better to omit all work involving the second power of a, so as to devote more time to simplifications, harder factors, and equation problems. There is much to be said on both sides, but taking into consideration the additional subjects in the shape of history, geography, and divinity now imposed on candidates for public school entrance, the writer is quite confident that more sound work can be done by limiting equations, &c., to the first power of x, granted always that in special cases the range can be extended to meet the special case. Though there may not be examinations on the additional work, yet no one would keep a boy within the ordinary bounds on that account. It is quite clear that a more advanced course in the case of any boys capable of profiting by it would lead to greater facility in attacking more elementary work, but to the general herd it would be detrimental in producing a rate of progression more rapid than they could adopt with benefit. Some boys of thirteen will easily reach and master the Binomial Theorem, but these are exceptions, and it is not wise to legislate for exceptional cases. They get their advantage in being placed in a higher division on entering their public school, and in scholarship examinations the style of the more advanced mathematician will in all probability attract attention, even in a more elementary examination.

There is no royal road to algebra. There is at first a certain amount of dulness and drudgery to be got through before the joys of lighter work can be reached, but it must be always borne in mind that factors are of the utmost importance, and it is impossible to make a boy too familiar with them in every form. Factors, identities and simplifications are the backbone of Algebra, as every teacher will agree. Equations and equation problems are another and scarcely less important point, and these should be multiplied almost indefinitely.

Perhaps it may be asked, at what age should algebra be commenced? Well, the answer cannot be given quite in this form; there is no special age, any more than there is a special age for beginning Greek Each should be begun as soon as the boy is ready, and the sooner the better. Any boy that has gone through an elementary course of vulgar fractions should begin algebra at once and he will find no difficulty. It will be a year before he has got beyond the rudiments, by which time he will have made a considerable advance in his arithmetic; but it will be a year's gain, and one that if deferred can never be made up.

It is only necessary to add that as in the elementary arithmetic it was advised to accustom the beginner to easy fractions, so in the four simple rules of algebra it is also of advantage to introduce fractional coefficients and indices, and to vary the form of questions in division, so that there may be occasional remainders. It is unwise to graft the idea into a child's mind that all divisions must necessarily come out exactly. In fact, as a general rule, questions occurring in actual practice do not come out exactly, and the notion that a sum must be wrong because there happens to be a remainder is better avoided in toto.

It is a good plan after one term's algebra to alternate arithmetic and algebra in successive weeks. In this way a more substantial advance can be made than by alternating the lessons, and boys feel that they are making good progress.

EUCLID.

As soon as a boy is old enough to read easily, and to grasp, of course with a reasonable amount of explanation, the definitions of Euclid, he should make a beginning. Euclid, as taught in the present day, is no longer the grim bugbear that it was a generation ago. It can be made, and should be made, a very interesting subject, and one which little boys like immensely. From the very outset, easy problems can be attempted, and many of the definitions will suggest them to the teacher. The very first definition supplies a mine of material when taken in conjunction with those that follow it immediately, and there are plenty more that will suggest themselves, such as the construction of two equilateral or isosceles triangles on opposite sides of the same base, or a square with hinged corners, producing a rhombus, which should be proved as a proposition from the definition of a circle. The first proposition may here be asked as a problem. The axioms give also much opportunity for example and thought, and the eighth can be made eminently easy by the explanation of equality by superposition."

Personally I always proceed at once to the fourth proposition, and it is seldom that there is any difficulty to be encountered. Problems should be worked at once upon this, one of the most useful of elementary propositions, and as a general rule it should be the practice to work problems on every proposition as it is done. As soon as a boy can solve an easy problem for himself, and it is not a long period of waiting, he will find no difficulty in understanding, and therefore remembering, any proposition, and the rest of the course of geometrical training is simply a question of time.

In consequence of the large number of problems that must be worked, if success is to be achieved, the first book takes rather a long time in proportion to those that follow, but it is not time wasted. It is better to confine a first course to quite simple problems, and to leave the more difficult to a second, or even a third course of reading.

Experience shows that in a first term a class of six or eight boys can easily learn thoroughly about six or eight propositions, as well as the definitions, axioms and postulates. It is well not to attempt too many, as the demonstrations must be carefully and exactly mastered—and as has been said above a large number of riders must also be done. During the next term it is not possible to determine any rigid limit, for it is now that the more mathematically disposed will leave the average boy behind. In fifteen minutes preparation one boy will easily do two or even three new propositions, others will find one as much as they can do thoroughly,

*Fuller suggestions on the "Teaching of Elementary Euclid" will be found in a paper contributed by the writer to the Preparatory Schools' Review, March, 1897.

256 The Teaching of Mathematics in Preparatory Schools.

but if it is borne in mind that every proposition successfully grasped is a step on the ladder, and that no steps are of any use at the top intermediate steps are unsound, real progress will be made, even though slow and steady.

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The enunciations and corollaries should always be thoroughly learned by heart, and clearly understood, for it is these that constitute the directions for the way, so to speak, and they are besides the only parts of the bookwork that are quoted in subsequent propositions.

The amount of Euclid that can be learned up to the limit of preparatory age depends entirely on the individual. With the majority three books form an amount that can generally be managed, while the more mathematically disposed will add the fourth and sixth books without much trouble in the same time. It is a good practice to work the greater part of the fourth book as problems, and this is certainly true of the first half of the book. A knowledge of the opening propositions of the sixth book gives a geometrical interpretation of ordinary proportion to which, in analytical form, by this time the boy is well accustomed in both arithmetic and algebra. Two lessons weekly of 50 minutes or one hour will be ample for ensuring a good knowledge. In this will be included preparation by beginners, but in the case of more advanced boys, an allowance of fifteen minutes twice a week in preparation will be of great value. Propositions should be written out neatly, all references put in the right hand margin, and the wording of the text insisted upon.

