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GEODESY GE EODESY (yî, the earth, Salw, to divide) is the science | two thermometers, and a level for determining the inclina
of surveying extended to large tracts of country, tion of the bar in measuring. The manner of transferring having in view not only the production of a system of maps the end of a bar to the ground is simply this : under the of very great accuracy, but the determination of the curva- end of the bar a stake is driven very firmly into the ture of the surface of the earth, and eventually of the figure ground, carrying on its -upper surface à disk, capable of and dimensions of the earth. This last, indeed, may be movement in the direction of the measured line by means the sole object in view, as was the case in the operations of slow-motion screws. A fine mark on this disk is conducted in Peru and in Lapland by the celebrated French brought vertically under the end of the bar by means of a astronomers Bouguer, La Condamine, Maupertuis, Clairaut, theodolite which is planted at a distance of 25 feet from and others; and the measurement of the meridian arc of the stake in a direction perpendicular to the base. Struve France by Mechain and - Delambre had for its end the investigates for each base the probable errors of the determination of the true length of the “ metre” which measurement arising from each of these seven causes :was to be the legal standard of length of France.
alignment, inclination, comparisons with standards, readThe basis of every extensive survey is an accurate tri- ings of index, personal errors, uncertainties of temperature, angulation, and the operations of geodesy consist in-the and the probable errors of adopted rates of expansion. measurement, by theodolites, of the angles of the triangles ; The apparatus used in the United States Coast Survey the measurement of one or more sides of these
consists of two measuring bars, each 6 me in length, the ground; the determination by astronomical observations supported on two massive tripod stands placed at oue of the azimuth of the whole network of triangles; the de- quarter length from each end, and provided, as in Colby's termination of the actual position of the same on the sur- apparatus, with the necessary mechanism for longitudinal, face of the earth by observations, first for latitude at some transverse, and vertical adjustment. Each measuring rod of the stations, and secondly for longitude.
is a compensating combination of an iron and a brass bar, To determine by actual measurement on the ground the supported parallel to one another and firmly connected at length of a side of one of the triangles, wherefrom to infer one end, the medium of connexion between the free ends the lengths of all the other sides in the triangulation, is not being a lever of compensation so adjusted as to indicate a tho least difficult operation of a trigonometrical survey. constant length independent of temperature or changes of When the problem is stated thus—To determine the num- temperature. The bars are protected from external influber of times that a certain standard or unit of length is ences by double tubes of tinned sheet iron, within which contained between two finely marked points on the surface they are movable on rollers by a screw movement which of the earth at a distance of some miles asunder, so that allows of contacts being made within totoo of an inch. the error of the result may be pronounced to lie between The abutting piece acts upon the contact lever which is certain very narrow limits, then the question demands very attached to the fixed end of the compound bar, and carries serious consideration. The representation of the unit of a very sensitive level, the horizontal position of which delength by means of the distance between two fine lines on fines the length of the bar. It is impossible here to give the surface of a bar of metal at a certain temperature is a full description of this complicated apparatus, and we nover itself free from uncertainty and probable error, owing must refer for details to the account given in full in the to the difficulty of knowing at any moment the precise United States Coast Survey Report for 1854. This appa. temperature of the bar; and the transference of this unit, ratus is doubtless a very perfect one, and the manipulation or a multiple of it, to a measuring bar, will be affected not of it must offer great facilities, for it appears to be possible, only with errors of observation, but with errors arising from under favourable circumstances, to measure a mile in one pacertainty of temperature of both bars. If the measuring day, 1:06 mile having been measured on one occasion in bar bo not self-compensating for temperature, its expansion eight and a half hours. In order to test to the utmost the must be determined by very careful experiments. The apparatus, the base at Atalanta, Georgia, was measured thermometers required for this purpose must be very care
twice in winter and once in summer 1872-73, at temperafully studied, and their errors of division and index error tures 51°, 45°, 90° F. ; the difference of the first and second determined.
