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The above table having been constructed, take out from the tables the sines and cosines of the angles formed by the bearings with the magnetic meridian, and multiply them by the lengths of the chain lines; in the column of northings, enter the products of the cosines, and in the column of eastings enter the products of the sines. The latter added up 165 06, will give the distance travelled eastward from the point A, and the total of the northings = 211.87, will give the distance travelled north of the point A. If now we were to set out in the direction due west, the length of the eastings = 165'06, we should return to a point due north of A, but the bearing of our line A B = angle of 15° with the magnetic meridian; so that whilst we have been travelling to f, a distance of 165-06 east, the line A B may be said to have travelled a certain distance east also, according to the angle NA B, and the length of N A. This latter easting will be equal to Ng, and if we can find the point g, we shall have returned exactly to the required position on the line А В.

Now observe that in the triangle N Ag, the cosine A h of the angle N A g, is to the sine h s, as the northing A N is to the easting N G, or,

9559 211-87: 2588: 57-36;

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therefore 165.06 57.36 107.70 = ƒ g, the distance due west from the point ƒ; and a line N 15° E from the point g, thus found, will restore us upon our original route in the direction A B. This bearing will have to be set out by making an angle B gf=75°; for 75° + 15° 90°, and the direction of A B produced will be determined by the line gf, instead of depending entirely on the needle in the compass box. In the same manner, when setting out the line fg, due west, instead of depending entirely on the magnetic needle, make the angle gfe equal to 57°, for the angles at e and ƒ are together equal to 90°, and 90° 33° 57°. Should there be any reason to fear any variation in the needle from local attraction, fix on any object as at O, and before leaving f, measure the angle gƒ 0; leave

POCKET TRIGONOMETRY.

107

some mark at f, and on reaching the point B measure the angle g Bf, and the angle f B B', which, together with the angle N' BB' formed by B B' with the meridian, should be equal to 180° + 15° = 195°; A g = AN × secant N A g = secant 15°, 211.87 × 1.0365 = 219'60 = A g.

The application, however, of this method, and of many others constantly in use in engineering field-work, presupposes a table of sines, cosines, &c., ready at hand, and which it is not always convenient to carry about; for the purpose of meeting this, which might be considered an objection, a table, consisting of a very few pages, may be readily copied and kept in the fieldbook, and which will be found to answer all general purposes for which trigonometrical tables may be required.*

Example.-Required the natural sine of 40° 2′; from the table of sines take the sines of 40° and of 40° 10′; subtract the less from the greater, the remainder will be 00223, which divide by 10; this being done by prefixing zero on the left hand, multiply by 2 for the two minutes, and add the product to the sine of 40°, which will give the sine of 40° 2′, sufficiently near for all practical purposes of field-work. Had the minutes been any other submultiple of 10, as for instance 7' instead of 2', then 7 would have been the multiplier of the difference divided by 10. Had we required the sine of 40° 25', then the first figures sought in the tables would have been the sines of 40° 20′ and 40° 30'.† The writer has for years carried a copy of these tables in his pocket-book, and has never required any others for general field purposes. With regard to using logarithms for such matters, there is only to observe that the divisions and multiplications effected are so short, that they are done in less time than a table of logarithms can be referred to. There are many persons who carry about with them a volume of logarithms of numbers, and logarithmic sines and cosines, and who might save themselves the inconvenience. With rare exceptions, such tables are only for the study and the office.

We close our remarks on traverses and bearings by observing that, in the case of circumstances obliging us to travel westwards instead of towards the east, as heretofore, we still take the bearing from the reading on the limb of the theodolite. In Fig. 76, let the east and west line be substituted for the former magnetic

*The reader totally unacquainted with the nature of sines and cosines will be so good as to refer to the chapter on "Setting out Curves," where he will find a full explanation of their meaning.

It is to be observed and remembered that the difference divided by 10, and multiplied by the number of odd minutes is to be added for sines, tangents, and secants, and subtracted for the cosines, cotangents, and cosecants, as will be more fully shown in the explanation to the table in question.

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meridian, so that the former north point shall be west, and the south shall be east; the angle NAc, instead of giving a bearing of 70° with the meridian, will now give one of 15°, being the angle N'A c, and the reading on the limb of the theodolite will be 345; and 360°-345° 15° the bearing N 15° W. When the theodolite is moved to station c, the bearing of the back angle vernier will be 345°- 180° 165°, before the bearing of cd is taken. The former bearing of c d was 35°, and its bearing with the present meridian will be 55°, for 35′ + 55° = 90° ; the reading on taking the bearing on the limb of the theodolite, will now be 305°; 360° — 305° = 55° = N 55° W. Similarly the bearing of de, as taken at d, will be 291°, which subtracted from 360°, leaves N 69 W.

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A little attention to this in the manner we have already described will fully explain it.

CHAPTER VIII.

Railway, Canal, and Road Surveys.-Setting out Bases, and General Management.—Exploring Surveys.-Canal Survey, and Tracing Levels.-General Features.

LAND surveys required for the above purposes generally consist of long but narrow ranges of land, twisting about in various directions, according to the route adopted; the width of survey required will generally also depend on local circumstances fixing the position of the line in one or more portions to some particular spots; where this is the case a width of about ten chains may be sufficient; at other times some uncertainty or other may render a much greater width necessary, as for instance, half-amile; the average width, however, generally surveyed does not exceed ten chains on each side of the centre line, excepting in town lands, marked on some general map, and which in England is usually now a sheet from the Ordnance map, by which the surveyor is guided in his route. Where such a map does not exist, then the best general map that can be procured will be used; at other times certain rivers, streams, roads, and villages will be named, and certain distances from these will be prescribed or certain bearings by compass combined with landmarks, but in these latter cases a much greater width of survey will be adopted, and a smaller scale than now used at home.

Whatever these circumstances may be, as soon as the surveyor reaches the scene of his labours, the first thing to be done will be to get two or three men as chainmen, who know something of the locality, and who will be able to point to some of the neighbouring objects, such as churches, villages, mills, or homesteads; anything, in short, as guiding points. Having satisfied himself on the ground as to the direction of a portion of his survey, it will be necessary to set out the base lines, whether these be long or short; if the direction of the proposed line of railway be pretty straight, then one or two long base lines, or more, according to requirements, may be made to run through the entire extent of the survey; whilst on the other hand, there

may be such short and sharp deflections that the bases will be proportionately short also. Where the bases are thus of inconsiderable lengths, as for instance half or three-quarters of a mile, they are generally very easily set out, and often do not require any setting out whatever, it being merely necessary to select some well-defined and prominent object to chain on, and to leave a flag-pole firmly fixed at the starting point;* in such cases the chaining may commence at once. But the base line may be of much greater extent, when, on account of undulating ground or tall vegetation, it will require setting out, an operation termed "boning." Under any circumstances, select some such welldefined object as we have already mentioned, if there be one, for the mark to chain on; it is not necessary that it be in the survey; rather the better if it be some distance beyond; neither is it necessary that the base should run exactly or quite through the centre of the survey; it is merely requisite that the fieldwork be based upon it; neither is it to be expected that it will be possible to bone such a base from either end of it; but with a theodolite it may often be done from some commanding point, from which the far object may be seen. We shall best illustrate our meaning from Fig. 77, which is a portion of a railway survey in which AB is a base line nearly three miles and a half long. A rough idea of the direction this base would take was ascertained from the commencement, but A, laying in a hollow, it was necessary to find some better spot for defining more accurately the direction of the base, and for boning it out. In the far end, beyond B, the centre of three tall poplar trees was selected as the object to chain upon; the point A was at first quite undetermined, and might fall up or down the road either way. It was only at some distance that high ground could be perceived. Taking our theodolite and poles we walked towards it across the fields, examining the enclosures right and left, and noticing as we went along how cross lines to take up the work would fall into what we supposed generally would be the direction of the base. On reaching the village, and observing the roads, knowing there was to be a junction of a branch with the main line at about this spot, it was determined to traverse these roads; passing onwards we only reached sufficiently high ground to command the view both ways, from the

*It will, of course, be understood from what has been already observed in these pages, that, long or short, these bases run through the survey.

The surveyor is supposed to have examined the length of his chain, and to have marked the correct length of it in some convenient place near his quarters.