arc of France by P. F. A. Méchain and J. B. J. Delambre had | P and Q are finely engraved dots 10 ft. apart. In practice the bars, for its end the determination of the true length of the “ metre when aligned, are not in contact, an interval of 6 in. being allowed which was to be the legal standard of length of France (sce measured by an ingenious micrometrical arrangement constructed between cach bar and its neighbour. This distance is accurately EARTH, FIGURE OF THE). on exactly the same principle as the bars themselves. The basis of every extensive survey is an accurate triangulation, The last base line measured in India had a length of 8913 ft. In and the operations of geodesy consist in the measurement, by consequence of some suspicion as to the accuracy of the compensation theodolites, of the angles of the triangles; the measurement of being conducted so as to determine the actual values of the probable apparatus, the measurement was repeated four times, the operations one or more sides of these triangles on the ground, the determin- errors of the apparatus. The direction of the line (which is at Cape ation by astronomical observations of the azimuth of the whole Comorin) is north and south. In two of the measurements the brass network of triangles; the determination of the actual position component was to the west, in the others to the east; the differences of the same on the surface of the earth by observations, first for +0.0017. -0.0049. -0.0015. +0.0045 ft. These differences are between the individual measurements and the mean of the four were latitude at some of the stations, and secondly for longitude; the very small: an elaborate investigation of all sources of error shows determination of altitude for all stations. that the probable error of a base line in India is on the average For the computation, the points of the actual surface of the #2.8 H. These compensation bars were also used by Sir Thomas earth are imagined as projected along their plumb lines on the Lacaille's arc at the Cape. The account of this operation will be Maclear in the measurement of the base line in his extension of mathematical figure, which is given by the stationary sea-level, found in a volume entitled Verification and Extension of Lacaille's and the extension of the sea through the continents by a system Arc of Meridian at the Cape of Good Hope, by Sir Thomas Maclear, of imaginary canals. For many purposes the mathematical published in 1866. A rediscussion has been given by Sir David surface is assumed to be a plane; in other cases a sphere of Gill in his Report on the Geodetic Survey of South Africa, &c., 1896. radius 6371 kilometres (20,900,000 ft.). In the case of extensive triangulations in Russia from 1817 to 1855. A very simple base apparatus was employed by W. Struve in his This consisted of four operations the surface must be considered as a compressed wrought-iron bars, each two toises (rather more than 13 st.) long; ellipsoid of rotation, whose minor axis coincides with the carth's one end of each bar is terminated in a small steel cylinder presenting axis, and whose compression, flattening, or ellipticity is about a slightly convex surface for contact, the other end carries a contact lever rigidly connected with the bar. The shorter arm of the lever 1/298 terminates below in a polished hemisphere, the upper and longer Measurement of Base Lines. arm traversing a vertical divided arc.. 'In measuring, the planc end of one bar is brought into contact with the short arm of the contact To determine by actual measurement on the ground the length of a lever (pushed forward by a weak spring) of the next bar. Each bar side of one of the triangles (“ base line "), wherefrom to inser the has two thermometers, and a level for determining the inclination lengths of all the other sides in the triangulation, is not the least of the bar in measuring. The manner of transferring the end of a difhcult operation of a trigonometrical survey. When the problem bar to the ground is simply this under the end of the bar a stake is stated thus-To determine the number of times that a certain is driven very firmly into the ground, carrying on its upper surface standard or unit of length is contained between two finely marked a disk, capable of movement in the direction of the measured line points on the surface of the earth at a distance of some miles asunder, by means of slow-motion screws. A fine mark on this disk is so that the error of the result may be pronounced to lie between brought vertically under the end of the bar by means of a theodolite certain very narrow limits,--then the question demands very which is planted at a distance of 25 ft. from the stake in a direction serious consideration. The representation of the unit of length by perpendicular to the base. Struve investigated for each base the means of the distance between two fine lines on the surface of a bar probable errors of the measurement arising from each of these seven of metal at a certain temperature is never itself free from uncertainty causes: Alignment, inclination, comparisons with standards, readand probable error, owing to the difficulty of knowing at any moment ings of index, personal errors, uncertainties of temperatare, and the the precise temperature of the bar; and the transference of this probable errors of adopted rates of expansion. He found that unit, or a multiple of it, to a measuring bar will be affected not +0.8 H was the mean of the probable errors of the seven bases only with errors of observation, but with errors arising from un- measured by him. The Austro-Hungarian apparatus is similar; certainty of temperature of both bars. If the measuring bar be not the distance of the rods is mcasured by a slider, which rests on one sell-compensating for temperature, its expansion must be