200 1 60 9-5543*9 9.970152 9-584I77 10.415823 Cosine. Diff. 1" Sine. Diff.l/ Cotang. Diff.l" Tang. M. 110 69 62 21 SINES AND TANOENTS. 158 M. Sine. Diff. 1" Cosine. Diff.l" Tang. Diff. V Cotang. 9-5543 2 9 5.48 9.970152 .81 9.584177 6.29 10.415823 60 1 2 4658 4987 5-47 0103 0055 4555 4932 6.29 6.28 544 J 5068 59 58 3 5315 5-47 9.970006 5309 6.28 4691 57 4 5 6 43 5.46 9.969957 5686 6.27 43H 56 5 597i 5.46 9909 6062 6.27 3938 55 6 6299 5-45 9860 6439 6.27 3561 54 7 6626 5-45 9811 6815 6.26 3 l8 5 53 8 6953 5-44 9762 7190 6.26 2810 52 9 10 7280 7606 5-44 5-43 9665 .81 7566 7941 6.25 6.25 M34 2059 51 50 11 9-55793J 5-43 9.969616 .82 9-588316 6.25 10.411684 49 12 8258 5-43 95 6 7 8691 6.24 1309 48 13 8583 5-4* 9518 9066 6.24 0934 47 14 8909 5-42 9469 944 6.23 0560 46 15 9*34 5-4 1 9420 9.589814 6.23 10.410186 45 16 9558 5.41 9370 9.590188 6.23 10.409812 44 17 9.559883 5-4 9321 0562 6.22 9438 43 18 9.560207 5-40 9272 935 6.22 9065 42 19 20 053 1 0855 5-39 5-39 9223 9173 1308 1681 6.22 6.21 8692 8319 41 40 21 9.561178 5.38 9.969124 9.592054 6.21 10.407946 39 22 1501 5-38 9075 2426 6. 20 7574 38 23 1824 5-37 9025 2798 6.20 7202 37 24 2146 5-37 8976 .82 6.20 6829 36 25 2468 5.36 8926 .83 3542 6.19 6458 35 26 2790 S-36 8877 39*4 6.19 6086 34 27 3112 5.36 8827 4285 6.18 5715 33 28 3433 5-35 8777 4656 6.18 5344 32 29 3755 5-35 8728 5027 6.18 4973 31 30 475 5-34 8678 5398 6.17 4602 30 31 32 9.564396 4716 5-34 5-33 9.968628 8578 9.595768 6138 6.17 6.16 10.404232 3862 29 28 33 5036 5-33 8528 6508 6.16 349 2 27 34 5356 5-3* 8479 6878 6.16 3122 26 35 5676 5-3 2 8429 7247 6.15 2753 25 36 5995 5.31 8379 7616 6.15 2384 24 37 6314 5.31 8329 7985 6.15 2015 23 38 6632 5-3 1 8278 83 8354 6.14 1646 22 39 6951 5-30 8228 .84 8722 6.14 1278 21 40 7269 5-3 8178 9091 6.13 0909 20 41 9-567587 5.29 9.968128 9459 6.13 0541 19 42 43 7904 8222 5.29 5.28 8078 8027 9.599827 9.600194 6.13 6.12 10.400173 10.399806 18 17 44 8539 5.28 7977 0562 6.12 9438 16 45 8856 5.28 7927 0929 6.ii 9071 15 46 9172 5- 2 7 7876 1296 6.ii 8704 14 47 48 9488 9.569804 5.27 5.26 7826 7775 1662 2029 6.1 1 6.10 8338 7971 13 12 49 9.570120 5-26 7725 2395 6.10 7605 11 50 0435 7674 2761 6.10 7^39 10 51 9-57075I 5- 2 5 9.967624 9.603127 6.09 10.396873 9 52 1066 5.24 7573 .84 3493 6.09 6507 8 53 1380 7522 .85 3858 6.09 6142 7 54 1695 5.23 4223 6.08 5777 6 55 2009 7421 4588 6.08 5412 5 56 2323 5- 2 3 737 4953 6.07 547 4 57 2636 5.22 73*9 53*7 6.07 4683 3 58 2950 5.22 7268 5682 6.07 4318 2 59 3263 5.21 7217 .85 6046 6.06 3954 1 60 9-573575 9.967166 9.606410 10.393590 Cosine. Diff. 1" Sine. Diff.l" Cotang. Diff. 1" Tang. M. 111 68 63 22 LOGA 157 RXTHMIC M. Sine. Diff. I" Cosine. Diff.l" Tang. Diff. V Cotang. I ~0 9-573575 5.21 9.967166 .85 9.606410 6.06 I-39359 60 1 3888 5.20 7115 6773 6.06 3227 59 2 4200 5.20 7064 7137 6.05 2863 58 ! 3 4 5 4512 4824 S 1 !^ 5-19 5-19 5.19 7013 6961 6910 7500 7863 8225 6.05 6.04 6.04 2500 2137 1775 57 56 55 6 5447 S .l8 6859 8588 6.04 1412 54 7 5758 5.18 6808 .85 8950 6.03 1050 53 8 9 10 6069 6379 6689 5-17 5-17 5.16 6756 6705 6653 .86 9312 9.609674 9.610036 6.03 6.03 6. 02 0688 10.390326 10.389964 52 51 50 11 9.576999 5 .l6 9.966602 0397 6. 02 10.389603 49 12 739 5.16 6550 0759 6.02 9241 48 13 7618 6499 1120 6.01 8880 47 14 7927 5- I 5 6447 1480 6.01 8520 46 15 8236 5.14 6 395 1841 6.01 8159 45 16 8545 5-H 6 344 2201 6.00 7799 44 17 8853 6292 2561 6.00 7439 43 18 9162 5- I 3 6240 2921 6.00 7079 42 19 9470 5- J 3 6188 3281 5-99 6719 41 20 9-579777 5.12 6136 .86 3641 5-99 6 359 40 21 9.580085 5.12 9.966085 .87 9.614000 5.98 10.386000 39 22 0392 5-" 6033 4359 5-98 5641 38 23 0699 5.11 4718 5-98 5282 37 24 1005 5928 5077 5-97 49 2 3 36 25 1312 5-io 5876 5435 5-97 45 6 5 35 26 1618 5.10 5824 5793 5-97 4207 34 27 1924 5.09 5772 6151 5.96 3849 33 28 2229 5.09 5720 6509 5.96 349 * 32 29 2 535 5-9 5668 6867 5.96 31 30 2840 5.08 5615 7224 5-95 2776 30 31 9-5 8 3 1 45 5.08 9.965563 9.617582 5-95 10.382418 29 32 3449 5.07 55" 7939 5-95 2061 28 33 3754 5-7 5458 8295 5-94 1705 27 34 4058 5.06 5406 .87 8652 5-94 1348 26 35 4361 5-06 5353 .88 9008 5-94 0992 25 36 37 4665 4968 S-o6 5.05 5248 e 93 6 4 9.619721 5-93 5-93 0636 10.380279 24 23 38 5272 5.05 9.620076 5-93 10.379924 22 39 5574 5-4 5*43 0432 5-9* 9568 21 40 5877 5.04 5090 0787 5-9 2 9213 20 41 9.586179 5-3 9.965037 9.621142 5.92 10.378858 19 42 6482 5-3 4984 H97 5-9 1 8503 18 43 6783 5-3 1852 5.91 8148 17 44 45 708) 7386 5.02 5.02 4879 4826 2207 2561 5-90 5-9 7793 7439 16 15 46 7688 5.01 4773 2915 5-9 7085 14 47 7989 5.01 4719 .88 1269 5-8 9 6731 13 48 8289 5.01 4666 .89 3623 5-89 6377 12 49 8590 5.00 4611 3976 5.89 6024 11 50 8890 5.00 4560 433 5.88 5670 10 51 9.589190 4.99 9.964507 9.624683 5.88 10.375317 9 52 9489 4-99 4454 5036 5-88 4964 8 53 9.589789 4-99 4400 5388 5-87 4612 7 54 9.590088 4.98 4347 5.87 4^59 6 55 0387 4.98 4294 6093 5.87 3907 5 56 0686 4-97 4240 6445 5.86 3555 4 57 0984 4-97 4187 6797 5.86 3203 3 58 1282 4-97 4133 7149 5.86 2851 2 59 1580 4.96 4080 .89 7501 5-85 2499 1 60 9.591878 9.964026 9.627852 10.372148 Cosine. Diff. I" Sine. Diff.1' Cotang. Diff. 1" Tang. M. 112 67 64. 23 SINES AND TANGENTS. 156 M. Sine. Diff. 1" Cosine. Diff.l" Tang. Diff. 1" Cotang. 9.591878 4.96 9.964026 .89 9.627852 5.85 10.372148 60 1 2176 4-95 397* .89 8203 1797 59 2 2 473 4-95 3919 .89 8554 5-85 1446 58 3 2770 4-95 3865 .90 8905 5-84 1095 57 4 3067 4-94 3811 9*55 5.84 0745 56 5 33 6 3 4-94 3757 9606 5-83 0394 55 6 7 3 6 59 3955 4-93 4-93 374 3650 9.629956 9.630306 5.83 10.370044 10.369694 54 53 8 4251 4-93 3596 0656 5-83 9344 52 9 4547 4.92 3542 1005 5.82 8995 51 10 4842 4.92 3488 1355 5.82 8645 50 11 9-595*37 4.91 9-9 6 3434 9.631704 5-82 10.368296 49 12 543 2 4.91 3379 2053 5 .8l 7947 48 13 57*7 4.91 33 2 5 2401 5 .8l 7599 47 14 6021 4.90 327! 2750 5 .8l 7250 46 15 63*5 4.90 3217 3098 5.80 6902 45 16 6609 4-89 3161 .90 3447 5 .8o 6 553 44 17 6903 4.89 3108 .91 3795 5.80 6205 43 18 7196 4.89 354 4H3 5-79 5857 42 19 749 4.88 2999 449 5-79 5510 41 20 7783 4.88 2945 4838 5-79 5162 40 n 9.598075 4.87 9.962890 9.635185 5.78 10.364815 39 22 8368 4.87 2836 553* 5.78 4468 38 23 24 8660 8952 4.87 4.86 2781 2727 till 5-78 5-77 4121 3774 37 36 25 9244 4.86 2672 6572 5-77 3428 35 26 9536 4-85 2617 6919 5-77 3081 34 27 9.599827 4.85 2562 7265 5-77 2735 33 28 9.600118 4.85 2508 7611 5.76 2389 32 29 30 0409 0700 4.84 4.84 H53 2398 .91 .92 7956 8302 5.76 5.76 2044 1698 31 30 31 9.600990 4.84 9.962343 9.638647 5-75 10.361353 29 32 1280 4.83 2288 8992 5-75 1008 28 33 1570 4.83 2233 9337 5-75 0663 27 34 1860 4.82 2178 9.639682 5-74 10.360318 26 35 2150 4.82 2121 9.640027 5-74 iQ-359973 25 36 2439 4.82 206 7 0371 5-74 9629 24 37 2728 4.81 2012 0716 5-73 9284 23 38 3017 4.81 1957 1060 5-73 8940 22 39 335 4.81 1902 i4-i 5-73 8596 21 40 3594 4.80 1846 1747 5.72 8253 20 41 9.603882 4.80 9.961791 9.642091 5.72 10.357909 19 42 4170 4-79 1735 2434 5.72 7566 18 43 4457 4-79 l68o .92 2777 5-7* 7223 17 44 4745 4-79 1624 93 3120 5-71 6880 16 45 ^5032 4.78 I56c 34 6 3 5-71 6 537 15 46 53*9 4-78 1511 3806 5-7 1 6194 14 47 5606 4.78 1458 4148 5-70 5852 13 48 5892 4-77 1402 4490 5.70 55 10 12 49 6179 4-77 I 34 6 4832 5-7 5168 11 50 6465 4.76 1290 5174 5.69 4826 10 51 9.606751 4.76 9.961235 9.645516 5.69 10.354484 9 52 7036 4.76 "79 5857 5-69 4'43 8 53 7322 4-75 1123 6199 5-69 3801 7 54 7607 4-75 1067 6540 5.68 3460 6 55 7892 4-74 IOII 6881 5.68 3"9 5 56 8177 4-74 0955 7222 5.68 2778 4 57 8461 4-74 0899 93 7562 5.67 2438 3' 58 8745 4-73 0843 94 7903 5.67 2097 2 59 9029 4-73 0786 94 8243 5.67 1757 1 60 9.60931-; 9.960730 9.648581 10.351417 Cosine. Diff. 1" Sine. Dial' Cotang. Diff. V Tang. M. 113 66 65 24 IiOGA 155 .RITHIMCIC M. Sine. Diff. 1" Cosine. Diff.l" Tang. Diff. 1" Cotang. 9.609313 4-73 9.960730 94 9.648583 5.66 10.351417 60 1 9597 4.72 0674 8923 5.66 1077 59 2 9.609880 4.72 0618 9263 5.66 0737 58 3 9.610164 4.72 0561 9602 5.66 0398 57 4 0447 4.71 0505 9.649942 5.65 10.350058 56 ' 5 0729 4.71 0448 9.650281 5.65 10.349719 55 6 JOI2 4-7 0392 0620 5.65 9380 54 7 1294 4.70 335 959 5.64 9041 53 8 I 57 6 4.70 0279 1297 5.64 8703 52 9 1858 4.69 0222 1636 5- 6 4 8364 51 10 2140 4.69 0165 94 1974 5.63 8026 50 11 9.612421 4.69 0109 95 9.652312 5.63 10.347688 49 12 2702 4.68 9.960052 2650 735 48 13 2983 4.68 9-959995 2988 5-63 7012 47 14 3264 4.67 9938 3326 5-6* 6674 46 15 3545 4.67 9882 3663 5.62 6337 45 16 38*5 4.67 9825 4000 5.62 6000 44 17 18 19 20 4105 4385 4665 4944 4.66 4.66 4.66 4.65 9768 97" 9654 9596 4337 4674 5011 5348 5.61 5.61 5.61 5.61 5663 5326 4989 4652 43 42 41 40 21 9.615223 4.65 9-959539 9.655684 5.60 10.344316 39 22 55 2 4.65 9482 6020 5.60 3980 38 23 5781 4.64 9425 6356 5.60 3 6 44 37 24 6060 4.64 9368 95 6692 5-59 3308 36 25 6338 4.64 9310 .96 7028 5-59 2972 35 26 6616 4.63 9*53 7364 5-59 2636 34 27 6894 4.63 9'95 7699 5-59 2301 33 28 7172 4.62 9138 8034 5.58 1966 32 29 745 4.62 9081 8369 5-58 1631 31 30 7727 4.62 9023 8704 5.58 1296 30 31 9.618004 4.61 9.958965 9.659039 5.58 10.340961 29 32 8281 4.61 8908 9373 5-57 0627 28 33 34 8558 8834 4.61 4.60 8850 8792 9.659708 9.660042 5-57 5-57 10.340292 10.339958 27 26 35 9110 4.60 8734 0376 5-57 9624 25 36 9386 4.60 8677 0710 5-56 9290 24 37 9662 4-59 8619 1043 8957 23 38 39 9.619938 9.620213 4-59 4-59 8561 8503 .96 97 1377 1710 5-56 5-55 8623 8290 22 21 40 0488 4.58 8445 2043 5-55 7957 20 41 0763 4-58 9-958387 9.662376 5-55 10.337624 19 42 1038 4-57 8329 2709 5-54 7291 18 43 1313 4-57 8271 3042 5-54 6958 17 44 1587 4-57 8213 3375 5-54 6625 16 45 1861 4.56 8154 377 5-54 6293 15 46 2135 4-56 8096 439 5-53 5961 14 47 2409 4.56 8038 437i 5-53 5629 13 48 2682 4-55 7979 473 5-53 5297 12 49 2956 4-55 7921 535 5-53 4965 11 50 3229 4-55 7863 5366 5-5* 4634 10 51 52 9.623502 3774 4-54 4-54 9.957804 7746 97 .98 9.665697 6029 5.52 5-5 2 10.334303 397 1 8 53 4047 4-54 7687 6360 5-5 1 3640 7 54 4319 4-53 7628 6691 339 6 55 459 1 4-53 7570 7021 5.51 2979 5 56 4863 4-53 7511 735 2 5-5 1 2648 4 57 5135 4.52 745* 7682 5-5 2318 3 58 5406 4.52 7393 8013 5-5 1987 2 59 5677 4-5* 7335 .98 8343 5-5 1657 1 60 9.625948 9.957276 9.668672 10.331328 Cosine. Diff. I" Sine. Diff.l" Cotang. Diff. 1" Tang. M. 114 65 66 25 SINES AND TANGENTS. 154 M. Sine. Diff. I" Cosine. Diff.l" Tang. Diff. 1" Cotang. 9.625948 4.51 9.957276 .98 9.668673 5-5 10.331327 60 1 6219 4.51 7217 9002 5-49 0998 59 2 6490 4.51 7158 933* 5-49 0668 58 3 6760 4-50 7099 9661 5-49 0339 57 4 7030 4-5 7040 9.669991 5.48 10.330009 56 5 7300 4-5 6981 .98 9.670320 5.48 10.329680 55 6 7 7570 7840 4.49 4-49 6921 6862 99 0649 0977 5-48 5-48 9351 9023 54 53 8 8109 4.49 6803 1306 5-47 8694 52 9 8378 4.48 6 744 1634 5-47 8366 51 10 8647 4.48 6684 1963 5-47 8037 50 11 9.628916 4-47 9.956625 9.672291 5-47 10.327709 49 12 9185 4-47 6566 2619 5.46 48 13 9453 4-47 6506 2947 753 47 14 9721 4.46 6447 3274 5-46 6726 46 15 9.629989 4.46 6387 3602 5.46 6398 45 16 9.630257 4.46 6327 39*9 5-45 6071 44 17 0524 4.46 6268 99 4*57 5-45 5743 43 18 0792 4-45 6208 1. 00 4584 5-45 42 19 1059 4-45 6148 4910 5-44 5090 41 20 1326 4-45 6089 5*37 5-44 4763 40 ~2l 9.631593 4-44 9.956029 9.675564 5-44 10.324436 39 22 1859 4.44 5969 5890 5-44 4110 38 23 2125 4-44 5909 6216 5-43 3784 37 24 2392 4-43 5849 6543 5-43 3457 36 25 2658 4-43 5789 6869 5-43 S'3 1 35 26 2923 4-43 57*9 7*94 5-43 2806 34 27 3189 4.42 5669 7520 2480 33 28 3454 4.42 5609 7846 5.42 2154 32 29 3719 4.42 5548 8171 1829 31 30 3984 4.41 5488 1. 