If I'm not mistaken, the Mercator projection is characterized by these two properties:
(1) It is conformal, and
(2) It maps curves of constant bearing on the sphere to straight lines in the plane. I.e. $\theta$ degrees east of north corresponds to a certain slope on the map regardless of where on earth you are.
In "An Application of Geography to Mathematics: History of the Integral of the Secant", by V. Frederick Rickey and Philip M. Tuchinsky, Mathematics Magazine, v. 53, No. 3, May 1980, pp. 162–166, we find this (the inside quote is from Edward Wright, who wrote some time around the year 1600):
"Consider a cylinder tangent to the earth's equator and imagine the earth to 'swal [swell] like a bladder.' Then identify points on the earth with the points on the cylinder that they come into contact with. Finally, unroll the cylinder; it will be a Mercator map. This model has often been misinterpreted as the cylindrical projection (where a light source at the earth's center projects the unswollen sphere onto its tangent cylinder), but this projection is not conformal."
The description involving "swelling" does seem to me to correspond to the cylindrical projection. What is the difference? How does "swelling like a bladder" give a result different from that given by the cylindrical projection?
(As the title suggests, the authors are saying a major motive for integrating the secant function is the Mercator projection.)
Note added later: I suppose the two conditions (1) and (2) could be regarded as more simply expressed by saying this: bearings on the globe equal bearings on the map, i.e. if north on the globe corresponds to upward on the map, then $\theta$ degrees east of north on the globe corresponds to $\theta$ degrees clockwise from straight upward on the flat map. If I'm not mistaken, the Mercator projection is the only projection of which that is true.
Still later note: I now find it being speculated in another forum where I asked about this, that the cylinder is viewed as something like a solid material, so that the swelling balloon clings to the sides of the cylinder rather than passing through it. I had imagined a spherical balloon expanding while retaining its spherical shape, so that progressively more of the sphere is outside the cylinder. Each point on the sphere would pass through ("touch") the cylinder once, and that point on the cylinder would be the image of that point on the sphere. It seems altogether unclear how that would determine the projection, and if that's what R & T had in mind, or what Wright had in mind, that's not at all clear from R & T's paper.
Best Answer
I'm not sure exactly what it means to "expand like a bladder", but it is true that the Mercator projection is different from the central cylindrical projection. The following picture shows a cross section of the Mercator projection, with the circle representing the spherical earth and the two vertical lines representing the cylinder. The blue lines indicate the projection of points on the circle to points on the cylinder: