IC-NR *B 272 TAYLOR LIBRARY OF THE UNIVERSITY OF CALIFORNIA. Class \ ; SURVEYOR'S HAND BOOK BY T. U. TAYLOR C. E,, University of Virginia, M. C. E., Cotnell University; Professor cf Civil Engineering in the University of Texas; Member American Society of Civil Engineers. CHICAGO AND NEW YORK THE MYRON C. CLARK PUBLISHING CO. 1908 Copyright 1908 By The Myron C. Clark Publishing Co. DEDICATION TO GEORGE W. BRACKENRIDGE SAN ANTONIO, TEXAS. A Civil Engineer and Patriot Who for a Quarter of a Century Has Been an Active and Useful Friend of Higher Education in Texas. 179777 PREFACE. It has been my object to prepare a book for the use of the surveyor in the field, of convenient size and scope, and one that contains all the essentials for ordinary surveying. It is too much to hope that there are no errors in these pages, in theory or example. The preliminary proof has been examined by twelve experienced surveyors and I am indebted to them for many valuable suggestions. I am under lasting obligations to my colleagues, R. A. Thompson, Expert Engineer of the Texas Railway Commission; Edward C. H. Bantel, Adjunct-Professor of Civil Engineering; and Stanley P. Finch, Instructor in Civil Engineering of the University of Texas. In addition to this assistance I have been aided by valuable suggestions from F. Lavis and Halbert P. Gillette, and from the following leading surveyors and engi- neers of Texas : C. F. H. von Blucher, Gustav Schleicher, B. F. Love, and W. D. Twichell. The thanks of the author are hereby expressed to W. & L. E. Gurley, Keuffel & Esser, Eugene Dietzgen Co., and A. Wissler for many illustrations of instruments. The traverse table has been omitted, as the ordinary ones are useless for angles not multiples of quarter degrees, and the large ones are books in themselves. As lands become more valuable, the transit survey is demanded where angles are read to the nearest minute, and for such surveys the small traverse tables are of no avail. Tables I, II, III and IV are taken by permission from Henck's "Field Book," while Table V is from Searles' "Field Engineering." T. U. TAYLOR. Austin, Texas, September 1, 1908. CONTENTS. CHAPTER I. CHAIN SURVEYING. Page 1. Gunter's Chain , 1 2. Engineer's Chain 2 3. Vara Chain 3 4. Steel Tapes 3 5. Standardized Tapes 4 6. Metallic Tapes 5 7. Pins 5 8. Range Poles 6 0. Plumb-bob , 10. Chaining 7 11. Chaining Over Hills or Across Valleys 12. Chain Survey 9 13. Chain Problems 10 14. Correction for Temperature 13 15. Stretch of Tape Due to Pull. 13 16. Correction for Sag ...... 14 17. Erroneous Lengths : 15 18. Erroneous Areas 16 19. Linear Units 17 20. Units of Land Measure 18 21. Area of a Triangle , 19 22. The 57.3 Rule 20 23. Applications of the 57.3 Rule 20 24. Pacing Survey 21 25. Location of Houses 22 26. Survey of Farm by Chain or Pace 23 CHAPTER II. COMPASS SURVEYING. 27. The Bearing of a Line 25 28. Azimuth 25 29. The Compass 25 30. Reading the Bearing , . 27 31. How to Use the Compass 27 32. The Vernier .-.',. 28 vii viii CONTENTS. Page 33. Declination of the Needle 20 '34. Compass Vernier 30 35. To Set Off Declination 31 36. Changes in Declination 31 37. Result of Changes 32 38. Old Lines 33 39. Magnetic Bearing 33 407 To Find the Declination for Any Special Farm. . . 34 41. Local Attraction 3;, 42. Witnessing a Line or Corner 35 43. Typical Field Notes 36 44. Compass Adjustments 36 CHAPTER III. TRANSIT SURVEYING. 45. The Transit 38 46. Compass Attachment 40 47. Vertical Circle ' . '40 48. Shifting Center ' 41 49. The Reticule '.'.'.'.'.'. '41 50. Setting Up the Transit ,41 51. Motions 42 52. Use of the Transit 42 53. The Transit as a Compass 43 54. Transit Surveying ' ' . 43 55. Transit Vernier 43 56. Example 44 57. Reference Lines .44 58. Repeating Method .. 45 59. To Adjust the Plate Levels 46 60. Line of Sight Adjustment 46 61. Peg Adjustment 48 62. Location of Meridian by Polaris .,.. , 48 63. Circumpolar Stars 51 64. Location of Meridian 52 65. PZS Triangle '.'.'.'.'.'.'.'.'.'. 52 66. Formulas 53 67. Observation on Sun 54 68. Refraction 55 69. Solar Attachment .55 70. Meridian Without Calculation 57 71. Example 58 72. Example '59 CONTEXTS. ix CHAPTER IV. CALCULATION OF AREAS. Page 73. Latitude and Departure of a Course 61 74. Traverse Tables 62 75. Example 63 76. Error of Closure 63 77. Balancing a Survey 64 78. The Double Meridian Distance 66 79. Area of a Farm 67 80. Area Table 68 81. Courses of No Latitude or Departure 72 82. Example 73 83. Area by Co-ordinates 75 84. Traversing 76 85. Example 76 8f>. Approximate Traversing 77 87. Irregular Boundaries 78 88. Discrepancies 79 CHAPTER V. DIVISION OF LAND. 89. Division of Triangle 81 90. Division Line Through Internal Point 82 91. Division of Quadrilateral 83 92. General Solution 84 93. Case 1 85 94. Case II. . -. 86 95. Case III 87 96. Case IV 89 97. Example 89 98. Dividing Land 90 99. Example 90 100. Example 92 CHAPTER VI. LEVELING. 101. The Y Level 96 102. The Telescope 96 103. Setting Up the Instrument 100 104. Rods 100 105. Theory of Leveling 102 106. Bench Marks 104 CONTENTS. Page 107. Profiles 106 108. Crosswire Adjustment 106 109. Bubble-Tube Adjustment 106 110. Adjustment of Wyes 107 111. The Radius of the Bubble-Tube 107 112. Curvature of Earth 109 113. Vertical Curves 110 114. Curve in Sag Ill 115. Vertical Circular Curves 112 CHAPTER VII. TOPOGRAPHIC SURVEY. 116. Topographic Survey 1 14 117. Topographic Methods 114 118. Stadia Formulas 115 119. Wire Interval 1 1(> 120. Inclined Sights 117 121. Stadia Rod 118 122. Field Work Ill) 123. Reduction Methods 122 124. Colby's Slide Rule ....'. 123 125. Usual Approximations 124 126. Topography by Hand Level 120 CHAPTER VIII. RAILROAD SURVEYING. 127. Railroad Surveying 129 128. Degree Formula 129 129. General Formula 12!) 130. To Lay Out Curve 130 .131. Obstacles " 132 132. Location by Offsets 132 133. Middle Ordinate 133 134. Approximate Formulas " 133 135. Reduction Tables .134 136. Metric Curves 135 137. Preliminary Survey 136 138. Location Survey '. 136 139. Field Book 138 140. Transit Party 140 141. Stakes 141 142. Hubs 141 143. Hand-Level . 142 COXTEXTS. xi Page 144. Slope Stakes in Excavation 144 145. Slope Stakes in Embankment 147 146. Berms 149 CHAPTER IX. EARTHWORK. 147. Prismoidal Formula , 151 148. Railroad Excavation ; 152 149. Level Sections 153 150. Two-Level Sections 153 151. Three-Level Sections ....... 154 152. Irregular Sections , . 154 153. Rules , i55 154. Side Hill Cuts .... 156 155. Average End Areas 157 156. Error of Average-End Area Formula 158 157. Examples 159 158. Preliminary Estimates 160 159. Earthwork" Note-Book 161 160. Special Case 163 161. Borrow Pits .164 162. End of Fill 166 163. Overhaul .166 164. Shrinkage 167 CHAPTER X. CITY SURVEYING. 165. The City Engineer . 171 166. Objects of Survey .171 167. Monuments 172 168. Additions 173 169. Kinds of Monuments 173 170. Location of Monuments 175 171 . Tapes 175 172. Transit '. 176 i 73. Datum 178 174. General Maps .....179 175. Water-Pipe Map 179 176: City Blocks 180 , 1 77. Rectangular Blocks .., 180 178. Rectangular Lots 181 179. Irregular Blocks and Lots 182 180. Private Notes , 183 xii CONTENTS. Page 181. Prescriptive Rights , ....184 182. Cross- Section of Streets 184 183. City Engineering Records 185 184. Field Note-Books ,, . 18(i 185. Detail Maps 187 186. Orders, Bids. Etc 189 CHAPTER XI. PLOTTING AND LETTERING. 187. Plots 190 188. Protractor Method. 190 189. Latitude and Departure Method 191 190. The Tangent Method 191 191. The Sine Method .'191 192. Co-ordinate Method 192 193. Correcting the Plot 194 194. Lettering 194 CHAPTER XII. GOVERNMENT SURVEYING. 195. Radii of Parallels. 190 196. Angular Convergence of Meridians 196 197. Linear Convergence 197 198. Off-Sets , 198 199. Running Parallels 198 200. Tangent Method 199 201. Secant Method 199 202. Intermediate Off-Sets 199 203. Example 200 204. Reference Meridians and Standard Parallels 200 205. Ranges 202 206. Townships 202 207. Dividing Up a Township 203 CHAPTER XIII. TRIGONOMETRIC FORMULAS. 208. Formulas for Right Triangle 208 209. Solutions for Right Triangle 208 210. Oblique Triangle 209 211. Right Spherical Triangle 210 212. Oblique Spherical Triangle 210 CONTEXTS, xiii CHAPTER XIV, TABLES, Page Table I. Logarithms of Numbers 211 Table II. Logarithmic Sines, Cosines, Tangents and Cotangents 227 Table III. Natural Sines and Cosines 275 Table IV Natural Tangents and Cotangents. 285 Table V. Cubic yards per 100 feet for slopes %:1, %:1, 1:1, 1%:1, 2:1,3:1,.., , ....;..... .299 CHAPTER I. CHAIN SURVEYING. 1. Gunter's Chain. This chain was invented in 1620 by Edmund Gunter. an English surveyor, and is now in use in a majority of the older states of America. Previous to its inven- tion, chains of irregular lengths had been in use, but there was no uniform system, and as soon as Gunter's chain was invented it was generally adopted. This chain is 66 ft. or 792 ins. in length, and is divided into 100 equal parts, called links, each link being 7.92 ins. long. Eighty of these chains make one mile. Also, we know 1 acre = 4,840 sq. yds. = 43,560 sq. ft. 1 sq. chain = 66 X 66 = 4,356 sq. ft. 43.560 ' . 1 acre -= r-- sq. chains, = 10 sq chains. Distances are measured in full chains and decimals. If the distance between two points is 9 full lengths and S3 links, we call the length 9.83 chains, as each link is one-hundredth part of a chain. Each link is composed of three parts a long wire with looped ends and two rings. These rings can be left open or soldered (brazed). If left open they soon become oval and elongated in the direction of the chain, and the chain thus be- comes lengthened. It is therefore best to have all joints brazed, as this makes the ring connections more stable and less liable to stretch. Figure 1 is an illustration of one form of Gunter's chain and the two rings at each joint can be seen in the upper right-hand part of the figure. At each end of the chain there are two brass handles, the measuring length cf the chain being from back to back of the handles. These loop handles are at- tached to the chain by nuts that are intended to be adjustable. When fixed in one position, it is important that they remain stationary till adjusted by the surveyor. The wearing surfaces make it necessary to remove links and this renders the chain in- 1 2 SURVEYOR'S HAND BOOK. accurate for fractional parts of a chain. There are in all about 600 wearing surfaces and if each surface is worn one-hundredth part of an inch, the chain will be lengthened 6 ins. and this would produce an error of 1 per cent in the calculation of areas. The tenth link from the end is marked by a one-point brass tag, the twentieth by a two-point tar, the thirtieth by a three- point, the fortieth by a four-point, and the fiftieth by a round tag, it being the middle of the chain. At the center there is generally a snap link for disconnecting the chain, so that a half-chain can Fig. 1. be used for steep hills and rough country. The chain is folded by commencing at the middle and folding two links at each time in the form of a warped surface, making when completed a shape something like an hour-glass. In unfolding the chain, take both handles in one hand and with the other throw the chain from you. With a little practice this can be done so that it will stretch its full half-length when thrown and the whole chain can then be opened out. 2. Engineer's Chain. This chain is made similar to Gun- ter's chain, is 100 ft. in length from back to back of the handles, CHAIN SURVEYING 3 and is tagged every 10 ft. Each link is 1 ft. long and it con- sists of one long wire and two or three rings whose joints are brazed. This chain is now rarely used in railroad or city sur- veying where great accuracy is required. The steel tape has al- most wholly superseded it where accurate work is desired. 3. Vara Chain. The vara chain is 20 varas long, and each vara is divided into five equal parts. Each vara is marked by a tag with its distance from one end stamped, and the tags are numbered from 1 to 19. The chain is thus divided into 100 equal parts, each part being one-fifth of a vara or one one- hundredth of 20 varas, and is, therefore, 0.2 of a vara. It is necessary to remember this, for in the Gunter and engineer chains the chain itself is the unit of length. If the distance be- tween two points is five full lengths, 16 varas and 2 links, then Fig. 2. the distance is 116.4 varas. In Texas a vara is 33^ ins. long by law. 4. Steel Tapes. For precise measurements the steel tape (Fig. 2) is used. It varies in length from 3 ft. to 1,000 ft., and is made of the best steel reasonably flexible. The tape has the ad- vantage of having no wearing surfaces, and is easily folded or looped up like a rope. The width of the tape varies from 3/16 to Vz in., the thickness being about 1/64 in It is marked every 5 ft. from one end and numbered on brass and copper plates bent around the tape from 5 to 95 ft., and every foot is marked by a brass rivet, and each foot from the end is divided into tenths of a foot. The even 5-ft. marks are usually made on a brass plate or sleeve, and the even 10-ft. marks are made on a copper sleeve. In order to assist in identifying the even 10-ft. 4 SURVEYOR'S HAND BOOK. marks when the figures have become so worn that they are illegible, rivets are driven through the plate close to the sleeve, one at the 10 and 90-ft. marks, two at the 20 and 80- ft. marks, three at the 30 and 70-ft. marks, and four at the 40 and 60-ft. marks. The rivets are always driven between the sleeve and the 50-ft. mark, so that, by noticing the position of the rivets, it is easy to distinguish the proper point. The 50-ft. mark is marked by two rivets, one on each side of the sleeve. One of the best fcrms of steel tapes for railroad or city engi- neers and surveyors is about 1 A in. wide and has the numbers for the different foot-marks stamped on solder which adheres to the tape. This form of tape has the advantage of not having a shoulder or projection to catch against the reel when the tape is being wound up or run out, or to catch on stones or other rough objects while in use. Fig. 3. 5. Standardized Tapes. For accurate base line measure- ments a steel tape from 100 to 300 ft. long is used (Fig. 3). Such tapes should be standardized; that is, the absolute length between the marked points under a certain pull at a known temperature should be determined. This is generally done in this country by the United States Coast and Geodetic Survey (Washington, D. C.) for a nominal price. If it is necessary to use any tape unsupported, its correct length when hanging free may be found by direct comparison. Lay the tape on a smooth straight support, give it the proper pull and mark the end points ; then, holding one of the ends directly over one of the marks just made, give it a known pull. Drop a plumb line from the other end of the tape and notice the amount by which it differs from the second mark. In this way the correct length of the unsup- ported tape under any given pull may be determined. CHAIN SURVEYING. 5 6. Metallic Tapes. The most serviceable tape for ordi- nary or common use is the metallic tape (Fig. 4), which is a cloth tape manufactured with very fine brass wires interwoven into it. This tape is generally % in. wide, and is made in lengths of 25, 50 tt nd 100 ft. It is conveniently inclosed in a leather case, and when it is rolled up it can easily be carried in the pocket. For light and irregular work it is much more con- venient than the larger steel tapes. It is largely used in building construction, cross-section work, and in railroad engineering, and in many places where its lightness, compactness, and flexibility commend it. It can not be used where accuracy is very im- Fig. 4. portant, for it stretches considerably under pull, but after a short period of use it will be found to have become permanently stretched. 7. Pins. Surveying pins are used to keep tally of the number of chains measured. They are made of pieces of round steel wire 3/16 in. in diameter and about 16 ins. long. One end is pointed and the other is bent to form a ring or handle about 2 ins. in diameter, Fig. 5. Eleven such pins form a set, and they are carried on a key ring, about 6 ins. in diameter, made of the same sized wire. Each pin usually has a small piece SURVEYOR'S HAND BOOK. of red flannel tied to its handle so as to make it more easy to be found when used in the field. In railroad chaining stakes are generally used instead of pins and these stakes are driven at every full station (every 100 ft.) and at intermediate points be- tween the stations. For a description of stakes see the chapter on Railroad Surveying. 8. Range Poles. Range poles are rods of steel, or wood- en rods shod in steel, varying from 6 to 10 ft. in length. Alter- nate foot lengths of the rod are painted red and white to make it more readily distinguishable against any background. They are used by the rear chainman to keep in a straight line when chaining. If the sun is shining and long sights are taken, the bright part of the range pole is seen as the other part is in shadow. To avoid this, a range pole with a flat face is used with the central longitudi- nal line clearly denned and alternate foot-lengths on each side of this line painted black or red. 9. Plumb-Bob. In chaining over rough or inclined ground it is often necessary to raise one end of the chain or tape to bring it to a horizontal. To locate a point on the ground directly un- der such elevated points or ends o-f chain, a plumb-bob will have to be used. The usual form of a plumb-bob is shown in Fig. G, (a) and (b), which consists of a conical shaped body rounded into a neck and head in the upper part. The bottom or apex of the cone is usually tipped with a steel point, while the cap-screw at upper end has a hole through its center for the in- sertion of the cord by which it is suspended. In the ordinary form (a) the cap screw is taken off, the cord is inserted, and a knot tied in the cord to prevent its slipping through the cap- screw when the bob is suspended. Fig. 6 (b) is a special form Fig. CHAIN SURy EYING, of plumb-bob which is provided with a spool on the inside by which the cord can be wound up and carried inside the bob in- stead of being wrapped around the outside as in the ordinary form. This winding is done by turning the cap-screw at the top. 10. Chaining. A line is measured or chained by two men, called the rear and head chainman. They should start with eleven pins, the rear chainman taking one pin and holding his end of the chain or tape over the intial point, and lines or ranges the head chainman in with the distant flag. The head Fig. 6. chainman sticks a pin at this point and advances to another sta- tion, the rear chainman following to the station just left by him. The rear chainman places his end of the chain or tape over this station, and again ranges in the head chainman. The rear chain- man must be careful to collect all pins, ^nd when the head chain- man calls "Out" he must drop his end of the chain and go to the head chainman, and should hand him 10 pins. The head chain- man should count the pins, and if there are not 10 pins the line 8 SURVEYOR'S HAND BOOK. should be chained over. The number of "Outs" is recorded by each chainman. If we are usinj the surveyor's chain, and have three "outs," and the head chainman has measured 3 chains and 23 links on the new out, the lenfth of the line is 33.23 chains. The head chainman always starts on a new out with 10 pins, and the head chainman or the rear chainman should never have more than 10 pins in his hand while measuring. The initial point and the end of each out should be carefully marked so that if a mis- take is made in a long line the chammen can return to the last out, and not have to go back to the beginning of the line. The methods of keeping track of the "outs" vary with different sur- veyors. In chaining long lines, a string tied in the button hole of the coat or shirt, with segments of unequal length, can be used by tying a knot in the long segment for an ordinary "out," and one in the short segment for every ten "outs." Another method is to have the chainmen make tally marks in a note book. In chaining up a hill which is too steep for one length to be brought horizontal, the head chainman stretches the chain to its full length, and then returns and takes a point on the chain suffi- ciently near the rear chainman to pull that part horizontal. He marks the point on the ground under the selected point with his pumb-bob, or places the point on the chain immediately on the ground, the rear chainman drops his end of the chain and takes up the point selected by the head chainman and raises it as high as he can over the point as tested by his plumb-bob. The head chainman in the meantime selects another point qn the ground in advance and marks that on the ground as before. This process is repeated until the length of the chain is exhausted. This is called "breaking the chain." In "breaking the chain" it is well to take sections of the chain that are multiples of ten. In measuring down a hill, the process is reversed, so that the rear chainman holds his end on the ground or near it, and the head chainman holds his point over his head as high as he can. The chain or tape should always be held level, because the horizontal distance between the two fixed points is constant, not- withstanding the fact that changes may be made on the surface of the ground. In the early days surveyors paid no attention to CHAIN SURVEYING. holding their chain level, and there has resulted, in consequence many discrepancies in their surveys, and much litigation. All good surveyors are now very careful in observing this rule. In using the tape in rough countries or thick underbrush, it is a good plan where great accuracy is demanded to attach the handle of the tape by a short loop of strong cord to allow twisting of the tape without breaking. 11. Chaining Over Hills or Across Val- leys. When it is impossible to see one sta- L/ tion from the initial station on account of an intervening hill or high timber, a series of range poles is used and a random line marked out so that at least three points can be seen from one station. Given the two points A and B (Fig. 7), to set the range poles in line AB. We start out from A and, guessing at the line, set enough range poles in a random line AD so that at least two can be seen from B. Then the man at B will have the flag pole at 3 set over in the line B-2 to the point 4, the man at 4 will have the Hag 2 set over in the line 4-1 at 5, the man at o will have the flag 1 set over in the line 5- A. Then again flag 4 will be set over to some point 'nearer AB, in line between B and 5, etc. This process is repeated until all the range poles are in the line A-B. In the preliminary ranging in the men themselves can act as range poles. Only one man is abso- p^g 7 lutcly necessary if he has plenty of range poles, but two can do it with reaspnable efficiency. 12. Chain Survey. When the area of a farm is wanted, or if it is desired to construct a map of same, it may be divided by stations into a system of triangles. All the sides are then measured carefully and a. map of the triangulation system can then be made to scale. The buildings and other topographical features, such as roads, fences, etc., can be tied in by measuring 10 SURVEYOR'S HAND- BOOK. AB = 240 from the nearest stations and a sufficient number of points on the building, and map can be completed to scale. In a recent survey (Fig. 8) the following measurements were made : EA = 180 = 300 In case it were impossible to measure the line EB, the area may still be found by a chain survey if use is made of two aux- iliary lines AF and BF, the point F being in AE produced. By means of these aux- iliary lines the triangle BAF may be calcu- lated and hence the angle BAE becomes known. From this angle and the sides AB and AE the length of BE can be calcu- lated and the area found as before. PROBLEM 1. Make a map of the chain survey ABCDE to a scale of 1 in. = 50 units. 13. Chain Problems. (a) To erect a perpendicular to a line at any point : We know that if the sides of a tri- D In K angle are 3, 4 and 5, or any multiple of these, the triangle will be right. This is apparent, as the sum of the squares of 3 and 4 equals the square of 5. If, in the triangle AKB, Fig. 9, the sides are 18, 24 and 30, it will be a right tri- angle. The rear chainman holds his end of the chain at B in the line BK so that the distance BK is equal to 18 links; he also holds the end of the seventy-second link at the same point ; the head chain- man passes the chain around a pin at K, which has been firmly driven or pushed &{% 9 30 CHAIN SURVEYING. 11 into the ground, then takes hold of the forty-second link and stretches the chain so that all parts are taut. A pin is then driven at A, which determines the perpendicular AK. In reality there are a great number of ways in which the problem can be solved, for" if 2w = first side M 2 1 = second side w 2 + 1 = third side the triangle is a right triangle, as (2n) 2 + (n 2 I) 2 = 4w 2 + w 4 2/r + l = * -f 2/r + 1 = (V 4- I) 2 . Therefore, we can make n equal to any number greater than unity. The following are some of the numbers actually used: 3, 4, 5, 5, 12, 13, 6, 8, 10, 8, 15, 17, 7, 24, 25, 12, 35, 37, 11, 60, 61, etc. Fig- 10. (b) Another easy method of erecting a perpendicular to the line AK at the poinj: K (Fig 10) is to let one of the chain- men hold the end of the chain at K, while a second chainman holds the other end of the chain at any point on AK so that the chain will be slack. The middle point of the chain is then car- ried away from the line AK until it occupies the position AEK. Tf the end of the chain at K is now swung around until it reaches a point C in the same straight line as A and E, the line CK will be the perpendicular to AK at K. 12 SURVEYOR'S HAND BOOK. (c) To find the distance across a marsh, river or pond by use of the chain : Suppose a line that we are chaining reaches a point A, Fig. 11, and a river intervenes wider than the length of one chain, and we wish to find the distance AB. At the point A by the former method we measure the distance to K on the perpendicu- lar AK, and at the ponit K set off the right angle BKC ' , and mark where KC produced crosses our original line. Measure AC. Fig. 11. In the right triangle BKC AK 2 = Fig. 12. X AC AK* BA-'- -JO- Caution. AK should be taken at least one-half of AB, oth- erwise AC will be so short that a slight error in measuring will produce a large error in the result. (d) Similar Triangles: To find the distance AB, Fig. 12, erect a perpendicular to AB at B with a chain and prolong it to some point C ; measure BC and set a flag pole at D in the line DC. Erect a perpendicular, DE, to BD and have the flag- man move along this perpendicular until he is in the line ACE. CHAIN SURVEYING. 13 Set the flag pole firmly in the ground and measure DC and DE. AB : DE : : BC : DC DE X BC AB = DC~ 14. Correction for Temperature. Steel tapes are stand- ardized by the Coast Survey by comparison with known stand- ards at Washington, and each standardized tape is marked some- what as follows : ''Length 100 feet at temperature 62 F., pull 12 pounds horizontal." The average coefficient of linear expansion is 0.0000065 for each degree Fahrenheit, and each unit length. Let L =: Length of tape. C Coefficient of tape. 7" Rise in temperature. Then the increased length of the tape LCT. Total length of tape = L + LCT = L(1 + CT~). EXAMPLES. A 300-ft. tape was standardized at 62 F., pull 12 Ibs. A base line was measured when the temperature of the tape was 102 F., find the length of the tape. Increase = LCT. .0000065 X 300 X 40 = .078. Total length = 300 + .078 = 300.078 ft. 15. Stretch of Tape Due to Pull. It is necessary to sub- ject all tapes to what is called a standardized pull for their true lengths. If it takes a 12-lb. pull to make a tape 100 ft. long, any pull greater than this will stretch the. tape, and has to be al- lowed for. Let P pull in pounds. A= cross-section in square inches. P _ Then the pull per unit area *j unit stress. If S = total stretch L = length of tape Unit stretch = - 14 SURVEYOR'S HAND BOOK. In ordinary pulls the unit stretch varies directly as the unit pull. Unit pull PL Therefore, Unit stretch = AS = Constant - This is Hook's law, which was published in the form "ut tcnsio sic vis." The unit pull divided by the unit stress is con- stant within the elastic limit and is called "the coefficient of elasticity,'' and is gcneraly represented by the letter E. For steel E = 30,000,000 Ibs. EXAMPLE: A bar I%"x%"x20" long was subjected to a pull of 18,000 Ibs. and produced a stretch of % in. Find E. Area = 9/8 sq. in. 18000 Unit pull = -T- = 16,000 Ib. Unit stretch = 240 = = liioon divided by 1/1920= 10000x1020 = 30,720,000. EXAMPLE: If a 100- ft. tape was standardized at a pull of 12 Ibs., and has a cross-section of .00371 sq. in., find how much it will be stretched by a pull of 26 Ibs. if = 30,000,000. The stretch over the standard length will be due to the extra pull of 14 Ibs. S = total stretch in feet. Unit stretch = ^QQ Unit pull - Iwn 1400 30,000,000 = 30,000,000 X .00371 16. Correction for Sag. The foregoing corrections for pull and temperature assume that the tape is horizontal, but hi field measurements it is never horizontal, although the two ends may be in the same horizontal plane. The tape hangs in a curve, which is practically a parabola, with which a circular CHAIN SURVEYING. 15 curve can coincide almost exactly. The effect is to shorten the chain. - If rf=sag L== length of tape or chain 8d 2 The correction for sag = ^ EXAMPLE : A 100-ft. tape, standardized at 62 F. and 12 Ibs. pull was used to measure a line when the temperature was 92 F., pull 25 Ibs. and sag 0.5 ft. Find the correct length of the tape if the cross-section is 0.00,3 sq. in. Correction for temperature = .0000065 X 100 X 30 = 0.0195 Correction for sag= 8/300 X 0.25 = 0.0067 13 X 100 X 1,000 Correction for pull 3 X 30,000,000 Length of tape = 300 + 0195 .0067 + .1444 = 300.1572 PROBLEM 2. A 100-ft. tape, cross-section 1/300 sq. in., was standardized at 62 F. and pull 12 Ibs. Find the length for a temperature of 96 F., pull 28 Ibs. and a sag of 0.5 ft. PROBLEM 3. A standardized tape is 100 ft. long between marks at 61 F., and a pull of 11 Ibs. Find the length when temperature is 97 F., 20 Ibs. pull, and a sag of 0.70 ft., if cross- section is 1/300 sq. in. 17. Erroneous Lengths. Chains become changed by the breaking of links, the loss of handles, and the wearing of the 600 rubbing surfaces. In the use of the chain two points on the ground, 66 ft. or 100 ft. apart, should be marked, and the chain should be compared with this at frequent intervals. The outer edge of one of the handles is placed over the zero and the 100-ft. mark is marked by a file if the chain is too long. If distances are measured by chains that are too long, we can find the true lengths of the lines by calculation without measure- ment. If the length of the chain used is 100 + a, and in the measurement we called it 100 ft, then the length of the line 'as measured will be too short. If the extra length of the chain is due to wear or stretch throughout the length, the true length of a line that has been 16 SURVEYOR'S I1AXD BOOK. measured with a tape of erroneous length may be found by multiplying the true length of the tape by the number of times it was applied to the ground in measuring the line. Afti-r a line 9.864 chains in length had been measured it was found that the chain was really 100.25 ft. long, find the true length of the line. The chain was applied to the line 9.804 times, con- sequently its true length must be 9.8G4 X 100.25 = 988.866 ft. However, it might happen that one link of an engineer's chain had been broken and t^ken out. thus making the chain 99 ft. long. Suppose an engineer's chain was used in measuring a line the length of which was recorded as 628 ft., and it \vu k - then discovered that 1 link was out of the 10-ft. section next to the head chainman. What is the true length of the line? Six full lengths were measured = 6 X 99 = 594 ft. If the 28 ft. was measured with the end of the chain next to the rear chainman the true length of the line was 622 ft., but if the 28 ft. was measured with the part of the chain that contained the unknown missing link, then the true length of the line was 621 ft. Let a = assumed length of chain, /=true length of chain, M = measured length of line as measured with chain of erroneous length, r = true length of line, ft = number of chain lengths in M (whole or fractional.) Then, A/ = na T = nt. 18. Erroneous Areas. If a farm is surveyed with a chain of erroneous length and the area is calculated by use of the er roncous data, we can find the area without rechaining. Let C = calculated area of farm, X = true area of farm, ..".;. To convert yards to varas multiply by 1.08. To convert Guntcr's chains to varas multily by 23.76. To convert poles to varas multiply by " !'!. To convert meters to varas multily by 1.1811. 20. Units of Land Measure. One acre = 4840 square yards, = 43560 square feet, 10 square chains. = 160 square poK-. = 5645.376 square varas, = 4046.87 square meters. Onevara = 33% inches. One yard = 36 inches. One foot = .36 vara. 10,000 One square vara= 1111.1 square inches= y square ins. One square yard = 1296 square inches. A Spanish league was denned as a square, 5, represents the al titude CD and c the base of any triangle. Fig. 13. In the right triangle ADC, p*=trx*. In the right triangle BDC, f>*=a 2 (c *= (^ + r 2 a 2 ) 2 = (2fcf + & 2 +c s a 2 ) (2bc + a 2 b 2 c = [(fr + O 2 fl 8 ) x (o 1 (& O f .l ^4 = (& + c + a) (& + r a) (a & + O ( Let 2J=(a + & + r) Then (b + c a)=2s 2a =2 (s a =2s 2c = 2(s r) ' ^ 45 5-a)(5- a) (56) (5 c) Therefore i(/v) = K" = V5(5 a) (5 6) (5 c) (S) PROBLEM 8. Calculate the areas of triangles ABE, EEC, and EDC in Fig. 8. PROBLEM 9. If the sides of a triangle are 520, 560, and 600 varas, find the area in acres. 20 SURVEYOR'S HAND BOOK. PROBLEM 10. If the sides of a triangle are 13, 20 and 21 chains (66 ft), find the area in acres. PROBLEM II. If a 750 varas, b = (>">0 varas, c = *200 varas, find area in acres. PROBLEM 12. If a = 50 poles. b = 4l poles, c = 39 poles, find area in acres. PROBLEM 13. If a = 300 poles, 6 = 240 poles, c = 180 poles, find area in acres. PROBLEM 14. If a= 280 poles, fc = 224 poles, c=168 poles, find area in acres. 22. The 57.3 Rule. Let EOA (Fig. 14) be a triangle where the angle x is less than 6, and the two arms OA and OE practically equal. If with O as a center and OA as a radius we describe a circle passing through E we have: Fig. 14. X :360 ::y : 2irr where y = AE 360 v 57.3y Then X= -^ X '- = ~ (4) That is, the small angle in degrees times the long side is equal to the short side times 57.3. PROBLEM 15. A straight roadway 1,320 ft. long has a rise of 21 ft. above the horizontal through the low end. Find its angle of elevation. 23. Applications of the 57.3 Rule. If the angle AOE Fig. 14, equals one-tenth of 57.3, then we have 5.73 AOB= ' Distance ' "' Distance = 10 X offset That is, when the small angle is 5.73 or 5 44', the -distance is ten times the small side or offset. CHAIN SURVEYING. 21 If the angle EOA is equal to 0.573, that is, 34'.38, the long side is one hundred times the offset. Hence OA = 100 X AE. This is generally expressed by saying that the distance is 100 times the offset. This principle is used in finding the approxi- mate area of a boundary. The angle that OA makes with some reference line is measured, and the distance OA is found by making the angle equal to 34.38 minutes. The assistant at A attaches one end of a tape or chain to the point A and then takes AE at right angles to AO and is sighted in the line OE by the distant transitman. When he is located, he reads on the tape the distance AE and records it in his note book. The dis- tance from A to the instrument man is 100 times this distance AE. PROBLEM 16. Make a drawing of the following area to a scale of 1 in. equals 100 ft., and find the area in acres, by divid- ing the boundary into triangles. Point. Angle. Offset. A 8.50 feet B - 45 10.00 C : 75 9.40 D 90 9.60 E 120 8.60 F ' 150 7.20 G 180 6.00 24. Pacing Survey. A rough approximate idea of the area of a farm can be .obtained by a pacing survey. With a little practice a man can train himself to step off a yard at each stride and in this way a fair approximation can be made to the area of a small farm or parcel of land. In a farm, ABCDF, Fig. 15, let AB 350 yds. ; BC 400 yds. ; CD 90 yds. ; DE 266.3 yds. ; EF 250 yds. ; FA 281.8 yds. Now the area of the farm can be found by dividing the field into triangles or by locating the points. CDE. etc.. by offsets from some reference line, AB. If the land is divided into triangles we pace the distance BD 410 yds., BE 300 and AE 211. This divides the land into four triangles, BCD, BDE. BEA and AEF. The area can be calcnbted by the use of formula (3). If it is desired to locate the corners by offsets, we adopt some 22 SURVEYOR'S HAND BOOK. reference line from which to take offsets. This reference line need not be a side of the farm, but can be some line assumed for convenience. However, in the case of Fig. 15, we shall assume AB as the reference line. As ABC is a right angle, the distance EC 400 yds., will locate C, and as angle BCD is also right, the distance, CD 90 yds., will determine the point D. Let DG be a perpendicular from D on line AB. If a perpendicular be dropped from E on AB cutting AB at H, where H = 240 PAB = 14,873, PBC = 54.577. PCD = 34.538, PDE = 23.546, PEF = 33,504, PFA = 25,422, or a total of 184,460 S q. yds. = 38.1 acres. If the distances are all carefully chained instead of paced the two methods should check within one-tenth of an acre. CHAPTER II. COMPASS SURVEYING. 27. The Bearing of a Line. The acute angle that a line makes with the meridian is called its true bearing. If the acute angle is made with that part of the meridian to the north of us it is called north, and if in addition it cuts to the right it is called North X East, where X equals the acute angle. If the acute angle is made with that part of the meridian to the south of us and cuts to the right it is called SXW. In Fig. 19 the bearing of AB is N 32 E; that of AD, N 54 W; that of EF, S 61 W, while that of EG is 5" 27 E. 28. Azimuth. The azimuth of a line is the angle made with the true meridian, and is measured from the south around by the west, north, and cast to the south again. If the bearing of a line is 5 30 W ', the azi- muth is .,n ; if the bearing is N 39 W, the .-izimuth is 141; if the bearing is .V 39 E, the n.ximuth is 219, and if the bearing is S 1 39 E, the azimuth is 321. In some states it is the practice to define "bearing" as the acute angle made by a line with the mag- netic meridian (that is, with the needle in its mean position). 29. The Compass. The essential parts of a surveyor's compass (Fig. 20) are a magnetic needle, a graduated horizontal circle, and a line of sights. These conditions can be fulfilled very crudely or elaborately. It is also convenient to have a declina- tion arc attached to the compass on which we can set off the declination of the needle. A magnetic needle when poised freely will not point towards the North Pole, but will dip towards the north an amount of x degrees. To make it horizontal in the compass it is mounted on an agate pivot and the South end is weighted by having an adjustable brass wire at that end. The 25 Fig. SURVEYOR'S HAND BOOK. accuracy of the compass depends largely upon the activity of the needle, which depends upon the intensity of the magnetic force, which must be kept alive. The pivot upon which the needle is mounted is in the center of a grad- uated circle which is generally raised to the level of the ends of the needle and is graduated on a silver plate. Inside the compass hox we find tin- letters , 5", W, and AT. If the com- pass has no declination arcs the zeros are in the line of sights as de- termined by the slots in the stand- ards or uprights. The graduated cir- cle* is mounted on a brass plan which has extended arms, to which the uprights are attached by me -in- of- mill-head screws. If the arms are not extended the uprights are at- tached to the graduated circle and fold down over the face when not in u-e. To set off the declination ac- curately, each compass should be pro- vided with a declination arc with a vernier attached. For the purpose of leveling, the compass is provided with two bubble tubes whose axes are at right angles to each other. It is leveled by a Fig. 20. ball and socket joint which affords easy and quick methods of setting up. It can be mounted on a Jacob's COMPASS SURVEYING. 27 staff or a tripod, but in most cases county surveyors use the Jacob's staff on account of its ease of transportation. The ball and socket joint is mounted on the Jacob's staff, which has a sharp conical iron shoe. In setting up, the staff is driven into the ground two or three times to get a firm footing so that there will be no vibration. The compass is then set on the staff leveled and it is now ready for use. When moving from sta- tion to station the compass should always be removed from the staff and carried under the arm, with the needle screwed reason- ably tight. In setting up always loosen the ball and socket joint and have the compass almost level and along the line of sights before tightening. If the tripod is used the compass can be taken off in moving from one station to another or it can be left on as with the transit. The tripod gives much more ac- curate work than the Jacob's staff because you can locate the points more accurately, and it gives a much more stable support. 30. Reading the Bearing. To read the bearing of a line, set up the instrument over any point on the line, turn the com- pass so that the arrow in the compass box points in the direc- tion in which you are running the line, and read the north end of the needle. The north end of the needle will lie between two letters, one of which will be N or S, while the other will be E or W . If it lies between N and W ', the bearing is northwest ; if between ^ and W , the bearing is southwest, etc. In sighting al- ways place the eye at, the end of the compass box marked S. 31. How to Use the Compass. Set up the tripod with the legs wide apart and firmly pressed into the ground. Place the compass on the brass spindle and then fasten the sights by means of the thumb screws provided for that purpose. This spindle is connected with the head of the tripod by a ball and socket joint, which gives it a limited range of motion. A groove about % in. wide and about the same depth is cut in the spindle, which engages a pin piercing the socket of the compass body which fits over the spindle and prevents the compass from fall- ing off the tripod. Take hold of the compass with both hands and level it by means of the motion available in the ball-and- socket joint. When both bubbles are in the center of their run 28 SURVEYOR'S HAND BOOK. that is, in the center of the tube the instrument is level. Do not lower the needle until the compass has been leveled. The compass may now be pointed in any direction by turning it on the spindle axis. In moving the instrument to another point, raise the needle by means of the screw controlling it, remove the compass from the tripod by pulling in the small pin in the socket mentioned above, at the same time lifting the compass from the tripod. Carry the compass under one arm and the tripod in the other hand, or on the other shoulder. If a Jacob staff is used instead of the tripod, the brass spindle connected to the ball-and-socket joint is connected with the staff by a tight fitting joint. When the compass is placed in its box to be stored away the needle should be left free. During some seasons of the year the compass will be affected by a charge of electricity due to atmospheric conditions. When this is the case one end of the needle will often adhere to the glass plate. If the glass is touched with a damp substance it will relieve this condition and release the needle. 32. The Vernier. The vernier is an auxiliary scale, either straight or circular, designed to read to a certain given part of the finest division on the limb. Thus in the Xew York rod (Fig. 21) the smallest division that can be read from the rod itself is one one-hundredth of a foot, but the vernier cuts this part into ten parts, so that we can read to one one-thousandth of a foot. In the ordinary transit the finest division is a half degree, but with the aid of the vernier we can read to minutes. If AB is the limb and CD is the vernier scale, let a equal the length of each part of the limb, and b equal the length of each part on the vernier, and n equal' the number of parts on the vernier, then (ft 1) will be the number of parts on the limb, so arranged Fig. 21. UNIVERSITY! COMPASS SURVEYING. that n parts on the vernier is equal to n 1 parts on the limb, consequently nb = a (n 1). If the lowest mark on the vernier agrees with a mark on the limb, then the highest point on the vernier will agree with a mark on the limb, also the second mark on the vernier will not agree by an amount of a b. If the vernier is moved a distance a b, then mark No. 1 on the vernier will agree with -a mark on the limb; if moved twice this dis- tance, then mark No. 2 will agree with a mark on the limb ; if moved three times this distance, then No. 3 will agree with a mark on the limb. If rrark No. 3 on the vernier agrees with a mark on the limb, it means that the zero at the vernier is 3 (a 6) from above the nearest point on the limb. But bn = a(n 1) a(n - i) o = n a(n 1) a a - 6 = a - ~^- ~ - a b is always one nth of the finest space on the limb and it is called the fineness of reading. If w = 10 parts and a = 1/100 of a foot, then the vernier reads to 1/1,000 of a foot. If w = 30 and a 30' then the vernier reads to minutes. This is the case in the ordinary transit; a = 30' or Vz and ;i = 30, and we can read to minutes. 33. Decimation of the Needle. The magnetic needle at any point when mounted on a pivot and weighted at one end so that it will rest in a horizontal position will make an angle with the true meridian. This angle is called the declination or varia- tion of the needle. In Texas the magnetic meridian cuts to the right of the true meridian passing through a point, and, there- fore, the declination is said to be east. In Austin the magnetic meridian makes an angle at the present date of about 8 with the true meridian, or the declination of the needle is said to be 8 east. The line of zero declination (called the agonic line) now passes near Charleston, S. C. ; Asheville, N. C. ; Knoxville, Tenn. ; 30 SURVEYOR'S HAND BOOK. Lima, Ohio; Battle Creek, Mich.; and passes through the re- mote corner of northeastern Indiana. All sections east of the line have west declinations, while all sections west of this agonic line have east declinations. The United States Coast and Geodetic Survey determines the magnetic declination at various points in each State at stated intervals; and by this means not only is the declination accurately determined, but its rate of change can be determined by a comparison of the declination for different dates. These results are placed on a map (called the Isogonic Chart) by the Coast Survey. This chart is issued at least every ten years and is of great use to surveyors, as it gives the declination for all parts of the United States with reasonable accuracy. It can be obtained by addressing a letter to the Coast and Geodetic Survey, Washington, D. C VERNIER 34. Compass Vernier. One form of compass vernier is shown in Fig. 22. This is the usual form of the vernier on the surveyor's compass. The vernier is divided into 30 equal parts anc these 30 parts cover 29 parts on the "limb" or graduated circle. The smallest division on the limb is one-half degree or 30 minutes and as the vernier can read to one-thirtieth of the smallest division on the limb, we can read to one-thirtieth of 30 minutes, or to 1 minute. We further notice that the vernier-zero is nearer the 5th division of the limb, and we find that the 5th division of the vernier to the left of the vernier-zero is opposite or coincides with a division on the limb. Hence the reading for the frac- COMPASS SURVEYING. 31' tional part is five minutes, which corresponds to this 5th divi- sion of the vernier that is opposite a divison of the limb. The whole reading should be 2 30' plus 5' or 2 35'. If the zero of the vernier is, as in Fig. 22, nearer the last division between the two zeros than it is to the division beyond the vernier-zero, the fractional part is read on the left half of the vernier. There are 15 divisions in this left half .and if the fractional reading is between zero and 15, one division of the left half of the vernier will coincide with one division on the limb, and the number of this division on the vernier is the frac- tional reading. Thus if the 5th division on the vernier agrees, as in Fig. 22, with a division on the limb, the reading is 5' ; if the 9th agrees, the fractional reading is 9', etc. However, if the vernier reading is greater than 15', this reading is obtained from the upper part of that half of the vernier that covers a section of the limb reading. 35. To Set Off Declination. This will be illustrated by an example. Suppose that the declination of the needle is 8 15' east. This means that if the needle was allowed to swing fre-ely it would come to rest in a line not pointing to the true .north, but in a line that makes 8 15' on the east side of the true meridian, or in a line whose bearing is N. 8 15' E. To set off the declination, level the instrument, lower the needle and allow it to come to rest. Turn the compass until the line of sight, through the slots in the standards, coincides- in direction with the needle. Clamp the instrument in this position. Since the neer iU when at rest points N. 8 15' E., the line of sight must how be N. 8 15' E., or make an angle of 8 15' with the true meridian. Then with the instrument clamped, and without disturb- ing the line of sight, turn the graduated circle in the compass box by means of the milled-head screw until the needle reads N. 8 15' E. The vernier scale that marks the declination arc should now read 8 1-V. The final and accurate test is the ver- nier arc where all declinations should be set off. 36. Changes in Declination. The declination of all points west of the agonic line has been decreasing, while that to the east of the agonic line has been increasing. In Texas the SURVEYOR'S HAND BOOK. declination has been decreasing at the rate of about three minutes per' year since the time of the first recorded land patents. This steady annual change goes through a large series of years and probably goes through a cycle. In addition to the annual change there is a daily change. In Texas the needle at about 6 p. m. is in its normal position ; at 8 a. m. the north end of the needle swings to the east about two to three minutes, and about 1 p. m. it swings about the same amount to the west of the normal position. 37. Result of Changes. An old survey was run in 1864 with the correct declination of 10 at the time the survey was made, and a surveyor in 1904, not knowing the present correct declination (which is 8), sets his com- pass on the old declination. The bear- ing of line was N. 42 E. ; that is, the line made 42 to the right of the true meridian and 32 with the magnetic meridian. When the correct declination was set off in the compass and the ends of the needle were brought to the zero marks on the graduated circle under the glass top, the line of sights pointed along the true meridian. But since the original survey was made the declina- tion has decreased to 8 and the mag- netic meridian has turned 2 to the left of its position in 18(54. Now if the surveyor of 1904 had set off the declination of 8 on the declination arc and brought the ends of the needle to agree with the zero marks of the graduated circle, the line of sights would have pointed along the true meridian. But instead he set off a declination of 10, and when he made the ends of the needle agree with the zero marks the line of sights marked out was 2 to the left of the true meridian. Now if AB, Fig. 23, were the original line that made 42 with the true meridian AN, and AM were the position of the magnetic meridian, the magnetic meridian of 1004 will occupy the position AM', two degrees to Fig. 23.. its position COMPASS SURVEYING. 33 the left of AM. The surveyor set off 10 on the declination which made the line of sights point to a position AN', which he assumed to be the true meridian, and from this he set off the bearing 42. As the angle NAB is 42, then the 42 measured from AN' will fall to the left of AB in some position AB'. In all cases in that section west of the agonic line, where the sur- veyor uses a declination greater than the correct declination, he, in effect, turns the assumed true meridian from which he locates bearings to the left, and all lines thus run will fall to the left of the old lines, if said old lines were surveyed with the correct declination. 34. Old Lines. In surveying old lands the great object is to ascertain a declination which, used with the bearings as ob- tained from the field notes, will retrace the old lines. This is the prime object. It may be the correct declination for the time and place, and it may not. If two points A and B can be found on any side of the tract, set the compass at one of these points A, run a random line AC with an assumed declination and the bearing of the line a distance AC equal to the distance AB. Measure the distance BC ' , multiply it by 57.3 and divide the product by the length of the line AB. The result is the error in degrees in the assumed declination. If the declination is cor- rected by this error the old bearings will trace out the lines as formerly marked. 39. Magnetic Bearing. In some of the older States the bearing of a line is defined as the angle it makes with the mag"- netic meridian. The result of this is in all of the States west of the agonic line where the declination has been decreasing for years that the northeast and southwest bearings will be increased over the old bearings by an amount equal to the change in declina- tion, while the northwest and southeast bearings will be de- creased by the change in declination since the old line was sur- veyed. In the country east of the agonic line the reverse of the above is true. PROBLEM 17. If the bearing of a line with reference to the magnetic merdian in the States west of the agonic line were N. 72 18' E., declination 8 45' east at the time of the old survey, 34 SURVEYOR'S IIAXD BOOK. find its magnetic bearing at the time \vhen its declination was 7 'IV east. Find the declination if the magnetic bearing of the line were S. 36 21' E. PROBLEM 18. The magnetic bearing of a line in a State east of the agonic line was, when the original grant was surveyed in 1806, S. C8 E., with a declination of 25' west. Find the magnetic bearing in 1806 when the declination had increased to 2 05'. Find the magnetic bearing if the true bearing was S. 29 42' W. PROBLEM 10. Find the magnetic bearing in the following : A. B True Bearing. N 26 54' N 74 12' W Declin- Magnetic ation. Bearing. 7 54' E 7 54' E r N 33 28' W 7 54' E o S >;) 6 36' E 7 54' E F N 2 14' E 8 17' E F S 87 14' E 8 17' E G N 5 20' W 8 17' E ' f{ N 88 ->2' W 8 17' E A PROBLEM 20. Find the true "Magnetic Bearing. N 3 14' W bearing for the following courses : DecliiiT True ation. Bearing. 2 8' E H S 5 18' W 6 12' E C 1\ 8 16' W 3 16' W /) S 74 6' W 3 18' W F N 17 23' W 3 l k >' E F S 74 -V \\ 4 02' W G N 17 23' W 5 43' E If N 9- 25' E 8 56' E 40. To Find the Declination for Any Special Farm. To resurvey an old farm or tract of land obtain the field notes from the county clerk's' office or from the deeds or grants. These papers should give the declination used in the original survey. This former declination (whether right or wrong) can not be used in a subsequent survey, and it is the surveyor's first duty to ascertain the proper declination to use in his own survey. If he can find one side of the tract marked by corners or trees, he can use these as a basis. If two corners at the end of a line can be found, all he has to do is to set off a declination on the declina- COMPASS SURl 'EYIXG. 3~> tion arc that will cause the compass when set on the line with the true bearing, to coincide with the line as defined by the trees or corners. However, if the corners can not be seen from each other, the surveyor must select a declination that he thinks will be correct. With this declination he runs a random line with the old bearing the full length of the line, and marks the end of the random line. If the end of the random line docs not agree with the corner, he measures the distance between the end of the random line and the true corner. This distance, multi- plied by 57.3, and the product 'divided by the length of the line, will give the correction to be applied to the assumed declination. 41. Local Attrac'i n. It often happens that ore in the ground, a wire fence, or a railroad track, etc., will pull the needle out of the magnetic meridian. When this is discovered, the only thing to do is to retrace o;ir steps to some point outside the limits of the attraction, set off the correct bearing and locate some point ahead. Then transfer to it, leaving a rear flagman ; set up at the point located, and sight on the rear flagman, and then prolong the line by locating the head flagman, transfer an:l backsirht, thus locating another point. This method will not apply when the whole line is within the field of attraction. We then have recourse to the transit and locate the line by internal angles. If the whole farm were within the field of attraction, it would all have to be surveyed with the transit by measuring in- ternal angles. 42. Witnessing a Line or Corner. All corners should have witness trees or some natural object to establish the corner, even though the stake disappears, thus: Begin at a stake from which a pecan tree 10 ins. in diameter marked "K" bears S. 32 E. 84 varas. To find the corner all we have to do is to find the witness tree, set the compass at it on the reverse bearing and chain off the distance. As a check it is well to have the witness line at the corner intersect at a large angle. The line is witnessed or marked by line trees. All trees that can lie reached with the; arm either way by a man standing on the lint should be marked with three hacks on the side next to the line, but these hacks should not cut into the flesh of the tree. It is often the case that 36 SURVEYORS HAND BOOK. the line passes through a tree ; such trees are marked on both sides with a hack, blaze, hack. These trees are called "fore and aft" trees. If trees are scattered some surveyors hack trees that are more than three feet from the line. 43. Typical Field Notes. (From Deed Book 185, p. 235, Travis County, Texas.) Ber inning at a stake, a corner to H. P. Sims and R. D. Rone, from which a hackberry marked "X" bears N. 61 30' W. 10 varas ; thence N. 3 W. 216 varas to a stone ; thence N. 16 W. 255.6 varas to a stone from which a live oak marked "A" bears N. 87 W. 31 varas; thence S. 28 30' W. 263.5 varas to a stone on side of hill; thence S. 5 16' W. 205.6 varas to a stone, from which a pecan 12 ins. in diameter marked T bears S. 63 W. 18 varas ; thence S. 85 6 15' E. 227.3 varas to the beginning, containing acres more or less. 44. Compass Adjustments. There are in all six adjust- ments of the compass that should be made. First. The axis of revolution should be perpendicular to the plane of the plate. This is done by the maker and if the adjust- ment becomes deranged, the instrument should be sent to the maker or some instrument house that has facilities for making such repairs or adjustments. Second. The plane of the plate bubbles should be parallel to the plane of the plate. If the first adjustment has been made, level the compass and then turn it through 180. If the bubble remains in the center of its run, no adjustment is necessary. However, if the bubble does not stay in the middle of its run after the compass has been turned 180, correct half the apparent error by the screws at the end of the bubble tube. Repeat the operation till the bubble remains in the middle of its run when the compass is turned 180. Third. If the needle is bent, its ends will not always read the same, but if the pivot is in the center, the difference of the read- ings of the ends will be constant. To straighten the needle, set 'one end at zero and read the other end. This reading will indi- cate the way the needle must be bent. Repeated trials will be necessary before the needle can be made straight. COMPASS SURVEYING. 37 If the difference of the readings is not constant, it shows that the pivot is also bent. Read the ends of the needle in any posi- tion and then turn the needle by hand till the north end is in the position formerly occupied by the south end. Read the- south end of the needle and note the difference of this reading and the first reading of the north end. The needle can then be bent till the north end when swinging free will bisect the space between the first reading of the north end and the second reading of the south end. Fourth. If the pivot is bent out of its central position, the ends of the needle will not have the same 'readings, and the dif- ference of the readings will be variable. After the needle is straightened, turn the compass till the difference of the end read- ings is the greatest. Remove the needle and bend the pivot to- wards the middle of the larger arc that was between the ends. Re- peat till the difference of the end readings is zero. Fifth. The plane of the sights can be made normal to the place of bubble tubes by leveling the compass and by sighting on some plumb line. If the slot-sight does not agree with the plumb line, the base of sight must be filed till a plumb line can be seen throughout the sights. Sixth. The diameter through the zero graduations should be made to coincide with the line of sights. This is an adjustment that is always made by reputable makers and the surveyor is rarely called upon to test his compass for this. A very fine wire stretched through the sights and over the compass box will indi- cate clearly whether-the line of sight agrees with the zero lines. Bibliography. "Davies' Surveying.'' By the late Charles Davies. This book has for several decades been one of the stand- ards for the school and camp, and its full discussion of the usual problems of land surveying, together with the traverse and trig- onometric tables, makes it a valuable assistant to the surveyor or a guide for the student. In addition to Davies', the works of the late J. B. Johnson, Wm. G. Raymond, Breed and Hosmer, etc., which are described at the end of the chapter on Transit Sur- veying, contain valuable data and suggestions for the compass surveyor. CHAPTER III. TRAXS! | SURVEYING. 45. The Transit. The essential parts of a transit (Fig. 24) are, mathematically, a line of sight and a graduated horizon- tal circle for reading horizontal angles. Mechanically, the essential parts are the telescope, the horizontal axis, the circular plates, the spindle, leveling head, tripod, and plumb-hob. The line of sight is determined and defined by the telescope mounted on the horizon- tal axis, the graduated circle by a horizontal circular plate ui>< n which the degrees and fractions of degrees are marked Tin- telescope is rigidly attached at right angles to two horizontal arms whose axes are in the same straight line, and whose outer ends rest in the standards. These standards consist of two diverging legs rigidly attached to the horizontal plate. Two small levels at right angles to each other are attached to the horizontal plate, and by means of these the plates can be brought to an absolute hori- zontal. Two verniers (T and D, Fig. 25, are attached to tin- plate with their zeros 180 apart and are provided with a gla-s < \ er for protection. These verniers are turned so as to fit the outer graduated circle called the limb. By pulling out the small clip S the whole upper part including the limb R can be taken off the head. The upper part of the transit, including telescope, plate. horizontal axis, standards, and verniers, is called the alidade and is supported on a spindle and can be turned on a vertical axis normal to the vernier plate. However, the limb B and alidade can be clamped tight together by a clamp DP operated by the milled-hcad screw, which is seen in the faint outline on the right of Fig. 25. When clamped the alidade and limb B can be turned around the interior spindle H by unclamping the lower clamp screw (not shown in Fig. 25 but which can be seen in Fig. 21 The transit is provided with a level head as in the Y-level, which has four leveling screws for bringing the limb B into a horizontal plane. T'he tripod is generally made of light, tough, straight grained wood, 1 the upper ends of the legs being connected by 38 TRANSIT SURVEYING. 39 40 SURVEYOR'S PIAND BOOK. pin-joints to the leveling head, while the lower ends are shod with metal shoes. The plumb-bob is one of the mechanical essentials of the transit as the instrument cannot be set over a point below without it. 46. Compass Attachment. Attached to and supported by the upper horizontal plate is a complete compass box, includ- ing graduated circle, needle, pivot, a declination arc inside the box and under the needle. The declination can be set off P Fig. 25. and the bearings read as in the compass, and the telescope simply helps to make the line of sight more exact. However, it has the disadvantage of having its line of sight confined to a single line, which a leaf, blade of grass, etc., can interrupt, while in the compass the line of sight is confined to a vertical plane passing through the slots and a slight interruption to the line of sight can be obviated by moving the eye. 47. Vertical Circle. For the purpose of reading angles of elevation a vertical circle is now generally attached to the TRANSIT SURVEYING. 41 end of the horizontal axis and is provided with a tangent screw and a vernier reading to minutes. It is not an essential part of the transit. To bring the line of sight to a horizontal a bubble tube is attached to the telescope whose axis is made parallel to the line of sight of the telescope. 48. Shifting Center. The modern transits are furnished with a shifting center. The lower part of the spindle to which the loop P is attached works in a ball and socket joint which is extended into circular, brim-like plate under the plate on which the leveling screws rest. If these are loosened so that the upper part of the transit can be moved, the point P can be moved a short distance in any direction. This is called the shifting center. 49. The Reticule. The line of sight in the telescope is defined by two cross-wires at right angles to each other, cemented into de- pressions in a metal ring, Fig. 26. This ring is inside the telescope and is controlled and operated by four capstan screws which can be seen in the view of the telescope of the level or transit. The whole arrangement is called the reticule and it is susceptible to slight mo- tions for the purpose of adjusting the line of sight of the telescope. The reticule is moved by loosening one capstan screw and by tightening the opposite one. 50. Setting Up the Transit. Set up the tripod with the legs widely apart and firmly pressed into the ground; take the transit out of the box by taking hold of the limb and lifting the entire weight with one hand, simply using the other as a guide Never grasp the transit by the telescope to lift it out, as such lifting springs the horizontal axis and otherwise in- jures the bearings. Set the transit on the tripod, turn it till the threads catch, revolve the telescope vertically and take hold of two legs of the tripod and straighten it up until all 4^ SURVEYOR'S HAND BOOK. the legs are together, and then place the tripod across the shoul- der and carry it to the place where the observations are to be made. When it is desired to set the tripod over a point, place the legs wide apart, and move them so that the plumb- bob will be practically over the point. Level up the instrument, and if the plumb-bob is not over the point loosen the leveling screws until the center can be shifted, then move the center until plumb-bob comes over the point below and rclevel. If there is not sufficient play in the shifting center to move the plumb-bob over the point the tripod will have to be moved in the direction necessary; then proceed as before. 