00 8496 5.42 1504 30 ~3T 9.634249 4.41 9.955428 I.OI 9.678821 5.41 10.321179 29 32 45H 4.40 5368 9146 5-4 1 0854 28 33 4778 4.40 537 f 947i 5.41 0529 27 34 35 5042 5306 4.40 4-39 5*47 5186 9.679795 9.680120 5.41 5-4 10.320205 10.319880 26 25 36 557 4-39 5126 0444 5-4 9556 24 37 5834 4-39 5 6 5 0768 5-4 9232 23 38 6097 4-39 55 1092 5-4 8908 22 39 6360 4-38 4944 1416 5-39 8584 21 40 6623 4.38 4883 1740 5-39 8260 20 41 9.636886 4-37 9.954823 9.682063 5-39 10.317937 19 42 7148 4-37 4762 2387 5-39 7613 18 43 7411 4-37 4701 2710 7290 17 44 7 6 73 4-37 4640 333 "I 6967 16 45 7935 4.36 4579 ' I.OI 335 6 6644 15 46 8197 4-36 4518 1. 02 3679 5-38 6321 14 47 8458 4-3 6 4457 4001 5-37 5999 13 48 8720 4-35 439 6 43 2 4 5-37 5676 12 49 8981 4-35 4335 4646 5-37 5354 11 50 9242 4-35 4274 4968 5-37 10 51 52 9503 9.639764 4-34 4-34 9.954213 9.685290 5612 5.36 10.314710 4388 9 8 53 9.640024 4-34 4090 5934 5-36 4066 7 54 0284 4-33 4029 6255 5.36 3745 6 55 0544 4-33 3968 6577 5-35 34 2 3 5 56 0804 4-33 3906 6898 5-35 3102 4 57 1064 4-3* 3845 7219 5-35 2781 3 58 1324 3783 1.02 7540 5-35 2460 2 59 60 1583 9.641842 4-3 2 3722 9.953660 1.03 7861 9.688182 5-34 2139 10.311818 1 Cosine. Diff. l" Sine. Diff.l" Cotang. Diff. I" Tang. M. 115 64 26 LOGARITHMIC 153 M. Sine. Difif. I" Cosine. Difif. 1" Tang. Difif. 1" Cotang. 9.641842 4-31 9.953660 1.03 9.688182 5-34 10.311818 60 1 2101 4.31 3599 8502 5-34 I 49 8 59 , 2 3 2360 26l8 4.31 4-3 3537 3475 8823 9H3 5-34 5-33 1177 0857 58 57 4 2877 4.30 3413 9463 5-33 0537 56 5 4-3 335* 9.689783 5-33 10.310217 55 6 3393 4.30 3290 9.690103 5-33 10.309897 54 7 3 6 5 4.29 3228 0423 5-33 9577 53 8 3908 4.29 3166 0742 5-3 2 9258 52 9 4165 4.29 3104 1062 5-3* 8938 51 10 4423 4.28 3042 1-03 1381 5-3* 8619 50 11 12 9.644680 493 6 4.28 4.28 9.952980 2918 1.04 9.691700 2019 5-3 1 5-3 1 10.308300 7981 49 48 13 4.27 2855 2338 7662 47 14 5450 4.27 2793 2656 5-3 1 7344 46 15 5706 4.27 2731 2975 7025 45 16 17 5962 6218 4.26 4.26 2669 2606 3*93 3612 5-3 5-3 6707 6388 44 43 18 6474 4.26 2544 393 5-3 6070 42 19 6729 4.26 2481 4248 5-3 575 2 41 20 6984 4.25 2419 4566 5434 40 21 9.647240 4.25 9.952356 9.694883 5.29 10.305117 39 22 7494 4.24 2294 5201 5.29 4799 38 23 7749 4.24 2231 1.04 5518 5 >2 9 4482 37 24 8004 4.24 2168 1.05 5836 5- 2 9 4164 36 25 8258 4.24 2106 5.28 3847 35 26 8512 4.23 2043 6470 5.28 353 34 27 8766 4.23 1980 6787 5.28 3213 33 28 9020 4-23 1917 7103 5.28 2897 32 29 9274 4.22 . 1854 7420 2580 31 30 9527 4.22 1791 773 6 5-*7 2264 30 31 32 9.649781 9.650034 4.22 4.22 9.951728 1665 9.698053 8369 5-*7 10.301947 1631 29 28 33 0287 4.21 1602 8685 5^6 1315 27 34 539 4.21 1539 9001 5-26 0999 26 35 0792 4.21 1476 9316 5.26 0684 25 36 1044 4.20 1412 1.05 9632 5.26 0368 24 37 1297 4.20 1349 1. 06 9.699947 5.26 10.300053 23 38 1549 4.20 1286 9.700263 5-25 10.299737 22 39 1800 4.19 1222 0578 5.25 9422 21 40 2052 4.19 1159 0893 5-25 9107 20 41 9.652304 4.19 9.951096 9.701208 5.24 10.298792 19 42 2555 4.18 1032 15*3 5-^4 8477 18 43 2806 4.18 0968 1837 5.24 8163 17 44 357 4.18 0905 2152 7848 16 45 33 8 4.18 0841 2466 5-^4 7534 15 46 3558 4.17 0778 2780 5- 2 3 7220 14 47 3808 4.17 0714 395 5- 2 3 6905 13 48 4059 4.17 0650 3409 5- 2 3 6591 12 49 4309 4.16 0586 1. 06 37*3 5-23 6277 11 50 4558 4.16 0522 1.07 4036 5.22 5964 10 51 9.654808 4.16 9.950458 9.704350 5.22 10.295650 9 52 5058 4.16 0394 4663 5.22 5337 8 53 537 4.15 0330 4977 5.22 5023 7 54 5556 4.15 0266 5290 5.22 4710 6 55 5805 4.15 02O2 5603 5.21 4397 5 56 6054 4.14 0138 5916 5.21 4084 4 57 6302 4.14 0074 6228 5.21 377* 3 58 6551 4.14 9.950010 6541 5.21 3459 2 59 6799 4.13 9.949945 I.0 7 6854 5.21 3146 1 60 9.657047 9.949881 9.707166 10.292834 Cosine. Diff. I" Sine. Difif. V Cotang. Difif. I" Tang. M. 116 63 68 27 SINES AND TANGENTS. 152 M. Sine. Diff. I" Cosine. Diff.l" Tang. Diff. V Cotang. 9.657047 4.13 9.949881 1.07 9.707166 5-20 10.292834 60 1 7295 4-13 9816 1.07 7478 5.20 2522 59 2 7542 4.12 9752 1.07 7790 5.20 2210 58 3 779 4.12 9688 1. 08 8102 5.20 1898 57 : 4 8037 4.12 9623 8414 1586 56 i 5 8284 4.12 9558 8726 5.19 1274 55 6 853J 4.11 9494 937 5.19 0963 54 7 8778 4.11 9429 9349 5.19 0651 53 8 9025 4.11 9364 9660 5-19 0340 52 9 9271 4.10 9300 9.709971 5.18 IO.290029 51 10 9517 4.10 9235 9.710282 5 .!8 10.289718 50 11 9.659763 4.10 9.949170 593 5.18 9407 49 B 9.660009 4.09 9105 0904 5.18 9096 48 13 0255 4.09 9040 1215 5.18 8785 47 14 0501 4.09 8975 1525 8475 46 15 0746 4.09 8910 1836 5-17 8l6 4 45 16 0991 4.08 8845 1.08 2146 5- 1 ? 7854 44 17 1236 ' 4.08 8780 1.09 2456 7544 43 18 1481 4.08 8715 2766 5.16 7^34 42 19 1726 4.07 8650 3076 5.16 6924 41 20 1970 4.07 8584 3386 5.16 6614 40 21 9.662214 4.07 9.948519 9.713696 5.16 10.286304 39 22 2459 4.07 8454 4005 5-i6 5995 38 23 2703 4-06 8388 43 '4 5-15 5686 37 24 2946 4.06 8323 4624 5-15 5376 36 25 3190 4.06 8257 4933 5-J5 5067 35 26 3433 4.05 8192 5242 5.15 4758 34 27 3 6 77 4-5 8126 5-H 4449 33 28 3920 4.05 8060 1.09 5860 5.14 4140 32 29 4163 4.05 7995 1. 10 6168 5-H 3832 31 30 4406 4.04 7929 6477 5.14 35^3 30 31 9.664648 4.04 9.947863 9.716785 5-H 10.283215 29 32 4891 4.04 7797 7093 2907 28 33 5'33 4.03 7731 7401 5.13 2599 27 34 5375 4.03 7665 7709 5.13 2291 26 35 4.03 7600 8017 5- x 3 1983 25 36 5859 4.02 7533 83^5 5.13 1675 24 37 6100 4.02 7467 8633 5.12 1367 23 38 6342 4.02 7401 8940 5.12 1060 22 39 6583 4.02 7335 9248 5.12 0752 21 40 6824 4.01 7269 9555 5.12 0445 20 41 9.667065 4.01 9.947203 1. 10 9.719862 5.12 10.280138 19 42 7305 4.01 7136 i. ii 9.720169 5.11 10.279831 18 43 7546 4.01 7070 0476 5-" 9524 17 44 7786 4.00 7004 0783 5.11 9217 16 45 8027 4.00 6937 1089 5- 11 8911 15 46 8267 4.00 6871 1396 5.11 8604 14 47 8506 3-99 6804 1702 5.10 8298 13 48 8746 3-99 6738 2009 5.10 7991 12 49 50 8986 9225 3-99 3-99 6671 6604 2315 2621 5.10 5.10 7685 7379 11 10 51 9464 3.98 9.946538 9.7229*7 5.10 10.277073 9 52 53 9703 9.669942 3-98 6471 6404 3*3* 3538 5.09 5.09 6768 6462 8 7 54 9.670181 3-97 6337 i. ii 3844 5.09 6156 6 55 0419 3-97 6270 1. 12 4149 5.09 5851 5 56 0658 3-97 6203 4454 5.09 5546 4 57 0896 3-97 6136 4759 5.08 5241 3 58 "34 3-9 6 6069 5 6 5 5.08 4935 2 59 1372 6002 1. 12 5369 5.08 463* 1 60 9.671609 9-945935 9.725674 10.274326 Cosine. Diff. V Sine. Diff.l" Cotang. Diff. I" Tang. M. ! 117 62 26 69 28 LOGARITHMIC 151 M. Sine. Diff. 1" Cosine. Diff.l" Tang. Diff. 1" Cotang. - ~~o 9.671609 3-9 6 9-945935 1. 12 9.725674 5-08 10.274326 60 1 1847 3-95 5868 5979 5.08 4021 59 2 2084 3-95 5800 6284 5-7 3716 58 3 2 3 2J 3-95 5733 6588 5-7 34 12 57 4 2558 3-95 5666 6892 5-07 3108 56 5 2795 3-94 5598 7197 5.07 2803 55 6 3 2o 3-94 553i 1. 12 7501 5-07 2 499 54 7 3268 3-94 5464 I.I3 7805 5.06 2195 53 8 3505 3-94 539 6 8109 5.06 1891 52 9 3741 3-93 53^8 8412 5.06 1588 51 10 3977 3-93 5261 8716 5.06 1284 50 11 9.674213 3-93 9-945I93 9.729020 5.06 10.270980 49 12 4448 3-9* 51*5 93*3 5-5 0677 48 13 4684 3-9* 55 8 9626 5-5 0374 47 14 4919 3-9* 499 9.729929 5-5 10.270071 46 15 5J55 3.92 4922 9.730233 5-5 10.269767 45 16 5390 3-9 1 4854 535 5-5 9465 44 17 5624 3-9 1 4786 0838 5-4 9162 43 18 5859 3-9 1 4718 1141 5.04 8859 42 19 6094 3-9i 4650 I.I3 1444 5.04 8556 41 20 6328 3-9 4582 I.I 4 1746 5-4 8254 40 ~21 9.676562 3.90 9.944514 9.732048 5-4 10.267952 39 22 6796 3-9 4446 235i 5-3 7649 38 23 7030 3-9 4377 2653 5-3 7347 37 24 7264 3-89 439 2955 5-3 7045 36 25 749 8 3-89 4.241 3*57 5-03 6 743 35 26 773 1 3-89 4172 3558 5-03 6442 34 27 7964 3.88 4104 3860 5.02 6140 33 28 8197 3.88 4036 4162 5.02 5838 32 29 8430 3.88 39 6 7 44 6 3 5.02 5537 31 30 8663 3.88 3899 4764 5.02 5236 30 31 9.678895 3-87 9.943830 9.735066 5.02 10.264934 29 32 9128 3.87 3761 I.I4 5367 5.02 4 6 33 28 33 9360 3.87 3 6 93 I.I5 5668 5.01 433 2 27 34 959^ 3-87 3624 5969 5.01 4031 26 35 9.679824 3-86 3555 6269 5.01 373 1 25 % 36 9.680056 3.86 3486 6570 5.01 343 24 37 0288 3.86 3417 6871 5.01 3129 23 38 5!9 3-85 3348 7171 5.00 2829 22 39 0750 3.85 3279 747i 5.00 2529 21 40 0982 3-85 3210 7771 5.00 2229 20 41 9.681213 3.85 9.943141 9.738071 5.00 10.261929 19 42 H43 3.84 3072 8371 5.00 1629 18 43 1674 3.84 3003 8671 4-99 1329 17 44 1905 3.84 2 934 8971 4-99 1029 16 45 2135 3-84 2864 I-I5 9271 4-99 0729 15 46 2365 3-83 2 795 1.16 9570 4-99 0430 14 47 2595 3-83 2726 9.739870 4-99 10.260130 13 48 2825 3-83 2656 9.740169 4-99 10.259831 12 49 355 3-83 2587 0468 4.98 953 2 11 50 3284 3.82 2517 0767 4-98 9233 10 51 9.683514 3.82 9.942448 9.741066 4-98 10.258934 9 52 3743 3.82 2378 i3 6 5 4.98 863*; 8 53 3972 3.82 2308 1664 4-98 8336 7 54 4201 3.81 2239 1962 4-97 8038 6 55 443 3-8i 2169 2261 4-97 7739 5 56 4658 3.81 2099 2 559 4-97 7441 4 57 4887 3.80 2029 2858 4-97 7142 3 58 5U5 3.80 1959 1.16 3156 4-97 6844 2 59 60 f 5343 9.685571 3.80 1889 9.941819 1.17 3454 9.743752 4-97 6546 10.256248 1 Cosine. Diff. 1" Sine. Diff.l" Cotang. Diff. 1" Tang. M. 118 61 70 29 SINES AND TANGENTS. 150 M. Sine. Diff. 1" Cosine. Diff.l" Tang. Diff. I" Cotang. 9.685571 3.80 9.941819 1.17 9-743752 4.96 10.256248 60 1 5799 3-79 1749 4050 4.96 5950 59 2 6027 3-79 1679 4348 4.96 5 6 52 58 3 6254 3-79 1609 4645 4.96 5355 57 4 5 6482 6709 3-79 3.78 1539 1469 4943 5240 4.96 4-95 557 4760 56 55 6 6936 3-78 1398 5538 4-95 4462 54 7 7163 3.78 1328 5835 4-95 4165 53 8 7389 3-78 "1258 6132 4-95 3868 52 9 7616 3-77 1187 6429 4-95 357 1 51 10 7843 3-77 1117 I.I7 6726 4-95 3274 50 11 9.688069 3-77 9.941046 1.18 9.747023 4-94 10.252977 49 12 8295 3-77 975 7319 4-94 2681 48 13 8521 3.76 0905 7616 4-94 2384 47 14 8747 3-76 0834 79 T 3 4-94 2087 46 15 8972 3.76 0763 8209 4-94 1791 45 16 9198 3.76 0693 8505 4-93 *495 44 17 9423 3-75 0622 8801 4-93 1199 43 18 9648 3-75 0551 9097 4-93 0903 42 19 9.689873 3-75 0480 9393 4-93 0607 41 20 9.690098 3-75 0409 9689 4-93 0311 40 21 0323 3-74 9.940338 9.749985 4-93 10.250015 39 22 0548 3-74 0267 9.750281 4-93 10.249719 38 23 0772 3-74 0196 1.18 0576 4.92 9424 37 24 0996 3-74 0125 1.19 0872 4.92 9128 36 25 1220 3-73 9-940054 1167 4.92 8833 35 26 1444 3-73 9.939982 1462 4.92 8538 34 27 1668 3-73 9911 1757 4.92 8243 33 28 1892 3-73 9840 2052 4.91 7948 32 29 2115 3-72 9768 2347 4.91 7 6 53 31 30 *339 3-72 9697 2642 4.91 7358 30 ~31 9.692562 3-72 9.939625 9-75*937 4.91 10.247063 29 32 2785 3.71 9554 3231 4.91 6769 28 33 3008 3-7i 9482 3526 4.91 6474 27 34 3231 3-7i 9410 3820 4.90 6180 26 35 3453 3-7i 9339 1.19 4"5 4.90 5885 25 36 3676 3-7 9267 1.20 4409 4.90 559 1 24 37 3898 3-70 9'95 4703 4.90 5 2 97 23 38 4120 3-70 9123 4997 4.90 5003 22 39 4342 3-7 9052 5291 4-9 4709 21 40 4564 3.69 8980 5585 4.89 44i5 20 41 9.694786 3- 6 9 9.938908 9-755878 4.89 10.244122 19 42 5007 3- 6 9 8836 6172 4.89 3828 18 43 5229 3.69 8763 6465 4.89 3535 17 44 545 3.68 8691 6 759 4.89 324 1 16 45 5671 3.68 8619 7052 4.89 2948 15 46 5892 3.68 8547 7345 4.88 2655 14 47 6113 3.68 8475 1.20 7638 4.88 2362 13 48 6 334 3-67 8402 I.2I 7931 4.88 2069 12 49 6 554 3.67 8330 8224 4.88 1776 11 50 6775 3.67 8258 8517 4.88 1483 10 51 9.696995 3.67 9.938185 9.758810 4.88 10.241190 9 52 7215 3-66 8113 9102 4.87 0898 8 53 7435 3.66 8040 9395 4-87 0605 7 54 7654 3.66 7967 9687 4.87 0313 6 55 7874 3.66 7895 9-759979 4.87 10.240021 5 56 8094 3.65 7822 9.760272 4.87 10.239728 4 57 8313 3-65 7749 0564 4-87 943 6 3 58 853^ 3-65 7676 0856 4.86 9144 2 59 8751 3-65 7604 I.2I 1148 4.86 8852 1 60 9.698970 9.937531 9.761439 10.238561 Cosine. Diff. V Sine. Diff.l" Cotang. | Diff. 1" Taug. M. 119 60 71 30 LOaARITHMIC 149 M. | Sine. Diff. I" Cosine. Diff.l" Tang. Diff. 1" Cotang. 9.698970 3.64 9-93753 1 1. 