The course of arithmetic as taught in the Pestalozzian school, Worksop. [With] Answers

1. A line admits of lengthening or shortening; a surface admits of extension or diminution; a weight allows its being made heavier or lighter; time allows of increase or decrease, so does motion.

2. Quantity or magnitude is anything which will admit of increase or decrease. For instance, lines, surfaces, weight, time, motion are quantities.

3. The science by which we become acquainted with the properties of quantity is called mathematics.

4. If we were told, that in a certain house, there is a room twenty feet long, sixteen broad, and twenty high, we should at once form a correct idea of it, with regard to its dimensions, because we had a previous knowledge of the length of the foot. One foot is here called the unit or unity.

Suppose a piece of cloth contain twelve yards; here one yard is the unit; in a basket there are eight pebbles, one pebble is the unit, and so on.

5. Therefore, in mathematics, the unit is a measure of any kind, arbitrarily taken, to which we refer every quantity of the same kind.

A collection of units of the same kind constitutes a number, thus ten horses, sixteen yards, forty houses, &c., are numbers.

6. When numbers are considered in a general sense, without referring them to any particular thing, they are called abstract, as four, seven, twelve, &c., but when applied to particular objects, as two pounds, ninepence, fourteen yards, they are termed concrete numbers.

7. To represent numbers, to express them, to give the means of computing by them, and to apply them to practical purposes, constitute a branch of mathematics called arithmetic.

8. Numbers may be expressed in two different ways, by words and by signs or characters, called figures or digits.

9. The method of expressing any number by figures is called notation, and that of reading numbers so expressed, numeration.

10. Words taken from the Saxon are used for the English names of numbers; they are: one, two, three, four, five, six, seven, eight, nine, ten.

11. It is agreed to consider the number ten as a unit of the second order, which is called tens, and to proceed with the tens as we did from one unit to nine units; but, for the sake of brevity, instead of these expressions: one tens, two tens, three tens, four tens, five tens, six tens, seven tens, eight tens, nine tens-we say ten, twenty, thirty, forty, fifty, sixty, seventy, eighty, ninety.

12. Between ten and twenty, there are nine numbers: ten one, ten-two, ten-three, ten-four, ten-five, ten-six, ten-seven, ten-eight, ten-nine, which appellations have been changed into eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen.

13. Likewise, between twenty and thirty, there are also nine numbers, and we say twenty-one, twenty-two, twenty-three, twenty four, twenty-five, twenty-six, twenty-seven, twenty-eight, twenty-nine; thirty-one, thirty-two...thirty-nine; and so on, as far as ninety-nine. Therefore, ninety-nine is the largest number containing tens and units only.

14 Now, if nine tens and nine units, or ninety-nine, be increased by one unit, we obtain ten tens, or one hundred, which commences a third order of units, called hundreds. The hundreds ascend also from one to ten; but, for shortness, we use one hundred, two hundred, and so on, to nine hundred. As there are ninety-nine numbers between any one hundred and the following, by adding successively the first hundred numbers to each, we have: one hundred and one, one hundred and two, &c.; two hundred and one, two hundred and two, &c.; three hundred and one, three hundred and two, &c.; and so on, to

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The course of arithmetic as taught in the Pestalozzian school, Worksop ...