IRLF 
 
 571 
 
r~\ B R A 
 
 u or THE 
 
 UNIVERSITY 
 
 PRESENTED TO THK LIBRARY 
 OF THE 
 
 UNIVERSITY 
 
 OF 
 
 CALIFORNIA 
 
 WITH THE COMPLIMENTS 
 OF THE 
 
 PUBLISHER 
 
I ft 
 
PLANE TABLE 
 
 By JOSEPH B. DAVIS 
 
 UNIVERSITY OF MICHIGAN 
 
 R 
 or THE 
 
 UNIVERSITY 
 
 ^LIFORHit- 
 
 GEORGE WAH R 
 
 University Publisher 
 Ann Arbor, Mich. 
 

PLANE TABLE. 
 
 1. The Plane Table is a drawing board in con- 
 nection with suitable devices for setting, and keeping 
 it level. The alidade is a straight edge, called the 
 rule and a line of sight. The line of sight is some- 
 times the edge of the rule, sometimes the line of 
 a pair of compass sights set in the line of the 
 edge of the rule, or it may be the line of sight of 
 a telescope so mounted upon the rule that this line of 
 sight revolves in a plane at right angles to the lower 
 face of the rule. When-a telescope is used, the plane 
 of revolution of its line of sight is set parallel to the 
 edge of the rule, or to contain the edge of the rule. 
 
 2. In the ordinary use of the Plane Table, a 
 piece of paper is fastened on the board and the lines 
 surveyed, and points that are determined, are platted 
 at once upon the paper. A good quality of drawing 
 paper is commonly used. This paper may be fas- 
 tened on the board in a variety of ways. If it is to 
 remain there long, and the board is well shrunk, it 
 may be best to shrink it on and stick the edges fast 
 with mucilage. It may also be tacked down with 
 very small curtain tacks. Thumb tacks are used. 
 Clamps are used, and rollers beneath the board are 
 used to some extent. If the paper is larger than the 
 board fasten the edges beneath the board. If the 
 paper is so large as to be shifted from time to time to 
 receive the extended drawing, it may be rolled and 
 fastened beneath the board by thumb tacks or suit- 
 
able clamps. In such a case the part of the paper on 
 the top of the board should be tacked fast to prevent 
 any shifting. When the paper is rolled under the 
 board, the roll, or rolls, and the part of the paper 
 where it passes over the edge of the board should be 
 protected by a covering of stout paper. It is difficult 
 to keep the paper clean. When the scale of the 
 map is small, so that but a small place on the paper 
 needs to be exposed at once, a newspaper may be 
 lightly fastened over the paper on the board and torn 
 away in small pieces to uncover the spots needed 
 for use at any time. After a place is finished and 
 no longer needed, it may be covered again with 
 newspaper, or the newspaper covering may be changed 
 at times, and the openings in it made to fit the work. 
 In some kinds of work an extra sheet of paper may 
 be useful for laying down lines upon, or for solving 
 certain problems, connected with the work, which 
 ought not to go upon the map or plat. 
 
 3. Center of the Sheet. The plane table 
 should be set up over an instrument point in the 
 same manner a transit is with a plumb bob from its 
 center standing over the mark. The observations 
 are then taken from the center of the sheet. The 
 center of the sheet is that point on the paper which 
 does not move when the board is turned about its 
 spindle axis. This point may be found on each 
 sheet, or a hole in the board made to mark it. To 
 find the center of the sheet revolve the board on its 
 axis and mark four fine marks on the board near its 
 corners by a pencil or needle held by a fixed support. 
 Find the center of the circle of which the four marks 
 are a part of the circumference. To do this the 
 
5 
 
 board may be taken off of its spindle and placed be- 
 side another board and so longer ranges be obtained 
 in drawing the lines to the center. Draw all four 
 lines. They should intersect at the same point. If 
 they do not, try again. Having found this center on 
 the board itself, a piece of brass may be set in the 
 board with a hole in it to permanently mark the cen- 
 ter. When a paper is fastened on the board, a few 
 trials with a needle will prick a hole in the paper cor- 
 responding to the hole in the brass plate. The 
 hole in the brass plate may be of the size to receive 
 a common brass pin, which being set firmly in the 
 hole may be a very useful aid in placing the alidade, 
 when sighting. The center of the sheet should be 
 found for use on every sheet or every time the sheet 
 is shifted. The hole in the brass plate should be 
 tested for position occasionally, to see that it stays 
 in its place. If, for any reason, the connections of 
 the board to the spindle are separated from the board, 
 this test should be made when they are restored. 
 
 4. Articles for use with the Plane Table. 
 
 1. Two good triangles, 30, 60, 90, from 9 to 
 12 inches, according to the size of the board, 
 and the scale of the map. These may be of 
 metal, rubber, celluloid, or wood, but should 
 have straight edges. A place to keep them in 
 may be made under the board. 
 
 2. Three legged dividers, with the third leg 
 running through a tubular clamp, so the 
 third leg may be varied in length, or removed. 
 
 3. Lead pencils, of the grade of No. 4 or 5 red 
 hexagon, Faber. 
 
6 
 
 4. Erasers typewriter, velvet rubber, and sponge 
 rubber. 
 
 5. Scale of equal parts for laying off distances. 
 
 6. Thumb tacks. 
 
 7. Curtain tacks No. i. 
 
 8. A little of Higgins' Drawing Board Mucilage. 
 
 9. Rubber cloth. A piece to cover the board 
 and contents and fasten beneath it for protec- 
 tion from rain and dirt. Some may prefer two 
 pieces, or one piece and a bag, the second 
 piece or the bag being for the alidade. 
 
 10. A haversack, or better, a skeleton coat to 
 carry the various tools and articles in. Even 
 an article on the principle of a mechanic's 
 apron with pockets on its front, is good. The 
 skeleton coat is best. 
 
 11. For some work, a compass needle in a 
 box. The box has a square or rectangular 
 base with the edges beveled like a rule and 
 parallel and perpendicular to the zero line of 
 the compass ring. The compass needle is 
 sometimes mounted on the rule of the alidade 
 but the separate compass is better. 
 
 12. Adjusting pins, common pins, needle pricker 
 and reading glass. 
 
 5. Setting up the Plane Table. Take the 
 table and tools to an instrument point where it is 
 desired that the table shall be set up. Lay the 
 alidade and other tools in a safe place. Tie on 
 the plumb bob. Set the table with its tripod, cen- 
 trally over the mark of the instrument point. 
 Spread the tripod legs so that when the table is 
 finally set it will come at the elevation of the bend at 
 
the middle of the body of the person who is to use 
 it. Crowd the legs into the eround till the table will 
 stand firm, and the plumb bob center the mark. 
 The table is usually set too high. The arms are 
 rested on the board, or the clothes rub against it, 
 thus stirring it from its proper position. By setting 
 it down to the bend in the body, one can lean over 
 it more conveniently to look through the telescope 
 or compass sights. Having attended to the placing 
 of the table, stand the alidade centrally on the board. 
 Turn the board, or the alidade, till the bubble tubes 
 are parallel to the leveling screws. By operating 
 the leveling screws make the bubble tubes both read 
 level. Place the alidade centrally on the board in a 
 new position. Level again, if needed. Proceed in 
 this way till the table is as perfectly leveled as it 
 can be. 
 
 After a little experience one may leave the 
 board off while setting up the tripod. 
 
 See that the spindle clamp and slow motion 
 screw are working well, that the leveling screws are 
 all bearing somewhat firmer than for a transit, and 
 that the tripod head (in some instruments) is screwed 
 down tight to the tripod cap. Leave the spindle un- 
 clamped. 
 
 The center of the sheet being known, stick up a 
 common pin there. Everything being found, or 
 made, secure, and the board standing steady, and 
 level, the plane table is ready for work. 
 
 6. The line of sight is set upon a point 
 very much as in the case of a needle compass, or 
 transit, with the difference that sometimes the alidade 
 is moved on the paper by the hands in order to do it, 
 
8 
 
 and sometimes the whole instrument is moved about 
 its spindle axis and the line of sight set by the spindle 
 clamp and slow motion screw, while the alidade 
 stands still on the paper. 
 
 Points are located and line given by sighting 
 with the telescope, the same as with a transit, or 
 with the compass sights, the same as with a compass. 
 
 7. First work at first station of a survey. 
 Set up the table at the station. Choose a point on 
 the paper to represent the station where the table 
 stands, so that the rest of the survey will come good 
 on the paper. Prick down the point and mark it 
 for that station. Choose the direction on the paper 
 in which the first line of the survey is to lie, when 
 platted. Turn the board around so the assumed di- 
 rection of the first line on the paper is, by estima- 
 tion, parallel to the line itself on the ground, and the 
 same end foremost. Clamp the spindle. Place the 
 working edge of the rule of the alidade to the pin at 
 the center of the sheet. With the hands, keep the 
 rule to the pin, and bring the line of sight to look 
 to another station of the first line of survey. By 
 more delicate handling set the line of sight upon the 
 mark at the distant station. Place one side of a 
 triangle against the edge of the rule. Place the 
 other triangle to the first, and slide the first over 
 the paper till the edge that was against the rule 
 comes to the mark on the paper for the first station. 
 Draw a line through the mark parallel to and in the 
 direction of the line on the ground. Slide the first 
 triangle back to the rule and see if it still fits. If 
 not review the work, and correct the errors. Make 
 the line drawn of the measured, or estimated, length 
 
9 . 
 
 of the line itself, to the scale of the plat, and mark, 
 at its end, the second station of the survey. 
 
 8. General directions. Follow the edge of 
 the rule, or triangle, carefully with the pencil held at 
 the same inclination to the board for the whole 
 length of the line. Keep the pencil sharp. Draw 
 no more of a line than is needed. Avoid erasures, 
 as much as possible. Be very particular when pro- 
 longing a line on the paper. Keep the edges of the 
 triangles straight. Then they may be used to pro- 
 long a line by setting one of them to the line and the 
 other to it from the opposite side of the line, when 
 they may be slid past each other, alternately, in the 
 direction the line is to go to the place the prolonga- 
 tion is to reach. This operation may be repeated 
 and the position of the prolongation tested'. If the 
 edges of the triangles are distrusted apply them to 
 the edge of the metal rule of the alidade, and try to 
 look through between the parts in contact against a 
 strong light. If the triangles are not true, they may 
 be worked down with emory or sand paper, or a 
 file, to fit the edge of the rule. They may also be 
 ground down on a grindstone. 
 
 9. Always place the alidade with the rule on 
 the same hand of the observer (either right or left, 
 as is most convenient) on all lines of the same sur- 
 vey, when looking along them in either direction. 
 Do not change hands with the rule when looking 
 back over a line. Keep the rule in the same hand 
 and bring it to the other side of the line. A right- 
 handed person will naturally take the alidade in the 
 left hand and always bring the rule to the left side of 
 any line. A left-handed person would reverse this. 
 
10 
 
 This would be the way to place the rule whether 
 looking forward, or backward, along the line. This 
 means that a telescope on an alidade would not be 
 reversed on its axis but remain the same side up. 
 
 10. The two edges of the rule should borh be 
 beveled, be straight and parallel, so that either can 
 be used to work from. 
 
 11. When carrying the table about, see that the 
 board is securely fastened to the tripod head so it 
 cannot fall off, and release the spindle clamp. The 
 plane table is a clumsy instrument to handle. It 
 must be carried with the more care and set down 
 gently. Otherwise, the connections with the spindle or 
 tripod head in many instruments can be easily injured 
 so as to make the table useless. The broad con- 
 nections below the board are much the best. Carry 
 the alidade, and handle it by a grasp upon some part 
 where a secure hold can be obtained, that will give 
 good control of it. Alidades are usually so heavy 
 that they should be made with a good hand hold. 
 There are so many things to carry, some of which 
 are awkward to handle, th.it it is best to consider 
 carefully, such features 
 
 12. Preparations at a second station of a 
 survey. These are to place the board in position 
 for making observations' at a station by means of a 
 line platted on the paper which runs to that station. 
 It is orienting the board by means of a platted line. 
 