21 9.761439 4-86 10.238561 60 1 9189 3- 6 4 7458 1.22 1731 4.86 8269 59 2 9407 3.64 7385 2023 4.86 7977 58 3 4 9626 9.699844 3- 6 4 3.63 7312 7238 2606 4.86 7686 7394 57 56 5 9.700062 3.63 7165 2897 4-85 7103 55 6 0280 3-63 7092 3188 4.85 6812 54 7 0498 3.63 7019 3479 4-85 6521 53 8 0716 3.63 6946 3770 4-85 6230 52 9 933 3.62 6872 4061 4.85 5939 51 10 "5 1 3.62 6799 435* 4.84 5648 50 TT 9.701368 3.62 9.936725 1.22 9.764643 4.84 10.235357 49 12 1585 3.62 6652 1.23 4933 4.84 5067 48 13 1802 3.61 6578 5224 4.84 4776 47 14 2019 3.61 6505 55H 4.84 4486 46 15 2*136 3.61 6431 5805 4-84 4195 45 16 2452 3.61 6357 6095 4.84 395 44 17 2669 3.60 6284 6385 4.83 3 6l 5 43 18 2885 3.60 6210 6675 4.83 33*5 42 19 3101 3.60 6136 6965 4.83 335 41 20 33*7 3.60 6062 7^55 4-83 2745 40 21 9-7 3533 3-59 9.935988 9.767545 4-83 10.232455 39 22 3749 3-59 59H 7834 4.83 2166 38 23 39 6 4 3-59 1.23 8124 4.82 1876 37 24 4 J 79 3-59 5766 1.24 8413 4.82 1587 36 25 4395 3-59 5692 8703 4.82 1297 35 28 4610 3-58 5618 8992 4.82 1008 34 27 28 4825 5040 3-58 5543 54 6 9 9281 957 4.82 4.82 0719 0430 33 32 29 30 5*54 5469 3-58 3-57 5395 5320 9.769860 9.770148 4.81 4.81 10.230140 10.229852 31 30 31 9.705683 3-57 9.935246 437 4.81 95 6 3 29 32 5898 3-57 5*71 0726 4.81 9274 28 33 6112 3-57 597 1015 4.81 8985 27 34 6326 3-56 5022 1303 4.81 8697 26 35 6 539 3.56 4948 1592 4.81 8408 25 36 6753 3 56 4873 1.24 1880 4.80 8120 24 37 6967 3.56 4798 1.25 2168 4.80 7832 23 38 7180 3-55 4723 2457 4.80 7543 22 39 7393 3-55 4649 2745 4.80 7255 21 40 7606 3-55 4574 333 4.80 6967 20 41 9.707819 3-55 9-934499 9.773321 4.80 10.226679 19 42 8032 3-54 4424 360? 4-79 6392 18 43 8245 3-54 4349 3896 4-79 6104 17 44 8458 3-54 4274 4184 4-79 5816 16 45 8670 3-54 4199 4-79 55 2 9 15 46 8882 3-53 4121 4759 4-79 5*41 14 47 9094 3-53 4048 5046 4-79 4954 13 48 9306 3-53 3973 1.25 5333 4-79 4667 12 49 9518 3-53 3898 1.26 5621 4-78 4379 11 50 973 3-53 3822 5908 4-78 4092 10 51 9.709941 3-52 9-933747 9.776195 4.78 10.223805 9 52 9.710153 3-5* 3671 6482 4.78 35*8 8 53 0364 3-5* 3596 6769 4.78 3231 7 54 575 3-5 2 3520 755 4-78 2945 6 55 0786 3-5i 3445 7342 4-78 2658 5 56 0997 3.51 33 6 9 7628 4-77 2372 4 57 1208 3.51 3*93 79*5 4-77 2085 3 58 14.19 3-5 1 3217 8201 4-77 1799 2 59 1629 3-5 1.26 8487 4-77 1513 1 60 9.711839 9.933066 9.778774 10.221226 Cosine. Diff. 1" Sine. Diff.]' Cotang. Diff. 1" Tang. M. 120 59 72 31 SINES AND TANGENTS. 148 M. Sine. Diff. 1" Cosine. Diff.l" Tang. Diff. V Cotang. 9.711839 3-50 9.933066 1.26 9.778774 4-77 10.221226 60 1 2050 3-50 2990 1.27 9060 4-77 0940 59 2 2260 3.50 2914 9346 4.76 0654 58 3 2469 3-49 2838 9632 4.76 03681 57 4 2679 3-49 2762 9.779918 4.76 10.220082! 56 5 2889 3-49 2685 9.780203 4.76 10.2197971 55 6 3098 3-49 2609 0489 4-76 9511 54 7 3308 3-49 2533 0775 4.76 9225! 53 8 3517 3-48 2457 1060 8940 i 52 9 3726 3-48 2380 1346 4-75 8654! 51 10 3935 3-48 2304 1631 4-75 8369 50 11 9.714144 3-48 9.932228 9-78I9I6 4-75 10.218084 49 12 435* 3-47 2151 1.27 22OI 4-75 7799 48 13 4561 3-47 2075 1.28 2486 4-75 75 J 4 47 14 4769 3-47 I 99 8 2771 4-75 7229 46 15 4978 3-47 1921 4-75 6944 45 16 5186 3-47 1845 3341 4-75 6659! 44 17 18 5394 5602 3$ 1768 1691 3626 3910 4-74 4-74 6374 6090 43 42 19 5809 3.46 1614 4'95 4-74 5805 41 20 6017 3.46 X 537 4479 4-74 55* 1 40 21 9.716224 3-45 9.931460 9.784764 4-74 10.215236 39 22 23 643* 6639 3-45 3-45 1383 1306 1.28 5048 533* 4-74 4-73 495* 4668 38 37 24 6846 3-45 1229 J.2 9 5616 4-73 4384 36 25 7053 3-45 1152 5900 4-73 4100 35 26 27 7*59 7466 3-44 3-44 1075 0998 6184 6468 4-73 4-73 3816 353* 34 33 28 29 7673 7879 3-44 3-44 0921 0843 6752 7036 4-73 4-73 3248! 32 2964! 31 30 8085 3-43 0766 4-7* 2681 30 31 9.718291 3-43 9.930688 9.787603 4.72 10.212397 29 32 8497 3-43 0611 7886 4.72 2114 28 33 8703 3-43 533 8170 4-7* 1830 27 34 8909 3-43 0456 8453 4-7* 1547! 26 35 9114 3-4* 0378 1.29 8736 4-7* 1264 25 36 9320 3.42 0300 1.30 9019 4.72 0981 24 37 95*5 3-4* 0223 9302 4.71 0698 23 38 9730 3.42 0145 9585 4.71 0415 22 39 9-7I9935 9.930067 9.789868 4.71 10.2101321 21 40 9.720140 3-4 1 9.929989 9.790151 4.71 10.209849 20 41 0345 "..S4* 9911 433 4.71 9567 19 42 0549 3-4 1 9833 0716 4.71 9284 18 43 0754 3-40 9755 0999 4.71 9001 17 44 0958 3-4 9677 1281 4.71 8719 16 45 1162 3-4 9599 X 5 6 3 4.70 8437 15 46 1366 3-4 95*i 1846 4.70 8154 14 47 1570 3-40 944* 1.30 2128 4.70 7872 13 48 1774 3-39 9364 2410 4.70 7590 12 49 1978 3-39 9286 2692 4.70 7308 11 50 2181 3-39 9207 2974 4.70 7026 10 51 9.722385 3-39 9.929129 9.793256 4-7 10.206744 9 52 2588 3-39 9050 3538 4.69 6462 8 53 2791 3-38 8972 3819 4.69 6181 7 54 2994 3-38 8893 4101 4.69 5899 6 55 3197 3-38 8815 4383 4.69 5617 5 56 3400 3.38 8736 4664 4.69 5336 4 57 3 6 3 3-37 8657 4945 4.69 555 3 58 3805 3-37 8578 5227 4.69 4773! 2 59 4007 3-37 8499 X -3 X 558 4.68 449* 1 60 9.724210 9.928420 9.795789 10.204211 Cosine. Diff. 1" Sine. Diff.l" Cotang. Diff. V Tang. M. 121 58 32 LOGARITHMIC 147 M. Sine. Diff. 1" Cosine. Diff.l' Tang. Diff. 1" Cotang. ~o 9.724210 3-37 9.928420 1.32 9.795789 4.68 10.204211 60 1 4412 3-37 8342 6070 4.68 393 C 59 2 461; 3-36 8263 6 35I 4.68 3 6 49 58 3 4816 3.36 iig; 6632 4.68 3368 57 4 5017 3-36 810; 6913 4.68 3087 56 5 5219 3-36 8025 7194 4.68 2806 55 6 5420 3-35 7946 7475 4-68 2525 54 7 5622 3-35 7867 7755 4.68 2245 53 8 5823 3-35 7787 8036 4.67 1964 52 9 6024 3-35 7708 8316 4.67 1684 51 10 6225 3-35 7629 8596 4.67 1404 50 11 9.726426 3-34 9.927549 1.32 9.798877 4.67 IO.2OI 123 49 12 6626 3-34 7470 '33 9157 4.67 0843 48 13 6827 3-34 739 9437 4.67 056-: 47 14 7027 3-34 7310 9717 4.67 028c 46 15 7228 3-34 7231 9-799997 4.66 I0.20000C 45 16 7428 3-33 7151 9.800277 4.66 10.199723 44 17 7628 3-33 7071 557 4.66 944; 43 18 7828 3-33 6991 0836 4.66 9164 42 19 8027 3-33 6911 1116 4.66 8884 41 20 8227 3-33 6831 1396 4.66 8604 40 21 9.728427 3-3 2 9.926751 9.801675 4.66 10.198325 39 22 8626 3-3 2 6671 J 955 4.66 8045 38 23 8825 3-3 2 6591 '33 2234 4-65 7766 37 24 9024 3-3* 6511 1.34 ^513 4.65 7487 36 25 9223 3-3 1 6431 2792 4.65 7208 35 26 9422 3-3 1 6 35* 3072 4.65 6928 34 27 28 9621 9.729820 3-3i 3-3 1 6270 6190 335i 3630 4.65 4.65 6649 6370 33 32 29 30 9.730018 0216 3-3 3-3 6no 6029 3908 4187 4-65 4.65 6092 58i3 31 30 31 0415 3-3 9.925949 9.804466 4.64 10.195534 29 32 0613 3-30 5868 4745 4.64 5 2 55 28 33 0811 3-3 5788 5023 4.64 4977 27 34 1009 3- 2 9 577 5302 4.64 4698 26 35 1206 3- 2 9 5626 i-34 558o 4.64 4420 25 36 1404 3- 2 9 5545 '35 5859 4.64 4141 24 37 1602 3- 2 9 54 6 5 6137 4.64 3863 23 38 1799 3- 2 9 5384 6415 4.63 3585 22 39 1996 3.28 533 6693 4- 6 3 337 21 40 2193 3.28 5222 6971 4.63 3029 20 41 9.732390 3.28 9.925141 9.807249 4- 6 3 10.192751 19 42 2587 3.28 5060 7527 4- 6 3 2 473 18 43 2784 3.28 4979 7805 4-63 2195 17 44 2980 3- 2 7 4897 8083 4-63 1917 16 45 3 J 77 3- 2 7 4816 i-35 8361 4.63 1639 15 46 3373 3- 2 7 4735 1.36 8638 4.62 1362 14 47 35 6 9 3- 2 7 4654 8916 4.62 1084 13 48 37 6 5 3- 2 7 457 2 9*93 4.62 0807 12 49 3961 3.26 4491 947i 4.62 0529 11 50 4i57 3.26 4409 9.809748 4.62 10.190252 10 51 9-734353 3.26 9.924328 9.810025 4.62 10.189975 9 52 4549 3.26 4246 0302 4.62 9698 8 53 4744 3- 2 5 4164 0580 4.62 9420 7 54 4939 3- 2 5 4083 0857 4.62 9143 6 55 5135 3- 2 5 4001 1134 4.61 8866 5 56 533 3- 2 5 39'9 1410 4.61 8590 4 57 5525 3- 2 5 3837 1.36 1687 4.61 8313 3 I 58 5719 3- 2 4 3755 i-37 1964 4.61 8036 2 1 59 59 J 4 3- 2 4 3 6 73 1.37 2241 4.61 7759 1 ! 60 9.736109 9.923591 9.812517 10.187483 Cosine. Diff. 1" Sine. Diff.l" Cotang. Diff. I" Tang. M. 122 57 74 33 SINES AlfD TANGENTS. 146 M. Sine. Diff. I" Cosine. Diff.l" Tang. Diff. V Cotang. ~T i 9.736109 6303 3.24 3.24 9.923591 3509 1.37 9.812517 2794 4.61 4.61 10.187483 7206 60 59 2 6498 3- 2 4 34^7 3070 4.61 6930 58 3 6692 3- 2 3 3345 3347 4.60 6653 57 4 6886 3- 2 3 3263 3623 4.60 6377 56 5 7080 3- 2 3 3181 3899 4.60 6101 55 6 7274 3-*3 3098 4'75 4.60 5825 54 7 7467 3-*3 3016 445 2 4.60 5548 53 8 7661 3.22 2 933 4728 4.60 5272 52 9 7855 3.22 2851 i-37 5004 4.60 4996 51 10 8048 3.22 2768 1.38 5279 4.60 4721 50 11 9.738241 3.22 9.922686 9-8I5555 4-59 10.184445 49 12 8434 3.22 2603 5831 4-59 4169 48 13 8627 3.21 2520 6107 4-59 3893 47 14 8820 3.21 2438 6382 4-59 3618 46 15 9013 3.21 *355 6658 4-59 3342 45 16 9206 3.21 2272 6 933 4-59 3067 44 17 9398 3.21 2189 7209 4-59 2791 43 18 9590 3.20 2106 7484 4-59 2516 42 19 9783 3.20 2023 7759 4-59 2241 41 20 9-739975 3.20 1940 1.38 8035 4.58 1965 40 21 9.740167 3.20 9.921857 i-39 9.818310 4.58 10.181690 39 22 0359 3.20 1774 8585 4.58 HI5 38 23 0550 3-*9 1691 8860 4.58 1140 37 24 0742 3-19 1607 9 r 35 4.58 0865 36 25 0934 3-19 1524 9410 4-58 0590 35 26 1125 3-'9 1441 9684 4.58 0316 34 27 1316 3.19 1357 9.819959 4.58 10.180041 33 28 1508 3.18 . 12 74 9.820234 4-58 10.179766 32 29 1699 3.18 1190 0508 4-57 9492 31 30 1889 3-i8 1107 0783 4-57 9217 30 31 9.742080 3.18 9.921023 i-39 9.821057 4-57 10.178943 29 32 2271 3.18 0939 1.40 1332 4-57 8668 28 33 2462 3-17 0856 1606 4-57 8394 27 34 2652 3-*7 0772 1880 4-57 8120 26 35 2842 3-i7 0688 2154 4-57 7846 25 36 333 3- J 7 0604 2429 4-57 757 1 24 37 3223 3-J7 0520 2703 4-57 7297 23 38 34 J 3 3.16 0436 2977 4.56 7023 22 39 3602 3.16 0352 3250 4.56 6750 21 40 379* 3.16 0268 35 2 4 4.56 6476 20 41 9.743982 3.16 9.920184 9.823798 4.56 10.176202 19 42 4171 3.16 0099 4072 4.56 5928 18 43 4361 3-*5 9.920015 1.40 4345 4.56 5655 17 44 455 S-'S 9.919931 1.41 4619 4.56 538i 16 45 4739 3-!5 9846 4893 4.56 5107 15 46 4928 3-'5 9762 5166 4.56 4834 14 47 5"7 3-'5 9677 5439 4-55 45 61 13 48 5306 3-H 9593 57i3 4-55 4287 12 49 5494 3-'4 9508 5986 4-55 4014 11 50 5683 3- J 4 9424 6259 4-55 374 1 10 51 9.745871 3-H 9.919339 9.826532 4-55 10.173468 9 52 6059 3-H 9 2 54 6805 4-55 3195 8 53 6248 3-'3 9169 7078 4-55 2922 7 54 6436 3-'3 9085 1.41 735i 4-55 2649 6 55 6624 3-'3 9000 1.42 7624 4-55 2376 5 56 6812 3-'3 8915 7897 4-54 2103 4 57 6999 3- J 3 8830 8170 4-54 1830 3 58 7187 3.12 8745 8442 4-54 1558 2 59 7374 3.12 8659 1.42 8715 4-54 1285 1 60 9.747562 9.918574 9.828987 10.171013 Cosine. Diff. I" Sine. Diff.l" Cotang. Diff. 1" Tang. M. ^123 56 75 34 XiOCtA 145 M. Sine. Diff.l" Cosine. Diff.l" Tang. Diff. 1" Cotang. o 9.747562 3.12 9.918574 1.42 9.828987 4-54 10.171013 60 1 7749 3.12 8489 9260 4-54 0740 59 2 7936 3.12 8404 953* 4-54 0468 58 3 4 8123 8310 3 .II 3- 11 8318 8*33 9.829805 9.830077 4-54 4-54 10.170195 10.169923 57 56 5 8497 3.11 8147 I. 4 2 0349 4-53 9651 55 6 8683 3-" 8062 1-43 0621 4-53 9379 54 7 8870 3-" 7976 0893 4-53 9107 53 8 9056 3.10 7891 1165 4-53 8835 52 9 3.10 7805 H37 4-53 8563 51 10 9429 3.10 7719 1709 4-53 8291 50 11 9.749615 3.10 9.917634 9.831981 4-53 10.168019 49 12 13 9801 9-749987 3.10 3-9 7548 7462 **53 4-53 4-53 7747 7475 48 47 14 9.750172 3-9 7376 2796 4-53 7204 46 15 0358 3-9 7290 3068 4-5* 6932 45 16 0543 3-9 7204 1.43 3339 4-5* 6661 44 17 0729 3-9 7118 1.44 3611 4-5* 6389 43 18 0914 3.08 7032 3882 4-5* 6118 42 19 1099 3.08 6946 4154 4-5* 5846 41 20 1284 3.08 6859 44*5 4-5* 5575 40 ~21 9.751469 3.08 9.916773 9.834696 4-5* 10.165304 39 22 1654 3.08 6687 4967 4-5* 533 38 23 1839 3.08 6600 5*38 4-5* 4762 37 24 2027 3.07 6514 559 4-5* 4491 36 25 2208 3-7 6427 5780 4.51 4220 35 26 2392 3-7 6341 6051 4.51 3949 34 27 2576 3-7 6254 1.44 6322 4.51 3678 33 28 2760 3-7 6167 1.45 6593 4.51 3407 32 29 2944 3.06 6081 6864 4-5 1 3136 31 30 3128 3.06 5994 7134 4-5i 2866 30 31 32 9-75331* 3495 3.06 3.06 9.915907 5820 9.837405 7675 4-1 1 10.162595 2325 29 28 33 34 3 6 79 3862 3.06 3-5 5733 5646 7946 8216 4-5 2054 1784 27 26 35 4046 3-5 5559 8487 4.50 1513 25 36 4229 3.05 5472 757 4.50 1*43 24 37 4412 3-05 5385 9027 4.50 0973 23 38 4595 3-5 5*97 9*97 4.50 0703 22 39 40 4778 4960 3.04 3.04 5210 5123 1.45 1.46 9568 9-839838 4.50 4.50 0432 10.160162 21 20 IT 9-755H3 3.04 9-9 I 535 9.840108 4-5 10.159892 19 42 5326 3-4 4948 0378 4.50 9622 18 43 5508 3.04 4860 0647 4.50 9353 17 44 5690 3.04 4773 0917 4-49 9083 16 45 5872 3-3 4685 1187 4-49 8813 15 46 6054 3-03 4598 J457 4.49 8543 14 47 6236 3-3 45 10 1726 4-49 8274 13 48 6418 3-03 4422 1996 4-49 8004 12 49 6600 3-3 4334 1.46 2266 4-49 7734 11 50 6782 3.02 4246 1.47 *535 4-49 7465 10 51 9.756963 3.02 9.914158 9.842805 4-49 10.157195 9 52 7144 3.02 4070 374 4-49 6926 8 53 7326 3.02 3982 3343 4-49 6657 7 54 7507 3.02 3612 4-49 6388 6 55 7688 3.01 3806 3882 4.48 6118 5 56 7869 3-oi 3718 4I5 1 4.48 5849 4 57 8050 3.01 3630 4420 4.48 3 58 8230 3.01 354 1 4689 4.48 53 11 2 59 8411 3.01 3453 1.47 4958 4.48 5042 1 60 9.758591 9.913365 9.845227 10.154773 CoKine. Diff. 1" Sine. Diff.l" Cotang. Diff. 1" Tang. ~MT 124 55 76 35 SINES AND TANGENTS. 