Archimedes, and Apollonius. Thales is said to have brought it into Greece, about six hundred years before the Christian era; he is also said, which indicates the state of the science at that time, to have himself discovered that the angle inscribed in a semicircle is a right angle, and to have testified his joy by a sacrifice to the Muses! Pythagoras and Plato also are said to have extended the bounds of geometrical knowledge; the former, who lived about 550 B. C., being understood to have discovered that elegant proposition, which still bears his name, respecting the squares described on the sides of a right-angled triangle. He also is said to have expressed his joy and gratitude to the gods, by the sacrifice of a hundred oxen. Whatever may have been the facts in regard to these discoveries and sacrifices, the manner of their mention sufficiently indicates the limited extent of the science at that time, even among the greatest philosophers. Euclid, who lived about 280 B. C., has left us more abundant evidence of the state of the science in his time, in his "Elements of Geome try," a work which held its ground as the principal, almost the only, text-book on the subject for more than two thousand years; until his name became a synonyme for the science, and men spoke of studying, not Geometry, but Euclid. Later still, Archimedes and Apollonius distinguished themselves in the higher departments of mathematical science; Apollonius, particularly, by a most valuable treatise on the Conic Sections. Archimedes," the most profound and inventive genius of antiquity,” is celebrated, not only for his mathematical science, but for his mechanical skill, by which he defended Syracuse, for a considerable time, against the utmost exertions of a Roman army, and for the boast, that, if he had a place on which to fix his lever, he would move the world.* Now, during all this time, and for many centuries after, the knowledge of Geometry was confined to the philosophers, -to the few. Out of the schools of philosophy, among the mass of the community, such science was utterly unknown. The universal diffusion of knowledge, as of all other blessings, * Δὸς ποῦ στῶ, καὶ τὸν κόσμον κινήσω. is the suggestion of Christianity. The properties of the circle and the triangle, at whose discovery Thales and Pythagoras are said to have been so elated, are now known to the tyro. Pascal, at the age of sixteen, composed a treatise on the Conic Sections, in which he gave, in a single proposition and four hundred corollaries, all that had come down from Apollonius, "the Great Geometer" of antiquity. And what one boy of sixteen may write, another boy of sixteen may learn. We believe that Geometry, instead of being confined, as formerly, to philosophers, or, as more recently, to an educated class, will, at a day not far distant, be introduced into the common schools all over our country, and brought within the reach of every boy and girl in the community. This is the proper sphere of the common school; not to communicate a fixed amount of knowledge, the same to our children as to our fathers, but to communicate continually additional knowledge, and to produce higher and higher degrees of intelligence; when men of science extend the bounds of knowledge, to diffuse that knowledge, till the world enjoys its benefits. This universal and progressive diffusion of knowledge constitutes the proper sphere of the common school. But if Geometry is to be studied in our common schools, in what shape shall it be presented? What principles should direct in the preparation of a text-book for this purpose? 1. The definitions should be perfectly clear and exact. A definition should equally avoid excess and defect. It should express neither too much nor too little. It should be so full as perfectly to identify the object defined, but should not include properties the possibility of whose combination is yet to be proved. Thus, Legendre's definition of the circumference of a circle as 66 a curved line, all the points of which are equally distant from an interior point, called the centre,' "'* does not * "La circonference du cercle est une ligne courbe, dont tous les points sont également distants d'un point intérieur qu'on appelle centre. identify the circumference of a circle; but is equally applicable to any line whatever drawn upon the surface of a sphere. It should be defined, "a curved line in a plane, all the points, &c." 66 Again, when, before any demonstration of the properties of quadrilaterals, the square is defined as a quadrilateral, which has all its sides equal, and all its angles right angles"; the rectangle, as one "which has its opposite sides equal, and its angles right angles"; the parallelogram, as one "which has its opposite sides and angles equal," or "its opposite sides equal and parallel," are examples of excess in definition. that a quadrilateral can have, at the sides equal and parallel, or its opposite sides and angles equal, or its opposite sides equal and its angles right angles, or all its sides equal, and its angles right? - these How do we yet know same time, its opposite Clearness and simplicity in the definitions are promoted by a natural order of succession. Of the quadrilaterals, for