51. Motions. If the lower clamp screw be clamped, and the upper loosened, the alidade can be turned on the vertical axis, and it will be noticed that the vernier plate moves with the alidade, and that the limb or graduated circle is stationary. This movement is called the upper motion. If the upper clamp screw be tightened and the lower one loosened, that part of the instrument above the leveling head can be turned around one of the spindles. This movement is called the lower motion. 52. Use of the Transit. After the transit has been set up over a point, make the zero of the vernier agree with the zero of the limb. Unclamp the upper motion and bring the two zeros as near together as possible; then clamp the upper motion and bring the zeros into exact coincidence by means of the tangent screw controlling the upper clamp. After the zeros have been brought together, loosen the lower motion clamp, take hold of the limb with both hands and turn the tele- scope till it points towards the object on which we wish to observe. The telescope can be brought approximately into the required direction by sighting over the telescope at the object and turning the instrument until the telescope points towards the point. The cross-wires are brought into the field of view by turning the screw that operates the eye-piece. The large milled- head screw on side of the telescope is then turned till the ob- served object is seen distinctly and clearly through the telescope. The tangent screws can then be turned till the vertical wire bi- sects the object. TRANSIT SURVEYING. 43 53, The Transit as a Compass. If it is desired to use the transit as a compass in regular surveying work, or to use the needle as a check on other work, the milled-head screw shown on the outside of the left leg of the front standard in Fig. 24 is loosened, and the milled-head screw that controls the declination arc, seen betw r een the rear standards, is turned until the proper declination is set off by the vernier inside the compass box. These screws are then clamped and the transit will then read angles with the true meridian. The needle is turned loose by means of the milled-head screw shown above the plate on the right of Fig. 24. 54. Transit Surveying. If the transit is not used as a compass, we must read the azimuth of each course or line in- stead of the bearing. As this azimuth is read from the south point around by the west, north and east, and on to the south again, we can have with a transit reading to minutes, an azi- muth of 3o9 59', -which could be a bearing of S. 01' E. These azimuths are read with reference to the true meridian and it is necessary to locate this very accurately if the abso- lute azimuth is desired. However, if it is only an accurate ex- pression for the area of the farm, a meridian can be assumed for the first course, and then carried around the farm by lo- cating this meridian from each course. 55. Transit Vernier. The transit vernier is a double ver- nier (Fig. 27) and has 30 divisions on each side of its zero. Each half of the vernier covers 29 parts or divisions on the limb. The smallest division on the limb is a half-degree, or thirty minutes, and hence the vernier can read to one-thirtieth of a half degree or to one minute. The angle may be meas- 44 SURVEYOR'S HAND BOOK. tired from the right or left, and the one we use depends upon the special problem under consideration. If read from the right we see that the zero of the Vernier is between 5 30' and 6 00'. The reading is 5 30' plus the vernier reading. As the reading of the transit is from the right, use the left half of the vernier. On examination, w^e find that the 14th di- vision of the vernier agrees with a division mark on the limb. The vernier reading is therefore 14'. The whole angle reading is therefore 5 44'. If the angle is to be read from the left, use the vernier on the right. The zero of the vernier lies between 354 and ?")4 30'. The 16th division of the vernier on the right agrees with a division on the limb and the vernier reading is therefore 16', and the whole angle reading is 354 16'. 56. Example. If a farm is surveyed with the transit, the field notes would be as follows: Course. Azimuth. Distance. AB 203 30' 255.72 varas BC 248 00' . 182.10 varas CD 3 47' 329.42 varas DA.... 84 15' 249.92 varas It will be observed that the shape and dimensions of the farm would not have been changed in the slightest if the first course AB had been taken at 202 instead of 203 30'. It simply would have amounted to a turning of all meridians in a clock- wise direction and the azimuths would have been as follows : 202, 246 30', 5 17', 82 45'. Then, if it is desired to obtain the area accurately, we can assume a meridian, and it is not necessary that this be the true meridian, but when this me- ridian is once assumed, the azimuth of all the courses must be with reference to it. 57. Reference Lines. The line to which the azimuth is referred can be assumed in any desired direction, and one of the sides is often taken as this reference line if only the area is required. Thus, in the example, if AB is assumed as the reference line, the azimuths with respect to this line are 180, 224 30', 340 17', 60 45'. In calculating the area the bear- TRANSIT SURVEYING. 45 ing can be taken with respect to the reference line. If AB were our reference line, the field notes would be as follows : Course. " Bearing. Distance. AB North 255.72 varas BC N 44 30' E 182.10 varas CD S 19 43' E . 329.42 varas DA N 60 45' W 249.92 varas PROBLEM 21. In a farm ABCDE, ,45 = 19.90 chains, BC 9.03 chains, CD = 9.77 chains, #=5.67 chains, ,4 = 13.24 chains; A=8 c 12', 5 = 73 37', C = 139 08', D = l(>3 40', = 74 24'. If the azimuth of AB = 18Q, find the azimuth and the bear- ings of the other lines. 58. Repeating Method. It is often desired to find the angle more accurately than it can be read by a single reading of the verniers. If we have a transit reading by vernier to one minute, we can, find any angle ABC to any desired fineness by the repeating method. Thus, if the one transit verniers read to one minute, we can find the angle to ten seconds by repeating the observation six times. The process is as follows : Telescope normal : 1. Set transit on point B, level up and set cross-wires on point A and read both verniers. 2. Unclamp upper motion and deflect to C, clamp upper mo- tion and read both verniers. 3. Unclamp lower motion, deflect to A and clamp. 4. Unclamp upper motion and deflect to C and clamp. 5. Unclamp lower motion, deflect to A, and set thereon. 6. Unclamp upper motion, deflect to C ' , set cross-wires there- on and read the angle as given by both verniers. This result is three times the angle ABC, etc., etc., etc. The process can be carried on till there have been five, ten, or twenty deflections by upper motion from A to C, thus measuring the an^le five, ten, or twenty times. Both verniers should be read in every case and . the average taken. Usually the angle is read a given number of times, as above, with the telescope normal beginning right (or left) station A, and then 46 SURVEYOR'S JIAXD BOOK. read the same number of limes with the telescope reversed, beginning on the left (or right) station C. Example. Vernier A. Vernier /< 1 : 31 42' 31 .!_>' 3 95 07' 95 07' 5 158 31' 158 32' The average of the five readings gives 158 31' 30", or an angle of 31 42' 18". 59. To Adjust the Plate Levels. The axis of the plate levels should be at right angles to the vertical axis or the axis of revolution. Set the transit up on the tripod, level it by the plate levels as near as possible, bring one of the. level tubes parallel to a pair of leveling screws, and bring the center of the bubble exactly to the center of its run. Then tarn the alidade 180 on its vertical axis, and if the bubble remains in the center of its tube, it is in adjustment. If not, lower the high end of the tube or raise the low end by means of the small capstan screws at the end of the tube a sufficient amount to correct half of the displacement of the bubble. Correct the re- mainder by means of the leveling screws and repeat as a check on your work. Usually it takes several trials to make this ad- justment. 60. Line of Sight Adjustment. To make the line of sight perpendicular to the horizontal axis, set up the instru- ment on some plane nearly level, bring the plate bubbles to the center of their run, and locate a point about 100 to 200 ft. from the instrument; 'turn the instrument on its horizontal axis and locate another point the same distance from the instrument, but in an opposite direction ; revolve the alidade and bring the ver- tical wire in coincidence with the point first located ; then turn the telescope on its horizontal axis and locate another point near the second point located in the intersection of the cross- wires. If this point last located coincides with the second point located, the line of sights is perpendicular to the horizontal axis. If it is not, correct one-fourth of the displacement and mark this point, and proceed as* before. Let AB, Fig. 28, be the position of the horizontal axis when the point 1 is located, and let the line of sights make an angle TRANSIT SURF EYING. 47 of 10.r = dr with the perpendicular xy to- the axis. Revolve the telescope on the horizontal axis and locate the point 2. Now the angle 102=180 2u. Turn the alidade around the vertical axis till the line of sights intersect 1. As 02 has been turned through the angle 102 or 180 2a, AB has been turned through the same angle and occupies the position A'B', where AO'A = 180 2o, orA'OB = 2a. The perpendicular has moved to the position of x'y' where xOx' = 180 2a or x'Oy = 2a. Let telescope point to 1, then revolve it on its horizontal axis and locate a point 3 in the line of sights near 2. The angle 103 = 180 2o. Therefore, the angle 203 = 4o; hence, all we have to do is to bring the line of sights into coincidence with x'y'. Divide the angle 203 into four parts and make 30,r' j, one-fourth of the angle 302. We can do this by setting a point x' at one-fourth of the distance from 3 to 2, and as 02 and 03 are several hundred times 2-3, this is as accurately as we can measure the angle 30;r' equal to one-fourth of 203. Now, keep the axis clamped in the position A'B' , and move the ver- tical wire by capstan screws till it coincides with the point x '. Repeat whole work until it checks. 48 SURVEYOR'S HAND BOOK. 61. Peg Adjustment. The axis of the bubble tube may be made parallel to the line of sights by the peg adjustment. Drive two pegs or stakes in the ground about 200 ft. apart, whose difference of level is less than 4 ft. Set the transit mur peg A, level the instrument, and turn the telescope so that the eye end is over the peg, while the bubble is in the center of its run; measure the height of the center of the eye-piece above the peg and call this distance h. Have an assistant hold the rod on top of peg B and measure from where the line of sights cuts the rod to the top of peg B and call this r. Transfer the. transit to peg B and set up as before, measuring the height of the center of the eye-piece from the top of peg B and call this distance h'. Have the rod placed on top of peg A and measure the distance from the line of sights to the top of peg A. and call FI *- 29 - this distance r. In Fig. 29, AK, CD and FG are the horizontal lines as de- termined by the bubble tube. Suppose the line of sights C/T cuts below the horizontal line an amount of DE = e', when the transit is transferred to B it will again cut below when the telescope is sighted to A an amount HG = e. Let AC = h BE = r, BF = h'^lH=r f . Then the true-dif- ference of level of A and B = BK = BD AC = r + c h. Also BK = BF AG = h'(r' + e) Therefore r + e h = h' r' e Therefore <=% [(h + //') (r + r')]. Rule: The double error is equal to the sum uf the instru- ment heights minus the sum of the rod heights. 62. Location of Meridian by Polaris. Table I gives the times when Polaris and the mean sun are on the meridian to- gether. For 1907 the "epoch" is 14.1 This means that the mean sun and Polaris are on the meridian together April 14, one- tenth of a day after the beginning of April 14 that is, 2.4 hours after the beginning of April 14. This would make the TRANSIT SURVEYING. 49 "epoch" occur on April 14 at 2:24 A. M. For 1909 the "epoch" is 13.8, or April 13, 7:12 P. M. The "epoch," then, is the time or date when Polaris and the mean sun are on a meridian at the same time. Table I Epochs equal date in April when Mean Sun and Polaris are on a Meridian together: Year. Epoch. Year. Epoch. Year. Epoch. Year. Epoch. Year. Epoch. 1907 14.1 1912 13.9 1917 14.6 1922 15.3 1927 15.9 1908 13.5 1913 14.2 1918 15.0 1923 15.6 1928 15.3 1909 13.8 1914 14.6 1919 15.3 1924 15.0 1929 15.6 1910 14.2 1915 14.9 1920 14.7 192.' 15.3 1930 15.9 1911 14.5 1916 14.3 1921 15.0 1926 15.6 1931 16.2 If Polaris and the mean sun are on a meridian together, the mean sun will reach the meridian 4 minutes later than Polaris on next day. The hour angle of the star will be more than that of the sun by 3.5)4 multiplied by the number of days after the epoch. EXAMPLE: Find the position of the star (/) in its orbit at 9 P. M. May 6, 1907. The "epoch" for 1907 is on April 14 at 2:24 A. M. The number of days from 2:24 A. M., April 14, to 9 P. M., May 6, is 22.775. Hence Polaris will be 22.775 X 3.94 = 89.73 min. ahead of the sun. At 9 P. M. the sun is 9 hrs. past the meridian of the observer, hence Polaris will be hrs. plus 89.73 min. or 10 hrs. 29.73 min. past the meridian. By using this time (/) in Table II we can find the angle Po- laris makes at that time with the true meridian. Table II. / = local mean time + 3.94 (date epoch). Hours. t Angle, a. Lat. cor., b. t 0' 74' 24 1 25' 72' 23 2 49' 64' 22 3 69' 52' 21 4 84' 37' 20 5 93' 19' 19 6 96' 18 7 92' +19' 17 8 82' +36' 16 9 67' +51' 15 10 47' +63' 14 11 24' +70' 13 12 0' +72' 12 1900. 1910. 1920. 1930. .82 .78 .75 .72 .88 .85 .81 .77 1.00 .96 .92 .87 1.10 1.14 1.09 1.04 50 SURVEYOR'S HAND BOOK. Table III. Azimuth Coefficients. Coefficients = K. Lat. 20 30 40 50 Table IV. Lat. correction Coefficient. Year. Coefficient, Q. 1900 1.00 1910 .!)< 1920 .92 1930 .87 EXAMPLE: Find the angle that Polaris makes \\ith the true meridian at 9 P. M. May 6, 1907, in latitude 3o. The time in- terval from epoch to date was. 22.775 days and the increase in time was 1 hr. and 29.77 mins. The value of t was found to he 10 hrs. and 29.73 mins. or 10.50 hrs. (which is near enough for our purposes). Looking in Table II under / for 1 ).-"> ho'r.s we find that we have to interpolate between 47' and 21', hence the angle is 35.5'. This must be multiplied by the a/imuth coefficients. For 1900, lat. 30, the coefficient, Table III, is 0.88, and for 1900 it is 0.85. For one year the decrease is 0.003, and for seven years it is 0.021. The coefficient is therefore 0.80. The angle or azimuth with the north meridian =35. 5 X .86 =30.6' west. The observed altitude of the star was 29 8'. The lati- tude coefficient for 1907 lies between 1.00 and 0.96 and an in- terpolation gives- .972. From Table II, lat. cor. (b) is 00.5'. Hence the correction for the altitude will = .972 X 60.5 = 04.04'. The latitude =29 8' + 64.04' = 30 12.64'. PROBLEM 22. Find the angle that Polaris makes with the true meridian 9 P. M. June 12, 1907, in latitude of 33. An- swer =20' east. PROBLEM 23. Given latitude of place = 36, find the angle Polaris makes with meridian on November 6, 1909, 10 P. M. Answer = 9'.2 east. PROBLEM 24. An observation was made on Polaris at 9:30 P. M. July 22, 1908, in latitude 30. Find the angle made with the meridian. Answer =71' east. TRANSIT SURVEYING. 51 NOIlVNIINYu X3MQ1 * * > Delta 63. Circumpolar Stars. A meridian can be located with sufficient accuracy for ordinary surveying by observations on the North Star (known as Polaris), which is about one and one- fifth degrees from the true North Pole, and if we would ob- serve it for a whole day it would appear to describe a circle about the North Pole in a direction contrary to the motion of the hands of a clock, /. c., contra-clockwise. On account of the invisibility of the true North Pole this motion can be best observed by select- ing some star in the Dipper. If we could noce exactly when one of the stars of the Dipper is directly above Polaris and could follow its motion throughout the bal- ^ ance of the night, the next ^ day and part of the next I- night, we would observe |- that the star would again ^ reach a point directly above || Polaris four minutes earlier than it did on the preced- ing day. If we observed it directly above Polaris at 10 P. M. on one night, the next night it would be at the same position at 56 mins. after 9 o'clock. Thus, each of these stars gains four minutes each day (exactly, 3.945 minutes). In one year it would gain 24 hours and would, therefore, make one more revo- lution than the earth makes on its axis. All stars that make an apparent complete revolution about the North Pole are called circumpolar stars, and any of them could be used for the location of a meridian when the selected star is directly above or below Polaris. Polaris Pole Major / \ ' 'The Dipper UPPER CULMINATION Fig. 30. SURVEYOR'S HAND BOOK. There are two groups of stars (called constellations) sit- uated opposite to each other with resp.ect to Polaris and the North Pole that afford favorable opportunity for the location of the meridian by surveyors. These constellations are those of the Great Bear (the Dipper) and of Cassiopeia (the Chair). By a glance at the outlines of these constellations, Fig. 30, it will be seen that the dotted lines outline the shape of a dipper and chair, respectively, hence the names. It must be remembered that Polaris is always opposite the Dipper with respect to the pole, and that it is on the same side as the Chair. When a star is directly above the pole it is said to be at its up per culmination, and when directly below, at its lower culmina- tion. When at the eastern! point of its orbit, it is said to be at its eastern elongation, and when at its western point, at its western elongation. 64. Location of Meridian. A line passing through the second star (Zeta) in the handle of the Dipper and the third in the back of the Chair (Delta Cassiopeia) passes through Polaris and the North Pole. When Zeta of the Dipper or Delta Cas- siopeia is directly above or below Polaris, Polaris is on the meridian, and is at its upper or lower culmination. If the Dip- per is above, Polaris is below the pole, and vice versa. But when the star is at either culmination, its horizontal motion is more rapid than at any other point in its path, and a slight error in time affects the result. When the star is at either elonga- tion, the direction of its motion is vertical, and a slight error in time does not have such decided influence on the azii.iuth. 65. PZS Triangle. The North Pole (P), the Zenith (Z) and the Sun (S) form a spherical triangle PZS, Fig. 31, where if Fig. 31. TRAXS1T SURVEYING. 53 /= latitude of observer hour angle a azimuth of sun d = declination of sun h altitude of sun. We have : PZ = co-latitude = 00 / ; PS = co-declination 90 d ; ZS = co-altitude = f h ; ZPS hour angle of sun ; SZM = azimuth 360 a. 66. Formulas. The usual problem is to locate a meridian at a certain place whose latitude and longitude are known. Drop a perpendicular from 6* on the meridian PZ of the observer, cutting it at M, and let ME = N where EQ is the celestial equa- tor, or earth's equator extended to the heavens. Now, ZE latitude /. :.ZM \ N. By the application cf Napier's Tangent Law, we have, in the right triangle PSM: cos f=tan PM cot PS cos t = tan (90 AO cot (90 -d) cor N tan d tan d In the right triangle, SZM sin ZM = -tan MS cot a or tan l\fS := siii ZM tan a. In the right triangle MPS, s i n PM = tan MS cot t or tan MS = sin PM tan t Equating the two values of tan MS , we get : tan a sin ZM = sin PM tan t sin PM tan t tan a * sin ZM But PM = M N, ZM = l N cos N tan t sin(l-N) 54 SCKl'liyOK'S //.LVD BOOK. 67. Observation on Sun. The best time of day to make an observation for azimuth on the sun is from c to 10 A. M. and from 3 to o P. M. Before an observation -.s made it is necessary to have mean local time and if a chronometer is not available, two watches should be set to agree with Western Union time. Thirty minutes before the observation is to be made the transit should be set over the station, the verniers should be brought to zero, and the transit be pointed to some definite terrestrial point, as a church spire. The transit should then be turned by upper motion to point approximately at the sun, and as soon as. the sun comes into the field of view of tin- telescope, the observer should clamp the upper motion and call "angle," when two men read the angles as given by the two opposite verniers. At the signal "angle" the timekeepers, of which there should be at least two, get ready to observe the time. As the disc of the sun approaches the vertical wire, the observer calls, "Get ready," and just as the edge of the sun's disc coincides with the vertical wire he calls "time" and im- mediately moves the vertical wire by aid of the tangent screw till the opposite edge of the sun's disc coincides with vertical wire, when he calls "time*' again. The time interval between the two calls of "time" should not be over six seconds. The timekeepers have noted both the hours, minutes and seconds at each call of "time," and the angle readers read both angles and record same. The data taken in the field therefore con- sist of reading the spire-station-sun angle for both discs of sun, and the times corresponding to these. The average of each is taken as the angle and time of the sun's center The local mean time is reduced to apparent time, and this to degrees, which gives the hour angle. The declination of the sun is found for the given time and N is found from Formula 5, and the substitution of values of N, t and / in Formula 6 will give the angle a. The second method of finding the angle a consists in meas- uring the altitude of the sun at the time of observation. To do this, the disc of the sun is brought to tangency with the vertical wire and on its left, so that the lower edge of disc TRAXSIT SURVEYING. -V> coincides with the horizontal wire. If we regard the cross-wires as axes, the sun would be in the second quadrant and tangent to both axes at the first observation. In this position we record time, the spire-station-sun angle, and the vertical angle. The disc is then brought into the fourth quadrant, so that it touches the two axes, when the same data are observed as before. The average of these is taken as the spire-station-sun angle, the angle of elevation, and the time of observation. Then the angle is corrected for refraction and this gives us the complement of ZS of the triangle PZS. The three sides of the triangle PZS are thus known whence the angle PZS can be calculated. Let* = i(PZ + ZS + PS). I ~sin (sPZ) sin (sZS) Then sin \PZS = -J s i n pz sin ZS 68. Refraction. The effect of refraction is to raise all bodies and make them appear higher than their true positions. Thus the sun can be seen wholly above the horizon, when in reality no part of it is above. If R represents the amount of refraction in seconds of arc and h is the altitude of the sun, we have: R = r>8" tan h Table V. Table of Refractions : Elevation. Refraction. Elevation. Refraction. Elevation. Refraction. 5 J 9' 52" 16 3' 20" 35 1' 23" 10 5' 19" 1V 3' 08" 40 1' 09" 11 4' 51" ' 18 2' 58" 45 0' 58" 12 4' 28" 19 2' 48" 50 0' 49" 13 4' 07" 20 2' 39" 60 0' 34" 14 3' 50" 25 2' 04" 70 0' 21" 15 3' 34" 30 1" 41" 80 0' 10" 69. Solar Attachment. There are various forms of solar attachments, but we shall here describe only two. Fig. 32 shows a diagonal prism, which "consists of a prism attached to the cap of the eye-piece, by which the object is presented to the eye when placed at right angles to the telescope. When the telescope is directed to the sun the slide or darkener con- taining the colored glass is moved over the opening. The cir- oilar plate with which the prism is connected is made to turn in the cap, so that when it is substituted for the ordinary cap of the eye-piece the opening of the prism can be easily adjusted 56 SURVEYOR'S HAND BOOK. to the position of the eye. Observations can be taken with the prism up to an angle of 60 of elevation." The other form of solar attachment consists of a second telescope, generally smaller in size, attached to the regular tele- scope of the transit. The second telescope is provided with colored glass to enable the observer to see the sun with dis- tinctness and definition. Fig. 33 illustrates a common form of this solar attachment which is provided with telescope level .-UK! tangent screws for horizontal and vertical motions. The line of sight of the solar telescope cin be made parallel to that of the transit by bringing both bubble tubes to the middle of their run, while the telescopes are pointed at a vertical line some '200 ft. away. This line should be marked on a white sheet of paper tacked to the side of a house on the same level practically with the telescopes. Draw two heavy horizontal lines on this sheet of paper at a distance apart equal to the distance between the axes of the telescopes. .Bring the cross wires of the transit telescope on the lower, of these lines, and if the lines of sights arc parallel the line of sight of the solar telescope will intersect the upper horizontal line. If it docs not, adjust its reticule till the line of sight as defined by the cross-wires intersect the upper line. Check till perfect agree- ment is secured. An error of 1-16 in. in the distance between the axes in 200 ft. would produce an error in the parallel align- ment of the lines of sight of dnly 5". A longer base would reduce the error. If the base is 507 ft. and the error in dis- tance between axes is 1-30 in., the lines of sight will make an angle of 1". To eliminate light errors in latitude and as a check on the work, observations can be taken in the forenoon and after- noon at about the same time from the meridian passage of the sun. In each set of observations the transit is set on a ter- restrial mark, the altitude of the sun, the angle mark- station-sun, and the times are taken and recorded. The angle TRANSIT SURVEYING. 57 PZS is calculated and the azimuth of the line from station to mark can be found by addition or subtraction. 70. Meridian Without Calculation. If the meridian is to be located directly by observation, some so- lar attachment like that of Fig. 33 is nec- essary. To locate a meridian by this method we proceed as follows : 1. Make the usual five adjustments for the transit, three for the ordinary transit and two for the solar attachment. 2. Bring the line of sight of solar tele- scope into the vertical plane of the line of sight of the transit telescope. 3. If declination of sun is south (north) depress (elevate) the transit telescope an amount equal to the declination corrected for refraction, then bring the solar tele- scope to a hori- Fig. zontal position by means of its bubble tube. The lines of sights of the telescope will now include an angle equal to the cor- rected declination. 4. Elevate the transit telescope till the vertical arc reads the co-latitude of the place. 58 SURVEYOR'S HAND BOOK. 5. Revolve both telescopes on their vertical axes till the image of the sun is bisected by the vertical wire of the solar telescope. When this bisection is secured the line of sight of the transit telescope will he in the plane of the meridian and will locate it. 71. Example: On April 15th, 1907, the following observa- tions were made on the sun at the magnetic sta-tion, Austin, Texas (latitude 30 17', longitude !>7 -44' 02") : Disc of Sun. W. U. Time. Mark, Station, Sun Angle. Right ........... Oh. 50m. 57s. 75 8' Left ....... ..... lOh. Om. 03s. 75 40' Average ......... lOh. Om. Os. 75 24' W. U. Time (00 meridian) = lOh O m Os .Correction 30 56 Local Mean Time Oh 20m 04s Time from Greenwich mean noon to Austin. Mean noon Oh 3<>s. Time interval from Greenwich noon to ohs." 4h. Declination at Greenwich mean noon = ( J 27' i" .00 N. Hourly increase = 53" .0(1. Total increase = 3' 35" .84. Declination at time of observation 30' 38" .7-1, " Equation of time at Greenwich, mean noon Orn 17.15:=. Hourly decrease 2i is. Total decrease = 2.504s. E. T. at time of ohs. = Om 14 .Cos. Apparent time of ohs. = Oh 28m 40.35s = 0.480375h. f = hour angle 57>Z = 2.510025h = 37 47' 40". tan d Log tan d = 9.224108 log cos t = 0.807745 log tan N 0.326363 N = ll 58' 12 / N = 18 18' 48" TRANSIT SURVEYING. cos N tan t tan a sn- // cos X = 9.990453 log tan t 9.889594 co-log sin (I N) = .502775. log tan (7 = 10.382822 a 07 -30' 8" Azimuth of sun = 292 29' 52" Azimuth of mark = 75 24' (.7 30' 8" = 7 53' 52" 72. Example: The following data were taken at a sta- tion where latitude = 29 8' .1 and longitude = 97 23' W. No. Sun. Alt. of Sun. M -b- s? & Sun W. U. Time = 90 M. 1 T) 31 53' 48" 5 58' 24" 8h. 53 .Om. A.M. 2 "(J| 32 3 JO 32. 22' 26' 00" 12" 5 4 33' 05' 12" 24" 8h. 8h. 55.5m. " 58.0m. " 4 JO Mean 32 55' 30" 3 30'' 12" 9h. 00.5m. " 32 24' 22.8" 4 48' 33" 8h. 56 .75m. " Declination of sun at Greenwich, me an noon = 2m . 19.3s. Hourly increase 58".4. Time interval from G. noon to Observation = 2h 56.8m. Total increase in Declination 2'.9. Declination at time of Observation = 2 17'. 2s. Observed altitude of Sun = 32 24'.3S. Correction for refraction and parallax = 1'.3. True altitude of Sun = 32 23'. 1. In the PZS triangle we have, PZ Go 3 51' 54" = co-lat. .P5 92 17' r2"= : co-deCo ZS $r 36' 54" = co-alt /. 2^ = 210 46' 00". ^ 105 23' .-=iy 5' 48" 60 SURVEYOR'S HAND BOOK. sin 5 sin (s codec) (Cos i pzsy- ' -- sin co _ alt sin co _ lat Log sin s = 9.084155 Loi sin (s-codec) = 9.355249 cologsin co-lat = 0.073417 colog sin co-lot = 0.058749 2 fog o>s iPZS = 19.471570 log cos I PZS = 9.735785. .'. \PZS=-- 57 1' 40" PZS '= 114 3' 20" Azimuth of sun at time of obs. = 294 3' 20" Angle Mk-Sta-Sun = . 4 48' 33" Azimuth of Mark = 298 51' 53" Bibliography. "Theory and Practice of Surveying." By J. B. Johnson. This is one of the best, most practical, and com- prehensive books upon higher surveying. It includes a discus- sion of the engineering instruments in their use in ordinary and higher surveying, leveling, topographic, hydrograohic, railroad, and earthwork surveying. "The Principles and Practice of Surveying " By Breed and Hosmer. 520 pages. This is a rather full treatment on the use, care, and adjustments of instruments, land surveying, traverse lines, meridians and latitude, city surveying, mine surveying, plotting, specimen note books and computations. "Plane Surveying." By Wm. G. Raymond. 485 pages. This is a full discussion of the construction and use of the engineer- ing field instruments, methods of land, city, hydrographic, etc., surveying, and an ample treatment of the slide rule (an un- usual feature of a work en surveying), and an excellent set of tables. "Surveying Manual." By W. D. Pence and Milo S. Ketchum. 252 pages. This is one of the most valuable hand-books or field manuals now in print. While it is modest in size, it covers in a satisfactory way the usual problems confronting the surveyor and engineer. A distinguishing feature is the sample pages of note books executed in freehand lettering. CHAPTER IV. CALCULATION OF AREAS. 73. Latitude and Departure of a Course. Given a course AB, Fig. 34, and a meridian through one end of the course, and a perpendicular B2 from the other end upon the meridian. Then A -2 is called the latitude of the course, and 2-B the de- parture. The latitude of BC is B-6 or 3-2. All the latitudes that go north are called plus and all those that go south are called minus. Thus in the figure the latitudes of AB and DA are plus, while those of BC and CD are minus. plus latitudes The sum of the The sum of the minus latitudes B6 + W = 2-4. The algebraic sum of all the lati- tudes is equal to zero. All east departures are plus and all west departures are minus. Thus the departure of AB and BC are plus, while the departures of CD and DA are minus, The sum of the plus departures 2B + 6C 3C, while the sum of the minus or west departures 5C + D4 = C3. Fig. 34. The algebraic sum of all the departures is equal to zero. In the triangle A2B, let the length AB = 1, and the angle BA2 == B (called the "bearing*'). But A'2 = AB cosine BA2, that is, Latitude ^ length X cosine of bearing.. . . L I cosine B ............. .... ____ ,..*.... ____ (7) Also, B2 = AB x sine of.BA2, that is, Departure = length X sine of bearing, . . D^l sim B . . . . . . . . , . , . . . . . . . . . , . . . . . . . . .(8) 61 SURVEYOR'S HAND BOOK. Squaring 7 and 8, and adding, we get, L 2 + >2 = P (Cos*B + Stn 2 B). But Sm 2 B + Cos 2 B = 1, _ . . L 2 + D* = P . ' . Z = VLT+~D* ............... (9) Dividing 8 by 7, we get, Departure Tangent 5= . ........................... (10) Example: The field notes of a farm are given in the fol- lowing table: Course. Bearing. Distance. AB ..................... N2737'E 48.6 chains BC. .................... S6714'E 6->.4 chains CD ..................... S382S'W 52.0 chains D/f ..................... N6515'W 55.0 chains To find the latitudes and the departures it is convenient to proceed by finding the natural sines and cosines of all the bear- ings, and arranging them under the latitudes and departures as follows : Latitudes. Departures. Cosine. Distance. Latitudes. Sine. Distance. Departures. .88674 48.6 43.10 .46226 IH.-i 22.47 .38698 65.4 25,31 .9220!) 65,1 00.30 .78297 :>2. 6 = Total dep. for 5(5.8 chains = 9.88 .As an exercise, find the latitude and departure for bearing of 12 and a distance of 37.48 chains. 76. Error of Closure. In surveying parties the surveyor is usually the only skilled man in the party. The chainmen are 10 64 SURVEYOR'S HAND BOOK. usually picked up in the locality and are not supposed to be trained in this work. It is assumed in balancing the survey that the errors are due to the chaining and that the surveyor reads the bearings correctly. Tf in balancing the error is greater than 1 in 500 the farm should be resurveyed. The error in latitude or departure is the amount that the algebraic sum of the lati- tudes or departures lacks of being zero. The error of closure is found by squaring the error in latitude and the error in de- parture and taking the square root of their sum and dividing this result by the perimeter of the farm. This is simply divid- ing the distance you miss the beginning corner by the length of the perimeter of the farm. Find the latitudes and departures for the following courses : Course. Bearing. ' Latitudes - Departures. AB N2330'E '255.72 234.49 101.96 BC N68E 182.1 68.22 168.84 CD S347'W 329.42 328.67 21.74 DA N8415'W 249.92 25.04 248.66 Thus in the example the error in latitude is .92 and the error in departure is + .40, that is, we went nonh 1527. To and south + 328.67, which leaves us + .92 south of A. We went east 270.80 and west 270.40, which leaves us + .40 west of A at some point A'. But A A' = v / (.92) 2 + (.40) 2 V(.92) g + (.4Q) 2 1 And the error of closure = 1Q17 ^ = JoJI 77. Balancing a Survey. Theoretically the algebraic sum of the latitudes is equal to zero, and the same is true of the de- partures. But in actual survey work these sums never are equal to zero, owing to unavoidable errors. These errors must be distributed in proportion to the length of the courses. We see that the error in departure is .40, which must be distributed among the courses in proportion to their lengths. The total distance around the farm (the perimeter) is 1017.16 varas, and the total error in departures is .40 and that CALCULATION OF AREAS. 65 for latitudes is .92. The error of any course is to the total error as the length of any course is to the perimeter. If the compass was used in making the survey this rule for balancing should be followed even if some of the courses are due north-south, or due east-west. The compass cannot define the angle accurately and there is as much probability of error in angle in a due north course as there is in a course whose bearing is N. 26 E. Again, in some of the older states the magnetic bearings are read and a course that is north at the present time could make one degree with the magnetic meridian twenty years hence. If the practice of distributing the errors in departure (or latitude) among those courses that have departure be fol- lowed in the calculation of the first survey, the above method would have to be followed in the last survey. Thus the same surveyor would get different results for the area of the farm. The usual rule should be followed in all cases for a compass survey. Therefore, the error for any course = total error perimeter X len ^ th of course ' Con AB BC CD DA 'ections for .92 Latitude of .23 .16 .30 .23 Cor AB BC CD DA rections for .40 Departure o f .10 .07 .13 .10 .40 ~ 1017.16 .92 X 255.72 " 1017.16 .40 X 255.72 x; 181.1 = X 329.42 = " 1017.16 .92 X 182.1 ~ 1017.16 .40 = 1017.16 .92 X 320.42 ~ 1017.16 .40 " 1017.16 Total for X 249.92 1017.16 A ^ y -^_ Total for Departure = Latitude = .92 fhese are arranged in the following table : Corrections. Lat. Dep. Course Cor. Lat. AB BC.. CD. DA, .23 .16 .30 .23 .10 .07 .13 .10 234.72 68.38 -328.37 25.27 Cor. Dep. 101.86 168.77 21.87 248.76 SURVEYOR'S HAND BOOK. The sum of the uncorrected plus latitudes b 327.75, and that of the minus latitudes is 328.67; all the plus latitudes must" be increased by their corrections, and the minus must be de- creased by their corrections. If these corrections are applied property we will get the numbers in the column "Cor. Lat.," which means corrected latitudes. The sum of the plus depart- ures is 270.80 and the sum of the minus departures is 270.40; the sum of the plus departures is greater by .40; therefore the minus departures must be increased and the plus departures decreased. The column headed "Cor. Dep.'' gives the corrected departures, 78. The Double Meridian Distance. - - The reference meridian generally passes through the most westerly corner of the land. The perpendicular from the mid point of the course upon this meridian is called the meridian distance. The meridian distance of MNj Fig. 35, is xy where x is the midpoint of MX. But if M3, A r 4 and O5 are perpendicular to the meridian, M3 + JV4 = 2*3', k O or double the meridian distance, and is called the D. M. D. That is, the DMD of any course is equal to the sum of the two per- pendiculars from its ends upon Fig. 35. the reference meridian. The DMD of A T = A'4 + O5 = A'4 + Ol + N6 + M3 = (N4 + M3) + N6 + 06 That is, the DMD of any course is equal to the DMD of the preceding course, plus* the departure of the preceding course plus the departure of the course itself. The DMD's of the first and last courses are always equal to their own departures. A sketch of the farm whose latitudes and departures were balanced in Art. 77 shows that A is the most westerly corner, and it will be convenient to take our reference meridian through "CALCULATION OF AREAS. this corner. Then the Double Meridian Distance of AB = De- parture of AB = 101.86. The D. M. D. of BC= 101.86 + 101 86 + 168.77 = 372.49. The D. M. D. of CD = 372.49 + 168.77 21.87 = 51f.39. The D. M. D. of >,4 = 519.39 21,87 248.70 = 248.76. The last result proves the correctness of our arithmetical work, as the DMD of the last course should equal the de- parture of that course. If the course AB does not happen to be in the first line of the table of notes, the DMD's can be calculated with reference to the most westerly comer without rearranging the table. 79. Area of a Farm. If we M drop perpendiculars from the ends of the courses upon the meridian NS. Fig. 36, we form trapezoids, or trian- gles. If we survey around the farm clockwise, all the areas determined by the courses and perpendiculars that have plus latitudes will be out- side the farm, while those that have minus latitudes will include part of the farm and part of the area be- tween the farm and the reference meridian. The algebraic sum of the "minus areas" and "plus areas" is equal to the area of the farm. The double area of AB2 = A'2 X B'2 Lat. XDMD. Double area 2# C62 3 X (B'l+ C'3) = Lat. X DMD. Double area BC D4 = 3 4 X (C3 + D4) Lat. X DMD. Double area W 5=5 4(774 + 5) = Lat. X DMD. Double area EA=5A (5E\=Lat. X DMD. The areas of 253C and 3CD4 have minus latitudes (2 3 and 3 4) and these areas are therefore called "minus areas," and they not only include the whole farm but also the areas between the farm and the reference meridian. The areas AB2, 4D.E5. and SEA have plus latitudes, and are called "plus areas." If Fig. 36. 68 SURVEYOR'S HAND BOOK. we add the "plus areas" to the "minus areas," there is left the area of the farm. ABODE. Co.r Be.,,-,, D.sr Cor Dp DMD Plus AK.O Minu* Area AB N38E. 26ch 2049 1601 20.40 16.15 1 6.15 -3294&00 BC S42E 34c, -25.27 2275 -2339 22.93 3523 1402.2637 CD S5IW 20t, -259 -15.54 -1266 -3.*3 62 73 794.1618 DA N72W 25h 775 -2378 765 2365 2365 1809285 28 22 -39 32 510 3825 2I904AIA -278G _ 38 76 5)0.3825 36 - -36 Z X Area = IS6.0690 Art a m S< Ch 6430343 Art to m Acres = 8430JT Err oca. La t Cor Dep Cor C,- - 3&*2< ; = .O9 C, t? *26* 14 c, x 34 J8 C.- X 2 5 07 c,. 1 20 = ji cv X2 3' 08 c* " 25- .13 Total = 36 Tetol - 50 Fig. 37. 80. Area Table. Placing the 1) M D's in the table and multiplying each by its corresponding latitude, we find the areas as given in the following table. Dividing the area in square yards by 4840 gives the area in acres : Course. Bearing. Dist, Lai, Corrections. Dep. Lat. Dep AB . BC .... CD .... DA .... N 23 30 E N 68 E S 3 47 W N84 15 W 255.72 182.1 329.42 249.92 234.40 68 22 328.67 25.04 101.96 168.84 21.74 248.66 .23 .16 .30 .23 .10 .07 .13 .10 CALCULATION OF AREAS. Cor. Lat. Cor. Dep. D. M. D. 234.72 101.86 101.80 68.38 1(58.77 372.41) -328.37 21.87 519.39 25.27 248.76 248.76 Plus Areas. Minus Areas. 23,905.5792 28,479.0662 170,552.0943 6,286.5652 55,671.2106 170,552.0943 55,671.5652 Double area .'. Area = 57,440.441 95 sq. yds. = 23.7357 acres. = 114,880.8837 The standard form of calculation of errors and areas is shown in Fig. 37. PROBLEM 25. William James Farm. Course. Bearing. Distance. AB S8445'E 19.73 chains BC S21E 15.85 chains CD S78W 19.53 chains DA N16W 21.51 chains Area = 35.01575 acres. PROBLEM 26. Cambria Farm. Course. Bearing. Distance. AB S 41 E 100 poles BC.. S 29 W 41 poles CD N 69 W 99 poles DA N 31 30 E 90 poles Area = 39.357 acres. PROBLEM 27. Oran Farm. Course. Bearing. Distance. AB N516'E 205 6 varas BC N2830'W 263,5 varas CD S16E 255.6 varas DE S315'E 210.6 varas EA N8415 W 227.7 varas Area =11. 958 acres. PROBLEM 28. Diego Blanco Farm. Course. Bearing. Distance. AB N56E 8.18 chains BC S16E 5.39 chains CD . . S5930'W 7.49 chains DA N1710'W 4.87 chains Area 3.97879 acres. 70 SURVEYOR'S HAND BOOK. PROBLEM 29. Bowie Blanca Farm. Course. Bearing. AB N56E BC S1(>E CD S5930'W ' DA N1710'W Area = 3.9786 acres. PROBLEM 30. Bantello Farm. Course. Bearing. AB N33E BC S67E CD S3840'W DA N48W Area = 27.5937 acres. PROBLEM 31. Leon Brooks Farm. Course. Bearing. AB N516'E BC S1710'E CD S5630'E DA N8515'W Area = 2.80 acres. PROBLEM 32. Francis Estell Farm. Course. Bearing. AB N5E BC S7015'E CD S16E DA S56W Area= acres. PROBLEM 33. Jnnn Viego Farm. Course. Bearing. AB N5930'E BC N78E CD S2330'W DA N5630'W Area = acres. PROBLEM 34. John Bruce Farm. Course. Bearing. AB.. ..N87E BC S5917'W CD S8445'W DA N16W Area = . . . acres. Distance. 540.0 feet 356.0 feet 524.0 feet 321.2 feet Distance. 14 chains 18 chains 19 chains 16 chains Distance. 205.6 varas 115.6 varas 207.7 varas 227.3 varas Distance. 750.0 feet 300.0 feet 356.8 feet 540.0 feet Distance. 7.94 chains 4.88 chains 10.77 chains 8.74 chains Distance. 376.0 varas 260.0 varas 117.3 varas 128.4 varas CALCULATION OF AREAS. 71 PROBLEM 35. H. Yandell Farm. Course. Bearing. AB N5917'E BC S915'W CD S68W DA N21W Area^ acres. PROBLEM 30. Sidney Dean Farm. Course. Bearing. AB N67W BC S5903'W CD S49E DA N4730'E Area = 23. 53 1 acres. PROBLEM 38. George Pierce Farm. Course. Bearing. AB N2645'E BC N1142'E CD N4005'E DE S4142'E EF S4015'W FG S2015'W GA N6934'W PROBLEM 38. Old Perry Farm. Course. Bearing. AB S7345'E BC S5330'E CD S4712'W DE '. ...S5500'E EF..... S4800'W FG.... N6357'W GH N5325'W HA N330'W Area = 48.15785 acres. PROBLEM 39. Course. Bearing. AB N5742'E BC S30E CD ..- S3840'W DE S23W EF S60E FA N3057'W Area = 34.2404 acres. Distance. 260.0 varas 151.6 varas 182.1 varas' 94.3 varas Distance. 11.52 chains 18.30 chains 14.30 chains 21.17 chains Distance. 60 feet 364 feet 1038 feet 1650 feet 341 feet 322 feet 1640 feet Distance. 21.60 chains 22.72 chains 6.95 chains 8.38 chains 5.55 chains 3.12 chains 41.60 chains 6.36 chains Distance. 17.47 chains 18.47 chains 2.27 chains 1.88 chains 13.05 chains 20.21 chains 72 SURVEYOR'S' HAND BOOK. PROBLEM 40. Course. Bearing. Distance. AB N353'E 7.70 chains BC S828'E 39.05 chains CD S8342'E 14.39 chains DE S569'W 14.26 chains EA N803'W 42.30 chains Area = 40.604 acres. PROBLEM 41. Course. Bearing. Distance. AB N6005'E 19.90 chains BC S13 C 32'W 9.03 chains CD S2720'W 9.77 chains DE S4340'W 5.7 chains EA N3043'W 13.24 chains Area = 16.3432 acres. 81. Courses of No Latitude or Departure. If a survey is made with the transit, the sum of the interior angles of the polygon should equal two right angles taken as many times as the polygon has sides less two. The error should not amount to more than three minutes, unless the number of sides is large. In a transit survey there can be very little error in the angular measurements and all errors in latitude and departure are largely due to errors in chaining. If a transit line is due north it is presumed that it is in the true meridian and there- fore has no departure. Similarly if the course is due east it has no latitude, and if the angles check within three minutes (3'), the errors must be distributed on the assumption that they were due 10 the chaining. The practice is to distribute the errors in latitude (departure) among those courses that have latitude or departure. Thus no north-south course would receive a correc- tion for departure as its original departure and also its balanced departure is zero. Similarly a due east-west course receives no correction for latitude. Hence if a course is north (east) its length is omitted in the perimeter of the field in calculating the errors in departure (latitude). The following rules are used in balancing : Rule No. 1. Distribute all errors in latitude (departure) in proportion to the length of the courses. If any course is north CALCULATION OF AREAS. 73 (east) its length is omitted from the perimeter of the Held. Error in latitude (departure) for any course is to the whole error in latitude (departure) as each course is to the corrected perimeter. Rule No. 2. The error in latitude (departure) in any course is to the whole error in latitude as the latitude of the course is to the sum of all the latitudes. The transit is rapidly becoming the surveyor's instrument, as there is greater demand for accuracy with the advanced price of land. The needle is inaccurate at best and when we consider the effect of barbed wire fences, telephone and telegraph wires, local attraction and other similar influences that render the needle unstable, its efficiency as an instrument of precision is rendered doubtful in the extreme. Rule No. 2 is by far the most logical in transit surveys and should be used in balancing, and it has the advantage that it is automatic in that it finds no error in departure for north-south courses or in latitude for east-west courses. 82. Example: In the following survey the errors were distributed in proportion to the length of those courses that have latitude or departure : Course AB BC CD DE EA Bearing. N47E East S32E South N7730'W D. M. D. 2966 Dist. Lat. Dep. 40.57 27.67 29.67 34.59 00. 34.59 27.10 22.98 14.36 22.01 22.01 .00 80.51 17.43 78.60 Plus Area. 819.8024 Cor- rections. Cor. Lat. Dep. Lat. .03 .01 27.64 .00 .00 .02 .00 23.00 .01 .00 22.02 .05 .01 17.38 Minus Area. Cor. Dep. 29.66 34.59 14.36 is.'e'i 9391 142.86 157 22 3,285.7800 3,461.9844 7861 13662418 2186.04-12 6,747.7644 2,184.0/42 Double area = 4,561.7202 Area = 2,280.8601 sq. chains = 228.08601 acres. 74 SURVEYOR'S HAND BOOK. If the errors are distributed in proportion to the latitudes and departures, the result is as follows: Correc- tions. Cor. Lat. Dep. Lat. .03 .00 .00 .00 .03 .00 .03 .00 .02 .02 27.64 0.00 23.01 23.04 17,41 Cor. Dep. D. M. D 29.67 29.67 34,59 93.93 14.36 0.00 78.62 142.88 157.24 78.62 820.0788 1368.7742 2188.8530 3287.0688 3465.5696 6753.2371 2188.8530 Double area = 4564.3844 .-. area = 2282.1922 = 228.22 acres Distance. 20.0 chains 8.0 chains 28.0 chains 23.3 chains Distance. 10 chains 11 chains 17 chains 20 chains PROBLEM 42. Course. Bearing. AB N369'E BC East CD South DA N592'W Area = 34,3779 acres. PROBLEM 43. Course. Bearing. AB N3930'E BC East CD South DA N61W Area = 19.158 acres. PROBLEM 44. Find the area of the following: Beginning at a stake in road 762.5 feet west from Chisholm's southwest cor- ner; thence N. 30' E. 661 feet; thence up branch S. 81 W. 117 feet, S. 22 W. 124 feet, S. 8 W. 87, S. 70 30' W. 162 feet, then S. 27 30' W. 153, S. 31 30' E. 62 feet, S. 34 "W. 94 feet, E. 304 feet, S. 5 W. 129 feet to middle of said road; thence E. along said road 116 feet to beginning. PROBLEM 45. Course. Bearing. Distance. AB N39E 20 BC East 8 CD South 28 DA.. ..N60W 23 CALCULATION OF AREAS. 75 83. Area by Co-ordinates. If the co-ordinates of each corner of the farm are given with reference to two axes OX and OY, we can find the area by dropping perpendiculars from each corner on either axis, as OX, Fig. 38. Let Oa, Ob, Oc and Od x&, x^, x c and A a, Bb, Cc and Dd = -y a , y b , y c and y&. repectively; and Now area aABb =- ab area bBCc == be area cCDd = cd area dDA a = da (Aa + Bo) 2 (Bb + Cc) 2 (Cc + Dd) (Dd + Aa) area of farm = aABb + bBCc cCDd dDAa = (# b x & ) O'a + y\>} . . Double area ( Vc >'a) - Similarly Double area = (y) + x\, (y & y c ) + x c (y b yd) + (x t \ x\>) + Tb (*& *e) + This can be crystallized into the following rule : To find the double area, multiply each abscissa (ordinate) by the difference of the adjacent ordinates (absc'issas) taken in order. 76 SURVEYOR'S HAND BOOK. EXAMPLE. Find the area of the farm whose co-ordinates are (2, 6), (6, 10), (12, 8), (4, 2). Diff. of X's. 2 10 2 n X. 2 6 12 4 Area, 7. 6 10 8 9 10 Fig. 39. Course. AB BC CD DA Bearing. N31E N33E N36E Diff. of Area. Y's. Area. 12 8 16 100 2 12 _16 8 96 _20 2 8 .76.0 76.0 PROBLEM 46. Find the area by both methods of the farm whose co-ordinates are (2, 4). (4. 8), (12, 12), (16, 4), (10; 0). Answer .96. PROBLEM 47. Find area of polygon whose co-ordinates are (0,0), (0,12), (10.9), (18, 14), (22, 13), (9, 0). 84. Traversing. When it is desired to find the bearing and distance of one point from another, a survey is run from the initial point to the final, making as many straight courses as desired. The lati- tudes and departures of these courses are calculated, and the closing course is a lost course whose bearing and length are desired and can be found by formulas 8 and 9. 85. Example. F i n d the bearing and length of AD in the following: Distance Latitude. Departure. 20 chains 17.14 10.30 24 chains 20.13 13.07 26 chains 21.03 15.28 CALCULATION OF AREAS. 77 38 65 The tangent of the bearing = ^^ = .66295. Therefore the bearing = N 33 32' E Length = v/(58.20) 2 + (38.6S) 2 = 69.86. 86. Approximate Traversing. Where the bearings of the different courses of a traverse do not differ by more than 6 the bearing can be found by an application of the 57.3 rule. Let ABCD, Fig. 39, be a traverse, and let the bearings be as in the preceding example. Take a reference line and let a, b, and c be the angles that AB, EC, and CD make with this line AG. IB "'' = WJ~3 HC^- ZO (-7 o 57.3 >- Let x = angle that AD makes with reference line AG. But AD = AB+BC + CD, nearly x (li + !+!.)= 67 i . 57.3 ali + bls + cl* li + l'2+la ali+blz + ck 57.3 If B = bearing of the reference line and we add B(\i + 1 2 H- 1 8 ) to each side, we get : /n v^ ._. I 1 + l 2 +l a That is, multiplying each bearing by its length, and dividing the sum of the results by the sum of the lengths of the courses gives the bearing required. Let a = 32, b == 33, c = 36, AB = 20, BC = 24, CD *** 26, find bearing of AD. 78 SURVEYOR'S HAND BOOK. PROBLEM 48. Find the approximate bearing of AD from the following notes : Course. Bearing. Distance. AB S28E BC S32E CD S30E DA.. 8 20 chains 18 chains 22 chains 87. Irregular Boundaries. It often happens that a creek or river is the boundary of a tract of land and the land fol- lows the meanders of the river. Thus the field notes of a cer- tain larm, Fig. 40, are as fol- lows : Beginning at a pecan tree marked X on Stone Creek, thence N. 36 9' E. to a stone in the prairie 29 chains ; thence E. 8 chains to a cottonwood marked H on the west bank of FI 4Q Mill Creek; thence with the meanders of Mill Creek to the junction of Stone Creek; thence up Stone Creek to the begin- ning. The following offsets were taken : CD DA Dist. Offset. Area. Dist. Offset. Area. 00 chains 00. chains .00 acres 00. chains 00. chains .00 acres 4 chains 2.0 chains .4 acres 5. chains 2.3 chains .575 acres 7 chains 2.5 chains .675 acres 9. chains 2.5 chains .960 acres 9 chains 2.2 chains .47 acres 14. chains 2.1 chains 1.15 acres 12 chains 1.0 chains .48 acres 17. chains 1.8 chains .32 acres 15 chains 1.4 chains .36 acres 19. chains 1.4 chains .07 acres 20 chains 1.8 chains .80 acres 20. chains .0 chains .05 acres 24 chains 2.0 chains .76 acres 21. chains 1.0 chains .09 acres 26 chains 1.7 chains .37 acres 22. chains .8 chains .09 acres 28 chains 0.0 chains .17 acres 23.3 chains .0 chains .052 acres 4.485 acres 3.852 acres CALCULATION OF AREAS. 79 Area of farm ABCD =34.3779 acres Area of offsets from C to D = 4.1250 acres Area of offsets from D to A = 3.852 acres Total area of farm with offsets = 42.7149 acres The land lines run up to the bank if the stream is navigable. PROBLEM 49. The following offsets were taken where R and L refer to right and left of the line being surveyed. Find the total area of farm if bounded by straight sides AB and BC and the meanders of the streams to which offsets were taken from points along CD and DA. Length along Offsets Length along Offsets CD DA 00 00 3 .6 R 3 .4 L 5 .8 R 5 .6 L 7 .7 R 7 .8 L 8 .3 R 10 .4 L 9 0.0 12 0.0 11 .3 L 14 .3 R 13 .5 L .16 .5 R 15 .4 L 18 .4 R 17 0.0 20 0.0 88. Discrepancies. It often happens that a survey is found where little care was exercised in the original survey when the grant or patent was taken up. If there are errors in the field notes of the original grant and there are no natural objects to which reference was made, it is very difficult, if not impossible, to re-establish the old lines. But if natural objects were re- ferred to in the original field notes, and these obj.ects can be found and identified, the re-establishment of the old survey is possible and, sometimes, comparatively easy. Corners are often defined or witnessed by natural objects, while the distances in the field notes do not agree with such witness objects. In such cases the "natural objects control and the corners must be located as called for by the natural object irrespective of the length of the lines in the notes. If a line begins at a well known tree and runs with a certain bearing to the middle of a certain 80 SURVEYOR'S HAND BOOK. stream, and thence with the meanders of the same, etc., the line must go to the center of the stream, although the distance of the line may fall short or exceed that called for in the recorded field notes. PROBLEM 50. The area was calculated to he 39.357 acres. Find the area of the farm if the line DA was a random line from which offsets were taken to a small creek on the left of DA, and completely outside the farm as given in problem 20. The follow- ing are the field notes for the offsets taken along DA : Dist. from D Offsets to left 16 8 28 12 40 6 48 12 68 4 90 Area = 3.55 acres. Tf this area is added to the area of prob- lem 26 we get for the whole area 42.907 acres, which is the area of the farm shown in the plot in Fig. 105. CHAPTER V. DIVISION OF LAND. 89. Division of Triangle. There are two cases which generally occur in practice. The first is to draw a line parallel to one side of a triangle to cut off a certain fraction of the whole area, or to divide the triangle into two parts whose areas shall have a certain ratio, while the second is to draw a line from one of the vertices of the triangle to divide it in a given ratio. First Case-' Given the triangle ABC, Fig. 41, the length of whose sides is known. The area of the triangle can be found from Formula 3. It is required to draw a line PQ parallel to BC so that \ B K C Fig. 41. Fig. 42. APQ : ABC :: m : n. Let AP=x, and A'Q=y. Then, APQ : ABC : : AP 2 ': AB 2 . .'. APQ : ABC : : x 2 : c 2 . UT In same way, y = o+l Example: Given a= 300, b= 240, c= 180. Find a line PQ that will cut off 4/9 of the triangle ABC. *=240- 4/9 = 240x2/3 = 160. y = 180X-2/3 = 120. 81 82 SURVEYOR'S HAND BOOK. Second Case: Given the triangle ABC, Fig. 42, to draw a line AK, so* that AK will cut off the triangle AKB equal to m/ n of the triangle ABC. The triangles ABK and ABC have the same altitude, and are therefore to each other as their bases. Hence, ABK : ABC : : m : n. But ABK : ABC : : BK : BC .'.BK : BC : : m :n. BK = BC X m/n EXAMPLE: Find BK in the foregoing example when B. IK is three-fifths of the triangle ABC. BK = 3/o X 300 = 1 So. PROBLEM 51. Given a = 340, 6 = 272, r = '_!'>!. Find the area of ABC and AP and AQ when PQ is parallel to BC and the triangle APQ is two-thirds of ABC. 90. Division Line Through Internal Point. It may be possible that it is desired that the dividing line shall pass through some point inside the triangle and divide the triangle in a cer- A Q D B Fig. 43. tain ratio. Let P be the internal point in the triangle ABC, Fig. 43, and let it be required to pass a line, HPQ, through P that will make the triangle AHQ have the ratio of m to n to the triangle ABC. The point P is known, and the perpendiculars PD and PE are known, or can be calculated. Let the area of the triangle ABC be represented by K, and PD = />, PE q, AQt=x, and AH = y. We have, Area APQ = U PD X AQ = U fix Area APH = U PE X AH = % qy. Area APQ + area ^P// = area AHQ = *& (px + qyl =m/n K (12) Also, we have, Area AHQ = % AQ X AH sin. A = % xy sin. A. Area ABC= AB X AC sin. A = % be sin. A. DIVISION OF LAND. 83 But Area AHQ = m/n area ABC -* \ x y s i- n - A = 2" be sin A . ' . xy = m/n be (13) Thus we have two equations in x and y, and these can be found and laid off on the sides AB and AC. EXAMPLE : Given AB = 420, AC = 400, BC = 260, PD = 100, PE = 60. Find x (AQ) and y (AH}, when triangle AHQ is four- tenths of ABC. By calculation we find area ABC 50,400. Then we have, 50 x + 30 3' = 4/10 50,400 = 20,160. A - y = 4/10 X 420 X 400 = 67,200. Solving for * and y, we get, .* = 219.57 or 183,63; y = 300.05 or 365.75. Fig. 44. PROBLEM 51. In the triangle, find x and y if the line HQ is to pass through P and bisect the triangle ABC. Answer, A- = 366.47, 3. = 229.21. 91. Division of Quadrilateral. Given a quadrilateral ABCD, Fig. 44. Required to find a line HQ through an inter- nal point P that will make ADHQ equal to m/n of ABCD. Let S = area of ADHQ and K = area ABCD. The point P is lo- cated by perpendiculars, PE and PF, on two sides of the quadri- lateral. Produce two opposite sides AB and CD to intersect in some point O. Let PF />, PE == q. The sides and angles of the quadrilateral ABCD are known, and from these the sides 84 SURVEYOR'S HAND BOOK. and area of OAD can be calculated. Adding area of OAD to ADHQ will give the required area of OHO, and adding the area of OAD to the area of ABCD will give the area of OBC. Find the ratio of OHQ to OBC. The problem is then re- duced to that of finding a line through P, dividing the triangle OBC into the ratio of m to n. The solution comes under the case of dividing a triangle by a line through an internal point. After the areas of AOD, OBC and OHQ are found we have, where, OA=a, OD = b. PE = q.PF = p, AQ =.r. DH = y, tt f>(a + x) + % q(b + y) = area OHQ, (a + x) (b + y) = m/n OB X OC. From these two equations, the values of x and y can be cal- culated. In the same way we can find the line passing through an internal point in a pentagonal field, dividing the field in a certain ratio. PROBLEM 52. If .45 = 300, C=192, CD=144, AD = 18Q, D = 240, P = 96 and PF = 60, find the values of .v ( AQ) and 3? ( = DH) when the area ADHQ is seven-twelfths of ABCD. 92. General Solution. There are many problems in land dividing that can be solved by special methods, and there are often short operations that can be applied at once. In the ma- jority of cases the line of division is not required to pass through an internal point. Where some certain point is given as the point of beginning of the division line, this point is gen- erally at a corner of the field or on one side at a given dis- tance from a corner. In such cases it is desired to find the bearing and length of the dividing line, and this problem is treated in a general way in the following articles. However, no attempt is made to solve problems of division in regard to the regular geometrical figures, as such solutions are raiher simple and offer no difficulties to the student. We have seen that the sum of the northings and the sum of the southings for a complete survey must each equal zero. Thus, we have two conditions to fulfill' and mathematically this gives us two equations. If we let A, h, /s, etc., represent the DIVISION OF LAND. lengths, xind Bi, B z , B 3 , etc., represent the bearings of the dif- ferent courses, we must have : /i Cos Bi + l* Cos 2 + /3 Cos B 3 etc. = ...... (14) li Sin 5i + I. Sin B 2 + / 3 Sin B 3 etc = ..... '. (15) Theoretical!}', if we know all the parts except two we can find these two unknown parts from equations 14 and 15. The lost or unknown parts can be: Case I. Bearing and length of one course. Case II. Length of two courses. Case III. Length of one course and bearing of another. Case IV. Bearing of two courses. 93. Case I. If the bearing and length of one course is unknown, the latitudes and departures of the known courses are first found. The algebraic sum of these must be the latitudes and departure of the unknown course with the signs changed. If we let L and D be the latitude and departure of the un- known course, respectively, then the length of the course Dep. 8.17 4.49 (7.20) 5.46 D And the tangent of the bearing y- EXAMPLE: Find the lost parts in the following: Course. Bearing. Dist. Lat. AB N62TE 9.24 4.32 EC S365'E 7.62 6.16 CD (S4529'W) (10.10) (7.08) DA N3128'W 10.46 8.92 L=4.32 + 8.92 6. 16=7.08 p=S. 17 + 4.495.46=7.20 Length CD = v/(7.08)- + (7.20) 2 =10.10 7 *>0 Tangent bearing^ ^==1. 1070 .-. Bearing=S4529'W PROBLEM 54. Find the lost parts in the following: Course. Bearing. AB N4622'E BC f CD S42W DA.. . N29W Distance. 38 chains 42 chains 54 chains 80 SURVEYOR'S HAND BOOK. 94. Case II. If. two lengths are unknown we first find the latitudes and departures of the known courses. Let x and y be the unknown lengths and M and A r be the bearings of these courses, respectively. Then from equations 14 and 15 we have: % Cos M+y Cos N + L=0 x Sin M +3 SinN + D=0 Multiply the first equation by Sin N and the second by Cos TV and we have : x Cos M Sin N + y Cos N Sin N + L Sin N=0 x Sin M Cos X+y Cos N Sin N + DCos N=*0 Subtracting and transposing, we get : x (Sin M Cos NCos M Sin N)=L Sin ND Cos N x Sin (MN)=L Sin ND Cos N LSinND CosN Sin (M-N) Example. Find the lost parts in the following survey : Course. Bearing. Dist. Lat. Dept. AB ... N472'E 31.30 21.33 22.90 BC ... S574'E 21.10 11.47 1771 CD ... S60W x x Cos 60 -x Sin 60 DA ... N40W y y Cos 40 y Sin 40 From formulas (14) and (15), we get, ' xCosQ() + y Cos 40+ 9.86=0 xSm 60 >> 5m 40 + 40. 6 1=0 Multiplying the first equation by Sin 40 and the second by Cos 40 we have : x Cos 60 Sin 40 + y 5*n 40 Cos 40+ 9.86 5m 40=0 x Sin 60 Cos 40 y Sin 40 Cos 40 + 40.61 Cos 40=0 Transposing and changing signs we have : x Cos 60 5m 40 y Sin 40 Cos 40= 9.86 5-m 40 x Sin 60 Cos 40 + y Sin 40 Cos 40=40.61 Cos 40 Adding : x (Sin 60 Cos + Cos 60 Sin 40)=40.61 Cos 40 + 9.86 Sin 40 x Sin 100=40.61 Cos 40 + 9.86 5m 40 40.61 Cos 40 + 9.86 Sin 40 5m 100 4061 X .76604 + 9.86 X .64279 .98481 =38.024 DIVISION OF LAND. 87 If we multiply the first equation by Sin 60 and the second by Cos 60 we get: x Cos 60 Sin 60 y Sin 60 Cos 40= 9.86 Sin 60 x Cos 60 Sin 60 + y Cos 60 Sin 40=40.61 Cos 60 Subtracting and changing the signs we have : y (Sin 60 Cos 40 + Cos 60 Sin 40)=40.61 Cos 60 9.86 Sin 60 y Sin 100=40.61 Cos 60 9.86 Sin 60 ' ^40.61 Cos 60 9.86 Sin 60 Sin 100 40.61 X.5--9. 86 X. 86603 PROBLEM irse. y 55. Find .98481 the lost parts. Bearing. N5E . S17E . . . . S56E (.. . N85W 9.58 Course. Bearing. Distance. \B N5E 8.68 />>r S17E x (7). DA .r = 4.087, .v = 8.937 95. Case III. The length of one course and the bear- ing of another lost. Find the unknown parts in the following example: Course. AB BC CD DE ... . EA . . . Bearing. X36E X S20E . S75W v N30W Distance 12 chains 8 chains 11 chains y chains 10 chains Latitude. 9.708 8 Cos X 10.337 y Cos 75 8.660 Departure. 7.054 8 Sin X 3.762 y sin 75 5.000 In all cases it is better to make a graphical solution in order to find the direction letters of the bearing. Lay off AB, Fig. 45, N. 36 E., equal to 12 chains to some scale, anc! EA S. 30 E. 10 chains. C will be somewhere on the circumference of a circle whose center is B and whose radius is 8 chains, while D will be somewhere on ED, N'T ere ED is ^.rawn with Pig. 88 SURVEYOR'S HAND BOOK. a bearing of N. 75 E. Through B draw BD' S. 20 E., and lay off CD" from D' equal and parallel to CD. Through C' draw CC parallel to ED and cutting the circle at C and C" and through C and C" draw CD and CD" parallel to BD'. There are two solutions, ABCDE being one and ADC"D"E being the other. From the figure we see that the bearing of BC is south- east, and that of BC" is southwest. Filling out the table for the southeast bearing and adding the latitude and departures, we get: 8 Cos X + y Cos 75 = + 8.031 8 Sin Xy Sin 75= 5.816 Multiplying the first equation by Sin 75 and the second by Cos 75, we have: 8 Cos X Sin 7.5 + y Cos 75 Sin 75 = 8.o31 Sin 75 8 Sin X Cos 75 y Cos 75 Sin 75 = 5.816 Cos 75 Adding, we have: 8 (Sin X Cos 75 + Cos X Sin 75) = 8.031 Sin 75 5.816 Cos 75 Sin bearings unknown. Let X and Y be the unknown bearings, a and b the lengths of these courses, and L and D be the latitude difference and the departure difference of these courses, respectively; then a Cos X + b Cos Y = L a SinX + b Sin Y=D D-aS*nX Then Cos Y* - ^ ^ yo Squaring and adding, we have : Let 27' = a 2 + L 2 + D 2 b z "But Cos 2 X= \-Siri 2 X Therefore 1 - 5,' X = Then a 2 L 2 P = a 2 (L 2 + D 2 ) Sin 2 X 2aDT Sin X From this quadratic in Sin X two values of Sin X will be found and there will be two solutions possible. 97. Example. Find the unknown parts in the follow- ing example : Course. Bearing. Distance. Latitude. Departure. AB..... N24E 26 chains 23.752 10.575 BC ..... SxE 28 chains 28 Cos X 28 Sin X CD ..... S38"E 24 chains 18.912 14.776 DE ..... SyW 36 chains 36 Cos Y 36 Sin Y EA ..... N44W 18 chains 12.948 12.504 To find the direction letters draw AB, Fig. 46, N. 24 E., and EA S. 44 E., move CD from its true position to some po- sition C'D' parallel and equal to itself where C' coincides with B. C has been moved 28 chains, because the length of BC is 28 chains. Now, D is 28 chains from D', but D is also 36 chains from E, hence with D' as a center and a radius of 28 chains describe an arc, and with E as a center and 36 chains as a radius describe an arc cutting the first arc at D. Draw DC N. 38 W. 24 chains. Draw BC and DE. Thus, we see that BC bears southeast and that DE bears southwest. Putting the di- 90 SURVEYOR'S HAND BOOK. rection letters in the table and filling out the latitude and de- parture columns, we have for our equations: 2S Cos X + 30 Cos F= 17.788 X 30 Sin 7 = 12.847 4.447 7 Cos X 3.2 12 + 7 Sin X Then Cos Y = - g - and S^n Y= -- - 81 = 30.092753 62.258 Cos X + 44.968 Sin X + 49 4.447 Cos X = 3.212 5m X .130232 Co* X = . 72228 Sm X .030035 1 Sm' X = .009285 + .04425409 Sin X + .521G884 SVn 8 X Sin 2 X + .02908 Sin X = .65655 SIM X= 82498 X = 5035'll" PROBLEM 50. Fnd the lost parts. Course. Rearing. Distance. AB ................................ X31E 14 chains BC ................................ N62E 20 chains CD ................................ x _'" chains DE ................................ S38W 23 chains EA ................................ y 24 chains 98. Dividing Land. It oft- en becomes necessary to di- vide farms among the differ- ent owners. A certain number of acres is sold from one part of a farm, and it becomes nec- essary to know the boundaries of the part cut off from the original survey. The partition is generally made in two ways, either by IG. a line starting at a certain point cutting off the required number of acres, or by a line that has a certain bearing. The following examples will serve to illustrate the methods. 99. Example. Find the bearing and length of a line AP that will cut off 40 acres from the farm ABCD, as given be- low in Fig. 47. DIVISION OF LAND. 01 Course. AB BC Bearing. N472'E S574'E S2842'W N4027'W Corrected Departure. 22.71 17.6-5 19.17 21.19 Distance. 31.0 chains 21.0 chains 40.0 chains 32.7 chains D. M. D. 22.71 63.07 61.5* 21.19 Lat. 21.13 11,12 -'35.09 24.88 Corrections. Dep. Lat. Dep. 22.68 .13 .03 17.63 .08 .02 19.21 .16 .04 21.22 .13 .03 CD DA Corrected Latitude. 21.26 11.34 34.93 25.01 46.51 + 46.01 40.43 40.31 .50 / 482.8146 .12 Lrea. 715.2138 2149.9415 529.9619 1012.7765 2865.1553 1012.7765 Double area = 1852.3788 sq. ch. Area= 92.61894 acres. Join the starting point A of division with the corner C near- est the final end of the required course. Find the area of the part thus cut off as follows : Course. Cor. Lat. Cor. Dep. D. M. D. Area. AB -21.26 2-2.71 22.71 482.8146 BC 11.34 17.65 63.07 715.2138 CA -9.92 40,36 40.36 -400.4712 Area = 31. 64352 acres. As the area of the triangle ABC is only 31.64 acres, the line AP that makes area ABCP equal to 40 acres must cut the side CD. hence P lies on side CD. Length CA = x/(9.92) 2 + (40.36) 2 = 41.561 40.36 Tan. bearing of CA = "9-92" ~ 4 - 06855 .'. Bearing of CA = 76 11 '28" Angle ACP = 4729'28" Now area ACP = 400 316.4352 = 83.5648 sq. chains. But area ACP = ^ CA, CP sin. ACP 2 area ACP 1671296 * ' CP ~ ^ 417561 x .73717 "" 5 ' 465 chaills 92 SURVEYOR'S HAND BOOK. . The latitude and departure of CP bear the same ratio to the corrected latitude and departure of CD that the length Cl 3 does to CD. .*. Lot. CP = 4.785 r>ct>. CP = 2.0-2' To find the length and bearing of PA, complete the table of ABCP. Course. Latitude. Departure. D. M. D. Area. AB 21.26 22.71 22.71 482.8140 BC 11.34 17.65 03.07 _ 715.2138 CP -4.785 - 2.02 78.10 374.4085 PA 5.135 37.74 37.74 103.7949 Double area = 800.6026 square chains. Area ABCP 40.03 acres. PROBLEM 57. In the 'example in Article 99, find the bearing and length of a line AP that will cut off an area ABP equal to nine acres. PROBLEM 58. Find the bearing and length of a line DK in the preceding problem that will make area ADK equal to six acres. 100. Example. Find the length of a line that bears N. 52 F. and cuts off 51 acres on the northwest side of the farm ABCD above. Draw a line CP, Fig. 47, through C that bears N. 52 E., and find the length CP and AP. Applying equations (14) and (15) we get: x cos. 4027' y cos. 52 = 9.92 (A) .r sin. 4027' + y sin. 52 =40,36 (B) Eliminating y 40.36 cos. 529.92 sin. 52 ~sin. 9227' Similarly, 40.36 cos. 4027' -f- 9.92 sin. 4027' ^ , fto - yr sin. 9227' =S7 ' 193 Find the area of A B C P, as follows: DIVISION OF LAND. Course. Latitude. AB 21.26 BC 11.34 CP 22.89 PA.. . +12.97 Departure. 22.71 17.65 29.30 1 1.06 D. M. D. Area. 22 71 482.S146 63.07 - 715.2138 51.42 - 1177.0038 11.06 143.4482 Area ABCP = 63.29724 acres. Fig. 47. The line CP cuts off 12.29724 acres in excess. Let the line MN, parallel to CP. cut off the required area. Hence the area MNCP is 122.9724 square chains. From C and P drop perpen- diculars on MN, cutting it at K and H. Angle MPH=2'Z?'; angle A'C/fT=19 4' Let r: = altitude of trapezoid MNCP = PH = CK Now, MNCP = HKCP NCK + MPH 94 SURVEYOR'S HAND BOOK. .'. 122.9724=37.193 z - ^ tan. 194'+ tan. 227' 2. 2i ^-(tan. 194' /aw. 227') 37.193 s= 422.9724 .15305s 2 37.1930-= 122.972* z 2 243. 01 2s= 803.48 .'. = 3.353 chains NC = 3.353 + cos. 194' =3.548 PM 3.353 -r- c u o (,_ uJ o * *- C a o Q. Q. O Z r a h ^ h h 1 \ \ [ l ' l' 1, Y I t / I \ \ \ \ I \ \ ': . f. ' [ ' ' / Q) u 16205 68.65 s 7O.4-8 CQ O rx tO cO I 75.93 N 7359 75 O6 17307 173.22 N 3 cu (X <0 8 g O g O o o g o 02 N 8 3 OJ O L * o fO < vD O 02 * "* ^ ^ ^ cO 01 cO g rx. K) o rO (D T^- tO ru fi 00 * <* c y 8 8 n 8 ^0 0] C 8 n o 8 o O o 1 V 0. H | ; cv cu 9 * OQ K cO ? 106 SURVEYOR'S HAXD BOOK. 107. Profiles. A profile is a drawing that shows the rise and fall of the ground on which the line was surveyed. The surveyed line may be straight, curved, or broken. To make a profile elevations of points on the line at short regular intervals must be found, as well as the points where there is a sudden change in the surface. Profiles are usually drawn to a horizontal scale of 1" = 400', and a vertical scale of 1" = 20'. Paper properly divided into squares by horizontal and vertical lines can be purchased by the roll or sheet. 108. Crosswire Adjustment. To make the intersection of the cross wires intersect in the axis of the telescope or the line of collimation, set up the instrument, level, and bring the cross wires into view by turning the telescope to clear sky. Focus the objective on some wall, and then have an assistant mark a spot on the wall at the intersection of the cross wires with a soft pencil; loosen the clips or loops that control the telescope, note that it still points to the spot on the wall, then turn the telescope in the wyes with the right 'hand until the bubble tube is on top. If the cross wires still intersect on the spot the instrument is in adjustment; if it intersects above or below, loosen the small capstan screws that control the wire ring and turn them so that the cross wire will be moved back one-half of the displacement. Bring it back to the spot by the leveling screws, and check by repeating the process. To correct t"he vertical wires turn the telescope so that the bubble is to the right or left of the instrument and in the same horizontal plane, and bring the cross wires on the spot by the leveling screws, then turn the telescope on its horizontal axis 180, and if there is any displacement correct one-half by the capstan screws that control the vertical wire and the other half by the leveling screws. Check by repeating the process. 109. Bubble-Tube Adjustment. To make the axis of the bubble tube parallel to the line of collimation, loosen the clips and level accurately, then take the telescope in the hand and turn it end for end in the wyes. If the bubble remains LEVELING. 107 in the center of the tube it is in adjustment, but if it does not, raise or lower one end of the bubble tube by means of the small capstan screws to correct one-half of the displacement. The rest is corrected by the leveling screws. Repeat until it checks. Level accurately, revolve the telescope slowly in the wyes and watch the bubble. If it has a tendency to move towards one of the ends, the bubble tube will have to be moved hori- zontally by the small horizontal capstan screws at one end of the tube. In some instruments the screws at one end of the bubble tube are to raise it vertically, while the screws at the other end move it horizontally. ^ 110. Adjustment of Wyes. To make the axis of the bub- ble tube and the line of collimation perpendicular to the ver- tical axis, level accurately over a pair of screws and then turn the telescope "**" 180. If there is any dis- placement of the bubble raise or lower the wyes by the capstan screws at the end of the horizontal bar Fig. 56. and correct one-half' of the displacement. Repeat the process until it checks. As a general check, repeat all the adjustments. 111. The Radius of the Bubble-Tube. Let TB, Fig. .'56, the tangent to the interior of the bubble tube cut the rod at B, distant d from the level, say, 100 ft. or over, turn the level- ing screws until the bubble travels a space s = n divisions to some point E. The tangent at E intersects . the rod at some point C ; take the difference in the readings of B and C, which gives us BC (r) in feet, measure the distance s the bubble trav- els, TE, in inches and reduce to feet. The two tangents TB and EC are perpendicular to the radii consequently. 108 SL'Rl'EYOR'S HAXD BOOK. Angle TOE = Angle BKC = 9 As the angle 9 in the sectors is very small we have TO : TE :: KB : BC or R : s :: d : r * = T (16) Where R=TO, the radius of the bubble tube. Now, KB is not exactly equal to TB, but when TB is loo ft., KB will be something like 99 ft. 11 ins., so they n im- practical ly equal. To find the angular value of one space on the bubble tube. note how many spaces n the bubble travels in the first opera- tion. In fbE we have 57 3 x s R By division H _ 573 X s n = nR After finding one angular division of the bubble tube, or better the angle subtended between two special marks, we can use the level for measuring distances across swamps, rivers. etc. Thus,, bring the end of the bubble to one of the end marks and locate the flag on the level, and have the rod read, shift the bubble until the end reaches the other mark and read the rod again, take the difference in the rod readings arid call this r, the angular division of the shift is O; then in the triangle BKC we have 9 = Ffg. 57. PROBLEM 64. An 18-in. Gurley level gave the following results : Distance (d) = 100 ft., rod reading (r;= 0.071 ft., shift of bubble = 0.7 in., corresponding to seven divisions on the bubble tube scale. Find radius of bubble tube and the angle sub- tended by one division of the scale. PROBLEM 65. If one angular division f tne bubble tube scale subtends an angle LEVELING. 109 of 21" at center of bubble tube circle, find the distance when dif- ference of rod readings was 1.28 ft, when the bubble was shifted five divisions. 112. Curvature of Earth. Let AB, Fig. 57, be. a horizon- tal line of sight, ACR the surface of the earth. Let distance AB = D, BC^c and radius of earth r. In right triangle OAB, OB*= OA* +~AB* Now the term c 1 is very small in comparison with D* and can be omitted without sensible error. 7~)2 /. c = , nearly If we wish the correction in feet while D is in miles, we get If D = l mile, c = 2/3 of 1 ft. = 8 ins. If D = 2 miles, c = 32 ins. If D = 3 miles, c = 72 ins. Effect of Refraction. Refraction has a tendency to make all bodies near the horizon appear higher than their natural positions. Thus if in Fig. 57 the level is at A, the line of sight will be the curved line AK, the radius of which is about seven times the radius of the earth. In formula for curvature, r be- comes lr. . BK _S_ L ~~ 2(7r) "~l4r If r is in feet while D is in miles, 6280<5280 2D^ _ BK ~ 14 X 3926 X 5280"" 21 If D = l mile, BD = 2/21 ft. = 1.14 ins. If D = "% mile, BK = 0.07 in. If D = 3.2o miles, BK = l ft., i. e., under ordinary condi- tions of atmosphere all points 3^4 miles from the observer ap- pear 1 ft. higher than their natural positions. 110 SURVEYOR'S HAND BOOK. 113. Vertical Curves. If two grades meet at a summit B, Fig. 08, it becomes necessary to round off this summit by uniting the two grades by a curve tangent to each. The simplest vertical curve that can be adopted for this purpose is a common 'parabola that touches the grade lines at A and C where the horizontal distance AK = KE, and AM = MC. Hence BM is a diameter of the parabola of which BA and BC are tangents. Then Fig. 58. PQ : BV = AT- : AK* . . PQ = BV x ~ PQ : DC = AT* : AE* ,' . PQ = DC ) Let g = grade of AB, rise per station, g' = grade of BC, fall per station, and ;i = number of stations in AB and BC. Now BK = ng. Draw #F parallel to horizontal line AE But FC = DC = DF + FC = But PQ - DC AT* -= Now, AE '= In, and if AT one station = PQ is the change of grade for the first station. Let this change = a '' a= 4M Change for 2nd station = Change for 3rd station = (21) LEVELING. Ill Example: Given g = 1.0, g' = 0.8, w = 3, elevation of B = 70.8', find the elevation of different points on the curve. Elevation of R, P, and A are 75.8, 74.8, 73.8 respectively, and the decrease in grade (or elevation) to bring road-bed to curve at points P, and R, and B are .15, 4 X .15, 9 X .15 or .15, .60, 1.36. Hence the elevations of points on the curve are 73.8 (74.8 .15), (75.8 .60) (76.8 1.35) or 73.8, 74.65, 75.20, 75.45. Original Change of Grade on Station. ,4 = 56 57 58 5 = 59 60 61 C = 62 Grade. 73.8 74.8 75.8 76.8 76.0 75.2 74.4 Grade. .00 .15 .60 ' 1.35 .60 .15 .00 Curve. 73.80 74.65 75.20 75.45 75.40 75.05 74.40 r>9. PROBLEM 66. If two grades at a summit are 1.4 and 1.0 and elevation of summit is 94.6, find elevation of points on curve if = 3. 114. Curve in Sag. If the curve occurs at a sag the same formulas will apply in finding the change for each station, but we must remember that the tangents are below the curve and that all elevations must be increased instead of diminished. Thus in Fig. 59, if grade of AB = .7, and of 5C=.5, eleva- tion of 5 = 54.8, and w = 3, we have a== 4n 4X3 Then we have the results as follows: 112 SURVEYOR'S HAND BOOK, Station. A = 22 23 24 5 = 25 26 27 Original Change of Grade on Grade. Grade. Curve. 56.9 .00 56.9 56.2 .10 56.3 55.5 54.8 55.3 ,55.8 56.3 .40 .90 .40 .10 .00 55.9 55.7 55.7 :,:, n 56.3 115. Vertical Circular Curves. If two tangents AB and BL meet at summit B, Fig. 60, a circular curve can be used to unite the two grades. Let O be the center of circular curve. Now g = grade of AB or the amount of rise of AB per station, or 100 ft. If the distance is measured in stations, g is the tangent of the angle the first line, AB, makes with the horizontal. In the rt-triangle AOB, angle AOB equals half of grade angle DEL. AB = OAtan AOB-'.T = Rtan % DEL, where AB = T, and OA=R. The angle DBL is very small, usually less than 4. .'. we can write: Tan AOB=tan V. T=R By geometry, AP'=PQ(2R + PQ) =2R x PQ + PQ* 2 Now PQ is so small in comparison with 2R x PQ that it can A p% AP* ( be omitted. . . AP 8 = 2R x PQ or PQ = = ^ LEVELING. 113 But PQ=a, and AP=y Now TAB=n, number of stations in AB The last formula is the same one we found for parabolic curve. The curve is really so flat that it can be regarded as a circle or parabola without error. CHAPTER VII. TOPOGRAPHIC SURVEY. 116. Topographic Survey. A compass or transit sur- vey will locate points with reference to each other in a hori- zontal plane. In other words such surveys show the geographic location of points with respect to each other, but they do not show how such points are situated in elevation with respect to each other. A topographic survey will give not only the rela- tive position of points with respect to their geographic posi- tions, but will also give their elevation vertically. A glance at the map will show the positions of the different objects in the geographic relations, but certain other data must be placed on thc^ maps to indicate the configuration of the terrain. Fig. 61. 117. Topographic Methods. There are four general methods of making a topographic survey: (1) By transit and level; (2) by stadia; (3) by plane table; (4) by hand level. The first method is costly, laborious, and slow. With the exer- cise of care, however, it is the most accurate method, but its cost and the labor required render its use almost prohibitive ex- cept for small tracts. The third method is coarse but rapid, and for large areas is by far the most practicable. It is suffi- ciently accurate for geologic purposes, and a survey by this method is a valuable adjunct to a more detailed survey by either of the other methods. It is useless to discuss here the methods of making a topographic survey by the transit and level, as the use of these instruments is fully discussed in the chapters de- 114 TOPOGRAPHIC SURVEY. 115 voted to their consideration. We shall in this chapter consider the stadia method only. 118. Stadia Formulas. The two stadia wires are placed in the reticule of the telescope of the transit above and below the horizontal cross-wire and parallel thereto. If these wires be represented by A and B in Fig. 61, and lines be drawn from A and B through the optical center O of the objective, these lines will cut the stadia rod at A' and B'. The lines A A' and BB' are called secondary axes. If we let t represent AB and r repre- sent A'B', then from the similar triangles, OAB and OA'B', we have, i:r: :f:d. But by the law of lenses, where F is the "principal focal distance." If parallel rays of light impinge on a lens they will 1 e brought to a focus at some point l\ which is called the "Principal Focus" of the lens, and the distance OV is called the principal focal distance. This dis- tance can be found for any given lens by holding the lens so that the central plane of the lens will be perpendicular to the sun's rays. The rays of sunlight will be brought to a focus, which can be found by moving a white sheet of paper parallel to the central plane of the lens. If the sheet of paper is beyond the focus from the lens the circular disc of light will be fringed with blue, while if between the focus and lens it will be fringed with red or yellow. When the sheet of paper is at the focus the rays of light will be concentrated into a very small circular disc of intense light. To find F for the object glass of the telescope, point the telescope to the clear sky and focus on the cross-wires, and then measure from reticule to center of object glass. From the first of the above equations, we have, 1 r J-Td and from the second, . 1 d-F 116 SURVEYOR'S HAND BOOK. Equating and reducing, we get, d = -j r + F. Now the "principal focal distance,"/ 7 , is fixed for any lens and (the distance between the stadia wires on the reticule) can be so adjusted that the ratio of F to i will be made an> value desired. From the last equation we have, But (d F) is the distance from the "principal focus" V tc the stadia rod, and as F ~=~ / is constant, we see that, in reality the distance from the 'principal focus to the stadia rod varies directly as the intercept r on the stadia rod. If we wish to obtain the distance D from the center of in- strument to the rod, we have, D = d + c = j r + F + c, ......... .... (23) where c is the horizontal distance from center of objective tc plumb-bob. In the majority of transits the distance F + c varies fron .80 to 1.25 and 1.00 can be assumed as a fair average withoui sensible error. 119. Wire Interval. To fix the stadia wires in a transit we must first find F, and then decide on some distance from the rod to the principal focus, say 400 ft. After this has been dom we focus on the rod, then measure the principal focal distance from the lens of the objective, which establishes the principa focus in the line of sight, and from this distance we measure the 400 ft. and set up the rod exactly at the end of this 400 ft Or we can measure from objective to the rod 400 ft. plus the principal focal distance. We now adjust the stadia wires so tha while one of them (the lower) reads 2.00 the upper will reac 6.00, the difference being 4.00. Then, F F 400 = 4 .-. = 100. TOPOGRAPHIC SURREY. 117 If the wires are fixed, find F, c, D and r for a given reading, then i = Fr-h (D F c). 120. Inclined Sights. If the line of sights OC is in- clined to the horizon at an angle v, as in Fig. 62, we shall for the purpose of mapping have to find the horizontal distance O and the vertical distance CE: The rod AB is always held ver- tically. The lines of sight as determined by the stadia wires are OA and OB. Draw A'B' perpendicular to OC, the line of sight as determined by the cross-wires, and let A'B'' = r'. The angle BCB' = v and the angles at A' and B' differ so slightly from a right angle that for all practical purposes we can assume them equal to 90. y Fig. 62. ' B'C = -BC cos. v. 2B'C = 2BC cos. v. or r' = r cos.' v. But OC=^r' + F + c F =^7 r c0s. v + (F + c) Then D=OE=OC cos. r cos. 2 v+ (F + c)cos. v. HCE=OC sin. v~ r sin. v cos. v sn. v 118 SURVEYOR'S HAND BOOK. = r sin - 2v + (F + c) sin. v .'.D = K cos.'' v+(F + c) cos. r H = Vfc K sin. 2v + (F + c) sin. v - Now the last terms in the formulas for D and H arc insig- nificant in comparison with the first term and unless refined ac- curacy is required these terms can be omitted. If F + = 540 X .9875 + (1 X .9937) =534.24. // = HX540X.222 + (1 X.1118)=60.0B. If the last terms are omitted we have D = 533.25 and // = :>'.' '"I. the errors being 1 in 538 and 1 in 537 respectively. For ordinary maps one-fiftieth of an inch is about as fine as we can indicate on the drawing paper. Thus, if we adopt a scale of 1 in. equals 10 ft., or one-tenth of an inch to the foot, the distance (D) above will be represented by a line 53.4 ins. But if we adopt a scale of 1 in. equal to 1 ft., which is the usual scale in railway topography, we would have, D = 5.34 ins. and the error com- mitted by the omission of the last term in the formula for dis- tance would be one hundredth part of an inch. 121. Stadia Rod. The essentials of a good stadia rod are that it should be clearly, accurately and distinctly graduated and that the graduations should be sufficiently clear to be read to the extreme limits of its longest range. There are many special rods on the market, each possessing special merits in the opinion of the designer, but the Philadelphia rod can be used while the marks are new and clear cut. Fig. 63 shows one form of stadia rod that is extensively used. It is 3.5 ins. wide, % in. thick in the body where the graduations are placed, and % in. thick on the edges. The rod is made of straight grained wood, is 12 ft. long over all and is hinged in the middle so that it can be folded for convenient transport. The raised flango (Hxl/16 in.) afford excellent and effective protection to the TOPOGRAPHIC Sl'Rl'EY. 119 graduation-; The foot marks are indicated in red figures. J/J-") to 0.75 in., while the tenths are indi- cated by black figures, 0.75 in. high by 0.5 in. width. The space is divided in alternate black and white strips one-hundredth of a foot in width. Each red fignre is opposite a black strip 2.5 ins. long, and the liiiure refers to the toh edge of the strip and indi- cates its distance from the bottom of the rod. In the same way each black figure is opposite a black strip of sanu width but only 1.25 in. in length, the black figures indicating the distance in length of a foot of the top of its strip from the top of the strip through the red figure below. The space between the black figures (the top through the black lines) is divided into ten equal spaces alternately painted black, while the white background forms another strip of the same width. If the wire reads between the Fed 3 and 4, between the black G and 7, and is at the top of the third black strip, the reading is 3. HO. It is well to remember that the top of the short black strips (about % in. long) indicate even hundredths, i. e., .02, .04, .06, etc., while the bot- tom of the black strip indicates the odd hundredths. These remarks apply (except as to lengths of the black strips) to the Philadelphia rod, which for dis- tances under fioO ft. forms an excellent stadia rod. 122. Field Work. When it is desired to make a topographic survey of a certain district by the stadia method, certain base lines or lines of refer- ence are adopted as a basis to tie into. If the dis- trict has been surveyed by triangulation, the trian- gulation stations form the points from which the survey proceeds. The transit is set up over one of these triangulation stations and sighted to an- other station of the triangulation survey. The azimuth of this line has been previously deter- mined and the transit can be adjusted by upper Fig. 63. 120 SURVEYOR'S HAND BOOK. motion so that the zeros of the verniers point north and south. When the transit has been set and adjusted so that the zeros will mark out the true meridian, the instrument man can send his rod man to certain strategic points in the terrain. The distance, azimuth and angle of elevation must be read and recorded. To obtain the distance the lowest stadia wire is brought preferably on some even foot-mark, as the 1 or 2, and the upper wire is then read 7.42. The difference is 5.42 and the distance by stadia 542 ft. To obtain the angle of elevation, the middle cross- wire must be brought on the mark on the rod that indicates the height of the center of the horizontal axis of the telescope. It is necessary for the transit man at every set up to take the height of the telescope above the surface under the plumb-bob. The azimuth is read from the south by west, north, east and on to south again. The primary triangulation stations are indicated by the sym- bol A, while the stadia stations are marked [T]with a number following to define it, as [V] 3, [][] 7, etc. If there has been no triangulation survey the topographic survey proceeds from the same local point to which the stadia stations are connected or TOPOGRAPHIC < SURVEY, 121 "tied in." Other points are variously described in the "object" column as "house," "tree," "cor. fence." If a reading is taken simply for a contour point it is marked C. P. Smith, Instrument. Henry, Recorder. Fox, Rod. Oct. 14, 1907. At [J J Ht. of Inst. = 5'.l Object. Azimuth. T CP 229 15' )istancc Ft. 99 206 332 370 387 281 294 181 81 163 401 754 Mean : 401 90 171 204 445 280 78 150 250 331 Mean 755 227 250 294 250 103 175 331 Elevation = 500'.0( >. Vert. Angle. Diff. of El. 152' 3'.2 048' 2'.9 1_24' 8'.1 1 4' G'.9 140' ll'.O 1_50' g'.o 2 8' 10'.9 056' 3'.0 8 38' 12'0 0_42' 2'.0 5_58' 41'.40 147' 23'.45 = 41'.52 541' _5_59' 41'.58 432' 13'"5 350' 17'.0 _2_ 50' 22'.7 _o22' 20'. 1 _858' 12'.0 _6 30' 10'.9 518' _ 30''5 = 23'.69 523' 149' 23'.93 _1_44' _f/.9 _0_l38' 2'.8 __!_ 8' 5'.8 342' 10'is _1_26' 4'.4 1O/V O '" 7 .""^DU O 4 ) El. 496'.8 502'. 9 508'. 1 5oe;.9 509''.i) 510'.9 503'.0 488;..0 219 12' 210 00' u 228 45' a 218 5' (C 254 10' 1C 283 30' it 290 - 8' ". 320 15' 64 38' .157 17' [T] 3 246 51' .52 At [T] 2 Ht. of Inst. = 4'.8 387 17' CP 229 12' 535'.0 528'.0 523'.9 5.1S'.8 515'.4 529'.5 524'.0 513'.9 .09 510'.8 520'.9 517'.9 509'. 1 513'.2 519'.3 532'.4 244 30' ..'.. .252 30' it 266 00' a 269 38' u 297 -18' u 18 - 5' it 316 15' At [T]' 3 Ht. HI 356 -10' 66 51' x CP . . 66 30' 1C 26 47' 00 35' (C .... 97 . 8' . . . . 133 20' M ,...162 40' U ..117 5' 122 SURVEYOR'S HAND BOOK. 123. Reduction Methods. The formula for finding the elevation of a point above the instrument, // = inclined distance X sin. v. When v is less than 6, we can find // readily by the application of the 57.3 rule. But to save time several labor-saving devices have been invented. Two of these make use of the principle of COX'S STADIA COMPUTER. Directions for Use. Pet the arrow, ir reading of the rod Opposite the vertical angle of the transit telescope find the Difference of Elevation, and opposite the same angle rked zero on the disc, oppos in the outer scale. the Distance t >le rind the II. EXAMPLE: al angle 12 ' 30'. reading of the Rod 537 feet. Set the o of the disc opposite W7, and opposite 12' 30- of the scale at the left rend 113) feet Difference of Klevation, and opposite 12" 30', of the Scale at the right read 512 feel Distance. Copyr.gh Detigncd by Wm. Cox. Fig. 65. the slide rule, Colby's -Slide Rule, which can be obtained from the leading dealers in drawing supplies and mathematical instru- ments, and Cox's "Stadia Computer," manufactured by W. & L. E. Gurley, Troy, N. Y. This "Stadia Computer," Fig. 65, is simply a circular slide rule about 15 ins. in effective length. It consists of a mounted card board, 6%x6}4 ins., upon which scale TOPOGRAPHIC SURVEY. is laid off the logarithm of numbers from 1 to 1,000 on the cir- cumference of a circle 5 ins. in diameter. Mounted on this scale is a circular disc concentric with the o-in. circle on the limb, on which is laid off the logarithm of the sines of angles from 3' up to 45. To find the difference of elevation for any distance and angle of elevation, turn the moving disc till the zero of the disc is oppo- site the required distance. Hold the disc in this position and opposite the given angle of the disc read the number on the limb. This is the required difference in height. The horizontal distance is read opposite the angle in the space marked "Hor. Distance." EXAMPLE: Given distance 480, angle of elevation = 5 10', find the difference of ele- vation. Turn the disc till the zero is oppo- site 480 on the limb and then opposite 5 10' on the disc read 43 ft. The whole computer can be carried in the coat pocket and its con- venient size makes it a very effective calcula- tor. No correction for horizontal distance is necessary for this angle of elevation. 124, Colby's Slide Rule. Colby's Slide Rule as shown in Fig. .66 consists of a base piece of trapezoidal cross section on which is laid off the logarithm of the numbers repre- senting the distance read by the stadia, and a sliding runner on which is laid off the angles of elevation to 18 30'. On the sliding run- ner is a mark labeled "same unit index," which can be seen on the right on the run- ner above the space between the numbers 3 and 4. To find the vertical distance between i! I Fig. the instrument and rod, set the mark under "same unit index" to agree with the distance read by the stadia, and then opposite the angle of elevation on the slide read the vertical distance on the log scale below. 124 SURVEYOR'S HAND BOOK. EXAMPLE: Given distance 600 and angle of elevation 3 10', to find the difference of elevation. Set index on slide opposite <>00 on log scale, and opposite 3 10' on the slide, read 33.1 on log scale, which is the difference of elevation. 125. Usual Approximations. The cosine of all angles less than 18 is greater than 0.95 and we may assume F+f 1 and (F + c) cos > .95. Now, if the horizontal distances are to be read to the nearest tenth of a foot, we can assume (F + c) cos v = 1. The following approximations may be made : (1) If the last term =1 and D = K. in the formula, D = K cos.\< + (F + c) cos.v, we have D = K cos. V + 1 or K = K cos.\< + 1 Cos-v Now if # = 200, _)( M ) = 200 C os.*r + 1 ' 7' = 404' If /C 700, v = 2W Thus, if the angle of elevation is 2 10' and the inclined distance 700, we can omit the last term and make the horizontal dis- tance equal to the inclined. The two approximations or as- sumptions balance each other. Check : D = 700'f J 2 2 W + cos2 W = 700 X .1)086 + .9993 = 699.02 + .9993 = 700.02 For an agle of elevation of 2 10' and a distance of less than 7oo (say, 500) we have D 500 X .9986 +.9993 = 500.3 For all distances less than 700 and a given angle of 2 10' the horizontal distance D will be greater than A', but the error is less than 1 foot. For all distances above 700 the horizontal distance (D) is less than K, but the error is less than one foot when K is les than 1,400'. The following table gives the values of v for certain distances when D == K: TOPOGRAPHIC SURVEY. 125 K Angles K Angle v 100 5 44' 700 2 10' 200 4 04' 800 2 02' 300 3 20' 900 155' 400 2' 52' 1,000 1 49' 500 2 34' 1,100 144' 600 2 20' 1,200 140' (2) When D is 1' less than K, \. e., for error of 1 ft. when the last term = 1'. we have, D = K 1; D = K cos.*v + 1 ; or AT 1 = K cos*v + 1, 2 .-. cos.' z v= 1 jp: Solving for the different values of K, we can fill out the fol- lowing table : K Angles/ K Angle v 100 8 08' 700 3 04' 200 5 44' 800 2 52' 300 4 41' 900 2 42' 400 4 03' 1,000 2 34' 500 3 38' 1,100 2 27' 600 3 20' 1,200 2 20' For any angle given in table and distance less than the cor- responding value of K, the error in D will be less than 1 ft. (3) When last tgrin-!* and there is a total error of 1 per cent in horizontal distance, we have D = .99K, D = K cos*v + 1 or .99AT = K cos.*v + I .' . cos*v = .99 J? A This formula gives the following : K Angles A' Angle v 100 8 08' 700 6 08' 200 7 02' 800 6 05' 300 6 38' 900 6 03' 400 6 25' 1,000 6 OK 500 6 17' 1,100 6 00' 600 6 12' 1,200 5 59' To find D from table, subtract 1 per cent. 126 SURVEYOR'S HAND BOOK. EXAMPLE: If A' = 800, we get D = 800 8 =792. (4) If the last term be omitted and there is an error of 1 per cent, i. e,, if there is a total error of 1 per cent minus 1 ft., or if > = .99A' + 1, we get, D = K cos.\> + \ But D = .MK + 1 .99 K + 1 = K cos.-v + 1 ..cos. s v = .99.'.v = 5 44' That is, if the angle of elevation be 5 44'. the horizontal dis- tance (D) will be less than the inclined (K) by 1 per cent of K less 1' or K . . Error = - 1 B L p Fig. D = K Error. 126. Topography by Hand- Level. The hand level can be used economically to obtain the data for a topographic map of any- small area. A base line should be adopted from which the survey proceeds, and lines perpendicular to this base line should be drawn at known intervals. Thus, if in Fig. 67, A BCD represents a section of area, adopt a base line PQ and at points P, 1, 2, 3, and Q locate lines normal to PQ. These lines should be marked out by stakes so they can be easily followed. In order to leave all elevations positive, assume some datum be- low the lowest point and refer the elevations of all points to this datum. Begin at some point as P and find the elevation of points along this line. The notes should be kept so the height of any point will appear as the numerator of a frac- tion, while its distance out from base, line will appear as the denominator. The height of the eye should first be deter- mined and rod readings should be taken at a sufficient number of points to determine the configuration of the landscape. The TOPOGRAPHIC SURVEY. 127 bench mark should be located somewhere below the point C, and from this the levelman makes his observation on the rod held on some point in line DC. The difference of the rod read- ing and height of eye will give the elevation of the point of rod above the observer. Thus, if h = height of eye of observer, r rod reading, then, h r elevation of rodman above observer. If h r is negative, the rodman is below the observer. The following notes were taken on a hand-level survey of a rectangular area : Line Left of PQ Base Line Right of PQ 33 28 DC. . 200 100 34 2.9 25 1 200 100 50 35 29 26 2 200 100 50 36 31 27 3 200 100 50 37 33 31 24 19 20 15 11 14 100 150 200 250 300 23 18 15 14 16 20 100 150 200 250 300 24 22 21 22 23 25 100 150 200 250 300 26 25 26 27 28 30 100 150 200~ 250 300 29 30 31 32 33 35 IT 100 150 200 250 300 200 100 50 BIBLIOGRAPHY. "A Manual of Topographic Methods/' by Henry Gannett. This work is published by the United States Geological Survey and its title indicates- its scope, as it deals only with the theory of topography, but gives also the illustrated methods as practiced by the engineers of the Survey, the most expert topographers in the world. "Topographic Surveying," by Herbert M. Wilson, 910 pages. Fully illustrated, having 18 engraved colored plates, 181 half- tone plates and many smaller figures. In addition to the ex- cellent illustrations of the best executed topography, the field instruments and other equipments for field parties are described and the methods explained. 128 SURVEYOR'S HAXD BOOK. "Elevation and Stadia Tables," by Arthur P. Davis. These tables are for use -in reducing inclined sights to the horizontal and for rinding the difference of elevation of observer and points. CHAPTER VIII. RAILROAD SURVEYING. 127. Railroad Surveying. By railroad surveying is meant the use of transit and level in selecting and locating the center lines of the track. The location of the straight sections of the track is a matter easily accomplished, but it becomes necessary to unite two straight sections of track that intersect at a defi- nite angle. That a train may pass gently from one straight line to another, making an angle with the first, the two must be connected with each other by an intermediate curve to which each straight line is tangent. On account of the ease of lo- cation circular curves are universally used to connect two straight sections of track whose directions are not the same. These straight portions may be joined by a curve of either great or small radius, depending upon the character of the ground. The magnitude of the curve is defined by the size of the angle that a 100-ft. chord sub- tends at the center of the circle. Thus, in a 4 curve the 100-ft. chord subtends an angle of 4 at the center of the circle. In a 3 curve, 3 at the center, etc. 128. Degree Formula. In Fig. 68 let AEB be a circular arc with O as center, and let ^5 100 ft. and angle AOB = 1). Then, if OC is perpendicular to AB, AC = CB = 50 ft. and AOC = BOC = D AC Now, Sin. AOC AO . * . Sin. W = - (24) 129. General Formula. In any curve AKB, Fig. 69, let AB chord c; AP' == - tangent T, AO^=- radius R, FK == mid. ordinate M, PK = External E t I = angle of intersection GPB = ^05. 129 130 SCRl'ID'OR'S BOOK. \\\ the right triangle AOP AP Tan. AOP = . Tan. .-. T = R Tan. In rt. triangle AFO, AF >in. AOF = T7S . ' . sn. $1 = 2R 2R sin. In rt. triangle AFK, Tan.FAK = ^~ F .-.rn:i/~g . ' . M = $c tan. \ I . In the triangle A KP, PK AP (27) sin. PAK PK If 7 is known and it is desired to pass a curve through some point on the bisector PO, we measure the distance PK E, and from formula (28) calculate 7'. Then find R from (25) and D from (24). 130. To Lay Out Curve. Let QA, Fig. 70, be a straight line or tangent from which a curve turns off at A. The point A where the curve begins is called the "Point of Curve" or P. C, while the point B, where we pass from the curve to the new tangent is called the ''Point of Tangent," or P. T. To lav out curve, set up the transit over the station at A, level up RAILROAD SURVEYING. 131 and back sight on a tack point in tangent line AQ. Revolve the telescope and turn off the angle of deflection, which is half the degree of curve. The rear chaimr.an holds end of the chain (the zero of chain or tape) on the tack point at A, and the head chainman swings his end of the chain around until the transit- man catches the flag pole in field of view. The flag pole is brought accurately to coincide with the line of sight and when the head chainman has the chain or tape straight, a peg is driven at the point /, which is a point on the curve. The chain- men now advance until the rear chainman reaches point 1, the transitman, in the meantime, having set the deflection angle again. The rear chainman holds the end of chain or tape on Q Fig. 70. point 1, while the head chainman is ranged in the line of sight A'2. When the chain is straight and the flag pole is in the line of sight, a peg is driven at this point 2. In the same way the other full station points on the curve are located. Example. Given D=2 3 30' and /=15 3 54' 50 50 Now > R= ,^r^ = ^r^= 2292/ - 1.V54' X 100 Length ot curve= rp on/ =b3o le-jt. - The total angle to deflect will be % / or 7' 57'. The angle of deflection is 1 1-V and there will be six full deflections of 1 15' each, making 7 30', and a partial deflection of '27', cor- responding to a chord of 36 ft. The usual curve is so flat that 132 SURVEYOR'S HAND BOOK. the angle of deflection for fractions of 100 ft. is proportional to the length of chord. Thus, if the deflection angle for 100 ft. is 1 15', then the deflection for 3(5 ft. should be M X 1 15' = -7', which checks the result found by subtraction. 131. Obstacles. It often happens that some object will interfere with our line of -sight and we cannot locate all the stations from the P. C. Suppose that there were a house or some other obstruction interfering with the line of sight from the P. C. to station 5. In this case the transit must be trans- ferred to station 4, where it is set up, leveled and a back sight taken on the rear flag at A, the P. C. Now, if G4 is a tangent to the curve at 4, the angle G4A = GA4. Hence, if we turn the telescope through an angle equal to the angle GA4, the amount deflected from the tangent AP, the line of sight will define the tangent 46". Set t he- transit at 4, level up, bring the verniers to zero, reverse the telescope and set on A. Plunge the telescope and set the vernier to read 6 15', and the line of sight will de- fine the line 45. This is more fully ex- plained and exemplified in Article 139. 132. Location by Offsets. Let ABC, Fig. 7 1 be a cir- cular curve when AB = BC = C, and where OA = OB R. Through B draw BE parallel to OA to cut the tangent AE at E. Draw OK perpendicular to AB. Then the triangles OAK and ABE are similar. .'.EB : AB = AK : AO. Now, EB is called the offset from the tangent to curve or simply tangent offset. Let EB=d .-.d : C=\C : R C 2 d=xi .....(29) IMB = C7=100, 5000 RAILROAD SURVEYING. 133 Let CF be drawn parallel to OB, to cut chord AB produced at F, and let BG be the tangent at B, cutting CF at G. Then triangle BCG = BGF. But BCG = ABll. .'-CG = BE, But CF 2 X CG = 2X5E = 2 d. C 2 . . chord offset CF -^r- If C = 100, 10,000 chord offset = ^ -- The formula for the chord offset may be written ,-.*-l.7> ............. (30) Thus, for a 1 curve the chord offset is 1.75, and that for any other curve can be found by multiplying 1.75 by the degree of the curve*. 133. Middle Ordinate. In Fig. 69 we have by Geometry, KF (2RKF)=AFxFB. .-.M (27?-M)= -Cx~C *J> Now M 2 is small in comparison with R, and in all practical cases can be omitted. & 134. Approximate Formulas. We have established the formula, 1 50 sin. ~9~ )= ~^' Now if D is no larger than 8 we can substitute the circular measure of the angle for its sine, that is J_ _* ' D sin. 2 D 2 ; 57 2965 D _50 ' ''2x57.2965""^ 5729.65 . ' . D = 5 134 SURVEYORS HAND BOOK. This is usually written, n _ 573" '.K-2ff-! , (32) Now if D=l, R=5730 ft. We have the general formula, T = R tan. = - tan. U. C = 2R sin. \l = '2 -- sin Let / remain fixed and 7\ and d be the tangent and chord for 1 -degree curve. Then, Ti = 5730 tan. \l Ci = 2 x 5730 sin. %I _" Ti .-. r D Again, we have, M =\C tan. %I jj- tan. $1 sin. 5730 = 7" tan. -\I = -- tan. U tan. \l. For a 1 curve these become, .l/i 57 30 tan. \l sin. $L E l = 5730 tan. \1 tan. \l. Then for all curves for- a fixed I } we have, D X T = 7\ = a constant, D X C CY a constant, D X M=M, = a constant, D X E=Ei = n constant. 135. Reduction Tables. The value of the tangent 7\, the long chord C\, the mid-ordinate Al\, and the external J5i have been calculated for a 1-degree curve, corresponding .to RAILROAD SURVEYING. 135 Value of / from to 117, for intervals of two minutes. To obtain the values of T, C , M, or E, it is only necessary to look for these for a 1-degree curve for the proper I, and then to di- vide by the value of D. EXAMPLE : Find T, C, M, and E, for a 4 curve when / = 21. For a 1-degree curve, we get 7! = 1062.0. CV= 2088.5, Mi =95.95, 1 = 97.58. . ' . T = M. X 1062 = 265.50, C=V4 X 2088.5 = 522.125, M= 1 A X 95.95 = 23.988, 14 X 97.58 = 24.395. 136. Metric Curves. Tn Mexico and the South American countries a chain or tape of 20 meters is used instead of the JOO-ft. tape that is used in the United States. The degree of the curve is the angle at the center of the circle subtended by a chord of 20 meters. Thus, in Fig. 72 if ,45 = 20 meters, and .405 = 7}, the number of degrees in the angle D gives tl 2 degree of curve. Sin.AOK = TTA Fig. 72. Sin 10 // D one degree, we have, But s^ne 30' = .10 Sin 30' = -F>- 1 2 x 57.3 = 1146 meters. __!_ _ 10 *' 114. (3 ~ R Now, the radius of a 1-degree curve for the foot system (pre- vailing in the United States) is 5730 ft = 5 X 1146. In the same way all the functions of a 1-degree metric curve are one-fifth of the corresponding functions of a 1-degree curve of the foot system. Thus, if 7 = 12 7 = 602.2', = 31.56', C 1197.9', for a 1-degree foot curve. Then 7 = 120.4 meters, E = 6.3 meters, C = 239.6 meters, which were obtained by dividing the former values of 7, E and C for the foot curve by 5. 13G SURVEYOR'S HAND BOOK. Again, if we have 7 = 14 30', and wish to find T, E, and C for a 3 metric curve, we can find T, E, and C from the usual tables for the foot curve and divide the results by live times the degree of curvature for the metric system. Thus, for 7 = 14 30' we have for a 1-degree curve 7 = 728.87, E = 46.18, C=144(>.2. Then for a metric curve of 3 we divide these values of T, E and L. C. by 3 X 5 = 15, as follows : T = Jg (728.87) = 48.59 meters, E = ^s (46.18) = 3. 08 meters, C = 1*5 (1446.2) = 96.41 meters. 137. Preliminary Survey. The first instrumental survey on a projected railway line is called the preliminary survey and consists in running a traverse line, staking the line out by means of pegs or stakes, which are driven at the hundred-foot marks, or "stations," as they are called, or at fractional parts thereof. When the survey is finished these stakes mark out a polygonal traverse or survey. There may be two or more preliminary surveys between the same termini, and a comparison of these as to cost of construction, revenue to be derived from probable traffic, and operating expenses will decide the most advantageous route. Fig. 73 is a double page illustration of the form of notes used in the field in preliminary survey. 138. Location Survey. When one of the preliminary surveys or routes has been adopted, the center line of the pro- posed track is then located. The different tangents must be connected by curves and the whole line must be surveyed by transit, running in the curves and driving new stakes or chang- ing the position of the old ones. As the curve is shorter than the sum of the two tangents, the first P. T. will be less in dis- tance from the beginning, that is, all stakes after the first P. C. will be moved forward. Those on the tangents (from P. C. to P. I. and from P. I. to P. T.) will be moved over to the curve and all those on the part of tangent from the P. T. to the next P. C. ahead will be moved forward so that the number of each stake will give its distance from the beginning as measured along the proposed center of track. Thus, if the angle of intersection 7=:]6 00' and we unite the two tangents by a 4 curve, the RAILROAD SURVEYING. 137 value of Ti = 805.2, and for a 4 curve 7 = 201.3. Now, if the distance from the beginning to P. I. was 3346 ft., i. e.. the P. I. was at station 33 + 46, the P. C. will be located at (3346 201.3) LU 3144.7, that is, at station 31 + 44.7. The P. T. will be located an equal distance from the P. I. or at 3547.3, according to the preliminary survey. Now, length of curve = 16 -f- 4 = 400 ft. 138 SCRl'EYOR'S HAND BOOK. Then, according to the location survey, the P. T. will be located at 3144.7 -f- 400 = 3544.7, or 2.6 ft. nearer the beginning by the curve route than by the P. I. route. Station 30 was 52.7 ft. from this P. T. according to the preliminary survey, but by the location chaining, the point, instead of being 3,600 ft. from the beginning, will be at 3597.4 ft., and hence the station stake 36 will be taken up and moved forward 2.6 feejt, so that it will really be 3600 ft. from the beginning. PROBLEM 67. The P. I. in the preliminary was 2614 ft. and 7 = 24. Find the positions of the P. C. and P. T. for a 3 curve. PROBLEM 68. The second P. I. in the previous problem was 0(554 ft. Find the position of P. C. in the location survey for a 3 curve if 7 = 18. 139. Field Book. It is important that the note book or field book should be neat and accurate and should show all the necessary data for the location of a curve and how it is connected to the tangent points, where it begins and where it ends. The supreme test of note taking and note keeping is that ANY engineer can understand fully and accurately exactly what the data mean. Fig. 74 is an illustration of both pages (left and right) of a location survey notebook where a curve has been run in to connect two intersecting tangents. The angle of in- tersection of the tangents 7 12 54', and the tangents are united by a 2 30'. The length of tangent for a 1-degree curve for 7 = 12 54' is 647.8 and for a 2 30' curve the length of tan- gent = 647.8 -=-2.5 = 259.1. This length of tangent can be cal- culated from the following formula : 50 tan $ 1 50 tan 6 27' T=R tan \ i^-^^^-^^^- = 259 ' 1 ' The curve is to begin at station 64 + 13.3 and the transit is set up at this point (the P. C.), the verniers brought to zero, and a back sight taken on the last hub. The next station in advance of the P. C. to locate is 65, which is (6500 6413.3) 86.7 ft. from the P. C. For a full 100 ft. the deflection is half the degree of curve or 1 15', and for 86.7 it is 86.7 -r- 100 of RAILROAD SURVEYING. 139 f 1 1 1 in I l\t\ 111 \ ill 3 (0 ri T3 \ 1 |L i < m i. \ \\ \ J \v v \JP r \ O P- . \ 4 -Q \ 1 \\ r* J 7 a i 1 h V O OJ ^ ^ MM id 1 J ^1 O 4- cu \ "O^ 0! O ytvi "O Q (U cu "0 cu S) C\j -o CXi Q S 1 ^ ^ *cO V' CJ h Jl; ' J i a- 0: K a.- * a- ' H a.- \ i\V \ \ \ I \ \ I I I I I \ I \ V i \ \ 1 / / f / 1 1 / ! i / / i / / / / 1 / c O) o fl u (M OJ u "0 10 (V UJ '0 M A ~ y "^ kl UJ "CO 'CO "(U v (M (U 3 O "cn OJ ni i o tO iO UJ "0 - , .0 CO CM X I _u en O U U u 02 o s O q 8 c V b 02 - b vD V> ^ >0 w '- c Q iO N <5 R it OJ K s R 4- CO N ^D ^ N vO 140 SURVEYOR'S HAND BOOK. 1 15'. Hence the deflection =.867X75' = ! 05', and this should be recorded opposite the station 65 that it locates. For the full stations 66, 67, 68 and 69 the record in the "index" column should be 2 20',' 3 35', 4 50' and 6 05', respectively, which are obtained by adding 1 15' to the record of the last full station in the index column. Now, the length of the curve = 100 X 12 54' -7-2 30' = 516 ft. Adding this 516 to the 64 + 13.3 (the station number of the P. C), we get 69 + 29.3, which is the station number for the P. T. The deflection angle for the 29.3 = . 293-X 75' = 22', which is added to the index of the last full station, 6 05', gives an index of 6 27'. Now, the read- ing of 6 27' on the P. T. should be half of I, that is, if we double the index for the P. T., we should get the value of /. or 2(6 27') = 12 54', which affords an easy and effective check. It may happen that in running the curve the transit has to be moved from the P. C. to some station as 67, the index of which is 3 35'. Now, after setting up the transit over 67, we can back sight on ANY station, provided we set the vernier to read the index of the station sighted at. Thus, if we backsight on 65, with telescope reversed, the vernier must read 1 05' (on the correct side of the vernier). Then to locate station 68, all we have to do is to revolve the telescope and set the vernier at 4 50', the index of the station sighted at, and have the stake driven at this point. However, if we. should set up the instru- ment at 67 and backsight on 64 + 13.3 (P. C.) with telescope reversed, we must set the vernier at 00', the index of the P. C, and then to locate 68 we again make the vernier read 4 50', the index of the station sighted at. Thus, wherever we set up the transit on the curve, the back sight on any station must read the index opposite the station sighted at, and to lo- cate any other station ahead, revolve the telescope and set the vernier to read the index for that station. 140. Transit Party. The transit party in the field should consist of transitman, rear chainman, head chainman, rear flagman, stakeman. and axmen. The transitman has charge of the party and should provide himself with the transit. tripod, plumb-bob, reading glass, notebook and pencil. RAILROAD SURVEYING. 141 The rear chainman should have charge of chain or tape and be responsible for it. The head chainman should provide and take care of the flag or range pole. The stakeman provides bag of stakes, keel for marking same, ax or hatchet for driving stakes, and tacks for hub-points. The rear ilagman has the silent duty of remaining ever in readiness to be called upon to give a sight at a signal or call from the transit man, and the axmen should have good 4.5-lb. axes to clear the way. It is poor economy to be restricted in the number of men that are to do the clearing. 141. Stakes. The stakeman should provide a sufficient number of stakes for each day's supply at least. The stakes . vary in size (Fig. 75), but sawed ~~M stakes are 2x1 ins. by 18 ins. in length, while "hubs" should be 2x2 ins. by 18 ins. in length. The flat shaped stake is used to facilitate V marking, as the broad surface offers sufficient space for the number of station and the letter indicating the -, line to be written or printed on the 1 i stake. The figures or letters are V' printed with keel (red chalk), which can be secured from dealers in draw- l ^' ' ing supplies or from local hardware dealers. 142. Hubs. At every angle point or transit station a "hub" is located. This consists of a stake (Fig. 76), 2x2 ins., driven flush w r ith the surface of the ground. A tack is driven in the top of the hub, where the range pole or flag rested in the line of sight. After the tack is driven partly in the hub it should be checked by the transitman so that any error in location can be corrected before it is driven too far to be with- drawn. After it has been checked, it is driven flush with the surface of the hub. About 1 ft. to the left of the hub a "guard" stake is driven with the number of the station marked on it. This guard stake is inclined towards the hub and is left project- 142 SURVEYOR'S HAND BOOK. ing from the ground several inches,- as shown in Fig. 76. The number of the station of the hub should be marked on the guard with a good system of letters. These figures should be printed with red keel, and in no case should they be written with a rough figure or letter. With care and a little practice the stakes- man can soon learn how to mark these in a standard and sys- tematic way. Fig. 76. 143. Hand-Level. This instrument. Fig. 77, is about 6 ins. long and has a level tube or vial on top. Across one half of the clear glass at object end a horizontal line "is drawn. The image of the bubble tube can be seen on half of the glass at object end of tube, as it is reflected by a prism. The ends of the tube are closed with plane glass and a semi-cir.cular convex lens at end of eye-piece or eye-tube magnifies level bubble and Fig. 77. the cross wire. The cross wire is fastened to a framework under the level tube and adjusted to its place by the screw shown on end of level case. To use the level, hold it with the hands so that the eye-end is next the eye, then move it until it is approximately hori- zontal. The image of the bubble can then be seen on half of the object-end glass. When the bubble appears on the horizontal RAILROAD SURVEYING. 143 mark or wire, the line of sight is horizontal. To use the hand- level it is necessary to know the height of your eye. Sight through the hand-level and bring the bubble on the horizontal wire and note the point on the ground indicated by the line of sight. Unless unusual refinement is necessary in taking to- pography,, the hand-level will subserve all necessary requirements and it is an economical, efficient and expeditious instrument for this purpose. In railroad surveying the line of survey affords a base line from which all transverse measurements can be made. "The topographer determines the height of his eye when stand- ing in his usual attitude and then taking a position on the line of sur- vey. A BCD (Fig. -78), he selects a direction at right angles to the line of survey. Bringing the level to its horizontal position and noting where the line of sight strikes the earth at point 1, he paces the distance from line to point (48 ft., say). At point 1 he notes that the next line of sight strikes ground at point 3, etc. This process is continued until the terri- tory 200 ft. on each side of the line covered. If the height of the eye Fig. 78. the position of is 5.2 ft., then each point of inter- section of horizontal line of sight with ground is 5.2 ft. higher than the observer. The elevation of the observer's position is 'known, or can be ascertained from the levelman's notes, and hence the elevation of each point located can be determined by adding or subtracting height of eye. On the lower side of the line it is well to have a rodman provided with a rod, graduated to half-feet, at least 12 ft. in length. If it is desired to have all contour points, the uniform height of eye above or below the adjacent points in any one normal line, the topographer can have his rodman walk away from the base line in a normal direction till the rod reads double 144 SURVEYOR'S HAND BOOK. the eye height. If other points are located, the rod is read by the hand level and the reading recorded. The topographer ad- vances to the rodman's i position and sends him on further out to locate other points. If no rodman is used the topographer can pace the distance in the normal direction to some point which he guesses is about the eye-height below his position. If the line of sight from his point strikes below the surface in the normal line, he must go toward the base line till the line of sight strikes the point on the base line. The distance from the base line is found by subtracting or adding the distance between the final location and the assumed point to the distance from base line to assumed point. With a little practice a topographer will soon be able to select a point within a foot or so of the correct point. 144. Slope Stakes in Excavation. In excavation in earth- work the cross section is defined by the roadbed AB, Fig. 70, and the side slopes AE and BC. The amount of slope of BC is determined by the ratio of BG to CG, and is designated by s. '.$ = BG H- CG = tan.BCG. .'. BG = s.CG = sfa, where h, = height of point C above roadbed AB =CG, and h = EF height of E above roadbed AB. Fig. 79. I== AB width of roadbed. Now .'.Distance out of stake point C half width of roadbed plus slope times height of point above roadbed. The center cut DK = c is already known before any attempt is made to set the slope stakes. The level is set up in some RAILROAD SURVEYING. 145 convenient position and a rod reading taken on the station at D. Let 1234 represent the horizontal line of sight, in D\ = rod readjng on station D, .: H = DK + Dl = c + m. Now, to locate the slope stake at C at a horizontal distance d from D, try some point as P and find the rod reading P2 /'. Then 2Q P2 = H r. Now, H r is the "surface height" of the trial point above roadbed AB. Calculated distance out d c =-b + s (H r}. But Measured distance out d m = b + s QT. Hence, we see that when the trial point is too near the center the measured distance out is less than the calculated dis- tance out. Try some point S. d m = measured distance out = b + s UX. d calculated distance out = b + s US. ' The measured distance out is greater than the calculated distance out when the trial point is too far out, and vice versa. Hence, if d m > d c , come in ; d m < d c , go further out. Rule: If the measured distance out to the trial point is greater than the calculated distance out, come in, and vice versa. Slope-Stakes in Level Sections. If DT ', the surface of the ground, is horizontal, then DK = TQ. In this case the point T will be the stake point. Its distance out, KQ = KB+ BQ = b + s TQ = b + s DK = b +sc. Thus, in level sections the distance out is found by multi- plying the center cut by the slope and adding the half width of roadbed. EXAMPLE: If center cut 14.6 and slope 3:2 and width of roadbed = 18 feet, then 3 Distance out = 9 + - 14.6 = 30.9. 140 SURl'RyQR'S HAND BOOK. Field Methods. In the field, if the ground is inclined, the usual practice is first to find the distance out on the assump- tion that the ground is level. This simply serves as a guide and useful help. If the ground slopes, the distance out on the upper side of the center of roadway is always greater than the distance out in a level section, if the ground slopes uniformly. While on the lower side the distance out of the slope stake is less than the level d. o. 3 EXAMPLE. Given.?- =9-- 26=18;c=14. The level was set up r,nd the rod reading on the center was 7.2. For a level sec- 3 tion, the distance out = 9 + 77 X 14 = 30. On the upper side the trial point was selected at 32 ft. from the center where the rodreading was 5.4 ft. H = U +7.2 = 21.2 //_,- = 21.2 5.4 15.8 3 Calculated d. o.=9 + -^ x 15.8=32.7 Now, the calculated d. o. is greater than the true, hence the trial point is too near center. Try a point 34 ft. out when rod reads 5.0. H r = 2l.2 5.0=16.2 g Calculated d. o.=8 + -^x 16.2=32.3 ft. The calculated d. o. is less than true d. o. The second trial point is too far out. Try point 33, where rod reads 5.2. H r = 21.2 5.2 = 16.0 3 Calculated d. o.=9 + -^ x 16=33.0 This location is correct. On the lower side the distance out must be less than 30, the d. o. for a level section. Try a point 29 ft. out, where rod reads r = SA. , '' H r = 21.2 8.4 = 12.8 RAILROAD SURVEYING. 147 Calculated d. o. = I) + V >' 12.8 = 28.2 Hence, the trial point was too far out. Try a point at 28.4, where rod reading = 8.3. Hr = 21. 2 8.3 12.9 3 Calculated d. o. =* 9 + j X 12.9 = 28.35 The' location is sufficiently accurate. PROBLEM 69. Center cut = 16.6, 2b = 18', s = 3 -^ 2. Rod reading on center =6.2. A trial point was taken at 35, where a rod reading was 5.0. Is the trial point too far out or in ? Answer : If the trial point was at 39.0 and the rod reading was 4.9, is it too far out or in ? Answer : If the point was 36 and the rod reading 4.8, how is it? Answer : PROBLEM 70. In the following table:' c = center cut, ' in = rod reading on center, d m == true distance out of trial point, i" = rod reading on trial point, d c = calculated distance out. Find the results as to accuracy of location point* r d c Result 5.2 4.8 5.0 7.6 8.0 7.8 6.4 4.0 4.2 4.1 3.6 ^ 3.8 145. Slope Stakes in Embankment. In embankments the road bed AB is usually for single track roads 14 ft. wide and the slope varies from 1 :1 to 2:1. However, on levees the slope is as flat as 5:1, Number c m s b d m A 12.8 6.6- 3/2 9 30.2 B 12.8 6.6 3/2 9 32.0 C ...12.8 6.6 3/2 9 30.6 n . . . 12.8 6.6 3/2 9 26.0 E 12.8 6.6 3/2 9 27.0 17 . . . 12.8 6.6 3/2 9 26.4 G ..... . . . 8.6 5.4 1/1 9 18.0 II 8.6 5.4 1/1 9 19.5 I ... 8.6 5.4 1/1 9 17.0 . 8(1 5.4 1/1 9 18.9 K 11.4 4.8 2/1 9 35.6 L . ..11.4 4.8 2/1 9 33.8 148 SURVEYOR'S HAND BOOK. The roadbed is AB, Fig. 80, and the side slopes BC and AE. Slope stakes must be set at the foot of the slopes at E and C. The center fill KD c is known and it is required to locate these slope stakes. The level is set up. 1, 2, 4, 3 is the hori- zontal line of sight, the rod reading (m) on the center is Dl. The height of instrument (H. /.) above roadbed AB is KI. Now, BG s X CG and AB = '2b, DI=m H.I. = KI = m c Distance out = KG = KB + BG = b + s. CG b + sh The rod reading (r) on C is 4C, But 4C = 4C + GC .'.,- = H. I. + h or h r H I' == r m + c Distance out = b + s (r m + r) Suppose we try a point P that is too close to the center. Rod reading ( ; -)=P2 = r Fig. 80. d c = calculated distance out = b + s(r m + c) = b+s.PQ. But d m = true distance out = KQ = KB + BQ = b + sQT. Thus d c is greater than d m . 'Hence the calculated distance out is too great and the trial point is too near the center. Try a point 5* where rod reading = 35". d c = b + s(r m. + c) = b+s.US But d m = KU = b + s.UX , . ' . d c is less than d m . RAILROAD SURVEYING. 149 Hence the calculated distance out is less tlian the true dis- tance out, or the trial point is too far out. Thus we see that the same rule applies to fills that applies to cuts. In deep fills the line of sight 1243 may he below A 13 and the height of instrument (H. /.) will be negative. In this case H. I. c m Distance out = b +s (c m + r~) Example. Given 2b=U: j = 3/2; center fill =14.8 ft.; rod reading on center = 5.4. If the ground is level the distance otit=7 + 3/2 (14.8) =29.2 ft. On the lower side the distance out will be greater than this, while it will be less on upper side. Try a point 31 out where the rod reading = 7. 20. d c = 7 + |(14.8 5.4 + 7.2) = 31.9. :. Point was too far out. Try a point 32 ft. out where r = 7.3. d = 7 + 3/2 ( 14.8 5.4 + 7.3) = 32.05. The location is sufficiently accurate for practical or ordinary requirements. PROBLEM 71. In the following table determine the results of the trials, i. e., whether trial point is too far, too near, or cor- rect : d m r d e Results 36 7.4 27.2 7 5 38 7.6 22.0 5.6 20.5 5.5 20.7 5. 23.0 3.4 146. Berms. It is often necessary to excavate the earth near the foot of the slope of the embankment to secure enough dirt to make the embankment. When such is the case it is necessary to leave a strip of unbroken original surface at least 4 ft. in width between the borrow pit and the foot of slope to afford a break for earth that washes down or off the slope. Thus in Fig. 80 FE is the berm, a strip of undisturbed natural earth, between the embankment CBAE and the borrow pit NF. In cuts it is often of the utmost importance to have an un- disturbed natural surface on each side of the cut. To do this it Number / c ...178 m 52 s 3/2 b 7 B .... C .... D .... E ....17.8 ....17.8 ....14.4 144 5.2 5:2 4.8 4.8 3/2 3/2 1/1 1/1 7 7 7 p 14.4 4.8 1/1 7 C .... .... 9.2 , 4.6 2/1 7 150 SURVEYOR'S HAXD BOOK. is necessary to prevent the deposition of any excavated mate- rial within 6 ft. of the edge of the side slope. If the loose earth is piled near the edge of the slope, heavy rains will wash it down the slope into the cut. Bibliography. "Railroad Location Surveys and Esti- mates," by F. Lavis. Published by the Myron C. Clark Publish- ing Co. This book is a complete epitome of actual field engineer- ing and includes a history of the preliminary survey from the organization of the party to the completion of the line. No better description can be applied to this work than to say that its theme is to tell and show "how to do things." In many re- spects it covers a territory heretofore not traversed, and is re- plete with valuable suggestions gained by experience as a field engineer. "Field Manual for Railroad Engineers.'' By James C. Nagle. Published by John Wiley & Sons, 403 pp. One of the leading field books of the country, containing full directions, suggestions, tables for the solution of the usual problems met with in field operations in preliminary and location surveys. A full set of tables of trigonometric functions, of a 1 curve, transition curve, coordinates, squares and cube roots. "Railroad Curves and Earthwork." By C. Frank Allen. Pub- lished by Spon & Chamberlain. 490 pp. Contains discussion of the usual railroad curves including the transition curve, rather full treatment of slope stakes and earthwork problems, with diagrams to facilitate the calculation for earth work; field and office tables. "The Field Engineer." By W. F. Shunk. Published by 1). Van Nostrand Company. 389 pp. This work treats of the prob- lems of preliminary and location surveys, many illustrative ex- amples, the essentials of slope stake setting, and the usual tables necessary for an engineer in the field. "Field Engineering." By Wm. H. Searles. This has been for years one of the standard manuals for field and office engi- neers, and it covers the problems of railway surveying, location and construction. The book is f.illy illustrated and has many valuable tables to shorten the labor of calculation. CHAPTER IX. EARTHWORK. 147. Prismoidal Formula. Let Fig. 81 represent ; solid bounded by two parallel planes and whose side faces are triangles. Draw the mid-section 12345678 and join any point P in this mid-section with ABCDEFGH, 1, 2, 3, 4, 5, 6, 7, and 8. This di- vides the solid into three kinds or types of pyra- mids. The first class has P for a vertex and A BCD for a base; the second has P for a ver- tex and EFGH for a base, while the third class has P for a vertex and for bases the side face triangles, as P EDC. Let 5, = area A BCD B, = area EFGH h = perpendicular distance between parallel planes A BCD and EFGH. 1. Volume PABCD = J ABCD x \h = \h B l 2. Volume P-EFGH = J EFGH x %h = \h B, 3. To find the volume of the pyramids of the third class, consider P EDC as a type of the third class. The pyramids P 12 and P EDC have the same vertex P and bases in the same plane EDC. Hence they are to each other as their bases, 151 152 SURVEYOR'S HAND BOOK. :.P EDC : P 12 : : EDC : 12. As 1 and 2 are the mid-points of the sides ED and EC, EDC = 4X12. :.P EDC = \ X P 12. But the volume of the Pyramid P 12 = $ x Area P12 X \h /. Volume PEDC=4x^X P12= -^ x P12. Similarly, Volume PEFC = -% X P23. 6 .-. Total volume of pyramids of third class= = -^- (P12 + P23-fP34 + P45 + P56 + P67 + P78 + P18) = ~Q- M, where M =area of mid-section 1234567S. Adding the volumes of the three types we get for total vol- ume V = Volume of solid = ^ (B { + 4A/ + B 2 ) ........... (33) 148. Railroad Excavation. In railroad earthwork, cross-sections at right angles to the center line of track are taken every 100 ft. Slope stakes are set and data obtained for calculating the volume to be exca- vated between the two sections 100 ft. apart. Such n solid is bounded by a plane roadbed, two parallel end areas, whose planes are perpendicular to the planes of the side slopes, while the upper surface is terminated by planes that are either triangular areas or that can be divided into tri- angles by drawing the diagonals as D'C. The prismoidal form- ula applies to such a solid. Fig. 82 represents the part of the excavation on one side of the central plane of roadbed. BKK'B' represents half of the roadbed between cross-sections DKBC and D'K'B'C'. To find the volume of the excavation by the prismoidal formula given above, it is necessary to find the areas of the ends or bases and of the mid-section. EARTHWORK. 