Editing Mercator projection (section)

==== On a meridian ====
A meridian of the map is a great circle on the globe but the continuous scale variation means ruler measurement alone cannot yield the true distance between distant points on the meridian. However, if the map is marked with an accurate and finely spaced latitude scale from which the latitude may be read directly—as is the case for the [[Mercator 1569 world map#Basel map|Mercator 1569 world map]] (sheets 3, 9, 15) and all subsequent nautical charts—the meridian distance between two latitudes ''φ''<sub>1</sub> and ''φ''<sub>2</sub> is simply
:<math>m_{12}= a|\varphi_1-\varphi_2|.</math>

If the latitudes of the end points cannot be determined with confidence then they can be found instead by calculation on the ruler distance. Calling the ruler distances of the end points on the map meridian as measured from the equator ''y''<sub>1</sub> and ''y''<sub>2</sub>, the true distance between these points on the sphere is given by using any one of the inverse Mercator formulae:
:<math>m_{12} = a\left|\tan^{-1}\left[\sinh\left(\frac{y_1}{R}\right)\right] -\tan^{-1}\left[\sinh\left(\frac{y_2}{R}\right)\right]\right|,</math>

where ''R'' may be calculated from the width ''W'' of the map by ''R''&nbsp;=&nbsp;{{sfrac|''W''|2{{pi}}}}. For example, on a map with ''R''&nbsp;=&nbsp;1 the values of ''y''&nbsp;=&nbsp;0, 1, 2, 3 correspond to latitudes of ''φ''&nbsp;=&nbsp;0°, 50°, 75°, 84° and therefore the successive intervals of 1&nbsp;cm on the map correspond to latitude intervals on the globe of 50°, 25°, 9° and distances of 5,560&nbsp;km, 2,780&nbsp;km, and 1,000&nbsp;km on Earth.