22 Dr. Roget’s description of a new instrument for 
calculation of chances, involve the investigation of powers, 
which may be facilitated by this instrument. Examples of its 
application also occur in considering the reduction of tempera- 
ture which bodies undergo by the communication of heat to 
surrounding bodies, the quantities of light transmitted through 
different thicknesses of a transparent medium : the diminution 
of density which the air in a receiver undergoes during its 
exhaustion by the air pump, and the relation of the density 
of the atmosphere with its elevation. The interpolation of a 
given number of mean proportionals between two given num- 
bers, is sometimes required for the solution of a problem, and 
is easily effected by the rule above described. Thus, in divid- 
ing the musical octave into twelve equal semi-tones, the fol- 
1 2 
lowing series of numbers must be calculated, viz. 2 1T , s T L 
2 Tz, 2 T^ 2 T2 } 2 T1^ 2 Tz, 2 Tz . can reac Jj]y fog 
clone in one position of the slider, for when the 1 2 marked on 
it is placed under 2 on the rule, the 1 of the slider will point 
1 -JL. 
to 1.0595 = a T % the 2 of the slider will indicate 1.1225 = 2 T % 
the 3, 1.1892 — 2 t % &c. 
When the first term of a progression is less than unity, all 
the succeeding terms, that is, all the powers of that fraction, 
continually decrease. Now all the numbers contained on the 
rule are above unity : but the terms of such a decreasing 
progression may yet readily be found by assuming, instead 
of the first term, its reciprocal, which, being above unity, will 
of course be contained on the scale. The powers of this reci- 
procal, will, in like manner, be the reciprocals of the required 
series, which will accordingly be determined without diffi- 
culty. Let the following question, for example, be proposed. 
