In reality the definition of an Infinite Quantity is not negative merely, but contains a positive part as well. We assume a quantity of a certain kind which may be augmented by carrying onward its limits in one or more directions: this is a finite quantity of a given kind. We then-when we have thus positively determined the kind of the quantity-suppose the limit in one or more directions to be annihilated, and thus we have an infinite quantity. But in this infinite quantity there remain the positive properties from which we began, as well as the negative property, the negation of a limit; and the positive properties joined with the negative property may and do supply grounds of reasoning respecting the infinite quantity.

4. This is lore so elementary to mathematicians that it appears almost puerile to dwell upon it; but this seems to have been overlooked, in the proof that we can have no knowledge concerning infinites. In such proof it is assumed as quite evident, that all infinites are equal. Yet, as we have seen, infinites may differ infinitely among themselves, both in quantity and in kind. A German writer is quoted for an "ingenious" proof of this kind. In his writings, the opponent is supposed to urge that a line BAC may be made infinite by carrying the extremity C infinitely to the right, and again infinite by carrying the extremity B infinitely to the left; and thus the line infinitely extended both ways would be double of the line infinite on one side only. The supposed reply to this is, that it cannot be so, because one infinite is equal to another: and moreover that what is bounded at one end A, cannot be infinite: both which assumptions are without the smallest ground. That one infinite quantity may be double of another, is just as clear and certain as that one finite quantity may. For instance, if one leaf of the book which the reader has before him were produced infinitely upwards it would be an infinite space, though bounded at the bottom and at both sides. If

♦ Werenfels in Mr. Mansel's Bampton Lectures, lect. ii. Note 15.

the other leaf were in like manner produced infinitely upwards it would in like manner be infinite; and the two together, though each infinite, would be double of either of them.

5. As I have said, infinite quantities are conceived by conceiving finite quantities increased by the transfer of a certain limit, and then by negativing this limit altogether. And thus an infinite number is conceived by assuming the series 1, 2, 3, 4, and so on, up to a limit, and then removing this limit altogether. And this shows the baselessness of another argument quoted from Werenfele. The opponent asks, Are there in the infinite line an infinite number of feet? Then in the double line there must be twice as many; and thus the former infinite number did not contain all the (possible) unities; (numerus infinitus non omnes habet unitates, sed præter eum concipi possunt totidem unitates, quibus ille careat, eique possunt addi). To which I reply, that the definition of an infinite number is not that it contains all possible unities: but this—that the progress of numeration being begun according to a certain law, goes on without limit. And accordingly it is easy to conceive how one infinite number may be larger than another infinite number, in any proportion. If, for instance, we take, instead of the progression of the natural numbers 1, 2, 3, 4, &c. and the progression of the square numbers 1, 4, 9, 16, &c. any term of the latter series will be greater than the corresponding term of the other series in a ratio constantly increasing, and the infinite term of the one, infinitely greater than the corresponding infinite term of the other.

6. In the same manner we form a conception of infinite time, by supposing time to begin now, and to go on, after the nature of time, without limit; or by going back in thought from the present to a past time, and by continuing this retrogression without limit. And thus we have time infinite a parte ante and a parte post, as the phrase used to run; and time infinite both ways includes both, and is the most complete notion of eternity.

7. Perhaps those who thus maintain that we cannot

conceive anything infinite, mean that we cannot form to ourselves a definite image of anything infinite. And this of course is true. We cannot form to ourselves an image of anything of which one of the characteristics is that it is, in a certain way, unlimited. But this impossibility does not prevent our reasoning about infinite quantities; combining as elements of our reasoning, the absence of a limit with other positive characters.

8. One of the consequences which is drawn by the assertors of the doctrine that we cannot know anything about Infinity, is that we cannot obtain from science any knowledge concerning God: And I have been the more desirous to show the absence of proof of this doctrine, because I conceive that science does give us some knowledge, though it be very little, of the nature of God: as I shall endeavour to show hereafter.