 Set up the table at the station. Take out the 
 pin at the center of the sheet. Place the edge of a 
 triangle to the platted line. Slide the triangle by 
 means of the other as if to draw a line at the cen- 
 ter of the sheet, nearly covering the center of the 
 
11- 
 sheet. Hold the first triangle down steady with the 
 fingers near the edge on the center of the sheet. 
 With the other hand, bring the edge of the rule of 
 the alidade up to the edge of the triangle at the cen- 
 ter of the sheet. With a sliding movement of the 
 alidade fit the edge of the rule to the triangle. Be 
 careful not to shift the triangle, as it is easily moved 
 by the heavier alidade. After setting the rule see if 
 it stands parallel to the line it was set by. 
 
 If the spindle happens to be clamped, release it. 
 Turn the board to direct the line of sight to the dis- 
 tant station on the line used in placing the alidade 
 at the center of the sheet. Clamp the spindle. By 
 the spindle slow motion screw set the line of sight on 
 the mark at the distant station. 
 
 In placing the triangle to the line in the first 
 place, and the alidade at the center of the sheet, have 
 regard to the relative position of things on the table 
 and on the ground, so that when the line of sight is 
 finally set on the mark at the distant station, the line 
 joining the plats of the stations on the board shall be 
 parallel to the corresponding line on the ground and 
 lie the same end foremost. 
 
 The board, or table, now stands parallel to its 
 former position. It is oriented, and ready for work. 
 
 13. Measuring a vertical angle. Set up the 
 table. Place the alidade central on the table, with 
 the edge of the rule at the center of the sheet and the 
 line of sight on a mark at the station on the line from 
 which the vertical angle is to be measured. Set the 
 horizontal cross wire on the mark that shows one 
 side of the angle to be read, by means of the tele- 
 scope axis clamp and slow motion screw. Read the 
 
12 
 
 vertical circle. Record the reading. Release the 
 telescope axis. By the means used before set the hori- 
 zontal cross wire on the mark on the other side of the 
 angle. Read the vertical circle. Record the read- 
 ing. Release the telescope axis. By the same 
 means set the horizontal cross wire again upon the 
 first mark. Read the vertical circle. Record the 
 reading. See if the first and last readings agree suffi- 
 ciently well. If not, review the work and correct 
 the errors. The difference of the readings will show 
 the angle. 
 
 14. If the vertical angle is an angle of elevation 
 or depression, set the horizontal cross wire on the 
 mark as first above directed. Read the vertical cir- 
 cle. Record the reading. Release the telescope 
 axis. By its clamp and slow motion screw set the 
 telescope so its level will read level. Read the verti- 
 cal circle. Record the reading. Reverse the alidade 
 on the board and bring it up to the line with the rule 
 on the opposite side of the center of the sheet. Set 
 the telescope level to read level, if it needs it, by 
 means of its slow motion screw. Read the vertical 
 circle. Record the reading. Release the telescope 
 axis. Set the horizontal cross wire on the mark 
 again by means of the clamp and slow motion screw 
 to the telescope axis. Read the vertical circle. 
 Record the reading. See if the two middle readings 
 (taken when the telescope was level) agree well. If 
 they are very different, the board is not flat, or it is 
 unsteady. It may be set firmer, but if it is uneven, 
 the angle value will be poor. Having found the 
 readings to be as good as practicable, find the angle by 
 the first pair of readings, and also by the second pair. 
 
13 
 
 Take the half sum of the two values for the correct 
 one. 
 
 15. To measure the angle of elevation or 
 depression of a line, measure the height of the 
 telescope axis from the ground and sight to a mark 
 at the same height from the ground at the other end 
 of the line. If the line is not on the ground, meas- 
 ure the distance of the telescope axis from it in a 
 similar manner, and sight to a mark at a correspond- 
 ing distance from the line at another place on it. 
 Measure the angle of elevation or depression of the 
 line from the transit axis of the telescope to the cor- 
 responding mark at the station by the method de- 
 scribed. 
 
 16- Rem. The work that can be done with the 
 plane table depends directly upon the methods of as- 
 certaining the position of a point with respect to 
 other known points. These methods come first. 
 
 The expression " set the alidade" upon a line or 
 mark means that the edge of the rule is to be placed 
 at the center of the sheet and the line of sight set 
 upon that line, or mark. 
 
 Capital letters denote points on the ground, and 
 the small letters the plats of the same points on the 
 sheet. 
 
 The work of the plane table can be done with a 
 transit and drawing board, by platting the work as it 
 goes on. 
 
 17- Determining a point by direct meas- 
 urement. Orient the table at a known station. Set 
 the alidade upon the line to the required point. Plat 
 a line from a plat of the instrument station, parallel 
 to the edge of the rule and running towards the re- 
 
14 
 
 quired point. Measure the distance from the instru- 
 ment station to the required point, by chain, steel 
 tapes, or stadia. If the stadia is used, the telescope 
 axis should stand at the center of the sheet. Lay 
 off this distance, to the proper scale, from the station 
 on the plat along the line drawn. Prick down the 
 point thus reached, and mark it for the required 
 point. 
 
 18. Determining a point by intersections. 
 Fig. 1. Let A and B be determined and platted at 
 Set up the table at A and orient it by a b. 
 Set the alidade on a line to 
 C. Draw from a, towards C, a 
 line parallel to the edge of the 
 rule, and continue it to the esti- 
 mated position of c on the sheet. 
 Set the alidade again upon A B. 
 See if the edge of the rule is 
 
 f A / 
 
 o parallel to a b. If not, review 
 
 the work and correct the errors. Take up the 
 plane table. Set it up at B. Orient it by b a. 
 Set the alidade on B C, a line to C. Draw from b, 
 towards C, a line parallel to the edge of the rule till 
 it intersects the line from a, parallel to A C. Set 
 the alidade again upon A B. See if the edge of the 
 rule is parallel to b a. If not, review the work and 
 correct the errors. Prick down the intersection of 
 the two lines drawn from a and b and mark this 
 point c. 
 
 But a short piece of a c, or b c, or of both, may 
 be needed, near c. Any number of points may be 
 determined from a base A B. a c and b c should in- 
 tersect as nearly as may be at right angles. If a c 
 
15 
 
 and b c intersect at a small angle, the result cannot be 
 trusted. Other lines to c from d, c, or/, other known 
 points, will help to fix its position. A number of 
 lines to c \vi\\ not all intersect at the same point on 
 the paper. The proper location of the point on pa- 
 per is theoretically a question of mathematical ad- 
 justment, but there are usually some circumstances 
 connected with the work which will enable the selec- 
 tion to be made properly. 
 
 19. Determining points by resection. This 
 is in reality a method by intersections. The differ- 
 ence between this and the last case is, that one of the 
 two lines necessary to determine the point is drawn 
 on the paper, while the table is set up at the point 
 whose position is sought. In the previous case ob- 
 jects may be located, or determined, without going 
 to them. In the case of resection the table is set up 
 at the station, or object, whose position is sought. 
 This is an advantage in carrying out work, as the ta- 
 ble may be advanced to a station before it is on the 
 map, if there is a line on the paper that runs to it 
 from some known station that is in sight from it. By 
 referring to Fig. i the principle of this method may 
 be stated. Let A and B be known and platted, at a 
 and b. Set up the table at A and orient it by a b. 
 Set the alidade on a line to C. Draw from a, to- 
 wards C, a line parallel to the edge of the rule, and 
 continue it to where c is thought to be on the paper, 
 or beyond. Set the alidade again upon A B. See if 
 the edge of the rule is parallel to a b. If not, review 
 the work, and correct the errors. Take up the plane 
 table. Set it up at C. Orient it by the indefinite line 
 from a, towards c. Set the alidade on C B. Draw 
 
16 
 
 from b (towards C were the table at B), a line inter- 
 secting the indefinite line from , and parallel to the 
 edge of the 'rule. Set the alidade again upon C A. 
 See if the edge of the rule is parallel to the indefinite 
 line from a towards c. If not, review the work, and 
 correct the errors. In a similar manner lines may be 
 drawn on the map, or plat, from any number of sta- 
 tions in sight -from C, whose positions have been 
 platted. Several lines thus drawn, running to c, will 
 help fix its position with much greater precision. 
 
 20. Hem. A single intersection is not sufficient 
 to fix a station on the plat. A direction line and a 
 measured distance are not alone sufficient. When a 
 point is to be determined by intersection, there should 
 be a number of lines observed that run to it. When 
 a point is to be determined by measurements, there 
 should be two or more. A combination of intersect- 
 ing lines and measurements may be used. The po- 
 sition of any station on the plat should not be used to 
 determine the position of any other station, or object, 
 unless its own position has been well checked. 
 Every available check upon the position of a station 
 should be used. Neglect no chance to tie up a 
 station. A check which shows an error in the position 
 of the station tested, may really reveal an error in the 
 position of the station from which the test comes. A 
 check works at both ends, and tests the position of 
 both stations concerned. What constitutes a good 
 determination of a point? This can be ascertained 
 only by experience upon any particular work, and 
 with the instruments in use. The stadia may be used 
 with the best plane table to get distances and heights 
 no much work. For general topographical surveys 
 
17 
 
 the stadia will probably give good results. When 
 filling in final details to a large scale, it may be found 
 unsuitable, especially as to elevations. Any general 
 statement relating to such matters must be materially 
 modified by the person doing the work and the cir- 
 cumstances of the case. These facts render such 
 statements of little value. 
 
 21. In orienting the table use more than a sin- 
 gle line, where practicable, no matter how well that 
 line may have been determined. That is, orient by 
 a line, and test the orientation by sights to other 
 stations, if possible. This serves to further check the 
 plat. 
 
 22. The advantage of resection is, that after' one 
 or more lines to the new station have been platted, it 
 is possible to set up the table at that station, and by 
 observing to other stations get a number of lines run- 
 ning to the new station with a single setting of the ta- 
 ble, when by intersections, each of the stations ob- 
 served would have to be occupied. This is often a 
 very great advantage. With some patience, it is con- 
 ceivable that the table might be oriented at a new 
 station by persistent trial in taking pointings on other 
 stations, at the same time the new station was de- 
 termined. There are better ways of doing this, given 
 below. 
 
 23. Determining a point by the "Three 
 Point Problem" and orienting the table. Fig. 
 
 2. Let the three points, A, B, and C, be well de- 
 termined upon the plat. Let the position of D, from 
 which A, B, and C, can be seen, be required, without 
 going to them. 
 
18 
 
 F*asten an extra piece of paper over the middle of 
 the board. Find the center of the sheet on this pa- 
 per. Call the center -of the sheet on this paper m. 
 
 Set up the table at D. 
 Clamp the table so the 
 extra piece of paper will 
 receive the lines D A, D 
 B, and D C, when platted 
 
 from the center of the 
 * 
 sheet. Set the alidade on 
 
 DA and draw a line, by 
 
 fig 2 t ^ ie ec te e f tne ru l e > from 
 
 m towards A. Set the 
 
 alidade on D B and D C, in succession, and draw 
 lines from m towards B and C, respectively. Notice 
 the position of D with respect to A B C, that is, see 
 if D lies on the convex or concave side of the broken 
 line ABC. Fit the three legged dividers to the 
 platted positions, on the plane table sheet, of the 
 points A, B, C. Make the fit carefully. Apply 
 the points of the dividers to the lines from m 
 towards A, B, and C, each point of the dividers 
 upon the line running towards the point it was 
 fitted to the plat of. By careful adjustment, make 
 the points of the dividers fit the three lines. 
 Prick down the points on the lines where the 
 dividers fit. The center of the sheet, m, on the 
 extra piece of paper, is in the same relative posi- 
 tion with respect to the three points, on the three 
 lines, that were pricked down with the points of the 
 three legged dividers, that D is with respect to A, B, 
 and C, on the ground, and may be transferred to the 
 plane table sheet with the three legged dividers, 
 
-19 
 thus obtaining the correct plat of all four points. 
 
