changes of number with them, until the pupil has a perfectly clear idea of the numbers and of their quantities.

9. What next?

The teacher advances to the use of figures.

10. What is the treatment of the number, with and without figures?

The latter always precedes the former; the written or slate arithmetic every where follows mental arithmetic. Not only does the cultivating power of arithmetic lie in the insight into the relations of number, but also the wants of practical life demand preeminently skill in mental arithmetic.

11. Upon what chiefly depends that skill?

First on the ability in handling the decimal principle, (Zehnergesetz;) then on the ability to compare and analyze numbers.

12. How do the exercises with so-called "pure," and with applied numbers, compare?

The former always precede; application presumes ability in treating the pure number. This being attained, questions, problems and exercises follow; together with denominate numbers, and their application to life.

13. Are the exercises with numbers from 1 to 100 to come in order after the four rules, (addition, subtraction, multiplication, division ?)

No. All operations ought to be performed successively with these numbers; the regulated uniformity of the operations comes later. (Grube, Schweitzer, etc.) 14. Shall fractional arithmetic be entirely separated from instruction in whole numbers?

No. No. 13 forbids it, and makes it impossible; even considered in itself it would be improper.

15. Which points must be distinguished in practical problems?

First, the understanding of the words.

Second, the relation of the question to the statement, or of the thing required to the thing given.

Third, the understanding of the way in which the unknown number depends on the number given.

Fourth, the finding of the unknown number from the given number; that is, the calculation, oral or written.

16. What has the teacher to do in these four processes, when the pupil can not proceed of his own strength?

In the first, the understanding of the words and things in their relations must be explained, and often directly given.

In the second, what is required must be well distinguished from what is given; the propriety of the question must be accurately considered.

The third point is to be brought out by means of questions from the teacher. The fourth is an affair by itself, and is the pupil's concern.

An exercise is not complete and satisfactory, until the pupil is able to explain these four points, one after another, orally, and without any aid.

The teacher leads by questions, (by analysis;) the pupil proceeds by synthesis. The former proceeds from what is sought, the latter from what is given.

17. How is talent for arithmetic to be recognized?

Besides what has been said in No. 16,-by the independent invention of new methods of solving the problems, of peculiar processes, etc.

18. In what way may uniformity in arithmetical instruction be gained? By solving each problem rationally, according to the peculiar nature of the

Tumerical relations occurring in it, and consequently, without admitting any external rule or formula, which on the contrary ought to result from the subject itself. Uniformity lies in the rational, transparent treatment, and, therefore, in the mind, not in the form. Good rules, etc., are not indifferent, but they must follow the observation of the thing.

19. Which is the most simple, natural and appropriate form of managing the problems externally?

Not the doctrine of proportions; it is too artificial, and too difficult for the common school; nor the chain rule, etc. The best form in slate arithmetic for the common school is the so-called "Zweisatz," the fractional form, (bruchform,) which every where requires reflection. (Scholz.)

20. What is the value of the so-called "proofs" and abbreviations?

The proofs are, with a rational method, superfluous; the latter are of little value. A well guided pupil finds them out himself, and if, in the highest class, some of them are pointed out to him, their origin, and thus their correctness, must be demonstrated at the same time.*

IV. GEOMETRY, (Raumlehre,) BY A. DIESTERWEG.

1. Is geometry required in the common school?

No doubt, for it teaches the forms in which every thing appears; the shape of matter and the laws of those forms; the laws of space and of extent in space; the dependence of magnitudes and forms on each other.

2. Why is such knowledge considered as a requisite for general cultivation ? Because the whole mass of bodies, the universe, as well as man, exists in space; because without the knowledge of the qualities of space, man would be ignorant of that appearance of things which belong to their inmost nature; because geometry teaches how to measure lines, surfaces and bodies, which knowledge is very necessary; because without it man could not divine, that the distance and size of the sun, moon and stars, could be determined; and because he would even have no idea of the extent of his own abode, and of the mathematical, i. e., fundamental qualities of the same. All this is consequently requi

site for general human cultivation, not to speak of its practical value, as well for female as male education, and therefore for the common school, the school of the people. Without it, not the most indispensable part, but an essential part, of education is wanting.

3. What elements of geometry are to be taught in the common school? and in general what parts of it may be considered there?

