. . . Binomial Surds and Impossible Quantities Nature and Solution of Equations of all Degrees Biquadratic Equations-Simpson's method Nature and Properties of Logarithms APPLICATION OF ALGEBRA TO GEOMETRY Definitions and First Principles Trigonometrical Functions of an Angle Functions of two or more Angles Inverse Trigonometrical Functions. Changing the Radius in Trigonometrical Equations On the Use of the Trigonometrical Tables The Application of Trigonometry to the Determination of Heights and Expansions and Developments of Trigonometrical Expressions Construction of Trigonometrical Tables Definitions and First Principles The Right Angled Spherical Triangle The Oblique Angled Spherical Triangle Solution of Spherical Triangles The Area of a Spherical Triangle COORDINATE GEOMETRY OF TWO DIMENSIONS . Exposition of Principles and Fundamental Theorems The Straight Line referred to Rectangular Coordinates Investigation of Properties for illustration relative to the Point and 301 301 ELEMENTS OF THE DIFFERENTIAL AND INTEGRAL CALCULUS:- Elementary Principles and Illustrations Differentiation of Algebraic Functions Differentiation of Transcendental Functions Maxima and Minima Functions of One Variable Differentiation of Functions of Several Variables Maxima and Minima Functions of Two Variables Singular Values of Functions . Curves referred to Polar Coordinates Differential of an Arc of a Curve The Direction of Curvature of a Curve Cóordinates of the Centre of Curvature Locus of the Centre of Curvature Change of the Independent Variable . Integration of Rational Fractions Integration of Irrational Differentials Integration of Binomial Differentials Integration by Successive Reduction . Integration of Transcendental Functions A COURSE OF 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. NUMERATION. 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 VOL. I. B a 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. > one. |