NUMERATION.] 



MATHEMATICS. ARITHMETIC. 



433 



that teaches us how to perform calculations by means of 

 numbers. The rules which direct the various opera- 

 tions constitute the art of arithmetic : the reasons and 

 principles on which these are founded belong to the 

 theory of arithmetic ; and the theory and practice united, 

 form the science of arithmetic. It is this that we are 

 now going to explain. You are aware that the symbols, 

 or marks, employed in this subject are called figures, 

 and that they are as follow : 1, 2, 3, 4, 5, 6, 7, 8, 9, 

 together with the mark 0, called nought, or cipher, or 

 zero, and which stands for nothing. This is also called 

 &fii/ure, so that there are ten figures in arithmetic: the 

 number of units, or one*, which each stands for, is here 

 written 



nothing, one, two, three, four, five, six, seven, eigrht, nine. 

 1234567 89 



Each of these figures is also called a number: but the 

 word number has a wider meaning. Thus 26, 43, 57, 

 <fec., are all numbers, each of which consists of 

 fyures; the first number is twenty-six, the second forty- 

 three, the third fifty-seven, and so on ; so, that you see, 

 the 2 in the first stands for two tens, the 4 in the second 

 for four tens, and the 5 in the third for five tens. In 

 like manner, 368 is a number of three figures. The first 

 figure, 3, stands for three hundred; the second, 6, for 

 ' ns, or sixty; and the third, 8, for eight ones, or 

 : the number itself standing for three hundred and 



! !ius perceive that a figure which, when written 

 . , stands merely for so many units, changes its 

 meaning, or value, according to the place it occupies in 

 a number of several figures. If it occupy the last, or 

 right-hand place, it still stands for units ; but if it be in 

 the next place, to the left, it stands for so many tens ; if 

 in the place next to that, for so many hundreds ; and if 

 it occupy the fourth place from the end, it stands for so 

 many thousands : the number 7352, for instance, is seven 

 thousand three hundred and fifty-two. 



Figures thus have a local value ; that is, a value de- 

 pending upon the places they occupy in a number. The 

 following is a number of twelve figures ; and when the 

 local values, of these figures are written against them, it 

 supplies what is usually called the 



NCMERATION TABLE. 



73169 



And this number is read thus : two hundred and forty- 

 six thousand eight hundred and seventy-three MILLION, 

 one hundred and fifty-nine THOUSAND, four hundred 

 and thirty-eight. 



A person beginning to learn arithmetic will be enabled, 

 by means of the above table, to read any number that 

 is, to express its value in words. In a large number 

 like that here given, the easiest way to proceed is this. 

 Cut off the last three figures, then the next three, then 

 the noxt, and so on ; thus dividing the figuras into sets 

 of three as far as po.- .'<!. \ glance at the table shows 

 that the leading figure of each set is hundreds of some- 

 thing ; that of the first set, on the right, is hundreds of 

 or simply hundreds ; that of the next set is hundreds 



The mark*, or symbols, made UM of in any science, constitute the 



rice. 



>'i' frequently throw Hutu on the oriirin of Hiingt : it is interesting 

 , that the name digit is plainly significant of the early rude 



of thousands ; that of the next, hundreds of millions .' 

 and so on. And by thus finding out the local value of 

 the leading figure in each period, as it is called, you 

 may read the number with ease. For example, the 

 number 68547329, when divided into periods, as here 

 proposed, is 68,547,329: pointing to the 3, you sa,y hun- 

 dreds, and passing to the 5, hundreds of thousands ; the 

 incomplete period, 68, must therefore be 68 millions; 

 and the entire number 68 million, 547 thousand, 329 ; 

 or, expressing the value wholly in words, it is sixty-eight 

 million, five hundred and forty-seven thousand, three 

 hundred and twenty-nine. In a similar way we find the 

 number 42638572613, or 42,638,572,613, to be 42 thou- 

 sand 638 million, 572 thousand 613. If you wished to 

 put this wholly into words, all you would have to do 

 would be to write forty-two for 42, six hundred and 

 thirty-eight for 638, five hundred and seventy-two for 

 572, and six hundred and thirteen for 613 . The leading 

 figure of a complete period, you know, is always hun- 

 dreds, according to the plan we have proposed ; and when 

 you have found by the table what these hundreds are, 

 or from practice can recollect what they are, you can 

 have no difficulty in reading the number. When any 

 of the figures are noughts, a little extra care is, however, 

 necessary. Thus, the number 460305007, which, divided 

 into periods, is 460,305,007, is read four hundred and 

 sixty million, three hundred and five thousand, and seven. 



From what has now been said, you clearly see what is 

 meant by the local value of a figure. If it occupy the 

 place of units, its value is so many ones ; if it be in the 

 place of tens, its value is ten times as many ones that is, 

 it represents so many tens ; if it be in the place of hun- 

 dreds, its value is ten times as many tens ; if in the place 

 of thousands, it is ten times as many hundreds ; and so 

 on. So that, as you advance a figure, place after place, 

 towards the left, you increase its value tenfold at every 

 remove : thus, 8 is simply eight ones ; 80, where the 8 is 

 now in the second place, and nothing, or nought, in the 

 first place, is eighty, or ten times eight ; 800, where the 8 

 is in the thinl place, and noughts in the first and second 

 places, is eight hundred, or ten times eighty ; and so on. 

 This tenfold increase in the value of a figure, when it is 

 removed one place from right to left, explains why our 

 system of numeration is called the decimal system ; the 

 word decimal being derived from a Latin word, meaning 

 ten. It was a beautiful contrivance thus to give a loud, 

 as well as an absolute value, to the symbols, or figures, 

 used in the nutation of arithmetic.* You see that by 

 this happy idea we are enabled to express all numbers 

 whatever by the help of only ten different marks, or 

 symbols : whether we owe it to the Arabs, to the Greeks, 

 or Romans, is a question on which there is still some 

 doubt. 



It ought, perhaps, to be mentioned, that although the 

 numbers considered above do not extend beyond twelve 

 figures, numbers with more figures than these may occur, 

 ami tliat there are words to_ express the additional periods. 

 If the number have thirteen figures, the leading figure on 

 the left would stand for so many billions ; and if a com- 

 plete additional period were joined to a number of twelve 

 figures, making a number of fifteen figures, then the 

 leading figure on the left would, of course, be hundreds 

 of billions. But billions, trillions, quailrMions, <fec. , are 

 names so seldom employed or wanted, that the nume- 

 ration table need not be encumbered with them. It'may 

 be worth a passing notice, too, that no distinct ideas are 

 conveyed by any of these terms : beyond a very moderate 

 extent our notions of the value of numbers become con- 

 fused. The number of ones in a million, even, is hard 

 to conceive : it is a thousand thousand, and would take 

 more than twenty-three days to count, though you kept 

 at it for twelve hours a-day, and counted one every 

 second. Our ten figures, or digits, as they are often 

 called (digitus being Latin for finger, and our ten fingers 

 suggesting the wordt) our ten figures thus enable us 

 accurately to express on paper, without the error of a 



method of counting on the fingers ; and that the name calculation as 

 plainly refers to the primitive practice of reckoning with pebblft [calculus, 

 a pr-bble). In a similar way electricity derives its name from electron 

 amher; maynMtm, from magneiia. Sic. 



3 K 



