Frank asked:

What is the wax, given, as Descartes says, that it is the same thingbefore and after it has melted?
What faculty, i.e., ability does Descartes use to acquire the knowledge that it is the same wax before
and after it has melted?

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You are referring to the famous Beeswax passage from Descartes' Second Meditation.Descartes has
been describing the essential properties that belong to his 'I':

I am a thing which thinks...I am not this assemblage of limbs called the human body; I am not
a thin and penetrating air spread through all these members; I am not a wind, a breath of air, a
vapour, or anything at all that I can invent or imagine...Meditations on First PhilosophyF.E.
Sutcliffe tr. p. 105

But this thought is difficult to hold onto:

I cannot help believing that corporeal objects, whose images are formed by my thoughts, and
which come under the senses, are more distinctly known to me than that, I know not what,
part of me which does not fall within the grasp of the imagination.ibid. p. 108

So he conducts an experiment. He places a piece of beeswax near the fire and observes the changes
that take place. What is it, he asks, that gives him the confidence, and the right to judge that the same
wax is still there, despite the radical alterations in its appearance?

It is the understanding, Descartes argues, and not the imagination which is the source of our
knowledge that it is the same wax before and after it has melted. Just as it is the understanding, not
the imagination, that reveals what I myself truly am.

The wax is a substance,a thing with qualities. According to this philosophical usage, 'substance'
applies not only to kinds of material stuffbut also to a table, or a human body. The table is the same
table, even after it has been painted white and the legs shortened by six inches. A human body
remains the same body, even though, over the passage of time, every living cell is replaced.

A contemporary philosopher would say that our concept of the identity of a material object through
space and time, and through changes in outward appearance, is tied to the notion of a sortal concept.
The material stuff of the table, before and after it has been chopped up for firewood is the same
wood.But it is not the same table,because the table has ceased to exist. The piece of beeswax is
the same waxbefore and after it has melted, even though the cells which the bees made out of the
wax have ceased to exist. If the wax is heated to a sufficiently high temperature the wax, too, ceases
to exist, and all that remains are the same chemical constituentsof the former wax now broken down
into carbon and water.

You will find the seminal discussion of identity and spatio-temporal continuity in David Wiggins
Sameness and SubstanceOxford 1980. The basic idea behind Wiggins' discussion, however, goes
back to Aristotle.

A supporter of the view that identity is always identity under a sortal concept would certainly agree
with Descartes' claim that perception and imagination are not sufficient for judging that an object, or a
stuff, is the same despite changes in outward appearance, if we lack the conceptof the thing in
question. Note that I said, 'judging that an object...is the same'. The dog that recognizes its former
owner, when the owner returns after many years unrecognizable to his former friends, does not judge
that its owner has returned, but merely exhibits a finely tuned capacity for recognizing smell, and
possibly other perceptual qualities. The dog reacts to the perception.

There are, however, ingredients in Descartes' notion of material substancethat a contemporary
philosopher would not agree to, notably the idea that the essence of material substance is completely
accounted for by the pure geometrical notion of extension. Leibniz criticised Descartes on this point.
And Descartes attempt to derive the science of mechanics from his analysis of substance as
extension was decisively refuted by Newton.

Geoffrey Klempner