CHAMBERS'S INFORMATION FOR THE PEOPLE. 



simplicity and its cumbrousness. The following 

 seems to be the most probable theory of its devel- 

 opment. A simple series of strokes was the basis 

 of the system ; but the labour of writing and read- 

 ing large numbers in this way would soon suggest 

 methods of abbreviation. The first and most 

 natural step was the division of the strokes into 

 parcels of tens, thus, Wffl\ QttttttQ, a plan which 

 produced great facility in the reading of numbers. 

 The next step was to discard these parcels of ten 

 strokes each, retaining only the two cross strokes, 

 thus, X as the symbol for 10. Continuing the 

 same method as larger numbers came to be used, 

 they invented a second new symbol for 100, thus, 

 C (which was at first probably the cancelling 

 stroke for ten Xs, in the same way as X was 

 originally the cancelling stroke for ten units) ; and 

 for the sake of facility in writing, subsequently 

 employed the letter C, which resembled it, in its 

 place. The circumstance that C was the initial 

 letter of the word centum, 'a hundred,' was doubt- 

 less an additional reason for its substitution in 

 place of the original symbol for 100. An exten- 

 sion of the same process produced M, the symbol 

 for loco, which was also written A> CO > an ^ very 

 frequently Cl3. This symbol was probably sug- 

 gested by the circumstance that M was the initial 

 letter of the Latin word mille, signifying a thou- 

 sand. The early Roman system went no higher. 

 But though the invention of these three symbols 

 had greatly facilitated the labour of writing down 

 and reading off numbers, further improvements 

 were urgently required. The plan of ' bisection of 

 symbols ' was now adopted ; X was divided into 

 two parts, and either half, V or A, used as the 

 symbol for 5 ; C was similarly divided, f", or 

 L, standing for 50 ; and K, CI, or 13, was 

 obtained in the same manner, and made the 

 representative of 500. The resemblance of these 

 three new symbols to the letters V, L, and D, 

 caused the substitution of the latter as the numeri- 

 cal symbols for 5, 50, and 500. A final improve- 

 ment was the substitution of IV for 4 (in place of i 

 IIII), IX for 9 (in place of VIIII), XC for 90 

 (instead of LXXXX), and similarly XL for 40, CD 

 for 400, CM for 900, &c. ; the smaller number, 

 when in front, being always understood as sub- 

 tractive from the larger one after it This last im- 

 provement is the sole departure from the purely 

 additional mode of expressing numbers ; and if 

 the symbols for 4, 9, 90, &c. be considered as 

 single symbols, which they practically are, the 

 deviation may be looked upon as merely one of 

 form. In later times, the Roman notation was 

 extended by a multiplication of the symbol for 

 loco, thus CCI33 represented 10,000 ; CCCI333 

 represented 100,000, &c. ; and the bisection of j 

 these symbols gave them 133 and ID 33 as 

 representative of 5000 and 50,000 respectively. 

 This, in all probability, is the mode according to 

 which the Roman system of notation was con- 

 structed. To found a system of arithmetic upon 

 this notation would have been well-nigh impos- 

 sible ; and so little inventive were the Romans, 

 that the attempt seems never to have been made. 

 They performed what few calculations they re- 

 quired by the aid of the Abacus. 



Greek System. The Greeks at first used a ; 

 method similar to the Romans, though at the ! 

 same time they appear to have employed the ! 

 letters of the alphabet to denote the first 24 num- ! 



594 



bers. Such a cumbrous system was naturally 

 distasteful to so fastidious a race, and they hit 

 upon the happy expedient of dividing their alpha- 

 bet into three portions using the first to symbol- 

 ise the 9 digits, the second the 9 tens, and the 

 third the 9 hundreds. But as they possessed only 

 24 letters, they had to use three additional 

 symbols ; their list of symbols of notation then 

 stood as follows : 



By these symbols, only numbers under 1000 

 could be expressed, but by putting a mark, called 

 iota, under any symbol, its value was increased a 

 thousandfold, thus f- = 1000, * = 20,000 ; or by 

 subscribing the letter M, the value of a symbol 

 was raised ten-thousandfold, thus, " = 80,000. For 



M 



these two marks, single and double dots placed 

 over the symbols were afterwards substituted. 

 This improvement enabled them to express with 

 facility all numbers as high as 9,990,000, a range 

 amply sufficient for all ordinary purposes. Fur- 

 ther improvements were made upon this system 

 by Apollonius, who also, by making 10,000 the 

 root of the system, and thus dividing the symbols 

 into tetrads, greatly simplified the expression of 

 very large numbers. Both Apollonius and Archi- 

 medes had to a certain extent discovered and 

 employed the principle of giving to symbols values 

 depending on their position and multiplicative of 

 their real value, but this principle was applied to 

 tetrads or periods of four figures only, and the 

 multitude of symbols seems to have stood in the 

 way of further improvement. Had Apollonius, 

 who was the chief improver of the system, dis- 

 carded all but the first nine symbols, and applied 

 the same principle to the single symbols which he 

 applied to the ' tetrad ' groups, he would have 

 anticipated the decimal notation. 



The Decimal System. The first nine numbers 

 are represented by nine separate symbols, or 

 figures, known as the Arabic numerals ; namely : 



i> 2, 3, 4, 5, 6, 7, 8, 9. 



one, two, three, four, five, six, seven, eight, nine. 



The nine figures, with the addition of the symbol 

 o, enable us to write any number, however large 

 or however small it may be. In the first place, 

 we suppose these figures to represent not only 

 simple units, but units of any order whatever ; 

 3 may represent three simple units, three tens, three 

 hundreds, &c. ; and so with the other figures. As 

 the figures then represent units of any order, we 

 must see when they represent units of a particular 

 order. Taking several columns, as is here shewn 

 and let it be understood that the simple units are 

 placed in the column nearest to the right, the tens 

 in the column next on the left, the hundreds in 

 the column next again, and so on then we may 

 write down any number; as, for example, three 



