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 ENGINEEE8 
 
 SURVEYING INSTRUMENTS; 
 
 THEIR CONSTRUCTION AND USE. 
 
 BY I. O. BA^JJR, C. B., 
 
 Professor of Civil Engineering, Univereity of Illinois. 
 
 NEW YORK. 
 Engineering News Publishing Co. 
 
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 Messrs. HELLER & BEIGHTLY publish a book containing tables and c bps 
 useful to Civil Engineers and Surveyors : also contains much valuable inforfiu- 
 tion respecting the selection, proper care, and use of field instrument- A ebpy 
 of this book they send by mail, post-paid, to any Civil Engineer or Biuveyof in 
 any part of the world, on receipt of a postal card containing the name- ■> > . <f p ->st - 
 office address of the applicant. 
 
ENGINEERS' 
 
 SURVEYING' INSTRUMENTS 
 
 THEIR CONSTRUCTION AND USE. 
 
 BY I. O. BAKER, C. E.,'i 
 
 PEOFESSOE OP OIVIL EKGINBEEING, UN^YEEBITY OF ILLINOIS. 
 
 NEW YORK 
 
 ENGINEERING NEWS PUBLISHING CO. 
 
 ■ 1888. 
 
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 Bt<> ■ 
 
 t^] 
 
 pebss or 
 
 ENGINEERING NEWS PUBLISHING CO., 
 
 TEIBUNE EUILDDTG, 
 
 NEW YORK. 
 
TABLE OF CONTENTS. 
 
 oductory. page. 
 
 Plane Table. 
 
 Its Uses and Advantages 1 
 
 Construction 2 
 
 Best Form of Plane Table 3 
 
 Adjustments 3 
 
 Field Work 4 
 
 Limits of Precision 4 
 
 Telescope. 
 
 What it Consists of 4 
 
 Construction. Two kinds . * 5. 
 
 A Seeing Telescope 5> 
 
 A Measuring Telescope % 
 
 Objective. Its Defects and how Eliminated 5 
 
 Eye Piece 6 
 
 Spherical and Chromatic Aberration 6 
 
 Huyghen or Negative Eye-Piece 6. 
 
 Kainsden or Positive Eye-Piece T 
 
 Difference between the two Eye-Pieces 7' 
 
 Erecting Eye-Piece <» 
 
 Telescope Slide 8- 
 
 Cross Hairs. Different Kinds lf- 
 
 Stretching the Cross Hairs 9- 
 
 The best Spider Lines and how to Stretch 9« 
 
 Testing the Telescope 10 
 
 For Chromatic and Spherical Aberration 10 
 
 Flatness of Field 10 
 
 Size of the Field 11 
 
 Aperture of Objective 11 
 
 Illumination 11 
 
 Definition. To test for 12 
 
 Magnification and Methods of Measuring it 13 
 
 Parallax and Adjustments for 13 
 
 Errors of Instruments due to Parallax 14 
 
 Care of Instrument 4 
 
 Optical Center of a Lens 15 
 
 Definition of a Lens 15 
 
TABLE OF CONTENTS. 
 
 Verniers. 
 
 Principles and Definitions i5 
 
 Illustrated Examples 16 
 
 Direct and Betrograde Verniers 16 
 
 Xieast Count of a Vernier 16 
 
 "To Bead a Vernier *. 17 
 
 ^Examples Illustrated 1 -8 
 
 Which Vernier to Bead -8 
 
 Practical Hints :9 
 
 The Most Frequent Error & 
 
 The Transit. 
 
 Definition. Difference from Theodolite 20 
 
 When First Invented 
 
 What Constitutes its Chief. Feature of Value 21 
 
 The Tripod. How Connected 21 
 
 How to Set a Tripod 21 
 
 Leveling Screws 22 
 
 SSerious Defects in Instruments 22 
 
 How Bemedied 23 
 
 Graduation. The Most Common One 23 
 
 Centers or Vertical Axes 24 
 
 Tangent Screws 24 
 
 Clamps 24 
 
 A Perfect Tangent Screw 24 
 
 The Oldest and Worst 25 
 
 One of the Best Forms. The Gambey 25 
 
 The Latest Tangent Movement 26 
 
 Belation of Magnifying Power of Telescope to Least 
 
 Count of Vernier 26 
 
 Taking Care of an Instrument 27 
 
 Practical Hints 27 
 
 Measuring Angles Accurately 29 
 
 By Series 29 
 
 By Eepetition 30 
 
 Comparison of Methods 30 
 
 Transit Surveying „ 31 
 
 Angle Method 31 
 
 Quadrant Method 31 
 
 Traversing. How to Bun 31 
 
 Keeping the Notes 32 
 
TABLE OF CONTENTS. 
 
 Comparison of Methods 33 
 
 Sources of Error 34 
 
 Errors of Manipulation 34 
 
 Errors of Sighting 34 
 
 Errors of Adjustment 34 
 
 Errors of Beading 35 
 
 Limits of Precision 35 
 
 Determining Areas by the Transit 35 
 
 -HE Level. 
 
 Qualities Desired in a Leveling Instrument 36 
 
 Stability of the Instrument 36 
 
 Sensitiveness of Bubble *. . 36 
 
 To Measure the Sensitiveness '. . 36 
 
 Different Forms of Level 37 
 
 The Y Level , 37 
 
 The Dumpy Level 38 
 
 Comparative Methods of the Y and Dumpy 38 
 
 Target Bods. Illustrated 39 
 
 Self Beading Bod 40 
 
 The Philadelphia Bod 40 
 
 The Francis Bod 40 
 
 Favorite English Bod 42 
 
 The Texas Bod 42 
 
 How to Make a Self Beading Bod 42 
 
 Target vs. Self Beading Bod 42 
 
 Adjustments 43 
 
 Two Peg Method 43 
 
 Sources of Error 44 
 
 Importance of Clear Comprehension of them 44 
 
 Instrumental Errors 45 
 
 Bod Errors -. 46 
 
 Errors of Observation 46 
 
 Moving of -the Bubble Error 47 
 
 Keeping the Horizontal Hair Horizontal 47 
 
 Curvature of the Earth and Befraction 47 
 
 Limits of Precision 48 
 
 Theory of Probabilities 48 
 
 A Day's Work in Leveling 49 
 
 Practical Hints 50 
 
 Best Length of Light 50 
 
TABLE OF CONTENTS. 
 
 Beciprocal Leveling 50 
 
 Sunshades 51 
 
 Precautions in Closing at Noon or Night 52 
 
 The Stadia. 
 
 Its Origin 52 
 
 Application in Gunnery 52 
 
 Introduction into America 52 
 
 Principle of the Stadia 52 
 
 Telescope Attachment 53 
 
 Adjusting the Hairs 53 
 
 Objections and Better Methods 54 
 
 Maximum Distance between Hairs 56 
 
 Formula for a Horizontal Line of Sight and a Vertical 
 
 Bod 56 
 
 The Eod 57 
 
 Self Beading and Target Bods 58 
 
 Position of Bod For Inclined Line of Light 59 
 
 Formulas for Inclined Line of Sight and Vertical Bod 61 
 
 Horizontal Distance 62 
 
 Vertical Distance ' 62 
 
 Beducing the Field Notes 62 
 
 Geometrical Tables 63 
 
 Beduction Diagram for Horizontal Distances 64 
 
 Stadia Beduction Tables 65, 66, 67, 68 
 
 Diagram for Vertical Distances 69 
 
 Diagram for Horizontal Distances 
 
 Sources of Error in Stadia Work 70 
 
 Inclination of Bod 70 
 
 b 
 
 Value of — 71 
 
 i 
 
 Errors of Observation 71 
 
 Limits of Precision 72 
 
 Practical Hints 73 
 
 APPE ND I X. 
 
 Local Attraction in Land Surveying 7± 
 
 When Only the Area is Wanted 75 
 
 When the Area and also the True Bearings are Desired 76 
 
TABLE OF CONTENTS. 
 
 Sources of Error 77 
 
 Limits of Precision 79 
 
 Balancing the Work 80 
 
 Accurate Chaining. 
 
 Compensating Errors 82 
 
 Cumulative Errors 82 
 
 Sources of Error 83 
 
 Remedies for These 84 
 
 Limits of Precision 86 
 
 Method of Least Squares : 87 
 
 Keal and Apparent Errors 87 
 
 To Determine a True Meridian. 
 
 Polaris and Mizar Method 89 
 
 Method by Equal Shadows of the Sun 91 
 
 Method by Elongation of Polaris 91 
 
ENGINEERS 
 
 SURVEYING INSTRUMENTS, 
 
 THEIR CONSTRUCTION AND USE. 
 
 BY I. O. BAKEE, C, E. 
 
 Professor of Civil Engineering, University of Illinois. 
 
 INTRODUCTORY. 
 
 It is proposed, to give a series of articles on the best methods of 
 constructing, adjusting, and using different engineering field in- 
 struments. The intention is to make the language so simple and 
 clear that all can understand, and at the same time make it some- 
 thing more than an elementary discussion, and not a re-statement of 
 matter found in ordinary text books and engineering manuals. It 
 is not necessary to dwell upon the importance to the engineer of a 
 knowledge of the best forms of construction and of a thorough 
 understanding of the principles which govern the adjustments of 
 his instruments, and of course the' engineer should be an expert in 
 handling his' instruments. 
 
 THE PLATTE TABLE. 
 
 This instrument is not properly appreciated by the engineering 
 profession. In a general way it is understood to be employed in 
 the surveys, geodetical, topographical and geological, made by the 
 general government, but it seems to be very well understood that 
 the plane table is both useful and available in many cases in ordi- 
 nary practice. It is frequently the best means of obtaining the 
 
2 THE PLANE TABLE. 
 
 plat and area of the irregular tracts that occur in ordinary land 
 and city surveying ; it is valuable in making plats of parks, ceme- 
 teries, mining property, system of ditches, etc. It has the great 
 advantage of dispensing with all notes and records of the measure- 
 ments, since they are platted as they are surveyed: it thus saves 
 time and lessens the possibility of making mistakes. 
 
 It will be shown later how the simplest form of the plane table, 
 combined with an ordinary transit or level, affords an easy, rapid, 
 and accurate method o*f "making preliminary surveys for railroads, 
 for the tile or open ditch drainage, for water supply, etc. 
 
 1. Constbuction.— " In its simplest form, the plane table con- 
 sists of a drawing board fixed on a rod, on which lines are drawn 
 to represent the direction of any object as indicated by a ruler 
 placed so as to point to the object. Any other parts are mere ad- 
 ditions to render the operations more convenient and precise." 
 The ruler is usually a scale of equal parts with a feather edge ; at 
 each end it carries a sight. The ruler and sight together are 
 called the alidade. Fig. 2. 
 
 A very good plane table can be made very cheaply, if the engi- 
 neer has a transit or 
 
 I — — | leveling instrument, in 
 
 which the upper part 
 is detachable from the 
 leveling screws. This 
 requires a good draw- 
 ing board, say 18 by 20 
 inches, to the under side 
 of which is screwed a 
 casting of iron or brass 
 similar to Fig. l. The 
 conical portion, a, 
 should be made to fit 
 the socket from which 
 the axis of the level or 
 transit is taken; the 
 cylindrical portion, ce 
 should at in the lower 
 clamps. 
 
 Fie. l 
 
THE PLANE TABLE. 3 
 
 2. In the best form of the plane table, such as are used on the 
 elaborate topographical surveys conducted, by the general govern- 
 
 D 
 
 Fig. 2. The Alidade. 
 
 ment, a telescope with vertical arc and stadia hairs is substituted 
 for the sights. These modifications add very considerably to the 
 range and precision of the instrument, but it is 'too elaborateland 
 expensive for many of the uses suggested above. A plane table of 
 this form is virtually a transit in which the horizontal circle is re- 
 placed by the drawing board, the lines being drawn upon the paper 
 instead of being read from the graduated limb. The principaljad- 
 vantages of this form of plane table may be'obtained by using a 
 transit with stadia hairs in connection with a drawing board and 
 protractor. The form of the instrument used by the U. S. Coast 
 and Geodetic Survey is illustrated and fully described, and the 
 method of adjusting and using it explained in the reports of the 
 above survey for 1865 and 1880. The article] is also published 
 separately, and is for sale by book dealers under the title, The 
 Plane Table and its uses in Topographical Surveying. 
 
 3. Adjustments.— The adjustments arejfew and easily made,, 
 and having once been made are not likely to getjout. They are ; 1,. 
 the feather-edge of the ruler should be a straight line ; 2, the sights, 
 should be perpendicular to the top of the board ;[3, the table should, 
 be horizontal when the level bubble is in the middle of its race. 
 The third is the rigorous requirement, but as the simpler form of 
 plane table can be sufficiently, accurate without any level, it is. 
 almost unnecessary to describe a method for 'testing this con- 
 dition. However, to make the test, place the alidade on the table, 
 and tip the table until the bubble is in* the middle ; reverse the 
 alidade end for end; if the bubble is not in the middle the above 
 condition is not satisfied. Even if the above_condition is not satis- 
 
4 THE TELESCOPE. 
 
 fled, it may be adjusted so as to do perfectly accurate work. To 
 adjust it, mark a point half way between its first and second 
 position ; when the bubble stands at this point the top of the board 
 is horizontal. 
 
 4. Field Work.— Since there is such variety in the kind of 
 work that may be done with the plane table, and in the method of 
 doing it, it will not be possible to give any explanation of the 
 method of doing field work in a short article. It is enough 
 to say that the instrument is very flexible in its adaptation 
 to different kinds of work, one valuable feature being that the 
 work can generally be checked at every step. Gillespie's Land 
 Surveying gives a few hints on the determination of areas 
 with the plane table, and other methods will readily suggest 
 themselves after a little practice. 
 
 5. Limits or Precision.— Since there is such variety in the kind 
 of work, in the method of doing it, and in the conditions under 
 which it is done, it is scarcely possible to state definitely the de- 
 gree of precision to be obtained with the plane table. 
 
 The ordinary class work of the author's students in measuring 
 the three angles of a triangle and four angles around a point, using 
 a wooden alidade with slit and string for sights, gives an average 
 error, for sighting, marking, and scaling off, of 1 minute and 44 
 seconds per angle; this is equivalent to a probable error of 88 
 seconds per angle. Under fair conditions the maximum error of 
 an angle ought not to exceed one or two minutes. An angle can be 
 measured by repetitions with a plane table with surprising 
 accuracy. 
 
 A plat taken by progression or radix progression, to a scale of 
 two inches to a chain, should close with a maximum error of about 
 J of a link per chain ; with a little care this error may be much re- 
 duced and is often inappreciable. Areas should be found by the 
 plane table with an average error of one in 5,000 or 6,000. 
 
 An engineer will find an instrument made as above very useful 
 in many ways and worth many times all it costs. 
 
 II. 
 
 THE TELESCOPE. 
 
 1. A telescope consists, optically, of certain lenses which assist 
 the eye in seeing distant objects ; and, mechanically, of certain 
 
THE TELESCOPE. 5 
 
 parts which facilitate the use of the optical parts. The mechanical 
 parts can best be discussed in connection with the instrument to 
 which the telescope is applied, and in the present chapter only the 
 optical parts will be considered. 
 
 No attempt at an elaborate discussion of the theory of the 
 optical workings of the telescope will be made, but attention will 
 be confined to such points as are needed by the engineer for the 
 intelligent use of his instruments. 
 
 2. Construction.— There are two forms of the simple refracting 
 telescope, usually known as the Galilean, and the astronomical. 
 The last term is not a happy one, and measuring is suggested as 
 being more appropriate, as will appear farther on. 
 
 The Galilean telescope consists of a double convex lens, called 
 the object-glass, or simply objective, placed next to the object, and 
 a double concave lens, called the eye-piece or ocular, placed near 
 the eye. This form shows the object erect ; an opera-glass is a good 
 example. 
 
 The chief purpose of the objective is to increase the amount of 
 light which reaches the eye from the object viewed. The sole object 
 of the eye-piece is to magnify the thing looked at. The above form 
 of telescope then assists the eye by its magnifying and light gather- 
 ing power; but such a telescope would be useless for making pre- 
 cise measurements, since there is no means of indicating the exact 
 point at which the telescope is sighted. It would not be inappro- 
 priate to call it a seeing telescope. The first telescope was of this 
 form and took its name from the inventor, Galileo. 
 
 The measuring telescope consists of three essential parts ; 1, a 
 convex objective, which collects the rays of light and forms a 
 bright inverted image of the object ; 2, a convex eye-piece which is 
 essentially a microscope, for viewing the image formed by the ob- 
 jective ; and 3, two fine wires or spider threads, placed in the plane 
 of the image, the intersection of which indicates the precise point 
 sighted at. The objective collects the light, the eye-piece magni- 
 fies, and the cross-hairs indicate the point at which the telescope 
 is directed. If such a telescope neither magnified nor increased 
 the illumination, it would still be of great advantage in making 
 measurements. This form shows the object inverted. 
 
6 THE TELESCOPE. 
 
 Both of the above skeleton forms of telescope are very imper- 
 fect. The methods of improving the optical qualities of these 
 elementary forms will now be considered. 
 
 3. Objective.— A single lens used as an objective has the follow- 
 ing defects. 1. The rays of light, which traverse a spherical lens 
 near the edge, are refracted to a point nearer the lens than the 
 rays which pass through the central portion ; consequently the 
 image is blurred. This deviation of the rays from the focus is called 
 spherical aberration. 2. Rays of white light, after being sepa- 
 rated by a single lens, are resolved into the colors of the prismatic 
 spectrum; consequently the image will be disfigured by colored 
 light. This deviation of the different colored rays is called chro- 
 matic aberration. 
 
 The objective of a telescope is rendered almost free from these 
 defects by substituting for the single lens, a compound one, com- 
 posed of a double-convex crown glass, and a concavo-convex flint 
 glass lens ; the two components have different refractive and dis- 
 persive powers, and by giving the four spherical surfaces proper 
 curvatures, these defects are nearly eliminated. 
 
 Eye-Piece. A single lens used as an eye-piece possesses the 
 same defects— spherical and chromatic aberration— as. when used 
 as an objective, but in a less degree. An eye-piece as a single lens 
 has also the following defects ; 1. The image of a flat object formed 
 by a lens does not lie in a plane, but is concave towards the lens. 
 This deviation of the image from a plane is termed aberration of 
 sphericity, which is wholly separate and distinct from spherical 
 aberration. The objective possesses this defect, but in so slight a 
 degree as to be inappreciable ; but in an eye-piece of a single lens, 
 it is very serious. 2. A telescope with a single lens for an eye-piece 
 has a limited field of vi6w. 
 
 In both forms of telescope, the chromatic and spherical aberra- 
 tion, and the aberration of sphericity are nearly eliminated by sub- 
 • stituting two plano-convex lenses for the single lens, which also 
 increase the field of view. 
 
 Htjtghen or Negative Eie- Piece.— This is a modification of the 
 concave eye-piece of the Galilean telescope. It consists of two 
 plano-convex lenses with their convex sides turned towards the 
 objective. The relations of the eye lenses to each other, and to the 
 objective, are shown in Fig. 3. 
 
THE TELESCOPE. 7 
 
 P is the object ; 0, the objective ; a and b constitute the eye- 
 piece which is placed between the objective and its principal faces ; 
 .Fa is known as the field lens, and 6 as the eye lens. The large 
 
 Fig, 
 
 arrow at I represents the object as it appears through' the tele- 
 scope, the small one representing it as it would appear without the 
 instrument. The angle which the large arrow at /subtends at the 
 eye divided by the aDgle which' the object subtends at the eye is 
 equal to the magnifying power ; that is, the magnifying power is 
 equal to the ratio of the larger arrow at J to the small one. 
 
 Eamsden or Positive Eye-Piece.— This is a modification of the 
 convex eye-piece of the measuring telescope. It consists of two 
 plano-convex lenses with their flat sides towards each other. The 
 relations of the eye lenses to the objective are shown in Mg. 4. 
 
 Pig. 4. 
 
 The nomenclature is as before. Notice that the eye lenses are 
 farther from the objective than its principal focus. The objective 
 forms an image at F which is viewed with the eye-piece. The 
 magnifying power is the ratio of the two arrows at J, as before. 
 
 Notice that the essential difference between the positive and 
 negative eye-piece is that only with the former is a real image of 
 the object formed, and hence it is the only eye-piece with which 
 cross-hairs can be used. Spider lines are sometimes placed in 
 negative eye-pieces, for example in sextants, simply to indicate the 
 middle of the field of view ; they are but indirectly concerned in the 
 
8 THE TELESCOPE. 
 
 accuracy of the observations. The negative eye-piece is better for 
 simply seeing, while the positive is absolutely necessary for mak- 
 ing precise measurements ; the latter is always used in transits, 
 levels, etc. Modifications of Eamsden's eye-piece, are frequently 
 used which consist in the substitution of compound lenses instead 
 of the single lenses shown in Pig. 
 
 4. Erecting Eye-Piece.— The erecting or terrestial eye-piece, in 
 its most elementary form, consists of a convex lens placed between 
 the eye and eye-piece of the measuring telescope, which inverts the 
 image formed by the objective and shows the object erect. In its 
 common form, it consists of a pair of plano-convex lenses, instead 
 of the single lens as above ; this pair is called the erecting piece. 
 
 The erecting eye-piece is inferior to the ftamsden or inverting eye- 
 piece, owing to the loss of light occasioned by the two extra lenses. 
 The inconvenience of the inversion of the object is easily overcome 
 with a little practice. Furthermore, other things being equal, the 
 telescope is shorter with inverting eye-piece ; this is quite an im- 
 portant advantage in the transit. Most of the American engi- 
 neering instruments, have an erecting eye-piece ; but it would be a 
 great improvement, if all were provided with the inverting eye- 
 piece. It is said that the inverting eye-piece is more common in 
 Europe than America. 
 
 Telescope Slide. —To assist in focusing the objective for dif- 
 ferent distances, the object glass is fastened in a tube which slides, 
 with a rack and pinion, into the end of the main tube of the tele- 
 scope. In instruments, provided with an inverting eye-piece, the 
 objective is fixed and the cross hairs and eye-piece move together, 
 to and from the objective. It is immaterial which form is used ; 
 the principle is the same in both, it being only important that the 
 slide shall be straight and fit snugly. 
 
 To facilitate the focusing of the eye-piece upon the cross hairs, 
 the ocular is provided with a similar slide. In some instruments 
 the ocular is moved by a rack and pinion ; but this is unnecessary 
 and even worse than useless, for, having once adjusted the ocular 
 for distinct vision of the cross hairs, it needs no change except for 
 different observers, and it is better that it should not be easily 
 moved. If no rack and pinion is provided, the ocular is moved in 
 and out by hand with a screw-like motion. 
 
THE TELESCOPE. 9 
 
 Ckoss Hairs.— These are usually two spider threads, one vertical 
 and the other horizontal, fastened to a ring which is adjusted in the 
 tube by three or four screws. Lines ruled or etched upon a piece 
 of thin glass, are sometimes used. The cross hairs are sometimes 
 made of very fine platinum wire; can be drawn to the re- 
 quired fineness only by being previously surrounded by silver, 
 which is removed by acid after the wire is drawn. Both spider 
 threads and platinum wires have their advantages or disadvantages 
 and as a consequence their advocates and opponents. 
 
 For the platinum wires, it is claimed that they are best because 
 they are opaque ; this is a desirable property when the cross hairs 
 must be illuminated as in astronomical and mine surveying. Spider 
 lines, particularly dark colored ones, can be illuminated pretty 
 well. It is' claimed also that wires are unaffected by the humidity 
 of the atmosphere, and hence the line of collimation is not liable 
 to change from this cause, an advantage which does not exist if the 
 spider threads are properly stretched. Spider threads, on account 
 of their fineness and cheapness, and the facility with which they 
 are applied, will continue in the future, as in the past, to be used 
 almost universally for cross hairs in engineering instruments. 
 
 Stretching the Cross Hairs.— It does not require much time 
 or skill to replace the cross hairs, the belief of many to the con- 
 trary notwithstanding. The ring which carries the cross hairs can 
 be taken out, (having first taken out the eye-piece) by removing two 
 opposite screws, and inserting a soft wooden stick of suitable size 
 into one of the holes thus left open in the ring, which is turned 
 sideways for that purpose, and then removing the other screws. 
 Two scratches, at right angles to each other, will be found upon the 
 face of the ring, into which the hairs are to be fastened. 
 
 The best spider lines are those of which the spider makes its 
 nest. These nests are yellowish-brown balls which may be found 
 hanging on the shrubs, etc., in the late fall or early winter. When 
 found, the nest should be torn open and the eggs thrown out; if 
 this is not done, the young spiders when hatched will eat the 
 threads. The fibers next to the eggs are to be preferred on account 
 of their darker color. 
 
 Draw out a single fiber and attach each end of it to as heavy a 
 tack, brad, etc., as experiment shows it will support. Dampen the 
 thread by breathing upon it, by holding it in a current of steam, or 
 
10 THE TELESCOPE. 
 
 better, by dipping it in hot water. Support the ring in such a way 
 that, when the thread is laid across it, the weights may hang freely 
 down the sides, thus stretching the thread. The thread may be 
 moved easily with a pin, and when in the proper position it can be 
 fastened with wax, gum, etc., shellac varnish being the best for this 
 purpose. The main point is to stretch it thoroughly and fasten 
 it tightly. If the work is well done, the threads will remain 
 straight, when the reticule is placed in a current of steam or even 
 in hot water. 
 
 12. Testing the Telescope.— In buying an instrument it is very 
 desirable to test the optical qualities of the telescope ; and the 
 following directions are sufficient for such examination. 
 
 Chromatic Aberration.— To test for chromatic aberration, focus 
 the telescope upon some bright object, either a celestial body or a 
 white disk, and then move the object-glass slowly in and out. If, in 
 the first instance, a light yellow ring is seen at the edge of the ob- 
 ject, and in the second a ring of purple light, the object glass may 
 be considered perfect ; this proves that the most intense colors of the 
 prismatic spectrum, orange and blue, are corrected. 
 
 13. Spherical Aberration.— To test this, cover the object-glass 
 ■with a ring of black paper, so as to reduce the aperture about one- 
 half; focus on some small and well defined object for distinct vision. 
 
