ENGINEERS' 
 
 SURVEYING INSTRUMENTS, 
 
 THEIR CONSTRUCTION, ADJUSTMENT, 
 AND USE. 
 
 IRA O. BAKER, C.E., 
 
 PROFESSOR OF CIVIL ENGINEERING, UNIVERSITY OF ILLINOIS; 
 AUTHOR OF A TREATISE ON MASONRY CONSTRUCTION. 
 
 SECOND EDITION, 
 REVISED AND GREATLY ENLARGED. 
 
 THOUSAND. 
 
 NEW YORK : 
 JOHN WILEY & SONS 
 
 LONDON : 
 CHAPMAN & HALL, LTD. 
 
 1897. 
 
COPYRIGHT, 1892, 
 
 BY 
 IRA O. BAKER. 
 
 75 
 
 ROBKRT DRUMMOND, ELECTROTYPER AND PRINTER, NEW YORK. 
 
PREFACE. 
 
 THIS volume was prepared for the author's own 
 students, and has been used in his classes for twelve 
 years past, in the form of a blue-print manuscript text- 
 book. Two series of extracts were published in two 
 engineering journals and afterwards reprinted in book 
 form. As one of these little books had the same title 
 as this volume, the latter is gratuitously called a second 
 edition. 
 
 The object of this volume is to acquaint the student 
 with the construction, adjustment, and use of surveying 
 instruments. In no degree is it intended for a treatise 
 on surveying, since there is no lack of good books on 
 the various branches of that subject. It has not, how- 
 ever, always been possible to draw a sharp line between 
 instruction in the use of the instruments, and methods of 
 surveying ; but, as a rule, only such of the latter have 
 been given as are common to all surveys made with the 
 particular instrument under consideration. In a few 
 cases, methods of surveying not given in the common 
 text-books and manuals are briefly discussed. 
 
 The author began the preparation of this volume 
 after becoming convinced, by his own observation as 
 well as by that of others, that students, as well as many 
 practicing engineers, would be benefited by a more 
 thorough study of the instruments. The knowledge of 
 the instruments and the practice in their use gained in 
 the study of subjects in which the instruments have 
 
 iii 
 
PREFACE. 
 
 but subordinate consideration, are not sufficient to 
 secure either economy of time and effort, or maxi- 
 mum accuracy. The experience of the author, the tes- 
 timony of his students after leaving college, and the 
 commendations the first edition received from engineers 
 in practice, have confirmed his opinion of the value of 
 a detailed study of the instruments themselves, the 
 sources of error in using them, the methods of eliminat- 
 ing the errors, and the degree of accuracy attainable. 
 
 In Appendix IV will be found the problems which 
 the author assigned in connection with a study of the 
 text. These problems are designed to familiarize the 
 student with the methods of using the instruments, to 
 acquaint him with the qualities of good instruments, 
 and also to teach him the degree of accuracy attainable. 
 Some of the problems are solved several times with 
 different instruments and under different conditions as 
 to distance, weather, experience, etc. 
 
 In the preparation of this volume great care has been 
 taken to present the subject clearly and concisely, and 
 to make the book convenient for daily use or ready 
 reference. The volume is divided into chapters and 
 articles, and it may be helpful to the reader to notice 
 that successive subdivisions of the latter are indicated 
 by capital black-face side-heads, by lower-case black- 
 face side-heads, by italic side-heads, and by simply 
 the serial section number. In some cases the major 
 subdivisions of the sections are indicated by small nu- 
 merals. The running title at the head of the pages 
 will be of assistance in finding the different parts of the 
 book. An index at the end of the volume makes every- 
 thing in the book easy of access. 
 
 CHAMPAIGN, ILL., Nov. 19, 1892. 
 
CONTENTS. 
 
 PAGE 
 
 INTRODUCTION. . i 
 
 CHAPTER I. CHAIN AND TAPE. 
 
 Art. i. CONSTRUCTION. Common Chain. Steel Tapes. 
 
 Linen Tapes . . .3 
 
 Art. 2. TESTING THE CHAIN AND TAPE. Standards. 
 Testing the tape. Testing the chain. Correcting for 
 error of chain 10 
 
 Art. 3. USING THE CHAIN. How to chain. Chaining on 
 a slope. Compensating vs. cumulative errors. Sources 
 of error. Limits of precision . . . . .14 
 
 CHAPTER II. TRIPOD, LEVELING SCREWS, 
 AND PLUMB-BOB. 
 
 Art. i. THE TRIPOD. Construction. Setting the tripod 31 
 Art. 2. LEVELING SCREWS. Defects of ordinary construc- 
 tion . . . .. . . . . . .33 
 
 Art. 3. PLUMB-BOB . . 'S* 35 
 
 CHAPTER III. MAGNETIC COMPASS. 
 
 Art. i. CONSTRUCTION. Prismatic compass . . . 37 
 
 Art. 2. TESTS. The needle. Metal of compass-box. 
 Sights. Zero of vernier . . . . .41 
 
 Art. 3. ADJUSTMENTS. Levels. Sights. Needle. Cen- 
 ter-pin . . . . . . . . . -43 
 
 Art. 4. USING THE COMPASS. Care. Practical hints. 
 Sources of error. Limits of precision. Balancing the 
 
 latitudes and departures .46 
 
 v 
 
VI CONTENTS. 
 
 CHAPTER IV. SOLAR COMPASS. 
 
 PAGE 
 
 Principle of the Solar Apparatus 57 
 
 Art. i. CONSTRUCTION OF THE SOLAR COMPASS . . 58 
 Art. 2. ADJUSTMENT OF THE SOLAR COMPASS. . . 61 
 Art. 3. USING THE SOLAR COMPASS. Merits and de- 
 fects. History , . . . . . . .62 
 
 CHAPTER V. VERNIERS. 
 
 Principles. To read a vernier. Practical hints. Microm- 
 eter . . . 64 
 
 CHAPTER VI. OPTICAL PARTS OF THE 
 TELESCOPE. 
 
 Art. i. CONSTRUCTION. Kinds of telescope. Objective. 
 
 Eye-piece. Telescope slide. Cross hairs . . .72 
 
 Art. 2. TESTING THE TELESCOPE. Chromatic aberration. 
 Spherical aberration. Definition. Flatness of field. 
 Size of field. Aperture of objective. Magnification. 
 Illumination .80 
 
 Art. 3. USING THE TELESCOPE. Adjustment for paral- 
 lax. Care of telescope 88 
 
 CHAPTER VII. THE TRANSIT. 
 
 Art. i. CONSTRUCTION. Graduation, Verniers. Centers. 
 Levels. Clamp and tangent screw. Object-glass slide. 
 Gradienter. Shifting plates. Compass. Various 
 extras 91 
 
 Art. 2. TESTS OF THE TRANSIT. Graduation. Eccen- 
 tricity. Magnifying power vs. vernier. Magnifying 
 power vs. level under telescope. Magnifying vs. plate 
 levels. Parallelism of vertical axes. Limb perpendic- 
 ular to axis. Object-glass slide . . . .103 
 
 Art. 3. ADJUSTMENTS OF THE TRANSIT. Levels. Cross 
 hairs. Centering the eye-piece. Standards. Level 
 under telescope. Zero of the vertical circle . .109 
 
CONTENTS. Vll 
 
 PAGE 
 
 Art. 4. USING THE TRANSIT. Practical hints. Measuring 
 angles : ordinary method, by series, by repetition. 
 Transit surveying : angle method, quadrant method, 
 traversing. Sources of error. Limits of precision. 
 Care of the transit . . . . . . . .115 
 
 CHAPTER VIII. SOLAR TRANSIT. 
 
 Art. i. CONSTRUCTION. Saegmuller's form. Other forms 130 
 Art. 2. ADJUSTMENT. Polar axis. Crosshairs . .134 
 Art. 3. USING THE SOLAR TRANSIT. To determine a 
 meridian. To determine the latitude. Sources of 
 error. Limits of precision. Solar transit in mine sur- 
 veying 135 
 
 CHAPTER IX. PLANE TABLE. 
 
 Art. i. CONSTRUCTION. Different forms : complete, light, 
 
 home-made . . . . r . . . . 146 
 Art. 2. TESTS AND ADJUSTMENTS. Sights. Edge of ruler. 
 
 Levels on alidade. Board. Telescope. Zero of vernier. 
 
 Level on telescope . . . . -. . . .153 
 Art. 3. USING THE PLANE TABLE. Methods. Three-point 
 
 problem. Two-point problem. Sources of error. 
 
 Limits of precision. Practical hints . . . .155 
 
 CHAPTER X. TELEMETERS. 
 
 Art. i. STADIA. Principles. Placing the hairs. The 
 rod. Formula for horizontal line of sight and vertical 
 rod. Position of rod for inclined line of sight. For- 
 mulas for inclined line of sight and vertical rod. Re- 
 ducing field notes : diagrams, tables. Sources of error. 
 Limits of precision. Practical hints . . . . 173 
 
 Art. 2. GRADIENTER. As a leveling instrument. As a 
 telemeter. Limits of precision. Vertical circle as a 
 gradienter . . 209 
 
 Art. 3. VARIOUS TELEMETERS . .... 216 
 
Vlll CONTENTS. 
 
 CHAPTER XL SPIRIT LEVELS. 
 
 PAGE 
 
 Art. i. CONSTRUCTION OF THE INSTRUMENT. Qualities 
 
 desired. Classification .218 
 
 Art. 2. CONSTRUCTION OF THE ROD. Target rods. Self- 
 reading rods . . . . ... . . 227 
 
 Art. 3. TESTING THE LEVEL. Bubble tube. Sensitive- 
 ness vs. magnifying power. Telescope slide. Rings 
 and wyes of wye level . ... . . . 236 
 
 Art. 4. ADJUSTMENTS OF WYE LEVEL. Level tube. 
 Cross hairs. Centering eye-piece. Wyes. Test level 241 
 
 Art. 5. ADJUSTMENTS OF DUMPY LEVEL. Collimation. 
 Wyes . . . . .247 
 
 Art. 6. ADJUSTMENTS OF LEVEL OF PRECISION . . 249 
 
 Art. 7. USING THE LEVEL. Differential leveling. Profile 
 leveling. Precise leveling. Sources of error. Limits 
 of precision. Practical hints. Care of the level . . 252 
 
 CHAPTER XII. BAROMETERS. 
 
 Art. i. MERCURIAL BAROMETER. Construction. Clean- 
 ing. Filling. Reading. Transporting . . . 292 
 
 Art. 2. ANEROID BAROMETER. Common or Vidi's. Gold- 
 schmid's . ... . . . ... . > . 301 
 
 Art. 3. THE PRACTICE. Sources of error : gradient, tem- 
 perature, humidity, instrumental, observational. Limits 
 of precision. Methods of observing: single observa- 
 tion, simultaneous observations, observations at se- 
 lected times, Williamson's method, Whitney's, Planta- 
 mour's, Riihlmann's, Gilbert's . . . . 308 
 
 Art. 4. BAROMETRIC FORMULAS. A. Statical Formulas. 
 Assumptions. Fundamental Relations. Typical 
 formulas : Laplace's, Babinet's, Bailey's, Plantamour's, 
 Williamson's. Altitudes cales of aneroids. B. Dynam- 
 ical Formulas. Ferrel's, Gilbert's, Weilenmann's. . 327 
 
 APPENDIX I. ELIMINATION OF LOCAL 
 ATTRACTION IN SURVEYING WITH 
 
 THE MAGNETIC COMPASS. 
 Art. i. MINE SURVEYS . . . ... . .349 
 
 Art. 2. LAND SURVEYS . ..." . . . . 356 
 
CONTENTS. IX 
 
 APPENDIX II. AREA BY TRAVERSING WIT 
 TRANSIT. 
 
 PAGE 
 
 Taking the notes. Computing the area .... 360 
 
 APPENDIX III. PROBABLE ERROR. 
 Principles. Formulas. Examples . . . . 366 
 
 APPENDIX IV. PROBLEMS. 
 
 Chain. Magnetic Compass. Transit. Solar Transit. Plane 
 Table. Stadia. Level. Barometer .... 373 
 
ENGINEERS' SURVEYING INSTRUMENTS. 
 
 INTRODUCTION. 
 
 THE importance to the engineer of a knowledge of 
 the best forms of construction of his instruments and 
 of a thorough understanding of the principles which 
 should govern their adjustments, is self-evident ; and it 
 is no less evident that he should be expert in handling 
 his instruments. The general plan of this discussion 
 will be (i) to call attention to the principal points in the 
 common forms of construction of each instrument, not- 
 ing the advantages and disadvantages of each ; (2) to 
 consider certain relations which the parts of a perfect 
 instrument should bear to each other, and which are 
 supposed to be adjusted by the maker once for all, ex- 
 plaining how the engineer may test them for himself 
 although he may not be able to correct them ; (3) to 
 explain the method of making the several adjustments ; 
 (4) to describe the method of using the instrument 
 which will secure the most accurate results in the short- 
 est and easiest way, and to note the various sources of 
 error to which the work is liable, and also to give data 
 to show the degree of accuracy attainable ; and (5) to 
 add a few hints on the proper care of the instrument. 
 It will be unnecessary to describe minutely the different 
 
INTRODUCTION. 
 
 parts of the several instruments and the objects of each, 
 except as is requisite in carrying out the above outline ; 
 for more can be learned by a moment's inspection of 
 the instrument than from any printed description. No 
 attempt will be made to describe any of the special 
 devices proposed by the different instrument-makers, 
 the desire being to state the general principles which 
 will enable the reader to judge of the merits of any 
 modification of the ordinary forms. 
 
 
CHAPTER I. 
 CHAIN AND TAPE. 
 
 ART. 1. CONSTRUCTION. 
 
 1. COMMON CHAIN. The ordinary chain consists of 
 one hundred pieces of wire, called links, bent into rings 
 at the ends and connected together by two (sometimes 
 three) rings. In using the chain a " link " includes a 
 ring at each end. Iron chains are made of two sizes of 
 wire, Nos. 8 and 10, the former being about five thirty- 
 seconds of an inch in diameter and the latter nearly 
 one eighth of an inch. Steel chains are made of No. 
 10 or 12 wire, the latter being about seven sixty-fourths 
 of an inch in diameter. The best chains are made of 
 No. 12 tempered-steel wire. All joints in the rings 
 and links should be brazed to prevent their opening, 
 and the consequent lengthening of the chain. The 
 chain is divided decimally by brass tags numbered from 
 each end to the middle. 
 
 The surveyor's chain (Gunter's chain) is 66 feet long, 
 each "link" being 7.92 inches. It is used only in find- 
 ing the area of land where f he acre is the unit of 
 measure, and is much less frequently used for this pur- 
 pose now than formerly. The 66-foot chain is used on 
 all the U. S. public-land surveys ; and in all deeds of 
 conveyance and other documents, when the word chain 
 is used, it is Gunter's chain that is meant. 
 
CHAIN AND TAPE. [CHAP. 
 
 An engineer's chain is 100 feet long, each "link" be- 
 ing i foot. This chain is used in surveying railroads, 
 canals, and where extensive line surveys are being con- 
 ducted. It is not infrequently employed in finding 
 areas in acres. It is preferred to the surveyor's chain 
 on account of its greater length, which enables one to 
 work more rapidly and more accurately. 
 
 When the chain is folded up the links should not be 
 parallel to each other, but should be crossed in such a 
 manner as to touch each other in the middle, thus pre- 
 venting the bending of the links in tying up the chain. 
 See Fig. i. 
 
 FIG. i. 
 
 2. Advantages and Defects of the Common Chain. The 
 principal advantages of the chain are its flexibility and 
 the ease with which its subdivisions are distinguished. 
 
 The ordinary chain is defective on account of (i) un- 
 avoidable wearing of the numerous points of contact, (2) 
 opening of the rings, (3) lengthening by stretching and 
 by flattening of the rings, (4) shortening by mud, ice, or 
 grass getting into the joints, (5) varying in length with 
 
ART. l] CONSTRUCTION. 
 
 temperature, and (6) kinking. Some of these defects 
 are so small as to be appreciable only in very careful 
 work ; and some exist even in the most elaborate ap- 
 paratus that can be devised. If each of the six hundred 
 points of contact of the common loo-link chain wears 
 only one hundredth of an inch, the length of the chain 
 is increased 6 inches. Mud and ice in the joints have a 
 still greater effect in the opposite direction. 
 
 3. STEEL TAPES. Steel ribbons are now made of any 
 length up to 1,000 feet without joint or splice from 
 end to end. In width these ribbons vary from an 
 eighth of an inch to a half inch, and in thickness from 
 one hundredth to four hundredths of an inch. For gen- 
 eral work a tape about a quarter of an inch wide and 
 two hundredths of an inch thick, blued and polished or 
 nickel-plated, is generally preferred. The wider and 
 thinner tapes are nearly useless in general field-work, 
 owing to the ease with which they are broken. 
 
 The divisions of the tape are marked in several ways, 
 viz.: (i) the numbers and graduation marks are stamped 
 in a piece of brass which is soldered on the tape ; (2) 
 the marks are countersunk in a lump of solder attached 
 to the tape for that purpose ; (3) the foot divisions are 
 marked by single rivets and the lo-foot divisions by a 
 brass burr riveted on, the 10, 20, etc., marks being distin- 
 guished either by numbers countersunk in the brass 
 burr or by i, 2, etc., small rivets ; or (4) the face of 
 the tape is etched with acid so that the division marks 
 and figures stand out in relief while the etched surface 
 appears dull. No method of indicating the graduation 
 is entirely satisfactory, but the first and second are prob- 
 ably best. The first is probably more durable than the 
 second, although the graduation marks are liable to 
 wear or tear off, particularly on gravel or stony ground. 
 In either case there should be two series of numbers, 
 one counting from each end. It is not desirable to have 
 
CHAIN AND TAPE. [CHAP. I 
 
 the graduation indicated by rivets, since the tape is 
 weakened by the rivet-hole, and also by the accumula- 
 tion of moisture under the rivet head. One advantage 
 of this method is that the graduation can be read from 
 either side of the tape. The first and second methods 
 are sometimes employed upon the same tape. The last 
 method is employed only for pocket tapes graduated to 
 feet and inches, owing to the liability of weak places 
 from over-etching. 
 
 Steel tapes graduated by any of the first three 
 methods mentioned above are frequently called band 
 chains and sometimes chain tapes, to distinguish them 
 from tapes graduated by the fourth method, /.<?., gradu- 
 ated fully throughout. 
 
 Steel tapes can be had graduated to feet and inches, 
 to feet and tenths, to links, or metrically. The wide 
 thin tapes graduated to feet and inches by etching are 
 very convenient in city surveying, or where accurate 
 measurement of irregular distances is frequently re- 
 quired. The tape ordinarily employed in general field- 
 work is a narrow ribbon graduated to feet, with each end 
 foot to tenths. This form of tape would be greatly im- 
 proved if, instead of subdividing the first and last foot 
 of the hundred, an additional foot were added at each 
 end, and so divided. When the end foot of the hundred 
 is subdivided, if one desires to lay off, say, 28.3 feet, one 
 chain-man must hold the tape at 29 feet and the other 
 at 7 tenths from the end of the graduation, which oper- 
 ation produces mental confusion and is apt to cause 
 error, since neither chain-man holds the tape at the 
 number to be laid off. With an extra foot at the end of 
 the tape, one chain-man holds the tape at the whole 
 number of feet to be laid off, while the other chain-man 
 holds at the number corresponding to the fraction to be 
 laid off. 
 
 Frequently the end graduation of a steel tape is 
 
ART. l] CONSTRUCTION. 7 
 
 simply a line across the face of the tape or a rivet in the 
 middle of its width, in which case it is not possible to 
 stick the pin quickly or exactly opposite the mark. 
 The end graduation mark should be indicated by the 
 square shoulder of a piece of brass soldered or riveted 
 to the tape for that purpose. These shoulders, if put 
 on at all, generally face opposite ends of the tape. 
 They should both face the same end (preferably the rear 
 end) of the tape, as shown in Fig. 2 ; in which case the 
 
 FIG. 2. 
 
 same side of the pin can be used by both the fore and 
 hind chain-man. 
 
 4. Reels. For convenience of transportation steel 
 tapes are wound up either on an open reel or in a 
 tight case, the narrower thicker ones on the former and 
 the wider thinner ones in the latter. In one respect the 
 open reel is better than the tight case, since with the 
 former the tape has a better chance to dry off. It is 
 desirable that the reel should be strong, durable, and 
 convenient, and at the same time be light and of such 
 a form as to be carried in the pocket when the tape is 
 in use. No reel that the writer has seen fulfills this 
 condition fairly well. Generally the axis about which 
 the tape is wound is so small as to break the tape near 
 the end. 
 
 Most of the reels require that one handle be re- 
 moved before beginning to wind up the tape, which 
 necessitates the handles being easily detached without 
 liability to come off when in use. Most of the handles 
 on the market are deficient in one or the other of these 
 
8 CHAIN AND TAPE. [CHAP. I 
 
 respects. On the whole, it is probably best to wind the 
 tape up in the hands in the form of a figure eight (8) 
 about two feet long, and tie with a string at the 
 center. 
 
 5. Advantages and Disadvantages of Steel Tapes. Steel 
 tapes have superseded chains for nearly all kinds of 
 work for the following reasons: (i) Tapes are lighter 
 than chains of equal strength. (2) Tapes being smooth 
 and having no projections are easier to drag. (3) Tapes 
 do not alter their length by wear. 
 
 The disadvantages are: (i) The steel tape will not 
 stand as rough handling as the chain. (2) It can not 
 be repaired in the field when broken, while the chain 
 can lose one or more of its links and still be of service. 
 (3) The graduation is not as legible as that of the 
 chain, and becomes obliterated with use. 
 
 Notwithstanding its defects, a steel tape is vastly 
 better than a chain for nearly all kinds of work. 
 
 6. Wood Rods vs. Steel Tapes. Before the introduc- 
 tion of steel tapes short wood rods were employed 
 when great accuracy was necessary. They were free 
 from some of the imperfections of the common chain, 
 but were defective in other and generally more im- 
 portant respects. Short rods should be used only with 
 the accurate alignment and elaborate means of secur- 
 ing delicacy of contact employed in geodetic measure- 
 ments ; and recent experience* seems to show that in 
 measuring geodetic base lines, work can be done with 
 greater speed and accuracy with steel tapes than with 
 the most elaborate and expensive base apparatus yet 
 devised. 
 
 * See Annual Report of the Missouri River Commission for 1886, Execu- 
 tive Document No. 28, 49th Congress, 2<d Session, pp. 31-35. For method of 
 accurate measurement with two steel tapes or wires used as a metallic ther- 
 mometer, see Annual Report of the Chief of Engineers, U. S. A., for 1890, 
 pp. 1838-45. 
 
ART. l] CONSTRUCTION. 
 
 7. Steel Wires. A steel wire makes a good substitute 
 for a chain. Fig. 3 shows a method of indicating the 
 end of the distance to be laid off. ab is a small rod 
 with a hole through it, into which the wire is fastened; 
 c is a V-shaped groove around the rod. The pin may 
 be set in the groove c without any danger of the back 
 pin's being displaced by the jerking of the wire by the 
 fore chain-man in getting it into line; or the distance 
 may end at a or , in which case it can be more accu- 
 rately marked with a knife in a board. The ends a 
 
 FIG. 3. 
 
 and b should be exactly at right angles to the direction 
 of the length of a b. A handle can be formed by pass- 
 ing the wire through a piece of wood to protect the 
 hand, and then bending the wire into a loop. Unfor- 
 tunately fractional parts of the unit are not easily ob- 
 tained; but in many cases only units are required. 
 
 8. LINEN TAPES. Although accurate work can not 
 be done with linen tapes, they are useful in many kinds 
 of work, as, for example, in cross-sectioning for rail- 
 road earthwork. A linen tape contracts in wet 
 weather and expands in dry, and it can easily be 
 permanently elongated by overstraining. 
 
 9. Metallic Tapes are a species of linen tapes hav- 
 ing fine brass wires woven through their entire length. 
 Metallic tapes do not vary in length with changes in 
 the hygrometric state of the atmosphere as much as 
 linen tapes, and are not so easily elongated by over- 
 straining. 
 
10 CHAIN AND TAPE. [CHAP. I 
 
 ART. 2. TESTING THE CHAIN AND TAPE. 
 
 10. STANDARDS. Each instrument-maker claims that 
 the chains and tapes which he sends out are true U. S. 
 standard; but it is certain that chains and tapes from 
 different makers, purporting to be of the same standard, 
 differ in length. Even if correct in the beginning, chains 
 wear and tapes break; and therefore neither are likely to 
 be of the same length after being used some time. Hence 
 every engineer should have some means of testing the 
 length of his chain or tape. The simplest way to get a 
 reliable standard is to send a new tape to the Superin- 
 tendent of the U. S. Coast and Geodetic Survey, Wash- 
 ington, D. C., who will compare it with the standard 
 and place the government stamp on it to show its de- 
 gree of accuracy. For a degree of accuracy sufficient 
 for ordinary engineering a fee of 50 cents is charged; 
 but when extreme accuracy is desired the fee is higher. 
 
 The tape so compared should be reserved for testing 
 other tapes, or the standard distance should be carefully 
 laid off on the floor of a building, or on a stone water- 
 table, or on two stone or iron posts firmly set in the 
 ground for that purpose, or in any way that shall per- 
 manently preserve the exact distance. When laying off 
 the standard distance the tape should be supported 
 throughout its entire length; should be perfectly 
 straight, both horizontally and vertically, and should be 
 stretched with the same tension as that employed in 
 testing it at Washington. In laying off this distance 
 the temperature of the tape should be carefully noted, 
 and if it is not the same as that at which the tape is a 
 standard, a correction (see 19, paragraph e) should be 
 applied before making the permanent mark. 
 
 11. It is sometimes held that since the chain when in 
 use is seldom if ever stretched perfectly straight, it 
 
ART. 2] TESTING THE CHAIN AND TAPE. II 
 
 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 the U. S. Land Office 
 to Surveyors-General states * that " the 66-foot chain 
 shall be 66.06 ft.," for the above reason. The French 
 have, or at least had,f a similar practice, the addition 
 being from r in 1,000 to i in 2,000. In all the arts de- 
 pending in any way upon accuracy of measurements 
 there is great confusion on account of differences be- 
 tween so-called standards. In some cases, particularly 
 that of iron work, this diversity arose in a manner sim- 
 ilar to that referred to above. The chain should be 
 exactly as long as the standard. 
 
 12. Notice that in many cases the standard by which 
 the engineer is to test his chain or tape is not an abso- 
 lute one. For example, the law under which the U. S. 
 public lands were surveyed says " all the corners marked 
 in the surveys returned by the Surveyor-General shall 
 be established as the proper corners," etc., and " the 
 length of such lines as returned shall be held and considered 
 as the true length thereof." This law establishes a stand- 
 ard of measure between every pair of adjacent corners 
 of the government survey, and this standard is the only 
 one that can legally be used in measuring that line. As- 
 sume, for example, that a surveyor being called upon to 
 establish the corner at the middle of the west side of 
 sec. i, measures the west side of that section and finds 
 it to be 79.26 chains by his chain, while the recorded 
 distance is 79.83 chains. Then, since he must consider 
 the recorded distance as the true distance, the true 
 length of his chain is 79.83 -=- 79.26 = 1.007 times the 
 standard. Since the surveyor is to establish a corner 40 
 chains north of the south-west corner of sec. i, the dis- 
 
 * In the Instructions for 1880, for the first time. 
 t Gillespie's Land Surveying, page 18, foot-note. 
 
12 CHAIN AND TAPE. [CHAP. I 
 
 tance as measured by his chain is 40 -f- 1.007 = 39.72. 
 A similar principle applies in city surveying when the 
 land is described as being a certain lot in a particular 
 block of a recorded plat. 
 
 If the land to be surveyed is described by metes and 
 bounds, then it is important that the surveyor shall lay 
 off true standard distances. 
 
 13. TESTING THE TAPE. In comparing a tape with 
 the standard distance laid off as above, the tape should 
 be under the same conditions as to temperature, tension, 
 etc., that it is to have when in use. The men who are 
 to use the tape should test it that they may better under- 
 stand the proper tension required. 
 
 14. TESTING THE CHAIN. The same precautions are 
 to be observed in testing the chain as for the tape. The 
 length of the chain should be considered as the distance 
 from the inside of one handle to the outside of the other 
 (see 16). In any case, the points considered as the 
 ends of the chain depend upon the manner of setting 
 the pins. 
 
 The custom of taking up the wear of the chain by a 
 screw at one end is wrong in principle, although it does 
 not produce much error in practice. The increased 
 length is produced by wear at every joint ; taking it up 
 at the ends destroys the equality of the scale of equal 
 parts, and when only a fractional part of the chain is 
 used an error is produced. The better method is to 
 compare frequently the chain with the standard, and 
 apply a correction to the measured distance or computed 
 area. 
 
 15. CORRECTING FOR ERROR OF CHAIN. Owing to wear 
 
 it frequently happens that the chain is not of stand- 
 ard length, and in using a tape in making re-surveys 
 the tape often does not agree with the unit used in lay- 
 ing off the line which is being re-measured. Conse- 
 quently it is often necessary to correct a measured dis- 
 
ART. 2] TESTING THE CHAIN AND TAPE. 13 
 
 tance for error of chain. If in testing the chain by the 
 method of 10 or 12 it is found to be four tenths of a 
 foot (or link) too long, then the chain is 1.004 times the 
 true standard ; and if the distance as measured is 273.8 
 ft., the true distance is 273.8 X 1.004 = 2 74-9 ft. 
 
 If the area is required, make the computations as 
 though the chain were correct ; and then the true area 
 is equal to the computed area multiplied by the square 
 of the length of the chain in terms of the standard. For 
 example, assume that the computed area is 10.875 acres, 
 and that the chain is 1.002 times the standard. Then 
 the true area is 10.875 X (i.oo2) a = 10.919 acres. Notice 
 that the length of the chain can be expressed thus : i 
 + .002 ; and that the square of that quantity can be ex- 
 pressed thus : (i + .oo2) a =1-1-2 (.002) + (.oo2) 2 . Since 
 the square of .002 is very small, it may be omitted, and 
 then (i + .002)* = 1 + 2 (.002) nearly. Hence the cor- 
 rection to the area is equal to the computed area multi- 
 plied by twice the correction of the chain. For illustra- 
 tion, the true area for the above example is 10.875 ~i~ 
 (10.875 X 2 X 0.002) = 10.918 acres. In this example 
 the results by the two methods differ by less than one 
 thousandth of an acre, and in this class of problems the 
 error will always be inconsiderable. If the chain is, say, 
 .002 too short, then its length is expressed by the for- 
 mula i .002, and the correction to the area is found 
 as above, except that it must be subtracted. 
 
 The student is cautioned to remember that if the chain 
 is too long the distance and the area are too small ; and, 
 vice versa, if the chain is too short the distance and the 
 area are too large. Very frequently errors are made by 
 applying this correction in the wrong way. 
 
14 CHAIN AND TAPE. [CHAP. I 
 
 ART. 3. USING THE CHAIN. 
 
 16. HOW TO CHAIN. The following is the general 
 method" oTsgrocedure in chaining, but is frequently 
 modified as circumstances require. To measure a dis- 
 tance with a chain, two men are required, a fore chain- 
 man and a hind chain-man. The hind chain-man has the 
 more responsible position. They should be provided 
 with eleven marking pins. 
 
 Supposing the chain to be tied up, the fore chain- 
 man throws it out in the direction opposite to that in 
 which the chaining is to be done, gives the hind chain- 
 man a pin, takes nine in his left hand and the end of 
 the chain and one pin in his right hand, and draws the 
 chain in the direction of the line. The hind chain-man 
 examines the chain as it passes, to see that there are 
 no kinks or bent links; or, if a tape is used, to see that 
 there are no loops in it. 
 
 When the fore chain-man has gone the proper dis- 
 tance, he stops, rests his right elbow on his right knee, 
 and extends his right hand, in which he holds the 
 handle and the pin,* as far as possible from his body, 
 so that the hind chain-man may have an unobstructed 
 view of the pin and the farther end of the line. The 
 fore chain-man is to keep the chain straight and taut, 
 and obey the signals of the hind chain-man. 
 
 The hind chain-man places his end of the chain at 
 the point of beginning, and, by placing himself behind 
 the point, with a motion of his arm directs the fore 
 chain-man where to place his pin. For example, if the 
 pin ought to be moved a considerable distance to the 
 
 * On a steel tape the handle extends beyond the end graduation, and hence 
 the fore chain-man should grasp the handle of the tape in his left hand, rest 
 his left elbow on his left knee, and hold the pin in his right hand, instead of as 
 described above for the chain. 
 
ART. 3] USING THE CHAIN. 15 
 
 right, the right arm is held far out from that side of his 
 body; if it should be moved only a little, the arm is 
 held nearly vertical. The signal thus indicates both 
 the direction and the amount of motion required. As 
 the pin approaches the proper position, the arm comes 
 more nearly vertical; and when the pin is at the proper 
 place, the hind chain-man calls out " stick." The fore 
 chain-man then brings his left hand to bear on the top 
 of the pin, and forces it vertically into the ground.* 
 After the pin is set, he should test it to see that the 
 pin, at the surface of the ground, is just in contact 
 with the front face of the handle. When the position of 
 the pin is satisfactory to the fore chain-man, he calls out 
 " stuck." At this signal the hind chain-man loosens his 
 end of the chain, and both move forward the length of 
 the chain. 
 
 The leader should keep his eye steadily on the far- 
 ther end of the line, so that he may keep near the line. 
 When the follower reaches the pin already set, he calls 
 " halt," and the leader prepares to set a pin. After 
 the fore chain-man has placed his pin in line, the fol- 
 lower drops his end of the chain over the pin,f at the 
 same time placing his hand on the pin to hold it firm,J 
 and calls out " stick." At the reply " stuck," he re- 
 moves his pin and the work proceeds as before. 
 
 When the leader has set his last pin, he calls " tally; " 
 and the hind chain-man comes up, and gives the fore 
 
 * When the tape is used, the end of it is held in the left hand and the pin 
 is forced into the ground with the right hand. 
 
 t Notice that both men measure to the same side of the pin. 
 
 % The liability of the back pin's being pulled over in this operation is an 
 objection to this method ; but it can be eliminated by having the fore chain- 
 man set the pin on the inside of his handle, while the back chain-man simply 
 brings the outside of his handle against the front side of the pin. However^ 
 it is very difficult for the fore chain-man to set the pin on the inside of the 
 handle. Notice that with a properly made steel tape (see last paragraph uf 
 3) neither of these difficulties will occur. 
 
l6 CHAIN AND TAPE. [CHAP. I 
 
 chain-man the ten pins which he has, both men count- 
 ing them to be sure that none have been lost. The 
 follower then makes note of the tally, and the work 
 proceeds as before. 
 
 When the leader reaches the end of the line, he 
 stops, holds his end of the chain against it and calls 
 " stuck." The follower then comes forward and counts 
 the distance beyond the last pin, being careful to 
 notice on which side of the middle the pin is. Each 
 tally represents ten chains, each pin held by the fol- 
 lower (not including the one in the ground) represents a 
 chain, and the feet just counted make up the total dis- 
 tance. Notice that the pin last set is not counted. It 
 should always remain in the ground until the distance 
 is recorded. 
 
 17. Chaining on a Slope. In nearly all cases, it is 
 the horizontal distance which is required. Therefore 
 it is necessary to determine the flattest slope that must 
 be taken into account. The difference between the dis- 
 tance measured on the slope and the true horizontal 
 distance is given very nearly* by the formula 
 
 in which d is the difference sought, and / the rise of the 
 slope in any distance b. The quantities a 7 ,/, and b all 
 must be in the same unit. For example, if the slope 
 
 * To determine the difference between the base and perpendicular of a right- 
 angle triangle, represent the base by , the hypotenuse by ^, and the perpen- 
 dicular by p. Then we have 
 
 b = \h*-p*=(fr-p*)*= h -, nearly. 
 Similarly, 
 
 h = !/$+/ = (i>* +/ 2 }* = b + , nearly. 
 Therefore we may derive this simple rule : The difference between the base 
 
ART. 3 ] 
 
 USING THE CHAIN. 
 
 rises 2 feet in 100 feet, then d=~- ,= ~^ = 0.02 ft. 
 
 zb 200 
 
 Hence, if the slope is 2 in 100, measuring on the slope 
 causes an error of i in 5,000. Disregarding a slope of 
 3 in 100 causes an error of nearly i in 2,000, and disre- 
 garding a slope of 4 in 100 causes an error of i in 1,200. 
 
 If the slope is too great to be disregarded, the ques- 
 tion then arises as to the best method of eliminating 
 the error. Either of two methods may be employed : 
 (i) one end of the chain may be raised to a level with 
 the other, or (2) the measurement may be made on the 
 slope and a correction applied to the result. 
 
 i. To measure by keeping the chain level, the chain- 
 men should be provided with a small plumb and line, 
 so that the end of the chain may be held vertically over 
 the proper point.* If the slope is not very steep, the 
 whole length of the chain can be laid off at once by 
 raising the lower end until the chain is horizontal and 
 transferring the end to the ground with the plumb ; but 
 
 and hypotenuse is equal to the square of the perpendicular divided by twice 
 the known side. 
 
 The degree of approximation involved in the preceding relation is shown 
 by the following table : 
 
 SLOPE. ERROR. 
 
 5 vertical to 100 horizontal i in 1,000,000 
 
 10 
 
 20 
 
 30 
 40 
 
 50 
 60 
 80 
 
 IOO 
 
 100 
 IOO 
 IOO 
 IOO 
 IOO 
 IOO 
 IOO 
 IOO 
 
 
 C.OOO 
 
 
 1,000 
 
 
 I OOO 
 
 6 
 
 1,000 
 
 .,;, 12 
 
 I OOO 
 
 30 
 
 1,000 
 
 . 57 
 
 1,000 
 
 This approximation is frequently very convenient, and the student should get 
 it well fixed in mind. 
 
 * The common practice of dropping a chaining-pin when one end of the 
 chain is elevated only two or three feet, although recommended by many text- 
 books, is objectionable. The error due to the pin's not dropping vertical is 
 probably greater than the error to be eliminated. 
 
l8 CHAIN AND TAPE. [CHAP. I 
 
 when the slope is so steep that the two ends of the chain 
 can not conveniently be brought to the same horizontal 
 line, then only part of the chain can be laid off at a time, 
 in which case some aliquot part should be used. When 
 only part of the chain is used, great care must be taken 
 not to confuse the count. 
 
 This method of chaining involves three difficulties : 
 (a) keeping the chain horizontal ; (b) transferring the 
 elevated end of the chain vertically to the ground ; and 
 (c) making the stretch from the pull equal to the short- 
 ening from sag. a. It is very difficult to determine a 
 level line, particularly when one is standing at one end 
 of it and looking up or down hill. In chaining up steep 
 slopes it is a great advantage to have a third man, who 
 shall stand at one side of the line and tell when the, 
 chain is horizontal. Even then one is liable to be. 
 greatly deceived as to the position of a horizontal line. 
 Generally the apparently horizontal line is too nearly 
 parallel with the slope, b. It is nearly impossible to 
 hold the plumb-line exactly at the end of the chain and 
 keep the chain both horizontal and sufficiently stretched, 
 and at the same time hold all so steady that the plumb- 
 line will hang still, c. The amount of pull required to 
 overcome the shortening from sag can be determined 
 only by trial. To do this, stretch the chain between 
 two points at the same elevation, having it supported 
 its entire length, and remove the supports, noting how 
 strong a pull is required to bring the ends of the chain 
 to the marks again. This should be done by the chain- 
 men themselves, to enable them to judge how hard 
 to pull it when it is off the ground. If only part of the 
 chain is to be laid off at once, this test must be applied 
 to that portion also. 
 
 Obviously it is not possible to perform all of the 
 preceding operations with any considerable degree of 
 accuracy; and it is generally more expeditious and also 
 
ART. 3] USING THE CHAIN. 19 
 
 more accurate to measure on the slope and apply a cor- 
 rection. 
 
 2. To compute the difference between the distance on 
 the slope and that on the horizontal requires the deter- 
 mination of the rate of slope. This can be found by 
 estimation, by means of a pocket level or a clinometer, 
 or by estimating the horizontality of a flag-pole and 
 measuring down. The quantity to be subtracted from 
 the distance on the slope is then computed by equation 
 (i), page 16. 
 
 Notice that the accuracy of the second method is de- 
 pendent upon the exactness with which the rate of slope 
 can be determined; but as this is only one of the three 
 difficulties encountered in the first method, we may con- 
 clude that the second is the more accurate. It is also 
 the more expeditious. 
 
 18. COMPENSATING vs. CUMULATIVE ERRORS. 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 be- 
 tween 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 impor- 
 tant. The observer should avoid errors which usually 
 occur in a single direction, but he need not always take 
 so great 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 ob- 
 servations 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. 
 
20 CHAIN AND TAPE. [CHAP. I 
 
 The uncertainty in the length of a line due to compen- 
 sating or accidental errors varies as the square root of 
 the number of units in the line, while the effect of cu- 
 mulative or constant errors varies directly as the length. 
 The whole amount of the cumulative errors remains un- 
 corrected, while only the square root of the compensat- 
 ing errors is uncompensated. For example, if the chain 
 is o.i of a foot too long, a line 25 chains long will be re- 
 corded 2.5 feet too short; but if the pin is sometimes 
 set o.i of a foot beyond the end of the chain and some- 
 times the same amount behind, the final error at the end 
 of the line due to this error is probably only o.i ^25 or 
 0.5 foot. If the head chain-man has a fixed habit of 
 setting the pin beyond the end of the chain, then this 
 becomes a cumulative error, and varies as the dis- 
 tance. 
 
 This illustrates that in the prosecution of any work it 
 is desirable that the operator should be cognizant of the 
 nature and importance of every source of error. The 
 more thorough and complete his knowledge in this re- 
 spect, the more readily and accurately will he 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 both to greater accuracy and to economy of 
 time and effort; for the observer, knowing that what 
 might otherwise have been attended to with consider- 
 able care may be neglected, is free to give all his atten- 
 tion to the weakest link in the chain of observations. 
 It enables the observer to correctly 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 observation. All this calls 
 
ART. 3] USING THE CHAIN. 21 
 
 for a clear understanding of the causes of error, and the 
 ability to determine their effect upon the final result. 
 
 19. SOURCES OF ERROR. The sources of error in chain- 
 ing are (a) incorrect length of chain, (b) kinking of the 
 chain and bending of the links, (c) the chain's not toeing 
 in a vertical plane, (a) unequal tension of the chain, (<?) 
 expansion and contraction with changes of temperature, 
 (/) errors of lining the fore chain-man, (g) not placing 
 the pin at the end of the chain, (h) drawing the pin by 
 hanging the back handle over it, (/) chain not being 
 level, and (/) such errors as miscounting tallies or chains, 
 counting from wrong end of chain, making a mistake 
 of 10 in the number of links, reading 18 for 22, 37 for 43, 
 etc. Errors of the last class are much too common, 
 but can be obviated by care and thoughtfulness. 
 
 a. An incorrect length of chain is a constant or cumu- 
 lative error, and may be plus or minus. When one is 
 desirous of attaining the last degree of accuracy, this 
 is the most difficult error to eliminate; and even for 
 the accuracy required in ordinary surveying, it is an 
 important element. Recent experience of the author 
 will illustrate the difference Which exists between so- 
 called standards. He had occasion to compare a 100- 
 foot steel tape just from the shop of a reputable manu- 
 facturer, which was " guaranteed to be true to U. S. 
 standard," with a 2o-foot pole said to have been pro- 
 nounced correct by the " U. S. A. engineers," and also 
 with a standard derived with inappreciable error from 
 two 2-foot steel rules made by the best tool-makers in 
 the United States. The errors of the intercomparisons 
 were inappreciable. The tape was 0.95 of an inch short 
 by the first, while by the second it was 0.45 of an inch 
 short. A subsequent comparison at the office of the 
 Mississippi River Commission showed the tape to be 
 0.256 0.07 of an inch short. The above differences 
 were practically independent of temperature correction. 
 
22 CHAIN AND TAPE. [CHAP. I 
 
 After the chain has once been adjusted to the stand- 
 ard, it should be tested frequently. The common chain 
 has 600 wearing surfaces, and if each wears only o.oi of 
 an inch, the length is increased 6 inches. A tempered- 
 steel chain with brazed links lengthened half an inch 
 in chaining 70 miles. It is not uncommon to find chains 
 differing i, 2, or even 3 inches. 
 
 A steel tape is liable to have its length changed each 
 time it is repaired, and is also liable to be permanently 
 lengthened by excessive pull in using, and by hammering 
 it to straighten out short bends. Therefore the steel 
 tape, as well as the chain, should be tested occasion- 
 ally. 
 
 For the method of applying a correction to reduce 
 the chain to the true standard, see 15. 
 
 b. Kinking of the chain and bending of the links are 
 sources of plus cumulative errors; i.e., they tend to make 
 the recorded distance too great. Mud and ice in the 
 joints produce an error in the same direction, while the 
 opening of the rings produces an error in the opposite 
 direction. The net result may be either plus or minus. 
 This source of error is easily avoided by substituting a 
 steel tape or wire for the chain. 
 
 C. If the chain is not stretched straight, the resulting 
 error is cumulative, and tends to make the recorded 
 distance too great. If the center of a roo-foot chain 
 is i foot out of line, the error is 0.02 ft.,* and varies as 
 the square of the error of alignment; that is, if the 
 middle point is half a foot out of line, the error in dis- 
 tance is only 0.005 ft.f The time required to get the 
 chain straight between pins can be greatly lessened by 
 
 * See foot-note page 16. 
 
 t Let the student show that if a point not the middle of the chain is out 
 given amount, the error is a little greater than if the center were out a li 
 amount, 
 
ART. 3] USING THE CHAIN. 
 
 the fore chain-man's being careful to walk on the line 
 being measured. 
 
 d. The effect of a difference in the pull on the tape 
 or chain is compensating. For the tape this error varies 
 directly as the difference of pull, directly as its length, 
 and inversely as its cross-section. To find the elonga- 
 tion of a steel tape, represent its length in inches by Z, 
 and its cross-section in square inches by S; then the 
 
 elongation for a pull of one pound is --- ~ . For 
 
 30,000,0000 
 
 a loo-foot tape 0.2 inch wide by 0.02 inch thick, this 
 gives an elongation of o.oi of an inch per pound of 
 pull. Tapes are sometimes made with a spring balance 
 attached at one end, whereby this source of error can 
 be practically eliminated. 
 
 The elongation of a chain will depend mainly upon 
 the form and size of the rings which connect the sev- 
 eral links together, and can be determined only by 
 trial with a spring balance. 
 
 e. The error due to expansion and contraction is 
 generally cumulative, and may be either plus or minus. 
 Tapes are usually made standard at 60 Fahr. For a 
 loo-foot tape a change of 30 F. makes a difference of 
 one fourth of an inch.* By remembering this relation 
 it is easy to determine the tape correction at any tem- 
 perature. For example, if the tape is true at 60, then 
 at 90 the tape will be 0.02 ft. (one quarter of an inch) 
 too long; i.e., at 90 the tape is 1.0002 times the stand- 
 ard length. At 30 the length of the tape will be 
 0.9998 (= i .0002) times the standard. In ordi- 
 nary work it is only the extremes of temperature that 
 will cause the expansion and contraction of the tape to 
 
 * The co-efficient of expansion varies with the kind and quality of the steel, 
 but closely approximates 0.0000065 per unit of length per i F, 
 
24 CHAIN AND TAPE. [CHAP. I 
 
 be appreciable; but in accurate work this is one of 
 the chief sources of error, and one very difficult to 
 eliminate. Tapes intended for city surveying and 
 other accurate work sometimes have a thermometer 
 attached, by means of which to determine the cor- 
 rections for temperature; but even this is not a perfect 
 remedy, since when the sun is shining the temperature 
 of the tape is often quite a good deal higher than 
 that of the atmosphere. There are several other, but 
 minor, difficulties to be overcome in eliminating this 
 error. 
 
 f. The error due to not setting the fore pin exactly 
 in line is generally overestimated in proportion to the 
 other errors. An undue amount of time is therefore 
 given to lining in the fore chain-man. With a loo-foot 
 chain, if the pin is i foot out of line, the error is 0.005 
 ft. (one sixteenth of an inch). The error varies 
 directly as the square of the error of alignment and 
 inversely as the length of the chain. Consequently the 
 longer the tape the less the error from this cause, and 
 the greater the speed. The error is cumulative and 
 plus. 
 
 Vertical inequalities in the ground produce errors 
 similar to those in aligning the fore chain-man, and 
 unfortunately in many cases this element limits the 
 degree of accuracy attainable. Supporting the tape by 
 hand and plumbing down is of very little advantage. 
 This source of error can be eliminated by suspending 
 the tape between fixed supports. The attempt has fre- 
 quently been made to introduce such devices for city 
 surveying, but they have not met with general favor. 
 They are complicated, cumbersome, expensive to oper- 
 ate, and contribute little, if any, to accuracy. The 
 chief source of error in all such devices that the 
 writer has seen is in plumbing down from the end 
 of the tape. The effect of the wind upon the sag is 
 
ART. 3] USING THE CHAIN. 25 
 
 another important source of error. Such devices are 
 appropriate for the measurement of geodetic base lines. 
 For an illustrated account of a simple, but accurate, 
 method of using a suspended tape, see Annual Report 
 of the Missouri Commission for 1886 Executive Doc- 
 ument No. 28, 49th Congress, 2d Session, pp. 31-35. 
 
 g. Many chain-men hold the pin, while setting it, in 
 such a manner that the point enters the ground consid- 
 erably in front of the end of the chain; then when the 
 rear end is brought forward it is laid on the ground, 
 thus introducing considerable error. This error may 
 occur with a tape, but is usually in the opposite direction. 
 The remedy is obvious. 
 
 The very general practice of placing the forward 
 handle of the chain against one side of the pin and the 
 back handle against the other side, can not be consid- 
 ered precise. The same side of the pin should be used 
 both times; then the length of the chain is from outside 
 of one handle to the inside of the other. A similar crit^- 
 icism is applicable to the way many steel tapes are used 
 (see the last paragraph of 3). 
 
 h. The error due to pulling the pin over after it is set 
 is usually cumulative, and may be either plus or minus. 
 It may be entirely eliminated by care. The back handle 
 should not be dropped over the pin until the chain is in 
 position preparatory to laying off the distance. With a 
 properly made tape (see the last paragraph of 3) this 
 particular source of error can be entirely eliminated; 
 but even then the rear chain-man must be careful that he 
 does not push the back pin over when holding the tape 
 against it. 
 
 i. The amount of error due to the chain's not being 
 horizontal and the method of reducing the resulting 
 error have already been discussed in 17. 
 
 20. Let us see if we can determine the final error 
 due to an assumed value for each of the above sources 
 
26 CHAIN AND TAPE. [CHAP. I 
 
 of error. Notice that, under ordinary conditions, all the 
 errors are cumulative except d and g. If the line to be 
 measured is n chains long, 
 
 the final error = 
 
 n( ab + ce+fht)+Vn(dg). (i) 
 
 According to the theory of probability, equation (i) 
 becomes, 
 
 the probable final error = 
 
 W (a* + & + S + S+f + # + ,*) + n (<? +/). (2) 
 
 To illustrate the method of using this formula, assume 
 that a line 1,000 feet long is to be measured with a steel 
 tape. Assume that the several partial errors are as fol- 
 lows: 0=0.01; ^ = 0.0; *=o.oi; */=o.o2; *=o.oi: 
 /= 0.005; -=0.01; ^ = 0.0; /=o.oi. Substituting 
 these values in equation (2), and solving, 
 
 the probable final error 0.22 ft ... (3) 
 
 The student should carefully consider the degree of 
 care necessary to reduce the several errors to the values 
 assumed above. Are the above values correct relatively? 
 i.e., do they represent equal care in the several opera- 
 tions ? Are the errors correctly classified as cumulative 
 and compensating ? 
 
 Notice that if several measures of a line made with 
 the same tape are to be compared, the error a should not 
 be included in the above equations. Notice also, that 
 if the reduction for grade is computed by the second 
 method of 17 (page 19), the error / should not be in- 
 cluded. Notice again, that if the same men measure the 
 line with the same appliances under the same condi- 
 tions, the probable error deduced from equation (2) 
 
ART. 3] USING THE CHAIN. 27 
 
 should be greater than that deduced from the measure- 
 ments of the distance, for the latter involves only the 
 variation in the errors, while the former depends upon 
 the errors themselves. See Problem No. i, Appendix 
 IV. 
 
 21. LIMITS OF PRECISION. It is very desirable that each 
 engineer should know the uncertainty of his ordinary 
 work. If this could be definitely stated for each method 
 of measuring and for the different kinds of ground, it 
 would be very instructive; but an engineer can learn 
 only by experience the amount of care and time required 
 to attain any particular degree of accuracy. The labor 
 required increases more rapidly than the degree of pre- 
 cision attained, and the more accurate the work the 
 greater the difference. 
 
 A few words are needed as to 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 measure- 
 ments represents the difference between the accidental 
 errors each time, and does not show the real error in the 
 observed length. Any constant or cumulative error, as 
 an incorrect length of the chain, would be the same in 
 each. This distinction is very important in discussing 
 the errors of any series of observations. 
 
 22. The following data are given to assist the 
 student in forming his standard of good work. Unfor- 
 tunately, there is very little on this subject to be found 
 in engineering literature. 
 
 Burt* found in the early surveys of the U. S. public 
 lands, that for common timber-land "with two sets of 
 chainmen instructed alike in the proper manner of 
 keeping their chain level and straight on the line, and 
 of setting the tally pins plumb, as well as holding the 
 
 * Key to the Solar Compass, p. 35. 
 
28 CHAIN AND TAPE. [CHAP. I 
 
 ends of the chain to them, the average difference was 
 i in 500, and sometimes i in 220, and under the most 
 favorable conditions it was i in 1,600.*' 
 
 At present the standard for good chaining in survey- 
 ing the public lands of Canada is a difference of i in 
 5,300 for two sets of men chaining the same line under 
 the same conditions.* " Experience with very careful 
 men shows that chaining on level prairie is about two 
 links to a mile (i in 4,000) longer than over hilly and 
 broken prairie, or over windfall and brush in the 
 woods." * 
 
 The author's students, in ordinary class work in land 
 surveying, measure lines from 200 to 1,000 feet long, 
 with a 66-foot chain, and repeat with a difference of 
 i in 15,000 to i in 20,000 for the same men; for dif- 
 ferent men on different days with different chains 
 (compared, however, with the same standard), the max- 
 imum difference is i in 800 to i in 1,000, the average 
 difference being i in 3,000 to i in 4,000. The ground 
 is favorable, but the work is done with the ordinary 
 expedition of actual practice.f The same students, 
 after a term's practice, with a loo-foot steel tape meas- 
 ure lines 500 to 1,000 feet long and repeat with a 
 maximum probable error for the surface distance of i 
 in 40,000, and an average of i in 103,000 for the same 
 men; and for different men on different days with dif- 
 ferent tapes (compared with the same standard), the 
 maximum error is i in 5,000, and the average error i 
 
 * Report of the Proceedings of the Association of Dominion Land Survey- 
 ors for 1890, p. 58. 
 
 t All values of the degree of precision credited in this volume to the au- 
 thor's students are results attained in the ordinary class work of the first or 
 second term's field practice. Usually the value given is the mean for the 
 whole class. The results are obtained from a specially prepared area the di- 
 mensions of which are accurately known. The lines vary from 30 to 800 feet, 
 and the areas from 0.5 to 7 acres, the longer lines and the larger areas occur- 
 ring more frequently in the problems. 
 
ART. 3] USING THE CHAIN. 29 
 
 in 14,300. For the purpose of this record, eleven pairs 
 of chainmen on different days measured a line 1,000 
 feet long with a probable error for a single measure- 
 ment of the surface distance of i in 7,800; omitting one 
 result, the probable error is i in 10,800. The difference 
 of elevation was 17 feet, and the reduction for grade 
 was 0.288, as determined from an accurate profile of 
 the line; and the probable error of a single estimate of 
 the correction for level (by equation (i), page 16, the 
 student having had no experience in leveling) was i in 
 1 1, 600. The probable error of a single determination 
 of the horizontal distance was i in 6,500, and omitting 
 one result the error was i in 7,400. 
 
 Seventy miles of the finished track of the Illinois 
 Central R. R. was measured in 1876 with a loo-foot steel 
 chain, and re-measured in 1885 with a loo-foot steel 
 tape, with a difference, after correcting for the differ- 
 ence of standard and applying a correction for wear of 
 the chain, on the average for the different miles of i in 
 2,360. 
 
 On the Atchison, Topeka and Santa Fe R. R., 
 through Western Kansas, the difference between the 
 preliminary survey and the government land survey 
 averaged about i in 1,000; and the difference between 
 the preliminary survey and the location was about i in 
 2,500. 
 
 The U. S. army engineers in re-measuring the Union 
 and Central Pacific railroads with steel chains found 
 the error of their own work to average i in 23,600, the 
 maximum being i in 10,000, as determined by retracing 
 distances varying from 2,000 to 26,000.* 
 
 On the U. S. Lake Surveyf base lines for topograph- 
 ical surveys were measured with a chain 20 meters 
 
 * House Executive Document, No. 37, 2d Session, 44th Congress, pp. 14, 
 21, 37- 
 t Report of Chief of Engineers U. S. A., 1876, part III, p. 9. 
 
30 CHAIN AND TAPE. [CHAP. 1 
 
 long, which differed from the ordinary chains only in 
 being made of heavier wire and having links 20 inches 
 long. In 15 lines, the maximum error was i in 5,700, 
 the mean being i in 17,500, and the minimum i in 
 45,100. 
 
 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 i in 32,000. 
 
 The U. S. Coast and Geodetic Survey Report for 
 1882, page 191, contains an account of the preliminary 
 measurement of two base lines with an iron wire J- of 
 an inch in diameter, 200 feet long, in which the differ- 
 ence, as compared with the measurement of the geo- 
 detic base apparatus, was i in 30,000 and i in 28,000. 
 
CHAPTER II. 
 
 TRIPOD, LEVELING SCREWS, AND PLUMB-BOB. 
 
 ART. 1. THE TRIPOD. 
 
 23. CONSTRUCTION. The manner of connecting the 
 leg with the head needs attention. The leg should not 
 be placed between two lugs or ears fastened to the 
 plate, for in case the leg wears or shrinks there is no 
 adequate means of making it fit. Drawing the ears 
 together by means of the screws through the top of the 
 leg bends the plate, and even then only partially reme- 
 dies 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 with a slot 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 thumb-nut. Sometimes the leg is in- 
 serted in a metal cap, which is then fastened to the tri- 
 pod head. This is better than putting the leg directly 
 between two metal ears, but is inferior to all the other 
 forms. No instrument can stand firmly if there is any 
 looseness in fitting of legs or shoes. 
 
 The best instrument makers construct the tripod legs 
 of two pieces braced together, or of a solid piece cut 
 out in such a way as to lighten it without materially 
 affecting its strength. 
 
 Some instrument manufacturers make a tripod with 
 
 31 
 
32 TRIPOD, LEVELING SCREWS, AND PLUMB-BOB. [CHAP. II 
 
 legs that may be made longer or shorter as circum- 
 stances require. In surveying in a mine or tunnel this 
 is a great convenience, if not a necessity. 
 
 24. SETTING THE TRIPOD. In setting the tripod notice 
 that to alter the position of the plumb-line the legs 
 must be swung on their pivots, but not sidewise; and 
 also, that to get the plate level the legs must be swung 
 sidewise, but not on their pivots. These two points, 
 though seemingly trivial 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, which last, owing to faulty construction 
 of the leveling appliances, is frequently very important. 
 Attention to the following principle also will save 
 much time and hard labor in setting the tripod. Let 
 the lines radiating from c y Fig. 4, represent lines join- 
 ing the feet of the tripod legs with 
 the point over which the plumb-bob 
 is to be placed, and b the position 
 of the plumb-bob. It is desired to 
 move the plumb-bob from b to c. 
 Press the leg,J? into the ground until 
 the bob swings to a in the line 3^ pro- 
 longed, and then force leg 3 into the 
 ground until the bob swings to c. In general, press one 
 of the legs into the ground until the plumb-bob swings 
 to the side of the point opposite 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 needlessly. Setting the tripod is an 
 operation that must be repeated many times> and the 
 beginner should learn to do it quickly and easily. 
 
ART. 2] LEVELING SCREWS. 33 
 
 ART. 2. LEVELING SCREWS. 
 
 25. Most instruments are provided with four leveling 
 or foot screws, while others have only three. The in- 
 strument can be leveled more quickly with three screws 
 than with four, since by using both hands the instru- 
 ment can be leveled in both directions at the same 
 time. Three screws are also more sensitive, and 
 are less liable to strain and damage the instrument 
 than four an important consideration with many in- 
 struments, owing to faulty construction of the level- 
 ing appliances. Although the best American instru- 
 ments, and many if riot most European ones, have 
 only three foot screws, nearly all American engineering 
 instruments have four. Doubtless this is partly due to 
 custom, and partly to the fact that for mechanical 
 reasons it is a little easier to arrange the shifting plate 
 of the transit ( in) with four than with three foot 
 screws. 
 
 Whatever the number, there should be no looseness 
 between the screw and the nut. This looseness is par- 
 ticularly objectionable in a leveling instrument or in a 
 transit used to measure vertical angles. To insure 
 steadiness the leveling screws should work in the arms 
 of a solid star-shaped casting, instead of in a thin 
 round plate into which the nuts are simply stuck as is 
 usual. The end of the arm should be split and pro- 
 vided with a clamp screw by which to adjust for wear. 
 
 26. 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 (see Fig. 24, page 95). Then when 
 one pair of screws is being used to level the plate, the 
 upper part of the instrument must revolve about a line 
 
34 TRIPOD, LEVELING SCREWS, AND PLUMB-BOB. [CHAP. II 
 
 parallel to, but below, a line joining the feet of the pair 
 of screws not in use; therefore the instrument can not 
 revolve without causing the screws not in use to bind, 
 and it can turn only by causing the feet of these 
 screws to slip on the lower plate. Besides the annoy- 
 ance in leveling the instrument, this binding tends to 
 bend the screws and warp the plates; and the slipping 
 defaces the instrument by cutting spherical holes in 
 the lower plate. Placing the feet of the screws in 
 small cups prevents the holes in the upper face of the 
 lower plate, but increases the objections in the other 
 and more important respects. Providing the foot of 
 the leveling screw with a ball-and-socket joint is no 
 improvement over the simple cup, except to prevent the 
 cup (socket) from getting lost. 
 
 The above defect may be remedied by bringing the 
 center of the ball-and-socket joint into the plane of the 
 feet of the leveling screws. Since with three leveling 
 screws the instrument can not be attached to the tripod 
 by a ball-and-socket joint, this defect can not exist in 
 that form of construction. With three leveling screws 
 the instrument is fastened to the tripod by a spiral spring. 
 
 Since nearly all instruments have the above defect, 
 it is very necessary that they should be leveled approxi- 
 mately by manipulating the tripod legs as already de- 
 scribed ( 24). 
 
 27. Some instruments are provided with an arrange- 
 ment for approximately leveling the instrument very 
 quickly without manipulating the foot screws, of which 
 there are several forms on the market. A quick-leveling 
 device is specially suitable for leveling instruments, but 
 by using a little care in setting the tripod it can be dis- 
 pensed with easily. All such additions are an advantage 
 in the greater convenience they afford, but a disadvan- 
 tage in the increased weight, complexity, and cost in- 
 volved, 
 
ART. 3 ] 
 
 PLUMB-BOB. 
 
 35 
 
 ART. 3. PLUMB-BOB. 
 
 28. The string employed should be small but strong. 
 The woven ones are best, since they do not twist or 
 untwist. The string should be connected to the in- 
 strument by a hook, so as to be easily attached and de- 
 tached ; but it should be suspended in such a way that 
 the vertical of the plumb-line will always pass through 
 the center of the instrument, however much the lower 
 plate may be out of the horizontal. 
 
 K 
 
 a be 
 
 FIG. 5. METHODS OF ADJUSTING LENGTH OF PLUMB-LINE. 
 
 To allow for an adjustment in the length of the 
 plumb-line, the free end of the line may be passed 
 through the attachment on the instrument and then 
 knotted around the suspended portion, the adjustment 
 being obtained by sliding the knot up and down the 
 string. A better method, particularly with a small string, 
 is to pass it through a small strip of wood, leather, or 
 metal, as shown at a, Fig. 5; or, the end may be fastened 
 
36 TRIPOD, LEVELING SCREWS, AND PLUMB-BOB. [CHAP. II 
 
 to a wire as shown at b, Fig. 5. The latter is the better, 
 since less surface is exposed to the action of the wind. 
 The adjustable plumb-bob shown at c, Fig. 5, is still 
 better but more expensive. This plummet has a con- 
 cealed reel ^, around which the string is wound by turn- 
 ing the milled head K. The friction of the cord through 
 the guide / holds the bob at any desired point on the 
 line. Another form of the last consists of a short spool 
 on top of the plumb-bob, around which the plumb-line 
 is wound. The spool is turned by hand, and held in 
 any position by friction. 
 
 The body of the plumb-bob should be a long slender 
 cone, rather than a short thick one, so that the point 
 may be seen without stooping. The tip of the bob 
 should be of steel, for durability. In case the pointed 
 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, have an assistant mark a line 
 under the instrument in this plane. Repeat the opera- 
 tion at 90 from the first position. The intersection of 
 these two lines is the desired point. 
 
 Owing to defective casting, the plumb-bob may not 
 hang vertically, although of true form and apparently 
 solid. In very accurate work this might cause error. 
 A rough method of testing this is to hold the string in 
 the hand and twist it a little, and while the string is 
 untwisting, lower the point into a basin of water. If 
 the weight is not truly distributed and consequently 
 the plummet not true, the eccentric motion of the point 
 will scatter the water. 
 
 If it is desired to plumb down from a high point, in- 
 stead of using a plumb-line, which is liable to be dis- 
 turbed by the wind, it is better to use a transit accord- 
 ing to a method to K ~ ^nlained farther on ( 125). 
 
CHAPTER III. 
 MAGNETIC COMPASS. 
 
 ART. 1. CONSTRUCTION. 
 
 29. THE ordinary form of the magnetic compass is 
 shown in Fig. 6. 
 
 The value of the magnetic compass as an engineer- 
 ing instrument depends chiefly upon (i) the delicacy of 
 the needle, and (2) the constancy with which it assumes 
 the direction of the magnetic meridian. The first de- 
 pends upon the ease with which the needle turns on the 
 pivot, and the second upon the intensity of the directive 
 force. To satisfy the first condition, there is attached 
 to the center of the needle an agate cup to receive the 
 point on which it turns; and the pivot is made of the 
 hardest steel ground to a smooth, round, sharp point. 
 The second condition is satisfied by making the needle 
 of shear steel and strongly magnetizing it. 
 
 The needle should be of such a length that it shall 
 just clear the graduation, otherwise there will be diffi- 
 culty in reading. Beyond a certain length (about 5 or 6 
 inches) no additional power is gained by increasing the 
 length of the needle, owing to the formation of second- 
 ary poles in long needles. It should not come to rest 
 too quickly; for if it does, it indicates either that the 
 needle is weakly magnetized or that the friction on the 
 pivot is great. The needle should be so sensitive that 
 when it is drawn to one side by the attraction of a piece 
 
 37 
 
MAGNETIC COMPASS. 
 
 [CHAP, in 
 
 of iron, it will settle to the same reading several times 
 in succession. 
 
 A compass is usually provided with a vernier (Chap- 
 ter V) for setting off the magnetic declination. This is 
 
ART. l] CONSTRUCTION. 39 
 
 indispensable in re-surveying land originally laid out 
 with reference to the magnetic meridian; but it is not 
 requisite where the land is laid out according to the 
 U. S. public land system. 
 
 Ordinarily compasses have upon the edges of the 
 sights graduations such that angles of elevation and 
 depression can be measured. This arrangement is par- 
 ticularly valuable for measuring the angle of slopes in 
 applying the correction to reduce slope measurements 
 to the horizontal distance. 
 
 30. Extras. Compasses are sometimes provided with 
 a graduation for reading angles independently of the 
 needle. This is an important addition; but the degree 
 of accuracy attainable is limited by the precision of 
 sighting, which with the open sights is not very great. 
 
 Magnetic compasses are sometimes provided with a 
 telescope instead of sights. This reduces the error of 
 sighting, and increases the length of sight ; but on the 
 whole the advantage is not very great. The chief mer- 
 its of the compass are cheapness, portability, and rapid- 
 ity and facility with which it may be used. It never 
 can be an instrument of great accuracy; and hence the 
 telescope adds cost, weight, and complication, without a 
 compensating accuracy. If greater accuracy is desired 
 than that attainable with the ordinary magnetic com- 
 pass, use the transit. 
 
 Some British instruments have the graduation on a 
 ring attached to the needle and moving with it, the 
 angles being read by a point projecting from the com- 
 pass box. There is no advantage, but great disadvan- 
 tage, in this arrangement, owing to increased weight on 
 the pivot. 
 
 31. PRISMATIC COMPASS. The peculiarity of this in- 
 strument is that a triangular glass prism is substituted 
 for one of the sights. In looking through the prism the 
 distant point and the graduation of the compass are 
 
MAGNETIC COMPASS. [CHAP. Ill 
 
 visible at the same time. The graduation is upon a 
 ring attached to the needle and moving with it, and the 
 bearing is read by a point projecting from the compass- 
 box, under the prism. A mirror is attached to the 
 sight in such a way as to reflect into the prismatic 
 eye-piece points above or below the horizontal plane 
 of the instrument. The prism is sometimes worked 
 
 FIG. 7. PRISMATIC COMPASS. 
 
 convex on the two faces at right angles to each other, 
 so as to magnify the graduation, and is moved up and 
 down to focus it (see Fig. 7). The instrument is some- 
 times provided with colored glass shades for observing 
 the sun. 
 
 Prismatic compasses vary from 2^ to 6 inches in 
 diameter. They are generally held in the hand, al- 
 though sometimes mounted upon a Jacob's staff. The 
 prismatic compass is used in preliminary reconnois- 
 sances, in clearing out lines, in " filling in " in topo- 
 graphical surveying, etc. 
 
ART. 2] TESTS OP THE COMPASS. 41 
 
 ART. 2. TESTS OF THE COMPASS. 
 
 32. THE NEEDLE. The needle should be strongly magnet- 
 ized. If the needle is not strongly magnetized, the 
 directive force will not overcome the friction on the 
 pivot, and hence the needle will not take its proper di- 
 rection. Needles may be re-magnetized with bar mag- 
 nets, as described in the text-books on physics, but such 
 methods are tedious, and give unsatisfactory results. A 
 needle may be quickly and thoroughly saturated with 
 magnetism by putting it into the magnetic field of an 
 electric light or electric railway dynamo. In doing this 
 there is a liability of reversing the poles; and therefore, 
 after having magnetized the needle, place it in the com- 
 pass box or rest it upon a pin held in the hand, and 
 notice if the poles have been reversed. If so, put the 
 needle back into the magnetic field the otherend about. 
 A good needle loses its magnetism very slowly if prop- 
 erly cared for ( 43). 
 
 33. The magnetic axis of the needle should coincide with 
 the axis of figure or line connecting the two ends. An error 
 in this respect affects only the declination to be set off; 
 therefore it produces no error, provided the declination 
 is determined by observing with the instrument on a 
 true meridian. The amount of this error could be de- 
 termined by inverting the needle on the cap and reading 
 in both positions. " The magnetic axis is liable to have 
 its position changed by shocks, in using and transport- 
 ing the compass. This is especially true of freshly 
 magnetized needles." For the above reason, as well as 
 for others which will be discussed in Art. 4, it is best to 
 eliminate the above error by observing the declination 
 with each instrument used. 
 
 34. METAL OF COMPASS-BOX. The metal of the compass- 
 box should contain no magnetic substance. If the metal is 
 
42 MAGNETIC COMPASS. [CHAP. Ill 
 
 not pure, the effect on the needle will be different for 
 different positions of the box, and consequently cause 
 error. Iron is liable to get mixed with the brass in 
 casting. To detect impure metal, set the sights upon 
 some well-defined object and read the needle. Move 
 the vernier (Chapter V), say 10, sight on the object, 
 and note whether the needle has changed the same 
 amount. This operation should be repeated all the 
 way around the circle. If the vernier and the needle 
 change like amounts, the metal of the box is pure. 
 When reading the needle be sure that nothing which 
 will affect it is carried on the person. Watch-chains, 
 buttons, stiffening-wire in the hat rim, and iron rivets 
 in the frame of the magnifier for reading the vernier, 
 etc., may produce error. 
 
 35. SIGHTS. The line of sight should pass through the 
 center of graduation. If this condition is not satisfied, 
 the angles will be sighted from one point and measured 
 at another. The error will vary inversely as the length 
 of sight. There is no probability that the error will be 
 sufficient to affect the work done. However, it could be 
 tested by stretching a fine thread through the sights 
 and observing whether it covers divisions on the limb 
 180 apart. 
 
 36. ZERO OF VERNIER. The zero of the vernier should 
 coincide with the line of sights. If it does not, the proper 
 declination will not be set off, and an error will be pro- 
 duced in the bearings. To test this condition, set the 
 vernier at zero, stretch a fine thread through the sights, 
 and observe whether it covers the zeros of the gradua- 
 tion. If it does not, it will produce no error, provided 
 the declination is determined with the instrument by an 
 observation on a true meridian. 
 
 This condition should be satisfied for the compass on 
 a transit, but there is no rigorous way of making the 
 test. Hence it is more important with the transit than 
 
ART. 3] ADJUSTMENT OF THE COMPASS. 43 
 
 with the simple compass that the declination should be 
 observed with each instrument compare the first line 
 of Table I, page 50, with the other line. 
 
 ART. 3. ADJUSTMENT OF THE COMPASS. 
 
 37. The study of the adjustments of engineering in- 
 struments is a very important part of our subject, as no 
 one is competent to handle an instrument who is not 
 able to determine when it is in adjustment, to adjust it 
 in every particular, and to discuss the effect of any error 
 of adjustment upon the work in hand. Instruments 
 should be examined frequently, for the adjustments, 
 though properly made, are liable to become deranged. 
 
 Nearly all of the adjustments of engineering instru- 
 ments consist in placing certain parts either perpen- 
 dicular or parallel to each other. The usual method in 
 making the various adjustments is that of reversions, 
 which doubles all errors and places them on opposite 
 sides, so that if there is no difference after reversal, there 
 is no error. If there is a difference, the mean of the two 
 positions is the true one. 
 
 As far as possible, the adjustments should be made 
 in such a manner as to be independent of each other. 
 The student should consider carefully the principles 
 involved, and also determine the effect upon subsequent 
 work of an error in the adjustment. The different 
 methods of construction may modify the manner of 
 making the adjustments of any instrument, but the same 
 general principles apply in all cases; and hence the 
 great importance of understanding the principle in- 
 volved, instead of performing the adjustments by rou- 
 tine simply. 
 
 3 8 . LEVELS. The axes of the levels should be perpendicular 
 to the vertical axis of the instrument. The chief reason for 
 
44 MAGNETIC COMPASS. fcHAP. Ill 
 
 demanding this condition is that the needle may play 
 freely when the bubbles are in the center, whatever the 
 direction of the line of sight. 
 
 To make this adjustment, bring the bubble to the 
 middle of the tube (any other point would do equally 
 well, but the center is most convenient) by turning the 
 instrument on the ball-and-socket joint. Turn the in- 
 strument half-way round; then, if the bubble does not 
 stand in the middle, correct one half of the difference 
 by means of the screws at the end of the level tube, and 
 the other half by turning the instrument on the ball- 
 and-socket joint. This operation should be repeated 
 until the bubble will remain stationary in the tube dur- 
 ing a complete revolution of the instrument. If the 
 levels are much in error, it is best to adjust each ap- 
 proximately before completing the adjustment of either. 
 Of course, before pronouncing an adjustment in error, 
 it should be carefully tested. 
 
 39. The above adjustment of the levels is sometimes 
 erroneously called an adjustment " to cause the circle 
 to be horizontal in every position." The adjustment of 
 the levels does not in any way involve the horizontal- 
 ity of the plate. " Leveling the instrument " is really 
 bringing the vertical axis vertical. It is assumed that 
 the plate is perpendicular to the vertical axis, and that 
 consequently when the latter is vertical the former is 
 horizontal. A method of testing the perpendicularity 
 of limb and axis will be given in the chapter on the 
 transit. For the compass the above assumption will 
 produce no appreciable error. 
 
 The compass should not be leveled by the needle. 
 The instrument should be leveled by the levels, and 
 then the needle should be balanced by sliding the coil 
 of wire, which is around the south half, in or out. 
 
 40. SIGHTS. The sights should be in the plane of the ver- 
 tical axis. This is the rigorous requirement, but for the 
 
ART. 3] ADJUSTMENT OF THE COMPASS. 45 
 
 compass it is sufficiently exact to put the sights in a 
 vertical plane, *>., in a plane parallel to the plane of the 
 vertical axis. If the slits are not vertical, sighting 
 through the bottom of one and the top of the other 
 will give a different direction from that obtained by 
 sighting through the bottoms or tops of both. 
 
 To make this adjustment, bring the vertical axis ver- 
 tical by the method of the previous adjustment. Re- 
 move one sight, and range two points, on the side of 
 a building, in the plane of the remaining sight. The 
 farther the points are apart the better. Reverse the 
 instrument on the vertical axis and bring the bottom of 
 the sight in range with the lower point; if the upper 
 point is then in range with the top of the sight, the 
 sight is in a vertical plane. If the upper point is not 
 in range, correct half the error by filing off one side of 
 the bottom of the sight, or by putting paper under the 
 other side. Adjust the other sight in the same way. 
 The plane of the sights is now vertical, but it may not 
 coincide with the plane of the vertical axis. The above 
 method of correcting the error is best, but in case of 
 necessity it may be done by holding both sights and 
 bending the plate as needed. With fair use the sights 
 should not get out of adjustment. 
 
