rigour, but the beginner cannot enter into this refinement. And, in fact, the order of the propositions is not necessary to correct reasoning. This consists in forning the conclusions rightly from the premises, no matter what these last may be. It would not be contrary to good logic to assume the whole of the first book of Euclid, and from it to prove the second; provided that afterwards the first book were proved, without the necessity of taking for granted any proposition in the second. The argument, or collection of arguments, would then stand thus: If the first book be true, the second is true. Therefore the second is true. The order in which the premises come does not affect the soundness of the conclusion, and provided the pupil understands that the conclusion depends equally on the premises and the reasoning grounded upon them, which are two distinct things, an error in one not necessarily affecting the other, he is perfectly safe, and takes a view of the process of reasoning not generally given to the young. We should then recommend the following principles in teaching geometry : Never to state a definition without giving ocular demonstration of one or more facts connected with the term employed. To defer every axiom, until that point is arrived at, where it becomes necessary. To impress upon the mind of the pupil that the reasoning is not affected by the assumption of an axiom to be proved afterwards, provided the proof of it is independent of the proposition which it was used in proving, and its consequences. To accustom the beginner to retrace his steps, and going backwards from any proposition, to continue the chain, until he arrives at the point which he set out by assuming. To supply a proof that "all right angles are equal," and to deduce the axiom on which Euclid grounds the theory of parallels, from this more simple and obvious one, viz., " through a given point not more than one parallel can be drawn to a given straight line." To omit those propositions which are not subsequently useful, among which may be reckoned many in the second book, and all in the fourth. In order to accustom the pupil to correct statement of propositions, he should be made to write all that he reads. But here is a probability that he will trust entirely to the book. This may be prevented by requiring him to use numerals instead of letters throughout, and to arrange the whole in the following manner. Let a sheet of paper have two vertical columns, ruled on the left, and let the whole enunciation, construction, and demonstration of the problem be divided into distinct paragraphs, each containing only one assertion. Number these paragraphs in the first ruled column, and, in the second, opposite to each paragraph, enter the numbers of the preceding ones from which it follows. Where a previous proposition, or an axiom is required, write its enunciation at the end, with a letter before it, and enter that letter opposite to the paragraph in which it is assumed. If the pupil does this correctly, the instructor may be well assured that he understands the proposition. In the application of algebraical symbols to geometry, misconceptions usually prevail which are coun tenanced by the looseness of expression of many elementary works. They are mostly founded upon the analogy existing between the algebraical expression a xa, or a2, called a square, and the geometrical square described upon the line which contains a units. Against this confusion of terms the teacher must be on his guard, and should carefully avoid that symbolical notation recommended in some books, by which A B2 is made to stand for the square described on A B.. If a short symbol for this be used it might be [A B], and the necessity of proving that the number of is contained in a is a x a, would not be concealed under a petitio principii. With regard to the fifth book of the Elements, we recommend the teacher to substitute for it the common arithmetical notions of proportion. Admitting that this is not so exact as the method of Euclid, still, a less rigorous but intelligible process is better than a perfect method which cannot be understood by the great majority of learners. The sixth book would thus become perfectly intelligible. It would much benefit the pupil if solid geometry were introduced at a more early period. There is nothing in the elementary propositions which requires more than the first book of Euclid; and by a judicious use of the real figures, instead of perspective drawings, the subject might be amazingly simplified. We come now to the subject of algebra, regretting that the limits of this article will not permit us to discuss the subject upon the scale which it deserves. The great drawback to the proper attainment of this science is the miserable previous instruction in arithmetic. When this defect is remedied, and not till then, can we expect any better results. It is the practice not to let the pupil proceed to the principles of equations until he can work questions in all the previous rules of a nature which very rarely occur in practice. And of these rules themselves it must be observed that, in order to preserve analogies with arithmetic, their meaning is usually distorted. To the unintelligible way in which the negative sign is used we have already alluded. We shall now explain our views as to the manner of proceeding. The new symbols of algebra should not be all explained to the student at once. He should be led from the full to the abridged notation in the same manner as those were who first adopted the latter. For example, at this period he should use aa, aaa, &c., and not a2, a3, and should continue to do this until there is no fear of that confusion of 2 a and a2, 3 a and a3, &c., which perpetually occurs. Whenever any new symbol is introduced, not a step should be made until it has been rendered familiar by finding its arithmetical value in particular cases. This indeed is the first exercise; algebraical expressions increasing in complexity are given, and also certain values for the letters, and the student is left to find the corresponding arithmetical value of the expression. Whenever a negative result occurs it should be thrown aside as an impossibility, the pupil being told at the same time that use will be afterwards made of such expressions when he can understand what they mean in the solution of a problem. The leading principles of the solution of equations of the first degree might then be easily established, and applied to some numerical equations. The four rules should follow, the principles being previously explained, and all negative results avoided. The student is then in possession of the means of solving an equation of the first degree in which some of the given quantities are literal, and may be supplied with examples a little more likely to aid his future studies than the conundrums about posts and saddles which we have instanced. At this stage of his progress the pupil should be set to work a problem in which a negative result occurs. It should then be pointed out to him that there is a misconception of the problem itself, and the manner of rectifying that error will show, in the course of several examples, what is the meaning of the negative answer. At the same time it will be easy to explain by examples the nature of the wrong suppositions which lead to results of the form and He should then examine a 0 0 for himself what change is produced in a process which sets out with some assumption as to b-a, when this has been incorrectly written for ab. By comparing the true and false processes he will deduce the rules according to which negative quantities must be treated, in order that their introduction may not affect the soundness of the conclusion. He is thus placed in the same condition as to results with the pupil who has pursued the common method; with this difference however, that he can explain conclusions which the other cannot, and has never believed that, a b meaning a diminished by b, there can be such a thing as - a, or a quantity less than nothing. - The view which is generally taken of expressions of the first and second degree is too confined for the future purposes of the mathematical student. It is this: what |