In 1 \ TREATISE S U ET" NG: IN "WHICH THE THEORY AND PRACTICE ARE FULLY EXPLAINED. PRECEDED BY A SHORT TREATISE ON LOGARITHMS: AND ALSO BY A COMPENDIOUS SYSTEM OF PLANE TRIGONOMETRY. ®Ije bfyok lllnstraleb bg *§nmzxoms dkamples. BY SAMUEL ALSOP, PHILADELPHIA: E. C. & J. BIDDLE & CO., No. 508 MINOR ST. (Between Market and Chestnut, and Fifth and Sixth Sts.) 1860. ~Tt\5J\5 . A46 \8 6 Entered according to act of Congress, in the year 1S57, by E. C. & J. DIDDLE, in the Clerk's Office of the District Court of the United States for the Eastern District of Pennsylvania. STEREOTYPED IiT L. J0HN8ON & CO. PHILADELPHIA. A Key to this -work has been published by E. C. & J. B. Let /g'7XSO 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. 6 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 bearings 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 bo 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 maybe conveniently employed in the field. Chapter YI. 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 YII. 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 Number? has been carefully compared with those of Babbage, Hutton, and other standard authors. That of Sines and Tangents was 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. pagb. 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 Srotion 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 9 10 CONTENTS. CHAPTER III. PLANE TRIGONOMETRY. Section 1. Definitions. PAQE 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. Rule 1 70 Rule 2 71 The three Sides being given, to find the Angles. Rule 1 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 Eve 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 93 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 CHAPTEE 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. PAGB 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... 102 The Magnetic Meridian • 163 CONTENTS. 13 PAGE The Magnetic Bearing 163 The Compass 164 The Sights ,, 166 The Verniers 166 The Pivot 168 The Divided Circle 168 Adjustments , ; 160 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 inaccesi'ble 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 bf 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 through a 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 285 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 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. 11 2 1 64 6 2048 4 2 128 7 4096 12 8 3 256 8 8192 13 16 4 512 9 16384 14 32 5 1024 10 32768 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 extracted, 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. L] 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 $ some one base, we will have, if a be the base, and n tide *number, <2* = 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, 12 3 4 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 -j- 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. Mantissae. 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 mantissae 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 mantissae of their logarithms, so arranged that Sec. I.] THE NATURE OF LOGARITHMS. 21 one can be readily determined from the other. In the table of logarithms appended to this treatise, the mantissas 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 mantissas of the logarithms. The first column, headed N", contains the numbers, from 100 to 999 ; and the second, headed 0, the mantissas 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 mantissas 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. l 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 the 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 !N", 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. L] THE NATURE OF LOGARITHMS. 23 tiplier ; then add the remainder to the logarithm first taken out : the sum will be 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 fi.ve 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 ^.ve 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 Next less .649378 .649335 cor. num. 4460 Difference 43 Next greater logarithm Next less .649432 .649335 Difference 97)4300(44 388 420 388 Hence, the number is 446.044. 32 Examples. Required the natural numbers corresponding to the fol- lowing logarithms. Sec. II.] ON THE USE OF LOGARITHMS. 25 1. 2.467415 2. —1.396143 3. 2.04163T 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 aflirmative 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. 47.25 log. 1.674402 397.3 " 2.599119 Product, 18772.5 4.273521 Ex. 2. Required the product of 764.3, .8175, .04729, and .00125. 764.3 log. 2.883264 .8175 " —1.912488 .04729 « —2.674769 .00125 « —3.096910 Product, .0369344 —2.567431 Ex. 3. Required the product of 87.5 and 6.7. Ans. 586.25. 26 THE NATURE AND USE OF LOGARITHMS. [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 fey 87 log. 1.939519 Quotient, 77.471 1.889141 Ex. 2. Divide 86.47 log. 1.936865 fey .0124 log. - -2.093422 Quotient, 6973.4 3.843443 Ex. 3. Divide .0642 log. - -2.807535 fey 87.63 log. 1.942653 Quotient, .00073263 -4.864882 Ex. 4. Divide .0642 log. - -2.807535 fey .008763 log. - -3.942653 Quotient, 7.3263 0.864882 Ex. 5. Divide 407.3 by 27.564. Ans. 14.7765 Ex. 6. Divide . 80743 by 63.87. Ans. .012642 Sec. II. ] ON THE USE OF LOGARITHMS. 27 Ex. 7. Divide 963.T 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. Eequired the fourth power of 5.5. 5.5 log. 0.740363 4 915.065 2.961452. Ex. 2. Eequired the fifth power of .63. .63 log. —1.799341 5 .099244 —2.996705. Ex. 3. Eequired the fourth power of 7.639. Ans. 3405.24. Ex. 4. Eequired 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 Result, 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 be pays $215, to E be pays $75, from F be re- ceives $437, and, finally, pays to G $137. How mucb has he left? 375^ 104 —215 — 75 437 —137 Ans. 489 r which are added as though they were f375 104 785 925 437 863 v. | 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 — : -j=- 125 : x. 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 „ 5o x 475 x 125 u ^ *-'- L 27 x 17 log. 27 / 1.431364") \ 1.230449 which are fA. C. 8.568636 " 17 added as A. c. 8.769551 " 55 1.740363 \- though [ 1.740363 " 475 2.676694 they were 2.676694 " 125 2.096910 J written < 2.096910 Result, 7114.66 3.852154 3.852154 30 THE NATURE AND USE OF 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 Geometr}^. 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 point, 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- I-) A. 30. An angle may be read either by the single letter at Sec. I.] 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 BAC. 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, Fig# 2 . 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. 3 34 PRACTICAL GEOMETRY. [Chap. II. 41. 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 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. L] DEFINITIONS. 35 51. All quadrilaterals that are not parallelograms 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. Fig. 5., 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. 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 oft* by the diameter. ABC and AEB (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. 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. 68. If two straight lines inter- sect each other, the angles verti- cally opposite are equal. Thus, AEC (Fig. 8) = BED, and AED = BEC. (15.1.) Fig. 8. 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- rallel lines, the angles similarly situ- ated in respect to these lines, and also those alternately situated, will be equal to each other (29.1.) Thus, (Fig. 9,) EFB = FGD, BFG = DGH, AFE = CGF, and AFG = CGH, being similarly situated ; and AFE = DGH, EFB = CGH, AFG = FGD, and BFG = FGC, being alternately situated. B 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 DGII 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 + BAC, and 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. 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, describe arcs cutting in C and D. Join CD cutting AB in E, and the thing is done. (10.1.) Fig. 12. s^ 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. (ii.i) Fig. 13. ,-¥' F 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- ~~ 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 CG be the perpendicular. Fig. 14. d/ E N \ ^ >'o, Fig. 15. >Cg E v .^ — 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 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.) A Fig. 16. C \ G D\ X E V <* B 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.) A- C E\ -f— B ^ 93. Second Method.— Lei 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. 42 PRACTICAL GEOMETRY. [Chap. II. Problem 5. — To bisect a given angle. 95. Let BAC (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.) 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) 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 EF equal to AD. Through A and F draw AF, which will be the parallel required. Fig. 21. B E D 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. 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.) Problem 8. — To inscribe a circle in a given triangle. 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. Fig. 26. Problem 10. — To find a third proportional to two straight lines. 102. Let M and S" (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 N. Join DF, and draw EG parallel to it. AG will be the third proportional re- quired. (11.6.) E D M- N- Problem 11. — To find a fourth proportional to three given straight lines. 44 PRACTICAL GEOMETRY. [Chap. IL 103. Let M, N, 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- rallel to it : then AG is the fourth proportional required. (12.6.) Fig. 27. M N Fig. 28. Problem 12. — To find a mean proportional between two straight lines. 104. First Method. — Place the lines 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 M*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.) Make AE = AD. Note. — This is a very convenient construction, and is often employed in the Division of Land. Fig. 30. Problem 13. — To divide a given line into parts having same ratio as two given numbers M and N". 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.) the 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 35 is oFa of the circumference. An arc of 180° is half the 45 46 PLANE TRIGONOMETRY. [Chap. III. Fig. 31. 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 the point A (Fig. 31) two circles BCD and EFGr 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 EFG. 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 formulae. The radius adopted in the 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. HO. 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. 1U. 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 + 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 = E 3 .) This is evident from the right-angled triangle CFB, (Fig. 32.) (47.1.) 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 : E : : E : cotan. a, or tan. a. cot. a = E 2 . This is evident from the similarity of the triangles ACT and DKC, (Fig. 32,) which give (4.6) AT : AC : : CD : DK. J- 1 ; .' 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 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. n. 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 V 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.) >kV.\ 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 should, 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- Fig. 33. 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. Fig. 34. C Sec. II.] DRAFTING OR PLATTING. 51 Fig. 35. 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 whether it is so, bring one leg 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 with 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. 3T. 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 BAC of 47°. Fie. 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 BAC 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. Bight 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 Guntcr'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. ! \ i .. s 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. Q A 2 4 6\ 8 B i D 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 ofT 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. 1 60 "in . [ran™ J- 2 3 ' 4f 5 6| 7I s| 9| a |p 1 2 3| 4 5| 6| 7 50 m_ it. »i si * 5] 6| r.l s| 9 l|o l| 2| s| 4 - 45 ■.V'l'M'J l| 2,] . Z.\ 4-! 5| 6[ 7|' 1 8 9t . lid ll 2I 3l ii)lJ!J||] - 40 njff|tt|fli ll sl 3J 4,| 5l. _ 6.1 . 7 sl . 9|< i|o l\ 35 ij Z\ Z\ 4r\ b\ 6 7 1 & 1 9 1 aio IJ II ! 11 1 1 1 '• 1 ° 1 "1. - L l u 40 Li.juj.'iji. 2 3, 4 5 *4 T »4 1 Ill 1 |-ll|| 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. 100 2100 i in it 88 66 144 II is a vernier. ISTow, 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 Q of ^ = ^ 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 176. 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 b^> 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 OF 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 V 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. 1" 4.35 25 2175 87 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 7 71 128 5 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. Eequired 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" 227 4 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 17 80 14 28 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" 76 8 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. 54 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 RIGHT-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 leg 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 similar triangles we have — 1. AC 2. AC 3. AB CB AB BC AE AE AD EF AF DG A r : sin. A. Rule 1 ; r cos. A. Rule 2 ; r tan. A. Rule 3. ¥ D 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 Fig. u. from a scale of equal parts. At the point A make the angle BAC = 35° 27'. Erect the perpendicular BC, meeting AC in C, and ABC is the triangle required. Calculation. Eule 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° 1' 3" 5 66 PLANE TRIGONOMETRY. [Cn EULE 1. As rad. 10.000000 : sin. A 41° 58' 57" 9.825363 :: AC 63.90 1.805501 : CB „ 42.74 1.630864 Ex. 3. Given the two legs AB = 59.47 yards, and BC = 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, to find 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. Rule 1. AB = n/AC 2 - BC 2 = ^2025 - 841 = s/1184 = 34.41. or, AB = •(AC + BC).(AC - BC) = x/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 Rule. 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 + 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. Calculation. As sin. C 74° 52' 10" A. C. 0.015322 : sin. B 39° 47' 20" 9.806154 :: AB 123.5 2.091667 : AC 81.87 1.913143 As sin. C A. C. 0.015322 : sin. A 65° 20' 30" 9.958474 :: AB 2.091667 : BC 116.27 2.065463 Ex. 2. Given the side AB = 327, the side BC = 238, and the angle A = 32° 27', to determine the rest. 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. Fig. 46. Calculation. Bule 2. AsBC 238 A. C. 7.623423 : AB 327 2.514548 : : sin. A 32° 47' 9.733569 : sin. C 48° 4' 6" 9.871540 or 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 433.97 C obtuse. 2.637459 As sin. C 131° 55' 54" A. c. 0.128460 : sin. B 15° 17' 6" 9.420979 :: 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 7 , in which it is obtuse. If BC were greater than BA, the point C' 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. BULE 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 pla?ie triangle 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 -j- ACF = ABC 4-ACB. Whence AFC = J (ABC -f AC B) ; 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 + ACB) : tan. £ (ABC — ACB). Examples. Ex. 1. Given AB = 527 yards, AC = 493 yards, and the angle A =37° 49'. Here C + B = 180° - 37° 49' = 142° 11', and Sec. IV.] NUMERICAL SOLUTION OF TRIANGLES. 71 As AB + AC 1020 A.C. 6.991400 : AB-AC 34 1.531479 C+B ::tan. g 71° 5' 30" 5° 33' 29" 10.465290 C-B : tan. — - — 2 8.988169 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. Eule 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. ,2 PLANE TRIGONOMETRY. [Chap. III. Demonstration. — Let ABC (Fig. 48) be any p ^ig. 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 = BDE ; and since EBD is a right angle, BDE = 45°. But BED + BDE = 2 BDE = BAD -f 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 -f- BD : AB — BD : : tan. J (BDA -|- BAD) : tan. £ (BDA— BAD) : : tan. BDE : tan. ADE; but BDE being equal to 45°, its tangent == rad. And ADE = (ADB — 45°) . •. AB + BD : AB — BD : : r : tan. (ADB — 45°) ; but AB + BC : AB — BC : : tan. J (ACB -f- BAC) : tan. £ (ACB — BAC) ; whence r : tan. (ADB — 45°) : : tan. £ (ACB + 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. As AB A. C. 7.403613 : BC 2.846392 : : Rad. 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" A 89° 57 ; 58" 9.723372 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. Rule 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 Rule 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 + 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. IIL 1 As AB 467 Ar. Co. 7.330683 J : AC + BC 807 2.906874 1 ::AC -BC 19 1.278754 I : AD -DB 32.833 1.516311 I J(AD- -DB) 16.4165 1 JAB 233.5 1 AD 249.9165 1 BD 217.0835 1 As AC 413 Ar. Co. 7.384050 1 : AD 249.9165 2.397794 J : :r 10.000000 J : cos. A 52° 45' 44" 9.781844 1 As BC 394 Ar. Co. 7.404504 1 : 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 f 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". Eule 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 EH is equal and parallel to CB, BH is jjV 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. describe a circle, and it will pass through F. On EH Now,2AK = 2AG4-2GK=AC+AD+2DG + 2GK=AC + AB-f-BC; or AK == J (AC -f AB + BC), and AI = AK— KI = J (AC + AB -f BC) — BC. But, (Rule 2, Art. 137,) As AD : AE and AB : AF whence (23.6) AB . AD : AE . AF : r : cos. DAE (cos. J BAC),' : r : cos. \ BAC ; : r» : cos. 3 \ BAC. But (36.3, Cor.) AE . AF = AK . AI = \ (AC + AB + BC) . \ (AC -f AB + BC) — BC; whence AB . AC : \ (AC + AB -f BC) • {\ (AC -f AB -f BC)— BC) cos. 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 5 — AB = 170; whence . n p ,fAC 413 S AC - BC {bC 394 A.C. 7.384050 A.C. 7.404504 . (s 637 : 5 .( 5 -AB)j 5 _ AB170 2.804139 2.230449 ::E 2 20.000000 : cos. 2 JBCA 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 AB = 167, AC = 214, and BC = 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 V. INSTRUMENTS AND FIELD OPERATIONS. 144. The Chain. Gunter'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 #9.37079 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 563.67 , foot. Thus, 563 feet 8 inches = 563.67 feet= eh. = 8.54 chains. bb 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 Pins. 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 fixe 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 his iron pins ; he then goes on one chain, and, sticking a 80 PLANE TRIGONOMETRY. [Chap. III. 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. III. Slope, Correction Correction Correction 1 Correction An S le - ner of base, in of hypoth. Angle. Slope. of base, in of hypoth. pei links. in links. links. 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 5.45 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 5.7 1.52 1.54 24° 1 2.2 8.65 9.46 11° 1 5.1 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 IY. 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. 84 PLANE TRIGONOMETRY [Chap. III. THE TRANSIT. Fig. 51. Sec. V.] INSTRUMENTS AND FIELD OPERATIONS. 85 THE THEODOLITE. Fig. 52. 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 giving 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 N". 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 screw 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 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- tinct vision. Fig. 53. D Fig. 54. 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 Sec. V.] INSTRUMENTS AND FIELD OPERATIONS. 89 towards each, other. This arrangement is known as " Ramsden'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,) some of which are invisible in the figure. These screws 00 PLANE TRIGONOMETRY. [Chap. III. serve to hold the ring in position, as 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 (gowo) 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 fixed by a little gum. To iix 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 must be remedied by removing the support which is too high and filing some of! 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 rilled 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 Gr 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 O 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 TJ. 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 R, 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 360 decrees there will be =19.1 divisions in the space 6 18.849 * 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 1 J times the natural size. Fig. 56. / S 10 . o ~i — r i D | C TTT 3 1 2 9 1 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 the vernier 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 will fall ^ of an inch below, the next jjfo, and so on. The 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 he placed as in Fig. 57, the reading will be 28.74 inches. Fig. 57. 1 5 V | 3 2 9 2|8 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 Reading 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, — observe how 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 nine on the scale : the reading is therefore to ^ of ^ = ^ 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 DQ PLANE TRIGONOMETRY. [Chap. III. spaces on the arc, the reading would be to ^ of \° = ^° = 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. Fier. 58. 97 / 1 , 1 I 1 I 3 2 9 2 8 The value of one division of the vernier is y 1 ^ 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 ^ below; and so on. If the vernier is raised so that the 1 on the vernier is in line, it is raised i^o i n ch ; if 2 is in line, it is raised ^ ; and so on. The reading in Fig. 58 is 29.7 inches, and in Fig. 59, 29.53 inches. Fig. 59. V I 6 1 1 1 | 1 3 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 r 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. 99 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. Fig. 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. III. 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 ^ 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 EI : EG : : number A of divisions on the vernier : the number f 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, b, 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 FIELD OPERATIONS. 101 division on AB ; then, keeping it pressed against the needle, bring it successively to the other points on GIF, 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 oh = 1.2, the degrees of ah 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. 103 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, by the screws a, a, and verify again. 188. Fourth Adjustment. The line 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- 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 c acute, when the telescope 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. D- G F 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 a, a, (Fig. 51,) move the diaphragm horizontally till the vertical line passes through II. Verify the adjustment; and, if not precisely correct, repeat the operation. 189. The above method is inapplicable to the theodolite, as its telescope does not turn over. For the means of detecting and correcting the error, see Art. 190. 190. Fifth Adjustment. The horizontal and the vertical axes should be perpendicular. Verification for the Transit. Suspend a long plumb-line from some elevated point, allowing the plummet to swing in a bucket of water ; then level carefully, and bisect the line accurately by the vertical wire. If, on elevating and depressing the telescope, the line is still bisected, the ad- justment is good. If not, the error may be corrected by filing one of the frames. Instead of a plumb-line, any ele- vated object and its image, as seen reflected from the surface of mercury or of molasses boiled to free it from bubbles, may be employed. Verification for the Iheodolite, If the instrument, treated as above, shows a defect, the error may be either in the axis, or in the position of the Y's. To determine which, turn the plates half round, and reverse the telescope. If the deviation is now on the same side as before, the Y's are in fault. Their position in most instruments may be cor- rected by screws which move one of them laterally. If the line deviates to the opposite side from before, the position of the axis may be corrected by filing, as directed for the transit. This adjustment may also be examined by directing the telescope to some well defined elevated object, and then to Sec. V.] INSTRUMENTS AND FIELD OPERATIONS. 105 another on or near the ground. If none such can be found, let one be placed by an assistant ; then reverse the telescope in its Y's if the instrument is a theodolite, or turn it over if the instrument is a transit, and direct it to the upper object. If the cross-wires still intersect upon the lower point when the tube is depressed, the adjustment is perfect. 191. Adjustments of the Vertical Limb. Having verified the various adjustments for horizontal motion, as described in the preceding articles, and rectified them if defective, the instrument is ready for use for horizontal work. To take angles of elevation, or to use the instru- ment for levelling, the following adjustments must also be examined : — 1. The level beneath the telescope must be parallel to the line of collimation. 2. The zero of the vernier must coincide with the zero of the vertical limb when the plates are level and the tele- scope horizontal. 192. First Adjustment. The level must be parallel to the line of collimation. Verification. Select a piece of level ground, and drive two stakes, A and B, (Fig. 68,) four or &ve chains apart. At C, equidistant from them, set the instrument. Level the plates, and bring the bubble in the telescope level, to the middle of its run ; then let an assistant hold a graduated staff on A. Note exactly the point in which the line of sight meets the staff: then let the assistant remove the staff to B, and drive the stake B until the telescope points Fisr. 68. to the same spot on the staff. The tops of A and B are then level, whether the instrument is in adjustment or not. 106 PLANE TRIGONOMETRY. [Chap. III. !Now remove the instrument to G, and level as before. Direct the telescope to the staff on B, and note the point I of intersection. Let the assistant carry the staff to A. Again note the intersection K. If the instrument is properly adjusted, these two points will coincide. If they do not, the line of collimation points too high or too low. Take the difference between BI and AK: This differ- ence will be LK, the difference of level as given by the instrument at G. Then say, As the distance between the stakes (BA) is to the distance from the instrument to the far stake (GA), so is the difference of apparent level of the stakes (LK) to the correction on the far staff (MK). This correction — either taken from the height AK if too great, or added to it if too small — will give AM, the height of a point on the same level as the instrument. Direct the telescope to this point, and rectify the level, by raising or lowering one end by means of the capstan screw until the bubble is in the middle of its run. If the operation has been carefully done, the adjustment is perfect. Verify again ; and, if needful, repeat the operation. 193. Second Adjustment. The zeros of the vernier and of the vertical limb should coincide when the telescope is level. When the first adjustment is perfected, and the telescope is still level, examine the reading on the vertical limb care- fully: if the zeros coincide, the vernier is properly ad- justed ; if they do not, note the error, and have it marked somewhere on the instrument under the plates, that it may not be forgotten. It must be applied to all angles of eleva- tion taken by the instrument. If the index-arm is movable, as is frequently the case with transits, it should be adjusted before taking vertical angles. 194. When all the preceding adjustments have been exa- mined, and rectified if necessary, the instrument is ready for work. It would be well, however, to examine carefully the reading of the verniers, to see that they are properly divided. However placed, no two lines of the vernier Sec. V.] INSTRUMENTS AND FIELD OPERATIONS. 107 except the first and last should coincide with divisions on the arc. If two are found to do so in any position, there is an imperfection in the graduation. If the division is very fine, a number of lines in the immediate neighbor- hood of the coincident lines will differ very slightly from coincidence; but, when carefully examined with a good magnifier, they should recede gradually. Place the instrument where a good view of a fine point, some eight or ten chains distant, can be obtained. Level carefully, direct the line of sight to the point, and note the reading on the horizontal limb. Reverse the telescope in its Y 's, or, if the instrument is a transit, turn it over ; turn the vernier-plate till the line of sight passes again through the point, and note the reading. It should differ by 180° from that before obtained. If it does not, the divisions are not perfect, or the telescope is not over the centre of the plates. Either defect should condemn the instrument, as it can be remedied only by the maker. This verification should be tried in various positions of the divided plate. If these tests, and those formerly mentioned, are found to detect no imperfection, the instrument may be pronounced a good one. 195. Taking Angles. Set the instrument precisely over the angular point, and level it, being careful to have the levelling screws pressed tightly against the plates, that the instrument may be steady. Set the index to zero, and clamp the plates, and, if there be more than one vernier, note the minutes and seconds of the others. Loosen the lower clamp, and bring the telescope so that the wires may intersect on the left-hand object; clamp, and perfect the ad- justment by the tangent screw. If there is a watch-tele- scope, set it upon some well-defined object, — such as a light- ning-rod or the corner of a chimney, — and clamp it tightly. Loosen the vernier-plate, and turn the telescope to the other object, perfecting the adjustment by the tangent screw. Examine the watch telescope, and, if the instru- ment has shifted, bring it back by the tangent screw, and readjust the telescope by moving the vernier-plate. 108 PLANE TRIGONOMETRY. [Chap. III. Now read the arc by the same index as before, noting the minutes and seconds by the other verniers. Take the mean of the minutes and seconds of each position for the true reading. Then the true reading in the first position taken from that in the second will give the angle required. It is convenient to have a table prepared, with the requisite number of columns, in which to set down the readings of the different verniers. Thus, suppose there were three verniers, 120 degrees apart : rule a table, with six columns, as below : — Occd. Sta. A A Obs. Sta. B C A B c 0° 0' 0" 75° 8' 15" 0' 30" 8' 0" 59' 45" 8' 30" Mean. 0° 0' 7|" 75° 8' 15" The first column is the occupied station; the second, the observed station ; the next three the readings of the verniers, and the sixth the mean. In the case above, the angle BAC would be 75° 8 r 7J". The instrument is supposed to read to 30", the 15" being taken when two lines on the vernier appear equally near coincidence. 196. Repetition of Angles. The following method of observation is sometimes employed. Suppose the angle ABC is to be measured, A being the left-hand object : direct to A, and turn to B as above directed. Clamp the vernier- plate and loosen below, and bring the telescope again to A. Clamp below, loosen the vernier, and bring the telescope again to B. The index has now traversed an arc measuring twice ABC. The operation may be repeated as often as desired, noting the number of whole revolutions the tele- scope has made. Then divide the whole number of degrees by the number of repetitions. The result will be the degrees of the angle required. If there is a watch-telescope, it should be set carefully before each observation. When this is done, and proper care is taken to avoid deranging Sec. V.] INSTRUMENTS AND FIELD OPERATIONS. 109 the instrument, the result may be depended on as more accurate than any single reading. Any error in the final reading, being divided by the number of observations, will affect the result by but a small part of its value. 197. Verification of the Angles. When it is possible to do so, all the angles of a triangle should be measured. If their sum does not make 180°, there must be an error somewhere. Should the error be considerable, the work ought to be reviewed. But if it does not exceed two or three minutes, providing the instrument only reads to minutes, it may be distributed equally among the three angles, should there be no reason to suppose one is more accurate than another. But if more observations have been taken for some angles than for others, their determination should be most depended on, and a proportionally less part of the correction assigned to them. Suppose, for example, the angle A is the mean of five observations, B of three, while at C but one was taken, the error being 1/ 45" : we would proceed thus : — As J + J + 1 : f : : V 45" : 14", the correction for A. In the same manner the correction for B would be found to be 23", and for C, V 08". 198. Reduction to the Centre. Where the objeet that has been observed is a spire or other portion of a building, it is impossible to set the instrument underneath the signal. In such cases, the observed angle must be reduced to what it would have been had the station been at the proper point. Thus, let C (Fig. 69) be the correct Fig. 69. station, and D the occupied station, which should be taken as near as possible to C. Take the angle ADB. Then if A, C, D, and B are all in the circumference of a circle, this will be equal to ACB. The station should be assumed as near this as possible. Calculate BC and AC from the distance AB and the angles observed at A and B. Also measure DC, either directly or by trigonometrical methods to be explained hereafter, and take ADC. 110 PLANE TRIGONOMETRY. [Chap. III. Then, (Art. 139,) As CA : CD : : sin. ADC : sin. CAD. And as CB : CD : : sin. BDC : sin. CBD. Hence, ACB - AEB — CAD = ADB + CBD — CAD, becomes known. Example. Let CA = 9647 ft. ; CB = 8945 ft. ; ADB = 68° 45'; DC = 150 ft,; and ADC = 97° 37'. As CA 9647 ft. A. C. 6.015608 : CD 150 ft. 2.176091 : : sin. ADC 97° 37' 9.996151 : sin. CAD 0° 52' 59" 8.187850 Ls CB 8945 ft. A. C. 6.048420 : CD 150 ft. 2.176091 : sin. CDB 166° 22' 9.372373 : sin. CBD 0° 13' 35" 7.596884 Whence ACB = ADB + CBD — CAD = 68° 5' 36". 199, Angles of Elevation. In measuring angles of ele- vation, the instrument must first be levelled; the telescope being then directed to the object, the reading of the vernier corrected for the index-error will be the angle of elevation. SECTION VI. MISCELLANEOUS PROBLEMS TO ILLUSTRATE THE RULES OF PLANE TRIGONOMETRY. Problem 1. Being desirous of determining the height of a fir-tree standing in my garden, I measured 100 feet from its base, the ground being level. I then took the angle of elevation of the top, and found it to be 47° 50' 30". Required the height, the theodolite being 5 feet from the ground. Sec. VL] MISCELLANEOUS PROBLEMS. Ill Fig. 70. Solution. Make AB (Fig. 70) equal to 100 feet; draw AD and BC perpendicular to AB, making the former five feet from the same scale. Draw DE parallel to AB, and make EDO = 47° 50', the given angle. Then will CB be the height of the tree. Calculation. As rad. : tan. EDO : : DE : EC = 110.45 feet whence BC = 110.45 + 5 = 115.45. Problem 2. One corner C (Fig. 71) of a tract of land being inaccessible, to de- termine the distances from the adjacent corners A and B, I measured AB = 9.57 chains. At A, the angle BAC was 52° 19' 15", and at B, the angle ABC was 63° 19' 45". Eequired the distances AC and BC. B Calculation. As sin. ACB (64°21') : sin. A (52° 19' 15") : : AB (957) : BC = 840.2 links. As sin. ACB (64° 21') : sin. B (63° 19' 45") : : AB : AC = 948.7 links. Problem 3. In measuring the sides Fi s- 72 - of a tract of land, one side AB (Fig. 72) was found to pass through a swamp, so that it could not be chained. I there- fore selected two stations, C and D, on fast land, and took the distances and angles as follows, — viz.: AC = 37.56 chains; CD = 50.25 chains; BAC = 65° 27' 30"; ACD = 123° 46' 20"; CDB = 107° 29' 15": the corner B being inaccessible, the distance BD could not be measured. Required AB. The angle CDA could not be taken, owing to obstructions. 112 PLANE TRIGONOMETRY. [Chap. III. Solution. Join AD. Then, from the triangle ACD. we have, (Art. 140,) CAD 4- CDA As CD + CA (87.81) : CD - CA (12.69) : : tan. 1 (28° 6' 50") : tan. CAD - CDA = 4° 24' 54"; whence CAD = 28° 6' 50" + 4° 24' 54" = 32° 31' 44", and CDA = 28° 6' 50" - 4° 24' 54" = 23° 41' 5G" ; then, sin. CDA : sin. ACD : : AC : AD = 77.68. Now, in ADB we have AD = 77.68, the angle DAB = CAB __ CAD = 32° 5b' 46", and the angle ADB = BDC - ADC = 83° 47' 19", to find AB; thus, As sin. B : sin. ADB :: AD : AB = 86.455 chains. Fig. 73. Problem 4. To determine the position of a point D on an island, I ascertained the distances of three objects on the main land as follows:— AB = 248.75 chains, BC = 213.25 chains, and AC = 325.96 chains. At D the angle ADB was found to be 29° 15', and BDC 20° 29' 30". Eequired the distance of D from each of the objects. Construction. With the given distances construct the triangle ABC. At C and A make the angles ACE = 29° 15', and CAE = 20° 29' 30". About AEC describe the circle ACD. Join EB, and pro- duce it to D, which will be the point required. For (21.3) ADB = ACE = 29° 15', and CDB = CAE = 20° 29' 30". Calculation. 1. In ABC we have the three sides to find the angle BAC = 40° 51' 30". 2. In CAE we have the angles and side AC to find the side AE = 208.705. 3. In BAE we have BA, AE, and the included angle BAE, to find ABE = 50° 55 f 48", AEB = 67° 43' 12". Sec. VI.] MISCELLANEOUS PROBLEMS. 4. In ABD we have the angles and side AB, to find AD = 395.24 and BD = 188.07. 5. In A CD we have the angles and sides AC, to find CD = 379. Problem 5.— Wishing to obtain the distance between tw trees, C and D, situated on Fig. 74. the side of a bill, and not being able to find level ground for a base, I select- ed a gradual slope, on which I measured the distance AB (Fig. 74) 400 yards. I then took the horizontal and ver- tical angles as follow: — At A, the angle BAD was 101° 47' 15", BAC 39° 25' 45". The elevation of B was 5° 32' 45", of C, 8° 19' 30", and of D, 12° 29'. At B, the angle ABD was 59° 13' 15", and ABC 125° 36' 45". Required the distance CD, and the elevations of C and D above A. Conceive a horizontal plane to pass through A, meeting vertical lines through B, C, and D in the points E, F, and G. Then, since the angular distances are measured horizontally, we have the following angles given, — viz. : EAG = 101° 47' 15", EAF = 39° 25' 45", AEG = 59° 13' 15", and AEF = 125° 36' 45". Calculation. 1. To find AE, we have r : cos. BAE (5° 32' 45") : : AB (400) : AE = 398.13. 2. To find AG. As sin. AGE : sin. AEG : : AE : AG = 1051.07, log. 3.021631. 3. To find AF. As sin. AFE : sin. AEF : : AE : AF - 1253.96, log. 3.098284. 4. TofindFG,(Art.l41.)AsAG:AF::r:tan.z==50°l'49". And, as rad. : tan. (x — 45°) : : tan. J (AGF + AFG) : tan. J (AGF - AFG) = 8° 16' 34"; then AGF = 58° 49' 15" + 8° 16' 34" = 67° 5' 49", and AFG = 58° 49' 15" - 8° 16' 34" = 50° 32' 41". 8 114 PLANE TRIGONOMETRY. [Chap. III. Then, as sin. AGF : sin. FAG : : AF : GF = 1205.9. 5. To find GD and CF. As r : tan. GAD : : AG : GD - 232.69 = Elevation of D. And as r : tan. CAF : : AF : FC = 183.49 = Elevation of C. 6. To find CD. CD = y/ CH 2 + HD 3 = 1206.9 = Dis- tance of CD. Problem 6. — Being desirous to determine the height of a tower standing on the summit of a hill, I measured 75 yards from its base down the declivity, which was a regular slope. I then took the elevation of the top, 49° 37' 45", and of the bottom, 8° 19', the height of the instrument being 5 feet. What was the height of the tower ? Ans. 76.44 yds. Problem 7. — To determine the height of a tree in an inaccessible situation, I took a station, and found the ele- vation of the top to be 38° 45' 15" ; then, measuring back 100 feet, the elevation was found to be 24° 18'. Required the altitude of the tree and its distance from the first sta- tion, the instrument being 4 feet 9 inches high. Ans. Height, 107.95 feet; distance, 128.57 feet. Problem 8. — To determine the distance of two objects A and B, I took two stations C and D, distant 35.75 chains, from which both could be seen. At C, the angle ACD was found to be 103° 47', and BCD 45° 29' 30" ; at D, the angle BDC was 110° 23' 30", and ADC 60° 21' 15". Required the distance AB. Ans. 99.236 ch. Problem 9.— The side AB (Fig. 75) of a tract of land being inaccessible, and not being able to find two stations from which both ends were visible, Fig. 75. I measured two lines, CD, 7.75 ch. "> and DE, 7.92 ch., and took the angles as follow : At C, the angle ACD was 68° 15'. At D, CDA was 50° 27', ADB 112° 46', and BDE 43° 30'. At E, DEB was 75° 10'. What was the length of AB ? Ans. 14.10 ch. Sec. VI.] MISCELLANEOUS PKOBLEMS. 115 Problem 10. — To determine the position of a point D, situated on an island, I took the angles to three objects, A, B, and C, situated on the shore, and found them to be ADB, 19° 14' 30", CDB, 24° 19'. I subsequently deter- mined the distances AB = 4596 yards, AC = 5916 yards, and BC = 4153 yards. Required the distance of D from each of the objects, it being nearest to B. Ans. AD = 828T.2 yards ; BD = 4127.7 yards ; CD = 7550.8 yards. Problem 11. — To determine the height of a mountain rising abruptly from the water of a lake, I selected a station C on the slope of the hill rising from the opposite shore, and took the angle of elevation of the summit, 47° 22' 15", and depression of the water's edge at the base of the mountain in the vertical plane through the summit, 12° 30'. Then measuring up the slope, directly from the rock, a distance of 800 yards, to a station D, the elevation of the summit was 25° 33' 30", the depression of the water's edge, 18° 15', and of the top of a staff left at C to mark the height of the instrument, 24° 15'. Required the height of the mountain. Ans. Height, 1390.7 yds. Fig. 76. Problem 12. — To determine the heights and distance of two trees C and D, standing on a hill side, I measured on level ground a base line AB 252.28 feet long, and took the following angles : At A, the angle of position of C from B was = 82° 54' 30", and of D from B = 89° 24'; the elevation of the base of C = 3° 45' ; of top of do. = 9° 25' ; of the base of D = 3° 54' ; of top of do. = 10° 29' 30". At B, the angle of position of D from C was = 6° 14' 30" ; and of A from C = 80° 51' 30", and for verification the elevations at B were of base of C = 3° 44', of top of do. = 9° 22' 15" ; of base of D = 3° 46', and of top of do. - 116 PLANE TRIGONOMETRY. [Chap. in. 10° 7' 30". Eequirecl the heights of the trees, and the dis- tance between their bases. Ans. Height of C = 89.37 ft. ; of D = 103.37 ft. ; dis- tance, 1.00.7 ft. "With the angles of verification ; height of C = 103.29 ft.; of D = 89.36 ft. Fig. 77. Problem 13.— One side EF (Fig. 77) of a tract of land being inaccessible, and there beiDg no station from which the two ends conld be seen, I selected four stations, A, B, C, D ; A and D being in the adjoining sides, and B and C between them. The following measurements were then taken,— viz. : AB = 7.37 ch. ; BC = 8.95 ch., and CD = 9.33 ch. ; at A, the angle EAB was 64° 37'; at B, ABE was 72° 43', and EBC 149° 32'; at C, BCF was 139° 47', and FCD 69° 38' ; and at D, CDF was 82° 35'. Kequired AE, EF, FD, and the angles AEF and EFD. Ans. EF = 33.50; AE = 10.38; DF = 18.77; AEF = 86° 39' ; EFD = 54° 29'. Fig 78. Problem 14. — Being desirous of finding the elevation and distance of an elevated peak C (Fig. 78) of a mountain rising abruptly from the shore of a river, and not being able to find a level place for a base line, or a regular slope as- cending in a line from the point to be measured, I selected two stations, the one A nearly opposite the base D of a rock jutting into the water, and which was so situated that A, C, and D were in the same vertical plane, and the other station B farther up the stream, the slope between them being regular. I then took the following Sec. VI.] MISCELLANEOUS PROBLEMS. 117 measurements, — viz. : AB, 850 yards. At A, the angle of position of B and C was 87° 49'; elevation of C, 35° 27'; depression of J), 3° 25' 45"; elevation of top of a staff at B of same height as the instrument, 3° 14' 30". At B, the angle of position of A and D was 47° 39', and of A and C, 70° 43' 30". Depression of A, 3° 14' 30" ; of D, 4° 48' 30" ; elevation of C, 33° 6'. Required the horizontal distance of C and D from A and B, and the elevation of A, B, and C above the water. Ans. Horizontal distance of C from A, 2189.8 yds. ; from B, 2318.1 yds.; of D from A, 894.3 yds. ; from B, 1209.2 yds. Elevation of C, 1612.7 yds. ; of A, 53.6 yds. ; and of B 101.7 yds. CHAPTER IV. CHAIN SURVEYING, SECTION I. . DEFINITIONS. 