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the operations involved in it, then, and not till then, it will be of advantage to the child to introduce him to those signs, by which he will be enabled to abridge his proceedings, in cases where he has no other object than to arrive at the result by the shortest way. We have not thought it necessary to say any thing concerning the mode of introducing the pupil to the knowledge and use of the different arithmetical and algebraic signs, the representations of known and unknown quantities, because the rules by which they are worked, are to be found in every work on the subject, and because we are perfectly sure that they will not offer the least difficulty to a teacher, who has with only a tolerable degree of ability and attention initiated his pupils in the nature of number according to the plan proposed by us. All that we have to add, therefore, is, that for the application of the laws of number to practical purposes, such questions ought to be selected, as are founded upon data, in themselves interesting and instructive, such as will relieve the pupil from the dulness of dead ciphering. The different sciences present inexhaustible treasures of this kind, and if we ever find leisure to publish a manual of number, we shall not fail to add so essential an appendage.
Method of Teaching Form ;-Geometry and Drawing.
We have determined upon connecting these two subjects together in one chapter, because the remarks of Pestalozzi, which we wish to bring under the notice of our readers, apply to them both as comprehended under the head form. He subdivides that head, it is true, into three sections, “the art of measuring,” “the art of drawing,” and “the art of writing;” but, abstracting from the latter, which has already found its place in our arrangeinent, the two former are so intermingled in his view, that he says as much on measuring in the section on drawing, as he says on drawing in the section on measuring. This arises from his attention not being properly directed to the distinction between real and apparent form, the one falling under the province of geometry, and the other of perspective. To him there was no other difference between measuring and drawing, than that which exists between the first and second step of the same operation. Measuring he considered as the art of apprehending, and drawing as that of representing, correctly the outline of any given object; but it did not strike him, at least not forcibly, that the outline of an object, such as it appears to the eye, and is represented on paper, is a very different outline from that which forms the subject of investigation in geometry. Notwithstanding the want of clearness on that particular head, the following remarks will not be read without interest:
"It is obvious, but altogether overlooked in general, that practical facility
MEASURING, THE FOUNDATION OF DRAWING.
in measuring things ought to precede every attempt at drawing; or, at least, that we can draw successfully so far only as we are capable of measuring. The common mode of proceeding, on the contrary, is to begin with an incorrect view, and a crooked representation of the object; to expunge and draw again, and to repeat this tedious process, until by degrees an instinctive sort of feeling of the proportions is awakened. Then, at length, we proceed to what we ought to begin with, viz. measuring.
“Our artists have no elements of measure; but by long practice they acquire a greater or less degree of precision in seizing and imitating outlines, by which the necessity of measuring is superseded. Each of them has his own peculiar mode of proceeding, which, however, none of them is able to explain. Hence it is, that if he comes to teach others, he leaves his pupils to grope in the dark, even as he did himself, and to acquire, by immense exertion and great perseverance, the same sort of instinctive feeling of proportions. This is the reason why art has remained exclusively in the hands of a few privileged individuals, who had talent and leisure sufficient to pursue that circuitous road. And yet the art of drawing ought to be an universal acquirement, for the simple reason that the faculty for it is universally inherent in the constitution of the human mind. This can, at all events, not be denied by those, who admit that every individual born in a civilized country has a claim to instruction in reading and writing. For let it be remembered, that a taste for measuring and drawing is invariably manifesting itself in the child, without any assistance of art, by a spontaneous impulse of nature; whereas the task of learning to read and write is, on account of its toilsomeness, so disagreeable to children, that it requires great art, or great violence, to overcome the aversion to it which they almost generally evince; and that, in many instances, they sustain a greater injury from the means adopted in gaining their attention, and enforcing their application, than can ever be repaired by the advantages accruing to them from the possession of those two mechanical acquirements. In proposing, however, the art of drawing, as a general branch of education, it is not to be forgotten, that I consider it as a means of leading the child from vague perceptions to clear ideas. To answer this purpose it must not be separated from the art of measuring. If the child be made to imitate objects, or images of objects, before he has acquired a distinct view of their proportions, his instruction in the art of drawing will fail to produce upon his mental development that beneficial influence which alone renders it worth learning.”
No one that has seen the drudgery and bad taste of common drawing lessons, or has attempted to penetrate the mysteries of perspective by the aid of our "standard works” on that subject, will deny the truth of these remarks; and as Pestalozzi's account of his own mode of proceeding in the
PESTALOzzi's COURSE OF MEASURING AND DRAWING.
joint-instruction of measuring and drawing is very compendious, we may venture to insert it at full length.
“The pupil,” he says, “is first made acquainted with the straight line, by itself, in the various positions in which it can be placed, and the different views that can be taken of it; he is taught to denominate it accordingly as a perpendicular, an horizontal, a slanting line, and the latter as slanting upwards and downwards to the right and to the left. Two lines are then placed parallel with each other, and by varying their position he learns to distinguish perpendicular parallels, horizontal parallels, and different sorts of slanting parallels. The next step is to place two lines converging, so as to form an angle, and he has again to learn the distinction of right angles, acute angles, and obtuse angles. After this the square is laid before him, and divided into halves, fourths, sixths, &c.; the circle is drawn next, with its oblong modifications, and these likewise are divided in a variety of ways.
“All this is to be done, as an exercise for the eye, without having recourse to mathematical instruments, and the following names are to be learned along with the respective figures and their divisions: the square, the horizontal, and perpendicular rectangle; the curve, the circle, the semicircle, the quadrant, first oval, second oval, third oval, fourth oval, &c. halves of the ovals, quarters of the ovals, &c.
“This being accomplished, the child is to be introduced to the relative proportions of these forms, and to learn to use them for the purpose of measuring. To this the mother's book contains preparatory exercises, as a variety of objects are there presented to the child's view, exemplifying their outlines the square, the rectangle, the circle, the oval, &c. After this the different figures of the alphabet of forms are put into his hands, cut out of cardboard, with their names attached to them, in order to render him familiar with each particular form, and to enable him to institute comparisons.
“The next step is to make the application of that knowledge of language and number, which the pupil has acquired by the course prescribed in the mother's manual, to the combination of the different figures of the alphabet of forms, and the determination and expression of their relative numerical value.
“This is to be followed by the exercise of drawing himself the different figures, which will not only render his idea of them more clear and distinct, but also give him a practical ability in the general elements of drawing. This must be connected with exercises of language on the proportions of the different figures; for instance, the height of this perpendicular rectangle is twice its breadth; the length of this horizontal rectangle is twice its height, and so on through all the figures and their divisions. This presupposes, of course, that they should all be executed upon one fundamental scale, and that the divisions should be so made as to affor a medium of comparison
PESTALOzzi's COURSE OF MEASURING AND DRAWING.
for the most dissimilar figures. In this course the attention is also to be directed to the different directions of the lines, and the nature of the angles arising out of their combination, as well as to the relation between the circle and oval, their different sections, and the parts of the square or rectangle in which they are enclosed.
“By these progressive exercises, the intuitive faculties are developed in conformity to the laws of form, or what means the same, educated in the art of measuring, which, as an elementary preparation, ought to precede the usual methods of drawing. Every child is thus enabled, by the simplest means that can be imagined, to form a correct idea of the outline, and the position of any object in nature, and to express his view of it in precise terms. He has the means of comparing, not only the different dimensions of every outline that occurs to him, with each other, but also the whole outline with the square, the circle, or their essential divisions and modifications, so as to determine its deviations from the standard form by the nature of its angles and curves. The alphabet of forms, moreover, furnishes him with terms, by means of which he may clearly describe such deviations. The further cultivation of the art of drawing, of which this course only contains the first rude attempts, leads to a corresponding progress in the art of measuring, by which the pupil will at last acquire the greatest facility in determining the proportions even of the most complicated objects, without having recourse to the actual process of measuring.
“It is hardly credible to what degree of mental development this proceeding leads even children of middling capacities. On this subject I will not be called a visionary. I have taught children upon this plan, and my theory is nothing else than the result of my successful experiments. “Come and see.' My children are not, it is true, much past the threshold of this method; but the short progress they have made is so decisive, that it requires a peculiar turn of mind to watch my pupils, and yet to resist conviction. And this is, after all, but very natural.”
In pursuance of the plan which we have laid down for ourselves, we shall now proceed to furnish our readers with some details illustrative of the general principles on which the instruction in geometry should be founded; excluding, from reasons which we have stated at the beginning of this chapter, the subject of drawing for the present. In order to have a perfectly clear view of his task, the teacher of geometry should bear in mind, that the objects of his instruction must necessarily be presented in a double aspect. In arithmetic, the science of calculable quantities, he had only to do with numerical proportions; but in geometry, which, taking the