isie.j 



AND HORTICULTURIST. 



229 



its several properties ; but a single diagram Avill 

 suffice to show one way in which it may be 

 located. 



Tangent or straight lines are first located upon 

 the ground, approximating as nearly as may be 

 to the final location of the drive or walk. In 

 ornamental work it is better to lay out one side 

 rather than the centre. These straight lines and 

 their angles of intersection are measured; or, if 

 no instrument of measuring angles is available, 

 auxiliary lines are also measured, by the help of 

 which they may be mapped, if need be, and 

 which will also furnish the necessary elements 

 for constructing the curve. These tangent lines 

 are then connected by means of some kind of 

 curve ; and it is here where the parabola is so 

 available. 



Tangent points at which to start and end the 

 curve are selected when most convenient. In 

 circular curves the tangents must be of equal 

 length, unless we take the trouble to introduce a 

 compound curve — that is, a curve of two or more 

 centres ; but a parabola can be introduced be- 

 tween either equal or unequal tangents, although 

 in most cases the longer tangent should not exceed 

 the shorter by more than one-half the length of the 

 latter. The length of the tangents and their 

 angles of intersection are all that are wanted to 

 compute the elements of the curve ; but if we 

 do not have the means of measuring the angle, 

 our tape-line will do very well, for all we will have 

 to do will be to measure the straight line across 

 from one tangent point to the other, instead of 

 the angle. These remarks are elementary, but 

 they may help further on. 



Let us now suppose that we find it necessary 

 to join two tangents, that is, two straight lines by 

 means of a curve, which are at present connected 

 by an angle, and that all we have to work with 

 is a tape-line, some pegs, and a hatchet. 



Measure the tangent AB— SO; B C— 120, and 

 the chord A C— 184 feet. Find D, the middle 

 point of A C ; then measure B D^^44.8 feet, and 

 find its middle point E. E is a point in the re- 

 quired cure, and B E of course equals 22.4 feet. 

 Now one property of the parabola is this : If we 

 divide its tangents A B and B C into any num- 

 ber of equal parts, the distance from any one of 

 these points to the curve is as the square of the 

 distance of that point from its tangent point A 

 or C. Thus, the distance from the point marked 

 (2) is four times the distance that (1) is from the 

 curve ; at (3) the distance is nine times, and at 

 B it is sixteen times that of (1) to the curve. 



Now we know the distance B E, because we have 

 measured it. It is the half of C D, that is, 22.4 

 feet. Hence, if we, for the sake of convenience/ 

 call the distance from (1) to curve a, we have 

 16 a— 22.4, and a— 22.4-16— 1.4 feet. The distance 

 at (2) will be 4 a— 5.6 feet, and at (3) it will be 

 9 a — 12.6 feet. Then divide each of the tangents 

 A B and B C into four equal parts, numbering 



them (1), (2), (3), (fee, counting from each tangent 

 point, and at (1) set a peg 1.4 from it, at (2) set 

 another at 5.6 feet from it, and at (3) one at 12.6 

 feet from it, and the curve will be located at 

 these points. If more points are wanted, we 

 must increase the number of points into which 

 we divide the tangents, always remembering that 

 the distance of the point nearest the tangent 

 point from the curve is found by dividing the 

 distance of the intersection of the two tangents 

 from the curve by the square of the number of 

 parts into which we have divided the tangents, 

 and that each succeeding distance is found by 

 multiplying the first distance (which we called 

 a), by the square of its number from the tangent 

 point. Thus, if in the example given we had 

 divided ovir tangent into six equal parts instead 

 of four, a would have been found by dividing 

 22.4 feet by 36, instead of by 16, and the other 

 distances would have been 4 a, 9 a, 16 a, 25 a. 



But there is one thing which we have not yet 

 considered, and which we must not forget, and 

 that is the direction in which the distances are to 

 be laid off". They must all be in one direction, par- 

 allel to that middle line B E, and therefore parallel 



