CHAPTER XII. SURVEYING BY OBSERVATION. A few observations on the great "practical" value and extreme simplicity of the elements of plane trigonometry.-Solution of right angled and oblique angled triangles, with examples showing application to numerous practical purposes, with the use of the tables.-General observations.Maxims.-Base of operations.-Measurement and position of Base. Primary and Secondary Bases. - Practical examples. THE title of "Surveying by Observation " has been adopted for this chapter, in preference to that of "Trigonometrical Surveying,' as being more suitable to such engineering field-work surveys, as require some few stations to be fixed by observation and calculation. The expression of trigonometrical surveying appears more applicable to operations connected with geodæsy, for the formation of a territorial or county map, extensive military plans, and long lines of coast, where details become secondary matters to those more important of position and distance as regards towns, villages, hamlets, hills, mountains and valleys, headlands and bays, and the general directions of roads and rivers. Besides, also, this latter expression would convey an idea of "matters mathematical," which to many readers would be sufficient to make them throw down our poor book, in which we have striven with no little pains to divest every page of all appearance of scientific attempts, and to give such modest information as we have to offer in such manner that any one possessing an average knowledge of arithmetic may understand and practise it. Such information, and indeed much more, is now very generally possessed by the best surveyors connected with engineering works, and it is more or less indispensable for conducting the surveys of river, estuaries, harbours, marshy districts, lands to be recovered, extensive gathering grounds for waterworks; it is often also very applicable where a long and tedious process of many miles of chain triangulation is, we must say, submitted to, in preference to taking, from a well-measured base, perhaps a dozen angles, and calculating a few sides of triangles. But it will be observed by many persons "it is not practical;" those who have put it in practice know better; "it is difficult to learn;" in one week the whole may be mastered. "It is troublesome to remember;" no more so than remembering how to measure the brickwork in a bridge, or the excavation of a cutting, when once it has been practically applied, and this remark may be made of every operation whatever connected with engineering field-work. Perhaps, however, no expression in the language has been more thoroughly abused than that of "practical man," which may, or should be very often taken as synonymous with "quack." We very well remember an illustration of this which occurred to our own knowledge some years since; an "eminently practical man" of the name of Mr. A-, who was in charge of some railway works, had to set out a tunnel on a reverse curve; after keeping the contractor with his miners and bricklayers waiting for about three weeks or a month, he abandoned the task, and sent for his friend Mr. B, the engineer in charge of the next length, to extricate him from his difficulties, and the tunnel was set out; Mr. B's assistant, who was another practical man, afterwards related that the men on the works inquired where that "stranger's peculiar instrument " was, when he replied, "in his head, I suppose-why he has taught me so plainly that I think I could do it myself," and so he could. But we have another very far more remarkable instance at this present time of an educated man becoming practical, in the person of one of our professional men, and whom we would name but that "comparisons are odious;" certainly a few years ago he was not a practical man, but he brought much knowledge of a high order to bear upon the subject he was engaged on, and, besides very soon mastering the practice, he is now one of our most eminent iron bridge engineers. We should certainly very much regret if any reader should suppose from this that we deprecate that knowledge which is practical, and which at starting in life very great sacrifices must be made to obtain; but we have been plain in the above remarks because in our experience we have often seen a good method abandoned for a bad one in mere deference to the cabalistic "practical;" the other extreme error or shoal is equally to be avoided, which is to imagine that by mathematical formulæ alone practical results can be carried out; here again it may be truly observed that extremes meet, and that they must always be avoided. Our profession, however, is based upon mathematical PRACTICAL VALUE OF ELEMENTARY TRIGONOMETRY, 153 knowledge, however obtained, and it is ridiculous to suppose that the elementary truths of the science are not always useful and applicable; every year the study of these truths is becoming more common, the really practical man, who is induced to apply himself to such study, eagerly seeks for more information on the subject, and if in the course of early professional practice, the reader is led to imagine that these elementary truths are less valuable than he at first supposed them to be, we would advise him to delay judgment, and to believe, on the contrary, that he himself has not yet acquired sufficient practical knowledge to know how, when, and where to apply them. In the chapter on "Ranging Curves" we have endeavoured to explain in the most simple manner the nature of the trigonometrical signs as regards an angle not exceeding 90°, and we hope it has been made quite plain and intelligible that the