FIOURE OF THE EARTH. 3*27 



logarithm of the base measured at Clifton, as an arc gives us 

 that of its sine in feet or in fathoms, so that by means of this 

 latter logarithm, and the spherical angles of the series of tri- 

 angles, we obtain at once^ and as easily as in plane trigonome- 

 try, the logarithms of the sines of all their sides in fathoms. 



After this, it is extremely easy to convert them into loga- 

 rithms of chords or of arcs, for the purpose of applying them 

 to the computation of the arcs on the meridian or azimnih,. 

 I give the preference to taking the logarithms of the sides as 

 arcs, because the computations become in that case mnch more 

 simple and expeditious. 



Near to Clifton, which is the northern extremilv of the arc, R«^'it'cticB of 

 ' - tne northern 



in a situation elevated 35 feet above the level of the sea, a base to toiscs. 



base was measured of 26342,7 feet in length, the chains 



being supposed at the temperature of 6V Fahrenheit, or 13^° 



Reaumur. 



For reducing this base to toises, we have the proportion of 

 ihe English foot to that of France, as 4 : 4,263, so that if p be 

 taken to express the fractional part of the French foot, corres- 

 ponding to English measure, then ]og.p=g,g7 234,46587, 



and then log. of 2(5,342,7 = 4,42066,02800, 

 and hence the log. of the base in toises will be found equal to 

 3,.6l485,36943, and the number of toises corresponding is 

 41 ; 9,5 taken at the same temperature, which corresponds to 

 164." of the centigrade thermometer. 



This base we must consider as an arc of a circle, and it is 

 easy to reduce it to the sine of the same arc, according to 

 the method given in a note at the end of this memoir. The 

 logarithm of the sine of the base in toises is found to be 

 3,61485,35800. 



With this quantity as base, and by means of the spherical fro-n which _ 

 T^i-i- YT r J and the spheri- 



triangles given by Lieut. Col. Mudge m his paper, 1 have found ^a! triangles 

 the logarithmic sines in toises of all the sides of his series of the portions or 



° , , 1 1 , 1 • 1 • intervals ot the 



triangles, and have subsequently reduced them to logarithmic ^^^^-^^^ ^j^i, 

 arcs of the same, which enable me to complete the rest of the their azimuths 

 calculation. With these we may compute any portions of the ^^^""^^ «-ompu' 

 meridian^ or successive intervals of different stations expressed 

 in toises, and in parts of the circle, or their respective azi- 

 muths. 



