4b6 On Refraction, 



3d. For the thermometer without. 



269 



58"-1192 X 1-0945 t 9 (Z - - y r) 



80 ' 29-6 



(1 — h -001919) = R." 

 Or, in logarithms: 

 1st. From zero to 49° within, 



L. t, (Z - - 8 q r) + 0-3413143 -{- L.# + L.(l -h 



•002147) = L.R." 

 2dly. Above 49° within. 



L. t, (Z ~-r) -f 0*3394060 -f L.l -f-L.(l — h 



•002067) = L. R." 

 3dly. For the thermometer without. 



L. t, (Z — -^T r ) + 0-3322437 + Li-fL,(l-i 

 * oO ' 



^001919) =L.R." 

 In these equations, although very simple, there is 

 some little arithmetic trouble in computing the first and 

 last part of each expression. As to the first, it will be 

 easiest done by a table of the product of 3-36-25 into each 

 of the nine digits: by means of which the product of this 

 number into any other will readily be obtained, by only 

 adding that of its several digits, into 3-3625. Or it may 



be done by multiplying r by — ■ ; or, by the following ex- 



3x0 1 



pression 3*3625 r = — - r — ftn **> which will only re- 

 quire contracted multiplication and division. 



The last may easily be effected by a very ingenious and 

 simple method pointed out by M.Cagnoli*. Thus it is 

 well known that cos. 1 = rad. 2 — sin. 3 ; therefore putting 

 x = 1 — lm f and comparing it with the latter of these 

 two expressions, \\e have 



c\ A = r 1 — s 1 , A as 1 — hn ad x: 



consequently s, A = s/ hn and the square of the cosine 

 corresponding will be equal to x. Whence we have this 

 rule. To the logarithm of h add the logarithm of ?/, and 

 divide the sum by 2. Seek this number in the Table of 

 Logarithmic Sines, and take out the logarithmic cosine cor- 



• Trigonometry, 2d edit. 4to. Paris 1808, p. 95, or 1st edit. 1787, p. 102. 



responding 