determined of the ends of each rod. Twenty-two base lines were measured in by very careful experiments. The thermometers required for this 1840-1899. purpose must be very carefully studied, and their errors of division General Carlos Ibañez employed in 1858-1879, for the measureand index error determined. ment of nine base lines in Spain, two apparatus similar to the In order to avoid the difficulty in exactly determining the tempera- apparatus previously employed by Porro in Italy: one is complicated, ture of a bar by the mercury thermometer, F. W. Bessel introduced the other simplified. The first, an apparatus of the brothers Brunner in 1834 near Königsberg a compound bar which constituted a of Paris, was a thermometric combination of two bars, one of platinum metallic thermometer. Å zinc bar is laid on an iron bar two toises and one of brass, in length 4 metres, furnished with three levels and long, both bars being perfectly planed and in free contact, the zinc four thermometers. Suppose A, B, C three micrometer microscopes bar being slightly shorter and the two bars rigidly united at one end. very firmly supported at intervals of 4 metres with their axes vertical, As the temperature varies, the difference of the lengths of the bars, and aligned in the plane of the base line by means of a transit as perceived by the other end, also varies, and affords a quantitative instrument, their micrometer screws being in the line of measurement. correction for temperature variations, which is applied to reduce the The measuring bar is brought under say A and B, and those microlength to standard temperature. During the measurement of the meters read; the bar is then shifted and brought under B and C. By base line the bars were not allowed to come into contact, the interval repetition of this process, the reading of a micrometer indicating the being measured by the insertion of glass wedges. The results of the end of cach position of the bar, the measurement is made. comparisons of four measuring rods with one another and with the Quite similar apparatus (among others) has been employed by the standards were elaborately computed by the method of least-squares. French and Germans. Since, however, it only permitted a distance The probable error of the measured length of 935 toises (about of about 300 m. to be measured daily, Ibañez introduced a simplif. 6000 st.) has been estimated as 1/863500 or 1.2 H (u denoting a cation; the measuring rod being made simply of steel, and provided millionth). With this apparatus fourteen base lines were measured with inlaid mercury thermometers. This apparatus was used in in Prussia and some neighbouring states; in these cases a somewhat Switzerland for the measurement of three base lines. The accuracy higher degree of accuracy was obtained. is shown by the estimated probable errors. 0.2 H to 0-8 H. The principal triangulation of Great Britain and Ireland has seven The distance mcasured daily amounts at least to 800 m. base lines: five have been measured by steel chains, and two, A greater daily distance can be measured with the same accuracy more exactly, by the compensation bars of General T. F. Colby, an by means of Bessel's apparatus; this permits the ready measureapparatus introduced in 1827-1828 at Lough Foyle in Ifeland. Ten ment of 2000 m. daily. For this, however, it is important to notice base lines were measured in India in 1831-1869 by the same apparatus. that a large staff and favourable ground are necessary. An im This is a system of six compound-bars self-correcting for temperature. portant improvement was introduced by Edward Jäderin of StockThe bars may be thus described: Two bars, one of brass and the holm, who measures with stretched wires of about 24 metres long: other of iron, are laid in parallelism side by side, firmly united at these wires are about 1.65 mm. in diameter, and when in use are their centres, from which they may freely expand or contract; at stretched by an accurate spring balance with a tension of 10 kg.? the standard temperature they are of the same length. Let AB be The nature of the grou has a very trilling effect on this method. one bar, A'B' the other; draw lines through the corresponding the dishculty of temperature determinations is removed by employ: extremities AA' (to P) and BB' (to Q), and make A'P = B'Q. AA' ing wires made of invar, an alloy of steel (64%) and nickel (36) being equal to BB'. If the ratio A'PAP equals the ratio of the co which has practically no lincar expansion for small thermal changes efficients of expansion of the bars A'B' and AB, then, obviously, the distance PQ is constant (or nearly so). In the actual instrument Geodetic Surrey of South Africa, vol. iii. (1905). p. viii; Les Nouveaux An arrangement acting similarly had been previously introduced Appareils pour la mesure rapide des bases géod., par J. René Benoit by Borda. et Ch. Ed. Guillaume (1906). metres. at ordinary temperatures, this alloy, was discovered in 1896 by the measures. Leaving the base line, the sides increase up to 10. Benổit and Guillaume of the International Bureau of Weights and 30 or 50 miles occasionally, but seldom reaching 100 miles. The Measures at Breteuil. Apparently the future of base-line measure-triangulation points may either be natural objects presenting themments rests with the invar wires of the Jäderin apparatus; next selves in suitable positions, such as church towers; or they may be comes Porro's apparatus with invar bars 4 to 5 metres long. objects specially constructed in stone or wood on mountain tops Results have been obtained in the United States, of great im- or other prominent ground. In every case it is necessary that the portance in view of their accuracy, rapidity of determination and precise centre of the station be marked by some permanent mark. economy. For