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ALGEBRAIC PARADOX.

"1. Let a=x, then,

2. multiplying by x, ax=x2,

3. adding-a, ax-a-x2-a2,

4. resolving into factors, a(x—a)=(x+a)(x—a),
5. dividing by x—a, a=x+a,

6. substituting a for x, a=a+a=2a, and

7. dividing by a, 1—2.”

In the October number, J. S. E. very properly decides that the fallacy is in passing from the fourth to the fifth equation. He might also, with equal propriety, have objected to the preparatory step of resolving into factors, as equation three is, by equation one, 0=0, and zero has no factors.

I have been requested by a teacher, to add a word on the actual meaning of equation six, and the determinate values of g.

Although O renders every quantity into which it enters as factor or divisor, an absurdity, so that the two steps in passing from equation three to equation five are both absurd, yet the physical or geometrical question may admit a rational solution when the algebraic solution fails. To obtain this true geomet rical or physical meaning of an absurd algebraic expression, we must substitute for 0 a very small quantity, and when the result is obtained, again substitute O for the infinitesimal. Thus equation seven is an absurdity, but its two members are in the ratio which the two members of equation three will have when x is very little larger than a.

Suppose that I should draw such a line on the surface of the earth, that a man travelling in it should always find his latitude equal to the cube of his longitude, (both being expressed in miles,)—at what angle would this line cross the equator?

Solution. The line would evidently cross the equator at the point where the first meridian crosses; and when the latitude was small, the line might be considered as the hypothenuse of a triangle, of which the latitude and longitude were legs. Hence its direction depends on the ratio of these legs, that is, on the fraction lat which at the equator would be 8. But by the question lat.onlon.2-0. That is, the longitude would, when very small, be very much greater than the latitude, and when both were zero, the line would be absolutely parallel with the equator; a result perfectly rational and true, though derived from the algebraic absurdity 8.

lon.

lon.

lon.

Again, at what angle would it cross the parallel of 8' north? Here the legs of our infinitesimal triangle would be lat.-8' and lon.-2', and their ratio, lat, would become at the required parallel. But in this case we should have lat.—8—

lon.-29

3

lon.3-8', which divided by lon.-2' gives lon.2+2 lon. +4, which, when lon. is 2', 12. That is, the parallel of 8' crosses this line at the same angle which a hypothenuse makes with a leg of 1, the other leg being 12. Here is then again a definite actual result, in a physical question, which in the algebraic form is absurd.

These questions are simple, but the principle is applicable to very intricate questions, and leading to very singular results. H. T.

Waltham, Dec. 1850.

The above remarks upon the effect of 0 in connection with numerals, suggest the expediency of a remark upon the simplest method of proving the value of fractional expressions when O is the denominator, e. g. 7. We have been told even, that the value of such a fraction is 0; because, say they, whenever 0 is a factor, 0 is the answer. The reverse of this is, however, the true answer. The expression is an unexecuted division. Division is but abbreviated subtraction, and in dividing, we simply inquire how many times we can subtract the divisor from the dividend. The question is, then, How many times can we subtract 0 from 7 without exhausting 7? The answer is an infinite number, and the value of is infinity. Many seemingly difficult mathematical points may be easily explained by performing the multiplying and dividing processes in addition and subtraction.-Ed.

x2+xy=8.

PROBLEMS.

x2+y=6. Required value of x and y.

It is said that the above problem cannot be solved by the use of quadratic equations alone.-Ed.

Suppose A to start from Boston on Monday noon, and travel west with the same rapidity as the sun. Suppose him to ask every man on his way, "What day is it?" and to receive from each an answer. Where will he meet the first man who will tell him that it is Tuesday?

Again, suppose two persons to start from Boston exactly at noon on Monday, and travel with electric speed on the same parallel of latitude, one east, the other west, and meet in, say one minute. Each reckoning time according to the general rule of adding one hour for every remove of fifteen degrees east, and subtracting one hour for every remove of fifteen degrees west, will it be, at the place of meeting, one minute after twelve of Monday or Tuesday morning? And if we decide it to be either, say Monday, what day is it at the antipodes of any other place, say London, when it is Monday noon at London ?-Ed.

NORFOLK COUNTY TEACHERS' ASSOCIATION.

REPORT OF THE PROCEEDINGS AT THE SEMIANNUAL MEETING HELD AT DEDHAM, DEC. 23d and 24th, 1850.

THE Norfolk County Teachers' Association held its sixth semiannual meeting, at Temperance Hall, Dedham, on Monday and Tuesday, December 23d and 24th, 1850.

Monday morning, at 10 1-2 o'clock, the Association was called to order by the President, George Newcomb, Esq., of Quincy, and the throne of grace was addressed by Rev. Dr. Lamson, of Dedham.

The secretary read the report of the last meeting; after which the president and secretary being called on for written communications from teachers, on educational topics, reported that none had been received.

The lecture of Mr. Hagar, which was in order for 11 o'clock, on motion, was deferred to the same hour of Tuesday.

On motion, it was voted to invite all present, not members, to take part in the deliberations of the Association; it was also voted to restrict gentlemen in their remarks to ten minutes.

The subject of spelling was then taken up and discussed by Messrs. Colburn, Reed, Dodge, Capen, Woodbury and Butler. All were of the opinion that the method of obliging scholars to write their words from dictation, was the best that could be adopted. The importance of classifying derivative words under their respective Latin or English roots, and making this an aid to the scholar, was referred to, and Mr. Colburn explained the method taught in the Bridgewater Normal School: he also alluded to the necessity incumbent on teachers of aiming at correctness in their own orthography, and mentioned instances in which, in applying for situations, they had failed of success, merely on account of inaccuracy in this respect.

