Book___^ CoipghtN?. f^ COPXRIGHT DEPOSIT THE PRACTICAL SURVEYOR'S GUIDE. CONTAINING THE NECESSARY INFORMATION TO MAKE ANT PERSON OF COMMON CAPACITY A FINISHED LAND SURVEYOR, WITHOUT THE AID OF A TEACHER. BY ANDREW DUNCAN", Land Surveyor and Civil Engineer, A NEW, REVISED AND GREATLY ENLARGED EDITION. ILLUSTRATED BY SEVENTY-TWO ENGRAVINGS. /^6 PHILADELPHIA : HBNEY CABBY BAIRD & CO., Industrial Publishers, Booksellers and Importers, 810 Walnut Street. 1892. Copyright, by HENRY CAREY BAIRD & CO., 1892. i^ 6^ ,-^n PRINTED AT COLLINS PRINTING HOUSk, PlllLADBLFMIA, U. S. A. ^-::?$?-7i j ir'^i- sofficiently concise, and instructive in the several details, necessary to qualify properly the practical surveyor. Many of the works already published contain subjects not neces- sary in such treatises ; such as Geometry, Plane Trigo- nometry, &c., which subjects, it is taken for granted, all who intend to become proficients have studied prior to reading Surveying. They are also found not to contain instruction that in recent improvements the surveyor re- quires to know. Many of these things the compiler of this short treatise, will endeavour to supply ; also, many other necessary things, which, in his long experience, he has found indispensable to the correct practitioner. He has collected the most necessary instruction in leveling and profiling, with a new and speedy plan of setting grades on rail and plank roads. The method of inflecting curves, not hitherto sufficiently explained. The description and design of a new instrument whereby distances are found at once without any calculation. A new method of sur- veying any tract of land by measuring one line through it, with a geometrical demonstration of the same. A geo- metrical method of correcting surveys taken with the Com- (V) VI PREFACE TO THE FIRST EDITION. pass, to fit them for calculation, with a table of corrections for certain distances, but applicable to all. A short naethod of finding the angles from the courses, and vice versa. The method of surveying with the Compass through any mine or iron works, and to correct the deflections of the needle by attraction. Description of an instrument by the help of which any gentleman may measure a map by inspec- tion, without calculation. A new and short method of calcu- lation, wherein fewer figures are used than in the common method ; also, the Pennsylvania method. Tables of diflfer- ence of Latitude and Departure, made expressly for two pole chains, but which can also be used with four poles. The method of correcting the diurnal variation of the needle, most useful in tracing the boundaries of surveys, a complete description of which is given with the reason for using 57-3° and how it is found. Various methods of plotting and embellishing maps. The most correct method of laying off lots with a pole, plummets, &c. Description of a new Compass which the compiler has contrived for that purpose, and which is made by Eeid & Sons, Smith- field street, Pittsburgh. The compiler does not deny that he lias borrowed from many authors those things he has found best adapted to the completion of a work adequate to make a finished American Surveyor, of which an unprejudiced and en- lightened public are the best judges. ANDREW DUNCAN, Land Surveyor and Civil Engineer, Office, Odeon Building, Fourth St., Pittshurgh. CONTENTS INTRODUCTION. PAGB Object of surveying; What a survey should include. 17 Definition of points ; Determination of a straight line and of a plane surface ; Map of a survey 18 Scales according to which maps of surveys are drawn . . 19 INSTRUMENTS FOE MEASURING DISTANCES AND THEIR USE. Mode of indicating a point in the field; Manner of mark- ing a point in the field; Boundary stones 20 Pins and poles for marking; Establishment of a straight line in the field; To find a point lying in the prolonga- tion of a straight line • • . . . 21 Two points, A and B, marked by poles, being given, to find a point of the straight line AB, which lies be- tween A and B 23 Signals and their meaning 24 Manner of setting poles ; To find the point C of the straight line AB, which lies between A and B, with- out the help of an assistant ; How to determine a point in the straight line AB, when it is impossible to take position in the prolongation of AB 25 Length of a straight line 27 Measuring rods and their use 28 How to measure with the measuring rods a line which is not horizontal 29 (vii) vill CONTENTS. PAGE The tape measure and its appurtenances 30 Mode of measuring with the tape 31 Marking pins 32 Determination of the distance A B when measuring is prevented by an obstacle between A and B 33 To find the length A B when the straight line A B passes through a pond or forest 34 How to find the distance A B when the straight line A B continues across a river 36 Two points A and B are given; between A and B lies a forest. The direction of the straight connecting line " of these points is to be determined in A and B . . . . 38 Chains; Gunter's link 39 Ordinary surveyor's chain ; Grumman's patent chains . . 40 Vara chains; Metre chains; Marking pins 41 Content of an acre; Manner of reducing the contents of a piece of land, when given in square links, to acres, roods and perches 42 Various methods of measuring distances; Wheel pedome- ters; Pacing and pedometer 43 Different kinds of pedometers ; Mode of pacing 44 Guessing or judging distance , 45 INSTRUMENTS FOR SETTING-OUT RIGHT ANGLES, AND THEIR USE. Surveyor's cross, or cross staff 46 The cross staff head 47 Problems which may be solved with the assistance of either of the above-mentioned instruments 48 How to erect a perpendicular at the point C in the straight line A B ; How to let fall a perpendicular to a straight line A B from a point C 49 CONTENTS. IX PAGE How to determine an intermediate point C in the straight line A B 50 To test whether the visual planes of the two sights stand perpendicularly one upon the other 51 Surveyor's angle-mirror , "" Laws of the reflection of light by plane mirrnrs ^4 Axis of incidence; Angle of incidence; ^\Tjgle of reflec- tion; Explanation of the mode of oj ■'.■ ■ ■ ' • angle-mirror , . . . . 55 Use of the angle-mirror 58 How to erect with the assistance of the angle-mirror a perpendicular to the straight line A B at a point C in the latter 59 How to let fall with the assistance of the angle-mirror a perpendicular C D from a point C to a straight line AB 60 How to find with the assistance of the angle-mirror an intermediate point in the straight line A B 62 Testing the angle- mirror as to its accuracy 64 Prism for right angles 65 Use of the prism 66 To set out right angles with the tape measure alone. . . 67 Given, two points A and B, which are separated by an obstacle which cannot be seen through, for instance, a foiest. To determine their connecting line in the points A and B, as well as the length of A B 69 Given, a straight line A B passing across a lake; its length is to be determined 71 A given straight line A B strikes in its prolongation an obstacle, which cannot be seen through, for instance, a house ; to find the prolongation on the other side of this obstacle 72 A straight line A B is given. From a point C a perpen- X CONTENTS. PAGE. dicular is to be let fall to the straight line, the direct execution of the operation being, however, rendered impossible by an intervening building *13 SURVEY OP SMALLER TRACTS WITH THE ASSISTANCE OF THE PREVIOUSLY-DESCRIBED INSTRUMENTS. To survey a three-sided field 16 To survey a many-sided field or polygon 77 Noting the results of the measurements 79 PROBLEMS. To reduce two pole chains and links to four pole ones ; To reduce four pole chains and links to two pole ones . 81 Areas; To find the content oi" a square 82 To find the area of a parallelogram 83 The content of an oblong piece of ground and one side being given, to find the other 84 To find the content of a trapezium 86 Various rules for finding the content of a triangle. ... 87 To find the area of a trapezoid, rule 99 To find the area of a trapezium when each side and the angle of intersection of the diagonals are given. . . . 101 To find the area of a trapexium, when the four sides are severally given, and also the sum of any two opposite angles 102 To find the area of a circle having the diameter given ; To find the area of an ellipsis ; To find the area of a parabola ; To find the area of a segment of a parabola ; To find the area of a field or lot, which is found to be the frustum or zone of a parabola, included by two parallel lines, and the intercepted curves of the para- bola 105 CONTENTS. XI TKiaONOMETRICAL SURVEYINS. PAGK Mode of executing a trigonometrical survey 107 JDetermination of the area of a triangle 109 How to measure a tract of land by measuring a base line through it, and not departing from that line, and yet finding all the distances round the land, their course'^ and angles of the field, and the area :';> Correction of the difference of latitude and depai '.re m surveys taken with the compass 120 To find what may be the error in the difi"erence oi rati- tude and departure of a given station arising from the inaccuracy of practice 121 Table of errors in links and decimals ; Rules for altering the legs of stations in the correcting of surveys. . . . 124 Description of an instrument by which any person, though unskilled in surveying, may measure a map, or part of a map, almost at one view 126 Description and design of a new instrument by which distances can be found at once, without any calcula- tion 127 The use of tlie instrument in measuring distances. . . . 132 Example in measuring distance 133 Given the bearings of any two stations of a survey, thence to determine the angle made by those stations . 136 Method of calculation 141 Another method of calculation wherein fewer figures are used 144 A new and concise method of calculation, wherein fewer figures are used than in the common methods .... 146 Noble's method of calculation 147 Example in numbers 151 To survey with the compass through any mine, or other cause for drawing the compass needle of its parallelism; Mean diurnal variation for every month in the year. . 153 Xll CONTENTS. PAGE Manner of plotting the notes 155 Lotting or laying out towns, etc 156 Methods of keeping field-books 157 Tracing of old mearings , 161 Of levelling 163 Field-book distances measured with a one hundred Ceet chain 165 To lay out a road on a regular grade up a liill 166 To inflect in curves on railrords and others 168 Problems ITI To find the perpendicular ordinates from the chord G of any arc of a railroad, in order to set off the curve cor- rectly and speedily, without the help of an instrument. 173 Tables of latitude and departure 180 TABLES OF SURVEYS. The use of the foregoing tables in relation to surveys; Examples • 192 SYSTEMS OP RECTANGULAR SURVEYING FOR SURVEYING THE PUBLIC LANDS OF THE UNITED STATES. Townships and sections ; What constitutes a range. . . 194 Size of townships 195 Establishment of standard parallels; Instruments to be used in surveying 196 Measurement of township lines and subdivision lines; Tally pins; Process of chaining 197 Levelling the chain and plumbing the pins 198 Marking lines; Sight trees or line trees 199 Marking trial or random lines 200 Establishing corners; Points for perpetuating corners . 201 Definition of and instructions for meandering; Meander corners 202 Meandering of lakes, deep ponds and navigable bayous . 203 Surveying ; Base line 204 Principal meridian ; Standard parallels j Guide meridians. 205 Exterior of township lines . , 206 Index 209 PRACTICAL SURVEYOR'S GUIDE. INTRODUCTION. The object of surveying consists in taking the measurements and observations of larger or smaller parts of the earth's surface, so that a map may be drawn of the part surveyed, and its content calcu- lated. A survey should include all the natural and artificial features of the ground, for instance, rivers, roads, railroads, fences, buildings, limits of cultiva- tion, boundaries of estates, etc. It is, however, not always necessary to take the measurement of all these objects, it depending mainly on the par- ticular object of the survey, which of them are to be included. Thus, for instance, in surveys for economical purposes, the quality of the soil and its producing capacity have to be taken into consider- ation, which is not necessary in topographical sur- veys. 2 (17) 18 THE surveyor's GUIDE. The *' points" which are to be determined in surveying are not the mathematical points treated of in geometry, but the corners of houses, fences, stones, etc., which are mere points in comparison with the extensive surfaces and areas which they are the means of determining. Strictly speaking, their centres should be regarded as the points alluded to. A straight line is determined^ that is, has its length and its position fixed, when the points at its extremities are determined ; and a plane surface has its form and dimensions determined, when the lines which bound it are determined. Consequently the determination of the relative position of points is all that is necessary for the principal objects of surveying, which are to make a map of any surface, such as a field, farm, etc. A map of a survey is a miniature copy of the field, farm, etc., as it would be seen by an eye moving over it ; or as it would appear, if, from every point of its irregular surface, plumb lines were dropped to a level surface under it, forming what is called in geometrical language its horizon- tal projection. INTRODUCTION. 10 The scale, according to which the map is drawn, varies according to the object of the survey. Thus, for economical surveys it is tb-u-b to Wtjtt ; for city and village surveys about sh ; while for topograph- ical surveys it is considerably smaller. INSTRUMENTS FOR MEASURING DISTANCES, AND THEIR USE. A POINT in the field may be indicated either by a natural or artificial mark, the vertical corner of a building serving, for instance, for the former. Such corner of a building marks the points at which it inter- sects the ground. In the same manner a lightning rod or a vertical flag pole may determine a point. The mode of artificially marking a point in the field depends on the time the mark is to last. If it is to be permanent, as, for instance, the boundaries of estates, stones are used. The intersecting point of two lines cut in the head of the stone indicates the point in the field which is to be marked by the stone. Boundary stones must be set suflBciently deep in the ground to insure the permanency of their position. In taking measurements in cities, cast iron posts, planted perpendicularly in the ground, are frequently used. These posts are pro- vided on the top with a conical aperture for the in- (20) MEASURING DISTANCES. 21 sertion of a marking pole, so that in measuring the point may be marked ieven at a greater distance. Temporary artificial marking may be effected by wooden pins. For temporary marking, whilst tak- ing measurements, painted marking poles shod with iron are used. If the distances are not very long, eight-feet poles, and for longer distances, 12-, and even 20- or 25-feet poles may be necessary, which then require rope-stays. The shorter poles are painted in alternate lengths of black and white, or red and white, the longer have the top painted and a red flag attached. A straight line in the field is established by two points marked in the above-described manner. The determination of further points in the straight line may, however, be effected in various ways and, hence, examples referring to definite cases will here be given. Let the points A and B, Fig. i, indicate a straight li7ie marked in the field hy two poles. To find the point lying in the prolongation of the straight line AB. Proceed with a marking pole to the neighborhood of the point C to be sought, and walk in a direction 22 THE surveyor's guide. perpendicular to AB, until to the eye, looking in Q^LZZZIZ... Q-! O ABC Fig. 1. the direction of BA^A appears to be covered by the pole B. Then plant the marking pole, which had been taken along, so that it covers the poles A and B. The pole then marks a point C from A B. However, the poles possess a thickness, which must not be neglected, and hence the accurate execution of the operation is not as easy as it may appear from the above description. After planting the pole (7, step back a few paces and examine its posi- tion. From Fig. 2 it will be seen that an angular space PoF is hid from the eye at o, by the pole (7, this angular space being the greater, the nearer the eye p o e --^^h3s^ ^ B C E' Fig. 2. IS to the pole. To the eye at o, the poles A and B MEASURING DISTANCES. 23 appear simultaneously covered, as long as they lie in the angular space P o P, even if O is not exactly in the straight line AB. Hence, to reduce this angular space as much as possible, it is recommended to step back at least a few paces, in order to exam- ine whether the poles A^ B^ Care in a straight line. If the poles used have the same diameter, the position of the pole Q may also be tested by bring- ing the eye in such a position that now the right, and then the left edges of the poles A and C cover each other. If this is feasible, the pole C stands in the straight line AB ; because if C is not in a straight line as shown in Fig. 3, the pole B pre- vents the right edges of A and C from being cov- ered. a::zzz::z2zz::z:zD - ^ B c Fig. 3. The pole C is now said to be ranged in the direc- tion ABy or aligned in the straight line AB. Two points^ A and B, marked by poles, being given, to find a point C of the straight line AB, which lies between A and B. 24 THE surveyok's guide. Place yourself a few steps behind B^ so that to the eye the pole A appears to be covered by B. Then by signals direct an assistant to shift the pole (7> held in a vertical position, in the direction oi AB until it is also covered by B. The pole then marks a point Q in the straight line AB. In this case it is also said the pole is ranged or aligned in the straight line AB. In order to maintain the pole (7 in a perpendicu- lar position, the assistant should grasp it above the middle, and hold it pendent between the thumb and index finger. The pole then assumes by itself a perpendicular position. It is, of course, necessary to explain to the assist- ant the meaning of the signals. He must further be instructed to constantly turn the face towards the surveyor, and hold the pole in a perpendicular position with the arm extended. As regards signals, the shifting of the pole to the right or left by the assistant is indicated by lifting the right or left arm. Making signs to the right or left for this purpose should be avoided, as they might be readily misunderstood, especially at a greater distance. At great distances, or with an MEASURING DISTANCES. 25 unfavorable light, it is also advisable to hold a read- ily visible object, for instance, a pocket handker- chief, in the hand. To indicate to the assistant that the pole held by him is in the right position, move the hand up and down in a vertical direction. The assistant then plants the pole in the indicated place. The poles should be set truly vertical, and so fixed as to remain so. The loss of time in having to send men long distances to reset poles on windy days may be very great. The vertical position of a pole may be readily tested by a plumb line. The point C of the straight line A B, which lies between A and B^ may also be found without the help of an assistant as follows : First determine according to the directions given on p. 21, a point i> in the prolongation of AB (see A ^ 5 . ^ Fig 4. Fig. 4) and then the point in the prolongation from BB. How to determine a point in the straight line AB, when it is impossible to take position in the prolongation of AB. 26 THE surveyor's guide. In fixing an intermediate point in the straight line AB according to the preceding methods, it was supposed that the surveyor could place himself in the prolongation of AB, for instance, behind B, and look towards A. This, however, is sometimes not possible if, for instance, the two points A and B are indicated, as in Fig. 5, by the corners of rig. 5. two buildings. The mode of procedure is then as follows : From any desired point aj, which should be located as near as possible to the straight line AB place a pole b^, in the straight line a^B. Next from 5i, place a pole, a^, in the direction of Ab^ Then again from a^, place a pole, b^, in the direc tion of a^B, and so on. The poles a^, b^, aa, h, con stantly come closer to the straight line AB. After a few repetitions of this operation, a point a or 5, lying in the straight line AB, will be reached. MEASURING DISTANCES. 27 The above described process is also employed for determining an intermediate point, when the straight line AB passes over a hill, so that it is impossible to see B from A. The assistant must, however, be careful to select the points designated in Fig. 5, bi, hi, etc., so that A is visible from them, and the points «i, otj, as, so that B can be seen from them. Points in the straight line AB may also be deter- mined if a building stands between A and B. Since, however, measurements of distances are re- quired for the purpose, the solution of this problem will be given later on. Length of a straight line AB. By the length of the straight line AB is understood the length of its horizontal projection, as for instance, the length Fig. 6. CB in Fig. 6. The length AB of the straight line connecting the points A and B, is called the ob- lique length. 28 THE sueveyor's guide. The most important instruments for measuring distances are the measuring rods, the tape-measure and the chain. The measuring rods, of which at least two are required for measuring a distance, are wooden rods or bars, protected on their ends by metal, and se- cured against moisture by being several times satu- rated with hot oil. They are painted aud graduated to feet and tenths, the feet marked on it being, as a rule, alternately painted black and white. For measuring a horizontal straight line AB, two measuring rods are required. The rod bearer places one end of the first rod at A, and aligns it in the straight line AB. He then takes the sec- ond rod, carefully places it against the end of the first and also aligns it in the straight line. He then takes up the first rod and lays it against the end of the second in the direction of AB. The measure- ment is thus continued until one of the rods extends over the terminal point B of the line to be meas- ured. To guard as much as possible against errors in counting, each rod when taken up should be counted aloud. The distance of the point B from the end of the last rod but one, is then read off MEASURING DISTANCES. 29 from the graduation of the last rod. Now, sup- pose the measuring rods are each 20 feet long and the entire distance is 12 measuring rods and 3J feet, then the length of AB is (12 X 20 + SJ feet) = 2431 feet. IToiv to measure ivith the measuring-rods a line AB, ivhich is not liorizontal. As previously stated, by the length of AB is understood the length of the horizontal projection AB' of AB (Fig. 7). For taking the measure- ment of this length two persons are required. Fig. 7. One person, by the eye., and for very accurate measurements, with the assistance of the plumb line, sets a prismatic rod perpendicularly at the point A. The other person lays the measuring rod in a truly horizontal position, so that one end 30 THE surveyor's GUIDE. touches the prismatic rod and the other the ground, and aligns it in the straight line AB. The first person then again plants the prismatic rod perpen- dicularly at hi, and the second person again lays the measuring rod horizontally on the prismatic rod and aligns it in the direction of AB, and so on. By now counting the number of lengths of measur- ing rods and measuring the horizontal distance of the terminal point b of the last measuring rod from B, the length of the line AB is found. The tape-measure and its appurtenances. The steel tape is from 25 to 100 feet or more in length J to f inch wide, and about yV i^ich thick. It is graduated to feet and tenths or twelfths. It is convenient for measurements made at any height from the ground, and also useful for long offsets. But its liability to twist and kink renders it easily broken, while few persons can anywhere be found to repair it. With care one may be safely used for a long time, but it should not be left, even for a moment, in inexperienced hands. Woven tapes strengthend with cords of catgut or wire are also used for the same purpose, but are far less accu- rate, and are unsuitable for measuring more than MEASURING DISTANCES. 31 two tape-lengths in one continuous distance, without appreciable error. Common measuring tapes are altogether untrustworthy. They stretch to an ex- tent visible to the naked eye, and shrink after wet- ting. On the ends the tape is provided with stout metal rings, which serve for the reception of stakes. The latter are wooden rods 3 to 4 feet long, with a diameter somewhat smaller than that of the termi- nal rings of the tape. Their lower end is shod with iron, and provided with an iron cross pin. The object of the pin is a two-fold one : it prevents the ring of the tape from sliding off the stake, and serves for forcing the latter by means of the foot into the ground. To take measurements with the tape two per- sons are required, the forward and the hind man, each of them carrying one of the two stakes over which the terminal rings of the tape are pushed. The forward man. Walking in the direction of AB, pulls the tape along until the hind man arrives at A. The latter then draws the attention of the for- ward man to the fact by calling out " halt." The hind man now sets his stake at A^ and ranges the 32 THE surveyor's guide. stake of the forward man in the direction of AB. The forward man, after noting the correct position of his stake, pulls the tape tight, so that it covers the point previously marked with the stake. The terminal point of the tape is then marked by the forward man by means of a marking or tally pin. The marking pins of stout steel wire are about 15 inches long, and provided below with a point and above with an eye. At the commencement of measuring the forward man receives ten such mark- ing pins strung through the eyes on a wire ring. After marking the first length of the tape with one of the pins, he calls to the hind man " go on," The tape having become slack by the hind man taking up the stake, is again pulled forward by the forward man until the hind man notifies him, by calling out " halt," that he has arrived at the marking pin. The hind man then takes up the marking pin, replaces it by his stake, adjusts the stake of the forward men, and so on. This process is repeated until the forward man has passed the terminal point B of the line to be measured. The number of marking pins taken up by the hind man then gives the number of tape-lengths. By multiply- MEASURING DISTANCES. 33 ing the number of marking pins thus taken up with the length (say 100 feet) of the tape, the dis- tance from A to the last marking pin is obtained. To this has to be added the distance from the last marking pin to B, which is read off from the tape itself, in order to obtain the entire length of AB. If the ten marking pins, with which tlie forward man sets out when commencing to take the meas- urement, do not suffice, those collected by the hind man in the course of measuring have to be delivered as often as necessary to the forward man. How- ever, to avoid mistakes, this should be done only after the hind man has taken up the tenth pin. An account of how often the ten marking pins have been delivered to the forward man must, of course, be kept by some convenient arrangement. Determination of the distance AB when meas- uring is prevented by an obstacle between A and B. This problem may be solved in various ways according to special conditions, and hence, a few solutions with reference to definite cases, will here be given. 34 THE SURVEYOR S GUIDE. To find the length AB when the straight line AB passes through a po7id or a forest. Choose a convenient point (Fig. 8) from which A and B can be seen, and the distances Fig. 8. measured. Mark the point (7 by a stake, extend the lines AC md BO beyond (7, measure ^(7 and BO, and make A,0 = Aa B,Q==BQ then AyB,=AB MBASUKING DISTANCES. 35 Hence it is only necessary to measure A^Bi. If, however, J-i^j are not accessible, make, in accordance with Fig. 9 or 10, then is A,0=~AO B,Q=~BQ A,B.^ ^AB and consequently AB = n A,B,. 36 THE surveyor's guide. Hence it is only necessary to measure A2B2 and to multiply this line by n. Fig. 10. How to find the distance AB when the st7'aight line AB continues across a river. Fix first a point Q (Fig. 11) in the extension of AB; choose a convenient point Z), from which Jl, jB, C, can be seen, and measurements be taken toward B and Q. Measure BD and (7Z>, and make B,B=~BD n CD = --CD MEASURING DISTANCES. 