144 M. Sine. Diff. 1" Cosine. Diff.l" Tang. Diff. 1" Cotang. 9-75859 1 3.01 9.913365 1.47 9.845227 4.48 10.154773 60 1 8772 3.00 3276 1.47 5496 4.48 4504 59 2 8952 3.00 3^7 1.48 5764 4.48 4236 58 3 9132 3.00 3099 6033 4.48 39 6 7 57 4 9312 3.00 3010 6302 4.48 3698 56 5 949 2 3.00 2922 6570 4-47 343 55 6 9672 2.99 2833 6839 4-47 3161 54 7 9.759852 2.99 2744 7107 4-47 2893 53 8 9.760031 2.99 2655 7376 4-47 2624 52 9 021 1 2.99 2566 7644 4-47 2356 51 10 0390 2.99 2477 79'3 4-47 2087 50 11 9.760569 2.98 9.912388 1.48 9.848181 4-47 10.151819 49 12 0748 2.98 2299 1.49 8449 4-47 i55i 48 13 0927 2.98 22IO 8717 4-47 1283 47 14 1106 2.98 2121 8986 4-47 1014 46 15 1285 2.98 2031 9254 4-47 0746 45 16 1464 a. 9 8 1942 9522 4-47 0478 44 17 1642 2.97 1853 9.849790 4.46 10.150210 43 18 1821 2.97 1763 9.850058 4.46 10.149942 42 19 1999 2.97 1674 0325 4.46 9675 41 20 2177 2.97 1584 593 4.46 9407 40 21 22 9.762356 2534 2.96 9.9II495 1405 1.49 9.850861 1129 4.46 4.46 10.149139 8871 39 38 23 2712 2.96 1315 1.50 1396 4.46 8604 37 24 2889 2.96 1226 1664 4.46 8336 36 25 3067 2.96 1136 I93i 4.46 8069 35 26 3*45 2.96 1046 2199 4.46 7801 34 27 3422 2.96 0956 2466 4.46 7534 33 28 3600 2.95 0866 *733 4-45 7267 32 29 3777 2.95 0776 3001 4-45 6999 31 30 3954 2.95 0686 3268 4-45 6732 30 ~31 9.764131 2.95 9.910596 9-853535 4-45 10.146465 29 32 4308 2.95 0506 1.50 3802 4-45 6198 28 33 34 4662 2.94 2.94 WS 3^5 I. 5 I 4069 433 6 4-45 4-45 s n l 5664 27 26 35 4838 2.94 0235 4603 4-45 5397 25 36 5i5 2.94 0144 4870 4-45 5 ^ 24 37 5191 2.94 9.910054 5137 4-45 4863 23 38 53 6 7 2.94 9.909963 5404 4-45 4596 22 39 5544 2.93 9873 5671 4-44 43*9 21 40 5720 2.93 9782 5938 4.44 4062 20 41 9.765896 2.93 9.90969! 9.856204 4.44 10.143796 19 42 6072 2.93 9601 6471 4.44 35 2 9 18 43 6247 2.93 9510 6737 4.44 3263 17 44 6423 2 -93 9419 1.51 7004 4.44 2996 16 45 6598 2.92 9328 1.52 7270 4-44 2730 15 46 6774 2.92 9237 7537 4.44 2463 14 47 6949 2.92 9146 7803 4.44 2197 13 48 7124 2.92 955 8069 4.44 I9JI 12 49 7300 2.92 8964 8336 4.44 1664 11 50 7475 2.91 8873 8602 4-43 1398 10 51 9.767649 2.91 9.908781 9.858868 4-43 10.141132 9 52 7824 2.91 8690 9134 4-43 0866 8 53 7999 2.91 8599 9400 4-43 0600 7 54 8177 2.91 8507 1.52 9666 4-43 334 6 55 8348 2.90 8416 '53 9.859932 4-43 10.140068 5 56 8522 2.90 8324 9.860198 4-43 10.139802 4 57 8697 2.90 8^33 0464 4-43 9536 3 58 8871 2.90 8141 0730 4-43 9270 2 59 9045 2.90 8049 i-53 0995 4-43 9005 1 60 9.769219 9.907958 9.861261 10.138739 Cosine. Diff. I" Sine. Diff.l" Cotang. Diff. 1" Tang. M. 125 64 77 36 IiOGARJTHXttJC 143 M. Sine. Diff. 1" Cosine. Diff.l" Tang. Diff. I" Cotang. ~0^ 9.769219 2.90 9.907958 1-53 9.861261 4-43 10.138739 60 1 9393 *- 8 9 7866 1527 4-43 8471 59 2 9566 2.89 7774 1792 4.42 8208 58 3 9740 2.89 7682 2058 4.42 7942 57 4 5 9.769913! 2.89 9.770087 2.89 759 749 8 2323 2589 4.42 4.42 7677 7411 56 55 6 0260 2.88 7406 1-53 2854 4.42 7146 54 7 8 0606 2.88 2.88 73H 7222 1.54 3"9 3385 4.42 4.42 6881 6615 53 52 9 0779 2.88 7129 3 6 5o 4.42 6350 51 10 0952 2.88 7037 39!5 4.42 6085 50 IT 9.771125 2.88 9.906945 9.864180 4.42 10.135820 49 12 1298 2.88 6852 4445 4.42 5555 48 13 1470 2.87 6760 4710 4.42 5290 47 14 1643 2.87 6667 4975 4.41 5 2 5 46 15 1815 2.87 6575 5240 4.41 4760 45 16 1987 2.87 6482 1.54 5505 4.41 4495 44 17 2159 2.87 6389 i-55 5770 4.41 4230 43 18 2331 2.86 6296 6035 4.41 39 6 5 42 19 2503 2.86 6204 6300 4.41 3700 41 20 2675 2.86 6m 6564 4.41 343 6 40 ~2l 9.772847 2.86 9.906018 9.866829 4.41 10.133171 39 22 3018 2.86 59 2 5 7094 4.41 2906 38 23 3190 2.86 5 8 32 7358 4.41 2642 37 24 3361 2.85 5739 7623 4.41 2377 36 25 3533 2.85 5645 7887 4.41 2113 35 26 374 2.85 5552 8152 4.40 1848 34 27 3875 2.85 5459 i-55 8416 4.40 1584 33 28 4046 2.85 5366 1.56 8680 4.40 1320 32 29 42 oZ 2.85 5272 8945 4.40 1055 31 30 4388 2.84 5179 9209 4.40 0791 30 81 9-774558 2.84 9.905085 9473 4.40 0527 29 32 4729 2.84 4992 9.869737 4.40 10.130263 28 33 4899 2.84 4898 9.870001 4.40 10.129999 27 34 5070 2.84 4804 0265 4.40 9735 26 35 5240 2.84 4711 0529 4.40 947i 25 36 5410 2.83 4617 0793 4.40 9207 24 37 55 8 o 2.83 45*3 1.56 i57 4.40 8943 23 38 575 2.83 4429 i-57 1321 4.40 8679 22 39 5920 2.83 4335 1585 4.40 8415 21 40 6090 2.83 4241 1849 4-39 8151 20 ~i 9.776259 283 9.904147 9.872112 4-39 10.127888 19 42 6429 2.82 453 2376 4-39 7624 18 43 6598 2.82 3959 2640 4-39 7360 17 44 6768 2.82 3864 2903 4-39 7097 16 45 6937 2.82 3770 3167 4-39 6833 15 46 7106 2.82 3676 343 4-39 6570 14 47 7275 2.8 1 358i 3 6 94 4-39 6306 13 48 7444 2.81 3487 i-57 3957 4-39 6043 12 49 7613 2.81 339 2 1.58 4220 4-39 578o 11 50 7781 2.81 3298 4484 4-39 55i 6 10 61 9.777950 2.81 9.903203 9.874747 4-39 10.125253 9 52 8119 2.81 3108 5010 4-39 4990 8 53 8287 2.80 3014 5*73 4-38 4727 7 54 8455 2.80 2919 5536 4.38 4464 6 55 8624 2.80 2824 5800 4-38 4200 5 56 57 8792 8960 2.80 2.80 2729 2634 6063 6326 4-38 4-38 3937 3 6 74 4 3 58 9128 2.80 *539 1.58 6589 4-38 34" 2 59 9295 2.79 2444 1.59 6851 4.38 3H9 1 60 9.779463 9.902349 9.877114 10.122886 Cosine. Diff. 1" Sine. Diff.l" Cotang. Diff. I" Tang. M. 126 53 78 37 SI!f E3 AND TANGENTS. 142 M. Sine. Diff. 1" Cosine. Diff.l" Tang. Diff. 1" Cotang. 9.779463 2.79 9.902349 1.59 9.877114 4-38 10.122886 60 1 2 9631 9798 2.79 2.79 2253 2158 7377 7640 4.38 4-38 2623 2360 59 58 3 9.779966 2.79 2063 793 4-38 2097 57 4 9.780133 2.79 1967 8165 4-38 1835 56 5 0300 2.78 1872 8428 4-38 1572 55 6 0467 2.78 I 77 6 8691 4.38 1309 54 7 0634 2.78 1681 8953 4-37 1047 53 8 0801 2.78 I5 8 5 9216 4-37 0784 52 9 10 0968 "34 2.78 2.78 1490 1394 1.59 1. 60 9478 9.879741 4-37 4-37 0522 10.120259 51 50 11 9.781301 2.77 9.901298 9.880003 4-37 10.119997 49 12 1468 2.77 1 202 0265 4-37 9735 48 13 1634 2.77 1106 0528 4-37 9472 47 14 18,00 2.77 1010 0790 4-37 9210 46 15 1966 2.77 0914 1052 4-37 8948 45 16 2132 2.77 0818 I3'4 4-37 8686 44 17 2298 2.76 0722 1576 4-37 8424 43 18 2464 2.76 0626 l8 39 4-37 8161 42 19 2630 2.76 0529 1. 60 2101 4-37 7899 41 20 2796 2.76 433 1.61 2363 4.36 7 6 37 40 21 9.782961 2.76 9.900337 9.882625 4.36 10.117375 39 22 3127 2.76 0240 2887 4.36 ?o 13 38 23 3292 2.75 0144 3148 4.36 6852 37 24 3458 2.75 9.900047 3410 4.36 6590 36 25 3623 2.75 9.899951 3672 4.36 6328 35 26 3788 2.75 9854 3934 4.36 6066 34 27 3953 2.75 9757 4196 4.36 5804 33 28 4118 2.74 9660 4457 4-36 5543 32 29 4282 2.74 95 6 4 1.61 47i9 4.36 5281 31 30 4447 2.74 9467 1.62 4980 4.36 5020 30 ~3l 9.784612 2.74 9.899370 9.885242 4-3 6 10.114758 29 32 4776 2.74 9 2 73 553 4.36 4497 28 8? 4941 2.74 9176 5765 4-3 6 4 2 35 27 34 5 I0 5 2.74 9078 6026 4-36 3974 26 35 5269 2.73 8981 6288 4-3 6 3712 25 36 5433 2-73 8884 6549 4-35 345 i 24 37 5597 a-73 8787 6810 4-35 3190 23 38 576i a-73 8689 7072 4-35 2928 22 39 59 2 5 2.73 8592 1.62 7333 4-35 2667 21 40 6089 2-73 8494 1.63 7594 4-35 2406 20 41 9.786252 2.72 9.898397 9.887855 4-35 10.112145 19 42 6416 2.72 8299 8116 4-35 1884 18 43 6579 2.72 8202 8377 4-35 1623 17 44 6742 2.72 8104 8639 4-35 1361 16 45 6906 2.72 8006 8900 4-35 1 100 15 46 7069 2.72 7908 9160 4-35 0840 14 47 7232 2.71 7810 9421 4-35 0579 13 48 7395 2.71 7712 9682 4-35 0318 12 49 7557 2.71 7614 9.889943 4-35 10.110057 11 50 7720 2.71 7516 1.63 9.890204 4-34 10.109796 10 51 9.787883 2.71 9.897418 1.64 0465 4-34 9535 "V 52 8045 2.71 7320 0725 4-34 9275 8 53 8208 2.71 7222 0986 4-34 9014 7 54 8370 2.70 7123 1247 4-34 8753 6 55 8532 2.70 7025 J57 4-34 8493 5 56 8694 2.70 6926 1768 4-34 8232 4 57 8856 2.70 6828 2028 4-34 7972 3 58 9018 2.70 6729 2289 4-34 7711 2 59 9180 2.70 6631 1.64 2549 4-34 745 * 1 60 9.789342 9.896532 9.892810 10.107190 Cosine. Diff. 1" Sine. Diff.l" Cotang. Diff. I" Tang. M. 127 52 79 38 LOGARITHMIC 141 M. Sine. Diff.l" Cosine. Diff.l' Tang. Diff. 1" Cotang. I ~0 9.789342 2.69 9.896532 1.64 9.892810 4-34 10.107190 60 1 9504 2.69 6 433 1.65 3070 4-34 6930 59 2 9665 2.69 6335 333 1 4-34 6669 58 3 4 9827 9.789988 2.69 2.69 6236 6137 Ss! 4-34 4-34 6409 6149 57 56 5 9.790149 2.6 9 6038 4111 4-34 5889 55 6 7 0310 0471 2.68 2.68 5939 5840 4371 4632 4-34 4-33 5629 54 53 8 0632 2.68 4892 4-33 5108 52 9 793 2.68 5641 4-33 4848 51 10 954 2.68 554* I.6 S 54^ 4-33 4588 50 11 9.791115 2.68 9.895443 1.66 9.895672 4-33 10.104328 49 12 1275 2.67 5343 5932 4-33 4068 48 13 I43 6 2.67 5*44 6192 4-33 3808 47 14 1596 2.67 5H5 6452 4-33 3548 46 15 1757 2.67 545 6712 4-33 3288 45 16 1917 2.67 4945 6971 4-33 3029 44 17 2077 2.67 4846 7231 4-33 2769 43 18 2237 2.66 4746 749 1 4-33 2509 42 19 2 397 2.66 4646 7751 4-33 2249 41 20 *557 2.66 4546 1.66 8010 4-33 1990 40 22 9.792716 2876 2.66 2.66 9.894446 4346 1.67 9.898270 853 4-33 4-33 10.101730 1470 39 38 23 335 2.66 4246 8789 13X1 37 24 3195 2.66 4146 9049 4.32 0951 36 25 3354 2.65 4046 938 4-3* 0692 35 26 27 28 35H 3673 3832 2.65 2.65 2.65 3946 3846 3745 9568 9.899827 9.900086 4-3* 4-3* 4.32 0432 10.100173 10.099914 34 33 32 29 399 1 2.65 3 6 45 0346 4-3* 9654 31 30 4150 2.64 3544 1.67 0605 4.32 9395 30 ~3T 9.794308 2.64 9.893444 1.68 9.900864 4.32 10.099136 29 32 2.64 3343 1124 4-3 2 8876 28 33 4626 2.64 3*43 1383 4-3 2 8617 27 34 4784 2.64 1642 8358 26 35 4942 2.64 3041 1901 4-3* 8099 25 36 5101 2.64 2940 2160 4-3* 7840 24 37 5 2 59 2.63 2839 2419 4.32 7581 23 38 39 5417 5575 2.63 2.63 2739 2638 2679 2938 4-3* 4.32 7321 7002 22 21 40 5733 2.63 2536 1.68 3*97 4.31 6803 20 41 9.795891 2.63 9.892435 1.69 9-93455 4-3 1 10.096545 19 42 6049 2.63 2334 37H 4.31 6286 18 43 6206 2.63 3973 4.31 6027 17 44 6364 2.62 2132 4232 4.31 5768 16 45 6521 2.62 2030 4491 4.31 559 15 46 6679 2.62 1929 4750 4-3 * 5250 14 47 6836 2.62 1827 5008 4-3 * 4992 13 48 6993 2.62 1726 5267 4-3 * 4733 12 49 7150 2.62 1624 1.69 55*6 4.31 4474 11 50 7307 2.61 1523 1.70 5784 4-3 1 4216 10 51 9.797464 2.61 9.891421 9.906043 4-3 10.093957 9 52 7621 2.61 1319 6302 4.31 3698 8 53 7777 2.61 1217 6560 4.31 344 7 54 7934 2.61 1115 6819 4.31 3181 6 55 8091 2.61 1013 7077 4.31 2923 5 56 8247 2.6 1 0911 733 6 4.31 2664 4 57 8403 2.60 . 0809 7594 4.31 2406 3 58 8560 2.60 0707 7852 4-3 1 2148 2 59 8716 2.60 0605 1.70 8m 4.30 1889 1 60 9.798872 9.890503 9.908369 10.091631 Cosine. Diff. I" Sine. Diff.l" Cotang. Diff. 1" Tang. If. 128 51 80 39 SINES AND TANGENTS. 140 M. Sine. Diff. 1" Cosine. Diff.l" Tang. Diff. 1" Cotang. 9.798872 2.60 9.890503 1.70 9.908369 4.30 10.091631 60 1 9028 2.60 0400 1.71 8628 4-3 1372 59 2 9184 2.60 0298 8886 4.30 : i i ; 58 3 9339 2.59 0195 9144 4-3 0856 57 4 9495 2.59 9.890093 9402 4.30 59 8 56 5 9651 2.59 9.889990 9660 4.30 0340 55 6 9806 2.59 9888 9.909918 4.30 10.090082 54 7 9.799962 2.59 9785 9.910177 4-3 10.089823 53 8 9.800117 2.59 9682 435 4.30 9565 52 9 0272 2.58 9579 0693 4.30 9307 51 10 0427 2.58 9477 1.71 0951 4.30 9049 50 11 9.800582 2.58 9.889374 1.72 9.911209 4-3 10.088791 49 12 0737 2.58 9271 1467 4-3 8533 48 13 0892 2. 