PYTHAGORAS TO PLATO does not undergo any such change and is a quite diVerent thing In the fourth place we have knowledge, understanding, and true opinion on these matters— these, collectively, are in our minds and not in sounds or bodily shapes, and thus are clearly distinct from the circle itself and from the three entities already mentioned Of all these items, it is understanding that is closest to the Wfth in kinship and likeness; the others are at a greater distance What is true of round is also true of straight, of colour, of good and beautiful, and just; of natural and manufactured bodies; of Wre, water, and the other elements; of all living beings and moral characters; of all that we and undergo In each case, anyone who totally fails to grasp the Wrst four things will never fully possess knowledge of the Wfth (342a–d) If I follow Plato, then, I will begin by distinguishing four things: the word ‘circle’, the deWnition of circle (a series of words), a diagram of a circle, and my concept of a circle The importance of being clear about these four items is to distinguish them from, and contrast them with, a Wfth thing, the most important of all, which he calls ‘the circle itself’ It is this that is one of the Ideas of which Plato’s celebrated theory treats The theory is a wide-ranging one, as is clear from the sentence at the end of the paragraph that lists the Welds in which the theory applies In his other writings Plato uses many other expressions to refer to Ideas ‘Forms’ (eide) is probably the most common, but the Idea or Form of X may be called ‘the X itself’, ‘that very thing that is X’, or ‘Xness’, or ‘what X is’ It is important to note what is absent from Plato’s list in the Seventh Letter He does not mention, even at the lowest level, actual material circular objects such as cartwheels and barrels The reason for his omission is clear from other passages in his writings (e.g Phd 74a–c) The wheels and barrel we meet in experience are never perfectly circular: somewhere or other there will be a bend or bump which will interfere with the equidistance from the centre of every point on the circumference This is true too, for that matter, of any diagram we may draw on paper or in the sand Plato does not stress this point here, but it is the reason why he says that the diagram is at a greater distance from the circle itself than my concept is My subjective concept of the circle—my understanding of what ‘circle’ means—is not the same as the Idea of the circle, because the Idea is an objective reality that is not the property of any individual mind But at least the concept in my mind is a concept of a perfect circle; it is not merely an imperfect approximation to a circle, as the ring on my Wnger is 50