

GEODESY. 



OBODKSY. 



Iways be 07 mull ; in fact, they will amount only to a few him- 

 dredtht of a Mooad eren in the largest triangles Nevertheless when 

 the angle* are considered a* baring a mutual dependence, and the 

 correction* for a whole aerie* are determined simultaneously from the 

 equation* of condition, the corrected angle* will be given (if great pre- 

 rirv^ U required) to three or four decimal* of a second, and the above 

 formulae will have a practical application. They were uaed by Besael 

 in the recalculation of the triangle* at the southern extremity of the 

 French arc of meridian ; and we give them a place the more willingly 

 a* they hare not hitherto, so far a* we know, found their way into any 

 English work. 



Professor Buxengeiger ha* also given a formula for the calculation of 

 the aide* of a geodetical triangle, which may be substituted for the 

 method of Legendre, or used for verification. Let a be the known 

 side, A, B, c, the corrected spherical angle*, = 0-4342945, the modulus 

 of the common logarithms, and n = m-:-3 = 0-0000007, then for com- 

 puting 4 and c we have 



log 6 = log a + log sin B log sin A + n K (cot A cot B) ; 



log c= log o + log *in c log sin A + n E (cot A cot c). 



With the help of a small table of natural tangents, this method is 

 scarcely more troublesome than Legendre's. The chord method U 

 more tedious than either, and doe* not appear to be attended with any 

 corresponding advantage. 



Latitude, Lonyitiutet, and Azimnth. Having ascertained the ter- 

 restrial distance* between the several stations, the next step is to 

 determine their geographical positions, or situations with respect to 

 the equator and an assumed first meridian. For this purpose the 

 latitude and longitude of one station at least, and the azimuth of a side 

 of one of the triangles, must be accurately determined by astronomical 

 mean* ; we have then the data that are necessary for computing the 

 geographical position of every other angular point, and the bearing of 

 every other side, through the whole series of triangles, assuming the 

 earth to be a spheroid of rotation of known dimensions and ellipticity. 

 The uncertainty, however, which always exists respecting the exact 

 form and curvature of any particular portion of the earth's surface, or 

 rather the irregularities of local configuration, require independent 

 astronomical observations, particularly of azimuth, to be made at more 

 stations than one, when the triangulation extends over a considerable 

 tract of country. 



If the country included in the survey contains a fixed observatory, 

 this will of course either form one of the principal stations or be con- 

 nected with the principal triangles, and may be taken as the point of 

 departure. In this case the astronomical position of the fundamental 

 point is known with the greatest certainty ; and the meridian-mark of 

 the transit-instrument affords the surest means of determining the 

 bearing of any signal visible from the observatory. At any other 

 station the best method of determining the azimuth is probably to 

 imitate this proceeding, by setting up a temporary mark as nearly in 

 the meridian as may be, and determining its deviation with a transit- 

 instrument by aome of the methods used in practical astronomy for 

 the purpose. [TBANsrr-lssTBUMKST.] The angle between the mark 

 and the signal at another of the principal stations is then measured 

 with the theodolite, whence the azimuth of the signal becomes known. 

 But as this method cannot always be conveniently followed in geode- 

 tical operations, the usual practice is to make the surveying instrument 

 itself subservient to the determination of azimuths. In the English 

 survey the method commonly adopted was, to observe with the theodo- 

 lite the angle between a Hag-staff and the pole-star at its extreme 

 digressions east and west, and to take half the sum of the two angles 

 a* the azimuth of the staff. In determining the azimuth in this way 

 a very accurate adjustment of the instrument is necessary. The 

 method usually followed on the Continent has been to observe the 

 angle between a referring signal and the sun, or some star whose place 

 is well known, when near the horizop ; and a* the azimuth of the sun 

 or star at a given instant of time can be computed with great precision, 

 the observation is liable to no particular cause of error, unless in the 

 determination of the exact clock-time. The result is usually made to 

 depend on the mean of a great number of observations. 



The problem on the solution of which the calculation of the geode- 

 tical latitudes and longitudes of the stations, and the azimuths of the 

 sides of the triangles, depends is this : Let A and B be two stations 

 whose distance has been determined, and suppose the latitude and 

 longitude of A to be known, together with the azimuth of B as seen 

 from A ; it is required to find the latitude and longitude of B, together 

 with the azimuth of A as seen from u. The azimuthal angles are sup- 

 posed to commence at the south point of the horizon, and to be 

 reckoned towards the west (or right) from to 360. Assume 



' J = the latitude of A, f = the latitude of B, 

 X = the longitude of A, A' = the longitude of B, 

 = the azimuth of A B at the station A, 

 f = the azimuth of A B at the station B, 

 if = the distance on the spheroid between A and B, in feet. 