200. Definition. Land Surveying is the art of mea- suring the dimensions of a tract of land, so as to furnish data for calculating the content and determining the area. 201. The position of the angular points of a tract may be determined either by measuring the lines of the survey, the diagonals, offsets, &c, or by linear measures in connection with angular distances. These different methods of fixing the points give rise to different modes of surveying, — the first of which, as it is performed principally by the chain, may be called chain surveying. 202. Advantages. As the chain, or some substitute, such as a tape-line or a cord, is readily procured by every one, surveying by this method may be performed where the more expensive instruments cannot readily be procured. To every farmer it may be important to know the content of a particular field, or of several fields, that he may divide them properly, or that he may know the value of crops which he is about to buy or to sell ; or for various other purposes that need not be mentioned. He should, there- fore, not be under the necessity of calling in a professional man to do for him what he himself, with a pair of carriage lines, can do, if not as well, yet fully well enough for all practical purposes. 118 Sec. II.] FIELD OPERATIONS. 119 In order that this very simple method may be fully understood, we shall treat of it somewhat at length. It must not be inferred from this that it is recommended in preference to the other methods to be explained here- after, but only as a substitute to be used, when, from the circumstances of the case, these are inapplicable or incon- venient. 203. Area Horizontal. It must be remembered that, in land surveying, it is the horizontal area that is required, and not the actual surface of the ground. Every measure- ment must, therefore, be made horizontally, as explained in Art. 149, et seq., and, where angles are taken, they must be horizontal angles. As the method of chaining has been fully explained in the articles above referred to, it will be unnecessary to repeat the directions here. There are, however, certain preliminary operations to be performed, which will form the subject of the next section. SECTION II. FIELD OPERATIONS. A.— TO RANGE OUT LINES, AND TO INTERPOLATE POINTS. 204. Hanging out Lines. This requires three persons, each of whom should be provided with a rod some ten or twelve feet long, one end being pointed with iron, that it may be thrust in the ground. He should also have a plumb-line, that he may set his rod upright. The lirst, 120 CHAIN SURVEYING. [Chap. IV. whom we shall call A, takes his station at the point of be- ginning. Looking in the direction of the line, he places B in the proper direction, signalling him to the right or left as may be required. "When the position is determined, B sets his rod firmly in the ground. C then goes forward, and looking back, by ranging with the rods of B and A, he puts his rod in line. A then comes forward, and, going ahead of C, puts himself in line, by ranging with C and B. They thus continue, the hindmost always coming forward, until the other end of the line is reached. At the point at which each rod was erected a stake should be driven for future reference; Lines may be prolonged in the same manner to any extent that may be desired. If the operation is carefully done, the rods being set plumb, the line will vary very slightly, if at all, from a straight line, even when extended several miles. 205. To interpolate points in a line. The men in chaining should keep themselves exactly in line. This may readily be done by a careful follower, when the end of the line can be seen. If, however, one end is not visi- ble from the other, and from every point in the line, there will be nothing by which the follower can range his leader, unless there are staves set up for that purpose, at points along the line. The fixing of such points is called inter- polation. 206. On level ground. If, for any purpose, such points were needed in a line on level open' ground, a person, stationing himself at one end, can signal another into the proper position. As many points as are wanted can thus be determined. 207. Over a hill. If a hill intervenes, from the top of which both points may be seen, let two persons, provided with rods, put themselves as near in line as possible. Then, by alternately signalling to each other, their proper Sec. II.] FIELD OPERATIONS. 121 places can be found. Thus, let XY (Fig. 79) be the Fig. 79. line to be interpolated. A will take his station in the supposed position of the line, and signal B until he ranges with X. B then places A in line with Y at C ; A again signals B to D, in line with X ; and so they proceed till they are both in the line XY. 208. If an assistant is not at hand, or if but bi[d| one point can be found from which both ends of the line can be seen, one person can put himself in line by having a rule with a sight at each end ; wires, set upright, will do very well : lay this on some support, and then go to each end in turn, sighting to the end of the line ; he can thus deter- mine whether it is the proper position, and alter it until he finds himself rightly placed. 209. By a Random Line. When the ends cannot be seen from each other, nor from any intermediate point, it is necessary to run a random line. This is done as directed in Art. 204, following a course as near that of the line to be interpolated as possible. "When the foremost person has come opposite the end of the line, measure the whole length, noting the distance to each stake, (the stakes, for convenience, being set as nearly as possible at equal distances ;) also measure the distance by which the end of the line is missed, then say : — As the whole distance is to the distance to any stake, so is the whole deviation to the correction for that stake. Measure the distance thus determined, in the proper di rection, and set the stake, or a stone, accordingly. 122 CHAIN SURVEYING. Tims, let AB (Fig. 80) be the line to be inter- polated. Run the random line AC, setting stakes at D, E, F, &c. Measure CB and the distance from A to D, 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, — — -= .057, the correction for 1 chain. 27.56 ' 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 OPERATIONS. 123 2H. 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 P (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 c one of the modes to be pointed out. (See Arts. 227-229.) Divide BD in F, so that BF : FD : : AP : PC; that is, BD . AP make BF = — ^-- — . Then PF will be the required linei AC B.— PERPENDICULAKS. 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 : — D 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°,andAandACD = 120°; whence DCE and DEC = 60°. But DE = DC : therefore DCE = 30°, and ACE = 90°. Fig. 83. Sec. II.] FIELD OPERATIONS. 125 Fig. 84. 216. (6.) 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 G. 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. Fig. 85. "*-^ C Then take EF = EC 2 -, and F a • f / / \ \ / / '. \ / / '» \ / \ \ 1 / \ » ' / \l 2.ED will be the foot of the perpendicular. Describe the semicircle EC A. 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 Hie 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 AEC, 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 be seen, and range out FC and EC. To these draw the perpendiculars EG and FH cut- ting in I: then CID will be the perpendicular required. A E 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. Fi From E, any point in CD, run a line cutting AB in F. Then make EG a fourth proportional EF.EC toDE,EF,andEC,orEG lei to AB. ED -, and GC will be paral- Froblem 2. — To draw a parallel to an inaccessible line, two points of which 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 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 = — - — =—3 and CG will be parallel to AB. AB — DE DE . BF For, since FG = — - — — - >wehaveAB-DE:DE::BF:FG. AB — DE 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 IX 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. K 182 CHAIN SURVEYING. [Chap. IV. and GI equal to CD. Then HIK 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 GFL Bisect FI in H. Then HG will be the prolongation of BC. Fig. 96-j. 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. Ill,] OBSTACLES IN RUNNING AND MEASURING LINES. 133 EF ; and in that of 232, BH = BE. If the method Art. 233 is employed, BG = 2 BE. 2. When one end is inaccessible. 236. First Method.— Rim 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 = aU^be Fig. 97. 237. Second Method.— Set off AC (Fig. 98) in any direction, and CD parallel to AB. Eun DE towards B. Measure AE, AE.CD 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. Bun DC perpendicular to DB. CD 2 Measure DC and CA : then CB = Fig. 99 CA' or AB = AD 2 ACT 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 lT _ BD.DC -jandDE = Fig. 100. FC CF 240. Second Method.- 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 and HE = -Through H, (Fig. 101,) any point Fig. 101. FH 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. Fig. 102. Then CE : CA : : ED : AB. D- 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 may be 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, f 567. ) Whatever plan is adopted should be scrupulously adhered to, — changes in the notation beiug 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-XOTES. 137 Thus, a station on a line AB at 597 links would be called F. S. 597 AB, or (597 ) 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 T raay 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 versa. Sta. B 947 742 _3££- — ' ri27 F. S. Sta. A K 15° E. The following notes will illustrate all these directions: They belong to the tract Fig. 103. 1.38 CHAIN SURVEYING. [Chap. IV. 1 Sta. D 2440 2020 (1395) Sta. A "^ 1 Sta. A 1135 Sta. C ^~^T Sta. C 1760 ( 950) Sta.B ^'- Sta. B 2492 1445 1170 Sta. A N.45°E. ^ Brook. (1395) in AD "*--^^ 1440 770 -"" 425 ^•"**"Ci ^■^ ( 950) In BC sout Sta.B 1760 ,Bi s^~ 515 ^ -\ Sta.D Brook. 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, f 95(A 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 (1395 j : 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, ( 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. For 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. Y.] 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., IE., 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. x 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 R., 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 Mi MM . . Area = = = 21.46975 chains = 2 A., 2 2 ' 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, DF, 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 -|- CM will be equal to the sum of the sides AB, AC, and BC ; therefore CK or CM = £ (AB -f AC + BC) = J S, if S stand for the sum of the three sides. But CE -f AE -f- BG = \ S ; therefore CK = CM = CA + BG, and AK = AL = BG; whence AG = AE = BL = BM, and EK == AB. Now, since CK = CM = \ S, we have AK = \ S — AC, EC = \ S — AB, and AE = BM = \ 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 AE : ED AE . EC : ED 2 v/AE . EC : ED : CK : KH, HK : KA, CK : KA : : CK a : CK . KA. CK : ^/CK . KA, CK . ED = v'CK . KA . AE . EC. Now, ABC = ACD+ BCD + ABD == \ AC . ED+ \ BC . ED + \ AB . ED = JS . ED = CK. ED. Wherefore, ABC = y/CK . KA. AE . EC. 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. Required 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. 672 + 875 + 763 2310 [Chap. IV. = 1155 = half-sum of sides. J sum = 1155 | sum - 672 = 483 J sum - 875 = 280 J sum - 763 = 392 Area, 247449 square links, = 2 A., 1 E., 35.9 P. log. 3.062582 log. 2.683947 log. 2.447158 log. 2.593286 2) 10.786973 5.393486 Ex. 2. Required 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 rig.105. distance AE, (Fig. 105,) may be mea- sured. Multiply the sum of the parallel sides by half the perpendicular : the product is the area. Demonstration. — ABCD = ABD + BCD = \ AB . DE + \ DC . DE = (AB + 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 beinsr 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 chains, AE = 7.84 chains, AF = 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. Examples. Fig, 107. C 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 EC = 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., OK., 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 : — perp. 936 perp. 825 Ans. 7 A., R., 30.3 P. Ex. 3. Calculate the area from the following field- notes : — perp. 892 perp. 568 Ans. 6 A., 2 R., 2 P. ~~ Stat. B. 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 C © c "3 690 *<§ 770 <6 915 a o fcu 570 510 C a o fcfi 510 250 B | 585 Brook. a E. 350 280 e3 s 360 Brook. ^ 365 AD ©A N.15°E. ©A E. of AD © E The construction is plain. Calculation. Twice trapezium ACDE = AD x (Ea + bC) = 6.90 x 8.60 = 59.34; twice triangle ABC = AC x Be = 7.70 x 2.50 = 19.25; 59.34 + 19.25 Fig. 108. whence ABCDE = 2 = 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 o F ©g 440 D230 150 ©C 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 bB 1.80 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 ABCDEFGbe the corners of the tract. Calculation. 2 ABCG = AC (Ga + Bb) = 5.50 x 3.10 = 17.05; 2 GCD = GC . cD = 4.40 x 2.30 = 10.12; 2 GDEF = FD (Ge + dE) = 5.20 x 2.00 = 10.40. Therefore area = 3 K., 20.56 P. 37.57 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 (3) 772 284 (5) KE. (2) 395 715 (6) K 10° E. Area 10 A., 2 E., 18.576 P. © 7 1150 675 0(8) 432 (11) 0(9) 1285 (8) 565 1000 960 0(7) 155 (10) L. of (7,5) 0(7) 1315 (4) 562 390 282 313 (10) 0(5). E. of (4) Area 12 A., 3 E., 18.1 P. Ex. 4. Eequired 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 -N^^ 648 540 (1) V ^^ 510 ->« v _^ Brook. 365 —Brook + (1.5) (6) 380 130 0(5) r 0(5) 1255 853 765 (1) (4) 500 440 0(3) r 0(3) 1150 Brook + (2.3)— 850 490 ^u Brook. (2) 482 200 0(1) Area 14 A., 3E., 28.24 P. Area 15 A., 2 E., 7 P. Note. — In the above field-notes the marginal references, such as "Brook 4 6.7," means to the point in -which the brook crosses the line (6.7.) 150 CHAIN SURVEYING. [Chap. IV. Fig. 110. 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, 6B, anddD. 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 : — C 1185 D420 840 760 200 B E280 590 F220 250 ©A East. Dist. 250 590 840 1185 1185 x 200 Int. Sum of Double Perp. Dist. Perp. Areas. 220 250 220 55000 280 340 500 170000 420 250 700 175000 345 420 144900 2 AF/ 2/FE* 2eEDd 2BdC 544900 237000 Left-hand areas. Eight " " 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 : — 0G 312T 2590 476 F H375 2145 2070 642 E 1400 1920 1485 523 D 840 516 C K600 790 200 465 B ©A E. ©F 4025 3617 792 G- 3254 826 H E594 2846 D435 2137 1548 319 1 C729 1026 429 623 K B237 175 ©A K 15° E. Area 25 A., 1 E.,5P. Area 38 A., 3R, 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 SURVEYING. [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 K10°E. 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 R, 26 P. 9.12875 ch. Sec. V.] SURVEYING FIELDS OF PARTICULAR FORMS. 153 Ex. 1. Kequired the area and plan from the following notes : — *N A 4830 \ 2040 ***»«»^ F r "^ F 21T5 E355 1428 D r D on creek-bank 41T5 C665 3335 55 (2160) B 270 1929 396 1408 310 1015 340 610 50 A K56|°E. E 60 14T1 95 930 140 485 60 D D 60 1072 130 750 85 390 55 C 55 55 1 C 1350 (2160) D 5000 y' y 3585 y s G B on A.D A 3000 G r G 4241 F r F 75 826 100 420 60 ). E r Fig. 112 is a plat of this tract. 154 CHAIN SURVEYING. [Chap. IV. || Calculation. 1 To find AGF, Art. 251. 1 AG 3000 1 FG 4241 1 FA 4830 2)12071 | sum 6035.5 3.780713 I J s - AG 3035.5 3.482230 J J s - FG 1794.5 3.253943 1 J5 -AF 1205.5 3.081167 1 2)13.598053- AGF = 6295435 To find AFD. 6.799026 AF 4830 AD 4175 FD 2175 2)11180 J sum 5590 3.747412 Js-AF 760 2.880814 Js -AD 1415 3.150756 Js-FD 3415 3.533391 2)13.312373 AFD = 4530917 6.656186 f.] SURVEYING FIELDS OF PARTICULAR FORMS. To find BCD. BC 1350 BD •2015 CD 10T2 2)4437 J sum 2218.5 3.346059 Js-BC 868.5 2.9387T0 |s-BD 203.5 2.308564 Js -CD 1146.5 3.059374 2)11.652767 155 BCD = 670475 5.826383 To find DEF. DE 1471 EF 826 DF 2175 2)4472 J sum 2236 3.349472 |s -DE 765 2.883661 J 5 -EF 1410 3.149219 is-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 BC 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 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 AFD BCD DEF - 128 A., 2 E., 21.5 P. 6295435 4530917 670475 383567 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. Eequired the map and the content of the whole: — D CM CO tH 55 1054 72 896 97 739 75 480 C % 255 C CN CO CM 1622 1081 B E CO CO B o 63 1414 35 1237 87 1016 45 824 50 652 551 452 295 D D CM Oi o CM 1310 992 A Diagonal. toD C o 1030 A A CO 1—1 752 E S71°E Area, 34 A., 3 R. To find the contents of the several enclosures, other lines would be required : these might be measured on the plat, if it were drawn with neatness and accuracy. 158 CHAIN SURVEYING. [Chap. IV. SECTION VI. TIE-LINES. 258. Tie-Lines. The external boundaries of a tract of land having more than three sides are not sufficient either for making a plat or calculating the area. In the methods heretofore laid down, diagonals were also used. In some cases, however, owing to obstructions, such as ponds, close woods, or buildings, it is difficult to run the diagonals. When this is the case, a line measured across one of the angles of a quadrilateral will determine the direction of two sides, and thus fix the relative position of all the lines of the tract. Such lines are called tie-lines. For example, suppose it pi g . 113. were required to survey the £ tract represented in Fig. 113, / -? c €V>?^ the interior of which is filled 1 wT-f^^JSM-M} with such thick woods that / ^ C^S'^^X the diagonals cannot be mea- / Md^'j^ ■'^uf'' sured : the external lines AB, BC, CD, and DA might be measured as before. Then / ..'''' on the lines adjacent to one / ,,.*-'' angle, as C, measure carefully f CE and CF ; also measure EF. These measures should be made with the greatest accuracy, as a slight error here will very materially affect the result. On the same account, the distances CE and CF should be taken as large as circum- stances will allow. If the tie-line cannot be run within the tract, the points may be taken at E and F in the sides produced. To plat such a tract, commence with the triangle. This being formed, the direction of CB and CD is known. 259. To calculate the Area. First find in ECF the angle ECF, whence by trigonometry BD is found, and then the area of the triangles. Sec. VI.] TIE-LINES. 159 If CE = CF, EF will be the chord of the arc to the EF radius CE, whence the chord to radius 1 = — — . This EC quotient being found in the table of chords the correspond- ing arc will give the degrees and minutes of the angle ECF : or CE : } EF : : rad. : sin. J ECF. 260. Inaccessible Areas. By a combination of tie-lines and offsets, tracts that cannot be entered, such as a pond or a swamp, may be measured. For this purpose, surround the tract by a system of lines bound at the angles by tie- lines, and take offsets to the prominent points in the bound- ary of the tract. 261. Defects of this Method. Every system of measure- ment or drafting should commence with the longer lines and end with the shorter. By this means the errors that are unavoidable are diminished as we proceed. If, for example, a diagonal of thirty chains were measured, this would fix the distance of the ends to a degree of certainty precisely equal to that of the measurement ; and if from this measurement the length of an inferior line joining two points in the sides were to be determined, the errors in the length of the diagonal would affect this length to a degree exactly proportional to its length, the error in a line of live chains long being one-sixth of that of the diagonal. Precisely the reverse is the case when the shorter line is measured : the error is magnified as we proceed. On this account, the method explained above should never be employed when it can be avoided. By the use of the compass, transit, or theodolite, this can always be done. The mode of using them for surveying purposes forms the subject of the next chapter. CHAPTER V. COMPASS SURVEYING. SECTION I. DEFINITIONS AND INSTRUMENTS. 262. In chain surveying, the position of any point is determined either by directly measuring to it from other known points, or by determining its distance from such points by the indirect methods explained in last chapter. In the method about to be explained, its position is ascer- tained by angular measurements taken from known stations, or by its distance from a known point and the angle which it makes with the meridian. All those methods, which have a direct reference to the meridian as the base of angular distance, are known under the head of compass surveying; whether the instrument used to determine the angle is a theodolite, a transit, or a compass. 263. The Meridian. If the heavens are examined during a clear night, the stars to the north will be perceived to revolve around a star elevated about 40°. This is called the pole-star, and is very nearly in the point in which the axis of the earth if produced would meet the heavens. This point is called the north pole of the heavens. The north star is not exactly at the pole, but revolves around it in a small circle. If a transit or theodolite be levelled, and the telescope directed to the centre of this circle (see chap, ix.) it will point exactly north. Depress it, and run 160 Sec. L] DEFINITIONS AND INSTRUMENTS. 161 out a line in the direction of the line of collimation. This will be a meridian line. 264. The Points of the Compass. If through any station a line be drawn perpendicular to the meridian it will run east and west. If we face the south, the west will be to the right hand and the east to the left. These four points — north, east, south, west — are called the cardinal points of the compass, and are used as reference for all angular distances from the meridian. Fig. 114. For nautical purposes, each of the quadrants into which the horizon is divided is further divided into eight parts called points, and named as in Fig. 114, commencing at the north and going to the east. North, 1ST.; North by East, (N.5E.;) North Northeast, (N.N.E. ;) Northeast by North, (N.E.6N. ;) Northeast, (N.E. ;) Northeast by East, (N.E.6E. ;) East Northeast, (E.N.E. ;) East by North, (E.&N. ;) East, (E.) and so on, E.6S.; E.S.E.; S.E.6E.; S.E.; S.E.6S. ; S.S.E.; S.&E.; S. For land surveying only the cardinal points are men- tioned, the direction being determined by the angular dis- tance from the meridian. 265. Bearing. The bearing of a line is the angle which it makes with a meridian through one end. It is expressed either by naming the points, as N.6E., S.S.E. J E., as is ll 162 COMPASS SURVEYING. [Chap. V. done in navigation, or by mentioning the number of degrees in the angle accompanied by the cardinal points between which it runs. Thus, if a line runs between north and west and makes an angle of 37° 25' with the meridian, its bearing is K 37° 25' W. It deflects 37° 25' from the north towards the west, and is therefore sometimes said to run from north towards the west. This expression, though convenient, is not strictly correct. 266. The Reverse Bearing. If the bearing of a line of moderate length is determined at one end, and then again at the other end, the latter is called the reverse bearing. It will be found to be of the same number of degrees as the bearing, but with the opposite points. Thus, if the bearing of a line be K 27J° E, its reverse bearing is S. 27|° W. If the line be long, there will be a continual variation from the initial course. Thus, if a line run 1ST. 45° E. through its whole course, it will be found to deviate to the left from a straight line. A true east and west line in latitude 40° is a curve with a radius of about 4800 miles. 267. The Magnetic Needle. A magnetic needle is a light bar of magnetized steel suspended on a pivot, so that it may turn freely in a horizontal direction. Such a needle will always place itself in nearly the same direction, one end of it being northward and the other southward. The needle should move very freely on its pivot, so that it may always assume its proper position. The pivot should there- fore be of very hard steel ground to a fine point. In the centre of the needle there should likewise be a cup of agate or some other hard material inserted for it to rest upon. As the needle is generally balanced before being magnet- ized, the north end in northern latitudes will always "dip" after the magnetic force has been communicated to it. To restore the balance, a coil of fine brass wire is wrapped around the south end. This may be slipped along the bar so as perfectly to restore the balance. It serves also to dis- tinguish the two ends of the needle. A good needle will vibrate for a considerable time after Sec. L] DEFINITIONS AND INSTRUMENTS. 163 having been disturbed. If it settles soon, it is defective in magnetic power, or the pivot is imperfect. To preserve the pivot in good order, the needle should always be lifted from it when not in use. 268. The Magnetic Meridian. The line upon the sur- face of the earth in the direction of the needle, when unin- fluenced by disturbing causes, is called the magnetic me- ridian. If -the needle pointed steadily to the north pole, the magnetic meridian would coincide with the true. This is, however, far from being the case. Throughout the east- ern part of the United States and Canada it points west of north, the amount of the deviation (called the variation of the compass) being different in different places. This amount is subject to a gradual secular change. (See chap, x.) 269. The Magnetic Bearing. The bearing of a line from the magnetic meridian is called the magnetic bearing. This has generally been used in land surveying. Its con- venience is such as to have heretofore counterbalanced its defects in the opinion of a large number of surveyors. The attention of scientific surveyors and legislators has of late been called to the difficulties arising from the use of such a false and varying standard. In Pennsylvania, by a late law, the bearings of all lines inserted in the title-deeds of real estate are required to be from the true meridian line. The surveys of United States public lands have always been made on this principle. 270. There are two modes in which the needle may be employed to enable us to determine the bearing of a line. First. Attached to the needle may be fixed a card divided as in Fig. 114, or subdivided into degrees, — the north point of the needle being directly under the north point of the card. Such a card would always place itself in the same position with respect to the cardinal points. To determine the bearing of a line, it would only be necessary to have a pair of sights in the line of a diameter of the card, with an index between them to show at what 164 COMPASS SURVEYING. [Chap. Y. point of the card the line crossed. The degrees between this point and the north or south point of the card would be the bearing required. Thus, the bearing of AB would be about !N". 67° E. The cardinal points on the card show the points between which the line runs. The great defect in this plan is that, in consequence of the weight of the card, the needle settles slowly, and the pivot is very liable to wear. The card, too, must be made of some light material, which cannot be divided so accurately as metal. This form is therefore never used except for the mariner's compass. Second. The sights may be connected with a circular box in the centre of which is the pivot, — the circumference of the box being appropriately divided. This is the plan em- ployed in the surveyor's compass or circumferentor. 271. The Compass. The compass consists of a stiff brass plate A, (Figs. 115, 116,) carrying the circular box B, and furnished at the ends with two brass sights C, perpen- dicular to its plane. In the centre of the box is the pivot to support the magnetic needle. The circumference of the box is divided into 360°, and these in the larger instruments are subdivided into halves. The zero-points are in the line joining the sights, one being marked for the north, and the other for the south. The degrees are counted from zero to 90° each way. If we stand opposite the south point looking towards the north, the 90° on the left hand is marked E. and that on the right "W. The cardinal points thus follow each other in an inverted order. The reason why this should be so will appear from con- sidering the difference between the mariner's compass and the circumferentor. In the former, the card is stationary, while the index moves; in the latter, the index, which is the needle, is stationary, while the divided circle moves: while, then, the north point of the box is moving towards the east, the north point of the needle will traverse it towards the ivest. In order, then, that the index should not only point to the number of degrees, but also show the cardinal points Sec. L] DEFINITIONS AND INSTRUMENTS 165 between which the line runs, those points must be engraved in a reverse order. Thus, supposing the instrument to be in the position, (Fig. 115,) the north point of the needle at L shows the magnetic north, and the south point the magnetic south; the point midway between these to the right is east. The line from C to C is therefore south of east. If then the north point of the needle is to be used as the index, it should be found between the letters S. and E. The bearing in the figure is S. 80° E. 166 COMPASS SURVEYING. [Chap. V. 272. The Sights. These consist of two plates of brass about an inch wide set at right angles to the plate. Each plate has a vertical slit cut in it, with larger openings at intervals, as seen in Fig. 116 at H. The faces of the sights are seen at G. The slits should be perfectly straight, and as narrow as is consistent with distinct vision. The largei openings enable the surveyor to see the object more readily than he could through the fine slits. Instead of the sights, a telescope that can be elevated or depressed in a plane perpendicular to that of the plate A is sometimes employed. It has the advantage of giving more distinct vision at great distances, and, when connected with a vertical arc, of determining the angle of elevation of a hill up or down which the line may run. This object may be obtained with the sights, by having at the lower end of one of them a projection pierced with a small hole, and upon the face of the other the angles of elevation engraved. By looking through the hole at an object on the summit of the hill, the angle of elevation may be read on the face of the engraved sight. If such a scale is not on the instrument, it may be put on by the surveyor himself; a mark being made on one sight near the bottom, or a small plate with a hole being screwed to it ; on the other, at the same distance from the plate, the zero mark should be made. The distance from zero to the other marks will be the tangent of the angle of elevation to a radius equal to the distance between the sights. Measure therefore accurately the distance between the sights, and say, As rad. : tangent of the number of degrees : : the distance between the sights : the distance from the zero point to the mark for that number of degrees. 273. Attached to the plate there are generally two levels at right angles to each other, as in the transit and theodolite. 274. The Verniers. In some instruments, the compass- box is movable about its centre for a few degrees, the amount of deflection being determined by the vernier V. The purpose of this arrangement will appear hereafter. Sec. L] DEFINITIONS AND INSTRUMENTS. 167 G =t (m s^> m» Fig. ne. II 3 *D N H" c 0=0 H =0 C= 168 COMPASS SURVEYING. [Chap. V. 275. In the figures 115, 116, the different parts described above are lettered as below. Different makers, however, arrange the parts differently. A is the principal plate, which bears all the other parts. B is the compass-box, sometimes movable about its centre by means of a pinion connected with the milled head I, and capable of being clamped in any position by the screw K. D is the needle, resting on a pivot in the middle of the compass-box. The needle can be raised from its pivot by the screw F. C and C are the sights, which are fastened to the plate by the screws !N\ M, M are the levels. 276. The Pivot. This should, as remarked above, be extremely hard and very sharp. It should likewise be placed exactly in the centre of the box and in the line join- ing the slits in the sights. To discover whether it is properly centred, and likewise whether the needle is straight, turn the compass until the north point of the needle coincides with any given number of degrees. The south point must be 180° distant. If it is so in all positions, or, in four, distant 90°, as for instance the 0's and 90's, the needle is straight and well centred. Draw a hair or fine silk string through the slits in the sights. If this passes over the zero-points, the centre is in line. Or, sight to a very near object, and note the reading. Turn the instrument half round, and again note the reading : if these do not agree, the pivot is not on the line of sight. Half the difference is the actual error. 277. The Divided Circle. The accuracy of the division may be tested by turning the plate into different positions. If in all cases the opposite ends of the needle point to the same number of degrees, the probability is that the circle is correctly divided. If the compass has a vernier, set the instrument in any direction. Then move the box through any number of degrees, and see whether the needle traverses the same number of degrees as the vernier. If it does in all posi- tions, the arc is properly divided. Sec. I.] DEFINITIONS AND INSTRUMENTS. 169 278. Adjustments. The levels may be adjusted as directed for the transit and theodolite. The sights should be perpendicular to the plane of the instrument. To verify this, suspend a long plumb-line: level the plate, and sight to this line. If it appears equally distinct through all parts of the slit, the sight is perpen- dicular. Turn the instrument half round and test the other sight in the same manner. If either is found incorrect, the maker should rectify it. 279. The compass, as already remarked, is very generally used for surveying purposes, though it is fast giving place to the transit. The latter is furnished with a compass-box, which was not described with the instrument, as it was not needed at that stage of the work. It is in all respects similar to the box attached to the compass itself. The theodolite likewise has a compass. It is, however, so small as to be of very little use in accurate work. 280. The compass is generally supported on an axis in- serted in the socket 0. This axis terminates in a ball, which works freely but firmly in a socket. This arrange- ment admits of the axis being placed in any direction. The compass-plate may thus be made level. Instead of a tripod, many surveyors prefer a single staff pointed with iron. This is called a "Jacob's Staff." Its chief defects are the difficulty of setting in hard ground or among stones, and the want of steadiness in windy weather. 281. Defects of the Compass. Though a very con- venient and useful instrument, the compass is deficient in two very important particulars: — its indications are neither correct nor precise. It is not correct, because, as already remarked, the needle (which is the standard) does not do what it professes : it does not point to the north. This would be of compara- tively little importance if its direction were fixed or paral- lel; but neither of these is the fact. It not only varies 170 COMPASS SURVEYING. [Chap. V. from year to year, but from season to season, and even during the same day. These variations will be the subject of a future chapter. The presence of ferruginous matter in the earth, or the too great proximity of the chain, or of any other piece of iron, may deflect it very seriously from its normal position. It is not precise. The divisions on the arc are rarely smaller than half-degrees ; and if they were finer it would be difficult to read to less than a quarter of a degree. A little calculation will convince one that this is a serious defect where accuracy is desired. An error of 5' in the bearing would cause a deviation of nearly one foot in ten chains, or about seven feet eight inches in a mile. SECTION II. FIELD OPERATIONS. 282. Bearings. To take the bearing of a line, set the compass directly over one end ; level it, and turn the plate till the other end of the line — or a rod set up in the direc- tion of the line at a distance as great as is consistent with distinct vision — can be seen through the slits. Then, when the needle has settled, notice the number of degrees to which the end of the needle points, and the cardinal points between which it is situated: the result will be the bearing of the line. If the north end of the compass is ahead, the north end of the needle should be used, and vice versa. If you are running with the north end of the compass ahead, and the north point of the needle is between S. and E. and points to 45J°, the bearing is S. 45J° E. In reading, the eye should be placed opposite to the other Sec. II. ] FIELD OPERATIONS. 171 end of the needle; otherwise, owing to the parallax of the point, it will appear to stand at a different point of the arc from what it really does. Any iron about the person will be less likely to affect the needle than when in another position. 283. Use of the Vernier. When the needle does not point to one of the divisions of the arc, it is usual to esti- mate the fraction. Some surveyors, however, after the needle has come to rest, notice between which divisions the needle points, and then move the compass-box, by turning the milled head I, until the point of the needle is op- posite one of the divisions. The amount by which the box is turned, as indicated by the vernier, will' give the fraction. This plan, though theoretically correct, adds really nothing to the correctness of the work. The liability to derangement, from handling the instrument, is so great as to neutralize any advantage it might otherwise possess. 284. Reverse Bearing. The reverse bearing of every line should be taken. To do this, set the compass at the position of the rod, and sight back to the former station. The bearing found should be the reverse of the former. If it is not, the work at the former station should be reviewed ; if found correct, the difference between the two must arise from some local cause. 285. Local Attraction. "When the back sight does not agree with the forward sight, some cause of derange- ment exists about one of the stations. This is called local attraction. It is generally caused by ferruginous matter in the earth. It is said that any high object, such as a building or even a tree, will slightly deflect the needle. In situations in which trap rocks abound, the local attraction is often very great. The author has known a variation of more than 10° in a line of two and a half chains long, produced by this cause alone. In such regions, running by the needle is very troublesome, and ma}^ cause 172 COMPASS SURVEYING. [Chap. V. very serious errors unless great care is taken to allow for the effect produced. To discover where the attraction exists, select a number of positions in the neighborhood of the suspected points, and note their bearings from these stations, and also from each other. The agreement of several of these will prove their probable correctness. The points thus found to be void of local attraction may be taken as the starting points. In surveying a farm, a very good way is to note the forward and back sights of every line. If these are found to agree on any line, they may be presumed to be right, and the others corrected accordingly. 286. To correct for back sights. When the back sight is greater than the fore sight, sub- tract the difference from the next bearing, if the two lie between the same points of the compass or between points directly opposite, but add it in all other cases. If the back sight is the less, add the difference in the former case, and subtract it in the latter. "Where the local attraction is great, or the line runs nearly in the direction of one of the cardinal points, a diffi- culty may occur in the application of the preceding rule. A little reflection will enable the surveyor to modify it to suit the case. 287. By the Vernier. It is more convenient in practice to turn the box by the vernier until the reading for the back sight corresponds with the fore sight. The needle will then give the true bearing of the new line as though no attraction existed. 288. To survey a Farm. Commence by goiug round it, and verifying, so far as can be done, the land- marks, fixing stakes at the corners, so that the assistant may readily find them if he is not already familiar with their position. Then, placing the compass at one corner, Seo. II.] FIELD OPERATIONS. 173 send the flag-man ahead to the next corner ; note the bearing of his pole; and so proceed with the sides, in succession, taking a back sight at each station. If the end of the line cannot be seen from the bes;in- ning, let the flag-man erect his pole, in the line, at a point as distant from the beginning as possible. Sight to the pole, as before; then, going forward, set the compass by sighting to the last station. The flag-man should now be placed, exactly in line, at another station. So proceed until the end of the line has been reached. 289. Random Line. If the first position of the flag- staff were not exactly in line, the course run will deviate to the right or left of the corner. Where such is' the case, measure the perpendicular distance to the corner, and de- termine the correction by the following rule : — As the length of the line is to the deviation found as above, so is 57.3 degrees, or 3438 minutes, to the correction in the bearing.* In running through woods, it is very frequently necessary to correct the bearing in this manner. In all cases, how- ever, where back sights are taken, the compass should be allowed to stand at the last station on the random line, since the local attraction often varies very considerably in a short distance. If it is desired to run the next line precisely on its location, the corner should be sighted to from the end of the random line, and a back sight taken. * This rule is founded on the ordinary rule for the solution of right-angled triangles, — the length being the hypothenuse, and the deviation the perpen- dicular, an arc of 57.3 degrees being equal in length to the radius. Thus, supposing, in running a line N. 35° 30 / E. 27.53 chains, the corner is found 35 links to the right hand : the calculation would be 27.53 : 35 : : 57.3° : 0° 43'. The proper bearing would therefore be N. 36° 13' E. 174 COMPASS SURVEYING. [Chap. V. 290. When the far end of the line cannot be seen, it will sometimes be found convenient to run to a station as near the middle of the line as possible, if one can be found from which both ends can be seen. Then, instead of con- tinuing on in the same course, sight to the corner. The chain-men should note the distance to the assumed station. A very obtuse-angled triangle will thus be formed, and the correction in bearing may be readily calculated. Thus, supposing the line were AB, (Fig. 117,) Fig. 117. passing over an elevation at C. At A the bearing of AC was found to be N. 43f° W., distance 10.50 chains. At C, CB was K 43° W., distance 7.36 chains. We have AC : BC :: sin. B : sin. A; or, as the angles are small, AC : BC :: B : A; whence AC + BC : BC : : B+A : A. That is, 17.86 : 7.36 :: 45' : A = 19', the required correction. The true bearing of AB is therefore K 43|° W. Where the deviation from the correct line is not much greater than in the example given, AB is sensibly equal to AC -f CB. Where the deviation is considerable, the angles and side should be calculated by Trigonometry. The above rule may be expressed thus : — As the sum of the distances is to the last distance, so is the whole deviation to the correction to be applied at the first station. 291. Proof Bearings. In the course of the survey, bearings or angles should be taken to prominent objects. These form a test of the accuracy of the work. Three bearings are necessary to each object: two of these, being required to fix its position, will aflbrd no check on the inter- mediate measurements; but their coincidence with a third will determine the probable correctness of all, and of the connecting measurements. Diagonal bearings and dis- tances may likewise be taken as proof lines. Sec. II.] FIELD OPERATIONS. 175 292. Angles of Deflection. In surveying with the transit or theodolite, it is most convenient to record the angles of deflection; that is, the angle by which the new course deviates to the right or to the left from that of the last line. This is always done in surveying roads, rivers, &c. From the angles of deflection the bearings are very readily deduced, by rules to be given hereafter. As checks to the work, the bearings of some of the lines may likewise be taken. In a closed survey the whole deflection must equal 360°. To determine whether it is so, arrange the deflections to the left in one column, and those to the right in another. Sum the numbers in each column : the difference of these sums should equal 360°. In practice this will rarely occur; though in open ground, where the angles can readily be taken, the error should not exceed four or five minutes in a tract of ten or twelve sides, provided a good transit or theodolite is employed. Example. The following are the notes of a survey taken by the author:— 1. S. 53° 10' W.; 2. Deflect 97° 3' to the right; 3. 97° 45' to the right; 4. 81° 14' to the right; 5. 30° 12' to the left; 6. 12° 14' to the left; 7. 27° 48' to the right. Whence the first line def Lects 98° 34' to Right hand. Left hand. 97° 3' 30° 12' 97° 45' 12° 14' 81° 14' 42° 26' 27° 48' 98° 34' 402° 24' 42° 26' 359° 58', differing but two minutes from 360°. 