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Library

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T

PLANE SURVEYING

1

I f

FOR USE IN THE CLASSROOM AND HELD

^ BY '

WILLIAM C'^RAYMOND, C.E., LL.D.

MEMBER AMERICAN SOCIETY OF CIVIL ENGINEERS

DEAN OF THE COLLEGE OF APPLIED SCIENCE

STATE UNIVERSITY OF IOWA

AMERICAN BCX)K COMPANY

NEW YORK ONONNATI CHICAGO

WILUAM G. RAVUOND. I COrmCHT, I^M, IN GUAT Butaim.

I

k

PREFACE TO THE SECOND EDITION

This book has not been changed since it was first written eight- een years ago. When a revision was proposed the author was advised by friends to put the book in pocket book form. The publishers being willing to do this, advantage has been taken of the necessity for resetting to rewrite almost the entire book and to rearrange parts of it for greater convenience in use.

Such errors as have been discovered have been corrected. The description of the adjustment of instruments has been wholly re- written, and it is believed that the collimation adjustn^ent of the ^ transit is correctly explained for the first time in a textbook. The

1^ number of examples and exercises has been increased and these

have been included in the articles they are intended to illustrate. ^ Authors differ as to what should be included in a text on survey-

^ ing. This author feels that minute details of various survey

H operations are likely to prove cumbersome a^d confusing in a text.

^ Some things the teacher should expect to tell the student, some

things the student should discover for himself, and no surveyor is worthy the name who needs to have the details of every kind of survey outlined in a reference book. But the author does think that all principles should be carefully explained, and that exercises should be planned to show the student, by his own experience, not only the possibilities, but also the limitations of instruments, meth- ods, and individuals. An attempt has been made to do these things. In the author's judgment Crockett's logarithmic tables are the best for the use of surveyors. These were adopted for the former edition, and the same form, slightly modified to improve them still further for the surveyor's use, has been used in this revised edition. The chapter on Mine Surveying has been carefully read by Mr. Robert M. Black, Professor of Mining Engineering in Pennsyl- vania State G>llege. Mr. F. C. Young, Instructor in Civil

3

358881

0 THE SECOND EDITION

University of Iowa, has given valuable ohn H. Dunlap, Assistant Professor of Engineering in the same institution, has given the entire manuscript a thorough critical reading to its great improvement. To these the author expresses his indebtedness. He is also under obhgation to the following instrument makers for permission to use cuts from their catal<^ues :

Messrs. W. & L. E. Gurley, Troy, N.Y. ; Keuffel & Easer Com- pany, New York ; Eugene Dietzgen Company, Chicago, III. ; C. L. Berger & Sons, Boston, Mass. ; and the Universal Drafting Ma- chine Company, Cleveland, Ohio. Other indebtednesses are noted in the text.

WILLIAM G. RAYMOND.

CONTENTS

PAGB

Intkodvction 7

CHAFTBR

- I. Mkasuixmbnt or Lxvu. and Hobkontal Limes . ii InstnunentB Used .11

Methods 18

Errors Livolved 24

II. Vekniek and Lkvkl Bubble ....... 38

Vernier .......... 38

Level BubUe 43

III. Measuring DirpBRSNCBs or Altitudb, ob Lbvxling ... 49

InstnunentB ......... 49

Use of the Levd 59

AdjustmentB of the Level ....... 80

Minor Instruments ........ 91

Leveling with the Barometer ....... 94

IV. Dbtbrmination of Direction and Measurement or Angles 98

The Compass 99

Use of the Compass ........ loi

Magnetic Declination . .112

Compass Adjustments . .116

The Transit i»o

Use of the Tranat 127

Adjustment of the Transit ....... 142

V. Land Survey Computations . . ^5*

Balancing the Survey ^57

Areas 161

Model Examples . . . . '^5

Supplying Omisaons . . . . 1 74 Coordinates . .". . . .*. .181

Dividing Land . 190

VI. Some Surveying Problems ^93

VII. Some Special OrricE Instbuments ...... 202

The Planimeter 202

The Slide Rule 210

The Pantograph 220

5

VIII. I

IX. MiUDiAN, Latitude, and Tihi ....

The Solar TnnBl

AdjuUmenis of the S6\ai Tnnat .... SaegniuUer Solar Tnniit

X. GlNKHAL SUBVIVING MlTHODl

Firm Survey] in the Olda Pam of the United Statei . United Slata Public Land Sorreyi ....