37 then is Bid parallel to BQ, or, what is the same, to AB. Now determine the point of intersection Ai from AD, with the elongation of B^Ci, by sim^ Fig.n. uitaneously ranging a stake A^ in AB and BiO^. Then A,Bi = ^AB n consequently AB = n.AxBi. Hence to find AB it is only necessary to meas- ure AiBi^ and multiply by n. 38 THE SURVEYOR S GUIDE. Two points A and B are give'ri ; between A and B lies a forest. The direction of the straight connecting line of these poi7its is to be determined in A and B. The direction of AB (Fig. 12) may be estab- lished by determining on each side of the forest a point which lies in the straight line AB. If i) and E are such points, AB and 5 £' indicate the di- rections of the straight line ^^ in the points A and B. To obtain the points B and E^ choose laterally from AB a point C, which allows of looking and MEASURING DISTANCES. measuring towards A and B. Then measure A C and B C, and make A,0=-^AO then is B,0=^BO; A,B, =: —AB. Now, in two points (^ and e, so situated that they can be seen from C along the edge of the forest, set marking poles, and determine the points of in- tersection, Bi and Bi, of the lines Cd and Ce^ with AiBi. By then making OB=nCB^ OB==nCB,, B and ±J are two points of the straight line AB. Chains. These are made of various patterns, and are galvanized, painted or plain. The latter are most liable to rust, while the first are liable to be less correct from the additional process they undergo. The long-linked pattern, having each link a foot or a Gunter's link in length, is consid- ered advantageous, it being lighter with the same amount of similar material, and its form showing most readily any accidental kink or derangement 40 THE surveyor's GUIDE. during chaining. Furthermore, its links can be readily hammered straight after being accidentally bent. The curb chain and such small-linked pat- terns are modifications introduced with the use of superior metal, the amount of which is correspond- ingly reduced. They can be made very light and convenient, but when damaged are less readily rec- tified than the old pattern. In any case, the length of a new chain should not only be tested with a good standard when simply laid straight, but also again after stretching it by a weight. During the progress of survey work, the length of the chain should be daily tested by comparison with a temporary standard marked for the purpose and kept invariable. Th» ordinary surveyor's chain is 66 feet, or four poles long, composed of 100 links, each connected to the other by two rings, and furnished with tally-marks at the end of every ten links. The tallies should be read from the beginning to the end of the chain, and not- from both ends to the middle. Grumman'' s Patent Ohains. These chains, in- vented and patented by J. M. Grumman, of Brook- lyn, N. Y., are made of very light steel wire, the MEASURING DISTANCES. ' 41 links being finely tempered and so formed at the ends as to fold together readily and thus dispense with the use of rings. Vara Chains. The Spanish or Mexican Vara, which is in general use in Texas, Mexico, Cuba and South America, is 33|^ inches long. The chains are made of ten or twenty varas, each vara being usually divided into five links, each link, including a ring at each end, is, therefore 6f inches. A chain of ten varas has titty links ; of twenty varas one hundred links. Each vara is marked by a round brass tally, numbered from one to nine in the ten-vara chain, and from one to ten, each way, in the twenty-vara chain. Sometimes, but rarely, the vara is divided into four links ; a ten-vara chain then has forty links, and a twenty-vara, eighty links. Met7'e Chains. The French metre is very gen- erally used as a standard in South America, the West Indies, etc. The number of links to a metre and the tallies are similar to those of the vara. Marking Pins. In chaining there are needed ten marking pins or chain stakes, made of iron, steel or brass wire, as may be preferred. The 42 THE surveyor's guide. pins are about 14 inches long, pointed at one end, to enter the ground, and formed into a ring at the other, for convenience in handling. They are sometimes loaded with a little mass of lead around the lower end, so as to answer as a plumb when dropped to the ground, from the suspended end of the chain. In land measurement, the acre is the unit. It contains 4 roods, and a rood contains 40 perches. A perch is a square rod, otherwise called a pole. A rod is 5| yards or 161 feet. Hence,! acre =4 roods = 160 perches =4,840 square yards = 43,560 square feet. One square mile = 5,280 X 5,280 feet = 640 acres. Since the ordinary surveyor's chain is 66 feet long, a square chain contains 4,356 square feet, and, consequently, 10 square chains make one acre. Care should be taken not to confound 10 square chains with 10 chains square. The former make one acre ; the latter space contains 10 acres. When the contents of a piece of land is given in square links, as is customary, cut off four figures on the right (i. e., divide by 10,000); to get it MEASURING DISTANCES. 43 into square chains and decimal parts of a chain, cut off the right hand figure of the square chains, and the remaining figures will be acres. Multiply the remainder by 4, and the figure, if any, outside of the new decimal point will be roods. Multiply the remainder by 40, and the outside figures will be perches. The nearest round number is usually taken for the perches, fractions less than half a perch being disregarded. Thus— 86.22 square chains = 8 acres, 2 roods, 20 perches; Also, 64.1818 « =6 « 1 « 27 « Also, 43.7564 " =z 4 " 1 '' 20 « Various methods of measuring distances. Wheel-pedometers record on a dial-face the num- ber of revolutions of, or distance passed over, by a wheel rolling on the surface of the ground. On very smooth and even ground the results are mod- erately fair, the error being chiefly due to slip- ping. Facing and pedometer. Next to guessing, 44 THE surveyor's guide. simple pacing is the worst method of obtaining dis- tance, the inequality of the paces under various con- ditions, combined with the errors in counting, ren- dering it very inaccurate. A pedometer may be used to register the number of paces taken without any attention on the part of the person wearing it. It is made in the form of a watch, and carried in the pocket. The number of steps given by the pedometer, multiplied by the length of the step, will give approximately any distance walked over. In another form the instrument is intended to be regulated according to the length of the step of the person carrying it, and then the distance is registered on the dial in miles. The pacing adopted should be even, but natural, and not strained with the mistaken object of conforming to any arbitrary length, for instance, three feet. Some paced distance, on the level, up-hill and down- hill, should be tested by careful measurement to obtain the ratio for reduction to yards, feet or miles, under these three conditions of any future pacings. The ratio should again be frequently checked. It is also necessary to acquire the art of walking in a straight direction. To do this, fix MEASURING DISTANCES. 45 the eye on two objects in the desired line, such as two trees, bushes, or stones, or tufts of grass. Walk forward, keeping the nearest of these objects steadily covering the other. Before getting up to the nearest object, choose a new one in line further ahead, and then proceed as before, and so on. Guessing or judging distance may, with con- tinued practice and checking, be carried to an accuracy that is surprising. The main points to be noticed are the rise or fall of the ground and the direction from which the light falls on any object at the distant point. Some persons estimate in yards, others in their own paces, which are more readily available for testing such guises. INSTRUMENTS FOR SETTING-OUT RIGHT ANGLES, AND THEIR USE. The simplest instrument for this purpose is the surveyor'' s cross, or cross staff. It consists of a block of hard grained wood, 3 or 4 inches square, and 1| to 2 inches thick, having two saw-cuts more than half through its thickness, and intersecting each other at right angles at the centre of the block. This block is fixed on a pointed staff, on which it can turn freely. Another form of the surveyor'' s cross (Fig. 13) consists of two pairs of sights (diopters) placed at the ends of two bars at right angles to each other. The slit, and the opening with a hair stretched from its top to its bottom, are respectively at the top of one sight and at the bottom of the opposite sight. The cross sits by means of a socket upon a pointed staff shod with iron. (46) SETTING-OUT RIGHT ANGLES. 47 Fig. 13. Another form of the surveyor's cross, the so- called cross staff head, is shown in Fig. 14. It is octagonal in shape. Four sides of the head are provided with sights (diopters), the oculars and objectives of which consist of slits (like A and _S), while upon each of the other four sides a slit CO' serves as the ocular, and a hair DD' as the objec- tive, the oculars and objectives upon the parallel sides belonging, of course, together. The visual planes AB, A'B', as well as OB and O'B' stand perpendicularly opposite to one another and inter- sect in the axis of the head, which is also the axis of the socket H and of the staff stuck in the latter. 48 THE SURVEYOR S GUIDE. With the assistance of either of the above de- scribed instruments the following problems may be solved : Fig. 14. 1. To erect a perpendicular at the point C in the straight line AB. 2. To let fall a perpendicular to a straight line AB from a point C. 3. A straight line through the points AB is SETTING-OUT RIGHT ANGLES. 49 given. An intermediate point C in the straight line is to be found without entering the prolonga- tion of AB. How to erect a perpendicular at the point O in the straight line AB. It is supposed the point C (Fig. 15) has been found by ranging a stake in the direction AB. Set oD J Fig. 15. the instrument at C, and turn its head until the stake A is seen through the sight ab from b. By- no w setting a stake D in the visual plane of the other sight ed, then CD is perpendicular upon AB, because the visual planes of the two sights stand perpendicularly one upon the other. How to let fall a perpendicular to a straight line AB from a point C. 50 THE SURVEYOR'S GUIDE. First determine a point E in the prolongation of AB (Fig. 16). Then set up the instrument at a point of the straight line AB, which seems to the eye to be about the bottom of the perpendicular. Now turn the head of the instrument until B is seen through one of the sights, and note whether 9i> W Fig. 16. the stake C is in the visual plane of the other sight. If such is the case, the staff of the instrument marks the bottom I) of the perpendicular CD. However, if C lies to the right or left of the visual plane, move the instrument to the right or left, and repeat the operation until the correct spot is found. How to determine an intermediate point in the straight line AB, SETTING-OUT RIGHT ANGLES. 51 Set the instrument in what is supposed to be about the center of AB, and sight through the sight ab the point B. If the point chosen lies in the straight line AB at about C (Fig. 17), A can also be sighted from b, provided the position of the in- strument remains unchanged. If, on the other d_ g;^ ^B Fig. 17. hand, the point chosen lies outside the straight line AB at about 0^, then when sighting through ab from a to B, the line of sight from bi on, will pass A in the opposite direction, and the distance AAi, at which it passes A, will be about that of CCi, when C lies nearly in the centre of AB. Now, therefore^ move the instrument about half the distance of AAi, and repeat the operation until the back-sight covers A. To test whether the visual planes of the two sights Qdiopters) stand perpendicularly one upon the other. 62 THE surveyor's guide. Suppose AB (Fig. 18) is a straight line marked by two stakes A and B^ and C an intermediate point in AB. Erect a perpendicular CD at the point Q. For this purpose sight through the sight ah the stake A^ and then set the stake D in the visual plane of the sight cd. The straight line (^ n Fig. 18. (7Z>i stands perpendicular upon AB only when the angle of the visual planes of the two sights is a right one. If it is not exactly a right angle, but as in Fig. 18, somewhat smaller, the angle ACD will also to the same extent be smaller than a right angle, because by the operation just described the angle aCd has actually been laid out at the point GmAB. By now turning the head of the instrument (see SETTING-OUT RIGHT ANGLES. 53 Fig. 19) until the visual plane ah covers the stake Z>, and again setting in the visual plane of the other sight a stake E, then the angle BiCE is equal to the angle aCd^ and consequently the ■OiTT-D 6 Fig. 19. angle A QE is double the angle of the visual planes of the two sights. Hence the instrument is correct only when A CE is an obtuse angle, A Q and E lie in a straight line, and the plane of sight ed covers, in the last position of the instrument, the point B. Surveyor's angle-mirror. This instrument (Fig. 20), consists of two mirrors. Si and Si, so placed as to form an angle of 45°. The mirrors are secured in a brass box closed towards the side of the vertex of the angle formed by them, and open on the other side. Above the mirrors the 54 THE surveyor's GUIDE. sides of the box are provided with rectangular apertures, the so-called windows Fx and F^. To the bottom plate G- is secured a handle which, as a rule, is provided with a small hook, to which a plumb bob may be suspended. Fig. 20. The construction of the angle-mirror is based upon the laws of the reflection of light by planft mirrors, which are as follows : 1. The reflected ray lies in the plane determined by the axis of incidence and the incident ray. 2. The angle of reflection is equal to the angle of incidence. SETTING-OUT RIGHT ANGLES. 65 By the axis of incidence of a ray of light is understood the perpendicular upon the plane of the mirror at the point at which the ray of light strikes it ; and by the angle of incidence^ the angle which the incident rays form with the axis of incidence. By the angle of reflection is understood the angle which the reflected ray of light makes with the angle of incidence. The mode of operation of the angle-mirror will be explained by the scheme, Fig. 21, in which the two mirrors, S^ and ♦S'a are represented as simple lines in ground plan. Suppose, in the direction AB, a ray of light falls upon the mirror S-^. The axis of incidence for this ray is the perpendicular Bh in the point B upon the mirror Si. The angle of incidence is the angle ABh. If the ray AB is reflected towards BCy then the angle CBh is the angle of reflection. Now, as previously stated, according to the laws of reflection, the angle of reflection is equal to the angle of incidence, and hence, O5 = 90°. Thus, the angle B OB, which the ray passing out forms with the entering ray, is constant; it is i'nde- 58 THE SURVEYOR S GUIDE, pendent of the angle of incidence A and of the ray AB. From this it follows that the direction of the ray passing out is not changed by turning the an- gle mirror around its axis. Therefore, the eye looking in the direction AB into the mirror, will see a quiescent picture, though the mirror may be turned around its axis. This property renders the angle-mirror very suitable for use, it requiring no support, it being simply held in the hand. Fig. 22. Use of the angle-mirror. By holding the angle- mirror before the eye vl, Fig. 22, so that both mirrors are vertical, and the eye on looking past the edge tj, of the mirror S^ through the window of the mirror *S'i, sees a stake P, while at the same SETTING-OUT EIGHT ANGLES. 59 time it looks in the same direction into the mirror Si, the visual ray, according to the explanation pre- viously given, takes the course ABQD. If a stake stands in the straight line (7Z>, the eye A sees this stake in the mirror S^ at B. Hence the stake D appears to the eye at A exactly in the same direction as P. The stake P, seen directly, forms the straight continuation of the picture of the stake Z), which lies sideways. The point is the vertex of a right angle, whose sides pass throu^ the points B and 1). The angle-mirror is of such small dimensions that in the field it may be taken as a point ; hence, may be considered coinci- dent with the point obtained by plumbing the an- gle-mirror. If, therefore, a point P, seen through the mirror aS'^, lies perpendicularly over the pic- ture of another point Z> seen in the mirror S-^, the directions from the location of the mirror to these two points are perpendicular to one another. How to erect with the assistance of the angle- mirror a perpendicular to the 'straight line AB at a point Q in the latter. Place yourself before the point C (Fig. 23) with the face turned towards the side where the 60 THE surveyor's GUIDE. perpendicular is to be erected. Now hold the angle- mirror, with the open end turned towards A (or 5), in a truly perpendicular position over C, and look with the eye for the picture of the stake A in the mirror. When this has been found, set the qG /d Fig. 23. stake P, so that when seen through the window of the angle-mirror, it is the direct continuation of A. Then AQD is a right angle and, hence, GB the desired perpendicular. ]n order to be convinced at any moment that the angle-mirror is in the straight line AB^ it is recom- mended to previously range in the direction AB^ a stake E in the prolongation of AB. The angle- mirror is then in J.-S, when the pictures of the two stakes A and E cover one another. How to let fall with the assistance of the angle SETTING-OUT RIGHT ANGLES. 61 mirror a perpendicular CD from a point O to a straight line AB. Determine first a point E^ Fig. 24, in the pro- longation of the straight line AB. Then place yourself with the face towards A in the neighbor- hood of the bottom of the perpendicular sought, so that the eye is in the straight line AB^ which Fig. 24. is recognized by the two stakes A and E covering one another. Hold the angle -mirror close before the eye, with the opening turned towards (7, and note whether the picture of O appearing in the mirror is the straight continuation of the directly seen stake A. If such is the case, the position of the angle-mirror indicates the bottom of the perpen- 62 THE surveyor's GUIDE. dicular ; if not, walk back or forward in the straight line until the picture of is the continuation of A. How to find with the assistance of the angle- mirror an intermediate point C in the straight line AB. This operation is based on the following facts, which will be readily understood : If a perpendicular is first erected at (7, Fig. 25, Fig. 25. upon AQ^ and then one upon 5 (7, these perpendic- ulars coincide. If (7j is not a point in the straight line AB, but has the position indicated in Fig. 26, the two perpendiculars to AQ^ and ^Ci will devi- ate from each other, the two right angles AQJ) and BCiD\ being separated by an angular space, DxCiD'i. If now the point (7 occuDies the position indicated by C-i in Fig. 27, the perpendiculars to SETTING- OUT RIGHT ANGLES. 63 AGi and BG% also do not cover one another, while the two right angles ACJ)i and BQJ)'^ partially cover one another. Hence, it will be seen that the surveyor is at a point C in AB only when the perpendiculars to J.C and BQ coincide, and fur- ther, that in order to reach the straight line AB^ he must walk forward if, as in Cj, the two right angles partially cover one another, or backward 64 THE surveyor's guide. if the two right angles are separated by an angular Testing the angle-mirror as to its accuracy. The two mirrors of the instrument should form an angle of 45°. For the purpose of testing it, range in the given line AB marked by stakes, two stakes C and B, Fig. 28, and laterally plant a stake B in ' -"'it the centre oi'CB. Now, with the assistance Fig. 28. of the angle-mirror, let fall a perpendicular to A, according to the directions previously given. This may be eflfected either by walking in the direction from B to 0, until the directly seen stake O is the straight continuation of the picture JS' seen in the mirror, or by walking in the direction from to B until the directly seen stake B forms the prolonga- tion of the picture of B. If the mirror is correct, SETTINa-OUT EIGHT ANGLES. 65 the operator will both times strike the same point F. If, however, the angle of the two mirrors is not exactly 45°, he will first strike the point #1 and then the point F^. Both points F^ and F^ will lie at the same distance in opposite directions from the correct point F. (Fig. 28 illustrates a case in which the angle is smaller than 45°.) If, now, Fx and F^ have been found, the centre of the dis- tance FxFi is the bottom of the perpendicular sought. Fig. 29. Prism for right angles. This instrument (Fig. 29) consists of a glass prism, the cross section of which is a right-angled isosceles triangle. The plane of the hypothenuse of the prism has a smooth reflecting surface. "The instrument is pro- G6 THE SURVEYOR S GUIDE. vided with a handle, on the lower end of which is, as a rule, a hook or eye for suspending a plumb- bob. The prism serves as a substitute for the an- gle-mirror, and, generally speaking, is used in the same manner. It is smaller, so that it can be con- rig. 30. veniently carried in the pocket, and, being invaria- ble, does not require to be constantly tested. Use of the prism. By looking towards the stake P (Fig. 30), and holding the prism before the eye J., so that the axis of the prism stands SETTING-OUT EIGHT ANGLES. 67 vertical, and the visual ray AB strikes tlie prism in the neighborhood of the sharp edge, and the plane of the hypothenuse stands about perpendicu- larly upon the visual ray, the latter in passing through the prism takes the course ABOD^F. Hence, a stake standing at F would appear to the eye A, looking through the prism, in the direction AP. Thus, while the eye looking over the frame of the prism sees the stake P, it perceives, in the straight continuation of the latter, the picture of the stake F in the prism. The angle POF is a right angle, and its vertex always lies so close to the prism, that in the field, the point obtained by plumbing the prism, may be designated 0. Hence, it may be said that the field-point obtained by •plumbing the prism is the vertex of a right angle, the sides of which pass through P and through F. If therefore, the stake P, which is directly seen, forms the straight continuation of the picture of the stake F, seen in the prism, then the visual rays from the position of the prism towards P and F stand perpendicularly one upon the other. To set out right angles with the tape measure alone. There are various methods for setting out 68 THE SURVEYOR S GUIDE. right angles with the tape measure alone ; two of them will here be given. Let AB he the straigJit line upon wliich at the point C a perpendicular is to be erected. 1. Measure off equal distances CD and OE (¥ig. 31) on each side of the point C. Then after fas- tening the ends of the tape with two pins at D and JE, grasp the tape in the centre, and walk sideways until both halves of the tape are stretched tight. 2 D C JE P Fig. 31. The centre F of the tape then indicates a point of the perpendicular to be erected in C ; because the triangle DFE is an isosceles triangle, and in such a triangle the line connecting the summit with the centre of the base stands perpendicularly upon the latter. 2. Measure off from C (Fig. 82), along the given SETTING-OUT RIGHT ANGLES. 69 line a distance CD = 6 metres, then let two as- sistants hold in OD the ends of a portion of the tape, 18 metres long, grasp the tape at a point E A Fig. 32. at a distance of 8 metres from C and 10 metres from D, and draw it tight. CE is then perpendic- ular to A Bj because the triangle D CE is right- angled at E, since DW=W=100. and OF' + UE = 6' + 8' = 36 +64 = 100, consequently DE'= dW+ CE. In the following a few examples of the application of setting out right angles are given. 1. Given, two points A and B, which are sepa- rated hyan obstacle which cannot he seen through. 70 THE surveyor's GUIDE. for instance, a forest. To determine their con- necting line in the points A and B, as well as the length of AB. Set out through the point A (Fig. 33) any con- venient straight line, which passes close by the forest, and determine upon it the bottom Q of the Fig 33. perpendicular let fall to it from the point B. By now measuring .4 (7 and BC, the length is obtained according to the Pythagorean proposition AB = ^AC'-i-BC To determine the direction of the straight line AB, erect upon A C two perpendiculars Avhich cut the straight line AB outside the forest. Now if SETTING-OUT RIGHT ANGLES. 71 D and E are the bottoms of these perpendiculars and F and (r their still unknown points of intersec- tion with AB, it follows from the similarity of the triangles ABF and AOB that FB: BQ=AB'. AC. Consequently FB =^AB. AC From the similarity of the two triangles AECr and A OB, it further follows that as : BC=AE: AC, and therefore BO GE= jjyAE, By now measuring AB and AE, FB and GE can be calculated with the assistance of the two formulas. By now transferring the lengths thus calculated to the perpendiculars erected in B and E, the points F and Gr lying in AB are obtained. Bj AF the direction of the straight line at A is determined by BGr a.t B, and the problem solved. 2. Criven, a straight line AB passing across a lake ; its length is to he determined. 72 THE SURVEYOR'S GUIDE. Erect in convenient points C and i^(Fig. 34) of the straight line AB perpendiculars and lay out upon them convenient, but equal lengths, GB and I'E. The connecting line BE oi the terminal points of these lengths is then parallel to AB, and in length equal to OF ; consequently AB = AC+BE^^FB, Hence it is only necessary to measure the three lengths J.(7, i>^, i^5; their sum total gives the length AB. 3. A given straight line AB strikes in its pro- longation an obstacle, which cannot he seen through, for instance, a house; to find the pro- longation on the other side of this obstacle. Erect upon AB (Fig. 85) at B, a perpendicular BO oi such length that the line through parallel SETTING-OUT RIGHT ANGLES. 73 to AB passes the obstacle. Then erect at B upon D (7 a perpendicular QD of such length that a per- pendicular erected at I) upon CD passes the obsta- cle. Make the latter perpendicular DE equal BO ; then ^ is a point in the straight line AB, and the perpendicular ^i^ erected at EmponBU, the prolongation of AB sought. Fig. 36. 4. A straight line AB is given. From a point C, a perpendicular is to he let fall to the straight 74 THE SURVEYOR'S GUIDE. line, the direct execution of the operation heing, however, rendered impossible by an intervening building. Erect at a convenient point I) m AB (Fig. 36) a perpendicular upon AB, and let fall to this per- pendicular from C the perpendicular CE. Now measure the distances CE and BE and make BF= CE. Then F is the bottom of the perpendicular sought, and BE its length. The problem maj also be solved as follows: ("t Set out through C (Fig. 37) any convenient straight line and measure off upon it from C equal lengths CB and CE. From the points B and E SETTING-OUT* RIGHT" ANGLES. 75 let fall perpendiculars to- the straight line AB. Suppose the bottoms of these perpendiculars are F and G ; then the centre ff of the length FCr is the bottom of the perpendicular sought, and the length of the perpendicular is C'ff=l{DF+ FG). To prevent inaccuracies, the erection of one per- pendicular upon another must in this method be avoided. SURVEY OF SMALLER TRACTS WITH THE ASSISTANCE OF THE PREVIOUSLY DE- ^ SCRIBED INSTRU- . MENTS. To survey a three-sided field. A triangle being determined by the length of its sides, it is only necessary to measure the latter. c Fig. 38. A triangle may also be determined by its base AB (Fig. 38), its altitude CD, and the bottom I) of its altitude. Hence a three-sided field may be surveyed by determining with the assistance of the (76) SURVEY OF SMALLER TRACTS. 77 angle-mirror or the prism the bottom D of the perpendicular let fall from (7 to AB^ and then measuring with the chain, tape or rods, the lengths AB, AB and CB. To survey a many-sided field or polygo'ri. Di- vide the polygon (Fig. 89) by diagonals into tri- angles, and measure the three sides of each trian- gle. Since each triangle is determined by its sides, the polygon also is thereby determined. Fig. Another mode of surveying a polygon is as fol- lows : Mark out by several stakes a straight line ab (Fig. 40), the so-called base-line and axis of the abscissa, which cuts the polygon as nearly as pos- sible in the direction of its greatest dimension, and 78 THE SURVEYOR'S GUIDE. let fall to it perpendiculars from all angles of the polygon. A convenient angle of the polygon, for instance the point (7, is then determined, according to its position, by the length of the perpendicular CCi let fall from it to the base-line ah, and by the distance of the bottom Ci of the perpendicular from a convenient point, for instance A^, upon the base- line ah, which has been chosen as the initial point for taking the measurement. The distance A^d is called the abscissa of the point (7, and the length C(7i, the ordinate of point C. Hence, every angle of the polygon is determined by its abscissa and ordinate. The abscissa and ordinate taken together are called the coordinates of the point O. This SURVEY OF SMALLER TRACTS. 79 method of surveying is also called the coordinate or normal' method. Noting the results of the measurements. To note the results of the measurements, so that there cannot be any doubt about their meaning, make first a rough sketch of the tract to be surveyed, and in this sketch, mark the measurements found. The marking must be done according to a definite princi- ple, so that the position of the figures at once shows to which distances they refer. Write all measure- ments on the terminal points of the distances verti- cally to the distance to which they refer, so that, when looked at from the initial point of the dis- tance, they stand upright, i. e., can be read (see Fig. 41). The abscissas are measureed by adher- ing throughout to the same initial point, in Fig. 38, 80 THE surveyor's GUIDE. they being, for instance, all counted from the point 0. "The abscissa of the point VII. is 84.6 metres, i. e., the bottom of the perpendicular from the point VII. to the base line is at a distance of 84.6 metres from ; the ordinate of the point VII. is 35.8 metres, i. e., the length of the perpendicular from the point VII. to the basis is 35.8 metres. Athough the polygon is completely determined by its corner points, it is also advisable to measure the sides. These measurements are written in the sketch either perpendicularly to the sides, in the centre, or on the end of the sides, or parallel to the sides in the centre of the latter. PROBLEMS. Problem First. To reduce two pole chains and links to four pole ones. If the number of chains be even, the half of them will be four pole ones, to which annex the given links. Thus : 1. In 16 chains, 37 links of two pole chains how many four pole ones : 2)16. 