Space admits of "intuitive," (anschauliche,) and a demonstrative, (begriffsmuessige,) observation.

The intuitive faculty of man perceives immediately objects in space, bodies in their qualities and forms; with the sense of touch he perceives what opposes him in space, the body and its external form; the sense of sight assists him, by determining extent and distance, and by comparing and measuring them. These are operations of external intuition. The intellect abstracts the differentia of the bodies, and fixes the pure, mathematical form; and thus aids the interior pure, or mathematical intuition. Moreover, the logical intellect, perceiving the

No school can do without an arithmetical text-book. Hence it sufficed to give here the principles. These contain the measure by which we have to judge of the value of the text-book.

dependence of magnitudes on each other, their mutual conditions, the inference of the one from the other, deduces and concludes.

The intuitive part of geometry is that elementary part which is proper for the common school. But thereby is not meant, that the pupils should not learn the dependence of one thing on the other; this even can not be avoided, it comes of itself; but according to the degree of ability, quicker and deeper with one than with another, and one school will make more progress in it than another. But the power to be immediately employed is the faculty of observing—first, the exterior, and then, and preeminently, the interior. The conclusions connected with that observation result therefrom spontaneously; the intellect works without being ordered. Therefore, in geometry, as every where-a fact, ignorance of which, causes much merely repetitious and lifeless teaching, as well as intellectual dependence and immaturity-the teacher ought to lead the scholar to immediate, true and vivid perceptions.

The strict or Euclidean geometry, with its artificial proofs, is not fit for the common school, nor does it prosper there.

4. What is more particularly the subject of geometrical instruction in the peoples' school?

The qualities of (mathematical) lines, surfaces and solids.

5. What method is to be pursued with it?

The point of starting is taken in the physical body; and from this the mathematical one is as it were distilled.

The order of single precepts or propositions is, as has been said, as much as possible genetical. Pedantry and anxiety are here, as every where, prejudicial. The method, always intuitive, requires originality, i. e., the evolving of every thing learned from some thing preceding; aims at immediate spontaneous understanding of one thing through the other.

6. What is the immediate purpose of this instruction?

To understand the qualities of lines, plains and bodies; to measure and calculate them.

7. What instruments are used by the pupil?

Pen and pencil, for drawing; compass and scales, for measuring; the usual measures of lines, surfaces and bodies, for calculating.

V. NATURAL HISTORY, BY ED. HINTZE.

1. What method should be used in teaching natural history?

The method of instruction is the mental development of the pupil by means of the material development of the object. The method is, therefore, essentially a process made by the teacher. Since there can be but one such development, there can be but one method.

2. Which is that true method?

The one true method is named from the principle contained in it; it is the developing method.

3. Wherein consists this developing method?

In development there are three steps; observation, (anschauung,) conception, (vorstellung,) and generalization, (begriff.) Such is the progress of the method. Every where teaching begins with facts, and therefore in this case with the observation of natural objects. Of these, individual action and growth must be shown, and the general law of nature thence inferred. In this way and only in this, the pupil is taught according to nature, since he proceeds from immediate observing and knowing to perceiving and understanding.

4. What mode of teaching is to be used?

That one which develops by questioning, (die fragend-entwickeldnde.)

5. Is this mode practicable in all three courses, (set down by Hintze elsewhere with regard to the capability of the scholars)?

In the first course, questioning is predominant; on the second, "der vortrag," i. e., proper teaching and explaining must be joined with it; on the third again, questioning predominates. In all good instruction questioning is predominant, and with it conversation with the whole class.

6. What have we to think of lecturing?

Lecturing is no form of instruction at all; it is a rocking chair for teacher and pupils; the former has easy work, whilst the latter stare and dream.

7. What ought to be required of the pupils?

Their first and chief object must be to learn to see right; then follows right reproduction; and the necessary result is right understanding.

8. What is the value of learning by heart?

In all instruction nothing must occur which is not understood, and merely learnt by words. One fact well understood by observation, and well guided development, is worth a thousand times more than a thousand words and sentences learnt by heart without understanding. A well guided pupil has nothing to learn by heart particularly; what is understood, is remembered for life. 9. Shall the pupil use a text-book?