It is not advisable to attempt the solution of complicated problems; easy work alone is suited to minds of this age. In this way it is manifest that nothing is to be gained by unduly limiting, as some would limit, the number of books to be read, for even though the time that would ensure a good elementary knowledge of six books be devoted to three books only, there remains the incontestable fact that problems are limited to those suitable to the age. Too difficult problems defeat their object; and in a recent scholarship examination, where the candidates, who expected, as usual, six books, were confined to three without notice, the differentiation of the better mathematicians was almost defeated by the fact that the increased difficulty of the problems tended to reduce all to the same level.

CONCLUSION.

To sum up, quality rather than quantity is the essential of good teaching. The aim should be to develop thinking power, and this is best attained by careful explanation being followed by plenty of practical examples, varied as much as the ingenuity of the teacher will permit. Allow no hard and fast rules; let method depend entirely on the interpretation that is to be placed on the definitions; cultivate style, and the result will be the development of a really mathematical mind, as opposed to a memory that is likely to be treacherous in the hour of need.

C. G. ALLUM.

NATURAL SCIENCE IN PREPARATORY SCHOOLS.

Of late years much has been written and said in favour of the more extensive teaching of scientific subjects. And to such an extent has this been the case that some of the advocates of science teaching appear to regard a boy, educated wholly on these lines, and illiterate in every other way, as a desirable product.

But the reaction from this early specialisation is sufficiently strong, in the majority of cases, to counteract the over-zealous advocacy; and in connection with Preparatory Schools there is probably no danger of its occurrence. In their case a more general education is the object, and there is little prospect of a small boy being induced to give so much of his time to science as to interfere with his general education.

In Public Schools the teaching of science has only recently begun to take reasonable shape, and ceased to be a series of fireworks, or isolated physical phenomena, presented in a casual and indigestible manner to the pupil; while there has been so little of it in Preparatory Schools that its past and present state in these institutions does not require any long exposition.

Nevertheless, now that the large number of subjects included under the head of Science are more reasonably taught to elder boys and others, there has arisen a fairly widespread feeling, amongst both parents and schoolmasters, that some elementary information on scientific subjects should be given to boys whilst still at Preparatory Schools, and that these subjects afford valuable material for educating the minds of such boys. To their credit be it said, Board Schools and Girls' Schools have for sometime realised this fact, and in many of them scientific subjects find a place in the curriculum.

In Preparatory Schools the result of this inclination has been that several tentative efforts in scientific instruction have been made, and are still in progress at many of them, though nothing approaching the systematic "nature-study nature-study" of the young American has as yet been achieved.

The following short account seems to represent the various schemes at present in force, and, as will be seen, they appear to afford possibilities of much success with a slight amount of direction and co-ordination.

The practice which has found most favour is probably the occasional lecture. Either one of the staff or a stranger gives a lecture, with or without lantern slides, on some more or less scientific subject.

R

The next place is occupied by Botany of some sort-but, unfortunately, mere Systematic Botany, consisting of the finding and naming of various flowers and weeds, is the rule.

After these two efforts the instruction is of an even more vicarious nature, consisting of scraps tacked on to geography or some other work, ranging from cyclones and thermometers to the distribution of animals. Lastly, in one or two places systematic attempts are made at teaching some given part of Chemical or Physical Science, such as the properties of Air or the Laws of Heat

Now, it will be seen that such attempts as the above, in most cases, are singularly lacking in those essentials which are supposed to constitute good teaching. There is no uniformity, no continuity-in fact, in their nature they too much resemble the "General Information" column of the modern cheap newspaper. And yet some good results have been produced, for these courses have tended to stimulate the mind and improve the reasoning powers of those boys who have had sufficient intellect to select the good from the chaos offered to them. So that for this reason alone one is tempted to consider whether there are not claims for, and advantages in, the teaching of scientific subjects such as to justify their inclusion in the curriculum of Preparatory Schools.

If properly managed, there seems to be little doubt that scientific work tends to truly educate the minds of even quite young boys. Certainly, their powers of manipulation and dexterity are visibly improved by a small amount of practical work entailing the use of their fingers and eyes.

As regards the use of the latter, the difference between a small boy's powers of seeing the features of some given natural object, when he has been taught to use his eyes, and his inability with an untrained eye to see the same things, until they are pointed out to him, is worthy of more than passing notice.

In the same connection this ability to see more leads to a wider range of thought and a greater knowledge of the powers of language for descriptive purposes. Moreover, the powers of reasoning are given fuller play in this manner than in the majority of taught subjects, if it be so arranged that the pupil has to suggest explanations and to arrive at conclusions for himself, subject to the correction of the master.

So that the advocates of this teaching of science would maintain that in the sum the advantages of increased powers of observation and manual dexterity gained from it justify it as a convenient and teachable subject for those ends. This leads to perhaps the most debatable part of the question, viz., the subjects to be taught and the methods to be employed.

Considering the various possibilities in turn, Chemistry, in virtue of its long-standing position, as the subject most taught in Public Schools, naturally suggests itself.

But the teaching of Chemistry involves a considerable amount of apparatus and. a room more or less adapted to the purpose, and it