A

F

B

C

D

Solution of Problem No. 32.-The angle FDA, by the conditions of the problem, is a right angle, and as the angle FBD is the complement of the angle FAD, the angle AFD is also the complement of the angle FAD, and the angles AFD and FBD are equal, and the two remaining angles of the triangles, AFD and BFD, are equal, and the two triangles are equiangular. Again, let the angle A=2, then, by the conditions of the problem, angle FCD=22, but the exterior angle FCD=the sum of the angles AFC and FAC. Subtracting the angle A from the angle FCD we find the angle AFC=z. Then the triangle ACF is isosceles, the side AC=the side CF. Let CD=y, FD=x, and by the conditions of the problem AB=a, BC=b, AD=a+b+y; and from the isosceles triangle we have a+b=CF. From the two equiangular triangles we have the following proportion: AD: FD::FD: BD, or AD.BD= FD2. Substituting for AD, BD, and FD their values, and we have (a+b+y)(y+b)=2(1). From the right angle triangle FCD we have (a+b)2—y2=r2 (2). Eliminating a, and reducing, we find y=2

a

Substituting for y in e. g. (2) and we find_x=√3a2+2ab+b2, and the

a 3

area of the triangle FCD=V 4a2+2ab+b2, and the area of BFD

Sotution of Problem No. 30.-It is evident that the distance 4 traveled in 221 days the distance B traveled in 221 days+twice the distance B traveled in 9 days, because, after 9 days A traveled back as far as B traveled in 9 days, and then again pursued his journey and overtook B 221 days from the time they first set out. We learn that A traveled 18 miles each day, hence 18 miles × 221-405 miles the distance A traveled in 221 days the distance B traveled in 221⁄2 days+the distance B traveled in 18 days. Therefore, 405÷(221+18)=10, the number of miles B traveled per day. L. CAMPBELL..

General Solution of Problem No. 19.-The space enclosed by the circumferences of three equal circles which touch each other externally is always equal to the area of an equilateral triangle formed by lines joining the centers of the three circles, minus half the area of one of the circles.

3.141592
2

Hence, A=√3 R2. R2, or R= √ A÷.161255, in which R represents the radius of each circle, and A the area of the space inclosed by the three circumferences. The side of an equilateral triangle described about the three circles is equal to the diameter of the circles, plus twice the perpendicular of a triangle formed by lines joining their centres. Hence, S=2R+2√3 R=5.4641R, in which S represents the side of an equilateral triangle described about the three circles. When the radius of a circle-1, the side of its inscribed equilateral triangle will be 3. 2.S Hence, 3:1:: S: D. Therefore, D=- =1.1547S, in which D √3 represents the diameter of a circle circumscribing the triangle. In problem 19 4-160 sq. rods. Hence, R=V160÷.161255-31.499+ rods. S 31.499 × 5.4641172.1136+ rods, and D=172.1136 x 1.1547= 198.7397 rods. L. CAMPBELL.

LAWRENCE UNIVERSITY.

EXAMINATION POINTS IN THE OUTLINES OF ELEMENTARY ALGEBRA.

The following will be interesting to teachers in our schools, and, perhaps, useful in their own class drilling:

SECTION I.-(1.) Axioms. State and illustrate all the axioms of common use in algebraic reasoning.

(2.) Symbols. Show by examples the use of each of the following signs: equality, inequality; addition, subtraction, multiplication, division; vinculum, or parenthesis. If a sign has several forms, give cach form.

(3. Equation. Define equation, write an example, and show what is meant by members. Define and illustrate transposition, clearing of denominators and clearing of coefficients, and give in full the reasoning applicable to each operation.

(4.) Miscellaneous. Define each of the following, and write examples: term, monomial, polynomial; additive number, subtractive number; numerical value of a polynomial; reduction of similar ters Give the reasoning applicable to the latter operation.

(5.) Addition. Define al braic sum, and state the distinction between it and arithmetical sum. Get the algebraic sums of several quantities,

numerical and literal, additive and subtractive; and in each case explain the operation.

(6.) Subtraction. Define algebraic difference, and explain the distinction between it and arithmetical difference. Get algebraic differences between several quantities, numerical and literal, additive and subtractive, and in each case explain the operation.

(7.) Multiplication. Define algebraic product, and give the distinction between it and arithmetical product. Form products of monomials, and explain fully the signs and coefficients and the literal part, in each result. State and prove the result when one factor is zero. Prove that the order of arrangement of the factors has no influence upon the value of the product. Extend the above principles to multiplication of polynomial by monomial, binomial by binomial, binomial by residual, and residual by residual.

(8.) Division. Define algebraic quotient, and give the distinction between it and arithmetical quotient. Divide a monomial by a monomial, and show clearly how you get the sign, the coefficient, and the literal part in the quotient. Also, divine a polynomial by a monomial, and explain as above. Show what is the effect, upon quotient, of increasing or diminishing divisor; of making dividend n times greater,- -n times less; of making divisor n times greater,-n times less; of making both dividend and divisor n times greater,- --n times less.