Topogripbj 311

Mapping 319

The Plane Table 340

XIII. EAiTHwomt Computation! 34!

Ordinary Method! 34S

Enimating Voluma liom a Map 357

Smple Railroad Earthwork 361

XIV. Cm SuiviviNO 36S

XV. HTDIOOtArHIC SotVITINO 386

Sotnding! 3S8

The Sextant 393

Meaiuring Velocity and Diacharge 397

XVI. Mini Su»yi»iho 413

Surfece Surrey! 413

Underground Survey! 415

Connecting Suiiace and Underground Work . ; . . 414

Mapinng the Survey 4*7

APPENDIX

I. EiAMPL.. AND P.oBL!«. m S«xv.n«<: 433

II. The JvaiaAL Fuhctiohs or SuxviYoag 440

III. The OwNufHir or Sdkiey! 449

IV. GaoctAfHicAL Po!maK> or Baie Limu and Piihcipal Muidiani

coviatJiNC THi Public Snavtvt 455

V. UNiTin S1AT11 Public Land Sdivivi 459

VI, THMK-FOIHT PtOILEMI WITH THI pLANI TAILU 467

VII. FiiLD Method foe DrruHiHiKG RirEAcnoH .... 471

VIII. Tabi.!! 473

INTRODUCTION

1. PreHminary conceptions. An ellipse of axes AB and CD (Fig. i), being revolved around its shorter axis CZ), will gener- ate the surface of an oblate spheroid of revolution.

If we imagine the sea to extend underneath the surface of the earth so that the visible solid portions of the earth wili be, as it were, floating on a ball of water, the shape of that ball will be, approximately, that of an oblate spheroid of revolution. The surface of this ball is called the mean surface of the earth.

The visible solid parts of the earth above the mean surface and the invisible solid parts below, make up a very irregular body. It is customary to speak of the visible parts of the earth's surface, both fluid and solid, as the " surface of the earth."

The shorter axis of the mean surface is that connecting the poles ; the Ic^nger is the diameter of the circle called the equator. In the case of the earth these two axes do not difl^er much in length, and, hence, the earth is usually spoken of as a " sphere slightly flattened at the poles." It may seem strange to the student that a difference of twenty- seven miles should be spoken of as a slight difference. But when it is said that this difference is about one third of one per cent of the length of the longer axis, the meaning is clearer.

The lengths of the two axes according to the latest deter- minations ^ are :

Fig. I.

Shorter or polar axis Longer or equatorial axis

41,709,790 feet. 41,852,404 feet.

If a plane is passed through an oblate spheroid of revolu- tion, parallel to its shorter axis, it will cut an ellipse from the

^ Claike's spheroid of 1880. The values as found for Clarke's spheroid of 1866 are those generally used by geodesists. They are: shorter, 41,710,242 feet; longer, 4X,852,X24 feet.

7

( INTRODUCTION

1 parallel to the longer axis, it will cut a

( larth : a plane passed parallel to the polar

axis cuts an ellipse from the mean surface of the earth, while one passed parallel ,to the equator cuts a circle. Hence, me- ridians of longitude are ellipses, and parallels of latitude are circles.

The surface of the sea, or that surface extended as before mentioned, forms what is called a level surface, and a line lying in this surface is a level line.

