Two small relations that make the earth's shape and spin useful on a working day.

How far can I see? $D = \sqrt{h\left(2R + h\right)}$, where $h$ is your height of eye above the water, $R$ is the earth's radius, and $D$ is the distance to the horizon. It is Pythagoras wearing a disguise: the sight line is tangent to the sphere, so it meets the radius at a right angle, and $(R+h)^{2} - R^{2}$ is exactly $h(2R+h)$. Since $h$ is tiny against $R$, it collapses to $D \approx \sqrt{2Rh}$ — in working units, 3.57 times the square root of the height in metres, giving kilometres.

The root is the lesson. Doubling your height of eye buys you only about 41 % more horizon, not double — which is why crow's nests got tall slowly and why an extra metre on a bridge wing is worth so little. Two footnotes worth carrying: this is the geometric horizon, and the one you actually see is roughly a tenth further out because the atmosphere bends light down along the curve (Bowditch gives 1.169 times the root of the height in feet, in nautical miles). And to find how far off a light is visible, add its own horizon distance, computed from its charted height, to yours.

Where am I, east to west? $\Delta t = 4\,\Delta\lambda$ — delta t equals four delta lambda. $\Delta\lambda$ is a longitude difference in degrees and $\Delta t$ is the solar time difference in minutes. The earth turns 360° in 24 hours, so 15° per hour, so 4 minutes per degree. Those two constants are reciprocal cousins and swapping them is the house trap of this lesson.

This is the relation that made a clock into a navigation instrument. Carry Greenwich time in a chronometer, watch for local apparent noon where you are, and the difference is your longitude — which is what Harrison's clocks were for, and why latitude was easy for two thousand years and longitude was not. Two honest limits: this is SOLAR time, not what any clock in a civil time zone reads, and the equation of time (the earth's tilt and its elliptical orbit) shifts true noon by up to a quarter of an hour through the year. The rule is exact about the geometry and silent about the politics.