Fig. 80.Larger illustration(31 kB).
Fig. 80.
Larger illustration(31 kB).
To use this scale, suppose a length of 24 miles 5 furlongs is required. Place one leg of the dividers upon the point in which the fourth diagonal intersects the fifth parallel, and extend the other to the point in which the primary division marked 20 intersects the same parallel. In like manner, if the distance required be 33 miles 3 furlongs, it must be taken from the intersection of the third diagonal with the third parallel, to the intersection of the primary division marked 30 with the same parallel.
It is obvious that if a scale of feet showing inches diagonally be required, twelve equidistant parallel lines must be drawn instead of eight as in the foregoing example where furlongs are required. The diagonal scale possesses the important advantages of accuracy and distinctness of division which render it very suitable as a scale of construction. Another practical advantage is that it is less rapidly defaced by use than the other kinds, in consequence of the measurements being taken on so many different lines.
The construction of the vernier scale is similar to that of thegraduated arcs of surveying and astronomical instruments. The principle of the vernier is as follows. If a line containingnunits of measurement be divided intonequal parts, each part will, of course, represent one unit; and if a line containingn+ 1 of these units be also divided intonparts, each part will be equal ton+ 1nunits; and the difference between one division of the latter and one of the former will bex+ 1n- 1 =1⁄nof the original unit. Similarly, the difference between two divisions of the one and two of the other will be2⁄nof a unit, between three of the one and three of the other,3⁄n, and so on. Hence, to obtain a length ofx⁄nof a unit, we have only to make a division on one scale coincide with one on the other scale; the space between the two correspondingxth divisions from this on both scales will be the required length of2⁄nof a unit. The same reasoning will evidently hold good if a length equal ton- 1 be taken.
Fig. 81.Larger illustration(23 kB).
Fig. 81.
Larger illustration(23 kB).
To show how the foregoing principle is applied in practice, we will take an example. It is required to construct a scale of1⁄100to show feet and tenths of a foot. Construct a scale in the ordinary way, and subdivide it throughout its whole length, as shown inFig. 81; then each division will show one foot. Above the first primary division, draw a line parallel to the scale and terminating at the zero point. From the zero point, set off on this line towards the left a distance equal to eleven subdivisions, and divide this distance into ten equal parts. Now, as eleven divisions of the plain scale have been divided into ten equal parts on the vernier, each division on the latter will represent11⁄10= 1·1 of that on the former; and as the divisions of the plain scale represent feet, those of the vernier will represent 1·1 foot. Consequently, the distances from the zero of the scale to the successive divisionson the vernier are 1·1, 2·2, 3·3, 4·4, 5·5, 6·6, 7·7, 8·8, 9·9, and 11 feet. It will be seen that the divisions of the two scales are made to coincide at the zero point.
The mode of using this scale will be seen from the following example. Let it be required to take off a distance of 26·7 feet. From zero to the 7th division of the vernier is, as we have seen, 7·7 feet. Therefore, to ascertain how far to the right of zero we must go to obtain the distance of 26·7 feet, we must subtract 7·7 from that distance, which gives 19. Thus to take off the distance, one leg of the dividers must be placed on the 7th division of the vernier, and the other on the 19th division of the plain scale. If the distance to be taken were 27·6 feet, one leg of the dividers would have to be placed on the 6th division of the vernier, and the other on the (27·6) - (6·6) = 21st division of the plain scale.
To construct a scale to show feet and inches, make the vernier equal to thirteen divisions of the plain scale and divide it into twelve equal parts. Each of these divisions will then represent13⁄12= 11⁄12of a foot.
Scales of construction may be purchased upon box-wood or ivory, but where great accuracy is important, it is best to lay down the scale upon some part of the drawing, as in such a case it expands and contracts with the drawing under the influence of moisture.
Examples of scales of distances will be found onPlates 8and9.
The transference of the measurements determined by the survey from the field-book to the paper is termedplotting. The operations of plotting are very simple, and the ability to perform them properly may be acquired with a little practice. But their due performance demands the same extreme care and attention as that of the operations in the field, for it is obvious that the precautions taken to ensure accuracy in the latter may be rendered nugatory by inaccurateplotting. The angular instrument used in plotting is the protractor, and to ensure correct results this instrument must be accurately divided. When, however, the survey has been made without the aid of an angular instrument, the protractor is not required in laying down the results. In such a case, which frequently occurs in surveys of small extent, the lines, having all been chained and registered in the field-book, are laid down directly from the scale by means of an ordinary straight-edge and a pair of compasses. The several methods of plotting and the various operations involved have now to be considered.