A line perpendicular to this surface at any point is a vertical line for that point. (Subject to slight deviations at some points, negligible by the surveyor, a plumb line at any point is a verti- cal line for that point. Except at a few points, the plumb line extended does not pass through the geometric center of the earth.)

A line perpendicular to a vertical line is a horizontal line.

A tangent to the earth's mean surface at any point is per- pendicular to the vertical line at that point, and hence is a hori- zontal line for that point.

An inclined line is a straight line that is neither vertical nor horizontal.

A vertical plane at any point is a plane including the verti- cal line at that point.

A horizontal plane at any point is a plane perpendicular to the vertical line at that point.

A vertical angle is an angle formed by lines in a vertical plane.

A horizontal angle is an angle formed by lines in a horizon- tal plane.

If water collects upon the earth's surface in some depression above the mean surface, as in a lake or pond, or even as in a small glass, and if the water is still, its surface will be nearly parallel to that portion of the mean surface of the earth that is vertically below it; hence it will be a level surface, and a line drawn on it will be a level line. Such a line will be longer than the corresponding line drawn on the mean surface of the earth between the verticals through the cxtremiues of the upper line.

Qutry. At any point on the earth's surface how many vertical lines may there be f How many horizontal lines f How many horizontal planes F How many vertical planes ?

INTRODUCTION 9

Surveying defined. Surveying is the art of finding the contour, dimensions, position, etc., of any part of the earth's surface, and of representing on paper the information found.

The operations involved are the measurements of distances, level, horizontal, vertical, and inclined, and of angles, horizontal. Vertical, and inclined; and the necessary drawing and computing to represent properly on paper the information obtained by the field work.

The drawn representation is called a map. It may be a map showing by conventional signs the shape of that part of the earth's surface that has been mes>sured ; or it may be simply an outline showing the linear dimensions of the bounding lines, together with the angles that they make with the north and south line, or with each other, and sometimes the position within the tract of structures, roads, or streams.

A map of the former kind is called a topographical mapy and the operations necessary to its production constitute a topographical survey.

A map of the latter kind is a land mapy and the operations necessary to produce it constitute a land survey.

Either one of these surveys is a geodetic survey if the tract is so large that the curvature of the earth's surface must be taken into account. This limit is supposed to be reached when the tract is greater than one hundred square miles, but many surveys of tracts of much greater area than this are made without considering the mean surface to be other than plane. Such surveys are of course inaccurate, but may be sufficiently correct for the purpose they are to serve.

A plane survey is one made on the assumption that the mean surface of the earth is a plane, above which is the irregular visible surface broken by hills and valleys. Almost all land surveys are plane surveys.

The operation of leveling is considered usually under plane surveying, though the work itself is done with reference to a level surface rather than to a plane surface.

In geodetic surveying the mean surface of the earth is as- sumed to be a sphere or spheroid according to the magnitude of the survey and the precision required. The irregular portion above the mean surface is supposed to be projected down ver- tically on the mean surface, and the lines measured on the ac- tual surface are reduced to the lengths of their projections on

lO INTRODUCTION

the There is nt> such reduction in plane

sun

Ii in mind that no physical measurements

arc exact. The art of surveying maLes it possible to deter- mine that a field of land contains a certain area, more or Ujj; that a mountain is so many feet high, more or Usj; that a mine is so many feet deep, marr or less; etc. That is to say, it is physically impossible to measure exaedy either distance or angles. The precision attainable or desirable in any survey- ing operations will be discussed elsewhere in this book.

PLANE SURVEYING

CHAPTER I

MEASURBMBNT OF LEVEL AND HORIZONTAL LINES

8. The line to be measured. The distance between two points on the surface of the earth is the length of the level line joining the verticals through the points. If one of these points is much higher than the other (further from the mean surface), confusion may arise as to which of several lines is meant by the above definition. In geodetic surveying it is customary to reduce the distance, when measured, to the length of the level line lying in the mean surface and contained between the verticals

^^^^ievel

^nSurfa^^ ^/^

through the points. The distance as measured will always be approximately the length of the level line lying midway as to altitude between the two points. In plane surveying the ap- proximate length of this line is obtained by meas- uring a series of short prac- tically horizontal lines ; the sum of these lines approxi- mates to the length of the

required level line somewhat as the regular polygon of number of sides approximates to the circle.

