In another direction it is possible to give the child algebraic ideas much beyond anything we would imagine. Suppose, for example, we want to give him a conception of addition. He easily realizes that objects--material bars, for example--can be selected so as to represent numbers by their length. He can be readily made to understand that if he has one bar three and another five inches long he can obtain the sum of these lengths, in what we might call a material way, by placing them lengthwise, one at the end of the other--an essentially practical notion and easily carried into effect. If we take a line and mark a starting point on it, calling it zero, then measure off segments on it representing the bars we have been talking about one after another, we can get the sum represented by the length of the two segments. If, instead of measuring three plus five inches I measure three plus two I reach another point. If, instead of adding two and three, I wish to take one of the bars or numbers away (3-2), or subtract, the operation will be easily performed by measuring the two in the opposite direction. The difference will be represented by the length that is left. If we try to form the quantity 3-5 in arithmetic we can not do it; but in proceeding in this method and measuring back on the bar we get to a point back of the original starting point which represents this difference--say two inches behind where we began. Here we have in the germ the whole theory of negative quantities, concerning which thousands and thousands of pages have been written. Yet we find that by carefully graduating our lines we can make it intuitive and accessible to a child who has learned that the common operations of addition and subtraction can be represented with material objects. The generation of negative and positive quantities follows quite naturally. These examples, I think, are sufficient to show that we might considerably enlarge the field of the investigations within reach of the child. For this purpose a small amount of very simple material, which we can vary as we please, is needful. The first element of this material is paper ruled in squares, a wonderful instrument, which everybody dealing with mathematics or with science generally should have. It is of special pedagogic use in giving children their first ideas of form, size, and position, without which their early instruction is only a delusion. Add to this paper dice, buttons, beans, and match-sticks--things always easy to get--and we have all the material we need.