—The lines chained over in a survey and recorded in the field-book are not usually the actual lines existing on the ground, but imaginary straight lines chosen for the purpose of referring other lines and points to them. They are, therefore, appropriately termedreference lines, and all points situate in them to which other lines are referred, in other words, all points in a reference line in which other reference lines intersect it, are termedreference points. Reference lines are generally made to form triangles for facility of computation, and these triangles enclose the area to be surveyed. But to determine the details included within them, it is necessary to form other and smaller triangles within the larger ones first laid down. The latter are, therefore, distinguished as Primary and the former as Secondary triangles, and the lines of which they are composed are called primary and secondary reference lines.
—In laying down a line of definite length upon paper, the positions of its extremities are determined and marked by pencil dots; such dots, or rather the points indicated by such dots, are termedplotted points. The line is drawn by joining the plotted points.
—To plot a reference line of a given length when the position of neither of its extremities is given, a light dot must be made upon the paper in a convenient part to indicate the position of one extremity. The pencil-point mark should be as light and well defined as possible, and hence it is essential that the pencil used should be hard, and always kept pared to a fineconical point. The scale must then be applied, in the direction of the line to be drawn, with its zero point coincident with the plotted point. The scale should be lightly but firmly held in this position with the left hand. The distance of the other extremity of the line must then be found upon the scale, and the eye placed directly over the line of the graduation; this is necessary to the correct placing of the point, and it is well to train the eye to trace accurately the prolongation of the line of graduation upon the paper. A dot must be placed in the prolongation of this line and close to the edge of the scale, to mark the position of the other extremity. The reference line is then to be drawn between these two plotted points with a sharp chisel-pointed pencil. Reference lines, like reference points, should be well defined, but drawn as fine and light as possible. The degree of fineness and lightness should be such that when the detail is finely but firmly outlined, the reference lines and points may not be visible, except on a close inspection of the surface.
To plot secondary reference lines, as, for example, a number of offset lines, apply the scale so that its edge may be parallel to and almost over the pencil trace of the primary line and its zero point coincident with the point at which the line begins. Care must always be taken to place the zero of the scale at the beginning of the line, and not at the end of it. At the distances recorded in the field-book as those at which the offsets were taken, plot upon the line, in the manner described above, the points indicating the extremities of the offset lines. All other points, such as stations and intersection of fences, roads, and streams, should be plotted at the same time. Around all stations, a light, hand-drawn circle should be placed, and intersections marked by small cross lines. This being done, place the offset scale so that the zero may coincide with the plotted point in the reference line and the edge be perpendicular to the line. To bring the edge into this position, the end of the scale should be placed parallel to the reference line; this is, of course, assuming the scale to be perfectly rectangular, as it ought to be. The other extremities of the lines may then be plotted in the same manner as those of the primary lines.
To illustrate the foregoing remarks, let it be required to plot the following portion of a field-book.
001346or 000 Line 1.751170961000056506640000000Line 3.From1946last.arrow19460014208812001441000110600755205000000Line 2.From2504last.arrow25040020006517909516101151440871220110100085028420100Line 1.00000or 1346Line 3.Begin at south corner and range N.
