If we do, then we enounce, at the very beginning, a notorious falsehood; and no conclusions drawn from it can be worth attending to. The boasted science of Mathematics becomes like the ravings of a madman, who first fancies himself a king, and then reasons well accordingly.

But if what we have above said, as to the nature of Quantity, be correct, then Mathematical demonstration is quite independent of Matter; and consequently we do not assume that the points, lines, triangles, circles, &c. as defined in books of Geometry, have any existence in the material universe. No doubt it was by the material world that we first became acquainted with points, lines, triangles, &c. approximating to those defined; but having once got these ideas, through our sensations, we can afterwards detach them from Matter, and consider them as modifications of pure space. Look at the arch of a bridge. Without the stone and lime which form the arch I might never have conceived a curve; but, having once seen a curve in Matter, I can imagine one immaterial. What is included between the arch, the piers, and the water below, forms a definite figure in empty space, the air being invisible. From constant association with Matter it is, no doubt, difficult, if not impossible, to avoid think

ing of Matter when we wish to think of figure only, and sensible diagrams are even put before us; but in framing or following the reasoning, we can attend so little to what is material, as not to be at all disturbed thereby in our calculations. I have, then, a notion of a Mathematical line, triangle, circle, &c., and a notion sufficiently clear to conduct me through the longest chain of reasoning without any confusion; and what more can I wish?

But, if such points, lines, and circles, &c., exist not in any material object, and if spirit be altogether inconsistent with extension and figure, can these points, lines, and circles, be said to exist at all? and if not, is the science purely imaginary? The only answer to this is, that if space can be said to exist, then do the figures of Geometry. They rest upon the same foundation, they must stand or fall together; and if Space and Time be not imaginary, neither is the science of Mathematics. Surely no one will say that the terms, Space and Time, have either no meaning at all, or mean what exists only in fancy, as the words Centaur, Mermaid. The same exactly may be said of the lines and figures of Geometry. That these terms have a meaning is evident, for otherwise how could we reason about them? and if they have a meaning, then we have notions

corresponding to them, for these are but different phrases for the same thing. And will any one pretend that those notions are fanciful, like the notions of Centaur and Mermaid? We have then notions, and notions which are not fantastic, what more can we desire for Truth?

With all respect for the abilities of the abovementioned author, I cannot but think that his doctrine, with respect to necessary truths, is fundamentally erroneous. Certain it is that philosophers have long made a distinction between necessary and contingent truth, a distinction which Mr. Mill would confound. Hume clearly marked out the difference, under the names of relations of ideas, and matters of fact, the latter known by experience, the former not. Whewell's account of this, as quoted by Mr. Mill, is as follows: "Necessary truths are those in which we not only learn that the proposition is true, but see that it must be true; in which the negative of the truth is not only false but impossible, in which we cannot, even by an effort of imagination, or in a supposition, conceive the reverse of that which is asserted. That there

are such truths cannot be doubted.

Dr.

We may

take for example all relations of number. Three and two, added together, make five; we cannot conceive it to be otherwise; we cannot by any

From this passage Mr. Mill deduces that, according to Dr. Whewell, a necessary truth may be defined to be a proposition, the negative of which is not only false, but inconceivable. Starting from this idea, Mr. Mill goes on to show that as the power of conception depends very much upon association, and as many things formerly supposed inconceivable, are now not only conceived but believed; for instance, the action of bodies on each other at a distance; of Matter on Mind, &c., he thence infers that inconceivability of the contrary is a very poor test of truth, and that what is called necessary truth rests, like every other, solely on experience.

Is there then no difference in the evidence on which these two propositions rest? The sun will rise to-morrow; the three angles of a triangle are equal to two right angles. Are they both contingent, or both necessary? Do they both rest upon experience? Consult your own mind. Why do we believe that the sun will rise to-morrow? Because as far as I know personally, or can learn from the testimony of others, alive or dead, it always has in time past. But can you see in that

any irresistible reason

why it should rise to

с

Philosophy of the Inductive Sciences, Part i. Book i. Chap. 9.

morrow? Must you not allow, that, for aught you know, the sun may be dissolved, and scattered throughout boundless space before another day? Ere you can say positively it will not, your knowledge must be far far more extended than it is at present, it must approximate to the knowledge of Him who created the sun and all things. But why do you believe that the three angles of a triangle are equal to two right angles? Because I have seen it demonstrated; that is, starting from some self-evident truth, I have followed a chain of reasoning, each link of which was an irresistible inference from the preceding, until I arrived at the conclusion, which was the last irresistible inference. The demonstration finished, I can no more doubt the truth in question, than I can doubt the existence of that feeling of which at the moment I am conscious. I see clearly that the conclusion holds good, and always will hold good; in short, that it must be true.

But how do you know that? All I can answer is that I see it to be so. I assert that to me the first proposition is self-evident, and that the inferences flow from it irresistibly, even to the conclusion. If you deny this, I can only bid you to study the theorem. Should you still persist in your doubts I can say no more, for I cannot give a demonstration of a demonstration. What is