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work, especially with regard to the position of the decimal point. The use of 5.204X105 for 520400, and of 5.204X103 for .005204. The meaning of a common logarithm; the use of logarithms in making calculations involving multiplication, division, involution and evolution; calculation of numerical values from all sorts of formulæ however complex.

The principle underlying the construction and method of using a common slide rule; the use of a slide rule in making calculations. Conversion of common logarithms into Napierian logarithms. of square roots by the ordinary arithmetical method. formulæ in working questions in ratio and variation. fractions. Calculation of percentages, etc.

The calculation Using algebraic Simplification of

Algebra. To understand any formula so as to be able to use it if numerical values are given for the various quantities. Rules of indices. Being told in words how to deal arithmetically with a quantity, to be able to state the matter algebraically. . . . Problems leading to easy equations in one or two unknowns. Easy transformations and simplifications of formulæ, and in easy cases finding any one of several quantities in a formula when the others are given. . The determination of the numerical values of constants in equations of known form when particular values of the variables are given. The meaning of the expression 'A varies as B.' Factors of such expressions as x2-a2, x2-11x+30, x2-5x-66."

From the paragraph on mensuration I can quote only a few statements, showing how Professor Perry would use experimental methods to test the accuracy of rules. Thus, "Testing experimentally the rule for the length of the circumference by using strings round cylinders, or by rolling a disc or sphere. Inventing methods of measuring the lengths of curves. Testing rules for the areas of a triangle, parallelogram, etc., by use of scales and squared paper." The determination of the areas of irregular plane figures by five different methods, including the use of Simpson's rule and the planimeter. . "Rules for volumes of prisms, cylinders, cones, spheres and rings, verified by actual experiment; for example, by filling vessels with water or by weighing objects of these shapes made of material of known density, or by allowing such objects to cause water to overflow from a vessel. Stating a mensuration rule as an algebraic formula, etc."

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"Use of Squared Paper.-The use of squared paper by merchants and others to show at a glance the rise and fall of prices, of temperature, of the tide, etc. The use of squared paper should be illustrated by the working of many kinds of exercises, but it should be pointed out that there is a general idea underlying them all." Among others he mentions such as the following: "Plotting of statistics of any kind whatsoever, of general or special interest. What such curves teach. Rates of increase.

Interpolation. . Probable errors of observation. The calculatation of a table of logarithms. Finding an average value. The method of fixing the position of a point in a plane. Plotting of functions, such as y=ax", y=ae"x, for various values of a, b, and n. The straight line, meaning of its slope, slope of a curve at any point in it.

Determination of maximum and minimum values. The solution of

equations."

"Geometry.-Dividing lines into parts in given proportions, and other experimental illustrations of the sixth book of Euclid. Measurements of angles in degrees and radians. The definitions of the sine, cosine and tan

gent of an angle; determination of their values by graphical methods; setting out of angles by means of a protractor when they are given in degrees or radians, also when the value of the sine, cosine or tangent is given. Use of tables of sines, cosines and tangents. The solution of a rightangled triangle by calculation and by drawing to scale. The construction of a triangle from any given data; determination of the area of a triangle. The more important propositions of Euclid may be illustrated by actual drawing; if the proposition is about angles, these may be measured by means of a protractor; or if it refers to the equality of lines, areas or ratios, lengths may be measured by a scale, and the necessary calculations made arithmetically. This combination of drawing and arithmetical calculation may be freely used to illustrate the truth of a proposition. A good teacher will occasionally introduce demonstrative proof as well as mere measurement." Then follow some elementary uses of analytic geometry of space and descriptive geometry.

In the very interesting discussion which followed the reading of the paper and the presentation of the syllabus decided objection was made to the strongly utilitarian tendency of a large part of Professor Perry's remarks. Professor Forsyth, in particular, said: "I must point out what would be a platitude if we were not in discussion, that scientific subjects do not progress necessarily on the lines of direct usefulness. Very many of the applications of the theories of pure mathematics have come many years, sometimes centuries, after the actual discoveries themselves. The weapons were at hand, but the men were not ready to use them. Take the case of medicine, which surely is a practical subject. It owes immense debts to the study of sciences like physiology and bacteriology; yet these have been developed and continue to be developed, along their own lines, without being guided in the direction of immediate application at every turn. Yet independent as has been their development, it is notorious that, perhaps all the more because of their freedom in growth, they have provided new knowledge that is of the utmost importance in the conduct of living processes. Take one last example, the X rays. If any one had been set down, as a practical problem, to take a photograph through solid things, I think the common answer would have been that he was being told to solve an insoluble problem. Yet its solution came from the physicists, indirectly as it were, in the course of researches made to obtain knowledge for its own sake. The knowledge so obtained has subsequently led to wonderful results in its application. Influenced by these examples and by others more directly mathematical upon which I shall not enter, I must decline to accept utility as the main or the sole discriminating test, either in the study or the teaching of mathematics."