measurements was +0:30 in., of the second and third The base apparatus of Bessel and that of Colby have been +0:34 in.,--the actual length and computed probable error described in FIGURE OF THE Eartu (vol. vii. p. 598). The expressed in metres being 9338-4763 +0.0166. It is to average probable error of a single measurement of a base be noted that in the account of a base recently measured in line by the Colby apparatus is, according to the very elab- the United States Lake Survey, some doubt is expressed as orate investigations of Colonel Walker, C.B., R.E., the Sur- to the perfection of the particular apparatus of this de veyor-General of India, + 1.5je (u meaning "one millionth”). scription there used, on account of a liability to permanent . Struve gives + 0-8 as the probable error of a base changes of length. line measured with his apparatus, being the mean of the
The last base line measured in India with Colby's comprobable errors of seven hases measured by him in Russia ; pensation apparatus had a length of 8912 feet only, and in but this estimate is probably too small. Struve's appa- consequence of some doubts which had arisen as to the ratus is simple : there are four wrought iron bars, each two accuracy of this compensation apparatus, the neasurement toises (rather more than 13 feet) long; one end of was repeated four times, the operations being conducted in each bar is terminated in a small steel cylinder presenting such a manner as to indicate as far as possible the actual a slightly convex surface for contact, the other end carries magnitudes of the probable errors to which such measures a contact lever rigidly connected with the bar. The shorter are liable. The direction of the line (which is at Cape arm of the lever terminates below in a polished hemisphere, Comorin) is north and south, and in two of the measure
upper and longer arm traversing a vertical divided arc. ments the brass component was to the west, in the other In measuring, the plane end of one bar is brought into two it was to the east. The differences between the indi. contact with the short arm of the contact lever (pushed vidual measurements and the mean of the four are Forward by & weak spring) of the
next bar. Each bar bas + .0017, - 0049, - .0015, + 0045 in feet. The measure,
ments occupied from seven to ten days cach, -the average be natural objects presenting themselves in suitable posi rate of such work in India being about a mile in five days. tions, such as church towers; or they may be objects • The method of M. Porro, adopted in Spain, and by the specially constructed in stone or wood on mountain tops Frenca in Algiers, is essentially different from those or other prominent ground. In every case it is necessary just described. The measuring rod, for there is only that the precise centre of the station be marked by some one, is a thermometric combination of two bars, one of permanent mark. In India no expense is spared in making platinum and one of brass, in length 4 metres, furnished permanent the principal trigonometrical stations-costly with three levels and four thermometers. Suppose A, towers in masonry being erected. It is essential that every B, C three micrometer microscopes very firmly sup. trigonometrical station shall present a fine object for ob
ported at intervals of 4 metres with their axes vertical, servation from surrounding stations.
Horizontal Angles. line of measurement. The measuring bar is brought In placing the theodolite over a station to be observed under say A and B, and those micrometers read; the bar from, the first point to be attended to is that it shall rest is then shifted and brought under B and C. By repetition upon a perfectly solid foundation. The method of obtainof this process, the reading of a micrometer indicating the ing this desideratum must depend entirely on the nature of end of each position of the bar, the measurement is made. the ground; the instrument must if possible bé supported The probable error of the central base of Madridejos, which on rock, or if that be impossible a solid foundation must has a length of 14664.500 metres, is estimated at # 0·17m. be obtained by digging. When the theodolite is required This is the longest base line in Spain; there are seven to be raised above the surface of the ground in order to others, six of which are under 2500 metres in length; of command particular points, it is necessary to build two scafthese one is in Majorca, another in Minorca, and a third in folds, -the outer one to carry the observatory, the inner one Ivica. The last base just measured in the province of Bar- to carry the instrument,--and these two edifices must have celona has a length of 2483.5381 metres according to the no point of contact. Many cases of high scaffolding have · first measurement, and 2483.5383 according to the second. occurred on the English Ordnance Survey, as for instance
The total number of base lines measured in Europe up at Thaxted Church, where the tower, 80 feet high, is surto the present time is about eighty, fifteen of which do not mounted by a spire of 90 feet. The scaffold for the obexceed in length 2500 metres, or about a mile and a half, servatory was carried from the base to the top of the spire; and two-one in France, the other in Bavaria-exceed that for the instrument was raised from a point of the spiro 19,000 metres. The question has been frequently discussed 140 feet above the ground, having its bearing upon timbers whether or not the advantage of a long base is sufficiently passing through the spire at that height. Thus the instru. great to warrant the expenditure of time that it requires, ment, at a height of 178 feet above the ground, was or whether as much precision is not obtainable in the end insulated, and not affected by the action of the wind on the by careful triangulation from a short base. But the answer observatory. cannot be given generally ; it must depend on the circum. At every station it is necessary to examine and correct stances of each particular case.