 , Remove the ring of the paper and cover that part of the objective 
 previously left open ; then notice how much the object-glass must 
 be moved in or out for distance vision. The amount of shift meas- i 
 ures the spherical aberration of the objective ; very little if any 
 motion should be required to obtain a distinct view. Another test 
 is to focus sharply upon some object, where the least motion of the 
 objective should render the object indistinct. This last is not so 
 good a test as the former, for the eye will change focus slightly to 
 accommodate itself to the change of distance. 
 
 14. Flatness oe Field.— The flatness of the field depends 
 mainly upon the aberration of sphericity of the eye-piece. To test - 
 a telescope in this respect, draw a heavy lined square, six or eight 
 inches on a side, on a sheet of white paper and fasten the paper to 
 a flat board. Place this object at such a distance that the square 
 shall nearly fill the field. Focus the telescope on it; then, if the 
 sides appear perfectly straight, the telescope is perfect in respect 
 to flatness of field. 
 
THE TELESCOPE. 11 
 
 A telescope which distorts the image perceptibly causes no 
 error in common use, but is decidedly objectionable for stadia 
 measurements, where two points of the field are used at the same 
 time. 
 
 15. Size of the Field.— By the field of view is meant all those 
 points which are visible at the same time through the telescope. 
 The field of view depends upon distance between the objective and 
 the image, and upon the diameter of the lenses of the eye-piece ; but 
 is independent of the size of the objective. Of course the wider the 
 field of view the better, since width of field facilitates rapid work- 
 ing. The greater the magnifying power the less the field of view. 
 
 To determine the angular width of field, sight upon a rod, house, 
 etc., and mark two opposite sides of the field ; the distance between 
 these points multiplied by 57.3 and divided by the distance from the 
 rod to the objective is the angular width of field in degrees. Since 
 the field varies with the distance, the latter should be stated. 
 
 16. Aperture or Objective. — By the aperture of the objective is 
 meant the effective diameter of the object-glass, that is, that rart of 
 the objective which transmits light which finally reaches the eye. 
 Usually, diaphragms are placed in the tube of the telescope, with a 
 view to improve the optical workings of the lines ; if the diaphragm 
 is placed near the object-glass, it will cut off those rays which pass 
 through the objective near the edge, thus improving the definition, 
 without diminishing the field, but with a loss of illumination. If 
 the diaphragm is placed in or near the eye-piece, it may diminish 
 both illumination and field of view. 
 
 To find the real aperture, direct the telescope towards the sky, 
 place a pointer in contact with the objective so that che pointer may 
 be seen in the small illuminated circle which will be noticed at the 
 small opening of the eye end when the head is drawn back a short 
 distance from the telescope ; move the pointer over the face of the 
 object-glass until the point just appears, and measure the distance 
 from the pointer to the edge of the object-glass. The real or clear 
 aperture of the objective is equal to the diameter of the object-glass 
 minus twice the above distance. 
 
 17. Illumikation.— The brightness of an object as seen 
 through a telescope depends upon: 1, the illumination of the 
 object ; 2, the relative sizes of pupils of the eye and the objective ; 
 
12 THE TELESCOPE. 
 
 3, the polish and transparency of the lenses ; and, 4, the magnifying 
 power. In all telescopes light is lost by reflection from the surface 
 of the lenses, and by imperfect transparency. The object-glass 
 transmits a certain amount of light, which the eye-piece distributes 
 over a larger or smaller area, according to the magnifying power 
 of the telescope ; therefore, the brightness of the image varies 
 inversely as the square of the magnifying power. Since the bright- 
 ness of view is an indispensable requisite of a good telescope, the 
 magnification should never be excessively large, as it frequently 
 happens that the telescope is used in viewing objects only faintly 
 illuminated. 
 
 Since there can be no unit by which to measure the illumina- 
 ting power of a telescope, we can only find the relative illumina- 
 tion. If two telescopes are to be compared as to light, they should 
 stand side by side and be looked through at the same time, so as to 
 be under the same atmospheric conditions. If, as night approaches, 
 the two telescopes be placed side by side, and the same object be 
 viewed through each, the one through which the object is longest 
 visible has the best illumination. 
 
 Definition.— The definition of a telescope depends upon the 
 accuracy of the curvature of the surfaces of the several lenses, and 
 upon the coincidence of the axes of the component lenses of the 
 objective and ocular. The reader should distinguish between 
 illumination and definition. The lack of the former causes the 
 image to be faint, the lack of the latter causes the image to be 
 indefinite. 
 
 To test a telescope for definition, focus upon some small, well 
 defined object; small, clear print is the best. If the print appears 
 clear and well defined and fully as legible at 40 or 50 feet as if 
 viewed with the naked eye at 6 or 8 inches (the best distance for 
 distinct vision), the surfaces of the lenses are correct and well 
 finished. Indistinctness may be caused by spherical aberration ; 
 therefore that should be tested for before definition. 
 
 Magnification. The power of a telescope, or degree of magni- 
 fication, depends upon the relative focal length of the objective and 
 eye-piece. Mathematically any power can be given to any tele- 
 scope ; but, in practice, it is limited by the effects of loss of light, 
 size of field and imperfection of the lenses. For rapid work, the 
 exact focusing necessary with high powers is a drawback, since a 
 
THE TELESCOPE. 13 
 
 small change in distance requires a corresponding change in focus. 
 The magnifying power varies slightly with the distance to the ob- 
 ject ; but, fortunately, the exact magnifying power of the telescopes 
 on engineering instruments is not required. Several methods of 
 measuring the magnifying power are given in Ohauvenet's Practical 
 Astronomy, Vol. I. Articles 7-13, of which the two following are the 
 simplest. 
 
 1st Method. " Direct the telescope in daytime towards the open 
 sky; near the eye-piece and a little beyond it, a small illuminated 
 circle will be seen, which is nothing more than the image of the 
 objective opening of the telescope. Let the diameter of this circle 
 be measured by a very minutely divided scale of equal parts ; then 
 the magnifying power is equal to the quotient arising from divid- 
 ing the diameter of aperture of the object glass by the diameter of 
 this illuminated circle." The chief difficulty in this method lies in 
 the exact measurement of the diameter of the small illuminated 
 circle. 
 
 2nd Method. "Let a staff which is very boldly divided into 
 equal parts of heavy lines, be placed vertically at any convenient 
 distance from the telescope, for example, 150 feet. While one eye 
 is directed towards the staff through the telescope, the other may 
 observe the staff by looking along the outside of the tube. One 
 division of the staff will be seen by the eye at the eye-piece to be 
 magnified, so as to cover a number of divisions of the staff, and this 
 number, which is the magnifying power required may be observed 
 by the other eye looking along the tube." A little difficulty may 
 be experienced on the first trial of this method, but with a few 
 trials it becomes very easy. 
 
 Parallax. — This is an apparent movement of the cross-hairs in 
 reference to the object sighted at, caused by a real movement of 
 the eye of the observer. It shows that the image and cross-hairs 
 are not in the same plane. 
 
 To make this adjustment, direct the telescope toward the sky, 
 or throw it out of focus so that no object can be distinguished in 
 the field, then move the ocular in or out until the cross-hairs 
 can be seen very distinctly. When the cross-hairs are properly 
 focused little specks of dust will be seen on them. Next 
 direct the telescope to the object, keeping the attention fixed upon 
 the cross-hairs, so that the eye shall not change focus to accom- 
 modate itself to the position of the object ; move the objective in or 
 
14 THE TELESCOPE. 
 
 cut until the object appears very sharply defined ; then move the 
 eye back and forth sidewise and note whether the cross-hairs and 
 object alter their relative position. If the cross-hairs appear to 
 move with the eye, they are farther from the eye than the image, 
 and therefore the objective should be moved nearer the object ; for 
 remember, first, that the farther object appears to move with the 
 eye, and second that the farther the object from the objective, the 
 nearer is the image. In practice this is more easily done by trial 
 than described. When properly focused there should be absolutely 
 no movement of the cross-hairs with reference to the object. Of 
 course a telescope should be accurately adjusted for parallax be- 
 fore it is used in making precise measurements. 
 
 The writer has incontestable evidence for believing that many 
 of the errors of instrument work are due to parallax ; this source of 
 error is more serious as the work becomes more accurate. The 
 difficulty is usually in not adjusting the hairs for distinct visions 
 for the normal or natural focus of the eye. The eye perceives 
 things best at a distance of 6 or 8 inches, and this is known as 
 the normal focus. When using a telescope tiie eye should be at the 
 normal focus, the vision then being better and the fatigue less. The 
 eye is continually changing focus to accommodate itself to the 
 different distances of the objects viewed, hence care should be 
 taken that the cross-hair focused for distant visions be at normal 
 focus, which is secured by following the directions as above, 
 particularly if the eye be closed for a moment before pronouncing 
 the adjustment correct. Having secured the proper focus, it need 
 not be changed except for the change of focus of the eye with ad- 
 vancing age and for different observers; the rack and pinion for 
 focusing the cross-hairs is worse than useless. 
 
 Caee. If the objective becomes dusty, brush it off with a fine 
 camel hair brush, or rub it off with a piece of soft, clean chamois 
 skin or a piece of old linen or silk, taking care to use a clean spot 
 for each rub. Unnecessary rubbing of the lenses should be avoided, 
 since it will destroy the fine polish upon which depends the sharpl 
 news and brilliancy of the image. Dust upon the glasses is not as 
 objectionable as a thin, almost imperceptible film of grease ; there- 
 fore, the lenses should never be touched with the fingers. ' When 
 the lenses become>ery dirty, wash them with alcohol. 
 
 In removing the cell containing the objective, care should 
 be taken to screw it back exactly to its former position, else the 
 
VEHNIEKS. 15 
 
 adjustment of the line of sight is thereby destroyed. The com- 
 ponent lenses of the objective should never be taken apart, nor 
 removed from the cell containing them, since they may not be re- 
 turned to their former position, thus disturbing the adjustments. 
 Dust should carefully be kept from the inside of the telescope tube, 
 as it will get on the lenses and cross-hairs. When not in use the 
 eye-piece and object-glass should be covered by their caps. 
 
 The following definition and explanations will be found useful 
 farther along. 
 
 The Optical Center of a lens is, " A point so situated that any 
 ray of light passing through it will undergo equal and opposite re- 
 flection on entering and leaving the lens. It will therefore, be 
 found where a line joining the extremities of two parallel radii of 
 the opposite surfaces, intersects the optical axis of the lens." For 
 a double-convex lens it is always within the surface of the lens. For 
 a planor-convex, or a plauo-concave, the optical center will be at 
 the inte section of the axis with the curved surface. 
 
 In lenses of long focal length, such as are used for the objec- 
 tives of telescopes, the deviation caused by the thickness of the 
 lens may be neglected, as the thickness is very small compared 
 with the distance to the object; therefore, it is assumed that the 
 rays which pass through the optical center are not deviated at all. 
 This leads to the simple definition that a lens is a mathematical 
 point which allows a great deal of light to go through it; the better 
 the lens the more nearly is this condition realized. This "point " 
 is the optical center, as defined ab ove. 
 
 VERNIERS. 
 
 hi 
 
 I. Principles.— A vernier is a short scale moveable by the side 
 of a longer scale, by which subdivisions of the longer scale may be 
 measured. The scale to be subdivided is called a limb. A division 
 of the vernier is a little shorter or longer than a division of .the 
 main scale or limb. The small difference is the space which is 
 measured by the vernier. 
 
 A vernier may be constructed by taking a length equal to any 
 number of parts of the limb, and dividing it into a number of equal 
 
16 
 
 VERNIERS. 
 
 r 
 
 « 
 
 ' 
 
 3 
 
 / — / 
 
 J 
 
 ' / 
 
 fi- 
 
 3 ~ 
 
 parts, one more or one less than the number into which the same 
 
 length on the limb is divided. 
 
 Tor example, the limb shown in Fig. 5, is a 
 scale of inches divided into tenths, the vernier 
 beside it is capable of measuring to hundredths. 
 Notice that the . spaces of the vernier are 
 equal to nine of the limb. Therefore each 
 space of the vernier is equal to 0.1 of 0.9 = 0.09 
 of an inch ; it is 0.01 of an inch shorter than a 
 space of the main scale. 
 
 The first mark of the vernier falls short of a 
 mark on the limb by 0.01 of an inch ; the second 
 falls short by 0.02 of an inch, and so on. There- 
 fore if the vernier be moved slowly forward the 
 — '. I successive coincidence of a line on the vernier 
 with one on the limb will indicate successive 
 advances of the vernier, each equal to 0.01 of 
 an inch. If the lines of the vernier are num- 
 bered as in Fig. 5, the number on the vernier 
 of the line which coincides will indicate the 
 amount that the zero of the vernier has passed 
 a division of the limb. 
 
 2. Direct and Retrograde Verniers.— In the above illustra - 
 tion the spaces on the vernier are shorter than those on the limb ; 
 the supposed motion was in the direction of the graduation of the 
 limb, and the successive lines of the vernier came into coincidence 
 also in the direction of the graduation ; the numbering on the limb 
 and also on the vernier increased in the same direction. Therefore 
 Fig. 5, is a direct vernier. 
 
 If the spaces on the vernier are larger than those of the limb 
 and the vernier is moved in the direction of the graduation of the 
 limb, the successive coincidence will occurin the direction opposite 
 to the motion and also opposite to the direction of the graduation 
 on the limb ; therefore.the lines on the vernier should be numbered 
 in an opposite direction to those on the limb. Such an arrange- 
 ment is a retrograde vernier ; see Fig. 7. 
 
 The direct vernier is much more common ; it is shorter and 
 more convenient to read. A retrograde vernier is used when the 
 line3 of a direct vernier would be inconveniently close together. 
 
 3. Least Count.— The least count of a vernier is the difference 
 
 Fig. 5. 
 
VERNIERS. 17 
 
 in length between a space on the limb and one on the vernier. To 
 find the least count of a vernier, i. e. to determine how small a dis- 
 tance it can measure, let I = the leDgth of a division on the limb ; 
 v = a division on the vernier ; and n = the total number of spaces 
 on the vernier ; then by the principle of the vernier, nl = nv + I, 
 
 I 
 solving which gives, I — v = — , the least count, 
 
 n 
 
 For example in Fig, 5, I = 0.1, and n = 10 ; 
 
 0.1 
 
 hence the least count equals — = = 0.01 
 
 n 10 
 
 The above formula expresses a very important relation ; it is 
 the key to reading all verniers. Notice that the least count of the 
 vernier is equal to the smallest division on the limb divided by the 
 number of spaces on the vernier. In practice, it is not necessary to 
 <5ount n ; it is indicated by the numbering on the vernier itself. For 
 example, if the limb is divided to half degrees and there are thirty 
 spaces on the vernier, which would be indicated by the end line 
 toeing numbered thirty, the vernier reads to one-thirtieth of a half 
 degree or to minutes. 
 
 Notice that the above relation is true for both direct and retro- 
 grade verniers. 
 
 4. To Read a Vernier. — Look at the zero line of the vernier ; 
 if it coincides with a division of the limb.the number of that line on 
 the limb is the correct reading, the vernier divisions not being re- 
 quired. But if, as usually happens, the zero of the vernier comes 
 between two divisions of the scale, note the nearest division on the 
 limb next less, and then look along the yernier till a line is found 
 which exactly coincides or forms a straight line with some line on 
 the limb. The number of this line on the vernier is the distance 
 between the zero of the vernier and the next lower division of the 
 limb, and must be added to the reading taken from the limb. 
 
 A number of examples will now be given to further illustrate 
 these general principles. The student should draw the limb and 
 scale on separate slips of paper of card board, and move one beside 
 the other until he can read them in any position. 
 
 5. • The vernier shown in Fig. 6, i3 the one used on the New 
 York levelling rod. 
 
18 
 
 VERNIERS. 
 
 1 
 
 ^^---^^ 
 
 
 1 
 
 
 1 
 
 
 1 
 
 
 A 
 
 1 
 1 
 
 
 1 
 
 
 
 
 
 
 1 
 
 1 
 
 
 1 
 
 
 
 
 The main scale is divided to feet, tenths and one hundredths; 
 ten divisions of the vernier are equal to nine of the limb ; there- 
 fore, the vernier reads to tenths of one hundreth, or to one thou- 
 
 sandths. The vernier as 
 
 drawn reads 3 feet, 
 tenths, three hundreths, 
 and seven thousands, 
 or 3,037 feet. 
 
 6. Fig. 7, shows part 
 of a circle graduated to 
 degrees and half de- 
 grees ; the vernier has 
 thirty parts, therefore it 
 reads to single minutes. 
 The reading is 210° 30' 
 + 8' = 210° 38'. This is 
 a very common vernier 
 for engineering instru- 
 ments. 
 
 7. The graduation of 
 transits usually has 
 two rows of numbers, 
 
 one increasing to the right and the other to 
 the left ; in this case there are two verniers, 
 one for each series of numbers. Such an ar- 
 rangement is shown in Fig. 8. Each vernier 
 is like the one described in the preceding 
 paragraph, and is read in the same way. 
 The reading of the left hand vernier is 181° «. 
 Pig. 7. 30' + 12' =181° 42' with the outer numbers, 
 
 and 1° 42' with the inner. The right hand 
 vernier reads 167° 12'. 
 
 There is sometimes a doubt as to which vernier should be read. 
 First notice whether the vernier divisions are larger or smaller 
 than those on the limb. If the spaces on the vernier are the 
 smaller, it is a direct vernier, and that vernier should be read the 
 numbers of which increase in the same direction as those on 
 the limb. If the spaces on the vernier are the larger, it is a 
 retrograde vernier, and that vernier should be read the numbers 
 of which increase in the opposite direction to those on the limb. 
 
 Fig. 6. 
 
VERNIERS. 
 
 19 
 
 Or the vernier to be read can always be determined as follows ; 
 move the vernier, until say, the ten mark of one vernier and the 
 twenty of the other coincides each with a line on the limb ; look 
 at the zero, estimate the reading, and then read the vernier that 
 agrees most nearly with the estimated reading: 
 
 In Fig. 8, the numbers on the vernier are inclined in the same 
 direction as the numbers on the limb to which the vernier belongs. 
 Instruments are not made in this way, but such an innovation 
 would be a great improvement. 
 
 8. Fig. 9, is another form of double vernier ; it is often applied 
 to the compass, to be used in setting the declination. Notice 
 that this form is only half as long as the double vernier of Fig. 8. 
 
 It is used where there is 
 
 Fig. 8. 
 
 Fig. 0. 
 
 not space for the longer one and 
 is called a double folded 
 ! vernier. Thelower 
 ' numbers on one side of 
 the zero and the upper 
 ones on the other side 
 constitute a vernier. The 
 proper vernier to be 
 read, in any given case, 
 can be determined by 
 either of the rules of the 
 preceding article. The 
 vernier as drawn, reads 
 2° 30' + 12 = 2° 42' to the 
 right. Notice that the 
 reading of the vernier 
 is 12' not 17'. 
 
 9. Practical Hints.— 
 To determine exactly 
 which lines coincides 
 best, notice the next 
 line on either side and 
 see whether they fall 
 short equal amounts. 
 When several lines on 
 the vernier appear to 
 coincide equally with 
 
20 VERNIERS. 
 
 several lines of the limb, take the middle line. When no line coin- 
 cides, but one line on the vernier is on one side of a line on the 
 limb, and the next line on the vernier is as far on the other side 
 of it, the true reading is midway between the readings indicated 
 by these two lines. If the graduation is very accurate 
 and the lines fine, it is possible, by this method, to estimate 
 the reading to half, or even to thirds, of the least count. 
 
 It frequently happens that the instrument is to be used to lay 
 off a number of equal angles, as for example, 1° ; the vernier is then 
 to be set each time at some particular mark. If it is a double ver- 
 nier, set the zero line to coincide each time, and notice whether the 
 lines next on either side fall short an equal amount ; if it is a single 
 vernier, set, say, the fifteen line to coincide, and note the agree- 
 ment on both sides. This process is considerably more accurate 
 than setting the end line of the vernier to coincide each time. 
 
 The most frequent error in reading a vernier is to omit part of 
 the readiD.s of the limb ; for example, in Pig. 7, forgetting to record 
 the half de,;! ee from the limb. 
 
 IV. 
 
 THE TRANSIT. 
 
 1. This is the name usually given to instruments in common 
 use, among American engineers for measuring horizontal and verti- 
 cal angles. It is sometimes, but incorrectly, called a theodolite. 
 The theodolite is the name given by British engineers to their 
 favorite portable instrument, which is capable of performing the 
 same work as the transit. The essential difference between the 
 transit and the theodolite is that in the former the telescope can 
 transit or turn completely over, while in the latter it cannot. The 
 telescope of the theodolite can be reversed only by lifting it out of 
 its supports and replacing it end for end, which is a very imperfect 
 substitute for the revolution of the telescope of the transit. The 
 transit is sometimes called an engineer's transit, or railroad tran- 
 sit, to distinguish it from an astronomical transit. The transit was 
 invented and first made by a Philadelphia firm in 1831, previous to 
 which time the English theodolite and the magnetic compass, 
 sometimes provided with a full circle graduation by which angles 
 could be read independently of the needle, were the common angle 
 Instruments. 
 
THE TRANSIT. 
 
 21 
 
 The great value of the transit as an instrument of precision is 
 due to the telescope, by which great precision in sighting is ob- 
 tained, and to the graduated circle with its vernier, by which 
 angles can be read with ease and accuracy. All other parts are to 
 facilitate the use of these two. 
 
 2. The Tripod.— The manner of connecting the leg with the 
 head needs attention. The leg should not be placed between two 
 lugs or ears which are fastened to the plate, for in case the leg 
 wears or shrinks there is no adequate means of making it fit. 
 Drawing the lugs together by a screw, bends the plate, and even 
 then only partially remedies the evil. An excellent form is that 
 in which the leg is made of two pieces that bear upon opposite 
 sides of a lug cast upon the plate. Sometimes the leg is in one 
 piece and slotted at the top, which is also very good. Another 
 good form is that in which the leg is not open at the top but bears 
 upon only one side of the lug. In the three forms last mentioned, 
 any looseness is taken up by a thutnb-nut. Sometimes the leg is 
 inserted in- a metal cap which is then fastened to the tripod head; 
 this is better than putting the leg directly between two metal ears, 
 but it is not as good as any of the others. 
 
 To set the tripod, notice 1st, that to allow the position of the 
 plumb-liae, the legs must be swung on their pivots but not side- 
 wise; and 2nd, that to put the 
 plate level, the legs ' must be 
 swung sidewise but not on their 
 pivots. These two points, though 
 small in themselves, are worth 
 remembering as a means of 
 saving time and labor and also of 
 preventing unnecessary wear 
 and strain on the leveling screws 
 owing to faulty construction of 
 the leveling appliances. "With 
 many instruments the last point 
 is a very important one. 
 
 Attention to the following 
 principles also will save much 
 time and hard labor in setting the tripod. Let the lines radiating 
 from c,Fig. 10, represent lines joining the feet of the tripod legs 
 
 Pig. 
 
22 THE TBANSIT. 
 
 with the point over which the plumb bob is to be placed ; b is 
 the position of the plumb bob. It is desired to move the plumb 
 bob from 6 to c. Press the leg 1 into the ground until the bob 
 swings into the line 3 c prolonged; then force 3 in until it 
 swings to c. In general, press one of the legs into the ground 
 until the plumb bob swings to the opposite side of the point 
 from one of the other legs ; then press that leg in until the plumb 
 bob arrives at the center. 
 
 The tripod should be set firmly, but a great deal of time and 
 effort is frequently wasted in forcing the legs into the ground need- 
 lessly. Setting the tripod is an operation that must be repeated 
 many times, and the beginner should learn to do it quickly and 
 easily. 
 
 In case the plumb-bob is lost and only a rough piece of some 
 heavy substance can be had, the instrument may still be plumbed 
 down accurately by holding a second plumb line before the eye in 
 such a position that the eye shall be in the same plane with the 
 two lines ; then, without moving the eye, mark a line under the 
 instrument in the plane. Kepeat the operation at 90 degrees from 
 the first position. 
 
 The intersection of these two is the desired point. The writer 
 heard of a railroad surveying party waiting ten hours while the 
 transitman sent for a plumb bob, which was almost as bad as being 
 •' delayed two days by a hornet's nest." 
 
 3. Leveling Screws. —Some [instruments are provided with 
 three leveling or foot screws, and others with four. The instru- 
 ment can be leveled more quickly with three than with four ; be- 
 sides three are more delicate, and are less liable to strain and 
 damage the instruments. The best instruments never have but 
 three screws. 
 
 A very serious defect in many instruments with four leveling 
 screws, is that the lower ends of the screws are above the center of 
 the ball and socket joint which fastens the upper part of the instru- 
 ment to the tripod. When the one pair of screws is being 
 used to level the plate, the upper part of the instrument 
 must revolve about a line parallel to, but below a line joining 
 the feet of the pair not in use ; therefore the instrument cannot 
 
THE TRANSIT. 23 
 
 revolve without causing the screws not in use to bind, and it can 
 turn only by causing the feet of the screws to slip on the lower 
 plate. Besides the annoyance in leveling the instrument, this 
 binding tends to bend the screws and warp the plates and the slip- 
 ping defaces the instrument by cutting spherical holes in the lower 
 plate. Placing the feet of the screws in small cups obviates the 
 holes in the upper face of the lower plate but increases the objec- 
 tions in the other and more important respects. 
 
 The defect may be remedied by bringing the center of the ball 
 and socket joint into the plane of the feet of the foot screws. As 
 there is no ball and socket joint with three leveling screws, this 
 defect can not exist in that form of construction. A spiral spring 
 fastens the instrument to the tripod. Since nearly, all instruments 
 possess the above defect, it is very necessary that the instrument 
 should be leveled approximately by manipulating the tripod legs 
 as already described. 
 
 Whatever the number, there should be no looseness between 
 the screw and the nut. This looseness is particularly objectionable 
 in a leveling instrument or in a transit used to measure vertical 
 angles. The best instruments have a split nut with a clamp-screw 
 by which to adjust for wear. 
 