Figure 2 will serve to make the above statements clearer.

Fig. 2.

large

INSTRUMENTS USED

4. Chains. The instruments used for the direct measure- ment of distances are chains, tapes, and wooden or metallic rods. Chains are of two kinds: Gunter's chain and the engineer's chain. These chains are alike in form, but vary in the length

II

12 OF LEVEL AND HORIZONTAL LINES

of t): ength of the entire chain. In Gunter's

cbaii t inches long, and in the engineer's chain

they or one foot, long. With this exception,

one description will apply to both.

A chain consists of one hundred "links" made of iron or steel wire. Number 12 steel wire is best. Figure 3 shows the form of the links. A link includes one of the long pieces and two or three rings, according as there are two or three rings used to con- nect the long pieces. The rings are introduced to enable one to handle the chain more readily. Brass tags with the proper number of points mark the ten-link divisions from each end toward the center, and a round tag marks the center of Bfty- link division. The handles are of brass, and are usually made adjust- able, 'SO that slight changes in the length of the chain may be corrected. A special form of handle is some- times used, having a knife edge on which a notch is filed indicating the zero of the chain for the day, the chain being compared daily with a standard kept for the purpose. Fio^ J, The Gunter's chain, having 100

links of 7.92 inches each, is 66. 00 feet long, and the engineer's chain is 100.00 feet long. The former was devised by Mr. Edward Gunter, for the purpose of facilitating the computations of areas that have been meas- ured. Its length was so taken that 10 square chains make one acre. It is the chain referred to in the table of surveyor's measure, which should be carefully memorized.

There are 7.92 inches in one link 100' links In one chain

10 squire chains in one acre 640 acres in one iquare mile

l6i feet in one rod, perch, or pcde 160 square tods in one acre-

INSTRUMENTS USED 1 3

In all surveys of the public lands of the United States the Gunter's chain is used, and all descriptions of land, found in deeds or elsewhere, in which the word ''chain" is used, are based on this chain. It is not convenient for use in connection with engineering works, such as railroad construction, canal building, bridge building, etc., where the unit of measure is the foot, and hence, in such work, the engineer's chain was formerly used, but has been almost entirely superseded by the steel tape.

Examples

1. How many rods are there in a Gunter's chain ? ^

2. How many square feet are there in an acre ?

3. What is the length of the side of a square acre in feet f In Gunter's chains ?

4. What is the length of the side of a square ten-acre plot in feet ? In chains ?

' 5. What is the difference in area between two plots respectively ten chains square and ten square chains ?

6. Reason as Gunter probably did in devising his chain and show that the length of his chain Iras correctly determined for its purpose.

5. Tapes and Reels. Steel tapes are better than any kind of chain for engineering work and for all fine surveying. These tapes are made in various forms, from thin ribbons half an inch wide to flat wires about one eighth of an inch wide and one fiftieth of an inch thick. The ribbon tapes are graduated on the front to feet, tenths, and hundredths of a foot, or to feet, inches, and eighths, and on the back to links of 7.92 inches. They come usually in box reels and are from twenty-fivt feet to one hundred feet long. They are suitable for very nice work of limited extent, and particularly for measurements for struc- tures, such as bridges and buildings, both in the shop and in the field. They are not suitable for ordinary field operations of surveying, because they are easily broken. For such work the narrower, thicker tapes are preferable. These may be ob- tained in any lengths up to a thousand feet or more; but the lengths usually kept in stock are fifty feet and one hundred feet. They are graduated variously, but may be graduated to suit the purchaser. For surveying work the following graduation is recommended :