example, the square is, in many of the books, defined before the rectangle, and the rectangle before the parallelogram, that is, the species before its genus. Whereas the true order of science requires us to proceed from the more to the less general, adding, at each step, only the necessary limitations or specifications. An object should be defined by means of that distinguishing property, from which its other properties may be most easily and satisfactorily deduced. The fact that parallels never meet seems to be less their distinguishing property, than a consequence of some other property. This property furnishes no convenient means of drawing parallels, nor any practical test of parallelism. It is much more satisfactory and convenient to define them as having the same direction. 2. The propositions should be enunciated with the utmost precision. A defect in this respect sometimes amounts to a gross error. Thus, in some of the books, we find this prop "Le cercle est l'espace terminé par cette ligne courbe.”— Éléments de Géométrie, 11me Éd., p. 33. 66 = osition. "If the product of two quantities be equal to the product of two other quantities, two of them " (any two, of course) may be made the extremes, and the other two the means, of a proportion"; e. g. 4 X 8 = 2 × 16; then, making 4 and 2 the extremes, we have 4 : 8 = 16:2; or, as the product of the extremes is equal to that of the means, 4 X 28 X 16, or 8 = 128. It should be, the two factors of one product may be made the extremes, &c. Nor is this a mere captious objection. I have repeatedly known students to make the mistake, and find it out only by trying to verify the result. Again, a good enunciation distinguishes, and marks the distinction with great care, between hypothesis and conclusion. 3. The most rigorous exactness of demonstration must be preserved. We want no tentative or experimental methods of proof. Empiricism is as bad in mathematics as in medicine. We want no practical results to be learned by rote, without proof. We must have proof,- infallible proof, demonstration. The reasoning may be simplified, and reduced to the comprehension of the young, by multiplying and shortening the steps, if need be; but still it must be demonstration. 4. The memory is aided, fresh interest awakened, and the whole mind invigorated, by the generalization of geometrical truths; a process which connects under one enunciation several apparently distinct propositions, and shows them to be only particular applications of a more general principle, — specific forms of a generic truth. 5. A book designed for elementary schools should abound in minute and familiar illustration, both of terms and principles. The definitions and propositions should be expressed as concisely as possible, so that they may be easily remembered, and conveniently quoted. But the terms used should be explained with great care, and such remarks added as will connect and show the relation between the rigorous expressions of Geometry, and the looser language of ordinary con versation. The practical application of principles should be set forth, not to strengthen the proof of a proposition,- for the infallible nature of demonstration, and the impossibility of increasing its certainty by additional evidence should be constantly insisted on, - but rather to illustrate the meaning of the abstract principle, to show how readily it applies itself to practical results, and connects itself with common things, and so, at the same time, to aid, to interest, and to benefit the pupil. It may be thought, that the teacher should supply the necessary illustrations. He should, indeed, so far as his time permits; but yet, for various reasons, besides the want of time, the burden should not be wholly thrown upon him. In the first place, many will be called upon to teach Geometry who are not particularly interested in it; some, perhaps, who are not very familiar with it; and some, possibly, like a teacher I once knew, who thought that "Euclid and English Grammar could be learned only by committing them to memory." To such teachers, illustrations, unless suggested by the book, will not be likely to occur. Another difficulty will be experienced by the most accomplished teachers. Explanations, if first suggested during the recitation, will not generally be appreciated or remembered. The pupil should study them with his lesson; he should reflect upon them, and see for himself their connection with the subject. He will then be prepared to appreciate any additional remarks from his teacher, and to ask intelligent questions of his own. The more abundantly illustrations are furnished by the book, the more readily will additional illustrations occur to both teacher and pupils. It may not be out of place here to remark, that a difficulty, of far greater magnitude, indeed, but of the same nature, is occasioned by faulty and inelegant definitions and propositions. If the fault be pointed out before the lesson have been learned, the subject is strange, and the correction not understood or remembered; if afterwards, the faulty expressions will then have been fixed in the mind, and cannot easily be eradicated. "But after all this simplification, illustration, and improve с |