Having laid down in a convenient part of the paper the beginning of Line 1, place the edge of the scale so that the zero may coincide with the plotted point and the edge be parallel to one of the meridians. Holding the scale firmly in this position, plot the distance 2504 links, and join the plotted points. Then, without removing the scale, plot upon this reference line the distances at which the secondary or offset lines were taken, that is at 420, 850, 1220, 1440, 1610, 1790, and 2000 links. In this case we have nothing but offsets; had there been stations or intersections of fences, roads, or streams, these would have had to be plotted at the same time, and distinguished by an appropriate mark. Line 2 begins at the end of Line 1, and returns on the left at an acute angle, as indicated by the arrow in the field-book. But as the magnitude of this angle is not known, the survey having been taken with the chain alone, the exact direction of Line 2 must be ascertained in the following manner. Take the length of the line in the compasses, and with this distance as a radius, strike an arc from the end of Line 1. Also with the length of Line 3 as a radius, strike an arc from the beginning of Line 1, at which point Line 3 closes, intersecting the former. The point of intersection will be the end ofLine 2. Join this point to the end of Line 1, and plot the offset reference points upon this line in the manner already described for Line 1. Line 3 begins at the end of Line 2, and terminates at the beginning of Line 1; hence both extremities being determined, we have only to join these points, and to plot the offset reference points as before. Had a split line been taken for the sake of accuracy from the angle A C B to the base Line 1, the intersection of this line should have been plotted upon Line 1, and the position of the end of Line 2 found by striking an arc from this point also with a radius equal to the length of the split line. If the distances have been correctly measured in the field and correctly taken from the scale, the three arcs will intersect at the same point. If they do not so intersect, the error must be noted in an error-sheet, and corrected in the field before proceeding further. It may be remarked here that on account of the irregularities of the surface chained over, all measured lines are liable to be recorded a little too long. One link in ten chains may be allowed for this source of error. Assuming, however, that the primary reference lines “close” properly, the offset lines may be plotted in the manner already described, and the detail or boundary lines drawn in, which in the present instance are hedges.Fig. 82shows the survey as laid down in the foregoing notes.
When the survey has been made with the aid of an angular instrument, the method of plotting the primary reference lines differs somewhat from the foregoing. In this case, the paper should be first covered with a number of parallel straight lines ruled about an inch and a half apart to represent magnetic meridians. The first station may then be marked upon one of these meridians in a convenient part of the paper. To lay down the first reference line, apply the protractor to this meridian with its centre point coincident with the plotted point, and from the bearing recorded in the field-book, lay off the given angle. Join the two plotted points and produce the line indefinitely; and upon this line lay off a distance equal to the length of the measured line. The second reference line must be drawn in the same manner, from the end of the first, by laying off from that pointthe recorded angle. But as the end of the first line will probably not fall upon a meridian, the protractor will have to be moved up to the point by means of a parallel ruler adjusted to the nearest meridian. A more convenient and a more accurate method, however, is to make the left of the drawing represent the north while plotting, and to use theT-square instead of the parallel ruler. All subsequent lines are plotted in the same way. Instead of covering the paper with meridians before commencing to plot, it may, in some cases, be found more convenient to draw with the set andTsquares a short meridian through the point as required.
Fig. 82.
Fig. 82.
To lay downFig. 82in this manner, having fixed the first station A, the length of the first primary reference line A B may be laid off upon the meridian, because in this case the bearing being due north, the reference line willbe coincident with the meridian. This done, the protractor is to be placed over the plotted point B and the second bearing laid down. Having plotted the length of this line in the point C, the third reference line C A will be determined both in length and direction by the plotted points C A, which should be joined and the line measured to ascertain whether its length corresponds with the measured distance. If these do not correspond, the angle must be replotted and the lengths laid off anew to discover the source of the error. Assuming, however, that the lines close properly, the offsets and other secondary points may be next plotted in the manner previously described.
Angles may be more accurately laid down by means of a table ofnatural sinesandcosinesand a linear scale than by means of a protractor. This is especially true when the angles are subtended by long lines, as, for example, lines of 3, 4, and even 6 feet. In such cases, a protractor is of little use. This mode of laying down angles is also convenient in some cases where angles have been taken, but some of the sides not measured. In using the table, it must be remembered that the radius of the sines and cosines is taken as unity; therefore, to find the sine and cosine for any other radius, the sine and cosine in the tables must be multiplied by that radius. To lay down the angle A B C inFig. 82, the reduced cosine of the angle should be plotted from B in the pointa, according to some scale. The scale length of the reduced sine should then be scribed froma, and the scale length of the radius scribed from B. A line drawn from B through the point of intersection of the scribes will lay down the angle corresponding to the sine and cosine in the table. Suppose the radius chosen to be 5 chains, the angle being 32° 30′. The cosine of 32° 30′ is ·8434, which multiplied by 5, the assumed radius, = 4·2170. Lay off this distance from B on the base A B. The sine of 32° 30′ is ·5373, which multiplied by 5, = 2·6865. From the pointa, which is distant 4·2170 chains from B, with a radius equal to 2·6865 chains, describe an arc; and from the point B, with a radius equal to 5 chains, describe another arc. From B draw a line through the intersection of these arcs, and lay off upon it the measured length of 1946 links as recorded in the field-book.