 Orient the table by means of a line from the 
 plat of D, on the plane table sheet, to the plat of 
 either A, B, or C, and test its position by lines to the 
 other points. Use the longest line to orient by. 
 
 When the plats of A, B, and C cannot be em- 
 braced by the three legged dividers, use the same 
 equal parts of the lines from the plat of B to the plats 
 of A and C, and take up the plat of B, and the two 
 auxiliary points with the dividers. Fit the dividers 
 to the three lines from m as before. The point m 
 will form the fourth angle of a quadrilateral whose 
 sides bear the same ratio to the sides of the one re- 
 quired on the plat, that the parts first used to do the 
 whole lines used on the plat. Fix a third auxiliary 
 point on the plat with respect to the plat of B and 
 the two first auxiliary points with the dividers. Draw 
 auxiliary lines from this third auxiliary point to the 
 others and to the plat of B. Extend the line to the 
 plat of B beyond the third auxiliary point. Draw 
 lines from the plats of A and C, respectively parallel 
 to the near sides of the quadrilateral of auxiliary lines 
 and produce them to intersect. They should inter- 
 sect on the line from the plat of B. If they do not, 
 review the work, and correct the errors. This in- 
 tersection is the plat of D. 
 
 24. Determining a point by the "Two 
 Point Problem," and orienting the table. Fig. 
 3. Let two points, A and B, be well determined 
 on -the plat. Let the position of C, from which A 
 and B can be seen, be required, without going to 
 them. This is possible when a fourth point D can 
 be occupied which is in sight from C, and from 
 
20 
 
 which A, B, and C, can be seen. D may be 
 made use of in continuing the survey, or not. As- 
 sume a station, D. 
 
 Fasten an extra piece of paper over the center 
 
 of the board. Find the center of the sheet on this 
 
 paper, and call it m. Set up 
 
 f- -7?^ the table at D. Clamp the 
 
 \, s'' i table so the extra paper will 
 
 I 'x v X ; receivea plat of the quadri- 
 > I / I lateral A B C D. Set the 
 
 ; /'' X ; alidade on D C, and draw a 
 
 jj C line, by the edge of the rule, 
 
 from m towards C. Set the 
 F'$ 3 alidade on D A and D B, in 
 
 succession, and draw lines from m towards A and B 
 respectively. Remove the table to C, and set it up 
 there. Orient the table by the line from m towards 
 C. Assume any convenient length from m on this 
 line to represent the length of D C, and prick 
 down, on the extra paper, a temporary plat of 
 C. Set the alidade on C A and C B, in succes- 
 sion, and complete, upon the assumed length for 
 D C, a temporary plat of A B C D. By this 
 means obtain a temporary plat of A B upon the 
 extra piece of paper. The angle between the plat of 
 A B on the plane table sheet and the temporary plat 
 of A B on the extra sheet is the amount the board is 
 to be turned to correctly orient it with respect to A 
 B on the ground and the plat of A B on the plane 
 table sheet. Set the alidade at m parallel to the 
 temporary plat of A B upon the extra sheet. 
 Find or fix a mark in range of the line of sight. 
 Set the alidade at m parallel to the correct plat 
 
21 
 
 of A B on the plane table sheet. Release the 
 spindle clamp. Set the line of sight on the mark by 
 means of the spindle clamp and slow motion screw. 
 The table is oriented correctly with respect to A B. 
 Resect upon A and B to determine C. 
 
 After orienting the table its position may be 
 tested. Set the alidade on m and parallel to the tem- 
 porary plat of either C D or C B on the extra sheet. 
 Release the spindle clamp. Set the line of sight on 
 D, or B, by means of the spindle clamp and slow 
 motion screw. Set the alidade on m and parallel to 
 the temporary plat of A B on the extra sheet. The 
 line of sight should strike the mark formerly used. 
 
 Particular notice must be taken to observe which 
 side of A B, the station C is on the ground and to 
 conform the temporary plat to the position of the 
 stations on the ground so as to prevent reversing the 
 table when finally orienting it, or of using the supple- 
 ment of the proper angle when making that orien- 
 tation. 
 
 If D is to be used in continuing the survey ; after 
 resecting upon A and B to get C, set the alidade on 
 C D and draw a line from the plat of C to get the 
 plat of D. Move the table. Set it up at D. Orient 
 it by the line from the plat of C towards D. Resect 
 on A and B to get D. It should fall on the line from 
 the plat of C. 
 
 The quadrilateral on the plane table sheet, or 
 the triangle A B C, in case D is not platted on the 
 plane table sheet, may be compared with the tempo- 
 rary plat upon the extra sheet. The angles should 
 be equal, and the sides proportional. 
 
22 
 
 C D may lie in any relation to A B cross it if 
 necessary. 
 
 25. When it is possible to set the table in line 
 with A B, by trial, as at D, Fig. 4, orient it by a b 
 A at once. Set the alidade on D C 
 and draw a line towards C, anywhere 
 on the sheet. Move the table to C. 
 Set it up. Orient it by the last 
 named line. Resect on A and B to 
 get C. Set the alidade on C D. 
 jj Draw a line from c parallel to C D 
 to intersect a b produced for d. 
 
 26. Elevations. Differences 
 in elevation as obtained with the 
 j. plane table may be good enough for 
 f}rt * some topographical surveys. They 
 
 may be obtained in several ways. 
 Except those of the rudest character, they require a 
 telescope on the alidade, a vertical circle or arc, a 
 level on the telescope, or both. The telescope level 
 should be a striding level, of the Berger pattern, 
 standing on collars turned on the body tube of the 
 telescope. These levels are readily attached and 
 detached, and cannot drop off. The telescope may 
 be furnished with stadia wires and the stadia used to 
 get heights and distances both. The distances may be 
 used to check those obtained by platting on the sheet. 
 27- If a telescope is used, the height of the 
 transit axis from the under side of the rule should be 
 measured and kept in mind in taking heights. This 
 distance added to the height of the table above the 
 ground, at any station will give the height of the line 
 of sight of the telescope above the ground when hor- 
 
23 
 
 izontal, ' or the height of the transit axis from the 
 ground. 
 
 28. If compass sights are used and rough levels 
 are wanted let a line be drawn parallel to the under 
 side of the rule near the top of the sights on one 
 side for depressions, and near the bottom of the 
 sights on the other side, for elevations. On the rear 
 sight place a peep sight on each of these lines. On 
 the front sight mark graduations, or a scale of feet 
 per hundred, like the tangent scales on compass 
 sights, for elevations on one side and depressions on 
 the other. Find the height of each level line from 
 the under side of the rule to be added to the height 
 of the table from the ground, to get the height of 
 the level line of sight from the ground. 
 
 29. Taking differences of elevation. With 
 the plane table set up at a station, to get the differ- 
 ence in elevation from the ground under it to the 
 ground at some other station, or place. Place the 
 alidade central on the table. Release the spindle 
 clamp. Level the table carefully. 
 
 If the difference is small, it may be taken as 
 levels are taken, for short distances. 
 
 Measure the height of the table from the ground 
 and add this to the height of the level line of sight 
 from the under side of the rule to get the height of 
 the line of sight from the ground. Call this H. I. It 
 is plus. Have a rod held at, the place where the dif- 
 ference in elevation is to be taken. Sight by the level 
 line of sight to the rod, and take a rod reading. Call 
 this S. It is minus. If compass sights are used, 
 hold a pencil point at the forward mark, and be sure 
 the table is well leveled. Subtract the rod reading 
 
24 
 
 from H I (the height of the instrument) to get the 
 difference in elevation. If the result is plus, the 
 place where the rod was held is the higher; if minus, 
 it is lower. If a telescope is used, it is set level by its 
 striding level and the rod reading taken even if the 
 table does not stand quite level. If better results are 
 sought, have the table well leveled. If much nicety 
 is required, lift the telescope from its transit axis 
 bearings, turn it over, replace it with the transit axis 
 reversed and the telescope the other side up, level it 
 again, and take a second rod reading. Take the half 
 sum of the two rod readings for the correct one. It 
 will be easier to keep the telescope, its line of sight, 
 and its striding level, in good adjustment where much 
 of this is to be done, than it will to take double 
 rod readings. 
 
 30. If the differences in elevation are too large 
 to be reached in the way described, proceed as fol- 
 lows : 
 
 Get H. I. as before. Sight to a mark on the rod 
 held up at ithe place, the same distance from the 
 ground the line of sight is. With compass sights, 
 hold a pencil point to the forward sight so it is in 
 range from the peep sight to the mark on the rod. 
 Take the reading of the scale on the forward com- 
 pass sight at the pencil point. Multiply this by the 
 distance in hundreds of feet to the rod for the dif- 
 ference in elevation. Record the scale reading 
 with a note of the place where the rod was held, and 
 do the multiplying at a convenient time. With a 
 telescope, set the line of sight on a mark on the 
 rod held at the place at the same height from the 
 ground, the line of sight is. Read the vertical circle. 
 
25 
 
 Record the reading with a note of the place the 
 rod is held at. Set the telescope level, by means of 
 its level. Read the vertical circle. Record the read- 
 ing HZ with the former reading. The difference be- 
 tween these two readings will be the angle of eleva- 
 tion, -f , or depression, , of a line joining the sur- 
 face of the ground at the station where the table 
 stands with the surface of the ground at the place 
 where the rod is held. The distance between the 
 two places may be taken from the plane table sheet, or 
 measured. The difference in elevation equals this dis- 
 tance multiplied by the tangent of the angle. For 
 greater refinements, the telescope may be turned over, 
 and the observations repeated, taking 'the half sum of 
 the two resulting angles for use. If the angle of ele- 
 vation is not over 5 or 300', the difference in eleva- 
 tion may be computed by the following equation : 
 
 Biff, in El. =d| 3 A " Vl A 1 ^ A ~ V300 A + .00000002d| 
 
 d = distance in feet. A = the angle in minutes. If 
 A is an angle of elevation, the result will be plus, if an 
 angle of depression, the result will be minus. The 
 last term covers the correction for curvature and re- 
 fraction and is always plus, as shown. The terms in 
 the numerator give the tan. A at 5, correct to 6 
 places. For A < 5 tan. A will be somewhat 
 large, but only .00000043 f r A =0 01'. This 
 equation can be used within the limits named, in 
 place of tables. 
 
 31. When a stadia is used, the telescope must 
 have cross wires in it to give stadia readings, and a 
 stadia made to fit the cross wires and telescope. 
 With this outfit, take and record the height of the 
 
-26 
 
 transit axis of the telescope from the ground at the 
 instrument station. Hold up the stadia at the place 
 whose distance and elevation is sought. Set the mid- 
 dle horizontal cross wire on a reading of the stadia, 
 equal to the height of the transit axis from 
 the ground. Read the vertical circle. Record the 
 reading, . Read the stadia at both the other 
 horizontal cross wires. Record these readings. 
 Set the telescope level, by means of its level. 
 Read the vertical circle. Record this reading, , 
 with the former reading. For greater refine- 
 ments, the telescope may be turned over, and 
 the observations duplicated, taking the half sum of 
 all readings for their final values, for use, with a 
 note of the place the stadia was held at. The stadia 
 notes may be reduced by any suitable method 
 amongst those "in use. For the determination of the 
 constants relating to the particular telescope in use 
 the methods given in Searle's Field Engineering 22$d 
 section, are good. The stadia may be used to de- 
 termine contours, also, by the methods designated in 
 such books as Johnson's Surveying, for such work. 
 32. Running a Straight Line. If the line is 
 already platted, set up the plane table at a station on 
 the line, and orient it. The alidade being at the cen- 
 ter of the sheet, mark its position on the paper by 
 drawing fine lines at the ends of the work edge of 
 the rule, and at the ends of the rule, so that the 
 slightest movement of the rule on the sheet can be 
 readily detected. Be particular to see that the levels 
 read level. The line may then be run, as with a 
 transit, reversing the plane table on its spindle as a 
 transit is reversed, leaving the alidade undisturbed on 
 
27 
 
 the sheet. Extreme care is required to keep from 
 moving the alidade on the sheet. 
 