 Some instruments are provided with an arrangement for ap- 
 proximately leveling the instrument very quickly without manipu- 
 lating the foot-screws. This device is especially suitable for level- 
 ing instruments ; but by using a little care in setting the tripod, it 
 can be dispensed with easily. All such additions are an advantage 
 in the greater convenience, and a disadvantage in the increased 
 weight, complexity and cost. 
 
 4. Graduation.— The lines of thegraduation should be uniform, 
 and as small as possible and still be legible. In the best instru- 
 ments the graduation is upon solid silver. 
 
 The most common graduation for the horizontal circle of engi- 
 neers' transits, is a circle about 6 inches in diameter divided to one- 
 half degree with a vernier reading to minutes. The degrees are 
 usually numbered in two rows, one like the compass and another 
 from 0° to 360°. The field work is most simple and least liable to 
 error if only the latter numbering is used. See remarks on the 
 transit verniers. 
 
24 THE TBANSir. 
 
 The verniers on the horizontal circle are sometimes placed 90 
 from the line of sight, in which case the observer must change his 
 place between sighting the telescope and reading the vernier ; 
 sometimes the verniers are placed immediately under the tele- 
 scope, in which case the observer can read the vernier and sight 
 the telescope without changing his position, although the telescope 
 must be revolved before the vernier can be read. The latter posi- 
 tion is preferable, especially in confined positions as in under- 
 ground surveying, etc. An intermediate position would be still 
 better. For convenience of reference, the verniers should have- 
 some distinguishing mark, say A. B. C. etc., upon their faces. 
 
 5. Centers or Vertical Axes.— Usually there are two concen- 
 tric vertical axes, the verniers, telescope, etc., turning about the 
 inner, and the graduation revolving about the outer one. As ordi- 
 narily used, the outer axis is useful only in enabling the observer 
 to shift the graduation so as to begin each time at zero. The inner 
 one is made more carefully than the outer. These two axes are 
 of ten called " centers " and sometimes "compound centers." For 
 work not requiring great accuracy, but demanding a light portable 
 instrument, these axes are made quite short and are called " flat 
 centers." The more accurate instruments have " long centers." 
 
 6. Tangent Screws.— With the unaided hand the telescope 
 cannot easily be made to cover or bisect the exact point sighted at ; 
 to assist in thus directing the telescope, the instrument is provided 
 with a clamp and tangent or slow motion screw. 
 
 Clamps are made in a variety of ways, but all consist essentially 
 of a contrivance by which a piece may be connected with the axis 
 or rim of the graduation by simply tightening a screw. The clamp 
 is connected with the vernier plate or the tripod, as the case may 
 be, by a screw which is always nearly tangent to the direction of 
 motion. When the screw is turned, the two parts of the instrument 
 rotate slowly with reference to each other. No description can give 
 an adequate understanding of these parts ; they must be seen and 
 examined to be comprehended. 
 
 A perfect tangent screw should have a very smooth motion,, 
 free from^ost motion or " back lash," a great durability and ao 
 even wear so that the screw will never have lost motion in one part 
 and move hard in another. Lost motion is the common defect of > 
 
THE TRANSIT. 25- 
 
 tangent screws and also the source of great annoyance to the engi- 
 neer, besides often being the cause of serious errors in the held. 
 Several forms of tangent screws will now be described. 
 
 The oldest and most imperfect of all is that known as the 
 English or stiff tangent screw, in which the screw works through a- 
 post in the clamp and against a collar in a post attached to the 
 plate, both parts being free to turn about a vertical axis. The de- 
 fects of this form of construction, are, 1, if the screw is not perfectly 
 straight and true, it will bind during one part of the revolution ; 2,. 
 the nut and collars should be the same height above the plate, and 
 to allow for errors of workmanship, the hole in the nut is oftea 
 made too large and then tightened until it fits, thus touching in- 
 only two points, and therefore soon wears loose ; and 3, the parts- 
 are so arranged that, if the movements are adiusted so as to pre- 
 vent looseness, the friction will be so great as to interfere with the 
 freedom of the motion. ' 
 
 Another form has a long spirnl spring between a collar on the- 
 screw and the nut, which takes up the back lash. On trial this ha* 
 proved defective, since the spring must be made long enough and 
 strong enough to act in every portion of the length of the screw, 
 and the alternate opening and closing weakens it so that in a short- 
 time it fails to remove the back play. 
 
 Another form, which has the same objections, allows the screw 
 to abut against the clamp, a spring keeping the clamp in contact, 
 with the screw. Even under the most favorable conditions, tangent 
 movements of this class are objectionable, because the spring- 
 causes a strain between the parts which may change with changes- 
 of temperature and which is liable to cause motion or derange the; 
 levels. 
 
 One of the best forms of tangent screws is that commonly known? 
 as the Gambey tangent movement, first made by the celebraled in- 
 strument maker, Gambey, of Paris. As commonly made, the nut 
 is a split sphere which is confined like any ball joint. Instead of 
 the collar, as in the English form, another split spherical nut, with. 
 a thread of a different pitch, is used. . This construction provide* 
 for all necessary motions and all probable imperfections of the? 
 screw. The only objection to this form is that, since the nut is 
 short the screw will wear most near the middle, and when the nut 
 
26 THE TBANSIT. 
 
 is tightened to remove the lost motion from the small part of the 
 screw, it will work too hard on portions less worn, therefore the 
 screw can be turned only a few revolutions without having lost 
 motion in one part or a great friction in another. 
 
 In the genuine Gambey tangent movement, the halves of the 
 spherical nuts are confined between springs ; consequently, it does 
 not possess the above defect, and is all that could be desired. 
 
 The latest tangent movement is an improvement of the counter- 
 fe't Gambey. It consists of " a Ions; spiral cylinder nut, from the 
 interior of which two-thirds of the threads have been removed ; 
 into half the recess thus left in the nut is nicelj' fitted a cylinder 
 follower, with the same length of screw thread as the nut; this 
 follower is fitted with a key, which prevents it from turning in the 
 recess, but allows motion in the direction of its length. A strong 
 spiral spring is placed in the remaining half of the recess, between 
 "the fixed nut and the movable follower ; this spring always has 
 tension enough to force the follower and fixed thread in contrary 
 directions, and thus remove any lost motion that may occur in the 
 screw. It will be observed that with this construction the spring 
 remains in a state of rest, instead of closing and opening, as has 
 been the case in all other application of springs." The spiral 
 spring takes up the wear of the screw, and the great length of the 
 nut secures nearly uniform wear over the whole screw. These long 
 nuts may be confined like any ball joint, or may be attached by 
 gimbals, thereby allowing free motion even though the screw 
 should not be straight. 
 
 Sometimes two opposing or butting screws are used to get rid 
 of lost motion ; but these are objectionable since two hands must 
 be employed in using them. For several reasons they are not at 
 all suitable for the upper motion, but on account of their steadi- 
 ness are often used for the lower motion. If a stiff, flat spring 
 •were attached to the side of the lug against which the tangent screw 
 bears, the setting could be. finished with one screw. Such a spring 
 is a very useful addition. 
 
 7. The magnifying power of the telescope and the least count 
 of the vernier should be proportioned to each other, so that the 
 least perceptible movement of the vernier will cause sufficient 
 motion of the cross-hairs on the object to be easily noticed through 
 
THE TRANSIT. 27 
 
 the telescope ; and vice versa, the least noticeable motion of the 
 cross-hairs on the object should cause a perceptible change of the 
 vernier. Since the horizontal circle is usually (and properly) the 
 more accurate, this condition applies more especially .to the hori- 
 zontal circle than to the vertical. A higher power, or a smaller 
 count of the vernier, than required by the above conditions is 
 detrimental, the former causes an unnecessary loss of light; and 
 the latter a waste of time in reading the vernier. A similar relation 
 should exist between the magnification and the level under the 
 telescope, and also between the magnification and the plate levels. 
 Many instruments are very faulty in some or all of these par- 
 ticulars. 
 
 8. An engineer should have a good instrument and then take 
 delight in keeping it in good condition. It is not desirable to take 
 the instrument apart unnecessarily, for even if the fittings are per- 
 fect, it requires considerable care to put them together properly. A 
 little dust or dirt in a joint or bearing, or a screw left loose or 
 tightened too hard, may damage the instrument and cause errors in 
 its use. 
 
 There are a great many small screws about a transit and the 
 general tendency is to overstrain them. This is especially true of 
 the cross-hair screws ; all straining of these screws, beyond that 
 necessary to insure a firm seat is more apt to cause the instrument 
 to lose than retain the adjustment. Over-straining the leveling 
 screws bends the plates and wears the screws unnecessarily. If 
 the leveling or tangent screws get to working hard, take them out 
 and brush with soap and water ; the nuts can be cleaned by screw- 
 ing a thin piece of soft wood through them. The tangent screws 
 are only intended to complete the settings ; they should never be 
 used except to give a slight movement. 
 
 The telescope slide and the center should be examined occa- 
 sionally ; if there is any fretting or cutting, take the piece out and 
 burnish down the rough place with any smooth hard tool, as the 
 back of the blade of a pocket-knife. Very little, if any, oil or 
 grease should be used upon bearings exposed to the dust ; probably 
 powdered plumbago is best for exposed bearings, and rendered ox- 
 marrow, or watch oil, for bearings not exposed. 
 
 8. Pbactical Hints.— The beginner should not neglect the pre-, 
 liminary matters of planting the tripod, bringing the plumb-bob 
 
28 THE TRANSIT. 
 
 over the point, and leveling the instrument. There is great differ- 
 ence in the skill and rapidity with which different persons will set 
 up the transit. ; generally there is an unnecessary waste of time and 
 hard labor. A very neat way of doing it is as follows : tighten the 
 screws in the upper end of the legs, until friction will just hold 
 them wheneyer placed, open them until they make an angle of 
 about 30° with the vertical ; let down the plumb-bob, and set the 
 instrument over the point, a gentle pressure on the legs of the tri- 
 pod brings the plummet over the point, and a slight movement of 
 the leveling screws brings the plate level. 
 
 Some engineers seem to think that the harder the tripod legs 
 are focused into the ground, the tighter the leveling screws are, 
 and the tighter the instrument is clamped, the more accurate the 
 work, but the contrary is more nearly true. An instrument keeps 
 it adjustments better and works more kindly, when handled 
 delicately. 
 
 If the instrument is not firm, examine the tripod-head, and the 
 iron shoes on the legs, to see that they are not loose ; no instru- 
 ment can stand firm with any looseness in these parts. The clamps 
 and tangent screws should be examined to see that they are not 
 loose. The instrument may slip on the lower plate, owing to the 
 leveling screws not being tight enough. 
 
 If the observer has clearly in mind what he is trying to do, and 
 thoroughly comprehends the effect of possible errors, in his instru- 
 ment, he can save much time and trouble, thus leaving himself 
 free to give his attention where it will do the most good. For ex- 
 sample, a great deal of time is often wasted in getting the instru- ■ 
 ment precisely over the point; the case should vary inversely as 
 the distance of the object sighted at; an inch may cause an error 
 of three seconds if the object is a mile away, but an error of three 
 minutes at 100 feet. The position of the instrument from the point, 
 with reference to the object sighted at, affects the value of the re- 
 sulting erro.r 
 
 It is a waste of time to level up accurately, each time, regard- 
 less of the kind of work to be done; if only the horizontal angles 
 between points on the same level, are to be found the instrument 
 can be leveled with sufficient accuracy by the eye alone ; if only 
 vertical angles between points in the same vertical are desired. 
 
THE TRANSIT. 29 
 
 only the level parallel to the telescope need be read. On the other 
 hand,"in measuring a horizontal angle between a high and a low 
 point, the levei perpendicular to the telescope should be very 
 carefully attended to. So in all instrumental work, there are cer- 
 tain operations which should be carefully attended to while to at- 
 tend equally carefully to others would be only a waste of effort. 
 
 10. Nothing need be said here concerning the adjustment of 
 the instrument or its ordinary manipulation. 
 
 Measuring Angles Accurately.— It sometimes happens that an 
 engineer desires an angle with the utmost accuracy. There are 
 two methods of making the observations, when extreme accuracy 
 is desired ; they are by series and by repetition. One or the other of 
 these methods is always used in measuring the principal angles of 
 the geodetic triangulation. The principles involved are useful in 
 less accurate work. 
 
 By Series.— Sight upon the first station, and read both ver- 
 niers ; this eliminates eccentricity. Sight upon the next station to 
 the right, and read as before ; continue on around the horizon, 
 reading upon each station, and close by reading upon the first 
 station again ; if the last reading is the same as the first, it proves 
 that the instrument did not slip or get moved. Reverse in altitude 
 and azi muith, move the lower motion a little to eliminate personal 
 bias, an read upon the first station ; proceed around the horizon 
 towards the left, reading upon each station and closing upon the 
 first. The reversal in azimuith and altitude eliminated eccentricity 
 of line of sight, error of telescope slide, and inclination of hori- 
 zontal axis. Reversing thedirectionaround the horizon eliminated 
 any twist of the tripod. Shifting the horizontal circle diminished 
 the posibilities of accidental error of graduation. The above ob- 
 servations constitute one " set." 
 
 To secure greater accuracy by increasing the number of obser- 
 vations and also to eliminate periodic errors of graduation, shift 
 the horizontal circle an aliquot part of the distance between ver- 
 niers, and take another set. The amount that the circle should be 
 shifted between sets is equal to the distance between verniers 
 divided by the number of sets to be taken. The mean of the ob- 
 served values is true angle. 
 
30 THE TKANSIT. 
 
 By Eepetition.— Sight upon the first station, read both verniers ; 
 with the upper motion, turn to the next station, and read as before, 
 (this last reading is only for a check) ; with the lower motion turn 
 back to the first station, the reading remaining unchanged ; then 
 unclamp above, and turn forward again to the second station. The- 
 angle will now have been measured a second time, but on a part of 
 the circle adjoining that on which it was first measured, the second 
 beginning where the first ended. This operation may be repeated, 
 any number of times ; read the circle after the last sighted upon 
 the second point: the difference of the first and last readings, 
 divided by the number of repetitions gives the angles more pre- 
 cisely than a single observation would. Notice that the vernier 
 need be read only at the begining and end, although the second 
 reading, as above, is a valuable check in determining how many 
 time 360° should be added to the last reading, in case the vernier 
 has passed 0°. Next reverse in altitude and azimuth, and measure 
 the angle as before, beginning however at the second station. This 
 eliminates all error of adjustments and reduces the error of obser- 
 vation by increasing their number. Of course, the mean of the ob- 
 served values is assumed to be the true angle. 
 
 Comparison of Methods. — Both methods seem be to about per- 
 fect, as far as the elimination of errors of adjustments, of gradua- 
 tion, and of observation is concerned. The method by series is 
 preferred by most observers, for triangulation work ; its peculiar 
 advantages can be fully realized only with the precise instruments- 
 used in that kind of work. With ordinary engineering instru- 
 ments, there is an obvious limit beyond which it is useless to- 
 multiply observations by this method. 
 
 The method by repetition was once a great favorite with the ' 
 best engineers, especially the French, for triangulation work ; but 
 the improvements in the manufacture of angle instruments, has 
 given preference for the most accurate work, to the method by 
 series. However, the method by repetition is peculiarly suited to 
 the precise measurement of angles with a coarsely-divided circle, 
 as, for example, a common engineer's transit. The principle of 
 this method is certainly very beautiful, but its accuracy is largely 
 dependent upon the freedom with which the instrument turns on 
 its centers, and upon the stability of the clamp and tangent screws. 
 Under ordinary conditions, the limit of this method is reached 
 after a few repetitions. 
 
THE TBANSIT. 31 
 
 Transit Surveying.— Unfortunately, there is no generally 
 accepted method of doing transit work, either for the field work or 
 for recording the results. Three methods of more or less general 
 use among engineers will be considered. The first, for want of a 
 better name, we will call the angle method ; the second can appro- 
 priately be called the quadrant method ; and the third could appro- 
 priately be called the full circle method ; but as this method of sur- 
 veying has been called traversing, we will use that term and not 
 attempt to introduce a new one. 
 
 1. Angle Method. — This method consists in measuring and 
 recording the angle which each line makes with the preceding one. 
 The angle measured may be the one included between the two lines, 
 or the angle between the second line and the first produced. In the 
 latter case, the telescope is sighted along the first course, the ver- 
 nier read, and the telescope transited ; the telescope now points in 
 the direction of the first line produced, it is next turned to the 
 second line, and the vernier read ; the difference of the readings, 
 i.e. the angle swept over, is equal to the angle between the first 
 course produced and the second. This angle is sometimes called 
 the angle of deflection. 
 
 2. Quadrant Method.— The distinguishing characteristic of 
 this method is the manner of designating the observed horizontal 
 angles with reference to the quadrant in which the enclosing lines, 
 belong. The angles are read and recorded as bearings, just as in. 
 compass surveying. The meridian through the first station is 
 obtained by reading the needle, or by sighting upon some line 
 whose bearing is known, or it may be assumed ; the object of such 
 surveys is generally not so much to get the true bearings, as to get 
 the relative bearings accurately. The bearing of any succeeding 
 line is found by measuring the angle which it makes with the pre- 
 ceding line produced, and adding it to. or subtracting it from th e 
 bearing of the preceding line. This method is doubtless a relic of 
 the time when the compass was the common angle instrument. 
 The 0° to 90° numbering on the horizontal circle is especially useful 
 in this system. 
 
 3. Traversing.— Traverse surveying, or running a traverse, or 
 simply traversing, is conducting a survey in such a way that the 
 readings of the plate will show the angles which each line of the 
 survey makes with any chosen reference line. Although the essen- 
 tial principles of this method are used to a limited extent, there is 
 
32 
 
 THE TRANSIT. 
 
 no agreement as to the details oE the process. Even the term 
 traversing is not always used as denned above. This method could 
 appropriately be called surveying by the back azimuth, or survey- 
 ing by the back angle, or surveying by carrying the azimuth. 
 
 D -7 I 
 
 L J 
 
 Traversing. 
 
 To run a traverse, set the instrument up over the first station, 
 i.e., the end of the first course. Eor the present we will assume 
 that the first side is to be the reference line, i.e., the meridian of 
 the survey. Set vernier A at 0°, clamp the upper motion, and turn 
 the telescope upside down, with the lower motion, direct the line of 
 sight to the other end of the first course ; transit the telescope, 
 loosen the upper motion ; sight upon the next station, and read 
 vernier A. Move to the next station, loosen the lower motion, 
 
 Station. 
 
 Back-Sights 
 
 Fore-Sights 
 
 Dist. 
 
 Remarks. 
 
 A 
 B 
 
 G 
 D 
 
 E 
 
 0° 00' 
 
 35° 52' 
 340° 81' 
 
 41° 08' 
 270° 00' 
 
 35° 52' 
 310° 31' 
 
 41° 08' 
 270° 00' 
 180° 00' 
 
 
 Angles read 
 from ver. A. 
 
 level up, invert the telescope, and glance at the reading to see that 
 It has not changed; sight upon the last station; using the lower 
 motion. Invert the telescope, loosen the upper movement, sight 
 upon the next station, and read. Proceed in like manner for any 
 number of lines. Each reading is the angle between its corres- 
 
THE TKANS1T. 33 
 
 ponding course and the first, one. Below are the notes and pjan of 
 a traverse survey, on preceding page. 
 
 The instrument was first at A, the line 0^4 being regarded as 
 the first course. The vernier was set at zero, and a back sight 
 taken to 0' ; it is recorded opposite A, in the back sight column. 
 The telescope was then transited, the upper motion loosened, and 
 the telescope directed to B ; the reading of this line, 35° 52', being 
 recorded as in the table. After the instrument is removed to B, 
 and the back sight taken up on A, the vernier is to be read to make 
 sure that it has not been changed ; the record of this reading is 
 made in the back sight column. The writing of this down will be 
 evidence that the reading of the vernier was checked; in actual 
 work this will be found to be an important check against errors, 
 as, for example, turning the wrong tangent screw or reading the 
 wrong vernier. The angles marked in the diagram are the corres- 
 ponding angles of the fore sight column in the table. 
 
 Instead of making the first course the reference line, as above, 
 we may take any other line, ?s the meridian or reference line. In 
 this case, the instrument is first set up at the point of beginning, 
 the vernier set as zero, and the first back sight made in the direc- 
 tion of the meridian ; the remainder of the field work and the re- 
 cord is as before. 
 
 It makes little or no difference whether the fiist back sight is 
 made towards the north or south end of the meridian; but a note 
 in the remarks column should show which way it was made. 
 
 This method of using the transit is specially convenient in rail- 
 road surveying. 
 
 Comparison of Methods.— In the time required for the field 
 work, there is very little difference between the three methods ; the 
 second and third require less than the first. The quadrant method 
 is simple and easy enough to understand ; but, in field work, in the 
 computations, and in the plotting, there will be found many draw- 
 backs and many opportunities for errors, which do not exist in the 
 full circle system. One of the chief objections to the quadrant 
 method is that four different directions are designated by the same 
 angle numerically. As the advantages of the full circle system are 
 better understood, it will be more fully adopted; as yet it is only 
 used to a limited extent. 
 
34 THE TKANSir. 
 
 Sources of Errors.— The sources of the errors of transit work 
 may be classified as lollows: 1. Errors in setting up; 2. of sighting; 
 3. of manipulation ; 4. of adjustment ; 5. of reading. 
 
 1. The plates of the instrument may not be placed horizontal 
 and the instrument may not be set up over the point about which 
 the angle is desired or over the point previously sighted at ; in any- 
 thing like good work, these errors will be inappreciable. 
 
 2. Errors or Sighting.— If a flag-pole is sighted at, it may not 
 be vertical ; therefore, sight as low upon it as possible. The inter- 
 section of the cross-hairs may not exactly cover the point ; this is 
 due to lack of care or to parallax in the instrument, and in either 
 case, the remedy is obvious. It is not enough to bring some point 
 of the vertical hair over the point, for the hair may nQt be truly 
 vertical. 
 
 Errors op Manipulation.— The wrong tangent screw may be 
 turned ; this is a fruitful source of error, and one difficult to dis- 
 cover and impossible to correct. If the tangent screw has back 
 lash, the instrument must be handled so as to prevent it; error 
 from this cause sometimes is produced by the instrument turning 
 on the bull and socket joint owing to the foot screws not. being 
 screwed up tight enough. The upper part of the instrument should 
 move so freely as not to twist the tripod. 
 
 Errors of Adjustment.— The points to be considered are eccen- 
 tricity, equality of standards, straightness of telescope slide, and 
 adjustment of cross-hairs. The eccentricity is always small and 
 easily eliminated by reading two verniers. The equality of the 
 standards can effect only the horizontal angles between high and* 
 low points, and in ordinary practice it will be inappreciable. 
 
 The straightness of the slide and the adjustment of the cross- 
 hairs are closely related; neither will produce error either in 
 measuring horizontal angles between points equal distances from 
 and equal distances above vertical angles between points or below 
 the instrument, or in measuring equally distant from the instru- 
 ment. An error of adjustment of the line of collimation produces 
 an error of double amount in traversing, or in prolonging a line by 
 back and foresights. If great accuracy is desired in either of these 
 operations, the telescope should be reversed on each alternate back- 
 
THE TRANSIT. 35 
 
 sight and fore-sight. All errors of adjustment can be eliminated 
 by reversing, making a second observation and taking the mean. 
 
 Errors or Reading. — Errors may be produced by reading the 
 wrong vernier, the wrong end of a double vernier, the wrong row of 
 numbers, or by reading 28° instead- of 32°, etc., or by forgetting to 
 add the £ degree of the line to the reading of vernier and recording 
 .20', instead of 50' etc. 
 
 Limits or Precision.— It is a little difficult to get sufficient data 
 for a satisfactory discussion of this subject, without making obser- 
 vations specially for this purpose, which is not desirable. The 
 sophomores, '86, in the prosecution of the ordinary class work in 
 topographical surveying, measured the angles of ten triangles ; the 
 sums of the three angles should in each case equal 180°. The length 
 of the sides varied between 400 and 1,200 feet ; the conditions as to 
 time, targets, etc., were about those of actual practice. The instru- 
 ment read to minutes, and certainly was not the best, 
 errors enumerated in the preceding article were involved. This 
 work gives the probable error of a single direction = 32 seconds; 
 the probable error of an angle = 50" ; the probable error in the sum 
 of three angles of a triangle = 86 seconds. The maximum error in 
 the closing of a triangle was 3J minutes. 
 
 The same class in measuring the four angles around a point 
 and the three angles of a triangle, using chaining pins for targets, 
 with sights about 100 feet long, obtained results about half as large 
 as those above. The results of ^traversing, with flag-poles for tar- 
 gets and sights varying between 200 feet and 800 feet, gave results 
 a little greater than half those of the preceding paragraph. This 
 is all the data at hand to show the degree of accuracy attainable, 
 but in a general way it is believed that the above results are not 
 exceptional. 
 
 A transit is sometimes used to determine areas. The legitimate 
 errors in the balancing of the latitudes and departures in transit 
 surveying can be discussed as in compass surveying; the formulas 
 deduced for that case are applicable to transit surveying. Such a 
 discussion shows that ordinary work with the tansit is proportion- 
 ally as accurate as the best chaining. Therefore, in finding areas 
 by the transit, the greatest care must be given to the chaining. 
 
33 THE LEVEL. 
 
 V. 
 
 THE LEVEL. 
 
 1. Qualities Desired in a Leveling Instrument. — The main 
 qualities to be secured in a leveling instrument are delicacy of 
 level, stability and defining and magnifying power in the telescope. 
 The optical qualities of the telescope have already been discussed, 
 but it may not, be amiss to repeat that the optical qualities should 
 be correctly proportioned to the sensitiveness of the level bubble. 
 Of two levels in the writer's possession, one had a magnification of 
 17 and a radius of the bubble of 84 feet, and the other a magnifica- 
 tion of 27 and a radius of 22 feet ; the simple expedient of changing 
 the bubbles improved both instruments very much. A third in- 
 strument is inferior to the other two in definition, although it has 
 a bubble of 165 feet radius. 
 