the measurement of the arc of meridian in longitude In India no expense is spared in making permanent the principal 98° E., in 1900, nine base lines of a total length of 69.2 km. were trigonometrical stations-costly towers in masonry being erected. measured in six months. The total cost of one base was $1231. It is essential that every trigonometrical station shall present a fine At the beginning and at the end of the field-season a distance of object for observation from surrounding stations. exactly 100 m. was measured with R. S, Woodward's" 5-m. icebar" (invented in 1891); by means of the remeasurement of this Horisontal Angles. length the standardization of the apparatus was done under the same In placing the theodolite over a station to be observed from, the conditions as existed in the case of the base measurements. For first point to be attended to is that it shall rest upon a perfectly the measurements there were employed two steel tapes of 100 m. solid foundation. The method of obtaining this desideratum must long: provided with supports a distances of 2.5two of 50 m depend entirely on the nature of the ground the instrument must and the duplex apparatus of Eimbeck, consisting of four 5-m. rods. if possible be supported on rock, or if that be impossible a solid Each base was divided into sections of about 1000 m.; one of these, foundation must be obtained by digging. When the theodolite is the "test kilometre," was measured with all the five apparatus, required to be raised above the surface of the ground in order to the others only with two apparatus, mostly tapes. The probable command particular points, it is necessary to build two scaffoldserror was about #0-8 m, and the day's work a distance of about the outer one to carry the observatory, the inner one to carry the 2000 m. Each of the four rods of the duplex apparatus consists of instrument,--and these two edifices must have no point of contact. two bars of brass and steel. Mercury thermometers are inserted Many cases of high scaffolding have occurred on the English Ordnance in both bars; these serve for the measurement of the length of the Survey, as for instance at Thaxted church, where the tower, 80 ft. base lines by each of the bars, as they are brought into their con- high, is surmounted by a spire of go ft. The scaffold for the obsecutive positions, the contact being made by an elastic-sliding servatory was carried from the base to the top of the spire; that contact. The length of the base lines may be calculated for each for the instrument was raised from a point of the spire 140 ft. above bar only, and also by the supposition that both bars have the same the ground, having its bearing upon timbers passing through the temperature. The apparatus thus affords three sets of results, spire at that height. Thus the instrument, at a height of 178 ft. which mutually control themselves, and the contact adjustments above the ground, was insulated, and not affected by the action of permit rapid work. The same device has been applied to the older the wind on the observatory. bimetallic-compensating apparatus of Bache-Würdemann (six At every station it is necessary to examine and correct the adbases, 1847-1857) and of Schott. There was also employed a single justments of the theodolite, which are these: the line of collimation rod bimetallic apparatus on F. Porro's principle, constructed by the of the telescope must be perpendicular to its axis of rotation; this brothers Repsold for some base lines. 'Excellent results have been axis perpendicular to the vertical axis of the instrument; and the more recently obtained with invar tapes. latter perpendicular to the plane of the horizon. The micrometer The following results show the lengths of the same German base microscopes must also measure correct quantities on the divided lines as measured by different apparatus: 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 Base at Berlin 1864 Apparatus of Bessel 2336-3920 telescope is first directed to A and the cross-hairs of the telescope 1880 Brunner •3924 Base "at Strehlen 1854 made to bisect the object presented by A, then the microscopes or Bessel 2762.5824 verniers of the horizontal circle (also of the vertical circle if necessary) 1879 Brunner •5852 Old base at Bonn are read and recorded. The telescope is then turned to B, which 1847 Bessel 2133.9095 is observed in the same manner; then C and the other stations. 1892 -9097 New base at Bonn Coming round by continuous motion to A, it is again observed, and 1892 2512.9612 this second reading with the first is some test of 1892 Brunner -9696 the stability of the instrument. In taking this round of anglesIt is necessary that the altitude above the level of the sea of every or " arc," as it is called on the Ordnance Survey-it is desirable part of a base line be ascertained by spirit levelling, in order that that the interval of time between the first and second observations the measured length may be reduced to what it would have been of A should be as small as may be consistent with due care. Before had the measurement been made on the surface of the sea, produced taking the next arc the horizontal circle is moved through 20° or in imagination. Thus if ! be the length of a measuring bar, h its 30°; thus a different set of divisions of the circle is used in each height at any given position in the measurement, , the radius of arc, which tends to eliminate the errors of division. the earth, then the length radially projected on to the level of the It is very desirable that all arcs at a station should contain one sea is l(1-h/r). In the Salisbury Plain base line the reduction to point in common, to which all angular measurements are thus the level of the sea is - 0:6294 ft. referred. ---the observations on each arc commencing and ending The total