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The inverse of w could be called a "lunar day" and is equal to ~1.0350 mean solar days (MSD). In half of such a lunar day, the declination can be expected to change by approximately

= 12.3 arc minutes at major standstill, and

A8 = 30.2(>4 x 1.035)2 = 8.1 arc minutes at minor standstill. For the Sun, the relations are simpler:

@ 3 arc arc seconds over a half-day interval.

The change in azimuth arising from a such a change in declination is obtainable through differencing (6.7), keeping the latitude and zenith distance (the complement of the altitude, h) constant:

cos 8 • A8 = - sin z x cos j x sin A x AA. (6.10)

Therefore, solving for the change in A due to the change in 8 alone, we get

so that we may write the change in azimuth as

Any change in azimuth can be matched by stepping a number of paces perpendicular to the direction of the distant foresight. If we call the distance to be stepped off in, say, feet, AS, then the relation between AS and AA is

where d is the distance to the foresight, and in the same unit as AS, and AA is in radian measure.

24 The apparent variation in declination of the Moon, whether in the 18.y61, the 27d32 or 173d3 cycles, or that of the Sun in the 365d2422 cycle, is sinusoidal and at the peak of the cycle can be approximated by an expression such as AS = Smax - Smax x cos(2p x At/P), where At is the time interval from the maximum value of the declination, S. From this, AS = Smax x [1 - cos(2p x t/P)] = Smax x [2sin2 (p x At/P)], which for short intervals from the moment of maximum (i.e., for At << P), becomes AS = 2Smax x (p x At/P)2 = k(At)2.

To distant foresight

time of max. decl.

+ sideways placement of stakes along a straight line X sideways placement of stakes with time-uniform forward steps time of max. decl.

+ sideways placement of stakes along a straight line X sideways placement of stakes with time-uniform forward steps

Figure 6.33. The geometry of the "step" analysis shows the parallactic shift in azimuth needed to observe a given shift in declination. The foresight is in the direction of time's arrow. Drawing by E.F. Milone.

We may then write for the size of step to observe the Moon's rising or setting at the same distant feature of the horizon:

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