 41. NEEDLE, a. The ends of the needle and the point of 
 the pivot should be in the same horizontal plane. This con- 
 dition is required so that the ends of the needle may 
 remain stationary even though the needle may roll or 
 quiver on the pivot. To make this adjustment, bend 
 the ends of the needle up until they remain steady even 
 though the middle of the needle may swing. 
 
 b. The ends of the needle and its center should be in the 
 same vertical plane. If this condition is not satisfied, the 
 readings of the two ends of the needle will not agree. 
 To test it, read both ends, and revolve the box until the 
 north end reads what the south end did. If there is a 
 
46 MAGNETIC COMPASS. [CHAP. Ill 
 
 difference between the second reading of the south end 
 and the first reading of the north end, correct half of 
 the difference by bending the needle. 
 
 42. CENTER-PIN. The pivot on which the needle swings 
 should be in the center of the graduated circle. If the pivot 
 is not in the center, either the needle will strike the 
 graduation and not turn freely, or the end will be so 
 far from the graduation that it can not be read pre- 
 cisely. Furthermore, if this condition is not satisfied, 
 a bearing read from the north end of the needle will 
 not agree with the value obtained from the south end. 
 To test this adjustment read one end of the needle in 
 four positions 90 apart ; if the readings of the other 
 end differ by 90, the pivot is in the center. If the 
 readings of the second end do not differ by 90, read 
 both ends of the needle again, move the box until one 
 end has passed over 90 and then bend the pivot until 
 the other end shall have passed over 90. Turn the 
 box about 90 and repeat the last operation. Test the 
 work by trying points half-way between those pre- 
 viously used. 
 
 This adjustment can be made, but less elegantly, by 
 using half the length of the needle to measure the dis- 
 tance from the pivot to the graduation. 
 
 This adjustment should be examined every time the 
 pivot is taken out. Notice that the adjustment of the 
 needle and of the center-pin are entirely independent, 
 text-books to the contrary notwithstanding. 
 
 ART. 4. USING THE COMPASS. 
 
 43. CAKE. It is quite important that the engineer 
 should understand the means by which he can prolong 
 the usefulness of his instrument. In the compass the 
 main thing is to avoid dulling or breaking off the fine 
 point of the pivot; consequently never jolt or carry 
 
ART. 4] USING THE COMPASS. 47 
 
 the compass without being sure that the needle is 
 lifted off the pin. Remember that the harder and 
 more perfect the point, the more liable it is to injury. 
 In lowering the needle do not let it fall upon the pivot. 
 To prevent unnecessary wear, check the vibrations of 
 the needle on letting it down, by lifting it off the 
 point a little; or, better, turn the needle in about the 
 proper direction before letting it down. 
 
 When the pivot becomes dulled it can be sharpened 
 on an oil-stone. Care should be taken to obtain a coni- 
 cal and not a pyramidal point. The sharpness of a 
 needle is easily ascertained by sliding the thumb-nail 
 over the point, at an angle of about 30 to it. If the 
 point sticks and holds the nail, it is sharp; if it glides 
 upon it, it is dull. Unnecessary grinding of the pivot 
 should be avoided, for if it becomes much shortened, 
 the ends of the needle will come below the graduation, 
 thereby producing parallax in reading. This parallax 
 could be avoided by bending up the ends of the needle, 
 but this would destroy the condition that the two ends 
 of the needle and the pivot should be in the same hori- 
 zontal plane (see a, 41). 
 
 Never allow the needle to be played with by at- 
 tracting it with a knife, piece of iron, etc.; for every 
 passage of a piece of steel or iron removes a portion of 
 the magnetism. When the compass is not in use, it is 
 best to let the needle assume its normal position in the 
 magnetic meridian, for in this position it will longer re- 
 tain and even increase its polarity. After the needle has 
 assumed this position it should be raised against the 
 glass. 
 
 44. PRACTICAL HINTS. If the needle is sluggish in its 
 movements and settles quickly, either it has lost its 
 magnetic force or it has a blunt pivot; and in either 
 case it is likely to settle considerably out of its true 
 position. The longer a needle is in settling, the more 
 
48 MAGNETIC COMPASS. [CHAP. Ill 
 
 accurate will be its final position. It can be quickly 
 brought very near to its true position by checking its 
 motion by means of the lifting screw; but in its final 
 settlement it must be left free. 
 
 The glass cover may become electrified from friction 
 and attract the needle. This electricity can be dis- 
 charged by touching the glass with a wet finger, or by 
 breathing upon it. 
 
 45. SOURCES OF ERROR.* The errors of compass work 
 may be classified as (i) local attraction, (2) instrumental 
 errors, and (3) observational errors. 
 
 46. Local Attraction. A common method of taking 
 the bearings with the compass is to set it at each cor- 
 ner or station in succession, and take the forward bear- 
 ing. If there is any local attraction of the needle, i.e., 
 if the needle is deflected from its normal direction by 
 iron, etc., the bearings will be incorrect. A method of 
 eliminating all local attraction is described in Appendix 
 I, and hence this source of error need not be discussed 
 here. 
 
 47. Instrumental Errors. Errors of this class are due 
 either to imperfect adjustment of the instrument or to 
 sluggishness of the needle. It is a good rule to adjust 
 an instrument 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 in adjustment. 
 
 Guard against errors of coincidence of magnetic and 
 geometrical axes of the needle ( 33), and also against 
 errors in straightness of needle ( 41), by reading al- 
 ways the same end of the needle. 
 
 48. In work involving the magnetic declination, to 
 guard against the possibility (i) that the magnetic axis 
 of the needle may not coincide with its geometric axis, 
 
 * See discussion ot Cumulative vs. Compensating Error, 18, 
 
ART. 4] USING THE COMPASS. 49 
 
 (2) that the zero of the vernier may not coincide with 
 the line of sight, and (3) that the needle is not straight, 
 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 generally nearly zero. 
 
 It is very important that the three sources of error 
 just mentioned should be eliminated. For one or the 
 other or all of these reasons, the bearing of a line as read 
 from several instruments at the same time and place, 
 by the same person, will often differ considerably. In 
 one instance,* the magnetic declination as read from 
 five instruments by four men at the same time and 
 place differed 20 minutes. As nearly as can be deter- 
 mined from so few observations, the probable error (see 
 Appendix III) of a single determination of the declina- 
 tion is 5^ minutes. In another instance,* the readings 
 of four men from four instruments read at the same 
 time and place differed 31 minutes, with a probable 
 error for each observation of 9.1 minutes. The follow- 
 ing is from a prominent instrument maker :f "I made 
 six needles as near alike as I could. I then set up a 
 compass and directed it toward a certain fixed object. 
 Then I placed these needles, made at the same time of 
 the same kind of material, on the center-pin one after 
 another. Three of these gave the same reading, but the 
 other three varied from 5 to 10 minutes. The result 
 was indeed surprising, for I had taken great pains to 
 have the needles all alike." 
 
 * Report of the Ohio Society of Surveyors and Engineers for 1884, p. 71. 
 fid., p. 74. 
 
5 
 
 MAGNETIC COMPASS. 
 
 [CHAP, in 
 
 In the course of ordinary class instruction, the author 
 had his class in land-surveying, consisting of eleven 
 members, determine the magnetic declination by ob- 
 serving upon a true meridian with six instruments at 
 the same time, each man observing with each instrument. 
 At the time of making the reading no one knew what 
 the others had read. The instruments were in good 
 adjustment in every particular. The following table 
 shows the mean declination for each instrument, and 
 also the probable error* of observation. 
 
 TABLE I. 
 
 DIFFERENCE IN MAGNETIC DECLINATIONS OBTAINED WITH DIFFER- 
 ENT INSTRUMENTS. 
 
 No. 
 
 Instrument. 
 
 Length of 
 Needle. 
 
 Mean 
 Declination. 
 
 Probable 
 Error of a 
 Single 
 Observation. 
 
 I 
 
 Black transit 
 
 5 inch 
 
 5 42' 
 
 I .5 
 
 2 
 
 Yellow " 
 
 c ' 
 
 4 27' 
 
 2 .8 
 
 q 
 
 Mining " 
 
 3 ' 
 
 4 33' 
 
 I .6 
 
 
 Black compass 
 
 6 ' 
 
 4 4O 
 
 i .6 
 
 5" 
 
 Yellow " 
 
 6 ' 
 
 4 m' 
 
 I .2 
 
 6 
 
 Old " 
 
 5 
 
 4 37' 
 
 l'.2 
 
 
 
 
 
 
 
 
 
 4 484-' 
 
 i'.7 
 
 
 
 
 
 
 Since each value of the declination in the above table 
 is the result of eleven observations, the error of obser- 
 vation is practically eliminated, and hence the difference 
 in the declinations is due almost solely to a difference 
 in the instruments. The above results show that the 
 declination determined by a single observation has a 
 probable error ( 2 of Appendix III) of 1.7 minutes due 
 
 * Since the readings were made, as a rule, only to the nearest 5 minutes, it 
 is not strictly correct to compute the probable error of such results. The prob- 
 able errors are given in this case to show that the difference in the declination 
 was due to the instruments, and not to the reading. 
 
RT. 4] USING THE COMPASS. 5 I 
 
 to the observer and 13.4 minutes due to the instrument; 
 or a total probable error of 13.5 minutes. In other 
 words, if both instruments are in good working condi- 
 tion and skilfully used, we may expect a difference in 
 the [bearing of the line due to differences in the in--" 
 struments, of 13.4 minutes (21 feet in a mile), as 
 read by two instruments at the same time and place. 
 Notice that the greatest difference between the declina- 
 tions is i 15' (120 feet in a mile). This source of error 
 is generally disregarded, and frequently it is impossible 
 to do otherwise; but it should be continually borne in 
 mind as a possible source of error. 
 
 49. Observational Errors. These may be divided into 
 (i) errors in sighting and (2) errors in reading. There 
 is a possibility of error owing to the compass or flag- 
 pole's not being set at exactly the right point; but this 
 can only occur on short sights, and with reasonable care 
 will not occur at all. 
 
 1. To eliminate any residual errors due to the plate's 
 not being level ( 38) or to the sights' not being perpen- 
 dicular to the plate ( 40), sight through the top or bot- 
 tom of both sights. To insure the slits' being vertical, 
 give the most care to the bubble perpendicular to the 
 line of sight. Care must be taken that the flag-pole is 
 seen through both slits, and not through one slit and 
 one hole. The flag-pole should be set vertical ; but, as 
 a precaution, sight as low on it as possible. 
 
 2. After the needle has come to rest it should be 
 tapped gently to destroy the effect of any adhesion to 
 the pivot. In reading the needle beware of magnetic 
 substances on the person as wire in the hat-brim, watch- 
 chains, rivets in the handle of the magnifying-glass, etc., 
 which may affect the needle. The most common errors 
 made in practice are such as reading 28 for 32, 30^ 
 for 29^, etc.; and reading N. for S., etc. The remedy 
 is obvious. 
 
52 MAGNETIC COMPASS. [CHAP. Ill 
 
 The common practice is to read the bearings to the 
 nearest quarter-degree. The accuracy of the work can 
 be appreciably increased by estimating the fraction of 
 a degree to the nearest five minutes. Since compasses 
 are ordinarily graduated to half-degrees, this necessi- 
 tates the estimation of sixths of a division. ' With a 
 little thoughtful attention this can be done with con- 
 siderable precision, and the increased accuracy is well 
 worth the extra time required. 
 
 It is sometimes recommended that the vernier be 
 used to read the fractions of a degree ; but a trial will 
 show that this is no advantage. The reasons are obvious. 
 It has also been proposed to place a light paper vernier 
 on the south end of the needle. Concerning all such 
 devices remember that if the needle were read without 
 any error at all, the magnetic compass would not be an 
 instrument of any considerable precision. 
 
 50. LIMITS OF PRECISION. With a compass graduated 
 to half-degrees, the angles can be estimated to the 
 nearest 5 minutes, in which case the maximum error of 
 the bearing should not exceed 10 minutes. The average 
 error of a bearing by the author's students, as deduced 
 from the errors of closing the angles around a field 
 ( 8 of Appendix I), the average length of sight being 
 about 300 feet, is 3 minutes.* The average error for ten 
 selected men, using a highly magnetized needle and a 
 sharp pivot, was 2.5 minutes. The same men read, for 
 the purpose of this record, the bearing of a flag-pole at 
 100, 200, and 300 paces, with average errors of 4, 5, and 
 5.2 minutes, respectively, the sun and wind being in the 
 observer's face while sighting. 
 
 51. The error of an area found by compass surveying 
 is made up of the errors of chaining, of reading the 
 
 * The error per sight is equal to the total angular error in closing a field, 
 divided by the square root of twice the number of sides, 
 
ART. 4] USING THE COMPASS. 53 
 
 bearings, and of the computations. With proper care, 
 the error of chaining should be much less than the error 
 of the bearings. Since i in 57.3 corresponds to i, the 
 above maximum error of 10' in reading an angle corre- 
 sponds to an error of i in 344 ; and the average error of 
 3', as above, corresponds to i in 1,146, which is a greater 
 error than that of chaining under the same conditions 
 (see fourth paragraph of 22). Since the computations 
 are self-checking at nearly every step, there is little 
 probability of any material undetected error in this part 
 of the work ; and as the work may be carried to any 
 desired number of decimals, the inaccuracies of the 
 computations can be made much less than those of the 
 field-work. We conclude, therefore, that the accuracy 
 of the area depends mainly upon the accuracy of read- 
 ing the needle. 
 
 The average error in area of eighty problems solved 
 by thirty-two (the best of thirty-six) of the author's 
 students in ordinary class-work* was i in 1,530.! 
 
 52. BALANCING THE LATITUDES AND DEPARTURES. An 
 
 answer can now be given to the following question, 
 which is frequently asked : How great a difference 
 between the sums of the + and latitudes and de- 
 partures is admissible ? 
 
 Let C represent the linear error of closure due to the 
 chaining, P the perimeter of the field, d the distance 
 in which the error of chaining is a unit ; then 
 
 * See foot-note, p. 28. 
 
 t For the sake of comparisons, it may be interesting to know that under 
 the same conditions the error of areas obtained with the chain alone was i in 
 1,520; and with a home-made plane table ( 172) the error was by radiation 
 ( 182) i in 586, by traversing ( 184) i in 826, and by radio-progression ( 187) 
 i in 1,111. 
 
54 MAGNETIC COMPASS. [CHAP. Ill 
 
 Let A represent the lineal error due to the measure- 
 ment of the angles ; and a the angular error of meas- 
 uring the angles, i.e., the difference between the last 
 fore-sight and the first back-sight,* in minutes. It 
 can not be known how this error occurred, whether all 
 in one sight, or equally among the sides, or among the 
 sides in proportion to their length ; but as the last as- 
 sumption 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 equiva- 
 lent, we must multiply it by the length of the perimeter 
 (= P) and divide it by the distance at which a unit 
 subtends an angle of one minute, or 
 
 A = a -- , nearly. ... (2) 
 3,438 10,000' 
 
 Let E = the total error due to chaining and to 
 measuring the angles. Notice that the error E is the 
 hypotenuse of a right-angled triangle of which the 
 differences of the latitudes and departures are the other 
 sides. Hence, if L = the difference of the -j- and 
 latitudes, and D = the difference between the -}- 
 departures, E V D l -f L\ 
 
 By the theory of probabilities we know that 
 
 -+ V nearl y- 
 
 1 2, 000,0007 
 
 See 8 of Appendix I. 
 
ART. 4] USING THE COMPASS. 55 
 
 I Equating these two values of , we get 
 
 12000000 
 
 As a rule, equation (4) is the one to be used in prac- 
 tice ; but 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 computations, or it can 
 be estimated in the field or from the plat with sufficient 
 accuracy. Then, from the theory of probabilities, 
 L = D Vn. Equation (4) then becomes 
 
 (s) 
 
 12,000,0007 
 
 53. To illustrate the method of applying the preced- 
 ing formulas, let us assume that in surveying a field 
 whose perimeter is 10 chains the difference between 
 the first back-sight and the last fore-sight was 10 
 minutes. We will also assume that the conditions 
 were such that we might expect an error in chaining of 
 i in 2,000. Equation (4) then becomes 
 
 = 1 = .00086. . . (6) 
 
 40,000 1,200 
 
 If the field is twice as long north and south as east 
 and west, then n = 2. Substituting in equation (6) and 
 reducing, we get 3 D* = 0.00086 chains ; therefore, 
 Z> = 0.017 chains = 1.7 links. This shows that the 
 
56 MAGNETIC COMPASS. fcttAP. Ill 
 
 error in the departures should be about 1.7 links. The 
 error in the latitude should be 1.7 Vn = 1.7 ^2 = 2.4 
 links. Results much greater than these show an error 
 in the work other than the usual inaccuracy. 
 
 Finally, knowing the error of closing the angles and 
 the errors in balancing the latitudes and departures, we 
 may reverse the problem and compute the error of 
 chaining. To do this, notice that all quantities in equa- 
 tion (4) would then be known except d> which could 
 then be computed. 
 
CHAPTER IV. 
 SOLAR COMPASS.* 
 
 54. THE solar compassf is the result of an attempt to 
 overcome the defects of the magnetic compass due to 
 the variation in the magnetic declination. The solar 
 compass determines lines with reference to the sun; 
 and hence the bearings are independent of any changes 
 of the magnetic needle. The solar compass consists of 
 an ordinary magnetic compass, to which is attached an 
 apparatus for sighting at the sun, briefly called the solar 
 apparatus. 
 
 55. PRINCIPLE OF THE SOLAR APPARATUS. In its most 
 
 elementary form the solar apparatus consists of a right 
 line parallel to the axis of the earth, to which is 
 pivoted another right line which is free to move only 
 in the plane of the first line. For convenience of 
 explanation, call the first line the polar axis, and the 
 second the solar sight-line. 
 
 If the polar axis is parallel to the earth's axis, and 
 the solar sight-line is directed towards the sun, then, as 
 the solar sight-line is revolved about the polar axis, it 
 will trace on the sky the diurnal path of the sun. But, 
 if the polar axis is not parallel to the axis of the earth 
 and the angle between the axis and the sight line re- 
 mains the same as before, it will not be possible to 
 
 * For a discussion of the Solar Transit, see Chapter VIII. . 
 t Invented by Wm. A. Burt, a U. S. public-land surveyor of Michigan, in 
 1836. 
 
 57 
 
58 SOLAR COMPASS. [CHAP. IV 
 
 make the sight line point to the sun.* In other words, if 
 the polar axis of the solar apparatus is parallel to the 
 axis of the earth, and if the sight line makes an angle 
 with the perpendicular to the polar axis equal to the 
 declination of the sun, then the sight line can be 
 brought to bear upon the sun only when the plane of the 
 two lines coincides with the true meridian. This is the fun- 
 damental principle of all solar apparatuses, however 
 much they may differ in detail. 
 
 The solar apparatus, then, consists (i) of a device for 
 making the polar axis parallel to the axis of the earth, 
 i.e., for giving the polar axis an angle of elevation 
 equal to the latitude of the place of observation; (2) of 
 a device for setting off an angle, between the perpen- 
 dicular to the polar axis and the solar sight-line, equal 
 to the declination of the sun; and (3) of some device 
 for revolving the solar sight-line about the polar axis. 
 If the solar apparatus is attached to a common compass 
 or transit in such a manner that the plane of the polar 
 axis is parallel to the plane of the terrestrial sight-line 
 of the instrument, then when the latitude of the place 
 of observation and the declination of the sun are cor- 
 rectly set off, and the solar sight-line is directed to the 
 sun, the terrestrial sight-line will indicate a true me- 
 ridian. 
 
 ART. 1. CONSTRUCTION OF THE SOLAR COMPASS. 
 
 56. The usual form of the solar compass is shown in 
 Fig. 8. It consists essentially of three arcs of circles 
 by which the latitude of a place, the declination of the 
 sun, and the hour of the day may be set off. These 
 
 * Strictly, if these conditions are not fulfilled the solar sight-line can be 
 made to point toward the sun but once, or at most only twice, during the 
 twenty-four hours, and then only for an instant. 
 
ART. l] CONSTRUCTION OF THE SOLAR COMPASS. 59 
 
 arcs are designated in the cut by the letters a, b, and c, 
 respectively. 
 
 The latitude arc a has its center of motion in two 
 
 FIG. 8. SOLAR COMPASS. 
 
 pivots, one of which is shown in Fig. 8, at d. The lati- 
 tude arc is moved either up or down within a hollow 
 arc, seen in the cut, by a tangent screw at /, and may 
 be fastened in any position by a clamp screw e. The 
 
60 SOLAR COMPASS. [CHAP, iv 
 
 vernier of the latitude arc usually reads to single min- 
 utes. 
 
 The declination arc b has a range of about 25, and 
 is read by a vernier to minutes. The arm carrying 
 the vernier is moved over the surface of the declination 
 arc, and its vernier set to any reading, by turning the 
 head of the tangent screw k. The vernier is clamped in 
 any position by a screw, not seen in Fig. 8. 
 
 At each end of the collimation arm h is a rectangular 
 block of brass. In the upper of these blocks is set a 
 small convex lens, having its focus on the surface of a 
 little silver or ivory plate fastened to the inside of the 
 opposite block. On the surface of the plate are two 
 pairs of lines intersecting each other at right angles. 
 The interval between the lines of each pair is just 
 sufficient to include the circular image of the sun as 
 formed by the solar lens. 
 
 Each end of the arm h has both a lens and a silver or 
 ivory plate. As shown in Fig. 8 the compass is set for 
 south declination; and when the sun has north declina- 
 tion, the declination arc and the solar line of sight are 
 turned 180 on the polar axis, and the observation is 
 made with the lens and plate on the end of the arm // 
 opposite, respectively, to those used for a south declina- 
 tion of the sun. Thus there are really two solar sight- 
 lines one for north and one for south declination. 
 
 57. When the instrument is leveled, and the co-lati- 
 tude is set off on the arc a, and the declination of the 
 sun* is set off on the arc b t if the whole instrument is 
 revolved about its polar axis,/, until the image of the 
 sun formed by the lens in the upper end of the collima- 
 tion arm, ^, falls in the middle of the lines on the plate 
 on the lower end of the collimation arm, then the terres- 
 trial line of sight of the instrument will be on a true meridian. 
 
 * After being corrected for refraction ( 159). 
 
ART. 2] ADJUSTMENT OF THE SOLAR COMPASS. 6l 
 
 The angle between any course and the true meridian 
 may be read by the graduation on the horizontal plate 
 of the instrument. 
 
 58. As solar work can be performed only during clear 
 weather, the instrument is provided with an ordinary 
 magnetic needle shown at n in Fig. 8 by which to 
 run lines in cloudy weather. The needle may be used 
 also to determine the magnetic declination. Instead of 
 sights the solar compass is sometimes provided with a 
 telescope; but this addition is of little, if any, advan- 
 tage. 
 
 ART. 2. ADJUSTMENT OF THE SOLAR COMPASS. 
 
 59. The following adjustments of the solar compass 
 must be carefully attended to: i. The plane of the 
 plate bubbles should be perpendicular to the vertical 
 axis. 2. The two solar sight-lines should be parallel. 
 3. The declination arc should read zero when the solar 
 sight-line is perpendicular to the polar axis. 4. The 
 latitude arc should read the latitude when the polar 
 axis is parallel to the axis of the earth.* 5. The terres- 
 trial line of sight and the solar line of sight should lie 
 in the same vertical plane. 
 
 The method of making these adjustments will not be 
 described here, for the following reasons: i. It will not 
 be difficult for any one familiar with the methods of 
 adjusting engineers' field-instruments to devise ways of 
 making these adjustments. 2. These adjustments are 
 similar to those of the solar transit to be described 
 presently. 3. On account of the defects of the solar 
 compass it is better to use either the magnetic compass 
 (Chapter III) or the solar transit (Chapter VIII). 
 
 * If, however, the value of the latitude used with the instrument be that 
 obtained by it, then no attention need be paid to this adjustment. It is im- 
 portant only when the true latitude is used in finding the meridian, or when 
 the true latitude of the place is to be found by the instrument. 
 
62 SOLAR COMPASS. [CHAP. IV 
 
 ART. 3. USING THE SOLAR COMPASS. 
 
 60. MERITS AND DEFECTS OF THE SOLAR COMPASS. 
 
 Merits, i. The chief merit claimed for the solar com- 
 pass is that lines run by it are not affected by vari- 
 ations of the magnetic needle ; but the common com- 
 pass can be used (Appendix I) so as to be absolutely 
 independent of variations of the needle, and hence this 
 claim has no force. 2. A second merit claimed is that 
 the solar compass enables the surveyor to obtain a true 
 meridian very easily ; but as there are several other 
 ways of finding a true meridian, this advantage is not 
 very great. 3. It is also claimed that the solar compass 
 is peculiarly valuable in the U. S. public-land surveys 
 in running parallels of latitude ; but as the transit is 
 more accurate and more convenient for this purpose, 
 this point is not well taken. 
 
 61. Defects. The solar compass has the following 
 defects: i. It can be used only when the sun is shining. 
 At other times it can be no more accurate than the mag- 
 netic compass, while it is much heavier and more com- 
 plicated. 2. Owing to its many adjustments and the 
 difficulty of making and keeping them, the solar com- 
 pass is not an exact instrument even in clear weath- 
 er. " A state boundary-line run with the solar compass 
 and intended to be straight was found to deviate ten 
 minutes when re-run with a transit." 3. To use the 
 solar compass, the engineer must know his latitude, the 
 declination of the sun for each hour of the day, the time 
 of day, and the refraction for all altitudes; and the 
 inevitable inaccuracies, not to consider large errors, in 
 these data render work done with the solar compass 
 unreliable. For example, an error of one minute in the 
 declination may cause an error of twelve minutes in the 
 direction of the supposed meridian. 
 
ART. 3] USING THE SOLAR COMPASS. 63 
 
 62. HISTORY. The solar compass was invented by 
 Wm. A. Burt, a deputy U. S. land-surveyor in the Lake 
 Superior mineral region, in 1836 about the time of the 
 introduction of the American transit. Previously the 
 magnetic compass and the heavy and inconvenient 
 English theodolite were the only angle instruments in 
 general use in this country. The solar compass was 
 much lighter and more convenient than the theodolite, 
 and more accurate than the magnetic compass as *^n 
 made and used. This probably accounts for the favor 
 with which the solar compass has been received in the 
 past. In popular estimation it is an instrument of great 
 precision, while in reality it is little, if any, more accu- 
 rate than the magnetic compass. It is certainly neither 
 so accurate nor capable of so great a variety of work 
 as the engineer's transit, which is properly employed 
 where formerly the solar compass was used. 
 
 The solar compass is certainly a very ingenious in- 
 strument, and reflects more credit upon the inventor than 
 upon the modern user. The form shown in Fig. 8 is 
 the one in use at the present time, and is essentially the 
 form originally given to it by the inventor. 
 
 We will not further discuss this instrument, but refer 
 the reader to Burt's " Key to the Solar Compass," in 
 which the inventor fully describes the adjustment and 
 use of his instrument. 
 
CHAPTER V. 
 
 VERNIERS. 
 
 63. A VERNIER* is a short scale movable by the side 
 or 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 a little longer than a division of the main 
 scale or limb. This small difference is the unit of the 
 space measured by the vernier. 
 
 64. PRINCIPLES. 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 parts, one more or one less than 
 the number into which the same length 
 on the limb is divided. For example, 
 the limb shown in Fig. 9 is a scale of 
 inches divided into tenths, and the di- 
 visions of the vernier are of such a 
 length that ten spaces on the vernier 
 are equal to nine on the limb or main 
 scale. Therefore each space on the 
 vernier is equal to o.i of 0.9, or 0.09, of 
 an inch; that is, each space on the 
 vernier is o.oi of an inch shorter than 
 a space on the main scale. Line i 
 on the vernier falls short of a line on 
 the limb by o.oi of an inch, line 2 
 falls short by 0.02 of an inch, and so on. If the vernier 
 
 FIG. 9. 
 
 * Invented about 1631 by Pierre Vernier of Burgundy. 
 
 64 
 
PRINCIPLES. 65 
 
 be moved slowly forward, the successive coincidences 
 of a line on the vernier with one on the limb will indi- 
 cate successive advances of the vernier, each equal to 
 o.oi of an inch. If, then, the lines of the vernier are 
 numbered as in Fig. 9, the number on the vernier of the 
 line which coincides will indicate the distance that the 
 zero of the vernier has passed a division of the limb. 
 
 65. Direct and Retrograde Verniers. In the above 
 illustration the spaces on the vernier were shorter than 
 those on the limb, the supposed motion was in the 
 direction of the graduation of the limb, the successive 
 lines of the vernier came into coincidence in the 
 direction of the graduation, and the numbering on the 
 limb, and also on the vernier, increased in the same 
 direction. A vernier fulfilling these conditions is 
 called a direct vernier. 
 
 If the spaces on the vernier are larger than those on 
 the limb, and the vernier is moved in the direction of 
 the graduation of the limb, the successive coincidence 
 will occur in the direction opposite to the motion, and 
 also opposite to the direction of the graduation on the 
 limb; and therefore the lines on the vernier should be 
 numbered in an opposite direction to those on the limb. 
 A vernier fulfilling these conditions is called a retrograde 
 vernier. For an example, see Fig. n, page 67. 
 
 66. Least Count. The least count of a vernier is the 
 difference in length between a space on the limb and 
 one on the vernier. To find the least count of a vernier, 
 z.e., Jo determine how small a distance it can measure, 
 let 7= the length 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, n I = n v /, solving which gives / v =. = 
 
 the least count. For example, in Fig. 9, /= o.i, and 
 
 / o.i 
 
 n = 10: hence the least count equals = -- = o.oi. 
 
 n 10 
 
66 VERNIERS. [CHAP, v 
 
 The preceding formula expresses a very important 
 relation, since it is the key to reading all verniers, 
 whether direct or retrograde. Notice that the lea^t 
 count of the vernier is equal to the smallest division 
 on the limb divided by the number of spaces on tne 
 vernier. In practice, it is not necessary to count n 
 it is indicated by the numbering on the vernier itser. 
 For exarnple, if the limb is divided to half-degrees aua 
 there are thirty spaces on the vernier (which would be 
 indicated by the end line being numbered thirty), the 
 vernier reads to one thirtieth of a half-degree or to 
 minutes. 
 
 67. To READ A VERNIEE. 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 read- 
 ing, the vernier divisions not being required; 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 vernier till 
 a line is found which exactly coincides in direction 
 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. 
 The particular division on the limb that may be in coin- 
 cidence is of no consequence. 
 
 68. Examples. A number of examples will now be 
 given to further illustrate these general principles. 
 The student should draw the limb and the scale on sep- 
 arate slips of paper or card-board, and move one beside 
 the other until he can read them in any position. 
 
 The vernier shown in Fig. 10 is the one used on the 
 New York leveling rod. The main scale is divided to 
 feet, tenths, and hundredths, and ten divisions of the 
 vernier are equal to nine of the limb ; therefore the 
 vernier reads to tenths of a hundredth, or to thou- 
 
TO READ A VERNIER. 
 
 sandths of a foot. The vernier as drawn reads 3 feet, 
 o tenths, 3 hundredths, and 6 thousandths, or 3.036 
 feet. 
 
 Fig. ii is a retrograde vernier which reads to hun- 
 aredths of an inch, the limb being divided to inches 
 
 FIG. 10. 
 
 FIG. ii. 
 
 and tenths. It is sometimes applied to barometers. 
 The reading is 15.64 inches. 
 
 Fig. 12 shows part of a circle graduated to degrees 
 and half-degrees. The vernier has thirty parts, and 
 therefore it reads to minutes. The reading is 212 30' 
 -f- ii' 212 41'. This is a very common vernier for 
 engineering instruments. 
 
 The graduation of transits usually has two rows of 
 numbers, one increasing to the right and the other to 
 the left, in which case there are two verniers one for 
 each series of numbers. Such an arrangement is shown 
 in Fig. 13, and is called a double vernier. The reading 
 of the left-hand vernier and the inside row of numbers 
 is 182 40', while that of the right-hand vernier and the 
 outer row of numbers is 177 20'. 
 
68 
 
 VERNIERS. 
 
 [CHAP, v 
 
 With a double vernier there is sometimes a doubt 
 as to which vernier should be read. To settle this, 
 notice whether the vernier divisions are larger or 
 smaller than those on the limb. If the spaces on the 
 
 FIG. 12. 
 
 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 
 
 FIG. 13. 
 
 vernier, and that vernier should be read the numbers 
 of which increase in the opposite direction to those on 
 the limb. Or the vernier to be read can always be de- 
 termined as follows: Move the vernier, until, say, the 
 ten line of one vernier and the twenty of the other 
 each coincide 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. 13 the numbers on the vernier are inclined in the 
 same direction as the numbers on the limb to which the 
 
TO READ A VERNIER. 
 
 6 9 
 
 vernier belongs. Instruments are not generally made 
 in this way, but such an innovation would be a great 
 improvement. 
 
 FIG. 14. 
 
 Fig. 14 is another form of double vernier. It is 
 called a double folded-vernier, and is often applied to 
 the compass, to be used in setting off the declination. 
 Notice that this form is only half 
 as long as the double vernier of 
 Fig. 13. It is used where there is 
 not space enough for the longer 
 The lower numbers on one 
 
 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 paragraph. The 
 vernier as drawn reads 2 2' to the 
 right. Notice that the reading of 
 the vernier is 2', and not 28'. 
 
 Fig. 15 represents the scale and 
 vernier of the mountain barometer. 
 The limb is .divided into inches, 
 tenths, and half-tenths or five- 
 hundredths. There are 25 spaces 
 on the vernier, and therefore it 
 reads to (.05 25 = .002) two thousandths of an inch. 
 The reading is 28.75 + 0.020 = 28.770 inches. Notice 
 
 FIG. 15. 
 
VERNIERS. 
 
 [CHAP, v 
 
 the method of numbering the vernier. The numbers 
 on it indicate hundredths. This form is much more 
 convenient to read than if the numbers ran from o 
 to 25. 
 
 Fig. 16 shows the usual scale and vernier of the bet- 
 ter sextants. The limb is graduated to 10 minutes, the 
 vernier has 60 spaces, and therefore it reads to sixtieths 
 of 10 minutes, or to 10 seconds. The numbers on the 
 
 FIG. 16. 
 
 vernier indicate minutes. The reading is 70 oo' on the 
 limb, plus 5' 10" on the vernier, or 70 5' 10". 
 
 69. PRACTICAL HINTS. The most frequent error in 
 reading a vernier is to omit part of the reading of the 
 limb ; as, for example, in Fig. 12, forgetting to record 
 the half-degree from the limb. With circular arcs hav- 
 ing two rows of figures there is great danger of read- 
 ing the wrong row. 
 
 To determine whether a particular line on the vernier 
 coincides exactly with one on the limb, notice the next 
 line on either side, and see whether both fall short 
 (or overreach) equal amounts. When several lines of 
 the vernier appear to coincide equally with several 
 lines of the limb, take the middle line. When none of 
 the lines exactly coincide, 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 a line on the 
 limb, the true reading is midway between the readings 
 indicated by these two lines. If the graduation is very 
 
PRACTICAL HINTS. 71 
 
 accurate and the lines fine, it is possible by this method 
 to estimate the reading to a half, or even to a third, of 
 the least count ( 66). 
 
 It frequently happens that the instrument is to be 
 used to lay off a number of equal angles, as, for ex- 
 ample, i. The vernier is then to be set each time at 
 some particular line. If it is a double vernier, set the 
 zero line to coincide each time, and notice whether the 
 next lines on either side fall short equal amounts ; if 
 it is a single vernier, set, say, the fifteen line to coincide, 
 and note the agreement on both sides. This method is 
 considerably more accurate than reading the vernier by 
 looking at only one line. 
 
 70. Transits are sometimes made with two verniers, 
 one of which reads to decimals of a degree and the 
 other to minutes, as usual. A decimal vernier is very 
 convenient in laying out railroad curves, besides having 
 some minor advantages in other kinds of work. 
 
 71. MICROMETER. When the highest degree of accu- 
 racy is desired, a micrometer is used. A micrometer 
 consists of a fine screw, having a graduated head, which 
 moves a frame on which is a fine line or spider's thread. 
 The movement of the line or thread is observed through 
 a microscope. The distance is determined by the revo- 
 lutions and fractions of a revolution of the screw. 
 
 Micrometers are not used on ordinary engineering 
 tield-instruments. 
 
CHAPTER VI. 
 OPTICAL PARTS OF THE TELESCOPE. 
 
 ART. 1. CONSTRUCTION. 
 
 72. A TELESCOPE consists, optically, of certain lenses 
 which assist the eye in seeing distant objects ; and, me- 
 chanically, of certain 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 hence in the present chap- 
 ter only the optical parts will be considered. No at- 
 tempt at an elaborate discussion of the theory of the 
 optical workings of the telescope will be made; but at- 
 tention will be confined to such points as need to be 
 understood by the engineer for the intelligent use of 
 his instruments. 
 
 73. KINDS OF TELESCOPE. 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 objective, placed next to 
 the object, and a double concave lens, called the eye- 
 piece or ocular, placed near the eye. The chief pur- 
 pose of the objective is to increase the amount of light 
 
 72 
 
ART. l] CONSTRUCTION. 73 
 
 which reaches the eye from the object viewed. The 
 sole object of the eye-piece is to magnify the thing 
 looked at. The Galilean telescope assists the eye by 
 its magnifying and light-gathering power ; but such 
 a telescope would be useless for making precise meas- 
 urements, since there is no means of indicating the 
 exact point at which the telescope is sighted. It would 
 not be inappropriate to call it a seeing telescope. The 
 first telescope was of this form, and took its name from 
 the inventor, Galileo. It shows objects erect. An 
 opera-glass is a good example. 
 
 The measuring telescope consists of three essential 
 parts : i, 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 objective ; and 3, 
 two fine wires or spider threads, placed in the plane of 
 the image, the intersection of which indicates the pre- 
 cise point sighted at. The objective collects the light, 
 the eye-piece magnifies, and the cross hairs indicate the 
 point at which the telescope is directed. If such a tele- 
 scope neither magnified nor increased the illumination, 
 it would still be of great advantage in making measure- 
 ments. This form shows the object inverted. Nearly 
 all telescopes belong to this class. 
 
 Both of the above skeleton forms of telescope are 
 very imperfect. The methods of improving the optical 
 qualities of these elementary forms will now be con- 
 sidered. 
 
 74. THE OBJECTIVE. A single lens used as an objec- 
 tive has the following defects: i. The rays of light 
 which traverse a spherical lens near the edge, are re- 
 fracted 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 
 
74 OPTICAL PARTS OF THE TELESCOPE. [CHAP. VI 
 
 light, in passing a single lens, are resolved into the 
 colors of the prismatic spectrum ; consequently the 
 image will be disfigured by colored light. This devi- 
 ation of the different colored rays is called chromatic 
 aberration. 
 
 The objective of a telescope is rendered almost free 
 from these defects by substituting for the single Jens, a 
 compound one composed of a double-convex crown- 
 glass, and a concavo-convex flint-glass, lens. The two 
 components have different refractive and dispersive 
 powers, and by giving the four surfaces proper curva- 
 tures the above defects may be nearly eliminated. 
 
 75. THE EYE-PIECE. A single lens used as an eye- 
 piece possesses the same defects spherical and chro- 
 matic aberration as when used as an objective, but in 
 a less degree. A single lens as an eye-piece has also the 
 following defects : i. 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 aberrations ( 74). The 
 objective also possesses this defect, but in so slight a 
 degree as to be inappreciable ; while in an eye-piece of a 
 single lens it is very serious. 2. A telescope with a sin- 
 gle lens for an eye-piece has a very limited field of view. 
 
 In both forms of telescope, the chromatic and spheri- 
 cal aberration, and also the aberration of sphericity of 
 the eye-piece, are nearly eliminated by substituting two 
 plano-convex lenses for the single lens, which also in- 
 creases the field of view. 
 
 76. Huyghen or Negative Eye-piece. This is a modifi- 
 cation of the concave eye-piece of the Galilean tele- 
 scope. 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 ob- 
 jective, are shown in Fig. 17. 
 
ART. l] 
 
 CONSTRUCTION. 
 
 75 
 
 P is the object; O is the objective ; and a and b con- 
 stitute the eye-piece, a being known as the field lens 
 and b as the eye lens. Notice that the eye-piece is 
 
 FlG. 17. HUYGHEN OR NEGATIVE EYE-PIECE. 
 
 placed between the objective and its principal focus, 
 F. The large arrow at / represents the object as it 
 appears through the telescope ; the small one repre- 
 sents it as it would appear without the instrument. 
 The angle which the large arrow at / subtends at the 
 eye, divided by the angle 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 /to the small one. 
 
 77. Ramsden or Positive Eye-piece. This is a modifi- 
 cation of the convex eye-piece of the measuring tele- 
 scope. It consists of two plano-convex lenses with 
 their convex sides towards each other. The relations 
 of the eye lenses to the objective are shown in Fig. 18. 
 
 FIG. 18. RAMSDEN OR POSITIVE EYE-PIECE. 
 
 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 /, as before. 
 
76 OPTICAL PARTS OF THE TELESCOPE. [CHAP. VI 
 
 Notice that the essential difference between the posi- 
 tive and negative eye-pieces 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 ; but they are only indirectly 
 concerned in the accuracy of the observations. The 
 negative eye-piece is better for simply seeing, while the 
 positive is absolutely necessary in making precise 
 measurements. The positive eye-piece is always used 
 in transits, levels, etc. 
 
 Modifications of Ramsden's eye-piece, consisting in 
 the substitution of compound lenses instead of the 
 single lenses shown in Fig. 18, are frequently employed. 
 The principal ones are Kellner's, which consists of an 
 achromatic combination instead of lens a of Fig. 18, 
 and Steinheil's, which consists of two achromatic com- 
 binations instead of the single lenses a and b oi Fig. 18. 
 These eye-pieces are better than Ramsden's in that they 
 are free from color and have a perfectly flat field. Of 
 the two, Kellner's permits the larger field. 
 
 78. Erecting Eye-piece. The erecting or terrestrial 
 eye-piece, in its most elementary form, consists of a 
 
 <i ; i i! 11 
 
 FIG. 19. ERECTING EYE-PIECE. 
 
 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 ; 
 but in its common form it consists of a pair of plano- 
 convex lenses instead of the single lens. Fig. 19 shows 
 the usual arrangement of the lenses and diaphragms of 
 an erecting eye-piece. The pair of lenses next to the 
 
ART. l] CONSTRUCTION. 77 
 
 eye is called the erecting piece ; the pair next to the 
 objective is the ordinary positive eye-piece. For a 
 Kellner's erecting eye-piece, see Fig. 27 (page 100) or 
 Fig. 62 (page 222). 
 
 The erecting eye-piece is inferior to the 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 the inverting eye-piece which is quite 
 an important advantage in the transit. Most Amer- 
 ican engineering instruments have an erecting eye- 
 piece ; but it would be a great improvement if all 
 were provided with the inverting eye-piece. The 
 latter is much more common in Europe than in 
 America. 
 
 79. TELESCOPE SLIDES. To assist in focusing the ob- 
 jective for different 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 telescope. In in- 
 struments 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 im- 
 material 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. 
 
FIG. 20. 
 
 \ OPTICAL PARTS OF THE TELESCOPE. [CHAP. VI 
 
 80. CROSS HAIRS.* These are usually two spider 
 
 threads, one vertical and the 
 other horizontal, fastened to 
 a ring which is adjusted in 
 the tube by four screws. 
 Fig. 20 shows a front view of 
 the ring in place. BB, Figs. 
 27 (page 100) and 62 (page 
 222) show side views of the 
 ring. Lines ruled or etched 
 upon a piece of thin glass 
 are sometimes used. The 
 cross hairs are sometimes made of very fine platinum 
 wire. Platinum can be drawn to the required fine- 
 ness 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 ad- 
 vantages and disadvantages, and as a consequence their 
 advocates and opponents. 
 
 For the platinum wires, it is claimed that they are 
 best because they are opaque ; and this is a desirable 
 property when the cross hairs must be illuminated, as 
 in astronomical work and mine surveying. Spider 
 lines, particularly dark-colored ones, can be illuminated 
 fairly 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. On the other hand, it is 
 claimed that platinum wires lose their elasticity and 
 sag, and also that they oxidize or corrode. Spider 
 threads, on account of their fineness and cheapness. and 
 the facility with which they may be applied, will con- 
 
 * Cross hairs were first used by Picard, in 1669. 
 
ART, l] CONSTRUCTION. 79 
 
 tinue in the future, as in the past, to be used almost 
 universally for cross hairs in engineering instruments. 
 
 81. Stretching the Spider Webs. It does not require 
 much time or skill to replace spider webs, the belief of 
 many to the contrary notwithstanding. The ring 
 which carries the cross hairs can be taken out (with 
 some telescopes only after having 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 last 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 weight as experiment shows it will sup- 
 port. Dampen the thread by breathing upon it, by 
 holding it in a current of steam, or, better, by dipping 
 it in clean 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, 
 varnish, or the like, shellac varnish being the best for 
 this purpose. The main point is to stretch the web 
 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. 
 
8o OPTICAL PARTS OF THE TELESCOPE. [CHAP. VI 
 
 ART. 2. TESTING THE TELESCOPE. 
 
 82. In buying an instrument it is very desirable to 
 test the optical qualities of the telescope ; and the fol- 
 lowing directions are sufficient for such examination. 
 
 83. CHROMATIC ABERRATION. To test for chromatic 
 aberration (i, 74), 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 object, and in the second a ring of purple 
 light, the object-glass may be considered perfect, as 
 this proves that the most intense colors of the prismatic 
 spectrum (orange and blue) are corrected. 
 
 84. SPHERICAL ABERRATION. To test for spherical 
 aberration (2, 74), cover the object-glass with a ring 
 of black paper, so as to reduce the area of the aperture 
 about one half, and focus on some small and well- 
 defined object for distinct vision. Then remove the 
 ring of the paper and cover that part of the objective 
 previously left open, and notice how much the object- 
 glass must be moved in or out for distinct vision. 
 The amount of shift measures the spherical aberra- 
 tion 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, when the least 
 motion of the objective should render the object indis- 
 tinct. This last is not as good a test as the former, for 
 the eye will change focus slightly to accommodate itself 
 to the change of distance. 
 
 85. DEFINITION. The definition of a telescope de- 
 pends upon the accuracy of the curvature of the sur- 
 faces of the several lenses, and upon the centering, i.e., 
 the coincidence of the axes of the component lenses of 
 
ART. 2] TESTING THE TELESCOPE. 8l 
 
 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. Indistinctness 
 may be caused by spherical aberration ; and therefore 
 the test for that should precede the test for definition. 
 
 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 8 or 10 inches (the best distance for distinct vision), 
 the surfaces of the lenses are correct and well finished. 
 
 To determine whether the lenses are well centered, 
 fix a white-paper disk having a diameter of, say, an 
 eighth of an inch and a sharp outline, in the middle of a 
 piece of black paper or cloth. Place this in a good 
 light 30 or 40 feet from the telescope; then if the 
 image of the disk, when a little out of focus, is equally 
 surrounded on all sides by a uniform haze, the center- 
 ing is good. 
 
 Other things being the same, the lower the magnify- 
 ing power the better the definition. 
 
 86. FLATNESS OF FIELD. The flatness of the field 
 depends mainly upon the aberration of sphericity of 
 the eye-piece ( 75). To test a telescope in this 
 respect, draw with India ink a heavy-lined square, 6 
 or 8 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. 
 
 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. 
 
82 OPTICAL PARTS OF THE TELESCOPE. [CHAP. VI 
 
 87. SIZE OF THE FIELD. By the field of view is meant 
 all those points which are visible at the same time 
 through the telescope. In Fig. 21 O is the objective, 
 E the ocular, DABC the object, ab the image of AB. 
 The two extreme rays from A and B just strike the 
 edge of the ocular, and rays from C and D do not 
 enter the ocular ; therefore the angle AOB is the field 
 
 FIG. 21. 
 
 of view. Notice that it is independent of the size 
 of the object-glass, and varies (i) inversely as the 
 distance between the objective and the ocular, and 
 (2) directly as the size of the eye-piece. The greater 
 the magnifying power the less the field of view. The 
 wider the field of view the better, since width of field 
 facilitates rapid working. One of the advantages an 
 eye-piece composed of two plano-convex lenses has 
 over a single lens is the larger field of view. 
 
 To determine the angular width of field, sight upon 
 any object, and mark the two opposite sides of the 
 field. The distance between these points multiplied 
 by 57.3 and divided by the distance from the object to 
 the objective, is the angular width of field, in degrees. 
 Since the field varies with the distance (particularly if 
 it be short), a considerable distance, say 200 or 300 
 feet, should be employed in making this test. 
 
 The field of view varies with the maker, but for the 
 telescopes on ordinary engineering instruments it is 
 about as follows : for a magnifying power of 20, i 30'; 
 
 25, i 15'; 30, i; 35> 5'- 
 
 88. APERTURE OF OBJECTIVE. By the aperture of the 
 objective is meant the effective diameter of the object- 
 glass, t,e. t that part of the objective which transmits 
 
ART. 2] TESTING THE TELESCOPE. 83 
 
 light that finally reaches the eye. Usually, diaphragms 
 are placed in the tube of the telescope with a view to 
 improve the optical qualities of the lenses. 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 ; 
 but if the diaphragm is placed in or near the eye-piece, 
 it may diminish both the illumination and the field of 
 view. 
 
 To find the real aperture, direct the telescope towards 
 the sky, and place a pointer in contact with the ob- 
 jective so that the 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 telescopy. Then move the 
 pointer over the face of the object-glass until the point 
 just disappears, 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. 
 In making this observation a hand magnifier is of great 
 assistance in viewing the image of the objective. 
 
 A better method of finding the real aperture is as 
 follows: Fix, with water or thin gum, several small 
 pieces of opaque paper, say 0.05, o.io, and 0.15 of an 
 inch square, in a row around the very outer edge of 
 the objective. Then turn the instrument up to the sky, 
 and observe as before. If all the pieces can be seen, 
 then the aperture is equal to the entire objective ; but 
 if any piece is invisible, the margin of the glass is cut 
 off to this extent. 
 
 As it is more difficult to get the outer portion of 
 the objective to correct curvature, it is not uncom- 
 mon to cut off this portion with a diaphragm, which 
 renders the objective equal only to a smaller glass. 
 
84 OPTICAL PARTS OF THE TELESCOPE. [CHAP. VI 
 
 Even an apparently small margin cut off makes a 
 considerable difference in the light-collecting power. 
 For example, a i^-inch objective will collect 56 per 
 cent more light than a i-inch glass. Not infrequently 
 objectives are reduced in as great a proportion as in 
 this example. 
 
 The aperture of the telescopes on ordinary engineer's 
 transits is i to if inches, and on levels ij to i inches. 
 
 89. MAGNIFICATION. The power of a telescope, or 
 degree of magnification, depends upon the relative 
 focal lengths of the objective and of. the 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 small change 
 in distance requires a corresponding change in focus. 
 The magnifying power varies slightly with the distance 
 to the object ; but, fortunately, the exact magnifying 
 power of the telescopes on engineering instruments is 
 not required. There are several methods of measuring 
 the magnifying power, of which the two following are 
 the simplest. 
 
 90. First Method. If the telescope be directed toward 
 the open sky in the day-time and the eye be held 
 8 or 10 inches back from the eye end, a small bright 
 illumined circle will be seen, which is nothing more 
 than the image of the objective opening of the tele- 
 scope. Measure the diameter of this circle with a 
 finely graduated scale of equal parts. Also measure 
 the real aperture of the objective ( 88). Then the 
 magnifying power is equal to the quotient arising from 
 dividing the diameter of the aperture of the object-glass 
 by the diameter of the illuminated circle. The chief 
 difficulty in this method lies in the exact measurement 
 of the diameter of the small illuminated circle, 
 
ART. 2] TESTING THE TELESCOPE. 85 
 
 91. Second Method. Let a staff which is very boldly 
 divided into equal parts by heavy lines be placed 
 vertically at any convenient distance from the tele- 
 scope 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 looking with the other eye along the outside 
 of the tube. A little difficulty may be experienced on 
 the first trial of this method, but after a few attempts it 
 becomes very easy. 
 
 The magnifying power of the telescopes on ordinary 
 engineer's transits varies from fifteen to thirty, twenty 
 to twenty-five being the more common on good instru- 
 ments ; and for levels it varies from twenty to fifty, 
 twenty-five to thirty being the more common. The 
 higher the power the greater the minimum distance at 
 which an object can be seen through the telescope. It 
 is frequently desirable, particularly with a transit, to 
 focus on a point only 4 to 6 feet from the instrument. 
 This can usually be accomplished with a magnifying 
 power of twenty. 
 
 92. ILLUMINATION. The brightness of an object seen 
 through a telescope depends upon (i) the size of the 
 objective, (2) the polish and transparency of the lenses, 
 (3) the magnifying power of the telescope, and (4) the 
 size of the pupil of the eye. 
 
 1. Obviously, the quantity of light entering a tele- 
 scope is dependent upon the size of the aperture of the 
 object-glass, for the larger its area the more rays of 
 light proceeding from any point of the object will be 
 intercepted and transmitted. 
 
 2. In ever telescope light is lost by reflection from 
 
86 OPTICAL PARTS OF THE TELESCOPE. [CHAP. VI 
 
 the surfaces of the lenses, and by imperfect trans- 
 parency. It is probable that no engineering instrument 
 transmits over 85 per cent* of the light striking the 
 objective, and many of them transmit much less. 
 
 3. 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 
 brightness 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 illu- 
 mined. 
 
 4. In order that all the light passing through the 
 telescope may be received by the eye, the beam that 
 emerges from the eye end must not exceed the diameter 
 of the pupil of the eye; for if the emergent beam is 
 larger, part of it can not enter the eye, and conse- 
 quently part of the light intercepted by the objective 
 will be lost. The average size of the pupil is about 
 one tenth of an inch; and hence the diameter of the 
 emergent beam should not be more. The best possible 
 effect will be when the beam is of exactly the same 
 diameter as the pupil, for the brightness of the object 
 will then be the same (except for the losses referred to 
 in paragraph 2, above) as when viewed with the naked 
 eye. If the emergent beam is smaller than this, the 
 brightness will be less than when viewed with the 
 naked eye. 
 
 93. Since there can be no unit by which to measure 
 the illuminating power of a telescope, we can find only 
 the relative illumination. If two telescopes are to be 
 compared as to light, they should stand side by side and 
 
 * Chauvenet's Practical Astronomy, vol. ii, p. 17, 
 
ART. 2] TESTING THE TELESCOPE. . 87 
 
 be looked through at the same time, so as to be under 
 the same atmospheric conditions. 
 
 If two telescopes have the same aperture, and also 
 the same magnifying power, they may be compared by 
 placing them side by side, and, as night approaches, ob- 
 serving the same object through each. The one through 
 which the object is longest visible has the better illu- 
 mination. Of course, both observations must be made 
 by the same person. Instead of waiting for the ap- 
 proach of night, observe the distance at which fine 
 print can be read with each, or the distance at which 
 the time can be read from the second hand of a watch. 
 Notice that the last method involves the definition 
 ( 85) and spherical aberration ( 84) of the telescope, 
 as well as the illumination. 
 
 If the two telescopes do not have equal apertures nor 
 equal magnifying powers, a numerical expression for 
 the illuminating power may be obtained as follows: By 
 observation determine the maximum distance at which 
 time can be told from the face of a watch, through each 
 telescope. Then, if the lenses are equally transparent, 
 these distances should be to each other directly as the 
 clear apertures, and inversely as the magnifying pow- 
 ers. That is, if d^ and d^ represent the distances, a l and 
 # a the apertures, and m^ and m % the magnifying power, 
 then 
 
 The error of observation by this method is considerable, 
 owing to the inability of the eye to judge accurately of 
 equal illuminations, and because the. method also in- 
 volves the defining powers of the two instruments. 
 
88 OPTICAL PARTS OF THE TELESCOPE. [CHAP. VI 
 
 ART. 3. USING THE TELESCOPE. 
 
 94. ADJUSTMENT FOE PARALLAX. Parallax is an appar- 
 ent movement of the cross hairs in reference to the 
 object sighted, 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. Of course, if the 
 object changes its position for different positions of the 
 eye, it will be impossible to do accurate work with the 
 .telescope. Therefore a telescope should be accurately 
 adjusted for parallax before being used in precise 
 measurements. All measuring telescopes require this 
 adjustment. It consists in bringing the cross hairs and 
 the image exactly into the same plane. In making this 
 adjustment two steps are required: (i) focusing the 
 cross hairs, and (2) focusing the object. 
 
 1. To focus the cross hairs, direct the telescope to- 
 ward the sky, or throw it out of focus so that no object 
 can be distinguished in the field, and move the ocular 
 in or out until the cross hairs can be seen very dis- 
 tinctly. When the cross hairs are properly focused, 
 little specks of dust may be seen on them. No object 
 should be in the field of view while this adjustment is 
 being made, for the eye is continually changing its 
 focus to accommodate itself to the distance of the object 
 viewed. Before pronouncing the adjustment correct, 
 close the eye for a moment, and then sight again. 
 Having adjusted the eye-piece, it need not be changed 
 except for the change of focus of the eye with advanc^ 
 ing age, or for different observers. The rack and pinion 
 frequently provided for focusing the cross hairs is 
 worse than useless. 
 
 2. To focus the objective, direct the telescope to the 
 object, keeping the attention fixed upon the cross hairs 
 so that the eye shall not change focus to accommodate 
 
ART. 3] USING THE TELESCOPE. 89 
 
 itself to the position of the object, and move the objec- 
 tive in or out 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 ap- 
 pears to move with the eye, and, second, that the farther 
 the object from the objective the nearer is the image. 
 This test is more easily made than described. When 
 the objective is properly focused there should be abso- 
 lutely no movement of the cross hairs with reference 
 to the object. 
 
 Many of the errors of instrument work are due to 
 parallax ; and this source of error is more serious as the 
 work becomes more accurate. The higher the power 
 the more difficult it is to eliminate parallax. 
 
 95. CAKE or THE TELESCOPE. If the objective or the 
 lens next to the eye becomes dusty, brush it with a fine 
 camel-hair brush, or rub it 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 sharpness and 
 brilliancy of the image. Dust upon the glasses is not 
 as objectionable as a thin, or even almost imperceptible, 
 film of grease ; therefore the lenses should never be 
 touched with the fingers. When the lenses become 
 very dirty, wash them with alcohol. 
 
 The interior face of the objective and the interior 
 lenses of the eye-piece will seldom need cleaning, unless 
 water should find access to the inside of the tube. Having 
 removed the cell containing the objective, care should 
 be taken to screw it back exactly to its former position, 
 else the adjustment of the line of sight may be de- 
 
90 OPTICAL PARTS OF THE TELESCOPE. [CHAP. VI 
 
 stroyed. The component lenses of the objective should 
 never be taken apart nor removed from the cell contain- 
 ing them, since they may not be returned to their for- 
 mer position, thus disturbing their adjustments ; but if 
 they should accidentally become separated, be sure to 
 replace them with the double-convex crown-glass out- 
 ward. Dust should be carefully kept from the inside 
 of the telescope tube; if not, 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. 
 
 If the telescope slide should get to fretting or cutting, 
 take it out and smooth the rough places. The blade of 
 a penknife forms a very good instrument for this pur- 
 pose. Scrape with the edge, slightly inclining it, and 
 burnish with the back of the knife. Wipe out the in- 
 side of the tube, and if possible burnish and scrape that 
 smooth. Grease the slide slightly, and wipe off the 
 grease before restoring the slide to its place. Too much 
 grease, however, causes dust to adhere. If this does 
 not remove the trouble, a little grinding with finely 
 powdered pumice-stone and oil may help the difficulty; 
 but great care should be taken to wipe off all abrading 
 material. Emery in any form should never be used. 
 If a slide once commences to fret, it rapidly grows 
 worse, and may get beyond repair. 
 
CHAPTER VII. 
 THE TRANSIT. 
 
 ART. 1. CONSTRUCTION. 
 
 97. THE instrument in common use among American 
 engineers for measuring horizontal and vertical angles 
 is usually called a transit. Fig. 22 shows the general 
 form. It is sometimes, but incorrectly, called a theodo- 
 lite. The theodolite is the name given by British en- 
 gineers to their favorite portable instrument (see Fig. 
 23, page 93), 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 can not. 
 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 and sometimes a railroad 
 transit, to distinguish it from an astronomical transit. 
 
 The transit was invented and first made by a Phila- 
 delphia firm in 1831, previous to which time the English 
 theodolite and the magnetic compass sometimes pro- 
 vided with a full circle graduation, by which angles 
 could be read independently of the needle were the 
 common angle instruments. 
 
 98. A modern engineer's transit has more than 350 
 distinct pieces. Although it appears quite complicated, 
 this impression disappears when each part is examined 
 
THE TRANSIT. [CHAP. Vll 
 
 FIG. 22. AMERICAN TRANSIT. 
 
ART. l] CONSTRUCTION. 93 
 
 in turn, and its uses and relations to the rest carefully 
 studied. The great value of the transit as an instru- 
 ment of precision is due to the telescope, by which great 
 precision in sighting is attained, 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. 
 
 FIG. 23, ENGLISH THEODOLITE. 
 
 The tripod, leveling screws, telescope, and vernier 
 all very important parts of the transit have been dis- 
 cussed in preceding chapters. 
 
 99. THE GRADUATION. The lines of the graduation 
 should be uniform, and as small as is consistent with 
 legibility. In the best instruments the graduation is 
 upon solid silver. 
 
 The most common graduation for the horizontal limb 
 of transits is a circle about 6 inches in diameter divided 
 to half-degrees having a vernier reading to minutes. 
 The better transits read to 30", and some to 20". The 
 degrees are usually numbered in two rows, one like the 
 compass and another from o to 360. The field work 
 is most simple and least liable to error, if only the latter 
 
94 THE TRANSIT. [CHAP. VII 
 
 numbering is used. Or, if there are two rows, they 
 should be like those shown in Fig. 13 (page 68). 
 
 The vertical circle should be numbered from o to 
 90; for then angles of either elevation or depression 
 may be read with the telescope both direct and inverted, 
 and the mean of the two will be independent of errors 
 of adjustment of the vertical circle. 
 
 100. THE VEBNIERS on the horizontal circle are some- 
 times 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 telescope, 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 position is preferable, especially 
 in confined places, as in underground 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. 
 
 The vernier and limb should be exactly in the same 
 plane, to avoid parallax in reading ; and the space be- 
 tween them should always have the appearance of a 
 uniform, fine, black line. The verniers should be pro- 
 vided with ground-glass or ivory shades for illuminat- 
 ing the graduation. 
 
 Verniers reading 20 seconds should have the reading- 
 glasses permanently attached in such a manner that 
 they can be moved radially along the entire length of 
 the vernier. The tube in which these reading-glasses 
 are mounted should also have, at the end nearest the 
 graduation, a fine pointer which will just reach the end 
 of the lines. This pointer, being in the center of the 
 field, will serve as a guide in moving the reading-glass 
 exactly opposite the coinciding line. The center of the 
 lens is thus used in reading, and parallax is avoided. 
 
ART. l] 
 
 CONSTRUCTION. 
 
 95 
 
 101. CENTERS, OR VERTICAL AXES. Usually there are 
 two concentric vertical axes, the verniers and telescope 
 turning about the inner, and the graduated circle revolv- 
 ing about the outer one. Fig. 24 shows the arrange- 
 ment of these axes. Ordinarily, the outer axis is useful 
 only in enabling the observer to shift the graduation 
 so as to begin each time at zero. The inner axis is 
 
 FIG. 24. SECTION OF THE TRANSIT. 
 
 made more carefully than the outer. These two axes 
 are often 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." The 
 instrument should turn on either axis without the slight- 
 est play, and yet with very little friction. 
 
 102. LEVELS. Since a transit is to be used in measur- 
 ing horizontal and vertical angles, it is necessary that 
 levels be attached to the instrument to determine the 
 plane of these angles. Two level vials, perpendicular 
 
g6 THE TRANSIT. [CHAP. VII 
 
 to each other, are sufficient to bring the vertical axis 
 vertical, which then serves as a datum to which to ad- 
 just the other parts of the instrument. If the instru- 
 ment is to be used mainly for measuring horizontal 
 angles, the level perpendicular to the telescope should 
 be the more sensitive. In ordinary practice vertical 
 angles are seldom required, and when they are less 
 precision is demanded than in horizontal ones; hence 
 it is very common to attach only two short levels to 
 the plate, in which case the level parallel to the tele- 
 scope should be the more sensitive. If the instrument 
 is to be used for the precise measurement of vertical 
 angles, a sensitive level should be attached to the tele- 
 scope to indicate a horizontal line the zero for vertical 
 angles. With a level under the telescope, a transit can 
 be used as a leveling instrument, but is not capable of 
 very reliable work, owing to a lack of stability. 
 
 103. CLAMP AND TANGENT SCREW.* With the unaided 
 hand, the telescope can not easily be made to cover or 
 bisect the exact point sighted at; and to assist in thus 
 directing the telescope, the instrument is provided with 
 a clamp, and a 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 
 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 under- 
 standing of these parts; they must be seen and examined 
 to be comprehended. 
 
 It is common to clamp the vernier and the graduated 
 
 * Invented by Helvetius, a celebrated astronomer of Dantzic, about 1650, 
 
ART. l] CONSTRUCTION. 97 
 
 plates at their circumferences; but as this is liable to 
 bend the plates, it is far better to clamp the axis of the 
 plate. 
 
 104. A perfect tangent screw should have a uniform 
 motion, and be free from lost motion or " back-lash," so 
 as to respond quickly to the touch and yet hold the 
 plate and vernier exactly as set. Lost motion the com- 
 mon defect of tangent screws is a source of great an- 
 noyance to the engineer, and often causes serious errors 
 in the field. 
 
 The tangent screw should have great durability, or at 
 least an even wear, so that the screw will never have lost 
 motion in one part and move hard in another. The 
 wear is greatly lessened by covering the thread with a 
 dust-guard, as is now quite common. The tangent 
 screw should be so made as to work at least fairly well 
 if it should get bent. Several forms of tangent screws 
 will now be described. 
 
 105. English Tangent Movement. 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 rn the clamp and against a collar in a post attached 
 to the plate, each post being free to turn about a verti- 
 cal axis. Fig. 25 shows the English tangent movement 
 
 FIG, 25. ENGLISH TANGENT MOVEMENT. 
 
 as applied to the horizontal circle of a transit. A is the 
 head of the clamp screw and B that of the tangent 
 screw, This form is defective in three particulars: i. 
 
98 THE TRANSIT. [CHAP. VII 
 
 Since the posts are movable only about a vertical axis, 
 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; 
 but to allow for errors of workmanship in satisfying this 
 condition, the hole in the nut is often made too large, 
 and the nut is then tightened until it fits, thus causing 
 it to touch the screw along only two lines, and it there- 
 fore soon wears loose. 3. If the posts are tightened up 
 so as to prevent looseness, the friction is so great as to 
 interfere with the freedom of their motion. 
 
 106. Gambey Tangent Movement. One of the best 
 forms of tangent screws is the Gambey movement, first 
 made by the celebrated instrument-maker, Gambey, of 
 Paris. The nut is a split sphere which may be confined 
 like any ball-joint. Instead of the collar, as in the 
 English form, another split spherical nut which works 
 in a series of grooves is used. This form of tangent 
 movement is shown at g, Fig. 8, page 59. As shown in 
 Fig. 8, the pieces which clamp the balls are too heavy. 
 They should be made light enough to have a little 
 flexibility up and down, thus permitting a free move- 
 ment of the balls. This construction provides for all 
 necessary motions and all probable imperfections of the 
 screw. 
 
 The objections to this form are: i. The balls are ex- 
 posed to flying dust and sand, which tends to roughen 
 their surfaces and to cause an uneven motion. Con- 
 fining the balls between springs remedies this in part. 
 2. As ordinarily made, the nut is much shorter than the 
 screw, and consequently the latter wears mainly in the 
 middle, and after a little use the screw can be turned 
 only a few revolutions without having lost motion in 
 one part or undue friction in another. This defect is 
 removed by making the thread in the nut nearly as 
 Jong as the thread on the screw, the long nut being con- 
 
ART. l] CONSTRUCTION. 99 
 
 fined as the ball of the ordinary form. There are sev- 
 eral slightly different ways of making this modification, 
 all of which are very satisfactory. For one form see 
 Fig. 22, page 92. 
 
 107. Spring and Abutting Screw. In a very common, 
 and one of the best, forms of tangent movement a 
 screw abuts against the clamp, and a spring on the op- 
 posite side keeps the clamp in contact with the screw. 
 To prevent excessive local wear the nut through which 
 the tangent screw works should be long; and to com- 
 pensate for wear the nut should be made adjustable 
 and have a little elasticity. An objection urged against 
 this form is that the spring causes a strain between the 
 parts, which may change with changes of temperature, 
 and which is liable to move the plates or derange the 
 levels. This objection has but little weight, particu- 
 larly if there is some means of tightening the spring. 
 For an example of this form of tangent movement see 
 Fig. 28 (page 101) and also Fig. 32 (page 131). 
 
 108. Two Abutting Screws. Sometimes two opposing 
 or abutting 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 steadiness 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 oc- 
 casionally so added. 
 
 109. OBJECT-GLASS SLIDE. The object-glass should 
 
 move in a right line perpendicular to the horizontal 
 axis of the telescope. Some instruments have an ad- 
 justment to alter the direction of this motion. This is 
 accomplished by causing the inner end of the slide to 
 work through a ring which is adjustable like the ring 
 carrying the cross hairs (see CC, Fig. 27). This device 
 
JOO 
 
 THE TRANSIT. 
 
 [CHAP, vii 
 
 FIG. 27. SECTION OF TELE- 
 SCOPE OF TRANSIT. 
 
 is quite objectionable, since the 
 ring is liable to get loose, be- 
 sides wearing too large. Fur- 
 thermore, the adjustment is 
 quite difficult to make, and no 
 reason can be assigned why the 
 ring should not be fastened per- 
 manently when once in the 
 proper position. A better way 
 is to make the whole length of 
 the slide fit the inside of the 
 telescope tube. The latter 
 method requires more care and 
 greater skill in the manufacture. 
 In either case the slide should 
 be perfectly straight. 
 
 110. GEADIENTEE. This is a 
 simple device which adds very 
 much to the convenience and 
 usefulness of the transit. It is 
 nothing more nor less than an 
 accurate tangent screw with a 
 micrometer head working beside 
 a scale from which the number 
 of complete revolutions is as- 
 certained.* It can be applied 
 to the vertical or horizontal 
 limb, and may be used in estab- 
 lishing grades, determining 
 horizontal distances (see Art. 2, 
 Chap. X), and measuring small 
 angles very accurately without 
 an arc or a vernier. 
 
 * Invented by Professor Stampfer, of 
 Vienna, in 1873. 
 
ART. l] 
 
 CONSTRUCTION. 
 
 IOI 
 
 Fig. 28 shows the gradienter as applied to the verti- 
 cal circle of a transit. A is a leg attached to one of the 
 standards which support the horizontal axis of the 
 telescope. The little scale immediately above the grad- 
 uated head is to register the complete revolutions 
 of the screw, the. fractions of a revolution being read 
 from the graduated head. The screw is so cut that one 
 revolution of it moves the telescope through an angle 
 whose tangent at 100 feet from the instrument is i foot 
 or 0.5 foot. In the former case the head is divided 
 
 FIG. 28. GRADIENTER. 
 
 into one hundred parts, and in the latter into fifty; 
 and hence in either case a unit of the graduation corre- 
 sponds to o.oi of a foot on a rod 100 feet from the 
 instrument. For hints on the use of the gradienter see 
 237-47- 
 
102 THE TRANSIT. [CHAP. VII 
 
 111. SHITTING PLATES. This is a device for bringing 
 the plummet exactly over the point, after having 
 brought it approximately into position by manipulating 
 the tripod legs (24). The usual arrangement allows 
 the whole instrument to be shifted laterally on the 
 tripod head about an inch. This is a very great con- 
 venience. 
 
 112. COMPASS. It frequently happens that it is abso- 
 lutely necessary to run lines by the magnetic compass, 
 even though a transit is at hand ; and therefore a 
 needle and compass graduation are usually added to 
 a transit. The compass is also valuable as a check 
 against gross errors in reading the vernier of the 
 transit. 
 
 The tests and adjustments of the compass on the 
 transit are essentially the same as for the simple com- 
 pass (see Chapter III, particularly 36). 
 
 113. VARIOUS EXTRAS. There are a few additions and 
 modifications of the transit which, though not common, 
 are sometimes made, and which increase the range and 
 convenience of the instrument for certain kinds of 
 work. A few of the most important of these extras 
 will now be described briefly. 
 
 When it is desired to take greater vertical angles 
 than is possible with the ordinary eye-piece, the little 
 cap on the eye end of the telescope is unscrewed and 
 replaced by another containing a small prism which 
 reflects the beam at right angles and brings it out to 
 the eye of the observer. When the telescope is used to 
 look at the sun, a colored shade must be placed over 
 the object-glass or be interposed between the eye and 
 the telescope. A piece of smoked window-glass, held 
 in the hand, will serve the purpose; but caps contain- 
 ing colored shades are made by the various instrument- 
 manufacturers, which are much more convenient, and 
 Comparatively inexpensive. 
 
ART. 2] TESTING THE TRANSIT. IOJ 
 
 When the telescope is used in a mine, or to look at 
 a star, unless some diffused light enters the telescope, 
 the cross hairs will not be visible. Caps are made 
 which support an annulus in front of the objective, at 
 an angle of 45 with the telescope; and the cross hairs 
 are then illumined by holding a lamp or candle at the 
 side of the objective. The same thing may be accom- 
 plished, though less easily, with a piece of paper held in 
 the hand or fastened to a board. 
 
 In mine surveying it is often necessary to sight verti- 
 cally up or down a shaft. For this purpose an extra 
 telescope is sometimes attached to the end of the hori- 
 zontal axis; or sometimes the standards are inclined, 
 so that the ordinary telescope may sight vertically 
 downward past the plates. See 168 for still another 
 method. 
 
 ART. 2. TESTING THE TRANSIT. 
 
 114. GRADUATION. The graduation may have two 
 classes of errors accidental and periodic. Accidental 
 errors are those which follow no regular law, and are 
 equally liable to occur at any given division. There is a 
 variety of causes which may produce them. Periodic 
 errors are those which follow some law, and are proba- 
 bly caused by some peculiarity of the graduating en- 
 gine. 
 
 Since the accuracy of the graduation is a vital point, 
 it is very unfortunate that there is no easy or simple 
 method of testing it. An imperfect test is to notice 
 whether the extreme lines of the vernier span the same 
 number of divisions in all parts of the circle. But with 
 the present graduating engines it is very poor gradua- 
 tion indeed whose errors may be detected in this way. 
 
 With astronomical and geodetic instruments very 
 elaborate methods are employed for detecting errors of 
 
IO4 THE TRANSIT. [CHAP. VII 
 
 graduation; but these methods are too complicated to 
 be explained here, and, moreover, they are not applicable 
 to common engineering instruments. 
 
 115. ECCENTRICITY. The center of graduation should 
 lie in the axis of rotation. To test this, read two 
 verniers 180 apart (or any number of equidistant 
 verniers), then move the circle preferably 90 and read 
 again. If the verniers have changed the same amount, 
 the circle is well centered. If the two have not 
 changed the same amount, the mean of the two 
 differences is the actual angle through which the in- 
 strument has been moved. Eccentricity is, therefore, 
 wholly eliminated by reading two opposite verniers 
 each time, and taking the mean. The horizontal circle 
 of the common engineer's transit seldom has more than 
 one minute of eccentricity. There is usually but one 
 vernier on the vertical limb, and hence there is no way 
 of testing or eliminating its eccentricity. 
 
 It is frequently said that if the readings of the two 
 verniers differ by 180, the graduation and centering 
 are perfect; but this is no criterion. The difference 
 between the verniers depends upon the accuracy of 
 the graduation, the eccentricity, and the angular dis- 
 tance between verniers. The readings might differ by 
 180, and still the graduation and centering be very 
 much in error. 
 
 Assuming the graduation to be perfect, the difference 
 between the two verniers can be expressed by 180 
 If e remains constant, the centers coincide, but the 
 verniers are not opposite; if e varies, the verniers are 
 opposite, but the centers do not coincide. 
 
 116. MAGNIFYING POWER vs. VERNIER. The magnify- 
 ing power of the telescope and the least count of the 
 vernier should be so proportioned that the least per- 
 ceptible movement of the vernier will cause sufficient 
 movement of the cross hairs on the object to be easily 
 
ART. 2J TESTING THE TRANSIT. 
 
 noticed through the telescope; and, vice versa, the least 
 noticeable motion of the cross hairs on the object 
 should cause a barely perceptible change of the vernier. 
 Since the horizontal circle is usually (and properly) the 
 more accurate, this condition applies more especially to 
 the horizontal circle. A higher power, or a smaller 
 count of the vernier, than that required by this condi- 
 tion is detrimental, the former causing an unnecessary 
 loss of light and the latter a waste of time in reading 
 the vernier. 
 
 117. MAGNIFYING POWER vs. LEVEL UNDER TELESCOPE. 
 
 If the transit is to be used as a leveling instrument, the 
 magnifying power of the telescope and the delicacy of 
 the level under the telescope should be so proportioned 
 that a barely perceptible movement of the cross hairs 
 on the object will cause a just perceptible movement of 
 the bubble. 
 
 118. MAGNIFYING POWER vs. PLATE LEVELS. The re- 
 lation which should exist between the magnifying 
 power of the telescope and the sensitiveness of the plate 
 levels depends wholly upon the kind of work to be done. 
 The following method of testing this relation suffices 
 for all kinds of work. Bring the bubbles of both 
 plate levels to the middle, and sight upon a well-defined 
 point at as great an angular elevation as the instrument 
 will permit ; then slightly derange the levels by manip- 
 ulating the foot screws, and carefully level up again. 
 If, on again looking through the telescope, the cross 
 hairs cover the same point as before, the sensitiveness 
 of the levels is proportional to the magnifying power of 
 the telescope. If the cross hairs are out horizontally, 
 it shows that the level perpendicular to the telescope is 
 not sensitive enough. If the cross hairs are out verti- 
 cally, the level parallel to the telescope is too sluggish. 
 This relation is much less important than the two pre- 
 ceding ones. 
 