 In case the line is not platted and it is to be 
 prolonged, set the line of sight on a mark, at a sta- 
 tion of the line, in the direction from the plane table, 
 opposite to that in which the line is to go ; using the 
 hands to move the alidade over the sheet ; not dis- 
 turbing -the board, as it is oriented ; and keeping the 
 edge of the rule at the center of the sheet. Reverse 
 the telescope on its axis and locate a line staff in ap- 
 parent line at the place where the next station is 
 needed. Reverse the telescope on its axis, and see 
 if the line of sight strikes the back sight mark. If 
 it does not, repeat the pointings, correcting the 
 leveling, or setting of the plane table till it will. Do 
 not disturb the orientation of the table if the set- 
 ting, or leveling, requires correction, or if it is dis- 
 turbed, orient the table again. Having located the 
 staff in apparent line, drive a stout stake, called a 
 plug, in its place. While the plug is being driven, 
 try the back sight again, see that the table is level, 
 and be ready to give line for a point on the plug. 
 Set the staff in line, on the plug, with the same 
 tests by back sighting, as before. Mark the point, 
 thus found on the plug, by a temporary mark, such 
 as may be made by sticking the point of the line staff 
 into the top of the plug a little. Release the spindle 
 clamp. Reverse the plane table on its spindle. 
 Level it again if needed. Set the line of sight on the 
 back sight mark by means of the spindle clamp and 
 slow motion screw, the alidade not being moved on 
 the sheet. The table will no longer be oriented, be- 
 ing reversed, and may not be precisely in a reversed 
 
28 
 
 position, that is, not just 180 from its first position, 
 unless all parts of the alidade are in adjustment. Re- 
 verse the telescope on its axis and locate a second 
 temporary mark on top of the plug, with the same 
 tests, by back sighting as before. Drive a nail equi- 
 distant from the two temporary marks to mark the 
 station on the line. 
 
 Set up a picket a little behind the plane table, and 
 on line for a mark to back sight to from the forward 
 station. A picket is a stick 3 or 4 feet long, about an 
 inch or so thick, having a foot or so of its top straight 
 and blazed off white, with its bottom end sharpened for 
 driving into the ground. The picket is set so as to 
 stand firm, with its white top end plumb and in line. 
 If close to the table, it may be lined by looking back 
 over the telescope tube, or, if the telescope is invert- 
 ing, by looking through it from the object end. If a 
 little distance away, the picket may be lined by 
 sighting through the telescope in the ordinary man- 
 ner. 
 
 Move the plane table to the forward station. Set 
 it up and orient it. Use the station where it stood 
 for a back sight, and continue the line as before. 
 
 Some telescopes do not reverse on their axes. 
 They may be made in such cases to be lifted readily 
 from the wyes or standards. With such a telescope 
 the above directions need modification. Open the 
 wye clips so the telescope can be lifted out before 
 beginning to make the pointings. To reverse 
 the telescope, lift it from the wyes ; turn it over, end 
 for end ; and replace it with each end of its axis in 
 the same wye it was in before. Otherwise than this, 
 the pointings are made the same as with a telescope 
 
that reverses on its axis. The chances of disturbing 
 the alidade on the sheet are greater with this form of 
 mounting, and corresponding care and pains are re- 
 quired. 
 
 If the telescope does not reverse, and does not 
 lift out of the wyes, run the line by foresights. 
 
 To run a line by foresights, set up the plane ta- 
 ble at a station of the line and orient it. Mark a sta- 
 tion in the direction in which the line is to go to 
 sight to, to get the line called a foresight. Set the 
 line of sight on the foresight mark by moving the 
 alidade over the sheet with the hands ; not disturb- 
 ing the board, as it is oriented ; and keeping the 
 edge of the rule at the center of the sheet. Mark 
 the position of the rule on the sheet. Locate a 
 point on a plug beyond the foresight, and where it 
 can be seen from the foresight station. See if the line 
 of sight still strikes the foresight. If not, repeat these 
 operations till it will. Release the spindle clamp. 
 Reverse the plane table on its spindle. Reverse the 
 alidade on the sheet by the marks showing where the 
 rule first rested. Level the plane table again if 
 needed. Set the line of sight on the foresight by 
 means of the spindle clamp and slow motion screw. 
 The plane table is no longer oriented. Locate a sec- 
 ond point on the plug beside the first, with the same 
 tests. Drive a nail equidistant from the two marks 
 on the plug to mark the station on the line. 
 
 Move the plane table to the foresight station. 
 Set it up and orient it. Use the new station for a 
 foresight, and continue the line as before. 
 
 When the line is not platted, or it is not neces- 
 sary to orient the plane table to do other work, the 
 
30 
 
 line may be prolonged by placing the alidade to the 
 center of the sheet without orienting the table, and 
 the first pointing made with the spindle clamp and 
 slow motion screw. 
 
 The plat of the line as prolonged is found on the 
 sheet by carefully continuing the piece that is there, 
 platting the new stations located, and testing their po- 
 sition on the sheet by shots at well determined points, 
 previously platted. 
 
 33. Intersecting Two Straight Lines. 
 Extend the lines on the ground and set two plugs, 
 with nails to mark one of the lines, the plugs stand- 
 ing one on each side of the other line, as nearly as 
 can be judged. Stretch a string between the nails. 
 Set a third plug under the string and on the other 
 line. Drive a nail in this plug where the other line 
 crosses the string. This nail will mark the intersec- 
 tion of the lines. 
 
 The two first plugs may, by miss-judgment, be 
 on the same side of the second line. In that case 
 extend the string to cross the second line and tie it to 
 a stake. Or the first line may be marked by another 
 plug which shall stand on the other side of the second 
 line. 
 
 Sometimes the three plug method is inconvenient 
 because the ground is so rough. Then set two plugs 
 on each line so as to straddle the other ; draw two 
 strings ; and set a nail in a fifth plug to mark the 
 intersection. 
 
 Sometimes the last plug comes considerably be- 
 low the string, or strings, too far below for good 
 sighting. In such a case hold a plumb line so as to 
 look past it, and the string being used to line by. 
 
31 
 
 Make the plumb-line cover the string and the sight 
 will be in a vertical plane with the string. A weight 
 tied on a string will do. Steady the hand holding 
 the plumb line on a staff it needed. 
 
 Sometimes strings cannot be stretched between 
 the line plugs. Set up the plane table over the nail 
 in one of the plugs ; set the line of sight on the nail 
 in the other and do the sighting with the line of sight 
 of the alidade. This takes more time. This plan 
 may be used in case the intersection plug comes in a 
 hole. 
 
 It may be sufficient for the purposes of the work 
 to plat the lines on the paper and produce them there 
 till they meet. 
 
 3i. Measuring the Angle Between the 
 Two Lines. Intersect the lines. Measure the 
 smaller angle between them. 
 
 Set up the plane table at the intersection of the 
 lines. Measure off on the sides of the smaller angle 
 between the lines from their intersection two hundred 
 feet or more on each. Make the measurement care- 
 fully, and set a nail in a plug on each line to mark 
 the ends of the measurements, using the alidade to 
 line the marks with. Measure the distance between 
 the two nails as precisely as practicable. Divide this 
 distance in feet by twice the number of feet measure- 
 ment off on the two lines from their intersection. 
 The quotient will be the sine of one-half the angle 
 between the lines. Look up the corresponding angle 
 in the table of natural sines and multiply it by two to 
 get the angle between the lines. 
 
 The same thing may be done with a scale on the 
 drawing, using the plat of the lines thereon. If five 
 
32 
 
 units of some convenient scale are used in laying off 
 the distances on the two sides of the angle, one- 
 tenth of the distance between the ends of the distances 
 measured in the same unit will be the sine of half the 
 angle sought. 
 
 One of the lines may need to be prolonged pas 
 their intersection to get the smaller angle. On the 
 ground this will be done with the plane table as in 
 running a straight line. On the drawing, do it with 
 the rule. 
 
 The measurements made on the ground may be 
 used at once with the scale and dividers to lay the 
 angle down on the drawing without finding the value 
 of the angle. 
 
 The process applies to the angle between two 
 lines of a survey, or to the deflection angle from one 
 to the next, since these angles are supplements of 
 each other. 
 
 35. Laying out a Given Angle. Plat the 
 angle on the sheet of paper with its vertex at the 
 center of the sheet. Draw the lines which form the 
 sides of the angle out sufficiently to fit the rule of 
 the alidade to, or mark pieces of them so the outer 
 end of the rule can be fitted to them, if it is not de- 
 sirable to draw the whole line. Or a separate extra 
 piece of paper may be fastened on the board for this 
 work. If the extra paper is used fasten it on first, 
 find and mark the center of the sheet on it, and plat 
 the lines from that. Set up the plane table at the 
 plug or station, that is to be the vertex of the angle. 
 Orient the table by one of the platted lines. Set the 
 alidade to the other line. Range out this line by the 
 
33 
 
 line of sight of the alidade, and mark it with plugs 
 and nails on the ground. 
 
 If the given angle is much over 90 it will be 
 better to plat its supplement on the plane table sheet, 
 and produce one side of the plotted angle to get a 
 plat of the given angle. This refers to the ordinary 
 case in the field where there is no protractor. If a 
 protractor is used, the angle can be laid down on the 
 sheet at once with that. Without a protractor use 
 the table of natural tangents or natural sines. 
 
 To plat an angle with natural tangents measure 
 off from the vertex of the angle on the given or as- 
 sumed side ten parts as large as can be conveniently 
 used, as 10 or 20 inches. Here draw a perpendicular 
 to the given side in the direction the angle is to lie. 
 Take from the table the natural tangent of the angle. 
 Move its decimal point one place to the right. Meas- 
 ure off on the perpendicular from the given side the 
 resulting number of the same parts, ten of which 
 were used in locating the perpendicular. Draw a line 
 from the end of this measurement on the perpendic- 
 ular to the vertex of the angle for the other side of 
 the angle. 
 
 To plat an angle with natural sines take out the 
 natural sine of half the angle and multiply this sine 
 by two. With a radius of ten parts draw the arc of 
 a circle with the vertex as a center and cutting the 
 given side. With a radius of a number of the same 
 parts equal to twice the sine of half the given angle 
 with its decimal point moved one place to the right, 
 and the intersection of the arc and the given side as a 
 center, draw an arc cutting the first arc. Draw the 
 
-34- 
 
 other side of the required angle from its vertex to the 
 intersection of the two arcs. 
 
 30. Running a Traverse. Run and plat the 
 successive lines of the survey in the manner previ- 
 ously described for the various items of elementary 
 work, or determine the vertices of the angles of the 
 lines of the survey by any of the methods given. The 
 result will be a plat of a continuous survey which is 
 often called a traverse. Strictly speaking the plane 
 table does not run a traverse in the sense of that term 
 in transit work, but produces a plat of the lines tra- 
 versed or surveyed. If computations are to be based 
 on a plane table plat, the lengths of the sides should 
 be measured on the ground with a steel tape and the 
 plat checked by these measurements. The plat 
 should also be to a large scale. 
 
