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SUEVEYING 
 
 AND 
 
 TRAVERSE TABLE 
 
 .^' 
 
 G. A. WENTWORTH, A.M. 
 
 AUTHOR OF A SERIES OF TEXT-BOOKS IN MATHEMATICS 
 
 REVISED EDITION 
 
 FFR 221896 
 
 Boston, U.S.A., and London 
 
 GINN & COMPANY, PUBLISHERS 
 
 1896 
 
 V 
 

 Entered, according to Act of Congress, in the year 1882, by 
 
 G. A. WENTWORTH 
 in the Oflfice of the Librarian of Congress, at Washington. 
 
 Copyright, 1895, by G. A. Wentworth. 
 
 
CONTENTS. Vll 
 
 SURVEYING. 
 
 CHAPTER I. Definitions. Instrujients and Their Uses: 
 
 Definitions, 193 ; instruments for measuring lines, 194 ; chaining, 
 194 ; obstacles to chaining, 196 ; the surveyor's compass, 198 ; uses 
 of the compass, 200 ; verniers, 203 ; the surveyor's transit, 207 ; uses 
 of the transit, 208 ; the theodolite, 208 ; the railroad compass, 208 ; 
 plotting, 211. 
 
 CHAPTER II. Land Surveying: 
 
 Determination of areas, 213 ; rectangular surveying, 217 ; field 
 notes, computation, and plotting, 218 ; supplying omissions, 222 ; 
 irregular boundaries, 222 ; obstructions, 222 ; modification of the 
 rectangular method, 225 ; variation of the needle, 226 ; methods of 
 establishing a true meridian, 227 ; dividing land, 231 ; United States 
 public lands, 236 ; Burt's solar compass, 237 ; laying out the public 
 lands, 239 ; Plane-table surveying, 241 ; the three-point problem, 246. 
 
 CHAPTER III. Triangulation : 
 
 Introductory remarks, 247 ; the measurement of base lines, 248 ; 
 the measurement of angles, 249. 
 
 CHAPTER IV. Levelling: 
 
 Definitions, 250 ; the Y level, 251 ; the levelling-rod, 251 ; differ- 
 ence of level, 252 ; levelling for section, 256 ; substitutes for the 
 Y level, 260 j topographical levelling, 263. 
 
 CHAPTER V. Railroad Surveying : 
 
 General remarks, 264 ; cross-section work, 265 ; railroad curves, 
 265. 
 
SUEYEYII^G. 
 
 CHAPTER I. 
 
 DEFINITIONS. INSTRUMENTS AND THEIR USES. 
 
 § 1. Definitions. 
 
 Surveying is the art of determining and representing dis- 
 tances, areas, and the relative position of points upon the 
 surface of the earth. 
 
 In plane surveying, the portion surveyed is considered as a 
 plane. 
 
 In geodetic surveying, the curvature of the earth is regarded. 
 
 A Plumb-Line is a cord with a weight attached and freely 
 suspended. 
 
 A Vertical Line is a line having the direction of the plumb- 
 line. 
 
 A Vertical Plane is a plane embracing a vertical line. 
 
 A Horizontal Plane is a plane perpendicular to a vertical 
 line. 
 
 A Horizontal Line is a line in a horizontal plane. 
 
 A Horizontal Angle is an angle the sides of which are in a 
 horizontal plane. 
 
 A Vertical Angle is an angle the sides of which are in a 
 vertical plane. If one side of a vertical angle is horizontal, 
 and the other ascends, it is an angle of elevation ; if one side 
 is horizontal, and the other descends, it is an angle of depression. 
 
 The Magnetic Meridian is the direction which a bar-magnet 
 assumes when freely supported in a horizontal position. 
 
194 SURVEYING. 
 
 The Magnetic Bearing of a line is the angle it makes with 
 the magnetic meridian. 
 
 Surveying commonly comprises three distinct operations ; 
 viz. : 
 
 1. The Field Measurements, or the process of determining 
 by direct measurement certain lines and angles. 
 
 2. The Computation of the required parts from the measured 
 lines and angles. 
 
 3. The Plotting, or representing on paper the measured 
 and computed parts in relative extent and position. 
 
 THE MEASUREMENT OF LINES. 
 
 § 2. Instruments for Measuring Lines. 
 
 The Gunter's Chain is generally employed in measuring land. 
 It is 4 rods, or 66 feet, in length, and is divided into 100 links. 
 Hence, links may be written as hundredths of a chain. 
 
 The Engineer's Chain is employed in surveying railroads, 
 canals, etc. It is 100 feet long, and is divided into 100 links. 
 
 A Tape Measure, divided into feet and inches, is employed 
 in measuring town-lots, cross-section work in railroad sur- 
 veying, etc. 
 
 In the United States Coast and Geodetic Survey, the meter 
 is the unit ; and, when great accuracy is required, rods placed 
 end to end, and brought to a horizontal position by means of 
 a spirit-level, are employed in measuring lines. 
 
 § 3. Chaining. 
 
 Eleven tally-pins of iron or steel are used in chaining ; also, 
 one or more iron-shod poles called flag-staffs or range poles. 
 
 A forward chainman, or leader, and a hind chainman, or 
 follower, are required. A flag-staff having been placed at the 
 farther end of the line, or at some point in the line visible 
 
CHAINING. 
 
 195 
 
 from the beginning, the follower takes one end of the chain, 
 and a pin which he thrusts into the ground at the beginning 
 of the line. The leader moves forward in the direction of 
 the flag-staff, with the other end of the chain and the remaining 
 ten pins, until the word "halt" from the follower warns him 
 that he has advanced nearly the length of the chain. 
 
 At this signal he stops, and the follower, meanwhile having 
 placed his end of the chain at the beginning of the line, directs 
 the leader by the words "right" or "left" until the chain is 
 exactly in line with the flag-staff. This being accomplished, 
 and the chain stretched tightly in a horizontal position, the 
 follower says, " down." The leader then puts in a tally-pin 
 exactly at the end of the chain, and answers, "down"; after 
 which the follower withdraws the pin at the beginning of the 
 line, and the chainmen move forward until the follower nears 
 the pin set by the leader. The follower again says, " halt," 
 and the operation just described is repeated. This process is 
 continued until the end of the line is reached. 
 
 If the tally-pins in the hands of the leader are exhausted 
 before the end of the line is reached, when he has placed the 
 last pin in the ground, he waits until the follower comes up to 
 him. The follower gives the leader the ten pins in his hand, 
 and records the fact that ten chains have been measured. The 
 measuring then proceeds as before. If the distance from the 
 last pin to the end of the line is less than a chain, the leader 
 places his end of the chain at the end of the line, and the 
 follower stretches tightly such a part of the chain as is 
 necessary to reach to the last pin, and the number of links is 
 counted. The number of whole chains is indicated by the 
 number of pins in the hands of the follower, the last pin 
 remaining in the ground. 
 
 In measuring, the chain must be held in a horizontal 
 position. If the ground slopes, one end of the chain must be 
 raised until the horizontal position is attained. By means of 
 
196 SURVEYING. 
 
 a plumb-line, or a slender staff, or, less accurately, by drop- 
 ping a pin (heavy end downwards), the point vertically under 
 the raised end of the chain may be determined. If the slope 
 is considerable, half a chain or less may be used. 
 To construct a perpendicular with a chain: 
 
 1. When the point through which the perpendicular is to 
 pass is in the line : 
 
 Let AB (Fig. 1) represent the line, and P the point. Measure from 
 
 P to the right or left, PC = 40 links, 
 
 ^ and place a stake at C. Let one end of 
 
 v^y'' I the chain be held at P, and the end of 
 
 ^V'^ xj 5 the eightieth link at G ; then, taking the 
 
 y ■ , •^j chain at the end of the thirtieth link from 
 
 ~ '• ' P, draw it so that the portions DC 
 
 AC P B 
 
 Pj^ - and DP are tightly stretched, and place 
 
 a stake at D. DP will be the perpen- 
 dicular required. (Why ?) 
 
 2. When the point is without the line : 
 
 Let AB (Fig. 1) be the line, and D the point. Take C any point in 
 the line, and stretch the chain between D and C ; then, let the middle of 
 the part of the chain between C and D be held in place, and swing the 
 end at D around until it meets the line in P. DP will be the perpen- 
 dicular required. (Why ?) 
 
 § 4. Obstacles to Chaining. 
 1. When a tree, building, or other obstacle is encountered 
 in measuring or extending a line, it may be passed by an offset 
 in the following manner : 
 
 To prolong the line AB' past a building (Fig. 2). At B erect BE per- 
 
 ^ pendicular to AB \ 
 
 ^ S- ?^ — /o"^ — ? ^ P at E erect EF per- 
 
 . I _ ^"^V ; ! pendicular to BE ; 
 
 E E^ F F' B,t F erect FC = BE 
 
 ^^^' ^- perpendicular to 
 
 EF: then, CD perpendicular to FC will be in the required line, and 
 
 AB 4- EF -\- CD = AD. By constructing two other perpendiculars, 
 
 B'E' and P'C, the accuracy of the work will be increased. 
 
OBSTACLES TO CHAINING. 
 
 197 
 
 2. To measure across a body of water : 
 
 Let it be required to measure the line ABCD (Fig. 3) crossing a river 
 between B and C. Measure BE — 400 links ; at E erect the perpendic- 
 ular EF =■ 600 links ; at B erect the perpendicular BG = 300 links. 
 Place a stake at C, the intersection of AD and FG beyond the river. 
 
 Fig. 3. 
 Then BC - 400 links. For, by similar triangles, EF .BG..CE: CB. 
 But EF = 2BG; hence, CE = 2 CB, and CB = BE = 400 links. EG 
 and FG should be measured, in order to test the accuracy of the work. 
 EG = FG = 500 liiiks. 
 
 Instead of the above distances, any convenient distanc es may be ta ken. 
 For, itEF = 2 BG, then CB = BE, and EG = FG= \JeB^ + BG^. 
 
 3. To measure a line the end of which is invisible from the 
 beginning, and intermediate points unknown : 
 
 F 
 A^ 'iG B 
 
 E ^---^^^ ; 
 
 Fig. 4. 
 Let AB (Fig. 4) represent the line. Set up a flag-staff at D 
 B and visible from A. From B let fall BC perpendicular to AD. 
 ure^CandJ5C. Then 
 
 AB 
 
 beyond 
 Meas- 
 
 = \/aC^ + BC^. 
 To find intermediate points on AB : 
 
 At any point ^ on ^ C erect EF perpendicular to A C, and determine 
 EG by the proportion AC -.CB ::AE : EG. G will be a point on AB. 
 The line AD is called a Random Line. 
 
198 SURVEYING. 
 
 THE MEASUREMENT OF ANGLES. 
 
 § 5. The Surveyor's Compass.* 
 
 The Surveyor's Compass is shown on the following page. 
 
 The compass circle is divided into half-degrees, and is fig- 
 ured from 0° to 90° each way from the north and south points. 
 In the centre of the compass circle is the pivot which supports 
 the magnetic needle. The needle may be lifted from the pivot 
 by a spring and pressed against the glass covering of the 
 compass circle, when the instrument is not in use. The main 
 plate moves around the compass circle through a small arc, 
 read by the vernier, for the purpose of allowing for the varia- 
 tion of the needle (§ 23). The sight standards at the extremi- 
 ties of the main plate have fine slits nearly their whole length, 
 with circular openings at intervals ; on the edges of the north 
 standard are tangent scales for reading vertical angles ; and 
 on the outside of the south standard are two eye-pieces at the 
 same distance from the main plate as the zeros of the tangent 
 scales, respectively. The telescopic sight (a recent improve- 
 ment by the Messrs. Gurley), consists of a small telescope 
 attached to the south standard. The main plate is furnished 
 with two spirit levels at right angles, and turns horizontally 
 upon the upper end of the hall spindle, the lower end of which 
 rests in a spherical socket in the top of the tripod or Jacobs 
 staff which supports the instrument. From the centre of the 
 plate at the top of the tripod a plummet is suspended by 
 which the centre of the compass can be placed directly over a 
 definite point on the ground. 
 
 *The instruments described on this and the following pages are 
 adjusted by the maker. If they should require re-adjustment, full 
 directions will he found in the manual furnished with the instruments. 
 
 The manual published by Messrs. W. & L. E. Gurley, Troy, N. Y., 
 has been freely used, by permission, in describing these instruments. 
 
THE SURVEYOR'S COMPASS. 
 
 Note. The letters E and W on the face of the compass are reversed 
 from their true positions. The reason for this is that if the sights are 
 turned towards the west, the north end of the needle is turned towards 
 the letter W, and if the north end of the needle is turned towards E, the 
 sights are turned towards the east. 
 
 If the north end of the needle poii^ts exactly towards E or W, the 
 sights will range east or west. 
 
INSTRUMENTS AND THEIR USES. 
 
 201 
 
 § 6. Uses of the Compass. 
 
 To take the bearing* of a line. Place the instrument so that 
 the plummet will be directly over one end of the line, and level 
 by pressing with the hands on the main plate until the bubbles 
 are brought to the middle of the spirit levels. Turn the south 
 end of the instrument toward you, and sight at the flag-staff 
 at the other end of the line. Eead the bearing from the north 
 end of the needle. First, write N. or S. according as the north 
 end of the needle is nearer N. or S. of the compass circle ; 
 secondly, write the number of degrees between the north end 
 of the needle and the nearest zero mark; and thirdly, write 
 E. or W. according as the north end of the needle is nearer 
 E. or W. of the compass circle. 
 
 In Fig. 5 the bearing would be N. 45° W. 
 In Fig. 6 the bearing would be S. 45° W. 
 ' In Fig. 7 the bearing would be S. 30° E. 
 In Fig. 8 the bearing would be N. 60° E. 
 
 If the needle coincides with the N.S. or E.W. line, the bear- 
 ing would be N., S., E., or 
 W., according as the north 
 end of the needle is over 
 N., S., E., or W. 
 
 As the compass circle is 
 divided into half-degrees, 
 the bearing may be deter- 
 mined pretty accurately to 
 quarter-degrees. 
 
 When a fence or other 
 obstruction interferes with 
 placing the instrument over 
 the line, it may be placed 
 
 at one side, the flag-staff fig. 7. fig. a 
 
 being placed at an equal distance from the line. 
 
 * The magnetic bearing is meant unless otherwise specified. 
 
202 SURVEYING. 
 
 Local Disturbances. Before a bearing is recorded, care 
 should be exercised that the chain, pins, and other instruments 
 that would affect the direction of the needle, are removed from 
 the vicinity of the compass. Even after the greatest care in 
 this respect is exercised, the direction of the needle is often 
 affected by iron ore, ferruginous rocks, etc. 
 
 Reverse Bearings. When the bearing of a line has been 
 taken, the instrument should be removed to the other end of 
 the line and the reverse bearing taken. The number of degrees 
 should be the same as before, but the letters should be reversed. 
 
 To take the bearing of a line one end of which cannot be seen 
 from the other. Eun a random line (§ 4, 3); then (Fig. 4), 
 
 whence the angle CAB may be found. This angle combined 
 with the bearing of the random line will give the 'bearing 
 required. 
 
 Another method will be given in § 19. 
 
 To measure a horizontal angle by means of the needle. Place 
 the compass over the vertex of the angle, take the bearing of 
 each side separately, and combine these bearings. 
 
 To measure angles of elevation. Bring the south end of the 
 compass towards you, place the eye at the lower eye-piece, 
 and with the hand hold a card on the front side of the north 
 sight, so that its top edge will be at right angles to the divided 
 edge and coincide with the zero mark; then, sighting over the 
 top of the card, note upon a flag-staff the height cut by the 
 line of sight ; move the staff up the elevation, and carry the 
 card along the edge of the sight until the line of sight again 
 cuts tlie same height on the staff; read off the degrees of the 
 tangent scale passed over by the card. 
 
 To measure angles of depression. Proceed in the same man- 
 ner as above, using the eye-piece and tangent scale on the oppo- 
 site sides of the sights, and reading from the top of the sight. 
 
INSTRUMENTS AND THEIR USES. 
 
 203 
 
 T^o = To'o of a foot. 
 
 § 7. Verniers. 
 
 First form. Let AB (Fig. 9) represent a portion of a rod 
 for measuring heights (§ 32). The graduation to feet and 
 hundredths of a foot begins at the lower end, which rests on 
 the ground when the rod is in use. The line 
 extending nearly across the rod at the bottom 
 of the portion shown marks the beginning of 
 the fourth foot. The face of the rod is 
 divided into four columns : in the first is 
 written the number of feet ; in the second, 
 the number of tenths ; and in the third, the 
 number of hundredths. 
 
 It is evident that, with the arrangement 
 just described, heights could be measured 
 only to hundredths of a foot. To enable us 
 to find the height more precisely, a contri- 
 vance called a Vernier is used. This is shown 
 at the right of the rod. It consists of a piece 
 of metal or wood, the graduated part of 
 which is yy^ of a foot in length; and this 
 is divided into ten equal parts. Hence, one 
 division of the vernier = yL of yLy = y^^o 
 of a foot ; and one division of the vernier 
 exceeds one division of the rod by yii^ ~ 
 
 The vernier slides along the face or side 
 of the rod. 
 
 To use the vernier, place the lower end of the rod upon the 
 ground, and move the vernier until its index or zero mark is 
 opposite the point whose distance from the ground is desired. 
 In the figure, the height of the index of the vernier is evidently 
 4.16 feet, increased by the distance of the index above the next 
 lower line (4.16) of the rod. We shall now determine this 
 distance. 
 
204 
 
 SURVEYING. 
 
 Observe which line of the vernier is exactly opposite a line 
 of the rod. In this case, the line of the vernier numbered 7 
 is opposite a line of the rod. Then, since each division of the 
 vernier exceeds each division of the rod by j J^ ^ of a foot, 
 
 C of the vernier is y^^o ^ of a foot above the next lower line of the rod. 
 5 of the vernier is ^-^^-q of a foot above the next lower line of the rod. 
 
 2 of the vernier is j-q^^q of a foot above the next lower line of the rod. 
 1 of the vernier is xo^o o ®^ ^ ^^ot above the next lower line of the rod. 
 of the vernier is j-q^^q of a foot above the next lower line of the rod. 
 
 Hence, the required reading is 4.16 + 0.007 = 4.167 feet. 
 
 In general, the following rule is evident: 
 Move the vernier until its zero line is at the 
 required height; read the height to the 7iearest 
 hundredth beloio the index, and write in the 
 thousandths^ place the number of the division 
 line of the vernier which stands opposite any 
 line of the rod. 
 
 Second form. In this form (Fig. 10) the 
 graduated part of the vernier is y§^ of a foot 
 in length, and is divided into ten equal parts. 
 Hence, one division of the vernier = y^ of 
 j|^ = y o»^^ of a foot ; and one division of the 
 vernier is less than one division of the rod 
 ^y tU — To%o = ToV^ of a foot. 
 
 The height of the index of the vernier in 
 Fig. 10 is 4.16 feet, increased by the distance 
 of the index from the next lower line (4.16) 
 of the rod. We shall now determine this 
 distance. 
 
 We observe that the line of the vernier 
 numbered 7 stands exactly opposite the line 
 of the rod numbered 3. H«nce, 
 
INSTRUMENTS AND THEIR USES. 
 
 205 
 
 6 of the vernier is j-^^^q of a foot above the next lower line of the rod. 
 5 of the vernier is j-^^^ of a foot above the next lower line of the rod. 
 4 of the vernier is j-^^qq of a foot above the next lower line of the rod. 
 3 of the vernier is j-^q-q of a foot above the next lower line of the rod. 
 2 of the vernier is j-q%q of a foot above the next lower line of the rod. 
 1 of the vernier is jQ%jf of a foot above the next lower line of the rod. 
 of the vernier is j-q\-q of a foot above the next lower line of the rod. 
 
 Hence, the required reading is 4.16 + 0.007 = 4.167 feet ; 
 and the rule is evidently the same as for the first form. 
 
 Fig. 11. 
 
 Compass Verniers. Let LL' (Fig. 11) represent the limb of 
 the compass graduated to half-degrees, and VV the vernier 
 divided into thirty equal spaces, equal to tv^enty-nine spaces 
 of the limb. Then one space of the vernier is less than one 
 space of the limb by 1', and the reading may be obtained to 
 single minutes. 
 
 In Fig. 11 the index or zero of the vernier stands between 
 32° and 32° 30', and the line of the vernier marked 9 coincides 
 with a line of the limb. Hence, the reading is 32° 9'. 
 
 When the index moves from the zero line of the limb in a 
 direction contrary to that in which the numbers of the limb 
 run, the number of minutes obtained as above must be sub- 
 tracted from 30' to obtain the minutes required. 
 
 If, however, the vernier be made double, that is, if it have 
 thirty spaces on each side of the zero line, it is always read 
 
206 
 
 SUKVEYING. 
 
 directly. The usual form of the double vernier, shown in 
 Fig. 12, has only fifteen spaces on each side of the zero line. 
 When the vernier is turned to the right less than 15' past a 
 division line of the limb, read the lower figures on the left of 
 the zero line at any coincidence ; if moved more than 15' past 
 a division line of the limb, read the upper figures on the right 
 of the zero line at any coincidence ; and vice versa. 
 
 25 
 
 30 
 
 25 
 
 20 
 
 10 
 
 5 z 
 
 ero 5 
 
 t 
 
 
 
 
 mil 
 
 
 
 
 
 __ 
 
 
 
 
 
 ^ 
 
 z 
 
 ero 
 
 
 Fig. 12. 
 
 Uses of the Compass Vernier. The most important use of the 
 vernier of the vernier compass is in setting off the variation 
 of the needle (§ 23). If the variation of the needle at any 
 place is known, by means of the vernier screw the compass 
 circle may be turned through an arc equal to the variation. 
 If the observer stands at the south end of the instrument, the 
 vernier is turned to the right or left according as the variation 
 is west or east. The compass will now give the bearings of 
 the lines with the time meridian. 
 
 In order to retrace the lines of an old survey, turn the sights 
 in the direction of a known line, and move the vernier until 
 the needle indicates the old bearing. The arc moved over by 
 the vernier will indicate the change of variation since the 
 time of the old survey. If no line is definitely known, the 
 rhange of variation from the time of the old survey will give 
 the arc to be set off. 
 
instruments and their uses. 207 
 
 § 8. The Surveyor's Transit. 
 
 This instrument is shown on page 209. 
 
 The compass circle is similar to that of the compass. The 
 vernier plate which carries the telescope has two verniers and 
 moves entirely around the graduated limb of the main plate. 
 The axis of the telescope carries a vertical circle which measures 
 vertical angles to single minutes by means of a vernier. Under 
 the telescope, and attached to it, is a spirit level by which hori- 
 zontal lines may be run, or the difference of level between two 
 stations found. The cross wires are t^^o fine fibres of spider's 
 web, or fine platinum wires, which extend across the tube of 
 the telescope at right angles to each other ; their intersection 
 determines the optical axis or line of collimation of the tele- 
 scope. The transit is levelled by four levelling screivs which 
 pass through a plate firmly fastened to the ball spindle, and 
 rest in depressions on the upper side of the tripod plate. 
 
 A quick centring head enables the surveyor to change the 
 position of the vertical axis horizontally without moving the 
 tripod ; and a quick levelling head enables him to bring the tran- 
 sit quickly to an approximately level position by the pressure 
 of the hands, after which the levelling screws are used ; also, 
 to change the position of the transit without changing the 
 positron of the tripod legs, so as to bring the plummet exactly 
 over any point. 
 
 To level the transit by the levelling^ screws. Turn the instru- 
 ment until the spirit levels on the vernier plate are parallel to 
 the vertical planes passing through opposite pairs of levelling 
 screws. Take hold of opposite screw heads with the thumb 
 and fore-finger of each hand, and turn both thumbs in or out 
 as may be necessary to raise the lower side of the parallel 
 plate and lower the other until the desired correction is made. 
 
 To use the telescope. Both the eye-piece and the object 
 glass may be moved in and out by a rack-and-pinion move- 
 ment. The eye-piece must be moved until the cross wires are 
 
208 SURVEYING. 
 
 perfectly distinct ; then a slight movement of the eye of the 
 observer, from side to side, will produce no apparent change 
 in the position of the threads upon the object. The object 
 glass must be moved until the object is distinctly visible; and 
 this must be repeated, if the distance of the object is changed. 
 
 § 9. Uses of the Transit. 
 
 The transit may be used for all the purposes indicated in 
 § 6, but with much greater precision than the compass. The 
 principal use, however, of the transit is in measuring horizontal 
 angles by means of the graduated limb and verniers. 
 
 To measure a horizontal angle with the transit. Place the 
 transit over the vertex of the angle ; level, and set the limb at 
 zero. Turn the telescope in the direction of one of the sides of 
 the angle, clamp to the spindle ; unclamp the main plate, and 
 turn the telescope until it is in the direction of the other side 
 of the angle, and read the angle by the verniers. The object 
 of the two verniers on the vernier plate is to correct any mis- 
 take that might arise from the want of exact coincidence in 
 the centres of the verniers and the limb. The correct reading 
 may be obtained by adding to the reading of one vernier the 
 supplement of the reading of the other, and dividing by 2. 
 
 By turning off a right angle by this method, perpendiculars 
 may be constructed with greater facility than by the chain. 
 
 § 10. The Theodolite. 
 The telescope of the transit can perform a complete revo- 
 lution on its axis ; whence the name transit. The theodolite 
 differs from the transit chiefly in that its telescope cannot be 
 so revolved. It is not much used in this country. 
 
 § 11. The Kailroad Compass. 
 This instrument has all the features of the ordinary com- 
 pass, and has also a vernier plate and graduated limb for 
 measuring horizontal angles. 
 
THE SURVEYOR'S TRANSIT. 
 
INSTRUMENTS AND THEIR USES. 
 
 211 
 
 § 12. Plotting. 
 The principal plotting instruments are a ruler, pencil, 
 straight-line pen, hair-spring dividers, diagonal scale, a right 
 triangle of wood, and a circular protractor. A T-square will 
 also be found convenient. 
 
 The Diagonal Scale. A portion of this scale is shown in 
 Fig. 13. AB is the unit. AB and A'B' are divided into ten 
 equal parts, and B is joined with h, the first division point to 
 the left of B' ; the first division point to the left of B is 
 joined with the second to the left of B', etc. 
 
 The part of the horizontal line numbered 1 intercepted 
 between BB' and Bh is evidently J^ of J^ = yi^ of the unit ; 
 the part of the horizontal line numbered 2 intercepted between 
 BB' and Bh is yf ^ of the unit, etc. 
 
 The method of using this scale is as follows : 
 
 Let it be required to lay off the distance 1.43. 
 
 Place one foot of the dividers at the intersection of the horizontal Hne 
 numbered 3 and the diagonal numbered 4, and place the other foot at 
 the intersection of the vertical line numbered 1 (CC) and the horizontal 
 line numbered 3 ; the distance between the feet of the dividers will be 
 the distance required. For, measuring along the horizontal line num- 
 bered 3, from CC to BIV is 1 ; from BB' to Bh is 0.03 ; and from Bh to 
 the diagonal numbered 4 is 0.4 ; and 1 -F 0.03 -H 0.4 = 1.43. 
 
212 
 
 SURVEYING. 
 
 The Circular Protractor. This instrument (Fig. 14) usually 
 consists of a semi-circular piece of brass or german silver, 
 having its arc divided into degrees and its centre marked. 
 
 To lay off an angle with the proti^actor, place the centre over 
 the vertex of the angle, and make the diameter coincide with the 
 given side of the angle. Mark off the number of degrees in the 
 given angle, and draw a line through this point and the vertex. 
 
 Fig. 14. 
 
 Some protractors have an arm which carries a vernier, by 
 which angles may be constructed to single minutes. 
 
 To draw through a giuen point a line parallel to a given 
 line, make one of the sides of a triangle coincide with the 
 given line, and, placing a ruler against one of the other sides, 
 move the triangle along the ruler until the first side passes 
 through the given point ; then draw a line along this side. 
 
 To draw through a given point a line perpendicular to a 
 given line, make the hypotenuse of a right triangle coincide 
 with the given line, and, placing a ruler against one of the 
 other sides of the triangle, revolve the triangle about the ver- 
 tex of the right angle as a centre until its other perpendicular 
 side is against the ruler ; then move the triangle along the 
 ruler until the hypotenuse passes through the given point, 
 and draw a line along the hypotenuse. 
 
CHAPTEE TI. 
 
 LAND SURVEYING. 
 
 § 13. Definitiox. 
 
 Land Surveying is the art of measuring, laying out, and 
 dividing land, and preparing a plot. 
 
 § 14. Determinatiox of Areas. 
 The unit of land measure is the 
 
 acre=: 10 square chains 
 = 4 roods 
 = 160 square rods, perches, or poles. 
 
 Areas are referred to the horizontal plane, no allowance 
 being made for inequalities of surface. 
 
 For convenience of reference, the following rules for areas 
 are given : 
 
 Let A, B, and C be the angles of a triangle, and a, b, and c 
 the opposite sides, respectively; and let s=^^(a-\-b-\-c). 
 
 Area of triangle ABC = ^ base X altitude [a] 
 
 = ^&csin^ . [b] 
 
 J a^ sin B sin C ^ -. 
 
 ~^ sin(i^+6') ^^^ 
 
 = ^s(s — a)(s — b)(s — c). [d] 
 
 Area of rectangle = base X altitude. 
 
 Area of trapezoid = ^ sum of parallel sides X altitude. 
 
 Problem 1. To determine the area of a triangular field. 
 
 Measure the necessary parts with a Gunter's chain, or with a chain 
 and transit, and compute by formula [a], [b], [c], or [d]. 
 
214 
 
 SURVEYING. 
 
 Problem 2. To find the area of a field having any number 
 of straight sides. 
 
 {a\ Divide the field into triangles by diagonals ; find the area of each 
 triangle and take the sum. 
 
 (6) Run a diagonal, and perpendiculars from the opposite vertices to 
 this diagonal. The field is thus divided into right triangles, rectangles, 
 and trapezoids, the areas of w^hich may he found and the sum taken. 
 
 Fig. 15. 
 
 Fig. 16. 
 
 Problem 3. To find the area of a field having an irregular 
 boundary line. 
 
 (a) Let AGBCB (Fig. 15) represent a field having a stream AEFG^ 
 HKB as a boundary line. Run the line AB. From E, F, G^ H, and K, 
 prominent points on the bank of the stream, let fall perpendiculars EE\ 
 FF\ etc., upon AB. Regarding AE^ EF, etc., as straight lines, the por- 
 tion of the field cut off by AB is divided into right triangles, rectangles, 
 and trapezoids, the necessary elements of which can be measured and 
 the areas computed. The sum of these areas added to the area of ABCD 
 will give the area required. 
 
 (6) When the irregular boundary line crosses the straight line joining 
 its extremities, as in Fig. 16, the areas of AEFTI and HGB may be found 
 separately, as in the preceding case. Then the area required = ABCD + 
 IIGB-AEFH. . 
 
 Problem 4. To determine the area of a field from two 
 interior stations. 
 
 Let ABCD (Fig. 17) represent a field, and P and P' two stations within 
 it. Measure PP' with great exactness. Measure the angles between PP' 
 and the Hues from P and P' to the corners of the field. 
 
DETERMINATION OF AREAS. 
 
 215 
 
 In the triangle PP'I), PP' and the angles P'PD and PP 
 hence, PJD may be found. In like manner, 
 PC may be found. Then in the triangle 
 PDC, PD, PC, and the angle DPC are 
 known ; hence, the area of PDC may be 
 computed. 
 
 In like manner, the areas of all the 
 triangles about P and P' may be deter- 
 mined. 
 
 Area ABCD = PAD + PI)C + PCB 
 + PBA. Also 
 
 Area ABCD = P'AD + P'DC + P'CB 
 + P'BA. 
 
 D are known 
 
 Problem 5. To determine the area of 
 exterior stations. 
 
 Let ABCD (Fig. 18) represent the field, and 
 P and P' the stations. Determine the areas of 
 the triangles PAD, PDC, PCB, and PBA, as 
 in the preceding problem. 
 
 Area ABCD = PAD + PDC -\- PBC - 
 PBA. Also, 
 
 Area ABCD = P'AD + P'DC + P'BA - 
 P'BC. 
 
 Exercise I. 
 
 a field from two 
 
 Fig. 18. 
 
 1. Eequired the area of a triangular field whose sides are 
 respectively 13, 14, and 15 chains. 
 
 2. Eeqnired the area of a triangular field whose sides are 
 respectively 20, 30, and 40 chains. 
 
 3. Eequired the area of a triangular field whose base is 
 12.60 chains, and altitude 6.40 chains. 
 
 4. Eequired the area of a triangular field which has two sides 
 4.50 and 3.70 chains, respectively, and the included angle 60°. 
 
 5. Eequired the area of a field in the form of a trapezium, 
 one of whose diagonals is 9 chains, and the two perpendiculars 
 upon this diagonal from the opposite vertices 4.50 and 3.25 
 chains. 
 
216 
 
 SURVEYING. 
 
 6. Kequired the area of the field ABCDEF (Fig. 19), if 
 ^^ = 9.25 (ih^\n.s^FF' = ^M chains, BE= 13.75 chains, DT)' 
 
 = 7 chains, DB = 10 chains, CC = 
 4 chains, and AA' = 4.75 chains. 
 
 7. Eeo[uired the area of the field 
 ABCDEF (Fig. 20), if 
 AF^ = 4 chains, FF' = 6 chains, 
 ^'^' = 6.50 chains, AE' = 9 chains, 
 AD = 14 chains, ^C" = 10 chains, 
 ^^'=6.50 chains, BB' = 7 chains, 
 C(7'=rr 6.75 chains. 
 Required the area of the field AGBCD (Fig. 15), if the 
 diagonal AC = 5, BB' (the perpen- 
 dicular from ^ to ^C) = l, DD' (the 
 perpendicular from D to ^C) = 1.60, 
 P ^^^' = 0.25, i^i<^' = 0.25, GG'== 0.60, 
 I£IF = 0.52, KK' = OM, AE' = 0.2, 
 E'F' = 0.50, F'G' = 0A5, G'R' = 
 0.45, ^'ir'=:0.60, and ^'^ = 0.40. 
 9. Required the area of the field 
 AGBCD (Fig. 16), if ^D==3, ^C 
 = 5, AB = 6, angle DAC = 4:5°, angle B AC = 30°, AE' = 
 0.75, ^i^' = 2.25, AH = 2.53, AG' = 3.15, EE' = 0.60, FF' = 
 0.40, and GG' = 0.75. 
 
 10. Determine the area of the field ABCD from two interior 
 if FP' = 1.50 chains, 
 89° 35', FP'D = 349° 45', F'PD = 165° 40', 
 
 
 F 
 
 E 
 
 4 
 
 L 
 
 B' [ C\ 
 
 B 
 
 Fig. 20. 
 
 PPB = 3° 35', P'PC = 303° 15'. 
 
 stations, P and P' 
 
 PP'C = 
 
 PP'B = 1S5°30', 
 PPA = 309° 15', P'PA = 113° 45', 
 
 11. Determine the area of the field ABCD from two exterior 
 stations, P and P', if PP' = 1.50 chains, 
 
 P'PB= 41° 10', P'Pi) = 104° 45', PP'B = 132° 15', 
 P'PA= 55° 45', PP'D= 66° 45', PP'.^ = 103° 0'. 
 P'PC= 77° 20', PP'C= 95° 40', 
 
RECTANGULAR SURVEYING. 
 
 217 
 
 RECTANGULAR SURVEYING. 
 § 15. Definitions. 
 An East and West Line is a line perpendicular to the 
 magnetic meridian. 
 
 The Latitude of a line is the distance between the east 
 and west lines through its extremities. 
 
 The Departure of a line is the distance between the 
 meridians through its extremities. 
 
 Note. When a line extends north of the initial point the latitude is 
 called a northing ; when it extends south, a southing ; when it extends 
 east the departure is called an easting ; when it extends west, a westing. 
 
 The Meridian Distance of a point is its distance from a 
 meridian. 
 
 The Double Meridian Distance of a course is double the 
 distance of the middle point of the course from the meridian. 
 
 Let AB (Fig. 21) represent a line, and NAS the magnetic 
 meridian. Let BB' be perpendicular to NS. jy 
 
 The bearing of the line AB is the angle 
 BAB'. 
 
 The latitude of the line AB is AB'. 
 
 The departure of the line AB is BB'. 
 
 The meridian distance of the point B is BB'. 
 
 In the right triangle ABB', 
 
 AB' = ABX cos BAB', 
 and BB' = ABX sin BAB'. 
 
 Hence, latitude = distance X cos of hearing, 
 and departure =■ distaiice X sin of bearing. 
 
 The latitudes and departures corresponding 
 to any distance and bearing may be found 
 from the above formulas by means of a table 
 of natural sines and cosines, or from " The 
 Traverse Table.''* fig. 21. 
 
 * See Table VII. of Wentworth & Hill's Five-Place Logarithmic and 
 Trigonometric Tables. 
 
218 
 
 SURVEYING. 
 
 § 16. Field Notes, Computation, and Plotting. 
 
 The field notes are kept in a book provided for the purpose. 
 The page is ruled in three columns, in the first of which is 
 written the number of the station ; in the second, the bearing 
 of the side ; and in the third, the length of the side. 
 
 Example 1. To survey the field ABCD (Fig. 22). 
 
 Field Notes. 
 
 1 
 
 X. 20° E. 
 
 8.66 
 
 2 
 
 S. 70° E. 
 
 5.00 
 
 3 
 
 S. 10° E. 
 
 10.00 
 
 4 
 
 N. 70° W. 
 
 10.00 
 
 {a) To obtain the field notes. 
 
 Place the compass at A, the first station, 
 and take the bearmg of AB (§ 6); suppose 
 it to be N. 20° E. Write the result in the 
 second column of the field notes opposite 
 the number of the station. Measure AB 
 = 8.66 chains, and write the result in the 
 third column of the field notes. 
 
 Place the compass at B, and, after testing 
 the bearing of AB (§ 6), take the bearing of 
 EC, measure 5C, and write the results in the 
 field notes ; and so continue until the bearing 
 
 
 Fig. 
 
 - 
 
 
 and len 
 
 gth of each side havejbeer 
 
 I recordec 
 
 Q)) To compute 
 
 the area. 
 
 
 I. 
 
 n. 
 
 in. 
 
 IV. 
 
 V. 
 
 W. 
 
 VII. 
 
 VIII. 
 
 IX. 
 
 X. 
 
 XI. 
 
 Side. 
 
 Bearing. 
 
 Dist. 
 
 N. 
 
 s. 
 
 E. 
 
 w. 
 
 M.D. 
 
 D.M.D. 
 
 N.A. 
 
 S.A. 
 
 AB 
 
 N.20°E. 
 
 8.66 
 
 AB' 
 8.14 
 
 
 BB' 
 
 2.96 
 
 
 BB' 
 2.96 
 
 BB' 
 
 2.96 
 
 2 ABB' 
 24.0944 
 
 
 BC 
 
 S. 70° E. 
 
 5.00 
 
 
 B'C 
 1.71 
 
 C"C 
 4.70 
 
 
 CC 
 7.66 
 
 BB'+CC 
 10.62 
 
 
 2C'CBB' 
 18.1602 
 
 CD 
 
 S. 10° E. 
 
 10.00 
 
 
 C'T>' 
 
 9.85 
 
 D'D 
 
 1.74 
 
 
 DD' 
 
 9.40 
 
 CC'+DD' 
 
 17.06 
 
 
 ID'DCC 
 
 168.0410 
 
 DA 
 
 N.TO-W. 
 
 10.00 
 
 D'A 
 3.42 
 
 
 .... 
 
 DD- 
 
 9.40 
 
 
 
 DD' 
 
 9.40 
 
 2 ADD' 
 
 32.1480 
 
 .... 
 
 
 
 33.66 
 
 11.56 
 
 11.56 
 
 9.40 
 
 9.40 
 
 
 
 56.2424 
 
 186.2012 
 
FIELD NOTES. 219 
 
 The survey may begin at any corner of the field ; hut in computing the 
 area, the field notes should be arranged so that the 
 
 ' . .1, , 186.2012 
 
 most eastern or most western station will stand 56.2424 
 
 first. For the sake of uniformity, we shall always 2 1 129.9588 
 
 begin with the most western station, and keep the ^^ |_64^98_sq. ch. 
 
 ^ fj .1 • 7.4 • • A -^ 6.498 acres, 
 field on the right m passing around it. 
 
 The field notes occupy the first three of the eleven columns in the 
 above tablet. Columns IV. , V. , VI. , and VII. contain the latitudes and 
 departures corresponding to the sides, and taken from the Traverse Table. 
 The lines represented by these numbers are indicated immediately above 
 each number. Column VIII. contains the meridian distances of the points 
 B, C, Z), and ^, taken in order. Column IX. contains the double meridian 
 distances of the courses. Their composition is indicated by the letters 
 immediately above the numbers. Column X. contains the products of 
 the double meridian distances by the northings in the same line. The 
 first number, 
 
 24.0944 = 2.96 X 8.14 = BB' X AB' = 2 area of the triangle ABB'; 
 32.1480 = 9.40 X 3.42 = DJK x AIT = 2 area of the triangle ADIT. 
 Column XI. contains the products of the double meridian distances by 
 the southings in the same line. The first number, 
 
 18.1602 = 10.62 X 1.71 = {BB' + CC) x B'C 
 
 = 2 area of the trapezoid C'CBB'; 
 168.0410 = 17.06 X 9.85 = {CC + BIY) X lyC 
 
 — 2 area of the trapezoid lYBCC'. 
 The sum of the north areas in column X. 
 
 = 56.2424 = 2 {ABB' + ABIT). 
 The sum of the south areas in column XI. 
 
 = 186.2012 = 2 {C'CBB' + B'DCC). 
 But {C'CBB' + B'DCC) - {ABB' + ABB') = ABCB. 
 
 Hence, 2{C'CBB' + B'BCC) - 2{ABB' + ABB') = 2ABCB; 
 that is, 186.2012 - 56.2424 = 129.9588 = 2 ABCB. 
 
 Hence, area ABCB = i of 129.9588 = 64.9794 sq. ch. = 6.498 acres. 
 
 (c) To make the plot. 
 
 The plot or map may be drawn to any desired scale. If a line one 
 inch in length in the plot represents a line one chain in length, the plot is 
 said to be drawn to a scale of one chain to an inch. In this case the plot 
 (Fig. 22) is drawn to a scale of eight chains to an inch. 
 
 Draw the line NAS to represent the magnetic meridian, and lay off 
 the first northing AB' = 8.14 (§ 12). Draw the indefinite line B'E per- 
 
220 SURVEYING. 
 
 pendicular to NS and lay off B'B, the first easting = 2.96. Join^ and B; 
 then the line AB will represent the first side of the field. Through B 
 draw BC perpendicular to BB', and make BC" — 1.71, the first southing. 
 Through C" draw C"C perpendicular to BC", and equal to 4.70, the 
 second easting. Join B and C. The line BC will represent the second 
 side of the field. 
 
 Proceed in like manner until the field is completely represented. The 
 extremity of the last line IX^, measured from JK, should fall at A. This 
 will be a test of the accuracy of the plot. 
 
 By drawing the diagonal A C, and letting fall upon it perpendiculars 
 from B and D, the quadrilateral ABCD is divided into two triangles, the 
 bases and altitudes of which may be measured and the area computed 
 approximately. 
 
 Other methods of plotting will suggest themselves, but the method just 
 explauied is one of the best. 
 
 Balancing the Work. 
 
 In the survey, we pass entirely around the field ; hence, we 
 move just as far north as south. Therefore, the sum of the 
 northings should equal the sum of the southings. In like 
 manner, the sum of the eastings should equal the sum of the 
 westings. In this way the accuracy of the field work may be 
 tested. 
 
 In Example 1, the sum of the northings is equal to the sum 
 of the southings, being 11.56 in each case ; and the sum of the 
 eastings is equal to the sum of the westings, being 9.40 in 
 each case. Hence, the work balances. 
 
 In actual practice the work seldom balances. When it does 
 not balance, corrections are generally applied to the latitudes 
 and departures, by the following rules : 
 
 The perimeter of the field : any one side 
 
 : : total error in latitude : correction ; 
 : : total error in departure : correction. 
 
 If special difficulty has been experienced in taking a par- 
 ticular bearing, or in measuring a particular line, the correc- 
 tions should be applied to the corresponding latitudes and 
 departures, 
 
FIELD NOTES. 
 
 221 
 
 The amount of error allowable varies in the practice of dif- 
 ferent surveyors, and according to the nature of the ground. 
 An error of 1 link in 8 chains would not be considered too 
 great on smooth, level ground; while, on rough ground, an 
 error of 1 link in 2 or 3 chains 
 might be allowed. If the error 
 is considerable, the field meas- 
 urements should be repeated. 
 
 Example 2. Let it be re- 
 quired to survey the field AB 
 CDEF (Fig. 23). 
 
 Field Notes. 
 
 1 
 
 N. 73° 30' W. 
 
 5.00 
 
 2 
 
 S. 16° 30' W. 
 
 5.00 
 
 3 
 
 N. 28°30' W. 
 
 7.07 
 
 4 
 
 N. 20° 00' E. 
 
 11.18 
 
 6 
 
 S. 43°30'E. 
 
 5.00 
 
 6 
 
 S. 13° 30' E. 
 
 10.00 
 
 243.0888 
 
 81.4955 
 
 2 1 161.5933 
 
 10 1 80.7967 
 
 8.0797 acres. 
 
 Explanation. The first station 
 in the field notes is D, but we re- 
 arrange the numbers in the tablet so 
 that A stands first. The northings 
 and southings balance, but the east- 
 ings exceed the westings by 1 link. 
 We apply the correction to the west- 
 ing 4.79 (the distance BE being in 
 doubt), making it 4.80, and rewrite 
 all the latitudes and departures in the next four columns, incor 
 the correction. In practice, the corrected numbers are written in 
 
 
 ^ t?3 b Q to !j^ 
 !^ ^ tq b Q ta 
 
 05 
 1 
 
 
 t2j CQ !^ OQ CB tz| 
 
 % s s| b b g 
 g g g g g § 
 
 ^ ^ ^' H H H 
 
 r 
 1 
 
 
 -J en Oi O en M 
 
 S § g S 8 S 
 
 7^ 
 
 
 
 ^ 
 
 ^ 
 ^ 
 
 *.. * CO CO • 
 
 93 
 
 to 
 
 ': '. \ I I I 
 
 ri . 
 
 to 
 
 05 M vl^ . 
 CO tl^ -fl • 
 
 ~^ to o • 
 
 ^ 
 
 
 
 ^ 
 
 
 *-tq . oC5 cots . 
 
 : 3^ : Sb Sq : 
 
 Co 
 
 
 l^^b coO 03 to 
 
 : : : £^b'^c^^t«' 
 
 rn 
 
 
 oshq mSj *.b ; ; ; 
 i^ ^ to Ky o ; ; ; ; 
 
 5 
 
 
 _ wh3 f-tg pb r^o Mb3 
 
 CD 
 
 
 eg b q b3 
 w^q ootq !^b S^ ^^ wto 
 
 »^ tg b Q 
 
 
 22 
 
 1 
 
 en 
 
 1^ 
 
 
 
 
 ?: 
 
 ^ 
 
 CO 
 
 
 to 
 
 
 CO 
 
 porating 
 red ink. 
 
222 
 
 SURVEYING. 
 
 The remainder of the computation does not require expla- 
 nation. 
 
 It will be seen that this method of computing areas is 
 perfectly general. 
 
 iV § 17. Supplying Omissions. 
 
 If, for any reason, the bearing 
 and length of any side do not appear 
 in the field notes, the latitude and 
 departure of this side may be found 
 in the following manner : 
 
 Find the latitudes and departures 
 of the other sides as usual. The 
 difference between the northings 
 and southings will give the north- 
 ing or southing of the unknown 
 side, and the difference between 
 the eastings and westings will give 
 the easting or westing of the un- 
 known side. 
 
 If the length and bearing of the 
 unknown side are desired, they may 
 be found by solving the right tri- 
 FiG. 23. angle, whose sides are the latitude 
 
 and departure found by the method just explained, and whose 
 hypotenuse is the length required. 
 
 § 18. Irregular Boundaries. 
 
 If a field have irregular boundaries, its area may be found 
 by offsets, as explained in § 14, Prob. 3. 
 
 § 19. Obstructions. 
 
 If the end of a line be not visible from its beginning, or if 
 the line be inaccessible, its length and bearing may be found 
 as follows : 
 
OBSTRUCTIONS. 
 
 223 
 
 1. By means of a random line (§ 4, 3). 
 
 2. When it is impossible to run a random line, which is 
 frequently the case on account of the extent of the obstruc- 
 tion, the following method may be used : 
 
 N 
 
 Let AB (Fig. 24) represent an inaccessible line 
 whose extremities A and B only are known, and B 
 invisible from A. 
 
 Set flag-staffs at convenient points, C and D. 
 Find the bearings and lengths of A C, CD, and DB, 
 and then proceed to find the latitude and departure 
 of AB, as in § 17. 
 
 Fig. 24. 
 
 Example. Suppose that we have the following notes (see 
 Fig. 24): 
 
 Side. 
 
 Bearing. 
 
 DIST. 
 
 N. 
 
 S. 
 
 E. 
 
 w. 
 
 AC 
 CD 
 DB 
 
 S. 45° E. 
 
 E. 
 
 N. 30° E. 
 
 3.00 
 3.50 
 4.83 
 
 4.18 
 
 2.12 
 
 2.12 
 3.50 
 2.42 
 
 
 
 4.18 
 
 2.12 
 
 8.04 
 
 
 
 4 1g The northing of AB is 2.06, and the easting, 8.04; which 
 
 2.12 numbers may be entered in the tablet in the columns N. and E., 
 2.06 opposite the side AB. 
 
 If the bearing and length of AB are required, construct the 
 right triangle ABC (Fig. 25), making AC = 8.04 and BC = 2.06. 
 
 tan^^C = 
 
 BC 2.06 
 
 AC 8.04 
 Hence, the angle BAG = 14° 22'. 
 Also, 
 
 = 0.256. 
 
