10^ INVESTIGATION 0¥ CERTAIN THEOREMS 



extent, but becaufe there is not any vilible mark by which its 

 that the denfity exiftence can always be dillinguilhed. The difference be- 

 °ear^th^"rth*s tween the primary and fecondary ftrata is probably one of the 

 fuiface muft be chief circumftances on which this inequality depends. The 

 "^" j^ greater to- primary ftrata, efpeclally if we include among them the granite, 

 gionsthan ^^X often have three times the fpecific gravity of water, 



others, whereas the fecondary, fuch as the marly and argillaceous, 



frequently have not more than twice the fpecific gravity of 

 that fluid. Suppofe, then, that a degree is meafured in a 

 country where the ftrata are all fecondary, and happens to ter- 

 minate near the junction of thefe with the primitive or denfer 

 flrata, the line of which junction we (hall fuppofe to lie neaiiy 

 eaft and weft; the fuperior attraction of the denfer ftrata muft 

 draw the plummet toward them, and make the zenith retire 

 in the oppofite diredtion ; thus diminiftiing the amplitude of 

 the celeftial arch, and increafing, of confequence, the geode- 

 Thls error may tical meafure affigned to a degree. From fuppofitions, no 

 feconds*"^'^^^* way improbable, concerning the denlity and extent of fuch 

 mafles of ftrata, I have found, that the errors, thus produced, 

 may eafily amount to ten or twelve feconds. 

 Accurate refults 3. While we continue to draw our conclufions, about the 



are therefore figure of the earth, from the meafurement of fingle degrees, 

 onlvtobehad ° ' „ . ,. * ^ ,. ° .^ 



from large arcj j there appears to be no way ot avoidmg, or even oi diminitn- 



ing, the effeCls of thefe errors. But if the arches meafured 

 are large, and confift each of feveral degrees, though there 

 (hould be the fame error in determining their celeftial ampli- 

 tudes, the effect of that error, with refpefl to the magnitude 

 and figure of the earth, will become inconfiderable, being 

 fpread out over a greater interval ; and it is, therefore, by 

 the comparifon of two fuch arches that the moft accurate refult 

 and thefe require is likely to be obtained. But, in purfuing this method, fince 



new modes of ^\^q arches meafured cannot be treated as fmall quantities, or 

 computation. ^ . . . ,,. r iii- 



mere fluxions or the earth s circumference, the calculation 



muft be made by rules quite different from thofe that have been 



hitherto employed. Thefe new rules are deduced from the 



following analyfis. 

 InTcftigatlon of 4. Let the ellipfis ADBE reprefent a meridian pafling 

 formulab for through the poles D and E, and cutting the equator in 

 figursofthc A and B. Let C be the centre of the earth, AC, the 

 tutth, radius of the equator, = a, and DC, half the polar axis, 



= b. Let FG be any very fmall arch of the meridian, having 



its 