the adjustments of the theodolite, which are these :—the line It is necessary that the altitude above the level of the of collimation of the telescope must be perpendicular to its sea of every part of a base line be ascertained by spirit axis of rotation; this axis perpendicular to the vertical levelling, in order that the measured length may be reduced axis of the instrument; and the latter perpendicular to the to what it would have been had the measurement been plane of the horizon. The micrometer microscopes must made on the surface of the sea, produced in imagination. also measure correct quantities on the divided circle or Thus if I be the length of a measuring bar, h its height circles. The method of observing is this
. Let A, B, C.... at any given position in the measurement, r the radius of be the stations to be observed taken in order of azimath; the earth, then the length radially projected on to the level the telescope is first directed to A and the cross-hairs of the of the sea is 1
telescope made to bisect the object presented by A, then
the microscopes or verniers of the horizontal circle (also of reduction to the level of the sea is - 0.6294 feet.
the vertical circle if necessary) are read and recorded. The In working away from a base line ab, stations c, d, e, f telescope is then turned to B, which is observed in the same are carefully selected so as to obtain from well-shaped tri- manner; then C and the other stations.
Coming round by angles gradually increasing sides.
continuous motion to A, it is again observed, and the agreeBefore, however, finally leaving
ment of this second reading with the first is some test of the base line it is usual to verify it
the stability of the instrument. In taking this round of by triangulation thus: during the
angles—or "arc," as it is called on the Ordnance Surveymeasurement two or more points,
it is desirable that the interval of time between the first as P, q (fig. 1), are marked in the
and second observations of A should be as small as may be base in positions such that the
consistent with due care. Before taking the next arc the lengths of the different segments
horizontal circle is moved through 20° or 30°; thus a difof the line are known; then,
ferent set of divisions of the circle is used in each arc, which taking suitable external stations,
tends to eliminate the errors of division, as h, k, the angles of the triangles
It is very desirable that all arcs at a station should thp, phq, hak, kqa are measured.
contain one point in common, to which all angular measureFrom these angles can be com.
ments are thus referred,--the observations on each arc computed the ratios of the seg.
mencing and ending with this point, wbich is on the Ordnients, which must agree, if all
nance Survey called the "referring object." It is usual for operations are correctly per
this purpose to select, from among the points which have formed, with the ratios resulting
to be observed, that one which affords the best object for from the measures. Leaving the
precise observation. For mountain tops a "referring obbase line, the sides increase up
ject” is constructed of two rectangular plates of metal in to ten, thirty, or fifty miles, occasionally, but seldom, reach. the same vertical plane, their edges parallel and placed at ing a hundred miles. The triangulation points may either such a distance apart that the light of the sky seen through
as a vertical line about 10" in width. The best to employ a heliostat. In its simplest form this is a plane ce for this object is from oue to two miles.
mirror 4, 6, or 8 inches in diameter, capable of rotation clear that no correction is required to the angles round a horizontal and a vertical axis. This mirror is ured by a theodolite on account of its height above placed at the station to be observed, and in fine weather
98-level ; for its axis of rotation coincides with the it is kept so directed that the rays of the sun 'reflected le mal to the surface of the earth, and the angles measured by it strike the distant observing telescope. To the between distant points are those contained between the observer the heliostat présents the appearance of a star of vertical planes passing through the axis of the instrument the first or second magnitude, and is generally a pleasant and those points.