 AB = ^AC^ + BC'^ = V8.042 + 2.062 = 8.29. 
 Therefore, the bearing and length of AB are N. 75° 38' E. and 8.29. 
 
 Note. Keep all the decimal figures until the result is obtained ; then 
 reject all decimal figures but two, increasing the last decimal figure 
 retained by 1, if the third decimal figure is 5 or greater than 5. 
 
224 
 
 SURVEYING. 
 
 Exercise II. 
 
 In examples 5 and 6 detours were made on account of inaccessible 
 sides (§ 19, 2). The notes of the detours are written in braces. 
 
 5. 8. 
 
 
 1. 
 
 
 Sta. 
 1 
 
 Bearings, 
 
 Dist 
 
 S. 75° E. 
 
 6.00 
 
 2 
 
 S. 15° E. 
 
 4.00 
 
 3 
 
 S. 75° W. 
 
 6.93 
 
 4 
 
 N.45°E. 
 
 5.00 
 
 5 
 
 N.45°W. 
 
 5.19^ 
 
 Sta. 
 1 
 
 Bearings. 
 
 Dist. 
 
 N.45°E. 
 
 10.00 
 
 2 
 
 S. 75° E. 
 
 11.55 
 
 3 
 
 S. 15° W. 
 
 18.21 
 
 4 
 
 N.45°W. 
 
 19.11 
 
 3. 
 
 Sta. 
 
 Bearings. 
 
 Dist. 
 
 1 
 
 N.15°E. 
 
 3.00 
 
 2 
 
 N.75°E. 
 
 6.00 
 
 3 
 
 S. 15° W. 
 
 6.00 
 
 4 
 
 N.75°W. 
 
 5.20 
 
 4. 
 
 Sta. 
 
 Bearings. 
 
 Dist. 
 
 1 
 
 N.89°45'E. 
 
 4.91 
 
 2 
 
 S. 7°00'W. 
 
 2.30 
 
 3 
 
 S. 28°00'E. 
 
 1.52 
 
 4 
 
 S. 0°45'E. 
 
 2.57 
 
 5 
 
 N.84°45'W. 
 
 5.11 
 
 6 
 
 N. 2°30'W. 
 
 5.79 
 
 Sta. 
 1 
 
 Bearings. 
 
 Dist. 
 
 S. 2°15'E. 
 
 9.68 
 
 r 
 
 N.51°45'W. 
 
 2.39 
 
 2] 
 
 S. 85°00'W. 
 
 6.47 
 
 
 S. 55°10'W. 
 
 1.62 
 
 3 
 
 N. 3°45'E. 
 
 6.39 
 
 4 
 
 S. 66°45'E. 
 
 1.70 
 
 5 
 
 N.15°00'E. 
 
 4.98 
 
 6 
 
 S. 82°45'E. 
 
 6.03 
 
 6. 
 
 sta. 
 
 Bearings. 
 
 Dist. 
 
 ■{ 
 
 S. 81°20'W. 
 
 4.28 
 
 N.76°30'W. 
 
 2.67 
 
 2 
 
 N. 5°00'E. 
 
 8.68 
 
 3 
 
 S. 87°30'E. 
 
 5.54 
 
 
 S. 7°00'E. 
 
 1.79 
 
 4- 
 
 S. 27°00'E. 
 S. 10°30'E. 
 
 1.94 
 5.35 
 
 
 N.76°45'W. 
 
 1.70 
 
 7. 
 
 Sta. 
 1 
 
 Bearings. 
 
 Dist 
 
 N. 6°15'W. 
 
 6.31 
 
 2 
 
 S. 81°50'W. 
 
 4.06 
 
 3 
 
 S. 5°00/E. 
 
 5.86 
 
 4 
 
 N.88°30'E. 
 
 4.12 
 
 Sta. 
 1 
 
 Bearings. 
 
 Dist 
 
 N. 5°30'W. 
 
 6.08 
 
 2 
 
 S. 82°30/W. 
 
 6.51 
 
 3 
 
 S. 3°00'E. 
 
 5.33 
 
 4 
 
 E. 
 
 6.72 
 
 9. 
 
 sta. 
 
 Bearings. 
 
 Dist. 
 
 1 
 
 N.20°00'E. 
 
 4.62^ 
 
 2 
 
 N.73°00'E. 
 
 4.16^ 
 
 3 
 
 S. 45°15'E. 
 
 6.18^ 
 
 4 
 
 S. 38°30nV. 
 
 8.00 
 
 5 
 
 Wanting. 
 
 Wanting. 
 
 10. 
 
 sta. 
 
 Bearings, 
 
 DIat. 
 
 1 
 
 S. 3°00'E. 
 
 4.23 
 
 2 
 
 S. 86°45'W. 
 
 4.78 
 
 3 
 
 S. 37°00'W. 
 
 2.00 
 
 4 
 
 N.81°00'W. 
 
 7.45 
 
 5 
 
 N.61°00'W. 
 
 2.17 
 
 6 
 
 N.32°00'E. 
 
 8.68 
 
 7 
 
 S. 75°50'E. 
 
 6.38 
 
 8 
 
 S. 14°45/W. 
 
 0.98 
 
 9 
 
 S. 79°15'E. 
 
 4.52 
 
rectangular method. 225 
 
 § 20. Modification of the Eectangular Method. 
 
 The area of a field may be found by a modification of the 
 rectangular method, if its sides and interior angles are known. 
 
 Let A, B, C, D, represent the inte- 
 rior angles of the field ABCD (Fig. / / 
 26). Let the side AB determine the 
 direction of reference. 
 
 The bearing of AB, with reference 
 to AB, is 0°. 
 
 The bearing of BC, with reference 
 to AB, is the angle h = 180° — B. 
 
 The bearing of CD, with reference A^ 
 to AB, is the angle c=C-b. ^'''' ^^■ 
 
 The bearing of DA, with reference to AB, is the angle d = A. 
 
 The area may now be computed by the rectangular method, 
 regarding AB SbS the magnetic meridian. 
 
 In practice, the exterior angles, when acute, are usually 
 measured. 
 
 As the interior angles may be measured with considerable 
 accuracy by the transit, the latitudes and departures should 
 be obtained by using a table of natural sines and cosines. 
 
 Exercise III. 
 
 1. Find the area of the field ABCD, in which the angle 
 ^ = 120°, B = 60°, (7=150°, and D = 30°; and the side 
 ^5 = 4 chains, J5C = 4 chains, CD = 6.928 chains, and Z>^ 
 = 8 chains. 
 
 Keep three decimal places, and use the Traverse Table. 
 
 2. Find the area of the farm ABODE, in which the angle 
 ^ = 106° 19', ^ = 99° 40', C=120°20', i) = 86°8', and JS' = 
 127° 33'; and the side ^^=79.86 rods, ^(7=121.13 rods, 
 CD = 90 rods, D^= 100.65 rods, and EA = 100 rods. 
 
 Use the table of natural sines and cosines, keeping two decimal places 
 as usual. 
 
226 surveying. 
 
 § 21. General Kemarks on Determining Areas. 
 
 Operations depending upon the reading of the magnetic 
 needle must lack accuracy. Hence/ when great accuracy is 
 required (which is seldom the case in land surveying), the 
 rectangular method (§§ 16-19) cannot be employed. 
 
 The best results are obtained by the methods explained in 
 §§14 and 20, the horizontal angles being measured with the 
 transit, and great care exercised in measuring the lines. 
 
 § 22. The Variation of the Needle. 
 
 The Magnetic Declination, or variation of the needle, at 
 any place, is the angle which the magnetic meridian makes 
 with the true meridian, or north and south line. The variation 
 is east or west, according as the north end of the needle lies 
 east or west of the true meridian. Western variation is indi- 
 cated by the sign -|-, and eastern by the sign — . 
 
 Irregular Variations are sudden deflections of the needle, 
 which occur without apparent cause. They are sometimes 
 accompanied by auroral displays and thunder storms, and are 
 most frequent in years of greatest sun-spot activity. 
 
 Solar-Diurnal Variation. North of the equator, the north 
 end of the needle moves to the west, from 8 a.m. to 1.30 p.m., 
 about 6' in winter and 11' in summer, and then returns 
 gradually to its normal position. 
 
 Secular Variation is a change in the same direction for 
 about a century and a half ; then in the opposite direction for 
 about the same time. 
 
 The line of no variation, or the Agonic Line, is a line join- 
 ing those places at which the magnetic meridian coincides with 
 the true meridian. In the United States, this line at present 
 (1895) passes through Michigan, Ohio, Eastern Kentucky, 
 the extreme southwest of Virginia, and the Carolinas. It is 
 moving gradually westward, so that the variation is increasing 
 
TO ESTABLISH A TRUE MERIDIAN. 
 
 227 
 
 at places east of this line, and decreasing at places west of 
 this line. East of this line the variation is westerly, and west 
 of this line the variation is easterly. 
 
 The table on pages 234 and 235, which has been prepared by 
 permission from data furnished by the United States Coast 
 and Geodetic Survey, shows the magnetic variation at different 
 places in the United States and Canada for several years ; 
 also, the annual change for 1895. 
 
 § 23. To Establish a True Meridian. 
 
 This may be done as follows : 
 
 1. By means of Burt'' s Sola?' Compass (§ 25). 
 
 2. By observations of Polaris. 
 
 The North Star or Polaris revolves about the pole at present 
 at the distance of about 1^-° ; hence, it is on the meridian twice 
 in 23 h. 56 m. 4 s. (a sidereal day), once above the pole, called 
 the upper culmination, and 11 h. 58 m. 2 s. later below the pole, 
 called the lower culmination. It attains its greatest eastern 
 or western elongation, or greatest distance from the meridian, 
 5 h. 59 m. 1 s. after the culmination. 
 
 The following table gives the mean local time of the upper 
 culmination of Polaris for 1895 at Washington. The time is 
 growing later at the rate of about one minute in three years. 
 
 Month. 
 
 First Day. 
 
 Eleventh Day. 
 
 Twenty-first Day. 
 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 January . . . 
 
 6 35 P.M. 
 
 5 55 P.M. 
 
 5 16 P.M. 
 
 February 
 
 
 
 4 32 P.M. 
 
 3 53 P.M. 
 
 3 14 P.M. 
 
 March . . 
 
 
 
 2 42 P.M. 
 
 2 03 P.M. 
 
 1 23 P.M. 
 
 April . . 
 
 
 
 12 40 P.M. 
 
 12 00 M. 
 
 11 17 A.M. 
 
 May . . . 
 
 
 
 10 38 A.M. 
 
 9 59 A.M. 
 
 9 20 A.M. 
 
 June . . . 
 
 
 
 8 37 A.M. 
 
 7 57 A.M. 
 
 7 18 A.M. 
 
 July . . . 
 
 
 
 6 39 A.M. 
 
 6 00 A.M. 
 
 5 21 A. 31. 
 
 August. . 
 
 
 
 4 38 A.M. 
 
 4 00 A.M. 
 
 3 19 A.M. 
 
 September 
 
 
 
 2 36 A.M. 
 
 1 57 A.M. 
 
 1 18 A.M. 
 
 October . 
 
 
 
 12 39 A.M. 
 
 11 59 P.M. 
 
 11 20 P.M. 
 
 November 
 
 
 
 10 37 P.M. 
 
 9 57 P.M. 
 
 9 18 P.M. 
 
 December 
 
 
 
 8 39 P.M. 
 
 7 59 P.M. 
 
 7 20 P.M. 
 
* Polaris. ^ * * ^ 
 
 Pole. * I 
 
 228 SXTRVEYING. 
 
 The time of the upper culmination of Polaris may be found 
 by means of the star Mizar, which is the middle one of the 
 three stars in the handle of the Dipper (in the constellation 
 of the Great Bear). It crosses the meridian at almost exactly 
 the same time as Polaris. Suspend from a height of about 
 20 feet a plumb-line, placing the bob in a pail of water to 
 lessen its vibrations. About 15 feet south of the plumb-line, 
 upon a horizontal board firmly supported at a convenient 
 height, place a compass sight fastened to a board a few inches 
 square. At night, when Mizar by estimation approaches the 
 meridian, place the compass sight in line with Polaris and the 
 
 plumb-line, and move 
 I ^ * it so as to keep it 
 
 in this line until the 
 plumb-line also falls 
 on Mizar (Fig. 27). 
 Note the time; then 
 (1895) fifty-one sec- 
 onds later Polaris will 
 be on the meridian. 
 
 This interval is 
 gradually increasing 
 at the rate of 21 sec- 
 onds a year. 
 
 If the lower culmination takes place at night, the time may 
 be found in a similar manner. 
 
 When Mizar cannot conveniently be used, as in the spring, 
 8 Cassiopeiae may be employed. This is the star at the 
 bottom of the first stroke of the W frequently imagined to 
 connect roughly the five brightest stars in Cassiopeia. In 1895 
 it culminates 1.75 minutes before Polaris, with an annual 
 increase of the interval of 20 seconds. 
 
 Instead of the compass sight, any upright with a small 
 opening or slit may be used. 
 
 Pole 
 Polaris. ^ 
 
 Fig. 27. 
 
TO ESTABLISH A TRUE MERIDIAN. 229 
 
 {a) To locate the true meridian by the position of Polaris 
 at its culmination. 
 
 1. By using the ai^jjaratus described in finding the time of 
 culmination. At tlie time of culmination bring Polaris, the 
 plumb-line, and the compass sight into line. The compass 
 sight and the plumb-line will give two points in the true 
 meridian. 
 
 2. By means of the transit. Bring the telescope to bear on 
 Polaris at the time of culmination, holding a light near to 
 make the wires visible, if the observation is made at night. 
 The telescope will then lie in the plane of the meridian, 
 which may be marked by bringing the telescope to a horizontal 
 position. 
 
 (h) To locate the meridian by the position of Polaris at 
 greatest elongation. 
 
 The Azimuth of a star is the angle which the meridian 
 plane makes with a vertical plane passing through the star 
 and the zenith of the observer. 
 
 A star is said to be at its greatest elongation, when its 
 vertical circle ZN (Fig. 28) is tangent to its diurnal circle, 
 that is, perpendicular to the hour circle PN. 
 
 Let Z (Fig. 28) represent the zenith of the place, P the pole, and N 
 Polaris at its greatest elongation ; that is, when its vertical 
 circle ZN is perpendicular to the hour circle PN. Let ZP^ 
 ZN., and PN be arcs of great circles ; then N will be a right 
 angle. 
 
 sin PN - cos (90° - ZP) cos (90° - Z). 
 
 [Spher. Trig. § 47.] 
 But ZP— the complement of the latitude. 
 
 Hence, 90° — ZP = the latitude of the place. 
 
 Hence, sin PN — cos latitude x sin Z. 
 
 sin PN 
 
 Hence, sin Z 
 
 cos latitude 
 
230 
 
 SURVEYING. 
 
 Hence, Z (the azimuth of Polaris) can be found if the 
 latitude of the place and the greatest elongation of Polaris 
 (PiV) are known. 
 
 The following table gives the mean value of the latter 
 element for each year from 1895 to 1906. 
 
 Greatest Elongation of Polaris. 
 
 1895 
 
 i°i5.r 
 
 1899 
 
 1°13.8' 
 
 1903 
 
 1°12.6' 
 
 1896 
 
 1° 14.8' 
 
 1900 
 
 1°13.5' 
 
 1904 
 
 1°12.3' 
 
 1897 
 
 1°14.5' 
 
 1901 
 
 1° 13.2' 
 
 1905 
 
 1° 12.0' 
 
 1898 
 
 P14.r 
 
 1902 
 
 1°12.9' 
 
 1906 
 
 1°11.7' 
 
 The greatest elongation of Polaris, or the polar distance, is 
 given in the Nautical Almanac. The table gives this element 
 for Jan. 1. It may be found for other dates by interpolation. 
 
 To obtain a line in the direction of Polaris at greatest 
 elongation. 
 
 1. Bij using the apparatus for finding the time of culmina- 
 tion. A few minutes before the time of greatest elongation 
 (5 h. 59 m. 1 s. after culmination), place the compass sight in 
 line with the plumb-line and Polaris, and keep it in line with 
 these, by moving the board in the opposite direction, until the 
 star begins to recede. At this moment the sight and plumb- 
 line are in the required line. 
 
 2. By means of the transit. A few minutes before the time 
 of greatest elongation, bring the telescope to bear on the star, 
 and follow it, keeping the v^ertical wire over the star until it 
 begins to recede. The telescope will then be in the required 
 line. 
 
 To establish the meridian. Having the transit sighted in the 
 direction of the line just found, turn it through an angle equal 
 to the azimuth in the proper direction. 
 
DIVIDING LAXD. 
 
 231 
 
 § 24. DiviDixG Laxd. 
 
 The surveyor must; for tlie most part, depend on his general 
 knowledge of Geometry and Trigonometry, and his own inge- 
 nuity, for the solutions of problems that arise in dividing land. 
 
 Problem 1. To divide a triangular field into two parts 
 having a given ratio, by a line through a ^ 
 
 given vertex. 
 
 Let ABC (Fig. 29) be the triangle, and A the 
 given vertex. 
 
 7? J) 
 Divide BC at D, so that -=— equals the given 
 JJO 
 
 ratio, and join A and D. ABD and ADC will 
 be the parts required ; for 
 
 ABD : ADC : : BD : DC. 
 
 Problem 2. To cut ofi from a 
 triangular field a given area, by a 
 line parallel to the base. 
 
 Let ABC (Fig. 30) be the triangle, 
 and let DE be the division line re- 
 quired. 
 
 ^/ABC : \/ADE ::AB: AD. 
 
 , „ JADE 
 
 Fig. 30. 
 
 Problem 3. To divide a field into 
 two parts having a given ratio, by a line 
 through a given point in the perimeter. 
 
 Let ABCDE (Fig. 31) represent the field, 
 P the given point, and PQ the required divi- 
 sion line. 
 
 The areas of the whole field and of the 
 required parts having been determined, run 
 the line PD from P to a comer 2>, dividing 
 the field, as near as possible, as required. 
 Determine the area PBCD. 
 
 Fig. 31. 
 
232 
 
 SURVEYING. 
 
 The triangle PDQ represents the part which must be added to PBCB 
 to make the required division. 
 
 Hence, BQ 
 Note. DQ = 
 
 Area PDQ = i X PD X DQ X sin PDQ. 
 _ 2 area PDQ 
 ~ PDx sin PDQ 
 2siYea.PDQ 
 
 perpendicular from P on DE 
 P on DE may be run and measured directly. 
 
 Problem 4. 
 
 This perpendicular from 
 
 To divide a field into a given number of parts, 
 so that access to a pond of water is 
 given to each. 
 
 Let ABODE (Fig. 32) represent the field, 
 and P the pond. Let it be required to divide 
 the field into four parts. Find the area of 
 the field and of each part. 
 
 Let AP be one division line. Run PE, 
 and find the area APE. Take the difference 
 between APE and the area of one of the 
 required parts ; this will give the area of the 
 triangle PQE, from which QE may be found, 
 as in Problem 3. Join P and Q ; PA Q will 
 be one of the required parts. In like manner, 
 PQR and PAS are determined ; whence, 
 PSR must be the fourth part required. 
 
 Exercise IV. 
 
 1. From the square ABCD, containing 6 a. 1 r. 24 p., part 
 off 3 A. by a line EF parallel to AB. 
 
 2. From the rectangle ABCD, containing 8 a. 1 r. 24 p., 
 part off 2 A. 1 R. 32 p. by a line JSF parallel to AD = 7 ch. 
 Then, from the remainder of the rectangle, part off 2 a. 3 r. 
 25 P., by a line GR parallel to EB. 
 
 3. Part off 6 A. 3 r. 12 p. from a rectangle ABCD, con- 
 taining 15 A., by a line FF parallel to AB ; AD being 10 ch. 
 
 4. From a square ABCD, whose side is 9 ch., part off a 
 triangle which shall contain 2 a. 1 r. 36 p., by a line BF 
 drawn from B to the side AD. 
 
EXAMPLES. 233 
 
 5. From ABCD, representing a rectangle, whose length is 
 12.65 ch., and breadth 7.58 ch., part off a trapezoid which 
 shall contain 7 a. 3 r. 24 p., by a line BjE from B to JDC. 
 
 6. InthetriangleJ^(7,^^ = 12ch.,^(7=10ch.,^C=8ch.; 
 part off 1 A. 2 R. 16 p., below the line DU parallel to AB. 
 
 7. In the triangle ABC, AB = 26 ch., AC == 20 ch., and 
 BC=16 ch. ; part off 6 a. 1 r. 24 p., below the line DU 
 parallel to AB. 
 
 8. It is required to divide the triangular field J 5 6' among 
 three persons whose claims are as the numbers 2, S, and 5, so 
 that they may all have the use of a watering-place at C ; AB 
 = 10 ch., ^C = 6.85 ch., and CB = 6.10 ch. 
 
 9. Divide the five-sided field ABC HE among three persons, 
 X, Y, and Z, in proportion to their claims, X paying ^500, 
 Y paying $750, and Z paying $1000, so that each may have 
 the use of an interior pond at P, the quality of the land being 
 equal throughout. Given ^P = 8.64 ch., BC =S.27 ch., 
 CB:= 8.06 ch., HE=Q>M ch., and EA = 9.90 ch. The per- 
 pendicular PB upon AB = 5.60 ch., FB' upon BC= 6.08 ch., 
 FD" upon C^=4.80 ch., FD'" upon HE=6A4. ch., and 
 FD"" upon EA = oAO ch. Assume FR as the divisional fence 
 between X's and Z's shares ; it is required to determine the 
 position of the fences F3f and Pi\" between X's and Y's shares 
 and Y's and Z's shares, respectively. 
 
 10. Divide the triangular field ABC, whose sides AB, AC, 
 and BC are 15, 12, and 10 ch., respectively^ into three equal 
 parts, by fences EG and DE parallel to BC, without finding 
 the area of the field. 
 
 11. Divide the triangular field ABC, whose sides AB, BC, 
 and AC are 22, 17, and 15 ch., respectively, among three 
 persons. A, B, and C, by fences parallel to the base AB, so 
 that A may have 3 a. above the line AB, B, 4 a. above A's 
 share, and C, the remainder. 
 
234 
 
 SURVEYING. 
 
 
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 . . . _ . . _ 
 
 (4 
 
 
 Halifax N.S. . . . 
 Eastport, Me. . . . 
 Bangor, Me. . . . 
 Provincetown, Mass. 
 Portland, Me. . . . 
 Portsmouth, N.II. . 
 Boston, Mass. . . . 
 Cambridge, Mass. 
 Quebec, Canada . . 
 Providence, 11. 1. . . 
 Hartford, Conn. . . 
 New Haven, Conn. . 
 Burlington, Vt. . . 
 Williamstown, Mass. 
 Montreal, Canada 
 Albany, N.Y. . . . 
 New York, N.Y. . . 
 New Brunswick, N.J. 
 Cape Henlopen, Del. 
 Philadelphia, Pa. 
 Cape Henry, Va. . . 
 Ithaca, N.Y. . . . 
 Baltimore, Md. . . 
 Williamsburg, Va. . 
 Harrisburg, Pa. . . 
 Washington, D.C. . 
 Newbern, N.C. . . 
 Buffalo, N.Y. . . • 
 Toronto, Canada . . 
 
VAEIATIOX OF THE NEEDLE. 
 
 235 
 
 11 
 
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 1 
 
 
 Charleston, S.C. . . 
 rittsburg, I'a. . . . 
 
 Frie, Fa 
 
 Savannah, Ca. . . 
 (Cleveland, O. . . . 
 K(\y West, Fla. . . 
 Detroit, Mich. . . 
 Sanlt Ste. Marie, Mich. 
 Cincinnati, (). . . . 
 (irand Uaven, Mich. 
 Nashville, Tenn. . . 
 Michigan City, Ind. . 
 ren.sa(M)la, Fla. . . 
 Chicago, 111. . . . 
 Milwaukee, Wis. . . 
 Mobile, Ala. . . . 
 New Orleans, La. 
 St. Louis, Mo. . . . 
 Duluth, Minn. . . 
 (Jalveston, Tex. . . 
 Omaha, Neb. . . . 
 Austin, Tex. . . . 
 San Antonio, Tex. . 
 Denver, Col. . . . 
 Salt Lake City, Utah 
 San Diego, Cal. . . 
 Seattle, Wash. . . 
 San Francisco, Cal. . 
 C. Mendocino, Cal. . 
 
236 SURVEYING. 
 
 § 25. United States Public Lands. 
 Burfs Sola)' Comj^ass. 
 
 This instrument, which is exhibited, on the following page, 
 may be used for most of the purposes of a compass or transit. 
 Its most important use, however, is to run north and south 
 lines in laying out the public lands. 
 
 A full description of the solar compass, with its principles, 
 adjustments, and uses, forms the subject of a considerable 
 volume, which should be in the hands of the surveyor who 
 uses this instrument. The limits of our space will allow only 
 a brief reference to its principal features. 
 