I06 THE TRANSIT. [CHAP. VII 
 
 119. PARALLELISM OF VERTICAL AXES. The vertical 
 
 axes should not only be parallel to each other, but 
 should also be concentric. To test the first condition, 
 adjust the most sensitive level about the instrument 
 perpendicular to one axis ( 38), clamp that axis, and 
 revolve the instrument about the other; then if the 
 Bevels are in adjustment about the second axis also, the 
 axes are parallel. If the axes are not parallel, no error 
 will be produced except when the instrument is used to 
 measure angles by repetition ( 132), provided the plate 
 levels are adjusted perpendicular to the irmer axis. 
 
 120. LIMB PERPENDICULAR TO Axis. The plane of the 
 graduation should be perpendicular to the axis about 
 which it revolves. If the limb is not horizontal, angles 
 measured on one part of it will be too great, and an- 
 gles 90 therefrom will be as much too small. How- 
 ever, the error would almost certainly be inappreciable 
 with common engineering instruments. 
 
 If desired, this condition can be tested as follows: 
 Bring the vertical axis vertical ( 38), place a block 
 level on the horizontal limb, read the position of the 
 bubble and reverse the limb ; then reverse the block 
 level to eliminate its error, and read the position of the 
 bubble again. If the bubble reads alike both times, 
 the horizontal limb is perpendicular to the vertical axis. 
 
 121. OBJECT-GLASS SLIDE. The optical center* of the 
 objective should be projected in the line of collimation.f 
 
 * The optical center is a point so situated that any ray of light passing 
 through it will undergo equal and opposite refractions on entering and leav- 
 ing 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 plano-convex or a plano-concave lens, the optical center will be at the 
 intersection of the axis with the curved surface. 
 
 A lens has been defined as a mathematical point which allows a great deal 
 of light to go through it. The better the lens the more nearly is this con- 
 dition realized. This " point" is the optical center. 
 
 t The line joining the intersection of the cross hairs and the optical center. 
 
ART. 2] TESTING THE TRANSIT. 1O7 
 
 If it is not so projected, and the instrument be colli- 
 mated for one distance ( 123), it will be out of colli- 
 mation for every other distance. This is equivalent to 
 requiring that the slide shall be straight and move in 
 the plane of the horizontal axis and perpendicular to it. 
 If the objective is fixed and the cross hairs are mova- 
 ble, the same conditions should be satisfied, and the 
 method of testing is the same in both cases. To make 
 this test, collimate the instrument ( 123, or 278, or 
 285) and test (i) for a deviation from a vertical plane, 
 (2) for a deviation from a plane at right angles to the 
 vertical plane, (3) for a deviation from the perpendic- 
 ular to the horizontal axis, and (4) for a deviation from 
 the plane of the horizontal axis. 
 
 i. To test for a deviation from a vertical plane. Se- 
 lect smooth ground and lay off any number of points, 
 say five or six, 100 feet apart, having one as near the 
 instrument as possible, and line the points carefully 
 with the telescope. Reverse in altitude and azimuth, 
 sight upon the first point, and clamp the instrument ; 
 then locate points in line with the first one, near each 
 of the others. Measure the distances between the sev- 
 eral pairs of points. If these distances vary as the dis- 
 tance from the first point, the slide is probably straight 
 and the optical center is projected in a vertical plane 
 but not perpendicular to the horizontal axis. To cause 
 the optical center to move in a plane perpendicular to 
 the horizontal axis, sight upon the first point and clamp 
 the vertical axis ; then direct the telescope to the second 
 point, and move the back end of the slide to correct 
 half the error. Moving the back end of the objective 
 slide may have destroyed the adjustment for collima- 
 tion ; therefore recollimate the instrument, and again 
 test for straightness of slide. If the instrument can not 
 be adjusted to bisect a row of points before and after 
 reversal, the slide is not straight, and no good work 
 
108 THE TRANSIT. [CHAP. VII 
 
 can be done with the instrument. Of course, due con- 
 sideration must be given to the errors of observation. 
 An error may be introduced by the slide's being loose ; 
 but this may be avoided by finishing every setting of 
 the object-glass by motion always in the same direction, 
 
 2. To test for a deviation from a plane at right angles fo 
 a vertical plane. The method is similar to the preceding. 
 Drive several stakes into the ground at equal intervals, 
 the first being near the instrument. Read a leveling 
 rod upon each, reverse in altitude and azimuth, and 
 bring the line of sight to the previous reading upon 
 the first stake; then read the rod upon all the other 
 stakes. If the differences of readings vary as the dis- 
 tance from the first station, the object-glass is pro- 
 jected in a plane perpendicular to the vertical plane. 
 Since it is projected in two planes at right angles to 
 each other, it must be in their intersection; therefore 
 the optical center is projected in a straight line, and the 
 slide is straight. 
 
 3. To test whether the motion is perpendicular to the hori- 
 zontal axis. Collimate the vertical hair (i 123) for a near 
 distance; then, if it is in collimation for a greater dis- 
 tance, the optical center of the objective is projected in 
 a line perpendicular to the horizontal axis of the tele- 
 scope. If it is not in collimation for the farther point^ 
 move the back end of the object-glass slide until it is; 
 but if there is no means of adjusting the slide, send the 
 instrument to the maker. See 109. 
 
 4. The rigorous condition is that the line of motion of 
 the optical center should pass through the horizontal axis. It 
 would be impossible to certainly secure this relation; 
 but for any cases likely to occur in practice, including 
 those requiring astronomical accuracy, it is sufficient to 
 require that the line of collimation be in the line of 
 motion of the optical center. Therefore it is only 
 necessary to place the horizontal hair in the line of 
 
ART. 3] ADJUSTMENTS OP THE TRANSIT. 109 
 
 motion of the optical center, which is an adjustment, 
 and will be discussed in Art. 3 of this chapter. 
 
 ART. 3. ADJUSTMENTS OF THE TRANSIT.* 
 
 122. LEVELS. The levels should be perpendicular to 
 the vertical axis, This is the same adjustment as that 
 described for the compass ( 38 and 39, which see). 
 As in the compass, this condition is important only in 
 certain kinds of work, as will be discussed in 128. 
 
 123. CROSS HAIRS. This adjustment is ordinarily 
 known as the adjustment of the line of collimation.f 
 For obvious reasons, it is better to call it the adjust- 
 ment of the cross hairs. It is made in two steps, (i) 
 the adjustment of the vertical hair, and (2) that of the 
 horizontal hair. 
 
 1. Vertical Hair. The line of collimation should be 
 perpendicular to the horizontal axis of the telescope. 
 Notice that if this condition is satisfied, the line of colli- 
 mation will describe a plane during the revolution of 
 the telescope; but if it is not, it will describe the surface 
 of a cone. To make this adjustment, level the instru- 
 ment, and sight upon some well-defined point; then 
 reverse the instrument in altitude, and fix a point in 
 line at an equal distance from the instrument. (If the 
 standards have not been adjusted, or if the horizontal 
 axis is not level, these points must be in the same hori- 
 zontal line, or nearly so.) Reverse the instrument in 
 azimuth and sight at the first point; then reverse in 
 altitude, and mark a point in line at the same distance 
 as before. If the two points do not agree, move the 
 vertical hair one fourth of the difference, and repeat the 
 whole operation to test the accuracy of the adjustment. 
 
 * For general remarks upon adjustments, see 37. 
 
 t The line joining the intersection of the cross hairs and the optical center 
 of the objective is the line of collimation. 
 
HO THE TRANSIT. [CHAP. VII 
 
 If there is any back-lash in the tangent screw, the 
 instrument must be manipulated very carefully, to pre- 
 vent any change of position on the vertical axis. There 
 are several methods of making this adjustment, but 
 this is the most simple, as well as the most accurate. 
 Notice that this method is independent of all instru- 
 mental errors, with the possible exception that the line 
 of collimation does not lie in the plane of the vertical 
 axis; but any possible error in this respect will be 
 inapppreciable except at very short distances. 
 
 Next proceed to make the vertical hair vertical. This 
 adjustment is useful only in sighting at a flag-pole, to 
 tell whether it is vertical. The simplest way to make 
 this adjustment is to level the horizontal axis of the 
 telescope ( 126), and see whether the hairs will coin- 
 cide with the corner of some building. Or, move 
 rigorously, having the axis of the telescope horizontal, 
 sight upon some well-defined point, and elevate or 
 depress the telescope. If the hair is vertical, the point 
 will seem to travel up and down the hair; if it does 
 not, loosen the screws a trifle and turn the diaphragm 
 slightly. 
 
 Before pronouncing upon any adjustment, repeat the 
 test to discover the error of observation. The greater 
 the accuracy of the instrument, the more important this 
 precaution. 
 
 2. Horizontal Hair. The horizontal hair should be 
 in the plane of motion of the optical center of the ob- 
 jective. If this condition is not satisfied, the line of 
 collimation will change with every change of the object- 
 glass, rendering the instrument useless for leveling, or 
 for measuring vertical angles. In discussions of the 
 adjustments of the transit no mention is ever made of 
 an adjustment of the horizontal hair; although such 
 an adjustment is necessary if the instrument is to be 
 
ART. 3] ADJUSTMENTS OF THE TRANSIT. Ill 
 
 used for anything except the measurement of hori- 
 zontal angles. 
 
 To make this adjustment, drive a stake near the in- 
 strument and read a level-rod upon it; then, without 
 moving the telescope in altitude, read a rod upon a 
 second stake 200 or 300 feet distant. Reverse in alti- 
 tude and azimuth, and bring the telescope to the 
 former reading upon the first point; then read upon the 
 other stake. If this reading is not the same as before, 
 correct half the difference by moving the horizontal 
 hair. After having adjusted the horizontal hair, test 
 the vertical hair, for moving one may have changed the 
 other. 
 
 It would be a great improvement if telescopes were 
 so made as not to require an adjustment of the hori- 
 zontal hair of the transits or vertical hair of levels. 
 This improvement could be most easily applied to in- 
 verting telescopes, and is another reason for using such 
 telescopes. In case the hair is non-adjustable, the en- 
 gineer should test the adjustment for himself. 
 
 124. CENTERING THE EYE-PIECE. After having ad- 
 justed the cross hairs, it may be that their intersection 
 will not be in the middle of the field of view. This 
 does not affect the accuracy of the work, but does affect 
 the seeing power of the telescope. Some instruments 
 are provided with two pairs of screws (A A, Fig. 27, 
 page 100), like the screws which move the cross-hair 
 ring, by which the inner end of the eye-piece may be 
 moved so that the cross hairs shall appear in the center 
 of the field. In inverting telescopes, no means is pro- 
 vided for making this adjustment. 
 
 125. STANDARDS. The line of collimation should re- 
 volve in a vertical plane when the vertical axis is verti- 
 cal ; or, in other words, the horizontal axis should be 
 horizontal when the instrument is leveled up, i.e., the 
 standards should be of the same length. 
 
112 THE TRANSIT. [CHAP. VII 
 
 To make this adjustment, having adjusted the cross 
 hairs ( 123), level the instrument very carefully, direct 
 the telescope to some high and well-defined point, and 
 clamp the vertical axis. Establish a low point in the 
 line of the telescope ; reverse in altitude and azimuth, 
 direct the telescope to the high point, and clamp the 
 vertical axis. If the telescope, when turned down, does 
 not cover the low point, correct one half the difference 
 by lowering the end of the axis towards which the line 
 of sight diverges. 
 
 In making this adjustment, it is only necessary to 
 level the bubble perpendicular to the telescope. It is 
 sometimes desirable to throw the instrument into a po- 
 sition that will command a large vertical angle, when 
 the level parallel to the telescope can not be used. 
 If the line of collimation has not been adjusted previ- 
 ously, the points used must be equally above and below 
 the instrument, and at the same distance from it; for 
 it must be borne in mind that if the line of collimation 
 is not perpendicular to the axis of the telescope, it de- 
 scribes the surface of a cone. 
 
 Another method of making this adjustment is to 
 cause the line of sight to follow a plumb-line ; or, what 
 is in effect the same thing, require it to cover a high 
 point and its reflection as seen in a basin of mercury 
 or water. Of these two, the second is the better, owing 
 to the vibrations of the plumb-line ; but both are really 
 tests of the accuracy of the adjustment of the level as 
 well as of the standards, and therefore neither of them 
 is as good as the one first described. 
 
 126. LEVEL UNDER TELESCOPE. The tangent of the 
 level should be parallel to the line of collimation when 
 the latter is horizontal. To make this adjustment, 
 bring the bubble to the middle (any point will do 
 equally well, but the middle is most convenient), and 
 clamp the telescope. Read a leveling-rod on a point, 
 
ART. 3] ADJUSTMENTS OF THE TRANSIT. 
 
 say 200 feet from the instrument ; reverse in azimuth, 
 bring the bubble to its former position by moving the 
 tangent screw of the vertical circle, and establish a 
 second point at the same distance from the instrument 
 as the first one, and read the rod upon it. A line join- 
 ing the two positions of the target is a horizontal line, 
 and the difference of the readings is the true difference 
 of level, however much the instrument may be out of 
 adjustment. Move the instrument very near to one 
 point say, 10 feet beyond it, and read a rod upon the 
 first point ; then, without changing the altitude of the 
 telescope, read upon the second. If the difference of 
 these two readings is the same as the difference of level, 
 the line of collimation is horizontal. If the observed 
 difference is not equal to the difference of level, correct 
 a little more than all the error (for the exact amount 
 see the next paragraph) by moving the farther target. 
 
 Let A and B, Fig. 29, be the two points. When the 
 instrument is at C, midway between A and B, the tar- 
 
 FIG. 29. 
 
 gets are at a and b, and ab is a level line ; and the true 
 difference of level between A and B = Aa Bb. When 
 the instrument is at Z>, the targets are at c and d\ and 
 the apparent difference of level is de = (Be Bb) 
 (Ad Aa). If the line of sight were horizontal, the 
 target would be at/; and therefore df is the correction. 
 AD _ ^ (zAC+BD\ * 
 
 a) 
 
 * This formula is not strictly correct, since (i) it was assumed that a line 
 which is horizontal at C, i.e., perpendicular to the radius of the earth at C, is 
 
1 14 THE TRANSIT. [CHAP. VII 
 
 If D is between A and B, the quantity BD must be 
 subtracted. 
 
 In making this adjustment, take two or three pairs 
 of observations each time for a check, and be careful 
 to see that the instrument is exactly level at the mo- 
 ment of sighting. 
 
 When the line of sight is brought horizontal, bring 
 the bubble to the middle by raising one end of the 
 level tube, and the adjustment is complete. In moving 
 the level tube, be very careful not to alter the inclina- 
 tion of the telescope. Notice that it will not do to 
 move the horizontal cross hair to bring the line of colli- 
 mation parallel to the level, as is sometimes recom- 
 mended, for moving the cross hair will destroy its ad' 
 justment for collimation (2, 123), and may destroy 
 that of the vertical hair. 
 
 127. ZERO OF THE VERTICAL CIRCLE. The vertical 
 circle should read zero when the vertical axis is vertical 
 and the line of sight is horizontal. If the instrument 
 has a level under the telescope, first adjust it ( 126) ; 
 then bring the bubble to the middle, and shift the 
 vernier until its zero coincides with the zero of the 
 limb. If the vernier is not movable, note the difference 
 and apply it as a correction to each angle of elevation 
 or depression. Notice that this adjustment is not nec- 
 essary if only the vertical angle between two points is 
 desired. 
 
 If the instrument has no level under the telescope, 
 the method of making this adjustment is nearly the 
 
 parallel to a line which is horizontal at Z>, and (2) the effect of refraction was 
 omitted. Therefore the observations at D should be corrected for curvature 
 and refraction ( 317-19). To make this correction, subtract o.oc*(BD -H 2oo) 2 
 ft. from the reading at B, and o.ooi[(2 AC -\- BD) -4- 220]* ft. from that at 
 A\ or, with sufficient accuracy, simply subtract 0.001(2 A C -s- 220) * ft. from 
 de before using it in equation (i). Ordinarily this correction is less than the 
 error of observation, and consequently may generally be omitted. 
 
ART. 4] USING THE TRANSIT. 115 
 
 same as that given in 126. Set the instrument mid- 
 way between two points, level it carefully, taking special 
 care with the level parallel to the vertical circle, clamp the 
 telescope nearly horizontal, and determine the differ- 
 ence of level between the points. Move the instrument 
 near one of the points, level it carefully, sight upon 
 each point, and take the difference of reading. If this 
 difference is equal to the true difference of level, as 
 found when the instrument was midway between the 
 points, the line of sight is horizontal. If the differ- 
 ence is not equal to the difference of level, alter the 
 telescope in altitude (see the second paragraph of 126) 
 until the differences are the same. Finally, when the 
 line of sight has been placed horizontal, move the ver- 
 nier to coincide with the zero of the circle. 
 
 ART. 4. USING THE TRANSIT. 
 
 128. PRACTICAL HINTS. The beginner should not 
 neglect the preliminary matters of planting the tripod, 
 bringing the plumb-bob over the point, and leveling 
 the instrument. There is great difference in the skill 
 and rapidity with which different persons will set up 
 the transit; but 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 wher- 
 ever placed, and 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 ; then a gentle 
 pressure on the legs of the tripod will bring the plum- 
 met over the point, and a slight movement of the level- 
 ing screws will bring the plate level. 
 
 If the observer has clearly in mind what he is trying 
 to do, and thoroughly comprehends the effect of possi- 
 ble errors in his instrument, he can save much time and 
 
Il6 THE TRANSIT. [CHAP. VII 
 
 trouble, thus leaving himself free to bestow his atten- 
 tion where it will do the most good. For example, a 
 great deal of time is often wasted in getting the instru- 
 ment precisely over the point. The care 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 if 100 feet 
 away. The direction of the instrument from the point, 
 with reference to the object sighted at, also affects the 
 value of the resulting error. 
 
 It is a waste of time to level up accurately, each time, 
 regardless 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 suffi- 
 cient accuracy by the eye alone ; if only vertical angles 
 between points in the same vertical are desired, only 
 the level parallel to the telescope need be read. On 
 the other hand, in measuring a horizontal angle be- 
 tween a high and a low point, particular attention 
 should be given to the level perpendicular to the tele- 
 scope. In all instrumental work there are certain oper- 
 ations which should be carefully attended to, while to 
 give equal care to others would be only a waste of effort. 
 
 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 instrument can stand firmly with any 
 looseness in these parts. The clamps and tangent 
 screws also should be examined to see that they fit 
 snugly. The instrument may slip on the lower plate, 
 owing to the leveling screws' not being tight enough. 
 
 Some engineers seem to think that the harder the 
 tripod legs are forced 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 its adjustments 
 better and works more kindly, when handled delicately. 
 
ART, 4] USING THE TRANSIT. II 7 
 
 129. MEASURING ANGLES. Although it is scarcely 
 necessary, a brief description of the method of measur- 
 ing a horizontal angle will now be given. Set the in- 
 strument over the point, and level it. Then clamp the 
 upper movement, and read one of the verniers, say A, 
 using the o-36o graduation. Focus the eye-piece 
 on the cross hairs, turn the telescope by hand until it 
 nearly bisects one of the points, and clamp the lower 
 motion. Next focus on the object and turn the lower 
 tangent screw until the intersection of the cross hairs 
 exactly covers the point. Then loosen the clamp of 
 the upper motion, direct the telescope to the other 
 point, and read as before. The difference of the two 
 readings will be the angle between the two points. 
 
 It is immaterial whether the instrument is turned to 
 the right or to the left. If there is only one row of num- 
 bers running from o to 360, and if the vernier passes 
 the zero point in turning to the second station, 360 
 must be added to the smaller reading before taking the 
 difference. If there are two rows of numbers running 
 from o to 360, it is immaterial which way the tele- 
 scope is turned, since that graduation may be read which 
 increases in the direction of the motion. It is con- 
 venient, though not necessary nor quite as accurate, to 
 set the vernier to read zero at the beginning, instead of 
 reading it where it may chance to be. 
 
 130. Angles Measured More 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; 
 viz., by series and by repetition. One or the other of these 
 methods is always used in measuring the principal an- 
 gles of a geodetic triangulation. The principles in- 
 volved are useful in less accurate work. 
 
 131. By Series. Sight upon the first station, and read 
 both verniers to eliminate eccentricity. Sight upon 
 
Il8 THE TRANSIT. [CHAP. VII 
 
 the next station to the right, and read as before. Con- 
 tinue 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 azimuth, turn on the lower axis a little to 
 eliminate personal bias, and read upon the first station. 
 Proceed around the horizon towards the left, reading 
 upon each station and closing upon the first. The re- 
 versal in azimuth and altitude eliminates eccentricity of 
 line of sight, error of telescope slide, and inclination of 
 horizontal axis. Reversing the direction around the 
 horizon eliminates any twist of the tripod ; and shifting 
 the horizontal circle diminishes the possibilities of ac- 
 cidental errors of graduation. The above observations 
 constitute one "set." 
 
 To secure greater accuracy by increasing the number 
 of observations, and also to eliminate periodic errors of 
 graduation, shift the horizontal circle an aliquot part 
 of the distance between verniers, 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. For example, if the 
 angle is to be read three times, and if there are two op- 
 posite verniers, shift the circle 60. 
 
 The arithmetical mean of the observed values is to 
 be considered the true angle. 
 
 132. By Repetition. Sight upon the first station, read 
 both verniers ; and with the upper motion turn to the 
 next station, and read as before. The 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 meas- 
 ured a second time, but on a part of the circle adjoin- 
 ing that on which it was first measured, the second 
 
ART. 4] USING THE TRANSIT. 119 
 
 beginning where the first ended. This operation may 
 be repeated any number of times, the circle being read 
 after the last sight upon the second point. The differ- 
 ence of the first and last readings divided by the num- 
 ber of repetitions gives the angle more precisely than 
 would a single observation. Notice that the vernier 
 need be read only at the beginning and end, although 
 the second reading, as above, is a valuable check in de- 
 termining how many times 360 should be added to the 
 last reading in case the vernier has passed o. Next 
 reverse in altitude and azimuth, and measure the an- 
 gle as before, beginning however at the second station. 
 
 This method eliminates all errors of adjustment, and 
 reduces the error of observation by increasing the num* 
 ber of observations. Of course, the mean of the ob* 
 served values is assumed to be the true angle. 
 
 133. Comparison of Methods. Both methods seem to 
 be about perfect, as far as the elimination of errors of 
 adjustment, of graduation, and of observation is con- 
 cerned. The method by series is preferred by most ob- 
 servers for triangulation work. Its peculiar advantages 
 can be fully realized only with the precise instruments 
 used in that kind of work. With ordinary engineering 
 instruments, there is a limit beyond which it is useless 
 to multiply observations by this method. For exam- 
 ple, if the instrument reads to minutes and the serial 
 readings agree to minutes, the multiplication of obser- 
 vations adds nothing to the accuracy of the result. 
 
 The method by repetition was once a great favorite 
 with the best engineers, especially the French, for tri- 
 angulation work ; but the improvements in the manu- 
 facture of angle instruments has given precedence to 
 the method by series for the most accurate work. How- 
 ever, 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 
 
120 THE TRANSIT. [CHAP. VII 
 
 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 
 screw. Under ordinary conditions, the limit of this 
 method is reached after a few repetitions. 
 
 134. TRANSIT SURVEYING. Under this head will be 
 discussed methods of doing transit work which are 
 more or less applicable in running a line survey as for 
 a railroad, or in finding areas, or in topographical sur- 
 veying. There is no generally accepted method of do- 
 ing transit work. The three methods in more or less 
 general use will now be considered. The first, for want 
 of a better name, will be called the angle method j the 
 second may be appropriately named the quadrant method; 
 and the third has been called traversing. The last 
 might well be named the full-circle system. 
 
 135. Angle Method. This method consists in measur- 
 ing 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 be- 
 tween the second line and the first produced. In the 
 latter case, the telescope is sighted along the first 
 course, the vernier read, and the telescope transited.. 
 The telescope then 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, 
 />., 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, and is re- 
 corded as a deflection to the right or to the left. 
 
 136. Quadrant Method. The distinguishing charac- 
 teristic of this method is that the angles are read 
 and recorded as bearings, just as in compass sur- 
 veying. The meridian through the first station is 
 obtained by reading the needle, or by sighting upon 
 
ART. 4] USING THE TRANSIT. 121 
 
 some line whose bearing is known, or it may be as- 
 sumed since generally the object of such surveys is 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 preceding line produced, and adding it to or 
 subtracting it from the bearing of the preceding line. 
 This method is doubtless a survival from the time when 
 the magnetic compass was the instrument ordinarily 
 used in measuring horizontal angles. The o-to-9o 
 numbering on the horizontal circle of transits is made 
 especially for this system. 
 
 137. 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. 
 
 To run a traverse, set the instrument up over the first 
 station the end of the first course. For the present 
 we will assume that the first course is the reference 
 line, i.e., the meridian of the survey. Set vernier A at 
 o, clamp the upper motion, turn the telescope upside 
 down, and with the lower motion direct the line of 
 sight to the other end of the first course, then clamp 
 the instrument, and transit the telescope. Now loosen 
 the upper motion, sight upon the next station, and read 
 vernier A again. Move to the next station, loosen the 
 lower motion (to prevent the possibility of disturbing 
 the reading in leveling up), level up, invert the tele- 
 scope, and glance at the reading to see that it has not 
 changed. Sight upon the last station (using the lower 
 motion), and clamp. 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 corresponding course 
 and the first one. 
 
122 
 
 THE TRANSIT. 
 
 [CHAP, vii 
 
 Fig. 30 shows the positions of the several lines of a 
 A 
 
 \ L 
 
 FIG. 30. 
 
 traverse survey, and the following table shows the notes 
 for the same. 
 
 Stations. 
 
 Back-Sights. 
 
 Fore-Sights. 
 
 Remarks. 
 
 A 
 
 o oo' 
 
 35 52' 
 
 Azimuths read from 
 
 B 
 
 35 52' 
 
 340 31' 
 
 vernier A, and 
 
 C 
 
 340 31' 
 
 41 08' 
 
 counted from the 
 
 D 
 
 41 08' 
 
 270 oo' 
 
 south toward the 
 
 E 
 
 270 oo' 
 
 180 oo' 
 
 west. 
 
 The instrument is first set at A, the line O A being 
 regarded as the first course. The vernier is set at 
 zero, and a back-sight taken to O, which is recorded 
 opposite A in the back-sight column. The telescope 
 is then transited, the upper motion loosened, the tele- 
 scope directed to B, and the reading of this line, 35 52', 
 recorded opposite A in the fore-sight column of the 
 table. After the instrument is removed to B and the 
 back-sight taken upon A, the vernier is to be read to 
 make sure that it has not been changed, and this 
 reading is recorded in the back-sight column. Writing 
 this down will be evidence that the reading of the 
 vernier was checked; and in actual work this wilt bo 
 found to be an important check against such errors as 
 turning the wrong tangent screw or reading the wrong 
 vernier. The angles marked in the diagram are the 
 
ART. 4] 
 
 USING THE TRANSIT. 
 
 123 
 
 corresponding angles of the fore-sight column in the 
 table. 
 
 138. Instead of making the first course the reference 
 line, as above, any other line, the meridian for exam- 
 ple, may be taken as the reference line. In this case 
 
 to 
 
 FIG. 31. 
 
 the instrument is first set up at the point of beginning, 
 the vernier set as zero, and the first back-sight made in 
 the direction of the meridian. The remainder of the 
 field work and the record is as before. 
 
 It makes but little difference whether the first back- 
 sight is made towards the north or the south end of the 
 meridian, but a note in the remarks column should show 
 which way it is made. However, it is more common, 
 and therefore better, to count azimuths from the south 
 toward the west. 
 
 139. Comparison of Methods. In the time required for 
 the field work, there is very little difference between the 
 three methods, the second and third requiring slightly 
 less than the first. 
 
 The quadrant system is the oldest and the one most 
 frequently used. Its chief advantage is in the facility 
 of checking the vernier by the compass needle. One of 
 the chief objections to this system is that four different 
 directions are designated numerically by the same angle. 
 
124 THE TRANSIT. [CHAP. VII 
 
 The principal disadvantage of the full-circle system 
 is the greater labor required in checking the vernier by 
 the compass needle; but even this could be removed by 
 having the compass graduation of the transit run from 
 o to 360. However, this is more than overbalanced 
 by the greater facility in computing and platting, and 
 in the freedom from ambiguity and error. This system 
 is very convenient in railroad surveying, and in surveys 
 to find the area.* The full-circle system is indispensa- 
 ble in topographical surveying. In fact, the latter is 
 the only kind of surveying in which it is employed to 
 any considerable extent; but as its advantages become 
 better understood, it will be more fully adopted. 
 
 140. SOURCES OF ERROR.t The errors of transit work 
 may be classified as errors (i) of position, (2) of sighting, 
 (3) of manipulation, (4) of adjustment, and (5) of reading. 
 
 141. Errors of Position. The instrument may not be 
 set up over the point about which the angle is desired, 
 nor over the point previously sighted at. An error of 
 an inch can not make an error of more than 3 seconds 
 if the object sighted at is a mile away ; but it may 
 make an error of 3 minutes if the object is only 100 feet 
 away. The position of the instrument with reference 
 to the true point and the object sighted at affects the 
 values of the resulting error. 
 
 142. Errors of Sighting. The flag-pole may not be 
 vertical, therefore sight as low upon it as possible. The 
 intersection of the cross hairs may not exactly cover 
 the point, owing to lack of care or to parallax in the 
 instrument ; but in either case the remedy is obvious. 
 Always bring the intersection of the cross hairs to bear 
 upon the point sighted at, for the vertical hair may not 
 be vertical. 
 
 * See Appendix II. 
 
 t For a discussion of Compensating vs. Cumulative Errors, see 18, 
 
ART. 4] USING THE TRANSIT. 125 
 
 143. Errors of Manipulation. The wrong tangent 
 screw may be turned. This is a fruitful source of error, 
 and one difficult to discover and impossible to correct. 
 If the tangent screw has back-lash, the instrument must 
 be handled so as to prevent it. Error is sometimes 
 produced by the instrument's turning on the ball-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. 
 
 144. Errors of Adjustment. The points to be consid- 
 ered are eccentricity, 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 affect only the horizontal angles between high and 
 low points, and in ordinary practice the resulting error 
 will be inappreciable. The straightness of the slide and 
 the adjustment of the cross hairs are closely related ; 
 but neither will produce error either in measuring hori- 
 zontal angles between points equal distances from, and 
 equal distances above or below, the instrument, or in 
 measuring vertical angles between points equally dis- 
 tant from the instrument. An error of adjustment of 
 the line of collimation produces an error of double the 
 amount in traversing, or in prolonging a line by back- 
 sights and fore-sights. All errors of adjustment can 
 be wholly eliminated by reversing, making a second 
 observation, and taking the mean. 
 
 145. Errors of 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 half- 
 degree of the limb to the reading of the vernier and 
 recording 20' instead of 50'. 
 
 146. LIMITS OF PRECISION. It is a little difficult to 
 get sufficient data for a satisfactory discussion of this 
 
126 THE TRANSIT. [CHAP. VII 
 
 subject without making observations specially for this 
 purpose, which is not desirable. Work is done under 
 very different conditions as to weather, speed, instru- 
 ments, etc., and results without full information on all 
 points are not very valuable. 
 
 The author's students, in the prosecution of the ordi- 
 nary class work in topographical surveying, measured 
 the angles of ten triangles. The length of the sides 
 varied between 400 and 1,200 feet. The conditions as 
 to time, targets, etc., were about those of actual prac- 
 tice. The instrument read to minutes, and certainly was 
 not the best qualit)^. All of the errors enumerated in 
 140-45 were involved. This work gave 32 seconds 
 for the probable error of a single direction ; 50 seconds 
 for the probable error of an angle ; and 86 seconds 
 for the probable error of the sum of the three angles 
 of a triangle. The maximum error in the closing of a 
 triangle was 3^ minutes. 
 
 The same class in measuring the four angles around 
 a point and the three angles of a triangle, using chain- 
 ing 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 targets 
 and sights varying between 200 feet and 800 feet, gave 
 results a little greater than half those of the preceding 
 paragraph. 
 
 Ten measurements of an angle with an engineer's 
 transit reading to minutes (the reading was estimated 
 to 30 seconds), and " distinct signals at about 400 feet," 
 gave a probable error of 19 seconds for a single value 
 of the angle. Under the same conditions, the probable 
 error of an angle as measured with a transit reading 
 to thirds of minutes was 12 seconds.* 
 
 In locating the piers of the bridge across the Ohio 
 
 * Prof. Mansfield Merriman, in Engineering News^ Vol. 10, p. 621. 
 
ART. 4] USING THE TRANSIT. 127 
 
 River, at Cairo, 111., by triangulation each angle was 
 measured fifteen times with an engineer's transif read- 
 ing to 10 seconds, the maximum sight being about 
 5,000 feet and the average about 2,500 feet. The aver- 
 age error of closure of the triangles was 1.5 seconds.* 
 
 In the topographical survey of St. Louis, Mo., the 
 angles were measured with an engineer's transit reading 
 to 10 seconds. Each series of courses began and ended 
 at triangulation stations, which were about a mile apart. 
 The azimuth was checked by comparing it with the 
 azimuth of the lines of the triangulation. The average 
 error of closure of azimuth was about i' 5" for each 
 series of courses, or about n seconds per line.f 
 
 147. 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 
 for compass surveying ( 52), the formulas deduced for 
 that case being applicable to transit surveying. Such 
 a discussion shows that ordinary work with the transit 
 is proportionally as accurate as the best chaining. 
 Therefore, in finding areas by the transit, as much care 
 must be given to the chaining as to the measurement 
 of the angles. 
 
 148. CARE OF THE TRANSIT. 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. 
 Overstraining the leveling screws bends the plates and 
 wears the screws unnecessarily. A very common fault 
 is applying too much force in clamping the instrument. 
 The operator frequently fails to appreciate the power of 
 
 * Journal of the Associated Engineering Societies, Vol. 9, p. 292, 
 f The Technograph, No. 5, p. 12, 
 
128 THE TRANSIT. [CHAP. VII 
 
 a screw. Some instrument makers invite this abuse of 
 their instruments by making the heads of the clamping 
 screw too large. To avoid overstraining the clamp, it 
 is best to ascertain by trial the minimum force required 
 to produce sufficient hold for the action of the tangent 
 screw, and then try to clamp only slightly in excess of 
 this amount. 
 
 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 screwing a thin piece of soft 
 wood through them. The tangent screw is intended 
 only to complete the setting, and should never be 
 used except to give a slight movement. The tele- 
 scope slide and the centers should be examined occa- 
 sionally; and if there is any fretting or cutting, take 
 the piece out and burnish the rough place with some 
 smooth hard tool, as the back of the blade of a pocket- 
 knife. 
 
 To lubricate bearings exposed to the air, first thor- 
 oughly clean, apply a little watch-oil, vaseline, or ren- 
 dered ox-marrow, and then wipe it off. The value of 
 these three lubricants is in the order enumerated, vase- 
 line and ox-marrow being too stiff in cold weather, and 
 the ox-marrow also wearing out rapidly. Finely pow- 
 dered plumbago is very good for exposed bearings. It 
 is a wise precaution to carry in the field a gossamer 
 water-proof bag to throw over the instrument in case 
 of a shower, or in case the instrument must be left stand- 
 ing for any length of time exposed to dust. 
 
 To preserve the outer appearance of an instrument, 
 never use anything in dusting it except a fine camel's- 
 hair brush. To remove dust spots, first use the camel's- 
 hair brush, and then rub with fine watch-oil, and wipe 
 dry. If the oil is allowed to remain on, it will catch 
 dust and dirt, and thus do more harm than good. On 
 reaching the office, after using the instrument, dust it 
 
ART. 4] USING THE TRANSIT. 1 29 
 
 off generally with a fine camel's-hair brush, examine the 
 centers and all other principal movements to see if they 
 run perfectly free and easy, and oil them if necessary. 
 
 To remove dirt and oxide that may have accumulated 
 on the surface of a silver graduation, apply some fine 
 watch-oil, and allow it to remain for a few hours; then 
 take a soft piece of old linen and lightly rub until dry, 
 but without touching the edge of the. graduations. 
 If, after cleaning, the silver surface should show dark 
 and bright spots (which would interfere somewhat 
 with the accurate reading of the graduation), barely 
 moisten the finger with vaseline and apply the same 
 to the surface; then wipe the finger dry and lightly 
 rub it once or twice around the circle, touching the 
 graduation as little as possible. Such cleaning, how- 
 ever, must be resorted to only when absolutely neces- 
 sary, and then only with the greatest care, as it is too 
 apt to spoil the sharpness of the lines, which will de- 
 crease the accuracy of the reading. 
 
 It is not desirable to take the instrument apart unnec- 
 essarily, for even if the fittings are perfect, 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 much, may damage the instrument 
 or cause errors in its use. As long as an instrument 
 works well and the centers revolve freely, it is best not 
 to disturb it. 
 
 For hints on the care of the telescope, see page 89. 
 
CHAPTER VIII. 
 SOLAR TRANSIT. 
 
 149. THE solar transit is simply an engineer's transit 
 to which is attached a solar apparatus ( 55). There is 
 great diversity in the form of the solar apparatus and 
 in the method of attaching it to the transit. 
 
 ART. 1. CONSTRUCTION OF THE SOLAR TRANSIT. 
 
 150. SAEGMULLEE'S SOLAR TRANSIT. The latest form 
 of solar transit is shown in Fig. 32.* It consists sim- 
 ply of a telescope and level attached to an ordinary 
 transit in such a manner as to be free to revolve in two 
 directions at right angles to each other. When the 
 transit telescope is horizontal, the auxiliary telescope 
 and its level revolve in horizontal and vertical planes. 
 
 If the transit telescope be brought into the meridian 
 and the vertical circle of the transit be set at the co- 
 latitude, the polar axis of the solar telescope will then 
 be parallel to the axis of the earth ; then if the solar 
 telescope be turned on the polar axis, the solar sight- 
 line will describe a line parallel to the equator. If the 
 solar sight-line is perpendicular to the polar axis, the 
 line of sight will describe the equator when the solar 
 
 * Invented by G. N. Saegmuller, in 1881. This form is made by G. N. 
 Saegmuller, Washington, D. C., and by Keuffel & Esser Co., New York 
 City ; and somewhat the same form is made by Buff & Berger, Boston, 
 Mass. 
 
 130 
 
ART. l] CONSTRUCTION OF THE SOLAR TRANSIT. 131 
 
 FIG. 32. SAEGMUI.LER'S 
 
 ATTACHMENT. 
 
132 SOLAR TRANSIT. [CHAP. VIII 
 
 telescope is revolved on the polar axis. If the solar 
 sight-line makes an angle with the perpendicular to the 
 polar axis equal to the declination of the sun, then 
 when the solar telescope is revolved the line of sight, 
 would follow the sun's path in the heavens for the given 
 day, provided the sun did not change its declination. 
 Therefore, if the solar telescope is set at an angle with 
 the perpendicular to the polar axis equal to the dec- 
 lination of the sun, and the solar telescope is directed 
 to the sun, then the terrestrial sight-line will indicate a 
 true meridian. 
 
 151. With the form of solar apparatus shown in Fig. 
 32 it is difficult to sight the solar telescope upon the sun 
 accurately, owing to there being no clamp and slow- 
 motion screw for the polar axis. The latest form of 
 solar attachment has a clamp and tangent screw for the 
 polar axis. See Fig. 33. 
 
 FIG. 33. SAEGMULLER'S IMPROVED SOLAR ATTACHMENT. 
 
 Two pointers are attached to the solar telescope to 
 aid in directing it to the sun. They are so adjusted 
 
ART. l] CONSTRUCTION OF THE SOLAR TRANSIT. 133 
 
 that when the shadow of the one is thrown on the other, 
 the sun will appear in the field of view. 
 
 The solar telescope is provided with colored glass 
 shades to protect the eye when observing. The objec- 
 tive and the cross hairs are focused in the usual way. 
 There is a pair of horizontal cross hairs and also a pair 
 of vertical ones, between which the image of the sun 
 is brought. 
 
 152. OTHER FORMS. Many solar transits consist of 
 an ordinary engineer's transit to which is attached the 
 solar apparatus of the solar compass ( 56). The solar 
 attachment is fastened to the center of the horizontal 
 axis of the telescope, or to the end of the horizontal 
 axis, or below the horizontal plate. All of these forms 
 are inferior to the solar transit shown in Fig. 32 for one 
 or more of the following reasons: 
 
 i. Most of them are more complicated. 2. All of 
 them are more difficult to adjust. 3. All are deficient 
 in precision. The solar sights of a solar compass con- 
 sist merely of a small lens and a piece of silver with 
 lines ruled on it placed at the focus of the lens, the 
 coincidence of the sun's image with the lines being 
 determined by the unaided eye- or, at best, with a 
 simple magnifying lens. This primitive telescope is 
 probably about as accurate as the terrestrial sights of 
 the common compass, but is much less accurate than 
 the terrestrial telescope of the solar transit. It is obvious 
 that the substitution of a telescope for the lens and the 
 lines of the solar compass greatly increases the precision 
 attainable. Obviously the power of the solar telescope 
 should be in keeping with that of the terrestrial telescope. 
 4. The solar telescope can be used when the sun is 
 partly obscured by clouds, at which time the ordinary 
 solar apparatus fails altogether. 
 
134 SOLAR TRANSIT. [CHAP. VIII 
 
 ART. 2. ADJUSTMENTS OF THE SOLAR TRANSIT.* 
 
 153. The adjustments of the solar transit consist of 
 two distinct operations the adjustment of the transit 
 and that of the solar apparatus. 
 
 The transit should be in perfect adjustment, particu- 
 larly the plate levels ( 38 and 39), the standards 
 ( 125), and the zero of the vertical circle ( 127). 
 
 154. ADJUSTMENT OF THE POLAK Axis. The polar axis 
 should be vertical when the line of collimation and the 
 horizontal axis of the transit are horizontal. To make 
 this adjustment set the vernier of the vertical circle to 
 read zero, and level the instrument by means of the 
 plate levels. If there is a level under the telescope, the 
 verticality of the vertical axis can be tested by revolving 
 the transit on its vertical axis, and noticing whether the 
 bubble of the level under the telescope remains station- 
 ary during the entire revolution. If it does not remain 
 stationary, correct half the error by turning the foot 
 screws. Since the level under the telescope is usually 
 more sensitive than the plate levels, it is better to level 
 the instrument by the latter than by the former. When 
 these steps have been correctly made, the plane of the 
 horizontal axis and the line of collimation are hori- 
 zontal. 
 
 Then, to bring the polar axis vertical, revolve the 
 solar telescope about the polar axis and notice if the 
 bubble on the small telescope maintains a constant 
 position. If it does not, correct half the movement by 
 revolving the solar telescope on its horizontal axis, and 
 the other half by means of the adjusting screws at 
 the base of the solar apparatus (see Fig. 33, page 132). 
 
 * For general remarks upon adjustments, see 37. 
 
ART. 3] USING THE SOLAR TRANSIT. 135 
 
 Notice that these adjusting screws are analogous to the 
 foot screws of the main instrument. 
 
 155. ADJUSTMENT OF THE CROSS HAIRS. The line of 
 collimation of the solar telescope and the axis of its 
 level should be parallel. Bring the terrestrial telescope 
 horizontal by setting the vernier of the vertical circle, 
 or by reading the bubble under the telescope. Place 
 the solar telescope as nearly as possible in the plane of 
 the terrestrial telescope, and make the former horizon- 
 tal by means of its bubble, and clamp it. Measure the 
 distance between the axes of the two telescopes, and 
 draw at this distance from each other two heavy black 
 lines on a piece of paper. Set this piece of paper up, 
 with the lines horizontal, at a convenient distance from 
 the instrument and on about the same level as the 
 telescope. Bring the terrestrial line of sight upon the 
 lower mark, and see if the solar line of sight falls upon 
 the upper mark. If it does not, move the cross hairs of 
 the solar telescope until it does. Since moving the 
 cross hairs is very liable to revolve the solar telescope 
 on its horizontal axis, test the adjustment by revolving 
 back to the horizontal position, and see If both bubbles 
 come to the middle simultaneously. 
 
 ART. 3. USING THE SOLAR TRANSIT. 
 
 156. To DETERMINE A MERIDIAN. Notice that an 
 observation with the solar transit involves four quan- 
 tities, as follows: (i) the time of day, i.e., the hour angle 
 of the sun; (2) the declination of the sun; (3) the lati- 
 tude of the place of observation; and (4) the direction 
 of the meridian. In a general way, if any three of these 
 are known, the fourth may be found by observation. 
 The prime object of the solar transit is to find the true 
 meridian when the other three elements are known. 
 
136 SOLAR TRANSIT. [CHAP. VIII 
 
 157. The Time. The declination of the sun is tabu- 
 lated for Greenwich or Washington noon, and hence a 
 correction must be applied to find the declination at the 
 time of the observation. An error of 30 minutes in the 
 time can not make an error of more than 30" in the 
 declination, and the average error will be about 15", 
 which is less than can be laid off on the instrument. 
 Therefore it is sufficient if the error in the time does 
 not exceed 30 minutes; and ordinarily there will be no 
 difficulty in finding it much closer. 
 
 158. The Decimation. The declination of the sun, and 
 also the hourly change in the declination, is given for 
 each day of the year, in the "American Ephemeris and 
 Nautical Almanac,"* for both Greenwich and Washing- 
 ton mean noon. Hence, to find the declination at 
 any given time, it is necessary to know the time of 
 day and the longitude from either Greenwich or Wash- 
 ington; and since the time generally employed in this 
 country, *>., standard time, differs from Greenwich 
 time by an integral number of hours, it is better to 
 use the Greenwich declination. Therefore, for a point 
 on, say, the meridian of New Orleans (90 west from 
 Greenwich), the declination given in the ephemeris for 
 Greenwich noon is the declination at 6 a.m. at the 
 place of observation. For a point either side of the 
 90th meridian, the standard time does not indicate 
 the true hour angle of the sun, i.e., there is a difference 
 between standard and local time; and hence for a place 
 west of the 90th meridian the declination at noon at 
 Greenwich is the declination at some time before 6 a.m., 
 and for a place east of the 90th meridian the declina- 
 tion at Greenwich noon is the declination at some time 
 after 6 a.m. 
 
 * Issued several years in advance by the U. S. Government, and for sale by 
 book dealers. Price $1.25. 
 
ART. 3] USING THE SOLAR TRANSIT. 137 
 
 Since the difference between standard and local time 
 is ordinarily less than 30 minutes, and since the hourly 
 change in declination is less than 60 seconds of arc, it 
 is sufficiently exact to assume that the declination of 
 the sun given in the ephemeris for Greenwich noon for 
 any day is the declination at 7, 6, 5, or 4 a.m. of the 
 same date according as the point is situated in the Eas- 
 tern, Central, Mountain, or Western time belt. Thus if 
 the point of observation is in the Central time belt, the 
 declination given in the ephemeris may be assumed to be 
 the declination at 6 a.m. at the place of observation. 
 Of course, this is strictly true only for points on the 
 governing meridian of each time belt. 
 
 Knowing the declination for a given time of day at 
 the point of observation, to find the declination for any 
 other time of day it is only necessary to multiply the 
 hourly change by the difference between the time for 
 which the declination is given and the time for which 
 it is required, and add it (algebraically) to the tabu- 
 lated value. For example, if the declination at Green- 
 wich noon on November 19, is 19 28' 24", and the 
 hourly change is 35", what is the declination at 
 ii a.m. at Chicago? The declination at Greenwich 
 noon corresponds (nearly) to the declination at 6 a.m. at 
 Chicago; and hence a change for 5 hours (from 6 a.m. 
 to ii a.m.) must be allowed for. Therefore the required 
 declination is 19 28' 24" plus ( 35" X 5), which 
 equals 19 31' 19". In a similar way the declination 
 for any hour may be found. 
 
 159. To Correct the Declination for Refraction. Owing 
 to refraction, all celestial objects appear higher than 
 they really are. The declination to be set off on the 
 solar apparatuses the apparent and not the real decli- 
 nation, and hence the declination as found above, before 
 being set off on the instrument, must be corrected for 
 refraction. 
 
138 SOLAR TRANSIT. [CHAP. VIII 
 
 The refraction is zero for a point in the zenith, about 
 i' for an altitude of 45, and about 34" at the horizon. 
 It varies greatly with the temperature, pressure, and 
 hygrometrical condition of the atmosphere, particu- 
 larly near the horizon. It is for this reason that all 
 astronomical observations made near the horizon are 
 very uncertain. Tables of mean refraction are fre- 
 quently given in text-books on astronomy,* in loga- 
 rithmic tables, etc. 
 
 Notice that refraction changes the apparent altitude, 
 which is measured perpendicular to the horizon, while 
 we desire to know the effect upon the apparent decli- 
 nation, which is measured perpendicular to the equa- 
 tor of the heavens. When the sun is on the meridian, 
 the change in altitude has its full effect in changing the 
 declination, but at other times the change in declination 
 is less than the change in altitude. The correction to 
 the declination due to refraction is 
 
 C" = 57" cot (d + N),\ . . , . (i) 
 
 where d = declination, plus when north and minus 
 when south. N is an auxiliary angle such that 
 
 tan -A 7 " = cot cos /, .... (2) 
 
 where is the latitude, and / is the hour angle. 
 
 160. By means of equations (i) and (2) the refraction 
 correction can be computed for any latitude, hour angle, 
 and declination. Messrs. W. and L. E. Gurley, Troy, 
 N. Y., publish a table giving this correction for each 
 2-J of latitude from 30 to 57^, for each 5 of declina- 
 tion, and for integral hour angles from o to 5. 
 
 * For example, Loomis's Practical Astronomy, p. 364, and Chauvenet's 
 Spherical and Practical Astronomy, vol. ii, pp. 604-7. 
 
 t See Chauvenet's Spherical and Practical Astronomy, vol. i, p. 171. 
 
ART. 3] USING THE SOLAR TRANSIT. 1^9 
 
 G. N. Saegmuller, Washington, D. C., publishes an- 
 nually for gratuitous distribution a small pamphlet giv- 
 ing the declination of the sun and its hourly change for 
 each day of the year. In the margin of the table is 
 given the refraction correction for 40 of latitude from 
 o hours to 5 hours of hour angle. In a supplemental 
 table is given a series of co-efficients by which the re- 
 fraction correction for 40 of latitude must be multi- 
 plied to obtain the refraction correction for any other 
 latitude. As very complete explanations accompany 
 these tables, and as it is necessary to have a table of 
 declinations for the current year, it is not thought wise 
 to consider this subject further here, but the reader 
 who has any considerable solar work to do is advised 
 to send for this little pamphlet, the title of which is 
 " Solar Ephemeris and Refraction Tables." 
 
 161. The Latitude. The latitude is required that the 
 polar axis may be set parallel to the axis of the earth. 
 To find the latitude, a few minutes before the meridian 
 passage of the sun level the transit carefully and point 
 the telescopes toward the south. Then incline the ter- 
 restrial telescope if the declination is north depress 
 it, and if south elevate it, until the reading of the 
 vertical circle is equal to the declination of the sun 
 uncorrected for refraction ; and bring the solar telescope 
 into the vertical plane of the terrestrial telescope, level 
 it carefully, and clamp it. By moving the terrestrial 
 telescope in altitude and azimuth, bring the image of 
 the sun between the hairs of the solar telescope, and 
 keep it there until the sun ceases to rise. Then the 
 reading of the vertical circle, corrected for refraction, 
 is the co-latitude. 
 
 162. The Observation. First focus the solar telescope 
 for distinct vision of the cross hairs, and then care- 
 fully focus it upon the sun. It is necessary to make 
 these adjustments before setting off the declination, 
 
140 . SOLAR TRANSIT. [CHAP. VIII 
 
 since the solar telescope is liable to be displaced if they 
 are made afterwards. 
 
 Set off the declination by inclining the transit tele- 
 scope until the vertical circle reads the declination if 
 the sun- is south of the equator elevate the transit tele- 
 scope, and if north depress it. Without disturbing the 
 position of the transit telescope, bring the solar tele- 
 scope into the vertical plane of the transit telescope, 
 and also to a horizontal position by means of the level 
 on the solar telescope. The two telescopes will then 
 make an angle with each other equal to the declination, 
 and the inclination of the solar telescope to its polar 
 axis will be equal to the polar distance of the sun. 
 Without disturbing the relative position of the two 
 telescopes, set the vernier of the vertical circle to the 
 co-latitude of the place of observation. 
 
 By revolving the transit on its vertical axis and the 
 solar apparatus about its polar axis, taking great care 
 not to revolve either telescope on its horizontal axis, 
 bring the image of the sun into the solar telescope, and 
 then the transit telescope must be on the meridian. If 
 the vertical motion of the transit be clamped, the tran- 
 sit telescope may be turned down and the meridian 
 marked; or the horizontal circle may be read, and the 
 angle which any line makes with the true meridian can 
 be found. 
 
 163. SOURCES OF ERROR.* There are three sources 
 of error in addition to those of adjustment and manip- 
 ulation of the instrument, viz.: (i) error of declination, 
 (2) error of latitude, and (3) error of refraction. 
 
 164. Declination and Latitude. Table II f (page 141) 
 shows the errors in the direction of the meridian due 
 to an error of i minute in the latitude or declination. 
 
 * For a discussion of Cumulative vs. Compensating Errors, see 18. 
 t From " Theory and Practice of Surveying," by J. B. Johnson, by per- 
 mission. 
 
ART. 3] USING THE SOLAR TRANSIT. 
 
 TABLE II. 
 
 ERRORS IN AZIMUTH DETERMINED BY SOLAR TRANSIT FOR i MINUTE 
 ERROR IN DECLINATION OR LATITUDE. 
 
 HOUR. 
 
 FOR i MIN. ERROR IN 
 DECLINATION. 
 
 FOR i MIN. ERROR IN 
 LATITUDE. 
 
 Lat. 30. 
 
 Lat. 40. 
 
 Lat. 50. 
 
 Lat. 30. 
 
 Lat. 40. 
 
 Lat. 50. 
 
 1.1.30 a.m. 1 
 12.30 p.m. ) 
 
 n a.m. . . , / 
 i p m , . C 
 
 Min. 
 
 8.85 
 
 4.46 
 2.31 
 1.6 3 
 1-34 
 
 1.20 
 
 I. IS 
 
 Min. 
 10.00 
 
 5.05 
 
 2.61 
 
 1.85 
 
 1.51 
 1.35 
 1.30 
 
 Min. 
 12.90 
 
 6.OI 
 
 3-n 
 
 2. 2O 
 I. 80 
 
 1.61 
 
 1.56 
 
 Min. 
 8.77 
 
 4-33 
 2.00 
 
 I-I5 
 0.67 
 
 0.31 
 0.00 
 
 Min. 
 9.92 
 
 4.87 
 2.26 
 1.30 
 0.75 
 0-35 
 
 0.00 
 
 Min. 
 II.80 
 
 5-80 
 2.70 
 I. 5 6 
 O.gO 
 
 0-37 
 0.00 
 
 10 a.m. . . . ) 
 
 2 n m . C 
 
 Q a m . ) 
 
 3 p.m. ... f 
 8 a.m ) 
 
 4n m f 
 
 7 a.m . . ) 
 
 5p.m f 
 
 6 a.m ) 
 6 p.m f 
 
 Several important conclusions may be drawn from this 
 table and the equations from which it was deduced. 
 
 1. " The solar apparatus should never be used between 
 ii a.m. and i p.m., and preferably not between 10 a.m. 
 and 2 p.m., if the best results are desired. 
 
 2. " At 6 a.m. and 6 p.m., when the line of collima- 
 tion lies in a plane at right angles to the plane of the 
 meridian, no small error in the latitude will affect the 
 accuracy of the result. 
 
 3. " The best times of day for using the solar apparatus 
 are from 7 to 10 a.m. and from 2 to 5 p.m. So far as 
 the instrumental errors are concerned, the greater the 
 hour angle the better the observation; but when the 
 
142 SOLAR TRANSIT. [CHAP, vni 
 
 sun is near the horizon, the uncertainties in the refrac- 
 tion may cause unknown errors of considerable size. 
 
 4. " For a given error in the setting for declination 
 or latitude the resulting error in azimuth will have op- 
 posite signs in forenoon and afternoon. If, therefore, a 
 jo-o'clock azimuth is in error 5' in one direction from 
 erroneous settings, a 2-o'clock observation with the 
 same instrument should give an azimuth 5' in error in 
 the opposite direction. 
 
 5. " If the declination angle be erroneously set off, 
 and the latitude angle be also affected by an equal error 
 in the opposite direction, then the two resulting errors in 
 azimuth will nearly balance each other. 
 
 6. " If the instrument is out of adjustment, the lati- 
 tude found by a meridian observation will be in error; 
 but if this observed latitude be used in setting off the co- 
 latitude, the instrumental error is eliminated. There- 
 fore always use for the co-latitude that given by the 
 instrument itself in a meridian observation." 
 
 165. Refraction. The refraction correction to the de- 
 clination ( 159) is computed on the assumption that 
 the refraction is a mean, whereas the actual refraction 
 at any time and place may differ considerably from the 
 mean or average. This difference is liable to be very 
 much greater at low than at high altitudes. For this 
 reason no observations should be made within 20 of 
 the horizon, and preferably not within 30. Fortu- 
 nately most solar work is done in the summer, when the 
 sun is high in the heavens, and this limitation is less 
 serious then than in the winter. 
 
 Any error in the refraction correction to the declina- 
 tion has the same effect as an equal error in the decli- 
 nation. The refraction correction may be computed by 
 means of equations (t) and (2), page 138; and the effect 
 of any assumed per cent of error in this correction may 
 be determined by an inspection of Table II, page 141. 
 
ART. 3] USING THE SOLAR TRANSIT. 143 
 
 For example, if the observation be made in latitude 40 
 on September 23 at 7 a.m., the refraction correction 
 will be a little more than 3'; and if we assume that the 
 refraction may be in error 20 per cent (probably a fair 
 assumption for this altitude), the refraction correction 
 may be in error nearly 40", and from Table II we see 
 that the corresponding error in azimuth would be a 
 trifle over 50". 
 
 If the observations are not made within 20, or, 
 better, 30, of the horizon, and if the limitations in 
 164 are observed, the error due to refraction can not 
 be serious. 
 
 166. LIMITS OF PRECISION. The author's students in 
 ordinary class work adjust their own instruments, and 
 determine a meridian with the Saegmuller solar attach- 
 ment (the latitude and time being known accurately) 
 with a maximum error of 4' to 5' between observations 
 made in quick succession, with an average error of 2' 
 to 3'. Eight students made three observations each, 
 and the greatest difference between the mean of each 
 three and the true meridian (as determined by obser- 
 vations with an astronomical transit) was 4'. 6. The 
 average difference was i'.6, the probable error (Appen- 
 dix III) of the mean of three observations was i'.8, and 
 the probable error of a single observation was I'.o. 
 
 167. The following results by Professor J. B. John- 
 son, of Washington University, St. Louis, Mo., may be 
 taken as the best than can be done : * 
 
 " In order to determine just what accuracy was possi- 
 ble with a Saegmuller solar attachment, I spent two 
 days in making observations on a line whose azimuth 
 had been determined by observations on two nights on 
 Polaris at elongatioa, the instrument being reversed to 
 eliminate errors of adjustment. Forty-five observations 
 
 * Journal of the Association of Engineering Societies, Vol. 5, p. 35. 
 
144 SOLAR TRANSIT. [CHAP. VIII 
 
 were made with the solar attachment on October 24, 
 1885, from 9 to 10 a.m., and from 1.30 to 4 p.m., and on 
 November 7 forty-two observations between the same 
 hours. 
 
 " On the first day's work the latitude used was that 
 obtained by an observation on the sun at its meridian 
 passage, being 38 39', and the mean azimuth was 
 20" in error. On the second day, the instrument hav- 
 ing been more carefully adjusted, the latitude used 
 was 38 37', which was supposed to be about the true 
 latitude of the point of observation. It was afterwards 
 found this latitude was 38 37' 15", as referred to Wash- 
 ington University Observatory, so that when the mean 
 azimuth of the line was corrected for the 15" error in 
 latitude it agreed exactly with the stellar azimuth of 
 the line, which might have been 10" or 15" in error. 
 On the first day all the readings were taken without a 
 reading glass, there being four circle readings to each 
 result. On the second day a glass was used. On the 
 first day the maximum error was 4', the average error 
 was o'.8, and the probable error of a single observation 
 was also o'.8. On the second day the maximum error 
 was 2'.7, the average error was i', and the probable 
 error of a single observation was o'.86. The time re- 
 quired for a single observation is from three to five 
 minutes." 
 
 168. THE SOLAR TRANSIT IN MINE SURVEYING. Since 
 the standards of the solar telescope, as ordinarily made, 
 are long enough to allow the small telescope to sight 
 past the horizontal plates of the transit, the solar at- 
 tachment can be used for oblique or vertical sighting, 
 as is frequently required in mine surveying. 
 
 If the solar transit is used for this purpose, it should 
 be adjusted as described in 153-155, and the lines of 
 sight of the two telescopes should lie in the same 
 vertical plane. The last adjustment may be made by 
 
ART. 3] USING THE SOLAR TRANSIT. 145 
 
 leveling the instrument and sighting both telescopes 
 upon a plumb-line. When this adjustment has been 
 made the lines of collimation of the two telescopes are 
 parallel, and any angle, say 90 from the horizontal, 
 may be set off on the vertical circle, and the sight made 
 through the small telescope. All instrumental errors 
 are eliminated if, after making one observation, the 
 transit is reversed on its vertical axis and another ob- 
 servation is made, the mean of the two points being the 
 correct one. 
 
CHAPTER IX 
 PLANE TABLE. 
 
 ART. 1. CONSTRUCTION. 
 
 169. In its simplest form, the plane table consists of 
 a drawing board mounted on a tripod, 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 additions to render the oper- 
 ations more convenient and precise.* 
 
 170. COMPLETE PLANE TABLE. Fig. 34 (page 147) 
 shows one of the most elaborate forms of plane table. 
 An instrument of this form is virtually a transit in 
 which the horizontal circle is replaced by the drawing 
 board, the lines being drawn upon the paper instead of 
 being read from the graduated limb. The lower plate 
 of the transit is expanded, and becomes the drawing 
 board; and the vernier or index is replaced by a 
 straight-edge, which with the telescope and vertical 
 arc is commonly called the alidade. 
 
 The alidade is usually about 20 inches long and 2-J 
 inches wide. At the center is a standard which sup- 
 ports the telescope and which serves as a handle for 
 the alidade. In the best plane tables the telescope is 
 
 * " The invention of the plane table is ascribed to Praetorious in 1537, but 
 the first published description appears to be that of Leonhard Zubler in 1625, 
 who ascribes the ' beginning ' of the instrument to one Eberhart, a stone- 
 mason." 
 
 146 
 
ART * l ] CONSTRUCTION. 
 
 147 
 
 ' 
 
 FIG. 34. COMPLETE PLANE TABLE. 
 
148 PLANE TABLE. [CHAP. IX 
 
 equal to that on an ordinary transit. In s,ome forms 
 the telescope does not transit on its horizontal axis, but 
 is reversed in azimuth by lifting it out of its bearings. 
 The telescope has no lateral movement with respect to 
 the ruler, but both may be moved at pleasure on the 
 table. The telescope is movable in a vertical plane, and 
 is provided with a vertical circle. In the form shown 
 in Fig. 34, the telescope is mounted centrally over the 
 standard. In some forms the line of sight is placed 
 over the beveled edge of the ruler, though this is not 
 essential. It is only necessary that they should have a 
 fixed horizontal angle with each other. 
 
 The mechanism connecting the board with the tripod 
 is the most important part of the plane table. The 
 board must be supported rigidly in a horizontal plane, 
 and also be free to move in that plane. The diffi- 
 culties of satisfying these conditions are increased by 
 the weight and possible eccentric position of the ali- 
 dade. The instrument should also have a substantial 
 clamp and tangent screw for the motion in azimuth. 
 In addition, portability must be considered. The form 
 shown in Fig. 34 secures great stability with little 
 weight, and allows the facilities necessary for manipu- 
 lation. Plane tables of this class are usually supported 
 upon three leveling screws. 
 
 A declinator, a small box carrying a needle which 
 can swing 10 or 15 either side of the zero line, should 
 accompany the instrument for use in determining the 
 magnetic meridian. The zero line being parallel to one 
 edge of the box, the magnetic meridian may at once be 
 marked down on any portion of the map, and the bear- 
 ing of any intersecting line may be determined by a 
 protractor. Sometimes the needle is capable of swing- 
 ing through 360, in which case the magnetic bearing 
 of any line may be read with the compass-box alone. 
 
 The board varies greatly in size, but usually is not 
 
ART. l] CONSTRUCTION. 149 
 
 larger than 24 by 30 inches. It is very important that 
 the board should be made of well-seasoned wood and 
 be so constructed as not to warp. The securing of a 
 plane upon which to work is one of the most difficult 
 conditions to fulfill in the construction of a plane table. 
 Brass and plate glass have been used for this purpose 
 in Europe, but make the table excessively heavy. The 
 paper is fastened to the board in any of several ways : 
 (r) by thumb-tacks screwed into a metal socket set in 
 countersunk holes so as to place the head of the tack 
 or screw out of the way of the ruler ; (2) by wrapping it 
 around rollers at the ends of the board ; (3) by pressing 
 the end of the sheet against the end of the board with 
 a strip of wood or metal connected to the board by 
 screws ; (4) by movable spring clamps gripping the edge 
 of the board; (5) by pasting it down. 
 
 171. LIGHT PLANE TABLE. The form shown in Fig. 
 
 34 is quite heavy and difficult to handle on rough or 
 obstructed ground. The mechanism connecting the 
 board to the tripod is the chief source of weight. To 
 reduce the weight of these parts, the form shown in 
 Fig- 35 was invented.* In the figure, a represents a 
 
 FIG. 35. GURLEY'S PLANE TABLE. 
 
 hemispherical concave metal cup fastened by screws to 
 the wood top of the tripod, and b a convex hemispheri- 
 cal cup fitting into the cup a and being clamped to it at 
 
 * Patented and manufactured by W. & L, E, Gurley, Troy, N, Y, 
 
150 PLANE TABLE. [CHAP. IX 
 
 will by the clamping pieces and nut d. A strong spiral 
 spring in the hollow cylinder between c and d holds the 
 two spherical surfaces of the two pieces together, and 
 also allows the easy movement of the one within the 
 other. The flange of the cup b supports the board and 
 is connected with its under surface by three segments 
 of brass, two of which are shown at e, e. These are 
 brought down firmly upon the shoulder of the flange 
 by capstan-head screws as shown, or released at will, 
 thus allowing the board to be moved horizontally when 
 desired. 
 
 Fig. 36 shows an improvement of the preceding form, 
 
 FIG. 36. JOHNSON'S PLANE TABLE. 
 
 which has been adopted by the U. S. Geological Sur- 
 vey. The arrangement is essentially the same as in 
 Fig. 35, with the addition of the winged nut g. When 
 it is desired to level the table, the clamping nut d is 
 released, the board is brought into position, and then 
 securely clamped by the same nut. When it is desired 
 to turn the table in azimuth, the nut g is loosened, 
 which leaves the hemispherical surface b free to move 
 around the concave part a of the tripod head. The 
 only objection to this form of plane table is the impos- 
 sibility of leveling the board sufficiently for the accu- 
 rate determination of vertical distances with the vertical 
 circle of the alidade. For work to which this limita- 
 tion does not apply, this is a very excellent form of 
 instrument, 
 
ART. l] 
 
 CONSTRUCTION. 
 
 172. HOME-MADE PLANE TABLE. A very fair plane 
 table can be made very cheaply, if the engineer has a 
 transit or leveling instrument in which the upper part 
 is detachable from the leveling screws. This requires 
 a good drawing board, say 18 by 20 inches, to the under 
 side of which is screwed a casting of iron or brass simi- 
 lar to Fig. 37. The conical portion, 0, should be made 
 to fit the socket from which the axis of the level or 
 
 a 
 
 FIG. 
 
 37- 
 
 transit is taken ; and the cylindrical portion, c c, should 
 fit the lower clamp. The alidade may be made of wood, 
 similar to that shown in Fig. 38. One slit is narrow, 
 
 FIG. 38. ALIDADE. 
 
 the other is wide and has a thread through the middle. 
 The scale can be obtained by attaching a printed paper 
 
152 PLANE TABLE. [CHAP. IX 
 
 scale. It is convenient to have a separate scale also. 
 A level vial may be obtained of an instrument maker for 
 a few cents, and may be fastened in the alidade with 
 plaster of Paris. In fixing the vial in position, remem- 
 ber that, as usually made, level tubes are convex on one 
 side and concave on the opposite. The former should 
 be up, else the bubble will run to one end or the other, 
 and never stop in the middle. 
 
 It is sometimes necessary to place a point on the 
 board exactly over the corresponding point on the 
 ground, which requires that the plumb-bob shall be sus- 
 pended from the under side of the board immediately 
 below the point given on the upper surface.* To aid in 
 doing this, a plumbing bar, or frame, similar to Fig. 39, 
 
 FIG. 39. PLUMBING BAR. 
 
 is very convenient. It is made of two light bars of 
 wood fastened together by a hinge at C. A is a. piece 
 of metal sharpened to a vertical edge. B is a piece of 
 metal fastened into the end of the arm B C, and shaped 
 to carry a plumb-line. The point A is placed upon the 
 point on the paper, and the plumb-line is suspended from 
 B. The plumbing bar is so made that if the under side 
 of the upper arm is horizontal, the line joining^ and B is 
 vertical. The length of the arm A C should be a little 
 greater than half the greatest dimension of the board, 
 and A B should be a little greater than the distance from 
 the top of the board to the bottom of the tripod head. 
 
 * For a method of obviating this difficulty, see 188. 
 
ART. 2J TESTS AND ADJUSTMENTS. 
 
 ART. 2. TESTS AND ADJUSTMENTS.* 
 
 173. THE SIGHTS. The sights of the elementary ali- 
 dade ( 172) should be perpendicular to its base. This 
 can be tested sufficiently with a try-square. If this con- 
 dition is not satisfied, the slits will not be vertical when 
 the board is level, and sighting through the top of 
 one and the bottom of the other will give a different 
 line from that given by sighting through the tops or 
 the bottoms of both. 
 
 174. EDGE OF RULER. The edge of the ruler should 
 be a straight line. To test this, place the rule upon a 
 smooth surface and draw a line along the edge ; then 
 reverse the rule end for end, place the edge upon the 
 line, and again draw a line. If the two lines coincide 
 the edge is straight. f 
 
 175. LEVELS ON ALIDADE. The bubble should be in 
 the middle when the table is level. To make this ad- 
 justment, place the alidade in the middle of the table 
 and bring the bubble to the center by means of the 
 leveling screws of the table. Draw lines along the edge 
 of the rule to show its exact position, and then reverse 
 it 180. If the bubble remains in the center, it is in ad- 
 justment. If it does not, correct one half by means of 
 the leveling screws of the table, and the other half by 
 the adjusting screws attached to the level. 
 
 It is next to impossible to make this adjustment ac- 
 curately if the table is not a perfect plane. Great care 
 
 * For general remarks upon the adjustments, see 37. 
 
 t " There is one deviation from a straight line, which, by a very rare possi- 
 bility, the edge of the ruler might assume, and yet not be shown by the above 
 test. It is when a part is convex, and a part similarly situated at the other 
 end concave, in exactly the same degree and proportion. In this case, on 
 reversal, a line drawn along the edge of the rule would be coincident with the 
 other, though not a true right line." To determine whether this defect exists, 
 move the alidade endwise and draw a third line. 
 
154 PLANE TABLE. [CHAP. IX 
 
 is necessary not to disturb the table in making the ad- 
 justment. 
 
 176. BOARD, i. The top surface should be a plane. 
 Test it in all directions by a straight-edge ( 174). If 
 it is not perfectly flat, work it down with a smooth 
 plane, a scraper, or sand-paper. 
 
 2. The face of the table should be perpendicular to 
 the vertical axis of the instrument. To make this ad- 
 justment, set up the instrument, place the alidade on 
 the table, and bring one of the bubbles, preferably the 
 one on the telescope, to the middle ; then reverse the 
 table on its axis. If the bubble has not moved, the 
 portion of the table covered by the alidade is perpendic- 
 ular to the vertical axis ; if it has moved, correct half 
 the error by inserting washers between the table and 
 the arms connecting it to the tripod head. 
 
 Turn the alidade 90 on the face of the board and 
 repeat the test. 
 
 177. TELESCOPE, i. The line of sight of the telescope 
 should be perpendicular to the horizontal axis. This 
 adjustment is the same as that described in 123 
 (page 109). 
 
 2. The horizontal axis of the telescope should be 
 parallel to the top of the table. This is the same ad- 
 justment as that discussed in 125 (page in). 
 
 3. To make the line of collimation coincide with the 
 fiducial edge of the alidade, level the table, set two 
 needles in its face in range with some object, say 10 
 feet away; then place the fiducial edge against the 
 needles, and direct the telescope toward .the object. If 
 the cross hairs bisect it, the adjustment is correct ; but 
 if they do not, it can be corrected by means of the 
 screws attaching the standard to the rule. 
 
 The line of sight of the telescope is usually in the 
 plane of the fiducial edge, although it is not necessary 
 that it be either in the plane of the edge or parallel to 
 
ART. 3] USING THE PLANE TABLE. 155 
 
 it; but it is necessary that the two should have a fixed 
 horizontal angle with each other. 
 
 178. ZERO OF VERNIER. The vernier of the vertical 
 arc should read zero when the line of sight is horizontal. 
 This adjustment is the same as that of 127 (page 114). 
 This adjustment is important when elevations are to be 
 determined by vertical angles. 
 
 179. LEVEL ON TELESCOPE. The bubble should be in 
 the middle when the line of sight is horizontal. This is 
 the same adjustment as that of 126 (page 112). It 
 is important only when elevations are to be determined 
 by using the telescope as a level and measuring the 
 difference of heights by a level rod. Since the level on 
 the telescope is ordinarily more sensitive than the plate 
 levels, difference of elevations can be determined more 
 accurately by horizontal lines of sight than by vertical 
 angles. 
 
 ART. 3. USING THE PLANE TABLE. 
 
 180. For many kinds of work the plane table is a very 
 useful instrument. Even the simplest form ( 172) may 
 be used to advantage in obtaining the plat and area of 
 the irregular tracts that occur in ordinary land and city 
 surveying ; and it is valuable in making plats of parks, 
 cemeteries, mining property, etc. It has the great ad- 
 vantage of dispensing with all notes and records of the 
 measurements (since they are platted as they are sur- 
 veyed), and thus saves time and lessens the possibility 
 of making mistakes. 
 
 181. METHODS. Points may be located with respect 
 to each other by any of four methods, viz., (i) radia- 
 tion, (2) traversing, frequently called progression, (3) 
 a combination of the first and second, here called radio- 
 progression, and (4) intersection. The explanation of 
 these methods will be worded for the determination of 
 
PLANE TABLE. 
 
 [CHAP, ix 
 
 a plat of a field, because that process includes all the 
 operations involved in plane-table work with one excep- 
 tion viz., placing the table in position at an undeter- 
 mined point by observations upon three known points 
 (see 191). 
 
 With a little study the operator can discover many 
 modifications and combinations of these methods. The 
 facility with which plane-table work can be modified to 
 suit circumstances is one of its excellencies. 
 
 182. Radiation. This is the most common method of 
 using the plane table, particularly in topographical 
 surveying. It is the simplest, though not the most 
 accurate, method of finding the plat of a field. 
 
 Let A, B) C, D, E, F, Fig. 40, represent the corners 
 
 FIG. 
 
 of a field the plat of which is to be found, and the rect- 
 angle near its center the plane table.* 
 
 Set the instrument at some point 6>, near the center of 
 the field, from which all the corners are visible. Level 
 
 * The size of the table in this and the following figures is greatly exagger- 
 ated. 
 
OF THK 
 
 ART. 3] USING THE PLANE fcXHW^VERSHTY 
 
 the table, and stick a pin or needle havrrrg^^eafing-wax 
 head in th.e prolongation of the plumb-line.* To deter- 
 mine whether the pin is set exactly right, revolve the 
 board and notice whether the pin is stationary. Place 
 the alidade against the pin, direct it to any corner of 
 the field, as A, and draw a line. Measure OA, and set 
 off this distance, to any convenient scale, from the 
 needle along the line just drawn to a. In the same way 
 plat b, c, d t etc., to represent the corresponding corners 
 of the field. Join a and , b and c, etc., and a complete 
 plat of the field is obtained. 
 
 Notice that this method is deficient in checks upon 
 the accuracy of the work. Any movement of the table 
 during the progress of the work may be detected by 
 sighting upon the first station. 
 
 183. The position of trees, houses, etc., may be deter- 
 mined in the same way. By using an alidade having 
 a telescope with stadia hairs, the distance to the object 
 may be determined from the stadia reading and the 
 elevation from the vertical circle (see Chapter X.). Thus 
 the map may be drawn in the field, which is a decided 
 advantage in some respects, but seriously objectionable 
 in others. 
 
 184. Traversing. This method is frequently called 
 progression / but as it is strictly analogous to traversing 
 with the transit ( 137), the term traversing seems pref- 
 erable. Let A BCD be the field to be surveyed. Select 
 some point, a, Fig. 41, on the table to represent the first 
 corner of the field, A. Estimate the dimensions of the 
 field and so locate a that the plat will fall upon the 
 board. Draw a line through a to represent AB\ then 
 measure AB, and lay off its length along this line to b. 
 
 * The center of the table should be permanently marked by setting flush 
 with the face of the board a piece of brass, say one-fou'rth inch in diameter, 
 having in it a hole just large enough to admit a common pin. 
 
'58 
 
 PLANE TABLE. 
 
 [CHAP, ix 
 
 Set the instrument at JS y so that the point b on the 
 board shall be exactly over the corresponding point on 
 the ground and the line ab on the plat shall have the 
 
 FIG. 41. 
 
 same direction as AB on the ground. To satisfy this 
 condition with any considerable degree of accuracy* 
 requires great patience and care. The difficulty is that 
 having placed the point on the board over the point on 
 the ground, it is necessary to destroy this condition in 
 turning the board to make the direction of ab coincide 
 with that of AB.\ 
 
 Having placed b over the corresponding point on the 
 ground and made the direction of ab coincide with that 
 of AB, place the alidade against the needle at b and 
 sight to C. Measure the distance BC> and lay it off 
 
 * The degree of accuracy required in setting the point on the board over the 
 corresponding point the on ground depends upon the scale of the map and 
 the distance to the object sighted at. 
 
 t A plane table with a shifting center would be very convenient for this pur- 
 pose ; but such tables are not made, owing to mechanical difficulties in their 
 construction. A German plane table is provided with a double slide-rest 
 motion for this purpose. 
 