37 Ans. 8.37 But if the number of chains be odd, take half of them and add 50 to the links. Thus : 2)131.40 Ans. 65.90 Problem Second. To reduce four pole chains and links to two pole ones. Double the chains and annex the links if 6 (81) 82 THE surveyok's guide. they be less than 50, but if they exceed 50, add one to aouble the chains and take 50 from the links. C. L. Thus: 16.25 of four poles, how many two pole chains. 16.25 2 Ans. 32.25 C. L. 2d. In 19.87 four pole chains how many two pole ones. 19.87 2.50 Ans. 39. 37 To reduce two pole chains and links to perches and decimal of a perch, multiply the chains by C. L. two and the links by four, thus: In 16.37 how many perches. 16.37 2. 4 Ans. 33.48 Article First — of Areas. A square is a plane figure having four equal sides and four right angles. To find the content, PROBLEMS. 83 multiply the side into itself and the product is the content. Example. Kequired the area of the square A B C D, one of whose sides is 25 chains 95 links. Fig. 42. 25.95 25.95 A c A. R. P. 67. 1. 14.44 12975 23355 12975 5190 A. 67.34025 4 25.95 R. 1.36100 40 P. 14,44000 A parallelogram is a four sided figure whose opposite sides and angles are equal. To find the area multiply one of the sides by the perpendicular demitted from one of its opposite angles. Example. Required the area of the parallelogram A B 84 THE surveyor's guide. C Dj the length of which is 15 chains, and height 12 64. 12.64 A. R. p. 18. 3, 33.6 15 00 C 12.64 15 6320 1264 Acres, 18,960 4 Roods, 3.840 40 Fig. 43. Perches, 33,600 The content of an oblong piece of ground and cue side are frequently given to find the other. Divide the area in perches by the given side, gives the side required which is easily reduced into chains and links. If a lot contains 507 perches and is 14.25 long, what is its width. PROBLEMS. 85 29)507 'l7.4827 29 8.25+12.06=8 37.0G 217.0000 203 140 116 240 232 80 58 220 203 To draw maps of these figures is too obvious to require any explanation. 5th. When the sides of the above figures are given in feet and inches, reduce the inches to deci- mal of a foot. Then multiply the length by the breadth and divide the product by 43-^60, the number of feet in an acre, the quotient will be the acres and decimal of an acre, which may be redu- ced to roods and perches by multiplying by 4 for the roods and 40 for the perches, pointing oif the proper number of decimal places each time, thus : A lot of land is 600 feet 4 inches long and 240 86 THE SURVEYOR S GUIDE. feet 3 inches wide, how many acres does it contain. 600.333.X240.25=144230.00325 Acres. A. R. P. This divided by 43560. gives 3.31106 or 3 1 09.T6 4 1.24424 40 9.76960 Ans. 6th. A trapezium is a four sided figure the oppo site sides of which are neither equal nor parallel. To find the content, measure a diagonal and two perpendiculars to the opposite corners, multiply the diagonal by half the sum of the perpendicu- lars, and the product will be the area. Example. Let A B C D be any trapezium, having A C 80 perches, and the perpendiculars as in the figure. Fig. 44. PROBLEMS. 87 25 20 2)45 22.5 80 160)1800,0(li.l.00 An8. 160 200 160 40)40(1 40 7th. A triangle is a figure having three sides and three angles, any side may be called the base, having the base and perpendicular given. Multi- ply the base by half the perpendicular, or the base by the whole of the perpendicular, and take halt the sum. Example. Let A B C be any triangle whose base IS 100 two pole ch's and 15 links, and perpendicular 40 Fig. 45. 88 THE surveyor's guide. chains and 20 links, required the content m acres. 100 15 2 4 40 20 200.6 perches and decimal. 2 4 40.4 half the perpendicular. 2)80 8 8024 8024 40 4 A. K. P. 160)8104.24(50 2 24.24 A71S. 800 40)104 80 24 8th. Having the three sid-es given to find the area rule, add the three sides together and take half the sum, from which subtract each side severally, multiply the half sum and three remainders con- tinually into each other, and the square root of the product will be the area. The most satisfactory proof of the above rule is the following: Let A B C be any triangle, B C its base, A B the greatest side, and A C the least, and let P be half PROBLEMS. »y the perimeter. In A B take A D=A C, joiii D and draw A E perpendicular to D C and E G pa- allel to B C cutting A B m G, with the centre G and radius G E describe a circle cutting A B in L and A B and E G pro- duced in K and H. Join H B and produce A E, H B till they meet in M. Since A D=A C and the angles at E are right, the squares of A E, E D are equal 2 A E,+2 E C and .-. E D=E C or D C=2 D E. Hence by similar triangles D G E, D B C, B C=2 G E=E H, and B C is also paral- lel to E H .-. H B M is parallel to C E D and (Euc. 1st 29th) the angle B M E=D E A, viz : a right angle, and H E being a diameter, M is a point in the circle. But from similar triangles D G E, D B C, D B=2 D G, to each of these equals add A D+A 0=2 A D, and B A+A C-2 A G, to each of which equals add B 0=2 G E=2 G K .-. A B+A 0+B 0=2 A K or A K is =P, half the perimeter. Now the area of the triangle A B 0=area A J) C+area B D 0=A EXD E+ 90 THE surveyor's GUIDE. M EXD E (B M being parallel to D C)=A MXD E But by similar triangles A D E, A M B : A E : E D :: A M to M B, and by equi-multiples the first multiplied by third : the second multiplied by the third :: the second X by the third : the second X by the fourth. Hence, A EXA M : E DXA M :: E DXA M : E DXM B, i. e. the area of the triangle is a mean proportional between A EXA M, and E BXM B. Now E DXM B ,= P— A BXP— A C, and A EXA M-A LXA K= PXP — B C. Hence the area of the triangle is : ^^TPXP— A BXP— A CXP— B C, which is the rule. Example. 9th. Suppose the sides to be measur- ed by a four pole chain and be AB 10.64) A C 12. 28 y B C 9. 00 j Sum 31.92 Fig. 47, I sum 15. 96 5. 32 first remainder. 3. 68 second do. 6. 96 third do. PROBLEMS. 91 15.96X5.32X3.68X6.96=2174.71013216(46.()387 16 86)574 516 926)5871 5556 And since 9323)31501 10 square 27969 4 pole ch's make one 93263)853232 acre, this 279789 becomes ACRRS. 932667)7344316 4.66337 6528669 4 2.65348 40 A R P 26.13920 The content is 4 2 26. If the sides are in perches and decimal, divide the square root of the products of the half sum and three remainders by 160, and the quotient will be the acres, and the remainder divided by 40 will be the roods. The same may be more readily done by loga- rithms, for as the addition of logarithms serves for 92 THE surveyor's guide. the multiplication of their corresponding numbers, and that the number answering to the half of a lo- garithm will give the square root of the number of mat logarithm, it follows that half the sum of the logarithms of half the sum of the sides, and the three remainders will give the area, thus : Half sum, 15.96 log. 1.20303 First remainder, 5.32 " 0.72591 Second " 3.68 " 0.56585 Tnird " 6.96 " 0.84261 2)3.33740 Square four poles 46.63 1.66870 Or, 4.663 4 2.652 40 A. R. P. 26.080 4 2 26 as before. 10th. When the three sides are given and the angles are required, call either side on which the perpendicular will fall from the opposite angle the base, then as the base is to the sum of the other two sides so is the difference of those sides to the difference of the segments made by the perpen-. dicular, then half that difference added to half the PROBLEMS. 93 sum gives the greater, and substracted the less, by which means it is divided into two right angled triangles, the hypothenuse and one leg of each being given, the angles are easily found by plane trigonometry. Example. Let A B C be / \ ng. 48. any triangle hav- ing the sides giv- en as follows, viz; A B 88, B C 54 and A C 108 to find the angles. A B=88 Then as 108 : 142 :: 34 B C=54 34 142 sum 34 difference. 568 426 108)4828.000(44.703 diff. of the 432 segments at the base 22.351 half diff. 508 432 760 756 400 324 94 THE surveyor's guide. Then half the base 54+22.351=76.351, the greater segment A D, and 54—22.351=31.649 the less segment. The triangle is now divided into two right angled triangles, the hypothenuse and base in each being given to find the angles, as follows : As A B 88 1.944482T : Rad. 90° 10.0000000 :: A D 76.351 1.8828147 11.8828147 i.9444827 : Sine A B D 60°.ll' 9.9383320 Ana 90— 60°.ll'=29°.49'.r-Angle BAD. In the same way C B D is found to be 35°. 53' its com- plement 54°.07"=/' B CD. Now A B D=60°.ll' B D=35 .53 Angle A B C-- =96 .04 t A= 29 .49 L C= 54 .07 180 .00 Proof as the three angles of every plane triangle are equal to 180° per 32d of the 1st of Euclid. 11th. Many things occur to the practical sur- PROBLEMS. 95 veyor in the triangle, some of which I shall take notice of in this place. It often happens in prac- tice that the two sides and their included angle are given to find the other angles and side. KuLE. — As the sum of the sides is to their differ- ence so is the tangent of half the sum of the oppo- site angles to the tangent of half the difference ; this half difference added to half the sum of the angles at the base gives the greater, and sub- tracted the less. Then as sine of either of the base angles is to its opposite side, so is sine of the contained angle to the required side. A Example. Let AC =80, B C Fig. 49. 110,and L A.QB 102°.30'to find A B and the angles A and B. Side B 110 From 180 Side AC 80 take /. C= 102.30 Sum 190 2)77.30 sum of bas< angles. Diff. of sides 30 I sum =38.45 96 THE surveyor's GUIDE. Then as 190 log. 2.2787536 38°.45' : 30 " 1.4771213 7 .13 :: Taff't 38°.45' 9.9044910 45 .58 ^ A 11.3816123 2.2787536 31 .32 ^ B !:Tag'tof|difr.7°.13' 9.1028587 Then as sine B 31°.32' 9.7184971 : A C 80 1.9030900 :: sine C 102.30 9.9895815 Or its supplem't 77.30 11.8926715 9.7184971 To A B 149.34 2.1741744 12th. Again, it often happens that the area must be found from the foregoing data, in that case multiply the two sides together, and that pro- duct by the natural sine of ^ the contained angle, gives the area. Example. Fig. 50. Let ABC be a triangle having the side A C 13 chains, A B t 7c. 501. and /- B A C 42° to find the area. PROBLEMS. 9T 7.5 13 225 75 ^T.5X.384565 half the nat'l sine of 42<»= A. R. P. tJ2.SiJ square four pole chains = 32 19.2 An». Demonstration. Let fall the perpendicular B. D. A B : B D :: rad : sine A .-. B D=--A BXsine A Ead But rad.^1. ,•■ B D=A B=sine. Multiply each side by A C ai.d B D. A C=A B. Sine A.XA C But A. C. B D— the area. Hence, 2 A B. A C. Sine A=area, which is the rule. 13th. Let B A C be a triangular farm, and P a well of water. It is required to draw a line or fence from the well that will divide the farm equally between two partners : 7 98 THE surveyor's GUIDE. E D Find D the middle of the base, B C, and from P take a course of P D. Again set your instrument at A, and take the same course A E ; cause a pole to be set at E, a line or fence from E to P will oisect the farm, which is easily demonstrated from the figure. See Bland. 14th. Again, suppose the well P, to be situated within the farm, and it be required to divide it equally between three occupants, so that each may have the use of the well. a In fig. 52 di- vide the base B C, into three 1 . • X/ / I '■ \\ Fig. 52. equal parts in D and E. Set your instrument at P, and take ^ PROBLEMS. 99 the courses P D and P E. Remove your instru- ment to A, and take A F the same course as P D, and A G the same as P E. Cause stakes to be driven at F and E in a straight line between B and C. Fences from F, G, and A, to P, trisect the farm, which is plain from the figure. 15th. To find the area of a Trapezoid Rule, multiply half the sum of the parallel sides by the perpendicular distance between them, and the pro- duct is the area. Let figure 53 be a Trapezoid ; if A D be g v bisected in E, and E F drawn parallel to A B or C D, it also ! ij^ig.ss. bisects B C in F. — Through F draw G H parallel to A D. It is evident the triangles, B F G, and F C H, are similar and equal. (26th Euclid, 1st.) .-. E F, half the sum of the sides, multiplied by the per- pendicular distance between them, A D, gives the area. F 100 THE SURVEYOR S GUIDE. Being surveying on the side of a bog, and want- ing four acres to make up a division, and seeing A Fig. 54, A 50 B would pass through a pond, I found A C fifty chains, and L C 56*^ ; how far must I measure from C towards B, so that the triangle ABC, may contain four acres. Since A C X C B X i the natl. sine of 56°=4 acres, it follows that 4 acres divided by the pro- duct of one half the natl. sine of 56° into A C, gives B C the required side. Thus : 50X4=200 perches X, 4145188=82.9 ; and 640 perches in 4 acres, divided by 82.9=7.72 per the length of B C, and in like manner any other simi- lar case can be done. 17th. Sometimes it is found necessary to ob- tain the area of a trapezium from having the diagonals and the angle of intersection given. PROBLEMS. 101 Rule — Half the product of the diagonals multi- plied by the natural sine of the angle of intersec- tion, will be the area. Example. If the two diagonals of a trapezium be 40.15, and 60.13 chains the /- of intersection 75° 45', what is the area, i of 40.15X60.13=1207.1097= half the product of the diagonals, and 1207.1097X 96923=(natural sine of 75°45')=1169.96693453l-- A. R. P. the area, in square four pole chains, or 116. 3. 39. 47. Answer. 18th. To find the area of a trapezium, when each side and the angle of intersection of the diagonals are given. Mule — Square each side of the trapezium ; add together the squares ot each pair of opposite sides ; subtract the less from the greater; multiply the difference by the tangents of the angle of intersection. One fourth of the product will be the area. Example. What is the area of a trapezium, the sides of which are 10, 13, 7.16, 8.32, and 10.05 chains respectively, and the V of intersection of the diagonals 52° 15'. 102 THE surveyor's GUIDE. (10.13)=102.6169 ^ 8.32)--= 69.2224 171.8393=Sum of sqs. of opposite sidea (15.05)=226.5025 ( 7.16)= 51.2656 277.7681=Sum of sqs. of other sides. 105.9288 Difference, Multiplied by .32288=^ the natural tangent, 34.20290944 or A 3. 1 .27,23 perches. For a demonstration of the foregoing, see Crib- (son's Surveying, hy Trotter. 19th. To find the area of a trapezium, when the four sides are severally given, and also the sum of any two opposite angles. Rule — From half the sum of the four given sides, subtract each seve- rally ; multiply the four remainders continually together ; from the result subtract one half the continual product of the four sides, multiplied by unity, increased by the natural cosine of the sum of the given angles. The square root of the result will be the area. PROBLEMS. 103 Remark. In the application of this theorem, it must be carefully remembered that the cosine uf an angle is positive when that angle is in either the first or fourth quadrants, and negative when it is in the second or third quadrants. For a demonstration of this beautiful theorem, see also, Grihson, by Trotter. N. B. When the sum of the opposite angles is 180°, that is, when the trapezium can be inscribed in a circle, the above rule is simply : from half the sum of the given sides, subtract each side seve- rally ; multiply the four remainders continually together, and extract the square root, gives the area. Example. " One morning in May I went to survey. As soon as bright Sol I espied ; I measured round a four cornered ground. In the margin see the length of each side ; The angle at B, together with D, An hundred and fifty degrees ; The meadow's content is all that I want, Assist me kind youths, if you please." 104 THE SURVEYOR'S GUIDE. A B 15.60 B C 13.20 C D 10.00 D A 26.00 ch'ns. 2)64.80 sum. 32.4=1 sum. Fig. 55. 16.80=lst remr. S=half the sum. 19.20=2d do. of the sides. 22.40=3d do. 6.40=4th do. Whence (s-A B)X(s-B C)X(s-C D)X(s-D A)- 32.4X16.8X19.2X22.4X6.4=46242.2016=46242. 2016 And A B. B C. C D. D A.X(l+cos. 150°) That is 15.60X13.20X10.00X26.00 -X0.1339746=3586.44C4 DifFerence=42655.7552 The square root of 42655.7552 is 206, 5327= A. R. P. area in square four pole chains, or 20. 2. 24,55232. N. B. This problem is taken from Deighan's Arithmetic, vol. second, page 148, and the answer A. R. p. there given is 21. 2. 00,64, which is obtained by taking the trapezium to be inscribed in a circle, which is not the case. PROBLEMS. 105 When the opposite angles of a quadralateral are equal to two right angles, a circle can be descriljed about it. The rule to find the area, then, is : mul- tiply the half sum, and four remainders continually together, and extract the square root, for in that case l+cos.(A+B)=0. 21st. To find the area of a circle having the diameter given. Rule — Square the diameter, and multiply by .7854, and you have the area. 22d. To find the area of an ellipsis. Rule — Multiply the transverse and conjugate diameters together, and that product by .7854, and you have the area. 23d. To find the area of a parabola. Rule — Multiply the height by the breadth, and take two- thirds of the product ; you have the area. 24th. To find the area of a segment of a para- bola. Rule — Multiply the base of the segment by the altitude thereof, and two-thirds of the product gives the area. 25th. To find the area of a field or lot, which is found to be the frustum or zone of a parabola, included by two parallel right lines, and the inter- cepted curves of the parabola. Rule — Add the two 106 THE surveyor's GUIDE. parallel ends, divide the square of either of these ends by this sum, add the quotient to the other end, multiply this sum by the altitude of the frus- tum or distance of the ends, take two-thirds of the product, and it gives the area. TRIGONOMETRIGAL SimYEYlNG, 26th. It was not my intention to say any thing concerning this branch of surveying, as it is too extensive a subject for this small work ; but as some young readers may not have met with any thing on that subject, I will present them with an outline of how that grand operation is conducted. When an entire country, or part of a country, containing one or more counties is to be surveyed, it is done by triangulation, and the application of the rule given in the 12th section of this work. A line of some miles in length is measured and re- measured in order to prove its accuracy, on some plane or heath which is nearly level, first havmg been traced by a transit instrument, and poles placed in an exact straight line, to guide the meas- urers, as A B in the annexed figure, which is assu- med as the base of the operations. A number of hills and elevated spots are selected, on which sig- nals can be placed, suitably distant and visible (107) 108 THE SURVEYOR S GUIDE. one from another. Thus, ifACDEBHG F, &c., be several objects, the situations of which are to be laid down on a map, and they are within the lines, ACDEBHGF, accu- rately Calculated. It is supposed that the stations A and B are chosen such as that all the others can be seen from each of them. Then from the ex- tremity A, measure the angles E A B, D A B, CAB, &c., H A B, G A B, F A B, &c. And from the other extremity B, measure the angles, C B A, D B A, E B A, &c., F B A, G B A, H B A, &c. And as the common base, A B, and the seve- ral angles of all the triangles are now known, the sides, A C, A D, A E, &c. may be determined by simple proportion, for as the natural sine of TRIGONOMETRICAL SURVEYING. 109 ACB:AB::sineC AB: C B andso issine ABC to C A, and so through all the triangles, the three sides being thus found in each triangle, the area is easily found, as shown in section 8th of this trea- tise. But to insure accuracy the objects C D E, etc., should be all intersected from some third station, O in the base A B, otherwise the figure may ap- pear in the plotting to be right when it is not so, and there will be no means of knowing whether the angles have been correctly taken without going over the work again. 27th. Here follows an example of a triangle con- A. K. p. taining a mean area of 1135.2.12.79. The sides of which were traced by a transit instrument, and poles placed at the several points marked thus ; this being done, the respective distances of the sides were ascertained by a mean of measures as follows, viz : B A 14643 links, or 9664.38 feet, A C 17814 links, or 11777.24 feet, B C 16588 links, or 10948. 08 feet. The angles were taken by a theodolite as they are marked in the figure. Now to determine the area of the triangle, A BO: 110 THE surveyor's GUIDE. 14643 links, or 9664.38 feet. 1st. From the data, A B, and the three angles of. the known formula A B^Xsine BXsine A, a r. p =1135.2.27.18 2 sine C. 2d, by B C, and the three angles, the area will be 1135.3.029 3d, by C A, and the three angles, the area will be 1135.0.38.6 TRIGONOMETRICAL SURVEYING. Ill dth, by data A B, and the two adjacent angles, we have by the known formula, A B^Xsine BXsine A, 2 sine (B+A) The area will be A. R. V 1135.2.25.7 5th, and by B C, and the two ad- jacent angles 6th, by a similar formula from A 1135.3.01.9 C, and the two adjacent angles, the area will be 1135.0.37.99 7th, by data A B, and the adja- cent angle A, and the remote an- gle C, we have by the known formula, (A B)^X8ine AXsine (C+A) 2 sine C area, 1135.2.27.8 8th, by a similar formula from hav- ing A B, and the angles, B and C ; area 1135.2 28.2 9th, by having C B and the angles, C and A; area 1135.3.03.58 10th, by having C B and the angles Band A; area 1135.3.04.38 112 . THE surveyor's GUIDE. 11th, by a similar formula data C A, and the angles, C and B, gives the area 1135.0.39.66 12th, by a similar formula from da- ta C A, and the angles A and B; area 1135.1.00.12 13th, by data A BXB C, and the contained angle, we have A BXB CXsine B=1135.2.35.06 2 14th, by A CXA B, and the con- tained angle 1135.1.32.92 15th, by A CXB C, and their con- tained angle C 1135.2.00.79 16th, by data, A BXB C, and the angle, A, we have by a known formula, B AXsine A BC. -sine C, and A BXB C, sine (A+C) 2 area 1135.2.394 17th, by the application of similar formula to the data, A BXB C, and angle, C ; area 1135.2.30.4 TRIGONOMETRICAL SURVEYING. 118 18th, by A CXC B,and angle, A, 1136-0.19.51 19th, by A CXB C, and angle, B, the area will be 1135.0.19.89 20th, by A CXB A, and angle, B, the area will be 1135.1.10.5 21st, by A CXB A, and angle, C, the area will be 1135.3.05.16 22d, by the usual rule from the three sides, s. s — a. s — b. s — c. 1135.2.14.7 Now the various data exhibited in this triangle have been ascertained with the same relative de- gree of precision ; and the different areas deduced therefrom have been subjected to the same loga- rithmic process, till the figure has been exhausted ; there is no reason to suppose that any one of them is nearer to the truth than another ; and taking a A. R. p. mean of the results we have 1135.2.12.97 for the nearest approximation to the true area. But suppose we consider the triangle as spheri- cal, and the admeasurement of the sides as the lengths of three arcs of three great circles of the sphere ; and, according to Sir Isaac Newton, the diameter of the earth to be 41,798,177 feet, we will then have, as the circumference of a great 114 THE surveyor's GUIDE. circle of the earth is to 360°, so is the length of C B to the number of degrees or minutes, &c.; contained in the arch, C B, viz : As 131312964.37: 360°: And do. : do. : And do. : do. : 11757.24 : l'.56".03868=arch C A. 9664.38 : l'.35".38309= " AB 10948.08 : l'.48".05263= " B C. Now let b a c, represent the sides of any spheri- oal triangle, and e the spherical excess, we have hy Lhuiller's theorem. Tangent ^ E= Tan. a+b+c. tan. a+b — c. tan. a — b+c. tan. — a+b+c. 4 4 4 4 And by restoring to a b and c, their deter- mined values, we find a+b+c=0° .1' .19". 8686 4 a+b— 0-0° .0' .32" .1771 a— bj-c=0° .0' .21" .8493 4 And, — a+b+c=0° .0' .25" .8423 Whence the log. tangt. of 0° .1' .19" .8686=: 6.5879531 of .0' .32" .1771= 6.1931205 of .0' .21" .8493= 6.0250065 of .0' .25" .8423= 6.0979010 2)24.9039811 Log. of i the spherical excess= 2.4519905 The arc corresponding to this log. will be found TRIGONOMETRICAL SURVEYING. 115 to be ,00584 parts of a second, consequently the spherical excess is 02336 of a second, and by a well known theorem, As 180° : the area of one- quarter the surface of the sphere : : the spherical excess to the area of the spherical triangle, viz : As 180° 31500428420,3 the area of a great cir- cle of the earth in statute acres : : 023360 to A. R. p. 1135.2.11.3 being | perch less than the mean area, which is in defect, but should be in excess ; but this is accounted for by the hills on the land not being taken into account ; the difference, how- ever, is insignificant, and shows that the diflference between a plane and spherical triangle of conside- rable dimensions is very inconsiderable. See Gib- son s Surveying hy Trotter. 28th. How to measure a tract of land by measu- ring a base line through it, and not departing from that line, and yet finding all the distances round the land, their courses, and angles of the field, and the area, never before published. In order to do this expeditiously, the surveyor should be provided with an instrument having two telescopes, one of which is movable, and the other fixed, by which he can at any time take half a right 116 THE surveyor's GUIDE. angipj from the base line, and also a right angle ; he must also have an active assistant with a flag- staff, to hold at the corners as he proceeds with the measurement on the base line. Let ABODE T M G H A be any tract of land that is to be sur- veyed, let the base K L, be traced through it with a transit instrument, and poles set perpen- dicularly, to be visible from one to another. Set your instrument at L, on the base line, which in this survey bears N 40 E. A theodolite and com- pass attached is the best instrument for this method ; adjust your instrument, and let L be the point where 45° inflected from the base L K wiU cut the flag-staff; at the corner H, commence chaining towards K, and five chains you find 45° degrees deflected from the base line to the flag- staff at B, on the left, will bisect it, which note in your field-book by an oblique line to the left, ma- king an angle as near 45° as the eye can judge ; at 9.30 half a right angle to the right will cut a pole at G, and at 12.00 came the fence ; at 13.20 half 0. L. a right angle will cut C, and at 19.15 you find a right angle will intersect H. Now it is evident c. L. that you are 19.15 distant from H, for H 19.15. L TRIGONOMETRICAL SURVEYING. 117 is an isosceles triangle, and .*. you mark 19.15 on the perpendicular. The next perpendicular is at ^J5.oo /-$■• 20, and the half right angle having been taken ar 0. 5 on the chain line, 20 — 5=15=the distance to B. c. L. Again at 32.35 you find half a right angle bisects 118 THE surveyor's GUIDE. ^he pole at D, and at 33.20 a right angle inter- c. L. sects at G, and 33.20— 9.30=80.40=the length of the perpendicular which set on it. At 34.20 you find the next perpendicular on the left to 0, and the one-half right angle having been taken at 13. 20 .-. 34.20—13.20=21.00 the distance to C ; pro- ceeding in this way you have 43.35 — 32.35=11 chains to D, and 51.30— 35,40=15.40=the dis- tance to M, and 57.40—38.30=19.10 to F, and 60— 57.40=2.10=the perpendicular of the last A within the fence on the right and 62.30—60=2.30== the perpendicular without the fence ; also, 62.30 — 41=21. 30=the distance to E, which A is to be ducted out of the area of the last trapezoid on the left. Thus you have found with very little trouble all the requisites for calculating ihe area of the land, and it may be remarked, that you might have commenced at the corner B and noted where the two perpendiculars fell at 19.15 and 20 and as you proceeded on your base line take back sights at the proper distances to intersect the poles at B and H, and the distances from where the perpendicu- h-^rs would fall to these several points would be the chains and links to be placed thereon. The dis- TRIGONOMETRICAL SURVEYING. 119 tances all round the land, can be accurately found, for in the present case -/(A a^)+(a H-)= A II, and r G, and r H being given V(r G'')-f(r H^)=H G and so on all round the land, and seeing that the courses of A a and a H are given, the course of A H may be readily found, for having the distance and difference of latitude and departure, the course is given in the tables ; also, the internal angles can be easily found, for in the ^ A a H A a : Had. : : a H : tang't a A H, and so with the A B b A. Hence, the angle B A H, is known, and it is evident the same holds good all round the land, the bases and perpendiculars of all the right angled triangles being found from the base line and can be marked on the sketch as the sur- veyor proceeds. The same may be done with a good compass, for having the course of the base line, the courses of the normals to right and kft are known, and the course of J a right angle beiug once ascertained on the right and left of the base will always serve to find the points on the base where they are to be taken ; but this would require many trials and waste time, whereas, an instru- ment showing i a right angle will save much time. 120 THE surveyor's GUIDE. Thus, in a plane country, the scientific reader will acknowledge the plan completely available, and the surveyor can calculate the content of the land on the margin of his book while his needle is set- tling, and be able to answer the farmer satisfac- torily, who thinks a surveyor should be able to tell the content the moment he has the last distance measured. The plotting and calculation of a survey taken on the above plan is so obvious as to require no explanation, seeing all the figures are either right angled triangles or trapezoids, to find the area of which is shown in figure 63. 