 The stability of the instrument depends very much upon the 
 leveling screws and the manner of connecting the instrument with 
 the tripod-head. This point was discussed under the transit ; but 
 we repeat that three screws give greater stability than four ; and, 
 in either case, the distance between opposite screws should be as 
 great as possible. This item is of greater importance in the level 
 than in the transit. It seems to be self-evident that the center of 
 gravity of the instrument should be as near to the tripod-head as 
 possible, but this is a common defect. 
 
 However good the other parts of the instrument may be, the 
 accuracy of the work depends upon the sensitiveness of the bubble. 
 It must be remembered that, although a sensitive bubble may not 
 remain exactly stationary, it will still give better results tnan a 
 sluggish one, which shows no movement when the instrument is 
 slightly displaced. 
 
 2. Sensitiveness of Bubble.— The sensitiveness depends upon 
 the radius of curvature, or what amounts to the same thing, upon 
 the distance the bubble moves for any change of inclination. 
 
 To measure the sensitiveness, proceed as follows :— Bring the 
 bubble nearly to the center and sight upon a rod held vertically. 
 Raise or lower one end of the level, by operating the foot screws, 
 until the bubble moves about as far to the other side of the center, 
 and sight at the rod again. Let h = the difference of rod readings ; 
 
THE LEVEL. 37 
 
 D = the distance from the instrument to the rod ; m = the distance 
 the bubble moved ; d = the length of one division of the scale : 
 n = the number of divisions bubble moved, m = nd; 1= the change 
 of inclination of the line of sight; R = radius of curvature of the 
 level tube. Then, in Fig. 1, we have, from the approximately 
 
 D D 
 
 similar triangles ABC and P Q. R = — m = — nd, 
 
 h h 
 
 h h 
 
 and tan I = — ; approximately I" tan l" = — 
 D D 
 
 h 
 
 hence I" = I" is the value in seconds 
 
 D tan 1" 
 of arc of n division of the scale, consequently the angular value of 
 one division in seconds 
 
 h h 
 
 n D tan 1" 0.000005 n D. 
 
 The sensitiveness of a bubble is generally stated by giving the 
 angle corresponding to a movement of an inch. Good engineer's 
 levels have a motion of 200" to 120" per inch, or a radius of curva- 
 ture of 80 to 140 feet. Levels of precision have a motion of about 
 75" to 100" per inch, or a radius of curvature of 400 to 160 feet. 
 
 3. Different Forms of Levels. — Leveling instruments may be 
 grouped in three classes. The first includes all instruments that 
 can be adjusted by reversals; the common wye or Y -level is a re- 
 presentative of this class. The second includes all that can not be 
 adjusted by reversals; the "dumpy level," is a type form. The 
 third includes all instruments whose errors of adjustment may be 
 wholly eliminated by a system of double observations ; the instru- 
 ments of this class are usually called " levels of precision," and are 
 used only where extreme accuracy is aimed at. 
 
 The distinguishing characteristic of the Y-level is that the tele- 
 scope may be revolved about its own axis, and turned end for end 
 in its bearings ; the only advantage of this construction being to 
 facilitate the adjustment of the instrument. " The Y-level is easily 
 adjusted and nearly always needs it." It is the favorite with 
 American engineers. 
 
38 
 
 THE LEVEL. 
 
 The " dumpy level " is the name given to that form of leveling 
 instrument in which the telescope is attached to the bar injaueh a 
 way as not to admit a rotation around the axis jQf the telescope, 
 nor allow of reversion end for end. TheJ»leseope is usually invert- 
 ing, and therefore shorter than the one commonly used on leveling 
 
 instruments, hence the name dumpy; however, this construction 
 can be employed with an erecting as well as with an inverting tele- 
 scope. English engineers use this form exclusively. 
 
 "Without doubt the American engineers could profitably follow 
 the English in using the dumpy to the exclusion of the Y-level. 
 As ordinarily made, the former has the advantage of an inverting 
 telescope as has been discussed ; it is also more simple and com- 
 pact. If equally well made, it is capable of doing as accurate 
 work as the more elaborate and more expensive Y-level. The only 
 disadvantage charged against the dumpy is that it is not so easily 
 adjusted, although it is admitted that when once adjusted it retains 
 the adjustment very much better than the Y-level. The last is a 
 very important advantage, for unless an instrument is in adjust- 
 ment the work is nearly certain to be inaccurate. 
 
 The writer firmly believes that the dumpy is as quickly and as 
 easily adjusted as the Y-level. Finally there are a few particulars 
 connected with the manufacture of the instrument in which the 
 
THE LEVEL. 
 
 39 
 
 general design of the dumpy is superior to the Y-level ; as these 
 would probably be inappreciable, except in very accurate work, 
 they will not be discussed here. 
 
 Target Eods.— There are two classes of leveling rods : 1, target 
 rods, these in which the line of sights is always directed to a mov- 
 able target, the position of which is read by the rod-man ; and 2, 
 
 J L 
 
 Fig. 2. 
 
 Fig. 3. 
 
 self -reading or speaking rods, those in which the graduation is 
 such that the position of the line of sight can be read with the tele- 
 scope. Toe well known New York rod is an example of the first- 
 clats. The Philadelphia rod may be used either as a target or a 
 self reading rod. 
 
 The target is a piece of brass or iron which can be moved up or 
 down the rod, or clamped in any position. It carries a scale or 
 vernier to sub-divide the least space on the rod. The face of the 
 target should be painted of such a pattern that it may be precisely 
 bisected by the horizontal cross-hairs. Some of the many varieties 
 are given in Figs. 2, 3 and 4. 
 
 Fig. 2. is the target of the New York rod ; the design is not 
 good, because the cross-hairs may be above or below the middle of 
 the target by its full thickness, as magnified by the eye-piece, with- 
 out the error being perceptible. 
 
40 
 
 THE LEVEL. 
 
 Fig. 3, depends upon the nicety with which the eye can deter- 
 mine whether a line bisects an angle ; this can be done very accur- 
 ately by noticing the length and position of the two points formed 
 on either side of the hair. 
 
 Fig. 4, depends upon the accuracy with which the eye can bi- 
 sect a space. 
 
 Black and white are best for visibility ; but red and white are 
 
 most easily distinguished 
 among trees, shadows, 
 etc., and red gives the 
 stronger contrast with 
 the cross-hairs. Probably 
 red and white are the 
 best for Fig. 2, and black 
 and white for Fig. 4. 
 
 5. Self-Reading Bods. 
 The most common rod of 
 this class is the Philadel- 
 phia rod. The gradua- 
 tion is in feet and tenths, 
 the figures indicating 
 feet are red, the tenths 
 black. The figures are six-hundredths in height, placed with 
 centers over the marks ; sometimes the bottom of the figures is 
 placed on the line, in which case the figures may be eight- 
 hundredths high. To make a reading greater than 7 feet, without 
 a target, the rod must be extended to its greatest length. 
 
 A few of the many patterns vhich have been proposed for self- 
 reading rods are shown in Figs. 5 to 8. 
 
 The advantage of the form of graduation of which Fig. 5 is a 
 type, is distinctness and visibility. The angles of the figures indi- 
 cate fractions of tenths ; the figures may be placed with centers or 
 with bottoms, on the lines. The graduation given is suitable for 
 short sights ; for longer ones the figures are made larger, and only 
 those for the even tenths are marked. 
 
 Fig. 6 shows the Francis rod, which is claimed to be a complete 
 graduation to hundredths without visual division, and can be read 
 without counting. The foot marks are red and larger than the 
 tenths. It is best adapted to short sights. 
 
 Fig. 4. 
 
THE LEVEL. 
 
 41 
 
42 THE LEVEL. 
 
 In its essential features. Fig. 7 is the favorite British self-read- 
 ing level rod. The graduation is to hundredths. Although the one 
 shown is better than the typical English rod, it cannot be con- 
 sidered a good one for the same reason for which the usual target 
 to the New York rod is condemned in a previous paragraph. 
 
 The principle of the Texas rod, Pig. 8, is of frequent application 
 in making self-reading rods. It is superior to that of Fig. 7 in that 
 it obviates the error due to the thickness of the cross-hairs. The 
 points indicate tenths, but it can be read by estimation to hun- 
 dredths; it is possible to make the reading much more accurately 
 by dividing the bblique.side of the triangle, than could be done 
 when the rod is graduated according to the principle of Fig. 7. 
 This form of graduation reduces the difficulty of counting a number 
 of smali divisions, and also the possibility of gross mistakes. 
 
 Any pattern suitable for a stadia rod, samples of which will be 
 given in the next article, can be used in making a self-reading 
 leveling rod. The pattern may be painted or stenciled directly 
 upon the wood, or it may first be drawn or painted upon paper, and 
 then fastened on the rod with varnish or any glue not soluble in 
 water. -Strips of cloth or paper containing scales for this purpose 
 are sold by dealers in engineering stationery. The graduation will 
 keep cleaner and last much longer, if the face of the rod is recessed 
 slightly to receive it. The rod can be made of a light board hinged 
 in the middle, so that it may be folded to protect the face and for 
 convenience of transportation. It is held open by a button or bolt 
 on the back. Stencils can be cut out of a sheet of writing paper 
 which has been varnished or oiled. 
 
 6. Target vs. Self-Beading Eods.— Probably the former are 
 the more common now, but as the advantages of the latter are be- 
 coming better understood, they are beine more generally used. The 
 chief advantage of self-reading rods is the saving of time. Setting 
 the target to some exact point in accordance with directions given 
 from a distance, is a tedious process at best. After a little famil- 
 iarity with the pattern of the self-reading rod, the height of the 
 line of sight upon the rod can be read very quickly. 
 
 On ; the other hand, target rods must of necessity be capable of 
 greater accuracy than self-reading ones, but the difference in 
 accuracy is not so great as might at first seem ; the accuracy of a 
 level determination depends upon a number of thiugs of which the 
 
THE LEVEL. 43 
 
 reading of the position of the line of sight uron the rod is one of 
 the least important, all of the others being independent of the kind 
 of rod. 
 
 Beading the position of the target to 1,000th is unnecessary and 
 useless, unless all the other parts of the work are equally precise. 
 The thing to be sought is corresponding accuracy in all parts of the- 
 work. If several Independent readings of a rod be made upon the- 
 same point, the difference between the various readings will prob- 
 ably be considerably larger than the probable error of reading a 
 self-reading rod. 
 
 In precise leveling self-reading rods are used, and the position of 
 two or three hairs, generally the latter, are read to reduce the 
 error of reading; the mean of these is used as the rod reading. 
 For a single observation, a target rod is more accurate than a self- 
 reading one : but three observations, as above, are probably more 
 accurate than a single observation upon a target, and can be made 
 in about- the same time. 
 
 7. Adjustments.— JSothing need be said of the ordinary adjust- 
 ments ; they are fully described in all text-books and manuals, but 
 the following section, which is specially applicable to the dumpy 
 level, is not so common but that it may be giyen here. 
 
 8. Two Peg Method.— To adjust the dumpy level, proceed a& 
 follows : Drive two pegs into the ground, say 400 feet apart, set the 
 instrument exactly half way between them, and carefully deter- 
 mine the difference of level in the ordinary way. A line joining 
 the two positions of the target is a level line, and the difference 
 between the readings is the true difference of level however much 
 the instrument may be out of adjustment. Next set the instrument 
 very near one of the pegs and re-determine the difference of level. 
 If the second difference is the same as the first, the instrument is 
 in adjustment. If the differences are not the same, raise or lower 
 the line of sight, by manipulating the foot-screws, until the differ- 
 ence between the two readings is the same as the difference first 
 obtained; the line of sight is then horizontal. Without altering 
 the inclination of the line of sight, raise or lower one end of the 
 level tube until the bubble is in the middle ; the instrument is then 
 in adjustment. 
 
44 
 
 THE LEVEL. 
 
 For example, let A and B, Fig. 9, 
 be the two points. When the instru- 
 ment is at c, midway between A and 
 B, the targets are at a and b- ab is a 
 level line. The true difference of 
 level between A and B = Aa—Ab. 
 When the instrument is at D, the 
 targets are at c and d, and the ap- 
 parent difference of level is d e = 
 (Cc—Bb)—(Ad—Aa). If the line of 
 sight were horizontal the target 
 would be at f; therefore df is the con- 
 nection. 
 
 AD / 1ac + BI)\ 
 
 df =de =de ( ) 
 
 AB \ 2Ac / 
 
 If D is between a and B, the quantity 
 B D must be subtracted. 
 
 The above is the only adjustment 
 absolutely necessary with the dumpy 
 level ; but to save the trouble of lev- 
 eling up every time the instrument 
 is turned on its vertical axis, it is de- 
 sirable to adjust the supports of the 
 telescope so that the bubble will 
 stand in the middle when the instru- 
 ment is revolved on its vertical axis. 
 This can be done by the method of 
 adjusting plate levels. 
 
 9. Sources or Ebrob.— For con- 
 venience of discussion, we will classify 
 errors of leveling as follows :— 1, In- 
 strumental errors ; 2, Bod errors ; and 
 3, Errors of observations. In general 
 a clear comprehension of all the 
 sources of error, their amounts and 
 the means of avoiding them, will be 
 of great service in indicating the 
 
THE LEVEL. 45 
 
 care necessary to secure a given degree of accuracy, and particu- 
 larly in leveling is this true. In no other form of surveying, with 
 the possible exception of chaining; is it as necessary to distinguish 
 between cumulative and compensating errors, as in leveling. An 
 apparently inappreciable cumulative error may in the course of a 
 series ofobservations amount to more than a much larger, but 
 compensating error. The effect of compensating errors is reduced 
 nearly to zero simply by multiplying the number of observations, 
 but cumulative errors should be avoided entirely, or observations 
 made by which they may be corrected. The observer should avoid 
 errors which usually occur in a single direction ; but he need not 
 always take the greater care to avoid errors which are as liable to 
 be negative as positive. A clear comprehension of all the sources 
 of error, their amounts, and the means of avoiding them, will be of 
 great service in indicating the care necessary to secure a given de- 
 gree of accuracy. 
 
 Instrumental Errors.— The chief error of this class is lack of 
 adjustment in the instrument; the one essential adjustment is that 
 the line of sight should be riarallel to the level. The ordinary 
 method of adjusting the Y-level is too well known to be repeated 
 here, but every Y-level should be tested by the two peg-method as 
 already described before being regarded as a Y-level; that is, a 
 Y-level should be carefully adjusted in every particular and then 
 tested by the two peg-method, and if this test shows it to be in 
 adjustment then it may be regarded as a Y-level and ever after- 
 wards adjusted as such, but if this test shows it not to be in adjust- 
 ment the instrument is nothing more than a dumpy level and must 
 be adjusted accordingly, and the telescope must not be reversed in 
 the Y's- If a Y-level in adjustment by the ordinary method, is 
 found to be out when tested by the two peg-method, the error may 
 be due to any one of several things ; the rings may not be of the 
 same size, the Y's may not have the same angle or may not squarely 
 face each other, the optical center of the objective may not lie in 
 the axis of the rings, the direction of motion of the objective may 
 not be in the line of the axis of the rings. The relation of these 
 elements is too complicated to be discussed here, but it is only 
 proper to say that none of these defects are likely to occur to an 
 extent appreciable in ordinary work for which the engineer should 
 thank the accuracy of American machinery and the good workman- 
 ship of American mechanics, rather than the design of the instru- 
 
46 THE LEVEL. 
 
 meat. All errors from any of the above sources are compensating 
 and will be entirely eliminated by setting the instrument midway 
 between the turning points. 
 
 There are a few minor instrumental errors, which though small 
 in themselves au.d not occuring frequently may have an appreciable 
 •effect in very accurate work as has been demonstrated practically ; 
 they are (1) the setting of the instrument on its vertical axis, (2) 
 the setting of the tripod legs into the ground in sandy ground or in 
 ground which is thawing, and the heaving of the legs in spongy or 
 clayey soil, (3) the sun heating and expanding one Y more than 
 ■other. They are all cumulative errors. 
 
 Bod Ereobs.— The principal rod error is in not holding the rod 
 vertical, and ia greater for a large rod reading than for a small one ; 
 it is compensating, and may bo eliminated by attaching a level to 
 the rod or by waving the rod. With a good rodman this error 
 should be inappreciable in ordinary work. With telescoping 
 target-rods when extended, the slipping of the upper piece after the 
 target has been pronounced correct and before the vernier has been 
 read, is a source of error, The target itself may slip, but this is not 
 so probable, because of its less weight. 
 
 Another source of error is the setting of the turning point, due 
 in loose or sandy soil, to its own weight or to the impact of setting 
 the rod upon it. The resulting error is cumulative. The remedy, 
 in the first case, is to use a long peg, or rest the rod upon a triang- 
 ular plate with the corners turned down slightly or with spikes on 
 the underside. This foot plate with a convex button attached, on 
 which to place the rod, is better for all cases than a peg. What- 
 ever the turning point, the rod should never be dropped upon it. 
 
 Finally, another small rod error is the error in the graduated 
 length. This affects only the total difference of elevation between 
 the two points. This is a much more important source of error with 
 the numerous home-made self-reading rods now in use, than with 
 the rods made by regular instrument-makers. 
 
 Errors of Observation.— The principal error is in reading the 
 position of the bubble ; an error may be made in reading it, or it 
 may be read before it has stopped moving. If the sensitiveness of 
 the bubble has been determined as described the levelman can 
 
THE LEVEL. 47 
 
 easily find the error on the rod corresponding to a slight move- 
 ment of the bubble ; he knows then how carefully he must read the 
 result to obtain the particular degree of accuracy aimed at. 
 
 Another important source of error is the moving of the bubble 
 after it has been centered and before the sighting has been made. 
 This movement of the bubble may be caused by its being read be- 
 fore it has come to rest, by disturbing the instrument by stepping 
 near the tripod legs, by turning the instrument slightly in azimuth, 
 or by raising or lowering one end of the telescope in focussing, of 
 by the action of the sun or wind. The bubble should be re-read 
 after the target is nearly adjusted, or, with a self-reading rod, after 
 the reading has been made and before the rodman is signaled to 
 move on. These two probably constitute the chief sources or error 
 in leveling operation. 
 
 In many Y-levels there is no adequate means of keeping^the 
 horizontal hair horizontal and if it is not horizontal any observa- 
 tion not taken at the intersection of the cross-hairs will be erro- 
 neous. Since it is very tedious to bring the vertical hair to coincide 
 with the rod at each sighting, and since the telescope is liable to 
 get revolved in the Y's, it is almost certain that this error is fre- 
 quently a great source of inaccuracy in leveling. It is compensa- 
 ting. To eliminate this error some instiument-makers fasten the 
 telescope in the Y's, in such a manner as to prevent any rotation ; 
 others place a mark upon the collar of the telescope and another 
 upon the Y ; by noticing to see that these lines coincide, the leveler 
 may be certain that the hairs are correct. This improvement can 
 be added to any instrument by riveting a lug on the side of one of 
 the clips and cutting a notch to match in the lip of the ring or 
 collar on the barrel of the telescope. The work can be done in any 
 machine shop. It is impossible to tell when the target is exactly 
 behind the target or the point on the rod covered by the hair. Be- 
 cause a target rod is read to thousandths is no evidence that it is 
 correct to that limit. This error is compensating, aid varies with 
 the form of target or kind of rod, distance and size of cross-hairs. 
 
 With long sights there is another source of error, the curvature 
 of the earth and the effect of the refraction of the atmosphere. 
 ;The combined effect of these two items is to make the target when 
 225 feet distant appear one-thousandth of a foot high ; this correc- 
 tion varies as the square of the distance. If the line of sight passes 
 
■48 THE LEVEL. 
 
 near the earth or over a body of water, the effect of refraction may- 
 vary considerably and consequently the above correction is only 
 approximate. Ordinarily this error is compensating; and will 
 usually be eliminated by setting the instrument midway between 
 turning points. 
 
 Such blunders as an error of one foot or one-tenth in reading 
 the rod, or recording the foresight in the back-sight column, or 
 vice versa do sometimes occur ; unfortunately they are not small, 
 and in ordinary leveling there is no check against them which is 
 practicable. 
 
 Limits of Precision. — A great diversity of opinion exists as to 
 what should be called accurate leveling, so diverse that nothing 
 would be gained by citing them. Let us first consider the degree 
 of precision attained in what is know as "precise leveling," by 
 skillful observers, with the best instruments andplenty of time. 
 
 According to the theory of probabilities, the final error of a 
 series of observations each of which is subject to error, and as liable 
 to be greater as to be smaller than the true result, will vary as the 
 square root of the number of observations. For example, if in 
 measuring a mile the error is 1 foot, it is probable that the error 
 for four miles will be 2 feet. This is true only when the error of 
 each observation is as liable to be plus as minus, when the only 
 errors are compensating ones. In the above illustration, if the 
 chain were too short, the error in the second distance due to this 
 cause would be four times as much as in the first distance. In 
 leveling, a method should be adopted which will eliminate all 
 cumulative errors ; aDd therefore, since only compensating errors 
 remain, the final error should vary as the square root of the dis- 
 tance. The final error due to cumulative errors will vary as the 
 distance, and as it is highly improbable that all cumulative 
 errors will be entirely eliminated, the final error will be a little 
 larger than proportional to the square roof of the distance. The 
 precision of leveling is usually assumed to vary as the square root 
 of the distance. 
 
 It is pretty well established that the error of the most accurate 
 leveling will be 0.005 feet y distance in miles, and even with'the 
 greatest care it may be as much as 0.020 feet */ distance in miles. 
 The first result may be taken as the extreme degree of accuracy at- 
 
THE LEVEL. 49 
 
 tainable. To accomplish this, requires skillful observers, the best 
 instruments, and plenty of time : ordinarily there are not more than 
 three or four hours of the day on which this class of work can be 
 done, and as a general average not more than one or two miles can 
 be made per day. 
 
 It is impossible to establish a limit for work less accurate than 
 the best, the conditions under which it may be done are too diverse. 
 Besults of leveling are often given of apparently greater accuracy 
 than the above, but an occasional accurate result, probably more 
 largely due to good fortune than good management, gives no indi- 
 cation as to what results may be regularly expected ; naturally it 
 is the most accurate result that is reported. However, the differ- 
 ence in precision between ordinary leveling and precise leveling, 
 is not as great proportionally as the difference in care, time, etc. ; 
 a little increase in accuracy costs a very great increase of effort. 
 
 A line of ordinary levels was run on the bank of the Mississippi 
 river, * and'cheeked upon the benches of the precise levels, with an 
 average error of .011 feet J distance in miles. This was about the 
 conditions under which a preliminary railroad survey is made, but 
 is probably more accurate than such surveys usually are. 
 
 eSi. In some of the branches of the A. T. & S. P. E. B., the instruc- 
 tions were to re-run the line whenever the difference between the 
 levels on construction and location was .03 feet between benches 
 about 2,000 feet apart ; this is equivalent to limiting the maximum 
 admissible error to .048 feet y distance in miles. 
 
 The amount of work that can be done by an observer in a day, 
 is a point-about which there has been much debate, and which will 
 never be settled. Leveling is of different kinds, for different pur- 
 poses, with instruments of different powers and delicacy, on dif- 
 ferent ground, and with varying atmospheric conditions. Professor 
 J. B. Johnson, who has had large experience in levels of precision 
 on the U. S. Lake Survey and on the Mississippi river, states, that 
 •" with a wye level and a target rod, a single instrument should 
 duplicate 30 miles per month, with no greater error than 0.05 feet 
 J distance in miles, or with a level of ^preci sion and speaking rod, 
 do the same work with a limit of 0.02 feet J distance in mile*:" 
 
 * Communicated privately to the writer. 
 
50 THE LEVEL. 
 
 Practical Hints.— Best Length ots Sight.— The length, of sight 
 is limited by the power of the telescope and the atmospheric con- 
 dition. It has just been seen that some errors increase directly and 
 others indirectly as the number of sights taken in a given distance. 
 In view of these facts, it is generally assumed that, for the most 
 accurate work, the rod should be at least 100 feet and never more 
 than 400 feet, from the instrument, for ordinary work ; the best 
 length of sight is thought to be 300 to 400 feet, while the greatest 
 should never exceed 500 to 600 feet. 
 
 It is very desirable that at each setting of the instrument the 
 length of the back-sight and the fore-sight should be equal; 
 for, as has been seen, there are a number of errors which 
 are proportional to the length of sight, but which cancel 
 each other when the distances are equal. This is a very 
 important point and should always be kept in mind. When 
 stakes are set at regular intervals, there is no difficulty in 
 determining the length of sights, and making them equal ; in other 
 cases, the distance can be easily determined by the principle of the 
 stadia, as will be explained in the next chapter. All levels should 
 be provided with two extra horizontal cross-hairs for this purpose. 
 
 In ascending or descending a hill, it is nearly impossible, and 
 always very tedious, to make the back-sight and fore sight equal. 
 As the rod is about twice as high as the instrument, the down-hill 
 sights will be about twice the length of the up-hill ones. When the 
 ground renders sights of unequal length unavoidable, keep notes of 
 the distance, and, as soon as possible, take sights with correspond- 
 ing inequalities in the contrary direction. When approaching a 
 long incline make part of this compensation in advance. 
 
 * 
 
 Eecipeocal Leveling.— In crossing a river, it is absolutely 
 necessary that the back-sight and fore-sight should differ consider- 
 ably ; also other somewhat similar cases occur, to which the prin- 
 ciples of reciprocal leveling are applicable. The method of pro- 
 cedure is very simple. 
 
 Establish a bench upon both sides' of the river and det ermine 
 the difference of level ; move the instrument to the other side, and 
 re-determine the difference of level. If the sights were taken in 
 quick succession, the mean of the two results. is the true difference 
 of level. Simultaneous observations with two instruments would 
 be still better. 
 
THE LEVEL. 51 
 
 The surface of the water can not be assumed to be level, except, 
 there is no current, or the line joining the two benches is per- 
 pendicular, to the current and there is no wind. 
 
 In ascending or decending a steep hill, it is desirable, for speed, 
 that the line of sight should strike as near as possible to the bottom 
 of the rod on the up-hill side, and to the top of the rod on the down- 
 hill side. In selecting the position of the instrument corresponding 
 to this condition, set the instrument up lightly, turn the telescope 
 in the right direction, bring the bubble approximately to the middle 
 by manipulating the tripod legs, and sight along the outside of the 
 telescope. Even this rude observation will be valuable as showing 
 whether the instrument should be moved up or down the hill. It 
 will save considerable time. 
 