And, u before, let a denote half the greater axis of the meridian, t the 

 excentricity, p' = a-r (! sin' 1), the radius of curvature of the arc 

 perpendicular to the meridian at A, and = 206264''8. Then, putting 

 (for brevity) u tl+f', the differences of the latitudes, longitudes, and 



azimuth* of the two stations, in seconds of arc, are given by the 

 following formula', namely : 



r-f =-.( cos 8-1 sin 1 9 tan I) (1 -r CDS' 0, 



\' A = a> u sin 8 sec T, 



ff-e = 180*-(X'-X) sin } (J' + f) sec 4 (I'-l). 



These formula) are only approximative ; but they are sufficiently exact 

 for every case that can arise in practical geodesy. Even in the case of 

 the greatest distances between the stations, 100 mile* for example, the 

 results which they give will agree with those which are computed n< m 

 the exact formula} of spheroidical trig<'ii'>int tr\ to within a small ii 

 of a second. 



By means of the above formula;, the geographical positions of the 

 principal stations are successively deduced from each other ; but 

 a chain of triangles runs nearly north and south, or east and west, the 

 differences of latitude and longitude are more readily computed by 

 referring the different stations to an assumed meridian by means of 

 parallels and perpendiculars. Let A, B, c, D, I, F be the summits of a 



B 



F 



chain of triangles, and x T the direction of the meridian passing through 

 the first point A; and let perpendiculars B 6, cc, Ac., be drawn from 

 each of the other points to x T. Suppose the angle x A B (the azimuth 

 of B on the horizon of A) to be determined by astronomical observations : 

 then, as the angles at B, c, D, Ac., are all known from the geodetical 

 observations, the angles which the several sides of the triangles make 

 with x Y are easily computed ; and the distance on the meridian 

 between the perpendiculars through the extremities of any side is 

 found by multiplying the length of the side into the cosine of its 

 inclination. Hence the distance on the meridian from the point of 

 departure A, to the foot of the perpendicular through any other point, 

 F, is equal to the sum of the sides which join A and F. each multiplied 

 into the cosine of its inclination. Thus, the sum of the products of 

 the three sides A B, B D, D F, by the cosines of their respective inclina- 

 tions, gives Aft +bd + df=t.f; or the sides AC, CE, EF, reduced in 

 the same manner, give AC + ce + tf A/. Hence the distance from A 

 to the foot of the perpendicular through each point becomes kimwii. 

 In like manner, on multiplying the length of any side into the tint of 

 its inclination, we have the difference of the distances of its two extre- 

 mities from x T, and the distance F/, of any point F, is the sum of 

 those differences (taken with their proper signs) in respect of all the 

 intermediate sides between A and F. These relations are more shortly 

 expressed by algebraic formula;. Let the inclinations of three sides 

 intermediate between A and F, for example, A B, BD, D r (the angles at 

 the different points being all reckoned in the same direction), be respec- 

 tively a, /3, f ; then, attention being given to the algebraic signs of the 

 trigonometrical lines, we have 



A/= A B cos a + B D cos /3 + D F cos y, 

 rf= A B sin a + B n sin $ + D F sin y. 



By this means, all the angular points of the series are referred to the 

 meridian of the first, exactly in the same manner as the different points 

 of a curve are referred to its axis by their co-ordinates. 



We have now to determine the differences of latitude. Taking the 

 point F for example, let F A be the arc of the parallel circle on the 

 spheroid, passing through F, and h its intersection with the meridian 

 of A, then A A (which is always less than A/) is the arc of meridian 

 corresponding to the difference of latitudes. To compute / A we have, 

 from the properties of the spheroid, the formula / h = F / tan /'-j- 2 R, 

 in which I' is the approximate latitude of F, and R the distance from 

 the centre of the earth ; and as / A is always very small, instead of 

 computing the value of R, it will be sufficiently accurate to use p, the 

 radius of curvature of the meridian corresponding to the latitude. 

 The distance / k thus found is expressed in feet ; to convert it into 

 seconds of arc we must multiply by u-r-p ; hence the difference of the 

 latitudes of F and A, expressed in seconds, is 



The latitude of F and its distance F/ from the meridian of A being 

 known, its longitude, or the are F A of the parallel circle, is found from 

 the formula already given. Let />' be the radius of curvature of the 

 perpendicular are F/, then (-=-(>') F/is the number of seconds in F/, 

 and hence the difference of longitudes of F and A in seconds is 



V A = (a-^-ff) x F/*ec P. 



When the position* of a considerable number of points are to be 

 determined, the calculations may be facilitated by forming a table of 

 the values of /A corresponding to every value of F/ proceeding by 



