examines the work will agree with us so far, and shall therefore not lose space in substantiating points upon which we feel we risk nothing by very positive assertion. We shall, therefore, proceed at once to the less agreeable, but more useful part of our task more useful, because laxity of reasoning is very often excessively agreeable, where strictness would be perhaps somewhat dull at first; and we would rather that books like the present should sink into obscurity, with all their good points, than be made an instrument of banishing the soundness of logic from the establishment of geometrical truths. No reasoning at all, and welcome, but-no bad reasoning. It appears that the author has agreed with us in feeling that geometrical results ought to be taught at an earlier age than is usually the case, and might be so, were it not for the difficulties of establishing them strictly. We have already recommended to dismiss the proofs entirely at first, and teach the properties of the angles of a triangle by precisely the same method which is used to prove mechanical truths, namely, ocular demonstration. Our author follows another plan. Where the reasoning which connects one proposition with another is very simple, it is correctly given, and often very happily; but where this is not the case, either good use is made of the terms obviously' or 'must necessarily be,' or else a set of words is substituted for proof, by which, if necessary, the most incongruous propositions might be also proved. In defence of this, the following very curious argument is given: curious, as we shall see when we come to trace its consequences. 'Another reason why I have avoided a formal demonstration of the converse of a proposition is, that the proof is most frequently of that kind which is called indirect, or a reductio ad absurdum, that is, showing it would be absurd to suppose the contrary. Pupils have a dislike to this kind of demonstration, and for this as well as other reasons it should be used very sparingly in geometry. Besides, in every case the real converse is necessarily true, the demonstration of the proposition establishing the truth of the converse. Thus, for example, if it be proved that the equality of two of the angles of a triangle depends essentially on the equality of the opposite sides, it follows that the equality of the sides depends essentially on the equality of the angles.'-p. 85. To take the preceding points separately, we have found that pupils have no great liking for any sort of demonstration, and the manner in which they show their dislike, by being always ready to contract good reasoning into bad, is to us sufficient proof of the necessity of inducing them to mend their habits. he We imagine all are agreed that the reductio ad absurdum should not be used more often than necessary; but no method has yet been found of avoiding it entirely for we cannot call our author's plan of assuming converses, a method. What says about the real converse, we agree to in one sense; but we suspect, from the latter paragraph, that we cannot agree as to what is the real converse.' We can imagine any one saying that the common logical converse is not the real converse, in universal affirmative propositions at least; but that in interchanging the subject and predicate the latter ought to be limited as in the original proposition. For example, that of 'all right angles are equal angles,' (1) the real converse is not 'all equal angles are right angles; (2) but some equal angles are right angles' (3). Let then (3), which is named by logicians a conversion per accidens, be called the real converse, which we hold to be very rational, and let (2), commonly called the converse, be entitled the extended converse, since the subject of (2) is more generally spoken of than the predicate of (I). This being supposed, we admit with our author that every proposition proves its real converse: and will never require proof from him. But it does not therefore follow, that every proposition proves its extended converse; and no one could imagine that our author would assume that it does. What then shall we say to the following? 6 For when ABD is a straight line, these angles amount to two right angles, therefore conversely, when they amount to two right angles, A BD must be a straight line.' 6 Here is the extended converse assumed as a consequence of the original proposition. We beg leave to parody the preceding for the assertion A B D is a straight line,' substitute an author makes a mistake,' and for these angles amount to two right angles,' read he is still a living man.' Mutatis mutandis, we have the following:— 6 When an author makes a mistake, he is still a living man, therefore conversely, so long as he is a living man, he must make mistakes.' We feel convinced that our author, if he be a living man long enough, will be able to confute the latter in his second edition; but how he is to do it without confuting the former, we are not able to guess. 6 The last part of our preceding quotation, beginning, Thus, for example, &c.' we are not able to understand. We do not see the consequence asserted; perhaps this arises from our not knowing what is meant by the emphatic word essentially,' as there applied. We may be dull perhaps, but the essences' of metaphysics do not strike our perceptions half so strongly as those which are sold with directions for use in the druggists' shops. In the theory of parallels the same sort of conversion appears, and we have to add one more to the many methods of establishing the disputed axiom of Euclid. Is it not strange that an author, who professedly throws away much of the reasoning of Euclid, should make compensation by furnishing a proof of what Euclid never attempted to prove? Yet so it is; the pupil is shown that when two right lines make, with a third, angles together equal to two right angles, they do not meet in either direction, and consequently that, when they do not meet in either direction, they must make angles together equal to two right angles with any third line-(page 33). Again, our author says (page 13), that a pupil who cuts out two triangles as described in Euclid i. 