A graduation mark at every foot; every fifth foot numbered

OF LEVEL AND HORIZONTAL LINES

; end of the tape to the other and not h end to the middle; the tape one foot ■xt the zero end than its nominal length, I foot divided into tenths or tenths and :hs. If the graduation of the end foot ths and it is required to maice measure- I hundredths, a pocket steel tape from five feet long graduated in feet, tenths, dredths may be used for the fractional In country surveys hundredths can be estimated with sufficient precision, city surveys, and other surveys re- accuracy, the hundredths must be 1.

lition to the steel tapes, linen and "me-

apes are used for rough work. The or-

- uiiiary iinen tape is well known Co every one. The

' ^ metallic tape is a linen tape with a few strands

of fine brass, wire woven through it. The linen tape is subject

to great change in length with changes of moisture in the

atmosphere, is soon stretched, and is easily worn out. The

metallic tape is not so subject to change in length with change

INSTRUMENTS USED 15

of atmospheric ctHtditions ; stretched, but u not neaii; worn out as is the linen ta these tapes, being easily stretched, soon becomequite inaccurate for any but the commonest kinds of work where the measurements are short and need not be closer than to the nearest tenth of a foot. They are graduated in feet, tenths, and half- tenths, and on the reverse side in links of 7.92 inches. Sometimes they are gradu- ated in inches. Theyaresold in paper or leather box reels.

For the narrow steel tapes there are a number of patterns of reels, most of them aiming to furnish an open reel, of form convenient to go in the pocket when not in use, and so con- structed as to enable the surveyor to reel the tape easily. There is but one

Fio. 7. V.

reel that has come to the author's at- Fk- 8.

tention that combines all three of these

requisites. This is shown in Fig. 6, and a modified form in Fig. 4. Other forms of reels are shown in Figs. 5, 7, 8, and 9. Figure 5 is a reel for a tape from 300 feet to 1000 feet in length.

l6 OF LEVEL AND HORIZONTAL LINES

Figi fitted with a spring balance for measuring

the pull on the tape when in use, a level to show when the tape is held horizontal, and a thermometer to give the tem- perature. The necessity for these attachments will, appear

hereafter. Such a tape is used for land surveys in the city of New York.

Tapes should always be Lept dry, and if wet by use, should be wiped dry and rubbed with a cloth or leather that has the smallest possible Quantity of mineral oil on it.

1. Show how to apply a tape graduated as recommended when measurine a distance that proves to be 39. 82 feet long. How would it be applied if the distance xs 51.27 feet ?

2. If the extra foot graduations are numbered, should they be numbered from the lero of the cape toward the end or in the opposite direction ?

INSTRUMENTS USED

17

8. Rods. While some rough measurements are made with the ordinary ten-foot pole or a similar arrangement, no other surveying measurements are made now with wooden or metallic rods, except measurements of base lines in connection with important geodetic surveys ; in these the rods, usually metallic, are arranged with other devices into a very elaborate piece of apparatus. It is believed that the narrow steel and brass tapes will entirely supersede the elaborate base apparatus now

in use. So-called 'Snvar" tapes made of an alloy of nickel and steel are now largely used in base line work.

Fig. 10.

7. Pins. These are used with the tape for the purpose of marking tape lengths. They are about fourteen inches long, made of steel or iron wire somewhat less than a quarter of an inch thick (No. 4 to No. 6 wire gauge), with a ring at one end into which a strip of cloth is fastened to insure ready finding of the pin when stuck in tall grass or brush. The other end is pointed. For nice work and work in the open, smaller pins from 6 inches to 9 inches long are used. Eleven of these pins constitute a set. They may be carried on a hook attached to the belt or loose in the hand. They are sometimes car- ried in a leather quiver or pouch (Fig. 10). .

