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FIRST NORMAL SCHOOL BUILDING OF MINNESOTA.

The above cut will give the reader some idea of this magnificent edifice It is not only beautiful and massive in architectural appearance, but is most admirably planned.

The main edifice is 63 by 78 feet, and is intersected at each

NORMAL SCHOOL BUILDING, WINONA, MINN.

extremity by a wing 50 by 85 feet. The front and rear walls of the main building are supported by turrets, and the east wing is surmounted by a tower, 130 feet high. The walls are of red bricks, with facings and trimmings of a drab colored sandstone, making a very pleasing contrast. The roof is of the French or Mansard style.

The basement, which is 10 feet in the clear, contains the janitor's rooms, a laboratory and lecture room, with elevated seats, exercise and calisthenic rooms, and furnace and store rooms.

The first story has a passage-hall or corridor, 10 by 166 feet, running through the entire building and intersected by transverse corridors, eight large school rooms, four for model classes and four for the schools of practice, a reception room, etc.

The second story contains, in the main building, the "Assembly Room," 63 by 78 feet, occupying its entire area; in the cast wing, the principal's office, the library, and two large recitation rooms, and, in the west wing, two recitation rooms and two ladies' wardrobe rooms.

The third story contins, in the main building, "Normal Hall," capable of seating 800 to 1000 persons; in the west wing, four large recitation rooms; and, in the east wing, a suite of rooms for a museum of natural history. The fourth story of the west wing contains two rooms, 33 by 35 feet, with skylights, for drawing classes.

The building is heated and ventilated by Ruttan's system, by which warm pure air is brought into each room and the impure air carried out by a system of circulation. The turrets and ventilating shaft are the exit tubes. The system is giving good satisfaction.

The entire cost of the building, including grounds, furniture, and apparatus, is about $150,000.

THE UNIVERSAL METHOD OF APPROXIMATION.

BY THOMAS HILL.

In White's new series of Arithmetics, the best by far which I have seen since Chase's was out of print, the last book is entitled "A Complete Arithmetic." Yet it omits Double Position, the universal method of approximation.

In every reverse process we are frequently obliged to be con

tent with approximate results. Thus in dividing ten by three, in the decimal system, or in extracting the square root of twenty, only approximate results are possible.

The most general axioms for obtaining approximate results are the assumptions, 1st, that the errors of results are in some degree proportionate to the error of the data; 2nd, that this proportion is more close as the errors are smaller. On these assumptions a very large part of our inductive investigations proceed, not only in mathematics, but in physics, and in the social sciences.

All arithmetical problems involving one unknown quantity, x, can be so arranged that certain operations upon x will yield a given constant, c. If now we assume x to have successively the values p and q, and find that these yield the results c+a and c+b, the first axiom gives us a means of approximating to x. For by that axiom,

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and by the second axiom this approximate value of x being substituted for p or q (whichever was most erroneous) gives us the means of nearer approximation. This is precisely the rule of Double Position in Daboll's Arithmetic, and, from its generality and universal usefulness, it certainly deserves a place in a complete treatise on Arithmetic.

It may be easier to explain to a child a secondary proportion from our first axiom, giving a correction for p instead of the corrected value. Thus, the difference of the errors a and b is to a as the difference between p and q is to the error of p. Regard must be had to the algebraic signs in both these formulas. These rules in an arithmetical form would be as follows:

RULE I. Arrange the question so that certain operations upon the required number shall yield a fixed result. Make two suppositions (or positions) for the number, and perform the operations, noting the errors of result. Multiply each error by the other position. If the errors are both in the same direction, divide the difference of these products by the difference of the errors, otherwise divide the sum of the products by the sum of the errors ;-the quotient is a corrected position.

RULE II-Multiply the first error by the difference of positions, and divide by the difference of results; the quotient is a correction to be added to or subtracted from the first position.

EXAMPLE.-Find the number which is double the difference between its square and its cube.

This question must be rearranged thus: From the cube of the number subtract successively the square and the half number; the remainder must be zero.

We make the two positions, 1.4, and 1.5. Subtracting from the cubes of these numbers, the squares, and half the numbers, the remainders are .085 and .375, which are the errors, because the remainders should be zero.

By the first rule, (.085×1.5—.375×1.4)÷(.085—.375)=1.37. By the second rule, (.085X.1)+(.085-375)=.029, which is to be subtracted from 1.4, because increasing 1.4 to 1.5 increased the error.

On testing 1.37 the error of result is +.009, showing that 1.367 is nearer, and by using 1.37 and 1.367, a third approximation is obtained toward the root 1.3660254. The root -.3660254 could be found in the same way; and also the root 0.

[We are too well pleased with the very high commendation, which Dr. Hill gives our arithmetics, to be sensitive respecting his criticism. It may, however, be proper to say, that we omitted Double Position because we did not consider the method of sufficient practical importance to be included in a school arithmetic. We learned it when a boy, but have never used it either in solving arithmetical problems or in practical life. It is found in very few school arithmetics, issued since Daboll's, and it is not included in most of the Higher" arithmetics. The opinion of so eminent a mathematician and teacher as the ex-President of Harvard College, will, however, cause us to reconsider the subject before a second edition of our series is issued.-EDITOR.]

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GRADED SCHOOLS.

BY J. L. PICKARD.*

A few words in favor of a graded system, guarded against serious objections, may not be amiss. The objections to any one system lie against all systems. The discussion rests between some system and no system. We must either have some prescribed course of study or leave all to chance or to the whim of the pupil. Public money can not be expended for private purposes. Our public schools, therefore, should not have for their aim the gratification of private caprice, but must, if true to

*Superintendent of Public Schools, Chicago, Ill.

themselves, seek the public good through the best possible training of future citizens. Let every pupil in our schools select his own studies and pursue only such as happen to strike his fancy, and we shall have a mass of crooked, mis-shapen material out of which to build a State. Certain kinds of knowledge are generally conceded to be necessary in such an education as will serve the State well. All States, having any schools at all supported at public expense, have given the sanction of law to these fundamental studies. All pupils educated at public expense are expected to pursue these studies. Pupils are not generally of an age to determine what is absolutely best for themselves, nor will wise men permit them to take this matter into their own hands. All men, whether in favor of graded schools or not, will admit the propriety of laying down certain fundamental studies that must be pursued if an education is to be of any value whatever. So far, then, choice is made for the pupil by others, and his course of study is determined to that extent. The right to determine a course of study, whether of greater or less extent, is, as I think, almost universally conceded.

The question at issue, therefore, is confined to the expediency of arranging the order of the selected studies, and of dividing up the work of instruction. To some extent, nature has fixed the order of the most important studies. Reading must come before study of printed books. The natural development of the child's mind also directs to a proper regulation of the order of studies. It will not answer for any one to set himself against the Maker of the child's mind, with any hope of success. The simple process of counting pebbles, or blocks, or any other concrete objects, must of necessity precede the mastery of conic sections, and no sane man would yield to the whim of any child who might aspire to a knowledge of the laws governing the motions of the heavenly bodies, unless it be attained through the successive steps leading to it.

So far, then, all are agreed-first, as to the propriety of selecting certain studies as the basis of an education; and secondly, as to the absolute necessity of yielding the order of each of these to the nature of the mind of the pupil. Every well arranged course of study must meet with universal approval, so far as the subjects and the order of different departments of the same subject are concerned.

We are left, then, to nothing but the question of the expediency of carrying on several subjects at the same time, and of

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