which is equal to a mean of the readings, may be obtained by subtracting half their 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 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 dif ferent objects, and a mean of the errors will be more cor rect than the result of a single observation. 26. 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 a mean of the readings for the angle required. MEASUREMENTS WITH THE TAPE OR CHAIN ONLY. 27. It often happens that instruments for the measur ment 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. 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 AB, and Then remove the end which place one extremity at B, and with the other trace the arc of a circle on the ground. 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. 29. Having made AB= AC, take any portion of the tape or chain considerably greater than the distance between B and O. Mark the middle point of it, and fasten its two extremities, the one at B and the other at C. Then, taking the chain by the middle point, stretch it tightly on either side of BC, and place a staff at D or E: DAE will be the perpendicular required. E THIRD METHOD. A -B 30. Let AB be the given line, and C the point at which the perpendicular is to be drawn. From the point measure a distance CA equal to 8. 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 intersecting at E, the one before described: then draw the line EC, and it will be perpendicular to AB at C. REMARK. Any three lines, having the ratio of 6, 8, and 10, form a right-angled triangle, of which the side corre FOURTH METHOD. 31. 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 A Place a at some point, as E, between the two lines which are to form the right angle. 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 at F. Then place a staff at F; ADF will be a right angle, being an angle in a semicircle. 32. There is a very simple instrument, used exclusively in laying off right angles on the ground, which is called the SURVEYING CROSS. Pl. 2, Fig. 1. This instrument consists of two bars, AB and CD, 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 the same 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. 6 II. From a given point without a straight line, to let fali a perpendicular on the line. 33. 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. on AB E any From A, measure FE perpendicular to AB. F B D Having stationed a person at A, measure along the per pendicular FE until the forward staff is aligned on the line AC: then measure the distance AE. angles, we have, Now, by similar tri AE : AF :: AC AD, in which all the terms are known except AD, which may, therefore, be found. The distance AD being laid off from A, the point D, at which the perpendicular CD meets AB, becomes known. If we wish the length of the perpen dicular, we use the proportion, AE : EF :: AC: CD, in which all the terms are known, excepting CD: therefore, CD may be determined. III. To determine the horizontal distance from a given point to an inaccessible object. FIRST METHOD. 34. Let A be an inaccessible object, and E the point from which the distance is to be measured. A At E lay off the right angle AED, and measure in the direction ED, any convenient distance to D, and place a staff at D. Then measure from E, directly towards the object A, a distance EB of a convenient length, and at B lay off a line D B E until a person at D shall range the forward staff on the line DA. Now, DF is known, being equal to the difference between the two measured lines DE and CB. Hence, by similar triangles, DF : FC :: DE: EA, in which proportion all the terms are known, except th fourth, which may, therefore, be found. SECOND METHOD. 35. At the point E lay off EB perpendicular to the line EA, and measure along it any convenient distance, as EB. At B lay off the right angle EBD, and measure any distance in the direction BD. Let a person at D align a staff on B E DA, while a second person at B aligns it on BE: the staff will thus be fixed at C Then measure the dis tance BC. The two triangles BCD and CAE being similar, we have, BC : BD :: CE: EA, in which all the terms are known, except the fourth, which may, therefore, be found. THIRD METHOD. 36. Let B be the given point, and A the inaccessible object; it is required to find BA. Measure any horizontal base line, as BC. Then, having placed staves at B and C, measure any convenient distances BD and CE, such that the points D, B, and A, shall be in the same right line, as also, the points E, C, and A; then measure the diagonal lines C |