EXAMPLE XLVI. If the Peak of Teneriffe be 2 miles high, and the angle CDB made by a plumb line, and a line DB meeting the surface of the sea in the farthest visible point B, be equal to 87°. 58′. 13′′, it is required to find the diameter CE of the earth, supposing it to be a perfect sphere, and the utmost distance DB that can be seen from the top D of the mountain. (rad.+sine CDB) X DC Answer. 7964 miles. cosine CDB =DB=141·12 miles, and cɛ CHAP. IV. OBSERVATIONS ON THE ADMEASUREMENT OF A BASE LINE. (G) Where the ground is perfectly level, the manner of measuring a straight line from one object to another appears to be simple and easy; yet, on account of the curvature of the earth, no two points on its surface can be exactly situated in the same horizontal line; the chord of the arc, and not the arc itself being the horizontal distance. Now the radius of one circle is to the radius of any other circle, as any arc of the former is to a similar arc of the latter. If we take, for instance, the base line measured on Hounslow-heath (D.67.) 27404°2 feet, the radius of the earth 3982 miles, or 21024960 feet, we shall have 21024960 feet: 27404.2 feet :: 1 : 27404.2 21024960 feet, the length of the measured arc in terms of the radius 1. But the difference between any arc and its chord, the radius being 1, is 4 of the cube of the length of the arc *; hence 27404-2 3 21024960 I X 24 ='000000000092258 will express the difference in terms of the radius 1, which multiplied by 21024960 feet, the radius of the earth, produces 001939, the extent in feet by which a terrestrial arc of 27404.2 feet exceeds the chord of the same arc, a difference scarcely worth notice, even where the greatest accuracy is required. (H) When the ground on which a base line is to be measured is sloping, it will be necessary, in some cases, to reduce it to a horizontal level. Thus, after having determined the direction of the base AF †, by poles LA, Im, Hn, Go, pointed at one end and fixed perpendicularly in the ground by means of a plumb * See Chapter V. following. The point A, and the summits of the hills m, n, o, r, should be connected, so as to form a regular slope, AF. line; rizontal distance AE; and, if the heights AL, mI, nH, Go be successfully measured, their sum will give the whole height EF. (I) If the ground be irregular, or if it rise and descend alternately, it is evident that the difference between the heights of the poles must be added when ascending, and subtracted when descending, in order to determine the different elevations and depressions of the ground. (K) Surveyors generally ascertain the altitudes of irregular hills by the assistance of a spirit-level, and perpendicular poles placed at convenient distances from each other. This practice is called levelling. (L) A base line on a sloping ground may likewise be measur ed by taking angles at its extremities with a theodolite. Thus, let Im represent a theodolite, AL a pole fixed perpendicular to the horizon and equal in length to the height of the instrument; also, let KI be a horizontal line (which may be ascertained by the bubble of air in the spirit-level of the telescope resting in the middle) and KIL the angle of depression between the top of the pole AL and the horizontal line KI. Then, because KI is parallel to AB, the angle KIL is equal to the angle m AB; then (E 35), rad. : Am :: cosine KIL: AB, or AB= Am X cosine KIL rad. (M) If Am=400 yards, and KIL=4°, AB will be 399-025 yards, hence the difference between Am and AB is less than 1 yard. It appears from this example, that when the measured base is inclined to the horizon in a small angle, a reduction of this kind will be unnecessary, except in cases where great accuracy is required. OF THE ERRORS WHICH OCCUR IN TAKING ANGLES OF ELEVATION AND DEPRESSION WITH A THEODOLITE. (N) When the observer is at a considerable distance from the object, the altitude taken with a theodolite will require correction. In the first place the horizon of the observer and that of the object observed are not the same. Let c be the centre of the earth, D the summit of a mountain, and HPOв the horizon of the observer. Through D draw EDF perpendicular to DC, and it will be the horizon of the point D. Now SD will be the true height of the mountain above the horizontal line ps; DPS the (0) The French academicians measured an inclined base at Peru, the length PD of which was found to be 6274-057 toises, the angle of elevation DPB was 1°. 5. 43" (the effect of refraction being deducted). Now, rad. : PD :: sine DPB: DB, PD X sine DPB =119.93 toises. In order to cal hence DB= rad. Culate BS; PD may be used instead of PB, and cs and CB may be considered as equal to each other without sensible error. And as (2Cs+BS) X BS=PB2 (Euclid III. and 36.), it will follow that PB2 2cs X BS=PB2, and hence BS= 2cs (P) If the diameter of the earth be taken=6543373 toises, as deduced from the admeasurements in Lapland, Paris, and Peru; Bs will be found. 