153 149. Level Sections. Where the intersection of the cross-section plane with the surface of the earth is horizontal, the section is said to be level, or a one-level section. In Fig. 83 AB 26, DK c, CG EF. Now, BG = sCG Area EABC = V 2 (EC + AB)DK = i (2b + 2sc + 26) c, (34) Fig. 83. SA, j = 3/2, find area of EXAMPLE: Given 26 = 18', section. Area = 2bc + sc a = 18 X 8,1+ 3/2 X (8.4) 2 = 257.04 sq. ft. 150. Two Level Sections. When the surface of the ground slopes uniformly transverse to the roadway, two points established on the surface will be sufficient to determine the cross-section. Then area ABCE ECGF BCG AEF = % (A, + h z ) (26 + s!h.+ s!h) Vish, V z sh* = b (7u + A 2 ) + sfhh, (35) The center cut is used only in locating the slope stakes at C and E, but is not used in the calculation of the area. 154 SURVEYOR'S HAND BOOK. EXAMPLE: Given 25 = 18, j = 3/2, /i, = 8.4, /z, = 6.6. Area of section = 9 (8.4 + 6.6) + 3/2(8.4 X 6.6) 135 + 83.16 = 218.16 sq. ft. 151. Three Level Sections. By far the most common and usual section is the one where the two side heights and the center cut are used in calculating the area. O J! Fig. As usual, CG = h lt EF = h->, Ib, BG = sh l} FA = sh,, KG = Area DKBC DKC + CKB In the same way, DKAE Total area = r/2(d! + d,) + = d,, KF d,, DRc, AB + sh, f KF + q _ sh*. Vzcd, + %5/i,. , + Vzbh,. + M .... ........ (36) Thus, in the three-level section, the double area is equal to the center cut multiplied by the sum of the distances out, plus the half roadbed multiplied by the sum of the side heights. Fig. 152. Irregular Sections. When the surface of the ground is very irregular, rod readings must be taken at every change in slope of surface. Thus, in Fig. 86 rod readings must be taken at seven different places, and this section would be called a seven-level section. In the field we would locate N by meas- uring its distance out Kn, and by its elevation Nn above AB the roadbed. Thus, for any point or the surface, we have its co-ordinates, i. e., distance above AB (roadbed) and the dis- EARTHWORK. 155 tancc from K (center of roadbed) to foot of perpendicular. To find the area of the section, we find first the area on the right of the central plane DK and then on the left. BKDMNPC = KDMm + mMNn + nNPp + pPCG BCG Let c t h m , h n , V h\ be the heights of D, M, N, P, C above AB and d m , d n , d v and d\ equal the distance out of M, N, etc. Area KDMm=\ (c + h m ) d m Area mMNn=b (h m + &n) (d n dm) Area nNP p=% (h n + h p ) (d v d n ) Area $PCG=\ (hp + hj ( on the lower. The grade-point is found 3' to left of center. Area in cut = ^ (2 + 8) + *( 2 + 3) 48 = 60 sq. ft. Area in fill = 1/2 6 X 4 12 sq. ft. PROBLEM 72. If BK = S', DK = 2 f , KA~r, PK = X, slope in cut 1 :1. slope in fill == 3:2, find area in cut and fill if A 1= =8, h, = 4. 155. Average End Areas. In practice, the volume is calculated by the average end area formula. Fig. 89 represents a form of a three-level section. The cen- tral plane DK divides the solid of excavation into two parts that 158 SURVEYOR'S HAND BOOK. can be treated separately. Let the center and side cuts at one station be c and hi and those at the next station 100' away be Ci and hi and let both sections be three-level sections, as in the figure. Let B t = area DKBC B 2 area at the next station corresponding to DKBC, We have, B,= y 2 (d,' d' + bh,') Now, if the solid is bounded by plane faces, we have center cut, side height, and distance out at mid-section. SM=(d, + ) .................. (37) 100 100 Let V e = ~2~ (Bi + 5,) = -^ (3diC 1 + 3d l 'c l f + 3bh l + 3bh l f ) 156. Error of Average-End Area Formula. The average end area formula generally gives an excess of volume. Let E be the excess in volume by end-area formula. ... E=V & -V = -^-[(cj-c/) (d l -d l ')] ......... (38) In the majority of cases, d c/, and di rf/ have the same sign -'-excess is positive, that is, there is really an ex- cess. But in passing over a saddle, o can be greater than c/ and di less than d\. In such cases the excess is negative that is, the volume calculated by the average-end-area formula is smaller than the true volume. By common consent among engineers, contractors and sur- veyors, practically all volumes in railway practice are calculated by the average-end-area (AEA) formula. In fact, it is highly probable that for the real earth solid, the AEA formula gives results as near the actual cubic contents as the true prismoidal formula. EARTHWORK. 159 157. Examples. The stations 1, 2, 3, etc., in the following table are 100 ft. apart. The numerators in each case show the depths of cuts and the denominators the distances out at the different points. Width of roadbed = IS'.O, slope = 3/2. Cut or Fill Cubic btation Areas Yards Left c Right 1.. 11 78 A 2 221.31 17.1 1003.3 2.. _Li QQ 320.46 20.1 27.3 1 381 . 2 3 JL-L !5J? 9 ^ 1?.8 HJ? 4>\5 4 21.3 10.0 1LO 30.3 181 9 7 4., . L 2 ^ 13 8 1*2 1^4 27.3 12.0 33.6 2 33l 2 5 m 16 - 6 is 7054 In calculating the areas (as at Station 3) we arrange the data as follows : 'JL _?J: 10.2 12.0 13^8 14.2 _0 2lT3 TOO ~0~ TITO 3O3 "9" and for positive terms work from the center outward, multiply- ing each numerator jy the next denominator ahead as we pass out from center, and for negative terms multiplying each nu- merator by the next denominator towards the center. Calculacion: Area on right=H12.0x 11.0+ 13.8 < 30.3+ 14.2x9.0 14.2 x 11.00H260.9 Area on lef t = H12 x 10 + 10- 2 X 21.3 + 8.2 x 98.2x10] = 164.53 Check calculation: Area on right = [ 25.8 X 11.0 + 28 X 19.3 14.2 x 21,3] -=260.9 sq. ft. Area on left [22.2 x 10 + ' 8A x H-3 8.2 X 12.3] =164.53 sq. ft. . Total area = 260. 9 + 164.53=425.4 160 SURVEYOR'S HAND BOOK. Area at Sta. 4 = [13.8 X 12+15 X 33.6+ 16.4 X + 13.8 X 27.3+12. ! 2 X 9 12.0 x 16.4] = 553.47. Area at Sta. 5 = i[16.6 X 67.2 + 32.8 X 0] = 705.4. 100 Volume 1-2 = -^ [221.31 + 320.46] = 1003.3 cubic yds., Volume 2-3 = -gy [320.46 + 425.4 ] = 1381.2 cubic yds., 1 no Volume 3-4 = -^ [425.4 + 553.47] = 1812.7 cubic yds., Volume 4-5 .= -^ [553.47 + 705.4 ] = 2331.2 cubic yds. Total volume 1-5 = 6528.4 cubic yards. PROBLEM 73. Find the areas, volumes and total volume from the following field notes: Cut or Fill. Cubic Station. Areas Yards Left c Right 15.8 20.2 32.7 18 ' 4 39.3 14.8 18.8 3L2 16 ' 9 37.2 12.8 14.7 16.3 17.4 28.2 13.0 15 ' 12.0 35.1 11.4 14.4 16.2 J.O.O 11 / \ o~T .26.1 11.0 33.3 Total volume = 7533.7 cubic yards. 158. Preliminary Estimates. Tn comparing preliminary surveys of several lines, it is necessary that we know the num- ber of cubic yards of excavation required on each line. The preliminary profile will give the cut or fill at the different sta- tions, and if we assume that the cross-section is level we can obtain a close approximation to the true areas and hence to the volumes without going to extra expense of setting slope stakes to determine the true cross-section. From article 149 the area of B of a level cross-section is given by EARTHWORK. 161 Where 2b ! = width of roadbed, r = center cut, s Aslope If 2fe = 18', j = 3:2, then # = 18c + 1.5r. Now, if we make c = l, C, 3, 4, 5, etc., we get areas of 19.5, .42, 67.5, 96, etc. It is assumed that any of these areas is the average of the two sections, 50 ft. on each side of it. R*it Volume = ^ cubic yards Making B equal to the areas above, we get the volumes in cubic yards to be 72, 156, 250, 356, 472 cubic yards, etc. In the same way we can find the volumes for any width of roadbed and any slope. The usual widths are 12, 14, 16, etc. Table V. gives the volumes in cubic yards, slopes 1 :4, 1 :2, 1 :1, 3:2, 2:1 and 3:1 and for the various widths. EXAMPLE: If 2b l8, ,y = l:l, and it is desired to find the volume in cubic yards from stations 5 to stations 10, where the center cuts are 6; 8, 10, 12, 11, we look in the table headed "Slopes 3:2" under "base" and opposite 6 we find 600, oppo- site 8, 889, etc. These are read from Table V. and recorded as below: Station. Center Cut. Volume. 5 6 600 6 8 889 7 10 1,222 8 12 1,600 9 11 1,406 10 9 1,050 Sum of volumes = 6,767 cu. yds. From this we must subtract half the end volumes, or 825. Volume between Sta. 5 and Sta. 10 = 5942 cu. yds. PROBLEM 74. If the center cuts at Stations 17, 18, 19, 20 and 21 are 12, 14, 15, 16, 15, find the number of cubic yards between Stations 17 and 21 for level sections by use of Table V. 159. Earthwork Note-Book. The preliminary estimate of the amount of earthwork is for a basis of comparison with other preliminary lines, but the final estimate is based on the actual notes taken in v the field in setting the slope stakes. The level 162 SURVEYOR'S HAND BOOK. notebook, as commonly used, has a left-hand page ruled into six columns, as shown in Fig. 90. The grade column (marked "Gr." / ^ ^ _ op (0 ^ i O ^ ,3 5^ CO >fl s O _ cO ^ cO ^ (U lO *o K^ fx ^f O o ? >o ' * (0 ni 1 ! . i (M -I (VI ^ ^ * 10 (U ^ vfl CO (U >0 oo 4 LJ * rw CO* CO vfl' CO CO CO CO N o 5 H H j- D - UiT L. x s N 10 s ed 3 o 0* K s o ^ s ^ ^ o to w o 91 - I N CO N CO JO " \0 + o (O M CO r tfi 3 CJ t- N (U (U 10 N (U 10 ears will shrink about 2 r /f more, often less than *l% in the year following the completion of the work, and very little in subsequent years. ~. The height of the embankment appears to have little ef- fect on its subsequent shrinkage. 8. By the proper mixing of clay or loam and gravel, fol- lowed by sprinkling and rolling in thin layers, a bank can be made weighing 1% times as much as loose earth, or 133 Ibs. per cu. ft. 9. The bottom lands of certain river valleys and banks of cemented gravel or hardpan are more than ordinarily dense and will occupy more space in the fill than in the cut unless rolled. Earthwork is paid for by the cubic yard, usually measured "in place," that is, in the natural bank, cut, or pit before loosen- ing; but there is no good reason why it should not be measured in the fill or embankment, and it often is so measured where it is very difficult to measure the borrow pits. In either case the specifications should distinctly state how the 'measurements are to be made. Sand or gravel for mortar and concrete are usually paid for by the load in the w r agon. Bibliography. "Railway and Earthwork Tables." By C. L. Crandall. It is sufficient to say that this book bears out its title, where the tables are arranged by which we can read the volume for railroad cuts and fills for any of the usual dnta given in the field notebooks for cross-sectioning. "Railway Earthwork." Parts I and IT. By the late A. M. Wellington. Part I discusses the volumes of the various solids in railway earthwork, while Part II consists of a series of dia- grams from which the volume corresponding to the field notes can be read at once. "Railroad Curves and Earthwork" (with Tables). By C. Frank Allen. In the section on earthwork the theorv and use of EARTHWORK. 10!) graphical diagrams are treated and the methods of using these diagrams to obtain the volumes are illustrated by many ex- amples. "Primoidal Formulas and Earthwork." By T. U. Taylor. The history of the different formulas that apply to the earth- work solid and their application to railway cuts and fills are given. A chapter is devoted to the two-term formula wherein it is shown that there is an indefinite number of two-term form- ulas that give the exact volume of the prismoid ; that if we take the average of two sections, these sections must be 21.14 feet from each end of the solid 100 ft. in length. "Manual of Road Making." By W. M. Gillespie. Contained in appendix some 40 pages upon the subject of earthwork, in which, in addition to the treatment of the ordinary cases, he showed that the prismoidal formulas applied to give the exact volume of the earthwork solid when the upper surface was a warped surface. "Earthwork and Its Cost." By H. P. Gillette. 244 pages. This work has taken up and considered actual examples, giving date, size of contract, conditions under which constructed, kind of earth, how handled, etc. The author has winnowed from many a contract the -essentials as to shrinkage, classification, loosening, cost when carried by wheel barrows, wagons, buck and drag scrapers, wheel scrapers, by elevating grader, steam shovels, cars, etc. "Rock Excavation. Methods and Cost.'' By H. P. Gillette. 375 pages. Its title abundantly indicates its scope. Its estimates of cost are from concrete examples where actual conditions are given. "Handbook of Cost Data." By H. P. Gillette. One of the most valuable books for the engineer that has appeared in many years, and it comes nearer filling a long existing void than any book before the engineering public. It includes a great deal of the material in the two books mentioned above and much additional matter. It deals directly from the ground with such questions as co^t of earth and rock excavation, roads and pave- 170 SURVEYORS HAND BOOK. ments, stone masonry, concrete construction, water works, sew- ers, piling, trestling, erecting buildings, steam and electric rail- ways, bridge erection, railway and topographic surveys and mis- cellaneous structures. This book should be a valuable Vadc Me cum for any engineer who has to deal with the cost of struc- tures. CHAPTER X. CITY SURVEYING. 165. The City Engineer. The most important factor and vital unit in all city surveying is the city engineer. A careless engineer means a careless, loose, inaccurate, conflicting and liti- gous survey. The city engineer is the supreme court and all the lower courts with respect to the accuracy of city survey- ing. As the city engineer, so is the survey. The engineer should be the first instrument of precision selected, and it is supreme folly to have a standardized steel tape and a highly sensitive transit in the hands of a careless operator. We apply corrections for sag, temperature and pull to our tape-line meas- urements, but these are mockeries if the engineer can be sagged from 1ii 's true course, or if he allows a "pull" to draw him from the straight line. The accurate, just, and fearless performance of his duty should be his pjatform. To this end should he be born, for this cause came he into the world, and he should bear witness to the truth. The surveying dernanded.of a city engineer does not involve any principles, operations, or intricacies that may not be easily overcome by any person who understands thoroughly the use of the ordinary instruments and theory of surveying heretofore de- scribed, but as land is much more valuable in cities than in the country it follows that the measurement of city property must be made much more carefully than the survey of a farm. The accuracy of the survey should increase with the value of the property. Small errors that may be neglected now may involve perplexing difficulties in years to come. It is always wise and safe to be considered a little too fine-haired rather than a little too careless. 166. Objects of Survey. The prime object in a city sur- vey is to establish the points and boundaries of city property with absolute accuracy. To do this, it is necessary to establish certain reference lines or points which will remain permanently 171 172 SURVEYOR'S HAND BOOK. fixed and which, like a reference library, are of easy access and of undisputed authenticity. Property is valuable, and to prevent litigation it is necessary to have all property lines authoritative- ly established beyond the shadow of a doubt. Chains with their many hundred wearing surfaces are unfit for such work, and as it lacks accuracy the compass can not be used. As the ordinary transit measures to the nearest minute and as an angle of 1' is subtended by an arc of 18 ins. at a distance of one mile its use" 1 should be precluded where accurate work is demanded. The primary object of a city survey should be the accurate loca- tion of all property lines in accordance with recorded notes or maps, and complete provision for the rapid, convenient and ac- curate re-establishment of these at any time. The most accurate instruments and greatest care should be used. 167. Monuments. It is of fundamental importance that lasting monuments be established to which all city lines, points and buildings can be referred. Eternal monuments is the price of accurate work in city surveying. While engineers and sur- veyors are liable to rail at and descant sneeringly at the loose methods pursued in making the original land surveys, many of such land surveys are monuments of accuracy when compared with the surveys of many of our cities. In fact, although our original land surveys were loosely made, all transfers of property- have been based on such surveys. These surveys have many monuments in the shape of trees to stand as silent witnesses to be called upon. The land at least had an original survey, while the original part of a majority of our cities has expanded with- out the semblance of an original survey. It is worthy of re- mark that more care and accuracy are displayed in surveying the "additions" and "out-lots" than obtained in the original sur- vey of the nest-egg of the town. But whether or not monuments were established in the origi- nal survey of the town, it is of the utmost and urgent importance that they be established at the earliest possible moment. In some cities a very loose habit has prevailed of using old build- ings for reference points. Such a practice should be condemned as a make-shift, for with the enhanced value of property, such CITY SURVEYING. 173 buildings arc liable to be razed to make room for mocUrn struc- tures. 168. Additions. The map of every "addition" or pro- jected town should when filed in the county clerk's office show clearly the location of all monuments and no map should be ad- mitted to record that does not give these data. Not only should such a map show the location of such monuments, but a full de- scription of such monuments should be made a matter of record. Such requirements should not be a matter of custom, ethics, or taste of the surveyor, but should be a matter of law; and there is no more reason for a law authorizing the employment of skilled surveyors to locate state lands and file a complete set of field notes for the same than there is for a similar law requiring every city to have a similar map or set of notes filed and made a matter of record. These notes should be so clear and include such a number of sketches that they may be readily understood by any person of average intelligence ; and such notes should be capable of only one interpretation. Litigation has always fed fat on loose and inaccurate surveying and an unmonumented city. Monuments should be set and established by the original sur- veyor. He it is that made the surveys with respect to such monuments and it is his duty to finish his survey. Tt can be truthfully said, "An unmonumented city has no survey." There is a certain respect paid to the County Surveyor and his work should command respect. So it should be with the work of the city engineer, but while our laws provide for "witness trees," "fore and aft trees," for land surveying, there are in many states no adequate laws for enforcing or establishing imperish- able witnesses to the city lines in a city survey. 169. Kinds of Monuments. Monuments should be con- structed of permanent material and the special kind will be de- cided by the question of economy. The materials most com- monly used are stone, concrete, wood, and iron rods or pipes. If a stone is used it should be imbedded in the ground with its upper part well underneath the surface, so that the big end will be down and so that it will rest solidly in its bed and have no tendency to change its position. A small hole from % to 1 in. 174 SURVEYOR'S JIAXD BOOK. in diameter should be drilled in the upper surface of the stone TO a depth of G to 8 ins. Into this hole a copper bolt should he inserted and melted lead or babbit metal run around it to hold it securely in position. The upper end of the bolt should be flush with the surface and two normal diametral lines should be marked across the bolt, their intersection forming the reference point over which the plumb-bob of the transit is suspended, or a {'UK poie set when other points are to be established. A concrete block, Fig. 0-">, can 1>e constructed as a monument rind it has many advantages over the stone monument, as it can he formed into any desired shape. For economy, the concrete monument should he built in the form of the frustrum of a cone or pyramid, ami its upper surface should be kept well K low the surface of the street. The copper bolt can be imbedded in the concrete before it hardens and it can be located in any de- sired position in the concrete. If wood is used, the most dur- able available wood should be se- lected. The important monument should be at least 6x6 ins. i>y J ft in length and should be imbedded Fig. 9;>. 6 on hard soil or preferably en a flat rock or a concrete mixture. Cedar is an excel lent material, while osage orange (boisdarc) has no superior. The young mountain locust, 10 ins. in diameter, is the most dura- ble in the east, while mesquite would be practically the only locally available wood of the southwest. An- iron rod or pipe is often driven with maul or sledge for a monument, but these do not make very satisfactory monu- ments and are not to be recommended, but it must be said that they are infinitely better than none at all and greatly superior to a small wooden stake. Wooden stakes are very easily disturbed or destroyed and unless they are immediately replaced by other monuments of a more permanent character the work will be wasted. CITY SURVEYING. 175 If the street is already graded and paved the monument should be set with its top below the foundation of the pavement and should be protected and made easily accessible by means of an iron jacket and cover plate such as are provided for the valves of the city water supply. If the street is neither graded nor paved, some thought should be given to the probable final street level and the monu- ment should be located to cor.form therewith if possible. It is the duty of the city engineer to establish suitable per- manent monuments wherever needed, to indicate the same clearly and correctly on the proper maps, to deposit in the office a complete set of all field notes, to leave his work in such a state that- it may all be intelligible and useful to his successor. 170. Location of Monu- ments. These should, if pos- sible, be located in the center lines of cross streets and should be on high points. They should be of easy access ; a few well located monuments are more valuable than many to which ready reference cannot be made. The fundamental re- quisites of good monuments are that their location is known and that their distance and azimuth are matters of record. Sometimes it is impracticable to set monuments in the center of the street. When this is the case, they should be placed as near the center as convenient, but they should always be refer- enced in to the four corners of the street. Wherever the monuments are located, the four corners of the streets should be marked by sub-monuments whose distances from the main monument are recorded. 171. Tapes. It is useless to have an excellent system of monuments unless this excellence prevails throughout the whole 17G SURVEYOR'S HAND BOOK. organization of the city survey. All lines should be measured with standardized steel tapes. The material of the tape should be of the best steel and its own individual constants should be determined. It should be sent to the U. S. Coast and Geodetic Survey, Washington, D. C, to be standardized. It is there com- pared .with an absolute standard, its coefficient of expansion ascertained, its pull and temperature for standard length de- termined. These data are returned with the tape and in all im- portant measurements should be used and corrections should be made for temperature, pull, sag, and grade. But accurate work can not be performed with accurate instruments unless accurate methods are used. In chaining, if the street is graded uniformly and the tape can be made straight, the correction for sag would thus be eliminated. If in addition to this, the standardizing pull be applied, the only correction remaining would be that for temperature and grade, and if the street is horizontal, the only correction to be applied would be that due to temperature. 172. Transit. After the monuments have been located with accuracy and the exact point of these monuments marked by the intersection of lines on the copper bolt head, it becomes necessary to use the most accurate and refined instruments in the prosecution of the further surveying of the city. As the ordi- nary transit reading to one minute of arc would produce an error of 18 ins. in one mile, its unfitness for accurate city surveying is at once seen. It is useless to locate monuments accurately and to use an accurate standardized tape in connection with a transit that has such possibilities of error as the ordinary engineer's transit. For this reason a special transit (Fig. 97) is constructed with minuter graduations. The same reason that precludes the use of the engineer's transit in refined city work, of course, would ex- clude the surveyor's compass to a greater degree. In the mod- ern transit constructed for accurate city surveying, the needle and the needle box are omitted and the standards are construct- ed in one U-shaped piece that gives greater rigidity of bearing to the horizontal axis that supports the telescope, and consequently greater accuracy. The horizontal circle is much larger and the graduations can be made as small as ten seconds of arc. The CITY. SURVEYING. 177 Fig. 97. 178 SURVEYOR'S HAND BOOK. horizontal circle is protected from view by a cover plate except where the slot is made for the reading by the verniers. The verniers are read by special reading glasses, which are often at- tached to the instrument itself. Whatever the fineness of the reading may be, whether it reads to 10" or 2<>", we can by the repeating method read the angle five times and thus reduce the fineness of the reading to one-fifth of that given by the verniers. Thus if the transit is graduated to 30", we can by repeating the observation five times get a reading of 6", and if it reads to 1<>", we can by the repetition of five times get a reading to 2". In the length of one mile a reading of 2" would mean about a half an inch error. The transit can be provided with stadia wires and complete vertical circle and a heavy tripod. The complete vertical circle and stadia wires are auxiliaries that are added for the purpose of making topographic survey. The transits ful- filling these requirements cost from $300 to $700 and if it is desired the stadia wires and vertical circle can be omitted. 173. Datum. There should be established in every city bench marks to which all elevations should be referred. In the majority of cases, the elevation of the bench marks can be re- ferred to the sea level or mean low tide. In many cities the U. S. Coast and Geodetic Survey has bench marks with reference to sea level that have been established by a system of precise levels run and checked from the coast to the interior. These are by far the most reliable and accurate bench marks that can be ob- tained. The U. S. Geological Survey has also a chain of bench marks established in certain sections of the country. The bench marks established by these two surveys are often copper bolts set vertically in the cap stone of bridge piers, or horizontal bolts set inside of stone buildings. Another form is a circular disc, Fig. 98, from the center of which a bolt 3 ins. long projects at Fig. CITY SURVEYING. 179 right angles to the surface of the disc. Two diametrical lines normal to each other are marked across the face of the disc and the elevation, is stamped on the horizontal line of the disc. A bed or setting is cut out of the stone for the disc and in the center of this bed a hole is drilled to receive the bolt. The bolt is then leaded into the stone. 174. General Maps. There is generally a small scale map made of the whole city, but this shows few engineering features and except in the case of small cities it can not show the dimen- sions of lots and the field notes for the location of monuments. In addition to the map of the whole city there should be a map of certain sections to a scale sufficiently large to show all lengths of all lines and angles made by intersecting lines. It is the practice in many cities to have block maps containing from one to four blocks with the position of all monuments marked with distance from street corners and angles made by such tie lines. These maps should show the center line of street, angles of intersection of center lines, and the location of monuments on street corners. The map should contain the following data : 1. Length of all lines. 2. Angles made by intersecting lines. 3. The exact position of all monuments. 4. The number of each block and lot. 5. The names of all streets and streams. f>. Water pipes and fire plugs. 7. Sewer pipes. 8. The true meridian. 9. Width of streets. 10. The position of adjoining property lines. 11. A complete title to map. 12. The scale. 175. Water-Pipe Map. If the city owns the water-works and sewerage systems, it should possess an up-to-date, accurate and distinct map of both the water-pipe lines and the sewer-pipe lines. If the city is small and pipe connections are not intricate 180 SURVEYOR'S HAND BOOK. nor numerous, one map will suffice for both systems, by adopting a different legend for the two systems. A water-pipe map should show clearly the position of all mains, valves, connections, fire hydrants, size of pipe, and all side connections. Such a map usually pays for itself many times over and it is a very loose city government that does not keep such a map. Without a pipe-line map all extensions and repairs have to be made some- what upon the temporary makeshift basis. In some cases, the city authorities depend upon the memory of a day laborer to lo- cate sub-mains, and these often have to spend hours in search of the pipe, all of which time could be saved by an accurate map. If a private company owns the water-works, an accurate map is part of its equipment because it is simply a part of good busim -< to have such a map. However, there often seems to be some- fatality about municipal ownership in regard to proper records The city records, covering expenditures of millions of dollars for public improvements are often thrown aside or dumped into boxes, or cases that cannot be used for any other purpose. The proper keeping of engineering data is a weak spot of municipal ownership, an indictment that cannot obtain in the same degree against private ownership. When city streets are improved by paving, it is of the utmost importance in making repairs or connections to know the exact distance of the main or sub-main from the sidewalk or property lines, as it is a matter of economy in time and renders the tear- ing up of a large area of paving unnecessary. 176. City Blocks. The size and shape of city blocks vary in different sections of the country and, in fact, in different sections of the same city. It is difficult to set any limits, but the regular rectangular blocks vary in length from 400 to 900 ft. With a width of street of 80 ft. there will be 5% to 11 blocks to the mile, and of course if the streets are narrower there would be from 6 to 12 to the mile, etc. 177. Rectangular Blocks. In ordinary cases, a rectangular block consists of two rectangular sections with an alley between. Thus if ABFG, Fig. 99, is a rectangular block, there are two sec- tions, ABCD and EFGH, with an alley DCEH. If the length of CITY SURVEYING. 181 the block is 300 ft. and if each section contains five lots, these should be 60 ft. wide. The length of the lot is 125 ft. and width of alley 16 ft., the block being 266 ft. wide. Each lot is described (1) by its number, (2) by the number of the block, (3) by the sub-division or addition, (4) by the name of the city, county, and state. Thus we should write : "Lot number (3) three in Block thirty-nine (39), Borden Addition, in City of Austin, Travis County, Texas." This de- scription is sufficient if an official map of this "Borden Addition" is on record in the city or county clerk's office, showing all di- mensions of such lot. However, if it is desired to insert the metes and bounds, this can be done as follows : "Lot number three (3) in block thirty-nine (39), Borden Addition, in the City of Austin, County of Travis, State of Texas, and bounded as follows : Beginning at the northeast corner of lot num- ber two (2) in said block, addition and city, one hundred and twenty (120) feet from the northwest cor- ner of said block, thence S 9 W, one hundred and twenty-five (125) feet with the east line of lot num- Ross ST: ber two (2), to a corner on the alley, thence S 81 E sixty (60) feet to the SW corner of lot number four (4) ; thence N 9 E, with west line of lot number four (4) one hundred and twenty-five (125) feet to a point on the north side of block, the northwest corner of lot number four (4), thence N 81 W with the north line of said block and with the south line of Adams St., sixty (60) feet to the beginning." 178. Rectangular Lots. The size of lots runs the scale from the narrow business property lot 25 ft in width to that of the broad frontage, merging into the suburban property defined by the acre and metes and bounds. The lots in the regular resi- dence section vary from 40 to 100 ft. in frontage, but there is 18.! SURyEYOR'S 11AXD BOOK. infinite variety to the special dimensions and the foregoing fig- ure are approximate only. In regard to the depth of the regular rectangular residence lot, it can be said that the depths are approximately double the frontage, varying from 90 to 200 ft. unless some irregular boundary, stream or hill intervenes to modify the general plan by which the lots are laid off. 179. Irregular Blocks and Lots. Tt often happens that the topography, old roads or streams force the engineer to make a block of irregular shape, the flat-iron, horse-shoe, triangular or oval. In such a case no rules can be laid down for cutting such a~ block up into lots, and the engineer can have only one guide, and that is to make each lot wide enough for the build- ings of that locality (busi- ness or residence) and of the ordinary depth. If ABCD, Fig. 100, rep- resents the apex block be- tween two converging streets it is often difficult to OT: divide this up into lots to the best advantage. The simplest method is to run the lot side lines perpendicular to the street line. This is shown by the side lines of lots 1, 2, 3, 4, and 5, all of which lines are perpendicular to the street line on Shaw St. However, it may hap- pen that for some substantial reason the lot lines are parallel to the alley or some other line. Again the lines may be drawn ac- cording to no system whatever. In the latter case, the opposite sides of the lot will not be parallel, and it will be necessary to describe each lot by the metes and bounds. In addition to this the corners should be marked by some permanent marks, as gal- vanized pipe, stones or concrete blocks. CITY SURI'EYIXG. 183 In the flatiron form of blocks, as in Fig. 100, a dead-end alley can be provided for at the big end of the block, and this can extend as far as the line of lots will permit. A lot in an irregular shaped block should have a rather full description. Thus lot 9 should be described as follows : "Lot number nine (9) in Block thirty-five (35), Division A, in the City of Austin, County of Travis, State of Texas, which is bounded as follows : beginning at an iron pipe in line of Fox Street 70 ft. from north- west corner of said block 35, thence along Fox Street S 6 W 40 ft. to corner of lot number 8, thence S 87 E 64 ft. to a copper bolt in a stone which is a corner to lots number 2 and 9 of said block, thence north 46 ft', to a stone corner to lots 3, 9, and 10, thence S 87 15' W 54 ft. to the beginning." 180. Private Notes. The careful engineer will mark the length of all lines, the angles made by the boundary lines of lots, give the full number of lot, the name of "addition," and all other data necessary to define clearly and distinctly the lot so that another engineer, years later, will have no trouble in tracing the steps of the former. Every modern engineer experiences a genuine appreciation of the original engineer, when he finds that the recorded map shows clearly all distances and angles, and the modern does not hesitate to commend the former when map dimensions, when applied to the field, are found to be true. Too many engineers are stingy with their data when it comes to put- ting it on the map. The question often arises as to how much data should be placed on the map, and this can be answered by saying that sufficient data should- be placed on the map to enable another engineer to go upon the ground and re-locate any lot without doubt or shadow of turning. Until this condition is fulfilled the map is incomplete; the claim of the engineer that his notes are private cannot be set up or maintained. The city engineer "is a public officer and should keep complete records af all work done in his official capacity during his incumbency. If he walks out of his office and retains notes the lack of which would embarrass his successor, he is practical^" a thief." (Ernest McCullough.) 184 SURVEYOR'S HAND BOOK. 181. Prescriptive Rights. Owners in new and sparsely settled additions arc often permitted to locate their own lots, and in doing so they get the side lines of the lots shifted a few feet. A fence is usually erected on the lot lines erroneously lo- cated and this fence stands as the visible mark of the lot lines for many years. The adjacent lots are not improved and the result is that the owner of the improved lot, although his fence lines are wrongly located and though there may be an excess in his frontage, has been in peaceable possession, undisturbed for a sufficient time to constitute a prescriptive right. This gives him the right of possession and when the owners of adjacent lots want the amount their deeds call for, they find the prescriptive right set up as a bar to moving fence lines. The result is that legal mills have to be set to grinding with no assurance of the quality of the grist. Fig. 101. Where the prescriptive right obtains it is of the highest im- portance to property owners to see that their lots are located properly and accurately by an official engineer, and that perma- nent corners are established. 182. Cross-Section of Streets. After the blocks and lots have been laid off and accurately marked, it then often falls within the province of the surveyor or engineer to establish the form of cross-section of the street. This cross-section is usually a curve having a certain rise or crown, depending on the mate- rial out of which the surface of the street is constructed. If the street is paved with vitrified brick the crown should be from % to % in. per foot of half width. Thus for a street width of 96 ft. between side walks there should be a crown of 6 to 18 ins., preferably the latter. If the side walks are at different eleva- tions, local conditions may demand that the cross-section shall consist of two curves tangent to each other at the crown or crest and that the amount of their descent shall be different. CITY SCRl'EYIXG. 185 Thus in Fig. 101 the cross-section can be formed by the two curves OA with a fall of Ol~ and OB with a fall of OC. Let VA = distance from curb to crest = b ; OV" == v, OK = .r; PK = y, fall from O to P. Then if curve OA is a parabola. Or y = F* 2 ................................. (40) If b = 48 feet, and v = 1 8" ^. 1 .5', y = "4^^ ** - J^T)' 3t^ If y equal fall in inches and x = distance in feet, j = y-jg By making x = 0, 4, 8, etc., the falls at these distances are found below. .r v x y x y :<><> 16 2.00 36 10.12.") 4 .12--, 24 4.5 40 12,5 8 .5 32 8.00 48 18.0 Formula (40) is a general formula and will apply to any con- ditions, and does not assume that the crest O is in the center of street. Circular Curve. Some engineers prefer to treat the curve OA as a circle and specify the amount of curvature by the radius of the circle. Let r.4=half of chord of circular arc OA ; r = rise = OF. As the arc is very flat, KP can be treated as a secant from P to circle. Then if R = radius of circle, OK^ KP (2R + PK), or, x* y (-2R + 3 .) = 2Ry + y": The last term is so small in comparison with the first that it can be omitted. If the crown is % or Vs in. per horizontal foot, then j? = 192 ft., or 576 fl. respectively. 183. City Engineering Records. There are three differ- ent kinds of records that should -be kept by the City Engineer: 18<> SURl r EYOR'S HAND BOOK. I Field Note-Books. II Detail maps. Ill Orders, letters of correspondence, bids, prices, contracts, specifications, results of tests, etc. 184. Field Note-Books. For simplicity one kind of style of book that is applicable to all kinds of surveying should be adopted and used exclusively. It should have stiff covers, should be leather bound, and be as large as the average coat pocket will accommodate. If the left hand page is ruled with "horizontal blue lines V in. apart and the page divided by vertical red lines into six columns, the right page being divided into small squares by horizontal and vertical blue lines, with a verti- cal red line in the center of the page, the book will be found to answer admirably for all-round work. In this book, level notes, transit notes, notes on earth-work, sewer-pipe, water-pipe, tri- angulation, land surveying, etc., can be recorded with clearness and neatness. The measurements can all be placed on the K'ft page, while sketches can be placed on the right page to an ap- proximate scale. Proper provision should be made for storing or filing all the note-books, preferably in a fire-proof vault. The books should be numbered consecutively and arranged in order on the shelves, and the Chief Engineer should require every note-book to be put in its proper place on the shelves or in the vault over night. Books should be assigned to certain classes of work rather than to particular assistants or transit men. - For example, all mis- cellaneous work relating to. property lines should be kept in one book, all work relating to grades of streets in another, etc. Each new book should be immediately given a number, the class of work for which it is intended being plainly lettered on the outside of the cover, thus : "Street Grades and Profiles, 5th, Oth and 7th Wards," and the first half dozen pages should be left blank for an index to its contents. Every new piece of work should be indexed in the book, and also in the general index of all the note-books kept in the office. The Chief Engineer should see that each assistant enters his notes in the proper book so neatly, completely and correctly that at the end of any day's CITY SURVEYING. 187 work the hook ma}- he handed to any other assistant who would be able to continue the work without the least possible duplica- tion or loss of time. ICach assistant should be required to carry with him 'the proper note-book, and to make in it the original notes of the work. If this is done, the field-book may be presented as evi- dence in case of law suits, but it could not be presented as evi- dence had the notes been copied in it from other books or from scraps of paper. Note-books should not be permitted to litter the draughting tables or desks of the office. When not in use they should be in their proper places on the shelves, or in cases. Kach member of the office staff should be impressed with the fact that surveys are expensive and that the data contained in these note-hooks arc valuable. Books should not be carelessly thrown about, but on the contrary should be carefully pre- served and everything should be done to make the records readily available for future reference. 185. Detail Maps. Tn addition to the large wall map of the city there should be smaller maps to a larger scale, showing all essential details of lines, angles, monuments, distances, etc. The wall map may be divided into sections by lines at right angles to each other,' or by streets and streams into sections corresponding to the smaller maps. This enables the detailed map of any section of the city to be found with the least loss of time and trouble. On these detail sheets the water, gas, sewer, and steam mains, telephone conduits, etc.," should be rep- resented by different colored inks or by specially dotted lines. If there are many of these pipe lines, it may be necessary to have several copies of each sheet, one devoted exclusively to water service (called the water-pipe map), one to sewerage, etc. Such maps should be made on the best quality of mounted egg-shell paper and should be service maps on which every change in pipe lines should be noted immediately. If it is con- sidered necessary to have records of conditions at different dates i. e., on the first of January each year tracings of these service sheets may be made, dated and filed. 188 SURVEYOR'S HAND BOOK. An excellent plan for standard sizes for drawings is to acY-pt the full sheet, half sheet, quarter sheet, and eighth sheet plan, and the dimensions of these can be for full sheets, 24x36 ins. ; for half sheets, 24x18 ins.; for quarter sheets. 12x18 ins.; and for eighth sheets, 12x9 ins. Each sheet should be trimmed % to 1 in. outside the border except on the left, where a double margin should be left for binding purposes. However, it is useless to have or to demand accurate city maps and drawings and not at the same time provide safe and secure repositories for such records. Substantial cases should be constructed with a set of drawers (say 40x27 ins. inside dimen- sions) for the full size drawings. For the half size drawings the 40 by 27 drawer can be divided by a thin partition across the middle, dividing it into two compartments about 27x10% ins. Another set can be provided for the quarter size drawings where the 40x27-in. drawer has two divisions or partitions at right angles to each other ; and in a similar way- the eighth size drawings can be provided for. The drawers should be numbered consecutively and if divided into compartments for fractional sizes each compartment should be given a letter and the draw- ings in it numbered in a special place on the drawing in addition to the general number that it must bear. Thus the drawing should be labeled "Drawer 26 D, Sheet 14, v in one corner, while the general number 76 will indicate that it is the 76th drawing made by the city. The legend "Drawer 26 D, sheet 14," indi- cates that it is to be replaced in drawer 26 in compartment D, between sheets 13 and 15. In addition, a systematic record should be kept showing clearly what each numbered drawing refers to in the general series. An alphabetical list should be made of these drawings, where the leading word in title or location will indicate the character of the drawing. Better than this, however, is a card catalogue where every map is cross-indexed in such a manner that it may be readily found. The card catalogue has many advantages over the book catalogue, in that references can be made with greater dispatch, and corrections and new insertions can be made without disturbing the other records. CITY SURVEYING, 180 186. Orders, Bids, Etc. It is doubtful whether it is necessary to mention the necessity of keeping a record of all correspondence, orders, etc., as this is the usual practice of every good business man, and every engineer should be a good business man, as far as the city is concerned at least. Contracts and specifications are important documents in con- nection with large undertakings or important works, and these should be kept in a fire proof safe, to which only the trusted members of the staff have access. Specifications, results of tests, and other data on miscellaneous matters should be indexed and may be filed in a manner similar to that for drawings. Bibliography. "Theory and Practice of Surveying." By the late J. B. Johnson. This book has long been a standard work for the surveyor and engineer. Its chapter on City Sur- veying was prepared by William Bouton, City Engineer of St. Louis, Mo., and gives the conditions necessary for high grade, accurate city surveying. "Principles and Practice of Surveying." By Breed and Hosmer. An excellent book for the city engineer, containing full directions, discussions, and illustrations of many problems that confront the city engineer. "Engineering Work in Towns and Cities." By Ernest Mc- Cullough. While the author disclaims any intention of writing for city engineers of cities over 10,000 population, the limit should have been placed at 50,000 instead of 10,000. The book is a history of city surveying. With gloves off it deals with the qualifications necessary for the position of city engineer, the compensation he should receive, the problems he has to solve, the difficulties he has to meet, how to keep city records, the necessary theory and principles for the various duties of the position, including the location of monuments, roads, walks, pavements, sanitation, drainage, sewerage, water supply, con- crete, contracts and specifications, office system, city engineer's records and field work. It ranks as possibly the best book be- fore the public for the use of the city engineer and especially for that city engineer who wishes to learn the best methods. CHAPTER XI. PLOTTING AND LETTERING. 187. Plots. After a farm is surveyed a line map of the farm or land should be made to some convenient scale, for the purpose of showing the shape of the farm or body of land, its connections with adjoining properties, and its location with re- spect to natural objects. Such a plot should contain the follow- ing data : 1. Boundary lines. 2. Bearing and distance printed on each line. 3. All corners described, as "a hickory 1 ft. diam., marked H," "a stone." 4. Names of adjoining property owners. .">. Meridian, or north and south line. G. Owner's name printed inside plot. 7. Number of acres printed under owner's name. 8. Complete set of field notes printed below plot. Fig. 105 illustrates the plot, description and style of letters. There are various methods used in making a plot from the field notes. These are generally known as the protractor, the tangent, the sine, or the co-ordinate method. 188. Protractor Method. A protractor, Fig. 102, is a semicircle of horn, celluloid, German silver, etc., graduated to half degrees. A diameter line is marked at one end and at the other end 180. A bearing is laid off by placing the center of the protractor over the point and the diameter along the meridian and the protractor to the right or left of the meridian as indicated by the last letter of the bearing; that is, east for the right and west for the left. A point is made on the circum- ference of the protractor at the point of the correct bearing, the protractor is moved and this point joined by a line to the be- ginning line or point. The length of the course is then laid off on this line to the scale of the map. Through the point thus located another meridian is located and the bearing is laid off as before. 190 PLOTTING AND LETTERIXG. 191 189. Latitude and Departure Method. Begin at some point A as in Fig. 103, and lay off the latitude AB due north and south from A, and through the point thus located draw an east and west line and lay off the departure on this \ line, and join the point thus located to A. Lay off the lati- tude of next course on line through C, and through the point thus located draw an- other east and west line and lay off the departure on this line, thus locating the point D. Proceed as above until all the points are located. 190. The Tangent Method. To lay off a line making a given angle with a given line at a given point A, Fig. 103. by the tangent method, lay off AB equal to ten parts on some scale, and at B erect a per- dicular to the given line, and on this perpendicular lay off CB equal to ten times the nat- ural tangent of the angle de- sired; join C to A. Thus, to lay off an angle of 29 41', we find from the table that the natural tangent of '29 41' .5701). Make AB equal to ten parts and lay off CB equal to 5.7 parts, thus locating C ; then join C to A and you have the angle required. 191. The Sine Method. To lay off a given angle at a given point by the natural sine method, take a radius equal to 192 SURVEYOR'S HAND BOOK. ten parts and with the given point as a center describe a circle. On a perpendicular to the given line at the given point lay off A3, Fig. 104, equal to ten times the natural sine of the angle required.. Through 3 draw a line parallel to the given line cutting the cir- cumference of the circle at B, join B to A and BAN is the angle required. Example : To construct an angle of 33 22' we find that the natural sine of the angle 33 22' is .5500. After de- scribing the circle whose radius is ten parts, lay off A3 equal to 5.5 parts, and draw B3 parallel to the line AN and join B, where it cuts the circumference of the circle, to A, and BAN then will be an angle of 33 22'. 192. Co-ordinate Method. Plotting can be done by the Co-ordinate Method : Determine the co-ordinates of each point Fig. 103. Fig. 104. with respect to axes (through the initial point, if convenient) and plot from .the axes each time. This method will ?void carrying forward any error, as each corner of the survey is found by returning to the original axes. The Y ordinate of any point is equal to the sum (algebraic) of the latitude of the pre- vious points and its own latitude. The X ordinate is equal to the sum of the previous departures plus its own. Using this table of corrected latitudes and departures insures the closing of the plot. This is most accurate method for any large plot, as previous to plotting the sheet can be checked off in squares accurately, say 1,000 ft. on each side, and table of ordinates computed, etc. PLOTTING AND LETTERING. 193 Scale Beginning at a stone on Bull CreeK a corner to R A Jones and John Cusler Thence with Ouster's line S.4-I*E. IOO poles To a blacK oaK a corner To John CusTer ond D.F?. Thomas ,- Thence with Thomas' line SS9W f -41 poles To a hicKovy a corner To D.TR.ThornaS ;- thence with ThomasMine N.G9* W 99 poles To a gum on Bull CreeK o. corner To JD.K.Thomas onct TC.G ore;- Th ence up the creeK with the meanders of the Sanne to the point of beginning ;- containing A- S.9 acres. Fig. 105. 194 SURVEYOR'S HAND BOOK. 193. Correcting the Plot. For the very same reasons that the latitudes and departures very rarely balance, the plot when completed to scale will very rarely close by an amount equal to AA' , Fig. 106. In balancing we really shift each corner in the direction of AA' , a distance proportional to its length from the beginning corner. To some scale lay off on a straight line the length of the courses ABCDA', and on a line at right angles to this line lay off AA' and through the points B, C, and D draw parallels to AA'. Through B, C , and D on the plot draw lines parallel to A A' and on these lines lay off distances equal to the amount of cor- rection, locating the II points B' , C' and D' in the direction that A' has to be moved to close. Then connect these points and close the plot. 194. Lettering. Ev- ery surveyor or engi- neer should, learn some one system of free-hand letters, similar to that in Fig. 107, or some other standard system. Many conclude before trial that they can not letter well, or even make a decent letter. While there is no royal road to good lettering, it is possible for every surveyor or engineer, not afflicted with palsy or extreme nervous- ness, to learn and execute a good, plain system of letters. But it requires care and implicit obedience to rules. Eternal vigilance and constant practice are required till a system of letters is once learned. After an experience of over twenty years in teaching, it can be asserted that the special books on lettering are far supe- rior to the ordinary alphabets printed as an appendix to works on surveying. If the young engineer will get "Lettering," by S Fig. 106. make PLOTTING AND LETTERING. 195 C. W. Reinhardt, published by D. Van Nostrand Company, New York, and will follow instructions faithfully, he can, without doubt, become a good letterer. There is no necessity for fancy letters in a drawing, as neatness, legibility, and clearness are the fundamental requisites. One of the most effective systems of lettering is shown in Fig. 107. Guide lines should always be drawn before the lettering is commenced and the student should adhere strictly to rules. Bibliography. "Lettering." By Chas. W. Reinhardt. Pub- lished by D. Van Nostrand & Co. This book explains in a clear abcdefqhijklmnopqrstuvwxyz. ABCDEFOHIJKLMNOPOR5TU V WXYZ. / 23456 78910. CROSS SECTION SECTION Extended Lettering Ordinary Compressed Type. INTERSTATE BRIDbE. Spvrmul t %DiamJ'Rn a bcdefghijklmnopqrstuvwxyz. ABCDEFGHIJ KLMNOPQRSTUV WXYZ. I 23456789 10. Ordinary Lettering Compressed, NEW YORK CENTRAL Fig. 107. and concise manner the system of letters devised by the author and shows by concrete examples how each letter should be formed and how constructed. In addition to this a well selected set of examples of title, heading, and detail lettering is given "Mechanical Drawing." By F. E. Giesecke. Part I. Pub- lished by Eugene Dietzgen Company. This book has grown out of the necessities of the office and class room and gives an ex- cellent system of free-hand letters for detail work and full in- structions are given for the construction of each letter. This book meets all the demands that a learner of lettering can make. CHAPTER XII. GOVERNMENT SURVEYING. 195, Radii of Parallels. Government lands are bounded by meridians and parallels of latitude. If AB, Fig. 108, is pare of a parallel of latitude, its latitude is the arc BO or the angle BOQ, which we will call L. Let the radius of the earth he R and the radius of the parallel be r or BH. Then in the right triangle OBH, That is, Or r = R Cos L . 196. Angular Convergence of Meridians. The two merid- ians PA and PB, Fig. 108, at the points A and B have the direc- tion AK and BK, respectively, the tangents to these meridians. The amount of convergence is the angle that they lack of being- parallel; that is, the angle AKB or their angle of intersection. Let 6 = the difference of longitude of A and B = angle AHB = EOQ. In the sector AHB we have: AB = -==-z x BH *)< .o In AKB we ha^e : AB = 3 x BK, where X = AKB. Consequently : X 6 X BK = -TO X BH 57.3 f 7.3 BK X = e w. L ...... .,......,..,. (41) 196 GOVERNMENT SURVEYING. 197 197. Linear Convergence. In the two similar sectors ABH and EOQ we have : AB : EQ :: BH : OQ BH AB = EQ x 0g EQ Cos L Fig. 108, Fig. 109. If DC is a part of a parallel between the same meridians in latitude L' we have : DC = EQCosL' Let c = Convergence ~ AB DC EQCosL EQCosE' = EQ (Cos L Cos L') DO Cos L' AB ^UoTT: Therefore : AB(CosL CosL'.) c = AB-DC = z^j- ,.-'..;. (42) Generally we do not know the difference of longitude of A and B, but know the length of AB in miles, and it is nec- essary to find 9 from the data given. The length of one degree on the equator is 69,16 miles. 198 SURVEYOR'S HAND BOOK, If D = length of AB in miles, then D AB in degrees = 69 . 16 Cos L But X = 9 Sin L D Sin L Therefore A = H 69.16 Cos L X"= gg-jg D X tan L=52.05 D tan L 198. Off-Sets. If we set the transit at B, Fig. 109, and set the zero on the meridian and turn off a right angle from this meridian, this last line will cut to the left of A. Draw the sector AKB as in the figure and make the angle KBR equal to 90. The amount the line BR misses A is called the off-set The angle ABR :== one-half X. IfAB=D, we have: AR^Off-srt-^-fir^ XD But X = B Sin L = Q( j 16 tan L W D Therefore, AR = tan L x This is the off-set in miles. If D is in miles and we wish the off-set it feet, we have : D 2 Off - set = 69.16X57.3X2 ian L x fc80 Therefore off-set = 59 16x57 3x2 tan LxD * = .66618 tan L D* = f D 2 /aw L, w^arZ^ ............. (43) 199. Running Parallels. It is impossible to run out the parallel of latitude with the transit directly. We can locate the GOVERNMENT SURVEYING. 199 secant BA or the tangent BR, and then take off-sets to the curve of latitudes at different points, which are generally one-half mile apart. There are two methods of locating points on the parallel of latitude, the secant method and the tangent method. 200. Tangent Method. Set up the transit at B and sight along the meridian BK. Then turn off an angle K'BR equal to 90. The line of sight will now locate the line BR, which is tan- gent to the latitude curve. To obtain the off-sets from this tan- gent line to the curve at any point on BR, let d distance from the point to B. Then we have from formula (43) : Off-SCt =% r f r^T&tunL. If the offset is in feet 2 K Cot. L, 2K and d in miles we have, Offset TC=%d 2 tan. L. The secant-curve off-set can be found by subtracting the tangent- curve off-set from the secant-tangent off-set. '.SC=%Dd tan. L. % d' 2 tan.L = 2/3 d (D d) tan.L. The secant-curve off-set is equal to two-thirds of the tangent of the latitude multiplied by the segments into which S divides AB. 203. Example. If a line AB is six miles in length and is a parallel of latitude where L = 45 the different off-sets for each mile can be found as follows : A. Tangent-curve off -set = % d~ tan. L. = % d 2 tun 45 = % d~. E. Secant-tangent off-set = % Dd tan. L = %Dd tan 45 = % Dd. C. Secant-curve off -set = % d (D d} tan. 45 % d (D d). Off-sets. Distance d. Secant-tangent. Tangent-curve. Secant-curve. 1 4 .667 3.333 2 8 2.667 5.333 3 12 6.000 6.000 4 16 10.667 5.333 5 20 16.667 3.333 6 24 24.000 0.000 PROBLEM 75. Fill out a similar table when latitude = 36. 204. Reference Meridians and Standard Parallels. In those states where public lands were surveyed by government surveyors, meridians were located very accurately at certain in- G O VBRNMEN T S UR VE YIXG. 201 tervals and parallels of latitude were also accurately located at certain distances apart. As an example the two meridians BC and AD, Fig. Ill, called "reference meridians," were located 24 miles apart, and the "Standard Parallels," AB and CD, were also located 24 miles apart. This makes a spherical trapezoid whose sides are nearly 24 miles each. The six-mile points on these sides are marked and joined by meridians and parallels, thus dividing the area into smaller trapezoids, with sides 6 miles each way approximately. These trapezoids are called "Town- ships." x w z Y X Fig. 111. Fig. 112. The south base of a trapezoid is 24 miles on a standard parallel and the next standard parallel is 24 miles to the north. If the latitude of the south base is 40, find the amount of con- vergence. To find L' the latitude of the north base we have : One degree 69. 16 miles. 24 24 miles = ~ degrees 69.16 =4020' 49" 24 [Cos 40 Cos (40 20' 49")] c= convergence - Cos 40 24(. 76604 -.76549) .76604 --12219 miles PROBLEM 76. A trapezoid is 24 miles each way, and the lati- tude of the mid-parallel is 46 30'. Find the amount of converg- ence, 202 SURVEYOR'S HAND BOOK. PROBLEM 77. Find the convergence of a trapezoid with 6-mile base and 24 miles north and south, if the latitude of the south base is 36. 205. Ranges. In each State or Territory a principal meridian was located as BC, Fig. 112. It received a name clue to some locality, as the Fayetteville or Butte meridian. Also a principal parallel is located as AB. The country is then divided into townships on either side of these axes and they serve as co- ordinates in locating the townships. Thus in the figure all ranges are west and north. Any row of townships running 6 5 4 5 2 1 7 a 9 JO n 12 J& 17 It. 75 74 73 73 20 21 22 23 24 30 29 28 27 26 25 37 32 33 34 35 36 Fig. 113. north and south is called a Range, while that running east and west is called a Tier. Each township is defined as in Range 1, 2, 3, or 4, Tier 1, 2, 3, or 4, as the case may be, number- ing from the Principal Meridian and Principal Parallel. Thus the township crossed will be Range 3 west, Tier 4 north. 206. Townships. The trapezoid in Fig. 112, 24 miles each way, was surveyed between reference meridians and standard parallels. If the six-mile points on the north and south lines are marked the spherical trapezoid would be divided into ap- proximate squares six miles each way, called townships. GOVERNMENT SURVEYING. -jo:i Each township is divided into 30 approximate squares, about one mile on each side, called sections. The sections in each township are numbered as shown in Fig. 113. Section number 1 is in the northeast corner of the township, while number 30 is in the southeast corner. 207. Dividing Up a Township. All township lines on the south base, on the standard parallels, are full G miles, as are all township lines on the meridians. In the range of townships EBCF, Fig. 112, there would really be only one east and west line that was fully miles long, as all the others are reduced by the convergence. In dividing the first township X into sections we mark off full miles on the south base EB, 80 chains each, and also full miles on the north and south lines BC and EF. If we made the north and south division lines true meridians the sec- tions would decrease materially in size as we proceeded north. To counteract this and keep them approximately 1 mile each way we make the south base of each section bordering on the town- ship lines 80 chains as far as possible. On each east and west township line we commence at the meridian on the east side of the township and measure off 5 full miles, marking the corners ; thus out of 144 sections in a Range we would have 21 sections with a full mile for the south base instead of sections if it were divided by true meridians on the mile points of the standard parallel AB. The amount of convergence of the town- ships X, Y , Z } and W will be practically the same if they have equal south bases. On the outlines of the townships the corners are marked with stones or posts as indicated by the small circles in Fig. 113. On the township lines the full mile points are all established and marked by corners. In making the survey of sub-division, we begin on the south base of the township at cor- ner to sections 35 and 30, and then run the line between sections 35 and 36 so that it will be parallel to the east line of 30. In the same way all the north-south lines are run parallel to the east line of the township except for sections from 1 to inclusive. From the corner of 1, 2, 11, and 12 the line between 1 and 2 is run directly to the established corner on the north base of the township. The lines between sections 2 and 3, 3 and 4, 4 and 5, 204 SURVEYOR'S HAND BOOK. and 5 and 6 are run in a similar way. On all north-south lines five full miles are measured from the south base of township, set- ting a post or stone at the end of each full mile for a section corner. The east-west lines join the corners on the north-south lines. A random line is first run from the section corners to the eastward and if it does not hit the corner, the correction is Fig. 114. made and the true line run. The east-west lines of sections 31, 30, 19, 18, 7, and 6 receive practically all the effect of conver- gence of the township ; and, if these sections are divided into quarter sections, the shortage in length is thrown into the west halves. GOVERNMENT SURVEYING. 205 The township is subdivided, Fig. 114, as follows: Beginning at corner 1-2-35-36 on the south base, thence Nl'W between sections 35 and 36. Wire fence, bears E. and W. Scattering cottonwood bears east and west. F. G. Alexander's house bears N28W. Leave cottonwood timber bears east and west. Enter road bears north. Southeast corner Alexander's field. Thence along west side of road. Cross roads. Bears east to Mound City. Bears north to Link City. Quarter section corner point falls in the road. Enter dense cottonwood timber; bears N54E. Set locust post 4"x4" 2' in the ground for corner- sections 25, 26, 35, and 36. Thence S89 57'E on a random line between sections: 25 and 36. Set temporary quarter section corner post. Intersect east line of township 3 links north of corner of sections 25, 30, 31, and 36, which is a sandstone 5"x8" set 5" above the ground, marked and wit- nessed. Thence N8956'W on a true litre between sections 25 and 36 'Over level bottom land. Cherry Creek, 12 links wide, clear water, 1 ft. deep, gentle current, sandy bottom, course northwest. Heavy timber, bears north and south. Leave heavy timber bearing north and south. Deposit a quart of charcoal 12 ins. in the ground as a quarter section corner. Dig pits 18x18x12 ins. east and west 4 ft. and raised a mound of earth 3V& ft base by Wz ft. high over the deposit. Enter heavy timber bears north and south. Leave heavy timber, enter scattering timber bears N25E. Comer sections 25, 26, 35, and 36. 