 With a little practice, the same observation may be made by 
 drawing the tripod legs together and using them as a Jacob's staff ; 
 then the bubble can speedily be brought to its proper position by 
 simply inclining the whole instrument. 
 
 If the up-hill rod is too near to be focused on, set a little to one 
 side. In short intermediate sights for which the telescope cannot 
 be focused, it is sufficient to sight by the bottom of the Y's or by 
 the side of the telescope. 
 
 If the line passes over a stream with steep high banks, or over 
 a narrow, deep gorge or valley, establish a turning point on the 
 farther side by reciprocal leveling ; then, to find the depth of the 
 opening, level down the bank without much regard to equality in 
 length of sights or other refinements. This will usually be all that 
 is necessary and is much quicker than leveling down one bank and 
 up the other. 
 
 Instruments are often provided with sun shades to prevent the 
 sun from troubling by shining into the telescope ; if the metallic 
 shade is not at hand, make one by rolling up a piece of paper and 
 gumming, pinning or tieing it together, or springing a rubber band 
 around it ; it is easier and better than holding the hat or note-book 
 over the objective. 
 
 If the instrument has once been leveled, and the bubble is 
 found to have moved a little, bring it back with a slight pressure 
 of the fingers, 
 
52 THE STADIA. 
 
 Finally, in closing at noon or night, be careful to set halfway 
 between the last two turning points ; on resuming work, set near 
 one of these points, and re-determine their difference of level; the 
 same difference of level each time, affords an excellent check upon 
 the adjustments of the instrument. 
 
 VI. 
 
 THE STADIA. 
 
 1. An Italian engineer (Porro) in about 1820, was the first to 
 suggest the determination of distance in surveying by a visual 
 angle and a rod. He proposed the name stadia; this term is ap- 
 plied to the instrument as a whole, although its derivation would 
 indicate that it referred more properly to the rod. The instrument 
 is more properly, but less frequently, called a telemeter. 
 
 The principles involved have long been well known, and have 
 been applied in gunnery and military reconnoisance ; but it is only 
 latelv that they bave been used in engineering. Even now. the 
 stadia has not come into as general use as its merits demand. It 
 was used first in Switzerland in 1836 in making a topographical 
 survey of that country. It seems not to have been introduced into 
 America until nearly thirty years afterward; and has not been 
 employed to any considerable extent except in topographical 
 surveys carried on by the TJ. S. Government. It is more generally 
 used in Europe than in the United States, although it is rapidly 
 coming into use here. 
 
 It is peculiarly adapted to topographical surveying, for it 
 possesses the double advantage of giving both the horizontal and 
 vertical co-ordinates, and this, too, by the most rapid method. 
 
 2. Pbinciple.— In all its forms the stadia depends upon the 
 principle of the similarity of triangles. A simple form of the stadia 
 is used in the familiar method of determining the distance of a man 
 by measuring on a rule held at arm's length the space covered by 
 his height. There are several forms of this simple device, but 
 none of them are of any practical value in surveying, owing to the 
 impossibility of focusing the eye for two distances at the same time, 
 and second, the indistinctness of the farther object. 
 
 To employ the principle of the stadia with a telescope, it is 
 
THE STADIA. 
 
 53 
 
 necessary to introduce two parallel cross-hairs and observe the 
 amount of the rod intercepted between the two hairs. The distance 
 from the instrument to the rod varies as the space on the rod inter- 
 cepted between the two hairs. The hairs may be horizontal aDd 
 the rod vertical, or vice versa ; the former is generally preferred. 
 Also the hairs may be fixed, the variable distance between them 
 being measured on the rod ; or the hairs may be movable and be 
 set to cover always the same two points on the rod, the varying 
 distance between the hairs being measured by a graduated screw. 
 The fixed hairs are cheaper, more simple, more accurate, and in 
 every way better ; it is the one generally used. 
 
 Adjusting the Haies.— Any measuring telescope can be 
 used as a stadia by adding a horizontal hair, 
 but it is much better to add two extra ones, 
 one on each side of tne ordinary one. With 
 the stadia hairs thus placed, the field of 
 view is symmetrical about the center, and 
 the added hairs interfere less with the ordi- 
 nary uses of the telescope. 
 
 In fixing the hairs, three conditions must 
 receive attention; they should be parallel, 
 equally distant from the central one, and at 
 a suitable distance from each other. The 
 various instrument makers have different 
 ways of making the hairs adjustable for 
 convenience in satisfying the above con- 
 ditions. 
 
 One of the most common of these is shown 
 in the accompanying sketches. The first 
 shows the reticule in place in the telescope 
 tube; the others show details. The screws 
 d and/ can be operated from the outside of 
 the telescope tube, and move (with reference 
 to the ring a), the slides b and c, which carry 
 the stadia hairs, e is a bent spring to take up 
 lost motion in the screws d and /. The ring 
 a, and all parts connected therewith, is ad- 
 justed in the telescope tube by the ordinary cross-hair screws, 
 which are not shown. By means of the screws d and /, the distance 
 of the stadia wires from the central wire, and from each other, can 
 
.54 
 
 THE STADIA. 
 
 be varied at will. Lines are made upon the slides b and c to assist 
 in placing the hairs parallel to each other. 
 
 The objections to this construction are, first, that it is expen- 
 sive; second, that the projecting heads of the adjusting screws are 
 liable to be struck and turned in handling the instrument or in 
 carrying it through brush, and so produce a serious error with no 
 adequate means of detecting it; and third, since a spring must be 
 inserted to take up lost motion, the screws have a tendency to work 
 loose. 
 
 4. Fig. 2, shows a much better way of making the hairs adjus- 
 table ; a, b, c, and d are small wire 
 plugs which are free to turn, being 
 held only by friction ; the shaded por- 
 tion is in the plane of the face of the 
 ring, the unshaded portion projects, 
 say i of an inch above. The cross- 
 hairs are to be stretched in the line of 
 the centers of a and d, and cand b, and 
 fastened at the outer edge of the ring ; 
 then, by turning the plugs, the hairs 
 will be moved toward or from the cen- 
 tral one, according to which side of 
 the -plug is toward the center hair. 
 The wires may easily be made to fit 
 
 tight enough not to work loose, and still turn freely enough for the 
 
 above adjustment. 
 
 With telescopes having inverting eye-pieces, which are much 
 the best for stadia work, the wires can be turned by removing the 
 eye-piece and without taking the reticule out of the telescope tube. 
 "With an erecting eye-piece, the hairs can be adjusted without re- 
 moving the ring from the telescope tube, by making a little wrench 
 (by cutting a kerf with a hacksaw in the end of a small strip of 
 brass) especially for the purpose, and operating it through a hole 
 in the telescope tube, also made for the purpose and which can be 
 closed by a band ; or the ring may be removed from the tube to ad- 
 just the stadia hairs. The telescope does not need to be collimated 
 to test the relative position of the stadia hairs. 
 
 This method provides a way of making the hairs parallel to 
 each other, and allows a variation in the distance between the cen- 
 
 Fig. 2. 
 
THE STADIA. 55 
 
 tral and each outside hair equal to the diameter of the plug. It is 
 comparatively inexpensive. 
 
 Although the conditions specified above, for the position of the 
 hairs, are the rigorous ones, farther on it will be shown that no 
 appreciable error will be produced if they are only approximately 
 satisfied. For example, the condition that the hairs should be 
 parallel, is only necessary in case the observation is not made ex- 
 actly on the vertical hair. Also if the stadia hairs are not sym- 
 metrically about the central one, it only produces an error in the 
 vertical angle. Finally, it is immaterial what distance the hairs 
 are apart, provided the rod is graduated to correspond. Hence it 
 is possible to attach simply two extra hairs in the ordinary way to 
 any measuring telescope, aDd thus fit it for stadia work. "We will 
 produce a formula to meet the case in which the rod is already 
 graduated, and does not agree with the distance between the 
 hairs. 
 
 5. The maximum distance between the hairs is limited by the 
 size of the field of view. A magnifying power of ten will generally 
 allow a field of view of 10° and for greater powers proportionally 
 less. Hence for a telescope which magnifies ten times, the 
 aperture may be 10° as viewed through the eye-piece, the real 
 visual angle being 1° and the distance between the hairs gVof the 
 focal length of the objective. This would be.about equivalent to a 
 ratio of space on the rod to the corresponding distance from the 
 
 instrument of 2 to 100. 
 This ratio might be 
 used for short sights, 
 but for long sights it 
 would require a rod 
 too long to be easily 
 4 ,....•■ handled. 
 
 :&:: 
 
 J.-C---— f- 
 
 On the other hand, 
 
 A' ••"- *> ■• — '"" i if the hairs are very 
 close together, the 
 Pig. 3 interception on the 
 
 rod will be too small 
 to be read accurately. All things considered, the best width is 
 probably that which makes 1 ft; on the rod correspond to 100 ft. on 
 the ground. 
 
56 THE STADIA 
 
 6. Formula for a Horizontal Line of Sight and a Vertical 
 Rod.— In Fig. 3, let a and b represent the stadia hairs, and i = the 
 distance between them ; let s = the distance on the rodp g, inter- 
 cepted between the hairs ; let/ = the focal distance of the objective ; 
 let e represent a point at a distance/ in front of the optical center 
 of the objective, or e is the outer focus of the objective ; c = the dis- 
 tance from the plumb line .of the instrument to the optical center: 
 of the objective; y = the distance from the outer focus e, to the 
 rod : D = the distance from the instrument to the rod. 
 
 From the principles of optics, all rays of light which pass 
 through e, become parallel to each other after passing through the 
 objective ; therefore, there ia some point p, which will emit a single 
 ray of light that will pass through e and after traversing the objec_ 
 tive, will strike the cross hair a. If the telescope is focused for th 
 point p, the objective will bring all rays emitted by p to a focus at a ; 
 hence it is immaterial whether we consider the real course of the 
 rays, or assume that all the light from p passes along the lines g e 
 a. Similarly for the point g. Therefore, the diagram shows cor- 
 rectly the geometrical relations between the variables involved in 
 the determination of distance by a visual angle with a telescope. 
 
 From Fig. 3 we easilv get s:y::i:f, from which 
 
 f 
 y = -s (l) 
 
 / 
 and £> = --s + c+/ (2) 
 
 i 
 / 
 Notice that - is a constant co-efficient peculiar to each instru- 
 i 
 
 ment, also that s varies with tte distance to be determined, and 
 that it is proportional to the distance from the rod to the outer 
 focus of the objective. 
 
 It is now necessary to show how the numerical value of the 
 / 
 terms - c and / can be found. To find / focus the instrument on an 
 1 
 
 object, say half a mile away, and measure with an ordinary rule 
 the distance from the cross-hairs to the middle of the thickness of 
 the objective. To find c measure the horizontal distance from the 
 vertical of the instrument to the middle of the thickness of the ob- 
 
THE STADIA. 57 
 
 jective ; c is not strictly constant in instruments in which the cross- 
 hairs are fixed and the objective movable, but if the values of c and 
 / are found within an Inch it is more than sufficient. 
 
 / 
 To find the co-efficient - mark a point on the ground under e 
 i, 
 (Fig. 3), by measuring a distance = (c -f f) from the plumb line of 
 the instrument, and set a rod, say 100 ft., in front of this point; 
 this distance = y in equation (1). Sight through the instrument 
 and have an assistant mark the point on the rod covered by each 
 hair, the distance between these points correspond to s (1), we have 
 
 / 
 
 - = 100 s. 
 
 i 
 
 If the space on the rod be then divided into this number of spaces, 
 and the same graduation be carried throughout the length of the 
 rod, then the number of divisions on the rod included in the visual 
 angle will always be equal to the distance from the rod to the front 
 focus of the objective. 
 
 If B = the number of divisions included in the visual angle, 
 equation (2) then becomes 
 
 D=B+f+c (3) 
 
 The use of the preceding equation demands that each value 
 
 / 
 of- shall have a rod graduated to correspond. It frequently 
 i 
 
 happens, from one cause or another, that it is desired to use a rod 
 whose divisions are too great or too small ; it then becomes neces- 
 sary to multiply the reading by a co-efficient to correct it. Let 
 K = this co-efficient. To find K, measure D — (f + c) for som& 
 particular case, and observe the corresponding B : then K should 
 have such a value as will make K B numerically equal to 
 
 £>-(/+ c 
 
 D-(f+c),0TK= (3> 
 
 B 
 
 then becomes 
 
 D = KB(f+c). 
 
58 
 
 THE STADIA. 
 
 8. The Bod.— The rod must be light and handy for transpor- 
 tation. A board about 1-in. thick and 4 or 5 wide does very well; 
 
 2 S 
 
 a 
 a 
 
 o ,, 
 
 w 
 
 « a 
 
 * s 
 
 1 <° 
 
 A o 
 
 ■3 -a 
 
THE STADIA. 59 
 
 it may be stiffened by screwing a strip edgewise on the back. The 
 rod may be hinged in the middle, folded for ease of transportation 
 and for the protection of the graduation ; when in use, it may be 
 kept upright by a button or a bolt on the back. It should be pro- 
 Tided with a disk level, or a short plumb, for keeping it vertical. 
 A handle is very useful in holding the rod steady. 
 
 As in leveling, there are two kinds of stadia rods, self-reading 
 and target rods. The latter require two targets; one of them is 
 sometimes permanently fixed to the rod to save the trouble of 
 setting two targets, but, when the vertical co-ordinate is desired, 
 this adds more complication than it saves. Target rods may be 
 a little more accurate, but they are certainly very much less con- 
 venient. Self-reading stadia rods are generally preferred. 
 
 9. The principles involved in the graduation of stadia rods are 
 much the same as those discussed for self-reading level rods ; the 
 only difference is that the latter are generally read at short dis- 
 tances, while it is frequently necessary to read the former at long 
 distances. Hence, in graduating a stadia rod, visibility is of first 
 importance. 
 
 It is generally stated that a red disk on a white ground can be 
 seen by the unaided eye at about 6,000 times its own diameter. A 
 black disk can be seen farther, and a black line still farther. A 
 good telescope would increase this distance nearly in proportion to 
 its magnifying power. To be on the safe side and allow for the loss 
 of light in the telescope and for unfavorable atmospheric con- 
 ditions, we will assume that a black line on a white ground will 
 •always be visible, to the unaided eye, at 4,000 times its least dimen- 
 sions ; experiments seem to show that this is safe. Therefore, the 
 greatest distance, L, at which a mark of width, w, will be visible 
 through a telescope magnifying M times is 
 
 L = 4,000 w M; (5) 
 
 and the length of rod, I, required for the sight, Z, is determined by 
 
 8 Si 
 
 l=L — = 4,000 w M — = L — very nearly, (6) 
 
 D D f 
 
 in which D is the distance on the ground corresponding to the 
 8 1 
 
 space, 8, on the rod ; — is usually about (7) 
 
 D 100 
 
60 THE STADIA. 
 
 D 
 
 Also the smallest difference in distance perceptible = — d in which 
 
 8 
 d is the value of the smallest division of the graduation. These 
 formulas, together with a knowledge of the field of view, the 
 magnifying power of the telescope, and the longest probable sight, 
 are useful in determining the distance apart of the hairs and in 
 graduating the rod. 
 
 10. Any self-reading leveling rod can be used as a stadia rod ; 
 but it is best to omit the numbers, for it is easier to count the divi- 
 sions included in the visual angle than to read both hairs and sub- 
 tract. If the graduation of the rod consists of a great number of 
 very small divisions, they become wholly indistinct at a distance 
 greater than L as above, and at even shorter distances, it is very 
 confusing to count them. Also, it is best to keep the colors as much 
 together as possible, else when the air is unsteady the figures will 
 run together to such an extent that it will be impossible to dis- 
 tinguish them. For these reasons, it is much better to make the 
 graduation marks larger, and secure the smaller divisions by sub- 
 dividing the larger ones by estimation ; the rod can then be used at 
 very great distances, although at a sacrifice of precision. This 
 method by estimation is almost, or quite, as accurate as when the 
 smaller divisions are marked ; it is often claimed that very small 
 subdivisions can be estimated more exactly than read by a direct 
 graduation. 
 
 The adjoining figures show a few of the many forms proposed 
 for stadia rods. Fig. 4 is a rod used on the U. 8. Coast Survey ; the 
 vertical distance from a to b corresponds to 10 ft. on the ground 
 from b to d = 4 ft. and by dividing b d into quarters by estimation,, 
 the rod may be read to single feet. The other designs are read in 
 a similar manner. Fig. 5 was used on the late U. S. Lake Survey. 
 The other figures are added to show the facility with which designs 
 may be made. Nearly every one who uses a stadia, has a design of 
 his own which he thinks is the best. 
 
 11. Position of Eod fob Inclined Line of Sight. — Formulas 
 (1) and (2) were deduced on the assumption that the central visual 
 ray was horizontal and the rod vertical, but this is not sufficiently 
 general. These formulas would be sufficient, if the observations 
 were made with a leveling instrument ; but the telescope of a level 
 is too limited in its range to secure the full advantage of the prin- 
 
THE STADIA. 
 
 61 
 
 ciple of the stadia 
 transit is used. 
 
 Formula (3), D 
 of sight, provided 
 visual ray ; in this 
 
 . In all that follows, it will be assumed that a 
 
 = B -f- (f + c), may be used with an inclined line 
 the rod is held perpendicular to the central 
 case D is no longer horizontal distance from the 
 
 instrument to the 
 rod; but is the 
 oblique distance 
 from the horizontal 
 axis of the telescope 
 to the point on the 
 rod covered by the 
 central visual ray. 
 It th»n becomes a 
 question whether, 
 with an inclined line 
 _;of sight, the rod 
 ^should be held per- 
 v pendicular to the 
 central visual ray, or 
 vertical. 
 
 The position of the 
 rod perpendicular to 
 the line of sight may 
 be determined by a 
 telescope, or a pair 
 of sights, attached 
 at right angles to the 
 rod, which is direoted 
 toward the observing 
 telescope by the rod- 
 man. The verti- 
 cality of the rod may- 
 be determined by at- 
 taching a plumb-line 
 ■ or a level. 
 
 I Some prefer the 
 
 Pig. 4. Fig. 5. Pig. 6. Pig. 7. rod perpendicular to 
 
 the line of sight, but this position involves serious difficulties ; 
 
 first, it is not easy to hold the rod steady in this position ; second 
 
62 
 
 THE STADIA. 
 
 it is not always possible for the rodinan to see the telescope, especi- 
 ally at long distances or great vertical angles, or when under- 
 growth, etc., intervenes. Without going into the details it is suffi 
 cient to state that the formulas for computing the horizontal and 
 vertical co-ordinates of the point are more simple when the rod is 
 held vertical than when it is perpendicular to the line of sight. 
 
 Formulas foe Inclined Line of Sight and Veetical Rod.— Let 
 
 = the angle of the 
 central visual ray 
 with the horizontal. 
 To measure 6, a third 
 horizontal hair 
 should be placed 
 half-way between the 
 other two, or rather 
 stadia hairs are to be 
 added on opposite 
 sides and equally 
 distant from the 
 ordinary horizontal one; is then determined as any other vertical 
 angle by reading the vertical circle. will generally be small. 
 Let 2 oo = the visual angle, = B ' o D, Fig. 8 ; 2 oo is always small, 35 
 may be taken as its maximum value. A E is the actual intercept, 
 and B D the value it would have if the rod were held perpendicular 
 to the central visual ray. It is desired to find a relation between 
 A E and B B. 
 
 Fig. 8. 
 
 The angle C B A = 90° + x\ and, since x is very small, B C = 
 A C cos nearly. Similarly C D E = 90° — x', and C D = C E-cos 9 
 Hence ( A C + C E) cos 8 = A E cos = B D. Notice that the two 
 approximations tend to neutralize each other ; the final error in- 
 volved is much less than the error of observing A E 2 . 
 
 Since B D is the intercept perpendicular to the line of sight, it 
 is the rod reading corresponding to the distance I C. If we repre- 
 
 2 If the side hairs are equally distant from the central one, the true relation 
 is B D = A E cos 6 (1 — tan? 8 tan 1 d); and if the side hairs are not equally dis- 
 tant, the upper angle being x' and the lower, x the true relation is 
 
 cos a;' cos x 
 
 AE=OB + CD- 
 
 cos (6 + o°') 
 
 cos (9 — oo) 
 
THE STADIA 63 
 
 sent the reading on the vertical rod by B, we have for the oblique 
 distance I C. 
 
 D = RcosQ +(/+c) (7) 
 
 13. Horizontal Distance.— Let H = the horizontal distance 
 from the center of the instrument to the vertical through the foot 
 of the rod: H= IP, Fig. 8. IP = I Coos 8 = D cos 6, and 
 
 H= R cos 2 e + (c + f) cose (8) 
 
 or H=R— Bsin* + (c+/)cos0 (8') 
 
 The second form is preferable, for it is always better to compute 
 a correction to a quantity than the quantity itself. Notice that 
 since (c +/) is always small, and cos 6 nearly 1', (c +/) cos 8 may 
 always be taken equal to (o+f), and often be omitted entirely. 
 Finally, sin 2 fi is so small that the whole operation of reducing an 
 observation by (8') may be performed mentally. 
 
 14. Vertical Distance.— To determine the vertical co-ordinate 
 of the point upon which the rod is set, the middle hair must be 
 sighted upon a point of the rod at a known distance from its foot. 
 The simplest way of accomplishing this is to provide the rod with a 
 movable target, which is to be set at a distance from the foot of the 
 rod equal to the height of the horizontal axis of the telescope above 
 the reference point.. If the instrument is set over the reference 
 point, the target can easily be set by placing the rod at the side of 
 the instrument ; if the instrument is not over the point, set the rod 
 on it, bring the line of sight horizontal, and sight the target in. To 
 determine the height of any subsequent point, set the middle hair 
 upon the target, and read the angle from the vertical circle. This 
 method of pointing, besides being very simple in itself, very much 
 simplifies the formula for the vertical co-ordinate. 
 
 Let V = the height of the point on which the rod is placed 
 above the reference point ; notice that for the above method of ad- 
 justing the target, Fis equal to the distance of the target above the 
 axis of the telescope. Then C P, the distance sought, = V=IG 
 sin 8 ; substituting the value of I C from (7) 
 
 F=-Rcos8sin8 + (c+/)sin8 (9) 
 
 V = Bi sin 2 8 + (c +/) sin 6 (9) 
 
 (c +/) sin 8 may generally be omitted. Notice that Fwill be + qr 
 —j according as 8 isan angle of elevation or depression. 
 
 15. Keducing the Field Notes.— This consists in finding the 
 horizontal and vertical distance from the"ob"s"ervF6r"readThg~and 
 
64 THE STADIA. 
 
 angle of observation. This can be done by using formulas (8) and 
 (9). Although the formulas are in a very convenient form for com- 
 putation, it would be very tedious and slow to solve both equations 
 for each observation. It is a great saving of labor and time, if the 
 results are computed once for all and tabulated. 
 
 Notice that B cos 2 8 and B\ sin 2 8 are independent of the instru- 
 ment and of the unit of linear measurement used, and can be tabu- 
 lated for all cases ; (c +/) cos 8 and (c +/) sin 8 depend upon both 
 the instrument and the unit, and must be computed for each special 
 case, but as they are so small they can be given in a brief supple- 
 mental table. 
 
 The result may be expressed in an arithmetical table or a 
 geometrical diagram. 
 
 Arithmetical tables are capable of greater accuracy ; but they 
 must be either very extended and therefore inconveniently large, 
 or brief and give results by interpolation which is slow and tedious. 
 On the other hand, it is urged against geometrical diagrams, that, 
 to be accurate, they must be drawn to a large scale, and are there- 
 fore large and unwieldy. It is believed that, with properly con- 
 structed diagrams, the reductions can be made without sacrificing 
 much, if any, accuracy and with greater facility than by the use of 
 arithmetical tables. 
 
 In constructing a table, we may tabulate B cos 2 8 from equation 
 (8') and B^ sin 2 from equation (9') for different values of B and 8, 
 in which case the horizontal and vertical distances can be taken 
 directly from the table ; or, we may tabulate only cos 2 8 and I sin 
 2 8 for different values of 8, in which case the horizontal and verti- 
 cal distances are found by multiplying the reading B, by the tabu- 
 lated factor. The first method would be the better, if it did not 
 require such voluminous tables. 
 
 16. Geometrical Tables.— There is a variety of methods of 
 constructing diagrams for reduction of stadia observations, but as 
 modifications of those proposed by Prof. S. W. Bobinson 3 are be- 
 lieved to be the best they only will be explained. The tables on 
 pages 65, 66, 67, 68, give the value of sin 2 9 in the column marked 
 H, and the value J sin 2 8 in the column marked V. 
 
 3 Journal Franklin Institute, Vol. 49. pp. 80-1 ; and do. Vol. 61, pp. 16-6. 
 
STADIA REDUCTION TABLES. 
 
 65 
 
 e 
 
 0° 
 
 1° 
 
 2° 
 
 3? 
 
 4° 
 
 6° 
 
 
 H. 
 
 V. 
 
 H. 
 
 T. 
 
 H. 
 
 V. 
 
 H. 
 
 V. 
 
 H, 
 
 V. 
 
 H. 
 
 V. 
 