 37. Obstacles on Straight Lines. Obstacles 
 are not frequently met with on ordinary plane table 
 surveys. The lines of the survey are chosen so as to 
 avoid them. It is, however, quite possible that they 
 may be met with. If the scale of the plat on the 
 board is so small that the object to be passed would 
 appear insignificant, or the lines used to pass it too 
 short for setting the alidade to, use an extra piece of 
 paper, and plat the work to a sufficiently large scale 
 while passing the object. This will permit the work 
 to be checked, as well as to be done with sufficient 
 precision. 
 
 38. Passing an Object on a Straight Line by 
 Deflections. Fig. 5. Fasten an extra piece of 
 paper over the middle of the sheet. Find and mark 
 the center of the sheet on the extra piece of paper. 
 Set up the table and orient it at some point of A B, 
 
35 
 
 as at B, where a line B C can be laid out clear of ob- 
 structions. Start B C so as to pass the object. 
 Draw lines from nt, the center of the sheet, towards 
 C and towards A, of suitable length to set the alidade 
 by. Set the alidade on B C. Release the spindle 
 clamp. Turn the table round on its spindle and set 
 the line of sight on B 
 A by means of the 
 spindle clamp and slow 
 motion screw, leaving 
 the alidade undisturbed 
 on the sheet. Set the 
 alidade again upon B 
 C and draw a second 
 line from m towards C. 
 This is done for the 
 purpose of drawing a 
 line from m that shall 
 make an angle with the 
 line originally drawn 
 from m towards A 
 equal to twice DEC, 
 or to lay down 2 D B 
 C on the sheet. Run 
 B C to C, measuring 
 its length, or till a sec- 
 ond line C D, run so that C D B = D B C, will pass 
 the object. Move the table to C. Set it up. Orient 
 it on C B by the second line drawn from m towards C 
 when at B. The line drawn from m towards A, 
 when at B, will now show the direction in which to run 
 C D. Run out C D accordingly and make it equal 
 by measurement to C B, thus locating D. Release 
 
36 
 
 the spindle clamp. Re-orient the table by the first 
 line drawn from m towards C when at B, so as to 
 orient the table with respect to A B. Set the alidade 
 on C D and draw a line from m towards D. Move 
 the table to D. Set it up. Orient it by the last line 
 drawn. 
 
 Check Locate E, supposed to be in A B, 
 produced. Place the alidade on the line first drawn 
 from m towards C when at B. Release the spindle 
 clamp. Turn the table round its spindle axis and set 
 the line of sight on E by means of the spindle clamp 
 and slow motion screw, leaving the alidade undis- 
 turbed on the sheet. See if the line drawn from m 
 towards A when at B now points to C. If it does 
 not, review the work and correct the errors. 
 
 If it is considered to be sufficiently accurate, the 
 point C may be determined by resection from A B, 
 and B C found from the plat. Good practice would 
 make use of C in determining other points and ob- 
 jects required by the survey. In such a case B C and 
 C D would be platted on the sheet with other sta- 
 tions. The distance B D may be found by platting 
 B C and C D to a large scale from the center of the 
 sheet on the extra sheet of paper and measuring the 
 .distance B D between their ends to the same scale. 
 Or measure the angle B C D on the ground with the 
 steel tape or on the plat with the scale (34) and find 
 B D = 2 B C sin ^ BCD. 
 
 For a check upon such work run past the object 
 more than once, locating C in a different place, or 
 prolong B C, and set another point on line beyond D. 
 Run past on opposite sides where practicable. Set 
 up the table at any one of the points beyond the ob- 
 
H 
 
 37 
 
 ject. Orient it by the line to the station where the 
 table stands. Locate E as before. See if the other 
 points beyond the object are in the line to E. Ap- 
 ply the check above given at any of the points be- 
 yond the object and see if it fits. 
 
 39. Passing an Object on a 
 Straight Line by Traversing. Fig. 
 
 6. Run a traverse past the object, 
 platting the lines of the traverse. The 
 stations on the traverse may be deter- 
 mined by measurement, resection, or any 
 of the usual methods. Extend the plat 
 of the original line on the sheet to the 
 forward part of the plat of the traverse. 
 Draw a line on the plat from the last 
 station of the traverse so as to intersect 
 the straight line produced past the ob- 
 ject, and in a position to be conveniently 
 run out on the ground to reach the 
 original line at a point beyond the object. 
 Measure the length of this line on the 
 sheet with the scale. Having set up 
 the plane table at the last station of the 
 traverse on the ground and oriented it, 
 run out the above described line, meas- 
 ure out its length and locate a station 
 beyond the object on the original line 
 produced. Continue the line by means 
 of the plat of it on the sheet. 
 
 The above method is comparatively rough, but 
 may be used in some cases. It is without a check. 
 To improve the results, the traverse may be platted 
 
 C 
 
38 
 
 on an extra sheet to a large scale, and the closing 
 distance more carefully measured. More than one 
 traverse may be carried past the object and the re- 
 sults compared. All the stations on all the traverses 
 riiay be useful in extending or perfecting the sur- 
 vey. The points beyond the object may be 
 checked by resection on other platted stations. 
 
 The traverse may be carried a sufficient dis- 
 tance beyond the object, as to D, E, and F, to 
 permit of lines being drawn from E, and F, to H, 
 the point where the closing line D H intersects A B, 
 produced past the object. Locate H by intersections 
 from D and E, and check it by resection on F. A 
 second point K may be located by similar means. A 
 number of these stations may be used to check the 
 locations of one another, as well as form the basis for 
 extending the survey. 
 
 40. Passing an Object on a Straight Line 
 With the Plane Table Alone. Fig. 7: Set up 
 the plane table at B. Orient it by B A. Run out 
 two lines B C and B D, making such equal angles 
 with A B that they will pass the object. Set a station 
 D on B D, so that D C perpendicular to A B will 
 pass the object. Set plugs as near C as can be es- 
 timated preparatory to intersecting D C with B C. 
 Move the table to D. Set it up. Orient it by D B. 
 Assume a point d for the plat of D, on the plat of D 
 B, so as to give a good length to the lines of the dia- 
 gram being developed on the sheet to set the rule 
 of the alidade by. From d lay out a line on the 
 sheet to represent the plat of D C, perpendicular to a b . 
 Run this line out on the ground and intersect it with 
 B C at C. Continue the line on the plat and find c. 
 
3D 
 
 Start the lines D E and D F for locating E and F at 
 convenient angles from D 
 
 B, and draw d e and dfon 
 the plat to intersect a b 
 produced. Set plugs on 
 D E and D F, as near E 
 and F as can be estimated 
 preparatory to intersect- 
 ing these lines with C E 
 and C F. Lay out c e 
 and c f from c on the plat, 
 making B C E = B D E J) 
 and B C F B D F. 
 Move the plane table to 
 
 C. Set it up at C. 
 Orient it by C B. Test 
 its position by C D. 
 Run out C E and C F. 
 Intersect them with D E 
 and D F, locating E and 
 
 F. Move the plane table and set it up at E or F. as 
 may be most desirable. Orient it by the line to C. 
 Test its position by F E and the line to D. Con- 
 tinue the line to H, or any point beyond, using F E 
 ( r A B, as they should be the same. 
 
 This work may require an extra sheet of paper, 
 but the practitioner will readily decide such matters 
 for himself. C, D, E, and F, may be used as regu- 
 lar stations on the survey. The object may be plat- 
 ted as this work proceeds. 
 
 41. Passing an Object on a Straight Line 
 by Resection. Fig. 8. Locate E in A B pro- 
 duced beyond the object. Choose D as anv con- 
 
40 
 
 venient station on the same side of the object that E 
 is. Set up the table at E and orient it by A B. Set 
 the alidade on E D, and 
 draw a line toward D, any- 
 where on the sheet. Move 
 the table to D. Set it up. 
 Orient it by the last-named 
 line. Resect on B and C 
 to get D. Set the alidade 
 on D E. Draw a line from 
 d, parallel to D E to inter- 
 sect a b produced to get e. 
 Measure c e on the plat. 
 As a check, measure D E 
 on the ground and d e on the 
 plat. See if they agree. 
 
 The larger the scale 
 used, the more perfectly will 
 the result be accomplished. 
 
 42. Inaccessible Lines. Problems relating 
 to inaccessible horizontal distances may be solved 
 with the plane table without calculation. This is in- 
 dicated by Figs. 6. 7, and 8. The relative elevation 
 of the points between which the required distance is 
 found in these cases has theoretically nothing to do 
 with the application of these principles. The same is 
 true of Fig. 3. A detailed account of problems re- 
 lating to inaccessible lines is not given, because the 
 methods illustrated by these figures may be adapted 
 to all the ordinary cases to be met with. Problems 
 of this class are rare in plane table work. 
 
 43. Running Circular Curves. Fig. 9. In- 
 tersect the tangents A I and B I at I (33). Set up the 
 
41 
 
 plane table at I. Measure the angle D I B= A (34)- 
 Choose a suitable curve. The curve may be desig- 
 nated by its degree, like a railroad curve when the 
 station chords, as P t 
 P 2 , P 2 P 3 ,etc., are 100 
 feet. When the station 
 chords are not 100 feet, 
 the curve may be 
 named by using the sta- 
 tion chord for the nu- 
 merator of a fraction 
 and the angle the sta- 
 tion chord covers at the 
 center of the curve,/? 
 as the denominator, f 
 thus 
 
 20 
 
 which signifies that a 
 chord of 20 feet covers 
 an angle at the center 
 of the curve of 10. 
 This device may be 
 called the ' l Name' ' of 
 the curve. It corres- 
 ponds to the conception used in the term ' l Degree' ' 
 used in railroad curves, which means the angle at 
 the center covered by a station chord of 100 feet, or 
 the numerator of the above fraction becomes 100. 
 If the curve is designated by its degree, take its radius 
 R, with log. R, from a curve table. If the curve 
 is "named," as above, compute 
 
 L 
 
42 
 
 R= , where c = the 
 sin ^2 d 
 
 station chord and d the station angle it covers. Com- 
 pute the tangents I A = I B = T=R tan ^ A, and 
 the vertex distance I K = V = T tan % A. Set the 
 alidade on I M. Measure from I towards M, I B 
 T, and locate B. Set a plug or stake and nail at B. 
 Bisect the angle a i b on the sheet, denning the di- 
 rection of i k. The plane table should be oriented 
 with respect to I A and I B, using i a or i b to orient 
 the table by, and check its position by the other. 
 Set the alidade by i k. Range out the line I K on 
 the ground. Measure from I to K, I K = V, and 
 locate K. Set a stake and nail at K. Set the ali- 
 dade on I L. Measure from I towards L, I A = T, 
 and locate A. Set a plug, or stake, and nail at A. 
 Move the plane table to A. Put on an extra sheet of 
 paper, if not already done at I. Set up the table at 
 A. Measure from the last preceding station, as L, to 
 A, as L A. Mark a witness stake for A, bearing the 
 line mark, as Q ; the number of the regular station 
 at L, plus the distance from L to A, as 147 -f 62.37 ; 
 the letters P. C. , point of curve ; the degree or 
 name of the curve as 4 or 
 
 20 
 
 20 c 
 
 the direction of the curve, as L. for left, or R. for 
 right ; and the angle D I B = A, as 70 34' 30' ', 
 
-43 
 to read from the top of the stake downwards, as 
 
 Q 
 
 147 + 62.37 
 
 P. C. 
 
 20' 
 
 ^0~ 
 
 C. R. 
 
 C. R. for 70 34' 30". Drive this stake about a 
 foot to the right of A, with its station number fac- 
 ing L. 
 
 Subtract the distance L A from the regular sta- 
 tion distance, P l P 2 to get the first short chord, as A 
 P l =c lm Find one-half the angle this chord covers 
 at the center of the curve, by 
 
 sin 
 
 R 
 
 Subtract % d^ from ^ A- Divide the result by ^ d. 
 The remainder found by this division will be ^ df 2 , 
 one-half the central angle, covered by the short 
 chord P 3 K = r 2 . Compute r 2 by 
 
 <: 2 = 2 R sin ^ d 
 
 Subtract ^ d l from ^ LJ . Divide the result by ^ d. 
 The remainder found by this division will be ^ ^ 3 , 
 one-half the central angle covered by the short clos- 
 ing chord P. B 8 . Compute c^ ' y 
 
 <: 3 = 2 R sin ^ ^ 3 . 
 
 Plat the angle I A P t = ^ ^! with its vertex 
 at the center of the sheet and with long sides for set- 
 
44 
 
 ting the alidade by in running the curve. On the 
 same side of this angle, supposing the observer to be 
 at its vertex, that the curve is to run. plat ^ d, one- 
 half the regular station angle. Number these lines 
 
 0. i, 2, in the order they come in, laying down the 
 angles platted. Still on the same side of the last an- 
 gle platted, or continuing from line 2, plat ^ d^ and 
 number this line 3. From line 2 also plat ^ d^ in 
 the same direction and number this final line 4. 
 
 Set the alidade on o. Set the line of sight on 
 A I by means of the spindle clamp and slow motion 
 screw. Set the alidade on i. Take a distance on 
 the chain, or tape, equal to c 1 = A P r Hold one 
 end of this distance at A. Swing the other end of 
 this distance on the tape into line at P l . Drive a 
 stake at P lt marked for the number of that station, 
 with the number facing A. Test this stake for line 
 and distance, and when it is adjusted for both, drive a 
 nail in it to mark the station Pj. Set the alidade on 
 2. Hold one end of a distance on the tape, equal to 
 the regular station chord, at P 1 . Swing the other end 
 into line at Pj. Set a stake at P 2 marked for that 
 station, with the same tests that were used in setting 
 Pj. Set the alidade back on line o. See if the line 
 of sight still ranges the tangent -A I. If not, review 
 the work and correct the errors. Set the alidade on 
 
 1. By means of the spindle clamp and slow motion 
 screw set the line of sight on P 2 . Set the alidade on 
 
 2. Measure a regular station chord from P 2 and lo- 
 cate P 3 as P 2 was located. Set the alidade again on 
 i, and see if the line of sight strikes P 2 . If not, 
 review the work from P 2 , and correct the errors. 
 The alidade being set on i , set the line of sight on 
 
45 
 
 P 3 and continue the curve as before, making the 
 various checks and tests to insure trustworthy work. 
 So continue the work till the station next preceding 
 K, the middle of the curve, is reached. This is P 3 in 
 Fig. 9. After P 3 is set, and before shifting the table 
 on its spindle preparatory to locating the next regular 
 station, set the alidade on line 3. See if the line of 
 sight strikes K. Measure the distance from P 3 to K. 
 See if this equals <r 2 . If either of these checks fail, 
 review the work and correct the errors. Continue 
 the curve from P 3 , not from K, till the station, as P 5 , 
 next preceding the P. T. at B is reached. After P 5 is 
 set and checked, set the alidade on line 4. See if the 
 line of sight strikes the P. T. at B. Measure from P 5 to 
 B. See if this distance is ^ 3 . If either of these 
 checks fail, review the work and correct the errors. 
 Mark a witness stake for the P. T. with the line 
 mark ; the station number of P 5 plus the distance 
 P 5 to B ; P. T. ; the name of the curve ; L or R ; 
 and the value of A- Drive this stake about a foot 
 to the right of B, with the number facing P.. Sub- 
 tract c^ from the regular station distance to get the 
 distance from B to M, the next regular station be- 
 yond B. 
 
 With the plane table oriented so the line 4 is di- 
 rected to B, set the alidade on A I, and draw a line 
 from m by the edge of the rule. Move the table to 
 B. Set it up over B. Set the alidade on m i, the 
 last line drawn. By means of the spindle clamp and 
 slow motion screw, set the line of sight on A. Set 
 the alidade on line 4. See if the line of sight ranges 
 on the tangent B I. If not, review the work and cor- 
 rect the errors. 
 
Range out the tangent I B. Measure c r 3 from 
 B to M, and locate the first regular station on the tan- 
 gent beyond B, at M. Continue the tangent as a 
 straight line (32). 
 
 In case the whole curve cannot be seen from A, 
 run as far as practicable, as to P 2 in Fig. 9. After 
 checking the location of the last station, set the ali- 
 dade on line i, and make the line of sight strike the 
 last station by means of the spindle clamp and slow 
 motion screw, just as if preparing to locate the next 
 regular station. Set the alidade on A I. Draw m i 
 from m towards I by the edge of the rule. Move the 
 plane table to P 2 . Set it up over P 2 . Turn the table 
 so the same side of the sheet is forward the way the 
 curve is going, that was forward when at A. Set 
 the alidade on i m, the last line drawn, to look back 
 to A. By means of the spindle clamp and slow mo- 
 tion screw set the line of sight on A. Set the ali- 
 dade on line 2. Locate the next regular station in 
 the usual manner, measuring a station chord from P 2 . 
 Check its location by a back sight on A. Continue 
 the curve as before directed for the table standing at 
 A ; also check on K as before directed. 
 
 Should it be necessary to move up again, let P 4 
 Fig. 9 be the station to be used as the next instru- 
 ment station. Set the alidade on i m again. By means 
 of the spindle clamp and slow motion screw, set the 
 line of sight again on A. Set the alidade on line 1. 
 Make or find a mark in apparent line on the tangent 
 E F. Call this mark F. Release the spindle clamp 
 and set the line of sight on P 4 by means of the spin- 
 dle clamp and slow motion screw, the alidade 
 being still set on line i. Set the alidade on P 2 
 
F. Draw a line m f by the edge of the rule. Move 
 the plane table and set it up over P 4 . Turn the same 
 part of the sheet to be forward as before. Set the ali- 
 dade on f m to look back to P 2 . Set the line of 
 sight on P 2 by means of the spindle clamp and slow 
 motion screw. Continue the curve as before. 
 
 Having moved up any number of times on a 
 curve, let P 4 be the instrument station from which B 
 can be seen and the curve closed. Before beginning 
 work from P 4 ' after the table is oriented, set the ali- 
 dade on line i and make or find a mark in appar- 
 ent line on the tangent G H. Call this point H. 
 Having closed the curve on B by line 4, leave the 
 table with 4 pointing to B, and set the alidade on P 4 
 H. Draw m h, by the edge of the rule. Move the 
 table to B. Set it up over B and orient it by h m, and 
 a back sight on P 4 . Set the alidade on line 4. See 
 if the line of sight ranges the closing tangent B I. If 
 not, review the work and correct the errors. 
 
 To lay out a tangent at any station as P 4 and set 
 up the plane table there, orient it preparatory to 
 continuing the curve. Set the alidade on line i , or 
 whatever line was pointing to the instrument station 
 from the previous instrument station when a point- 
 ing was made on the tangent there, preparatory to 
 moving up. The line of sight then ranges on the 
 tangent. 
 
 Obstructions to running the curve that cannot 
 be removed may often be passed by a long cnord. 
 Compute the distance from the station where the 
 plane table stands to the station beyond the obstruc- 
 tion which can be seen, by 
 
 C, =2 R sin y 2 Dj, 
 
-48- 
 
 where C l is the required distance, or long chord, 
 and Dj is the central angle covered by C 15 found by 
 the number of station chords on the curve between 
 the two ends of C l . Run the curve as far as practi- 
 cable towards the obstruction. Having checked the 
 location of the last station, and without shifting the 
 plane table, plat from line 2 an angle equal to one- 
 half the central angle included between the last sta- 
 tion located and the station at the far end of C 1 . 
 Call the line thus laid down 5. Set the alidade on 
 2. See that the table is level and the line of 
 sight strikes the last station located. If need be, 
 level the table and set the line of sight on the last 
 station located by the spindle clamp and slow motion 
 screw. Set the alidade on 5. Run this line a dis- 
 tance C l from the station where the table stands and 
 locate a station beyond the obstruction. Mark the 
 tangent at the instrument station, as directed in the 
 last paragraph. Orient the table so that 5 will point 
 to the station located beyond the obstruction. Set 
 the alidade on the tangent and draw a line by the 
 edge of the rule from the center of the sheet. Move 
 the table to the station located beyond the obstruc- 
 tion. Set it up there. Turn the table so the same 
 side is forward as before. Set the alidade on the last 
 line drawn, to look back at the station the table was 
 moved from. Set the line of sight on that station 
 by means of the spindle clamp and slow motion 
 screw. Set the alidade on 5. Mark, or find a point 
 in apparent line for a mark on the tangent at the new 
 station. Set the alidade on i. Set the line of sight 
 on the tangent by the spindle clamp and slow motion 
 screw. Set the alidade on 2. Measure from the in- 
 
OF THE 
 
 UNIVERSITY 
 
 strument station a station chord, and loCatC the next 
 regular station on the curve. Continue the curve as 
 before. 
 
 To run the curve backwards to the obstruction, 
 set the alidade on 2, instead of i. Set the line of sight 
 on the tangent by the spindle clamp and slow motion 
 screw. Set the alidade on i, so as to look back on 
 the curve. Measure from the instrument station a 
 station chord and locate the next regular station 
 backwards on the curve. Continue the curve back- 
 ward the same as forward, except that lines i and 2 
 exchange places in the order of their being used. 
 
 Where a plus station occurs, one-half the partial 
 angle covered by the chord to it may be laid off on 
 either side of one-half the regular angle, i m 2, as 
 may be required, as was done for starting the curve. 
 
 Wishing to know the plus station that would 
 come on a designated line running from the instru- 
 ment station, measure the angle (34) between it and a 
 line to the last station located, or between it and the 
 tangent at the instrument station. Call this angle 
 
 Find n = 
 
 Reduce the decimal part of n to feet, or the 'units the 
 station distance is measured in. Add the resulting 
 number and plus to the number, and plus (if any), of 
 the station where the plane table stands to get the 
 number and plus of the station at the end of the line 
 if \ d was measured from the tangent at the instru- 
 ment station, or the last station located, if reckoning 
 from there. 
 
50 
 
 Lay the angles out on the sheet with great care. 
 Be especially careful in setting the alidade to the lines 
 on the sheet. 
 
 44. Laying Out Parabolic Curves. Fig. 10. 
 
 These curves may be used where the tangents I A 
 and I B are equal or unequal. The curve cuts the 
 equally spaced breadths i-i 2-2 etc., drawn parallel 
 to the line I M, joining I with M, the middle point 
 of A B, so as to make the following proportions. The 
 distances 3~P 3 ; 5~P 5 ; etc., from the tangents I A 
 and I B, and P 2 -2 ; P 6 -6; et ., from the chord A B 
 are equal, if equally removed from I M. P 4 is in 
 the middle of I M. i-Pj ; 2-P 2 ; 3~P 3 ; etc., are in 
 the ratios of the squares of the natural numbers. P r -i ; 
 P 2 -2 ; P 3 - 3, etc. , are in the ratios of the products of the 
 natural numbers written from A to M, the one at M 
 repeated, and continued back to A, as in the figure. 
 This may be represented as follows : 
 
 o A A o 8 o 
 
 1 i I-P! P^i i 7 7 
 
 2 4 2-P 2 P 2 ~2 26 12 
 
 3 9 3-P 3 p 3 -3 35 15 
 
 4 16 I-P 4 P 4 -M 44 16 
 
 Make an even number of divisions between A 
 
 and I M. 
 
 Intersect the tangents A I and B I at I (33). 
 Survey and plat these tangents. Set up the plane 
 table at I. Locate the tangent points A and B at 
 equal or unequal distances from I, as may be desired. 
 It may be well to vary the positions of A or B slightly 
 in order to make the spacings on the tangents, I~3 ; 
 3-2 ; or 1-5 ; 5-6 ; etc. , of a convenient length to lay 
 
7 
 
 S 
 
 -51- 
 
 out. Lay out and locate 1,2, 3, etc., on A I ; and 
 5, 6, 7, etc., on I B, so as to divide each tangent 
 into the same even number of equal parts. Move the 
 plane table to A. Set it up at A and orient it by A 
 I. Set the alidade on A B. 
 Measure A B and plat it. See if 
 it checks on b. If not, review the 
 work and correct the errors. Lo- 
 cate M, the middle point of A B. 
 Locate points i ; 2 ; etc., on A 
 B, so as to divide A B into the 
 number of equal parts in both 
 tangents A I and I B. Measure 
 
 1 M on the ground and on the 
 plat. See that they agree. Divide 
 ^ I M by the square of the num- * 
 ber of divisions in the tangent A 
 
 I, to get the unit measure for the 3 
 work. Multiply this unit by the 
 ratio numbers formed as above 
 noted to get the measurements 
 for locating points on the curve. 
 Figure out both sets. Add them/ 
 in succession for the various 
 points P : ; P 2 ; etc. See if the 
 results are in proportion to the 
 natural numbers i, 2, 3, etc. If 
 not, review the figures and cor- 
 rect the errors. Stretch^ a cord, or tape, from i 
 to i. Locate P l on this line by the measure- 
 ments for P l . Stretch a cord, or the tape, between 
 
 2 and 2. Locate P 2 on this line by the measure- 
 ments for P,. So continue for each of the several- 
 
52 
 
 stations. Plat them on the sheet in the same man- 
 ner. Measure from A to P 1 ; P : to P 2 ; P 2 to P 8 ; 
 etc., around the curve. Record these measure- 
 ments. 
 
 To lay out a tangent at any point, as P 3 . Di- 
 yide the number of the point by 2. The quotient 
 will show the number of spaces on A I, from A to E, 
 the intersection of the required tangent with A I ; and 
 the number of spaces on I B, from I to F the inter- 
 section of the required tangent with I B. Measure off 
 these spaces, and locate E and F. Run a line from 
 E to F. See if it passes through P 3 . If not, re- 
 view the work and correct the errors. 
 
 Any number of tangents may be laid out as above 
 noted, the measurements computed from them to 
 points on the curve ; and these used in locating sta- 
 tions ; either with respect to A B, or the chord join- 
 ing the tangent points of these tangents with each 
 other, or with A or B. By this means obstructions 
 may be passed or difficulties overcome. 
 
 The station as P 3 , at which a tangent as E F is 
 laid out, divides that tangent into two parts whose 
 ratios are expressed by the number of station distances 
 between the station and A and B. E P 3 is j^sths 
 and P 3 F is ^ths of E F. This principle may be 
 used to avoid obstructions. 
 
 45. Direction of a Line. Fig. 11. The 
 direction of a line may be found by observing stars at 
 their elongations. The stars near the pole are the 
 most useful for this purpose. Their names and rel- 
 ative positions can be readily learned from a star map 
 or a cheap planisphere. This will enable one to' find 
 the stars required for observation, or to tell the stars 
 
-53- 
 
 that may be observed at any time. The apparent 
 daily motions of the stars are in arcs of circles. When 
 a star is at that part of its daily motion path, where it 
 appears to move vertically, or follow the range of a 
 plumb line, it is at its elongation. At the eastern 
 elongation it is going upwards, at the western down : 
 wards. Western elongations are more convenient 
 for observation with ordinary instruments. The ob- 
 servation is an easy one to make, as the star moves 
 in a nearly vertical direction for some time. The ob- 
 servation consists in watching the star near its elonga- 
 tion, and keeping the range of a line to it. Finally 
 get the direction of a line to the star when at its elon- 
 gation. This is done by following the star with the 
 vertical cross wire in a telescope, or by moving a 
 compass sight on a level board, even keeping the 
 star behind a suspended plumb line, when watching 
 it through the compass sights, or .by any similar 
 device. For further information regarding this kind 
 of work consult an article on the ' ' Direction of a 
 Line, " in the TECHNIC for 1892. 
 
 Examine the plane table and find how large an an- 
 gle of elevation can be observed with the rule stand- 
 ing on the board when sighting. Unless this angle 
 considerably exceeds the latitude of the place of ob- 
 servation, a diagonal eyepiece will be required to the 
 telescope. If compass sights are used the observa- 
 tion will need to be made on an entirely different 
 plan. 
 
 To illuminate the cross-wires, tie a piece of un- 
 glazed white paper (common newspaper) over the 
 objective by a string around the telescope tube back 
 of the lens, with a hole in the paper centrally over 
 
54 
 
 the objective. This hole may be from %" to y&" 
 broad. Light shining upon the outside of this paper 
 will make the cross-wires visible. A lantern with a 
 lens throwing light in one direction is best. A com- 
 mon lantern may be made use of by nearly surround- 
 ing it with something to shut off the light. The light 
 must not shine in the observer's face. It must there- 
 fore shine obliquely from the side upon the paper 
 over the objective. The light may be toned to any 
 faintness within the telescope by withdrawing the 
 lantern, or changing the obliquity of the beam of 
 light. 
 
 Choose some point of the line whose direction is 
 sought, or reference line, from which a clear view of 
 the stars to be observed may be obtained for an in- 
 strument point. Fasten a lantern at some other 
 point of the reference line to sight to. A common 
 lantern about a mile away will do very well. If nec- 
 essary to have it nearer a fainter light may be made 
 by covering part of the lantern globe. This light 
 should be faint enough to look about like a faint 
 star, so that no change of focus of the objective will 
 be required. 
 
 In good season for beginning .the observations, 
 set up the plane table over the instrument point of 
 the reference line, with a sheet of paper on for this 
 work. Set a needle at the center of the sheet as a 
 guide for the edge of the rule. Turn the board so 
 there will be room on the paper, on the side towards 
 the stars to be observed, and a good place for the plat 
 of the reference line. Clamp the spindle. Set the 
 alidade on the reference line, sighting to the distant 
 lantern, the cross-wires being illuminated. Use the 
 
55 
 
 hands in setting the alidade, and keep the edge of 
 the rule to the needle at the center of the sheet. 
 Draw a line from the center of the sheet by the edge 
 of the rule, m r, Fig. n. Place the alidade to look 
 at a star, the edge of the rule being kept to the nee- 
 dle. Direct the telescope to a star. Bring the star 
 into the field of view of the telescope by the hands. 
 Illuminate the cross-wires. Set the vertical wire on 
 the star, using the hands and keeping the edge of the 
 rule to the needle at the center of the sheet. Keep 
 this wire on the star very carefully as it approaches 
 its elongation, which may be known by the increased 
 time taken by the star to cross the wire. Follow the 
 star, moving the alidade with the hands, keeping 
 the vertical wire exactly on it till it appears to pass di- 
 rectly along this wire. Presently it will leave this wire 
 and go to the same side it came from. Follow it no 
 longer than till it passes directly along the wire. Draw 
 a line from the center of the sheet by the edge of the 
 rule. Mark this line for the star observed with a note 
 of its elongation, east or west, as m, s, Fig. n. 
 Set the alidade again on the reference line. See if 
 m r fits the rule. If it does not fit well this observa- 
 tion must be rejected. The nicety of the fit depends 
 upon the character of the work. 
 
 To save confusion on the sheet in case the rule 
 does not fit m r, set the edge of the rule to m r, and 
 set the line of sight on the distant lantern, with the 
 spindle slow motion screw as a preparation for the 
 next observation. 
 
 In case the rule fits well enough, but not ex- 
 actly, draw a second line by the edge of the rule and 
 mark both for the star observed. If the table stands 
 
56 
 
 still -till time for the next observation, mark the sec- 
 ond line for next star. 
 
57 
 
 Observe a second star as the first was observed.' 
 Draw a line marked for it as m s'. Take a pointing 
 again on the reference line as before. " Mark the ref- 
 erence line drawn on the board if it differs from the 
 last, otherwise note the fact that the lines checked. 
 
 If there are two lines m r drawn on the sheet 
 for the observation of a star, draw a third line bisect- 
 ing the angle between them for the line to be used in 
 making the reductions which follow, unless there is 
 good reason to take one of the original lines drawn, 
 if there is, use that line. 
 
 Observe three or more stars in this way, noting 
 the line for each, and the elongation of each. 
 
 The lines to the stars are to be taken in pairs 
 and the position of the meridian line found graphi- 
 cally. There may be as many results found as there 
 are pairs that can be formed from the number of 
 stars observed. Three stars will give 3 pairs and 3 
 results. Four stars will give 6 pairs and 6 results. 
 And so on. 
 
 Let Fig. 1 1 represent the lines needed in finding 
 the position, m p, of the meridian. 
 
 Take from the Nautical Almanac the apparent 
 declinations of the stars for the date of the observa- 
 tion. 
 
 If A = the azimuth of the star s and 8 =its 
 apparent declination, and A l = the azimuth of the 
 star s' and 8 1 = its apparent declination, the follow- 
 ing equations may be written. 
 
 Tan^(A +A,)= tan ^( A o~ A i) ( cos S +cos8,) 
 
 cos 8 cos 8 l 
 
 A ) tan ^ < A Q + A i) (cos 8 Q 
 
 cos cos 
 
58 - 
 
 These equations may be solved graphically, being 
 simple proportions. 
 
 Bisect the angle ^ m s' by a line m c. Lay off 
 m c equal to ten parts of some scale, as large as may 
 be. Draw ^ s' perpendicular to m c at c. Draw two 
 indefinite lines a b and a d, making any convenient 
 angle with each other. Lay off a f an daf' = 
 natural cosine of 8 , or any multiple of cos 8 (in this 
 case 4 was used ) to the same scale m c was meas- 
 ured by. Lay off f e equal to the same multiple of cos 
 8j, to the same scale, and/" e' =fe. Lay off a i = 
 s c = s' c. Also i , 2 = 2, 3 = 3, 4 = -r c, if desired. 
 Join e and e' . Draw parallels to e e' through i, 2, 3, 
 and 4, intersecting a dm /z 15 A 2 , A 3 , and A 4 ; a h l should 
 equal A lt A 2 ; h^ h^ ; and h^ h. From c on s s', 
 lay off c p = a k l , towards the line which marks the 
 star having the greater declination, because the star 
 having the greater declination will have the smaller 
 azimuth from the north at its elongation. Draw m p. 
 p m r is the angle between the reference line and the 
 meridian 
 
 By taking the stars observed in pairs, a separate 
 result may be obtained for each pair, and the final 
 location of m p made more trustworthy. If the 
 lines become too numerous and the drawing too com- 
 plicated, separate drawings may be made for each 
 pair by pricking the line m r and the lines for a 
 pair of stars through onto a separate sheet, and do- 
 ing the drawing on that sheet. After the angle p 
 m r is found it can be compared with the value on 
 the original sheet. 
 
 There are four cases of this graphical solution. 
 The reference line m r may be between m s and m s' t 
 
59 
 
 or not, as in Fig. n. The stars may be at the same 
 elongation or at different elongations, as in Fig. n. 
 In case the stars are at the same elongation the dis- 
 tance corresponding to c p will represent tan ^ 
 (A -f- AJ) and must be laid off from c towards the 
 place of the meridian m p, that is, to the left of c if 
 the 'stars are both at the eastern elongation, or to 
 the right of c if at the western elongation, when the 
 direction m p is to the N. The circumstances of each 
 case will dispose of all ambiguity. 
 
