how do early cartographers make maps?

Answer 1

The general answer is: Poorly!
The French king Louis XIV said that he lost more territory to cartographers than to war. The reason being that early French maps where showing France much bigger than it really was.
Before Harrison and his invention of the chronometer, there was no certain way to assess the longitude. But latitude was easily found from the elevation of the north star. Early maps were relatively correct in latitude but not longitude. This is how Columbus thought that the route to India was shorter by sailing to the west than the east.

Today, at the age of satellite navigation, we find that maps drawn one hundred years ago can still be some hundred meters off the now standard reference called WGS-84.

A recent triangulation in England discovered that the prime meridian of Greenwich (marked as stones in front of the observatory) was slightly off what is the longitude 0. Note that this prime meridian was arbitrarily defined as such. Before, places like Amsterdam, Paris, Rome and the Canary Islands were also used as prime meridian, i.e. the place from which longitudes are counted as east or west of.

Answer 2

Pen and paper.