96 Olservations on the Measuremenl of 



in plane trigonometry, the logarithms of the sines of all 

 tl)cn" sides in fathoms. 



Aftei: this, it is extremely easv to convert them into lo- 

 garithrtis of chords or of arcs, for the purpose of applying 

 them to the computation of the arcs on the meridian or 

 azimuths. I give the preference to taking the logarithms 

 of the sides as arcs, because the compulations become in 

 that cnse much more simple aiid expeditious. 



Near to Clifton, which is the northern extremity of the 

 arc, in a situation elevated 35 feet above the level of the 

 sea, a base was measured of 26342,7 feet in length, the 

 chains being supposed at the tenijicraiurcoi 62" Fahrenheit, 

 or 134-' Reaumur. 



For reducing this base to tf)!ses, u'e have the proportion 

 of the 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, corresponding to English measure, then log. p = 

 9,y7234,46jS7, and then log. of 26,3-12,7 = 



4.42()g6,02S60 ; and hence the log. of the base in toises 

 vill be found equal to 3,01435,36943, and the nimiber of 

 toises corresponding is 4119,3 taken at the same teuipera- 

 ture, wliich corresponds to 16J^ of the centigrade thermo- 

 meter. 



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 loaariihni of the sine of the base in toises is found to 

 be 3,61485,35500. 



With this quantify as base, and by means of the spherical 

 triangles given by Lieut. Col. Mudge in his paper, I have 

 found the logarithmic sines in toises of all the sides of his 

 series of triangios, ai^.d have subsequently reduced lliein to 

 ]r)garilhiTiic arcs of the same, \\ hich enable me to complete 

 the rest of the calculation. With these we may ccnnpute 

 anv portions of the meridi<in, or successive intervals of 

 different stations expressed in toises, and in parts nf the 

 circle, or their respective azimuths, having regard always 

 to the rc-lativc convergence of different niendians. 



The author ha> made observations for detertniniirg tho 

 latitude o! the two cxiremitics of his arc, and has also de- 

 termined the azinniihs of the extt;rior sides in his series of 

 trianoles by means of the greatest elongation of the pole 

 star. 



In the calmlatioiis that 1 have made, I began at Clifton 

 in yorkshire, the nothern exiremity of the arc, and lor this 



purpose 