176 COMPASS SURVEYING. [Chap. V. Where the difference amounts to several minutes, it is best to distribute it among the angles. The rule which is sometimes given: to determine the angles from the bearings, and ascertain whether the sum of the internal angles is equal to twice as many right angles as the figure has sides, less four right angles — proves nothing in regard to the correctness of the field work. Any set of bearings will prove in this way. SECTION III. OBSTACLES IN COMPASS SURVEYING.* A.— PROBLEMS IN RUNNING LINES. 293. Many of the obstacles that occur in angular sur- veying have already been alluded to. These, and all others which the operator will meet with, may be over- come by the principles of Trigonometry. As, however, there is frequently a choice in the means to be used, the following methods are given, as being perhaps the most simple : — 294. Problem 1. — To run a line making a given angle with a given line from a given point within it Place the instrument at the point, and sight along the line. Tarn the plate the required number of degrees, and the sights or telescope will be in the required line. >; Many more such methods may be found in Gillespie's "Land Surveying." Sec. III.] OBSTACLES IN COMPASS SURVEYING. 177 295. Problem 2. — To run a line making a given angle with a given inaccessible line at a given point in that line. Let AB (Fig. 118) be the given rig. us. line, and A the given point. Take two points C and D from which A and some other point B in AB may be seen, and measure CD. Then take the angles ACD, BCD, ADC, and BDC. The dis- tance AC and the angle CAB may be calculated. Run CE, making ACE = CAB : CE will then be parallel to AB. Now, if we suppose AE to be drawn, ' we shall have in the triangle ACE all the angles and side AC to find CE. Lay off this distance from C to E, and run the line EF towards A. If A cannot be seen from E, calculate CEF, and run the line from E, making the proper angle with CE. Problem 3. — From a given point out of a line, to run a line making a given angle with that line, 296. Where the line is accessible. If the compass is used. Take the bearing of the given line. Then place the compass at the given point, and set it to same bearing. Deflect the compass the number of degrees required, and run the line. If a transit or theodolite is used. Fi g- 119 « Set the instrument at some point A (Fig. 119) in the line, and take the angle BAC. Move the instru- \ ment to C, and make the angle ACB = B - A, or = 180° - (B + A), and CB or CB' will be the line required. In all cases, unless the line is to be a perpendicular, there will be two lines that will answer the conditions. 12 178 COMPASS SURVEYING. [Chap. V. 297. If the line is inaccessible. Let m %- 12 °- A F AB (Fig. 120) be the given line, and — *r *; — C the given point. Run any con- venient base CD, and take the angles of position of two visible points A and B in the given line. Then, in the triangle ADC, we shall have DC and the angles, to find CA. Similarly, in CBD, find CB. Then, in ACB, we shall have AC, CB, and ACB to find ABC. Run CF, making BCF = B - F, or 180° - (B +F), and it will make the required angle with AB. 298. If the point be inaccessible. From any convenient stations A and B (Fig. 121) in the line AB, take the angles of position of the point C, and measure AB. Then, in the triangle ABC, we shall have the angles and the side AB to find BC. In BCD we then have the angles and side BC to find BD. BD may be found by a single proportion, thus : — Sin. ACB . sin. BDC : sin. BAC . sin. BCD : : AB BD. For we have sin. ACB : sin. BAC : : AB : BC, and sin. BDC : sin. BCD : : BC : BD. Whence (23.6) sin. ACB . sin. BDC : sin. BAC . sin. BCD : : AB BD. Having found BD, DC may be run towards C; or by the angle, if C be invisible from D. If C is visible from the point D, the latter may be found by trial, thus : — Set the instrument at a station as near the proper posi- tion as possible, and deflect the given angle. Notice whether the line passes to the right or left of the point, and Sec. III.] OBSTACLES IN COMPASS SURVEYING. 179 move the instrument accordingly. A few trials will put it in its proper place. Fig. 122. G 299. If the point and the line both be inaccessible. Take any convenient station D, (Fig. 122,) and run DE parallel to AB, by Art. 302. Then run CFG, making the required angle with ED, by Art. 298 ; or the dis- tance on the base DC (Fig. 125) may be calculated. Problem 4. — To run a line parallel to a given line through a given point. 300. If the line he accessible. With the compass. Take the bearing of the given line, and through the given point run a line with the same bearing. With the transit or theodolite, At any point A (Fig. 123) in the given line take the angle BAC. Eemove the instrument to C, and make ACD = d " BAC. CD will be parallel to AB. 301. If the point be inaccessible. Fi s- 124. At A and B, (Fig. 124,) any two points in the given line, take the angles BAC and ABC. Measure AB, and calculate AC. Make CBD = ACBandBD=AC. Through D run DE in the line CD : it will be the parallel required. Fig. 123. A 302. If the line be inaccessible. From C (Fig. 125) run any base- line CD ; and at C and D take the angles of position of two visible points A and B in the given line. Calculate the angle 180 COMPASS SURVEYING. [Chap. V. CAB. Bun EOF, making ACE = CAB, and EF is the parallel required. If the line and the 'point both be inaccessible. Fig. 126. Fig. 127. 303. First Method.— Assume any station D, (Fig 126,) and run a line DE parallel to AB, by Art. 302, and towards C run Fa parallel to DE, by Art. 301. 304. Second Method. — Take any convenient base DE, (Fig. 127,) and take the angles of position of C, A, and B at D and E. Calculate BE, CE, and EBA. Then CFB = 180° - EBA. In CEF, we then have the angles and CE to find EF. Lay off EF the calculated distance, and run the line from F to C. B — PROBLEMS FOR THE PROLONGATION AND INTER- POLATION OF LINES. 305. In running a line, obstacles are often met with which it requires some ingenuity to overcome, and which will perplex the surveyor unless he has prepared himself by previous study of all cases which are likely to occur. If the total length of a line were all that it was necessary to determine, the two points at its extremity might be con- nected by a series of triangles, and that length calculated by Trigonometry; but it is generally desirable to have the line marked out so that the exact position of the dividing fence, if one is placed, or of the division if there be no fence, may be indicated by stakes or by marked trees. To do this, the line itself must be traced, or another run Sec. Ill] OBSTACLES IN COMPASS SURVEYING. 181 in its neighborhood, so related to that in question that the surveyor can at any time pass from the one to the other to set his landmarks. "We shall treat of the different kinds of obstructions likely to occur; and, as the prolongation and interpolation of the lines are generally closely con- nected with the determination of their lengths, the two will be considered together. Problem 1. — To prolong a line beyond a building or other obstruction. 306. First Method. — At a point of the line erect a per- pendicular of such length as to pass beyond the obstacle. Through the extremity of this run a parallel to the given line : after passing the obstacle, pass back to the required line by an equal perpendicular. The distance will be equal to that of the parallel. 307. Second Method. — At B (Fig. 128) deflect 60°, and mea- sure BC. At C deflect 120°, and measure CD = BC. Deflect 60°, and run DE, which will be in line with AB. BD = BC ; for BDC is an equilateral triangle. Pig. 128. 308. Third Method.— At B (Fig. 129) deflect 60°, and measure BC. At C deflect 90°, and measure CD =1.732 times BC. At D deflect 30°, and DE will be in line with AB. BD = 2BC. Pis:. 129. 309. Fourth Method.— At B (Fig. 130) deflect 45°. A Measure BC. At C turn 90°, and make CD = BC. Pig. 130. At D turn 45°, and DE will be in line. BD = 1.414 BC. 182 COMPASS SURVEYING. [Chap. V. Problem 2. — To interpolate points in a line. 310. If one end be visible from the other. Set the instru- ment at one end and sight to the other: an assistant can then be signalled to place stakes directly in line. In crossing a valley, determine a station, as above, on the borders, from which the valley can be seen; and, placing the instrument at this point, sight to a similarly deter- mined station on the other side. Stations may thus be determined down a very considerable declivity. With the transit almost any slope may be sighted down. In this operation, the instrument must be very carefully levelled sideways ; otherwise, the points determined in the valley will be out of line. 311. By a Random line. If a wood, or other ob- Fi s- 131 « A struction, prevents one end of the line, as B, (Fig. 131,) from being seen, run a line AC as nearly in the given course as possible, and drive a stake every five or ten chains, or oftener if desirable. "When you have arrived opposite the end of the line, note the distance. Also measure the distance CB to the end. The correction of the bearing may be found as in Art. 289, and the points be inter- polated as in Art. 209. 312. If the line cannot be run from the first station. Lay off AC (Fig. 132) as nearly perpendicular Fig.^132. to the line as possible, and run the random line CD. On arriving opposite the end, measure DB. Then say, — As CD is to the difference between BD and AC, so is 57.3°, or 3438', to the correction of J_j bearing. c i ; 1 1 / I ( I i I .' I / Li To interpolate points — Say, as CD is to the distance Ca to any station on the random line, so is the difference between BD and AC to a fourth d 1 -- ~--k \a \ \ \ Ah \ \ \ - V D Sec. III.] OBSTACLES IN COMPASS SURVEYING. 18o term. This fourth term added to AC if BD is greater than AC, but subtracted if it be less, will give the correction for the point a. If the random line crosses the other, as in Fig. Fi s- 133 - G A 133, say, As CD is to the sum of AC and BD, so is 57.3°, or 3438', to the correction of the bearing. Points may be interpolated by the following rule : — Say, As CD is to the sum of AC and BD, so is the distance Ca to any point in the random line to a fourth term. Take the difference between this fourth term and AC. c \~d' Then if AC is the greater of the two, lay off j the difference on the same side of the random e'"~~_ line that A is; but if AC be the less, lay off the remainder on the opposite side. Where a point in the line at a given distance from the beginning is required, measure that distance on the ran- dom line, and determine the offset as above. If the random line comes out very distant from the far station, it is better to run another than to depend on that as a basis for interpolation. C.— PROBLEMS FOR THE MEASUREMENT OF INAC- CESSIBLE DISTANCES. 313. The various methods of determining the lengths of inaccessible lines are merely applications of the rules of Trigonometry, and might, therefore, be applied by the stu- dent without further instruction. There is, however, always a choice in the method to be employed: the fol lowing are therefore given, that all that is needful in the case may be brought together. Problem 1. — To determine the distance between two points which are accessible and visible from each other. 184 COMPASS SURVEYING. [Chap. V. Fig. 134. Fig. 135. 314. First Method. — Select any station C, (Fig. 134.) Measure BC, and take the angles BAC and b ABC. Thence we can calculate AB. 315. Second Method. — Measure CA and CB (Fig. 134) and the angle ACB; whence, having two sides and the included angle, AB may be determined. 316. Third Method.— Where the angles can be taken to the ex- tremities of an inaccessible but known base CD, (Fig. 135,) the dis- tance AB may be calculated thus : — In ABD we have AD : AB : : sin. ABD : sin. ADB, and in ABC we have AB : AC : : sin. ACB : sin. ABC. Whence (23.6) AD : AC : : sin. ABD . sin . ACB : sin. ADB . sin. ABC. Then, in CAD having the ratio of AC to AD and the angle CAD, we may find the other angles by Art. 141, thus : — As AD : AC, or sin. ABD . sin. ACB : sin. ADB . sin. ABC : : r : tan. x, and as rad. : tan. (^^45°) : : tan. j- (ACD + ADC) : tan. J (ACD *» ADC.) Having now the angles and one side of ACD, AD is found ; whence, in ADB, AB may be determined. Thus, and sin. CAD : sin. ACD : : CD : AD, sin. ABD : sin. ADB : : AD : AB. Whence (23.6) sin. CAD . sin. ABD : sin. ACD. sin. ADB : CD : AB. Sec, III.] OBSTACLES IN COMPASS SURVEYING. 185 Examples. To determine the distance AB, accessible at its extremi- ties, I took the angles to the ends of a line CD 10.75 chains long, as follows:— B AC = 100° 35'; BAD, 48° 19'; ABC, 46° 15'; and ABD, 85° 23'. Required the distance AB. ACB = 180° - (BAC + ABC) = 33° 10'. ADB = 180° - (BAD + ABD) = 46° 18'. A _ r sin. ABD AsADor isin.ACB 85° 23' A. C. 0.001411 33° 10' " " 0.261952 , _. f sin. ADB : AC or | sim ABC 46° 18' 46° 15' 9.859119 9.858756 : : rad. 10.000000 : tan. x 43° 45' 46" 45 9.981238 tan. 45° — x 1° 14' 14" 8.334392 ACD + ADC tan. 2 63° 52' 10.309258 ACD - ADC tan. 2 2° 31' 14" 8.643650 ACD 66° 23' 14" k f sin. CAD Then, As < . A x^ ' t sin. ABD 52° 16' A. C. 0.101896 85° 23' " " 0.001411 ( sin. ACD \ sin. ADB 66° 23' 14" 9.962025 46° 18' 9.859119 :: CD 10.75 ch. 1.031408 : AB 9.034 ch. 0.955859 Problem 2. — To determine the distance on a line to the in- accessible but visible extremity. 317. This may be done by the methods explained in Arts. 236, 237, and 238, using the transit or theodolite in running the lines, or by the following method : — 318. Run a base line from a point in the line making any 186 COMPASS SURVEYING. [Chap. V. angle therewith, and at its extremity take the angle of posi- tion of the point. A triangle is thus formed of which the angles and one side are known. In this operation the triangle should be made as nearly equilateral as possible. Problem 3. — To determine the distance when the end is in- visible and inaccessible. 319. First Method.— De- Vig.ue. fleet at B (Fig. 136) by any angle, and measure BD to a point from which C is visi- ble. TakeBDC. Thencalcu- A late BC. The angle C should be made as large as possible. If AB will not certainly pass through C, operate by the second method. ¥&um 320. Second Method. — Run EBD making any angle with AB, (Fig. 137.) Take the angles D and E. In DEC find DC. Then in DCB we have two sides DC and DB and the included angle to find BC and DBC. If DBC is equal to ABE, C is in AB produced. Fig. 137. -vc Problem 4. — To determine the distance to the intersection of two inaccessible lines. Sec. III.] OBSTACLES IN COMPASS SURVEYING. 187 321. Let AB and CD (Fig. 138) be the lines, their intersec- tion E being both in- visible and inaccessi- ble. It is required to run a line from a given point G, that shall pass through E, and to determine GE. Run any base line GH, and take the angles of position of the points A, B, C, and D on the given lines. Find GC, GD, and GDC ; also GA, GB, and GBA. Then, in GBD, we have GB, GD, and BGD, to find GBD, GDB, and BD. In BDE we then have BD and the angles to find BE. Finally, in GBE we have GB, BE, and the included angle, to find BGE and GE. If the lines AB and CD were accessible, the line GE might be run by Art. 212, and the distance determined by taking the angles C and G, (Fig. 139.) sin. GCE ^ Then GE = — — - - GC. sin. GEC Fig. 139. ~:^--"e Problem 5. — To determine the distance between two inac- cessible points. 322. First Method.—. .Select if . Fi s- uo - possible a point C, in the direc- ^ 1 tion of the line AB, (Fig. 140.) N >» % From a station D, take ADB v *^ and BDC, and measure DC. Then in CDB we have CD and the angles to find CB, and in CDA we have CD and the angles to find CA. AB = CA - CB. D 188 COMPASS SURVEYING. [Chap. V. 323. Second Method.— Take a base line CD, (Fig. 135,) which, if possible, should be chosen nearly parallel to AB, and not much shorter than it. From C and D take the angles of position of A and B, whence AB may be calcu- lated. 324. Third Method. — If no two points can be found whence A and B can both be seen, the distance can be found as in Prob. 9, p. 114. 325. Fourth Method. — If A and B can both be seen from no one station, the distance may be found by Prob. 13, p. 116. 326. Examples illustrative of the preceding rules. Ex. 1. It being necessary to run a parallel to a given in- accessible line AB, so as to pass through a given point C, also inaccessible and probably invisible from any point in the proposed line, I took a base line DE (Fig. 127) of 18 chains, and at D and E determined the following angles of position,— viz. : EDO = 106° 35'; EDA = 72° 5'; EDB = 21° 20'; DEC = 26° 50'; DEA = 61° 20'; and DEB = 120° 45'. Required the distance DG and the angle DGF ; also the distance GC to the given station. Ans. DG 8.48 ch., GC 13.47 ch., and DGF = 124° 8' 17". Ex. 2. One side AB of a tract of land being inaccessible, and it being required to run from a given station C a line which shall make an angle of 67° 35' with that side, I measured a base line CD of 7 chains, and took the angles CDA = 100° 25'; CDB = 47° 29'; DCA = 32° 17'; and DCB = 90° 3'. Required the angle DCF which the required line makes with DC ; also the distance on CF to the line AB, and the distance of the point of intersection from A. Ans. DCF = 49° 10' 20", CF = 7.84, AF = 2.94. Ex. 3. The line AB not being accessible except at its ex- tremities, which were, however, visible from each other, I took the angles as follow to the points C and D, whose dis- tance I had previously found to be 10.78 chains, and found Seg. III.] OBSTACLES IN COMPASS SURVEYING. 189 them to be BAD = 46° 30'; BAC = 81° 43'; ABC = 37° 23'; and ABD = 80° 47'. Required AB. Ans. AB = 13.76 ch. Ex. 4. To a given inaccessible line AB it being required to run a perpendicular which shall pass through a point P also inaccessible, I took a base CD of 15 chains, and mea- sured the angles as follow,— viz. : DCP = 105° 30'; DCA = 256° 50'; DCB = 326° 42'; PDC = 38° 50'; PDA = 79° 38'; PDB = 131° 7'. Eequired the distance on DC from D to the proposed line. Ans. DF = 14.36. Ex. 5. One side AB of a tract of land being inaccessible, and it being required to locate the adjoining side AE, which makes with the former an angle BAE of 98° 17', a base CD of 10 chains was measured. At C, the angle DCA was 95° and DCB = 37° 20'. At D, CDA was 43° 45', and CDB = 87° 39'. Required the angle between CD and a parallel to AB ; also the distance on that parallel to the point E in AE, and the distance AE. Ans. The parallel makes with CD the angle DCE = 163° 57', CE = 5.19 ch., and AE = 9.89 ch. Ex. 6. In running a random line AB N". 87° E. towards a point C, after proceeding 7.50 chains I came to an impass- able swamp. I therefore measured on a perpendicular K 3° W. 4.25 chains, and S. 3° E. 5 chains to the points D and E from which C could be seen. At D, the angle CDE was 66° 39', and at E, DEC was 67° 25'. Required the dis- tance BC, the true course and distance of AC. Ans. BC = 10.93 ch. ; AC = 18.42 ch. ; True course K 88° 26' E. 190 COMPASS SURVEYING. [Okap. V. SECTION IV. FIELD-NOTES. 327. The field-notes, when the bearings are taken, are recorded in various modes. First Method. — The simplest method is to write them after each other, as ordinary writing, thus : — Beginning at a limestone corner of James Brown's land, "N. 27J° E. 7.75 chains, to a marked white-oak. Thence, S. 60J° E. 10.80 chains, to a limestone, &c. In recording the boundaries, it is well to name the pro- prietors of the adjoining properties. These are always inserted in deeds of conveyance. 328. Second Method. — Rule three columns, aa in the ad- joining plan: in the first, insert the station; in the second, the bearing; and, in the third, the distance: the margin to the right will serve for the landmarks, adjoining proprietors, &c. The left-hand page of the book may be reserved — as directed in Chain Surveying — for remarks, subsidiary calculations, &c. Sta. 1 2 3 4 5 6 Bearing. N.27|°E. S.62|°E. S. 80° E. S.47i°E. S. 54| W. K37J°W. Distance. Landmai-ks, &c. 7.75 10.80 9.50 9.37 8.42 23.69 to a marked white-oak. " limestone. " do. " forked white-oak. u limestone. " do. the place of beginning. 329. Third Method. — Where there are subsidiary mea- surements, — such as offsets, intermediate distances, &c, — the above method is not convenient, as it requires a new table for each line along which such measurements are Sec. IV.] FIELD-NOTES. 191 made. In such, cases, the method by columns, with mar- ginal sketches of fences, streams, &c, is perhaps the best. The notation for " False Stations," the crossing of lines, streams, &c, (adopted in Art. 244,) may be employed here. The bearing should be inserted diagonally in the columns, and the bearings of cross fences, proof bearings, with the offsets, should be recorded in the right or left-hand margin, according as the lines or points to which they refer are to the right or left of the line being run. Sketches of the adjoining fences may likewise be inserted in the margin, with the distances to the intersections. By this combination of the columns and sketches, all the field- work may be recorded concisely, luminously, and accu- rately. The following notes of a survey will illustrate the above : — P5 e3 a © H o Sta. 4 B 1132 55 1054 CO 72 896 o 97 739 75 480. to s, nn * o Sta. 3 Sta. 3 Limestone on 1450 bank of run 1030 <*°, ^^ Sta. 2 Sta. 2 a limestone. 1344 N. 59°10'E. Sta. 1 a limestone. d o Sta. 5 1740 63 1414 35 1237 87 1016 45 824 50 652 551 452 295 ^.tf* Sta. 4 a marked tree, corner of Wm. Phillips's. 192 COMPASS SURVEYING. [Chap. V. Fig. 141 is a plat of this tract. Fig. 141. SECTION V. 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 ; hut where the bearing is east or west, the latitude is zero, Sec. V.] LATITUDES AXD 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, T U2 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. !N"o notice is, therefore, taken of the rotundity of the earth in "Land i Surveying." i 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 mag 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 !N". 56^° \V. 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 !N\, 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 N"., 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. 33J° 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. Erom 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. Eor 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 K 37|° 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 ep . 62.645 5 " 2.231 4.475 33.465 67120 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 35}° 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. 63|° W. 27.49 chains ? Ans. Lat. 12.27 S.; Dep. 24.60 W. Ex. 5. What are the latitude and departure of a line "N". 55 j- ° E. 27 chains ? Ans. Lat. 15.20 1ST.; 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 K; Dep. 123.04 E. Ex. 7. What are the latitude and departure, the bearing 24f° W. 97.56 chains? Ans. Lat. 88.60 S. ; Dep. 40.84 W. and distance being S. 24f ° W. 97.56 chains ? 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 41J\ and say, — As 15' is to the difference between 41J° and 41° 18', or 3', so is the difference between the latitudes to the correction for 3'. This correction subtracted from the latitude for 41J° 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 41i° 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 "K. 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 L 29° 17' E. 123.75 chains. Ans. Lat. 107.937 S. ; Dep. 60.529 E. of a line running S. 29° 17' E. 123.75 chains. 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 X. 41° 18' E. 43.27 chains. 198 COMPASS SURVEYING. [Chap.V. 41° 18' C( )sine .75126 Si ne 66000 Dist. DiflF. 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. Required the latitude and departure of a line N. 29° 38' E. 26.47 chains. 29° 38' Cosine .86921 Sine.49445 Och. 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' 8' Cosine .86921 Sine i .49445 ch. 17.384 9.889 6 " 5.215 2.967 .4 348 198 .07 61 34 Lat. 23.008 K Dep. 13.088 E. Ex. 3. Required the latitude and departure of a line bear- ing S. 56° 7' E. 63.48 chains. Ans. Lat. 35.39 S. ; Dep. 52.70 E. Ex. 4. Required the latitude and departure of a line bear- ing N". 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 that 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. Kew columns are, however, to be preferred. Ex. 1. Given the bearing and distances as follows, to find the corrected latitudes and departures. K43|°"W". E". 29f ° E. S. 80° E. East. S. 10J° E. S. 64° W. K 63f ° W. S. 57i° W. 28.43 30.55 28.74 40.00 23.70 25.18 20.82 31.65 T ~2 IF T "5 7T T Bearings. Dist. N. S. E. W. Cor. N. Cor. W. .01 N. S. E. W. N.43^°W. 28.43 30.55 20.62 — "l5l6~ 19.57 20.62 ~15JL4~ "2T28 19.58 N.29%°E. 26.52 .02 S. 80° E. 28.74 40.00 4.99 28.30 .01 .02 .02 .01 4.99 East. 23.32 11.04 40.00 .01 39.98 S.10i^°E. 23.70 4.22 23.32 4.21 22.64~ S. 64° W. 25.18 _ 9T21" 22.63 .01 .01 .02 11.04 N.63%°W. 20.82 18.67 9.21 18.68 8 S.57^°W. 31.65 17.01 26.69 17.01 26.71 229.07 56.35 56.36 56.35 Er. N. .01 87.68 87.56 .01 .12 87.56 .12 Er. W. 56.36 56.36 87.61 87.61 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 25|° W. 14.09 ch. ; 4. K 43|° E. 14.70 ch. ; 5. K 12|° W. 17.95 ch. ; 6. K 88f ° 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. COMPASS SURVEYING. [Chap. V. SECTION VI. PLATTING THE SURVEY.* 339. With the Protractor. First Method.— Dn&w a line NS, 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- 1*3. 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 " Land Surveying." 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 :— 2. S. 60J° E. 10.80; 3. S. 8° E. 9.50; 1. K 27|° E. 7.75 4. S. 47J° E. 9.37 5. S. 54J° W. 8.42 6. K 37J° "V7. 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 Kg. 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 O (Fig. 145) draw a line NS to represent the meridian. Lay off from the north and south points the different bearings, 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 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. Fig. 146. N 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. Describe a circle (Fig. 146) about the centre of the paper with a radius equal to 5 on a scale of equal parts. This scale should be taken as large as con- w - venient. Through its centre A draw NS 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 NS. 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 North, 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\ 27| E., divide it by 2, and from the table of natural sines take out the sine of the quotient 13° 45'. It is fouud 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 1ST 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. 47^° E., which exceeds 30° by 17|° : 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 !N\, 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., jN"., 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 Sec. VI.] PLATTING THE SURVEY. 207 The latitude of the fourth side is .01 N". 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 N. 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, NS, (Fig. 147,) and take any point E for the starting point. From this 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., and the deflection of the next side 43° 1 5 r to the right. BD=K 37° E. w . DBC = 43° 15' Whence BC is K 80° 15' E. Ex. 2. Given AB K 37° E., and the deflection of BC 43° 15' to the left. BD = N. 37° E. DBC = 43° 15 ' Whence BC' is K 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. 41° 20' "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., ^149. 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. K 46° W. S. 79° E. Fig. 150. N AB BC ABC DBC 33° 180° 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 K 39° E. BC K 63° W. DBC 102° = deflection to the left. Fig. 151. N Ex. 4. Given AB (Fig. 152) S. 82° E., and BC K 67° E., to find the de- flection. AB S. 82° E. BC K 67° E. Fig. 152. N 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° V 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 K 82i W.to one K 29J° W.? Ans. 53|° 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° \Y. 5 and DE X. 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. N". 57° E.; 212 COMPASS SURVEYING. [Chap. V. 2. K 89° E.; 3. S. 49|° E.; 4. South; 5. S. 27|° W.; 6. S. 53i° W. ; 7. JNT. 89° W. ; 8. K 37° W. ; 9. K 43° E. to the place of beginning. Kequired to change the bearings, so that the ninth side may be a meridian. 1. K 57° E. K 43° E. 2. K 89° E. K 43° E. 3. S. 49J° E. K43° E. If. 14° E. IF. 46° E. 92J° 180° K 87J° E. 4. s. o°w. 5. S. 27f ° W. 6. S. 53J° W K 43° E. K43° E. K 43 ° E. S. 43° E. S. 15J° E. S. 10i° w 7. K 89° W. K 43° E. 132° 180° 8. K". 37° W. K 43° E. K 80° W. 9. North. S. 48° W. Ex. 2. Change the bearings in the following notes, so that the second side may be a meridian : — 1. N. 43° 25' W. ; 2. 1ST. 29° 48' E. ; 3. 8. 80° E. ; 4. K 89° 55' E. ; 5. S. 10° 13' E.; 6. K 63°55'W.; 7. S.63°45'W.; 8. N. 57°35'W. Ans. 1. K 73° 13' W.; 2. North; 3. N. 70° 12' E.; 4. K 60° 7' E. ; 5. S. 40° V E.; 6. S. 86° 17' W.; 7. 5. 33° 57' "W. ; 8. N. 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. 5. 47° E. ; 3. S. 59i° W. ; 4. N. 84i° W. ; 5. 25T. 12° W. ; 6. N. 17i° E., and 7. S. 29f ° W. Ans. S. 21i° W.; % S. 37i°"W.; 3. N. 36| W.; 4. North; 5. K 72J° E.; 6. S. 78° E.; 7. N. 65 j° W. Sec. VIIL] SUPPLYING OMISSIONS. 213 SECTION VIIL 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 3 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 F.G 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. N. 56 J° "W. 15.35 chains; 2. K 9° W. 19.51 ch.; 3. Unknown; 4. S. 39}° E. 13.35 ch.; 5. K 82 J° E. 12.65 ch.; 6. S. 6|° W. 12.18 ch.; 7. S. 52i° W. 20.95 ch. Sec. VIII.] SUPPLYING OMISSIONS. 215 Sta. Bearing. Distance. N. s. E. w. 1 ~2~ K 56i° "W. 15.35 8.53 12.76 K 9° "W. 19.51 19.27 3.05 3 4 S. 39|° E. 13.35 10.26 8.54 5 K 82J° E. 12.65 1.65 1L2JL0" 12.54 6 S. 6£° W. 12.18 1.43 7 S. 52J° W. 20.95 12.75 16.62 29.45 35.11 21.08 33.86 29.45 5.66 E". 21.08 12.78E. Diff. Lat. Departure, Bearing, Bearing, Diff. Lat. Distance, 5.66 log. 0.752816 12.78 log. 1.106531 HT. 66° V E. tang. 10.353715 66° V cos. 9.607322 log. 0.752816 13.98 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, 1ST. 11° 5' W. 15.53 ch.; EF, K 23° E. 9.72 ch., and EB, K. 75° 12 r E. 14.00 chains. Eequired the bearing and distance of AB. 216 COMPASS SURVEYING. [Chap. V. Sta. Bearing. Distance. N. s. E. w. A K 47° W. 16.55 11.29 12.10 C 1ST. 19° 5' E. 11.48 ~15J53 10.85 ~1572T _ 8795" 3.75 D K11°5'W. 2.99 K 23° E. 9.72 3.80 ~137oT K75°12'E. 14.00 3.58 (49.91) B ~2L09~ (6.00) 49.91 15.09 15.09 6.00 Diff. Lat. 49.91 log. 1.698188 Departure, 6.00 log. 0.778151 Bearing AB, K 6° 51' E. tang. 9.077963 Bearing, 6° 51' cos. 9.996889 Diff. Lat. 50.27 1.698188 Distance, 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. 37i° W. 5.86 ; 3. S. 39J° E. 11.29 ; 4. H". 53° E. 19.32; 5. Unknown; 6. S. 60}° W. 7.12; 7. S. 29J°E. 2.18; 8.S. 60J o 'W.8.12; to find the bearing and dis- tance of the fifth side. Ans. K". 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 54J E. 13.45 chains; 3. Unknown; 4. S. 74° W 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. 49i° W. 9.30 chains ; CD S. 32|° E. 10.25 chains ; DE S. 5J° W. 6.75 chains ; and EB K 79|° 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. 10|° 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 beariDg 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. V. Examples. Ex. 1. Given the courses and distances as below, to find the unknown bearing and distance. Sta. 1 2 3 4 5 6 7 Bearing. Changed Bearing. Dist. N. s. E. w. K 56% W. S. 57f W. 15.35 8.19 12.98 N. 9 W. 1ST. 75 "V7. "T^orth. 19.51 5.05 18.85 1ST. 66 E. (14.00) S. 39 j E. N. 74J E. 13.35 3.62 12.85 1ST. E. ~S. 6f W. "sr"52JW7 12.65 (12.12) (3.62) S. 59£ E. 12.18 20.95 6.23 10.47 S. 13J e. 20.37 4.89 31.83 34.79 34.79 31.83 Dist., fifth side, 12.65 Dep. " 3.62 Ch. bear. " 2SL 16° 38' E. 66° A. C. 8.897909 0.558709 sin. 9.456618 N". 82° 38' E., bearing of fifth side. Ch. bear., fifth side, 16° 38' Dist. " Diff. Lat. " 12.12 Dist., third side, 14.00 ch. cos. 9.981436 1.102091 "1.083527 Ex. 2. Given— 1. K 47° W. 16.55 chains; 2. K 19° 5' W. 11.48 chains; 3. I". W. 15.53 chains ; 4. K 23° E. 9.72 chains; 5. K 75|° 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,— 1. K 56%° W. 15.35 chains ; 2. K 9° W., unknown; 3. K 66° E. 14.00 chains; 4. S. 39 j° E. 13.35 chains; 5. K 82}° E., unknown ; 6. S. 6f¥, 12.18 chains; 7. S. 52J° W. 20.95 chains; to find the distances of the second and fifth sides. Sta. 1 Bearing. Changed Bearing. Dist. N. s. E. w. K56JW. K 47JW. 15.35 10.42 11.27 2 3 K 9 W. North. (19.54) (19.54) 1ST. 6Q E. K 75 E. 14.00 3.62 13.52 4 5 S. 39f E. S. 30f E. 13.35 11.47 6.83 K82fE. S. 88| E. .39 (12.64) 6 7 S. 6f W. S. 15} W. 12.18 11.72 3.31 18.41 32.99 S.52JW. S. 61J W. 20.95 10.00 33.58 33.58 32.99 220 COMPASS SURVEYING. [Chap. V. Ch. bear., fifth side, 88° 15' A. C. sin. 0.000203 Dep. " 12.64 1.101747 Dist. " 12.65 1.101950 Ch. bear. cos*. 8.484848 Dist. 1.101950 Diff. Lat. 0.39 S. — 1.596798 Ex. 2. Given— 1. S. 29f° 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. Rule. 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 56J° W. 15.35 chains; BC K 9° W. 19.51 chains; CD K — E. 14 chains ; DE S. 39|° E. 13.35; EF 1ST. 82|° 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. AB BC Ce ef GA Bearing. Dist. N. s. E. w. N". 56i w. 15.35 8.53 12.76 K". 9¥. 19.51 19.2T 3.05 S. 39| E. 13.35 12.65 1.65 10.26 8.54 K 82J e. 12.54 ' S. 52i w. 20.95 12.75 16.62 29.45 23.01 21.08 32.43 23.01 21.08 6.44 11.35 Diff. Lat. 6.44 A. C. 9.191114 Dep. , 11.35 1.054996 Tang, closing line, S. 60° 26' E. 10.246110 Cos. bear. 60° 26' A.C. 0.306769 Diff. Lat. 0.808886 Dist. closing line, 13.05 1.115655 FG 12.18 /G 13.05 A. C. 8.884388 /F 14.00 2)39.23 " " 8.853872 19.615 1.292588 7.435 0.871281 2)19.902129 JF/G 26° 41' cos. 9.951064 F/G 53° 22' 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 6° 51' 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. K 53° E. 19.32 chains; 5. K — W. 16.26 chains; 6. S.60f°W. 7.12 chains; 7. S. 29J° E. 2.18 chains ; 8. S. 60J° 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. _. t _. Fig. 154: Let ABCDEFG (Fig. 154) be the d plat of the tract, the bearings of CD ,^^^\ and FG being supposed unknown. f\ If Ce and ef be drawn parallel to 1 \^^**_ the sides DE and EF, and /G be 1 ^"" joined, then will ABCe/G form a B V closed figure, the bearings and dis- ^^ J yS tances of all the sides except /G *a being known. The course and dis- * tance of this side, which is the closing line, are found as directed in the rale. 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 OP 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 and 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 the 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, — As rad. : sin. A : : AC : CD : : AB . AC : AB . 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. sin. B AB BC 2 ABC { 47° 35' 12.36 ch. 14.36 2)131.033 AC. 0.000000 9.868209 1.092018 1.157154 2.117381 65.5165 ch. = 6 A., 2 R., 8.26 P. Sec. IX.] CONTENT OF LAND. 225 Ex. 2. Given AB K 37° 14' W. 17.25 chains, and BC K 74° 29' W. 10.87 chains, to determine the area of the triangle ABC. Ans. 5 A., 2 K,, 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) t : sin. A : : AC : CD (Art. 137), and sin. 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. As Trad, jsin. C A.C . 0.000000 55° 49' tc 0.082366 rsin. A (sin. B 47° 56' 9.870618 76° 15' 9.987372 JAB Iab 21.62 ch. 1.334856 21.62 1.334856 2 ABC 2 ) 407.444 2.610068 Area = 203.722 ch. = 20 A., 1 E., 19.5 P. c 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. Rule. 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 perpendicular to AB, and CF perpendicular to ED. Then will DCF = 180° so (B + C.) Also, draw AH parallel to CB, and join DH. Then will 2 ABD = AB . DE = AB (EF ± DF) = AB.EF±AB.DF = 2 ABC =fc 2 CDH. Fig. 156. Fig. 157. Whence 2 ABCD = 2 BDC + 2 ADB = 2 BCD + 2 ABC it 2 CDH : the plus sign being used (Fig. 157) when the sum of the angles is greater than 180°. 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 AB 6.95 0.841985 BC 8.37 0.922725 { : 2ABC ■ 57.975 1.763237 As r 0.000000 : sin. 180° - (B + C) 40° 31' 9.812692 f AB 6.95 0.841985 " 1 CD 5.43 0.743800 : 2CDH 25.031 1.398477 As r 0.000000 : sin. C 54° 12' 9.909055 (BC 8.37 0.922725 "(CD 5.43 0.734800 : 2 BCD 36.862 57.975 94.837 25.031 2)69.806 1.566580 34.903 ch.= = 3A.,1R.,38.45P 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 P. 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 Eig. 156, we have 2ABCD=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 (Eig. 158) we shall 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.[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.]+ CD. sin. [C.D.]+ C.E.sin. TC.E.1 + D.E. sin. [D.E]. Fig. 159. Sec. IX.] CONTENT OF 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 HSrS be a meridian anywhere a on the map. Through the corners draw the perpendicu- *\ lars Aa, Bb, &c. Then, it is evi- dent that ABCDEFG = AagQ + Qgf¥ + DdeE - AabB - BbcC - GcdD - EefF. Now, these various figures being trapezoids, their areas will be found by multiplying their perpendiculars by the 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 N"S being arbitrary, the sum Aa + Bb, corresponding to the first side AB, may be taken at pleasure. Now, if from Aa + Bb we take Ah, the whole departure of the two sides AB and BC, we have Bb 4- Cc, the sum of the parallel sides of BbcC. Similarly, if to Bb + Cc we add z'D, 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. Dl E. qV IE mE W. AJc Bp E. D. D. W. D. D. Multipliers. N. Areas. S. Areas. AB BA; Afc + Go Aa + Bb, E. 2 AakB BC Afc+Bp Bb + Cc. E. Cc + Dd, E. 2 BbcC CD DE Cq qV — Bp 2 CcdD qD + lE Dd-j-Ee, E. 2 DdeB EF Em ZE + mF Ee + F/. E. 2Ec/F FG nG En Go mF — Fn F/+ Qg, E. 2 FfgG GA En + Go Gg + Aa. E. 2 G^aA 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; -f 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: + Bp, is the sum of those of AB and BC, and is west ; the third is Do — Bp, and is east, because D is east of B ; the fourth, Dq + El, 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 E"S being to the west of AB, the multiplier is east. The double de- parture AA: + Bp = Ah being west, subtract it from Aa + B6, and we have B6 + Co. To Bo + Co add the next double departure, qD — pB = zT>, and we have Cc + Dd; qD + IE added to Co + Dd gives Dd + Ee ; IE + mF added to Dd-h Ee gives Ee -f F/; mF — En added to Ee + F/ gives F/ -f Gg ; and, lastly, En + Go taken from Ef + G# leaves G# + 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 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 AabB + 2 BbcC + 2 CcdD - 2Ddr + 2reE -2EefE + 2 EfgQ + 2 Ggs - 2 saA = 2 AabB + 2BbcC + 2 CcdD - 2 (Ddr - reE) - 2 Ee/F + 2 EfgGc + 2 (Qgs - saA.) iS 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 + B6) ao + (B6 + Cc) 6c + (Ce + Dd) ed - (Dd - Ee) de - (Ee + /F) e/ + (/F + G#)/# -f- (G^ — Aa) ag. The following table exhibits the whole. Sides. N. S. E. Cz W. pB qC E. D. D. W. D. D. Multipliers. N. Areas. S. Areas. AB Ap pB+Go Bo + Act, W. 2 Aa&B BC B? pS + qG Bo+Cc, W. 2BoCc CD Di Gi — qG Cc + m, w. Dd — Ee, W. 2 CcdD DE Ei Ci + Dt 2 (DdV — Eer) - EF Em mF Di + Fm Ee + F/, E. 2 (Ee/F) FG Grc Go Fm — Frc F/+G£,E. 2F/#G GA A.0 Fw + Go Gg — Aa, E. 2(Ggs — Aas) Here the first multiplier is west, the meridian being to the east of the line AB. The subsequent multipliers are found as follow :— (B6 + Aa) + (p~B + qC) = B& + Cc ; (Bo + Cc) - (0* - qC) = Cc + Dd; (Cc + Dd) - (Gi + Bt) = Dd - Ee ; (Bt + Fm) - (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/) + (Pm-Fn) = F/ + Gg; and (F/ + Gg) — (Fw + Go) = G^ — 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 : — Kule. Eule 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 56%° W. 15.35 ch. ; 2. K 9° W. 19.51 ch. ; 3. K 66° E. 14.01 ch. ; 4. S. 39|° E. 13.35 ch. ; 5. K 82|° E. 12.65 ch.; 6. S. 6f° W. 12.18 ch. ; T. S. 52J°W. 20.95 ch. ; to find the area. Sw. IX.] CONTENT OF LAND. 233 £ £ H OS CO oq 00 OS »0 Cq rH IrH I OS I rH CO CO I CO CO j rH o w £ 00 © CO H H w o o Oq 00 CO CM rH CO rH o o 1—1 1— 1 CO CO rH T— 1 © 1 y* © P . j rH f iH T— 1 © i— i © 1 <£> 1 t- © CO rH oq 1 y ' CO 1— 1 1 CO 1 oq | 1 © 1 1 1 ~ H 1 1 1 © i— i 1 e^ 1 rH 1 00 OS CO CO f£ £ w w p4 £ fh)tJ< CD CD wh* r-|M CO OS OS oq >o CO 00 CO 525 a fc 02 £ 02 rH © l>- i—i rH CO O OS iO OS rH IQ rH CO CO CO CO oq rH oo oq »o © COrH oq »o rH 1>. oq © CO CO £3 1-4 t~ oq r CO < CO OS CD 00 00 e3 CO H CO <1 CO CO CO co oq i— i iO CO oq i—i iO CO oq © rH -*3 © 02 03 CD £ CO to o CO u CO CO CO oq co co © iO CO IO iH rH l—l 1-H o kffl iO * CO CO A © © o CO OQ o T-l o *4 Sh W 234 COMPASS SURVEYING. [Chap. ? Ex. 2. Given the bearings and distances as in the ad- joining table, to calculate the area. o CO tH o vO o o © 1 «* OS OS T* l^ o -tf »>• CO OS l—\ CO t>- - © CO CO 1—t CO T— 1 1 £ vO 'DO o CO to CI CO I "* co o o CO CS CO o OS CI vO o CG o o ci oq © O OS © I CM © OS OS CO co 00 o o CI CI H w £ & ^ O CO OS V O VO vO o CO CD o CM o OS vO o vO o >o CO fc DQ m go !Z5 H 3 i I— 1 OS « 55 CO to .2 o o o o vO GO CO t— 1 CM I— 1 i— i CO i— < OS CO I-H © CO T— 1 CM vffl CO CO I— 1 © rH T— 1 CO OS © CO I— 1 © GO rH CM CO "H VO © t^ CO OS © r— 1 ©rH OS ,-( CM OS rH . CO © © vo CM i— I © rH © © CO CM VO id co vd co CM CO vd CM CM CO »d CM CM l>- ■<* CM CM tH CM vO © © © CM vd CM Tt< OS CO CM to o CM CO VO rH Tin tj5 CM CM PR vO © O PR rU © Ph © CO Sec. IX.] CONTENT OF LAND. 235 Ex. 3. Given the bearings and distances as follow, to calculate the area:— 1. K 27° 15' E. 7.75 ch.; 2. S. 62° 25' E. 10.80 ch.; 3. S. 7° 55 f E. 9.50 ch.; 4. S. 47° 25' E. 9.37 ch. ; 5. S. 54° 25' W. 8.42 ch.; 6. K 37° 35' W. 23.69 ch. Ans. 22 A., 1 E., 26.17 P. Ex. 4. Calculate the area from the following notes: — 1. K 46° 40' W. 18.41 ch. ; 2. K 54° 30' E. 13.45 ch. ; 3. K 5° 30' W. 12.65 ch.; 4. S. 74° 55 f E. 17.58 ch; 5. S. 47° 50' E. 15.86 ch. ; 6. S. 47° 25' W. 16.36 ch. ; 7. S. 62° 35' W. 14.69 ch. Area, 66 A., 2 E., 21 P. Ex. 5. Given the bearings and distances of the sides of a tract of land, as follow,— viz. : 1. JST. 43° 25' W. 28.43 ch. ; 2. K 29° 48' E. 30.55 ch.; 3. S. 80° E. 28.74 ch.; 4. K 89° 55' E. 40 ch. ; 5. S. 10° 13' E. 23.70 ch. ; 6. S. 63° 55' W. 25.18 ch. ; 7. K 63° 45' W. 20.82 ch. ; 8. S. 57° 25' W. 31.70 ch. : to determine the area. Area, 262 A., 2 E., 31 P. Ex. 6. Calculate the distances of the third and fourth sides, and the area of the tract, from the following notes : — 1. S. 64° 5' W. 11.18 ch. ; 2. K". 49° 45' "W. 12.91 ch. ; 3. K 35° 20' E., distance unknown; 4. S. 82° 25' E., distance unknown; 5. K 87° E. 13.82 ch. ; 6. K 49° 30' E. 4.95 ch. ; 7. S. 33° 25' E. 10.80 ch. ; 8. S. 0° 55' E. 9.22 ch. ; 9. S. 79° 10' W. 14.30 ch. ; 10. H". 52° 15' W. 8.03 ch. Ans. 3d side, 12.13 ch. ; 4th, 9.71 ch. ; Area, 57 A., 1 E., 12 P. Ex. 7. One corner of a tract of land being in a swamp, but visible from the adjacent corners, I took the bearings and distances as follow:— 1. S. 45° E. 13.65 ch.; 2. K 38|° E. 17.28 ch. ; 3. K 19° W. 23.43 ch. ; 4. S. 58° W. 14 ch. ; 5. K 87° W. 8.14 ch. ; 6. K 45J° ~W. 9.23 ch. ; 7. S. 28J° W. 14.60 ch. ; 8. S. If ° E. ; 9. K 79J°E. Eequired the distances of the last two sides and the area of the tract. Ans. 8th side, 16.44 ch. ; 9th, 20.51 ch. ; Area, 92 A., 1 E., 7 P. 362. Offsets. If any of the sides border on a water- course, or are very irregular, stationary lines may be run as 236 COMPASS SURVEYING. [Chap. V. near the boundary as possible, and offsets be taken as directed in chain surveying. The area within the stationary lines may then be calculated as above. That of the spaces included between those lines and the true boundary is to be calculated as in Art. 256. These areas added to or subtracted from the former, according as the stationary lines are within or without the tract, will give the content required. "When the tract bounds on a stream, it is usual to con- sider the boundary as the middle of the stream, except in tide waters or large rivers which are navigable and are thus considered public highways. In these cases the boundary is low-water mark. In reciting the boundaries in title-deeds, the offsets are not generally given. The description usually runs thus: — Thence S. 43J° E. 10.63 chains to a stone on the bank of Ridley Creek, and thence on the same course 1.05 chains to the middle of said creek. Thence along the bed of said creek, in a southwesterly direction, 37.63 chains; thence N". 47° "W., by a marked white-oak on the banks of the creek, 25.63 chains to a limestone, corner of John Brown's land, &c. Examples. Ex. 1. Calculate the area from the following field-notes : — 55 (4) 1350 55 (3) N.26°45'E. 55 (3) 2160 270 1929 396 1408 310 1015 340 610 50 (2) N.56°30'E. 3050 Mid. of do. 3000 (2)on r.bank (1) N.36°30'W. 60 (6) 1471 95 930 140 485 60 (5) S.51°30'E. 60 (5) 1072 130 750 85 390 55 (4) S.84°45'E. (1) 4316 Middle 75 of river. PL S.45°15'W. 75 V) 826 100 420 60 (6) S.11°45'E. Sec. IX. J CONTENT OF LAND. 237 Sta. i~r~ \~ 3 ~T~ 5 6 7 Bearings. Dist. N. S. E. W. E. D. D. W.D.D. Mult. N.Areas. S. Areas. N.36>^W. 30.00 24.12 17.84 47.96 .00E. N.56%E. 21.60 11.92 18.01 .17 .17E. 2.0264 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 S.ll^E. 8.26 8.09 1.68 13.19 76.40E. 618.0760 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.S444 294.6020 Area of offsets calculated as in Ex. 1, Art. 257. 