206 SURVEYOR'S HAND BOOK. Thence Nl'W between sections 25 and 26. 25.36 Right bank of Yellowstone river. Set locust post 4"x4" 24" in the ground for meander corner for sections 25 and 26, marked MC on north side. Entered shallow water 1 to 2 ft. deep. Across shallow channel 64 links wide to sand bar. 32.12 To right bank of main channel, course east. 40 Quarter section corner falls in the river. 49.46 Left bank of Yellowstone river, 12 ft. high, deposited a marked stone 12 ins. in the ground. 55.70 Wire fence bears east and west. 62.80 Telegraph line bears east and west. Set cedar post for corner sections 23, 24, 25, and 26. 40 79.98 39.99 58.00 79.98 Thence S8956'E on a random line. Set temporary quarter section corner. Intersect east line of township 3 links north of section corners 25, 24, 30, and 19; which is a sandstone 5x9 ins. 4 ins. above ground marked and witnessed. Thence back N8955'W on a true line between sec- tions 25 and 24. Set a cedar post 3 ft. by 3 ins. square with a marked stone 24 ins. in the ground for a quarter section corner. Short creek, 3 links wide. Cor. of sees. 23, 24, 25, and 26. The survey progresses in this way till we reach the corner of sections 1, 2, 11, and 12, when we continue as follows: Beginning at corner 1, 2, 11, 12. Thence Nl'W on a random line between sections 1 and 2. 40 Set temporary quarter section corner. 79.77 Intersect north line of township at corner of sections 1, 2, 35, and 36, which is a limestone 6"x6" 5' above ground, marked and witnessed. Thence Sl'E on a true line between sections 1 and 2 39.77 Set marked stone for quarter section corner. In the next Range of sections we begin at corner on south base 2, 3, 34, and 35, and proceed as before. In this case, after, the surveyor -has located the corner 2, 3, 10, 11 he runs a random line N. 2' W. between sections 2 and 3 and misses the corner ofi GOVERNMENT SURVEYING. 207 sections 2, 3, 34, and 35, five links to the west, and thence runs due south on a true line between sections 2 and 3. Bibliography. "A Manual of Land Surveying." By F. Hodgman. 374 pages. A very valuable book for the surveyor or field engineer in surveying the public lands. A unique and very important feature is a digest of the legal decisions ,by the different State and Federal courts in regard to U. S. Lands, sur- veys, conflicts, etc. "A Manual of Surveying Instructions." Prepared under di- rection of the Commissioner of the General Land Office of the United States, Washington, D. C. It contains full and minute directions for the execution of surveys in the field in conformity to the laws of the United States. CHAPTER XIII. TRIGONOMETRIC FORMULAS. 208. Formulas for Right Triangle. In the right triangle ABC, Fig. 115, where C is the right angle, and a, b, and c are the sides, we have the following expressions for the different trigonometric functions : Fig. 115. Fig. 116. . w sin A=; cos A=, tan A=-j-; esc A=-, sec A=r-j-; cot A=~; Also, sin A cos A tan A-- __ cot A esc A' sec The following relations are sometimes useful : sin* A + cos*A=l; 1 + tan z A = sec z A; 1 + cot* A = csc*A; 209. Solutions for Right Triangle. There are four gen- eral cases that can occur, according to the data given, which may be I. The hypotenuse and one leg ; II. The two legs; III. The hypotenuse and one of the acute angles ; IV. A leg and an acute angle. The data given, the data required, and the solutions are given in the following tabular statement. It is assumed that if angle B is known, A is also known. 208 TRIGONOMETRIC FORMULAS. 209 Given Required Solutions a, c b,A,B sinA=> b = c cos A ; B = 90 A. <*,b A,c,B tanA=y; c= --^p B = 90 A. c, A a, b, B a = c sin A; b=c cos A ; B = 90 A . f.-> ^-^ T ; b=~ B = W-A. 210. Oblique Triangle. In the general triangle ABC, Fig. 110, threq parts, one of which must be a side, have to be given to find the other three. There are four general cases according to the data given. Thus we may have : I. Two angles and the included side ; II. Two sides and the included angle; III. Three sides ; IV. Two sides and an angle opposite one of them. The given parts, the required parts, and the formulas for solution are given in the following table : Given Required. Formulas for Solutions A,C, b B ,c, a ^ 18 b, c, A B, C, a B + C=180A ; Ian %(B-C) = ~ b sin A = \ fiz^itf) \ 5 (5-a) C/t?cfe : ,4 " 6 s/;/ . 1 A B,C,c sin 210 SURVEYOR'S HAND BOOK. a sin C c=s ~wTA" Case IV is sometimes ambiguous. We may have the follow- ing conditions and results : If A is obtuse, and a>b there-is one solution; If A is acute and a = or > b, there is one solution; If A is acute and ab sin A, there are two solutions; If A is acute and a < b and a=b sin A , there is one solution ; If A is acute and a8 6599 6790 6981 7172 7363 7554 7744 191 8 7935 8125 8316 ssoe 8696 8886 9076 9266; 9456 9646 190 9 9835 360025 360215 360404360593 360783 360972 361161 361350 361539 189 230 361728 361917 362105 362294362482 363671 362859 36304P 3*53236 363424 188 1 3612 3800 3938 4176; 4363 4551 4739 4926 5113 6301 188 2 6488 5675 5862 6049 6236 6423 6&0 6796 6983 7169 187 3 7356 7542 7729 7915 8101 8287 8473 8659 8845 9030 186 4 9216 9401 9587 9772 9958 370143 370328 370513 370698 370883 186 6 371068 371253 371437 371622371806 1991 2175 2360 2544 2728 184 6 2912 3096 3280 3464 3647 3831 4015 4198 4382 4565 184 7 4748 4932 5115 5298 6481 6664 5846 6029 6212 6394 183 8 6577 6759 6942 7124 7306 7488 7670 7852 8034 6216 182 9 8398 8580 8761 8943 9124 9306 9487 9668 9849 380030 181 240 380211 380392 380573 380754 380934 381115 381296 381476 381656 381837 181 1 2017 2197 2377 2557 2737 2917 3097 3277 3456 3636 180 2 3815 3995 4174 4353 4533 4712 4891 6070 6249 5428 179 3 5606 5785 5964 6142 6321 6499 6677 6856 7034 7212 178 4 7390 7568 7746 7923 8101 8279 8456 8634 8811 8989 178 6 9166 9343 9520 9693 9875 390051 39022.5 390405 390582 390759 177 6 390935 391112 391288 391464 391641 1817 1993 2169 2345! 2521 176 7 2697 2873 3048 3224 3400 3575 3751 3926 4101 4277 176 6 4452 4627 4802 4977 6152 5326 6501 6676 6850 6025 176 9 6199 6374 6548 6722 6896 7071 7245 7419 7592 7766 174 260 397940 398114 398287 398461 398634 398808 398981 399154 399328 399501 173 1 9674 9847 400020 400192 400365 400538 400711 400883 401056 401228 173 2 401401 401573 1745 1917 2089 2261 2433 2605 2777 2949 172 3 3121 3292 3464 3635 3807 3978 4149 4320 4492 4663 171 4 4834 5005 6176 6346 6517 6688 5858 6029 6199 6370 171 6 6540 6710 6881 7051 7221 7391 7561 7731 7901 8070 170 6 8240 8410 8579 8749 8918 9087 9257 9426 9595 9764 169 7 9933 410102 410271 410440 410609 410777 410946 411114 411283 411451 169 8 411620 1788 1956 2124 2293 2461 2629 2796 2964 3132 168 9 3300 3467 3635 3S03 3970 4137 4305 4472 4639 4806 167 200 414973 415140 415307 415474 415641 415808 415974 416141 416308 4i6474 167 1 6641 6807 6973 7139 7306 7472 7638 7804 7970 8135 166 2 8301 8467 8633 8798 8964 9129 9295 9460 9625 9791 165 3 9956 420121 420286 420451 420616 420781 420945 421110 421275 421439 165 4 421604 1768 1933 2097 2261 2426 2590 2754! 2918 3082 164 5 3246 3410 3574 3737 3901 4065 4228 4392 4555 4718 164 6 4882 5045 5?08 5371 5534 5697 6860 6023 6186 6349 163 7 6511 6674 683o 6999 7161 7324 7486 7648 7811 7973 162 8 8135 8297 8459 86al 8783 8944 9106 9268 9429 9591 162' 9 9752 9914 430075 430236 430398 430559 43072C 430881 431042 431203 161 '^70 431364 431525 431685 431846 432007 132167 432328 432488 432649 432809 161 1 2969 3130 3290 3450 3610 3770 3930 4090 42494 4409 160 2 4569 4729 4888 5048 5207 5367 5526 5685 6845 6004 159 3 6163 6322 6481 6640 6799 6957 7116 7275 7433 7592 159 4 7751 7909 8067 8226 8384 8542 8701 8859 9017 9175 158 5 9333 9491 9648 9806 9964 440122 440279 440437 440594 440752 158 6 440909 441066 441224 441381 441538 1695 1852 2009 2166 2323 157 7 2480 2637 2793 2950 3106 3263 3419 3576 3732 3889 157 8 4045 4201 4357 4513 4669 4825 4981 6137 6293 6449 156 9 5604 5760 5915 6071 6226 6382 6537 6692 6848 7003 165 No. 1 2 3 A 5 6 T 8 9 Dltf. TABLE I. LOGARITHMS OF NUMBERS. 215 Ho O 1 3 3 4 5 6 7 8 9 Diff. 280 447158 447313 147468 447623 447778 47933 4-nss 48242 448397 48552 155 1 8706 8861 9015 9170 9324 9478 9633 9787 9941 50095 154 2 450219 4504(13 50557 45071 1 450SG5 51018 51172 51326 451479 1633 154 3 1786 1940 2093 2247 2400 2553 2706 2859 3012 3165 153 4 3318 3471 3624 3777 3930 4082 4235 4387 4540 4692 153 5 4845 4997 5150 5302 5454 5606 5758 6910 6062 6214 152 6 6366 6518 6670 6821 6973 7125 7276 7428 7579 7731 152 7 7882 8033 8184 8336 8487 8638 8789 8940 9091 9242 151 8 9392 9543 9694 9845 9995 60146 60296 60447 60597 60748 151 9 460898 461048 61198 461348 461499 1649 1799 1948 2098 2248 150 290 462398 462548 62697 462847 162997 63146 63296 63445 463594 463744 150 j 3893 4042 4191 1 4340 4490 4639 4788 4936 5085 5234 149 2 6383 5532 5680 5829 5977 6126 6274 6423 6571 6719 149 3 6868 7016 7164 7312 7460 7608 7756 7904 8052 8200 148 4 8347 8495 8643 8790 8938 9085 9233 9380 9527 9675 148 5 9S22 9969 70116 470263 470410 70557 470704 70851 470998 471145 147 6 471292 471438 1585 1732 1878 2025 2171 2318 2464 2610 146 7 2756 2903 3049 3195 3341 3487 3633 3779 3925 4071 146 8 4216 4362 4508 4653 4799 4944 5090 5235 5381 5526 146 9 6671 5816 5962 6107 6252 6397 6542 6687 6832 6976 145 300 477121 477266 77411 477555 477700 77844 477989 478133 478278 478422 145 1 8566 8711 8855 8999 9143 9287 9431 9575 9719 9863 144 2 480007 480151 80294 480438 480582 80725 480869 481012 481 156 481299 144 3 1443 1586 1729 1872 2016 2159 2302 2445 2588 2731 143 4 2874 3016 3159 3302 3445 3587 3730 3872 4015 4157 143 5 4300 4442 4585 4727 4869 6011 5153 5295 6437 6579 142 6 5721 5863 6005 6147 6289 6430 6572 6714 6855 6997 142 7 7138 7280 7421 7563 7704 7845 7986 8127 8269 8410 141 8 8551 8692 8833 8974 9114 9255 9396 9537 9677 9818 141 9 9958 490099 490239 490380 490520 490661 490801 490941 491081 491222 140 310 491362 491502 491642 491782 491922 492062 492201 492341 492481 492621 140 1 2760 2900 3040 3179 3319 3458 3597 3737 3876 4015 139 2 4155 4294 4433 4572 4711 4850 4989 5128 5267 5406 139 3 5544 5683 6822 5960 6099 6238 6376 6515 6653 6791 139 4 6930 7068 7206 7344 7483 7621 7759 7897 8035 8173 138 6 8311 8448 8586 8724 8862 8999 9137 9275 9412 9550 138 6 9687 9824 9962 500099 500236 500374 500511 500648 500785 500922 137 7 601059 501196 501333 1470 1607 1744 1880 2017 2154 2291 137 8 2427 2564 2700 2837 2973 3109 3246 3382 3518 3655 136 9 3791 3927 4063 4199 4335 4471 4607 4743 4678 6014 136 320 605150 505286 505421 505557 505693 505828 505964 506099 506234 506370 136 1 6505 6640 6776 6911 7046 7181 7316 7451 7586 7721 135 2 7856 7991 8126 8260 8395 8530 8664 8799 8934 9068 135 3 9203 9337 9471 9606 9740 9874 510009 510143 510277 510411 134 4 510545 510679 510813 510947 511081 511215 1349 1482 1616 1750 134 5 1883 2017 2151 2284 2418 2551 2684 2818 2951 3084 133 6 3218 3351 3484 3617 3750 3883 4016 4149 4282 4415 133 7 4548 46S1 4813 4946 5079 5211 5344 5476 5609 5741 133 e 5874 6006 6139 6271 6403 6535 6668 6800 6932 7064 132 9 7196 7328 7460 7592 7724 7855 7987 8119 8251 8382 132 330 & 8514 518646 518777 518909 51904( 519171 519303 519434 519566 519697 131 11 9828! 9959 520090 520221,520353 520484 520615 520745 520876 621007 131 2621138 521269 1400 1530 166 1792 1922 2053 2183 2314 131 a 2444 2575 2705 2835 2966 3096 3226 3356 3486 3616 130 4 3746 3876 4006 4136 4266 4396 4526 4656 4785 4915 130 I 5045 5174 5304 5434 5563 5693 5822 6951 608 6210 129 e 6339 6469 6598 6727 6856 6985 7114 7243 7372 7501 129 7 7630 7759 7888 8016 8145 8274 8402 8531 8660 8788 129 1 1 8917 9045 9174 9302 9430 9559 9687 9815 9943 530072 128 > 53020C 530328 530456 530584 530712 530840 530968 531096 531223 135 128 Na 1 J8 3 Wff. 216 TABLE I. LOGARITHMS OF NUMBERS, No. o 1 2 Dlff. 340 531479 531607 631734 531862 531990 532117 532245 532372 532500 632627 128 1 2754 2882 3009 3136 3264 3391 3518 3646 3772 3&99 127 2 4026 4153 4280 4407 4534 4661 47*7 4914 5041 5167 127 3 5294 5421 6547 6674 6800 5927 6053 6IM3 6306 6432 (26 4 6558 6685 6811 6937 7063 7189 7315 7441 7567 7693 126 6 7819 7945 8071 8197 8322 8448 8574 8699 8825 8951! 126 6 9076 9202 9327 9452 9578 9703 9829 9954 540079 '540204 126 7 540329 54W55 540580 C40705 640S30 640956 641080641206 1330! 1454 125 8 1579 1704 1829 1953 2078 2203 2327 2452 2576 2701 125 9 2825 2950 3074 3199 3323 3447 3571 3696 3820 3944 124 3ft: 544068 644192 644316 644440 644564 M46Q6 644812 644936 645060 645183 124 1 6307 5431 6555 6678 6802 6925 6049 6172 6296 6419 124 2 6543 DODO 6789 6913 7036 7159 72.V2 7405 7529 7652 123 3 7775 7898 8021 6144 8267 8389 651-2 S635 87f8 8881 123 4 9003 9126 9249 9371 9494 9616 9739 9S6I 9984 560106 123 6 660228 650351 650473 660595 650717 660840 650962 651084 651206 1328| 122 6 1450 1572 1694 1816 1938 WOO 2181 2303 2425 2547 122 7 2668 2790 2911 3033 3155 3276 3398 3519 3640 3762 121 8 3883 4004 4126 4247 4368 4489 4610 4731 4852 4973 121 9 6094 6215 6336 6457 6678 6699 6820 6940 6061 6182 121 360 666303 656423 666544 666664 6667S5 566905 657026 667146 657267 657387 120 7607 7627 7748 76631 7988 8108 8228 63491 6469 8589 120 2 8709 8S29 837 6006-46 600755 600-64 109 9 600973 601 0& 1191 1299 1408 1517 1625 1734, 1843, 1961; 109 No 1 9 3456 7 1 8 9 Diff. TABLE I, LOGARITHMS OF NUMBERS. 217 No.| 1 3 3 | 4 5 | 6 7 | 9 Diff 400602060 602169 602277 '602336 '6024 94 602603 602711 602819602928 603036 108 1 3144 3253 3361 34691 3577 3686 3794 3902 4010 4118 108 21 4'226 4334 4442 4550 4658 4766 4874 4932 5039 5197 108 3> 53(15 5413 5521 : 5628 5736 5844 5951 6059 6166 6274 103 4 6331; 6439 6596 6704 .6811 6919 7026 7133 7241 7348 107 6! 7455 7562 7669 7777 7884 7991 8098 8205 8312 8419 107 6 8526 SG33 8740 8847 8954 9061 9167 9274 9331 9488 107 7 9594! 9701 9808 9914 610021 610128 610234 610341 610447 610554 107 8 610660:610767 610.373 610979 1036 1192 1298 1405 1511 1617 106 9 1723 1829 1936 2042 2148 2254 2360 2466 2572 2678 106 410 612784 612390 612996 613102 613207 613313 613419 613525 613630 613736 106 1 3842 3947 40531 4159 4264 4370 4475 4581 4686 4792 106 2 4897 5003 5108 5213 5319 5424 5529 5634 5740 5845 105 3 5950 6055 6160 6265 6370 6476 6581 6636 6790 6895 105 4 7000 7105 7210 7315 7420 7525 7629 7734 7839 7943 105 5 8048 8153 8257 8362 8466 8571 8676 8780 8884 8989 105 6 9093 9198 9302 3406 9511 9615 9719 9824 9928 620032 104 7 620136 620240 620344 620443 620552 620656 620760 620864 620968 1072 104 8 11761 1280 1384 1488 1592 1695 1799 1903 2007 2110 104 9 2214 2318 2421 2525 2623 2732 2835 2939 3042 3146 104 420 623249 623353 623456 623559 623663 623766 623869 623973 624076 624179 103 1 4232 4335 4438 4591 4695 4798 4901 5004 5107 5210 103 2 5312 5415 5518 5621 5724 5827 5929 6032 6135 6238 103 3 6340 6443 6516 6643 6751 6853 6956 7058 7161 7263 103 4 7366 7463 7571 7673 7775 7878 7980 8082 8185 8287 102 5 8339 8491 8593 8695 8797 8900 9002 9104 9206 9308 102 6 9410 9512 9613 9715 9817 9919 630021 630123 630224 630326 102 7 630423 630530 630631:630733 630835 630936 1038 1139 1241 1342 102 8 1444 1545 1647 1743 1849 1951 2052 2153 2255 2356 101 9 2457 2559 2660 2761 2862 2963 3064 3165 3266 3367 101 430 63346S 633569 633670 633771 633872 633973 634074 634175 634276 634376 101 1 4477 4578 4679 4779 4880 4981 5081 5182 5283 5383 101 2 5484 5584 5635 5785 5886 5986 6087 6187 6287 6388 100 8 6483 6533 6638 6789 6889 6989 7089 7189 7290 7390 100 4 7490 7590 7690 7790 7890 7990 8090 8190 8290 8389 100 5 8439 8539 6639 8789 8888 8988 9038 9188 9287 9387 100 6| 9436 9586 9686 9785 9885 9984 640084 640183 640283 640382 99 7 1 640431 640581 640680 640779 640379 640978 1077 1177 1276 1375 99 8 1474 1573 1672 1771 1871 1970 2069 2168 2267 2366 99 9 2465 2563 2662 2761 2860 2959 3058 3156 3255 3354 99 440643453 643551 643650 643749 643347 643946 644044 644143 644242 644340 98 I 4439 4537 4636 4734 4832 4931 5029 5127 5226 5324 98 2 5422 5521 5619 5717 5815 5913 6011 6110 6208 6306 98 3 6404 6502 6600 6698 6796 6894 6992 7089 7187 7285 98 4 1 7333 7481 7579 7676 7774 7872 7969 8067 8165? 8262 98 6 8360 8458 8555 8653 8750 8848 8945 9043 9140 9237 97 6 9335 9432 9530 9627 9724 9821 9919 650016 650113 650210 97 7 650303 650405 650r>02 630599 650696 650793 650890 0987 1084 1181 97 8 1278 1375 1472 1569 1666 1762 1859 1956 2053 2150 97 9 2246 2343 2440 2536 2633 2730 2826 2923 3019 3116 97 460 653213 653309 653405 653502 653598 653695 653791 653888 653984 654080 96 1 4177 4273 4369 44651 4562 4658 4754 4850 4946 5042 96 2 5133 5235 5331: 5427 5523 5619 5715 5810 5906 6002 96 3 6093| 6194 6290 1 6336 6482 6577 6673 6769 6864 6960 96 4 7056 7152 72471 7343 7438 7534 7629 7725 7820 7916 96 5 8011 8107 8202 8293 8393 8488 8584 8679 8774 8870 95 6 8965 9060 9155 9250 9346 9441 9536 9631 9726 9821 95 7 9916 66001 1 660106 660201 660296 660391 660486 660581 660676 660771 95 8 660865 0960 1055 1150 1245 1339 1434 1529 1623 1718 95 _9 1813 1907 2002 2096 2191 j 2236 2380 2475 2569 2663 95 a. ~O 1 a 3 Dlff 218 TABLE 1. LOGARITHMS OF NUMBERS. No. 1 3 9 IAS. 460 562758 562852 662947 63041 G63135 663230 663324:663418 663512 663607 94 1 3701 3795 3889 3983 4078 4172 4266 4360 4454 4548 94 2 4642 4736 4830 4924 5018 5112 5206 5299 5393 5487 94 3 5581 5675 6769 6862 5956 6050 6143 6237 6331 6424 94 4 6518 6612 6705 6799 6892 6986 7079 7173 7266 7360 M 5 7453 7546 7640 7733 7826 7920 8013 8106 8199 8293 93 6 8386 8479 8572 8665 8759 8852 8945 9038 9131 9224 93 7 9317 9410 9503 9f96 9689 9782 9875 9967 670060 670153 93 8 570246 70339 670431 670524 670617 670710 670802 670895 0988 1080 93 9 1173 1265 1358 1451 1543 1636 1728 1821 1913 2005 93 470 672098 672190 672283 672375 672467 672560 672652 672744 672836 672929 92 1 3021 3113 3205 3297 3390 3482 3574 3666 3758 3850 92 2 3942 4034 4126 4218 4310 4402 4494 4586 4677 4769 92 3 4861 4953 5045 6137 .5228 5320 5412 6503 5595 '5687 92 4 5778 5870 5962 6053 6145 6236 6328 6419 6511 6602 92 5 6694 6785 6876 6968 7059 7151 7242 7333 7424 7516 91 6 7607 7698 7789 7881 7972 8063 8154 8245 8336 8427 91 7 8518 8609 8700 8791 8882 8973 9064 9155 9246 9337 91 8 9428 9519 9610 9700 9791 9882 9973 680063 680154 680245 91 9 680336 680426 680517 680607 680698 680789 680879 0970 1060 1151 91 480 681241 681332 681422 681513 681603 681693 681784 681874 681964 682055 90 1 2145 2235 2326 2416 2506 2596 2686 2777 2867 2957 90 2 3047 3137 3227 3317 3407 3497 3587 3677 3767 3857 90 3 3947 4037 4127 4217 4307 4396 4486 4576 4666 4756 90 4 4845 4935 5026 6114 5204 5294 6383 6473 6563 5652 90 6 5742 6831 5921 6010 6100 6189 6279 6368 6458 6547 89 6 6636 6726 6815 6904 6994 7083 7172 7261 7351 7440 89 7 7529 7618 7707 7796 7886 7975 8064 8153 8242 8331 89 8 8420 8509 8598 8687 8776 8865 8953 9042 9131 9220 89 9 9309 9398 9486 9576 9664 9753 9841 99301690019 690107 89 490 890196 690285 690373 690462 690550 690639 690728 690816 690905 690993 69 I 1081 1170 1258 1347 1435 1524 1612 1700 1789 1877 68 2 1965 2053 2142 2230 2318 2406 2494 2583 2671 2759 88 3 2847 2935 3023 3111 3199 3287 3375 3463 3551 3639 88 4 3727 3815 3903 3991 4078 4166 4254 4342 4430 4517 88 6 4605 4693 4781 4868 4956 5044 5131 5219 5307 6394 88 6 5482 5569 5657 5744 5832 6919 6007 6094 6182 6269 87 7 6356 6444 6531 6618 6706 6793 6880 6968 7055 7142 87 8 7229 7317 7404 7491 7578 7665 7752 7839 7926 8014 87 9 8101 8188 8275 8362 8449 8536 8622 8709 8796 8883 87 500 698970 699057 699144 699231 699317 699404 699491 699578 699664 699751 87 1 9838 9924 700011 700098 700184 700271 700358 700444 700531 700617 87 2 700704 700790 0877 0963 1050 1130 1222 1309 1395 1482 86 j 1568 1654 i741 1827 1913 1999 2086 2172 2258 2344 86 4 2431 2517 2603 2689 2775 2861 2947 3033 3119 3205 86 5 3291 3377 3463 3549 3635 3721 3807 3893 3979 4065 86 6 4151 4236 4322 4408 4494 4579 4665 4751 4837 4922 86 7 5008 5094 6179 5265 6350 5436 6522 5607 5693 6778 86 8 5864 5949 6035 6120 6206 6291 6376 6462 6547 6632 85 9 6718 6803 6888 6974 7059 7144 7229 7315 7400 7485 85 510 707570 707655 707740 707826 707911 707996 708081 708166 708251 708336 85 ] 8421 8506 8591 8676 8761 8846 8931 9015 9100 9186 85 ,< 9270 9355 9440 9524 9609 9694 9779 9863 9948 710033 85 | 710117 710202 710287 710371 710456 710540 710625 710710 710794 0879 85 I 0963 1048 1132 1217 1301 1385 1470 1554 1639 1723 84 5 1807 1892 1976 2060 2144 2229 2313 2397 2481 2566 S4 6 2650 2734 2818 2902 2986 3070 3154 3238 3323 3407 84 j 3491 3575 3659 3742 3826 3910 3994 4078 4162 4246 84 8 4330 4414 4497 4581 4665 4749 4833 4916 5000 5084 84 9 5167 5251 53351 5418 5502 5536 5669 5753 5836 5920 84 Ho 1 334 6 7 6 9 JMff. TABLE I. LOGARITHMS OF NUMBERS. 219 No.l 1 3 i 3 4 ,5 6 7 8 9 Difl. 620 716003 16087 716170 716254 716337 16421 16504 16588 716671 716754 ~K 1 6838 6921 7004 7088 7171 7254 7338 7421 7504 7587 83 2 7671 7754 7837 7920 8003 8086 8169 8253 8336 8419 83 3 8502 8585 8668 8751 8834 8917 9000 9083 9165 9248 83 4 9331 9414 9497 9580 9663 9745 9328 9911 9994 720077 83 5 720159 720242 720325 720407 720490 20573 20655 720733 720321 0903 83 6 0986 1068 1151 1233 1316 1398 1481 1563 1646 1728 82 7 1811 1893 1975 2058 2140 2222 2305 2387 2469 2552 82 8 2634 2716 2798 2881 2963 3045 3127 3209 3291 3374 82 9 3456 3533 3620 3702 3784 3866 3948 4030 4112 4194 82 630 724276 724358 724440 724522 724604 24685 724767 24849 ~24931 725013 82 1 5095 5176 5258 5340 5422 5503 5585 5667 5748 5830 82 2 5912 5993 6075 6156 6233 6320 6401 6483 6564 6646 82 3 6727 6809 6890 6972 7053 7134 7216 7297 7379 7460 81 4 7541 7623 7704 7785 7866 7948 8029 8110 8191 8273 81 6 8354 8435 8516 8597 8678 8759 8841 8922 9003 9084 81 6 9165 9246 9327 9408 9489 9570 9651 9732 9813 9893 81 7 9974 730055 730136 730217 730298 730378 730459 ~30540 730621 730702 81 8 730782 0363 0944 1024 1105 1186 1266 1347 1428 1508 81 9 1589 1669 1750 1830 1911 1991 2072 2152 2233 2313 81 640 732394 732474 732555 732635 732715 732796 732876 732956 733037 733117 80 1 3197 3278 3358 3438 3518 3598 3679 3759 3839 3919 80 2 3999 4079 4160 4240 4320 4400 4480 4560 4640 4720 80 3 4800 4880 4960 5040 6120 5200 5279 6359 5439 5519 80 4 5599 5679 6759 5838 5918 5998 6078 6157 6237 6317 80 6 6397 6476 6556 6635 6715 6795 6874 6954 7034 7113 80 6 7193 7272 7352 7431 7511 7590 7670 7749 7829 7908 79 7 7987 8067 8146 8225 8305 8384 8463 8543 8622 8701 79 8 8781 8860 8939 9018 9097 9177 9256 9335 9414 9493 79 9 9572 9651 9731 9310 9889 9968 740047 740126 740205 740284 79 550 740363 740442 710521 740600 740678 740757 740836 740915 740994 741073 79 1 1152 1230 1309 1388 1467 1546 1624 1703 1782 1860 79 2 1939 2018 2096 2175 2254 2332 2411 2489 2568 2647 79 3 2725 2804' 2332 2961 3039 3118 3196 3275 3353 3431 78 4 3510 3533 3667 3745 3823 3902 3930 4058 4136 4215 78 6 4293 4371- 4449 4528 4606 4684 4762 4840 4919 4997 78 6 5075 6153 5231 5309 5337 5465 5543 5621 5699 5777 78 7 5855 5933 6011 6089 6167 6245 6323 6401 6479 6556 78 8 6634 6712 '6790 6363 6945 7023 7101 7179 7256 7334 78 9 7412 7489 7567 7645 7722 7800 7878 7955 8033 8110 78 r>60 748183 748266 748343 748421 748498 748576 748653 748731 748808 748885 77 11 8963 9:^4 306451 806519 806587 06655 306723 806790 68 1 6858 69 6 6994 7061 7129 7197 7264 7332 7400 7467 68 2 7535 7603 7670 7738 7806 7873 794; 8008 8076 8143 68 3 8211 8279 8346 8414 8481 8549 861 ; 8684 8751 8818 67 4 8886 8953 9021 9088 9156 9223 9290 9358 9425 9492 67 5' 9560 9827 9694 97621 9829 9896 9964 810031 810098 810165 67 6;810233;810300 10367 810434 810501 810569 810636 0703 0770 0837 67 7 0904 0971 1039 1106 1173 1240 1307 1374 1441 1508 67 . 8 1575 1642 1709 1776 1843 1910 1977 2044 2111 2178 67 9 2245 2312 2379 2145 2512 2579 2646 2713 2780 2847 67 650 812913 812980 313047 813114 813181 813247 813314 813381 13448 813514 67 1 3581 3648 3714 3781 3848 3914 3981 4048 4114 4181 67 2 4248 4314 4381 4447 4514 4581 4647 4714 4780 4847 67 3 4913 4980 5046 5113 5179 5246 5312 5378 5445 5511 66 4 5578 5(544 5711 5777 5843 5910 5976 6042 6109 6175 66 5 6241 6:308 6374 6440 6506 6573 6639 6705 6771 6838 66 6 6904 6970 7036 7102 7169 7235 7301 7367 7433 7499 66 71 7565 7631 7698 ' 7764 7830 7896 7962 8028 8094 8160 66 8 8226 82' 12 8358 84241 8490 85-56 8622 8638 8754 8820 66 9 8885 8951 9017 9083 9149 92i5 9281 9346 9412 9478 66 660' 8 19544 819610 819676819741 819807 819873 819939 820004 820070 820136 66 1 8202(11 2 0858 820267 0924 820333820399 0939' 1055 820464 1120 320530 1186 820595 1251 0661 1317 0727 1382 0792 1448 66 66 3 1514 1579 1645 1710 1775 1841 1906 1972 2037 2103 65 4 2168 2233 2299 2364 2430 2495 2560 2626 2691 2756 65 5 2822 28S7 2952 3018 3((83 3148 3213 3279 3344 3409 65 6 3474 3539 3605 3670 3735 3800 3865 3930 3996 4061 65 7 4126 4191 4256 4321 4386 4451 4516 4581 4646 4711 65 8 4776 4841 4906 4971 5036 5101 5166 5231 5296 5361 65 9 5426 5491 5556 5621 5686 5751 5815 5880 5945 6010 66 670 826075 826140 326204 826269 326334 326399 326-16-1 826528 826593 826658 66 1 6723 6787 6852 6917 6981 7046 7111 7175 7240 7305 65 2 7369 7434 7499 7563 7628 7692 7757 7821 7886 7951 65 3 8015 8080 8144 8209 8273 8338 8402 8467 8531 8595 64 4 8660 8724 8789 8853 8918 8982 9046 9111 9175 9239 6-1 5 9304 9368 9432 9497 9561 9625 9690 9754 9618 9882 64 6 9947 830011 330075 830139 330204 830268 830332 330396 830460 830525 64 7 830589 0653 0717 0781 0845 0909 0973 1037 1102 1166 64 8 1230 1294 1358 1422 1486 1550 1614 1678 1742 1806 64 9 1870 1934 1998 2062 2126 2189 2253 2317 2381 2445 64 680 832509 832573 832637 832700 832764 832328 832892 832956 833020 833083 64 1 3147 3211 3275 3338 3402 3466 3530 3593 3657 3721 64 2 3784 3848 3912 3975 4039 4103 4166 4230 4294 4357 64 3 4421 4484 4548 4611 4675 4739 4802 4866 4929 4993 64 4 5056 5120 5183 5247 5310 5373 5437 5500 5564 5627 63 5 5691 5754 5817 5881 5944 6007 6071 6134 6197 6261 63 6 6324 6387 6451 6514 6577 6641 6704 6767 6830 6894 63 7 6957 7020 7083 7146 7210 7273 7336 7399 7462 7525 63 8 7588 7652 7715 7778 7841 7904 7967 8030 8093 8156 63 9 8219 8282 8345 8408 8471 8534 8597 8660 8723 8786 63 690 838849 833912 833975 839038 839101 839164 839227 839289 839352 839415 63 l| 9478 9541 9604! 9667 9729 9792 9855 9918 9981 840043 63 21840106 840169 840232840294 840357 840420 840482 840545 840608 0671 63 3 0733 0796 0859 0921 0984 1046 1109 1172 1234 1297 63 4 135S 1422 1485' 1547 161C 1672 1735 1797 1860 1922 63 e 193 2047 2110; 2172 223S 2297 2360 2422 2484 2547 62 6 2602 2672 2734 2796 285S 2921 2983 3046 3108 3170 62 7 3235 3295 3357 3420 3482 3544 3606 3669 373 3793 62 385 3918 398C 4042 4104 4166 4229 4291 4353 4415 62 8 4475 4539 4601 4664 4726 478S 4850 4912 4974 5036 62 Ho. 1 a 3 4 5 6 7 8 e Dtff. 222 TABLE I. LOGARITHMS OF NUMBERS. Mo. O 1 2 3 4 5 6 7 8 i 9 Dlff. 700 345098 345160 845222 845284 845346 845408 345470 845532 845594:845656 ~62 1 5718 5780 5842 5904 5966 6028 6090 6151 6213 6275 62 2 6337 6399 6461 6523 6585 6646 6708 6770 6832 6894 62 3 6955 7017 7079 7141 7202 7264 7326 7388 7449 7511 62 4 7573 7634 7696 7758 7819 7881 7943 8004 8066 812& 62 5 8189 8251 8312 8374 8435 8497 8559 8620 8682 8743 62 6 8805 8866 8928 8989 9051 9112 9174 9235 9297 9358 61 7 0419 9481 9542 9604 9665 9726 ~9788 9849 9911 9972 61 8 850033 350095 850156 850217 850279 85034(1 850401 850462 850524 850585 61 9 0646 0707 0769 0830 0891 0952 1014 1075 1136 1197 61 710 851258 851320 851381 851442 851503 851564 851625 851686 851747 851809 61 . 1 1870 1931 1992 2053 2114 2175 2236 2297 2358 2419 61 2 2480 2541 2602 2663 2724 2785 2846 2907 2968 30*9 61 3 3090 3150 3211 3272 3333 3394 3455 3516 3577 3637 61 4 3698 3759 3820 3881 3941 4002 ^063 4124 4185 4245 61 6 4306 4367 4428 4488 4549 4610 4670 4731 4792 4852 61 6 4913 4974 5034 5095 5156 5216 5277 5337 5398 5459 61 7 5519 5580 5640 5701 5761 5822 5882 5943 6003 6064 61 8 6124 6185 6245 6306 6366 6427 6487 6548 6608 6668 60 9 6729 6789 6850 6910 6970 7031 7091 7152 7212 7272 60 720 857332 857393 857453 857513 857574 857634 857694 857755 857815 857875 60 7935 7995 8056 8116 8176 8236 8297 8357 8417 8477 60 2 8537 8597 8657 8718 8778 8838 8898 8958 9018 9078 60 3 9138 9198 9258 9318 9379 9439 9499 9559 9619 9679 60 4 9739 9799 9859 9918 9978 860038 860098 860158 860218 860278 60 6 860338 860398 860458 860518 860578 0637 0697 0757 0817 0877 60 6 0937 0996 1056 1116 1176 1236 1295 1355 1415 1475 60 7 1534 1594 1654 1714 1773 1833 1893 1952 2012 2072 60 8 2131 2191 2251 2310 2370 2430 2489 2549 2608 2668 60 9 2728 2787 2847 2906 2966 3025 3085 3144 3204 3263 60 730 863323 863382 863442 863501 863561 863620 863680 863739 863799 863858 69 3917 3977 4036 40% 4155 4214 4274 4333 4392 4452 59 J 4511 4570 4630 4689 4748 4808 4867 4926 4985 5045 69 5104 5163 5222 52S2 5341 5400 5459 5519 5578 5637 59 4 5696 5755 5814 5874 5933 5992 6051 6110 6169 6228 69 6 6287 6346 6405 6465 6524 6583 6642 6701 6760 6819 69 6 6878 6937 6996 7055 7114 7173 7232 7291 7350 7409 69 7467 7526 7585 7644 7703 7762 7821 7880 7939 7998 69 8 8056 8115 8174 8233 8292 8350 8409 8468 8527 8586 69 9 8644 8703 8762 8821 8879 8938 8997 9056 9114 9173 59 740 869232 869290 869349 869408 869466 869525 869584 869642 869701 869760 59 9818 98771 9935 9994 870053 870111 870170 870228 870287 870345 69 I 870404 870462870521 870579 0638 0696 0755 0813 0872 0930 68 3 0989 1047 1106 1164 1223 1281 1339 1398 1456 1515 58 4 1573 1631 1690 1748 1806 1865 1923 1981 2040 2098 58 5 2156 22i6 2273 2331 2389 2448 2506 2564 2622 2681 68 6 2739 2797 2855 2913 2972 3030 3088 3146 3204 3262 68 ; 3321 3379 3437 3495 3553 3611 3669 3727 3785 3844 68 i 3902 39601 4018 4076 4134 4192 4250 4308 4366 4424 68 I 4482 4540 4598 4656 4714 4772 4830 4888 4945 5003 68 760 875061 875119 875177 875235 875293 875351 875409 875466 875521 875582 ts 5640 5698 5756 5813 5871 5929 5987 6045 6102 6160 58 6218 6276 6333 6391 6449 6507 6564 6622 6680 6737 68 ] 6795 6853 6910 6968 7026 7083 7141 7199 7256 7314 68 , 7371 7429 7487 7544 7602 7659 7717 7774 7832 7889 68 , 7947 8004 8062 8119 8177 8234 8292 8349 .8407 8464 57 i 8522 8579 8637 8694 8752 8809 8866 8924 8981 9039 57 9096 9153 9211 9268 9325 9383 9440 9497 9555 9612 57 9669 9726 9784 9841 9898 9956 880013 880070 880127 880185 67 880242 880299 880356 880413 880471 880528 0585 0642 0699 0756 67 No O i i a 3 4 6 6 7 8 9 DIft TABLE I. LOGARITHMS OF NUMBERS. 223 No. 1 3 4 5 6 7 8 9 Dlff. 760 880814 880871 880928 880985 881042 881099 881156 881213 881271 881328 ~67 1 1385 1442 1499 1556 1613 1670 1727 1784 1841 1898 57 2 1955 2012 2069 2126 2183 2240 2297 2354 2411 2468 57 3 2525 2581 2638 2695 2752 2809 2866 2923 2980 3037 57 4 3093 3150 3207 3264 3321 3377 3434 3491 3548 3605 57 6 3661 3718 3775 3832 3888 3945 4002 4059 4115 4172 57 6 4229 4285 4342 4399 4455 4512 4569 4625 4682 4739 57 1 7 4795 4852 4909 4965 5022 5078 5135 5192 5248 5305 57 l 8 5361 5418 5474 5531 5587 5644 5700 5757 5813 5870 57 9 5926 5983 6039 6096 6152 6209 6265 6321 6378 6434 56 770 866491 886547 886604 886660 886716 886773 886829 886885 886942 886998 66 1 7054 7111 7167 7223 7280 7336 73921 7449 7505 7561 56 2 7617 7674 7730 7786 7842 7898 7955 8011 8067 8123 56 3 8179 8236 8292 8343 8404 8460 8516 8573 8629 86S5 56 4 8741 8797 8853 8909 8965 9021 9077 9134 9190 9246 66 5 9302 9358 9414 9470 9526 9582 9638 9694 9750 9806 56 6 9862 9918 9974 890030 890086 890141 890197 890253 890309 890365 56 7 890421 890477 890533 0589 0645 0700 0756 0812 0868 0924 56 8 0980 1035 1091 1147 1203 1259 1314 1370 1426 1482 56 9 1537 1593 1649 1705 1760 1816 1872 1928 1983 2039 56 780 892095 892150 892206 892262 892317 892373 892429 892484 892540 892595 56 1 2651 2707 2762 2818 2873 2929 2985 3040 3096 3151 56 2 3207 3262 3318 3373 3429 3484 3540 3595 3651 3706 56 3 3762 3817 3873 3928 3984 4039 4094 4150 4205 4261 55 4 4316 4371 4427 4482 4538 4593 4648 4704 4759 4814 55 5 4870 4925 4980 5036 5091 5146 5201 5257 5312 5367 65 6 5423 5478 5533 5588 5644 5699 5754 5809 5864 5920 55 7 5975 6030 6085 6140 6195 6251 6306 6361 6416 6471 55 8 6526 6581 6636 6692 6747 6802 6857 6912 6967 7022 55 9 7077 7132 7187 7242 7297 7352 7407 7462 7517 7572 56 790 897627 897682 897737 897792 897847 897902 897957 898012 898067 898122 55 1 8176 8231 8286 8341 8396 M51 8506 8561 8615 8670 55 2 8725 8780 8835 8890 8944 8999 9054 9109 9164 9218 55 3 9273 9323 9383 9437 9492 9547 9602 9656 9711 9766 65 4 9821 9875 9930 9985 900039 900094 900149 900203 900258 900312 55 5 900X7 900422 900476 900531 0586 0640 0695 0749 0804 0859 55 6 0913 0968 1022 1077 1131 1186 1240 1295 1349 1404 55 7 1458 1513 1567 1622 1676 1731 1785 1840 1894 1948 54 8 2003 2057 2112 2166 222! 2275 2329 2384 2438 2492 54 9 2547 2601 2655 2710 2764 2818 2873 2927 2981 3036 54 800 903090 903144 903199 903253 903307 903361 903416 903470 903524 903578 54 1 3633 3687 3741 3795 3849 3904 3958 4012 4066 4120 54 2 4174 4229 4283 4337 4391 4445 4499 4553 4607 4661 54 3 4716 4770 4824 4878 4932 4986 5040 5094 5148 5202 54 4 5256 5310 5364 5418 5472 5526 5580 5634 5688 5742 54 5 5796 5850 5904 5958 6012 6066 6119 6173 6227 6281 54 6 6335 63S9 6443 6497 6551 6604 6658 6712 6766 6820 54 7 6874 6917 6981 7035 7039 7143 7196 7250 7304 7358 54 8 7411 7465 7519 7573 7626 7680 7734 7787 7841 7895 54 9 7949 8002 8056 8110 8163 8217 8270 8324 8378 8431 54 810 908485 908539 908592 908646 908699 908753 908807 908860 908914 908967 54 1 9021 9074 9128 9181 9235 9289 9342 9396 9449 9503 54 2 9556 9610 9663 9716 9770 9823 9877 9930 9984 910037 53 3 910091 910144 910197 910251 910304 910358 910411 910464 910518 0571 53 4 0624 0678 0731 0784 0838 0891 0944 0998 1051 1104 53 5 1158 1211 1264 1317 1371 1424 1477 1530 1584 1637 53 6 1690 1743 1797 1850 1903 1956 2009 2063 2116 2169 53 7 2222 2275 2328 2381 2435 2488 2541 2594 2647 2700 63 8 2753 2806 2859 2913 2966 3019 3072 3125 3178 3231 63 9 3284 3337 3390 3443 3496 3549 3602 3655 3708 3761 53 No. 1 3 3 Dltf. 224 TABLE I. LOGARITHMS OF NUMIUIRS. 1 913814 4343 913367 43% 913920 4449 913973 45(12 914026 4555 914079 4608 914132 4660 914I84J914237 4713 4766 914290 63 4819 53 2 4872 4925 4977 6030 6083 5136 5i-'j 5241 5294 5347 63 3 5400 5453 6505 6558 6611 5664 6716 6769 f-22 5875 53 4 6927 5980 6033 6085 6138 6191 6243 62% 6349 6401 63! 6 6454 6507 6559 6612 6664 6717 6770 6si2. 6-75 6927 53 6 6930 7033 7085 713S 7190 7'243 7295 7348 7400 7453 63! 7 7506 7658 7611 7663 7716 7768 7820 7873 7925 797SI '62 8 8030 8083 8135 8188 8240 8293 8345 8397 8450 8502; 62 9 8565 8607 8659 8712 8764 8816 8ti69 8921 8973 902C 68 j 830 919078 919130 919183 919235 919287 919340 919392 919444 9194% 91 9549' 68 1 9601 %53 9706 9758 9810 0603 9914 9967 020019 92IKI71 53 2 180123 920176 920223 y-rr-ii WKU 92/13X4 9204 : 990488 0541 0593 62 3 0645 0697 0749 0801 0853 0906 0958 1010 1062 1114! 52 4 1166 1218 1270 1322 1374 1426 1478 1630 1582 1634 63 6 1686 1738 1790 1842 1 -94 1946 1998 2050 2102 2154' 6* 6 2206 2258 2310 2362 2414 2466 2518 2570 2622 2674 6ft 7 2725 2777 2S29 21 2933 81 3037 m 3140! 3litt 53; 8 3244 32% 3348 3399 3451 3503 3655 3607 3658J 3710 63 9 3762 3S14 3865 3917 3%9 4021 4072 4124 4176J 4228 i 69 840 924279 924331 924383 924434 9244JJ6 924538 924689 924641 924693924744* 5' 1 47% 4848 IBM 4951 5(03 6054 6K 6157 5209 6261 Bl 2 6312 5364 6415 6467 6518 6670 6621 6673 6725 6776 53) 3 6828 6879 6931 6982 6034 Gn-f, 6137 6188 6240! 6291 51 4 6342 6394 6445 6497 6548 6600 6651 6702 67f>4 6805 51 6 6857 6908 6959 7011 7062 7114 7165 7216 7*r 7319 51 6 7370 7422 7473 7524 7676 7627 7678 7730 7781 7832 61 7 7883 7936 net 8037 BOB! 8140 8191 8242 8293 8345 61 8 83% 8447 MM 8649 8601 8652 8703 8754 8SOC- 8857 51 8908 81 9010 9061 9112 9163 9215 9266 9317 9368 61 $60 1 929419 9930 J2yi7u 9981 92'.) ",21 JXf.U 'j-j;::.:^ 930083 '.i2'/;2.J 930194 929674 000166 92''. ~' 930236 929776 WWWJ7 9302K7 93<>33* 929879 9309 61 61 2 930440 93M91 0642 0592 0643 0694 0746 07% 0647 0898 61 3 0949 1000 1061 1102 1153 1204 1254 1305 1356 1407 61 4 1458 1609 1660 1610 1661 1712 1763 1814 1865 1915 51 6 1966 2017 2063 2118 2169 2220 2271 2322 2372 2423 51 6 2474 2524 2575 2626 2677 2727 2778 2829 vn 2930 61 7 2931 3031 3082 3133 3183 3234 3285 3335 33S6 3437 61 8 3487 3538 3689 3639 3690 3740 3791 3841 3892 3943 61 3993 4044 4094 4145 4195 4246 4296 4347 4397 4448 61 860 934498 934549 934599 934660 934700 934751 934801 934852 934902 934953 60 5003 6054 6104 6164 6205 6255 6306 6356 5406 5457 60 2 6507 6558 6603 6658 6709 6759 6809 6860 6910 5%0 60 3 6011 6061 6111 6162 6212 6262 6313 6363 6413 6463 60 4 6514 6564 6614 6665 6715 6765 6815 6865 6916! 6966 60 6 7016 7066 7117 7167 7217 7267 7317 7367 74 IS 7468 60 6 7618 7568 7618 7668 7718 7769 7819 7869 7919 7969 60 7 8019 8069 8119 8169 8219 8269 8320 8370 ; 8420 6470 60 8 8520 8570 6620 8670 8720 8770 8820 8870 8920 8970 60 9 9020; 9070 9120 9170 9220 9270 9320 9369 9419 9469 60 870 939519 939569 939619 9400181940068940118 939669 940163 939719 940218 939769939819 940-267940317 939869 939918 940367 940417 939968 940467 60 50 2 0516 0566 0616 0666 0716 07C5 0815 0,^65 0915 0964 60 3 1014 1064 1114 1163 1213 1263 1313 1362 1412 1462 60 4 1511 1561 1611 1660 1710 1760 1809 1859 1909 l'J5S 60 5 2008 2058 2107 2157 2207 2256 2306 2355 2405 2455 60 6 1504 2554 2603 2653 2702 2752 2801 2851 2901 2950 50; 7 3000 3.ns) 3.NM! 3148 3193 . 3217 3297 3346 33% 3445 49! 8 3495 3514 i 35^8 3643 3692 3742 3791 3841 3890 3939 49 9 3989; 4038 4088 4137 4186 4236 4285 4335 4384 4433; 49 Wo 1 a 3 | 5 6 7 i 8 9 Dlfl i TAIU.I. I. l.OCAKITHMS OF NTMHEKS. No.' 1 8 9 Diff. 880 <141483 W4532 944531 944631 M46SO M4729 944779 944828 944877 944927 49 1 4976 5026 5074 5124 5173 5222J 5272 5321 6370 5419 49 a 5469 55! 8 6567 5616 6665 5715 6764 6813 6362 5912 49 3 5981 6010 6059 6103 6157 6207 6256 6305 6354 6403 49 4 6452 6/501 6551 6600 6649 6698 6747 6796 6845 6894 49 6 6943 6992 7041 7090 7140 7189 7238 7287 7336 7385 49 7434 7483 7532 7581 7630 7679 7728 7777 7826 7876 49 7 7924 7973 8022 8070 8119 8168 8217 8266 8315 8364 49 8 8413 8462 8511 8560 8609 8657 8706 8765 8804 8853 49 9 8902 8951 8999 9048 9097 9146 9196 9244 9292 9341 49 390! 949390 949439 949488 949536 949585 949634 949683 949731 949780 949829 49 I 9878 9926 9975 950024 950073 50121 950170 950219 950267 950316 49 2!>o0365 950414 950462 0511 0560 0608 0657 0706 0754 0803 49 3 0851 0900 0949 0997 1046 1095 111 : 1192 1240 1289 49 41 1338 13^6 1435 1483 1532 1580 1629 1677 1726 1775 49 6 1823 1S72 1920 1969 2017 2066 *in 2163 2211 2260 48 6 2308 2356 24M5 2453 2502 2550 2599 2647 2696 2744 48 7 2792 2841 2889 2938 2936 3034 3083 3131 3180 3228 48 8 3276 3325 3373 3421 3470 3518 3566 3615 3663 3711 48 9 3760 3808 3856 3905 3953 4001 4049 4098 4146 4194 48 900 954243 954291 954339 954387 954435 954484 954532 954580 954628 954677 48 I 4725 4773 4S21 4S69 4918 4966 6014 6062 6110 6158 48 2 6207 5255 5303 6351 5399 5447 6495 6543 6592 6640 48 3 5688 5736 5784 5832 5380 6928 6976 6024 6072 6120 48 4 6168 6216 6265 6313 6361 6409 6457 6505 6653 6601 48 6 6C49 6697 6745 6793 6840 6888 6936 6984 7032 7080 48 6 7128 7176 7224 7272 7320 7363 7416 7464 7512 7559 48 7| .7607 7655 7703 7751 7799 7847 7894 7942 7990 8038 48 8 8036 8134 8181 8229 8277 8325 8373 8421 8468 8616 48 9 8664 8612 8659 8707 8755 8803 8850 8898 8946 8994 48 910 959041 959089 959137 959185 959232 95923C 969328 959375 959423 959471 48 1 US 1 8 9566 9614 9661 9709 9757 9804 9852 9900 9947 48 2 9995 960fU2 960090 9601 33 960185 960233 9602SO 960328 960376 960423 48 3 %0471 0518 0566 0613 0661 0709 0756 0804 0861 0899 48 4 0946 0994 1041 1039 1136 1184 1231 1279 1326 1374 48 5 1421 1469 1516 1563 1611 1658 1706 1753 1801 1849 47 6 1895 1943 1990 2038 2035 2132 2180 2227 2275 2322 47 7 2369 2417 2464 2511 2559 2606 2653 2701 2748 2795 47 8l 2843 2890 2937 2935 3032 3079 3126 3174 3221 3268 47 9 3316 3363 3410 3457 3504 3552 3599 3646 3693 3741 47 920963783 963835 963882 963929 963977 964024 964071 9641 IS 964165 964212 47 1 426C 4307 4354 4401 4448 4495 4542 4590 4637 4684 47 2 473 4778 4825 4372 4919 4966 5013 5061 6108 5155 47 3 5202 5249 5296 5343 5390 6437 5484 6531 6678 5625 47 4 5672 5719 5766 5313 6S60 5907 5954 6001 6048 6095 47 5 6142 6189 6236 6233 6329 6376 6423 6470 6517 6564 47 6 661 6658 6705 6752 6799 6845 6892 6939 6986 7033 47 7 7080 7127 7173 7220 7267 7314 7361 7408 7454 7501 47 8 7548 7595 7642 7638 7735 7782 7829 7875 7922 7969 47 9 8016 8062 8109 8156 8203 8249 8296 8343 8390 8436 47 930 968483 968530 968576 968623 968670 963716 968763 968810 968856 968903 47 11 8950 8996 9043 9090 9136 9183 9229 9276 9323 9369 47 1 9416 9463 9508 9556 960? 9649 9695 9742 9789 9835 47 3 9882 9928 9975 970021 970063 9701 14 970161 970207 970254 970300 47 4 97034 970393 970440 0486 0533 0579 0626 0672 0719 0765 46 5 0812 0858 0904 0951 0997 1044 1090 1137 1183 1229 46 6 127 1322 1369 1415 146 1508 1554 1601 1647 1693 46 7 1740 1786 1832 1879 192o 1971 2018 2064 2110 2157 46 8 220 2249. 2295 2342 2388 2434 2481 2527 2573 2619 46 2666 2712] 2768 2304 285 2897 2943 2989 3035 3082 46 No. 9 Diff. TABLE I. LOGARITHMS OF NUMBERS. No. 940 973128 973174 973220 973266 973313 973359 973405 973451 973497 973548 950977724 8181 BOS? 1 1 8 4 ;, 9NNN..", t; oi> 7 0912 1366 1619 4051 4512 4972 5432 SIM 6350 Wis 2723 3175 8888 4077 4:,27 J9i 7 6488 :,s;:, 6324 970986772 7219 roM 8113 BBBO BOOCS 9696 90339 o;s: 5 2111 2.V>1 8906 3436 8877 4317 4757 5196 6074 6512 8259 8 9131 9 9565 Noj 4097 5478 5937 7312 977769 97781.-) 977861 977906 977962 977998 978043 978089 978135 9BM 980049980094 0503 09:,; 1411 1864 2709 3220 3671 4122 4572 r.n-,'2 6471 72; 7711 8157 8604 904U '.I'.' I 9989 S <>90339 990883 990428 ON] 980 KM22T,.t!1270 991315 1713 8136 3480 3921 4:161 4801 5240 611 6555 7867 8739 9174 4143 4604 5m; 4 5524 6868 6142 6900 7358 8272 9184 9639 0649 1008 i :,; 1909 2814 3365 8716 4167 4617 6097 6616 6868 6418 7809 7758 8809 6646 '.*>-. I 9689 1758 221 HI 2642 4405 4*45 6161 6599 7037 7474 7910 8347 8782 9218 9652 3728 4169 4650 51101 ;V)7(i 8089 8466 8846 7408 3774 42*-. 4281 4696! 4742 6156 5202 5616 8076 1533 t;w2 7449 8317 8774 8819 980140 980185 980881 1114* 1501 1954 8810 8789 4212 4662 6112 6561 6010 6458 0640 lu'..:5 1547 2000 290 8868 3NC 42.".: 4707 6157 .v ;,*, 6066 6608 168905968951 7:Vi3 7896 7800. 7845 8247 8291 8693 8737 9138 9183 9672 990028990072990117 "i;2 0516 0561 0916! 0960 1004 6121 e:,; '.i 7037 7495 S4"9 *N;:, 9881 97W 1139 1898 201.-. .'iloi 6868 4808 4;.vj .YJ02 6851 C,].N) ? I J.", 87X2 1802 2211 8187 8686 4449 990 995635 995679 995723 995767 995811 996654 995898 995942 99K966 996030 6205 6643 7080 7517 7954 S:MI 1846 1890 2288 2338 2730 2774 8172 8216 3613 365' 4053 4097 4493 4537 4933 4977 5372, 5416 7124 7561 7998 S43I 6731 7168 7605 8041 8477 8869 8913 9305 9348 9739 9783 4327 47!* 6707 6187 7088 7541 8454 8911 9868 8881 yuy 0730 1184 1637 8090 (5 980322 980367 980412 8448 8897 4847 47 '.-7 6847 6696 6144 7984 '..'72 9717 990161 0605 KM!) 3913 4374 4*34 5753 6212 6671 7129 7586 6500 6958 9412 0776 1899 16S3 2135 8481 8942 4998 JM'J 6898 6741 6189 7588 7'..;;. 6496 6871 9816 9761 4420 4880 5340 5799 6856 8717 7175 7632 4005 44 6 6763 7220 7678 K>4G 9002 U457 0681 127:. 1728 2181 8586 8987 4437 4-T 6837 6786 6284 7577 N.-JJ 6470 6916 3C,I 9606 0650J 0694 1098! 1137 991359 991403 991448 991492 991530 991580 991625 8591 9047 9968 0689 1880 1773 8681 4088 41S2 4932 1888 Win 027 9 6727 KM! 9406 9660 0738 1182 19a"j 1979 2377 2421 2819 2863 3260 3304 8701 374')' 4141 4185 4581 4625 5021 5065: 5460, 5504, 2023 2465 2907 3348 8789 4229 2067 2509 29. r >l 4273 4Wi 4713 5108; 6152 5547! 6591 6337, 6380 6774 6818, 7212' 7255 7648 7692 8085 ! 8129 8581 8564! 8956; 9000! 9392 9435 9826 9870 ~6| 7 1 6424 6468 6862 6906 7299| 7343 7736 7779 sS "-' 9043| 9087 9479! 9522 9913 9957 Diff. 44 41 44 41 41 41 14 41 41 J3 Diff. TABLE II. LOGARITHMIC SINES, COSINES, TANGENTS AND COTANGENTS. 227 TABLE II. LOGARITHMIC SI NOTE. THE table here given extends to minutes only. The nsual method of extending such a table to seconds, by proportional parts of the difference between two consecutive logarithms, is ac- curate enough for most purposes, especially if the angle is not very sinalL When the angle is very small, and great accuracy is required, the following method may be used for sines, tangents, and cotangents. I. Suppose it were required to find the logarithmic sine of 5' 24 . By the ordinary method, we should have log. sin. 5 = 7.162690 diff. for 24 ' = 31673 log. sin. 5' 24" = 7. The more accurate method is founded on the proposition in Trigo- nometry, that the sines or tangents of very small angles are pro- portional to the angles themselves. In the present case, there- fore, we have sin. 5' : sin. 5 24' =5:5 24' = 300 : 324' . Hence sin. 5' 24 = 324 ^', or log. sin. 5' 24 = log. sin. 5 + log. 324 - tJUU log. 300. The difference for 24" will therefore, be the difference between the logarithm of 324 and the logarithm of 300. The operation will stand thus : log. 324 = 2.510545 log. 300 = 2.477121 diff. for 24 ' = 33424 log. sin. 5' = 7.162696 log. sin. 5 24 = 7.196120 Comparing this value with that given in tables that extend to seconds, we find it exact even to the last figure. IL Given log. sin. A = 7.004438 to find A. The sine next less than this in the table is sin. 3' = 6.940847. Now we have sin. 3 : AXD OOTJLXGE3GTR. '_*_ log. sin. -1 log. sin. 3. Henoe it appeals, that, to find the loga- rithm of A in minutes, we must add to the logarithm of 3 the difference between log. sin. A and log. sin. 3". log. sin. A = 7.004438 log. an. V = 61940847 MU kg. 3 = 0.477121 A = 3.473 0510712 or A = 3* 3&38 . By the common method we should hare found A = 3 30.54 . The same method applies to tangents and cotangents, except that in the case of cotangents the differences are to be subtracted. % * The radios of this table is unity, and the 8, 7, and 6 stand respectively lor t, 2, 3, and 230 TABLE II. LOGARITHMIC SINES, M. Sine. D.I . Cosine. D. 1". Tang. D. 1". Cotang. 11. 1 2 3 4 5 Inf. neg. 6.463726 .764756 .940847 7.065786 .162696 5017.17 2934.85 2082.31 1615.17 0.000000 .000000 .000000 .000000 .000000 .000000 .00 .00 .00 .00 .00 r*n Inf. neg. 6.463726 .764756 .940847 7.065786 .162696 5017.17 2934.85 2((82.31 161517 Infinite. 3.536274 .235244 .059153 2.934214 .837304 60 59 58 67 56 65 6 7 .241877 .308824 1I15>8 Qfifi W 9.999999 999999 .uu .00 HA .241878 .308825 1319.69 1115.78 n/ &A .758122 .691175 54 63 8 .366S16 yoo.oo 999999 .uu .366817 yt>o.o4 QCO CC .633183 62 9 .417968 762.62 .999999 .01 .417970 OO.4.55 762.63 .582030 61 10 7.463726 fiQQ QQ 9.999998 111 7.463727 AQQ QQ 2.536273 50 11 12 13 14 15 16 17 18 19 .505118 .542906 .577668 .609853 639816 .667845 .694173 .718997 .742478 Doy.oo 629.81 579.37 536. 4 j 499.38 467.14 438.81 413.72 391.35 371.27 .999998 999997 .999997 .999996 .999996 999995 .999995 999994 999993 .Ul .01 .01 .01 .01 .01 .01 .01 .01 .01 .505120 .542909 .577672 .609857 639820 .667849 .694179 .719003 .742484 DCHP.OO 629.81 579.37 536.42 499.39 467.15 438.82 413.73 391.36 371.28 .494880 .457091 .422328 .390143 .360180 .332161 .305821 .280997 .257516 49 48 47 46 45 44 43 42 41 20 21 22 23 7.764754 .785943 806146 .825451 353.15 336.72 321.75 QflQ ft* 9.999993 999992 999991 999990 .01 .01 .01 7.764761 .785951 .806155 .825460 353.16 336.73 321.76 QAQ H7 2.235239 .214049 193845 .174540 40 39 38 37 24 25 .843934 .861662 oUo.Uo 295.47 999989 999989 02 OQ .843944 861674 oUo.U/ 295.49 ooo on .156056 138326 36 35 26 .878695 273! 17 999988 .u* .02 .878708 -coo.yu 273.18 121292 34 oo 28 29 '.910879 .926119 263.23 253.99 245.38 999987 999986 999985 02 .02 .02 .910894 926134 263.25 254.01 245.40 1089106 00 32 31 30 31 32 33 34 35 36 37 38 39 7.940842 .955082 .968870 .982233 .995198 S.007787 .020021 .031919 .043501 .054781 237.33 229.80 222.73 216.08 209.81 203.90 198.31 193.02 188.01 183.25 9.999983 999982 999981 999980 999979 .999977 .999976 .999975 .999973 999972 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 7.940858 .955100 968889 982253 .995219 8.007809 .020044 .031945 .043527 .054809 237.35 229.82 222.75 216.10 209.83 203.92 19833 193.05 188.03 183.27 2.059142 .044900 .031111 .017747 .004781 1.992191 .979956 .968055 .956473 .945191 30 29 28 27 26 25 24 23 22 21 40 41 42 43 44 45 46 47 48 49 8.065776 .076500 .086965 .097183 .107167 .116926 .126471 .135810 .144953 .153907 178.72 174.42 170.31 166.39 162.65 159.08 155.66 152.38 149.24 146.22 9.999971 .999969 999968 .999966 .999964 .999963 .999961 .999959 .999958 .999956 .02 .03 .03 .03 .03 .03 .03 .03 .03 .03 8.065806 .07653! .086997 .097217 .107203 116963 .126510 .135851 144996 .153952 178.75 17444 170.34 16642 162.68 159.11 155.69 15241 149.27 146.25 1.934194 .923469 .913(103 .902783 .892797 .883037 .873490 .864149 .855004 .846048 20 19 18 17 16 15 14 13 12 11 50 51 52 53 54 55 56 57 58 59 60 8.162681 .171280 .179713 .187985 .196102 .204070 .21 1895 .219581 .227134 .234557 .241855 143.33 140.54 137.86 135.29 132.80 130.41 128.10 125.87 123.72 121.64 9.999954 .999952 999950 .999948 999946 999944 999942 999940 999938 999936 .999934 .03 .03 .03 .03 .03 .03 .03 .04 .04 .04 8.162727 .171328 .179763 .188036 .196156 .204126 .211953 .219641 227195 .234621 241921 14336 140.57 13790 135.32 132.84 13044 129.14 125.91 123.76 121.68 1.837273 .828672 .820237 .811964 .803844 .795874 .788047 .780359 .772805 .765379 .768079 10 9 8 7 6 6 4 3 2 M. Cosine. D. 1". Biaa. . 1". Oofiaug. D. l'. Tang. M. 89 , TANGENTS, AND COTANGENTS. M. Sine D. !. Cceine. D 1". Tang. D. 1". Cotang. M. 8.241855 .243033 119.63 9.999934 .999932 .04 8.241921 .249102 119.67 1.758079 .750898 lib" 59 .256094 .263042 .269881 .276614 .283243 117.69 115.80 113.98 112.21 110.60 .999929 .999927 .999925 .999922 .999920 .04 .04 .04 .04 .04 .256165 .263115 .269956 .276691 .283323 1 17.72 115.84 114.02 112.25 110.54 .743835 .736885 .730044 .723309 .716677 58 57 56 55 54 .289773 .296207 .302546 108.83 107.22 105.66 104.13 .999918 .999915 .999913 .04 .04 .04 .04 .289856 .29S292 .302634 108.87 107.26 105.70 104.18 .710144 .703708 .697366 53 52 51 10 U 12 13 14 15 16 17 13 19 8.308794 .314954 .321027 .327016 .332924 .338753 .344504 .350181 .355783 .361315 102.66 101.22 99.82 98.47 97.14 95.86 94.60 93.38 92.19 91.03 9,99*910 .999907 .999905 .999902 .999899 .999897 .999894 .999891 .999888 .999885 .04 .04 .04 .05 .05 .05 .05 .05 .05 .05 8.308884 .315046 .321122 .327114 .333025 .338856 .344610 .350289 .3C5895 .361430 102.70 101.26 99.87 98.51 97.19 95.90 94.65 93.43 92.24 91.08 1.691116 .684954 .678878 672886 666975 .661144 .655390 .649711 .644105 .638570 50 49 48 47 46 45 44 43 42 41 20 8.366777 9.999882 AC 8.366395 1.633105 40 21 .372171 QQ Qf\ .999879 tvO .372292 89.95 .627708 39 23 24 26 26 27 29 .377499 .382762 .387962 .393101 .398179 .40311)9 .408161 .413063 oo.oU 87.72 86.67 85.64 84.64 83.66 82.71 81.77 80.86 .999876 .999873 .999870 .999867 .999864 .999861 .999858 .999854 '.05 .05 .05 .05 .05 .05 .05 .05 .377622 .382889 .388092 .393234 .398315 .403338 .408304 .413213 88.85 87.77 86.72 85.70 84.69 83.71 82.78 81.82 80.91 .622378 .617111 .611908 .606766 .601685 .596662 .591696 .686787 38 37 36 35 34 33 32 31 30 31 32 33 34 36 38 37 38 39 8.417919 .422717 .427462 .432156 .436800 .441394 .445941 .450440 .454893 .459301 79.96 79.09 78.23 77.40 76.58 75.77 74.99 74.22 73.47 72.73 9.999851 .999848 .999844 .999841 .999833 1999831 .999827 .999824 .999820 .06 .06 .06 .06 .06 .06 .06 .06 .06 .06 8.418068 .422869 .427618 .432315 .436962 .441560 .446110 .450613 .455070 .459481 80.02 79.14 78.29 77.45 76.63 75.83 75.05 74.28 73.53 72.79 1.581932 .677131 .572382 .567685 .563038 558440 .653890 .549387 .544930 .540519 30 29 28 27 26 25 24 23 22 21 40 41 42 43 8.463665 .467985 .472263 .476498 72.00 71.29 70.60 9.999816 .999813 .999809 .999805 .06 .06 .06 rut 8.463849 .468172 .472454 .476693 72.06 71.35 70.66 1.536151 .531828 .527546 .523307 20 19 18 17 44 .480693 69.91 .999801 .Uo ne .480892 69.98 .519108 16 45 46 47 48 49 .484848 488963 .493040 .497078 .501080 68^59 67.94 67.31 66.69 66.08 .999797 .999794 .999790 .999786 .999782 .UB .06 .07 .07 .07 .07 .4,85050 .489.70 .493250 .497293 .501298 68!65 68.01 67.38 66.76 66.15 .514950 .510830 .506750 .502707 .498702 15 14 13 12 11 50 ft 505045 9 999778 AT 8.505267 CC EC ' 1.494733 10 51 .508974 65.48 .999774 .U/ AT .509200 QQUOO .490800 9 52 53 .512867 .516726 64.32 CO 7C .999769 .999765 .U7 .07 AT .513098 .516961 64.39 CO QO .486902 .483039 8 7 54 55 .520551 .524343 oo. 7o 63.19 .999761 .999757 .tl/ .07 AT .520790 .524586 DO.O4 63.26 .479210 .475414 6 5 58 .528102 CO | | .999753 .07 AT .528349 62.72 CO I Q .471651 4 57 58 59 80 .531328 .535523. .539186 .542819 (XG. I ! 61.58 61.06 60.55 .999748 .999744 .999740 .999735 .U/ .07 .07 .07 .532080 .535779 .539447 .543084 O4. lo 61.65 61.13 60.62 .467920 .464221 .460553 .456916 3 2 M. Corfne. D.I'. Sine. D. 1". Cotang. P. 1". Tang. M. 910 88* 232 TABLE II. LOGARITHMIC SINES. 8 Iff 3 M. Sine. D. 1". Cosine. D 1". Tang. D. 1". Cotang. M. 8.542819 .546422 60.04 9.999735 .999731 .07 8.543084 .546691 60.12 1.456916 .453309 60 59 2 3 .549995 .553539 59.06 .999726 .999722 .08 .550268 .553817 69.14 .449732 .446183 68 67 4 .657054 .999717 .557336 .442664 66 6 6 8 9 .560540 .563999 .567431 570836 .574214 58.11 67.65 57.19 56.74 56.30 55.87 .999713 .999708 .999704 999699 .999694 .08 .08 .08 .08 .08 .560828 .564291 567727 571137 .574520 57.73 57.27 66.82 56.38 65.95 .439172 .435709 .432273 .428863 .425480 55 64 53 62 61 10 11 12 13 14 15 16 17 18 19 8.577566 .580892 .584193 .587469 .590721 .593948 .597152 .600332 .603489 .606623 55.44 55.02 54.60 54.19 53.79 53.39 53.00 52.61 52.23 51.86 9.999689 .999685 .999680 999675 999670 999665 .999660 .999655 .999650 .999645 .08 .08 .08 .08 .08 .08 .08 .08 .08 .09 8.677877 .681208 .684514 587795 .591051 .594283 .597492 .600677 .603839 .506978 55.52 55.10 64.68 64.27 63.87 53.47 53.08 52.70 52.32 51.94 1.422123 .418792 .415486 .412205 .408949 .405717 .402508 ,399323 .396161 .393022 60 49 48 47 46 46 44 43 41 20 8.609734 9.999640 8.610094 1.389906 40 21 99 .612823 51.12 .999635 .09 .613189 51.21 .386811 39 OQ 23 24 25 26 27 28 29 .618937 .621962 .624965 .627948 .630911 .633854 .636776 60.77 60.41 50.06 49.72 49.38 49.04 48.71 48.39 .999624 .999619 .999614 .999608 .999003 .999597 .999592 .09 .09 .09 .09 .09 .09 .09 .09 .619313 .622343 .625352 .628340 .631308 .634256 .637184 50.85 50.50 50.15 49.81 49.47 49.13 48.80 48.48 380687 377657 .374648 .371660 .368692 .365744 .362816 37 35 35 34 33 32 31 30 31 32 33 34 35 36 37 33 39 8.639680 .642563 .645428 .648274 .651102 .653911 .656702 .659475 .662230 .664968 48.06 47.75 47.43 47.12 46.82 46.52 46.22 45.93 45.63 45.35 9.999586 .999581 .999575 .999570 .999564 .999558 .999553 .999547 .999541 .999535 .09 .09 .09 .09 .09 .10 .10 .10 .10 .10 8.640093 .642982 .645853 .648704 .651537 .654352 .657149 .659928 .662689 .665433 48.16 47.84 47.53 47.22 46.91 46.61 46.31 46.02 45.73 45.45 1.359907 .357018 .354147 .351296 .348463 .345648 .342851 .340072 .337311 .334567 30 29 28 27 26 25 24 23 22 21 40 41 42 43 44 45 46 47 48 49 8.667689 .670393 .673080 .675751 .678405 .681043 .683665 .686272 .688863 .691438 45.07 44.79 44.51 44.24 43.97 43.70 43.44 43.18 42.92 42.67 9.999529 .999524 .999518 .999512 .999506 .999500 .999493 .999487 .999481 .999475 .10 .10 .10 .10 .10 .10 .10 .10 .10 .10 8.668160 .670870 .673563 .676239 .678900 .681544 .684172 .686784 .689381 .691963 45.16 44.88 44.61 44.34 44.07 43.80 43.64 43.28 43.03 42.77 1.331840 .329130 .326437 323761 .321 100 .318456 .315828 .313216 .310619 .308037 20 19 18 17 16 15 14 13 12 11 50 61 52 53 54 65 66 57 58 59 8.693998 .696543 .699073 .701589 .704090 .706577 .709049 ,711507 .713952 .716383 42.42 42.17 41.93 41.68 41.44 41.21 40.97 40.74 40.51 9.999469 .999463 .999456 .999450 .999443 .999437 .999431 .999424 .999418 .999411 .10 .11 .11 .11 .11 .11 .11 .11 .11 8.694529 .697081 .699617 .702139 .704646 .707140 .709618 .712083 .7H534 .710972 42.52 42.28 42.03 41.79 41.65 41.32 41.08 40.85 40.62 4O 40 1.305471 .302919 .300383 .297861 .295354 .292860 .290382 .287917 .285466 .283028 10 9 8 7 5 4 3 a 60 .718800 .999414 .719396 .280604 M. Cosine. D. 1". Sine D.I". Cotang. Tang. M. COSINES, TANGENTS, AND COTANGENTS. 233 30 M. Sine. D. 1". Cosine. D. 1". Tang. D. 1". Cotang. M. 1 2 8.718800 .721204 .723595 40.06 39.84 OQ CO 9.999404 .999393 .999391 .11 .11 8.719396 .721806 .724204 40.17 39.95 1.280604 .278194 .275796 60 59 58 3 4 .725972 .728337 VwCHI 39.41 on i Q .999384 .999378 .11 .726588 .728959 3^52 OQ qi .273412 .271041 57 56 5 .730688 oy. i y OQ QQ .999371 .731317 oy.oi .268683 55 6 .733027 oo.yo OQ 77 .999364 .733663 OQ Q .266337 54 7 .735354 oo. / / OQ C7 .999357 .735996 00* CO .264004 53 8 .737667 OO.O/ .999350 10 .738317 OO.OO .261683 52 9 .739969 sa 16 999343 . 1< .12 .740626 38.'27 .259374 51 10 11 12 3.742259 .744536 .746802 37.96 37.76 07 cc 9.999336 .999329 .999322 .12 .12 8.742922 .745207 .747479 38.07 37.88 Q7 CQ 1.257078 .254793 .252521 50 49 43 13 14 15 .749055 .751297 .753528 o/.oo 37.37 37.17 .999315 .999308 .999301 !l2 .12 .749740 .751989 .754227 O/.OO 37.49 37.29 07 m .250260 .24801 1 .245773 47 46 45 16 17 .755747 .757955 36.'80 .999294 .999237 iia 1 O .756453 .758668 oY.lU 36.92 OC ~"> .243547 .241332 44 43 18 19 .760151 .762337 36.61 36.42 36.24 .999279 999272 .1* .12 .12 .760872 .763065 ob.76 36.55 36.36 .239128 .236935 42 41 20 21 8.764511 .766675 36.06 9599265 .999257 .12 8.765246 .767417 36.13 ofi nn 1.234754 .232583 40 39 22 23 24 25 26 27 23 29 .768828 .770970 .773101 .775223 .777333 .779434 .781524 783605 35! 70 35.53 35.35 35.18 35.01 34.84 34.67 34.51 .999250 .999242 .999235 .999227 999220 999212 999295 999197 !l2 .12 .13 .13 .13 .13 .13 .13 .769578 .771727 .773866 .775995 .778114 .780222 .782320 .784408 oo.UU 35.83 35.65 35.48 35.31 35.14 34.97 34.80 34.64 .230422 .228273 .226134 .224005 .221886 .219778 .217680 .215592 38 37 36 36 34 33 32 31 30 31 32 8.785675 .787736 .789787 34.34 34.18 9.999189 999181 999174 .13 .13 8.786486 .788554 .790613 34.47 34.31 1.213514 .211446 .209387 30 29 23 33 34 35 36 37 33 .791828 .793859 .795881 .797894 .799897 .801892 sase 33.70 33.54 33.39 33.23 oo no .999166 .999153 .999150 .999142 .999134 .999126 .13 .13 .13 .13 .13 .13 10 .792662 .794701 .796731 .798752 .800763 .802765 34. 15 33.99 33.83 33.63 33.52 33.37 .207338 .205299 .203269 .201248 .199237 .197235 27 26 25 24 23 22 39 .803876 oo.uo 32.93 .999113 .Id .13 .804758 33.22 33.07 .195242 21 40 41 42 43 44 45 46 47 43 49 8.805852 .807819 .809777 .811726 .813667 .815599 .817522 .819436 .821343 .823240 32.78 32.63 32.49 32.34 32.20 32.05 31.91 31.77 31.63 31.49 9.999110 .999102 .999094 .999086 .999077 .999069 .999061 .999053 .999044 .999036 .14 .14 .14 .14 .14 .14 .14 .14 .14 .14 8.806742 .808717 .810683 .812641 .814589 .816529 .818461 .820384 .822298 .824205 32.92 32.77 32.62 32.48 32.33 32.19 32.05 31.91 31.77 31.63 1.193258 .191283 .189317 .187359 .185411 .183471 .181539 .179616 .177702 .175795 20 19 18 17 16 15 14 13 12 11 50 51 52 53 54 55 56 57 58 59 60 8.825130 .827011 .828884 .830749 .832607 .834456 .836297 .838130 .839956 .841774 .843585 31.36 31.22 31.08 30.95 30.82 30.69 3056 30.'30 30.17 9.999027 .999019 .999010 .999002 .993993 .993984 .993976 .998967 .993958 .998950 .998941 .14 .14 .14 .14 .14 .14 .15 .15 .15 .15 8.826103 .827992 .829374 831748 .833613 .835471 .837321 .839163 .840998 .842825 .844644 3J 50 31.23 31.09 30.96 30.83 30.70 30.57 30.45 30.32 1.173897 .172008 .170126 .168252 .166387 .164529 .162679 .160837 .159002 .157175 .155356 10 9 8 7 6 5 4 3 2 M. Cosine. D.I" Slue. D. 1". Cotang. D.l". Tang. M. 93" 234 TABLE II. LOGARITHMIC SINES, M Sine. D. 1". Cosine. D. 1". Tang. D 1". Cotang. M. 1 2 3 4 5 6 7 8 9 8.843585 .845387 .847183 .848971 .850751 .852525 .854291 .856049 .857801 .859546 30.05 2992 29. SO 2963 29.55 29.43 29.31 29.19 29.08 28.96 9.998941 .998932 .998'.t23 .998914 .998905 993896 .998887 .998873 .998869 .99886C .15 .15 .15 .15 .15 .15 '5 .15 .15 .15 8.844644 .846455 .848260 .850057 .851846 .853628 .855403 .857171 .858932 860686 30.20 30.07 29.95 29.83 29.70 29.58 29.46 29.35 29.23 29 11 1.155356 .153545 .151740 .149943 .148154 .146372 .144597 .142829 .141068 .139314 60 59 58 57 56 55 54 53 52 51 10 11 12 13 14 15 16 17 18 19 8.861283 .863014 .864738 .866455 .863165 .869863 .871565 .873255 .874933 .876615 23.84 28 ~3 23.61 23.50 23.39 28.2* 28.17 28.06 27.95 27.84 9.998851 .998841 .998832 .998823 .998813 .998804 998795 .998785 '. 998766 .15 .15 .15 .16 .16 16 .16 16 .16 .16 8.862433 .864173 .865906 .867632 .869351 .871064 .872770 .874469 .876162 .877849 29.00 128.88 28.77 28.66 28.55 28.43 28.32 28.22 28.11 28.00 1.137567 .135827 .134094 .132368 .130649 .128936 127230 .125531 .123838 .122151 50 49 48 47 46 45 44 43 42 41 20 21 22 23 24 25 26 27 28 29 8.878285 .879949 .881607 .883253 884903 .886542 883174 .889801 891421 .893035 27.73 2763 27 52 2742 2731 27.21 27.11 27. 00 26.90 26.80 9.998757 .998747 .998733 .99*728 99^719 91)8703 9i)>699 99*689 .99^79 .998669 .16 .16 .16 .6 .16 .16 .16 .16 .16 .17 8.879529 .881202 .882869 .884530 .886185 .887833 .889476 .891112 892742 .894366 27.89 2779 27.68 27.58 2747 27.37 27.27 27 17 2707 26.97 1.120471 .118798 .117131 .115470 .113815 .112167 .110524 .108888 .107258 .105634 40 39 38 37 36 35 34 33 32 31 30 31 8.894643 .896246 26.70 26.60 9.998659 .998649 .17 17 8.895984 .897596 26.87 2677 1.104016 .102404 30 29 33 34 35 36 37 38 39 .899432 .901017 .902596 .904169 .905736 .907297 .908853 26.51 26.41 26.31 2622 26.12 26.03 25.93 25.84 .998639 .998629 998619 .998609 .998599 998589 998578 993568 .17 .17 .17 .17 .17 .17 .17 .17 .899203 .900803 .902398 903937 .905570 .907147 .908719 .910235 26.67 26.58 2648 26.39 26.29 26.20 26.10 26.01 .100797 .099197 .097602 .096013 .094430 .092853 .091281 .089715 27 26 25 24 23 22 21 40 41 42 8.910404 .911949 .913488 25.75 2566 9.998558 .993548 .998537 .17 .17 8.911846 .913401 .914951 25.92 25.83 1.088154 .086599 .085049 20 19 IS 43 44 45 46 47 43 49 .915022 .916550 .918073 .919191 .921103 .922610 .924112 25.47 25.33 25.29 25.21 25.12 25.03 24.94 998527 .998516 .998506 .998495 .9984^5 .998474 .998464 .17 .17 .18 .18 .18 .13 .18 .916495 .913034 .919568 .921096 .922619 .924136 .925649 25.65 25.56 25.47 25.38 25.29 25.21 25.12 .083505 .081966 .080432 .078904 .077381 .075864 .074351 17 16 15 14 13 12 11 50 51 8.925609 .927100 2486 9.993453 .998442 .18 8.927156 .923658 25.04 1.072844 .071342 10 9 52 53 54 .928587 .930068 2469 24.60 .998431 .998421 .998410 .18 18 .930155 .931647 .933134 24.87 2473 .069845 .068353 .066866 8 7 6 55 56 57 58 59 60 .933015 .934481 .935942 .937398 .938350 .940296 24.43 24.35 24.27 24.19 24.11 .998399 .998388 .998377 .998366 .998355 .998344 .18 .18 .18 .18 .18 .934616 .936093 .937565 .939032 .940494 .941952 24.62 24.53 24.45 24.37 24.29 .065384 .063907 .062435 .U60968 .059506 .058048 5 4 3 2 1 M. Ooelue. D.I'. Sine. D. 1". Cotang. D. 1' . Tug. M 94 85 COSINES, TANGENTS, AND COTANGENTS. 235 BO 17*0 11 Sine. D.P. Cosine. D. 1". Tang. D. I'. Gotang. M. 1 2 8.940296 .941738 .943174 24.03 23.95 9.993344 .998333 .998322 .18 .19 8.941952 .943404 .944352 24.21 24.13 1.058048 .056596 .055148 60 59 58 3 4 5 .944606 .946034 .947456 23.79 2371 .998311 .998300 .998289 .19 .19 .916295 .947734 .949168 23.97 23.90 .053705 .052266 .050832 57 66 65 6 7 6 9 .948874 .950287 .951696 .953100 23.55 23.48 23.40 23.32 .998277 .998266 .998255 .998243 .19 .19 .19 .19 .950597 .952021 953441 .954856 23.74 23.67 23.59 23.51 .049403 .047979 .046559 .045144 54 63 52 61 10 11 12 13 14 15 8.954499 .955394 .957284 .958670 .960052 .961429 23.25 23.17 23.10 23.02 22.95 9.998232 .998220 .993209 .993197 .998186 .998174 .19 .19 .19 .19 .19 8.956267 .957674 .959075 .960473 .961866 .963255 23.44 23.36 23.29 23.22 23.14 1.043733 .042326 .040925 .039527 .038134 .036745 50 49 48 47 46 45 16 17 18 19 962301 .964170 .965534 .966393 22.81 22.73 22.66 22.59 .993163 .998151 .993139 .998128 .19 .20 .20 .20 .964639 .966019 .967394 .968766 23.00 22.93 22.86 22.79 .035361 .033981 .032606 .031234 44 43 42 41 20 21 22 23 24 25 26 27 28 8.963249 .969600 .970947 .972239 .973623 .9749f,2 .976293 .977619 .978941 .980259 22.52 22.45 22.38 22.31 22.24 22.17 22.10 22,03 21.97 21.90 9.998116 .998104 .993092 .993080 .998068 .998056 .998044 .993032 .998020 .998008 .20 .20 .20 .20 .20 .20 .20 .20 .20 .20 8.970133 .971496 .972855 .974209 .975560 .976906 .978243 .979586 .930921 .932251 22.72 22.65 22.58 22.51 22.44 22.37 22.30 22.24 22.17 22.10 1.029867 .023504 .027145 .025791 .024440 .023(194 .021752 .020414 .019079 .017749 40 39 38 37 36 35 34 33 32 31 30 31 32 33 8.981573 .93S333 .984189 .985491 2183 21.77 21.70 9.997996 .997984 .997972 .997959 .20 .20 .20 8.983577 .934899 986217 .987532 22.04 21.97 21.91 1.016423 .015101 .013783 .012468 30 | 29 28 27 34 35 36 37 38 39 .986789 .988033 .989374 .990660 .991943 .993222 21.57 21.51 21.44 21.38 21.31 21.25 .997947 .997935 .997922 .997910 .997897 .997885 .21 .21 .21 .21 .21 .21 .988842 .990149 .991451 .992750 .994045 .995337 21.78 21.71 21.65 21.59 21.52 21.46 .011158 .009351 .008549 .007250 .005955 .004663 26 25 24 23 22 21 40 41 42 43 44 45 46 47 48 49 8.994497 .995768 .997036 .998299 .999560 9.000316 .002069 .003318 .004563 .005805 21.19 21.12 21.06 21.00 20.94 20.88 20.82 20.76 20.70 20.64 9.997872 .997860 .997847 .997835 .997822 .997809 .997797 .997734 .997771 .997758 21 .21 .21 .21 .21 .21 .21 .21 .21 .21 8.996624 .997903 .999138 9.000465 .001738 .003007 .004272 .005534 .006792 .003047 21.40 21.34 21.27 21.21 21.15 21.09 21.03 20.97 20.91 20.85 1.003376 .002092 .000812 0.999535 .993262 .996993 .995728 .994466 .993208 .991953 20 19 18 17 16 15 14 13 12 11 50 51 52 53 54 55 56 57 58 59 60 9.007044 .003278 .009510 .010737 .011962 .013182 .014400 .015613 .016824 .018031 .019235 20.58 20.52 20.46 20.40 20.35 20.29 2023 20.17 20.12 20.06 9.997745 .997732 .997719 .997706 .997693 .997690 .997667 .997654 .997641 .997623 .997614 .22 .22 .22 .22 .22 .22 .22 .22 .22 .22 9.009298 .010546 .011790 013031 .014263 .015502 .016732 .017959 .019183 .020403 .021620 20.80 20.74 20.68 2062 20.56 20.51 20.45 20.39 20.34 20.28 0.990702 .989454 .988210 .986969 .985732 .984498 .983*68 .982041 .9*0817 .979597 .978380 10 9 8 7 6 6 4 3 2 M. Oosina | D. 1". Sine. D. 1". Gotang. D. 1". Tang. M. 236 TABLE II. LOGARITHMIC SINES, M. Sine. D. 1". Cosine. D 1". Tang. D. 1". Cotang. M. 1 2 9.019235 .020435 .021632 20.00 19.95 9.997614 .997601 .997588 .22 .22 9.021620 .022834 .024044 20.23 20.17 on i o 0.978380 .977166 .975956 lo" 59 58 3 4 6 6 7 8 9 .022325 .024016 .025203 .026386 .027567 .028744 .029918 19.89 19.84 19.78 19.73 19.67 19.62 19.57 19.51 .997574 .997561 .997547 .997534 .997520 .997507 .997493 !22 .22 22 .23 .23 23 23 .025251 .026455 .027655 .028852 .030046 .031237 .032425 SRf.l* 20.06 20.01 19.95 19.90 19.85 19.79 19.74 .974749 .973515 .972345 .971148 .969954 .968763 .967576 67 56 65 64 53 52 61 10 11 12 13 14 15 9.031089 032257 .033421 .034582 .035741 .036896 19.46 19.41 19.36 19.30 19.25 ici on 9.997480 .997466 .997452 .997439 .997425 .997411 23 23 .23 23 .23 oo 9.033609 .034791 .C35969 .037144 .038316 .039485 19.69 19.64 19.58 1953 19.48 1Q Al 0.966391 .965209 .964031 .962856 .961684 .960515 60 49 48 47 46 45 16 17 13 19 .038048 .039197 .040342 .041485 Iw.SHI 19.15 19.10 19.05 19.00 .997397 .997333 .997369 .997355 JM .23 .23 .23 .23 .040651 .041813 .042973 .044130 llf.M 19.38 19.33 19.28 19.23 .959349 .958187 .957027 .955870 44 43 42 41 20 21 22 23 9.042625 .043762 .044895 .046026 18.95 18.90 18.85 9.997341 .997327 .997313 .997299 .23 .23 .24 9.045284 .046434 .047582 .048727 19.18 19.13 19.08 0.954716 .953566 .952418 .951273 40 39 38 37 24 25 26 .047154 .048279 .049400 18.80 18.75 18.70 .997285 .997271 .997257 .24 .24 .24 .049869 051008 .052144 19.03 18.98 18.93 .950131 .948992 .947856 36 35 34 27 28 29 .050519 .05M35 .052749 18.65 18.60 18.65 18.50 .997242 .997228 .997214 .24 .24 .24 .24 .053277 .054407 .055535 18.89" 18.84 1879 18.74 .946723 .945593 .944465 33 32 31 10 31 32 33 34 36 36 37 38 39 9.053859 .054966 .056071 .057172 .058271 .059367 .06(M60 .061551 .062639 .063724 18.46 1841 18.36 18.31 1827 18.22 18.17 18.13 18.08 18.04 9.997199 .997186 .997170 .997156 .997141 .997127 .997112 .997098 .997(183 .997068 .24 .24 .24 .24 .24 .24 .24 .24 .25 .25 9.066659 .057781 .058900 .060016 .061130 .062240 .063348 .064453 .065556 .066655 18.70 18.65 18.60 1856 18.51 1846 18.42 18.37 18.33 18.28 0.943341 .942219 .941100 .939984 .938870 .937760 .936652 .935547 .934444 .933345 30 29 28 27 26 25 24 23 22 21 40 41 42 43 44 45 46 47 48 9.064806 .065885 .066962 .06S036 .069107 .070176 .071242 .072306 .073366 17.99 17.95 17.90 17.86 17.81 17.77 17.72 17.68 9.997053 .997039 .997024 .997009 .996994 .996979 .996964 .996949 .996934 .25 .25 .25 .25 .25 .25 .25 .25 9.067752 .068846 .069938 .071027 .0721 13 .073197 .074278 .075356 .076432 18.24 18.19 18.15 18.10 18.06 18.02 17.97 17.93 0.932248 .931154 .930(62 .928973 .927887 .9268(0 .925722 .924644 .923568 20 19 18 17 16 15 14 13 12 49 .074424 17.64 17.59 .996919 .25 .25 .077505 17.89 17.84 .922495 11 50 51 3.075480 .076533 17.55 9.996904 .996889 .25 9.078576 .079644 17.80 . .*! 