 The main plate and standards resemble these parts of the 
 compass. 
 
 a is the latitude arc. 
 
 b is the declination arc. 
 
 h is an arm, on each end of which is a sola)' lens having its 
 focus on a silvered plate on the other end. 
 
 c is the hour arc. 
 
 n is the needle-hox, which has an arc of about 36°. 
 
 To run a north and south line with the solar compass. Set 
 
 off the declination of the sun on the declination arc. Set oif 
 the latitude of the place (which may be determined by this 
 instrument) on the latitude arc. Set the instrument over the 
 station, level, and turn the sights in a north and south direc- 
 tion, approximately, by the needle. Turn the solar lens 
 toward the sun, and bring the sun's image between the 
 equatorial lines on the silvered plate. Allowance being made 
 for refraction, the sights will then indicate a true north and 
 gouth line, 
 
BURT'S SOLAR QOMPASS. 
 
LAYING OUT THE PUBLIC LANDS. 
 
 239 
 
 Laying Out the Public Lands. 
 
 The public lands north of the Ohio Kiver and west of the 
 Mississippi are generally laid out in townships approximately 
 six miles square. 
 
 A Principal Meridian, or true north and south line, is first 
 run by means of Burt's Solar Compass, and then an east and 
 west line, called a Base Line. 
 
 Parallels to the base line are run at intervals of six miles, 
 and north and south lines 
 at the same intervals. Thus N 
 
 the tract would be divided 
 into townships exactly six 
 miles square, if it were not 
 for the convergence of the 
 
 meridians on account of the * I I I I I I I ' I E 
 
 curvature of the earth. 
 
 The north and south 
 lines, or meridians, are 
 called Range Lines. The 
 east and west lines, or 
 parallels, are called Town- 
 ship Lines. 
 
 Let NS (Fig. 33) represent a principal meridian, and WE 
 a base line ; and let the other lines represent meridians and 
 parallels at intervals of six miles. 
 
 The small squares. A, B, C, etc., will represent townships. 
 
 A would be designated thus: T. 3 N., K. 2 W.; that is, 
 township three north, range two west ; which means that the 
 township is in the third tier north of the base line, and in the 
 second tier west of the principal meridian. B and C, respec- 
 tively, would be designated thus : T. 4 S., R. 3 W. ; and 
 T. 2 N., R. 2 E. 
 
 3 
 Fig. 33. 
 
240 
 
 SURTETING. 
 
 6 
 7 
 18 
 19 
 30 
 31 
 
 5 
 8 
 17 
 
 20 
 29 
 32 
 
 4 
 9 
 16 
 21 
 28 
 33 
 
 3 
 10 
 15 
 
 22 
 27 
 34 
 
 2 
 11 
 14 
 23 
 26 
 35 
 
 1 
 
 12 
 13 
 24 
 25 
 36 
 
 Fig. 34. 
 
 The townships are divided into sections approximately one 
 mile square, and the sections are divided 
 into quarter-sections. The township, 
 section, and quarter-section corners are 
 permanently marked. 
 
 The sections are numbered, beginning 
 at the northeast corner, as in Fig. 34, 
 which represents a township divided 
 into sections. The quarter-sections are 
 designated, according to their position, 
 as K E., N.W., S. E., and S. W. 
 
 Every fifth parallel is called a Standard Parallel or Correc- 
 tion Line. 
 
 Let NS (Fig. 35) represent a principal meridian; WU a 
 
 base line; rp, etc., meridians; 
 ^ and ms the fifth parallel. If 
 
 Oj) equals six miles, mr will 
 be less than six miles on 
 account of the convergence 
 of the meridians. Surveyors 
 are instructed to make Op 
 such a distance that mr shall 
 be six miles; then mh, hk, 
 etc., are taken similarly. At 
 the correction lines north of 
 ,-^ ms the same operation is 
 repeated. 
 
 The township and section 
 lines are surveyed in such 
 an order as to throw the errors on the north and outer town- 
 ships and sections. 
 
 If, in running a line, a navigable stream or a lake more 
 than one mile in length is encountered, it is meandered by 
 
 W- 
 
 m 
 
 r 
 
 h 
 
 Jc \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 O 
 
 J 
 
 P 
 
 
 Fig. 35. 
 
PLANE-TABLE SURVEYING. 241 
 
 marking the intersection of the line with the bank and 
 running lines from this point along the bank to prominent 
 points which are marked, and the lengths and bearings of the 
 connecting lines recorded. 
 
 Six principal meridians have been established and con- 
 nected. In addition to these there are several independent 
 meridians in the Western States and Territories which will 
 in' time be connected with each other and with the eastern 
 system. 
 
 § 26. Plane -Table Surveting."* 
 
 After the principal lines of a survey have been determined 
 and plotted, the details of the plot may be filled in by means 
 of the plane-table ; or, when a plot only of a tract of land is 
 desired, this instrument affords the most expeditious means 
 of obtaining it. 
 
 An approved form of the plane-table, as used in the United 
 States Coast and Geodetic Survey, is shown in the plate on 
 page 243. 
 
 The Table-top is a board of well-seasoned wood, panelled 
 with the grain at right angles to prevent warping, and sup- 
 ported at a convenient height by a Tripod and Levelling 
 Head. 
 
 The Alidade is a ruler of brass or steel supporting a 
 telescope or sight standards, whose line of sight is parallel to 
 a plane perpendicular to the lower side of the ruler, and 
 embracing its fiducial edge. 
 
 The Declinatoire consists of a detached rectangular box 
 containing a magnetic needle which moves over an arc of 
 about 10° on each side of the point. 
 
 * In preparing this section the writer has consulted, by permission, 
 the treatise on the plane-table by Mr. E, Hergesheinier, contained in the 
 report for 1880 of the U.S. Coast and Geodetic Survey. 
 
242 SURVEYING. 
 
 Two spirit levels at right a^ngles are attached to the ruler or 
 to the declinatoire. In some instruments these are replaced 
 by a circular level, or by a detached spring level. 
 
 The paper upon which the plot is to be made or completed 
 is fastened evenly to the board by clamps, the surplus paper 
 being loosely rolled under the sides of the board. 
 
 To place the table in position. This operation, which is 
 sometimes called orienting the table, consists in placing the 
 table so that the lines of the plot shall be parallel to the 
 corresponding lines on the ground. 
 
 This may be accomplished approximately by turning the 
 table until the needle of the declinatoire indicates the same 
 bearing as at a previous station, the edge of the declinatoire 
 coinciding with the same line on the paper at both stations. 
 
 If, however, the line connecting the station at which the 
 instrument is placed with another station is already plotted, 
 the table may be placed in position accurately by placing it 
 over the station so that the plotted line is by estimation over 
 and in the direction of the line on the ground ; then making 
 the edge of the ruler coincide with the plotted line, and turn- 
 ing the board until the line of sight bisects the signal at the 
 other end of the line on the ground. 
 
 To plot any point. Let ah on the paper represent the line 
 AB on the ground ; it is required to plot c, representing C on 
 the ground. 
 
 1. By intersection. 
 
 Place the table in position at A (Fig. 36), plumbing a over A^ and 
 
 p making the fiducial edge of the ruler 
 
 :t-v^ pass through a; turn the alidade 
 
 / "-.^ about a until the line of sight 
 
 / ""v^ bisects the signal at C, and draw a 
 
 line along the fiducial edge of the 
 ruler. Place the table in position 
 at JB, plumbing h over B, and repeat 
 the operation just described, c will 
 ^ be the intersection of the two lines 
 Fig. 36. thus drawn. 
 
 -^'c 
 
THE PLANE-TABLE. 
 
tLANE-TABLE SURVEYING. 
 
 245 
 
 / 
 
 / 
 
 / 
 
 N 
 
 
 \ 
 
 
 / 
 
 / 
 
 "-V 
 
 
 / 
 
 / 
 
 ■'-^ 
 
 
 b 
 
 — a 
 
 B 
 
 
 A 
 
 2. By resection. 
 
 Place the table in position at A (Fig. 37), and draw a line in the direc- 
 tion of C, as in the former case ; then remove the instrument to C, place 
 it in position by the line drawn from 
 a, make the edge of the ruler pass 
 through 6, and turn the alidade about 
 b until B is in the line of sight. A 
 line drawn along the edge of the 
 ruler will intersect the line from 
 a in c. 
 
 3. By radiation. 
 
 Place the table in position at A 
 (Fig. 38), and draw a line from a 
 toward 0, as in the former cases. 
 Measure A C\ and lay off ac to the >v, 
 
 same scale as a6. ^^^ 
 
 To plot a field ^^Ci) 
 
 1. By radiation. 
 
 Set up the table at any point P, 
 
 and mark p on the paper over P. B 
 
 Draw indefinite lines from p to- _ 
 
 Fig 
 ward A, B, C Measure PA, 
 
 PB, , and lay off pa, pb, , to a suitable scale, and join a and 6, 
 
 b and c, c and d, etc. 
 
 Fig. 37. 
 
 2. By progression. 
 
 Set up the table at A, and draw a line from a toward B. Measure 
 AB, and plot ab to a suitable scale. Set up the table in position at B, 
 and in like manner determine and plot 6c, etc. 
 
 3. By intersection. 
 
 Plot one side as a base line. Plot the other corners by the method of 
 intersection, and join. 
 
 4. By resection. 
 
 Plot one side as a base line. Plot the other corners by the method of 
 resection, and join. 
 
246 
 
 SURVEYIXG. 
 
 The Three Point Prohlem. 
 
 Let A, B, C represent three field stations plotted as a, h, c, 
 respectively (Fig. 39) ; it is required to plot d representing a 
 fourth field station Z>, visible from A, B, and C. 
 
 Fig. 39. 
 
 Place the table over D, level and orient approximately by 
 the declinatoire. Determine d by resection as follows : Make 
 the edge of the ruler pass through a and lie in the direction 
 aA, and draw a line along the edge of the ruler. In like 
 manner, draw lines through b toward B and through c toward 
 C. If the table were oriented perfectly these lines would 
 meet at the required point d, but ordinarily they will form 
 the triangle of error ^ ah, ac, he. In this case, through a, b, and 
 ab', a, G, and ac; and b, c, and be, respectively, draw circles; 
 these circles will intersect in the required point d. Eor at 
 the required point the sides ab, ae, be must subtend the same 
 angle as at the points ab, ae, be, respectively. Hence, the 
 required point d lies at the intersection of the three circles 
 mentioned. The plane-table may now be oriented accurately. 
 
 Note. The three point problem may be solved by fastening on the 
 board a piece of tracing paper and marking the point d representing J), 
 after which lines are drawn from d toward A, B, and C. The tracing 
 paper is then moved until the lines thus drawn pass through a, 6, c, 
 respectively, when by pricking through d the point is determined on the 
 plot below. 
 
CHAPTER III. 
 
 TRIANGULATION.* 
 
 § 27. Introductory Remarks. 
 
 Geographical positions upon tlie surface of the earth are 
 commonly determined by systems of triangles which con- 
 nect a carefully determined base line with the points to be 
 located. 
 
 Let F (Fig. 40) represent a point whose position with refer- 
 ence to the base line AB is 
 required. Connect AB with 
 F by the series of triangles 
 ABC, ACD, ADE, and DBF, 
 so that a signal at C is visible 
 from A and B, a signal at D 
 visible from A and C, a signal ^^^- ^• 
 
 at E visible from A and D, and a signal at F visible from 
 D and E. In the triangle ABC, the side AB is known, 
 and the angles at A and B may be measured ; hence, AC 
 may be computed. In the triangle ACD, AC i^ known, 
 and the angles at A and C may be measured ; hence, AD 
 may be computed. In like manner DE and EF or DF 
 may be determined. DF, or some suitable line connected 
 with DF, may be measured, and this result compared with 
 the computed value to test the accuracy of the field meas- 
 urements. 
 
 * In preparing this chapter the writer has consulted, by permission, 
 recent reports of the United States Coast and Geodetic Survey. 
 
248 SURVEYING. 
 
 Three orders of triangulation are recognized, viz. : 
 
 Primary, in which, the sides are from 20 to 150 miles in 
 length. 
 
 Secondary, in which the sides are from 5 to 40 miles in 
 length, and which connect the primary with the tertiary. 
 
 Tertiary, in which the sides are seldom over 5 miles in 
 length, and which bring the survey down to such dimensions 
 as to admit of the minor details being filled in by the compass 
 and plane-table. 
 
 § 28. The Measurement of Base Lines. 
 
 Base lines should be measured with a degree of accuracy 
 corresponding to their importance. 
 
 Suitable ground must be selected and cleared of all obstruc- 
 tions. Each extremity of the line may be marked by cross 
 lines on the head of a copper tack driven into a stub which is 
 sunk to the surface of the ground. Poles are set up in line 
 about half a mile apart, the alignment being controlled by a 
 transit placed over one end of the line. 
 
 The 'preliminary measurement may be made with an iron 
 wire about one-eighth of an inch in diameter and 60™ in 
 length. In measuring, the wire i's brought into line by means 
 of a transit set up in line not more than one-fourth of a mile 
 in the rear. The end of each 60°" is marked with pencil lines 
 on a wooden bench whose legs are thrust into the ground after 
 its position has been approximately determined. If the last 
 measurement exceeds or falls short of the extremity of the 
 line, the difference may be measured with the 20"" chain. 
 
 The final meas%irement is made with the base apparatus^ 
 which consists of bars 6"^ long, which are supported upon 
 trestles when in use. These bars are placed end to end, and 
 brought to a horizontal position, if this can be quickly accom- 
 plished ; if not, the angle of inclination is taken by a sector, 
 or a vertical offset is measured with the aid of a transit, so 
 that the exact horizontal distance can be computed. 
 
MEASUREMENT OF ANGLES. 249 
 
 A thermometer is attached to each bar, so that the tempera- 
 ture of the bar may be noted and a correction for temperature 
 applied. 
 
 The method of measuring lines varies according to the 
 required degree of accuracy in any particular case, but the 
 brief description given above will give the student a general 
 idea of the methods employed. 
 
 § 29. The Measurement of Angles. 
 
 Angles are measured by the transit with much greater accu- 
 racy than by the compass, since the reading of the plates of 
 the transit is taken to minutes, and by means of microscopes 
 to seconds, while the reading of the needle of the compass is 
 to quarter or half-quarter degrees. 
 
 In order to eliminate errors of observation, and errors 
 arising from imperfect graduation of the circles, a large 
 number of readings is made and their mean taken. Two 
 methods are in use ; viz., repetition and series. 
 
 The method of repetition consists, essentially, in measuring 
 the angles about a point singly, then taking two adjacent 
 angles as a single angle, then three, etc. ; thus " closing the 
 horizon," or measuring the whole angular magnitude about a 
 point in several different ways. 
 
 The method of series consists, essentially, in taking the 
 readings of an angle with the circle or limb of the transit 
 in one position, then turning the circle through an arc and 
 taking the readings of the same circle again, etc. ; thus read- 
 ing the angle from successive portions of the graduated circle. 
 
 On account of the curvature of the earth, the sum of the 
 three angles of a triangle upon its surface exceeds 180°. This 
 spherical excess, as it is called, becomes appreciable only when 
 the sides of the triangle are about 5 miles in length. To 
 determine the angles of the rectilinear triangle having the 
 same vertices, one-third of the spherical excess is deducted 
 from each spherical angle. 
 
CHAPTER IV. 
 
 LEVELLING. 
 
 § 30. Definitions, Curvature, and Refraction. 
 
 A Level Surface is a surface parallel with the surface of 
 still water ; and is, therefore, slightly curved, owing to the 
 spheroidal shape of the earth. 
 
 A Level Line is a line in a level surface. 
 Levelling is the process of finding the difference of level of 
 two places, or the distance of one place above or below a level 
 line through another place. 
 
 The Line of Apparent Level of a place is a tangent to the 
 level line at that place. Hence, the line of apparent level is 
 perpendicular to the plumb-line. 
 
 The Correction for Curvature is the deviation of the line of 
 apparent level from the level line for any distance. 
 
 Let t (Fig. 41) represent the line of apparent level of the 
 place P, a the level line, d the diame- 
 ter of the earth ; then c represents the 
 correction for curvature. To compute 
 the correction for curvature : 
 
 ^2=.c(c+f^).(Geom.§348.) 
 
 Therefore, c = — ; — ; = — 
 G-\-d d 
 
 approximately, since c is very small 
 
 compared with d. and t = a without 
 Fig. 41. . , , 
 
 appreciable error. 
 
 Since d is constant (= 7920 miles, nearly), the correction for 
 
 curvature varies as the square of the distance. 
 
THE LEVELLING ROD. 
 
 251 
 
 81 
 
 Example. What is the correction for curvature for 1 mile ? 
 By substituting in the formula deduced above, 
 a^ 1^ . „ . 
 ^=^ = 7920^^'^^ ^^- 
 Hence, the correction for curvature for any 
 distance may be found in inches, approximately, 
 by multiplying 8 by the square of the distance 
 expressed in miles. 
 
 Note. The effect of curvature is to make an object 
 appear lower than it really is ; and the effect of refraction 
 of light, caused by the greater density of the atmosphere 
 near the surface of the earth, is to make an object appear 
 higher than it really is. AVhen both effects are taken into 
 
 account c is more correctly expressed by c = f of — • 
 
 § 31. The Y Level. 
 
 This instrument is shown on page 253. 
 
 The telescope is about 20 inches in length, and 
 rests on supports called F's, from their shape. 
 The spirit level is underneath the telescope, and 
 attached to it. The levelling-head and tripod are 
 similar to the same parts of the transit. 
 
 § 32. The Levelling Rod. 
 
 The two ends of the Philadelphia levelling rod 
 are shown in Fig. 42. The rod is made of two 
 pieces of wood, sliding upon each other, and held 
 together in any position by a clamp. 
 
 The front surface of the rod is graduated to 
 hundredths of a foot up to 7 feet. If a greater 
 height than 7 feet is desired, the back part of the 
 rod is moved up until the target is at the required 
 height. When the rod is extended to full length, 
 the front surface of the rear half reads from 7 to 
 13 feet, so that the rod becomes a self -reading rod 13 feet long. 
 
 |gS 
 
 Fig. 42. 
 
252 
 
 SURVEYING. 
 
 The target slides along the front of the rod, and is held in 
 place by two springs which press upon the sides of the rod. 
 It has a square opening at the centre, through which the 
 division line of the rod opposite to the horizontal line of the 
 target may be seen. It carries a vernier by which heights 
 may be read to thousandths of a foot (§ 7). 
 
 § 33. Difference of Level. 
 
 To find the difference of level between two places visible from 
 an intermediate place, when the difference of level does not 
 exceed 13 feet. 
 
 Let A and B (Fig. 43) represent the two places. Set the 
 Y level at a station equally distant, or nearly so, from A and 
 
 Fig. 43. 
 
 B, but not necessarily on the line AB. Place the legs of the 
 tripod firmly in the ground, and level over each opposite pair 
 of levelling screws, successively. Let the rodman hold the 
 levelling rod vertically at A. Bring the telescope to bear 
 upon the rod (§ 8), and by signal direct the rodman to move 
 the target until its horizontal line is in the line of apparent 
 level of the telescope. Let the rodman now record the height 
 AA' of the target. In like manner find BB'. The difference 
 between AA' and BB' will be the difference of level required. 
 If the instrument is equally distant from A and Bj or nearly 
 so, the curvature and the refraction on the two sides of the 
 instrument balance, and no correction for curvature or refrac- 
 tion will be necessary. 
 
ilTT"^^^ '-W I " ' H III ^^ 
 
 THE Y LEVEL. 
 
DIFFERENCE OF LEVEL. 
 
 255 
 
 If the instrument be set up at one station, and the rod at 
 the other, the difference between the heights of the optical 
 axis of the telescope and the target, corrected for curvature 
 and refraction, will be the difference of level required. 
 
 To find the difierence of level of two places, one of which 
 cannot be seen from the other, and both invisible from the same 
 place ; or, when the two places difier considerably in level. 
 
 Let A and D (Fig. 44) represent the two places. Place the 
 level midway between A and some intermediate station B. 
 
 Fig. 44. 
 
 Find AA' and BB', as in the preceding case, and record the 
 former as a back-sight and the latter as a fore-sight. Select 
 another intermediate station C, and in like manner find the 
 back-sight BB" and the fore-sight CC ] and so continue until 
 the place D is reached. 
 
 The difference between the sum of the fore-sights and the sum 
 of the back-sights will be the difference of level required. 
 For, the sum of the fore-sights 
 
 = BB'-\-CC^-{-DD' 
 = BB" -^ B'B" -\- CC" -{- C'C" -\- BD'. 
 The sum of the back-sights 
 
 = AA'-\-BB"-\-CC". 
 Hence, the difference = B'B" + C"(7" + BB' — AA' 
 = AA"-AA' = AA". 
 
256 surveying. 
 
 § 34. Levelling for Section. 
 
 The intersection of a vertical plane with the surface of the 
 earth is called a Section or Profile. The term profile, how- 
 ever, usually designates the Plot or representation of the 
 section on paper. 
 
 Levelling for Section is levelling to obtain the data neces- 
 sary for making a profile or plot of any required section. 
 
 A profile is made for the purpose of exhibiting in a single 
 view the inequalities of the surface of the ground for great 
 distances along the line of some proposed improvement, such 
 as a railroad, canal, or ditch, and thus facilitating the estab- 
 lishment, of the proper grades. 
 
 The data necessary for making a profile of any required 
 section are, the heights of its different points above some 
 assumed horizontal plane, called the Datum Plane, together 
 with their horizontal distances apart or from the beginning of 
 the section. 
 
 The position of the datum plane is fixed with reference to 
 some permanent object near the beginning of the section, 
 called a Bench Mark, and, in order to avoid negative heights, 
 is assumed at such a distance below this mark that all the 
 points of the section shall be above it. 
 
 The heights of the different points of the section above 
 the datum plane are determined by means of the level and 
 levelling-rod ; and the horizontal length of the section is 
 measured with an engineer's chain or tape, and divided into 
 equal parts, one hundred feet in length, called Stations, 
 marked by stakes numbered 0, 1, 2, 3, etc. 
 
 Where the ground is very irregular, it may be necessary, 
 besides taking sights at the regular stakes, to take occasional 
 sights at points between them. If, for instance*, at a point 
 sixty feet in advance of stake 8 there is a sudden rise or fall 
 in the surface, the height of this point would be determined 
 and recorded as at stake 8.60. 
 
LEVELLING FOR SECTION. 
 
 257 
 
 The readings of the rod are ordinarily taken to the nearest 
 tenth of a foot, except on hench marks and points called 
 turning 2^oints, where they are taken 
 to thousandths of a foot. 
 
 A Turning Point is a point on 
 which the last sight is taken just 
 before changing the position of the 
 level, and the first sight from the 
 new position of the instrument. A 
 turning point may be coincident with 
 one of the stakes, but must always 
 be a hard point, so that the foot of 
 the rod may stand at the same level 
 for both readings. 
 
 To explain the method of obtain- 
 ing the field notes necessary for 
 
 making a profile, let 0, 1, 2, 3, 
 
 11 (Fig. 45) represent a portion of a 
 section to be levelled and plotted. 
 Establish a bench mark at or near 
 the beginning of the line, measure 
 the horizontal length of the section, 
 and set stakes one hundred feet apart, 
 numbering them 0, 1, 2, 3, etc. Place 
 the level at some point, as between 2 
 and 3, and take the reading of the rod 
 on the bench = 4.832. Let PP^ rep- 
 resent the datum plane, say 15 feet 
 below the bench mark, then 
 
 15 + 4.832 = 19.832 
 
 will be the height of the line of sight 
 AB, called the Height of the Instru- 
 ment, above the datum plane. Now take the reading at 
 = 5.2 = 0^, and subtract the same from 19.832, whicli 
 
258 SURVEYING. 
 
 leaves 14.6 = OP, the height of the point above the datum 
 plane. Next take sights at 1, 2, 3, 3.40, and 4 equal respec- 
 tively to 3.7, 3.0, 5.1, 4.8, and 8.3, and subtract the same 
 from 19.832 ; the remainders 16.1, 16.8, 14.7, 15.0, and 11.5 
 will be the respective heights of the points 1, 2, 3, 3.40, 
 and 4. Then, as it will be necessary to change the position 
 of the instrument, select a point in the neighborhood of 4 
 suitable as a turning point (t.p. in the figure), and take a 
 careful reading on it = 8.480 ; subtract this from 19.832, and 
 the remainder, 11.352, will be the height of the turning point. 
 Now carry the instrument forward to a new position, as 
 between 5 and 6, shown in the figure, while the rodman 
 remains at t.p. ; take a second reading on t.p. = 4.102, and 
 add it to 11.352, the height of t.p. above FF' ; the sum 15.454 
 will be the height of the instrument CD in its new position. 
 Now take sight upon 5, 6, and 7, equal respectively to 4.9, 
 2.8, and 0.904; subtract these sights from 15.454, and the 
 results 10.6, 12.7, and 14.550 will be the heights of the points 
 5, 6, and 7 respectively. The point 7, being suitable, is made 
 a turning point, and the instrument is moved forward to a 
 point between 9 and 10. The sight at 7 = 6.870 added to 
 the height of 7 gives 21.420 as the height of the instrument 
 EF in its new position. The readings at 8, 9, 10, and 11, 
 which are respectively 5.4, 3.6, 5.8, and 9.0, subtracted from 
 21.420, will give the heights of these points, namely, 16.0, 
 17.8, 15.6, and 12.4. 
 
