The New Art of Memory/Chapter 8

In this chapter, we have given faithfully the substance of Mr. Feinaigle's Lecture on Arithmetic, without any attempt at illustration; and as this Lecture has been accurately detailed in a recent publication,^[1] it is extracted from that work, but without any of the reporter's commentaries and observations.

"We have now to see how our methods will apply to Arithmetic.

"In this subject we think we have, or may have evidence, for every particular proposition. But let us think a little; in many cases we have certainty: but is certainty and evidence the same thing? For instance, we know that 6 multiplied by 6 gives 36: this is certain; but is it evident? All we can say is that we have learned so: but ​where is the evidence that 6 × 6 gives just 36? When you say that 6 × 6 is 36, you answer that it is three tens and six units; but see we this? How are we convinced that it is just 36, and no other number? It is only in our machine; but how it comes we know not.

We have these products given us in our multiplication tables, which we all know how difficult it is for children to learn; nay, many grown persons cannot learn it, because it is founded only upon the poor natural memory, upon which we can never depend. We make it only an object of memory instead of presenting it to the intellect, and we have no evidence, because we want the first evidence. To find the first evidence we must consider the figures themselves. Let us see then what is in the figures: we have

What comes after 9? Is it 10? No; 10 is a higher unit, and must therefore be placed before the 1; now what comes after 10? Is it 11? No; it is 20. Thus we find those nations did who employed letters for numbers: after having used the first 9 they went on thus, 10, 20, 30, ​&c. and not 10, 11, 12, &c. Thus change these numbers as you please, you will always find they go from 1 to 9, and by considering the numbers in this way, the child sees at once that the rapport of 10 to 60 is exactly the same with that of 1 to 6; and all the relations of these numbers are at once in his mind. The first thing then must surely be to give the evidence of those figures; after this every thing will be easy. In problems, the greatest difficulty is to understand the question; when we do that, the problem is half solved; the mind then acts like an algebraical formula. O! we see—put this here, and that there; do this, and do that, and it is done.

"Let us see then how we are to get the true idea of number.

Let this be one,

- - - - - - - - - O

Let it be one something, an apple, or an orange, or whatever, and let this be another,

- - - - - - - - - O

Now what have we here? Is this two? I see only two ones; and we say that these are equal to one two: But how know we this? Have we evidence in the thing itself that two ones are the same with one two ? I should see two things in one thus;

- - - - - - - - - Φ
- - - - - - - - - O

and the child sees at once that two halves are ​equal to one; and that two halves and one are equal to one two. In the same manner I have for three, a circle divided into three sectors; and the child sees at once that three thirds are equal to one; and that the half of 3 thirds is one half; and that three thirds and two halves and one are equal to three. And so on for the higher numbers.

"Thus the child sees at once the proportions between the fractions: those things which are most difficult to be learned by the common way are here the first to be acquired, because they go with the first conception. If I say give me one half of three thirds, or one third of one half, or one half of one third, or one third together with one half of one third, he gives me them at once, because he has a clear conception of their meaning. I give not these things to the child, he must give them to me; and it is wonderful what calculations many children will make when they go on with their reason; but all this is gone when they begin with the usual methods, because evidence is taken away, and commonly we find that the more instruction they receive, the difficulty is the greater. But in our method they proceed with pleasure, because they continue to have evidence; and I will engage that any child instructed in this method, would in one fortnight ​perform calculations of which you have no idea. So true is this, that if we were to unknow all that we have learned, and begin from the foundation, it would be better.

"We can go on with the same principles to Mathematics; in them we have three things, weight, measure, and number; but all are reducible to number.

"If we now represent our succession of units thus, and divide ten into two halves, as we have seen that this is necessary for representing two in one. If I ask what is 6 to 8, or give me one half of eight, and one fourth of eight, the child finds this at once, which is sometimes the solution of a difficult problem. If we go on to Addition, and ask what is 7 and 8? the child sees at once that 7 is equal to 5 and 2, and that 8 is equal to 5 and 3; so that 7 and 8 are equal to two fives and five, or one ten and five, or fifteen. In the same manner 6 and 6 are one ten and two, 8 and 8 are one ten and six, &c. &c. so that we see addition is certainly demonstrated thus; and subtraction is as evident.

"Let us go on then to Multiplication. Say that we ask how much is eight taken six times, the answer must be in tens and units, the child sees that
and 8 is equal to 5 and 3,
and 6 is equal to 5 and 1.

​

But let us see if this is not in our dots also,

We have certainly above a and below b and c four dots, which are the tens; above b we have four, and above c two; two multiplied by four give eight for the units, so that we have 48.

"In the same manner 7 multiplied by 9,
we have 6 tens, and one multiplied by 3,
or 8 units, that is 63.

And so 8 multiplied by 9, we have 7 tens, and one multiplied by two units or 72, and so in every other case; only the rule must be changed when we change the object of the question.

So that we see a child has no need of the multiplication table; he burdens not his mind with it; he sees not only the relation of the different numbers, but he sees all how they affect and combine with each other; all is in the nature of the thing; the evidence is before him.

"Let us now go on to Division. Suppose we have to divide 63 by 7; let us see if this is not included in the nature of the thing. We have

If we subtract the 7 from 10, we have 3; and ​if we add this to 6, we have 9 the quotient. Divide 54 by 9.

Subtracting the 9 from 10, and adding the remainder to 5, we have 6 the quotient.

And in cases where the dividend does not exactly contain the devisor, as in

we find by multiplication, that 8 multiplied by 9 gives 72, we have then 4 over, which is consequently 4 ninths.

"Thus in every case we have always the answer to the nearest whole number. Here also we have no need of the multiplication table, which, as I said before, is so difficult to learn, as the numbers themselves give us the answer; it is in their nature. You see then how easy it is to advance by our method, and we charge not the memory with what it is so difficult to fix."