128 A., 2 R., 14.76 P. 2 )2375.2424 1187.6212 = 98.S0145 = 128.592265 Ex. 2. Given the field-notes as below of a rneadow bounding on a small brook, to calculate the area: — (2) 1132 55 1054 72 896 97 739 75 480 On brook. (3) 1740 63 1414 35 1237 87 1016 § 45 824 1 50 652 si 551 452 75 295 75 **> -(2) (l) 1450 (5) (5) 1344 3 (1) 9.12 1- 98° 34' (7) f- 27° 46' (7) 2.40 corner 14 2.26 2.00 1.75 6 1.50 32 12° 14' -J (6) In calculating the area, it will be necessary first to calcuS late the bearings from the observed angles. . v 4rea, 15 A., 2 R., 11.5 P v Fig. 162. F B 363. Inaccessible Areas. When it is desired 4;o de- termine the area of a tract of difficult access, such as a pond, a thick copse, or a swamp, it should be surrounded by a system of lines as near the boundaries as they can be run without multiplying the number of sides unnecessarily. Offsets should then be taken to different points of the boundary, so as to determine its sinuosities. The areas of the parts determined by these offsets, taken from the area enclosed in the base lines, will leave the content required. Where two base lines make an angle with each other, the first offset on each should be taken to the same point in the irregular boundary. Thus, if AB and BC (Fig. 162) are two adjacent baselines enclosing an irregular boundary HDI, the first offsets should be. taken at F and E, so situated that the offsets FD and ED should meet at the same point D of the boundary. The triangular spaces BDF and BDE will then be included with the areas belonging to the lines AB and BC respectively. Sec. IX.] CONTENT OF LAND. 239 The following examples of the field-notes and calculation for the area of a pond will illustrate this subject: — Fig. 163 is a plat of Ex. 1 on a scale of 1 inch to 10 chains. Fig. 163. (5) 1866 155 1805 25 1675 1475 10 1250 55 950 22 800 75 475 r78°55' (4) 115 55 90 105 22 42 42 r23°51' (1) 1140 i- f 52° 52' on 1 1(1). (2). 1100 90 875 10 750 10 500 60 250 112 >1' 75 112 (6) T 56° 35' (6) 920 870 122 750 32 575 17 300 73 85 97 V (5) f 69° 39' 240 COMPASS SURVEYING. [Chap. V. 1 Sta, 1 ~2~ 3 4 5 6 Bearings. Dist. N. S. E. W. E.D.D. W.D.D.j 29.88 J Multipli'r. S. Areas. N.88°35'W. 22.80 .55 22.78 ~2i3~ .00 E. N. 9° 40' W. 13.85 13.65 25.11 25.11 W. 342.7515 N. 68° 28' E. 11.52 4.23 10.72 8.39 16.72 W. 12.64 E. 70.7256 S. 87° 41' E. 18.66 .76 18.64 29.36 9.6064 S. 18° 2' E. 9.20 8.75 2.85 21.49 34.13 E. 298.6375 S. 38°33'W. 11.40 8.92 7.10 4.25 29.88 E. 266.5296 18.43 18.43 32.21 32.21 Content within the hase-lines, 59.24 59.24 2)988.2506 494.1253 ch. Base. Dist, Offsets. Inter. Dist. Sum of Offsets. Areas. 0.00 0.55 .82 .55 .82 .4510 3.12 .55 2.57 1.37 3.5209 5.55 .10 2.43 .65 1.5795 7.05 .32 1.50 .42 .6300 10.00 2.20 2.95 2.52 7.4340 12.40 2.91 2.40 5.11 12.2640 (1)(2) 14.80 1.75 2.40 4.66 11.1840 17.70 .33 2.90 2.08 6.0320 20.15 .07 2.45 .40 .9800 22.15 .60 2.00 .67 1.3400 22.80 .65 .60 .3900 45.8054 .47 .75 .47 .75 .3525 1.55 .22 1.08 .97 1.0476 4.30 .55 2.75 .77 2.1175 (2) (3) 7.75 .10 3.45 .65 2.2425 9.75 2.00 .10 .2000 11.25 .25 1.50 .25 .3750 12.95 1.55 1.70 1.80 3.0600 13.85 .90 1.55 1.3950 10.7901 1.32 1.20 1.32 1.20 1.5840 2.50 .70 1.18 1.90 2.2420 (3) (4) 5.25 11 2.75 81 2.2275 7.75 2.50 11 .2750 9.50 1.75 .0000 11.35 42 1.85 42 .7770 11.52 17 42 .0714 7.1769 Sec. IX.] CONTENT OF LAND. 241 Base. Dist. Offset. Inter. Dist. Sum of Offset. Areas. 3.9900 1.3650 .3300 3.8100 4.3875 2.9000 2.2100 .7015 (4) (5) (5) (6) (6)(1) .00 4.75 8.00 9.50 12.50 14.75 16.75 18.05 18.66 .42 .42 .00 .22 1.05 .90 M 1.15 .00 4.75 3.25 1.50 3.00 2.25 2.00 1.30 .61 .84 .42 .22 1.27 1.95 1.45 1.70 1.15 19.6940 .00 .85 3.00 5.75 7.50 8.70 9.20 .97 .73 .17 .32 1.22 .00 .85 2.15 2.75 1.75 1.20 .50 .97 1.70 .90 .49 1.54 1.22 .8245 3.6550 2.4750 .8575 1.8480 .6100 10.2700 .00 .75 2.50 5.00 7.50 8.75 11.00 11.40 1.12 1.12 .60 .10 .10 .90 .00 .75 1.75 2.50 2.50 1.25 2.25 .40 1.12 2.24 1.82 .70 .20 1.00 .90 .8400 3.9200 4.5500 1.7500 .2500 2.2500 .3600 13.9200 Area within base lines, A. 49.41253 Double area , cut off by (1) (2) 4.58054 (2) (3) 1.07901 (3) (4) .71769 (4) (5) 1.96940 (5) (6) 1.02700 (6) (1) 1.39200 i of 10.76564 = 5.38282 Area of pond, 44.02971 = 44 A., OR., 4.75 P. 16 242 COMPASS SURVEYING. [Chap. V. The following are the field-notes taken for the survey of a pond. The area is required. Fig. 164 is the plat, to a scale of 1 inch to 10 chains : — Fig. 164. (1460 ) AB 580 560 70 300 15 150 20 48' -| Sta.F Sta.F 627 475 65 250 20 90 onAB Q 627) f 44° 5' Sta. A r f 87° l' 1 l on AB. 950 900 20 750 70 500 400 300 30 150 25 27 70 Sta.E [-70° 29'. Area, 24 A., 3 R., 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 fonnd 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 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', 27° 33', GFC = 35° 35', GFD = 58° 25', GFE = 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. GFB 92° 10', FGA = 26° 5', Calculation. 1. To find FA. As sin. FAG 90° 40' .000029 : sin. FGA 26° 5' 9.643135 : : FG 12.25 1.088136 : FA 0.731300 244 COMPASS SURVEYING. [Chap. V. To find FB. As sin. FBG 93° 57' .001033 : sin. BGF 58° 30' 9.930766 : : FG 1.088136 : FB To find FC. 1.019935 As sin. FCG 47° 13' 0.134347 : sin. FGC 97° 12' 9.996562 :: FG . 1.088136 : FC 1.219045 To find FD. As sin. FDG 49° 7' 0.121453 : sin. FGD 72° 28' 9.979340 :: FG 1.088136 : FD 1.188929 To find FE. As sin. FEG 50° 18' 0.113848 : sin. FGE 37° 32' 9.784776 :: FG 1.088136 : FE 0.986760 • To find 2 FAB. sin. AFB 35° 42' 9.766072 FA 0.731300 FB 32.9084 1.019935 2 FAB 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 To find 2 FCD. sin. CFD 22° 50' 9.588890 CF 1.219045 FD 1.188929 2FCD 99.2805 To find 2 FDE. 1.996864 sin. DFE 33° 45' 9.744739 DF 1.188929 FE 0.986760 2FDE 83.2585 To find 2 FEA. 1.920428 sin. AFE 28° 55' 9.684430 FE 0.986760 FA 0.731300 2FEA 25.2633 1.402490 2FBC / 24.2286 2FCD 99.2805 2FDE 83.2585 206.7676 2 FAB 32.9084 2FAE 25.2633 ' 58.1717 2)148.5959 74.29795 sq.ch = 7 A., 1 K., 28.76 P. 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 K 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, IC, ID, 1 &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. V. 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 maybe 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 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° IV 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, Fig# 167# 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 3, and C of 2 : T 2 _ = i of the error should be applied to A, r 8 o 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 38407.5 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. \ t 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 mz and nz represent the sides, and A the area, then /A will mnz* = A. Whence z = / — . sjmn 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* = FG a = GE* -\- EF a = area -\- square of half the difference of the sides. Then, AB = AF + FB, BC = FB — FE. Fig. 168. G <\ 4 ' I \ / , N / ^ I I \ 1 | s 1 1 i \ \ E F A Sec. L] 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. Eequired to lay out 17 A., 3 B., 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. Fig. 169. Demonstration. — By Art. 357 we have (Fig. 169) ^ c ^ r : sin. A : : AB . AC : 2 ABC, or (1.6) r . AB : sin. A [ /\ ~~7 . AB : : AB . AC : 2 ABC; whence sin. A. AB : 2 ABC ; / \ / : : r . AB : AB . AC : : r : AC. '> / \ / Examples. Ex. 1. Eequired to lay out 43 A., 2 R. in the form of a parallelogram, one side AB being 54 chains, and the adja- cent angle BAC 63°. 254 LAYING OUT AND DIVIDING LAND. [Chap. VII. As AB . sin. I sm. A 54 63° A . C. 8.267606 " 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£° 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 1ST. 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. Fig. 170. c ----. 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 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. I.] 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 = — — - — . Draw the * AD parallel EF cutting AF in F. Eun FG the given course. Take AB a mean pro- portional between AD and AG or = N /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 cut 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. 1 / f i \ i y i \ 1 s i i \ A D B G 256 LAYING OUT AND DIVIDING LAND. [Chap. VII. Here the angles are A = 69° 45', B = 63°, and C = 47° 15'. Whence A fsin.A I sin. B 69° 45' Ar. Co . 0.027709 63° u a 0.050119 ( rad. \ sin. C 47° 15' 10.000000 9.865887 : : 2 ABC 140 chains 2.146128 : AB 2 2)2.089843 AB 11.09 1.044921. Ex. 2. Given the bearings of two adjacent sides, taken at the same station, K 57° 15' "W. and K 45° 30' E., to deter- mine the distance from the angular point of a station on the first side from which a line running JT. 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. 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-f- CDK. To Sec. L] 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 -r- PL. Then in DFC we have DF = PI and FC = DE to find DC, which added to AD will give AC. L? 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 PI. As sin. I 56° A. C. 0.081426 : sin. PAI 7° 9.085894 : : AP 18.85 1.275311 : PI As rad. : sin. PAL :: PA : PL Given area, AD 12.65 1.102182; whence ED = AD - PI = 12.65 - 2.77 = 9.88. To find DC. FC + FD = 12.65 1.102182 FC - FD = 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 2.77 0.442631 To find PL and AD A. C 0.000000 49° 9.877780 18.85 1.275311 1.153091 180 ch. 2.255273 258 LAYING OUT AND DIVIDING LAND. [Chap. VII. of which from AB and AC are 3.25 chains and 7.92 chains respectively. Eequired 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 ET. 46° 15' E. and N. 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 = 140 = 5.98. ^-.-v 23.42 Draw PE parallel to AB. Erect the perpendicular DF N \ j = AE, and make FC = ED. \ \ / Then CB will be the divi- • ^ sion-line. For, as before, AIHD = the given area; but PTTFT = PBI + CKD ; .-. HIBK = CKD, and AIHD = ABC. r : tan. 30°: : AP (23.42) : AE = DF = 13.52; whence CF = DE = AE + AD = 19.50, and DC = N /CF 2 -FD 2 = v/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 % line running from the first side to the third. Sec. I.] LAYING OUT LAND. 259 CASE 1. 383. The division line to be parallel to the second, side. Fie. 175. Conceive the lines CB and Fig. m. 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. Then say, As EAB is to ECD, so is AB 2 to CD 2 . And, as sin. E is to sine of B, so is AB^CDtoAD. The following is a neat construction : — Produce HB 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 = x/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. [Chap. VII. CD 3 : AB 3 ; CD 3 *cAB 3 : AB°; CD 3 «nsAB 3 : AB a , 2 ABCD : CD'cssAB 9 . Demonstration. — ECD : EBA Whence, by division, ABCD : EBA consequently, 2 ABCD : 2 EBA and 2 EBA : AB 3 But (Art. 380) sin. A. sin. B : rad. sin. E : : 2 EBA : AB 3 ; whence sin. A. sin. B : rad. sin. E :: 2 ABCD : CD 3 «>»AB 3 . Examples. Ex. 1. Given— 1. K 62° 15' E. ; 2. K 19° 12' W. 7.92 chains ; 3. S. 87° W., to cut off 5 acres by a line parallel to the second side. Bequired the length of the division line, and the distance on the first side. First Method.— To find ABE, (Art.358.) ( rad. I sin. E A. C. 0.000000 24° 45' a " 0.378139 ( sin. A 98° 33' 9.995146 1 sin. B 106° 12' 9.982404 ..J AB 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 2 (7.92 (7.92 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 : AB 5.539 0.743460 Sec. I.] LAYING OUT LAND. 261 Second Method. f sin. A AS i ' "R (_ sm. B 98° 33' A. C. 0.004854 106° 12' 0.017596 J rad. 10.000000 1 sin. E 9.621861 : : 2 ABCD 100 ch. 2.000000 : CD 2 -AB 2 44.087 1.644311 AB 2 62.7264 10.33 Whence CD = s/ 106.8134 = 5, as before. Ex. 2. Given— 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 parallel to the second side. The distance on the first side and the length of the division line are required. Ans. Division line, 10.72 chains; distance, 5.98 ch. Ex. 3. Given the bearing of three sides of a tract of land, and the length of the middle one, as follow, — viz. : 1. K 15° 30' W. ; 2. K 74° 30' E. 11.60 chains ; 3. S. 45° E. : to cut off 12 acres by a line parallel to the second side. The division line and distance on the first side are re- quired. Ans. Division line, 16.44 chains; distance, 8.555 ch. ;k £2 DJ LI 384. If AD and BC (Fig. 176) are &g.m. nearly parallel, the following method may be employed with advantage : — Divide the area by AB : the quotient will give the approximate length of the perpendicular AI. Draw FE parallel to AB, and AK parallel to BH. In AIK and ALF find IK and IF. FK = FI ± IK, and FE = AB ± FK. If the sum of the angles is greater than 180°, the area cut off by EF will be too great by the small triangle AFK = FK . AI _ __ AFK FK . AI a Make IL = Then will AL be 2 FE 2 FE the corrected perpendicular : AD may thence be found. 262 LAYING OUT AND DIVIDING LAND. [Chap. VII. Examples. Ex. 1. Given GA K 87° W., AB N". 5° W. 14.25 chains, and BH S. 89° E., to lay off 10 acres by a line parallel to AB. Here the angles are A = 98° and B = 84° : AK will therefore lie between I and F. 100 AI = tttt^ = <-02 chains, nearly. 14.25 J In IAF we have IAF = 8° and IA = 7.02; whence IF = .987. In IAK we have IAK = 6° and IA = 7.02 ; whence IK = .738. Whence KF = .25 and EF = 14.50. KF.AI - , . ilence IL = - -— = .06 chains, 2EF ' and AL = 7.02 - .06 = 6.96 chains ; whence AD = 7.03 chains. The above method is very convenient for field operations. EF may be measured directly on the ground; whence FK is known, and IL = - ^^ , as before. > 2FE ' Ex. 2. Given GA North, AB K 89° E. 7.86 chains, and BC S. 1° 30' W., to cut off 10 acres by a line parallel to AB. Required the distance of the division line from A. Ans. 13.00 ch. CASE 2. 385. By a line running a given course. Fig. m. Construct, as in last case, ABG to contain the given area. Draw BL according y\ \ to the given course. Take y'' ED a mean proportional B '" "aw l Sec. L] LAYING OUT LAND. 263 between EL and EG: CD p Fig 178. parallel to BL will be the ^ "-^ ra division line. Eor, by the \ »^ B •\0 lemma, ECD = EBG; ' \ / \ / \ whence ABCD = ABG, \ J i 1 A * *• the reqi lired area. \ / \ / \/ A*^ I \ » , N. . W \i> g i \ • 'A N - % a»E The calculation may be performed by the finding AE 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 EraW = EAB, EW is a mean proportional between EA and EL. Whence nW is a mean proportional between AP and BL ; there- fore AP . BL = wW 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« : AP . BL = wW a ; and, by demonstration to last case, sin. G . sin. D : rad. sin. E : : 2 rcWDC : CD 3 **rcW 3 . 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. K 62° 15' E. ; 2. K 19° 12' W. 7.92 chains ; 3. S. 87° "W\, to cut off 5 acres by a line perpen- dicular to the first side. Eequired 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. Cc . 0.378139 : sin. B 106° 12' 9.982404 :: 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,) c sin. E ( sin. D 24° 45' Ar. Co. 0.378139 90° « u 0.000000 ( rad. 10.000000 ( sin. C 65° 15' 9.958154 : : 2 ECD 242.278 2.384314 : ED 2 t 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 Seo. L] LAYING OUT LAND. 265 As ■■{ Second Method. sin. C 65° 15' Ar. Co. 0.041846 sin. D 90° u a 0.000000 sin. A 98° 33' 9.995146 sin. B AB 106° 12' 7.92 chains 9.982404 0.898725 AB u 0.898725 riW 2 65.5913 1.816846 As r sin. C \ sin. D Ar. Co. 0.041846 a a 0.000000 f rad. 10.000000 ( sin. E 24° 45' . 9.621861 :: 2ABCD 100 chains 2.000000 : CD 2 -riW 2 46.1006 1.663707 nW 2 65.5913 CD = >/111.6919 = 10.57. As sin. C 65° 15' Ar. Co. 0.041846 : sin. B 106° 12' 9.982404 :: AB 7.92 0.898725 : BM 8.375 0.922975 CD 10.57 DX 2.195 As sin. E 24° 45' Ar. Co. 0.378139 : sin. C 65° 15' 9.958154 :: DF 2.195 0.341435 : AD 4.76 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 = >/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. H". 78° 17' E; 2. K 5° 13' E. 15.62 chains; and 3. 2sT. 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, N". 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.L] 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 ECD PL Lay off EF = ——. j PI 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 + v/FE? - EK 2 , and AD = ED - EA. Examples. Ex. 1. Given DA West, AB K 16° 15' W. 6.30 chains, BC IS". 57° E., to cut off 3 acres by a line through a P, situated K 25° 30' E. 6.09 chains from the corner A. spring ., To find EA, EAB, and ECD. As sin. E 33° : sin. B 73° 15' :: AB 6.30 : EA 11.077 AB 6.30 sin. A 73° 45' 2 EAB 66.994 2 ABCD 60. 2 ECD = 126.994. Ar. Co. 0.263891 9.981171 0.799341 1.044403 0.799341 9.982294 1.826038 268 LAYING OUT AND DIVIDING LAND. [Chap. VII. To find PI and EF. As rad. Ar. Co. 0.000000 : sin. PAI 64° 30' 9.955488 :: AP 6.09 0.784617 : PI 5.497 0.740105 ECD 63.497 1.802753 EF 11.552 1.062648 To find AK, EK, and EF. As sin. K 33° At. Co. 0.263891 : sin. APK 31° 30' 9.718085 :: AP 6.09 0.784617 : AK 5.842 0.766593 AE 11.077 EK = FG = 5.235 mce KF = GD = EF - EK = 6.317. To find FD. • GD + GF 11.552 1.062648 GD- GF 1.082 0.034227 2)1.096875 FD = 3.535 .548437 Whence AD = EF + FD — EA = 4.01. Ex. 2. The bearings of three adjacent sides of a tract of land are as follow,— viz. : DA 1ST. 47° E., AB 1ST. 35° 16' W. 15.23 chains, and BC S. 36° W., to cut off 15 acres by a line running through a spring P 9.22 chains distant from the first, and 10.55 chains from the second, side. The dis- tance of the division line from the end of the first side is required. Ans. 10.82 chains from A. Sec. I.] LAYING OUT LAND. CASE 4. 387. By the shortest line. Produce the lines CB and DA (Fig. 180) to meet in E, and calcu- late ABE and AE, whence ECD is known. £s~ow, the shortest line cut- ting off a given area will make equal angles with the sides. Therefore EC ™ -r. ~™^ EC.ED.sin.E = ED. But 2 ECD = — 269 Fig. 180. 1<„ E ED 2 , sin E E whence we must have AD = EA c* . GO CD CM o Tt< CO 00 • lO ■* CO 1^ T* o I— 1 I-l r-l T~i 1—1 CO I— 1 1—1 I— 1 CO CO rH CO co* CD I— 1 o o CD l-H l-H CD* M i-i r-i £ w w £ o o o A C0|<*< W|«J( i— (,kj< P 03 O t~ T* •^ t^ i— t W fc £ 0Q m CD CO CO r- o CD r?< CO CD r-t CM W H W pq o |o |^ [ CO O CO 1 h cq |h J a 'B ft 5 o i—l co o cq 3 o i—l CO GQ o bb .3 B 1 pq o T— 1 CD .-to CO iO o T— 1 io o H(M O !— 1 02 P P*3 |H? 1 o cq' IO o co cq O io cq ,-i lO CO CO io v_^ IO i-H IO 00 o CO © o o © T-! T-H CO CO CO cq i—i id CO CO iO CO © Tti © t— CO iO "* CO © rH CD -+| © r? co vd ^d CO o o CO O CO i-l CO o CO © o © CO •— 1 »d + lid > fc II ci P Eg P ft tf rd Q d c3 PQ .2 h3 P 278 LAYING OUT AND DIVIDING LAND. [Chap. VII. I Ex. 2. Required to straighten the north boundary of the tract the field-notes of which are given Ex. 1, Art. 389, the new line to run from a point five chains from the be- ginning of the tenth side. The bearing and distance of the new line, and the position of the point where it strikes the fourth side, are desired. Ans. Division line, S. 83° 14' E. 40.41 chains to a point 3.51 chains from the beginning of the fourth side. 392. Third Method. — When the old lines do not vary- much from the position of the new, and are crooked, it will frequently be found most convenient to run a "guess-line," and take offsets from this to different points of the bound- ary. Then calculate the contents of the parts cut off on each side of this line. These, if the assumed line were correct, must be equal ; if they are not so, divide the dif- ference of the areas by half of the length of the "guess- line/' and set the quotient off perpendicular to that line. Through the extremity of that perpendicular run a parallel to the "guess-line," meeting the side of the tract. The division line will run through this point, very nearly, if the "guess-line" did not differ much from the true one. If greater accuracy is required, the operation may be repeated, using the line determined by the first approximation as the basis of operations. 393. Fourth Method. — Run a random line from the start- ing point to the side on which the new line will fall, and calculate the area contained between this line and the original boundaries. Then, by Art. 378, run a new line to cut off the same area : this will be the line required. Thus, (Ex. 1, Art. 390,) the bearing of EG (Fig. 184) being K 10}° E: run BA S. 79J E. 45.45 chains, falling on GE at A, distant .69 chains from E. in GE produced. Sec. L] LAYING OUT LAND. 279 pq ^ £ CO c3 00 '30 b- © CM I— CM © CI o to to o b- CM ^ CD t- CD rH tH CD CD O to r-t ^H tr- CD r-i tO r^ CD r-i CD r-i CD c3 O f-4 CD O ;- tS m o3 O CM CD © © CO O o S3 © © CO* to cm b- ^. O tO td 3 280 LAYING OUT AND DIVIDING LAND. [Chap. VII. Problem 12. — To run a new line between two tracts of dif- ferent prices, so that the quantities cut off from each may be of equal value. 394. This problem is in general a very complicated one, and can be best solved by approximation. Thus, run a "guess-line," and calculate the area cut off from each tract. If these areas are in the inverse ratio of the prices, the line is a correct one; if not, run a new line near this, and repeat the calculation: a few judicious trials will locate the line correctly. 395. The following cases admit of simple solutions : — CASE 1. When the old line is straight, and the new line is to run a given course. Fig. 185. The method of solution will best be shown by an ex- ample. Let the bearings of the lines be LA (Fig. 185) K 46° 45' E., AE S. 71° 20' E., 24.10 chains, and BM K". 10° 35' E., the land to the north of AB being estimated at $80 per acre, and that to the south at $100 per acre. It is required to run a new division line, running due east, so as not to alter the value of the two tracts. Through B and A draw BD and AC parallel to the division line, and CF parallel to AB, meeting LA pro- duced in F. Take AL = \p AD = { AD, and FI a mean proportional between AL and AF. Join LB, and draw FE Bec.L] laying out land. 281 parallel to it, meeting AB in E. Then the division line will run through E. Demonstration.— AL : FI : : FI : AF; .-. AL : AF : : FI a : AF 2 ; but AB = | AL ; .-. AD : AF : : f FP : AF 2 : : f BE 2 : AE* : : BE a : f AE a . But AD : AF : : ADB : AFB (1.6) : : ADB : ABC : : BE a : £ AE a ; (A) and ABC : BEH :: AB 3 : BE 3 ; ... (23.5) ADB : BEH : : AB a : f AE a ; but ADB : f AEK : : AB> : £ AE a , (Cor. 2, 19.6.) .♦.BEH = | AEK. The operations in the above construction may readily be done on the ground. Thus : Eun BD, AC, and CF. Measure AF and AD. Calcu- late v/| AD . AF, which call M. Then say, As AF + M : AF : : AB : AE. Through E run the division line. Calculation. To find AD. Say, As sin. ADB (43° 15') : sin. ABD (18° 40') : : AB (24.10) : AD = 11.26. To find AF. Say, As sin. ACB . sin. BAF : sin. BAC . sin. ABC : : AB : AF ; that is, As sin. 79° 25' . sin. 61° 55' : sin. 18° 40' . sin. 81° 55' : : 24.10 : AF = 8.81 ; FI = V\ AD . AF = 11.13. Then, As AF + FI (19.94) : AF (8.81) : : AB (24.10) : AE = 10.64; Or, As AF + FI (19.94) : AF (8.81) : : AD (11.26) : AK = 4.97. CASE 2. 396. The division line to run through a given point E in AB. Let the bearings be as in last case. To run the division line through a point E in AB 10.64 chains distant from A. 282 LAYING OUT AND DIVIDING LAND. [Chap. VIL Fig. 186. Construction. — Take AI (Fig. 186) a third proportional to BE and AE. Let AK = f AI and AL = BE. Draw LM parallel to BC, cutting AB in E ; and KM parallel to AB. Make LO = ME. Join AO, and draw GEH parallel to it. Then the thing is done. Demonstration. — Conceive BC and AL to meet in P. Then we have BE : EA : : EA : AL .-. (Cor. 2, 20.6) BE : AI : : BE 3 : EA 3 , and LA : AK : : BE 3 : f EA 3 . Again : PB : PC : : PD : PA : : PA : PF : : AD : AF; but PB : PC : : LN : LO : : LN : NM : : LA : AK : : BE 3 : f EA 3 ; whence AD : AF : : BE 3 : £ EA 3 , which agrees with (A) in the demonstration of last case. Then, following the steps of that demonstration, we find BEH = £ AEG. This, like the last case, may readily be done on the AE 2 ground, thus ; Calculate AI = — — -, and make AK = j AI. EB Lay off on DA produced AL = BE : run 1EM and KM. Lay off LO = EM, and run GEH parallel to AO. Calculation. „ 5 AE 2 AK - 4EB = 10 " 51 - Then sin. M (81° 55') : sin. AKM (61° 55') : : AK (10.51) : EM = 9.37 = LO ; and, As LA + LO (22.83) : LA - LO (4.09) : : tan. ^PAJ^_42 (71° 550 : tan - L0A ~ LA0 = 28° 45'; LAO = 71° 55' - 28° 45' = 43° 10'. But AF bears E. 46° 45' E. ; hence GH bears K 89° 55' E. Bec. I.] laying out land. 283 CASE 3. 397. When the starting point is in the line AD. Given as before to run the line from a point G in AD at 4.97 chains from A. Produce DA and BC (Fig. 186) to meet in P. Calcu- late AP : let the given ratio f be represented by r : then, As sin. P (36° 10') : sin. ABC (81° 55') : : AB (24.10) : AP = 40.432. r.AG 2 Put __ =.7636 = A; AP and M 2 = A . PG = 34.67. Lay off GD = } A ± V\ A 2 + M 2 = .382 + 5.900,= 6.282, (the lower sign being used when G is between A and P.) Then GH parallel to DB will be the division line. Demonstration.— Since GD = £ A -f- y/\ A 3 + M 3 , we have GD — \ A = ^/\ A* + M 3 , and GD 3 — A . GD == M 3 , or GD (GD — A) = A . PG ; whence PG : DG : : DG — A : A, /r AG 3 \ and composition, PD : DG : : DG : A ( ' ) : : AP . DG : r . AG 8 ; whence r . PD . AG 3 = AP . DG 3 , and r . AG 3 : DG 3 : : AP : PD : : PC : PB : : PF : PA : : AF : AD, or, r . AE 3 : EB 3 : : AF : AD. As this agrees with (A) in the demonstra- tion to Case 1, the truth of the work is clear. Having found AD, the bearing of DB, which is parallel to GH, may be found by calculating the angle ADB ; thus : As (AB + AD) 35.352 : (AB - AD) 12.848 : : tan. ADB + ABD „ „ ADB - ABD 30° 57J' : tan. = 12° 17' 55 ,f . 2 2 2 Whence the angle ADB is 43° 15' 25", and the bearing of DB or GH is S. 89° 59' 35" E. The whole of the preceding construction might be made geometrically, but some of the lines required would be so small that no dependence could be had on the work ; the method is therefore omitted. If the given point were not on one of the lines, the pro- blem becomes very complicated. It may, however, be solved by running " guess-lines." 284 LAYING OUT AND DIVIDING LAND. [Chap. VII. SECTION II. DIVISION OF LAND. Problem 1. — To divide a triangle into two parts having a given ratio. CASE 1. 398. By a line through one of the corners. Divide the base into two parts having the same ratio as the parts into which the triangle is to be divided, and draw a line from the point of section to the opposite angle, (1.6), Examples. Ex. 1. A triangular field ABC contains 10 acres, the base AB being 22.50 chains. It is required to cut off 4J acres towards the point A by a line CD from the angle C. What is the distance AD ? Calculation. As 10 : 4J : : AB (22.50) : AD = 10.125 chains. ♦ Ex. 2. The area of a triangle ABC is 7 acres, the side AC being 15 chains. To determine the distance AD to a point in AC, so that the triangle ABD may contain 3 acres. Ans. AD = 6.43 chains. CASE 2. 399. By a line through a given point in one of the sides. Say, As the whole area is to the area of the part to be cut off, so is the rectangle of the sides about the angle towards which the required part is to lie, to a fourth term. This fourth term divided by the given distance will give the distance on the other side. Sec. II.] DIVISION OF LAND. 285 Demonstration. — Let ABC (Fig. 187) be the given tri- angle, and ADE the part cut off. Then we shall have (Art. 357) rad. : sin. A : : AB . AC : 2 ABC, and rad. : sin. A : : AD . AE : 2 ADE ; wherefore 2 ABC : 2 ADE : : AB . AC : AD . AE, or ABC : ADE : : AB . AC : AD . AE. Examples. Ex. 1. Given the side AB = 25 chains, AC = 20 chains, and the distance AD = 12 chains, to find a point E in AB, snch that the triangle cut off by DE may be to the whole triangle as 2 is to 5. Calculation. As 5 : 2 : : AB . AC (500) : AD . AE (200) ;' 200 whence AE 12 = 16.66 chains. Ex. 2. Given AB = 12.25 chains, AC = 10.42 chains, and the area of ABC = 5 A. 3 E. 8 P., to cut off 3 acres to- wards the angle A by a line running through a point E in AB 8.50 chains from the point A. Required the distance on AC. Ans. 7.77 chains. Fig. 188. CASE 3. 400. By a line parallel to one of the sides. Since the part cut off will be similar to the whole, say, As the whole area is to the area to be cut off, so is the square of one of the sides to the square of the correspond- ing side of the part. The problem may be constructed thus : Let ABC (Fig. 188) be the given triangle. Divide AB in F, so that AF may be to FB in the ratio of the parts into which the triangle is to be divided. Take AD A a mean proportional between AF and AB. Then, DE parallel to BC will divide the triangle as required. For AFC : FCB : : AF : FB, and (lemma) ADE = AFC ; therefore ADE : DECB : : AF : FB, 286 LAYING OUT AND DIVIDING LAND. [Chap. VII. Examples. Ex. 1. The three sides of a triangle are AB = 25 chains, AC = 20 chains, and BC = 17 chains, to divide it into two parts ADE and DECD, having the ratio of 4 to 3, by a line parallel to BC. Say, As 7 : 4 : : AB 2 (625) : AD 3 = 357.1428; whence AD = 18.90 chains. Ex. 2. The three sides of a triangle are AB = 25 chains, AC = 20 chains, and BC = 15 chains, to divide it into two parts ADE and DECB, which shall be to each other as 2 to 3, by a line parallel to BC. What is the distance on AC to the division line ? Ans. 12.65 chains. CASE 4. 401. By a line running a given course. Construction. — Divide AB in G, (Eig. 189,) so that AG- may be to GB in the ratio of the parts of the triangle. Run CE according to the given course. Take AD a mean proportional be- tween AF and AG. Then DE paral- lel to CF is the division line. Fig. 189. d B F x B For ACG : CGB : : AG : GB, and, by the lemma, ADE = ACG. ADE : DECB : : AG : GB. Calculation. In ACF find AF. Then AD = ^AG . AF ; or say, As the rectangle of the sines of D and E is to the rectangle of the sines of B and C, so is the square of BC to a fourth term. Then, if the ratio of the parts is to be as m to ft, m cor- responding to the triangular portion, multiply this fourth term by m, and divide by m -f n : the quotient will be the square of DE. Whence AD is readily found. Sec. II.] DIVISION OF LAND. 287 Demonstration. — Draw xy parallel to CF, making kxy = ABC, and draw BR parallel to xy. Then, as was shown in Art. 385, sin. D . sin. E : sin. B . sin. C ; : BC 2 : xy% and (Cor. 2, 20.6) Axy : ADE or m + n : m : : xy* : DE* Examples. Ex. 1. The bearings and distances of the sides of a tri- angular plat of ground are AB K 71° E. 17.49 chains, BC S. 15° W. 12.66 chains, and CA K 63f° W. 14.78 chains, to divide it into two parts ADE and DECB, in the ratio of 2 to 3, by a line running due north. The distance AD is required. First Method. As sin. F 71° A. C. 0.024330 sin. ACF 63° 45' 9.952731 AC 14.78 1.169674 AF 1.146735 AG = f AB = 6.996 0.844850 2)1.991585 AD = 9.904 ch. .995792 Second Method. 71° A. a 0.024330 63° 45 r 0.047269 56° 9.918574 78° 45' 9.991574 12.66 1.102434 a 1.102434 153.68 2.186615 2 5)307.36 DE = V 61.472 = 7.841. As sin. A 45° 15' A. C. 0.148628 : sin. E 63° 45' 9.952731 : : DE 7.841 0.894371 : AD 9.902 0.995730 288 LAYING OUT AND DIVIDING LAND. [Chap. VII. Ex. 2. Given AB K 63° W. 12.73 ch., BC S. 10° 15' W. 8.84 ch., and CA K 77° 15' E. 13.24 ch., to determine the distance AD on AB so that DE perpendicular to AB will divide the triangle into two equal parts. Ans. AD = 8.049 ch. CASE 5. Fig. 190. 402. By a line through a given 'point Let ABC (Fig. 190) be the tri- angle to be divided into two parts CLK and ABKL, which shall be to each other as the numbers m and n : the division line to run through a given point P. Construction. Bisect BC in D ; divide CA in F, so that CF : FA : : m : n. Through P draw HPE parallel to BC. Join ED ; draw FG parallel to it, and complete the parallelogram CH. Make GI perpendicular to BC and equal to EP. With the centre I and the radius PH, describe an arc cutting BC in K ; then KPL will be the division line. If IG is greater than IK, the question is impossible in the terms proposed. The triangular part will then be adjacent to one of the other angular points, and a construction alto- gether analogous to the above will fix the position of the division line. Demonstration. — Conceive DA, DF, and EG to be joined. Then, since CD = \ BC, ADC = \ ABC, and, because CF : FA : : m : n, we have by composition CA : CF : : m -f n : m; whence CFD = m-\-n CAD. But CDF = CEG, andCH = 2 CEG .-. CH = CAB, and by demonstration (Art. 381) CKL = CH ; m -\-n therefore CKL = m -f- n CAB. Sec. II] DIVISION OF LAND. 289 Calculation. Find PE, EC, and FC = AC ; then CE : CF : : CD m -f n (I BC) : CG, and KG = s/ EI 2 - IG 2 = V PH 2 — PE 2 . Finally, CK = CG ± GK. Examples. Ex. 1. Given the bearings and distances of the adjacent sides of a triangular tract,— viz. : CA K 10° 17' W. 13.25 ch., CB N". 82° 5' W. 13.75 ch.,— to divide it into two por- tions ABEL and KLC in the ratio of 4 to 5, by a line through a point P 3S". 28 W. 7.85 chains from the corner, C. The distance CK is required. Calculation. To find PE and EC. As sin. PEC 108° 12' A. C. 0.022289 : sin. PCE 17° 43' 9.483316 :: PC 7.85 0.894870 : PE 2.515 0.400475 As sin. PEC 108° 12' A. C. 0.022289 : sin. CPE 54° 5'- 9.908416 :: PC 0.894870 : CE 6.692 0.825575 - To find CG. AsCE 6.692 A. C. 9.174425 : CF = fCA 7.361 0.866937 : : CD = J CB 6.875 0.837273 : CG = EH 7.562 0.878635 EP 2.515 PH = IK = 5.047 19 290 LAYING OUT AND DIVIDING LA ND. [Chap. VII. To find KG and CK. KI + IG 7.562 0.878635 KI-IG 2.532 0.403464 2)1.282099 KG = 4.376 .641049 CG = 7.562 CK = 11.938 Ex. 2. Given AB JST. 46° 15' E. 8.80 ch., AC S. 65° 15' E. 11.87 ch., to determine the distance AK to a point K in AB so that a line from K through a spring P K". 80° E. 5.90 ch. from A may divide the triangle into two equal parts. Ans. AK = 8.58 ch., or 6.244 ch. Problem 2. To divide a trapezoid into two parts having a given ratio. CASE 1. Fig. 191. D G F 403. By a line cutting the parallel sides. a. Divide DC and AB (Fig. 191) in F and E so that the parts may have the same ratio as the parts into which the trapezoid is to be divided: join EF and the thing is done. b. If the division line is to pass through a given point G in one of the parallel sides. Determine F and E as before ; then lay off EH = FG, and GH will be the division line. c. If the division line is to pass through a point P (Fig. 192) not in AB or CD. Determine EF as before. Bisect it in I. Through P and I draw the division line GH. Should GH cut either of the non- a eh b parallel sides before it does both of these, one of the por- tions will be a triangle. It will then be necessary to calcu- late the area of the whole tract, whence that of each por- tion is found. Then, by Art. 381, lay off a triangle by a line through P so as to contain the required area. Fig. 192. D G F c m/...V j L \ / \ \ 1/ ; \ \ Seo. IL] DIVISION OF LAND. Calculation. 291 Through P draw MPL parallel to AB, and from the data given find AM and MP. Then DA : AM : : AE — DF : AE - LM ; whence LM and PL are known. J AD : : PL : GF = EH; and DG = But AM — J AD DF - FG. Examples. Ex. 1. Given AB E. 9.10 ch., BC K 14° 20' W. 4.40 ch., CD W. 6.95 ch., and DA S. 14° W. 4.39 ch., to divide the tract into two parts having a ratio of 3 to 4 by a line HG through a spring !N". 47° E. 4.40 ch. from the corner A; the smaller division to be next to AD. Required the distances of the division line from A and D. Calculation. To find AM and MP. As sin. M 76° A. C. 0.013096 : sin. APM 43° 9.833783 :: AP 4.40 0.643453 : AM 3.093 0.490332 And As sin. M A. C. 0.013096 : sin. PAM 33° 9.736109 :: AP 0.643453 : PM 2.470 0.392658 To find EH, AH, and DG. DF = f DC = 2.979, and AE = f AB = 3.90. Then, As AD (4.39) : AM (3.093) : : AE — DF (.921) : A& — ML = .649; whence ML = 3.251, and PL = 3.251 — 2.470 = .781. As AM — | AD (.898) : J AD (2.195) : : PL (.781) : FG = EH = 1.909. Finally, AH = AE + EH = 5.81, and DG = df — FG = 1.07. Ex. 2. Given AB S. 62° 50' E. 14.93 ch., BC K 7° 30' W. 6.29 ch., CD K 62° 50' W. 11.88 ch., DA S. 21 W. 5.18 ch., 292 LAYING OUT AND DIVIDING LAND. [Chap. VII. to determine DG and AH so that a line joining G and H will pass through P K 75° 50' E. 6.20 ch. from A, and cut off one-third of the area of the tract towards AD. Ans. AH = 3.40 ch. ; DG = 5.53 ch. CASE 2. 404. The division line to be parallel to the parallel sides. Fig. 193. Let ABCD (Fig. 193) be the trape- zoid to be divided into two parts AEFD and FEBC having the ratio of two numbers m and n by a line EF parallel to AD or BC. Construction. l Join CA, and draw DH parallel to it. Join CH. Divide HB in I so that HI : IB : : m : n. Produce CD and BA to meet in G, and take GE a mean proportional between GI and GB. Join CI, and draw EF parallel to AD : then will EF be the division line required. Demonstration. — Because DH is parallel to CA, AHC = ADC (37.1) ; .*. ABCD = BCH, and, since HB is divided in I so that HI : IB : : m : n, we have CHI : CIB : : m : n (1.6.) These triangles are therefore equal to the parts into ■which the trapezoid is to be divided. But (lemma) GEF = GIC : therefore EBCF = ICB, and EF is the division line. Calculation. EF may be found by the formula EF 2 = ; ; then BC svs AD : EF (Fig. 196 ;) „.„ m . xy 2 -+• n . AB 2 „,. or EF 2 = ^— L , (Tig. 197.) m + n Demonstration. — Draw AM and BN (Fig. 196) parallel to EF. Then sin. M . (sin. E) : sin. B : : AB : AM, and sin. N . (sin. F) : sin. A : : AB : BN ; (23.6) sin. E .sin. F : sin. A . sin. B : : AB a : AM . BN. Now, since Gxy = GAB, Gx is a mean proportional between GA and GN. Wherefore xy is a mean proportional between AM and BN. Hence, AM . BN consequently, sin. E . sin. F : sin. A . sin. B : : AB a : xy\ If EF is parallel to AB, (Fig. 197,) the demonstration will be precisely similar to the above. Examples. Ex. 1. Given the bearings and distances as follow, — viz. : AB K 25|° E. 4.65 chains, BO N". 77° E. 6.30 chains, CD South 7.30 chains, and DA K 78J° W. 8.35 chains,— to divide the trapezium into two parts ABEF and FECD, having the ratio of 2 to 3, by a line EF parallel to AB. AF and EF are required. Calculation. First Method.— As in Ex. 1 of Art. 405, we find GA = 8.662, GB = 10.777, GC = 17.077, GD = 17.012, GH - 5.466, and GI = GH + |HD = 10.084. 300 LAYING OUT AND DIVIDING LAND. [Chap. VII. To find GK and GF. AsGB 10.777 A. C. 8.967504 : GA 8.662 0.937622 :: GO 17.077 1.232412 : GK 1.137538 GI 10.084 1.003633 2)2.141171 GF = V GI . GK = 11.765 1.070585 GA = 8.662 AF = 3.103 To find EF. AsGA 8.662 A. C. 9.062378 : AB 4.65 0.667453 :: GF 11.765 1.070585 : EF 6.316 Second Method. 1.800416 f sin. E A a J 128° 45' A. C. 0.107970 \ sin. F 76° ^ a " 0.013096 f sin. C 77° 9.988724 \ sin. D 78° 15' 9.990803 J CD 7.30 0.863323 :: [CD 0.863323 : xy 2 67.18 2 134.36 1.827239 3 AE 2 64.8675 5)199.2275 EF = V 39.8455 = To find AF. 6.312. 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 OF 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 BO. 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. Let ABCD (Fig. 198) be Fig> 198> ' the trapezium to be divided into two parts ABEF and FECD, in the ratio of m to yY ^ n, by a line EF running any ^''^\\\ course. s*** \\\ The construction of this ^___ V-.Y case is the same as that of G H axnif k«d 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 vw 2 by the proportions sin. E . sin. F : sin. A . sin. B : : AB 2 : xy 2 , and sin. E . sin. F : sin. C . sin. D : : CD 2 : vuf, the truth of which has been proven in the demonstration to rule for Art. 407. m . vw 2 + n . xy 2 Then (Art. 404) EF 2 = Dra^ and P. m + n Draw AOP parallel to BC, meeting BIsT and EF in O 302 LAYING OUT AND DIVIDING LAND. [Chap. VII. Then sin. BOA (sin. E) : sin. BAO (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 — v'GTTGK. Example. Ex. 1. The bearings and distances being as in the ex- amples in last case, it is required to divide the trapezium into two parts ABEF and FECD, having the ratio of 2 to 3, by a line perpendicular to AD. To find AF and EF. Ans. AF = 3.84; EF = 5.76. CHAPTER VIII. MISCELLANEOUS EXAMPLES. Ex. 1. Two sides of a triangle are 32 and 50 parches respectively. Required the third side, so that the area may be 3 acres. Ans. 31.05 P. or 78 P. Ex. 2. A gentleman has a garden in the form of a rect- angle, the adjacent sides being 120 and 100 yards respec- tively. There is a walk half round the garden, which takes up one-eighth of the ground. "What is its width ? Ans. 7.05 yards. Ex. 3. The three sides of a triangle are in the ratio of the numbers 3, 4, and 5. What are their lengths, the area being 2 A., 1 R., 24 P.? Ans. 6 chains, 8 chains, and 10 chains. Ex. 4. The diameter of a circular grass-plat is 150 feet, and the area of the walk that surrounds it is one-fourth of that of the plat. Required the width. Ans. 8.85 feet. Ex. 5. To determine the height of a liberty-pole which had been inclined by a blast of wind, I measured 75 feet from its base, the ground being level, and took the angle of elevation of its top 67° 43' 30", the angle of position of the base and top being 5° 37'. Then, measuring 100 feet farther, I found the angle of position of the bottom and top to be 2° 29'. Required the length of the pole. Ans. 194 feet. Ex. 6. The distances from the three corners of a field in the form of an equilateral triangle to a well situated within it are 5.62 chains, 6.23 chains, and 4.95 chains respectively. "What is the area ? Ans. 4 A., R., 6 P. 303 304 MISCELLANEOUS EXAMPLES. [Chap. VIII. Ex. 7. At a station on the side of a pond, elevated 30 feet above the water, the elevation of the summit of a cliff on the. opposite shore was found to be 37° 43' and the de- pression of the image 45° 26'. Required the elevation of the cliff. Ans. 221.8 ft. Ex. 8. To find the altitude of a tower on the brow of a hill, I measured, on slightly-inclined ground, a base-line AB 157 yards, A being on a level with the base of the hill. At A the angle of position of B and C was 87° 45'; elevation of B, 2° 17'; of base of tower, 39° 43', and of top, 52° 13'. At B the depression of A was 2° 17'; the angle of position of A and C, 54° 23' ; elevation of base of tower, 33° 4', and of top, 45° 42'. Eequired the height of the hill and also of the tower. Ans. Height of hill, 172.5 ft.; of tower, 95.5 ft. Ex. 9. To determine the height of a tree C standing on the opposite shore Of a river, I measured a base-line AB of 100 feet. At A the angle BAC was 90°, and the angle of depression of the image of the top of the tree was 39° 48'. At B the angle of depression was 32°. Required the height, the instrument having been 10 feet above the water at each station. Ans. 84.47 feet. Ex. 10. Not being able to measure directly the three sides of a triangle, the corners of which were visible from each other, I took the angles as follow, — viz. : A = 57° 29', B = 72° 41', and C = 49° 50'. I also measured the dis- tances from the corners to a point within the triangle, and found them to be AD = 7.56 chains, BD = 9.43 chains, and CD = 8.42 chains. Required the lengths of the sides. Ans. AB = 12.63 chains, AC = 15.78 chains, and BC =s 13.94 chains. Ex. 11. The base of a triangle being 50 perches, and the area 5 acres, what are the other sides, their sum being S5 perches ? Ans. 33.3785 P. and 51.6215 P. Ex. 12. It is required to lay out 7 acres in a triangular form, one side being 20 chains, and the others in the ratio of 2 to 3. MISCELLANEOUS EXAMPLES. 305 Ans. The other sides are 9.86 and 14.79 chains, or 39.58 and 59.37 chains. Ex. 13. The bearings of the dividing lines of two farms being as follow,— viz. : 1. K 83}° E. 2.37 chains ; 2. S. 47° E. 6.25 chains; 3. K 62f° E. 5.17 chains; 4. S. 56^° E. 3.92 chains, and 5. "N. 14 J° E., — it is required to straighten the boundary, the new line to start from the beginning of the first side and fall on the last. The bearing of the new line is required, and also the distance on the last side. Ans. Bearing, S. 74° 40' E. to a point .25 chains back from the commencement of the last side. Ex. 14. One side of a tract running through a thick copse, I took a station S. 26J° E. 1.53 chains from the corner, and ran a "guess-line" bearing N". 60 J° E. 19.37 chains, w T hen the other end bore K". 28 J° W. 3.27 chains. What is the course and distance of the line, and what must be the course and distance of an offset from a point 8.53 chains on the random line, that it may strike a stone in the side 8.53 chains from the point of beginning? Ans. Side, K 55° 22' E. 19.42 chains ; Offset, K 28° 8' W. 2.29 chains. Ex. 15. Three observers, A, B, and C, whose distances asunder are AB = 1000 yards, BC = 1180 yards, and AC = 1690 yards, take the altitude of a balloon at the same instant, and find it to be as follow, — viz. : At A, 53° 43', at B, 46° 40', and at C, 52° 46'. Eequired the height of the balloon. Ans. 1461.4 yards or 2411 yards. Ex. 16. The bearings and distances of the sides of a tract of land are,— 1. 1ST. 61° 20' W. 22.55 chains; 2. K 10° W. 16.05 chains ; 3. K 60° 45' E. 14.30 chains ; 4. S. 66° 40' E. 17.03 chains ; 5. S. 86° E. 22.40 chains ; 6. S. 31° 40' E. 19.10 chains, and 7. S. 76° 35' W. 39 chains,— to divide it into two equal parts by a line running due north. The position of the division line is desired. Ans. The division line runs from a point on the 7th side 3.77 chains from the end thereof. 20 306 MISCELLANEOUS EXAMPLES. [Chap. VIII. Ex. 17. Not being able to run a line directly, on account of a projecting cliff, I took the angles of deflection and the distances as follow, — viz. : 1. to the right, 67° 35' 10 chains ; 2. to the left, 48° 43' 7.25 chains; 3. to the left, 11° 45' 5.43 chains, and 4. to the left, 65° 17'. How far on the last course must I run before coming in line again? at what angle must I deflect to continue the former direction ? and what is the distance on the first line ? Ans. Distance on the last course, 14.42 chains ; on the first, 23.67 chains ; deflection, 58° 10' to the right. Ex. 18. To find the length of a tree leaning to the south, I measured due north from its base 70 yards, and found the elevation of the top to be 25° 10' ; then, measuring due east 60 yards, the elevation of the top was 20° 4'. What was the length and inclination of the tree ? Ans. Length, 35.1 yards ; inclination, 83° 11'. Ex. 19. The bearings and distances being as in Ex. 16, it is required to divide the tract into two equal parts by a line running from the first corner. The bearing of the division line is required. Ans. K 14° 59' E. 27.66 chains to a point on the fifth side 1.61 from beginning. Ex. 20. The boundaries of a quadrilateral are, — 1. "N. 35J° E. 23 chains ; 2. K 75|° E. 30.50 chains ; 3. S. 3J° E. 46.49 chains, and 4. N". 