If only the length of the base A B and the magnitudes of the angles A B C and B A C were given, the lengths of the sides B C and A C would have to be calculated by trigonometrical formulæ. This method of calculating the lengths of the sides of triangles and plotting them with the beam compasses, like chained triangles, is the most accurate for laying down the great or primary triangles of a survey.
When it is required to plot according to this principle a solitary angle, as, for example, that between a station line and the meridian, a circle should be drawn with as large a radius as practicable round the station at which the angle is to be laid down. The distance between the points at which the two lines enclosing the angle cut that circle is then found by multiplying the radius by thechordof the angle, that is, by twice the sine of half the angle.
It sometimes happens, particularly in extensive surveys, that all the angular points of some triangles cannot be plotted upon the same sheet of paper. In such cases, the plot of the outlying points and the sides of the triangles may be laid down in the following manner. Plot the intersected triangles independently and trace them on tracing paper. Then, having drawn a fine line upon both sheets to represent the sheet edge, lay the points on the trace corresponding to those already plotted on the first sheet down upon, and make them to coincide with, the latter. Secure the trace in this position and trace the sheet edge line upon it. The intersected lines may now be plotted on the fair sheet with a pricker at points outside the sheet edge line. Next apply the trace to the second sheet and make the sheet edge lines coincide. Having secured the trace in this position, the points and the intersected lines on this second sheet may be plotted upon the fair paper by means of the pricker.
—In plotting a traverse survey in which the angles have been measured from a fixed line of direction, the magnetic meridian, the direction of the lines may be all laid down at the first angular point. An example will best show the method employed in this case. It is required to lay down the traverse shown inFig. 83.
Fig. 83.Larger illustration(75 kB).
Fig. 83.
Larger illustration(75 kB).
In a convenient part of the paper draw the straight line N S to represent the magnetic meridian, and plot upon it the first station A. Set the protractor with its centre accurately placed over this point and its 360th and 180th divisions coinciding with the meridian. Holding the instrument securely in this position, lay off around it all the bearings as entered in the field-book, numbering them in the order in which they were taken. Against each of these numbers it is well to place the page of the field-book on which the measurement of the angle and the survey of the line are entered. The plotting must now be commenced by laying down the first line through the first bearingand determining its length from the recorded measurements. The direction of the second line has next to be transferred from the first station A to the extremity of the first line, or the second station B. This is accomplished by means of the parallel ruler, by placing the edge of the ruler through the plotted point and the point marked as bearing 2, and extending it till the same edge intersects the point B. A line is then to be drawn from this point and its length laid off from the field-book as before. The direction of the third line will then be transferred from the first angular point to the end of the second line, or station C, in the same manner. This will be continued for all the lines in the traverse, and if all the measurements have been correctly laid down, not only will the last line pass through the point A, but it will be of the same length as the chained line. Also the bearings taken from A to E and H will pass through these latter stations. These proof line bearings should be laid down at the same time as the reference line bearings, from which they should be distinguished by some sign. The directions of the reference lines should be consecutively transferred, and the length of each line should be plotted in its proper place before the direction of the next is transferred. To ensure the work closing properly, great care must be taken to plot the points accurately and to draw the pencil lines fine.
Fig. 84.
Fig. 84.
The degree of accuracy to be attained will depend in a great measure upon the extent of the traverse. With long lines the difficulties increase, and with a great number of angles the chances of error are multiplied. If the angles are carefully taken, it is probable that seconds have been read off in several instances, and these if neglected, especially upon long chain lines, may lead to an error of some importance. Also when the lines are long, the parallel ruler becomes practically useless, and some other system has to be adopted. One way of overcoming these difficulties is to draw a parallel to the first meridian through every third or fourth angle; in such a case, great care must be observed in drawing the parallels. A more easily practicable method, however, is to use theT-square in the manner already described. If the left-hand edge of the drawing board be made the north, the blade will determine meridional lines, and bylaying the straight side of a semicircular protractor against the edge of the blade, its zero will be adjusted to the fixed line of direction. The first bearing having been laid down, the line is drawn and made the scale length of the chain line; the blade of the square is then pushed to the station thus found, and the next bearing set off. This operation is repeated until all the lines have been laid down. If the work closes properly, the plotting of the secondary lines may be proceeded with.