0.921424 .920356 10 9 62 53 54 55 66 57 58 .077583 .078631 .079676 .080719 .081759 .082797 .083832 17.51 17.46 17.42 17.38 17.34 17.29 17.25 .996874 .996858 .996843 .996828 .996812 .996797 .996782 .25 .25 .25 .26 .26 .26 .26 .080710 .081773 .082833 .083891 .084947 .086000 .087050 17.76 17.72 17.67 17.63 17.59 17.55 17.51 .919290 .918227 .917167 .916109 .915053 .914000 .912950 8 7 6 6 4 3 2 59 60 .084864 .085894 17.21 17.17 .996766 .996751 .26 .26 .088098 .089144 17.47 17.43 .911902 .910S56 HI. Codne. D. 1". Sine D.I. Cotang. D.I". Tang. M 83 COSINES, TANGENTS, AND COTANGENTS. 237 70 M. Blue. D. 1". Cosine. D.1*. Tang. D. i. Coiaiig U. 1 2 3 4 5 8 7 8 9 9.085894 .086922 .087947 .088970 .089990 .091003 .092024 .093037 .094047 .095056 17.13 17.09 17.05 17.00 16.96 16.92 16.88 16.84 16.80 16.76 9.996751 .996735 .996720 .996704 996688 996673 996657 .996641 .996625 .996610 .26 .26 .26 .26 .26 .26 .26 .26 .26 .26 9.089144 .090187 .091228 .092266 .093302 094336 .095367 096395 .097422 .098446 17.39 17.35 17.31 17.27 17.23 17.19 17.15 17.11 17.07 17.03 0.910856 .909313 .908772 .907734 .9UK93 .905664 .9046:53 .903605 .902578 .901554 80 69 68 67 56 65 51 63 52 61 10 11 12 13 14 15 16 17 9.096062 .097065 .093066 .099065 100062 .101056 102048 .103037 16.73 16.69 16.65 16.61 16.57 16.53 16.49 9.996594 996578 996562 996546 996530 996514 996498 996482 .27 .27 .27 .27 .27 .27 .27 9.099468 .100487 101504 102519 103532 .104542 105550 .106556 16.99 16.95 16.91 16.88 16.84 16.80 16.76 0.900532 .899513 .893496 .897481 .896468 .895458 .894450 .893444 60 49 48 47 46 46 44 43 18 .104025 99G4C5 27 .107559 16 69 .892441 42 19 .105010 16.38 .996449 .27 .108560 1&6S .891440 41 20 21 22 23 24 25 26 27 23 29 9.105992 106973 107951 .108927 .109901 .110873 .111842 .112809 .113774 .114737 16.34 16.30 16.27 16.23 16.19 16.16 16.12 16.08 16.05 16.01 9.996433 996417 .996400 9963GS .996351 996335 996318 .996302 996285 .27 .27 .27 .27 .27 .27 .23 .28 .28 .28 9.109559 .110556 .111551 .112543 .113533 .114521 .115507 .116491 .117472 .118452 16.61 16.58 16.54 16.50 16.47 16.43 16.39 16.36 16.32 16.29 0.890441 .889444 .888449 .887457 .886467 .885479 .884493 .883509 .882528 .881548 40 39 38 37 36 36 34 33 32 31 30 31 32 33 34 35 36 37 38 39 9.115698 .116656 .117613 .118567 .119519 .120469 .121417 .122362 .123306 .124248 T5.98 1594 15.90 15.87 15.83 15.80 15.76 15.73 15.69 15.66 9.996269 .996252 .996235 .996219 .996202 996185 996168 996151 996134 .996117 .28 .28 .28 .28 .23 .23 .23 28 .28 .28 9.119429 .120404 .121377 .122.-M8 .123317 124284 125249 126211 127172 128130 16.25 16.22 16.18 16.15 16.11 16.03 16.04 16.01 15.93 15.94 0.880571 .879596 .878623 .877652 .876683 .875716 .874751 .873789 .872823 .871870 30 29 28 27 26 26 24 23 22 21 40 41 42 43 44 45 48 47 48 49 9.125187 .126125 .127060 .127993 .128925 .129854 .130781 .131706 .132630 .133551 15.62 15.59 15.56 15.52 15.49 15.45 15.42 15.39 15.35 15.32 9.996100 996083 996066 996049 .996032 .996015 .995998 .995980 995963 995946 .28 .28 .28 .29 .29 .29 .29 .29 .29 .29 9.129087 130041 .130994 .131944 .132893 .133339 .134784 .135726 .136667 .137605 15.91 15.87 15.84 15.81 16.77 15.74 16.71 15.68 15.64 16.61 0.870913 .869959 .869006 .868056 .867107 .866161 .8652t6 .864274 .863333 .862395 20 19 18 17 16 16 14 13 12 11 60 51 52 9.134470 .135337 .136303 15.29 15.26 9.995928 .995911 995894 .29 .29 9.138542 139476 140409 15.58 15.55 IE C1 0.861453 .860524 .859591 10 9 8 53 64 65 56 57 137216 .138128 .139037 .139944 .140S50 15.19 1516 15.13 15.09 995876 995.359 995841 995,323 995806 .29 .29 .29 .29 141340 142269 .143196 144121 .14S044 15.43 15.45 15.42 15.39 .858660 .857731 .856804 .855879 .854956 7 6 6 4 3 68 59 141754 142655 15.03 995788 995771 .29 .145966 146885 15.32 .854034 .853115 2 60 .143555 .995753 .147803 .852197 M. Cosine. D. i. Blue. D. 1". Ootaug. D. 1' . Ring. M 238 TABLE II. LOGARITHMIC SINES, M. Sine D. 1". Ccelne. D.I' Twig. D. 1". Cotang M. 1 2 3 4 5 6 7 8 9 9 143555 .144453 .145349 .146243 .147136 .148026 .148915 .149802 .150686 .151569 14.97 14.93 14.90 14.87 14.84 14.81 14.78 14.75 14.72 14.69 9.995753 .995735 .995717 .995699 995681 .995664 .995646 .995628 .995610 .995591 .30 .30 .30 .30 .30 .30 .30 .30 .30 .30 9. 147803 .148718 .149632 .150544 .151454 152363 .153269 154174 155077 .155978 15.26 15.23 15.20 15.17 15.14 15.11 15.08 15.05 15.02 14.99 0.852197 .851282 .850368 .849456 .848546 .847637 .846731 .845826 .844923 .844022 60 59 58 67 56 55 54 53 52 61 10 11 9.152451 .153330 14.66 9.995573 .995555 .30 9.156877 .157775 14.96 0.843123 .842225 50 49 12 13 14 15 16 17 18 19 154208 155(183 155957 .156830 .157700 .158569 .159435 .160301 14.60 14.57 14.54 14.51 14.48 14.45 14.42 14.39 .995537 .995519 .995501 .995482 .995464 .995446 .995427 .995409 .30 .30 .30 .30 31 .31 .31 .31 .31 .158671 .159565 .160457 .161347 .162236 .163123 .164008 .164892 14.93 1490 14.87 14.84 14.81 14.78 14.75 14.73 14.70 .841329 .840435 .839543 .838653 .837764 .836877 835992 .835108 48 47 46 45 44 43 42 41 20 21 22 23 24 25 26 27 28 29 9.161164 .162025 .162885 .163743 .164600 .165454 .166307 .167159 168008 .168856 14.36 14.33 14.30 14.27 14.24 14.22 14.19 14.16 14.13 14.10 9.995390 .995372 .995353 .995334 .995316 .995297 .995278 995260 .995241 .995222 .31 .31 31 .31 .31 .31 .31 .31 .31 .31 9.165774 .166654 .167532 .168409 .169284 .170157 .171029 .171899 .172767 .173634 14.67 14.64 14.61 14.58 14.56 14.53 14.50 14.47 14.44 14.42 0.834226 .833346 832468 .831591 .830716 .829843 .828971 .828101 .827233 .826366 40 39 38 37 36 35 34 33 32 31 30 31 9.169702 .170547 14.07 9.995203 .995184 .31 9.174499 .175362 14.39 0.825501 .824688 30 29 32 33 34 35 36 37 38 39 .171389 .172230 .173070 .173908 .174744 .175578 176411 .177242 14.02 13.99 13.96 13.94 13.91 13.88 13.85 13.83 .995165 .995146 .995127 .995108 .995089 .995070 .995051 .995032 .32 .32 .32 .32 .32 .32 32 .32 .32 .176224 .177084 .177942 .178799 .179655 .180508 .181360 .182211 14.33 14.31 14.28 14.25 14.23 14.20 14.17 14.15 .823776 .822916 .822058 .821201 .820345 .819492 .818640 .81778? 23 27 26 25 24 23 22 21 40 41 42 43 9.178072 .178900 .179726 .180551 13.80 13.77 13.75 9.995013 .994993 .994974 .994955 .32 .32 .32 9.183059 .183907 .184752 .185597 14.12 14.09 14.07 0.816941 .816093 .815248 .814403 20 19 18 17 44 '15 46 47 48 49 .181374 .182196 .183016 .183834 .184651 .185466 13.69 13.67 13.64 13.61 13.59 13.56 .994935 .994916 .994896 .994877 .994857 .994838 .32 .32 .33 .33 .33 .33 .186439 .187280 .188120 .188958 .189794 .190629 14.02 13.99 13.97 13.94 13.91 13.89 .813561 .812720 811880 .811042 .810206 .809371 16 16 14 13 12 11 60 51 52 53 54 55 66 67 58 59 9.186280 187092 .187903 .188712 .189519 .190325 .191130 .191933 .192734 .193534 1364 13.51 13.48 13.46 13.4J 13.41 13.38 13.36 13.33 9.994818 .994798 .994779 .994759 .994739 .994720 994700 .994680 994660 994640 .33 .33 .33 .33 .33 .33 .33 .33 .33 9.191462 .192294 .193124 .193953 .194780 .195606 .196430 .197253 .198074 .198894 13.86 13.84 13.81 13.79 13.76 13.74 13.71 13.69 13.66 0.808538 ' .807706 .806876 .806047 .805220 .804394 .803570 .802747 .801926 .801106 10 9 8 7 6 6 4 3 2 60 .194332 .994620 199713 .800287 M. Conine. D. 1". Sloe. D. 1". Cotang. D. 1". Taug. M. 980 COSINES, TANGENTS, AND COTANGENTS. 239 90 1TO" M. Sine. D. 1". Cosine. D. 1". Tung. D. i: Cotang. M. 1 2 3 4 5 6 7 8 9 9.194332 .195129 .195925 .196719 .197511 .198302 .199091 .199379 .200666 .201451 13.28 13.26 13.23 13.21 13.18 1316 13.13 13.1 13.08 13.06 97994620" .994600 .994580 .994560 .994541) .994519 .994499 .994479 .994459 .994438 .33 .33 .34 .34 .34 .34 .34 .34 .34 .34 9.199713 .200529 .201345 .202159 .202971 .2113732 .204592 .205400 .206207 .207013 J3.62 13.59 13.57 13.54 13.52 13.49 13.47 13.45 13.42 13.40 0.800287 .799471 .798655 .797841 .797029 .796218 .795403 .794600 .793793 .792987 60 59 58 57 56 55 54 53 52 51 10 11 12 9.202234 .203017 .203797 13.04 13.01 1O Oil P 9944 18 994398 .994377 .34 .34 04 9.207817 .208619 .209420 13.38 13.35 10 qq 0.792183 .791331 .790530 50 49 43 13 14 15 16 17 18 19 204577 205354 206131 206906 207679 203452 209222 IX. M) 1296 12.94 12.92 i2.89 12.87 12.85 12.32 .994357 .994336 994316 .994295 .994274 .9942.54 .994233 .<>* .34 .34 .34 .34 .34 .35 .36 .210220 .211013 .211815 .212611 .213405 .214193 .214989 ia.oo 13.31 13.23 13.26 13.24 13.21 13.19 13.17 .789780 .788982 .788185 .787.3S9 .78659.1 .785802 .785011 47 46 45 44 43 42 41 20 21 22 23 24 9.209992 210760 211526 .212291 .213056 12.80 12.78 1275 1273 9.994212 .994191 .994171 .994150 .994129 .35 .35 .35 .35 oc 9.215780 216568 .217356 .218142 .218926 13.15 13.12 13.10 13.08 i o nc 0.784220 .783432 .782644 .781858 .781074 40 39 38 37 36 25 26 27 28 29 21381S 214579 215338 216097 216854 12.71 1268 12.66 12.64 12.62 12.59 .994108 .994037 .994066 .994045 .994024 ID .35 .35 .35 .35 .35 .219710 .220492 221272 2220.12 222330 lo.UO 13.03 13.01 12.99 12.97 12.95 .780290 .779508 .778728 .777948 .777170 35 34 33 32 31 30 31 32 33 34 35 36 37 38 39 9.217609 218363 819116 219368 220618 221367 222115 222861 .223606 .224349 12.57 12.55 12.53 12.50 12.48 12.46 12.44 12.42 12.39 12.37 9.994003 9939S2 993960 993939 .993918 .993397 .993375 .993354 .993332 .993811 .35 .3.1 .35 .35 .36 .36 .36 .36 .36 .36 9.223607 224332 225156 225929 226700 .227471 223239 .229007 .229773 .230539 12.92 12.90 12.88 12.86 12.84 12.82 12.79 12.77 12.75 12.73 0.776393 .775618 .774844 .774071 .773300 .772529 .771761 .770993 .770227 .769461 30 29 28 27 26 25 24 23 22 21 40 41 42 43 9.225092 225833 .226573 .227311 12.35 12.33 12.31 9.993789 993768 993746 .993725 .36 .36 .36 9.231302 232065 232326 233586 12.71 12.69 12.67 0.768698 .767935 .767174 .766414 20 19 18 17 44 45 46 47 .228048 .228784 .229518 .230252 12.29 12.26 12.24 12.22 .993703 .993681 .993660 993633 .36 .36 .36 .36 234345 235103 235359 .236614 12 65 12.63 12.60 12.58 .765655 .764397 .764141 .763386 16 15 14 13 48 49 .230934 .231715 12.20 12.18 12.16 .993616 .36 .36 .36 .237363 233120 12.56 12.54 12.52 .762632 .761880 12 11 50 9.232444 9.993572 9.238872 0.761 128 10 51 52 53 54 55 56 57 233172 233399 234625 235349 .236073 .236795 .237515 12.14 12.12 12.10 12.07 12.05 12.03 12.01 993550 993528 .9935(16 .993484 .993462 .993440 .993413 .37 .37 .37 .37 .37 .37 37 .239622 240371 241118 .241865 242610 .243354 .244097 12.50 12.48 12.46 12.44 12.42 12.40 12.33 .760378 .759629 .758882 .758135 .757390 .756646 .755903 9 8 7 6 5 4 3 58 59 .2382.'6 .233953 11.99 11.97 993396 .993374 .37 244339 .245579 12.36 12.34 .755161 .754421 2 I 60 .23967** 11.95 .993351 .37 .216319 12.32 .753681 M. Cosine. D. 1". Sine. D.1". Cotaiig. D. 1" Tang. M. 99 800 240 HP TABLE II. LOGARITHMIC SINES, .239670 .240386 .241101 .941814 .842686 .243237 .243947 .244656 .245363 9.246775 .247478 .248181 .261677 .252373 .253067 9.253761 .254463 .255144 .855834 .257211 .259268 .259951 .261314 .264027 M. Cosine. 1000 .265377 .266051 9.267395 .268734 .270069 .270735 .271400 .272064 .272726 9.2T4049 .274708 .275367 .276025 .276681 .277337 .277991 .278645 .279948 D. 1". Cosine. 11.93 11.91 11.89 11.87 11.85 11.83 11.81 11.79 11.77 11.75 11.73 11.71 11.69 11.67 11.66 11.63 11.61 11.59 11.58 11.56 11.54 11.52 11.60 11.48 11.46 11.44 11.42 11 41 11.39 11.37 11.35 11.33 11.31 11.30 11.28 11.26 11.24 11.22 11.20 11.19 11.17 11.16 11.13 11.12 11.10 11.08 11.06 11.05 11.03 11.01 10.99 10.98 10.96 10.94 10.92 10.91 10.89 10.87 10.86 10.84 D. 1". 9.993351 .993329 .993307 .993284 993240 .993217 .993195 .993172 .993149 9.993127 .993104 .993081 .993059 .993036 .993013 .992967 .992375 .992783 .992736 .992713 .992572 .992335 .992287 .992263 .992214 9.992190 .992166 .992142 .992118 .992093 .992044 .992020 .991996 .991971 .991947 D. 1". .39 .40 .40 .40 .40 .40 .40 .40 .40 .40 .40 .40 .40 .40 .40 .40 .41 .41 .41 .41 .41 .41 .41 D. 1". 9.246319 .247057 .247794 .248530 .249264 .249998 .250730 .251461 .252191 9.253648 .254374 .255100 .255824 .256547 .257269 .257990 .258710 .260146 9.260863 .261578 .262292 .263005 .263717 .264428 .265138 .265847 .266555 .267261 9.267967 .268671 .269375 .270077 .270779 .271479 .272178 .272876 .273573 9.274964 .275658 .276351 .277043 .277734 .278424 .279113 .279801 .230488 .281174 .282542 .283225 .283907 .285947 .287301 .287977 CoUng. D. 1". Cotang M. 12.30 12.28 12.26 12.24 12.22 12.20 12.18 12.17 12.15 12.13 12.11 12.09 12.07 12.05 12.03 12.01 12.00 11.98 11.96 11.94 11.92 11.90 11.89 11.87 11.85 11.83 11.81 11.79 11.78 11.76 11.74 11.72 11.70 11.69 11.67 11.65 11.64 11.62 11.60 11.58 11.57 11.55 11.53 11.51 11.50 11.48 11.46 11.45 11.43 11.41 11.40 11.38 11.36 11.35 11.33 11.31 11.30 11.28 11.26 11.25 D. 1". 0.753681 .752943 .752206 .751470 .750736 .750002 .749270 .748539 .747809 .747080 0.746352 .745626 .744900 .744176 .743453 .742731 .742010 .741290 .740571 .739854 0.739137 .738422 .737708 .736995 .736283 .735572 .734862 .734153 .733445 .732739 0.732033 .731329 730625 .729923 .729221 .728521 .727822 .727124 .726427 .725731 0.725036 .724342 .723649 .722957 .722266 .721576 .720887 .720199 .719512 .718826 0.718142 .717458 .716775 .716093 .715412 .714732 .714053 .713376 .712699 .712023 .711348 Tang. M. COSINES, TANGENTS, AND COTANGENTS. 241 HO 1090 M. Sine D.I". Cosine. D, 1". Tang. D. 1". Cotang. M. 1 2 3 4 5 9.280599 .281248 .281897 .282544 .283190 .283836 10.82 10.81 10.79 10.77 10.76 9.991947 .991922 .991897 -991873 99184S 991823 .41 .41 .41 .41 .41 9.288652 .289326 .289999 .290671 .291342 .292013 11.23 11.22 11.20 11.18 11.17 0.711348 .710674 .710001 .709329 .7086C9 .707987 60 59 58 57 1 66 65 6 7 8 9 .284480 .285124 .285766 .286408 10.74 10.72 10.71 10.69 10.67 .991799 .991774 .991749 .991724 .41 .41 .41 42 .292682 .293350 .294017 .294684 11.14 11.12 11.11 11.09 .707318 .706650 .705983 .705316 54 63 62 61 10 11 12 13 14 9.287048 .237688 .288326 .288964 289GOO 10.66 10.64 10.63 10.61 9.991699 .991674 .991649 .991624 .991599 .42 .42 .42 .42 9.295349 .296013 .296677 .297339 .298001 11.07 11.H6 11.04 11.03 0.704651 .703987 .703323 .702661 .701999 60 49 48 47 46 15 16 17 18 19 .290236 .290870 291504 .292137 .292768 10.59 10.58 10.56 10.55 10.53 10.51 .991574 .991549 .991524 .991498 .991473 .42 .42 .42 .42 .42 .298662 .299322 .299980 .300638 .301295 11.00 10.98 10.97 10.95 10.93 .701338 .700678 .700020 .699362 .698705 46 44 43 42 41 20 21 22 23 24 25 26 27 28 29 9.293399 .294029 294658 295286 295913 296539 ..297164 .297788 298412 .299034 10.50 10.48 10.47 10.45 10.43 10.42 10.40 10.39 10.37 10.36 9.991443 .991422 .991397 .991372 .991346 .991321 .991295 .991270 .991244 .991218 .42 .42 .42 .42 .42 .43 .43 .43 .43 .43 9.301951 .302607 .303261 .303914 .304567 .305218 .305869 .306519 .307168 .307816 10.92 16.90 10.89 10.87 10.86 10.84 10.83 10.81 10.80 10.78 0,698049 697393 .696739 .696086 .695433 .694782 .694131 .693481 .692832 .692184 40 39 38 37 36 35 34 33 32 31 30 31 32 33 34 35 36 37 38 39 S 299655 300276 300S95 .301514 302132 .302748 .303364 .303979 .304593 .305207 10.34 10.33 10.31 10.30 10.28 10.26 10.25 10.23 10.22 10.20 9.991193 .991167 .991141 .991115 .991090 .991064 .991038 .991012 .990986 .990960 .43 .43 .43 .43 .43 .43 .43 .43 .43 .43 9.308463 .309109 .309754 .310399 .311042 .311685 .312327 .312968 .313608 .314247 10.77 10.76 10.74 10.73 10.71 10.70 10.68 10.67 10.66 10.64 0.691537 .690891 .690246 .689601 .688958 .688315 .687673 .687032 .686392 .685753 30 29 28 27 26 25 24 23 22 21 40 41 42 43 44 45 46 '47 48 49 9.305819 .306430 .307041 .307650 .308259 .308867 .309471 .310080 .310685 .311239 10.19 10.17 10.16 10.14 10.13 10.12 10.10 10.09 10.07 10.06 9.990934 .990908 .990882 .990355 .990829 .990803 .990777 .990750 .990724 .990697 .44 .44 .44 .44 .44 .44 .44 .44 .44 .44 9.314885 .315523 .316159 .316795 .317430 .318064 .318697 .319330 .319961 .320592 10.62 10.61 10.60 10.58 10.57 10.55 10.54 10.53 10.51 10.50 0.685115 .684477 683841 .683205 .682570 .681936 .681303 .680670 .680039 .679408 20 19 18 17 16 15 14 13 12 11 60 61 62 53 64 65 66 67 9.311893 .312495 .313097 313693 .314297 .314897 .315495 .316092 10.04 10.03 10.01 10.00 9.98 9.97 9.96 9.990671 .990645 .990618 .990591 .990565 .990538 .990511 .990485 .44 .44 .44 .44 .44 .44 .45 9.321222 .321851 .322479 .323106 .323733 .324358 324983 .325607 10.48 10.47 10.46 10.44 10.43 10.41 10.40 0.678778 .678149 .677521 .676894 .676267 .675642 675017 .674393 10 9 8 7 68 69 60 316689 .317284 .317879 9.94 9.93 9.91 .99(1458 .990431 .990404 .45 .45 .45 32H231 326853 .327475 10.39 10.37 10.36 .673769 .673147 .672525 M. Ocdne. D.l. 8b. D.l. Cotang. D. 1". Tang. M. 1010 242 TABLE II. LOGARITHMIC SINES, M. Sine D.I" Cosln*. D.l". Tang. D. 1". Cotaug. M. 1 9.317879 .318473 9.90 9.990404 .990378 .45 9.327475 .328095 10.35 in OQ 0.672525 671905 ~60~ 59 2 3 4 .319066 '.320249 9.87 9.86 .990351 .990324 .990297 A5 .45 328715 .329334 .329953 lU.oo 10.32 10.31 .671285 .670666 .670047 58 67 66 6 6 7 8 .320840 .321430 .322019 .322607 9.S3 9.81 9.80 n TO .990270 .990243 .990215 .990188 !45 .45 .45 .330570 .331187 .331803 .332418 m28 10.27 10.25 .669430 .668813 .668197 .667582 55 63 52 9 .323194 y. /y 9.77 .990161 '.45 .333033 10!23 .666967 51 10 11 12 13 14 9.323780 .324366 .324950 .325534 326117 9.76 9.75 9.73 9.72 Q 7fl 9.990134 .990107 .990079 .990052 990025 .45 .45 .46 .46 9.333646 .334259 .334871 .335482 .33(5093 1021 10.20 10.19 10.17 in IA 0.666354 .665741 .665129 .664518 .663907 60 49 48 47 46 16 .326700 y. / u O G .989997 .46 .336702 I''. ID in i K. .663298 45 16 17 .327281 .327862 y.oy 9.68 Q A 989970 989942 .46 .46 AH .33731 1 .337919 lU.lo 10.14 .662689 .662081 44 43 18 19 .328442 .329021 y. DO 9.65 9.64 .989915 .989887 .46 .46 .46 .33S527 .339133 lo'.n 10.10 .661473 .660867 42 41 20 21 22 23 9.329599 .330176 .330753 .331329 9.62 9.61 9.60 9.989860 .989832 .989804 .989777 .46 .46 .46 9.339739 .340344 .340948 .341552 10.08 10.07 10.06 in nK 0.660261 .659656 .659052 .658448 40 39 38 37 24 25 26 27 28 .33 '903 .332478 .333051 .333624 .334195 9^57 9.56 9.54 9.53 .989749 .989721 .989693 .989665 .989637 .46 .46 .46 .46 .47 At .342155 .342757 343358 343958 .344558 lu.Uo 10.03 10.02 10.01 10.00 Q QQ .657845 .657243 .656642 .656042 .655442 36 36 34 33 32 29 .334767 9^50 .989610 .47 .47 .345167 y.yo 9.97 .654843 31 30 9.335337 Q AQ 9.989582 9.345755 Q QA 0.654245 30 31 32 33 34 35 30 .33591 .336475 .337043 .337610 .338176 .338742 y.^y 9.48 9.46 9.45 9.44 9.43 Q .1 1 989553 939525 989497 989469 989441 989413 .47 .47 .47 .47 .47 .47 .346353 .346949 .347545 .348141 .348735 .349329 y.yo 9.95 9.93 9.92 9.91 9.90 O fifi .653647 .653051 .652455 .651859 .651265 .650671 29 28 27 26 25 24 37 38 39 .339307 339871 .340434 y.ii 9.40 9.39 9.37 989385 989356 989328 .47 .47 .47 .47 .349922 .350514 .351106 y.oo 9.87 9.86 9.85 .650078 .649486 .648894 23 22 21 4C 41 9.340996 .341558 9.36 9989300 .969271 .47 9.351697 .352237 9.84 O DO 0.648303 .647713 20 19 42 43 44 45 46 47 48 49 .342119 .342679 .343239 .343797 .344355 .344912 .345469 .346024 9.35 9.34 9.32 9.31 9.30 9.29 9.27 9.26 9.25 .989243 989214 989186 989157 989128 989100 .989071 .989042 .47 .47 .48 .48 .48 .48 .48 .48 .48 .352876 .353465 .354053 .354640 .355227 .355813 .356398 .356982 LIB 9.81 9.80 9.79 9.78 9.76 9.76 9.74 9.73 .647124 646535 .645947 .645360 .644773 .644187 643602 .643018 18 17 16 15 14 13 12 11 60 61 62 53 54 55 66 9.346579 .347134 .347687 .348240 .348792 349343 .349893 9.24 9.22 9.21 9.20 9.19 9.17 9.989014 .988985 .988956 .988927 .988898 .988869 .988840 .48 .48 .48 .48 .48 .48 9.357566 .358149 .358731 .359313 .359893 .360474 .361053 9.72 9.70 9.69 9.68 9.67 9.66 0.642434 .641851 .641269 .640687 .640107 .639526 .638947 10 9 8 7 6 6 4 67 68 .350443 .350992 9.16 9.15 91 A .988811 .988782 .48 .48 .361632 362210 9.66 9.63 .638368 .637790 3 2 59 6C .351540 .352088 .14 9.13 988753 .988724 .49 .49 362787 363364 9^61 .637213 .636636 1 M. Coelue. D. 1". Sine. D. 1". Cotaug. D.l. ~~Ttag~ "T 103 770 COSINES, TANGENTS, AND COTANGENTS. 243 M. Sine. D. 1". Cosine. D.1". Tang. D. 1". Cotang. M. Q 9.352038 Q 1 1 9.938724 49 9.363364 o An 0.636636 60 2 3 4 5 6 r 8 9 .352635 .353131 .3537?fi .354271 .354315 .355353 .355901 .356443 .356934 9*11 9.10 9.09 9.03 9.07 9.05 9.04 9.03 9.02 9.01 .938695 .933666 .983636 .938607 .938578 .988548 .938519 .938489 .938460 149 .49 .49 .49 .49 .49 .49 .363940 .364515 .365090 .365664 .366237 .366810 .367382 .367953 .363524 y.ou 9.59 9.58 9.57 9.55 9.54 9.53 9.52 9.51 9.60 .636060 .635485 .634910 .634336 .633763 .633190 .632618 .632047 .631476 59 58 67 66 55 54 53 52 51 10 9.357524 9.938430 9.369094 0.630906 60 11 12 13 14 15 16 17 18 19 .358064 .358603 359141 .359678 .360215 .360752 .361287 .361822 .362356 a 98 8.97 8.96 8.95 894 8.92 8.91 8.90 8.89 .938401 .983371 .983342 .988312 .988282 .988252 .938223 .988193 .983163 !49 .49 .50 .50 .50 .60 .50 .60 .60 .369663 370232 .370799 .371367 .371933 .372499 .373064 .373629 .374193 &48 9.47 9.45 9.44 9.43 9.42 9.41 9.40 9.39 .630337 .629768 .629201 .628633 .628067 .627501 .626936 .626371 .625807 49 48 47 46 45 44 43 42 41 20 9.362889 Q OQ 9.988133 en 9.374756 O OQ 0.625244 40 21 22 23 24 26 '26 27 28 29 .363422 .363954 .364485 .365016 .365546 .366075 .3G6604 .367131 .367659 O.oo 8.87 8.36 8.84 8.83 8.82 8.81 8.80 8.79 8.78 .988103 .988073 .988043 .988013 .937983 .S37953 .987922 987892 987862 .OiJ .50 .60 .50 .50 .60 .50 .50 .60 .61 .375319 .375831 .376442 .377003 .377563 .378122 .378681 .379239 .379797 y. oo 9.37 9.36 9.35 9.33 9.32 9.31 9.30 9.29 9.28 .624681 .624119 .623558 .622997 .622437 621878 .621319 .620761 .620203 39 38 37 36 35 34 33 32 31 30 9.363185 o 70 9.987832 9.380354 0.619646 30 31 368711 o. /o Q -YE 987801 C| .380910 O OA .619090 29 32 33 369236 369761 O. /D 8.74 Q P*O 987771 987740 .01 .61 Cl .381466 .332020 y.to 9.25 .618534 .6179.30 28 27 34 35 36 37 370285 370308 371330 371852 O.ld 8.72 8.71 8.70 .987710 .987679 987649 .987618 .51 .51 .51 .61 C| .382575 .383129 .383632 .384234 9/23 9.22 9.21 Q on .617425 .616871 .616318 .615766 26 25 24 23 38 39 .372373 .372394 8'63 8.66 .987588 .987557 .91 .61 .61 .384786 .385337 7.6 7.76 7.75 7.74 7.74 .530720 .530254 .529789 .529324 .528859 35 34 33 32 31 30 9.453342 9.981737 9.471605 7 7O. 0.528395 30 31 32 33 34 35 .453768 .454194 .454619 .455044 .455469 7.10 7.10 709 7.08 7.07 .931700 .981662 .981625 .981587 .931549 !62 .63 .63 .63 CQ .472069 .472532 .472995 .473457 .473919 /. to 7.72 7.71 7.71 7.70 .627931 .527468 .527005 .526543 .526081 29 28 27 26 25 36 37 38 39 .455893 .456316 .456739 .457162 7.07 7.06 7.05 7.04 7.04 .981512 .931474 .931436 .931399 .DO .63 .63 .63 .63 .474381 .474842 .475303 .475763 7.69 7.63 7.67 7.67 .525619 .625158 .524697 .624237 24 23 22 21 40 41 9.457584 .458006 7.03 9.931361 .981323 .63 9.476223 .476633 7.66 0.523777 .523317 20 19 42 .458427 7.02 .981235 .63 .477142 7.65 .522358 18 43 44 .458348 459268 7.01 701 .981247 .981209 '.63 /Q .477601 .478059 7.65 7.64 T O .522399 .521941 17 16 45 46 .459688 .460108 7.00 6.99 .981171 .981133 .DO .63 CO .478517 .478975 7.6,3 7.63 .521483 .521025 16 14 47 48 49 .460527 .460946 .461364 6.98 6.98 6.97 6.96 .981095 .981057 .981019 .OO .64 .64 .64 .479432 .479889 .480345 7^61 7.61 7.60 .520568 .520111 .519655 13 12 11 50 51 9.461782 .462199 6.96, 9.980981 .9,30942 .64 CA 9.480801 .481257 7.59 ff c<\ 0.519199 .618743 10 9 52 .462616 6.95 .980904 .64 .481712 7.59 .518288 8 53 64 55 56 57 58 59 60 .463032 .463448 .463364 .464279 .464694 .465108 .465522 .465935 6.94 6.93 6.93 692 6.91 6.90 690 6.89 .980366 .980827 .980789 .980750 .980712 .980673 .980635 .980596 .64 .64 .64 .64 .64 .64 .64 .64 .482167 .482621 .483075 '.483982 .484435 .484887 .485339 "*.58 7.57 7.57 7.56 7.55 755 7.54 7.53 .517833 .617379 .516925 .516471 .616018 .515565 .515113 .514661 7 6 6 4 3 2 I M. Cosine. D. 1". Sin* D.I". Cotang. D. 1". Tang. M. COSINES, TANGENTS, AND COTANGENTS. 247 M. Sine. D.I". Ooalne. D. 1". Tang. D. I". Cotang. M. 1 9.465935 466348 6.83 A QQ 9.980596 .930558 .64 fid 9.485339 .485791 7.53 ry CO 0.514661 .514209 60 59 2 3 4 6 466761 .467173 467585 467996 o.oo 6.87 6.86 6.85 A QK. .930519 .980480 .980442 .980403 .l>4 .65 .65 .65 .486242 .486693 .487143 .487593 7.54 7.51 7.51 7.50 .513753 .513307 .512357 .512407 58 57 56 55 6 7 8 9 .468407 .463817 .469227 .469637 D.ou 6.84 6.83 6.83 6.82 .980364 .930325 .980236 .980247 '.65 .65 .65 .65 .483043 .488492 .488941 .489390 7.50 7.49 7.43 7.48 7.47 .511957 .511508 .511059 .510610 54 53 52 51 10 11 12 9.470046 .470455 .470863 6.81 6.31 on 9.980208 .980169 .980130 .65 .65 AC 9.439838 .490286 .490733 7.46 7.46 0.510162 .509714 .509267 50 49 48 13 14 .471271 .471679 D.oU 679 A TO .980091 .930052 .DO .65 .491180 .491627 7.45 7.44 .508820 .508373 47 46 15 16 17 .472086 .472492 472898 O-/O 6.78 677 .9,30012 .979973 .979934 .65 .65 .492073 .492519 .492965 7.44 7.43 7.43 .507927 .507481 .507035 45 44 43 18 19 473304 473710 6.76 676 6.75 .979895 .979855 .66 .66 .66 .493410 .493354 7.42 7.41 7.41 .506590 .506146 42 41 20 21 22 23 9.474115 474519 474923 475327 6.74 6.74 6.73 9.979316 .979776 .979737 .979697 .66 .66 .66 9.494299 .494743 .495186 .495630 7.40 7.39 7.39 0.505701 .505257 .504814 .504370 40 39 33 37 24 25 26 475730 476133 476536 6.72 6.72 671 .979658 .979618 .979579 .66 .66 .66 .496073 .496515 .496957 7.38 7.38 7.37 .503927 .503485 .503043 36 35 34 27 .476938 6.70 .979539 .66 497399 7.36 .502601 33 28 29 .477340 .477741 6.69 6.69 6.68 .979499 .979459 .66 .66 .66 .497841 .498282 7.36 7.35 7.34 .502159 .501718 32 31 30 9.478142 9.979420 9.498722 0.501278 30 31 .478542 6.67 .979380 66 499163 7.34 .500837 29 32 33 34 35 36 37 38 .478942 .479342 .479741 .480140 .480539 .430937 .481334 6.67 6.66 665 6.65 6.64 6.63 6.63 .979340 .979300 .979263 .979220 .979180 .979140 .979100 .66 .67 .67 .67 .67 .67 .67 .499603 .500042 .500481 .500920 .501359 .501797 .502235 7.33 7.33 7.32 7.31 7.31 7.30 7.30 .500397 .499958 .499519 .499080 .498641 .498203 .497765 28 27 26 25 24 23 22 39 .481731 6.62 6.61 .979059 .67 .67 502672 7.29 7.28 .497328 21 40 9.482128 9.979019 9.503109 0.496391 20 1 41 .432525 6.61 .978979 .67 503546 7.28 .496454 19 42 43 .432921 .483316 6.60 6.59 .978939 .973898 .67 .67 503982 .504418 7.27 7.27 .496018 .495582 18 17 44 .483712 6.59 .973858 .67 .504854 7.26 .495146 16 45 46 47 .484107 .484501 .434395 6.58 6.57 6.57 .978817 .978777 .978737 .67 .67 .67 .505239 .505724 .506159 7.25 7.25 7.24 .494711 .494276 .493341 15 14 13 48 .485239 6.56 .978696 .68 .506593 7.24 .493407 12 49 .485682 6.55 6.55 .978655 .68 68 .507027 7.23 7.23 .492973 11 50 9.486075 9.978615 9.507460 492540 10 51 .486467 6.54 .978574 .68 .507893 7.22 .492107 9 52 .486860 6.54 .973533 .63 .503326 7.21 .491674 8 53 .437251 6.53 .973493 .68 .503759 7.21 .491241 7 54 55 56 .437643 .438034 .48,3424 6.52 6.52 6.51 .973452 .978411 .978370 68 .68 .63 .509191 .509622 .510T54 7.20 7.20 7.19 .490809 .490378 .439946 6 5 4 57 53 59 .488814 .489204 .489593 6 50 6.50 6 19 .978329 .978288 .978247 .68 .68 .63 .510435 .510916 .511^46 MS 7.17 .489515 .489034 .488654 3 2 1 60 .489982 6.48 .973206 .68 .511776 7.17 .4^8224 M. Cosine. D. 1". Sloe. D. 1" Cotang. D. 1". Tang. 1ST 73 248 TABLE 11. LOGARITHMIC SINES, 180 161 M. Sine. D. 1". Cosine. D. 1". Taog. D. 1". Cotang. M. f 9.489982 .490371 6.48 9.978206 .978165 .68 CO 9.511776 .5122(16 7.16 71 ft 0.488224 .487794 ~60~ 59 2 .490759 ii.47 .978124 .oy .512635 lo 71 r .487365 58 3 .491147 6.46 .978083 .69 .513064 .15 7 U 486936 57 4 .491535 6.46 .978042 CO 513493 i-jj 186507 56 6 6 7 8 9 .491922 .492308 .492695 .493081 .493466 6.4o 645 644 643 643 6.42 .973001 .977959 .977918 .977877 .977835 .oy .69 .69 .69 .69 .69 .513921 .514349 .514777 .615204 .515631 7.14 7.13 7.13 7.12 7.12 7.11 186079 .485651 485223 .484796 .484369 66 64 53 52 61 10 9.493851 & A\ 9.977794 9.516057 7 in 0.483943 50 11 .494236 0.41 .977752 .69 .616484 .111 .483516 49 12 13 14 15 16 17 18 19 .494621 .495005 .495383 .495772 .496154 .496537 .496.H9 .497301 6.41 6.40 6.39 6.39 6.33 6.38 6.37 636 6.36 .977711 .977669 .977623 .977586 .977544 .977503 .977461 .977419 .69 .69 69 .69 .69 .70 .70 .70 .70 .61-6910 517335 .517761 .518186 .518610 .519034 .619458 .519882 7. 10 7.09 7.09 7.08 7.08 7.07 7.07 7.06 7.05 .483090 .482665 .482239 .481814 .481390 .480966 .480542 .480118 48 47 46 45 44 43 42 41 20 21 22 23 9.497682 .498064 .498444 .498825 6.35 6.34 6.34 A oo 9.977377 .977335 .977293 .977251 .70 .70 .70 9.520305 .520723 .521151 .621573 7.05 7.04 7.04 T fiQ 0.479695 .479272 .478849 .478427 40 39 38 37 24 25 .499204 .499584 D.OO 6.33 QO .977209 .977167 .70 .521995 .522417 /.Uo 7.03 T ftf) .478005 .477583 36 36 26 27 28 29 .499963 .500342 .500721 .501099 O.O* 6.31 6.31 6.30 6.30 .977125 .977083 .977041 976999 .70 .70 .70 .70 .522838 .523259 .523680 .524100 7.IW 7.02 7.01 r.oi 7.00 .477162 .476741 .476320 .475900 34 33 32 31 30 31 9.501476 .501854 629 9.976957 .976914 .70 9.524520 .524940 6.99 0.476480 .476080 30 29 32 .502231 6.23 .976872 .525359 QQ .474641 28 33 34 36 36 37 38 39 .502607 .502984 .503360 .503735 .504110 .504485 .504860 6.23 6.27 6.27 626 625 6.25 6.24 6.24 .976830 .976787 .976745 .976702 .976660 .976617 .976574 .71 .71 .71 .71 .71 .71 .71 .71 .625778 626197 .526615 .527033 .527451 .627863 .528285 6 yy 6.98 6.97 6.97 6.96 6.96 6.C5 6.95 .474222 .473803 .473385 472967 472549 .472132 .471716 27 26 25 24 23 22 21 40 41 42 43 44 45 46 47 48 49 9.505234 .505608 .505981 .506354 .506727 .507099 .507471 .607843 .508214 .508585 6.23 622 622 6.21 6.21 6.20 6.19 6.19 6.18 6.18 9.976532 .976489 .976446 .976404 .976361 .976318 .976275 .976232 .976189 .976146 .71 .71 .71 .71 .71 .72 .72 .72 .72 .72 9.528702 .529119 .529535 .529951 .530366 .630781 .531196 .531611 .532025 .632439 6.94 6.94 6.93 6.93 6.92 6.91 6.91 6.90 6.90 6.89 0.471298 .470881 .470465 .470049 .469634 .469219 .463804 .468389 467975 .467561 20 19 18 17 16 15 14 13 12 11 50 9.508956 9.976103 9.532853 on 0.467147 10 61 52 53 54 55 56 57 63 59 60 .509326 .50%96 .510065 .510434 .510803 .511172 .511540 .511907 .512275 .512642 6.17 6.16 6.16 6.15 6.15 6.14 6.14 6.13 6.12 6.12 .976060 .976017 .975974 .975930 .975887 .975844 .975800 .975757 .975714 .975670 .72 .72 .72 .72 .72 .72 .72 .72 .72 .72 .533266 .533679 .534092 .534504 .534916 .535328 .535739 .536150 536561 .536972 o. By 6.88 6.88 6.87 6.87 6.86 6.86 6.85 6.85 6.84 .466734 .466321 465908 .465496 .465084 .464672 464261 .463850 .463439 .463028 9 8 7 6 5 4 3 2 1 M. Coeiuo. D.1". Bine. D. F. Cotang. D. !'. Tang. M. 108 COSINES, TANGENTS, AND COTANGENTS. 24f 190 160 U Sine. D. 1". Coelne. D.l" Tang. D. 1". Cota.ig ! M. 9.512642 61 1 9.975670 70 9.536972 QA 0.46?028 60 .513009 .513375 .513741 .514107 .1 1 6.11 6.10 6.09 .97.0627 .975583 .975569 .975496 ./O .73 .73 .73 73 .537382 .537792 .538202 .538611 D.B4 6.83 6.S3 6.82 a QO .462618 .462208 .461798 .461389 59 58 57 56 7 8 9 .614472 .514837 .515202 515566 615930 6.08 6.08 6.07 6.07 6.06 .975452 .975408 .975365 .975321 .975277 '.73 .73 .73 .73 .73 .539020 .539429 .539837 .540245 .540653 O.O'G 6.81 6.81 6.80 6.80 6.79 .460980 .460571 .460163 .459755 .459347 55 54 53 52 51 10 11 12 13 14 15 16 17 13 19 3.516294 .516657 .517020 .517382 .617745 .518107 .518468 .518829 .519190 .519551 6.05 6.05 6.04 6.04 6.03 6.03 6.02 6.02 6.01 6.00 9.975233 .975189 .975145 .975101 .975057 .975013 .974969 .974925 .974880 .974836 .73 .73 .73 .73 .73 .74 .74 .74 .74 .74 9.541061 .541468 .541875 .642281 .542688 .543094 .543499 .543905 .544310 .644715 6.79 6.78 6.78 6.77 6.77 6.76 6.76 6.75 6.75 6.74 0.458939 .458532 .458125 .457719 .457312 .456906 .456501 .456095 .455690 .455285 50 49 48 47 46 45 44 43 42 41 20 21 22 23 24 25 26 27 23 29 9.519911 .520271 .520631 .520990 .521349 .521707 .522066 .522424 .522781 .523138 6.00 5.99 6.99 5.98 5.98 5.97 5.97 5.96 5.95 6.95 9.974792 .974748 .974703 .974659 .974614 .974570 .974525 .974481 .974436 .974391 .74 .74 .74 .74 .74 .74 .74 .74 .74 .75 9.545119 .545524 .545928 .546331 .546735 .547138 .517540 .547943 .548345 .548747 6.74 6.73 6.73 6.72 6.72 6.71 6.71 6.70 6.70 6.69 0.454881 .454476 .454072 .453669 .453265 .452862 .452460 .452057 .451655 .451253 40 39 38 37 36 35 34 33 32 31 30 31 32 33 34 35 36 9.523495 ,523852 .524208 .524564 .524920 .525275 .525630 5.94 5.94 5.93 6.93 6.92 5.92 e ni 9.974347 .974302 .974257 .974212 .974167 .974122 .974077 .75 .75 .75 .75 .75 .75 9.549149 .549550 .549951 .550352 .550752 .551 153 .551552 6.69 6.68 6.63 6.67 6.67 6.67 0.450851 .450450 .450049 .449648 .449248 .448847 .448448 30 29 23 27 26 25 24 37 .525984 o.yi e Qrt .974032 75 .551952 6.66 C fi .448048 23 38 39 .526339 .526693 o.yu 5.90 5.89 .973987 .973942 >5 .75 .552351 .552750 6. DO 6.65 6.65 .447649 .447250 22 21 40 9.527046 C OQ 9.973897 9.553149 A /f 0.446851 20 41 42 43 44 .527400 .527753 .528105 .523458 VtCfV 5.88 5.88 6.87 E Q7 .973852 .973807 .973761 .973716 '.75 .75 .75 .553548 .553946 .554344 .554741 6.64 6.64 6.63 6.63 .446452 .446054 .445656 .445259 19 18 17 16 45 .528810 O.O/ fi flft .973671 7A .555139 6.62 a Q .444861 15 46 47 46 49 .529161 .529513 .529864 .530215 o.oo 5.86 5.85 6.85 5.84 .973625 .973580 .973535 .973489 . /O .76 .76 .76 .76 .555536 .555933 .556329 .556725 O.O4 6.61 6.61 6.60 6.60 .444464 .441067 .443671 .443275 14 13 12 11 50 9.530565 e QQ 9.973444 9.557121 0.442879 10 51 5? 63 54 55 56 57 58 69 60 .530915 .531265 .531614 .531963 .532312 .532661 .533009 .533357 .533704 .594052 O.OO 5.83 5.82 5.82 5.81 5.81 5.80 5.80 6.79 5.79 .973398 .973352 .973307 .973261 .973215 .973169 .973124 .973078 .973032 .972986 !76 .76 .76 .76 .76 .76 .76 .77 .77 .557517 .557913 .558308 .558703 .559097 .559491 .559885 .560279 ,560673 .561066 6.59 6.59 6.59 6.58 6.53 6.57 6.57 6.56 6.56 6.55 .442483 .442087 .441692 .441297 .440903 .440509 .440115 .439721 .439327 .438934 9 8 M. Oofllne. D.l. Slue. D. 1". Cotuug. D. 1". Ikng. M. 100 70Q -50 . TABLE II. LOGARITHMIC SINES, 900 160C M. Sine. D. 1". Coeine. D. I". Taug. D. 1". Cotang. M, 1 2 3 4 6 6 7 8 9 9.534052 .534399 .534745 .535092 .535438 .535783 .536129 .536474 .536818 .637163 5.78 5.78 5.77 5.77 5.76 5.76 5.75 6.75 6.74 6.74 9.972986 .972940 .972894 .972848 .972802 .972755 .972709 .972663 .972617 .972570 .77 .77 .77 .77 .77 .77 .77 .77 .77 .77 9.561066 .561459 .561851 .562244 .562636 .563(128 .563419 .563811 .564202 .564593 6.55 6.54 6.54 6.54 6.53 6.53 6.52 6.52 6.51 6.51 0.438934 .438541 .438149 .437756 .437364 .436972 .436581 .436189 .435798 .435407 ~60~ 59 58 67 56 55 64 53 52 51 10 11 12 13 14 9.537507 .537851 .538194 .538538 .538880 5.73 5.73 5.72 6.71 9.972524 .972478 .972431 .972385 .972338 .77 .77 .78 .78 9.564983 .565373 .565763 .566153 .566542 6.50 6.50 6.50 6.49 0.435017 .434627 .434237 .433847 .433458 60 49 48 47 46 15 16 17 18 .539223 .539565 .639907 .640249 5.71 6.70 5.70 6.69 .972291 .972245 .972198 .972151 .78 .78 .78 .78 TO .566932 .667320 .567709 .568098 6.49 6.48 6.48 6.47 H Af .433068 .432680 .432291 .431902 45 44 43 42 19 .640590 6.68 .972105 .7o .78 .668486 6.47 6.46 .431614 41 20 21 22 23 24 25 26 27 28 29 9.540931 .541272 .641613 .641953 .542293 .642632 .542971 .543310 .543649 .543987 6.68 5.67 6.67 5.66 5.66 6.65 6.65 6.64 664 6.63 9.972058 .97201 1 .971964 .971917 .971870 .971823 .971776 .971729 .971682 .971635 .78 .78 .78 .78 .78 .78 .78 .79 .79 .79 9.568873 .569261 .569648 .570035 .570422 .670809 .571195 .671581 .571967 .672352 646 6.46 6.45 6.45 ti.44 6.44 G.43 6.43 6.43 6.42 0.431127 .430739 .430352 .429965 .429578 .429191 .428806 .428419 .428033 .427648 40 39 38 37 36 36 34 33 32 31 30 31 32 9.544325 .544663 .645000 6.63 6.62 c co 9.971588 .971540 .971493 .79 .79 9.572738 .673123 .673507 6.42 6.41 A A 1 0.427262 .426877 .426493 30 29 28 33 34 35 36 37 38 545338 .545674 .646011 646347 .546683 .547019 O.D/ 6.61 6.61 6.60 6.60 6.59 C en .971446 .971398 .971351 .971303 .971256 .971208 >9 .79 .79 .79 .79 .673892 .674276 .674660 .676044 .676427 .675810 D.31 6.40 6.40 6.40 6.39 6.39 d QU .426108 .425724 .425340 .424956 .424573 .424190 27 26 26 24 23 22 39 .647354 o.&y 6.58 .971161 '.79 .676193 b.JC 6.88 .423807 21 40 41 42 43 44 45 46 47 48 49 9.547689 .548024 .548359 .648693 .549027 .649360 .549693 .650026 .650359 .650692 6.58 6.57 5.57 6.56 6.56 6.55 6.55 555 6.54 6.54 9.971113 .971066 .971018 .970970 .970922 .970874 .970827 .970779 .970731 .970683 .79 .80 .80 .80 .80 .80 .80 .80 .80 .80 9.676576 .676959 .577341 .677723 .678104 .678486 .578867 .579248 .579629 .680009 6.37 6.37 6.37 6.36 6.36 6.35 6.35 6.34 6.34 6.34 0.423424 .423041 .422659 .422277 .421896 .421514 .421 133 .420752 .420371 .419991 2(1 19 18 17 16 16 14 13 12 11 6G 9.551024 9.970635 9.680389 fi oo 0.419611 10 51 52 53 54 65 56 .551356 .551687 .552018 .552349 .552680 .553010 6.53 5.53 652 552 5.51 5.51 .970586 .970538 .970490 .970442 .970394 .970345 .80 .80 .80 .80 .80 .81 Ol .580769 .681149 .581528 .681907 .682286 .682665 DEM 6.33 6.32 6.32 6.32 6.31 Ol .419231 .418851 .418472 .418093 .417714 .417335 9 8 7 6 5 4 57 68 59 60 .553341 .653670 .654000 .654329 6.50 6.50 5.49 6.49 .970297 .970249 .970200 .970152 .ol .81 .81 .81 .683*44 .683422 .583*00 .584177 b.dl 6.30 6.30 6.90 .416956 .416578 .416200 .415823 3 2 1 M. Ooetao. D. 1'-. ffliift D. 1". Ootaiig. D. 1". Tang. M. 1100 60 COSINES, TANGENTS, AND COTANGENTS. a io M. Slue. D. 1". Cosine. D. 1". Tang. D. 1". Cotang. M. 1 2 3 4 5 9.554329 .554658 .664US7 .655315 .55r,643 .655971 5.48 5.43 5.47 6.47 6.46 9.970152 .970103 .970055 .970006 .969957 .969909 .81 .81 .81 .81 81 Q 1 9.584177 .584555 .584932 .585309 .585686 .586062 6.29 6.29 6.28 6.28 6.28 0.415823 .415445 .415068 .414691 .414314 .413938 60 69 68 67 56 55 6 7 8 .556299 .556626 556953 5.46 5.45 5.45 .969860 .969P11 .969762 01 .81 .81 Ql .586439 .5868 1 5 .587190 6.27 6.27 6.26 .413561 .413185 .412810 54 53 52 9 .557280 5.44 6.44 .969714 ,ol .81 .587566 6.26 6.26 .412434 61 10 11 12 13 14 15 16 17 18 19 9 557606 .557932 .558258 .55.3583 .558909 .559234 .559558 .559883 .560207 .560531 5.44 5.43 5.43 5.42 5.42 5.41 5.41 6.40 5.40 6.39 9.969665 .%9616 .969567 .969518 .969469 .969420 .969370 .969321 .969272 .969223 .82 .82 .82 .82 82 .82 .82 .82 .82 .82 9.587941 .588316 .588691 .589066 .589440 .589814 .590188 .590562 .590935 .591308 6.25 6.25 6.24 6.24 6.24 6.23 6.23 6.22 6.22 6.22 0.412059 .411634 .411309 .410934 .410560 .410186 .409812 .409438 .409065 .408692 60 49 48 47 46 45 44 43 42 41 20 9.560855 9.969173 9.591681 0.408319 40 21 .561178 5.39 .969124 .82 QO .592054 6.21 .407946 39 22 23 .561501 .561824 6.38 5.38 .969075 .969025 CM .82 .592426 .592799 6.21 6.20 .407574 .407201 38 37 24 25 .562146 .562468- 5.37 6.37 .968976 .968926 '.83 .593171 .593542 6.20 6.20 .406829 .406458 36 35 26 27 28 29 .562790 .563112 .563433 .563755 5.37 6.36 6.36 6.35 5.35 .968377 .968827 .968777 .968728 .83 .83 .83 .83 83 .593914 .594285 .594656 .595027 6.19 6.19 6.18 6.J8 6.18 .406086 .405715 .405344 .404973 34 33 32 31 30 31 32 9.564075 .5643% .564716 5.34 5.34 9.96S678 .968623 .968573 .83 83 9.595393 .595763 .596138 6.17 6.17 0.404602 .404232 .403862 30 29 28 33 34 35 .565036 .565356 .565676 5.33 6.33 5.32 .968523 .968479 .968429 .83 .83 .83 .596508 .596878 .597247 6.16 6.16 6.16 .403492 .403122 .402753 27 26 25 36 37 38 39 .565995 .566314 .566632 .566951 5.32 6.32 6.31 6.31 6.30 .968379 .968329 .968278 .968228 .83 .83 .83 .84 .84 .597616 .597985 .598354 .698722 6.15 6.15 6.15 6.14 6.14 .402384 .402015 .401646 .401278 24 23 22 21 40 41 42 43 44 45 48 47 48 49 9.567269 .667587 .567904 .568222 .568539 .568856 .569172 .569488 .569304 .670120 5.30 6.29 5.29 6.28 6.28 6.28 6.27 5.27 6.26 6.26 9.968178 .968123 .968078 968027 .967977 .967927 .967876 .967826 .967775 .967725 84 .84 .84 .84 .84 - .64 .81 .84 .84 .84 9.599091 .599459 .599827 .600194 .600562 .600929 .601296 .601663 .602029 .602395 6.13 613 6.13 6.12 6.12 6.12 6.11 6.11 6.10 6.10 0.400909 .400541 .400173 .399806 .399438 .399071 .398704 .398337 .397971 .397605 20 19 18 17 16 15 14 13 12 11 50 51 9.570435 .570751 5.25 9.967674 .967624 .84 9.602761 .603127 6.10 0.397239 .396873 10 9 52 .671066 6.25 .967573 .^4 .603493 6.09 .396507 8 53 54 55 .571380 .671695 : 572009 5.24 6.24 5.24 .967522 .%7471 .967421 .85 .85 .85 .603858 .604223 .604588 6.09 6.09 6.08 .396142 .395777 .395412 7 6 6 56 57 53 .572323 672636 572950 6.23 6.23 6.22 .967370 .967319 .967268 .85 .85 .85 .604953 .605317 .605682 6.08 6.07 6.07 .395047 .394683 .394318 4 3 2 59 60 .673263 .673575 6.22 6.21 .967217 .967166 .85 .85 .6fl6046 .606410 6.07 6.06 .393954 .393590 M. Oosino. D.l". Sine. D. 1". Gotang. "D^ Tang. M. 68 252 TABLE II. LOGARITHMIC SINES, 330 isr M Bine. D. 1". CoPine. D.I". Taiig. D. 1". Cotang. M. 1 9573575 .673888 5.21 9.967166 .967115 .85 QC 9.606410 .606773 6.00 c rt/s 0.393590 .393227 60 59 2 3 4 5 6 .574200 .574512 .574824 .575136 .675447 5.2( 5.2( 5.20 5.19 5.19 .967064 .967013 .966961 .966910 .966859 ,OO .85 .85 .85 .85 .607137 .607500 .607863 .608225 .60a r x38 o.Uo 6.05 6.05 6.05 6.04 .392863 .392500 .392137 .391775 .391412 58 57 56 56 54 7 .575758 5.18 .966808 .86 .608950 6.04 no .391050 63 8 9 .576069 .576379 5. 18 5.17 5.17 .966756 .966705 'M .86 .609312 .609674 b.Uo 6.03 6.03 .390688 .390326 61 10 11 9.576889 .576999 5.17 9.966653 .966602 .86 9.610036 .610397 6.02 c no 0.389964 .389603 50 49 12 13 .577309 .577618 .16 5.16 .966550 .966499 .86 .86 .610759 .611120 o.uz 6.02 * ni .389241 .383880 48 47 14 .577927 5.15 .966447 .86 .611480 O.U1 .388520 46 15 16 .578236 .578545 5.15 5.14 .966395 966344 .86 .86 611841 .612201 6.01 6.01 nn .388159 .387799 45 44 17 .678853 5.14 966292 .86 612561 O.IKJ .387439 43 18 19 579162 .679470 5.14 5.13 5.13 966240 .966188 .86 .86 .86 612921 .13281 6.00 6.00 5.99 .387079 .386719 42 41 20 21 9.579777 .680035 '5.12 9.966136 .966035 .87 9.613541 .614000 5.99 0.336359 .386000 40 39 22 23 .580392 .580699 5.12 5.11 .966033 .965981 .87 .87 .614359 .614718 5.98 d QQ .335641 .385282 38 37 24 25 26 27 .581005 .581312 .581618 .531924 5. 1 1 511 5.10 5.10 .965929 .965376 .965324 .965772 .87 .87 .87 .87 .615077 .615435 .615793 .616151 5.9o 6.97 5.97 6.97 .334923 .384565 .384207 .383849 36 35 34 33 28 29 .582229 .532535 5.09 5.09 5.09 .965720 .965663 .87 .87 .87 .616509 .616867 5.96 6.96 5.96 .383491 .383133 32 31 30 31 32 33 34 9.582840 .583145 .583449 .583754 .584058 5.08 5.08 5.07 5.07 9.965615 .965563 .965511 .965458 .965406 .87 .87 .87 .87 9.617224 .617582 .617939 .61 8295 .613652 5.95 5.95 5.95 5.94 0.382776 .382418 .382061 .381705 .381348 30 29 28 27 26 35 .584361 5.06 .965a53 .88 .619008 5.94 .380992 26 36 37 33 .584665 .534968 .535272 5.06 5.06 5.05 .965301 .965248 .965195 .88 .88 .88 .619364 .619720 .620076 5.94 5.93 5.93 .380636 .380280 .379U24 24 23 22 39 .585574 6.05 5.04 .965143 .88 .88 .620432 6.93 5.92 .379568 21 40 9.585377 9.965090 9.620787 0.379213 20 41 .586179 5.04 .965037 .88 .621 142 5.92 378858 19 42 536482 5.04 .964984 .88 .621497 5.92 r ni .378503 18 43 536783 5.03 .964931 .88 .621652 5.91 378148 17 44 45 46 .537085 587336 .587638 5.03 5.02 5.02 .964379 .964326 .964773 .88 .88 .83 .622207 .622561 .622915 5.91 6.91 5.90 .377793 .377439 377085 16 15 14 47 .587939 5.01 .964720 .88 .623269 5.90 .376731 13 48 .583289 5.01 .964666 .88 623623 5.90 n QQ 376377 12 49 .588590 5.01 5.00 .964613 .89 .89 .623976 5.H9 5.89 .376024 11 50 9.58,3890 9.964560 9.624330 0.375670 10 51 .589190 5.00 .964507 .89 .624633 C. QQ .375317 9 52 53 54 45 539439 589789 .590088 .590387 4.99 4.99 4.99 4.93 .964454 .964400 .964347 .964294 .89 .89 .89 .89 .625036 .625388 .625741 .626093 o.bo 5.88 5.88 5.87 C QT .374964 .374612 .374259 .373907 8 7 6 5 56 .590686 4.98 .964240 .89 .626445 o.o/ .373555 4 67 58 .590984 .591282 4.97 4.97 .964187 .964133 .89 .89 .626797 .627149 5.87 5.86 C 00 .373203 .372851 3 2 59 .591530 4.97 .964080 .89 .627501 o.oo .372499 1 60 .591878 4.96 .964026 .89 .627852 5.86 .372148 M. Oodne. D. 1". Sine. D. 1". Cotaug. D.I". Tang. M. COSINES, TANGENTS, AND COTANGENTS. 253 380 166 H. Bine. D. 1". Cosine. D. 1" Tng. D. 1". Cotang. M. 9.691878 9 96402o QQ 9.627852 0.372148 60 .692176 1 461 4.74 4.74 .960955 .960S99 .93 .93 647222 .647562 O.DO 5.68 .352778 .352438 4 3 68 69 60 .608745 .609029 .609313 4.74 473 4.73 .960843 .960786 .960730 .94 .94 .94 .647903 .64S243 .648583 5.67 5.67 5.67 .352097 .351757 .351417 2 M. Cosine. D. 1". Sine D. 1". Co tang D.I". Timg. M." 1130 '254 TABLE II. LOGARITHMIC SINES, 156> M. Sine. D.l". Codue. D. 1". Tang. D.I'. Cotang. M. 1 2 3 4 6 6 7 8 9 9609313 .609.397 .609380 .610164 .610447 .610729 .611012 611294 .61 1576 .611868 4.73 4.72 4.72 4.72 4.71 4.71 4.71 4.70 4.70 4.69 9.960730 .960674 .960618 .960561 .960505 .96(448 .960392 .960335 .960279 .960222 .94 .94 .94 .94 .94 .94 .94 .94 .94 .94 9.648583 .64-1923 .649263 .649602 .649942 .651 (231 .650620 .650959 .651297 .651636 6.67 6.66 5.66 6.66 665 6.66 5.65 6.64 5.64 5.64 0.351417 .351077 .350737 .350398 .350058 .349719 .349380 349041 .348703 .348364 59 58 67 56 55 64 63 52 51 10 9.612140 9.960165 9.651974 K. G.A 0.343026 60 11 .612421 A Q .960109 .95 .652312 6.64 .34763.3 49 12 13 14 15 16 17 18 .612702 .612933 .613264 613545 .613825 .614105 .614335 4. by 4.63 4.63 4.63 4.67 4.67 4.67 j CO .96)062 .959995 .959933 .959382 .959325 .959763 .959711 .95 .95 .95 .95 .95 .95 .95 O' .652650 .652933 .6;53326 .653663 .654000 .654337 .654674 5.63 6.63 6.63 662 6.62 5.62 6.62 C , I .347350 .347012 .346674 .346337 .346000 345663 .345326 48 47 46 46 44 43 42 19 .614665 4. DO 4.66 .959654 .o .96 .655011 O.Dl 6.61 .344989 41 20 21 22 23 24 26 26 9.614944 615223 .615502 .615781 616060 616333 .616616 4.65 4.65 4.65 4.64 4.64 4.64 1 O 9 959596 .9.59539 .959482 .9f>9425 .9.59368 .959310 .959263 .95 .95 .95 .95 .96 .96 9.655348 .655634 656020 .666356 .6.56692 .657028 .6.57364 6.61 6.61 6.60 6.60 6.60 6.59 t CO 0.344652 344316 .343930 .343644 .343303 .342972 .34*636 40 39 38 37 36 35 34 27 28 29 .616394 .617172 .617450 4. DO 4.63 463 4.62 .959195 .9.59133 .959080 .96 96 .96 .98 .657699 .658034 .658369 O.5B 6.59 6.58 6.68 .342301 .341966 .341631 33 32 31 30 9.617727 9.959023 9.658704 e eo 0.341296 30 31 .618004 4.62 .953965 .96 .659039 5. 5o .340%! 29 32 33 36 36 37 38 39 .618231 .618553 .618334 .619110 619336 .619662 .619933 .620213 4.61 4.61 4.61 4.60 4.60 4.60 4.69 4.59 4.59 .953903 .9598(0 .953792 .958734 .953677 .953619 .953561 .958503 .96 .96 .96 .96 .96 .96 .97 .97 .97 659373 659708 660042 660376 660710 661043 .661377 .661710 6.58 6.57 6.67 6.67 5.56 6.56 656 6.56 6.65 .340627 .340292 .339624 .339290 .333957 .33^623 .338290 28 27 26 25 24 23 22 21 40 41 42 43 44 45 46 47 48 9.620488 .620763 .621033 .621313 .621537 .621361 .622135 .622409 .622632 4.53 4.53 4.58 4.57 4.57 4.57 4.56 4.56 9.958445 .953337 .953329 .953271 .958213 .958154 .953096 .953033 .957979 .97 .97 .97 .97 .97 .97 .97 .97 9662043 662376 .662709 .663042 .663375 .663707 664039 .664371 664703 6.55 655 6.54 654 654 6.54 653 5.53 0.337957 .337G24 337291 .336958 336625 336293 335961 33:5629 335297 20 19 18 17 16 16 14 13 12 49 .622956 4.56 4.55 .957921 .97 .97 665035 5.53 653 334965 11 60 9.623229 9.957863 9.665366 C CO 0.334634 10 51 .623502 4.55 957804 .97 665698 6 52 334302 9 52 63 .623774 .624047 4.54 4.54 .957746 957687 .98 .98 .666029 666360 5.52 5.52 .333971 .333640 8 7 64 .624319 4.54 957628 .98 666691 5.61 .333309 6 55 .624591 4.53 .957570 .98 667021 5.51 332979 5 66 .624363 4.53 .95751 1 .93 .667352 5.51 .332648 4 57 68 69 .625135 .625106 625677 4.53 4.52 4.52 .9-57452 9H7393 .957335 .98 .98 .93 667682 668013 .663343 5.51 5.50 6.60 e cfi .332318 .331987 .331657 3 2 1 60 .625948 4.52 .957276 .98 .663673 O.oU .331327 M. Coetaa. D. 1". Slue. D. 1". Ootang. D. 1". Ifcng. if- COSINES, TANGENTS, AND COTANGENTS. 164 M Blue. D.l". Cosine. D 1" Tiuig. D.l". Ootaug. M. 1 i 9.625948 .626219 .626490 4.61 461 9.957276 .957217 .957168 .98 .98 9.66S673 .669002 .669332 6.60 6.49 0.331327 .330998 .330668 60 69 68 3 .626760 4.61 .957099 flN .669661 fila .330339 67 4 6 6 7 8 9 .627030 .627300 .627670 .627840 .628109 .628378 i.K 460 4.19 4.49 4.49 4.48 .967040 .956981 .956921 .956662 .956803 .956744 '99 .99 .99 .99 .99 99 .669991 .670320 .670649 .670977 .671306 .671636 6^49 6.48 6.48 6.48 6.47 6.47 .330009 .329680 .329351 .329023 .328694 .328365 66 65 64 63 62 61 ic 11 9.628647 .628916 4.48 9.956684 .956625 .99 oo 9.671963 .672291 6.47 0.328037 .327709 60 49 IS .629185 4.48 .956566 .yy .672619 6.47 .327381 48 13 14 .629453 .629721 4.47 4.47 .956506 .956447 .99 .99 .672947 .673274 6.46 6.46 C 4 .327053 .326726 47 46 16 16 .629989 .630257 446 A Ati .956387 .956327 '99 .673602 .673929 O.48 6.46 .326398 .326071 46 44 17 18 19 .630524 .630792 .631059 4.45 446 4.45 4.45 956268 .956208 .966148 .99 .99 1.00 1.00 .674257 .674584 .674911 6.46 6.46 6.46 6.46 .325743 .325416 .325089 43 42 41 22 13 24 *6 26 27 28 29 9.631326 .631593 .631859 .632 25 .632392 .632658 .632923 .633189 .633454 .633719 4.46 4.44 444 4.44 443 443 443 442 442 4.42 9.956089 .956029 .955969 .955909 .955S49 .955789 .955729 .955669 .955609 .955648 l.CO 1.00 1.00 1.00 1.00 1 00 1.00 100 100 1.00 9.675237 .6.~5564 .676890 .676217 .676543 .676869 .677194 .677620 .677846 .678171 6.44 6.44 6.44 6.44 6.43 643 643 642 6.42 6.42 0.324763 .324436 .324110 .323783 .323457 .323131 .322806 .322480 .322154 321629 40 39 38 37 36 36 34 33 32 31 30 31 32 33 34 35 36 9.633984 .634249 .634514 .634r8 .635042 .635306 .6355^0 4.41 441 4.41 4.40 4.40 4.40 A on 9.955488 .955428 .955368 .955307 .955247 .955186 .955126 1.00 1.01 1.01 1.01 1.01 1.01 1 II] 9.678496 .678821 .679146 .679471 .679795 .680120 .680444 642 6.41 6.41 6.41 5.41 6.40 0.321504 .321179 .320854 .320529 .320205 .319880 .319556 30 29 28 27 26 25 24 37 38 39 .635834 .636097 .636360 4.o9 4.39 4.39 4.38 .955(*5 .955005 .954944 I.UI 1.01 1.01 1.01 .680768 .681092 .681416 6.40 6.40 6.40 6.39 .319232 .316908 .318584 23 22 21 40 41 42 9.636623 .6368S6 .637148 4.38 4.38 A ** 9.954883 .954823 .954762 1.01 1.01 9.681740 .682063 .682387 6.39 5.39 c on 0.316260 .317937 .317613 20 19 18 43 44 .637411 .637673 4. J7 4.37 A or .954701 .954640 1.01 101 .682710 .683033 6. oil 5.38 e oo .317290 .316967 17 16 45 46 47 48 49 .637935 .638197 .638458 .638720 .638981 4.o7 4.36 4.36 4.36 4.35 4.35 .954579 .954518 .954457 .954396 .954335 L02 1.02 1.02 1.02 1.02 .683356 .683679 .684001 .684324 .684646 D.OO 6.38 5.38 6.37 6.37 6.37 .316644 .316321 315999 .315676 .315354 15 14 13 12 11 60 61 62 53 9.639242 .639503 .639764 .640024 4.35 4.34 4.34 9.954274 .954213 .954152 .954090 1.02 1.02 1.02 9.684968 .685290 .685612 .685934 6.37 6.36 6.36 0.315032 .314710 .314388 .314066 1C 9 8 7 64 .640284 4.34 400 .954029 .02 .686255 5.36 .313745 6 65 .640544 .00 A 9Q .953968 .02 .686577 6.36 e oe .313423 6 56 67 68 .640804 .641064 .641324 4.OO 433 4.32 .953906 .953845 .953783 .02 .02 .03 .6S6898 .687219 687540 O.OO 6.35 6.35 .313102 .312761 .312460 4 3 2 69 60 .641583 .641842 4.32 4.32 .953722 .SC3660 .03 1.03 687861 .638182 6.35 6.35 .312139 .311818 I M Cosine D 1" Slue D. 1". Cotaag- D.I'. TMg. "if TABLE II. LOGARITHMIC SINES, 460 1680 M. Sine. 1 D. 1". Cosine. D. 1". Tang. D. 1". Cotwg. M. ~0 1 2 3 4 5 6 7 9.64184V .642101 .642360 .642613 .642877 .643135 .643393 .643650 4.32 4.31 4.31 4.31 4.30 4.30 4.30 9.963660 .953599 .953537 .953475 .953413 .953352 .953290 .953228 1.03 1.03 1.03 1.03 1.03 1.03 1.03 9.688182 .688502 .688823 .689143 .689463 .689783 .690103 .690423 5.34 5.34 5.34 5.34 5.33 6.33 6.33 K QO 0.311818 .31149? .311177 .310857 .310537 .310217 .309897 .309577 60 59 68 67 66 66 64 63 8 9 .643908 .644165 4^29 4.29 .953166 .953104 1.03 1.03 .690742 .691062 D.OO 6.32 6.32 309258 .308938 62 61 10 11 9.644423 .644680 4.28 4 OQ 9.953042 .952980 1.03 1 1 Li 9.691381 .691700 6.32 e oo 0.308619 .308300 60 49 12 13 14 15 16 17 18 19 .644936 .645193 .645450 .645706 .645962 .646213 .646474 .646729 4.-6O 4.28 4.27 4.27 4.27 4.26 4.26 4.26 4.26 .952918 .952855 .952793 .952731 .952669 .952606 .952544 .952481 1 .'rl 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04 .692019 .692338 .692656 .692975 .693293 .693612 .693930 .694248 O.'fi 5.31 6.31 6.31 6.31 5.30 6.30 6.30 6.30 .307981 307662 & (7344 .307025 .306707 .306388 .306070 .306752 48 47 16 45 14 43 42 41 20 21 22 23 24 25 26 27 28 29 9.646984 .647240 .647494 .647749 .648004 .648258 648512 .648766 .649020 .649274 4.25 4.25 4.25 4.24 4.24 4.24 4.23 4.23 4.23 4.22 9.952419 .952356 .952294 .952231 .952168 .952106 .952043 .951980 .951917 .951854 1.04 1.04 1.04 1.04 1.05 1.05 1.05 1.05 105 1.05 9.694566 .694,833 .695201 .695518 .695836 .696153 .696470 .696787 .697103 .697420 6.29 6.29 6.29 6.29 6.29 6.28 6.28 6.29 5.23 6.27 0.305434 .305117 .304799 .304482 .304164 .303847 303530 .303213 .302897 .302530 40 39 33 37 36 35 34 33 32 31 30 31 32 33 34 35 36 37 38 39 9.649527 .649781 .650031 .650287 .650539 .650792 .651044 .651297 .651549 .651800 4.22 4.22 4.22 4.21 4.21 4.21 4.20 4.20 420 4.19 9.951791 .951728 .951665 .951602 .951539 .951476 .951412 .951349 .951286 .951222 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.06 1.06 1.06 9.697736 .698053 .698369 .698685 .699001 .699316 .699632 .699947 .700263 .700578 6.27 6.27 6.27 6.26 6.26 6.2ft 6.26 6.26 5.25 6.25 0.302264 .301947 .301631 .301316 .300999 .300684 .300363 .300053 .299737 .299422 30 99 98 97 96 96 24 93 22 21 40 41 42 43 9.652052 .652304 .652555 .652806 4.19 4.19 .18 9.951159 .951096 .951032 .950968 1.06 1.06 1.06 9.700893 .701208 .701523 .701837 6.25 6.25 6.24 0.299107 .298792 .298477 .298163 90 19 18 17 44 .653057 .18 .950905 1.06 1f\t .702152 6.24 .297848 16 45 46 .653308 .653558 .18 .18 .950841 .950778 .Uo 1.06 1 HA .702466 .702781 6.24 6.24 e o>4 .297534 .297219 16 14 47 .653808 .17 .950714 J.Ub i rus .703095 5.4* e io .296905 13 48 49 654059 654309 .17 .17 .16 .950650 .950586 l.Uo 1.06 1.06 .703*^9 .703722 5.23 6.23 6.23 .296591 .296278 19 11 50 51 52 53 54 55 56 57 9.654558 .654808 .655053 .655307 .655556 .655805 .656054 .656302 .16 .16 .15 .15 .15 .15 .14 9.950522 .950453 .950394 .950330 .950266 .950202 .950138 .950074 1.07 1.07 1.07 1.07 1.07 1.07 1.07 9.704036 .704350 .704663 .704976 .705290 .705603 .705916 .706228 6.23 6.22 6.22 6.22 5.22 5.22 5.21 0.295964 .295650 .295337 .295024 .994710 .294397 .294084 .293772 10 9 8 7 6 6 4 3 58 59 60 .656551 .656799 .657047 4.14 4.14 4.13 .950010 .949945 .949381 1.07 1.07 1.07 .706541 .706854 .707166 5.21 6.21 6.21 .293459 .293140 .292834 9 M. Cosine. D. 1". Bine. D. 1". Ootang. D.1". Tang M. COSINES, TANGENTS, AND COTANGENTS. 15* M. Slue. D. 1". Cosine. D. 1". Tang. D. 1". Co tang. M. 9.657047 9.949831 fY7 9.707166 0.292834 60 1 2 3 4 6 6 7 5 .657295 .657542 .657790 .658037 .658234 .658531 .658778 659025 4.13 4.13 4.12 4.12 4.12 4.J2 4.11 4.11 .919816 .949752 .949688 .949623 .949494 .949429 .949364 .if/ .07 .07 .08 .08 .03 .08 .08 .707478 707790 .7081 02 .708414 .708726 .709037 .709349 .709660 5>20 5.20 5.20 6.20 6.19 6.19 5.19 .292522 .292210 .291893 .291586 .291274 .290963 .290651 .290340 69 58 57 66 65 64 63 52 9 .659271 4.11 4.10 .949300 .08 .08 709971 5.19 6.18 .290029 51 10 11 12 13 14 16 16 9659517 .659763 .660009 .660255 .660501 .660746 .660991 4.10 4.10 4.10 4.09 4.09 4.09 A AQ 9.949235 .949170 .949105 .949040 .948975 .948910 .948345 1.08 1.08 1.08 1.08 1.08 1.08 9.710282 .710593 .710904 .711215 .711525 .711836 .712146 6.18 6.18 6.18 6.18 5.17 6.17 61 T 0.289718 .289407 .289096 .288785 .288475 .288164 .287854 60 49 48 47 46 45 44 17 18 .661236 .661481 3.UCJ 4.03 A it~ .948780 .948715 .09 1.09 Ino .712456 .712766 . 17 6.17 617 .287544 .287234 43 42 19 .661726 .Uo 4.08 .948650 .uy 1.09 .713076 . 1 / 6.16 .286924 41 20 9.661970 4 O7 9.948584 9.713386 fi Iti 0.286614 40 21 22 23 24 .662214 .6624;">9 .662703 .662946 .U/ 4.07 4.07 4.06 .948519 .943454 .943338 .943323 L09 .09 .09 .713696 .714005 .714314 .714624 O. IO 6.16 6.16 6.15 61 e .2863(4 .285995 .2856,36 .285376 39 38 37 36 25 26 27 23 29 .663190 .663433 .66:77 .663920 .664163 4.06 406 4.05 4.05 4.05 4.05 .943257 .948192 .943126 .943060 .947996 .09 .09 .09 .09 .09 .10 .714933 .715242 .715551 .715860 .716168 . 15 6.15 6.15 6.15 (.14 6.14 .285067 .284758 .284449 .284140 .283832 36 34 33 32 31 30 31 32 33 34 35 36 37 38 39 9.664406 .664643 .664391 .635133 .665375 .665617 .665359 .666100 .666*42 .666533 4.04 4.04 4.04 4.03 4.03 4.03 4.03 4.02 4.02 4.02 9.947929 .947363 .947797 .947731 .917665 .947600 .947533 .947467 .947401 .947335 .10 .10 .10 .10 .10 .10 .10 .10 .10 .10 9.716477 .716785 .717093 .717401 .717709 .718017 .713325 .718633 .718940 .719243 6.14 6.14 6.14 6.13 5.13 5.13 5.13 5.13 5.12 5.12 0.283523 .283215 .282907 .282599 .282291 .281983 .281675 .281367 .281060 .280752 30 29 28 27 26 25 24 23 22 21 40 41 42 i 43 9.666824 .667065 .667305 .667546 4.01 4.01 4.01 9.947269 .947203 .947136 .947070 .10 .11 .11 9.719555 .719362 .720169 720476 5.12 6.12 6.11 0.280445 .280138 .279331 .279524 20 ! 19 13 17 44 45 46 .667786 .663027 .663267 4.01 4.00 4.00 A Of\ .947004 .946937 .946871 .11 .11 .11 .720783 .721089 .721396 5.11 6.11 5.11 C 1 I .279217 .278911 .278604 16 15 14 47 43 49 .668506 .663746 .668936 4.UU 3.99 3.99 3.99 .946304 ^946671 , .11 .11 .11 .721702 .722009 .722315 O.I 1 5.10 6.10 6.10 .278298 .277991 .277685 13 12 11 50 51 52 53 64 65 9.669225 .669464 669703 .669942 .670181 .670419 3.99 3.98 3.93 3.98 3.98 9.946604 .946533 .946471 .946404 .946337 .946-270 .11 .11 .11 .11 .12 9.722621 .72*87 .723232 .723533 .72;i344 .724149 6.10 6.10 6.09 6.09 5.09 0.277379 .277073 .276768 .276462 .276156 .275351 10 9 8 7 6 6 56 57 .670658 .670396 3^97 O QT .946203 .946136 .12 .12 .724454 .724760 5.09 5.09 .275.546 .275240 4 3 68 5S 60 .671134 671372 .671609 O.97 3.96 3.96 .946069 .946002 .945935 .12 .12 12 .725065 .725370 .72r674 5.08 5.08 5.03 .274935 .274630 .274326 2 M Cosine. D.I". Sine. D 1". Cotaug. D.I'. Tang. M. LOGARITHMIC M. Sine. D. 1". Codne. D.I" Tang. D. I". Cotang. M. I 9.671609 .671847 .672084 .672321 3.96 3.96 3.95 9.945935 .945868 .945800 .945733 1.12 1.12 1.12 9.725674 .725979 .726284 .726588 6.08 5.08 6.07 0.274326 .274021 .273716 .273412 59 58 57 4 5 6 7 9 .672558 .672795 .673032 .673268 .673505 .673741 3 95 395 394 394 3.94 3.94 3.93 .945666 .945598 .945531 .945464 .945396 .945328 1.12 1.12 1.12 1.12 1.13 1.13 1.13 .726892 .727197 .727501 .727805 .728109 .728412 5.07 6.07 6.07 5.07 6.06 6.06 6.06 .273108 .272803 .272499 .272195 .271891 .271588 56 CC 54 53 52 51 10 9.673977 9.945261 11 O 9.728716 f\M 0.271284 60 11 19 .674213 .674448 393 .945193 .945125 .Id 1.13 11 'J .729020 .729323 5.06 6.06 K. ne .270980 .270677 49 48 13 14 .674684 .674919 3.92 .945058 .944990 .19 1.13 .729626 .729929 o.Uo 6.05 .270374 .270071 47 46 1C 16 17 IS 19 .675155 .675390 .675624 .675859 .676094 3.92 3.92 3.91 3.91 391 3.91 .944922 .944854 .944786 .944718 .944650 i'i3 1.13 1.13 1.13 1.13 .730233 .730535 .730838 .731141 .731444 5.05 6.05 6.05 6.05 6.04 C.04 .269767 .269465 .269162 .268859 .268556 46 44 43 42 41 90 91 99 93 94 96 9.676328 .676562 .676796 .677030 .677264 .677498 3.90 3.90 3.90 390 3.89 o on 9.944582 .944514 .944446 .944377 .944309 .944241 1.14 1.14 1.14 1.14 1.14 9.731746 .732(V48 .732351 .732653 .732955 .733257 6.04 6.04 6.04 6.03 6.03 K no 0.268254 .267952 .267649 .267347 .267045 .266743 40 37 36 36 96 97 98 .677731 .677964 .678197 o 8y 389 3.83 .944172 .944104 .944036 1.14 1.14 1.14 .733558 .733860 .734162 O.Uo 6.03 6.03 .266442 .266140 .265838 34 33 39 99 .678430 3-88 3.88 .943967 1.14 1.14 .734463 6.02 6.09 .965537 31 30 9.67S663 9.943899 9.734764 K. no 0.265236 30 31 .678895 3.88 .943830 1.14 .735066 Q.U* .264934 29 39 33 34 36 36 37 38 39 .679128 .679360 .679592 .679824 .680056 .680288 .680519 .680750 3.87 3.87 3.87 3.87 3.86 386 386 386 3.85 .943761 .943693 .943624 .943555 .943486 .943417 .943348 .943279 1.14 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.16 .735367 .735668 .735969 .736269 .736570 .736870 .737171 .737471 6.02 6.02 6.01 6.01 6.01 6.01 6.01 6.01 6.00 .264633 .264332 .264031 .263731 .263430 .263130 .262829 .262529 28 27 26 26 24 23 22 21 40 9.680982 9.943210 9.737771 0.262229 90 41 42 43 44 45 46 47 18 49 .681213 .681443 .681674 .681905 682135 .682365 .682595 682825 .683055 3.85 3.85 3.84 3.84 3.84 3.84 3.83 3.83 3.83 .943141 .943072 .943003 .942934 .942864 .942795 .942726 .942656 .942587 1.15 1.15 1.15 1.15 1.15 1.16 1.16 1.16 1.16 .738071 .738371 .738671 .738971 .739271 .739570 .739870 .740169 .740468 6.00 6.00 6.00 6.00 4.99 4.99 4.99 4.99 4.99 .261929 .261629 .261329 .261029 .260729 .260430 .260130 .25983! .25953* 19 18 17 J6 15 14 13 12 3.83 1.16 4.98 60 61 9.683284 .683514 3.82 9 00 9.942517 .942448 1.16 9.740767 .741066 4.98 0.259233 .258934 V 1 62 .633743 3. ox .942378 1.16 Iia .741365 A QQ .258635 8 63 64 65 66 67 68 59 60 .683972 .684201 .684430 .'684887 .685115 .685343 .685571 a82 3.81 3.81 3.31 3.80 3.80 3.80 .942308 .942239 .942169 .942099 .949029 .941959 .941889 .941819 .18 1.16 1.16 1.16 1.16 1.17 1.17 1.17 .741664 .741962 .742261 .742559 .742858 .743156 .743454 .743752 4.99 4.98 4.98 497 497 497 497 4.97 .258336 .258038 .257739 .257441 .257142 .256S44 .256546 .256248 7 6 6 4 3 2 1 K. Oodue. D 1". Slue. D. 1". Cotang. D.I". Tang. M. 61 COSINES, TANGENTS, AND COTANGENTS. M State. D. 1". Cosine. D. 1". Tang. D. 1". Cotang. M. 2 3 4 5 I 7 8 9 9.685571 635799 .686027 686254 .6-86482 .6S6709 .686936 687163 .637339 .637616 3.80 3.79 3.79 3.79 3.79 3.73 3.73 3.78 3.73 3.77 9.941819 .941749 .941679 .941609 .941539 .941469 .941398 .941328 .941258 .941137 1.17 1.17 1.17 1.17 1.17 1.17 1.17 1.17 1.17 1.17 9.743752 .744050 .744343 .744645 .744943 .745240 .745538 .745835 .746132 .746429 4.96 4.96 4.96 4.96 496 4.96 4.95 4.95 4.95 4.95 0.256248 .255950 .255652 .255355 .255057 .254760 .254462 .254165 .253868 .253571 60 69 68 67 66 66 64 53 52 61 IG il 12 9.637843 . 68-3069 .633295 3.77 3.77 9.941117 .941046 .910975 1.18 1.18 9.716726 .747023 .747319 4.95 4.95 0.253274 .252977 .252681 60 49 48 13 14 15 16 17 18 19 .638521 .688747 .633972 .689193 .639423 .689648 .639373 3.77 3.76 3.76 3.76 3.76 3.75 3.75 3.75 .940905 .9408.34 .940763 .940693 .940622 .940551 .940480 1.18 1.18 1.18 1.13 1.18 1.18 1.18 .747616 .747913 .748209 .748505 .748801 .749097 .749393 4.94 4.94 4.94 494 4.93 4.93 4.93 .252334 .252087 .251791 .251495 .251199 .250903 .250607 47 46 45 44 43 42 41 20 21 i 24 25 26 27 28 29 9690098 .690323 .690548 .690772 .690996 .691220 .691444 .691663 .691392 .6921 15 3.75 3.74 3.74 374 3.74 373 3.73 3.73 373 3.72 9.940409 .940338 .940267 .940196 .940125 .940054 .939932 .939911 .939-^40 .939768 1.18 1.18 1.19 1.19 1.19 1.19 1.19 1.19 1.19 1.19 9.749689 .749935 .750281 .750576 ' .750872 .751167 .751462 .751757 .752052 .752347 4.93 4.93 4.93 4.92 4.92 492 492 4.92 492 4.91 0.250311 .250015 .249719 .249424 .249128 .243833 .248538 .24*243 .247948 .247653 40 39 38 37 36 35 34 33 32 31 30 31 32 33 34 9.692339 .692562 .692735 .693003 .693231 3.72 3.72 3.72 3.71 9.939697 .939625 .939554 .939482 .939410 1.19 1.19 1.19 1.19 9.752642 .752937 .753231 .753526 .753820 4.91 4.91 4.91 4.91 0.247358 .247063 .246769 .246474 .246180 30 29 28 27 26 35 36 37 38 39 .693453 .69:3676 .693398 .694120 .694342 3.71 3.71 3.71 3.70 3.70 3.70 .939339 .939267 .939195 .939123 .939052 1.20 1.20 1.20 1.20 1.20 .754115 .754409 .754703 .754997 .755291 4.90 4.90 4.90 4.90 4.90 .245885 .245591 .245297 .245003 .244709 25 24 23 22 21 40 41 42 43 44 45 9.694.564 .694736 .695007 .695229 .695450 .695671 3.70 3.69 3.69 3.69 3.69 9.933980 .938903 .933336 .938763 .938691 .938619 1.20 1.20 1.20 1.20 120 1 2ft 9.755585 .755878 .756172 .756465 .756759 .757052 4.89 4.89 4.89 4.89 4.89 4 SQ 0.244415 .244122 .243328 .243535 .243241 .242948 20 19 18 17 16 16 46 47 48 .695892 .6961 13 .696334 3.68 3.68 .933547 .938475 .938402 1.20 1.21 .757345 .757638 .757931 4.88 4.88 .242655 .242362 .242069 14 13 12 49 .696554 3.67 .933330 1.21 .758224 4.88 .241776 11 50 51 52 53 64 55 56 57 68 59 60 9.696775 .696995 .697215 .697435 .697654 .697874 .693094 .693313 .698532 .698751 .6981*70 3.67 3.67 3.67 3.66 3.66 366 366 3.65 365 3.65 9.933258 .938185 .938113 .938040 .937967 .937895 .937822 .937749 .937676 .937604 .937531 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.22 9.758517 .753810 .759102 .759395 .759687 .759979 .760272 .760564 .760856 .761148 .761439 4.88 4.88 4.87 4.87 4.87 4.87 4.87 4.87 4.86 4.86 0.241483 .241190 .240893 .240605 .240313 .240021 .239728 .239436 .239144 .238852 .238561 10 9 8 7 6 6 4 3 3 M. Ooelne. ' J>. 1". Blue. D. I". Coteng. D. 1". Tnng. M. 1190 260 TABLE II. LOGARITHMIC SINES, 80 149= M. Sloe. D.1". Cosine. D. 1". Tang. D.l. Cotang. M 9.698970 9.937531 1 99 9.761439 0.238561 60 1 3 .699189 .099407 .699626 3.64 3.64 .937458 .937385 .937312 1 22 1.22 .761731 .762023 .762314 4.86 4.86 .23->269 .237977 .2376M6 69 68 67 4 6 .699844 .700062 .700280 3.64 363 .937238 .937166 .937(92 1.22 1.22 1 V9 .7626(6 .762897 .763188 4.86 4.85 .237394 .237103 236^12 66 55 54 7 8 9 .700498 .700716 .700933 363 3.63 3.62 .937019 .936946 .936872 1.22 1.22 1.22 .763479 .763770 .764061 485 4.85 4.86 .236521 .236230 .2:*S'J39 53 62 61 10 11 12 13 14 15 16 17 9.77*151 .701368 .7015X5 .701802 .702019 .702236 .702452 .702669 362 362 362 361 361 3.61 3.61 9.936799 .936?25 .936652 .936578 .936.105 .936431 .936357 .936284 1.22 1.23 1.23 123 1.23 1.23 1.23 9.764352 .764643 .764U33 .76T>224 .765514 .765805 .766095 .766385 446 4.84 4.84 4.84 4.84 4.84 4.84 0.23fi648 .23f>.r>7 .23.*.' "67 .234776 .21*4486 .234195 .2XWC5 .233615 60 49 48 47 46 45 44 43 18 19 .702885 .703101 3.60 3.60 .936210 .936136 1.23 1.23 .766675 .766965 4.83 4.83 4.83 .233325 .233035 42 41 20 21 22 23 24 25 9.703317 .703533 .703749 .703964 .704179 .704395 3.60 359 359 3.59 359 9.936062 .935988 .935914 .935840 .935766 .935692 1.23 1.23 1 23 123 1.24 9.767255 767545 .767834 768124 788414 768703 4.83 483 483 4.82 4*2 0.232745 .232455 .232166 231876 .231586 .23I2'J7 40 39 38 37 36 36 26 27 28 29 .704610 .70425 .705f40 .705254 3.68 358 358 3.58 .935618 .935543 .935469 .935395 1.24 1.24 1 24 1.24 .768992 .769281 .769571 .769860 4 82 ua 482 4.82 4.82 .23IU03 .23(1719 .23(1429 .230140 34 33 32 31 30 31 32 33 34 35 36 37 38 39 9.705469 .705683 .705898 .7061 12 .706326 .705539 .706753 .706967 .707180 .707393 3.67 357 367 3.67 3.56 356 356 3.56 3.55 3.65 9.935320 .935246 .935171 .935097 .935022 .934948 .934873 .934798 .934723 .934649 1.24 1.24 1.24 1.24 1 24 1.24 1.25 1.25 1.25 l.'*5 9.770148 .770437 .770726 .771015 .771303 .771592 .771880 .772168 .772457 .772745 4.81 4.81 4.81 481 481 4.81 4.80 4.80 4.80 4.80 0.229852 .22i63 .229274 .228985 .22^697 .25M08 .22H20 .227832 .227543 .227255 30 29 28 27 26 25 24 23 22 21 40 41 42 43 44 45 46 47 43 49 9707606 .707819 .708032 .708245 .7&S453 .708670 .708882 .70')094 .709306 .709518 3.55 3.55 3.54 3.54 3.54 3.54 3.54 3.53 3.53 3.53 9.934574 .934499 .934424 .934349 .934274 .934199 .934123 .934048 .933973 .933898 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.26 1.2B 9773033 .773321 .773608 .773896 .774184 .774471 .774759 .775046 .775333 .775621 4.80 4.80 4.80 4.79 4.79 4.79 4.79 4.79 4.79 4.78 0.226967 .226679 .226392 .226104 .225816 .225529 .225241 .224954 .224667 .224379 20 19 18 17 16 15 14 13 12 11 50 51 52 53 54 9.709730 .709941 .710153 .710364 .710575 3.53 352 3.52 3.52 9.933822 .933747 .933671 .933596 93aS20 1.26 1.26 1.26 1.26 9 775908 .776195 .7764*2 .776768 .777055 4.78 4.78 4.78 4.78 224092 .2238(15 .223518 .223232 .222945 10 9 8 7 6 55 56 .710786 .710997 3.51 933445 933369 1.26 .777342 .777628 4.78 .222658 .222372 6 4 57 68 59 60 .711208 .711419 .711629 .711839 3.51 3.51 3.61 3.51 933293 933217 .933141 .933066 1.26 1.26 1.26 .777915 .778201 .778488 .778774 4.77 4.77 4.77 .222085 .221799 .221512 .221226 3 2 1 M. Oofiine. i D.I*. Sine. D.l Cotang. D.I" { Tang. M. UK* COSINES, TANGENTS, AND COTANGENTS. 261 lio 148 M Bine. | D.I*' Cosine. D. 1". I*ng. D. 1". Cotang M, 2 3 4 6 6 7 8 9 9.711839 1 .7121.50 ! .712260 .712469 .712679 .712889 .713093 .713308 ,713517 .713726 3.50 3.50 3.50 3.50 3.49 349 349 349 343 348 9.933066 932990 932914 932838 . 932762 932685 932609 .932533 932457 .932330 1.27 1.27 1.27 1.27 1.27 1.27 1.27 1.27 1.27 1.27 9.778774 .779060 .779346 .779632 .779913 .780203 .780489 .780775 .781060 .781346 4.77 4.77 4.77 4.76 4.76 4.76 4.76 4.76 4.76 4.76 0.221226 .220940 .220654 .220368 .221 M (32 .219797 .219511 .219225 .218940 .218654 60 59 58 57 56 55 54 53 52 51 10 11 12 13 14 9.713935 .714144 .714352 .714561 .714769 348 3.43 3.43 3.47 9.932304 .932228 .932151 .932075 .931993 1.27 1.27 1.28 1.28 9.781631 .781916 .782201 .782136 .782771 4.75 4.75 4.75 4.75 0.218369 218034 .217799 .217514 .217229 50 49 48 47 46 15 16 17 18 19 .714978 .715136 .715394 .715602 .716809 347 347 346 3.46 3.46 .931921 .931345 .931763 .931691 .931614 1.28 1.23 1.28 1.28 1.28 .783056 .783341 .783626 .783910 .784195 4.75 4.75 4.74 4.74 4.74 .216944 .216659 .216374 .216090 .215805 45 44 43 42 41 20 21 9.716017 .716-224 3.46 9.931537 .931460 1.28 9 784479 784764 4.74 0.215521 .215236 40 39 22 23 24 716432 716639 716-^6 345 345 .931333 931306 931229 1.28 1.2.3 785048 .785332 .785616 4.74 4.74 .214952 .214668 .214384 38 37 36 25 27 28 29 .717053 717-259 .717466 .717673 .717879 3.45 345 344 344 344 3.44 93! 