object for observing. The theodolites used in geodesy vary in pattern and in
Astronomical Observations. size—the horizontal circles ranging from 10 inches to 36 inches in diameter. In Ramsden's 36-inch theodolite the The direction of the meridian is determined either by a telescope has a focal length of 36 inches and an aperture theodolite or a portable transit instrument. In the former of 2-5 inches, the ordinarily used magnifying power being case the operation consists in observing the angle between 54; this last, however, can of course be changed at the a terrestrial object-generally a mark specially erected and requirements of the observer or of the weather. The pro- capable of illumination at night—and a close circumpolar bable error of a single observation of a fine object with this star at its greatest eastern or western azimuth, or, at any theodolite is about 0".2.
rate, when very near that position. If the obsarvation be Fig. 2 represents an altazimuth theodolite of an im- made t minutes of time before or after the time of greatest proved pattern now used on the Ordnauce Survey. The azimuth, the azimuth then will differ from its maximum
(4501)* sin 1" + in seconds of angle, omitting smaller terms. Here the symbol 8 is the star's declination, - 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 2 + c cosec 2, 6 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 in the resulting value of tlie azimuth,- an uncer. tainty 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 ou 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 is the German diagonal form, in which the Fio. 2.-Altazimuth Theodolito.
rays of light after passing through the object glass are
turned by a total reflexion prism through one of the trang. horizontal circle of 14 inches diameter is read by three verse arms of the telescope, at the extremity of which arm micrometer microscopes; the vertical circle has a diameter is the eye-piece. The unused half of the ordinary telescope of 12 inches, and is read by two microscopes.
being cut away is replaced by a counterpoise. In this inIn the Great Trigonometrical Survey of India the theo strument there is the advantage that the observer without dolites used in the more important parts of the work have moving the position of his eye commands the whole been of 2 and 3 feet diameter,—the circle read by five meridian,
and that the level may remain on the pivots equidistant microscopos. Every angle is measured twice whatever be the elevation of the telescope. But there is in each position of the zero. of the horizontal circle, of the disadvantage that the flexure of the transverse axis which there are generally ten; the entire number of causes a variable collimation error depending on the zenith measures of an angle is never less than 20. An examin- distance of the star to which it is directed, and moreover ation of 1407 angles showed that the probable error of an it has been found that in some cases the personal error.of observed angle is on the average = 0·28.
an observer is not the same in the two positions of the For the observations of very distant stations it is usual | telescope.
To determine the direction of the meridian, it is well to
tan o- tan 8, erect two marks at nearly equal angular distances on either
tand, -tand, side of the north meridian live, so that the pole star crosses Of course this is still only approximative, but it will enable the vertical of each mark a short time before and after the observer (who by the help of a table of natural tangents attaining its greatest eastern and western azimuths. If now the instrument, perfectly levelled, is adjusted to placing at the proper time, which he now knows approxi
can compute e in a few minutes) to find the meridian by have its contre wire on one of the marks, then when ele-mately, the centre wire of his instrument on the first star vated to the star, the star will traverse the wire, and its that passes—not near the zenith. exact position in the field at any moment can be measured
The transit instrument is always reversed at least once by the micrometer wire. Alternate observations of the in the course of an evening's observing, the level being star and the terrestrial mark, combined with careful level
frequently read and recorded. It is necessary in most readings and reversals of the instrument, will enable one, instruments to add a correction for the difference in size even with only one mark, to determine the direction of the of the pivots. meridiau in the course of an hour with a probable error of The transit instrument is also used in the prime vertical less than a second. The second mark enables one to com- for the determination of latitudes. In the preceding figure plete the station more rapidly, and gives a check upon the let q be the point in which the northern extremity of the work. As an instance, at Findlay Seat, in latitude 57 axis of the instrument produced meets the celestial sphere. 35', the resulting azimuths of the two marks were 177° Let nZq be the azimuthal deviation = a, and l being the 4537"-29 + 0"-20 and 182° 17' 15":61 = 0":13, while level error, 2q = 90° – b; let also nPg=q and Pq = 4. Let the angle between the two marks directly measured by a
S' be the position of a star wlien observed on a wire whose theodolite was found to be 4' 31' 37"-43 & 0".23.
