"Double the first figure, set it down, and use it as a trial divisor for the two first figures. Place the quotient thus found to its right, and then divide as usual. (Set down 6 after the 4 and multiply 46 by 6.)

"26 is the number sought, and is the square root of 676."

Really, as I recite it, the rule sounds more like a riddle, or a series of instructions in numerical legerdemain, than an appeal to the understanding. Whatever be the accuracy or worth of the result produced, it is certain that the process so described will do more to deaden than to invigorate the thinking faculty of any one who practises it. Moreover, as the rule appears utterly arbitrary, the memory will have great difficulty in retaining it, and without constant and toilsome practice, will probably not retain it at all.

thetical

Now before describing to you the rational process of The Synattaining this result, I may remind you that in the earlier before the part of arithmetic the rules came in pairs. Thus, in Analytical Rule. Addition, you have the parts given, and are required to find the whole; and this rule is followed by Subtraction, in which you have the whole given and one of the parts, and are required to find the other part. So also in Multiplication, the factors are given, and you have to find their product; and then there is the inverse process of Division, in which the product and one of the factors are given, and you are required to find the other factor. In each case the former process is one of synthesis, or putting parts together, and the latter process one of analysis or decomposition of parts. But we all feel this order to be a natural and proper one. You would not teach Subtraction before Addition, nor Division before Multiplication; because unless a learner in this science first knows how to put the parts together to make the

F. L.

22

Involution.

result, he is not in a position, with the result before him, to find out how that result is produced. Now the rule for finding the square root of a number is obviously a rule of decomposition or analysis, and is one of a pair of rules, analogous to Multiplication and Division, of which the one shews how to form the second power of a number out of the multiples of its parts, and the other shews how, when the second power of a number is given, to find the parts of that number of which it is the second power. But this rule for Evolution presupposes the rule of Involution; and cannot, in fact, be properly understood, unless that rule has first been learned. Yet in text-books of arithmetic, no mention is generally made of Involution, but the pupil is introduced at once to the Extraction of the Square Root.

Instead therefore of departing from the analogy of the earlier rules of arithmetic, and plunging at once into the rule for the extraction of roots, before we examine the formation of squares, let us begin by trying to find the second power of an easy number composed of two parts. Thus :

Because II = 7+4; then 11 × 11 = (7 + 4) × (7 +4).

But on multiplying each of these parts of eleven by each of the parts of eleven successively, and adding them together, I find I have four distinct products, of which the first is the square or second power of 7, the last is the square or second power of 4; and the remaining two are alike, each being the product of 7 and 4.

And in this way we may easily arrive at this general truth:

"If a number consists of two parts, the second power of the whole number consists of the second power of the first part, together with the second power of the second part, together with twice the product of the first and second parts."

I will suppose that you have, by the help of varied illustration, made your pupils perfectly familiar with this proposition-and led them to recognise it under the general abstract formula ::

If a = b + c then a2 = b2 + c2 + 2bc.

You are now in a position to deal with the problem Evolution. originally proposed: Find the square root of 676.

But if so, the remainder 276 must contain not only the square of that other number, but twice the product of the number 20 and that number. With a view to find that number, try how many times twice 20 are contained in the remainder. The number 6 appears to fulfil this condition. See now, if 276 contains six times 40, together with 6 times 6, or 6 times 46 in all. If so, 6 is the unit figure of the required root. It has now been shewn

that 676 contains the square of 20, and the square of 6, and twice the product of 20 and 6,

The whole explanation of this inverse process is evidently deducible from the simple law of involution first described. The reason of the pupil follows every step, and acquiesces in a rule, otherwise primâ facie absurd, and therefore hard to remember. All this is of course very familiar and simple to the student of algebra; but I have never been able to understand why it should be postponed to algebra, or why the principles of arithmetic, requiring as they do for their elucidation no use. of symbols, no recondite language, nothing but simple numerical processes, should not be taught on their own merits, and in their own proper place.

Analogous An appeal to the eye will greatly help the undertruths in Arithmetic standing of the rule for the extraction of the square root. and A square may be erected on a line divided into two unGeometry. equal parts, and it will be seen to be separable into four spaces whose dimensions correspond to the products. just given. Afterwards a square on a line divided into three or more parts may be shewn, and the dimensions of the several parts may be expressed in numbers. In like manner every proposition in the Second Book of Euclid may be compared with some analogous proposition respecting the powers and products of numbers. But it is important here not to mistake analogy for identity. Some teachers seem to think they have proved the theorems in geometry when they have expressed the corresponding truths in algebraic symbols. The use of the word 'Square,' both for a four-sided figure and for the second power of a number, is a little misleading; and

Arithmetical and Geometrical Analogies. 341

Euclid's use of the terms Plane and Solid numbers in his Seventh Book would have further mystified students had it been commonly accepted. But since Geometry is founded entirely on the recognition of the properties of space, and Algebra and Arithmetic on those of number, it is necessary to preserve a clear distinction in the reasoning applicable to the two subjects. Except as shewing interesting analogies, the two departments of science should be kept wholly separate; and while the truths about the powers and products of numbers should be investigated by the laws of number alone, geometrical demonstrations should be founded rigorously on axioms relating to space, and should not be confused by the use of algebraic symbols.

Our attention to-day has been necessarily confined Algebra to the consideration of a rational way of treating Arith- and Geometry. metic, the one department of mathematics with which, in a school, the teacher is first confronted. But the same general design should be in the mind of the teacher, through Geometry, Algebra, Trigonometry, the Calculus, and all the later stages of mathematical teaching. While constantly testing the success of his pupils by requiring problems to be worked, he will nevertheless feel that the solution of problems is not the main object of this part of his school discipline, but rather the insight into the meaning of processes, and the training in logic. If Algebra and Geometry do not make the student a clearer and more accurate and more consecutive thinker, they are worth nothing. And in the new revolt against the long supremacy of Euclid, as represented in the Syllabus. of the "Association for the Improvement of Geometrical Teaching," the one danger we have to fear is that the demonstrative exercises will be cut up into portions too small to give the needful training in continuity of thought.