29th. The most correct method of correcting the difference of latitude and departure in surveys taken with the compass, to fit them for calculation, some authors divide the differences proportionally "among all the stations ; but as there may be some stations in a survey really correct, any alteration in them would make them incorrect, so that the altering of the legs of stations in surveys where land is of great value, is a matter of considerable importance. TRIGOJlOxMETRICAL SURVEYING. 121 Fig 59. Problem. Vo find what may be the error in the difference of latitude and departure of a given station arising from the inacu- racy of practice : Let the right angled triangle A B D, fig. 59, represent a station with its diiFerence of latitude and departure; if the angle A be the bearing, then will the leg A D, be the difference of latitude, and the leg B D, the departure ; but if the angle at B be the bearing, then will the leg B D, be the differ- ence of latitude, and D A the departure. Let the small angle B A b represent the error committed in taking the bearing, which may amount TJ min- utes, and the small part B e or E b, the error com- mitted in chaining, in proportion to the whole line A B, or A e, as 0,5 is to 5.00, (for in measuring the 122 THE surveyor's guide, lengtn of lines, there may be an error committed of half a link in 10 chains ; (this is found by expe- rience), and let e a, b d, and E c be drawn parallel to B D, and B n o, and res, parallel to D A. Case 1st. Suppose A B to be the true bearing and length of a station, and A b the one found by observation. Now it is plain that instead of the triangle A B D, we shall have by observation the triangle A b d, so that there is an error of the (quantity n b, by which the leg B D is increased, and an error of the quantity B n, by which the leg A D is decreased, and the contrary may be sup- posed, if A b be the true distance and bearing and A B that found by observation ; but when the angle at A is very small, D d may be supposed equal to (0). Case 2d. Suppose the true length and bearing of a station to be A e, and that found by measur- ment to be A B the bearing exact. Now it is plain that the leg e a is increased by the error r B, and that the leg A a, is increased by the error r e or a D, and the quantities or errors by which each leg is increased are in proportion to the legs TRIGONOMETKICAL SURVEYING. 123 themselves, that is, B r : e a : : r e : A a, and as B e is to A e. Case 3dt. Suppose A e the true bearing and length of a station, and A b the same, found by observation. This supposes a compound error both in chaining and bearing, and that the error in the bearing increases the smallest angle in respect of the bearing and its complement. Here we see that when the leg A a is increased to A D by the error in chaining, as in the last case, it shall, at the same time, be brought back to A d by the error in the bearing, as in case 1st. Therefore, the leg A a will be increased by the quantity r e — B n, or de- creased by the quantity B n — r e ; but r e is greater than B n, when the angle at A is small ; and B n is greater than r e, when the angle is near 45°; for they become equal when the angle, is about 25°; but at the same time the leg e a will be increased to d b, by the error b S=b n+B r. Case 4:th. Suppose A E the true distance and bearing, and A B that found by observation ; this supposes the error in the bearing to decrease the smallest angle. Now it is evident that the longer leg A c is increased by the error B o or D c, and 124 THE SURVEYOR S GUIDE. the shorter leg decreased by the error E o. But B o=B n+r e (for r e=n o) and E o=b n- -B r. These errors are easily found in numbers by con- sidering the figure, and that they are always pro- portional to the length of the stations. Here follows a table of errors in links and deci- mals, calculated for a station of 30 two pole chains, and for the different angles and their com- plements, under which they are placed, but which can be changed to any other length, by altering them in the same proportion as are the stations. B A b=4°error in bearing. 2 88 3.2 0.0 12 78 23 67 3°2 58 42 48 45 B e 1.6 links error in chain- ing. b n= Bn= 3.1 0.7 3.0 1.4 2.8 1.7 2.4 2.2 1.0 1.1 2.3 2.3 1.0 1.0 Error in short Leg. ) Case Error in long Leg. J 1st. Br= r e= 0.0 1.5 0.3 1.5 0.6 1.4 .^8 1.3 Error in short Leg. ) Case Error in long Leg. J 2d. bs=(bn+Br) ad=(B nj"r e) 3.2 1.5 3.4 0.8 3.6 0.0 3.6 0.4 3.4 1.1 3.3 1.3 Error in short Leg. ] Cixse Error in long Leg. J 3d. Eo-^(br.-B r) 3.2 Bo=(Bu-t-re) 1.5 2.8 2.2 2.4 2.8 2.0 3.0 1.4 3.3 1.3 3.3 Error in short Leg. ( Case Error in long Leg. j 4th. CORROLLARY. Hence we may adopt the following rules for altering the legs of stations in the correcting of surveys : trigonometrical surveying. 125 Rule First. When the course, or angle, is either great or small ; or when the difference of latitude and de- parture are found in the beginning of the tables, then the shortest leg may be increased or decreav«ed by any quantity not greater than 3.2 links, and the longest leg increased by any quantity not greater than 1.5 links. Rule Second. When the latitude and departure are found about the middle of the tables, or when the angle is about 20° under or over 45°, then the shortest leg may be increased by any quantity not greater than 3.6, or rather 4 links, and the longest leg left unaltered, which is, when the error in the bearing increases the angle opposite the smallest side ; but when contrary, the longer leg may be increased by any quantity not greater than 3 links, and the shorter leg decreased by 2 links. Rule Third. When the difference of latitude and departure are found in the latter part of the tables, or when the bearing is about 45°, then either of the legs 126 THE surveyor's guide. (they being nearly equal) may be increased or de- creased by any quantity not greater than 3 links, and the other leg by 1.4 links, but when one leg is increased the other must be decreased. These rules are on the supposition that the chaining is always too long, which, in practice, I have nearly always found to be the case ; but when a surveyor has reason to think otherwise, he may alter the rules to his opinion, not only in respect to this, but also relative to the quantity of the errors. A description of an instrument by which any person, though unskilled in surveying, may measure a map, or part of a map, almost at one view : Get a piece of good glass about 8 or 9 inches long, and 6 or 7 inches broad, and divide it into small oblong rectangles of eight-tenths of an inch by 5 five-tenths, as fig. 60th. By laying this in- strument (which I call a computor) on a map you can tell with very few figures, sometimes with the eye only, how many of the rectangles are con- tained in the map, and consequently, how many acres. When the map is laid down by a scale of 20 perches to an inch, then each rectangle will be TRIGONOMETRICAL SURVEYING. 127 16 perches by 10, or one acre; and if the map be 40 perches to an inch, then each rectangle will be 32 perches by 20, or 4 acres ; and if by 80 per- ches to an inch, then each rectan- gle will contain 16 acres. This instrument would be useful to gentlemen and others not very well skilled in surveying, to measure a map, or part of a map that they wished to know the content of nearly. It is easily used. The sides of the glass must be made to coincide with as many of the lines on the map as possible, and the broken squares can be estimated by the eye, or a square inch horn. Fig. 60. Description and design of a new instrument by which dis- tances can be found at once, without any calculation : Let a brass semi-circle (fig. 61) of about 9 inches radius, have its inner edge or limb, divided into 90 equal parts, beginning at N and counting up- wards 10, 20, 30, &c., to 90 at Z, and each of these divisions subdivided into 6 equal parts. Let 128 THE surveyor's GUIDE. the outer limb be divided into degrees and 6th parts of a degree, marking the degrees from the mi Idle of the limb, both ways, 10, 20, 30, &c., to 90 at N and Z. Let also, the middle space be- tween the outer and inner limbs, be marked from Z to N, 10, 20, 30; 40, &c., to 180 at N. Let this semi-circle be fixed to the middle of a box ruler B D, about S^ feet long, an inch and a half broad, and of a convenient thickness. The inner breadth of half this rule must be level with the surface of the semi-circle, but the outer half must be higher about two-tenths of an inch. On the outer half there must be fixed a thin brass scale of an equal length and breadth with the box ruler, the breadth of which scale is to be divided, by lines drawn from end to end, into three equal parts, and the length into inches, half inches, and tenth of an inch ; the inches are to be drawn di- rectly across the whole breadth, and marked 1, 2, 8, 4, &c., both ways to B and D ; the half inches are to be drawn across the middle and innermost third, and the lOths only across the inner third. Let th/jre be on one end of this scale an inch, and in the other end half an inch, each divided very TRIGONOMETRICAL SURVEYING. 129 exactly into 10 equal parts diagonally, that the lOths and centesms which may happen in the operations, on the square and indices hereafter to be described, may be exactly measured on them by a pair of dividers. The reason for raising the outer half of the box ruler above the inner half two-tenths of an inch, is to make room for the in- dices A b and A d, which are to be fixed to the centre of the semi-circle, and there to open and shut as occasion requires, like the legs of a sector. Those indices are about 26 inches long, three- fourths of an inch broad, and about two-tenths thick ; their breadth is to be divided into three equal parts, and their length into inches, half inches, and tenths, as the brass scale before men- tioned. The inches are to be marked from the center A, with 1, 2, 3, 4, &c., to b and d, and the tenths drawn across the inner third. Each of those indices must have a small screw nut with a pin or bit of wire upon it, which pin may, by the screw nut, be fixed exactly to any division on them in order to suspend the label, or ruler T Y, which has a thin piece of brass with a small hole in it, exactly fitting the aforesaid pin, and is to be fixed 9 130 THE surveyor's GUIDE. also to any division of the ruler, as occasion re- quires. Let this label, or ruler, be about two feet long, and of the same breadth and thickness as the indices A b and A d, and divided after the same manner as they are, only the tenths are to be drawn across the inner edge, as well as across the inner third of the breadth, and the inches are to be marked 1, 2, 3, 4, &c., from C to T and Y, making C T eighteen inches, and C Y six. The like divisions are to be made on the side of the square K X, beginning at the inner edge of the brass ruler at K, marking the full inches on the upper side, 1, 2, 3, 4, &c., to 24; the tenths are to be drawn across the upper third and the upper edge. Let this instrument be fixed on a tripod with a ball and socket like those of a common sur- veying instrument, but very strong, in order to have it very firm ; and let there be sights which may, as occasion requires, be fixed on the diame- ter, indices, and ruler T Y, of the the same kind with those of a surveying instrument. N. B, The ball and socket must not be fixed exactly under the center of the semi-circle, but some distance from it, on the cross-bar which goes from TRIGONOMETRICAL SURVEYING. 131 the center to the middle of the limb, as well to sup- port the head of the instrument more easily by be- ing nearer its center of gravity, as to make room for an air level, which must be fixed exactly under the diameter or ruler A B, so that when the semi- circle is turned vertically the diameter may be fixed horizontally. 132 THE surveyor's guide. The use of the Instrument in measuring distances i" Example. Let it be required to find the distance from the house at A to the castle, (fig. 62) or to any part thereof, as the weather-cock on the top of the spire at C. Having set up your instrument at A, turn it about till through the sights on the diameter, you see a mark set up at B, and having fixed the di- ameter in that position, turn the moving index till through the narrow slit of a small sight fixed on the center, you see the hair in the other sight cut the spire at C, then fixing the index in that posi- tion to the limb of the semi-circle, measure with a four pole chain in a straight line from A to B ; and having marked the chains and links of that distance on the diameter and placed the ruler with the sights on it exactly to that distance, by means of the small pin and hole mentioned before, set up your instrument at the end of the distance you measured (which you may make full chains if you please) and turn it about till through the sights on the diameter you see a pole at the first station A, and having fixed it in that position, turn the ruler on the pin which is fixed at the former distance on TRIGONOMETRICAL SURVEYING. 133 the diameter, till through the sights on it you see the vane at C ; then will the part of the index a c, cut by the inner edge of the ruler, give the dis- tance A C from the house to the spire at C, which was to be found ; and if there be occasion, the dis- tance from the mark at B to the spire will be found on the ruler at the intersection of the index ; all of which is plain from the similarity of the trian- gles ABC and a. pin. c, or that formed by the diameter, index, and ruler, from Cor. 1st 4 Euc. Book 6th. Thus the surveyor can find the distance of any or all the particular objects he can see and may wish to set down in his map, and by turning the instrument vertically by means of a notch in the socket, inaccessible heigths can, in like manner, be readily ascertained in the same manner. Example in Measuring Distance. Let it be required to find the distance from the house at A to the castle, (fig. 62) or to any part thereof, as the weather-cock at the top of the spire at C. Having set up your instrument at A, turn it about till through the sights on the diameter, you Bee a pole at B, and having fixed the diameter in 134 THE SURVEYOR S GUIDE. that position, turn the moving index till through the narrow slit of a small sight fixed on the center, you see the hair in the other sight cut the spire at C ; then fix the index in that position to the limb of the semi-circle and measure with your chain of 100 links in a straight line from A to B, which mark on the diameter, and place the ruler, having the sights on it exactly on that distance by means of the small pin and hole before mentioned ; set up the instrument at the end of the measured dis- tance, and turn it about till through the sights on Fig. 62. the diameter you bisect the pole at A, and having fixed it in that position, turn the ruler on the pin which is fixed at the former distance on the diame- ter, till through the sights you see the vane at C ; then will the part of the index, a c, cut by the in- TRIGONOMETRICAL SURVEYING. 135 ner edge of the ruler, give the distance A C from the house to the spire at C. And in like manner by directing the ruler to any other objects from A, and noting the degrees cut by the ruler on the limb, and directing from B to each object, the distance from A will be shown as before explained, and thus the surveyor fur- nished with such an instrument, can from the end of his first station, tell the length of his diagonals to as many corners as he can see from that point. Also, by turning the instrument vertically, heights can be determined in the same manner. I would recommend the surveyor to use a com- pass, having the limb divided into 360°, and the bottom of the box into four 90's ; then in taking the courses, if N. W., the limb and quarter com- pass are the same ; but if in the S. W. quarter, the sum of the degrees on the limb and quarter compass are 180° ; and in S. E. quarter, the dif- ference of the degrees on the limb and quarter compass make 180° ; lastly, if in the N. E. quar- ter, the sum of the quarter compass and limb make 360. A surveyor should prove all his courses by this rule before he quits his instrument. 136 THE surveyor's guide. Problem. Given the bearings of any two stations of a survey, thence to determine the angle made by those stations. Rule — Deduct the preceding bear- ing from the succeeding, according as the remain- der is greater or less than 180°. Add — or+180° (as the case may be) and you have the required angle. N. B. The angle found by the above rule will be internal if the polygon lie towards the right hand in the traverse ; and external, if toward the left. Example First. Required the several angles of the polygon A B C D E F G, the courses of the sides being, viz . Fig. f" E 1 A B 26'9|° or S. E. 891° 2 B C 2511 or S. E. ni 3 C D 252| or S. E. 72| 4 D E 1621 or S. W. 171 5 E F 77|- or N. W. 77| 6 FG 30| or N. W. 30f 7 GA 5f or N. W. 5| TRIGONOMETRICAL SURVEYING. 137 From 251^ From 77| take 269| take 162 J —18 — 84J +180 +180 Sum 162= Ang. ABO. Sum 951= Z. D E F From 252| From 30| take 2511 take 77| +li —47 +180 +180 Sum 1811=/. BCD. Sum 133=/. E F G From 162^ From 5| take 252| take 30| —901 _25 +180 +180 Sum 891=/. CDE. Sum 155=/. F G A From 269J take 5| Bem. 263J —180 Sum 831=/. GAB Now 180" multiplied by the number of sides in 138 THE surveyor's guide. any polygon minus 360°, equals the sum of the in- ternal angles .-. 180X7= 160 and 1260—360=900 So 83|+162+181^+89i+95|+133+155=900°. Proof. Next. Having the bearing of any station and all the* internal angles of any polygon, thence to determine the courses of each of the other stations in the regular order of succession, viz ; the land lying to the right hand as you surround it. Rule : According as the given angle is+or — than 180° ; add the preceding bearing, succeeding angle, and +or — 180° (as the case may be ;) their sum will be the succeeding bearing or course. Note. — It sometimes happens that the result will be more than 360° ; in this case take 360° from it and the remainder will be the course of ihe succeeding station. Example. Take the course of A B 269| or S. 89| E, in the preceding figure, and the angles as there found, TRIGONOMETRICAL SURVEYING. I'jQ 269J 1621- 1G2 95i +180 +180 611^ 4371 Deduct 360 360 Cou. of B C 25 1| or S 711 E. Cou. of E F 77| or N 77| ^^ 251J 77f 18U 133 —180 +180 Cou. of C D 252f or S 72f E. 390| 252f ^ _^-^gQ^ Cou. of F G 30f or N BOf W 30f 5221 155 Deduct 360 +180 Cou. of D E 162i or S 17f W. 365f 360 Cou. of G A 5| or N. 5f W. 51 83| +180 Cou. of A B 269Jor S89|E. being the same as that given ; therefore, a proof of the correctness of the work. And thus the sur- veyor has a sure method of avoiding the inconve- nience of the needle being drawn from its true position by mines or other causes, and also correct the diurnal variation ; for no matter how much 140 THE surveyor's GUIDE. the needle may be attracted at any station, the angle will be correct by taking a back and fore sight at every station, and having the true course of the first station. All the others can be found by the foregoing rules. And to know if any at- traction exists at the first station, take a course in a different direction from your chain line ; go to the object bisected, or to some convenient distance in that direction, and take a back sight ; if that agree with the fore sight, you may safely conclude that no attraction exists at either ; but should it differ, make trial in some other direction, in like manner, till you find what station the attraction ia in ; but by using a good theodolite all such trouble is avoided. In every survey that is truly taken, the sum of the Northings is equal to the sum of the Southings, and the sum of the Eastings to the sum of the VVestings. Let a b c e f g h represent a plot or parcel of land; let a be the first station, b the second, c the third, and so on. Let N S be a meridian line, then will all lines parallel thereto, which pass through the TRIGONOMETRICAL SURVEYING. 141 several stations, be a meridians also, as a o, b s, c d, &c., and the \ lines b 0, c s, d e, i &c., perpendiculars to these, will be east or west lines or de- parture. The northings e i+g o+h g=a o+b s-f c d+f r, the southings ; for let the figure be com' pleted, then it is plain that g o-fh g+r k=a o-f b s+c d and e i — r k=f r ; if to the former part of this first equation e i — r k, be added, and f r to the latter, then g o+h g+e i=a o+b s+c d+f r ; that is, the sum of the northing is equal to the sum of the southings. The eastings c s+q a=o b+d e+i f+r g+o h, the westings for a q+y o (a z)=d e+i f+r g+o h, and h o=c s — y o. If to the former part of this first equation c s — y o, be added, and b o to the latter, then c s+a q— o h+d e+i f+r g+o h ; that is, the sum of the eastings is equal to the sum of the westings. Now, as there is many methods of calculation, and every man chooses one in preference to all 142 THE surveyor's GUIDE. others, I shall here show the method which I have always practiced, being, I think, least liable to mistakes, although not the shortest, as shall bo hereafter shown. ^ i i :::::: 'i'.oo 2.00 2.00 2.00 1.00 o o §§§§ !§§§§ : : i : i § CO.-tOO.-l :r-l(>irH.-, ; ; ; ; ; 3 i If 32 o : : : '^ ': : ? : : :o o o o o CO O CO CO : c<- iiS n 1 i i § o c oggoogoooooo .-; c«5 o ,« rH- .-; o ^: o =o t^ o^ liislPSHis ooooo :ooo :o : o sii ^ ^ ^ liwum c^'cocscoc<;co--HC;r4c^c^irqc<^l :2 1 p^ pq H w ^ W pq p4 a ^' ^' ^' ^' ^ '50^'^^^ur5coo'iooco''Oco^ ;^' :^ «3 oQ M M M iz; aj M ai ;zh' oQ ^ ^'1 -^(MCO'^OOIt-OOOJOi-KNMtXiO o o •=> o o o o o a g S ttiO! In the above method the northings and south ings, eastings and westings, being corrected by the TRIGONOMETRICAL SURVEYING. 143 foregoing rules, set the sum of the northings, or southings at the top of the column titled latitude, then continually add the northings and subtract the southings, or add the southings and subtract the northings, and the last number -will always be the same as the first, which is a proof of so much of the work. Then add the first and last latitudes together, and place their sum opposite to the first station in the column under latitudes, added, and so continue to add every two adjoining latitudes, and place their sum in a line with the latter, then mul- tiply each of these numbers by the particular easting or westing belonging to that station, and place the product in the column of east or west area, as the case may be, and the difference of these two columns divided by two, will be the con- tent of the survey. In this method there is no danger of making mistakes from indirect stations, and by using the eastings, and westings, in the same manner as you did the northings, and the south- ings, you can prove the work, and find the area four different ways. 144 THE SURVEYOR S GUIDE. ANOTHER METHOD, WHEREIN FEWER FIGURES ARE USED, NEVER BEFORE PUBLISHED: The Eastings and Westings, Northings and South- ings, are here corrected according to the foregoing rules, and placed as usual, as follows : CALCULATION OF THE NOTES ON THE SUCCEEDING PAGE. I I Lats. iDoub. Semi L. N.I L.S. 'added.! Rectangle. I B. I 2.60 0..36 3.35 3.34 2.11 1.75 1.77 2.76 4.44 0.77 4.38 0.40 2.53 15.28 15.28 Double the sum of the indirect, 6.04 2.70 0.59 0.99 3.52 1.77 0.00 2.76 9.96 9.53 4.38 2.60 6.56 9.27 8.74 3.29 1.58 4.51 5.29 1.77 +4.4712 -1-11.9620 -f-20.5848 — 1.9272 14.3520 —10.3416 -f-59.1426 -j- 17.4800 —4.4086 -{-5.6564 -h5.4120 +5.0784 +4.4958 1.62 1.20 1.86 Ex.W 2.00 3.58 1.20 0.96 2.54 165.3026 15.40 15.40 33.3548 Ex.K 2.16 5.621 .9739 Angular spacoB. TRIGONOMETRICAL SURVEYINa. 145 11.76 Parallel breadth. 12.35 Meridianal breadth. 5880 3528 2352 1176 145.2360 Content of parallelogram. 65.9739 7,9.2621 4 3.70484 40 28.19360 A.R. P. '7.3.28.19, the same as on next page. 1 l\ i j \ i y 1 / 1 j i 1 if /l3 i j i ..... \ i 1 ^ 10 Fig. 65. 146 THE surveyor's GUIDE. The foregoing plot and calculation may not be unacceptable to the reader, being as complicated a figure as could be easily met with. A new and concise method of Calculation, wherein fewer figures are used than in the common methods: 2.60 0.36 3..35 3.34 2.11 1.75 1.77 2.76 4.44 0.77 4.38 0.40 2.53 1.62 1.20 6.44 L86 '2.06 3.58 1.20 0.96 2.54 3. 4 2, 2, 5.52 7 5. 1.34! 6 M. J). 162 24 E. 44 E. 28 E. 72 E. 80 W. 94 W. 32 W. 32 W. 66 W. 08 W. 88 W. 92 W. 2 E. 86 E. 68 E. 72 E, 00 E. 08 W. 74 W. 26 W. 64 W. 98 W. 74 W. 96 AV. 80 W. 70 E. Area. Deduction. 13.4136 34.0992 5.1744 21.9000 2080 1.3464 27.6710 42 2176 25 2778 3.8960 12.5488 4.9000 15.28 15.28 15.40 15.40 A. R. 7 3 176.2080 17. 17.6838 2,0)15,8.5242 7,9.2621 4 3,7.0484 40 28.19360 This method may be called a compound of Burgh's and Gibson's, without being intimately connected with either. It allows the first meridian to pass at any distance from the first station not less than the first latitude or first departure. TRIGONOMETRICAL SURVEYING. 147 This example supposes the first meridian to pass at the distance of the first Easting from the first station of the survey, and the M. D. column is completed by one single addition of the Eastings, or one single subtraction of the Westings, to or from each preceding one, agreeably to the nature of the signs. The D. D., or double distance col- umn, is completed by adding the first and last, and placing their sum in a line with the first East- ing or Westing, and then adding every two ac- cording to the signs, and placing their sum in a line with the latter, marking E. or W. as the case may be. Then the Eastings X by the Southings, and the Westings X by the Northings, must be put into the area column ; but, the Westings X by the Southings, and the Eastings X by the Northings, must be put into the deduction column, the differ- ence is double the area of the survey. The following is a method of calculation first published by Noble, the inventor, and is a very superior plan when well understood, but requires considerable attention to distinguish the indirect stations, as the areas belonging to them must be deducted. A little practice will enable the learner to know both them and the four extremes, viz : 148 THE surveyor's guide. N. S. E. and W. That author's description of a semi-rectangle is a figure limited by the latitudes of both ends of the station, the station itself, and a section of the parallel from which the latitudes are measured, equal to the departure ; and when the last mentioned is indirect, the semi-rectangle is indirect also, viz : Indirect or retrograde sta- tions are those stations, in respect of the rest, which bear backward or contrary to the natural succession of the four quarters of the compass. If, in proceeding Southerly from the extreme point North, there happen a station to turn North- erly, or, in proceeding Northerly from the extreme point South, there happen a station to turn South- erly, such stations are indirect or retrograde sta- tions. The same may be said of stations that turn after the like manner in proceeding from the extreme points E. and W. of the survey. The extreme points, N. S. E. or W. of a survey, are the ends of those stations which run more to the N. S. E. or W. than any other stations in the survey. Though most surveys have those four extreme points, yet there are some where one and the same station may be the greatest extreme N., and at the same time the greatest extreme East or West ; TRIGONOMETRICAL SURVEYING. 149 or one and the same station may be the extreme South, and likewise the extreme East or West. The circumscribing parallelogram of a survey is a rectangle or parallelogram circumscribing the body of the land, whose four sides, passing through the four extremes N. S. E. and W. of the survey, are two meridians and two parallels of latitude. The angular spaces are the areas contained between the sides of the circumscribing paralello- gram, and the stations of the land surrounded, which, deducted from the area of the second parall- elogram, leaves the content of the survey. Now in order to find the area of those angular spaces, the four extremes must first be ascertained. This an experienced hand can see at once by examining his field-book, which, being known, you must find the latitude of each station in the survej' from the extreme points North and South ; thus, having found and corrected your latitudes and departures, and placed them as in the following table, write in a line with N, and also the South extreme as in the following table. Now, begin- ning at each of these extremes, North and South, continue to add the Northings, and subtract the Southings to find the latitude of each station to 150 THE surveyor's GUIDE. the extreme point West, but you must still add the Southings and subtract the Northings to the extreme point East. When the latitude of every station is thus found, and placed in their proper columns, add every two latitudes next each other, and put their sum in a line with the latter station in the column marked L. A., and each sum or number in this column is the length of a rectangle, which is double the semi-rectangle of each station. It is no matter at which of the two latitudes you begin, so that you place their sum in a line with the latter or succeeding station ; but it is common to begin by adding the first and last stations to- gether, and placing their sum in a line with the first station ; then add the first and second, and place it in a line with the second, and so on till the column is filled. Then each number must be multiplied by its corresponding Easting or Westing, and the products put in the column marked D. S., or double semi-rectangle of each station. If the Easting or Westing be direct then this product must be marked f ; but if it be indirect, with the negative sign — , and the sum of all the affirmatives, abating the sum of all the negatives, will be the content of all the angular TRiaONOMETRICAL SURVEYING. 