distance from the collimation centre is c, positive when to We now come to the consideration of the determination
the south, and lot h be the observed hour angle of the star, of time with the transit instrument. Lct fig. 3 repro- viz., Zps'. Then the triangles qPS', qrZ give sent the sphere stereographi
- Sin c - sin 8 cos w - cos 8 sin x cos (h+7), cally projected on the plane
Cosy - sin b sin 0 + cos o cos o cos ft, of the horizon,-ns being the
Sin y sin 1 = cos d sin ito meridian, we the prime verti
Now when a and b are very small, wo see from the last cal, Z, P the zenith and the
two equations that y=$-6, a=q sin y, and if we calcupole. Let p be the point in w which the production of the
late $ by the formula cot d' = cot & cos h, the first equa
tion leads us to this result axis of the instrament meets
a sin + o cos etc the celestial sphere, S the posi
+, tion of a star when observod on a wire whose distance from the
the correction for instrumental error being very similar collimation centre is c.
to that applied to the observed time of transit in the case be the azimuthal deviation, namely, the angle wzp, 6 the of meridian observations. When a is not very small and : level error so that Zp=90° - b. Let also the hour angle is small, the formulæ required are more complicated. corresponding to p be 90° — », and the declination of the same = m, the star's declination being 8, and the latitude . prime vertical has the disadvantage of being a somewhat Then to find the hour angle ZPS = =r of the star when slow process, and of requiring a very precise knowledge of observed, in the triangles pPS, pPZ we have, since the time, a disadvantage from which the zenith telescope is PPS = 90+- n,
free. In principle this instrument is based on the proposi-Sin (= sin m sin 8+ cos m cos 8 sin (n-1),
tion that when the meridian zenith distances of two stars Sin m - sin b sin o-coscos sin a,
at their upper culminations--one being to the north and the Cos i sin n-sin b cas o + cos b sin o sin a.
other to the south of the zenith-aro equal, the latitude is And these equations solvo the problem, however large bc the mean of their declinations; or, if the zenith distance of the errors of the instrument. Supposing, as usual, a, b, a star culminating to the south of the zenith be Z, its dem, n to be small, we have at once r = n + c sec 8+ m tan 8, clination being 8, and that of another culminating to the which is tho correction to the observed time of transit. north with zenith distance Z' and declination 8, then Or, eliminating m and n by means of the second and third clearly the latitude is (8+8) + }(2-2). Now the equations, and putting z for the zenith distance of the star, zenith telescope does away with the divided circle, and subi for the observed time of transit, the corrected time is stitutes the measurement micrometrically of the quantity
Z'-Z. a sin + b cos z+C
The instrument (fig. 4) is supported on a strong tripod,
fitted with levelling screws; to this tripod is fixed the aziAnother very convenient form for stars near the zenith is muth circle and a long vertical steel axis. Fitting on this this
axis is a hollow axis which carries on its upper end a short Tab sec +c sec 8 + n (tan 8 – tan o).
transverse horizontal axis. This latter carries the telescope, Suppose that in commencing to observe at a station the which, supported at the centre of its length, is free to error of tho chronometer is not known; then having se- rotate in a vertical plane. The telescope is thus mounted cured for the instrument a very solid foundation, removed excentrically with respect to the vertical axis around which as far as possible level and collimation errors, and placed it it revolves. An extremely sensitive level is attached to by estimation Dearly in the meridian, let two stars dif- the telescope, which latter carries a micrometer in its eyefering considerably in declination be observed—the in- piece, with a screw of long range for measuring differences strument not being reversed between them. From these of zenith distance. For this instrument stars are selected two stars, neither of which should be a close circumpolar in pairs, passing north and south of the zenith, culminating star, a good approximation to the chronometer error can be within a few minutes of time and within about twenty obtained ; thus let ej, co be the apparent clock errors given minutes (angular) of zenith distance of each other. When by these stars, if 81, 8, be their declinations the real error a pair of stars is to be observed, the telescope is set to the is
mean of the zenith distances and in the plane of the
** The method of determining Tatitude by-transite in the
peridian. The first star on passing the central meridional, the circumstance that its requirements prevent the selection wire is bisected by the micrometer; then the telescope is of stars whose positions are well fixed; very frequently it rotated very carefully through 180° round the vertical axis
, is necessary to have the declinations of the stars selected and the second star on passing through the field is bisected for this instrument specially observed at fixed observatories.