 Proceed in like manner until the entire section is levelled, 
 establishing bench marks at intervals along the line to serve 
 as reference points for future operations. 
 
 As wind and bright sunshine affect the accuracy of levelling, 
 for careful work a calm and cloudy day should be chosen ; and 
 great pains be taken to hold the rod vertical and to manipu- 
 late the level properly. 
 
 A record of the work described above is kept as follows .• 
 
LEVELLING FOR SECTION. 
 
 259 
 
 Station. 
 
 + s. 
 
 H.I. 
 
 -s. 
 
 H.S. 
 
 Remabks. 
 
 B 
 
 4.832 
 
 
 
 15. 
 
 Bench on rock 20 ft. 
 
 
 
 
 19.832 
 
 5.2 
 
 14.6 
 
 south of 0. 
 
 1 
 
 
 
 3.7 
 
 16.1 
 
 
 2 
 
 
 
 3.0 
 
 16.8 
 
 
 3 
 
 
 
 5.1 
 
 14.7 
 
 3 to 3.40 turnpike road. 
 
 3.40 
 
 
 
 4.8 
 
 15.0 
 
 
 4 
 
 
 
 8.3 
 
 11.5 
 
 
 tp. 
 
 4.102 
 
 
 8.480 
 
 11.352 
 
 
 5 
 
 
 15.454 
 
 4.9 
 
 10.6 
 
 
 6 
 
 
 
 2.8 
 
 12.7 
 
 
 7 
 
 6.870 
 
 
 0.904 
 
 14.550 
 
 
 8 
 
 
 21.420 
 
 5.4 
 
 16.0 
 
 
 9 
 
 
 
 3.6 
 
 17.8 
 
 
 10 
 
 
 
 5.8 
 
 15.6 
 
 
 11 
 
 
 
 9.0 
 
 12.4 
 
 
 B 
 
 
 
 
 
 Bench on oak stump 
 
 12 
 
 
 
 
 
 27 ft. N.E. of 12, 
 
 etc. 
 
 
 
 
 
 etc. 
 
 The first column contains the numbers or names of all the 
 points on which sights are taken. The second column con- 
 tains the sight taken on the first bench mark, and the sight on 
 each turning point taken immediately after the instrument 
 has been moved to a new position. These are called Plus 
 Sights {-\- S.) because they are added to the heights of the 
 points on which they are taken to obtain the height of the 
 instrument given in the third column (H.I.). The fourth 
 column contains all the readings except those recorded in the 
 second column. These are called Minus Sights (— S.) because 
 they are subtracted from the numbers in the third column to 
 obtain all the numbers in the fifth column except the first, 
 which is the assumed depth of the datum plane below the bench. 
 The fifth column (H.S., height of surface) contains the required 
 heights of all the points of the section named in the first column 
 together with the heights of all benches and turning points. 
 
260 SURVEYING. 
 
 To find the difference of level between any two points of 
 the section, we have only to take the difference between the 
 numbers in the fifth column opposite these points. 
 
 The real field notes are contained in the first, second, fourth, 
 and last columns ; the other columns may be filled after the 
 field operations are completed. The field book may contain 
 other columns; one for height of grade (H.G.), another for 
 depth of cut (C.) and another for height of embankment or 
 fill {F.). 
 
 To plot the section. Draw a line PP' (Fig. 45), to repre- 
 sent the datum plane, and beginning at some point as F, lay 
 off the distances 100, 200, 300, 340, 400 feet, etc., to the 
 right, using some convenient scale, say 200 feet to the inch. 
 At these points of division erect perpendiculars equal in 
 length to the height of the points 0, 1, 2, 3.40, 4, etc., 
 given in the fifth column of the above field notes, using in 
 this case a larger scale, say 20 feet to the inch. Through the 
 extremities of these perpendiculars draw the irregular line 
 
 0, 1, 2, 3 11, and the result, with some explanatory figures, 
 
 will be the required plot or profile. 
 
 The making of a profile is much simplified by the use of 
 profile paper ^ which may be had by the yard or roll. 
 
 If a horizontal plot is required, the bearings of the different 
 portions of the section must be taken. 
 
 A plot should be made, if it will assist in properly under- 
 standing the field work, or if it is desirable for future reference 
 in connection with the field notes. 
 
 § 35. Substitutes for the Y Level. 
 
 For many purposes not requiring accuracy, the following 
 simple instruments in connection with a graduated rod will be 
 found sufficient. 
 
 The Plumb Level (Fig. 46) consists of two pieces of wood 
 joined at riglit angles. A straight line is drawn on the 
 
SUBSTITUTES FOR THE Y LEVEL. 
 
 261 
 
 upright perpendicular to the upper edge of the cross- 
 head. 
 
 The instrument is fastened to a support by a screw through 
 the centre of the cross-head. The upper edge of the cross- 
 head is brought to a level by making the line on the upright 
 coincide with a plumb-line. 
 
 Fig. 46. 
 
 -feEE-EEi 
 
 Fig. 47. 
 
 Fig. 48. 
 
 A modified form is shown in Fig. 47. A carpenter's square 
 is supported by a post, the top of which is split or sawed so as 
 to receive the longer arm. The shorter arm is made vertical 
 by a plumb-line which brings the longer arm to a level. 
 
 The Water Level is shown in Fig. 48. The upright tubes 
 are of glass, cemented into a connecting tube of any suitable 
 material. The whole is nearly filled with water, and sup- 
 ported at a convenient height. The surface of the water in 
 the uprights determines the level. 
 
 By sighting along the upper surface of the block in which 
 the Spirit Level is mounted for the use of mechanics, a level 
 line may be obtained. 
 
 Exercise V. 
 
 1. Find the difference of level of two places from the fol- 
 lowing field notes : back-sights, 5.2, 6.8, and 4.0 ; fore-sights, 
 8.1, 9.5, and 7.9. 
 
 2. Write the proper numbers in the third and fifth columns 
 of the following table of field notes, and make a profile of the 
 section : 
 
262 
 
 SURVEYING. 
 
 Station. 
 
 + s. 
 
 H.I. 
 
 -s. 
 
 H.S. 
 
 Remakks. 
 
 B 
 
 0.944 
 
 
 
 20 
 
 Bench on post 22 ft. 
 
 
 
 
 
 7.4 
 
 
 north of 0. 
 
 1 
 
 
 
 5.6 
 
 
 
 2 
 
 
 
 3.9 
 
 
 
 3 
 
 
 
 4.6 
 
 
 
 tp. 
 
 3.855 
 
 
 5.513 
 
 
 4 
 
 4 
 
 
 
 4.9 
 
 
 
 5 
 
 
 
 3.5 
 
 
 
 6 
 
 
 
 1.2 
 
 
 
 •3. Stake of the following notes stands at the lowest point 
 of a pond to be drained into a creek; stake 10 stands at the 
 edge of the bank, and 10.25 at the bottom of the creek. Make 
 a profile, draw the grade line through and 10.25, and fill 
 out the columns H. G. and C, the former to show the height 
 of grade line above the datum, and the latter, the depth of 
 cut at the several stakes necessary to construct the drain. 
 
 Station. 
 
 + S. 
 
 H.I. 
 
 -s. 
 
 H.S. 
 
 H.G. 
 
 c. 
 
 Remarks. 
 
 B 
 
 6.000 
 
 
 
 25 
 
 
 
 Bench on rock 
 
 
 
 
 
 10.2 
 
 
 20.8 
 
 0.0 
 
 30 feet west of 
 
 1 
 
 
 
 5.3 
 
 
 
 5.3 
 
 stake 1. 
 
 2 
 
 
 
 4.6 
 
 
 
 
 
 3 
 
 
 
 4.0 
 
 
 
 
 
 4 
 
 
 
 6.8 
 
 
 
 
 
 5 
 
 4.572 
 
 
 7.090 
 
 
 
 
 
 6 
 
 
 
 3.9 
 
 
 
 
 
 7 
 
 
 
 2.0 
 
 
 
 
 
 8 
 
 
 
 4.9 
 
 
 
 
 
 9 
 
 
 
 4.3 
 
 
 
 
 
 10 
 
 
 
 4.5 
 
 
 
 
 
 10.25 
 
 
 
 11.8 
 
 
 
 
 
 Horizontal scale, 2 ch. = 1 in. 
 Vertical scale, 20 ft. = 1 in. 
 
TOPOGRAPHICAL LEVELLING. 
 
 263 
 
 § 36. Topographical Levelling. 
 
 The principal object of topographical surveying is to show 
 the contour of the ground. This operation, called topographi- 
 cal levelling, is performed by representing on paper the curved 
 lines in which parallel horizontal planes at uniform distances 
 apart would meet the surface. 
 
 It is evident that all points in the intersection of a horizontal 
 plane with the surface of the ground are at the same level. 
 Hence, it is only necessary to find points at the same level, 
 and join these to determine a line of intersection. 
 
 The method commonly employed will be understood by a 
 reference to Fig. 49. The ground ABCD is divided into 
 equal squares, and a numbered stake driven at each intersec- 
 tion. By means of a level and levelling rod the heights of the 
 other stations above m and D, the lowest stations, are deter- 
 mined. A plot of the ground with the intersecting lines is 
 then drawn, and the height of each 
 station written as in the figure. 
 
 Suppose that the horizontal 
 planes are 2 feet apart; if the 
 first passes through m and D, 
 the second will pass through p, 
 which is 2 feet above m ; and 
 since n is Z feet above m, the 
 second plane will cut the line 
 mn in a point s determined by 
 the proportion mn : ms : : 3 : 2. 
 In like manner the points t, q, 
 and r are determined. 
 
 The irregular line tsp qr represents the intersection of 
 
 the second horizontal plane with the surface of the ground. 
 In like manner the intersections of the planes, respectively, 
 4, 6, and 8 feet above m are traced. The more rapid the change 
 in level the nearer these lines will approach each other. 
 
CHAPTEE V. 
 
 RAILROAD SURVEYING. 
 
 § 37. General Remarks. 
 
 When the general route of a railroad has been "determined, 
 a middle surface line is run with the transit. A profile of this 
 line is determined, as in § 34. The levelling stations are com- 
 monly 1 chain (100 feet) apart. Places of different level are 
 connected by a gradient line, which intersects the perpendic- 
 ulars to the datum line at the levelling stations in points 
 determined by simple proportion. Hence, the distance of each 
 levelling station, above or below the level or gradient line 
 which represents the position of the road bed, is known. 
 
 § 38. Cross Section Work. 
 G' f ly 
 
 Excavations. If the road bed lies below the surface, an 
 excavation is made. 
 
 Let A CDB (Fig. 50) represent a cross section of an excava- 
 tion, / a point in the middle surface line, /' the corresponding 
 point in the road bed, and CD the width of the excavation at 
 the bottom. The slopes at the sides are commonly made so 
 that AA' = iA'C, and BB' = ^DB'. ff and CD being known, 
 the points A, B, C, and D' are readily determine'd by a level 
 and tape measure. 
 
RAILROAD CURVES. 
 
 265 
 
 If from the area of the trapezoid ABB' A' the areas of the 
 triangles AA'C and BB'I) be deducted, the remainder will be 
 the area of the cross section. 
 
 In like manner the cross section at the next station may be 
 determined. These two cross sections will be the bases of a 
 frustum of a quadrangular pyramid whose volume will be the 
 amount of the excavation, approximately. 
 
 Embankments. If the road bed lies above the surface, an 
 embankment is made, the cross section of which is like that of 
 the excavation, but inverted. 
 
 Fig. 51 represents the cross section of an embankment 
 which is lettered so as to show its relation to Fig. 50. 
 
 39. Eailroad Curves. 
 
 When it is necessary to change the direction of a railroad, 
 it is done gradually by a 
 curve, usually the arc of 
 a circle. 
 
 Let AF and AO (Fig. 
 52) represent two lines to 
 be thus connected. Take 
 any convenient distance 
 AB = AE= t The inter- 
 section of the perpendicu- 
 lars BC and EC deter- 
 mines the centre C, and the radius of curvature BC =^r. 
 The length of the radius of curvature will depend on 
 
266 
 
 SURVEYING. 
 
 the angle A and the tangent AB. For, in the right 
 triangle ABC, 
 
 tan BAC = -r^ > or tan ^A = -• 
 Hence, r=^t tan ^A. 
 
 The degree of a railroad curve is the angle subtended at the 
 centre of the curve by a chord of 100 feet. If D is the degree 
 of a curve and r its radius, 
 
 50 
 sin^i) = — and r = 50csc-^i>. 
 
 For example, a 6° curve has a radius of 955.37 feet. 
 
 To Lay out the Curve. 
 
 First Method. Let Bm (Fig. 53) represent a portion of the 
 tangent. It is required to find mP, the 
 perpendicular to the tangent meeting 
 the curve at P. 
 
 mP = Bn=^CB—Cn. 
 But CD = r. 
 
 and 
 
 Cn = ^'CP'-Pn 
 
 Fig. 54. 
 
 Hence, mP = r — V/^ — t^. 
 
 Second Method. It is required to find 
 mP (Fig. 54) in the direction of the 
 centre. 
 
 mP = mC — PC. 
 
 But mC='^~BC''-\-lBm=-J^i^^. 
 Hence, 
 
 mP = VrM^ — n 
 
RAILROAD CURVES. 
 
 267 
 
 Fig. 55. 
 
 Third Method. Place transits afc B and E (Fig. 55). Direct 
 the telescope of the former 
 to E, and of the latter to A. 
 Turn each toward the curve 
 the same number of degrees, 
 and mark P, the jjoint of 
 intersection of the lines of 
 sight. P will be a point in 
 the circle to which AB and 
 AE are tangents at B and E, respectively. 
 
 Fourth Method. If the degree D of the curve is given and 
 the tangent BA at B (Fig. 
 56), place the transit at B 
 and direct toward A. Turn 
 off successively the angles 
 
 ABP, PBP', P'BP'\ each 
 
 equal to -J-Z), and take DP, 
 
 PP\ P'P", each 100 ft., 
 
 the length of the tape. Then 
 
 P, P', P", lie on the 
 
 required curve. 
 
 If the angle A and the tangent distance BA = t are given, 
 D can be found from the formulas 
 
 Fia. 56. 
 
 sin^i>== — J r = ttain^A, 
 
 50 
 '. sin|-i>^ — cot -J- J. 
 
TRANSIT WITH SOLAR ATTACHMENT. 
 
 The circles shown in the cut are intended to represent in miniature 
 circles supposed to be drawn upon the concave surface of the heavens. 
 
SURVEYING. 
 
 1. 8 A. 64 p. 
 
 2. 29 A. 7f p. 
 
 3. 4 A. 5^^ P. 
 
 4. 115^ p. 
 
 1. 2 A. 26 p. 
 
 2. 20 A. 12 p. 
 
 3. 2 A. 54 p. 
 
 4. 2 A. 151 p. 
 
 
 Exercise 
 
 I. 
 
 
 
 5. 
 
 3 A. 78 p. 
 
 
 9. 
 
 13.0735, 
 
 6. 
 
 13 A. 6rV p. 
 
 
 10. 
 
 2 A. 58^ p. 
 
 7. 
 
 11 A. 157 p. 
 
 
 11. 
 
 4 A. 35 p. 
 
 8. 
 
 7.51925. 
 Exercise 
 
 II. 
 
 
 
 5. 
 
 8 A. 54 p. 
 
 
 8. 
 
 3 A. 122 p. 
 
 6. 
 
 5 A. 42 p. 
 
 
 9. 
 
 6 A. 2 p. 
 
 7. 
 
 2 A. 78 p. 
 
 
 10. 
 
 9 A. 40 p. 
 
 Exercise III. 
 1. 2 A. 12^ p. 2. 98 A. 92 p. 
 
 Exercise IV. 
 
 1. AE = 3.75 ch. 9. EM {on EA) = 2.5087 ch. 
 
 2. AE = 3.50 ch. ; AN (on AB) = 6.439 ch. 
 
 10. LetEQ>DF, 
 
 EG = 3.42 ch. 
 AE = 4.55 ch. 
 
 rAE = 12.247 ch. 
 
 4. ^£' = 5.50ch. \aG= 9.798 ch. 
 
 5. CE = 4.456 ch. *^^^ 1 AD = 8.659 ch. 
 
 6. AD = 2.27 bch.; [ AF = 6.928 ch. 
 BE = 1.S2 ch. 11. Let DG > EF, 
 
 7. AD = 4.51 ch. ; rCG= 14.862 ch. 
 ii£: = 3.61ch. I Ci)= 13.113 ch. 
 
 8. The distances on AB are 2, 3, ^^^^ ] CF = 11.404 ch. 
 and5ch. I C^= 10.062 ch. 
 
ANSWERS. 
 
 25 
 
 Exercise V. 
 
 1. 9.5 feet. 
 
 2. Third column : 26.944 opposite ; 25.286 opposite 4. 
 
 Fifth column: 20, 19.5, 21.3, 23, 22.3, 21.431, 20.4, 21.8, 24.1. 
 
 
 
 
 
 
 r'" 
 
 
 il 
 
 Datum Level. 
 
 
 
 V. 
 
 1 
 
 2 3 4 
 
 5 
 
 6 
 
 3. Column a^.G^. 20.8, 20.4, 20.0, 19.6, etc. 
 Column C. 0.0, 5.3, 6.4, 7.4, 5.0, 5.1, etc. 
 
 cq lo o o c<« O 
 
 8 9 10 10.25 
 
70 
 
 
 TABLE VII.- 
 
 -TRAVERSE TABLE. 
 
 
 
 Bearing. 
 
 Distance 1. 
 
 Distance 2. 
 
 Distance 3. 
 
 Distance 4. 
 
 Distance 5. 
 
 Bearing. 
 
 o f 
 
 Lat. 
 