6274-0572 6543373 =60158 toises. By adding Bs to BD the true height SD of the mountain=125.9458 toises. Had the height Ds been determined from the triangle Dps, by the most exact calculation, it would have been=125.97 toises. Hence it appears that in most cases the angle at c may be rejected, that PB may be taken=PD, the PBD a right angle, and Cs=CB without material error. (Q) Angles of elevation or depression taken with a theodolite may be corrected thus: Let D be the place of the telescope when the theodolite stands on the vertical line CD; P the situa * Bouguer, Figure de la Terre. A toise 6 French feet, and 107 French feet 114 English feet. tion Then if the tele tion of the telescope on the vertical line cr. scope at p be directed to an object at D, the elevation of that object above the horizontal line HPB is the DPB; and when the telescope is at D and directed to an object at P, the angle of depression, below the horizontal line EDF, is the FDP. Now, because PB touches the circle, and ps cuts it, the BPS (measured by half the arc PS*)=1c; therefore 2DPS=1⁄2¿C+ the of elevation DPB. Through D draw DG parallel to SP, then the (Euclid I. and 29.), hence GDP = C + the DPB; and because the triangles CPS and CDG are thecis common to both of them, the CPS= angles CPB and CDF are right angles, therefore the =4FDG. / GDP=/DPS of elevation isosceles, and CDG, but the BPS (= <C) Again, FDP/GDP+FDG=c+the 4 of elevation DPB +c; therefore FDP /c+the 2 of elevation DPB; to each of these equals add the DPB, then the of depression FDP+ the of elevation DPB=2c+twice that of elevation DPB; (≤ of depr. FDP+≤ of elev. DPB) — ≤C hence elevation. 2 the true of When the angles are both elevations or both depressions, their difference must be diminished by the c, and half the remainder will be the true of elevation of the higher of the two objects. The c is generally very small, and where the measured base does not exceed six or seven hundred yards, it may be rejected. EXAMPLE XLVII. (R) Suppose D and P to be two objects fixed exactly at the same height above the ground as the height of the telescope of the theodolite; now if the FDP of depression be 26', and the DPB of elevation 14', what will be the error in observation? The arc Ps, or distance of the stations, being 8000 feet. The length of a degree in latitude 51°. 9'. is 364950 feet +; 364950 feet: 60':: 8000 feet: 1'. 19" nearly = 4C. Then (26' 14')-1'. 19" = 19. 20" the true of elevation DPB, hence 2 The angle formed between the tangent of any arc and its chord, is measured by half that arc. The TBF (Plate 1. fig. 1.) is measured by half the arc BF. For the TBF is equal to the Fun in the alternate segment (Euclid III. and 33.), and the ▲ FbB FCB (Euclid III. and 20.) therefore the TBF=FCB; but the FCB is measured by the arc Br, therefore the TBF is measured by balf the arc Br. Q. E. D. ↑ Trigonometrical Survey of England and Wales, Vol. II. Part ff. page 113. 19. 20'-14' 5'. 20" is the error of the instrument, or the quantity by which the z of elevation was too small, or the of depression too large. THE NATURE OF TERRESTRIAL REFRACTION AND ITS EFFECTS ON ANGLES OF ELEVATION*." (S) As terrestrial refraction arises from the gross vapours, and exhalations of various kinds, which are suspended in the air near the surface of the earth, and which are perpetually changing, it is very difficult to ascertain the exact quantity of it at any particular time. (T) The course of a ray of light in its passage through the atmosphere is, in general, that of a curve which is concave towards the earth, and the observer views the object in the direction of a tangent to this curve; hence the apparent, or observed angle of elevation is always greater than the true angle. (U) The altitudes of the heavenly bodies when within 5° or 6° of the horizon, should never be used where a very accurate result is required. The figures of the sun and moon, when near the horizon, are sometimes elliptical, having the minor axis perpendicular to the horizon, and the major axis parallel to the horizon. This change of figure arises from the refraction of the under limb being greater than that of the upper. But a perpendicular object, situated on the surface of the earth, will not have its length altered by refraction, the refraction of the bottom being the same as that of the top. (W) The allowances usually made for refraction are too uncertain for any reliance to be placed on them, as scarcely two writers agree on this subject. Dr. Maskelyne makes it of the intermediate arc ps between the observer and the object : Bouguer; Legendre; General Roy from to; and in the second volume of the Trigonometrical Survey, the variation is found to be from to of the intermediate arct. This difference does not arise from inaccuracy of observation, but from circumstances which cannot be avoided, as the evaporation of rains, dews, &c. which produce variable and partial refractions. (X) The following method is used in the Trigonometrical Survey for ascertaining the quantity of refraction. Let c be the centre of the earth, P and s two stations on its *See a paper by Mr. Huddart, in the Philosophical Transactions for 1797, page 29; and another by the Rev. S. Vince, 1799, page 13. Also the Trigonometrical Survey of England and Wales, vol. i. page 175. Page 177 and 178, Part I. M surface; |