c2 = a2+b2.—If c2 = a2 + b2 and a, b, c have no common factor, then c exceeds a or b by an odd square.

No two of a, b, c must have common factor, or all three would.

The following problem was suggested by a difficulty in distinguishing between the two hands of a clock.

The difficulty in teaching Multiplication and Division of Fractions consists solely in the fact that the ideas formerly associated with these operations when whole numbers were concerned no longer fit the case when we come to deal with fractions, e.g., 9 × 7 can be intelligibly interpreted as meaning the number of articles in 9 bundles, each containing 7 articles. In this respect, multiplication is simply contracted addition. When we come to a case like , we cannot frame a question on the lines of the former. But if we present the problem in a concrete form we are more likely to be successful in teaching young pupils the method of multiplying and dividing vulgar fractions.

A big stumbling-block in the way of the child beginning Algebra is the transition from the conception of definite numbers in Arithmetic to that of indefinite quantities in Algebra, and the performance on these of the fundamental operations with facility and certainty. It is a truism that our algebraic teaching must grow out of our arithmetical: in its initial stages it may with advantage be based on some such method as the following. From a box of counters, or little wooden cubes, or in fact any handy articles, ask several pupils—better still, all the pupils, if the class be of convenient size—to pick out respectively five counters, eight counters, and so on. Their possessions can be exhibited on the black-board thus:—“A has five counters,” “B has eight counters,” etc. The counters being replaced, now ask others to take out some counters each. So long as the boys hold these in their closed hands no one can tell what quantity of counters any boy has; all that can be said is that each boy has some counters. Hence on the black-board is written:—“L has some counters,” “M has some counters,” and so on for each boy. Now the pupils have been accustomed in their arithmetical work to use various shorthand symbols in order to avoid writing out in full frequently occurring words or phrases. Thus, instead of the word “one” we use 1; the expression “multiplied by” is represented by ×, and so on. We therefore rub out the words “five” and “eight” that we wrote at first, and write in place of them their symbols “5” and “8.”

The following is meant as a set of working rules for applying the definitions of the trigonometrical ratios of any angle, where numerical evaluation is required. Let the initial arm of the angle be horizontal.

All these identities may be proved after one method as follows:—

Draw two straight lines at right angles and mark off parts a and b as indicated in the figures. At the points of section draw perpendiculars to the lines. All the figures thus formed are rectangles, and therefore have their opposite sides equal. The proofs should then be obvious.

(1) Let A, B, C represent Aberdeen, Glasgow, Edinburgh respectively on a map of Scotland. Let D, E, F represent the same places in order on a map of Scotland on a different scale. The straight line AB represents the road from Aberdeen to Glasgow, the straight line AC represents the road from Aberdeen to Edinburgh. The angle A represents the angle between these roads. The angle D represents the same angle.

I. To draw quickly a good figure of A, B, C, I, I1, I2, I3, etc. Draw a circle, and a chord BC. Mark L the middle point of the arc below BC by the “engineer's method,” viz., with B as centre and a radius as near BL as can be judged by the eye, make a mark on the arc, with C as centre and the same radius make another mark on the arc, judge by the eye the middle point of the arc between these marks; this is L. With centre L, radius LB or LC, describe a circle: I and I1 lie on this circle. Mark M the middle point of the arc above BC. With centre M, radius MB or MC, describe a circle: I2 and I3 lie on this circle. Now I and I1 lie on AL; I2 and I3 on AM. Mark A on the first circle, so that MA lies conveniently on the paper. The various collinearities and perpendicularities justify the figure to the eye; the properties of the mid-points of II1 etc., I2T3, etc., and the loci of I, I1 I2 I3 as A varies, are emphasised.