 Dep, 
 
 Lat. Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 O f 
 
 015 
 
 1.000 
 
 0.004 
 
 2.000 0.009 
 
 3.000 
 
 0.013 
 
 4.000 
 
 0.017 
 
 5.000 
 
 0.022 
 
 89 45 
 
 30 
 
 1.000 
 
 0.009 
 
 2.000 0.017 
 
 3.000 
 
 0.026 
 
 4.000 
 
 0.035 
 
 5.000 
 
 0.044 
 
 30 
 
 45 
 
 1.000 
 
 0.013 
 
 2.000 0.026 
 
 3.000 
 
 0.039 
 
 4.000 
 
 0.052 
 
 5.000 
 
 0.065 
 
 15 
 
 1 
 
 1.000 
 
 0.017 
 
 2.000 0.035 
 
 3.000 
 
 0.052 
 
 3.999 
 
 0.070 
 
 4.999 
 
 0.087 
 
 89 
 
 15 
 
 1.000 
 
 0.022 
 
 2.000 0.044 
 
 2.999 
 
 0.065 
 
 3.999 
 
 0.087 
 
 4.999 
 
 0.109 
 
 45 
 
 30 
 
 1.000 
 
 0.026 
 
 1.999 0.052 
 
 2.999 
 
 0.079 
 
 3.999 
 
 0.105 
 
 4.998 
 
 0.131 
 
 30 
 
 45 
 
 1.000 
 
 0.031 
 
 1.999 0.061 
 
 2.999 
 
 0.092 
 
 3.998 
 
 0.122 
 
 4.998 
 
 0.153 
 
 15 
 
 2 
 
 0.999 
 
 0.035 
 
 1.999 0.070 
 
 2.998 
 
 0.105 
 
 3.998 
 
 0.140 
 
 4.997 
 
 0.174 
 
 88 
 
 15 
 
 0.999 
 
 0.039 
 
 1.998 0.079 
 
 2.998 
 
 0.118 
 
 3.997 
 
 0.157 
 
 4.996 
 
 0.196 
 
 45 
 
 30 
 
 0.999 
 
 0.044 
 
 1.998 0.087 
 
 2.997 
 
 0.131 
 
 3.996 
 
 0.174 
 
 4.995 
 
 0.218 
 
 30 
 
 45 
 
 0.999 
 
 0.048 
 
 1.998 '0.096 
 
 2.997 
 
 0.144 
 
 3.995 
 
 0.192 
 
 4.994 
 
 0.240 
 
 15 
 
 3 
 
 0.999 
 
 0.052 
 
 1.997 0.105 
 
 2.996 
 
 0.157 
 
 3.995 
 
 0.209 
 
 4.993 
 
 0.262 
 
 87 
 
 15 
 
 0.998 
 
 0.057 
 
 1.997 0.113 
 
 2.995 
 
 0.170 
 
 3.994 
 
 0.227 
 
 4.992 
 
 0.283 
 
 45 
 
 30 
 
 0.998 
 
 0.061 
 
 1.996 0.122 
 
 2.994 
 
 0.183 
 
 3.993 
 
 0.244 
 
 4.991 
 
 0.305 
 
 30 
 
 45 
 
 0.998 
 
 0.065 
 
 1.996 0.131 
 
 2.994 
 
 0.196 
 
 3.991 
 
 0.262 
 
 4.989 
 
 0.327 
 
 15 
 
 4 
 
 0.998 
 
 0.070 
 
 1.995 0.140 
 
 2.993 
 
 0.209 
 
 3.990 
 
 0.279 
 
 4.988 
 
 0,349 
 
 86 
 
 15 
 
 0.997 
 
 0.074 
 
 1.995 0.148 
 
 2.992 
 
 0.222 
 
 3.989 
 
 0.296 
 
 4.986 
 
 0.371 
 
 45 
 
 30 
 
 0.997 
 
 0.078 
 
 1.994 0.157 
 
 2.991 
 
 0.235 
 
 3.988 
 
 0.314 
 
 4.985 
 
 0.392 
 
 30 
 
 45 
 
 0.997 
 
 0.083 
 
 1.993 0.166 
 
 2.990 
 
 0.248 
 
 3.986 
 
 0.331 
 
 4.983 
 
 0.414 
 
 15 
 
 6 
 
 0.996 
 
 0.087 
 
 1.992 0.174 
 
 2.989 
 
 .0.261 
 
 3.985 
 
 0.349 
 
 4.981 
 
 0.436 
 
 85 
 
 15 
 
 0.996 
 
 0.092 
 
 1.992 0.183 
 
 2.987 
 
 0.275 
 
 3.983 
 
 0.366 
 
 4.979 
 
 0.458 
 
 45 
 
 30 
 
 0.995 
 
 0.096 
 
 1.991 0.192 
 
 2.986 
 
 0.288 
 
 3.982 
 
 0.383 
 
 4.977 
 
 0.479 
 
 30 
 
 45 
 
 0.995 
 
 0.100 
 
 1.990 0.200 
 
 2.985 
 
 0.301 
 
 3.980 
 
 0.401 
 
 4.975 
 
 0.501 
 
 15 
 
 6 
 
 0.995 
 
 0.105 
 
 1.989 0.209 
 
 2.984 
 
 0.314 
 
 3.978 
 
 0.418 
 
 4.973 
 
 0.523 
 
 84 
 
 15 
 
 0.994 
 
 0.109 
 
 1.988 0.218 
 
 2.982 
 
 0.327 
 
 3.976 
 
 0.435 
 
 4.970 
 
 0.544 
 
 45 
 
 30 
 
 0.994 
 
 0.113 
 
 1.987 0.226 
 
 2.981 
 
 0.340 
 
 3.974 
 
 0.453 
 
 4.968 
 
 0.566 
 
 30 
 
 45 
 
 0.993 
 
 0.118 
 
 1.986 0.235 
 
 2.979 
 
 0.353 
 
 3.972 
 
 0.470 
 
 4.965 
 
 0.588 
 
 15 
 
 7 
 
 0.993 
 
 0.122 
 
 1.985 0.244 
 
 2.978 
 
 0.366 
 
 3.970 
 
 0.487 
 
 4.963 
 
 0.609 
 
 83 
 
 15 
 
 0.992 
 
 0.126 
 
 1.984 0.252 
 
 2.976 
 
 0.379 
 
 3.968 
 
 0.505 
 
 4.960 
 
 0.631 
 
 45 
 
 30 
 
 0.991 
 
 0.131 
 
 1.983 0.261 
 
 2.974 
 
 0.392 
 
 3.966 
 
 0.522 
 
 4.957 
 
 0.653 
 
 30 
 
 45 
 
 0.991 
 
 0.135 
 
 1.982 0.270 
 
 2.973 
 
 0.405 
 
 3.963 
 
 0.539 
 
 4.954 
 
 0.674 
 
 15 
 
 8 
 
 0.990 
 
 0.139 
 
 1.981 0.278 
 
 2.971 
 
 0.418 
 
 3.961 
 
 0.557 
 
 4.951 
 
 0.696 
 
 82 
 
 15 
 
 0.990 
 
 0.143 
 
 1.979 0.287 
 
 2.969 
 
 0.430 
 
 3.959 
 
 0.574 
 
 4.948 
 
 0.717 
 
 45 
 
 30 
 
 0.989 
 
 0.148 
 
 1.978 0.296 
 
 2.967 
 
 0.443 
 
 3.956 
 
 0.591 
 
 4.945 
 
 0.739 
 
 30 
 
 45 
 
 0.988 
 
 0.152 
 
 1.977 0.304 
 
 2.965 
 
 0.456 
 
 3.953 
 
 0.608 
 
 4.942 
 
 0.761 
 
 15 
 
 9 
 
 0.988 
 
 0.156 
 
 1.975 0.313 
 
 2.963 
 
 0.469 
 
 3.951 
 
 0.626 
 
 4.938 
 
 0.782 
 
 81 
 
 15 
 
 0.987 
 
 0.161 
 
 1.974 0.321 
 
 2.961 
 
 0.482 
 
 3.948 
 
 0.643 
 
 4.935 
 
 0.804 
 
 45 
 
 30 
 
 0.986 
 
 0.165 
 
 1.973 0.330 
 
 2.959 
 
 0.495 
 
 3.945 
 
 0.660 
 
 4.931 
 
 0.825 
 
 30 
 
 45 
 
 0.986 
 
 0.169 
 
 1.971 0.339 
 
 2.957 
 
 0.508 
 
 3.942 
 
 0.677 
 
 4.928 
 
 0.847 
 
 15 
 
 10 
 
 0.985 
 
 0.174 
 
 1.970 0.347 
 
 2.954 
 
 0.521 
 
 3.939 
 
 0.695 
 
 4.924 
 
 0.868 
 
 80 
 
 15 
 
 0.984 
 
 0.178 
 
 1.968 0.356 
 
 2.952 
 
 0.534 
 
 3.936 
 
 0.712 
 
 4.920 
 
 0.890 
 
 45 
 
 30 
 
 0.983 
 
 0.182 
 
 1.967 0.364 
 
 2.950 
 
 0.547 
 
 3.933 
 
 0.729 
 
 4.916 
 
 0.911 
 
 30 
 
 45 
 
 0.982 
 
 0.187 
 
 1.965 0.373 
 
 2.947 
 
 0.560 
 
 3.930 
 
 0.746 
 
 4.912 
 
 0.933 
 
 15 
 
 11 
 
 0.982 
 
 0.191 
 
 1.963 0.382 
 
 2.945 
 
 0.572 
 
 3.927 
 
 0.763 
 
 4.908 
 
 0.954 
 
 79 
 
 15 
 
 0.981 
 
 0.195 
 
 1.962 0.390 
 
 2.942 
 
 0.585 
 
 3.923 
 
 0.780 
 
 4.904 
 
 0.975 
 
 45 
 
 30 
 
 0.980 
 
 0.199 
 
 1.960 0.399 
 
 2.940 
 
 0.598 
 
 3.920 
 
 0.797 
 
 4.900 
 
 0.997 
 
 30 
 
 45 
 
 0.979 
 
 0.204 
 
 1.958 0.407 
 
 2.937 
 
 0611 
 
 3.916 
 
 0.815 
 
 4.895 
 
 1.018 
 
 15 
 
 12 
 
 0.978 
 
 0.208 
 
 1.956 0.416 
 
 2.934 
 
 0.624 
 
 3.913 
 
 0.832 
 
 4.891 
 
 1.040 
 
 78 
 
 15 
 
 0.977 
 
 0.212 
 
 1.954 0.424 
 
 2.932 
 
 0.637 
 
 3.909 
 
 0.849 
 
 4.886 
 
 1.061 
 
 45 
 
 30 
 
 0.976 
 
 0.216 
 
 1.953 0.433 
 
 2.929 
 
 0.649 
 
 3.905 
 
 0.866 
 
 4.881 
 
 1.082 
 
 30 
 
 45 
 
 0.975 
 
 0.221 
 
 1.951 0.441 
 
 2.926 
 
 0.662 
 
 3.901 
 
 0.883 
 
 4.877 
 
 1.103 
 
 15 
 
 13 
 
 0.974 
 
 0.225 
 
 1.949 0.450 
 
 2.923 
 
 0.675 
 
 3.897 
 
 0.900 
 
 4.872 
 
 1.125 
 
 77 
 
 15 
 
 0.973 
 
 0.229 
 
 1.947 0.458 
 
 2.920 
 
 0.688 
 
 3.894 
 
 0.917 
 
 4.867 
 
 1.146 
 
 45 
 
 30 
 
 0.972 
 
 0.233 
 
 1.945 0.467 
 
 2.917 
 
 0.700 
 
 3.889 
 
 0.934 
 
 4.862 
 
 1.167 
 
 30 
 
 45 
 
 0.971 
 
 0.238 
 
 1.943 0.475 
 
 2.914 
 
 0.713 
 
 3.885 
 
 0.951 
 
 4.857 
 
 1.188 
 
 15 
 
 14 
 
 0.970 
 
 0.242 
 
 1.941 0.484 
 
 2.911 
 
 0.726 
 
 3.881 
 
 0.968 
 
 4.851 
 
 1.210 
 
 76 
 
 15 
 
 0.969 
 
 0.246 
 
 1.938 0.492 
 
 2.908 
 
 0.738 
 
 3.877 
 
 0.985 
 
 4.846 
 
 1.231 
 
 45 
 
 30 
 
 0.968 
 
 0.250 
 
 1.936 0.501 
 
 2.904 
 
 0.751 
 
 3.873 
 
 1.002 
 
 4.841 
 
 1.252 
 
 30 
 
 45 
 
 0.967 
 
 0.255 
 
 1.934 0.509 
 
 2.901 
 
 0.764 
 
 3.868 
 
 1.018 
 
 4.835 
 
 1.273 
 
 15 
 
 16 
 
 0.966 
 
 0.259 
 
 1.932 0.518 
 
 2.898 
 
 0.776 
 
 3.864 
 
 1.035 
 
 4.830 
 
 1.294 
 
 75 
 
 o r 
 
 Dap. Lat. 
 Distance 1. 
 
 Dep. Lat. 
 Distance 2. 
 
 Dep. Lat. 
 Distance 3. 
 
 Dep. Lat. 
 Distance 4. 
 
 Dep. Lat. 
 Distance 5. 
 
 O f 
 
 Bearing. 
 
 Bearing. 
 
 76°-90^ 
 

 
 
 
 
 o°- 
 
 -16° 
 
 
 
 
 
 71 
 
 Bearing. 
 
 Distance 6. 
 
 Distance 7. 
 
 Distance 8. 
 
 Distance 9. 
 
 Distance 10. 
 
 Bearing. 
 
 o f 
 
 Lat. 
 
 Dep, 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 O f 
 
 15 
 
 6.000 
 
 0.026 
 
 7.000 
 
 0.031 
 
 8.000 
 
 0.035 
 
 9.000 
 
 0.039 
 
 10.000 
 
 0.044 
 
 89 45 
 
 30 
 
 6.000 
 
 0.052 
 
 7.000 
 
 0.061 
 
 8.000 
 
 0.070 
 
 9.000 
 
 0.079 
 
 10.000 
 
 0.087 
 
 30 
 
 45 
 
 5.999 
 
 0.079 
 
 6.999 
 
 0.092 
 
 7.999 
 
 0.105 
 
 8.999 
 
 0.118 
 
 9.999 
 
 0.131 
 
 15 
 
 1 
 
 5.999 
 
 0.105 
 
 6.999 
 
 0.122 
 
 7.999 
 
 0.140 
 
 8.999 
 
 0.157 
 
 9.999 
 
 0.175 
 
 89 
 
 15 
 
 5.999 
 
 0.131 
 
 6.998 
 
 0.153 
 
 7.998 
 
 0.175 
 
 8.998 
 
 0.196 
 
 9.998 
 
 0.218 
 
 45 
 
 30 
 
 5.998 
 
 0.157 
 
 6.998 
 
 0.183 
 
 7.997 
 
 0.209 
 
 8.997 
 
 0.236 
 
 9.997 
 
 0.262 
 
 30 
 
 45 
 
 5.997 
 
 0.183 
 
 6.997 
 
 0.214 
 
 7.996 
 
 0.244 
 
 8.996 
 
 0.275 
 
 9.995 
 
 0.305 
 
 15 
 
 2 
 
 5.996 
 
 0.209 
 
 6.996 
 
 0.244 
 
 7.995 
 
 0.279 
 
 8.995 
 
 0.314 
 
 9.994 
 
 0.349 
 
 88 
 
 15 
 
 5.995 
 
 0.236 
 
 6.995 
 
 0.275 
 
 7.994 
 
 0.314 
 
 8.993 
 
 0.353 
 
 9.992 
 
 0.393 
 
 45 
 
 30 
 
 5.994 
 
 0.262 
 
 6.993 
 
 0.305 
 
 7.992 
 
 0.349 
 
 8.991 
 
 0.393 
 
 9.991 
 
 0.436 
 
 30 
 
 45 
 
 5.993 
 
 0.288 
 
 6.992 
 
 0.336 
 
 7.991 
 
 0.384 
 
 8.990 
 
 0.432 
 
 9.989 
 
 0.480 
 
 15 
 
 3 
 
 5.992 
 
 0.314 
 
 6.990 
 
 0.366 
 
 7.989 
 
 0.419 
 
 8.988 
 
 0.471 
 
 9.986 
 
 0.523 
 
 87 
 
 15 
 
 5.990 
 
 0.340 
 
 6.989 
 
 0.397 
 
 7.987 
 
 0.454 
 
 8.986 
 
 0.510 
 
 9.984 
 
 0.567 
 
 45 
 
 30 
 
 5.989 
 
 0.366 
 
 6.987 
 
 0.427 
 
 7.985 
 
 0.488 
 
 8.983 
 
 0.549 
 
 9.981 
 
 0.611 
 
 30 
 
 45 
 
 5.987 
 
 0.392 
 
 6.985 
 
 0.458 
 
 7.983 
 
 0.523 
 
 8.981 
 
 0.589 
 
 9.979 
 
 0.654 
 
 15 
 
 4 
 
 5.985 
 
 0.419 
 
 6.983 
 
 0.488 
 
 7.981 
 
 0.558 
 
 8.978 
 
 0.628 
 
 9.976 
 
 0.698 
 
 86 
 
 15 
 
 5.984 
 
 0.445 
 
 6.981 
 
 0.519 
 
 7.978 
 
 0.593 
 
 8.975 
 
 0.667 
 
 9.973 
 
 0.741 
 
 45 
 
 30 
 
 5.982 
 
 0.471 
 
 6.978 
 
 0.549 
 
 7.975 
 
 0.628 
 
 8.972 
 
 0.706 
 
 9.969 
 
 0.785 
 
 30 
 
 45 
 
 5.979 
 
 0.497 
 
 6.976 
 
 0.580 
 
 7.973 
 
 0.662 
 
 8.969 
 
 0.745 
 
 9.966 
 
 0.828 
 
 15 
 
 5 
 
 5.977 
 
 0.523 
 
 6.973 
 
 0.610 
 
 7.970 
 
 0.697 
 
 8.966 
 
 0.784 
 
 9.962 
 
 0.872 
 
 85 
 
 15 
 
 5.975 
 
 0.549 
 
 6.971 
 
 0.641 
 
 7.966 
 
 0.732 
 
 8.962 
 
 0.824 
 
 9.958 
 
 0.915 
 
 45 
 
 30 
 
 5.972 
 
 0.575 
 
 6.968 
 
 0.671 
 
 7.963 
 
 0.767 
 
 8.959 
 
 0.863 
 
 9.954 
 
 0.959 
 
 30 
 
 45 
 
 5.970 
 
 0.601 
 
 6.965 
 
 0.701 
 
 7.960 
 
 0.802 
 
 8.955 
 
 0.902 
 
 9.950 
 
 1.002 
 
 15 
 
 6 
 
 5.967 
 
 0.627 
 
 6.962 
 
 0.732 
 
 7.956 
 
 0.836 
 
 8.951 
 
 0.941 
 
 9.945 
 
 1.045 
 
 84 
 
 15 
 
 5.964 
 
 0.653 
 
 6.958 
 
 0.762 
 
 7.952 
 
 0.871 
 
 8.947 
 
 0.980 
 
 9.941 
 
 1.089 
 
 45 
 
 30 
 
 5.961 
 
 0.679 
 
 6.955 
 
 0.792 
 
 7.949 
 
 0.906 
 
 8.942 
 
 1.019 
 
 9.936 
 
 1.132 
 
 30 
 
 45 
 
 5.958 
 
 0.705 
 
 6.951 
 
 0.823 
 
 7.945 
 
 0.940 
 
 8.938 
 
 1.058 
 
 9.931 
 
 1.175 
 
 15 
 
 7 
 
 5.955 
 
 0.731 
 
 6.948 
 
 0.853 
 
 7.940 
 
 0.975 
 
 8.933 
 
 1.097 
 
 9.926 
 
 1.219 
 
 83 
 
 15 
 
 5.952 
 
 0.757 
 
 6.944 
 
 0.883 
 
 7.936 
 
 1.010 
 
 8.928 
 
 1.136 
 
 9.920 
 
 1.262 
 
 45 
 
 30 
 
 5.949 
 
 0.783 
 
 6.940 
 
 0.914 
 
 7.932 
 
 1.044 
 
 8.923 
 
 1.175 
 
 9.914 
 
 1.305 
 
 30 
 
 45 
 
 5.945 
 
 0.809 
 
 6.936 
 
 0.944 
 
 7.927 
 
 1.079 
 
 8.918 
 
 1.214 
 
 9.909 
 
 1.349 
 
 15 
 
 8 
 
 5.942 
 
 0.835 
 
 6.932 
 
 0.974 
 
 7.922 
 
 1.113 
 
 8.912 
 
 1.253 
 
 9.903 
 
 1.392 
 
 82 
 
 15 
 
 5.938 
 
 0.861 
 
 6.928 
 
 1.004 
 
 7.917 
 
 1.148 
 
 8.907 
 
 1.291 
 
 9.897 
 
 1.435 
 
 45 
 
 30 
 
 5.934 
 
 0.887 
 
 6.923 
 
 1.035 
 
 7.912 
 
 1.182 
 
 8.901 
 
 1.330 
 
 9.890 
 
 1.478 
 
 30 
 
 45 
 
 5.930 
 
 0.913 
 
 6.919 
 
 1.065 
 
 7.907 
 
 1.217 
 
 8.895 
 
 1.369 
 
 9.884 
 
 1.521 
 
 15 
 
 9 
 
 5.926 
 
 0.939 
 
 6.914 
 
 1.095 
 
 7.902 
 
 1.251 
 
 8.889 
 
 1.408 
 
 9.877 
 
 1.564 
 
 81 
 
 15 
 
 5.922 
 
 0.964 
 
 6.909 
 
 1.125 
 
 7.896 
 
 1.286 
 
 8.883 
 
 1.447 
 
 9.870 
 
 1.607 
 
 45 
 
 30 
 
 5.918 
 
 0.990 
 
 6.904 
 
 1.155 
 
 7.890 
 
 1.320 
 
 8.877 
 
 1.485 
 
 9.863 
 
 1.651 
 
 30 
 
 45 
 
 5.913 
 
 1.016 
 
 6.899 
 
 1.185 
 
 7.884 
 
 1.355 
 
 8.870 
 
 1.524 
 
 9.856 
 
 1.694 
 
 15 
 
 10 
 
 5.909 
 
 1.042 
 
 6.894 
 
 1.216 
 
 7.878 
 
 1.389 
 
 8.863 
 
 1.563 
 
 9.848 
 
 1.737 
 
 80 
 
 15 
 
 5.904 
 
 1.068 
 
 6.888 
 
 1.246 
 
 7.872 
 
 1.424 
 
 8.856 
 
 1.601 
 
 9.840 
 
 1.779 
 
 45 
 
 30 
 
 5.900 
 
 1.093 
 
 6.883 
 
 1.276 
 
 7.866 
 
 1.458 
 
 8.849 
 
 1.640 
 
 9.833 
 
 1.822 
 
 30 
 
 45 
 
 5.895 
 
 1.119 
 
 6.877 
 
 1.306 
 
 7.860 
 
 1.492 
 
 8.842 
 
 1.679 
 
 9.825 
 
 1.865 
 
 15 
 
 11 
 
 5.890 
 
 1.145 
 
 6.871 
 
 1.336 
 
 7.853 
 
 1.526 
 
 8.835 
 
 1.717 
 
 9.816 
 
 1.908 
 
 79 
 
 15 
 
 5.885 
 
 1.171 
 
 6.866 
 
 1.366 
 
 7.846 
 
 1.561 
 
 8.827 
 
 1.756 
 
 9.808 
 
 1.951 
 
 45 
 
 30 
 
 5.880 
 
 1.196 
 
 6.859 
 
 1.396 
 
 7.839 
 
 1.595 
 
 8.819 
 
 1.794 
 
 9.799 
 
 1.994 
 
 30 
 
 45 
 
 5.874 
 
 1.222 
 
 6.853 
 
 1.425 
 
 7.832 
 
 1.629 
 
 8.811 
 
 1.833 
 
 9.791 
 
 2.036 
 
 15 
 
 12 
 
 5.869 
 
 1.247 
 
 6.847 
 
 1.455 
 
 7.825 
 
 1.663 
 
 8.803 
 
 1.871 
 
 9.782 
 
 2.079 
 
 78 
 
 15 
 
 5.863 
 
 1.273 
 
 6.841 
 
 1.485 
 
 7.818 
 
 1.697 
 
 8.795 
 
 1.910 
 
 9.772 
 
 2.122 
 
 45 
 
 30 
 
 5.858 
 
 1.299 
 
 6.834 
 
 1.515 
 
 7.810 
 
 1.732 
 
 8.787 
 
 1.948 
 
 9.763 
 
 2.164 
 
 30 
 
 45 
 
 5.852 
 
 1.324 
 
 6.827 
 
 1.545 
 
 7.803 
 
 1.766 
 
 8.778 
 
 1.986 
 
 9.753 
 
 2.207 
 
 15 
 
 13 
 
 5.846 
 
 1.350 
 
 6.821 
 
 1.575 
 
 7.795 
 
 1.800 
 
 8.769 
 
 2.025 
 
 9.744 
 
 2.250 
 
 77 
 
 15 
 
 5.840 
 
 1.375 
 
 6.814 
 
 1.604 
 
 7.787 
 
 1.834 
 
 8.760 
 
 2.063 
 
 9.734 
 
 2.292 
 
 45 
 
 30 
 
 5.834 
 
 1.401 
 
 6.807 
 
 1.634 
 
 7.779 
 
 1.868 
 
 8.751 
 
 2.101 
 
 9.724 
 
 2.335 
 
 30 
 
 45 
 
 5.828 
 
 1.426 
 
 6.799 
 
 1.664 
 
 7.771 
 
 1.902 
 
 8.742 
 
 2.139 
 
 9.713 
 
 2.377 
 
 15 
 
 14 
 
 5.822 
 
 1.452 
 
 6.792 
 
 1.693 
 
 7.762 
 
 1.935 
 
 8.733 
 
 2.177 
 
 9.703 
 
 2.419 
 
 76 
 
 15 
 
 5.815 
 
 1.477 
 
 6.785 
 
 1.723 
 
 7.754 
 
 1.969 
 
 8.723 
 
 2.215 
 
 9.692 
 
 2.462 
 
 45 
 
 30 
 
 5.809 
 
 1.502 
 
 6.777 
 
 1.753 
 
 7.745 
 
 2.003 
 
 8.713 
 
 2.253 
 
 9.682 
 
 2.504 
 
 30 
 
 45 
 
 5.802 
 
 1.528 
 
 6.769 
 
 1.782 
 
 7.736 
 
 2.037 
 
 8.703 
 
 2.291 
 
 9.671 
 
 2.546 
 
 15 
 
 15 
 
 5.796 
 
 1.553 
 
 6.761 
 
 1.812 
 
 7.727 
 
 2.071 
 
 8.693 
 
 2.329 
 
 9.659 
 
 2.588 
 
 75 
 
 o r 
 
 Dep. Lat. 
 Distance 6. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 o f 
 
 Bearing, 
 
 Distance 7. 
 
 Distance 8. 
 
 Distance 9. 
 
 Distance 10. 
 
 Bearing. 
 
 76° -90' 
 
72 
 
 
 
 
 
 15°- 
 
 -30° 
 
 
 
 
 
 Bearing. 
 
 Distance 1. 
 
 Distance 2. 
 
 Distance 3. 
 
 Distance 4. 
 
 Distance 5. 
 
 Bearing. 
 
 o f 
 
 Lat. 
 
