but for this last equation is not necessarily equivalent to

and hence from this it does not follow that the values (0, 0) do

not satisfy our given system.

The following graph represents the equations

The solution (2/3, 2) is the one ordinarily given in our textbooks, while (0, 0) is not usually mentioned.

BY STELLA E. MYERS,

719 Washington Ave., Kansas City, Kan.

The ancient abacus affords an intermediary process between full object work and the symbolism of Arabic numerals. In using objects our custom is to retain tens in groups; with the

abacus a ten is discarded and is symbolized by one in the next higher order.

In the abacus here illustrated, the patent for which is pending, balls colored in groups of ten are used upon wires and in the form of a series of compound curves. The right-hand wire is units, the next to the left is tens, and the third from the right is hundreds. The small wire on the left is for thousands in the result, these having been carried over from hundreds. A number is represented from left to right upon a series of horizontal curves, one on each wire. The small size of the abacus is designed for individual work and may rest upon the top of the school desk. On the large size, to be hung map fashion on the school-room wall, six numbers with orders up to hundred thousands may be indicated and a result reaching millions.

When an addition problem is represented with the balls the full groups of color upon units-wire are, in work with beginners, pushed upon the upper scroll; in more advanced work this is unnecessary. The part of a group remaining, if any, upon unitswire is pushed down to the lowest horizontal curve.

For every group of color or ten balls pushed upon the upper scroll of units-wire one ball is pushed from the lower scroll of tens-wire to the lowest horizontal curve of this wire. The process is continued in the same manner with the other wires and the whole result is read from the lowest horizontal curves of all the wires.

The children should do no counting of objects but should recognize at sight parts of the groups of ten and make combinations readily in two colors up to nine. Five is the largest part

necessary to recognize, as larger parts of ten are learned from the number, one, two, three, or four, that is missing from ten.

Before representing a problem in subtraction or division one group of ten beads is placed on the upper scroll of units-wire and another on the upper scroll of tens-wire to be used if necessary in place of one in the columns to the left.

The idea of multiplication may be given by using successive addends or, after the table is learned, by representing the products in each order and transferring tens. Object work in division is not feasible where more than one is to be transferred from one column to another.

This form of the abacus deals with the orders in the same relative position as in the written problem. Any other position confuses the concrete and the symbolical processes. As an illustration of the simplification of addition by this method, a child under four years of age that had been taught nothing of numbers was permitted to manipulate the beads according to the rules of the game for an average time of thirty minutes each day during one week. Then she could show the sum of six numbers in hundred-thousands in one half the time required by a fourth grade pupil working on paper. In the work on the frame there were no errors, while the written solution was not reliable.

There is naturally suggested by the use of this abacus a system of written addition which prevents mental fatigue in cases of continuous work.

236 236 843

843

7425

657

65

X25

318

534 268 546 4,827

In looking down units column the first number that helps with the preceding ones to form a ten has drawn through it an oblique line. With beginners or where an accountant is apt 318 to be interrupted the part of the number not used in forming the ten is written to the right. It is used in forming the second ten. The last sum less than ten is written for the result in the same column. The tens are marked out 4,827 of tens column in the same manner and the part of a ten remaining at the foot of the column together with a number of corresponding to the number of lines drawn in units column, the number of tens in that column, is written for the result in this column. The same method is then continued

5426

throughout the problem. Time is saved as well as energy. There is no occasion for inaccuracy, and consequently none for a test of the solution.

A COMMUNICATION CONCERNING THE TYPE px2 + qx + r. Essex High School, Essex, Ontario, May 27, '07.

Editors of SCHOOL SCIENCE AND MATHEMATICS,

Dear Sirs:

440 Kenwood Terrace, Chicago.

The question of precedence of the method of factoring the type px2 + qx + r, as given in the April number of SCHOOL SCIENCE AND MATHEMATICS, has been raised by Prof. Smith in the June number. Whether or not this method was first published in the United States in 1900 as stated, it certainly was published some 14 years earlier in America. In "The Elements of Algebra,” by J. A. McLellan, and published in Toronto, 1886, will be found on page 110, article 106, the identical method so well demonstrated by Mr. Toan in the April number.

The second method illustrated by Prof. Smith, or one very similar, is, I believe, quite generally taught to 3rd and 4th year High School students throughout Ontario.

The solution of the quadratic equation is taught about as follows:

In the process of this solution the factors of the quadratic expression have been obtained, and when a formula becomes desirable the one will answer a dual purpose.

Thus to factor 6a+ 11x-10 the student writes first the roots of 6x2+11x-10=0

6

(x+ 11−19) × (x+11+19) = 6(x − 2)(x+ 5) = (3x − 2) (2x+5)

Of course, multiplication by 4 p may be used instead of division by P; but, having used both, I prefer the latter. Again referring to Mr. Toan's method of factoring 12-11a-5 compare this:

The great majority of such examples are solved readily by the average student by trial. The above work indicates the method by trial reduced to a science. Write in order all the possible pairs of factors of the end coefficients repeating one set reversed i. e. § and . Starting with 12 try cross products first with then with looking for a difference of 11. This failing 12 may be crossed out. Proceed in this manner until the coefficients of the factors are found. If every pair fails then the expression is not factorable.

Once more experience in teaching both Mr. Toan's method and this makes me use the latter almost exclusively.

Yours truly,

R. W. ANGLIN, Prin.

A COMMUNICATION.

Jersey City High School, Jersey City, N. J.

Prof. Ira M. DeLong, University of Colorado, Boulder, Colo.

My Dear Sir:-I was very much interested in the solution of the Frustum Theorem given by Dr. Blakslee in the December, 1906, number of SCHOOL SCIENCE AND MATHEMATICs, especially since I have for some time used a similar proof with my classes in Solid Geometry. I enclose my solution. Yours very respectfully,

NELSON L. RORAY.

Lemma-In a series of equal ratios the sum of the mean proportionals of each antecedent to its consequent is equal to the mean proportional of the sum of the antecedents to the sum of the consequents.