4, and applies them to one another, will clearly see that they not only do coincide, but they must necessarily do so.' What the pupil will see, without the reasoning of Euclid, more than that they do coincide, we cannot imagine: our only idea of necessary propositions is applied to those which we can show by reasoning not to be sometimes true and sometimes false, but always true. We like popular illustrations very much when they are correct, but not in the contrary case. Our author illustrates the proposition, that the area of a circle is equivalent to that of a triangle, having the circumference for its base, and the radius for its altitude, in the following manner. He supposes a roll of ribbon half cut through, in which case the divisions will fall down on both sides, and the circular side of the roll will present the appearance of a triangle, having what was the circumference for a base, and the radius for an altitude. This is very ingenious, and might seduce any one into a belief of the proposition as of one proved. But if the roll were in the form of an elliptic, instead of a circular, cylinder, the same process would prove that an ellipse is equal to a triangle, having the circumference for a base, and either axis for its altitude. Should this unluckily turn out to be true, alas for Legendre and Jacobi! that they should so long have worn on their breasts enough to prove more than they ever could get out of their heads. We might notice some little inaccuracies which require the author's attention in a second edition. The definition of a straight line, even as corrected in the erratum, is hardly necessary when the property implied in the definition is not to be subsequently made use of. For beginners in geometry, a straight line is a straight line all the world over. Again, it is not true (page 75) that a plane surface is a figure bounded by lines,' even when line' means straight line. Neither is 6 it correct to say (page 135), that an interminate decimal fraction is incommensurable with the unit, at least there is an infinitude of ratios, which are incommensurable (if this be the meaning of the term), which are not so according to any other author. In the exercises also, which are in general very good, there are a few which might be omitted, either as little curiosities, in which the wonder only arises from the vagueness with which they are put, or as being fairly beyond, not only all pupils, but many teachers. Such, for example, as the following: ( Prove that no more trees can grow at the same distance from each other on a mountain, than on the horizontal base on which it stands.' 'Why did the bee select the regular hexagon for constructing its cell, rather than the equilateral triangle or square? The first of these, if by distance' be meant distance measured on the hill, as one might naturally suppose, is not true; and if horizontal distance be meant, there is hardly any exercise, and the diagram, which accompanies the question, will surely mislead the pupil. As to the second, we think we remember to have seen it proved very satisfactorily, that the hexagonal form of the cell is a consequence of the bee's number of legs, and not at all of any selection, or attention to the theory of maxima and minima. But even according to the common notion, it is a very hard question. In conclusion, we recommend this book to teachers for the facts and the illustrations; but we put it to their discretion whether they will teach their pupils to reason by it, after the instances which we have selected. We all have heard of a student who read Euclid in an afternoon, that is, 'read the large print, and looked at the pictures of scratches and scrawls, but left out all the tiresome A's and B's and C's.' This latter part our author has partially done; if any intelligent teacher will complete the work, he will, in very little time, be able to give his pupils a great deal of the useful and the entertaining combined, and will, we doubt not, inspire considerable interest in the study. While this article was going through the press, we have ascertained that the author has, since the publication of his work, made some additions to the preface. From a passage which we now cite, we should imagine that some friend had suggested what might possibly be said upon his method of reasoning. 'We could easily point out the very pages where a purely geometrical critic, who scarcely knows anything of the "delightful task," will pick and cull, and find fault because these pages are not what they never were intended to be; but we shall refrain from doing so, lest we should fall into the error of writing a long preface to a small volume.' This amuses us exceedingly; for the 'delightful task,' though from the context it might appear to be picking and culling, and finding fault,' which has been suspected by some to be delightful to reviewers, is no such thing. We remember well what it is; for in our first spelling-book, at a part yclept the frontispiece, was an awful picture of an old lady and a birch rod, and a dozen children, who looked dismally cognizant of the probability of nearer acquaintance with the latter, and underneath was written : 'Delightful task! to rear the tender thought On looking at the tender thoughts' which this work inculcates, and seeing that it is now the avowed object to teach that, every bird being an animal, 'therefore conversely,' every animal is a bird, which is our author's method of reasoning, we are of opinion that the young idea is made to shoot dreadfully wide of the mark. We cannot but suppose that the author, when he wrote the preceding passage, was aware of the points on which any criticism must turn, and knew that, in his methods of reasoning, he differed from the universal sense of mankind. If so, well might he say that he knew the very pages (aye, and lines and words), on which a purely geometrical critic' would throw his censure; but we will go even further, and say that it is not necessary to be any great geometer to find him out. That the work was not intended to be correct is no answer, but a great aggravation; and changes our verdict from logical manslaughter into wilful murder. |