vour to plot it to a small scale, you will find you cannot get all the details in ; they are simply too small to be shown. The surveyor must learn to discriminate between such cases, so as not to waste his time, on the one hand, or under one state of things, and not to neglect his work on the other; a little thought and observation, after a few days' field-work and plotting, will make these remarks obvious enough. If, during the perusal of the above pages, the reader has taken the trouble to practice the instructions here given, and has endeavoured gradually to rely upon his own intelligence and discrimination, equally as upon our advice, there is every reason to hope that he will by this time have obtained considerable insight into the business of Land Surveying, including the use of all the instruments required ; but inasmuch as it is much more an art of experience than of theory, if we may be allowed the expression, we must strongly impress on his mind that it is only by practical study on the ground that he can attain skill combined with readiness in the performance of his work; without practice no amount of reading or teaching will ever make a surveyor. CHAPTER X. Ranging Curves.-General Observations.—Tangent, Secant, Šine, &c.—Practical Ecplanation and Application of Trigonometrical Tables.- Practice of Ranging Curves on the Ground. The author feels that he will be accused of pursuing a very unusual course, in placing this chapter amongst the remarks on the different systems of surveying; he can only submit as an excuse for doing this, that he considers the subject as affording the means of introducing the non-mathematical reader to the understanding and practical application of a little plain trigonometry; such as many circumstances in field engineering operations render at times of every-day use, when, if the reader happens not to possess such elementary knowledge, he has only to follow some rules which he only partially understands, and is consequently apt sometimes to misapply; moreover, if he happens to meet with an unusual difficulty, and he may expect to meet with such a condition at any time, he is certain to have double and treble labour to overcome it, from the want of understanding a few rules as simple as the arithmetical calculations he has to perform every day. A sound knowledge of the rationale of the system to be pursued in setting out or ranging curves is not only requisite in railway work, but to engineering field-work generally, and this rationale is based on "right ; angled trigonometry,” which it is as simple to understand as the easiest rules of geometry. Circumstances constantly arise where one method is preferable to another, and not unusually cases occur when the engineer has to construct some system to meet special difficulties; to effect this he must understand the principles on which all the systems are based. We are quite aware, that many practical men will at once turn away from the mere term "trigonometry,” for they appear to entertain an intuitive dislike to everything approaching to mathematics, little aware, as they consequently must be, that they are every day performing some mathematical operations on the ground. As a practical man myself, I am not without hopes that such readers will now glance over and examine the following half-dozen pages, when I think that much of this feeling will be removed ; "plane rightangled trigonometry” will then be easily understood as well as the use of the tables ; this will form an introduction to obliqueangled trigonometry, or at least as much of it as we require for “surveying by observation,” or for harbour and coast work, and occasionally under other circumstances. Such is the writer's excuse for placing this subject where he has, and also for treating it in the following manner, very un. mathematical and unsymmetrical, but such that any practical man will understand. In Fig. 81, let A B be a curve, which it is required to trace on the ground ; let AC be a tangent (so called because it touches the curve without cutting it), from which at the point C it is required to measure off the offsett CD, so that D shall be a point on the curve. Observe that the length CD is the only line which, measured off at right angles to A C, from the point C, will exactly intersect the curve at D; also that C D is equal to the trigonometrical sign A E, which is called coversine, and that A C is equal to the line 0 I, called cosine of the angle DOB; also that the end D of the line C D, intersects the curve at the point where this curve is also intersected by the line O F, called the secant, and which forms one side of the angle D O B. Observe also that C D, plus the line called the sine, is equal to O E, plus the line called the coversine, and that the sine, plus C D, or plus the coversine, is equal to the radius of the Observe again, that whatever fraction or portion of the radius AC may be, these proportions must always exist; see Figs. 82 and 83 also. The angle DO B has been mentioned above, as also some of the trigonometrical lines; it will now be time to refer to the others, premising, however, that we are not going at first beyond the right angle, or an angle of 90 degrees, because this will be sufficient for our present purposes, and because