 0' 
 2 
 4 
 
 .0000 
 .0000 
 .0000 
 
 .0000 
 .0006 
 .0012 
 
 .0003 
 .0003 
 .0003 
 
 .0174 
 0180 
 .0186 
 
 .0012 
 .0013 
 .0013 
 
 .0319 
 .0356 
 .0360 
 
 .0027 
 .0028 
 .0029 
 
 .0523 
 .0528 
 0534 
 
 .0049 
 .0049 
 .01150 
 
 .0696 
 .0702 
 .0707 
 
 .0076 
 .0077 
 .0078 
 
 .0868 
 .0874 
 .0880 
 
 6 
 
 8 
 
 10 
 
 .0000 
 .0000 
 .0000 
 
 .0017 
 .0023 
 .0029 
 
 .0004 
 .0004 
 .0004 
 
 .0192 
 .0197 
 .0203 
 
 .0013 
 .0014 
 .0014 
 
 .0366 
 .0372 
 .0378 
 
 .0029 
 .00 
 .0030 
 
 .0540 
 .0546 
 .0552 
 
 .OO^l 
 .0052 
 .0053 
 
 .0713 
 .0719 
 .0725 
 
 .0079 
 .0080 
 .0081 
 
 .0886 
 .0--<91 
 .0897 
 
 12 
 14 
 16 
 
 .0000 
 .0000 
 .00-0 
 
 .0035 
 .0041 
 .0046 
 
 .0004 
 .0004 
 .0005 
 
 .0209 
 .0215 
 '.0221 
 
 .0015 
 .0015 
 .0016 
 
 .0384 
 .0390 
 .0395 
 
 .H031 
 .0032 
 .0032 
 
 .0557 
 .l'563 
 .0569 
 
 .0054 
 .0054 
 .0055 
 
 .0730 
 •0736 
 ■9742 
 
 .0082 
 .0083 
 .0084 
 
 .0903 
 .0908 
 .0914 
 
 18 
 20 
 22 
 
 .0000 
 .0000 
 .0000 
 
 .0052 
 .0058 
 .0064 
 
 .0005 
 .0005 
 .0005 
 
 .0227 
 '.0232 
 .0238 
 
 .0016 
 .0017 
 .0017 
 
 .0401 
 .0407 
 ,0413 
 
 .0033 
 .0034 
 .0035 
 
 .0575 
 .0^80 
 .06 -;6 
 
 .0056 
 .0057 
 .0058 
 
 .0748 
 .0753 
 .0759 
 
 .0085 
 .0086 
 .0087 
 
 .0920 
 .0926 
 .0931 
 
 24 
 26 
 28 
 
 .0000 
 .0000 
 .0001 
 
 .0070 
 .0076 
 .0081 
 
 .0006 
 .0006 
 .0007 
 
 .0244 
 :0250 
 .0256 
 
 .0017 
 .0018 
 10018 
 
 .0418 
 .0424 
 .0430 
 
 .0035 
 .0036 
 .0037 
 
 .0592 
 .0598 
 .0604 
 
 .0059 
 .0060 
 .0061 
 
 .0765 
 .0771 
 .0776 
 
 .0088 
 .0089 
 .0090 
 
 .0937 
 .0943 
 .0948 
 
 30 
 32 
 34 
 
 .0001 
 .0001 
 
 ;oooi 
 
 .0087 
 .0093 
 :0099 
 
 :ono7 
 
 .0007 
 
 :ooo8 
 
 .0261 
 ;0267 
 .0273 
 
 :ooi9 
 
 .0019 
 .0020 
 
 .0438 
 ."442 
 .0148 
 
 .0037 
 .0038 
 .00 J9 
 
 .0609 
 .0615 
 .0621 
 
 .0062 
 .0062 
 .0063 
 
 .0782 
 ;0788 
 .0794 
 
 .0091 
 .0092 
 .0093 
 
 .0^54 
 .0960 
 .0965 
 
 36 
 38 
 
 40 
 
 .0001 
 .0001 
 .0001 
 
 :oi05 
 
 .0110 
 
 .xiie 
 
 .'01108 
 .0008 
 .0009 
 
 .0279 
 .0285 
 .0291 
 
 .0021 
 .0021 
 .0022 
 
 .0453 
 .0459 
 .0465 
 
 .0039 
 .0040 
 .0041 
 
 ,0627 
 .0633 
 .0638 
 
 .0064 
 .0065 
 .0066 
 
 .0799 
 .0805 
 .0811 
 
 .0094 
 .0095 
 .0097 
 
 .0971 
 .0977 
 .0983 
 
 42 
 44 
 46 
 
 :ooo2 
 
 .O0'i2 
 
 :ooo2 
 
 .0122 
 .0128 
 .0131 
 
 .0009 
 .0009 
 .0010 
 
 .0297 
 0302 
 .0308 
 
 .0022 
 .0023 
 ,0023 
 
 .0471 
 .0476 
 .0482 
 
 .0042 
 .0042 
 .0043 
 
 .0644 
 .0650 
 .0656 
 
 .0067 
 .0068 
 .0069 
 
 .0817 
 .0822 
 .0828 
 
 .11098 
 ,0100 
 .0101 
 
 .0988 
 .0994 
 .1000 
 
 48 
 50 
 52 
 
 .0002 
 .0002 
 .0003 
 
 .0139 
 .0145 
 .0151 
 
 .0010 
 .1)010 
 .0011 
 
 .0314 
 .0320 
 .0326 
 
 .0024 
 ,0024 
 .0025 
 
 .0488 
 .0494 
 .0499 
 
 .0044 
 .0045 
 .0045 
 
 .0661 
 .0667 
 .0673 
 
 .0070 
 .0071 
 .0072 
 
 .0834 
 .0840 
 .0845 
 
 .0102 
 .0103 
 .0104 
 
 .1005 
 .1011 
 .1017 
 
 54 
 56 
 
 58 1 
 60 1 
 
 .0003 
 .0003 
 .0003 
 .0003 
 
 .0157 
 .0163 1 
 .01 63 1 
 
 .01741 
 
 .0011 
 .0011 
 .0011 
 .0012 
 
 ■0331 
 .0337 
 .0343 
 ,0349 
 
 .0026 
 .0026 
 .0027 
 .0027 
 
 .0505 
 .0511 
 .0617 
 .0522 
 
 .0046 
 .0047 
 .0048 
 .0049 
 
 •0S78 
 .0684 
 .0690 
 .0696 
 
 .0073 
 .0074 
 .0075 
 .0076 
 
 .0851 
 .0857 
 .0863 
 .0868 
 
 .0105 
 .0107 
 .0108 
 .0109 
 
 .1022 
 .1028 
 .1034 
 .1040 
 
 For all values oE a . found on this page. we may take ( ( -)- /) S j n fl — 
 
STADIA REDUCTION TABLES. 
 
 e 
 
 6" 
 
 7° 
 
 8° 
 
 9° 
 
 10° 
 
 11° 
 
 
 H. 
 
 V. 
 
 H. 
 
 V. 
 
 H. 
 
 V. 
 
 H. 
 
 V. 
 
 H. 
 
 V. 
 
 H. 
 
 V. 
 
 
 2 
 4 
 
 .0109 
 .0110 
 .0112 
 
 .1040 
 .1045 
 .1051 
 
 .0149 
 .0150 
 .0151 
 
 .1210 
 .1215 
 .1221 
 
 .0194 
 .0195 
 .0197 
 
 .1378 
 .1384 
 .1389 
 
 .0245 
 .0247 
 .0248 
 
 .1545 
 .1551 
 .1556 
 
 .0302 
 .0304 
 .0306 
 
 .1710 
 .1716 
 .1721 
 
 .0364 
 .0366 
 .0368 
 
 .1873 
 .1878 
 .1884 
 
 6 
 8 
 10 
 
 .0113 
 .0114 
 .0115 
 
 .1057 
 .1062 
 .1063 
 
 .0153 
 .0154 
 .0156 
 
 .1226 
 .1232 
 .1238 
 
 .0199 
 .0200 
 .0202 
 
 .1395 
 .1401 
 .1406 
 
 .0250 
 .0252 
 .0254 
 
 .1562 
 .1667 
 .1673 
 
 .0308 
 .0310 
 .0312 
 
 .1726 
 .1732 
 .1737 
 
 .0371 
 .0373 
 .0375 
 
 .1889 
 .1895 
 .1900 
 
 12 
 14 
 16 
 
 .0117 
 .0118 
 .0119 
 
 .1074 
 .1079 
 .1085 
 
 .0157 
 .0158 
 .0160 
 
 .1243 
 .1249 
 .1265 
 
 .0203 
 .0205 
 .0207 
 
 .1412 
 .1417 
 .1423 
 
 .0256 
 .0257 
 .0259 
 
 .1678 
 .1584 
 .1589 
 
 .0314 
 .0316 
 .0318 
 
 .1743 
 .1748 
 .1754 
 
 .0377 
 .0379 
 .0382 
 
 .1906 
 .1911 
 .1916 
 
 18 
 20 
 
 22 
 
 .0120 
 .0122 
 .0123 
 
 .1091 
 .1096 
 .1102 
 
 .0161 
 .0163 
 .0164 
 
 .1260 
 .1266 
 .1272 
 
 .0208 
 .0210 
 .0212 
 
 .1428 
 .1434 
 .1440 
 
 .0261 
 .0263 
 .0265 
 
 .1595 
 .1600 
 .1606 
 
 .0320 
 .0322 
 .0324 
 
 .1759 
 .1765 
 .1770 
 
 .0384 
 .0386 
 .0388 
 
 .1921 
 .1927 
 .1932 
 
 21 
 26 
 28 
 
 .0124 
 .0126 
 .0127 
 
 .1108 
 .1113 
 .1119 
 
 .0166 
 .0167 
 .0169 
 
 .1 77 
 .1283 
 .1288 
 
 .0218 
 .0215 
 .0217 
 
 .1445 
 .1451 
 .1456 
 
 .0267 
 .026 J 
 .0271 
 
 .1611 
 .1617 
 .1622 
 
 .0326 
 .0328 
 .0330 
 
 .1776 
 .1781 
 .1786 
 
 .0391 
 .0393 
 .0395 
 
 .1938 
 .1943 
 .1948 
 
 30 
 32 
 34 
 
 .0128 
 .0129 
 .0131 
 
 .1125 
 .1130 
 .1136 
 
 .0170 
 .0172 
 .0173 
 
 .1294 
 .1300 
 .1305 
 
 .0218 
 .0220 
 .0222 
 
 .1462 
 .1467 
 .1473 
 
 .0272 
 .0274 
 .0276 
 
 .1628 
 .1633 
 .1639 
 
 .0332 
 .0334 
 .0336 
 
 .1792 
 .1797 
 .1803 
 
 .0897 
 .0400 
 .0402 
 
 .1954 
 .1959 
 .1964 
 
 36 
 38 
 40 
 
 .0132 
 .0133 
 .0135 
 
 .1142 
 .1147 
 .1153 
 
 .0175 
 .0176 
 .0178 
 
 .1311 
 .1317 
 .1322 
 
 .0224 
 .0225 
 .0227 
 
 .1479 
 .1484 
 .1490 
 
 .0278 
 .0280 
 .0282 
 
 .1644 
 .1650 
 .1655 
 
 .0338 
 .0340 
 .0342 
 
 .1808 
 .1814 
 .1819 
 
 .0404 
 .0407 
 .0409 
 
 .1970 
 .1975 
 .1980 
 
 42 
 44 
 46 
 
 .0136 
 .0137 
 .0139 
 
 .1159 
 .1164 
 .1170 
 
 .0180 
 .0181 
 .0183 
 
 .1328 
 .1331 
 .1339 
 
 .0229 
 .0231 
 .0231 
 
 .1495 
 .1501 
 .1606 
 
 .0284 
 .0286 
 .0283 
 
 . 661 
 .1666 
 .1672 
 
 .0345 
 .0347 
 .0319 
 
 .1824 
 .1830 
 .1835 
 
 .0411 
 .0414 
 .0136 
 
 .1986 
 .1991 
 .1996 
 
 48 
 
 50 
 
 , 62 
 
 .0140 
 .0142 
 .0143 
 
 .1176 
 .1181 
 .1187 
 
 .0181 
 .0186 
 .0187 
 
 .1345 
 .1350 
 .1356 
 
 .0234 
 .0236 
 .0238 
 
 .1512 
 .1517 
 .1523 
 
 .0290 
 .0292 
 .0294 
 
 .1677 
 .1683 
 .1688 
 
 .0351 
 .0353 
 .0355 
 
 .1841 
 .1846 
 .1851 
 
 .0418 
 .0421 
 .0423 
 
 .2002 
 .2007 
 .2012 
 
 54 
 66 
 68 
 60 
 
 .0144 
 .0146 
 .0147 
 .0149 
 
 .1193 
 .1198 
 .1204 
 .1210 
 
 .0189 
 .0190 
 .0192 
 .0194 
 
 .1361 
 .1367 
 .1373 
 .1378 
 
 .0239 
 .0241 
 .1243 
 .0245 
 
 .1528 
 .1534 
 .1540 
 .1545 
 
 .0296 
 .0298 
 .0300 
 .0302 
 
 .1694 
 .1699 
 .1706 
 .1710 
 
 .0358 
 .0360 
 .0362 
 .0364 
 
 .1857 
 .1862 
 .1868 
 .1873 
 
 .0425 
 .0428 
 .0430 
 .0432 
 
 .2018 
 .2023 
 .2028 
 .2034 
 
 For V£ 
 
 dues of 9- oi 
 
 1 this 
 
 Daee j(«+/)cose 
 
 page i, c+/)si „e 
 
 = (c + /) 
 
 varies between o.l (c +-/) &0.2 (e+f 
 
STADIA BEDUOTION TABLES. 
 
 67 
 
 
 
 12° 
 
 13° 
 
 14° 
 
 16° 
 
 16° 
 
 17° 
 
 
 H. 
 
 V. 
 
 H. 
 
 V. 
 
 H. 
 
 V. . 
 
 H. 
 
 V. 
 
 H. 
 
 v.. 
 
 H. 
 
 V. 
 
 I 
 
 4 
 
 .0432 
 .0435 
 .0137 
 
 .2034 
 .2039 
 
 .2044 
 
 .0506 
 .0509 
 .0311 
 
 .2192 
 .2197 
 .2202 
 
 .0535 
 .0588 
 .0591 
 
 .2347 
 .2352 
 .2358 
 
 .0670 
 .0673 
 .0676 
 
 .2500 
 .2505 
 .2510 
 
 .0760 
 .0763 
 .0766 
 
 .2650 
 .2656 
 .2659 
 
 .0855 
 .0858 
 .0861 
 
 .2796 
 .2801 
 .2806 
 
 6 d 
 8 
 10 
 
 .0139 
 .0442 
 .0444 
 
 .2050 
 .2055 
 
 .2060 
 
 .0513 
 .0516 
 .0519 
 
 .2208 
 .2213 
 .2218 
 
 .0594 
 .0596 
 .0599 
 
 .2363 
 ,2368 
 .2373 
 
 .0679 
 .0682 
 .0685 
 
 .2515 
 .2520 
 .2525 
 
 .0769 
 .0772 
 .0775 
 
 ,2664 
 .2669 
 .2674 
 
 .0865 
 .0*68 
 .0871 
 
 .2810 
 .2815 
 .2820 
 
 12 
 11 
 16 
 
 .0447 
 .0449 
 .6451 
 
 .2066 
 .2071 
 .2076 
 
 .0521 
 .0524 
 .0527 
 
 .2223 
 .2228 
 .2234 
 
 .0602 
 .0605 
 .0607 
 
 .2378 
 .2383 
 .2388 
 
 .0687 
 .0690 
 .0693 
 
 .2530 
 .2535 
 .2540 
 
 .0778 
 .0782 
 .0785 
 
 .2679 
 .2684 
 .2689 
 
 .0874 
 .0878 
 .0881 
 
 .2825 
 .2830 
 .2834 
 
 18 
 20 
 22 
 
 .0454 
 .0456 
 .0459 
 
 .2081 
 .2087 
 .2092 
 
 .0529 
 .05d2 
 .0534 
 
 .2239 
 .2244 
 .2249 
 
 .0610 
 .0613 
 .0616 
 
 .2393 
 .2399 
 .2404 
 
 .0696 
 .0699 
 .0702 
 
 .254. 
 .2550 
 .2555 
 
 .0788 
 .0791 
 .0794 
 
 .2694 
 .2699 
 .2704 
 
 .0884 
 .0888 
 .0891 
 
 .2839 
 .2844 
 .2849 
 
 24 
 26 
 28 
 
 .0461 
 .0164 
 .0466 
 
 .2097 
 . 103 
 .2108 
 
 .0537 
 .0540 
 .0542 
 
 .2254 
 
 .2260 
 .2265 
 
 .0619 
 .0621 
 .0621 
 
 .2409 
 .2114 
 .2419 
 
 .0705 
 .0708 
 .0711 
 
 .2560 
 .2565 
 
 .2510 
 
 .0797 
 .0800 
 .0804 
 
 .2709 
 .2713 
 .2718 
 
 .0894 
 .0898 
 .0901 
 
 .2854 
 .2858 
 .2863 
 
 30 
 32 
 34 
 
 .0469 
 .0471 
 .0473 
 
 .2113 
 .2118 
 .2121 
 
 .0545 
 .0548 
 .0550 
 
 .2270 
 .2275 
 .2280 
 
 .0627 
 .0630 
 /633 
 
 .2124 
 .2429 
 .2434 
 
 .0714 
 .0717 
 .0720 
 
 .2575 
 .2580 
 .2585 
 
 .0807 
 .' 810 
 .0813 
 
 .2723 
 .2728 
 .2733 
 
 .0904 
 .0908 
 .0911 
 
 .2868 
 .2873 
 .2877 
 
 36 
 38 
 40 
 
 .0476 
 .0478 
 .0481 
 
 .2129 
 .2134 
 
 .2139 
 
 .0553 
 .0556 
 .0558 
 
 .2285 
 .2291 
 .2296 
 
 .0635 
 .0618 
 .0641 
 
 .2439 
 .2444 
 .2419 
 
 .0723 
 .0726 
 .0729 
 
 .2590 
 .2595 
 .2600 
 
 .0816 
 .0819 
 .0822 
 
 .2738 
 .2743 
 
 .2748 
 
 .0914 
 .0918 
 .0921 
 
 .2882 
 .2887 
 .2892 
 
 42 
 44 
 46 
 
 .0483 
 .0486 
 .0488 
 
 .2145 
 .2150 
 .155 
 
 .0561 
 .0564 
 .0566 
 
 .2301 
 .2306 
 .2311 
 
 .0644 
 .0647 
 .0650 
 
 .2455 
 .2460 
 .2465 
 
 .0732 
 .0735 
 .0738 
 
 .2605 
 .2610 
 .2615 
 
 .0826 
 .0829 
 .0832 
 
 .2752 
 .2767 
 A762 
 
 .0924 
 .0928 
 .0931 
 
 .2896 
 .2901 
 .2906 
 
 48 
 50 
 52 
 
 .0491 
 .0493 
 .0196 
 
 .2160 
 .2166 
 .2171 
 
 .0669 
 .0572 
 .0575 
 
 .2316 
 .2322 
 .2327 
 
 .0653 
 .0656 
 .0658 
 
 .2470 
 .2475 
 .2480 
 
 .0741 
 .0744 
 .0747 
 
 .2620 
 .2625 
 .2630 
 
 .0835 
 .0839 
 .0842 
 
 .2767 
 .2772 
 .2777 
 
 .0936 
 .0938 
 .0941 
 
 .2911 
 .2915 
 .2920 
 
 54 
 56 
 58 
 60 
 
 .0498 
 .0501 
 .0504 
 .0506 
 
 .2176 
 .2181 
 .2187 
 .2192 
 
 .0577 
 .0580 
 .0583 
 .0585 
 
 .2332 
 .2337 
 .2342 
 ,2347 
 
 .0661 
 .0664 
 .0667 
 .0670 
 
 .2485 
 .2490 
 .2495 
 .2500 
 
 0750 
 .0754 
 .0757 
 .0760 
 
 ,2635, 
 .2640 
 .2645 
 .2650 
 
 .0845 
 .0848 
 ,0852 
 .08f5 
 
 .2781 
 .2786 
 ,2791 
 .2796 
 
 .0945 
 .0948 
 .095! 
 .0955 
 
 .2925 
 .2930 
 .29*4 
 .2939 
 
 „ « , - t, iV< i(c + /)C08 6>.95(c+/) = (c+/) 
 
 For.values oi 8 on this page} (c+/ , sin fl varIeB t^een 0.2 (c+/) <fco,3 {<,+/>. 
 
68 
 
 STADIA REDUCTION TABLES. 
 
 8 
 
 18" 
 
 19° 
 
 20° 
 
 21° 
 
 22° 
 
 23° 
 
 H. 
 
 Y. 
 
 H. 
 
 Y. 
 
 H. 
 
 Y. 
 
 H. 
 
 Y. 
 
 H. 
 
 Y. 
 
 H. 
 
 v. 
 
 
 2 
 4 
 
 .0955 
 .0953 
 .0962 
 
 .2939 
 .2944 
 .2948 
 
 .1060 
 .1064 
 .1067 
 
 .3078 
 .3083 
 .3087 
 
 .1170 
 .1174 
 .1177 
 
 .3214 
 .3218 
 .3223 
 
 .1284 
 .1288 
 .1292 
 
 .3346 
 .3360 
 .3361 
 
 .1403 
 .1407 
 .1411 
 
 .3473 
 .3177 
 ,3482 
 
 .1527 
 .1531 
 .1535 
 
 .3597 
 .3601 
 .3606 
 
 6 
 
 8 
 
 10 
 
 .0965 
 .0969 
 .0972 
 
 .2953 
 .2958 
 .2962 
 
 .1071 
 
 .1074 
 .1078 
 
 ,3092 
 .3097 
 .3101 
 
 .1181 
 .1185 
 .1189 
 
 .3227 
 .3232 
 .3236 
 
 .1296 
 .1300 
 .1304 
 
 .3359 
 .3363 
 .3367 
 
 .1416 
 .1120 
 .1121 
 
 .3486 
 .3490 
 ,3491 
 
 .1639 
 ,1644 
 .1518 
 
 .3609 
 -.3613 
 .3617 
 
 12 
 11 
 16 
 
 .0976 
 .0979 
 .0982 
 
 .2967 
 .2972 
 .2976 
 
 .1082 
 .1085 
 .1089 
 
 .3106 
 .3110 
 .3115 
 
 .1192 
 .1196 
 .1200 
 
 .3241 
 .3245 
 .3219 
 
 .1308 
 .1312 
 .1316 
 
 .3372 
 .3376 
 .3380 
 
 .1428 
 .1132 
 .1436 
 
 .3198 
 .3502 
 .3507 
 
 .1552 
 .1556 
 .1560 
 
 .3621 
 .3625 
 .3629 
 
 18 
 20 
 22 
 
 .0986 
 .0989 
 .0993 
 
 .2981 
 .2986 
 .2990 
 
 .1092 
 
 .loa6 
 
 .1099 
 
 .3119 
 .3124 
 .3128 
 
 .1204 
 .1207 
 .1211 
 
 .3254 
 .3258 
 .3263 
 
 .1320 
 .1323 
 .1327 
 
 .3384 
 .3389 
 .3393 
 
 .1410 
 .1144 
 .1148. 
 
 .3511 
 .3511 
 .3519 
 
 .1565 
 .1569 
 .1573 
 
 .3633 
 .3637 
 .3641 
 
 24 
 26 
 28 
 
 .0996 
 .1000 
 .1003 
 
 .2995 
 .3000 
 .8004 
 
 .1103 
 .1107 
 .1111 
 
 .3133 
 .3138 
 .3142 
 
 .1215 
 .1219 
 .1223 
 
 .3267 
 .3272 
 .3276 
 
 .1331 
 .1335 
 .1339 
 
 .3397 
 .3401 
 .3406 
 
 .145! 
 .1456 
 .1160 
 
 .3523 
 .3627 
 .3531 
 
 .1577 
 .1582 
 .1586 
 
 .3645 
 .3649 
 .3653 
 
 30 
 32 
 34 
 
 .1007 
 .1010 
 .1014 
 
 .3009 
 .3014 
 .3019 
 
 .1114 
 .1118 
 .1122 
 
 .3147 
 .3151 
 .3156 
 
 .1227 
 .1230 
 .1234 
 
 .3280 
 .3285 
 .3289 
 
 .1343 
 .1347 
 .1351 
 
 .3410 
 .3411 
 .3418 
 
 :1465 
 .1469 
 .1473 
 
 .3536 
 .3510 
 .3544 
 
 -.1590 
 .1591 
 ;1599 
 
 .e657 
 .3661 
 .3665 
 
 36 
 38 
 40 
 
 .1017 
 .1021 
 .1024 
 
 .3023 
 .3028 
 .3032 
 
 .1126 
 .1129 
 .1133 
 
 .3160 
 .3165 
 .3169 
 
 .1238 
 .1242 
 .1246 
 
 .3293 
 .3298 
 .3302 
 
 .1355 
 .1J59 
 .1363 
 
 .8423 
 .3427 
 .3131 
 
 .1477 
 
 .1481 
 .1185 
 
 .3548 
 .3552 
 .3556 
 
 .1603 
 ;1607 
 .1611 
 
 .3669 
 :3673 
 .3677 
 
 ' 42 
 44 
 46 
 
 .1028 
 .1032 
 .1035 
 
 .3037 
 .3041 
 .3046 
 
 .1136 
 .1140 
 .1144 
 
 .3174 
 .3178 
 .3183 
 
 .1249 
 .1253 
 .1257 
 
 .3307 
 .3311 
 .3315 
 
 .1367 
 .1371 
 .1376 
 
 .3435 
 .3440 
 .3441 
 
 .1489 
 .1493 
 .1498 
 
 .3560 
 .3564 
 .3568 
 
 .1616 
 .1620 
 ;1624 
 
 .3680 
 .3684 
 .3688 
 
 48 
 50 
 52 
 
 .1039 
 .1042 
 .1046 
 
 .3051 
 .3055 
 .3060 
 
 .1147 
 .1150 
 .1155 
 
 .3187 
 .3192 
 .3196 
 
 .1261 
 .1265 
 .1269 
 
 .3320 
 .3324 
 .3328 
 
 .1379 
 .1383 
 .1387 
 
 .3448 
 .3452 
 .3457 
 
 .1502 
 .1506 
 .1510 
 
 .3572 
 .3576 
 .3580 
 
 .1629 
 .1633 
 .1637 
 
 .3692 
 .3696 
 .3700 
 
 54 
 56 
 58 
 60 
 
 .1049 
 .1053 
 .1056 
 .1060 
 
 .3065 
 .3069 
 .3074 
 .3078 
 
 .1159 
 .1163 
 .1166 
 .1170 
 
 .3201 
 .3205 
 .3209 
 .3214 
 
 .1273 
 .1277 
 .1280 
 .1284 
 
 .3333 
 .3337 
 .3341 
 .3346 1 
 
 .1391 
 .1395 
 .1400 
 .1403 
 
 .3461 
 .3465 
 .3469 
 .3473 
 
 .1514 
 .1518 
 .1523 
 .15i7 
 
 .3585 
 .3589 
 .3593 
 .3597 
 
 .1641 
 .1046 
 .1660 
 .1664 
 
 .3704 
 .3708 
 .3712 
 .3716 
 
 For va 
 
 . ic +/) cos (y>o.9 (c +/); therefore we may take 
 lues of on this page ) (c + ) cos 6 = ( + f) 
 
 
 ' (c+f) sin fl varies between 0.3 (e+/) & 0.4 
 
 O+A 
 
T. 
 