66^° W. 49.64 chains, — to divide the tract into four equal parts by two straight lines, one of which shall be parallel to the third side. Required the distance of the parallel line from the first corner, the bearing of the other division line and its distance from the same corner, measured on the first side. Ans. Distance of parallel division, 32.50 chains ; bear- ing of the other, S. 88° 22' E. ; distance from the first corner, 5.99 chains. CHAPTER IX. MERIDIANS, LATITUDE, AND TIME. SECTION I. MERIDIANS. 409. The meridian of a place is a true north and south Hue through that place ; or it may be denned to be a great circle of the earth passing through the pole and the place. 410. As it is of great importance to the surveyor to be able to trace accurately a meridian line, the following methods are given. Any one of these is sufficiently accu- rate for his purposes. Those which require the employ- ment of the transit or the theodolite are to be preferred, if one of these instruments is at hand. When the obser- vations are performed with the proper care, and the instru- ments are to be depended on, the line may be run within a few seconds of its proper position. 411. Although the methods to be explained in the follow- ing articles are in theory perfectly accurate, yet the results to which they lead cannot be relied on with the same cer- tainty when the observations are made with surveyors' instruments, as if the larger and more expensive instru- ments to be found in fixed observatories were employed. These instruments generally rest on permanent supports: their positions and adjustments may therefore be tested, and corrected when found defective, and thus their proper posi- tion be finally obtained with almost perfect accuracy. Not 307 308 MERIDIANS, LATITUDE, AND TIME. [Chap. IX. so with the theodolite or the surveyors' transit. The ad- justments in their position must be made at the time, and renewed for every fresh observation. The results alone are to be corrected by subsequent observation, and not the 'position of the instrument. Notwithstanding these diffi- culties, which must always prevent his attaining the pre- cision of the astronomer, yet, with ordinary care, the sur- veyor may run his lines with all the accuracy which is necessary for his operations. Problem 1. — -To run a meridian line. 412. First Method. — By equal altitudes of the sun. Select a level surface, ex- Fi s- im- posed to the south, and erect an upright staff upon it. Around the foot of this staff A (Fig. 199) as a centre de- scribe a circle. Observe care- fully the point B at which the end of the shadow crosses this circle in the morning, and likewise the point C where it crosses in the evening. Bisect the angle BAC by the line NS, which will be a meridian. If a number of circles be de- scribed around A, several observations may be made on the same clay, and the mean of the whole taken. If the staff' is not vertical, let fall a plumb-line from the summit, and describe, the circles around the point in which this line cuts the surface. A piece of tin, with a small cir- cular hole through it for the sun's rays to pass through, is better than the top of the staff, the image being definite. Where much accuracy is not required, the above method is sufficient. It supposes the declination of the sun to re- main unchanged during the observation. This is not true except at the solstices,— 21st of June and 22d of December. Sec. L] MERIDIANS. 309 Those days — or at least a time not very remote from them — should therefore be chosen for determining the meridian by this method. 413. Second Method. — By a meridian observation of the North Star. Tlie Pole Star {Polaris, or a Ursce Minoris) is situated very nearly at the North Pole of the heavens. If it were exactly so, all that would be necessary to determine the direction of the meridian would be to sight to the star at anytime. The North Star, being, however, about 1J° from the pole, is only on the meridian twice in twenty-four hours. There is another star, {Alioth,) in the tail of the Great Bear, ( Ursai If aj oris,) which is on the meridian very nearly at the same time as the Pole Star. The constellation in which Alioth is situated is one of the most generally known. It is often called the Plough, the Dipper, the Wagon and Horses, or Charles's Wain. The two stars in the quadrangle farthest from the handle, or tail, are called the Pointers, from the fact that the line joining them will, when produced, pass nearly through the Pole Star. The star in the handle of the dipper, nearest the quadrangle, is Alioth. 414. To determine the direction of the meridian. Suspend a long plumb-line from some fixed elevated point. If a window can be found properly situated, a staff may be projected from it to afford a support. The plum- met should be heavy, and be allowed to swing in a vessel of water, so as to lessen the effect of the currents in the air. At some distance to the south of the line set two posts, east and west from each other, making their tops level, and nail upon them a horizontal board. To another board screw a compass-sight. This may be moved steadily to the east or west upon the other board. Then, some time before Polaris is on the meridian, place the compass- 310 MERIDIANS, LATITUDE, AND TIME. [Chap. IX. sight so that by looking through it Alioth may be hidden by the plumb-line. As the star recedes from the line, move the sight, so as to keep the line and star in the same direction ; at last Polaris will also be covered by the line. The eye and plumb-line are then very nearly in the me- ridian. If the time is noted, and Polaris sighted to seven- teen minutes after the former observation, the meridian will be much more accurately determined. The compass-sight may now be firmly clamped till morning. In making the above described observation, it will generally be necessary for an assistant to illuminate the line if the night is dark. 415. To determine the time Polaris is on the meridian. 1. Take from the American Almanac, or other Ephemeris, the sun's right ascension, or sidereal time of mean noon, for the noon preceding the time for which the transit is wanted. The sidereal time is given in the American Al- manac for mean noon at Greenwich (England) for every day in the year, and may be calculated for any other meridian by interpolation, thus : — The difference between the sidereal times for two suc- cessive days being 3 minutes 56.555 seconds, say, As twenty- four hours is to the longitude expressed in time, so is 3 minutes 56.555 seconds to the correction to be applied to the sidereal time at noon of the given day at Greenwich. This correction — added to the sidereal time taken from the almanac if the longitude be west, but subtracted if it be east — will give the sidereal time at mean noon at the given place. The above correction, having been once determined for the given place, will serve for all the calculations that may be wanted. Example. Ex. 1. Let it be required to find the sidereal time at mean noon, at Philadelphia, long. 5 h. m. 40 sec. W., on the 11th of August, 1855. The sidereal time at mean noon, Greenwich, August 11, Sec. L] MERIDIANS. 311 is 9 hours, 17 minutes, 32.74 seconds, as taken from the American Almanac of that year. And, As 24 h. : 5 h. m. 40. s. : : 3m. 56M5 s. : 49.391. h. m. sec. Then, sidereal time at Greenwich, mean noon 9 17 32.74 Correction for difference of long. 49.39 Sidereal time at Philadelphia, mean noon 9 18 22.13 2. Subtract the sidereal time above determined from the right ascension of the star, taken from the same almanac, increasing the latter by 24 hours, if necessary to make the subtraction possible. The remainder is the time of the transit expressed in sidereal hours. To convert these into solar hours. Say, As 24 hours is to the number of hours in the above time, so is 3 minutes 55.9 seconds to the correction. This correction, subtracted from the sidereal time, will give the mean solar time of the upper transit. The time thus determined will be astronomical time. The astronomical day begins at noon, the hours being counted to twentvrfour. The first twelve hours, therefore, correspond with the hours in the afternoon of the same civil day ; but the last twelve agree with the hours of the morning of the next succeeding day. Thus, August 11, 8h. 15 m., astronomical time, corresponds with August 11, 8h. 15m. p.m., civil time; but August 11, 16 h. 15 m., astronomical time, agrees with August 12, 4 h. 15 m. a.m., civil time. If, therefore, the number of hours of a date expressed in astronomical time be greater than twelve, to convert it into civil time the days must be increased by one and the hours diminished by twelve. Required the time of the upper transit of Polaris, Sep- tember 11, 1855, for Philadelphia. 312 MERIDIANS, LATITUDE, AND TIME. Sidereal time at mean noon, Greenwich, September 10 Correction for Philadelphia Sidereal time, mean noon, atPhila. (A) Right ascension of Polaris, Sept. 11 (B) (B) - (A) Correction for 13 h. 50 m. 24 sec. Astronomical time, September 10 agreeing with, civil time, Sept. 11 [Chaj>. IX. h. m. sec. 11 15 49.38 49.39 11 16 38.77 1 7 2.71 13 50 23.94 2 16.04 13 48 7.90 1 48 7.90 a.m. 416. The times of the upper transit of Polaris for every tenth day of the year is given in the following table. The calculation is made for the meridian of Philadelphia, the year 1855. As a change of six hours, or 90° of longi- tude, will only make a change of one minute in the time of the transit, the table is sufficiently accurate for any place within the United States : — Time of Polaris crossing the meridian, upper transit. Months. January. . . . February . . March April May June July August September. October November. December.. 1st. h. 6 4 2 10 8 6 4 2 10 8 m. 22 20 29 27 30 a 28 30 29 27 30 24 26 M. M. M. 11th. 5 43p 3 40 150 1148 a 9 50 7 49 5 51 3 50 148 11 46 p 9 44 7 46 M. M. M. 21st. 5 3p 3 1 111 11 9a 9 11 7 5 3 1 11 9 7 10 12 11 9 7p 5 7 M. M. M. If the time of the passage of the star for any day not given in the table be desired, take out the time of passage for the day next preceding, and deduct four minutes for Sec. L] MERIDIANS. 313 each clay that elapses between the date in the table and that for which the time of transit is required ; or, more ac- curately, thus : — Say, As the number of days between those given in the table is to the number between the preceding date and that for which the time of transit is desired, so is the difference between the times of { transit given in the table to the time to be subtracted from that corresponding to the earlier of the two days. Let the time of transit, August 27, be desired. Time. Aug. 21, 3 h. 11 m. Sept. 1, 2 27 , Difference 44 As ' 11 d. : 6 d. : : 44 : 24 ; therefore 3 h. 11 m. — 24 m. = 2 h. 47 m. is the time re- quired. 417. If the time of the lower transit be desired, it may be obtained from the table by changing a.m. into p.m. and diminishing the minutes by 2, or changing p.m. into a.m. and increasing the minutes by 2. 418. The above table is calculated for the year 1855. It will, however, serve for the observation described in Art. 414 for many years, the time of the meridian passage being determined in that method by the time of Polaris and Alioth being in the same vertical. "When the time is more accurately needed, as in Method 3 (Art. 419) for deter- mining the meridian, it will be necessary to correct the numbers in the table for the years that elapse between 1855 and the current year. The Pole Star passes the meridian about 21 seconds — more accurately, 20.6 seconds — later every year than the preceding one, so that in 1860 the time will be 1 minute, 43 seconds later than those given in the table ; in 1870, 5 minutes ; in 1880, 8 minutes 35 seconds ; and, in 1890, 12 minutes later. 314 MERIDIANS, LATITUDE, AND TIME. [Chap. IX. 419. Third Method. — By a meridian passage observed with a transit or theodolite. Having accurately levelled the instrument, sight to Po- laris when on the meridian. Then, depressing the telescope, set up an object in the line of sight : a line drawn from the instrument to that object will be a meridian. In observing with the transit or theodolite at night, it is needful that the wires should be illuminated. This may be done by an assistant reflecting the rays of a lamp into the tube by a sheet of white paper. An error of 5 minutes in the time of the transit of Po- laris will make an error of about 1J' in the bearing of the star, so that if the observation is not made near the proper time, it must be corrected. This may be done thus : — Deduct the star's polar distance from the complement of the latitude. Then say, As sine of this difference is to the sine of the polar distance of the star, (1° 28' at present,) so is sine of the error in time (expressed in degrees) to the sine of the bearing of the star. East if the time be too early, but west if it be too late. The time is reduced to degrees by multiplying by 15 : thus, 5 minutes = 1° 15'. Example. Required the bearing of Polaris 5 minutes after the upper meridian passage, the latitude of the place being 40°. 50° - 1° 28' = 48° 32' As sine of 48° 32' Ar. Co. 0.125320 : sine of star's polar distance 1° 28' 8.408161 : : sine of time, in degrees, 1° 15' 8.338753 : sine of star's bearing 1' 37" W. 6.872234 420. Fourth Method. — By an observation of Polaris at its greatest elongation. As a circumpolar star revolves round the pole, it gradu- ally recedes from the meridian towards the west until it Sec. I.] MERIDIANS. 315 attains its most remote point: here it apparently remains stationary, or at least appears to move directly towards the horizon for a few minutes, and then gradually moves east- ward towards the meridian, which it crosses below the pole. Continuing its course, in about six hours it reaches its greatest eastern deviation, when it again becomes sta- tionary. When most remote from the meridian, it is said to have its greatest elongation. As the star is apparently stationary at the time of its greatest eastern or western elongation, this time is a very favorable one for observing it. A variation of a few minutes in the time will then make no appreciable error in the bearing of the line. 421. The subjoined table contains the times of the great- est eastern or western elongations, according as the one or the other occurs at a time of day favorable for observa- tion. The times of greatest elongations are calculated thus : Take from one of the almanacs mentioned in Art. 415 the polar distance of the star at the given time, and call it P. Call the latitude of the place L. Then find the semi- diurnal arc by the following formula: — H . cosine x = tan. P . tan. L. Reduce x to time by dividing by 15, calling the degrees hours, and correct for the sidereal acceleration : the result will be the semidiurnal arc expressed in time. Call it t Then, if T be the time of greatest elongation, and T r be the time of the upper meridian passage of the star, T = T' + t or T' — t, according as the time of the western or eastern elongation is desired. The hour angle for Polaris at its greatest elongation, July 1, 1855, in lat. 40° N"., was 5 hours 54 minutes ; but, as the polar distance of the star is diminishing at the rate of 19. 23" per annum, the semidiurnal arc is slowly in- creasing. The change is so small, however, — being about one second per year, — that it may be entirely neglected. As the time of the meridian passage of the star is later by 20.6 seconds each year than the preceding one, the times 316 MERIDIANS, LATITUDE, AND TIME. [Chap. IX. of greatest eastern and greatest western elongation will be similarly affected: in 1860 they will be 1 minute 43 seconds later than the times given in the table; in 1870, 5 minutes ; and, in 1880, 8 minutes 35 seconds later. 422. Table of Times of Greatest Elongation of Polaris for 1855. Latitude, 40° K Months. 1st. nth. 21st. h. m. h. m. h. m. January... "West 16 A.M. 11 37 p.m. 10 57 p.m. February.. West 10 14 p.m. 9 35 " 8 55 " March West 8 23 " 7 44 " 7 4" April East 6 33 a.m. 5 54 a.m. 5 15 A.M. May East 4 35 " 3 56 " 3 17 « East 2 34 " 155 " 1 15 " July East 36 " 11 53 p.m. 11 14 P.M. August.... East 1 10 31 p.m. 9 51 " 9 12 " September East 8 29 " 7 50 " 7 11 " October ... West 6 24 a.m. 5 44 a.m. 5 5 A.M. November West 4 22 " 3 42 " 3 3" December West 2 24 " 145 " 15" The above table is calculated for lat. 40°, for which lati- tude the hour angle is 5 h. 54 m. 6 sec. ; for latitude 50° the hour angle is 5 52 2, and for lat. 30° " " " 5 55 38 ; therefore, for lat. 50° the eastern elongation occurs two minutes later, and the western two minutes earlier, than those given in the table ; for lat. 30° the times of the eastern elongation must be diminished, and those of the western increased, by 1 minute 32 seconds. 423. The observation for the meridian is made as directed Art. 414. Suspend the plumb-line, and, having placed the compass-sight on the table, as the star moves one way move the sight the other, so as to keep the star always hid by the line. At the time of greatest elongation the star will appear stationary behind the line. Clamp the board to which the compass-sight is attached. If the plumb-line is suspended from a point that is not liable to derangement, Sec. L] MERIDIANS. 317 the remainder of the work may be left till daylight ; other- wise, let an assistant take a short stake, with a candle attached to it, to a distance of 8 or 10 chains. He may then be placed exactly in line with the plumb. When the stake has been so adjusted, it should be driven firmly into the ground and its position again tested. Measure accurately the distance between the compass- sight and the stake. Call it D. Take the azimuth of the star from the following table and call it A. D . tan. A Calculate x = — , and set off the distance x to the east or west of the stake, according as the western or eastern elongation was' observed. The point thus determined will be on the meridian passing through the compass-sight. Permanent marks may then be fixed at any convenient points in this line. If a transit or theodolite is at hand, direct the telescope to the stake first set up. Turn it through an angle equal to the azimuth : it will then be in the meridian : or direct the telescope to the star when at its greatest elongation, and then turn the plate through an angle equal to the azimuth. 424. The azimuth of a star is its bearing, and may be determined by the following formula, — A being the azi- muth, L the latitude of the place, and P the polar distance of the star: — a . A R . sin. P Sin. A = — . cos. L Azimuths of the Pole Star at its Greatest Elongation. Lat. 1855. I860. 1865. 1870. o o / // O 1 II O 1 II o r ii 30 1 41 21 1 39 32 1 37 42 1 35 49 35 1 47 11 1 45 14 1 43 16 1 41 19 40 1 54 37 1 52 32 1 50 27 1 48 20 45 2 4 11 2 1 55 1 59 35 1 57 18 318 MERIDIANS, LATITUDE, AND TIME. [Chap. IX. The above are calculated from the mean place of the star as given in Loomis's "Practical Astronomy." 425. Fifth Method. — By equal altitudes of a star. If a theodolite or a transit with a vertical arc is at hand, the meridian may be run very accurately by observing a star when at equal altitudes before and after passing the meridian. For this purpose select a star situated near the equator, and, having levelled the instrument with great care, take the altitude of the star about two or three hours before it passes the meridian, and notice carefully the horizontal reading. When the star is about as far to the west of the meridian, set the telescope to the same elevation, and fol- low the star by the horizontal motion until its altitude is the same as before, and again notice the reading. Then if the zero is not between the two observed read- ings, take half their sum, and turn the telescope until the vernier is at that number of degrees and minutes : the tele- scope will then be in the meridian. If the vernier has passed the zero, add 360 to the less reading before taking the sum. Thus, if the first reading were 150° 37' 30", and the 431° 2 f 30" second 280° 25', the half sum = 215° 31' 15" ' 2 would be the reading for the meridian. Instead of taking the readings, a stake may be set up at any distance — say ten chains — in each observed course : then bisect the line joining the stakes, and run a line from the instrument to the point of bisection. The mean of a few observations taken in this manner will determine the meridian with considerable precision. Sec. II.] LATITUDE. 319 SECTION II. LATITUDE. The latitude of a place may be determined in various modes. 426. First Method. — By a meridian altitude of the Pole Star. The altitude of the pole is equal to the latitude of the place. Take the altitude of Polaris when on the meridian, and from the result subtract the refraction taken from the following table. Increase or diminish the remainder by the polar distance of the star according as the lower or upper transit was observed : the result will be the latitude. 427 . Refraction to be taken from the apparent latitude. App. Alt. o Kef. App. Alt. Kef. App. Alt. o Kef. App. Alt. o Kef. App. Alt. o Kef. / // o / n 1 / II / // 20 2 39 30 1 40 40 1 9 50 49 60 34 21 2 30 31 1 37| 41 1 7 51 47 61 32 22 2 23 32 1 33 42 1 5 52 45 62 31 23 2 16 33 1 29| 43 1 2 53 44 63 30 24 2 10 34 1 26 44 1 54 42 64 28 25 2 4[ 35 1 23 45 58 55 41 65 27 26 1 59 36 1 20 46 56 5Q 39 m 26 27 1 54 37 1 17 47 54 57 38 67 25 28 1 49 38 1 14 48 52 58 36 68 24 29 1 45| 39 1 12 ! 49 50 59 35 69 22 428. Second Method. — Take the altitude of the star six hours before or after its meridian passage. The result, corrected for refraction, will be the latitude. 429. Third Method. — By a meridian altitude of the sun. Take the meridian altitude of the upper or the lower limb of the sun, and correct for refraction. The result, 320 MERIDIANS, LATITUDE, AND TIME. [Chap. IX. increased or diminished by the semidiameter of the sun according as the lower or the upper limb was observed, will be the altitude of the sun's centre. (The apparent semi- diameter of the sun is given in the American Almanac for every day of the year.) To the altitude of the sun's centre, add his declination (taken from the same almanac) if south, but subtract it if north: the result subtracted from 90° will give the latitude. Instead of the sun, a bright star, the declination of which is small, may be observed. 430. If the exact direction of the meridian is not known, the telescope must be fixed on the body some time before it is south. As the sun or star approaches the meridian its altitude increases, and it will therefore rise above the hori- zontal wire. Move the telescope in altitude and azimuth so as to follow the body until it ceases to leave the wire. The reading will then give the observed meridian altitude. The altitude alters very slowly for some minutes before and after its meridian passage, thus affording ample time to direct the telescope accurately towards the object. 431. Fourth Method. — By an observation of a star in the prime vertical. Any great circle passing through the zenith is called a vertical circle. All such circles are perpendicular to the horizon. That vertical circle which is perpendicular to the meridian is called. the prime vertical: it cuts the horizon in the east and west points. Level the plates of the transit or theodolite carefully, and direct the telescope to the east or west, so that it may move in the prime vertical or nearly so. Then, having selected some bright star which passes the meridian a little south of the zenith, (the declination of such a star is rather less than the latitude of the place,) observe the time of its 'crossing the vertical wire of the telescope before passing the meridian, and again, when in the west, after its meridian passage. Let Sec. II.] LATITUDE. 321 these times be called T and T'. Let tlie interval between T and T' be called x, which must be reduced to sidereal time by adding to the solar time 3 minutes 56.55 seconds for 24 hours, or 9.85 seconds per hour; also, let L be the latitude of the place, and D be the declination of the star. _ _ R. tan. D Then tan. L = COS. f X Thus, for example, the transit of a Lyrce over the prime vertical was observed July 1, 1855, at 10 h. 43 m. 4 sec, and again at 13 h. 3 m. 48 sec, mean solar time. Re- quired the latitude, — the apparent right ascension of the star (as given in the American Almanac) being 18 h. 32 m. 4 sec, and the declination 38° 39' 0.4". Here the interval is 2 h. 20 m. 44 sec, solar time. Reduction 23 2)2 21 7~ 1 h. 10 m. 33.5 sec. = 17° 38' 2 Cos. \x 17° 38' 22" A. C. 0.020915 tan. D 38° 39' 0.4" 9.902940 tan. L 40° 0'4" 9.923855 432. Half the sum of the observed times is the time of meridiem passage in mean solar time. If this is reduced to sidereal time and increased by the sidereal time of mean noon at the given place, the result should be equal to the right ascension of the star. In the example before us the times of observation are h. m. sec. 10 43 4 and 13 3 48 Sum 2) 23 46 52 Half sum 11 53 26 Reduction for sidereal time 1 57 (A) 11 55 23 21 322 MERIDIANS, LATITUDE, AND TIME. [Chap. IX. Sidereal time, mean noon, at Greenwich 6 h. 35 m. 54 sec. Add for difference of meridians 49 6 36 33 18 18 31 32 56 4 Add (A) Right ascension of star Error in position of the instrument 8" A slight error in the position of the instrument will make no appreciable error in the result. Hence, this method affords perhaps the best means of determining the latitude. SECTION III. TO FIND THE TIME OF DAY. 433. First Method, — If a good meridian line has been run, the transit or theodolite may be placed in that line, and, being well levelled, the telescope, if adjusted by being directed to the meridian mark, will, when elevated, move in the meridian. Observe the time that the western limb of the sun comes to the vertical wire, and also when the eastern limb leaves it. The mean between these will be the time that the centre of the sun is on the meridian, or apparent noon. Increase or diminish the observed time of the passage of the centre by the equation of time according as the sun is too slow or too fast, and the result will be the time of mean noon as given by the watch. The difference between this and twelve hours will be the error of the watch. 434. Second Method. — Calculate the time that a fixed star having but little declination will pass the meridian as directed for Polaris, Art. 415. Then the difference between the observed and the calculated time will be the error of the watch. Sec. III.] TO FIND THE TIME OF DAY. 323 435. Third Method. — If the meridian line has not been determined, the time may be obtained by an altitude of the sun or of a star when out of the meridian. Take the altitude of the sun when three or four hours from the meridian, noting the time by the watch, and correct it for refraction and semidiameter. The altitude of the upper limb should be taken in the afternoon, and the lower in the morning, as the wire then crosses the face of the sun before the observation, and may be distinctly seen. Call the altitude of the sun A, the polar distance D, the latitude L, and the hour angle H. Then sin.* J H = °°«- * (A + L + D) sin. j(L ,+ D - A^ sin. D . cos. L or, if S = | (A + L + D), then S - A = \ (L + D - A), and sin.* J H = cos. 8 .sin. (8 -A^ sin. D . cos. L Rule. Call the corrected altitude A. From the Ephemeris take the sun's declination at the time of observation, (the watch- time will be sufficiently accurate) ; if north, subtract it from 90°, but if south, add it to 90° : the result will be the sun's polar distance, which call D. Call the latitude of the place L. Let S = J (A + L + D). Add together Ar. Co. sin. D, Ar. Co. cos. L, cos. S, and sin. (S — A), divide the result by 2, and the quotient will be the sine of half the hour angle of the sun at the time of observation. If the obser- vation is made in the afternoon, the hour angle reduced to time is the apparent time ; but, if the observation is in the morning, the hour angle subtracted from 12 is the apparent time. To the apparent time apply the equation of time, and the result is the mean time of the observation. The difference between the calculated time and that shown by the watch is the error of the watch. Several observations may be made in the course of a few minutes, and the mean of the results taken. If the obser- vation is carefully made with a good transit or theodolite, 324 MERIDIANS, LATITUDE, AND TIME. [Chap. IX. the time obtained by this method will not differ more than a small fraction of a minute from the true time. 436. If a star is observed instead of the sun, the mode of calculation is the same. The hour angle will then be in sidereal hours, which must be converted into solar hours. The result, added to or subtracted from the time of the meridian passage of the star, according as the observation was made after or before the star had passed the meridian, will give the mean time of observation. 437. If two altitudes of the sun or a star be taken, and the times noted by a watch, the true time and the latitude may both be found. But, as other and preferable methods have already been given for finding the latitude, it is un- necessary to give the rule here. CHAPTER X. VARIATION OF THE COMPASS.* 438. It has been mentioned (Art. 268) that the magnetic and the geographical meridian do not generally coincide ; the difference between the directions of the two being called the variation of the compass. If this variation were constant, it would be of no practical importance to the sur- veyor. A line run by the compass at one time could be retraced on the same bearing at any other. The variation is, however, subject to continual changes, — some of them having a period of many years, perhaps several centuries, others being annual or diurnal, and some accidental or tem- porary. 439. Secular Change. From the time of the earliest observations made in this country on the position of the magnetic needle till about the commencement of the pre- sent century, the north point was gradually moving to the west. Since then, the direction of its motion has been re- versed. This motion constitutes what is called the secular change. To give an idea of the extent of this deviation, the following table of observations, made at Paris, is pre- sented : — Tear. Variation. 1541 7° East. 1580 11 30' " 1618 8 1663 1700 8 10 West. 1780 19 55 " 1805 22 5 " 1814 22 34 " Year. Variation- 1816 22° 25' West 1823 22 23 1827 22 20 1828 22 5 1829 22 12 1835 22 3 1853 20 17 u it u * This subject, in its connections with Land Surveying, was first fully developed by Professor Gillespie, in his Treatise before referred to. 325 32G VARIATION OF THE COMPASS. [Chap. X. From this table, it appears that in 1580 the needle had attained its greatest eastern deviation. From that time to about the year 1814 it moved towards the west, the great- est deviation being 22° 34'. Since 1814 it has been moving to the east. From observations made at various places in Europe and America, it appears that similar changes have been going on throughout all these countries. 440. The following table, mostly taken from the "Report of the Superintendent of the United States' Coast Survey" for 1855, gives the variation and secular change for some of the more important places in this country : — Change Locality. Lat. Lon. Date. Variation. in 1850. Montreal, C.W 45° 30' 73° 35' 1850 + 9° 28' 4-4' 49° 40' 79° 21' 1850 1° 36' Burlington, Vt 44° 27' 73° 10' 1855 9° 57'. 1 4'.9 43° 39' 70° 16' 1851 11° 41/ 44° 20' 71° 2' 1854 9° 31' 5'.2 Providence, R.I — 41° 50' 71° 24' 1855 9° 31'.5 6'.0 New Haven, Conn. 41° 18' 72° 54' 1845 6° 17'.3 4'. 8 40° 43' 74° 0' 1845 6° 25'.3 5'.2 Albany, N.Y 42° 39' 73° 44' 1836 6° 47' 7'.2 Philadelphia, Pa... 39° 58' 75° 10' 1855 4° 31'.7 6'. 8 40° 26' 79° 58' 1845 33' 3'.5 Wilmington, Del... 39° 45' 75° 34' 1846 2° 30'.7 Baltimore, Md 39° 16' 76° 34' 1847 2° 18'. 6 Washington, D.C... 38° 53' 77° 1' 1855 2° 25' 5'.0 Petersburg, Va 37° 14' 77° 24' 1852 0° 26'. 5 34° 81° 2' 1854 — 3° 1'.7 32° 5' 81° 5' 1852 — 3° 40'. 3 39° 6' 84° 22' 1845 — 4° 4' 4' 39° 49' 84° 47' 1845 — 4° 52' 4' Detroit, Mich 42° 24' 82° 58' 1840 — 2° 0' V San Francisco, Cal. 37° 48' 122° 27' 1852 — 15° 27' The above are derived from the best data that could be procured ; but many of the observations are doubtless very imperfect. 441. Line of no Variation. From a map published by Professor Loomis, it appears that in 1840 the lines of equal variation crossed the United States in a direction to the east of south, tending more to the east in the ISTew England States. At that date, the line of no variation passed a little VARIATION OF THE COMPASS. 327 to the west of Pittsburg and to the east of Kaleigh, N.C., — all those portions of the country to the east of that line having western variation. From a similar map, published in the Report above referred to, it appears that the line of no variation had shifted to the west a few miles since that time. It also results from the calculations in the same report, that the rate of change in variation has now attained its maximum, and is beginning to diminish. 442. As it is frequently of importance to know the former variation, the following information is added : — The variation in Burlington, Vt, in 1792 7° 38' W. Salem, Mass., 1781 7° 2' W. New Haven, Ct., 1761 5° 47' W. " 1819 4° 35' W. New York, 1686 8° 45' W. " " 1789 4° 20' W. Philadelphia, 1710 8° 30' W. 1793 1° 30' ~W. 1818; 7° 30' W. 1805, 5° 57' W. 1775, 5° 25' W. 1750, 6° 22' W 1824, 4° 40' W 1750, 5° 45' \Y 1837, 3° 52' \V 443. Prom the table, (Art. 440,) the variation for any time not far remote from those given may readily be found. This will also apply for places not very far distant from the line of equal variation passing through that place. As, however, the rate of change varies, calculations based on such a table can only be considered correct when the interval of time is comparatively small. In all cases, when it can be done, the variation should be found by direct ob- servation by the methods explained in the next article. 444. To determine the change in variation by old lines. As the rate of change varies, the above rule can only be considered as true when the interval of time has not been great. If a number of years have elapsed since the prior survey, and no observations can be found relating to the immediate neighborhood, the change of variation can be 328 VARIATION OF THE COMPASS. [Chap. X. found, nearly, by comparison with other places where such observations have been made. "When any well-established marks can be found, the change may be determined by taking the bearings of these and comparing them with the records. The difference will give the change that has taken place between the dates of the two surveys. If the two marks are not on the same line, they may still be used for this purpose. Thus, according to an old deed, the bearings of three adjacent sides of a tract were as follows, — viz. : 1. Beginning at a marked locust, N". 60J° E. 200 perches to a chestnut ; 2. E". 25J° E. 183 perches to a post; 3. K 45° E. 105.3 perches to a white-oak. The locust is gone, but the stump remains, and the white-oak is still standing. The intermediate corners are entirely lost. Setting the instrument over the stump, run "N. 60J° E. 200 perches ; thence N". 25J° E. 183 perches ; and thence N". 45° E. 105.3 perches. If no change had taken place in the Fig. 200. variation, and both surveys had been accurately made, the last distance / y> D would have been terminated at the / / white-oak. Instead of this, however, / / the tree bears S. 54° 25' E. 2.93 perches. Fig. 200 is a draft of the above. From the bearings of AB, BC, and CD, calculate that of AD, which (Art. 350) will be found to be N". 43° 59' E. 470.38 perches. This, therefore, was the bearing and distance of AD at the time of the former survey. It is now the bearing and dis- tance of AD'. With the latitude and. departure of AD' and that of DD', calculate the present bearing and distance of AD (Art. 350.) It will be found to be 1ST. 47° 54' E. 476.25 perches. The change of variation has therefore been 3° 55 f W. There is likewise a variation of 5.87 perches in the measurement, from which it is inferred that the chain used in the former survey was 101.25 links in length, or 1J links too long. VARIATION OF THE COMPASS. 329 In order, therefore, correctly to trace the lines of the tract, the vernier of the compass must be set 3° 55' W., and all the distances be increased 1J- links per chain, or 1J perches per hundred. The magnetic bearings and the distances of the three sides are now, — 1. 1ST. 64° 25' E. 202.5 perches; 2. K 29° 10' E. 185.3 perches; 3. K 48° 55' E. 106.6 perches. 445. Diurnal Change. If the position of the needle be accurately noted at sunrise on a clear summer day, and the observation be repeated at intervals, it will be found that the north pole will gradually be deflected to the west, attain- ing its maximum deviation about 2 or 3 o'clock. During the afternoon it will gradually return towards its former position, which it will regain about 8 or 9 o'clock in the evening. This deviation from the normal position is known as the diurnal change. It amounts sometimes to as much as a quarter of a degree, being greater in a clear day than when the sky is overcast, and not being perceptible if the day is entirely cloudy. It is likewise greater in summer than in winter. In consequence of this diurnal change, it is evident that a line run in the morning cannot be retraced with the same bearings at noon. The surveyor should therefore record not merely the date at which a survey is made, but also the time of day at which any important line was run, and also the state of the weather, whether clear or other- wise. 446. Irregular Changes. Besides the seculat and diurnal changes, the needle is subject to disturbance from the passage of thunder storms, or from the occurrence of aurora boreali. It is likewise sometimes violently agitated when no apparent cause exists. Such disturbances pro- bably result from the occurrence of a distant magnetic storm, which would otherwise be unperceived, or from the passage of electric currents through the atmosphere. 447. From the preceding articles it will be apparent that 330 VARIATION OF THE COMPASS. [Chap.X. the needle, though an invaluable instrument for many pur- poses, is little to be depended on where precision is re- quired. It would be very desirable that prominent marks, the bearings of which were fully known, were established over the country, and that all important lines should be determined, by triangulation, from these. The true bear- ings should always be recorded. There would then be no difficulty in retracing old lines. In the State of Pennsyl- vania, and perhaps in some others, this is now required by law, though it is very doubtful whether the law is yet car- ried out in a way to be of much practical benefit, owing to the want of scientific knowledge on the part of much the larger number of those who undertake the business of surveying. Until there is a more general diffusion of theoretical as well as practical science among those whose business it is to settle the boundaries of estates, cases will continually occur in which confusing lines will be found to exist. This could never occur if all the bearings were made to the true meridian, the surveyor being careful to determine the local attraction and to allow for it in making his record. In no instance should a station be left before the back-sight has been taken, since, even in those regions where but little such influence exists, it will sometimes be found at par- ticular points. It sometimes likewise extends, without any change, over a considerable space, and thus may deflect the needle similarly at a number of stations. An instance of this kind was related to the author, a short time since, by a surveyor of great practical experience. A line was in dispute. One of the parties called in a surveyor, whom we shall call A., who ran the line, coming out at a stone. The other party, not being satisfied, called upon B., who traced a line agreeing exactly with the one run by A. until he came to a certain point: he then deviated from the former line some 4° to the west. He likewise ar- rived at a stone. Both parties were now dissatisfied. The first called on A. again, who retraced his line, following exactly his former course. B. was again employed. His course de- viated at the same point as before from A.'s. It was then VARIATION OF THE COMPASS. 331 concluded to have them together. B., being the older hand, went ahead. When they arrived at the point at which their lines separated, B. called on A. to look through the sights, saying, "Is not this right, Mr. A. ?" " It looks very well," he replied: "but look back, Mr. B." On doing so, he found he was really running 4° to the west of his former course. The attraction was first manifest at that point, and continued, without change, at all the sub- sequent stations along the line he had traversed. APPENDIX. The following demonstration of the rule for finding the area of a triangle when three sides are given is more concise than that given in Art. 251. As the former, however, develops some important properties respecting the centre of the inscribed circle, it was thought best to retain it : — Let ABC (Fig. 201) be the triangle, the construction being the same as in Fig. 50, p. 75. Then, as was proved in the demon- stration of the Rule in Art. 143, AK = £ (AB -f BC + AC) = £ s. AI = J s — BC. Fig. 201. We have also KD = Bl = £ s — AC, and KB = J s — AB. Now, from similar triangles, ADE and AFB, we have AE : ED : : AF : FB. But AF : ED : : AF : ED ; whence (23.6) AE . AF : ED 2 : : AF 2 : ED . FB. But AE . AF = AK . AI (Cor. 36.3), and ED . FB = HB . FB = IB . BK (35.3) ; AI . AK : ED a : : AF 3 : IB . BK, and y/ AI . AK . IB. BK = ED . AF = ED . (AE -f EF) = ADC + BDC = ABC. 332 MATHEMATICAL TABLES. MATHEMATICAL TABLES. PAGE I. Table of Latitudes and Departures * 3 II. Table of Logarithms of Numbers »........<,... 17 III. Table of Logarithmic Sines and Tangents Bo IY. Table of Natural Sines and Cosines 87 V. Table of Chords 97 TRAYERSE TABLE; OR, DIFFERENCE OF LATITUDE AND DEPARTURE. _ • — — LATITUDES AZffD DEFAB.TUB.ES. D. | 1 i Deg. j 1 1 i Deg. I Deg. 1 Deg. D. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. I.OOOO .0044 ! I.0000 .0087 •9999 1 .0131 .9998 .0175 1 1 2 2.0000 .0087 I.9999 .0175 1.9998 .0262 I.9997 .0349 2 ! 3 3.0000 .0131 2.9999 .0262 2.9997 •0393 2.9995 .0524 3 4 4.0000 .0175 3.9998 .0349 3-9997 .0524 3-9994 .0698 4 5 5.0000 .0218 4.9998 .0436 4.9996 .0654 4.9992 .0873 5 6 5.9999 .0262 5.9998 .0524 5-9995 .0785 5-9991 .1047 6 ! 7 6.9999 .0305 6.9997 .0611 6.9994 .0916 6.9989 .1222 7 8 7.9999 •°349 7.9997 .0698 7-9993 .1047 7.9988 .1396 8 9 8.9999 •°393 8.9997 .0785 8.9992 .1178 8.9986 .1571 9 10 1 1 9.9999 .0436 9.9996 .0873 9.9991 .1309 9.9985 •1745 10 89f Deg. 89 J Deg. 89 £ Deg. 89 Deg. U Deg. 1* Deg. If Deg. 2 Deg. .9998 .0218 •9997 .0262 •9995 .0305 •99'94 .0349 1 2 1.9995 .0436 1.9993 .0524 1.9991 .0611 1.9988 .0698 2 3 2.9993 .0654 2.9990 .0785 2.9986 .0916 2.9982 .IO47 3 4 3.9990 .0873 3.9986 .1047 3.9981 .1222 3-9976 .1396 4 5 4.9988 .1091 4.9983 .1309 4-9977 .1527 4.9970 •1745 5 6 5.9986 .1309 5-9979 .1571 5-9972 .1832 5.9963 .2094 6 7 6.9983 .1527 6.9976 .1832 6.9967 .2138 6.9957 .2443 7 8 7.9981 •1745 7.9973 .2094 7.9963 .2443 7-995 1 .2792 8 9 8.9979 .1963 8.9969 .2356 8.9958 .2748 8.9945 •3 J 4i 9 10 1 9.9976 .2181 9.9966 .2618 9-9953 •3°54 9.9939 •349° 10 1 88f Deg. 88| Deg. 88i Deg. 88 Deg. 2i Deg. 2i Deg. 2f Deg. 3 Deg. .9992 •°393 .9990 .0436 .9988 .0480 .9986 .0523 2 1.9985 .0785 1. 9981 .0872 1.9977 .0960 1.9973 .1047 2 3 2.9977 .1178 2.9971 .1308 2.9965 .1439 2.9959 •157° 3 4 3.9969 .1570 3.9962 •1745 3-9954 .1919 3-9945 .2093 4 5 4.9961 .1963 4.9952 .2181 4.9942 •2399 4-993 1 .2617 5 6 5-9954 .2356 5-9943 .2617 5-993 1 .2879 5.9918 .3140 6 7 6.9946 .2748 6.9933 •3°53 6.9919 •335 8 6.9904 .3664 7 8 7.9938 .3140 7.9924 •349° 7.9908 .3838 7.9890 .4187 8 9 8.9931 •3533 8.9914 .3926 8.9896 .4318 8.9877 .4710 9 10 1 9-99*3 .3926 9.9905 .4362 9.9885 .4798 9.9863 .5234 10 1 871 Deg. 87i Deg. 87i Deg. 87 Deg. 3 i Deg. 3* Deg. 3| Deg. 4 Deg. .9984 .0567 .9981 .0610 •9979 .0654 .9976 .0698 2 1.9968 •"34 1.9963 .1221 1-9957 .1308 1.9951 •1395 2 3 2.9952 .1701 2.9944 .1831 2.9936 .1962 2.9927 .2093 3 4 3.9936 .2268 3.9925 .2442 3-99*4 .2616 3.9903 .2790 4 5 4.9920 .2835 4.9907 .3052 4.9893 .3270 4.9878 .3488 5 6 5.9904 .3402 5.9888 .3663 5.9872 •39 2 4 5«9 8 54 .4185 6 7 6.9887 .3968 6.9869 .4273 6.9850 .4578 6.9829 .4883 7 8 7.9871 •4535 7.9851 .4884 7.9829 .5232 7.9805 •558i 8 9 8.9855 .5102 8.9832 •5494 8.9807 .5886 8.9781 .6278 9 10 9.9839 .5669 9.9813 .6105 9.9786 .6540 9.9756 .6976 10 D. D. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 86f Deg. 86£ Deg. 861 Deg. 86 Deg. 22 LATITUDES AND DEPARTURES. D. 1 41 Deg. 4£ Deg. 4f Deg. 5 Deg. I D. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. •9973 .0741 .9969 •0785 .9966 .0828 .9962 .0872 1 ! % 1.9945 .1482 I.9938 .1569 I.9931 .1656 I.9924 •1743 2 1 3 2.9918 .2223 2.9908 .2354 2.9897 .2484 2.9886 .2615 3 1 4 3.9890 .2964 3-9877 .3138 3.9863 •33 12 3.9848 .3486 4 ! 5 4.9863 •3705 4.9846 •392-3 4.9828 .4140 4.9810 •4358 5 1 6 5-9^35 •4447 5-98I5 .4708 5-9794 .4968 5.9772 .5229 6 ! 7 6.9808 .5188 6.9784 •5492 6.9760 •5797 6.9734 .6101 7 8 7.9780 .5929 7-9753 .6277 7.9725 .6625 7.9696 .6972 8 9 8-9753 .6670 8.9723 .7061 8.9691 •7453 8.9658 •7844 9 10 1 9.9725 .7411 9.9692 .7846 9.9657 .8281 9.9619 .8716 10 1 85f Deg. 85J Deg. 851 Deg. 85 Deg. 51 Deg. 5| Deg. 5f Deg. 6 Deg. .9958 .0915 •9954 .0958 .9950 .1002 •9945 .1045 2 1. 9916 .1830 1.9908 .1917 1.9899 .2004 1.9890 .2091 2 3 2.9874 .2745 2.9862 .2875 2.9849 .3006 2.9836 .3136 3 4 3-9 8 3 2 .3660 3.9816 •3834 3-9799 .4008 3.9781 .4181 4 5 4.9790 •4575 4.9770 .4792 4.9748 .5009 4.9726 .5226 5 6 5-9748 .5490 5.9724 •5751 5.9698 .6011 5.9671 .6272 6 7 6.9706 .6405 6.9678 .6709 6.9648 .7013 6.9617 .7317 7 8 7.9664 .7320 7.9632 .7668 7-9597 .8015 7.9562 .8362 8 9 8.9622 .8235 8.9586 .8626 8.9547 .9017 8.9507 .9408 9 10 1 9.9580 .9150 9.9540 •9585 9.9497 1. 0019 9.9452 1.0453 10 84f Deg. 84£ Deg. 841 Deg. 84 Deg. 1 61 Deg. 6* Deg. 6f Deg. 7 Deg. -9941 .1089 .9936 .1132 .9931 .1175 .9925 .1219 2 1.9881 .2177 1.9871 .2264 1. 