Let ab (Fig. 370) be the chord, and cd the height of the segment. Through c draw e f parallel to a b; draw bf at right angles to cb; make ce equal to cf; draw ag and bh at right angles to a b; divide ce, c f, da, db, ag, and bh, each into a like number of equal parts, as four; draw the lines 1 1,22, etc., and from the points 0, 0, and 0, draw lines to c; at the intersection of these lines trace the curve, acb, which will be the segment required.
In very large work, Or in laying out ornamental gardens, etc., this will be found useful; and where the centre of the proposed arc of a circle is inaccessible it will be invaluable. (To trace the curve, see note at Art. 550.)
The lines ea, c d, and fb, would, were they extended, meet in a point, and that point would be in the opposite side of the circumference of the circle of which acb is a segment. The lines 1 1, 2 2, 3 3, would likewise, if extended, meet in the same point. The line cd, if extended to the opposite side of the circle, would become a diameter. The line fb forms, by construction, a right angle with be, and hence the extension of fb would also form a right angle with be, on the opposite side of be; and this right angle would be the inscribed angle in the semi-circle; and since this is required to be a right angle (Art. 352), therefore the construction thus far is correct, and it will be found likewise that at each point in the curve formed by the intersection of the radiating lines, these intersecting lines are at right angles.
520, - Ordinates. - Points in the circumference of a circle may be obtained arithmetically, and positively accurate, by the calculation of ordinates, or the parallel lines 0 1,
02, 03, 04 (Fig, 371). These ordinates are drawn at right angles to the chord-line a b, and they may be drawn at any distance apart, either equally distant or unequally, and there may be as many of them as is desirable; the more there are the more points in the curve will be obtained. If they are located in pairs, equally distant from the versed sine e d, calculation need be made only for those on one side of ed, as those on the opposite side will be of equal lengths, respectively; for example: 0 1, on the left-hand side of cd, is equal to 01 on the right-hand side, 0 2 on the right equals 0 2 on the left, and in like manner for the others.
The length of any ordinate is equal to the square root of the difference of the squares of the radius and abscissa, less the difference between the radius and versed sine (Art. 445). The abscissa being the distance from the foot of the versed sine to the foot of the ordinate. Algebraically,
where t is put to represent the ordinate; x, the abscissa; b, the versed sine; and r, the radius.
Example. - An arc of a circle has its chord ab (Fig. 371) 100 feet long, and its versed sine cd, 5 feet. It is required to ascertain the length of ordinates for a sufficient number of points through which to describe the curve. To this end it is requisite, first, to ascertain the radius. This is readily done in accordance with Art. 517. For (c/2)2 + v2 becomes
502 + 52 = 252.5 = radius. Having the radius, the curve
might at once be described without the ordinate points, but for the impracticability that usually occurs, in large, flat segments of the circle, of getting a location for the centre, the centre usually being inaccessible. The ordinates are, therefore, to be calculated. In Fig. 371 the ordinates are located equidistant, and are 10 feet apart. It will only be requisite, therefore, to calculate those on one side of the versed sine cd. For the first ordinate 01, the formula
= 252.3019.247.5. = 4.8019 = the first ordinate, 01.
For the second -
= 251.7066 - 247.5. = 4.2066 = the second ordinate, 02. For the third -
= 250.7115 - 247.5.
= 3.2115 = the third ordinate, 0 3.
For the fourth -
= 249.3115 - 247.5.
= 1 .8115 = the fourth ordinate, 04.
The results here obtained are in feet and decimals of a foot. To reduce these to feet, inches, and eighths of an inch, proceed as at Reduction of Decimals in the Appendix. If the two-feet rule, used by carpenters and others, were decimally divided, there would be no necessity of this reduction, and it is to be hoped that the rule will yet be thus divided, as such a reform would much lessen the labor of computations, and insure more accurate measurements.