5 8 9168 1724 4-3 8276 47 14 1047 2. 5 8 9064 1982 4-3 8018 46 15 1201 2. 5 8 8961 2240 4.30 7760 45 16 1356 *-57 8858 2498 4-3 7502 44 17 2.57 8755 2756 4.30 7244 43 18 1665 2.57 8651 3 OI 4 4.29 6986 42 19 1819 2.57 8548 1.72 3*7i 4.29 6729 41 20 1973 2.57 8444 1.73 35*9 4.29 6471 40 21 22 9.802128 2282 2.57 2.56 9.888341 8237 9.913787 4044 4.29 4.29 10.086213 5956 39 38 L>3 2436 2.56 8134 4302 4-*9 5698 37 24 2589 2.56 8030 4560 4.29 544 36 25 2743 2.56 7926 4817 4.29 5183 35 26 2897 2.56 7822 575 4.29 49*5 34 27 3050 ,*.56 7718 533* 4-*9 4668 33 28 3204 2.56 7614 559 4,29 4410 32 29 3357 2.55 7510 1.73 5847 4.29 4153 31 30 3511 2.55 7406 1.74 6104 4.29 3896 30 31 9.803664 2.55 9.887302 9.916362 4.29 10.083638 29 32 3817 2.55 7198 6619 4.29 338i 28 33 34 3970 4'*3 2.55 2.55 6989 6877 7134 4.29 4.29 mi 27 26 35 4276 2.54 6885 7391 4.29 2609 25 36 4428 a. 54 6780 7648 4.29 2352 24 37 458i *-54 6676 7905 4.29 2095 23 38 2.54 6571 8163 4.28 1837 22 39 4886 2.54 6466 1.74 8420 4.28 1580 21 40 5039 2.54 6362 J-75 8677 4.28 1323 20 41 9.805191 2.54 9.886257 9.918934 4.28 10.081066 19 42 5343 2.53 6152 9191 4.28 0809 18 43 5495 2.53 6047 9448 4.28 0552 17 44 5 6 47 2,53 5942 9705 4.28 02 9 C 16 45 5799 4.53 5837 9.919962 4.28 10.080038 15 46 5951 *-53 573* 9.920219 4.28 10.079781 14 47 6103 2.53 5627 0476 4.28 95*4 13 48 6254 2.53 55** 0733 4.28 9267 12 49 60 6406 6 557 2.52 2.52 5416 53" '75 1.76 0990 1*47 4.28 4-28 9010 8753 11 II ~5T .02 9.806709 6860 2.52 2.52 9.885205 5100 9.921503 1760 4.28 4.28 10.078497 82AO 9 8 53 7011 2.52 4994 2017 4.28 7983 7 54 7163 2.52 4889 2274 4.28 7726 6 55 73H 2.52 4783 2530 4.28 7470 5 56 .07 7465 7*766 2.51 2.51 2.51 4677 457* 4466 1.76 1.77 2787 344 3300 4.28 4.28 4.28 6956 6700 4 3 2 59 791 7 2,51 4360 1.77 3557 4.27 6443 1 GO 9.808067 9.884254 9.923813 10.076187 Cosine. Diff. V Bine. Diff.1" Cotang. Diff. 1" Tang. M. 129 60 81 40 XIOGAR.ITHRXXC 139 M. Sine. Diff. V Cosine. Diff.l" Tang. Diff. V Cotang. 9.808067 2.51 9.884254 1.77 9.923813 4.28 10.076187 60 1 8218 2.51 4148 4070 4.27 593 59 2 8368 2.51 4042 43 2 7 4.27 5 6 73 58 3 8519 2.50 393 6 4583 4.27 54*7 57 4 8669 2.50 3829 4840 4.27 5160 56 5 8819 2.50 37^3 5096 4.27 4904 55 6 8969 2.50 3617 5352 4.27 4648 54 7 9119 2.50 35io 5609 4.27 439 1 53 8 9269 2.50 3404 1.77 5865 4.27 4*35 52 9 9419 2.49 3*97 I. 7 8 6l22 4.27 3878 51 10 9569 2.49 3191 6378 4.27 3622 50 11 12 9718 9.809868 2.49 2.49 9.883084 2977 9.926634 6890 4.27 4.27 10.073366 3110 49 48 13 9.810017 2.49 2871 7147 4.27 * 8 53 47 14 0167 2.49 2764 7403 4.27 2597 46 15 0316 2.48 2657 7659 4.27 *34' 45 16 0465 2.48 *55 79'5 4.27 2085 44 17 0614 2.48 *443 I. 7 8 8171 4.27 1829 43 18 0763 2.48 2336 1.79 8427 4.27 1573 42 19 0912 2.48 2229 8683 4.27 1317 41 21 1061 2.48 2121 8940 4.27 1060 40 ~2l 9.811210 2.48 9.882014 9.929196 4.27 10.070804 39 22 1358 2.48 1907 945 2 4.27 0548 38 23 i57 3.47 1799 9708 4.27 0292 37 24 1655 2.47 1692 9.929964 4.27 10.070036 36 25 1804 2.47 I5 8 4 9.930220 4.26 10.069780 35 26 1952 2.47 *477 475 4.26 952"; 34 27 2100 2. 47 1369 1.79 0731 4.26 9269 33 28 2248 2.47 1261 1.80 0987 |L&6l 0017 32 29 2396 2.46 "53 i*43 4.26 8757 81 30 2544 2.46 1046 1499 4.26 8501 30 31 9.812692 2.46 9.880938 9-93'755 4.26 10.068245 29 32 2840 2.46 0830 2OIO 4.26 799 28 33 2988 2. 4 6 0722 2266 4.26 7734 27 34 3135 2.46 0613 2522 4.26 7478 26 35 3283 2. 4 6 055 2 77 8 4.26 7222 25 36 343 2.46 397 i. 80 333 4.26 6967 24 37 3578 a-45 0289 1.81 3289 4.26 6711 23 38 3725 2.45 0180 3545 4.26 6 455 22 39 3872 2.45 9.880072 3800 4.26 6200 21 40 4019 2.45 9.879963 4056 4.26 5944 20 41 9.814166 2.45 9855 9-9343" 4,26 10.065689 19 42 4313 2.45 9746 45 6 7 4.26 5433 18 43 4460 2.44 9 6 37 4823 4.26 S*77 17 44 4607 2.44 95 2 9 57 8 4.26 4922 16 45 4753 2.44 9420 5333 4.26 4667 15 46 4900 2.44 93 11 1.81 5589 4.26 4411 14 47 5046 2.44 9202, 1.82 5844 4.26 4i5 6 13 48 5193 2.44 993 6100 4.26 3900 12 49 5339 2.44 8984 6 355 4.26 3 6 45 11 ! 50 5485 ^43 8875 6610 4.26 339 10 "31 9.815631 2-43 9.878766 9.936866 4-25 10.063134 9 52 5778 2.43 8656 7121 4-25 2879 8 53 59 2 4 2-43 8547 7376 4.25 2624 7 54 6069 2.43 8438 7632 4-25 2368 6 55 6215 2.43 8328 1.82 7887 4- 2 5 2113 5 56 6361 ^43 8219 1.83 8142 4.25 1858 4 57 6507 2.42 8109 8398 4.25 1602 3 58 6652 2.42 7999 8653 4.25 1347 2 59 6798 2.42 7890 1.83 8908 4.25 1092 I 60 9.816943 9.877780 9.939163 10.060837 Cosine. Diff. V Sine. Diff.l" Cotang. Diff. V Tang. M. 130 49 41 SI3KTES AND TAZUaZUXTTS. 138 ] M. Sine. Diff. 1" Cosine. Diff.l" Tang. Diff. V Cotang. 9.816943 2.42 9.877780 ^83 9.939163 4-25 10.060837 60 1 7088 2.42 7670 9418 4.25 0582 59 2 7233 2.42 7560 9 6 73 4.25 0327 58 3 7379 2.42 745 9.939928 4.25 10.060072 57 4 7524 2.42 734 1.83 9.940183 4.25 10.059817 56 5 7668 2.41 7230 1.84 0438 4.25 9562 55 6 7813 2. 4 I 7120 0694 4-25 9306 54 7 795* 2.41 7010 0949 4.25 95! 53 8 8103 2. 4 I 6899 1204 4.25 8796 52 9 8247 2.41 6789 1458 4-25 8542 51 10 8392 2.41 6678 1714 4.25 8286 50 11 9.818536 2.40 9.876568 9.941968 4.25 10.058032 49 12 8681 2.40 6457 2223 4.25 7777 48 13 8825 2.40 6347 1.84 2478 4.25 7522 47 14 8969 2.40 6236 1.85 2733 4.25 7267 46 15 9113 2.40 6125 2988 4.25 7012 45 16 9257 2.40 6014 3243 4.25 6757 44 17 9401 0,40 594 3498 4.25 6502 43 18 9545 2.40 5793 3752 4.25 6248 42 19 9689 2-39 5682 4007 4.25 5993 41 20 9832 2-39 557i 4262 4.25 5738 40 21 9.819976 2.39 9-*75459 9.944517 4.25 10.055483 39 22 9.820120 2-39 534 8 477 * 4.24 5229 38 23 0263 2-39 5237 1.85 5026 4.24 4974 37 24 0406 2-39 5126 1.86 5281 4.24 47*9 36 25 55 *. 3 8 5014 5535 4.24 4465 35 26 0693 *. 3 8 493 579 4.24 4210 34 27 0836 2.38 47 9 * 6045 4.24 3955 33 28 0979 2.38 4680 6299 4.24 3701 32 29 1122 a. 3 8 4568 6 554 4.24 344 6 31 30 1265 2.38 4456 6808 4.24 3192 30 31 9.821407 2.38 9-874344 i.S6 9-947063 4.24 10.052937 ~29~ 32 155 2.38 4232 1.87 73i* 4.24 2682 28 33 1693 2-37 4121 7572 4.24 2428 27 34 1835 2.37 4009 7826 4-24 2174 26 35 1977 2-37 3896 8081 4.24 1919 25 36 2I2O 2.37 3784 8336 4.24 1664 24 37 2262 2.37 3672 8590 4.24 1410 23 38 2404 2-37 3560 8844 4.24 1156 22 39 2546 2-37 3448 9099 4.24 0901 21 40 2688 2.36 3335 9353 4.24 0647 20 41 9.822830 2.36 9.873223 1.87 9607 4.24 393 19 42 43 2972 3"4 2.36 2.36 3110 2998 1.88 9.949862 9.950116 4.24 4.24 10.050138 10.049884 18 17 44 3*55 2.36 2885 0370 4.24 9630 16 45 3397 2.36 2772 0625 4.24 9375 15 46 3539 2.36 2659 0879 4.24 9121 14 47 3680 z-35 2547 JI 33 4.24 8867 13 48 3821 2-35 2434 1388 4.24 8612 12 49 39 6 3 2.35 2321 1642 4.24 8358 11 50 4104 2-35 2208 1.88 1896 4.24 8104 10 61 9.824245 2-35 9.872095 1.89 9.952150 4.24 10.047850 9 52 4386 2-35 1981 2405 4.24 7595 8 53 45*7 ^35 1868 2659 4.24 734i 7 54 4668 2-34 1755 2913 4.24 7087 6 55 4808 2-34 1641 3167 4.23 6833 5 56 4949 2 -34 1528 3421 4-23 6579 4 57 5090 2-34 1414 3 6 75 4-23 63*5 3 58 5230 2-34 1301 3929 4.23 6071 2 59 537 1 2-34 1187 1.89 4183 4.23 5817 1 60 9.825511 9.871073 9-954437 10.045563 Cosine. Diff. I" Sine. Diff.l" Cotang. Diff. 1" Tun-. M. 131 48 83 42 ZiOCtARITHimC 137 M. Sine. Diff. 1" Cosine. Diff. 1" Tang. Diff. 1" Cotang. 9.825511 2.34 9.871073 I. 9 o 9-954437 4.23 10.045563 60 1 5651 2.33 0960 4691 4.23 539 59 2 2-33 0846 4945 555 58 3 593 1 2.33 0732 5200 4.23 4800 57 4 6071 2-33 0618 5454 4.23 4546 56 5 6211 *-33 0504 577 4-23 4293 55 6 6351 2.33 0390 5961 4-23 439 54 7 6491 2.33 0276 6215 4-23 3785 53 8 6631 2.33 0161 1.90 6469 52 6770 2.32 9.870047 1.91 6723 4.23 3277 51 10 6910 2.32 9.869933 6977 4-23 3023 50 11 9.827049 2.32 9818 9.957231 4.23 10.042769 49 12 7189 2.32 9704 7485 4.23 2515 48 13 7328 2.32 9589 7739 4.23 2261 47 14 7467 2.32 9474 7993 4.23 2007 46 15 7606 2.32 9360 8246 4-23 J754 45 16 7745 2.32 9245 8500 4.23 1500 44 17 7884 2.31 9130 1.91 8754 4-23 1246 43 18 8023 2.31 9015 1.9-2 9008 0992 42 19 8162 2.31 8900 9262 4.23 0738 41 20 8301 2.31 8785 9516 4-23 0484 40 21 9.828439 2.31 9.868670 9.959769 4-23 10.040231 39 22 8578 2.31 8555 9.960023 4-23 10.039977 38 23 8716 2.31 8440 0277 4.23 9723 37 24 8855 2.30 8324 0531 4.23 9469 36 25 8993 2.30 8209 0784 4.23 9216 35 26 27 9*3' 9269 2.30 2.30 8093 7978 1.92 1038 1291 4-23 4.23 8962 8709 34 33 28 9407 2.30 7862 '545 4-23 8455 32 29 9545 2.30 7747 1799 4.23 8201 31 30 9683 2.30 7631 2052 7948 30 31 9821 2.29 9.867515 9.962306 4.23 10.037694 29 32 9.829959 2.29 7399 2560 4.23 7440 28 33 9.830097 2.29 7283 2813 4.23 7187 27 34 0234 2.29 7167 3067 4.23 6933 26 35 0372 2.29 7051 I. 93 3320 4-23 6680 25 36 0509 2.29 6 935 1.94 3574 4-23 6426 24 37 0646 2.29 6819 3827 4.23 6173 23 38 0784 2.29 6703 4081 4.23 59'9 22 39 0921 2.28 6586 4335 5665 21 40 1058 2.28 6470 4588 4.22 5412 20 "If 9.831195 2.28 9.866353 9.964842 4.22 10.035158 19 42 1332 2.28 6237 595 4.22 4905 18 43 1469 2.28 6120 1.94 5349 4.22 4651 17 44 1606 2.28 6004 1.95 5602 4.22 4398 16 45 1742 2.28 5887 5855 4.22 4H5 15 46 1879 2.28 5770 6109 4.22 3891 14 47 2015 2.27 5653 6362 4.22 3638 13 48 2152 2.27 553 6 6616 4.22 3384 12 49 2288 2.27 5419 6869 4.22 3*3! 11 50 2425 2.27 5302 7123 4.22 2877 10 51 9.832561 2.27 9.865185 9.967376 4.22 10.032624 9 52 2697 2.27 5068 7629 4.22 2371 8 53 54 2833 2969 2.27 2.26 4950 4833 1.95 1.96 7883 8136 4.22 4.22 2117 1864 7 6 55 3 I0 5 2.26 4716 8389 4.22 1611 5 1 56 3*41 2.26 4598 8643 4.22 1357 4 1 57 3377 2.26 4481 8896 4.22 1104 3 58 35 12 2.26 4363 9149 4.22 0851 2 ! 59 3648 2.26 4245 1.96 4.22 0597 1 I 60 9.833783 9.864127 9.969656 10.030344 Cosine. Diff. 1" Sine. Diff.]" Cotang. Diff. 1" Tang. M. 132 47 43 SINES AND TANGENTS. 136 Jkl. Sine. Diff. 1" Cosine. Diff.l" Tang. Diff. 1" Cotang. ~0 9^33783 2.26 9.864127 1.96 9.969656 4.22 10.030344 60 1 3919 2.25 4010 1.96 9909 4.22 0091 59 2 4054 2.25 3892 1.97 9.970162 4.22 10.029838 58 3 4189 2.25 3774 0416 4.22 9584 57 4 43*5 2.25 3656 0669 4.22 9331 56 5 4460 2.25 3538 0922 4.22 9078 55 6 4595 2.25 3419 1175 4.22 8825 54 7 473 2.25 33i 1429 4.22 8571 53 8 4865 2.25 3183 1682 4.22 8318 52 9 4999 2.24 3064 1.97 *935 4.22 8065 51 10 5*34 2.24 2946 I. 9 8 2188 4.22 7812 50 ~TT 9.835269 2.24 9.862827 9.972441 4.22 10.027559 49 12 5403 2.24 2709 2694 4.22 7306 48 13 5538 2.24 2590 2948 4.22 7052 47 14 5672 2.24 2471 3201 4.22 6 799 46 15 5807 2.2 4 *353 3454 4.22 6546 45 16 594i 2.24 2234 377 4.22 6293 44 17 6075 2.23 2115 3960 4.22 6040 43 18 6209 2.23 1996 4213 4.22 5787 42 19 6 343 2.23 1877 I. 9 8 4466 4.22 5534 41 20 6477 2.23 1758 1.99 4719 4.22 5281 40 21 9.836611 2.23 9.861638 9-974973 4.22 10.025027 39 22 6745 2.23 1519 5226 4.22 4774 38 23 6878 2.23 1400 5479 4.22 4521 37 24 7012 2.22 1280 573* 4.22 4268 36 25 7146 2.22 1161 5985 4.22 4015 35 26 7279 2.22 1041 6238 4.22 3762 34 27 7412 2.22 0922 6491 4.22 359 33 28 754 6 2.22 0802 1.99 6744 4.22 3256 32 29 7679 2.22 0682 2.00 6997 4 .22 3003 31 30 7812 2.22 0562 7250 4.22 2750 30 31 9-837945 2.22 9.860442 9-9775 3 4.22 10.022497 29 32 8078 2.21 0322 7756 4.22 2244 28 33 8211 2.21 O2O2 8009 4.22 1991 27 34 8344 2.21 9.860082 8262 4.22 1738 26 35 8477 2.21 9.859962 8515 4.22 1485 25 36 8610 2.21 9842 2.00 8768 4.22 1232 24 37 8742 2.21 9721 2.01 9021 4.22 0979 23 38 8875 2.21 9601 9274 4.22 0726 22 39 9007 2.21 9480 9527 4.22 0473 21 40 9140 2.20 9360 9.979780 4.22 IO.O2O22O 20 41 9.839272 2.20 9.859239 9.980033 4.22 10.019967 19 42 9404 2.20 9119 0286 4.22 9714 18 43 953 6 2.20 8998 0538 4.22 9462 17 44 9668 2.20 8877 2.01 0791 4.21 9209 16 45 9800 2.20 8756 2.02 1044 4.21 8 95 6 15 46 9.839932 2. 