In a large survey such an obstacle as this happening in the middle of the base line, it should not be continued upon such a slender foundation as from c to e; better abandon the line or change the diagram. If it happened very near the end of the line, it may then be adopted if executed with peculiar care. But if the obstacle lies in a valley, and the ground on each side so elevated as to see over it and pole the line, then this process would be sufficiently accurate.

Problem 22.

In the four following examples the field-book is dispensed with, and the lines, with their lengths, and such angles as are requisite to plot and prove the accuracy of the survey, are inserted on the plan.

To survey a field with several sides, Fig. 1, Plate 18.

First fix flags or marks to all the stations; commencing at a, chain the lines regularly round the field, leaving marks at 173 in (2), and 176 in (4); then take the angles at a, b, c, which is all that is required.

It must be noticed that by extending line (6 to 173 in 2) it saves taking a very obtuse angle at station (2); as line 1 is fixed by the measured angle, so also will line 2 be fixed. The angle taken at c fixes line 4, consequently line

becomes a proof line, provided the lines and angles have been accurately taken; line (7) is similar to that of (6) by crossing part of the field from c to b; the angle taken at b determines line 6, therefore line 5 is a proof, and the survey is finished; by extending 6 and 7 two angles less are required. Supposing the lines 3), and (6) were extended to d, the intersection of 6 and 7), and line 2 fixes every line in the survey, line 5 being the only one not plotted and the last proof; comparing this and the example, Fig. 2, Plate 10, with Fig. 3, Plate 18, will clearly show which of the three examples are best; in the one instance there is the care and trouble with

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the instrument and time in taking the angles and plotting them, the others require only the chain and offset staff.

Note. When surveys are executed by the chain with the theodolite it is not advisable to take angles at every station; a multiplicity of angles would only create mistrust and confusion, nor is it possible to take angles with that great degree of nicety to be proved by the sum of the whole, as will be shown in the next example; and the method adopted in practice to prove angles taken on large surveys will be further explained.

Problem 23.

Fig. 2, Plate 18. Euclid, Book I., Prop. 32, Cor. 2: "All the exterior angles of any rectilineal figure made by producing the sides successively in the same direction, are together equal to four right angles."

Because the interior angle, A B C, and its adjacent exterior angle, A B D, are together equal to two right angles, therefore all the interior angles, together with all the exterior angles of the figure, are equal to twice as many right angles as the figure has sides.

It has been proved by the foregoing corollary that all the interior angles, together with the four right angles, are equal to twice as many right angles as the figure has sides. Therefore all the interior angles, together with all the exterior angles, are equal to all the interior angles and four right angles, and all the exterior angles are equal to four right angles.

The given figure has five sides; the sum of all the interior angles will be = 5 × 2 — 4 = 6 right angles = 90° x 6 = 540, viz. :

To survey three fields divided by a road, Fig. 3, Plate 18. Although the boundary of this plot is very irregular adjoining the stream, the number of chain lines are few and effective, requiring only three angles by the instrument, as A B C. The

necessary flags being fixed, commence chaining at A; on this line leave two station marks at 867 and 1662, length of line 2210; 2 commencing at the end of 1, leaving off at 550, there fix a flag; now return to 1662 in (1) and chain line (3 ; at 338 the end of (2) completes the triangle on base (1); at 400 put down a peg and continue the line to 586; this line by the triangle is fixed; return to 400 in (3) and chain line (4); at 192 put down a peg; leave the pin in the ground at 200, and proceed to measure the two short lines of the trapezium and enter them in the margin of the field-book, which finishes up that part of the survey; for these short lines take the angle at C 96° 10' with the box-sextant; the trapezium is then proved; now commence again from the 200, at 740 have a peg, and at 1042 have a peg or mark, finishing the line at 1618, there fix a flag. Return to (1), stopping at the last flag 450, and finish the triangle A (which enter in the margin of the field-book): continue the line to 740. Although this line is fixed by line 4), it is possible an error may occur in chaining, therefore to guard against casualties take the angle at A; proceed with (5 and 6, take an angle at B; the line is a proof to this tra (5) pezium; line (7) completes the survey; this is also a proof line.

The plotting of this survey is extremely simple. First plot the angle A, next the triangle D; set off the station and length of (12), and the station and length of (3); the lines (12) and 3 are fixed; apply the scale to line 4 if it agrees with the length entered in the field-book the survey is correct. is correct.

Now

plot the angle B and set off the length; this line is fixed; apply the scale to 5 and 7; the chain lines being all proved, plot the offsets, and all is complete.

This example shows the great advantage of combining the two systems, and not to depend wholly on the use of instrumental angles.

Problem 25.

To survey three fields peculiarly situated, Fig. 4, Plate 18. This example possesses more difficulties in projecting the

chain lines than any preceding it. By comparing it with Fig. 4, Plate 10 (the same plot), it is at once very evident that the theodolite reduces labour and time by the reduction of chain lines.

Commencing at line (1), at 239 measure up the small triangle and enter it in the margin of the field-book; at 762 leave a flag, to be seen from the road, for line 6, which, by taking an angle, will be a fixed line; finish chaining line (1) and the small triangle at the left; next proceed with line (2), which passes through the hedge near 575. It will be observed this line divides a very crooked fence, making the offset short on both sides, and avoiding the wood; at the end of this line take an angle to fix line (3), supposing permission was not given to cross the field next the road, otherwise the angle would be useless, as lines 3, 4, and 5 would be proof lines. The cot(5) tage-garden in the one case must be taken up by a small trapezium; in the latter, it can be taken up by offsets from the two lines (3) and (6 An angle must be taken at the end of 5 which determines the position of that line; in fact, by measuring the whole of 6), and the two angles on that line, determines the whole survey.

mark off the different

To plot this survey first draw line 1 lengths, and the angle at 762; draw line

6, mark off the length 507 links, and prick off the angle 108° 0'; draw line

mark off the station 383, and the length 766 links, proved

by the scale, as also line (4

In every kind of survey the diagram or chain lines should always be kept in the mind, to determine on certain angles by which the work is to be plotted, as it is only then that the correctness or defects of the survey can be discovered, which shows the necessity of plotting the chain lines immediately.

It is always better to have a line or angle too much, than to want one, as frequently a very trifling line or angle would prevent many difficulties.

Problem 26.

To survey a small estate, Plate 19.

The great irregularity in the boundary of this estate shows that the survey by the chain only increases the labour, though that would not be so in every case; much depends on the fences within the boundary.

The base line (1) is similar to that in the former example, Fig. 1, Plate 11, and has the same effect in fixing the position of the small field at the extent.

The triangle formed by 4 and 42 is confirmed by the small tie line; all the other lines are fixed by the angles taken with the instrument, and are chiefly on the base line; when plotted the remaining lines are all proof lines.

Problem 27.

To survey a road within the fences, Plate 20.

This example is well adapted for the Prismatic Compass, full description of which is given in Part V.

The centre of the instrument is fixed directly over the station point in the same manner as the theodolite; the engraved card is divided into degrees and half degrees, and attached to the magnetic needle. It must be observed that the card reads reversed—that is, north for south.

In taking an angle by this instrument, place the box containing the needle as level as possible, look through the prism to the forward object, and bring the thread in the slide to coincide with it and the flag in front; when the card is perfectly steady take the angle.

In a survey of this kind, where extreme accuracy is not required, the prismatic compass, from its portability and readiness in fixing, makes it preferable to any other instrument.

To plot this, draw a meridian line through every station, as at A B, &c.; lay the straight edge of the protractor along the line with 360° at the top and the centre at the station, then prick