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CURVILINEAR PLANES AND BODIES.
At what part of that distance is the point b placed?
If I draw the line fg, as it appears at the distance of the point b, how much must I reduce its length.
Answer: to one third.
To draw the line di, as it appears at the distance of the point c, by how much must I reduce its length?
Answer: by one fourth, &c.
Being questioned in this manner, the pupil will soon perceive the analogy between the distance and the decrease of length, and without much difficulty discover the general rule, that in order to ascertain the apparent length of a line, you must divide its length by the quotient, obtained by dividing its real distance by the distance of the point at which it is drawn; or, divide its length by its distance, and multiply the quotient by the distance of the point at which it is drawn. In a similar manner, the ratio of decrease of the two sorts of horizontal lines ought to be investigated, which being done, the pupil will be able to draw any rectangular body in what is termed parallel perspective,—that is, the front of the body being at right angles with the line of sight,—with mathematical correctness, even without having the body placed before him, simply from the data of its dimensions, distance, elevation, &c.
After this no farther difficulty will be experienced, as every oblique line lies either between the angular points of a square or parallelogram, or between those of a cube or parallelopiped; and by the same means it is easy to determine any
number of points which may be deemed necessary for the perspective of curves and curvilinear bodies. If, for instance, a cone be given, all the pupil has to do, is to imagine the square which would enclose its basis, and erect upon it, with the front parallel to the horizon, a parallelopiped, whose sides would be equal in height to the sectional height of the cone. The square of the base being then drawn in its perspective appearance, and the perspective centre of the top square determined,
the former will give the ellipse which forms the perspective base of the cone, and a straight line drawn from the centre of the top square, to each end of the longest diameter of that ellipse, completes the perspective outline. It is not intended, of course, that the pupils should always proceed with the rule and compass in their hands, in drawing; they are to be accustomed to determine distances and proportions with their eye; but, it is obvious, that they will do so with greater success, and above all with infinitely more intelligence, if they have previously ascertained the mathematical rules which are to guide them, than if they are proceeding on a mere guess, which, though by practice the character of “a rough guess” may gradually wear off, yet can never become an intelligent act, until the difference between real and apparent outline be thoroughly understood. And so far from cramping the hand by such a proceeding, it is clear that the teacher cannot more effectually promote its freedom, than by removing the cause of that timidity which must ever be consequent upon utter ignorance of the point to which any given line is to be drawn.
Having said thus much as regards the method of teaching perspective, we shall only add, with reference to the effect of light and shade, that in this also the pupil's self-observation should be called forth. He should be made to draw the same object in different positions, and under different lights, and these exercises continued until he have acquired such a familiarity with the laws of light, that he would be enabled to draw any given object in a stated position, and under a certain light, entirely from his mental conception of it, under the circumstances described. Lastly, the pupil should from time to time be allowed to exercise his own imagination in original composition, the teacher interfering no farther than by a progressive enlargement of the sphere within which he would permit him to chose his subjects.
Concerning the third element of the art of drawing, which we have termed “the spirited touch,” we know too well that
NO METHOD TO PRODUCE GENIUS.
genius alone can teach, and genius alone learn it, ever to dream of bringing it within the rules of system. Deeply as we are convinced of the advantages resulting from a methodical progress in instruction, we acknowledge that there is, with reference to every faculty of the mind, and every branch of knowledge or art, a lofty something, a gift from above, which no education can instil or draw forth, but only prepare the way for it, that its bright beams may not be obstructed by an opaque medium, when the time of its spontaneous effulgence shall have arrived.
Method of Teaching Geography ;-Branches of Instruction
connected with it.
We have already hinted, in an earlier part of this volume, that on the subject of geography the ideas put forth by Pestalozzi in his work, “How Gertrude Teaches her Little Ones," formed a complete contrast with his own principles: and we should, therefore, not swell the bulk of our publication by any extracts on the subject, were it not for the predilection which our age evinces on all occasions for superficial mechanical contrivances. If the letters from which we have selected the most practical parts, and embodied them in this sketch of the Pestalozzian method, should fall into the hands of any of those transcendental engineers, who are busying themselves in the construction of “new railways of intellect,” and they should happen to alight upon a passage like that which we are about to quote, what an outcry would they not raise against us, for suppressing exactly those few solitary passages, in which Pestalozzi proved himself to be not “a mere theorist,” but “a practical man,” the only passages from which any "useful information” can be derived. Our wisdom, therefore, is, to be exceedingly honest, and produce ourselves what evidence there might be against us, if evidence it be, by inserting the following specimen of the manner in which our author proposed to initiate his children in the treasures of language,” which, as may be recollected from the twenty-second chapter, he divided under "four heads: geogra
GEOGRAPHY AS A MATTER OF SOUND.
phy, history, physical science, and natural history,” and which he brought first under the cognizance of his pupils, by giving them series of tables, with the words belonging to each head written on them in “alphabetical order.”
“In the first instance,” he says, “the words contained in these tables are to be laid before the child in merely alphabetical order, without the admixture of any opinion whatever, nor even in any order dictated by certain opinions. This being done, the question arises: “What arrangement does the mind suggest, according to the peculiar nature of each subject? A new task then begins. The same seventy or eighty tables, the words of which were, at first, presented and impressed upon the memory in merely alphabetical order, are now to be subdivided according to different scientific points of view, and the children are to be exercised in assigning to each word the place to which it belongs. For this purpose the different subdivisions may be marked by ciphers, abbreviations, or any other arbitrary signs. These being put against the different words of each table, according to the subdivisions to which they belong, the child is made to read them together with the words, and thus to convert the alphabetical nomenclature into a scientific one.
“It seems hardly necessary to give an example; on account of the novelty of the plan, however, I will add the following instance for illustration. England* is one of the subdivisions of Europe. The child is then to learn the division of England into forty counties, with a number attached to each county. After this the child is supplied with an alphabetical list of the towns and cities of England, every town in the list being marked with the number of the county to which it belongs. The child having first been exercised in reading the names of them without the numbers, and being afterwards made acquainted with the signification of each number, he will soon be able to arrange them in their respective counties.
“Suppose the following table of towns and cities be laid before the child:
• We have substituted England instead of Germany, to make the illustration more intelligible.