Most explanations miss a very simple point. Mercator projection becomes simple if we use complex numbers. It is the stereographic projection which sends the North pole to $\infty$ and the South pole to $0$, followed by the complex logarithm. All formulas immediately follow from this observation.

For the details and historical discussion see Osserman's paper Mathematical Mapping from Mercator to the Millennium,

http://www.msri.org/people/staff/osserman/papers/ROssermanPart_V.pdf

Most explanations miss a very simple point. Mercator projection becomes simple if we use complex numbers. It is the stereographic projection which sends the North pole to $\infty$ and the South pole to $0$, followed by the complex logarithm.

Indeed, Mercator's projection is characterized by two properties a) it is conformal, and b) meridians and parallels are mapped to perpendicular lines. Stereographic projection maps meridians to lines through the origin, and parallels to circles centered at 0. Complex logarithm maps them to horizontal and vertical lines respectively.

All formulas immediately follow from this observation.

For the details and historical discussion see Osserman's paper Mathematical Mapping from Mercator to the Millennium,

http://www.msri.org/people/staff/osserman/papers/ROssermanPart_V.pdf