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On the use of the telemeter in topographical surveys / by C. Schenk.

On the use of the telemeter in topographical surveys / by C. Schenk. Schenk, Charles. 400dpi TIFF G4 page images University of Kentucky, Electronic Information Access & Management Center Lexington, Kentucky 2002 b96-13-34924027 Electronic reproduction. 2002. (Beyond the shelf, serving historic Kentuckiana through virtual access (IMLS LG-03-02-0012-02) ; These pages may be freely searched and displayed. Permission must be received for subsequent distribution in print or electronically. On the use of the telemeter in topographical surveys / by C. Schenk. Schenk, Charles. Stereotyped for the Survey by Major, Johnston & Barrett, Yeoman Press, Frankfort, Ky. : [1880] 19 p. : ill. ; 28 cm. Coleman Pages also numbered 23-41. Microfilm. Atlanta, Ga. : SOLINET, 1996. 1 microfilm reel ; 35 mm. (SOLINET/ASERL Cooperative Microfilming Project (NEH PS-21089) ; SOL MN06012.15 KUK) Printing Master B96-13. IMLS This electronic text file was created by Optical Character Recognition (OCR). No corrections have been made to the OCR-ed text and no editing has been done to the content of the original document. Encoding has been done through an automated process using the recommendations for Level 1 of the TEI in Libraries Guidelines. Digital page images are linked to the text file. Telemeter (Physiological apparatus) Topographical surveying Kentucky. GEOLOGICAL SURVEY OF KENTUCKY. N. S. SHALER, DIRECTOR. ON THE USE OF THE TELEMETER IN TOPOGRAPHICAL SURVEYS, BY C. SCHENK. PART I11. VOL. V. SECOND SERIES. rERB E' - FORH5URVEY BY.-JI, J....... . ....ET, YEOMAN P-ESS, PEAEFORT, Y. 23 A 24 This page in the original text is blank. ON THE USE OF THE TELEMETER IN TOPO- GRAPHICAL SURVEYS. As I have already said in my topographical report, I made use of the instrument called the telemeter for measuring lengths during the survey I made of a part of Greenup county and of Lawence county. I was principally moved to apply this means by the topographical relations of the coun- try I had to deal with ; for there are in this region but few plains over which a direct measuring of lengths, by means of a measuring-staff or chain, is practicable, with anything like the rapidity which was a necessity with me. Owing to the many bends of the roads, it seemed to me pretty difficult and too inaccurate to use an odometer for measures of length and a compass with sights for measuring the angles. Moreover, an odometer cannot be used in the case of rivers, both of whose banks are covered with brushwood, because the apparatus will not work among brushwood. An odometer consists, for the most part, of a wlhe l Wvhich rolls on the ground, and connected with an indicator, and, NOhc1n puLshled by the operator, rolls fur- ther. From the number of revolutions, which correspond to a certain amount of road gone over by the instrument, the dis- tance is read off by means of the indicator. On straight level roads, and especially on railroads, the odometer is a good instrument for measuring lengths, and is more accurate than a chain. But where the roads are winding and bordered by fences, the use of an odometer must entail a great inaccuracy. The telemeter, used along with an instrument for measuring angles, offered me the following advantage: from the point where the instrument rested I could measure the distances at the same time that I measured the angles, and I could meas- ure the distances through the air above or beneath the many obstacles that were in the way; in the same way I was able to measure very successfully across a surface of water in the 25 4 ON THE USE OF THE TELEIETER case of streams, a thing which cannot be done with chains, staffs, or odometer. These remarkable advantages threw the balance completely on the side of the telemeter. As many looked on with heads shaking disapproval on the method of measurement which I use(l, an d as the telemeter is not sufficiently well known, even by specialists, and in general is not so much used as it deserves to be, I have decided, in accordance with the wish of my chief, to give in the following pages a short explanation of the theory, and its practical ap- plication. THE THEORY. Theoretically, telemeters are divided, first of all, into two classes. In one class a staff and telescope are used in such a way that the operator looks through the latter towards the former; in the other class of telemeters a staff is altogether dispensed with. We must therefore distinguish between tel- emeters with and without a staff. Many telemeters without a staff have been proposed, and even constructed, but so far they have firnished less satisfac- tory results than those obtained from telemeters with a staff, and are of less value for practical geometry. Consider the annexed sketch, which is inten(led to repre- sent two telescopes provided with diaphragms, which can be moved to and fro on a staff in such a way that the angle which their optical axes make with each other remains constant. It is easy" to see that the distance of the two telescopes from each other must be exactly proportional to the distance of a point sighted through both telescopes, assunling that the instrument is not moved. This contrivance would be very excellent if only the condition of keeping the angle of sight constant could be ful- filled with sufficient accuracy. This would require a staff of an uncommon grade to move the telescopes on, and one that would not warp; in short, almost mathematical accuracy would be required in the making of the different parts. 