ABBREVIATIONS. + Increased by. Diminished by. X Multiplied by. • Divided by. Equal to. Since, or seeing that. .. Hence, or therefore. : Indicates the quotient of one divided by the other of the quantities it connects, called sometimes the ratio of the quantities. :: Indicates an equality of ratios, and connects equal ratios in a proportion. Thus, a :b::c: d indicates that a b=0 id; or it may be read, a is to b as c is to d. ( ) Brackets indicate that the operations embraced by them shall first be performed, and the result treated as a single factor in the remaining processes required by a formula. Thus, (a X 0) = (a + b) requires that the product of a and b shall be divided by their sum. A2. A small secondary figure annexed thus to an expression is called its exponent. It requires the principal to which it is attached to be used as many times in continued multiplication as there are units in the exponent. Thus, A2 = A X A; A? = A X A X A, which is called the cube, or third power, of A. ✓ This is called the square root sign: it signifies that the square root of the quantity covered by it is to be taken. ♡ If preceded by a small secondary figure, called the index, as in the marginal figure, it indicates that the cube root of the quantity covered by it shall be taken; and so on. v If the index be fractional, as in the marginal figure, it requires that the square root of the thirıl power of the quantity covered shall be taken. B. M. Benchmark : any fixed reference point for the level, ix as outcropping ledge, water-table of building, or other permanent object. Usually a blunt conical seat for the rod, hewn on a buttressed tree-base, having a small nail sometimes driven flush in the top of it, and a blaze opposite, on which the elevation is marked with kiel. T. P. Turning-point : usually marked O in the field-book. P. I. Point of intersection; as of tangents, which are to be connected by a curve. A. D. Apex distance : i.e., the distance from the P. I. to the point where a curve merges in the tangent. P. C. Point of curve : the stake-mark at the beginning of à curve. P. T. Point of tangent: the stake-mark at the end of a curve. P. C. C. Point of compound curvature : the stake-mark where a curve merges in another of different curvature, turning in the same direction. P. R. C. Point of reverse curvature: the stake-mark where a curve merges in another turning in the opposite direction. B. S. Backsight, in transit work; or the reading of the rod to ascertain the instrument height in levelling. F. S. Foresight, in transit work; or the reading of the rod to ascertain elevations in levelling. H. I. Height of instrument : elevation of the level above the datum or zero plane, H. W. High water. TABLE OF CONTENTS. PAGE II. Manner of using the tables To find the logarithm of any number To find the number corresponding to a given Multiplication by means of logarithms Division by means of logarithms To raise a number to any power by means of To extract roots by means of logarithms VII. Solution of plane triangles VIII. Right-angled plane triangles ADJUSTMENT AND USE OF INSTRUMENTS. IX. General remarks on adjustment To bring the intersection of the cross-hairs into . . To adjust the vertical hair so that it shall re- volve in a plane, and mark backsight and fore- sight points in the same straight line PROPOSITIONS AND PROBLEMS RELATING TO THE CIRCLE. XVI. Propositions relating to the circle XVII. Circular curves on railroads XVIII. To find the radius, the apex distance, the length, the degree, &c., of a curve Given the intersection angle I and radius R, to Given the intersection angle I and tangent T, to Given the intersection angle I and chord AB=C, connecting the tangent points, to find the Given the intersection angle I and the degree of curvature or deflection angle D, with 100-feet chords, to determine the length of the long chord C, the versed sine V, the external secant Given C, V, S, or T, of any curve, and D, the degree of curvature, to find the intersection Given the intersection angle I and deflection angle D, to find the length of the curve. Given any radius R and chord C, to find the de- Given any radius R and chord C, to find the de- . . . Given any radius R and chord C, to find the Given the radius R, chord C, and middle ordi- nate M, to find any other ordinate Ordinates of a 1o curve, chord 100-feet TRACING CURVES AND TURNING OBSTACLES IN THE FIELD. XX. To trace a curve on the ground with the chain XXI. To trace a curve on the ground with transit and XXII. Turning obstacles to vision in tangent XXIII. Turning obstacles to measurement in tangent 73 SUGGESTIONS AS TO FIELD-WORK AND LOCATION-PROJECTS. XXIV. Suggestions concerning field-work XXV. The curve-protractor and the projecting of loca- Table showing the distance, D, in feet, at which a straight line must pass from the nearest point of any curve struck with radius r, in order that a terminal branch having a radius R=2 r, and consuming a given angle, x, may Table showing the distance, d, in feet, at which curves of contrary flexure must be placed asunder, in order that the connecting tangent, XXVI. How to proceed when the P. C. is inaccessible . 93 XXVII. How to proceed when the P. C. C. is inaccessible, 95 XXVIII. To shift a P. C. so that the curve shall termi- XXIX. To substitute for a curve already located one of different radius, beginning at the same point, containing the same angle, and ending in a XXX. Having located a curve A B C, to find the point B at which to compound into another curve of given radius, which shall end in tangent EF, parallel to the terminal tangent of the |