Understanding Charles Sanders Peirce's cartography Charles Sanders Peirce wrote$^\dagger$ about

an orthomorphic or conform projection formed by transforming the stereographic projection, with a pole at infinity, by means of an elliptic function.

("Conform projection" seems to mean what today we would call a conformal projection.)

For that purpose, $l$ being the latitude and $\theta$ the longitude, we put $$ \cos^2\varphi = \frac{\sqrt{1 - \cos^2 l \cos^2\theta} - \sin l}{1 + \sqrt{1-\cos^2 l\cos^2\theta}},$$ and then $\dfrac 1 2 F\varphi$ is the value of one of the rectangular coördinates of the point on the new projection.

Here two questions arise: $(1)$ What is $F$? $(2)$ What does this have to do with elliptic functions?

This is the same as taking $$ \cos am (x+y\sqrt{-1}) = \text{(angle of mod. $=45^\circ$)} = \tan \frac p 2 (\cos\theta + \sin\theta\sqrt{-1}), $$ where $x$ and $y$ are the coördinates of the new projection, $p$ is the north polar distance.

I take this to mean $p$ is the great-circle distance from the north pole to the point being mapped (where the great-circle distance from the north pole to the south pole is $\pi$), and $\tan(p/2)$ is the distance from the image of the north pole in the stereographic projection to the image of the point being mapped. But what does $\text{“angle of mod.''}$ mean? And what is $am$? And where are the elliptic functions here?
(For now, I'm omitting the one occurrence in the history of human language of the phrase "orthomorphic potential". )
Despite not knowing the answers to the questions above, I know where Peirce is going: He has a multiple-valued mapping from the sphere to the plane that maps the whole sphere in a periodic way to each of infinitely many non-overlapping copies of a  $2\times1$ rectangle, and this mapping is conformal except at isolated points.

$\dagger$ C. S. Peirce, "A Quincuncial Projection of the Sphere", American Journal of Mathematics, volume 2, number 4, December 1879, pages 394–6
 A: Here is the long-overdue follow-through.
As I mentioned in the comments, the $F$ in the expression $\frac12F\varphi$ is exactly the incomplete elliptic integral of the first kind, which in modern notation is rendered as $F(\varphi\backslash \alpha)=u$, where $\varphi$ is the amplitude, and $\alpha$ is the modular angle (referred to as "angle of mod" in the OP). (See this for a long rant discussion on argument conventions for elliptic integrals.)
A little confusingly, "amplitude" is also the term used for the inverse function of $F(\varphi\backslash \alpha)$. The Jacobian amplitude is represented as $\operatorname{am}(u\backslash \alpha)=\varphi$, in consistency with the notation for its inverse function.
On top of this, the Jacobian elliptic functions were classically defined as trigonometric functions of the amplitude. In particular, one of Jacobi's functions is termed as "cosinus amplitudinis", whose classical notation is $\cos\operatorname{am}u$ (where the modular angle is often suppressed for brevity). Due to the unwieldy original notation of Jacobi, Gudermann (and much later, Glaisher) came up with the currently accepted notation for these elliptic functions; in particular, $\cos\operatorname{am}u$ would now be rendered as $\operatorname{cn}(u\backslash\alpha)=\cos(\operatorname{am}(u\backslash \alpha))$.
In effect, then, Peirce is relating a complex number represented as a Riemann sphere point (equivalently, its stereographic projection) with cosinus amplitudinis, where the modular angle $\alpha$ is set to $45^\circ$.
Here is how the mapping looks like in the complex plane:


For those interested in an implementation of the quincuncial projection, I wrote a Mathematica implementation here. Here's the result of applying it to an actual map of the Earth:

