As regards what is "of necessity", we must ask whether the necessity
is "hypothetical", or "simple" as well. The current view places what
is of necessity in the process of production, just as if one were to
suppose that the wall of a house necessarily comes to be because
what is heavy is naturally carried downwards and what is light to
the top, wherefore the stones and foundations take the lowest place,
with earth above because it is lighter, and wood at the top of all
as being the lightest. Whereas, though the wall does not come to be
without these, it is not due to these, except as its material cause:
it comes to be for the sake of sheltering and guarding certain things.
Similarly in all other things which involve production for an end; the
product cannot come to be without things which have a necessary
nature, but it is not due to these (except as its material); it
comes to be for an end. For instance, why is a saw such as it is? To
effect so-and-so and for the sake of so-and-so. This end, however,
cannot be realized unless the saw is made of iron. It is, therefore,
necessary for it to be of iron, it we are to have a saw and perform
the operation of sawing. What is necessary then, is necessary on a
hypothesis; it is not a result necessarily determined by
antecedents. Necessity is in the matter, while "that for the sake of
which" is in the definition.

    Necessity in mathematics is in a way similar to necessity in
things which come to be through the operation of nature. Since a
straight line is what it is, it is necessary that the angles of a
triangle should equal two right angles. But not conversely; though
if the angles are not equal to two right angles, then the straight
line is not what it is either. But in things which come to be for an
end, the reverse is true. If the end is to exist or does exist, that
also which precedes it will exist or does exist; otherwise just as
there, if-the conclusion is not true, the premiss will not be true, so
here the end or "that for the sake of which" will not exist. For
this too is itself a starting-point, but of the reasoning, not of
the action; while in mathematics the starting-point is the
starting-point of the reasoning only, as there is no action. If then
there is to be a house, such-and-such things must be made or be
there already or exist, or generally the matter relative to the end,
bricks and stones if it is a house. But the end is not due to these
except as the matter, nor will it come to exist because of them. Yet
if they do not exist at all, neither will the house, or the saw-the
former in the absence of stones, the latter in the absence of
iron-just as in the other case the premisses will not be true, if
the angles of the triangle are not equal to two right angles.

    The necessary in nature, then, is plainly what we call by the name
of matter, and the changes in it. Both causes must be stated by the
physicist, but especially the end; for that is the cause of the
matter, not vice versa; and the end is "that for the sake of which",
and the beginning starts from the definition or essence; as in
artificial products, since a house is of such-and-such a kind, certain
things must necessarily come to be or be there already, or since
health is this, these things must necessarily come to be or be there
already. Similarly if man is this, then these; if these, then those.
Perhaps the necessary is present also in the definition. For if one
defines the operation of sawing as being a certain kind of dividing,
then this cannot come about unless the saw has teeth of a certain
kind; and these cannot be unless it is of iron. For in the
definition too there are some parts that are, as it were, its matter.