26 IN TOPOGRAPHICAL SURVEYS. One improvement is to use the so-called angle-mirror or prism instead of the telescope and staff; or, as with the case- sextant mirror, one can deduce the distance from the eccen- tricity of the alhidade and the measured angle. Angle-mirrors or prisms can be easily put in position, by means of which the engineer, working on a short basis, can determine geometrically a third (inaccessible) point. An in- strument of this kind can even be put in one's vest pocket. The relation of the base, which is to serve for measuring the distance of the point which is to be determined, must be decided before getting ready the prisms, since this relation determines the angle B, according to which the prisms must be drawn. Let D equal the distance from the point where the operator D stands to the object; let b equal the base, and be D = m; so is Cos 1 = I. The quotient or may go up to 3, 4, 5, etc. Military men have already frequently made experiments with telemeters without staffs. One may claim already to measure accurately with prisms, especially if mi is not taken too large. So much for telemeters without a stafl. I now pass to tel- eieters with a staff. TELEMETERS WITH A STAFF. These are divided into two kinds. In one kind the length of staff is constant-i. e., the length of staff that can be used for compnutation (mostly by ineanis of points whose distance from each other is known) is constant; the anlle betwveen the two extreme lines of sight is measured, and from this angle, together with the known length of the staff, the distance of the staff from the point where the instrument stands is com- puted. In the other kind a portion of staff proportional to the distance is used, and the distance is comp)uted from this portion of staff and a constant, which latter depends on the construction of the instrument used. 27 5 d ON THE USE OF THE TELEMETER TELEMETERS WITH A CONSTANT LENGTH OF STAFF. This contrivance consists of a telescope for measuring dis- tances with cross-hairs on one side and a staff of determined length on the other side. After setting the staff and tele- scope the latter is used to sight the former, and thus the angle, which is wanted in order to bring the cross-hairs from one point of the staff (the blank) to another, is menasured. This angle is measured either by moving the telescope, or turning it, or else the horizontal thread itself is moved (Mleier- stein's telemeter). It is easy to perceive that the accuracy of measurements depends entirely on the accuracy with which the angle is taken, if one leave out of account the correct position of the staff; and in practice this condition is com- monly fulfilled by means of finely-cut screws, on which are heads divided so as to determine with accuracy the whole and fractional parts of the necessary rotations, which take place, when the cross-hairs are moved fromn one target to another. The telemeters in which, in order to obtain the above result, the telescope is turned through the necessary angle by means of a fine screw, are called Stampfer's telemeters. As has been already mentioned, the angle which must be known in order to move the optical axis of the telescope during the measuring of the distance from one blank to an- other, is measured by means of a screw. 'Furthermore, it is easy to see, that if u equal the necessary number of rotations of the screw, u is inversely proportional to the distance of the staff from the telescope itself; and hence, from the quan- tity u the distance itself can be determined. Let I equal the distance between the two blanks on the staff. Let D equal the distance from the staff to the stand- point of the observer. Then, owing to the smallness of the angle of inclination which the lower line of sight drawn to the target makes with the upper one, it is accurate enough to say D = t x ,where x denotes this angle of inclination. 28 IN TOPOGRAPHICAL SURVEYS. Now the angle is to be expressed by the number of screw- threads that are used in order to bring the telescope through the arc x; and from this one writes down Ianz x =c u, where c denotes a constant depending of the disposition of the in- strument itself, and one obtains D = I = I tan x CUg The constant c can be determined as follows: A length D is accurately measured on horizontal ground; the instrument is placed at one end of this distance, the staff at the other, and one calculates how great u is when I and D are known, and it will be good to determine u several times. Further, we have - D U = a coefficient which we can call k, and then we write- = It =- k, hence D = k Stampfer's instruments have the constant k = 324, and therefore D / 324. We now see the results of this equa- z tion: / is constant, and different values assumed for u are put together in tables from which one can immediately read off the length D, corresponding to the quantity ue. A very great accuracy is claimed for this telemeter by Prof. Staml)fer. I have only touched lightly on this telemeter be- cause, unlike the kind I am going to describe, it. cannot be adapted to a theodolite which is not specially constructed for the purpose. I now turn to the other kind of telemeters known as Reichenbach's telemeters. REICHENBACH S TELEMETER. This telemeter can easily be adapted to any telescope by drawing two parallel threads across the diaphragm ring of the telescope at any distance from each other. 