He found the sextant had changed its error considerably, and that the glasses had lost their parallelism from the contraction of the brass.  In measuring the error he perceived that the diameter of the sun’s image was considerably short of twice the semi-diameter, a proof of the uncertainty of celestial observations made during these intense frosts.  The results of this and another similar observation are given in the footnote.*

(Footnote.  The observed meridian altitude of sun upper limb was 2 degrees 52 minutes 51 seconds.  Temperature of the air minus 45.5 degrees.  By comparing this altitude, corrected by the mean refraction and parallax with that deduced from the latitude which was observed in autumn, the increase of refraction is found to be 6 minutes 50 seconds, the whole refraction therefore for the altitude 2 degrees 52 minutes 51 seconds is 21 minutes 49 seconds.  Admitting that the refraction increases in the same ratio as that of the atmosphere at a mean state of temperature the horizontal refraction will be 47 degrees 22 seconds.  But the diameter of the sun, measured immediately after the observation, was only 27 minutes 7 seconds, which shows an increase of refraction at the lower limb of 3 minutes 29 seconds.  The horizontal refraction calculated with this difference and the above-mentioned ratio is 56 minutes 3 seconds at the temperature minus 45.5 degrees.  So that in the parallel 68 degrees 42 minutes where, if there is no refraction, the sun would be invisible for thirty-four days, his upper limb with the refraction 56 minutes 3 seconds is in fact above the horizon at every noon.

The wind was from the westward a moderate breeze and the air perfectly clear.  January 1st, 1821.  Observed meridian altitude of sun lower limb 2 degrees 35 minutes 20 seconds, sun apparent diameter 29 degrees 20 minutes.  For apparent altitude 2 degrees 35 minutes 20 seconds the mean refraction is 16 minutes 5 seconds (Mackay’s Tables) and the true, found as detailed above, is 20 minutes 8 seconds which, increasing in the same ratio as that of the atmosphere at a mean state of temperature, is 41 minutes 19 seconds at the horizon.  But the difference of refraction at the upper and lower limbs increasing also in that ratio gives 55 minutes 16 seconds for the horizontal refraction.  Temperature of the air minus 41 degrees.  Wind north, a light breeze, a large halo visible about the sun.  January 15th, 1821.  Observed an apparent meridian altitude sun lower limb 4 degrees 24 minutes 57 seconds.  Sun apparent diameter 31 minutes 5 seconds.  For apparent altitude 4 degrees 24 minutes 57 seconds the mean refraction is 10 minutes 58 seconds (Mackay’s Tables) and the true, found as detailed above, is 14 minutes 39 seconds which, increasing in the same ratio as that of the atmosphere at a mean state of temperature, is 43 minutes 57 seconds at the horizon.  But the difference of refraction between the upper and lower limbs increasing also in that ratio gives 48 minutes 30 seconds for the horizontal refraction.