vernier at its extremity, adapted to the divided circle, and the other a milled-head, d, which turns a pinion, working in a toothed rack round the exterior circle of the instrument; sometimes a third arm is applied, at right angles to the other two, to which the pinion is attached, and a vernier can then (if required) be applied to each of the other two, and it also prevents the observer disturbing that part of the instrument with his hand when moving the pinion. The rack and pinion give motion to the arms, which can be thus turned quite round the circle for setting the vernier to any angle that may be required. Upon a joint near the extremity of the two arms (which form a diameter to the circle) turns a branch, e e, which, for packing, may be folded over the face of the instrument, but when in use, must be placed in the position shown in the figure; these branches carry, near each of their extremities, a fine steel pricker, the two points of which, and the centre of the protractor, must (for the instrument to be correct) be in the same straight line. The points are prevented from scratching the paper as the arms are moved round, by steel springs, which lift the branches a small quantity, so that, after setting the centre of the protractor over the angular point, and the vernier in its required position, a slight downward pressure must be given to the branches, and each of the points will make a fine puncture in the paper; a line drawn through one of these punctures and the angular point will be the line required to form the angle. "Any inaccuracy in placing the centre of the protractor over the angular point may easily be discovered, for, if incorrectly done, a straight line drawn through the two punctures in the paper will not pass through the angular point; which it will do, if all be correct. "The face of the glass centre-piece on which the lines are drawn is placed as nearly even with the under surface of the instrument as possible, that no parallax may be occasioned by a space between the lines and the surface of the paper. "By help of the vernier the protractor is graduated to single minutes, which, taking into consideration the numerous sources of inaccuracy in this kind of proceeding, is the smallest angular quantity that we can pretend to lay down with certainty* If, however, for the sake of greater accuracy, it be preferred to lay down the triangles by means of the sides, beam compasses with vernier scales attached should be used in this operation. The meridian must then also be plotted by means of measures of length. A ready way offers itself, by calculating the lengths of the sides in a right-angled triangle, having for one of its angles the azimuthal distance of the observed side, and the said side for the hypotenuse. But the following method will be found more convenient. Let AB be the side of the triangle, the azimuthal angle of which has been ascertained with reference to NS, the meridian line. Take from an accurately divided diagonal scale, exactly 5 inches as a radius, and from A, as a centre, describe an arc CD; now the chord of an arc being equal to twice the sine of half the arc, the chord CD is equal to twice CE, the sine of half the angle CAD. Take a radius AF equal to twice A C, and * SIMMS on Mathematical Instruments. describe the arc F G intersecting the radius AB in F, draw the sine F H, then by similar triangles: AF: AC:: FH: CE, but AF FH 2 AC by construction, therefore that is, the chord of a given arc is equal to the sine of half the arc with double the radius. The radius of the tables of natural sines is equal to 1 or 10; and having taken the half of 10 or 5 inches for the radius A C, the natural sine of half the given angle taken from the tables will correspond to F H, the sine of half the given angle with double the radius; but F H was proved equal to CD; the natural sine therefore of half the given angle to a radius 10, will be equal to the chord of the whole angle to a radius 5. Having taken that distance from the same scale of inches as the radius, place one foot in the point C, and with the other mark the point D on the arc CD, then through D and A draw the line N S, which will be the direction of the meridian. When the operations of a Trigonometrical Survey are extended, in eastern or western directions, beyond spaces of about 60 miles from a fixed meridian, it is expedient to observe new meridians, in order to avoid errors which would otherwise take place as the result of computations made on the supposition of the earth's surface being a plane. Within a limit of about 60 miles such a supposition produces no sensible error*. Interior detail of a Trigonometrical Survey. The triangulation for a survey being accomplished, the filling in of the interior detail, such as roads, streams, legal and ecclesiastical boundaries, towns, villages, houses, woods, &c., presents little difficulty. The larger triangles being * Trigonometrical Survey., vol. ii., page 4. subdivided into others of a smaller size, the sides of these are measured with the chain, and the field-book is kept according to the form given in the first Chapter, the surveyor entering into such detail as the object of the work may demand, even to the minute tracing of all fields and enclosures. His object, however, may not always be to make detailed property plans, but simply to lay down the roads, rivers, boundaries of woods, and other great lines of artificial or natural demarcation. In this case, the survey of the roads, rivers, woods, &c., is made with the chain and theodolite, according to a process to which the term traversing" is applied, and for the description of which the reader is referred to Chapter VI. While measuring the sides of the triangles or station-lines, the surveyor takes the angles of elevation and depression, both for the purpose of reducing the inclined lines more correctly to the horizonal plane, as also to obtain as many altitudes as possible over the surface of the district surveyed; and as these levelling operations form an essential part of the trigonometrical survey, the subject of levelling must next be considered before detailing more fully the practical operations connected therewith. 66 Concluding Remarks. Frequent reference has been made in the course of this Chapter to the work describing the operations of the "Trigonometrical Survey for England and Wales." The reader who is desirous to study this branch of the subject more fully is recommended to consult that work. There has been as yet no description published of the operations adopted in the course of the Irish survey, which the advanced state of science has made more perfect. With respect to it, the Rev. Dr. Robinson, professor of astronomy at Armagh, makes the following observations: "In respect to triangulation, it is unmatched in the world; there is nothing like it in existence; the details of that part of the work in fact cannot, as I conceive, in the present state of knowledge, be exceeded; indeed, some of the facts I know respecting it are almost marvellous. I was some years since engaged in verifying the position of my own observatory in Armagh, by means of rocket signals. The spot on which those signals were to be fired was a mountain not visible from my observatory, because of an intervening ridge of hills. The officers of the survey furnished me, however, with the direction in which that mountain lay, although invisible from the observatory, and a telescope was placed in that position. I went to the intervening ridge with a theodolite, which I shifted until by signal it was placed in the line of the telescope; it ought to have been then exactly in the line of the mountain, although that was invisible from the observatory; and on taking by the instrument the angle between the mountain and the telescope, I found it to be accurately two rightangles; they were in a straight line, without the error of a hair's breadth. When those experiments were completed, we ascertained the distance in feet between the two observatories of Armagh and Dublin, deduced from astronomical observations. We derived from the officers of the survey the distance in feet given by the operation of that survey: it is, I think, about 70 English miles, and the difference between the astronomical determination and the trigonometrical was not four feet-a degree of coincidence that I believe has never been equalled since such operations have been carried on. The manner in which the engraving and publication of the result of that triangulation have been executed, are of the same high degree of excellence*.” * Evidence given before the Commissioners to inquire into the Facts relating to the Ordnance Memoir of Ireland, 1844. |