number of base lines measured in Europe up to the with this point, which is on the Ordnance Survey called the “referring present time is about one hundred and ten, nineteen of which do object." It is usual for this purpose to select, from among the not exceed in length 2500 metres, or about if miles, and three-points which have to be observed, that one which affords the best one in France, the others in Bavaria – object for precise observation. For mountain tops a "referring exceed 19.000 metres. The question object " is constructed of two rectangular plates of metal in the has been frequently discussed whether same vertical planc, their edges parallel and placed at such a distance not the ad tage of a long base is apart that the light of the sky seen through appears as a vertical line sufficiently great to warrant the ex- about 10' in width. The best distance for this object is from penditure of time that it requires, or I to 2 miles. whether as much precision is not obtain- This method seems at first sight very advantageous; but if, able in the end by careful triangulation however, it be desired to attain the highest accuracy, it is better, from a short base. But the answer as shown by General Schreiber of Berlin in 1878, to measure only cannot be given generally: it must single angles, and as many of these as possible between the directions depend on the circumstances of each to be determined. Division-errors are thus more perfectly eliminated, particular case. With Jäderin's appa- and errors due to the variation in the stability, &c., of the instruments ratus, provided with invar wires, bases are diminished. This method is rapidly gaining precedence. of 20 to 30 km. long are obtained with- The theodolites used in geodesy vary in pattern and in size-the out difficulty. horizontal circles ranging from 10 in. to 36 in. in diameter. In In working away from a base line ab, Ramsden's 36-in. theodolite the telescope has a focal length of stations c, d, e, f are carefully selected so 36 in. and an aperture of 2.5 in., the ordinarily used magnifying as to obtain from well-shaped triangles power being 54; this last, however, can of course be changed at the gradually increasing sides Before, how- requirements of the observer or of the weather. The probable ever, finally leaving the base line, it is error of a single observation of a fine object with this theodolite usual to verify it by triangulation thus: is about 0".2. Fig. 2 represents an altazimuth theodolite of an during the measurement two or more improved pattern used on the Ordnance Survey. The horizontal Fig. 1. in positions such that the lengths of the vertical circle has a diameter of 12 in., and is read by two micro the different segments of the line are scopes. In the great trigonometrical survey of India the theodolites known; then, taking suitable external stations, as h, k, the angles of used in the more important parts of the work have been of 2 and the triangles bhp, phq, hqk, kqq are measured. From these angles 3 ft. diameter--the circle read by five equidistant microscopes. can be computed the ratios of the segments, which must agree, if all Every angle is measured twice in each position of the zero of the operations are correctly performed, with the ratios resulting from horizontal circle, of which there are generally ten; the entire number of measures of an angle is never less than 20. An examina. sponding uncertainty in the resulting value of the azimuth. --an tion of 1407 angles showed that the probable error of an observed uncertainty, which increases with the latitude and is very large angle is on the average * 0":28. in high latitudes. This may be partly remedied by observing in For the observations of very distant stations it is usual to employ connexion with the star its reflection in mercury. In determining a heliotrope (from the Gr. vjcos, sun; pótos, a turn), invented by the value of one division" of a level tube, it is necessary to bear Gauss at Göttingen in 1821. In its simplest form this is a plane in mind that in some the value varies considerably with the temperamirror, 4. 6, or 8 in. in diameter, capable of rotation round a horizontal ture. By experiments on the level of Ramsden's 3-foot theodolite, and a vertical axis. This mirror is placed at the station to be ob- it was found that though at the ordinary temperature of 66® the served, and in fine weather it is kept so directed that the rays of the value of a division was about one second, yet at 32° it was about sun reflected by it strike the distant observing telescope. To the five seconds. observer the heliotrope presents the appearance of a star of the In a very excellent portable transit used on the Ordnance Survey, first or second magnitude, and is generally a pleasant object for the uprights carrying the telescope are constructed of mahogany, observing. cach upright being built of several pieces glued and screwed together; Observations at night, with the aid of light-signals, have been the base, which is a solid and heavy plate of iron, carries a reversing repeatedly made, and with good results, particularly in France apparatus for lifting the telescope out of its bearings, reversing it by General François Perrier, and more recently in the United and letting it down again. Thus is avoided the change of temperaStates by the Coast and Geodetic Survey; the signal employed ture which the telescope would incur by being lifted by the hands being an acetylene bicycle-lamp, with a lens 5 in. in diameter. of the observer. Another form of transit is the German diagonal Particularly noteworthy are the trigonometrical connexions of form, in which the rays of light after passing through the objectSpain and Algeria, which were carried out in 1879 by Generals glass are turned by a total reflection prism through one of the