152 931075 930993 930921 .930343 1.29 1.29 1.29 1.29 1.29 785900 .786184 .786463 .786752 .787036 4.73 4.73 4.73 4.73 4.73 4.73 .214100 213816 .213532 .213248 .212964 35 34 33 32 31 30 31 32 33 34 35 36 37 38 39 9.718085 .718291 .718497 .718703 .713909 .719114 .719320 .719525 .719730 .719935 343 343 343 3.43 343 342 342 342 342 9.930766 .930638 930611 .930533 .930456 930378 .930300 .930223 .930145 .930067 1.29 1.29 1.29 1.29 1.29 129 1.30 1.30 1.30 9.787319 .787603 .78788fi .788170 .788453 .788736 .789019 .789302 .789585 .789868 4.73 4.72 4.72 4.72 4.72 4.72 4.72 4.72 4.71 0.212681 .212397 .212114 .211830 .211547 211264 .210981 .210698 .210415 .210132 30 29 28 27 26 25 24 23 22 21 40 41 42 43 44 45 46 47 48 49 9.720140 .720345 .720549 .720754 .7'20958 .721162 .721366 .721570 .721774 .721978 3.41 3.41 341 341 3.40 340 340 340 3.39 3.39 9.929989 .92991 1 .929333 .9-29755 .929677 .929599 .929521 .929442 .929364 .929286 1.30 1.30 1.30 1.30 1.30 1.30 1.30 1.31 1.31 1.31 9.790151 .790434 .790716 .790999 .791-231 .791563 .791846 .792123 .792410 .792692 4.71 4.71 4.71 4.71 4.71 4.70 4.70 4.70 4.70 4.70 0.209849 .209566 .209284 .209001 .208719 .203437 .208154 .207872 .207590 .207308 20 19 18 17 16 15 14 13 12 11 5C 61 52 63 54 55 9.722131 .722385 .7225,38 .722791 .722994 .723197 3.39 3.39 3.39 3.38 3.38 9.929207 .9-29129 .9'29050 .928972 .928893 .923815 1.31 1.31 1.31 1.31 1.31 9.792974 .793256 .793538 .793819 .794101 .794383 4.70 4.70 4.70 469 4.69 0.207026 .2(6744 .206462 .206181 .205*99 .205617 10 9 8 7 6 5 56 .723400 .928738 1.31 .794664 4.69 .20533* 4 57 58 .723603 .723305 337 .928657 .923578 1.31 794946 .79*227 4.69 .205054 .204773 3 2 59 60 .72401)7 .724210 3.37 3.27 923499 .926420 1.31 1.32 .795508 .795789 4.69 4.S9 .2W492 .204211 t M Oofltae. L. i". Slue. D.I" Cotaug. D. 1". Tang M 580 TABLE LOGARITHMIC SIXES, M. Bine. D. 1". Cosine. D. 1". Tang D. 1". Cotang. M. 9.724210 q qy 9.928420 9.795789 A Q 0.204211 ^0~ 1 .724412 u.o/ o 07 .9-28342 1 "W .796070 4.OO A CO .203930 59 2 .724614 .724*16 o.oi 3.36 O Ofl .928263 L32 .?%351 .796632 4 Do 469 A CQ .203649 .203368 58 67 .725017 O.oO o oc !928.04 IOO .796913 4. bo A CO .203087 56 .725-219 .725420 U.OO 3.36 O Oft .92Si5 .927946 . O* 1.32 .797194 .797474 4.OO 4.68 A Q .202806 .202526 55 54 .725622 O.OD Q QC 927^67 , Cj .797755 4. DO .202245 63 .725823 .726024 O.oO 3.35 3.35 >27787 .927708 L32 1.32 .798036 .798316 4.68 4.67 4.67 .201964 .201684 52 61 10 11 12 13 14 16 16 17 18 19 9726225 .726426 .726626 .726827 .727027 .727228 .727428 .727628 .727828 .728027 3.35 3.34 334 3.34 3.34 3.34 3.33 333 3.33 3.33 9.927629 .927549 .927470 .927390 .927310 .927231 .927151 .927071 .926991 .926911 1.32 1.33 1.33 1.33 1.33 1.33 1.33 1.33 1.33 1.33 9.798596 .798877 .799157 .799437 .799717 .799997 .800277 .81X1557 .800836 .801116 4.67 4.67 4.67 4.67 4.67 466 4.66 466 466 4.66 0.201404 .201123 .200843 .200563 .200283 .200003 .199723 .199443 .199164 .198884 60 49 48 47 46 45 44 43 42 41 20 9.728227 o oo 9.926831 IOO 9.801396 0.198604 40 21 22 23 24 26 26 27 28 29 .728427 .728626 728825 .729024 .729223 .729422 .729621 .729820 .730018 OnK) 3.32 3.32 332 3.32 331 3.31 3.31 3.31 3.31 .926751 .926671 .926591 .926511 .926431 .926351 .926270 .926190 .926110 .OO 1.33 1 33 1.34 1.34 134 1.34 1.34 1 34 1.34 .801676 .801955 .802234 .802513 .802792 .803072 .803351 1803909 4.66 4.66 4.66 465 465 465 4.65 4.65 465 465 .198325 .198045 .197766 .197487 .197208 .196928 .196649 .196370 .196091 39 38 37 36 35 34 33 32 31 80 31 32 33 34 36 36 37 88 89 9.730217 .730415 .730613 .730811 .731009 .731206 .731404 .731602 .731799 .731996 3.30 3.30 3.30 3.30 3.30 329 3.29 3.29 3.29 3.28 9.926029 925949 925S68 .925788 925707 .925626 .925545 .925465 .925384 .925303 1.34 1.34 1.34 1.34 1.35 1.35 1.35 1 35 1.35 1.35 9.804187 .804466 .804745 .805023 .805302 .805580 .805859 .806137 .806415 .806693 4:65 4.64 464 4.64 4.64 4.64 4.64 464 4.64 4.63 0.195813 .195534 .195255 .194977 .194698 .194120 .194141 .193863 .193586 .193307 30 29 28 27 26 25 24 23 22 21 40 41 42 9.732193 .732390 .732587 3.28 3.28 9.925222 .925141 .925060 1.36 1.35 1 OK W 806971 .807249 .807527 4.63 4.63 A O 0.193029 .192761 .192473 20 19 18 43 .732784 O OQ .924979 l.oo 1 OR 807805 4.O.J A O .192195 17 44 46 16 47 .732980 .733177 .733373 .733569 o./o 3.27 3.27 3.27 O OT .924897 .924816 .924735 .924654 I.JO 1.35 1.35 1 36 .8(18083 .808361 .80*638 .808916 4. DO 463 4.63 463 .191917 !l913f2 .1910H4 16 15 14 13 48 IS .733766 .733961 6.4.7 3.27 3.26 .924572 .924491 1.36 1.36 1.36 .809193 .809471 4.62 462 4.C2 .190807 .190529 12 11 60 9.734157 Q O A 9.924409 IOC 9.809748 0.190252 10 61 62 63 64 .734353 .734549 .734744 .734939 O. <6O 3.26 3.26 3.26 O OK .924328 .924246 .924164 .9'24083 .OO 1.36 l .36 1.36 1JC .610025 .810302 .810580 .810857 4.62 4C2 462 .189975 .189698 .189420 .189143 9 8 7 6 66 .7a r >l35 O.X) .924001 .JO .811134 4.62 .188866 6 66 67 68 59 60 .735330 .735525 .735719 .735914 .736109 3.25 3.25 3.25 325 3.24 .923919 .923*37 .923755 .923673 .923591 1.36 1.36 1.37 137 137 .811410 .811687 .811964 .812241 .812517 4.61 4.61 4fll 461 4.61 .188590 .188313 .188036 187759 .187483 4 3 2 1 M. Cosine. D. 1". 8U. D. 1". Ootaug D.l" Tang H 67 COSINES, TAN<;KNTS, AND c OTAN<;KNTS. 263 33 140 M. Sine. D. 1". Cosine. D. 1". Tang. D. 1". Cotang. M. | 1 2 3 4 5 6 7 8 9 9.736109 .736303 .736498 .736692 .736886 .737080 .737274 .737467 .737661 .737855 324 3.24 3.24 3.23 3.23 3.23 3.23 3.23 3.22 3.22 9.923591 .923509 .923427 .923345 .923263 .923181 .923098 .923016 .922933 .922851 1.37 1.37 1.37 1.37 1.37 1.37 1.37 1.37 1.37 1.38 9.812517 .812794 .813070 .813347 .813623 .813899 .814176 .814452 .814728 .815004 4.61 4.61 4.61 4.61 4.60 4.60 4.60 4.60 4.60 4.60 0.187483 .187206 .186930 .186653 .186377 .186101 .185824 .185548 .185272 .184996 60 59 58 57 56 55 54 53 52 51 10 11 12 13 14 9.738048 .738241 .738134 .738627 .738820 3.22 3.22 3.22 3.21 9.922768 .922686 .922603 .922520 .922438 .38 .38 .38 .38 9.815280 .815555 .815831 .816107 .816382 4.60 4.60 4.59 4.59 0.184720 .184445 .184169 .183893 .183618 50 49 48 47 46 15 .739013 3.21 .816658 .183342 45 16 17 18 .739206 .739398 .739590 3.21 3.21 92227*2 [922189 .922106 .38 .38 00 .816933 .817209 .817484 4.59 4.59 .18:3067 .182791 .182516 44 43 42 19 .739783 3.20 .922023 .38 .817759 4.59 .182241 41 20 9.739975 9.921940 9.818035 0.181965 40 21 22 23 24 .740167 .740359 .740550 .740742 3.20 3.20 319 .921857 .921774 .921691 .921607 .39 .39 .39 1 SQ .818310 .818585 .818860 .8191&5 4.58 4.58 4.58 4 K8 .181690 .181415 .181140 .180865 39 38 37 36 [ 25 .740934 .921524 .819410 .180590 H5 26 i 27 .741125 .741316 3.19 .921441 .921357 1.39 .819684 .819959 4.58 4.58 .180316 .180041 34 33 28 29 .741508 .74161)9 3.18 3.18 .921274 .921190 1.39 1.39 .820234 .820508 4.58 4.58 .179766 .179492 32 31 30 9.741889 9.921107 9.820783 0.179217 30 31 .742080 Q JO .921023 .821057 .178943 29 : 32 .742271 .920939 .821332 .178668 2H 34 j 35 .742462 .742652 .742842 3.17 3.17 o 17 .920856 .920772 .920688 .40 .40 40 .821606 .821880 .822154 4.57 4.57 .178394 .178120 .177846 27 26 25 36 | 37 .743033 .743223 3.17 .920604 .920520 .40 .822429 .822703 4.57 .177571 .177297 24 23 38 39 .743413 .743602 3.16 3.16 .920436 .920352 .40 .40 .822977 .823251 4.57 4.56 .177023 .176749 22 21 40 41 42 43 44 45 46 47 9.743792 .743982 .744171 .744361 .744550 .744739 .744928 .745117 3.16 3.16 3.16 3.15 3.15 3.15 3.15 9.920268 .920184 .920099 .920015 .919931 .919846 .919762 .919677 .40 .40 .40 .41 .41 . .41 .41 9.823524 .823798 .824072 .824345 .824619 .824893 .825166 .825439 4.56 4.56 4.56 4.56 4.56 4.56 456 0.176476 .176202 .175928 .175655 .175381 .175107 .174834 .174561 20 19 18 17 16 15 14 13 1 48 49 .745306 .745494 3.14 3.14 .919593 .919508 .41 1.41 .825713 .825986 4.55 4.55 .174287 174014 12 11 50 51 52 53 9.745683 .745371 .746060 .746248 3.14 3.14 3.14 9.919424 .919339 .919254 .919169 .41 1.41 .41 9.826259 .826532 .826805 .827078 4.55 4.55 4.55 0.173741 .173468 .173195 .172922 10 9 8 7 54 .746436 S'iS .919085 .827351 .172649 fi 55 56 57 .746624 .746812 .746999 3.13 3.13 .919000 .918915 .918830 1.42 1.42 .827624 .827897 .828170 4.55 4.55 .172376 .172103 .171830 5 4 8 58 59 60 .747187 .747374 .747562 3.12 3.12 .91*745 .918659 .91&574 1.42 1.42 .828442 .828715 .828987 4.54 4.54 .171558 .171285 .171013 3 1 M. Cosine. D. I". Sine. D. I". Cotang. D. 1". Tang. M. 133= 56' -!(>4 TABLE II LOGARITHMIC SiXfcS. 340 !4 ac M. Sine D. J. Cosine. D. 1". Tang. D. 1". Cotang. M. 1 2 3 4 5 6 7 8 9 9.747562 .747749 .747936 .748123 .743310 .743497 .748683 .748870 .749066 .749243 3.12 3.12 312 311 311 3.11 311 3.11 3.10 3.10 9. 9 18574 .918489 .918404 .918318 .918233 .918147 .9IR062 .917976 .917391 .917805 1.42 1.42 1.42 1.42 1.42 143 143 1.43 1.43 1.43 9.828987 .829260 .829532 .829sd5 .830077 .830349 .830621 .830893 .831165 .831437 4.54 4.54 4.54 454 454 4.64 4.53 4.53 4.53 4.53 0.171013 .170740 .17H468 .170196 .169923 .169651 .169379 .169107 .163835 .163563 60 59 68 67 66 66 64 53 62 61 10 11 12 13 14 16 16 9.749429 .749615 .749801 .749987 .750172 .750358 .750543 3.10 3.10 3.10 3.10 3.09 3.09 9.917719 .917634 .917548 .917462 .917376 .917290 .917204 1.43 1.43 1.4:5 1.43 1.43 1.43 9.831709 .831981 .832253 .832."25 .832796 .833063 .833339 4.53 4.53 4.53 4.53 4.63 4.53 0.168291 .168019 .167747 .167476 .167204 .166932 .166661 50 49 48 47 46 45 44 17 18 19 .750729 .750914 .751099 3.09 3.09 3.08 .917118 .917032 .916946 1.44 1.44 1.44 .83361 1 .833382 .834154 4.62 4.52 4.52 4.52 .166389 .166118 .165846 43 42 41 20 21 22 23 24 25 26 27 28 29 9.751231 .7GKf>9 .7516C4 .751839 .7620X3 .752*03 .752392 .752576 752760 .752944 3.08 308 3.08 303 307 3.07 3.07 307 307 3.06 9.916859 .916773 .916687 .916600 .916511 .916427 .91C341 .916254 .916167 .916081 1.44 144 1.44 1.44 1.44 1.44 1.44 1.44 1.45 1.45 9.834425 .834696 .834967 .835233 .835509 .835780 .836051 .836322 .836593 .836864 4.52 4.52 452 4.52 4.52 4.52 4.51 4.51 4.51 4.51 0.165575 .165304 .165033 .164762 .164491 .164220 .163949 .163678 .163407 .163136 40 39 38 37 36 36 34 33 32 31 80 31 32 9.753123 .753312 .753495 3.06 3.06 9.915994 .915907 .915820 1.45 1.45 9.837134 .837405 .837675 4.61 4.61 0.162866 .162595 .162325 30 29 28 33 34 36 36 37 38 39 .7536/9 .753862 .754046 .754229 .754412 .764595 .754778 3.06 3.05 3.05 3.05 305 3.05 3.05 .915733 .915646 .915559 .915472 .915aS5 .915297 .915210 1.45 1.45 1.45 1.45 1.45 1.45 1.45 1.46 .837946 .838216 .838487 .838757 .839027 .839297 .839568 4.51 4.61 4.61 4.51 4.50 4.60 4.50 4.60 .162054 .161784 .161513 .161243 .160973 .160703 .160432 27 26 25 24 23 2tt 21 40 41 42 43 44 45 46 47 48 49 9.754960 .755143 .755326 .755508 .755690 .755872 .756054 .756236 .756418 .756600 3.04 3.04 3.04 3.04 3.04 3.03 3.03 3.03 3.03 3.03 9.915123 .915035 .914948 .914S60 .914773 .914685 .914598 .914510 .914422 .914334 1.46 1.46 1.46 1.46 1.46 1.46 1.46 1.48 1.46 1 46 9.839838 .840108 .840378 .840643 .840917 .841187 .841457 .841727 .841998 .642266 4.60 4.50 4.60 450 4.50 4.49 4.49 4.49 4.49 4.49 0.160162 .159892 .159622 .159352 .159083 .158813 .158543 .158273 .158004 .157734 20 19 18 17 16 15 14 13 12 11 60 61 62 63 64 66 66 67 68 9.756782 .756963 .757144 .757326 .757507 .757688 .757869 .753050 .758230 3.02 3.02 3.02 3.02 3.02 3.02 3.01 3.01 9.914246 .914158 .914070 .913982 .913894 .9I38GA .913718 913630 .913541 1.47 1 47 1.47 1.47 1.47 1.47 1.47 1.47 9.842535 .842805 .843074 .84a343 .843612 .843382 .844151 .844420 .844689 4.49 4.49 449 449 4.49 449 4.48 4.48 0.157465 .157195 .156926 .156657 .156388 .156118 .155849 .155580 .155311 10 9 8 7 6 5 4 3 2 69 60 .753411 .758591 3.01 .913453 .913365 1.47 1.47 .844958 .845227 4.48 .155042 .154773 1 M. Oodne. D. 1". Sine. D. 1". Cotang D,l. Tang. M. 1*40 COSINES, TANGENTS, AND COTANGENTS. M. Sine. D.I" Codna. D. 1". Tang. D. 1". Cotang. M 1 2 3 4 6 6 7 8 9 9.758591 .758772 .768952 .759132 .769312 .7594'.2 .759672 .759852 .760031 .760211 3.01 3.00 3.00 3.f0 3.00 3.00 2.99 2.99 2.99 2.99 9.913365 .913276 .9131*7 .9130H9 .913010 .912922 .912833 .912744 .912655 .912566 1.47 1.48 1 48 1.48 148 1.48 1.43 1.48 1.43 1.48 9.845227 .845496 .845764 .846033 .846302 .846570 .846839 .847108 .847376 .847644 4.48 4.48 4.48 4.48 4.48 4.48 4.48 4.47 447 4.47 0.154773 .154504 .154236 .153967 .153698 153430 .153161 .152*92 .152624 .162356 60 69 68 67 66 66 64 63 62 61 10 11 9.760390 .760569 2.99 9.912477 .912338 1.48 9.847913 .848181 4.47 0.152087 .151819 50 49 12 13 14 15 16 17 18 19 .760748 .760927 .761106 .761285 .761464 .761642 .761821 .761999 2.98 2.98 2.98 2.98 2.98 2.97 2.97 2.97 .912299 .912210 .912121 .912031 .911942 .911853 .911763 .911674 1.49 1.49 1.49 1.49 1.49 1 49 1.49 1.49 .848449 .848717 .848986 .649254 .849522 .849790 .850057 .350325 447 4.47 4.47 4.47 4.47 4.46 4.46 4.46 .151651 .151283 .151014 .160746 .150478 .160210 .149943 .149675 48 47 46 45 44 43 42 41 4) 21 22 23 24 25 26 17 28 29 9.7C2177 .762356 .7625:34 .762712 .762889 .763067 .763245 .763422 .763600 .763777 2.97 2.97 2.97 2.96 296 2.96 2.96 2.96 2.95 2.95 9.911584 .911495 .911405 .911315 .91 1226 .911136 .911046 .910956 910866 910776 1.49 1.49 1.49 1.60 1.60 1.60 1.60 1.60 1.60 1.60 9.850593 .850861 .851129 851396 .851664 851931 .852199 .852466 .852733 .853001 4.46 4.46 4.46 4.46 4.46 4.46 4.46 4.46 4.46 4.46 0.149407 .149139 .148871 .148604 .148336 .148069 .147801 .147534 .147267 .146999 40 39 38 37 36 36 34 33 32 31 30 31 9.763954 764131 2.95 9.910686 910596 1.50 9.853268 .853535 4.45 0.146732 .146465 30 29 32 33 34 35 36 37 38 764308 764485 .764662 .764838 .765015 .765191 .765367 2.95 295 2.94 2.94 2.94 2.94 910506 910416 .910325 .910235 910144 .910054 .909963 1.60 1.61 1.61 1.61 1.61 1.61 .853802 .854(69 .854336 .854603 .854870 .855137 .855404 4.45 4.45 4.45 4.45 4.45 4.45 .146198 .145931 .145664 .146397 .145130 .144863 .144596 28 27 26 26 24 23 22 39 .765544 2.03 .909873 1.51 .856671 4.44 .144329 21 40 41 42 9.765720 .765896 .766072 2.93 2.93 9.909782 .909691 .909601 1.51 1.C1 9.855938 .856204 .856471 4.44 4.44 0.144062 .143796 .143529 20 19 18 43 .766247 .909510 1.51 .856737 .143263 17 44 45 46 47 48 49 .766423 .766593 .766774 .766949 .767124 .767300 2.93 292 2.92 2.92 2.92 2.92 909419 909328 .909237 .909146 .909055 .9089^4 1 51 1 52 1.6& 1.62 1.62 1.62 1.52 .857004 .857270 .857537 .867803 .858069 .858336 4.44 4.44 4.44 4.44 444 4.44 .142996 .142730 .142463 .142197 .141931 .141664 16 15 14 13 12 11 60 61 62 53 64 65 56 67 63 59 60 9767475 .767649 .767824 .767999 .768173 .768348 .76>522 .76*897 .768871 .769045 .769219 2.91 291 291 2.91 291 2.91 2.90 290 2.90 2.90 9.908873 .90S781 .908690 .'90*507 .90S4I6 .90S324 .908233 .90S141 .908049 .907958 1.62 1.52 1.62 1.62 1.52 1.53 1.53 1.53 1.53 1.53 9.858602 .858868 .859134 .859400 .859666 !860198 .860464 .860730 .860995 .861261 4.44 443 443 443 4.43 443 443 4.43 4.43 4.43 0.141398 .141132 .140866 .140600 .140334 .140068 .139802 .139536 .139270 .139005 .138739 10 9 8 7 M. OOBUM. D.1". Blue. D.I". CoMng. D. 1". Tang; M. 5*4 266 TABLE II. LOGARITHMIC SINES, M. Sloe. D. 1". Ooeine. D. 1". Tkmg. D. 1". Ootang. M 9.769219 .769393 .769566 .769740 2.90 2.90 2.89 9.907958 .907866 .907774 .907682 1.53 1.53 1.53 9.861261 .861527 .861792 .862058 4.43 4.43 4.43 0.138739 .138473 .138208 .137942 60 59 58 57 .769913 .907590 .862323 .137677 56 .770087 .770260 .770433 .7706(* .770779 2.89 2.89 2.88 2.88 2.88 .907498 .907406 .907314 .907222 .907129 1.53 1.54 1.54 1.54 1.54 .862589 .862854 .863119 .863385 .863650 4.42 4.42 4.42 4.42 4.42 .137411 .137146 .136881 .136615 .136350 55 54 53 Si 51 10 11 9.770952 .771125 2.88 9.907037 .906945 1.54 9.863915 .864180 4.42 0.136085 .135820 50 49 12 13 14 15 18 17 18 19 .771298 .771470 .771643 .771815 .771987 .772159 .772331 .772503 2.88 2.87 2.87 2.87 2.87 2.87 2.87 2.86 .906852 .906760 .906667 .906575 .906482 .906389 .906296 .906204 1.54 1.54 1.54 1.54 1.55 1.55 1.55 1.55 .864445 .864710 .864975 .865240 .865505 .865770 .866035 .866300 4.42 4.42 4.42 4.41 4.41 4.41 4.41 4.41 .135555 .135290 .135025 .134760 .134495 .134230 .133965 .133700 48 47 46 45 44 43 42 41 20 21 22 23 24 25 26 27 28 29 9.772675 .772847 .773018 .773190 .773361 .773533 .773704 .773375 .774046 .774217 2.86 2.86 2.86 2.86 2.85 2.85 2.85 2.85 2.85 2.85 9.906111 .906018 .905925 .905832 .905739 .905645 .905552 .905459 .905366 .905272 1.55 1.55 1.55 1.55 1.55 1.55 155 1.56 1.56 1.56 9.866564 .866829 .867094 .867358 .867623 .867887 .868152 .868416 .868680 .868945 4.41 4.41 4.41 4.41 4.41 4.41 4.41 4.41 4.40 4.40 0.133430 .133171 .132906 .132642 .132377 .132113 .131848 .131584 .131320 .131055 40 39 38 37 36 35 34 33 32 31 30 31 32 33 34 9.774388 .774558 .774729 .774899 .775070 2.84 2.84 2.84 2.84 9.905179 905085 .904992 .904898 .904804 1.56 .56 .56 .56 9.869209 .869473 .869737 .870001 .870265 4.40 4.40 4.40 4.40 0.130791 .130527 .130263 .129999 .129735 30 29 28 27 26 35 36 37 38 39 775240 .775410 .775680 .775750 .775920 2.84 2.83 2.83 2.83 .904711 .904617 .904523 .904429 .904335 .56 .56 .57 .57 .870529 .870793 .871057 .871321 .871585 4.40 4.40 4.40 4.40 .129471 .129207 .128943 .128679 .128415 25 1 24 23 22 21 40 41 42 43 44 9.776090 .776259 .776429 .776598 .776768 283 2.83 2.82 2.82 9.904241 .9. t O 2.73 2.73 .893639 .898592 l'62 1.62 .887072 .887333 4.35 4.35 4.35 .112928 .112667 22 21 40 41 42 43 44 45 9 786039 .786252 .786416 .786579 .786742 .786906 2.73 2.73 272 272 272 Q 7O 9.898494 .893397 .898299 .898202 .893104 .898006 1.63 1 63 1.63 1.63 1.63 9.887594 .887855 .888116 .888378 .888639 .888900 4.35 4.35 4.35 4.35 4.35 0.112406 .112145 .111884 .111622 .111361 .111100 20 19 18 17 16 15 43 47 48 49 .787069 .737232 .787395 .787557 c. t ft 2.72 2.72 2.71 2.71 .897908 .897810 .897712 .897614 1.63 1.63 1 63 1.63 .889161 .889421 .889632 .889943 4.35 4.35 4.35 4.35 4.35 .110839 .110579 .110318 .110057 14 13 12 11 60 61 9.787720 .787883 271 9.897516 .897418 1.64 9.890204 .890465 4.35 O.f09796 .109535 10 9 62 53 54 55 66 57 68 59 60 .788045 .788208 .788370 .788532 783694 .783856 789018 .789180 .789342 2 71 271 271 270 270 2.70 270 2.70 2.70 .897320 .897222 .897123 .897025 .896926 .896>*iS .896729 .896631 .896532 1 64 1 64 1.64 1 64 1 64 1.64 1.64 1 64 1.64 .890725 .890936 891247 .891507 891763 892028 .892289 .892549 .892810 4.35 434 4.34 4.34 4.34 4.34 4.34 4.34 4.34 .109275 .109014 .108753 .10*193 .103232 107972 .107711 .107451 .107190 8 7 6 5 4 3 2 1 M. Oodno. D.I". Sine. D. 1". Cotaug D. 1". Tang. M. 1587' 268 TAJiLE II. LOGARITHMIC SINES. 1410 M. Blue. D. 1". Conine. D. 1". Tang. D. 1". Cotang. M. 1 2 3 4 5 8 7 8 9 9.789342 .7895(>4 .789665 .789827 .789988 .790149 .790310 .790471 .790632 .790793 2.69* 2.69 269 269 269 2.69 268 268 263 2.68 9.696532 .896433 .896335 .896236 .896137 .8960:38 .895939 .895*40 .895741 .895641 1.65 1.65 1.65 165 .65 .65 .65 .65 65 .65 9.892*10 .893070 .693331 .893591 !6941I1 .694372 .894632 .894392 .895152 434 4.34 434 434 434 434 434 434 4.33 4.33 0.107190 .106930 .106669 106409 .106149 .105*89 .10628 .105368 .105108 .104848 60 59 58 57 56 55 54 53 52 51 10 11 9790954 .791115 268 9.895542 .895443 .66 9.895412 .895672 4.33 0.104588 .104328 60 49 12 13 14 15 16 17 18 19 .791275 .791436 .791596 .791757 .791917 .792077 .792237 .792397 267 267 267 267 2.67 267 267 2.66 .895343 .895244 .895145 .895(H5 .894945 .894S46 .894716 .894646 .66 .66 .66 .66 .66 66 66 .66 .895932 .896192 .896452 .896712 .696971 .897231 .897491 .897751 433 433 433 433 433 433 433 4.33 .1041168 .103808 .103548 .1032*8 .103029 .102769 . 102509 .102249 48 47 46 45 44 43 42 41 20 22 23 24 26 26 27 28 29 9.792557 .792716 .792876 .793035 .793195 .793354 .793514 .793673 .793832 .793991 266 266 2H6 266 266 265 265 265 265 2.65 9.89-1546 .8'J4! 46 .894346 .894246 .894146 .694046 .893946 .893846 .893745 .893645 .67 67 .67 .67 .67 67 67 .67 .67 .67 9.898010 .898270 .898530 .898789 .899049 .891)303 .699563 .899827 .9000H7 .900346 433 433 433 433 433 4 32 4 32 43* 4 32 432 0.101990 .101730 .101470 .101211 .looar.i .10061)2 .100432 .100173 .099913 .099654 40 39 38 37 36 a 33 32 31 30 31 82 33 34 35 36 37 38 39 9794150 .794308 .794467 .794626 .794784 .794942 .795101 .795259 .795417 .795575 265 2.64 2.6-1 264 264 264 264 264 2f,3 2.63 9.893544 .693414 .693343 .893243 .693142 .893041 .892U40 .892*39 .892739 .892638 .68 .63 .68 .63 68 .68 .63 68 66 .68 9.900605 .900SG4 .901124 .9013*3 .901642 .901901 .902160 .902120 .902679 .902933 432 432 4 32 432 432 432 432 432 432 4.32 0.099395 !09**76 .09*617 .098358 .09*099 .01)7^0 !09732I .097062 30 29 28 27 26 25 24 23 22 21 40 41 9.795733 .795891 263 9.892536 .692435 69 9.903197 .903456 432 ioyr.544 20 19 42 43 .796049 .796206 263 .6923:34 .892233 .69 .903714 903973 4 31 18 17 44 45 46 .796364 .796521 .796679 262 2.62 .892132 .692030 .891929 .69 .69 .904232 .904491 .904750 431 431 .095768 .095509 16 15 14 47 48 .796336 .796993 262 ^891726 69 .905003 .905267 4 31 >>94733 IB 12 49 .797150 2.61 .891624 .69 .905526 4.31 .094474 i. 60 51 52 53 9.797307 .797464 .797621 .797777 2.61 261 261 9.89IT.23 .891421 .891319 .891217 .70 .70 .70 9.90575 .906043 .X3i >2 .9f'G56( 431 431 4.31 0.094215 .093957 .093698 .093440 10 9 8 7 54 55 .797934 .79^091 261 .891115 .891013 70 .906s 1 9 .907077 431 .093 HI .0921)23 6 5 1 56 57 68 .79^247 .79403 .798560 261. 260 .890911 .890^09 .890707 .70 .70 .907336 .907594 .907853 4.31 431 431 .0926B4 .01)2406 .092147 4 3 2 59 .793716 ~ 5!; .890605 .908111 .091*89 60 .793872 .890503 .908369 4.31 .091631 M. (Cosine. D.I bine. D. 1". Cotang. D. 1". Tang. M. 51 COSINES, TANGENTS, AND COTANGENTS. '269 890 140 M. Sine. D. 1". Cosine. D. 1". Tang. D. 1". Cotang. M. I 2 9.798872 .799028 .799134 2.60 2.60 9.890503 .890400 .890298 1.71 1.71 9.90.8369 .9US628 4.30' 4.30 0.091631 .091372 .091114 60 59 58 3 4 5 6 7 8 9 .799339 .799495 .799651 .799806 .799962 .800117 .800272 2.60 2.59 2.59 2.59 2.59 2.59 2.59 2.59 .890195 .890093 .889990 .8898,88 .889785 .889682 .889579 1.71 1.71 1.71 1.71 1.71 1.71 1.71 1.71 !909144 .9UJ402 .909660 .910177 .910135 .910693 430 4 30 430 4.30 . 4.30 4.30 4.30 .090356 .090598 .090340 .090082 .089823 .089565 .089307 57 56 55 54 53 52 51 10 ll 12 13 14 16 10 17 18 19 9.800427 .800582 .800737 .800892 .W1047 .SO 1201 .801356 .801511 .801665 801319 2.58 2.58 2.58 2.58 2.58 2.53 2.57 2.57 2.57 2.57 9.889477 .889374 .889271 .889168 .889064 .883961 .888853 .888755 .888651 .888543 1.72 1.72 1.72 1.72 1.72 1.72 1.72 1.72 1.72 1.72 9.910951 .911209 .911467 .911725 .911982 .912240 .912493 .912756 .913014 .913271 4.30 4.30 4.30 4.30 4.30 4.30 4.30 4.30 4.30 4.30 0.089049 .088791 .088533 .088275 .088018 .087760 .087502 .087244 .086986 .086729 50 49 48 47 46 45 44 43 42 41 20 21 22 23 24 25 9801973 .802123 .802282 .802436 .802589 .802743 2.57 2.57 2.57 256 2,50 9.888444 .888341 .888237 .888134 .888030 .887926 1.73 1.73 1.73 1.73 1.73 9.913529 .913737 .914044 .914302 .914560 .914817 4.29 4.29 429 4.29 4.90 0.086471 .086213 .085956 .085693 .085440 .085183 40 39 38 37 36 36 26 27 23 29 '.803050 .803204 .803357 2.56 2.56 2.56 2.56 2.55 .887322 .887718 .837614 .887510 1.73 1.73 1.73 173 1.74 .915075 .915332 .915590 .915547 429 4.29 429 4.29 .084925 .0*4*68 .084410 .084153 34 33 32 31 30 98035U 9.887406 If A 9.916104 0.083396 30 31 32 33 34 35 36 .803664 .803-117 .803970 .804123 .804276 .804428 2.55 2.55 255 2.55 255 2.55 .887302 .887193 .887093 .886989 .886886 .886780 .74 1.74 1.74 1.74 1.74 1.74 .916362 .910619 .916877 .917134 .917391 .917648 429 429 4.29 429 4.29 .083633 .083381 .083123 .082*56 .052609 .082352 29 28 27 26 25 24 37 38 39 .8IM581 .804734 .804386 2 54 254 2.54 2.54 .886676 .886571 .886466 1.74 1.74 1.74 1.75 .917906 .918163 .918420 4' 29 4.29 4.29 .082094 .031837 .081580 23 22 21 40 41 a 43 44 45 46 47 48 49 9.805039 .805191 .805343 .805495 .805647 .805799 .805951 .806103 .806254 .806406 254 254 254 2.53 2.53 2.53 2.53 253 253 2.52 9.&S6362 .886257 .886152 .886047 .8859-42 .8*5837 .885732 .8S5627 .885522 .885416 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.76 9.918677 .918934 .919191 .919443 .919705 .919962 .920219 .92IM76 .920733 .920990 4.28 423 4.28 4.23 4.28 4.28 4.28 4.28 423 4.28 0.081323 .081066 .080809 .080552 .080295 .0*0038 .079781 .079524 .079267 .079010 20 19 18 17 16 15 14 13 12 11 50 51 9.806557 .800709 2.52 9.885311 .8ft5205 1.76 9.921247 .921503 428 0.078753 .07^497 10 9 52 .806S60 2 52 .885100 1.76 .921760 4 28 .07-^240 8 53 54 .807011 .807163 2.52 2.52 .8*4994 .884889 1.76 1.76 .922017 .92-2-274 4.28 4-23 .0779S3 .077726 7 6 55 56 57 53 59 80 .807314 .807465 .807615 .807766 .807917 808067 2.52 2.52 2.51 2.51 251 2.51 .884783 .884677 .884572 .884466 .884360 .884254 1.76 1.76 1.76 1.76 1.77 1.77 .922530 .922787 .923044 .923300 .923567 .OU3R14 4.28 423 423 428 4.28 4.28 .077470 .077213 .076956 .076700 .076443 .076186 5 4 3 2 M. Codno D. 1". Sloe. D.1". Cotang D. 1". Ofcng M. 189 50 270 OC.ARITH MIC SINES. M. Sine. D. 1". Cosine. D. I". Tang. D.I . Cotang. M. 9808067 o Ci 9.884254 9.923314 0.076186 60 1 .808213 V.OI 9 ci .884148 77 .924070 A OQ .075930 59 2 3 4 .803363 .808519 .808669 B.OI 2.51 2.50 o n .884(142 .883936 .883829 '.77 .77 77 .924327 '.924840 V.J0 4.27 4.27 A O7 .075673 .075417 .075160 58 57 56 6 6 7 .808819 803969 .809119 VkOU 2.50 250 O /k .883723 .883617 .883510 . // .77 .77 .925(196 .925352 .925609 *. .1 4.27 4.27 .074904 .074648 .074391 55 54 53 8 .809269 3LMJ .883404 .77 .925865 4.27 .074135 52 9 .809419 2.50 .883297 .78 .73 .926122 4^27 .073878 51 10 11 12 13 14 15 16 17 18 19 9.809569 .809718 .809368 .810017 .810167 .810316 .810465 .810614 .810763 .810912 2.49 2.49 2.49 2.49 2.49 2.49 2.43 2.43 2.48 2.43 9.883191 .883084 .882977 .882371 .832764 .882657 .832550 .882443 .832336 .882229 .78 .78 .78 .78 .78 .78 .78 .79 .79 .79 9.926378 .926634 .926390 .927147 .927403 .927659 .927915 .923171 .923427 .92S684 4.27 4.27 4.27 427 4.27 4.27 4.27 4.27 427 4.27 0.073622 073366 .073110 .072853 .072597 .072341 .072085 .071329 .071573 .071316 50 49 43 47 46 45 44 43 42 41 20 21 22 23 24 25 26 27 28 9.811061 .811210 .811358 .811507 .811655 .811804 .811952 .812100 .812243 243 2.48 248 2.47 2.47 2.47 2.47 247 9882121 .882014 .881907 .881799 .881692 .88I5H4 .881477 .881369 .881261 .79 .79 .79 .79 .79 .79 .79 .80 9.92S940 .929196 .929-152 .929703 .929964 .930220 .930475 .930731 .93U937 427 4.27 4.27 4.27 4.27 4.27 4.26 4.26 0.071060 .070804 .070543 .070292 .070036 .069780 .069525 .069269 .069013 40 39 38 37 36 35 34 33 32 29 .812396 2.47 2.47 .881153 .80 .80 .931243 4 26 4.26 .063757 31 30 31 32 33 9.812544 .812692 .812840 .812988 246 246 2.46 9.881046 .880933 .880330 .880722 .80 .80 .80 9.931499 .931755 .932010 .932266 4:26 4.26 4.26 0.068501 .068245 .067990 .067734 30 29 28 27 34 35 .813135 .813283 2.46 2.46 .830613 .880505 .80 .80 .932522 .932778 4.26 4.26 .067478 .067222 26 25 36 .813430 2.46 O Ifi .880397 .80 Dl .933033 4.26 .066967 24 37 .813578 2.4O .880289 Jot .933239 4.26 066711 23 38 39 .813725 .813872 2.45 2.45 2.45 .880180 .880072 .81 .81 .81 .933545 .933800 4.26 4.26 4.26 .066455 .066200 22 21 40 41 9.814019 .814166 2.45 9.8^9963 .879855 .81 9.934056 .934311 4.26 0.065944 .065689 20 19 42 43 44 45 46 .814313 .814460 .814607 .814753 .814900 2.45 2.45 2.45 2.44 2.44 .879746 .879637 .879529 .870420 .879311 .81 .81 .81 .81 .81 .934567 .934822 .935078 .935333 .935589 4.26 4.26 4.26 4.26 4.26 .065433 .065178 .064922 .064667 .064411 18 17 16 15 ' 14 47 .815046 2.44 .879202 .82 .935844 4.26 .064156 13 48 .815193 2.44 .879093 .82 .936100 4.26 .063900 12 49 .815339 2.44 2.44 .878984 .82 .82 .936355 4.26 4.26 .063645 11 50 9.815485 9.878875 OO 9.936611 0.063389 10 51 52 53 54 55 .815632 .815778 .815924 .816069 .816215 2.44 2.43 2.43 2.43 2.43 .878766 .878656 .878547 .878438 .878328 .04 .82 .82 . .82 .82 .936866 .937121 .937377 .937632 .937887 4 26 4.26 4.26 4.25 4.25 .063134 .062879 .062623 .062368 .062113 9 8 7 6 6 66 57 58 59 .816361 .816507 .816652 .816798 2.43 2.43 2.43 2.42 .878219 .878109 .877999 .877890 .83 .83 .83 .83 .933142 .938398 .93*653 .938908 4.25 4.25 4.25 4.25 .061358 .061602 .061347 .061092 4 3 3 60 .816943 2.42 .877780 1.83 .939163 4.25 .060837 M. Cosine D. 1". Sine D. 1''. Cotang. D.I" Tang. M COSINES, TANGENTS, AND COTANGENTS. 271 410 138 M. Blue. D.I". Cosine. D 1". fcrng. D.1'. Ootaog. M. a !817088 .817233 2.42 2.42 O /1O 9.877780 .877670 .877560 1.83 1.83 IQQ 9.939163 .939418 .939673 4.26 4.25 A OK 0.060837 .060582 .060327 60 69 68 3 4 6 6 7 8 9 .817379 .817524 .817668 .817813 .817958 .818103 .818247 <6.4<6 2.42 2.42 2.41 2.41 2.41 2.41 2.41 .877450 .877340 .877230 .877120 .877010 .876899 .876789 .OO 1.83 1.84 1.84 1.84 1.84 1.84 1.84 .939928 .940183 .940139 .940694 .940949 .9412^ .941459 4.<6W 4.25 4.25 4.25 4.25 4.25 4.25 4.25 .060072 .059817 .059561 .059306 .059051 .058796 .058541 67 56 55 54 53 52 61 10 11 12 13 14 15 16 17 18 9.818392 .818536 .818681 .818825 .818969 .819113 .819257 .819401 .819545 2.41 2.41 2.40 2.40 2.40 2.40 2.40 2.40 9.876678 .876568 .876457 .876347 .876236 .876125 .876014 .875904 .876793 1.84 1.84 1.84 1.84 1.85 1.85 1.85 1.85 9.941713 .941968 .942223 .942478 .942733 .942988 .943243 .943498 .943752 4.25 4.25 4.25 4.25 4.25 4.25 4.25 4.25 0.058287 .058032 .057777 .057522 .057267 .057012 .056757 .056502 .056248 60 49 48 47 46 45 44 43 42 19 .819689 2.40 2.39 .875682 1.85 1.85 .944007 4^25 .055993 41 20 21 22 23 24 25 26 27 28 9 819832 .819976 .820120 .820263 .820406 .820550 .820693 .820836 .820979 2.39 2.39 2.39 2.39 2.39 2.39 2.38 2.38 9.875571 .875459 .8753-18 .875237 .875126 .875014 .874903 .874791 .874680 1.85 1.85 1.85 1.86 1.86 1.86 1.86 1.86 9.944262 .944517 .944771 .945026 .945281 .945535 .945790 .946045 .946299 4.25 4.25 4.24 4.24 4.24 4.24 4.24 4.24 0.055738 .055483 .055229 .054974 .054719 .054465 .054210 .053955 .053701 40 39 38 37 36 35 34 33 32 29 .821122 2.38 2.38 .874568 1.86 1.86 .946554 4.24 4.24 .053446 31 30 31 9.821265 .821407 2.38 9.874456 .874344 1.86 9.946808 .947063 4.24 0.053192 .052937 30 29 32 33 34 35 36 37 38 39 .821550 .821693 .821835 .821977 .822120 .822262 .822404 .822546 2.33 2.33 2.37 2.37 2.37 2.37 2.37 2.37 2.37 .874232 .874121 .874009 .873896 .873784 .873672 .873560 .873448 1.86 1.87 1.87 1.87 1.87 1.87 1.87 1.87 1.87 .947318 .947572 .947827 .948081 .948335 .948590 .948844 .949099 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 .052682 .052428 .052173 .051919 .051665 .051410 .051156 .050901 28 27 26 25 24 23 22 21 40 41 42 43 44 45 46 47 48 49 9.822688 .822330 .822972 .823114 .823255 .823397 .823539 .823680 .823821 .823963 2.37 2.36 2.36 2.36 2.36 2.36 2.36 2.36 2.35 9.873335 .873223 .873110 .872998 .872885 .872772 .872659 .872547 .872434 .872321 1.87 1.88 1.88 1.88 1.88 1.88 1.88 1.88 1.88 9.949353 .949608 .949862 .950116 .950371 .950625 .950879 .951133 .951388 .951642 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 0.050647 .050392 .050138 .049884 .049629 .049375 .049121 .048867 .048612 .048358 20 19 18 17 16 15 14 13 12 11 2.35 1.88 4.24 60 61 62 9.824104 .824245 .824386 2.35 2.35 9.872208 .872095 .871981 1.89 1.89 9.951896 .952150 .952405 4.24 4.24 0.048104 .047850 .047595 10 9 8 53 .824527 2.35 .871868 1.89 .952659 4.24 .047341 7 54 55 .824668 .824808 2.35 2.35 .871755 .871641 1.89 1.89 .952913 .953167 4.24 4.24 .047087 .046833 6 6 56 67 63 69 .824949 .825090 .825230 .825371 2.34 2.34 2.34 2.34 .871528 .871414 .871301 .871187 1.89 1.89 1.89 1.89 .953421 .953675 .953929 .954183 4.24 4.24 4.23 4.23 .046579 .046325 .046071 .045817 4 3 a i 60 .825511 2.34 .871073 1.90 .954437 4.23 .045563 M. Cosine. D.I". Slue. D. 1". Cotang. D. 1". Tang. M. 48 .OGARITHMIC SIXES, 137* M. Sine. D. 1''. Codne. D. 1". Tang. D. 1". Cotang. M. 2 3 4 5 6 7 8 9 9.8255 1 1 .825651 .825791 .825931 .826071 .82621 1 .826351 .826491 826631 .826770 2.34 2.34 2.33 2.33 2.33 2.33 2.33 2.33 2.33 2.33 9.871073 .870960 .870846 .870732 .870618 .870504 .870390 .870276 .870161 .870047 .90 .90 .90 .90 .90 .90 .90 .90 .91 .91 9.954437 .954691 .954946 .955200 .953454 .9557(13 .955%! .956215 .956469 .956723 4.23 4.23 4.23 4.23 4.23 4.23 4.23 4.23 4.23 4.23 0.045563 .045309 .045054 .044800 .044546 .044292 .044039 .043785 .043531 .043277 60 59 58 57 56 55 54 53 52 51 ie 11 9.826910 .827049 2.32 O '}.) 9.869933 .869318 .91 9.956977 .957231 4.23 A OQ 0.043023 .042769 50 49 12 13 14 15 16 17 18 .827139 .82732-3 .827467 .827606 .827745 .827884 .828023 A.&A 2.32 2.32 2.32 2.32 2.32 2.31 O Ql .869704 .869589 .869474 .869360 .869245 .869130 869015 .91 .91 .91 .91 .91 .91 .92 QO .957435 .957739 .957993 .958247 .958500 .958754 .959008 4.-6O 4.23 4.23 4.23 4.23 4.23 4.23 A O1 .042515 .042261 .042007 .041753 .041500 .041246 .040992 48 47 46 45 44 43 42 19 .828162 .&{ 2.31 .868900 ,Wm .92 .959262 %JH 4.23 .040738 41 20 21 9.828301 .828439 2.31 9.868785 .863670 .92 9.959516 .959769 4.23 0.040484 .040231 40 3S 22 .823578 2.31 .868555 .92 .960023 4.23 .039977 38 23 24 .828716 .828855 2.31 2.31 .868440 .868324 .92 .92 .960277 .960530 4.23 4.23 .039723 .039470 37 36 25 26 27 28 29 .828993 .829131 .829269 .829407 .829545 2.31 2.30 2.30 2.30 2.30 2.30 .8GS209 .863093 .867978 .867862 .867747 .92 .92 .93 .93 .93 .93 .960784 .961033 .961292 .961545 .961799 4.23 4.23 4.23 4.23 4.23 4.23 .039216 .03^962 .038708 .038455 .038201 35 34 33 32 31 30 31 32 33 34 35 36 9.829683 .829821 .829959 .830097 .830234 830372 .830509 2.30 230 2.29 2.29 2.29 2.29 9.867631 .867515 .867399 .867233 .867167 .867051 .866935 .93 .93 .93 .93 .93 .94 9.962052 .962306 .962560 .962813 .963067 .963320 .963574 4.23 4.23 4.23 4.23 4.23 4.23 0.037948 .037694 .037440 .037187 .036933 .036630 .036426 3C 29 28 27 26 24 37 38 39 .83(1646 .830784 .830921 2.29 2.29 2.29 2.29 .866819 .866703 .866586 .94 .94 .94 .94 .963828 .964081 .964335 4.23 4.23 4.23 4.23 .036172 .035919 .035665 23 22 21 40 41 42 43 44 45 46 47 48 9.831058 .831195 .831332 .831469 .831606 .831742 .831879 .832015 .832152 2.23 2.2l? 2.23 2.28 2.23 2.28 2.23 2.27 O O7 9.866470 .866353 .866237 .866120 .866004 .865387 .865770 .865653 .865536 .94 .94 .94 94 .95 .95 .95 .95 nc 9.964588 .964842 .965095 .965349 .965602 .965855 .966109 .966362 .966616 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 A QO 0.035412 .035158 .034905 .034651 .034393 .034145 .033891 .033633 .033384 20 19 18 17 16 15 14 13 12 49 .832288 22 4.22 4.22 4.22 4.22 4.22 4.22 .029584 .029331 .029078 .023825 .023571 .028318 .023065 57 56 55 54 53 52 51 10 9.835134 9.86-2946 9.972188 0.027812 50 11 12 13 14 15 16 17 18 19 .835269 .8354(13 .83.KJ.i8 .835672 .835807 .835941 .836075 .836209 .836343 2.24 2.24 2.24 2.24 2.24 2.24 2.24 2.23 2.23 2.23 .862827 .862709 .862590 .862471 .862353 .862234 .862115 .661996 .861877 ^93 .98 .93 .93 1.93 1.98 1.93 1.98 1.99 .972441 .972695 .972948 .973201 .973454 .973707 .973960 .974213 .974466 4'22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 .027559 .027305 .027052 .026799 .026546 .026293 .026040 .025787 .025534 49 48 47 46 45 44 43 42 41 20 21 9.836477 .836611 2.23 9.861758 .861638 .99 9.974720 .974973 4.22 A * 0.025280 .025027 40 39 22 1 23 24 .836745 .836878 .837012 2.23 2.23 2.23 .861519 .861400 .861280 .99 .99 .99 .975226 .975479 .975732 4. & 4.22 4.22 .024774 .024521 .024268 38 37 36 25 26 27 28 '29 .837146 .837279 .837412 .837546 .837679 2.23 2.22 2.22 222 2.22 2.22 .861161 .861011 .860922 .860802 .860632 .99 .99 .99 2.00 2.00 2.00 .975985 .976238 .976491 .976744 .976997 4>22 4.22 4.22 4.22 4.22 .024015 .023762 .023509 .023256 .023003 35 34 33 32 31 30 31 32 33 34 35 36 37 38 39 9.837812 .837945 .838078 .83821? .838344 .838-177 .838610 .838742 .838875 .839007 222 222 2.22 2.21 221 2.21 2.21 2.21 2.21 2.21 9.860562 .86TH42 .860322 .860202 .8600.32 859962 .859.342 .859721 .859601 .859480 2.00 2.00 2.00 2.00 2.00 2.00 2.01 2.01 2.01 2.01 9.977250 .977503 .977756 .978009 .978262 .978515 .978763 .979021 .979274 .979527 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 0.022750 022497 .022244 .021991 .021738 .021485 .021232 .020979 .020726 .020473 30 29 23 27 26 25 24 23 22 21 40 41 42 43 44 45 46 47 48 49 9.839440 .839272 .839404 .839536 .839668 .839800 .839932 .840064 .840196 .84C328 221 2.20 2.20 2.20 2.20 2.20 2.20 2.20 2.19 2.19 9.859360 .859239 .859119 .858998 .858877 .858756 .858635 .858514 .853393 .858272 2.01 2.01 2.01 2.01 2.02 2.02 2.02 2.02 2.02 2.02 9.979780 .980033 .980286 .980538 .980791 .981044 .981297 .981550 .981603 .932056 4.22 4.22 4.22 4.22 4.22 4.21 421 4.21 4.21 4.21 0.020220 .019967 .019714 .019462 .019209 .018956 .018703 .018450 .018197 .017944 20 19 18 17 16 15 14 13 12 11 60 61 52 63 64 65 56 9.840459 .840591 .840722 .840.854 .840985 .841116 .841-247 2.19 2.19 219 '.19 2.19 2.! 9 9.858151 .858029 .857908 .857786 .857665 .857543 .85?422 2.02 2.02 2.02 2.03 203 203 9.982309 .982562 [983067 .9-^3320 .9S3573 .933826 4.21 421 4.21 421 421 421 0.017691 .017433 .017186 .016933 .016680 .016427 .016174 10 9 8 7 67 58 59 60 .841378 .841509 .841640 .841771 2. 18 2.18 2.18 2.18 .85?30() .857178 .857056 .856934 2.03 2.03 2.03 2.03 .9S4U79 .984332 .984584 .984837 4.21 4.21 4.21 4,21 .01592! .015668 .015416 .015163 M. Coetra. D. 1". Bine. D. 1". Cotung. D.I". Tang. M. 1330 46 274 TABLE II. LOGARITHMIC SINES, M. Sine. D. 1". Cosine. D. 1". Tng. D. 1". Cotang. M. 7 8 9.841771 .841902 .842033 .842163 .842294 .842424 .842555 .842685 .842815 2.13 218 218 2.18 2.17 217 217 217 9.856934 .856812 .856690 .856568 .856446 .856323 .856201 .856078 .855956 203 204 204 204 204 2.04 2.04 2.04 9.984837 .985090 .985343 .985596 .985843 .986101 .986354 .986607 .936860 4.21 421 4.21 4.21 4.21 4.21 421 4.21 0.015163 .014910 .014657 .014404 .014152 .013899 .013646 .013393 .013140 60 69 68 67 58 55 54 63 52 9 .842946 2.17 .855833 2.04 .987112 4.21 .012388 51 10 11 12 13 14 15 16 17 18 19 9.843076 .843206 .843336 .843466 .843595 .843725 .843355 .843984 .844114 .844243 2.17 2.17 216 2.16 216 2.16 216 216 216 2.16 9.855711 .855538 .855465 .855342 .855219 .855096 .854973 .854850 .854727 .854603 2.05 2.05 205 205 205 205 2.05 2.05 2.06 2.06 9.937365 .987618 .987871 .988123 .988376 .988629 .988882 .989134 .989387 .989640 4.21 4.21 4.21 4.21 4.21 4.21 4.21 4.21 4.21 4.21 0.012635 .012382 .012129 ,011877 .011624 .011371 .011118 .010866 .010613 .010360 60 19 48 47 46 46 44 43 42 41 20 21 22 23 24 25 26 27 28 29 9.844372 .844502 .844631 .844760 .844.889 .845018 .845147 .845276 .845405 .845533 2.15 215 215 2.15 215 2 15 215 2 15 2 14 2.14 9.854480 .854356 .854233 .854109 .853936 .853362 .853738 .853614 .853490 .853366 206 206 2.06 2.06 2.08 206 2.06 207 2.07 2.07 9.989393 .990145 .990393 .990651 .990903 .991156 .991409 991662 .991914 .992167 4.21 421 421 4.21 421 4.21 4.21 4.21 4.21 4.21 0.010107 .009855 .009602 .009349 .009097 .008844 .008591 .008338 .008088 .007833 40 39 38 87 36 36 34 33 31 31 30 31 32 33 34 35 36 37 38 39 9.845662 .845790 .845919 .846047 .846175 .846304 .846132 .846560 .846688 .846316 2.14 214 214 214 214 2.14 213 2.13 213 2.13 9.853242 .853118 .852994 .852369 .852745 .852620 .852496 .852371 .852247 .852122 2.07 2.07 2.07 2.07 2.07 2.08 2.08 2.08 2.08 2.03 9.992420 .992672 .992925 .993178 .993431 .993683 .993936 .994189 .994441 .994694 4.21 4.21 4.21 421 4.21 421 421 4.21 4.21 4.21 0.007580 .007328 .007076 .006,822 .006569 .006317 .006064 .005811 .005559 .005308 30 29 28 27 26 26 24 23 22 21 40 41 42 43 44 45 46 47 48 49 9.846944 .847071 .847199 .847327 .847454 .847582 .847709 .847836 .847964 .843091 2.13 2.13 2.13 213 2.12 2.12 2.12 2.12 2.1'2 2.12 9.851997 .851872 .851747 .851622 .851497 .851372 .851246 .851121 .a r >0996 .850870 2.08 2.03 2.08 2.09 2.09 2.09 209 2.09 2.09 2.09 9.994947 .995199 .995452 .995705 .995957 .996210 .996463 .996715 .996968 .997221 4.21 4.21 4.21 4.21 4.21 4.21 4.21 4.21 4.21 4.21 0.005053 .004801 .004548 .004295 .004043 .003790 .003537 .003285 .003032 .002779 20 19 18 17 16 16 14 13 19 11 60 61 52 63 64 65 68 57 58 59 60 9.848218 .848345 .848472 .848599 .848726 .848852 .848979 .849106 .849232 .849359 .849485 2.12 2.12 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 9.850745 .850619 .850493 .850368 .850242 .8501 16 .849990 .849864 .849738 .84961 1 .849485 209 2.10 2.10 2.10 2 10 2.10 2.10 2.10 2.10 2.11 9.997473 .997726 .997979 .998231 .998484 .998737 .998989 .999242 .999495 .999747 0.000000 4.21 4.21 4.21 4.21 421 4.21 4.21 4.21 4.21 4.21 0.002527 .002274 .002021 .001769 .001516 .001263 .001011 .000758 .000505 .000253 .000000 10 9 8 7 6 6 4 9 1 M. Ooelne. D.1". Sine. D. 1". Cotang. D. 1". Tang. M. 134 TABLE III. NATURAL SINES AND COSINES. 275 276 TABLK III NATURAL SIXES AND COSTXES. GO 10 30 30 4o M Sine! Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Coslu M .00000 One" .01745 .99985 .03490 .99939 .05234 .99863 .06976 1J9756 80 1 .000:29 One. .01774 .999.84 03519 .99938 .05263 .99>61 .070115 .99754 50 2 00058 One. .01803 .999S4 .03548 .99937 .05292 .99-560 .07034 .99752 58 . 3 .OUD87 One. .01832 .99983 .03577 .99936 05321 .99858 .07063 .99750 37 4 .00116 One. .01862 .99983 .036116 .99935 .05350 .99857 .07092 .99748 56 5 00145 One. .01891 .99982 .03635 99934 .05379 .99855 .07121 .99746 56 6 .00175 One. .01920 .99932 .(^664 .99933 .05408 .99854 .07150 .99744 54 7 00201 One. .01949 .99981 .03093 .99932 .05437 .99852 .07179 .99742 53 8 00233 One. .01978 .99980 .03723 .99931 .05466 .99851 .07208 .99740 52 9 00*62 One. .02007 .99980 .03752 .9993U .05495 .99849 07237 .99738 51 10 00291 One. .02036 .99979 .03781 .99929 .05524 .99847 .07266 .99736 50 11 00320 .99999 .02065 .99979 .03-510 .99927 .05553 .99846 .07295 .99734 49 12 00341) .99999 .02094 .99978 .03839 .99926 .05582 .99844 .07324 .99731 48 13 .00378 .99999 .02123 .99977 .03368 .99925 .0561 1 .99842 .07353 .99729 47 14 00407 .99999 .02152 .99977 .03897 .99924 .05640 .99,841 .07382 .99727 46 15 .00436 .99999 .02181 .99976 .03926 .99923 .05669 .99839 .07411 .99725 45 16 .0046--, .99999 .02211 .99976 .03955 .99922 .05698 .99838 .07440 .99723 44 17 .00495 .99999 .0224' .,j*76 .03984 .99921 .05727 .99336 .07469 .99721 43 18 .00524 .99999 .0226y .00974 |. 0401 3 .99919 .U5756 .99834 .07498 .99719 42 19 .00553 .99993 .02298 .99974 .04042 .99918 .05785 .99833 .07527 .99716 41 20 .00532 .99993 .02327 .99973 .04071 .99917 .05814 .99831 .07556 .99714 40 21 .00611 .99998 .02356 .99972 .04100 .99916 .05844 .99329 07585 .99712 39 22 .00640 .99998 .02385 .99972 .04129 .99915 .05873 .99827 07614 .99710 38 23 00669 .99993 .02414 .99971 .04159 .99913 .05902 .99326 07643 .99708 37 24 .00698 99998 .02443 .99970 .04188 .99912 .05931 .99824 07672 .99705 36 25 .00727 .99997 .02472 .99909 .04217 .99911 .05960 .99822 07701 .99703 35 26 .00756 .99997 . 02501 .99969 .04246 .99910 .05939 .99821 .07730 .99701 34 27 .00785 .99997 .02530 .99963 .04275 .99909 06018 .99819 .07759 .99699 33 28 .00814 .999-7 .02560 .99967 .0430-1 999i 17 .06047 .99817 .07788 .99696 32 29 .00844 99W6 .02589 .99966 .01333 .99906 .06076 .99815 .07817 .99694 31 30 OaS73 .90996 .02618 .99966 .04362 .99905 .06105 .99813 07846 9%92 30 31 oiwa .99996 .02647 99965 .04391 .99904 .06134 .99812 07875 .9Sti8i> 29 32 .00931 .99996 .02676 .99964 .04420 .99902 .06163 .99310 07904 .99687 IS 33 .00960 .99995 .02705 .99963 .04449 .99901 .06192 .99808 07933 .99685 17 34 .00939 .99995 .02734 .99963 .04478 .99900 .06221 .99806 07962 .99683 26 35 .01018 .99995 .02763 .99962 .04.307 .99898 .06251* .99804 .07991 .99680 25 36 .01047 99995 .02792 .99961 .04536 .99897 .06279 .99303 .08020 .99678 24 37 .01076 99994 02821 .99960 .04565 .99396 .06308 .99301 .08049 .99676 23 38 .01105 .99994 02350 .99959 .04594 .99894 .06337 .99799 .03078 .9%73 22 39 .01134 .99994 02879 .99959 .04623 .99893 .06366 .99797 .08107 .99671 21 40 01164 .99993 02908 .99958 .04653 .99892 .06395 .99795 .08136 .99668 20 41 .01193 .99993 .02938 .99957 .046812 .99890 .06124 .99793 .08165 .99666 19 42 .01222 .99993 02967 .99956 .04711 .99889 .06453 .99792 .08194 .99664 18 43 .01251 .99992 .02996 .99955 .04740 .99888 .06482 .99790 .08223 .99661 17 44 .01280 .99992 .03025 .99954 .04769 .99886 06511 .99788 08252 .99659 16 45 .01309 .99991 .03054 .99953 .04798 .99885 .06540 .99786 .08281 99657 15 46 .01338 .99991 03083 99952 .04827 .99883 .06569 .99784 .08310 .99654 14 47 .01367 .99991 .03112 .99952 .04856 .99882 .06598 .99782 .08330 .99652 13 48 .01396 .99990 .03141 .99951 .04885 .99881 .06627 .99780 .08368 .99649 12 49 .01425 .99990 .03170 .99950 .04914 .99879 .06656 .99778 .08397 .99647 11 50 .01454 .999S9 .03199 .99949 .04943 .99378 .06685 .99776 .08426 .99644 10 51 .01483 .99989 .03228 .99948 .04972 .99876 .06714 .99774 08455 .99642 9 52 .01513 .99989 .03257 .99947 .051(01 .99875 .06743 .99772 08484 .99639 8 53 .01542 .99988 03286 .99946 .05030 .99873 .06773 .99770 08513 .99637 7 54 01571 .999-18 .03316 .99945 .05059 .99872 .06802 .9976^ .03542 .99635 6 55 .01600 .99937 03345 .99944 .05088 .99870 .06831 .99766 08571 .99632 5 56 01629 .99937 .03374 .99943 .05117 .99369 .08860 .99764 .086 r )0 .99630 4 57 01658 .99986 .03403 .99942 .05146 .99*67 .06389 .99762 .08629 .99627 3 58 .01687 .99986 .03432 .99941 .05175 .99866 .06918 .99760 08658 .99625 2 59 01716 .99985 .03461 .99940 .05205 .99864 .06947 .99758 .08687 .99622 60 .01745 .99985 .03490 .99939 .05234 .99863 .06976 .99756 .08716 .99619 M: Ooein. Slue Cosin. Sine. Cosin! Sine. Cosin. Sine. Coein. Sine. M. 89 883 87 86 850 TABLE III. NATURAL SIXES AND COSINES. 50 GO 70 8 9 M Sine. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Couin. M. 08716 .99619 .1O453 .99452 .12137 .99255 .13917 .99027 .1564c 1)8769 60 1 .0874') .99617 .10482 .994* 9 .12216 .99251 .13946 .99023 .15672 98764 59 2 08774 .99614 10511 .99446 .12245 .99.^43 .13975 .99019 .15701 .93760 58 3 .08*13 99612 . 10.i4<) .9944.1 . 12274 .99244 .14004 .99015 .15730 .98755 57 4 .08831 99&I9 10569 .99440 .12302 .99240 .14033 .99011 .15753 .98751 56 5 .08860 99607 10597 .09437 .12331 .99237 .14061 .99006 .15787 .98746 56 6 0888 99rt.ll 10626 .J..134 .12360 .99233 .14090 .99002 15816 .98741 54 7 08918 .996(12 10655 .99431 .12389 .99230 .14119 9899s .15,845 .98737 53 6 08947 .99599 10684 99428 .12418 .99226 .14143 .98994 .15873 .98732 52 9 .08976 .99596 10713 .91)121 .12447 .99222 .14177 .98990 .15902 .98728 51 10 .09(105 .99594 10742 .99421 .12476 .99219 .14205 .98986 .15931 .98723 50 11 09034 .99591 10771 .99418 .12504 .99215 .14234 .98932 .15959 .98718 49 12 09063 .9953- 10800 .99415 .12533 .99211 .14263 .98978 .15938 .98714 48 13 .09092 .99586 10829 .99112 .12562 .99208 .14292 .98973 .16017 .98709 47 14 09121 .9958:5 10358 .99409 .12591 .9920-1 .143-20 .98969 16046 .98704 46 15 .09150 .99580 .10*87 .99406 .12620 .99200 .14349 .98965 .16074 .98700 45 16 .09179 .99573 10916 .99402 .12649 .99197 .14378 .93961 .16103 .98695 44 .09208 .99575 10945 .99399 .12678 .99193 .14407 .98957 16132 .98690 43 18 (19237 .99572 10973 .99396 .12706 .99189 .14436 .98953 16160 .98686 42 19 09266 99570 11002 .99393 .12735 .991,86 .14464 .98WS 161 89 : 98681 41 20 0929;) .99567 1 1031 .99390 .12764 .99182 .14493 .98944 16218 .98676 40 21 09324 .99564' 11060 .993-6 .12793 .99178 .14522 .98940 .16246 .9,8671 39 2*2 09353 .99562 1 1089 99383 .12822 .99175 .14551 .98936 16275 .98667 38 23 09382 99559 11118 .99380 .12851 .99171 .14580 .98931 16304 .98662 37 24 09111 .99556 i!147 .99377 .12880 .99167 . 14603 .98927 16333 .98657 36 25 09440 .99553 1 1176 .99374 .1290S .99163 .14637 .98923 16361 .98652 35 26 09469 .99551 .11205 99370 .12937 .99160 14666 98919 16390 .98643 34 27 09498 .99548 .I12: .99367 .12966 .99156 .14695 98914 16419 .98643 33 28 095-27 .99545 11263 .99364 12995 99152 14723 .98910 16447 .98638 32 29 (19556 99542 11291 .99360 .13024 .99148 14752 .98906 16476 .98633 31 30 09585 .99540 11320 .99357 13053 .99144 .14781 .98902 . 16505 .98629 30 31 .09614 .99537 11349 99354 .13031 .99141 14810 .98897 16533 .98624 29 32 (19642 .99534 11378 .99351 .13110 .99137 .14838 .98893 16562 .98619 28 33 (19671 .99531 .11407 .99347 .13139 .99133 .14867 98889 16591 .98614 27 34 .09700 .9952S 11436 99344 13168 .99129 . 14896 .98834 16620 .98609 26 35 (19729 .99526 11465 99341 .13197 .99125 .14925 .98830 16643 .93604 25 36 09758 .99523 11494 99337 .13226 .99122 14954 .98876 16677 .98600 24 37 1)9787 .99520 11523 .99334 .13254 .99118 14982 .98871 16706 93595 23 3 09816 99517 11552 99331 .13283 .99114 .15011 .98867 16734 .98590 22 39 09^45 .99514 .11580 99327 .13312 .99110 15040 .98863 16763 98585 21 40 .093*4 .99511 11609 .99324 .13341 .99106 15069 .98853 16792 .98530 20 41 09903 .99508 .11638' .99320 .13370 .99102 .15097 .98854 16820 .98575 19 42 .09932 .99506 11667 .99317 .13399 .99098 15126 .98849 16849 .98570 18 43 .09961 .99503 .11696 .99314 . 13427 .99094 15155 .98845 16378 .98565 17 44 .09990 .99500 .11725 .99310 .13456 .99091 15184 .98841 16906 .98561 16 45 .10019 .99497 .11754 .99307 .13485 .99087 .15212 .98836 .16935 .98556 15 46 .10048 99494 .11783 .99303 .13514 .99083 15241 .98332 16964 .98551 14 47 10077 99491 .11812 .99300 .13543 .99079 .15270 .98327 16992 .98546 13 48 10106 99488 .11840 .99297 .13572 .99075 .15299 .98323 17021 .98541 12 49 10135 .99485 11869 .99293 .13600 .99071 .15327 .98818 .17050 .98536 11 50 .10164 994*2 .11898 .99290 .13629 .99067 .15356 .98814 .17078 .98531 10 5i 10192 99479 .11927 .99286 .13658 .99063 .15385 .93309 .17107 .98526 9 52 10*21 .99476 11956 .99283 .13687 .99059 .15414 .98305 .17136 .93521 8 53 10250 99473 1 1985 .99279 13716 .99055 .15442 .98800 .17164 .98516 7 54 KI279 .99470 12014 .99276 .13744 .99051 .15471 .98796 .17193 .9^*511 6 55 10318 99467 12043 .99272 13773 .99047 15500 .98791 .17222 98506 5 56 111337 99164 12071 .99269 13802 .99(^3 15529 .98787 .17250 .98501 4 57 10366 .99461 12100 .99265 13831 .99039 15557 .98782 .17279 98496 3 53 IK395 .994.V 12129 99-262 13860 .99035 15586 .98778 17303 .98491 2 59 10424 .99455 12153 .99258 13839 .99(131 15615 .98773 17336 9*486 1 60 10453 99452 12187 .99255 13917 .99027 15643 .98769 .17365 .9M431 M: C^IiT Sine. Cosin. Sine. Costa. SineT CoainT Sine. Cosin Sin*. ML 8*0 830 83 810 800 278 TABLE III. NATURAL SINES ANI1 COSINES. 100 110 120 130 14* M. Sb. Corfn- Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Coain. M. ~0 17365 .98481 19081 .9*163 .20791 .97815 .22495 .97437 .24192 .97030 60 I 17393 .9>476 19109 .98157 20.-S20 .97809 .2*333 .97430 .24220 97023 59 2 17422 .98471 .19138 .98152 .20S48 .97-43 .95536 .31206 .95006 .32859 .94447 49 12 .26219 .96502 .27899 96029 .29571 .9552S .31233 .94997 .32887 .94438 48 13 .26247 .96494 .27927 .96021 .29.-i99 .95519 .31261 .94933 .32914 .94428 47 14 .26275 .964 -iff .27955 .96013 .29626 .95511 .31239 .94979 .32942 .94418 46 15 .26303 .96479 .27983 .96005 .29654 .95502 .31316 .94970 .32969 .94409 45 16 26331 .96471 .23011 .95997 .29632 .95493 .31344 .94961 .32997 .94399 44 17 26359 .96463 .2.3039 959-0 .29710 .95485 .31372 .94952 .33024 .94390 43 IS 26337 .96456 .23067 .95931 .29737 .95476 .31399 .94943 .33051 .94380 42 19 .26115 96448 .23095 .95972 .29765 .95467 .31427 .94333 .33079 .94370 41 20 26443 .96440 .28123 .95964 .29793 .95459 .31454 .94924 .33106 .94361 40 21 26471 .96433 .28150 .95956 .29821 .95450 .31482 .94915 .33134 94351 39 22 .26500 .96425 .28178 .95943 .29849 .95441 .31510 .94906 33161 .94342 33 23 .26523 .96417 .28206 .95940 .29876 .95433 .31537 .94897 33189 .94332 37 24 .26556 .96410 .23234 .95931 .29904 95424 .31565 .94838 33216 .94322 36 25 265S4 .964i 12 .28262 .95923 .29932 .95415 .31593 .94878 33244 .94313 35 26 26612 .96394 .88290 .95915 .29960 .95407 .31620 .94869 33271 .94303 34 27 26640 .963s6 .23318 .95907 .29987 .95398 .31648 .94860 33298 .94*93 33 28 26668 .96379 .28346 .95393 .30015 .95389 .31675 .94851 33326 .94284 32 29 26696 96371 2.3374 .95890 .30043 .95330 .31703 94842 33353 .94274! 31 30 26724 .96363 .28402 .95882 .30071 .95372 .31730 .94832 33381 .94264 30 31 26752 .96355 .28429 .95874 .30098 .95363 .31758 .94323 33408 .94254 29 32 26780 .96347 23457 .95,365 .30126 .95354 .31786 .94814 33436 .94245 28 33 26308 .96340 23135 .95357 .30154 .95345 .31813 .94805 33463 .94*35. 27 34 26836 .963.52 23513 .95349 .30182 .95337 .31841 .94795 3490 .94225 26 35 26364 .96324 28541 .95841 .30209 .95323 .31863 .94786 33518 .94215 25 36 26392 .96316 28569 .95332 30237 .95319 .31396 .94777 33545 .94206 24 37 26920 .96308 .23597 .95324 30265 .95310 .31923 .94763 33573 94196 23 38 26943 .96301 23625 .95316 .30292 .95301 .31951 .94758 .33600 .94186 22 39 26976 96-293 28652 .95307 .30320 .95293 .31979 .94749 33627 .94176121 40 27004 .96235 23630 .95799 .30348 .95284 .32006 .94740 33655 .94167 20 41 .27032 96277 23708 .95791 .30376 .95275 .32034 .94730 33682 .94157 19 42 .27060 .96269 23736 .95782 .30403 .95266 .32061 .94721 33710 .94147 18 43 .27088 96261 23764 .95774 .30431 .95257 .32039 .94712 .33737 .94137 17 44 27116 .96253 28792 .95766 .30459 .95248 .32116 .94702 .33764 .94127 16 45 .27144 .96246 .28820 .95757 .30486 .95240 .32144 .94693 33792 .94118 15 46 27172 .96233 .28847 .95749 .30514 .95231 .32171 .94684 33819 .94108 (4 47 27200 .96230 .28375 .95740 .3' 1542 .95222 .32199 .94674 .33846 .94098 13 43 27228 .913222 23903 .95732 .30570 .95213 .32227 .94665 33874 .94088 12 49 27256 .96214 .28931 .957*4 .30597 .95204 32254 .94656 33901 .94078 II 50 27*34 .96206 .23959 .95715 .30625 .95195 .32282 .94646 33929 .94068 10 61 27312 .96193 .28937 .93707 .30653 .95186 .32309 .94637 .33956 .94058 9 52 27340 .96.90 .29015 .95693 .30680 .95177 .32337 .94627 .33983 .94049 8 53 27368 .96182 .29042 .95690 .30708 .95163 .32364 .94618 .34011 .94039 7 54 27396 96174 .29070 .95631 .30736 .95159 32392 .94609 .34033 .94029 6 55 27124 96166 .29099 .95673 .30763 .95150 32419 .94599 .34065 .94019 6 56 27452 96158 .29126 .95664 30791 .95142 32447 .94590 .340931.94009 4 57 27430 .96150 29154 .95656 30819 .95133 .32474 .94580 .34120 .93999 3 58 .27508 .96142 29182 .95647 .30346 .95124 .32502 .94571 .34147 .939^9 2 59 27536 .96134 .29209 .95639 .30874 .95115 .32529 .94561 .34175 .93979 I 00 .27564 .96126 .29237 .95630 .3,0902 .95106 .32557 .94552 34202 .93969 M. Corfu. Sine. Cosin. Sine. Cosin. ' Sine. Coain. Sine. Cosin. Sine. M. 74 730 73 710 700 280 TAULK NATURAL SINKS AND COSINES- 300 310 330 330 M Bine. Coein Sine. Coein Sine. jCoein 8in7~ Coein Sine Coein M ~0 34202 .93989 358:37 .933 .IS 37461 927|x .39073 92050 4n674 .91366 60 1 34229 93D5'J .35*61 .9334- .37488 .92707 .39100 .92039 4o7i KJ 91343; 59 2 34257 9394 '. 37515 .92697 .39127 9202- 4 f i7*^7 91331 1 58 3 342-<4 93931* 35918 .93:i27 .37648 .926-<; 311 153 92016 40753 91319 67 4 34311 .93929 3.-.9I5 .9a3i<> .37569 .92675 3'M*o .92005 }(l7-0 .913JC 56 5 343-39 93919 a~>973 .9331 Ri .37595 .92664 !39207 91994 .40*06 91-295 56 6 34366 .93909 36000 .93-295 .37622 .92653 .39234 40*33 912*3! 54 7 34393 .93899 36027 .37649 19*642 .39260 91971 40*60 91272 53 8 34421 .93**9 36054 ! 93274 .37676 .92631 3-J2S7 91959 4o**6 91260 52 9 34148 93879 36081 .93264 37703 .92620 .39314 9194- 40913 91248 51 10 34475 93*6:* .36108 .93253 .3773d .92609 .39341 919.% 40939 91236150 11 34.503 93859 .36135 .93243 .37757 .9259* .39367 .91925 40966 91-224 49 12 34630 36162 .9323-2 .37784 .92587 .39394 91914 40992 .91212 4* 13 34557 ! 93*39 36190 .93222 37811 .92576 .39421 9l9(r2 41019 91-200 47 14 34584 .93>29 36217 .93211 .37838 .92565 3944* 91891 41045 46 15 .34612 .93319 .36244 .93201 .37865 .92554 .39474 .91879 41072 .91176 45 16 34639 93-09 36271 .93190 .37892 .92543 .39501 .9136* 41098 .91164 44 17 31666 93799 36298 .93180 37919 .92532 .39528 .91966 41125 .91152 43 18 34694 937S9 .36325 .93169 .37946 .92.V21 39555 91845 41151 .91140 42 19 34721 .93779 36352 .93159 .37973 .92510 39581 91*33 41178 .91 1-28 41 20 .34748 .93769 .36379 .93148 .37999 .92499 .39608 91822 412(4 .91116 40 21 34775 .93759 .364% .93137 .3S026 .9248.-! .39635 .91810 41231 .91 KM 39 22 34803 .9374* 36434 .93127 .33053 .92477 .39661 91799 41 57 .91092 3 s 23 34830 .93738 .36461 .93116 .38080 .92466 .39688 .91787 .412*4 .910SO 37 24 34857 .93728 .364S8 .93106 .33107 92455 .39715 91776 .41310 9 1 o6 36 25 34884 .93718 .36515 .93095 .3^134 92444 .39741 91764 .41337 .91056 35 26 34912 .93703 .36648 .930s4 .38161 .92432 39768 .91752 .41363 9KM4 34 27 31939 .93698 365(59 .93074 .38188 .92421 .39795 .91741 .41390 91032 33 28 .34966 .9368* 3651H5 .93063 .3S2I5 .92410 .39*22 .91729 .41416 . 9 1 02; i 32 29 .34993 ' 93677 .36623 .93052 .33241 .92399 .39848 9171* 41443 JWOflH 31 30 .35021 .93667 .36669 .93042 .3S-26* .9238.8 .39876 ! 91 706 .41469 .90996 30 31 .aws .93657 36677 .93031 .3*295 .92377 .3990T. 91694 41496 909*4 29 32 .35075 .93617 ,36704 .93020 .38322 .92366 .39988 .916*3 41522 .90972 2* 33 .35102 .9:5637 .36731 .93010 .38349 .923;", .39955 .91671 41549 .90960 27 34 .35130 .93626 36758 .92999 .3*376 .92343 .39982 .91660 .41575 .90948 26 35 a5 157 .93616 .36786 .9298S .3*403 .92332 .40008 .9164* .41*08 .90936 25 36 a>l84 .93606 .36812 .92978 .38430 .92321 .40035 .91636 .41628 .90924 24 37 !av2ii .93596 .36839 .92967 .92310 .40062 .91625 41655 .90911 23 38 .35239 .9:3585 .36367 .92956 !Ws3 92299 .40088 .91613 41641 .90*99 22 39 aV266 .93575 .36S94 .92915 .3*510 .922*7 4"l 15 .91601 .41707 90**7 21 40 .35293 .93565 .36921 .92935 38537 92276 '40141 .91590 .41734 90876 2ft 41 .35320 .93555 369 H .92924 .38564 .92265 .4016* .91578 41760 90863 19 42 .35347 .93544 .36975 .92913 .38591 .92254 .40195 .91566 .41787 .90851 IS 43 .35375 .93531 37002 .92902 .38617 .92213 .40221 .91555 .41813 .90839 17 44 .a5402 .9*524 .37029 .92S92 .38644 .92231 .40248 .9? 543 .41840 .90826 16 45 .35429 .93514 .37056 .92881 .38671 .92220 .40875 .91531 .41866 .908141 15 46 35456 .93503 37083 .92^70 3*69* .92209 .40301 .91519 41892 90802 14 47 .35484 .93493 37110 .92859 .38785 .9219S .40328 ,9l5os 41919 90790 13 48 .35511 .934 S3 37137 .92849 38752 .921*6 .40355 .91496 .41945 .90778 12 49 .35538 .93472 37164 .92838 18778 .92175 4i 1331 .914.84 41972 .90766 11 50 .35565 .9.3462 37191 .92S27 .38305 .92164 .40408 .9147-2 41998 .90753 10 51 35592 .93452 .37218 .92816 .39833 .92152 .40434 91461 48034 .90741 9 52 .35619 .93411 37245 .92.S05 .3*859 .92141 .40461 91449 42061 .90729 8 53 .3">647 .93431 37272 .92794 .38886 .92130 .404.88 91437 42077 .90717 7 54 .35674 .93420 .37299 .927S4 .3*912 92119 .40514 91425 48104 907(M 6 55 .35701 93410 37326 .92773 .38939 .92107 .40541 91414 42130 .90692 5 56 .35728 .93400 .37353 .92762 .38966 .92096 40567 9140* 42156 906*0 4 57 .35755 933*9 .37380 .92751 .3*993 .92085 .40594 .91390 42 183 9; tf.rtf 3 63 35782 .93379 .37407 .92740 92073 .40621 .9137* 42809 90655 o 59 35*10 9336* 37434 .92729 .' 391*46 .9-2062 .40647 91366 .42836 .9064.3 1 60 35S37 .37461 .92718 ,39073 .92050 .40674 91355 42*62 .9(1631 if Coein. Sine. Oosin. Sine. Codn. Stair Coflin. Sine. Coein. Slue. M 690 680 670 660 65- TABLE III. XATl'RAI. SIXES AXI) COSIXKS. 353 36" 87 28 29 M. Sine. CoBln. Sine. Cosin. Blue. Cosin. Sine. Cosin. Sine. Ocein M. .42262 .90631 43337 89379 .45399 .89HU .46947 .88295 .48481 .87462 60 1 .422-W .9(1618 .43363 .89367 .45425 .89087 46973 .88281 .48506 .87448 59 2 .4231 5 .90*16 43889 .89854 .46451 .89074 .46999 .S8267 .48532 .87434 58 3 .42341 .90194 .43916 .89841 .45477 .89061 .47(124 .88254 .48557 .87420 57 1 42:167 .90.-)32 .43942 .89323 .455(13 .891*48 .47050 .8824'! .48583 .87406 56 5 42394 90569 .4:1963 .89816 .45529 .89035 .47076 .88226 .48608 .87391 55 6 44420 .90557 .43994 .89803 46664 .89021 .47101 .88213 .48634 S7377 54 7 42446 .9054:. 44020 .89790 45580 .89008 .47127 .88199 .48659 .87363 53 9 42473 . 9(632 .44046 .89777 .15606 .88995 .47153 .88185 .48684 87349 52 9 42499 .90520 44072 .89764 .45632 .88981 .47178 .88172 .48710 .87335 51 10 .42525 .90507 .44098 .89752 .45658 .88968 .47204 .88158 .48735 .87321 50 11 .42552 .9049o .44124 .89739 .45684 .88955 .47229 .88144 .48761 .87306 49 12 42573 .90433 44151 .89726 .45710 .88942 .47255 .88130 .48786 .87292 48 13 .42604 .90470 .44177 .89713 .45736 .88928 .47231 .88117 .48811 .87278 47 14 .42631 .90458 442f3 .89700 .45762 .88915 .47306 .88103 .48837 .87264 46 15 .42657 .90446 44229 .89687 .45787 .88902 .47332 .88089 .48862 .87250 45 16 .42683 .90433 44255 .89674 .45313 .88888 .47358 .88075 .48883 .87235 44 17 .42709 .90421 44281 .8966-2 .45,839 .88875 .47383 .88062 .48913 .87221 43 18 .42736 .90408 44307 .89619 .45865 .88862 .47409 .88048 .48933 .87207 42 19 42762 .90396 44333 .89636 .45891 .88848 47434 .88034 .48964 .87193 41 20 .42738 .90333 .44359 .89623 .45917 .88835 .47460 .88(120 .48989 .87178 40 21 .42*15 .90371 44335 89610 .45942 .88822 .47486 .88006 .49014 .87164 39 22 .42841 .90358 .44411 .89597 .45963 .88808 .47511 .87993 .49040 .87150 38 23 .42367 90346 44437 .895.84 45994 .88795 .47537 .87979 .49065 .87136 37 24 .42-894 .9(1334 44464 .89571 .4602') .88782 .47562 .87965 .49090 .87121 36 25 .42920 .90321 44490 .89553 46046 .88763 .47538 .87951 .49116 .87107 35 26 .42948 .90309 44516 .89515 .46072 .88755 .47614 .87937 .49141 .87093 34 27 .42972 .90290 44542 .895:12 46097 .S3741 .47639 .87923 49166 .87079 33 23 .42999 .902*4 44563 .89519 .46123 .88728 .47665 .87909 .49192 .87064 32 29 .43025 .90271 44594 .89506 .46149 .88715 .47690 .87896 .49217 .87050 31 30 .43051 .90259 44620 .89493 .46175 .88701 .47716 .87882 .49242 .87036 30 31 .43077 .90246 44646 .89480 .46201 .88688 .47741 .87863 .49268 .87021 29 32 43104 .9(233 41672 .89467 .46226 .&867J .47767 .87854 .49293 .87007 28 33 .43130 .90221 44693 89454 .46252 .88661 .47793 .87840 49318 .86993 27 34 43156 j. 90203 44724 .89441 .46278 .88647 .47818 .87826 .49344 .86978 26 35 .431 32 . 90196 44750 .89428 46304 .836:14 .47844 .87812 49369 .86964 25 36 .4:1209 .901 S3 44776 .89415 46330 .88620 .47369 .37798 491194 .86949 24 37 .4-1236 .90171 44802 .89402 46355 .88607 .47895 .87784 .49419 .86935 23 3* .43261 .90158 44828 893*9 46:i*i .88593 .47920 .8777(1 49445 .86921 22 39 .4.')287 .90146 44854 ,89376 46-407 .88580 .47946 .87756 49470 .86906 21 40 .43313 .90I33 44880 .89363 46433 .88566 .47971 .87743 49-495 .86S92 20 41 .43340 .90120 .449(t6 .893.V) 46453 .88553 47997 .87729 .49521 .86878 19 42 .4.3366 .901118 44932 .89337 46484 .88539 .480-22 .87715 49546 .86863 18 43 .43392 .90095 .44953 .89324 46510 .88526 .48043 .87701 49571 .86349 17 44 .43-J13 .90082 44984 .8931 1 46536 .88512 .48073 .87687 49596 .86834 16 45 .43445 .90071) .45010 .89293 .46561 .88499 .48099 .87673 .49622 .86820 15 46 .43471 .90057 45036 .89285 .46587 .8.8435 .48124 .87659 .49647 .86805 14 47 .43497 .90045 45(162 .89272 .46613 .88472 .48150 .87645 .49672 .86791 13 43 .43523 .90032 .46088 .892*9 .4C.639 .88458 .48175 .87631 49697 .86777 12 49 43549 .90019 45114 .89245 .4f/,fi4 .88445 .48201 .87617 49723 .86762 11 50 43575 90007 45HO .89232 .46690 .88431 48226 .87603 49748 .86748 JO 61 43602 .89994 15166 .89219 .46716 .88417 .48252 .87539 49773 .86733 9 52 .43628 89981 45192 .89206 .46742 .884m .48277 87575 49798 .86719 8 53 43654 .8996* 4/V2I3 .89193 .46767 .88.390 .48:103 .87561 49824 .867(4 7 54 .43680 . 89956 45243 .89180 .46793 .88377 .48328 .87546 49*49 86690 6 55 43706 .8994.'! 45269 .89167 ,46>H9 .88363 .483*4 .87532 49*74 .86675 5 56 43733 .89930 45295 .89153 46844 .88349 .43379 .87518 4 aao 330 840 M. Sine. CotiD Si.i* Cosin. Sine jCorin. Sine. Coeln. 8ine. Gotdn M. 50000 86603 51504 .85717 52992,. 84805 .54464 ."83867 55919 .*2904 60 1 .50J25 H65SN 51629 .85702 5 it 7 1.84 789 .544^s .8.3.S51 55943 82N87 69 2 .50)50 .86573 51554 .8MH7 63041|.84774 .54513 .83835 55968 .*87l 58 3 .50076 86559 .51679 . 85*72 53066 .o475y .64537 .;i-5iy 55992 .82,855 57 4 50101 86544 61004 .85057 wwi .84743 .64561 .H3H 56016 .82*39 56 5 .50126 8653 1 Mfl2H .3664 X! .63115 .84725 .:-4.> o .8373 .83724 .56136 .82767 51 10 .50252 .86457 81753 .a'wfi? 53. -38 .846^J .54708 .837(K .56160 .82741 60 11 50277 86442 SITTb .86651 53263 .S46:i .54732 .83692 .561*4 .82724 49 12 .50302 86127 51803 .85536 .53288 .84619 .54756 .83676 .56208 .S2708 48 13 .50327 86113 51828 .85521 .53312 .816.4 .54781 .83661' .56232 .82692 47 14 .50352 86398 51852 .8650.; .63337 .8L r r^S .64S06 .83645 .66256 82675 46 15 .60377 .86384 .5187; .86491 .53.361 .84573 .64829 .83629 .66280 .82659 45 16 .60403 .86369 .51902 .8.', 4 76 .53386 .845o7 .64854 .83613 .56305 .82643 44 17 .6042* 86354 .61927 .85461 .6.3411 .84542 .54^78 .83597 .56329 .82626 43 13 .50453 .86340 .61952 .854 46 .5:3435 .84526 .54902 .835;! 1 56353 .82610 42 19 .5(1478 .86326 ,61977 .8-Vj.il .53460 .84511 .54927 .83565 66377 .82593 41 20 .50503 .86310 .62002 .854 16 .534*1 .84495 .54951 .83549 66401 .82677 40 21 .50528 .862:* .52026 .awi .eaw .844 SO .54975 .83533 56425 82561 39 22 .50553 .862*1 .52' 151 .85.V5 .5::).u .84464 .64999 .83517 .56449 .82544 38 23 .50578 .86266 .62076 .85370 .5355,8 .844 H .55024 .83501 56473 .82528 37 24 .50603 86251 .6*2101 .85355 .83683 .84433 .55048 .834 >5 .56497 .62511 36 25 50628 86237 .62126 .85.40 .5:5607 .84417 .55072 .83469 56521 .82495 35 26 50654 .Wf .62151 .a r >325 .53632 .84402 .55097 .83453 .56545 .8247S 34 87 50679 88207 .62175 .85310 .53656 .P-13^6 .55121 .83437 56569 .82462 33 2* .60704 86192 .62200 .9521*4 .63681 .84370 .5514.') .83421 56593 .82446 32 29 60729 8178 . 52225 .85279 .53705 .84355 .5. r )l6'J .83405 56617 .82429 31 30 .60754 .86163 52250 .85264 .63730 .84339 .55194 .83389 .56641 .82413 30 31 .50779 8614- 52275 .86249 ,83754 .84324 .55218 .83373 56665 .82396 29 32 .50804 .86111 62299 .*:.2;n .53779 .843 H .55212 .83356 666*9 .82380 28 33 .50829 .86119 52324 .852 IS .68804 .84292 .55266 .83340 66713 .82363 27 34 .50854 .86101 52349 .a<5203 .53838 .84277 .55291 .83324 56736 .82:347 26 35 .50K79 .860*9 T.2371 .85188 .r,353 .84261 .55315 .83308 56760 .82330 26 36 .60904 .86074 52399 .85173 .53*77 .84215 55339 .83292 567S4 .82314 24 37 50929 ;. 86069 52423 .85157 .53902 .84^50 .55363 .83276 56^08 .82297 23 3S .60954 86U45 88449 .85142 .53926 .84214 .55588 .83260 .56832 .82281 22 39 .50979 8fl030 52473 .85127 .53951 .84193 .55412 .83244 56856 .82264 21 40 61004 .860(5 ->4>49* .85112 .53975 .84182 .66436 .83228 5#0 .82248 20 41 .61029 .96000 52522 .850% .54000 .84167 55460 83212 569f4 .82231 19 42 51054 859Xft 52547 .86081 .54024 .84151 554 s4 .a? 195 .56928 .82214 18 43 .61079 .8597(1 52572 .85066 .54049 .84135 55509 .83179 .56952 .82198 17 44 .61104 .85956 52597 .850ol .54073 .84120 55533 .83 W3 56976 .82181 16 45 .51129 .85941 52621 .85035 .54097 .84104 .55557 .83147 .67000 .82165 '6 46 61154 .85926 52646 .85020 .54122 .84088 ,650)1 .83131 .67024 .82148 14 47 .51179 .85911 52671 .85005 .54146 .84072 .55605 .831 15 67047 .82132 13 4* .i!204 .85*96 52696 .849*9 .64171 .84057 65S30 8309 57071 .82115 r? 49 .51229 85RS | 52720 .84974 54195 .84041 55654 .83082 .67095 .82008 11 60 51254 83e 52745 .84959 .54220 .84025 .6567* .83066 57119 .89089 10 51 .51279 85H51 52770 .84943 54244 .84009 55702 .83050 67143 .824165 9 52 .61304 a^3 52794 .8492* 54269 .83994 55726 .83034 .67167 .82048 8 53 51329 85^21 MS 19 .84913 54293 .8397* 65750 .83017 .57191 .82032 54 51354 85soR 52*44 .84*97 54317 .83962 55775 .83001 .57215 .82015 55 51379 85792 52*69 .8488V! .-4342 .R3946 55799 82986 .67238 .81999 5 56 514^ .8577? 52*93 .84*66 .W66 .8393*1 65823 82969 .57262 8I9R3 4 57 51429 .857H2 52918 84*51 .54391 .83915 55S47 82953 57286 91966 3 58 51454 .85747 52943 .84S3fi 54415 .83*99 .55871 .829:36 67310 81949 59 51479 .85732 52967 .84*20 54440 .83K.S3 .56896 82920 57334 .81932 1 60 51504 .85717 32992 .84805 54464 .83^57 55919 82904 57358 81915 M: Oodn. Blno. Cosin. Sice. Cosin. Sine. Coflin. Bine Cosin. Sine. M. 590 583 67 56 550 T A I;].K Jil. X.VIt'K.M. AXD COSINES. 283 aao 360 370 380 300 M. Slue. Oosln Bine. Corfu. Sine. Cosin Sine. Corin Sine. Cceiu M. 67358 .81915 58779 .80902 .60182 .79*64 .61566 178801 ~62932 TTTlfi 60 573*1 .81899 .68809 .80**f> 60205 .79846 .615*9 .787*3 62955 .77696 69 '4 57405 .81882 .58826 .80867 .60228 .79S29 .61612 .78765 62977 .77676 68 3 57429 .81865 .58*49 .80850 602'> 1 .79*11 .61635 .78747 63000 .77660 67 4 57453 .81848 .58873 .80833 60274 .79793 .61658 .78729 63022 .77641 66 5 67477 .81832 .68896 .80*16 .60298 .7977C .61681 .78711 6*45 .77623 65 6 .57501 .81815 .68921 .80799 60321 .79758 61704 .78694 63068 r, eos 64 7 .57524 .81798 5*943 .807M2 63 .77125 2H 33 58141 81361 59552 30334 .60945 792*2 .62320 .78206 63675 .771071 27 34 58165 81344 59576 80316 6(1968 79264 .62312 .78188 63698 .77088 26 35 58189 81327 59599 80299 .60991 79247 .62365 .78170 63720 .770701 25 36 58212 81310 59622 802*2 .61015 79229 .62388 .78152 63742 77051 24 37 58236 .81293 59646 80264 61038 79211 .62411 .78134 .63765 77033i 23 38 58260 .81276 59669 80247 .61061 79193 .62433 .78116 63787 77014 22 39 58283 .81250 59693 80230 61084 79176 .62456 7809 .63810 76996 21 40 .58307 .81242 59716 80212 61 107 79158 62479 78079 63*32 76977 20 41 .58330 .81225 59739 80195 61130 79140 62502 78061 .63*54 76959 19 42 .58354 .812(18 59763 80178 61153 79122 62524 78m3 .63877 76940 18 43 58378 .81191 59786 80160 61176 79105 62547 78025 63899 76921 17 44 58401 .81174 59809 80143 61199 79087 62570 78007 .63922 76903 16 45 58425 .81157 59832 80125 61222 79069 88592 77988 63944 76884 16 46 58449 .81140 59856 801 OR 61245 79051 62615 77970 63966 76866 14 47 .58472 .81123 59879 80091 61268 79033 62638 77952 63989 76847 13 481.58496 .81106 69909 80073 61291 79016 62660 77934 64011 76828 12 49 .58519 .810*9 59926 S0056 61314 78998 626*3 77916 64033 76810 11 60 .58543 .81072 59949 8003* 61337 789*0 62706 77897 64056 76791 10 61 5a%7 .8H65 59972 80021 61360 7*962 62728 77879 64078 76772 9 62! 58590 .8103- 59995 80003 6(383 78944 62751 77861 64100 76754 8 63 ,58614 .81021 60(119 799*6 61406 7*926 62774 77*43 64123 76735 7 M .60637 .8|otf 6fH42 795)6* 61429 7*908 62796 77824 64145 76717 6 55 58661 801^7 60065 79951 61451 78891 62*19 77806 64167 76698 6 56 5*6*4 .b0970 6flQ89 799:14 61474 78873 62842 777*8 64190 76679 4 57 58708 .80953 601 12 79916 .61497 78855 62864 77769 6421? 76661 3 53 58731 .80936 60136 79*99 61520 78837 62887 77751 64234 76642 2 69 58755 80919 60158 79** 1 61543 78819 62909 77733 642. r 6 76623 1 60 .58779 .80902 60182 79864 .61566 78801 62932 77715 64279 7604 jjE Coedn. Sine. Coulri. Sine. Corfu. Slue. Cosin. Sine. Cosin. Sin*. fit 540 680 | 6 51 600 284 TABLE III, NATURAL SIXES AND COSINES. 