ceding assertions; that both may exist consistently with the triangle being either wholly or partly in or out of the square, and that the only relation inconsistent with the two preceding is that which places the whole square in the triangle. The only deduction is, that some of the square is not in the triangle. A verbal analysis of this conclusion should now be sought, and it may be found in the following reasoning. The first assertion states, that some of the circle is common to the square; we must, therefore, stand prepared to admit that anything which can be asserted or denied of all the circle as to position, can be asserted or denied of certain points of the square. But by the second assertion it is denied that the triangle and circle have any points in common, consequently there are certain points of the square which are not in the triangle. This same reasoning should now be applied to other instances of the same form, drawn from various subjects, on which we do not think it necessary to dilate. To take an example from among those combinations which are without conclusion, -let some of the square be in the circle, and some of the triangle not in the circle. This will be found by trial to be perfectly consistent with either of the two, the triangle or square being wholly or partially contained in the other. To make this intelligible, the student must be made to understand that the phrases of common conversation are stripped of some of their implied meaning in all strict reasoning, and most particularly in that of geometry. The literal extent of the fact stated is all that is allowed, nor must anything be concluded from one simple proposition. For example, if in ordinary conversation we say that some men have a mechanical genius, it is always implied that the rest of mankind

have not the same, or it is considered as signifying that some men, and some men only, have that endowment. And by the word "some" is generally understood the smaller number, in opposition to the greater. None of this is supposed to hold in geometrical reasoning: to assert that some equilateral triangles are equiangular is perfectly correct, though, at the same time, it is true that all equilateral triangles are equiangular. In the same manner, to avoid the necessity of repeating two propositions instead of one, the word "some" is considered as standing for one, as well as for more than one. When, therefore, we say, that " some of the square is in the circle," we mean that one point at least of the square is in the circle, or it may be the whole square. But at the same time, though the truth may be more general than the proposition, the conclusion must not be so.

Returning now to the proposition given, the student must be made to see that no two assertions can give us the right to make a third, unless there is something common to both, expressed in strict reasoning, but often only implied in ordinary conversation. In the pair of propositions which we are now considering, namely, that " some of the square is in the circle," and " some of the triangle is not in the circle," we can only say, that a particular part of the triangle (that which is out of the circle) has nothing in common with a particular part of the square (that which is in the circle), which may be true, whether the one is wholly or in part contained in the other. Again, from the propositions, "some of the circle is in the triangle," 66 some of the circle is in the square," nothing can be drawn, for no deduction can follow, unless there is something in common in the two assertions which may be made to furnish the means of

comparing the two. Here, though the word circle is mentioned in both, it is partially mentioned; the "some parts" of the first may not be the "some parts" of the second; at least it is not so expressed, and is therefore not to be assumed. We should recommend that the sixty-four possible combinations of propositions above alluded to should each be separately considered, and either a deduction drawn, or a reason against drawing one pointed out. This seems to us perfectly necessary, but will not universally appear so. But certainly either the beginner is competent to detect the truth of each instance immediately, in which case a couple of hours will suffice for the whole, or there are some combinations which he does not understand, and then we presume it will be universally admitted that attention should be paid to these. Is so much time to be spent on mere etymology, and accurate combination of singulars with singulars, and plurals with plurals, and this, too, when practice would produce a sufficiently correct habit; and is no attention to be paid to that structure of a sentence on which its truth or falsehood depends, so far as structure can influence one or the other, when all people must reason more or less, and very few do it correctly? It matters little, as we have said before, whether the language of the schools be adopted or not; those who know it can use their discretion, and those who do not may be informed that correct logic is in their power, and if they will adopt the preceding process, must be a necessary result of their experimental researches on the structure of a simple argument. Of course we confine the term logic to its strict meaning, not supposing it to have any reference to the truth or falsehood of assertions themselves, but only to the circumstances

under which two of them give us a right to deduce a third.

The preceding course having been adopted, it remains to put it in practice in following the order of Euclid. The definitions claim the first place; and these should be explained without the ambiguity which commonly attends them. As they usually stand, there is nothing to distinguish them from the propositions which follow. It must be remembered that the perversity of human nature, we suppose, lends beginners a really wonderful tact in choosing the wrong idea, if there be two to choose between, that is, in every possible case. Hence "a parallelogram is a four-sided figure, whose opposite sides are parallel," is too nearly allied in form to "the opposite angles of a parallelogram are equal." The development of the first should be, "let it be agreed to call every four-sided figure whose opposite sides are parallel, by the name of parallelogram, so soon as it shall have been shown that such figures can be drawn, and let no other figures be called by that name." The indefinite idea which exists in the mind of a beginner as to the distinction between the component parts of a demonstration, must be well known by all teachers who have any clear ideas on the subject themselves. The first answer, when pressed on any point, is, "Of course it is so," which must always be interpreted to mean that belief outstrips knowledge. But when further questioned, the useful terms " by definition," or "by hypothesis," are applied, according to which of the two hard words first comes into the memory. To avoid this, it would be advisable to draw a perpetual verbal distinction between the application of a definition and the deduction of a consequence: thus, instead of allowing the pupil to

say, "A B is parallel to C D, and A D to B C, therefore A B C D is a parallelogram," it might be worded thus, "A C is parallel, &c., that is A B C D, being a four-sided figure whose opposite sides are parallel, is one of those to which we have agreed to give the name of parallelograms." The mischief arises, in a great measure, from the mixed character of our language, which, instead of coining words as they are wanted, runs to an ancient tongue for the way of expressing the most common ideas. Thus we are not permitted to say lines run "side by side;” the idea must be expressed by the Greek word parallels, Tapáλλŋλa:" nor must there be a "parallelsided figure;" we again have the Greek "parallelogram, Tаparλópapμor." Hence the fallacy, noticed by Dr. Whately, of supposing that an assertion is proved by repeating it in other words, borrowed from another language, is never out of the mouths of beginners in geometry.

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There is no difficulty in the " axioms," except in the Greek name, which is apt to convey a notion of mysterious power, and except in the celebrated axiom by which the properties of parallel lines are established. The pupil who has gone through the preparatory system of logic which we have recommended, can be easily made aware that all strict reasoning consists in the comparison of two simple assertions, and the deduction of a third. It will be evident that there must be some assertions with which to begin, and as these can have no demonstration, they should require none. With regard to the axiom on parallel lines, we may refer the readers to the Journal of Education, vol. I. p. 276, and vol. II. p. 341.