Next we would take thirds and ninths; and after that, comparing halves and thirds, we would take up the sixths as the medium of comparison. This would be done by laying before the pupil three fractional squares of the same dimensions, but differently divided, as follows:

Here the pupil at once perceives, that one half is equal to three sixths, and one third to two sixths, from which, if he have gone through the exercises of mental arithmetic which we have pointed out, he will at once abstract that one half is equal to one third, and the half of a third.

In like manner the system of decimal fractions ought to be taught upon the ground of intuition, by using squares divided into ten, a hundred, a thousand, &c. parts, and comparing them to every other species of fractions. The pupil having by these means acquired a perfectly clear notion of the nature of fractions generally, and of each kind of fraction in particular, a course of mental calculation in fractions, and afterwards in numbers mixed of integers and fractions, ought to be sketched out, analogous to that of which we have given an outline as regards integers.

The same principles which we have illustrated with reference to integral and fractional calculation, apply also to the farther pursuit of the science of calculable quantities through the different operations of algebra. In each case the pupil ought first to be made thoroughly acquainted with the nature of the subject in hand, by illustrations which appeal to the evidence of his senses; and when this has been accomplished, and the mind has, by practice, been familiarized with

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.

CHAPTER XXVI.

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 arrangement, 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

332

MEASURING, THE FOUNDATION OF DRAWING.

in measuring things ought to precede every attempt at drawing; &, a
that we can draw successfully so far only as we are capable di
The common mode of proceeding, on the contrary, is to begin with
rect view, and a crooked representation of the object; to expunge
again, and to repeat this tedious process, until by degrees an
sort of feeling of the proportions is awakened. Then, at length,
to what we ought to begin with, viz. measuring.

"Our artists have no elements of measure; but by long prac
acquire a greater or less degree of precision in seizing and imitating
by which the necessity of measuring is superseded. Each of the
own peculiar mode of proceeding, which, however, none of them i
explain. Hence it is, that if he comes to teach others, he leaves his pape
grope in the dark, even as he did himself, and to acquire, by immense
tion and great perseverance, the same sort of instinctive feeling of prope
This is the reason why art has remained exclusively in the hands of a
privileged individuals, who had talent and leisure sufficient to pursue
circuitous road. And yet the art of drawing ought to be an un
acquirement, for the simple reason that the faculty for it is unive
inherent in the constitution of the human mind. This can, at all events
be denied by those, who admit that every individual born in a C
country has a claim to instruction in reading and writing. For let
remembered, that a taste for measuring and drawing is invariably manife
itself in the child, without any assistance of art, by a spontaneous imp
of nature; whereas the task of learning to read and write is, on account s
toilsomeness, so disagreeable to children, that it requires great art, of
violence, to overcome the aversion to it which they almost generally
and that, in many instances, they sustain a greater injury from the me
adopted in gaining their attention, and enforcing their application, than
ever be repaired by the advantages accruing to them from the possess

evi

of those two mechanical acquirements. drawing, as a general branch of education, it is not to be forgotten, I consider it as a means of leading the child from vague perceptions to clez ideas. To answer this purpose it must not be separated from the art In proposing, however, the art measuring. before he has acquired a distinct view of their proportions, his instruction If the child be made to imitate objects, or images of objects, 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 mys teries 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

2

THE FOUTCO

PESTALOZZI'S COURSE OF MEASURING AND DRAWING.

333

precede instruction of measuring and drawing is very compendiso far yawe may venture to insert it at full length.

, on the

is awake

he pupil," he says, "is first made acquainted with the straight line, by tation in the various positions in which it can be placed, and the different spres that can be taken of it; he is taught to denominate it accordingly as pendicular, an horizontal, a slanting line, and the latter as slanting meas rds and downwards to the right and to the left. Two lines are then s of mered parallel with each other, and by varying their position he learns [recision stinguish perpendicular parallels, horizontal parallels, and different sorts ng is santing parallels. The next step is to place two lines converging, so as to which, an angle, and he has again to learn the distinction of right angles, acute es to teaches, and obtuse angles. After this the square is laid before him, and self, added into halves, fourths, sixths, &c.; the circle is drawn next, with its ort of itong modifications, and these likewise are divided in a variety of ways. ed excise All this is to be done, as an exercise for the eye, without having recourse and anathematical instruments, and the following names are to be learned along drawh the respective figures and their divisions: the square, the horizontal, at the perpendicular rectangle; the curve, the circle, the semicircle, the quadrant, mind. Est oval, second oval, third oval, fourth oval, &c. halves of the ovals, quarters ery in the ovals, &c.

ding "This being accomplished, the child is to be introduced to the relative dramoportions of these forms, and to learn to use them for the purpose of aeasuring. To this the mother's book contains preparatory exercises, as a d and ariety of objects are there presented to the child's view, exemplifying in eir outlines the square, the rectangle, the circle, the oval, &c. After this he 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 amiliar with each particular form, and to enable him to institute comparisons.

it it re

iter:

reater D

the

DOS

402

"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 afford a medium of comparison