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 ease a few premises to the bottom, and cannot in the least penetrate those matters in which there are many premises.

There are then two kinds of intellect:  the one able to penetrate acutely and deeply into the conclusions of given premises, and this is the precise intellect; the other able to comprehend a great number of premises without confusing them, and this is the mathematical intellect.  The one has force and exactness, the other comprehension.  Now the one quality can exist without the other; the intellect can be strong and narrow, and can also be comprehensive and weak.

3

Those who are accustomed to judge by feeling do not understand the process of reasoning, for they would understand at first sight, and are not used to seek for principles.  And others, on the contrary, who are accustomed to reason from principles, do not at all understand matters of feeling, seeking principles, and being unable to see at a glance.

4

Mathematics, intuition.—­True eloquence makes light of eloquence, true morality makes light of morality; that is to say, the morality of the judgment, which has no rules, makes light of the morality of the intellect.

For it is to judgment that perception belongs, as science belongs to intellect.  Intuition is the part of judgment, mathematics of intellect.