them in pairs, which two are so connected that both cannot be true, but one must be true. Let a careful distinction be made between those which are contradictory and those which are contrary, that is, which cannot both be true, but which may both be false. Thus, that “ all equilateral triangles are equiangular," and that "some one equilateral triangle is not equiangular," are contradictory; but the contrary of the first is "no equilateral triangle is equiangular." This distinction had better be not only explained, but, so to speak, also made fast to the terms employed; the want of it may create as much confusion in geometry as in common conversation, one of the prevailing fallacies of which is the misapplication of these two ideas. The learner can now see that it is the same thing whether we prove a proposition, or disprove the contradictory proposition, for one of the two must be true. Now let the following method of expression be explained: "If A is B, C is D," using various instances, as in the preceding illustrations. This is not a syllogism, but a method of stating that there is an argument or chain of arguments, which, it is admitted, only want the proposition "A is B" added to their number, to complete the proof that C is D. Now, though it does not follow that if we disprove "A is B," we therefore disprove "C is D," because that which might follow from certain premises, if true, may also follow from others which are true; yet it does follow, that if we disprove "C is D," we disprove "A is B," for if A were B, C would be D. To suppose that a false proposition could be legitimately deduced from correct premises, would be a contradiction: if, therefore, a false proposition be deduced from premises all of which, except one, are true, that one, must be false. This is the

method of Euclid in indirect demonstration, as we shall show by putting the first case in which it occurs into the form in which it might be taught; substituting only the deduction of the corollary instead of the proposition, as the necessary phrases are thereby rendered more simple. It has been proved, and is admitted, that all equilateral triangles are equiangular, which is to be remembered throughout: it is to be shown that all equiangular triangles are equilateral. He who admits the first and denies the second, maintains that the first, and a proposition contradicting the second, may be true together: that is, he affirms that " some one equiangular triangle is not equilateral" may be true. The proposition of Euclid may then be thrown into a direct form as follows: "If any one equiangular triangle be not equilateral, then a whole is equal to its part," which is accordingly proved by the assistance of the admitted proposition. But the whole is greater than its part; therefore it is false that "some one equiangular triangle is not equilateral," that is, the contradiction, that "all equiangular triangles are equilateral, is true. It would be a good exercise to accustom the student to add indirect demonstrations to some of the direct demonstrations given by Euclid, as also to prove simple derivative propositions of such a form as this-"If two triangles have two sides of the one respectively equal to two sides of the other, but the included angles unequal, the remaining sides will be unequal, &c."

On the question whether the fifth book of Euclid, which is on proportion, can be made intelligible to beginners in general, we must suspend our opinion: we are very certain that it is not so to nine out of ten at least. We e may perhaps resume this subject at a future

time; but as it is, we can only recommend instructors either to reject this book, and substitute the numerical definition of proportion, or, if they retain the book, to make the numerical definition accompany it. Many will not agree with us in the first recommendation, and we feel, as much as they do, the hiatus which would be made. in the system by attending to it. But surely it is no compensation to the pupil for an employment of time which brings in no knowledge, (which is his case if he read the fifth book without understanding it,) that his instructor can appreciate the superior completeness and rigour with which all the demonstrations of the sixth book might be given, if the student could only comprehend that which he does not comprehend, namely, the fifth book. We would say to all, teach the fifth book, if you can; but we would have all remember that there is an if.

We now come to solid geometry, which we may observe is seldom or never taught before plane trigonometry: that is, a purely conventional arrangement has placed a very easy part of the subject after one of much greater difficulty; so that, in fact, access to the easier part is practically forbidden to all who do not first master the harder. The propositions contained in the first and second books are sufficient for the establishment of as much of the eleventh as is necessary for the purposes of spherical trigonometry, that is, of the first elements of astronomy; and the same pains which are taken with the fourth book, which is of very little use, would, if applied to the eleventh, most materially increase the power of the student to comprehend popular works on physics.

The main difficulty is one which is not in the subject,

but in the manner of treating it, namely, the substitution of drawings upon paper instead of the solid objects which are considered. Yet a few pieces of card, or even of the very paper on which the student looks with despair at right angles which are acute, and lengths the relative magnitudes of which have changed places, would be sufficient for the formation of bonâ fide prototypes of these perspective anomalies. We should like to know to how many mathematical teachers per cent. it has occurred, instead of drawing one plane inclined to another on a paper, to fold the paper itself, and place the two folds at the required angle? Would it give too much trouble? Does the pupil say his proposition as well without it?

The eleventh book of Euclid may, in our opinion, be abandoned with advantage in favour of more modern works on solid geometry, particularly that of Legendre, which the English reader will find in Sir David Brewster's Translation. If Euclid be adhered to, the first twenty-three propositions of the eleventh book are sufficient for common purposes. We need hardly repeat, that the ocular demonstrations should be made to precede all others, which cannot, of course, be done without lines which are really in different planes.

So long as geometry is made a mere exercise of memory, it is idle to expect that the pupil should make any step for himself in the solution of a problem which is not in the book; and as there are no rules by which such a thing can be done, we find accordingly that this is an exercise almost unknown in the geometrical classes of schools. But supposing the pupil to be taught on a rational system, there is nothing to prevent his being tried with easy deductions from time to time, except the

difficulty of procuring the problems in cases where the teacher cannot invent them. The works of Messrs. Bland and Creswell, published at Cambridge, bearing the titles of "Geometrical Problems," and "Deductions from Euclid," contain problems perhaps of too difficult a cast; but from them a judicious teacher might select some which would suit the capacity of his pupils. Previous to this, the pupils should have been accustomed to retrace the steps of the several propositions of Euclid from the end to the beginning, whenever this inversion will not affect the reasoning. This will accustom them to the analytical method, by which alone they can hope to succeed in the solution of problems. But great care must be taken not to introduce sophisms in this reverse process. For example, in the forty-seventh proposition of the first book, it is shown that the square on each side of the right-angled triangles is equal to one of the rectangles into which the square on the hypotenuse is divided. The concluding argument is therefore of this form: A is equal to B, and C is equal to D; therefore, the sum of A and C is equal to the sum of B and D. Assuming the result, that A and C together are equal to B and D together, it cannot, therefore, be assumed that A is equal to B, and C to D, but only that if A be equal to B, C is equal to D. Of all the exercises which we have proposed, this is the one which requires most care on the part of the instructor.

Our readers will see that we have throughout advocated the union of the forms of logic with the reasoning of geometry. We are convinced that it would be advantageous to make the former science systematically a part of education. If we except Oxford, there is no place in this country where it is still retained; and unfortunately