(2) Three numbers are given - it is required to find a fourth,
which shall be to the third as the second is to the first.
(23:3) Tradesmen will at once tell us that they know what is required
to find the fourth number, for they have not yet forgotten the rule
which was given to them arbitrarily without proof by their masters;
others construct a universal axiom from their experience with simple
numbers, where the fourth number is self-evident, as in the case of
2, 4, 3, 6; here it is evident that if the second number be
multiplied by the third, and the product divided by the first,
the quotient is 6; when they see that by this process the number
is produced which they knew beforehand to be the proportional,
they infer that the process always holds good for finding a fourth
number proportional. 

[24]  (1) Mathematicians, however, know by the proof of the nineteenth
proposition of the seventh book of Euclid, what numbers are
proportionals, namely, from the nature and property of proportion
it follows that the product of the first and fourth will be equal
to the product of the second and third: still they do not see the
adequate proportionality of the given numbers, or, if they do see it,
they see it not by virtue of Euclid's proposition, but intuitively,
without going through any process. 

[25] (1) In order that from these modes of perception the best may
be selected, it is well that we should briefly enumerate the means
necessary for attaining our end.

I.  (2) To have an exact knowledge of our nature which we desire to
    perfect, and to know as much as is needful of nature in general. 
            
II. To collect in this way the differences, the agreements, and the
    oppositions of things.
            
III. To learn thus exactly how far they can or cannot be modified.
            
IV. To compare this result with the nature and power of man.
    (4) We shall thus discern the highest degree of perfection
    to which man is capable of attaining.

[26] (1) We shall then be in a position to see which mode of
perception we ought to choose.  (2) As to the first mode, it is
evident that from hearsay our knowledge must always be uncertain,
and, moreover, can give us no insight into the essence of a thing,
as is manifest in our illustration; now one can only arrive at
knowledge of a thing through knowledge of its essence, as will
hereafter appear.  (3) We may, therefore clearly conclude that
the certainty arising from hearsay cannot be scientific in its
character.  (4) For simple hearsay cannot affect anyone whose
understanding does not, so to speak, meet it half way.

[27]  (1) The second mode of perception [i] cannot be said to
give us the idea of the proportion of which we are in search.
(2) Moreover its results are very uncertain and indefinite,
for we shall never discover anything in natural phenomena by its
means, except accidental properties, which are never clearly
understood, unless the essence of the things in question be
known first.  (3) Wherefore this mode also must be rejected. 

[28] (1) Of the third mode of perception we may say in a manner
that it gives us the idea of the thing sought, and that it 
us to draw conclusions without risk of error; yet it is not by
itself sufficient to put us in possession of the perfection we
aim at. 

[29] (1) The fourth mode alone apprehends the adequate essence of
a thing without danger of error.  (2) This mode, therefore, must be
the one which we chiefly employ.  (3) How, then, should we avail
ourselves of it so as to gain the fourth kind of knowledge with
the least delay concerning things previously unknown?  (4) I will
proceed to explain. 

[30] (1) Now that we know what kind of knowledge is necessary for
us, we must indicate the way and the method whereby we may gain
the said knowledge concerning the things needful to be known.
(2) In order to accomplish this, we must first take care not to
commit ourselves to a search, going back to infinity - that is,
in order to discover the best method of finding truth, there is
no need of another method to discover such method; nor of a third
method for discovering the second, and so on to infinity.  (3) By
such proceedings, we should never arrive at the knowledge of the
truth, or, indeed, at any knowledge at all.  (30:4) The matter stands
on the same footing as the making of material tools, which might
be argued about in a similar way.  (5) For, in order to work iron,
a hammer is needed, and the hammer cannot be forthcoming unless
it has been made; but, in order to make it, there was need
of another hammer and other tools, and so on to infinity.
(6) We might thus vainly endeavor to prove that men have no
power of working iron.

[31]  (1) But as men at first made use of the instruments supplied
by nature to accomplish very easy pieces of workmanship, laboriously
and imperfectly, and then, when these were finished, wrought other
things more difficult with less labour and greater perfection;
and so gradually mounted from the simplest operations to the making
of tools, and from the making of tools to the making of more complex
tools, and fresh feats of workmanship, till they arrived at making,
complicated mechanisms which they now possess.  (31:2) So, in like
manner, the intellect, by its native strength, [k], makes for itself
intellectual instruments, whereby it acquires strength for performing
other intellectual operations, [l], and from these operations