transIbañez and Perrier (over a distance of 270 km.), of Sicily and Malta verse arms of the telescope, at the extremity of which arm is the in 1900, and of the islands of Elba and Sardinia in 1902 by Dreye-piece. The unused half of the ordinary telescope being cut away Guarducci (over distances up to 230 km.); in these cases artificial 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 clevation 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 the star to which it is directed; and moreover it has been found that in some cases the personal error of an observer is not the same in the two positions of the telescope. To determine the direction of the meridian, it is well to erect two marks at nearly equal angular distances on either side of the north meridian line, so that the pole star crosses the vertical of each mark a short time before and after attaining its greatest castern and western azimuths. If now the instrument, perfectly levelled, is adjusted to have its centre wire on one of the marks, then when elevated to the star, the star will traverse the wire, and its exact position in the field at any moment can be measured by the micrometer wire. Alternate observations of the star and the terrestrial mark, combined with careful level readings and reversals of the instrument, will enable one, even with only one mark, to determine the direction of the meridian in the course of an hour with a probable error of less than a second. The second mark enables one to complete the station more rapidly and gives a check upon the work. As an instance, at Findlay Seat, in latitude 57° 35', the resulting azimuths of the two marks were 177° 45' 37":29 +0":20 and 182° 17' 15"-61 +0":13, while the angle between the two marks directly measured by a theodolite was found to be 4° 31' 37"-430":23. We now come to the consideration of the determination of time with the transit instrument. Let fig. 3 represent the sphere stereo graphically projected on the plane of The horizon,-ns being the meridian, we the prime vertical, Z,P the zenith and the pole. I © Let p be the point in the celestial be 90°-11, the star's declination being 0, and the FIG. 2.-Altazimuth Theodolite. latitude 0. Then to find the hour light was employed: in the first case electric light and in the two angle ZPS=r of the star when observed, in the triangles pPs, PPZ others acetylene lamps., we have, since pPS=90+1-n, -Sin c=sin m sin 8+cos m cos o sin (n-1), Sin m=sin b sin -cos b cos o sin a, Cos m sin n = sin b cos +cos b sin o sin a. or a portable transit instrument. In the former case the operation And these equations solve the problem, however large be the errors consists in observing the angle between a terrestrial object-generally of the instrument. Supposing, as usual, a, b, m, n to be small, a mark specially erected and capable of illumination at night we have at once r=n+c sec 8+m tan s, which is the correction to and a close circumpolar star at its greatest eastern or western the observed time of transit. Or, eliminating m and n by means azimuth, or, at any rate, when very near that position. If the of the second and third equations, and putting 2 for the zenith observation be made i minutes of time before or after the time of distance of the star, t for the observed time of transit, the corrected greatest azimuth, the azimuth then will differ from its maximum time is 1+(a sin a+b coss+c) cos 8. Another very convenient form value by (4501)? sin i sin 28/ sin z, in seconds of angle, omitting for stars near the zenith is ?=b sec +c sec 8+m (tan 8-tano). smaller terms, 8 being the star's declination and z its zenith distance, Suppose that in commencing to observe at a station the error of the The collimation and level errors are very carefully determined chronometer is not known; then having secured for the instrument before and after these observations, and it is usual to arrange the a very solid foundation, removed as far as possible level and colliobservations by the reversal of the telescope so that collimation mation errors, and placed it by estimation nearly in the meridian, error shall disappear. If b, c be the level and collimation errors, let two stars differing considerably in declination be observed the the correction to the circle reading is b cot 2+¢ cosec 2, b being instrument not being reversed between them. From these two positive when the west end of the axis is high. It is clear that any stars, neither of which should be a close circumpolar star, a good uncertainty as to the real state of the level will produce a corre-l approximation to the chronometer error can be obtained; thus n meets Z P to letą, o be the apparent clock errors given by these stars if & op the telescope, which latter carries a micrometer in its eye-piece, be their declinations the real error is too scivanjem with a screw of long range for measuring differences of zenith dis (=e+(er-ea) (tan 4-tan 81)/(tan 8-tan 8). 1. 2001 tance. Two levels are employed for controlling and increasing the Of course this is still only approximate, but it will enable the observer accuracy. For this instrument stars are selected in pairs, passing (who by the help of a table of natural tangents can compute e in a north and south of the zenith, culminating within a few minutes few minutes) to find the meridian by placing at the proper time, tance of each other. When a pair of stars is to be observed, the of time and within about twenty minutes (angular) of zenith diswhich he now knows approximately, the centre wire of his instrument telescope is set to the mean of the zenith distances and in the plane on the first star that passes-not near the zenith. The transit instrument is always reversed at least once in the of the meridian. The first star on passing the central meridional wire is bisected by the micrometer; then the telescope