ARITHMETIC. 317 



marks, meeting in the shape of the letter V. Ten, being expressed 

 by two V's, one of them inverted, would form the letter X. Fifty, 

 being expressed by two straight marks in a new position, would 

 form the letter L, which might also have been derived from the lower 

 half of the letter C : but C and M were more probably abbreviations 

 of centum and mille, the Latin words for 100 and 1000. 



Early in the second century of our Era, Ptolemy introduced the 

 Sexagesimal notation; which probably originated in India, where 

 it is still used. From 1 to 59, the numbers were expressed by the 

 Roman numerals ; but 60 was represented by I with a dash on the 

 right, (F) ; twice 60 by (II'), and so on to 60 times 60, which was 

 represented by I with two dashes, (I") ; and so onward. Hence came 

 our division of the hour and degree into 60 minutes, and of the minute 

 into 60 seconds. The Decimal notation, already noticed, as intro- 

 duced by the Arabians, is proved by Montucla, almost conclusively, 

 to have been invented in India, before the Arabian conquests. The 

 first European writer who is known to have employed this system, 

 was Jordanus Nemorarius, A. D. 1230 : but it was not applied to 

 fractions till nearly 300 years thereafter. The rules of False Posi- 

 tion, were also borrowed from the Arabians. 



We shall treat further of Arithmetic, under the heads of, 1, Ground 

 Rules ; 2, Denominate Numbers ; 3, Fractions ; 4, Proportion ; 5, 

 Mercantile Rules ; and 6, Powers and Progressions. 



1. The Five Ground Rules of Arithmetic, constituting the Ara- 

 bic Jllgorithm, are Numeration, Addition, Subtraction, Multiplication, 

 and Division. Numeration, is the art of writing or reading num- 

 bers, expressed by means of figures ; whereas Notation rather sig- 

 nifies the kind of figures used, as the Roman, or Arabian. The nine 

 digits and cypher, of the latter notation, express directly any number 

 less than ten : but every ten is regarded as a distinct quantity, and ex- 

 pressed by a figure in the second place ; that is, with another figure 

 on the right of it, to make it stand in the tens' place, and to express 

 the remaining units. Every ten of these tens, forms one hundred ; 

 which is written in the third place, with two other figures necessarily 

 on the right of it, for the surplus tens and units. Thus, the great 

 principle of Numeration, is, to regard every number as composed of 

 units of different kinds ; each higher unit being equal to ten of the 

 next lower, throughout the scale. For example, the number 1841, 

 is written as if it were the sum of the numbers, 1000, 800, 40, and 1, 



Addition, is the putting together of two or more numbers, and 

 finding of their sum. For this purpose, we write the numbers, one 

 under another, so that all the right hand figures or units may be in 

 the same column, or vertical line ; after which, we add all the units 

 together, and find how many tens they make, setting down the 

 remainder for the odd units of the sun. We then carry the tens 

 thus found, that is add them, to the column of tens ; and find how 

 many tens of tens, or hundreds this column contains, setting down 

 the surplus tens as the odd tens of the sum. In like manner, we 

 carry the hundreds to the third column, and so repeat the process till 

 we come to the last column ; the whole sum of which being written 

 on the left, completes the total sum required. 



2D2 