The directors were appointed a Committee to furnish subjects for discussion. At one o'clock, the Association adjourned, and met again at two, P. M.

The Committee on questions reported in favor of discussing either the subject of Arithmetic or Geography: the former subject was taken up and discussed by Messrs. Dodge, Alden, Colburn, Reed, and Capen. At three o'clock, the hour appointed for the lecture of Mr. Smith, it was voted to defer the lecture until half past three, and the discussion on Arithmetic was resumed, and continued by Messrs. Reed, Hagar, Colburn, and Capen. The discussion rested chiefly on the importance of thoroughness in teaching the elements of arithmetic. The use of keys, and of printed answers in any form by the pupil, as an

One gen

aid, was denounced in the most unqualified terms. tleman stated that his practice, on finding a key in the hands of the pupil, was invariably to burn it up. We believe that all who heard him, concurred in the propriety of such a course. Various short methods of calculation, of multiplication, and of obtaining the least common multiple, were explained by several of the speakers.

At half past three o'clock, a lecture was delivered by Rev. Increase S. Smith, of Dorchester, on the subject, "Qui docet, discit "" He who teaches, learns." Not having taken notes of this most excellent lecture, we copy a short abstract of it from the "Norfolk County Journal." The lecturer commenced with the motto, "The teacher learns." When he begins, he has no fixed methods, but looks at things somewhat as his pupils do. He has a feeling that he must appear to know everything; he is then full of enthusiasm, and is often a better teacher than when he becomes old and fixed in his methods, and grinds out the same old tunes. Teachers should be the literary and scientific enthusiasts of the land, should not drill less, but should study and investigate more. They need out-door exercise, and should go out as naturalists, and collect specimens in Botany and Mineralogy, and bring them in to interest their pupils. They are in danger of rusting out, therefore they should study, study, study. The lecture was interesting and instructive, and received the hearty approbation of those present.

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Mr. Kneeland, of Dorchester, spoke on the subject of the lecturer's appeal to teachers to devote themselves to the acquisition of other knowledge than such as is required merely in the sphere of their usual duties, and closed his remarks by moving that the thanks of the Association be presented to Rev. Mr. Smith, for his able, interesting, and instructive address, which motion unanimously passed. After the appointment of a committee to attend to the more effectual heating of the hall, at 1-2 past 4 o'clock, the Association adjourned, and met again at 7 P. M., to listen to a lecture from Rev. Mr. Dean, of Quincy.

The lecture of Mr. Dean was a very able, interesting, and methodical exposition of the duties arising out of the teacher's profession. He made many earnest appeals to teachers on subjects of interest to them, and was listened to with evident satisfaction on the part of his audience. On motion, it was voted, that the thanks of the Association be presented to Rev. Mr. Dean for his excellent and interesting lecture.

After recess, the importance of affording a liberal patronage to the "Massachusetts Teacher," was introduced by Mr. Capen, and additional remarks were made by Messrs. Reed, Kneeland, and Colburn.

A long and interesting discussion followed, on various subjects: Mr. Barrows, of Dorchester, spoke upon the difficulty the teacher labored under, in not being independent in his profession. Mr. Colburn spoke on the idea thrown out in the lecture, that our public schools were fast supplanting colleges in public estimation. Further remarks on the relative position and influence of public schools, academies, and colleges, were made by Rev. Mr. Dean, Dr. Cutter, and Messrs. Colburn, Barrows, and Hagar. At 10 o'clock, the Association adjourned to meet at 9 A. M., of Tuesday.

Tuesday, Dec 24th, 1850. The Association met according to adjournment, and was called to order by the president. The secretary read the records for Monday; after which, on motion, it was voted, that the morning session close at a quarter before one o'clock, and that the time of meeting in the afternoon be at two o'clock.

A resolution on the subject of school-books was offered, which, after some amendments, and an animated debate, sustained by Messrs. Kneeland of Dorchester, Morse of Quincy, Capen of Boston, Colburn and Capen of Dedham, Reed of Roxbury, Woodbury of Dorchester, and Slafter of Dedham, was unanimously adopted, as follows:

Resolved, That, in our opinion, it would secure a better selection of text-books in our public schools, if the teachers were permitted to meet with the committee, and discuss with them the merits of all books proposed.

The subject of Mr. Smith's lecture was then taken up and discussed by Messrs. Kneeland, Barrows, and Capen of Boston.

Voted to take a recess of five minutes. After which, at 11 o'clock, according to appointment, a lecture was delivered by D. B. Hagar, Esq., of the Jamaica Plain High School, Roxbury, upon the subject of the " Supervision of Schools." "The lecturer's views were radical, and were delivered with great beauty and energy, and it may be said, without any disparagement of others, that it was the crowning glory of the meeting.' The lecturer made an able argument in favor of the plan of having a general superintendent of schools in our cities and large towns.

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On motion of Mr. Reed, of Roxbury, it was unanimously voted, that the thanks of the Association be presented to Mr. Hagar, for his very able, interesting, and valuable lecture; and that a copy be requested for the press. Remarks on the subject were made by Messrs. Reed and Dodge, and Dr. Cutter.

"Mr. Kneeland, of Dorchester, then introduced to the Association Mrs. Haines, of Milwaukie, Wisconsin, an associate of Miss Catherine Beecher, who is engaged in the laudable work of employing teachers for the West. Mrs. Haines spoke in a

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