4O 410 4? o 43 44 M. Sine. Cosin Sine. Cosin. Sine. Cosin. Sine. Cosin. Siue. | Cosin. M. ~0 64279 .76604 .65606 .75471 .66913 .74314 .68200 .73135 .69466 .71934 60 .64301 .76.M6 .65648 .75452 .66935 .74295 .6*^21 .73116 6-J487 .71914 59 2 .643*3 .76.->67 6.-.6M .75433 .66956 .74276 .6*-242 .73098 .69503 .71*94 58 3 64.346 .7654* .6.-.072 .75414 .66978 .74*56 .6*204 73076 .69529 .71873 57 4 64368 .765.30 .65694 .75395 .66999 .74237 .6*'2*r .73' 156 6 65759 65781 .75337 .75318 .67064 .670S6 .74178 .74159 .6S349 .6S37I .72996 .72976 .69012 69033 .7I792J 53 .71*72 52 9 .64479 .76436 .65803 .75299 .67107 .74139 .68:391 .7*957 0%:>4 .71752151 10 .64501 .76417 .65825 .75280 .67129 .7412' .6*412 .72937 69676 .71732 50 11 .64.V24 .76398 65847 .75261 .67151 .74100 .6*4:34 .72917 09696 .71711 49 12 .64546 .763SH 65869 .75241 .67172 .74080 .6*455 .72897 .69717 .71691 48 13 .64568 .76361 65891 .75222 .67194 .74061 .6S476 .72877 .69737 .71671 47 14 .64.-.9U .76342 .65913 .75203 67215 .74041 .6*497 .72857 .69758 .7165( 46 15 64612 .76323 65935 .75184 .67237 .7402-2 .68518 .72837 .69779 .7163( 45 16 64635 76304 65956 .75165 67258 74002 6.8539 .72817 .69800 .716K 44 17 64657 .762S6 65978 .75146 672.80 .739.83 68561 .72797 .69821 .71590 43 13 .64679 .76267 66000 .75126 .67301 .73963 .68582 .72777 .69842 .715691 42 19 64701 .76248 66022 .75107 .67323 .73944 68603 .72757 69862 .715491 41 20 64723 .76229 .66044 .75088 . 67344 .73924 68621 .72737 .69883 .71529 40 21 64746 .76210 66066 .75069 67366 .739(M 6,8645 .72717 69904 .71508 39 22 61768 .76192 66088 .75050 .67387 .73885 .68666 .72097 69925 .71483 38 23 64790 .76173 66109 .75030 .67409 .73S65 .68688 .72077 09946 .71468 37 24 64S12 .76154 66131 .75011 .67430 .73846 68709 .72657 09966 .71447 36 25 64*34 .761,35 66153 .74992 .67452 .7*826 6*73f .72637 89987 .71427 35 26 64S56 .76116 60175 .74973 .67473 .73806 .68751 .72617 .70008 .71407 34 27 64878 .76097 66197 .74953 .67495 .73787 .6S772 .72597 70029 .71386 33 28 64901 .7607?- 66218 .74934 67516 73767 68793 .72577 70049 .71366 32 29 649-23 .760.->< 00-240 .74915 .67538 .73747 68814 .72557 70O70 .71345 31 30 64945 .76041 66262 .74896 67559 .73728 68835 .72537 .70091 .71325 30 31 64967 .76022 662*4 .74876 .67580 .73708 6,8857 .72517 70112 .71onJ5 29 32 6-1989 .766 .721:36 .70505 .70916 10 51 65408 75642 60718 .744S9 68008 7:0 14 69277 .72116 .70525 .70*96 9 52 .69430 758*23 66740 .74470 6*029 73294 .69-298 .72(195 70546 .70875 8 53 65452 75ft M 66762 .74451 68051 7:}'274 69319 .72"75 70567 .70855 7 54 65474 75585 60783 .74431 68(172 732."v4 69: HO 72055 70587 .70S34 6 55 65496 75506 66805 .74412 68093 .73234 69361 . 7'2035 70608 .70813 5 56 65518 75547 66H27 .74392 681 15 .73215 693S2 72015 .70628 70793 4 57 .65540 75528 66S48 .74373 .68136 .73195 69403 71995 .70649 70772 3 58 65562 75509 66870 74353 .69157 .73175 .69424 71974 .70670 .70752 2 59 65584 75490 66891 743:<4 .68179 .73155 .69445 71954 .70690 70731 I 60 .65606 .75471 66913 74314 .68200 ,7,'31 35 s .69466 719134 .70711 70711 M. Cosin. Sine. Cosin. SineT Cosin. Sine. Cosin. "sin^r Cosin. Sine. M. 49 48 47 40 45 TABLE IV. NATURAL TANGENTS AND COTANGENTS. 285 286 TABLE IV. NATURAL TANCiEXTS AXJ) COTANGENT C 1 o 3 jo }0 M. Tiuig. Cotang. Taug. Cotang. Tang. Cotang. Tang. Cotang. M. "OOOOO Infinite. .01746 67.29110 .03492 28.6363 (15241 19.0811 60 1 .00029 343? 75 .01775 56.3T.li6 .03521 28.3994 05270 18.9755 59 2 .00058 1718.67 .01804 55.4415 .03550 28.1664 .05299 18.8711 58 3 .OOOS7 1145.593 .05941 16.8319 36 26 (10727 137.507 .02473 40.4358 .04220 236945 .05970 1674% 36 26 .(Xf756 132.219 .025*12 39.%:. 5 .04250 23.5321 05999 16 66*1 34 27 .OU785 127.321 .02531 39.5059 .04279 23371^ 06029 16.5S74 33 28 00815 122.774 .02560 39.0568 .04308 232137 .06058 16.5075 32 29 .00844 118.540 .02589 3S.6177 .(M337 230577 .06087 164283 31 30 .00873 114.589 .02619 38.1885 .04366 22.903* .06116 16.3499 80 31 .00002 110R92 .02648 37.76*6 .04395 227519 .06145 16.2722 29 32 .00931 107.426 .02677 37.3579 .04424 22 6020 .06175 16.1952 28 33 .00960 101 171 .02706 36.9560 .04454 224541 .06204 16.1190 27 34 .009*9 101.107 .02735 36.5627 .(U4S3 22.3081 06233 16.0435 26 35 .01018 9S.2179 .02764 36.1776 .04512 22.1640 (16262 15.9687 25 36 .01(47 954*95 .02793 35.8006 .04541 22.0217 .06291 15.8945 24 37 .01076 92.90S5 .(tt*22 35.4313 .04570 21.8813 .06321 15>21 1 23 38 .01105 90.4633 .02-?5 1 35.0695 .04599 21.7426 061 r >0 I5.74S3 22 39 .01135 88.1436 .02S81 34.7151 .04628 21.61156 .06379 15 6762 21 40 .01164 85.9398 .02910 34.3678 .04658 21 47(H .06408 156018 20 41 .01193 83.8435 .02939 34.0273 .046-47 21.3369 .06437 155310 19 42 .01222 81.8470 .02968 33.6935 .04716 21 204'J .06467 154638 18 43 .01251 79.9434 .02997 33.3662 .04745 21.0747 .06496 153943 17 44 .01280 7S.1263 .03026 33.W52 M774 20 9460 .06525 153254 16 45 .01309 76.39(10 .03056 32.7303 .048(3 20.8188 .06554 15 2571 15 46 .01338 74.7292 .03084 32.4213 .04833 20.6932 065R4 15.1893 14 47 .01367 73.1390 .03114 32.11*1 .04*62 20.5691 .06613 15.1222 13 48 01396 71.6151 .03143 3I.R205 .04*91 20.4465 .06642 1 5 ,'57 12 49 .01425 70.1533 03172 31.62*1 .04920 20.3253 .06671 14.9K98 1! 5( .01455 68.7501 .03201 31.2416 .04949 20.2056 .06700 14.9244 10 61 .01484 67.4019 .03230 30.9599 .04978 20.0*72 .06730 14.S596 62 .01513 66.1055 .03259 306*33 .a r >007 19.9702 06759 147954 63 .01542 64.85^0 .03288 30.4116 .05037 19.8546 .06788 14 7317 64 .01571 636567 .03317 30 1446 .05066 197403 06817 146686 65 .01600 62.4992 03346 298823 .05095 19.6273 06S47 146059 66 .01629 61.3*29 01176 296245 (15124 19.5156 .06*76 14.5438 67 .01658 60.305* (134(15 29.3711 .05153 19.4051 06905 144823 3 58 .016-!7 59.2659 .03434 29.1220 .051*2 192959 .06934 14.4212 2 59 .01716 58.2612 (13463 28.*77I .05212 19 1*79 06963 143607 1 60 .01746 57.290(1 .03492 2*. 6363 .05241 19.0811 .065193 14.3007 M. Cftaug. Taug. Cotang. Taug. Co tang. Tang. Cotang. Tang. M. 8 90 8 go 8 70 1 160 TABLE IV. NATURAL TANGENTS AND COTANGENTS. 287 *o 50 6<> T | Ttuig Cotang Tang. Cotaug. Tang. Cotaug. Ttrng. Ootan*. M. -$ ^06993 14.3IJ07 .08749 11.4301 .10510 9.51436 12278 8.14436 60 1 .07022 14.2411 .08778 11.3919 .10540 9.4*761 .12308 8.124S1 60 2 .07051 14.1821 .08807 11.3540 10569 9.46141 12338 8.10536 68 3 07080 14.1235 .08837 11.3163 .10599 9.43515 12367 8.06600 57 4 07110 14.0655 .08666 11.2769 .10628 9.40904 12397 8.06674 66 6 .07139 14.0079 .08895 11.2417 106.37 9.36307 12426 8.04756 56 6 .07168 13.9507 .08925 ll.M" .10687 9.35724 12456 8.02848 54 7 .07197 13.8940 .06954 11.1681 .10716 9.33155 .12486 8.00948 53 8 .07227 13.8378 .08983 11.1316 .10746 9.30599 12515 7.99058 52 9 .07256 13.7821 .09013 11.0954 .10775 928056 12544 7.97176 61 10 07265 13.7267 .06049 11.0594 .10805 9.25530 12574 7.95302 50 11 .07314 13.67J9 .09071 11.0237 .10834 9.23016 12603 7.93438 49 12 .07344 13.6174 .09101 10.9S.S2 .10863 9.20516 12633 7.915S2 48 13 .07373 13.5634 .09130 10.9529 .10893 9.18026 12662 7.89734 47 14 .07402 13.509^ .09159 10.9178 .10922 9.15554 12692 7.87895 46 15 .07431 13.4566 .09189 10.8829 10952 9.13093 12722 7.86064 45 16 07461 13.4039 .09218 10.8483 .10981 9.10646 12751 7.84242 44 17 07490 13.3515 .09247 10.8139 .11011 9.0821 1 12781 7.82428 43 18 .07519 13.2996 .09277 10.7797 .11 WO 9.05789 12810 7.80622 42 19 .07548 13.24SO 093U6 10.7457 .11070 9.03379 12840 7.7S825 41 20 .07578 13.1969 09335 10.7119 .11099 9.00983 12669 7.77036 40 21 07607 13.1461 09365 10.6783 .11128 8.98598 12899 7.75254 39 22 .07636 13.0956 09394 10.6450 .11158 8.96227 12929 7.73480 38 23 .07665 13.0458 09423 10.6118 .11187 8.93667 12958 7.71715 37 24 .07695 12.9962 09453 10.5789 .11217 8.91520 129S3 7.69957 36 25 .07724 12.9469 .09482 10.5462 .11246 8.89185 13017 7.68208 36 26 .07753 12.8981 09511 10.6136 .11276 8.86362 13047 7.66466 34 27 .07782 12.8496 09541 10.4813 .11305 8.84551 13076 7.64732 33 23 .07812 12.8014 09570 10.4491 .11335 8.82252 13106 7.630(15 32 29 .07841 12.7536 .06600 10.4172 .11364 8.79964 13136 7.61287 31 30 .07870 12.7062 09629 10.3854 .11394 8.77689 13165 7.59576 30 31 .07899 12.8591 .09658 10.3538 11423 8.75425 13195 7.67872 29 32 .07929 12.6124 09688 10.3224 .11452 8.73172 13224 766176 28 33 .07258 12.5660 .09717 10.2913 .11482 8.70931 13254 7.54487 27 34 .07987 12.5199 09746 10.2602 .11511 8.68701 13284 7.52806 26 35 .08017 12.4742 09776 10.2294 .11541 8.664S2 13313 7.51132 26 36 .08(146 12.4288 09805 10.1988 .11570 8.64275 13343 7.49465 24 37 .08075 12.3838 09834 10.1683 .11600 8.62078 13372 7.47806 23 38 .08104 12.3390 .09864 10.1381 .11629 8.59693 13402 7.46164 22 39 .08134 12.2W6 .09893 10.1080 .11659 8.57718 13432 7.44509 21 40 .08163 12.2505 09928 10.0780 .11688 8.55555 13461 7.42871 20 41 .08192 12.2067 09952 10.0483 .11718 8.53402 13491 7.41240 19 42 | .08221 12.1632 .09981 10.0187 .11747 8.51259 13521 7.39616 18 43 .08251 12.1201 .10011 9.98931 .11777 8.49128 13550 7.37999 17 44 .08280 12.0772 .10040 9.96007 11806 8.47007 13580 7.36389 16 45 .08309 12.0346 .10069 9.93101 .11836 8.44896 13609 7.34786 16 46 .08339 11.9923 .10099 9.90211 .11865 8.42795 13639 7.33190 14 47 .08368 11.9504 .10128 9.87338 .11995 8.40705 13669 7.31600 13 48 .08397 11.9087 .10158 9.84482 .11924 8.36625 13698 7.30018 12 49 .08427 11.8673 .10137 9.81641 .11954 8.36555 13728 7.26442 11 50 .08456 11.8262 .10216 9.76817 .11983 8.34496 13758 7.26873 10 51 .08485 11.7853 .10246 9.76009 .12013 8.32446 13787 7.25310 9 52 .08514 11.7448 .10275 9.73217 .12H42 8.30406 13817 7.23754 8 53 .08544 11.7045 .10305 9.70441 .12072 8.28376 13846 7.22204 7 54 .08573 116645 .10334 9.67680 .12101 8.26355 13876 7.20661 6 81 .08602 11.6248 .10363 9.64935 .12131 8.24345 13943 5.60452 10 5.08139 52 9 .14321 T.9826S .16107 6.20S&I .17903 5.&s;,73 19710 5.07360 61 10 .14351 6.96823 16137 6.19703 .179.53 55763S 19740 5.06584 50 II .14381 6. 9538; 16167 6.18559 17'J63 5.56? 16286 6.14023 .18083 5.53007 19891 5.02734 46 16 14529 6.R8278 16316 6.12S99 18113 6.52090 19921 6.01971 44 17 14559 6.86874 16346 6.11771* .18143 6.51176 19952 5.01210 43 18 14538 6.85475 16376 6.I06G4 .18173 5.50264 19'J^2 500451 42 19 .14618 6.840S2 16405 6.09552 .18203 5 49356 2(K)12 4.99695 41 20 14648 6.82694 16435 6.084-14 .18233 5.-4M.';! .2rt2 4.98940 40 91 14678 6.81312 16165 6.07340 IS263 5.4754^ 20073 498188 39 22 14707 6. 79936 1&495 6.062411 18293 6.4664H 20103 497438 38 23 14737 6.78564 16525 6.05143 .18323 5.45751 .20133 4 96690 37 24 14767 6.77199 16555 6.04: 151 .18353 5.44>57 20164 495945 36 26 14796 6.75->38 16585 6.02962 It>3~i4 5.43-J66 20194 4.yfi20| 35 2fi M-<26 6. 744 S3 .16615 60l8?s 1K4I4 6.4>77 2ir224 494460 34 27 14866 673133 16645 6.007-J7 .18444 5.42192 2ir2.-v| 493721 33 28 14HS6 6.71789 16674 6.99720 .18474 6.41309 2ir/K5 492984 32 29 14915 67W50 16704 5.9S6-16 1-fKH 540429 20315 4 92249 31 'JO 14945 6.69116 16734 5.97576 .18534 6.39552 20345 491516 30 31 14976 667787 16764 596510 .18564 53^577 20376 4.90785 29 32 15015 6 66-163 16794 5.9544 s * .1-S5W G. 37*05 2W( 4 .90056 28 33 15034 665144 16824 5.94:WO .l-24 5.36y: .2TM36 4 893:*) 27 34 15064 663S3I 16854 6.93335 i^:>4 5.36070 2(Mn6 4.S.X06 26 35 15094 6.62523 10884 6.922S3 1<6>4 6.35206 .2IM97 4.878W2 25 36 15124 661219 16914 5.9123G .18714 6.34.115 20527 4.87162 24 37 15153 6.59921 16944 6.9f19l 18745 5.3:M>7 20557 486444 23 38 15183 6.&S62? 16974 5.89151 .18775 5.32631 20.'>88 485727 22 39 15213 6.57339 17004 6.88114 .18805 5.31 77s 20613 4.S50I3 21 40 .15243 6561)55 17033 6.8708O .18*35 5.3092S .20648 484300 20 41 15272 6.54777 17063 5.860r>l 1 <-*65 530(K) 20679 4.83590 19 42 .15302 6.C35TI3 17093 5.85024 1 -i-yr, 5.29235 .20709 4.82882 18 43 15332 6.52234 17123 5.84(101 .H92. r > 5.2.S393 .20739 482176 17 44 15362 6.5W70 .17153 6.829^ .18955 5.27553 .20770 481471 16 46 .15391 6.49710 .17183 6.81966 .18986 6.26715 .20800 4.80769 16 46 .15421 6.4*456 17213 6.809r<3 .190)6 6.25880 .20830 4.80088 14 47 .15451 6472116 .17213 5.79iM4 1W*16 6.25(M8 20861 4.79370 13 48 154S1 6.4f>96l 17273 5.7^9.'^ .19076 524218 .20891 4.7H673 18 49 .15511 6.44720 17303 5.779.16 19106 5.23391 20921 4.77978 11 60 .15540 6434^4 .17333 6. 76937 19136 5.22566 20952 4.7726 10 51 .15570 642253 .17363 5.75U4I 19106 5.21744 20982 4.76596 52 .15600 6.41026 17393 5.74949 19197 6 20925 2KU3 4.76910 53 .156.10 639804 17423 5.73J60 .19227 5.20107 21043 475219 64 .15660 6.38M7 17453 5.72974 19257 6.19293 .21(173 474534 65 15689 637374 17463 5.71992 I92H7 5.IH4SO 21104 4.73851 66 .15719 6.36165 17513 5.71013 19317 6.17671 21134 4.73171' 57 .15749 6.349fil 17543 6.70037 19:M7 5.I&H63 21164 4.72490 3 58 .15779 6.337RI 17573 5.69064 19378 5 lOfis 21195 471813 2 69 .15X19 6.3*566 .17603 5.6S<94 19408 6.15256 .21225 4.71137 1 60 .15838 6.31375 17633 5.6712 19438 5.14455 21256 470463 M Cotang. Tmug. Ootang. Tang. Cotang' Tang. Ootuug. Tang. M. 8 10 8 OQ 7! 9ft 7 RO TABLE IV. NATURAL TANGENTS AND COTANGENTS. 289 I 90 1 3 1 * 1 50 M Tang. Cotang. iTwig. Cotang. Tang. Cotang. Tang. Cotung M. 21256 4.70463 23087 4.33148 7124933 4.01078 .26795 3.73205 60 1 21 486 4697'JI .23117 4.32573 .24964 4.00582 .26826 3.72771 59 2 21316 4.69121 .23148 4.321 101 .24995 4.D0086 .26657 3.72338 58 3 21347 168452 23179 4.31430 .25026 3.99592 .26888 3.71907 57 4 21377 467766 23*19 4.30S60 .25056 3.9909'J .26920 3.71476 56 5 214(18 4.67121 .23240 4.30291 .25087 3.93607 .26951 3.71046 65 6 21438 4.66458 .23271 4.29724 .25118 3.98117 .26982 3.70616 54 7 21469 465797 23301 4.29159 25149 3.97627 .27013 3.70188 53 8 21499 465138 23332 4.28595 .25180 3.97139 .27044 3.69761 52 9 21529 464480 23363 4.28032 25211 3.96651 .27076 3.69335 51 10 21560 463825 23393 4.27471 .25242 3.96165 .27107 3.68909 50 11 21590 463171 23424 4.26911 .25273 3.95680 .27138 3.68485 49 12 21621 462518 23455 4.26352 .25304 3.95196 27169 3.68061 48 13 21651 461868 234 85 4.25795 .25335 3.94713 27201 3.67638 47 14 21682 461219 23516 4.25239 .25366 3.94232 .27232 3.67217 46 15 21712 4.60572 23547 4.24685 .25397 3.93751 .27263 3.66796 45 16 21743 459927 23578 4.24132 .25428 3.93271 .27294 3.66376 44 17 21773 4 592*3 23608 4.23580 .25459 3.92793 27326 3.65957 43 18 21804 456641 23639 4.23030 .25490 3.92316 27357 3.65538 42 19 21834 4.58001 23670 4.22481 25521 3.91839 27388 3.65121 41 20 21864 457363 23700 4.21933 .25552 3.91364 27419 3.64705 40 21 21*95 4 56726 23731 4.21387 .25583 3.90890 27451 3.64289 39 22 21925 456091 23762 4.20842 .25614 3.90417 27482 3.63874 38 23 21956 455458 23793 4.20298 .25645 3.89945 27513 3.63461 37 24 21986 454826 23823 .19756 .25676 3.89474 27545 3.63048 36 25 22017 454196 .23864 .19215 .25707 3.89004 27576 3.62636 36 26 22047 453568 .23885 .18675 .25738 3.88536 27607 3.62224 34 27 .22078 4.52941 23916 .18137 25769 3.88068 27638 3.61814 33 28 22108 4.52316 23946 .17600 .25800 3.87601 27670 3.61405 32 29 .22139 451693 .23977 .17064 25831 3.87136 27701 3.60996 31 30 22169 4.51071 24008 .16530 25862 3.86671 27732 3.60588 30 31 .22200 4.50451 24039 .15997 25893 3.86208 27764 3.60181 29 32 22231 4.49832 24(169 .15465 .25924 3.85745 27795 3.59776 28 33 22261 4.49215 24100 .14934 .25955 3.85284 27826 3.59370 27 34 22292 4.48600 24131 .14405 .25986 3.84824 27888 3.58966 26 35 22322 4.47986 24162 .13877 .26017 3.84364 27889 3.58562 25 36 22353 4.47374 24193 .13350 26048 3.83906 27921 3.5816U 24 37 22383 4.46764 24223 .12825 .26079 3.83449 27952 3.57758 23 33 22414 4.46155 24254 .1230! .26110 3.82992 27983 3.57357 22 39 22444 4.45548 24285 .11778 .26141 3.82537 28015 3.56957 21 40 22475 4 44942 24316 .11256 .26172 3.82083 28046 3.56557 20 41 22505 4.44338 24347 .10736 .26203 3.81630 28077 3.56159 19 42 22536 443735 24377 .10216 .26235 3.81177 .28109 3.55761 18 43 22567 443134 24408 4.09699 .26266 3.80726 28140 3.55364 17 44 22597 442534 24439 4.09182 .26297 3.80276 28172 3.54968 16 45 22628 4.41936 24470 4.08666 .26328 3.79827 .28203 3.54573 15 46 22658 4.41340 24501 4.08152 .26359 3.79378 28234 3.54179 14 47 226S9 4.40745 24532 4.07639 26390 3.78931 28266 3.53785 13 48 22719 4.40152 24562 4.07127 .26421 3.78485 .28297 3.53393 12 49 22750 439560 24593 4.06616 .26452 3.78040 .28329 3.53001 11 50 22781 4.38969 24624 4.06107 .26483 3.77595 28360 3.52609 10 51 22811 4.aS381 24655 4.05599 26515 3.77152 28391 3.52219 9 52 22842 4.37793 246^6 4.05092 26546 3.76709 28423 3.51829 8 53 22872 4.37207 24717 4 04586 26577 3.76268 .28454 3.61441 7 54 22903 436623 24747 4.040H1 26608 3.75828 28486 3.51053 6 65 22934 4 36(140 24778 4.03578 26639 3.75388 28517 3.50666 6 66 22964 4>I59 24S09 4.03076 26670 3.74950 28549 3.50279 4 57 22995 434879 2440 4.02574 26701 3.74512 .28580 3.49894 3 58 23W8 434300 24871 402074 26733 3.74075 .28612 3.49509 f 69 .23056 433723 24902 4.01576 26764 378640 28643 3.49125 1 60 23H87 4.33148 24933 4.01078 26795 3.73205 .28675 3.48741 M. Totting. Tang. Cotang. Tang Coteng: Tang. Cotang. Tang. M. 1 T 7 60 7 50 7 EO 290 TABLE IV. NATURAL TANGENTS AND COTANGENTS. i GO 1 to ] go 1 90 M. Ttoig. Cotang. Tang. Cotang. Tang. Cotang. Tang. Cotang. M. ~0 .28675 3.48741 .30573 3.27085 .32492 ~3707768 .34433 2.90421 60 1 .28706 3.48359 .30605 3.26745 .32524 3.07464 .34-165 2.90147 59 2 .28738 3.47977 .30637 3.26406 .32556 3.07160 .34498 2.89873 58 3 .28769 3.47596 .30669 3.26067 .32588 3.06857 .34530 2.89600 57 4 .28800 3.47216 .30700 3.25729 .32621 3.06554 .34563 2.89327 56 6 .28832 3.46S37 .30732 3.25392 .32653 3.06252 .34596 2.89055 55 6 .28864 3.46458 .30764 3.25055 .32685 3.05950 .34628 2.88783 64 7 .28895 3.46080 .30796 3.24719 .32717 3.05649 .34661 2.8851 1 53 8 .28927 3.45703 .30823 3.24383 .32749 3.05349 .34693 2.88240 52 9 .28958 ?.45327 .30860 3.24049 .32782 3.05049 .34726 2.87970 51 10 .28990 3.44951 .30891 3.23714 .32814 3.04749 .34758 2.87700 50 11 .29021 3.44576 .30923 3.23381 .32846 3.04450 .34791 2.87430 49 12 .29053 3.44202 .30955 3.23048 .32878 3.04152 .34824 2.87161 48 13 .29084 3.43329 .309S7 3.22715 .32911 3.03854 .34856 2.86892 47 14 .29116 3.43456 .31019 3.22384 .32943 3.03556 .34889 2.86624 46 15 .29147 &43084 .31051 3.22053 .32975 3.03260 .34922 2.86356 45 16 29179 3.42713 .31083 3.21722 .33007 3.02963 .34954 2.86089 44 17 29210 3.42343 .31115 3.21392 .33040 3.02667 .34987 2.85822 43 18 .29242 3.41973 .31147 3.21063 .33072 3.02372 .35020 2.85555 42 19 .29274 3.41601 .31178 3.20734 .33104 3.02077 .35052 2.85289 41 20 .29305 3.41236 .31210 3.204% .33136 3.01783 .35085 2.85023 40 21 .29337 3.40869 .31242 3.20079 .33169 3.01489 .35118 2.84758 39 22 .29363 3.40502 .31274 3.19752 .33201 3.01196 .35150 2.84494 38 23 .29400 3.40136 .31306 3.19426 .33233 3.00903 .35183 2.84229 37 24 .29432 3.39771 .31338 3.19100 .33266 3.00611 .35216 2.83965 36 25 .29463 3.39406 .31370 3.18775 .33298 3.00319 .35248 2.83702 36 26 .29495 339042 .31402 3.13451 .33330 3.00028 .35231 283439 34 27 .29526 3.38679 .31434 318127 .33363 2.99738 .35314 2.83176 33 28 .29558 3.38317 .31466 3.17804 .33395 299447 .35346 2.82914 32 29 .29590 3.37955 .31498 3.17481 .33427 299158 .35379 2.82653 31 30 .29621 3.37594 .31530 3.17159 .33460 2.98668 .35412 2.82391 30 31 .29653 3.37234 .31562 3.16838 .33492 2.98580 .35445 2.82130 29 32 .29685 3.36875 .31594 3.16617 .33521 2.98292 .35477 2.81870 2S 33 .29716 3.36516 .31626 3.16197 .33557 2.98004 .35510 281610 27 34 .29748 3.36158 .31653 3.15877 .33589 2.97717 .35543 2.81350 26 35 .29780 3.35800 .31690 3.15558 .33621 2.97430 35576 2.81091 25 36 .29811 335443 .31722 3.15240 .33654 2.97144 35608 2.80833 24 37 .29843 3.35087 .31754 3.14922 .336S6 2.96858 35641 2.80574 23 38 .29875 3.34732 .31786 3.14605 .33718 2. 965~3 35674 2.80316 22 39 .29906 3.34377 .31818 3.14288 .33751 2.96288 .35707 2.80059 21 40 .29933 3.34023 .31850 3.13972 .33783 2.960'U 35740 2.79802 20 41 .29970 333670 .31882 3.13656 .33816 2.95721 35772 2.79545 19 42 .30001 3.33317 .31914 3.13341 .33848 2.95437 .35805 2.79289 18 43 .30033 3.32965 .31946 3.13027 .33881 2.95155 .35838 2.79033 17 44 .30065 3.32614 .31978 3.12713 .33913 2.94872 .35871 2.78778 16 45 .30097 3.32264 .32010 3.12400 .33945 2.94591 .35904 2.78523 15 46 .30128 3.31914 .32042 3.12087 .33978 2.94309 .35937 2.78269 14 47 .30160 3.31565 .32074 3.11775 .34010 2.94028 .35969 2.78014 13 48 .30192 3.31216 .32106 3.11464 .34043 2.93748 .36002 2.77761 12 ' 49 .30224 3.30863 .32139 3.11153 .34075 2.93468 .36035 2.77507 11 50 .30255 3.30521 .32171 3.10842 .34108 2.93189 .36068 2.77254 10 51 .30287 3.30174 .32203 3.10532 .34140 2.92910 .36101 2.77002 52 .30319 3.29829 .32235 3.10223 .34173 2.92632 36134 2.76750 53 .30351 3.29483 .32267 3.09914 .34205 2.92354 .36167 2.76498 54 .30382 3.29139 .32299 3.09606 .34238 2.92076 .36199 2.76247 55 .30414 3.28795 .32331 3.0929S .34270 2.91799 36232 2.75996 56 .30446 3.28452 .32363 3.08991 .34303 2.91523 .36265 2.75746 57 .30478 3.28109 .32396 3.08685 .34335 2.91246 .36298 2.75496 58 .30509 3.27767 .32428 3.08379 .34363 2.90971 .36331 2,75246 59 .30541 3.27426 .32460 3.08073 .34400 2.90696 .36364 2.74997 60 .30573 3.27085 .32492 3.07768 .34433 2.90421 .36397 274748 M: Cotaug. Tang. Cotang. Tang. Cotang Tang. Cotuug. Tang. M. V 30 7 o y 10 T OQ TABLE NATURAL TANGENTS AND COTANGENTS. 291 20 31 aa 23 M. Tang. Cotang. Tang Gotang. Tang. Cotang. Tang. Cotang. M. .36397 2.74748 .38386 2.60509 .40403 ~!2.47509 .42447 2.35585 60 1 .3&430 2.74499 38420 2.60283 .40436 2.47302 .42482 2.35395 59 2 .36463 2.74251 38453 2.60057 .40470 2.47095 .42516 2.35205 68 3 .36496 2.74004 3S4*7 2.59h3l .40504 2.46888 .42551 2.35015 67 4 .36529 2 73756 38520 2.59606 .40538 2.46682 .42585 2.34825 56 5 .36562 2.73509 38553 2.593S1 .40572 2.46476 .42619 2.34636 55 6 .36595 2.73*63 .38587 2.59156 .40606 2.46270 .42654 2.34447 54 7 .36628 2.7U7 38620 2.58932 .40640 2.46065 42688 2.34258 53 8 .36661 2.72771 38654 2.58708 .40674 2.45860 .42722 2.34069 52 9 .36694 2.72526 38687 2.58484 .40707 2.45655 .42757 2.33881 51 10 .36727 2.72281 38721 2.58261 .40741 2.45451 42791 233693 50 11 .36760 2.72036 38754 2.58038 .40775 2.45246 42826 233505 49 12 .36793 2.71792 38787 2.57815 .40809 2.45043 42860 233317 48 13 36826 2.71548 38821 2.57593 .40843 2.44839 42894 2.33130 47 14 .36859 2.71305 38854 2.57371 .40877 2.44636 42929 2.32943 46 15 .36892 2.71062 38888 2.57150 .40911 2.44433 42963 2.32756 45 16 36925 2.70819 38921 2.56928 .40945 2.44230 42998 2.32570 44 17 .36958 2.70577 38955 2.56707 .40979 2.44027 43032 2.3-2383 43 18 36991 2.70335 38988 2.56487 .41013 243825 43f67 2.32197 42 19 37024 2.70094 39022 2.56266 .41047 2.43623 43101 2.32012 41 20 37057 2.69853 39055 2.56046 .41081 243422 43136 2.31826 40 21 37090 269612 39039 2.55827 .41115 2.43220 43170 2.31641 39 22 37123 2.69371 39122 2.55608 .41149 2.43019 43205 2.31456 38 23 37157 2.69131 39156 2.553S9 41183 2.42819 43239 2.31271 37 24 37190 2.68892 .39190 2.55170 41217 2.42618 43274 2.31086 36 25! 37223 2 68653 39223 2. 54952 .41251 2.42418 43308 2.30902 35 26 37256 2.68414 39257 2.54734 .41285 242218 43343 2.30718 34 27 37239 2.68175 39290 2.54516 .41319 2.42019 43378 230534 33 28 37322 2.67937 39324 2.54299 .41353 2.41819 43412 2.30351 32 29 37355 2.6770U 39357 2.54082 .41387 2.41620 43447 2.30167 31 30 373S8 2.67462 39391 2.53865 .41421 2.41421 43481 2.29984 30 31 37422 2.67225 W25 2.53648 .41455 241223 43516 2.29801 29 32 37455 2.66989 3^458 2.53432 .41490 2.41025 43550 2.29619 28 33 37488 2.66752 3iH^< 2.53217 .41524 2.40827 43585 2.29437 27 34 37521 2.66516 395VK 2.53001 .41558 2.40629 43620 2.29254 26 35 37554 2 66-2*1 39559 2 52786 .41592 2.4043-2 43654 2.29073 25 36 37588 2.66(1-46 39593 2.52571 41626 2.40235 43689 2.28891 24 37 37621 2.65811 39626 252357 .41660 2.40038 43724 2.28710 23 1 38 37654 265576 39660 2.52142 41694 2.39841 43758 2.28528 22 39 . 37687 265.'542 39694 2.51921* 41728 2.39645 43793 2.28348 21 40 ; 37720 2. 651 (19 39727 2.51715 41763 2.39449 43828 2.28167 20 41 .37754 2.64875 39761 2.51502 41797 2.39253 43862 2.27987 19 42) 37787 264642 39795 2.51289 41831 2.39058 43897 2.27806 18 43 i .37820 264410 39829 2.51076 41865 2.3S863 43932 2.27626 17 44 37S53 264177 39862 2.50864 .41899 2.38668 43966 2.27447 16 45 .37887 2.63945 39896 2.50652 .41933 2.38473 44001 2.27267 15 16 37920 263714 39930 2.50440 41963 2.38279 44036 2.27088 14 47 37953 2 634*3 39963 2.502-29 42002 2.38084 44071 2.26909 13 48 37986 263252 39997 250018 42036 2.37891 .44105 2.26730 12 49 .38020 263021 40031 2.49807 42070 2.37697 .44140 2.26552 11 50 .38053 262791 40065 2.49597 42105 2.37504 .44175 2.26374 10 51 .38086 2 62561 40098 2.49386 42139 2.3731 1 44210 2.26196 9 52 38120 2 62fi 40132 2.49177 4-2173 2.371 18 44244 2.26018 8 53 38153 262103 40166 24S967 42207 2.36925 .44279 2.25840 7 54 .38186 261874 4021 K) 2.48758 4-2242 2.36733 .44314 2.25663 6 55 .3^220 26IR46 40-23-1 2 48549 42-276 236541 .44349 225486 6 56 .3-1253 261418 40267 2.48340 42310 2.36349 .44384 225309 4 57 38286 261190 41 1301 248132 42345 2.36158 44418 2.25132 3 58 .38320 260963 40335 2 479'24 .4-2379 2.35967 .44453 2.24956 2 59 .3^353 260736 .40369 247716 4-2413 2.35776 .44488 2.24780 1 60 .38386 2.60509 40403 2.47509 .42447 2.35585 .44523 2.24604 IT Gotang. Tang. Cotang. Tang. Gotang. Tang. Cotang. Tang. M. | 69 68 67 66 2!)j! TABLE IV. NATURAL TANGEXTS AND COTAXGKX'I : i 40 1 J5 a fto \ 570 1 M. Tang. Cotang. Tang. Cotaug Tang. Cotang. Tung Cotaug. M. If 744523 2.24604 .46631 2.14451 .48773 2.05030 .50953 1.96261 60 I .44558 2.24428 .46666 2.14288 .48809 2.04879 .50989 1.961-20 59 2 .44593 2.24:452 .467)12 2.14125 .48845 2.04728 .51026 1.95979 58 3 .44627 2.24077 .467!{7 2.13963 .48881 2.04577 .51063 1.95838 57 4 .44662 2.23902 46772 2.13801 .48917 2.04426 .51099 1.95698 56 5 .44697 2.23727 46808 2.13639 .48953 2.04276 .51136 1.95557 55 6 .44732 2.23553 .46>t43 2.13477 48989 2.04125 .51173 1.95417 54 7 .44767 2.23378 46879 2.13316 .49026 2.03975 51209 1.95277 53 8 .44802 2.23204 46914 2.13154 .49062 2.03825 .51246 1.95137 52 9 .44837 2.23030 46950 2.12993 .49098 2.03675 .51283 1.94997 51 10 .44872 2.22857 46985 2. 12832 .49134 2.03526 .61319 1.94858 50 11 .44907 2.22683 47021 2.12671 .49170 203376 51356 1.94718 49 12 .44942 2.22510 47056 2.12511 49206 2.03227 51393 1.94579 48 13 44977 2.22337 47092 2.12350 49242 2.03078 51430 1.94440 47 14 .45012 2.22164 47128 2.12190 .49273 2.02929 51467 1.94301 46 15 45047 2.21992 47163 2.1203*1 49315 2.02780 51503 1.94162 45 16 45082 2.21 SI 9 47199 2.11871 .49351 2.02631 51540 1.94023 44 17 45117 2.21647 47234 2.11711 .49387 2.02483 51577 1.93866 43 18 45152 2.21475 47270 2.11552 49423 2.02:}:35 51614 1.93746 42 19 45187 2.21304 47305 2.11392 .49459 2.02187 51651 1.93608 41 20 45222 2.21 132 47341 2.11233 .49495 2.02039 51688 1.93470 40 21 45257 2.20961 47377 2.11075 49532 2.01891 51724 1.93332 39 22 45292 2.2079ft 47412 2.10916 .49563 2.01743 51761 1.93195 38 23 45327 2.20619 .47443 2.10758 49604 2. 01 596 51793 1.93007 37 24 45362 2.20449 47483 2.10600 49640 2.0144'J 51835 1.92020 36 25 45397 2.20278 47519 2.10442 .49677 2.01302 51872 1.92782 35 26 45432 2.20108 47555 2.10284 49713 2.01155 51909 1 92645 34 27 45467 2.19938 47590 2.10126 49749 2.01008 51946 i. 92508 33 28 45502 2.19769 47626 2.09969 49786 2.00862 51983 1.92371 32 29 46538 2.19599 47662 2.09311 49822 2.00715 52020 1.92235 31 30 45673 2.19430 47698 2.09654 .49858 2.00569 52057 1.92098 30 31 45608 2.19261 47733 2.09498 49894 2.00423 52004 1.91962 29 32 45643 2.1909-2 47769 2.09341 49931 2.00277 52131 1.91826 28 33 45678 2.18923 47805 2.09184 49967 2.00131 52168 1.91690 27 34 45713 2.18755 47840 2.09028 50004 1.999*6 52205 1.91554 26 35 45748 2.18587 47876 2.08872 50(MO 1.99H41 52242 1.91418 25 36 45784 2.18419 47912 2.08716 50076 1.99695 62-279 1.91282 24 37 45819 2.18251 47948 2.08560 50113 1.9955(1 52316 1.91147 23 38 45854 2.18084 47984 2.08405 50149 1.99406 52353 1.91012 22 39 45889 2.17916 48019 2.08250 .50185 1.99261 52390 1.90876 21 40 45924 2.17749 48055 2.0S094 .50222 1.99116 52427 1.90741 20 41 45960 2.17682 48091 2.07939 .50258 1.98972 62464 1.90607 (9 42 45995 2.17416 48127 2.07785 50295 .98828 52501 1.90472 18 43 46030 2.17249 48163 2.07630 50331 .98684 52538 1.90337 17 44 46065 2.17083 48198 2.07476 60368 .98540 52575 1.90203 16 45 46101 2.16917 48234 2.07321 50404 .98396 52613 1.90069 15 46 46136 2.16751 48270 2.07167 50441 .98253 52650 1.89935 14 47 46171 2.16585 48306 2.07014 50477 .98110 52687 1.89801 13 48 46206 2.16420 48342 2 06860 50514 .97966 52724 1.89667 12 49 46242 2 16255 .48378 2.06706 .60550 .97823 52761 1.89533 11 50 46277 2.16090 .48414 2.06553 .50587 .97681 52798 1.89400 10 51 46312 2.15925 .48450 2.06400 50623 .97538 52836 1.89266 9 52 46348 2. 15760 .48486 2.06247 50660 97395 52873 1.89133 8 53 46383 2. 15596 48521 2.06094 50696 .97253 52910 1.89000 7 54 46418 2.15432 48557 2.05942 50733 .97111 .52947 1.88867 55 46454 2 15268 48593 2.05790 .50769 .96969 529S5 1.88734 56 46489 215104 48629 2.05637 50806 .96.S27 63022 1.88602 57 46/525 2.14940 48665 2.05485 50843 1.96685 59159 1.88469 58 46560 2.14777 48701 2.05333 50879 .96544 53096 1.88337 59 46*95 2.14614 48737 2.05182 50916 1.96402 53134 1.88205 I 60 .46631 2.14451 48773 2.05030 50953 .96'26I 53171 1.88073 M. Cutang. Tang. Cotang. Tang. Co tang. Tang. Cotang. Tung. M. G [JO 6 1 6 JO O! 8 TARLI-; IV. NATURAL TANGENTS AND COTANCKX 380 393 300 310 M Tang Cotang. Tang. Cotang. Tang. Cotang. Tang. Cotang M. ~o "63171 1.88073 55431 1.80405 .57735 .73205 60086 1.66428 6U 63208 1.87941 55469 1.802S1 .57774 .73089 60126 1.66318 59 63246 1.87809 55507 1.8W58 .57813 .72973 60165 1.66209 68 53283 .87677 55545 1.80034 .57851- .72857 60205 1.66099 67 63320 .87546 55583 1.79911 .57890 .72741 60245 1.65990 56 63358 .87415 65621 1.79788 .57929 .726*5 80384 1.65881 66 63395 .87283 55659 1.79665 .57968 .72509 60324 1.65772 64 63432 .87152 55697 1.79542 .58007 .72393 60364 .65663 53 .63470 .87021 55736 1.79419 .58046 .72278 60403 .65554 62 .63507 .86891 55774 1.79296 .58085 .72163 60443 .65445 51 10 63545 .86760 55812 1.79174 .58124 .72047 60483 .65337 60 11 53582 .86630 55850 1.79051 .58162 .71932 60522 .65228 49 12 53620 .86499 55888 1.78929 .58201 .71817 60562 .65120 48 13 63657 .86369 55926 1.78807 .58240 .71702 .60602 .65011 47 14 53694 .86239 55964 1.78685 .58279 .71588 .60642 .64903 46 15 .63732 .86109 56003 1.78563 .68318 .71473 .60681 .64795 45 16 63769 .85979 56041 1.78441 .58357 .71358 .60721 .64687 44 17 53807 .85850 56079 1.78319 .58396 .71244 60761 .64579 43 18 53844 .85720 56117 1.78198 58435 .71129 60801 .64471 42 19 53882 .85591 56156 1.78077 .58474 .71015 60841 .64363 41 20 53920 1.85462 56194 1.77955 .58513 .70901 60881 .64256 40 21 63957 1.85333 56232 1.77834 .58552 .70787 60921 .64148 39 22 53995 1.85204 56270 1.77713 .68591 .70673 60960 .64041 38 23 54032 1.85075 56309 1.77592 .58631 .70560 61000 .63934 37 24 64070 1.84946 56347 1.77471 .58670 .70446 61040 .63826 36 25 54107 1.84818 56385 1.77351 .58709 .70332 61080 .63719 36 26 64145 1.84689 56424 1.77230 .58748 .70219 61120 .63612 34 27 54183 1.84561 56462 1.77110 .58787 .70106 61160 63505 33 28 54220 1.84433 56501 1.76990 .58826 1.69992 61200 63398 32 29 54258 1.84305 56539 1.76869 .58865 1.69879 61240 .63292 31 30 64296 1.84177 .56577 1.76749 .58906 1.69766 61280 1.63165 30 31 64333 1.84049 56616 1 76629 .58944 1.69653 61320 1.63079 29 32 64371 .83922 56654 1.76510 .58983 1.69541 .61360 1.62972 28 33 .64409 .83794 56693 1.76390 .59022 1.69428 .61400 1.62866 27 34 64446 .83667 56731 1.76271 .59061 1.69316 61440 1.62760 26 35 64484 .83540 56769 1.76151 .59101 1.69203 61480 1.62654 06 36 54522 .83413 56S08 1.76032 .69140 1.69091 61520 1.62548 24 37 .64560 .83286 56S46 1.75913 .59179 1.68979 61561 1.62442 23 38 54597 .83159 56885 1.75794 .59218 .68866 61601 1.62336 22 39 64635 .83033 56923 1.75675 .59258 .68754 61641 1.62230 21 40 64673 .82906 56962 1.75556 .59297 .68643 .61681 1.62125 20 41 64711 .82780 57000 1.75437 .59336 .68531 .61721 1.62019 19 42 64748 .82654 57039 1.75319 .59376 .68419 .61761 1.61914 18 43 64786 1.82528 57078 1.75200 .59415 .68308 .61801 1.61808 17 44 64824 1.82402 57116 1.75082 .59454 .68196 .61842 1.61703 16 45 64862 1.82276 .57155 1.74964 .59494 .68085 .61882 1.61598 15 M .64900 1.82150 .57193 1.74846 .59533 .67974 61922 161493 14 47 64933 1.82025 .57232 1.74728 .59573 .67863 61962 1.61388 13 48 .64975 1.S1899 .57271 1.74610 .59612 .67752 62003 1.61283 12 it .65013 1.81774 .57309 1.74492 .59651 .67641 62043 1.61179 11 5C .65051 1.81649 .57348 1.74375 .59691 .67530 62083 1.61074 10 61 55089 1.81524 .57386 1.74257 59730 .67419 62124 1.60970 9 52 .65127 1.81399 .57425 1.74140 59770 .67309 62164 1.60865 8 53 .55165 1 81274 57464 1.74022 .59809 .67198 62204 1.60761 7 54 55203 1.81150 57503 1.73905 .59849 .67088 .62245 1.60657 56 55241 1.81025 .57541 1.73788 .59888 .66978 62285 .60553 66 6527* 1.80901 .57580 1.73671 59928 .66867 .62325 .60449 57 55317 1.80777 57619 1.73555 59967 .66757 62366 .60345 68 55355 1.80653 57657 173438 60007 .66647 .62406 .60241 69 55393 1.80529 57696 1 73321 .60046 1.66538 62446 .60137 60 .55431 1.80405 .57735 1.73205 .60086 .66428 62487 .60033 M: Cotaug. Taug. Cotang. Tang. Cotang. Tang. Cotang. Tang. M. 610 600 590 680 204 TARLE IV. NATURAL TANGENTS AND COTANGKXTS. i o 9 13 3 40 1 IC M JUang. Cotaug. Taug. Cotaog. Tang. Cotaug. Taug. Cotang. M. ~o .62487 .60033 .64941 1.53986 .67451 1.43256 .70021 1.42816 60 I .62527 .59930 .64982 1.53888 .67493 1.48163 .70064 .42726 69 2 .62568 . 59HJ&6 .65024 1.53791 .67536 1.48070 .70107 .42638 63 3 .62608 .59723 .65065 1.53693 .67578 1.47977 .70151 .42550 67 4 .62649 .59620 .65106 1.53595 .67620 1.47885 .70194 .42462 66 6 .62689 .69517 .65148 1.53497 .67663 1.47792 .70238 .42374 66 6 .62730 .69414 .65189 1.53400 .67705 1.47699 .70281 .42286 64 7 .62770 1.59311 .65231 1.633(12 .67748 1.47607 70325 1.42198 63 8 .62811 1.59208 65272 1.63205 .67790 1.47514 .70368 1.42110 62 9 .62852 1.69105 65314 1.63107 .67832 1.47422 .70412 142022 61 10 .62892 1.59002 .65355 1.53010 .67875 1.47330 70455 1.41934 60 11 .62933 1.68900 65397 1.62913 .67917 1.47238 70499 1.41847 49 12 .62973 .58797 65438 1.52816 .67960 1.47146 .70542 1.41759 48 13 .63014 1.58695 65480 1.62719 .68002 1.47053 .70586 1.41672 47 14 .63055 1.58593 65521 1.52622 .68045 1.46962 .70629 1.41684 46 16 .63095 1.68490 65563 1.62525 .68088 1.46870 .70673 1.41497 46 16 .63136 1.68388 65604 1.52429 .68130 1.46778 .70717 1.41409 44 17 .63177 1.58286 65646 1.52332 .68173 1.46686 .70760 1.41322 43 18 .63217 1.58184 65688 1.62235 .68215 1.46595 .70804 1.41235 42 19 .63258 .58083 65729 1.52139 .68258 1.46503 .70848 1.41148 41 20 .63299 1.67981 65771 1.52043 68301 1.46411 .70891 1.41061 40 21 .63340 1.57879 .65813 1.61946 68343 1.46320 .70935 1.40974 39 22 .63330 1.67778 .65854 1.61850 .68386 1.46229 70979 1.40887 3S 23 .63421 .67676 65896 1.61764 .68429 1.46137 71023 1.40800 37 24 .63462 .67575 65938 1.61658 .68471 1.46046 .71066 .40714 36 26 .63503 .67474 65980 1.61662 .68514 1.45955 .71110 1.40627 35 26 63544 .67372 66021 1.61466 .68567 1.45864 71164 1.40540 34 27 .63584 .67271 66063 1.61370 .68600 1.45773 .71198 1.40454 33 28 .63626 .67170 66106 1.61276 .68642 1.45682 .71242 1.40367 32 29 .63666 .67069 .66147 1.61179 .68685 1.45592 .71286 1.40281 31 30 .63707 .66969 66189 1.61084 .68723 1.45501 .71329 1.40196 30 31 .63748 .66868 66230 1.60988 .68771 1.45410 .71373 1.40109 29 32 .63789 .66767 66272 1.60893 .68814 1.45320 .71417 1.40022 28 33 .63830 .56667 66314 1.60797 .68857 1.45229 .71461 1.39936 27 24 .63871 .66566 66356 1.60702 .68900 1.45139 .71606 1.39850 26 35 .63912 .56466 66398 1.50607 .68942 1.45049 .71649 1.39764 26 36 .63953 .56366 66440 1.60512 .68985 1.44958 .71593 1.39679 24 37 .63994 .56265 664 S2 1.50417 .69028 1.44868 .71637 1.39593 23 38 .64035 .66165 66524 1.50322 .69071 1.44778 .71681 1.39507 23 39 .64076 .5606o 66566 1.50228 .69114 1.44688 71725 1.39421 21 40 .64117 .55966 66608 150133 .69157 1.44598 71769 1.39336 20 41 .64158 .55866 66650 1.50038 .69200 1.44508 71813 1.39250 19 42 .64199 .65766 666D2 1.49944 .69243 1.44418 .71867 1.39165 18 43 .64240 .65666 .66734 1 49849 .692S6 1.44329 .71901 1.39079 17 44 .64281 .65567 66776 1 49755 .69329 1.44239 .71946 1.38994 16 45 .64322 .66467 .66818 1.49661 .69372 1.44149 .71990 1.38909 16 46 .64363 .65368 .66860 1.49566 .69416 1.44060 .72034 1.38824 14 r .64404 .55269 .66902 1.49472 .69459 1.43970 .72078 1.38738 13 48 .64446 .65170 .66944 1.49378 .69502 1.43881 .72122 1.38653 11 49 .64487 .65071 .66986 1.49284 .69545 1.43792 .72167 1.38568 11 60 .64528 .54972 .67028 1.49190 .69588 1.43703 .72211 1.38484 10 61 .64569 .54873 .67071 1.49097 .69631 1.43614 72255 1.38399 9 62 .64610 .54774 .67113 1.49003 .69675 1.43525 72299 1.38314 8 63 .64652 .54675 .67155 1.48909 .69718 1.43436 .72344 1.38229 7 54 .64693 .64576 .67197 1.48816 .69761 1.43347 .72388 1.38145 6 66 .64734 .64478 .67239 1.48722 .69804 1.43258 .72432 1.38060 6 66 .64775 .54379 .67282 1.48629 .69847 1.43169 .72477 1.37976 4 67 .64817 .64281 .67324 1.48536 .69^91 1.43080 .72521 1.37891 3 68 .64868 1.64183 .67366 1.48442 .69934 1.42992 .72565 1.37807 a 69 .64899 1.64085 .67409 1.48349 .69977 1.42903 .72610 1.37722 i 60 .64941 1.53986 .67451 1.48256 .70021 1.42815 .72654 1.37638 IT Ootaiig. Tang. Cotang. -TS5T Cotaug. Tang. Cotaug. ?*" s: 6 70 54 JO 5 BO 5' 10 TABLE IV. NATURAL TANGENTS AND COTANC.EN' 3 GO 3 T 3 8 31 DO M. Tmng. Cotang. Tang. Cotang. Tung. Cotang. Tang. Cotang. M. ~0 .72654 .37638 .75355 .32704 .78129 1.27994 .80978 1.23490 60 .72699 .37554 .75401 .32624 .78175 1.27917 .81027 1.23416 69 .72743 .37470 .75447 .32544 .78222 1.27841 .81075 1.23343 68 .72788 .37386 .75492 .32464 .78269 1.27764 .81123 1.23270 67 .72832 .37302 .75538 .32384 .78316 1.27683 .81171 1.23196 56 .72877 .37218 .75584 .32304 .78363 1.27611 .81220 1.23123 55 .72921 .37134 .75629 .32224 .78410 1.27535 .81268 1.23050 64 .72966 .37050 .75675 .32144 .78457 1.27458 .81316 1.22977 63 .73010 .36967 .75721 .32064 .78504 1.27382 .81364 1.22904 62 .73055 .36883 .75767 31984 .78551 1.27306 .814(3 1.22831 51 10 .73100 .36800 .75812 .31904 .78598 1.27230 .81461 1.22758 60 11 .73144 .36716 .75858 .31825 .78645 1.27i53 .81510 1.22685 49 12 .73189 .36633 .75904 .31745 .78692 1.27077 .81558 1.22612 48 13 .73234 .36549 .75950 .31666 .78739 1.27001 .81606 1.22539 47 14 .73278 .36466 .75996 .31586 .78786 1.26925 .81655 1.22467 46 15 .73323 .36383 .76042 .31507 .78834 1.26849 .81703 1.22394 45 16 .73368 .36300 .76088 .31427 .78881 1.26774 .81752 1.22321 44 17 .73413 .36217 .76134 .31348 .78928 1.26698 .81800 1.22249 43 18 .73457 .36134 .76180 .31269 .78975 1.26622 .81849 1.22176 42 19 .73502 .36051 .76226 .31190 .79022 1.26546 .81898 1.22104 41 20 .73547 .35968 .76272 .31110 .79070 1.26471 .81946 1.22031 40 21 .73592 .35885 .76318 .31031 .79117 1.26395 .81995 1.21959 39 22 .73637 .35802 .76364 .30952 .79164 1.26319 .82044 1.21886 38 23 .73681 .35719 .76410 .30873 .79212 1.26244 .82092 1.21814 37 24 .73726 .35637 .76456 .30795 .79259 1.26169 .82141 1.21742 36 25 .73771 .35554 .76502 .30716 79306 1.26093 .82190 1.21670 36 26 .73816 .35472 .76548 .30637 .79354 1.26018 82238 1.21598 34 27 .73861 .35389 .76594 .30558 .79401 1.25943 .82287 1.21526 33 28 .73906 .35307 .76640 .30480 .79449 1.25867 .82336 1.21454 32 29 .73951 .35224 .76686 .30401 .79496 1.25792 .82385 1.21382 31 30 .73U96 .35142 .76733 .30323 .79544 1.25717 .82434 1.21310 30 31 74041 1.35060 .76779 .30244 .79591 1.25642 .82483 1.21238 29 32 .74086 1.34978 .76825 .30166 .79639 1.25567 82531 1.21166 28 33 74131 1.34896 .76871 .30087 .79686 1.25492 .82580 1.21094 27 34 74170 1.34814 .76918 .30009 .79734 1.25417 82629 1.21023 26 35 .74221 1.34732 .76964 .29931 .79781 1.25343 .82678 1.20951 26 36 .74267 1.34650 .77010 1.29853 .79829 1.25268 .82727 1.20879 24 37 74312 1.34568 .77057 1.29775 .79877 1.25193 .82776 1.20808 23 38 74357 1.34487 .77103 1.29696 .79924 1.25118 .82825 1.20736 22 39 .74402 1.34405 .77149 1.29618 .79972 1.25044 .82874 1.20665 21 40 .74447 1.34323 .77196 1.29541 .80020 1.24969 82923 1.20593 20 41 .74492 1.34242 .77242 1.29463 .80067 1.24895 .82972 1.20522 19 42 .74538 1.34160 .77289 1.29385 .80116 1.24820 .83022 1.20451 18 43 .74583 1.34079 .77335 1.29307 ,80163 1.24746 .83071 1.20379 17 44 .74628 1.33998 .77382 1.29229 .80211 1.24672 .83120 1.20308 16 45 .74674 1.33916 .77428 1.29152 ,80258 1.24597 .83169 1.20237 16 46 74719 1.33835 .77476 1.29074 .80306 1.24523 .83218 1.20166 14 47 .74764 1.33754 .77521 1.28997 .80354 1.24449 .83268 1.20095 13 48 .74810 1.33673 .77568 1.28919 .80402 1.24375 .83317 1.20024 12 49 .74855 1.33592 .77615 1.28842 .80450 1.24301 .83366 1.19953 11 60 .74900 1.33511 .77661 .28764 .80498 1.24227 .83415 1.19882 10 51 .74946 1.33430 .77708 .28687 .80546 1.24153 .83465 1.19811 9 62 .74991 1.33349 .77754 .28610 .80594 1.24079 .83514 1.19740 8 63 .75037 1.33268 .77801 .28533 .80642 1.24005 .83564 1.19669 7 64 .76082 1.33187 .77848 .28456 .80690 1.23931 .83613 1.19599 6 65 .75128 1.33107 .77895 .28379 .80738 1.23858 .83662 1.19528 6 66 .75173 1.33026 .77941 .28302 .80786 1.23784 .83712 1.19457 4 57 .75219 1.32946 .77988 1.28225 .80834 1.23710 .83761 1.19387 3 68 .76264 1.32865 .78035 1.28148 .80882 1.23637 .83811 1.19316 2 69 .75310 1.32785 .78082 1.28071 .80930 1.23563 .83860 1.19246 1 60 .75355 1.32704 .78129 .27994 .80978 1.23490 .83910 1.19175 M. Ootang. Tang. Cotaiig. Tang. Cotaug. Tang. Cotang. Tang. M. e 30 58 5 1 5 QO I'lKI TAKLE IV. NATURAL TANGENTS AND COTANGENTS. 4O 410 4o 430 M. Tang. Cotang. Tang. Cotang. Tang. Cotang. Tang. Cotang. M. ~0 .83910 1.19176 .86929 .15037 .90040 " .11061 .93259 1.07237 60 j .83960 .19105 .86980 .14969 .90093 .10996 .93306 .07174 59 .84009 .19035 .87031 .14902 .90146 .10931 .93360 .07112 58 .84069 .18964 .87082 .14834 .90199 .10867 93415 .07049 57 .84108 .18894 .87133 .14767 .90251 .10602 93469 .06987 56 .84158 .18824 .87184 .14699 .903(4 .10737 93524 .06925 55 .84208 .18754 .87236 .14632 .90357 .10672 93578 .06862 54 .84258 .18684 .87287 .14565 .90410 .10607 93633 06800 53 .84307 .18614 .87338 .14498 .90463 .10543 93688 .06738 52 .84357 .18544 .87389 .14430 .90516 .10478 93742 .06676 51 10 .84407 .18474 .87441 14363 .90569 .10414 93797 .06613 50 11 .84457 .18404 .87492 .14296 .90621 .10349 93852 .06551 49 12 .84507 .18334 .87543 .14229 .90674 .10285 .93906 .06489 48 13 .84556 .18264 .87595 .14162 .90727 .10220 .93961 .06427 47 14 .84606 .18194 .87646 .14095 .90781 .10156 .94016 .06365 46 15 .84656 .18125 .87693 .14028 .90834 .10091 .94071 .06303 45 16 .84706 .18055 .87749 .13961 .90887 .10027 .94125 .06241 44 17 84756 .17986 .87801 .13894 .90940 .09963 .94180 .06179 43 18 .84806 .17916 87852 .13828 .90993 1.09899 94235 .06117 42 19 .84856 .17846 87904 .13761 .91046 1.09834 94290 .06056 41 20 84906 .17777 87955 .13694 .91099 1.09770 .94345 .05994 40 31 .84956 .17708 .88007 .13627 .91153 1.09706 94400 .05932 39 22 .85006 .17638 .88059 .13561 .91206 1.09642 94455 .05870 38 23 .85057 .17569 .88110 .13494 .91259 1.09578 .94510 .05809 37 24 .85107 .17500 .88162 .13428 .91313 1.09514 .94565 .05747 36 25 .85157 .17430 .88214 .13361 .91366 1.09450 94620 .05685 36 26 85207 .17361 .88265 .13295 .91419 1.09386 .94676 .05624 34 27 .85257 .17292 .88317 .13228 .91473 1.09322 94731 .05562 33 28 .85308 .17223 .883^9 .13162 .91526 1.09258 .94786 .05601 32 29 85358 .17154 .88421 .13096 .91580 1.09195 .94341 .05439 31 30 .85408 .17085 .88473 .13029 .91633 1.09131 .94896 1.05378 30 31 .86458 .17016 .88524 .12963 .91687 1.09067 94952 .05317 29 32 .85509 .16947 .88576 .12897 .91740 1.09003 95007 .05255 28 33 .85559 .16878 .88628 .12831 .91794 1.08940 95062 .05194 27 34 .85609 .16809 .88680 .12765 .91847 1.08876 .95118 .05133 26 35 .85660 .16741 .88732 .12699 .91901 1.08813 .95173 .05072 25 36 .85710 .16672 .88784 .12633 .91955 1.08749 95229 .05010 24 37 .85761 .16603 .88836 .12567 .92008 1.08686 .95234 .04949 23 38 .85811 .16535 .88888 .12501 .92062 1.08622 .95340 1.04888 22 39 .85862 .16466 .88940 .12435 .92116 1.08559 95395 1.04827 21 40 85912 .16398 .88992 .12369 .92170 1.08496 95451 1.04766 20 41 .85963 .16329 .89045 .12303 .92224 1.08432 .95506 1.04705 19 42 .86014 .16261 .89097 1.12238 .92277 1.08369 95562 1.04644 18 43 .86064 .16192 .89149 .12172 .92331 .08306 .95618 1.04583 17 44 .86115 .16124 .89201 .12106 .92385 1.08243 .95673 1.04522 16 45 .86166 .16056 .89253 .12041 .92439 1.08179 .95729 1.04461 15 46 .86216 .15987 .89306 .11975 .92493 1.08116 95785 1.04401 14 47 .86267 .15919 89358 .11909 .92547 1.08053 .95841 1.04340 13 48 .86318 .15851 .89410 .11844 .92601 1.07990 95897 1.04279 12 49 .86368 .15783 .89463 .11778 .92655 1.07927 95952 1.04218 11 50 .86419 .15715 .89515 .11713 .92709 1.07864 .96008 1.04158 10 51 .86470 .15647 .89567 .11648 .92763 1.07801 .96064 1.04097 52 .86521 .15579 .89620 .11582 .92817 1.07738 96120 1.04036 53 .86572 .15511 .89672 .11517 .92872 1.07676 .96176 1.03976 54 .86623 .15443 .89725 .11452 .92926 1.07613 .96232 1.03915 55 .86674 .15375 .89777 .11387 .92980 1.07550 .96288 1.03855 56 .86725 .15308 .89830 .11321 .93034 1.07487 .96344 1.03794 57 .86776 .15240 .89883 .11256 .93088 1.07425 .96400 1.03734 58 .86827 .15172 .89935 .11191 .93143 1.07362 .96457 1.03674 59 .86878 .15104 .89988 .11126 .93197 .07299 .96513 1.03613 60 .86929 1.15037 .90040 .11061 .93252 1.07237 .96569 1.03553 IT Gotaug. Tang. Cotaug. Tang. Cotang. Tang. Cotang. Tang. M~ 490 48 470 46 TABLE IV. NATURAL TANGENTS AND COTANGENTS. 4 4 1 4 ' 1 4^ 1 M. Tans. Cotang. M. M. Tang. Cotang. M. M. Tang. Cotang. M. .96569 1.03553 HO 20 .97700 1.02355 40 40 .98843 1.01170 20 1 .9(5625 1.03493 59 21 .97756 1.02295 39 41 .98901 1.01112 19 2 .96681 1.03433 58 22 .97813 1.02236 38 42 .98958 1.01053 18 3 .96738 1.03372 57 23 .97870 1.02176 37 43 .99016 1.00994 17 4 .96794 1.03312 56 24 .97927 1.02117 36 44 .99073 1.00935 16 5 .96850 1.03252 55 25 .97984 1.02057 85 45 .99131 1.00876 15 6 .96907 1.03192 54 213 .98041 1.01998 34 46 .99189 1.00818 14 .96963 1.03132 53 27 .98098 1.01939 33 47 .99247 1.00759 13 8 .97020 1.03072 52 28 .98155 1.01879 32 48 .99304 1.00701 12 9 .97076 1.03012 51 29 .98213 1.01820 31 49 .99362 1.00642 11 10 .97133 1.02952 50 30 .98270 .01761 30 50 .99420 1.00583 10 11 .97189 1.02892 49 31 .98327 .01702 M 51 .99478 .00525 9 12 .97246 1.02832 48 32 .98384 .01642 28 52 .99536 .00467 8 13 .97302 1.02772 47 33 .98441 .01583 27 53 .99594 .00408 7 14 .97359 1.02713 46 34 .98499 .01524 26 54 .99652 .00350 6 15 .97416 1.02653 45 35 .98556 .01465 25 55 .99710 .00291 5 16 .97472 1.02593 44 3fi .98613 1.01406 ?4 56 .99768 .00233 4 17 .97529 1.02533 37 .98671 1.01347 23 57 .99826 .00175 3 18 .97586 1.02474 42 38 .98728 1.01288 23 58 .99884 .00116 2 19 .97643 1.02414 41 39 .98786 1.01229 21 59 .99942 .00058 1 20 .97700 1.02355 40 40 .98843 1.01170 20 60 1.00000 1.00000 M. Cotang. Tang. 31. M, Cotang. Tang. M. M. Cotang. Tang. M. 4 5 4. > 41 i TABLE V. CUBIC YARDS PER 100 FEET. SLOPES %:!;%:!; 1 : 1; iVu : 1; 2 : 1; 3 : 1. 299 loo TABLE V. CUBIC YARDS PER 100 FEET. SLOPES */ : 1. Depth Base 12 Base H Base 16 Base 18 Base 22 Base 24 Base 26 Baso 20 1 45 53 60 68 82 90 97 105 2 93 107 122 137 167 181 196 211 3 142 163 186 208 253 275 297 319 4 193 222 252 281 341 370 400 430 5 245 282 319 356 431 468 505 542 6 300 344 389 433 522 567 611 656 7 356 408 460 512 616 668 719 771 8 415 474 533 593 711 770 830 889 9 475 542 608 675 808 875 942 1008 10 537 611 685 759 907 981 1056 1130 11 601 682 764 845 1008 1090 1171 1253 12 667 756 844 933 1111 1200 1289 1378 13 734 831 926 1023 1216 1312 1408 1505 14 804 907 1010 1115 1322 1426 1530 1633 15 875 986 1096 1208 1431 1542 1653 1764 16 948 1067 1184 1304 1541 1659 1778 1896 17 1023 1149 1274 1401 1653 1779 1905 2031 18 1100 1233 1366 1500 1767 1900 2033 2167 19 1179 1319 1460 1601 1882 2023 2164 2305 ^0 1259 1407 1555 1704 2000 2148 2296 2444 21 1342 1497 1653 1808 2119 2275 2431 2586 22 1426 1589 1752 1915 2241 2404 2567 S730 23 1512 1682 1&53 2023 2364 2534 2705 2875 24 1600 1778 1955 2133 2489 2667 2844 3022 25 1690 1875 2060 2245 2616 2801 2986 3171 26 1781 1974 2166 2359 2744 2937 3130 3322 27 1875 2075 2274 2475 2875 3075 3275 3475 28 1970 2178 2384 2593 3007 3215 3422 3630 29 2068 2282 2496 2712 3142 asse 3571 3786 30 2167 2389 2610 2&S3 3278 3500 3722 3944 31 2268 2497 2726 2956 3416 3645 3875 4105 32 2370 2607 2844 3081 3556 3793 4030 4267 33 2475 2719 2964 3208 3697 3942 4186 4431 34 2581 2833 3085 3337 3841 4093 4344 4596 35 2690 2949 3208 3468 3986 4245 4505 4764 36 2800 3067 3333 3600 4133 4400 4667 4933 37 2912 3186 3460 3734 4282 4556 4831 5105 88 3026 3307 3589 3870 4433 4715 4996 5278 39 3142 3431 3719 4008 4586 4875 5164 5453 40 3259 3556 3852 4148 4741 5037 5333 5630 41 3379 3682 3986 4290 4897 5201 5505 5808 42 3500 3811 4122 4433 5056 5367 5678 5989 43 3623 3942 4260 4579 5216 5534 5853 6171 44 3748 4074 4400 4726 5378 5704 6030 6356 45 3875 4208 4541 4875 5542 5875 6208 6542 46 4004 4344 4684 5026 5707 6048 6389 6730 47 4134 4482 4830 5179 5875 6223 6571 6919 48 4267 4622 4978 5333 6044 6400 6756 7111 49 4401 4764 5127 5490 6216 6579 6942 7305 50 4537 4907 5278 5648 6389 , 6759 7130 7500 51 4675 5053 5430 5808 6564 6942 7319 7697 52 4815 5200 5584 5970 6741 7126 7511 7896 53 4956 5349 5741 6134 6919 7312 7705 8097 54 5100 5500 5900 6300 7100 7500 7900 &300 55 5245 5653 6060 6468 7282 7690 8097 8505 56 5393 5807 6222 6637 7467 7881 8296 8711 57 5542 5964 6386 6808 7653 8075 8497 8919 58 5693 6122 6552 6981 7841 : 8270 8700 9130 59 5845 6282 6719 7156 : 8031 8468 8905 9342 60 6000 6444 6889 7333 8222 8667 9111 9556 TABLE V. CUBIC YARDS PER 100 FEET. SLOPES 301 Depth Base Base Base Base Base Base Base Base 12 14 16 18 22 24 26 23 j 46 54 61 09 83 91 98 106 g 96 111 126 141 170 185 200 215 150 17'2 194 217 261 283 306 328 4 207 237 267 296 356 385 415 444 5 269 306 343 380 454 491 528 565 6 333 378 422 467 556 600 644 689 7 402 454 506 557 661 713 765 317 8 474 533 593 652 770 830 889 948 9 550 617 683 750 883 950 1017 1083 10 630 704 778 852 1000 1074 1148 1222 11 713 794 876 957 1120 1202 1283 1365 12 800 889 978 1067 1244 1333 1422 1511 13 891 987 1083 1180 1372 1469 1565 1661 14 985 1089 1193 1296 1504 1607 1711 1815 15 1083 1194 1306 1417 1639 1750 1861 1972 . 16 1185 1304 1422 1541 1779 1896 2015 2133 17 1291 1417 1543 1669 1920 2046 2172 2298 18 1400 1533 1667 1800 2067 2200 2333 2467 19 1513 1654 1794 1935 2217 2357 2498 2639 20 1630 1778 1926 2074 2370 2519 2667 2815 21 1750 1906 2061 2217 2528 2683 2839 2994 22 1874 2037 2200 2363 2689 2852 3015 3178 23 2002 2172 2343 2513 2854 3024 3194 3365 24 2133 2311 2489 2667 3022 3200 3378 3556 25 2269 2454 2639 2824 3194 3380 3565 3750 26 2407 2600 2793 2985 3370 &5C3 3756 3948 27 2550 2750 2950 3150 &550 3750 3950 4151 28 2696 2904 3111 3319 3733 3941 4148 4356 29 2846 3061 3276 3491 3920 4135 4350 4565 30 3000 3222 3444 3667 4111 4333 4556 4778 31 3157 3387 3617 3846 4306 4535 4765 4994 32 3319 3556 3793 4030 4504 4741 4978 5215 33 3483 3728 3972 4217 4706 4950 5194 5439 34 3652 3904 4156 4407 4911 5163 5415 5667 35 3824 4083 4343 4602 5120 5380 5639 5898 36 4000 4267 4533 4800 5333 5600 5867 6133 37 4180 4454 4728 5002 5550 5824 6098 6372 38 4363 4644 4926 5207 5770 6052 6333 6615 39 4550 4839 5128 5417 5994 6283 6572 6861 40 4741 5037 5333 5630 6222 6519- 6815 7111 ' 41 4935 5239 5543 5846 6454 6757 7061 7365 42 5133 5444 5756 6067 6689 7000 7311 7622 43 5335 5654 5972 6291 6928 7246 7565 7883 44 5541 5867 6193 6519 7170 7496 7823 8148 45 5750 6083 6417 6750 7417 7750 8083 8417 46 5963 6304 6644 6985 7667 8007 8348 8689 47 6180 6528 6876 7224 7920 8269 8617 8965 48 49 6400 6624 6756 6987 7111 7350 7467 7713 ffi m 50 6852 7222 7593 7963 87G4 9074 9444 9815 51 7083 7461 7839 8217 8972 9350 9728 10106 52 7319 7704 8089 8474 9244 9630 10015 10400 53 7557 7950 I 8343 8735 9520 9913 10306 10698 54 7800 8200 8600 9000 9800 10200 10600 11000 55 8046 8454 8861 9269 10083 10491 10898 11306 56 8296 8711 9126 9541 10370 10785 11200 11615 57 8550 8972 9394 9817 10661 11083 11506 11928 58 8807 9237 9667 10096 10956 11385 ! 11815 12244 59 9069 9506 9943 10380 11254 11691 | 12128 12565 60 9333 9778 10222 10667 11556 12000 | 12444 12889 302 TABLE V. CUBIC YARDS PER 100 FEET. SLOPES 1 : 1. Depth Base 12 Base j Base 14 16 Base Base 18 20 Base Base 28 30 Base 32 1 48 56 63 ! 70 78 107 115 122 2 104 119 133 148 j 163 222 1 237 252 3 167 189 211 233 256- 344 367 889 4 237 267 296 326 356 474 504 533 5 315 352 389 426 463 611 648 685 6 loo 444 489 533 578 756 800 844 7 493 544 596 648 700 907 959 1011 6 593 652 711 770 830 1067 1126 1185 9 700 767 833 900 967 1233 1300 1367 10 815 889 963 1037 1111 1407 1481 1556 11 937 1019 1100 1181 1263 1589 1670 1752 12 1067 1156 1244 1333 1422 1778 1867 1956 13 1204 1300 1396 1493 1589 1974 2070 2167 14 1348 1452 1556 1659 1763 2178 2281 2385 15 1500 1611 1722 1833 1944 2389 2500 2611 16 1659 1778 1896 2015 2133 2607 2726 2844 17 1826 1952 2078 2204 2330 2833 2959 3085 18 2000 2133 2267 2400 2533 3067 3200 3333 19 2181 2322 2463 2604 2744 3307 3448 3589 20 2370 2519 2667 2815 2963 3556 3704 3852 21 2567 2722 2878 3033 3189 3811 3967 4122 22 2770 2933 3096 3259 3422 4074 4237 4444 23 2981 3152 3322 3493 3663 4344 4515 4685 24 3200 3378 3556 3733 3911 4622 4800 4978 25 3426 8611 3796 3981 4167 4907 5093 5278 26 3659 3852 4044 4237 4430 5200 5393 5585 27 3900 4100 4300 4500 4700 5500 5700 5900 28 4148 4356 4563 4770 4978 5807 6015 6222 29 4404 4619 4833 5048 5263 6122 6337 6552 80 4667 4889 5111 5333 5556 6444 6667 6889 31 4937 5167 5396 5626 5856 6774 7004 7233 32 5215 5452 5689 5926 6163 7111 7348 7585 33 5500 5744 5989 6233 6478 7456 7700 7944 34 5793 6044 6296 6548 6800 7807 8059 8311 35 6093 6352 6611 6870 7130 8167 8426 8685 36 6400 6667 6933 7200 7467 8533 8800 90G7 37 6715 6989 7263 7537 7811 8907 9181 9456 38 7037 7319 7600 7881 8163 9289 9570 9852 39 7367 7656 7944 8233 8522 9678 9967 10256 40 7704 8000 8296 8593 8889 10074 10370 10667 41 8048 8352 8656 8959 9263 10478 10781 11085 42 8400 8711 9022 9333 9644 10889 11200 11511 43 8759 9078 9396 9715 10033 11307 11626 11944 44 9126 9452 9778 10104 10430 11733 12059 12385 45 9500 9833 10167 10500 10833 12167 12500 12833 46 9881 10222 10563 10904 11244 12607 12948 13289 47 10270 10619 10967 11315 11663 13C56 13404 13752 48 10667 11022 11378 11733 12089 13511 13867 14222 49 11070 11433 11796 12159 12522 13974 14337 14700 50 11481 11852 12222 12593 12963 14444 14815 15185 51 11900 12278 12656 13033 13411 14922 15300 15678 52 12326 12711 13096 13481 13867 15407 15793 16178 53 12759 13152 13544 13937 14330 15900 16293 16685 54 13200 13600 14000 14400 14800 16400 16800 17200 55 13648 14056 14463 14870 15278 16907 17315 17722 56 14104 14519 14933 15348 15763 17422 17837 18252 67 14567 14989 15411 15833 16256 17944 18367 18789 58 15037 15467 15896 16326 16756 18474 18904 19333 59 15515 15952 16389 16826 17263 19011 19448 19885 60 16000 16444 16889 17333 17778 19556 20000 20444 TABLE V. CUBIC YARDS PER 100 FEET. SLOPES 1% : 1. 303 Depth Base 12 Base 14 Base 16 Base . 18 Base 20 Base 28 Base 30 Base 32 I 50 57 65 72 80 109 117 124 2 111 126 141 156 170 230 244 259 3 183 206 228 250 272 361 383 406 4 267 296 326 356 385 504 533 563 5 361 398 435 472 509 657 694 731 6 467 511 556 600 644 822 867 911 r* 583 635 687 739 791 998 1050 1102 g 711 770 830 889 948 1185 1244 1304 9 C50 917 983 1050 1116 138t 1450 1517 10 1000 1074 1148 1222 1296 1593 1667 1741 11 1161 1243 1324 1406 1487 1813 1894 1976 12 1333 1422 1511 1600 1689 2044 2133 2222 13 1517 1613 1709 1806 1902 2287 2383 2480 14 1711 1815 1919 2022 2126 2541 2644 8748 15 1917 2028 2139 2250 2361 2806 2917 3028 16 2133 2252 2370 2489 2607 3081 3200 3319 17 2361 2487 2613 2739 2865 3369 3494 3620 18 2600 2733 2867 3000 3133 3667 3800 3933 19 2850 2991 3131 3272 3413 3976 4117 4257 20 3111 3259 3407 3556 3704 4296 4444 4592 21 3383 3539 3694 3850 4005 4628 4783 4939 22 3667 3830 3993 4156 4318 4970 5133 5296 23 3961 4131 4302 4472 4642 5324 5494 5665 24 4267 4444 4622 4800 4978 5689 5867 6044 25 4583 4769 4954 5139 5324 6065 6250 6435 26 4911 5104 5296 5489 5681 6452 6644 6837 27 5250 5450 5650 5850 6050 6850 7050 7250 28 5600 5807 6015 6222 6430 7259 7467 7674 29 5961 6176 6391 6606 6820 7680 7894 8109 30 6333 6556 6778 7000 7222 8111 8333 8555 31 6717 6946 7176 7406 7635 8554 8783 9013 32 7111 7348 7585 7822 8059 9007 9244 9482 33 7517 7761 8006 8250 8494 9472 9717 9962 34 7933 8185 8437 86S9 8941 9948 10200 10452 35 8361 8620 8880 9139 9398 10435 10694 10954 36 8800 9067 9333 9600 9867 10933 11200 11467 37 9250 9524 9798 10072 10346 11443 11717 11991 38 9711 9993 10274 10556 10837 11963 12244 12526 39 10183 10472 10761 11050 11339 12494 12783 13072 40 10667 10963 11259 11556 11852 13037 13333 13630 41 11161 11465 11769 12072 12376 13591 13894 14198 42 11667 11978 12289 12600 12911 14156 14467 14778 43 12183 12502 12820 13139 13457 14731 15050 15369 44 12711 13037 13363 13689 14015 15319 15644 15970 45 13250 13583 13917 14250 14583 15917 16250 16583 46 13800 14141 14481 14822 15163 16526 16867 17207 47 14361 14709 15057 15406 15754 17146 17494 17843 48 14933 15289 15644 16000 16356 17778 18133 18489 49 15517 15880 16243 16606 16968 18420 18783 19146 50 16111 16481 16852 17223 17592 19074 19444 19815 51 16717 17094 17472 17850 18228 19739 20117 20494 52 17333 17719 18104 18489 18874 20415 20800 21185 53 17961 18354 18746 19139 19531 21102 21494 21887 54 18600 19000 19400 19800 20200 21800 22200 22600 56 19250 19657 20065 20472 20880 22509 22917 23324 56 19911 20326 20741 21156 21570 23230 23644 24059 57 20583 21006 21428 21850 22272 23961 24383 24805 58 21267 21696 22126 22556 22985 24704 25133 25563 59 21961 22398 22835 23272 23709 25457 25894 26332 60 82667 23111 23556 24000 24444 26222 26667 27111 TABLE V. CUBIC YARDS PICK 1600 27200 27600 28000 55 24852 25-259 25667 2(5074 26481 28111 28519 28926 56 25719 26133 26548 26963 27378 29037 29452 29867 57 23600 27^22 27444 27867 28289 23978 30400 30822- 58 27496 r .^26 28356 23785 29215 30933 31363 31793 59 28407 ^8844 29281 29719 30156 31904 32341 32778 60 29333 29778 30222 30667 31111 83889 38333 33778 TAIM.I-: V. CUI'.IC VARHS I'KU 1 0( ) FKKT. SLOl'F.S '>:!. Depth Base Base Base Base Base Base Base Base 12 14 16 18 20 28 30 I 32 1 56 63 70 78 85 115 122 130 1:33 148 163 178 193 252 267 281 8 233 9M 278 300 322 4lf 433 456 4 356 385 415 444 474 693 622 652 5 500 537 574 611 648 796 888 870 6 667 711 756 800 844 1022 1067 1111 7 856 907 959 1011 1'HB 1270 1322 1374 8 1067 1126 1185 1244 1304 1541 1600 1659 9 1300 1367 1433 1500 1567 1833 1900 1967 10 1556 1630 1704 1778 1852 2148 2222 2296 n 18.33 1915 1996 2078 2159 2485 2567 2648 12 2133 222 2311 2100 2489 2844 2933 3022 13 2456 2552 2648 2744 2811 3226 3322 3419 14 2800 2904 3007 3111 3215 3630 3733 3837 15 3167 8378 3389 3500 3611 4056 4167 4278 16 .3556 3674 3793 3911 4030 4504 4622 4741 17 3967 4093 4219 4344 4470 4974 5100 5226 18 4400 4533 4667 4800 4933 5467 5600 5733 19 4856 4996 5137 5278 5419 5981 6122 6263 20 5333 5481 5630 5778 5926 6519 6667 6815 21 5833 5989 6144 6300 6456 7078 7233 7389 22 6356 6519 6681 6844 7007 7659 7822 7985 23 6900 707'0 7241 7411 7581 8263 8433 8504 24 7467 7644 7822 8000 8178 8889 9067 9144 25 8056 8241 8426 8611 8796 9537 9722 9807 26 8667 8859 9052 9244 9437 10207 10400 10593 27 9300 9500 9700 9900 10100 10900 11100 11300 28 9956 10163 10370 10578 10785 11615 11822 12030 29 10633 10H48 11063 11278 11493 12352 12567 12781 30 11333 11556 11778 12000 12222 13111 13333 13556 31 12056 12285 12515 12744 12974 13893 14122 14352 32 12800 13037 13274 13511 13748 14696 14933 15170 33 13567 13811 14056 14300 14514 15522 15767 16011 34 14356 14607 14859 15111 15363 16370 16622 16874 35 15167 15426 15685 15944 16204 17241 17500 17759 36 16000 16267 16533 16800 17067 18188 18400 18667 37 16856 17130 .17404 .17678 17952 19048 19322 19596 38 17733 18015 18296 18578 18859 19985 20267 20548 39 18633 18922 19211 19500 19789 20944 212&3 21522 40 19556 19852 20148 20444 20741 21926 22222 22516 41 20500 20804 21107 21411 21715 22930 23233 23537 42 21467 21778 22089 22400 22711 23956 24267 24578 43 23456 22774 23093 23411 23730 25004 25322 25641 44 2S467 23793 24119 24444 24770 26074 26400 26726 45 24500 24833 25167 25500 35888 27167 27500 27833 46 25556 25896 26237 26578 26919 28281 28622 28963 47 26038 26981 27:330 27678 28026 29419 29767 30115 48 27733 , 28089 28444 28800 29156 30578 30933 31289 49 28856 89219 28581 29944 30307 ! 31759 32122 32485 50 30000 30370 30741 31111 31481 32963* 33333 33704 51 31167 31544 31922 32300 32678 34189 34567 34944 52 32356 32741 33126 33511 33896 35437 35822 36207 53 33567 3-3959 34352 34744 35137 36707 37100 87493 54 34800 35200 35600 36000 36400 38000 38400 38800 55 36056 31)463 36870 , 37278 37685 39315 39722 40180 56 37333 37748 38163 38578 38993 40652 41067 41481 57 38633 39056 39478 39900 40322 42011 42433 42856 58 39956 40385 40815 41244 41674 4a393 43822 44252 59 41300 41737 42174 42611 43048 44796 45233 45670 60 42667 43111 43556 44000 44444 46222 46667 j 47111 INDEX. Acre in square chains 18 in square feet 18 in square meters 18 in square poles 18 in square varas 18 in square yards 18 Additions, city 173 Adjustments, axis of revolu- tion 36 bubble tube 106 compass 36 compass needle 36 compass pivot 37 compass plate bubble 36 cross-wire 106 plane of sights 37 transit line of sights 46 transit plate levels 46 wyes 107 Agonic line 29 Alidade 38 Angles by repetition 45 Angular convergence 196 Application of 57.3 rule 20 Approximate traversing 76 Approximations in stadia 124 Area, by coordinates 75 of farm 67 of triangle 19 table 68 Attachment, compass , 40 Attraction, local 35 Average end areas 157 formula 158 Azimuth 25 by sun 54 formula 53 Back sights 102 Balancing a survey 64 rule, no latitude 72 Bearing . 25 and length lost ; 85 how read 27 lost 89 magnetic 33 of line 25 true 25 Bench marks 104 Horm 148 Bibliography, city surveying. 189 compass surveying 37 division, of land 94 earthwork 168 lettering 195 topography 127 Blocks, city 180 irregular 182 Borrow pits 148, 164 Boundaries, irregular 78 Breaking chair 8 Bubble tube adjustment 106 radius 107 Cabinets for drawings 188 Cases for city drawings 182 Chain, breaking 8 engineer's ; . . . 2 Gunter's 1 problems 10 surveying .' . . . 9 vara ' Chaining 7 over hills 9 over valleys 9 Chainman, head 8 rear 8 Chainmen 7 Changes in declination 31 Chart, isogonic 30 Circular curve cross-section. 185 Circular curves, vertical 112 Circumpolar stars 51 City additions 173 blocks 180 contracts 189 datum 178 engineer 171 engineer's notes 183 engineer's records 185 field notes 186 orders 189 surveying 171 Colby's slide rule 123 Compass 25 adjustments 36 attachment 39 bibliography 37 tripod 26 use of 27 vernier 30 Concrete monuments 174 Convergence, angular .196 linear . . . 197 of meridians 196 Correction for erroneous areas 16 lengths 16 plot 194 pull 13 sag 1 f 1 temperature 13 307 308 1XDEX. Courses of no departure 72 no latitude 72 Cox's stadia computer 122 Cross-section, streets 184 railroad 152 Cross-wire adjustment 106 Culmination, lower 52 upper 52 Curvature of earth 109 refracted rav 109 Curve at sag Ill summit 110 Curves, parabolic 110 vertical 110 Cuts, side-hill 1 57 Data for city map 179 on land plots 190 Datum for city 1 <8 piano 103 Declination changes 31 for farm 34 how set off 31 on vernier 29 Degree formula 129 of curve 129 Departure, definition 61 how found 61 Detail maps for city 1 .187 Diagonal prism 56 Dipper, the 51 Discrepancies in survey.... 79 Dividing land 90 Division of quadrilateral 83 of township 203 of triangle 81 Double meridian distance... 66 Drawing cabinets 188 Earthwork , 151 bibliography 1 68 examples ' 159 note book : . . . 162 special 164 Effect df refraction 109 Elasticity, modulus of 14 Elevation 103 Elongation, cast 52 west 52 End areas 157 End of fill 166 Engineer's chain 2 Erroneous areas 16 lengths 15 Error, of closure 63 External 130 Eye-piece of telescope 98 Feet to varas 18 Field notes for city 186 for farm 36 for U. S. survey 205 Foot curves 135 Foresights 102 Formula for area of triangle. 19 for azimuth 53 for length correction 16 for oblique triangle 209 1'or offsets 198 prismoidal 15.! Formulas, approximate 133 for right triangle 208 Freehaul 167 General formulas, railroad. . .130 maps for city 179 solution for division 84 Government surveying 196 Grade 163 point 164 Great Bear 52 Gunter's chain 1 Gunter's chain to varas 18 Hand-level 142 topography 126, 143 Height of instrument 102 Hook's law 14 Hubs 141 Inclined sights 117 Intersections 23 Irregular boundaries 78 blocks ...- 182 section 154 Isogonic chart 30 .Jacob's staff 27 Labor 18 Land plots 190 Latitude, definition 61 how found 61 Laying- out curve 130 League, Spanish 18 Length of two courses 86 of curve 131 Lenses of eye-piece 98 Lettering 194 books 195 Level note books 105 sections 153 the wye 96 Leveling, theory of 102 Linear convergence 197 units 17 Local attraction 35 Location field book 138 by off-sets 132 of houses 22 of meridian 48 of meridian by Polaris 52 1 survey 136 INDEX. 309 Lost parts 85 Lots, rectangular .181 Lower motion of transit 42 Magnetic bearing 33 needle 25 Maps, detail for city 187 for city 179 Meanders 78 Meridian by Polaris 48 by sun 54 distance 66 reference 200 without calculation.; 57 Metallic Tapes 5 Meters to varas 18 Methods of plotting 190 Metric c-irves 135 Middle ordinate 133 Modulus of elasticity 14 A'fonuments, kinds .173 for city surveying 172 location of 175 necessity for 172 Motion, lower 42 upper 42 Napier's laws 210 Needle, magnetic 25 New York rod 100 Note-book for earthwork. .. .162 for level 105 for transit 137, 139 Object glass 98 Objective of telescope 97 Objects of city survey. ...;. .171 Oblique trianglo 208 spherical triangle 210 Obstacles 132 Off-sets, examples 200 in government surveying. .198 intermediate ' 199 location by 132 Old lines, how run 33 Outs IS Overhaul 166 Pacing survev 21 Parallels, how run 198 of latitude 196 standard 200 Parts of compass 25 of level 96 of transit 38 Peg adjustment of transit... 48 Philadelphia rod 100 Pins, surveying 5 Pit, borrow 164 Pivot adjustment 37 Plot correction 194 of fsirni . -.193 Plots 190 and lettering 190 Plotting by co-ordinates. .. .192 by latitudes and depar- tures 191 by sines 191 by tangents 191 Plumb-bob 6 Point of curvature (P.C.) 130 of intersection (P. I.) 130 of tangency (P.T.) 136 Polaris 51 Poles to varas 18 range 6 Preliminary earthwork esti- mates 160 note-book 136 survey 136 Prescriptive rights 184 Primary triangulation 120 Principal focus 115 focal distance 115 Prismoid 151 Prismoidal formula 151 Private notes 183 Profile ; 106 Protractor 188 Protractor plotting 190 Pull on tape 13 Radius of bubble tube 101 of curve 129 of parallels 196 of street cross-section. . . .185 Railroad curve 129 excavation 152 surveying 129 Range lines 23 poles . 6 Ranges 202 Reading bearings 27 compass vernier 31 of transit vernier 30 rod vernier 29 Records, city engineers 185 Rectangular blocks 180 lots 181 off-sets 23 Reduction method 122 tables 134 Reference lines 44 meridians 200 Refraction 55 effect of 109 Reinhr.rdt's lettering 195 Repeating method for angles 45 Result of declination changes 32 Reticule 41 Right angle by chain 10 plane triangle 208 spherical triangle 210 INDEX. Rights, prescriptive 184 Rod, New York ..100 Philadelphia 100 self -reading 1-02 stadia 118 Kuie tor balancing 65 for borrow pits 165 for D.M.D 66 for earthwork 155 for finding area 67 for setting slope stakes... 145 of 57.3 20 Running parallels 198 Sag correction 15 Secant method 199 Sections, irregular 154 level 153 three-level 154 two-level 153 Self-reading rod ......102 Setting declination 31 up level 100 up transit 41 Shifting center 41 Shrinkage in earthwork 167 Side-hill cut 157 Slide rule. Colby's 123 Slope-stakes in cut 144 in fill 145 on level 145 Solar attachment 55 Spanish labor 18 league 18 Special case of earthwork. . .164 Square chains in acre 18 feet in acre 18 poles in acre 18 varas in acre 18 yards in acre 18 Stadia computer 123 formulas 115 rod 118 stations 120 Stakes for railroad survey. 141 Standard parallels 200 Standardized tapes 4 Steel tapes 3 Stone monuments 174 Street cross-section 184 Survey by pacing 21 discrepancies 79 of farm by pace 23 topographic 114 Surveying by transit 43 city 171 Surveyor's compass 25 pins 5 Table for area 68 for level sections 161 traverse Tangent method 62 199 of plotting ................ 191 Tapes for city surveying . ..175 metallic 5 standardized 4 steel 3 Telescope 101 Temperature correction 73 The 57.3 rule 19 Theory of leveling 1 02 Three-level sections ir>4 Tiers 202 Topographic field work 119 survey 114 Topography by hand level ..126 by stadia 115 Township division 203 Townships 202 Transit 38 as compass 43 essential parts 38 for city use 176 party .140 plate levels adjusted 4fi surveying 43 topography 114 vernier 43 Traverse tables 62 Traversing 7(5 approximate 7H Trees, fore and aft .,f> line 3f : witness 35 Triangle area of 19 oblique 209 PZS : 52 Trigonometric formulas 208 Tripod, compass 26 Two-level sections 153 Unit pull 13 stress 14 stretch 13 Units of land measure 18 Upper motion of transit 42 Use of compass 27 of transit 42 Vara 18 Vara chain 3 Vernier, compass 30 rod 28 transit 43 Vertical circle 40 circular curves 112 curves 110 Wire interval 116 Witness trees 35 Witnessing a corner 35 Wye adjustment 107 level .' 96 Yard 17 Yards to varas 18 96 I UNIVERSITY) THIS BOOK O ^ " XB II 059 179777