cosine, cotangent, and cosecant of an arc or angle are so called, because whatever angular or arcual measure it may be deficient of 90°, is called the complement of the angle, for the cosine, cotangent, and cosecant of the angle are the sine, tangent, and secant of the complement. Exactly as the term complement is applied to that which an arc or angle wants of 90°, or to the difference between any arc or angle and one of 90°, so is the term supplement applied to the difference between any arc or angle and 180°; thus in Figures 81, 82, and 83, as DO A is the complement of the angle DOB, or as the arc DA is the complement of the arc BD, So is the angle DO B' the supplement of the angle BOD, and consequently so is the arc D B', the supplement of the arc BD. It has already been shown that a right line falling from the one extremity of an arc perpendicularly to the radius which passes through the other extremity of the arc, is the sine of such arc, or the cosine of its complement. The sines increase in length from 0° up to 90°, so that the sine of a right angle is equal to radius, and a glance at the figures will show that the sine of an obtuse angle or of the arc subtending it is the same as the sine of its supplement; for in the two arcs BD and B'D', let B D = B' D'; then BD+B D', or B D + B'D= 180°; therefore B'D=BD' is the supplement of B D, or BO D' = B'OD is the supplement of BOD; and sine D I = sine D' I'. Therefore the sine of an obtuse angle is the same as the sine of the supplement, and consequently in looking in the tables for the sine of an angle above 90°, we must find the sine of the supplement. It also follows that because the sine increases from 0° to 90°, so it decreases from 90° to 180°. The cosine of an arc or of an angle is also equal to the sine of its supplement in length, but opposite in sign; thus IO=0I', but lays in an opposite direction; or as the angle increases from 0° to 90° the cosine decreases in length, but from 90° to 180° it increases in length. From the above definitions it follows that the tangent of an obtuse angle is the same or equal to the tangent of the supplement; thus the tangent of B'OD is the same as the tangent of BOD=BF. As the arc or angle extends from 0° to 90° the tangent is positive, increasing from 0 to infinity, but from 90° to 180° the tangent is negative, decreasing from infinity to 0. To find in the tables the tangent of an obtuse angle we must look for the tangent of the supplement. As regards the derivation of the numerous trigonometrical formulæ, which are all based upon geometrical properties, we shall not proceed beyond the following demonstrations, but refer such investigations to mathematical works on trigonometry, and proceed as speedily as possible to the practical application of the elements of the science to engineering field-work. Given the sine of any arc or angle, to find the cosine, the tangent, the cotangent, and the secant.-In the Figures 81, 82, and 83, given the sine D'I, to find the cosine OI. In the right-angled triangle DIO the square of OI = the square of OD-the square of DI; therefore I O, the cosine = √O DID. Or from the square of the radius subtract the square of the sine; the square root of the remainder gives the cosine required. To find the tangent; in the similar triangles DOI, FOB, OI is to ID as OB to B F; then multiplying the sine by the radius, and dividing by the cosine gives the tangent required. And multiplying the cosine by the radius, and dividing by the sine gives the cotangent required. And √O B2 + BF2 = OF; or extracting the square root of the sum of the squares of the radius and the tangent of any angle gives the secant of such angle. And the square root of the sum of the squares of the radius and the cotangent gives the cosecant. From the above it may be deduced that the cotangents of angles are inversely as the tangents of such angles, or that as the tangents increase in length, the cotangents decrease, or the reverse, as may be seen in Figures 81, 82, and 83. Given the sine and the cosine of an angle, to find the sine and cosine of half the angle.-Let DŎ B, Fig. 84, be the angle of which DI is the sine, and IO the cosine; TRIGONOMETRICAL SIGNS. 155 required the sine and the cosine of half the angle DOB = BOL. Draw the chord D B, and bisect it on H, and draw OL through H vertical to D B. The definition of a sine being a line drawn from one extremity of arc perpendicular to the radius passing through the other extremity, it is evident that DH and BH are respectively the sines of the arcs D L, BL, or of the angles DOL, BOL; and OH is the cosine of each of these angles also by definition. Draw the chord B'D, which will be equal to twice O H. Now in the right-angled triangles BDI, and B D B', DH or B H = sine of D O L or BOL=/BB'x IB. Also because, BB: B'D:: B'D: BI', we shall have B'D2=BB' x B' I, and and B'D B' D = √B B' × B' I, OH= =cosine of half the angle DOB. Therefore the sine of half any angle is a mean proportional to half the radius, and the difference between the radius and the cosine of the whole arc; and the cosine of half an arc is also a mean proportional to half the radius, and the sum of the radius and the cosine of the whole arc. Of course by the tables the above is done at once by inspection. Given the sine and the cosine of an arc; to find the sine and the cosine of twice this arc.-In Fig. 95 let the angle LOB be called A, of which the sine is B H, and the cosine O H. It will be seen that DI and IO are the sine and the cosine of twice the angle A. The right-angled triangles, B HO and BDI, with the angle B common to each, will be similar triangles, and |