Geometry lesson: cool
Skipping the fundamentals of cartography: not so cool
I wonder if the MC is going to invent the compass like in MaoYuu?
Without a magnetic compass, one might set up a measuring station on a prominent hilltop and adjust it to be plumb. The top would be a circular table divided into degrees (Arthur: "Why 360 divisions?"). At dawn, mark the direction of the sun's rising, and at dusk mark the direction of it's setting. Halfway between the two is geographic north.
Now orient zero degrees to the north, and take sightings to other reliable landmarks with as much precision as possible. These angles can be recorded in association with the particular measuring station. If possible, drive a long iron rod into the ground with a marking to geographic north for future reference and a number or name designation for that reference point. This would be a 'benchmark'.
Travel to another high location and repeat, setting and recording another benchmark and angles to the same and to different sets of landmarks. Doing this many times will allow one to use the compiled data to make a relative map of the locations of the benchmarks. It will also become apparent that the map must correct for/account for the curvature of the planet. This would be a map that is geometrically correct even without knowing absolute distances between landmarks 'as the crow flies'. Theoretically, a single precise and accurate distance measure would be sufficient to give scale to the final map.
Such maps are of strategic military importance. If the government were to realize this, the maps and the method of creating them might be restricted as state secrets. But they are also of commercial significance, and so money could certainly be made from merchant guilds for them. Both such parties could be persuaded to fund an institute of cartography to train and send out surveyors and to archive, compile, and reconcile data. Such an institution would be given a royal charter, and would develop a system of standard measures of distance. This could spin off an institute of weights and measures or bureau of standards which would create and impose standard weights and measures that would be affixed with a royal seal and sent to provincial governors to set the same standard throughout the kingdom. Being 'sealed', the act of altering such standard weights would be
lese majeste, and punishable by death (This was the actual level of punishment for altering standard weights in ancient times. The Seal meant that the standard weights sent out from the capital were the 'word of the king', and so changing them was treason.).