= great deal for himself. With here and there an apt suggestion, he has been able to name the numbers from ten to twenty; to name the tens (three-ty thirty, etc.) up to ten tens (the child will probably call it ten-ty), when he is given a new name for the new unit of measureone hundred; and so on with the other numbers that he has been using. The Symbols. From idea to name, and name to symbol, is the order. As he names with but little aid from suggestion, so he needs but little assistance in notation. He is given the digits and the zero, and knows their significance. He knows that 1 denotes any one unit of measure whatever, and that O denotes no unit, no quantity. He is told that the expression for a 10-unit quantity is 10, meaning one 10-unit and no one-unit. The unit of reference, the one-unit, being called simply the unit, he will pass immediately to expressions for one ten and one unit, one ten and two units, one ten and three units, . . . one ten and ten units-i. e., two tens, or twenty. He will also express without any help two tens and no units, three tens and no units, etc.; then two tens and one unit, two tens and two units, up to ten tens and no units, which he will write at once as 100; counting up and expressing, that is, one ten (10), two tens (20), three tens (30), ten tens (100), just as he has counted up and expressed the one-units, 1, 2, 3, 4 . . . 10. But he knows that the ten tens make a new unit of measure, viz., one hundred; he sees the significance of the 1 here, as in 1 (one-unit) and in 10, and counts and expresses his counts in exactly the same way-one hundred (100), two hundred (200), three hundred (300), etc., to ten hundred (1000). He has learned also that ten of the hundred units make a new unit of measurethe one-thousand unit-and he now sees the significance. of the 1 in this place (the fourth place), and can go on counting and expressing his counts of the ten-thousand units. Proceeding thus to any desired extent, he has almost, unaided, mastered the principles of the decimal system-the use of the zero, the absolutely unchanging values of the digits as numbers, the values of the units of measure denoted by any digit according to its place in the series, the single figure denoting so many one-units (so many units of reference-yard, dollar, pound, etc.), the second figure to the left so many ten-units, the third so many hundred-units, the fourth so many thousandunits, the fifth so many ten-thousand units, etc. He will see that 7, 75, 754, always denote seven, seventy-five, seven hundred and fifty-four respectively, the position of the figure or figures in each case giving the unit; and will note that, in reading the numbers expressed by the figures, the figures are always taken in groups of three-for example, in the number 745,745,745, each 745 is read seven hundred and forty-five, the difference being in the unit only: 745 of the million-unit, 745 of the thousand-unit, and 745 of the one-unit, the primary unit of reference. The Decimal Point.-Since the pupil knows, if rightly taught upon the idea of measurement, that the unit of reference-the metre, the yard, the dollar, etc.— may itself be measured off in ten parts, or a hundred parts, etc., he will be curious to learn how the parts may be expressed. Knowing that 1 denotes 1 unit, or 1 tenunit, or 1 hundred-unit, etc., according to its position, he will be eager to learn how it may be used to denote the one-tenth unit, ten of which make up the one unit (of reference); the one-hundredth unit, one hundred of which make up the unit; the one-thousandth unit, one thousand of which make up the unit, etc. He actually sees, for example, that the metre is divided into ten equal parts (tenths), each of these into ten parts (hundredths), etc. How are these to be expressed? He will have but little difficulty with the problem. In the quantity expressed by 111 metres (or dollars), he knows that the 1 on the right denotes one-metre, the next 1 one 10metre unit, the third 1 one 100-metre unit; and passing from left to right, he knows that the second 1 denotes one tenth of the first, and the third one tenth of the second. Can we place another 1 to the right of the third, to denote one tenth of a metre, then another to denote 1-hundredth of the metre, etc.? Yes, if we in some way mark off the figures representing the subdivisions of the unit (metre) from the multiples of the metre. We might distinguish the 1's denoting metres from those denoting parts of the metre by drawing a vertical line between them-thus: 111 | 111-the figures to the left of the line denoting, respectively, 1 metre, 1 ten-metre, 1 hundred-metre; and those to the right denoting 1-tenth metre (decimetre), 1-hundredth metre (centimetre), and 1-thousandth metre; and both constituting one series governed by the same law, namely, increasing throughout from right to left by using ten as a multiplier, and decreasing throughout from left to right by using ten as a divisor-i. e., one tenth as a multiplier. But, instead of such a separating line, it is more convenient to use a dot, called the decimal point, to mark the place of the figure expressing the single unit-i. e., the unit of reference. Thus the number expressed by the ones previously given will be expressed by 111 111. The first figure to the left of the unit-figure whose position is thus marked, for example, in 453•453 metres, denotes tens; the first figure to the right, tenths; the second figure to the left, hundreds; the second figure to the right, hundredths; the third figure to the left, thousands; the third figure to the right, thousandths, etc. It will be readily observed, too, that the figures to the right of the decimal point are read in groups of three, just as those to the left are. As denoting a number, 453 is always four hundred and fifty-three. In this example it is on the left side of the decimal point, 453 metres (primary units); on the other it is 453 millimetres, etc. Here, as everywhere, there must be a good deal of drill, in order that the pupil may acquire perfect facility in reading and writing numbers; this means, ability to read automatically any number and its unit of measure, and similarly to express any quantity that may be named. For example: naming each period according to its unit of measure, name the first period (group of three figures) to the left-the units period; the second-the thousands (thousand-unit) period; the third to the left-the millions period; the first period to the right the thousandths period; the second to the right-the millionths period. Make the figure 7 express billionths, hundred thousands, tens, tenths, billionths; make 45 express tens, hundreds, thousandths, millionths, etc. Care is to be taken to name correctly the measuring units in the periods to the right-for example, 00573 is five hundred and seventy-three hun dred-thousandths; 0006734 is six thousand seven hundred and thirty-four ten-millionths. ADDITION AND SUBTRACTION. Addition. In addition, as we have seen, we work from and within a vague whole of quantity for the purpose of making it definite. If a quantity is measured by the parts-2 feet, 3 feet, 4 feet, 5 feet—we do not arrive at the definite measurement by simply counting the number of the parts; we have to count the number of the common unit of measure in all the parts, and so find the whole quantity as so many times this common unit. In learning addition, the countings are associated with intuitions of groups of measuring units, and the results stored up for practical use. The pupil who has been properly trained does not, in the foregoing example, start with 2, count in the 3 by ones, then the 4 by ones, etc., though this counting was part of the ini tial stage even when aided by the best arrangements of objects, by which he at last perceives that 5 + 4 = 9, without now counting by ones. Addition may therefore be considered as the operation of finding the quantity which, as a whole, is made up of two or more given quantities as its parts. The parts are the addends (quantities to be added), and the result which explicitly defines the quantity is the sum. It follows that in every addition, integral or fractional, all the addends and the sum must be quantities of the same kind--i. e., each and all must have the same measuring unit. Not only is it impossible to add 5 feet to 4 minutes; it is impossible to add 5 feet to 4 rods-i. e., to express the whole quantity by a number (denoting so many units of measurement)— |