7 ON THE USE OF THE TELEMETER In order to perceive the way in which this contrivance (the hair micrometer) works in connection with a staff, let us con- sider, for our purposes, a simple (Kepler's) telescope. Let us assume, further, that the staff is in a vertical position, and that the telescope is directed at right angles to it-i. e., that it is itself horizontal. Let 0 denote the object glass. Let o denote the ocular glass. Let m denote the distance between the threads. ss, of the hair micrometer. I-et a denote the distance of the position of the image of the staff from the object-glass. Let A denote the distance of the staff itself from the object glass. Then from the similarity of the two triangles T x 7 and s x s, we have the following simple relation: I=-/- . (1). Further, according to a formula in dioptrics, = X + whereby f is meant the focal distance of the object lens. This formula will be -ouind developed in any good text-book on physics. 'faking now this latter equation with respect to a, we get a = _ Substituting the value of a in the first equation, we have finally, A f / . . I. (1). That is to say. the distance from the anterior focus of the object-glass to the staff is proportional to the length of the portion of the staff cut off on the staff by the threads. Reichenbach's telemeter is constructed with this proportion as a basis. The proportion between the focal distance f of an object- glass and the distance m between the spider-webs, which 30 8 IN TOPOGRAPHICAL SURVEYS. latter are drawn tightly across the diaphragm ring in the tel- escope is a constant quantity. viz: f i. Furthermore, in m the same telescope the distance between the axis of revolu- tion and the object-glass is also constant. But the middle point of the axis of revolution is also the middle point of the instrument from which one wishes to measure the distance to the staff. If now, for our lurther investigations, we adopt a telescope whose object-glass stands 1St from the axis of revo- lution of the tube, which is approximately right, then, first, the distance of the staff from the middle point of the instrument is expressed by A + i/f If i ,f is added to both sides of equation (11), and if for f its valtue i is written, we have .4 + 4 f = i / - i.5 f (111). If, then, in a telescope the constant quantities i and f are known, then by means of (111) for every distance of the staff fromt the middle of the instruiment-that is, for every value of I I-- Yzf-a quantity which we will hereafter denote by AE- tile corresponding portion of the staff can be calculated and tile staff divided accordingly. Inl the case of Ramsden's eye-piece, which is here most Comm01onlly used, and in that of Kepler's telescope, or by the excellent Kellner ocular, which I now use, this formula holds true without exception. In the case of Hughen's telescope, however, the intervention of the collecting lens causes the rays coming from the objective glass to be drawn together; and in this case, to complete our investigation, further consid- eratiois are necessary. Since, however, Hughen 's eye-piece is but little used in this country, I will not stop to investigate it here; still, I will add, that, for my own part, I have hitherto uised a Hughen's eye-piece, and that I shall be glad to help ally one who desires explanations about it. TIhe eye-piece. mostly used with measuring-telescopes here in America, is. so far as my experience extends, the so-called terrestrial or Rheitas eye-piece, which may be considered as arising from a combination of Ramsdemn's and Hughen's. This 31 9 ON THE USE OF THE TELEMETER eye-piece is like the simple one, or like Ramsden's, in that there is no collecting lens between the cross-hairs and the object-glass. When one wishes to make a division of the staff according to the formula given above, it should be observed that if the threads are placed much apart, a long staff is needed, while if the threads are placed close to each other, a small staff suffices; small staffs, however, give less accurate results than long staffs, and many persons hold that one should not take f larger than m 70. This value is not very convenient, especially because it requires a long staff that must be divided in a special way. If loo is used as unit of division, one has the advantage of being able to use every leveling staff for measuring distances, although, indeed, the accuracy of measurement is somewhat less. I use ioo as the unit of division, and, therefore, for a dis- tance of iooo feet, I need a staff about lo feet long. Let us now investigate the