Had I the time I should like to quote from the remarks of several of the other participants, but I shall have to content myself with a few selections.

Lord Kelvin wrote: "I am overdone with work which must not be postponed, and I am sorry therefore not to be able to write anything on the subject. I think your syllabus was good indeed. It is very like the teaching I had from my father."

Sir John Gorst, chairman of the joint session, told a unique experience he bad in New Zealand in his younger days. He said: "I taught, or attempted to teach, mathematics to the Maori boys and men. As far as the teaching of arithmetic went I taught on a sort of embryo Sonnenschein

principle, and I found them remarkably apt and quick pupils. They learned the practical arithmetic, which was useful to them in actual life, and they learned it with extraordinary rapidity- far faster than boys or men would generally learn it in this country. But when in my youthful enthusiasm I proceeded to try to teach some of them Euclid, or rather geometry after the Euclid fashion, I absolutely and entirely failed. There was not one of them that could grasp or understand the simplest of the propositions of Euclid. . Had I had the advantage of the discussion to which I have listened to-day, I should have abandoned teaching in the ordinary way until they had been familiarized with angles, lines, areas, and geometrical figures, of which the Maori youth was absolutely ignorant. I suppose by a method of that kind even the least developed intellect of the uncivilized native of New Zealand might have been brought to take in some of the very simple propositions of geometry."

Professor Everett said: "The teaching of geometry has been too prosaic. The minds of boys and girls are not ripe for dealing with abstractions. The way in which Euclid begins (especially if the whole body of definitions is taken first) gives the learner the impression of a castle in the clouds.

A moderate amount of practical geometry should come first, including methods of bisecting angles and lines, drawing lines at right angles, making a triangle with sides of prescribed lengths, and inscribing a regular hexagon in a given circle. This will give the learner definite conceptions, and help him to feel that he is on solid ground.

Side by side with Euclid, or a substitute for Euclid, verification by actual measurement of carefully drawn figures should be encouraged. It is useful as a test of the accuracy with which measurements can be made by the methods employed, and also useful as a check against mistakes-which are liable to be made in abstract reasoning as well as in other matters. One of the most important habits in scientific investigation of all kinds is the habit of testing the correctness of one's conclusions by independent methods; and this habit should be inculcated by assiduous practice, as an important element in personal character-an element inseparably associated with the honest pursuit of truth.

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The learner should be taken on, as quickly as is consistent with intelligent progress, to the higher branches of mathematics.

The elementary conceptions of the infinitesimal calculus and its simpler processes, should be introduced at an early stage in mathematical teaching.

Another subject that is too long postponed is solid geometry. It is postponed so long that most boys do not get it at all. Considering that we live and move in space of three dimensions, it is unreasonable and impractical to confine all accurate thinking and teaching for three or four years to two-dimensional space. The result is to produce an instinctive shrinking from three-dimensional thinking, as if it involved some terrible mystery." I wonder what Professor Everett would think of the method now coming into vogue in Italy of teaching plane and solid geometry together from the beginning! Professor Everett's remarks are so good throughout that I should like to quote further, but time will not permit. I will hasten on to Professor Perry's summing up of the discussion. He says: "We who have taken part in this discussion have been criticized hy some educationists because we have only been expressing well-known educational truths. They forget that, however well-known these truths

may be, they have never yet-never till now-been expressed publicly by more than two or three mathematical teachers. They forget that a reform in the teaching of mathematics was absolutely impossible without the consent and advice of the mathematicians.

It will be found that my syllabus contains almost all the new suggestions which were made by speakers who had no time to study it. (1) Experimental geometry to precede demonstrative. (2) Some deductive reasoning to accompany experimental geometry. (3) Mathematics to enter into the experimental science syllabus as much as possible. (4) Rough guessing at lengths. weights, etc., to be encouraged. (5) Recognition of the incompleteness of any external examination. (6) The importance of familiarizing a boy with problems in three-dimensional space. (7) A hard and fast syllabus undesirable; even the sequence of subjects to be left to a good teacher's initiative.