9861 .2351 1.9851 •2437 2 3 2.9822 .3266 2.9807 •339 6 2.9792 .3526 2.9776 •3656 3 4 3.9762 •4355 3-9743 .4528 3-97^3 .4701 3.9702 •4875 4 5 4.9703 •5443 4.9679 .5660 4-9 6 53 •5877 4.9627 .6093 5 6 5-9 6 43 .6532 5.9614 .6792 5.9584 .7052 5-9553 .7312 6 7 6.9584 .7621 6.9550 .7924 6.9515 .8228 6.9478 •853i 7 8 7.9524 .8709 7.9486 .9056 7-9445 •9403 7.9404 .9750 8 9 8.9465 .9798 8.9421 1.0188 8.9376 1.0578 8.9329 1.0968 9 10 1 9.9406 1.0887 9-9357 1. 1320 9.9307 1. 1754 9-92-55 1. 2187 10 1 83| Deg. 83 £ Deg. 831 Deg. 83 Deg. 71 Deg. 7£ Deg. 7f Deg. 8 Deg. .9920 .1262 .9914 •i3°5 .9909 .1349 .9903 .1392 2 1.9840 .2524 1.9829 .2611 1.9817 .2697 1.9805 .2783 2 3 2.9760 .3786 2.9743 .3916 2.9726 .4046 2.9708 •4175 3 4 3.9680 .5048 3-9 6 58 - .5221 3-9635 •5394 3.9611 •5567 4 5 4.9600 .6310 4.9572 .6526 4-9543 •6743 4-95I3 .6959 5 6 5.9520 .7572 5-9487 .7832 5-945 2 .8091 5.9416 .8350 6 7 6.9440 .8834 6.9401 •9 J 37 6.9361 .9440 6.9319 .9742 7 8 7.9360 1.0096 7.9316 1.0442 7.9269 1.0788 7.9221 1.1134 8 9 8.9280 I-I358 8.9230 1. 1747 8.9178 1.2137 8.9124 1.2526 9 10 D. 9.9200 1.2620 9.9144 1.3053 9.9087 1.3485 9.9027 1. 3917 Lat. 10 D. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 82| Deg. 82 i Deg. 821 Deg. 82 Deg. LATITUDES AND DEPARTURES. D. 1 81 Deg. 81 Deg. 8f Deg. 9 Deg. D. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. .9897 •1435 .9890 .1478 .9884 .1521 .9877 .1564 1 2 1-9793 .2870 I.9780 .2956 I.9767 .3042 1-9754 .3129 At 3 2.9690 .4305 2.9670 •4434 2.9651 .4564 2.9631 .4693 3 4 3.9586 .5740 3-95 6 l .5912 3-9534 .6085 3.9508 .6257 4 5 4.9483 •7175 4.9451 .7390 4.9418 .7606 4.9384 .7822 5 6 5-9379 .8610 5-9341 .8869 5.9302 .9127 5.9261 .9386 6 7 6.9276 I.0044 6.9231 1.0347 6.9185 1.0649 6.9138 I.0950 < 8 7.9172 "479 7.9121 1. 1825 7.9069 1. 2170 7.9015 I.2515 8 9 8.9069 1. 2914 8.9011 1.3303 8.8953 1. 3691 8.8892 I.4079 9 10 i i 1 1 9.8965 1-4349 9.8902 1.4781 9.8836 1. 5212 9.8769 I.5643 10 1 81f Deg. 811 Deg. 811 Deg. 81 Deg. 91 Deg. 91 Deg. 9f Deg. 10 Deg. .9870 .1607 .9863 .1650 .9856 .1693 .9848 .1736 2 1.9740 •3 2I 5 1.9726 .3301 1.9711 .3387 1.9696 •3473 2 3 2.9610 .4822 2.9589 .4951 2.9567 .5080 2.9544 .5209 3 4 3.9480 .6430 3-9451 .6602 3.9422 .6774 3-9392- .6946 4 5 4.9350 .8037 4-93I4 .8252 4.9278 .8467 4.9240 .8682 5 6 5.9220 .9645 5-9*77 .9903 5-9*33 I.0161 5.9088 1. 0419 8 7 6.9090 1. 1252 6.9040 «553 6.8989 I-I854 6.8937 1.2155 7 8 7.8960 1.2859 7.8903 1.3204 7.8844 I-3548 7-8785 1.3892 8 9 8.8830 1.4467 8.8766 1.4854 8.8700 I.5241 8.8633 1.5628 9 10 1 9.8700 1.6074 9.8629 1.6505 9.8556 I.6935 9.8481 I-7365 10 80f Deg. 801 Deg. 801 Deg. 80 Deg. 1 101 Deg. 101 Deg. lOf Deg. 11 Deg. .9840 .1779 •9 8 33 .1822 .9825 .1865 .9816 .1908 2 1. 9681 •3559 1.9665 •3 6 45 1.9649 •373° i-9 6 33 .3816 2 3 2.9521 •533* 2.9498 .5467 2.9474 .5596 2.9449 •57M 3 4 3.9362 .7118 3-933° .7289 3.9298 .7461 3.9265 .7632 4 5 4.9202 .8897 4.9163 .9112 4.9123 .9326 4.9081 .9540 5 6 5.9042 1.0677 5- 8 995 1.0934 5-8947 1.1191 5.8898 1. 1449 6 7 6.8883 1.2456 6.8828 1.2756 6.8772 i-3°57 6.8714 1-3357 7 8 7.8723 1.4235 7.8660 1-4579 7.8596 1.4922 7.8530 1.5265 8 9 8.8564 1. 6015 8.8493 1.6401 8.8421 1.6787 8.8346 1. 7173 9 10 1 9.8404 1.7794 9.8325 1.8224 9.8245 1.8652 9.8163 1. 9081 10 791 Deg. 791 Deg. 791 Deg. 79 Deg. Ill Deg. Ill Deg. Ill Deg. 12 Deg. .9808 .1951 •9799 .1994 .9790 .2036 •9781 .2079 1 2 1. 9616 .3902 1.9598 •39 8 7 1.9581 •4°73 1-9563 .4158 2 3 2.9424 •5853 2.9398 .5981 2.9371 .6109 2.9344 .6237 3 4 3-923 1 .7804 3.9197 •7975 3.9162 .8146 3.9126 .8316 4 5 4.9039 •9755 4.8996 .9968 4.8952 1. 0182 4.8907 1.0396 5 6 5.8847 1.1705 5-8795 1. 1962 5-8743 1. 2219 5.8689 1.2475 6 7 6.8655 1.3656 6.8595 1.3956 6.8533 1.4255 , 6.8470 1-4554 7 8 7.8463 1.5607 7.8394 1.5949 7.8324 1. 6291 7.8252 1.6633 8 9 8.8271 1.7558 8.8193 1-7943 8.8114 1.8328 8.8033 1. 8712 9 10 9.8079 1.9509 9.7992 1.9937 9.7905 2.0364 9.7815 2.0791 Lat. 10 - D. D. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 781 Deg. 781 Deg. 781 Deg. 78 Deg. LATITUDES AND DEPARTURES. 1 1 121 Deg. 12* Deg. 12| Deg. 13 Deg. D. 1 Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. •9772 1 .2122 •97 6 3 .2164 •9753 .2207 •9744 .2250 ; « 1-9545 .4244 1.9526 •43 2 91 1.9507 •44 1 4 1.9487 •4499 2 3 2.9317 .6365 2.9289 • 6 493 2.9260 .6621 2.9231 .6749 3 4 3.9089 .8487 3.9052 .8658 3.9014 .8828 3- 8 975 .8998 4 5 4.8862 I.0609 4.8815 1.0822 4.8767 1. 1035 4.8719 1. 1 248 5 6 5.8634 I.273I 5-8578 1.2986 1 5.8521 1.3242 5.8462 r -3497 6 7 6.8406 I.4852 6.8341 1.5151 6.8274 1-5449! 6.8206 J -5747 7 8 7.8178 I.6974 7.8104 i-73 x 5 7.8027 1.7656 1 7.7950 1.7996 8 9 8.7951 1.9096 j 8.7867 1.9480 8.7781 1.9863 8.7693 2.0246 9 10 1 9.7723 2.1218 9.7630 2.1644 9-7534 2.2070 9-7437 2.2495 10 1 77f Deg. 77* Deg. 77i Deg. 77 Deg. 13i Deg. 13* Deg. 13f Deg. 14 Deg. •9734 .2292 .9724 • 2 334 •9713 •2377 •9703 .2419 2 1.9468 .4584 1.9447 .4669 1.9427 •4754 1.9406 .4838 2 3 2.9201 .6876 2.9171 .7003 2.9140 •7I3 1 2.9109 .7258. 3 4 3- 8 935 .9168 3.8895 •933 8 3.8854 .9507 3.8812 .9677 4 5 4.8669 1. 1460 4.8618 1. 1672 4.8567 1. 1884 4- 8 5*5 1.2096 5 6 5.8403 I-375 2 5.8342 1.4007 5.8281 1. 4261 5.8218 1.4515 6 7 6.8137 1.6044 6.8066 1. 6341 6.7994 1.6638 6.7921 I-6935 7 8 7.7870 1.8336 7.7790 1.8676 7.7707 1.9015 7.7624 1-9354 8 9 8.7604 2.0628 8 -75 J 3 2.1010 8.7421 2.1392 8.7327 2.1773 9 < 10 1 9-733 8 2.2920 9.7237 2 -3345 9-7*34 2.3769 9.7030 2.4192 10 76| Deg. 76* Deg. 76i Deg. 76 Deg. 1 14i Deg. 141 Deg. 14| Deg. 15 Deg. .9692 .2462 .9681 .2504 .9670 .2546 .9659 .2588 2 1.9385 •4923 1.9363 .5008 1-9341 .5092 1. 9319 •5176 2 3 2.9077 .7385 2.9044 .7511 2.9011 .7638 2.8978 •7765 3 4 3.8769 .9846 3.8726 1.0015 3.8682 1. 0184 3.8637 I -°353 4 5 4.8462 1.2308 4.8407 1.2519 4.8352 1.2730 4.8296 1.2941 5 6 5- 8l 54 1.4769 5.8089 1.5023 5.8023 1.5276 5-7956 1.5529 6 7 6.7846 1. 7231 6.7770 1.7527 6.7693 1.7822 6.7615 1.8117 7 8 7.7538 1.9692 7.7452 2.0030 7.7364 2.0368 7.7274 2.0706 8 » 8.7231 2.2154 8 -7!33 2.2534 8.7034 2.2914 8.6933 2.3294 9 9.6923 2.4615 9.6815 2.5038 9.6705 2.5460 9-6593 2.5882 10 1 75f Deg. 75* Deg. 75i Deg. 75 Deg. 15i Deg. 15* Deg. 15f Deg. 16 Deg. .9648 .2630 .9636 .2672 .9625 .2714 .9613 .2756 ! 2 1.9296 .5261 1.9273 •5345 1.9249 .5429 1.9225 •55*3 2 3 2.8944 .7891 2.8909 .8017 2.8874 .8143 2.8838 .8269 3 1 4 3- 8 59* 1. 0521 3- 8 545 1.0690 3.8498 1.0858 3- 8 45° 1. 1025 4 1 5 4.8239 1. 3152 4.8182 1.3362 4.8123 I-357 2 4.8063 1.3782 5 6 5.7887 1.5782 5.7818 1.6034 5-7747 1.6286 5.7676 1.6538 6 1 7 6-7535 1. 8412 6.7454 1.8707 6.7372 1. 9001 6.7288 1.9295 7 8 7-71*3 2.1042 7.7090 2.1379 7.6996 2.1715 7.6901 2.2051 8 9 8.6831 2.3673 8.6727 2.4051 8.6621 2.4430 8.6514 2.4807 9 10 9.6479 2.6303 9.6363 2.6724 9.6246 2.7144 9.6126 2.7564 10 D. D. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 74| Deg. 74* Deg. 74£ Deg. 74 Deg. XiATXTUDES AD7D DEPARTURES. D. 1 16i Deg. 16i Deg. 16f Deg. 17 Deg. 1 D. S 1 Lat. Dep. Lat. Dep. Lat. Dep. 1 Lat. Dep. .9600 .2798 .9588 .2840 •9576 .2882. •9563 .2924 2 1. 9201 •5597, 1. 9176 .5680 1.9151 •5764: 1. 9126 •5847 2 ; 3 2.8801 .8395. 2.8765 .8520 2.8727 .86461 2.8689 .8771 3 : 4 3.8402 1.1193, 3- 8 353 I-I36I, 3.8303 1.152,8} 3.8252 1. 1 695 4 5 4.8002 1. 3991 1 4-7941 1. 4201 4.7879 1 .4410 4-78I5 I.4619 5 6 5.7603 1.6790 5-75*9 I.7041 5-7454 I.7292 5-7378 I.7542 6 7 6.7203 1.9588 6.7117 1. 9881 6.7030 2.0174 6.6941 j 2.0466 1 8 7.6804 2.2386 7.6706 2.2721 7.6606 2.3056 7.6504 2.3390 8 i 9 8.6404 2.5185! 8.6294 2.5561 8.6181 2.5938] 8.6067I 2.6313 9 ;10 i j i 1 9.6005 2.7983 9.5882 2.8402 9-5757 2.8820 9-563o| 2.9237 10 1 73f Deg. 73 i Deg. 73i Deg. 73 Deg. 17i Deg. 17 i Deg. 17f Deg. 18 Deg. .9550 .2965 •9537 .3007 .9524 .3049, •95" .3090 2 1. 9100 •593 1 1.9074 .6014 1.9048 .6097 1. 9021 .6180 2 3 2.8651 .8896; 2.8612 .9021 ■ 2.8572 .9146 2.8532 .9271 3 4 3.8201 1. 1862! 3.8149 I.2028 3.8096 I.2195 3.8042 1. 2361 4 5 4-775 1 1.4827 j 4.7686 i-5°35 4.7620 i-5 2 43 . 4-7553 I.5451 5 6 5-73 01 1.779*1 5.7223 1.8042; 5-7I44 1.8292 5.7063 1. 8541 6 1 7 6.6851 2.0758; 6.6760 2.1049 6.6668 2.1341! 6.6574 2.1631 7 | 8 7.6402 2.3723 7.6297 2.40561 7.6192 2.4389 7.6085 2.4721 8 9 8.5952 2.6689! 8.5835 2.7064 ' 8.5716 2.7438 8-5595 2.7812 9 ! 10 1 9.5502 2,-9654 j 9.5372 3.0071 ; 9.5240 3.0486 1 9.5106 3.0902 10 72f Deg. T2i Deg. 72i Deg. 72 Deg. i 1 i 18 i Deg. ■18i Deg. 18f Deg. 19 Deg. •9497 •3 I 3 2 .9483 •3 x 73l •9469 .3214 •9455 •3*56 2 1.8994 .6263 1.8966 .6346 1.8939 .6429 ! 1. 8910 .6511 2 1 3 2.8491 •9395 2.8450 •95191 2.8408 .9643 I 2.8366 .9767 3 ! 4 3.7988 1.2527 3-7933 1.2692 , 3.7877 1.2858 3.7821 I.3023 4 5 4-74 8 5 1.5658 4.7416 1.5865 4-7347 1.6072 4.7276 I.6278 5 6 5.6982 1.8790 5.6899 1.9038 5.6816 1.9286 5.6731 1-9534 6 ! 7 6.6479 2.1921 6.6383 2.2211 6.6285 2.2501 ; 6.6186 2.2790 7 2 7.5976 *-5°53 7.5866 2.5384 7-5754 2.5715 i 7.5641 2.6045 8 , 9 8-5473 2.8185 8-5349 2.8557 8.5224 2.8930 8.5097 2.9301 9 I 10 1 1 9.4970 3.1316 9.4832 3.1730 9.4693 3.2144 ! 9.4552 3-*557 10; 1 1 71| Deg. 71* Deg. ! 7H Deg. 71 Deg. 19 £ Deg. 19 i Deg. 19f Deg. 20 Deg. .9441 •3*97 .9426 .3338 .9412 •3379 •9397 .3420 2 1.8882 .6594 1.8853 .6676 1.8824 •6758 1.8794 .6840 2 3 2.8323 .9891 2.8279 1. 0014 2.8235 1.0138 i 2.8191 1. 0261 3 4 3-77 6 4 1. 3188 3.7706 I-335 2 3-7647 I-35I7 1 3-7588 1. 3681 4 5 4.7204 1.6485 4.7132 1.6690 4-7°59 1.6896 | 4-6985 1.7101 5 6 5.6645 1.9781 5-6558 2.0028 5.6471 2.0275 1 5.6382 2.0521 6 7 6.6086 2.3078 6.5985 2.3366 6.5882 2.3654 6.5778 2.3941 < 8 7-55*7 2-6375 7.541 1 2.6705 7-5*94 2.7033 ! 7-5 I 75 2.7362 8 9 8.4968 2.9672 8.4838 3.0043 8.4706 3-°4!3 1 8.4572 3.0782 9 10 9.4409 3.2969 9-4*64 3-338i Lat. j 9.41 1 8 ■ 3-379* . 9-3969 3.4202 10 D. D. Dep. Lat. Dep. Dep. Lat. Dep. Lat. 70| Deg. 7(H Deg. 1 70i Deg. 70 Deg. LATITUDES AEfD DEPARTURES. D. ! i 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 202- Deg. 20£ Deg. 20f Deg. 21 Deg. D. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. .9382 I.8764 2.8146 3-7528 4.6910 5.6291 6.5673 7-5°55 8-4437 9.3819 .3461 .6922 I.0384 I-3845 I.7306 2.0767 2.4228 2.7689 3.1151 3.4612 •9367 I.8733 2.8100 3-7467 4.6834 5.6200 6.5567 7-4934 8.4300 9.3667 .3502 .7004 I.0506 1.4008 1.7510 2.1012 2-45I5 2.8017 3- J 5i9 3.5021 •9351 I.8703 2.8054 3-74°5 4-6757 5.6108 6.5459 7.48 1 1 8.4162 9-35I4 •3543 .7086 1.0629 1.4172 1.7715 2.1257 2.4800 2.8343 3.1886 3-5429 .9336 I.8672 2.8007 3-7343 4.6679 5.6015 6-5351 7.4686 8.4022 9-3358 .3584 .7167 I.0751 1-4335 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 9 10 69| Deg. 69 1 Deg. 69 \ Deg. 69 Deg. 21 1 Deg. 21JDeg. 21f Deg. 22 Deg. •9320 1.8640 2.7960 3.7280 4.6600 5.5920 6.5241 7.4561 8.3881 9.3201 .3624 .72491 I.0873 I.4498 1. 8122 2.1746 2.5371 2.89951 3.2619 3.6244 •9304 1.8608 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 3-7152 4.6440 5-5729 6.5017 7-43°5 8-3593 9.2881 .3706 .7411 1.1117 1.4822 1.8528 2.2233 2-5939 2.9645 3-335° 3.7056 .9272 1.8544 2.7816 3.7087 4.6359 5-5631 6.4903 7-4175 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£ Deg. 68 \ Deg. 68 Deg. 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 D. 22 \ Deg. 22 i Deg. 22f Deg. 23 Deg. •9*55 1.8511 2.7766 3.7022 4.6277 5-553^ 6.4788 7.4043 8.3299 9-2-554 .3786 •7573 *- I 359 1. 5146 1.8932 2.2719 2.6505 3.0292 3.4078 3.7865 •9239 1.8478 2.7716 3-6955 4.6194 5-5433 6.4672 7.3910 8.3149 9.2388 .3827 •7654 1.1481 i-53°7 I-9I34 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 1.1601 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 •39°7 •7815 1. 1722 1.5629 1-9537 2.3444 2-735 1 3-1258 3.5166 3-9°73 671 Deg. 67 \ Deg. 67i Deg. 67 Deg. 23 i- Deg. 23 } Deg. 23f Deg. 24 Deg. .9188 1.8376 2.7564 3.6752 4.5940 5.5127 6-43 1 5 7-35°3 8.2691 9.1879 •3947 •7895 1. 1842 1.5790 1-9737 2.3685 2.7632 3.1580 3-5527 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 •9 I 53 1.8306 2.7459 3.6612 4.5766 5.4919 6.4072 7.3225 8.2378 9.1531 .4027 .8055 1.2082 1.6110 2.0137 2.4165 2.8192 3.2220 3.6247 4.0275 •9 J 35 1. 8271 2.7406 3-6542 4-5677 5.4813 6.3948 7.3084 8.2219 9- I 355 .4067 .8135 1.2202 1.6269 2.0337 2.4404 2.8472 3-2539 3.6606 4.0674 D. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 66| Deg. | 66* Deg. 66 \ Deg. 66 Deg. 10 _ _ = — =a LATITUDES^ AIT D BEFAET¥EES. D. 1 2 3 4 5 6 7 8 9 10 i 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 24£ Deg. 24J Deg. 24| Deg. 25 Deg. 1 D. 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. .9118 I.8235 2 -7353 3.6470 4.5588 5.4706 6.3823 7.2941 8.2059 9.1176 .4107 .8214 I.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 .4147 .8294 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 I.2560 I.6746 2.0933 2.5120 2.9306 3-3493 3.7679 4.1866 .9063 1. 8126 2.7189 3.6252 4-53I5 5-4378 6.3442 7-2505 8.1568 9.0631 .4226 .8452 I.2679 I.6905 2.II31 2-5357 2.9583 3.3809 3.8036 4.2262 65f Deg. 65 J Deg. 65 1 Deg. 65 Deg. 25i Deg. 25J Deg. 25f Deg. 26 Deg. •9045 1.8089 2.7*34 3.6178 4.5223 5.4267 6.3312 7.2356 8.1401 9.0446 .4266 .8531 I.2797 1.7063 2.1328 2.5594 2.9860 3-4I25 3- 8 39* 4.2657 .9026 1.8052 2.7078 3.6103 4.5129 5-4155 6.3181 7.2207 8.1233 9.0259 •4305 .8610] 1.2915 1.7220 2.1526 2.5831 3.0136 3-444 1 3.8746 4.3051 .9007 1. 8014 2.7021 3.6028 4-5035 5.4042 6.3049 7.2056 8.1063 9.0070 •4344 .8689 i-3°33 I-7378 2.1722 2.6067 3.041 1 34756 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-3I5 1 J-7535 2.1919 2.6302 3.0686 3.5070 3-9453 4.3837 64f Deg. 64£ Deg. 64i Deg. 64 Deg. 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 D. 26i Deg. 26* Deg. 26f Deg. 27 Deg. .8969 1-7937 2.6906 3-5 8 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- J 595 8.0544 8.9493 .4462 .8924 1.3386 1.7848 2.2310 2.6772 3- I2 34 3.5696 4.0158 4.4620 .8930 1.7860 2.6789 3-57I9 4.4649 5-3579 6.2509 7-I438 8.0368 8.9298 .4501 .9002 1.3503 1.8004 2.2505 2.7006 3- I 5°7 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 •454o .9080 1.3620 1. 8160 2.2700 2.7239 3.1779 3.6319 4.0859 4-5399 63| Deg. 63 i Deg. 63 i Deg. 63 Beg. 27i Deg. 27i Deg. 271 Deg. 28 Deg. .8890 1.7780 2.6671 3-556i 4-445 1 5-334J 6.2231 7.1121 8.0012 8.8902 •4579 •9 I S7 1.3736 1.8315 2.2894 2.7472 3.2051 3.6630 4.1209 4-5787 .8870 1.7740 2.6610 3.5480 4-435 1 5.3221 6.2091 7.0961 7-983 1 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 5-3°99 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 3.5318 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 * Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 62f Deg. 62* Deg. 62 i Deg. 62 Deg. 11 LATITUDES AND DEPARTURES, D. 1 2 3 4 5 6 7 8 9 10 1 2 3 i 4 5 G 7 8 9 10 i 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 I 10 28 £ Deg. 28£ Deg. 28| Deg. 29 Deg. D. 1 2 3 4 5 6 r 8 9 10 1 2 3 4 5 6 7 8 9 10 Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. .8809 1. 7618 2.6427 3-5236 4.4045 5-2853 6.1662 7.0471 7.9280 8.8089 •4733 .9466 1.4200 1-^933 2.3666 2.8399 3-3 x 3 2 3.7866 4.2599 4.7332 .8788 I.7576 2.6365 3-5I53 4.3941 5.2729 6.1517 7.0305 7.9094 8.7882 •4772; •95431 i-43 x 5 1.9086 2.3858 2.8630 3.3401 3-8i73 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 I.4430 I.9240 2.4049 2.8859 3.3669 3.8479 4.3289 4.8099 .8746 I.7492 2.6239 34985 4-3731 5-2477 6.1223 6.9970 7.8716 8.7462 .4848 .9696 I.4544 I.9392 2.4240 2.9089 3-3937 3-8785 4.3633 4.8481 61 f Deg. 61 J Deg. 61 i Deg. 61 Deg. 29 i Deg. 29 J Deg. 29f Deg. 30 Beg. .8725 1.7450 2.6175 3.4900 4.3625 5.2350 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 2.6111 3.4814 4-35i8 5.2221 6.0925 6.9628 7.8332 8.7036 .4924 .9848 1-4773 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 I.4886 I.9849 2.48 1 1 2.9773 3-4735 3.9697 4.4659 4.9622 .8660 1.7321 2.5981 3.4641 4-33°i 5.1962 6.0622 6.9282 7.7942 8.6603 .5000 1. 0000 1.5000 2.0000 2.5000 3.0000 3.5000 4.0000 4.5000 5.0000 60f Deg. 60 i Deg. 60i Beg. 60 Deg. 1 2 3 4 5 6 7 I 1 2 3 4 5 6 7 8 9 10 D. 30i Deg. 30 J Deg. 30f Deg. 31 Deg. .8638 1.7277 2.5915 3-4553 4.3192 5.1830 6.0468 6.9107 7-7745 8.6384 .5038 1.0075 2.0151 2.5189 3.0226 3.5264 4.0302 4-534° 5.0377 .8616 1.7233 2.5849 3-4465 4.3081 5.1698 6.0314 6.8930 7-7547 8.6163 •5°75 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 1.7143 2.5715 3.4287 4.2858 5.1430 6.0002 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 59| Deg. 591 Deg. 59 i Deg. 59 Deg. 3U Deg. 311 Deg. 31f Deg. 32 Deg. .8549 1.7098 2.5647 3.4196 4.2746 5-1295 5.9844 6.8393 7.6942 8.5491 .5188 i-o375 I-5563 2.0751 2-5939 3.1126 3.6314 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-i35o 3-6575 4.1800 4.7025 5.2250 .8504 1.7007 2.5511 3.4014 4.2518 5.1021 5-9525 6.8028 1 7.6532 1 8.5035 1 .5262 1.0524 1.5786 2.1049 2.6311 3-J573 3-6835 4.2097 4-7359 5.2621 Lat. .8480 1. 6961 2-5441 3.3922 4.2402 5.0883 5-9363 6.7844 7.6324 8.4805 •5299 1.0598 1.5898 2.1197 2.6496 3- I 795 3.7094 4-2394 4.7693 5.2992 D. Dep. Lat. Dep. Lat. Dep. Dep. Lat. 58| Deg. 58 * Deg. 1 581- Deg. 58 Deg. 12 LATITUDES AND DEPARTURES, D. 1 i 2 3 1 4 5 ! 6 7 8 9 ! io 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 32 i Deg. 32J Deg. 32| Deg. 33 Deg. D. | 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. •8457i I.6915 2.5372 3-3829 4.2286 5.0744 5.9201 6.7658 7.6116 8-4573 .5336 I.0672 I.6008 2.1345 2.6681 3.2017 3-7353 4.2689 4.8025 5-336i •8434 1.6868 2.5302 3.3736 4.2170 5.0603 5-9°37 6.7471 7-59°5 8-4339 •5373 1.0746 1.6119 1 2.1492 2.6865 3.2238 3-76ii 4.2984 4-8357 5-373° .8410 1. 6821 2.5231 3.3642 4.2052 5.0462 5.8873 6.7283 7.5694 8.4104 .5410 1. 0819 I.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 I.6339 2.1786 2.7232 3.2678 3.8125 4-3571 4.9018 5.4464 571 Deg. 57i Deg. 57i Deg. 57 Deg. . 33i Deg. 33 i Deg. 33f Deg. 34 Deg. .8363 1.6726 2.5089 3-3451 4.1814 5- OI 77 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-o°33 5-8372 6.6711 7.5050 8.3389 •55*9 1.1039 1-6558 2.2077 2.7597 3.3116 3.8636 4-4I55 4.9674 5.5194 .8315 1.6629 2.4944 3-3259 4- J 573 4.9888 5.8203 6.6518 7.4832 8.3147 •5556 I.IIII I.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 I.6776 2.2368 2.7960 3-3552 3-9*44 4-4735 5.0327 5-59*9 56f Deg. 56J Deg. 56i Deg. 56 Deg. 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 34i Deg. 34£ Deg. 34| Deg. 35 Deg. .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 6.5930 7-4171 8.2413 .5664 1. 1328 1.6992 2.2656 2.8320 3-3984 3.9648 4-53 12 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 4.560O 5.I3OO 5.7OOO .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 55 1 Deg. 55 i Deg. 55? Deg. 55 Deg. 35 \ Deg. 35J Deg. 35| Deg. 36 Deg. .8166 1.6333 2.4499 3.2666 4.0832 4.8998 5.7165 6.5331 7-3498 8.1664 •577i I-I543 i 1-73*4 2.3086 2.8857 3.4629 4.0400 4.6172 5-1943 5-77I5 .8141 1.6282 2.4423 3-2565 4.0706 4.8847 5.6988 6.5129 7.3270 8.1412 •5807 1 1.1614 i-742i 2.3228 2.9035 3.4842 4.0649 4.6456 5.2263 5.8070 Lat. .8116 1. 6231 2-4347 3.2463 4-°579 4.8694 5.6810 6.4926 7.3042 j 8.1157 .5842 1. 1685 1.7527 2.337O 2.9212 3-5055 4.0897 4.674O 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- "45 4.7023 5.2901 5-8779 D. Dep. Lat. 1 Dep. Dep. Lat. 1 ! Dep. Lat. 54f Deg. 54£ Deg. ; 54 \ Deg. 54 Deg. 13 LATITUDES AND DEPARTURES. D. 1 2 3 4 5 6 7 8 : 9 1 10 l 2 3 4 5 6 7 8 9 10 1 1 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 361 Deg. 36* Deg. 36f Deg. 37 Deg. D. 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 Lat. Dep. •5913 1. 1826 1-7739 2.3652 2.9565 3-5479 4.1392 4.7305 5.3218 5.9131 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 .8039 I.6077 2.4116 3-2154 4.0193 4.8231 5.6270 6.4309 7-2347 8.0386 •5948 1. 1896 I.7845 2.3793 2.9741 3-5689 4.1638 4.7586 5-3534 5.9482 .8013 I.6025 2.4038 3.2050 4.0063 4.8075 5.6088 6.4100 7-2II3 8.0125 .5983 1. 1966 I.7950 2-3933 2.9916 3-5899 4.1883 4.7866 5.3849 5.9832 .7986 1-5973 2-3959 3-1945 3.9932 4.7918 5-59°4 6.3891 7.1877 7.9864 .6018 I.2036 I.8054 2.4073 3.0091 3.6109 4.2127 4.8145 5.4163 6.0181 53f Deg. 53 J Deg. 531 Deg. 53 Deg. 371 Deg. 37* Deg. 37f Deg. 38 Deg. .7960 1.5920 2.3880 3.1840 3.9800 4.7760 5-5720 6.3680 7.1640 7.9600 .6053 1. 2106 1. 8159 2.4212 3.0265 3.6318 4-237I 4.8424 54476 6.0529 •7934 1.5867 2.3801 3.1734 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 5-5i6i 6.3041 7.0921 7.8801 .6157 I-23I3 I.8470 2.4626 3- 783 3.6940 4.3096 4-9253 5-54IO 6.1566 52| Deg. 52* Deg. 521 Deg. 52 Deg. 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 D. 381 Deg. 38* Deg. 38 f Deg. 39 Deg. •7853 1.5706 2.3560 3-HI3 3.9266 4.71 19 5.4972 6.2825 7.0679 7.8532 .6191 1.2382 1.8573 2.4764 3-°955 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-H95 3.8994 4-6793 5.4592 6.2391 7.0190 7.7988 .6259 1.2518 1.8778 2.5037 3.1296 3-7555 4-38i5 5.0074 5-6333 6.2592 .7771 1-5543 2-33H 3.1086 3.8857 4.6629 5.4400 6.2172 6-9943 7-77I5 .6293 I.2586 I.8880 2-5I73 3.1466 3-7759 4.4052 5.0346 5.6639 6.2932 51| Deg. 51* Deg. 511 Deg. 51 Deg. 39* Deg. 39* Deg. 39f Deg. 40 Deg. •7744 1.5488 2.3232 3.0976 3.8720 4.6464 5.4207 6.1951 6.9695 7-7439 •6327 1.2654 1. 8981 2.5308 3.1635 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 6.9446 7.7162 .6361 1.2722 1.9082 2-5443 3.1804 3.8165 4.4525 5.0886 5-7247 6.3608 .7688 1-5377 2.3065 3-°754 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 5.1423 5.7851 6.4279 L Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 50f Deg. 50* Deg. 501 Deg. 50 Deg. 14 ! LATITUDES AETO DEPARTURES. 1 D. 1 1 2 1 3 4 5 6 7 8 9 10 1 2 ! I 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 i 1 2 3 4 5 6 7 8 9 10 40£ Deg. 40J Deg. 40| Deg. 41 Deg. D. 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. .7632 1.5265 2.2897 3.0529 3.8162 4-5794 5.3426 6.1059 6.8691 7.6323 .6461 I.2922 I.9384 2.5845 3.2306 3.8767 4.5229 5.1690 5.8I5I 6.4612 .7604 I.5208 2.2812 3.0416 3.8020 4.5624 5.3228 6.0832 6.8437 7.6041 •6494 I.2989 I.9483 2.5978 3.2472 3.8967 4.5461 5-I956 5.8450 6.4945 •7576 I.5151 2.2727 3.0303 3-7878 4.5454 5-3030 6.0605 6.8181 7-5756 .6528 L3055 I-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 I.3121 I.9682 2.6242 3.2803 3-9364 4.5924 5.2485 5.9045 6.5606 49f Deg. 49 \ Beg. 49 i Deg. 49 Deg. 41 1 Deg. 41 £ Deg. 41 f Deg. 42 Beg. .7518 1.5037 2.2555 3.0074 3-7592 4,5110 5.2629 6.0147 6.7666 7.5184 •6593 I.3187 I.9780 2.6374 3.2967 3.9561 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 I.3252 I.9879 2.6505 3-9757 4.6383 5.3010 5.9636 6.6262 .7461 1.4921 2.2382 2.9842 3-73°3 4-4763 5.2224 5.9685 6.7145 7.4606 .6659 I-33I8 I.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 J 4 ,6691 I-3383 2.0074 2.6765 3-3457 4.0148 4.6839 5-353° 6.0222 6.6913 48f Deg. 48 h Deg. 48 1 Deg. 48 Deg. 1 2 3 4 5 6 7 8 9 10 1 3 4 5 6 7 8 9 10 D. 42 \ Deg. 42£ Deg. 42f Deg. 43 Deg. .7402 1.4804 2.2207 2.9609 3-7011 4-4413 5-1815 5.9217 6.6620 7.4022 .6724 I.3447 2.0171 2.6895 3.3618 4.0342 4.7066 5-3789 6.0513 6.7237 •7373 1.4746 2.2118 2.9491 3.6864 4.4237 5.1609 5.8982 6.6355 7.3728 •6756 1. 3512 2.0268 2.7024 3.3780 4-°535 4.7291 5-4047 6.0803 6-7559 •7343 1.4686 2.2030 2-9373 3.6716 4.4059 5- J 403 5.8746 6.6089 7.3432 .6788 I-3576 2.0364 2.7152 3.3940 4.0728 4.7516 5-43°4 6.1092 6.7880 •73 J 4 1.4627 2.1941 2.9254 3.6568 4.3881 5.1195 5.8508 6.5822 7-3J35 .6820 1.3640 2.0460 2.7280 3.4100 4.0920 4.7740 5.4560 6.1380 6.8200 471 Deg. 47£ Deg. 47i Deg. 47 Deg. 43 £ Deg. 43 £ Deg. 43f Deg. 44 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 34259 4.1IH 4.7963 5-48I5 6.1666 6.8518 •7254 1.4507 2.1761 2.9015 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 .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 34576 4.1491 4.8406 5.5321 6.2236 6.9151 •7193 1.4387 2.1580 2.8774 3.5967 4.3160 5-°354 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 D. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 46f Deg. 46£ Deg. 46£ Deg. 46 Deg. 15 LATITUDES AWTD DEPARTURES. D. J 1 2 3 4 5 6 7 8 9 10 44 i Deg. 44| Deg. 44f Deg. 45 Deg. D. 1 2 3 4 5 6 7 8 9 10 D. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. .7163 1.4326 2.1489 2.8652 3-58I5 4.2978 5.0141 5-73°4 6.4467 7.1630 .6978 I.3956 2.0934 2.7912 3.4890 4.1867 4.8845 5-5823 6.2801 6.9779 •7133 I.4265 2.1398 2.8530 3.5663 4.2795 4.9928 5.7060 6.4193 7.1325 .7009 1. 4018 2.1027 2.8036 3-5°45 4.2055 4.9064 5.6073 6.3082 7.0091 .7102 I.4204 2.1306 2.8407 3-55°9 4.2611 4-97I3 5.6815 6.3917 7.1019 .7040 1.4080 2.1120 2.8161 3.5201 4.2241 4.9281 5.6321 6.3361 7.0401 .7071 I.4142 2.1213 2.8284 3-5355 4.2426 4-9497 5.6569 6.3640 7.0711 .7071 1.4142 2.1213 2.8284 3-5355 4.2426 4.9497 5.6569 6.3640 7.071 1 D. Dep. Lat. • Dep. Lat. Dep. Lat. Dep. Lat. 45f Deg. 45£ Deg. 451 Deg. 45 Deg. TABLE OF USEFUL NUMBERS. Logarithms. Ratio of circumference to diameter n = 3.1415926536 0.4971499 Area of circle to radius 1 = " " Surface of sphere to diameter 1 = " " Area of circle to diameter 1 = 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 lbs. avoir- dupois of water at 39. 8° Fahr. U. S. standard bushel contains 2150.42 c. in., or 77.627413 lbs. av. of water at 39. 8° • Fahr. British imperial gallon contains 277.274 c. in., = 1.2003 wine gallons of 231 c. in. French metre = 39.37079 in. == 3.28089917 feet. " toise = 6.39459252 feet. " are = 100 sq. metres = 1076.4299 sq. ft. " hectare = 100 ares = 2.471 143 acres = 107642.9936 sq. ft. " litre = 1 cubic decimeter = 61.02705 c. in. = .2641 8637 gallons of 231 c. in. u hectolitre = 100 litres = 26.418637 gallons. I pound avoirdupois = 7000 grs. = 1. 21 5277 pounds Troy. 1 " Troy = 5760 grs. = .822857 pounds avoir. I gramme = 15.442 grains. 1 kilogramme = 1000 grammes = 15442 grs. = 2.20607 lbs. avoir. Tropical year = 365 d. 5 h. 45 m. 47.588 sec. 16 TABLE or THE LOGARITHMS OF NUMBERS, FROM 1 to 10,000. 17 A TABLE OP THE LOGARITHMS OF NUMBERS FKOM 1 TO 10,000. N. Log. N. Log. N. lAg. N. Log. 1 o.oooooo 26 i-4 x 4973 51 1.707570 76 1 .880814 2 0.301030 27 1.431364 52 1.716003 77 1 .886491 1 3 0.477121 28 1.447158 53 1.724276 78 1 .892095 4 0.602060 29 1.462398 54 1.732394 79 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 i-755 8 75 82 1 .913814 8 0.903090 33 1.518514 58 1.763428 83 1 .919078 9 0.954243 34 i-SlWV 59 1.770852 84 1 924279 10 1. 000000 35 36 1.544068 60 1.778151 85 1 929419 11 1.041393 1.556303 61 1.785330 86 1 934498 12 1.079181 37 1.568202 62 1.792392 87 1 939519 13 1.113943 38 1.579784 63 I-79934I 88 1 944483 14 1.146128 39 1. 591065 64 1. 806180 89 1 949390 15 1.176091 40 1.602060 65 1.812913 90 1 954H3 16 1. 204120 41 1. 612784 66 1. 819544 91 1 959041 17 1.230449 42 1.623249 67 1.826075 92 1 963788 18 1.255273 43 1.633468 68 1.832509 93 1 968483 19 1.278754 44 I-643453 69 1.838849 94 1 973128 20 1. 301030 45 1. 653213 70 1.845098 95 1 977724 21 I. 322219 46 1.662758 71 1. 851258 96 1 982271 22 1.342423 47 1.672098 72 1.857332 97 1 986772 23 1. 361728 48 1.681241 73 1 863323 98 1. 991226 24 1.380211 49 1. 690196 74 1 869232 99 1. 995635 25 1.397940 1 50 1.698970 75 1. 875061 100 2. 000000 19 N. 100. LOGARITHMS. Log. 000. N. 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 1 2 3 1301 5609 9876 4100 8284 2428 6533 °6oo 4628 8620 2576 6495 O380 4230 8046 1829 558o 9298 2985 6640 ~°a66 3861 7426 0963 447i 795i 1403 4828 8227 1599 4944 8265 1560 4830 8076 1298 449 6 7671 °822 3951 7058 °I42 3205 6246 9266 2266 5244 8203 1141 4060 6959 9839 2700 5542 8366 1 171 3959 6729 9481 2216 4 5 6 2598 6894 ^47 5360 9532 3664 7757 !8l2 5830 9811 3755 7664 ! 538 5378 9185 2958 6699 0407 4085 _77 3 i J 347 4934 8490 2 oi8 55i8 8990 2434 5851 9241 2605 5943 9256 2544 5806 9045 2260 545 1 8618 1763 4885 7985 ^63 4120 7 J 54 °i68 3161 6134 9086 2019 4932 7825 %99 3555 6391 9209 2010 4792 7556 °3°3 3°33 7 8 9 oooooo 4321 8boo 012837 7033 021189 5306 9384 033424 7426 o434 475 1 9026 3 2 59 745 1 1603 5715 9789 3826 7825 1787 57H 9606 34 6 3 7286 1075 4832 8557 2250 59" 9543 3144 6716 0258 3772 7257 0715 4146 7549 0926 4277 7603 0903 4178 7429 0655 3858 7037 °i94 33 2 7 6438 95 2 7 2594 5640 8664 1667 4650 7613 0555 3478 6381 9264 2129 4975 7803 0612 34°3 6176 8932 1670 1 . 0868 5181 945i 3680 7868 2016 6125 °i95 4227 8223 2182 6105 9993 3846 7666 > 1452 5206 8928 2617 6276 9904 3503 7071 °6n 4122 7604 1059 4487 7888 1263 46 1 1 7934 1231 4504 7753 0977 4177 7354 0508 3 6 39 6748 9835 2900 5943 8965 1967 4947 7908 0848 3769 6670 9552 2415 5259 8084 0892 3681 6 453 9206 1943 2 J 734 6038 °3oo 4521 8700 2841 6942 '004 5029 9017 2l66 6466 °7 2 4 4940 9116 3252 7350 1408 543° 9414 3029 7321 1570 5779 9947 4°75 8164 2216 6230 2O7 4148 8053 1924 5760 95 6 3 3333 7071 0776 445 1 8094 1707 5291 8845 2 37° 5866 9335 2777 6191 9579 2940 6276 9586 2871 6131 9368 2580 5769 8934 2076 5i9 6 8294 1370 4424 7457 o 4 6 9 3460 6430 9380 2311 5222 ~87i3 ° 9 86 3839 6674 9490 2289 5069 7832 °577 33°5 7 3461 7748 1993 6197 0361 4486 8571 2 6i 9 6629 %02 4540 8442 2309 6142 9942 3709 7443 1145 4816 8457 2067 5647 9198 2721 6215 9681 3 IJ 9 653 1 9916 3 2 75 6608 99 J 5 3198 6456 9690 2900 6086 9249 2 389 55°7 8603 ] 6 7 6 4728 7759 ° 7 6 9 3758 6726 9674 2603 5512 8401 I272 4123 6956 9771 2567 5346 8107 0850 _3577 8 3891 8i74 2415 6616 °775 4896 8978 3 02I 7028 o 99 8 041393 53 2 3 9218 053078 6905 060698 4458 8186 071882 5547 2969 6885 0766 4613 8426 2206 5953 9668 3352 7004 3362 7275 'i53 4996 8805 2582 6326 0038 37i8 7368 0987 4576 8136 '667 5169 8644 2091 55i° 8903 2270 4932 8830 2694 6524 0320 4083 7815 1514 5182 8819 2426 6004 9552 3 o7i 6562 O26 3462 6871 °253 3609 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 079181 082785 6360 99°5 093422 6910 100371 3804 • 7210 110590 "3943 7271 120574 3852 7io5 i3°334 3539 6721 9879 I43°i5 146128 9219 152288 533 6 8362 161368 4353 7317 170262 3186 °6 2 6 4219 778i V5 4820 8298 1747 5169 8565 1934 5278 8595 1888 5I5 6 8399 1619 4814 7987 1136 4263 5611 8926 2216 5481 8722 1939 5i33 8 3°3 *45° 4574 7676 0756 3815 6852 9868 2863 5838 8792 1726 4641 6940 0245 3525 6781 °OI2 3219 6403 9564 2702 5818 89II ^82 5032 8o6l I068 4055 7022 9968 2895 5802 8689 I558 4407 7239 °o5i 2846 5623 8382 ] I24 3848 140 141 142 143 144 145 146 147 148 149 7367 0449 35io 6 549 9567 2564 554i 8497 1434 435i 150 151 152 153 154 155 156 157 158 159 176091 8977 1 81 844 4691 7521 190332 3125 5900 8657 201397 7248 °I26 2985 5825 8647 1451 4237 7005 9755 2488 753 6 V3 3270 6108 8928 1730 45*4 7281 O29 2761 N. 3 4: 5 6 9 20 N. L60. LOGARITHMS Log. 204. i N. 1 2 3 4 5 6 5746 7 6016 8 "6286 9 1 160 204120 439 1 4663 4934 5204 5475 6556 161 6826 7096 7365 7 6 34 7904 8173 8441 8710 8979 9 2 47 162 9515 9783 0051 °3 J 9 0586 °853 J I2I '388 '654 ! 92I i 163 212188 2454 2720 2986 3252 35i8 3783 4049 43 J 4 4579 ! 164 4844 5109 5373 5638 5902 6166 643O 6694 6957 7221 | 165 7484 7747 8010 8273 8536 8798 9060 9323 9585 9846 166 220108 0370 0631 0892 1153 1414 1675 1936 2196 2456 167 2716 2976 3236 3496 3755 4015 4274 4533 4792 5051 168 53°9 55^8 5826 6084 6342 6600 6858 7115 7372 7630 169 7887 8144 8400 8657 8913 9170 9426 1979 9682 9938 2488 °i93 170 230449 0704 0960 1215 1470 1724 2234 2742 171 2996 3250 3504 3757 401 1 4264 45*7 4770 5023 5276 172 55*8 5781 6033 6285 6537 6789 7041 7292 7544 7795 173 8046 8297 8548 8799 9049 9299 955° 9800 °o5o °3oo 174 240549 0799 1048 1297 1546 1795 2044 2293 2541 2790 175 3038 3286 3534 3782 4030 4277 4525 4772 5019 5266 176 5513 5759 6006 6252 6499 6745 6991 7237 7482 7728 177 7973 8219 8464 8709 8954 9198 9443 9687 9932 0176 178 250420 0664 0908 1151 1395 1638 1881 2125 2368 . 2610 179 2853 3096 55H 3338 3580 5996 3822 4064 6477 4306 "6778 4548' 6958 4790 7198 5°3 X 180 255 2 73 5755 6237 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 3 6 3 6 3873 4109 4346 4582 184 4818 5°54 5290 552-5 57 6x 599 6 6232 6467 6702 6937 185 7172 7406 7641 7875 8110 8344 8578 8812 9046 9279 186 9513 9746 9980 0213 0446 0679 O912 I144 i 377 x 6o9 187 271842 2074 2306 2538 2770 3001 3233 3464 3696 3927 188 4158 4389 4620 4850 5081 53 11 5542 5772 6002 6232 189 6462 6692 8982 6921 7151 9439 7380 7609 7838 °I23 8067 °35i 8296 8525 190 278754 9211 9667 9895 °8o6 191 281033 1261 1488 1715 1942 2169 2396 2622 2849 3°75 192 3301 35*7 3753 3979 4205 4431 4656 4882 5107 5332 193 5557 5782 6007 6232 6456 6681 6905 7130 7354 7578 194 7802 8026 8249 8473 8696 8920 9H3 9366 9589 9812 195 290035 0257 0480 0702 0925 1 147 1369 1591 1813 2034 196 2256 2478 2699 2920 3H 1 33 6 3 3584 3804 4025 4246 197 4466 4687 4907 5 I2 7 5347 55 6 7 5787 6007 6226 6446 198 6665 6884 7104 73 2 3 7542 7761 7979 8198 8416 8635 199 8853 9071 1247 9289 1464 9507 1681 9725 9943 2114 °i6i 233 1 u 378 2547 u 595 2764 °8i3 2980 200 301030 1898 201 3196 3412 3628 3844 4059 4275 449 1 4706 4921 5136 202 535i 5566 578i 599 6 6211 6425 6639 6854 7068 7282 203 7496 7710 7924 8i37 8351 8564 8778 8991 9204 9417 204 9630 9843 0056 °268 ° 4 8i 0693 0906 I118 1330 ! 542 205 3"754 1966 2177 2389 2600 2812 3° 2 3 3234 3445 3656 206 3867 4078 4289 4499 4710 4920 5i3 534° 555i 5760 207 5970 6180 6390 6599 6809 7018 7227 743 6 7646 7854 208 8063 8272 8481 8689 8898 9106 93M 9522 973° 9938 209 210 320146 0354 2426 0562 2633 0769 2839 0977 1184 I39 1 3458 1598 3665 1805 2012 322219 3046 3252 4077 211 4282 4488 4694 4899 5105 53 10 5516 5721 5926 6131 8176 212 633 6 6541 6745 6950 7155 7359 7563 7767 7972 213 8380 8583 8787 8991 9194 9398 9601 9805 °oo8 °2II 214 33°4i4 0617 0819 1022 1225 1427 1630 1832 2034 2236 215 2438 2640 2842 3°44 3246 3447 3 6 49 3850 4051 4253 216 4454 4655 4856 5057 5257 5458 5<>5« 58S9 6059 6260 217 6460 6660 6860 7060 7260 7459 7659 7858 8058 8257 218 8456 8656 8855 9054 9253 945 1 9650 9849 °°47 °2 4 6 219 340444 0642 0841 IQ 39 3 1237 1435 5 1632 1830 7 2028 8 2225 N. 1 2 4 6 9 23 21 = — _ — _ N. 220. ZiOGvAJtfTHMS. Log. 342. N. 1 2620 4589 6549 8500 0442 2 375 4301 6217 8125 O2 5 2 3 4 5 6 7 3802 5766 7720 9666 1603 353* 5452 7363 9266 »i6i 3048 4926 6796 8659 V3 2360 4198 6029 7852 9668 1476 3 2 77 5070 6856 8634 °4°5 2169 3926 5676 7419 9*54 0883 2605 4320 6029 773 1 9426 x ii4 2796 447^ 6141 7804 9460 2754 4392 6023 7648 9268 °88i 8 9 220 221 222 223 224 | 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 34 2 4 2 3 4392 6353 8305 350248 2183 4108 6026 7935 9835 2817 4785 6744 8694 0636 2568 4493 6408 8316 °2I5 2105 3988 5862 7729 9587 .1437 3280 5"5 6942 8761 0573 2377 4*74 5964 7746 9520 1288 3048 4802 6548 3014 4981 6939 8889 0829 2761 4685 6599 8506 °4°4 2294 4176 6049 79*5 9772 1622 3464 5298 7124 8943 0754 2557 4353 6142 7923 9698 1464 3224 4977 6722 8461 °I92 1917 3635 5346 7051 8749 0440 2124 3803 5474 7139 8798 0451 2097 3737 537i 6999 8621 0236 "1846 345° 5048 6640 8226 9806 1381 2950 4513 6071 3212 5178 7135. 9083 1023 2954 4876 6790 8696 0593 3409 5374 7330 9278 1216 3 X 47 5068 6981 8886 0783 3606 557o 7525 9472 1410 3339 5260 7172 9076 °972 2859 4739 6610 8473 0328 2175 4015 5846 7670 9487 1296 3°97 4891 6677 8456 °228 I993 375 1 55 QI 7245 8981 °7ii 2433 4149 5858 7561 9257 0946 2629 A^A 5974 7638 9295 °945 2590 4228 5860 7486 9106 O720 2328 393° 5526 7116 8701 °279 1852 3419 4981 6537 3999 5962 79*5 9860 1796 3724 5643 7554 9456 J 35° 3236 5"3 6983 8845 0698 2544 4382 6212 8034 9849 1656 3456 5249 7034 8811 0582 2345 4101 5850 J7592 9328 1056 2777 4492 6199 7901 9595 '283 2964 4639 6308 7970 9625 l *7S 2918 4555 6186 7811 9429 ! 042 2649 4249 5844 7433 9017 °594 2166 373 2 5293 6848 8 4196 6157 8110 0054 1989 3916 5834 7744 9646 J 539 361728 3612 5488 7356 9216 371068 2912 4748 6577 8398 380211 2017 3815 5606 739° 9166 39°935 2697 4452 6199 1917 3800 5675 7542 9401 1253 3096 4932 6759 8580 0392 2197 3995 5785 7568 9343 1112 2873 4627 6374 8114 9847 1573 3292 5°°5 6710 8410 IO2 I788 3467 5140 6807 8467 °I2I I768 34IO 5°45 6674 8297 9914 1525 3*3° 4729 6322 7909 949 J 1066 2637 4201 576o 1 2482 4363 6236 8101 9958 1806 3647 5481 7306 9124 2671 455i 6423 8287 °i43 1991 3831 5664 7488 9306 1115 2917 4712 6499 8279 °o5i 1817 3575 5326 7071 8808 0538 2261 3978 5688 739 1 9087 °777 2461 4137 5808 7472 9129 O781 2426 4065 5697 7324 8944 °559 3424 53 QI 7169 9030 0883 2728 4565 6394 8216 o3o 1837 3636 5428 7212 8989 °759 2521 4277 6025 7766 0934 2737 4533 6321 8101 9875 1641 3400 5152 6896 250 251 252 253 254 255 256 , 257 [ 258 259 39794° 9674 401401 3121 4 8 34 6540 8240 9933 411620 3300 8287 O2O 1745 3464 5176 688l 8579 O27I 1956 3635 8634 o 3 6 5 2089 3807 5517 7221 8918 °6o9 22q3 397° 9501 J 228 2949 4663 637O 807O 9764 I45I 3 J 3 2 4806 260 261 262 263 264 265 266 | 267 268 269 4*4973 6641 8301 9956 421604 3246 4882 6511 8i35 9752 431364 2969 4569 6163 775i 9333 440909 2480 4045 5604 5307 6973 8633 O286 1933 3574 5208 6836 8459 °°75 1685 3290 4888 6481 8067 9648 1224 2793 4357 59 J 5 5641 7306 8964 °6i6 2261 3901 5534 7161 8783 0398 6474 8i35 9791 M-39 3082 4718 6349 7973 959 1 1203 270 271 272 273 274 275 276 277 278 279 2007 3610 5207 6799 8384 9964 1538 3106 4669 6226 2167 377° 5367 6957 8542 °I22 1695 3263 4825 6382 5 2488 4090 5685 7275 8859 °437 2009 3576 5137 6692 2809 4409 6004 759 2 9*75 0752 2323 3889 5449 7003 2 3 4 6 7 9 22 — — 1 N. 280. LOGARITHMS. Log. 447. N. 1 2 3 4 5 6 8088 9633 1172 2706 4235 5758 7276 8789 0296 1799 7 8242 9787 1326 2859 4387 59io 7428 8940 °447 1948 3445 4936 6423 7904 9380 0851 2318 3779 5235 6687 8 9 280 281 282 283 284 285 286 287 288 289 447158 8706 450249 1786 33i8 4845 6366 7882 939 2 460898 7313 8861 0403 1940 347i 4997 6518 8033 9543 1048 2548 4042 553 2 7016 8495 9969 1438 2903 4362 5816 7266 8711 0151 1586 3016 4442 5863 7280 8692 °o99 1502 2900 4294 5683 7068 8448 9824 1196 2564 3927 5286 6640 7991 9337 0679 2017 335i 4681 6006 7328 8646 9959 1269 2575 3876 5174 6469 7759 9045 0328 7468 9015 0557 2093 3624 5i5o 6670 8184 9694 1198 7623 9170 07 1 1 2247 3777 5302 6821 8336 9845 _i348 2847 4340 5829 7312 8790 0263 1732 3*95 4653 6107 7555 8999 0438 1872 33° 2 4727 6147 7563 8974 0380 7778 9324 0865 2400 3930 5454 6973 8487 9995 1499 7933 9478 1018 2553 4082 5606 7125 8638 °i 4 6 1649 8397 9941 1479 3012 4540 6062 7579 9091 °597 2098 3594 5085 6571 8052 9527 o 99 8 2464 3925 538i 6832 8552 ; °°95 j 1633 j 3 l6 5 1 4692 | 6214 773 1 9242 C748 2248 3744 5234 6719 82OO ! 9675 1145 26lO 407I 5526 6976 290 291 292 293 294 295 296 297 298 299 462398 3S93 5383 6868 8347 9822 471292 2756 4216 5671 2697 4191 5680 7164 8643 °n6 1585 3°49 4508 5962 741 1 8855 0294 1729 3*59 4585 6005 7421 8833 °239 1642 3040 4433 5822 7206 8586 9962 1333 2700 4 o6 3 5421 6776 8126 947i 0813 2151 3484 4813 6139 7460 2997 4490 5977 7460 8938 °4io 1878 334i 4799 6252 3146 4639 6126 7608 9085 °557 2025 3487 4944 6397 3296 4788 6274 7756 9233 °7°4 2171 3633 5090 6542 7989 9431 0869 2302 373° 5*53 6572 7986 9396 °8oi 2201 3597 4989 6376 7759 9137 °5ii 1880 3246 4 6o 7 5964 7316 8664 °oo9 1349 2684 4016 5344 6668 7987 300 301 302 303 304 305 306 307 308 309 477121 8566 480007 1443 2874 4300 572i 7138 8551 995 8 491362 2760 4 J 55 5544 6930 8311 9687 501059 2427 379 1 7700 9143 0582 2016 3445 4869 6289 7704 9114 52O 7844 9287 0725 2159 3587 5011 6430 7845 9255 °66i 2062 3458 4850 6238 7621 8999 °374 1744 3109 447 ! 8i33 9575 1012 2445 3872 5295 6714 8127 9537 °94i 2341 3737 5128 6515 7897 9275 0648 2017 3382 4743 6099 745i 8799 0143 1482 2818 4149 5476 6800 8119 9434 °745 2053 335 6 4656 595i 7243 8531 9815 1096 7 8278 9719 1156 2588 4015 5437 6855 8269 9677 ^81 2481 3876 5267 6653 8035 9412 0785 2154 3518 4878 6234 7586 8934 °277 1616 2951 4282 5609 6932 8251 9566 °8 7 6 2183 3486 4785 6081 7372 8660 9943 1223 8 8422 9863 I299 273I 4157 5579 6997 8410 9818 ! 222 310 311 312 313 314 315 316 317 318 319 ~32F 321 322 323 324 325 326 327 328 329 1782 3*79 4572 5960 7344 8724 °o99 1470 2837 4199 5557 6911 8260 9606 0947 2284 3617 4946 6271 7592 8909 °22I 153° 2835 4136 5434 6727 8016 9302 0584 1922 33 J 9 4711 6099 7483 8862 0236 1607 2973 4335 2621 40I5 5406 679I 8173 955° °922 229I 3655 5 OI 4 6370 7721 9068 °4ii 1750 3084 4415 5-4i 7064 8382 5°5i5° 6505 7856 9203 5 IQ 545 1883 3218 4548 5874 7196 5693 7046 8395 9740 1081 2418 375° 5079 6403 7724 5828 7181 8530 9874 1215 2551 3883 5211 6535 7855 330 331 332 333 334 335 336 337 338 339 518514 9828 521138 2444 3746 5°45 6339 7630 8917 530200 8777 o9o 1400 2705 4006 53°4 6598 7888 9174 0456 9040 °353 1661 2966 4266 5563 6856 8145 9430 0712 9171 0484 1792 3096 4396 5 6 93 6985 8274 9559 0840 5 93°3 °6i5 1922 3226 4526 5822 7114 8402 9687 0968 9697 •007 23H 3616 49*5 6210 7501 8788 O072 1351 9 N. 1 2 3 4: 6 23 N. 340. XiOG-ARXTHXMES. Log. 531. 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IiO&JZLTLlTHM&. Log. 880. N. 1 "0871 1442 2012 2581 3150 3718 4285 4852 5418 5983 2 3 4 5 6 1156 1727 2297 2866 3434 4002 4569 5*35 5700 6265 7 8 9 760 761 762 763 764 765 766 767 768 769 880814 1385 1955 2525 3°93 3661 4229 4795 53 61 5926 0928 1499 2069 2638 3207 3775 4342 4909 5474 6039 6604 7167 773° 8292 8853 9414 9974 °533 1091 _i649 2206 2762 3318 3873 4427 4980 5533 6085 6636 7187 0985 1556 2126 2695 3264 3832 4399 4965 553i 6096 ~666o 7223 7786 8348 8909 9470 °o3o 0589 1 147 1705 2262 2818 3373 3928 4482 5036 5588 6140 6692 7242 1042 1613 2183 2752 33 21 3888 4455 5022 5587 6152 1099 1670 2240 2809 3377 3945 4512 5078 5644 6209 6773 733 6 7898 8460 9021 9582 °i4i 0700 1259 1816 1213 1784 2354 2923 349i 4°59 4625 5192 5757 6321 1271 1841 241 1 2980 3548 4115 4682 5248 5813 6378 6942 7505 8067 8629 9190 9750 °3°9 , 0868 1426 !9 8 3 2540 3096 3651 4205 4759 5312 5864 6416 6967 7517 8067 8615 9164 9711 0258 0804 1349 1894 2438 2981 3524 4066 4607 5H8 5688 6227 6766 7304 7841 8378 1328 j 1898 ! 2468 3037 j 3605 4172 4739 53°5 5870 6434 770 771 772 773 774 775 776 777 778 779 886491 7054 7617 8179 8 74I 9302 9862 890421 0980 1537 6547 7111 7674 8236 8797 9358 9918 0477 1035 1593 2150 2707 3262 3817 437i 4925 5478 6030 6581 7132 7682 8231 8780 9328 9875 0422 0968 1513 2057 2601 3*44 3687 4229 4770 53 10 5850 6389 6927 7465 8002 ~«539 9074 9610 0144 0678 1211 1743 2275 2806 3337 1 6716 7280 7842 8404 8965 9526 °o86 0645 1203 1760 6829 7392 7955 8516 9077 9638 °i97 0756 1314 1872 2429 2985 354° 4094 4648 5201 5754 6306 6857 7407 6885 7449 8011 8573 9*34 9694 °253 0812 1370 1928 2484 3040 3595 4150 4704 5257 5809 6361 6912 7462 8012 8561 9109 9656 2O3 0749 1295 1840 2384 2927 347° 4012 4553 5094 5 6 34 6173 6712 7250 7787 8324 8860 939 6 993° 0464 0998 153° 2063 2594 3125 3 6 55 6998 7561 8123 8685 9246 9806 0365 0924 1482 2039 2595 3151 3706 4261 4814 53 6 7 5920 6471 7022 7572 780 781 782 783 784 785 786 787 788 789 892095 2651 3207 3762 43 l6 4870 5423 5975 6526 7077 897627 8176 8725 9 2 73 9821 900367 0913 1458 2003 2547 903090 3 6 33 4*74 4716 5 2 5 6 5796 6 335 6874 741 1 7949 908485 9021 9556 910091 0624 1158 1690 2222 2753 3284 2317 2873 3429 3984 4538 5091 5644 6195 6747 7297 2373 2929 3484 4039 4593 5M 6 5699 6251 6802 _7j52 7902 8451 8999 9547 °°94 0640 1186 1731 2275 2818 790 791 792 793 794 795 796 797 798 799 ~800" 801 802 803 804 805 806 807 808 809 7737 8286 8835 9383 993° 0476 1022 1567 2112 2655 3199 374 1 4283 4824 53 6 4 5904 6 443 6981 7519 8056 8592 9128 9663 0197 0731 1264 1797 2328 2859 339° 2 7792 8341 8890 9437 9985 0531 1077 1622 2166 2710 32-53 3795 4337 4878 5418 5958 6497 7°35 7573 8110 H646 9181 9716 0251 0784 1317 1850 2381 2913 3443 3 7847 8396 8944 9492 0039 0586 1131 1676 2221 2764 7957 8506 9054 9602 °i 49 . 0695 1240 1785 2329 2873 8122 8670 9218 9766 0312 0859 1404 1948 2492 3036 33°7 3849 439i 4932 5472 6012 6551 7089 7626 8163 336i 39°4 4445 4986 5526 6066 6604 7143 7680 8217 3416 3958 4499 5040 558o 6119 6658 7196 7734 8270 8807 9342 9877 041 1 0944 1477 2009 2541 3072 3602 6 3578 4120 4661 5202 5742 6281 6820 7358 7895 8431 8967 95°3 °°37 0571 1 1 04 1637 2169 2700 3231 3761 ! 9 810 811 812 813 814 815 816 817 818 | 819 8699 9^35 9770 0304 0838 1371 1903 2435 2966 3496 8753 9289 9823 0358 0891 1424 1956 2488 3019 3549 5 8914 9449 9984 o5i8 1051 1584 2116 2647 3178 3708 P*!- 4 7 8 31 N. 820. LOGARITHMS. Log. 913. N. ^820' 1 786t~ 2 3 3973 4 5 6 4*3 2 7 4184 8 4 2 37 9 913814 3920 4026 4079 4290 821 4343 4396 4449 4502 4555 4608 4660 47i3 4766 4819 822 4872 4925 4977 5030 5083 5136 5189 5241 5 2 94 5347 823 5400 5453 5505 5558 5611 5664 5716 5769 5822 5875 824 5927 5980 6033 6085 6138 6191 6243 6296 6 349 6401 825 6454 6507 6559 6612 6664 6717 6770 6822 6875 6927 826 6980 7033 7085 7138 7190 7243 7295 7348 7400 7453 827 7506 7558 7611 7663 7716 7768 7820 7873 7925 7978 828 8030 8083 8i35 8188 8240 8293 8345 8397 8450 8502 829 830 8555 8607 9130 8659 9183 8712 9235 8764 8816 8869 939 2 8921 9444 8973 9496 9026 9549 919078 9287 9340 831 9601 9653 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 IOIO 1062 1114 834 1166 1218 1270 1322 1374 1426 1478 1530 1582 1634 835 1686 1.738 1790 1842 1894 1946 1998 2050 2102 2154 836 2206 2258 2310 2362 2414 2466 2518 2570 2622 2674 837 2725 2777 2826 2881 2933 2985 3°37 3089 3140 3192 838 3 2 44 3296 3348 3399 345i 35°3 3555 3607 3^58 3710 839 924279 3814 433 1 3865 4383 3917 4434 3969 4021 4538 4072 4589 4* 2 4 4641 4176 4693 4228 4744 840 4486 841 4796 4848 4899 495i 5003 5°54 5106 5*57 5209 5261 842 53i 2 5364 5415 5467 55i8 557° 5621 5 6 73 5725 5776 843 5828 5879 5931 5982 6034 6085 6137 6188 6240 6291 844 6342 6394 6445 6497 6548 6600 6651 6702 6754 6805 845 6857 6908 6959 701 1 7062 7114 7165 7216 7268 7319 846 737° 7422 7473 75M 7576 7627 7678 773° 7781 7832 847 7883 7935 7986 8037 8088 8140 8191 8242 8293 8345 848 8396 8447 8498 8549 8601 8652 8703 8754 8805 8857 849 8908 8959 9470 9010 9061 9572 9112 9' 6 3 9674 9215 9725 9266 9776 _9ji7 9827 9368 9879 850 929419 9521 9623 851 993° 9981 O032 °oS3 °i34 °i85 0236 0287 0338 0389 852 930440 0491 0542 0592 0643 0694 °745 0796 0847 0898 853 0949 1000 1051 1 1 02 1153 1204 1254 1305 1356 1407 854 1458 1509 1560 1610 1661 1712 1763 1814 1865 1915 855 1966 2017 2068 2118 2169 2220 2271 2322 2372 2423 856 2474 2524 2575 2626 2677 2727 2778 2829 2879 2930 857 2981 3031 3082 3 J 33 3183 3*34 3285 3335 3386 3437 858 3487 3538 3589 3 6 39 3690 3740 379 1 3841 3892 3943 859 3993 4°44 4549 4°94 4599 4'45 4650 4195 4246 4296 4801 4347 4852 4397 4902 4448 860 934498 4700 475i 4953 861 5003 5°54 5 io 4 5154 5 2 °5 5*55 5306 5356 5406 5457 862 5507 5558 5608 5658 5709 5759 5809 5860 5910 .5960 863 6011 6061 6111 6162 6212 6262 6 3 J 3 6363 6413 6463 864 65H 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 ~87T 9020 939519 9070 9569 9120 9610 9*7° 9669 9220 9270 _932o 9819 9369 9869 9419 9918 9469 9719 9769 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 *3*3 1362 1412 1462 874 1511 1561 1611 1660 1710 1760 1809 1859 1909 1958 875 2008 2058 2107 2157 2207 2256 2306 2355 2405 2455 876 2504 2554 2603 2653 2702 2752 2801 2851 2901 2950 877 3000 3°49 3099 3148 3198 3 2 47 3297 3346 339 6 3445 878 3495 3544 3593 3643 3692 3742 379i 3841 3890 3939 879 3989 1 4088 4H7 3 4186 4 2 3 6 5 4285 6 _433_5 7 4384 8 4433 N. 2 4 9 32 N. 880. LOGARITHMS. Log. 944. N. 880 1 2 3 4631 4 5 6 7 8 9 944483 453* 4581 4680 4729 4779 4828 4877 4927 881 4976 5° 2 5 5°74 5 I2 4 5*73 5222 5272 53^1 537o 54i9 882 5469 5518 5567 5616 5665 5715 5764 5813 5862 5912 883 5961 6010 6059 6108 6157 6207 6256 6 3°5 6354 6403 884 6452 6501 6 55I 6600 6649 6698 6747 6796 6845 6894 885 6943 6992 7041 7090 7140 7189 7238 7287 733 6 7385 886 7434 7483 753 2 758i 7630 7679 7728 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 9439 8999 9048 9536 9097 9146 9195 9683 9244 9292 9780 9341 890 949390 9488 9585 9 6 34 9731 9829 891 9878 9926 9975 O24 °°73 °I2I O170 °2I9 0267 0316 892 95°3 6 5 0414 0462 05 1 1 0560 0608 0657 0706 0754 0803 893 0851 0900 0949 0997 1046 1095 "43 1192 1240 1289 894 1338 1386 H35 1483 I53 2 1580 1629 1677 1726 1775 895 1823 1872 1920 1969 2017 2066 2114 2163 2211 2260 i 896 2308 2356 2405 2 453 2502 2 55° 2599 2647 2696 2744 j 897 2792 2841 2889 2938 2986 3034 3 o8 3 3!3i 3180 3228 1 898 3276 33 2 5 3373 3421 347° 35i8 3566 3615 3663 37ii ! 