(9.) Use of Vinculum. Prove each of the following principles relating to use of vinculum, or parenthesis, and illustrate each by an example: 1st, Parenthesis preceeded by plus may be omitted without affecting the inclosed expression. 2d, Parenthesis preceded by minus may be omitted, provided we change sign of each inclosed term. 3d, Any quantity may be inclosed within a parenthesis and preceded by minus, provided the sign of each inclosed term be changed. The above principles are of great use in giving different equivalent forms to polynomials, in addition, and in subtraction. Write an example of each application.

(10.) Powers. Define power of a number, and give examples. Show how to indicate a power by use of sign of multiplication, and show how this expression may be shortened by use of an exponent. Explain clearly the precise meaning of this exponent.

Different powers of a same number are to be multiplied together; show how we may at once form the the exponent of the required product. A power of a number is also to be divided by a less power of the same number; show how we may at once form the exponent of the required quotient.

To be Continued.

Editorial Miscellany.

THE account of the proceedings of the series of Institutes held under the call and direction of Dr. Barnard, is finished in this number, and we shall hereafter give our usual variety of matter. The importance of the movement is our apology for occupying so much space with the proceedings, and our contributors must have patience, as their articles will appear in due time.

THE following summary of facts, collected in reference to the teachers in at endance upon the Institutes, will be of interest to all: Whole number in a tendance whose names were recorded, 1425; No. who filled blanks, 992; No less than 16 years of age, 35; No, between 16 and twenty, 360; No. between 20 and 25, 377; No. between 25 and 30, 121; more than 30, 85. Born in New-York, 439; Wisconsin, 89; Vermont, 76; Oh o, 65; Pennsylvania, 44; Massachusetts, 41; Maine, 40; England, 33; New-Hampshire, 25; Illinois, 21; Colnecticut, 16; Indiana, 13; Canada, 12; Michigan, 10, New-Brunswick, 6; Wales, Nova Sco ia and Scotland, each 5; New-Jersey, Ireland, and Norway, each 4; Mississippi, Germany, Siam, Prussia, Missouri, District of Columbia, and Spain, each 1-964. Students of Universities, 98; Colleges, 64; Seminaries, 126; Academies, 343; High Schools, 203-834. Have taught less than 1 year, males, 105, females, 159 -264; from 1 to 2 years, males, 79, females, 102—181; from 2 to 5 years, males 89, females, 83--172; from 5 to 10 years, males 27, females, 23--50; more than 10 years, males 30, females, 24-54. Have taught for less than $10 per month last year, males, 1, females, 88; from $10 to $15, males, 10, females, 138; from $15 to $25, males, 153, females, 106; from $25 to $35, males, 96, females, 30; more than $35, males, 19, females, 0. Who intend to make teaching a profession, males, 129, females, 170; have attended Normal Schools, 16; attended Normal Classes, 204; attended Institutes previously, 195; who own educational works, 213; who own Teachers' Library, 18; who have read one educational work, 206; more than one, 220; who subscribe for Journals of Education, 279.

FOND DU LAC.-E. C. Johnson, late of the High School of this city, has accepted a Professorship in Chicago University, and entered upon the discharge of his duties. Mr. J. has done a good work at Fond du Lac, and has left behind him warm friends, both in the schoolroom and the community. A faithful and efficient teacher, and one of the Board of Editors of this Journal, we are sorry to part with him, but give him our best wishes for his success in his new position. We have his promise that he will continue to take an interest in the Journal, and furnish matter for its columns. He is succeeded by Mr. S. H. Peabody, formerly of Philadelphia Polytechnic College, and late of Eau Claire, in this State, who has the reputation of a finished scholar, and successful teacher.

I think I may say, in the outset, that no study of such vital importance has ever been neglected so universally in our common schools as this. All of us remember where we first began to study "the science of computation by numbers," in that little book whose first lesson was to learn to count the stars in a certain triangular arrangement, where every side of the triangle counted ten, and how we studied, and counted our fingers to make correct answers to those progressive questions which came afterwards.

Well, so far, all right. But what followed? As soon as we were capable of telling how many times eight make forty-eight, we must have a slate and pencil, and another book, whose rules and principles were Greek and Latin to us, out of which to "cipher" like our large brothers. Here is the fatal step!

The idea of a scholar beginning to "cipher" when he is so young that he must have his pencil tied to his slate for fear he will lose it, in my humble opinion, is the height of folly, and should be tolerated by no teacher, under any circumstances. If scholars could never see a practical arithmetic until they were adepts in mentalizing, we should not see our seminaries and academies choked with stumbling, blundering mathematicians.

The greatest obstacle to be overcome in teaching any advanced branch of mathematics, is the almost universal dullness of scholars upon this particular point. In Algebra we find a wide field for mental operations, but so much neglected have been their mentalizing powers, that unless every part of an example is put in "black and white," even to its most minute