ART. 3] USING THE PLANE TABLE. 159 
 
 from b to c. Move the instrument to C, and proceeds as 
 before. At the last station, D, determine the position 
 of the first station, A, as though it were not already 
 platted. The agreement of the two determinations of 
 a is a check upon the accuracy of the work. The work 
 may be checked as it progresses, by seeing whether any 
 line, as ca, on the plat agrees with the corresponding 
 line, CA> on the ground. Tlxe^ahility-to check the work 
 at every step is a valuable feature of this method. 
 
 185. Instead of trying to place the point on the paper 
 exactly over the point on the ground, it is better to set 
 the table level and approximately over the point, and 
 then sight a line through the point on the paper to a 
 temporary point having a corresponding position with 
 reference to the point to be determined. For example, 
 if in setting the table over K to determine the direc- 
 tion of Z, the point k is 6 inches perpendicularly to the 
 right from the line KL, set near L a point 6 inches per- 
 pendicularly to the right and sight at the point so 
 marked. The line : drawn on the paper is then parallel 
 to the line on the ground. The true distance K L is to 
 be measured and laid off on the paper. 
 
 186. This method is especially suited to the survey 
 of a road, stream, or the like. Often the offsets required 
 may be sketched in by the eye with sufficient accuracy. 
 
 When the paper is filled, put on a new sheet, and begin 
 by fixing on it two points which were on the former 
 sheet, and from them proceed as before. The sheets 
 can then afterwards be united, so that all the points 
 shall be in their true relative positions. 
 
 187. Radio-progression. This method combines the 
 simplicity of the method by radiation with the checks 
 of the method by traversing (progression). The chief 
 advantage is the convenience with which the table is 
 set up at each station, since the center of the table is. 
 always set over the station occupied. 
 
i6o 
 
 PLANE TABLE. 
 
 [CHAP, ix 
 
 Place a needle in the center of the board and set the 
 table over any corner of the field, as A, Fig. 42. Take 
 a sight to one of the adjacent corners, as E. Next sight 
 upon the other adjacent corner, as B. To avoid con- 
 fusion, mark the lines so as to indicate the side of the 
 field to which they correspond. Move the table to the 
 station last sighted at, set it up, and turn it until the 
 alidade, when placed upon the last line drawn, bears 
 upon the station previously occupied. Sight at the 
 next station in order around the field, and proceed in a 
 similar manner at all the corners. Fig. 42 represents 
 
 FIG. 42. 
 
 the table at each corner in succession, commencing at A^ 
 The sides of the field and the corresponding lines on the 
 table are numbered similarly for convenience of refer- 
 ence. If the sighting is correctly done, the fore-sight 
 from the last station will coincide with the back-sight 
 from the first station. The work may be checked as it 
 progresses by sighting at some other than the two adja- 
 cent points used above, ad, Fig. 42, is such a check line. 
 The lengths of the several sides are to be measured 
 as usual. 
 
ART. 3] 
 
 USING THE PLANE TABLE. 
 
 161 
 
 It is obvious that the lines radiating from the center 
 of the table are respectively parallel to the sides of the 
 proposed plat. To draw the plat, it is only necessary to 
 assume an initial point and draw lines through it par- 
 allel to any two adjacent sides of the field, and then lay 
 off on these lines the lengths of the respective sides, 
 to any convenient scale (see Fig. 43). Two other cor- 
 
 FlG. 43- 
 
 ners of the plat are thus determined. Through each of 
 tnese draw a line parallel to the next side of the field ; 
 and do likewise for all the remaining corners. 
 
 The drawing can be checked as it progresses, by plat- 
 ting the check lines and noticing whether they pass 
 through the corresponding point in the map. The 
 closing of the plat checks the accuracy of all the work. 
 
 188. The principle of having the center of the table 
 always'over the point occupied may be applied in locat- 
 ing the position of trees, corners of buildings, points on 
 streams, etc., by drawing through the point on the table 
 corresponding to the point over which the instrument 
 is set a line parallel to the edge of the alidade. This 
 requires the use of two large and similar triangles in 
 
162 PLANE TABLE. [CHAP. IX 
 
 drawing the line parallel to the edge of the alidade. 
 The point occupied by the instrument being located on 
 the paper, the alidade is placed against the needle stand- 
 ing in the center of the board, and the sight turned 
 upon the point to be platted. Then without disturbing 
 the alidade place one triangle against the edge of the 
 alidade and the second triangle against the first, and 
 slide one against the other until the edge of the second 
 passes through the point on the paper corresponding to 
 the point occupied. Next draw a line along the edge 
 of the triangle, and on this line lay off the distance from 
 the instrument to the point sighted at. Proceed in like 
 manner for each point to be located. 
 
 Before moving the table to the next station, it is well 
 to check the position of the table by re-determining the 
 direction of the first line. For greater accuracy and 
 speed in making this test, it is better to draw a line 
 through the center of the board to mark the first position 
 of the alidade. Having checked the position of the 
 table, sight to the next station to be occupied, and 
 mark the position of the alidade by a line through the 
 center of the board. This line must also be transferred. 
 Then move the instrument to the next station, set the 
 tripod over the point, level the board, place the alidade 
 on the last line drawn, through the center and turn the 
 table until the lineof sight covers the point formerly 
 occupied. Observations may now be made as at first. 
 
 If a considerable number of points are to be deter- 
 mined at each setting of the instrument, it is easier to 
 spend some time in setting the point on the paper over 
 the corresponding point on the ground, than to transfer 
 all the lines; but if there are only a few points to be 
 determined at each setting, it is easier to set the center 
 of the board over the point and transfer the lines 
 One advantage of keeping the center of the board over 
 the point on the ground is that the alidade, which has 
 
ART. 3] USING THE PLANE TABLE. 163 
 
 considerable weight particularly one carrying a tele- 
 scope always stands centrally on the board and never 
 near one edge. Notice that the two triangles take the 
 place of the plumbing bar. As ordinarily made, plane 
 tables have no means of hanging a plumb-line from the 
 center of the instrument; but the deficiency is easily 
 and cheaply supplied. If the two edges of the alidade 
 are parallel, the triangle may be placed against either 
 edge, as is most convenient. 
 
 189. Intersection. Measure any line, preferably cen- 
 tral in the tract to be surveyed, on the ground, as M JV, 
 Fig. 44. Select some point on the paper to represent 
 
 -4- 
 
 *' 
 
 c/ 
 
 FIG. 44. 
 
 My draw a line through it to represent the direction of 
 the line MN, and lay off the distance MN. Then set 
 the point on the paper representing M over the corre- 
 sponding point on the ground, and orient the table so 
 that the alidade when placed upon the line MN will 
 sight to N, and clamp the instrument. Then sight to 
 all the points whose location is desired, as A, B, C, >, 
 etc., and draw lines to show the direction of these 
 points. Each line and each point should be so marked 
 that they can be ejj^hil^_identified. Notice that if 
 
764 PLANE TABLE. [CHAP. IX 
 
 there are any considerable number of lines it is not 
 enough simply to put a letter or figure on the line, as in 
 Fig. 44, since another line may be subsequently drawn 
 through the number. Under these conditions the 
 designation of the line should be placed in a small 
 rectangle, one side of which coincides with the line to 
 be identified. If necessary, the points sighted at may 
 be marked by driving a numbered stake beside them. 
 
 Next, set the instrument over N, placing the point on 
 the paper over the corresponding point on the ground, 
 and orient the table so that the direction of the line 
 on the paper corresponding to MN coincides with that 
 line on the ground. Then sight at all the points and 
 draw lines to correspond with their new directions. 
 The intersection of the two lines of sight to each point 
 will determine its position. 
 
 If there are other points not visible from M, as will 
 be the case in a line survey, sight at them from JV and 
 measure a new base line, and proceed as befcre. Or 
 the end of the new base line may be determined, as any 
 other point, by tne intersection of two sight lines from 
 the ends of the original base line. Obviously, measur- 
 ing the new base directly is the more accurate. 
 
 190. Before the introduction of the stadia, the 
 method of intersection was the most usual and most 
 rapid method of using the plane table. But this 
 method affords no check upon the accuracy of the 
 work, is often deficient in precision on account of 
 oblique intersections, and requires so many lines upon 
 the board as to cause confusion and error. It is well 
 adapted to the mapping of harbors, shore lines, and 
 generally to the plotting of inaccessible points. Of 
 course, in this as in all triangulations, weH^conditioned 
 triangles give more satisfactory results; or in other 
 words, avoid, if possible, angles less than 30 or greater 
 than 150. 
 
ART. 3] USING THE PLANE TABLE. 165 
 
 191. THE THREE-POINT PROBLEM. In this problem 
 
 three stations A> B, C* are plotted, as a, b, c, on the 
 table, and the instrument being set up over a fourth 
 point Z>, it is required to find the position of this point 
 on the map. This problem is indeterminate when the 
 point D lies in the circumference of a circle passing 
 through A, B, and C, in which case the two-point prob- 
 lem ( 194) may be applied. The three-point problem, 
 which also occurs in the use of the sextant in locating 
 soundings, has been much discussed, and many solu- 
 tions have been proposed. f Only two will be given 
 here. 
 
 192. Mechanical Solution. Fasten a she.et of tracing- 
 paper on the board, and fix a point d to represent the 
 station at which the instrument is set. With the ali- 
 dade centring on </, direct the telescope successively to 
 A, B, and C, and draw lines of indefinite length along 
 the ruler's edge towards these stations. Then, if the 
 tracing-paper be shifted until the three lines thus drawn 
 pass through the points , b y and c, the point d will indi- 
 cate the position of D. The position of this point may 
 now be transferred to the map by pricking through. 
 The tracing-paper is then removed, and, the table 
 oriented. 
 
 193. Graphical Solution. J Let a, b, and <:, Fig. 45, be 
 the points on the sheet representing the signals A, B, 
 and C, on the ground. Set the table at the point to be 
 determined, D, and level it. Set the alidade upon the 
 line ca, and by revolving the table, sight upon the 
 signal A, and clamp. Then with the alidade centring 
 
 * The capital letters refer to the points on the ground, and the lower-case 
 letters to the corresponding points on the board. 
 
 t The U. S. Coast and Geodetic Survey Report for 1880, pp. 180-84, gives 
 a number of solutions elaborately illustrated. 
 
 I Known as Bessel's method by inscribed quadrilateral see U. S. Coast 
 and Geodetic Survey Report, 1880, p. 181. 
 
j66 
 
 PLANE TABLE. 
 
 [CHAP. IX 
 
 a 
 
 on <r, sight upon the middle signal B, and draw the line 
 ce along the edge of the ruler. Set the alidade upon 
 the line ac, direct the telescope to the signal C by 
 
 revolving the table, and 
 clamp. Then with the ali- 
 dade centring on , direct 
 the telescope to the middle 
 signal B, and draw the line 
 ae along the edge of the 
 ruler. The point e (the 
 intersection of these two 
 lines) will be in the line 
 passing through the mid- 
 dle point and the point 
 sought. Set the alidade 
 upon the line be, direct b 
 to the signal B by revolv- 
 ing the table, and the table 
 will then be in position. 
 Clamp it, center the ali- 
 dade upon a, direct the telescope to the signal .4,, and 
 draw along the ruler the line ad. This will intersect 
 the line be at the point sought. To verify its position, 
 center the alidade on f, and sight to C. 
 
 The opposite angles of the quadrilateral adce being 
 supplementary, the angles ace and ade are subtended by 
 the same chord ae, and cae and cde are subtended by 
 the same chord ce ; and, consequently, the intersection 
 of ae and ce at e must fall on the line db. Or, the 
 segments of two intersecting chords in a circle being 
 reciprocally proportional, the triangles adf and cef are 
 similar, as also the triangles cdf and aef\ and there 
 fore </,/, and e must be in a right line passing through . 
 194. THE TWO-POINT PKOBLEM. There are several 
 solutions* of this problem, only one of which will be 
 
 FIG. 45. THREE-POINT PROBLEM 
 
 * U. S. Coast Survey Report, 1880, pp. 184-85. 
 
ART. 3] USING THE PLANE TABLE. 167 
 
 r\ 
 
 given. Two points, A and B, not conveniently accessi- 
 ble, being given by their projections a and ^, it is re- 
 quired to put the plane table in position at a third! 
 point, C. Select a fourth point, D, Fig. 46, such that 
 the intersections of lines from C and D upon A and B 
 make sufficiently large angles for good determinations. 
 Put the table approximately In position at Z>, by esti- 
 mation or by compass, and draw the lines Aa and Bb, 
 
 intersecting in d 9 . Through d' a\\^ [_ 
 
 draw a line directed to C, and 
 
 on this line lay off, from </', the 
 
 estimated distance CD, and 
 
 mark the point thus found c'. 
 
 Set the instrument on C with c' 
 
 over the point, and orient on > 
 
 by the line c'd'. Draw lines 
 
 from c r to A and to B. These 
 
 will intersect the lines d' A and 
 
 d'B at points a 9 and ', which FlG - < 6 - -TWO-POINT PROBLEM. 
 
 form with c' and d' a quadrilateral similar to the true 
 
 one, but erroneous in size (since the distance c'd 9 was 
 
 assumed), and in position (since the table was not 
 
 properly oriented at either station). 
 
 The angles which the lines ab and a f b' make with 
 each other is the error in position. By constructing 
 now through c' a line ?a* t making the same angle with 
 c f d f as that which ab makes with a'b' and directing the 
 line c'd' to D, the table will be brought into position, 
 and the true point c can be found by the intersection of 
 a A and bB. 
 
 Instead of transferring the angle of error by construc- 
 tion, we may conveniently proceed as follows, observ- 
 ing that the angle which the line a'b' makes with ab is 
 the error in the position of the table. As the table now 
 stands, a'b 9 is parallel with AB ; but we want to turn it 
 
l68 PLANE TABLE. [CHAP. IX 
 
 so that ab shall be parallel to that line. Therefore if 
 we place the alidade on a'b' and set up a mark in that 
 direction, and then place the alidade on ab and turn 
 the table until it again points to the mark, ab will be 
 parallel to AJ3, and the table is in position. 
 
 195. SOURCES OF ERROR.* The sources of error in 
 plane table work, in addition to those of chaining 
 ( 19) or stadia measurements (229-32) are as follows: 
 (i) error of position of instrument; (2) error of sighting; 
 (3) movement of the board between sights; (4) errors 
 of adjustment of the instrument; (5) the inclination of 
 the board; (6) error in marking the line upon the 
 paper; (7) error in scaling off; and (8) the hygrometric 
 effect of the atmosphere on the paper. The effect of 
 any of these errors will depend upon the scale of the map. 
 The first four are essentially the same as the corre- 
 sponding errors in transit work, for discussion of which 
 see 140. If the board is not level, an error is pro- 
 duced in determining horizontal angles between points 
 not in the same horizontal plane, and also in determin- 
 ing vertical angles between points not in the same ver- 
 tical plane. The sixth and seventh depend upon the 
 skill and care of the draughtsman. The changes in the 
 dimensions of the drawing, due to the hygrometric 
 state of the atmosphere, will vary with the weather and 
 the kind of paper. The U. S. Coast Survey,f to de- 
 termine this variation, cut several strips two meters 
 long from three samples of drawing paper, and ob- 
 served the variations daily for six months. For strips 
 cut longitudinally the average variation was 9.0 milli- 
 meters, 13.0 millimeters, and 15.7 millimeters, and the 
 maximum was n.i, 15.5, and 20.9, respectively; and for 
 
 * For a discussion of Compensating vs. Cumulative Errors, see 18. 
 t U. S. Coast Survey Report for 1862, p. 255. 
 
ART. 3] USING THE PLANE TABLE. 169 
 
 strips cut transversely the average variation was 8.4, 8.1^ 
 and 12. i millimeters, and the maximum was 10.3, 9.6, and 
 15.0, respectively. 
 
 196. LIMITS OF PRECISION. Since there is so much 
 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 in general the degree of pre- 
 cision to be obtained with the plane table. Further- 
 more the plane table is designed more for rapidity than 
 accuracy, although the best modern instruments are 
 capable of a considerable degree of precision. 
 
 197. With Plain Alidade. In ordinary class work of 
 the first term of surveying the author's students meas- 
 ured three angles of a triangle and four angles around 
 a point, using a wooden alidade with slit and string for 
 sights, with an average error, for the best thirty-two 
 out of thirty-six results, of i minute and 44 seconds per 
 angle (a probable error of 88 seconds). Under fair 
 conditions the maximum error per angle should not 
 exceed 3 minutes. 
 
 Under the above conditions the error of finding 
 areas* was as follows: for radiation ( 182), an average 
 error of i in 586 and a maximum error of i in 274; for 
 traversing ( 184), an average error of i in 826 and a 
 maximum error of i in 450; and for radio-progression 
 ( 187), an average of i in 1,111 and a maximum error 
 of i in 640. For the sake of comparison it may be 
 interesting to know that for the same students under 
 the same conditions the average error with a magnetic 
 compass ( 51) was i in 1,220 and the maximum i in 
 580; and with a chain alone, the average error was i in 
 1,520 and the maximum i in 500. 
 
 198. With Telescope Alidade. Rarely, if ever, would 
 the most elaborate form of plane table be employed in 
 
 * See second foot-note, p. 28. 
 
170 PLANE TABLE. [CHAP. IX 
 
 finding areas; and, for reasons before stated, it is im- 
 possible to give a summary of the precision attainable 
 with this form of plane table in topographical survey- 
 ing. Therefore this subject will be concluded with a 
 few references to descriptions of surveys made with the 
 plane table, which contain data on precision, cost, and 
 speed. 
 
 1. Professional Papers of the Engineer Department, 
 U. S. A., No. 18, Report of the Geological Explora- 
 tion of the Fortieth Parallel, by Clarence King, Wash- 
 ington, D. C., 1878, Vol. I, p. 762. 
 
 2. U. S. Geographical Surveys West of icoth Meridian, 
 Wheeler, 1883, p. 47. 
 
 3. Report on the Third International Geographical 
 Congress and Exhibition at Venice, Italy, 1881, Geo. 
 M. Wheeler, Washington, 1885, pp. 79-81. 
 
 4. Science (New York City), July 29, 1887, p. 49. 
 
 5. Plane Table Methods used by the U. S. Geological 
 Survey in Western Massachusetts, Louis F. Cutter, in 
 Journal of the Association of Engineering Societies 
 (Chicago), Vol. X, pp. 356-69. 
 
 199. PRACTICAL HINTS. The board should be placed 
 so low as to be readily reached, even at the most remote 
 corners, and yet high enough to enable the observer 
 to sight with comfort. This will bring it a little below 
 the elbow. All beginners are apt to set the table 
 too high. Care must be taken that no part of the body 
 touch or rest against the edge of the board. In using 
 the alidade, steady the standard with the left hand, 
 while the right swings the rear end of the ruler in the 
 proper direction. 
 
 Manilla paper is easier on the eyes in the sunshine 
 than white, and also shows dirt less. Thumb-tacks 
 and rollers for holding down the sheet are both ob- 
 jectionable, especially in high winds. The edges may 
 be pasted underneath, or spring clamps may be used 
 
ART. 3] USING THE PLANE TABLE. 17 1 
 
 to advantage. The sides of the sheet where they are 
 turned under the table and come more or less in contact 
 with the coat of the observer, should be protected by 
 strips of paper about 4 inches wide, and 6 inches 
 longer than the sides of the table, so as to fold under it 
 and clamp on with the sheet itself. Tracing vellum is 
 good for this purpose, as the points near the edge of the 
 sheet can be seen through it. 
 
 A short linen coat with large pockets, in the style of 
 a hunting-coat, probably affords the best means for 
 carrying the requisite accessories for plane-table work; 
 viz., scale, triangles, pencils, rubber, note-book, etc. By 
 this means the weight of these articles is distributed in 
 separate pockets, and they are always at hand when 
 needed. 
 
 On beginning the work, set off at some point near 
 the middle of the sheet, the magnetic meridian for the 
 purpose of putting the table in approximate position at 
 any subsequent station with the declinator. 
 
 Use as hard a pencil, and make as few lines, as possi- 
 ble. In locating points on contours, plot the distance 
 at once along the edge of ruler by a detached scale, 
 making only a dot at the point which should receive 
 the number of the contour. Objects on a straight line 
 may be quickly located by plotting the ends of the line 
 and determining the intermediate points by the inter- 
 sections of this line and lines of sight to the several 
 points. 
 
 Always before leaving a station, and also at intervals 
 when not otherwise employed, take sights to determine 
 whether the board has been displaced. 
 
 It is well to have ready a light india-rubber cloth 
 cover to slip over the board in case of a sudden shower, 
 as well as to protect the paper from dust on the roads, 
 mud in swampy ground, etc. A metal chart case 
 should always accompany the table to protect the 
 
172 PLANE TABLE. [CHAP. IX 
 
 sheet from sudden rain and other injury liable to occur 
 in the transportation of the sheet to and from the field, 
 and for its safe keeping when not in use. Its diameter 
 should be not less than 3 inches, for no sheet can be 
 rolled to a less diameter without serious rupture of the 
 fiber of the paper. 
 
CHAPTER X. 
 TELEMETERS. 
 
 200. TELEMETER is a term variously employed to 
 designate some form of an instrument for determining 
 distances by means of the visual angle subtended by a 
 short base. There are a great number of instruments 
 of this class, but the only ones of any considerable value 
 in engineering practice are the stadia and the gradi- 
 enter. These instruments will be discussed at consid- 
 erable length in Arts, i and 2, respectively, and in Art. 3 
 various forms of telemeters will be briefly described. 
 
 ART. 1. THE STADIA. 
 
 201. The stadia is an instrument for determining the 
 distance of a point from the observer by the visual 
 angle subtended -by an object of known size placed at 
 the point. Ordinarily not only the distance but also 
 the horizontal and vertical angles are observed, these 
 three being sufficient to determine the direction, dis- 
 tance, and elevation of the point upon which the rod is 
 placed. In 1820 Parro, an Italian engineer, first sug- 
 gested the determination of distances in surveying by a 
 visual angle and a rod.* He used the word stadia to 
 designate the rod, but the term is now generally ap- 
 plied to the instrument as a whole. On the U. S. Coast 
 and Geodetic Survey the term telemeter is used in- 
 stead of stadia. In Great Britain the instrument by 
 which the observation is made is called a tacheometer, 
 and the rod is called a stadia. 
 
 * William Green, a London optician, in 1778 published a pamphlet giving 
 a very full account of the stadia, but the credit of its practical introduction 
 seems to be due to Parro. 
 
 173 
 
TELEMETERS. [CHAP, x 
 
 The principles involved have long been well known, 
 and have been applied in gunnery and military recon- 
 noissance ; but it is only lately that they have been 
 used in engineering. The first notable use of the stadia 
 was in 1836., in making a topographical survey of 
 Switzerland. It seems not to have been introduced 
 into America until nearly thirty years afterward, and 
 has not been employed here to any considerable extent 
 except in topographical surveys carried on by the U. S. 
 Government. It has not come into as general use as 
 its merits warrant, although it is more generally used 
 in Europe than in the United States. The stadia is 
 peculiarly adapted to topographical surveying, for it 
 possesses the double advantage of giving both the hori- 
 zontal and vertical co-ordinates, and this, too, by the 
 most rapid method. 
 
 202. PEINCIPIES. In all its forms the stadia is an ap- 
 plication of the principle of similarity of triangles. The 
 simplest form of the instrument is used in the familiar 
 method of determining the distance of a man from an 
 observer, 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 impossi- 
 bility of focusing the eye for two distances at the same 
 time, and to the indistinctness of the farther object. 
 
 To employ the principle of the stadia with a tele- 
 scope, it is necessary to introduce two parallel cross 
 hairs and observe the amount of the rod intercepted 
 between them. The hairs may be horizontal and the 
 rod vertical, or vice versa; but the former is usually pre- 
 ferred, since the rod is then more steady, and there is 
 less liability of its being obscured by brush, etc. The 
 hairs may be fixed, the intercept on the rod being 
 variable ; or the hairs may be movable and be set to 
 cover always the same two points on the rod, the vari- 
 
ART. l] THE STADIA. 175 
 
 able distance between the hairs being measured by a 
 graduated screw. The fixed hairs are cheaper, more 
 simple, more accurate, and in every way better than 
 the movable ones ; and are generally used. 
 
 In the stadia with a telescope the two similar trian- 
 gles have a common apex at the optical centre (see first 
 foot-note, page 106) of the objective, the base of one 
 being the distance between the cross hairs, and that of 
 the other the intercept on the rod. Since in focusing 
 the telescope, the distance from the cross hairs to the 
 objective varies with the distance of the object sighted 
 at, the relation between the distance to the rod and the 
 intercept is not as easily found as with the simple de- 
 vice previously mentioned. The formula for the stadia 
 with telescope will be deduced presently. 
 
 203. PLACING THE HAIRS. In placing the stadia hairs 
 in the telescope three conditions must receive attention : 
 they should be parallel, equally distant from the cen- 
 tral one, and at a suitable distance from each other. 
 
 These conditions are the rigorous ones, but farther on 
 it will be shown that no appreciable error will be pro- 
 duced if they are only approximately satisfied. It is 
 necessary that the hairs should be parallel only in case 
 the observation is not made exactly on the vertical hair. 
 If the stadia hairs are not equally distant from the 
 central one, it produces only a small error in the vertical 
 angle. Finally, the distance between the hairs is im- 
 material, provided the rod is graduated to correspond. 
 A formula will be deduced to meet the case in which the 
 rod is already graduated and does not agree with the 
 distance between the hairs ( 215). 
 
 Any measuring telescope can be used as a stadia by 
 adding a second horizontal hair ; but it is much better 
 to add two extra ones, one on each side of the ordinary 
 one. With the stadia hairs thus placed, the field of 
 view is symmetrical about the center, and the added 
 
i 7 6 
 
 TELEMETERS. 
 
 [CHAP, x 
 
 hairs interfere less with the ordinary uses of the tele- 
 scope. 
 
 204. Instrument makers not infrequently fasten three 
 horizontal cross hairs on the reticule ( 80), the two 
 outer ones being at such a distance apart that each foot 
 of the intercept on the rod corresponds, at least approx- 
 imately, to a hundred feet of the distance from the 
 instrument to the rod. The space between these hairs 
 is computed and fine grooves are engraved at the proper 
 distance, into which the hairs are laid and fastened. If 
 these hairs get broken the engineer can replace them 
 ( 81) with a little care. 
 
 Fig. 47 shows a common method of making the dis- 
 tances between the stadia hairs adjustable. The upper 
 portion of Fig. 47 shows the reticule 
 in place in the telescope tube, and 
 the other shows details. The screws 
 d and f 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. The screws d and / 
 are shown in Fig. 22, page 92, im- 
 mediately in front of the ordinary 
 cross-hair screws, e is a bent spring 
 to take up lost motion in the screws 
 d and /. The ring a and all parts 
 connected therewith are adjusted 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 be varied at will. Lines 
 are made upon the slides b and c to assist in placing the 
 hairs parallel to each other. The following objections 
 
ART. l] 
 
 THE STADIA. 
 
 77 
 
 are sometimes offered to this construction : i. The pro- 
 jecting heads of the screws d and / 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. 2. Since 
 a spring must be inserted to take up lost motion, the 
 screws have a tendency to work loose. 3. It is expensive. 
 205. Fig. 48 shows an inexpensive method of in- 
 serting adjustable .stadia hairs, in an instrument 
 provided with them. The diagram 
 represents the fro.nt view of the ordi- 
 nary reticule, a, b, c, and d are small 
 wire plugs which are free to turn, be- 
 ing held only by friction. The dark 
 portion is in the plane of the- face of 
 the ring, and the light portion projects, 
 say an eighth of an inch, above. The 
 
 not 
 
 FIG. 48. 
 
 hairs are to be stretched in the line of the centers of a 
 and c, and d and ^, and fastened at the outer edge of the 
 ring. Then by turning the plugs the hairs will be moved 
 toward or from the central hair, according to which 
 side of the plug is toward it. The wire plugs may easily 
 be made to fit so tight as 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, after 
 removing the eye-piece, without taking the reticule out 
 of the telescope tube. With an erecting eye-piece, the 
 hairs can be adjusted without removing the ring from 
 the telescope tube, by the use of a little wrench (made 
 especially for this purpose by cutting a kerf with a hack- 
 saw in the end of a small strip of brass), and operating 
 it through a hole in the telescope tube (also made for 
 the purpose), which can be closed by a metal or rubber 
 band ; or the ring may be removed from the tube to 
 adjust the hairs. The telescope does not need to be 
 
TELEMETERS. [CHAP. X 
 
 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 dis- 
 tance between the central and each outside hair equal 
 to the diameter of the plug, which may be made of any 
 size to suit the circumstances. 
 
 206. The maximum distance between the hairs is 
 limited by the size of the field of view ( 87). The field 
 of view of the telescopes on ordinary engineering instru- 
 ments differs with different makers, but is about as fol- 
 lows : for a magnification of twenty, i 30'; of twenty- 
 five, i 15'; of thirty, i; and of thirty-five, 50'. With 
 the smallest power mentioned above the distance be- 
 tween the stadia hairs can not be greater than one 
 fortieth of the focal length of the objective, and for the 
 largest power one seventieth. Since it is not possible 
 to have the hairs at the extreme edge of the field of view, 
 and since the outer portion of the field is not as good 
 optically as the central portion, it is safe to say that 
 with the higher powers the distance between the stadia 
 hairs should not be more than a hundredth of the focal 
 length of the objective. With the lower powers the 
 distance between the hairs might be greater, but for 
 long sights the rod would be inconveniently long. 
 
 On the other hand, if the hairs are very close together 
 the intercept on the rod will be too small to be deter- 
 mined accurately. All things considered, it is probably 
 best to make the distance between the hairs a hundredth 
 of the focal length of the objective. A method of accu- 
 rately making this adjustment will be described pres- 
 ently ( 214-15). 
 
 207. THE ROD. In stadia work there are two kinds of 
 rods self-reading and target. A self-reading rod is one 
 having a graduation such that the value of the intercept 
 can be read through the telescope. A target rod is one 
 
ART. l] THE STADIA. 179 
 
 having a sliding target moved by the rod-man, in re- 
 sponse to signals from the instrument-man, until it is 
 in the plane of sight, when its position is read by the 
 rod-man. The latter rod requires two targets, one of 
 which 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 convenient. Self- 
 reading stadia rods are generally preferred.* 
 
 Figs. 49-52 (page 180) show a few of the many gradua- 
 tions proposed for self-reading stadia rods. Fig. 49 is 
 a graduation formerly much used on the U. S. Coast 
 and Geodetic Survey, and is on the whole the best of 
 the four shown. The distance from a to b corresponds 
 to 10 feet on the ground. By dividing the oblique side 
 of the triangle into fifths by estimation, the rod may be 
 read to single feet. Notice that the division of the 
 oblique side of the triangle is the principle of the well- 
 known diagonal scale. Fig. 50 was used on the late 
 U. S. Lake Survey, and is the standard on the surveys 
 conducted by the Mississippi River Commission. It is 
 better for short distances than Fig. 49, but not so good 
 for long ones. The other figures are added to show 
 the facility with which designs may be made. Fig. 52 is 
 very good for short distances, and very poor for long 
 ones. As a rule the graduation on self-reading stadia 
 rods is not numbered, since it is generally considered 
 easier to count the divisions included in the visual angle 
 than to read both hairs and subtract. 
 
 Any self-reading leveling rod ( 270) may be used as 
 a stadia rod ; but as a rule self-reading leveling rods 
 have a great number of very small divisions, which be- 
 come indistinct or wholly invisible at long distances ; 
 
 * For a discussion of the relative merits of self-reading and target leveling 
 rods, see 270. 
 
8o 
 
 TELEMETERS. [CHAP. X 
 
 FIG, 49. FIG. 50. FIG. 51 
 
ART. l] THE STADIA. l8l 
 
 and hence most self-reading leveling rods are more suit- 
 able for short than for long sights. Leveling rods are 
 generally read only at short distances, while stadia rods 
 are frequently read at long distances. 
 
 208. Apparently stadia rods are not sold by instru- 
 ment makers ; but the engineer can easily make his 
 own. 
 
 The rod must be light and convenient for transporta- 
 tion. Usually it is a board about i inch thick, 4 or 5 inches 
 wide, and from 10 to 14 feet long. It may be stiffened 
 by screwing a strip edgewise on the back. It may be 
 hinged in the middle, and folded for ease of transporta- 
 tion and for the protection of the graduation ; and when 
 in use, it may be kept upright by a button or a bolt on 
 the back. The rod should be provided with a disk 
 level, or a short plumb, for keeping it vertical, and a 
 handle by which to hold it. The graduation will keep 
 cleaner and last much longer if the face of the board 
 is recessed slightly to receive it. The face should be 
 made perfectly white by a number of thin coats of paint, 
 each being thoroughly dry before the succeeding one 
 is applied. 
 
 In graduating a stadia rod, visibility is of the first 
 importance. If the graduation consists of a number of 
 small divisions, they will become invisible at long dis- 
 tances, and even at short distances will be very confus- 
 ing. The graduation marks should therefore be made 
 large and the smaller divisions be obtained by sub- 
 dividing the larger ones by estimation. 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 read more exactly 
 by estimation than by a direct graduation, owing to the 
 liability of error in counting the graduation marks. 
 
 The pattern may be painted or stenciled directly upon 
 the wood, or it may first be drawn or painted upon 
 
 
l82 TELEMETERS. [CHAP. X 
 
 paper and then fastened on the rod with varnish or any 
 glue not soluble in water. Stencils may be made of 
 writing paper which has been varnished or oiled. In 
 either case the pattern should have a sharp outline, and 
 should be marked with jet-black paint that dries with a 
 dead, and not a shiny, surface. 
 
 209. To determine the vertical co-ordinates of the 
 point on which the rod is set, it is necessary that the 
 line of sight should be directed to a point at a known 
 distance from the foot of the rod. The point at which 
 the central visual ray is directed is most easily indi- 
 cated by a target, which may be either fixed or movable, 
 although the movable target is much the better. It 
 may be made of a piece of rolled brass 2\ to 4 inches 
 wide and about an eighth of an inch thick, bent as 
 shown in Fig. 53. The pieces a and b are made concave 
 
 F 
 
 FIG. 53. 
 
 on their inner faces to fit the edges of the rod. a is 
 soldered or riveted to the body of the target, and b is 
 held in position by two plates (not shown in Fig. 53) 
 attached to the top and bottom of it, which extend 
 over the edges of the body of the target. The target is 
 clamped at any point on the rod by the milled-head 
 screw. A diamond is painted, preferably in red, on the 
 face of the target (see 268). The back is left open so 
 the target may be moved up and down past the handle 
 and plummet on the back of the rod. 
 210. FORMULA FOE HORIZONTAL LINE OF SIGHT AND 
 
 VERTICAL ROD. In Fig. 54 let a and b represent the 
 stadia hairs ; / the distance between them ; s the dis- 
 tance,/^, on the rod intercepted between the hairs ; / 
 
ART. i] THE STADIA* 183 
 
 the principal focal distance of the objective ; e a point 
 at a distance /in front of the optical center of the ob- 
 jective, that is, e is the principal focus of the objective ; 
 c the distance from the plumb-line of the instrument to 
 the optical center of the objective ; y the distance from 
 the outer focus, ^, to the rod ; and D the distance from 
 the instrument to the rod. For convenience in print- 
 ing represent . by k. 
 
 
 FIG. 54. 
 
 From the principles of optics, we know that all rays 
 of light which pass through e are parallel to each other 
 after emerging from the objective. Therefore there is 
 some point ^, which will emit a single ray of light that 
 will pass through e, and, after traversing the objective 
 will strike the cross hair a. If the telescope is focused 
 for the point ^, the objective will bring all rays emitted 
 by q to a focus at a ; and hence it is immaterial whether 
 we consider the real course of the rays, or assume that 
 all the light from q passes along the line qea. 
 
 Similarly we may assume that all the rays from/ pass 
 along the line peb . 
 
 From Fig. 54 we easily get s \ y ::/:/, from which 
 
 (0 
 
 rto 
 each instrument, and also that the intercept s on the 
 
 Notice that k = 4, is a constant coefficient peculiar to 
 
184 TELEMETERS. [CHAP. X 
 
 rod varies as y the distance of the rod from the outer 
 focus of the objective. These relations may be seen 
 directly from Fig. 54. Since the two rays from/ and q 
 are parallel after entering the telescope, it is immaterial 
 where the cross hairs are; and, therefore, the distance 
 of the rod from e is always proportional to the intercept 
 s. In other words, the intercept on the rod is propor- 
 tional to the distance of the rod from the point ^, and 
 any change in the position of the cross hairs in focusing 
 upon the rod produces no change in this relation. 
 From Fig. 54 and equation (i) we get 
 
 D=ks+c+f. (2) 
 
 211. As equation (2) is ihe foundation of all stadia 
 formulas, it will be demonstrated by a slightly different 
 method. In Fig. 55 o represents the optical centre of 
 
 /f\ 
 
 - C -, 
 
 FIG. 
 
 the objective; x the distance from the cross hairs to 
 the optical center ; and z the distance from the optical 
 centre to the rod. The remainder of the nomenclature 
 is as in Fig. 54. Since a lens may be regarded as a 
 mathematical point which allows a very great amount 
 of light to pass through it,* we may consider the rays 
 of light as passing in a right line from q through o to a, 
 and also from / through o to b. Then by similar tri- 
 angles, 
 
 s : z : : i : x, 
 
 * See first foot-note on page 106. 
 
ART. l] THE STADIA. 185 
 
 from which 
 
 From the theory of optics, we have the well-known 
 relation for a convex lens, 
 
 Combining equations (3) and (4) gives 
 
 ...... (5) 
 
 Adding c to both members of (5) gives equation (2), 
 page 184. 
 
 212. There are other and less simple demonstrations 
 of the fundamental stadia formula equation (2); but 
 as they all arrive at the same final form, they must in- 
 volve the same approximations. Both of the preceding 
 methods involve slight approximations: i. Equation (i) 
 assumes that the line qe, Fig. 54, and the horizontal line 
 through a intersect in a vertical plane through the 
 optical center, whereas they do not so intersect. 2. 
 Equation (3) assumes that the lines qo and oa, Fig. 55, 
 are one and the same right line, whereas they are not. 
 
 In addition to these, several relatively unimportant 
 approximations are indirectly involved. Although the 
 formula for the stadia is not mathematically correct, 
 it is much more accurate than any observations that 
 can be made with an ordinary telescope, and it will be 
 shown later that the stadia is an instrument of con- 
 siderable precision. 
 
 213. To find c and/. To find c, focus the instrument 
 on a point 100 feet or more away, and measure with a 
 
1 86 TELEMETERS. [CHAP, x 
 
 pocket-rule the horizontal distance from the vertical 
 axis to the middle of the objective. Strictly, c is not 
 constant in instruments in which the cross hairs are 
 fixed and the objective movable ; but if it is found 
 within an inch it is more than sufficient.* The distance 
 found as above is practically the minimum, but it is also 
 the value corresponding nearly to the mean distance of 
 the rod from the instrument. 
 
 To find/, focus the instrument on a point 100 feet or 
 more away, and measure the distance from the cross 
 hairs to the middle of the thickness of the objective.! 
 
 214. To find k. Set the rod vertical, at any convenient 
 distance in front of the instrument, and having brought 
 the line of sight horizontal, determine the intercept s 
 by using a stadia rod ( 207) or a self-reading level rod 
 ( 270). Then measure the distance D with a band- 
 chain. From equation (2), page 184, we have 
 
 *=- -ir -> < 6 ) 
 
 from which it is an easy matter to compute k. For 
 greater accuracy, make several observations at different 
 distances, and take the mean of the corresponding 
 values of k. The error of this mean will be consider- 
 
 * The variation in c is the same as the variation in x in equation (4), 
 page 185. Assuming/" to be i foot a fair average, we find that if z = jo 
 ft., x i ft. 1.3 in.; if z = 20 ft., x = i ft. 0.6 in.; if z = too ft., x = i ft. 
 o.i in.; and if z = 1000 ft., x i ft. o.oi in. Therefore, since the distance to 
 the rod will ordinarily be more than 100 ft., the variation in c is less than one 
 tenth of an inch, which is wholly inappreciable in stadia surveying. 
 
 t It is sometimes desirable to know /"and the position of the optical center 
 accurately. To find them proceed as follows : Remove the lens from the 
 telescope tube, and set it up midway between two small movable white screens, 
 one of which has a square ruled upon it with black ink. Shift the positions 
 of the screens until the square and its image upon the other screen are of 
 exactly the same size. The optical center will then be exactly midway between 
 the two screens, and /is equal to one fourth of the distance between them. 
 
ART. ij THE STADIA. 187 
 
 ably less than the error of a single determination of the 
 intercept s. 
 
 The above method may be employed to determine 
 the k for each of the side intervals. If the stadia hairs 
 are not adjustable, the side intervals will probably not 
 be equal to each other ; but a method of overcoming 
 this will be explained later ( 235, second paragraph). 
 Of course, if the intervals are equal, the values of k will 
 be equal to each other ; and in any case the sum of the 
 reciprocals of k for the side intervals must be equal to 
 the reciprocal of k for the extreme interval, which 
 affords an excellent check on the accuracy of the work. 
 
 215. For convenience of computation, k is ordinarily 
 made 100 and s is measured in feet. This has the fur- 
 ther advantage of permitting the use of ari ordinary 
 level rod as a stadia rod. If the hairs are adjustable, 
 this condition may be readily attained. To adjust the 
 hairs, carefully draw three diamonds with black ink on 
 cardboard, and fasten two of these targets on a rod 
 exactly i foot apart, and place the third one midway 
 between the other two. Then from the plumb-line 
 measure a distance (100 -j- ^-j-/) feet in front of the 
 instrument, and set the rod vertical. With the tangent 
 screw of the vertical movement, bring the central hair 
 to bisect the middle target, and, next, with the stadia- 
 hair screws bring the side hairs to bisect the side 
 targets respectively. Test the adjustment by placing 
 the outside targets 2 feet apart and setting the rod 
 (200 -)-<:+/) feet from the instrument. In making 
 these observations, the line of^sight should be at le^st 
 nearly horizontal (see 219). When the hairs are ad- 
 justed as above, the stadia formula becomes 
 
 Z>ft. = 100 jft. + (*-}-/) ft. . . . (7) 
 
 If the stadia hairs are not adjustable, k can be made 
 equal to 100 (or any other number) by the following 
 
I&8 TELEMETERS. [CHAP X 
 
 method. Measure (loo-f-^H-/) feet in front of the in- 
 strument and set the rod vertical. Place the telescope 
 horizontal or nearly so, fasten a target on the rod at 
 about the height of the point covered by the upper 
 hair, and bring this hair exactly to bisect the target. 
 Then have an assistant place the other target so that it 
 will be bisected by the lower stadia hair. If now the 
 distance between the two targets is carefully measured 
 and divided into one hundred equal parts, and the 
 graduation is continued over the whole length of the 
 rod, the number of units in the intercept will be equal 
 to the number of feet the rod is from the front focus of 
 the objective ; that is to say, for this rod and this tele- 
 scope, k = 100. 
 
 Since it would probably be impossible to mark each 
 of the one hundred divisions upon the rod in such a way 
 as to make them visible at any considerable distance, 
 it is best to mark, say, each tenth one (see Fig. 49, 
 page 180) and estimate the units. When k is determined 
 as above, the stadia formula becomes 
 
 D ft. = 10 R + (c+f) ft, ... (8) 
 
 in which R is the number of figures (spots) included in 
 the intercept. 
 
 216. It is possible to determine both k and (c -f-/) by 
 the method explained in 214. Make two observations 
 for s at two known distances, Z>, and insert the results 
 in equation (2), page 184, which gives two equations with 
 two unknown quantities, from which k and (c -\- f) can 
 be determined by the ordinary rules of algebra. For 
 greater accuracy make several observations, each two 
 of which will give a value of each of the unknown quan- 
 tities. Strictly speaking, the resulting equations should 
 be solved by the method of least squares ; but as the 
 method of finding c, /, and k, as explained above, is very 
 
ART. l] THE STADIA. 
 
 much more simple and is abundantly exact, this method 
 will not be discussed further, even though it is practiced 
 by some prominent engineers. 
 
 Frequently in finding k, the intercept is assumed to 
 vary as the distance from the center of the instrument, 
 which is equivalent to omitting (c-\-f) from equa- 
 tion (6), page 186. In this case, the distance as deter- 
 mined by the stadia is always in error, except when 
 it is the same as the distance employed in finding k. 
 For shorter distances the result is too small, and for 
 greater distances it is too large. Sometimes observa- 
 tions are made at several distances and a mean value 
 for k is computed, which simply averages errors and 
 makes it impossible to find either the amount or direc- 
 tion of the error. This method is incorrect in principle, 
 and inaccurate in practice, and has nothing to commend 
 it. The method of 215 is simple and gives strictly 
 correct results ; and if approximate results are desired 
 the quantity (c -\- f) in equation (2) may be disregarded 
 in computing the distance, in other words, this quan- 
 tity may be omitted or inserted at will without intro- 
 ducing error into any other part of the work. 
 
 217. To explain a method of eliminating (c +/) from 
 the formula, conceive that the rod has been graduated 
 for the formula D = k s -\- (c-}-f) and that it is standing 
 (ioo-f-^+/) feet from the instrument. The intercept 
 may be considered as divided into 100 divisions, each of 
 which corresponds to a foot of distance on the ground. 
 Then if the rod is shortened by cutting out of the in- 
 tercept as many divisions, or units of the graduation, 
 as there are feet in the distance (c +/), the numbering 
 of the graduations remaining unchanged, the apparent 
 number of units between the stadia hairs will corre- 
 spond to the number of feet from the rod to the center of 
 the instrument. If care is taken that the point at which 
 a portion of the graduation is cut out is always between 
 
190 TELEMETERS. [CHAP, x 
 
 the stadia hairs, the reading will give the distance from 
 the center of the instrument. This method is objec- 
 tionable in that the point at which the graduation is 
 omitted can not always be made to fall between the 
 stadia hairs, owing to obstructions to sighting ; and, 
 further, in that the determination of the vertical co- 
 ordinate is complicated thereby. The advantage of 
 having the intercept proportional to the distance is so 
 slight that it does not seem wise to obtain it by com- 
 plicating both the graduation and the use of the rod. 
 Furthermore, the stadia is not intended for an instru- 
 ment of high precision, and the work for which it is 
 generally used does not require extreme accuracy ; 
 therefore, as a rule, the term (c +/) in the final formula 
 may simply be omitted. 
 
 Parro, in 1823, showed that by placing an auxiliary 
 lens between the objective and the cross hairs, the in- 
 tercept can be made proportional to the distance from 
 the center of the instrument. It is not known that such 
 a telescope has ever been made. 
 
 218. POSITION OF ROD FOR INCLINED LINE OF SIGHT. 
 
 The preceding formulas were deduced on the assump- 
 tion that the central visual ray was horizontal and tke 
 rod vertical, i.e., the central visual ray was assumed to 
 be perpendicular to the rod. These formulas would be 
 sufficient if the observations were made with a leveling 
 instrument; but a level is too limited in its range to 
 secure the full advantage of the principle of the stadia. 
 In all that follows, it will be assumed that a transit is 
 used. 
 
 The formula D = k s -f- c -\- f may be used with an 
 inclined line of sight, provided the rod is held perpen- 
 dicular to the central visual ray. In this case D is no 
 longer the 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 
 
ART. l] THE STADIA. 
 
 by the central visual ray. It then becomes a question 
 whether, with an inclined line of sight, the rod should 
 be held perpendicular to the central visual ray, or verti- 
 cal. 
 
 The perpendicularity of the rod 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 directed 
 toward the observing telescope by the rod-man. The 
 verticality of the rod may be estimated by the rod-man, 
 or it may be determined easily and accurately by at- 
 taching a plumb-line or a level vial. 
 
 Some prefer the rod perpendicular to the line of 
 sight, but this position involves serious difficulties: (i) 
 it is not easy to hold the rod steady in this position; 
 (2) it is not always possible for the rod-man to see the 
 telescope, especially at long distances or great vertical 
 angles, or when undergrowth intervenes; and (3) the 
 formulas for computing the horizontal and vertical co- 
 ordinates of the point are more simple when the rod is 
 vertical than when it is perpendicular to the line of 
 sight. Only the case of the vertical rod will be con- 
 sidered here. 
 
 219. FORMULAS FOE INCLINED LINE OF SIGHT AND VER- 
 TICAL ROD. Let the angle of the central visual ray 
 with the horizontal = C//(Fig. 56). To measure 8, 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; and 6 is then determined as any other 
 vertical angle by reading the vertical circle. 8 will 
 generally be small. Let 2tx = the visual angle BOD. 
 2a is always small, its maximum being about 35 min- 
 utes. AE is the actual intercept, and BD the value it 
 would have if the rod were held perpendicular to the 
 central visual ray. It is desired to find a relation be- 
 tween AE and BD, 
 
I 9 2 
 
 TELEMETERS. 
 
 [CHAP. X 
 
 The angle CBA = 90 -j- a\ and, since a is very small, 
 BC AC cos 6 nearly. Similarly CDE 90 a, and 
 CD=CE cos nearly. Hence BD-(AC-\-CE} cos 8 = 
 AE cos 6 nearly. Notice that the two approximations 
 
 FIG. 56. 
 
 tend to neutralize each other, and that the final error 
 involved is much less than the error of observing AE* 
 
 Since BD is the intercept perpendicular to the line of 
 sight, it is the value of s ( 210) corresponding to the 
 distance 1C. 
 
 Hence equation (2), page 184, becomes 
 
 D = k s cos 6 -\- (c -4- /), (o) 
 
 I \ I J /> \s/ 
 
 in which D is the oblique distance 1C (Fig. 56). 
 
 220. The Horizontal Distance. Let H the horizontal 
 distance from the center of the instrument to the verti- 
 cal through the foot of the rod = // (Fig. 56). H 1C 
 cos 9 D cos 6; and substituting the value of D from 
 equation (9), we have 
 
 0+(c+f)cosB, .;;.'. (10) 
 
 =s cos 
 
 or 
 
 J?= ks ks sin* 0+ (c+f) cos 0. 
 
 * If the side hairs are equidistant from the central one, the true relation is 
 BD = AE cos Q(i tan? Q tan? a) ; and if the side hairs are not equidis- 
 tant, the upper angle being a' and the lower a, , the true relation is 
 
ART. l] THE STADIA. 193 
 
 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 + f) is always small and 
 cos 9 nearly unity, (c -J- /) cos 6 may always be taken 
 equal to (^+/), and often may be omitted entirely. 
 Notice also that since si/i* B is small, the whole opera- 
 tion of reducing an observation by equation (n) may 
 be performed mentally. 
 
 221. 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 ( 209). 
 
 If the instrument is set over the reference point, the target 
 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 point. The proper position of the target 
 may readily be found by setting the rod up by the side 
 of the instrument. Then the difference in level be- 
 tween the point under the instrument and the one on 
 which the rod is placed is equal to the diffevence in ele- 
 vation between the horizontal axis of the telescope and 
 the target on the rod; and therefore to determine the 
 height of any subsequent point, set the rod upon it, 
 bisect the target, and read the angle from the vertical 
 circle. 
 
 If the instrument is not placed over the reference point, 
 bring the line of sight horizontal, set the rod on the 
 reference point, and move the target until it is bisected 
 by the middle cross hair. To determine the height of 
 any subsequent point, set the rod upon it and read ex- 
 actly as before. 
 
 222. To deduce a formula for computing the vertical 
 co-ordinate, let V = the height of the point on which 
 the rod is placed, above the reference point; F~the 
 
194 TELEMETERS. [CHAP, X 
 
 distance the target is above the axis of the telescope = 
 JC (Fig. 56). V 1C sin 6 = D sin 0; and substitut- 
 ing the value of D from equation (9) gives 
 
 V = k s cos sin 6 + (c +/) sin 0; . . (12) 
 V = \ks sin 26+ (c+/) sin 6. . . (13) 
 
 Notice that generally (c +/) sin 6 may be omitted. 
 
 223. REDUCING THE FIELD NOTES. This consists in 
 finding the horizontal and vertical distances from the 
 rod reading and the observed vertical angle. This can 
 be done by using formulas (10) and (13). Although the 
 equations are in a very convenient form for computa- 
 tion, it would be very tedious and slow to solve both 
 equations for every observation. 
 
 Notice that ks cos* and \ks sin 2 8 are independent 
 of the instrument and of the unit of linear measurement 
 used, and therefore they can be tabulated for all cases. 
 (c -{- f) cos 9 and (c -f- f) sin 8 depend upon both the in- 
 strument and the unit, and must be computed once for 
 each particular instrument and rod. Obviously it is a 
 great saving of labor and time if the results are com- 
 puted once for all and tabulated. 
 
 We may compute ks cos* 8 from equation (10), and 
 \ks sin 2 8 from equation (13), for different values of ks 
 and 0, and tabulate the results, in which case the hori- 
 zontal and vertical distances can be taken directly from 
 the table ; or, we may tabulate only cos* 8 and \ sin 2 8 
 for different values of 0, in which case the horizontal 
 and vertical distances are found by multiplying ks by 
 the tabulated factor. The first method would be the 
 better, if it did not require such voluminous tables. 
 
 In either case the results may be expressed in an 
 arithmetical table or in a geometrical diagram. Arith- 
 metical tables are capable of greater accuracy; but they 
 
ART. l] THE STADIA. 195 
 
 must be either very extended and therefore inconveni- 
 ently 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 that therefore 
 they are large and unwieldy. It is believed that with 
 properly constructed diagrams the reductions can be 
 made without sacrificing much, if any, accuracy, and 
 with greater facility than by the use of tables. 
 
 224. Arithmetical Tables. Pages 196 to 198 contain a 
 brief arithmetical reduction table. The column headed 
 H gives the value of sin 9 6 (see equation (n), page 192). 
 The column headed V gives the value of \ sin 2 6 (see 
 equation (13), page 194). The terms (c +/) cos and 
 (c -f-jO sin tt are seldom required ; but they are given in 
 a note at the bottom of the page, for use when great ac- 
 curacy is desired. 
 
 Ockerson and Teeple, assistant U. S. engineers, have 
 published tables * which give \ k s sin 2 -f~ 0.43 in metres 
 (for values of ks in feet) from o to 500, varying by 10, 
 for each minute of Q from o to 10. These tables give 
 also sin* B -f 0.43 for the even minutes from 2 to 20. 
 
 225. Geomatrical Diagrams. There is a variety of 
 methods of constructing diagrams for reducing stadia 
 observations, but a modification of those proposed by 
 Prof. S. W. Robinson f are believed to be the best. 
 
 226. Horizontal Distance. Fig. 57 (between pages 
 200 and 201) gives the term^j sin* (see equation (n), 
 page 192). To use the diagram, find the observed value 
 of ks on the lower line, and from this point follow 
 the inclined line to the radial line indicating the 
 observed value of 6 ; then the elevation of this point as 
 
 * For sale by J. A. Ockerson, Chief Engineer Mississippi River Commis- 
 sion, St. Louis, Missouri. 
 
 t Journal of the Franklin Institute, Vol. 49, pp. 80, 81 ; and Ibid., Vol, 
 51, pp. 15, 16. 
 
196 
 
 TELEMETERS. 
 
 [CHAP. X 
 
 TABLE III. 
 STADIA REDUCTION TABLES. 
 
 9 
 
 o' 
 
 2 
 
 o 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 H 
 
 v. 
 
 H 
 
 v 
 
 H 
 
 v. 
 
 H 
 
 v. 
 
 H. 
 
 v. 
 
 H. 
 
 v 
 
 
 
 
 
 
 
 .0027 
 .0028 
 
 0523 
 .0528 
 
 .0049 
 .0049 
 
 .0696 
 
 .0702 
 
 .0076 
 .0077 
 
 .0868 
 
 .0874 
 
 .0000 
 .0000 
 
 .0000 
 
 .0006 
 
 .0003 
 .0003 
 
 .0174 
 .0180 
 
 .OOIK 
 
 .0013 
 
 349 
 
 0355 
 
 I 
 
 .0000 
 .0000 
 
 .0012 
 
 .0017 
 
 .0003 
 .0004 
 
 .0186 
 
 .0192 
 
 .0013 
 0013 
 
 .0360 
 
 .0366 
 
 .0029 .0534 
 .0029 .0540 
 
 .0050 
 .0051 
 
 .0707 
 
 7*3 
 
 .0078 
 .0079 
 
 .0880 
 .0885 
 
 8 
 10 
 
 .0000 
 .0000 
 
 .0023 
 .0029 
 
 .0004 
 .0004 
 
 .0197 
 
 .0203 
 
 .0014 
 .0014 
 
 .0372 
 .0378 
 
 .0030 
 .0030 
 
 .0546 
 .0552 
 
 .0052 
 
 0053 
 
 .0719 
 .0725 
 
 .0080 
 .0081 
 
 .0801 
 
 .0897 
 
 12 
 *4 
 
 .0000 
 .0000 
 
 .0035 
 
 .0041 
 
 .0004 
 .0004 
 
 .0209 
 .0215 
 
 .0015 
 .0015 
 
 .0384 
 .0390 
 
 .0031 
 .0032 
 
 P5S7 
 
 0563 
 
 .0054 
 .0054 
 
 .0730 
 .0736 
 
 .0082 
 .0083 
 
 .0903 
 .0908 
 
 16 
 18 
 
 .0000 
 .0000 
 
 .0046 
 
 .0052 
 
 .0005 
 .0005 
 
 .0221 
 .0227 
 
 .0016 
 
 .0016 
 
 0395 
 
 .0401 
 
 .0032 
 .0033 
 
 .0569 
 0575 
 
 .0055 
 .0056 
 
 .0742 
 .0748 
 
 .0084 
 .0085 
 
 .0914 
 .0920 
 
 20 
 22 
 
 .0000 
 
 .0000 
 
 0058 
 
 .0064 
 
 .0005 
 
 .0005 
 
 .0232 
 
 .0238 
 
 .0017 
 .0017 
 
 .0407 
 .0413 
 
 .0034 
 .0035 
 
 .0580 
 
 .0586 
 
 .0057 
 .0058 
 
 0753 
 
 .0759 
 
 .0086 
 .0087 
 
 .0925 
 .0931 
 
 24 
 26 
 
 .0000 
 
 .0000 
 
 .0070 
 
 .0076 
 
 .0006 
 .0006 
 
 .0244 
 .0250 
 
 .0017 
 .0018 
 
 .0418 
 .0424 
 
 0035 
 .0036 
 
 .0592 
 .0598 
 
 .0059 
 
 .0060 
 
 .0765 
 
 .0771 
 
 .0088 
 .0089 
 
 0937 
 0943 
 
 28 
 30 
 
 .0001 
 .0001 
 
 .0081 
 .0087 
 
 .0007 
 
 .0007 
 
 .0256 
 .0261 
 
 .0018 
 
 .0019 
 
 .0430 
 .0436 
 
 .0037 
 .0037 
 
 .0604 
 .0609 
 
 .0061 
 .0062 
 
 .0776 
 .0782 
 
 .0090 
 .0091 
 
 .0948 
 
 OQ54 
 
 32 
 
 34 
 
 .0001 
 
 .0001 
 
 .0093 
 .0099 
 
 .0007 
 
 .0008 
 
 .0267 
 
 .0273 
 
 .0019 
 .0020 
 
 .0442 
 .0448 
 
 .0038 
 .0039 
 
 .0615 
 .0621 
 
 .0062 
 .0063 
 
 .0788 
 .0794 
 
 .0092 
 .0093 
 
 .0960 
 .0965 
 
 g 
 
 .0001 
 
 .0001 
 
 .0105 
 
 .OIIO 
 
 .0008 
 .0008 
 
 .0279 
 
 .0285 
 
 .0021 
 
 .0021 
 
 0453 
 0459 
 
 .0039 
 
 .0040 
 
 .0627 
 
 0633 
 
 .0064 
 .0065 
 
 .0799 
 
 .0805 
 
 .0094 
 0095 
 
 .0971 
 .0977 
 
 40 
 42 
 
 .0001 
 
 .0002 
 
 .0116 
 .0122 
 
 .0009 
 .0009 
 
 .0291 
 .0297 
 
 .0022 
 
 .0022 
 
 .0465 
 .0471 
 
 .0041 
 .0042 
 
 .0638 
 .0644! 
 
 .0066 
 
 .0067 
 
 .0811 
 .0817 
 
 .0097 
 .0098 
 
 .0983 
 .0988 
 
 8 
 
 4 8 
 5 
 
 .0002 
 
 .0002 
 
 .0002 
 .0002 
 
 .0128 
 .0134 
 
 .0139 
 
 .0145 
 
 .0009 
 
 .0010 
 .0010 
 
 .0010 
 
 .0302 
 .0308 
 
 .0314 
 .0320 
 
 .0023 
 .0023 
 
 .0024 
 .0024 
 
 .0476 
 .0482 
 
 .0488 
 .0494 
 
 .0042 
 .0043 
 
 .0044 
 .0045 
 
 .0650 
 
 .0656 
 
 .0661 
 .0667 
 
 .0068 
 .0069 
 
 .0070 
 .0071 
 
 .0822 
 .0828 
 
 .0834 
 
 .0840 
 
 .0100 
 .OIOI 
 
 .0102 
 .0101 
 
 .0994 
 
 . IOOO 
 
 . 1005 
 ion 
 
 52 
 
 54 
 
 .0003 
 .0003 
 
 .0151 
 
 01 57 
 
 .0011 
 .OOII 
 
 .0326 
 Q33 1 
 
 .0025 
 .0026 
 
 .0499 
 .0505 
 
 .0045 
 
 .0046 
 
 .0673 
 .0678 
 
 .0072 
 .0073 
 
 .0845 
 .0851 
 
 .0104 
 .0105 
 
 . 1017 
 
 .1022 
 
 56 
 58 
 
 .0003 
 
 .0003 
 
 .016} 
 
 .0168 
 
 .001 1 
 .oon 
 
 0337 
 0343 
 
 .0026 
 .0027 
 
 .0511 
 
 0517 
 
 .0047 
 .0048 
 
 .0684 
 
 .0690 
 
 .0074 
 0075 
 
 .0857 
 
 .0863 
 
 .0107 
 .0108 
 
 .1028 
 
 .1034 
 
 Co 
 
 .0003 
 
 *74 
 
 .0012 
 
 .0349 
 
 .0027 
 
 .0522 
 
 .0049 
 
 .0696 
 
 .0076 
 
 .0868 
 
 .OIOC 
 
 .1040 
 
 For all /alues of Q found on this page 
 
 (c +/)' 9 = 0. 
 
ART. l] 
 
 THE STADIA. 
 
 I 9 7 
 
 TABLE III. 
 
 STADIA REDUCTION TABLES. 
 
 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 
 H. 
 
 V. 
 
 H. 
 
 V. 
 
 H. 
 
 V. 
 
 H. 
 
 .0245 
 .0247 
 
 V. 
 
 1545 
 J55 1 
 
 H. 
 
 .0302 
 .0304 
 
 V. 
 
 .1710 
 
 . 1716 
 
 H. 
 
 364 
 .0366 
 
 V. 
 
 1873 
 
 .1878 
 
 o 
 
 2 
 
 .0109 
 
 .0110 
 
 . 1040 
 .1045 
 
 .0149 
 .0150 
 
 .1210 
 .1215 
 
 .0194 
 .0195 
 
 .1378 
 .1384 
 
 \ 
 
 .0112 
 
 .0113 
 
 1051 
 1057 
 
 0151 
 0153 
 
 .1221 
 .1226 
 
 .0197 
 .0199 
 
 1389 
 .1395 
 
 0248 
 .0250 
 
 .1556 
 
 .1562 
 
 .0306 
 .0308 
 
 .1721 
 .1726 
 
 .0368 
 0371 
 
 .1884 
 .1889 
 
 8 
 
 10 
 
 .0114 
 .0115 
 
 .1062 
 .1068 
 
 0154 
 .0156 
 
 .1232 
 
 .1238 
 
 .0200 
 .0202 
 
 .1401 
 . 1406 
 
 .0252 
 .0254 
 
 1567 
 I 573 
 
 .0310 
 .0312 
 
 1732 
 I 737 
 
 0373 
 375 
 
 .1895 
 
 .1900 
 
 12 
 
 14 
 
 .0117 
 
 .0118 
 
 .1074 
 .1079 
 
 OI 57 
 .0158 
 
 .1243 
 .1249 
 
 .0203 
 .0205 
 
 .1412 
 .1417 
 
 .0256 
 .0257 
 
 1578 
 .1584 
 
 .0314 
 .0316 
 
 1743 
 .1748 
 
 377 
 379 
 
 1905 
 .1911 
 
 16 
 18 
 
 .0119 
 
 .0120 
 
 -1085 
 .10911 
 
 .0160 
 .0161 
 
 .1255 
 . I26o 
 
 .0207 
 .0208 
 
 1423 
 .1428 
 
 .0259 
 .0261 
 
 .1589 
 1595 
 
 .0318 
 .0320 
 
 1754 
 1759 
 
 .0382 
 .0384 
 
 .1916 
 .1921 
 
 20 
 22 
 
 .OI22 
 .0123 
 
 ,0,6 
 
 . IIO2 
 
 .0163 
 .0164 
 
 .1266 
 .1272 
 
 .O2IO 
 .0212 
 
 1434 
 .1440 
 
 .0263 
 .0265 
 
 .1600 
 .1606 
 
 .0322 
 .0324 
 
 -1765 
 .1770 
 
 .0386 
 .0388 
 
 .1927 
 .1932 
 
 24 
 26 
 
 .OI24 
 .0126 
 
 . 1 108 
 "'3 
 
 .0166 
 .0167 
 
 .1277 
 .1283 
 
 .0213 
 .0215 
 
 1445 
 1451 
 
 .0267 
 .0269 
 
 .1611 
 .1617 
 
 .0326 
 .0328 
 
 .1776 
 .1781 
 
 .0391 
 0393 
 
 .1938 
 1943 
 
 28 
 
 3 
 
 .0127 
 .OI28 
 
 .1119 
 " 2 5 
 
 .0169 
 .0170 
 
 .1288 
 .1294 
 
 .O2I7 
 .02l8 
 
 1456 
 
 . 1462 
 
 .0271 
 
 .0272 
 
 . 1622 
 .1628 
 
 -0330 
 0332 
 
 .1786 
 .1792 
 
 0395 
 397 
 
 .1948 
 1954 
 
 32 
 34 
 
 .0129 
 .0131 
 
 II 3] 
 .1136 
 
 .0172 
 0173 
 
 .I 3 00 
 1305 
 
 .O22O 
 .0222 
 
 .1467 
 1473 
 
 .0274 
 .0276 
 
 .1633 
 .1639 
 
 0334 
 0336 
 
 .1797 
 .1803 
 
 .0400 
 .0402 
 
 1959 
 .1964 
 
 * 
 
 .0132 
 .0133 
 
 .1142 
 .1147 
 
 0175 
 .0176 
 
 .13" 
 -1317 
 
 .0224 
 .O225 
 
 *479 
 .1484 
 
 .0278 
 .0280 
 
 .1644 
 . 1650 
 
 0338 
 .0340 
 
 .1808 
 .1814 
 
 .0404 
 .0407 
 
 .1970 
 1975 
 
 40 
 42 
 
 0135 
 .0136 
 
 "53 
 "59 
 
 .0178 
 .0180 
 
 .1322 
 .1328 
 
 .0227 
 .0229 
 
 .1490 
 H95 
 
 .0282 
 .0284 
 
 1655 
 .1661 
 
 .0342 
 , -0345 
 
 .1819 
 .1824 
 
 .0409 
 1.0411 
 
 .1980 
 .1986 
 
 44 
 46 
 
 0137 
 .0139 
 
 . 1164 
 .1170 
 
 .0181 
 .0183 
 
 1333 
 1339 
 
 .0231 
 .0232 
 
 .1501 
 .1506 
 
 .0286 
 .0288 
 
 .1666 
 .1672 
 
 347 
 0349 
 
 .1830 
 1835 
 
 .0414 
 .0416 
 
 .1991 
 .1996 
 
 48 
 
 So 
 
 .0140 
 .0142 
 
 .1176 
 .1181 
 
 .0184 
 .0186 
 
 I 345 
 135 
 
 .0234 
 .0236 
 
 .1512 
 I 5 I 7 
 
 .0290 
 .0292 
 
 .1677 
 .1683 
 
 0351 
 0353 
 
 .1841 
 .1846 
 
 .0418 
 .0421 
 
 .2002 
 .2007 
 
 5 2 
 54 
 
 .0143 
 .0144 
 
 .1187 
 "93 
 
 0187 
 .0189 
 
 :$ 
 
 .0238 
 .0239 
 
 !523 
 
 .1528 
 
 .0294 
 .0296 
 
 .1688 
 .1694 
 
 355 
 .0358 
 
 .1851 
 .1857 
 
 .0423 
 .0425 
 
 .2012 
 
 .20l8 
 
 
 
 60 
 
 .0146 
 .0147 
 
 .0149 
 
 .1198 
 
 .1204 
 
 . M .o| 
 
 .0190 
 .0192 
 
 .0194 
 
 1367 
 1373 
 
 1378 
 
 .0241 
 .0243 
 
 .0245 
 
 1534 
 .1540 
 
 1545 
 
 .0298 
 .0300 
 
 .0302 
 
 .1699 
 I 75 
 
 .1710 
 
 .0360 
 .0362 
 
 .0364 
 
 .1862 
 .1868 
 
 1873 
 
 .0428 
 .0430 
 
 .0432 
 
 .2O23 
 .2028 
 
 .2034 
 
 For values of 6 on this page -j 
 
 sin varies between o.i(c-\-f) and o.z(c-}-/^ 
 
198 
 
 TELEMETERS. 
 
 [CHAP, x 
 
 TABLE III. 
 STADIA REDUCTION TABLES. 
 
 
 
 I2 8 
 
 13 
 
 14 
 
 '5 
 
 it 
 H. 
 
 .0760 
 .0763 
 
 V. 
 
 2650 
 2655 
 
 17 
 
 
 H. 
 
 V. 
 
 H. 
 
 V. 
 
 H. 
 
 V. 
 
 H. 
 
 V. 
 
 2500 
 .2505 
 
 H. 
 
 V. 
 
 o' 
 
 2 
 
 .0432 
 435 
 
 .2034 
 .2039 
 
 .0506 
 .0509 
 
 .2192 
 .2197 
 
 .0585 
 .0588 
 
 2347 
 2352 
 
 .0670 
 .0673 
 
 0855 
 0858 
 
 2796 
 2801 
 
 4 
 6 
 
 437 
 439 
 
 .2044 
 
 .2050 
 
 .0511 
 <>5 I 3 
 
 .2202 
 .2208 
 
 .0591 
 0594 
 
 .2358 
 2363 
 
 .0676 
 .0679 
 
 2510 
 2515 
 
 .0766 
 .0769 
 
 .2659 
 .2664 
 
 0861 
 0865 
 
 2806 
 
 2810 
 
 8 
 10 
 
 .0442 
 .0444 
 
 2055 
 
 .2060 
 
 .0516 
 .0519 
 
 .2213 
 .2218 
 
 .0596 
 0599 
 
 .2368 
 2 373 
 
 .0682 
 0685 
 
 .2520 
 2525 
 
 .0772 
 0775 
 
 .2669 
 .2674 
 
 0868 
 0871 
 
 2815 
 2820 
 
 12 
 
 M 
 
 .0447 
 .0449 
 
 .2066 
 .2071 
 
 .0521 
 .0524 
 
 .2223 
 .2228 
 
 .0602 
 .0605 
 
 .2378 
 2383 
 
 .0687 
 .0690 
 
 2530 
 2535, 
 
 .0778 
 .0782 
 
 .2679 
 .2684 
 
 0874 
 0878 
 
 2825 
 2830 
 
 16 
 
 18 
 
 Q45 1 
 454 
 
 .2076 
 
 .2081 
 
 .0527 
 .0529 
 
 .2234 
 .2239 
 
 .0607 
 .0610 
 
 .2388 
 2393 
 
 .0693 
 .0696 
 
 .2540 
 2545 
 
 .0785 
 .0788 
 
 .2689 
 .2694 
 
 0881 
 0884 
 
 .2834 
 
 .2830 
 
 20 
 22 
 
 .0456 
 0459 
 
 .2087 
 .2092 
 
 .0532 
 0534 
 
 .2244 
 .2249 
 
 .0613 
 .0010 
 
 2399 
 .2404 
 
 .0699 
 .0702 
 
 255 
 
 2555 
 
 .0791 
 .0794 
 
 .2699 
 .2704 
 
 .0888 
 .0891 
 
 .2844 
 .2849 
 
 24 
 26 
 
 .0461 
 .0464 
 
 .2097 
 
 .2103 
 
 0537 
 .0540 
 
 .2254 
 .2260 
 
 .0619 
 .0621 
 
 .2409 
 .2414 
 
 .0705 
 .0708 
 
 .2560 
 2565 
 
 .0797 
 .0800 
 
 .2709 
 .2713 
 
 .0894 
 .0898 
 
 2854 
 .2858 
 
 28 
 30 
 
 .0466 
 .0469 
 
 .2108 
 
 .2113 
 
 .0542 
 0545 
 
 .2265 
 .2270 
 
 .0624 
 .0627 
 
 .2419 
 .2424 
 
 .0711 
 .0714 
 
 .2570 
 2575 
 
 .0804 
 .0807 
 
 .2718 
 2723 
 
 0901 
 ,0904 
 
 .2863 
 .2868 
 
 3 2 
 34 
 
 IS 
 
 .0471 
 0473 
 
 0476 
 .0478 
 
 .2118 
 .2124 
 
 .2129 
 .2134 
 
 .0548 
 0550 
 
 0553 
 0556 
 
 .2275 
 .2280 
 
 .2285 
 .2291 
 
 .0630 
 0633 
 
 .2429 
 2434 
 
 2439 
 .2444 
 
 .0717 
 .0720 
 
 .0723 
 .0726 
 
 .2580 
 2585 
 
 .2590 
 2595 
 
 .0810 
 0813 
 
 .0816 
 .0819 
 
 .2728 
 2733 
 
 2738 
 2743 
 
 .0908 
 .0911 
 
 9'4 
 .0910 
 
 .2873 
 .2877 
 
 .2882 
 .2887 
 
 40 
 
 42 
 
 .0481 
 .0483 
 
 .2139 
 2145 
 
 0558 
 0561 
 
 .2296 
 .2301 
 
 .0641 
 .0644 
 
 .2449 
 2455 
 
 .0729 
 .0732 
 
 .2600 
 .2605 
 
 .0822 
 .0826 
 
 -2748 
 .2752 
 
 .0921 
 .0924 
 
 .2892 
 
 .2896 
 
 44 
 46 
 
 .0486 
 .0488 
 
 .2150 
 2155 
 
 .0564 
 .0566 
 
 .2306 
 .2311 
 
 .0647 
 .0650 
 
 .2460 
 .2465 
 
 .0735 
 .0738 
 
 .2610 
 .2615 
 
 .0829 
 .0832 
 
 2757 
 .2762 
 
 .0928 
 .0931 
 
 .2901 
 .2906 
 
 48 
 5 
 
 .0491 
 0493 
 
 .2160 
 .2166 
 
 .0569 
 .0572 
 
 .2316 
 .2322 
 
 .0653 
 .0656 
 
 .2470 
 2475 
 
 .0741 
 .0744 
 
 .2620 
 .2625 
 
 .0835 
 .0839 
 
 .2767 
 .2772 
 
 .0935 
 .0938 
 
 .2911 
 .2915 
 
 52 
 54 
 
 .0496 
 .0498 
 
 .2171 
 
 .2176 
 
 575 
 0577 
 
 .2327 
 .2332 
 
 .0658 
 .0661 
 
 .2480 
 .2485 
 
 .0747 
 .0750 
 
 .2630 
 .2635 
 
 .0842 
 .0845 
 
 .2777 
 .2781 
 
 .0941 
 0945 
 
 .2920 
 .2925 
 
 s 
 
 60 
 
 .0501 
 .0504 
 
 .0506 
 
 .2181 
 .2187 
 
 .2192 
 
 .0580 
 .0583 
 
 0585 
 
 2337 
 
 * 34 
 
 2347 
 
 .0664 
 0667 
 
 .0670 
 
 .2490 
 2495 
 
 .2500 
 
 0754 
 0757 
 
 .0760 
 
 .2640 
 .2645 
 
 .2650 
 
 .0848 
 0852 
 
 0855 
 
 .2786 
 .2791 
 
 .2796 
 
 .0948 
 .0952 
 
 9S5 
 
 .2930 
 2934 
 
 2939 
 
 For valuesjof Q on this page 
 
 i c+f) cos > o.9s(tf +/) = 
 
 \ (c +/") sin fl varies between o.2(c -f /) and 0.3(4: -j-/> 
 
ART. l] 
 
 THE STADIA. 
 
 I 99 
 
 TABLE III. 
 STADIA REDUCTION TABLES. 
 
 9 
 
 o' 
 
 2 
 
 iS 
 H. 
 
 JO 
 
 V. 
 
 19 
 
 20 
 
 21 
 
 22 
 
 23- 
 
 H. 
 
 V. 
 
 H. 
 
 V. 
 
 H. 
 
 V. 
 
 H. 
 
 .1403 
 .1407 
 
 V. 
 
 3473 
 3477 
 
 H. 
 
 V. 
 