 Taile for <c+fjsui0 j 
 
 ^ 
 
 s- 
 
 ±o- 
 
 IS' 
 
 Zo-\ 
 
 l-oo 
 
 ■<f 
 
 ■17 
 
 .zt 
 
 .3* 
 
 ISO 
 
 .13 
 
 ib 
 
 ■■>? 
 
 Si 
 
 o.uo 
 
 -04 
 
 »7 
 
 JO 
 
 -IS 
 
 JO" 
 
 10 20' \ <H> 
 
 A lr \ xr\ -ft 
 
 « So \ .0' u I w s y 
 
 I *_ » - > L 1 l L 
 
 ¥ ^ se. S- t V 
 
 /fl* ^5' 
 
 / -ZZ" 
 
 1 \ 1 t — -s 2 s »* ^» ^s. -j^ <i s ^. 
 
 Plate 2. -Stadia Reduction Diagram. 
 
THE STADIA.. 69 
 
 17. Diaqeam for Vebtical Distanoe.— We will employ equa- 
 tion (9,) and for the present consider only the term B$ sin 2 8. 
 Draw any line, as O B, Fig. 9, (Plate 2), and on it construct any 
 scale of equal parts to represent the observed value of the reading 
 B ; for convenience, we will assume 1,000 as the maximum value of 
 B, which will be sufficient for most cases. Through the point B, 
 B, draw another line B V, making the angle V B O about 60° ; on 
 this line lay off another scale of equal parts, making the unit of 
 this scale, say ten times, greater than that of B 0. Put B = 1,000 
 and 8=1° and compute the value of _KJ sin. 2 8 ; from the corres- 
 ponding point of the scale of B V, draw a line to 0, and mark it 1°, 
 as in the diagram. In a similar manner, draw lines corresponding 
 to 6 = 2°, 3°, 4°, etc., and such fractions of degrees as may be de- 
 sired. Complete the diagram by drawing lines, parallel to B O, 
 through the principal points of division of B V. 
 
 If the construction of the diagram, with the proportions as 
 above, be continued much beyond 5° or 6°, the intersections will 
 become too oblique for accuracy. This difficulty may be met by 
 continuing the scale of B V along V W, the angle O V W being 
 about 60°. The lines of the scale by which Fis determined will no 
 longer be parallel to the line along which B is measured. The 
 position where the line for V= 100 intersects the line V is found 
 by the diagram as already explained ; to find where the line for 
 V= 100 intersects W 0, use the equation V — Bi sin. 2 8, putting 
 Y= 100, 8 = 12° (the degree represented by W), and solve for B. 
 For the example cited B = 490.7; hence the line for V— 100 in the 
 triangle V WvausX be drawn from its intersection with FOtoa 
 point on W corresponding to B = 499.7. Parallel to the line 
 drawn as above, draw other lines through the intersections of the 
 elevation lines of the triangle B O Fwith V*. 
 
 This process maybe continued until the probable maximum 
 value of 8 has been reached, or until the diagram completes the 
 circle. 
 
 If the scale of B Fis made smaller, a greater number of de- 
 grees will be included, but the diagrams will be less accurate. For 
 convenience in using the sections V W, etc., repeat the numbering 
 of B along V 0, WO, etc. 
 
 4 This ingenious device was proposed by one of the author's students , 
 Alonzo Courtney ; Prof. Kobinson meets the difficulty by constructing separate 
 diagrams. 
 
70 THE STADIA. 
 
 Concerning the second term of (5'), (c +/) sin. 8, we notice that 
 it is always small and varies but slightly for a change of a whole 
 degree in 8; therefore calculate the values of (c +/) sin. 6, and 
 place them in the supplemental tables; as on the diagram. For 
 values of (c + /) which differ from those given, the correction may 
 easily be interpolated with ample accuracy. If the particular value 
 of (c + f ) is known, the correction may be written outside of the 
 diagram between the radii for the different degree. 
 
 To use this diagram, find on O B the number corresponding 
 to the reading of the stadia; then follow a line from this point 
 parallel to B For V W, etc., until it intersects the line drawn from 
 O, which represents the number of degrees in the observed vertical 
 angle; then follow a line parallel to B or V, etc., to the scale 
 B V W; the intersection on this scale, plus the small correction 
 (c +/) sin. 6, is the vertical co-ordinate sought. 
 
 18. Diagram fob Hobizontal Distance.— It is best to use equa- 
 tion (4'), and find the correction, B sin. 2 0, which is to be sub- 
 tracted from B. The construction of the diagram for this case is 
 similar to the previous one. The completed diagram is shown in 
 Fig. 10, (Plate 1). The correction, B sin. * 6, as found from this 
 table is to be subtracted from B, and the term (c + /) cos. 6, as 
 found from the little table on the diagram, added, to obtain the re- 
 quired horizontal distance. 
 
 These diagrams are very carefully drawn, and can be used in 
 reducing actual stadia readings. 
 
 19. Soubces of Eeboe in Stadia Wobk.— Stadia work is subject to 
 many of the errors discussed in the articles on the transit 5 and the' 
 level 6 ; the additional errors peculiar to the stadia are, 1, inclina- 
 
 / 
 tion of the rod; 2, error in the value of-; and 3, error of observa- 
 
 i 
 tion. 
 
 Inclination of Kod.— To investigate the effect of a slight in- 
 clination of the rod, let B = the reading when the rod is vertical ; 
 B' = the reading when the rod makes an angle v with the vertical. 
 
 "Engineebing News. Vol. 17, p. 305-6. 
 "Engineebihg News. Vol. 17. n. 197-9. 
 
THE STADIA. 71 
 
 Than R = B' cos. v very nearly, and the error = R — R' = R 
 
 v R 
 (1 — cos. v) = R sin. 8 — = ■ v 2 , in which v is in degrees. Notice 
 
 2 7000 
 that the error increases as the square of the inclination, which 
 shows the necessity of some means of plumbing the rod. Without 
 some means of keeping the rod vertical, this would undoubtedly be 
 the principal source of error in stadia work. On the U. S. Lake 
 Survey, an inclined support was used to steady the rod and prevent 
 its inclining to or from the instrument ; a light tripod was also 
 sometimes used for the same purpose. 
 
 / / 
 
 Value of -. If the value of -, or the corresponding unit on the rod 
 i i 
 
 is not correctly determined, the results will be incorrect, although 
 the relative position of points will be all right. The correct value 
 
 / 
 of - is easily found in the beginning, and with the methods of at- 
 i 
 
 taching the hairs described in Sec. 4 there is but little danger of its 
 changing. 
 
 Ebrobs or Observations.— The principal item under this head is 
 the inaccuracy in estimating the position of the hair on the rod ; 
 the amount of this error depends upon the size of the space on the 
 rod, corresponding to a unit on the ground, and upon the form of 
 graduation. For those rods with which the sides of the hairs are 
 observed, this error is still further increased by the uncertainty 
 due to the thickness of the hairs, and to any inequality in their 
 thickness, as magnified by the eye-piece ; this element of uncer- 
 tainty does not exist with the graduations shown in figs. 4—8. 
 
 Imperfect focusing, either of rod or hairs, is a source of error, be- 
 cause it is only when both are in focus at the same time that the 
 assumed relations exist. If the rod is not in focus, the image covers 
 too much space, which makes the distance too small. If the cross 
 hairs are not in focus, the distance will be read too small or too 
 great according as the hairs are on one side or the other of the 
 focus. If there is no parallax in the telescope, there will be no 
 error from this cause. 
 
 The indistinctness of the image due to an unsteadiness of the at. 
 mosphere produces an error hy making the image too large, and 
 
72 THE STADIA. 
 
 therefore, the distance too small. The only remedy is to wait for 
 better atmospheric conditions, 
 
 Another somewhat common error is miscounting the reading so 
 as to make errors of 10, 1000, etc., feet. These errors may be pre- 
 vented by care, or checked by double readings ; or, instead of mak- 
 ing an entirely new reading, the two halves of the visual angle may 
 be read and their sum used to check the reading of the two outside 
 hairs. 
 
 20. Limits op Precision.— With reasonable care a high degree of 
 accuracy can be attained in stadia measurement. The degree of 
 precision is dependent upon the magnifying power of the telescope, 
 the length of sight, and the ratio of the space on the rod to the cor- 
 responding space on the ground. It is sometimes claimed that 
 stadia measurements are more accurate than chaining, but from 
 the nature of the principles involved, it cannot be so for equally 
 favorable conditions ; nevertheless, under many circumstances, the 
 stadia is more accurate than chaining. 
 
 The following result seem to have been obtained with an ordin- 
 ary, engineer's transit ; they are not selected, but are all the results 
 of actual practice that could be found. In comparing results we 
 must distinguish between the error of simply finding the distance 
 by the stadia, and the final error of a series of courses, the lengths 
 of which were determined by the stadia; the latter involves the 
 error of measuring the angles as well as the distances. 
 
 To show the degree of precision obtained in measuring hori- 
 zontal distance, we have the following: Three measurements, each 
 of six distances, from 50 to 500 ft. made to test the accuracy of the* 
 stadia measurements, 7 the readings being made with targets, show 
 .an average probable error for a single sight of 1 in 4,000. 8 On the 
 U. 8. Lake Survey, three measurements of a base line, gave errors 
 >of 1 in 1000 ; 1 in 1635 and 1 in 1888. 9 
 
 Only the following can be found concerning the accuracy with 
 which vertical distances can be determined. " Courses have been 
 run one to six miles, over heights of 150 to 200 feet in which the 
 
 'Van Nostrand's Engineering Magazine. Vol. 30, p. 319. 
 "Van Nostrand's Engineering Magazine. Vol. 30, p. 476. 
 
 •Trmr Prnnlrlin Tnfit. Vol. 49. Tl. 74. 
 
 •Jour. Franklin Inst. Vol. 49, p. 74. 
 
THE STADIA. 73 
 
 final error in height ranged from to 1.5 feet, with no more than 
 ordinary care." 9 
 
 Concerning the final error of a series of courses, the lengths of 
 which were determined by the stadia, we have the following: 
 " The stadia was used for getting the topography of some densely 
 wooded timber land in the summer of 1863. The courses were so 
 run as to connect points of triangulatiori from one to four miles 
 apart. The distances from point to point along the courses, ranged 
 from 100 to 500 feet. The latitudes and departures were subse- 
 quently computed with the object of finding the error of stadia 
 measurement. The results obtained were about 1 in 800, 1 in 1,000. 
 and 1 in 1,100." 10 . On the U. S. Lake survey, in computing the co- 
 ordinates of stadia work for 1875, the average amount discovered in 
 HI lines, varying between 3,200 feet and 22,000 feet (mean 8,080 feet), 
 when compared with lines determined by triangulation or chaining, 
 was found to be 1 in 649 ; the maximum limit of error was put at 1 
 in 300. ll In a survey of Eed river, conducted by the U. S. Army 
 Engineers and extending over about 70 miles, the stadia work and 
 chaining were checked at 8 points along the line, and showed an 
 average difference of 1 in 430. ia 
 
 21. Practical Hints.— The many advantages of stadia measure- 
 ments in surveying need not be dwelt upon; they are self-evident 
 to those acquainted with the principles. The general ease and 
 quickness of telescopic measurement has always been recognized. 
 In broken country,, the stadia can be readily used where the use of 
 the train is not practicable at all. The stadia is specially applic- 
 able to topographical surveying, to the topographical work of a 
 railroad survey, and to determining lengths of sights in leveling. 
 
 It could be profitably employed in ordinary land surveying ; the 
 trouble of training chainmen, or of trusting work to inexperienced 
 chainmen, would be avoided, and the delays of chaining would be 
 saved. Instead of two chainmen only, one rodman is needed, and 
 his duties are very simple ; thus all the important work could be 
 done by the surveyor himself. This would secure greater accuracy 
 where most needed, in the measurement of the distances. It would 
 require the substitution of a telescope for the ordinary compass- 
 
 10 Jour. Franklin Inst. Vol. 49, p. 74. 
 "Finnl Keport U. S. Lake Survey, p- 34. 
 "Eeport of Chief Engineer. t T . S. A., 1873, p. 638. 
 
74 LOCAL ATTRACTION IN LAND SURVEYING. 
 
 sights, or, better the employment of a transit instead of a compass, 
 and the reading of the angles independent of the needle. 
 
 The stadia may also be used for under-ground surveying, where 
 chaining is peculiarly disagreeable and difficult; in under-ground 
 work, a box lighted on the inside, with a glass front on which the 
 graduation is painted, is substituted for the rod. 13 
 
 22. To obviate the difficulty of estimating a fraction for both 
 haiis, set one of them at the beginning of a division ; this will pro- 
 duce a little error in the vertical co-ordinate, but generally so little 
 as to be inappreciable. With long sights or small angles, it is still 
 more convenient and also sufficiently accurate, to set one of the 
 hairs upon one of the more prominent divisions ; as the even foot- 
 mark, etc. ; by remembering that each hair may be moved either of 
 two ways and moving the telescope so as to produce the least dis- 
 placement, this method of placing one of the hairs at an even di- 
 vision can frequently be used without any error, and thus greatly 
 acilitate the work. 
 
 APPENDIX. 
 
 LOCAL ATTEACTION IN LAND SURVEYING-. 
 
 It is very common to hear the remark that the common com- 
 pass could not be used because of local attraction. It is well known 
 that there are many localities in which the needle is defleoted by 
 the attraction of iron, etc. ; but the object of this article is to show 
 that no matter how much the local deflection, the common compass 
 can be made to give as accurate results as though there was no such 
 disturbing influence. 
 
 This method of using the compass seems not to be either under- 
 stood or practiced by surveyors generally; arid this, even though 
 the essential features of the method are given, rather obscurely, 
 perhaps, in a book which is very generally used, " Gillespie's Land 
 Surveying," page 143. This is the more surprising when it is known 
 that instead of using the simple compass in this very easy way, the 
 complicated, expensive, and it is believed, less accurate, solar com- 
 pass has been employed. The much maligned magnetic compass 
 is an instrument capable of much greater precision than it is gen- 
 erally credited with; with the solar compass the case is nearly re- 
 versed. 
 
 13 Jour. Franklin Inst. Vol. 65, pp. 384-7. 
 
LOCAL ATTBACTION IN LAND SUKVEYING. 
 
 75 
 
 Case I.— Wlien only the area is wanted.— Place the compass at 
 each corner of the field in succession and determine the bearing of 
 the two lines meeting in each station. Call the sight made in the 
 direction in which the surveyor goes around the field a fore-sight, 
 and that made in an opposite direction a back-sight, keep one end 
 of the box front on fore-sights and the other end front on back- 
 sights ; but if one sight of the compass consists of a slit and the 
 other of a hair, the same end must necessarily be kept next to the 
 eye, therefore read the letters from one end of the needle for fore- 
 sights, and from the opposite end for back-sights, and the degrees 
 from the same end. When all the bearings have been taken, the 
 record will be similar to the first three columns of the following 
 table, with the exception of the small figures written above each 
 bearing which will be explained presently. 
 
 Station. 
 
 Back Sight. 
 
 Fore Sight. 
 
 Correction. 
 
 JV 
 
 S.78 00 E 
 
 N 16 05' W 
 
 
 
 Q 
 
 „1G 5 ,„ 
 
 JV 16 45 W 
 
 „ SO 30 
 
 -IV 81 10 W 
 
 40' F 
 
 s 
 
 .... 8 3 
 
 JV80 05 W 
 
 „ 8 B 1 __ 
 
 S46 35 W 
 
 25' B 
 
 u 
 
 c *6 10 
 
 S46 35 
 
 „ 83 00 _ 
 
 S 82 35 -E 
 
 25' B 
 
 V 
 
 o 83 00 t, 
 
 S 82 25 -B 
 
 St! II E 
 
 35' B 
 
 Owing to changes in the declination, it is not probable that 
 every fore-sight will agree with its corresponding back-sight. How- 
 ever, if the two bearings at any station have been taken in a short 
 time of each other, the notes may be corrected so as to wholly 
 eliminate the effect of any change of the needle. 
 
 Since the assumption is that only the area is required, it is im- 
 material where we commence to correct the angles. Therefore we 
 may assume the bearings taken at any station, say Jv, sre correct. 
 The back-sight from the next station, say Q, (it is immaterial 
 whether the angles are connected in the order in which they were 
 surveyed or not), differs from the fore-sight from iVby 40', therefore 
 the bearings at Q are in error 40', to correct which we may conceive 
 that the needle should be moved 40' in a direction from the north 
 
76 LOCAL ATTRACTION IN LAND SURVEYING. 
 
 toward the east, i. e., in the direction in which the hands of a watch 
 move. It makes no difference, in this matter, whether we consider 
 N, E, Sand W in their true relations or in their reversed position, 
 as given by the face of the compass. We will assume them to be 
 in their true relations ; and briefly say that the correction at Q is ; 
 40' forward. This is the correction which is to be applied to the 
 fore-sight at Q, and is written in the column headed " correction." 
 The corrected fore-sight from Q is 80° 30' ; the back-sight from 8 
 should agree with this, but there is a difference of 25'. Therefore 
 the correction at S is 25' backward ; and the corrected fore-sight is 
 46° 10'. The corrections and corrected angles for the other stations 
 are formed in a similar manner, and the corrected bearings written 
 above the observed value, as in the table. 
 
 The agreement of the last corrected fore-sight with the first 
 corrected back-sight is a valuable check on the accuracy of the 
 work, and as a rule, this difference should not exceed five or ten 
 minutes. 
 
 Case II.— When the area and also the true bearings are desired.— 
 The case requires one of two things, either 1 : That several succes- 
 sive back-sights shall agree with their corresponding fore-sight 
 before either have been connected ; or 2, that there shall be a true 
 meridian which can be connected with a corner of the field by 
 lines whose bearings are to be found as those of the boundary lines. 
 In the first instance, it may safely be assumed that, since a num- 
 ber of the fore-sights and back-sights agree, there- has been no 
 change in the middle between those stations; therefore the ob- 
 served bearings may be assumed to be the true bearings. Having 
 some of the correct bearings, if there are aDy stations at which the 
 back-sight and fore-sights do not agree, they may be corrected as 
 in Case I. 
 
 If the declination is not set off on the compass, it may be ap- 
 plied as a correction to those stations at which the back and fore- 
 sights corresponded as above.and the correction carried on to those 
 stations at which a difference was found. 
 
 In the second instance as above, the true bearings of the 
 several lines can be found in succession beginning at the meridian. 
 The following example will illustrate this method : Z is not a corner 
 
LOCAL ATTRACTION IN LAND SURVEYING. 77 
 
 of the field but a point on a known meridian ; the line Z Q is run to 
 determine the declination at Q, a corner of the field. All of the re- 
 maining stations are corners of the field. 
 
 Stations. 
 
 Back Sights. 
 
 Fore Sights. 
 
 Corrections. 
 
 Z 
 
 NS° 50 W 
 
 , T 1 3 10 
 JV 69 20 £ 
 
 3° 50' F 
 
 Q 
 
 , T 7 3 1 
 
 Nm 15 E 
 
 
 3° 55 F 
 
 Q 
 
 , r 14 40 „ 
 
 N 10 45 E 
 
 93 40 
 S89 45 TF 
 
 
 T 
 
 c 93 40 .,_. 
 
 S89 35 TF 
 
 S39 00 W 
 
 4° 05 F 
 
 TT 
 
 r, 4 3 5 Trr 
 
 S39 10 T^ 
 
 „14 45 
 NW 40 £ 
 
 3° 55 F 
 
 Notice that the first two lines of the above record are prelimin- 
 ary to the survey of the field, and are required only to find the de- 
 clination at Q. Having found the declination or correction at Q, 
 the bearings are corrected as in the previous case. The back-sight 
 from Q to U5' from the fore-sight at U to Q, which shows that 
 there was an error or inaccuracy of 5' in reading the angles. 
 
 Undoubtedly the method of correcting with a meridian as just 
 explained, is more exact than the previous method, although the 
 latter is more convenient and shorter and is the one which will gen- 
 erally be used in practice. However if the bearings show local 
 attraction at a number of stations the first method of Case II 
 cannot be applied while the second will give strictly correct results 
 whatever the number of stations at which local attraction exists or 
 whatever its amount. 
 
 The back-sights should always be taken, although it may be 
 nearly certain that no local attraction exists, for this is a check 
 against possible local deflection of the needle, and also against 
 errors in reading the needle. 
 
 Soueces of Eeeob.— The error of compass work may be grouped 
 in one of the following classes : 1st. Instrumental errors. 2nd. In- 
 strument or flag-pole not being set at the right place. 3rd. Errors 
 of chaining. The first have been discussed already, f The second 
 
 Engineering News, Sept. 15, 1885. 
 
78 LOCAL ATTKACTION IN LAND SUBVEYING. 
 
 are small and easily remedied. The flag-pole should be set vertical 
 but, as a precaution, sight as low on it as possible. 
 
 The instrumental errors are due either to lack of adjustment, 
 or to errors in the practice. In using any instrument, it is a good 
 rule to adjust it carefully in all particulars and then use it in such 
 a way as to eliminate any residual errors of adjustment, or, in other 
 -words, adjust it carefully and then use it as though it were not 
 adjusted. In using a compass, guard against errors of coincidence 
 of magnetic and geometrical axes of the needle, and also of straight- 
 ness of needle, by reading always the same end of the needle. 
 Sighting nearly horizontal will reduce the possibility of error due 
 to lack of perfectness in the adjustment of the levels. 
 
 To guard against the possibility (1) that the magnetic axis of the 
 needle may not coincide with its geometric axis, (2) that the zero 
 of the vernier may not coincide with the line of sight, and (3) errors 
 in straightness of the needle, determine the declination by setting 
 the compass upon a true meridian, sighting along it, and then 
 moving the vernier until the needle reads zero. This also provides 
 against errors in charts or tables giving the declination, and also 
 against changes in the declination since the chart was made. This 
 observation should be made about 10 a. m., or 6 p. m., as then the 
 effect of the daily variation is nearly zero. 
 
 It is very important that the three sources of error just men- 
 tioned should be eliminated. For one or the other or all of these 
 reasons the bearing of a line as read by several instruments at the 
 same time and place, and by the same person will often differ 10 
 to 15 minutes. For an example of five instruments which differed 
 20 minutes, see the Fifth Annual Eeport of the Ohio Society of Sur- 
 veyors and Civil Engineers, page 71 ; the same page gives an account 
 of four instruments which differed 31 minutes. Notice that these 
 instruments were read at the same time and place, but by different 
 men. On page 74 of the above report, is an account of six needles 
 made specially for this purpose, as nearly alike as possible, read un- 
 der identically the same conditions in every respect, which differed 
 10 minutes. In all work in which the location of a line is to be deter- 
 mined by its recorded bearing, the above precaution is exceedingly 
 important. 
 
 The writer believes that we shall never be able to follow all the 
 excentricities of the variation of the declination, but he also 
 
LOCAL ATTEACTION IN LAND SUEVEYING. 79 
 
 believes that the method of using the compass here advocated, 
 together with the precautions mentioned above, is practically inde- 
 pendent of all such variations. 
 
 The most common error in the practice is reading N. for. S., 
 etc., and 28° for 32°, etc. The only check or remedy is carefulness. 
 After the needle has come to rest, it should be tapped gently to 
 destroy the effect of any adhesion to the pivot. 
 
 Limits of Pbecision.— The error of taking a bearing is made up 
 of the error of sighting, of sluggishness of the needle, and error of 
 reading the needle. The last is the greatest. With a compass 
 graduated to half a degree, the angles can be estimated to the near- 
 est rive minutes, and the maximum error of a bearing should 
 not exceed ten minutes. The average probable error of a bearing 
 for a class of ten, as deduced from the error in closing the angles 
 around a field, was 1.64 minutes. That the average probable error 
 is 1.6 minutes show that bearings can be read certainly tc the near- 
 est five minutes. For the purpose of this record, the same men 
 read the bearing of a flag-pole at one, two and three hundred 
 paces, with probable errors of 2f, 3J and 3£ minutes respectively, 
 the sun and wind being in the observer's face while sighting. This 
 would seem to indicate that, under favorable circumstances, the 
 probable error of a bearing should be less than two minutes, and 
 under unfavorable conditions it may be more than three minutes. 
 The angular error of closing a field is equal to the square root of 
 twice the. number of the sides, multiplied bv the error of a single 
 bearing. 
 
 The error of an area found by compass surveying is made up of 
 the error of chaining, of reading the bearings, and of computations. 
 Land surveyors in actual practice, find that the chaining as ordi- 
 narily done is much the greatest source of error. "With proper 
 care the error of chaining should be much less than the error of 
 the beaiings. Since one in fifty-seven corresponds to 1°, the 
 above maximum error of 10 minutes corresponds to an error of one 
 in three hundred and forty-four, and the probable error of 1.64 
 corresponds to one in two thousand, which is greater than that of 
 chaining under the same conditions. Since the computations are 
 self-checking at every step, there is little probability of any 
 blunder in this part of the work ;-and as the work may be carried 
 
80 LOCAL ATTRACTION IN LAND SURVEYING. 
 
 to any desired number of decimals, the inaccuracies of the compu- 
 tations can be made much less than those of the field work; 
 therefore we conclude that the accuracy of the area depends 
 mainly upon the accuracy of reading the needle. An area found 
 by compass surveying should be true to within three or four 
 thousand. 
 
 Balancing the Woek.— "When the angles are measured as 
 above, an answer can be given to the question, which is frequently 
 asked, viz : — How great a difference between the sums of the + and 
 — latitude and departure is admissible ? 
 