The most accurate method of plotting a traverse is by rectangular co-ordinates, or, as it is usually termed, Northings, Southings, Eastings, and Westings, because the position of each station is plotted independently, and is not affected by the errors committed in plotting previous stations. This method consists in assuming two fixed lines or axes crossing each other at right angles at a fixed point, computing the perpendicular distances or co-ordinates of each station from those two axes, and plotting the position of each station by means of theTand set squares and a linear scale. The meridian is usually made to represent one of the axes, and in this case the co-ordinates parallel to one axis will be the distances of the stations to the north or south of the fixed point, and those parallel to the other axis will be their distances to the east or west of the same point. Let N S,Fig. 84, represent the meridian, and A B the first bearing taken, and the first line measured. The angle in this case is N A B = θ. If θ is an acute angle, the second station B is to the north of the first station A; if it is an obtuse angle, B is to the south of A. If the angle θ lies to the right of the meridian, B is to the east of A; if to the left, to the west. Thus it will be seen that if the northernly and easternly directions are considered positive, the southernly and westernly directions will be negative. From the foregoing it is manifest that the co-ordinates of B are asfollows:—
Northing Aa=bB (or if negative, southing) = A B × cos. θ.Easting Ab=aB (or if negative, westing) = A B × sin. θ.
Northing Aa=bB (or if negative, southing) = A B × cos. θ.Easting Ab=aB (or if negative, westing) = A B × sin. θ.
To plot the point B, draw through the point A, with the aid of theT-square, a horizontal line. Multiply the chained length of the line A B by the sine of the angle N A B as entered in the field-book, and set off this distance along the horizontal line. From the point thus determined, erect, with the aid of the set square, a perpendicular, which will be parallel to the meridian. Multiply the chained length of A B by the cosine of the angle N A B, and set off this distance along the perpendicular line. The point thus determined will be the position of the second station B, which may then be joined to A by a straight line.
Fig. 85.
Fig. 85.
The mode of laying down the survey inFig. 85will now be obvious. Having determined the position of the second station B in the manner just described, draw a horizontal line through B and determine the third station C in the same way. The fourth station D being to the left of the meridian passing through C,c dis a westing and is to be considered as negative. Therefore the horizontal line through C must be drawn to the left of that station, and the station D determined in the same manner as the preceding stations.
The results of all these calculations should be entered in a book,in four columns, for northings, southings, eastings and westings respectively. Also in four other columns should be entered the total northing or southing, and easting or westing, of each station from the first station, computed by adding all the successive northings and subtracting the southings made in traversing to the station, the result being a northing if positive, and a southing if negative. The same treatment is applied to the eastings and westings. This affords a means of testing the accuracy of the work. It is also obvious that the position of the last or any station may be determined by this means without plotting the intermediate stations.
Let it be observed that both θ and sin. θ are positive or negative according as that angle lies to the east or to the west of the meridian; and that the cosines of obtuse angles are negative.
—By “detail” is meant outlines or objects whether natural or artificial, such as fences, walls, rivers, canals, roads, lakes, water margins, beach marks, seas, or imaginary boundary lines. In plotting from the entries of measurements for detail, these measurements should be laid down upon the paper in the order and manner indicated in the field-book. The mode of plotting the perpendicular reference lines by means of which the position of the detail is fixed has already been fully described and illustrated. The proper connections for detail, as shown by the field-book, should be made by drawing a firm pencil line through the detail points with the aid of an offset scale adjusted successively to the adjacent points. All such connections should be clearly and elegantly made. When all boundaries, roads, and streams have been drawn and inked in, tracings should be taken in small portions of all that has been laid down for the use of the “examiner.” The duties of the examiner are to make on the ground the necessary corrections for omissions and detail in error; to give, in position and character, woods, water, marsh, commons, vegetable and geological features, and permanent artificial structures; and to furnish the descriptive names of places and things, or any other desirable information. The topographical character of mountains, marshes, bogs, rough pasture, woods and water, should be drawn in characteron trace and tinted. If required, hill sketching should also be supplied on the examiner’s trace. On being returned to the office, the plotter should replot from the field notes the detail corrected, and transfer the details from the trace to the map.