151 spaces. But, to find the length and breadth of the circumscribing parallelogram, note that from the sum of all the Northings or Southings you must deduct the sum of all the Northings or South- ings that have indirect difference of latitude, which will give one side, and the same must be done with the Eastings and Westings to find the other side. The length and breadth of the parall- elogram being thus found, they must be multiplied together, and from their product take the content of the angular spaces, and the remainder will be the content of the survey. TAKE THE FOLLOWING EXAMPLE IN NUMBERS. 9.23 9.04 4.66 1 51 6.78 17.46 12.97 11.76 6.17 1.61 0.00 6.78 24.24 37.21 6.00 30.28 39.51 48.55 5.75 30.28 69.79 88.06 7.68 1.51 6.78 31.02 61.45 86.18 E. W. +076.5900 13.32 +468.7344 15.48 +687.4315 9.85 +626.1066 7.11 +088.9344 n.58 +020.2340 13.40 +049.8330 7.35 +268.3230 8.65 — 233.5100 3.80 +739.4244 8.58 54.72 54.72 2)2792.1013 49.56 49.; 1396.0506=Ang. spaces. In this example there are no indirect stations in the Northings or Southings, 54.72 is the me- ridianal breadth of the survey. But station 9th being indirect in the parallel breadth, must be 152 THE surveyor's guide. deducted from the sum of the Easting or Westing to find the other side of the circumscribing par- allelogram. Thus : 49.56 Sum E. or W. 3.80 Indirect. 45.76=Parallel breadth. 54.72=Meridianal breadth. 91.52 32032 18304 22880 2503.9872 Content of circum. parallelogram. 1396.0506 " of the angular spaces. 1107.9366 4 3.17464 40 6.98560 lib. 3.06.98, the content. In this example you may see that the four ex- tremes are the 6th, 1st, 10th, and 4th stations You can also see that the two latitudes of the extreme West is equal to the two latitudes of the extreme East, that is 6.17+48.55=48.97+5.75, which is a proof to so much of the work. TRIGONOMETRICAL SURVEYING. 153 If you begin with the Eastings and Westings, and proceed as you were directed, all along with the Northings and Southings, you can find the content of the survey in like manner, and so prove the work. To survey with the compass through any mine, or other cause for drawing the compass needle off its parallelism : The diurnal variation of the needle is known to every practical surveyor, but is easily cor- rected by examining the time of the day when the courses of long stations were taken ; as from about 8 o'clock in the morning till about 2 in the after- noon, the needle varies Westerly to from about 7'08" to about 13'21", as shown in the following table. The surveyor can make such allowance as will (all other errors apart) insure a complete clase. MEAN DIURNAL VARIATION FOR EVERY MONTH IN THE YEAR. January, 0" 7'08" July, 0'a3'14" February, 8'58" August, 12'19" March, IVll" September, 11'43" April, 12'26" October, 10'36" May, 13'00" November, 8'09" June, 13'21" December, 6'58" 154 ' THE surveyor's guide. Now, in surveying with the compass detached from a Theodolite, both back and fore sights should always be taken ; and to make sure that no attraction exists in the first station, take a course in a contrary direction to some object, go to that object and take a back sight ; if the fore and back sight agree you may be satisfied that no attraction is at your first station ; but should they not agree, you must then, from the latter station, repeat the like process till you find at which of them the attraction exists; if, at 'the first station, either note its quantity, which allow on the next course, as in tracing old boundaries ; or pay no attention to it at the starting, but continue to take the fore and back sights throughout, and as at any station the needle will be as much attracted at the fore as the back sight, the angles can all be truly found as formerly shown, and thence the true courses for calculation by latitude and departure. Thus may the expert surveyor traverse any city, mountain, or other place containing mines or other substances which attract the needle, about which I have heard many complaints. TRIGONOMETRICAL SURVEYING. 155 Now to plot the last given notes, and in like manner any other survey similarly prepared : — Having the length and breadth of the circumscri- bing parallelogram, let it be drawn by the same nc-le you intend to lay down your map by, and beginning at either of the extremes, as 1, lay off your latitude as l.a 5.575, and at right angles to that, the departure of that station or Westing a.2 13.32, and join their extremities with the line 1 2, 156 THE surveyor's guide. which is the distance. The next' station is Ic-:, W, Draw toward the North 2.b parallel to the sides of your parallelogram, and on it lay 30.28, your next Northing, and at right angles thereto toward the West 15.48, your next Westing, and join 2 and 3, which is your next distance, and so on all round, and as your Northings are equal to your Southings, and your Eastings to your Westings, your last departure, whether East or West, will fall into the point of beginning, as T.l. This is the most expe- ditious mode of plotting surveys, and can be made use of in the most extensive work, and is much superior to protraction by parallels and a metallic protractor. The mechanical methods of finding area, shown by many authors, I do not think well to notice, as none of them can be depended on for accuracy. OF LOTTING OR LAYING OUT TOWNS, &C. llegarding this kind of surveying, little can be said more than giving some general directions con- cerning the method of operation, as every man has mostly predetermined the manner in which he Intends to have his property cut up into lots. Provide yourself with a 20 or 25 feet pole, ten TRIGONOMETRICAL SURVEYING. 157 skivers with sharp points and thin edges, two brass plummets with steel points hung to fine cords ; then having fixed poles so as to direct you in a straight line, and set them perpendicular by the help of your plumb, direct your assistant to hold one end of your pole in the straight course, with his plummet hanging over the extremity, whilst you hold yours touching the end of the pole which you hold, and the point of your plummet exactly over the starting point ; when both plummets are steady, order your assistant to stick, and exactly where the point sticks, he sticks one of his skivers edgewise and slanting, so as that you can, when you arrive at it, hang the point of your plummet exactly over the edge of the skiver, and your assistant again sticks his plummet in the ground, and a skiver as before, and so on to the end. By measuring carefully in this manner, property can be laid out with great accuracy. Almost every man has his own method of keep- ing his field-book, but the following method, which I have always adopted, is, I think, best calculated to prevent confusion in extensive surveys, for as writing backward and chaining forward are con- trary, it is more congenial, and natural, to both 158 THE surveyor's guide. write and chain forward, by beginning at the bot- tom of the page. N. B. It may not be unacceptable to the reader CO see these notes, calculated by Noble's meinod, as on page 119. TRIGONOMETRICAL SURVEYING. 169 Maple 60.00 35.10 N.77fW. Pin Oak 0.5 i 1 W. 0. (5) i Stump. „^ 1 41.40 162} "a S. 17f W. 1 1^ (4) To a stone. 1 To the place ef 36.40 18.48 "" 0.6 33.00 beginning. @ 1 m 21.00 7.10 R. 0. m S. 0. 3.24 0.15 to a Pine N. 5f E. s 15.00 0.10 (7) 7.00 2521 S. 72|E. toChesnut m J^ 1 B.Oak (3) To a post. t 36.25 m 23.00 To a post. Chesnut. 41.00 m 26.00 4.00 Dogwood "2 25 If m S 10.00 0.10 to a Beech 2 w o S. 71fE. (2) 7.00 N.30-IW. (6) m 20.00 269J chains to a hickory 9 Shellbark Hickory. S. 89*E. (1) tree. 66.00 Stream North 62.00 36 West. Begins at a White Oak on Squire Hays' Estate. 160 THE SURVEYOR S GUIDE. The foregoing method of keeping a fiehi book, I think, is the most convenient I have seen. The following is the calculation of the notes corrected by the foregoing rules. 7^E. .Ti%W. F. Poles, 10.00 20.50 18.40 20.90 33.00 18.25 9.48 5.50 19.90 31.95 25.40 19.90 0.00 64.00 57.35 45.30 19.90 6.94 29.56 64.67 640.0000 1115.4575 795.9210 126.9620 223.8150 275.7948 2602.7683 626.5718 47.96 47.96 626.5718 2.0)1976.: in square four polo chains. A. R. P. M.3.0967 Lat. Lat. Lats. Double Semi- N. S. South. North. Added. rectangles. E. W. 0.10 0.10 0.10 —1.0000 10.00 6.55 6.65 6.75 +131.2875 19.45 5.50 19.90 12.15 18.80 +330.3160 17.57 Ex.E. Ex. S. 19.90 00.00 19.90 —128.9620 6.38 6.94 6.94 6.94 +223.8150 32.25 15 68 22.62 9.43 29.56 +275.7&48 Ex. W. 9.33 9.43 North. 00.00 9.43 +8.8&42 0.94 32.05 32.05 Content of the angular spaces, TRIGONOMETRICAL SURVEYING. 161 47.96 Parallel breadth. 32.05 23980 9592 14388 1537.1180 Content of circum. parallelogram. 549.0197 " of the angular spaces. 988.0983 4 3.23932 40 A. R. P. 98.3.95.73 the same as before. 9,57280 ^ There are no indirect stations in the above, but were the longitudes made use of instead of the latitudes, the last station would be indirect; and here also it may be seen that the sum of the oppo- site latitudes, against the extremes East and West, are equal, viz : 12.16+19.90=32.05 and 22.62+ 9.43=32.05. Of the Tracing of Old Hearings. Gummer, in his work on Surveying, gives the general number 57.3°, for doing this which many 11 162 THE surveyor's guide. work with, although it is not correct, but comes out pretty near the truth when the chain line is not very long. To find this number, say 6.2831853 (the circum- ference of a circle whose diameter is 2) : 360° :: 1 : 57.3° nearly. Now if two corners are known, and can be both seen, set your compass at one of them, and direct your sights to the other ; the dif- ference between that shown by your needle, and that shown in the deed, will be the variation to be allowed on each course round the land, supposing all those given in the deed to have been correctly taken at the time the survey was made, which fre- quently happens not to be the case. If the two corners cannot be seen from each other, run the course and distance given in the deed, and observe if the point you arrive at, joined to the corner, form an isosceles triangle, which will be the case if all be right ; otherwise some mistake has been made in the distances, which must be corrected. Then take the pendicular distance to the given corner, and say : As the measured distance is to the distance to the corner, so are 57.3° to the number of degrees, minutes, or seconds, as the case may be, wtiich will be the variation. Or, TRIGONOMETRICAL SURVEYING. 163 more accm-ately. As the distance to where the perpendicular was taken is to radius, so is the dis- tance to the corner to the tangent of the variation. In running your trial line, you will be told you are wrong, and that you don't understand your busi- ness, and all such stuff, will be sounded in your ears ; but pay no attention to such nonsense, for it is to be regretted that too many men are so igno- rant as to think that a Surveyor can, by some mysterious means, direct his compass on the exact line, and find all the courses as if by magic. It often happens that the corners runs through clumps of trees or other obstructions through which you cannot chain. In such a case I have often chosen an opening some degrees to right or left of the fence, and at certain distances driven posts till I found a perpendicular to the corner. Then, as the whole distance is to the perpendicular, so is each distance from the beginning to the perpen- dicular distance from the measured line to the fence, which, being correctly laid off, and posts driven at their extremities, will point out the true boundary. Of Levelling. The art of levelling consists in finding or tra- 164 THE surveyor's guide. cing a line on a given portion of the earth's sur- face, parallel to the horizon at all points. The subject is too extensive to be comprised in this small treatise. I shall give an example, which it is hoped will enable the reader to do anything of that nature that may come in his way. Any one desirous of being fully informed on that subject, should consult BrufF's Engineering, where every information on that subject can be obtained. Re- garding the adjustment of the level, which is a simple matter, let the practitioner always place his level in the middle, between the back and fore- sights, and keep the bubble in the middle of the divisions, and all will be right. TRIGONOMETRICAL SURVEYING. 165 riELD BOOK DISTANCES, MEASURED WITH A HUNDRED FEET CHAIN. Eleva- tion. Back sight. Fore sight. "rr Total elevat'n Datum 100 feet. Dis- tance. Eemarks. 1.11 6.84 5.73 8.10 8.15 5.80 5.00 5.01 4.05 4.98 6.12 2.25 7.77 3.95 6.30 1.60 4.24 6.74 2.17 4.60 5.36 5.73 8.10 8.15 5.80 5.00 4.55 4.05 4.98 6.12 6.67 7.77 13.52 6.30 10.80 4.24 6.74 10.20 4.60 5.36 5.99 100. 101.11 98.74 98.69 101.04 101.84 102.29 103.26 102.32 101.18 100.63 96.11 89.36 87.01 82.61 79.87 77.37 7.3.91 71.48 70.72 70.09 .00 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 "1 On cross road 50 J ft. W. N. 54 W. ) Side of ravine, 30 J feet deep. 10. bottom 10 ftwd. ] Top of bank on J other side. > Middle of stream, i N. 57 W. im at top, from which luced level is deducted, a third proof of the cy of the work. 2.37 0.05 2.35 0.80 0.45 0.96 0.93 1.14 .56 5.62 6.76 2.36 4.50 2.64 2.50 3.46 2.43 0.76 0.63 5.67 104.76 134.67 104.76 29.91 35.58 5.67 100.... 70.09 1 Dat. ! the ret [giving 29.91 29.91 J accura The fall in the following section from 1 to 21 is 29.91 feet; this divided into 2100 feet, the whole distance gives 1 in 70.21, the regular grade ; and to find the grade in degrees, it will be as 2100 is to radius :: 29.91 to the tangent of the angle in this case 0° 49' nearly. Here it will be observed 166 THE SURVEYOR'S GUIDE. that the difference between the datum line and any grade, is the height above or below the base line, running through the first station. If the ordinate be greater, the difference is above base ; if less, below. Some old fashioned levellers follow a more intricate plan. Thus 101.11—98.74=2.37—1.11 =1.26 above; again, 98.74—98.69=0.05, and 1.26+0.05=131 below, and so on. But this re- quires too much thought, when to add and when to subtract ; whereas the other method is done by one subtraction. Fig. 67. inn woXlOD ipa wo ioo\ioo WO lOO lon lUO mo looiioo lOoXioo ioo\ioo loo\ioo Scale of length, 800 feet to an inch— of height, 100 To LAY OUT A Road on a Regular Grade up A Hill. Set your instrument at the starting point, level it, and set the vane on your levelling rod to the exact height of the centre of your glass. Ele- TRIGONOMETRICAL SURVEYING. 167 Vate your grading instrument to the number of degrees you intend your road to be. Send forward your rod to any place where the cross wire will cut the middle of the vane, and there drive a post, and on it mark grade, and so on to the end of the road. And to find the cuttings and fillings, the following flan is the most convenient. Set your instrument on the starting point, measure very exactly the height of the centre of the glass, and send your rod to the first point where cutting or filling is required. Elevate your instrument to the grade, mark where it cuts the rod, and the dif- ference of the height of the instrument and height on the rod, will be the cutting or filling. If the height of the instrument exceeds that on the rod, the difference is cutting, and per contra. Example : Fig. 68. In the above cut the height of instrument is 4 feet ; height of rod, 2 ; difference, 2 cutting. 168 THE surveyor's guide. Again, height of instrument, 4, and back sight to rod 5 ; difference, 1 to be added to last, gives 3 of cutting at instrument. Fore sight, 3 ; differ- ence to be added to last, gives 4 feet cutting at the rod ; but now, height of instrument, 5; back sight, 2 ; difference, 3 ; which, deducted from 4, leavea 1, and so through the whole. To Inflect in Curves on Rail Roads and OtherSo The curves most in use at the present time, are those of a circle. The angle made at the angular point of the tangents is always given — the length of your tangent is also given. To find the radius, multiply the natural tangent of half the contained angle by the length of the tangent of your curve, and the product will be the radius of the curve. To find the degree of curvature, divide half the chord to be inflected by the radius of the curve, and it gives the natural sine of the degrees of cur- vature. Thus, in the annexed figure, where the radius is 140, and the cord to be inflected 100. 140)50, 000000 (.357142 is the natural sine of 20° 55' = the decrees of curvature. TRIGONOMETRICAL SURVEYING. 169 Demonstration. The angle A D C, is a right L=to A D B, and ^ C B is common to the two triangles, A B C and A D B. Hence LAG D=B A Di the L angle of deflection. Now set your instrument at A, direct your index to E, turn it towards the curve till 20° 55' are told on the limb, holding the end of your chain at A ; let the assistant hold the chain tight, and move round till the other end comes in the line of the perpendicular wire of tho 170 THE SUEVEYOR'S GUIDE. telescope at G, and then fix a pin. Again, if nothing intervene to prevent your seeing, inflect from A E, double the said /., and fixing one end of the chain at G, let the other be stretched to come in contact with the telescope at H, and so on through the whole. If II cannot be seen from A, move the instrument to G, and take a back sight to A, and inflect double the L of the degrees of 3urvature from G K, which will fall into H. I have met with some calling themselves engi- neers, who adopt the following plan. They divide 57.,3°X60=3438' by the radius of the circle, mul- tiplying the quotient by the number of feet in the chord, and divide by 60 for double the angle ; but this is erroneous. I remember having met Avith a person who declared that the angle found by this rule was the true angle of deflection. I gave him the tangent 100, and the radius 100 feet, and he did it by this rule, viz: 100)3438(34.38X100= 3438-1-60=57° 18'. In this instance the L made by the tangent and chord is only 45°, so that in- stead of inflecting in 100 feet, this 57° 18' would fall below the chord. Nor is the half of it cor- rect, viz: 28° 39'. For by the true method 100)50(=,5, the natural sine of 30° the true angle. TRIGONOMETRICAL SURVEYING. 171 It remains to find the length of the curve A G H F. The circumference of a circle whose di- ameter is 2, is 6.2832 nearly. Hence as 360° : 6.?832 :: 1 : .01745, &c. This number, multi- plijd by the degrees in the arc, and by the radius oi the curve, gives the length of the arc, thus : 01745 11 17450 268.£ the length of the arc, and 1745 so of any other. 1,91950 140 7678000 191950 268,73000 The two following problems may be amusing to some readers, viz : A gentleman has a lot 40 perches long and 30 perches wide. He thinks the ends may be so ap- plied, as that when their extremities are joined, the area may be the greatest possible. The perpcn- 172 THE surveyor's guide. dicular breadth, and the length of the unknowr side are required. Answer: Breadth, 26.815 nearly. Length of the unknown side, 66.904 nearly. Problem Second. A plank road is to be made from the city A to the town B, 20 miles asunder. A straight road is so situated that a perpendicular from A to it is 10 miles, and from B 6 miles. The plank road must touch the straight road in such a point as to be the shortest possible by that route, the length of the plank road, the point of contact, and radius of the curve havmg 200 feet tangent, are required. Answer: Length of the pJank road, 25.298 The distance of the point of contact from A, 12.2474 And from B, 7.3484 Radius of the curve havii^g 200 feet tangent, 245 feet nearly. This note to be placed after the calculation of the large triangle. It is there shown that the area of any plane triangle, the three sides of Tviiieh are given, IS A B^. Sine B. Sine A C 2 Sine C. Which it thus proved BAA TRIGONOMETRICAL SURVEYING. 173 It has been already shown that B C. B A. Sine B. equal area of the triangle, .'. B C. A C. Sine C. 2 Hence B C. B A. Sine B=B C. A C. Sine C. Multiply each side by B A, and we have B C. B A^ Sine B=B C. A C. B A. Sine 0. Divide this equation by B C, we have B A'^. Sine B=A C, B A. Sine C. Multiply each side by Sine A .*. B A^ Sine B. Sine A=A C. B A. Sine A. Sine C. Divide this by Sine C, and B A^ Sine B. Sine A. =A C. B A. Sine A.=twice SineC the area, and therefore A B^ Sine B. Sine A =Area 2 Sine C To find the perpendicular ordinates from the chord 6 of any arc of a railroad, in order to set off the curve correctly and speedily, without the help of an instrument, suppose it to be a 20° curve, the tangent 200. Find the radius, as formerly taught ; multiply the radius by the natural co. sine of half the vertical angle, and you have J the chord. 174 THE surveyor's GUIDE. Multiply the radius by the natural sine of the same angle, and you have the distance from the centre to the middle of the chord, a constant num- ber to be deducted. Now take any distance, sup- nose 10 feet, at which you choose to erect your ^ /I 'v '(In? Fig. 70. ordinates, and from the semi-chord subtract thia number, square the remainder, and subtract it from the square of the radius ; extract the square root, from which take the aforesaid constant num- ber, and the remainder is the ordinate to be rightly applied, and so proceed till you arrive at TRIGONOMETRICAL SURVEYING. 175 the middle of the chord ; then the dijBFerence be- tween the said constant number and the radius, is the versed sine or greatest ordinate, and now you are prepared to lay off the other side of your curve, and all this can be done in a few minutes in the field. Example. See last figure. Nat. tangent of 80°=5.67128 200 1134,25600=Radius, 1134. Nat. CO. sine of 80° =,17365 5671280 6805536 3402768 7939792 1134256 196,96355440=Semi -chord, 197. Nat. sine of 80° 1134 ,9848 9072 4536 9072 10206 lllfi 7fi5^9 Distance from centre to saM XL±V,^VO^ chord=m7. 176 THE surveyor's guide. From 1134 Take 1117 17= Versed sine FE— From 197 Take 10 (187)==34969 From (1134)^=1285956 Take ( 187)^= 34969 1250987=1118 From which take 1117 1 ft. the 1st ordinate. Again, for the 2d ordinate, 197— 20=(177)^ =31329 and 1285956 _ 31329 (1120 From 1120 ^/ 1254627 Take 1117 1 — 3=the next. 21)25 21 222)446 444 2240)227 Ordinate. All this is plain from the figure, and when the radius and constant subtrahend aie TRIGONOMETRICAL SURVE^'ING. 177 found (which is only the work of a minute) all tlic others are nearly had at sight. This I consider quite superior to any other method now^ in practice Otherwise thus : Let the radius, versed sine, chord, and constant quantity D E, be found as before, divide the semi-chord into any numbei of parts as e f g h i. From E C deduct one of the pnrts i C, leaves E i=m F, then D F=(radius^, squared — (m F)^=(m D)^ the square root of which, minus the constant quantity, E D, gives the ordi- nate i F, and in like manner all the others are found, and thus the curve can be laid off in a few minutes in the most accurate manner, (by the 47th of the first of Euclid.) 13 178 THE SURVEYOR S GUIDE. i'ROBLEM. Let A B C be a right angled triangle, the hy- pothenuse of which is 35, and the difference be- tween the area of the enscribed square (one of c whose angles coin- cides with the right angle of the trian- gle) and the area of the A is 150. Required the sides of the triangle. Fig. 72. Solution. Put A E=x and D F=y. Then per 4th Euc. bth, y : J :: X : f=CF,f+x j=Double the area of the As A D E and DEC .-.=300 or ^f -f 2 X y=600. Also f + Y=B C and x+y = A B. Now (ZVy)'+(x + y)'=35^ viz: l^+'^' + y* +y+2 X y=1225 Deduct '^' +2 X y=600 Root and Z^_l_2 y''+x^=625 Ex't the square. -^'+x=25 or ■x==25 X :A D==5 V X. TRIGONOxMETRICAL SURVEYINQ. 179 Hence 5 V x : x :: 35 : x+y, and by divi- ding the first and third by 5. x| : x :: y x+i and :: are their squares, x : x^ :: 49 . x^ fy^-" 2 X y. Multiply the extremes and means, and 49 x2=x"+x y-+2 x^ y-^by x and From 49 x =x^+y=2 x y ; but 25 x=y-+x^ .*. Take 25 x =x^+y^ 24 X ^ 2 X y or 24 = 2 y and y =12 the side of the square. .And the sides of the angle are 21 and 28. 28 21 28 56 2)588 294=area of A. Deduct, 144 150-.-area of the 2 A's. 180 THE surveyor's GUIDE. ;' TABLES OF LATITUDE AND DEPARTURE. ]' N. S. E. W. |N. S. |E. W. 1 N. S. E.W. N. S. E.W. 1 Oi 191 1 Oi 0.9999 89i Oi 89i 0.9999]0.0131 0.9999 0.0043 0.0087 f, 1.9999 0.0087' 1.9999 0.0174 1.9998 0.0262 3 2.9999 0.01311 2.9998 0.0261 2.9997 0.0392 4 3.9999 0.0174! 3.9998 0.0349 3.9996 0.0623 5 4.9999 0.0218 4.9998 0.0436 4.9995 0.06.64 fi 5.9999 0.0262 5.9997 0.0623J 5.9994 0.0785 7 6.9999 0.0305 6.9997 O.O6I1I 6.9993J0.0916 1 fi 7.9999 0.0349 7.9997 0.0698' 7.9992 0.1047 | 9 89° 8.9999 0.0393 8.9996 0.0785, 8.9991 li 0.1178 1° li I'i 88i 1 1 0.9998 0.0174 0.9997 0.0218 0.9996 0.0262 0.9995 0.0305 f, 1.9997 0.0349 jl.9995 0.0436 1.9993 0.0523 1.9990 0.0610 8 2.9995 0.0523 12.9993 0.0654 2.9989 0.0785 2.9986 0.0916 \ ■^ 9994 0.0698 13.9990 0.0872 3.9986 0.1047 3.9981 0.1221 ti 4.9992 0.0872 14.9988 0.1090 4.9982 0.1309 4.9976 0.1527 fi 5.9991 0.1047 15.9986 0.1309 6.9979 0.1570 6.9972 0.1832 7 6.9989 0.1221 16.9983 0.1527 6.9976 0.18321 6.9967 0.2137 8 7.9988 0.1396 17.9981 0.1745 7.9972 0.2094 7.9962 0.2443 9 1 8.9886 2° 0.1570 8.9978 0.9992 0.1963 0.0392 8.9969 0.2356 8.9968 0.2748 87i 88° 0.0349 2i 87i 0.9994 0.9990 0.0436 0.9988,0.0479 i 9 1.9987 0.0698 1.9984 0.0785 1.9981 0.0872 1.997710.0969 | S 2.9981 0.1047 2.9977 0.1178 2.9971 0.1308 2.9965 0.1439 4 3.9975 0.1396 3.9969 0.1570 3.9962 0.1745 3.9964 0.1919 4.9969 0.1745 '4.9961 0.1963 4.9962 0.2181 4.9942 0.2399 6 5.9963 0.2094 5.9954 0.2365 15.9943 0.2617 5.9931 0.2878 7 6.9957 0.2443 6.9946 0.2748 6.9933 0.3053 6.9919 0.3358 8 7.9951 0.2792 7.9938 0.3141 7.9924 0.3489 7.9908 0.3838 1 9^8.9945 0.3141 87° 0.0523 8.9930 3i° 0.9984 0.3533 861 0.0567 8.9914 0.3926 8.9896 0.9978 0.4318 86i 0.0654 1 3° 9986 86i 0.0610 0.9981 2 1 9973 0.1047 11.9968 0.11.34 11.9963 0.1221 1.9957 0.1308 3 2 9959 0.1570 12.9952 0.1701 2.9944 0.1831 2.9936;0.1962 1 /I 3.9945 0.2093 3.9935 0.2268 3.9926 0.2242 3.9914 0.2616 1 4.9931 0.2617 4.9919 0.2835 4.9907 0.3052 4.9893 0.3270 A 5.9918 0.3140 5.9903|0.3402 5.9888 0.3663 5.9871 0.3924 6.9904 0.3664 6.988810.3968 6.986910.4273 6.9860 0.4578 s 7.9890 0.41871 7.9871 0.4535 7.9851 0.4884 17.9829 0.5232 9 8.9877 E.W. 0.4710':8.9855|0.5102 9.9832 1 0.5494 '8.9807 e7w7 0.5886 N. S. lE.AY. IN. S. E.W. N. S. N. S. TABLES OF SURVEYS. 181 ■l -^ , 1 TABLES OF LATITUDE AND DEPARTURE. N. S. E.W. N. S. E.W. N. S. E.W. iN. S. |E. W. j 1 4° ^86° 0.0698 4i 0.9972 85i 0.0741 44 0.9969 85i 0.0784 ! 41" 0.9965 85i 0.0828 0.9976 2 1.9951 ).1395 1.9945 0.1482 1.9938 0.1569 ;1.9931 0.1656 3 2.9927 0.2093 2.9917! 0.2223 2.9907 0.2364 2.9897 0.2484 4 3.99021 0.2790 3.989010.2964 ,3.9977 0.313813.986310.3312 | 5 4.9878 0.3488 4.9S62 0.3706 4.9846 0..3923; 4.9828 0.4140 1 fi' 5.9854 0.4185 '5.983510.4446' I5.9S15 0.4707 5.9794 0.4968 7|6.9829 0.4883 ;6.9S07';0.5187 6.9784 0.5492; 6.9759 0.5796 8 7.9805 0.5580 i7.9780 0.5928 7.9753 0.6277; 17.9725 0.6626 9 1 8.9780 0.6278 8.9752 0.6670 8.9722 0.7061 8.9691 0.7453 0.1002 5° 85° ! 5i 0.0871 0.9958 84i 1 5i 0.9954 84i 6i 0.9961 0.0916 0.0958 0.9949 1.9923 0.1743! 1.9916 0.1830 1.9908 0.1917 1.9899 0.2004 3 2.9884 0.2615 ! 2.9874 0.2745' 2.9862 0.2875 2.9849 0.3006 4 3.9846 0.3486 3.9832 0.3660 3.9816 0.38.34 3.9799 0.4008 5 4.980S 0.4358 ,4.979010.4575 14.9770 0.4792 4.9748 0.5009 6 5.9769 0.5229 5.9748 0.5490 5.9724 0.5761 5.9698 0.6011 7 6.9731 0.6101 16.9706 0.6405 6.9678 0.6709 6.9648 0.7013 8 7.9692 0.6972 17.9664 0.7320 ,7.9632 0.7668 . 7.9597 0.8016 9 8.9654 0.7844 8.9622 0.8235 ;8.9586 0.8626 8.3J 8.9547 0.9017 6° 84 6i 83i 6i 1 61 83i 1 0.9945 0.1045 0.9940 0.1088 '0.9935 0.1132 ,0.9930 0.1175 2 1.9890 0.2090, 1.988l|0.2177|lL9871 0.2264 1 1.9861 0.2351 3 2.9836 0.3136 ,2.9821 0.3266 |2.9807 0.3396 2.9792 0.3526 4 3.9781 G4181 '3.9762 0.4355 13.9743 0.4528 3.9723 0.4701 5 4.9726 0.5226 i4.9703 0.5443 4.967810.6660 |4.9653 0.5877 fi 5.9671 0.6272 j 5.9643 0.6532 5.9614 0.6792 5.9584 0.7062 716.9616 0.7317 6.9584 0.7621 6.9560 0.7924 6.9515 0.8228 ! 817.9562 0.8362 17.9524 0.8709 7.9486 0.9066 7.9445 0.9403 9 8.9507 7° 0.9408 83° 18.9465 1 7i 0.9798 8.9421 1.0188 8.9376 7i 1.0678 82i 82i 1 7i 82i 1 0.9925 0.1218 0.9920 0.12621 0.9914 0.1306 0.9908 0.1348 ? 1.9851 0.2437 1.9840 0.2524, 1.9829 0.2610 1.9817 0.2697 3 2.9776 0.3656 2.9760 0.3786 2.9743 0.3916 2.9726 0.4045 4 3.9702 0.4874 ,.3.9680 0.5048, 3.9657 0.5221 3.9635 0.5394 ^ 4.9627 0.6093 4.9600 0.6310 4.9572 0.6626 4.9543 0.6742 fi 5.9553 0.7312 5.9520 0.7572 5.9487 0.7831i 15.9462 0.8091 7 6.9478 0.8531 6.9440 0.S834 6.9401 0.9137] 6.9361 0.9439 8 17.9404 0.9750 7.9360 1.0096 7.9316 1.0442 7.9269 1.0788 918.9329 1.0968 S.92S0 1.13.58 8.9i30 1.1747 8.9178 1.2136 L- 1 E. W. N. S. E.W. N. S. 1 E.W. 'n. S. I E.W. NTs. J 182 THE surveyor's GUIDE. TABLES OF LATITUDE AND DEPARTURE. N. S. E.W. 1 N. S. E. W.j N. S. E. W. iN. S. E. W. 1 8° (L99()2 82 1 8i 0.9896 811 0.1435 8i 0.9890 8H 0.1478 0.9883 81^ 0.1391 0.1521 2 1.9805 0.2783 1.9793 0.2870 1.9780 0.2956; 1.9767 0.3042 3 2.9708 0.4175 2.9689 0.4305 2.967010.4434112.9661 0.4664 4l3.961ll0.5567 3.9586 0.5740' 3.9560,0.6912 3.9634 0.6085 5I4.9513 0,6959 4.9483]0.7176 4.945110.7390 4.9418 0.7606 6 5.9416 0.8350 5.9379I0.86O5 5.934110.8868 6.9302 0.9127 7 6.9319 0.9742 6.9276 1.0045 6.9231! 