 Dep, 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 O f 
 
 15 15 
 
 0.965 
 
 0.263 
 
 1.930 
 
 0.526 
 
 2.894 
 
 0.789 
 
 3.859 
 
 1.052 
 
 4.824 
 
 1.315 
 
 74 45 
 
 30 
 
 0.964 
 
 0.267 
 
 1.927 
 
 0.534 
 
 2.891 
 
 0.802 
 
 3.855 
 
 1.069 
 
 4.818 
 
 1.336 
 
 30 
 
 45 
 
 0.962 
 
 0.271 
 
 1.925 
 
 0.543 
 
 2.887 
 
 0.814 
 
 3.850 
 
 1.086 
 
 4.812 
 
 1.357 
 
 15 
 
 16 
 
 0.961 
 
 0.276 
 
 1.923 
 
 0.551 
 
 2.884 
 
 0.827 
 
 3.845 
 
 1.103 
 
 4.806 
 
 1.378 
 
 74 
 
 15 
 
 0.960 
 
 0.280 
 
 1.920 
 
 0.560 
 
 2.880 
 
 0.839 
 
 3.840 
 
 1.119 
 
 4.800 
 
 1.399 
 
 45 
 
 30 
 
 0.959 
 
 0.284 
 
 1.918 
 
 0.568 
 
 2.876 
 
 0.852 
 
 3.835 
 
 1.136 
 
 4.794 
 
 1.420 
 
 30 
 
 45 
 
 0.958 
 
 0.288 
 
 1.915 
 
 0.576 
 
 2.873 
 
 0.865 
 
 3.830 
 
 1.153 
 
 4.788 
 
 1.441 
 
 15 
 
 17 
 
 0.956 
 
 0.292 
 
 1.913 
 
 0.585 
 
 2.869 
 
 0.877 
 
 3.825 
 
 1.169 
 
 4.782 
 
 1.462 
 
 73 
 
 15 
 
 0.955 
 
 0.297 
 
 1.910 
 
 0.593 
 
 2.865 
 
 0.890 
 
 3.820 
 
 1.186 
 
 4.775 
 
 1.483 
 
 45 
 
 30 
 
 0.954 
 
 0.301 
 
 1.907 
 
 0.601 
 
 2.861 
 
 0.902 
 
 3.815 
 
 1.203 
 
 4.769 
 
 1.504 
 
 30 
 
 45 
 
 0.952 
 
 0.305 
 
 1.905 
 
 0.610 
 
 2.857 
 
 0.915 
 
 3.810 
 
 1.220 
 
 4.762 
 
 1.524 
 
 15 
 
 18 
 
 0.951 
 
 0.309 
 
 1.902 
 
 0.618 
 
 2.853 
 
 0.927 
 
 3.804 
 
 1.236 
 
 4.755 
 
 1.545 
 
 72 
 
 15 
 
 0.950 
 
 0.313 
 
 1.899 
 
 0.626 
 
 2.849 
 
 0.939 
 
 3.799 
 
 1.253 
 
 4.748 
 
 1.566 
 
 45 
 
 30 
 
 0.948 
 
 0.317 
 
 1.897 
 
 0.635 
 
 2.845 
 
 0.952 
 
 3.793 
 
 1.269 
 
 4.742 
 
 1.587 
 
 30 
 
 45 
 
 0.947 
 
 0.321 
 
 1.894 
 
 0.643 
 
 2.841 
 
 0.964 
 
 3.788 
 
 1.286 
 
 4.735 
 
 1.607 
 
 15 
 
 19 
 
 0.946 
 
 0.326 
 
 1.891 
 
 0.651 
 
 2.837 
 
 0.977 
 
 3.782 
 
 1.302 
 
 4.728 
 
 1.628 
 
 71 
 
 15 
 
 0.944 
 
 0.330 
 
 1.888 
 
 0.659 
 
 2.832 
 
 0.989 
 
 3.776 
 
 1.319 
 
 4.720 
 
 1.648 
 
 45 
 
 30 
 
 0.943 
 
 0.334 
 
 1.885 
 
 0.668 
 
 2.828 
 
 1.001 
 
 3.771 
 
 1.335 
 
 4.713 
 
 1.669 
 
 30 
 
 45 
 
 0.941 
 
 0.338 
 
 1.882 
 
 0.676 
 
 2.824 
 
 1.014 
 
 3.765 
 
 1.352 
 
 4.706 
 
 1.690 
 
 15 
 
 20 
 
 0.940 
 
 0.342 
 
 1.879 
 
 0.684 
 
 2.819 
 
 1.026 
 
 3.759 
 
 1.368 
 
 4.698 
 
 1.710 
 
 70 
 
 IS 
 
 0.938 
 
 0.346 
 
 1.876 
 
 0.692 
 
 2.815 
 
 1.038 
 
 3.753 
 
 1.384 
 
 4.691 
 
 1.731 
 
 45 
 
 30 
 
 0.937 
 
 0.350 
 
 1.873 
 
 0.700 
 
 2.810 
 
 1.051 
 
 3.747 
 
 1.401 
 
 4.683 
 
 1.751 
 
 30 
 
 45 
 
 0.935 
 
 0.354 
 
 1.870 
 
 0.709 
 
 2.805 
 
 1.063 
 
 3.741 
 
 1.417 
 
 4.676 
 
 1.771 
 
 15 
 
 21 
 
 0.934 
 
 0.358 
 
 1.867 
 
 0.717 
 
 2.801 
 
 1.075 
 
 3.734 
 
 1.433 
 
 4.668 
 
 1.792 
 
 69 
 
 15 
 
 0.932 
 
 0.362 
 
 1.864 
 
 0.725 
 
 2.796 
 
 1.087 
 
 3.728 
 
 1.450 
 
 4.660 
 
 1.812 
 
 45 
 
 30 
 
 0.930 
 
 0.367 
 
 1.861 
 
 0.733 
 
 2.791 
 
 1.100 
 
 3.722 
 
 1.466 
 
 4.652 
 
 1.833 
 
 30 
 
 45 
 
 0.929 
 
 0.371 
 
 1.858 
 
 0.741 
 
 2.786 
 
 1.112 
 
 3.715 
 
 1.482 
 
 4.644 
 
 1.853 
 
 15 
 
 22 
 
 0.927 
 
 0.375 
 
 1.854 
 
 0.749 
 
 2.782 
 
 1.124 
 
 3.709 
 
 1.498 
 
 4.636 
 
 1.873 
 
 68 
 
 15 
 
 0.926 
 
 0.379 
 
 1.851 
 
 0.757 
 
 2.777 
 
 1.136 
 
 3.702 
 
 1.515 
 
 4.628 
 
 1.893 
 
 45 
 
 30 
 
 0.924 
 
 0.383 
 
 1.848 
 
 0.765 
 
 2.772 
 
 1.148 
 
 3.696 
 
 1.531 
 
 4.619 
 
 1.913 
 
 30 
 
 45 
 
 0.922 
 
 0.387 
 
 1.844 
 
 0.773 
 
 2.767 
 
 1.160 
 
 3.689 
 
 1.547 
 
 4.611 
 
 1.934 
 
 15 
 
 23 
 
 0.921 
 
 0.391 
 
 1.841 
 
 0.781 
 
 2.762 
 
 1.172 
 
 3.682 
 
 1.563 
 
 4.603 
 
 1.954 
 
 67 
 
 15 
 
 0.919 
 
 0.395 
 
 1.838 
 
 0.789 
 
 2.756 
 
 1.184 
 
 3.675 
 
 1.579 
 
 4.594 
 
 1.974 
 
 45 
 
 30 
 
 0.917 
 
 0.399 
 
 1.834 
 
 0.797 
 
 2.751 
 
 1.196 
 
 3.668 
 
 1.595 
 
 4.585 
 
 1.994 
 
 30 
 
 45 
 
 0.915 
 
 0.403 
 
 1.831 
 
 0.805 
 
 2.746 
 
 1.208 
 
 3.661 
 
 1.611 
 
 4.577 
 
 2.014 
 
 15 
 
 24 
 
 0.914 
 
 0.407 
 
 1.827 
 
 0.813 
 
 2.741 
 
 1.220 
 
 3.654 
 
 1.627 
 
 4.568 
 
 2.034 
 
 66 
 
 15 
 
 0.912 
 
 0.411 
 
 1.824 
 
 0.821 
 
 2.735 
 
 1.232 
 
 3.647 
 
 1.643 
 
 4.559 
 
 2.054 
 
 45 
 
 30 
 
 0.910 
 
 0.415 
 
 1.820 
 
 0.829 
 
 2.730 
 
 1.244 
 
 3.640 
 
 1.659 
 
 4.550 
 
 2.073 
 
 30 
 
 45 
 
 0.908 
 
 0.419 
 
 1.816 
 
 0.837 
 
 2.724 
 
 1.256 
 
 3.633 
 
 1.675 
 
 4.541 
 
 2.093 
 
 15 
 
 25 
 
 0.906 
 
 0.423 
 
 1.813 
 
 0.845 
 
 2.719 
 
 1.268 
 
 3.625 
 
 1.690 
 
 4.532 
 
 2.113 
 
 65 
 
 15 
 
 0.904 
 
 0.427 
 
 1.809 
 
 0.853 
 
 2.713 
 
 1.280 
 
 3.618 
 
 1.706 
 
 4.522 
 
 2.133 
 
 45 
 
 30 
 
 0.903 
 
 0.431 
 
 1.805 
 
 0.861 
 
 2.708 
 
 1.292 
 
 3.610 
 
 1.722 
 
 4.513 
 
 2.153 
 
 30 
 
 45 
 
 0.901 
 
 0.434 
 
 1.801 
 
 0.869 
 
 2.702 
 
 1.303 
 
 3.603 
 
 1.738 
 
 4.503 
 
 2.172 
 
 15 
 
 26 
 
 0.899 
 
 0.438 
 
 1.798 
 
 0.877 
 
 2.696 
 
 1.315 
 
 3.595 
 
 1.753 
 
 4.494 
 
 2.192 
 
 64 
 
 15 
 
 897 
 
 0.442 
 
 1.794 
 
 0.885 
 
 2.691 
 
 1.327 
 
 3.587 
 
 1.769 
 
 4.484 
 
 2.211 
 
 45 
 
 30 
 
 0.895 
 
 0.446 
 
 1.790 
 
 892 
 
 2.685 
 
 1.339 
 
 3.580 
 
 1.785 
 
 4.475 
 
 2.231 
 
 30 
 
 45 
 
 0.893 
 
 0.450 
 
 1.786 
 
 0.900 
 
 2.679 
 
 1.350 
 
 3.572 
 
 1.800 
 
 4.465 
 
 2.250 
 
 15 
 
 27 
 
 0.891 
 
 0.454 
 
 1.782 
 
 0.908 
 
 2.673 
 
 1.362 
 
 3.564 
 
 1.816 
 
 4.455 
 
 2.270 
 
 63 
 
 15 
 
 0.889 
 
 0.458 
 
 1.778 
 
 0.916 
 
 2.667 
 
 1.374 
 
 3.556 
 
 1.831 
 
 4.445 
 
 2.289 
 
 45 
 
 30 
 
 0.887 
 
 0.462 
 
 1.774 
 
 0.923 
 
 2.661 
 
 1.385 
 
 3.548 
 
 1.847 
 
 4.435 
 
 2.309 
 
 30 
 
 45 
 
 0.885 
 
 0.466 
 
 1.770 
 
 0.931 
 
 2.655 
 
 1.397 
 
 3.540 
 
 1.862 
 
 4.425 
 
 2.328 
 
 15 
 
 28 
 
 0.883 
 
 0.469 
 
 1.766 
 
 0.939 
 
 2.649 
 
 1.408 
 
 3.532 
 
 1.878 
 
 4.415 
 
 2.347 
 
 62 
 
 15 
 
 0.881 
 
 0.473 
 
 1.762 
 
 0.947 
 
 2.643 
 
 1.420 
 
 3.524 
 
 1.893 
 
 4.404 
 
 2.367 
 
 45 
 
 30 
 
 0.879 
 
 0.477 
 
 1.758 
 
 0.954 
 
 2.636 
 
 1.431 
 
 3.515 
 
 1.909 
 
 4.394 
 
 2.386 
 
 30 
 
 45 
 
 0.877 
 
 0.481 
 
 1.753 
 
 0.962 
 
 2.630 
 
 1.443 
 
 3.507 
 
 1.924 
 
 4.384 
 
 2.405 
 
 15 
 
 29 
 
 0.875 
 
 0.485 
 
 1.749 
 
 0.970 
 
 2.624 
 
 1.454 
 
 3.498 
 
 1.939 
 
 4.373 
 
 2.424 
 
 61 
 
 15 
 
 0.872 
 
 0.489 
 
 1.745 
 
 0.977 
 
 2.617 
 
 1.466 
 
 3.490 
 
 1.954 
 
 4.362 
 
 2.443 
 
 45 
 
 30 
 
 0.870 
 
 0.492 
 
 1.741 
 
 0.985 
 
 2.611 
 
 1.477 
 
 3.481 
 
 1.970 
 
 4.352 
 
 2.462 
 
 30 
 
 45 
 
 0.868 
 
 0.496 
 
 1.736 
 
 0.992 
 
 2.605 
 
 1.489 
 
 3.473 
 
 1.985 
 
 4.341 
 
 2.481 
 
 15 
 
 30 
 
 0.866 
 
 0.500 
 
 1.732 
 
 1.000 
 
 2.598 
 
 1.500 
 
 3.464 
 
 2.000 
 
 4.330 
 
 2.500 
 
 60 
 
 O f 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 O f 
 
 Bearing. 
 
 Distance 1. 
 
 Distance 2. 
 
 Distance 3. 
 
 Distance 4. 
 
 Distance 5. 
 
 Bearing. 
 
 60° - 75^ 
 

 
 
 
 
 16°- 
 
 -30 
 
 O 
 
 
 
 
 73 
 
 Bearing. 
 
 Distance 6. 
 
 Distance 7. 
 
 Distance 8. 
 
 Distance 9. 
 
 Distance 10. 
 
 Bearing. 
 
 o r 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 O f 
 
 15 15 
 
 5.789 
 
 1.578 
 
 6.754 
 
 1.841 
 
 7.718 
 
 2.104 
 
 8.683 
 
 2.367 
 
 9.648 
 
 2.630 
 
 74 45 
 
 30 
 
 5.782 
 
 1.603 
 
 6.745 
 
 1.871 
 
 7.709 
 
 2.138 
 
 8.673 
 
 2.405 
 
 9.636 
 
 2.672 
 
 30 
 
 45 
 
 5.775 
 
 1.629 
 
 6.737 
 
 1.900 
 
 7.700 
 
 2.172 
 
 8.662 
 
 2.443 
 
 9.625 
 
 2.714 
 
 15 
 
 16 
 
 5.768 
 
 1.654 
 
 6.729 
 
 1.929 
 
 7.690 
 
 2.205 
 
 8.651 
 
 2.481 
 
 9.613 
 
 2.756 
 
 74 
 
 15 
 
 5.760 
 
 1.679 
 
 6.720 
 
 1.959 
 
 7.680 
 
 2.239 
 
 8.640 
 
 2.518 
 
 9.601 
 
 2.798 
 
 45 
 
 30 
 
 5.753 
 
 1.704 
 
 6.712 
 
 1.988 
 
 7.671 
 
 2.272 
 
 8.629 
 
 2.556 
 
 9.588 
 
 2.840 
 
 30 
 
 45 
 
 5.745 
 
 1.729 
 
 6.703 
 
 2.017 
 
 7.661 
 
 2.306 
 
 8.618 
 
 2.594 
 
 9.576 
 
 2.882 
 
 15 
 
 17 
 
 5.738 
 
 1.754 
 
 6.694 
 
 2.047 
 
 7.650 
 
 2.339 
 
 8.607 
 
 2.631 
 
 9.563 
 
 2.924 
 
 73 
 
 15 
 
 5.730 
 
 1.779 
 
 6.685 
 
 2.076 
 
 7.640 
 
 2.372 
 
 8.595 
 
 2.669 
 
 9.550 
 
 2.965 
 
 45 
 
 30 
 
 5.722 
 
 1.804 
 
 6.676 
 
 2.105 
 
 7.630 
 
 2.406 
 
 8.583 
 
 2.706 
 
 9.537 
 
 3.007 
 
 30 
 
 45 
 
 5.714 
 
 1.829 
 
 6.667 
 
 2.134 
 
 7.619 
 
 2.439 
 
 8.572 
 
 2.744 
 
 9.524 
 
 3.049 
 
 15 
 
 18 
 
 5.706 
 
 1.854 
 
 6.657 
 
 2.163 
 
 7.608 
 
 2.472 
 
 8.560 
 
 2.781 
 
 9.511 
 
 3.090 
 
 72 
 
 15 
 
 5.698 
 
 1.879 
 
 6.648 
 
 2.192 
 
 7.598 
 
 2.505 
 
 8.547 
 
 2.818 
 
 9.497 
 
 3.132 
 
 45 
 
 30 
 
 5.690 
 
 1.904 
 
 6.638 
 
 2.221 
 
 7.587 
 
 2.538 
 
 8.535 
 
 2.856 
 
 9.483 
 
 3.173 
 
 30 
 
 45 
 
 5.682 
 
 1.929 
 
 6.629 
 
 2.250 
 
 7.575 
 
 2.572 
 
 8.522 
 
 2.893 
 
 9.469 
 
 3.214 
 
 15 
 
 19 
 
 5.673 
 
 1.953 
 
 6.619 
 
 2.279 
 
 7.564 
 
 2.605 
 
 8.510 
 
 2.930 
 
 9.455 
 
 3.256 
 
 71 
 
 15 
 
 5.665 
 
 1.978 
 
 6.609 
 
 2.308 
 
 7.553 
 
 2.638 
 
 8.497 
 
 2.967 
 
 9.441 
 
 3.297 
 
 45 
 
 30 
 
 5.656 
 
 2.003 
 
 6.598 
 
 2.337 
 
 7.541 
 
 2.670 
 
 8.484 
 
 3.004 
 
 9.426 
 
 3.338 
 
 30 
 
 45 
 
 5.647 
 
 2.028 
 
 6.588 
 
 2.365 
 
 7.529 
 
 2.703 
 
 8.471 
 
 3.041 
 
 9.412 
 
 3.379 
 
 15 
 
 20 
 
 5.638 
 
 2.052 
 
 6.578 
 
 2.394 
 
 7.518 
 
 2.736 
 
 8.457 
 
 3.078 
 
 9.397 
 
 3.420 
 
 70 
 
 15 
 
 5.629 
 
 2.077 
 
 6.567 
 
 2.423 
 
 7.506 
 
 2.769 
 
 8.444 
 
 3.115 
 
 9.382 
 
 3.461 
 
 45 
 
 30 
 
 5.620 
 
 2.101 
 
 6.557 
 
 2.451 
 
 7.493 
 
 2.802 
 
 8.430 
 
 3.152 
 
 9.367 
 
 3.502 
 
 30 
 
 45 
 
 5.611 
 
 2.126 
 
 6.546 
 
 2.480 
 
 7.481 
 
 2.834 
 
 8.416 
 
 3.189 
 
 9.351 
 
 3.543 
 
 15 
 
 21 
 
 5.601 
 
 2.150 
 
 6.535 
 
 2.509 
 
 7.469 
 
 2.867 
 
 8.402 
 
 3.225 
 
 9.336 
 
 3.584 
 
 69 
 
 15 
 
 5.592 
 
 2.175 
 
 6.524 
 
 2.537 
 
 7.456 
 
 2.900 
 
 8.388 
 
 3.262 
 
 9.320 
 
 3.624 
 
 45 
 
 30 
 
 5.582 
 
 2.199 
 
 6.513 
 
 2.566 
 
 7.443 
 
 2.932 
 
 8.374 
 
 3.299 
 
 9.304 
 
 3.665 
 
 30 
 
 45 
 
 5.573 
 
 2.223 
 
 6.502 
 
 2.594 
 
 7.430 
 
 2.964 
 
 8.359 
 
 3.335 
 
 9.288 
 
 3.706 
 
 15 
 
 22 
 
 5.563 
 
 2.248 
 
 6.490 
 
 2.622 
 
 7.417 
 
 2.997 
 
 8.345 
 
 3.371 
 
 9.272 
 
 3.746 
 
 68 
 
 15 
 
 5.553 
 
 2.272 
 
 6.479 
 
 2.651 
 
 7.404 
 
 3.029 
 
 8.330 
 
 3.408 
 
 9.255 
 
 3.787 
 
 45 
 
 30 
 
 5.543 
 
 2.296 
 
 6.467 
 
 2.679 
 
 7.391 
 
 3.061 
 
 8.315 
 
 3.444 
 
 9.239 
 
 3.827 
 
 30 
 
 45 
 
 5.533 
 
 2.320 
 
 6.455 
 
 2.707 
 
 7.378 
 
 3.094 
 
 8.300 
 
 3.480 
 
 9.222 
 
 3.867 
 
 15 
 
 23 
 
 5.523 
 
 2.344 
 
 6.444 
 
 2.735 
 
 7.364 
 
 3.126 
 
 8.285 
 
 3.517 
 
 9.205 
 
 3.907 
 
 67 
 
 15 
 
 5.513 
 
 2.368 
 
 6.432 
 
 2.763 
 
 7.350 
 
 3.158 
 
 8.269 
 
 3.553 
 
 9.188 
 
 3.947 
 
 45 
 
 30 
 
 5.502 
 
 2.392 
 
 6.419 
 
 2.791 
 
 7.336 
 
 3.190 
 
 8.254 
 
 3.589 
 
 9.171 
 
 3.988 
 
 30 
 
 45 
 
 5.492 
 
 2.416 
 
 6.407 
 
 2.819 
 
 7.322 
 
 3.222 
 
 8.238 
 
 3.625 
 
 9.153 
 
 4.028 
 
 15 
 
 24 
 
 5.481 
 
 2.440 
 
 6.395 
 
 2.847 
 
 7.308 
 
 3.254 
 
 8.222 
 
 3.661 
 
 9.136 
 
 4.067 
 
 66 
 
 15 
 
 5.471 
 
 2.464 
 
 6.382 
 
 2.875 
 
 7.294 
 
 3.286 
 
 8.206 
 
 3.696 
 
 9.118 
 
 4.107 
 
 45 
 
 30 
 
 5.460 
 
 2.488 
 
 6.370 
 
 2.903 
 
 7.280 
 
 3.318 
 
 8.190 
 
 3.732 
 
 9.100 
 
 4.147 
 
 30 
 
 45 
 
 5.449 
 
 2.512 
 
 6.357 
 
 2.931 
 
 7.265 
 
 3.349 
 
 8.173 
 
 3.768 
 
 9.081 
 
 4.187 
 
 15 
 
 25 
 
 5.438 
 
 2.536 
 
 6.344 
 
 2.958 
 
 7.250 
 
 3.381 
 
 8.157 
 
 3.804 
 
 9.063 
 
 4.226 
 
 65 
 
 15 
 
 5.427 
 
 2.559 
 
 6.331 
 
 2.986 
 
 7.236 
 
 3.413 
 
 8.140 
 
 3.839 
 
 9.045 
 
 4.266 
 
 45 
 
 30 
 
 5.416 
 
 2.583 
 
 6.318 
 
 3.014 
 
 7.221 
 
 3.444 
 
 8.123 
 
 3.875 
 
 9.026 
 
 4.305 
 
 30 
 
 45 
 
 5.404 
 
 2.607 
 
 6.305 
 
 3.041 
 
 7.206 
 
 3.476 
 
 8.106 
 
 3.910 
 
 9.007 
 
 4.345 
 
 15 
 
 26 
 
 5.393 
 
 2.630 
 
 6.292 
 
 3.069 
 
 7.190 
 
 3.507 
 
 8.089 
 
 3.945 
 
 8.988 
 
 4.384 
 
 64 
 
 15 
 
 5.381 
 
 2.654 
 
 6.278 
 
 3.096 
 
 7.175 
 
 3.538 
 
 8.072 
 
 3.981 
 
 8.969 
 
 4.423 
 
 45 
 
 30 
 
 5.370 
 
 2.677 
 
 6.265 
 
 3.123 
 
 7.160 
 
 3.570 
 
 8.054 
 
 4.016 
 
 8.949 
 
 4.462 
 
 30 
 
 45 
 
 5.358 
 
 2.701 
 
 6.251 
 
 3.151 
 
 7.144 
 
 3.601 
 
 8.037 
 
 4.051 
 
 8.930 
 
 4.501 
 
 15 
 
 2T 
 
 5.346 
 
 2.724 
 
 6.237 
 
 3.178 
 
 7.128 
 
 3.632 
 
 8.019 
 
 4.086 
 
 8.910 
 
 4.540 
 
 63 
 
 15 
 
 5.334 
 
 2.747 
 
 6.223 
 
 3.205 
 
 7.112 
 
 3.663 
 
 8.001 
 
 4.121 
 
 8.890 
 
 4.579 
 
 45 
 
 30 
 
 5.322 
 
 2.770 
 
 6.209 
 
 3.232 
 
 7.096 
 
 3.694 
 
 7.983 
 
 4.156 
 
 8.870 
 
 4.618 
 
 30 
 
 45 
 
 5.310 
 
 2.794 
 
 6.195 
 
 3.259 
 
 7.080 
 
 3.725 
 
 7.965 
 
 4.190 
 
 8.850 
 
 4.656 
 
 15 
 
 28 
 
 5.298 
 
 2.817 
 
 6.181 
 
 3.286 
 
 7.064 
 
 3.756 
 
 7.947 
 
 4.225 
 
 . 8.829 
 
 4.695 
 
 62 
 
 15 
 
 5.285 
 
 2.840 
 
 6.166 
 
 3.313 
 
 7.047 
 
 3.787 
 
 7.928 
 
 4.260 
 
 8.809 
 
 4.733 
 
 45 
 
 30 
 
 5.273 
 
 2.863 
 
 6.152 
 
 3.340 
 
 7.031 
 
 3.817 
 
 7.909 
 
 4.294 
 
 8.788 
 
 4.772 
 
 30 
 
 45 
 
 5.260 
 
 2.886 
 
 6.137 
 
 3.367 
 
 7.014 
 
 3.848 
 
 7.891 
 
 4329 
 
 8.767 
 
 4.810 
 
 15 
 
 29 
 
 5.248 
 
 2.909 
 
 6.122 
 
 3.394 
 
 6.997 
 
 3.878 
 
 7.872 
 
 4.363 
 
 8.746 
 
 4.848 
 
 61 
 
 15 
 
 5.235 
 
 2.932 
 
 6.107 
 
 3.420 
 
 6.980 
 
 3.909 
 
 7.852 
 
 4.398 
 
 8.725 
 
 4.886 
 
 45 
 
 30 
 
 5.222 
 
 2.955 
 
 6.093 
 
 3.447 
 
 6.963 
 
 3.939 
 
 7.833 
 
 4.432 
 
 8.704 
 
 4.924 
 
 30 
 
 45 
 
 5.209 
 
 2.977 
 
 6.077 
 
 3.474 
 
 6.946 
 
 3.970 
 
 7.814 
 
 4.466 
 
 8.682 
 
 4.962 
 
 15 
 
 30 
 
 5.196 
 
 3.000 
 
 6.062 
 
 3.500 
 
 6.928 
 
 4.000 
 
 7.794 
 
 4.500 
 
 8.660 
 
 5.000 
 
 60 
 
 o r 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 O f 
 
 Bearing. 
 
 Distance 6. 
 
 Distance 7. 
 
 Distance 8. 
 
 Distance 9. 
 
 Distance 10. 
 
 Bearing. 
 
 60° -76' 
 
74 
 
 
 
 
 
 30° -45 
 
 o 
 
 
 
 
 
 Bearing. 
 
 Distance 1. 
 
 Distance 2. 
 
 Distance 3. 
 
 Distance 4. 
 
 Distance 5. 
 
 Bearing. 
 