case where we wish to use for measuring distances a leveling staff that is already divided, and let us see what the practical results would be. We have given the leveling staff with decimal division, there- fore we must have i = ioo. We wish further to measure the focal distance of the object- ive glass; this can be done with sufficient accuracy by means of compasses; after a distant object-a star-has been sighted to that there is no parallax, one has only to measure the dis- tance between the object-glass and the diaphragm. We have now i = ioo, and we have also determined f We can, therefore, from equation (III) or equation (II) or from f- ioo-calculate the distance between the threads, m and then place the threads in position. It is necessary to cal- culate the distance between the threads as nearly as possible, in order that the threads may be placed in their proper posi- tion as nearly as possible, so that it may not be impossible to 33 TO IN TOPOGRAPHICAL SURVEYS. effect the small correction which is mostly required later, and which should be rendered possible by means of some contri- vance such as a screw. For, with ordinary means, it will be impossible to place the threads with the accuracy which does not show itself till under the magnifying power of the eye- piece. When we have placed the threads in position upon the diaphragm ring, and the latter in the telescope, we can go on to investigate what the telemeter accomplishes and with how much accuracy it works. First of all, we have equation (III), viz: E = il +- I.5 f to take into account, and to determine the portion which must be used for certain distances E. If, for example, we make E successively equal to loo, 200, 300, etc., the above equation solved with respect to / gives for =100 100- / =- i.5f i 100 for E= 200, = 200 -1 f/= 2 -5f 100 for E= 300, 1 3 - 1.5f =/ 31-5f 100 that is, when the staff is Ioo feet off the portion of staff used is i foot - L'5/f in length; when the staff is distant 200 feet 100 from the centre of the instrument the length of staff used is 2 .--I feet, and so on; so that a portion of staff less than I 100 foot corresponds to the first l oo feet, and exactly one foot more of staff is requiredfor every additionaloo feet. The point on the staff which is determined by the quantity l'Sf has been fitly called the zero point. In this connection 100 I should mention, that for this determination I supposed a telescope, whose pivoting axis was Y/f from the object-glass; if this quantity were different, it would have to be introduced with its proper value. For example, in the case of a tele- scope, the focal distance of whose object-glass was six, and VOL. V.-3 33 I I ON THE USE OF THE TELEMETER between the axis and object-glass three, decimal inches, the portion of staff corresponding to a distance of one hundred feet would be o.99i, and for two hundred feet, 1.99i, etc. When this zero point has been so computed it is marked on the staff itself, and, in measuring, the tipper thread is always directed to this zero point, while, by means of the lower thread, the distance is read off on the staff. If one has to work with a staff without zero point, it is necessary to add to the distance taken between aiiy niumbers the constant 1.5 f in order to get the correct distance. Accordingly, as I have here pointed out, it is entirely wrong to place one's threads in such a way that, for a distance of one hundred feet, they cover just one foot of the staff. This arrangement is assumed by many to be correct, as I found to my grief; it is even set forth as correct in instructions about the use of instruments. One has only to use an instrument with a considerable focal dis- tance in order to perceive, that if the threads cover one foot of staff for a distance of one hundred feet, the measurement of great distances becomes very inaccurate. XVith my tele- scope I would have made an error of fifteen feet in i,ooo' dis- tance if its threads had been set in the faulty way I have mentioned. When one's telemeter has been put in order, and the staff also is in order, the zero point having been determined, the next thing is to meastire on an even plane, as accurately as possible, a suitable extent of ground from five hundred-one thousand feet with a staff or chain. A straight railroad rail is peculiarly suitable for a good measurement, which can be accurately taken by means of a steel tape measure. Next the instrument should be placed at one end of the measured strip, the distance staff should be set vertically up at the other end, and the engineer should examine whether the threads have the separation which corresponds to this distance, and whether they cover exactly the zero point above, and-for example, for a distance of one thousand feet-the ten-foot partition counted from above, 34 I 2 IN TOPOGRAPHICAL SURVM. A variation from the present disposition of the instrument can be obtained, if necessary, by pushing the slits on which the threads rest, which movement is practically effected by adjusting screws. It will be good to try the telemeter on many lengths that have been accurately measured, and it is specially advisable to measure small distances with it. If all