1.0347 6.9185 1.0649 8 7.9221 1.1134 7.9172 1.1479 7.9121]1.1826|.7.9069 1.2170 9 1 8.9124 1.2626 8.9069 1.2914 8.9011 1.3303 8.8952 9S 1.3691 9° 81 07i664 n 80i 0.1607 n Tor 0.9877 0.9870 0.9863'0.1650i 0.9865 1 0.1 693 2 1.9754 0.3129 1.9740 0.3216 1.9726J0.3301, 11.9711,0.3387 3 2.9631 0.4693 2.9610 0.4822 2.9689!0.4951 |2.9566'0.6080 4 3.9508 0.6267 3.9480 0.6430 3.9451 0.6602 3.9422 0.6774 5 4.9384 0.7822 4.9360 0.S037 4.9314 0.8252 4.9278 0.8467 6 5.9261 0.9386 6.9220 0.9644 5.9177 0.9903 5.9133 1.0161 6.9138 1.0950 6.9090 1.1252 6.9040'1.1553l 6.8989 1.1864 8 7.9015 1.2515 7.8960 11.2869 7.8903 1.3204 7.8844 1.3648 9 8.8892 1.4079 80° 8.8830 1.4467 79i 8.8766 ,"ior 1.4854 8.8700 1.6241 10° 1 lOi 794 m J9i_ 1 0.9848 0.1736 0.9840'0.1779 1. 9832 j 0.1822 0.9824 0.1865 2 1.9696 0.3473 1.9681 10.3669 0.9666 0.3045 1.9649 0.3730 3 2.9544 0.5209 2.9521 0.5338 2.9497 0.5467 2.9473 0.5596 4 3.9.392 0.6946 3.9362 0.71181 3.9330;0.7289 3.9298 0.7460 5 4.9240'0.8682 4.9202 0.8897 4.9163 0.9112 4.9123 0.9325 6 5.9088 1.0419 5.9042 1.0676 5.8995 1.0933 6.8947,1.1190 7 6.8937 1.21561 6.8S83 1.2456 6.882811.2756 6.8772 1.3066 8 7.8785 1.3892 7.8723 1.4235 7.8660!l.4579 7.8696 1.4920 9 8.8633 1.6628 8.8664 1.6015 8.8493 1.6401 8.8421 1.6785 1 11° 0.9816 79 0:i908 ~iu' 0.9808 7SS Hi 0.9799 78-r 0.1993 111 0.9790 78r 0.1951 0.2036 2 1.9633 0.3816 1.9616 0.3902 1.9598 0.3987 1.9581 0.4073 3 2.9449 0.5724 2.9424 0.6863 2.9398 0.6981 2.9371 0.6109 4 3.9265 0.7632 3.9231 0.7804 *3.9] 97 1,0.7976 3.9162 0.8145 5!4.9081 0.9540 4.9039'0.9765 '4.8996 0.99(i,'- 4.8952 1.0182 616 8898 1.1449 5.884711.1705 5.8796 i 1.1 962 15.8743 1.2218 716.8714 1.3367 6.8666 1 1.3666 6.8596 1.3966 16.8533 1.4265 8 '7.853 3 1.5266 7.8463 1.6007 7.8394 1.6949 17.!- 324 1.6291 1 9 1 8.8346 11. 71 73 8.827l!l.7658 8.8193J.7943I 8,8114 1.8327 eTw? nTs'i Ie.~w; ¥.~s. I 1 eTw. I"n7^. E. W. N. S. u TABLES OF SURVEYS. 183 TABLES OP LATITUDE AND DEPARTURE. | N. S. E.W. N. S. ,E. W.I N. S. E. W. N. S. E. W. 12° 78° 12i 1 77i 12J 77i m 77i 1 0.9781 0.2079 0.9772 0.2122 0.9763 0.2164 0.9753 0.2207 2 1.95fi.S 0.4158 1.9544 0.4242 1.9526 0.4329 1.9507 0.4414 3 2.9344 0.6237 2.9317^0.6365 2.9289 0.6493 2.9260 0,6621 4 3.91^6 0.8316 3.9089 0.84871 3.9052 0.8657 3.9014 0.8828 ft 4.8907 1,0396 4.8861 1.0609 4.8815 1.0822 4.8767 L1035 6 .5.8689 1.2475 5.8634 1.2730 5.8578 L2986I 5.8520 1.3242 7 fi.8470 1.4554 6.8406 1.4852 6.8341 L5151||6.8274 1.5449 8 7.8252 1.6633 7.8178 1.6974 7.8104 1.7315 7.8027 L7656 9 1 8.8033 13° 1.8712 77° 0.2249 8.7951 0.9734 1.9096 8.7867 1.9479 76i 8.7781 13i 0.9713 1.9863 1 76i 0.2377 76i m 0.9744 0.2292 0.9724 0.2334 9. 1.9487 0.4499 L9467 0.4584 1.9447 0.4669 1.9427 0.4764 S ?..9m 0.6749 2.9201 0.68761 2.9171 0.7003 2.9140 0.7131 4 3.8975 0.8998 3.8934 0.91681 3.8895 0.9338 3.8864 0.9507 , 5 4.8718 1.1248 4.8669 1.1460 4.8619 L1672 4.8667 L1884 6 .5.8462 1.34971 i5.8403 1.3752 5.8343 1.4007 16.8280 1.4261 1 7 6.8206 1.5746! 6.8136 1.6044 6.8067 L6341 6.7994 1.0638 8 7.7950 L7996| 7.7870 1.83361 7.7790 1.8676 7.7707J 1.9015 !j 9 8.7693 2.0246! 76° 8.7604 14i 2.0628 75S 8.7515 14J 2.1010 75i 8.7421 14J 2.1392 14° I 1 0.9703 0.2419 0.9692 0.2461 0.9681 0.2604 0.9670 0.2546 I ?. 1.9406:0.4838 L9385'0.4923 L9363 0.500S L9341 0.5092 1 3 2.91090.7258 2.9077 0.7385 2.9044'0.7511 2.9011 0.7638 3.86S2 1.1084 4.8352 L2730 | 4 3.8812'0.9677 3.8769 0.9846 3.St27 1.0015 5 4.8515|l.2096 4.8461 1.2308 4.8407 L2519 fi 5.8218 1.4515 5.8154'L4769 5.8089 1.5023 5.802311.5276 | 7 6.7921 1 1.6935 6.7846 1.7231 6.7770 L7627 6.7693 1.7822 8 7.7624 1.9354 7.7538 1.9692 7.7452 2.0030 7.7364 2.0368 1 9 8.7327, 2.1773| 8.7231 "l5i~ 2.2154 T4i~ 8.7133 15,^ 2.2534 "74^ 8.7034 2.2914 74i 15 75 15i 1 0.9659 0.2588 0.9648 0.263O 0.9636 0.2672 0.9624 0.2714 1 9. 1.9319 0.5176 1.9296 0.5261 1.9273 0.5345 1.9249 0.5429 i 3 2.8978 0.7765, 2.8944 0.7891 2.8909 0.8017 2.8874 0.8143 , 4 3.8637 1.0353 3.8591 1.0521 3.854511.0689 3.8498 1.0858 5 4.8296 '1.29411 4.8239 1.3152 4.8182 1.3362 |4.8123|1.3572 1 6 5.7956|1,5529| 5.7887 1.5782 5.7818 L6034 15.7747 11.6286 7 6.7615 1.81171 6.7535 1.8412 6.7454 1.8707 6.7372 1.9001 ! 8 7.7274 2.0706 7.7183 2.1042 7.7090 2.1379 7.6996 2.1715 jj 9 8.6933 2.3294 8.6831 e"v~ 2.3673 1 8.6727 2.4051 8.6621 2 4430 E.W. N. S. N.S.I E. VV. N. S. JE. W. N. S. Ij 184 THE SURVEYOR S GUIDE. TABLES OF LATITUDE AND DEPARTURE. | N.S. 16° E. W. 74 0.2766 N.S. E.W. N.S. E.W. N;S. E.W. 73* 16i 0.9600 731 0.2798 16i 0.9588 73i 0.2840 161 0.9612 0.9576 0.2882 L9225 0.5513 1.9201 0.5596 1.9176 ).5680 1.9151 0.5764 2.8838(0.8269 2.8801 0.8395 2.8766 0.8520 2.8727 0.8646 3.8450 1.1025 3.8402 1.1193 3.8353 1.1361 3.8303 1.1528 4.S063I 1.3782 4.8002 1.3991 4.7941 1.4201 4.7878 1.4410 |! 5.767611.6638 5.7603 1.6790 5.7629 1.7041 5.7454 1.7292 j 6.728811.9295 6.7203 1.9588' 6.7117 1.9881 6.7030 2.0174 1 7.6901 2.2051 7.6804 2.2386 7.6705 2.2721 7.6606 2.3066 1 8.6513 2.4807 73 8.6404 17i 0.9550 2.5185 0.2965 8.6294 m 0.9537 2.6561 8.6181 171 0.9523 2.5938 72i 0.3048 17 0.9563 0.2924 0.3007 1.9126 0.5847 1.9100 0.5931 1.9074 0.6014 1.9048 0.6097 : 2.8689 0.8771 2.8651 0.8896 '2.8611 0.9021 2.8572 0.9U6 3.8252 1.1695 .3.8201 1.1862 3.8149 1.2028 3.8096 1.2195 4.7815 1.4619 4.7761 1.4827 14.7686 1.5035 4.7620 1.524.? 5.7378 1.7642 5.7301 1.7792 1 5.7223 1.8042 5.7144 1,8292 1 6.6941 2.0466 6.6851 2.0758 16.6760 2.1049 6.6668 2.1340 ' 7.650412.3390' 7.6402 2.3723 7.6297 2.4066 7.6192 2.4389 , 8.6067 18° 2.6313 72 8.5962 2.6689 8.6834 2.7063 7H 8.5716 ~^8f 2.7438 7U 711 18i 0.9510i 0.3090 0.9497 0.3131 0.9483 0.3173 0.9469'0.3214 1 1.9021 0.6180 1.8994 0.6263 1.8966 0.6346 1.8939,0.6429 J 2.8532 0.9271 2.8491 0.9395 2.8450 0.9519 2.8408 0.9613 • 3.8042 1.2361 3.7988 1.2527 3.7933 1.2692 3.7877 1.2857 i 4.7553 L6451 4.7485 1.5658 4.7416 1.6865 4.7346 1.6072 i 5.7063 1.8541 5.6982 1.8790 5.6899 1.9038 6.6816' 1.9286 | 6.6574 2.1631 6.6479 2.1921 6.6383 2.2211 6.6286 2.2601 7.6084 2.4721 7.6976 2.5053 7.6866 2.5384 7.5754 2.5715 8.5595 2.7812 8.5473 2. 8186 8.5349 2.8567 8.5224 2.S929 19= 0.9455 71° 0.3255 19i 70i 0.3297 19i 0.9426 70i 0.3338 191 70i 0.9412 0.3379 0.9441 1.S91U 0.6511 1.8882 0.6594 1.8853 0.6676 1.8823:0.6758 2.8366 0.9767 2.8323 0.9891 2.8279 1.0014 2.8233 1.0137 3.7821 1.3023 3.7764 1.3188 3.7706 1.3352 3.76471.3517 4.7276 16278 4.7204 1.6484 4.7132 1.6690 4.7059jl,6896 5.6731 1.9534 5.6645 1.9781 6.6558 2.0028 5.6471 [2.0276 6.6186 2.2700 6.6086 2.3078 6.5986 2.3366 6.588212.3654 7.5641 2.6045 7.6527|2.0375 7.6411 2.6705 7.6294 2.7033 1 8.5097 2.9301 8.496812.9672 8.4838 .3.0043 8.4706 3.0412 ;_ E. W. N. S. e.w.In. s. E.W.I N.S. 1 E.W.! N.S. TABLES OF SURVEYS. 185 TABLES OF LATITUDE AND DEPARTURE N. S. B.W. 70 N. S. 20i E. AV. N. S. 20i E. W. ~69r N. S. E. W. 20 691 69i 1 0.9397 0.3420 0.9382 0.3461 0.9366 0.3502 0.9351 0.3543 2 1.8794 0.6840 1.8764 0.6922 1.8733 0.7004 1.8703 0.7086 3 2.8191 L0261 2.8146 1.0383 2.8100 1.0606 •2.8064 1.0629 4 3.7588 1.3681 3.7628 1.3845 3.7467.|l.4008 4.6834ll.7510 3.7406 1.4172 6 4.6985 1.7101 4.6910 1.7306 4.6767 L7715 6 5.6381 2.0521 5.6291 2.0767 5.6200 2.1012 6.6108 2.1267 7 6.5778 2.3941 6.5673 2.4228 6.6567 2.4514 6.6459 2.4800 « 7.5175 2.7362 '..5055 2.7689 7.4934 2.8016 7.4811 2.8343 9 1 8.4572 21° 3.0782 A4437 3.1160 1 21i 68i i0.9320 0.3624 8.4300 _2H 0.9304 3.1519 8.4162,3.1886 69° m 21 i m 0.9336 0.358'l 0.3666 0.9288|0.3705 2 1.8672 0.7m(il.8640 0.7249 1.8608 0.7330 1.867610.7411 3 2.8008 1.P757!, 2.7960 1.0873 2.7913 '1.0996 2.7864|L1117 4 3.7343 1.433r 1 13.7280 1.4497 3.7217 1.4660 3.7152 1.4822 5 4.6679 L79181 |4.6600 1.8122 4.6621 1.8325 4.6440 L8528 6 5.6015 2.1502 5.5920 2.1746 5.5826 2.1990 5.5729 2.2233 7 6.5351 2.6086 6.6240 2.5371 6.5129I2.5665 16.6017 2.5939 8 7.4686 2.8669 7.4660 2.8995 i 7.44332.9320 7.43052.9644 9 8.4022 3.2253 68° 8.3880 3.2619 67i" 8.3738 3.2985 8.3593 3.3360 22° 22J 22i 67i 22i i 67i ! 1 0.9272 0.3746 0.9255! 0.3786 0.9239 0.3827i 0.922210.3867 2 1.8544 0.7492 L8511' 0.7573 1.8478 0.7664 1.84440.7734 3 2.7816 1.1238 2.7766|1.1359 2.7716 1.1480 12.7666 1.1601 4 3.7087 1.4984 3.7022 1.6146 3.6966 L5307 3.6888 1.5468 5 4.6359 1.8730 4.6277 1.8932 4.6194 1.9134 4.6110 1.9335 6 5.5631 2.2476 5.6532 2.2719 6.6433 2.2961 5.6332 2.3202 7 6.4903 2.6222 6.4788 2.6506 6.4671 2.6788 6.4654 2.7069 8 7.4175 2.9968 7.4043 3.0292 7.3910'3.0615 7.3776 3.0936 9 1 8.3447 3.3715 67 0.3907 8.3299 23i 0.9188 3.4078 66i 8.3149 3.4441 8.2998 3.4803 23 J3i 0.9170 66i 23i 66i 0.9205 0.3947 0.3987 10.9163 0.4027 2 1.8410 0.7816 L8376 0.7895 1.8341 0.7975 il.8306 0.8065 3 2.7615 1.1722 2.7664 1.1842 2.7512 L1962 2.7459 1.2082 4 3.6820 1.6629 3.6762 1.6790 3.6682 L5950 3.6612 1.6110 S '4.6026,1.9537 4.6939 11.9737 4.5863|1.9937 4.5766 0.0137 6 1 6.5230 2.3444 6.5127 2.3686 6.6024 2.3925 5.4919 2.4165 7 6.4435 2.7351 6.4315 2.7632 6.4194I2.7912 6.4072|2.8192 8 7.364013.1268 7.3503 3.1679 7.3366 .3.1900 7.2225|3.2220 9 8.2845{3.5166 8.2691 3.6527 8.2635 3.5887 18.2378 3.6247 ! E. W. 1 N. S. E. W 1 N. S. E. W N. S. 1 E. W !N. S. •_J 186 THE surveyor's GUIDE. TABLES OF LATITUDE AND DEPARTURE. j N. S. 24 E.W. N. S. 24i E. W. 65f" N. S. E. W.I N. S. 241 65i 1 66 24i 65i 1 0.9135 0.4067 0.9117 0.4107 0.9099 0.41471 0.9081 0.4186 2 11.827] 0.8135 1.8235 0.8214 1.8199 0.82941 1.8163 0.8373 3 12.7406 1,2202 2.7353 1.2322 I2.7299|1.244l! 2.7244 1.2560 4 3.G542 1.62691 3.6470.1.6429 3.6398 L6588 3.6326 1.6746 1 5 4.5677 2.03371 4.558812.0536 4.5498 2.0735 4.5407 2.0933 1 6 5.4813 2.4404 5.470612.4643 5.4598 2.4882 5.4489 2.5122 7 6.3948 2.8472 6.382312.8750 6.369712.9029 6.3570 2.9306 8 7.30S4 3.2539 7.294113.2857 7.2797|3.3175i 7.2651 3.3493 ! 9 8.2219 3.6606 8.2058 3.6965 8.1896 3.7322 18.1733 3.7679 1 25° 65 25i 64i 25i 64.i 25i 64i || 1 0.9063 0.4226 0.9044 0.4265 0.9026 0.4305 0.9007 0.4344 2 1.8126 0.8452 il.8089 0.8531! 1.8052 0.8610 1.8014 0.8688 3 2.7189 1.2679 12.7134 1.2797! 2.7077 12915 12.7021 1.3032 4 3.6252 1.6905 j3.6178 1.7063 3.6103 1.7220 13.6028 1.7376 i 5 4.5315 2.1131 4.5223 2.1328 4.5129 2.1525 14.5035 2.1720 6 5.4378 2.5357 15.4267 2.5594 5.4155 2.5831 15.4042 2.6004 7 6 3442 2.9583 6.3312|2.9860| 6.3181:3.0136 6.3049 3.0408 S 7.2505 3.3809 7.2356 3.4125 1 7.2207 3.4441 7.2056 3.4752 9 8.1568 3.8036 8.1401 3.83911 8.12333.8746 8.1063 3.9096 26° 64° 26.i 63i 26i 63i 26i 63i 1 0.8988 0.4384 0.8969,0.4423 0.8949 0.4462 0.8930 0.4501 2 1.7976 0.8767 L7937 0.8846 1.7899 0.8924 1.785910.9002 8 2.6964 1.3151 12.6906 1.3269' 2.6848, L33S6 2.67891.3503 4 3.5952'l.7535 13.5875 L7692 3.57971.7848 3.5719 1.8004 4.4940l2.1919 1 4.4843 2.2115 4.4746 2.2310 4.4649 2.2505 1 fi 5.3928 2.6302 1 5.3812 2.6537 5.3696 2.6772 5.3579 2.7006 1 7 6.2916 3.0686 ,6.2781 3.0960 6.2645 3.1234 6.2508 3.1507 ( s 7.1904 3.5070i. 7.1750 3.5383 7.1594 3.5696 7.1438 3.6008 ] 9 8.0891 3.9453 8.0718 3.9806 8.0544 4.0158 8.0368 4.0509 1 27° 63 27i 62| 27i 62i 271 62i 1 0.89100.4540 0.8890 0.4578 0.8870 0.4617 0.8850 0.4656 ?. 1.7820 0.9080 1.7780|0.9157 1.7740 0.9235 1.770010.9312 .S 2.6730;].3620 2.6670!l.3736 2.6610|l.3852 2.6650 1.3968 4 3.5640 1.8160 3.5561 |l.S315 3.54S0 1.8470 3.5400 1.8624 J 1 ■'' 4.4550 2.2699 4.445l!2.2894 4.435(1 2.3087 |4.4250 2.3281 1 fi 5 3460 12.7239 5.3341 2.7472 ! 5.3221 2.7705 15.3099 2.7937 1 7 6.2370:3.1779 6.2231 3.2051 16.2092 3.2322 6.1949 3.2593 s 7.1280 3.6319 7.1121 3.6630 [7.0961 3.6940 7.0799 3.7249 9 8.0191 4.0859 8.0011 4.1209 7.9831 4.1553 7.9649 4.1905 1 B. W. ' N. S. E. W.I N. S. E. \V. N. S. 'e. av.In. s. TABLES OF SURVEYS. 187 TABLES OP LATITUDE AND DEPARTURE 1 N. S. E. W. N. S. E.W.I N.S. E. W. N.S. E.W. 28 62 0.4694 28i 0.8809 61i 28^ 61i 0.4771 28} 0.8766 6U 0T48IO 1 0.8829 0.4733 0.8788 2 L7659 0.9389 1.7618 0.9466 L7576 0.9543 1.7634 0.9620 3 2.6488 1.4084 2.64271 1.4199 2.6364 1.4315 2.6302 1.443U ! 4 3.5318 1.8779 3.52361.8933 3.5153 1.90S6 .3.5069 1.9239 5 4.4147 2.3474 4.4045 3.3666 4.3941 2.3858 4.3836 2.4049 6 5.2977 2.8168 5.2S54I2.8399 5.2729 2.8629 5.2604 2.8859 7 6.1806 3.2863 6.1662 3.3132 6.1517 3.3401 6.1371 3.3669 8 7.0636 3.75581 7.0471 3.7866| 7.0305 3.8173 7.0138 3.8479 v» 7.9465 4.2252 61 7.9280 4.2599 29i 1 60i 7.9093 29,^ 4.2944 60i 7.8905 29i 4.3289 1 29 1 0.8746 0.4848 '0.8725J0.4886 0.8703 0.4924 0.8682 0.4962 ^ 1.7492 0.96961 ,1.7450,0.9772 1.7407 0.9848 L7364 0.9924 3 2.6239 1.45441 2.617511.4659 2.6111 1.4773 2.6046 1.48S6 4 3.4985 1.9392 3.4900 1.9545 3.4814 L9697 3.4728 1.9849 4.3731 2.4240 :4.3625 '2.4431 4.3518 2.4621 4.3410 2.4811 fi 5.2477 2.9089 ■5.2350'2.9317l 5.2221 2.9545 6.2092 2.9773 7 6.1223 3.3937 6.1075 3.4203 6.0925 3.4463 6.0774 3.4736 S 6.997013.8785 6.9800 3.90901 6.9628 3.9394 7.9466 3.9697 9 7.8716 4.3633 7.8525 4.3976 7.8332 4.4318 7.S138 4.4669 30 60 30i 59i 30i 59,^ 30i 59i 1 0.866010.5000 0.8638 0.5038 0.8616 0.5075 0.8594 0.5113 '>. 1.7320 1.0000 1.7277 1.0076 1.7232 1.0151 1.7188 1.0226 3 2.5981 1.5000 :2.59151.5]13 2.5849 1.5226 2.5782 1.5339 4!3.464ll2.2000 I3.4552 2.0151 3.4465 2.0301 3.4376 2.0462 5 |4.3301 2.5000 4.3192 2.5189 4.3081 2.537Y 4.2970 2.6564 6 5.1961 3.3000 5.1830 3.0226 5.1698'3.04o2 5.1564 3.0677 7 6.0622 3.5000 6.0468 3.5264 [6.0314 3.5528 6.0158 3.6790 8 6.9282 4.0000 6.9107 4.0302 6.8930 4.0603 6.8762 4.0903 9 [7.7942 4.5000 7.7745 4.5339 7.7547 4.6678 7.7346 4.6016 1 ^^ 59 3U 58i 3U 58.} 311 68i 1 1 |0.8571 0.5150 0.8549 0.5188 0.8526 0.5225 : 0.8603 0.5262 2 11.7143 1.0301 1.7098 1.0375 1. 7053,1. OloO 1.7007 !l.0524 3 12.5715 1 5451 2.5647^1.5563 2.5579 1.5675 2.6510'1.6786 11 4 3.4287 2.0602 3.4196 2.0751 3.4106 '2.0900 3.4014 2.1048 1 4.2618|2.6311 | 5 4.2858 2.5752 4.2745 2.5939 4.2632 2.6125 6 5.1430 3.0902 5.1295 3.1126 5.1158 .3.1350 5.1021 3.1573 1 7 6 0002 ,3.6053 5.9844 3.6314 5.9685 3.6575 5.9526 3.6835 i s 6.8573 4.1203 16.8393 4.1602 6.8211 4.1800 6.8028 4.2097 9 7.7145 4.6353 7.6942 J4.66S9 7.6738 4.7025 7.6532 4.7359 IeTw N. S. , E.W.I N.S. E.W. 1 N.S. E.W.iN.ri. , 188 THE SURVEYOR S GUIDE. TABLES OF LATITUDE AND DEPARTURE. | N. S. E.W. 58° N. S. 32i 0.8457 E.W. N. S. E. W.l -m\ 0.5373] N. S. E.W. "67i~ 32° 57i 0.5336 32i 0.8434 32« 1 9.8480 0.5299 0.8410 0.5409 ; 2 L.6961 1.0598 1.6914 1.06721 1.0868 1.0746 L6821 1.0819 3 2.5441 1.5897 2.537211.60081 2.5302 1.6119 2.6231 1.6229 4 i.3922 2.1197 3.3829 2.1344 3.373612.1492 3.3642 2.1639 5 4.2402 2.6496 4.228612.66^1 1 4.2169 2.6805 4.2052 2.7049 1 (5 5.0883 3.1795, 5.0744 3.2017 5.0603 3 2238! 5.0462 3.2458 I 7| 5.9363 3.70941 5.9201 3.7353 5.9037 3.76111 6.8873 3.7868 1 « 6.7S44 4.2394 6.7658 4.2689 6.7471 4.2984 6.7283 4.3278 9 1 7.0324 33 4.7693 7.6115 33i 0.8363 4.8025 5-6I 0.5483 7.5905 8339 4.8357 [7.5694 4.8688 56^ 0.5555 57 56i 0.5519 33} 0.8314 0.838(5 0.5446 2 1.0773 1.0893 1.6726 1.0966 1.6678 1.1039 1.6029 1.1111 3 2.516U 1.0339 12.5089 1.6449 '2.5017 1.65581 2.4944 1.6667 4 3.3547 2.17843 3.3451 2.1932 ,3.3355 2.2077 3.3259 2.2223 5 4.1934 2.7232 4.1814'2.7415| 4.1694 2.7597 4.1573 2.7778 (i 5.0320 3.2678 l5.0177|3.2898l 5.0033 3.3116, 4.9888 '3.3334 7 5,8707 3.8125 15.8510 3.8381 5.8372 3.8635 5.8203.3.8890 8 0.7094 4.3571 1 0.6903 4.3863 6.6711 4.4155 6.6518 4.4446 1 9 7.5480 4.9018 7.5200 4.9340 1 7.5059 4.9074 7.4832i5.0001 1 34 56° 34i 5.5.i 1 m 55i 343 551 \ 7 0.82901 0^5692 |o.8200 0.5028 '0.8241 0.56641 0.8210 0.5700 \ 2 L658ll].iis4 1.6532 1.1250 1.6482, 1.132N 1.0433 1.1400 1 -^ 2.4871 l.fi770 2.1798 1.6884 2.47241 1.0992 2.4049 1.7100 4 3.3162 2.23(iS 3.3063 2.2512 3.2965 '2.:i650 3.2806 2. 2800 ft 4.1452|2.7900l 4.i:-i29 2.8140 4.1200,2.8320 4.1082|2.8500 4.0742!3.y,552| '4.9595 3.3708 4.9447 3.3984 4.9299 1 3.4200 7 5.8033 3.9144' 16.7861 3 9390 5.7689 3.9048 6.7615 3.9900 8 6.6323 4.4735 6.6127 4.5024 6.5930 4.5313 6.5732 1 4. 5(;00 9 1 7.4013 35° 5.0327 17.4393 [ 35i 0.8166 5.0052 7.4171 "35"^ 0.8141 5.0977 54i 0.5807 7.3948 1 6. 1300 353 i 54i 1 55 543 0.5771 8191 0.5736 0.8116 0.5842 ?, 1.638;' 1.1472 1.6333 ! 1.1543 1.6282 1.1014 1.0231 1.1685 .S 2.4575 1.7207 2.4499 |l.7314ii2.4423 1.7421 2.4347 1.7527 1 ■* 3.2766 2.2943 3.2606 2.3086 3.2505 2.3228 3.2463 2.3370 ' ■ ft 4.0958 2.S679r4.0832,2.8857 4.0700 2.9035 4.0579 2,0212 ' 1 fi 4.9149 3.1415 4.8998 3.4629 4.8847 3.4842 4.8694 3.5055 '. ' 7 5.7341 1.0150 .5.7165 4.0400 5.6988 4.U049 5.681 C 4,0897 | 6.4926 14.6740 1 8 6.5532 4.5886 0.533114.6172 |6.5129 4.6456 1 ^ 7.3724 5.1622 7.3498 5.1943 1 7.3270 5.2263 7.3042 '5.2388 Erw.'iii.T' H 1 lE. W.'N. S. K W. N. S. E. ^Y. i N. S. TABLES OF SURVEYS. 189 TABLES OF LATITUDE AND DEPARTURE. 1 N. S. E.W. N. S. E.W. jN. S. E. W. IN. S. E. W. 36 54 36i 5.31 1 36i m 1 m 5.3i I 1 0.8090 0.5878 0.8064 0.5913 ' 0.8038 0.5948 osoia 0.5983 1 2 1.6181 1,1756 1.6129 1.1826 1.6077,1.1890 1.0-026 1.1966 1 3 2.4271 1.7634 2.4193 1.7739 '2.4116 1.7845 2.4038 1.7950 4 3.236l!2.3611 3.2258 2.3662 3.2154 2.3793 3.2050 2.3933 5 4,0451 2.9389 4.0322 2.9665 4.0193 2.9741 4.0063 2.9916 6 4.8541 3.5267 4.8387 3.5478 4.8231 '3.5689 4.8075 3.5899 7 5.6631 4.1145 5.6451 4.1391 5.6270 4.1638 6.6088 4.1883 8 6.4721 4.7023 6.4516 4.7304 6.4308 4.7586 6.4100 4.7866 9 7.2812 5.2901 7.2580 5.3217 7.2347 6.3534 7.2111 5.3849 37 63 37i 52i 37i 52i 37i m ] 0.7986 0.6018 ] 0.7960 0.605.3 0.7933,0.6087 0.7907 0.6122 2 1.5973 1.2036 1.5920 1.2106 1.5867 12176 1.6814 1.2244 3 2.3959 1.8054 '2.3880 1.8159 2.3801 1.8263 2.3721 1.8366 4 3.1945 2.4073, 3.1840 2.4212 3.1734 2.4360 13.1628 2.4489 6 3.9932 3.0091 3.9800 3.0265 3.966813.0438 13.9534 3.0611 6 4.7918 3.6109 4.7760 3.6318 4.7601 3.6626 .4.7441 3.6733 7 5.5904 4.2127: 15.6720 4.2371 0.5635 4.2613 5.5348 4.2856 8 6 3891 4.8145 6.3680 4.8424 6.3468 4.8701 6.3255 4.8977 ! 9 7.1877 5.4163 52 7.1640 5.4476 ~61i~ 7.1402 6.4788 51i 7.1162 381 5.5099 5U 38° 38i 1 0.7880 0.6156 0.7853 0.6191 0.7826 0.6225 0.7799 0.6259 2 1.5760 1.2313 1.5706 1.2382 1.6652 1.2450 L6698 1.2518 3 2.3640 1.8470 2.3559 1.8673 2.3478 1.8675 2.3397 1.8778 4 3.1520 2.4626 3.1413 2.4764 3.1304:2.4900 3.1195 2.5037 5 3.9401 '3.0783! 3.9266 3.0965 3.913013.1125 3.8994 3.1296 6 4.7281 3.6940 4.7119 3.7146 4.6966 3.7351 4.6793 3.7656 7 5.5161 4.3096 5.4972 4.3337 '6.478214.3576 5.4592 4.3815 8 6.3041 4.9253 6.2825 4.9528 6.2608 4.9801 6.2391 6.0074 1 9 1 7.0921 6.5409 j 51 0.62931 7.0678 "39r 0.7744 6.5718 ToF 7.0434 39i 0.7716 6.6026 502^ 0.6361 7.0190 39i 0.7688 5.6333 1 50i 0.6394 39° 0.7771 0.6327 2 1.5543 1.25S6 1.5488 1.2664 1.6432 1.2621 1.6377 1.2789 3 2.. 33 14 1.8880 2.3232 1.8981 2.3149 1.90821 2.3065 1.9183 4 13.1086 2.5173 3.0976 2.6308 3.0865 2.6443 3.0764 2.5578 5 3.8857 3.1466 3.8719 3.16351 3.8581 3.1804 3.8442 3.1972 1 fi 4.6629 3.7759 4.6463 3.7962 4.6297 3.81651 4.6130 3.8366 1 7 5 4400 4.4052 5.4207 4.4289 6.4014 4.45251 5.3819 4.4761 | 8 6.2172 5.0346 6.1951 5.0616 6.1730 6.08861 6.1507 5.1156 9 6.9943 5.6639 6.9695 6.0943 6.9446 5.72471 6.9196 5.7550 |E.W. N. S. kw. N. S. E. W. N. S. 1 E. W. N. S. 1 190 THE SURVEYOK S GUIDE. TABLES OF LATITUDE AND DEPARTURE. | 1 N.S. 40° 0.7660 E. W. N. S. 40i 0.7632 E. W. 49i N. S. 40i 0.7604 E. W. N.S. 40i 0.7575 E.W. 49i 0.6527 50 0.6428 49 i 0.6494 0.6461 2 1.5321 1.2856 1.5265 L2922 1.5208 1.2989 1.5151 1.3055 3 2.2981 L9284 2.2897 1.9384 '2.2812 1.9483 2.2727 1.9583 4; 3. 0642 2.5711 3.0529 2.5845 3.0416 2.5978 3.0303 2.6110 1 5 3.8302 3.2139 J3.S162 3.2306 3.8020 3.2472 3.7878 3.2638 6 4.5963 .3.8567 14.5794 3.8767 4.5624 3.8967 4.5454 3.9166 7 5.3623 4.4995 5.3425 4.5229 ,5.3228 4.5461i 5.3029 4.5693 8 6.1284 5.1423 6.1059 5.1690 '6.0832 5.1956! 6.0605 5.2221 9 6.8944 5.7851 6.8691 5.8151 6.8436 5.8450 6.8181 5.8748 41 49 41i 48i m 48i 1 41S 48i 1 0.7547 0.6560 10.7518 0.6593 :0.7489 0.6626 0.7460 0.6659 2 1.5094 1.3121 1.5037 1.3187 1.4979 1.3252 1.4921 1.3318 3 2.2641 1.9682 2.2555 1.9780 '2.2468 1.9879' 2.2382 1.9976 1 i 3.0188 2.6242 3.0074 2.6374 2.9958 2.65051 2.9842 2.6635 i 5 3.7735 .3.2803 3.7592 3.2967 ,3.7447 3.3131 3.7303 3.3294 6 4.5283 3.9364 4.5110 3.9560 ! 4.4937 3.9757 14.4764 3.9953 7 '5.2830 4.5924' '5.2629 4.6I54' '5.2426 4.6383 5.2224 4.6612 8 ,6.0377 5.2485. 6.0147 5.2747] 5.9916 5.3010 5.9685 5.3270 !| 9 6.7924 5.9045 '6.7666 5.9341 6.7405 5.9636 6.7145 5.9929 42 48 ■m 47i 42. V 47i 421 m 1 0.7431 0.6691 0.7402 0.6723 0.7373 0.675<3 0.7343 0.6788 2 1.4863 1.3383' 1.4804 1.3147 1.4746 1.3512 1.4686 1.3576 3 '2.2294 2.0074 2.2207 2.0171 2.2118 2.0268 2.2029 2.0364 4 2.9726 2.6765 2.9609 2.6895 2.9491 2.7024 2.9373 2.7152 5 3.7157 3.3457 3.7011 3.3618 3.6864 3.3779] 3.6716 3.3940 6 4.45894.0148 4.4413 4.0342 4.4237 4.0535 4.4059 4.0728 1 7 5.2020 4.6839 5.1815 4.7066 5.1610 4.7291 5.1402 4.7516 8 15.9452 5.3530 5.9218 5.3789, 5.8982 5.4047 5.8746 5.4304 9! 6.6883 6.0222 6.6620 6.0513 6.6355 6.0803 6.6089 6.1092 1 ^^ 47 1 m 461 43i 46i 43S 46i 1 0.7313 O.6S20I '0.7283 0.6852 0.7253' 0.68S3| 0.7223 0.6915 2 !l.4627 1.3640 1.4567 1.3704 1.4507 1.3767 1.4447 1.3830 a 2.1941 2.0460 2.1851 2.0555' 2.1761 2.0651 2.1671 2.0745 4 2.9254 2.7280 2.9135 2.7407 2.9015 2.7534 2.8894l2.766« || 5 3.6568 3.4100 3.6418 3.4259 13.6269 3.4418 3.6118 3.4576 i 6 14.3881 4.0920 4.3702 4.1111 I4.3522 4.1301 4.3342 4.1491 ! 1 7 5.1195 4.7740 5.0986 4.7963 5.0776 4.8185 5.0565 4.8406 i 8 6.8508 5.4560 5.8269 5.4814 I5.8O3O 5.5068 5.7789 5.5321 1 9 6.5822 6.1380 6.5553 6.1666 '6.5284 6.1952 6.5013 6.2236 1 1 E. W. N. S. 1 E. W. N. S. E. W. N. S. E.W. N.S. 1 TABLES OF SURVEYS. 191 TABLES OF LATITUDE AND DEPARTURE N. S. E.W.I N. S. E. W. N. S. 1 E. W. N. S. E. W. 44° 46 Uk 453 44i Abh 445 ~45r 1 0.7193'0.6946i 0.7163 0.6978 0.7132,0.7009 0.7102 0.7040 '2. L4387iL3893| L4326 1.3956 i 1.4265 ).401S ! 1.4204 1.4080 a 2.1580 2.0840' 2.14S9 2.0934 2.1397 2.1027 2.1305 2.1120 4 2.8774,2.7786, 2.8652 2.7912 2.8530.2.8036 |2.8407 2.8101 ;> 3.5967 3.4733 3.5815 3.4889 3.5662 3.5045 ,3.5509 3..5201 6 4.3160 4.1679 4.2978 4.1867 4.2795 4.2054 4.2611 4.2241 1 V 5.0354 4.8626 5.0141 4.8845 4.9927 4.9063 4.9713 4.9281 s 5.7547 5.5573 5.7304 5.5823 5.7060 5.6072 5.6S15 6.6321 1 6.4741 45 6.2519 6.4467 6.2801 6.41.92 6.30«1 6.3917 6.3361 45 0.7071 07071 2 1.4142 1.4142 3 2.121312.1213 4 2.82842.82S4 3.5355'3.5355 6 4.2426 4.2426 7 4.9497 4.9497 5.6569 5.6569 8 9 6.3640 6.3640 E W. N. S. 1 E. W. N. S. E.W. N. S. E.W. KS. TABLES OF SURVEYS. THE USE OF THE FOREGOING TABLES IN RELATION TO SURVEYS. They show, by inspection, the alteration of lati- tude and departure to every degree on the com- pass, and that for any distance not exceeding 100.000 links. In the uppermost rank of every division are placed the several angles and their complements, to 45°, including the quarter, half, and three- quarters of degrees ; and in the left-hand column are lengths of the measured lines of the field- work, and in the common areas are the difference of latitude and departure. Examples. Suppose the angle to be N. E. 27J degrees, and the line in the field measured to 6 chains, and it be required to find the Northings and Eastings of that station, under 27|^ degrees, and answering to 6 in left-hand column, the number in the com- (193) TABLES OF SURVEYS. 193 nion area, 5.3221, which shows the Northings ; and under 62J, (which is the complement to that angle) opposite the same number in the side col- umn, I find 2.7705, which shows the Easting of that station. If the course be the same, and dis- tance 60 chains, remove the decimal point one place to the right-hand, and the latitude and de- parture will be 53.221 27,705. And if the line were 600 chains, the course remaining the same, the Northings would be 532 chains, 21 links, and the Eastings 277 chains, 05 links. If the measured line doth not consist of an ex- act number of tens, as suppose its length to be 75 chains, 03 links, or 75 chains, 34 links, and the course 27J° ; then under this angle, and opposite c. 70 are 62.091 .32.322 5 " 4.435 5 " 2.308 0.30 links 0.266 0.30 links 0.04 " 0.138 0.04 " 0.035 0.018 NorthinsT 66.827 Eastinf 34.786 for 76 chains, 34 links. for 75 cliains, 34 links. And so for any other. N. B. These tables will answer to ^° or 7|', an arithmetical mean between J° and J°, or between r and f °. 13 SYSTEMS OF RECTANGULAll SUE- VEYING FOR SURVEYING THE PUBLIC LANDS OF THE UNITED STATES.* The public lands of the United States are ordi- narily surveyed into rectangular tracts, bounded by lines conforming to the cardinal points. This is effected by meridian lines and parallels of lati- tude, established six miles apart. The squares thus formed are called townships. They are bodies of land 6 miles square, as near as may be, containing as near as may be 23,040 acres. The townships are subdivided into 36 tracts, called sec- tions, each containing as near as may be 640 acres. Any number or series of contiguous townships, sit- uate north or south of each other, constitute a range . *This section is mainly taken from "Manual of Sur- veying Instructions for the Survey of the Public Lands of the United Sates and Private Land Claims, Washrng- tou, 1890." (194) SURVEYING THE PUBLIC LANDS. 195 The law requires that the lines of the public surveys shall be governed by the true meridian, and that the townships shall be six miles square — two things involving in connection a mathematical impossibility — for, strictly to conform to the meri- dian, necessarily throws the township out of square by reason of the convergency of meridians, and hence, by adhering to the true meridian, results the necessity of departing from the strict require- ments of law as respects the precise area of town- ships and the subdivisional parts thereof, the town- ships assuming something of a trapezoidal form, which inequality develops itself more and more as such, the higher the latitude of the surveys. In view of these circumstances the law provides that the sections of a mile square shall contain the quantity of 640 acres, as nearly as may he., and moreover provides that in all cases where the exterior lines of the townships, thus to be subdivided into sections or half sections, shall exceed or shall not extend 6 miles, the excess or deficiency shall be especially noted, and added to or deducted from the western or northern ranges of sections or half sections in such townships, according as the error may be in 196 THE surveyor's guide. running the lines from east to west, or from south to north ; the sections and half sections bounded on the northern and western lines of such townships shall be sold as containing only the quantity ex- pressed in the returns and plats, respectively, and all others as containing the complete legal quantity. Standard parallels are established at intervals of every 24 miles, north and south of the base line, and* guide meridians at intervals of every 24 miles, east and west of the principal meridian ; the object being to confine the errors resulting from convergence of meridians and inaccuracies in measurements, within the tracts of lands bounded by the lines so estab- lished. The survey of all principal base and meridian, standard parallels and guide meridians and town- ship lines, must be made with an instrument ope- rating independently of the magnetic needle. Burt's improved solar compass, or other instru- ment of equal utility, must be used of necessity in such cases ; and it is deemed best that such instru- ment should be used under all circumstances. Where the needle can be relied on, however, the ordinary compass, if provided with a revolving SURVEYING THE PUBLIC LANDS. 197 compass box and variation arc, may be used in sub- dividing and meandering. The township lines and the subdivision lines will usually be measured by a two-pole chain of 33 feet in length, consisting of 50 links, and each link being seven and ninety-two-hundredths of an inch long. On uniform and level ground, however, the four- pole chain may be used. The measurements will, however, always be represented according to the four-pole cTiain of 100 links. Tally-pins. Eleven tally-pins made of steel are to be used. They should not exceed 14 inches in length, be weighty enough toward the point to make them drop perpendicularly, and have a ring at the top, in which is to be fixed a piece of red cloth, or something else of conspicuous color to make them readily seen when stuck in the ground. Process of chaining. In measuring lines with a two-pole chain, every five chains are called a " tally ;" and in measuring lines with a four-pole chain every ten chains are called a tally, because at that distance the last of the ten tally-pins with which the forward chainman sets ont will have been stuck. He then cries " tally," which cry is 198 THE surveyor's guide. repeated by the other chainman, and each registers the distance by slipping a thimble, button or ring of leather, or something of the kind, on a belt worn for that purpose, or by some other convenient method. The hind chainman then comes up, and having counted, in the presence of his fellow, the tally-pins which he has taken up, so that both may be assured that none of the pins have been lost, he then takes the forward end of the chain and pro- ceeds to set the pins. Thus the chainmen alter- nately change places, each setting the pins that he has taken up, so that one is forward in all the odd, and the other in all the even tallies. Such proce dure, it is believed, tends to assure accuracy ia measurement, facilitates the recollection of the dis.- tances to objects on the line, and renders a, mis- tally almost impossible. Levelling the chain and 'plumbing the pins. The length of every line run is to be ascertained by precise horizontal measurements,, as nearly ap- proximating to an air-line as is possible in practice on the earth's surface. This all-important object can only be attained by a rigid adherence to the three following observances : SURVEYING THE PUBLIC LANDS. 199 1. Ever keeping the chain stretched to its ut- most degree of tension on even ground. 2. On even ground, keeping the chain not only- stretched as aforesaid, but horizontally levelled. And when ascending or descending steep ground, hills or mountains, the chain will have to be short- ened to one-half its length (and sometimes more), in order accurately to obtain the true horizontal measure. 3. The careful plumbing of the tally pins, so as to ascertain precisely the spot where they should be stuck. The more uneven the surface, the greater the caution needed to set the pins. Marking lines. All lines on which are to be established the legal corner boundaries are to be marked after this method, viz : Those trees which may intercept your line must have two chops or notches cut on each side of them without any other marks whatever. These are called " sight-trees''' or " line-trees^ A sufficient number of other trees standing; nearest to your line, on either side of it, are to be blazed, on two sides diagonally or quarter- ing toward the line, in order to render the line conspicuous and readily to be traced, the blazes 200 THE surveyor's guide. to be opposite each other, coinciding in direction with the line where the trees stand very near it, and to approach nearer each other the further the line passes from the blazed trees. Due care must ever be taken to have the lines so well marked as to be readily followed, and to cut the blazes' deep enough to leave recognizable scars as long as the trees stand. Bushes on or near the line should be bent at right angles therewith, and receive a blow of the axe at about the usual height of blazes from the ground, sufficient to leave them in a bent position, but not to prevent their growth. On trial or random lines, the trees are not to be blazed, unless occasionally from indispensable necessity, and then it must be done so guardedly as to prevent the possibility of confounding the marks of the trial line with the true. But bushes and limbs of trees may be lopped, and stakes set on the trial or random line, at every ten chains, to enable the surveyor on his return to follow and correct the trial lines and establish therefrom the true line. To prevent confusion, tne temporary stakes set on the trial or random lines must be pulled when the surveyor returns to establish the true line. SURVEYING THE PUBLIC LANDS. 201 EstahlisMng corners. After a true coursing and most exact measurements the establishment of corners is the consummation of the work. If, there- fore, the corner be not perpetuated in a permanent and workmanlike manner, the great aim of the sur- veying service wilL not have been attained. A boundary corner, in a timbered country, is to be a tree, if one be found at the precise spot ; and if not, a post is to be planted thereat ; and the posi- tion of the corner-post is to be indicated by trees adjacent (called bearing trees), the angular bear- ings and distances of which, from the corner, are facts to be ascertained and registered in the field- book. In a region where stone abounds, the corner boundary will be a small monument of stones along- side of a single marked stone, for a township cor- ner — and a single stone for all other corners. In a region where timber is not near, nor stone, the corner will be a mound of earth, of prescribed size, varying to suit th.e case. The following are the different points for perpet- uating corners, viz : 1. For township boundaries, at intervals of every six miles. 202 THE surveyor's guide. 2. For section boundaries, at intervals of every mile, or eighty chains. 3. For quarter section boundaries, at intervals of every half mile, or 40 chains. 4. Meander corners are established at all those points where the lines of the public surveys inter- sect the banks of such rivers, bayous, lakes or islands as are by law directed to be meandered. Meandering is a term applied to the usual mode of surveying with the compass, particularly as ap- plied to navigable streams. The instructions for this are, in part, as follows : Both banks of navigable rivers, as well as all rivers not embraced in the class denominated as " navigable," the right-angle width of which is three chains and upwards, will be meandered by taking the courses and distances of their sinuosities, and the same are to be entered in the field book. At those points where either the township or sec- tion lines intersect the banks of a navigable stream or any meanderable line, corners are to be estab- lished at the time of running these lines. These are called "meander corners ;" and in meandering you are to commence at one of those corners, coursing SURVEYING THE PUBLIC LANDS. 203 the banks or boundary line, and measuring the dis- tance of each course from your commencing corner to the next " meander corner." By the same method you are to meander the opposite bank of the river. The crossing distance between the meander cor- ners^ on the same line, and the true bearing and distance between opposite meander corners is to be ascertained by triangulation or direct measurement, in order that the river may be protracted with entire accuracy. The particulars to be given in the field- notes. You are also to meander, in manner aforesaid, all lakes and deep ponds of the area of 25 acres and upwards ; also navigable bayous. The precise relative position of islands in a township made frac- tional by the river in which the same are situated, is to be determined trigonometrically. Sighting to a flag or other fixed object on the island from a special and carefully measured base-line, con- nected with the surveyed lines, on or near the river bank, you are to form connection between the meander corners on the river to points correspond- ing thereto, in direct line, on the bank of the 204 THE surveyor's guide. island, and there establish the proper meander cor- ners, and calculate the distance across. Surveying. The initial point having been established, the lines of the public survey are to be extended therefrom as follows : Base line. The base line shall be extended east and west from the. initial point by the use of solar instruments or transits. The transit should be desig- nated for the alignment of all important lines. The proper corners shall be established at each 40 and 80 chains, and at the intersection of the line with rivers, lakes, or bayous that should be meandered, in accordance with the instructions for the estab- lishment of corners. In order to check errors in measurement, two sets of chainmen, operating inde- pendently of each other, must be employed. Where transits are used, the line will be run by setting off at the point of departure on the principal meridian a tangent to the parallel of latitude, which will be a line falling at right angles to said meridian. The line thus determined will be prolonged by two back and two fore sights at each setting of the instru- ment, turning the horizontal limb 180° in azimuth between the observations. The survey will be con- SURVEYING THE PUBLIC LANDS. 205 tinued on this line for 12 miles, but the corners ■will be established at the proper points by ofF-sets northerly from said line, at the end of each half mile. Principal meridian. The principal meridian shall be extended north and south from the initial point, by the use of solar instruments or transits. Where solar instruments are used, the line will be run in the same manner as prescribed for running the base line by solar instruments. Where transits are used, observations upon the polar star must be taken within each 12 miles of line run. Two sets of chainmen operating independently of each other must be employed. Standard parallels. Standard parallels which are also called correction lines, shall be extended east and west from the principal meridian, at inter- vals of every 21 miles north and south of the base line, in the same manner as prescribed for running the base line. Guide meridians shall be extended north and south from the base line at intervals of every 21 miles east and west from the principal meridian, in the same manner as prescribed for running the principal meridian. 206 THE surveyor's guide. Exterior of township lines. The east and west boundaries of townships are always to be run from south to north on a true meridian line ; and the north and south boundaries are to be run from east to west, or from west to east, on a random or trial line, and corrected back to a true line. The dis- tance north or south of the township, corner to be closed upon, from the point of intersection of these random lines with the east or west boundary of the township, must be carefully measured and noted. Should it happen, however, that such random line should fall short, or overrun in length, or intersect the east or west boundary more than three chains^ distance from the township corner thereon, as compared with the corresponding boundary on the south (due allowance being made for con- vergency) the line, and if necessary the entire exterior boundaries of the township, must be retraced, so as to discover and correct the error. In running random lines, temporary cor- ners are to be set at each 40 and 80 chains, and permanent corners established upon the true line as corrected back, in accordance with instructions, throwing the excess or deficiency on the west SURVEYING THE PUBLIC LANDS. 207 half as prescribed by law. Permanent corners are to be established in accordance with instructions on the east and west township boundaries at the the time they are run. Whenever practicable, the township lines within these tracts of 24 miles miles square must be surveyed in regular order from south to north, i. e., the exterior boundaries of the township in any one range lying immediately north of the south boundary of such tract of 24 miles square must first be surveyed, and the exte- riors of the other three townships in said range extended therefrom in regular order from south to north, and it is preferable to first survey the entire range of townships in such tract adjoining the east boundary or the west boundary, and the other three ranges in regalar sequence. In cases, however, where the character of the land is such that this rule cannot be complied with, the following will be observed : In extending the south or north boundaries of a township to the west, where the southwest or north- west corners cannot be established in the regular way by running a north and south line, such boun- daries will be run west on a true line, allowing; for 208 THE surveyor's guide. convergency on the west half mile ; and from the township corner established at the end of such boundary,, the west boundary will be run north or south as the case may be. In extending south or north boundaries of a township to the east, where the southeast or northeast corner cannot be estab- lished in the regular way, the same rule will be observed, except that such boundaries will be run east on a true line, and the east boundary run north or south, as the case may be. Il^DEX. ACRE, content of the, 42 Angle-mirror, its construc- tion and mode of op- eration, 53-58 Angle-mirror, testing, the, 64, 65 to erect a perpendicular with the assistance of the, 59, 60 to find an intermediate point in a straight line, with the assist- ance of the, 62-64 to let fall a perpendicu- lar, with the assist- ance of the, 60-62 use of the, 58, 59 Angle of incidence, 55 reflection, 55 to determine the, made by two stations of a survey, 136-143 Angles, right, examples of the application of setting out, 69-75 instruments for setting out, 46-75 prism for, and its use, 65, 66 to set out with the tape-measure, 67-69 Areas, 82-106 Axis of incidence, 55 (2 BASE line, the, of the public survey, 204, 205 Blazing, 199 Boundary corner, 201 stones, 20 Burt's improved solar compass, CALCULATION, methods of, 141-143 Calculation, new method of, 144- 147 Noble's method of, 147-153 Chain, levelling the, and plumb- ing the pins, 198, 199 Chaining, process of, 197-199 Chains, 39-41 Grumman's patent, 40, 41 metre, 41 vara, 41 Circle, to find the area of a, 105 Cities, posts used, in taking measurements in, 20, 21 Compass, to survey with the, through any mine, 153, 154 Corner, boundary, 201 Corners, establishment of, 201,202 points for perpetuating, 201 202 Cross statf-head, and its use, 47- 51 testing the, 51-53 9) 210 INDEX. Cross staff-head, or surveyor's cross, and its use, 46- 51 Curves on railroads, to inflect in, 168-179 DEPARTURE and latitude, tables of, 180-191 Departure and latitude, to find the error in the difference of, 120-126 Distance, determination of a, when measuring is pre- vented by an obstacle, 33-39 example in measuring, 133-135 guessing or judging, 45 to find the, when the straight line ;ontinues across a river, 36, 37 Distances, instrument for finding, without calculation, 127- 133 instruments for measuring, and their use, 20-45 various methods of meas- uring, 43-45 Diurnal variation, mean, for every month in the year, 153 ELLIPSIS, to find the area of an, 105 Examples of the application of setting out right angles, 69-75 FARM, triangular, division of a, 97-99 I'ield-book distances, measured with a 100 feet chain, 165 Field-book, methods of keeping, 157-161 Field, many-sided, to survey a, 77-79 or lot, to find the area of a, which is found to be the frustum or zone of a par- abola, 105, 106 three-sided, to survey a, 76, 77 Forest, determination of the di- rection of a straight line through a, 38, 39 or pond, to find the length when the straight line passes through a, 34-36 GRUMMAN'S patent chains, 40, 41 Guide meridians, 205 Gunter's link, 39 INCIDENCE, angle of, 55 j^ Incidence, axis of, 55 Instrument for finding distances, without calculation, and its use, 127-133 for measuring a map, 126, 127 Instruments for measuring dis- tances, and their use, 20-45 for setting out right angles, and their use, 46-75 LAND, how to measure a tract of, by measuring a base-line through it, 115-120 Land measurement, unit in, 42 reduction of the content of a piece of, to acres, rods and perches, 42, 43 INDEX. 211 Latitude and departure, tables of, 180-191 to find the error in the differ- ence of, 120- 126 Laying out or lotting towns, etc., 156, 157 Legs of stations, rules for altering the, in the correcting of sur- veys, 124-126 Length, to find the, when the straight line passes through a pond or a forest, 34-36 Levelling, 163-168 Light, laws of the reflection of, by plane mirrors, 54 Line, not horizontal, to measure a, with the measuring rods, 29, 30 straight, determination of a, 18 determination of the di- rection of a, through a forest, 38, 39 establishment of a, 21 length of a, 27 measuring a, with the measuring rods, 28, 29 to determine an inter- mediate point in a, 50,. 51 to determine a point in, a, when it is impossi- ble to take position in its prolongation, 25- 27 to erect a perpendicular at a point in a, 49 tO) find an intermediate point in a, with the as- sistance of the angle- mirror, 62-64 Line, straight, to find a point in a, 23-25 to find a point in the prolongation of a, 21- 23 to let fall a perpendicu- lar to a, from a point, 49, 50 trees or sight trees, 199 Lines, marking, 199, 200 township, exterior of, 206- 208 trial or random, 200 Lot or field, to find the area of a, which is found to be the frus- tum or zone of parabola, 105, 106 Lotting or laying out towns, etc., 156, 157 MAP, instrument for.measUjriag a, 126, 127 Marking lines, 199, 200.^ pins, 32, 41, 42 Meander corners,. 202 Meandering,, definition of, 202 instruction for, 202-204 Mearings,. tracing of old, 161-163 Measure, tape, and its appurten- ances, 3.0, 31 Measurements, noting the results of the, 79, 80 Measuring distances, various methods of, 43-45 rods, and their use, 28-30 Meridian, principal, 205 Meridians, guide, 205 Metre chains, 41 Mile, square, content of a, 42 Mine, to survey with the compass through any, 153, 154 Mirrors, plane, laws of the reflec- tion of light by, 54 212 N OBLE'S method of calcula- tion, 147-153 PACING and pedometer, 43-45 Parabola, to find the area of a, 105 to find the area of a seg- ment of a, 105 Parallelogram, to find the area of a, 83-85 Parallels, standard, 205 establishment of, 196 Pedometer and pacing, 43-45 Pedometers, wheel, 143 Perch, 42 Perpendicular, to erect a, at a point in a straight line, 49 to erect a, with the as- sistance of the angle- mirror, 59, 60 to let fall a, to a straight line from a point, 49, 50 to let fall a, with the as- sistance of the angle- mirror, 60-62 Pins, marking 32, 41, 42 plumbing the, and levelling the chain, 198, 199 tally, 197 wooden, for marking, 21 Point, indication of a, in the field, 20 marking a, 20 to determine a, in a straight line, when it is impossible to take position in its pro- longation, 25-27 to find a, in a straight line, 23-25 Point, to find a, in the prolongation of a straight line, 21-23 Points, definition of the, to be de- termined in surveying, 18 Pole, 42 chains and links, reduction of, 81, 82 Poles for marking, 21 Polygon, survey of a, 77-79 Pond or forest, to find the length, when the straight line passes through a, 34-36 Principal meridian, 205 Prism for right angles, and its use, 65, 6G Problems, 81-106 Public lands of the United States, systems of rectangular survey- ing for surveying the, 194-208 UARTER section boundaries, 202 RAILROAD, to inflect in curves on, 168-179 Random or trial lines, 200 Range, constitution of a, 194 Rectangular surveying, systems of, for surveying the public lands of the United States, 194-208 Reflection, angle of, 55 Right angles, examples of the ap- plication of setting- out, 69-75 instruments for setting- out, 46-75 prism for, and its use, 65, 66 to set out with the tape measure, 67-69 213 River, to find the distance when the straight line continues across a, 36, 37 Road, to find the cuttings and fillings of a, 167, 168 to lay out a, on a regular grade up a hill, 166-168 Roads, to inflect in curves on, 168-179 Rod, square, 42 Rods, measuring, and their use, 28-30 Rood, 42 SECTION boundaries, 202 Sections, 194 Sections, content of, 195, 196 Sight trees or line trees, 199 Signals, 24, 25 Solar compass, Burt's improved, 196. Square, to find the content of a, 82, 83 Stakes, 31 Standard parallels, 205 Stones, boundary, 20 Surface, plane, determination of a, 18 Survey, extent of a, 17 map of a. 18, 19 of a polygon, 77-79 of a three-sided field, 76,77 of smaller tracts, 76-80 plotting the notes of a, 155, 156 public, the lines of the, 204, 205 to determine the angle made by two stations oi a, 136-143 to, with the compass through any mine, 153, 154 Surveying, definition of the points to be determined in, 18 object of, 17 rectangular, systems of, for surveying the public lands of the United States, 194-208 trigonometrical, 107-191 Surveyor's angle-mirror, its con- struction and mode ' of operation, 53-58 chain, ordinary, 40 cross or cross staff, and its use, 46-51 Surveys, rules for altering the legs of stations in the correcting of, 124-126 tables of, 192, 193 TABLE of errors in links and decimals, 124 Tables of latitude and departure, 180-191 of surveys, 192, 193 Tally pins, 197 Tape measure, measuring with the, 31-33 the, and its appurten- ances, 30, 31 to set out right angles with the, G7-69 Township boundaries, corners for, 201 lines, exterior of, 206-208 measurement of, 197 Towns, lotting or laying out of, 156, 157 Townships, 194 Tracts, smaller, survey of, 76-80 Trapezium, to find the area of a, 101-105 to find the content of a, 86 214 Trapezoid, rule to find the area of a, 99 Trees, sight or line, 199 Trial or random lines, 200 Triangle, rules for finding the content of a, 8'7-99 the, considered as spherical, 113-115 to determine the area of a. 109-113 Triangulation, 107-109 Trigonometrical surveying, 107- 191 UNITED STATES, systems of rectangular surveying for surveying the public lands of the, 194-208 V ARA, chains, 41 Variation, mean diurnal for every month in the year, 153 TTTHEEL pedometers, 43 0-A.T-A.ILiOC3-TJE OF practical and Scientific Boo^^ PUBLISHED BY Henry Carey Baird & Co. INDUSTRIAL PUBLISHERS. BOOKSELLERS AND IMPORTERS. 810 Walnut Street, Philadelphia. ■ Any of the Books comprised in this Catalogue will he sent hy mail, fipee ti postage, to any address in the world, at the pnhlication prices, • A Descriptive Catalogue, 84 pages, 8vo,, will he sent free and free of postage, to any one in any part of the world, who will furnish his address, ■ Where not otherwise stated, all of the Books in this Catalogue are bound in muslin. AMATEUR MECHANICS' WORKSHOP: A treatise containing plain and concise directions for the manipula- tion of Wood and Metals, including Casting, Forging, Brazing, Soldering and Carpentry. By the author of the " Lathe and Its Uses." Seventh edition. Illustrated. 8vo. . . . ^3.00 ANDRES.— A Practical Treatise on the Fabrication of Volatile and Fat Varnishes, Lacquers, Siccatives and Sealing Waxes. From the German of Erwin Andres, Manufacturer of Varnishe* and Lacquers. With additions on the Manufacture and Application of Varnishes, Stains for Wood, Horn, Ivory, Bone and Leather. From the German of Dr. Emil Winckler and Louis E. Andes. The whole translated and edited by William T. Brannt. With 1 1 illustrations. i2mo. ^3.50 ARLOT.— A Complete Guide for Coach Painters : Translated from the French of M. Arlot, Coach Painter, for eleven years Foreman of Painting to M. Eherler, Coach Maker, Paris. By A. A. Fesquet, Chemist and Engineer. To which is added an Appendix, containing Information respecting the Materials and the Practice of Coach and Car Painting and Varnishing in the United States and Great Britain. i2mo. . . . $1.25 a) HENRY CAREY BAIRD & CO.'S CATALOGUE. ARMENGAUD, AMOROUX, AND JOHNSON.— The Practi- cal Draughtsman's Book of Industrial Design, and Ma' chinist's and Engineer's Drawing Companion : Forming a Complete Course of Mechanical Engineering and Archi- tectural Drawing. From the French of M. Armengaud the elder, Prof, of Design in the Conservatoire of Arts and Industry, Paris, and MM. Armengaud the younger, and Amoroux, Civil Engineers. Re- written and arranged with additional matter and plates, selections from and examples of the most useful and generally employed mechanism of the day. By William Johnson, Assoc. Inst. C. E. Illustrated by fifty folio steel plates, and fifty wood-cuts. A new edition, 4to,, half morocco ......... ^lo.oo ARMSTRONG.— The Construction and Management of Steam Boilers : By R. Armstrong, C. E. With an Appendix by Robert Mallet, C. E., F. R. S. Seventh Edition. Illustrated, i vol. l2mo. 75 aRROWSMITH.— Paper-Hanger's Companion : A Treatise in which the Practical Operations of the Trade are Systematically laid down : with Copious Directions Preparatory to Papering; Preventives against the Effect of Damp on Walls; the various Cements and Pastes Adapted to the Several Purposes oi the Trade ; Observations and Directions for the Panelling and Ornamenting of Rooms, etc. By James Arrowsmith. i2mo., cloth $1.2$ ASHTON. — The Theory and Practice of the Art of Designing Fancy Cotton and Woollen Cloths from Sample : Giving full instructions for reducing drafts, as well as the methods of spooling and making out harness for cross drafts and finding any re- quired reed; with calculations and tables of yarn. By Frederic T. AsHTON, Designer, West Pittsfield, Mass. With fifty-two illustrations. 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BAIRD. — Standard Wages Computing Tables : An Improvement in all former Methods of Computation, so arranged that wages for days, hours, or fractions of liours, at a specified rate per day or hour, may be ascertained at a glance. By T. Spanglek Baird. Oblong folio ^S-OO SAKER. — Long-Span Railway Bridges : Comprising Investigations of the Comparative Theoretical and Practical Advantages of the various Adopted or Proposed Type Systems of Construction ; with numerous Formulae and Tables. By B. Baker. i2mo. $1.30 BAKER. — The Mathematical Theory of the Steam-Engine ; With Rules at length, and Examples worked out for the use of Practical Men. By T. Baker, C. E., with numerous Diagrams. Sixth Edition, Revised by Prof. J. R. Young. i2mo. . 75 BARLOW. — The History and Principles of Weaving, by Hand and by Power : Reprinted, with Considerable Additions, from " Engineering," with a chapter on Lace-making Machinery, reprinted from the Journal of the " Society of Arts." By Alfred Barlow. 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By James Callingham. i2mo ;?i.5o CAMPIN. — A Practical Treatise on Mechanical Engineering: Comprising Metallurgy, Moulding, Casting, Forging, Tools, Work.- shop Machinery, Mechanical Manipulation, Manufacture of Steam- Engines, etc. With an Appendix on the Analysis of Iron and Iron Ores. By Francis Campin, C. E. To which are added. Observations on the Construction of Steam Boilers, and Remarks upon Furnaces used for Smoke Prevention ; with a Chapter on Explosions. By R. Armstrong, C. E., and John Bourne. Rules for Calculating th« Change Wheels for Screws on a Turning Lathe, and for a Wheel* cutting Machine. By J. La Nicca. Management of Steel, Includ- ing Forging, Hardening, Tempering, Annealing, Shrinking an^ Expansdim ; and the Case-hardening of Iron. By G. Ede. 8vo. Illustrated with twenty-nine plates and 100 wood engravings $S-Oa HENRY CAREY BAIRD & CO.'S CATALOGUE. CAREY.— A Memoir of Henry C. Carey. By Dr. Wm. Elder. With a portrait. 8vo., cloth . . 75 CAREY.— The Works of Henry C. Carey : Harmony of Interests : Agricultural, Manufacturing and Commer. cial. 8vo. . . ^1.50 Manual of Social Science. Condensed from Carey's " Principles of Social Science." By Kate McKean. i vol. i2mo. . $2.25 Miscellaneous Works, With a Portrait. 2 vols. 8vo. ;^io.oo Past, Present and Future. Svo ;?3.5o Principles of Social Science. 3 volumes, Svo. . . ^7.50 The Slave-Trade, Domestic and Foreign; Why it Exists, and How it may be Extinguished {1853). Svo. . . . |2.oo The Unity of Law : As Exhibited in the Relations of Physical, Social, Mental and Moral Science {1872). Svo. . . $3.S<^ CLARK. — Tramv/ays, their Construction and Working: Embracing a Comprehensive History of the System. With an ex' haustive analysis of the various modes of traction, including horse- power, steam, heated water and compressed air; a description of the varieties of Rolling stock, and ample details of cost and working ex- penses. By D. KiNNEAR Clark. Illustrated by over 200 wood engravings, and thirteen folding plates. 2 vols. Svo. . ^12.50 COLBURN.— The Locomotive Engine : Including a Description of its Structure, Rules for Estimating its Capabilities, and Practical Observations on its Construction and Man- agement. By Zerah Colburn. Illustrated. i2mo. . ^I.oa COLLENS.— The Eden of Labor ; or, the Christian Utopia. By T. Wharton Collens, author. of " Humanics," "The Historj of Charity," etc. l2mo. Paper cover, $1.00; Cloth . $1.2^ COOLEY. — A Complete Practical Treatise on Perfumery : Being a Hand-book of Perfumes, Cosmetics and other Toilet Articles. With a Comprehensiv» Collection of Formulae. By Arnold J. COOLEY. i2mo i^i.5& COOPER.— A Treatise on the use of Belting for tbe Trans* mission of Power. With numerous illustrations of approved and actual methods of ar- ranging Main Driving and Quarter Twist Belts, and of Belt Fasten ings. Examples and Rules in great number for exhibiting and cal- culating the size and driving power of Belts. Plain, Particular and Practical Directions for the Treatment, Care and Management o'' Belts. Descriptions of many varieties of Beltings, together witn chapters on the Transmission of Power by Ropes; by Iron and Wood Frictional Gearing; on the Strength of Belting Leather; and on the Experimental Investigations of Morin, Briggs, and others. Bf John H. Cooper, M. E. Svo $3-S^ CRAIK.— The Practical American Millwright and Miller. By David Craik, Millwright. Illustrated by numerous wood en- gravings and two folding plates. Svo $$,os HENRY CAREY BAIRD & CO.'S CATALOGUE. CREW. — A Practical Treatise on Petroleum : Comprising its Origin, Geology, Geographical Distribution, History, Chemistry, Mining, Technology, Uses and Transportation. Together with a Description of Gas Wells, the Application of Gas as Fuel, etc. By Benjamin J. Crew. With an Appendix on the Product and Exhaustion of the Oil Regions, and the Geology of Natural Gas in Pennsylvania and New York. By Charles A. Ashburner, M. S., Geologist in Charge Pennsylvania Survey, Philadelphia. Illustrated by 70 engravings. 8vo. 508 pages . . . . ^i^.oo CROSS.— The Cotton Yarn Spinner: Showing how the Preparation should be arranged for Different Counts of Yarns by a System more uniform than has hitherto been practiced; by having a Standard Schedule from which we make all our Changes. By Richard Cross. 122 pp. i2mo. . 75 CRISTIANI. — A Technical Treatise on Soap and Candles : With a Glance at the Industry of Fats and Oils. By R. S. Cris- TIANI, Chemist. Author of " Perfumery and Kindred Arts." Illus- trated by 176 engravings. 581 pages, 8vo. . . . ^15.00 CKISTIANI.— Perfumery and Kindred Arts: A Comprehensive Treatise on Perfumery, containing a History of Perfumes from the remotest ages to the present time. A complete detailed description of the various Materials and Apparatus used in the Perfumer's Art, with thorough Practical Instruction and careful Formulae, and advice for the fabrication of all known preparations of the day, including Essences, Tinctures, Extracts, Spirits, Waters, Vinegars, Pomades, Powders, Paints, Oils, Emulsions, Cosmetics, Infusions, Pastilles, Tooth Powders and Washes, Cachous, Hair Dyes, Sachets, Essential Oils, Flavoring Extracts, etc. ; and full details for making and manipulating Fancy Toilet Soaps, Shaving Creams, etc., by new and improved methods. With an Appendix giving hints and advice for making and fermenting Domestic Wines, Cordials, Liquors, Candies, Jellies, Syrups, Colors, etc., and for Perfuming and Flavor- ing Segars, Snuff and Tobacco, and Miscellaneous Receipts foi various useful Analogous Articles. By R. S. Cristiani, Con- sulting Chemist and Perfumer, Philadelphia. 8vo. . . ^10.00 DAVIDSON.