 Lay off the denominator of the fraction in the 
 second number of either equation and the distance 
 corresponding to s c on the same line. Lay off on 
 the other line that part of the numerator depending 
 for its value upon 8 andSj. Join the points located 
 by functions of 8 and 8 lf and draw a parallel or 
 parallels to this line as required. A multiple of the 
 cosines is taken in order to bring e and e f a good dis- 
 tance apart s' c was laid off four times as a test of 
 the precision of the drawing. The lines a b and a d 
 should be open enough to give good intersections at 
 Aj, h. 2 , etc., better than in Fig. n. 
 
 The angle p m r may be measured (34), and the 
 work oriented with respect to the meridian. The 
 meridian m p may be laid out on the ground if de- 
 sired. 
 
 The meridian may be laid out on the ground di- 
 rectly, without platting the lines on the sheet if de- 
 sired. After observing a star at its elongation, point 
 the telescope downwards and s t a stake marked for 
 the star with a point marked on it in line. Do the 
 same for two or more stars, setting the stakes, as far 
 away as convenient, yet near enough for the points 
 
60 
 
 on them to be well located. Reverse the telescope 
 about its axis ; shift the rule to the other side of the 
 needle at m ; and level the table anew after observing 
 and marking the line to each star. 
 d ^1 Make notes showing the position of 
 \2)' the telescope when observing each 
 star. In combining the observa- 
 tions in pairs, place together those 
 for two stars for which the notes 
 show the telescope to have been in 
 two different positions. By so 
 doing the imperfections of the 
 instrument will do less harm. 
 
 If I in Fig. 12 shows the place 
 of the plane table C and D, the 
 stakes for two stars, one at its E. 
 elongation and the other at its W. 
 elongation, and I K the meridian, 
 make the following measurements 
 carefully, as with a steel tape. C D = d; I C = a\ 
 and I D = b. Let x = C K, and y = K D, the seg- 
 ments of C D between the meridian and C and D. 
 
 = d 
 
 x y = 
 
 COS3 0~ b 
 
 a cos 8 -~ b cos 
 
 where 8 is the apparent declination of the star repre- 
 sented by C, and 8 X the apparent declination of the 
 star represented by D. Find x and y and locate K 
 with the plane table and steel tape. As a check on 
 the figuring use 
 
 a cos 8 
 
 y b cos 8, 
 
-01 
 
 when the stars are at the same elongation d = 
 
 x y and 
 
 , _ , a cos 8 -f- b ros 8, 
 a cos 8 () cos 8 l 
 
 In this case K will not come between C and D but to 
 the right of D if the stars are in the W. elongation, 
 or to the left of C if in the E. elongation. Should 
 D' be set in line from I to D so that I D' = I C = a, 
 the figuring would be simpler. Measure C D' = d' ', and 
 
 x'y' = - d' tan y 2 (8 + 8 X ) tan ^ (8 l 8J. 
 For the stars at the same elongation d' = x' y' and 
 
 x f +y=d' cot % (8 + 8 t ) cot ^ (8 -8i). 
 An inspection of this last equation shows that stars 
 having nearly the same declination should not be 
 used together, if observed at the same elongation. 
 
 Since A = C I K and A l = D I K, the azi- 
 muths of the stars at their elongations. C I D = A 
 -f- Aj. For the stars in the same elongation, C I D 
 = A Aj. Measure C I D with the plane table 
 or compute its value from the steel tape measure- 
 ments. Then 
 
 Tan % (A - A^ = - tan V 2 A + At) tan %(<, + *i) tan V 2 (* -<i) 
 Tan % (A +Ai) = - tan V 2 (A + AX) cot y 2 (S + 5i) cot V 2 (*o-i) 
 
 Observations on stars having nearly the same decli- 
 nation, made at the same elongation, should not be 
 combined. 
 
 The latitude of the place < may be computed 
 as follows : 
 
 cos 8 ft cos 8, 
 
 cos <> = 
 
 LO __ 
 
 sin A sin 
 
 Whence it appears that the direction of the me- 
 ridian and the latitude of a place may be found 
 
without reading angles. A and A 1 may be found 
 directly from the triangles C I K and D I K. 
 
 In case a plane table is used having compass 
 sights on its alidade the direction of the meridian 
 may be found as follows : Suspend a long plumb line 
 from a fixed support so high that the stars to be ob- 
 served will appear somewhat below the support Use 
 a large piece of stone or some bricks for the bob, and 
 let it hang in a vessel of water to steady the line. 
 Drive two stakes some distance S. of the plumb line, 
 and nearly in an E. and W. line. Saw off their tops 
 to the same elevation. Nail a smooth board on top 
 of the stakes. The stakes should stand firm, at such 
 a distance S. of the plumb line that with the eye a 
 little above the board, the stars to be observed will 
 appear below the support of the plumb line. The 
 length of the board should be 6 or 8 ft. if about 20 
 ft. S. of plumb line. Level the board carefully, both 
 ways. Remove one of the compass sights from the 
 rule of the alidade. As a star approaches its elonga- 
 tion place the rule on the board with the compass 
 sight to the south, the rule pointing to the plumb 
 line. Looking through the compass sight keep the 
 plumb line over the star by shifting the alidade with 
 the hands on the board. Follow the star till it ap- 
 pears to pass directly along the plumb line, and leave 
 it again on the same side it approached from. Illu- 
 minate the plumb line if need be by a light from be- 
 hind the observer. Follow the star no longer than 
 till it passes directly along the plumb line. Keep the 
 rule pointing as nearly as may be to the plumb line 
 all the time. Mark the place where the slit in the 
 compass sight stands on the board and number the 
 
--63 
 
 mark. Record in the note-book the same number 
 with the name of the star and a note of the elongation, 
 E. or W. Observe 2 or more stars in this manner. 
 Measure the exact distances of each of the marks on 
 the board from the plumb line. Measure very fully 
 the distances of each mark on the board from all the 
 others. Record these measurements. Reduce the 
 position cf the meridian by calculations similar to 
 those above given, remembering that the triangle C 
 I D, Fig. 12, lies S. of I. The point K may then 
 be marked on the board. The line from K on the 
 board to the plumb line should be very carefully 
 produced by repeated sightings. This may be done 
 with a field glass, if one is to be had, by driving a 
 nail at K in the board, standing back a sufficient 
 distance, and using the field glass to see that the 
 stake and nail beyond is set in line of K and the 
 plumb line. Better prolong the line some distance 
 S. of the board, and then N. of the plumb line, set- 
 ting stakes and nails in both directions. 
 
 SOME BOOKS 
 
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 Publisher and Bookseller to the University of Michigan, 
 Ann Arbor. 
 
 Any book in this list will be sent, carriage free, to any address in the 
 
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 CHEEVER. Select Methods in Inorganic Quantitative Analysis. By 
 Byron W. Cheeyer, A. M., M. D., late Acting Professor of Metallurgy 
 in the University of- Michigan. Revised and enlarged by Frank 
 Clemes Smith, Professor of Geology, Mining and Metallurgy in the 
 State School of Mines, Rapid City, S. D. Parts I and II. Third 
 edition. 12mo. $1.75. 
 
 aUTHE. Laboratory Exercises with Primary and Storage Cells 
 Cloth, 45 cents. 
 
64 
 
 CLASSEN-HARRIMAN. Quantitative Analysis. By Alexander Clas- 
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 Chemistry in the - oyal Technical chooi, Aachen. Authorized 
 Translation from the Fifth German Edition, with an Appendix on 
 the Qualitative Analysis of Minerals, Ores, Slags, Metals, Alloys, 
 Etc., Including the Rare Elements, by Norman F. Harriman, Assist- 
 ant in Chemistry in the University of Michigan. 79 Illustrations. 
 Handsomely bound in half leather, $4.00. 
 
 DZIOBEK. Mathematical Theories of Planetary Motions. By Dr.. 
 Otto Dziobek, Privatdocent in the Royal Technical High School of 
 Berlin, Charlottenburg. Translated by Mark W. Harrington, for- 
 merly Chief of the United States Weather Bur< au, and Professor of 
 Astronomy and Director ot the Observatory at the University of 
 Michigan, President of the University of Washington, and Wm.J. 
 Hussey, Assistant Professor of Astronomy in the Leland Stanford, Jr. 
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 GREENE. The Action of Materials Under Stress, or Structural Me- 
 chanics. With examples and problems. By Chas. E. Greene, A. M., 
 M. E., Professor of Civil Engineering in the Univer-ity of Michigan. 
 Consulting Engineer. Octavo. Cloth, $3.00. 
 
 NETTO. The Theory of Substitutions and Its Application to Algebra. 
 By Dr. Eugene Netto, Professor of Mathematics in the University of 
 Giessen. Revised by the author and translated with his permission, 
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 in the University of Michigan, Professor of Mathematics, Columbia 
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 REED-GUTHE. A Manual of Physical Measurements. By John O. 
 Reed, Junior Professor of Physics, University of Michigan, and Karl 
 E. Guthe, Assistant Professor of i hysics, University of Michigan. 
 185 pages 89 illustrations. Octavo. $1.50 
 
 WATSON. Tables for the Calculation of Simple or Compound Interest 
 and Discount and the Averaging of Accounts. The Values of Annui- 
 ties, Leases, Interest in Estates and the Accumulations and Values 
 of Investments at Simple or Compound Interest for all Rates and 
 Periods; also Tables for the Conversion of Securities and Value of 
 Stocks and Bonds. With full Explanation for Use. By James C. 
 Watson, Ph. D., LL. D. Quarto, cloth, $250. 
 
 WRENTMORE-GOULDING. A' Text-Book of Elementary Mechanical 
 Drawing for Use in Office or School. By Clarence G. Wrentmor^ , 
 R. S., C. E., and Herbert J. Goulding, B. S., M. E., Instructors in 
 Descriptive Geometry and Drawing at the University of Michigan. 
 Quarto. 109 pages and 165 cuts. $1.00. 
 
 WRENTMORB. Plain Alphabets for Office and School. Selected by 
 C. G. Wrenlmore, B. S., C. E., Instructor in Descriptive Geometry and 
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 PHYSICAL LABORATORY NOTE BOOK.- .4 Note Book for the 
 Physical Laboratory. Designed to be used in connection with any 
 Physical Laboratory Manual. ontains full directions for keeping a 
 Physical Laboratory Note Book. 112 pages of excellent ledgt-r ,v ril- 
 ing paper, ruled in cross sections, Metric System, size 7x9^ inches. 
 Bound in full canvass, leather corner;-. Price, by mail, 30 cents. 
 Special prices to Schools furnished on application. 
 
 BOTANICAL LABORATORY NOTE BOOK. A Note Book for the 
 Botanical Laboratory. 200 pag s of best writing paper, ruled with 
 top margins. Pocket on inside of front cover for drawing cards. 
 Bound in substantial cloth cover and leather back. Size, 6x!>^. 
 Price, by mail, 35 cents. Special prices to schools furnished on 
 
 application. 
 
 ENGINEERING LABORATORY NOTE BOOK. A Note Book for the 
 Engineering Laboratory, University of Michigan. Full sheep bind- 
 ing. Size, 5^x8. Contains 200 pages. (With general directions. 
 Cross section ruled.) Price, 75 cents. 
 
 FIELD ENGINEERING NOTE BOOK, SURVEYING.- 200 pages. Cross 
 section ruled. Full sheep binding, 50 cents.^ 
 
 'or THE' 
 UNIVERSITY 
 
UNIVE 
 
 CALIFORNIA LIBRARY 
 
 THIS BOOK IS DUE ON THE LAST DA 
 STAMPED BELOW 
 
 ME 
 
 9, 6 
 
 OCT 13 1916 
 
 
 MAR 10 1942 
 
 MAfi 31 1942 
 
 REC'D LE> 
 
 APR 1999 
 
 30m-l,'15 
 
YB 51917