How did medieval architects determine sizes and numbers of pillars, buttresses etc?
Mostly, by use of scale models, chalk floor drawings, and elaborate simulations of beam loading constructed by hanging weights from string (as with Gaudi's Hanging Chain Models).
"But where are the plans?" you ask.
The rectangular-grid mathematics on which such are based, analytic geometry, didn't yet exist; it's a contrivance of Rene Descartes (hence Cartesian plane) only developed in the 16th century. The printing processes (and presses) for mass reproduction blueprints or anything similar, and distributing them to workmen, likewise are centuries in the future. It's been proposed that the master masons appointed as architects constructed templates for the workmen in their tracing rooms, as that is how restoration is now done; but that is contested by Holton as the two surviving tracing rooms York Minster and Wells have no access large enough for the necessary templates to be transported out.
Consider the example of Cupola Firenze - the Dome of Florence, constructed by Brunelleschi overtop Cathedral of Saint Mary of the Flower. Conceived to be topped by the largest dome ever yet conceived, construction was begun in 1296 A.D. and "completed" in 1380; but with a 44m, or 150 foot, hole in the roof where the conceived dome was to have been.
"Yes - let's build a cathedral, seating 30,000 and (to be) topped by the world's largest dome; and leave a 180 foot diameter hole where the dome should be, for a generation, until our great grand children figure out how to construct it."
A design contest for the dome, announced in 1418, was won by Brunelleschi with a scale model of the to-be-completed dome with its lantern. But Brunelleschi refused to share either his design or means of construction without any scaffolding ("without either dirt or coins", referring to the legendary means by which the dome of the Pantheon had been constructed).
Unwilling to entrust the entire construction to a single mere goldsmith, construction was awarded jointly to Brunelleschi and his rival Ghibert. Construction began in 1420; but by 1423 Ghibert admitted that the it was beyond his comprehension and Brunelleschi was given sole authority.
The dome construction was made self-supporting by means of a clever six-fold spiral herringbone over the octagonal base. This design pattern then reappears elsewhere, despite Brunelleschi's draping of cloths over and around the construction site to maintain secrecy, so his better workmen recognized this innovation. However, his true masterpiece was in how he ensured the six octagonal walls would - and did - meet cleanly at the apex to support the lantern.
The trick of how he did that, with strings stretched out from a well obscured workspace beneath the center of the dome, was only rediscovered a few years ago by some clever scientific archaeology.
Update, from comments below:
Q: Did the Florentinians lack the entire dome design or "just" the method of construction?
A: The Florentines held a design contest in 1418-19, for completion of the basilica; won by Brunelleschi. The only (apparent) constraints were the (already built) octagonal footprint for the dome, and the thickness/strength of the (again, already built) walls supporting that footprint. Some commentary I read seem to imply that the supporting wall thickness was less than ideal for the size of dome envisaged (and thus possibly complicating and delaying the design and construction); but that's just a bit of reading between the lines and not official.
Q's: ... how did [medieval master masons] estimate the correct weight [in their string-and-weight load simulations)? Did the architects consider different densities and strengths of brick/Sandstone/Granite etc?
A: I have not researched that. In truth, that should be a separate question I think; or, at least, a full separate answer.
Translation of the Design Contest announcement:
Modern analysis of ??? herring-bone brickwork in the dome
