lies open to the view, and cannot be mistaken: it remains unchangeable, and may at leisure be considered and examined, and the demonstration be revised, and all the parts of it may be gone over more than once without any danger of the least change in the ideas. This cannot be thus done in moral ideas: we have no sensible marks that resemble them, whereby we can set them down."-(Book IV., chap. iii., § 19.)

It matters not that he is the best mathematician, or arithmetician, who needs least the sensible diagram, or the figure on the paper; nor to say, with Mr. Stewart, that the figure on paper cannot pretend to that precise exactness which is the object of our reasoning; that the line we draw will have some breadth, and the circle, however steady the instrument and the hand, may deviate in some point from equidistance. The most skilful reasoners can only have a certain idea of visible figure, and of the relation of its several parts present to their minds, which the less skilful require for facility and permanency of reference on the of reference on the paper. The

diagram approaches sufficiently to sensible exactness to keep before the mind that quality of the figure which is the sole object of the reasoning; and it is sufficient that the more nearly the specific figure before us approaches to exactness, the more applicable will the reasoning be to that figure, or, more correctly, it is only in so far as the figure fairly represents the mind's view of its qualities that the reasoning applies to it at all. There is no such mystery in the most obscure of the definitions as to make us deny their reference to a certain specific quality of objects, that is, to real existencies, in the only practical sense of the words. The constant application of mathematical reasonings to the various branches of natural philosophy, and the common use of mathematical terms in the mathematical sense, prove the contrary. We speak, for instance, of the line between one shade of colour and another, and length without breadth is the only object of the mind's contemplation in so speaking. Points and angles are words of perpetual occurrence: the

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former in the sense of the commencement or termination of lines, without being any decided parts or given portions of the line; and the latter in the sense of the meeting of two or more lines together, converging or diverging with more or less of rapidity or extension.

It is perhaps of little consequence to determine whence we get the notions or conceptions upon which mathematical reasoning turns, whilst it is certain we have the notions and defined terms appropriate to them, except in so far as it appears that in numerical calculations, and in the geometry of Euclid, there is a certain verification of the reasoning by an appeal to the evidence of the senses. In fact, it is hard to divine whence we get notions of figure or quantity if it be not from the sight and the touch, or from experience, a word of extensive signification, comprehending all the results of observation and reflection. Those who say we do not get these notions or conceptions from experience, would do well to tell us whence we do get them, or produce the mathematician

upon whom God has not bestowed the five senses with which he has happily blessed the rest of mankind.

I will here venture a remark upon Mr. Whewell's language, in his pamphlet on Mathematical Studies (second edition, p. 32):

"I mentioned it," says Mr. Whewell, "as likely to make the study of mathematics less beneficial as a mental discipline than it might otherwise be if the first principles of our knowledge be represented as borrowed from experience, in such a manner that the whole science becomes empirical only.

"I will not suppose that any person who has paid any attention to mathematics does not see clearly the difference between necessary truths and empirical facts,-between the evidence of the properties of a triangle and that of the general laws of the structure of plants. The peculiar character of mathematical truth is, that it is necessarily and inevitably true; and one of the most important lessons which we learn from our mathematical studies is a knowledge that there

are such truths, and a familiarity with their form and character.

'This lesson is not only lost, but read backwards, if the student is taught that there is no such difference, and that mathematical truths themselves are learnt by experience. I can hardly suppose that any mathematician would hold such an opinion with regard to geometrical truths, although it has been entertained by metaphysicians of no inconsiderable acuteness, as Hume. We might ask such persons how experience can show, not only that a thing is, but that it must be; by what authority he, the mere recorder of the actual occurrences of the past, pronounces upon all possible cases, though as yet to be tried hereafter only, or probably never. Or, descending to particulars, when it is maintained that it is from experience alone that we know that two straight lines cannot enclose space, we ask, who ever made the trial, and how? And we request to be informed in what way he ascertained that the lines with which he