— A Practical Manual of House Painting, Grain- ing, Marbling, and Sign- Writing : Containing full information on the processes of House Painting in Oil and Distemper, the Formation of Letters and Practice of Sign- Writing, the Principles of Decorative Art, a Course of Elementary Drawing for House Painters, Writers, etc., and a Collection of Useful Receipts. With nine colored illustrations of Woods and Marbles, and numerous wood engravings. By Ellis A. Davidson. i2mo. ^3.00 DAVIES. — A Treatise on Earthy and Other Minerals and Mining : By D. C. Davies, F. G. S., Mining Engineer, etc. Illustrated by 76 Eugraviugs, l2mo 15-°*^ HENRY CAREY BAIRD & CO.'S CATALOGUE. DAVIES. — A Treatise on Metalliferous Minerals and Mining: By D. C. Davies, F. G. S.; Mining Engineer, Examiner of MineSr Quarries and Collieries. Illustrated by 148 engravings of Geological Formations, Mining Operations and Machinery, drawn from thft practice of all parts of the world. 2d Edition, i2mo., 450 pages ^5.0d DAVIES.— A Treatise on Slate and Slate Quarrying: Scientific, Practical and Commercial. By D. C. Davies, F. G. S., Mining Engineer, etc. With numerous illustrations and folding plate*. i3mo ^2.03 PAVIS. — A Treatise on Steam-Boiler Incrustation and Meth4 ods for Preventing Corrosion and the Formation of Scale .' By Charles T. Davis. Illustrated by 65 engravings. 8vo. ;?i.5o DAVIS.— The Manufacture of Paper: Being a Description of the various Processes for the Fabrication, Coloring and Finishing of every kind of Paper, Including the Dif- ferent Raw Materials and the Methods for Determining their Values, the Tools, Machines and Practical Details connected with an intelli- gent and a profitable prosecution of the art, with special reference to the best American Practice. To which are added a History of Pa- per, complete Lists of Paper-Making Materials, List of American Machines, Tools and Processes used in treating the Raw Materials, and in Making, Coloring and Finishing Paper. By Charles T. Davis. Illustrated by 156 engravings. 608 pages, 8vo. ;!>6.oo OAVIS.— The Manufacture of Leather: Being a description of all of the Processes for the Tanning, Tawing, Currying, Finishing and Dyeing of every kind of Leather ; including the various Raw Materials and the Methods for Determining their Values; the Tools, Machines, and all Details of Importance con- nected with an Intelligent an-d Profitable Prosecution of the Art, with Special Reference to the Best American Practice. To which are added Complete Lists of all American Patents for Materials, Pro- cesses, Tools, and Machines for Tanning, Currying, etc. By Charles Thomas Davis. Illustrated by 302 engravings and 12 Samples of Dyed Leathers. One vol., Svo., 824 pages . . . $10.00 DAWIDOWSKY— BRANNT.— A Practical Treatise on the Raw Materials and Fabrication of Glue, Gelatine, Gelatine Veneers and Foils, Isinglass, Cements, Pastes, Mucilages, etc. : Eased upon Actual Experience. By F. Dawidowsky, Technical Chemist. Translated from the German, with extensive additions, including a description of the most Recent Ameriean Processes, by William T. Brannt, Graduate of the Royal Agricultural College of Eldena, Prussia. 35 Engravings. l2mo. . . . |!2.SO DE GRAFF. — The Geometrical Stair-Builders' Guide: Being a Plain Practical System of Hand-Railing, embracing all it3 necessary Details, and Geometrically Illustrated by twenty-two Steel Engravings; together with the use of the most approved pnnciplei til Practical Geometry. By Simon De Graff, Architect. 4X0. HENRi CARE\ Eaiki> 6: CO'.S CATALOd'Jxi-. DE KONINCK— DIETZ.— A Practical Manual of Chemical Analysis and Assaying : As applied to the Manufacture of Iron from its Ores, and to Cast Iron, Wrought Iron, and Steel, as found in Commerce. By L. L. De KoNiNCK, Dr. Sc, and E. Dietz, Engineer. Edited with Notes, by Robert Mallet, F. R. S., F. S. G., M. I. C. E., etc. American Edition, Edited with Notes and an Appendix on Iron Ores, by A. A. Fesquet, Chemist and Engineer. i2mo. . . . ^^2.50 DUNCAN.— Practical Surveyor's Guide: ^ Containing the necessary information to make any person of com- mon capacity, a finished land surveyor without the aid of a teacher By Andrew Duncan. Illustrated. lamo. . . . ^1.25 DUPLAIS. — A Treatise on the Manufacture and Distillation of Alcoholic Liquors : Comprising Accurate and Complete Details in Regard to Alcohol from Wine, Molasses, Beets, Grain, Rice, Potatoes, Sorghum, Aspho del, Fruits, etc.; with the Distillation and Rectification of Brandy, Whiskey, Rum, Gin, Swiss Absinthe, etc., the Preparation of Aro- matic Waters, Volatile Oils or Essences, Sugars, Syrups, Aromatic Tinctures, Liqueurs, Cordial Wines, Effervescing Wines, etc., the Ageing of Brandy and the improvement of Spirits, with Copioaa Directions and Tables for Testing and Reducing Spirituous Liquors, etc., etc. Translated and Edited from the French of MM. DuPLAis, Aine et Jeune. By M. McKennie, M. D. To which are added the United States Internal Revenue Regulations for the Assessment and Collection of Taxes on Distilled Spirits. Illustrated by fourteen folding plates and several wood engravings. 743 pp. 8vo. $10 oa aUSSAaCE.— Practical Treatiseon the Fabrication of Matches, Gun Cotton, and Fulminating Powder. By Professor H. DussAUCE. i2mo $300 OYER AND COLOR-MAKER'S COMPANION: Containing upwards of two hundred Receipts for making Colors, on the most approved principles, for all the various styles and fabrics now in existence; with the Scouring Process, and plain Directions for Preparing, Washing-ofF, and Finishing the Goods. i2mo. ^i 25 aDWARDS. — A Catechism of the Marine Steam-Engine, For the use of Engineers, Firemen, and Mechanics. A Practical Work for Practical Men. By Emory Edwards, Mechanical Engi- neer. Illustrated by sixty-three Engravings, including examples of the most modern Engines. Third edition, thoroughly revised, with much additional matter. 12 mo. 414 pages . . . ^2 GO CDWARDS. — Modern American Locomotive Engines, Their Design, Construction and Management. By Emory EdwaRDS. Illustrated i2mo ;^2.oo SDWARDS.— The American Steam Engineer: Theoretical and Practical, with examples of the latest and most ap- proved American practice in the design and construction of Steam Engines and Boilers. For the use of engineers, machinists, boiler- i)«'\kers, and engineering studenis. By Emory Edwards. Fully 41usirated, 419 pages. i2mo. .... ;?2.50 ra HENRY CAREY BAIRD & CO.'S CATALOGUE. EDWARDS. — Modern American Marine Engines, Boilers, an4 Screw Propellers, Their Design and Construction. Showing the Present Praaice of the most Eminent Engineers and Marine Engine Builders in the United States. Illustrated by 30 large and elaborate plates. 4to. ^S.oa EDWARDS.— The Practical Steam Engineer's Guide In the Design, Construction, and Management of American Stationary, Portable, and Steam Fire- Engines, Steam Pumps, Boilers, Injectors, Governors, Indicators, Pistons and Rings, Safety Valves and Steam Gauges. For the use of Engineers, Firemen, and Steam Users. By Emory Edwards. Illustrated by 119 engravings. 420 pages. i2mo J? 2 50 EISSLER.— The Metallurgy of Gold: A Practical Treaiise 011 the Metallurgical Treatment of Gold-Bear- ing Ores, including the Processes of Concentration and Chlorination, and the Assaying, Melting, and Refining of Gold. By M. Eissler. "With 132 Illustrations. l2mo $3-SO EISSLER.— The Metallurgy of Silver : A Practical Treatise on the Amalgamation, Roasting, and Lixiviation of Silver Ores, including the Assaying, Melting, and Refining of Silver Bullion. By M. Eissler. 124 Illustrations. 336 pp. i2mo M-2S ELDER, — Conversations on the Principal Subjects of Political Economy. By Dr. William Elder. 8vo. ... . . . ^2.50 ELDER.— Questions of the Day, Economic and Social. By Dr. William Elder. 8vo. . ^3.00 ERNL— Mineralogy Simplified. Easy Methods of Determining and Classifying Minerals, including Ores, by means of the Blowiiipe, and by Humid Chemical Analysis, based on Professor von Kobell's Tables for the Determination of Minerals, with an Introduction to Modern Chemistry. By Henry Erni, A.m., M.D., Professor of Chemistry. Second Edition, rewritten, enlarged and improved. i2mo. ..... $3°^ FAIRBAIRN.— The Prmciples of Mechanism and Machinery of Transmission • Comprising the Prmciples of Mechanism, Wheels, and Pulleys, Strength and Proportions of Shafts, Coupling of Shafts, and Engag- ing and Disengaging Gear. By SiR William Fairbairn, Bart C. E. Beautifully illustrated by over 1 50 wood-cuts. In one volume. i2mo .....•••• $2.^C FLEMING.— Narrow Gauge Railways in America. A Sketch of their Rise, Progress, and Success. Valuable Statistics as to Grades, Curves, Weight of Rail, Locomotives, Cars, etc. By Howard Fleming. Illustrated, 8vo $1 00 FORSYTH.— Book of Designs for Headstones, Mural, and other Monuments : Contaming 78 Designs. By James Forsyth. With an Introduction "hy Charles Boutell, M. A. 4 to., cloth . . - $5 ^ HENRY CAREY BAIRD & CO.'S CATALOGUE. 13 FRANKEL— HUTTER.— A Practical Treatise on the Manu» facture of Starch, Glucose, Starch-Sugar, and Dextrine: Based on the German of Ladislaus Von Wagner, Professor in the Royal Technical High School, Buda-Pest, Hungary, and other authorities. By Julius Frankel, Graduate of the Polytechnic School of Hanover. Edited by Robert Hutter, Chemist, Practical Manufacturer of Starch-Sugar. Illustrated by 58 engravings, cover- ing every branch of the subject, including examples of the most Recent and Best American Machinery. 8vo., 344 pp. . ^3.50 GARDNER.— The Painter's Encyclopaedia: Containing Definitions of all Important Words in the Art of Plain and Artistic Painting, with Details of Practice in Coach, Carriage, Railway Car, House, Sign, and Ornamental Painting, including Graining, Marbling, Staining. Varnishing, Polishing, Lettering, Stenciling, Gilding, Bronzing, etc. By Franklin B. Gardner. 158 Illustrations. i2mo. 427 pp ;^2.oa GARDNER.— Everybody's Paint Book: A Complete Guide to the Art of Outdoor and Indoor Painting, De- signed for the Special Use of those who wish to do their own work, and consisting of Practical Lessons in Plain Painting, Varnishing, Polishing, Staining, Paper Hanging, Kalsomining, etc., as well as Directions for Renovating Furniture, and Hints on Artistic Work for Home Decoration. 38 Illustrations. i2mo., 183 pp. . ^i.oo GEE. — The Goldsmith's Handbook : Containing full instructions for the Alloying and Working of Gold, including the Art of Alloying, Melting, Reducing, Coloring, Col- lecting, and Refining: the Processes of Manipulation, Recovery of Waste; Chemical and Physical Properties of Gold; with a New System of Mixing its Alloys; Solders, Enamels, and other Useful Rules and Recipes. By George E. Gee. i2mo. . . ^1-75 GEE. — The Silversmith's Handbook : Containing full instructions for the Alloying and Working of Silver, including the different modes of Refining atld Melting the Metal ; its Solders ; the Preparation of Imitation Alloys ; Methods of Manipula- tion ; Prevention of Waste ; Instructions for Improving and Finishing the Surface of the Work ; together with other Useful Information and Memoranda. By George E. Gee. Illustrated. i2mo. ^i-7S GOTHIC ALBUM FOR CABINET-MAKERS: Designs for Gothic Furniture. Twenty-three plates. Oblong ;^2.oo GRANT.— A Handbook on the Teeth of Gears : Their Curves, Properties, and Practical Construction, By George B. Grant. Illustrated. Third Edition, enlarged. 8vo, ^1.50 GREENWOOD.— Steel and Iron: Comprising the Practice and Theory of the Several Methods Pur- sued in their Manufacture, and of their Treatment in the Rolling- Mills, the Forge, and the Foundry. By William Henry Green- wood, F. C. S. With 97 Diagrams, 536 pages. i2mo. ;SS2.ao t4 HENRY eAREY BAIRD & CO.'S CATALOGUE. GREGORY, — Mathematics for Practical Men : Adapted to the Pursuits of Surveyors, Architects, Mechanics, and Civil Engineers. By Olinthus Gregory. 8vo., plates $3.00 GRIMSHAW.— Saws : The History, Development, Action, Classification, and Comparison of Saws of all kinds. IVM Copious Appendices. Giving the details of Manufacture, Filing, Setting, Gumming, etc. Care and Use of Saws; Tables of Gauges; Capacities of Saw-Mills; List of Saw- Patents, and other valuable information. By Robert Grimshaw. Second and greatly enlarged edition, with Supplement, and 354 Illustrations. Quarto ^S-OO GRISWOLD. — Railroad Engineer's Pocket Companion for tha Field : Comprising Rules for Calculating Deflection Distances and Angles, Tangential Distances and Angles, and all Necessary Tables for En- gineers; also the Art of Levelling from Preliminary Survey to the Construction of Railroads, intended Expressly for the Young En- gineer, together with Numerous Valuable Rules and Examples. By W. Griswold. i2mo., tucks ^i-75 GRUNER. — Studies of Blast Furnace Phenomena: By M. L; Gruner, President of the General Council of Mines ol France, and lately Professor of Metallurgy at the Ecole des Mines. Translated, with the author's sanction, with an Appendix, by L. D. B. Gordon, F. R. S. E., F. G. S. 8vo. , . . ^^2.50 Hand-Book of Useful Tables for the Lumberman, Farmer and Mechanic: Containing Accurate Tables of Logs Reduced to Inch Board Meas* ure. Plank, Scantling and Timber Measure; Wages and Rent, by Week or Month; Capacity of Granaries, Bins and Cisterns; Land Measure, Interest Tables, with Directions for Finding the Interest on any sum at 4, 5, 6, 7 and 8 per cent., and many other Useful Tables. 32 mo., boards. 186 pages .,,... .25 HASERICK.— The Secrets of the Art of Dyeing Wool, Cotton, and Linen, Including Bleaching and Coloring Wool and Cotton Hosiery and Random Yarns. A Treatise based on Economy and Practice. By E. C. Haserick. Illustrated by 323 Dyed Patterns of the Yami or Fabrics. 8vo %1-'^Q HATS AND FELTING: A Practical Treatise on their Manufacture. By a Practical Hatter. Illustrated by Drawings of Machinery, etc. 8vo. . . ^1.25 H OFFER. — A Practical Treatise on Caoutchouc and Gutta Percha, Comprising the Properties of the Raw Materials, and the manner nl" Mixing and Working them ; with the Fabrication of Vulcanized and Hard Rubbers, Caoutchouc and Gutta Pescha Compositions, Wate» proof Substances, Elastic Titesues, the Utilization of Waste, etc., etc From the German of Raimund Hoffer, By W. T. Brannt. Illustrated i2mo ^2.50 HENRY CAREY BAIRD & CO.'S CATALOGUE. 15 HAUPT— RHAWN.— A Move for Better Roads : Essays on Road-making and Maintenance and Road Laws, for which Prizes or Honorable Mention were Awarded through the University of Pennsylvania by a Committee of Citizens of Philadel- phia, with a Synopsis of other Contributions and a Review by the Secretary, Lewis M. Haupt, A. M., C. E. ; also an Introduction by William H. Rhawn, Chairman of the Committee. 319 pages. 8vo ^2.00 HUGHES. — American Miller and Millwright's Assistant: By William Carter Hughes. i2mo ^1.50 HULME. — Worked Examination Questions in Plane Geomet- rical Drawing : For the Use of Candidates for the Royal Military Academy, Wool- wich ; the Royal Military College, Sandhurst ; the Indian Civil En- gineering College, Cooper's Hdl ; Indian Public Works and Tele- graph Departments ; Royal Marine Li^ht Infantry; the Oxford and Cambridge Local Examinations, etc. By F. Edward Hulme, F. L. S., F. S. A., Art-Master Marlborough College. Illustrated by 300 examples. Small quarto .....<. ^2.50 JERVIS.— Railroad Property: A Treatise on the Construction and Management of Railways; designed to afford useful knowledge, in the popular style, to tha holders of this class of property ; as well as Railway Managers, Offi- cers, and Agents. By John B. Jervis, late Civil Engineer of the Hudson River Railroad, Croton Aqueduct, etc. i2mo., cloth ^2.0G KEENE.— A Hand-Book of Practical Gauging: For the Use of Beginners, to which is added a Chapter on Bistilla. tion, describing the process in operation at the Custom-House fol ascertaining the Strength of Wines. By James B. Keene, gf H. M. Customs. 8vo. $f-25 KELLEY. — Speeches, Addresses, and Letters on Industrial and Financial Questions : By Hon. William D. Kelley, M. C. 544 pages, 8vo. . $3.00 SELLOGG.— A New Monetary System : The only means of Securing the respective Rights of Labor and Property, and of Protesting the Public from Financial Revulsions.' By Edward Kellogg. Revised from his work on "Labor and ' other Capital." With numerous additions from his mpnuscvipt. Edited by MARY Kellogg Putnam, Fifth edition. To which i» added a Biographical Sketch of the Author. One volume, i2mo. Paper cover ^I.oo Bound in cloth 1.5a KEMLO.— Watch-Repairer's Hand-Book : Being a Complete Guide to the Young Beginner, in Taking Apart, Putting Together, and Thoroughly Cleaning the English Lever and other Foreign Watches, and all American Watches. By F. Kemlo, ■practical Watchmaker. With Illustrations. i2mo. . $l.2i i^ HENRY CARGY BAIRD & CO.'S CATALOGUE. KENTISH.— A Treatise on a Box of Instruments, And the Slide Rule ; with the Theory of Trigonometry and Log* rithms, including Practical Geometry, Surveying, Measuring of Tim. ber, Cask and Malt Gauging, Heights, and Distances. By Thomaj Kentish. In one volume. i2mo. . . . . Si 2=; KERL.— The Assayer's Manual: An Abridged Treatise on the Docimastic Examination of Ores, and Furnace and other Artificinl Products. By Bruno Kerl, Professor in the Royal School of Mines. Translated from the German by William T. Brannt. Second American edition, edited with Ex- tensive Additions by F. Lynwood Garrison, Member of the American Institute of Mining Engineers, etc. Illustrated by 87 en- gravings. 8vo. . ^3.00 KICK.— Flour Manufacture . A Treatise on Milling Science and Practice. By Frederick Kick, Imperial Regieningsmth, Professor of Mechanical Technology in the imperial German Polytechnic Institute, Prague. Translated from the second enlarged and revised edition with supplement by H. H. P. PowLES, Assoc. Memb. Institution of Civil Engineers. Illustrated with 28 Plates, and 167 Wood-cuts, 367 pages. 8vo. , ^10,00 KINGZETT,— The History, Products, and Processes of the Alkali Trade : Including the most Recent Improvements. By Charles Thomas KiNGZETT, Consulting Chemist. With 23 illustrations. 8vo. ^2.50 KIRK. — The Founding of Metals : A Practical Treatise on the Melting of Iron, with a Description of the Founding of Alloys ; also, of all the Metals and Mineral Substances used in the Art of Founding. Collected from original sources. By Edward Kirk, Practical Foundryman and Chemist. Illustrated, Third edition. 8vo. ^2.50 LANDRIN.— A Treatise on Steel : Comprising its Theory, Metallurgy, Properties, Practical Working, and Use. By M. H. C. Landrin, Jr., Civil Engineer. Translated from the French, with Notes, by A. A. Fesquet, Chemist and En- gineer. With an Appendix on the Bessemer and the Martin PrO' cesses for Manufacturing Steel, from the Report of Abram S. Hewitt United States Commissioner to the Universal Exposition, Paris, 1867J I2mo $3-OC LANGBEIN.— A Complete Treatise on the Electro-Deposition of Metals : Translated from the German, with Additions, by Wm. T. Brannt. 125 illustrations. 8vo ^4.00 LARDNER.— The Steam-Engine : Fer the Use of Beginners. Illustrated. lamo, ... 75 LEHNER.— The Manufacture of Ink: Comprising the Raw Materials, and the Preparation of Writing, Copying and Hektograph Inks, Safety Inks, Ink Extracts and Pow- ders, etc. Translated from the German of Sigmund Lehner, with additions by William T. Brannt. Illustrated. i2mo. ;g2.oo \ HENRY CAR^\ BAIRD & CO.'S CATALOGUE. 17 LARKIN. — The Practical Brass and Iron Founder's Guide: A Concise Treatise on Brass Founding, Moulding, the Metals and their Alloys, etc. ; to which are added Recent Improvements in the Manufacture of Iron, Steel by the Bessemer Process, etc., etc. By James Larkin, late Conductor of the Brass Foundry Department in Reany, Neafie & Co.'s Penn Works, Philadelphia. New edition, revised, with extensive additions. i2mo. . . . ^2,50 IJEROUX.—A Practical Treatise on the Manufacture of Worsteds and Carded Yarns : Comprising Practical Mechanics, with Rules and Calculations applied to Spinning; Sorting, Cleaning, and Scouring Wools; the English and French Methods of Combing, Drawing, and Spinning Worsteds, and Manufacturing Carded Yarns. Translated from the French of Charles Leroux, Mechanical Engineer and Superintendent of a Spinning-Mill, by Horatio Paine, M. D., and A, A. Fesquet, Chemist and Engineer. Illustrated by twelve large Plates. To which is added an Appendix, containing Extracts from the Reports of the International Jury, and of the Artisans selected by the Committee appointed by the Council of the Society of Arts, London, on Woolen and Worsted Machinery and Fabrics, as exhibited in the Paris Uni- versal Exposition, 1867. 8vo. . . . ■ . . ^5.00 LEFFEL. — The Construction of Mill-Dams : Comprising also the Building of Race and Reservoir Embankments and Head-Gates, the Measurement of Streams, Gauging of Water Supply, etc. By James Leffel & Co. Illustrated by 58 engravings. 8vo. ^2.50 LESLIE.— Complete Cookery: Directions for Cookery in its Various Branches. By Miss Leslie. Sixtieth thousand. Thoroughly revised, with the addition of New Receipts. i2mo IS1.50 LE VAN. — The Steam Engine and the Indicator : Their Origin and Progressive Development; including the Most Recent Examples of Steam and Gas Motors, together with the Indi- cator, its Principles, its Utility, and its Application. By William Barnet Le Van. Illustrated by 305 Engravings, chiefly of Indi- cator-Cards. 469 pp. Svo. ...... ^.00 tlEBER.— Assayer's Guide ; Or, Practical Directions to Assayers, Miners, and Smelters, for the Tests and Assays, by Heat and by Wet Processes, for the Ores of all the principal Metals, of Gold and Silver Coins and Alloys, and of Coal, etc. By Oscar M. Lieber. i2mo, . . . ^1.25 Lockwood's Dictionary of Terms : Used in the Practice of Mechanical Engineering, embracing those Current in the Drawing Office, Pattern Shop, Foundry, Fitting, Turn- ing, Smith's and Boiler Shops, etc., etc., comprising upwards of Six Thousand Definitions. Edited by a Foreman Pattern Maker, author of " Pattern Making." 417 pp. l2mo. . . . i^J.oo i8 HENRY CAREY BAIRD & CO.'S CATALOGUE. ' ' ' — — '■— - - ^ trUKIN.— Amongst Machines; Embracing Descriptions of the various Mechanical Appliances used in the Manufacture of Wood, Metal, and other Substances. J2mo. iUKIN.— The Boy Engineers : What They Did, and How They Did It. 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WARE.— The Sugar Beet. \ Including a History of the Beet Sugar Industry in Europe, Varietia of the Sugar Beet, Examination, Soils, Tillage, Seeds and Sowings Yield and Cost of Cultivation, Harvesting, Transportation, Conserva» tion, Feeding Qualities of the Beet and of the Pulp, etc. By Lewh S. Ware, C. E., M. E. Illustrated by ninety engravings. 8vo. WARN.— The Sheet-Metal Worker's Instructor: For Zinc, Sheet-Iron, Copper, and Tin-Plate Workers, etc. Contain- ing a selection of Geometrical Problems ; also. Practical and Simple Rules for Describing the various Patterns required in the different branches of the above Trades. By Reuben H. Warn, Practical Tin-Plate Worker. To vi^hich is added an Appendix, containing Instructions for Boiler-Making, Mensuration of Surfaces and Solids, Rules for Calculating the Weights of different Figures of Iron and Steel, Tables of the Weights of Iron, Steel, etc. Illustrated by thirty- two Plates and thirty-seven Wood Engravings. 8vo. . ^3.00 WARNER.— New Theorems, Tables, and Diagrams, for the Computation of Earth-work : Designed for the use of Engineers in Preliminary and Final Estimates, of Students in Engineering, and of Contractors and other non-profes. sional Computers. In two parts, with an Appendix. Part I. A Prac- tical Treatise; Part II. A Theoretical Treatise, and the Appendix. Containing Notes to the Rules and Examples of Part I. ; Explana- tions of the Construction of Scales, Tables, and Diagrams, and a Treatise upon Equivalent Square Bases and Equivalent Level Heights. The whole illustrated by numerous original engravings, comprising explanatory cuts for Definitions and Problems, Stereometric Scales and Diagrams, and a series of Lithographic Drawings from Models i Showing all the Combinations of Solid Forms which occur in Railroad Excavations and Embankments. By John Warner, A. M., Mining and Mechanical Engineer. Illustrated by 14 Plates. A new, revised and improved edition. 8vo ^4-00 WATSON.— A Manual of the Hand-Lathe : Comprising Concise Directions for Working Metals of all kinds. Ivory, Bone and Precious Woods; Dyeing, Coloring, and French Polishing; Inlaying by Veneers, and various methods practised to produce Elaborate work with Dispatch, and at Small Expense. By Egbert P. Watson, Author of " The Modern Practice of American Machinists and Engineers." Illustrated by 78 engravings. $l.SO WATSON. — The Modern Practice of American Machinists and Engineers : Including the Construction, Application, and Use of Drills, Latha Tools, Cutters for Boring Cylinders, and Hollow-work generally, with the most Economical Speed for the same ; the Results verified by Actual Practice at the Lathe, the Vise, and on the Floor. Together HENRY CAREY BAIRD & CO.'S CATALOGUE. 29 with Worksl^op Management, Economy of Manufacture, the Steam* Enguie, Boilers,, Gears, BeUing, etc., etc. By Egbert P. Watson. Illustrated by eighty-six engravings. i2mo. . . . ' $2.^0 UTATSON.— The Theory and Practice of the Art of Weaving by Hand and Power • With Calculations and Tables for the Use of those connected with the Trade. By John Watson, Manufacturer and Practical Machine- Maker. 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Together with a Model Specification involving a great variety of instructive and suggestive matter. By George Wight'wick, Architect. A new edition, revised and considerably enlarged; comprising Treatises on the Principles of Construction and Design. By G. HusKissoN Guillaume, Architect. Numerous Illustrations. One vol. i2mo $2.0(t 9\nLL>. — Tables of Qualitative Chemical Analysis. With an Introductory Chapter on the Course of Analysis. By Pro* fessor Heinrich Will, of Giessen, Germany. Third American* from the eleventh German edition. Edited by Charles F. HlMES^ Ph. D., Professor of Natural Science^ Dickinson College, CarHsle, Pa. 8vo. . . • ^1.50 WILLIAMS.— On Heal and Steam: Embracing New Views of Vaporization, Condensation, and Explo> sion. By Charles Wye Williams, A. I. C. E. Illustrated 8vo. $3 5<» WILSON.— A Treatise on Steam Boilers : Their Strength, Construction, and Economical Working. By RoBERt Wilson. 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Pocket-book form, gilt edge ^1.50 POSSELT.— Technology of Textile Design : Being a Practical Treatise on the Construction and Application of Weaves for all Textile Fabrics, with minute reference to the latest Inventions for Weaving. Containing also an Appendix, showing the Analysis and giving the Calculations necessary for the Manufac- ture of the various Textile Fabrics. By E. A. PosSELT, Head ■ Master Textile Department, Pennsylvania Museum and School of Industrial Art, Philadelphia, with over 1000 illustrations. 29a pages. 4to ' $5-oo POSSELT.— The Jacquard Machine Analysed and Explained: With an Appendix on the Preparation of Jacquard Cards, and Practical Hints to Learners of Jacquard Designing. By E. A. Posselt. With 230 illustrations and numerous diagrams. 127 pp. 4to. . ^3.00 POSSELT.— The Structure of Fibres, Yarns and Fabrics : Being a Practical Treatise for the Use of all Persons Employed in the Manufacture of Textile Fabrics, containing a Description of the Growth and Manipulation of Cotton, Wool, Worsted, Silk, Flax, Jute, Ramie, China Grass and Hemp, and Dealing with all Manu- facturers' Calculations for Every Class of Material, also Giving Minute Details for the Structure of all kinds of Textile Fabrics, and an Appendix of Arithmetic, specially adapted for Textile Purposes. By E. A. Posselt. Over 400 Illustrations, quarto. . ^10.00 RICH.— Artistic Horse-Shoeing: A Practical and Scientific Treatise, giving Improved Methods of Shoeing, with Special Directions for Shaping Shoes to Cure Different Diseases of the Foot, and for the Correction of Faulty Action in Trotters. By George E. Rich. 62 Illustrations. 153 pages. i2mo ^i°oo 32 HENRY CAREY BAIRD & CO.'S CATALOGUE. RICHARDSON.— Practical Blacksmithing : A Collection of Articles Contributed at Different Times by Skilled Workmen to the columns of "The Blacksmith and Wheelwright," and Covering nearly the Whole Range of Blacksmithing, from the Simplest Job of Work to some of the Most Complex Forgings. Compiled and Edited by M. T. Richardson. Vol.1. 2IO Illustrations. 224 pages. l2mo. . . ;gi.oo Vol. II. 230 Illustrations. 262 pages. l2mo. . . ^i.oo Vol. III. 390 Illustrations. 307 pages. i2mo. . . ;gi.oo Vol. IV. 226 Illustrations. 276 pages. i2mo. , . ^i.oo RICHARDSONr— The Practical Horseshoer: Being a Collection of Articles on Horseshoeing in all its Branchef which have appeared from time to time in the columns of " 1 he Blacksmith and Wheelwright," etc. Compiled and edited by M. T. Richardson, i 74 illustrations ^i.oo ROPER. — Instructions and Suggestions for Engineers and Firemen : By Stephen Roper, Engineer. i8mo. 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Illustrated by 422 engravings. 4to., 320 pages . . JS6.00 ROSE.— Steam Boilers: A Practical Treatise on Boiler Constmction and Examination, for the Use of Practical Boiler Makers, Boiler Users, and Inspectors; and embracing in plain figures all the calculations necessary in Designing or Classifying Steam Boilers. By Joshua Rose, M. E. Illustrated by 73 engravings. 250 pages. 8vo ^2.50 SCHRIBER.— The Complete Carriage and Wagon Painter: A Concise Compendium of the Art of Painting Carriages, Wagons, and Sleighs, embracing Full Directions in all the Various Branches, including Lettering, Scrolling, Ornamenting, Striping, Varnishing, and Coloring, with nurnei-ous Recipes for Mixing Colors. 73 Illus- trations. 177 pp. i2mo ;^i.oa Lr;