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In order to get the direction of the new tangent onwards from the sixth point, measure from the third point in the direction of the fourth the small extra distance ED, given in the tabular data: thus obtaining a point on the curve exactly half-way between the starting and the sixth point, or in other words, the middle point of that portion of curve. From this point measure the normal distance DF, also given, in such a way that the other end of it may fall in the original tangent. The point in the tangent thus obtained gives a direction, taken through the sixth point on the curve onwards, which is that of the new tangent.

This operation may be repeated for every new tangent, and will be found to be practically correct, the theoretical error caused by measuring the small extra distance ED as a chord instead of on the curve being insignificant in practice.

The check given in the table is the distance CM from the sixth point on the curve to a point on the original tangent, and is measured in the same way as the quantity DF. The point thus obtained gives a direction, taken through the twelfth point on the curve onwards, which is the direction of the third tangent used. And this will be found a useful check on the direction of this tangent, obtained by means of the second tangent in the manner previously described.

The tabular quantity (curve), gives the exact length of curve set out for every six points.

The few numbers required for setting-out a curve to any one radius may be copied on a slip of paper from the table, and after a little use, may even be borne in mind without that aid.

The first portion of the table will be found most

applicable for radii in Gunter's chains, up to a mile or 80 chains the intervals when the radius is more than 40 chains, being double or 2 chain distances: the latter part again is more suitable for radii in hundreds of feet up to 5280 feet or I mile, and for intervals of 50 or 200 feet but the numbers given have a sufficiently wide range for setting-out a curve, either with inches, yards, mètres, or any unit whatever.

For Example.

In setting out a curve to the radius 900 feet by offsets from six points, taken 50 feet apart on successive tangents.

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Referring to figure 31, the points marked 1, 2, 3, 4, 5, and 6, are the equidistant points 50 feet apart on the tangent, as given in the set above, in distances from A, viz., 50, 100, 150, 200, 250, 300; from these points the rectangular offsets given in the slip as 139, 5'57, 12:59, etc., are set off, giving as points on the curve the points marked I., II., III., IV., V., VI.

Now to obtain a new tangent at the point VI., first measure from point III. towards point IV. the short dis

tance ED, given on the slip just quoted as M=2:22 feet, and thus fix the point D. Now put the handle of the chain or tape on the peg at D, and taking the distance DF= 13.15 feet on the chain, let it rotate freely and range this length so that F may fall exactly in the tangential line AM; this can be easily done by an observer standing at 6, the sixth point on the tangent and looking in the direction of A. The point F being thus obtained the direction FCGH through VI., the sixth point on the curve, is the new tangent required.

From the new tangent six more points can be setout with the same foregoing figures, and after that a third tangent may be obtained in the same way as the last; and so on to the end of the curve. Each time that six points are set-out the length of curve set-out by them amounts to 305 85 feet, and the distances AD, DC, CL are each 152.92 feet. CM or IN is the quantity given as the check normal=54:59 feet; it is, when required, laid off from C or 7 in the same way that the normal DF was laid off from D; the extremity Л being in the tangential direction is marked by a pole; on arrival at f it will be found that the points G and M are in exactly the same direction, if the curve has been carefully setout, and this affords a useful check on the work.

The table and the figure are made to suit a sixpoint system, but the same principle might be applied to four, eight, or any even number of points, not requiring long offsets. The advantage of this system is that angular instruments are entirely dispensed with, a crosshead staff being sufficient for all alignment and settingout the right angles; the method is so simple that after explaining it once or twice in practice, good chainmen can do it by themselves.

TABLE FOR RADII AND DISTANCES IN CHAINS AND DECIMALS.

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