by attempting more it would perhaps only confuse the reader, and interfere with the speedy application to practice of what we are about to say, which speedy application will we trust lead the reader on to pursue this subject, as he must see at the very commencement how useful, the very simplest knowledge of the merest elements of trigonometry may be made. An angle is subtended by an arc, as D O B is subtended by the arc D B, or if a straight line be drawn from D to B, the angle DOB will be subtended by the chord D B. The circle being divided into 360 degrees, and each degree into 60 minutes, curve. and each minute into 60 seconds, the angle may be measured by the number of such divisions and subdivisions contained in the arc. TANGENT AND SECANT. The tangent of an are, or of the angle it subtends, is the straight line drawn from the beginning of the arc B, at right angles to the radius, which forms one side of the angle, and is terminated where it is intersected by the other side of the angle produced ; in the three figures already referred to, BF is the tangent of the angle DOB. The leg or side of the angle produced to intersect the tangent is called the secant, thus, o F is the secant of the angle DO B, of which BT is the tangent. SINE. The sine of an arc, or of the angle subtended by it, is a line drawn from one extremity of the arc at right angles to the radius, drawn from the other extremity of the arc, and through the centre of the circle, thus, DI is the sine of the angle DO B. Both the sine and the tangent are perpendicular to the radius, drawn from and to the same point, and they are therefore parallel. Each of the above three signs increases as the angle increases. COSINE. The cosine of an arc, or of the angle subtended by it, is a portion of the radius, intercepted at the centre by the secant at one end, and by the sine at the other end; it is at right angles to the sine and to the tangent. This sign decreases as the angle increases. OI is the cosine of the angle DO B. VERSINE. The versine of an arc, or of the angle subtended by it, is that portion of the radius intercepted between the sine and the tangent, to both of which it is perpendicular. The versine increases as the angle increases. I B is the versine of the angle DOB. Secant? = Radiuso + Tangent?, or the square of the Secant = the square of the Radius + the square of the Tangent. The square of the Radius = the square of the Sine + the square of the Cosine. Cosine : Sine :: Radius : Tangent. ♡ (Radius+ Tangentạ) = Secant. COMPLEMENT OF AN ANGLE. The complement of an angle is that portion of a right angle which an angle wants of a right angle; or the complement of an angle is that portion of a K + right angle which the angle wants of 90 degrees; thus, if the given angle be 50 degrees, then the complement of the angle is 40 degrees. DOB being the angle, A O D is the complement of the angle; and, therefore, an angle and its complement are always equal to 90 degrees. COTANGENT AND COSECANT.-The cotangent of an arc, or of the angle subtended by it, is the tangent of the complement. It is a straight line drawn from one end of an arc at right angles to the radius, such radius being one side of the complement, until intercepted by the other side produced ; this other side produced, until it intersects the cotangent, is called the cosecant, and is to the complement what the secant is to the given angle, that is the secant of the complement. The cotangent is parallel to the cosine or versine, and the cosecant is on the same line as the secant. Thus A G is the cotangent of the angle DOB, and . the tangent of A OB; O G is the cosecant of the angle DOB, and the secant of the complementary angle A OD. COVERSINE.-The coversine of an arc, or of the angle subtended by it, is to the complement that which the versine is to the given angle; thus A E is the coversine of the angle DOB, and it is the versine of the complementary angle A OD. The coversine of an angle plus the sine of that angle = radius, for I C = A 0=0 B = to radius; therefore radius minus sine = coversine. Thus much as to all we shall require from a knowledge of the trigonometrical signs as regards the setting out curves. Now, let it be supposed that in the figures 81, 82 and 83, the radius represents 1, 10, or 100 miles, chains, feet, inches, or any other measure, it will be seen at a glance that the signs we have above referred to, that is the tangents, secants, cosines, sines, &c., will be either less or greater than, or equal to, the radius, and that their proportions to the radius will increase or decrease as the angle DO B is more or less acute. Thus in Fig. 83, where the angle D OB is greater or more obtuse than in Figs. 81 and 82, the tangent BT, the secant OT, the sine I D, the versine I B, are greater in Fig. 83, in proportion to the radius, than these signs are in Figs. 81 and 82 ; but as regards the cosine 0 I, the cotangent A G, the cosecant OG, the coversine A E, these signs will be less in proportion to the radius in Fig. 83, where the angle DO B is greater than in Figs. 81 and 82, where the angle D O B is less. APPLICATION OF TRIGONOMETRICAL TABLES. If, therefore, we take the radius as equal to any given lineal measure, it is clear that the signs above referred to will have a |