The maximum value of these arcs can, however, be found by a simple proportion, by saying; as the arc AR, plus the inclination, is to the inclination, so is the inclination to the difference between them; and therefore, the inclination, plus half the difference, is equal the greater arc, and the inclination, minus half the difference, is equal the lesser; the greater being positive, and the lesser negative.
Having found the arc AR, and knowing the moon’s distance from either node, we must reduce these values of the arcs RK and R′K′ just found, in the ratio of radius to the sine of that distance, and apply it to the arc AR or A′R′, and we shall get the first correction equal to the arc AK or AK′.
| Call the | arc AR | = a |
| " | inclination | = n |
| " | distance from the node | = d |
| " | arc AK | = k |
and supposing the value of AK be wanted for the northern hemisphere when the moon is between her descending and ascending node, we have
If the moon is between her ascending and descending node, then
