NavList:

I rechecked my computation using Table 5 Meridional Parts and it is correct, only a small part of my explanation was left out and it is here in bold.

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If you are a perfectionist you would use Meridional Parts found in Table 5 of Bowditch. These are simply the number of nautical miles from the equator to a particular latitude, listed to the nearest 0.1 NM, as you would plot on a Mercator chart,  and takes into account the oblateness of the Earth. For example 34° latitude would is 2,040 NM from the equator but on a Mercator chart  it is plotted further away, Table 5 gives it as 2158.4 NM. So to construct a perfect Mercator plotting sheet, for example covering two degrees of latitude, 33° to 35°, and two degrees on longitude we look up the Meridional parts for 33° and 35°, and subtract to find the number of NM between those two parallels as plotted on the chart.. 2230.0 for 35° and 2086.8 for 33° makes a difference of 143.2 NM. Since this distance represents 120 minutes of latitude we divide 120 by 143.2 to find the ratio of parallels to degrees of longitude at a mid-latitude of 34°. 120/ 143.2 = 0.838 so we place the lines of longitude at this ratio to parallels (if we start with parallels spaced 60 NM apart, which is the normal way to do it. If you, for some reason you wanted to start with meridians then you would place the parallels at the inverse of this ratio 1.193.) So along the mid-latitude of 34° ,one degree of longitude will equal 50.279 NM. 

The easy was to construct the plotting sheet using the cosine method is to draw parallels equally spaced to represent 60 NM and the center horizontal line for the mid-latitude. Then from the intersection of the mid-latitude and the mid-longitude draw a line upward from the mid-latitude line by the number of degrees of mid-latitude. Measure along this angled line by 60 units of latitude (60 NM) and draw in the meridian from this point vertically and this will space the meridian at the cosine of the mid-latitude times 60 NM. then repeat the spacing for additional meridians.This is exactly what Hakel's method shows.