 3597 
 3601 
 
 95S 
 .0958 
 
 2939 
 2944 
 
 . 1060 
 .1064 
 
 3078, 
 3083 
 
 .1170 
 "74 
 
 .3214 
 
 .3218 
 
 .1284 
 .1288 
 
 3346 
 3350 
 
 1527 
 i53i 
 
 4 6 
 
 .0962 
 .0965 
 
 .2948 
 2953 
 
 .1067 
 . 1071 
 
 .3087 
 .3092 
 
 "77 
 .1181 
 
 .3223 
 3227 
 
 .1292 
 .1296 
 
 3354 
 3359 
 
 .1411 
 .1416 
 
 3482 
 3486 
 
 1535 
 1539 
 
 3605 
 3609 
 
 8 
 
 10 
 
 .0969 
 .0972 
 
 .2g=;8 
 .2962 
 
 .1074 
 .1078 
 
 3097 
 .3101 
 
 "85 
 .1189 
 
 .32321 
 
 .3236 
 
 .1300 
 .1304 
 
 3363 
 3367 
 
 .1420 
 .1424 
 
 '490 
 3494 
 
 3* 
 
 3613 
 36i7 
 
 12 
 Z4 
 
 .0976 
 .0979 
 
 .2967 
 .2972 
 
 .1082 
 .1085 
 
 .3106 
 .3110 
 
 . 1192 
 .1196 
 
 .3241 
 3245 
 
 1308 
 .1312 
 
 3372 
 3376 
 
 .1428 
 M32 
 
 3498 
 3502 
 
 1552 
 1556 
 
 3621 
 3625 
 
 16 
 18 
 
 .0982 
 .0986 
 
 .2976 
 .298. 
 
 .1089 
 .1092 
 
 3"5 
 3"9 
 
 .1200 
 .1204 
 
 3249 
 3254 
 
 1316 
 .1320 
 
 338o 
 3384 
 
 .1436 
 .1440 
 
 3507 
 35" 
 
 .1560 
 1565 
 
 3629 
 3633 
 
 20 
 
 92 j 
 
 .0989 
 0993 
 
 .2986 
 .2990 
 
 .1096 
 
 .1099 
 
 3124 
 
 .3128 
 
 . I2O7 
 .1211 
 
 .3258 
 
 3263 
 
 1323 
 I 3 2 7 
 
 3389 
 3393 ! 
 
 .1444 
 .1448 
 
 3515 
 3519 
 
 .1569 
 1573 
 
 3637 
 3641 
 
 3 
 
 .0996 
 
 . IOOO 
 
 2995 
 .3000 
 
 .1103 
 .1107 
 
 3*33 
 3138 
 
 .1215 
 . 1219 
 
 .3267 
 .3272 
 
 T 33i 
 1335 
 
 3397 
 34 01 
 
 .1452 
 I 45 6 
 
 3523 
 3527 
 
 1577 
 .1582 
 
 3645 
 3649 
 
 28 
 30 
 
 . 1003 
 
 .1007 
 
 .3004 
 .3009 
 
 .mi 
 .1114 
 
 3142 
 3*47 
 
 .1223 
 .T22 7 
 
 .3276 
 
 .3280 
 
 1339 
 1343 
 
 3406 
 34io 
 
 . 1460 
 1465 
 
 3531 
 3536 
 
 .1586 
 -159 
 
 3653 
 3657 
 
 32 
 34 
 
 i . 1010 
 
 .1014 
 
 .3014 
 .3019 
 
 .1118 
 
 .1122 
 
 3151 
 3156 
 
 .1230 
 1234 
 
 3285 
 .3289 
 
 1347 
 i35i 
 
 34H 
 34i8 
 
 .1469 
 1473 
 
 3540 
 3544 
 
 1594 
 
 J 599 
 
 3661 
 3665 
 
 S 
 
 .1017 
 
 .1021 
 
 3023 
 
 .3028 
 
 .1125 
 .1129 
 
 .3160 
 .3^65 
 
 .1238 
 .I2 4 2 
 
 3293 
 .3298 
 
 1355 
 1359 
 
 3423 
 3427 
 
 M77 
 .1481 
 
 3548 
 3552 
 
 1603 
 .1607 
 
 3669 
 3673 
 
 40 
 
 42 
 
 a 
 
 .1024 
 .1028 
 
 .1032 
 
 1035 
 
 -3032 
 3037' 
 
 .3041 
 3046 
 
 ."33 
 .1136 
 
 .1140 
 .1144 
 
 .3169 
 3174 
 
 3'7 8 
 3183 
 
 .1246 
 .12 4 9 
 
 1253 
 1257 
 
 33 2 
 3307 
 
 33" 
 33i5 
 
 1363 
 1367 
 
 *37i 
 1375 
 
 3431 
 3435 
 
 3440 
 3444 1 
 
 M85 
 .1489 
 
 J 493 
 .1498 
 
 3556 
 .3560 
 
 3564 
 .3568 
 
 .16" 
 .1616 
 
 .1620 
 .1624 
 
 3677 
 .3680 
 
 3684 
 .3688 
 
 4 8 
 50 
 
 .1039 
 .1042 
 
 35* 
 355 
 
 "47 
 ."50 
 
 3187 
 .3192 
 
 .I26l 
 .1265 
 
 3320' 
 3324 
 
 1379 
 1383 
 
 3448 
 3452 
 
 1502 
 .1506 
 
 3572 
 3576 
 
 .1629 
 1633 
 
 3692 
 .3696 
 
 52 
 
 54 
 
 .1046 
 .1049 
 
 .3060 
 .3065 
 
 "55 
 "59 
 
 .3196 
 .3201 
 
 .1269 
 
 "73 
 
 .3328 
 3333 
 
 1387 
 i39i 
 
 3457 
 346i 
 
 .iSJQ 
 i5H 
 
 .358o 
 3585 
 
 .,637 
 .1641 
 
 .3700 
 3704 
 
 fi 
 
 1053 
 .1056 
 
 .3069 
 374 
 
 .1163 
 .1166 
 
 3205 
 .3209 
 
 .1277 
 .1280 
 
 3337 
 3341 
 
 *395 
 .1400 
 
 34^5 
 3469 
 
 .1518 
 1523 
 
 .3589 
 3593 
 
 .1646 
 .1650 
 
 .3708 
 3712 
 
 60 
 
 .1060 
 
 .3078 
 
 .1170 
 
 .3214 
 
 .1284 
 
 3346 
 
 .1403 
 
 3473 
 
 1527 
 
 3597 
 
 .1654 
 
 37 l6 
 
20O TELEMETERS. [CHAP. X 
 
 given by the scale at the left of the diagram is the 
 value of ks sin* 6, which is to be subtracted from ks. 
 For extreme accuracy add (<:+/). For example, if ks 
 = 740 and 8 = 10, what is JfJ On the bottom line find 
 740, and follow the inclined line to an intersection with 
 the radial line marked 10. By the scale on the left 
 this point reads 22.4. Therefore Z> =: 740 22.44- 
 
 If 6 is more than 13, the method of using the dia- 
 gram differs slightly from that just described. For 
 example, if ks = 820 and 6 17, what is HI On th| 
 extreme right-hand line of the diagram, find 820 and 
 follow this horizontal line to its intersection with the 
 radial line marked 17, and then follow this oblique 
 line to the scale at the top, we see the value sought is 
 70. Therefore H= 820 70 + (c+f). 
 
 227. Vertical Distance. Fig. 58 (between pages 200 
 and 201) gives the ralue of ^kssin2 6 (see equation 
 (13), page 194). 
 
 This diagram is used in essentially the same manner 
 as Fig. 57 ( 226). If the observed value of 9 is found 
 in the left-hand triangle or section of the diagram, the 
 value of ks is to be found on the bottom line to the left 
 of the center, and the final result is found on the left 
 edge of the left-hand triangle. If the value of 6 is 
 found in the middle section, the value of ks may be 
 found as before, then following the inclined line up to 
 the 6 line, and from this point following a horizontal 
 line to the radial line corresponding to 0, and from 
 thence following the oblique line to the scale on the 
 upper edge of the drawing, find the value sought ; or if 
 preferred, the value of k s and also of the final result may 
 
 e found by the numbers written upon the face of the 
 diagram. If the given value of is found in the right- 
 hand triangle or section of the diagram, find the value 
 of k s on the bottom line to the right ot the center, and 
 
ART. l] THE STADIA. 2OI 
 
 follow the more inclined lines to the intersection with 
 the proper radial line ; and then the reading of this 
 point by the scale of the nearly vertical lines is the 
 value of \ ks sin 2 6. Fig. 58 was constructed very care- 
 fully, and can be used with confidence ; and ordinarily 
 will give results as accurate as the observations. 
 
 Concerning the term (c-\-f)sin 6 of equation (13), 
 page 194, notice that it is always small, and varies but 
 slightly for a considerable change in 6 ; and therefore 
 the supplemental table on Fig. 58 gives this term with 
 sufficient precision. The first two values of (c -f /) in 
 this table are to be used when the metre is the unit, 
 and the last two when the foot is the unit. For values 
 of (+/) 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 degrees. 
 
 228. Another Geometrical Diagram. If cross-ruled 
 paper is at hand, a diagram for reducing field notes 
 may easily be constructed as follows : Establish a zero 
 at one corner of the sheet and lay off 100, 200, etc., 
 along one side say, the bottom of the paper, to repre- 
 sent the several values of ks, and lay off 10, 20, etc., 
 parallel to the other side, to represent the values sought. 
 Obviously the scale of the latter should be much larger 
 say, ten or twenty times than that of the former. 
 
 If it is desired to construct a table for finding the 
 vertical co-ordinate, compute, either by the tables on 
 pages 196-98 or by ordinary trigonometric tables, the 
 values of 1,000 -J sin 2 6 for the various values of 0, 
 and lay off the distances vertically above the 1,000 point 
 on the scale representing the values of ks] and then 
 connect the points so determined with the zero point. 
 The completed diagram will appear something like 
 Fig. 59, page 202. Since large values of &sand # seldom 
 
202 
 
 TELEMETERS. 
 
 [CHAP. X 
 
 occur together, the diagram may be truncated as shown, 
 which greatly extends the range of a single diagram. 
 With cross-ruled paper it is very easy to construct a 
 diagram like that outlined in Fig. 59. 
 
 1000 
 
 m 
 
 FIG. 59. STADIA REDUCTION DIAGRAM. 
 
 229. SOURCES OF ERROR.* Stadia work is subject to 
 many of the errors of ordinary transit work (see 
 140-45). The additional errors peculiar to the stadia 
 are (i) inclination of the rod, (2) error in the value of 
 ks, and (3) error of observation. 
 
 230. Inclination of Rod. To investigate the effect of 
 a slight inclination of the rod, let ks = the intercept 
 when the rod is vertical, and k s' =. the intercept when 
 the rod makes an angle a with the vertical. Then k s = 
 k s' cos a very nearly (see 219), and the error = k s' 
 k s = k s' (i cos a) = ks' sin 1 % a =. 0.000076 k s' a' 2 , a' 
 
 * For general discussion of Cumulative vs. Compensating Errors, see 18. 
 
V 
 
 ART. l] THE STADIA. 203 
 
 being in degrees. It is convenient to remember that if 
 the rod inclines 12 the resulting error in the distance 
 is about one per cent. Notice, however, that ths error 
 increases as the square of the inclination, which shows 
 the necessity of some means of plumbing the rod. With- 
 out some device for keeping the rod vertical ( 208), 
 this would undoubtedly be the principal source of error. 
 On the U. S. Lake Survey, an inclined support was used 
 to steady the rod and prevent its leaning toward or 
 from the instrument. A light tripod is sometimes used 
 for the same purpose. 
 
 231. Value of ks. If the value of ks, or the correspond- 
 ing unit on the rod, is riot correctly determined the re- 
 sults will be incorrect, although the relative position of 
 points will be right. The correct value of ks is easily 
 found (see 214-16), and with the methods of attach- 
 ing the hairs described in 205 there is but little danger 
 of its changing. 
 
 232. Errors of Observation. 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 only one side of the hair is 
 observed, this error is still further increased by the un- 
 certainty due to the thickness of the hairs, arid to any 
 inequality in their thickness ( 268). This element of 
 uncertainty does not exist with the graduations shown 
 in Figs. 49-52 (page 180). 
 
 Imperfect focusing, either of rod or hairs, is a source 
 of error, because 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 
 
264 TELEMETERS. [CHAP. X 
 
 of the focus. If there is no parallax in the telescope 
 ( 94), there will be no error from this cause. 
 
 The indistinctness of the image due to an unsteadi- 
 ness of the atmosphere produces an error by making 
 the image too large and, therefore, the distance too 
 small. The only remedy is to wait for better atmos- 
 pheric conditions. 
 
 Another somewhat common error is miscounting the 
 reading so as to make errors of 10, 100, etc., feet. These 
 errors may be prevented by care, or checked by double 
 readings ; or, instead of making an entirely new read- 
 ing, the two halves of the visual angle may be read and 
 their sum used to check the reading of the two outside 
 hairs. 
 
 233. LIMITS OF PBECISION. The stadia is designed to 
 secure rapidity rather than accuracy ; but nevertheless 
 with reasonable care a considerable degree of accuracy 
 may be obtained. The claim is sometimes made that 
 the stadia is more accurate than the chain; but from the 
 nature of the principles involved, it can not be under 
 equally favorable conditions, although under some cir- 
 cumstances the stadia is more accurate than the chain. 
 The degree of precision is dependent upon the mag- 
 nifying power of the telescope, the length of sight, and 
 the ratio of the space on the rod to the corresponding 
 space on the ground. 
 
 To ascertain the effect of the magnifying pow r, fight 
 of the author's students determined a series of k/iown 
 distances with a telescope magnifying fifteen times and 
 also with a telescope magnifying twenty-five times, all 
 under essentially the same conditions. The average 
 error in the first case was i in 282, and in the second i 
 
 in 333- 
 
 To ascertain the relation between the length of sight 
 and the error, sixteen of the author's students deter- 
 mined three series of distances each, the first being less 
 
ART. l] tHE STADIA. 
 
 than 100 feet, the second between 100 and 200 feet, and 
 the third between 200 and 300 feet. The conditions 
 were essentially the same in all the observations. The 
 average error in the first series was i in 182, in the 
 second i in 263, and in the third i in 370. The relation- 
 ship between the error and the distance depends upon 
 the graduation, the magnifying power, and the atmos- 
 pheric conditions ; but for each particular case there is 
 probably a distance at which the ratio of the error to 
 the distance is a minimum. In a trial made by the 
 author for the purpose of this recor'd, with a graduation 
 like Fig. 49, page 180 (i foot on the rod corresponding 
 to 100 feet on the ground) and a transit magnifying 
 twenty-five times, this ratio was a minimum at about 
 700 feet. 
 
 234. The following data show the degree of accu- 
 racy attained in actual practice. They are not selected, 
 but are all that could be discovered by personal in- 
 quiry and by search through engineering literature. 
 Apparently the results were obtained with an ordinary 
 engineer's transit. In comparing results we must dis- 
 tinguish between the error of simply finding the dis- 
 tance 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 that of measuring the distances. 
 
 To show the degree of precision obtained in measur- 
 ing horizontal distance, we have the following : Thre 
 measurements, each of six distances, from 50 to 500 ft., 
 made to test the accuracy of the stadia measurements, 
 the readings being made with targets, show an average 
 error for a single sight of i in 1,100.* On the U. S. 
 Lake Survey, three measurements of a base line gave 
 errors of i in 1,000, i in 1,635, an< ^ T ' in i888.f On the 
 
 * Van Nostrand's Engineering Magazine, Vol. 30, pp. 319 and 476. 
 f Journal of the Franklin Institute, Vol . 49, p. 74. 
 
2o6 TELEMETERS. [CHAP. X 
 
 Mississippi River Survey the maximum discrepancy 
 permissible is i in 500, the maximum length of sight 
 being 1,600 feet. 
 
 Only the following data can be found concerning the 
 accuracy with which vertical distances can be deter- 
 mined. "Courses have been run with no more than 
 ordinary care i to 6 miles, over heights of 150 to 200 
 feet, in which the final error in height ranged from o to 
 1.5 feet."* In the topographical survey of St. Louis, 
 Mo., "the average error of elevations was less than 0.2 
 of a foot per mile.*' f The " Instructions for Topo- 
 graphical and Hydrographical Field Work on the Mis- 
 sissippi River," issued by the Mississippi River Com- 
 mission, limit the maximum permissible error to i foot, 
 the maximum length of sight being i, 600 feet. 
 
 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 triangulation from i to 4 miles apart. 
 The distances from point to point along the courses 
 ranged from 100 to 500 feet. The latitudes and de- 
 partures were subsequently computed with the object 
 of finding the error of stadia measurements. The re- 
 sults obtained were about i in 800, i in 1,000, and i in 
 1,100." I On the U. S. Lake Survey, in computing the 
 co-ordinates of stadia work for 1875, the length of one 
 hundred and forty-one lines varying between 3,200 feet 
 and 22,000 feet (mean 8,080 feet) were compared with 
 the lengths determined by triangulation or chaining, 
 and the average error was found to be i in 649. The 
 
 * Journal of the Franklin Institute, Vol. 49, p. 74. 
 
 t The Technograph, No. 5 (1890-91), p. 13. 
 
 } Journal of the Franklin Institute, Vol. 49, p. 74. 
 
ART. l] THE STADIA. 207 
 
 maximum error permissible was i in 300.* In a survey 
 of the Red River of the South, conducted by the U. S. 
 Army Engineers, extending over about 70 miles, the 
 stadia work and chaining were checked at eight points, 
 and showed an average difference of i in 430. f In the 
 topographical survey of St. Louis, Mo., "the error of 
 closure, after making corrections for inclination and 
 graduation of the rods, was about i in 800." J Without 
 these corrections the error was about i in 500. Eigh- 
 teen observations were made on two targets at dis- 
 tances from 50 to 500 feet, with a mean error for the 
 mean of three observations (the individual observa- 
 tions are not recorded) of i in 1,500.^ 
 
 235. PRACTICAL HINTS. To obviate the difficulty of 
 estimating a fraction of a division of the rod for each 
 stadia hair, set one of them at the beginning of a 
 division. This will produce a slight, but generally in- 
 appreciable, error in the vertical co-ordinate. With long 
 sights or small angles, it is still more convenient and 
 sufficiently accurate, to set one of the hairs upon one 
 of the more prominent divisions as, for example, the 
 even foot-mark. By remembering that either hair may 
 be moved up or down, and by moving the telescope so 
 as to produce the least displacement, this method of 
 placing one of the hairs at an even division can fre- 
 quently be used without any error, and will greatly 
 facilitate the work. 
 
 If not enough of the rod is visible to read both side 
 hairs, read first with one side hair and the middle one, 
 and then with the middle one and the other side hair; 
 and then add the two intercepts and compute the hori- 
 
 * Professional Papers, Corps of Engineers, U. S. A., No. 24 Primary 
 Triangulation U. S. Lake Survey p. 34. 
 
 t Report of Chief of Engineers, U. S. A., 1873, p. 638. 
 
 % The Technograph, No. 5 (1890-91), p. 12. 
 
 Tf Van Nostrand's Engineering Magazine, Vol. 30, p. 476. 
 
208 TELEMETERS. [cHAP. X 
 
 zontal and vertical co-ordinates with the mean of the 
 corresponding vertical angles. 
 
 In traversing with the stadia ( 137 or 184), check the 
 work by reading the rod and the vertical angle on both 
 fore-sights and back-sights. Frequently the work can 
 also be checked by locating from each instrument sta- 
 tion a point nearly in line between them ; when the 
 sum of the two distances to the object should be equal 
 to the observed distance between the two instrument 
 stations. Notice that this check is entirely independent 
 of the check by back-sights and fore-sights. 
 
 If the target is not visible, owing to brush, etc., sight 
 at any portion of the rod that is visible and note the 
 point covered by the central visual ray. Make a record 
 of this fact, and the next time the rod is visible through- 
 out its entire length determine the correction for the 
 former reading. 
 
 In observing for azimuth have the edge of the rod 
 turned toward the instrument. 
 
 2,36. The many advantages of stadia measurements 
 in surveying need not be dwelt upon, as they are self- 
 evident .to those acquainted with the principles. In 
 broken country, the stadia can be readily used where 
 the use of the chain is not practicable. The stadia is 
 specially applicable to topographical surveying, to the 
 topographical work of a railroad survey, and to deter- 
 mining the lengths of sights in leveling. 
 
 The stadia may also be used for underground sur- 
 veying, where the chaining is peculiarly disagreeable 
 and difficult. For this kind of work there is substituted 
 for the rod a shallow box, lighted on the inside, with a 
 glass front on which the graduation is painted.* 
 
 * Journal of the Franklin Institute, Vol. 55, pp. 384-87. 
 
ART. 2] THE GRADIENTER. 
 
 ART. 2. THE GRADIENTER. 
 
 237. The gradienter is a tangent screw having a 
 micrometer head, attached to the horizontal axis of the 
 telescope for the purpose of measuring a vertical angle 
 in terms of its tangent. For a figure and description of 
 the gradienter, see Fig. 28, page 101. 
 
 238. THE GRADIENTER AS A LEVELING INSTRUMENT. 
 As its name indicates, the most important use of the 
 gradienter is in locating grades in surveying railroads, 
 irrigating ditches, etc. To locate a grade of, for ex- 
 ample, 1.85 per cent, i.e., 1.85 feet per 100 feet, bring the 
 telescope level, and read the head of the gradienter 
 screw; and then, if the screw is graduated so that one 
 revolution corresponds to i foot at 100 feet, turn the 
 screw 1.85 revolutions. The line of sight will then have 
 a grade of 1.85 per cent up or down according to the 
 way the screw was turned; and by setting a target on 
 a rod at the height of the horizontal axis of the tele- 
 scope, the ground corresponding to this grade may be 
 easily found. By an obvious modification of the above 
 process this device may be employed to determine the 
 grade of any particular slope. 
 
 The gradienter is also very convenient in leveling up 
 and down hills, over logs, etc., when extreme accuracy 
 is not required. 
 
 Notice that to use the gradienter as a leveling instru- 
 ment requires the direct measurement of the horizontal 
 distances with a chain or tape. 
 
 239. THE GRADIENTER AS A TELEMETER. The gradi- 
 enter may also be used to measure distances, in either 
 of two ways: (i) By measuring the space on the rod 
 passed over by the line of sight for a given number of 
 revolutions of the screw; or (2) by observing the num- 
 
210 TELEMETERS. [CHAP. X 
 
 ber of revolutions required to carry the line of sight 
 over a constant space on the rod. The first is the more 
 rapid, particularly for short sights; and the second is 
 the more accurate, particularly for long sights, since the 
 observations may be made on targets. Notice that there 
 is not the same objection to the use of targets with a 
 fixed intercept as with a variable intercept ( 207). 
 
 240. Constant Number of Revolutions and Variable 
 Intercept. Line of Sight Perpendicular to Rod. Let D = 
 the distance from one extremity of the intercept to the 
 horizontal axis of the telescope; J7= the horizontal 
 distance to be determined ; V '= the vertical distance to 
 be determined; ^=the length of the measured hori- 
 zontal base; S= the space on the vertical rod passed 
 over by one revolution of the gradienter screw, at the 
 distance B ; s = the space for one revolution, at the 
 distance D. Then by the principle of similar triangles 
 
 B : D::S: j; or 
 
 -0 = 1* (i4) 
 
 Usually gradienters are so made that a single revolu- 
 tion of the screw carries the hair over i foot at a distance 
 
 T> 
 
 of ioo feet; that is, ordinarily -- in.equation (14) is equal 
 
 o 
 
 to ioo. With some forms of instruments two revolutions 
 are required to carry the line of sight over i foot at a 
 distance of ioo feet. In either case the gradienter for- 
 mula becomes 
 
 Z>ft. = loojft (15) 
 
 This is the fundamental equation for the gradienter, 
 and corresponds closely with the fundamental equation 
 for the stadia see equation (2), page 184. 
 
ART. 2] 
 
 THE GRADIENTER. 
 
 211 
 
 241. Inclined Line of Sight and Vertical Rod. It has 
 already been shown ( 218) that it is better to keep the 
 rod vertical even though the line of sight is inclined. 
 In Fig. 60, AE represents the intercept on the vertical 
 
 FIG. 60. 
 
 rod ; BE the intercept perpendicular to the lower visual 
 ray; 6 the angle of elevation of the lower visual ray; 
 and a = the visual angle. 
 
 From the triangle AEB we get 
 
 BE _ cos (6 -f- a) _ cos 6 cos a sin 6 sin a 
 AE ~ cos a cos a 
 
 (16) 
 
 BE AE (cos 6 - sin Q tan a). \. . (17) 
 BE corresponds to s of equation (15), and hence 
 D ft. = 100 BE = 100 AE (cos sin tan or). . (18) 
 
 Since ^fi" = J and tan = = ( 240), the preceding 
 
 Jj IOO 
 
 equation may be written thus: 
 
 D i t. = s ft. (100 cos sin 0). . . 
 
212 TELEMETERS. [CHAP. X 
 
 Notice that equation (19) was deduced for the case in 
 which 6 represented an angle of elevation. If 6 repre- 
 sents an angle of depression, equation (19) is still correct 
 provided the angle is measured to the upper extremity 
 of the intercept. But, for obvious reasons, it is better 
 to measure always to the same extremity of the in- 
 tercept say, the lower, in which case equation (19) 
 may be used, if for angles of depression 6 be considered 
 as the observed angle minus the visual angle, i.e., if 6 is 
 an angle of depression, subtract 34' from it before in- 
 serting it in equation (19). 
 
 242. From equation (19) we easily get 
 
 &it.=s ft. (100 cos 2 8 - i sin 2 0)* . (20) 
 H ft. = 100 s ft. 100 s ft. sin 8 6 % s f t. sin 2 6* (21) 
 
 Equations (20) and (21) correspond to equations (10) 
 and (n), page 192, and may be reduced in the same 
 way see 224 and 226. Frequently the last term 
 in each may be neglected. 
 
 243. To determine the vertical co-ordinate with the 
 gradienter, place a target on the rod at a distance from 
 its foot equal to the height of the horizontal axis of the 
 instrument above the reference point, and make the tar- 
 get the lower extremity of the intercept. 
 
 From equation (19) we easily get 
 
 . = s ft. (100 cos sin 6 sin 3 0).* . (22) 
 Fft. = 100 s ft. -J sin 2 6 s ft. sin 2 0* . (23) 
 
 Equation (23) corresponds to equation (13), page 194, 
 and may be reduced in the same way see 224 and 
 227. Frequently the last term can be neglected. 
 
 * If represents an angle of depression, subtract 34' from it before vising it 
 in th}s equation. 
 
ART. 2] THE GRADIENTER. 2 13 
 
 244. Constant Intercept and Variable Number of Revo- 
 lutions. Let D the perpendicular distance from the 
 center of the intercept to the horizontal axis of the tele- 
 scope ; H '= the horizontal distance from the rod to the 
 center of the instrument; V the vertical distance 
 from the point under the instrument to the point on 
 which the rod is placed ; S = the distance between tar- 
 gets ; N= the number of revolutions required to move 
 the line of sight from one extremity to the other of 
 the constant intercept, at the distance B\ n = the num- 
 ber of revolutions required to move the line of sight 
 over the constant intercept, at the distance D. 
 
 Then by the principle of the similarity of triangles, 
 B : D :: n : N, or 
 
 (24) 
 
 Ordinarily gradienters are so made that if B = 100 feet 
 and S i foot, JV = i ; and consequently for any value 
 of S the number of units in S and Wwill be the same. 
 Under these conditions, equation (24) becomes 
 
 This equation corresponds to equation (2), page 184, 
 and equation (15), page 210, and is the fundamental 
 equation for this method of using the gradienter. 
 Notice that *$*, the distance between the targets, can be 
 changed, as desired, to suit the nature of the work. 
 
 245. From Fig. 60, page 211, we may at once write 
 
214 TELEMETERS. [CHAP. X 
 
 in which 8 is measured to the lower of the targets, the 
 observed value being used for angles of elevation, 
 and for angles of depression N times 34' is subtracted 
 from the observed value before inserting it in the for- 
 mula (see 241). The numerator of equation (26) 
 may be found by Table III, pages 196-98, or by Fig. 57, 
 between pages 200 and 201. 
 
 246. If the bottom target is placed at a distance 
 from the foot of the rod equal to the height of the in- 
 strument above the reference point, then 
 
 in which # is measured as in 245. The numerator of 
 equation (27) can be found by Table III, pages 196-98, 
 or by Fig. 58, between pages 200 and 201. 
 
 247. Stadia vs. Gradienter. It is sometimes claimed 
 that the gradienter is superior to the stadia, but this is 
 at least doubtful. When the gradienter is employed 
 with a variable intercept ( 240-43), the labor and time 
 required for the observations are essentially the same 
 as with the stadia; but the formulas equations (20) or 
 (21), and (23) are more complicated, particularly the 
 correction of 6 for angles of depression. When the 
 gradienter is employed with a constant intercept 
 ( 244-46), the time required to make an observation is 
 greater than with the stadia, and the formulas equa- 
 tions (26) and (27) are considerably more complicated. 
 
 248. LIMITS OF PRECISION. The gradienter as a dis- 
 tance-measurer is not as accurate as the stadia. This 
 is chiefly due to the fact that with the former, two 
 observations must be made upon each point and the 
 instrument is liable to be disturbed between these 
 observations. Further, changes of atmospheric refrac- 
 tion occur quickly, so that there is more risk of error 
 
ART. 2] THE GRADIENTER. 
 
 from two separate observations than if they were made 
 simultaneously, as with the stadia. 
 
 In using the gradienter, to eliminate any back-lash or 
 lost motion in the screw it is best to finish the setting 
 with a motion always in the same direction. 
 
 249. VEKTICAL CIRCLE AS A GRADIENTER. The ver- 
 tical circle may be employed as a gradienter to lay off 
 grades, or as a telemeter to determine the horizontal 
 and vertical co-ordinates of a point. 
 
 250. To Measure Grades. To measure grades with the 
 vertical circle, it is only necessary to change the degrees 
 and minutes of arc into feet per hundred feet, by means 
 of a table of natural tangents. For example, if we de- 
 sired to run a grade line of one in a hundred, />., a i 
 per cent grade, we look in a table of natural tangents 
 and find the angle whose tangent is o.oi, and set it off 
 on the vertical circle. Similarly to run a 1.85 per cent 
 grade, i.e., a grade of 1.85 ft. per 100 ft., we would find 
 the angle whose tangent is 0.0185. 
 
 If the angle of inclination of a particular slope 
 has been measured, and we desire to express it in feet 
 per hundred, we have only to find the natural tangent 
 of the observed angle. For example, if the slope has 
 an inclination of 43', an inspection of a table of natural 
 tangents shows that the grade is 1.25 per cent. 
 
 251. To Measure Distances. To use the vertical circle 
 as a telemeter, notice that if the angle of elevation or 
 depression of two points in the same vertical be ob- 
 served, the distance between the points is the difference 
 between the natural tangents of the observed angles. 
 
 To deduce a formula for this case, let If = the 
 horizontal distance from the instrument to the rod ; 
 V '= the vertical distance from the point on the ground 
 under the instrument to the point upon which the rod 
 is placed ; a = the larger of the two observed angles ; 
 
 = the smaller of the two angles ; and s = the dis- 
 
2l6 TELEMETERS. [cttAP. X 
 
 tance on the rod between the two points sighted at. 
 Then, if a and ft have the same sign, t.e., if both are 
 angles of elevation or angles of depression, 
 
 H= - ~ -- -5. . . . (28) 
 
 tan a tan ft 
 
 If a and ft do not have the same sign, 
 
 H = - - --- 3. (29) 
 
 tan a -f- tan ft 
 
 If the target to which ft is measured is placed at a 
 distance from the foot of the rod equal to the height of 
 the instrument above the reference point, 
 
 F = ^tan/?; ...... (30) 
 
 tan a T tan / 
 
 ART. 3. VARIOUS TELEMETERS. 
 
 252. Eckhold's Telerneter consists of a telescope and 
 micrometer-microscope firmly connected at right angles 
 with each other, and both turning together in the 
 same vertical plane. Below the rotation axis, in the 
 plane of the telescope, is placed a finely graduated 
 scale, which is read through the microscope. The in- 
 tercept on the rod is of a constant length. To use this 
 telemeter the telescope is directed to the top of the rod 
 and the microscope read; the telescope is next directed 
 to the bottom of the rod and the microscope is read 
 again. The horizontal and vertical distances are com- 
 puted by formulas essentially the same as equations 
 (28) or (29), and (31) above. 
 
 , Extravagant claims are made for this instrument, but 
 they are not sustained by experience. It is inferior in 
 accuracy to the stadia. 
 
ART. 3] VARIOUS TELEMETERS. 21^ 
 
 253. Gautier's Telemeter consists of a combination of 
 two mirrors and a prism, such that when two succes- 
 sive observations are made from two points (a few feet 
 apart) upon a third, the distance to the third point is the 
 product of a factor read from the instrument, and the 
 distance between the two points of observation. 
 
 "The accuracy of this instrument is extraordinary. 
 With a base of 20 meters [66 feet], the error for dis- 
 tances below a kilometer [3,280 feet] is almost imper- 
 ceptible. Distances from 3 to 6 kilometers [roughly 2 
 to 4 miles], and even more, have been measured by it, 
 with bases of from 20 to 50 meters [66 to 164 feet], with 
 a maximum error not exceeding one fourth of one per 
 cent." Obviously such accuracy is impossible, i. Only 
 a low magnifying power is claimed for the telescope. 
 2. The observations are made by the coincidence of 
 images, as in the sextant. 3. The instrument is held in 
 the hand. 4. The error of measuring the base is multi- 
 plied in the result. 
 
 254. Adie's, Smyth's, Clarke's, and Struve's Telemeters 
 each consist of two mirrors mounted on the ends of 
 a rod, the direction of a third point being measured 
 by the coincidence of images, as with the sextant. In 
 Adie's telemeter the base is 36 inches, in Struvejs 75 
 inches, and the others mentioned have bases inter- 
 mediate between these two. 
 
 It is now generally conceded that the sextant is the 
 best instrument for determining a distance when a 
 measured base can be fixed. 
 
 255. Other Telemeters. There are many other forms 
 of telemeters, several of which were invented for 
 military use, but the above illustrate the principles of, 
 at least, the most important ones. Only the stadia with 
 fixed hairs and the gradienter have any value as engi- 
 neering instruments. 
 
CHAPTER XI. 
 SPIRIT LEVELS. 
 
 ART. 1. CONSTRUCTION. 
 
 257. QUALITIES DESIRED. The main qualities to be 
 secured in a spirit-leveling instrument are stability of 
 the instrument, defining and magnifying power of the 
 telescope, and delicacy of the level bubble. 
 
 The stability of the instrument depends mainly upon 
 the leveling screws and the manner of connecting the 
 instrument with the tripod head (see Chap. II, Art. 
 2). The ordinary leveling instrument, as well as the 
 transit, has four foot-screws; but as three give more 
 stability and greater delicacy, it is more important 
 to have three foot-screws on levels than on transits, 
 for it is seldom necessary that the latter be exactly 
 level^l 128). Whether three or four foot-screws are 
 used, the distance between opposite screws should be 
 as great as possible. The center, or spindle, should be 
 long and hard, and well fitted in the socket. Obviously 
 the center of gravity of the instrument should be as 
 near to the tripod head as possible, although many 
 instruments are needlessly defective in this particular. 
 
 The optical qualities of the telescope have already 
 been discussed (see Chap. VI). 
 
 However good the other parts of the instrument, the 
 accuracy of the work depends upon the sensitiveness of 
 the bubble. The bubble of a leveling instrument cor- 
 
 218 
 
ART. l CONSTRUCTION. 
 
 responds in importance with the graduation of a transit. 
 It must be remembered that although a sensitive bub- 
 ble may not remain exactly stationary, it will still give 
 better results than a sluggish one which shows no 
 movement when the inclination of the instrument is 
 slightly changed. The liquid employed in level vials 
 should have a minimum adhesion to the glass so as to 
 settle quickly and accurately. Pure ether is best in this 
 respect, but it has too large a co-efficient of expansion 
 to permit its use in field instruments, which are exposed 
 to great differences of temperature. Pure alcohol is 
 frequently employed, but a mixture of alcohol and 
 ether is generally considered best. The level vials 
 of astronomical instruments and of some of the best 
 engineering field instruments are provided with an air- 
 chamber for regulating the length of the bubble, in 
 which case pure ether can be used. A large air-bubble 
 is more sensitive than a small one. 
 
 The scale by which the position of the bubble is read 
 should be as close to the bubble as possible, to avoid 
 parallax in reading; and therefore the graduation should 
 be upon the glass tube. The glass tube should be pro- 
 tected by a metal case, but the former should be so 
 fastened into the latter as to be free to expand and 
 contract with changes of temperature. The common 
 method of fastening the vial in the metal case with 
 plaster of Paris is inadvisable for a sensitive bubble, 
 since changes of temperature cause changes in the cur- 
 vature of the tube and consequently in its sensitive- 
 ness. 
 
 258. CLASSIFICATION. Spirit-leveling instruments may 
 be grouped in three classes. The first includes all in- 
 struments that can be adjusted by reversals. The wye 
 level, Fig. 61, page 221, is the representative of this class. 
 The second includes all leveling instruments that can 
 not be adjusted by reversals. The dumpy level, Fig. 
 
226 SPIRIT LEVELS. [CHAP. XI 
 
 63, page 223, is the representative of this class. The 
 third includes all instruments whose errors of adjust- 
 ment may be eliminated by double observations. Fig. 
 
 64, page 225, is a representative. Instruments of this 
 class are commonly called levels of precision, sometimes 
 geodesic levels; and are used when extreme accuracy 
 is sought. 
 
 259. Wye Level. The instrument shown in Fig. 61, is 
 named the wye, or Y, level from the form of the supports 
 of the telescope. A section of a wye level, by a different 
 maker than that of Fig. 61, is shown in Fig. 62, page 
 222. The wye level is used far more than any other 
 by American engineers. Its distinguishing characteris- 
 tic is that the telescope may be revolved about its own 
 axis, and turned end for end in its bearings. The only 
 advantage of this construction is that it facilitates the 
 adjustment of the instrument; while, on the other hand, 
 owing to the nature of the construction the instrument 
 does not hold its adjustment well. " The wye level is 
 easily adjusted, and nearly always needs it." 
 
 Notice that the instrument shown in Fig. 61 has an 
 inverting eye-piece, and that in Fig. 62 an erecting eye- 
 piece. The latter is much more common, but the for- 
 mer is the better (see 78). 
 
 To adjust the telescope, it must be free to revolve 
 in the wyes about its own axis; but in using the in- 
 strument, it is necessary that the vertical hair should 
 be vertical. Therefore, when the telescope is fastened 
 in the wyes ready for use, there should be some means 
 of knowing that the vertical hair is truly vertical. 
 For this purpose some makers place a mark upon 
 the wye and another upon the collar on the telescope; 
 while others have a device (of which there are several 
 forms) such that the clip passing over the telescope at 
 the top of the wyes can not be fastened until the tele- 
 scope is in a particular position. One of these devices 
 
ART. 
 
 CONSTRUCTION. 
 
 221 
 
222 
 
 SPIRIT LEVELS. 
 
 [CHAP. XI 
 
ART. l] 
 
 CONSTRUCTION. 
 
 223 
 
 is shown in Fig. 61, page 221, on the inner side of each 
 clip. 
 
 260. Dumpy Level. This name is given to that form 
 of leveling instrument in which the telescope is attached 
 
 FIG. 63. DUMPY LEVEL. 
 
 to the bar in such a way as not to admit of its rotation 
 around its own axis, nor to allow of its reversion end for 
 end. The telescope is usually inverting, and therefore 
 shorter than the one commonly used on leveling instru- 
 ments; hence the name dumpy level. A very good form 
 
224 SPIRIT LEVELS. [CHAP. XI 
 
 of this instrument is shown in Fig. 63, page 223. English 
 engineers use the dumpy level almost exclusively. In 
 the English books it is sometimes called the Troughton 
 level, from the first manufacturer; and sometimes the 
 Gavatt level, in honor of Gavatt's improvements. 
 
 In construction the dumpy is more simple and com- 
 pact than the wye level, but is less convenient to adjust. 
 It retains its adjustments better than the wye level, 
 which is an important item in practice. If equally well 
 made, it will do as accurate work as the more elaborate 
 and more expensive wye level. The dumpy level usu- 
 ally has the inverting telescope, which gives it an advan- 
 tage over the ordinary wye level (see 78). 
 
 261. Levels of Precision. The distinguishing charac- 
 teristic of this class of levels is that the telescope may 
 be revolved about its own axis, and the level may be 
 reversed end for end independently of the telescope. 
 This construction enables the observer to eliminate all 
 errors of adjustment, by making a double observation. 
 
 There is considerable variety in the form of this 
 class of levels, but only two have been used to any 
 appreciable extent in this country: the Swiss or Kern 
 level, by the Lake Survey and Mississippi River Com- 
 mission; and a modification of the Vienna or Stampfer 
 level, by the Coast and Geodetic Survey. The con- 
 struction of the two is similar, hence only the latter 
 will be described here. The former is described in 
 Professional Papers, Corps of Engineers, U. S. A., No. 
 24 Primary Triangulation U. S. Lake Survey, p. 597, 
 and also in Report of Chief of Engineers, U. S. A., for 
 1877, p. 1190. 
 
 262. The instrument shown in Fig. 64 is the level 
 of precision employed on the U. S. Coast and Geo- 
 detic Survey.* The telescope may be reversed end 
 
 * Report for 1879, Appendix No. 15, pp. 202-11. 
 
ART. l] CONSTRUCTION. 225 
 
 for end and revolved about its optical axis, the two 
 positions in which the horizontal thread is horizontal 
 being definitely fixed by projecting pins. The level 
 may also be reversed end for end independently of 
 
 FIG. 64. U. S. COAST SURVEY LEVEL OF PRECISION. 
 
 the telescope. One end of the telescope and level 
 may be raised and lowered by the micrometer screw. 
 Near the micrometer is a cam hook, by which the 
 weight of the superstructure may be raised off the 
 micrometer during transportation. Under the telescope 
 are two false wyes on lever arms by which the telescope 
 may be raised out of the wyes for transportation. The 
 
226 
 
 SPIRIT LEVELS. 
 
 [CHAP, xi 
 
 whole instrument is secured to the tripod-head by a 
 brass plate which fits over the feet of the leveling 
 screws. 
 
 The aperture of the telescope is 43 mm. (if inches 
 nearly), focal length 410 mm. (16^ inches nearly), mag- 
 nifying power 37. The value of one millimeter of the 
 level scale is 1.5" (radius = 450 feet). The diaphragm 
 is glass, and has one vertical and two horizontal lines 
 ruled upon it. The two horizontal lines are used as 
 stadia hairs to determine the length of sight. 
 
 The instrument including the tripod weighs 45 
 pounds. A smaller size of this instrument, weight 23 
 pounds, is also used. 
 
 263. The following table contains some of the im- 
 portant information concerning the instruments used 
 on four national surveys.* 
 
 TABLE IV. 
 DATA CONCERNING STANDARD LEVELS OF PRECISION. 
 
 
 Great 
 Britain. 
 
 India. 
 
 Switzer- 
 land. 
 
 France. 
 
 BUBBLE TUBE : 
 
 
 
 
 
 Radius of curvature . 
 
 360 ft. 
 
 540 ft. 
 
 600 ft. 
 
 360 ft. 
 
 Movement of line of 
 
 
 
 
 
 sight correspond- 
 
 
 
 
 
 ing to a movement 
 
 
 
 
 
 of the bubble of o. I 
 
 
 
 
 
 inch 
 
 5" 
 
 3i" 
 
 3" 
 
 3" to 7" 
 
 TELESCOPE : 
 
 
 
 
 
 Focal length . . . 
 
 24 ins. 
 
 21 ins. 
 
 16 ins. 
 
 19 ins. 
 
 Diameter of objective 
 
 2^ ins. 
 
 2 ins. 
 
 ii ins. 
 
 ii ins. 
 
 Magnifying power . 
 
 50 
 
 42 
 
 42 
 
 36 
 
 RODS : 
 
 
 
 
 
 Length 
 
 10 ft. 
 
 1 6 ft. 
 
 9.8 ft. 
 
 
 Graduated to ... 
 
 I-IOO ft. 
 
 I-IOO ft. 
 
 3-100 ft. 
 
 6-100 ft. 
 
 Read to 
 
 I 1,000 ft. 
 
 11,000 ft. 
 
 31,000 ft. 
 
 3-1,000 ft. 
 
 LENGTH OF SIGHT : 
 
 
 
 
 
 Maximum distance 
 
 
 
 
 
 from instrument to 
 
 
 
 
 
 rod 
 
 aqo ft 
 
 660 ft. 
 
 OTQ ft 
 
 J.3O ft 
 
 
 JO** 1 ** 
 
 
 JJU 1L. 
 
 4O U * 
 
 * Proc. Inst, of C. E. r Vol. 44, p. 181. 
 
ART 2] LEVELING RODS. 227 
 
 ART. 2. CONSTRUCTION OF LEVELING RODS. 
 
 264. The line of sight of the level is horizontal, and 
 the distance of points below this line is measured by a 
 graduated rod held vertically. Rods are generally 
 graduated to read feet and decimals of a foot, and oc- 
 casionally to feet, inches, and fractions of an inch; and 
 rods graduated according to the metric system are 
 sometimes used in this country. 
 
 There are two classes of leveling rods: (i) target rods, 
 those having a target which is moved into the plane of 
 sight, its position being read by the rod-man; and (2) 
 self-reading or speaking rods, those having a graduation 
 such that the position of the intersection of the line of 
 sight and the rod can be read with the telescope. With 
 the self-reading rod the rod-man has only to hold the 
 rod vertical. 
 
 265. TAEGET RODS. New York Rod. The target rod 
 most frequently used in this country is the New York 
 rod, shown in Fig. 65, page 230. This rod usually con- 
 sists of two pieces of maple or satinwood sliding one 
 upon the other, the same end always being held on the 
 ground, and the graduations starting from this end. 
 The graduations are to tenths and hundredths of a foot, 
 the tenth figures being marked with a black figure 
 and the feet with a larger red figure. The target carries 
 a vernier reading to thousandths of a foot (see Fig. 10, 
 page 67). 
 
 The front surface, i.e., the one on which the target 
 moves, reads to 6.5 feet. When a greater height is 
 required, the horizontal line of the target is fixed at 6.5 
 feet, and the upper half of the rod, carrying the target, 
 is slid upward, and the reading is obtained by a vernier 
 on the side of the rod. The rod, when extended, may 
 
228 SPIRIT LEVELS. [CHAP. XI 
 
 be fastened in that position by means of a clamp at the 
 lower end of the upper piece. 
 
 This rod is also made in three and sometimes four 
 pieces. These forms are recent modifications to make 
 the rod shorter when closed and longer when extended. 
 The three-piece rod is 5 feet long when closed and 14 
 feet long when extended. The four-piece rod when 
 closed is 5 feet long, and when extended is 16 feet 
 long. 
 
 266. Boston Rod. The Boston Rod, Fig. 66, page 229, 
 is formed of two pieces of mahogany or baywood, each 
 about 6 feet long and sliding easily by each other in 
 either direction. One piece is furnished at each end 
 with a clamp and also a vernier ; the other, or front 
 piece, carries the target, and has on each edge a strip 
 of satinwood inlaid, upon which divisions of feet, 
 tenths, and hundredths are marked. The target is a 
 rectangle of wood fastened on the front half. 
 
 When a reading of less than 6 feet is desired, the rod 
 is placed target-end down, and the piece carrying the 
 target is raised. When a reading of more than 6 feet is 
 desired, the rod is placed target-end up, and the piece 
 carrying the target is raised, the reading being taken 
 from the other vernier. This rod is very convenient 
 owing to its extreme lightness, but the parts are too 
 frail to endure rough usage, and therefore engineers 
 have generally given the preference to heavier and more 
 substantial rods. 
 
 267. Philadelphia Rod. The Philadelphia Rod, Fig. 
 67, page 229, is a self-reading rod which is fitted with a 
 target. The rod is made of two strips of cherry, each 
 about | inch thick by ij inches wide and 7 feet long, 
 connected by two metal sleeves, the lower one of which 
 has a clamping screw for fastening the two parts to- 
 gether when the rod is raised for a greater reading 
 than 7 feet. Both sides of the back strip and one side 
 
ART. 2] 
 
 LEVELING RODS. 
 
 229 
 
 FIG. 65. 
 NEW YORK ROD. 
 
 FIG. 66. 
 BOSTON ROD. 
 
 FIG. 67. 
 PHILADELPHIA ROD. 
 
230 SPIRIT LEVELS. [CHAP. XI 
 
 of the front one are recessed one sixteenth of an inch 
 below the edges. These depressed surfaces are painted 
 white, and divided into feet, tenths, and hundredthsof a 
 foot, according to the general principle illustrated in 
 Fig. 72, page 233. The front piece is graduated from 
 the bottom upward to 7 feet, the front surface of the 
 rear half from 7 to 13 feet, also from the bottom up- 
 ward ; and the back surface of the rear half is figured 
 from 7 to 13 feet, from the top downward. 
 
 This rod may be used either as a target rod or as a 
 self-reading rod. The target, usually painted as Fig, 
 69, page 231, carries a scale (not a vernier) one tenth 
 of a foot long divided to hundredths and half-hun- 
 dredths of a foot, by which the rod is read to half- 
 hundredths of a foot. When used as a target rod, 
 for readings of less than 7 feet the target is moved up 
 and down the front piece ; and for readings greater 
 than 7 feet the target is set at 7 feet and the back 
 piece is run up, the reading then being obtained by 
 a scale upon the back piece. When used as a self- 
 reading rod, the observer notes, through the telescope, 
 the point on the rod covered by the cross hair. For 
 readings greater than 7 feet, the rear piece is fully ex- 
 tended, the whole front of the rod then becoming a self- 
 reading rod 13 feet long. 
 
 268. The Target. This 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 subdivide 
 the least space on the rod. The face of the target 
 should be painted of such a pattern that it may be .pre- 
 cisely bisected by the horizontal cross hair. Some of 
 the many varieties are shown in Figs. 68, 69, 70, and 71, 
 page 231. 
 
 Fig. 68 depends upon the nicety with which the eye 
 can determine whether a line bisects an angle, which 
 can be done very accurately by noticing the relative po- 
 
ART. 2] 
 
 LEVELING RODS. 
 
 sition of the two points formed on opposite sides of the 
 hair. For data on the relative accuracy of the targets 
 shown in Figs. 68 and 69,866 Tables I and II, Appendix 
 III. 
 
 Fig. 69 is the target of the New York rod, and 
 essentially that of the Philadelphia rod. The design is 
 
 FIG. 68. 
 
 FIG. 69. 
 
 FIG. 70. 
 
 FIG. 71. 
 
 not good, because the cross hair may be above or be- 
 low the middle of the target by its full thickness, as 
 magnified by the eye-piece, without the error's being 
 perceptible. 
 
 Fig. 70 is the same in principle as Fig. 68. For long 
 sights Fig. 68 is the better, but for very short sights 
 the diamond is larger than the field of view. Fig. 70 
 
232 SPIRIT LEVELS. [CHAP. Xi 
 
 was designed to obviate this defect, and is an excellent 
 target except for very long sights. 
 
 Fig. 71 depends upon the accuracy with which the 
 eye can bisect a space. The only objection to this form 
 of target is that for the greatest accuracy the width of 
 the white band should be proportional to the length of 
 the sight ; and therefore if the width is right for short 
 sights it will be too narrow for long ones. If the length 
 of a sight were constant this would probably be the 
 best form of target. 
 
 269. Level targets are usually painted red and white. 
 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. 69, and black and white for the other figures on 
 page 231. 
 
 270. SELF-HEADING RODS. A self-reading rod is one 
 so graduated as to enable the observer to note at once 
 the reading of the point which lies in the line of sight. 
 The rod-man has only to hold the rod vertical ; the 
 observer notes and records the reading. The most 
 common rod of this class is the Philadelphia rod, which 
 may be used as a target rod also ( 265). A few of the 
 many patterns which have been proposed for self-read- 
 ing leveling rods are shown in Figs. 72-75, page 233. 
 
 Fig. 72 illustrates the principle of the Philadelphia 
 rod ( 267). 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 figure is placed on the line, in which case 
 the figures are eight hundredths high. The angles of 
 the figures indicate fractions of tenths. The gradua- 
 tion given is suitable for short sights; and for longer 
 ones the figures are made larger, and only those for the 
 even tenths are marked. The advantages claimed for 
 
ART. 2] CONSTRUCTION OF LEVELING RODS. 
 
 233 
 
 FIG. 72. 
 
 FIG. 73. 
 
 FIG. 74. 
 
 FIG. 75. 
 
234 SPIRIT LEVELS. [CHAP. XI 
 
 this general form of graduation are distinctness and 
 visibility ; but in neither of these respects is it as good 
 as Figs. 74 or 75, or as the stadia rods shown in Figs. 
 49-52, page 1 80. 
 
 Fig. 73 shows the graduation of the Francis rod,* 
 which is 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. 74, in its essential features, is the graduation of 
 the favorite British level rod. The graduation shown is 
 to tenths, the hundredths to be estimated. The gradua- 
 tion of the standard British rod consists of a number of 
 black lines on a white ground, one hundredth of a foot 
 wide and one hundredth of a foot apart. The lines in- 
 dicating the tenths are longer than the others, and each 
 alternate tenth is numbered. This graduation is very 
 confusing to read. The one shown in Fig. 74 is de- 
 signed on the principle that it is better to estimate the 
 hundredths than to read them from a finely divided 
 scale. This form of graduation reduces the difficulty 
 of counting a number of small divisions, and also the 
 possibility of gross mistakes. Fig. 74 has greater dis- 
 tinctness and visibility than the standard British rod, 
 but it must be condemned for the same reason that the 
 quadrant target of the New York rod was condemned 
 in 268. 
 
 The principle of the Texas rod,f Fig. 75, is of fre- 
 quent application in making self-reading rods. It is 
 superior to that of Fig. 74 in that it obviates the error 
 due to the thickness of the cross hair. The points in- 
 dicate tenths, but it can be read by estimation to hun- 
 dredths. It is possible to make the reading much more 
 
 * Engineering News, Vol. 8, p. 415. 
 f Engineering News, Vol. 8, p. 289. 
 
ART. 2] LEVELING RODS. 
 
 accurate by dividing the oblique side of the triangle, 
 than can be done when the rod is graduated according to 
 the principle of Fig. 74. 
 
 Any pattern suitable for a stadia rod (see Figs. 49-52, 
 page 180) 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. For a few points 
 applicable to the manufacture of home-made self-read- 
 ing level rods, see 208 and 209. 
 
 271. Target vs. Self-reading Rods. Probably the 
 former are the more common now, but as the advan- 
 tages of the latter are becoming better understood 
 they are being more generally used. The chief advan- 
 tage 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 familiarity 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 precision than self-reading ones, but 
 the difference in accuracy is not so great as might at 
 first seem. The accuracy of leveling depends upon a 
 number of things ( 310-19), of which the reading of 
 the position of the line of sight upon the rod is one of 
 the least important, and all of the others are independ- 
 ent of the kind of rod. Reading the position of the 
 target to thousandths of a foot is unnecessary and use- 
 less, unless all other parts of the work are equally pre- 
 cise. The thing to be sought is proportionate accuracy 
 in all parts of the work. If several independent read- 
 ings of a rod be made upon the same point, the differ- 
 
236 SPIRIT-LEVELS. [CHAP. XI 
 
 ence between the various readings will probably be 
 considerably larger than the probable error of reading 
 a self-reading rod (see Tables I and II, Appendix III). 
 
 A self-reading rod is generally used in precise level- 
 ing, two or three hairs, usually the latter, being 
 read to reduce the error of reading. Three observa- 
 tions on a self-reading rod are probably more accurate 
 than a single observation upon an ordinary target, and 
 can be made in about the same time. The difference 
 between the readings of the central and the two side 
 hairs affords a check on the reading and also on the 
 length of sight. 
 
 ART. 3. TESTING THE LEVEL. 
 
 272. THE BUBBLE TUBE. Its Form. A bubble tube 
 is a glass tube bent or ground so that its inside 
 upper surface is circular on a longitudinal section. 
 Since the position of the bubble is determined by read- 
 ing the position of its ends, a longitudinal section of 
 the interior should be of uniform curvature, i.e., cir- 
 cular, and the tube should not be in the least conical. 
 
 These conditions are satisfied if both ends of the 
 bubble move over equal spaces for equal displacements 
 of the bubble tube in altitude. This condition is not 
 an absolute necessity, since the bubble should always 
 stand in the middle when an observation is made. But 
 in very delicate instruments it is nearly impossible to 
 keep the bubble in the middle ; and hence, if the above 
 condition is satisfied, the bubble need not be brought 
 exactly to the center each time, for its position may be 
 noted and a correction applied. 
 
 To test the uniformity of the curvature of the tube, 
 sight at the target of a leveling rod, and read the posi- 
 tion of the bubble; and then move the target over equal 
 
ART. 3] TESTING THE LEVEL. 237 
 
 spaces, sighting upon it and noting the movement of 
 both ends of the bubble for each position. 
 
 Or this test may be made without the aid of a tele- 
 scope, by fastening the bubble tube to a board hinged 
 so as to move in a vertical plane, and measuring the 
 tangent of the angle of elevation. A micrometer 
 screw* affords the easiest and best means of measur- 
 ing the tangent. Such a contrivance is known as a 
 level-trier. 
 
 A less delicate method of testing the above condition 
 is to note whether the bubble expands and contracts 
 equally both ways from the center during changes of 
 temperature. In making these observations great care 
 must be taken that the inclination of the level tube is 
 not changed by external forces or by a change of temper- 
 ature of the parts of the instrument. 
 
 273. Its Sensitiveness. The most important condition 
 to be fulfilled by the level tube is that the bubble shall 
 be sensitive. The sensitiveness depends upon the radius 
 of curvature, or, in other words, 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 read- 
 ings; 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 
 the bubble moved; I the change of inclination of the 
 line of sight; R radius of curvature of the level tube; 
 
 * A fair micrometer screw can be made of a tangent screw from a transit 
 or leveling instrument, to which is attached a graduated cardboard disk. 
 
238 
 
 SPIRIT-LEVELS. 
 
 [CHAP, xi 
 
 and F= the angular value of one division of the scale. 
 
 FIG. 76. 
 
 Then in Fig. 76, from the approximately similar tri- 
 angles ABC and OPQ,we have : 
 
 (0 
 
 tan 7= 
 
 Hence 
 
 tan \"nD 0.00000485 nD' 
 
 (3) 
 
 The sensitiveness of a bubble is generally stated by 
 giving the angle corresponding to a movement of one 
 inch. Good engineer's levels have a motion of 200 to 
 120" per inch, or a radius of curvature of 85 to 140 feet. 
 Levels of precision have a motion of about 50 to 30" 
 per inch, or a radius of curvature of 340 to 600 feet. 
 
 274. SENSITIVENESS vs. MAGNIFYING POWEK. The mag- 
 nifying power of the telescope and the sensitiveness of 
 the level should be so proportioned to each other that 
 
 * The engineer should carry the value of tan l'' in his mind, as it is of 
 frequent application. To assist the memory notice that the value is approx- 
 imately 5 zeros and a 5. 
 
ART. 3] TESTING THE LEVEL. 239 
 
 the least perceptible motion of the bubble will cause 
 sufficient motion of the cross hair on the rod to be 
 easily noticed; and vice versa, the least noticeable motion 
 of the cross hair on the rod should cause a perceptible 
 movement of the bubble. A higher power or a more 
 sensitive level than that required by the above con- 
 dition adds nothing to the accuracy of the instrument 
 and is even worse than useless, for the former causes a 
 loss of brilliancy of the object and the latter an annoy- 
 ance in leveling the instrument. Of two levels in the 
 writer's possession, one had a magnification of 17 and 
 a radius of the bubble of 84 feet, and th'e'bther a mag- 
 nification of 27 and a radius of 22 feet. The simple 
 expedient of changing the bubbles improved both 
 instruments very much. A third instrument is inferior 
 to the other two in definition, although it has a bubble 
 of 165 feet radius. 
 
 275. TELESCOPE SLIDE. The telescope slide should be 
 straight and the optical center should move in the line 
 of collimation. The wye level, after having been colli- 
 mated^ 278), may be tested by the method of paragraph 
 i, 121. The slide is first tested for deviation from a 
 vertical plane by setting a row of points with the tele- 
 scope in a chosen position, and then revolving it 180 
 in the wyes and setting a second row. An easy method 
 of accomplishing this is to set the instrument in a cut 
 or low place, and sight at a plumb-line, marking the 
 point on the head of a stake. Care must be taken not 
 to disturb the instrument when moving the slide. If 
 there is any play or looseness in the object-glass slide 
 it will be almost impossible to make this test satisfac- 
 torily. Having completed this test, revolve the tele- 
 scope 90 from its first position and repeat as above. If 
 the two rows of points do not coincide, either the slide 
 is not straight or it does not move in the right direction. 
 If the direction of motion of the slide is adjustable, the 
 
240 SPIRIT LEVELS. [CHAP. XI 
 
 engineer can determine the source of error only by 
 trial. In doing this remember that a movement of the 
 back end of the objective slide disturbs the adjustment 
 of the line of collimation. A better method of testing the 
 slide of the wye level will be given when considering 
 the method of collimating that instrument (see 280.) 
 
 The slide of the dumpy level can not be tested by the 
 method of paragraph 2, 121, since the telescope can not 
 be reversed about its own axis. It may be tested, after 
 having been collimated ( 285), by taking readings on a row 
 of stakes as described in paragraph 2, 121, and then 
 moving the instrument to the other end of the row and 
 sighting upon them again. If the differences of level 
 relative to the stake first sighted at are the same both 
 times, the slide is straight. This is not a very good test 
 owing to the errors of observation, but it is the only 
 one available. 
 
 276. RINGS AND WYES OF THE WYE LEVEL. The 
 
 rings should be cylinders of the same size, and the wyes 
 should present the same angle in the direction of the 
 telescope tube. The last is not likely to be in error an 
 appreciable amount. The wyes may be compared by 
 taking them off and placing them side by side, or by 
 carefully marking the angle of each upon a piece of 
 paper. 
 
 The equality of the rings is a very important matter, 
 but is often overlooked. Neglecting to examine this 
 point makes the accuracy of the engineer's work depend 
 upon that of an unknown workman. 
 
 The size of the rings can not be tested by any system 
 of reversals, and can be examined only by means of 
 some auxiliary instrument. They may be compared 
 by calipering or by means of a delicate striding level. 
 The student will do well to inquire into the accuracy of 
 these methods. But, for the engineer, the best method 
 is to take a test level, as will be described in 284. 
 
ART. 4] ADJUSTMENTS OF WYE LEVEL. 241 
 
 The test level should not be taken until all the adjust- 
 ments have been made. 
 
 If the rings are not the same size, the instrument is, 
 in effect, a dumpy level, and must be adjusted and used 
 as such, that is to say, the telescope must not be 
 reversed in the wyes end for end, as in the wye level. 
 
 ART. 4. ADJUSTMENTS OF THE WYE LEVEL.* 
 
 277. LEVEL TUBE. The tangent at the middle of the 
 level tube (the zero of the scale) should be parallel to 
 the bottom of the wyes. The adjustment is made in 
 two steps: the first is to bring the tangent and the axis 
 of the telescope into the same plane, and the second is 
 to bring the level parallel to the bottom of the wyes. 
 
 i. In making this adjustment, it is best to have the 
 sun-shade on and have the object-glass run out for the 
 mean length of sight, for then the weight will be 
 symmetrical about the vertical axis and the level will 
 not be affected by any unequal strain. 
 
 Clamp the vertical axis of the instrument, loosen the 
 clips that hold the telescope in the wyes, rotate the tele- 
 scope in the wyes until the level is about vertically 
 under the telescope, and bring the bubble to the middle 
 by moving the foot screws. Then rotate the telescope 
 in the wyes, so that the level tube swings a few degrees 
 to one side of a vertical plane. If the bubble changes 
 its position longitudinally, it shows that the axis of the 
 telescope and the level are not in the same plane. Cor- 
 rect all the error by moving the azimuth screws of the 
 level. 
 
 After having made the bubble tube at least approxi- 
 mately parallel to the bottoms of the rings, if the bubble 
 runs toward the same end when swung on both sides 
 
 * For general remarks on adjustments, see 37. 
 
242 SPIRIT LEVELS. [CHAP. XI 
 
 of the vertical, the tube is conical (see 272). Correct 
 work can not be done with a tube of this form, owing 
 to the effect of changes of temperature and to the un- 
 certainty of the level's being vertically under the tele- 
 scope. 
 
 2. Having made the first step, as above, bring the 
 bubble to the middle, and carefully reverse the telescope 
 in the wyes, end for end. If the bubble does not stand 
 in the middle after reversal, correct half the difference 
 by the altitude screws of the level, and the other half 
 by the foot screws. Notice that if the angles of 
 the wyes are the same and symmetrically placed, this 
 brings the level parallel to the bottom of the wyes, 
 whether the rings are of the same size or not. 
 
 If the rings are of the same size, this adjustment 
 brings the tangent of the level parallel to the axis of 
 the rings. The equality of the rings must be tested by 
 a test level ( 284). 
 
 278. CROSS HAIRS. The line of collimation should 
 coincide with the line of the centers of the rings. To 
 make this adjustment, /.<?., to collimate the instrument, 
 focus upon some well-defined point ( 94) 200 or 300 
 feet distant, and turn the telescope over in the wyes. 
 If the intersection of the cross hairs has moved from 
 the point sighted at, bring it half-way back by moving 
 the screws in the diaphragm which carries the cross 
 hairs (B B, Fig. 62, page 222), and correct the other half 
 by the foot screws and the tangent screw to the vertical 
 axis. 
 
 Since the cross hairs may not have been moved over 
 exactly half the difference, and since the instrument 
 may have been disturbed, the adjustment should be 
 repeated. The instrument need not be level while 
 making this adjustment.* 
 
 * The next two sections relate to the testing of the direction of motion of 
 the object-glass slide. If the instrument is non-adjustable and has been 
 
ART. 4] ADJUSTMENTS OF WYE LEVEL. 243 
 
 279. Since it is tedious to always bring the target 
 exactly to the middle of the field of view, the horizontal 
 hair should be as nearly horizontal as possible; and 
 therefore, after having adjusted the line of collimation, 
 bring the vertical axis vertical, close the clips at the top 
 of the wyes (or bring the mark on the ring to coincide 
 with the mark on the wye 259), and bisect the tar- 
 get of a leveling rod with the horizontal hair. Move 
 the telescope in azimuth, keeping it level, and then if 
 the target is not bisected throughout the length of the 
 hair, rotate the reticule until it is. The horizontal hair 
 will now be horizontal, and the other one will be vertical 
 or nearly so, since the two are supposed to have been 
 placed perpendicular to each other. It will be at least 
 nearly enough vertical to plumb the rod by. 
 
 Instead of adjusting the horizontal hair as above, it 
 is customary to place the vertical hair vertical by sight- 
 ing at a corner of a building. Since the rod may not 
 be brought exactly to the center of the field, it is more 
 important that the horizontal hair should be truly hori- 
 zontal than that the vertical one should be vertical. 
 For this reason, it is better to adjust the horizontal thaia 
 the vertical hair. 
 
 280. When the instrument has been collimated for 
 the above distance, focus upon a point very near the 
 instrument, and test the adjustment of the line of col- 
 limation for this distance. If the instrument is not in 
 adjustment for the second point, either the optical 
 center is not in the axis of the rings, or the line of 
 motion is not parallel to the axis of the rings. The 
 engineer has no means of determining which of these 
 conditions causes the error, except by trial. The former 
 may be caused by the optical center's not lying in the 
 
 tested in this respect, the operations described in the next two sections may be 
 omitted in making the adjustments. If the direction of motion is adjustable, 
 the engineer should occasionally test it. 
 
244 SPIRIT LEVELS. [CHAP. XI 
 
 center of figure of the lens, or by the telescope slides 
 not being straight. With an instrument which is pro- 
 vided with a means of adjusting the direction of 
 motion of the object-glass slide (C, Fig. 62, page 222), 
 the engineer may move the inner end of the slide until 
 half the error is corrected; and then collimate again 
 upon the first point, and test the adjustment upon the 
 second. If the instrument is in adjustment for the two 
 points at the same time, it is probable that the optical 
 center lies in the axis of the rings and that the direction 
 of motion coincides with that line. 
 
 281. It is generally held that if an instrument is in 
 collimation for two different distances at the same time, 
 the line of sight must be in the axis of the rings; but 
 this is not necessarily true. In general, if either the 
 intersection of the cross hairs or the optical center of 
 the objective does not lie in the axis of the rings, the 
 line of collimation will describe the surface of a cone 
 when the telescope is revolved in the wyes; and for any 
 given position of the optical center, it is possible to 
 have such a position of the cross hairs and correspond- 
 ing motion of the slide that the instrument may be in 
 collimation for two points at the same time and still 
 the line of collimation not be in the axis of the rings. 
 That is to say, if the instrument is not in collimation, 
 the line of sight will describe the surface of a variable 
 cone, the points collimated upon being two positions of 
 the vertex. Such a relation of parts could be obtained 
 only by a series of successive approximations. The 
 only check against the occurrence of this condition is a 
 test level ( 284). 
 
 Another particular case is that in which the line of 
 collimation describes a cylinder. This occurs when the 
 optical center and the intersection of the cross hairs are 
 on the same side of, and equally distant from, the axis 
 o the rings. This state of affairs will be revealed by 
 
ART. 4] ADJUSTMENTS OF WYE LEVEL. 245 
 
 the instrument's being out of collimation the same 
 amount for all distances. In this case the line of colli- 
 mation will be parallel to the level (assuming the rings to 
 have been found to be of the same size), and therefore 
 the instrument is in adjustment. If the optical center 
 does not lie in the line of the centers of the rings, the 
 cross hairs maybe moved, by a series of trials, until the 
 line of sight shall describe the surface of a cylinder. 
 Notice, however, that this can occur only in those in- 
 struments in which the line of motion is parallel to the 
 axis of the rings. 
 
 282. CENTERING THE EYE-PIECE. If after having 
 
 brought the intersection of the cross hairs into the axis 
 of the ring, the same field of view is not presented dur- 
 ing an entire revolution of the telescope in the wyes, 
 the fault is in the centering of the eye-piece; in other 
 words, the optical axis of the eye-piece does not coincide 
 with that of the objective. Some instruments are pro- 
 vided with an arrangement for correcting this by a 
 motion of the inner end of the eye-piece, see A A, Fig. 
 62, page 222. (The adjusting screws are covered by a 
 ring which can easily be slipped off.) Strictly, this ad- 
 justment does not place the two axes parallel; it only 
 makes the optical axis of the eye-piece intersect the 
 optical axis of the objective in the plane of the image. 
 However, the only effect is a slight blurring of opposite 
 sides of the image. 
 
 Inverting telescopes have no such adjustment; and 
 it would be an advantage if it were omitted in telescopes 
 with an erecting eye-piece; but this can not be done, 
 owing to mechanical reasons. 
 
 283. WYES. The line of the bottoms of the wyes 
 should be perpendicular to the vertical axis of the in- 
 strument. When the instrument is in adjustment, the 
 bubble tube is parallel to the bottom of the wyes 
 ( 2 77)> and therefore this adjustment is equivalent to 
 
246 SPIRIT LEVELS. [CHAP. XI 
 
 placing the bubble tube perpendicular to the vertical 
 axis of the instrument. Notice that this adjustment is 
 for convenience only, as it in no wise affects the accu- 
 racy of the work, but merely saves the labor of leveling 
 up every time the telescope is moved in azimuth. 
 
 To make this adjustment bring the bubble to the mid- 
 dle by turning the foot screws, and turn the telescope 
 180 in azimuth. If the bubble does not stand in the 
 middle after reversal, correct half the error with the 
 nuts on the lower ends of the wyes (Fig. 62, page 222), 
 and the other half with the foot screws. Since the tele- 
 scope may not have been revolved exacfe-ly 180 in azi- 
 muth, test the adjustment for a position 90 from the one 
 used as above, and then repeat in the first position. 
 
 284. TEST LEVEL. After having made the usual ad- 
 justments of the wye level, it is absolutely necessary 
 that they should be verified by taking a test level. 
 
 To take a test level, drive two pegs into the ground, 
 say, 400 feet apart, set the instrument exactly half-way 
 between them, and carefully determine the difference 
 of level. 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 instru- 
 ment may be out of adjustment ( 126). Make several 
 determinations of the difference of level for a check. 
 Next, set the instrument near one of the pegs, say, 10 
 feet beyond it, and at least nearly in line with the other, 
 and re-determine the difference of level, making several 
 pairs of observations. 
 
 If the difference is the same both times, it is certain 
 that all the conditions enumerated in the preceding dis- 
 cussion have been satisfied. That is, an agreement in 
 the test level proves (i) that the slide is straight, (2) 
 that the line of motion is parallel to the line of centers 
 of the rings, (3) that the rings are of the same size, (4) 
 that the level is parallel to the bottom of the wyes, and 
 
AkT. 5] ADJUSTMENTS OF 1>UMPY LEVEL. $tf 
 
 (5) that the line of collimation is parallel to the line of 
 the centers of the rings. 
 
 If the differences for the two positions of the instru- 
 ment are not the same, the error is due to defect in one 
 or more of the five particulars mentioned above. If the 
 slide is not straight, the only remedy is in a new instru- 
 ment. Some instruments are provided with a means of 
 changing the direction of motion of the object-glass 
 slide; but it is at least doubtful whether this is any ad- 
 vantage to an instrument. When no means of adjust- 
 ment is provided, the only remedy is a new instru- 
 ment. If the rings are not of the same size, the instru- 
 ment can be adjusted and used as a dumpy level. If 
 the fourth or fifth conditions are not satisfied, a re- 
 adjustment is sufficient. 
 
 ART. 5. ADJUSTMENTS OF THE DUMPY LEVEL.* 
 
 285. COLLIMATION. It is required to make the line of 
 collimation and the level parallel to each other. This 
 may be accomplished in either of two ways: (i) by bring- 
 ing the line of collimation parallel to the level, or (2) 
 by bringing the level parallel to the line of collimation. 
 In case both the level and the line of collimation are mov-. 
 able, the latter method is to be preferred; for then the 
 intersection of the cross hairs can first be placed in the 
 optical axis of the telescope, and not be disturbed by 
 this adjustment. The best form of instrument is that 
 in which the level is fixed, for then the maker can place 
 the level parallel to the optical axis before he fastens the 
 telescope to the standards; and when the line of collima- 
 tion is adjusted, it will coincide with the optical axis. 
 
 i. To place the line of collimation parallel to the level, 
 set the instrument upon a level piece of ground, and 
 
 * For general remarks on adjustments, see 37. 
 
248 SPIRIT LEVELS. [cHAP. XI 
 
 bring the bubble to the middle of its race, read the rod 
 upon a point, say, 200 feet distant, turn the instrument 
 about, bring the bubble to the middle, and sight in the 
 opposite direction upon a point at exactly the same dis- 
 tance as before. The difference of readings is the true 
 difference of level. Make several determinations of the 
 difference for a check. Then move the instrument near 
 one of the points, say, 10 feet beyond it, bring the bubble 
 to the middle, and sight upon each point. If the second 
 difference is not equal to the first, correct the error by 
 moving the cross hairs over a space on the farther rod 
 equal, for the case supposed above, to f-^ths of the ap- 
 parent difference of level (see the second paragraph of 
 126). When the difference of readings is equal to the 
 true difference of level, the line of collimation is horizon- 
 tal and therefore parallel to the level. 
 
 2. To place the level parallel to the line of collimation, 
 set the instrument midway between two points and 
 determine the true difference of level. Next move the 
 instrument near one of the points, and by means of the 
 foot screws change the inclination of the telescope until 
 the difference of the readings is the same as when the 
 instrument was in the middle (see 126). The line of 
 ^sightis then horizontal (see Fig. 29, page 113). Without 
 altering the inclination of the line of sight, raise or 
 lower one end of the level until the bubble is in the 
 middle. The level and line of sight are then parallel. 
 
 Notice that this adjustment does not necessarily make 
 the line of collimation coincide with the optical axis of 
 the telescope; but it can be shown* that if the direction 
 of motion of the slide is parallel to the optical axis, the instru- 
 ment will still give correct results when adjusted as 
 above. 
 
 286. WYES. For convenience, the level and line of 
 collimation should be perpendicular to the vertical axis. 
 
 * Rankine's Civil Engineering, p. 84. 
 
ART. 6] ADJUSTMENTS OF LEVEL OF PRECISION. 249 
 
 To make this adjustment, bring the bubble to the mid- 
 dle by means of the foot screws alone, and revolve the 
 instrument 180 in azimuth. If the bubble is in the 
 middle after reversal, it is adjusted; if not, correct half 
 the error by raising or lowering one of the standards 
 connecting the telescope with the tripod head,* and the 
 other half by the foot screws. Repeat the operation 
 until the bubble does not move in the tube when the 
 instrument is turned in any position on its vertical axis. 
 In some instruments there are no means of adjusting 
 the height of the standards or wyes, in which case it be- 
 comes necessary to make the level perpendicular to the 
 vertical axis before adjusting the line of collimation, 
 and to adjust the line of collimation by the first method. 
 
 ART. 6. ADJUSTMENTS OF LEVEL OF PRECISION. 
 
 287. As the adjustments of levels of precision do not 
 commonly differ materially from those of the ordinary 
 forms, the method need not be re-stated here. It is 
 customary to adjust the instrument as carefully as pos- 
 sible, and then determine the error of adjustment. 
 Each observation may then be corrected, thus affording 
 a check between the members of a double observation. 
 The tests and adjustments of the form shown in Fig. 64 
 (page 225) are the same as those of the wye level, with 
 the following tests in addition. 
 
 288. The telescope when raised or lowered by the 
 micrometer screw should move in a vertical plane. 
 This adjustment is necessary, since it is not always 
 practicable to have the line of sight exactly horizontal at 
 the moment of sighting. To test this adjustment, care- 
 
 * The screws for raising or lowering the standards are on the under side 
 of the cross-bar, and are to be turned with a screw-driver. Owing to the unex- 
 posed position of these screws, they are not liable to be turned accidentally, 
 which is one of the advantages of this form of level. 
 
SPIRIT LEVELS. [CHAP. XI 
 
 fully level the instrument, and sight upon a plumb-line; 
 and then move the telescope in altitude and see whether 
 the intersection of the cross hairs follows the plumb- 
 line. If it does not, the only remedy is to return the 
 instrument to the maker. 
 
 289. The angular value of one revolution of the mi- 
 crometer screw should be known approximately. This 
 can be determined only approximately, owing to the 
 eccentric position of the line of sight with reference to 
 the pivots on which the telescope turns. The value of 
 a revolution of the micrometer is useful only in correct- 
 ing an observation made when the line of sight is not 
 exactly horizontal, and hence an approximate value is 
 sufficient for any case arising in practice. To determine 
 an approximate value of one revolution, bring the line 
 of sight nearly horizontal, read the micrometer, and 
 sight upon a level target at a distance D. Then turn the 
 micrometer screw one revolution, and read the rod again. 
 Representing the difference of rod readings by d, the 
 angular value of one revolution in seconds of arc is 
 
 -- r/ . To reduce the error of observation, deter- 
 D tan i" 
 
 mine the value of several say, three or four revolu- 
 tions and divide the result by the number of revolutions. 
 
 290. The collars, or rings, on the telescope, /.<?., the 
 bearings in the wyes, should be of exactly the same 
 diameter. If they are not, the inequality must be deter- 
 mined and a correction computed. The inequality of 
 rings is the only instrumental error that can not be 
 eliminated by any system of double observations. It 
 may be eliminated from the final result by equal back- 
 and fore-sights (326); but to employ the check of 
 double leveling ( 306), a correction for inequality of 
 collars must be applied to the rod reading. 
 
 To find the inequality of collars, set the instrument 
 firmly, and level it carefully. Read both ends of the 
 
ART. 6] ADJUSTMENTS OF LEVEL OF PRECISION. 351 
 
 bubble, and take the mean; reverse the bubble tube 
 end for end, read again, arid take the mean. Half the 
 difference of these means is the inclination of the top of 
 the rings, expressed in divisions of the bubble scale. 
 Reverse the telescope end for end in the wyes, and re- 
 peat the entire process as above. To meet the possibil- 
 ity of the rings' not being perfectly round, revolve the 
 telescope about the optical axis and determine the in- 
 clination as above with the telescope both direct and 
 reversed. If 
 
 E d = the inclination when telescope is direct and bubble 
 
 tube erect, 
 E r = the inclination when telescope is reversed and 
 
 bubble tube erect, 
 I d = the inclination when telescope is direct and bubble 
 
 tube inverted, 
 f r = the inclination when telescope is reversed and 
 
 bubble tube inverted, 
 
 V = the value of one division of the bubble scale, 
 the correction to be applied to an observed reading at 
 a distance >> is equal to 
 
 sn 
 
 If the object-end ring is too large, the line of sight will 
 be inclined downward when the tops of the rings are 
 horizontal. 
 
 291. For the most accurate work, it is necessary to 
 determine the absolute value of the unit of the rod, 
 and also to test the uniformity of the graduation. 
 The coefficient of expansion of the rod and the gradu- 
 ation of the attached thermometer should likewise be 
 tested, as also the adjustment of the level by which the 
 rod is kept vertical. 
 
SPIRIT LEVELS. JCHAP. Xl 
 
 ART. 7. USING THE LEVEL. 
 
 292. A level line is a line parallel to the surface of 
 still water. A horizontal line is a straight line tangent 
 to a level line. Level and horizontal are frequently 
 used as meaning the same thing, and many times there 
 is no appreciable difference between the two (319, 
 equation 3). 
 
 The level is used (i) to find how much one point is 
 above or below a level line passing through another 
 point, (2) to obtain a profile of a line, and (3) to 
 locate contour lines, grade lines, boundaries of em- 
 bankments and excavations, etc. Whatever the ulti- 
 mate purpose of the work, the immediate object for 
 any setting of the instrument is to find how much one 
 point is higher or lower than another, and this is ascer- 
 tained by obtaining a horizontal line, and measuring 
 how far each point is below this line. The line of 
 sight of a properly adjusted leveling instrument is a 
 horizontal line; and as the instrument is revolved in 
 azimuth, this line marks a horizontal plane. The ver- 
 tical distance of any point below this plane is meas- 
 ured by the leveling rod. 
 
 293. DlFFEBENTIAL LEVELING. This consists in find- 
 ing how much one point is above or below a level line 
 passing through another point. If the two points are 
 not too far apart, either horizontally or vertically, the 
 difference of level may be found by setting the instru- 
 ment between the two and taking a reading of the 
 level rod on each point. The difference of the read- 
 ings will be the difference of level required. 
 
 If the two points are so far apart, either horizontally 
 or vertically, that the difference of level can not be 
 found by a single setting of the instrument, establish 
 one or more intermediate points, and determine the 
 
A.RT. 7] USING THE LEVEL. 25$ 
 
 difference of level between successive pairs of points. 
 The algebraic sum of the difference of level between 
 the successive pairs of points is the required difference 
 of level ; or, in other words, the sum of the rod-read- 
 ings for the odd-numbered sights minus the sum of the 
 readings for the even-numbered sights is equal to the 
 algebraic difference of level. 
 
 If the elevation of the first point is given with refer- 
 ence to any datum, as, for example, sea-level, the ele- 
 vation of the first point plus the reading of the rod on 
 that point gives the elevation of the plane of sight ; 
 and the height of the plane of sight minus the reading 
 of the rod on the second point gives the elevation of 
 that point above datum. The rod reading on the 
 known point is called the back-sight, and the reading 
 on the point whose elevation is to be determined is 
 called the fore '-sight. Notice that the terms back-sight 
 and fore-sight have no reference to directions. As 
 shown above, the back-sight reading is essentially posi- 
 tive, and the fore-sight is essentially negative ; and 
 therefore the back-sight is sometimes also called ^plus- 
 sight and the fore-sight a minus-sight. The intermediate 
 points established between two remote points for the 
 purpose of determining the difference of level of the 
 latter, are called turning points. Bench marks, or simply 
 benches, are points of more or less permanent character 
 whose elevations are determined and recorded for fu- 
 ture reference. In the notes, benches are indicated by 
 B. M. or B, with a subscript to show the number of the 
 bench. Thus B^ means the third bench from the be- 
 ginning of the line. 
 
 294. Field Routine. The rod-man holds the rod on 
 the -starting point. He should stand directly behind 
 the rod, and hold it vertically. The instrument-man, 
 or leveler, sets up the instrument 100 to 300 feet ( 325) 
 from the end of the line, in the direction in which the 
 
254 SPIRIT LEVELS. fcHAP. Xl 
 
 work is to proceed. It is not necessary that the instru- 
 ment be set on the line, unless the distance run is also 
 desired, in which case the instrument should be ap- 
 proximately on line. Having leveled the instrument, 
 the leveler sights upon the rod, and, if it is a self-read- 
 ing rod, notes the reading and writes it down as a 
 " -f- sight" ( 293, third paragraph). 
 
 If a target rod is used, the leveler signals the rod- 
 man the direction in which to move the target. There 
 is great variety in the manner of making these signals. 
 The important thing is that they should be easily seen 
 and readily understood. A very good system of sig- 
 nals is that in which the arm raised above the shoulder 
 indicates that the target should be moved up, the 
 arm coming more nearly horizontal as the target ap- 
 proaches the desired point ; when the target is right, 
 the arm is swung horizontally, or in a circle, at which 
 signal the rod-man clamps the target. Similarly, the 
 arm held below the shoulder indicates that the target 
 should be moved down. The rod-man should place 
 the target as near at the proper place as he can by 
 estimation, without waiting for signals from the leveler. 
 
 After the target has been lined in by the leveler,the rod- 
 man should wave the rod slightly to and fro, being sure 
 that it passes through the vertical position; and in the 
 mean time the man at the instrument notices whether the 
 target rises to the line of sight, i.e., whether the highest 
 position of the target coincides with the horizontal hair. 
 If it does not, the leveler directs the rod-man to move 
 the target. The instrument-man can tell whether the 
 rod leans to the right or left, by comparing it with the 
 vertical hair. If the rod is not vertical, he should hold 
 one arm up vertically, stand where the rod-man can see 
 him. and incline his body in the direction the rod should 
 be moved. A good rod-man will seldom, or never, need 
 this signal. 
 
ART. 7] USING THE LEVEL. 255 
 
 When the position of the target is satisfactory, the 
 rod-man calls out, first, the number of the station or 
 stake, which the leveler calls back as evidence that he 
 understands; and then the rod-man calls out the read- 
 ing, one figure at a time, which is repeated by the leveler 
 as he records it in his note-book. These numbers may 
 be read and called off while the rod-man is walking to 
 the next station. 
 
 Having obtained the reading at the first station, the 
 rod-man proceeds as far on the other side of the instru- 
 ment as the instrument was from the first station (see 
 326), and establishes a turning point by holding the 
 rod upon some fixed point or upon a stake driven for 
 the purpose. An iron pin i inch in diameter at the 
 top and 6 or 8 inches long, tapering to a point, makes a 
 good turning point; or a thin triangular plate of iron, 
 having sides about 8 to 12 inches long with about 2 
 inches of the corners turned down, and having a small 
 button attached to the middle of its upper surface, 
 makes an excellent turning point for hard ground where 
 there is no grass, weeds, etc. Whether the pin or plate 
 is employed, a suitable handle should be attached to it. 
 
 A reading is taken upon the turning point, and the 
 result is recorded as a " sight." The instrument is 
 then moved beyond the turning point, and set up again, 
 a second reading is taken upon the turning point, and 
 the result is recorded as a " -f- sight." Thus the work 
 proceeds, the rod and the instrument alternately being 
 ahead. 
 
 295. The Record. The record for differential leveling 
 is shown in Table V, page 256. The form is so simple 
 that it does not need much explanation. The sum of the 
 back-sights minus the sum of the fore-sights gives the 
 algebraic difference of elevation. The length of sight, 
 which may be obtained by the principle of the stadia 
 (pp. 173-208), is not required except to check the equality 
 
2 5 6 
 
 SPIRIT LEVELS. 
 
 [CHAP, xi 
 
 of the length of the back-sight and the fore-sight ( 326), 
 or to determine the distance run. 
 
 TABLE V. 
 
 FORM OF RECORD FOR DIFFERENTIAL LEVELING. 
 
 Sta. 
 
 Length 
 of 
 Sight. 
 
 + Sights. 
 
 Sights. 
 
 Elevation 
 of 
 Benches. 
 
 Remarks. 
 
 i 
 
 I 
 
 2 
 
 B* 
 
 3 
 4 
 
 #3 
 
 
 3.214 
 3-6I7 
 5-23I 
 
 2.431 
 4-733 
 3-214 
 
 O.OOO 
 + 1-879 
 
 -1.506 
 
 Corner of water-table 
 at N.-W. corner of 
 court-house. 
 Marked point on W. 
 abutment of White 
 River bridge. 
 
 3i8c 
 
 -f- 12.062 
 10.383 
 
 - 10.383 
 
 -f 1.879 
 4.215 
 2.831 
 
 3-217 
 
 5-272 
 3.821 
 
 8.925 
 
 12.310 
 
 + 8.925 
 
 J0 
 
 + 1-879 
 
 - 3-385 
 
 
 
 296. PROFILE LEVELING. The object of this form of 
 
 leveling is to obtain a profile of the surface along an 
 established line. Hence it is necessary to determine both 
 the horizontal and the vertical distances from the initial 
 point to the points at which rod readings are taken. 
 When the line is established, stakes are driven at regu- 
 lar intervals, usually 100 feet apart. The stakes are 
 numbered in order, the first being o, so that the num- 
 ber of any stake will indicate its distance from the be- 
 ginning. The leveler is to obtain the elevation of the 
 ground at each of these stakes and at as many interme- 
 diate points as may be necessary to enable him to draw 
 a fairly accurate profile. The loo-foot stakes are called 
 Stations, and the points between the i<?o-foot stakes are 
 
ART. 7] USING THE LEVEL. 
 
 called pluses. Thus the tenth zoo-foot stake is called sta- 
 tion 10, or simply 10; and an intermediate point 20 feet 
 beyond station 10 is referred to as station io-j- 20. 
 
 297. Field Routine. The field work is very much like 
 that for differential leveling (see 293-94). The rod is 
 first held on the ground at the foot of the first stake, 
 i.e., at station o, or on a bench near the beginning of the 
 line. The number or designation of the point upon 
 which the rod is held is entered in the first column of 
 the record (see Table VI, page 261); and a brief descrip- 
 tion of the location of the point is recorded in the last 
 column. The reading itself, which obviously is a -f- 
 sight, is recorded in the second column. 
 
 A rod reading is then taken at each of the stations in 
 order, the readings being recorded in the S. column 
 under the sub-head intermediates. The work thus 
 proceeds until a point is reached which is about as far 
 from the instrument as the point of beginning. If there 
 is no firm point upon which to place the rod, the rod- 
 man drives a level-peg ( 294, fifth paragraph), which 
 he carries for the purpose, until the top of it is nearly 
 level with the surface of the ground, and holds the rod 
 upon it. This is a turning point, and the reading is 
 recorded in the S. column under the sub-head P. S. 
 If the turning point is a regular station, it needs no 
 other designation in the station column ; if it is not a 
 regular station, but is in the line, then it can be desig- 
 nated as a plus ; and if it is not in the line, it is desig- 
 nated as peg or T. P. 
 
 The instrument is next moved forward, and a sight 
 is taken upon the turning point that the height of the 
 new line of sight may be determined with reference to 
 tne first station. The work then proceeds from the 
 nirning point as from the beginning. The first sight 
 taken each time after setting up the instrument is a 
 back-sight, or -j- sight; the last sight taken before re- 
 
258 SPIRIT LEVELS. [CHAP. XV 
 
 moving the instrument is a fore-sight, or sight; and 
 ail others are intermediate sights, or simply interme- 
 diates. 
 
 The back-sights and fore-sights require the greatest 
 care, since any error in them affects all subsequent work, 
 while an error in an intermediate affects only the height 
 of that single point. When the rod is set upon the 
 ground, as is usually the case, and a target rod ( 264) 
 is used, it is sufficient to read on turning points to hun- 
 dredths of a foot and on intermediates to the nearest 
 tenth; although frequently, and generally improperly, 
 the rod is read on turning points to thousandths and 
 on intermediates at least to hundredths. With the 
 Philadelphia rod ( 267), it is rarely necessary to use the 
 target except on turning points. 
 
 The rod-man is to place the rod at the foot of each 
 stake, and at any point in the line where there is any 
 considerable change of elevation. He is to determine 
 the position of such points by stepping from the last 
 station, and this distance is recorded in the first column 
 of the record (see Table VI, page 261). For example, 
 a reading having been taken on a point 50 feet from 
 station 8, the record is made as in Table VI, see the 
 line under the record for station 8. 
 
 298. The Record. A great variety of forms of record- 
 ing the notes have been proposed. The form shown in 
 Table VI is frequently used, but a slight modification 
 of it (see 300, second paragraph) is by far the most 
 common in this country. The common form and the 
 forms shown in Tables VI and VII, pages 261 and 
 263, are examples of the method by height of instrument. 
 The form shown in Table VIII, page 264, is an example 
 of the method by differences. The last is not much used 
 in this country, but apparently it is the favorite with 
 British engineers, 
 
ART. 7] USING THE LEVEL. 259 
 
 299. Modified Method by Height of Instrument. The 
 first,, second, fourth, and fifth columns of Table VI, 
 page 261, have already been explained ( 297). Since 
 the most convenient method of expressing the relative 
 heights of a series of points is to refer them to some 
 common datum, and since it is immaterial where this 
 reference plane is chosen; therefore, if the elevation of 
 the first stake station o or B^ has not been deter- 
 mined by previous operations, it is customary to assume 
 it to be 10, 100, or 1,000 feet; that is to say, the datum 
 is assumed at this distance below the first station. The 
 elevation of the first station is assumed at some number 
 supposed to be greater than any depression on the line 
 to be leveled, so as to obviate -f- and signs in the 
 column indicating elevations. The assumed elevation 
 of the first station is written in the first line of the 
 elevations column. see Table VI. The -j- S. on the first 
 bench indicates that the line of sight was 1.763 feet 
 above B^\ and this added to the elevation of B^ gives 
 the elevation of the plane of sight, or, briefly, the height 
 of instrument, which is recorded in the third column of 
 the table. The readings on stations o, i, 2, and 3 in- 
 indicate the respective distances of these points below 
 the line of sight; and these quantities subtracted from 
 the height of instrument give their elevations. The in- 
 strument is then moved, and a sight is made back to 
 station 3. The^. S. on station 3 added to the elevation 
 of that station gives the new height of instrument, from 
 which the succeeding minus sights are subtracted as 
 before. The table and Fig. 77, page 260, will now 
 mutually explain themselves. 
 
 300. After one page of the book has been filled, the 
 computations should be checked. When correct, the 
 difference of the sums of the back-sights and fore-sights 
 on the page will be equal to the difference between the 
 first and last elevation on the page. In this connection. 
 
260 
 
 SPIRIT LEVELS. 
 
 [CHAP, xi 
 
 the word "elevation " is to be taken as applying to the 
 elevation of a station or the height 
 of the instrument. The two eleva- 
 tions to be used in checking can 
 always be determined easily by a 
 moment's consideration of the na- 
 ture of the quantities. 
 
 It is to facilitate this method of 
 checking that the minus sights on 
 turning points and on intermediates 
 should be kept in separate columns; 
 but as a rule, in ordinary practice, 
 all the minus sights are recorded in 
 a single column. With the excep- 
 tion of the separation of the minus 
 sights, the form shown in Table VI 
 is the erne commonly employed in 
 this country. Books containing rul- 
 ings suitable for either of these 
 forms are regularly sold by dealers 
 in engineering stationery, the re- 
 marks column occupying the right- 
 hand page, and the remainder of the 
 table the left-hand page. 
 
 The computations of the eleva- 
 tions of the turning points and 
 bench marks should be checked by 
 the rod-man also, who should keep 
 the notes of these points for this 
 purpose. The elevations of the 
 turning points and benches ought 
 to be computed and checked when 
 taken. The rod-man has ample time 
 to do this while the instrument is 
 
 FIG. 77. PROFILE LEVEL- 
 
 ING - being moved forward, and the leveler 
 
 can, by watching his opportunity, compute his par* 
 
USING THE LEVEL. 
 
 26l 
 
 Remarks. 
 
 1* & 1 -S& 
 o o -a c <u 
 
 "? M 3 .. R S" 
 
 o+ || - 
 
 O rt C O cS Jj 
 4-.^ !> c-S^rt 0^ 
 
 w O *O O o *- "S 
 D ** CU O m O 
 
 G m ^ *^2ioo 
 "rt o 
 
 oq fc OQ 
 
 3 
 
 .2 
 
 O O N Tj- W M 
 
 rt 
 
 V 
 
 s 
 
 8OvO TWCOO mo co in "-I *3~<x> O O O^O r^co O co 
 O^ O^ O^ o^co co co co o^ ^ o^co co co co r^co co co o^ o* 
 
 M 
 
 tfl 
 
 aoco O in N ^t Oi-ia O ON M 
 
 c 
 CO 
 
 01 m f*** in t^co t^*" O co ^ t^ co O co M ^ M 
 
 M M 
 
 1 
 w 
 
 i>. in eo M o N TJ- 
 
 O M CO VO N 1- 
 
 t> P* N co in t^ co 
 
 s O O* O O *f m 
 
 "o 
 
 ffi 
 
 t> O CO O M 
 OO^O*OO Q> 
 
 t/J 
 
 PQ 
 fc 
 
 </i 
 
 M M CI CO in 
 
 C/} 
 
 a ^ 
 
262 SPIRIT LEVELS, [CHAP. X r 
 
 without delaying the work. The elevations of the in- 
 termediates may be filled in afterward. 
 
 Finally, notice that the above tests check only the 
 computations, and in no way prove the accuracy of the 
 field work. 
 
 301. Improved Method by Height of Instrument. Some 
 engineers object to the form of record shown in 
 Table VI, page 261, (i) because the station and ele- 
 vation columns are too widely separated for conve- 
 nience in platting the profile and in referring to the 
 notes, and (2) because the station column is too far 
 from the remarks. For a form which meets these ob- 
 jections see Table VII, page 263, a form occasionally 
 used in railroad surveying. The notes are the same as 
 in Table VI, the order of the columns only being 
 changed, and therefore need no explanation. One of 
 the merits of this form is that wherever it is necessary 
 to combine two numbers by addition or subtraction, 
 they are found in adjacent columns. 
 
 302. Form of Record by Differences. The following 
 objections are sometimes urged against the method of 
 keeping level notes by the height of instrument ( 299- 
 301): (i) The elevations of the intermediates are not 
 checked at all ; (2) one of the checks on the benches and 
 turning points requires the rod-man to keep a set of 
 notes, which he can not well do with a self-reading rod; 
 and (3) the notes should contain the data necessary 
 to check them thoroughly at any time. The form 
 shown in Table VIII, page 264, meets these objections, 
 and may be used with either a target or a self-reading 
 rod, and gives a check upon every point and a double 
 check upon turning points. 
 
 The first four columns are copied from Table VI, 
 page 261, and need no further explanation. The quan- 
 tities in the column headed " Differ." are the differences 
 between the readings on successive stations. If a read- 
 
ART. 7 ] 
 
 USING THE LEVEL. 
 
 263 
 
 
 
 2 
 
 i^ 
 
 
 
 w 
 
 ^ 2 
 
 
 
 in bfl 
 
 C 
 
 
 
 ** JO 
 
 2 -8 
 
 
 
 C c5 
 
 o S 
 
 
 
 
 v c X 
 
 
 
 2 g 
 
 ^ PQ o> 
 
 
 
 ^ ^ 
 
 D *t <U 
 
 
 C/3 
 
 On c 
 
 C O 'O *O 
 
 
 J4 
 
 .0 M J 
 
 O . nt 
 
 
 g 
 
 * 7 ^ 
 
 "C^ $x < 
 
 
 & 
 
 o *2 
 
 ^ S 
 
 
 
 O 5 'Q 
 
 P o ^ ^ 
 
 
 
 Pi 
 
 
 
 
 l*H 
 
 
 
 
 GO O W 
 
 I 
 
 
 
 
 "* *> o "^ *i- " Oh 
 
 
 
 -C p ^ 
 
 r^ ti ^ o^ ^ 
 
 
 
 .2r .2* 
 
 rt O 
 
 C9 
 
 
 E E 
 
 Jz; fQ 
 
 2 S' 
 
 
 QQ Q M C^ CO ^j" lOO r^* OC O O^ O rJ? w C^ in C^) rj- 9? lT>vO 
 
 ij ^ 
 
 w S 
 
 a 
 
 > 3 
 
 y!\ 
 
 w 
 
 
 J 5 
 
 
 SJ 
 
 . fc O 
 
 H 
 
 s ! CO O s t^\Q U^ OJ *O 
 
 en O c^ ^*vO O^ O O 
 
 - | 
 
 > OH -^ 
 
 |l 
 
 0^*0 Tt* oo O u~>vQ 10 ^-< ^oo vO O^O r^* O co 
 O^OO^ oooooooo o^ O^oo oo oo r^oo oo c^ O^ 
 
 J * ^ 
 
 5 
 
 
 
 ^s* 
 
 t a M 
 
 c/j 
 
 civOco Omw*3- Ow 
 m ^oo O c* co m M r^ 
 
 MC OOOCM MO 
 
 H P"* 1> 
 
 8^ 
 
 W ^ 
 
 1 
 
 mr- inr-oo^ O en 
 
 Mt^ enOenM Tf M 
 
 S 3 
 
 
 
 
 *1 
 
 * 
 
 
 
 en !> m 
 
 en vo 
 
 S 2 
 
 S| 
 
 K 
 
 M en m 
 
 O 0^ 
 
 % 
 
 o^ 
 
 
 M 
 
 
 PH 
 
 
 r^ in 
 
 en M o M rf 
 
 
 C/3 
 
 rJ. ^ 
 
 1-1 en *o w -q- 
 a oo m r^ co 
 
 
 fc, 
 
 O\ O 
 
 O^ O O ^ m 
 
 04 M 
 
 
 OpQ 
 
 \O CM 
 
 TtN 
 
 
 c-o 
 
 C 
 
 O M 
 
 en i^ 
 
 O M t^ 
 
 
 *J ^ 
 
 o w en 
 
 CO vO OO 
 
 
 S 
 
 
 oo oo oo 
 
 
 ui 
 
 en M en 
 vo cn ^^ 
 r O"> M 
 
 r> co c<< 
 rj- en r> - 
 
 
 " 
 
 MM e 
 
 cn in 4- 
 
264 
 
 SPIRIT LEVELS. 
 
 [CHAP, xi 
 
 ing is subtracted from a succeeding one, the difference 
 is minus ; and if from a preceding one, the difference is 
 
 TABLE VIII. 
 
 FORM OF RECORD FOR PROFILE LEVELING. 
 Method by Differences. 
 
 Sta. 
 
 + s. 
 
 or 
 B. S. 
 
 - S. 
 
 Differ. 
 
 Eleva. 
 
 Check. 
 
 Remarks. 
 
 Int's. 
 
 F. S. 
 
 o 
 
 !. 7 6 
 
 2.52 
 
 
 0.76 
 
 100.00 
 
 99.24 
 
 -9-73 
 + 1.76 
 
 1 highest point 
 of bowlder, 15 
 ft to K of Sta 
 
 i 
 
 
 546 
 
 
 2.94 
 
 96.30 
 
 - 7-97 
 
 + 10 ft. 
 
 2 
 
 
 7-83 
 
 
 - 2.37 
 
 93.93 
 
 IOO.OO 
 
 
 3 
 
 1-93 
 
 
 9-73 
 
 - 1.90 
 
 92.03 
 
 92.03 
 
 
 4 
 
 
 5.00 
 
 
 3-7 
 
 88.96 
 
 + 1-93 
 
 
 5 
 
 
 7-25 
 
 
 - 2.25 
 
 86.71 
 
 0.79 
 
 
 6 
 
 
 8.32 
 
 
 -1.07 
 
 85.64 
 
 + 1.14 
 
 
 7 
 
 
 7-54. 
 
 
 + 0.78 
 
 86.42 
 
 92.03 
 
 
 8 
 
 2.17 
 
 
 0.79 
 
 + 6-75 
 
 93-17 
 
 93-17 
 
 
 +50 
 9 
 
 
 O.IO 
 
 3-72 
 
 
 + 2.07 
 -3-62 
 
 95-24 
 91.62 
 
 -9.2! 
 + 2.I 7 
 
 Summit of Yan- 
 kee Ridge. 
 
 10 
 
 
 II. OI 
 
 
 7.29 
 
 84-33 
 
 7.04 
 
 
 2? a 
 
 
 7-3 2 
 
 
 + 3-69 
 
 88.02 
 
 
 Nail on root of 
 
 
 
 
 
 
 
 
 
 II 
 
 3-47 
 
 
 9.21 
 
 - 1.89 
 
 86.13 
 
 86.13 
 
 tree, 50 ft. to R. 
 
 12 
 
 
 3-40 
 
 IO.2O 
 
 
 + 0.07 
 - 6.80 
 
 86.20 
 79.40 
 
 + 3-47 
 -0.83 
 
 Bottom of Bone 
 Yard Branch, 10 
 
 13 
 
 
 3.00 
 
 
 + 7.20 
 
 86.60 
 
 + 2.6 4 
 
 ft. wide, water 
 
 
 
 1.72 
 
 
 + 1.28 
 
 87.88 
 
 86.13 
 
 3 ft. deep. 
 
 peg 
 
 5.38 
 
 
 0.83 
 
 + 0.89 
 
 88.77 
 
 88.77 
 
 
 is 
 
 
 4.II 
 
 (4-11) 
 
 + 1.27 
 
 90.04 
 9.96 
 
 
 
 
 
 
 - 2<t 67 
 
 
 
 
 
 
 
 
 + 14.71 
 
 
 100.00 
 
 
 
 
 
 
 9.96 
 
 1 
 
 
 
 
 plus. The elevations are determined by applying these 
 differences, each with its proper sign, to the elevation of 
 the preceding station. 
 
ART. 7] USING THE LEVEL. 265 
 
 The work is checked, as shown in the column headed 
 " Check," by taking the difference between the B. S. and 
 F. S. for each setting of the instrument, and applying 
 it to the elevation of the preceding turning point, which 
 should give the elevation of the succeeding turning 
 point. Checking the elevation of the turning point 
 checks the elevation of the preceding intermediate 
 points. Each page may be checked by adding the B. S. 
 and F. S., and applying their difference to the last ele- 
 vation, in essentially the same way as in the other 
 forms. In Table VIII the elevation of station 15 was 
 regarded as the "last elevation," and consequently the 
 reading on that station was considered as the last F. S., 
 which it would be if the work stopped at station 15. 
 
 Notice that by this method each point is checked 
 once and the turning points twice. Of course this 
 method of keeping the notes requires more computing 
 than the others, but it also more thoroughly checks the 
 work. If rapid work is required, the quantities in the 
 "difference" and "elevation" columns need not be 
 worked out in the field, in which case the turning points 
 will be checked only once. 
 
 303. Drawing the Profile. A profile is a vertical sec- 
 tion of the line leveled, showing the relative heights 
 and distances of the various points at which levels were 
 taken. It is really a graphical representation of the 
 station and elevation columns of the level notes, the first 
 being the horizontal and the latter the vertical co-ordi- 
 nates. The ground may be assumed to slope uniformly 
 between the points at which levels were taken, for they 
 were taken not only at the regular stations, but also at 
 every considerable change of elevation; consequently, a 
 line joining the points obtained by plotting the level 
 notes, represents in detail the rise and fall of the line, 
 as seen in a side view. 
 
 The profile may be made by drawing a horizontal 
 
266 SPIRIT LEVELS. [CHAP. XI 
 
 line to represent the datum, and laying off along it, to 
 any convenient scale, the distances or stations from the 
 first column of the field notes. The elevations corre- 
 sponding to these points are next to be laid off at right 
 angles to the datum line and above it. As the rise and 
 fall of a line is always very small in proportion to its 
 length, it is usual to make the vertical scale much 
 greater than the horizontal. By thus employing two 
 different scales, the irregularities of the surface are 
 made more apparent to the eye, and also any subse- 
 quent use of the profile is rendered much easier and 
 more accurate. 
 
 304. In practice, engraved profile paper is generally 
 used, which is ruled in rectangles, to which any arbi- 
 trary values may be assigned. Three styles of profile 
 paper are regularly sold by stationers. They are dis- 
 tinguished as A, B, and C, according to the scale of the 
 ruling. The first has four horizontal and twenty verti- 
 cal spaces to the inch ; the second, four horizontal and 
 thirty vertical; and the third, five horizontal and twenty- 
 five vertical. The paper is in rolls 10 yards long and 
 20 inches wide, and may be had printed on either 
 opaque or translucent paper. 
 
 It is customary to make one space horizontally equal 
 to one station, and one space vertically equal to one 
 foot. In using engraved profile paper it is not neces- 
 sary to plot the datum, but only to select a horizontal 
 line near the middle of the paper and mark it, say, 100 
 feet, and number the other lines accordingly. The 
 vertical lines being numbered to correspond with the 
 stations, the points are easily and speedily plotted. A 
 fine line drawn exactly through the plotted points com- 
 pletes the profile. 
 
 If, owing to the rise or fall of the ground, the surface 
 line runs off the paper at the top or bottom, it is only 
 necessary to change the numbering of the horizontal 
 
ART. 7] USING THE LEVEL. 267 
 
 lines and commence again at the other edge. The be- 
 ginner is liable to plot the readings on benches, turning 
 points, and plus stations as though they were regular 
 stations. The readings on benches and turning points 
 are not plotted, except when they are stations as well. 
 The plus stations are plotted in their proper relative 
 position, the distance on the profile being estimated. 
 
 305. PRECISE LEVELING. Precise, or geodesic, level- 
 ing is differential leveling ( 293) performed with the 
 utmost care. Leveling of the greatest accuracy possi- 
 ble is required in surveys to determine the figure of 
 the earth, the relative elevations of the surfaces of the 
 great lakes, the relative elevation of the water on 
 opposite sides of an isthmus, the fall of the great rivers, 
 etc. 
 
 306. Methods. There are two principal methods in 
 leveling, according to the sequence of the instrument 
 and rod, which may be called single and double leveling. 
 Each may be performed with one rod or with two, 
 which gives rise to four methods, according to the 
 sequence of the instrument and the rod. There are 
 also two methods of duplicating the work: viz., dupli- 
 cate leveling once over the line, and duplicate leveling 
 twice over the line. In all, then, there are six methods 
 of leveling. The first five are represented in Fig. 78, 
 and the sixth is simply the first method applied a 
 second time. 
 
 In Fig. 78, 7, , 7 2 , 7 3 , etc., indicate successive positions 
 of the instrument; A l9 A y , etc., successive positions of 
 one rod ; B^ , 7? 2 , etc., successive positions of the other 
 rod; and 7, //, ///, etc., the several methods of leveling. 
 
 /, single leveling with one rod, is the method employed 
 in ordinary leveling ( 293 and 296). 
 
 //, single leveling with two rods, is more accurate and 
 also more rapid than leveling with a single rod. By 
 the use of two rods the back-sight and the fore-sight 
 
268 
 
 SPIRIT LEVELS. 
 
 [CHAP, xi 
 
 may be taken in quick succession, which saves time. 
 Since less time intervenes between the sights, there is 
 less liability of change in the plane of sight; and hence, 
 for this reason also, this method is more accurate than 
 
 IT \ 
 
 A 
 
 I 
 
 /ft 
 
 A. 
 i 
 
 I. A t Z t A) Ij A* 
 
 JI 1 
 
 ~j( 
 
 /i\ 
 
 M 
 
 A, 
 
 i. B, /, , ft "a 
 
 M fr M 
 
 J/T 1 
 
 EZZOUJ 
 
 A 
 
 A. 
 
 t' 
 
 X, 4. /)j ^ >. ^ Z) ^ 
 
 1 
 
 
 m 
 
 
 
 B 
 
 /l\ 
 
 I, 
 * 
 
 FIG. 78. METHODS OF LEVELING. 
 
 the preceding one. This is the method used on the 
 U. S. Lake Survey * and on the Mississippi River Sur- 
 vey.f 
 
 ///, double leveling with one rod, affords a perfect check 
 against errors of adjustment and observation, since the 
 difference of the rod readings for the two fore-sights 
 should be the same as the difference for the two back- 
 sights following. 
 
 IV, double leveling with two rods, combines all the ad- 
 vantages of the second and third methods. The nu- 
 merals adjacent to the rods show the order in which 
 
 * Chief Engineer's Report, U. S. A., for 1880, pp. 2366 and 2429. 
 t Mississippi River Commission Report for 1881, p. 50. 
 
ART. 7] USING THE LEVEL. 269 
 
 they are sighted upon. This is the method employed 
 on the U. S. Coast and Geodetic Survey.* On level 
 ground or where the slope does not interfere, the dis- 
 tances A^A^, B^A^, etc., are about 220 meters (720 feet), 
 and the distances B^A^, A^B^, etc., are about 20 meters 
 (66 feet). 
 
 V, duplicate leveling once over the line, is simply dupli- 
 cating the work without the labor of going twice over 
 the line. Two rods are used, placed as in the figure. 
 After having observed upon A l and B^ the instrument 
 is pulled up and re-set a little to one side and the two 
 rods are sighted upon again. This method duplicates 
 the work as far as instrumental errors are concerned, 
 but is not as perfect a check as either the third or the 
 fourth method. 
 
 A sixth method, duplicate leveling twice over the line, 
 consists in an independent re-leveling of the line. The 
 most reliable results are obtained by repeating the work 
 in a direction opposite to that in which it was first done 
 
 (315). 
 
 307. Field Routine. After having planted the tripod 
 firmly and leveled the instrument, read both ends of 
 the bubble estimating the fraction of a division. If 
 there is a milled-head screw under one end of the tele- 
 scope, the bubble can easily be brought to the middle 
 each time ; but if there is not, it is better to bring it 
 nearly to the middle and apply a correction. It is not 
 enough to read only one end, since the bubble is liable 
 to change its length with a change of position or of 
 temperature. 
 
 Next read the position of the three wires on the rod; 
 and then read the bubble again for a check, and also to 
 detect any change. Reverse the level end for end, and 
 turn the telescope 180 about its optical axis, and repeat 
 
 * Report for 1879, P- 2O ^- 
 
270 SPIRIT LEVELS. [cHAP. XI 
 
 the operations as above. The first reversal eliminates 
 any inequality in the lengths of legs of the striding 
 level; and the second eliminates any error of collimation. 
 The mean of the several readings must be corrected for 
 the difference in position of the bubble, and for inequal- 
 ity of the collars ( 290). 
 
 Instead of reversing the level and telescope at the 
 same time, the observations are sometimes made as fol- 
 lows: read upon the rod, reverse the level, and read 
 again; reverse the telescope and read a third time; then 
 reverse the level and make a fourth reading. The first 
 method is the better. 
 
 308. On the Coast Survey the method of observing 
 differs slightly from that described above.* Errors of 
 level and of collimation are eliminated by reversing both 
 the bubble and the telescope on each back-sight and 
 fore-sight. Each observation is of a single wire on a 
 target. The target is set but once for each station, the 
 differential quantities being read by the micrometer 
 under the eye end of the telescope. This seems not to 
 be as good a method as the above; there are two objec- 
 tions to it, aside from the time and labor required to 
 set the target. First, there is no sufficient check 
 against errors in reading the positions of the target. 
 Second, the micrometer is read for a central position of 
 the bubble, the telescope is then moved to bisect the 
 target, and the screw is read again; therefore there is 
 no check on the stability of the instrument. 
 
 309. The Record. It is hardly wise to attempt, in this 
 volume, an explanation of the method of recording the 
 notes and making the computations. For a full expla- 
 nation of the method employed on the U. S. Coast and 
 Geodetic Survey, see the annual report of that Survey 
 
 * U. S. Coast and Geodetic Survey Report for 1879, PP- 2 6 an d 20 7' 
 
ART. 7] USING THE LEVEL. 271 
 
 for 1879, PP- 2I an( ^ 2II > or report for 1880, p. 139. 
 For a full explanation of the form recommended by the 
 U. S. Lake Survey, see annual report of the Chief of 
 Engineers, U. S. A., for 1880, Part III, p. 2431. 
 
 310. SOUKCES OF ERKOK. Probably in no other kind of 
 surveying, with the possible exception of chaining, is it 
 as important to distinguish between compensating and 
 cumulative errors ( 18) as in leveling. 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 care necessary to secure 
 a given degree of accuracy and particularly is this true 
 in leveling. 
 
 For convenience in discussing them we will classify 
 errors of leveling as (i) instrumental errors, (2) rod 
 errors, (3) errors of observation, (4) personal errors, 
 (5) errors in recording and computing, and (6) errors 
 of curvature and refraction.* 
 
 311. Instrumental Errors. The principal instrumental 
 error is that due to the line of sight's not being parallel 
 to the level. This may be caused either by imperfect 
 adjustment, or by unequal size of rings, or by both. If 
 the telescope slide is not straight, or does not fit closely, 
 
 * Owing to the unequal density of the earth's crust, the plumb-line does 
 not always point to the center of the earth; and this gives rise to another 
 source of error, i.e., the local deflection of the plumb-line. This source of 
 error has been suggested as the explanation of the discrepancy in the elevation 
 above sea-level of a point at St. Louis, Mo., as determined by two lines of 
 precise levels one run from mean-tide of the Atlantic Ocean near New York 
 City, and the other from mean-tide of the Gulf of Mexico near Mobile, Ala. 
 This source of error is too complicated for discussion here. For a brief refer- 
 ence to this subject, see U. S. Coast and Geodetic Survey Report for 1882, p. 
 
 5i7- 
 
 If a line of levels were run fron mean-tide at the equator to mean-tide at 
 the pole, what would the difference of elevation be? If the difference of ele- 
 vation were determined with a mercurial barometer, what would it be? If 
 with an aneroid barometer? 
 
272 SPIRIT LEVELS. [CHAP. XI 
 
 or does not move in the axis of the rings, it may produce 
 an error. Of course the instrument must be focused so 
 as to eliminate parallax. 
 
 All of the preceding errors are compensating, and the 
 elevations of turning points may be made entirely inde- 
 pendent of them, whatever their value, by always plac- 
 ing the instrument midway between the turning points. 
 
 312. Rod Errors. The principal rod error is caused by 
 not holding the rod vertical. It is greater for a large 
 rod reading than for a small one. It is compensating 
 on turning points, and may be entirely eliminated by 
 attaching a level or short plumb, or by waving the rod 
 ( 294, third paragraph). In waving the rod care must 
 be taken that the face is not lifted by the rod's resting 
 on its back edge when revolved backwards. To obviate 
 this source of error and save the time consumed in wav- 
 ing the rod, a " corner target " is sometimes used. This 
 consists of an ordinary target (Fig. 69 or Fig. 71, 
 page 231), whose right and left halves are in planes at 
 right angles to each other. The corner of the rod and 
 of the target face the instrument, and if the rod is not 
 vertical the central portion of the horizontal line of the 
 target will be either above or below the horizontal cross 
 hair. Another device for accomplishing the same pur- 
 pose is a target similar to Fig. 69 or Fig. 71, in which 
 the right and left thirds are, say, 2 inches behind the 
 plane of the middle third. If the rod is not vertical, 
 the horizontal line on the middle third of the target will 
 be above or below that on the side portions. For 
 several reasons neither of these devices is as rapid or as 
 accurate as a short plumb-line or level either a disk 
 level, or two level vials at right angles to each other 
 attached to the rod. 
 
 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 
 
ART. 7] USING THE LEVEL. 273 
 
 read, is a source of error. The target itself may slip, 
 but this is not so probable, because of its inferior weight. 
 
 Another source of error is the settling of the turning 
 point, due, in coarse 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 to rest the rod upon a triangular 
 plate having the corners turned down slightly ( 294, 
 fifth paragraph). Whatever 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. With the numer- 
 ous home-made self-reading rods now in use, this is a 
 much more important source of error than with the rods 
 made by professional instrument makers. 
 
 " An important source of error in spirit leveling, and 
 one very commonly overlooked, is the change in the 
 length of the leveling rod from variations of tempera- 
 ture. It is quite possible that errors from this source 
 may largely exceed the errors arising from the leveling 
 itself."* 
 
 313. Errors of Observations. The principal error of ob- 
 servation is in reading the position of the bubble. Even 
 if the bubble is kept in the middle, it is nevertheless 
 read. Every leveler should know the error on the rod 
 corresponding to a given difference of reading of the 
 bubble, since he then knows how accurately he must 
 read the bubble for the particular accuracy aimed at. 
 
 A small error arises from the fact that the adhesion 
 of the liquid to the sides of the glass tube prevents the 
 bubble from coming precisely to its true point of equi- 
 librium (see the third paragraph of 257). Even though 
 it may finally arrive at the true point, it is liable to be 
 
 * Wright's Adjustments of Observations, p. 372. 
 
274 SPIRIT LEVELS. [CHAT. XI 
 
 read before it has stopped moving. The bubble should 
 be re-read after the target is nearly adjusted, or, with a 
 self-reading rod, after the reading has been taken and 
 before the rod-man is signaled to move on. 
 
 Another small error is due to the effect of the sun in 
 raising one end of the telescope by the unequal expan- 
 sion of the^different parts of the instrument. In ordi- 
 nary leveling operations, the bubble is first brought to 
 the middle and then the target is sighted in, leaving an 
 interval for the sun to act. The error is greatest in 
 working toward or from the sun. Ordinarily it is cu- 
 mulative; for on the back-sight one wye is expanded, 
 which elevates the line of sight, while on the fore-sight 
 the other wye is expanded, which depresses the line of 
 sight, the two errors affecting the difference of elevation 
 in the same way. The error on the fore-sight is farther 
 increased by the cooling of the wye which was expanded 
 during the back-sight. The error due to the sun is 
 always small, and can be nearly eliminated by noticing 
 the position of the bubble after setting the target; and 
 can be still farther reduced by shading the instrument, 
 although " the Indian Geodetic Survey proved conclu- 
 sively that the error was appreciable, even when the 
 instrument was shaded." 
 
 Such a seemingly trivial thing as the unequal heating 
 of the bubble tube produces an appreciable effect. The 
 bubble always moves toward the warmer portion of the 
 tube, owing to a change in adhesion of the fluid to the 
 glass tube; and therefore, in accurate work, the leveling 
 instrument should be protected from the direct rays of 
 the sun, and the bubble tube should never be touched 
 with the finger nor be breathed upon. 
 
 If the observer is compelled to change his position 
 after reading the level and before sighting the telescope, 
 his movement about the instrument may cause a change 
 in the inclination of the telescope. " In some trials in 
 
ART. 7] USING THE LEVEL. 275 
 
 France, the inclination from this cause (one leg being 
 in the line of sight) varied from two to one hundred 
 seconds of arc according as one leg rested on the pave- 
 ment or on vegetable soil." * This error is a minimum 
 when two legs are placed in a line parallel to the 
 route and the third at right angles to it. Sometimes, to 
 eliminate this error, one man reads the bubble while a 
 second sights the telescope; but this is objectionable, 
 since it requires two skillful men and also divides the 
 responsibility. It is better to provide the instrument 
 with a mirror or a prism so that the bubble may 
 be read from the position from which the telescope 
 is sighted; for when the bubble is almost constantly in 
 motion, the same man should see it and the cross hairs 
 at the same time. 
 
 It has been found that appreciable errors are caused 
 by the settling of the instrument on its vertical axis, 
 and by the settling of the tripod legs into the ground. 
 In spongy or clayey soil the tripod legs are some- 
 times gradually lifted up. These errors, though small 
 in themselves, are more important than is generally 
 supposed, inasmuch as they are cumulative ( 18). They 
 can be eliminated by re-running the line in the opposite 
 direction. 
 
 If the reading is made on either side of the vertical 
 hair, there is a possibility of error, owing to the horizon- 
 tal hair's not being horizontal; and with wye levels not 
 provided with a means of preventing the rotation of the 
 telescope in the wyes, this possibility becomes a probabil- 
 ity. Any device which insures the horizontality of the 
 horizontal hair increases the rapidity and accuracy of 
 the work. This error is compensating. 
 
 The inaccuracy of telling when the hair exactly 
 covers the center of the target is a source of slight error, 
 
 * Wilfred Airy, in Proc. Inst. of C.E., Vol. 78, p. 458. 
 
276 SPIRIT LEVELS. [CHAP. XI 
 
 but not so slight as many think. In setting a quadrant 
 level target, the instrument having a telescope magni- 
 fying thirty-five times and a bubble tube with a radius 
 of 145 feet, the average probable error ( 2 of Appendix 
 III) for a class of fifteen was 1.4 thousandths of a foot 
 at 100 feet and 2.25 thousandths of a foot at 300 feet 
 (see Tables I and II, Appendix III). In running a line 
 of levels this error would be compensating. Owing to 
 this source of error, the difference in accuracy between 
 a target rod and a self-reading rod is not so great as 
 their difference in precision. 
 
 With a target rod, errors of one foot, one tenth, etc., 
 are not uncommon. In double and duplicate leveling 
 ( 36) these errors are easily discovered and corrected. 
 In ordinary leveling ( 293 and 296) they may be elimi- 
 nated by having the rod-man and instrument-man read 
 the rod independently and afterwards compare notes. 
 In single differential leveling with one rod both men 
 can read the rod without materially delaying the work, 
 by the rod-man's reading the rod and recording it in a 
 book carried for that purpose, and carrying the rod to 
 the leveler if it be a back-sight or waiting for the com- 
 ing of the leveler if it be a fore-sight, when the observer 
 also reads the rod and records it in his book, after which 
 the two records are compared. But in profile leveling 
 this can not be done; and the only way to check the 
 work is to duplicate it. With a self-reading rod the 
 liability of this class of errors is somewhat reduced by 
 reading three hairs. 
 
 314. Personal Errors. The errors previously described 
 are liable to occur with any observer. They are due 
 chiefly to the instruments and to the nature of the work, 
 and would probably not materially differ for equally 
 skilled observers. We come now to a class of errors 
 which depend mainly upon inaccuracies peculiar to the 
 individual. One observer may read a target higher or 
 
ART. 7] USING THE LEVEL. 277 
 
 lower than another of equal skill; or in reading the posi- 
 tion of the bubble he may have peculiar views as to 
 what constitutes the end of the bubble; or he may ha- 
 bitually read the bubble so as to get a distorted view of 
 it through the glass tube. With skillful observers, all 
 such errors are quite small and generally cancel them- 
 selves. In fact, the errors here classed as personal are 
 possible rather than demonstrated as actually occurring; 
 and yet there is nothing more certain than that in any 
 series of accurate observations there is a difference be- 
 tween the results obtained by different individuals. 
 This difference is known as the personal equation. In 
 long lines of accurate leveling, it has been found that 
 each man's way of performing the several operations has 
 an appreciable effect upon the final result. 
 
 315. It is a curious fact, but one abundantly verified, 
 that when lines are duplicated in opposite directions, 
 the discrepancies tend to one sign and increase with the 
 distance. This subject has been much discussed, and 
 various explanations of the fact have been offered; as 
 settling of instrument, settling of turning point, dis- 
 leveling effect of the sun, unequal illumination of the 
 target on back-sights and fore-sights, unequal illumina- 
 tion of the two ends of the bubble, effect of refraction 
 in reading the bubble, the change of the position of the 
 observer to read the bubble (see 313, fifth paragraph), 
 and personal bias in reading the target or bubble; but 
 none of these reasons are entirely satisfactory. These 
 discrepancies vary with different observers; are not even 
 constant for the same observer; are nearly proportional 
 to the distance; and seem to be independent of the 
 nature of the ground, the direction in which the work is 
 done, the season, or the manner of supporting the rod.* 
 
 * For a valuable discussion of this source of error, together with numerical 
 results derived from actual work, see Report of the Mississippi River Commis- 
 sion for 1883, pp. 141-62 ; or Report of Chief of Engineers U. S. A., for 1884, 
 PP- 2547-68, 
 
278 SPIRIT LEVELS. [CHAP. XI 
 
 The effect of this class of errors maybe eliminated by 
 each observer's duplicating his work in the opposite 
 direction under as nearly the same conditions as pos- 
 sible. The accuracy is increased by leveling alternate 
 sections in opposite directions, as is done in India; 
 and it may be still further increased by reading the 
 back-sight first each alternate time the instrument is 
 set up. 
 
 316. Errors of Recording and Computing. Such blun- 
 ders as recording the fore-sight in the back-sight column 
 and, vice versa, the back-sight in the fore-sight column, 
 although the result of gross carelessness, do neverthe- 
 less occur. To check against such errors, the rod-man 
 and the leveler should read the rod independently, as 
 explained in the last paragraph of 313. In profile 
 leveling, errors of this class are easily discovered, if the 
 rod-man keeps the notes and works out the elevations 
 of turning points and benches; and in any case, such 
 errors are more likely to be discovered if the elevations 
 are worked out immediately after the observations are 
 taken. 
 
 Errors of computation, and the method of checking 
 them, have already been discussed, incidentally, in 
 299-302, which see. 
 
 317. Errors of Curvature and Refraction. The object 
 of leveling is to find the distance that one point is above 
 or below a level surface ( 292) passing through some 
 other point. The line of sight of a properly adjusted 
 leveling instrument is a horizontal line, i.e., a tangent 
 to a level line. Therefore, if the two points sighted at 
 are not equidistant from the instrument, the difference 
 between the rod readings will not be the true difference 
 of level, owing to the curvature of a level surface, i.e., 
 to " the curvature of the earth." The difference between 
 a level and a horizontal line can be computed ( 319); 
 and hence if the length of sight is known, the effect of 
 
ART. 7] USING THE LEVEL. 
 
 curvature can be eliminated by applying a correction. 
 The error is compensating, and may be entirely elimi- 
 nated by setting the instrument midway between turn- 
 ing points. 
 
 318. Owing to the refraction of the atmosphere the 
 beam of light from the target to the telescope is slightly 
 concave downwards, and hence the line of sight is not 
 a truly horizontal line. The difference between a hori- 
 zontal line and the true line of sight can be computed 
 for an average condition of the atmosphere ( 319) ; and 
 therefore if the length of sight is known, and if the air 
 is in its normal condition, the effect of refraction can 
 be eliminated by applying a correction. 
 
 But the atmosphere is not always in its normal con- 
 dition; and hence if there is abnormal refraction or a 
 change of refraction between sights, there may be re- 
 sidual errors of refraction, even though the correction 
 for mean refraction be applied. As refraction varies 
 greatly with the nature of the surface over which the 
 light passes and also with the distance from the surface, 
 the effect of the refraction may vary considerably. The 
 most serious effect of variable refraction is the tremu- 
 lousness or "boiling" of the atmosphere caused by the 
 innumerable small currents of air of different tempera- 
 tures or densities which exist when the atmosphere and 
 the earth differ much in temperature. The beam of 
 light in proceeding from the target to the telescope 
 passes alternately through denser and rarer media, each 
 of which produces a slight refraction of the ray, thus 
 causing the target to " dance " and making it difficult 
 to determine just when it is properly bisected by the 
 cross hair. When this condition exists the only remedy 
 is to shorten the length of sight, or wait for better 
 atmospheric conditions. The atmosphere is usually in 
 the best condition for seeing just before sunrise and a 
 
SPIRIT LEVELS. 
 
 [CHAP, xi 
 
 little while before sunset, although the refraction is then 
 greater. A cloudy day is better than a clear one. 
 
 Ordinarily this error is compensating; and will usu- 
 ally be eliminated by setting the instrument midway 
 between the turning points. 
 
 319. Correction for Curvature and Refraction. To 
 compute the correction for curvature of the earth, let 
 AD, Fig. 79, represents the horizontal line ; AB the 
 level line ; and AE the radius of the earth. Then DB 
 is the correction for curvature. By geometry AD* = 
 DB(2AE -f DB]. Dropping DB from the parenthesis, 
 as it is very small in comparison with zAE, and repre- 
 senting the length of sight by k and the radius of curva- 
 
 k* 
 ture of the earth by p, BD = . If BD is expressed in 
 
 feet and k in miles, the correction for curvature becomes 
 BDit. = 0.66 7/fc a miles ..... (i) 
 
 To compute the correction for refraction, notice that 
 D' , Fig. 79, is the true position 
 of the target and D its apparent 
 position. Hence DD' is the 
 correction sought. The ratio of 
 the refraction angle, DAD' , to 
 the angle at the center of the 
 earth, AED, is known as the 
 coefficient of refraction, which 
 we will represent by m. 
 
 k 
 
 AED, in sec. of arc, = : --- r . . 
 psm i" 
 
 DD' = 
 
 FIG. 79. 
 
 = kD'AD"\.zm" = 
 
 The ordinary values of m vary between 0.06 and 0.08, 
 
ART. 7] USING THE LEVEL. 281 
 
 The value generally employed in computing the correc- 
 tion for refraction is 0.07. Then 
 
 D'D ft. 0.09/P miles ..... (2) 
 Combining equations (i) and (2), we have 
 
 BD' ft. = 0.57/fc 2 miles ..... (3) 
 
 In general, the total correction for curvature and re- 
 fraction, to be applied to the observed reading, is 
 
 ft* 
 BD' = BD DD' = (i - 2m) = 0.000,000,020,45^. 
 
 If k 220 ft, BD' o.ooi ft.; for k = 300 ft., BD' = 
 0.002 ft. Notice that the correction varies as the square 
 of the length of sight. Numerous tables have been 
 computed which give this correction directly for the 
 different lengths of sight ; or tables can be prepared 
 which will give the difference of the correction with the 
 difference of length of sight for an argument. 
 
 320. LIMITS OF PRECISION. The probable error per 
 unit of distance is generally adopted as a convenient 
 measure of the precision reached. According to the 
 theory of probabilities, the final error of a series of obser- 
 vations, affected only by accidental errors, will vary as 
 the square root of the number of observations. In 
 leveling, a method should be adopted which will elimi- 
 nate all cumulative errors ; and therefore, since only 
 compensating errors remain, the final error of leveling 
 a number of units of distance is assumed to vary as the 
 square root of the distance. 
 
 This assumption would be true if only compensating 
 errors remained uncorrected, and if the number of ob- 
 servations were strictly proportional to the distance 
 leveled, i.e., if the length of sight was constant and if 
 
282 SPIRIT LEVELS. [CHAP. XI 
 
 the inclination of the surface of the ground leveled over 
 was the same (see Table IX, page 283). Since it is 
 improbable that all cumulative errors will be entirely 
 eliminated, that part of the final error which is due to 
 cumulative errors will vary as the distance leveled. It 
 has frequently been noticed that, considered individu- 
 ally, the errors of a number of short lines were well 
 within the limits which were prescribed to vary as the 
 square of the distance ; yet when the sum for several 
 lines were considered, the total discrepancy would exceed 
 the limit. In other words, the error is not strictly pro- 
 portional to the square root of the distance. One part 
 of the error is proportional to the square root of the 
 distance, and another portion varies nearly as the dis- 
 tance; hence, the shorter the distance, the easier to attain a 
 limit prescribed to vary as the square root of the distance. 
 
 If the error was determined by duplicating the work 
 in the same direction, and especially if at the same time, 
 by methods III, IV, or V, Fig. 78, page 268, the differ- 
 ence will be the apparent error, and necessarily be too 
 small. The result obtained by the adjustment of a net 
 of lines by the method of least squares affords the best 
 means of determining the degree of precision. 
 
 321. According to the Geodetic Association of Europe, 
 levels of precision executed of late years in Europe show 
 that the probable error of a line of levels of precision 
 should never exceed 5 mm. ^distance in kilometers 
 (0.0208 ft. Smiles); that 3 mm. 4/dist. in kilometers is 
 tolerable, 2 mm. Vdist. in kilometers is a fair average, 
 and i mm. Vclist. in kilometers is high precision.* The 
 Coast Survey requires 5 mm. 4/2 dist. in kilometers 
 (0.030 ft. Vmiles). The Mississippi River Commission's 
 limit is 5 mm. 4/dist. in kilometers (0.021 ft. Vmiles). 
 
 Of late years the Coast Survey's and Mississippi 
 
 * Report of U. S. Coast and Geodetic Survey for 1882, page 522. 
 
ART. 7] 
 
 USING THE LEV2L. 
 
 283 
 
 River Survey's work are considerably within the limit 
 of 2 mm. Vdist. in kilometers. Table IX shows the 
 degree of accuracy attained in precise leveling on 
 the national surveys of Great Britain, India, and Switz- 
 erland.* 
 
 TABLE IX. 
 
 DATA SHOWING THE DEGREE OF ACCURACY ATTAINED IN PRECISE 
 LEVELING, AND THE EFFECT OF DIFFERENT INCLINATIONS OF 
 JHE GROUND. 
 
 Reference No. 
 
 KIND OF GROUND. 
 
 AVERAGE DIFFERENCE PER 
 MILE BETWEEN Two 
 OBSERVERS. . 
 
 Great 
 Britain. 
 
 India. 
 
 Switzer- 
 land. 
 
 I 
 2 
 
 3 
 4 
 5 
 
 Nearly level, with very favorable 
 weather . 
 
 foot. 
 0.0230 
 0.0238 
 0.0379 
 0.0566 
 
 foot. 
 0.0142 
 0.0168 
 0.0208 
 0.0350 
 
 foot. 
 0.0125 
 0.0148 
 0.0183 
 0.0308 
 0.0416 
 
 Slightly undulating gradients not ex- 
 
 Gradients entirely between I in 100 
 and i in 20 . 
 
 Gradient entirely between I in 20 and 
 i in 10 
 
 Steep and rough ground gradients 
 frequently steeper than i in 10 . . 
 
 
 
 To attain the preceding limits requires skillful ob- 
 servers, the best instruments, and plenty of time. Ordi- 
 narily there are not more than three or four hours of the 
 day in which this class of work can be done; and as a 
 general average not more than one or two miles can be 
 run in a day (see 323). 
 
 322. It is impossible to establish a limit for work less 
 accurate than the best, since the conditions under 
 which it may be done are too diverse. However, the 
 difference in precision between ordinary leveling and 
 precise leveling, is not as great proportionally as the 
 difference in care and time given. A little increase in 
 
 * Wilfred Airy in Proc. Inst. of C. E., Vol. 44, page 181. 
 
SPIRIT LEVELS. [CHAP. XI 
 
 accuracy costs a very great increase of effort. Results 
 of leveling of apparently greater accuracy than the fore- 
 going are often given,* but an occasional accurate result 
 probably more largely due to good fortune than good 
 management gives no indication as to what results 
 may be regularly expected. 
 
 Regularity of result and evenness of error is of more 
 importance than occasionally a small disagreement. 
 Naturally, it is usually the latter that is reported. 
 
 A line of "ordinary" levels was run on the bank of 
 the Mississippi River, and checked upon the benches of 
 the precise levels, with an average error of 0.034 feet 
 Vdist. in miles. f The conditions under which this work 
 was done were about the same as those of a prelimi- 
 nary railroad survey, but it is probably more accurate 
 than such surveys usually are. 
 
 In some of the branches of the A., T., & S. F. R. R., 
 the instructions were to re-run the line whenever the 
 difference between the levels on construction and loca- 
 tion was 0.03 foot, between benches about 2,000 feet 
 apart. This is equivalent to limiting the maximum 
 admissible error to 0.048 feet tMist. in miles. 
 
 In the topographical survey of the city of St. Louis, 
 Mo., " the limit of error allowed was 5 millimeters 
 V'dist. in kilometers (0.0208 feet 4/dist. in miles). The 
 average closure was 0.013 ^ eet ^dist. in miles. The 
 probable error in the determination of a single mile of 
 the work was o.ooi foot." J 
 
 323. Speed. The amount of work that an observer 
 
 * For example, several text-books contain the statement: "A French 
 leveler contracts to level the bench marks of a railroad survey to within 0.002 
 ef a foot per mile." Compare this with 321 and also with Tables I and 
 II of Appendix III. 
 
 t Report of Mississippi River Commission for 1882, p. 2269 ; or Report of 
 Chief of Engineers, U. S. A., for 1883, p. 2269. 
 
 J The Technograph, No. 5 (1890-91), p. n. 
 
ART. 7] USING THE LEVEL. 285 
 
 should do in a day can not be stated definitely. It 
 depends upon the accuracy required, the power and 
 delicacy of the instrument, the method pursued, the 
 ground, and very largely upon the atmospheric condi- 
 tions. For the very best work, not more than three or 
 four hours can be utilized, even in clear weather. The 
 average daily run for several seasons on the Mississippi 
 River, using a Kern level and method II ( 306), was 
 one and a half miles a day for the entire season, and 
 two and a half miles for the days on which work was 
 actually done.* On the U. S. Lake Survey, with the 
 same instrument and method, the distance was about 
 two miles a day for the days on which leveling was 
 done.f On the Swiss levels of precision, 3 kilometers 
 (1.8 miles) along railroads, and 2 kilometers (1.2 miles) 
 along highways in the plains, was considered a fair 
 day's work.J 
 
 Professor J. B. Johnson, who has had large experi- 
 ence 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 thirty miles per month, with no greater error 
 than 0.05 feet Vdist. in miles, or with a level of precision 
 and speaking rod, do the same work with a limit of 0.02 
 feet i^dist. in miles." 
 
 324. PRACTICAL HINTS. For a number of points ap- 
 plicable in leveling, particularly to setting the tripod, 
 see 128. 
 
 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 finger. 
 
 * Report of Chief of Engineers, U. S. A., for 1884, p. 2462. 
 t Professional Papers Corps of Engineers, U. S. A., No. 24 Primary 
 Triangulation U. S. Lake Survey, pp. 597-99. 
 % Proc. Inst. of C. E., Vol. 78, p. 456. 
 Jour. Association of Engineering Societies, Vol. 2, p. 160. 
 
286 SPIRIT LEVELS. [CHAP. XI 
 
 Instruments are usually provided with sun-shades to 
 prevent trouble from the sun's shining into the tele- 
 scope. If the metallic shade is not at hand, make one 
 by rolling up a piece of paper and gumming, pinning or 
 tying it together, or springing a rubber band around it. 
 This is easier and better than holding the hat or note- 
 book over the objective. 
 
 In ascending or descending a steep hill, it is desir- 
 able, 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 f the rod on the down-hill side. 
 In selecting the position of the instrument to satisfy 
 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 tele- 
 scope. Even this rude observation will be valuable as 
 showing whether the instrument should be moved up 
 or down the hill, and 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, when 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, move 
 either the rod or the instrument a little to one side. In 
 short intermediate sights for which the telescope can not 
 be focused, it is sufficient to sight by the bottom of the 
 wyes or by the side of the telescope. 
 
 In closing at noon or night, be careful to set half-way 
 between the last two turning points; and 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 adjust- 
 ments of the instrument ( 234). 
 
ART. 7] USING THE LEVEL. 287 
 
 325. Length of Sight. The length of sight is limited 
 by the power of the telescope, the atmospheric condi- 
 tions, the accuracy desired, the time available, etc. 
 Some errors increase directly, and others indirectly, as 
 the number of sights taken in a given distance. It is 
 generally assumed that for the most accurate work the 
 rod should be at least 100 feet from the instrument and 
 never more than 400 feet; and that for ordinary work 
 the rod should be 300 or 400 feet away and never more 
 than 500 or 600 feet. 
 
 On the U. S. Coast and Geodetic Survey the length 
 of sight ranges from 50 to 150 meters (164 to 492 feet) 
 according to the condition of ground and weather, the 
 average being no meters (360 feet), the distance be- 
 tween the two rods on the same side of the instrument 
 being 20 meters (see method IV, Fig. 78, p. 268). On 
 the U. S. Lake Survey* the maximum length of sight 
 was 100 meters (328 feet). On the Prussian Land Survey, 
 the maximum sight has not exceeded 50 meters (164 
 feet), except for river crossings.! In the Swiss levels 
 of precision, the length of sight on railroads with gradi- 
 ents under i in 100 was 100 meters (328 feet); on rail- 
 roads with steeper gradients, from 50 to 100 meters (165 
 to 328 feet); on highways in the plains, from 30 to 60 
 meters (100 to 200 feet); and on mountain roads, from 
 10 to 25 meters (33 to 82 feet).J 
 
 326. Equal Back-sight and Fore-sight. It is very desir- 
 able that at each setting of the instrument the lengths 
 of the back-sight and the fore-sight should be equal ; 
 for, as has been seen, there are then a number of errors 
 which cancel each other. This is a very important point, 
 
 * Professional Papers Corps of Engineers, U. S. A., No. 24 Primarjr 
 Triangulation U. S. Lake Survey, p. 598. 
 
 t Wright's Adjustment of Observations, p. 375. 
 ^ Proc, Jnst. of C, E,, Vol. 78, p. 456, 
 
288 SPIRIT LEVELS. [CHAP. XI 
 
 and should always be kept in mind. Leveling is the only 
 kind of surveying wherein the instrumental errors may 
 be thoroughly eliminated without duplicating the work. 
 This is done by making the back-sights and fore-sights 
 of equal length. 
 
 When stakes are set at regular intervals, there is no 
 difficulty in determining the length of sights, and mak- 
 ing them equal ; and in other cases the distance can be 
 determined by stepping. When extreme accuracy is 
 desired, the rod-man approximates the distance by 
 stepping, and the instrument-man measures it by the 
 principle of the stadia (Chap. X). All levels should be 
 provided with two extra horizontal cross hairs for this 
 purpose. 
 
 In the precise leveling on the U. S. Lake Survey, and 
 in surveys made under the direction of the Mississippi 
 River Commission, as well as in the Swiss levels of pre- 
 cision, the difference between corresponding back-sight 
 and fore-sight is not allowed to exceed 10 meters (33 
 feet). 
 
 In ascending or descending a hill, it is nearly impossi- 
 ble, 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 sight will be about 
 twice the length of the up-hill one. When the ground 
 renders sights of unequal length unavoidable, keep 
 notes of the distance; and, as soon as possible, take 
 sights with corresponding inequalities in the contrary 
 direction. When approaching a long incline make part 
 of this compensation in advance. 
 
 327. Reciprocal Leveling. In crossing a river, it is 
 absolutely necessary that the back-sight and fore-sight 
 should differ considerably ; and there are other some- 
 what similar cases in which they can not be made even 
 approximately equal. In such instances the principles 
 
ART. 7] USING THE LEVEL. 289 
 
 of reciprocal leveling are applicable.* The method of 
 procedure is as follows : 
 
 Establish a bench upon both sides of the river and 
 determine the difference of level ; move the instrument 
 to the other side, and re-determine the difference of 
 level. If the sights are 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 instructions on the U. S. Lake Survey were to (i) 
 read upon a bench on the nearer shore with the tele- 
 scope normal and also inverted, (2) read upon the 
 bench on the farther shore five times with the telescope 
 normal and five times with it inverted, and (3) read 
 upon the nearer bench as before. The rod on the 
 farther shore is provided with a target from 6 inches to 
 i foot square, according to the distance. The line of 
 sight should pass at least 10 or 12 feet above the water 
 to eliminate abnormal refraction ( 318). The observa- 
 tions must be corrected for curvature and refraction 
 (319). With a single Kern level ( 261), this process 
 has given for a river half a mile wide (the Ohio, at 
 Cairo, 111.) five results the mean of which had a proba- 
 ble error (App. Ill, 2) of 0.5 mm. (0.002 ft. nearly). f 
 Of course this must be regarded as an exceptionally 
 accurate result, but it gives an idea of the practice in 
 such matters. In another case, sixteen simultaneous 
 observations on each side gave a probable error of 
 i mm. (0.003 ft-) f r a river crossing 600 meters (2,000 
 ft.) wide. 
 
 * The surface of water can not be assumed to be level except (i) where 
 there is no current or wind, or (2) where the thread of the current is midway 
 between the banks, and the line joining the two points of observation is per- 
 pendicular to the current, and there is no wind. 
 
 t Report of the Chief of Engineers, U. S. A., for 1880, Part III, p. 2432, 
 
2QO SPIRIT LEVELS. [CHAP. XI 
 
 On the U. S. Coast and Geodetic Survey,* eleven 
 sets of observations, consisting of twelve to sixteen 
 pointings each, made at four different hours on three 
 different days, across a river 2,200 feet wide, gave a 
 probable error of 1.9 mm. for the mean of each set. 
 In the above observations the line of sight was about 
 12 feet above the water; but an equal number of ob- 
 servations, made at the same time as the above, with 
 the line of sight 5 feet above the surface, gave a result 
 for the difference of level between the two bench marks 
 on opposite sides of the river differing from the mean 
 of the above observations by 23 mm. This shows the 
 uncertainty of such observations, even though the re- 
 sults may not disagree among themselves. 
 
 328. If the line to be leveled 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 
 done than leveling down one bank and up the other. 
 
 329. Bench Marks. These are permanent objects, 
 natural or artificial, whose heights are determined and 
 recorded for future reference. Any object not easily 
 disturbed, and easily described and found, may be used 
 as a bench mark, as, for example, the highest point of a 
 bowlder, a nail in the root of a prominent tree, a stone 
 door-sill, etc. A stake driven to the surface of the 
 ground, with another projecting above the ground to 
 mark its position, is frequently used in railroad and 
 drainage surveying ; but it is not very reliable at best, 
 and is entirely unreliable after having stood over winter, 
 owing to the " heaving of the frost." The precise Joca- 
 
 * Annual report for 1879, pp. 212-13. 
 
ART. 7] USING THE LEVEL. 291 
 
 tion and description of every bench should be given 
 very fully and definitely in the " remarks " column of 
 the field notes. The description should be such that an 
 entire stranger could find the bench by the aid of the 
 notes alone. For convenience, the benches are num- 
 bered, the number, and the fact that it is a bench, being 
 marked on or near it. 
 
 Bench marks should be established at frequent inter- 
 vals along the line. They serve as points of beginning 
 in case of accident, as, for example, losing a turning 
 point, and are points on which to check in case the 
 line is re-run. They are usually put in at each mile 
 and half-mile from the beginning of the line ; and they 
 should also be put in at all places where they may be 
 necessary or convenient as, for instance, near where 
 the line crosses a prominent road, on both sides of a 
 river crossed, near the top or bottom of a high hill 
 crossed, etc. 
 
 330. CARE. For remarks on the care of instruments, 
 which are applicable to the level, see 95 and 148. 
 
CHAPTER XII. 
 BAROMETERS. 
 
 332. THE difference of level of two places may be de- 
 termined by ascertaining the difference in the height of 
 the atmosphere above the places. This may be found 
 in any of three ways; viz., (i) by determining how high 
 a column of mercury, or other liquid, the column of air 
 above it will balance, (2) by finding the pressure it 
 will exert against an elastic box from which the air has 
 been exhausted, or (3) by observing the temperature at 
 which a liquid boils, i.e., by observing the temperature 
 at which the pressure of the atmosphere just balances 
 the tension of the vapor. There are then three slightly 
 different methods of barometric leveling according as 
 the instrument used is a mercurial barometer, an ane- 
 roid barometer, or a thermo-barometer or boiling-point 
 apparatus. As the thermo-barometer has been super- 
 seded by the aneroid, only the methods of leveling by 
 the mercurial and the aneroid barometers will be con- 
 sidered here. 
 
 Barometric leveling is specially adapted to finding the 
 difference of level between points at considerable hori- 
 zontal or vertical distance apart. Under these condi- 
 tions, it is the most expeditious, but the least accurate, of 
 any of the methods of leveling. It is very valuable in 
 making geographical surveys of large areas for deter- 
 mining the elevation of stations to be occupied by the 
 topographer. It is also well suited to making a recon- 
 noissance for a railroad or for a scheme of triangulation. 
 
 292 
 
ART. l] MERCURIAL BAROMETER. 293 
 
 ART. 1. THE MERCURIAL BAROMETER. 
 
 333. CONSTRUCTION. There are two kinds of mercurial 
 barometers, the cistern and the siphon. The former is 
 the better and more reliable for hypsometrical purposes. 
 The general form of the cistern barometer needs no 
 description. Fig. 80, page 294, shows some of the details 
 of Green's barometer, which is generally considered one 
 of the best. The right-hand portion of Fig. 80 shows 
 the cistern and the details at the. lower end of the in- 
 strument, and the left-hand portion the vernier and 
 scale at the upper end of the mercury column. 
 
 The cistern consists of a glass cylinder F t which allows 
 the surface of the mercury to be seen, and a top plate 
 G, through the neck of which the barometer tube / 
 passes, and to which it is fastened by a piece of kid 
 leather, making a strong but flexible joint. To this 
 plate is attached also a small ivory point, h, the extrem- 
 ity of which marks the commencement or zero of the 
 scale. The lower part, containing the mercury into 
 which the end of the tube, t, is plunged, is formed 
 of two parts, i and /, held together by four screws 
 and two divided rings, / and m. To the lower piece,/, 
 is fastened the flexible bag N, made of kid leather, fur- 
 nished in the middle with a socket, /, which rests on the 
 end of the adjusting screw O. These parts, with the 
 glass cylinder, F, are clamped to the flange, B, by means 
 of four large screws, P, and the ring, J?. On the ring, R 
 screws the cap, S, which covers the lower parts of the 
 cistern and supports at the end the adjusting screw O. 
 G, i,j, and k, are of boxwood ; the other parts of brass or 
 German silver. The screw O serves to adjust the mer- 
 cury to the ivory point, and also, by raising the bag so 
 as to completely fill the cistern and tube with mercury, 
 to put the instrument in condition for transportation. 
 
204 
 
 BAROMETERS. 
 
 [CHAP. XL 
 
 The milled-head screw D moves, by means of a rack 
 and pinion, the vernier C, so as to bring the horizontal 
 
 FIG. 80. MERCURIAL BAROMETER. 
 
 line just below C level with the top of the mercury 
 column. 
 
 334. CLEANING THE BAROMETER. It frequently hap- 
 pens that the mercury in the cistern becomes so dirty 
 that the ivory point, or its reflection in the mercury, can 
 no longer be seen. This often occurs even though the 
 barometer is in good condition in every other respect. 
 
ART. l] MERCURIAL BAROMETER. $9$ 
 
 "The instrument can be taken apart and cleaned with 
 safety and without changing in the slightest degree the 
 zero of the instrument. Everything used in the opera- 
 tion must be clean and dry. Avoid blowing upon any 
 of the parts, as the moisture from the breath is in- 
 jurious. 
 
 " Turn up the adjusting screw at the bottom until 
 the mercury entirely fills the tube, carefully invert, 
 place the instrument firmly in an upright position, un- 
 screw and take off the brass casing which encloses the 
 wooden and leather parts of the cistern. Remove the 
 screws, and lift off the upper wooden piece to which the 
 bag is attached; the mercury will then be exposed. By 
 inclining the ins-trument a little, a portion of the mer- 
 cury in the cistern may be poured out into a clean ves- 
 sel at hand to receive it, when the end of the tube will 
 be exposed. This is to be closed by the gloved hand, 
 when the instrument can be inverted, the cistern 
 emptied, and the tube brought again to the upright 
 position. Great care must be taken not to permit any 
 mercury to pass out of the tube. The long screws 
 which fasten the glass portion of the cistern to the other 
 parts can then be taken off, the various parts wiped with 
 a clean cloth or handkerchief, and restored to their 
 former position. 
 
 *' If the old mercury is merely dusty, or dimmed by 
 the oxide, the cleaning may be effected by straining it 
 through chamois leather, or through a funnel with a 
 capillary hole at the end of a size to admit of the pas- 
 sage of but a small thread of the metal. Such a funnel 
 is conveniently made of letter paper. The dust will 
 adhere to the skin or paper, and the filtered mercury 
 will present a clean and bright appearance. If chemi- 
 cally impure, it should be rejected, and fresh, clean 
 mercury used. With such clean mercury the cistern 
 should l>e filled as nearly full as possible, the wooden 
 
296 BAROMETERS. fcHAP. XII 
 
 portions put together and securely fastened by the 
 screws and clamps, the brass casing screwed on, and 
 the screw at its end screwed up. The instrument can. 
 then be inverted, hung up, and re-adjusted. The tube 
 and its contents having been undisturbed, the instrument 
 should read the same as before." * 
 
 With the instrument before the operator, these in- 
 structions are easily understood. In this case, as in 
 using and caring for any instrument, a little care and a 
 thoughtful inspection of the method of construction is 
 worth more than any written description. If a little 
 mercury has been lost during the operation, and there 
 is none at hand to replace it, no serious harm has been 
 done; but if much is lost, the open end of the tube may 
 become exposed in inverting the instrument, in which 
 case air may enter. 
 
 335. FILLING THE BAROMETER. It is no slight matter 
 to properly fill a barometer. It can best be done by 
 the manufacturer, who has all the facilities; but as it 
 is sometimes necessary for the observer to re-fill his 
 barometer, the following hints are given. Tubes require 
 re-filling owing to the breakage of the glass or to the 
 entrance of a bubble of air. 
 
 The mercury should be chemically pure and free from 
 oxide, otherwise it will adhere to the glass and tarnish 
 it. Moreover, if it is not pure, the height of the baro- 
 metric column will not be correct. No mercury should 
 be used except that which has been purified by distilla- 
 tion. For the best results, the mercury should be boiled 
 in the tube to expel moisture and air; but this can not 
 always be done, and fair results may be obtained with- 
 out boiling. 
 
 In extended barometric operations in the field, a sup- 
 ply of extra tubes is carried, to be used in case a tube is 
 
 * On the Use of the Barometer on Surveys and Reconnoissances, Maj. R. 
 S. Williamson, U. S. A., New York City, 1868, pp. 136 and 137. 
 
ART. l] MERCURIAL BAROMETER. 297 
 
 broken. These tubes should be drawn out so as to be 
 a little longer than the) 7 are required to be when fitted 
 into the barometer. The open end should be cut off to 
 such a length that it shall always be immersed and yet 
 not interfere with the rise of the lower part of the cis- 
 tern. When the instrument is finally put together, the 
 cork in the upper end of the brass case should be ad- 
 justed so as to hold the closed end of the tube firmly. 
 
 336. By Boiling. " To fill a tube by boiling, an alco- 
 hol lamp is needed, although it can be done over a char- 
 coal fire. The lamp being filled and put in order, begin 
 to fill the tube by pouring in through the funnel as 
 much warm mercury as will occupy about 5 inches; then, 
 holding the tube with both hands above the mercury, 
 heat it gently, and let the mercury boil from the sur- 
 face downward to the end of the tube, and then back 
 again, chasing all of the bubbles of air upward. A 
 little practice will make this easy, the tube being held 
 a little inclined from the horizontal, and constantly and 
 rapidly revolved, always in the same direction, so that 
 every portion of the metal maybe heated gradually and 
 uniformly. After this has been done, let the tube cool 
 sufficiently to admit of its being held by the gloved hand, 
 and then pour in enough warm mercury to occupy sev- 
 eral inches more of the tube, which may now be held 
 with both hands, one above and the other below the 
 heated portion. After boiling this thoroughly free from 
 air, repeat the same operation with more mercury added, 
 until the tube is filled to the end. With care and prac- 
 tice the mercury may be boiled entirely free from air 
 up to within an inch or less of the end of the tube. A 
 tube filled in this way may have, in every respect, as 
 perfect a vacuum as one prepared by a professional 
 instrument maker." * 
 
 337. Without Boiling. The glass tube, which should 
 
 * Williamson's On the Barometer, p. 138. 
 
298 BAROMETERS. [CHAP. XII 
 
 be clean and dry, must have its open end ground straight 
 and smooth, so that it can be closed air-tight with the 
 finger, which should be covered with a piece of chamois 
 or kid skin. Warm well both mercury and glass tube, 
 and through a clean paper funnel with a very small hole 
 (about ^Q- of an inch) below, filter in the mercury to 
 within one fourth of an inch of the top. Shut up the 
 end and turn the tube horizontal, when the mercury will 
 form a bubble which can be made to run from end to 
 end by a change of inclination, and which will gather 
 all the small air bubbles that adhered to the inside of 
 the glass tube during filling. Let this bubble, which 
 has grown somewhat larger, pass to the open end; then 
 fill the tube completely with mercury, and shut it tightly. 
 Next reverse the tube over a basin, when, by slightly 
 relieving the pressure against the end, the weight of 
 the column of mercury will force some out, forming a 
 vacuum above, which ought not to exceed one half an 
 inch. Closing up again tightly, let this vacuum bubble 
 traverse the length of the tube on the several sides, that 
 it may absorb those minute portions of air that were 
 not drawn out at the first gathering, and which are now 
 greatly expanded from removed atmospheric pressure. 
 The complete absence of air is easily recognized by the 
 sharp concussions with which the column beats against 
 the sealed end, when the tube is held horizontally and 
 slightly moved. 
 
 " A barometer filled without boiling will probably 
 read lower by a few thousandths than if the tube had 
 been boiled; but in a stationary barometer its error will 
 probably not soon change, and carrying on horseback 
 will be apt to improve it rather than otherwise, as it is 
 then carried with the cistern uppermost and the bubbles 
 will be jolted toward the open end. If possible it should 
 be compared with a standard barometer."* 
 
 * Williamson's On the Barometer, p. 140. 
 
ART. l] MERCURIAL BAROMETER. 2!)9 
 
 338. READING THE BAKOMETEK. Read the attached 
 thermometer first. It is more sensitive than the barom- 
 eter, and is affected by the heat of the body, while the 
 barometer is not so affected. The thermometer should 
 be read as closely as possible, for a difference of i F. is 
 equivalent to about 3 feet in altitude. Parallax should 
 be carefully avoided in making this reading. 
 
 Then, by means of the adjusting screw at the lower 
 end of the instrument (<?, Fig. 80, page 294), bring the 
 ivory point just to the mercury in the cistern. If there 
 is a line of light visible between the point and mercury, 
 the mercury in the cistern is too low; and if the point 
 makes a depression, the mercury in the cistern is too 
 high. If neither a line of light nor a depression can be 
 seen, the adjustment has been correctly made. It is 
 usually best to lower the screw till a distinct line of 
 light can be seen, and then gradually raise it until the 
 light disappears. When the mercury is bright, a shadow 
 of the point can be seen, and if the shadow and 'the 
 point itself form a continuous unbroken line, the screw 
 at the bottom of the cistern has been properly ad- 
 justed. Before making the final adjustment, tap the 
 barometer a little just above the cistern, to destroy the 
 adhesion of the metal to the glass. Complete the con- 
 tact of the mercury and the ivory point, at the same time 
 being certain that the barometer hangs freely, /'.*., ver- 
 tically. 
 
 Next, tap the barometer gently in the neighborhood 
 of the top of the mercury column, to destroy the adhe- 
 sion of the mercury. This is very important, since 
 raising or lowering the mercury in the previous opera- 
 tion materially affects the form of the upper surface. 
 Then take hold lightly of the brass casing of the barom- 
 eter at a distance from the attached thermometer, that 
 neither the case nor the thermometer may be unneces- 
 sarily heated, and by means of the milled-head screw 
 
300 BAROMETERS. [CHAP. XII 
 
 near the middle of the tube (Z>, Fig. 80), bring the front 
 and back edge of the vernier into the same horizontal 
 plane with the top of the mercury in the tube, and 
 remove the hand to allow the instrument to hang 
 vertically. Move the eye about, and if, in any position, 
 a line of light can be seen between the mercury and the 
 vernier, the latter must be moved down; if there is no 
 line of light and a portion of the meniscusjs obscured, the 
 vernier must be moved up. As the top of the column 
 is more or less convex, when the adjustment is correctly 
 made a small place is obscured in the center while the 
 light is seen on either side. 
 
 Finally, having adjusted the instrument as above, it 
 may be read at leisure. On the best barometers the 
 scale is divided to inches, tenths, and half-tenths, and 
 the vernier reads to one twenty-fifth of a half-tenth 
 (sV X 0.05), or two thousandths (0.002) of an inch. (See 
 Fig. 15, page 69.) 
 
 339. TRANSPORTING THE BAROMETER. " In transporting 
 
 a barometer, even across a room, it should be screwed 
 up, and carried with its cistern uppermost. For travel- 
 ing it should be provided with a wooden and leather 
 case. On steam boats or steam cars it should be hung 
 up by a hook. In wheeled vehicles it should be carried 
 by hand, supported by a strap over the shoulder or held 
 upright between the legs; but it should not be allowed 
 to rest on the floor of the carriage, for a sudden jolt 
 might break the tube. If carried on horseback it 
 should be strapped over the shoulder of the rider, where 
 it is not likely to be injured unless the animal is sub- 
 ject to a sudden change of gait. When about to be 
 used, it should be taken from its case while screwed 
 up, gently inverted and hung up, when it can be un- 
 screwed. While it has its cistern uppermost the tube is 
 full, is one solid mass of metal and glass, and not easily 
 injured ; but when hung up, a sudden jolt might send 
 
ART. 2] ANEROID BAROMETERS. 3<M 
 
 a bubble of air into the vacuum at the upper end of 
 the tube, and the instrument would be useless until re- 
 filled." * 
 
 ART. 2. ANEROID 'BAROMETERS. 
 
 340. The aneroid barometer consists of a cylindrical 
 metallic box, from which the air is exhausted, having a 
 thin corrugated metal top which readily yields to alter- 
 ations in the pressure of the atmosphere. When the 
 pressure increases, the top is pressed inwards; when it 
 decreases, the elasticity of the lid tends to move it in 
 the opposite direction. There are two general forms of 
 aneroids, according to the method employed for read- 
 ing the movements of the top of^ the vacuum chamber. 
 In the common form these motions are transmitted by 
 delicate multiplying levers to an index which moves 
 over a scale. In the Goldschmid aneroid the move- 
 ments of the top of the vacuum box cause an index to 
 move through the field of a micrometer microscope by 
 which the movement is measured. 
 
 341. COMMON OK VIDI ANEROID. Fig. 81, page 302, 
 
 shows the mechanism of the ordinary form of aneroid 
 barometer. The outside case and the graduated dial 
 are not shown. The movement of the top of the 
 vacuum chamber, M, is communicated to the index 
 through the levers /, ;//, and r, and the watch chain S 
 which winds around the axis carrying the index. The 
 broad curved spring R keeps the lever/ in contact with 
 the post on the top of the vacuum chamber. 
 
 There are several modifications of this form which 
 differ in the mechanism employed to multiply the linear 
 motion of the end of the vacuous box, and to convert it 
 into angular motion. A spring is sometimes inserted 
 
 * Williamson's On the Barometer, p. 134. 
 
302 BAROMETERS. [CHAP. XII 
 
 between the two ends of the vacuum chamber to rein- 
 force the elasticity of the corrugated ends. 
 
 Sometimes the vacuous box is not entirely exhausted, 
 the claim being that the enclosed air renders the indi- 
 
 FIG. 81. COMMON ANEROID BAROMETER. 
 
 cations of the instrument independent of changes of 
 temperature. Such aneroids are said to be compen- 
 sated; but the theory is incorrect, and some "compen- 
 sated " aneroids are more affected by changes of tem- 
 perature than uncompensated ones (see 343, par. 2). 
 An aneroid should have a thermometer attached to it. 
 
 The mechanism shown in Fig. 81 is enclosed in a 
 brass or silver case, which varies from 2 to 6 inches in 
 diameter, and from 0.5 to 3 inches in thickness. The 
 smaller sizes are usually made of silver, in the form of 
 a watch, and are known as pocket aneroids. The 
 larger ones look like a short brass cylinder, and are 
 
ART. 2] ANEROID BAROMETERS. 303 
 
 carried in a leather case supported by a strap over the 
 shoulder. 
 
 342. The face of the instrument has a scale of inches, 
 and sometimes also a scale of elevations. The former 
 is graduated empirically by comparing its indications 
 under different pressures with those of a mercurial 
 barometer. The scale is marked to correspond to 
 inches of the ordinary barometer column, the inches 
 being divided into tenths, and the tenths usually into 
 four parts. At the back of the instrument is a little 
 screw which presses against one end of the exhausted 
 box; and by turning this screw the index can be moved 
 over the scale, and the instrument may thus be made 
 to agree at any time with a standard mercurial barom- 
 eter. 
 
 The altitude scale is obtained by converting the scale 
 of inches into elevations by the use of some barometric 
 formula (see Art. 4), and engraving the results upon 
 the face, adjacent to the scale of inches. The altitude 
 scale is at best of doubtful utility (see 403). 
 
 343. Defects. The aneroid is a very convenient in- 
 strument, and where nice readings are not required 
 it does very well; but for accurate hypsometrical re- 
 sults it is an inferior instrument. It has the following 
 defects: 
 
 1. The elasticity of the corrugated top of the vacuum 
 chamber is affected by repeated changes in pressure. 
 This will produce error in the scale readings. 
 
 2. It is usually claimed that, in consequence of not 
 completely exhausting the vacuum box, the indications 
 of the aneroid become independent of the effect of 
 changes of temperature of the instrument. The best 
 that can be hoped is that for small changes the tem- 
 perature correction is less than the error of observa- 
 tion. In instruments compensated for temperature, the 
 effect of a change is sometimes the same as that in the 
 
304 BAROMETERS. [CHAP. XII 
 
 mercurial barometer, and sometimes the reverse. The 
 effect of the temperature on any particular instrument 
 can be determined only by trial. 
 
 3. The different spaces on the scale of inches are sel- 
 dom correct relative to each other, owing probably to 
 errors of observation and graduation, and possibly to 
 differences of temperature and changes in elasticity. 
 As a matter of fact, the scale is often only a scale of 
 equal parts. The barometer scale is more accurate 
 than the elevation scale, since the latter has all the in- 
 accuracies due to the formula by which it is graduated 
 ( 403), in addition to those of the instrument itself. 
 Before using the aneroid, it should be compared with a 
 mercurial column under an air-pump, to determine the 
 errors of its scale for different temperatures and press- 
 ures. 
 
 4. The weight of the machine affects its indication, 
 i.e., the reading of the aneroid will differ when held in 
 different positions. In the best instruments this differ- 
 ence is sometimes as much as 0.008 of an inch, corre- 
 sponding to a difference of elevation of about 8 feet. 
 Usually the reading is higher with the dial is horizontal 
 (face uppermost), than when it is vertical. 
 
 5. Like all combinations of levers, screws, and springs, 
 the aneroid is liable to continual shifting of parts, 
 when subject to the jars and jolts encountered in 
 transportation and in use. The only remedy is fre- 
 quent comparisons with a mercurial barometer. 
 
 6. The aneroid is deficient in precision, since the 
 least reading is 0.025 f an i n ch, which corresponds 
 nearly to 25 feet of elevation. 
 
 7. With .most aneroids the spring ceases to act after 
 the pressure has been lowered somewhat, i.e., the in- 
 strument runs down. Before using it, experiments 
 should be made to determine the range of pressure to 
 which it may be exposed before the spring ceases to 
 
ART. 2] ANEROID BAROMETERS. 305 
 
 act. In case an aneroid is to be used in an elevated 
 region, if there is a mercurial barometer with the party, 
 screw up the anerord until the spring acts well, and set 
 the instrument by the mercurial barometer so that there 
 shall be a considerable difference, say 2 or 3 inches, be- 
 tween them. Of course this difference must be added 
 to each reading of the aneroid. 
 
 344. " With all these defects a good aneroid is of 
 much assistance on a survey or reconnoissance in 
 mountainous districts, on side trips of one, or even 
 several, day's duration, when the instrument has been 
 previously compared with a standard mercurial barom- 
 eter at various temperatures and in different elevations, 
 and proper tables of corrections made. It should be 
 compared before and after it is used in that way, to see 
 if the zero has not changed in the meantime, and if the 
 agreements are satisfactory the results can be relied 
 upon. It is evidently important that there should be 
 a good attached thermometer." * 
 
 345. Reading the Aneroid. In measuring heights 
 with the aneroid barometer, it should be shielded from 
 the direct rays of the sun, and care should be taken 
 that it is not unnecessarily influenced by the heat of 
 the body of the observer. If the instrument has a 
 thermometer attached, that should be read first, as it is 
 liable to be affected by the heat of the hand of the ob- 
 server. 
 
 The barometer should be held in the same position 
 for both observations preferably with the face hori- 
 zontal; and should be tapped gently with the finger 
 just before taking the reading. Considerable care i? 
 required to determine exactly where the index pointy 
 Some instruments are provided with a small lens for 
 this purpose (see last paragraph on page 94). Without 
 
 * Williamson's On the Barometer, p. 133. 
 
BAROMETERS. 
 
 [CHAP, xii 
 
 the lens, the chief error is due to parallax. Of course, 
 the index should move very close to the graduation 
 and yet not touch the face of the instrument. 
 
 346. GOLDSCHMID ANEROID. The common aneroid was 
 invented about the beginning of this century, but was 
 first made of a serviceable form by Vidi, of Paris, in 
 
 FIG. 82. GOLDSCHMID ANEROID. 
 
 1847. The defects of its complicated system of mul- 
 tiplying levers have long been recognized; and as early 
 as 1857, Goldschmid designed a form of aneroid in 
 which he dispensed with the transmitting and multiply- 
 ing mechanism of the Vidi form. 
 
 Figs. 82 and 83, pages 306 and 307, are two views of one 
 of the latest forms of the Goldschmid aneroid. Fig. 82 
 
ART. 2] 
 
 ANEROID BAROMETERS. 
 
 307 
 
 is an external view, showing the microscope Z, and the 
 micrometer M '; and Fig. 83 is a section showing the 
 compound vacuum chamber. Obviously, the greater 
 the number of boxes, the larger the motion of the index 
 a. The relative position of the movable index a and 
 the fixed point of reference , is observed by the tele- 
 scope Z, and the distance is measured by the micrometer 
 M. The instrument is very delicate in its indications, 
 
 FIG. 83. SECTION OF GOLDSCHMID ANEROID. 
 
 but is liable to serious disarrangement by ordinary 
 handling. Different manufacturers make slightly dif- 
 ferent forms of the Goldschmid type, but all have essen- 
 tially the same defect, i.e., are not able to stand ordinary 
 use. 
 
 It is doubtful if there is any advantage in an aneroid 
 as complicated as that shown in Figs. 82 and 83. It 
 seems probable that no form can be devised which 
 shall be both delicate in its indications and able to 
 stand rough handling. The chief advantage of the 
 
308 BAROMETERS. [CHAP. XII 
 
 common aneroid is its portability, combined with mod- 
 erate accuracy. The mercurial and the aneroid barom- 
 eters supplement each other ; the first is delicate and 
 the second is portable. These qualities can not be com- 
 bined in a single instrument, nor can one be obtained 
 more delicate or more reliable than the mercurial 
 barometer. 
 
 ART. 3. THE OBSERVATIONS. 
 
 347. OUTLINE OF METHOD. To determine a difference 
 of elevation with the barometer it is necessary to find at 
 each of the two stations (i) the reading of the barometer, 
 (2) the temperature of the barometer, and (3) the tem- 
 perature of the air. Sometimes observations are made 
 to determine (4) the amount of watery vapor in the 
 atmosphere. Inserting these data in a barometric 
 leveling formula (Art. 4) and reducing gives the differ- 
 ence of elevation. 
 
 Before discussing the method further, it is necessary 
 to consider the errors to which barometric leveling is 
 liable. 
 
 348. SOURCES OF ERROR. For convenience of discus- 
 sion, the sources of error will be considered under five 
 heads ; viz., (i) gradient, (2) temperature of air, (3) 
 humidity, (4) instrumental errors, (5) errors of observa- 
 tion, and (6) effect of the wind. 
 
 349. Gradient Errors. Let A, B, and C designate three 
 points at which the pressure is the same. The plane 
 passing through A, B, and C is then a surface of equal 
 pressure. If the air were in a state of equilibrium, this 
 plane would be level ; but under ordinary conditions it 
 will be inclined in some direction. The inclination of 
 this surface is called the barometric gradient. 
 
 Instead of considering only three points, we can in 
 imagination project through the air a surface contain- 
 
ART. 3] THE OBSERVATIONS. 309 
 
 ing all points which have the same pressure. If the 
 atmosphere were at rest, this surface would be a hori- 
 zontal plane ; but under the actual conditions, it is 
 never a plane and is ever undulating. For small areas 
 under ordinary conditions, this surface would probably 
 not differ much from a plane. 
 
 Conceive another surface passed through all points 
 at which the pressure differs from the preceding one by 
 any constant quantity. With atmospheric equilibrium 
 two such surfaces would be both level and parallel, but 
 in the actual case they are neither level nor parallel. 
 When widely separated surfaces are compared, the 
 variations from parallelism are often so great that their 
 inclinations above the same locality have opposite direc- 
 tions. The atmospheric gradient at the surface of the 
 ground may therefore differ greatly in amount and 
 direction from the simultaneous gradient at a consid- 
 erable altitude above that point. 
 
 350. That the atmosphere is not in static equilibrium 
 is shown by its being continually in motion and also by 
 the variation in the height of the barometer. A varia- 
 tion of the barometric pressure indicates a change in 
 the barometric gradient, and a gradient or a change in 
 the gradient may produce an error in the result ob- 
 tained by barometric leveling. For example, if A and 
 
 B, Fig. 84, page 310, are two stations and the atmos- 
 phere is at rest, the surface of equal pressure, BC, is a 
 horizontal plane, and AC is the difference of elevation 
 which would be obtained by applying any one of the 
 common barometric formulas. If the air is not in static 
 equilibrium, the pressure at A will be greater or less 
 than before, and the surface of equal pressure will lie 
 above or below BC. If the pressure at A is greater 
 than the average, the surface of equal pressure is above 
 
 C, say at E, and AE is the corresponding difference of 
 elevation ; and similarly, if BD is the surface of equal 
 
BAROMETERS. 
 
 [CHAP, xii 
 
 pressure, AD is the corresponding difference of eleva- 
 vation. 
 
 The problem is further complicated by the fact that 
 the air above B also is in a state of oscillation. If the 
 
 FIG. 84. 
 
 variations in pressure at the two stations were simulta- 
 neous and alike in amount, no error would be pro- 
 duced by the barometric gradient ; but these condi- 
 tions are seldom or never realized, and therefore there 
 is always a possibility of error in barometric leveling, 
 owing to the barometric gradient and also to a change 
 in the gradient between the observations at the two 
 stations. 
 
 351. There are four classes of barometric gradient ; 
 viz., (i) diurnal, (2) annual, (3) non-periodic, and (4) 
 permanent. 
 
 352. Diurnal Gradient. It is a fact familiar to meteor- 
 ologists that the pressure of the air everywhere under- 
 goes a daily oscillation. The gradient introduced by 
 this daily change is called the diurnal gradient. The 
 pressure has two maxima and two minima which are 
 easily distinguishable. Near the sea-level the barometer 
 attains its first maximum about 9 or 10 A.M. In the 
 afternoon there is a minimum about 3 to 5 P.M. It then 
 rises until 10 or n P.M., when it falls again until about 
 
ART. 3] THE OBSERVATIONS. 3ft 
 
 4 A.M., and again rises to attain its forenoon maximum. 
 The maxima occur when the temperature is about the 
 mean of the day, and the minima when the temperature 
 is at the highest and lowest respectively. The day 
 fluctuations are the larger. 
 
 The daily oscillation is subject to variations in char- 
 acter and magnitude. It is greater in summer than in 
 winter ; and is greatest at the equator and diminishes 
 toward the poles, but is not the same for all places of the 
 same latitude. Within the United States it varies be- 
 tween 40 and 120 thousandths of an inch. Changes of 
 altitude often cause a marked increase in the amount 
 of the diurnal oscillation. The difference which per- 
 tains to latitude does not materially affect the ordinary 
 hypsometric problem, but the difference depending on 
 the altitude has a very important effect. 
 
 A change of i thousandth of an inch in the height 
 of the barometer corresponds to a difference of eleva- 
 tion of from 0.8 to i.o foot. Therefore, if an observa- 
 tion were made at one station at about 10 A.M., and at a 
 second station at about 3 P.M., the difference of elevation 
 would probably be from 40 to 120 feet in error. 
 
 353. Annual Gradient. The annual progress of the 
 sun from tropic to tropic throws a preponderance of 
 heat first on one side of the equator and then on the 
 other, which produces an annual cycle of changes in 
 the pressure and gives rise to what has been called the 
 annual gradient. The amount of variation in the baro- 
 metric pressure is quite small near the equator, but 
 increases rapidly toward the poles. The mean annual 
 variation in the United States ranges from 120 to 200 
 thousandths of an inch, although the variation at any 
 particular station, or for any one year, may be very 
 much greater. For interesting diagrams showing the 
 annual variation for a great number of stations, see 
 Williamson's On the Barometer, pages 68-81. 
 
312 BAROMETERS. [cHAP. XIl 
 
 354. Non-periodic Gradient. In addition to the diurnal 
 and annual variations in the pressure, there are others 
 due to the same general cause the heat of the sun, 
 but so modified by the local conditions topography, 
 humidity, winds, storms, etc., as to make it impos- 
 sible to discover the law of their action. These non- 
 periodic variations are much greater in amount and 
 more rapid in action than any of the others. For ex- 
 ample, in a trial made for the purpose of this record, 
 under apparently favorable circumstances, the baro- 
 metrically-determined difference of elevation of two 
 points having a difference of elevation of about 100 
 feet and being 25 miles apart on a plain, was in error 
 about 400 per cent, owing mainly to non-periodic grad- 
 ient. Usually these variations are greater immediately 
 before and after a storm. 
 
 The errors arising from non-periodic gradients are 
 approximately proportional to the force of the wind 
 and to the horizontal distance between the two sta- 
 tions. 
 
 355. Permanent Gradient. Since the atmosphere, if 
 undisturbed, would have no gradients, and since every 
 disturbance produces them, it is easy to understand that 
 any continuous disturbance will be accompanied by 
 permanent gradients. The excess of solar heat received 
 in the tropics, as compared with the polar regions, is of 
 the nature of a continuous disturbance, and sets in 
 motion the great currents of the atmosphere and the 
 ocean. The joint action of these causes gives rise to 
 a great system of permanent gradients. The annual 
 gradients are only variations of the permanent gradient, 
 caused by the annual progress of the sun from tropic to 
 tropic. 
 
 The topographic conditions of the earth's surface also 
 probably cause a somewhat permanent gradient. 
 
 356. Conclusion as to Gradient Errors. The result ob- 
 
ART. 3] THE OBSERVATIONS. 
 
 tained by barometric leveling may be in error to almost 
 any degree owing to barometric gradient and to changes 
 in the gradient. A description will presently be given 
 (see 371-81) of several methods of making the obser- 
 vations which eliminate at least part of the errors due 
 to gradient. 
 
 357. Temperature of the Air. Variations in temperature 
 is the chief cause of changes in barometric pressure, but 
 the variation in the temperature of the air has another, 
 and generally a more serious, effect upon the results 
 obtained by barometric leveling. Let A and B, Fig. 84, 
 page 310, be two stations the difference of elevation of 
 which is to be obtained from observations of the 
 barometer and thermometer made at each. Assumed 
 that the pressure observed at B is the same as that at 
 C vertically over A and on a level with B. To use any 
 of the barometric leveling formulas, the temperature 
 of the column AC must be known; and in applying 
 any of these formulas it is assumed that the mean tem- 
 perature of this column is equal to the mean of the 
 temperatures observed at A and B. 
 
 How admissible this assumption is will appear at once 
 when the manner in which the air acquires and loses 
 heat is recalled. The body of the atmosphere is heated 
 directly by the sun, and gives off its heat by radiation 
 into space. The surface of the earth is heated and 
 cooled in the same manner, but many times more rapidly, 
 so that by day it is always much warmer than the body 
 of the air, and by night it is much cooler. A layer of 
 air next to the earth receives its warmth from the earth, 
 and hence differs widely in temperature from the re- 
 mainder of the atmosphere. Not only is the greater 
 part of the column inaccessible to us, but that portion 
 to which our observations are restricted is the portion 
 least representative of all. 
 
 " By measuring the difference of elevation of two 
 
314 fcAROMETERS. [CHAP, xii 
 
 points with the spirit level, reversing the barometer for- 
 mula, and computing the temperature of the air column, 
 it has been found that in middle latitudes the average 
 daily range of the temperature of the body of the air is 
 about 4, of the superficial layer from 10 to 20 near 
 the seashore, and from 20 to 35 in the interior of con- 
 tinents." * There is a stratum of air near the surface of 
 the earth which oscillates daily throirgh this wide range, 
 while the temperature of the upper and larger portion 
 of the column AC is relatively constant; and therefore 
 the mean of the observed temperatures absolutely fails 
 to give the mean temperature of the column AC as re- 
 quired in the formula. 
 
 358. Nor does the trouble end here. Whenever the 
 ground layer is cooler than the air above, it is of course 
 heavier, and, like any other heavy fluid, it flows down 
 hill and accumulates in valleys, forming lakes of cold 
 air. The nightly layer of abnormally cool air is there- 
 fore thinner on eminences than in valleys, and the con- 
 trast increases as the night advances. When the condi- 
 tions are reversed so that the lower layer is warmer 
 than the air above it, it has a tendency to rise, but ac- 
 complishes the change in an irregular manner, breaking 
 through the immediately superior layer here and there 
 and rising in streams which spread out in sheets wher- 
 ever the conditions of equilibrium are reached. Ob- 
 servers in balloons, as they ascend or descend, rarely 
 find an orderly succession of temperatures. If, there- 
 fore, we could in any way determine the temperature 
 of some point in the upper portion of the column AC, 
 we should still be unable to deduce the mean tempera- 
 ture of the column with any considerable degree of 
 accuracy. 
 
 359. Concision as to Error in the Temperature of the 
 
 * G. K. Gilbert, in the Report of the U. S. Geological Survey for 1880- 
 81, p. 421. 
 
ART. 3] THE OBSERVATIONS. 3lg 
 
 Atmosphere. For an error of 5 F. in the mean tempera- 
 ture of the level stratum of air between the two stations, 
 the resulting error is approximately i per cent of the 
 difference of elevation (see 390). This error may even 
 under favorable conditions be two or three times this 
 amount, and under unfavorable conditions five or six 
 times as large. 
 
 Several methods of eliminating the error due to the 
 uncertainty in the temperature of the air will be de- 
 scribed presently ( 371-81); but even with the utmost 
 care and the most elaborate system of observations, the 
 determination of the temperature of the air is the chief 
 source of error in barometric leveling. 
 
 360. Humidity. The barometer is influenced by the 
 elastic force of the invisible aqueous vapor suspended in 
 the atmosphere, in the same way that it is influenced 
 by the dry air; and hence the attempt has been made to 
 introduce a term into the barometric leveling formula, 
 which shall take account of the amount of this watery 
 vapor. The determination of the humidity of the at- 
 mosphere involves essentially the same difficulties as 
 the determination of its temperature (see 357-59). 
 The observations are made in the stratum next to the 
 earth, in which the amount of moisture is the greatest 
 and the most variable. A change of position of a few 
 feet, or a slight variation in the direction or force of the 
 wind, will often cause a very important difference in the 
 amount of watery vapor present. 
 
 The variations in the hygrometric state are still 
 further increased by vaporization and condensation. 
 Whenever a current of air moves upward its tempera- 
 ture is lowered by refraction, and a point may be 
 reached where the accompanying vapor can no longer 
 exist as such, and is condensed to cloud or even to rain 
 or snow. On the other hand, whenever a current of 
 air moves downward its capacity for moisture is in- 
 
BAROMETERS. [CHAP. Xll 
 
 creased, and it acquires the power to take up water 
 from whatever moist surface it comes in contact with. 
 At the surface of the earth there is an almost continu- 
 ous passage of moisture from ground to air, only a 
 part of which is returned as dew. The daily circulation 
 of the atmosphere incited by the heat of the sun carries 
 the moistened air upward, and eventually it is con- 
 densed and returned to the earth in the form of rain or 
 snow; but the condensation and succeeding precipita- 
 tion are exceedingly irregular. 
 
 The irregularities of humidity are greater proportion- 
 ally than the irregularities of temperature, but the 
 error in the difference of altitude due to humidity is 
 less than that due to temperature, because humidity is 
 a much smaller factor of the hypsometric problem. In 
 a very general sense, in temperate climates near the sea 
 level, the amount of vapor in the atmosphere varies 
 from 0.2 to 0.4 of an inch, or about one hundredth of the 
 whole. 
 
 361. Conclusion as to Humidity. As the methods em- 
 ployed to eliminate errors of gradient and temperature 
 ( 37 x ~8i) a ^ so eliminate errors due to humidity, the 
 subject will not be discussed further here. See 393- 
 
 95- 
 
 362. Instrumental Errors. Mercurial Barometer. The 
 errors that may exist in a mercurial cistern barometer 
 are (i) error in the position of the index, (2) imper- 
 fect graduation of the scale, (3) error in the position 
 of the zero and in the graduation of the attached ther- 
 mometer, (4) impure mercury, and (5) air in the tube. 
 The first is eliminated by comparing the instrument 
 with a standard barometer, and either re-adjusting the 
 zero of the scale or noting the correction to be applied 
 to the reading. With a good barometer properly 
 handled, the other errors will be inappreciable. 
 
 363. Aneroid Barometer. The index of the aneroid 
 
ART. 3] THE PRACTICE. 31 7 
 
 is easily displaced, and hence it should be frequently 
 compared with a mercurial barometer. The attached 
 thermometer also should be tested. See 343-45. 
 
 364. Errors of Observation. Mercurial Barometer. The 
 principal errors of observation are in making contact 
 between the ivory point and the mercury, and in adjust- 
 ing the vernier to coincide with the top of the mercurial 
 column (see 338). Gilbert, from a comparison of three 
 hundred and sixty pairs of observations made by the 
 Signal Service and by the Geological Survey, found 
 the average error of observation to be a trifle less than 
 three thousandths of an inch.* This difference does 
 not involve the personal equation between two ob- 
 servers, which, even for experts, may be nearly as much 
 more.f 
 
 365. Aneroid Barometer. Owing to parallax, there is 
 a liability of considerable error in reading the position 
 of the index. The amount of this error will depend 
 mainly upon the distance the index is from the scale; 
 but as the aneroid is not an instrument of any consid- 
 erable precision, this subject will not be discussed fur- 
 ther. See 343-45- 
 
 366. Thermometers. There is liability of error owing 
 to parallax in reading the attached and detached ther- 
 mometers. With the mercurial barometer an error of 
 o.i c 7. in reading the attached thermometer produces 
 an error of about 0.3 ft. in the elevation. Notice that 
 this source of error is independent of, and in addition 
 to, that discussed in 357-59. The latter is usually 
 very much the greater. 
 
 The chief source of error in determining the tempera- 
 ture of the air or of the instrument, is in the exposure 
 of the thermometer. It should be very carefully 
 shielded from both the direct and reflected rays of the 
 
 * U. S. Geological Report, 1880-81, p. 542. 
 t U. S. Coast Survey Report, 1870, p. 79. 
 
318 BAROMETERS. [CHAP. XII 
 
 sun, and the observer should be careful that the heat of 
 his own body does not affect the thermometer ( 338). 
 
 The brass scale is usually so thin that it undergoes 
 changes of temperature more rapidly than the mercury; 
 and therefore if the temperature of the surrounding 
 air be gradually raised, the brass scale responds more 
 promptly than the mercurial column and becomes rela- 
 tively too long, while the reverse takes place if the 
 temperature is lowered. The result is that a rising 
 temperature gives too low a barometric pressure, and a 
 falling temperature too high a one. If the change of 
 temperature is rapid, the error from this cause may 
 amount to ten or fifteen thousandths of an inch, which 
 corresponds to a difference of elevation of 10 or 15 feet. 
 The precaution generally recommended is to put the 
 barometer in position and leave it with unchanged con- 
 ditions for fifteen or twenty minutes before making the 
 observation. 
 
 367. Wind. The wind may cause either a condensa- 
 tion or a rarefaction of the air in the room in which 
 the barometer is hung, or even in the cistern of the 
 barometer itself. This effect will vary with the velocity 
 of the wind, the position of the openings with reference 
 to the direction of the wind, the size of the room, etc. 
 " On Mount Washington, a wind of 50 miles per hour 
 caused the barometer to read 0.13 of an inch too low. 
 The effect of the wind will vary as the square of its 
 velocity. It may be nearly, if not wholly, eliminated by 
 having two apertures, one each on the windward and 
 leeward side of the enclosed space." * 
 
 The wind has a similar effect when the instrument 
 is read in the immediate vicinity of any body which 
 obstructs the wind. For example, if the barometer 
 is observed on the windward side of a mountain, the 
 
 *G. K. Gilbert in Report U. S. Geological Survey for 1880-81, p. 562. 
 
ART. 3] THE PRACTICE. 319 
 
 reading will be too high; and if on the leeward, too 
 low. The only w r ay to eliminate this error is to select 
 stations not thus affected, although it is not always 
 possible to do so. 
 
 368. LIMITS OF PRECISION. It is sometimes stated that 
 "the barometer is the most accurate instrument for 
 determining differences of level;" but it needs only a 
 moment's reflection to see that this can not be true. 
 The following results, given by Professor Guyot,* are 
 frequently quoted as showing the great accuracy of 
 barometric leveling: 
 
 Mount Blanc, by barometer, 15,781 feet, 
 
 by spirit level, 15,780 feet. 
 Mount Washington, by barometer, 6,291.7 feet, 
 
 by spirit level, 6,293 feet. 
 In North Carolina, by barometer, 5,248 feet, 
 
 by spirit level, 5,246 feet. 
 In North Carolina, by barometer, 6,701 feet, 
 
 by spirit level, 6,711 feet. 
 
 A few examples showing a small error prove nothing 
 as to the accuracy of the method, for the agreement 
 may be wholly accidental. A fair inference from the 
 remarks of Professor Guyot accompanying the above 
 data, is that the mean of several observations taken 
 during one or two days will generally give as accurate 
 results as those above; but it is probable that he did 
 not intend to convey such an impression, for the very 
 observations on which he relies for his spirit level alti- 
 tude of Mount Washington resulted in a suggestion to 
 modify the constants in the barometric formula. J In 
 one instance his barometric results were in error 125 
 feet in a difference of altitude of 1,780 feet; and in 
 another 75 feet in a difference of altitude of 6,280 
 feet. 
 
 * Smithsonian Miscellaneous Collection, Vol. I, Art. II, Group D v p. 
 JUS. Coast Survey Report, 1854, p. 100*. 
 
320 BAROMETERS. [CHAP. XII 
 
 369. The difference of altitude computed from one, 
 or even several, day's observations can not be relied 
 upon as being more than a rough approximation. This 
 has been shown by Williamson,* who has computed 
 the difference of altitude between Geneva and St. Ber- 
 nard by the formula used in the last three examples 
 quoted above, for every day for two years, using the 
 daily means of simultaneous bi-hourly observations. 
 In several cases, the difference between the result by 
 the barometer and the spirit level was more than 3 
 per cent. Under less favorable circumstances the 
 errors were even more than twice as great. f The alti- 
 tude computed by the monthly means of bi-hourly ob- 
 servations for different months of the same year, and 
 also for the same month of different years, differ as 
 much as i per cent.J 
 
 The following differences between the results by the 
 barometer and the spirit level do not indicate that high 
 degree of accuracy in barometric hypsometry, even when 
 a long series of observations is used, which was formerly 
 supposed to be attainable by this means. The results 
 by the barometer were obtained by computing the dif- 
 ference of altitude from monthly means of the mean of 
 the daily observations, and taking the mean for the 
 time stated. 
 
 Vera Cruz and City of Mexico, I year's observations, 
 
 + 5 metres in 2,282 metres. 
 Sacramento and Summit, 3 years' observations, 
 
 24 feet in 6,989 feet. 
 Portland and Mt. Washington, 6 years' observations, 
 
 + 37 feet in 6,289 feet- 
 Geneva and St. Bernard, 12 years' observations, 
 
 2.6 metres in. 2,070 metres. 
 
 * Williamson's On the Barometer, pp. 194-205. 
 
 t U. S. Geological Survey Report for 1880-81, pp. 456-59. 
 
 \ Williamson's On the Barometer, p. 236. 
 
 S U. S. Coast and Geodetic Survey for 1881, p. 254. 
 
OF THK 
 
 UNIVERSITY 
 
 _. 
 
 ART. 3] THE PRACTIClL 12 1 
 
 For an interesting comparison of the absolute, and 
 also the relative, errors of the various methods, see 
 Chap. Ill of the Report of U. S. Geological Survey for 
 1880-81. For additional data concerning the accuracy 
 of barometric leveling, see Report of the U. S. Coast 
 and Geodetic Survey for 1870, p. 88; ibid., 1871, pp. 
 154-75; ibid., 1876, pp. 356-76; Williamson's On the 
 Barometer, p. 205. 
 
 370. Although the barometer can not be regarded as 
 a hypsometric instrument of great precision, yet with 
 care it can be made to give results with sufficient accu- 
 racy for reconnoissance or exploration. For this pur- 
 pose, it is unexcelled by any other instrument; and this 
 is about the only use the engineering profession can 
 make of it. 
 
 371. METHODS OF OBSEKVING. In 348-61 it was 
 
 shown that barometric leveling was liable to large 
 errors owing to barometric gradient, and to errors in 
 determining the temperature and humidity of the at- 
 mosphere. In the best work, the observations are made 
 in such a way as to eliminate part of these errors. The 
 several methods employed to accomplish this will now 
 be considered. 
 
 372. Single Observations. The simplest, and also the 
 most common, method consists in using only one ba- 
 rometer, which is carried from station to station. By 
 taking one or more observations at each station, the 
 errors of observation (364-67) are nearly eliminated; 
 but the vastly greater errors due to gradient, tempera- 
 ture, and humidity ( 349-61) are undiminished. Re- 
 sults obtained by a single observation, or even by 
 several in quick succession, are only the rudest kinds of 
 approximations (354), and the greater the horizontal 
 and vertical distances between the two stations the 
 greater the error. 
 
 Distant stations are sometimes connected by inter- 
 
322 BAROMETERS. [CHAP. XII 
 
 mediate ones to eliminate changes of gradient, tem- 
 perature, etc., during the time the exigencies of travel 
 require the barometer to remain at the intermediate 
 station. For example, to determine the difference of 
 elevation between A and C, make an observation at A 
 and proceed towards C, making an observation at B, an 
 intermediate point, on arrival at that station and a 
 second one on leaving it; and finally make an observa- 
 tion at C. The difference of elevation of A and B is 
 determined from the first two readings, and that of B 
 and C from the second two; and the difference of eleva- 
 tion of A and C is equal to the sum of the partial differ- 
 ences. 
 
 373. Simultaneous Observations. The errors due to 
 change of. gradient (349-56) are partially eliminated by 
 making simultaneous observations at the two stations. 
 If the phase and the amplitude of the variation were the 
 same at both stations, which probably seldom or never 
 occurs, simultaneous observations would give results 
 independent of changes in the gradient. 
 
 Errors due to gradient are still further reduced by 
 making a number of simultaneous observations and 
 using the mean. This may eliminate the variable ele- 
 ment, but fails to take account of permanent gradient 
 (see 369). 
 
 374. Observations at Selected Times. It is often recom- 
 mended that the observations be made at certain hours 
 of the day, at which time the diurnal gradient is supposed 
 to be zero. These times can be determined only by ob- 
 servation, and will vary greatly with the state of the 
 atmosphere, the season, the locality, the elevation, etc. 
 The U. S. Coast Survey recommend the following 
 times.* They were probably deduced for the middle 
 Atlantic coast. The hours refer to the middle of the 
 
 * Report for 1876, p. 349. 
 
ART. 3] THE PRACTICE. 323 
 
 month, and other times are to be determined by inter- 
 polation. 
 
 January I P.M. 
 
 February 10 A.M. and 4 " 
 
 March 8 " 6 
 
 April 7 30 " " 7 
 
 May 7 " 7 
 
 June 6.30 " " 9.30 " 
 
 July 6.30 " " 9.30 " 
 
 August 7 " " 7.30 " 
 
 September 8 " "6 
 
 October 10 " " 3.30 " 
 
 November 10.30 " " 2,30 " 
 
 December at no time. 
 
 Obviously, a single observation, even if taken in ac- 
 cordance with a table similar to the above, could be 
 considered only a rough approximation owing to the 
 liability of error due to non-periodic gradient ( 354). 
 For a long series of observations, the results would 
 probably be more accurate if the observations were 
 made in accordance with some such scheme; but the 
 determination of the best times at which to make the 
 observations would necessitate an extensive prelimi- 
 nary series of observations at short intervals. Such a 
 method is clearly inapplicable, since the time and expense 
 required to obtain approximate results by the barometer 
 would be nearly or quite as great as that required to 
 determine an equal number of more accurate results by 
 the stadia (Chapter X) or by spirit leveling (Chapter 
 XI). 
 
 375. Notice that none of the preceding methods 
 eliminate the error due to the fact that the mean of the 
 observed temperatures does not represent the mean 
 temperature of the air column ( 357-59). 
 
 376. Williamson's Method.* This method is specially 
 adapted to reconnoissances and topographical surveys. 
 
 * Williamson's On the Use of the Barometer, pp. 39-42, 150-59. 
 
324 BAROMETERS. [CHAP. XII 
 
 A centrally located station, called a base station, is 
 chosen, at which the barometer and thermometers are 
 read at stated hours each day for several days. In the 
 meantime, itinerary observers make observations at the 
 points whose elevations are to be determined, taking 
 pains to have each observation correspond in time with 
 one of the observations at the base station. In the 
 progress of the itinerary survey, a series of observa- 
 tions, similar to those at the base station, are made as 
 frequently as practicable at semi-permanent camps. 
 The object of the series at the base station and at the 
 semi-permanent camps is to ascertain the nature of the 
 diurnal variation of pressure and temperature. 
 
 The barometric readings at the base stations, after 
 being corrected for temperature of the instruments 
 ( 389)' are plotted upon ruled paper so as to exhibit 
 their curve, and all readings shown by inspection to 
 be influenced by abrupt and violent atmospheric dis- 
 turbances, such as thunder storms, are discarded, their 
 places being filled by interpolated values. From the 
 corrected observations at the base stations, a correction 
 is deduced, which, being applied to the several baro- 
 metric readings, reduces them to the daily mean. Ap- 
 plying this correction eliminates at least part of the 
 effect of the diurnal gradient (352). 
 
 Instead of determining the temperature of the air 
 column from the temperature at the time of observing, 
 the mean temperature of the day is used. This can be 
 quite accurately determined at the base stations, but is 
 only approximately known at the other stations. No- 
 tice, however, that the mean of the observed tempera- 
 tures will not be the mean temperature of the stratum 
 of air included between the two stations ( 357-59). 
 
 The difference of altitude can be computed from the 
 reduced barometric readings and the mean daily tem- 
 perature, by using any statical formula (Art. 4). Wil- 
 
ARTI 3] THE PRACTICE. 325 
 
 liamson himself used his translation of Plantamour's 
 formula ( 401). 
 
 377. Whitney's Method. From observations made in 
 connection with the Geological Survey of California, a 
 series of corrections were deduced for reducing the 
 barometric readings made at different hours of the day 
 of the different days of the different months, and for 
 different altitudes, to the daily mean for the year. 
 
 These corrections can only be used in the neighbor- 
 hood in which the observations on which they were 
 based were made. Similar tables made for different 
 climates differ materially from each other. For tables 
 constructed upon this principle for the climates of Ger- 
 many, Philadelphia, and Greenwich, respectively, see 
 Smithsonian Miscellaneous Collection, Vol. I (3d Edi- 
 tion), Art. Ill, Group D, pp. 80-82, 86, and 93-94. 
 
 378. Plantamour's Method. In the hypsometric survey 
 of Switzerland, Plantamour made simultaneous obser- 
 vations of the barometer, thermometer, and psychrom- 
 eter at Geneva, St. Bernard, and at the station whose 
 height was to be determined. The approximate differ- 
 ence of altitude between the new station and Geneva, 
 and between it and St. Bernard, and also between 
 Geneva and St. Bernard, were computed by Plan- 
 tamour's formula ( 401). The computed difference of 
 elevation between Geneva and St. Bernard compared 
 with the difference of altitude determined by the spirit 
 level, gave a correction to be applied to the computed 
 difference for the new station. The ingenious details 
 of the computation are too complex to be described 
 here. 
 
 Marshall employed, in the geographical surveys in 
 the Rocky Mountains, a method somewhat similar to 
 the above.* 
 
 * U. S. Geological Surveys West of the zooth Meridian, Vol. II, pp. 522, 
 523. 
 
326 BAROMETERS. [CHAP. XII 
 
 379. Riihlmann's Method. This method differs from 
 Plantamour's ( 378) chiefly in that Riihlmann makes 
 use of the pressure and the moisture factors observed 
 at two stations whose difference in level is known, to 
 compute the temperature of the intervening air column. 
 The temperature thus derived he afterwards applies in 
 the computation of the unknown difference in level of 
 two other stations. 
 
 This method is applicable only to a detailed topo- 
 graphical survey, and requires such an elaborate series 
 of observations at base stations that the time and ex- 
 pense are nearly, or quite, as great as that required to 
 determine the results with the stadia (Chap. X) or the 
 spirit level (Chap. XI). 
 
 380. Gilbert's Method.* This method is practically a 
 combination of Plantamour's and Riihlmann's, and 
 requires simultaneous observations of the barometer at 
 three stations, the vertical distance between two of 
 which is known, but does not require observations of 
 the temperature and humidity. From the known dif- 
 ference between two of the stations and the observations 
 at each, the actual density of the air, which is dependent 
 upon the pressure, temperature, and humidity, may be 
 determined by reversing the ordinary barometric for- 
 mula. The true density is then used to compute the 
 difference of elevation between either of these stations 
 and the third. For the formula employed in this 
 method, see 407. 
 
 This method has not proved as satisfactory in prac- 
 tice as was hoped when it was proposed. It is simply 
 another attempt to secure accurate results by a modifi- 
 cation of a general method which is inaccurate in its 
 essential features. 
 
 381. Conclusion as to Methods. Unfortunately all 
 methods of eliminating gradient and temperature 
 
 * Report oL U. S. Geological Survey *or ^080-81, pp. ^05-561. 
 
ART. 4] BAROMETRIC FORMULAS. 327 
 
 errors involve considerable time and expense, and even 
 then do not thoroughly accomplish the desired end, 
 which shows that when great accuracy is desired the 
 barometer should be dispensed with altogether, and the 
 difference of elevation be determined by some other 
 means. 
 
 ART. 4. BAROMETRIC FORMULAS. 
 
 382. Barometric leveling formulas may be divided 
 into two classes; viz., (A) those that assume the at- 
 mosphere to be in statical equilibrium, and (B) those 
 that take account of the fact that the air is more or 
 less in motion. The first will be designated statical 
 formulas, and the latter dynamical. The former are by 
 far much more frequently used. 
 
 A. Statical Formulas. 
 
 383. ASSUMPTIONS. All the barometric leveling formu- 
 las in common use are dependent upon the assumption 
 that the air is in a state of statical equilibrium.* If 
 this state were possible, we might suppose the whole 
 atmosphere to be arranged in a system of horizontal 
 layers, each of which is denser than the one above it 
 and rarer than the one below it, each being uniform 
 throughout in temperature and humidity. 
 
 But the air is never in a state of statical equilibrium, 
 but is perpetually undergoing local changes of pressure, 
 temperature, and humidity. For example, during the 
 day the sun imparts a certain amount of heat to the 
 whole atmosphere, but a much higher temperature is 
 given to the ground and by it communicated to the 
 contiguous layer of air. At night the atmosphere loses 
 heat by radiation to space, but the ground loses it more 
 
 * There are only three not thus included (see 406-8), and practically they 
 are not used. 
 
328 BAROMETERS. [CHAP. Xll 
 
 rapidly and imparts its low temperature to the lowest 
 stratum of air. The lower strata, therefore, have 
 exceptional warmth by day and exceptional coolness by 
 night. Again, if the air is moist, during the day it 
 intercepts a greater quantity of solar heat than if it 
 were dry, so that a less quantity reaches the ground; 
 but at night the moisture checks radiation from the 
 ground. The power of the earth's surface to receive or 
 part with heat varies with its character; naked rocks 
 and cultivated fields, bare earth and grass, forest and 
 snow are affected very differently by the heat rays of 
 the sun, and exert equally diverse influences on the 
 adjacent air, so that one tract of land is often in a con- 
 dition to heat the air while an adjacent tract is cooling 
 it. Then, too, the sun's heat is unequally distributed 
 through the year; outside the tropics there is a pro- 
 gressive accumulation of heat through summer and a 
 progressive loss through winter. The ocean undergoes 
 less change of temperature than the land, and its rate 
 of change is slower, so that there is frequent, and 
 indeed almost continuous, contrast of condition be- 
 tween it and the contiguous land. As a result of all 
 these influences, together with others that might be 
 enumerated, the equilibrium of the air is constantly 
 disturbed, and the winds, which tend to readjust it, 
 are set in motion. 
 
 The temperature of the air is continually modified 
 by external influences; the static order of densities is 
 broken and currents are set in motion; and the circula- 
 tion and the inequalities of temperature also conspire 
 to produce inequalities of moisture. Every element of 
 equilibrium is thus set aside, and the air is rendered 
 heterogeneous in density, temperature, and humidity. 
 
 384. All of the common barometric formulas are 
 dependent upon two assumptions; viz., (i) that a dif- 
 ference of pressure is due only to a difference of eleva- 
 
ART. 4] STATICAL FORMULAS. $2$ 
 
 tion ; and (2) that the mean of the temperatures of 
 the air at the two stations represents the mean tem- 
 perature of the level stratum of air between them. A 
 few of the formulas involve also a third assumption; 
 viz., that the mean humility of the air at the two sta- 
 tions is the same as the humidity of the layer between 
 them. 
 
 A consideration of the facts stated in 383 and dis- 
 cussed more fully in 348-61, will show that any 
 formula dependent upon the preceding assumptions 
 can be, at best, only approximate. In 371-81 were 
 explained several methods of reducing the errors due 
 to the above assumption; but notice that errors of 
 gradient, temperature, and humidity were 
 eliminated by the methods of making the 
 observations and computing the results, 
 rather than by the use of any particular 
 formula. -C 
 
 385. FUNDAMENTAL RELATIONS. Let A and 
 
 B, Fig. 85, represent the two stations, and 
 assume that it is required to determine the 
 vertical distance between them. A and B 
 are not necessarily, and not even usually? 
 in the same vertical line. Let C represent FlG - 8 s- 
 any point in AB, and D a point an infinitesimal dis- 
 tance below C. 
 
 Let a = the weight of a cubic inch of dry air at the 
 sea level, in latitude 45, at 32 P F., when 
 the barometer stands at 29.92 inches. 
 a the weight of a cubic inch of air under the 
 pressure, temperature, etc., existing be- 
 tween C and D. 
 
 c = the co-efficient of expansion of air. 
 jy = the height of the barometric column at the 
 sea level in latitude 45, 29.92 inches. 
 
330 BAROMETERS. [CHAP. XII 
 
 If = the height, in inches, of the barometer at the 
 
 upper point, reduced to 32 F. 
 H^ = the height, in inches, of the barometer at the 
 
 lower point, reduced to 32 F. 
 
 M O = the weight of a cubic inch of mercury at 
 the sea level, in latitude 45, when the ba- 
 rometer reads 29.92 inches. 
 m' = the weight of a unit volume of mercury at 
 
 the upper station. 
 m l = the weight of a unit volume of mercury at 
 
 the lower station. 
 
 = the mean latitude of the two stations. 
 P the pressure per square inch, at D. 
 dP = the difference in pressure between C and Z>, 
 
 />., dP is the differential pressure. 
 P 9 = the pressure per square inch, at the sea 
 
 level, in latitude 45. 
 P' = the pressure at the upper station. 
 P l = the pressure at the lower station. 
 R = the radius of the earth. 
 T the mean temperature of the layer of air 
 
 between A and B. 
 T f = the temperature of the air at the upper 
 
 station. 
 T t = the temperature of the air at the lower 
 
 station. 
 
 X = the difference in elevation. 
 Z the elevation of the lower station above the 
 
 sea level. 
 
 386. It is clear that the increase of pressure from C 
 to D is equal to the weight of a column of air having 
 a unit section and the height CD. Expressing this in 
 the language of the calculus, we have 
 
 dP a dx (i) 
 
ART. 4] STATICAL FORMULAS. 33! 
 
 By Marriotte's law, a : a : : P : P , from which 
 
 ( 2 ) 
 
 Equation (2) gives the weight of a unit of air at 32 
 F., and for any other temperature T 9 
 
 Equation (3) gives the weight of a unit of air at lati- 
 tude 45, and for any latitude 0, 
 
 P_ I _ I _ v 
 
 **/> 1 +CT I +0.002,60 COS 2 0* 
 
 Equation (4) gives the value of a at the sea level, and 
 for any other altitude (Z -\- X), 
 
 ~~ a 
 
 p o I J r cT i + 0.002,60 cos 2(p 
 
 Combining equations (i) and (5), integrating, and 
 transposing, 
 
 P P 1 * 
 
 TT~ '*&~D' ~ i T^r 
 d n f I H- 6 2 
 
 I -j- O.OO2,6o COS 20 ^ 2 
 
 A = -#o w o = 29-92 W . . . . (7) 
 
 - i *~-i i /o\ 
 
 "D7 ~ rr/T , V b ; 
 
332 
 
 BAROMETERS. 
 
 [CHAP, xii 
 
 The weight of a body varies inversely as the square 
 of the distance from the center of the earth, and there- 
 fore 
 
 m 
 
 (R + Z) a 
 
 (9) 
 
 Combining equations (6), (7), (8), and (9), we get 
 
 - 
 
 (l-}-0.002,6oCOS 20) 
 
 (10) 
 
 The quantities at the right of the brace are three 
 factors of the second member of the equation. 
 
 Substituting, in equation (10), the sum of the loga- 
 rithms for the logarithm of the product, passing to 
 common logarithms, and expressing X y Z, and -#, in feet ', 
 
 = 5-744.- log 
 
 (l -(- 0.002,60 COS 20) 
 
 X 
 
 2l g 
 
 i + 
 
 Notice that the preceding equation involves no ap- 
 proximations. 
 
 387. Equation (n) includes the principal relations 
 involved in determining differences of height with the 
 barometer. The formula to be used in practice has 
 
ART. 4] STATICAL FORMULAS. 333 
 
 been given differently by different investigators, accord- 
 ing to the values chosen for the constants, to the indi- 
 vidual preference for one form over another, and to the 
 degree of refinement desired. 
 388. The Constant. The value of the term 5.744 
 
 ... -, generally known as the barometric co-efficient, 
 will depend upon the values for the densities of air and 
 mercury which are used. Boit and Arago found - 
 
 
 
 = 10,467,* which makes the barometer co-efficient 
 60,096.3 ft. (18,317 meters). Regnault's values,* which 
 are the most recent and probably the most accurate, 
 give 60,384 feet (18,404.8 meters). 
 
 Raymond in 1803 found * the value of the barometric 
 co-efficient by determining the value it should have to 
 make the results by the formula agree with those fur- 
 nished by trigonometrical leveling. The value ob- 
 tained in this way is 60,158.6 ft. (18,336 meters). Even 
 under the most favorable circumstances, the observa- 
 tions, eight in all,f were too few to determine such a 
 co-efficient with sufficient accuracy, owing to the varia- 
 tions in the barometric pressure ( 349-56), the tem- 
 perature ( 357-59), the humidity ( 360), and the 
 refraction ( 318). The method employed by Ray- 
 mond is the least accurate of any, although his co-effi- 
 cient is more frequently used than any of the others. 
 
 One of the author's students determined the baro- 
 metric co-efficient by reversing Laplace's formula, 
 equation (13), page 339, and using the barometric 
 readings made by the U. S. Weather Bureau, together 
 with the difference of level of the stations. The 
 results are not very satisfactory owing to the uncer- 
 
 * Smithsonian Miscellaneous Collections, Vol. I, Guyot's Meteorological 
 Tables, Group D, page 9. 
 f Report U. S. Coast and Geodetic Survey, 1881, p. 235. 
 
334 BAROMETERS. [cHAP. XTt 
 
 tainty in the values for the differences of level. The 
 co-efficient obtained from twelve pairs of stations, 
 using the mean of tri-daily observations for three to 
 twelve years, is 60,156 feet.* 
 
 For a discussion showing that the errors due to 
 gradient and temperature are very much greater than 
 the errors due to the differences between the several 
 barometric constants, see Williamson's On the Use of 
 the Barometer, pages 221-33. 
 
 389. Temperature of the Barometer. Before insert- 
 ing the barometric readings in equation (u), page 332, 
 they must be reduced to the corresponding heights 
 they would have at a common temperature. This cor- 
 rection may be made in either of two ways; viz., (i) one 
 barometer may be* reduced to the temperature of the 
 other, or (2) both may be reduced to any other temper- 
 ature assumed as a standard. 
 
 i. The co-efficient of expansion of mercury is o.ooo,- 
 100,1 for i F.; and that of brass, of which the scales 
 are generally made, is 0.000,010,4. The difference the 
 relative co-efficient of expansion of mercury is o.ooo,- 
 089,7. For the centigrade scale this difference is 
 0.000,161,41. If h' represents the height of the barom- 
 eter at the upper station reduced to the temperature 
 of the lower, and f and /, the temperature of the 
 barometers at the upper and lower stations respectively, 
 
 h' = ff'[l - d(f - /,)] (12) 
 
 in which d stands for one of the above differences, 
 according to the kind of thermometer used. When 
 this method is employed, a term is inserted in the 
 formula to correct for the difference in temperature of 
 the barometers. For an example see 400. 
 
 * Bachelor's Thesis of Edward E. Ellison, Class of '88, University of 
 Illinois. 
 
ART. 4] STATICAL FORMULAS. 335 
 
 2. To reduce both readings to a common tempera- 
 ture, as for example the freezing point of water, apply 
 equation (12) to both readings, considering t l to repre- 
 sent the temperature of melting ice (32 F. or o C.) 
 and f the reading of the attached thermometer. Nu- 
 merous tables have been computed for facilitating this 
 reduction; for example, Smithsonian Miscellaneous 
 Collection, Vol. I, Guyot's Collections (3d ed.), Group 
 C, pp. 61-127; ibid., Group D, pp. 30, 46, 53, etc.; Lee's 
 Tables (3d ed.), pp. 152-59; Williamson's On the Use of 
 the Barometer, pp. 1-64 of the appendix. 
 
 390. Temperature of the Air. The term (i -f c T) is 
 frequently called the temperature term. The name is 
 not fortunate, since some barometric leveling formulas 
 have also a term to correct for the difference of tem- 
 perature of the barometers ( 389). 
 
 The co-efficient c is equal to 0.003,75 P er degree cen- 
 tigrade. It is usually approximated at 0.004, which 
 makes some allowance for the greater expansive power 
 of the watery vapor contained in the atmosphere. 
 
 The mean temperature of the layer of air between 
 the two stations is usually assumed to be equal to the 
 mean of its temperature at the two stations. Under 
 this assumption, which however may be greatly in 
 error (see 357-5 8 )> ?= i(^ + ^')> and the temper- 
 ature term becomes 
 
 2(2* -4- T) 
 
 j _| l H for centigrade degrees, 
 
 and 
 
 rrif I fji /- 
 
 i -| for Fahrenheit degrees. 
 
 900 
 
 391. Latitude Term. The term (i -f- 0.002,60 cos 20) 
 of equation (IT), page 332, is known as the latitude 
 
536 BAROMETERS. [CHAP. XII 
 
 term. A few formul as still contain an older and less 
 accurate co-efficient of cos 20 than the above. How- 
 ever, this correction is unimportant, for even at the 
 equator or the poles, where this term is a maximum, it 
 is only 0.002,60 of the difference of elevation. Further- 
 more, in comparison with the possible errors due to 
 gradient, temperature, and observation (see 349-67), 
 the error caused by the omission of the latitude term 
 is wholly inappreciable. 
 
 392. Altitude Term. The last term of equation (n), 
 page 332, takes account of the variation of gravity 
 with the altitude. This term is always relatively small, 
 and may be omitted without materially affecting the 
 result. Furthermore, owing to the matters considered 
 in 349-67, as well as the fact that the barometer at 
 best can be read only to a thousandth of an inch 
 (which corresponds to i foot of altitude), the appear- 
 ance of extreme accuracy by retaining the altitude and 
 latitude terms can be regarded only as a mathematical 
 illusion, inapplicable to ordinary practice. 
 
 393. Humidity Term. In deducing equation (n), page 
 332, it was assumed that the atmosphere was com- 
 posed exclusively of dry air; but it is really a mixture 
 of air (oxygen and nitrogen), watery vapor, and carbonic 
 acid. The carbonic acid is very small and nearly con- 
 stant, and hence need not be considered here ; but the 
 watery vapor is both large and variable. If dry air and 
 aqueous vapor had even nearly the same density under 
 the same conditions, the presence of the latter would 
 not affect the problem ; but as watery vapor is only five 
 eighths as dense as dry air, the weight of a unit of vol- 
 ume of the atmosphere will depend upon the amount 
 of vapor which it contains. Accordingly a humidity 
 term has been introduced into some barometric for- 
 mulas. 
 
 The introduction of a humidity term requires that 
 
ART. 4] STATICAL FORMULAS. 337 
 
 the hygrometric state of the air column shall be known, 
 and therefore an observation with a hygrometer must be 
 made at each station. For this purpose the wet bulb 
 hygrometer, or psychrometer, is generally preferred, 
 because of its greater accuracy and convenience. It 
 consists of a pair of accurate thermometers, one of 
 which is exposed to the air in the usual manner, while 
 the other is exposed with a moistened bulb. The 
 evaporation of moisture from the surface of the wetted 
 bulb has a cooling effect, and causes that thermometer 
 to indicate a lower temperature than the other. 
 
 Knowing the readings of the wet and dry bulb 
 thermometers, the barometric pressure due to the 
 aqueous vapor in the air may be determined from 
 tabtes,* which are the results of experiment. 
 
 "In a very general sense, in temperate climates near 
 the sea level, the amount of vapor in the atmosphere is 
 from 0.2 to 0.4 of an inch, or about one hundredth of 
 the total barometric pressure." 
 
 394. The correction for humidity may be applied in 
 either of two ways ; viz., (i) the observed heights of 
 the barometer may be corrected for the pressure of the 
 aqueous vapor before substituting them in the formula; 
 or (2) the observed heights may be used uncorrected, 
 and the resulting altitude be multiplied by a factor the 
 humidity term of the barometric formula to correct 
 for the effect of the aqueous vapor in the atmosphere. 
 For formulas containing a humidity term, see 401 
 402. Laplace slightly increased the co-efficient of expan- 
 sion of air ( 396) to partially compensate for the greater 
 expansive power of the aqueous vapor. This method of 
 correcting for humidity is incorrect in principle, and 
 
 * Williamson's On the Use of the Barometer, Table C of the Appendix ; 
 Smithsonian Miscellaneous Collections, Vol. I, Guyot's Tables, Group B, pp. 
 46-72, and pp. 102-6; Lee's Tables (3d ed.), pp. 172-76. 
 
338 BAROMETERS. [CHAP. XII 
 
 may at times give rise to considerable error, particu- 
 larly in case of an extremely dry or an extremely moist 
 atmosphere, or when the temperature of the air is at or 
 near 32 F. 
 
 395. " It Is doubtful whether any considerable in- 
 crease of accuracy is obtained by including a separate 
 correction for the aqueous vapor. The laws of the 
 distribution and transmission of moisture through 
 the atmosphere are too little known, and its amount, 
 especially in mountain regions, is too variable and 
 depends too much upon local winds and local condensa- 
 tion, to allow a reasonable hope of obtaining the mean 
 humidity of the layer of air between the two stations 
 by means of hygrometrical observations taken at each 
 of them. The observations for humidity are made in 
 the stratum of air next to the surface of the earth, 
 which probably contains the greatest amount of moist- 
 ure, and which is therefore least representative of the 
 layer of air between the two stations. At any rate, the 
 gain, if there is any, is not sufficient to compensate for 
 the extra trouble in making the observations and the 
 undesirable complication of the formula." * 
 
 At best the determination of heights by a barometer 
 is only an approximate method ( 368-69), and no 
 additional instruments, nor any refinements of compu- 
 tation, can make it a comparatively accurate method 
 of leveling ; and therefore it is not thought wise to 
 consider this phase of the subject further. 
 
 396. TYPICAL FORMULAS. Laplace's Formula. Laplace 
 was the first to give a rational formula for determining 
 heights by the barometer, and his formula has served 
 as a basis for several others. It is, for Fahrenheit de- 
 grees, 
 
 * Professor A. Guyot, in Smithsonian Miscellaneous Collections, Vol. I, 
 Art. Ill, Group D, p. 33. 
 
ART. 4] STATICAL FORMULAS. 339 
 
 X ft. = 60158.6 log < 
 
 (l+ : ^-~ 
 
 (l -j- 0.002,60 COS 20). (13) 
 
 , ( l + ~2^886,860 + 10,443,435]' 
 
 X in the last term is the value of the preceding part of 
 the formula. 
 
 The last term of equation (13) is derived from the 
 corresponding term of equation (n), page 332, as fol- 
 
 lows : Place lo i + - = 0.4 . . ---. Find 
 
 log (i + -^ J = 0.43 . . - 
 
 TT 
 
 log \ by omitting the terms in equation (u) after the 
 
 brace and solving, calling 5.744 - = 60,158. Substi- 
 
 tute these results in the last term of equation (n), per- 
 form the operations indicated, and omit the terms con- 
 taining R* in the denominator. See 392. 
 
 The true co-efficient of expansion of air is 0.003,75 
 per i C, but Laplace increased it to 0.004 " to allow 
 for the greater expansive force of the aqueous vapor in 
 the atmosphere." * See 394 and 395. 
 
 The second term in the last line of equation (13) is 
 usually called " the correction for the variation of grav- 
 ity on the mercury," and the third term in the same 
 line " the correction for the variation of gravity on the 
 air"; but an examination of the method by which the 
 last term was deduced will show that this explanation 
 of these terms is erroneous. 
 
 Numerous tables have been prepared to facilitate the 
 application of the above formula ; for example, see 
 
 * Prof. A. Guyot in Smithsonian Miscellaneous Collections, Vol. I, Art. 
 Ill, Group D, p. 9. 
 
340 BAROMETERS. [CHAP. XII 
 
 Smithsonian Miscellaneous Collections, Vol. I, Art. Ill, 
 Group D, Guyot's Table, pp. 35-48 ; ibid. t Loomis's 
 Table, pp. 49-53 ; Loomis's Practical Astronomy, pp. 
 390-91. 
 
 Tables are useful where a great number of observa- 
 tions are to be reduced; but they generally contain an 
 unnecessary number of figures, and hold forth a show 
 of extreme accuracy which the nature of the observa- 
 tions can not justify. 
 
 397. Babinet's Formula. Babinet's formula* for Fahren- 
 heit degrees is 
 
 X ft. = 60,334 log 1 + . ( I4 ) 
 
 Notice that this formula has no term for the variation 
 of gravity. It is sometimes claimed f that the baromet- 
 ric co-efficient is Laplace's ( 396) increased to include 
 the correction due to the variation of gravity with the 
 altitude; but owing to the nature of the case this 
 procedure is entirely indefensible 
 
 Searle's Field Engineering (p. 4 and pp. 307-9) con- 
 tains tables to facilitate the application of this formula. 
 
 398. Babinefs Approximate Formula. If H' and H v do 
 not greatly differ it can readily be shown that 
 
 H H H ,.. 
 
 in which M is the modulus of the common system of 
 logarithms. Making this substitution in equation (14) 
 gives Babinet's approximate formula, 
 
 * Guyot's Tables, Group D, p. 68. 
 
 t Prof. A. Guyot in Smithsonian Miscellaneous Collection, Vol. I, Art, III, 
 Group D, p. 9. 
 
ART. 4] STATICAL FORMULAS. 341 
 
 The mathematical approximation involved in equa- 
 tion (15) is inappreciable for elevations less than 3,000 
 feet. 
 
 399. The following formula* is essentially the same 
 as Babinet's approximate formula equation (16) ex- 
 cept the value of the barometric co-efficient and the 
 omission of the temperature term. Notice that the 
 barometric constant of equation (17) is larger than that 
 of equation (16), which was derived from a relatively 
 large value, i.e., that of equation (14). A similar 
 formula is freqr^ntly given with slightly different 
 constants ; for example, see Lee's Tables (3d ed.), 
 p. 151. 
 
 TT _ TTt } 
 
 X ft. = 54,500^ + ff , 10. . . . . (17) 
 
 The last two terms of equation (17) are supposed to 
 show the degree of reliance to be placed upon the 
 result. Formulas (16) and (17) are much used with 
 "compensated" aneroid barometers (see 342-43). 
 
 400. Bailey's Formula. All the preceding formulas 
 require that the barometric reading be reduced to 32 F. 
 before being inserted in the formula; but in Bailey's 
 formula, equation (i8),f the readings are used without 
 any reduction (see 389). 
 
 X ft. =60,346 log h -\ 
 
 h* i + o.oooi^j /) 
 
 r + r, -.jS4 
 
 (l + O.OO2,695 ^^ 20) 
 
 * U. S. Coast Survey Report, 1876, pp. 352-53. 
 t Guyot's Tables, Group D, p. 69. 
 
342 BAROMETERS. [CHAP. XII 
 
 in which h l and h* are the observed readings of the 
 barometers at the two stations, and t l and /' the tem- 
 peratures of the barometer in Fahrenheit degrees. 
 
 Tables for the application of Bailey's formula are 
 given in Lee's Tables (2d ed.), pp. 84, 85, and in 
 Smithsonian Miscellaneous Collections, Vol. I, Art. Ill, 
 Group D, pp. 70, 71. 
 
 401. Plantamour's Formula. None of the preceding 
 formulas takes account of the humidity of the at- 
 mosphere, except by the erroneous method of increasing 
 the co-efficient of expansion of air ( 396). Bessel was 
 the first to propose a separate humidity term. Planta- 
 mour's formula,* which differs from the one proposed 
 by Bessel only in the form and the value of the con- 
 stants, is 
 
 X (in meters) = log H^ - log H' + V 
 
 + V -r+* ; to) 
 
 I-JF44r 
 
 in which 
 
 = 398.25 jg, 
 
 397.25 - cT a 
 
 30.030,197,57- - 0.000,080,1701 
 
 397.25 - cT 
 q = i 0.002,625,7 cos 0, 
 
 s and $' = the fraction of saturation of the layer of air 
 between the two statrons. 
 
 Tables for the application of this formula are given in 
 Smithsonian Miscellaneous Collections, Vol. I, Guyot's 
 Tables, Group D, pp. 78, 79. 
 
 * Guyot's Tables, Group D, p. 75. 
 
AkT. 4] STATICAL FORMULA^ 343 
 
 402. Williamson's Formula. Williamson's formula,* 
 equation (20), is a translation of Plantamour's formula 
 into Laplace's form. 
 
 X ft. = 60,384 log -j f 
 
 7^4.7* -64 
 
 982.2647 
 
 (l 0.002,625,7 COS 20) 
 
 z x (20) 
 
 
 20,886,860 10,443,4307 
 
 ((i+M) 
 
 in which (i -j- ^O ' ls tne humidity term ( 393-95). 
 
 Williamson's On the Use of the Barometer pages 
 111-55 f the Appendix contains elaborate tables for 
 the application of this formula. The same tables are 
 also given in Lee's Tables (3d ed.), pp. 152-71. 
 
 403. Altitude Scales of Aneroids. The dials of many 
 aneroid barometers have a scale engraved upon them, by 
 which the elevation is read directly. The scale may be 
 graduated according to any of the preceding formulas, 
 but apparently equation (17), page 341, is most frequently 
 employed for this purpose. Sometimes the zero of the 
 altitude scale is placed to correspond with a pressure of 
 30 inches of the mercurial barometer, as though the 
 scale of elevations could be employed to determine ab- 
 solute elevations. But, obviously, the altitude scale can 
 not give absolute elevations with any considerable ac- 
 curacy, owing to the variations in pressure, temperature, 
 and humidity (349-61); and consequently it should 
 be used only to find differences of elevations, in which 
 case the difference of level is obtained by subtracting 
 
 * Williamson's On the Use of the Barometer, p. 102 of the Appendix, 
 
344 BAROMETERS. fcHAP. XII 
 
 the reading in feet at the lower station from that at the 
 upper. To prevent a misuse of the aneroid in this re- 
 spect, the zero of the altitude scale is sometimes placed 
 to correspond with 31 inches of pressure, which gives 
 such large numbers as to suggest, particularly when find- 
 ing small elevations, the proper use of the instrument. 
 
 Sometimes the altitude scale is engraved upon a mov- 
 able ring encircling the dial, so that the scale may be 
 set to agree with the known altitude of any station (see 
 343, paragraph 7) ; and then, as the instrument is 
 carried about, the pointer will indicate with a fair de- 
 gree of approximation, for some hours, the altitude of 
 stations at which it is read. This device is convenient, 
 and in the hands of an intelligent observer who requires 
 rapid work and desires only approximate results, it is a 
 valuable modification; but such a use of the movable 
 scale may at times involve large errors, as it is based 
 on the assumption that differences of pressure cor- 
 respond at all heights with the same differences of ele- 
 vation. 
 
 404. The altitude scale can be correct only for some 
 particular temperature of the air; and consequently, if 
 the most accurate results are desired, a correction for 
 the temperature of the air ( 390) must be applied. 
 With some instruments this correction is made by mov- 
 ing the altitude scale. In this case the scale is graduated 
 for some particular temperature, its zero is adjusted by 
 reading the instrument at some station of known eleva- 
 tion, and this position of the zero is marked by aline on 
 the dial numbered to correspond with the normal tem- 
 perature; then points are determined to quote the 
 words of the inventor of this method of applying this 
 correction, " partly by trial and partly by computa- 
 tions," at which the zero of the altitude scale should be 
 set for different temperatures of the air. With this form 
 of instrument, the scale must be set for the mean 
 
ART. 4] DYNAMICAL FORMULAS. 345 
 
 temperature of the air, before any readings are taken; 
 and it must not be shifted during the progress of the 
 work. 
 
 The method of applying the correction for the tem- 
 perature of the air by shifting the altitude scale, is not 
 theoretically correct; and the accuracy of the results 
 depend entirely upon the accuracy of the scale by which 
 the altitude scale is shifted. The best plan is to dis- 
 pense entirely with altitude scales, whether fixed or 
 movable, and calculate the heights. 
 
 405. Conclusion. The preceding do not comprise all 
 the formulas which have been proposed for barometric 
 leveling, but include the more common ones, and illus- 
 trate all the principles involved except those discussed 
 in 406-9. Some of the omitted formulas are ap- 
 proximate, some have empirically determined pressure 
 co-efficients, etc. Owing to the matters discussed in 
 349-67 it is comparatively unimportant which of the 
 generally recognized barometric formulas is used. 
 
 B. Dynamical Formulas. 
 
 406. FerreFs Formula. Ferrel has deduced a formula* 
 which is claimed to be independent of the defects of all 
 formulas founded upon an assumed statical condition 
 of the atmosphere. Although the formula is very care- 
 fully and ingeniously worked out, it is probably of little 
 use for ordinary hypsometrical work, since it requires 
 observations to be made for a long time over a consid- 
 erable area, to get the data by which to compute cor- 
 rections for gradient, temperature, and humidity. 
 Without the data for making these reductions, this 
 formula is essentially the same and has essentially the 
 
 * Report of U. S. Coast and Geodetic Survey, 1881, pp. 225-68, the for- 
 mula itself being given on page 235. 
 