2O 8635 1297 4.21 8703 14 47 9.840064 2.19 85H 1550 4.21 8 45 13 48 0196 2.19 8393 1803 4.21 8l 97 12 49 0328 2.19 8272 2056 4.21 7944 11 50 459 2.19 8151 2309 4.21 7691 10 51 9.840591 2.1 9 9.858029 9.982562 4.21 10.017438 ~9~ 52 0722 2.19 7908 2814 4.21 7186 8 53 0854 2.19 7786 2. 02 3067 4.21 6933 7 54 55 0985 1116 2.19 2.19 7665 7543 2.03 3320 3573 4 .2I 4.21 6680 6427 6 5 56 1247 2.18 7422 3826 4.21 6174 4 57 1378 2.18 7300 4079 4.21 59" 3 58 1509 2.18 7178 433 1 4.21 5669 2 59 1640 2.18 7056 2.03 4584 4.21 5416 1 60 9.841771 9.856934 9.984837 10.015163 : Cosine. 1 Diff. 1" Sine. Diff.l" Cotang. Diff. I" Tang. M. i 1 133 46 1 27 85 44 I,OCb 135 fLRITHIMEIC M. Sine. Diff. V Co.-iiue. Diff.l' Tang. Diff. 1" Cotang. 9.841771 2.l8 9.856934 2.03 9.984837 4.21 10.015163 60 1 1902 2.18 6812 2.0C 5090 4.21 4910 59 2 2033 2.18 6690 2.04 5343 4.21 4 6 57 58 3 2163 2.17 6568 5596 4.21 4404 57 4 2294 2.17 6446 5848 4.21 4152 56 5 2424 2.1 7 6323 6101 4.21 3899 55 6 2 ^ 5 2.17 6201 6 354 4.21 3646 54 7 2685 2.17 6078 6607 4.21 3393 53 8 2815 2.17 5956 6860 4.21 3140 52 9 2946 2.17 5833 2.04 7112 4.21 2888 51 10 3076 2.17 5711 3-05 73 6 5 4.21 2635 50 11 9.843206 2.16 9.855588 9.987618 4.21 10.012382 49 12 3336 2.16 5465 7871 4.21 2129 48 13 3466 2.16 534^ 8123 4.21 1877 47 14 3595 2.16 5219 8376 4.21 1624 46 15 37^5 2.16 5096 8629 4.21 I37i 45 16 3855 2.l6 4973 8882 4.21 1118 44 17 3984 2.16 4850 9 J 34 4.21 0866 43 18 4114 2.16 4727 2.05 9387 4.21 0613 42 19 4 2 43 2.I 5 4603 2.06 9640 4.21 0360 41 20 4372 2.15 4480 9.989893 4.21 10.010107 40 21 9.844502 2.15 9.854356 9.990145 4.21 10.009855 39 22 4631 2.15 4233 039 4.21 9602 38 23 4760 2.15 4109 0651 4.21 9349 37 24 4889 2.15 3986 0903 4.21 9097 36 25 5018 2.15 3862 1156 4.21 8844 35 26 5H7 2.15 3738 2.06 1409 4.21 8591 34 27 5276 2.14 3614 2.07 1662 4.21 8338 33 28 5405 2.14 349 1914 4.21 8086 32 29 5533 2.1 4 3366 2167 4.21 7833 31 30 5662 2.14 3242 2420 4.21 7580 30 ~3l 9.845790 2.14 9.853118 9.992672 4.21 10.007328 29 32 59i9 2.1 4 2994 2925 4.21 7075 28 33 6047 2.14 2869 3178 4.21 6822 27 34 35 6175 6304 2.14 2.14 ^745 2620 2.07 343 3683 4.21 4.21 6570 6317 26 25 36 6432 2.13 2496 2.08 393 6 4.21 6064 24 37 6560 2.13 2371 4189 4.21 5811 23 38 6688 2.13 2247 4441 4.21 5559 22 39 6816 2.13 2122 4694 4.21 5306 21 40 6944 2.13 1997 4947 4.21 5053 20 41 9.847071 2.1 3 9.851872 9.995199 4.21 10.004801 19 42 7199 2.13 1747 5452 4.21 4548 18 43 73*7 2.13 l622 2.08 575 4.21 4295 17 44 7454 2.12 M97 2.09 5957 4.21 4043 16 45 75^ 2.12 1372 6210 4.21 379 15 46 7709 2.12 1246 6463 4.21 3537 14 47 7836 2.12 II2I 6715 4.21 3285 13 48 7964 2.12 0996 6968 4.21 3032 12 49 8091 2.12 0870 7221 4.21 2779 11 50 8218 2.12 0745 7473 4.21 2527 10 51 9.848345 2.12 9.850619 2.09 9.997726 4.21 10.002274 9 52 8472 2.1 I 0493 2.10 7979 4.21 2021 8 53 8599 2.II 0368 8231 4.21 1769 7 54 8726 2.II 0242 8484 4.21 I5l6 6 55 8852 2.1 I 9.850116 8737 4.21 1263 5 56 8979 2. 1 I 9.849990 8989 4.21 IOII 4 57 9106 2. 1 I 9864 9242 4.21 0758 3 58 9232 2. 1 1 9738 9495 4.21 0505 2 59 9359 2. 1 1 9611 2.10 9.999747 4.21 0253 1 j 60 9.849485 9.849485 IO.OOOOOO 10.000000 Cosine. Diff. 1" Sine. Diff.l" Cotang. Diff. 1" Tang. M. 134 45 'TABLE OF NATURAL SINES COSINES. 7 NATURAL SINES AND COSINES. f 1 2 3 4 Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. ~b~ 00000 Unit. 01745 99985 03490 99939 05*34 99863 06976 9975 6 60 i 00029 Unit. 01774 99984 035*9 99938 05263 99861 07005 99754 59 2 00058 Unit. 01803 99984 03548 99937 05292 99860 07034 99752 58 3 00087 Unit. 01832 99983 03577 99936 05321 99858 07063 99750 57 4 00116 Unit 01862 99983 03606 99935 05350 99857 07092 99748 56 5 00145 Unit. 01891 99982 03635 99934 05379 99855 07121 99746 55 6 00175 Unit. 01920 99982 03664 99933 05408 99854 07150 99744 54 7 00204 Unit. 01949 99981 03693 99932 05437 99852 07179 99742 53 8 00233 Unit. 01978 99980 03723 9993 1 05466 07208 99740 52 9 00262 Unit. 02007 99980 03752 99930 05495 99849 07237 99738 51 10 00291 Unit. 02036 99979 03781 99929 05524 99847 07266 99736 50 11 00320 99999 02065 99979 03810 99927 05553 99846 07295 99734 ~49~ 12 00349 99999 02094 99978 03839 99926 05582 99844 07324 9973 1 48 13 00378 99999 02123 99977 03868 999*5 05611 99842 07353 99729 47 14 15 00407 00436 99999 99999 02152 02181 99977 99976 03897 03926 99924 99923 05640 05669 99841 99839 07382 07411 99727 99725 46 45 16 00465 99999 022X1 99976 03955 99922 05698 99838 07440 99723 44 17 00495 99999 02240 99975 03984 99921 05727 99836 07469 99721 43 18 00524 99999 02269 99974 04013 99919 05756 99834 07498 99719 42 19 00553 99998 02298 99974 04042 999x8 05785 99833 07527 99716 41 20 00582 99998 02327 99973 04071 99917 05814 99831 07556 997*4 40 21 00611 99998 02356 99972 04100 999x6 05844 99829 07585 99712 ~39~ 22 23 00640 00669 99998 99998 02385 024X4 99972 99971 04129 04159 99915 999*3 05873 05902 99827 99826 07614 07643 997*0 99708 38 37 24 00698 99998 02443 99970 04188 99912 05931 99824 07672 99705 36 25 00727 99997 02472 99969 04217 99911 05960 9982* 07701 99703 35 26 00756 99997 O25OI 99969 04246 99910 05989 99821 07730 99701 34 27 00785 99997 02530 99968 04275 99909 06018 99819 07759 99699 33 28 29 30 00814 00844 00873 99997 99996 99996 02560 02589 026l8 99967 99966 99966 04304 4333 04362 9997 99906 99905 06047 06076 06105 99817 99815 99813 07788 07817 07846 99696 99694 99692 32 31 30 31 00902 99996 02647 99965 04391 99904 06134 99812 07875 99689 29 32 00931 99996 02676 99964 04420 99902 06163 99810 07904 99687 28 33 00960 99995 02705 99963 04449 99901 06192 99808 07933 99685 27 34 00989 99995 02734 99963 04478 99900 06221 99806 07962 99683 26 35 01018 99995 02763 99962 04507 99898 06250 99804 07991 99680 25 36 01047 99995 02792 99961 4536 99897 06279 99803 08020 99678 24 37 01076 99994 02821 99960 04565 99896 06308 99801 08049 99676 23 38 01105 99994 02850 99959 04594 99894 06337 99799 08078 99673 22 39 01134 99994 02879 99959 04623 99893 06366 99797 08107 99671 21 40 01164 99993 02908 99958 04653 99892 06395 99795 08136 99668 20 41 01193 99993 02938 99957 04682 9989 06424 99793 08165 99666 19 42 01222 99993 02967 99956 04711 99889 06453 99792 08194 99664 18 43 OI25I 99992 02996 99955 04740 99888 06482 99790 08223 99661 17 44 01280 99992 03025 99954 04769 99886 06511 99788 08252 99659 16 45 01309 99991 03054 99953 04798 99885 06540 99786 08281 99657 15 46 01338 99991 03083 99952 04827 99883 06569 99784 08310 99654 14 47 01367 99991 03II2 99952 04856 99882 06598 99782 08339 99652 13 48 01396 99990 03X41 99951 04885 99881 06627 99780 08368 99649 12 49 01425 99990 03170 99950 04914 99879 06656 99778 08397 99647 11 50 01454 99989 03X99 99949 04943 99878 06685 99776 08426 99_ 6 _44 10 51 01483 99989 03228 99948 04972 99876 06714 99774 08455 99642 9 52 OI5I3 99989 03257 99947 05001 99875 06743 99772 08484 99639 8 53 01542 99988 03286 99946 05030 99873 06773 99770 08513 99637 7 54 OI57I 99988 03316 99945 05059 99872 06802 99768 08542 99635 6 55 Ol6oO 99987 03345 99944 05088 99870 06831 99766 08571 99632 5 56 01629 99987 03374 99943 05117 99869 06860 99764 08600 99630 4 1 57 01658 99986 03403 99942 05146 99867 06889 99762 08629 99627 3 58 01687 99986 03432 99941 05175 99866 06918 99760 08658 99625 2 59 01716 99985 03461 99940 05205 99864 06947 99758 08687 99622 1 60 01745 99985 03490 99939 05234 99863 06976 99756 08716 99619 Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. 89 88 87 86 85 NATURAL SINES AND COSINES. / 5 6 7 8 9 Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. e 08716 99619 *453 99452! 12187 99255 *39*7 99027 *5 6 43 98769 60 i 08745 99617 10482 99449 12216 99251 13946 99023 15672 98764 59 2 08774 99614 10511 99446 12245 99248 *3975 990*9 15701 98760 58 3 08803 99612 10540 99443 12274 99244 14004 990*5 15730 98755 57 4 08831 99609 10569 99440 12302 99240 14033 99011 15758 98751 56 5 08860 99607 10597 99437 12331 99237 14061 99006 15787 98746 55 6 08889 99604 10626 99434 12360 99233 14090 99002 15816 98741 54 7 08918 99602 10655 9943* 12389 99230 *4**9 98998 15845 98737 53 8 08947 99599 10684 99428 12418 99226 14148 98994 15873 98732 52 9 08976 99596 107*3 99424 12447 99222 14177 98990 15902 98728 51 10 09005 99594 10742 99421 12476 99219 14205 98986 1593* 98723 50 11 09034 9959* 10771 994*8 12504 992*5 14234 98982 15959 98718 49 12 09063(99588 10800 994*5 12533 99211 14263 98978 15988 987*4 48 13 09092(99586 10829 994*2 12562 99208 14292 98973 16017 98709 47 14 09121 99583 10858 99409 12591 99204 14320 98969 16046 98704 46 15 09150 99580 10887 99406 12620 99200 *4349 98965 16074 98700 45 16 09179 99578 10916 99402 12649 99*97 14378 98961 16103 98695 44 17 09208 99575 10945 99399 12678 99*93 14407 98957 16132 98690 43 18 09237 99572 *973 99396 12706 99*89 14436 98953 16160 98686 42 19 09266 99570 1 1 002 99393 12735 99186 14464 98948 16189 98681 41 20 09295 99567 II03I 99390 12764 99182 *4493 98944 16218 98676 40 21 09324 99564 11060 99386 12793 99*78 14522 98940 16246 98671 39 22 09353 99562 11089 99383 12822 99*75 *455* 98936 16275 98667 38 23 09382 99559 11118 99380 12851 99171 14580 98931 16304 98662 37 24 09411 99556 ii*47 99377 12880 99*67 14608 98927 98657 36 25 09440 99553 11176 99374 12908 99*63 14637 98923 16361 98652 35 26 09469 9955* 11205 99370 12937 99160 14666 98919 16390 98648 34 27 09498 99548 11234 99367 12966 99*56 14695 989*4 16419 98643 33 28 09527 99545 11263 99364 12995 99*52 *47 2 3 98910 16447 98638 32 29 09556 99542 11291 99360 13024 99*48 14752 98906 16476 98633 31 30 09585 99540 11320 99357 13053 99*44 14781 98902 16505 98629 30 ~B1 09614 99537 1*349 99354 13081 99141 14810 98897 16533 98624 29 32 09642 99534 H378 9935* 13110 99*37 14838 98893 16562 98619 28 33 09671 9953* 11407 99347 13139 99*33 14867 98889 16591 98614 27 34 09700 99528 11436 99344 13168 99*29 14896 98884 16620 98609 26 35 09729 99526 11465 9934* 13197 99*25 14925 98880 16648 98604 25 36 09758 99523 i*494 99337 13226 99122 *4954 98876 16677 98600 24 37 09787 99520 11523 99334 13254 99118 14982 98871 16706 98595 23 38 09816 995*7 i*552 9933* 13283 99114 15011 98867 16734 98590 22 39 09845 995*4 11580 99327 13312 991 10 15040 98863 16763 98585 21 40 09874 995** 11609 99324 13341 99106 15069 98858 16792 98580 20 41 09903 99508 11638 99320 13370 99102 15097 98854 16820 98575 19 42 09932 99506 1 1667 993*7 *3399 99098 15126 98849 16849 98570 18 43 09961 99503 11696 993*4 *3427 99094 *5*55 98845 16878 98565 17 44 09990 99500 1*725 993* 13456 99091 15184 98841 16906 98561 16 45 10019 99497 **754 9937 *34 8 5 99087 15212 98836 16935 98556 15 46 10048 99494 11783 99303 13514 99083 15241 98832 16964 98551 14 47 10077 99491 11812 99300 *3543 9979 15270 98827 16992 98546 13 48 10106 99488 11840 99297 13572 99075 15299 98823 17021 98541 12 49 10135 99485 11869 99293 13600 99071 *5327 98818 17050 98536 11 50 10164 99482 11898 99290 13629 99067 15356 98814 17078 98531 10 51 10192 99479 11927 99286 13658 99063 *53 8 5 98809 17107 98526 9 52 I022I 99476 11956 99283 13687 99059 15414 98805 17136 98521 8 53 IO25O 99473 11985 99279 13716 99055 15442 98800 17164 98516 7 54 10279 99470 12014 99276 *3744 99051 15471 98796 17193 98511 6 55 10308 99467 12043 99272 *3773 99047 15500 98791 17222 98506 5 56 10337 99464 12071 99269 13802 99043 15529 98787 17250 98501 4 1 57 10366 99461 I2IOO 99265 1383* 99039 *5557 98782 17279 98496 3 i 58 10395 99458 I2I29 99262 13860 99035 15586 98778 17308 98491 2 59 10424 99455 I2I58 99258 13889 99031 15615 98773 17336 98486 1 60 10453 99452 I2I87 99255 99027 *5643 98769 17365 98481 f Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. 