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M A THEMATICS.
PRELIMINARY DEFINITIONS. Mathematics is the science which treats of all quantities that can be numbered or measured. Its two great divisions are pure mathematics and mixed mathematics. Pure mathematics consists of the three following divisions :
1. Arithmetic, which treats of numbers or particular quantities; 2. Algebra, which treats of the relations of any quantities what
ever under particular conditions, and may properly be
termed Universal Arithmetic; and 3. Geometry, which treats of extended quantities, or continued
magnitudes, as possessing three dimensions, viz., length,
breadth, and thickness. This last division embraces a much greater compass and variety of reasoning than either of the other divisions, and all of them are founded on the simplest notions of abstract quantities. The applications of these three divisions, one to another, form other important parts of pure mathematics.
Miced mathematics is the application of the different parts of pure mathematics to those physical inquiries which are founded upon principles deduced from experiment or observation. It comprehends Mechanics, or the science of equilibrium and motion of bodies; Astronomy, in which the motions, distances, etc., of the celestial bodies are considered ; Optics, or the theory of light, besides various other important subjects. In all these branches of mixed mathematics, if the first prin. ciples be accurately determined by experiment or observation, the results which are deduced are as certain and indisputable as those which can be deduced by geometry, or by any other part of pure mathematics, from axioms and definitions.
PRINCIPLES OF ARITHMETIC.
Art. 1. Arithmetic is that division of pure mathematics which treats of numbers, and of the method of performing calculations by means of them.
Number is a collection of several objects of the same kind, or of many
separate parts. It is one of the forms of magnitude, an attribute or quality of objects by which they are conceived to be susceptible of increase or diminution. The other form of magnitude is distinguished by the connexion or continuity of the parts,--being an entire mass without distinction of parts ; whereas in number the consideration is merely how many parts it contains. The definition of number supposes the existence of one of the things or parts of which it is composed, taken as a term of comparison, and which, in that case, is denominated unity.
2. Some knowledge of numbers must have existed in the earliest ages of the world. The ten fingers with which man had been formed, the flocks and herds which he had acquired, and the variety of objects that surrounded him, would all contribute to impress his mind with a notion of number. While small numbers only were required, the ten fingers would furnish the most convenient way of reckoning them, since with his fingers any person could make those little calculations which his limited wants required. He would name all the different collections of his fingers, and frame appropriate words, in his own language, answering to one, two, three, four, five, six, seven, eight, nine, and ten. As his wants increased he would proceed to higher numbers, adding one continually to the former collection, as he advanced from lower numbers to higher. He would soon perceive that there is no limit to the different numbers that may be formed, and consequently that it would be impossible to express them in ordinary language by distinct names independent of each other. By arranging numbers in groups or classes, they might be expressed by a comparatively small number of words, still the continual repetition which unavoidably occurs in calculation would necessarily preclude the use of names of numbers, except in operations of the very simplest character.
3. The English names of numbers have been formed from the Saxon language, by combining the names of the first ten numbers mentioned in the preceding article. Thus, eleven, signifying that one is left after ten is taken, or ten and one.
twelve, signifying that two is left after tep is taken, or ten and two. thirteen, ten and three. twenty-three, two tens and three. fourteen, ten and four.
thirty, three tens. fifteen, ten and five. forty, four tens. sixteen, ten and six. fifty, five tens. seventeen, ten and seven. sixty, six tens. eighteen, ten and eight. seventy, seven tens. nineteen, ten and nine. eighty, eight tens. twenty, two tens.
ninety, nine tens. twenty-one, two tens and a hundred, ten tens.
a thousand, ten hundreds. twenty-two, two tens and a million, ten hundred thousand, or two.
one thousand thousand, etc. 4. For facilitating calculations it would be found necessary to substitute short and expressive signs for words, and when some few signs or characters had been chosen, to combine them so as to represent the names of all other numbers whatever. We shall here show how this has been done by the Greeks and Romans, and then advert to the admirable system of notation which so generally prevails among different nations of the world at the present time.