899 3760 3808 4291 3856 4339 3905 4387 3953 4001 4049 4098 4580 4146 4628 4194 4677 900 954243 4435 4484 453 2 901 4725 4773 4821 4869 4918 4966 5oi4 5062 5110 5158 902 5207 5255 53°3 535i 5399 5447 5495 5543 559 2 5640 903 5688 573 6 5784 5832 5880 5928 597 6 6024 6072 6120 904 6168 6216 6265 6313 6361 6409 6 457 6505 6 553 6601 905 6649 6697 6745 6 793 6840 6888 6936 6984 7032 7080 906 7128 7176 7224 7272 7320 7368 7416 7464 7512 7559 907 7607 7655 77°3 775 1 7799 7847 7894 7942 7990 8038 908 8086 8134 8181 8229 8277 8325 8373 8421 8468 8516 909 8564 8612 9089 8659 8707 9185 8755 8*03 9280 8850 9328 8898 9375 8946 9423 8994 910 959041 9137 9232 9471 911 9518 9566 9614 9661 9709 9757 9804 9852 9900 9947 912 9995 O42 °o9o "138 °i85 °233 °28o 0328 0376 °4 2 3 913 060471 o SINES AND TANGENTS. 178° M. Sine. Diff. 1" Cosine. Diff.l" .04 Tang. Diff. 1" Cotang. 60 8.241855 119.63 9-999934 8.241921 II9.67 II.758079 1 249033 117.68 93 2 249102 117.72 750898 59 2 256094 115.80 929 256165 115.84 743835 58 8 263042 113.98 927 263115 114.02 736885 57 4 269881 112. 21 925 269956 112,25 730044 56 5 276614 HO.50 922 276691 IIO.54 7233 9 55 6 283243 I08.83 920 283323 108.87 716677 54 V 289773 IO7.22 917 289856 107.26 710144 53 8 296207 IO5.65 9*5 296292 105.70 703708 52 9 302546 IO4.I3 912 302634 1 04. 1 8 697366 51 10 11 308794 8.314954 I02.66 IOI.22 910 308884 102.70 691116 50 49 9.999907 8.315046 101.26 11.684954 12 321027 99.82 905 321122 99.87 678878 48 13 327016 98.47 902 .04 327114 98.51 672886 47 14 332924 97.I4 899 .05 333° 2 5 97.19 * 666975 46 16 33 8 753 95 .86 897 338856 95.90 661144 45 16 3445°4 94.60 894 344610 94.65 655390 44 IV 350180 93-3 8 891 350289 93-43 64971 1 43 18 355783 92.19 888 355895 92.24 644105 42 iy 361315 91.03 885 361430 91.08 638570 41 20 21 366777 8.372171 89.90 882 366895 89.95 88.85 , 633105 40 39 88.80 9.999879 8.372292 II.627708 22 377499 87.72 876 377622 87.77 622378 38 23 382762 86.67 873 382889 86.72 617111 37 24 387962 85.64 870 388092 85.70 611908 36 25 393101 84.64 867 393 2 34 84.69 606766 35 26 398179 83.66 864 398315 83-71 601685 34 27 403199 82.71 861 403338 82.76 596662 33 28 408 1 61 81.77 858 408304 81.82 591696 32 29 413068 80.86 854 .05 41321^ 80.91 586787 31 30 31 417919 8.422717 79.96 79.09 851 .06 418068 80.02 581932 11.577131 30 29 9.999848 8.422869 79.14 32 427462 78.23 844 427618 78.29 572382 28 33 432156 77.40 841 43 2 3!5 77-45 567685 27 34 436800 76.57 838 436962 76.63 563038 26 35 441394 75-77 834 441560 75-83 558440 25 36 445941 74-99 831 446 1 1 75-°5 553890 24 3/ 450440 74.22 827 450613 74.28 549387 23 38 454 8 93 73-4 6 823 455070 73-52 54493° 22 39 4593oi 72.73 820 459481 72.79 540519 21 40 41 463665 8.467985 72.00 816 9.999813 463849 8.468173 72.06 536151 20 19 71.29 71-35 II. 531827 42 472263 70.60 809 472454 70.66 52754 6 18 43 476498 69.91 805 476693 69.98 523307 17 44 480693 69.24 801 .06 480892 69.31 519108 16 45 484848 68.59 797 .07 485050 68.65 5 J 495o 15 46 488963 67.94 793 489170 68.01 510830 14 47 493040 67.31 79° 493250 67.38 506750 13 48 497078 66.69 786 497 2 93 66.76 502707 12 49 501080 66.08 782 501298 66.15 498702 11 50 51 505045 8.508974 65.48 778 505267 8.509200 6 5-55 494733 11.490800 10 9 64.89 9-999774 64.96 52 512867 64.32 769 513098 64.39 486902 8 53 516726 63.75 765 516961 63.82 483039 7 54 520551 63.19 761 520790 63.26 479210 6 55 5M343 62.64 757 524586 62.72 4754H 5 56 528102 62.11 753 528349 62.18 471651 4 57 531828 61.58 748 532080 61.65 467920 3 58 535523 61.06 744 535779 61.13 464221 2 59 539186 60.55 74° •07 539447 60.62 460553 1 60 8.542819 Cosine. Diff. 1" 9-999735 Sine. 8.543084 Cotang. 11.456916 Tang. M. Diff.l" Diff. 1" 1 J n° 88° 43 2< i LOGARITHMIC 177° M. Sine. Diff. 1" Cosine. Diff.l" .07 Tang. Diff. 1" Cotang. ~60 _ 8.542819 60.04 9-999735 8.543084 60.12 11.456916 1 46422 59-55 73 1 .07 46691 59.62 533°9 59 2 49995 59.06 726 .07 50268 59.14 49732 58 3 53539 58.58 722 .08 53 8l 7 58.66 46183 57 4 57054 58.11 717 5733 6 58.19 42664 56 5 60540 57-65 713 60828 57-73 39172 55 6 63999 57-19 708 64291 57.27 35709 54 7 6743 1 56.74 704 67727 56.82 32273 53 8 70836 56.30 699 71137 56.38 28863 52 y 74214 55-87 694 74520 55-95 25480 51 10 n 77566 55-44 689 9.999685 77877 55-52 22123 50 49 8.580892 55.02 8.581208 55-io 11.418792 12 84193 54.60 680 84514 54.68 15486 48 13 87469 54-19 675 87795 54-27 12205 4JT 14 90721 53-79 670 91051 53-87 08949 46 15 93948 53-39 665 94283 53-47 05717 45 16 8.597152 53-°° 660 8.597492 53.08 11.402508 44 17 8.600332 52.61 655 8.600677 52.70 11.399323 43 18 03489 52.23 650 .08 03839 52.32 96161 42 19 06623 51.86 645 .09 06978 51.94 93022 41 20 21 09734 51.49 640 10094 51.58 89906 40 39 8.612823 51.12 9-999 6 35 8.613189 51.21 11.386811 22 15891 50.76 629 16262 50.85 83738 38 23 18937 50.41 624 i93 J 3 50.50 80687 37 24 21962 50.06 619 22343 50.15 77657 36 25 24965 49.72 614 25352 49.81 74648 35 26 27948 49.38 608 28340 49-47 71660 34 27 30911 49.04 603 31308 49-13 68692 33 28 33 8 54 48.71 597 3425 6 48.80 65744 62816 32 29 36776 4 8 -39 592 37184 48.48 31 30 31 39680 8.642563 48.06 586 40093 48.16 59907 30 29 47-75 9.999581 8.642983 47.84 11. 357017 32 45428 47-43 575 45853 47-53 54H7 28 33 48274 47.12 57o 48704 47.22 51296 27 34 51102 46.82 564 .09 5 J 537 46.91 48463 26 35 539 11 46.52 558 .10 54352 46.61 45648 25 36 56702 46.22 553 57H9 46.31 42851 24 37 59475 45.92 547 59928 46.02 40072 23 38 62230 45.63 54i 62689 45-73 373" 22 39 64968 45-35 535 65433 45-44 34567 21 40 41 67689 45.06 529 68160 45.16 31840 20 19 8.670393 44-79 9.999524 8.670870 44.88 11. 329130 42 73080 44.51 518 73563 44.61 26437 18 43 75751 44.24 512 76239 44-34 23761 17 44 78405 43-97 506 78900 44.07 21 100 16 45 81043 43.70 500 81544 43.80 18456 15 46 83665 43-44 493 84172 43-54 15828 14 47 86272 43.18 487 86784 43.28 13216 13 48 88863 42.92 481 89381 43-°3 10619 12 49 91438 42.67 475 91963 42.77 08037 11 50 51 93998 9 6 543 42.42 469 .10 .11 94529 42.52 o547i 10 9 42.17 9.999463 97081 42.28 02919 52 8.699073 41.92 456 8.699617 42.03 11.300383 8 53 8.701589 41.68 450 8.702139 41.79 11. 297861 7 54 04090 41.44 443 04646 41-55 95354 6 00 06577 41.21 437 07140 4.1.32 92860 5 56 09049 40.97 431 09618 41.08 90382 4 57 1 1 507 40.74 424 12083 40.85 87917 3 58 I395 2 40.51 418 14535 40.62 85465 2 59 16383 40.29 411 .11 16972 40.40 83028 1 60 8.718800 9.999404 Diff.l" 8.719396 Cotang. 11.280604 Tang. M. Cosine. Diff. 1" Sine. Diff. 1" ! )2° 87° 44 3 3 SINES AND TANGENTS, 176° M. Sine. DiflF. 1" Cosine. Diff.l" .11 Tang. Diff. 1" Cotang. II.280604 60 8.718800 40.06 9.999404 8.719396 40.17 1 21204 39- 8 4 9398 21806 39-95 78194 59 2 23595 39.62 9391 24203 39-74 75797 58 8 25972 39-41 9384 26588 39-52 73412 57 4 28337 39- J 9 9378 28959 39-3i 71041 56 5 30688 38.98 9371 .11 3I3I7 39-°9 68683 55 6 33027 3 8 -77 9364 .12 33663 38.89 66337 54 7 35354 3f.57 9357 35996 38.68 64004 53 8 37667 38.36 935° 3 8 3 J 7 38.48 61683 52 9 39969 38.16 9343 40626 38.27 59374 51 10 11 42259 37.96 9336 42922 38.07 57078 50 49 8.744536 37-76 9.999329 8.745207 37-87 "•254793 12 46802 37.56 9322 47479 37.68 52521 48 13 49°55 37-37 93*5 49740 37-49 50260 47 14 5 I2 97 37-17 9308 51989 37-29 48011 46 15 535 28 36.98 9301 54 22 7 37.10 45773 45 16 55747 3 6 -79 9294 56453 36.92 43547 44 17 57955 36.61 9286 58668 3 6 -73 41332 43 18 60151 36.42 9279 60872 36-55 39128 42 19 62337 36.24 9272 63065 36-36 36935 41 20 21 645 1 1 36.06 9265 65246 8.767417 36.18 . 34754 40 39 8.766675 35.88 9.999257 .12 36.00 11.232583 22 68828 35-7° 9250 •13 69578 35-83 30422 38 28 70970 35-53 9242 71727 35-65 28273 37 24 73101 35-35 9235 73866 35-48 26134 36 25 75223 35-i8 9227 75995 35-31 24005 35 26 77333 35-°i 9220 78114 35-H 21886 34 27 79434 34.84 9212 80222 34-97 19778 33 28 81524 34-67 9205 82320 34.80 17680 32 1 29 83605 34-5i 9197 84408 34.64 15592 31 30 31 85675 34-34 9189 86486 8.78*554 34-47 I35H 30 29 8.787736 34.18 9.999181 34-31 11.211446 32 89787 34.02 9 J 74 90613 34-15 09387 28 33 91828 33.86 9166 92662 33-99 07338 27 34 93 8 59 33-7° 9158 94701 33-83 05299 26 85 95881 33-54 9150 96731 33-68 03269 25 36 97894 33-39 9142 8.798752 33-52 11. 201248 24 87 8.799897 33-2-3 9 J 34 8.800763 33-37 11. 199237 23 38 8.801892 33.08 9126 02765 33.22 97235 22 39 03876 32-93 9118 04758 33-°7 95242 21 40 41 05852 32.78 9110 06742 32.92 93258 11.191283 20 19 8.807819 32.63 9.999102 •13 8.808717 32-77 42 09777 3 2 -49 9094 .14 10683 32.62 893^ 18 43 11726 3 2 -34 9086 12641 32.48 87359 17 44 13667 32.19 9077 14589 32-33 85411 16 45 15599 32.05 9069 16529 32.19 83471 15 46 17522 31.91 9061 1 846 1 32.05 8i539 14 47 19436 3 J -77 9053 20384 31.91 79616 13 48 21343 31.63 9044 22298 3 x -77 77702 12 49 23240 31.49 9036 24205 31.63 75795 11 50 51 25130 3 J -35 31.22 9027 26103 31-5° 73897 11. 172008 10 9 8.827011 9.999019 8.827992 31.36 52 28884 31.08 9010 29874 31.23 70126 8 53 3°749 3°-95 9002 3^48 31.09 68252 7 54 32607 30.82 8993 33613 30.96 66387 6 55 3445 6 30.69 8984 3547i 30.83 64529 5 56 36297 30.56 8976 .14 3732i 30.70 62679 4 57 38130 3°-43 8967 •15 39163 3°-57 60837 3 58 39956 3°-3° 8958 •15 40998 3°-45 59002 2 59 41774 30.17 8950 ■15 42825 3°-3 2 57175 1 60 8.843585 9.998941 Sine. Diff.l" 8.844644 Diff. 1" 11. 155356 M. Cosine. Diff. 1" Cotang. Tang. 9 3° 86° 45 4: C > XiOaARXTHMIC 175° M. Sine. Diff. 1" 1 Cosine. Diff.l" Tang. Diff. 1" Cotang. 60 8.843585 30.05 9.998941 •15 8.844644 30.19 U-I5535 6 i 453 8 7 29.92 932 46455 30.07 53545 59 2 47183 29.80 923 48260 29.95 5i74o 58 | 3 48971 29.67 914 5°°57 29.82 49943 57 | 4 5°75i 29.55 905 51846 29.70 48i54 56 i 5 52525 29.43 896 53628 29.58 46372 55 6 54291 29.31 887 55403 29.46 44597 54 V 56049 29.19 878 57i7i 29.35 42829 53 8 57801 29.08 869 58932 29.23 41068 52 9 59546 28.96 860 60686 29.II 393 J 4 51 10 11 61283 28.84 851 62433 29.00 37567 50 49 8.863014 28.73 9.998841 8.864173 28.88 11. 135827 12 64738 28.61 832 •15 65906 28.77 34094 48 13 66455 28.50 823 .16 67632 28.66 32368 47 14 68165 28.39 813 6935 1 28.54 30649 46 15 69868 28.28 804 71064 28.43 28936 45 16 71565 28.17 795 72770 28.32 27230 44 17 73^55 28.06 785 74469 28.21 25531 43 18 74938 27.95 776 76162 28.11 23838 42 19 76615 27.84 766 77849 28.00 22151 41 2U 21 78285 27.73 757 79529 8.881202 27.89 20471 40 39 8.879949 27.63 9.998747 27.79 11.118798 22 81607 27.52 738 82869 27.68 17131 38 23 83258 27.42 728 8453o 27.58 15470 37 24 84903 27.31 718 86185 27.47 13815 36 25 86542 27.21 708 87833 27-37 12167 35 26 88174 27.II 699 89476 27.27 10524 34 27 89801 27.00 689 91112 27.17 08888 33 28 91421 26.90 679 .16 92742 27.07 07258 32 29 93°35 26.80 669 •17 94366 26.97 05634 31 30 ~31 94643 26.70 659 95984 26.87 26.77 04016 30 29 96245 26.60 9.998649 97596 02404 32 97842 26.51 639 8.899203 26.67 11. 100797 28 33 8.899432 26.41 629 8.900803 26.58 11. 099197 27 34 8.901017 26.31 619 02398 26.48 97602 26 35 02596 26.22 609 03987 26.38 96013 25 36 04169 26.12 599 05570 26.29 9443° 24 37 05736 26.03 589 07147 26.20 92853 23 38 07297 25-93 578 08719 26.10 91281 22 39 08853 25.84 568 10285 26.01 89715 21 40 41 10404 8.911949 25-75 25.66 558 1 1 846 25.92 88154 11.086599 20 19 9.998548 8.913401 25.83 42 13488 25-56 537 14951 25.74 85049 18 43 15022 25.47 527 •17 16495 25.65 83505 17 44 16550 25-38 516 .18 18034 25.56 81966 16 45 18073 25.29 506 19568 25.47 80432 15 46 19591 25.20 495 21096 25.38 78904 14 4V 21103 25.12 485 22619 25.30 7738i 13 48 22610 25.03 474 24136 25.21 75864 12 49 24112 24.94 464 25649 25.12 74351 11 50 51 25609 8.927100 24.86 24.77 453 27156 8.928658 25.03 24-95 72844 11-071342 10 9 9.998442 52 28587 24.69 43i 3oi55 24.86 69845 8 53 30068 24.60 421 3 l6 47 24.78 68353 7 54 3*544 24.52 410 33*34 24.70 66866 6 55 33 OI 5 24.43 399 34616 24.61 65384 5 56 34481 24-35 388 36093 24-53 63907 4 57 3594 2 24.27 377 37565 24.45 62435 3 58 3739 8 24.19 366 39032 24.37 60968 2 59 38850 24.11 355 .18 40494 24.29 595o6 1 60 8.940296 Cosine. 9.998344 Sine. Diff.l" 8.941952 11.058048 M. Diff. 1" Cotang. Diff. 1" Tang. 9- 4° ■ 85° 46 5 SINES AND TANGENTS. 174° M. Sine. Diff. 1" Cosine. Diff.l" Tang. Diff. 1" Cotang. 60 8.940296 24.03 9.998344 .19 8.941952 24.21 II.058048 1 41738 23.94 333 43404 24.13 56596 59 2 43 J 74 23.87 322 44852 24-05 55148 58 3 44606 23.79 3 11 46295 23-97 53705 5/ 4 4 6o 34 23.71 300 47734 23.90 52266 56 6 4745 6 23-63 289 49168 a3.8a 50832 55 6 48874 ^3-55 277 5°597 23-74 49403 54 7 50287 23.48 266 52021 33.66 47979 53 8 51696 23.40 255 53441 23-59 4 6 559 52 y 53100 23-3 2 243 54856 23-5I 45 J 44 51 10 n 54499 8.955894 23.25 232 56267 23-44 43733 50 49 23.17 9.998220 8.957674 23-37 11.042326 12 57284 a3.r0 209 59°75 33.39 40925 48 13 58670 23.02 197 60473 23.22 395 2 7 47 14 60052 22.95 186 61866 23.14 38i34 46 15 61429 22.88 174 63255 23.07 36745 45 16 62801 22.80 163 64639 23.00 3536i 44 17 64170 22.73 151 .19 66019 22.93 3398i 43 18 65534 22.66 139 .20 67394 22.86 32606 42 19 66893 aa.59 128 68766 22.79 3 I2 34 41 20 i 21 68249 8.969600 22.52 116 7oi33 22.71 29867 40 39 22.45 9.998104 8.971496 22.65 11.028504 22 70947 22.38 092 72855 22.57 27145 38 23 72289 22.31 080 74209 22.51 25791 37 24 73628 22.24 068 7556o 22.44 24440 36 25 74962 22.17 056 76906 22.37 23094 35 26 76293 aa.io 044 78248 22.30 21752 34 27 77619 aa.o3 032 79586 22.23 20414 33 28 78941 21-97 020 80921 22.17 19079 32 29 80259 ai.90 9.998008 82251 22.10 17749 31 30 31 81573 21.83 9.997996 83577 22.04 16423 30 29 8.982883 21-77 984 8.984899 21.97 11.015101 32 84189 ai.70 972 86217 21.91 13783 28 33 85491 ai.63 959 87532 21.84 12468 27 34 86789 21-57 947 .20 88842 21.78 11158 26 35 88083 ai.50 935 .21 90149 21.71 09851 25 36 8 9374 a 1. 44 922 91451 21.65 08549 24 37 90660 21.38 910 92750 21.58 07250 23 38 9 J 943 21.31 897 94045 21.52 05955 22 39 93222 21.25 885 95337 21.46 04663 21 40 41 94497 21.19 21.12 872 96624 21.40 03376 20 19 8.995768 9.997860 97908 21.34 02092 42 97036 21.06 847 8.999188 21.27 11. 000812 18 43 98299 ai.oo 835 9.000465 21.21 io-999535 17 44 8.999560 ao.94 822 01738 21.15 98262 16 45 9.000816 ao.88 809 03007 21.09 96993 15 46 02069 ao.82 797 04272 21.03 95728 14 47 03318 20.76 784 05534 20.97 94466 13 48 04563 20.70 771 06792 20.91 93208 12 49 05805 20.64 758 08047 20.85 9*953 11 50 51 07044 ao.58 745 09298 20.80 90702 10 9 9.008278 20.52 9.997732 9.010546 20.74 10.989454 52 09510 20.46 719 11790 20.68 88210 8 53 10737 20.40 706 .ai 13031 20.62 86969 7 54 11962 20.34 693 .a2 14268 20.56 85732 6 55 1318a 30.29 680 1550a 20.51 84498 5 56 14400 20.23 667 16732 20.45 83268 4 57 15613 20.17 654 17959 20.40 82041 3 58 16824 20.12 641 19183 20.34 80817 2 ) 59 18031 20.06 628 .22 20403 20.28 79597 1 1 60 i 9.019235 9.997614 9.021620 10.978380 M. Cosine. Diff. 1" Sine. Diff.l" Cotang. Diff. 1" Tang. I 9. 5° 84° J 47 6 3 LOGARITHMIC j 173° M. Sine. Diff. 1" Cosine. Diff.l" .22 Tans. Diff. 1" Cotang. IO.978380 60 9.019235 20.00 9.997614 9.021620 20.23 1 20435 19.95 601 22834 20.17 77166 59 2 21632 19.89 588 24044 20.11 75956 58 3 22825 19.84 574 25251 20.06 74749 57 4 24016 19.78 561 26455 20.01 73545 56 5 25203 19-73 547 .22 27655 19.95 72345 55 6 26386 19.67 534 •23 28852 19.90 71148 54 7 27567 19.62 520 30046 19.85 69954 53 8 28744 19-57 5°7 31237 19.79 68763 52 9 29918 I9-5I 493 3 2 4 2 5 19.74 67575 51 10 11 31089 19.46 480 9.997466 33609 19.69 66391 50 49 9.032257 19.41 9.034791 19.64 10.965209 12 334 2 i 19.36 45* 35969 19.58 64031 48 13 345 82 19.30 439 37H4 19-53 62856 47 14 35741 19.25 425 38316 19.48 61684 46 15 36896 19.20 411 39485 19-43 60515 45 16 38048 19.15 397 40651 19.38 59349 44 IV 39197 19.10 383 41813 19-33 58187 43 18 40542 19.05 3 6 9 42973 19.28 57027 42 19 41485 18.99 355 44I3 19.23 55870 41 20 21 42625 18.95 34i •23 45284 19.18 547i6 40 39 9.043762 18.89 9.997327 .24 9.046434 19.13 10.953566 22 44895 18.84 3*3 47582 19.08 52418 38 23 46026 18.79 299 48727 19.03 51273 37 24 47154 18.75 285 49869 18.98 50131 36 25 48279 18.70 271 51008 18.93 48992 35 26 49400 18.65 257 52144 18.89 47856 34 27 5 5i9 18.60 242 53277 18.84 46723 33 28 5 l6 35 18.55 228 544°7 18.79 45593 32 29 5 2 749 18.50 214 55535 18.74 44465 31 30 31 53 8 59 18.45 199 56659 18.70 4334 1 30 29 | 9.054966 18.41 9.997185 9.057781 18.65 10.942219 32 56071 18.36 170 58900 18.60 41100 28 33 57*72 18.31 156 60016 18.55 39984 27 34 58271 18.27 141 61130 18.51 38870 26 35 593 6 7 18.22 127 62240 18.46 37760 25 36 60460 18.17 112 63348 18.42 36652 24 37 61551 18.13 098 .24 64453 18.37 35547 23 38 62639 18.08 083 •25 65556 18.33 34444 22 39 63724 18.04 068 66655 18.28 33345 21 40 41 64806 17.99 053 67752 18.24 32248 20 19 9.065885 17.94 9.997039 9.068846 18.19 10.931154 42 66962 17.90 024 69938 18.15 30062 18 43 68036 17.86 9.997009 71027 18.10 28973 17 44 69107 17.81 9.996994 72113 18.06 27887 16 45 70176 17.77 979 73 J 97 18.02 26803 15 46 71242 17.72 964 74278 17.97 25722 14 47 72305 17.68 949 75356 17.93 24644 13 48 73366 17.63 934 76432 17.89 23568 12 49 74424 17-59 919 77505 17.84 22495 11 50 "51 75480 9.076533 17-55 904 78576 17.80 21424 10 9 17-5° 9.-996889 9.079644 17.76 10.920356 52 775 8 3 17.46 874 80710 17.72 19290 8 53 78631 17.42. 858 8i773 17.67 18227 7 54 79676 I7-38 843 82833 17.63 17167 6 55 80719 17-33 828 .25 83891 17-59 16109 5 56 8i759 17.29 812 .26 84947 17-55 15053 4 57 82797 17.25 797 86000 17.51 14000 3 58 83832 17.21 782 87050 17-47 12950 2 59 84864 17.17 766 .26 88098 17-43 11902 1 60 9.085894 9.996751 Diff.l" 9.089144 Cotang. 10.910856 Tang. M. Cosine. Diff. 1" Sine. Diff. 1" 9 6° 83° 48 7 D SINES AND TANGENTS. 172° M. Sine. Diff. 1" Cosine. Diff.l" Tang. Diff. 1" Cotang. 60 9.085894 I7-I3 9.996751 .26 9.089144 I7-38 IO.910856 i 86922 17.09 735 90187 17-35 09813 59 2 8 7947 17.04 720 91228 17.30 08772 58 3 88970 17.00 704 92266 17.27 07734 57 4 89990 16.96 688 93302 17.22 06698 56 5 91008 16.92 673. 94335 17.19 05664 55 6 92024 16.88 657 95367 17.15 04633 54 7 93037 16.84 641 96395 17.11 03605 53 8 94047 16.80 625 97422 17.07 02578 52 9 95056 16.76 610 98446 17.03 01554 51 10 11 96062 16.73 594 9.996578 .26 99468 16.99 IO.900532 50 49 9.097065 16.68 .27 9.100487 16.95 IO.899513 12 98066 16.65 562 01504 16.91 98496 48 13 9.099065 16.61 546 02519 16.87 97481 47 14 9.100062 16.57 53o 03532 16.84 96468 46 15 01056 16.53 514 04542 16.80 95458 45 16 02048 16.49 498 05550 16.76 94450 44 17 03037 16.45 482 06556 16.72 93444 43 18 04025 16.42 465 °7559 16.69 92441 42 iy 05010 16.38 449 08560 16.65 91440 41 20 21 °599a 9.106973 16.34 433 °9559 9.110556 16.61 ' 90441 40 39 16.30 9.996417 16.58 10.889444 ■ 22 07951 16.27 400 11551 16.54 88449 38 23 08927 16.23 384 12543 16.50 87457 37 24 09901 16.19 368 13533 16.47 86467 36 25 10873 16.16 35i 14521 16.43 85479 35 26 1 1 842 16.12 335 15507 16.39 84493 34 27 12809 16.08 318 .27 16491 16.36 83509 33 28 13774 16.05 302 .28 17472 16.32 82528 32 29 14737 16.01 285 18452 16.29 81548 31 30 I- — 31 15698 15-97 269 19429 16.25 80571 30 29 9.116656 15.94 9.996252 9.120404 16.22 10.879596 ! 32 17613 15.90 235 21377 16.18 78623 28 1 33 18567 15.87 219 22348 16.15 77652 27 i 34 19519 15.83 202 23317 16.11 76683 26 35 20469 15.80 185 24284 16.08 75716 25 36 21417 15.76 168 25249 16.04 74751 24 37 22362 15-73 151 26211 16.01 73789 23 38 23306 15.69 134 27172 15-97 72828 22 39 24248 15.66 117 28130 15.94 71870 21 40 41 25187 15.62 100 .28 29087 15.91 70913 20 19 9.126125 15-59 9.996083 .29 9.130041 15.87 10.869959 42 27060 15.56 066 3°994 15.84 69006 18 43 27993 15.52 049 3*944 15.81 68056 17 44 28925 15.49 032 32893 15-77 67107 16 45 29854 15-45 9.996015 33 8 39 15-74 66161 15 46 30781 15.42 9.995998 34784 I5-7I 65216 14 47 31706 15-39 980 35726 15.67 64274 13 48 32630 15-35 963 36667 15.64 63333 12 49 33551 15.32 946 37605 15.61 62395 11 : 50 i 51 , 3447° 9- I 353 8 7 15.29 928 3 8 542 15.58 61458 10.860524 10 9 15-^5 9.995911 9.139476 15-55 o2 36303 15.22 894 40409 15.51 5959 1 8 53 37216 I5-I9 876 4 X 34° 15.48 58660 7 54 38128 15.16 859 42269 15-45 5773 1 6 1 55 39037 15.12 841 43196 15.42 56804 5 56 39944 15.09 823 441 2 1 '5-39 55879 4 57 40850 15.06 806 45°44 15-35 54956 3 58 41754 i5- 3 788 45966 I5-3 2 54034 2 59 42655 15.00 771 .29 46885 15.29 53II5 1 60 9-H3555 Cosine. 9-995753 9.147803 10.852197 M. Diff. 1" Sine. Diff.l" Cotang. Diff. 1" Tang. 9 7° 82° 49 8 J LOGARITHMIC 171° | M. Sine. Diff. 1" Cosine. Diff.l" Tang. 9.147803 Diff. 1" Cotang. 60 | 9-*43555 14.96 9-995753 .30 15.26 IO.852197 1 4453 14.93 735 8718 i5- 2 3 1282 59 j 2 5349 14.90 717 9.149632 15.20 IO.850368 58 1 8 6243 14.87 699 9.150544 15.17 10.849456 57 | 4 7136 14.84 681 1454 15.14 8546 56 5 8026 14.81 664 2363 15.11 7637 55 6 8915 14.78 646 3269 15.08 6731 54 7 9.149802 14-75 628 4174 i5-o5 5826 53 8 9.150686 14.72 610 5°77 15.02 4923 52 y 1569 14.69 59i 5978 14.99 4022 51 10 n 2451 14.66 573 6877 14.96 3123 50 49 9- I 5333° 14.63 9-995555 9- J 57775 14.93 10.842225 12 4208 14.60 537 8671 14.90 1329 48 13 5083 14-57 519 .30 9- J 595 6 5 14.87 10.840435 47 14 5957 14-54 501 •31 9.160457 14.84 IO.839543 46 lb 6830 H-5i 482 1347 14.81 8653 45 16 7700 14.48 464 2236 14.78 7764 44 17 8569 14.45 446 3123 14-75 6877 43 18 9-159435 14.42 427 4008 14-73 599 2 42 iy 9.160301 14-39 409 4892 14.70 5108 41 20 21 1 1 64 14.36 390 5774 9.166654 14.67 4226 40 39 9.162025 14-33 9-99537* 14.64 10.833346 22 2885 14.30 353 753 2 14.61 2468 38 23 3743 14.27 334 8409 14.58 1591 37 24 4600 14.24 316 9.169284 14-55 10.830716 36 2o 5454 14.22 297 9.170157 14-53 10.829843 35 26 6307 14.19 278 1029 14.50 8971 34 27 7i59 14.16 260 •31 1899 14.47 8101 33 28 8008 14.13 241 .32 2767 14.44 72-33 32 2y 8856 14.10 222 3 6 34 14.42 6366 31 30 31 9.169702 14.07 203 4499 14-39 55oi 10.824638 30 29 9- J 7°547 14.05 9.995184 9.175362 14.36 32 1389 14.02 165 6224 14-33 3776 28 33 2230 13.99 146 7084 H-3 1 2916 27 34 3070 13.96 127 7942 14.28 2058 26 35 3908 13-94 108 8799 14.25 1201 25 36 4744 13.91 089 9.179655 14.23 10.820345 24 87 5578 13.88 070 9.180508 14.20 10.819492 23 38 6411 13.86 051 1360 14.17 8640 22 3y 7242 13.83 032 2211 14.15 7789 21 40 41 8072 13.80 9.995013 3°59 14.12 6941 10.816093 20 19 8900 13-77 9.994993 9.183907 14.09 42 9.179726 13-74 974 475* 14.07 5248 18 43 9.180551 13.72 955 5597 14.04 44°3 17 44 1374 13.69 935 .32 6 439 14.02 356i 16 45 2196 13.67 916 •33 7280 13-99 2720 15 46 3016 13.64 896 8120 13.96 1880 14 47 3834 13.61 877 8958 13-94 1042 13 48 4651 13-59 857 9.189794 I3-9I 10.810206 12 4y 5466 I3-5 6 838 9.190629 13.89 10.809371 11 50 6280 13-53 818 1462 13.86 13.84 8538 10.807706 10 9 51 9.187092 I3-5 1 9.994798 9.192294 52 79°3 13.48 • 779 3 I2 4 13.81 6876 8 53 8712 13.46 759 3953 13-79 6047 7 54 9.189519 13-43 739 4780 13.76 5220 6 55 9.190325 13.41 719 5606 13-74 4394 5 56 1130 I3-38 700 6430 13.71 3570 4 1 57 1933 13.36 68c 7253 13.69 2747 3 58 2734 13-33 660 8074 13.66 1926 2 59 3534 13.30 640 •33 8894 13.64 1106 1 60 9.194332 9.994620 9.199713 10.800287 Tang. M. Cosine. Diff. 1" Sine. Diff.]" Cotang. Diff. 1" 98° 81° 50 i 9° SXCTES AND TANaENTS. 170° M. Sine. Diff. 1" Cosine. Diff.l" Tang. Diff. 1" Cotang. j 60 9.194332 13.28 9.994620 •33 9-I997I3 13.61 10.800287I 1 5129 13.26 600 •33 9.200529 13-59 10.799471 59 2 59^5 13.23 580 •33 1345 I3-5 6 8655 58 3 6719 13.21 560 •34 2159 13-54 7841 57 4 75" 13.18 540 2971 I3-52 7029 56 6 8302 13.16 5 J 9 3782 13-49 6218 55 6 9091 I3-I3 499 4592 r 3-47 5408 54 . V 9.199879 13.11 479 5400 13-45 4600 53 8 9.200666 13.08 459 6207 13.42 3793 52 y 1451 13.06 438 7013 13.40 2987 51 1U n 2234 13.04 418 7817 13.38 2183 50 49 9.203017 13.01 9-994397 8619 13-35 1 38 1 12 3797 12.99 377 9.209420 13-33 10.790580 48 13 4577 12.96 357 9.210220 I 3-3 I 10.789780 47 14 5354 12.94 33 6 1018 13.28 8982 46 15 6131 12.92 316 1815 13.26 8185 45 16 6906 12.89 295 •34 2611 13.24 7389 44 17 7679 12.87 274 •35 34°5 13.21 6595 43 18 8452 12.85 254 4198 I3-I9 5802 42 19 9222 12.82 233 4989 I3-J7 5011 41 20 21 9.209992 12.80 212 5780 i3-!5 4220 10.783432 40 39 9.210760 12.78 9.994191 9.216568 13.12 22 1526 12.75 171 7356 13.10 2644 38 28 2291 12.73 150 8142 13.08 1858 37 24 3°55 12.71 129 8926 i3-°5 1074 36 25 3818 12.68 108 9.219710 13.03 10.780290 35 26 4579 12.66 087 9.220492 13.01 10.779508 34 27 5338 12.64 066 1272 12.99 8728 33 28 6097 12.61 045 2052 12.97 7948 32 29 6854 12.59 024 2830 12.94 7170 31 30 31 7609 9.218363 12.57 9.994003 3606 12.92 6394 30 29 12.55 9.993981 9.224382 12.90 10.775618 32 9116 i 2 -53 960 5i5 6 12.88 4844 28 33 9.219868 12.50 939 5929 12.86 4071 27 34 9.220618 12.48 918 •35 6700 12.84 3300 26 35 1367 12.46 896 .36 747i 12.81 2529 25 36 2115 12.44 875 8239 12.79 1761 24 37 2861 12.42 854 9007 12.77 0993 23 38 3606 12.39 832 9.229773 12.75 10.770227 22 39 4349 12.37 811 9.230539 12.73 10.769461 21 40 41 5092 is-35 12.33 789 1302 12.71 8698 10.767935 20 9.225833 9.993768 9.232065 12.69 42 6573 12.31 746 2826 12.67 7174 18 43 7311 12.28 725 3586 12.65 6414 17 44 8048 12.26 703 4345 12.62 5 6 55 16 45 8784 12.24 681 Sio3 12.60 4897 15 46 9.229518 12.22 660 5859 12.58 4141 14 47 9.230252 12.20 638 6614 12.56 3386 13 48 0984 12.18 616 •36 7368 12.54 2632 12 49 1714 12.16 594 •37 8120 12.52 1880 11 50 _ 51 2444 12.14 572 8872 12.50 1128 10.760378 10 ~~ 9~ 9.233172 12.12 9-99355° 9.239622 12.48 52 3 8 99 12.09 528 9.240371 12.46 10.759629 8 63 4625 12.07 506 1118 12.44 8882 7 54 5349 12.05 484 1865 12.42 8135 6 5o 6073 12.03 462 2610 12.40 7390 5 56 6795 12.01 440 3354 12.38 6646 4 57 7515 11.99 418 4097 12.36 59°3 3 58 8235 11.97 396 4839 12.34 5161 2 1 59 8953 11.95 374 •37 5579 12.32 4421 1 1 60 9.239670 9-993351 9.246319 10.753681 Tang. M. Cosine. Diff. 1" Sine. Diff.l" Cotang. Diff. 1" 99° 80° i 51 1 10° LOOARXTHXftlC 169° M. Sine. Diff. 1" Cosine. Diff.l" Tang:. Diff. 1" Cotang. 60 9.239670 "•93 9-993351 •37 9.246319 12.30 IO.753681 1 9.240386 H.91 3 2 9 7057 12.28 2 943 59 ! 2 IIOI II.89 3°7 7794 12.26 2206 58 ■ 3 1814 II.87 285 8530 12.24 1470 57 4 2526 II.85 262 9264 12.22 0736 56 5 3*37 II.83 240 •37 9.249998 I2.20 10.750002 55 6 3947 II.81 217 •38 9.250730 I2.I8 IO.749270 54 7 4656 II.79 195 1461 12.17 8539 53 8 53 6 3 II.77 172 2191 12.15 7809 52 y 6069 II.75 149 2920 12.13 7080 51 10 n 6775 "•73 127 3648 12. II 6352 50 49 9.247478 11.71 9.993104 9- 2 54374 I2.O9 IO.745626 12 8181 11.69 081 5100 I2.O7 4900 48 13 8883 11.67 059 5824 I2.O5 4176 47 14 9.249583 11.65 036 6547 I2.03 3453 46 15 9.250282 11.63 9.993013 7269 I2.0I 2731 45 16 0980 11.61 9.992990 7990 I2.00 2010 44 17 1677 11.59 967 8710 II.98 1290 43 18 2373 11.58 944 9.259429 II.96 10.740571 42 19 3067 11.56 921 9.260146 II.94 10.739854 41 20 21 3761 11.54 898 0863 II.92 9*37 40 39 9.254453 11.52 9.992875 9.261578 II.90 10.738422 22 5 J 44 11.50 852 .38 2292 II.89 7708 38 2'6 5834 11.48 829 •39 3005 II.87 6995 37 24 6523 11.46 806 3717 II.85 6283 36 25 7211 11.44 783 4428 II.83 5572 35 26 7898 11.42 759 5138 II.8I 4862 34 27 8583 11.41 736 5847 11.79 4153 33 28 9268 "•39 713 6555 II.78 3445 32 29 9.259951 n-37 690 7261 II.76 2739 31 30 31 9.260633 n-35 666 7967 II.74 2033 30 29 I3H n-33 9.992643 8671 II.72 1329 32 1994 11. 31 619 9.269375 II.70 10.730625 28 33 2673 11.30 596 9.270077 II.69 10.729923 27 34 335 1 11.28 572 0779 II.67 9221 26 35 4027 11.26 549 1479 II.65 8521 25 36 4703 11.24 525 2178 II.64 7822 24 37 5377 11.22 501 •39 2876 11.62 7124 23 38 6051 11.20 478 .40 3573 II.60 6427 22 39 6723 11. 19 454 4269 II.58 573 1 21 40 41 7395 11. 17 430 4964 11.57 5036 20 19 9.268065 11. 15 9.992406 9.275658 11.55 10.724342 42 8734 11. 13 382 6351 "•53 3649 18 43 9.269402 11. 12 359 7043 11. 51 2957 17 44 9.270069 11. 10 335 7734 11.50 2266 16 45 0735 11.08 3" 8424 11.48 1576 15 46 1400 11.06 287 9"3 11.46 0887 14 47 2064 11.05 263 9.279801 11.45 10.720199 13 48 2726 11.03 239 9.280488 11.43 10.719512 12 49 3388 11. 01 214 1174 11.41 8826 11 50 51 4049 10.99 190 9.992166 1858 11.40 8142 10 9 9.274708 10.98 9.282542 11.38 10.717458 52 5367 10.96 142 .40 3225 11.36 6775 8 53 6024 10.94 117 .41 39°7 11.35 6093 7 54 6681 10.92 °93 4588 11.33 5412 6 55 7337 10.91 069 5268 11.31 473* 5 56 7991 10.89 044 5947 11.30 4°53 4 57 8644 10.87 9.992020 6624 11.28 3376 3 58 9297 10.86 9.991996 7301 11.26 2699 2 59 9.279948 10.84 971 .41 7977 11.25 2023 1 60 9.280599 9.991947 9.288652 10.711348 M. Cosine. Diff. 1" Sine. Diff.l" Cotang. Diff. 1" Tang. lioo° 79° 52 11° SINES AND TANGENTS. ■■,.,...—. — , 168° M. Sine. Diff. 1" Cosine. Diffi,l" Tang. Diff. 1" Cotang. 60 1 9.280599 IO.82 9.991947 .41 9.288652 II.23 10.711348 J 1248 10.81 922 9326 11.22 0674 59 j 2 1897 IO.79 897 9.289999 II.20 10.710001 58 1 8 2544 IO.77 873 9.290671 II. l8 10.709329 57 4 3190 IO.76 848 1342 II. 17 8658 56 5 3^6 IO.74 823 2013 II. 15 7987 55 6 4480 IO.72 799 .41 2682 II. 14 7318 54 7 5124 IO.71 774 .42 335° II. 12 6650 53 8 5766 IO.69 749 4017 II. II 5983 52 9 6408 IO.67 724 4684 II.09 53 l6 51 10 11 7048 10.66 699 5349 II.07 4651 50 49 9.287687 IO.64 9.991674 9.296013 II.06 10.703987 12 8326 IO.63 649 6677 II.04 3323 48 13 8964 IO.61 624 7339 II.03 2661 47 14 9.289600 IO.59 599 8001 II.OI 1999 46 15 9.290236 IO.58 574 8662 II.OO 1338 45 16 0870 IO.56 549 9322 IO.98 0678 44 IV 1504 IO.54 5 2 4 9.299980 IO.96 10.700020 43 18 2137 IO.53 498 9.300638 IO.95 10.699362 42 19 2768 IO.51 473 1295 IO.93 8705 41 2U 21 3399 9.294029 IO.50 448 1951 IO.92 8049 40 ~3.*9* 8486 33 28 5987 7.60 398i 2006 &.19 7994 32 29 6 443 7.60 3946 2497 8.18 7503 31 30 31 6899 9-4 2 7354 7-59 39 11 2988 8.17 7012 30 29 7.58 9.983875 •58 9-443479 8.16 10.556521 32 7809 7-57 3840 •59 3968 8.16 6032 28 33 8263 7.56 3805 4458 8.15 554* 27 34 8717 7-55 377° 4947 8.14 5°53 26 35 9170 7-54 3735 5435 8.13 45 6 5 25 36 9.429623 7-54 3700 59 2 3 8.12 4077 24 37 9.430075 7-53 3664 641 1 8.12 3589 23 38 0527 7.52 3629 6898 8.11 3102 22 39 0978 7-5 1 3594 7384 8.10 2616 21 40 41 1429 7.50 3558 7870 8.09 2130 10.551644 20 19 9.431879 7-49 9.983523 9.448356 8.09 42 2329 7-49 3487 8841 8.08 1159 18 43 2778 7.48 3452 9326 8.07 0674 17 44 3226 7-47 3416 9.449810 8.06 10.550190 16 4o 3 6 75 7.46 338i 9.450294 8.06 10.549706 15 46 4122 7-45 3345 0777 8.05 9223 14 47 4569 7-44 33°9 •59 1260 8.04 8740 13 48 5016 7-44 3273 .60 1743 8.03 8257 12 49 5462 7-43 3238 2225 8.C2 7775 11 50 51 5908 7.42 3202 2706 8.02 7294 10.546813 10 9 9-43 6 353 7.41 9.983166 9-453 l8 7 8.01 52 6798 7.40 3*3° 3668 8.00 633 2 8 53 7242 7.40 3°94 4148 7-99 585 2 t 54 7686 7-39 3058 4628 7-99 5372 6 65 8129 7-38 3022 5 io 7 7.98 4893 1 56 8572 7-37 2986 5586 7-97 4414 4 67 9014 7-3 6 2950 6064 7.96 393 6 3 58 9456 7.36 2914 6542 7.96 3458 2 59 9-439 8 97 7-35 2878 .60 7019 7-95 2981 1 60 9.440338 9.982842 9.457496 Diff. 1" 10.542504 Tang. M. Cosine. Diff. 1" Sine. Diff.l" Cotang. 105° 74° 57 16° LOGARITHMIC 163° | J M. Sine. Diff. 1" Cosine. Diff. 1" Tang. Diff. 1" Cotang. 60 9.440338 7-34 9.982842 .60 9.457496 7-94 10.542504 1 0778 7-33 2805 .60 7973 7-93 2027 59 2 1218 7-3 2 2769 .61 8449 7-93 1551 58 3 1658 7.31 2733 8925 7.92 1075 57 4 2096 7-3 1 2696 9400 7.91 0600 56 5 2535 7-3° 2660 9-459 8 75 7.90 IO.540125 55 6 2973 7.29 2624 9.460349 7.90 IO.539651 54 7 3410 7.28 2587 0823 7.89 9177 53 8 3 8 47 7.27 2551 1297 7.88 8703 52 9 4284 7.27 2514 1770 7.88 8230 51 10 11 4720 7.26 2477 2242 7.87 7758 50 49 9-445I55 7.25 9.982441 9.462714 7.86 10.537286 12 559° 7.24 2404 3186 7.85 6814 48 13 6025 7.23 2367 3658 7.85 6342 47 14 6459 7-^3 2331 4129 7.84 5871 46 15 6893 7.22 2294 4599 7-83 5401 45 16 7326 7.21 2257 .61 5069 7.83 4931 44 17 7759 7.20* 2220 .62 5539 7.82 4461 43 18 8191 7.20 2183 6008 7.81 3992 42 19 8623 7.19 2146 6476 7.80 35 2 4 41 20 21 9054 9485 7.18 7.17 2109 _6?45 9.467413 7.80 3°55 40 39 9.982072 7-79 10.532587 22 9-4499 x 5 7.16 2035 7880 7.78 2120 38 23 9-45°345 7.16 1998 8347 7.78 1653 37 24 °775 7-15 1961 8814 7-77 1186 36 25 1204 7.14 1924 9280 7.76 0720 35 26 1632 7-13 1886 9.469746 7-75 10.530254 34 27 2060 7-13 1849 9.47021 1 7-75 10.529789 33 28 2488 7.12 1812 0676 7-74 2P 4 32 29 2915 7.11 1774 1 141 7-73 8859 31 1 30 31 3342 9.453768 7.10 1737 .62 1605 7-73 8395 30 29 7.10 9.981699 .63 9.472068 7.72 10.527932 32 4194 7.09 1662 *53 2 7.71 7468 28 33 4619 7.08 1625 2995 7.71 7005 27 34 5044 7.07 1587 3457 7.70 6543 26 35 5469 7.07 1549 39 J 9 7.69 6081 25 36 5 8 93 7.06 1512 438i 7.69 5619 24 37 6316 7.05 1474 4842 7.68 5158 23 38 6739 7.04 1436 53°3 7.67 4697 22 39 7162 7.04 1399 5763 7.67 4237 21 40 41 7584 9.458006 7-°3 1361 6223 7.66 3777 20 19 7.02 9.981323 9.476683 7.65 10.523317 42 8427 7.01 1285 7142 7.65 2858 18 43 8848 7.01 1247 7601 7.64 2399 17 44 9268 7.00 1209 8059 7.63 1941 16 45 9.459688 6.99 1171 .63 8517 7- 6 3 1483 15 46 9.460108 6.98 "33 .64 8975 7.62 1025 14 47 0527 6.98 1095 9432 7.61 0568 13 48 0946 6.97 1057 9.479889 7.61 10.520m 12 49 1364 6.96 1019 9-4 8o 345 7.60 10.519655 11 50 ~51 1782 6.95 6.95 0981 0801 7-59 9199 10 9 9.462199 9.980942 9.481257 7-59 10.518743 52 2616 6.94 0904 1712 7.58 8288 8 53 3°3 2 6-93 0866 2167 7-57 7833 7 54 3448 6.93 0827 2621 7-57 7379 6 55 3864 6.92 0789 3°75 7.56 6925 5 56 4279 6.91 0750 3529 7-55 6471 4 57 4694 6.90 0712 3982 7-55 6018 3 58 5108 6.9c 0673 4435 7-54 5565 2 59 55 22 6.89 0635 .64 4887 7-53 5113 1 60 9465935 Cosine. 9.980596 Diff.]' 9.485339 10.514661 M. Diff. 1" Sine. Cotang. Diff. 1" Tang. 106° 73° 58 17° SINES AND TANCHSNTS. 162° M. Sine. Diff. 1" Cosine. Oiff.l" Tang. Diff. 1" 7-53 Cotang. 60 9.465935 6.88 9.980596 .64 9-4 8 5339 IO.514661 1 6348 6-88 0558 .64 579i 7.52 4209 59 2 6761 6.87 0519 .65 6242 7-5i 375 8 58 3 7173 6.86 0480 6693 7-5i 33°7 57 4 75^5 6.85 0442 7H3 7.50 2857 56 5 7996 6.85 0403 7593 7-49 2407 55 6 8407 6.84 0364 8043 7-49 1957 54 7 8817 6.83 0325 8492 7.48 1508 53 8 9227 6.83 0286 8941 7-47 1059 52 9 9.469637 6.82 0247 9390 7-47 0610 51 10 11 9.470046 6.81 0208 9.489838 7.46 10.510162 50 49 0455 6.80 9.980169 9.490286 7.46 10.509714 12 0863 6.80 0130 o733 7-45 9267 48 13 1271 6.79 0091 1180 7-44 8820 47 14 1679 6.78 0052 1627 7-44 8373 46 15 2086 6.78 9.980012 2073 7-43 7927 45 16 2492 6.77 9-979973 .65 2519 7-43 7481 44 17 2898 6.76 9934 .66 2965 7.42 7035 43 18 33°4 6.76 9895 3410 7.41 6590 42 19 3710 6.75 9855 3 8 54 7.40 6146 41 2U 21 4115 6.74 9816 4299 7.40 5701 40 39 9.474519 6.74 9.979776 9-494743 7-39 10.505257 22 49 2 3 6.73 9737 5186 7-39 4814 38 23 53 2 7 6.72 9697 5630 7-38 437° 37 24 5730 6.72 9658 6073 7-37 39 2 7 36 25 6l 33 6.71 9618 65!5 7-37 34 8 5 35 26 6536 6.70 9579 6957 7-3 6 3°43 34 27 6938 6.69 9539 7399 7.36 2601 33 28 734° 6.69 9499 7841 7-35 2159 32 29 774i 6.68 9459 8282 7-34 1718 31 30 31 8142 6.67 9420 8722 7-34 1278 0837 30 29 9.478542 6.67 9.979380 9163 7-33 32 8942 6.66 934° .66 9.499603 7-33 10.500397 28 33 9342 6.65 9300 .67 9.500042 7.32 10.499958 27 34 9.479741 6.65 9260 0481 7-3 1 95i9 26 35 9.480140 6.64 9220 0920 7.31 9080 25 36 °539 6.63 9180 1359 7.30 8641 24 37 0937 6.63 9140 1797 7-3° 8203 23 38 1334 6.62 9100 2235 7.29 7765 22 39 1731 6.61 9059 2672 7.28 7328 21 40 41 2128 6.61 9019 3109 7.28 6891 20 19 9.482525 6.60 9.978979 9.503546 7.27 10.496454 42 2921 6.59 8939 3982 7.27 6018 18 43 3316 6.59 8898 4418 7.26 55^ 17 44 3712 6.58 8858 4854 7.25 5H6 16 45 4107 6.57 8817 5289 7.25 471 1 15 46 4501 6.57 8777 5724 7.24 4276 14 47 4895 6.56 8736 .67 6159 7.24 3841 13 48 5289 6.55 8696 .68 6593 7.23 34°7 12 49 5682 6.55 8655 7027 7.22 2973 11 50 51 6075 9.486467 6-54 8615 7460 9.507893 7.22 2540 10 9 6-53 9.978574 7.21 10.492107 52 6860 6-53 8533 8326 7.21 1674 8 53 7251 6.52 8493 8759 7.20 1 241 7 54 7643 6.51 8452 9191 7.19 0809 6 55 8034 6.51 841 1 9.509622 7.19 10.490378 5 56 8424 6.50 8370 9.510054 7.18 10.489946 4 57 8814 6.50 8329 0485 7.18 9515 3 58 9204 6.49 8288 0916 7.17 9084 2 59 9593 6.48 8247 .68 1346 7.17 8654 1 60 9.489982 9.978206 Sine. Diff.l' 9.511776 10.488224 M. Cosine. Diff. 1" Cotang. Diff. 1" Tang. 107° 72° 59 18° LOGARITHMIC 161° M. Sine. Diff. 1" Cosine. Dift.l" .68 Tang. Diff. 1" Cotang. 60 9.489982 6.48 9.978206 9.511776 7.16 IO.488224 1 9.490371 6.47 8165 2206 7.16 7794 59 2 0759 6.46 8124 .68 2635 7-15 7365 58 3 1 147 6.46 8083 .69 3064 7.14 6936 57 4 1535 6.45 8042 3493 7.14 6507 56 5 1922 6.44 8001 3921 7.13 6079 55 6 2308 6.44 7959 4349 7.13 5 6 5i 54 7 2695 6.43 7918 4777 7.12 5223 53 8 3081 6.42 7877 5204 7.12 4796 52 9 3466 6.42 7835 5 6 3! 7.11 43 6 9 51 10 11 3851 6.41 7794 9.977752 6057 7.10 3943 50 49 9.494236 6.41 9.516484 7.10 10.483516 12 4621 6.40 7711 6910 7.09 3090 48 13 5°°5 6.39 7669 7335 7.09 2665 47 14 5388 6.39 7628 7761 7.08 2239 46 15 5772 6.38 7586 .69 8185 7.08 1815 45 16 6154 6.37 7544 .70 8610 7.07 1390 44 17 6537 , 6 -37 .7503 9034 7.06 0966 43 18 6919 6.36 7461 9458 7.06 0542 42 19 7301 6.36 7419 9.519882 7.05 10.480118 41 20 21 7682 6.35 7377 9.520305 7-05 10.479695 40 39 9.498064 6-34 9-977335 0728 7.04 9272 22 8444 6.34 7293 "5i 7.04 8849 38 23 8825 6-33 7251 1573 7-03 8427 37 24 9204 6.32 7209 1995 7.03 8005 36 25 9584 6.32 7167 2417 7.02 7583 35 26 9.499963 6.31 7125 2838 7.02 7162 34 27 9.500342 6.31 7083 3 2 59 7.01 6741 33 28 0721 6.30 7041 3680 7.OI 6320 32 29 1099 6.29 6999 4100 7.00 5900 31 30 31 1476 6.29 6957 4520 6.99 5480 30 29 9.501854 6.28 9.976914 .70 9.524939 6.99 10.475061 32 2231 6.28 6872 •71 5359 6.98 4641 28 33 2607 6.27 6830 5778 6.98 4222 27 34 2984 6.26 6787 6197 6.97 3803 26 35 3360 6.26 6745 6615 6.97 3385 25 36 3735 6.25 6702 7°33 6.96 2967 24 37 4110 6.25 6660 745 x 6.96 2549 23 38 4485 6.24 6617 7868 6.95 2132 22 39 4860 6.23 6574 8285 6.95 1715 21 40 41 5*34 6.23 6532 8702 6.94 1298 20 19 9.505608 6.22 9.976489 9.529119 6-93 10.470881 42 598i 6.22 6446 9535 6-93 0465 18 43 6 354 6.21 6404 9.529950 6 -93 10.470050 17 44 6727 6.20 6361 9.530366 6.92 10.469634 16 45 7099 6.20 6318 0781 6.91 9219 15 46 7471 6.19 6275 •71 1196 6.91 8804 14 47 7843 6.19 6232 .72 1611 6.90 8389 13 48 8214 6.18 6189 2025 6.90 7975 12 49 8585 6.18 6146 2439 6.89 7561 11 50 51 8956 9326 6.17 6103 2853 6.89 7H7 10 9 6.16 9.976060 9.533266 6.88 10.466734 52 9.509696 6.16 6017 3 6 79 6.88 6321 8 53 9.510065 6.15 5974 4092 6.87 5908 7 54 0434 6.15 593° 4504 6.87 5496 6 55 0803 6.14 .5887 4916 6.86 5084 5 56 1172 6.13 5844 53*8 6.86 4672 4 57 1540 6.13 5800 5739 6.85 4261 3 58 1907 6.12 5757 6150 6.85 3850 2 59 2275 6.12 57H .72 6561 6.84 3439 1 60 9.512642 Cosine. 9.975670 Diff.l" 9.536972 Cotang. 10.463028 M. Diff. 1" I Sine. Diff. 1" Tang. 108° 71° GO 19° SINES AND TANCtENTS. 160° M. Sine. Diff. 1" Cosine. Diff.r' Tang. Diff. 1" Cotang. 60 9.512642 6.11 9.975670 •73 9.536972 6.84 10.463028 1 3009 6.II 5627 7382 6.83 2618 59 2 3375 6.10 5583 7792 6.83 2208 58 3 374i 6.09 5539 8202 6.82 1798 57 4 4107 6.09 5496 86ll 6.82 1389 56 5 4472 6.08 545* 9020 6.81 0980 65 6 4837 6.08 5408 9429 6.81 0571 54 7 5202 6.07 53 6 5 9-539837 6.80 IO.460163 63 8 5566 6.07 53 21 9.540245 6.80 IO.459755 52 y 593° 6.06 5277 0653 6.79 9347 51 10 n 6294 6.05 5233 I061 6.79 8939 50 49 9.516657 6.05 9.975189 9.541468 6.78 10.458532 12 7020 6.04 5H5 1875 6.78 8125 48 13 7382 6.04 5101 2281 6.77 7719 47 14 7745 6.03 5°57 2688 6.77 7312 46 15 8107 6.03 5 OI 3 •73 3°94 6.76 6906 45 16 8468 6.02 4969 •74 3499 6.76 6501 44 17 8829 6.01 4925 3905 6.75 6095 43 18 9190 6.01 4880 4310 6.75 5690 42 19 955i 6.00 4836 4715 6.74 5285 41 20 21 9.519911 6.00 4792 5"9 6.74 ' 4881 40 39 9.520271 5-99 9.974748 9-545524 6-73 10.454476 22 0631 5-99 4703 5928 6.73 4072 38 23 0990 5-98 4659 6331 6.72 3669 37 24 1349 5.98 4614 6735 6.72 3265 36 25 1707 5-97 457o 7138 6.71 2862 35 26 2066 5-96 4525 754o 6.71 2460 34 27 2424 5-9 6 4481 7943 6.70 2057 33 28 2781 5-95 443 6 8345 6.70 1655 32 29 3138 5-95 4391 •74 8747 6.69 1253 31 30 31 3495 5-94 4347 •75 9149 6.69 0851 30 29 9.523852 5-94 9.974302 955° 6.68 0450 32 4208 5-93 4 2 57 9.549951 6.68 10.450049 28 33 4564 5-93 4212 9-55°35 2 6.67 10.449648 27 34 4920 5.92 4167 0752 6.67 9248 26 35 52-75 5.91 4122 1152 6.66 8848 25 36 5630 5.91 4077 1552 6.66 8448 24 37 59 8 4 5-9° 4032 1952 6.65 8048 23 38 6 339 5-9° 3987 2351 6.65 7649 22 39 6693 5-89 3942 2750 6.65 7250 21 40 41 7046 5.89 3897 3*49 6.64 6851 20 19 9.527400 5.88 9.973852 9-553548 6.64 10.446452 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 2 59 16 45 8810 5.86 3671 5 J 39 6.62 4861 15 46 9161 5.86 3625 5536 6.61 4464 14 47 9513 5.85 3580 5933 6.61 4067 13 48 9.529864 5.85 3535 6329 6.60 3671 12 49 9.530215 5.84 3489 6725 6.60 32-75 11 60 51 o5 6 5 5.84 3444 7121 6-59 2.879 10 9 9-53°9 I 5 5-83 9-973398 9-5575I7 6.59 10.442483 52 1265 5.82 335 2 7913 6.59 2087 8 53 1614 5.82 33°7 8308 6.58 1692 7 54 1963 5.81 3261 8702 6.58 1298 6 55 2312 5-8i 3 2I 5 9097 6.57 0903 6 56 2661 5.80 3169 9491 6.57 0509 4 57 3009 5.80 3 I2 4 9.559885 6.56 10.4401 1 5 3 58 3357 5-79 3078 .76 9.560279 6.56 10.439721 2 59 3704 5-79 3032 •77 0673 6.55 9327 1 60 9.534052 Cosine. 9.972986 Diff.l" 9.561066 Cotang. 10.438934 M. Diff. 1" Sine. Diff. 1" Tang. 109° 70° 61 20° LOGARITHMIC 159° M. ~0 Sine. Diff. 1" Cosine. Diff.1" Tang. Diff 1" Cotang. 1 60 9.534052 5-78 9.972986 •77 9.561066 6.55 IO -43 8 934 1 4399 5-78 2940 1459 6.54 8541 o9 2 4745 5-77 2894 1851 6.54 8149 58 3 5092 5-77 2848 2244 6-53 7756 57 4 5438 5-76 2802 2636 6-53 73 6 4 56 5 5783 5.76 2755 3028 6-53 6972 55 6 6129 5-75 2709 3419 6.52 6581 54 V 6474 5-74 2663 3811 6.52 6189 53 8 6818 5-74 2617 4202 6.51 5798 52 y 7163 5-73 2570 4592 6.51 5408 51 10 n 7507 5-73 2524 4983 9-5 6 5373 6.50 5 OI 7 50 49 9-53785I 5.72 9.972478 •77 6.50 10.434627 12 8194 5-7 2 2431 .78 57 6 3 6.49 4 2 37 48 13 8538 5-7i 2385 6i53 6.49 3847 47 14 8880 5-7i 2338 6542 6.49 3458 46 15 9223 5-7o 2291 6932 6.48 3068 45 16 9565 5-7° 2245 7320 6.48 2680 44 IV 9-5399°7 .5- 6 9 2198 7709 6.47 2291 43 18 9.540249 5- 6 9 2151 8098 6.47 1902 42 19 0590 5.68 2105 8486 6.46 1514 41 20 ~21 0931 5.68 2058 8873 6.46 1127 40 39 9.541272 5.67 9.972011 9261 6.45 0739 22 1613 5- 6 7 1964 9.569648 6.45 10.430352 38 23 1953 5.66 1917 9.570035 6.45 10.429965 3Y 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 5- 6 4 I729 •79 1581 6 -43 8419 33 28 3 6 49 5- 6 4 1682 1967 6.42 8033 32 29 39 8 7 5-63 1635 2352 6.42 7648 31 30 31 4325 5-63 1588 2738 9-573 I2 3 6.42 6.41 7262 30 29 9.544663 5.62 9.971540 10.426877 32 5000 5.62 1493 3507 6.41 6 493 28 33 533 8 5.61 1446 3892 6.40 6108 2V 34 5 6 74 5.61 1398 4276 6.40 57 2 4 26 3b 6011 5.60 1351 4660 6 -39 534° 25 36 6347 5.60 I303 5044 6-39 4956 24 37 6683 5-59 1256 5427 6.39 4573 23 38 7019 5-59 1208 5810 6.38 4190 22 39 7354 5-58 Il6l 6193 6.38 3807 21 40 41 7689 9.548024 5.58 III3 •79 .80 6576 6-37 3424 10.423042 20 19 5-57 9.971066 9.576958 6.37 42 8359 5-57 1018 734i 6.36 2659 18 43 8693 5.56 0970 77 2 3 6.36 2277 IV 44 9027 5.56 0922 8104 6.36 1896 16 45 9360 5-55 0874 8486 6-35 1514 15 46 9-549 6 93 5-55 0827 8867 6-35 "33 14 4V 9.550026 5-54 0779 9248 6-34 0752 13 48 0359 5-54 0731 9.579629 6-34 10.420371 12 49 0692 5-53 0683 9.580009 6-34 10.419991 11 50 51 1024 5-53 5-5* 0635 0389 6 -33 961 1 10.419231 10 9 9-55 J 35 6 9.970586 9.580769 6 -33 52 1687 5-52 0538 1 149 6.32 8851 8 53 2018 5.52 0490 1528 6.32 8472 7 54 2349 5-5i 0442 1907 6.32 8093 6 55 2680 5-5i . 0394 .80 2286 6.31 7714 6 56 3010 5-5° °345 .81 2665 6.31 7335 4 5V 334i 5.50 0297 3°43 6.30 6957 3 58 3670 5-49 0249 3422 6.30 6578 2 59 4000 5-49 0200 .81 3800 6.29 6200 1 60 9.554329 9.970152 Diff.l" 9.584177 10.415823 i Cosine. Diff. 1" Sine. Cotang. Diff. 1" Tang. 110° 69° 62 21° SINES AND TANGENTS. 158° M. Sine. Diff. 1" Cosine. Diff.l" Tang. Diff. 1" Cotang. 60 9-5543^9 5.48 9.970152 .81 9.584177 6.29 10.415823 1 4658 5-48 0103 4555 6.29 5445 59 2 4987 5-47 0055 4932 6.28 5068 58 3 53*5 5-47 9.970006 53°9 6.28 4691 57 4 5643 5.46 9.969957 5686 6.27 43 J 4 56 5 597i 5-4 6 9909 6062 6.27 3938 55 6 6299 5-45 9860 6439 6.27 35 61 54 7 6626 5-45 9811 6815 6.26 3185 53 8 6953 5-44 9762 7190 6.26 2810 52 y 7280 5-44 97H 7566 6.25 2434 51 10 n 7606 5-43 9665 .81 7941 9.588316 6.25 2059 50 49 9-S5793 2 - 5-43 9.969616 .82 6.25 10.411684 12 8258 5-43 9567 8691 6.24 1309 48 13 8583 5-4* 9518 9066 6.24 0934 47 14 8909 5.42 9469 9440 6.23 0560 46 16 9234 5.41 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 0531 5-39 9223 1308 6.22 8692 41 20 21 0855 5-39 9173 1681 6.21 ' 8319 40 39 9.561178 5.38 9.969124 9.592054 6.21 10.407946 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 3 J 7! 6.20 6829 36 25 2468 5-3 6 8926 .83 3542 6.I9 6458 35 26 2790 5-36 8877 39M 6.I9 6086 34 27 3112 5-3 6 8827 4285 6.18 5715 33 28 3433 5-35 8777 4656 6.18 5344 32 29 3755 5-35 8728 5027 6.l8 4973 31 30 31 4°75 5-34 5-34 8678 9.968628 5398 6.I7 4602 10.404232 30 29 9.564396 9.595768 6.I7 32 4716 5-33 8578 6138 6.16 3862 28 33 5036 5-33 8528 6508 6.l6 349 2 27 34 535 6 5-3 2 8479 6878 6.16 3122 26 35 5676 5-32 8429 7247 6.I5 2 753 25 36 5995 5-3i 8379 7616 6.I5 2384 24 37 6314 5-3 1 8329 7985 6.I5 2015 23 38 6632 5-3i 8278 .83 8354 6.I4 1646 22 39 6951 5-3° 8228 .84 8722 6.I4 1278 21 40 41 7269 5-3° 8178 9.968128 9091 9459 6.I3 0909 20 19 9.567587 5.29 6.I3 0541 42 7904 5.29 8078 9.599827 6.I3 10.400173 18 43 8222 5.28 8027 9.600194 6.12 10.399806 17 44 8539 5.28 7977 0562 6.12 9438 16 45 8856 5.28 7927 0929 6.11 9071 15 46 9172 5.27 7876 1296 6.11 8704 14 47 9488 5.27 7826 1662 6.11 8338 13 48 9.569804 5.26 7775 2029 6.10 7971 12 49 9.570120 5.26 7725 2395 6.10 7605 11 50 51 0435 9.570751 5- 2 5 7674 2761 9.603127 6.10 6.09 7239 ,10.396873 10 9 5- 2 5 9.967624 52 1066 5- 2 4 7573 .84 3493 6.09 6507 8 53 1380 5- 2 4 7522 .85 3858 6.09 6142 7 54 1695 5- 2 3 747i 4223 6.08 5777 6 55 2009 5- 2 3 7421 4588 6.08 5412 5 56 2 3 2 3 5- 2 3 737o 4953 6.07 5°47 4 57 2636 5.22 7319 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 Diff.l" 9.606410 Cotang. 