84 83 82 81 80 NATURAL SIBXES AKD CO3IBJES. 10 11 12 13 14 f Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. o 7365 98481 19081 98163 20791 97815 22495 97437 24192 97030 60 i 7393 98476 19109 98157 20820 97809 22523 97430 24220 97023 59 2 7422 98471 19138 98152 20848 97803 22552 97424 24249 97015 58 3 745* 98466 19167 98146 20877 97797 22580 974*7 24277 97008 57 4 7479 98461 *9*95 98140 20905 9779* 22608 974** 24305 97001 56 5 7508 98455 19224 98135 20933 97784 22637 97404 24333 96994 55 6 7537 98450 19252 98129 20962 97778 22665 97398 24362 96987 54 7 7565 98445 19281 98124 20990 97772 22693 97391 24390 96980 53 8 7594 98440 19309 98118 21019 97766 22722 97384 24418 96973 52 9 7623 98435 19338 98112 21047 97760 22750 97378 24446 96966 51 10 7651 98430 19366 98107 21076 97754 22778 9737* 24474 96959 50 11 7680 98425 *9395 98101 21104 97748 22807 97365 24503 96952 49 12 17708 98420 19423 98096 21132 97742 22835 97358 2453* 96945 48 13 *7737 98414 19452 98090 2Il6l 97735 22863 9735* 24559 96937 47 14 17766 98409 19481 98084 21189 97729 22892 97345 24587 96930 46 15 *7794 98404 19509 98079 2I2I8 97723 22920 97338 24615 96923 45 16 17823 98399 19538 98073 21246 977*7 22948 9733* 24644 96916 44 17 17852 98394 19566 98067 21275 977** 22977 97325 24672 96909 43 18 17880 98389 *9595 98061 21303 97705 23005 973*8 24700 96902 42 19 17909 9838, 19623 98056 21331 97698 23033 97311 24728 96894 41 20 *7937 98378 19652 98050 21360 97692 23062 97304 24756 96887 40 21 17966 98373 19680 98044 21388 97686 23090 97298 24784 96880 ~39~ 22 *7995 98368 19709 98039 2*4*7 97680 23118 97291 24813 96873 38 23 18023 98362 19737 98033 2*445 97673 23146 97284 24841 96866 37 24 18052 98357 19766 98027 2*474 97667 23*75 97278 24869 96858 36 25 18081 98352 *9794 98021 21502 97661 23203 97271 24897 96851 35 26 18109 98347 19823 98016 21530 97655 23231 97264 24925 96844 34 27 18138 98341 19851 98010 2*559 97648 23260 97257 24954 96837 33 28 18166 98336 19880 98004 2*587 97642 23288 9725* 24982 96829 32 29 18195 98331 19908 97998 21616 97636 23316 97244 25010 96822 31 30 18224 98325 *9937 97992 21644 97630 23345 97237 25038 96815 30 31 18252 98320 19965 97987 21672 97623 23373 97230 25066 96807 29 32 18281 983*5- *9994 97981 21701 97617 23401 97223 25094 96800 28 33 18309 98310 2OO22 97975 21729 97611 23429 97217 25122 96793 27 34 35 18338 18367 98304 98299 20051 20079 97969 97963 21758 21786 97604 97598 23458 23486 97210 97203 25151 25*79 9678^ 96778 26 25 36 i8395 98294 20108 97958 21814 97592 235H 97196 25207 96771 24 37 18424 98288 20136 97952 21843 97585 23542 97*89 25235 96764 23 38 18452 98283 20165 97946 21871 97579 23571 97182 25263 96756 22 39 18481 98277 20193 97940 21899 97573 23599 97*76 25291 96749 21 40 18509 98*72 2O222 97934 21928 97566 23627 97*69 25320 96742 20 41 18538 98267 20250 97928 21956 97560 23656 97162 25348 96734 19 42 18567 98261 20279 97922 21985 97553 23684 97*55 25376 96727 18 43 i8595 98256 20307 979*6 22013 97547 23712 97*48 25404 96719 17 44 18624 98250 20336 979*0 22041 9754* 23740 97141 25432 96712 16 45 18652 98245 20364 97905 22070 97534 23769 97*34 25460 96705 15 46 18681 98240 20393 97899 22098 97528 23797 97*27 25488 96697 14 47 18710 98234 20421 97893 22126 97521 23825 97120 255*6 96690 13 48 18738 98229 20450 97887 22155 975*5 23853 97**3 25545 96682 12 49 18767 98223 20478 97881 22183 97508 23882 97106 25573 96675 11 60 i8795 98218 20507 97875 22212 97502 239* 97100 25601 96667 10 51 18824 98212 20535 97869 2224O 97496 23938 97093 25629 96660 gT 52 18852 98207 2056^ 97863 22268 97489 23966 97086 25657 96653 8 53 18881 98201 20592 97857 22297 97483 23995 97079 25685 96645 7 54 18910 98196 20620 97851 22325 97476 2402-: 97072 257*3 96638 6 55 18938 98190 20649 97845 22353 97470 24051 97065 25741 96630 5 56 18967 98185 20677 97839 22382 97463 24079 97058 25769 96623 4 57 18995 98179 20706 97833 22410 97457 24108 97051 25798 96615 8 58 19024 98174 20734 97827 22438 9745 i 24136 97044 25826 96608 2 59 19052 98168 [20763 97821 22467 97444! 24164 97037 25854 96600 1 60 19081 98*63 20791 978*5 22495 97437 24192 97030 25882 96593 Cosine Sine. Cosine Sine. Cosine. Sine. Cosine Sine. Cosine. Sine. 79 78 77 76 75 90 NATURAL SINES AND COSINES. 15 16 17 18 19 Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. 25882 96593 27564 96126 29237 95630 30902 95106 32557 94552 60 i 25910 96585 27592 96118 29265 95622 30929 95097 32584 94542 59 2 25938 96578 27620 96110 29293 95613 3957 95088 32612 94533 58 3 25966 96570 27648 96102 29321 95605 30985 95079 32639 94523 57 4 25994 96562 27676 96094 29348 95596 31012 95070 32667 945*4 56 5 26022 96555 27704 96086 29376 95588 31040 95061 32694 94504 55 6 26050 9 6 547 27731 96078 29404 95579 31068 95052 32722 94495 54 V 26079196540 27759 96070 29432 9557* 3*095 95043 32749 94485 53 8 26107 96532 27787 96062 29460 95562 31123 950331 32777 94476 52 9 10 26135 96524 26163(96517 27815 27843 & 29487 29515 95554 95545 31151 31178 95024 950*5 32804 32832 94466 94457 51 50 11 26191 96509 27871 96037 29543 95536 31206 95006 32859 94447 49 1 12 26219 96502 27899 96029 29571 95528 3*233 94997 32887 94438 48 j 13 26247196494 27927 96021 29599 955*9 31261 94988 329*4 94428 47 14 26275 96486 27955 96013 29626 955** 31289 94979 32942 944*8 46 15 26303 96479 27983 96005 29654 9552 3*3*6 94970 32969 94409 45 16 26331 96471 28011 95997 29682 95493 3*344 94961 32997 94399 44 17 18 26359 96463 26387 96456 28039 28067 95989 95981 29710 29737 95485 95476 3*372 3*399 94952 94943 33024 33051 94390 94380 43 i 42 19 20 26415,96448 26443196440 28095 28123 95972 95964 29765 29793 95467 95459 3*427 3*454 94933 94924 33079 33106 94370 94361 41 40 21 22 26471 26500 96433 96425 28150 28178 9595 6 95948 29821 29849 9545 9544* 31482 3*5*o 949*5 94906 33134 33161 9435* 94342 39 38 23 26528 964*7 28206 95940 29876 95433 3*537 94897 33*89 94332 37 24 26556 96410 28234 9593* 29904 95424 3*565 94888 33216 94322 36 25 26584 96402 28262 95923 29932 954*5 3*593 94878 33244 943*3 35 26 26612 96394 28290 959*5 29960 95407 31620 94869 33271 94303 34 27 26640 96386 28318 95907 129987 95398 31648 94860 33298 94293 33 28 26668 96379 28346 95898 130015 95389 3*675 94851 33326 94284 32 29 26696 96371 28374 95890 30043 95380 31703 94842 33353 94274 31 1 30 26724 96363 28402 95882 30071 95372 3*73 94832 33381 94264 30 31 26752 96355 28429 95874 30098 95363 31758 94823 3348 94254 29 32 26780 96347 28457 95865 30126 95354 31786 94814 33436 94245 28 1 33 26808 96340 28485 95857 30154 95345 31813 94805 3346T 74235 27 j 34 26836 96332 285*3 95849 30182 95337 31841 94795 33490 94225 26 35 26864 96324 28541 95841 30209 95328 31868 94786 335*8 942*5 25 36 26892 96316 28569 95832 30237 953*9 31896 94777 33545 94206 24 37 26920 96308 28597 95824 30265 953*0 31923 94768 33573 94*96 23 38 26948 96301 28625 95816 30292 95301 3*95* 94758 33600 94*86 22 39 26976 96293 28652 95807 30320 95293 3*979 94749 33627 94176 21 1 40 27004 96285 28680 95799 30348 95284 32006 94740 33655 94167 20 41 27032 96277 28708 9579* 30376 95275 32034 94730 33682 94*57 19 42 27060 96269 28736 95782 30403 95266 jj 32061 94721 337*0 94*47 18 43 27088 96261 28764 95774 30431 95257 32089 947*2 33737 94*37 17 44 27116 96253 28792 95766 30459 95248 321 16 94702 33764 94*27 16 45 27144 96246 28820 95757 30486 95240 32*44 94693 33792 94118 15 46 27172 96238 28847 95749 30514 95231 32171 94684 338*9 94108 14 47 48 27200 27228 96230 96222 28875 28903 9574 95732 30542 95222 32199 30570 952131132227 94674 94665 33846 33874 94098 94088 13 1 12 49 '127256 96214 50 127284 96206 28931 28959 95724 957*5 30597 30625 95204 32254 95195 132282 94656 94646 33901 33929 94078 94068 11 | 10 51 27312 96198 28987 95707 30653 95186)132309 94637 33956 94058 9 52 53 54 55 27340196190 27368 96182 27396 96174 27424 96166 29015 29042 29070 29098 95698 95690 95681 95673 30680 30708 30736 30763 95*77 95168 95*59 95150 32337 32364 32392 1324*9 94627 33983 94618 34011 (94609 34038 J94599 34o65 94049 94039 94029 940*9 8 7 6 5 56 27452 96158 29126 95664 30791 95142 | 32447 94590 i 34093 94009 4 57 27480 96150 29*54 95656 30819 95*33 32474 94580 134120 93999 3 58 27508 96142 29182 95647 30846 95124 32502194571 134147 93989 2 j 59 60 27536 96134 27564 96126 29209 29237 95639 9563 30874 30902 95**5 95106 32529(94561 134175 32557 945521 34202 93979 93969 1 Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine Cosine. Sine. 1 I i 74 73 72 71 I 70 91 NATURAL SINES AND COSINES. 20 21 22 || 23 24 Sine. Cosine. Sine. Cosine. Sine. 'Cosine. ,,,. Cosine. Sine. Cosine. 34202 93969 35837 93358 3746i;927i8 39073 92050 40674 9*355 60 i 34229 93959 35864 93348 37488 92707 39*0 92039 40700 9*343 59 34257 93949 35891 93337 375*5 92697 39*27 92028 40727 9*33* 58 3 34284 93939 359*8 93327 37542 92686 39*53 92016 40753 9*3*9 57 4 J343*i 93929 35945 933*6 37569 92675 39180 92005 40780 9*307 56 5 34339 939*9 35973 93306 37595 92664 39207 9*994 40806 9*295 55 6 34366 93909 36000 93295 37622 92653! 39234 91982 40833 91283 54 7 34393 93899 36027 93285 37649 92642 39260 9*97* 40860 91272 53 8 3442* 93889 36054 93274 37676 92631 39287 9*959 40886 91260 52 9 10 34448 93879 34475 93869 36081 36108 93264 93253 37703 3773 92620 92609 393*4 3934* 9*948 9*936 40913 40939 91248 9*236 51 50 11 3453 93859 36*35 93243 37757 92598 39367 9*925 40966 91224 49 12 3453 93849 36162 93232 37784 92587 39394 9*9*4 40992 91212 48 13 34557 93839 36190 93222 37811 92576 39421 91902 41019 91200 47 14 15 34584 34612 93829 93819 36217 36244 932** 93201 37838 37865 92565 92554 39448 39474 91891 9*879 4*045 41072 91188 91176 46 45 16 34639 93809 36271 93190 37892 92543 395oi 91868 41098 91164 44 17 34666 93799 36298 93180 379*9 92532 39528 91856 4*125 91152 43 18 19 34694 93789 34721 93779 36325 36352 93*69 93*59 37946 37973 92521 92510 39555 3958i 9*845 9*833 41151 41*78 91140 91128 42 41 20 34748 93769 36379 93*48 37999 92499 39608 91822 41204 91116 40 21 34775 93759 36406 93*37 38026 92488 39635 91810 41231 91 104 39 22 34803 93748 36434 93*27 38053 92477 39661 9*799 4*257 91092 38 23 34830 9373 s 36461 93**6 38080 92466 39688 9*787 41284 91080 37 24 25 34857 34884 93728 93718 36488 365*5 93106 93095 38107 92455 92444 397*5 3974* 9*775 91764 4*3*0 4*337 91068 91056 36 35 26 27 349*2 34939 93708 93698 36569 93084 93074 38161 38188 92432 92421 39768 39795 9*752 91741 4*363 41390 91044 91032 34 33 28 34966 93688 36596 93063 382*5 92410 39822 9*729 41416 91020 32 29 34993 93677 36623 93052 38241 92399 39848 9*7*8 4*443 91008 31 30 35021 93667 36650 93042 38268 92388 39875 91706! 4*469 90996 30 31 35048 93657 36677 933* 38295 92377 39902 9*694 4*496 90984 29 32 3575 93647 36704 93020 38322 92366 39928 9*683! 