7. One of the simplest forms of recurrence is _alternation_, as when we have alternate strong and slight syllables. For instance,-- Awáke, aríse, or bé for éver fáll'n. Or without any subordination, as when we reckon numbers, and call them in succession, _odd_, _even_, _odd_, _even_. 8. But the simplest of all forms of recurrence is that which has no variety;--in which a series of units, each considered as exactly similar to the rest, succeed each other; as _one_, _one_, _one_, and so on. In this case, however, we are led to consider each unit with reference to all that have preceded; and thus the series _one_, _one_, _one_, and so forth, becomes _one_, _two_, _three_, _four_, _five_, and so on; a series with which all are familiar, and which may be continued without limit. We thus collect from that repetition of which time admits, the conception of _Number_. 9. The relations of position and figure are the subject of the science of geometry; and are, as we have already said, traced into a very remarkable and extensive body of truths, which rests for its foundations on axioms involved in the Idea of Space. There is, in like manner, a science of great complexity and extent, which has its foundation in the Idea of Time. But this science, as it is usually pursued, applies only to the conception of Number, which is, as we have said, the simplest result of repetition. This science is _Theoretical Arithmetic_, or the speculative doctrine of the properties and relations of numbers; and we must say a few words concerning the principles which it is requisite to assume as the basis of this science. {{138}} CHAPTER IX. OF THE AXIOMS WHICH RELATE TO NUMBER. 1. THE foundations of our speculative knowledge of the relations and properties of Number, as well as of Space, are contained in the mode in which we represent to ourselves the magnitudes which are the subjects of our reasonings. To express these foundations in axioms in the case of number, is a matter requiring some consideration, for the same reason as in the case of geometry; that is, because these axioms are principles which we assume as true, without being aware that we have made any assumption; and we cannot, without careful scrutiny, determine when we have stated, in the form of axioms, all that is necessary for the formation of the science, and no more than is necessary. We will, however, attempt to detect the principles which really must form the basis of theoretical arithmetic. 2. Why is it that three and two are equal to four and one? Because if we look at five things of any kind, we see that it is so. The five are four and one; they are also three and two. The truth of our assertion is involved in our being able to conceive the number five at all. We perceive this truth by _intuition_, for we cannot see, or imagine we see, five things, without perceiving also that the assertion above stated is true.