is rotated course of an evening's observing, the level being frequently read and recorded: It is necessary in most instruments to add a correction very carefully through 180° round the vertical axis, and the second star on passing through the field is bisected by the micrometer on The transit instrument is also used in the prime vertical for the of the zenith distances, and the calculation to get the latitude is the centre wire. The micrometer has thus measured the difference determination of latitudes. In the preceding figure let 9 be the point most simple. Of course it in which the northern extremity of the axis of the instrument observations are not necessarily confined to the centre wire. necessary to read the level, and the In produced meets the celestial sphere. Let nz be the azimuthal fact it n, s be the north and south readings of the level for the south »Pq=r and Pg=y. Let S be the position of a star when observed star, m', s' the same for the north star, 1 the value of one division on a wire whose distance from the collimation centre is c, positive of the level, m the value of one division of the micrometer, ?, r the when to the south, and let h be the observed hour angle of the star, refraction corrections, u, # the micrometer readings of the south viz. ZPS'. Then the triangles qPS', qPZ give and north star, the micrometer being supposed to read from the zenith, then, supposing the observation made on the centre wire, -Sin c=sin & cos -cos & sin v cos (k+r), 6-10 +8)+36-w')m+{(n+n'-s-3')} + {(r-1'). Cos v = sin b sin 6+cos o cos o cos a, It is of course of the highest importance that the value m of the Sin sin 7 = cos b sin a. screw be well determined. This is done most effectually by observing Now when e and b are very small, we see from the last two equa. the vertical movement of a close circumpolar star when at its greatest azimuth. tions that *-b, a=, sin y, and if we calculated by the formula cot d'=cot & cos h, the first equation leads us to this result In a single night with this instrument a very accurate result, say with a probable error of about 0":2, could be obtained for $=$+(a sin s+b cos z+c){cos , latitude from, say, twenty pair of stars; but when the latitude is the correction for instrumental error being very similar to that required to be obtained with the highest possible precision, two applied to the observed time of transit in the case of meridian nights at least are necessary. The weak point of the zenith telescope observations. When a is not very small and z is small, the formulae lies in the circumstance that its requirements prevent the selection required are more complicated. of stars whose positions are well fixed; very frequently it is necessary The method of determining latitude by transits in the prime to have the declinations of the stars selected for this instrument vertical has the disadvantage of being a somewhat slow process, specially observed at fixed observatories. The zenith telescope is and of requiring a very precise knowledge of the time, a disadvantage made in various sizes from 30 to 54 in. in focal length; a 30-in. from which the zenith telescope is free. In principle this instrument telescope is sufficient for the highest purposes and is very portable. is based on the proposiThe net observation probable-error for one pair of stars is only tion that when the me +0.1. ridian zenith distances of The zenith telescope is a particularly pleasant instrument to two stars at their upper work with, and an observer has been known (a sergeant of Royal culminations-one being Engineers, on one occasion) to take every star in his list during to the north and the other eleven hours on a stretch, namely, from 6 o'clock p.M. until 5 A.M., to the south of the zenith and this on a very cold November night on one of the highest points --are equal, the latitude of the Grampians. Observers accustomed to geodetic operations is the of their attain considerable powers of endurance. Shortly after ihe comdeclinations; or, if the mencement of the observations on one of the hills in the Isle of Skye zenith distance of a star a storm carried away the wooden houses of the men and left the culminating to the south observatory roofless. Three observatory roofs were subsequently of the zenith be 2, its demolished, and for some time the observatory was used without a declination being 8, and roof, being filled with snow every night and emptied every morning. that of another culminat. Quite different, however, was the experience of the same party when ing to the north with on the top of Ben Nevis, 4406 ft. high. For about a fortnight the zenith distance Z' and state of the atmosphere was unusually calm, so much so, that a declination s', then clearly lighted candle could often be carried between the tents of the men the latitude is }(+8') † and the observatory, whilst at the foot of the hill the weather was I(2-2'). Now the zenith wild and stormy. telescope does away with The determination of the difference of longitude between two the divided circle, and stations A and B resolves itself into the determination of the local substitutes the measure time at each of the stations, and the comparison by signals of the ment micrometrically of clocks at A and B. Whenever telegraphic lines are available these the quantity 2-Z. comparisons are made by telegraphy. A small and delicately-made In fig. 4 is shown a apparatus introduced into the mechanism of an astronomical clock zenith telescope by H. or chronometer breaks or closes by the action of the clock an electric Wanschaff of Berlin, circuit every second. In order to record the minutes as well as which is the type used seconds, one second in each minute, namely that numbered o or 60, (according to the Central is omitted. The seconds are recorded on a chronograph, which Bureau at