For natural history it is useless. The good teacher does not depend on it, the bad one has a good means to cover his inability, and the scholar has nothing but a dry skeleton.

The teacher must have mineralogical, botanical and zoological collections, and, if possible, a microscope.

10. What must the pupil do at home?

Write out and draw what has been treated in school-in proportion to his time-in a brief, concise and neat manner. Besides, the well directed pupil will voluntarily and eagerly occupy himself with nature, look with interest and intelligence at plants, stones, etc., and collect them.

11. How does an able teacher distinguish himself in this study?

The able teacher takes pains with his school every where, and particularly in this branch; all energy, punctuality and vivacity, must be applied here, if instruction is not to be a dead and dry mechanism.

12. What distinguishes a painstaking (strebsamen) teacher?

The able teacher is found out at school, the painstaking one at home. There are certain branches which are soon done with. But this is not the case with natural history; he who is devoted to it, must follow its own path of progress. The teacher must never cease to study, to make excursions, experiments, collections, etc., to search, to listen, to observe and investigate.

13. What characterizes the inspiring (geistanregende) teacher?

He is distinguished by a happy development of sound talents, love of study, and devotion to his vocation. By force of application every one may acquire the necessary knowledge, for nature is every where. If the able teacher shows himself at school, the painstaking teacher principally at home,-there flows from the inspiring teacher every where something that indeed can not be completely gained by study and application; but an earnest will accomplishes a great deal. Besides, it is true, that as under the hands of Midas every thing was changed into gold, so in the hands of an inspiring teacher every thing

becomes enlivened. As the creative mind every where works attractively, so particularly in natural history, zeal, application, love and devotion, spring up spontaneously in the pupils.

VI. NATURAL PHILOSOPHY, BY A. DIESTERWEG.

1. Should natural philosophy be studied in the common school?

Certainly. Shall the children in the common school learn nothing of weather and wind, of thermometer and barometer, of the phenomena of light and air, of rain and snow, dew and hoar-frost, fog and clouds, lightning and thunder? shall they see the aeronaut, travel by steam, and read telegraphic news, without knowing the how and the why? Shall they remain ignorant of the constituents of food, and of the process of their stomachs and their lungs? Or is it sufficient to read of all this in the Reader? He who answers those questions in the affirmative, is either himself an ignoramus or a misanthrope, and he who affirms the last knows nothing of the way in which real knowledge is acquired. 2. What do we begin with? and when does the proper instruction in natural philosophy commence?

As every where, with showing single phenomena, with intuitive contemplation, with oral representation of what has been observed, and reflection thereupon. We begin with it in the intuitional instruction of the lowest class. The instruction in geography and natural history develops further the faculty of intuition, and in the highest class the proper instruction in this branch commences. 3. On what portions of natural philosophy are we to lay stress?

On all such as belong to the knowledge of phenomena, within the pupil's sphere; the knowledge of the most common things is the chief point.

By this principle we make our choice; we omit, therefore, all that is remote, invisible, and incapable of being made visible; all that can be demonstrated only by mathematical proofs; and keep within the field of immediate observation, stops with those things which every one may know by observation and experience, and show such things, as are not obvious, by experiments with simple and cheap apparatus.

4. What method is to be used?

To say nothing of the regard for the individual quality of the pupil, the method depends on the nature of the subject, and on the way in which man naturally acquires his knowledge. Every where man is surrounded by natural phenomena; they happen before his eyes. These, therefore, must be opened, in order to observe apprehendingly, to remember what has been observed, to fix the succession of phenomena, and what is common in a series of similer ones; not only to learn the facts, but also the laws by which they happen, and finally, by reflection, to discover the hidden causes.

Natural philosophy belongs to the inductive sciences, i. e., to those which begin with the knowledge of single facts, abstract from them the law of the process, and then in inverse order, deduce the phenomena from the causes.

The way, therefore, prescribed by the nature, as well as the history of natural philosophy, is, that which proceeds from observation and experience to rule and law, if possible, advancing to the cause, (the so-called regressive method.) 5. What is the aim of this instruction?

The knowledge of the most essential phenomena, by which man is surrounded, and the ability to explain them, that is, to state in a simple way their causes. Most important is the knowledge of all that refers to weather, and we expect, therefore, from a graduating pupil, correct answers to the following questions: