HEATH'S TREATISE ON ARITHMETIC. BOOK I. . PRELIMINARY IDEAS-DEFINITIONS AND FORMATION OF NUMBERS- - NUMERATION, ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION OF WHOLE NUMBERS. SECTION I. Preliminary Ideas, Definitions, and Formation of Numbers. 1. A limited portion of any natural object; as of time, space, weight, heat, &c., or of any substance, fluid, solid, or æriform, is called quantity. 2. The object of all mathematical science is to measure quantity. 3. Quantity can only be measured by comparing it with some known quantity of the same kind acknowledged as a standard. 4. Standard Measures, to prevent error or variableness, are generally derived from nature. For example, measures of time, from the time of the revolution of the earth about its axis: of space, from the length of a barley-corn, taken from the middle of a full-grown ear; also, from the circumference of the earth: of weight, from the weight of a grain of wheat, taken as above; also, from the weight of a definite quantity of distilled water: of heat, from the temperature of boiling water, &c. 5. Quantity, mathematically considered, is only susceptible of increase and diminution; these are, therefore, called the attributes of quantity. 6. A quantity may be increased in two ways: either by 11 combining with it a quantity of the same kind, greater or less than itself, which is called Addition; or, by combining with it, or continuing to add to it, a quantity exactly equal to itself, called Multiplication: because, in the latter case, the quantity is repeated many times. 7. A quantity may also be diminished in two ways, either by taking from it a quantity not greater than itself, called Subtraction; or, by repeatedly taking from it a quantity less than itself, called Division: because, by this last operation, the quantity is evidently separated or divided into a number of equal parts. 8. The two attributes of quantity, viz., increase and diminution, are then the two fundamental principles, upon which are based the four operations-Addition, Subtraction, Multiplication, and Division. 9. Now, as there is no other method whereby a quantity can be increased and diminished, these four operations are called the four cardinal or fundamental operations of Arithmetic: all others being dependent on and performed by them. 10. Arithmetic, the science of numbers, then, is, in its most extended sense, neither more nor less than the knowledge of all the numerical operations based upon, and arising out of, the two simple attributes of quantity-increase and diminution. 11. The four cardinal operations are represented by the following signs:\ + Plus or more, the sign of Addition. Minus or less, X or ÷By (divided by) 12. That which we attempt to examine, discuss, determine, or prove, by one or more mathematical operations, is called the Proposition; commonly in Arithmetic, the Question. 13. The truth or conclusion arrived at is called the Result or Answer. 14. The general process by which a proposition or question is determined is called the Solution. 15. When, in solving a question, only one cardinal operaration is used, the answer has a distinctive name. In Addition, the answer is called the Sum. S Difference or Thus: a plus b (a + b) is a Sum, (a added to b.) a minus b (ab) is a Difference, (b taken from a.) a into b (ab) is a Product, (a multiplied by b.) a by b (ab) or a is is a Quotient (a divided by b.) 16. The quantity established as a Standard Measure is called a Unit. Thus, One Barley-corn is a Unit. Three barley-corns, placed lengthwise in contact, make one inch; twelve inches, one foot; three feet, one yard, &c. The time of the revolution of the earth about its axis is called one day. The day is divided into twenty-four equal parts, each of which is one hour. The hour is divided into sixty equal parts, each of which is one minute. The minute is divided into sixty equal parts, each of which is one second. 17. The Unit is represented by the Figure 1, one; which, in its general sense, signifies one thing of any kind; and serves as a Standard of Comparison for all things of the same kind. Hence the Figure 1 may represent 1 barley-corn, 1 inch, 1 foot, 1 yard, 1 day, 1 hour, or any other known standard measure or thing. 18. The figure 1 is still a unit, when not applied to any thing, in which sense it is called Abstract, (abstrahère, Lat., to separate.) When applied, it is called Concrete, (concretus, Lat., joined together.) 19. The Combination or Addition of two or more Units forms what is called a Number. 20. A sign made thus, called Equal to or Equals, is placed between two quantities or expressions to show their equality; the whole forming what is called an Equation. Thus, the Equation 1+1=2 is read, one plus one, equal to two; or, more commonly and perhaps better, one plus one, equals two. Hence, (Art. 19,) 2 is the first or lowest number. A sign made thus > is called greater than; reversed thus <, less than. Observe that 1+1 is a Combination or Addition, in fact a Sum or Product; and this sum or product equals 2. It is erroneous to say 1+1, one plus one, are equal to 2; or 1 and 1 are equal to 2, because the plural word (are) implies their disunion. We must therefore say 1 and 1 is 2, or twice 1 is 2. That is, the sum or product is 2. 21. The next Combination, formed by adding another unit, is expressed by each of the Equations 1+1+1=3; |