41522 90972 28 33 35102 93637 36731 93010 38349 92355 39955 91671 4*549 90960 27 34 35*3 93626 36758 92999 38376 92343 39982 91660 4*575 90948 26 35 35*57 93616 36785 92988 38403 92332 40008 91648 41602 90936 25 36 35*84 93606 36812 92978 38430 92321 40035 91636 41628 90924 24 37 352*1 93596 36839 92967 38456 923*0 40062 91625 4*655 90911 23 38 35239 93585 36867 92956 38483 92299 40088 9*6*3 41681 90899 22 39 35266 93575 36894 92945 385*0 92287 40115 91601 4*707 90887 21 40 35293 93565 36921 92935 38537 92276 40141 91590 4*734 90875 20 | 41 35320 93555 36948 92924 38564 92265 40168 9*578 41760 90863 19 42 35347 93544 36975 929*3 3859* 92254 40195 91566 41787 90851 18 43 35375 93534 37002 92902 38617 92243 40221 9*555 41813 90839 17 44 35402 93524 37029 92892 38644 92231 40248 9*543 41840 90826 16 1 45 35429 935*4 37056 92881 38671 92220 40275 9*53* 41866 90814 15 46 35456 93503 37083 92870 38698 92209 40301 9*5*9 41892 90802 14 1 47 35484 93493 37*10 92859 38725 92198 40328 91508 4*9*9 90790 13 48 355 1 * 93483 37*37 92849 38752 92186 40355 9*496 4*945 90778 12 49 35538 93472 37*64 92838 38778 92*75 40381 91484 4*972 90766 11 50 35565 93462 92827 38805 92164 40408 91472 4*998 90753 10 51 35592 93452 37218 92816 38832 92152 40434 91461 42024 90741 9 52 53 35619 93441 3564719343* 37245 37272 92805 92794 38886 92141 92130 40461 40488 9*449 9*437 42051 42077 90729 90717 8 7 i 54 35674193420 37299 92784 389*2 92119 40514 9*425 42104 90704 6 1 55 3570*193410 37326 92773 38939 92107 I454* 9*4*4 42130 90692 5 1 56 35728 93400 37353 92762 38966 92096 40567 91402 42156 90680 4 57 35755 93389 9275* 38993 92085 40594 9*390 42183 90668 3 1 58 35782 93379 37407 92740 39020 92073 40621 91378 42209590655 2 59 35810 93368 37434 92729 39046 92062 40647 91366 42235 90643 1 1 60 35837 93358 92718 39073 92050 40674 9*355 42262 90631 1 Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. 69 68 67 66 65 92 NATURAL SIEVES A1TO COSINES. / 25 26 27 28 29 / ~w 59 58 57 56 55 54 53 52 51 50 Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. i 2 3 4 5 6 7 8 9 10 42262 90631 42288 90618 42315 90606 42341190594 42367^0582 42394 90569 42420190557 42446 90545 42473 90532 42499:90520 42525 90507 43863 43889 43916 43942 43968 43994 44020 44046 44072 44098 89879 89867 898.54 89841 89828 89816 89803 89790 89777 89764 89752 45399 45425 45451 45477 45503 455 2 9 45554 45580 45606 45632 45658 89101 89087 89074 89061 89048 89035 89021 89008 88995 88981 88968 46947 46973 46999 47024 47050 47076 47101 47127 47153 47178 47204 88295 88281 88267 88254 88240 88226 88213 88199 88185 88172 88158 48481' 48506 48532 48557 48583 48608 48634 48659 48684 48710 48735 87462 87448 87434 87420 87406 87391 87377 87363 87349 87335 87321 11 12 13 14 15 16 17 18 19 20 42552 42578 42604 42657 42683 42709 42736 42762 42788 90495 90483 90470 90458 90446 90433 90421 90408 90396 90383 44124 44151 44177 44203 44229 44255 44281 44307 44333 44359 89739 89726 89713 89700 89687 89674 89662 89649 89636 89623 45684 45710 4573 6 45762 45787 45813 45839 45 86 5 45891 459*7 88955 88942 88928 88915 88902 88888 88875 88862 88848 88835 47229 47255 47281 47306 47332 47358 47383 47409 47434 47460 88144 88130 88117 88103 88089 88075 88062 88048 88034 88020 48761 48786 48811 48837 48862 48888 48913 48938 48964 48989 49014 49040 49065 49090 49116 49141 49166 49192 49217 49242 49268 49293 49318 49344 49369 49394 49419 49445 49470 49495 87306 87292 87278 87264 87250 87235 87221 87207 87193 87178 49 48 47 46 45 44 43 42 41 40 21 22 23 24 25 26 27 28 29 30 42815 42841 42867 42894 42920 42946 42972 42999 43025 43 5i 90371 90358 90346 9334 90321 90309 90296 90284 90271 90259 44385 44411 44437 44464 44490 445 1 6 44542 44568 44594 44620 89610 89597 89584 89571 89558 89545 89532 89519 89506 89493 45942 45968 45994 46020 46046 46072 46097 46123 46149 46175 88822 88808 88795 88782 88768 88755 88741 88728 88715 88701 47486 475 11 47537 47562 47588 47614 47639 47665 47690 47716 88006 87993 87979 87965 87951 87937 87923 87909 87896 87882 87164 87150 87136 87121 87107 87093 87079 87064 87050 87036 87021 87007 86993 86978 86964 86949 86935 86921 86906 86892 86878 86863 86849 86834 86820 86805 86791 86777 86762 86748 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 i 31 32 33 34 35 36 37 38 39 40 4377 43 I0 4 43130 43156 43182 43209 43235 43261 43287 43313 4334 43366 4339^ 434i 8 43445 43471 43497 43523 43549 43575 90246 90233 90221 90208 90196 90183 90171 90158 90146 90*33 90120 90108 90095 90082 90070: 90057 90045 ! 90032 i 9 OOI 9 i 90007 | 44646 44672 44698 44724 4475 44776 44802 44828 44854 44880 89480 89467 89454 89441 89428 89415 89402 89389 89376 89363 46201 46226 46252 46278 46304 4633 46355 46381 46407 46433 88688 88674 88661 88647 88634 88620 88607 88593 88580 88566 47741 47767 47793 47818 47844 47869 47895 47920 47946 47971 87868; 87854 87840 87826 87812 87798 87784! 87770 87756| 87743 41 42 43 44 45 46 47 48 49 50 61 52 53 54 55 56 57 58 ' 59 60 / 44906 44932 44958 44984 45010 45036 45062 45088 45114 45140 89350 89337 89324 89311 89298 89285 89272 89259 89245 89232 46458 46484 46510 46536 46561 46587 46613 46639 46664 46690 88553 88539 88526 88512 88499 88485 88472 88458 88445 88431 47997 48022 48048 48073 48099 48124 48150 48175 48201 48226 877291 87715 87701 87687 87673 87659! 87645! 87631 | 87617! 87603 49521 49546 49571 49596 49622 49647 49672 49697 49723 49748 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 I / 43602 89994 43628189981 1 43654 89968 43680 89956 43706 89943 43733 89930 43759i 8 99 l8 43785 89905 43811189892 43837 89879 45166 45192 45218 45243 45269 45295 45321 45347 45373 45399 89219 89206 89193 89180 89167 89153 89140 89127 89114 89101 46716 46767 46793 46819 46844 46870 46896 46921 46947 88417 88404 88390 88377 88363 88349 88336 88322 88308 88295 48252 48277 48303 48328 48354 48379 48405 48430 48456 48481 87589 87575 87561 87546] 8 753 2j 87518) 87504) 87490 87476 87462 49773 49798 49824 49849 49874 49899 49924 49950 49975 50000 86733 86719 86704 86690 86675 86661 86646 86632 86617 86603 Cosine. Sine. Cosine. Sine. Cosine. Sine. Co&ine. Sine. Cosine. Sine. 64 63 62 61 60 NATURAL SINES AND COSINES. l 2 3 4 5 6 7 8 9 10 30 31 32 33 34 ' Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. 50000 50025 50050 50076 50101 50126 50151 50176 50201 50227 50252 86603 86588 86573 86559 86544 86530 865*5 86501 86486 86471 86457 5*5*9 5*554 5*579 51604 51628 5*653 5*678 5*7*8 5*753 85717 85702 85687 85672 85657 85642 85627 85612 85597 85582 85567 5*99* 53017 53066 539* 53**5 53*40 53*64 53189 53**4 53*38 84805 84789 84774 84759 84743 84728 84712 84697 84681 84666 84650 54464 54488 545*3 54537 54586 54610 54635 54659 54683 54708 83867 83851 83835 83819! 83804 83788 83772 83756 83740 83724 83708 559*9 55943 55968 5599* 56016 56040 56064 56088 561 12 56136 56160 82904 82887 82871 82855 82839 82822 82806 82790 82773 82757 82741 60 59 58 57 ' 56 55 54 53 52 51 50 11 12 13 14 15 16 17 18 L9 20 50277 50302 50327 5035.2 50377 50403 50428 5453 50478 50503 86442 86427 86413 86398 86384 86369 86354 86340 86325 86310 5*778 5*803 51828 51852 5*877 51902 5*9*7 5*977 52002 8555* 85536 85521 85506 85491 85476 85461 85446 85431 85416 53*63 53*88 533** 53337 53386 534** 53435 5346o 53484 84635 84619 84604 84588 84573 84557 84542 84526 84511 84495 5473* 54756 5478i 54805 54829 54854 54878 54902 549*7 5495* 83692 83676] 83660 83645 83629 83613 83597 83581 83565 83549 56184 56208 56232 56256 56280 56305 56329 56353 56377 56401 82724 82708 82692 82675 82659 82643 82626 82610 82593 82577 49 48 47 46 45 44 43 42 41 40 38 37 36 35 34 33 32 31 30 21 22 23 24 25 26 27 28 29 30 ~31 32 33 34 35 36 37 38 39 40 11 43 44 45 46 47 48 49 50 52 53 54 55 56 57 58 59 60 50528 50553 50578 50603 5.0628 50654 50679 50704 50729 50754 86295 86281 86266 86251 86237 86222 86207 86192 86178 .86163 52026 52051 52076 52101 52126 5**5* 52175 52200 52225 52250 85401 85385 85370 85355 85340 853*5 853*0 85*94 85*79 85264 53509 53534 53558 53583 53607 5363* 53656 53681 53705 5373 84480 84464 84448 84433 844*7 84402 84386 84370 84355 54975 154999 55o*4 55072 55097 55121 55*45 55*69 55*94 55218 55242 55266 55291 553*5 55339 55363 55388 55436 83533 83517 83469 83453 83437 83421 83405 83389 564*5 56449 56473 56497 56521 56545 [56569 !5 6 593 56617 56641 82561 82544 82528 82511 82495 82478 82462 82446 82429 82413 50779 50804 50829 50854 50879 50904 50929 50954 50979 51004 86148 86133 86119 86104 86089 86074 86059 86045 86030 86015 5**75 52299 5*3*4 5*349 5*374 5*399 5*4*3 5*448 5*473 52498 85249 85234 85218 85203 85188 85173 85*57 85142 85127 85112 53754 53779 53804 53828 53853 53877 53902 539*6 5395* 53975 843*4 84308 84292 84*77 84261 84245 84230 84214 84198 84182 83373 83356 83340 833*4 83308 83292 83260 83*44 83228 156665 82396 56689 82380 56713 82363 [56736 82347 56760 82330 5678482314 56808 82297 56832182281 56856 82264 56880 82248 29 28 27 26 25 24 23 22 21 20 51029 5*054 5*079 51104 5*i*9 51179 51204 51229 51254 86000 35985 85970 85956 85941 85926 85911 85896 85881 85866 52522 5*547 5*57* 5*597 52621 52646 52696 5*7*o 5*745 85096 85081 85066 85051 85035 85020 85005 84989 84974 84959 54000 54024 5449 54073 5497 54*22 54*46 54*7* 54*95 5422Q 54244 54269 54*93 543*7 5434* 543 6 6 5439* 544*5 54440 54464 84167 84151 84*35 84120 84x04 84088 84057 84041 840*5 84009 83994 83978 83962 83946 83930 839*5 83899 83883 83867 55484 55509 55'533 55557 5563 55654 55678 83212 83*95 83*79 83163 83*47 83131 83115 83098 83082 83066 56904 56928 56976 57000 57024 i5747 57071 57095 57**9 57*43 157*67 57191 57215 57*38 57262 57286 573* 57334 57358 82231 82214 82198 82181 82165 82148 82132 82115 82098 82082 19 18 17 16 15 14 13 12 11 I 10 51279 85851 5130485836 5*3*9 85821 51354 85806 5*379 85792 51404 85777 5*4*9 85762 51454185747 51479 85732 51504 85717 5*770 5*794 52819 52844 52869 5*893 5*9*8 52943 5*967 5*99* 84943 84928 84913 84897 84882 84866 84851 84836 84820 84805 [55702 55726 [55750 55775 55799 55847 5587* 55895 1559*9 83050 83034 83017 83001 82985 82969 82953 82936 82920 82904 82065 82048 82032 82015 8*999 81982 81965 81949 81932 8*9*5 9 8 7 6 5 4 3 2 1 ' Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. ' 59 58 57 | 56 55 94 NATURAL SINES AND COSINES. ~b~ i 2 3 4 5 6 7 8 9 10 35 36 37 38 39 60 59 58 57 56 55 54 53 52 51 50 Sine. Cosine. Sine. Cosine. Sine. 6oT82 60205 60228 60251 60274 60298 60321 60344 60367 60390 60414 Cosine. Sine. Cosine. Sine. Cosine. 57358 57381 57405 57429 57453J 57477 575 01 57524 57548 57572 57596 8I9I5 81899 81882 81865 81848 81832 81815 81798 81782 81765 81748 58779 58802 58826 58849 58873 58896 58920 58943 58967 58990 59014 80902 i 80885! 80867 80850! 80833 80816 80799 80782 80765 80748 80730 79864 79846 79829 79811 79793 79776 79758 79741 79723 79706 79688 61566 61589 11612 61635 61658 61681 61704 61726 61749 61772 61795 78801 78783 78765 78747! 787291 78711 78694 78676 78658 78640 78622 &, 62977 63000 63022 63045 63068 63090 63113 63^58 77696 77678 77 66o 77641 77623 77605 77586 77568 7755 7753 1 11 12 13 14 15 16 17 18 19 20 57619 57643 57667 57691 57715 57738 57762 57786 57810 57833 81731 81714 81698 81681 81664 81647 81631 81614 81597 81580 59037 59061 59084 59108 59131 59 J 54 59178 59201 59225 59248 89713 80696 80679 80662 80644 80627 80610 80593 80576 80558 60437 60460 60483 60506 60529 60553 60576 60599 60622 60645 79671 79653 79635 79618 79600 79583 79565 79547 7953 79512 61818 61841 61864 61887 61909 61932 61955 61978 62001 62024 78586 78568 78550 78532 78514 78496 78478 78460 78442 63180 63203 63225 63248 63271 63293 63316 63361 6 _3.3l3 63406 63428 63451 63473 63496 63518 63585 63608 77513 77494 77476 77458 77439 77421 77402 77384 77366 ZZ.347 77329 77310 77292 77273 77255 77236 77218 77199 77181 77 l62 49 48 47 46 45 44 43 42 41 40 ~3 :i u ,i& 04951= DEC 26 1944 *v LIBRAPV jftN 28 1946 JUL 2 1 1955 NOV 9 1946 JUL211955LU icocr ! -i ; ^ ^i , if . - v ot k .. . ^ . icQi rM 7JW-" 10AprtSlC| F?Fpn i ^ 24Mav'5lLi -V,. a J . J^IJ tsn ' v -' * ~ , ffiPB w * LD 21-100m-7,'39(402s) F3525. U.C.BERKELEYLIBRA1R1ES - UNIVERSITY OF CALIFORNIA LIBRARY