NavList:

Increaseing shadow length will not improve the accuracy.  As has been noted the accuracy is limited by the sun not being a point source of light.  A sextant  or the observers eye in the case of a sextan sight taken without a telescope produces an image of the sun atop the horizon, so that the angle to a specific point on the sun is measured.

If you wish to achieve such accuracy using shadows, then build a pin hole camera. And position a small ball in the center of the image.  Then the angle of the ray from the pinhole to the ball relative to the horzontal will more accurately duplicate a sextant height measurement.

The reference to longitude is only due to John's taking prime vertical sights.  Apply to meridian transit sights and you have latitude.

DaveF