Whoever says 6 + 3 = 9, expresses the same as he who says 6 + 3 are identical with 9. Clearly in the affirmation of equality, no attention is paid to the form in which the quantities are expressed, but to the quantities themselves alone; otherwise we should be unable to affirm not only identity, but also equality; for it is evident that 6 + 3, as to their form, neither written, spoken, nor thought, are identical with, or equal to, 9. The equality is in the values expressed, and these are not only equal but identical; 6 + 3 are the same as 9. The whole is not distinguished from its united part; 9 is the whole, 6 + 3 its united parts. The different manner of conceiving 6 + 3 and 9 does not exclude the identity. The difference is in the intellectual form, and occurs not only here but also in the perceptions of the simplest things; there is nothing which we do not conceive under different aspects, and whose conception we may not decompose in various ways; but we do not therefore say that the thing ceases to be simple and identical with itself. What we have said of an arithmetical equation may be extended to algebraical and geometrical equations. If we have an equation whereof the first member is very simple, as Z, and the second very complicated, as the development of a series, we cannot say that the first expression is equal to the second; the equality is not in the expression but in the thing expressed, in the value designated by the letters; in this sense it is true, in the former it is evidently false. Two circumferences having the same radius are equal. Here we seem to treat solely of equality, since there are two distinct objects, the two circumferences, which may be traced on paper or represented in the imagination; yet not even in this case is the distinction true, it is only apparent, for here, as in algebraical and arithmetical equations, there is distinction and even diversity in form with identity at bottom. The principal argument, on which the distinction is founded, may be combatted by observing that the circumferences which may be traced or represented, are only forms of the idea, not the idea itself. Whether traced or represented they have a determinate size and a certain position on the planes seen or imagined; in the idea, and in the proposition containing it, there is nothing of this; we abstract all size, all position, and speak in a general and absolute sense. True, the representations may be infinite either externally or in the imagination; but this, so far from proving them identical, shows their diversity, since the idea is one and they are infinite; the idea is constant, they are variable; the idea is independent of them, they are dependent on the idea, and have the character and denomination of circumferences, inasmuch as they approach it by representing what it contains.