Some Investigations in the Theory of Map Projections

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This leads in our notation to the two equations ^ (« + X — ^) ~ sin (x — 8) = sin X - « (« + x) = '^ ('^ + X + ^) ~ sin (x + 8), whence n = cos x sin 8/8 = cos x ( i ~ t + ■ • • ) i a + X = tanxiScot|8 = tanx(i " i^"-^---)- The map extends from co-latitude 20° to co-latitude 50° and Euler finds by a calculation not so direct as ours « =0-8098, « = o'o853 = 4° 53' 22".
As this value of a is so nearly 5° = o'o873 Euler adopts the latter value and so fails to reproduce one of his conditions, viz.,
... that of equality of errors at middle and extremes. The following Tables give the particulars of this map calculated from the Minimum Error formula, from Euler's own con- stants, and from the correct values of Euler's constants.
3—2 36 MINIMUM ERROR Minimum Error Formulae.
« = cos X ( I + — =o' 30/ 82156, a = tan x ( ,.--^x= : 0-07927.
e 6 in cm. and r sin 9 nrjsm 6 nrjsm 8-1 V= Squares VAnB 20° 25° 30° 0^34907 ■42834 ■43633 •51560 •52360 •60287 •61086 •69013 .


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