There is not much mathematics immediately at hand here, but you might like to logically consider the advantages and disadvantages associated with the two types.
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Structural simplicity
The vertical generator has nice clean lines, less moving parts, and the drive shaft feeds directly to the turbine.
Independent of wind direction
The horizontal windmill will only work when the rotor faces into the wind. The vertical generator on the other hand accepts wind from any direction.
Needs priming to start
This is a disadvantage of the vertical generator. It requires winds of a certain minimal velocity in order to function, and must then be initially turned mechanically. On the other hand, the horizontal windmill begins to turn by itself with a small breeze.
Relatively cheap to produce
Of course the two illustrated windmills are not in the same class, but the vertical generator is by nature more sophisticated and expensive to build. The Aeolé generator at Cape Chat is in fact 110 metres high and cost $37 000 000 !