comes out right, one can trust with safety to his telemeter. The above remarks were made under the supposition that the staff was placed vertically and the telescope horizontally. In practice such a use is seldom made of the instruments, ex- cept in leveling; on the contrary, the sights are inclined either upwards or downwards. We have still to investigate in what way the results of measurement are modified by this departure from the previous hypothesis. We have further to consider whether the staff shall be placed vertically; or perpendicularly to the line of sight of the tele- scope. The staff can be placed vertically by hand, by a level, by a plumb-line, or by balancing the staff on a point placed under it. I make use both of balancing and plumbing, according to the nature of the work. The staff can be made perpendicular to the line of sight by placing a diot/er on the staff, and then sighting the instru- ment from the staff. REDUCTION OF OBLIQUE LENGTHS. If one sights a vertical staff under an inclined angle, then, on account of the oblique sighting, a larger part of the staff will come between the threads than corresponds to the direct distance. The length of the portion of staff so sighted can be read off directly. The angle x, under which the staff is sighted, can also be read off. Therefore, we have the data for reducing. The portion of staff a' b' corresponding to the distance J L is the correct one, while as a matter of fact the greater portion a 6 is read off, and we wish, therefore to deduce a' 6' from a b. 35 13 ON THE USE OF THE TELEMETER Owing to the smallness of the arc which a' 6' subtends, a' b' can be expressed with sufficient accuracy by ' I cosx, when ab = I and a' b' = 1 the value I' so obtained corres- ponds to the oblique distance J L, which is denoted by the expression, oblique length. This quantity, J L, however, must still be brought down level with the horizon; it cannot yet be called the correct horizontal measure. This quantity I cosx must itself be reduced, which is done by multiplying again by cosx, so that we have E = il tos'x. The quantity cos'x can be very easily deduced from .r itself. Thus, if it is wished to carry out the multiplication in a graphic way (which is a very convenient operation if one wishes to put the lengths on a scale, in order to introduce them into a map). one has merely to calculate the values of cos'x. to consider them as cos of an angle, and to set these angles down in a diagram, by means of which the whole reduction can be com- pleted with compasses. (Jordan's diagram.) If the staff is placed at right angles to the line of sight. the necessary reduction to horizontal valuies can be performed by the help of the following considerations. If a b again stands for the portion of the staff covered by the threads, then in both figures the horizontal correction will simply be: a b cos x = E. But it will be perceived that, owing to the oblique position of the staff, it rests at another place than that which E requires. This reduction depends also 36 14 IN TOPOGRAPHICAL SURVEYS. on whether x is an angle above or below the horizon, and on how high onl the staff the middle line of sight reaches. If we call this length, which changes for every distance, A, then, besides the above reduction, A silix will have to be added to E for an angle of elevation, and subtracted from E for an angle of depression, in order to determine the point where the staff touches the ground. The data of reduction for a given staff, and for different values of E and x, have been tabulated, and by means of these tables the reduction can easily be effected. With respect to the telescopes that are to be used for meas- uring with telemeters, they must be of the best quality, and must possess great clearness, and especially definition, to- gether with great magnifying power. ON THE ACCURACY OF LENGTH MEASUREMENTS. Experience shows that in all measurements a deviation occurs from the real length. I do not wish to include among errors so committed those which arise from inaccurate obser- vation and inaccurate reading; such mistakes can be avoided; I refer to mistakes arising from the imperfection of our tools and senses. An idea of this kind of error may be got by measuring several times a length of xoo' with a i-foot meas- ure, and finding that each result deviates slightly from the preceding one. 37 15 ON THE USE OF THE TELEMETER It is possible, by repeatedly measuring the same quantity, to get an approximation of its absolute value. It is advisable to measure the same length twice over, if for no other reason, because a tolerable agreement of both measurements gives the certainty that no grave error has been committed. By the method of the smallest square one can find the mean error out of several measurements of a quantity, or the mean error- of one single measurement. This method is of use to the practitioner, because he can make clear to himself the errors which he has to fear during his work. It is especially used where, after taking great care with the measurements, one wishes to bring the final result still nearer the probable true value. In the following tables are arranged, as far as I know them, the restilts which have been obtained by the method of the smallest square, from the most accurate and from less accu- rate measurements: MEASUREMENT OF BASES. -ean in a sieg -measuremen f theength of one ilom eter - -,oeo metee38oEnglish f.t Year. Millimeters. English inches. 