The zenith telescope is made in various sizes from 30 to 54 inches in focal length; a 30-inch telescope is sufficient for the highest purposes, and is very portable. The zenith telescope is a particularly pleasant instrument to work with, and an observer has bee kno (a sergeant of Royal Engineers, on one occasion) to take every star in his list during eleveu hours on a stretch, namely, from 6 o'clock P.M. until 5 A.M., and this on a very cold November night on one of tho highest points of the Grampians. Observers accustomed to geodetic operations attain considerablo powers of endurance. Shortly after the commencement of the observations on one of the hills in the Isle of Skye a storm carried away the wooden houses of the men and left the observatory roofless. Three observatory roofs were subsequently demolished, and for some time the observatory was used without a roof, being filled with snow every night and emptied every morning. Quite different, however, was the experience of the same party when on the top of Ben Nevis, 4406 feet high. For about a fortnight the state of the atmosphere was unusually calın, so much so, that a lighted candle could often be carried between the tents of the men and the observatory, whilst at the foot of the hill the weather was wild and stormy
Calculation nf Triangulation. The surface of Great Britain and Ireland is uniformly covered by triangulation, of which the sides are of various lengths from 10 to 111 miles. The largest triangle has one angle at Snowdon in Wales, another on Slieve Donard in Ireland, and a third at Scaw Fell in Cumberland; each side is over a hundred miles, and the spherical excess is 64".
The more ordinary method of triangulation is, however, that of chains of triangles, in the direction of the meridian and perpendicular thereto. The principal triangulations of France, Spain, Austria, and India are so arranged. Oblique chains of triangles are formed in Italy, Sweden, and Norway, also in Germany and Russia, and in the United States. Chains are composed sometimes merely of consecutive plain triangles ; sometimes, and more frequently in India, of coinbinations of triangles forming consecutive
polygonal figures. In this method of triangulating, the F19. 4.--Zenith Telescope.
sides of the triangles are generally from 20 to 30 miles in by the micrometer on the centre wire. The micrometer has length—seldom exceeding 40. thus measured the difference of the zenith distances, and The inevitable errors of observation, which are inseparthe calculation to get the latitude is most simple. Of able from all angular as well as other measurements, incourse it is necessary to read the level, and the observa-troduce a great difficulty into the calculation of the sides tions are not necessarily confined to the centre wire. In of a triangulation. Starting from a given base in order to fact if n, s be the north and south readings of the level for get a required distance, it may generally be obtained in the south star, n', s' the same for the north star, 1 the several different ways—that is, by using different sets of value of one division of the level
, m the value of one triangles. The results will certainly differ one from another, division of the micrometer, r, s, the refraction corrections, and probably no two will agree. The experience of the My po
' the microineter readings of the south and north star, computer will then come to his aid, and enable him to the micrometer being supposed to read from the zenith, say which is the most trustworthy result ; but no experithen, supposing the observation made on the centre wire, - ence or ability will carry him through a large network of
= 1(8+8) + 3(x-')m+ |(r + n'-5-8)2+1(r-'). triangles with anything like assurance. The only way to It is of course of the highest importance that the value
obtain trustworthy results is to employ the method of least m of the screw bo well determined. This is done most
squares, an explanation of which will be found in FIGURE effectually by observing the vertical movement of a close illustration of this method as applied to general triangula
OF THE EARTH (vol. vii. p. 605). We cannot here give any circumpolar "star when at its greatest azimuth.
In a single night with this instrument a very accurate tion, for it is most laborious, even for the simplest cases. result
, say with a probable error of about 0":3 or 0":4, We may, however, take the case of a simple chain-comcould be obtained for latitude from, say, twenty pair of mencing with the consideration of a single triangle in which staro; but when the latitude is required to be obtained
with all three angles have been observed. the highest possible precision, four or five fine nights are
Suppose that the sum of the observed angles exceeds the proper necessary. The weak point of the zenith telescope lies in rections to the angles, so as to cause this error to disajipear. 3 To
amount by a small quantity : it is required to assign proper cor