(1) Although this object looks circular, it is not circular (it's elliptical)
(2) Plates are not elliptical
(3) This object is not a plate
(4) We'll call it a sense-datum
(5) There is nonetheless an actual plate corresponding to the sense-datum.
The argument is valid, unlike the sense-datum arguments parodied and rejected by philosophers like Austin. It starts right away with the premiss that this thing, which you took to be a plate, is not circular at all. It looks circular, but isn't. And if plates are circular, it clearly can't be a plate.
Is the argument sound? Is the first premiss true? Why do we think that the sense-datum is not circular, even though it looks circular? Hume tells us (in connection with a different example, in the Treatise, Book I part iv sec. 2 "Of Scepticism with regard to the Senses") that we come to this 'by reflection' or by 'studied reflection'. Imagination or 'fancy' suggests that the datum is circular, that it is just the way it looks. Reason or reflection tells us that it is not circular. For example, we can look at the datum as an artist sees it and as he represents it in a flat picture, as an ellipse.
The final step of the argument is curious. Why do we suppose there is anything circular there at all? Hume has an interesting theory about this. I paraphrase him as follows. There is naturally an opposition between the two mental forces of imagination and reason, for they are telling us contradictory things. Imagination suggests the datum is circular. Reason and reflection tell us it is not. To set ourselves at ease, we invent a new theory which seems to reconcile both: the philosophical system (i.e. the representative theory of perception) that posits the double existence of the sense-datum and its external object. This satisfies our reason in allowing that the sense-datum is not circular, while agreeing with our imagination in attributing the circularity to something else, which we call an 'object'.
This philosophical system, therefore, is the monstrous offspring of two
principles, which are contrary to each other, which are both at once embrac'd by
the mind, and which are unable mutually to destroy each other.
The imagination tells us, that sense-data have all properties that we commonly suppose them to have (plate sense-data are round, sense-data of straight sticks in the water really are straight). Reflection tells us that they really do not have thise properties. We escape the contradiction between these opinions by a fiction which conforms to both reflection and fancy, by ascribing the contrary features to different things - such as the circularity to the object, the ellipticality to the datum.
Not being able to reconcile these two enemies (reason and fancy), we try "to set ourselves at ease as much as possible, by successively granting to each whatever it demands, and by feigning a double existence, where each may find something, that has all the conditions it desires". Were we fully convinced that the datum was circular we would never run into this idea of a double existence, since we would find satisfaction in the first supposition, without looking any further. Again, if we were fully convinced that the datum was elliptical, we would be as little inclined towards the theory of double existence, since in that case we would clearly perceive the error of the belief that it was circular, and would not bother with it any more.
Absolutely my favourite bit of Hume.
'Tis therefore from the intermediate situation of the mind, that this opinion
arises, and from such an adherence to these two contrary principles, as makes us
seek some pretext to justify our receiving both; which happily at last is found
in the system of a double existence.