Similarly, subtraction ought to be actually performed by drawing away, and the word explained—its connection with drag, traction, tray, dray, etc. Thus the common fault of writing “substraction” may be avoided. It should be thought of as undoing addition. The signs - and = may now be given.
Analysis of numbers.(4) We learn by analysis and synthesis, i.e., to see the parts in the whole, and the whole as made up of parts. It is very useful at this stage to get children to group numbers, to think of 2, e.g., as 1 + 1, of 3 as 1 + 1 + 1 and 1 + 2, of 8 as 1 + 7, 2 + 6, 3 + 5, 4 + 4, 2 + 2 + 2 + 2. This is much insisted on in Germany and America. In kindergartens there are many pictures which are used for grouping numbers, thus, e.g., a seven-branched candlestick.
We may give 7, as 3 + 1 + 3, as 1 + 2 + 2 + 2, as 1 + 6. This makes numbers, so to speak, easily fall into their constituents, which will be shown to be of use later. I knew a child who habitually thought of the written figures as picturing the number. Children might arrange the 9 digits in various ways, thus, giving also the written figures:—
| · | ·· | ··· | ···· | ····· | ······ | ······· |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| ·+· | ·+·· | ·+··· | ·+···· | ·+····· | ·+······ | |
| 1+1 | 1+2 | 1+3 | 1+4 | 1+5 | 1+6 | |
| 2+2 | 2+3 | 2+4 | 2+5 | |||
| etc. |
At this stage the question would naturally arise why there are only 9 figures, and an historical digression could be conveniently made. I give a sketch of such a lesson before coming to more difficult and abstract things.
Historical methods.Dogs are very clever. A collie will go with the shepherd and take care that none stray. Suppose one has disappeared over a cliff when he was not looking, would he know one was gone, would he count like the shepherd? No, he will track out a lost sheep, by scent, as we cannot, but I never heard of a shepherd setting the dog to count. If puss has 3 kittens and you take 1, she seems not to know. Some savage races can count only a few numbers, but man carries a ready-reckoner in his fingers, and most can easily count up to 5 or 10, or, if taking in the toes, up to 20; all the higher races are marked out by their greater power of doing long and difficult sums.
Now, suppose some great owner of sheep, as Abraham or Jesse, sent out a shepherd with many sheep, how would he know each day whether they were all right? Well, the simplest way would be to have two stones for each—the master could have one bag and the man another, and then they could calculate each night; calculus is the Latin for a stone. The shepherd would need a long bag for his stones. Was that how David happened to have the one which he used as a sling to kill Goliath?
Suppose, however, the flock was very large, a bag of stones would be heavy. Has a shepherd something else, which, instead of his exactly carrying, seems to help to carry him? The shepherd’s staff. Could he not put notches on this for his sheep? It would hold a good many; but in days when people had to use stones for knives, it was not so easy to cut a great many notches, and besides it would get used up with a large flock. Could he not make a sign like a hand, V, for every 5 sheep? That was what the Romans did, and next they said, why not have a sign for two hands, X, and let that stand for 10? So, if they wanted to write sixteen sheep, they would put XVI instead of sixteen strokes. You see in the Bible the Roman numbers. The Greeks used letters, too, as the Romans did, for numbers.
Money.When people began to trade they wanted something more than tally sticks and stones—something the value of which all knew. Amongst pastoral people the most ready things to calculate by were sheep or cattle. A piece of land would be sold for so many sheep, but it would be very inconvenient to have to drive your money about, and so people seem very early to have had pieces of metal which were reckoned to be equal in value to sheep or cattle, and to save weighing, each piece had, perhaps, a sheep scratched on it; and this was called in Latin (from pecus, cattle) pecunia, i.e., the piece of metal representing the value of cattle. This would be carried about and exchanged. Lawyers now put in our wills “goods and chattels”; by the first they mean houses and lands, which cannot be moved; by the latter, things which, like cattle, can be moved. Then people could have larger and smaller pieces of money, representing half or a quarter of a sheep, or many sheep.