10.393590 Tang. M. Cosine. Diff. 1" Sine. Diff. 1" 111° 68° 63 22° LOGARITHMIC 157° | |M.| Sine. Diff. 1" Cosine. Diff.l" Tang. Diff. 1" Cotang. i 60 ! 9-573575 5.21 9.967166 .85 9.606410 6.06 IO.393590 1 3888 5.20 7115 6773 6.06 3227 59 2 4200 5.20 7064 7137 6.05 2863 58 3 4512 5.19 7013 7500 6.05 2500 57 4 4824 5- J 9 6961 7863 6.04 2137 56 5 5136 5-J9 6910 8225 6.04 1775 55 6 5447 5.18 6859 8588 6.04 1412 54 7 5758 5.18 6808 .85 8950 6.03 1050 53 8 6069 5- J 7 6756 .86 9312 6.03 0688 52 9 6379 5-i7 6705 9.609674 6.03 10.390326 51 10 11 6689 5.16 6653 9.610036 6.02 10.389964 50 49 9.576999 5.16 9.966602 0397 6.02 10.389603 12 73°9 5.16 6550 0759 6.02 9241 48 13 7618 5-J5 6499 1 1 20 6.01 8880 47 14 7927 5-i5 6447 1480 6.01 8520 46 15 8236 5-H 6 395 1841 6.OI 8159 45 16 8545 5-H 6344 220I 6.00 7799 44 17 8853 - 5-i3 6292 2561 6.00 7439 43 18 9162 5-!3 6240 2921 6.00 7079 42 19 9470 5-i3 6188 3281 5-99 6719 41 20 21 9-579777 5.12 6136 .86 3641 9.614000 5-99 6359 40 39 9.580085 5.12 9.966085 .87 5.98 10.386000 22 0392 5-" 6033 4359 5-98 5641 38 23 0699 5" 598i 4718 5-98 5282 37 24 1005 5.11 5928 5077 5-97 4923 36 25 1312 5.10 5876 5435 5-97 45 6 5 35 26 1618 5.10 5824 5793 5-97 4207 34 27 1924 5-°9 5772 6151 5-9 6 3849 33 28 2229 5-°9 5720 6509 5-9 6 349i 32 29 2535 5-o9 5668 6867 5-9 6 3 I 33 31 30 31 2840 5.08 5615 7224 5-95 2776 30 29 9-5 8 3 J 45 5.08 9.965563 9.617582 5-95 10.382418 32 3449 5-°7 55" 7939 5-95 2061 28 33 3754 5.07 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 4665 5.06 53 01 93 6 4 5-93 0636 24 37 4968 5-o5 5248 9.619721 5-93 10.380279 23 38 5272 5-°5 5i95 9.620076 5-93 10.379924 22 39 5574 5-°4 5H3 0432 5.92 9568 21 40 41 5877 5.04 5090 0787 5.92 9213 20 19 9.586179 5-°3 9.965037 9.621142 5.92 10.378858 42 6482 5-°3 4984 1497 5-9i 8503 18 43 6783 5-°3 493i 1852 5-9 1 8148 17 44 7085 5.02 4879 2207 5.90 7793 16 45 7386 5.02 4826 2561 5-9° 7439 15 46 7688 5.01 4773 2915 5-9° 7085 14 47 7989 5.01 4719 .88 3269 5-89 6731 13 48 ' 8289 5.01 4666 ■89 3623 5.89 6377 12 49 8590 5.00 4613 397 6 5-89 6024 11 50 51 8890 5.00 4560 4330 5.88 5670 10 9 9.589190 4.99 9.964507 9.624683 5.88 i°-3753 I 7 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 574i 5.87 4259 6 55 0387 4.98 4294 6093 5.87 39°7 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 Cosine. 9.964026 Sine. 9.627852 10.372148 M. Diff. 1" Diff.l" 1 Cotang. Diff. 1" Tang. 112° 67° 64 23° SINES AtfD TANGENTS. 156° M. Sine. Diff. 1" Cosine. Diff.l" Tang. Diff. 1" Cotang. 60 9.591878 4.96 9.964026 .89 9.627852 5.85 IO.372148 1 2176 4-95 3972 .89 8203 5.85 1797 59 2 2473 4-95 3919 .89 8554 5.85 1446 58 3 2770 4-95 3865 .90 8905 5.84 I095 57 4 3067 4.94 3811 92-55 5.84 0745 56 5 33 6 3 4.94 3757 9606 5.83 0394 55 6 3659 4-93 37°4 9.629956 5-83 IO.370044 54 7 3955 4-93 3 6 5° 9.630306 5.83 IO.369694 53 8 4251 4-93 359 6 0656 5.83 9344 52 y 4547 4.92 354^ 1005 5.82 8995 51 10 n 4842 4.92 3488 1355 5.82 8645 50 49 9-595I37 4.91 9.963434 9.631704 5.82 10.368296 12 543 2 4.91 3379 2°53 5.81 7947 48 13 57 2 7 4.91 33 2 5 2401 5.81 7599 47 14 6021 4.90 3271 2750 5.81 7250 46 15 6315 4.90 3217 3098 5.80 6902 45 16 6609 4.89 3163 .90 3447 5.80 6553 44 17 6903 4.89 3108 .91 3795 5.80 6205 43 18 7196 4.89 3054 4H3 5-79 5857 42 19 7490 4.88 2999 4490 5-79 55io 41 20 21 7783 4.88 2945 4838 9.635185 5-79 ' 5162 40 39 9.598075 4.87 9.962890 5.78 10.364815 22 8368 4.87 2836 553 2 5.78 4468 38 23 8660 4.87 2781 5879 5.78 4121 37 24 8952 4.86 2727 6226 5-77 3774 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 0409 4.84 2-453 .91 7956 5-76 2044 31 30 31 0700 4.84 2398 .92 8302 5-76 1698 30 29 9.600990 4.84 9-9 62 343 9.638647 5-75 10.361353 32 1280 4.83 2288 8992 5-75 1008 28 33 1570 4.83 2233 9337 5-75 0663 27 34 i860 4.82 2178 9.639682 5-74 10.360318 26 36 2150 4.82 2123 9.640027 5-74 i°-359973 25 36 2439 4.82 2067 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 3305 4.81 1902 1404 5-73 8596 21 40 41 3594 4.80 1846 1747 9.642091 5-72 8253 20 19 9.603882 4.80 9.961791 5-72 io-3579°9 42 4170 4-79 1735 2434 5-72 7566 18 43 4457 4-79 1680 .92 2777 5.72 7223 17 44 4745 4-79 1624 •93 3120 5-7i 6880 16 45 5032 4.78 1569 34 6 3 5-7i 6 537 15 46 53*9 4.78 1513 3806 5-7i 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 1346 4832 5-7° 5168 11 50 51 6465 4.76 1290 5*74 5.69 4826 10.354484 10 9 9.606751 4.76 9.961235 9.645516 5.69 52 7036 4.76 1179 5857 5- 6 9 4H3 8 53 7322 4-75 1123 6199 5- 6 9 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- 6 7 2438 3 58 8745 4-73 0843 •94 7903 5- 6 7 2097 2 59 9029 4-73 0786 •94 8243 5- 6 7 1757 1 60 9.609313 Cosine. 9.960730 9.648583 10.351417 Tang. M. Diff. 1" Sine. Diff.l" Cotang. Diff. 1* 113° 66° 65 24° LOGARITHMIC 155° M. Sine. Diff. 1" Cosine. Diff.l" •94 Tang. Diff. 1" Cotang. 60 9.609313 4-73 9.960730 9.648583 5.66 10.351417 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 °5°5 9.649942 5.65 10.350058 56 5 0729 4.71 0448 9.650281 5.65 10.349719 55 6 1012 4.70 0392 0620 5.65 9380 54 7 1294 4.70 0335 0959 5- 6 4 9041 53 8 1576 4.70 0279 1297 5.64 8703 52 9 1858 4.69 0222 1636 5-64 8364 51 10 11 2140 4.69 0165 ■94 •95 1974 5-63 8026 50 ~49~ 9.612421 4.69 0109 9.652312 5- 6 3 10.347688 12 2702 4.68 9.960052 2650 5-63 7350 48 13 2983 4.68 9-959995 2988 5-63 7012 47 14 3264 4.67 9938 3326 5.62 6674 46 15 3545 4.67 9882 3663 5.62 6 337 45 16 3^5 4.67 9825 4000 5.62 6000 44 17 4105 4 ;66 9768 4337 5.61 5663 43 18 4385 4.66 9711 4674 5.61 5326 42 19 4665 4.66 9654 5011 5.61 4989 41 20 21 4944 4.65 9596 5348 5.61 4652 40 39 9.615223 4.65 9-959539 9.655684 5.60 10.344316 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 33°8 36 25 6338 4.64 9310 .96 7028 5-59 2972 35 26 6616 4.63 9 2 53 73 6 4 5-59 2636 34 27 6894 4- 6 3 9i95 7699 5-59 2301 33 28 7172 4.62 9138 8034 5.58 1966 32 29 7450 4.62 9081 8369 5.58 1631 31 30 31 7727 9.618004 4.62 9023 9.958965 8704 5.58 1296 30 29 4.61 9.659039 5-58 10.340961 32 8281 4.61 8908 9373 5-57 0627 28 33 8558 4.61 8850 9.659708 5-57 10.340292 27 34 8834 4.60 8792 9.660042 5-57 10.339958 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 i°43 5.56 8957 23 38 9.619938 4-59 8561 .96 1377 5.56 8623 22 39 9.620213 4-59 8503 •97 1710 5-55 8290 21 40 41 0488 4.58 8445 2043 5-55 7957 20 19 0763 4.58 9.958387 9.662376 5-55 10.337624 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 8i54 37°7 5-54 6293 15 46 2135 4.56 8096 4039 5-53 5961 14 47 2409 4.56 8038 437 1 5-53 5629 13 48 2682 4-55 7979 4703 5-53 5297 12 49 2956 4-55 7921 5°35 5-53 4965 11 50 51 3229 9.623502 4-55 7863 9.957804 5366 5-5* 5-5 2 4634 10 9 4-54 •97 9.665697 i°-3343°3 52 3774 4-54 7746 .98 6029 5-5 2 3971 8 53 4047 4-54 7687 6360 5-5i 3640 7 54 43 J 9 4-53 7628 6691 5-5i 3309 6 55 459 1 4-53 7570 7021 5-5i 2979 5 56 4863 4-53 7511 7352 5-5i 2648 4 57 5*35 4.52 7452 7682 5-5° 2318 3 58 5406 4.52 7393 8013 5-5° 1987 2 59 5677 4.52 7335 .98 8343 5-5° 1657 1 60 9.625948 9.957276 Diff.l" 9.668672 Cotang. 10.331328 M. Cosine. j Diff. V Sine. Diff. 1" Tang. 1 114° 65° 66 25° SIEVES AND TANGENTS. 154° M. Sine. Diff. 1" 4.51 Cosine. Diff.r' Tang. Diff. 1" Cotang. 60 i 9.625948 9.957276 .98 9.668673 5-5° 10.331327 J 6219 4.51 7217 9002 5-49 0998 59 2 6490 4.51 7158 9332 5-49 0668 58 3 6760 4.50 7099 9661 5-49 °339 57 4 7030 4.50 7040 9.669991 5.48 10.330009 56 5 7300 4.50 6981 .98 9.670320 5.48 10.329680 55 6 7570 4.49 6921 •99 0649 5-48 935 1 54 7 7840 4.49 6862 0977 5-48 9023 53 8 8109 4.49 6803 1306 5-47 8694 52 9 8378 4.48 6 744 1634 5-47 8366 51 10 11 8647 9.628916 4.48 6684 9.956625 1963 9.672291 5-47 8037 50 49 4-47 5-47 10.327709 12 9185 4-47 6566 2619 5-4 6 738i 48 13 9453 4-47 6506 2947 5.46 7053 47 14 9721 4.46 6447 3 2 74 5.46 6726 46 15 9.629989 4.46 6387 3602 5-4 6 6398 45 16 9.630257 4.46 6327 3929 5-45 6071 44 17 0524 4.46 6268 •99 4257 5-45 5743 43 18 0792 4-45 6208 1. 00 4584 5-45 5416 42 19 1059 4-45 6148 4910 5-44 5090 41 20 21 1326 9- 6 3 I 593 4-45 6089 5 2 37 9.675564 5-44 5-44 ' 4763 10.324436 40 39 4.44 9.956029 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 3 1 3 I 35 26 2923 4-43 5729 7194 5-43 2806 34 27 3 l8 9 4.42 5669 7520 5.42 2480 33 28 3454 4.42 5609 7846 5.42 2154 32 29 3719 4.42 5548 8171 5-4^ 1829 31 30 3984 4.41 5488 1. 00 1. 01 8496 9.678821 5.42 1504 30 29 31 9.634249 4.41 9.955428 5.41 10.321179 32 45H 4.40 5368 9146 5.41 0854 28 33 4778 4.40 53°7 947i 5-4i 0529 27 34 5042 4.40 5*47 9.679795 5.41 10.320205 26 35 5306 4-39 5186 9.680120 5.40 10.319880 25 36 557° 4-39 5126 0444 5.40 9556 24 37 5834 4-39 5 o6 5 0768 5.40 9232 23 38 6097 4-39 5005 1092 5.40 8908 22 39 6360 4-38 4944 1416 5-39 8584 21 40 41 6623 9.636886 4.38 4883 9.954823 1740 9.682063 5-39 8260 20 19 4-37 5-39 10.317937 42 7148 4-37 4762 2387 5-39 7613 18 43 741 1 4-37 4701 2710 5-38 7290 17 44 7673 4-37 4640 3°33 5.38 6967 16 4o 7935 4.36 4579 I.OI 335 6 5.38 6644 15 46 8197 4-3 6 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 4324 5-37 5676 12 49 8981 4-35 4335 4646 5-37 5354 11 50 51 9242 9503 4-35 4274 9.954213 4968 5-37 5032 10.314710 10 9 4-34 9.685290 5-3 6 52 9.639764 4-34 4i5 2 5612 5-3 6 4388 8 53 9.640024 4-34 4090 5934 5-3 6 4066 7 54 0284 4-33 4029 6255 5-3 6 3745 6 55 0544 4-33 3968 6577 5-35 34*3 5 56 0804 4-33 3906 6898 5-35 3102 4 57 1064 4-3 2 3845 7219 5-35 2781 3 58 I3 2 4 4.32 3783 1.02 754o 5-35 2460 2 59 1583 4.32 3722 1.03 7861 5-34 2139 1 60 9.641842 Cosine. 9.953660 Diff.l" 9.688182 Cotang. 10.311818 M. Diff. 1" Sine. Diff. 1" Tang. 115° 04° b7 26° LOGARITHMIC 153° M. Sine. Diff. 1" Cosine. Diff.l" Tang. Diff. 1" Cotang. 60 9.641842 4-3 1 9.953660 I.03 9.688182 5-34 IO.311818 1 2101 4.31 3599 8502 5-34 1498 59 2 2360 4-31 3537 8823 5-34 1177 58 3 2618 4.30 3475 9H3 5-33 0857 57 4 2877 4-3° 3413 9463 5-33 0537 56 5 3 J 35 4-3° 3352 9.689783 5-33 IO.310217 65 6 3393 4-3° 3290 9.690103 5-33 10.309897 54 7 3650 4.29 3228 0423 5-33 9577 53 8 3908 4.29 3166 0742 5-32 9258 52 9 4165 4.29 3104 1062 5-3 2 8938 61 10 11 4423 4.28 3042 I.03 I.04 1381 5-3 2 8619 50 49 9.644680 4.28 9.952980 9.691700 5-3 1 10.308300 12 4936 4.28 2918 2019 5-3 1 7981 48 13 5193 4.27 2855 2338 5-3i 7662 47 14 545° 4.27 2793 2656 5-3i 7344 46 15 5706 4.27 2731 2975 5-3i 7025 45 16 5962 4.26 2669 3 2 93 5-3° 6707 44 17 6218 4.26 2606 3612 5-3° 6388 43 18 6474 4.26 2544 393° 5-3° 6070 42 19 6729 4.26 2481 4248 5-3° 575* 41 20 21 6984 4.25 2419 4566 5- 2 9 5434 40 39 9.647240 4.25 9.952356 9.694883 5- 2 9 10.305 1 17 22 7494 4.24 2294 5201 5.29 4799 . 38 23 7749 4.24 2231 I.04 55i8 5- 2 9 4482 37 24 8004 4.24 2168 I.05 5836 5- 2 9 4164 36 25 8258 4.24 2106 6i53 5.28 3847 35 26 8512 4-^3 2043 6470 5.28 353° 34 27 8766 4- 2 3 1980 6787 5.28 3 2I 3 33 28 9020 4.23 1917 7103 5.28 2897 32 29 9274 4.22 1854 7420 5.27 2580 31 30 31 9527 4.22 1791 773 6 9.698053 5-^7 2264 30 29 9.649781 4.22 9.951728 5-*7 10.301947 32 9.650034 4.22 1665 8369 5.27 1631 28 33 0287 4.21 1602 8685 5.26 I3!5 27 34 0539 4.21 1539 9001 5.26 0999 26 35 0792 4.21 1476 9316 5.26 0684 25 36 1044 4.20 1412 I.05 9632 5.26 0368 24 37 1297 4.20 1349 I.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- 2 5 9422 21 40 41 2052 4.19 1159 0893 5.25 9107 20 19 9.652304 4.19 9.951096 9.701208 5.24 10.298792 42 2555 4.18 1032 1523 5- 2 4 8477 18 43 2806 4.18 0968 1837 5.24 8163 17 44 3057 4.18 0905 2152 5.24 7848 16 45 3308 4.18 0841 2466 5.24 7534 15 46 3558 4.17 0778 2780 5- 2 3 7220 14 47 3808 4.17 0714 3°95 5- 2 3 6905 13 48 4059 4.17 0650 34°9 5- 2 3 6591 12 49 4309 4.16 0586 I.06 3723 5- 2 3 6277 11 50 4558 4.16 0522 I.07 4036 5.22 5.22 5964 10 9 51 9.654808 4.16 9.950458 9.704350 10.295650 52 5058 4.16 0394 4663 5.22 5337 8 53 53°7 4.15 0330 4977 5.22 5023 7 54 5556 4.15 0266 5290 5.22 4710 6 55 5805 4.15 0202 5603 5.21 4397 6 56 6054 4.14 0138 5916 5.21 4084 4 57 6302 4.14 0074 6228 5.21 3772 3 58 655i 4.14 9.950010 6541 5.21 3459 2 59 6799 4.13 9.949945 I.07 6854 5.21 3146 1 60 9.657047 Cosine. Diff. 1" 9.949881 9.707166 10.292834 M. Sine. Diff.]" Cotang. Diff. 1" Tang. 116° 63° 08 27° SIKTES AND TANCfrENTS. 152° M. Sine. Diff. 1" Cosine. Diff.l" Tang. 9.707166 Diff. 1" Cotang. l 60 9.657047 4.13 9.949881 1.07 5.20 10.292834 1 7295 4-13 9816 I.07 7478 5- 20 2522 59 2 754 2 4.12 9752 I.07 7790 5 20 2210 58 8 7790 4.12 9688 I.08 8102 5 20 1898 57 : 4 8037 4.12 9623 8414 5 19 1586 56 8284 4.12 9558 8726 5 19 1274 55 6 853i 4.1 1 9494 9037 5 19 0963 54 7 8778 4.11 9429 9349 5 19 0651 53 8 9025 4.1 1 9364 9660 5 19 0340 52 y 9271 4.10 9300 9.709971 5 18 IO.290029 51 10 n 9517 4.10 9235 9.710282 °593 5 18 10.289718 50 49 9.659763 4.10 9.949170 5 18 9407 12 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 5 17 8475 46 15 0746 4.09 8910 1836 5 17 8164 45 16 0991 4.08 8845 I.08 2146 5 17 7854 44 17 1236 4.08 8780 I.09 2456 5 17 7544 43 18 1481 4.08 8715 2766 5 16 7234 42 19 1726 4.07 8650 3076 5 16 6924 41 20 21 1970 9.662214 4.07 8584 3386 5 16 6614 40 39 4.07 9.948519 9.713696 5 16 10.286304 22 2459 4.07 8454 4005 5 16 5995 38 23 2703 4.06 8388 4314 5 M 5686 37 24 2946 4.06 8323 4624 5 15 5376 36 25 3190 4.06 8257 4933 5 15 5067 35 26 3433 4.05 8192 5242 5 !5 4758 34 27 3 6 77 4.05 8126 555i 5 J 4 4449 33 28 3920 4.05 8060 I.09 5860 5 H 4140 32 29 4163 4.05 7995 I. IO 6168 5 x 4 3832 31 30 31 4406 4.04 7929 6477 5 H 35^3 30 29 9.664648 4.04 9.947863 9.716785 5 14 10.283215 32 4891 4.04 7797 7093 5 13 2907 28 33 5133 4-°3 7731 7401 5 !3 2599 27 34 5375 4.03 7665 7709 5 J 3 2291 26 35 5 6 i7 4.03 7600 8017 5 J 3 1983 25 36 5859 4.02 7533 8325 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 41 6824 4.01 4.01 7269 9555 5 .12 0445 20 19 9.667065 9.947203 I. IO 9.719862 5 .12 10.280138 42 73°5 4.01 7136 I. II 9.720169 5 11 10.279831 18 43 7546 4.01 7070 0476 5 .11 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 IO 8298 13 48 8746 3-99 6738 2009 5 IO 7991 12 49 8986 3-99 6671 2315 5 IO 7685 11 50 51 9225 3-99 6604 2621 5 IO 7379 10.277073 10 9 9464 3.98 9.946538 9.722927 5 .10 52 9703 3.98 6471 3232 5 09 6768 8 53 9.669942 3-98 6404 3538 5 .09 6462 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 3.96 6069 5065 5 .08 4935 2 59 1372 3.96 6002 1. 12 5369 5 .08 4631 1 60 9.671609 9-945935 9.725674 10.274326 M. ! Cosine. Diff. 1" Sine. Diff.l" Cotang. Diff. 1" Tang. 117° 62° | 26 69 28° LOGARITHMIC 151° M. Sine. Diff. 1" Cosine. Diff.l" Tang. Diff. 1" Cotang. 60 9.671609 3-9 6 9-945935 1. 12 9.725674 5.08 IO.274326 1 1847 3-95 5868 5979 5.08 4021 59 2 2084 3-95 5800 6284 5-°7 3716 58 3 2321 3-95 5733 6588 5.07 3412 57 4 2 55 8 3-95 5666 6892 5.07 3108 56 5 2795 3-94 5598 7197 5.07 2803 55 6 3°3 2 3-94 553i 1. 12 75oi 5.07 2499 54 7 3268 3-94 5464 1. 13 7805 5.06 2195 53 1 8 35°5 3-94 539 6 8109 5.06 1891 52 9 374i 3-93 53*8 8412 5.06 1588 51 10 11 3977 9.674213 3-93 5261 8716 5.06 1284 50 49 3-93 9-945I93 9.729020 5.06 IO.270980 12 4448 3.92 5125 93*3 5-05 0677 48 13 4684 3-9 2 5058 9626 5-°5 0374 47 14 4919 3-9 2 4990 9.729929 5-°5 IO.270071 46 15 5 J 55 3-9 2 4922 9-73° 2 33 5-o5 IO.269767 45 16 539° 3-9 1 4854 0535 5-°5 9465 44 17 5624 3-9 1 4786 0838 5.04 9162 43 18 5859 3-9 1 4718 1141 5-°4 8859 42 19 6094 3-9 1 4650 113 1444 5-°4 8556 41 20 21 6328 3-9° 4582 1. 14 1746 5-o4 8254 40 39 9.676562 3.90 9.944514 9.732048 5.04 IO.267952 22 6796 3.90 4446 2351 5-°3 7649 38 23 7030 3-9° 4377 2653 5-°3 7347 37 24 7264 3.89 4309 2955 5-°3 7045 36 25 7498 3-89 4241 3^57 5-^3 6743 35 26 773 1 3.89 4172 3558 5-°3 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 3967 44 6 3 5.02 5537 31 30 31 8663 3.88 3899 4764 5.02 5236 30 29 9.678895 3.87 9.943830 9.735066 5.02 10.264934 32 9128 3.87 3761 1. 14 53 6 7 5.02 4 6 33 28 33 9360 3.87 3 6 93 1. 15 5668 5.01 4332 27 34 9592 3-87 3624 59 6 9 5.01 4031 26 35 9.679824 3.86 3555 6269 5.01 373 1 25 36 9.680056 *!£ 3486 6570 5.01 343° 24 j 37 0288 3.86 34i7 6871 5.01 3129 23 I 38 0519 3.85 3348 7171 5.00 2829 22 39 0750 3-85 3*79 7471 5.00 2529 21 40 41 0982 3-85 3-85 3210 7771 5.00 2229 20 19 9.681213 9.943141 9.738071 5.00 10.261929 42 1443 3-84 3072 8371 5.00 1629 18 43 1674 3.84 3003 8671 4.99 1329 17 44 1905 3-84 2934 8971 4.99 1029 16 45 2135 3-84 2864 1. 15 9271 4.99 0729 15 46 2365 3.83 2795 1. 16 9570 4.99 0430 14 47 259S 3-83 2726 9.739870 4.99 10.260130 13 48 2825 3-83 2656 9.740169 4.99 10.259831 12 49 3°55 3-83 2587 0468 4.98 9532 11 50 51 3284 9.683514 3.82 2517 0767 4.98 9233 10 9 3.82 9.942448 9.741066 4.98 10.258934 52 3743 3.82 2378 i3 6 5 4.98 863s 8 53 3972 3.82 2308 1664 4.98 8336 7 54 4201 3.81 2239 1962 4-97 8038 6 55 443° 3.81 2169 2261 4-97 7739 5 56 4658 3.81 2099 2559 4-97 7441 4 57 4887 3.80 2029 2858 4-97 7142 3 58 5H5 3.80 1959 1. 16 3 J 5 6 4-97 6844 2 59 ™ 5343 3.80 1889 1. 17 3454 4-97 6546 1 60 9.685571 Cosine. 9.941819 Diff.l" 9-743752 Cotang. 10.256248 Diff. 1" Sine. Diff. l" Tang. M. 118° 61° 70 29° SINES AND TANGENTS. 150° M. Sine. Diff. 1" Cosine. Diff.l" Tang. Diff. 1" Cotang. 60 9.685571 3.80 9.941819 1. 17 9-74375 2 4.96 IO.256248 1 5799 3 79 1749 4050 4.96 595° 59 2 6027 3 79 1679 4348 4.96 5 6 5 2 58 3 6254 3 79 1609 4645 4.96 5355 57 4 6482 3 79 1539 4943 4.96 5°57 56 5 6709 3 78 1469 5 2 4° 4-95 4760 55 6 6936 3 78 1398 5538 4-95 4462 54 7 7163 3 78 1328 5835 4-95 4165 53 8 73 8 9 3 78 1258 6132 4-95 3868 52 9 7616 3 77 1187 6429 4-95 357i 51 10 11 7843 3 77 1117 I.17 1. 18 6726 4-95 3274 50 49 9.688069 3 77 9.941046 9.747023 4.94 10.252977 12 8295 3 77 0975 7319 4.94 2681 48 13 8521 3 76 0905 7616 , 4-94 2384 47 14 8747 3 76 0834 79*3 4.94 2087 46 15 8972 3 76 0763 8209 4.94 1791 45 16 9198 3 76 0693 8505 4-93 1495 44 IV 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 21 9.690098 0323 3 75 0409 9689 9.749985 4-93 4-93 03 1 1 40 39 3 74 9.940338 10.250015 22 0548 3 74 0267 9.750281 4-93 10.249719 38 23 0772 3 74 OI96 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 7653 31 30 31 2339 9.692562 3 72 9697 9.939625 2642 9-75 2 937 4.91 7358 10.247063 30 29 3 72 4.91 32 2785 3 7i 9554 3 2 3 J 4.91 6769 28 33 3008 3 7i 9482 3526 4.91 6474 27 34 3 2 3 J 3 7i 9410 3820 4.90 6180 26 35 3453 3 7i 9339 1. 1 9 4115 4.90 5885 25 36 3676 3 70 9267 I.20 4409 4.90 559 1 24 3V 3898 3 70 9 J 95 4703 4.90 5297 23 38 4120 3 70 9123 4997 4.90 5003 22 39 4342 3 70 9052 5 2 9! 4.90 4709 21 40 41 4564 9.694786 3 69 8980 9.938908 5585 9.755878 4.89 4.89 4415 20 ~i7S8 92999 38376 92343 39982 91660 4*575 90948 26 3D 35157 93616 36785 92988 38403 92332 40008 91648 41602 90936 25 36 35184 93606 36812 92978 38430 92321 40035 91636 41628 90924 24 37 35211 93596 36839 92967 38456 92310 40062 91625 4*655 90911 23 38 35239 93585 36867 92956 38483 92299 40088 91613 41681 90899 22 39 35266 93575 36894 92945 38510 92287 401 1 5 91601 4*707 90887 21 40 41 35293 35320 93565 93555 36921 36948 92935 92924 38537 38564 92276 j 92265 1 401 4 1 40168 91590 4*734 41760 90875 90863 20 l 19 91578 42 35347 93544 36975 92913 38591 92254| 40195 91566 4*787 90851 18 43 35375 93534 37002 92902 38617 92243 | 40221 9*555 4*813 90839 17 44 354° 2 935 2 4 37029 92892 38644 92231 i 40248 9*543 41840 90826 16 4o 354 2 9 935*4 37°5 6 92881 38671 92220 40275 9*53* 41866 90814 15 46 35456 935°3 37083 92870 38698 92209 40301 9*5*9 41892 90802 14 47 354H 93493 37110 92859 38725 92198 40328 91508 4*9*9 90790 13 48 355** 93483 37137 92849 .38752 92186 j 40355 9*496 4*945 90778 12 49 35538 93472 37164 92838 38778 92175 1 40381 9*484 4*972 90766 11 50 51 355 6 5 3559 2 93462 93452 37*9* 37218 92827 92816 38805 38832 92164 92152 40408 40434 91472 11998 42024 90753 90741 10 9 91461I 52 35619 93441 37245 92805 3*859 92141 40461 9*449 42051 90729 8 53 35647 9343 1 37272 92794 38886 92130 40488 9*437 42077 907*7 7 54 35674 93420 37299 92784 38912 92119 40514 9*425 42104 90704 6 55 357oi 93410 37326 92773 38939 92107 40541 9*4*4 42130 90692 5 j 56 35728 93400 37353 92762 38966 92096 40567 91402 42156 90680 4 57 35755 93389 3738o 92751 3^993 92085 40594 9 i 39 oi 42183 90668 3 58 35782 93379 37407 92740 39020 92073 40621 9*378 42209 90655 2 59 35810 93368 37434 92729 39046 92062 ; 40647 9*366 42235 90643 1 1 60 / 35837 Cosine. 93358 Sine. 3746i Cosine. 92718 Sine. 39073 Cosine. 92050' Sine. 40674 9*355 Sine. 42262 Cosine. 90631 1 / Cosine. Sine. 69° 68° 67° 66° 65° 92 NATURAL SINES AND COSINESS. | / 25° 26° 27° 28° 29° / Sine. Cosine. Sine. Cosine, j Sine. Cosine. Sine. Cosine. Sine. Cosine. 42262 90631 [ 43 8 37 89879 45399 89101 46947 88295 48481 87462 i 42288 90618 ' 43863 89867 45425 89087! 46973 88281 48506 87448 59 2 42315 90606 43889 89854 45451 89074 | 46999 88267 48532 87434 58 3 42341 90594 43916 89841 45477 89061 47024 88254 48557 87420 57 4 42367 90582 43942 89828 45503 89048 47050 88240 48583 87406 56 5 4 2 394 90569 43968 89816 45529 89035 47076 88226 48608 87391 55 6 42420 9°557 43994 89803 45554 89021 47101 88213 48634 87377 54 V 42446 9°545 44020 89790 45580 89008 47127 88199I 4865,9 87363 53 s 42473 90532 44046 89777 45606 88995 47153 88185 48684 87349 52 y 42499 90520 44072 89764 45632 88981 47178 88172! 48710 87335 51 10 42525 90507 44098 89752 45658 88968 47204 88158 48735 87321 50 n 42552 90495 44124 89739 45684 88955 47229 88144 48761 87306 49 12 42578 90483 441 5 1 89726 45710 88942 47255 88130, 48786 87292 48 13 42604 90470 44*77 89713 4573 6 88928 47281 88117 48811 87278 47 ! 14 42631 90458 44203 89700 45762 88915 47306 88103I 48837 87264 46 lb 42657 90446 44229 89687 457 8 7 88902 47332 88089 48862 87250 45 1 16 42683 90433 44255 89674 45813 88888 47358 88075 j 48888 87235 44 1 iv 42709 90421 44281 89662 45 8 39 88875 47383 88062! 4S913 87221 43 18 42736 90408 1 44307 89649 45865 88862 47409 880481148938 87207 42 1 iy 42762 90396 44333 89636 45891 88848 47434 880341I48964 87193 41 j 20 42788 90383 44359 89623 459 J 7 88835 47460 88020: 48989 87178 40 i 21 42815 90371 443 8 5 89610 45942 88822 47486 88006 j 49014 87164 39 1 22 42841 90358 4441 1 8 9597 45968 88808 475 11 87993 49040 87150 38 23 42867 90346: 44437 89584 45994 88795 47537 87979 ! 49065 87136 37 1 a 42894 9°334j 44464 89571 46020 88782 47562 879651 49090 87121 36 ! 2o 42920 90321 44490 89558 46046 88768 47588 87951 49116 87107 35 j 26 42946 90309! 44516 8 9545 46072 88755 47614 87937| 49141 87093 34 ; 2V 42972 90296 : 44542 89532 46097 88741 47639 87923 49166 87079 33 : 28 42999 90284: 44568 89519 46123 88728 47665 87909 ;49 I 9 2 87064 32 29 43025 9027I j 44594 89506 46149 88715 47690 87896:49217 87050 31 30 43°5i 90259 j 44620 89493 46175 88701 47716 87882J 49242 87036 30 31 43°77 90246 i 44646 89480 46201 88688 47741 87868^49268 87021 29 32 43104 90233: 44672 89467 46226 88674 47767 87854 149293 87007 28 1 33 43130 90221 44698 89454 46252 88661 r47793 87840 49318 86993 27 1 34 43156 90208 44724 89441 46278 88647 47818 87826 49344 86978 26 85 43182 9OI96 4475° 89428 46304 88634 47844 87812 49369 86964 25 j 36 43209 90183 44776 89415 46330 88620 47869 87798! 49394 86949 24 i 8V 43 2 35 9OI7I 44802 89402 4 6 355 88607 47895 87784- 49419 86935 23 38 43261 90158 j 44828 89389 46381 88 593 47920 87770 49445 86921 22 i 8y 43287 90146 ' 44854 89376 46407 88580 47946 87756; 4947o 86906 21 1 40 43313 90133 | 44880 89363 4 6 433 88566 4797 1 877431 49495 86892 20 j 41 4334° 90120 44906 89350 46458 88553 47997 87729 49521 86878 19 42 43366 90108 44932 89337 46484 88539 48022 87715' 49546 86863 IS 43 43392 9OO95 44958 89324 '46510 88526 48048 87701 j 4957i 86849 17 44 434* 8 90082 44984 89311 46536 88512 48073 87687 49596 86834 16 ! 4o 43445 90070' 45010 89298 46561 88499 48099 87673 49622 86820 15 i 46 43471 9OO57; 45036 89285 46587 88485 48124 87659! 49647 86805 14 1 4/ 43497 90045 45062 89272 46613 88472 48150 87645 49672 86791 13 48 43523 9OO3Z '45088 89259 46639 88458 48175 87631! 49697 86777 12 | 49 43549 90019 45"4 89245 46664 88445 48201 87617 49723 86762 11 1 50 43575 90007 45140 89232 46690 88431 48226 87603 49748 86748 10 ! 51 43602 89994 1 45166 89219 46716 88417 48252 87589! 49773 86733 9 i | 52 43628 89981 !45 I 9 2 89206 46742 88404 48277 87575 i 49798 86719 8 53 43 6 54 89968 145218 89193 46767 88390 48303 875611 49824 86704 7 o4 43680 89956 ! 45243 89180 46793 88377 48328 87546: 49849 86690 6 i 5o 43706 8 9943| 1 45269 89167 46819 88363 48354 875321 49874 86675 5 56 43733 89930 45295 8 9*53 46844 88349 48379 87518J 49899 86661 4 57 43759 89918 145321 89140 46870 883361 48405 87504' 49924 86646 3 58 437*5 89905 '45347 89127 46896 88322 48430 87490: 4995° 86632 2 o9 43811 89892; 145373 89114 46921 88308 48456 87476 : 49975 86617 1 60 / 43 8 37 Cosine. 89879 Sine. 45399 Cosine. 89101 Sine. 46947 Cosine. 88295 Sine. 48481 Cosine. 87462 ; Sine, j 50000 86603 / Cosine. Sine. 64° 63° 62° 61° 60° 93 NATURAL SINES AND COSINES. / 30° 31° 32° 33° 34° / 60 Sine. Cosine. 86603 Sine. Cosine. Sine. 52992 Cosine. Sine. Cosine. 83867 Sine. Cosine. 50000 51504 85717 84805 54464 559*9 82904 l 50025 86588 51529 85702 53017 84789 54488 83851 55943 82887 59 2 50050 86573 51554 85687 53041 84774 545*3 83835 55968 82871 58 3 50076 86559 5*579 85672 53o66 84759 54537 83819 55992 82855 57 4 50101 86544 5160J. 85657 5309* 84743 5456i 83804 56016 82839 56 5 50126 86530 51628 85642 53**5 84728 54586 83788 56040 82822 55 6 5 OI 5* 86515 5l 6 53 85627 53*4° 84712 54610 83772 56064 82806 54 V 50176 86501 51678 85612 53*64 84697 54635 83756 56088 82790 53 8 50201 86486 5*703 85597 53189 84681 54659 83740 56112 82773 52 y 50227 86471 51728 85582 53214 84666 54683 83724 56136 82757 51 10 50252 50277 86457 86442 5*753 51778 85567 53238 53263 84650 84~6~35 547o8 54732 83708 56160 56184 82741 50 49 85551 83692 82724 12 50302 86427 51803 85536 53288 84619 54756 83676 56208 82708 48 13 50327 86413 51828 85521 533*2 84604 54781 83660 56232 82692 47 14 50352 86398 51852 85506 53337 84588 54805 83645 56256 82675 46 15 50377 86384 51877 85491 5336i 84573 54829 83629 56280 82659 45 16 50403 86369 51902 85476 53386 84557 54854 83613 56305 82643 44 IV 50428 86354 51927 85461 534** 84542 54878 83597 56329 82626 43 18 5°453 86340 51952 85446 53435 84526 54902 83581 56353 82610 42 19 50478 86325 5*977 8543* 53460 845 1 1 54927 83565 56377 82593 41 20 21 5 503 50528 86310 86295 52002 52026 85416 53484 53509 84495 84480 5495* 54975 83549 83533 56401 56425 82577 82561 40 39 85401 22 50553 86281 52051 85385 53534 84464 54999 835*7| 56449 82544 38 23 50578 86266 52076 85370 53558 84448 55024 83501 56473 82528 37 24 50603 86251 52101 85355 53583 84433 55048 83485 56497 82511 36 25 50628 86237 52126 85340 53607 844*7 55072 83469 56521 82495 30 26 5° 6 54 86222 5**5* 85325 53632 84402 55097 83453 56545 82478 34 27 50679 86207 52175 853*0 53656 84386 55*2* 83437 56569 82462 33 28 50704 86192 52200 85294 53681 84370 55*45 83421 56593 82446 32 29 50729 86178 52225 85279 53705 84355 55*69 83405 56617 82429 31 30 31 50754 50779 86163 86148 52250 52275 85264 5373° 53754 84339 55*94 552i8 83389 83373 56641 56665 824J3 82396 30 29 85249 84324 32 50804 86133 52299 85234 53779 84308 55242 83356 56689 82380 28 33 50829 86119 52324 85218 53804 84292 55266 83340 567*3 82363 27 34 50854 86104 5 2 349 85203 53828 84277 55291 83324 56736 82347 26 35 50879 86089 5 2 374 85188 53853 84261 553*5 83308 56760 82330 25 36 50904 86074 5 2 399 85*73 53877 84245 55339 83292 56784 82314 24 37 50929 86059 5 2 4 2 3 85*57 53902 84230 55363 83276 56808 82297 23 38 50954 86045 52448 85142 53926 84214 55388 83260 56832 82281 22 39 50979 86030 5 2 473 85*27 5395* 84198 554*2 83244 56856 82264 21 40 iTf 51004 86015 86000 52498 52522 85112 85096 53975 54000 84182 84167 55436 5546o 83228 83212 56880 56904 82248 82231 20 19 51029 42 51054 «59«5 5 2 547 85081 54024 84151 55484 83195 56928 82214 18 , 43 51079 85970 52572 85066 54049 84*35 55509 83*791 56952 82198 17 44 5 1 104 8595b 52597 85051 54073 84120 55'533 83163 56976 82181 16 46 51129 85941 52621 85035 54097 84104 55557 83*47 57000 82165 15 46 5"54 85926 52646 85020 54122 84088 5558i 83131 57024 82148 14 47 5**79 85911 52671 85005 54146 84072 55605 83*15 57047 82132 13 48 51204 85896 52696 84989 54*7* 84057 55630 83098 1 57071 82115 12 49 51229 85881 52720 84974 54*95 84041 55654 83082 57095 82098 11 50 l"5T 5^54 51279 85866 8585I 52745 84959 84943 i 54220 54244 84025 84009 55678 55702 83066 83050 57**9 57*43 82082 10 9 52770 82065 52 51304 85836 52794 84928 [54269 83994 55726 83034 57167 82048 8 53 5*3 2 9 85821 52819 84913 54293 83978 5575° 83017 57191 82032 7 54 5*354 85806 52844 84897 543*7 83962 55775 83001 57215 82015 6 65 5*379 85792 52869 84882 54342 83946 55799 82985 57238 81999 5 56 51404 85777 52893 84866 54366 83930 55823 82969 57262 81982 4 57 51429 85762 52918 84851 5439* 839*5 55847 82953 57286 81965 3 58 5H54 8 5747 52943 84836 544*5 83899 5587* 82936 573*o 81949 2 59 5H79 8 573 2 52967 84820 54440 83883 55895 82920 57334 81932 1 60 / 51504 Cosine. 85717 Sine. 52992 84805 Sine. 54464 Cosine. 83867 Sine. 559*9 Cosine. 82904 57358 8*9*5 / Cosine. Sine. Cosine. Sine. 59° 58° 57° 56° 55° 94 '■■ -"■■"■"■'— .im— ■.-».■— —.—..— — p-^»a NATURAL SIEfES AETO COSINES. / 35° 36° 37° 38° 39° / 60 Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. 5735 8 81915 58779 80902 60182 79864 61566 78801 62932 77715 1 57381 81899 58802 80885 60205 79846 61589 78783 62955 77696 59 2 574°5 81882 58826 80867 60228 79829 61612 78765 62977 77678 58 | a 574 2 9 81865 58849 80850 60251 79811 61635 78747 63000 77660 57 4 57453 81848 58873 80833 60274 79793 61658 78729 63022 77641 56 5 57477 81832 58896 80816 60298 79776 61681 78711 63045 77623 55 6 57501 81815 58920 80799 60321 79758 61704 78694 63068 776o5 54 V 575M 81798 58943 80782 60344 7974 1 61726 78676 63090 77586 53 8 57548 81782 58967 80765 60367 79723 61749 78658 63113 7750* 52 9 57572 81765 58990 80748 60390 79706 61772 78640 63^5 7755° 51 10 11 5759 6 57619 81748 59014 59037 80730 89713 60414 60437 79688 79671 61795 61818 78622 63158 63180 7753 1 775*3 50 49 81731 78604 12 57643 81714 59061 80696 60460 79653 61841 78586 63203 77494 48 13 57667 81698 59084 80679 60483 79635 61864 78568 63225 77476 47 14 57691 81681 59108 80662 60506 79618 61887 78550 63248 77458 46 15 57715 81664 59 I 3 I 80644 60529 79600 61909 78532 63271 77439 45 16 5773 s 81647 59*54 80627 6o553 79583 61932 78514 63293 77421 44 17 57762 81631 59178 80610 60576 795 b 5 61955 78496 63316 77402 43 18 57786 81614 59201 8o593 60599 79547 61978 78478 63338 77384 42 19 57810 81597 59225 80576 60622 7953° 62001 78460 63361 77366 41 20 21 57833 57857 81580 81563 59248 59272 80558 60645 60T68 79512 79494 62024 62046 78442 78424 63383 63406 77347 77329 40 39 80541 22 57881 81546 59295 80524 60691 79477 62069 78405 63428 77310 38 23 57904 81530 593 l8 80507 60714 79459 62092 78387 63451 77292 37 24 57928 81513 59342 80489 | 60738 79441 62115 78369 63473 77273 36 2o 57952 81496 593 6 5 80472 60761 79424 62138 78351 63496 77255 35 26 5797 6 81479 59389 80455 1 60784 794o6 62160 78333 63518 77236 34 ti 57999 81462 59412 80438 60807 79388 62183 78315 63540 77218 33 28 58023 81445 59436 80420 60830 79371 62206 78297 63563 77199 32 29 58047 81428 59459 80403 60853 79353 62229 78279 63585 77181 31 30 31 58070 81412 81395 59_48_ 2 595o6 80386 60876 60899 79335 79318 62251 78261 78243 : 63608 63630 77162 77144 30 29 j 58094 80368 62274 32 58118 81378 59529 8035! 60922 79300 62297 78225 | 63653 77125 28 33 58141 81361 59552 80334 60945 79282 62320 78206 63675 77107 27 34 58165 81344 59576 80316 60968 79264 62342 78188 63698 77088 26 35 58189 81327 159599 80299 60991 79247 62365 78170 63720 77070 25 36 58212 81310J 59622 80282 61015 79229 62388 78152 63742 77051 24 37 58236 81293! 59646 80264 61038 79211 6241 1 78134 63765 77033 23 1 38 58260 812761 59669 80247 61061 79193 62433 78116 63787 77014 22 | 39 58283 81259 59 6 93 80230 61084 79176 62456 78098 63810 76996 21 ! 40 41 5 8 3°7 5 8 33° 81242 81225 59716 59739 80212 ! 61 I07 - 61130 79158 79140 62479 62502 78079 78061 63832 63854 76977 76959 20 19 80195 42 5 8 354 81208 59763 80178 6II53 79122 62524 78043 63877 76940 18 43 58378 81191 59786 80160 !6n 7 6 79105 62547 78025 63899 76921 17 44 58401 81 174 59809 80143 61199 79087 62570 78007 63922 76903 16 45 58425 8II57 59832 80125 61222 79069 62592 77988 63944 76884 15 i 46 58449 81140 59856 80108 61245 79051 62615 77970 63966 76866 14 47 58472 81123 59879 80091 61268 79033 62638 77952 63989 76847 13 48 58496 81106 59902 80073 61291 79016 62660 77934 6401 1 76828 12 1 49 58519 81089 59926 80056 I61314 78998 62683 77916 64033 76810 11 j 50 58543 81072 59949 80038 61337 78980 62706 77897 64056 76791 10 ! 51 58567 81055 59972 80021 \ 61360 78962 62728 77879 64078 76772 9 52 5859 81038 59995 80003 61383 78944 62751 77861 64100 76754 8 53 58614 81021 60019 79986 61406 78926 62774 77843 64123 76735 7 54 58637 81004 60042 79968 61429 78908 62796 77824 64145 76717 6 55 58661 80987 60065 79951 61451 78891 62819 77806 64167 76698 5 56 58684 80970 60089 79934 61474 78873 62842 77788 64190 76679 4 57 58708 80953 60112 79916 61497 78855 62864 77769 64212 76661 3 58 58731 80936 60135 79899 1 61520 78837 62887 7775 1 64234 76642 2 69 58755 80919 60158 79881 •61543 78819 ■ 62909 77733 64256 76623 1 60 / 58779 80902 60182 Cosine. 79864 Sine. 61566 Cosine. 78801 62932 77715 Sine. 64279 Cosine. 76604 Sine. * Cosine. Sine. Sine. ! Cosine. i 54° 53° 52° 51° 50° 95 NATURAL SINES AND COSINES. /. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 I 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 / 40° 41° 42° 43° 44° / 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 a 2 1 / Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. 64279 64301 64323 64346 64368 64390 64412 64435 64457 64479 64501 64524 64546 64568 64590 64612 64635 64657 64679 64701 64723 64746 64768 64790 64812 64834 64856 64878 64901 64923 64945 64967 64989 65011 65033 65055 65077 65100 65122 65144 65166 65188 65210 65232 65254 65276 65298 65320 65342 65364 65386 65408 65430 65452 65474 65496 65518 65540 65562 65584 65606 Cosine. 76604 76586 76567 76548 76530 76511 76492 76473 76455 76436 76417 76398 76380 76361 76342 76323 76304 76286 76267 76248 76229 76210 76192 76173 76154 76135 76116 76097 76078 76059 76041 76022 76003 75984 75965 75946 75927 759o8 75889 75870 75851 75832 75813 75794 75775 75756 75738 75719 757oo 75680 75661 75642 75623 75604 75585 75566 75547 75528 75509 75490 75471 Sine. 65606 65628 65650 65672 65694 65716 65738 65759 65781 65803 65825 65847 65869 65891 65913 65935 65956 65978 66000 66022 66044 66066 66088 66109 66131 66153 66175 66197 66218 66240 66262 66284 66306 66327 66349 66371 66393 66414 66436 66458 66480 66501 66523 66545 66566 66588 66610 66632 66653 '66675 66697 66718 66740 66762 66783 66805 66827 66848 66870 66891 66913 Cosine. 75471 75452 75433 75414 75395 75375 75356 75337 753i8 75299 75280 75261 75241 75222 75203 75184 75i65 75H6 75126 75107 75088 75069 75050 75030 75011 74992 74973 74953 74934 74915 74896 74876 74857 74838 74818 74799 74780 74760 74741 74722 74703 74683 74664 74644 74625 74606 74586 74567 74548 74528 74509 74489 74470 74451 74431 74412 74392 74373 74353 74334 74314 Sine. 66913 66935 66956 66978 66999 67021 67043 67064 67086 67107 67129 67151 67172 67194 67215 67237 67258 67280 67301 67323 67344 67366 67387 67409 67430 67452 67473 67495 67516 67538 67559 67580 67602 67623 67645 67666 67688 67709 67730 67752 67773 67795 67816 67837 67859 67880 67901 67923 67944 67965 67987 68008 68029 68051 68072 68093 68115 68136 68157 68179 68200 743H 74295 74276 74256 74237 74217 74198 74178 74159 74139 74120 74100 74080 74061 74041 74022 74002 73983 73963 73944 73924 73904 73885 73865 73846 73826 73806 73787 73767 73747 73728 737o8 73688 73669 73649 73629 73610 7359° 7357° 73551 73531 735" 73491 73472 73452 73432 73413 73393 73373 73353 73333 733H 73294 73274 73254 73234 73215 73195 73175 73155 73135 Sine. 68200 68221 68242 68264 68285 68306 68327 68349 68370 68391 68412 68434 68455 68476 68497 68518 68539 68561 68582 68603 68624 68645 68666 68688 68709 68730 68751 68772 68793 68814 68835 68857 68878 68899 68920 68941 68962 68983 69004 69025 69046 69067 69088 69109 69130 69151 69172 69193 69214 69235 69256 69277 69298 69319 69340 69361 69382 69403 69424 69445 69466 Cosine. 73135 73116 73096 73076 73056 73036 73016 72996 72976 72957 72937 72917 72897 72877 72857 72837 72817 72797 72777 72757 72737 72717 72697 72677 72657 72637 72617 72597 72577 72557 72537 72517 72497 72477 72457 72437 72417 72397 72377 72357 72337 72317 72297 72277 72257 72236 72216 72196 72176 72156 72136 72116 72095 72075 72055 72035 72015 71995 71974 71954 71934 Sine. 69466 69487 69508 69529 69549 69570 69591 69612 69633 69654 69675 69696 69717 69737 69758 69779 69800 69821 69842 69862 69883 69904 69925 69946 69966 69987 70008 70029 70049 70070 70091 70112 70132 70153 70174 70195 70215 70236 70257 70277 70298 70319 70339 70360 70381 70401 70422 70443 70463 70484 70505 70525 70546 70567 70587 70608 70628 70649 70670 70690 70711 Cosine. 71934 71914 71894 71873 71853 71833 71813 71792 71772 71752 71732 71711 71691 71671 71650 71630 71610 71590 71569 71549 71529 71508 71488 71468 7H47 71427 71407 71386 71366 71345 71325 71305 71284 71264 71243 71223 71203 71182 71162 71141 71121 71 100 71080 71059 71039 71019 70998 70978 70957 70937 70916 70896 70875 70855 70834 70813 70793 70772 70752 70731 70711 Cosine. Sine. 49° 48° 47° 46° 45° 96 TABLE OF CHOKDS. « A TABLE OF CHORDS. M. 0° 1° 2° 3° 40 5° 6° 7° 8° M. oooo .0175 .0349 .0524 .0698 .0872 .1047 .1221 •1395 5 0015 .0189 .0364 .0538 .0713 .0887 .1061 .1235 .1410 5 10 0029 .0204 .0378 •o553 .0727 .0901 .1076 .1250 .1424 10 15 0044 .0218 •°393 .0567 .0742 .0916 .1090 .1265 .1439 15 20 0058 .0233 .0407 .0582 .0756 .0931 .1105 .1279 •1453 20 25 0073 .0247 .0422 .0596 .0771 .0945 .1119 .1294 .1468 25 30 0087 .0262 .0436 .0611 .0785 .0960 •"34 .1308 .1482 30 35 0102 .0276 .0451 .0625 .0800 .0974 .1148 •i3 2 3 .1497 35 40 0116 .0291 .0465 .0640 .0814 .0989 .1163 •1337 .1511 40 45 0131 .0305 .0480 .0654 .0829 .1003 .1177 •1352 .1526 45 50 0145 .0320 .0494 .0669 .0843 .1018 .1192 .1366 .1540 50 55 0160 .0335 .0509 .0683 .0858 .1032 .1206 .1381 •1555 55 60 0175 .0349 .0524 .0698 .0872 .1047 .1221 •1395 .1569 60 9° 10° 11° 12° 13° 14° 15° 16° 17° 1569 .1743 .1917 .2091 .2264 .2437 .2611 .2783 .2956 5 1^4 .17555 .1931 .2105 .2279 .2452 .2625 .2798 .2971 5 10 1598 .1772 .1946 .2119 .2293 .2466 .2639 .2812 .2985 10 15 1613 .1787 .i960 .2134 .2307 .2481 .2654 .2827 .2999 15 20 1627 .1801 •1975 .2148 .2322 .2495 .2668 .2841 .3014 20 25 1642 .1816 .1989 .2163 .2336 .2510 .2683 .2855 .3028 25 30 .1656 .1830 .2004 .2177 .2351 .2524 .2697 .2870 .3042 30 35 1671 .1845 .2018 .2192 .2365 .2538 .2711 .2884 .3057 35 40 ( .1685 .1859 .2033 .2206 .2380 •2553 .2726 .2899 .3071 40 45 .j-00 .1873 .2047 .2221 .2394 .2567 .2740 .2913 .3086 45 50 1714 .1888 .2062 .2235 .2409 .2582 •2755 .2927 .3100 50 55 ,729 .1902 .2076 .2250 .2423 .2596 .2769 .2942 •3"4 55 60 1743 .1917 .2091 .2264 • 2 437 .2611 .2783 .2956 .3129 60 98 TABX.E OF CHORDS. M. T 18° 19° 20° •3473 21° 22° 23° 24° 25° 26° M. .3129 .3301 •3 6 45 .3816 •3987 •4158 .4329 •4499 5 1 •3*43 •33i5 •3487 •3 6 59 .3830 .4002 .4172 •4343 •4513 5 10 •3*57 .3330 .3502 •3 6 73 •3845 .4016 .4187 •4357 .4527 10 15 j .3172 •3344 .3516 .3688 •3859 .4030 .4201 .4371 .4542 15 20 .3186 •3358 •353° .3702 •3873 .4044 .4215 .4386 .4556 20 25 .3200 •3373 •3545 .3716 .3888 .4059 .4229 •4400 .4570 25 30 •3 2I 5 •3387 •3559 •373° .3902 .4073 .4244 .4414 •4584 30 35 .3229 .3401 •3573 •3745 .3916 .4087 .4258 .4428 •4598 35 40 .3244 .3416 .3587 •3759 •393° .4101 .4272 .4442 .4612 40 45 .3258 •343° .3602 •3773 •3945 .4116 .4286 .4456 .4626 45 50 .3272 •3444 .3616 .3788 •3959 .4130 .4300 .4471 .4641 50 55 .3287 •3459 •3 6 3° .3802 •3973 .4144 ■43 x 5 •4485 .4655 55 60 .3301 •3473 •3 6 45 .3816 •3987 .4158 •43 2 9 •4499 .4669 60 ~0~ 27° 28° 29° 30° •5!76 31° 32° 33° 34° 35° .4669 .4838 .5008 •5345 •55i3 .5680 •5847 .6014 5 .4683 •4853 .5022 .5190 •5359 •55^7 .5694 .5861 .6028 5 10 .4697 .4867 .5036 .5204 •5373 •554i .5708 •5875 .6042 10 15 •47 1 1 .4881 .5050 .5219 .5387 •5555 .5722 .5889 .6056 15 20 .4725 •4895 .5064 -52-33 .5401 •55 6 9 •573 6 •59°3 .6070 20 25 .4740 .4909 .5078 .5247 •5415 .5583 •575° •5917 .6083 25 30 •4754 .4923 .5092 .5261 •54^9 •5597 .5764 •593 1 .6097 30 35 .4768 •4937 .5106 •5 2 75 •5443 .5611 •5778 •5945 .6m 35 40 .4782 •49 5 " .5120 .5289 •5457 .5625 .5792 •5959 .6125 40 45 .4796 .4965 •5 J 34 •53°3 •547i •5638 .5806 .5972 .6139 45 50 .4810 •4979 .<5i 4 8 •53 x 7 •5485 .5652 .5820 .^986 .6153 50 55 .4824 •4994 .5162 •533 1 •5499 .5666 .5833 .6000 .6167 55 60 .4838 .5008 .5176 •5345 •5513 .5680 .5847 .6014 .6180 60 36° 37° 38° 39° 40° 41° 42° 43° 44° .6180 .6346 •6511 .6676 .6840 .7004 .7167 •733° .7492 5 .6194 .6360 .6525 .6690 .6854 .7018 .7181 •7344 .7506 5 10 .6208 .6374 ■6539 .6704 .6868 .7031 •7i95 •7357 •75 I 9 10 15 .6222 .6387 •6553 .6717 .6881 .7045 .7208 •737i •7533 15 20 .6236 .6401 .6566 .6731 .6895 .7059 .7222 •7384 .7546 20 25 .6249 .6415 .6580 .6745 .6909 .7072 •7235 •7398 .7560 25 30 .6263 .6429 .6594 .6758 .6922 .7086 .7249 •74" •7573 30 35 .6277 .6443 .6608 .6772 .6936 .7099 .7262 .7425 •7586 35 40 .6291 .6456 .6621 .6786 .6950 •7"3 .7276 .7438 .7600 40 45 .6305 .6470 .6635 .6799 .6963 .7127 .7289 .7452 .7613 45 50 .6319 .6484 .6649 .6813 .6977 .7140 •73°3 .7465 .7627 50 55 .6332 .6498 .6662 .6827 .6991 •7154 .7316 •7479 .7640 55 60 .6346 .6511 .6676 .6840 .7004 .7167 •733° .7492 .7654 60 45° 46° 47° 48° 49° 50° 51° 52° 53° .7654 .7815 •7975 .8135 .8294 .8452 .8610 .8767 .8924 5 .7667 .7828 .7988 .8148 .8307 .8466 .8623 .8780 •8937 5 10 .768! .7841 .8002 .8161 .8320 •8479 .8636 •8794 .8950 10 15 .7694 .7855 .8015 .8175 •8334 .8492 .8650 .8807 .8963 15 20 .7707 .7868 .8028 .8188 •8347 .8505 .8663 .8820 .8976 20 25 .7721 .7882 .8042 .8201 .8360 .8518 .8676 .8833 .8989 25 30 •7734 .7895 •8o<;s .8214 •8373 •8531 .8689 .8846 .9002 30 35 .7748 .7908 .8068 .8228 .8386 .8545 .8702 .88s 9 .9015 35 40 .7761 .7922 .8082 .8241 .8400 .8558 .8715 .8872 .9028 40 45 •7774 •7935 .8095 .8254 .8413 .8571 .8728 .8885 .9041 45 50 .7788 •7948 .8108 .8267 .8426 .8584 .8741 .8898 .9054 50 55 .7801 .7962 .8121 .8281 •8439 •8597 .8754 .8911 .9067 55 60 .7815 •7975 •8i35 1 -8294 .8452 .8610 .8767 | .8924 .9080 60 99 TABLE OF CHORDS. M. 54° 55° 56° 57° 58° 59° 60° 61° 62° M. .9080 .9235 .9389 •9543 .9696 .9848 1. 0000 1.0151 1. 0301 5 .9093 .9248 .9402 .955b •9709 .9861 1. 0013 1. 0163 1.0313 5 10 .9106 .9261 •94i5 .9569 .9722 .9874 1.0025 1. 0176 1.0326 10 15 .9119 .9274 .9428 .9581 •9734 .9886 1.0038 1.0188 i-°33 8 15 20 .9132 .9287 .9441 •9594 •9747 .9899 1.0050 1. 0201 1.0351 20 25 •9i45 .9299 •9454 .9607 .9760 .9912 1.0063 1. 0213 1.0363 25 30 •9i57 .9312 .9466 .9620 .9772 .9924 1.0075 1.0226 i-o375 30 35 .9170 •93*5 •9479 •9 6 33 •97*5 •9937 1.0088 1.0238 1.0388 35 40 .9183 -933 8 .9492 .9645 .9798 .9950 I.OIOI 1. 0251 1.0400 40 45 .9196 •935i •9505 . 9 b 5 8 .9810 .9962 1.0113 1.0263 1. 0413 45 50 .9209 •93 6 4 .9518 .9671 .9823 •9975 1. 0126 1.0276 1.0425 50 55 .9222 •9377 •953° .9683 .9836 .9987 1.0138 1.0288 1.0438 55 60 •9*35 .9389 •9543 .9696 .9848 67° 1. 0000 1.0151 1. 0301 1.0450 60 63° 64° 65° 66° 68° 69° 70° 71° 1.0450 1.0598 1.0746 1.0893 1. 1039 1.1184 1.1328 1. 1472 1.1614 5 1.0462 1.0611 1.0758 1.0905 1.1051 1.1196 1. 1340 1. 1483 1. 1626 5 10 1.0475 1.0623 1. 0771 1. 0917 1. 1063 1. 1208 1. 1352 1. 1495 1. 1638 10 15 1.0487 1.0635 1.0783 1.0929 1. 1075 1. 1220 1-1364 1. 1507 1. 1650 15 20 1.0500 1.0648 1.0795 1.0942 1. 1087 1. 1232 1-1376 1.1519 1.1661 20 25 1. 0512 1.0660 1.0807 1.0954 1. 1099 1. 1 244 1.1388 1.1531 1-1673 25 30 1.0524 1.0672 1.0820 1.0966 I. XIII 1. 1256 1. 1 400 1. 1543 1. 1685 30 35 i-o537 1.0685 1.0832 1.0978 1.1123 1. 1268 1.1412 I-I555 1. 1697 35 40 1.0549 1.0697 1.0844 1.0990 1.1136 1. 1280 1. 1424 1. 1567 1. 1709 40 45 1. 0561 1.0709 1.0856 1. 1002 1. 1148 1. 1292 1. 1436 1. 1579 1. 1720 45 50 1.0574 1. 0721 1.0868 1.1014 1.1160 1. 1304 1. 1448 1. 1590 1. 1732 50 55 1.0586 1.0734 1.0881 1. 1027 1.1172 1.1316 1. 1460 1. 1602 1. 1744 55 60 1.0598 1.0746 1.0893 1-1039 1.1184 1. 1328 1. 1472 1.1614 1. 1756 60 72° 73° 74° 75° 76° 77° 78° 79° 80° 1. 1756 1. 1896 1.2036 1. 2175 1. 2313 1.2450 1.2586 1.2722 1.2856 5 1. 1767 1. 1908 1.2048 1. 2187 1.2325 1.2462 1.2598 1-2733 1.2867 5 10 1. 1779 1. 1920 1.2060 1. 2198 1.2336 1-2473 1.2609 1.2744 1.2878 10 15 1.1791 1.1931 1. 2071 1. 2210 1.2348 1.2484 1.2620 1.2755 1.2889 15 20 1. 1803 1. 1943 1.2083 1. 2221 1-2359 1.2496 1.2632 1.2766 1.2900 20 25 1.1814 1. 1955 1.2094 1.2233 1.2370 1.2507 1.2643 1.2778 1.2911 25 30 1. 1826 1. 1966 1. 2106 1.2244 1.2382 1. 2518 1.2654 1.2789 1.2922 30 35 1. 1838 1. 1978 1.2117 1.2256 1.2393 1.2530 1.2665 1.2800 1.2934 35 40 1. 1850 1. 1990 1. 2129 1.2267 1.2405 1.2541 1.2677 1.2811 1.2945 40 45 1.1861 1. 2001 1.2141 1.2279 1. 2416 1.2552 1.2688 1.2822 1.2956 45 50 1. 1873 1.2013 1. 2152 1.2290 1.2428 1.2564 1.2699 1.2833 1.2967 50 55 1. 1885 1.2025 1. 2164 1.2302 1.2439 1-^575 1. 2710 1.2845 1.2978 55 60 1. 1896 1.2036 1.2175 1.2313 1.2450 1.2586 1.2722 1.2856 1.2989 60 i 81° 82° 83° 84° 85° 86° 87° 88° 89° 1.2989 1.3121 1.3252 i-33*3 1. 3512 1.3640 1.3767 i-3 8 93 1. 4018 5 1.3000 1. 3132 1.3263 1.3393 1.3523 1-3651 377* 1.3904 1.4029 5 10 1.3011 I-3H3 !-3 2 74 1.3404 I -3533 1. 3661 3788 I-39H 1.4039 10 15 1.3022 I-3J54 1.3285 1. 3415 J -3544 1.3672 3799 1.3925 1.4049 15 20 1.3033 1. 3165 1.3296 1.3426 1-3555 1.3682 3809 J -3935 1.4060 20 25 1.3044 1. 3176 i-33°7 1-3437 I-3565 1.3693 3820 1-3945 1.4070 25 30 i-3°55 1. 3187 1.3318 1-3447 I-3576 i-37°4 3830 1.3956 1.4080 30 35 1.3066 1. 3198 1.3328 I-345 8 i-35 8 7 I-37H 3841 1.3966 1. 409 1 35 40 1.3077 1.3209 1.3339 1.3469 1-3597 1.3725 3*5i 1.3977 1.4101 40 45 1.3088 1.3220 1-335° 1.3480 1.3608 J-3735 3862 1.3987 1.4m 45 50 1.3099 i-3 2 3 J i-336i 1.3490 1-3619 I-374 6 3*72 1.3997 1. 4122 50 55 1.3110 1.3242 1.3372 1. 3501 1.3629 1-3757 3883 1.4008 1.4132 55 60 1.3121 1.3252 i-33 8 3 1. 3512 1.3640 1.3767 i-3*93 1. 401 8 1.4142 60 100 V J 64 8 LIBRARY OF CONGRESS 029 942 51 4> 2 5 JHU fV BBP 4.H ami 'UK II mi IM U r I %■ i" «n H^H la ml MMM lIHl 11 KK D tii xmra an