A R I T H M E T I C-A L G E B R A. 



IN the present and succeeding sheets, an attempt 

 is made to convey to those who may not have 

 had the benefit of a regular course of instruction 

 in the subject, some knowledge of Mathematical 

 science, both as regards measurement by numbers 

 (ARITHMETIC) and measurement of dimensions 

 (GEOMETRY). The sketch we offer of each is 

 necessarily brief and imperfect ; but our end will 

 be gained if we afford that amount of information 

 on the subject which is generally possessed by 

 persons of moderately well-cultivated intellect 



A recognition of the value of numbers is coeval 

 with the dawn of mental cultivation in every com- 

 munity ; but considerable progress must be made 

 before methods of reckoning are reduced to a 

 regular system, and a notation adopted to express 

 large or complex quantities. An inability to reckon 

 beyond a few numbers is always a proof of mental 

 obscurity ; and in this state various savage nations 

 have been discovered by travellers. Some are 

 found to be able to count as far as five, the digits 

 of the hand most likely familiarising them with 

 that number ; but any further quantity is either said 

 to consist of so many fives, or is expressed by the 

 more convenient phrase, ' a great many.' Among 

 the North American Indians, any great number 

 which the mind is incapable of distinctly recog- 

 nising and naming, is figuratively described by 

 comparing it to the leaves of the forest ; and in 

 the same manner the untutored negro of Africa 

 would define any quantity of vast amount by point- 

 ing to a handful of sand of the desert 



On the first advance of any early people towards 

 civilisation, it would be found impossible to give 

 a separate name to each separate number which 

 they had occasion to describe. It would therefore 

 be necessary to consider large numbers as only 

 multiplications of certain smaller ones, and to 

 name them accordingly. This is no doubt what 

 gave rise to classes of numbers, which are different 

 in different countries. For instance, the Chinese 

 count by twos ; the ancient Mexicans reckoned 

 by fours. Some counted by fives, a number which 

 the fingers would always be ready to suggest. The 

 Hebrews, from an early period, reckoned by tens, 

 which would also be an obvious mode, from the 

 number of the fingers of the two hands, as well 

 as of the toes of the two feet. The Greeks adopted 

 this method ; from the Greeks it came to the 

 Romans, and from them to us and all civilised 

 nations. 



NUMERATION. 



Numeration spoken. As a number may be in- 

 creased to any extent (for however great it may 

 be, it is always possible to increase it by adding 

 another unit to it), it is evident that the number of 

 numbers is infinitely great, and consequently it 

 would be impossible to express them in any lan- 

 guage by means independent of each other. It is 

 necessary, then, to find the means of expressing all 

 numbers with a limited system of words combined 

 90 



in a convenient manner. The first numbers have 

 received names independent of each other, and 

 are one (or the unit by itself), two (or a unit more 

 than one), three, four, five, six, seven, eight, nine. 

 The number following is called ten. It is regarded 

 as a new unit, or a unit of a new order. Then we 

 count by tens, as we do by units, namely, one ten, 

 two tens or twenty, three tens or thirty, forty, 

 fifty, sixty, seventy, eighty, ninety. Between ten 

 and twenty there are nine other numbers, namely, 

 eleven, twelve, thirteen, fourteen, fifteen, sixteen, 

 seventeen, eighteen, nineteen. Between twenty and 

 thirty there are nine numbers, and also between 

 thirty and forty, forty and fifty, &c. In this way 

 we arrive at ninety-nine, and the addition of one 

 to this gives one hutidred. 



We reckon by hundreds as we do by tens, thus : 

 one hundred, two hundreds, three hundreds, &c. 

 Then by placing successively between the words one 

 hundred and two hundred, between two hundred and 

 three hundred, &c. the names of all the numbers 

 from one up to ninety-nine, we have at last the num- 

 ber nine hundred and ninety-nine, and the addi- 

 tion of one to this gives ten hundred or one thou- 

 sand. This, again, is taken as a new unit, and we 

 say one thousand, two thousand, three thousand, 

 &c. . . . ten thousand . . . nine hundred and 

 ninety-nine thousand. The number nine hundred 

 and ninety-nine thousand nine hundred and ninety- 

 nine can thus be formed, and the addition of a unit 

 to this gives ten hundred thousand, or a thousand 

 thousands, or one million. In the same manner a 

 million millions may be formed, and this is called 

 a billion, &c. 



Numeration written. The invention of arith- 

 metical notation must have been coeval with the 

 earliest use of writing, whether hieroglyphic or 

 otherwise, and must have come into use about the 

 time when it was felt that a mound, pile of stones, 

 or huge misshapen pillar, was insufficient as a 

 record of great events, and required to be supple- 

 mented by some means which would suffice to 

 hand down to posterity the requisite information. 

 The most natural method undoubtedly was to 

 signify ' unity ' by one stroke, thus : I ; ' two ' by 

 two strokes, 1 1 ; ' three ' by three strokes, 1 1 1, 

 &c. : and, as far as we know, this was the method 

 adopted by most of those nations who invented 

 systems of notation for themselves. It is shewn 

 on the earliest Latin and Greek records, and is 

 the basis of the Roman, Chinese, and other sys- 

 tems. We have thus a convenient division of 

 the different notational systems into the natural 

 and artificial groups, the latter including the 

 systems of those nations who adopted distinct 

 and separate symbols for at least each of the nine 

 digits. The Roman and Chinese systems are the 

 most important of the former, and the Hebrew, 

 later Greek, and 'decimal' systems of the latter 

 group. 



Roman System. The system adopted by the 

 Romans was most probably borrowed at first from 

 the Greeks, and was distinguished equally by its 



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