Lana goes on to assume that any large vessel can be entirely
exhausted of nearly all the air contained therein. Then he
takes Euclid's proposition to the effect that the superficial
area of globes increases in the proportion of the square of the
diameter, whilst the volume increases in the proportion of the
cube of the same diameter, and he considers that if one only
constructs the globe of thin metal, of sufficient size, and
exhausts the air in the manner that he suggests, such a globe
will be so far lighter than the surrounding atmosphere that it
will not only rise, but will be capable of lifting weights.
Here is Lana's own way of putting it:--

'But so that it may be enabled to raise heavier weights and to
lift men in the air, let us take double the quantity of copper,
1,232 square feet, equal to 308 lbs. of copper; with this double
quantity of copper we could construct a vessel of not only
double the capacity, but of four times the capacity of the
first, for the reason shown by my fourth supposition.
Consequently the air contained in such a vessel will be 718 lbs.
4 2/3 ounces, so that if the air be drawn out of the vessel it
will be 410 lbs. 4 2/3 ounces lighter than the same volume of
air, and, consequently, will be enabled to lift three men, or at
least two, should they weigh more than eight pesi each. It is
thus manifest that the larger the ball or vessel is made, the
thicker and more solid can the sheets of copper be made, because,
although the weight will increase, the capacity of the vessel
will increase to a greater extent and with it the weight of the
air therein, so that it will always be capable to lift a heavier
weight. From this it can be easily seen how it is possible to
construct a machine which, fashioned like unto a ship, will float