very clear sense is needed to perceive
them, and to judge rightly and justly when they are perceived, without
for the most part being able to demonstrate them in order as in
mathematics; because the principles are not known to us in the same way,
and because it would be an endless matter to undertake it. We must see
the matter at once, at one glance, and not by a process of reasoning, at
least to a certain degree. And thus it is rare that mathematicians are
intuitive, and that men of intuition are mathematicians, because
mathematicians wish to treat matters of intuition mathematically, and
make themselves ridiculous, wishing to begin with definitions and then
with axioms, which is not the way to proceed in this kind of reasoning.
Not that the mind does not do so, but it does it tacitly, naturally, and
without technical rules; for the expression of it is beyond all men, and
only a few can feel it.

Intuitive minds, on the contrary, being thus accustomed to judge at a
single glance, are so astonished when they are presented with
propositions of which they understand nothing, and the way to which is
through definitions and axioms so sterile, and which they are not
accustomed to see thus in detail, that they are repelled and
disheartened.

But dull minds are never either intuitive or mathematical.

Mathematicians who are only mathematicians have exact minds, provided
all things are explained to them by means of definitions and axioms;
otherwise they are inaccurate and insufferable, for they are only right
when the principles are quite clear.

And men of intuition who are only intuitive cannot have the patience to
reach to first principles of things speculative and conceptual, which
they have never seen in the world, and which are altogether out of the
common.

2

There are different kinds of right understanding;[2] some have right
understanding in a certain order of things, and not in others, where
they go astray. Some draw conclusions well from a few premises, and this
displays an acute judgment.

Others draw conclusions well where there are many premises.

For example, the former easily learn hydrostatics, where the premises
are few, but the conclusions are so fine that only the greatest
acuteness can reach them.

And in spite of that these persons would perhaps not be great
mathematicians, because mathematics contain a great number of premises,
and there is perhaps a kind of intellect that can search with