for greater radii, instead of using the mile of 5280 feet, the round numbers, 5000 feet, or 6000 feet would be preferred. The American system of nomenclature also requires explanation. It is one of great convenience in one respect, and for one system of setting-out, viz., that where an instrument is used, as it assumes for its basis of nomenclature the centre angle subtended by a chain of a hundred feet; thus naming curves, as one-degree curves, two-degree curves, three-degree curves, etc., when the centre angle subtended by a hundred feet on the curve is one degree, two, or three, etc. The mistake adopted in this system consists in naming the centre angle the angle of deflection; this not only creates confusion, but is obviously incorrect. It is most evident that the true angle of deflection is that made by the chord of the curve with its last or previous piece of straight, that is to say, the angular deviation of direction as observed by a person walking along the line; and this true angle of deflection or angle at the tangent point is exactly a half the centre angle subtended by the same arc. Again, a little consideration will show any one that it is quite unnecessary to deal with centre angles at all; no practical system of setting-out recommends one to proceed to the centre of the curve-circle. An improved angular nomenclature would deal only with the actual angle of deflection; the curves being named one-degree, two-degree, etc., curves in accordance with the angle subtended at the tangent-point by an arc of 100 feet. The following small Table gives the radii in feet corresponding to curves thus named. TABLE GIVING ANGLES DUE TO AN ARC OF 100 FEET FOR VARIOUS CURVES. This system of nomenclature, though preferable to the former, will doubtless soon become obsolete, as it is based on a single method of ranging curves now not exclusively adopted; and the other method of naming curves according to their radii in feet seems more likely to remain in vogue. Collection of Formula for Curves. (These angles are for conciseness in degrees and decimals.) Angle of intersection of tangents=a. Centre angle (B) = 180°-a. Angle of deflection (y) for the whole curve 90° — 1a. Angle of deflection (8) for an arc (7)= 28°64789 × R Angle of deflection to middle point of curve=45° — 1a. Length of curve-tangent, T=R tan. y. Distance of the middle point of curve from point of intersection, D=R sec y-I. Chord of the whole curve, C=2R sin y=2 √2 l.R. — V2. Distance from middle of chord to middle of curve, L2 √ (R2 _ C2) = 72, (approx.). V=R vers. y=R— √ √ R2 4 8R2 Total length of curve, L=0349066 x Ry=ln. δ The length of any arc (/) whose chord is (c) 1=}(8 × chord of 3/—c)or c=1[1 [2 24 R2 approximately. The ordinate (y) for any abscissa (x) measured from the tangent-point along the tangent=R- √R2—x2. A normal (d) taken at any point in a curve, when (†) is the intercepted distance along the tangent, d = √R2+t2-R. The ordinate (y) for any abscissa (r) measured from the middle of the whole chord along it Offsets from the production of a Series of equal chords. Rectangular offset = 2y; y being the first tangential ordinate. |