Potsdam) since consists of a cylinder revolving uniformly at the rate of one revolution about 1890 for the deter- per minute covered with white paper, on which a pen having a slow mination of the variations movement in the direction of the axis of the cylinder describes a of latitude due to different, continuous spiral. This pen is deflected through the agency of an Lotut the but as yet imperfectly electromagnet every second, and thus the seconds of the clock are understood, influences. recorded on the chronograph by offsets from the spiral curve. An To 0.75 siis The instrument is sup- observer having his hand on a contact key in the same circuit can ported on a strong tripod, record in the same manner his observed times of transits of stars. Al fitted with levelling | The method of determination of difference of longitude is, therefore, screws; to this tripod is virtually as follows. After the necessary observations for instru4. in fixed the azimuth circle mental corrections, which are recorded only at the station of obserFig. 4.—Zenith Telescope constructed and a long vertical steel vation, the clock at A is put in connexion with the circuit so as to for the International Stations at Mizu axis. Fitting this axis write on both chronographs, namely, that at A and that at B. sawa, Carloforte, Gaithersburg and hollow axis which Then the clock at B is made to write on both chronographs. It is Ukiah, by Hermann Wanschaft, Berlin. carries on its upper end a clear that by this double operation one can eliminate the effect of the short transverse horizon- small interval of time consumed in the transmission of signals, for ... 677/1365 bon tal axis with a level. This the difference of longitude obtained from the one chronograph latter carries the telescope, which, supported at the centre of its will be in excess by as much as that obtained from the other will be length, is free to rotate in a vertical plane. The telescope is thus in defect. The determination of the personal errors of the observers mounted eccentrically with respect to the vertical axis around in this delicate operation is a matter of the greatest importance. which it revolves. Two extremely sensitive levels are attached to as therein lies probably the chief source of residual error. mean is a 20 -0.108 -0.004 20 -0.027 B-S These errors can nevertheless be almost entirely avoided by using | For the sphere a=b=s, and making this simplification, we obtain the the impersonal micrometer of Dr Repsold (Hamburg, 1889). In theorem previously given by A. M. Legendre. With the terms of the this device there is a movable micrometer wire which is brought by fourth order, we have (after Andrae): hand into coincidence wi 's the star and moved along with it; at m-22. fixed points there are electrical contacts, which replace the fixed A *******). wires. Experiments at the Geodetic Institute and Central Bureau at Potsdam in 1891 gave the following personal equations in the case B-B=j+ + C-C=+* ***). -0".314 in which e=ok{1+(mok/8)), 3m2 = 0% ++0, 3k=a+b+r. For the ellipsoid of rotation the measure of curvature is equal to 1pn, B-G -0.225 +0.013 -0.086 e and ņ being the radii of curvature of the meridian and per. -0.023 pendicular. +o': 109 It is rarely that the terms of the fourth order are required. As a These results show that in the later method the personal equation rule spheroidal triangles are calculated as spherical (after Legendre). is small and not so variable; and consequently the repetition of i.e. like plane triangles with a decrease of each angle of about 3: longitude determinations with exchanged observers and apparatus e must, however, be calculated for each triangle separately with its entirely eliminates the constant errors, the probable error of such mean measure of curvature k. determinations on ten nights being sca cely *0901. The geodetic line being the shortest that can be drawn on any surface between two given points, we may be conducted to its most Calculation of Triangulalion, important characteristics by the following considerations: let P. 2 The surface of Great Britain and Ireland is uniformly covered by be adjacent points on a curved surface; through s the middle point triangulation, of which the sides are of various lengths from 10 to of the chord po imagine a plane drawn perpendicular to pg, and let I miles. The largest triangle has one angle at Snowdon in Wales, S be any point in the intersection of this plane with the surface another on Slieve Bonard in Ireland, and a third at Scaw Fell in then ps+Sq is evidently least when sS is a minimum, which is Cumberland; each side is over a hundred miles and the spherical when sS is a normal to the surface; hence it follows that of all excess is 64". The more ordinary method of triangulation is, however, plane curves on the surface joining p. 9. when those points are inthat of chains of triangles, in the direction of the meridian and definitely near to one another, that is the shortest which is made perpendicular thereto. The principal triangulations of France, by, the normal plane. That is to say, the osculating plane at any, Spain, Austria and India are so arranged. Oblique chains of tri- | point of a geodetic line contains the normal to the surface at that angles are formed in Italy, Sweden and Norway, also in Germany point. Imagine now three points in space, A, B, C, such that AB= and Russia, and in the United States. Chains are composed some BC=c; let the direction cosines of AB be l, m, n, those of BCI, times merely of consecutive plain triangles; sometimes, and more m', n', then x, y, z being the co-ordinates of B, those of A and C will frequently in India, of combinations of triangles forming consecutive be respectivelypolygonal figures. In this method of triangulating, the sides of the x-d:y-cm: triangles are generally from 20 to 