Having loosened the clamp-screw Q, of the lower telescope MN, direct it with the thumb-screw P to the same object at which the upper telescope is directed; then tighten the clampscrew Q. This being done, loosen the clamp-screw S of the vernier plate, and direct the telescope to the other object: the arc passed over by the 0 point of the vernier, is the measure of the angle sought. The lower telescope having been made fast to the limb, will indicate any change of the position of the limb, should any have taken place; and, as the accuracy of the measurements depends on the fixedness of the limb, the lower telescope ought to be often examined, and, if its position has been altered, the limb must be brought back to its place by the tangent-screw L. NOTE. It is not necessary to place the 0 point of the vernier at the 0 point of the limb, previously to commencing the measurement of the angle, but convenient merely; for, whatever be the position of this point on the limb, it is evident that the arc which it passes over is the true measure of the horizontal angle. If, therefore, its place be carefully noted for the first direction, and also for the second, the difference of these two readings will be the true angle, unless the 0 point of the vernier shall have passed the 0 point of the limb, in which case the greater reading must be subtracted from 360°, and the remainder added to the less. MEASUREMENT OF VERTICAL ANGLES. 28. We shall first explain the method of determining the index error. Having levelled the horizontal limb, direct the telescope to some distinctly marked object, as the top of a chimney, and read the instrument. Reverse the telescope in the Y's, and turn the vernier plate 180°, and having directed the telescope to the same object, again, read the instrument. If the two readings are the same, the limb is adjusted; that is, the 0 of the limb coincides with the 0 of its vernier, when the axis of the telescope is parallel to the horizontal limb. When the reading, found with the eye end of the telescope nearest the vernier, is greater than that obtained in the reversed position, the true elevation of the object, which is equal to a mean of the readings, may be obtained by subtracting half the difference from the first reading. If the first reading is less than the second, the half difference must be added to the first. Hence, To find the index error, take the reading of the limb when the telescope is directed to a fixed object, first with the eye end of the telescope nearest the vernier, and then with the telescope and vernier plate both reversed. Take half the difference of these readings, and affect it with a minus sign if the first is the greater, or a plus sign if the second is the greater; this is equal to the index error. Let the operation be repeated several times, using different objects, and a mean of the errors will be more correct than the result of a single observation. 29. Having determined the index error, let the axis of the telescope be directed to any point either above or below the plane of the limb, and read the arc indicated by the 0 of the vernier. To the arc so read, apply the proper correction, if any, and the result will be the true angle of elevation or depression. The angle of elevation may be more correctly found by taking the elevation of the object, and repeating the observation with the telescope and vernier plate reversed, and then taking MEASUREMENTS WITH THE TAPE OR CHAIN ONLY. 30. It often happens that instruments for the measurement of angles cannot be easily obtained; we must then rely entirely on the tape or chain. We now propose to explain the best methods of determining distances, without the aid of instruments for the measurement of horizontal or vertical angles. I. To trace, on the ground, the direction of a right line, that shall be perpendicular at a given point, to a given right line. FIRST METHOD. 31. Let BC be the given right line, and A the given point. Measure from A, on the line BC, two equal distances AB, AC, one on each side of the point A. Take a portion of the chain or tape, greater than B A. AB, and place one extremity at B, and with the other, trace the arc of a circle on the ground. Then remove the end which was at B to C, and trace a second arc intersecting the former at D. The straight line, drawn through D and A, will be perpendicular to BC at A. SECOND METHOD. = 32. Having made AB AC, take any portion of the tape or chain, considerably greater than the distance between B and C. Mark the middle point of it, and fasten its two B extremities, the one at B and the other at C. Then, taking the chain by the E middle point, stretch it tightly on either side of BC, and place a staff at D or E: DAE will be the perpendicular required. THIRD METHOD. 33. Let AB be the given line, and C the point at which the perpendicular is to be drawn. From the point C, measure a distance CA equal to S. With C as a centre, and a radius equal to 6, describe an arc on either side of AB: then, with A as a centre, and a radius equal to 10, describe a second arc C A -B intersecting the one before described, at E: then draw the line EC, and it will be perpendicular to AB at C. NOTE.-Any three lines, having the ratio of 6, 8, and 10, form a right-angled triangle, of which the side corresponding to 10 is the hypothenuse. FOURTH METHOD. 34. Let AD be the given right line, and D the point at which the perpendicular is to be drawn. Take any distance, on the tape or chain, and place one extremity at D, and fasten the other at some point, as E, between the two lines which are to form the right angle. Place a staff at E. Then, having stationed a person at D, remove that extremity of the chain and carry it round until it ranges on the line DA, at A. Place a staff at A: then remove the end of the chain at A, and carry it round until it falls on the line AE, prolonged, at F. Then place a staff at F; ADF will be a right angle, being an angle in a semicircle. NOTE.-There is a very simple instrument, used exclusively in laying off right angles on the ground, which is called SURVEYOR'S CROSS. 35. This instrument consists of two bars, AB and CD, Pl. 2, Fig. 1, permanently fixed at right angles to each other, and firmly attached at E, to a pointed staff, which serves as a support. Four sights are screwed firmly to the bars, by means of the screws a, b, c, and d. As the only use of this instrument is to lay off right angles, it is of the first importance that the lines of sight be truly at right angles. To ascertain if they are so, let the bar AB be turned until its sights mark some distinct object; then look through the other sights, and place a staff on the line which they indicate: let the cross be then turned until the sights of the bar AB come to this last line: if the other sights are directed to the first object, the lines of sight are exactly at right angles. The sights being at right angles, if one of them be turned in the direction of a given line, the other will mark the direction of a line perpendicular to it, at the point where the instrument is placed. II. From a given point without a straight line, to let fall a perpendicular on the line. 36. Let C be the given point, and AB the given line. From C, measure a line, as CA, to any point of the line AB. From A, measure on AB any distance as AF, and at F erect FE perpendicular to AB. E B F D Having stationed a person at A, measure along the perpendicular FE until the forward staff is aligned on the line AC: |