 O f 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat, Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 O f 
 
 30 15 
 
 0.864 
 
 0.504 
 
 1.728 
 
 1.008 
 
 2.592 1.511 
 
 3.455 
 
 2.015 
 
 4.319 
 
 2.519 
 
 59 45 
 
 30 
 
 0.862 
 
 0.508 
 
 1.723 
 
 1.015 
 
 2.585 1.523 
 
 3.447 
 
 2.030 
 
 4.308 
 
 2.538 
 
 30 
 
 45 
 
 0.859 
 
 0.511 
 
 1.719 
 
 1.023 
 
 2.578 1.534 
 
 3.438 
 
 2.045 
 
 4.297 
 
 2.556 
 
 15 
 
 31 
 
 0.857 
 
 0.515 
 
 1.714 
 
 1.030 
 
 2.572 1.545 
 
 3.429 
 
 2.060 
 
 4.286 
 
 2.575 
 
 59 
 
 15 
 
 0.855 
 
 0.519 
 
 1.710 
 
 1.038 
 
 2.565 1.556 
 
 3.420 
 
 2.075 
 
 4.275 
 
 2.594 
 
 45 
 
 30 
 
 0.853 
 
 0.522 
 
 1.705 
 
 1.045 
 
 2.558 1.567 
 
 3.411 
 
 2.090 
 
 4.263 
 
 2.612 
 
 30 
 
 45 
 
 0.850 
 
 0.526 
 
 1.701 
 
 1.052 
 
 2.551 1.579 
 
 3.401 
 
 2.105 
 
 4.252 
 
 2.631 
 
 15 
 
 32 
 
 0.848 
 
 0.530 
 
 1.696 
 
 1.060 
 
 2.544 1.590 
 
 3.392 
 
 2.120 
 
 4.240 
 
 2.650 
 
 58 
 
 15 
 
 0.846 
 
 0.534 
 
 1.691 
 
 1.067 
 
 2.537 1.601 
 
 3.383 
 
 2.134 
 
 4.229 
 
 2.668 
 
 45 
 
 30 
 
 0.843 
 
 0.537 
 
 1.687 
 
 1.075 
 
 2.530 1.612 
 
 3.374 
 
 2.149 
 
 4.217 
 
 2.686 
 
 30 
 
 45 
 
 0.841 
 
 0.541 
 
 1.682 
 
 1.082 
 
 2.523 1.623 
 
 3.364 
 
 2.164 
 
 4.205 
 
 2.705 
 
 15 
 
 33 
 
 0.839 
 
 0.545 
 
 1.677 
 
 1.089 
 
 2.516 1.634 
 
 3.355 
 
 2.179 
 
 4.193 
 
 2.723 
 
 57 
 
 IS 
 
 0.836 
 
 0.548 
 
 1.673 
 
 1.097 
 
 2.509 1.645 
 
 3.345 
 
 2.193 
 
 4.181 
 
 2.741 
 
 45 
 
 30 
 
 0.834 
 
 0.552 
 
 1.668 
 
 1.104 
 
 2.502 1.656 
 
 3.336 
 
 2.208 
 
 4.169 
 
 2.760 
 
 30 
 
 45 
 
 0.831 
 
 0.556 
 
 1.663 
 
 1.111 
 
 2.494 1.667 
 
 3.326 
 
 2.222 
 
 4.157 
 
 2.778 
 
 15 
 
 34 
 
 0.829 
 
 0.559 
 
 1.658 
 
 1.118 
 
 2.487 1.678 
 
 3.316 
 
 2.237 
 
 4.145 
 
 2.796 
 
 56 
 
 15 
 
 0.827 
 
 0.563 
 
 1.653 
 
 1.126 
 
 2.480 1.688 
 
 3.306 
 
 2.251 
 
 4.133 
 
 2.814 
 
 45 
 
 30 
 
 0.824 
 
 0.566 
 
 1.648 
 
 1.133 
 
 2.472 1.699 
 
 3.297 
 
 2.266 
 
 4.121 
 
 2.832 
 
 30 
 
 45 
 
 0.822 
 
 0.570 
 
 1.643 
 
 1.140 
 
 2.465 1.710 
 
 3.287 
 
 2.280 
 
 4.108 
 
 2.850 
 
 15 
 
 35 
 
 0.819 
 
 0.574 
 
 1.638 
 
 1.147 
 
 2.457 1.721 
 
 3.277 
 
 2.294 
 
 4.096 
 
 2.868 
 
 ^^ 
 
 15 
 
 0.817 
 
 0.577 
 
 1.633 
 
 1.154 
 
 2.450' 1.731 
 
 3.267 
 
 2.309 
 
 4.083 
 
 2.886 
 
 45 
 
 30 
 
 0.814 
 
 0.581 
 
 1.628 
 
 1.161 
 
 2.442 1.742 
 
 3.257 
 
 2.323 
 
 4.071 
 
 2.904 
 
 30 
 
 45 
 
 0.812 
 
 0.584 
 
 1.623 
 
 1.168 
 
 2.435 1.753 
 
 3.246 
 
 2.337 
 
 4.058 
 
 2.921 
 
 IS 
 
 36 
 
 0.809 
 
 0.588 
 
 1.618 
 
 1.176 
 
 2.427 1.763 
 
 3.236 
 
 2.351 
 
 4.045 
 
 2.939 
 
 54 
 
 15 
 
 0.806 
 
 0.591 
 
 1.613 
 
 1.183 
 
 2.419 1.774 
 
 3.226 
 
 2.365 
 
 4.032 
 
 2.957 
 
 45 
 
 30 
 
 0.804 
 
 0.595 
 
 1.608 
 
 1.190 
 
 2.412 1.784 
 
 3.215 
 
 2.379 
 
 4.019 
 
 2.974 
 
 30 
 
 45 
 
 0.801 
 
 0.598 
 
 1.603 
 
 1.197 
 
 2.404 1.795 
 
 3.205 
 
 2.393 
 
 4.006 
 
 2.992 
 
 15 
 
 37 
 
 0.799 
 
 0.602 
 
 1.597 
 
 1.204 
 
 2.396 1.805 
 
 3.195 
 
 2.407 
 
 3.993 
 
 3.009 
 
 53 
 
 15 
 
 0.796 
 
 0.605 
 
 1.592 
 
 1.211 
 
 2.388 1.816 
 
 3.184 
 
 2.421 
 
 3.980 
 
 3.026 
 
 45 
 
 30 
 
 0.793 
 
 0.609 
 
 1.587 
 
 1.218 
 
 2.380 1.826 
 
 3.173 
 
 2.435 
 
 3.967 
 
 3.044 
 
 30 
 
 45 
 
 0.791 
 
 0.612 
 
 1.581 
 
 1.224 
 
 2.372 1.837 
 
 3.163 
 
 2.449 
 
 3.953 
 
 3.061 
 
 15 
 
 38 
 
 0.788 
 
 0.616 
 
 1.576 
 
 1.231 
 
 2.364 1.847 
 
 3.152 
 
 2.463 
 
 3.940 
 
 3.078 
 
 52 
 
 15 
 
 0.785 
 
 0.619 
 
 1.571 
 
 1.238 
 
 2.356 1.857 
 
 3.141 
 
 2.476 
 
 3.927 
 
 3.095 
 
 45 
 
 30 
 
 0.783 
 
 0.623 
 
 1.565 
 
 1.245 
 
 2.348 1.868 
 
 3.130 
 
 2.490 
 
 3.913 
 
 3.113 
 
 30 
 
 45 
 
 0.780 
 
 0.626 
 
 1.560 
 
 1.252 
 
 2.340 1.878 
 
 3.120 
 
 2.504 
 
 3.899 
 
 3.130 
 
 15 
 
 39 
 
 0.777 
 
 0.629 
 
 1.554 
 
 1.259 
 
 2.331 1.888 
 
 3.109 
 
 2.517 
 
 3.886 
 
 3.147 
 
 51 
 
 15 
 
 0.774 
 
 0.633 
 
 1.549 
 
 1.265 
 
 2.323 1.898 
 
 3.098 
 
 2.531 
 
 3.872 
 
 3.164 
 
 45 
 
 30 
 
 0.772 
 
 0.636 
 
 1.543 
 
 1.272 
 
 2.315 1.908 
 
 3.086 
 
 2.544 
 
 3.858 
 
 3.180 
 
 30 
 
 45 
 
 0.769 
 
 0.639 
 
 1.538 
 
 1.279 
 
 2.307 1.918 
 
 3.075 
 
 2.558 
 
 3.844 
 
 3.197 
 
 15 
 
 40 
 
 0.766 
 
 0.643 
 
 1.532 
 
 1.286 
 
 2.298 1.928 
 
 3.064 
 
 2.571 
 
 3.830 
 
 3.214 
 
 60 
 
 15 
 
 0.763 
 
 0.646 
 
 1.526 
 
 1.292 
 
 2.290 1.938 
 
 3.053 
 
 2.584 
 
 3.816 
 
 3.231 
 
 45 
 
 30 
 
 0.760 
 
 0.649 
 
 1.521 
 
 1.299 
 
 2.281 1.948 
 
 3.042 
 
 2.598 
 
 3.802 
 
 3.247 
 
 30 
 
 45 
 
 0.758 
 
 0.653 
 
 1.515 
 
 1.306 
 
 2.273 1.958 
 
 3.030 
 
 2.611 
 
 3.788 
 
 3.264 
 
 15 
 
 41 
 
 0.755 
 
 0.656 
 
 1.509 
 
 1.312 
 
 2.264 1.968 
 
 3.019 
 
 2.624 
 
 3.774 
 
 3.280 
 
 49 
 
 IS 
 
 0.752 
 
 0.659 
 
 1.504 
 
 1.319 
 
 2.256 1.978 
 
 3.007 
 
 2.637 
 
 3.759 
 
 3.297 
 
 45 
 
 30 
 
 0.749 
 
 0.663 
 
 1.498 
 
 1.325 
 
 2.247 1.988 
 
 2.996 
 
 2.650 
 
 3.745 
 
 3.313 
 
 30 
 
 45 
 
 0.746 
 
 0.666 
 
 1.492 
 
 1.332 
 
 2.238 1.998 
 
 2.984 
 
 2.664 
 
 3.730 
 
 3.329 
 
 15 
 
 42 
 
 0.743 
 
 0.669 
 
 1.486 
 
 1.338 
 
 2.229 2.007 
 
 2.973 
 
 2.677 
 
 3.716 
 
 3.346 
 
 48 
 
 15 
 
 0.740 
 
 0.672 
 
 1.480 
 
 1.345 
 
 2.221 2.017 
 
 2.961 
 
 2.689 
 
 3.701 
 
 3.362 
 
 45 
 
 30 
 
 0.737 
 
 0.676 
 
 1.475 
 
 1.351 
 
 2.212 2.027 
 
 2.949 
 
 2.702 
 
 3.686 
 
 3.378 
 
 30 
 
 45 
 
 0.734 
 
 0.679 
 
 1.469 
 
 1.358 
 
 2.203 2.036 
 
 2.937 
 
 2.715 
 
 3.672 
 
 3.394 
 
 15 
 
 43 
 
 0.731 
 
 0.682 
 
 1.463 
 
 1.364 
 
 2.194 2.046 
 
 2.925 
 
 2.728 
 
 3.657 
 
 3.410 
 
 47 
 
 15 
 
 0.728 
 
 0.685 
 
 1.457 
 
 1.370 
 
 2.185 2.056 
 
 2.913 
 
 2.741 
 
 3.642 
 
 3.426 
 
 45 
 
 30 
 
 0.725 
 
 0.688 
 
 1.451 
 
 1.377 
 
 2.176 2.065 
 
 2.901 
 
 2.753 
 
 3.627 
 
 3.442 
 
 30 
 
 45 
 
 0.722 
 
 0.692 
 
 1.445 
 
 1.383 
 
 2.167 2.075 
 
 2.889 
 
 2.766 
 
 3.612 
 
 3.458 
 
 15 
 
 44 
 
 0.719 
 
 0.695 
 
 1.439 
 
 1.389 
 
 2.158 2.084 
 
 2.877 
 
 2.779 
 
 3.597 
 
 3.473 
 
 46 
 
 15 
 
 0.716 
 
 0.698 
 
 1.433 
 
 1.396 
 
 2.149 2.093 
 
 2.865 
 
 2.791 
 
 3.582 
 
 3.489 
 
 45 
 
 30 
 
 0.713 
 
 0.701 
 
 1.427 
 
 1.402 
 
 2.140 2.103 
 
 2.853 
 
 2.804 
 
 3.566 
 
 3.505 
 
 30 
 
 45 
 
 0.710 
 
 0.704 
 
 1.420 
 
 1.408 
 
 2.131 2.112 
 
 2.841 
 
 2.816 
 
 3.551 
 
 3.520 
 
 15 
 
 45 
 
 0.707 
 
 0.707 
 
 1.414 
 
 1.414 
 
 2,121 2.121 
 
 2.828 
 
 2.828 
 
 3.536 
 
 3.536 
 
 45 
 
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 Dep. 
 
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 Dep. 
 
 Lat. 
 
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 Lat. 
 
 o r 
 
 Bearing. 
 
 Distance 1. 
 
 Distance 2. 
 
 Distance 3. 
 
 Distance 4. 
 
 Distance 5. 
 
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 46° -60' 
 

 
 
 
 
 30°- 
 
 -45 
 
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 75 
 
 Bearing. 
 
 Distance 6. 
 
 Distance 7. 
 
 Distance 8. 
 
 Distance 9. 
 
 Distance 10. 
 
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 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
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 Dep. 
 
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 30 15 
 
 5.183 
 
 3.023 
 
 6.047 
 
 3.526 
 
 6.911 
 
 4.030 
 
 7.775 
 
 4.534 
 
 8.638 
 
 5.038 
 
 59 45 
 
 30 
 
 5.170 
 
 3.045 
 
 6.031 
 
 3.553 
 
 6.893 
 
 4.060 
 
 7.755 
 
 4.568 
 
 8.616 
 
 5.075 
 
 30 
 
 45 
 
 5.156 
 
 3.068 
 
 6.016 
 
 3.579 
 
 6.875 
 
 4.090 
 
 7.735 
 
 4.602 
 
 8.594 
 
 5.113 
 
 15 
 
 31 
 
 5.143 
 
 3.090 
 
 6.000 
 
 3.605 
 
 6.857 
 
 4.120 
 
 7.715 
 
 4.635 
 
 8.572 
 
 5.150 
 
 59 
 
 15 
 
 5.129 
 
 3.113 
 
 5.984 
 
 3.631 
 
 6.839 
 
 4.150 
 
 7.694 
 
 4.669 
 
 8.549 
 
 5.188 
 
 45 
 
 30 
 
 5.116 
 
 3.135 
 
 5.968 
 
 3.657 
 
 6.821 
 
 4.180 
 
 7.674 
 
 4.702 
 
 8.526 
 
 5.225 
 
 30 
 
 45 
 
 5.102 
 
 3.157 
 
 5.952 
 
 3.683 
 
 6.803 
 
 4.210 
 
 7.653 
 
 4.736 
 
 8.504 
 
 5.262 
 
 15 
 
 32 
 
 5.088 
 
 3.180 
 
 5.936 
 
 3.709 
 
 6.784 
 
 4.239 
 
 7.632 
 
 4.769 
 
 8.481 
 
 5.299 
 
 58 
 
 15 
 
 5.074 
 
 3.202 
 
 5.920 
 
 3.735 
 
 6.766 
 
 4.269 
 
 7.612 
 
 4.802 
 
 8.457 
 
 5.336 
 
 45 
 
 30 
 
 5.060 
 
 3.224 
 
 5.904 
 
 3.761 
 
 6.747 
 
 4.298 
 
 7.591 
 
 4.836 
 
 8.434 
 
 5.373 
 
 30 
 
 45 
 
 5.046 
 
 3.246 
 
 5.887 
 
 3.787 
 
 6.728 
 
 4.328 
 
 7.569 
 
 4.869 
 
 8.410 
 
 5.410 
 
 15 
 
 33 
 
 5.032 
 
 3.268 
 
 5.871 
 
 3.812 
 
 6.709 
 
 4.357 
 
 7.548 
 
 4.902 
 
 8.387 
 
 5.446 
 
 57 
 
 15 
 
 5.018 
 
 3.290 
 
 5.854 
 
 3.838 
 
 6.690 
 
 4.386 
 
 7.527 
 
 4.935 
 
 8.363 
 
 5.483 
 
 45 
 
 30 
 
 5.003 
 
 3.312 
 
 5.837 
 
 3.864 
 
 6.671 
 
 4.416 
 
 7.505 
 
 4.967 
 
 8.339 
 
 5.519 
 
 30 
 
 45 
 
 4.989 
 
 3.333 
 
 5.820 
 
 3.889 
 
 6.652 
 
 4.445 
 
 7.483 
 
 5.000 
 
 8.315 
 
 5.556 
 
 15 
 
 34 
 
 4.974 
 
 3.355 
 
 5.803 
 
 3.914 
 
 6.632 
 
 4.474 
 
 7.461 
 
 5.033 
 
 8.290 
 
 5.592 
 
 56 
 
 15 
 
 4.960 
 
 3.377 
 
 5.786 
 
 3.940 
 
 6.613 
 
 4.502 
 
 7.439 
 
 5.065 
 
 8.266 
 
 5.628 
 
 45 
 
 30 
 
 4.945 
 
 3.398 
 
 5.769 
 
 3.965 
 
 6.593 
 
 4.531 
 
 7.417 
 
 5.098 
 
 8.241 
 
 5.664 
 
 30 
 
 45 
 
 4.930 
 
 3.420 
 
 5.752 
 
 3.990 
 
 6.573 
 
 4.560 
 
 7.395 
 
 5.130 
 
 8.217 
 
 5.700 
 
 15 
 
 35 
 
 4.915 
 
 3.441 
 
 5.734 
 
 4.015 
 
 6.553 
 
 4.589 
 
 7.372 
 
 5.162 
 
 8.192 
 
 5.736 
 
 55 
 
 15 
 
 4.900 
 
 3.463 
 
 5.716 
 
 4.040 
 
 6.533 
 
 4.617 
 
 7.350 
 
 5.194 
 
 8.166 
 
 5.772 
 
 45 
 
 30 
 
 4.885 
 
 3.484 
 
 5.699 
 
 4.065 
 
 6.513 
 
 4.646 
 
 7.327 
 
 5.226 
 
 8.141 
 
 5.807 
 
 30 
 
 45 
 
 4.869 
 
 3.505 
 
 5.681 
 
 4.090 
 
 6.493 
 
 4.674 
 
 7.304 
 
 5.258 
 
 8.116 
 
 5.843 
 
 15 
 
 36 
 
 4.854 
 
 3.527 
 
 5.663 
 
 4.115 
 
 6.472 
 
 4.702 
 
 7.281 
 
 5.290 
 
 8.090 
 
 5.878 
 
 54 
 
 15 
 
 4.839 
 
 3.548 
 
 5.645 
 
 4.139 
 
 6.452 
 
 4.730 
 
 7.258 
 
 5.322 
 
 8.064 
 
 5.913 
 
 45 
 
 30 
 
 4.823 
 
 3.569 
 
 5.627 
 
 4.164 
 
 6.431 
 
 4.759 
 
 7.235 
 
 5.353 
 
 8.039 
 
 5.948 
 
 30 
 
 45 
 
 4.808 
 
 3.590 
 
 5.609 
 
 4.188 
 
 6.410 
 
 4.787 
 
 7.211 
 
 5.385 
 
 8.013 
 
 5.983 
 
 15 
 
 37 
 
 4.792 
 
 3.611 
 
 5.590 
 
 4.213 
 
 6.389 
 
 4.815 
 
 7.188 
 
 5.416 
 
 7.986 
 
 6.018 
 
 53 
 
 15 
 
 4.776 
 
 3.632 
 
 5.572 
 
 4.237 
 
 6.368 
 
 4.842 
 
 7.164 
 
 5.448 
 
 7.960 
 
 6.053 
 
 45 
 
 30 
 
 4.760 
 
 3.653 
 
 5.554 
 
 4.261 
 
 6.347 
 
 4.870 
 
 7.140 
 
 5.479 
 
 7.934 
 
 6.088 
 
 30 
 
 45 
 
 4.744 
 
 3.673 
 
 5.535 
 
 4.286 
 
 6.326 
 
 4.898 
 
 7.116 
 
 5.510 
 
 7.907 
 
 6.122 
 
 15 
 
 38 
 
 4.728 
 
 3.694 
 
 5.516 
 
 4.310 
 
 6.304 
 
 4.925 
 
 7.092 
 
 5.541 
 
 7.880 
 
 6.157 
 
 52 
 
 15 
 
 4.712 
 
 3.715 
 
 5.497 
 
 4.334 
 
 6.283 
 
 4.953 
 
 7.068 
 
 5.572 
 
 7.853 
 
 6.191 
 
 45 
 
 30 
 
 4.696 
 
 3.735 
 
 5.478 
 
 4.358 
 
 6.261 
 
 4.980 
 
 7.043 
 
 5.603 
 
 7.826 
 
 6.225 
 
 30 
 
 45 
 
 4.679 
 
 3.756 
 
 5.459 
 
 4.381 
 
 6.239 
 
 5.007 
 
 7.019 
 
 5.633 
 
 7.799 
 
 6.259 
 
 15 
 
 39 
 
 4.663 
 
 3.776 
 
 5.440 
 
 4.405 
 
 6.217 
 
 5.035 
 
 6.994 
 
 5.664 
 
 7.772 
 
 6.293 
 
 51 
 
 15 
 
 4.646 
 
 3.796 
 
 5.421 
 
 4.429 
 
 6.195 
 
 5.062 
 
 6.970 
 
 5.694 
 
 7.744 
 
 6.327 
 
 45 
 
 30 
 
 4.630 
 
 3.816 
 
 5.401 
 
 4.453 
 
 6.173 
 
 5.089 
 
 6.945 
 
 5.725 
 
 7.716 
 
 6.361 
 
 30 
 
 45 
 
 4.613 
 
 3.837 
 
 5.382 
 
 4.476 
 
 6.151 
 
 5.116 
 
 6.920 
 
 5.755 
 
 7.688 
 
 6.394 
 
 15 
 
 40 
 
 4.596 
 
 3.857 
 
 5.362 
 
 4.500 
 
 6.128 
 
 5.142 
 
 6.894 
 
 5.785 
 
 7.660 
 
 6.428 
 
 50 
 
 15 
 
 4.579 
 
 3.877 
 
 5.343 
 
 4.523 
 
 6.106 
 
 5.169 
 
 6.869 
 
 5.815 
 
 7.632 
 
 6.461 
 
 45 
 
 30 
 
 4.562 
 
 3.897 
 
 5.323 
 
 4.546 
 
 6.083 
 
 5.196 
 
 6.844 
 
 5.845 
 
 7.604 
 
 6.495 
 
 30 
 
 45 
 
 4.545 
 
 3.917 
 
 5.303 
 
 4.569 
 
 6.061 
 
 5.222 
 
 6.818 
 
 5.875 
 
 7.576 
 
 6.528 
 
 15 
 
 41 
 
 4.528 
 
 3.936 
 
 5.283 
 
 4.592 
 
 6.038 
 
 5.248 
 
 6.792 
 
 5.905 
 
 7.547 
 
 6.561 
 
 49 
 
 15 
 
 4.511 
 
 3.956 
 
 5.263 
 
 4.615 
 
 6.015 
 
 5.275 
 
 6.767 
 
 5.934 
 
 7.518 
 
 6.594 
 
 45 
 
 30 
 
 4.494 
 
 3.976 
 
 5.243 
 
 4.638 
 
 5.992 
 
 5.301 
 
 6.741 
 
 5.964 
 
 7.490 
 
 6.626 
 
 30 
 
 45 
 
 4.476 
 
 3.995 
 
 5.222 
 
 4.661 
 
 5.968 
 
 5.327 
 
 6.715 
 
 5.993 
 
 7.461 
 
 6.659 
 
 15 
 
 43 
 
 4.459 
 
 4.015 
 
 5.202 
 
 4.684 
 
 5.945 
 
 5.353 
 
 6.688 
 
 6.022 
 
 7.431 
 
 6.691 
 
 48 
 
 15 
 
 4.441 
 
 4.034 
 
 5.182 
 
 4.707 
 
 5.922 
 
 5.379 
 
 6.662 
 
 6.051 
 
 7.402 
 
 6.724 
 
 45 
 
 30 
 
 4.424 
 
 4.054 
 
 5.161 
 
 4.729 
 
 5.898 
 
 5.405 
 
 6.635 
 
 6.080 
 
 7.373 
 
 6.756 
 
 30 
 
 45 
 
 4.406 
 
 4.073 
 
 5.140 
 
 4.752 
 
 5.875 
 
 5.430 
 
 6.609 
 
 6.109 
 
 7.343 
 
 6.788 
 
 15 
 
 43 
 
 4.388 
 
 4.092 
 
 5.119 
 
 4.774 
 
 5.851 
 
 5.456 
 
 6.582 
 
 6.138 
 
 7.314 
 
 6.820 
 
 47 
 
 15 
 
 4.370 
 
 4.111 
 
 5.099 
 
 4.796 
 
 5.827 
 
 5.481 
 
 6.555 
 
 6.167 
 
 7.284 
 
 6.852 
 
 45 
 
 30 
 
 4.352 
 
 4.130 
 
 5.078 
 
 4.818 
 
 5.803 
 
 5.507 
 
 6.528 
 
 6.195 
 
 7.254 
 
 6.884 
 
 30 
 
 45 
 
 4.334 
 
 4.149 
 
 5.057 
 
 4.841 
 
 5.779 
 
 5.532 
 
 6.501 
 
 6.224 
 
 7.224 
 
 6.915 
 
 15 
 
 44 
 
 4.316 
 
 4.168 
 
 5.035 
 
 4.863 
 
 5.755 
 
 5.557 
 
 6.474 
 
 6.252 
 
 7.193 
 
 6.947 
 
 46 
 
 15 
 
 4.298 
 
 4.187 
 
 5.014 
 
 4.885 
 
 5.730 
 
 5.582 
 
 6.447 
 
 6.280 
 
 7.163 
 
 6.978 
 
 45 
 
 30 
 
 4.280 
 
 4.206 
 
 4.993 
 
 4.906 
 
 5.706 
 
 5.607 
 
 6.419 
 
 6.308 
 
 7.133 
 
 7.009 
 
 30 
 
 45 
 
 4.261 
 
 4.224 
 
 4.971 
 
 4.928 
 
 5.681 
 
 5.632 
 
 6.392 
 
 6.336 
 
 7.102 
 
 7.040 
 
 15 
 
 45 
 
 4.243 
 
 4.243 
 
 4.950 
 
 4.950 
 
 5.657 
 
 5.657 
 
 6.364 
 
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 Distance 6. 
 
 Distance 7. 
 
 Distance 8. 
 
 Distance 9. 
 
 Distance 10. 
 
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 46° -60^ 
 
SUEVEYIN(t 
 
 AND 
 
 TRAVERSE TABLE 
 
 G. A. WENTWOKTH, A.M. 
 
 AUTHOR OF A SERIES OF TEXT-BOOKS -IN MATHEMATICS 
 
 REVISED EDITION 
 
 Boston, U.S.A., ani) London 
 GINN^ & COMPANY, PUBLISHEKS 
 
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