To determine by actual measurement on the ground the length of a side of one of the triangles, wherefrom to infer the lengths of all the other sides in the triangulation, is not the least difficult operation of a trigonometrical survey. When the problem is stated thus—To determine the number of times that a certain standard or unit of length is contained between two finely marked points on the surface of the earth at a distance of some miles asunder, so that the error of the result may be pronounced to lie between certain very narrow limits,—then the question demands very serious consideration. The representation of the unit of length by means of the distance between two fine lines on the surface of a bar of metal at a certain temperature is never itself free from uncertainty and probable error, owing to the difficulty of knowing at any moment the precise temperature of the bar; and the transference of this unit, or a multiple of it, to a measuring bar, will be affected not only with errors of observation, but with errors arising from uncertainty of temperature of both bars. If the measuring bar be not self-compensating for temperature, its expansion must be determined by very careful experiments. The thermometers required for this purpose must be very carefully studied, and their errors of division and index error determined.

The base apparatus of Bessel and that of Colby have been described in Figure of the Earth (vol. vii. p. 598). The average probable error of a single measurement of a base line by the Colby apparatus is, according to the very elaborate investigations of Colonel Walker, C.B., R.E., the Surveyor-General of India, ± l ’ 5∕x,(μ, meaning “one millionth”). W. Struve gives ± 0'8μ, as the probable error of a base line measured with his apparatus, being the mean of the probable errors of seven bases measured by him in Russia; but this estimate is probably too small. Struve’s apparatus is simple: there are four wrought iron bars, each two toises (rather more than 13 feet) long; one end of each bar is terminated in a small steel cylinder presenting a slightly convex surface for contact, the other end carries a contact lever rigidly connected with the bar. The shorter arm of the lever terminates below in a polished hemisphere, the upper and longer arm traversing a vertical divided arc. In measuring, the plane end of one bar is brought into contact with the short arm of the contact lever (pushed forward by a weak spring) of the next bar. Each bar has two thermometers, and a level for determining the inclination of the bar in measuring. The manner of transferring the end of a bar to the ground is simply this: under the end of the bar a stake is driven very firmly into the ground, carrying on its upper surface a disk, capable of movement in the direction of the measured line by mean3 of slow-motion screws. A fine mark on this disk is brought vertically under the end of the bar by means of a theodolite which is planted at a distance of 25 feet from the stake in a direction perpendicular to the base. Struve investigates for each base the probable errors of the ,measurement arising from each of these seven causes:— alignment, inclination, comparisons with standards, readings of index, personal errors, uncertainties of temperature, and the probable errors of adopted rates of expansion.

The apparatus used in the United States Coast Survey consists of two measuring bars, each 6 metres in length, supported on two massive tripod stands placed at one quarter length from each end, and provided, as in Colby’s apparatus, with the necessary mechanism for longitudinal, transverse, and vertical adjustment. Each measuring rod is a compensating combination of an iron and a brass bar, supported parallel to one another and firmly connected at one end, the medium of connexion between the free ends being a lever of corhpensation so adjusted as to indicate a constant length independent of temperature or changes of temperature. The bars are protected from external influences by double tubes of tinned sheet iron, within which they are movable on rollers by a screw movement which allows of contacts being made within 10 ¾ 00 of an inch. The abutting piece acts upon the contact lever which is attached to the fixed end of the compound bar, and carries a very sensitive level, the horizontal position of which defines the length of the bar. It is impossible here to give a full description of this complicated apparatus, and we must refer for details to the account given in full in the United States Coast Survey Report for 1854. This apparatus is doubtless a very perfect one, and the manipulation of it must offer great facilities, for it appears to be possible, under favourable circumstances, to measure a mile in one day, 1Ό6 mile having been measured on one occasion in eight and a half hours. In order to test to the utmost the apparatus, the base at Atalanta, Georgia, was measured twice in winter and once in summer 1872-73, at temperatures 51°, 45°, 90° F.; the difference of the first and second measurements was +0 ’ 30 in., of the second and third + 0 ’ 34 in.,—the actual length and computed probableerror expressed in metres being 9338 ’ 4763 ± 0Ό166. It is to be noted that in the account of a base recently measured in the United States Lake Survey, some doubt is expressed as to the perfection of the particular apparatus of this description there used, on account of a liability to permanent changes of length.

The last base line measured in India with Colby’s compensation apparatus had a length of 8912 feet only, and in consequence of some doubts which had arisen as to the accuracy of this compensation apparatus, the measurement was repeated four times, the operations being conducted in such a manner as to indicate as far as possible the actual magnitudes of the probable errors to which such measures are liable. The direction of the line (which is at Cape Comorin) is north and south, and in two of the measurements the brass component was to the west, in the other two it was to the east. The differences between the individual measurements and the mean of the four are + Ό017, - , 0049, - ∙0015, + Ό045 in feet. The measurements occupied from seven to ten days each,—the average rate of such work in India being about a mile in five days.

The method of Μ. Porro, adopted in Spain, and by the French in Algiers, is essentially different from those just described. The measuring rod, for there is only one, is a thermometric combination of two bars, one of platinum and one of brass, in length 4 metres, furnished with three levels and four thermometers. Suppose A, B, C three micrometer microscopes very firmly supported at intervals of 4 metres with their axes vertical, and aligned in the plane of the base line by means of a transit instrument, their micrometer screws being in the line of measurement. The measuring bar is brought under say A and B, and those micrometers read; the bar is then shifted aud brought under B and C. By repetition of this process, the reading of a micrometer indicating the end of each position of the bar, the measurement is made. The probable error of the central base of Madridejos, which has a length of 14664 ’ 500 metres, is estimated at ÷ 0 ’ 17^. This is the longest base line in Spain; there are seven others, six of which are under 2500 metres in length; of these one is in Majorca, another in Minorca, aud a third in Iviça. The last base just measured in the province of Barcelona has a length of 2483 ’ 5381 metres according to the first measurement, and 2483 ’ 5383 according to the second.

The total number of base lines measured in Europe up to the present time is about eighty, fifteen of which do not exceed in length 2500 metres, or about a mile aud a half, and two—one in France, the other in Bavaria—exceed 19,000 metres. The question has been frequently discussed whether or not the advantage of a long base is sufficiently great to warrant the expenditure of time that it requires, or whether as much precision is not obtainable in the end by careful triangulation from a short base. But the answer cannot be given generally; it must depend on the circumstances of each particular case.

Itis necessary that the altitude above the level of the sea of every part of a base line be ascertained by spirit levelling, in order that the measured length maybe reduced to what it would have been had the measurement been made on the surface of the sea, produced in imagination. Thus if l be the length of a measuring bar, h its height at any given position in the measurement, r the radius of the earth, then the length radially projected on to the level of the sea is l - l. In the Salisbury Plain base line the reduction to the level of the sea is - 0 - 6294 feet.

In working away from a base line ab, stations c, d, e, f are carefully selected so as to obtain from well-shaped triangles gradually increasing sides. Before, however, finally leaving the base line it is usual to verify it by triangulation thus: during the measurement two or more points, as p, q (fig. 1), are marked in the base in positions such that the lengths of the different segments of the line are known; then, taking suitable external stations, as h, k, the angles of the triangles blip, plιq, hqk, kqa are measured. From these angles can be computed the ratios of the segments, which must agree, if all operations are correctly performed, with the ratios resulting from the measures. Leaving the base line, the sides increase up to ten, thirty, or fifty miles, occasionally, but seldom, reaching a hundred miles. The triangulation points may either be natural objects presenting themselves in suitable positions, such as church towers; or they may be objects specially constructed in stone or wood on mountain tops or other prominent ground. In every case it is necessary that the precise centre of the station be marked by some permanent mark. In India no expense is spared in making permanent the principal trigonometrical stations—costly towers in masonry being erected. It is essential that every trigonometrical station shall present a fine object for observation from surrounding stations.

Horizontal Angles.

In placing the theodolite over a station to be observed from, the first point to be attended to is that it shall rest upon a perfectly solid foundation. The method of obtaining this desideratum must depend entirely on the nature of the ground; the instrument must if possible be supported on rock, or if that be impossible a solid foundation must be obtained by digging. When the theodolite is required to be raised above the surface of the ground in order to command particular points, it is necessary to build two scaffolds,—the outer one to carry the observatory, the inner one to carry the instrument,—and these two edifices must have no point of contact. Many cases of high scaffolding have occurred on the English Ordnance Survey, as for instance at Thaxted Church, where the tower, 80 feet high, is surmounted by a spire of 90 feet. The scaffold for the observatory was carried from the base to the top of the spire; that for the instrument was raised from a point of the spire 140 feet above the ground, having its bearing upon timbers passing through the spire at that height. Thus the instrument, at a height of 178 feet above the ground, was insulated, and not affected by the action of the wind on the observatory.

At every station it is necessary to examine and correct the adjustments of the theodolite, which are these :—the line of collimation of the telescope must be perpendicular to its axis of rotation; this axis perpendicular to the vertical axis of the instrument; and the latter perpendicular to the plane of the horizon. The micrometer microscopes must also measure correct quantities on the divided circle or circles. The method of observing is this. Let A, B, C.... be the stations to be observed taken in order of azimuth; the telescope is first directed to A and the cross-hairs of the telescope made to bisect the object presented by A, then the microscopes or verniers of the horizontal circle (also of the vertical circle if necessary) are read and recorded. The telescope is then turned to B, which is observed in the same manner; then C and the other stations. Coming round by continuous motion to A, it is again observed, and the agreement of this second reading with the first is some test of the stability of the instrument. In taking this round of angles—or “arc,” as it is called on the Ordnance Survey— it is desirable that the interval of time between the first and second observations of A should be as small as maybe consistent with due care. Before taking the next arc the horizontal circle is moved through 20° or 30°; thus a different set of divisions of the circle is used in each arc, which' tends to eliminate the errors of division.

It is very desirable that all arcs at a station should contain one point in common, to which all angular measurements are thus referred,—the observations on each arc commencing and ending with this point, which is on the Ordnance Survey called the “referring object.” It is usual for this purpose to select, from among the points which have to be observed, that one which affords the best object for precise observation. For mountain tops a “referring object” is constructed of two rectangular plates of metal in the same vertical plane, their edges parallel and placed at such a distance apart that the light of the sky seen through appears as a vertical line about 10" in width. The best distance for this object is from one to two miles.

It is clear that no correction is required to the angles measured by a theodolite on account of its height above the sea-level; for its axis of rotation coincides with the normal to the surface of the earth, and the angles measured between distant points are those contained between the vertical planes passing through the axis of the instrument and those points.

The theodolites used in geodesy vary in pattern and in size—the horizontal circles ranging from 10 inches to 36 inches in diameter. In Ramsden’s 36-inch theodolite the telescope has a focal length of 36 inches and an aperture of 2 ’ 5 inches, the ordinarily used magnifying power being 54; this last, however, can of course be changed at the requirements of the observer or of the weather. The probable error of a single observation of a fine object with this theodolite is about 0" ’ 2.

Fig. 2 represents an altazimuth theodolite of an improved pattern now used on the Ordnance Survey. The horizontal circle of 14 inches diameter is read by three micrometer microscopes; the vertical circle has a diameter of 12 inches, and is read by two microscopes.

In the Great Trigonometrical Survey of India the theodolites used in the more important parts of the work have been of 2 and 3 feet· diameter,—the circle read by five equidistant microscopes. Every angle is measured twice in each position of the zero of the horizontal circle, of which there are generally ten; the entire number of measures of an angle is never less than 20. An examination of 1407 angles showed that the probable error of an observed angle is on the average ÷ 0 ’ 28.

For the observations of very distant stations it is usual to employ a heliostat In its simplest form this is a plane mirror 4, 6, or 8 inches in diameter, capable of rotation round a horizontal and a vertical axis. This mirror is placed at the station to be observed, and in fine weather it is kept so directed that the rays of the sun reflected by it strike the distant observing telescope. To the observer the heliostat presents the appearance of a star of the first or second magnitude, and is generally a pleasant object for observing.

Astronomical Observations.

The direction of the meridian is determined either by a theodolite or a portable transit instrument. In the former case the operation consists in observing the angle between a terrestrial object—generally a mark specially erected and capable of illumination at night—and a close circumpolar star at its greatest eastern or western azimuth, or, at any rate, when very near that position. If the observation be made t minutes of time before or after the time of greatest azimuth, the azimuth then will differ from its maximum value by

(4δ00 2 8 inΓ⅛¾ . ; . .

sin z in secouds of angle, omitting smaller terms. Here the symbol δ is the star’s declination, ε its zenith distance. The collimation and level errors are very carefully determined before and after these observations, and it is usual to arrange the observations by the reversal of the telescope so that collimation error shall disappear. If b, c be the level and collimation errors, the correction to the circle reading is b cot z ÷ c cosec z, b being positive when the west end of the axis is high. It is clear that any uncertainty as to the real state of the level will produce a corresponding uncertainty iu the resulting value of the azimuth,—an uncertainty which increases with the latitude, and is very large in high latitudes. This may be partly remedied by observing in connexion with the star its reflexion in mercury. In determining the value of “one division "of a level tube, it is necessary to bear in mind that in some the value varies considerably with the temperature. By experiments on the level of Ramsden’s 3~foot theodolite, it was found that though at the ordinary temperature of 66° the value of a division was about one second, yet at 32° it was about five seconds.

The portable transit in its ordinary form hardly needs description. In a very excellent instrument of this kind used on the Ordnance Survey, the uprights carrying the telescope are constructed of mahogany, each upright being built of several pieces glued and screwed together; the base, which is a solid and heavy plate of iron, carries a reversing apparatus for lifting the telescope out of its bearings, reversing it, and letting it down again. Thus is avoided the change of temperature which the telescope would incur by being lifted by the hands of the observer. Another form of transit Î3 the German diagonal form, in which the rays of light after passing through the object glass are turned by a total reflexion prism through one of the transverse arms of the telescope, at the extremity of which arm is the eye-piece. The unused half of the ordinary telescope being cut away is replaced by a counterpoise. In this instrument there is the advantage that the observer without moving the position of his eye commands the whole meridian, and that the level may remain on the pivots whatever be the elevation of the telescope. But there is the disadvantage that the flexure of the transverse axis causes a variable collimation error depending on the zenith distance of...