lum. A girls' school without lessons in the use of the needle could scarcely have been regarded otherwise than as something of a monstrosity, because wanting in that strictly womanly element which should mark it as a school for the gentler sex. These sug

gestions in respect to the spirit of "ye olden time" are dwelt upon to give pertinence to the evolution that has taken place in one brief century.

The eagerness with which girls devoted themselves to the acquisition of book knowledge, when they were given the opportunity to go beyond the scant pastures of a primary education, seems to have occasioned general alarm. Even learned men feared the consequences of learning in women. Nor were their facilities for anything like a higher education gained without a most painful struggle. The epithet "blue stocking" implied almost every unwomanly characteristic, and the possibility of almost every action from which a modest woman shrinks. Women to become learned in the "good old days," had to impale themselves upon the spikes of public opinion, or in milder terms, to endure a certain contempt, and even something like ostracism for their heroism. Educated women are now becoming so common that a few years hence a girl without a diploma of some kind, will probably be something of a phenomenon.

Woman's prompt, and, as a rule, good use of her facilities for education prove her innate love of learning. Her intellectual ability has been placed beyond question by the many learned women which this century has produced. Her greatly increased usefulness, especially since women's colleges and girls' schools of a high order have been founded, demonstrates, at least by implication, the great worth of the much talked of "higher education" to woman's character and influence.

Yet not a few contend that because of the usually different life duties of the sexes, their mental training should differ. There is some cogency in such reasoning, if we look upon education simply as a preparation, in a purely material, or technical sense, for the distinctive duties of men, and of women; that is, to make men bread-winners, and women home-keepers.

Doctor Lord, whose beautiful lectures really show an old-fashioned, chivalrous regard for women, says, "A woman should be educated to be interesting," "useful at home," etc. "She should be taught to become the friend and help-mate of man, never his

rival." He deplores the fact that women are sometimes forced to adopt the callings of men, and to prevent this catastrophe he advises all women "to pursue some one art-like music, or painting, or decoration," "for proficiency in these arts belongs as much to the sphere of women as to men, since it refines and cultivates them." Are these words what many will feel tempted to call them, merely the expression of obsolete ideas? By no means. Within a stone's throw of any one who reads this essay, it may be said most confidently, will be found persons with still more backward visions than Doctor Lord's.

It is a commonly, perhaps a usually accepted view, that the chief end of a woman's school education is refinement; that of a man, practical utility. How else can we account for the practice in many "well-to-do" families of sending the girls to high schools and colleges, while the boys are permitted to grow up in a great degree uneducated? For what other reason are art and music, or accomplishments, still thought more suitable for girls than for boys? Why else are girls still allowed to neglect solid, intellectual acquirements for the sake of these so-called accomplishments?

Ideal education is the development of the individual, and no doubt the best results can be attained only by the individual training of every boy and of every girl. Until the millennium and the perfection of all things come a little nearer, however, children will of necessity be educated in masses, and natural aptitudes can be only in a measure considered. But while an ideal standard in methods is so slowly approached, as to seem sometimes little more than a fair dream of a far distant future, it is consoling to know that the greatest aim of all systems of education is, after all, the formation of character. With this aim in view, it is difficult to understand why the question of sex should enter into education.

Is it not just as important to boys as to girls to be gentle, thoughtful, tender, and virtuous? Is it not as important to girls as to boys to be honest, prompt in keeping engagements, self-help. ful, and useful? Why should not the study of the classics give the same fine literary tastes, deep culture, and peculiar mental development to the one sex as to the other? Why should not the same discipline of mind, and development of practical sense, accrue alike to both sexes from the study of mathematics? Why should not girls as well as boys be given the benefit of lessons, so important to life, and acquire the same habits of accuracy, to be learned

from the thorough study of natural science, in all of its branches? Why, again, should not boys, whose usual life experiences make the saving influence of personal refinement peculiarly necessary, be taught music, French, drawing, decorative art, and other things supposed to be so important in the education of a refined young lady?

The only good objection to a similarity of teaching for both sexes, ever brought forward, is that girls have not the requisite strength for a thorough collegiate education; yet this objection is a sentimental one, and has no real existence in fact. It is controverted daily by experiences requiring of women the utmost physical endurance, which is, it seems superfluous to state, the special kind of strength necessary for the acquisition of book-knowledge.

THE TEACHING OF MATHEMATICS.1

VII.

THE FUTURE OF GEOMETRY TEACHING.

BY GEORGE WILLIAM EVANS, ENGLISH HIGH SCHOOL, BOSTON.

THE

HE aim of this article is not to present a novel or original plan for the teaching of this most important subject, but rather to note the attempts that have been made to change the methods now in vogue, and to remark upon the general tendency of such changes.

The recognized textbook throughout the civilized world is Euclid's Elements. American teachers have for the most part adopted the modifications of Legendre's School; English teachers aim at a similar result by appendices and notes to "the first six books with portions of the eleventh and twelfth." While English mathematicians and mathematical schools hold the rank that they hold at present, one certainly cannot say that experience has shown Euclid to be a less efficient textbook than Legendre. There is an advantage in having a standard numbering for classic theorems, and there is even some little also in having the Euclid diagrams,

1 Copyright, 1888, by Eastern Educational Bureau.

letters and all, available for reference in briefly summarizing or illustrating an argument or demonstration; although it may be said on the other hand, that when teaching dwells on numbering and lettering, and set diagrams, the pupil will be better able to pass examinations than to understand his own words. Geometry from such a standpoint is too much like classicism from the standpoint of the grammar and dictionary, it is laboriously correct,

but it is very dead.

An offset to rote learning is the great number of "riders". original exercises-set by English teachers. American books have followed this, too. The later books recognize the value of numerous easy exercises from the beginning, as an aid to developing the power of the student, while the earlier attempts in this direction aimed rather to test his power after it was developed.

It is natural to suppose that a textbook which met the views of the Greeks at Alexandria two thousand years ago would fail when tested by modern requirements; and that, too, not merely in lacking what has since been accomplished, but in conforming to the warped standards of a primitive development. One must wonder, then, that so little has been changed in it. We must own, too, that conservatism in this matter smacks somewhat of the Aristotleworship, so obstructive during the Middle ages. Euclid wrote thus and thus, and if it did not seem good to such or such an unquestioned authority, why did not he change it? This has been a successful means of instruction; why "fly to others that we know not of"?

Still, textbooks multiply, each with its own little attempt at repair.

A fault that has been tinkered with in our textbooks is the arrangement of the theorems, and their grouping in books. The remedy has led to a singular anomaly. Proportional lines and similar figures are treated of before areas, and, by consequence, products and squares of lines are introduced before their geometrical raison d'être. The answer to this objection is, of course, that linear magnitude is subject to algebraic laws, and any result of their application is valid so far as it can be geometrically interpreted. But such an answer confutes the argument for making proportions any part of a geometrical treatise; and all objection to the "algebraic" proof, that bugbear of pedagogues, must also go. It is a far-reaching answer, but it is a good one, as can readily

be shown. The entire system of the science of algebra rests upon three well-known laws,1 namely: -

abba (the commutative law),

a. bcab. c (the associative law),

a (b+c)=ac+ae (the distributive law),

and these laws are axiomatic for geometry also. Any expression involving geometric magnitudes may therefore be transformed according to the rules of algebra, without invalidating any argument which may be based upon such a transformation: because it is equivalent to a repetition or combination of the transformations implied in the fundamental theorems cited above. As for the geometric ratio, upon which the last stand is made, that is, of course, not a quotient of a line by a line: but it certainly is a quotient of their numerical measures. The limitations of arithmetic only forbid us to make divisor and quotient both concrete: quibbles aside, if any be raised, we actually do divide the numerical measure of one concrete quantity by the numerical measure of another as often as we figure the weight of a tub of butter from the price; and a geometrical ratio is not widely different from such a quotient. The incommensurability of the ratio is beside the mark, but even there the algebraic ratio meets all the require

ments.

Peirce's Geometry2 occurs to me as a book which recognizes this difficulty and discreetly puts the proportion theorems into an algebraic introduction. The same book has another feature worth noticing that is the introduction of the idea of direction. It is remarkable that so fundamental an idea as this, and one which contributes so much to brevity and clearness in demonstration. should be exiled from the textbooks. The influence of tradition

is to be seen here; and there are also criticisms of the term itself as well as of the simpler definitions growing from it. The conception of an angle as "difference in direction" has been particularly objected to; perhaps because difference implies subtraction, and there are no quantities resident in lines that by subtraction would give an angle. Perhaps to define an angle as "the change in direction from one straight line to another" would be more satisfactory, if it seems worth while to meet objections of this sort.

1 Chrystal's Algebra, Part I., Page 20.

2 An elementary treatise on plane and solid geometry, by Benjamin Peirce. Boston: 1866.