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135 | 22
| Page Line Faults.
READ 59 | 23 | 12 .. 4 :: 14 6. ..... 12 : 4 :: 14 : 6 105 -362 (in the Divisor)
-36 3 119 23 (* tate
repeated all along
with the third.
7 152 13
product. 159 27 S=(21-11) S=(21-d. 161 20 roportioni......
progression. 162 9 by Example....
for Example. 166 | 22 pass by such ...........
pass through. 170 3 2,22441..
2,262451 173 30 265 4800
2655214 174 1(& in two other places) 174 20 0,026109
See the note, page 275. 182 3 Ais. 1004
dns 1104 184 11 Product 884,630.
885,1030 188 27 (Quotient) 207 + 230 + 40 208 10 Ans 181116 30s. 13dwt. 181116 103 · 13dwt. f 219 d. (4th Example) įd. 241 10 Ad....
1d 242 14 Ans ..340
4d 245 cube root of 6983587... 65450827
GENERAL PRINCIPLES OF
NUMERATION. ARITHMETIC is the Science of Numbers. Numbers are expressed by means of ten characters called figures ; viz. 0,1,2,3,4,5,6,7,8,9.
The manner of using these ten figures, to express every number imaginable, is called the Art of Numeration, which is founded upon this principle, that, if several figures be written in an horizontal line, such as 5473, the units of any one of these figures, of 7 for instance, are each of them worth ten times a unit of the figure placed at its right, and ten times less than the units of the figure at its left. Thus, the first fig. 3 expressing three single units, the fig. 7 will express seven tens of the same units; the fig. 4, four tens of tens or four hundred, and so on; ten units of any fig. being equal to one unit of the fig. placed immediately at its left.
This principle being once settled, another convention has been necessary, for reading easily every sort of vumbers. Let us suppose a considerable number,
578 | 632 | 459 ; this number is mentally divided into members of three figures each, begina
billions. millions. thousands.
such as 4
ning by the right, and proceeding towards the left ; each of which members having its peculiar appellation ; namely, the first on the right, being called the member of units; the second, of thousands ; the third, of millions ; the fourth, of billions, &c. &c.The menibers being thus denominated, can be each of them read separately, considering that each complete member contains units, tens, and hundreds. Let us, for instance, take the member of thousands : If the first fig. 2, represents thousands, the 3, at its left, will express tens of thousands ; and the 6, at the left of the 3, hundreds of thousands ; so that this member contains Six hundred thirty two thousands. On this principle, the total number will be thus ready-four billions, five hundred and seventy eight millions, siz hundred and thirty two thousand, four hundred and fifty nine units.
Whereupon it is to be obseryed,--Firstly, that the 0, or cypher, which by itself has no value, answers nevertheless two important objects in a number. It not only indicates that there are no units of the order in which it is placed in that number, but also serves to give a tenfold value to the figures placed at its left. Thus, for instance, in the number 3704, the cypher shews, in the first place, that there are no tens ; then it is evident, that it makes all the figures, at its left, ten times greater than they would have been, had the cypher been omitted ; for, if we take it off, and write 374, the 7, which expressed hundreds, would now indicate only tens ; and the 3, which was of thousands, would now only be hundreds :
Therefore, Secondly ; that in order to make a number 10, 100, 1000, &c. times greater, it suffices to write one, two, or three, &c. cyphers at the right of that number. Let us take the number 25; if I add a cypher and write 250, the fig. 5, which was of units, becomes tens ; and the %, which expressed tens,
now indicates hundreds ; consequently each of these two figures has become ten times greater; therefore the whole number has been itself made ten times greater.
By a similar reasoning, we shew, that if we put two cyphers, and write 2500, the number 25 has been made one hundred times greater, and so on.It is sufficiently evident, that cyphers, placed at the left of a number, alter not its value.
Thirdly; that, if we suppress one, two, three, &c. cyphers at the right of a number, we make it 10, 100, 1000, &c, times smaller ; this is a necessary consequence of the foregoing remark.
Conformably to the above principles, if we wish to express the number ten, we write 10, since 1 being, by the accession of 0, made ten times greater, becomes a ten. To express eleven, we write 11, that is one ten, more one unit ; for twelve, we should write 12, which is one ten and two units ;-to express twenty fave, we write 25, or two tens and five units. Proceeding on the same way, we see that it is easy
to form the series of natural numbers, which will be infinite, since we may always add a figure to any number, however great it may be supposed.
OF DECIMAL PARTS.
If we conceive the unit divided into ten equal parts, then each of these into other ten, and so on, we shall have the notion of a series of new numbers, expressive of parts of the principal unit : these are what we called Decimals. But in the same way, in which, by progressively decupling the principal unit, we have formed tens, hundreds, thousands, &c. which compose an ascending suite from the right to the left, we may take a descending one, in the opposite order, which progressively decreases in a tenfold proportion, such