25 checked by such means as I have briefly described; but there are cases, such as woods or ponds, in which it is impossible to get through or across, where it is necessary to chain round, taking the exterior boundary, and fix the relative directions of the lines circumscribing the figure by means of what are called chain-angles. I have already explained that three sides of a triangle measured is no proof of accuracy, to ensure which a fourth or tie-line is required. This is all the more necessary in the case under consideration, where we have, as in Fig. 39, to run our lines all round outside, and have to prove our work. Here we have to tie our lines in such a manner as to comprehend the outline of the wood, through which it is quite impossible to survey. Briefly, to prolong lines 1 and 4 and to tie their extremities by the line A a would not be sufficient to ensure the angle, therefore a second tie a a' is necessary, and, similarly, lines 3 and 4 by means of the ties D d' and D d. The acute angle formed by lines 1 and 2, although tied by bb' (which serves the double purpose of a survey line), could hardly be trusted unless checked at the other extremity of 2 by the ties c c', c c, c c2. Fig. 39.-Chain Angles. I might give numbers of instances of how such figures may be circumscribed by means of lines and chain-angles, but in these days, when instrumental observations have superseded such methods, I deem it to be unnecessary to dwell upon the subject. CHAIN ANGLES. 4 (3) B Inaccessible Distances.-It rarely happens that a survey of any extent can be carried out without some difficulties being encountered, such as base or important chain-lines being interrupted by obstacles, in the form of rivers, arms of lakes, ponds, buildings, &c., when it is necessary to resort to some means of working round in the one case, or by geometric construction, or angular observation to ascertain the intervening distance. This strengthens my argument in favour of reconnoitre previous to commencing a survey, as in undulating ground a building or other obstacle which had been unobserved might come directly in the line, which by careful arrangement beforehand might have been avoided. In the absence of any instrument, such as a box sextant or optical square, a right angle may be approximately set out on level ground by the following simple method. Measure forty links on the chain-line, and put arrows, as at A and B (Fig. 40), then with the end of the chain held carefully at a take eighty links, and instruct another chain-man to hold the eightieth link at в; take the fiftieth link in your hand and pull from a and в until they are fairly tight, when an arrow at c will be perpendicular with the line A B, in other words A B will equal 40, в c 30, and c A 50 links. Line Chain 40 Links 50 Links 30 Links C Fig. 40. I have said this may be done approximately on level ground, but I do not recommend any reliance being placed upon a right angle set out in the manner above described if intended to overcome a difficulty such as is represented in Fig. 41, where the line A B is interrupted by a house. In this case it is assumed that if at a, on the line a B, a right angle be set out (as explained) and a sufficient distancé a c, say 60 links, measured, and C D (made perpendicular to a c) 80 links, and D b at right angles to C D measuring also 60 links, and b в made perpendicular to b D, then a b will be within the points A and B, in other words in the same line, supposing the building did not obstruct. Thus four right angles have to be set out and measured to carry the line A в past the building. I recommend the student to practise this problem on perfectly level ground, and I venture to think he will agree with me that, unless the line b в has been ranged. from A upon sufficiently high ground to see over the building, very little reliance must be placed in the prolongation of the line a a by such means as I have described, and yet there are numbers of works on surveying which give it as a practical example. C I can only say that I should observe the greatest care in checking with a theodolite such work before I should trust to such a prolongation. 1 !B 30 30 30 30 Fig. 41. I have selected one or two such examples of measuring over inaccessible distances, across rivers or ponds, by the chain only, as appear to me to be capable of satisfactory results, if great care and accuracy be observed, for, unlike the case of the building, you can command all points. Suppose, as in Fig. 42, the line A B is intercepted by a river, the width of which is too great to ascertain by measurement across. We must therefore proceed to set out such a figure on one side of the stream as will enable us to range across it a line which shall so intersect the line AB, that this point of intersection shall be equidistant from a given point to another point, to which we are able to measure on the ground. First, range the line AB across the stream, sending a man with rods to establish on the other side where directed in the first Bic instance at B. From any convenient point b measure towards a such a distance as judgment tells to be greater than that across the river, say 400 links, at a the extremity of 400, and b, set out right angles, and from b measure 300 links to b', and from A 600 links to a'. Place rods at a' and INACCESSIBLE DISTANCES. Fig. 42. (having previously checked the lines A b' and a' b' which should respectively be 500 links); now range through a' and b' the point c on the line A B, then cb' will equal a' b', viz. 500 links, and b c will equal a b, viz. 400 links. Measure from each edge of the stream to b and c, the sum of which deduct from 400, and you have the width of the river. Again, in Fig. 43 at c on the line A в set out the perpendicular c D, and make it some equal number of links, say 400; bisect C D in b, and at D set out the right angle C D c, make D c = 300 links, place rods at c and b and range the line through until it intersects A B in B, then C B will equal D C = 300 links. Similarly, if the line passes obliquely (Fig. 44), set out any line parallel (approximately) AB RSITY 600 Re 27 Fig. 43. Fig. 44. with the bank of the river, as c D, measure 200 links either way, at each end set off the perpendiculars D A, C B, then will C B = C A =540 links. Again, as in Fig. 45, measure off the perpendiculars B C, DE, ranging the point c in line with A E; then BCX BD All the foregoing are fairly good methods of determining in accessible distances, in the absence of instruments for taking angles, but I need hardly say that the right angles should be set out with an optical square or other reliable appliance, and even then the very greatest care must be observed. The simplest, quickest, and most reliable method of determining an inaccessible distance is as follows: (Fig. 46) at c, with a box sextant or theodolite set out the line C D at right angles with a B, measure any distance c D, and at D observe the angle E D C. Then nat. tan. E D C X C D. CE For example, the angle EDC is 51°, and the length cD450 links. Now nat. tan. of 51° = 1.2349. .. 1.2349 × 450 = 555·7050 links, which is the length c E. Should there be any doubt as to the accuracy of the observation or calculation, place the instrument at E and observe the angle c E D, which should equal 90° — 51° = 39. I now leave this branch of my subject, as in subsequent chapters I propose to treat the whole question of field work in greater detail. CHAPTER III. SURVEYING INSTRUMENTS. In the early days of surveying only very primitive instruments were available, but nowadays the science of surveying has attained such a state of perfection that we have instruments of all kinds for facilitating geodetical operations in the field. Cross Staff. In the foregoing chapters I have referred to the process of taking offsets with an offset staff, which for short lengths may generally be relied upon. Although I am bound in this division to refer to the cross-staff, I have no hesitation in condemning its use upon nearly every ground. I look upon such appliances as only an excuse for long offsets, against which I am very strongly opposed, and with such feelings I naturally discourage their use. Indeed, apart from this prejudice, I cannot see any feature of recommendation in the cross-staff except for approximation. The cross-staff is made either cylindrical or octagonal in shape, about three inches in diameter (see Figs. 47 and 48) and five inches deep. It has slots placed at right angles to Fig. 47. each other, in which are con- . tained fine wires strained very true and vertical. In the octagonal staff there are also slots on the other four faces, which may be used for approximating an angle of 45 deg. The staff is fixed. upon a rod (spiked at the end), and being Fig. 48. placed perfectly perpendicular at a point on the line A B (Fig. 49), at which it is desired to set out a right angle, the slots a and b are adjusted so that, looking from a to B and back from b to A, the wires are coincident with the points A and B. Many cross-staves have a compass fixed at the top, as in Fig. 47, which-provided the staff |