vinced that every work forming part of a schoolboy's education would be simply looked upon as a grind,” and that to secure real interest in a book, it inust be offered to those who would enter on the study as a pleasure and not as a “grind.” Colenso wrote a work on arithmetic, endeavouring to make the study pleasurable to boys ; but I recollect reading a remark by a schoolboy on hearing the name mentioned : “ Colenso; that's the fellow who gave us the 'grind' in arithmetic. I wish we could get him into our playground, and wouldn't we pay him out.” I fear that if Homer, or Virgil, or even Horace, could appear bodily in a boy's playground he would get, to use a common phrase, “ more kicks than halfpence;" and as to poor Euclid, I fear he would get more kicks than “grahams;" and I must admit that I think Euclid's work is a “regular grind.” Although disgusted with Euclid, I felt that geometry itself was so useful, and if properly studied might be made so interesting, that I determined to do what I would advise others to do-namely, to resort to self-instruction. As to English works, I found that they were mostly modifications of Euclid, and often not interesting modifications, and I consequently turned to French works, and found those works easy, interesting, and useful. A most important character in French works is, that whilst rudimentary points are never neglected, the several works treat the study from a special point of view, according to the ultimate object of the student being to apply the science to naval or military matters, to mensuration, or to other purposes. The consequence is, that not only do French works give an interest in the subject from whatever point viewed, but they show its utility, and how such utility may be practically developed ; whilst the student who “grinds” Euclid will find the work almost entirely unsuggestive of any practical use to be derived from it. Even in French works I never found what appeared to me to be an essential in a thoroughly scientific treatment of the subject-namely, a scientific base, from which the subsequent propositions would naturally follow, and from which the demonstrations might be deduced. From the almost unlimited power of parallels in geometrical operations, I am surprised that it has never occurred to writers on geometry to adopt parallels (or rather cross parallels) as the scientific base, and from that base to deduce, as may be done, and is done in this work, all the leading propositions in geometry. Such a course, however, as far as my experience goes, has not hitherto been adopted; for though Lacroix, Bezout, Legendre, and other French authors, have recognised the importance of parallels, they have not followed out that recognition to its proper extent. I am convinced, as well as satisfied, that "cross right-lined parallels,” or (only right-lined parallels being used in geometry) "cross parallels," form the scientific base of geometry (see Figs. 2 to 5 in Div. C), and that from cross parallels all the leading propositions in geometry may be deduced. From that base, the deductions, to state only a few of them, comprise the equality of opposite or corresponding ز angles, or alternate angles on same side of parallels; the proof that the three angles of a triangle equal two right angles, that the external angle equals the two opposed angles, and is supplementary to the adjacent angle; the equivalence of parallelograms or triangles with equal bases and equidistant parallels; the equivalence of the square of hypothenuse with sum of the squares of base and perpendicular; the proportion of the squares of the sides to the adjacent segments of the hypothenuse; the conversion of figures into equivalent figures with different bases, sides, altitudes, or angles, and, in fact, almost all the propositions in geometrical operations. The present work is arranged in the following Divisions : -A and B the introductory and mechanical portions of the work, or how to do it. C to G the scientific portion, or why it is done; and H the practical application, or when to do it. The information in Divisions A and B enable a student, without any strain on the mind, to acquire a considerable knowledge of the subject, and to have the advantage in the subsequent study of neatly prepared figures, a far from unimportant point in the scientific portion of the study. Divisions C to G give the scientific portion of the study in the following order :-C. Equivalents, or figures equal in area, whatever may be their difference in form. D. Differentials. E. Measurement of angles. F. Proportionals. And G. Partitionals. Div. H gives the practical application. By this arrangement a student may know at once to what Division to refer, to meet the subject of any geometrical problem to be solved. In going through the work, reference will often be found to subsequent Divisions, but never unless the information previously given will enable a student to fully understand the subject referred to. In fact, most of the demonstrations in geometry are deduced from Sectns. 17 and 27 in Div. C and a simple application of parallels. In Div. C (commencing the scientific portion of the work) the earlier Sections and figures introduce and show the effect and the importance of the scientific base “cross parallels," the subsequent Sections being deduced from that base. Sect. 17 gives the all-important proof that“parallelograms (or triangles) on equal bases and between equidistant parallels are equivalent." In Euclid no less than four Propositions (35 to 38) are given to attempt the results obtainable under Sect. 17, and yet leaving the most important case unprovided for, namely, equal bases and equidistant parallels. Sectns. 18 to 25 follow out the base started with, by numerous interesting applications of Sect. 17, and especially the conversion into equivalent figures with different altitudes, bases, &c. The Sections relative to conversion of figures in this and the other Divisions are not only interesting geometrically, but the operations are of great value as exercises of the mind in the most valuable mode of reasoning, being that of never attempting to grapple with difficulties in the lump, but always in detail. That mode of reasoning will not only directly render it easy to overcome difficulties, but indirectly the overcoming one difficulty will often suggest the mode of overcoming the next. The habit of so exercising the reasoning powers, will be found as valuable in the difficulties of active life as in those having relation to geometry, and it gives the key to the success of some of the most brilliant campaigns of the first Napoleon. Sect. 27 gives another all-important proposition (similar to 1, 47 of Euclid), that “In a right-angled triangle, the square on the side opposed to the right angle (the hypothenuse) is equivalent with the squares on the two sides which include the right angle (the base and perpendicular).” It is scarcely possible to over-estimate the value of this proposition and of the deductions which may be made from it, if properly followed out; but in Fuclid, the proposition and the deductions from it are only used in a small proportion of the cases to which they might be applied, and one of the most important deductious is never drawn. The figure and demonstration in Euclid may probably account for the comparatively little use made of the proposition and of the deductions from it. In Euclid the triangle is so placed that the hypothenuse appears at the base, and the two sides as slant lines, not obviously suggesting a right angle. The demonstration is also confined to one side of the figure, the length of that demonstration being such, that the student is left to work the other half at equal length himself, or take it as demonstrated. The proof is also attempted with triangles forming halves of the squares and rectangle to which the demonstration is applied. |