1736 Base of Yarotqui, in Peru, two measurements with wooden sta.es, from 15 to 20 feet long 16.4 five eighths. 1736 Base of Tornea, in Lapland, two measurements with wooden staves. 20.2 eleven six- 1739 Re-measuring of the Picard base under Juvisy teenths. by Cassitti. ....................... . 63.2 two and ahalf. 1819 -Schwverd's small Speyer base, two measurements, 859.4409 meters long... 1.5 one sixteenth. 834 Base line of the measurement of a degree in East Prussia.... .......... . . . 2.2 three thirty- 1846. Base line near Berlin for the coast survey, two seconds. measurements.......... .. . 1..6 one sixteenth. i858 . Spanish base of Madridejos, twice measured 0.4 one forty- I860. Small Spanish base of Ivice, measured four time 0.3 [eighth. .868 Austrian base in Dalmatia, two measurements. 0.7 MEASUREMENT WITH STAVES. The mean error of one measurement derived from measuring twicc. the lengs measured being diffct. Length in feet. Mean errors of a measurement in decimal inches. 300 1.098 6oo 1553 1,000 2.005 38 z6 IN TOPOGRAPHICAL SURVEYS. CHAIN MEASUREMENTS. This is derived from over 500 measurements taken twice of lengths going up to i,ooo feet, with chains of from 30 to 50 feet long: MEAN -ERROR OF ONE MEASUREMENT IN DECIMAL INCHES. Length in feet. On sandy ground. On loamy ground. 300 5 3 6oo 7 4 ,ooo to 6 Whence it is evident that measurement with a staff is three times as accurate as measurement with a chain. In ordinary measurements, which are made with somewhat less care, the mean error proves to be as follows: on hard ground, I . Iooo; on washy or soft ground, I 500; on com- mon ground, I 700. When a chain is used, its tension should carefully be ob- served in order to compensate it. The depression of the chain, owing to careless stretching, produces an error which increases with the square of the depression. OF THE MEASUREMENT OF DISTANCES BY MEANS OF REICHEN- BACH'S TELEMETER. I obtained the following results on a railroad track on which the distances had been accurately measured with a steel rib- bon; the distance was read off three times for each place where the staff was put up: WITH THE TELEMETER-MEASUREMENT. Length measured with the Steel Ribbon. First. Second. Third. 100 l oo I to zoo 200 200 200 2W0 400 399.5 400.5 400 6oo 599.5 600 800 799 799.5 1,000 999 1,000 1,000 1,200 1,199 1,201 1,198 act_3 I7 ON THE USE OF THE TELEMETER Whence the deviation from the accurately measured length is at the utmost X . 6oo. These measurements were made with great care, in quiet weather, and with good light. Such accuracy is surely not to be obtained by a day's work. Accord- ing to the results obtained by other observers, the accuracy amounts to somewhat less; the error is given as from i . 400 up to I . 300. But, taking merely the accuracy obtained with the last error, and the telemeter is still a very good in- strument for topographic work. Moreover, it is easy with a telemeter to take several read- ings instead of one, and so to increase the accuracy; so much so that four readings double the accuracy. It is of special importance in the measuring of distances that the staff, if it is used in a perpendicular position, should be satisfactorily held perpendicular; and this is pre-eminently true on sloping ground, since the very considerable errors are made by carelessness in placing the staff. If, for example, a ten-foot staff is held so that its deviation from the perpendicu- lar amounts to one foot, and if the staff is sighted at under an angle of five degrees, this would already bring about an error of one per cent. in the length. With a telemeter we have also to consider the state of the air with respect to rest or motion, the light, and the con- dition of equilibrium of the lower layers of air. In summer, during the hot part of the day, the sunlight brings about such a trembling of the images of the sighted object that sighting and reading off become very difficult operations. When the sky is clouded the objects are quietest, and one can then work very comfortably. The advantages of the telemeter are specially manifested in a favorable light in making topography which must be quickly completed, because the measurement of distances takes place from the same stand of instruments from which other objects, such as houses, etc., are placed in; as uneven- nesses, bushes, etc., are disregarded, as long as one can see through and over them. If one has to survey a stream, and is unable to see along on the shore, owing to weeds, 40 IN TOPOGRAPHICAL SURVEYS. X9 trees, or plantations, one has only to go to the water, -and if one can only get a place to stand upon, there is nothing to prevent the measurement along or across the water. The rapidity with which the work progresses depends natu- rally on the ground and material which one has to deal with. From six to eight miles is a good day's work. Sometimes I have measured ten miles in a day. 41 A42