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Our latitude in Shawnee is 35 degrees north, and we find Polaris 35 degrees above the northern horizon. Similarly, to sail west at a constant latitude Columbus kept the north star at a constant altitude above the horizon.
Any celestial object crossing the meridian is at its highest altitude (distance from the horizon) during that night (or day). The angle the star paths make with respect to the horizon as they rise up or set down = 90 degrees minus the observer's latitude.