8. Range poles. Poles are used to range out the line to be measured. They are usually of wood, round or hexagonal, six to eight feet long, tapering from the bottom to the top, shod with a pointed iron shoe, and painted ted and white in alternate strips one foot long. Gas pipe is sometimes used, but is not recom- mended, because, while it does not break, and while from its weight such a pole is easily balanced on its point, it is also very easily bent, and very difficult to hayiiond's sury. 2

Fio. II.

1 8 MEASUREMENT OF LEVEL AND HORIZONTAL LINES

straighten, and is too heavy t pole (Fig. ii) for nice work in made of hexagonal steel about tt inch thick, painted like the wood

METI

9. Ranging a line. Rangin) more than placing a pole or fla: which the measurement is to p e of sight, the f<

•ff pv9J 0} Sm}Wai » 9uo9tuos -ajqis -ioi sv uoos SB KjsjQt] atf} 0} }t umj3i inq 'pajtixa sstf jtuifj 3tmf atff jt)un ttviK' tou op ^ooq su/f Suipr)94 pat/sfutf 3tiv^ ttoK fta^M ''.\ usaiojjoq at{f Kq pajjttout 9q jfun. Kvp tai sfuaj (g) 911^ fo 9u^ B atvp 3aoqt> 9t{f 9J0f9q M) uo pmM%

FlQ. 13.

proper line by the rear tapeman, who " lines him in " with the forward flag when he has drawn out the tape and is ready to make a measurement. If the forward tapeman carries a pole, it is the pole that is put in line.

Sometimes it is necessary to range a line across a valley, hollow, or ravine between points on the two bounding ridges. It is impossible for the eye to carry a vertical plane down into a valley, particularly if the valley has a fairly steep slope across the line to be ranged. In such a case a plumb line is used as indicated in Fig. 12.

b-

*«■*■

METHODS 19

When a line is to be ranged over a hill intervening between the two ends of the line so that one end cannot be seen from the other, two men with flags station themselves as shown in Fig. 13 ; A lines B between himself and D; B then lines A between himself and C; they proceed similarly until no further move- ment of either is required ; both are then in the line CD.

The same method is used when one wishes -^

to place himself inline between two points and there is no one at ^ther point to direct ^^ him: If alone, he Fig. 13.

may use two poles, or

fi^Sy placing first one and then the other until both are in the required line.

Query, To secure a true line, which- is better, to use a foresight flag to- ward which the chaining is directed, or a back sight flag from which (after making the first measurement, which must be directed toward the distant point) the measurement proceeds ?

10. Doing up and undoing the chain and tape. It will be no- ticed when the chain is received from the maker that it is so folded together as to be compact in the center of the bundle and somewhat "bulky at the ends, in shape not unlike an hour glass. This remits from doing up the chain as follows :

Take the two links at the center- of the chain in the left hand, with the fifty-link tag on the left. Take the right hand ends of the pair of links next but one to those in the left hand, in the right hand, and lay the right hand pair and the intermediate pair in the left hand diagonally across the pair already there. In like manner proceed to the ends of the chain, being careful always to place the new links diagonally across the links already in the left hand and always diagonally the same way. It is better, however, to do up the chain from one end instead of from the middle. The method is the same except that the two end links are first taken in the left hand, the handle end to the right. A little more time is required, but the chain is more readily loosened for service.

To undo the chain the strap with which the chain is fastened is removed, and, if the chain has been done up from the middle.

20 MEAfflHnMENT OF LEVEL AND HORIZONTAL LINES

the two handles are taken in the left hand of the forward chain- man and the chain bundle in the right hand, allowing a few links next the handles to fall off. The chain bundle is then thrown out in a direction opposite to that in which the measure- ment is to be made, the chainman retaining the handles in his left hand. The chain should be thrown with sufficient force to straighten it out. If the chain is done up from one end, it should not be thrown out, but laid on the ground. The for- ward chainman then takes one end and carries it out, the rear chainman allowing the chain to slip through his hands and watch- ing for kinks or bent links.

When, as is now usually the case, a steel tape is used, the tape is looped in the form of a figure eight and usually is kept in this way instead of on a reel. It falls naturally into this shape and is about of the right size if an amount of tape about equivalent to the easy stretch of a man's arms, or, say about five feet, is used for each full loop.

11. Measuring on level ground. It will be assumed that the ground on which the line is to be measured is comparatively level, and that the ends of the line are visible, one from the other. If there is no visible object to mark the farther end of the line, a range pole is placed there, toward which the measure- ment is to be made. If the rear end of the line is also marked by a pole, or the direction fixed by some other visible object, the head tapeman will be able, without difiicul^y, to put him- self in approximate line, thus saving time. The forward man, usually called the head tapeman^ takes the forward end of the tape and ten of the pins, and starts toward the farther end of the line, while the rear tapeman allows the tape to slip through his hands to see that it is not kinked or bent. If he finds any bends, he straightens them.

One pin is left with the rear tapeman. When the tape is almost out the rear man calls, '' Chain." (The words, chain-* man and chain, are still quite commonly used even thcugh all work is done with a tape.) The head man then stops, turns, and straightens the tape while being put into approximate line by the rear man. The tape being taut, the rear man, holding the zero point at the pin, by motions of head or hand^ or the word " right " or " left," accurately aligns the pin held by the head man and cries '* Stick." The head man then forces the pin

METHODS . 21

into the ground, taking care that it marks exactly the gradu- ated end of the tape, and cries, " Stuck."

The rear man then, and not till then, draws his pin, keeping hold of the tape, and follows the head man, who moves on to- ward the forward end of the line, and the whole operation is repeated. After one^pin has been placed, the head man, on being stopped by the call of the rear man, can quickly put him- self in approximate line by sighting back over the pin last set to the flag left at the starting point or other object used for lining. The work thus proceeds till the farther end of the line is reached, when the head man notes the fraction of a tape be- tween the last pin and the point. This added to the number of tape lengths gives the distance required. If the distance is more than ten tape lengths the head man, when he sticks his last or tenth pin, calls " Stuck out '* or " Tally." He then waits by the pin till the rear man comes up with the pins he has col- lected, which, with the pin he started with, should be ten. He counts them, as does the head man as a check, and they note one " tally." At any time the number of tallies plus the num- ber of pins in the rear tapeman's hands gives the distance that has been measured in tens of tapes and tapes.

When the head man sets his pin he sets it slanting across the line rather than plumb, the point on the pin where it enters the ground being directly under the proper graduation mark on the tape. The tape is removed from the reel for the day or the sur- vey, and either metal handles or raw hide loops are used in the ends of the tape. The zero end of the tape should be carried by the head man.

12. Hints. The following hints may be of service to be- ginners :

The rear tapeman should not use the pin to brace himself. If a chain is used, he should hold the outside edge of the handle flush with the rear side of the pin, without moving the pin.

He should not stop the head tapeman with a jerk.

He should not sit down on the ground while holding the tape at the pin.

Motions and words should be sharp and distinct.

Motions and instructions should be proportionate to the distance that the pin is to be moved; for example, the arms should not be swung wildly when the pin is to be moved an inch.

22 OF LEVEL AND HORIZONTAL LINES

Tt hould learn to walk " on line " and should

Tl Id see that the rear man is looking when

he tries to straighten the chain or tape.

The chain should not be jerked in straightening it; it should . be straightened by an undulatory motion; the tape must be handled more carefully and never pulled unless it can be seen to be clear, as it may be kinked in the grass or weeds where a strong pull will break it.

In straining the chain or tape, the headman should pull steadily, an estimated proper amount learned by experience and discussed in Arts. 19 and 20.

Attention to these matters will greatly facilitate the work.

13. Meftsurmg oa slopes. In measuring up or down hill, one end of the tape is raised till both ends are as nearly as pos- sible in a horizontal line.

If the slope is so steep that one end of a full tape cannot be , raised enough to bring both ends in a horizontal line, the tape is " broken" ; that is, the distance is measured by using a part of the tape at each measurement. To do this, the tape should be carried out as if a full length were to be used, the head man returning to such a point on the tape (preferably a ten-foot or link point) that the portion of tape between himself and the rear man may be properly leveled. A measurement is made with this portion, then with the next succeeding portion, and so on till the whole tape has been used. Care must be taken not to get the pin numbering confused. The rear man should have but one pin for the whole tape.

The high end of the tape is transferred to the ground in one of several ways, according to the precision desired. If the work is to be done with care, a plumb line is used. If an error of a tenth of a link or foot in each tape is not impariant, a pin may be dropped from the Jiigh end, and stuck in the ground where it is seen to fall. The pin should be dropped ring down. A small pebble will serve the purpose for rough work. In care- ful work the plumb bob should not be dropped and the pin placed in the hole made ; but it should be noticed where the bob will drop, the ground should be made smooth with the foot, and the bob swung down till it is still and just clearing the ground ; then it should be carefully lowered till it touches. The tape-

METHODS 23

man should then lower his grasp on the string, hand over hand, keeping the bob steadily in its place, and place a pin in the ground at the point of the bob. The pin should be put in the ground in an inclined position across the line, so that the point where it enters the ground is that covered by the bob. The position should then be ^checked. In place of a pin a small wire brad may be used and left in the ground.

In measuring up hill, the rear man must hold the bob directly over the pin which has been set in an inclined position, and must at the same time align the head man and see that he sticks at a moment when the bob is directly over the point. It will be inferred at once that it is easier to measure down hill cor- rectly than up hill. Therefore, where close work is required on inclined ground, the measurements always should be made down hill if possible.

Where close work is required, it is considered best to measure along the slope, keeping the tape or chain supported throughout its entire length, and making the necessary reductions when the line has been measured. The reduction can be made exactly by the use of a table of versed sines if the angle of slope is known. It may be approximately obtained from Table I, page 473, by interpolating for the small angles, or it may also be approxi- mately obtained by the use of the following formula when the rise in a tape length or in the entire line, if it is of uniform slope, is known:

The square of the rise divided by twice the known side^ be it base or hypotenuse, gives the difference between the base and hypotenuse.

Demonstration (Fig. 14) : Let B he, the base, H the hypotenuse, and R the rise; C being the difference between B and H. Then 5 = i/- C and iJ = 5 + C. Assuming H known, there is written

IP -(H "C)^ = R";

whence C =

2H

Neglecting C as a very small quantity, there results

zH

F LEVEL AND HORIZONTAL LINES

there may be written ^^ + Q* - B' ~ S*,

C ;

xB

and, as betore, neglecting C, there results

Hence the rule already given.

ERRORS INVOLVED

14. Classes. The errors involved in the method of measuring just described, whether the work is done with a chain or a tape, are of two classes : (a) constant or cumulative errors, and (b) accidental or compensating errors.

(d) Cumulative errors are such as occur each time in the samf diiection. They are not equal necessarily, but may be so. Thus a line so long as to require that a tape one inch too short shall be applied to it ten times, will be recorded ten inches too long, the error of the tape being added each time the tape is applied. In this case the errors are equal.

(b) Compensating errors are such as tend to balance ; that is, they are as likely to be in one direction as in another. Thus the error that may be made in setting the pin, if it is attempted to set it just right, will be a compensating error, for it will be set ahead of the true point about as often as It will be set behind it. Error in plumbing is of the same character.

Cumulative errors arise From four causes :

1. Failure to straighten the tape for each measurement;

2. Erroneous Ungth of tape;

3. Sag of the chain or tape when not supported throughout its length (with a chain (bis is considerable);

4. Errors in judgment in making the tape horizontal in measuring up or down hill.

Compensating errors arise from accidental inaccuracies in setting the pin, from irregularities in the pull exerted on the chain or