Further on Professor Perry rather nonchalantly says: "On the whole, I think it may be said that I am in accord with every one of my critics, but of course I know that they cannot unreservedly agree to the adoption of my syllabus as it stands for every kind of student. At all events it is quite evident that there is unanimity in the desire for an immediate large reform in the teaching of mathematics.

I have long known that there is this unanimity among educationists generally, but it is unexpected to find it among the great mathematicians, and the most important teachers of mathematics. I take it that we are all agreed upon the following points:

1. Experimental methods in mensuration and geometry ought to precede demonstrative geometry, but even in the earliest stages some deductive reasoning ought to be introduced.

2. The experimental methods adopted may greatly be left to the judgment of the teachers; they may include all those mentioned in the elementary syllabus which I presented.

Some of the things for which I contend were put so prominently forward that, if speakers did not object to them specifically, they may almost be taken as agreed to. They are such things as these that follow. of them are agreed to specifically by about half my critics.

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3. Decimals ought to be used in arithmetic from the beginning. 4. The numerical evaluation of complex mathematical expressions may be taken up almost as part of arithmetic, or at the beginning of the study of algebra, as it is useful in familiarizing boys with the meaning of mathematical symbols.

5. Logarithms may be used in numerical calculation as soon as a boy knows that anXa"=a"+", and long before he is able to calculate logarithms. But a boy ought to have a clear notion of what is meant by the logarithm of a number.

6. In mathematical teaching, a thoughtful teacher may be encouraged to distinguish what is essential for education in the sequence which he employs, from what is merely according to arbitrary fashion, and to endeavor to find out what sequence is best, educationally, for the particular kind of boy whom he has to teach.

7. Examination cannot be done away with in England. Great thoughtfulness and experience are necessary qualifications for an external examiner. It ought to be understood that an examination of a good teacher's

pupils by any other examiner than the teacher himself is an imperfect examination.

I have not much doubt as to the unanimity with which everybody may be said to have agreed explicitly or implicitly to all the above statements. About these that follow I am in more doubt. More than half my critics will, I believe, agree to them all for all students. I think that every one of my critics will agree to allow a judicious teacher a free hand, especially when he knows that his pupils are likely to need the use of mathematics in their other studies, and especially if they are likely to become engineers, -i. e., men who apply the principles of natural science in their daily work. 8. A thoughtful teacher ought to know that by the use of squared paper and easy algebra, by illustrations from dynamics and laboratory experiments, it is possible to give to young boys the notions underlying the methods of the Infinitesimal Calculus.

9. A thoughtful teacher may freely use the ideas and symbolism of the calculus in teaching elementary mechanics to students.

10. A thoughtful teacher may allow boys to begin the formal study of the calculus before he has taken advanced algebra or advanced trigonometry, or the formal study of analytical or geometrical conics, and ought to be encouraged to use in this study not merely geometrical illustrations, but illustrations from mechanics and physics, and illustrations from any other quantitative study in which a boy may be engaged."

Since Professor Perry's report was published the discussion has gone merrily on. By invitation of a member of the British Association committee some twenty-two teachers in prominent English public schools sent in a sketch of the changes they would like to see made. In the treatment of geometry they are of opinion:

"1. That the subject should be made arithmetical and practical by the constant use of instruments for drawing and measuring.

2. That a substantial course of such experimental work should precede any attack upon Euclid's text.

3. That a considerable number of Euclid's propositions should be omitted, and in particular

4. That the second book should be treated slightly and postponed till III, 35 is reached.

5. That Euclid's treatment of proportion is unsuitable for elementary work."

As to arithmetic, they think it "might well be simplified by the abolition of a good many rules which are given in text-books. Elaborate exercises in vulgar fractions are of doubtful utility; the same amount of time given to the use of decimals would be better spent. Four

figure logarithms should be explained and used as soon as possible. It is generally admitted that we have a duty to perform towards the metric system. This is best discharged by providing all boys with a centimeter scale, and giving them continual exercise in verifying geometrical propositions by measurement. Probably it is right to teach square root as an arithmetical rule. Cube root is harder and should be postponed until it can be studied as a particular case of Horner's method of solving equations approximately."

Passing to algebra, we find that a teacher's chief difficulty is in the tendency of his pupils to use their symbols in a mechanical and unintelligent way. Elementary work in algebra should be made as far as

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