346 BAROMETERS. [CHAP. XII 
 
 same defects as the formulas depending upon an 
 assumed statical condition of the atmosphere. 
 
 407. Gilbert's Formula. This formula, unlike all those 
 in division A ( 383-405), does not require the observa- 
 tion of either the temperature or humidity of the air; 
 but requires observations of the barometer at three 
 stations, the difference of elevation of two of which is 
 known and the elevation of the third of which is to be 
 determined. The formula is 
 
 , . . 
 
 log/ log u 490,000 
 
 in which B is the difference in elevation of the two 
 stations, / is the reading of the barometer at the lower 
 station, reduced to 32 F., u the same for the upper sta- 
 tion, and n the same for the station whose elevation is 
 to be determined, and A is the value of the first term 
 of the second member. This formula is deduced on 
 the assumption that the three stations are in the same 
 vertical, and also that the station whose altitude is to be 
 determined is between the other two. The less nearly 
 these conditions are fulfilled, the more nearly this 
 formula is liable to the same errors as statical formu- 
 las. The denominator of the last term was determined 
 by experiment, and is therefore liable to error. See 
 33o. 
 
 408. Weilenmann's Formula. The Report of the U. 
 S. Coast and Geodetic Survey for 1876, pages 388-90, 
 contains a comparatively brief discussion of a baromet- 
 ric leveling formula based upon the dynamical theory 
 of heat. In the one case to which this formula is ap- 
 plied, it gives errors one half less than the best of the 
 ordinary barometric formulas; but as the errors in the 
 
 * G. K. Gilbert in U. S. Geological Survey Report for 1880-81, p. 448. 
 
ART. 4] DYNAMICAL FORMULAS. 347 
 
 single example to which it is applied, due to diurnal 
 variations of pressure, amount to one per cent, and as it 
 is much more complicated than the ordinary formulas, 
 it will not be considered further. 
 
 409. Conclusion. Owing to the possibility of errors 
 due to gradient, temperature, and humidity ( 348-67), 
 there is but little difference in accuracy between statical 
 and dynamical formulas. See 381. 
 
APPENDIX I. 
 
 ELIMINATION OF LOCAL ATTRACTION IN 
 
 SURVEYING WITH THE MAGNETIC 
 
 COMPASS. 
 
 1. IT is very common to hear the remark that the 
 magnetic compass can not be used because of local at- 
 traction. It is welt known that there are many locali- 
 ties in which the needle is deflected by the attraction 
 of iron, etc.; but the object of this article is to show 
 that no matter how much the local deflection, the com- 
 mon magnetic compass can be made to give as accurate 
 results as though there was no disturbing influence. 
 
 Since the conditions for mine surveying differ slightly 
 from those for land surveying, this subject will be consid- 
 ered under the heads of Mine Surveys and Land Surveys. 
 
 ART. 1. MINE SURVEYS. 
 
 2. Generally a survey of a mine is made to determine 
 the relative position of the underground workings 
 and the boundary lines of surface property. Conse- 
 quently it is necessary to find in the mine (i) a point 
 directly under a known point on the surface, and (2) 
 the direction of a line corresponding to a known line 
 on the surface. The first may be found by means of a 
 plumb-line, or its equivalent; and the second by the use 
 of two plumb-lines, or by a transit usually the former. 
 For the present it will be assumed that the true direc- 
 tion of the initial underground line is known. 
 
 For the sake of illustration, assume that a survey is 
 
 349 
 
35 
 
 ELIMINATION OF LOCAL ATTRACTION. 
 
 to be made, starting from some point, say A, on a line 
 whose direction is, say, S. 78 10' E. Set the compass 
 at A y and also at each corner or station of the survey, 
 successively, and read the bearing of the two lines 
 meeting in each station. Call the sight made in the 
 direction in which the survey progresses a fore-sight, 
 and that made in an opposite direction a back-sight. 
 Keep one end of the box next to the eye on fore-sights 
 and the other end 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, but the degrees from the same end all the time. 
 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. 
 
 TABLE I. 
 
 Station. 
 
 Back-sight. 
 
 Fore-sight. 
 
 Correction. 
 
 
 78 10 
 
 16 05 
 
 
 A 
 
 S. 78 oo' E. 
 
 N. 15 55' W. 
 
 io'B. 
 
 
 16 05 
 
 80 30 
 
 
 Q 
 
 N. 16 45' W. 
 
 N. 81 10' W. 
 
 40' F. 
 
 
 80 30 
 
 46 10 
 
 
 s 
 
 N. 80 05' W. 
 
 S. 46 35' W. 
 
 25' B. 
 
 
 46 10 
 
 83 oo 
 
 
 u 
 
 S. 46 35' W. 
 
 S. 82 35' E. 
 
 25' B. 
 
 
 83 oo 
 
 78 oo 
 
 
 V 
 
 S. 82 25' E. 
 
 S. 77 25' E. 
 
 35' B. 
 
 As explained above, the true bearing of the initial or 
 reference line is S. 78 10' E., and hence, if there had 
 been no local attraction at A, the back-sight at that sta- 
 tion, i.e., the bearing of the reference line, would have 
 been S. 78 10' E. But as it read S. 78 o' E., there 
 is a local attraction of 10', and therefore the bearings 
 
ART. l] MINE SURVEYS. 351 
 
 taken at A are in error 10'. The needle read S. 78 o' 
 E., but if there had been no local attraction it would 
 have read S. 78 10' E. The true reading is therefore 
 10' farther from the south toward the east than the 
 observed reading, and hence, to get the true bearing 
 the needle should be moved 10' in a direction from the 
 south towards the east. The fore-sight at A is N. 15 
 55' W.; but to get correct bearings at this station, we 
 must, in imagination, move the needle 10' in a direction 
 from the south towards the east, which is the same 
 direction as from the north towards the west, and hence, 
 the correct bearing of the line starting from A is N. 15 
 55' W., plus 10', which gives N. 16 5' W. 
 
 For simplicity of explanation, let us call a direction 
 from the south towards the west forward, and the op- 
 posite direction backward, these terms being assigned 
 to agree with the direction of the motion of the hands 
 of a watch. It makes no difference in this matter 
 whether we consider N., S., E., and 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 A is 10' backward. This is the correction which is 
 to be applied to the fore-sight at A, and is written in 
 the column headed " correction." The true bearing 
 of the initial line is written over the back-sight taken 
 at A, and the corrected fore-sight is written above the 
 fore-sight observed at Q. 
 
 At Q the back-sight, />., the bearing of the line from 
 A towards Q, is N. 16 45' W.; but as the true bearing is 
 N. 16 5' W., as found for the corrected fore-sight at A, 
 the error at Q due to local attraction is 40'. The cor- 
 rection at Q is therefore 40' forward, and the corrected 
 fore-sight is 80 30'. The corrections and corrected 
 bearings for the other stations are found in a similar 
 manner, 
 
35 2 
 
 ELIMINATION OF LOCAL ATTRACTION. [APPEN. I 
 
 3. As is easily seen, when the true bearing of the 
 initial line is known, the above method absolutely 
 eliminates all local attraction. On the other hand, if 
 the direction of the initial line is not known, this 
 method will give a fairly good determination of it. To 
 find the true bearing of all of the lines, including that 
 of the first one, set the compass at each station, and 
 read the back-sights and fore-sights at each as in the 
 preceding case. The record will then be as in Table II, 
 except for the small figures above the bearings. 
 
 TABLE II. 
 
 Station. 
 
 Back-sight. 
 
 Fore sight. 
 
 Correction. 
 
 A 
 
 78 10 
 
 S. 77 50' E. 
 
 16 05 
 
 N. i545'W. 
 
 20' B. 
 
 Q 
 
 16 05 
 
 N. 1 6 35' W. 
 
 80 30 
 N. 81 oo' W. 
 
 30' F. 
 
 S 
 
 N. 80 30' W. 
 
 S. 46 10' W. 
 
 
 
 U 
 
 S. 46 10' W. 
 
 S. 83 oo' E. 
 
 o 
 
 V 
 
 S. 83 oo' E. 
 
 S. 78 oo' E. 
 
 
 
 w 
 
 78 oo 
 
 S. 78 10' E. 
 
 N. 85 25' E. 
 
 10' F. 
 
 Notice that the fore-sight at S agrees with the back- 
 sight at Uj and also that the fore-sight at 7 agrees with 
 the back-sight at V. This makes it extremely probable 
 that there is no local attraction at these three stations, 
 and that the correction at these stations is o. To find 
 the correction at Q, write the back-sight at S above the 
 fore-sight at Q, and take the difference between the ob- 
 served and the corrected fore-sight. This difference is 
 30', and is the correction at Q. A moment's reflection 
 shows that this correction is " forward," as marked. 
 The remaining bearing at Q and the two at bearings A 
 
ART. l] 
 
 MINE SURVEYS. 
 
 353 
 
 may be found as already explained, and similarly those 
 at W\ that is to say, the bearings in the upper part of 
 the table are corrected by working up from S, and those 
 in the lower part by working down from V. 
 
 4. It may happen that not even two successive stations 
 are found at which the corresponding back-sight and 
 fore-sight agree. When this occurs, assume, for tempo- 
 rary purposes, that the correction at the first station is 
 
 zero, and correct all the bearings accordingly, 
 result will then be as in Table III. 
 
 TABLE III. 
 
 The 
 
 Station. 
 
 Back-sight. 
 
 Fore-sight. 
 
 Correction. 
 
 A 
 
 S. 78 00' E. 
 
 N. 75 5' E. 
 
 
 
 
 75 55 
 
 80 o 
 
 
 P 
 
 N. 75' 35' E. 
 
 N 80 LO' E. 
 
 20' F. 
 
 
 80 20 
 
 85 30 
 
 
 R 
 
 N. 80 10' E. 
 
 S. 85 40' E. 
 
 lo'F. 
 
 
 85 30 
 
 88 50 
 
 
 T 
 
 S. 85 50' E. 
 
 N. 88 30' E. 
 
 20' F. 
 
 
 88 50 
 
 78 oo 
 
 
 X 
 
 N. 88 20' E. 
 
 S. 78 30' E. 
 
 30' F. 
 
 
 78 oo 
 
 85 20 
 
 
 Y 
 
 S. 78 20' E. 
 
 N. 85 oo' E. 
 
 20' F. 
 
 Notice that there are three stations at which the cor- 
 rection is the same, i.e., 20' F. This indicates that the 
 apparent local attraction at these stations is the same; it 
 is therefore probable that the needle had its normal or 
 natural position at these stations, and that the local at- 
 traction was confined wholly to the other stations. If 
 this conjecture be correct (its validity will be examined 
 farther presently), then the corrections at P, T, and Y 
 are zero. By inserting this correction at these stations, 
 and correcting the notes as previously explained, we get 
 Table IV, which gives the most probable bearings of the 
 several 
 
354 
 
 ELIMINATION OF LOCAL ATTRACTION. [APPEN. I 
 
 5. For the preceding illustration the conjecture is 
 slightly defective, since the agreement of the corrections 
 at only three stations might possibly be due to an acci- 
 
 TABLE IV. 
 
 Station. 
 
 Back-sight. 
 
 Fore-sight. 
 
 Correction. 
 
 A 
 
 78 20 
 S. 78 oo' E. 
 
 N. 7 7 5 5 3 5' E. 
 
 20' B. 
 
 P 
 
 R 
 
 N. 75 35' E. 
 N. 80 io' E. 
 
 N. 80 oo' E. 
 
 85 50 
 S. 85 40' E. 
 
 o 
 10' B. 
 
 T 
 X 
 
 S. 85 50' E. 
 
 88 30 
 
 N. 88 20' E. 
 
 N. 88 30' E. 
 
 78 20 
 
 S. 78 30' E. 
 
 o 
 lo'F. 
 
 Y 
 
 S. 78 20' E. 
 
 N. 85 oo' E. 
 
 o 
 
 dental agreement of the local attraction at these stations. 
 If this is the case, then it is impossible by this method 
 to find the true bearings of the lines, unless proportion- 
 ally more stations can be found at which the apparent 
 local attractions agree. For example, if the next five 
 stations give an apparent local attraction agreeing with 
 that found in Table III for station X, then those stations 
 must be regarded as having the normal position of the 
 needle, and the bearings of the others must be corrected 
 accordingly. On the other hand, if it is impossible to 
 find a number of stations at which the apparent local 
 attractions agree, it is impossible to determine the true 
 bearings with the needle alone. What then ? 
 
 If the true direction of the initial line has not been 
 determined by plumbing the shaft, and if the apparent 
 local attraction does not agree at a proportionally large 
 number of stations, observe the back-sights and fore- 
 sights as previously described, and correct the bearings 
 with reference to the initial line as explained in Table 
 III. Of course this method does not certamiy deter- 
 
ART. l] MINE SURVEYS. 355 
 
 mine the true bearings, but it finds correctly the relative 
 directions of the several lines, and all the computations 
 and plats will be perfectly correct, except that they will 
 be turned around an amount equal to the difference 
 between the assumed and the true bearing of the initial 
 line. If the survey is made to determine the position of 
 the various parts of the mine relative to the land lines 
 above, then in many cases, particularly in the coal mines 
 of Illinois, this method would give fairly good results, 
 since the error would probably be slight, while the 
 value of the vein would be nearly uniform. Farther, if 
 at any subsequent time the true direction of the initial 
 line should be determined, then the correct bearings of 
 all the lines of the underground survey could be deter- 
 mined at once by applying, as a "correction," the dif- 
 ference between the true and the assumed direction of 
 the first line. 
 
 6. Finally, it may happen that the apparent local at- 
 tractions as found by the trial balance, Table III, pre- 
 ponderate in one direction. For example, in Table III 
 it will be noticed that all the corrections are " forward " 
 except that at A. As a general rule local attraction is 
 as likely to be in one direction as another, and hence, 
 unless the local conditions furnish some good reason to the 
 contrary ', the corrections found by the trial balance 
 should be about as much in one direction as the other, 
 />., the sum of the "forward" corrections should be 
 nearly equal to the sum of the " backward " corrections. 
 This is not so in Table III, which is an actual survey. 
 The local conditions gave no indications that the local 
 attraction was more in one direction than another, and 
 we may reasonably assume that the corrections are all 
 one way in the table because the true correction &\.A is 
 backward an amount about equal to the average of the 
 corrections at the other stations, i.e., about 18'. If this 
 correction were applied to the original bearings in 
 
356 ELIMINATION OF LOCAL ATTRACTION. [APPEN. I 
 
 Table III, the resulting directions would be nearly the 
 true ones; but if the correction at any station is con- 
 siderably larger than any of the others, it is evident 
 that the local attraction at this station is considerably 
 more than the average, and hence this method of cor- 
 recting the bearings should not be employed, at least 
 the excessive correction should not be included in find- 
 ing the average. 
 
 In Art. 2 a modification of the preceding methods will 
 be considered, which, under certain conditions, may be 
 of great importance in making mine surveys. 
 
 ART. 2. LAND SURVEYS. 
 
 7. Land surveys differ from mine surveys in that the 
 former close, i.e., return to the point of beginning, 
 while the latter, as a rule, do not. When the survey 
 closes, the above method of observing and correcting 
 the bearings possesses, in addition to finding the true 
 bearings as explained above, two important advan- 
 tages : i, it affords under every condition the means 
 of checking the accuracy of the bearings ; 2, it enables 
 the true area to be determined, even if the true bearing 
 of the sides can not be found. There are then two 
 cases, I, when only the area is wanted ; II, when the 
 area and also the true bearings are required. 
 
 CASE I. When only the area is wanted. 
 
 8. The angles are read as described in Art. 1, the 
 results being as in Table V (page 357). 
 
 Since the assumption is that only the area is required, 
 it is immaterial, where we commence to correct the 
 angles ; and therefore we may assume that the bearings 
 taken at any station, say Z, are correct, and that the 
 correction at Z is zero. Beginning then at Z, we 
 may correct the bearings by the method already ex- 
 
ART. 2] 
 
 LAND SURVEYS. 
 
 357 
 
 plained. It is immaterial whether the angles are cor- 
 rected in the order in which they were surveyed, or the 
 opposite. 
 
 TABLE V. 
 
 Stations. 
 
 Back-sight. 
 
 Fore-sight. 
 
 Correction. 
 
 z 
 
 S. 21 00' W. 
 
 N. 6 55' W. 
 
 
 
 
 6 55 
 
 55 25 
 
 
 Y 
 
 N. 7os'W. 
 
 N. 55 35' W. 
 
 10'F. 
 
 
 55 25 
 
 36 3 
 
 
 X 
 
 N. 55 35' W. 
 
 S. 36 20' W. 
 
 10' F. 
 
 
 36 30 
 
 2 2O 
 
 
 W 
 
 S. 36 50' W. 
 
 S. 2 oo' E. 
 
 20' B. 
 
 
 2 20 
 
 43 
 
 
 u 
 
 S. 2 40' E. 
 
 S. 43 20' E. 
 
 20' F. 
 
 
 43 
 
 69 25 
 
 
 T 
 
 S. 43 05' E. 
 
 S. 69 30' E. 
 
 5'F. 
 
 
 69 25 
 
 21 5 
 
 
 S 
 
 S. 69 30' E. 
 
 N. 21 oo' E. 
 
 5'F. 
 
 It will be noticed in the example cited that the back- 
 sight at Z differs from the corrected fore-sight at S. If 
 the angles had been correctly read, these two would 
 have agreed exactly. The difference between the first 
 corrected back-sight and the last corrected fore-sight 
 shows the error of reading the angles. This difference 
 should not exceed 5' or 10' (see 50, page 52). (The 
 example in Table V is an actual survey made by one of 
 the author's students.) 
 
 CASE II. When the area and also the true bearings are 
 
 required. 
 
 9. This case requires one of two things : either that 
 several successive back-sights shall agree with their 
 corresponding fore-sights before either have been cor- 
 rected, or that there shall be a true meridian which can 
 be connected with a corner of the field by lines whose 
 
358 
 
 ELIMINATION OF LOCAL ATTRACTION. [APPEN. I. 
 
 bearings are to be found in the same manner as are 
 those of the boundary lines. 
 
 10. In the first instance it may safely be assumed 
 that, since a number of the fore-sights and back-sights 
 agree, the observed bearings are the true ones. Hav- 
 ing some of the correct bearings, if there are any sta- 
 tions at which the back-sights do not agree with the 
 corresponding fore-sights, they may be corrected as in 
 Table III. If the declination is not set off on the com- 
 pass, it may be applied as a correction to those stations 
 at which the back-sight and fore-sight corresponded 
 as above, and the correction may then be carried on to 
 the remaining stations. 
 
 11. In the second instance, the true bearings of the 
 several lines can be found in succession, beginning at 
 the meridian. The example in Table VI will illustrate 
 
 TABLE VI. 
 
 Stations. 
 
 Back-sight. 
 
 Fore-sight. 
 
 Correction. 
 
 Z 
 
 
 73 10 
 
 N. 69 20' E. 
 
 3 50' F. 
 
 Q 
 
 73 I0 
 
 N. 69 15' E. 
 
 
 3 55' F. 
 
 Q 
 
 14 40 
 N. 10 45' E. 
 
 93 4 
 
 S. 89 45' W. 
 
 3 55' F. 
 
 T 
 
 93 4 
 
 S. 89 35' W. 
 
 43 5 
 
 S. 39 oo' W. 
 
 4 05' F. 
 
 U 
 
 43 05 
 
 S. 39 10' W. 
 
 M 35 
 
 N. 10 40' E. 
 
 3 55' F. 
 
 this case. Z is not a corner of the field, but a point on 
 a true meridian, the line ZQ being run to determine 
 the declination at Q, a corner of the field. All the re- 
 maining stations are corners of the field. 
 
 Notice that the first two lines of the above record are 
 preliminary to the survey of the field, and are required 
 
ART. 2] LAND SURVEYS. 359 
 
 only to find the declination or correction at Q. Hav- 
 ing found the correction at Q, the bearings are cor- 
 rected as in the previous case. The back-sight from 
 Q to ^/differs 5' from the fore-sight from 7 to Q, which 
 shows that there was an error or inaccuracy of 5' in 
 reading the angles. 
 
 12. Undoubtedly the method of connecting with a 
 meridian is more exact than the previous one, although 
 the former is more convenient and shorter, and is the 
 one which will generally be used in practice. However, 
 if the bearings show local attraction at a number of 
 stations, the first method of Case II can not be applied, 
 while the second will give strictly correct results what- 
 ever the number of stations at which local attraction 
 exists. 
 
 Notice that when the survey closes, the method ex- 
 plained above absolutely eliminates all local attraction, 
 and also that this method is frequently applicable in 
 mine surveying. Finally, the back-sight should always 
 be taken, even though it is probable that no local 
 attraction exists, since the back-sights afford a very 
 valuable means of checking the readings of the needle. 
 
APPENDIX II. 
 
 FINDING AREA BY TRAVERSING WITH THE 
 TRANSIT. 
 
 1. TAKING THE FIELD NOTES. Assume that not only 
 
 the area is desired, but also the angles which the 
 several sides make with the meridian. Place the instru- 
 ment over the first station, say A in Fig. 86, and make 
 the vernier read zero. Direct the telescope along 
 the meridian, clamp the instrument, take a back-sight 
 
 FIG. 86. 
 
 upon the last station, and proceed around the field ac- 
 cording to the process described in 137. On getting 
 to the last station and looking back to the first, the 
 fore-sight should be the same as the back-sight from 
 the first station. Fig. 86 and Table I (page 361) mutually 
 explain each other. 
 
 The angles in the column headed fore-sights are the 
 azimuths or angles which the several courses make with 
 
 360 
 
COMPUTATIONS. 
 
 36: 
 
 the reference line, in this case the meridian, and are 
 marked in Fig. 86. If only the area is desired, it is 
 immaterial whether the reference line be a meridian or 
 not. 
 
 TABLE I. 
 FIELD NOTES FOR TRAVERSE SURVEYING. 
 
 Sta 
 
 Back-sight. 
 
 Fore-sight. 
 
 Dist. 
 
 Remarks. 
 
 A 
 
 320 19' 
 
 63 48' 
 
 
 Vernier A read, zero 
 
 B 
 
 63 48' 
 
 123 2l' 
 
 
 being set on true 
 
 C 
 
 123 2l' 
 
 1 80 00' 
 
 
 meridian. 
 
 D 
 
 1 80 00' 
 
 260 35' 
 
 
 Azimuths counted from 
 
 E 
 
 260 35' 
 
 320 19' 
 
 
 north toward east. 
 
 It makes little or no difference which way the sur- 
 veyor goes around the field. 
 
 2. COMPUTING THE AREA. In computing the area by 
 the common or Rittenhouse method, there is no check 
 upon the work after the latitudes and departures have 
 been balanced. The principal object of the method 
 explained below is to overcome this defect. 
 
 With the following explanation of the nomenclature, 
 the method can readily be understood by an inspection 
 of Tables II and III, pages 362 and 363. 
 
 A^ A^, A 3l etc., are the azimuths of the several sides, 
 respectively. 
 
 C lt C 2 , C 3 , etc., are the lengths of the several sides, 
 respectively. 
 
 *i x i-> X ^ etc -> are tne latitudes of the several sides, 
 respectively. 
 
 y\9 y*>y> etc -> are t^ 6 departures of the several sides, 
 respectively. 
 
 X l , X^ , X 3 , etc., are the total latitudes of the several 
 corners; i.e., the total distance which any corner is 
 north or south of some other corner which is assumed 
 as an initial one for the purposes of computation. 
 
362 AREA BY TRAVERSING WITH TRANSIT. [APPEN. II 
 
 
 H H h 
 +++0 
 
 c a c c 
 3 
 
 82S 
 
 O 
 
 ^.S 
 
 
 1 
 
 +tt 2 J 
 
 
 S 
 
 ^^^^ i 
 
 
 g 
 
 *T H H H H 
 
 a 
 
 flj 
 
 * 
 
 MM i 
 
 1 
 
 
 
 3 
 
 
 ^^^^ 'J 
 
 1 
 
 1 
 
 7??7 + 
 ++++2 T 
 
 
 tJ 
 
 
 
 S 
 
 cls.S'sSo. w ^ 
 
 
 a 
 
 ^^"^^ ^ 7 
 
 
 
 M 1 1 ^ 
 
 
 
 1 
 
 g 
 
 
 ^^^ 
 
 s 
 
 
 1 1 1 
 
 tt 
 
 ?; 
 
 2 
 
 O) 
 
 Q 
 
 "rt 
 
 u 
 
 II II II II II 
 
 O 
 
 
 K *V** k" k 1 * k S 
 
 H 
 
 
 ^^^ ^ 
 
COMPUTATIONS. 
 
 363 
 
 M 
 
 I 
 
 s S 
 
 fc 
 
 c 
 
 Q 
 O 
 
 
 
 5 
 
 1 
 
 N r- rfr CO M 
 O O O co co t^ 
 
 M co m r>. co co 
 
 CO O M t^ ON Tt- 
 
 M ^- O CO 
 1 i 1 
 
 vO co 
 en O> 
 
 CO 
 C4 OO 
 
 f ? 
 
 1 
 
 Q 
 
 Q 
 
 in N M O i- ON 
 co t^ O M in co 
 
 + 1 1 + 1* + 
 
 c 
 
 S s 
 1 ~ 
 
 8 
 
 ,J 
 
 m o oo en m m 
 
 rj- m O r^~ co M 
 
 1, f 
 
 
 H 
 
 Tt- rj- ^- M N in 
 
 1 1 1 1 1 1 
 
 " .G 5 
 
 Q 
 
 X 
 
 | 
 
 3 
 
 in o -< M o O 
 r> N t>i i> in o 
 
 co t> ^ N r>. o 
 d ^- N in o 
 
 ^ M Q\ CO d 
 
 co 0^ M f"* en 
 ( 
 
 rx QO 
 
 ? cT 
 5 
 
 f ! 
 
 
 Q ' 
 
 co CO TJ- i-i rf- 
 00 00 O VO CO 
 
 N CO M M M 
 
 \O co in ^ M 
 
 + 7 i T i 
 
 nf ' 
 
 . " a ' 
 
 S 
 
 M 
 
 2 
 
 in in en O m O 
 
 ^- o o *> M o 
 
 Tt* O ^ l^* in O 
 m M en en r^ o 
 e< M en o 
 
 1 1 1 1 1 
 
 o a< e3 5^ 
 
 T3 w g 
 
 - | rt 
 
 H co 
 
 DEPART- 
 URE. 
 
 ; 
 
 W ^"* ^t O M 00 
 
 \O f^ 1 w M r^* oo 
 
 IT) d M O w O^ 
 
 en r-. o M in co 
 
 + 1* I + l* + 
 
 fj 
 
 LATITUDE. 
 
 C/) 
 
 m o oo r>. m m 
 
 ^ O O> vO in M 
 
 TJ- in en co a in 
 in in N O O t- 
 
 1 y 1 1* + + 
 
 |i 
 
 
 on 
 
 Q 
 
 c? 2> S ^ S 5" 
 
 M w tn ^t rf oo 
 ^ co vO O *O O 
 cT a M M eo 
 
 
 Q 
 
 .15 
 
 3 
 
 g 
 
 ^ fc ->* "* ** 
 O m co w N m 
 
 -s J 
 
 a 
 
 
 
 
 en *> co rj- w O 
 co rj- vO m M O 
 M en M N 
 
 t/i 
 
 u 
 
 
 1 
 
 ft- EN ^ij tsj !^ ^ 
 
 
364 AREA BY TRAVERSING WITH TRANSIT. [APPEN. II 
 
 Y l9 F 2 , F 3 , etc., are the total departures of the several 
 corners. 
 
 2 represents the sum of the quantities to which it is 
 prefixed. 
 
 n is the total number of sides. 
 
 T. Z., in the heading of Table III, stands for total 
 latitude, and A. T. L. for adjacent total latitude, that is for 
 
 x lf Ai + jr., ^ 2 + ^ 3 , etc. 
 
 A. D., in Table III, represents the sums^ -j-jy a , J 2 +^ 3 > 
 etc., of Table II, which are called adjacent departures. 
 
 In computing the latitudes and departures, attention 
 must be given to the signs of the trigonometrical 
 functions. The latitudes and departures are to be 
 balanced as in the method of ordinary land surveying 
 In applying this method, it will be necessary to pre- 
 pare a column in which to write the adjacent depar- 
 tures y l +jy 2 , y 9 -\-y 3 , etc., and another in which 
 to write the sums of the adjacent total latitudes 
 X^-^X^, X z -\-X 3l etc. These columns, and two 
 similar ones for the third and fourth methods, were 
 omitted from Table II for convenience in printing. 
 
 If only two computations of the area are to be made 
 (two will generally give a sufficient check), it is shorter 
 to use either the first and second method or the third 
 and fourth, than to pair them differently. For simply a 
 numerical check, the signs of the area may be dis- 
 regarded; but if the signs in the table are conformed 
 to, the areas will agree in sign as well as in amount. 
 
 Notice that in the first method, each partial area is 
 obtained by multiplying the total latitude of any corner 
 by the sum of the departures of the adjacent sides, or 
 briefly, each partial area is the product of the total 
 latitude and the sum of the adjacent departures; and 
 note also that in the second method each partial area 
 is the product of the departure and the sum of the 
 adjacent total latitudes. Note further, that the third 
 
COMPUTATIONS. 365 
 
 and fourth methods are the same as the first and second 
 excepting the substitution of latitudes for departures. 
 The fourth method is the usual one of double meridian 
 distances, except that the former deals with the co- 
 ordinates of the corners of the field instead of the 
 middle of the sides. 
 
 The example shown in Table III is a problem solved 
 by one of the author's students in the ordinary class 
 work. For reasons not necessary to explain, the com- 
 putations are carried to an unusual, and ordinarily an 
 indefensible, number of decimal places. 
 
APPENDIX III. 
 
 PROBABLE ERROR. 
 
 1. ALL quantities determined by observation are sub- 
 ject to error, and any one who deals with such quanti- 
 ties should recognize the certainty of error in his data. 
 This limitation is frequently disregarded as, for 
 example, when, in a report which recently came under 
 the author's eye, the cost per cubic yard of concrete 
 was figured out to the thousandth of a cent. Other 
 examples could be cited in which the result, although 
 less ridiculous, involved vastly greater consequences. 
 The object of this discussion is to present some of 
 the elementary principles involved in the comparisons 
 of results derived from observation. 
 
 All observations are liable to three classes of error; 
 viz., mistakes, constant errors, and accidental errors, 
 (i) Mistakes are errors due to inexperience, to lack of 
 care, to mental confusion, etc.; as, for example, reading 
 28 instead of 32, or reading the wrong vernier, or 
 counting from the wrong end of the tape. Frequently 
 such errors may be corrected by comparison with other 
 observations. (2) Constant errors are those produced 
 by well understood causes as, for example, the chain 
 may be too long, or it may be used at a temperature 
 differing from that at which it is of standard length, or 
 there may be phase in the target sighted at, etc. Such 
 errors can always be eliminated by the application of 
 computed corrections, and, strictly speaking, are not 
 errors at all. (3) Accidental errors are those still re- 
 maining after all evident mistakes and all constant 
 
 366 
 
PRINCIPLES. 367 
 
 errors have been eliminated as, for example, the errors 
 in leveling due to a movement of the bubble after its 
 inspection, to an imperfect bisection of the target, to 
 an inclination of the rod, etc. Only the last class of 
 errors will be considered here. 
 
 To many persons it seems strange, and even im- 
 proper, to speak of the law of irregular and unknown 
 errors ; but it is nevertheless true that even accidental 
 errors follow a rigorous mathematical law the law of 
 probability. 
 
 The whole theory of probable error depends upon 
 the three following theorems, derived from experience: 
 
 1. Small errors are more frequent than large ones. 
 
 2. Positive and negative errors are equally frequent. 
 
 3. Very large errors do not occur. 
 
 2. The probable error is such a quantity that there 
 is an even chance that the real error is greater or less 
 than it. Or, in other words, if the errors of a series of 
 observations were arranged in the order of their mag- 
 nitude, the middle error in that series would be the 
 probable error. Notice that the probable error would 
 occur in the middle of the series, but would be less 
 than the mean of the errors, because small errors are 
 more likely to occur than large ones. Or, again, the 
 probable error is taken by mathematicians as the limit 
 within which it is as likely as not that the truth 
 will fall. Thus, if 5.45 be the mean of a number 
 of determinations, and 0.20 be the probable error, 
 then the true result is as likely to lie between 5.25 
 (= 5-45 0.20) and 5.65 (= 5.45 + 0.20) as to lie out- 
 side of these limits. The probable error is usually 
 represented by writing it after the number, but preced- 
 ing it by the sign indicating//^ or minus. For exam- 
 ple : 5.45 o. 20 indicates that 5.45 is the mean of the 
 observed quantity, and that 0.20 is the probable error of 
 this value. 
 
368 PROBABLE ERROR. 
 
 3. When a number of separate observations of any 
 kind have been made with equal care, the mean or 
 average of them all is the most probable value of the 
 quantity sought, and is considered the true value. 
 Let n the number of the observations. 
 
 d the difference between any one observation 
 and the mean of the observations. (The 
 quantities represented by d are generally 
 called residuals.) 
 
 E l = the probable error of a single observation. 
 f = the mean of the errors. 
 m = the probable error of the mean of all the 
 
 observations. 
 
 q 0=6745 a constant determined by com- 
 putations according to the theory of 
 probability. 
 
 2 = a symbol signifying sum of* 
 Then 
 
 l = 
 
 E l = 0.84537, approximately. . (3) 
 
 Less labor is required in applying equation (3) than 
 in using (i). If the number of observations were in- 
 finite, the two would give exactly the same results. 
 
 Mathematicians agree far better as to the form of the 
 law of error, and also as to the method of computing 
 the probable error, than they do as to the manner in 
 which the law can be deduced. Therefore no demon- 
 stration will be attempted ; but to prevent misappre- 
 hension it may be well to remark that the law as stated 
 above has frequently been tested and found to agree 
 
EXAMPLES. 369 
 
 very closely with experience. It should be borne in 
 mind that the method is grounded upon the hypothesis 
 that a large number of observations have been taken. 
 However, when only a limited number of observations 
 have been made, the probable error, when computed 
 according to the above formulas, is sufficiently exact 
 for all purposes of comparison. It should not be for- 
 gotten that the probable error can be computed legiti- 
 mately only after all constant errors have been cor- 
 rected. Nor should it be forgotten that the probable 
 error considers only errors that are as likely to occur 
 on one side as on the other. 
 
 4. Tables I and II (pages 370 and 371) will illus- 
 trate the method of applying the preceding formulas. 
 The observations were made by leveling up the instru- 
 ment, sighting the target, and then reading the rod. 
 The instrument was then disleveled, the target was 
 moved, and another observation was made. The ob- 
 servations as recorded were consecutive, /'.*., no poor 
 ones were thrown away. The observations were made 
 by one of the author's students in ordinary class-work, 
 and are fairly representative of what an inexperienced 
 but very careful man can do under favorable conditions 
 as to wind, light, etc., with an ordinary wye level. 
 
 Incidentally the tables give some data concerning the 
 relative accuracy of two forms of level targets ( 268), 
 and also show the effect of increasing the length of 
 sight. All the observations were made by the same 
 man on the same day. For the distances in the table, 
 the error of setting the target seems to vary about as 
 the square root of the length of sight, but the error 
 probably decreases with the distance for a time and 
 then increases. Notice that the probable error of 
 setting the quadrant target (the one generally used in 
 practice) at 300 feet (about the ordinary distance) was 
 0.002 ft. Under the usual conditions this error would 
 
370 
 
 PROBABLE ERROR. 
 
 TABLE I. 
 
 ERROR OF READING LEVEL TARGET. 
 Rod 100 Feet from Instrument. 
 
 QUADRANT TARGET. 
 
 DIAMOND TARGET'. 
 
 Reading 
 in feet. 
 
 dm 
 thousandths. 
 
 d*. 
 
 Reading 
 in feet. 
 
 dm 
 
 thousandths. 
 
 d*. 
 
 3.169 
 
 
 
 o 
 
 3.172 
 
 I 
 
 I 
 
 3.170 
 
 I 
 
 i 
 
 3.174 
 
 I 
 
 I 
 
 3.170 
 
 I 
 
 i 
 
 3-174 
 
 I 
 
 I 
 
 3.170 
 
 I 
 
 i 
 
 3.172 
 
 I 
 
 I 
 
 3.169 
 
 
 
 
 
 3-173 
 
 
 
 
 
 3-168 
 
 I 
 
 I 
 
 3.173 
 
 O 
 
 
 
 3.I7I 
 
 2 
 
 4 
 
 3-173 
 
 o 
 
 o 
 
 3.168 
 
 I 
 
 i 
 
 3-173 
 
 
 
 
 
 3.169 
 
 O 
 
 o 
 
 3.175 
 
 2 
 
 4 
 
 3.168 
 
 I 
 
 i 
 
 3-174 
 
 I 
 
 i 
 
 3.169 
 
 
 
 
 
 3.173 
 
 
 
 
 
 3.170 
 
 I 
 
 I 
 
 3-174 
 
 I 
 
 I 
 
 3.169 
 
 0.75 
 
 II 
 
 3 173 
 
 O.67 
 
 IO 
 
 mean 
 
 mean 
 
 sum 
 
 mean 
 
 mean 
 
 sum 
 
 Prob. error of single obs. 
 
 Prob. error of single obs. 
 
 = 0.67 y = 0.00067 ft. 
 
 = 0.67 y = 0.00064 ft- 
 
 Approx. prob. error of single obs. 
 
 Approx. prob. error of single obs. 
 
 = 0.84 X 0.75 = 0.00063 ft. 
 
 = 0.84 X 0.67 = 0.00056 ft. 
 
 Prob. error of mean 
 
 Prob. error of mean 
 
 0.67 
 
 = = = 0.00019 ft- 
 
 4/12 
 
 0.64 
 = = = 0.00018 ft. 
 
 4/12 
 
 probably be considerably more than this possibly two 
 or three times as great. 
 
 5. To resume the discussion of the general principles 
 of the probable error, notice that according to equation 
 (2) the probable error of the mean of a number of 
 observations varies inversely as the square root of 
 their number. For example, if a compensating error 
 
EXAMPLES. 
 
 371 
 
 TABLE II. 
 
 ERROR OF READING LEVEL TARGET. 
 Rod 300 Feet from Instrument. 
 
 QUADRANT TARGET. 
 
 DIAMOND TARGET. 
 
 Reading 
 in feet. 
 
 din 
 
 thousandths. 
 
 d*. 
 
 Reading 
 in feet. 
 
 dm 
 thousandths. 
 
 rf a . 
 
 4-843 
 
 5 
 
 25 
 
 4.848 
 
 2 
 
 4 
 
 4-835 
 
 3 
 
 9 
 
 4.85I 
 
 
 
 4-836 
 
 2 
 
 4 
 
 4.851 
 
 
 
 4-837 
 
 I 
 
 i 
 
 4-852 
 
 
 
 4-834 
 
 4 
 
 16 
 
 4.849 
 
 
 
 4-834 
 
 4 
 
 16 
 
 4.849 
 
 
 
 4.837 
 
 i 
 
 i 
 
 4.851 
 
 
 
 4.839 
 
 i 
 
 i 
 
 4-852 
 
 2 
 
 4 
 
 4.838 
 
 
 
 o 
 
 4.848 
 
 2 
 
 4 
 
 4-835 
 
 3 
 
 9 
 
 4-850 
 
 
 
 
 
 4.842 
 
 4 
 
 16 
 
 4.846 
 
 4 
 
 16 
 
 4.841 
 
 3 
 
 9 
 
 4-852 
 
 2 
 
 4 
 
 4-838 
 
 2.6 
 
 107 
 
 4-850 
 
 1.6 
 
 41 
 
 mean 
 
 mean 
 
 sum 
 
 mean 
 
 mean 
 
 sum 
 
 Prob. error of single obs. 
 
 Prob. error of single obs. 
 
 = 0.67 JU^-L 0.0021 ft. 
 r ii 
 
 = 0.67 A/ $1 = 0.0013 ft. 
 
 Approx. prob. error of single obs. 
 
 Approx. prob. error of single obs. 
 
 = 0.84 X 2.6 = 0.0022 ft. 
 
 = 0.84 X 1.6 0.0013 ft. 
 
 Prob. error of mean 
 
 Prob. error of mean 
 
 
 21 
 
 36ft. 
 
 13 oo 
 
 x>4ft. 
 
 4/12 
 
 o.o 
 
 4/12 
 
 of o.oi of a foot per chain is made in measuring, 
 the error in a line 100 chains long would be only 
 O.OT X 1/100 = 0.1 feet, and not o.oi X 100 = i.o foot. 
 Again, if the probable error of a single reading of a 
 level target is 0.005 feet, then the error in determining 
 a difference of elevation by a single setting of the 
 instrument is V~2 X 0.005 feet = 0.007 feet nearly, since 
 
37 2 PROBABLE ERROR. 
 
 the determination requires two settings of the target. 
 If sixteen such pairs of observations were taken to 
 determine the difference in elevation of two remote 
 points, then the probable error of the difference of level 
 of the extreme points would be 0.007 X Vi6 = 0.028 
 feet. 
 
 The following illustrations show slightly different 
 forms of the above conclusion. If a represents the mean 
 of a measured distance having a probable error x, and 
 b represents another distance having a probable error 
 y, then the probable error of the sum of these distances 
 is W -|- y. Or, stating this algebraically we have 
 
 (a x) + (b y) = a + b Vx* +/. . . (^ 
 
 Again, if c x and d y represent the two sides of a 
 rectangle, the area will be represented by 
 
 If z represents the probable error per unit (say per 
 chain), then the area is represented by 
 
 cd zcd(c + d) (6) 
 
 To apply the last formula, assume that a lot 20 X 100 ft. 
 is laid out with a chain with such a degree of accuracy 
 that the probable error per chain is o.oi ft., then 
 
 cd z Vcd(c -{- d) 2,000 o.oi ^2,000 (20 -J- 100) 
 = (2,000 4.90) sq. ft. 
 
 There are various ways in which the preceding prin- 
 ciples may be made use of in practical work, some of 
 which will readily suggest themselves, but it is not wise 
 to discuss them here, 
 
APPENDIX IV. 
 
 PROBLEMS IN TESTING, ADJUSTING, AND 
 
 USING ENGINEERS' SURVEYING 
 
 INSTRUMENTS. 
 
 THE following problems are solved by the author's 
 students in connection with the study of the preceding 
 text. Some of them are solved several times with dif- 
 ferent instruments and under different conditions as 
 to distance, weather, experience, etc. A report of each 
 problem is made upon a sheet of standard co-ordinate 
 paper. 
 
 PROB. 1. ERROR OF MEASURING WITH A STEEL TAPE. 
 
 1. Measure a distance of about 1,000 feet with a steel 
 tape. Do not use a spring balance or a thermometer, 
 but take every other precaution to secure extreme ac- 
 curacy. Make at least four measurements, each man 
 being fore chain-man at least once in each direction. 
 Compare the tape with the standard, both before and 
 after taking the measurement. Report the true horizon- 
 tal distance between the ends of the line. 
 
 2. Compute the probable error of a single measure- 
 ment and also of the mean. 
 
 3. Estimate the values of the errors due to the several 
 sources mentioned in 19, i.e., assign values to a, b, c, 
 etc., in equation (2). page 26; and compare the estimated 
 probable error of a single measurement with the cor- 
 responding observed probable error. 
 
 373 
 
374 PROBLEMS. 
 
 4. Level the line and compute the correction for 
 inclination. How does the estimated quantity agree 
 with the true value ? 
 
 PROB. 2. ERROR OF MEASURING WITH TWO STEEL RULES. 
 
 i. To determine the degree of accuracy with which dis- 
 tances can be laid off by means of short rules, use two 
 2-foot steel rules and lay off 50 feet on the floor. The 
 rules may be used for either end or line measurement ; 
 /.*., the rules may be placed end to end, or they may be 
 lapped and the coincidence of lines observed. Make 
 three measurements of the distance by each method. 
 
 Report which method is considered the better, and 
 state the reasons therefor. 
 
 PROB. 3. ANGLES WITH A TAPE. 
 
 1. With a steel tape, measure four angles around a 
 point such that their sum = 360, no two of the angles 
 being of the same size. 
 
 2. Measure the three angles of a triangle having sides 
 about 250 to 300 feet long. 
 
 3. Report the sums in each case, and also state the 
 radius used, the form of target (flag pole or chaining 
 pin), weather, etc. 
 
 PROB. 4. ADJUSTMENT OF THE MAGNETIC COMPASS. 
 
 1. Test to see whether the center of graduation is in 
 the line of sight. 
 
 2. Test to see if the zero of the vernier is in the line 
 of sight. 
 
 3. Adjust the levels perpendicular to the vertical axis. 
 
 4. Place the sights (a) in the same plane, and (b) per- 
 pendicular to the plate. 
 
 5. Place the point of suspension of the needle (a) in 
 
MAGNETIC COMPASS. 375 
 
 the vertical plane of the ends, and (fr) in the horizontal 
 plane of the ends. 
 
 6. Sharpen the pivot and place it in the center of the 
 graduation. 
 
 7. If the instrument is provided with a means of set- 
 ting off a perpendicular, test it. 
 
 8. Test the magnetism of the needle. 
 
 PROB. 5. ERROR OF SIGHTING A MAGNETIC COMPASS. 
 
 1. Set up the magnetic compass and clamp the verti- 
 cal axis. At TOO feet from the instrument, set a flag 
 pole in the line of the slits, and mark its position on a 
 board laid on the ground for that purpose. Then 
 slightly change the position of the flag pole, and line it 
 in again. Make at least ten sightings on the pole. 
 Measure the position of the marks from one end of the 
 board, and compute the linear distance corresponding to 
 the probable error of sighting. Reduce the linear error 
 to angular error. 
 
 2. Repeat the above process at 300 and also at 600 
 feet. 
 
 PROB. 6. ANGLES WITH A MAGNETIC COMPASS. 
 
 i. With a compass measure four angles around a 
 point, such that their sum 360. Make the observa- 
 tions as follows : Sight upon the first line and read the 
 needle; sight the second line and read again. The dif- 
 ference of these readings is the first angle. Move the 
 vernier a little to eliminate personal bias, and turn the 
 compass slightly to eliminate any sticking of the needle; 
 then measure the second angle as before. Do similarly 
 for the other angles. 
 
 Try to have the conditions as to length of sight, tar- 
 gets, etc., as nearly as possible like those of Prob. 3. 
 
376 PROBLEMS. 
 
 2. Measure the three angles of a triangle having sides 
 250 to 300 feet long. 
 
 3. Report the sums in each case; and also state the 
 form of target, length of sight, and condition of wind, 
 sun, etc. 
 
 PROB. 7. ELIMINATION OF LOCAL ATTRACTION. 
 
 1. With a magnetic compass observe the back-sights 
 and fore-sights of the several sides of a field of at least 
 five sides. Read the bearings to the nearest five 
 minutes. 
 
 2. Correct the field notes to eliminate local attraction, 
 and to determine the angular error of closure. 
 
 PROB. 8. TRUE MAGNETIC BEARINGS. 
 
 1. Determine the true bearings of the several sides 
 of a field with a magnetic compass by connecting one 
 corner of the field with a true meridian. See n, Ap- 
 pendix II. 
 
 2. Find the true bearings and also the angular error 
 of closure. 
 
 PROB. 9. TESTING A TELESCOPE. 
 
 1. Test (a) spherical aberration, (b) chromatic aberra- 
 tion, (c) defining power, and (d) flatness of field. 
 
 2. Measure the magnifying power. Use two methods 
 and make three determinations by each. 
 
 3. Measure the angular width of the field of view. 
 
 4. Compare two telescopes (a) for illumination, and 
 (b) for definition. Are the illuminating powers of the 
 two telescopes in the direct ratio of the areas of real 
 apertures and in the inverse ratio of the squares of the 
 magnifying powers ? 
 
TRANSlf. 377 
 
 PROB. 10. STRETCHING SPIDER WEBS. 
 
 1. Attach three spider webs to a diaphragm, parallel 
 to each other and -g* th of an inch apart. Be sure that 
 all these are in the same plane. 
 
 2. Fasten a fourth web perpendicular to the others 
 and in the same plane. 
 
 3. After the webs are dry, breathe upon them or 
 hold them in a gentle current of steam. If they be- 
 come slack, they were not sufficiently stretched before 
 being fastened. 
 
 PROB. 11. MAKING A VERNIER. 
 
 i. Graduate a scale of equal parts on a piece of Bris- 
 tol board, using a right-line pen and India ink, and 
 make a vernier to read the scale. The unit of the scale 
 and vernier will be assigned. The accuracy of the 
 graduation is the most important feature of this 
 problem. 
 
 PROB. 12. ADJUSTMENT OF A TRANSIT. 
 
 1. Test the eccentricity of the graduation. 
 
 2. Adjust the plate levels perpendicular to the ver- 
 tical axis. 
 
 3. Adjust the line of collimation perpendicular to the 
 horizontal axis. 
 
 4. Adjust the horizontal axis perpendicular to the 
 vertical axis. 
 
 5. Test the motion of the objective in a vertical 
 plane. 
 
 6. Adjust the level under the telescope (if there is 
 one) parallel to the line of collimation, by the two-peg 
 method. 
 
 7. Adjust the zero of the vertical circle to read zero 
 when the line of sight is horizontal. 
 
PROBLEMS. 
 
 PROB. 13. ERROR OF SIGHTING A TRANSIT. 
 
 1. Set up the transit and clamp the vertical axis. At 
 100 feet from the instrument set a flag pole in the line 
 of sight of the telescope, and mark the position of the 
 pole on a board laid on the ground for that purpose. 
 Slightly change the position of the flag pole, and line 
 it in again. Make at least ten sightings on the pole. 
 Measure the position of the marks on the board, from 
 one end of it, and compute the linear distance corre- 
 sponding to the probable error of sighting. Reduce 
 the linear error to angular error. 
 
 2. Repeat the above process at 300 and also at 600 
 feet. 
 
 PROS. 14. ANGLES WITH TRANSIT. 
 
 1. With a transit and the other conditions about as in 
 Probs. 3 and 6, measure the four angles around a point. 
 Make the observations as follows: Sight upon one tar- 
 get and read the vernier, then turn to the second target 
 and read again. Next turn the instrument a trifle, with 
 the lower movement, to eliminate personal bias, and 
 sight upon the second target and read the vernier ; 
 then turn to the third target and read again. In a 
 similar manner measure all four of the angles. 
 
 2. Measure the three angles of a triangle. Measure 
 both the interior and exterior angle at each station as a 
 check. 
 
 3. Report the sum of the four and also of the three 
 angles, and state the length of sights, the kind of tar- 
 gets, the least count of the vernier, the condition of the 
 weather, etc. 
 
TRANSIT. 370 
 
 PROB. 15. ANGLES WITH A TRANSIT BY REPETITION. 
 
 1. With the conditions as nearly as possible the same 
 as in Prob. 14, measure the three angles of a tri- 
 angle by repetition. Repeat each angle three times. 
 Measure both the interior and exterior angles as a 
 check. 
 
 2. Report the sum of the three angles, and the details 
 of the work. 
 
 PROB. 16. TRAVERSING WITH A TRANSIT. 
 
 i. With a transit, determine the azimuths of the 
 several sides of a field, using one of the sides as the ref- 
 erence line. 
 
 What will be the effect of an error of collimation ? 
 
 PROB. 17. TRAVERSING WITH A TRANSIT. 
 
 i. With a transit, determine the azimuths of the several 
 sides of a field, using a true meridian as the reference 
 line. 
 
 PROB. 18. AREA WITH A TRANSIT. 
 
 1. Determine the area of a field having at least five 
 sides, using a transit and a roo-foot steel tape. 
 
 2. Compute the area by two methods (see Tables II 
 and III, Appendix III). 
 
 3. Report the field notes and the plat upon one sheet 
 of regulation co-ordinate paper, and the computations 
 upon another. State the area in acres and decimals 
 thereof. 
 
 4. Try to secure uniform accuracy throughout. How 
 many places of logarithms should be used ? As worked, 
 what is the weakest link ? 
 
380 PROBLEMS. 
 
 PROB. 19. MERIDIAN WITH SOLAR TRANSIT. 
 
 1. Determine the direction of the zero line of the sur- 
 veying spiral, with a solar transit. Make three obser- 
 vations in the forenoon. 
 
 2. Make an equal number of observations in the af- 
 ternoon. 
 
 3. What conclusions can be drawn from a comparison 
 of the forenoon and afternoon observations? What 
 conclusion can be drawn from a comparison of the 
 mean of all the observations and the true direction of 
 the meridian ? 
 
 PROB. 20. TESTING A HOME-MADE PLANE TABLE. 
 
 1. Test the straightness of the edge of the alidade. 
 
 2. Test the sights to see (a) if they are in a plane, (b) 
 if they are perpendicular to the board, and (c) if their 
 plane passes through the edge of the alidade. 
 
 3. Mark the point at which the centre of the bubble 
 should stand when the top of the board is level. 
 
 PROB. 21. ANGLES WITH A PLANE TABLE. 
 
 1. Lay down four angles on paper, such that their sum 
 = 360, no two being of the same size. Determine the 
 value of each by measuring the chords with a scale. 
 The difference between the sum of the observed values 
 and 360 shows the error of scaling off. Compute the 
 probable error of scaling off a single angle. Compare 
 this error with that of paragraph i, Prob. 3. 
 
 2. With the conditions as nearly as possible like those 
 in Probs. 3 and 6, measure the three angles of a triangle. 
 Compute the probable error of a single angle. Com- 
 pare this error with those of Probs. 3 and 6. 
 
 3. Report the results, and state the length of radius, 
 kind of scale used, etc., and also the form of target, the 
 condition of the weather, etc. 
 
PLANE TABLE. 3$] 
 
 PROB. 22. ANGLES WITH A PLANE TABLE BY REPETI- 
 TION. 
 
 1. Set out on the ground an approximately equi-angu- 
 lar triangle, having the length of sides, targets, etc., 
 about as in Prob. 3, 6, and 14. 
 
 2. Measure each angle with the plane table by "re- 
 peating " it six times. Scale off the difference between 
 six times the angle and 360. Make this measurement 
 with two different radii or two different scales, or both, 
 to reduce the error of measuring this angle. Subtract 
 this difference from 360, and divide the result by six to 
 get the observed value of the angle of the triangle. 
 
 3. Report the sum of the three angles of the triangle, 
 the length of radius and kind of scale used in scaling 
 off the angle, the length of sight, the kind of targets, 
 the weather, etc. 
 
 PROB. 23. AREA WITH A PLANE TABLE BY RADIATION. 
 
 1. Find the area of a field by the method of radia- 
 tion. 
 
 2. Present a reduced plat of the field, and state the 
 scale of the plat shown and also the scale of that from 
 which the area was determined. Show the directions 
 of the cardinal points. State the area in acres and 
 decimals. 
 
 PROB. 24. AREA WITH A PLANE TABLE BY TRAVERS- 
 ING. 
 
 1. Find the area of a field by traversing. 
 
 2. Make a report as in paragraph 2 of Prob. 23. 
 
 PROB. 25. AREA WITH A PLANE TABLE BY RADIO-PRO- 
 GRESSION. 
 
 1. Find the area of a field by radio-progression. 
 
 2. Make a report as in paragraph 2 of Prob. 23. 
 
PROBLEMS. 
 
 PROB. 26. THREE-POINT PROBLEM. 
 
 1. Knowing the lengths of the three sides of a tri- 
 angle located upon the ground, and having the position 
 of a fourth point given upon the ground, set the plane 
 table over the latter and determine its position on the 
 map by a mechanical solution. 
 
 2. Check the above work by a graphical solution. 
 
 3. Present a reduced plat of the work of the second 
 method, giving the scale of the plat shown and stating 
 that of the plat used in the field. State the difference 
 between the results by the two methods, 
 
 PROB. 27. TWO-POINT PROBLEM. 
 
 1. Knowing the distance between two " inaccessible " 
 points and having a third point given upon the ground, 
 determine the position of the latter upon the paper by 
 solving the two-point problem with the plane table. 
 
 2. Measure the distance from each of the "inacces- 
 sible" points to the point over which the instrument is 
 set, and plat the true position of this point. 
 
 3. Present a reduced plat of the work, stating the 
 scales used in the field and in making the reduced plat. 
 
 PROB. 28. STADIA CONSTANTS. 
 
 1. Establish eight or ten points approximately in line 
 at irregular intervals on nearly level ground. 
 
 2. Set the instrument near the end of the line, and 
 with the stadia determine the distance from the instru- 
 ment to each point. 
 
 3. With a chain or steel tape measure the distance 
 from the instrument to each point. Measure with the 
 stadia, first to avoid the possibility of any personal bias 
 in reading the stadia rod. 
 
 4. Measure c and /. 
 
STADIA. 383 
 
 5. Compute k in the formula D = ks-}-c-\-f, for 
 each distance. 
 
 6. Find the mean value of k, and compute the prob- 
 able error (in linear units on the ground) of a single 
 observation, and also the probable error of the mean. 
 
 PROB. 29. HORIZONTAL DISTANCES WITH THE STADIA. 
 
 1. Locate a number of points approximately in line 
 at irregular intervals. 
 
 2. Use the same instrument and rod as in Prob. 28, 
 and determine the horizontal distances with the stadia. 
 
 3. Rod-man and instrument-man change places, and 
 re-determine the distances. 
 
 4. Measure the distances with a steel tape. 
 
 5. Report the two series of stadia-determined dis- 
 tances, and also the true distances. Does the error 
 of observation vary with the distance ? How ? Why ? 
 
 PROB. 30. VERTICAL DISTANCES WITH THE STADIA. 
 
 1. Use the instrument and rod employed in Prob. 
 28, and determine the elevation of each point of the sur- 
 veying spiral, with reference to the central one. 
 
 2. Rod-man and instrument-man change places, and 
 re-determine the elevations. 
 
 3. Report the two series of results in tabular form. 
 
 PROB. 31. ERROR OF SETTING A LEVEL TARGET. 
 
 1. Take at least ten readings upon a rod at 100 feet. 
 Preserve at least ten consecutive results. 
 
 2. Take ten or more readings at 300 feet. Preserve 
 ten consecutive results. 
 
 3. Compute the probable error for a single reading 
 for each distance. 
 
 4. In the report state the magnifying power of the 
 telescope, the radius of curvature of the level vial, ancj 
 the kind of weathei, 
 
384 PROBLEMS. 
 
 PROB. 32. ADJUSTMENT OF THE WYE LEVEL. 
 
 1. Find (a) the radius of curvature of the level vial, 
 (J>) the value in arc of one division of the scale, and (c) 
 the value in arc of one inch of the scale. 
 
 2. Adjust the line of the bottoms of the rings per- 
 pendicular to the vertical axis. 
 
 3. Adjust the level tube parallel to the bottoms of 
 the rings. 
 
 4. Make the line of collimation to coincide with the 
 axis of the rings. In making this adjustment use a r.ear 
 point, and then test the adjustment for a remote point. 
 If it is in adjustment for the latter, the telescope slidfi is 
 correct in all particulars. 
 
 5. Test the accuracy ot the adjustments and the 
 equality of the rings, by a check level. 
 
 PROB. 33. ADJUSTMENT OF THE DUMPY LEVEL. 
 
 1. Find (a) the radius of the curvature of the level 
 vial, (b) the value in arc of one division of the scale, and 
 (<c) the value in arc of one inch of the scale. 
 
 2. Adjust the level tube perpendicular to the vertical 
 axis. 
 
 3. Adjust the line of sight parallel to the level tube, 
 by the two-peg method. 
 
 4. Repeat the adjustment for a check. 
 
 PROB. 34. DIFFERENTIAL LEVELING. 
 
 1. Level a circuit of about a half mile, returning to 
 the point of beginning. 
 
 2. State the error per mile, assuming it to vary as the 
 square root of the distance. State also the time given 
 to the field work, 
 
LEVELING. 385 
 
 PROB. 35. PROFILE LEVELING. 
 
 1. Determine the relative elevations of the several 
 points of the surveying spiral and close upon the point 
 of beginning. 
 
 2. Draw a profile on profile paper, assuming the dis- 
 tance between successive stations to be 100 feet. 
 
 3. Submit the original field notes with the profile. 
 
 PROB. 36. PRECISE LEVELING. 
 
 1. Level a circuit of about a half mile, returning to 
 the point of departure. Use the method of double lev- 
 eling with one rod see III, Fig. 78, page 268. 
 
 2. State the error per mile, assuming it to vary as the 
 square root of the distance. State also the time given 
 to field work. 
 
 PROB. 37. ERROR OF READING MERCURIAL BAROMETER. 
 
 1. Read the attached thermometer. Try to have the 
 conditions such that the temperature of the barometer 
 will not change during the observations. 
 
 2. Read the barometer, and preserve at least ten con- 
 secutive results. Alter the vernier and also the adjust- 
 ment of the mercury and the ivory point, after each ob- 
 servation, to eliminate unconscious bias. 
 
 3. Read the attached thermometer again, as a check 
 against a change of temperature of the instrument. 
 
 4. Compute the probable error of a single reading. 
 
 PROB. 38. LEVELING WITH A MERCURIAL BAROMETER. 
 
 i. Determine the elevation of the clock tower above 
 the observatory, by reading the barometer and the at- 
 tached and detached thermometers at each place. Take 
 three readings at the observatory, ihree in the tower, 
 and then three more in the observatory. 
 
 3. Reduce all the readings to 32 F. What do the 
 
386 PROBLEMS. 
 
 three readings at each point show as to the accuracy of 
 barometric leveling ? What do the means of the two 
 series at the observatory show as to gradient error ? 
 
 3. Compute the difference of elevation by at least 
 three standard formulas, using the mean of the six read- 
 ings at the observatory for the reading at the lower 
 station, and the mean of those in the tower for the 
 reading at the upper station. 
 
 4. After having performed the above operations, ask 
 the instructor for the true difference of level. What 
 does the difference between the true and the observed 
 result show as to the accuracy of barometric leveling? 
 Are the conditions of this problem favorable or unfavor- 
 able as a test of the accuracy of barometric leveling ? 
 
 PROB. 39. LEVELING WITH A MERCURIAL BAROMETER. 
 
 Repeat Prob. 38, using two points having a consider- 
 able horizontal distance between them. 
 
 PROB. 40. LEVELING WITH AN ANEROID BAROMETER. 
 
 1. Determine the difference of elevation between the 
 floor of the observatory and each floor of University 
 Hall, and also that of the clock tower. Read the aneroid 
 both the scale of inches and the scale of elevations 
 and the detached thermometer at the observatory, and 
 also at each floor on the way up, and repeat at each floor 
 on the way down and at the observatory. 
 
 2. Compute the elevation of each floor above the ob- 
 servatory, for each of the two readings, by at least two 
 formulas. 
 
 3. Compare the computed differences with the differ- 
 ences obtained from the scale of elevations. What do 
 the differences show? 
 
 4. Are the conditions of this problem favorable or un- 
 favorable as a test of the accuracy of the aneroid ? 
 
INDEX. 
 
 ABE BAR 
 
 Aberration, chromatic, 80 
 
 spherical. 80 
 
 of sphericity, 74 
 Adie's telemeter. 217 
 Adjustments, general principles, 43 
 
 see the instrument in question. 
 Alidade, denned, 146 
 
 construction, 146 
 home-made, 151 
 Altitude scale of aneroids, 313 
 Aneroid barometer, altitude scale, 343 
 
 common. 301 
 
 formula for, 341 
 
 Goldschmid, 306 
 
 scale of inches, 303 
 
 Vidi, 301 
 
 Aperture of objective. 82 
 Areas, accuracy of, with chain, 53 
 with compass, 52 
 with plane table, 169 
 with transit, 127 
 
 Babinet's barometric formula, 340 
 Bailey's barometric formula, 341 
 Balancing latitudes and departures, 53 
 Barometer, aneroid, 301 
 common, 301 
 defects, 303 
 formula for, 341 
 merits, 305 
 reading. 305 
 scale of elevations, 343 
 temperature correction for, 303 
 Goldschmid, 306 
 Vidi, 301 
 
 correct ion for temperature of, 334 
 forms. 292 
 mercurial, 293 
 cleaning, 294 
 filling, 296 
 reading, 299 
 transporting, 300 
 Vidi. 301 
 
 Barometric co-efficient, 333 
 Barometric formulas. 327 
 air, temperature of, 335 
 altitude term. 336 
 assumptions, 327 
 Babinet's, 340 
 Bailey's, 341 
 constants, 333 
 Ferrel's, 345 
 
 BAR CHA 
 
 Barometric formula, fundamental rela- 
 tions, 329 
 altitude term, 336 
 constants, 333 
 humidity, 336 
 latitude, 335 
 temperature of air, 335 
 temperature of barometer, 334, 303 
 Gilbert's formula, 346 
 humidity term, 336 
 instrument, temperature of, 334 
 Laplace's formula, 338 
 latitude term, 335 
 Plantamour's formula, 342 
 temperature of air, 335 
 temperature of instrument, 334 
 typical formulas, 338 
 Weilenmann's formula, 346 
 Williamson's formula. 343 
 Barometric gradient, 308 
 annual, 311 
 diurnal, 310 
 non-periodic, 312 
 permanent, 312 
 Barometric leveling, 292 
 limits of precision, 319 
 methods of observing, 321 
 Gilbert's, 326 
 
 observations at selected times, 322 
 Plantamour's, 325 
 Ruhlmann's, 326 
 simultaneous observations, 322 
 single observations, 321 
 Whitney's, 325 
 Williamson's. 323 
 sources of error, 308 
 gradient, 308 
 annual. 311 
 diurnal, 310 
 non-periodic, 312 
 permanent, 312 
 humidity, 315 
 instrument, 316 
 observation, 317 
 temperature of air, 313 
 wind, 318 
 
 Benches in leveling, 290 
 Boston leveling rod, 228 
 British leveling rod, 234 
 
 Chain, construction, 3 
 correction, 12 
 
 387 
 
3 88 
 
 INDEX. 
 
 CHA GOL 
 
 Chain, engineer's, 4 
 Gunter's, 3 
 merits and defects, 4 
 testing, 12 
 tying up, 4 
 using, 14 
 Chaining, 14 
 limits of precision, 27 
 on slope, 16 
 sources of error, 21 
 Chromatic aberration, 80 
 Clamp, 96 
 
 Clarke's telemeter, 217 
 Compass, magnetic, 38 
 adjustments, 43 
 care of, 46 
 construction, 37 
 needle, 41 
 adjustment, 45 
 test, 41 
 tests, 41 
 using, 46 
 
 limits of precision, 52 
 local attraction, 349 
 practical hints, 47 
 sources of error, 48 
 solar, 57 
 adjustments, 61 
 construction, 58 
 defects, 62 
 history, 63 
 merits, 62 
 principle of, 57 
 Corner target for leveling, 272 
 Cross hairs, material of, 78 
 
 stretching, 79 
 Curvature, correction for, 280 
 
 Declinator, defined, 148 
 Defining power of telescope, 80 
 Departures and latitudes, balancing 
 of, 53 
 
 level, adjustments, 247 
 collimation, 247 
 level, 248 
 wyes, 248 
 
 Eckhold's telemeter, 216 
 
 Error, probable see Probable error. 
 
 Errors, compensating vs. cumulative, 
 
 19 
 Eye-piece, 74 
 
 Ferrel's barometric formula, 345 
 Foot-plate, described, 255 
 Foot screws. 33 
 Francis leveling rod, 234 
 
 Galilean telescope, 73 
 
 Gautier's telemeter, 217 
 
 Geodesic leveling, 267 
 field routine, 270 
 methods, 268 
 record, 270 
 river crossings, 288 
 
 Gilbert's barometric formula, 346 
 
 Gilbert's method of barometric level- 
 ing, 326 
 
 Goldschmid aneroid, 306 
 
 GBA LEV 
 
 Gradient, barometric, 308 
 annual, 311 
 diurnal, 310 
 non-periodic, 312 
 permanent, 312 
 Gradienter, construction. 100 
 defined, 209 
 theory. 209 
 as a level, 209 
 as a telemeter, 209 
 constant intercept, 213 
 variable intercept, 210 
 vs. stadia, 214 
 
 vertical circle as agradienter, 215 
 Gunter's chain, 3 
 Gurley's plane table, 149 
 
 Hints, practical see the particular 
 
 instrument. 
 Huyghen's eye-piece, 74 
 
 Johnson's plane table, 150 
 Kellner's eye-piece, 77 
 
 Laplace's barometric formula, 338 
 Latitudes and departures, balancing, 
 
 Level, care of, 291 
 dumpy, adjustments, 247 
 collimation, 247 
 level, 248 
 wye, 218 
 
 notes, differential leveling, 252 
 precise leveling, 267 
 profile leveling, 256 
 height of instrument, 259, 262 
 differences, 262 
 peg, described, 255 
 precise, adjustment, 249 
 
 tests, 249 
 rod see Rod. 
 
 test by two-peg method, 246 
 tests, 236 
 bubble tube, 236 
 
 magnification vs. sensitiveness,238 
 using, 252 see also Leveling, 
 wye, adjusting, 241 
 collimation, 242 
 cross hairs, 242 
 level tube, 241 
 test level, 246 
 Leveling, barometric see Barometric 
 
 Leveling, 
 benches, 290 
 deflection of plumb, 271 
 differential, 252 
 field routine, 253 
 record, 255 
 
 instruments, adjustment, dumpy ,247 
 geodesic, 249 
 wye, 241 
 
 construction, 218 
 classified, 219 
 dumpy, 223 
 geodesic. 224 
 precise, 224 
 wye, 220 
 testing, 236 
 
INDEX. 
 
 339 
 
 LEV NOT 
 
 Leveling, testing bubble tube, 236 
 
 sensitiveness, 237 
 magnification vs. sensitiveness, 
 
 238 
 
 limits of precision, 281 
 reciprocal, 288 
 river crossings, 288 
 sight, length of, 287 
 sights, equality of, 28? 
 speed, 284 
 practical hints, 285 
 precise, 267 
 field routine, 269 
 methods, 267 
 record, 270 
 river crossings, 288 
 profile, 256 
 field routine, 257 
 record, 258 
 by differences, 262 
 by height of instrument, 259 
 improved, 262 
 modified, 259 
 reciprocal, 288 
 
 precision of, 289 
 river crossing, 288 
 screws, 33 
 sights, length of, 287 
 
 equality of, 287 
 sources of error, 271 
 computing, 278 
 curvature, 278 
 instrumental, 272 
 observational, 273 
 personal, 276 
 recording, 278 
 refraction, 278 
 rod, 272 
 
 settling of turning points, 273 
 speed, 284 
 waving rod, 254 
 
 Limits of precision see the instru- 
 ment in question. 
 Local attraction, land surveys, 356 
 mine surveys, 349 
 
 Magnetic compass, 38 
 adjustments, 43 
 cai-e of, 46 
 construction, 37 
 needle, adjustment, 45 
 
 test, 41 
 tests, 41 
 using, 46 
 
 limits of precision, 52 
 local attraction, 349 
 practical hints, 47 
 sources of error, 48 
 Magnetic declination, varies with the 
 instrument, 50 
 variation, elimination of, 349 
 Micrometer, 71 
 Mine surveys, local attraction in, 349 
 
 Needle, magnetic see Magnetic Corn- 
 
 New York leveling rod, 227 
 Notes, method of keeping see the 
 particular instrument. 
 
 OBJ HOD 
 
 Objective, aperture of, 82 
 Objective slide, dumpy level, 240 
 
 transit, 99 
 
 wye level, 239 
 
 Parallax in telescope, 88 
 Philadelphia leveling rod, 228 
 Plane table, adjustments, 153 
 construction, 146 
 complete, 146 
 Gurley's, 149 
 home-made, 151 
 Johnson's, 150 
 light, 149 
 
 limits of precision, 169 
 yractical hints, 170 
 sources of error, 168 
 tests, 153 
 
 using, method of, 155 
 intersection, 163 
 progression, 157 
 radiation, 156 
 radio-progression, 159 
 three-point problem, 165 
 Plane table, traversing, 157 
 two-point problem, 166 
 Plantamour's barometric formula, S42 
 Plan I amour's method of barometric 
 
 leveling, 325 
 
 Plate levels, to adjust, 43 
 Plumb bobs, 35 
 
 Plumbing bar, home-made, 152 
 Practical hints see the particular in- 
 strument. 
 Precise level, adjustments, 249 
 
 tests, 249 
 
 Precise leveling, 267 
 field routine, 269 
 methods, 267 
 record, 270 
 see also Leveling. 
 Prismatic compass, 39 
 Probable error, defined, 367 
 formulas for computing, 368 
 
 applications, 372 
 examples, 370 
 Problems, 373 
 Profile, drawing, 265 
 ' leveling see Leveling, 
 paper, kinds of, 266 
 
 Ramsden's eye-piece, 75 
 Refraction, correction for, 280 
 Rods, leveling, 227 
 Boston, 228 
 British, 234 
 classified, 227 
 Francis, 234 
 New York, 227 
 patterns for self -reading, 233 
 Philadelphia, 228 
 self-reading, 232 
 patterns for, 233 
 vs. target, 235 
 target, 227 
 
 target for, 230 
 Texas, 234 
 stadia, 178 
 patterns. 180 
 target, 182 
 
39 
 
 INDEX. 
 
 BUH TAP 
 
 Ruhlmann's method of barometric 
 leveling, 326 
 
 Smythe's telemeter, 217 
 Spherical aberration, 80 
 Solar compass, 57 
 adjustments, 61 
 construction, 58 
 defects, 62 
 history, 63 
 merits, 62 
 principle of, 57 
 
 Solar transit, adjustments, 134 
 construction, 130 
 using, 135 
 
 limits of precision, 143 
 mine surveys, 144 
 sources of error, 140 
 Sources of error see the instrument 
 
 in question. 
 
 Stadia, advantages of, 208 
 c, to find, 185 
 defined, 173 
 /, to find, 185 
 
 formula, fundamental, 182 
 horizontal Hue of sight, 182 
 inclined line of sight, 191 
 horizontal distance, 192 
 vertical distance, 193 
 hairs, placing of, 175 
 adjustable, 170 
 distance between, 178 
 fc, to find, 186 
 notes, reducing, 194 
 arithmetical tables, 196 
 geometrical diagrams, 200, 201 
 principles. 174 
 reducing the notes, 194 
 arithmetical tables, 196 
 geometrical diagrams, 200, 201 
 rods, kinds, 178 
 graduation, 179 
 home-made, 181 
 patterns for, 180 
 self-reading, 180 
 target for. 182 
 
 surveying, limits of precision, 204 
 practical hints, 207 
 sources of error, 202 
 inclination of rod, 202 
 observation, 203 
 using see Stadia surveying. 
 vs. gradienter. 214 
 Standard, for chain, 10 
 Surveys, land, local attraction in, 356 
 mine, local attraction in, 349 
 
 Tacheometer, defined, 173 
 Tangent screw, 96 
 
 abutting, 99 
 
 English, 97 
 
 Gam bey, 98 
 
 spring, 99 
 Tapes, linen, 9 
 
 metallic, 9 
 
 testing, 12 
 
 standard, 10 
 
 steel, 5 
 merits, 8 
 
 TAP TEA 
 
 Tapes, steel reel. 5 
 Target, corner, for level rod, 272 
 ordinary, for level rod, 230 
 
 for stadia rod, 182 
 Telemeter, Adie's, 217 
 Clarke's, 217 
 defined, 173 
 Eckhold's, 216 
 Gautier's, 217 
 gradienter, 209 
 Smythe's, 217 
 stadia, 173 
 Struve's, 217 
 tacbeometer, 173 
 vertical circle, 215 
 Telescope, aberration in, 80 
 care, b9 
 classified, 72 
 construction, 72 
 cross-hair ring, 78 
 defining power, 80 
 eye- piece, 74 
 erecting, 76 
 Huyghen's, 74 
 Kellner's. 77 
 negative, 74 
 positive, 75 
 Kamsden's, 75 
 flatness of field of, 81 
 Galilean, 73 
 magnification of, 84 
 measuring. 73 
 objective, 73 
 parallax. 88 
 size of field, 82 
 slide, 77 
 
 dumpy level, 240 
 transit, 99 
 wye level, 239 
 testing, 80 
 
 Tests see the instrument in question. 
 Texas leveling rod, 234 
 Transit, engineer's, adjustment, 109 
 cross hairs, 109 
 eye-piece, 111 
 levels, plate, 109 
 telescope, 112 
 standards, 111 \ 
 vertical circle, 114 
 care, 127 
 construction, 91 
 graduation, 93 
 accuracy, 103 
 eccentricity, 104 
 lines, 103 
 
 limits of precision, 125 
 measuring angles, 117 
 repeating, 118 
 series, 117 
 objective slide, 99 
 
 test of. 106 
 practical hints, 115 
 sources of error, 124 
 tests, 1C3 
 
 using, 115 see also Transit sur- 
 veying, 
 verniers, 94 
 vertical axes, 95 
 solar, adjustments, 134 
 
INDEX. 
 
 39* 
 
 TEA VER 
 
 Transit, solar, construction, 130 
 using, 135 
 
 limits of precision, 143 
 mine surveying, 144 
 sources of error, 140 
 surveying, 120 
 angle method, 120 
 comparison of methods, 128 
 quadrant methods, 120 
 traversing, 121 
 
 determining areas by, 360 
 Tripod, construction, 31 
 setting, 32 
 
 Vernier, denned, 64 
 direct, 65 
 least count, 65 
 principle, 64 
 reading, 66 
 barometer, 67, 69 
 
 VER WYE 
 
 Vernier reading level rod, 66 
 
 practical hints, 70 
 
 transit, 67 
 retrograde, 65 
 
 Vertical circles as a gradienter, 215 
 Vidi aneroid, 301 
 
 Weilenmann's barometric formula, 
 
 346 
 
 Whitney's method of barometric level- 
 ing, 325 
 Williamson's barometric formula, 
 
 343 
 Williamson's method of barometric 
 
 leveling, 323 
 
 Wire, to measure with, 9 
 Wye level, adjustments, 241 
 
 collimation, 242 
 
 cross hairs, 242 
 
 level tube, 241 
 
 test level, 246 
 
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