Oriani's theorem explains why Cassini's uniform-density model works well except near the horizon the atmospheric refraction from the zenith to a zenith distance of 70?is not dependent on the details of the distribution of the gas.
22.
If the declination of the body is in the opposite hemisphere ( ie if you are in the northern hemisphere and the declination is in the southern hemisphere ) then the declination must be subtracted from your true zenith distance, otherwise the declination is added.
23.
A number of solar panels may also be mounted vertically above each other in a tower, if the zenith distance of the Sun is greater than zero, and the tower can be turned horizontally as a whole and each panels additionally around a horizontal axis.
24.
The difference of the two zenith distances so read off is double the error of the zenith or horizontal point of the graduation, and their semi-sum is the true zenith distance of the collimating point, or the co-inclination of the axis of the collimating telescope to the horizon.
25.
The difference of the two zenith distances so read off is double the error of the zenith or horizontal point of the graduation, and their semi-sum is the true zenith distance of the collimating point, or the co-inclination of the axis of the collimating telescope to the horizon.
26.
Observer B is standing at one of the poles ( latitude 20�N on the same meridian, i . e . on the same longitude as observer A . His measured altitude would be 70?and by subtracting this from 90?gives a 20?zenith distance which in turn is " his " latitude.
27.
That is, one set of numbers begins at the left end of the arc and goes to 90 at the right end while the other set the zero is at the right and the 90 is at the left . This double numbering enables the instrument to measure either celestial altitude or zenith distance or both simultaneously.
28.
His measurements were subject to several inaccuracies : ( 1 ) though at the summer solstice the noon sun is overhead at the Tropic of Cancer, Syene was not exactly on the tropic ( which was at 23?43'latitude in that day ) but about 22 geographical miles to the north; ( 2 ) the difference of latitude between Alexandria ( 31.2 degrees north latitude ) and Syene ( 24.1 degrees ) is really 7.1 degrees rather than the perhaps rounded ( 1 / 50 of a circle ) value of 7?12'that Eratosthenes used; ( 3 ) the actual solstice zenith distance of the noon sun at Alexandria was 31?12' " 23?43'= 7?29'or about 1 / 48 of a circle not 1 / 50 = 7?12', an error closely consistent with use of a vertical gnomon which fixes not the sun's center but the solar upper limb 16'higher; ( 4 ) the most importantly flawed element, whether he measured or adopted it, was the latitudinal distance from Alexandria to Syene ( or the true Tropic somewhat further south ) which he appears to have overestimated by a factor that relates to most of the error in his resulting circumference of the earth.
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