The

For instance, I conceive that when we say that God is an eternal Being, this phraseology is not empty and unmeaning. It has been used by the wisest and most thoughtful men in all ages, and, as I conceive, may be used with undiminished, or with increased propriety, after all the light which science and philosophy have thrown upon such declarations. reader of Newton will recollect how emphatically he uses this expression along with others of a cognate character: "God is eternal and infinite,...that is, He endures from eternity to eternity, and is present from infinity to infinity...He is not eternity and infinity, but eternal and infinite. He is not duration and space, but He endures and is present. He endures always, and is present everywhere, and by existing always and everywhere He constitutes duration and space.' We shall see shortly that the view to which we are led may be very fitly expressed by this language.

But I will first notice some other aspects of this philosophy.

5 Scholium Generale at the end of the Principia.

CHAPTER XXVII.

SIR WILLIAM HAMILTON ON INERTIA AND WEIGHT.

IN Titor, to English readers, of

Na preceding chapter I have spoken of Sir William

modern German systems, and especially of the so-called "Philosophy of the Unconditioned." But the same writer is also noticeable as a continuator of the speculations of English and Scottish philosophers concerning primary and secondary qualities; and these speculations bear so far upon the philosophy of science that it is proper to notice them here.

1. In our survey of the sciences, we have spoken of a class which we have termed the Secondary Mechanical Sciences; these being the sciences which explain certain sensible phenomena, as sound, light, and heat, by means of a medium interposed between external bodies and our organs of sense. In these cases, we ascribe to bodies certain qualities: we call them resonant, bright, red or green, hot or cold. But in the sciences which relate to these subjects, we explain these qualities by the figure, size and motions of the parts of the medium which intervenes between the object and the ear, eye, or other sensible organ. And those former qualities, sound, warmth and colour, are called secondary qualities of the bodies; while the latter, figure, size and motion, are called the primary qualities of body.

2. This distinction, in its substance, is of great antiquity. The atomic theory which was set up at an early period of Greek philosophy was an attempt to account for the secondary qualities of bodies by means of their primary qualities. And this is really the scientific ground of the distinction. Those are primary

qualities or attributes of body by means of which we, in a scientific view, explain and derive their other qualities. But the explanation of the sensible qualities of bodies by means of their operation through a medium has till now been very defective, and is so still. We have to a certain extent theories of Sound, Light and Heat, which reduce these qualities to scales and standards, and in some measure account mechanically for their differences and gradations. But we have as yet no similar theory of Smells and Tastes. Still, we do not doubt that fragrance and flavour are perceived by means of an aerial medium in which odours float, and a fluid medium in which sapid matters are dissolved. And the special odour and flavour which are thus perceived must depend upon the size, figure, motion, number, &c. of the particles thus conveyed to the organs of taste and smell: that is, those secondary qualities, as well as the others, must depend upon the primary qualities of the parts of the medium.

3. In this way the distinction of primary and secondary qualities is definite and precise. But when men attempt to draw the distinction by guess, without any scientific principle, the separation of the two classes is vague and various. I have, in the History of Scientific Ideas', pointed out some of the variations which are to be found on this subject in the writings of philosophers. Sir William Hamilton' has given an account of many more which he has compared and analysed with great acuteness. He has shown how this distinction is treated, among others, by the ancient atomists, Leucippus and Democritus, by Aristotle, Galen, Galileo, Descartes, Boyle, Malebranche, Locke, Reid, Stewart, Royer-Collard. He then proceeds to give his own view; which is, that we may most properly divide the qualities of bodies into three classes, which he calls Primary, Secundo-primary, and Secondary. The former he enumerates as 1, Extension; 2, Divisibility; 3, Size; 4, Density or Rarity; 5, Figure;

1 B. iv. c. i

* Reid's Works, Supplementary Dissertation D.