30 miles in length-seldom exceed *+:ytcm': 2+cn'. ing 40. The inevitable errors of observation, which are inseparable from Hence the co-ordinates of the middle point M of AC are x + c(-1). all angular as well as other measurements, introduce a great difficulty +cm'-m), s+fcn'-n), and the direction cosineś of BM are into the calculation of the sides of a triangulation. Starting from a therefore proportional to l-1: m'-m: n'-n. If the angle made given base in order to get a required distance, it may generally be by BC with AB be indefinitely small, the direction cosines of BM obtained in several different ways-that is, by using different sets are as 8l : 8m : En. Now if AB, BC be two contiguous elements of of triangles. The results will certainly differ one from another, a geodetic, then BM must be a normal to the surface, and since al, and probably no two will agree. The experience of the computer om, on are in this case represented by &(dx/ds), 3(dy/ds), 8(dz/ds), will then come to his aid, and enable him to say which is the most and if the equation of the surface be x = 0, we have trustworthy result; but no experience or ability will carry him dix du dy /du_ d-zdu through a large network of triangles with anything like assurance. ds/dx=ds/ dyd-/ To' The only way to obtain trustworthy results is to employ the method of least squares. We cannot here give any illustration of this method which, however, are equivalent to only one equation. In the case as applied to general triangulation, for it is most laborious, even for of the spheroid this equation becomes the simplest cases. d? day Three stations, projected on the surface of the sea, give a spherical Idsax di or spheroidal triangle according to the adoption of the sphere or the ellipsoid as the form of the surface. A spheroidal triangle differs which integrated gives ydx-xdy=Cds. This again may be put in from a spherical triangle, not only in that the curvatures of the sides the form o sin a=C, where a is the azimuth of the geodetic at any are different one from another, but more especially in this that, point-the angle between its direction and that of the meridian while in the sphericaltriangle the normals to the surface at the angular and , the distance of the point from the axis of revolution. points meet at the centre of the sphere, in the spheroidal triangle From this it may be shown that the azimuth at A of the geodetic the normals at the angles A, B, C meet the axis of revolution of the joining AB is not the same as the astronomical azimuth at A of B spheroid in three different points, which we may designate a,, 8, or that determined by the vertical plane AaB. Generally speaking, respectively. Now the angle A of the triangle as measured by a the geodetic lies between the two plane section curves joining A and theodolite is the inclination of the planes BAa and CAa, and the angle B which are formed by the two vertical planes, supposing these points at B is that contained by the planes ABB and CBB. But the planes not far apart. If, however, A and B are nearly in the same latitude, ABa and ABB containing the line AB in common cut the surface in the geodetic may cross (between A and B) that plane curve which two distinct plane curves. In order, therefore, that a spheroidal lies nearest the adjacent pole of the spheroid. The condition of triangle may be exactly defined, it is necessary that the nature of the crossing is this. Suppose that for a moment we drop the consideralines joining the three vertices be stated. In a mathematical point tion of the earth's non-sphericity, and draw a perpendicular from of view the most natural definition is that the sides be geodetic or the pole Con AB, meeting it in between A and B. Then A being shortest lines. Ç. C. G. Andrae, of Copenhagen, has also shown that point which is nearest the pole, the geodetic will cross the plane that other lines give a less convenient computation. curve if AS be between AB and AB. IT AS lie between this last K. F. Gauss, in his treatise, Disquisitiones generales circa superficies value and AB, the geodetic will lie wholly to the north of both curvas, entered fully into the subject of geodetic (or geodesic) plane curves, that is, supposing both points to be in the northern triangles, and investigated expressions for the angles of a gcodetic | hemisphere. triangle whose sides are given, not certainly finite expressions, but The difference of the azimuths of the vertical section AB and of approximations inclusive of small quantities of the fourth order, the the geodetic AB, i.e. the astronomical and geodetic azimuths, is side of the triangle or its ratio to the radius of the nearly spherical very small for all observable distances, being approximatelysurface being a small quantity of the first order. The terms of the Geod. azimuth - Astr. azimuth fourth order, as given by Gauss for any surface in general, are very (cosa o sin ze+ 12 1-e pm . retain a). in which: e and a are the numerical eccentricity of the geodetic triangle, while A, B, C are those of a plane triangle and semi-major axis respectively of the meridian ellipse, o and a are having sides equal respectively to those of the geodetic triangle, the latitude and azimuth at A's=AB, and p and n are the radii of then, o being the area of the plane triangle and a, b, c the measures of curvature at the angular points, curvature of the meridian and perpendicular at A. For s=100 kilometres, only the first term is of moment; its value is o".028 A-Ato(2a+b+r) 1, cos o sin 2a, and it lies well within the errors of observation. If we B=B+ola+26+r) /12, imagine the geodetic AB, it will generally trisect the angles between 0-C+ola+b+26) /12. the vertical sections at A and B, so that the geodetic at A is near quantities of the second order only, and put A, B, C for the angles in 26 sin |