17S 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



[M. 



TO FIND THE MAGNITUDE AND FIGURE OF THE EARTH, 

 WITH THE ASSISTANCE OF RAILWAYS; 



Without presupposing it to he any particular form.^ 



By Oliver Btrnjs, Mathematician, &c. 



To find the lengtli of a degree in any position on the surface of the 

 earth, with the assistance of railroads, presents no obstacle whatever 

 to the mathematician. 



In England, America, and on the Continent, there are at present a 

 considerable number of railroads long enough for the purpose, so that 

 we can have a sufficient number of examples. The lengths, bearings, 

 elevations, and depressions, &c., of the several planes which compose 

 each railroad, may be had from different companies or their respective 

 engineers, and if the dimensions thus obtained, are not considered 

 accurate enough, they can be easily taken over again. When this in- 

 formation is acquired, the length of the arc on the surface of the earth 

 between any two given points in the line of road may be very readily 

 determined, on account of the approximant of the general line of most 

 railroads to levelness. By astronomical observations the latitudes of 

 both ends of the road, or any point in it may be determined, and also 

 the length of a degree upon any arc passing through either extremitv, 

 and a point in the line of road; as the bearings and length of such arc 

 can always be found. 



Then supposing a railroad selected of such a length, that from a 

 point in it the length of a degree is known on the arc passing through 

 that point and the beginning of the road, and also that of a degree on 

 the arc passing through the said point and the termination of the road, 

 and the arcs to be oblique ; or it n ill answer the purpose as well to 

 have two roads in the same latitude or nearly so, such that the length 

 of a degree on the arc passing through the extremities of each is as- 

 certained ; from which it is required to find the length of a degree on 

 the meridian, and also that of a degree on an arc perpendicular to it 

 at the same point. 



This may be done as follows : — 



EWA 



Let LAR be an arc of the meridian, SAP an arc of the curve at 

 right angles to it, AT one of the arcs upon which the length of a de- 

 gree is known = L ; and A Q the other upon which the length of a 

 degree =:/; also, let the bearings of these arcs, or the angles LAT 

 and Q A R, which are known = o and 6 respectively ; again, let g = 

 57'^'295779,5 the arc equal in length to the radius, therefore the radius 

 of curvature at A in the arc AT = Lg; and that at A in AQ = i^. 

 To find the radius of curvature of the arcs R A L and SAP, let the 

 radius of curvature of the meridian at A = .r, and ihat of its perpen- 

 dicular or of the arch S A P at the point A ^ y. Take A C, an inde- 

 finitely small portion of AT==s, and AE, an indefinitely small por- 

 tion of A Q = s'. 



Then A B =: s cos. a, and B C = s sin. a, as AC is supposed to be 

 indefinitely small. 



Now from the nature of the osculating circle at A, in the arc R AL, 

 the distance of the point B from the horizontal plane passing through 

 A, 



for A' B', fig. 2, stands in the same position as A B, fig. 1, and may be 

 considered as a perpendicular from the right angle at A or A', falling 

 between radius of cuivature O B, and the depression m B' of the 

 point B' below A', or that of B below the horizontal plane passing 



1 This articles is extracted from Mr. Byrne's extensive unpublished work, entitled, 

 * A New Theory of the Heavens and Earth." 



through A ; and the depression of C below B is =: ^ — ^'"' " for the 



y 



radius of curvature of the arc B C is ultimately the same with that of 

 SAP at the point A, the total depression of C below A is equal, 



n. 'a _ 5 /cos. 'a sin. -a\ 

 y ~ \ X y /■ 



Again, by the property of the circle above quoted, 



= the radius of curvature of the arc AT, at 



, /cos.^ a I sin', a \ 



the point A =z L^; 



1 



L.sr = 



xy 



y cos.' a ■\- X cos.- a 



(!•) 



(COS.- a sin.' a\ 

 ^ y ) 



By the same process of reasoning, we find, 



^y 7 



^ z^ I g 



y COS.' h -\- X sin.- h 



In (1) and (2) for the sine and cosine of a, substitute s, and c le- 

 spectively; and s' and c' for the sine and cosine of h; let L^ = r', 

 and I g= r'. Then equations (1) and (2) become 



(2.) 



xy 



and 



1/ c' -\- X »- 



xy 

 y' c\+ ;;;' s' 



(3.) 

 (4.) 



- from (3) 



T % X 



y = ;r - r' c" ^'■'"" ^'*^ 

 ;— p, hence we find 



x^ 



: c'« — c' s'2 _, , s'- c- ■ 



-; — ^T- X '■ r and y = - 



T s'' — r' s " ■ r c^ — r c - 



By restoring the values of r and r', we have 



sO c'= — c- s'- - , , 's'- c- — c" s' 



X r r. 



, the radius 



of curvature of the meridian, and that of the arc perpendicular to it 

 at the poinc A. 



- =: length of a degree on the meridian = -z — s p— ^ X L /. 



X 

 g 



y _ 



s c' — c s^ 

 - .= length of degree on arc perpendicular = , „ . ,., X h I. 



By restoring the values of s, s', c, c', we have 

 sin." a COS.- h — sin.' h cos.' a 



L sin.- a — I sin." A 



in." h cos.' a — cos.' A sin." a 



meridian : and. 



X L /, ^ the length of a degree on the 

 X L i = the length of 



L cos.' a — / COS." h 



a degree on an arc perpendicular to the meridian at the point A. 



By the help of this problem the true figure of the earth, with very 

 little trouble, might be ascertained, as we are not obliged to imagine 

 the earth to be any known form, because be it of whatever form it 

 may, the planes which compose a line of railroad must be in its sur- 

 face or very nearly so : nor is it very hard to find a degree or more 

 of road in the same direction. 



An Explosion of Subterkaneocs Water took place lately in the district 

 of Vizeu, in Portugal, by which the soil was torn up, and earth and stones flung to a 

 great height into the air, for the distance of more than a league, between the small river 

 Oleiros and the Douro. All the cultivated land over which the water flowed was de- 

 stroyed, and in many places it created ravines forty feet in depth, and thirty fathoms 

 wide. It carried away and shattered to fragments in its course, which was of extreme 

 rapidity, no fewer than fifty wind and water mills, choked the Douro with rubbish, and 

 caused the death of nine persons, including one entire family. On the same day a similar 

 explosion took place in the mountain of IMarcelim, in the same district, arising from the 

 same source, but branching off in the direction of the river Bastanza. It carried away a 

 farm-house, four cows, and some sheep and goats. A similar occurrence took place here 

 last year and the year before, and 18 months since in Madeira.— Times. 