—The student who has made himself familiar with the methods of laying down angles, and plotting reference lines and points, will find no difficulty in laying down contour traces. When the contour points have been surveyed with the chain, the contour is obtained by drawing a free line of feature through the plotted points. But when the contour points have been surveyed by measuring magnetic angles to known points, such angles must be laid down at these points, and produced to meet in the contour point. The drawing of contours differs from the drawing of ordinary detail insomuch as each contour point is shown by a small dot, and each carrying point by a similar dot surrounded by a small hand-drawn circle to distinguish it. The former should be so plotted as to be distinguishable in the trace or contour line, which should be readily traceable, but not conspicuous. The line joining adjacent points should be true lines of feature. Colour is usually employed for these lines, and it is well to give them a broken or somewhat undefined character. When the French system is adopted, contour lines are drawn continuous, a broad but faint line of colour.
—When the angles have been measured on dry land with the theodolite, these angles should be laid down at the dry-land points, and the lines produced to meet in the sounded point. But when the angles have been measured on the water with a sextant, a station pointer is required. The arcs of the pointer should be adjusted to read the measured angles, and the instrument applied to the plotted points of the observed objects so as to bring the hair lines accurately to their respective object points. The sounded point may then be correctly plotted through the centre of the pointer. If the angles have been measured by the magnetic compass, that is, if the angles are those made with the magnetic meridian, the angles should be laid down at the plotted points of theobserved land objects, and the lines produced to meet in the sounded point. Instead of the station pointer, a piece of tracing paper may be used in the following manner. Draw three straight lines radiating from one point so as to make with each other angles equal to the measured angles. Lay the paper on the plan and move it about till the three lines traverse the observed objects. The point from which they diverge will then mark the position of the sounded point, which may be plotted by being pricked off.
The sounded point determined by angles measured with the sextant may also be plotted by describing circles on the land-object lines as chords, to contain segmental angles equal to the measured angles. Such circles will intersect in the common land-object point and the sounded point. To plot the sounded point in this manner, requires the solution by construction of the problem, “to describe on a given line a segment of a circle that shall contain a given angle.” But this method is generally found too tedious in practice.
—There is a tendency, as we have previously remarked, for the measured lengths of lines to be a little too long, by reason of the irregularities of the surface. It is usual to allow for this source of error 1 in 1000 in fair open country, and 11⁄2in 1000 in close country. When the measurements differ by an amount exceeding these limits, the pencil trace should not be drawn between the reference points, but the line should be entered on an “office error-sheet.” The error-sheet should show the number of the plot-sheet, the triangle, the book and page in which the measurements are entered, and the scale and measured lengths of the line. To ascertain the source of the error, other lines referenced to the reference point or points of the line in error should be plotted, and the apparent source should be entered on the error-sheet. If the lines referred to the same point be found to plot to another point in the reference line, the scale measurement of this point should also be entered. And if the reference point in error be not directly surveyed in the survey of their respective lines, the measurements for reference and the arithmetical reductions will have to be examined. Besides the office error-sheet, there shouldbe a field error-sheet for each book and triangle, upon which should be entered the book, the page, and the line in error, and some indication of the source of the error. This sheet will be forwarded with the field-book to the surveyor for correction. The following are examples of a common and very good form of error-sheet, but it may be varied in many ways if thoughtdesirable:—
Office Error-sheet.
Field Error-sheet.
—In plotting a vertical section, a fine and firm horizontal line is first drawn to represent thedatum line. The reference points are then plotted upon this line from the level-book bymeans of a linear scale, in the manner already described for plotting such points. The reference points to be plotted upon the datum line are the chain lengths entered in the field-book in the column headed Distances. These distances are the points at which the levels were taken, and between them, unless otherwise stated in the field-book, the ground is supposed to slope uniformly. Moreover, these distances are assumed to be measured horizontally, and therefore care must be taken to ascertain whether or not they were so measured in the field; if not, they must be reduced before plotting, or the section will be too long. Having plotted the reference points on the datum line, a perpendicular must be erected from each of them, and a length laid off upon this perpendicular equal to the vertical height above the datum line indicated by the entry in the column of the level-book headed Reduced Levels against the distance to which the perpendicular corresponds. To render the differences of altitude more apparent, these vertical distances are plotted to a much larger scale than the horizontal, as explained in a preceding Section. To erect the perpendiculars, theTand set square furnish the most convenient means. The detail points thus determined upon the perpendiculars representthe points in the surface of the ground at which the levels were taken, and by joining these points we obtain the surface line. An example will clearly show the method pursued. Let it be required to lay down the section from the followinglevel-book:—