 Let C =the linear error of closing due to the chaining; C is 
 equal to the perimeter of the field (=P) divided by the distance 
 = d) in which the error of chaining s a unit, that is 
 
 P 
 
 c = — 
 a 
 
 Let .4= the linear error due to the measurement of the angles. 
 The difference between the last fore-sight and the first back-sight 
 is the angular error of measuring the angles ; we will assume this 
 difference to be in minutes and represent it by a. It cannot be 
 known how this error occurred ; whether it occurred all in one 
 sight, whether it occurred equally among the sides, or among the 
 sides in proportion to their length. But as the last is most probable, 
 and also as it gives the largest linear error in closing, we will 
 assume the error a to have occurred among the sides in proportion 
 to their lengths. Hence to reduce a to its linear equivalent, we 
 must multiply it by the length of the perimeter (= P) and divide it 
 it by the distance at which a unit subtends an angle of one minute, 
 or 
 
 P SaP 
 
 A = a = nearly. 
 
 3438 10,000 
 
 Let E = the total error, i. e. the error due to chaining and meas- 
 uring the angles. Notice that the actual error, E, is the hypothe- 
 nuse of a right angled triangle of which the differences of the lati- 
 tudes and departures are the other sides. Hence, if L = the 
 difference of the + an d — latitu des, and D = equals the same for 
 the departures, E= ^/ D 2 + L 2 . 
 
LOCAL ATTRACTION 1 1ST LAND SURVEYING. 81 
 
 By the theory of probabilities, we know that 
 
 X*~SA» + G»= \/ (-)+(—) =Pi/(- + ^ ") nearly. 
 ^d' \3438/ ' \d z 12000000' 
 
 Equating these two values of E, we get 
 
 /l a? v 
 
 D 2 + i 2 = PM - + ) 
 
 \i a 12000000 ' 
 
 We may simplify the matter a little further by finding a relation 
 between D and L. Assume that the sum of the latitude = n times 
 the sum of the departures, n is easily determined from the compu- 
 tations, or it can be estimated in the field or from the plot with 
 sufficient accuracy. From the theory of probabilities, we know 
 that L = S n D; then the above formula becomes 
 
 D* + £4 = (l + n ) D* = P 2 i—-\ ) 
 
 M 1 12000000 ' 
 
 To illustrate the method of applying this let us assume that in 
 surveying a field whose perimeter is ten chains the difference 
 between the first back-sight and the last fore-sight was 10' ; we will 
 also assume that the conditions were such that we might expect an 
 error of 1 in 2000 in chaining. The above formula then becomes. 
 
 r/H' 10* 1 1 
 
 D* + £* == io* ( ) -\ = 1 = .00086, or 
 
 L ^2000/ , 12000000 J 40000 1200 
 1 a 2 
 
 (1 + n) B* = P' (— -| ) = .00086 chains." 
 
 a 2 12000000 
 
 If the field is twice as long north and south as east and west, 
 n = 2, and hence D = .017 chains or 1.7 links. This shows that the 
 error in the departures should be about 1.7 links. The error in the 
 latitude should be 1.7 -/n = \.1«/% = 2.4 links. Results much 
 greater than these show an error in the work other than the usual 
 inaccuracy. 
 
 Finally, by reversing the problem, knowing the error of closing 
 the angle, and the errors in balancing the latitudes and departures, 
 we may reverse the above formulae and compute the error of 
 chaining. 
 
82 ACCURATE CHAINING. 
 
 The writer does not wish to be understood as advocating that 
 the compass is the best instrument for land surveying, or that it is 
 sufficiently exact; he only wishes to call attention to a better 
 method of using an instrument that is not without merits, but 
 which can never become an instrument of the highest precision. 
 
 ACCURATE CHAINING. 
 
 Chaining is a very simple operation, and doubtless, everyone 
 thinks he knows all about it. Many there are who will claim that 
 they can do absolutely accurate chaining. But the results of inac- 
 curate chaining in the past is a continual annoyance to the present 
 generation of surveyors and engineers, and the day of unnecessarily 
 inaccurate work is probably not yet passed. The following remarks 
 are intended to apply specially to land surveying. 
 
 It goes without saying that the means used in the past have 
 been rude, and the results correspondingly inaccurate; but, owing 
 to the increase in value of land, and the reflex action of the general 
 advancement in other directions, there seems to be a growing 
 desire for increased accuracy in the methods of land surveying. To 
 assist in this laudable reform is the purpose of this article, although 
 increased accuracy in chaining is but one of the steps leading to 
 the desired end. 
 
 Before considering the several errors to which chaining is 
 liable, it will be well to notice that, in all measuring operations, the 
 observer should carefully distinguish between two classes of errors, 
 viz : — compensating errors, or those which are as likely to be plus as 
 minus, and tend to balance each other ; and cumulative errors, or 
 those which always have the same sign, and affect the final result 
 in the same way. This distinction is very important. The observer 
 should avoid errors which usually occur in a single direction, but 
 he need not always take the greatest care to avoid errors which are 
 as liable to be negative as positive. An apparently inappreciable 
 but cumulative error may, in the course of a series of observations, 
 amount to more than a much larger but compensating error. The 
 effect of compensating errors is reduced nearly to zero simply by 
 multiplying the number of observations: but cumulative errors 
 should be avoided entirely, or observations made by which they 
 may be corrected. 
 
ACCURATE CHAINING. 83 
 
 The uncertainty in the length of a line due to compensating or 
 accidental errors, varies as the square root of the number of units 
 in the line, while the effect of cumulative, or constant errors varies 
 directly as the length. The whole number of cumulative errors re- 
 main uncorrected, while only the square root of the compensating 
 errors are uncompensated. For example, if the chain is 0.1 of a 
 link too long, a line 25 chains long will be recorded 2.5 links too 
 short ; but if the pin is sometimes set 0.1 of a link beyond the end of 
 the chain, and sometimes the same amount behind, the final error 
 at the end of the line, due to this error is 0.1^25 or only 0.5 link. 
 
 If the head chainman has a fixed habit of always setting the 
 pin beyond the end of the chain, then this becomes a cumulative 
 error, and varies as the distance. This illustrates that in the pro- 
 secution of any work, it is desirable that the operator should be 
 cognizant of the nature and importance of every source of error 
 The more thoroughand complete his knowledge in this respect, the 
 more readily and accurately will be able to decide what source of 
 error may be wholly neglected, what may be partially provided 
 against, and what must be carefully avoided or eliminated. This 
 knowledge is conducive of both greater accuracy and of economy 
 of time and effort, for the observer then knows what might other- 
 wise have been attended to with considerable care, may be neglec- 
 ted, thus leaving him free to give all his attention to the weakest 
 link in the chain of observations. It enables the observer to cor. 
 rectly proportion his pains to the degree of precision required. A 
 good observer is one who is able to take just care enough to attain 
 the desired accuracy, without wasting time and energy in uselessly 
 perfecting certain parts of the work. He must be able to discover 
 the relative accuracy required in different parts of a complete ob- 
 servation. All this calls for a clear understanding of the causes of 
 error, and the ability to reason out their effect upon the result. 
 
 Sources of Erbor, — The sources of error in chaining are : 1, in- 
 correct length of chain ; 2, kinking of the chain and bending of the 
 links; 3, the chain not being drawn straight; i, unequal tension of 
 the chain ; 5, expansion and contraction with changes of tempera- 
 ture ; 6, errors of lining the fore chainman ; 7, not placing the pin 
 at the end of the chain ; 8, drawing the pin by hanging the back 
 handle over it; and 9, such errors as miscounting tallies, or chains, 
 counting from wrong end of chain, making a mistake of 10 in the 
 
84 ACCUKATE CHAINING. 
 
 number of links, reading 18 for 22, 37 for 43, etc. This last class of 
 errors are much too common, but can be obviated by care and 
 thoughtfulness. 
 
 1. This is a constant or cumulative error ; when one is desirous 
 of attaining the last degree of accuracy, this is the most difficult 
 error to eliminate ; even for the degree of accuracy attained in land 
 surveying, it is an important element. In some classes of work, 
 relative accuracy is sufficient, in which case an arbitrary standard 
 may be chosen. In the location of permanent land lines, nothing 
 less than absolute accuracy should be aimed at. Probably the only 
 way to reach a reasonable degree of accuracy is to have the length 
 of a number of standards determined by some competent authority, 
 as the coast survey or an authorized representative. 
 
 Some recent experience of the writer will illustrate the differ- 
 ence which exists between supposed true standards. He had oc- 
 casion to compare a 100 foot steel tape just from the shop of a repu- 
 table manufacturer, which was "guaranteed to be true to U. S. 
 standard," with a 20 foot pole said to have been pronounced correct 
 by the "U. 8. A. engineers," and also with a standard derived, cer- 
 tainly with inappreciable error, from two 2 ft. steel rules made by 
 the best tool makers in the U. S. The errors of the intercompari- 
 sons were inappreciable. The tape was 0.95 of an inch short by the 
 first, by the second it was 0.45 of an inch short, A subsequent com- 
 parison at the office of the Mississippi Eiver commission shows the 
 tape to be 0.256 + 0.07 of an inch short. The above differences are 
 practically independent of temperature correction. 
 
 After the chain has once been adjusted to the standard, it 
 should be frequently tested. The common chain has 600 wearing 
 surfaces, and if each wears only 0.1 of an inch, the length is in- 
 creased 6 inches. Mud and ice in the joints have a similar effect in 
 the opposite direction. A tempered steel chain with brazed links, 
 lengthened half an inch by chaining 70 miles. It is nothing uncom- 
 mon to find chains differing 1, 2 or even 3 inches. 
 
 2. The simplest remedy for kinking of the chain and bending 
 of the links is to use a steel tape or wire. A tape or wire obviates 
 the effect of wearing of the links of a chain, as well as having some 
 other advantages. 
 
ACCUEATE CHAINING. 85 
 
 A steel wire makes an excellent substitute for a chain. The 
 end of the unit may be indicated by a small tube or rod with a hole 
 through it fastened near each end. The distance may count from 
 corresponding ends of the tube or rod or from a V-shaped groove 
 around the rod ; for accurate work the former is used, the end being 
 marked on the top of a stake or a board carried for that purpose ; 
 for less accurate work, the pin is to be set in the groove. There 
 are several advantages in this way of putting the pin, which will 
 appear on reflection. A handle can be formed by passing the wire 
 through a piece of wood to protect the hand, and then forming it 
 into a loop. The wire should be steel, at least J- of an inch in dia- 
 meter. The whole can be made in any shop at a trifling cost. 
 
 3. The time required to get the chain straight-between pins can 
 be greatly lessened by the fore chainman being careful to walk 
 on line. The error is cumulative. 
 
 It is sometimes advocated that, since the chain is seldom, if 
 ever, perfectly straight, it should be made a little longer than the 
 standard, so that the full length of the standard may be laid off 
 each time. The instructions issued by U. S. Land Office to Survey- 
 ors General states (report for 1880, page 20, for the first (?) time that 
 "the 66 feet chain shall be 66.06 feet," for the above reason. The 
 French have, or at least had (Gillespie's Land Surveying, page 18, 
 foot-note) a similar practice, the addition being from 1 in 1000 to 1 
 in 2000. The remedy is at best, very questionable. 
 
 4. The effect of a difference in the pull upon the chain or tape 
 is small, and varies as the length of the tape, inversely as the cross- 
 section, and is less for steel than iron. With a steel wire J in dia- 
 meter a difference of 10 pounds will make a difference of one 
 twenty-fifth of an inch. For a chain with bronzed links it would 
 be more owing to the flattening of the rings. It could be deter- 
 mined by trial with a spring balance. It is generally compensa- 
 ting. 
 
 5. Only with the extremes of temperature will the expansion 
 and contraction of the tape be appreciable in ordinary work. In 
 accurate work it is one of the chief sources of error, and one very 
 difficult to determine. The temperature of the chain, when the sun 
 is shining, is often quite a good deal higher than that of the atmo- 
 sphere. For a 100 feet tape, a change of 30° makes a difference in 
 
86 ACCUKATE CHAINING. 
 
 length of one-fourth of an inch. Generally this error is cumula- 
 tive. 
 
 6. The error due to not setting the fore pin exactly in line is 
 generally overestimated in proportion to the other errors. An un- 
 due amount of time is therefore given to lining in the fore chain- 
 man. With a 66 feet chain, if the pin is out of line 1 foot, the error is 
 A of an inch ; the error varies inversely as the length of the chain. 
 It is cumulative. There is no reason why a wire 200 or even 300 feet 
 long should not be used for most of the work where a 66 feet chain 
 is now employed ; such a wire would be both more accurate and 
 more speedy. 
 
 Vertical inequalities in the ground produce errors similar to 
 those in aligning the fore ehainman, and unfortunately in many 
 cases this element limits the degree of accuracy attainable. It is cu- 
 mulative. It can only be obviated by suspending the tape or wire 
 tween fixed supports, which is complicated and expensive. Sup- 
 porting the tape by hand and plumbing down is of very little ad- 
 vantage. 
 
 7. Many chainmen hold the pin, while setting it, in such a man- 
 ner that the point enters the ground considerably in front of the 
 end of the chain ; when> the rear end is brought forward, it is laid 
 on the ground, thus introducing considerable error. The remedy 
 is obvious. 
 
 The very general practice of placing the forward handle 
 against one side of the pin, and the back one against the other side, 
 can not be considered very precise. The same side of the pin 
 should be used both times ; the length of the chain is then from 
 outside of one handle to the inside of the other. 
 
 8. The back handle should not be dropped over the pin until 
 the chain is in position preparatory to lay off the distance, or some 
 device similar to the V-shaped groove, described above, should be 
 used. 
 
 Limits of Precision. — It is very desirable that each surveyor 
 should know the uncertainty of his ordinary work , if this could be 
 stated for each method of measuring and for the different kinds of 
 ground it would be very instructive. Each surveyor should learn by 
 repeated trials the amount of labor required to attain a given de- 
 gree of accuracy. Everybody knows that the labor required in- 
 
ACCURATE CHAINING. 87 
 
 creases more rapidly than the degree of precision attained ; the 
 more accurate the work the greater the differencr. 
 
 The method of least squares shows that in a series of observa- 
 tions affected only by accidental errors, the final error varies as the 
 square root of the number of observations. Now, many of the 
 errors of chaining are cumulative, that is, conspired to make the 
 recorded length too long; hence the method of least squares is not 
 applicable, although it is sometimes so applied. If the error is as- 
 sumed to vary as the square root of the distance, i.e. as the number 
 of times the unit is applied, other things being equal, the longer 
 lines will show apparently the more accurate work ; if it is taken 
 proportional to the distance, the longer lines will appear less accu- 
 rate as between two lines measured under the same conditions, the 
 errors should vary as the square root of the distance. That is, if 
 the difference between two measures is multiplied in a quarter of a 
 mile, it should be twice as much in one mile. 
 
 A few words are needed about the difference between real and 
 apparent errors. If a line is twice measured with the same chain 
 by the same men under the same conditions, the difference between 
 the two measurements represents the difference between the acci- 
 dental errors each time, and does not show the real error in the 
 observed length. Any constant or cumulative error, as an incor- 
 rect length of the chain, would be the same in each. This distinc- 
 tion is very important in discussing the error of any series of 
 observations. 
 
 The following data is given to assist the surveyor in deciding 
 what he will call accurate work. Unfortunately, there is very little 
 on this subject to be found in engineering literature. If any one 
 having reliable data on this subject will publish it with full details, 
 it will afford an opportunity for some instructive comparisons 
 which may ultimately lead to an increased precision. 
 
 Burt (Key to the Solar compass, page 35) found that " the aver- 
 age error in surveying the public lands to be 1 in 500, and some- 
 times 1 in 220 as between two sets of chainmen ; under the most 
 favorable conditions it was 1 in 1600. 
 
 The writer's students, in ordinary class work in land surveying, 
 measured lines from 200 to 1000 feet long, with a 66-foot chain, and 
 repeated with a difference of 1 in 15,000 to 1 in 20,000 for the same 
 
88 ACCURATE CHAINING. 
 
 men ; for different men on different days with different chains 
 (compared however with the same standard), the maximum differ- 
 ence was 1 in 800 to 1 in 1000, the average error being 1 in 3000 to 1 
 in 4000. The ground was favorable, but the work was done with 
 the ordinary expedition of actual practice. Seventy miles of the 
 finished track of the Illinois Central R. R., was measured in 1876 
 with a 100 foot steel chain, and re-measured in 1885 with a 100-foot 
 steel tape, with a difference, after correcting for the difference of 
 standard and applying a correction for wear of the chain, on the 
 average for the different miles of 1 in 2,360. 
 
 On the Atchison, Topeka and Sante Fe R. R„ through Western 
 Kansas, the difference between the preliminary survey and the 
 government land survey averaged about 1 in 1000 ; the difference 
 between the preliminary survey and the location was about 1 
 in 2500. 
 
 The U. S. A. engineers in remeasuring the Union & Central 
 Pacific R. R., found the error of their own work to average 1 in 
 23,600, the maximum being 1 in 10,000, as determined by retracing 
 distances varying from 2000 to 26,000 feet (See House Ex. Doc, No. 
 37, 2nd session, 44th Cong., pp. 10, 14, 21, 37, after correcting errors of 
 transcription). 
 
 On the U. S. Lake Survey, (See Rept. of Ohf. Eng. U. S. A., 1876, 
 Part III, page 9) base lines for topographical surveys were made 
 with a chain 20 meters long, differing from the ordinary chains 
 only in being made of heavier wire and having links 20 inches 
 long ; in 15 lines, the maximum error was 1 in 5700, the mean being 
 1 in 17,500, and the minimum 1 in 41,100. 
 
 In connection with the U. S. Lake Survey work, a line 12 miles 
 long was measured on the railroad track with a wire 150 feet long, 
 at the rate of about 10 miles per day, with a difference between two 
 measurements of 1 in 32,000. 
 
 The U. S. Coast Survey Report 1882, page 191, contains an 
 account of the preliminary measurement of two base lines with an 
 iron wire £ of an inch in diameter, 200 feet long, in which the differ- 
 ence, as compared with the measurement of the geodetic base 
 apparatus, was 1 in 30,000 and 1 in 28,000. 
 
89 
 
 TO DETERMINE A TRUE MERIDIAN. 
 
 In connection with the article on " Local Attraction in Land Sur- 
 veying" in Engineering News of Oct. 31, 1885, the writer has re- 
 ceived several requests for his opinion as to the best method of 
 determining a true meridian, and as in the past he has occasion- 
 ally received similar requests, he therefore ventures to offer the 
 following : 
 
 Polaris and Mizar Method.— For this time of year (December) 
 this is about the most convenient method, and is sufficient accurate 
 for compass work. The principle of the method is the same as 
 that described in the text-books as the Alioth and Polaris Method. 
 Mizar is the second star, counting from the fore end, in the handle 
 of the Great Dipper ; Alioth is the third one ; Polaris is the pole 
 star. "When Polaris and Mizar are in a vertical line, they are also 
 very nearly on a meridian, (see Fig. 1.) 
 
 POLARIS 
 + POLE 
 
 To make the observation, suspend a plumb 
 line, say 20 feet long, about the same distance 
 north of a board fixed horizontally near the 
 ground in an east and west direction ; let the 
 plumb bob swing in a pail of water to check 
 the vibrations ; hang a lantern overhead and 
 behind to make the line visible. Watch and 
 wait until Polaris and Mizar 
 aie exactly covered by theplumb 
 */ line ; see Figure 1. ; when the 
 
 ,v stars are in the same vertical 
 
 plane, mark a point on the 
 board in range with the stars and 
 plumb line. On the first of Jan- 
 uary, 1886, this line will lie about 
 2.7 seconds west of the true me- 
 
 ridian: the line will move west 
 
 horizon about 26" per year for several 
 
 years. The whole apparatus 
 ^rig.i may be left undisturbed until 
 
 morning, when the line can be permanently marked ; or it may 
 be marked at the time of observation by having an assistant hold 
 
 x? 
 
 "UAH 
 
 
90 
 
 TO DETERMINE A TRUE MERIDIAN. 
 
 u 
 
 POUE 
 POLARIS 
 
 a light, seen through a small aperture in a board, which is moved 
 according to signals until it is in line with the plumb line and the 
 point marked on the board. 
 
 On the 1st of January, Polaris and Mizar are in the same verti- 
 cal about 6 :30 p. m. ; the time for any day before that can be found 
 by subtracting four minutes for each day, and for any day after 
 January 1st, add a like amount. In the summer and early fall 
 these stars come into the required position at an inconvenient 
 hour, and can not be used at all in the spring on account of coming 
 in the desired position in daytime. At these time the same method 
 may be applied in observing a different pair, Polaris and 5 Cassio- 
 peia. (See Pig. 2.) 
 
 The constellation Cassiopeia is on the opposite side of the pole 
 from the Great Dipper ; it is on the meridian under the pole about 
 6.30 June 1st, and four minutes earlier each day later. The rela- 
 tive position of the stars at the time of 
 observation is shown in Fig. 2. The 
 line determined by observing these 
 stars June 1, 1886, will be within 1" or 
 2" of a true meridian and will move 
 east about 25" per year. 
 
 Neither of these pairs can be ob- 
 served from June 1st to August 1st, and ■ 
 from the latter date until November 
 1st at inconvenient hours ; at such 
 times one of the following methods 
 must be used : 
 
 (In passing, it may be well to cau- 
 tion the surveyor against following the 
 text- books in the matter of the accu- 
 racy of the Polaris-Alioth method ; it 
 is sometimes stated that when Polaris 
 and Alioth are in the same vertical, 
 they are " very nearly in a meridian." 
 and it is also stated more accurately 
 that " Polaris is precisely on a merid- 
 ian seventeen minutes after the two 
 are in the same vertical." The facts 
 
 \ 
 
 
 -y 
 
 Fig. 2. 
 
 are that the two stars were 12' 20" east of a true meridian January 1, 
 
TO DETERMINE A TRUE MERIDIAN. 
 
 91 
 
 1886, and Polaris came to a true meridian about twenty-eight min- 
 utes thereafter ; these quantities increase from year to year.) 
 
 Method Br Equal Shadows of the Sun: "On the south side 
 
 of any level surface, erect an up- 
 right staff, shown in horizontal 
 projection, at S, Fig. 3. Two or 
 three hours before noon, mark 
 the extremity, A, of its shadow. 
 Describe an arc of a circle with 
 S, the foot of the staff, for a cen- 
 ter, and SA, the distance to the 
 extremity of the shadow, for a 
 radius. About as many hours 
 after noon as it had been before 
 noon when the first mark was 
 made, watch for the moment when the end of the shadow touches 
 the arc at another point, B. Bisect the arc AB at N. Draw SN, 
 and it will be the meridian." 
 
 " For greater accuracy, describe several arcs before hand, mark 
 the point in which each of them is touched by the shadow, bisect 
 each and adopt the average of all. The shadow will be better de- 
 fined if a piece of tin with a hole through it be placed at the top of 
 the staff, as a bright spot will thus be substituted for the less defi- 
 nite shadow; nor need the staff be vertical, if from its summit a 
 plumb line tie dropped to the ground, and the point which this 
 strikes be adopted as the center of the arcs. " 
 
 This method is strictly correct, June 21st and December 21st ; 
 on March 21st, if the observations are made within two hours of 
 noon, the line thus determined is about 2|' west of a true meridian, 
 on September 21st, same amount to the east, and proportional for 
 intermediate dates. 
 
 This is a very simple method, can be applied at a convenient 
 hour, and with moderate care will give results sufficiently accurate 
 for the best compass work. Experience of the writer's students 
 shows that a meridian can be determined within 2 minutes by mak- 
 ing three or four pairs of observations and taking the mean. 
 
 Method by Elongation of Polaris.— The Pole Star is about 1 
 degree 20 minutes from the pole of the heavens, and describes a 
 circle about it once in a little less than 24 hours. The star then 
 apparently moves from the meridian for nearly 6 hours, and then 
 
92 TO DETERMINE A TRUE MERIDIAN. 
 
 returns to the meridian ; when at its greatest angular distance from 
 the meridian, east or west, it is said to be at its greatest eastern or 
 western elongation. One of the very best methods of determining 
 the meridian is by observing Polaris at elongation ; when the star 
 is in this position it seems only to move up and down, and neither 
 to the right nor' left, consequently the conditions are such as not 
 only to allow an accurate observation butalsoto give time to repeat 
 the observation, 
 
 The angle, or azimuth of elongation will vary with the latitude, 
 and also with the year, owing to a change in the position of the 
 star. Within the United States it varies from tj degrees to 2 
 degrees. Tables are often given which show the azimuth of elonga- 
 tion, but they are too voluminous to be quoted here ; besides it 
 takes only a moment to compute it. The formula is 
 
 sine of s Polar distance 
 
 sine of azimuth = 
 
 cosine of latitude 
 
 The polar distance of Polaris was 1 degree 17 minutes 57 seconds 
 for January i, 1886, and decreases 19 seconds per year. A change of 
 20 seconds in the polar distance of Polaris will make only about O.i 
 seconds in the angle of elongation, and a difference of 1 degree in 
 the latitude in the southern part of the United States will make 
 only about 0.8 seconds difference in the azimuth, and in the extreme 
 northern part about 2.4 seconds ; therefore it will not be difficult to 
 compute the azimuth of elongation as accurately as even a transit 
 will read. 
 
 The Polar distance of Polaris for any year can be determined 
 sufficiently accurate for many years to come by applying the 
 above annual variation ; or it could doubtless be obtained by sendj 
 iDg an addressed postal card to the Naval Observatory, Washing- 
 ton, D. O. 
 
 To make the observation, arrange the plumb line and other 
 apparatus as in the Alioth-Mizar Method ; a little before the time 
 of elongation, mark a point by a pin, knife, etc., in line with 
 Polaris and the plumb line. If the star departs from the plumb 
 line move the point in the opposite way, and keep moving one way 
 as the star goes the other, until the s.tar attains its greatest elonga- 
 tion, when it will continue behind the plumb-line for several min- 
 utes ; then it will recede from the line in the direction contrary to 
 its motion before its becomes stationary. 
 
Fred'k W. Devoe. Jas. F. Drummond. J. Beaver Page 
 
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