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This is commonly called reductio ad absurdum. The former is the more com. mon method of conducting a demonstrative argument, and is the most satisfactory to the mind.
17. A Lemma is a subsidiary truth or proposition, demonstrated for the purpose of using it in the demonstration of a theorem, or the solution of a problem.
18. A Corollary is an inference or principle deduced from a preceding proposition.
19. A Scholium is a remark made upon a preceding proposition, pointing out its connection, use, restriction, or extension.
20. An Hypothesis is a supposition, made either in the statement of a proposition, or in the course of a demonstration.
21. An Axiom is a self-evident proposition ; that is, a proposition whose truth is so evident at sight, that no process of reasoning can make it plainer. The following axioms are among the most common :
1. Quantities which are equal to the same quantity, are equal to each other.
2. If the same or equal quantities are added to equals, the sums will be equal.
3. If the same or equal quantities are subtracted from equals, the remainders will be equal.
4. If the same or equal quantities are added to unequals, the šums will be unequal.
5. If the same or equal quantities are subtracted from unequals, the remainders will be unequal.
6. If equal quantities are multiplied by the same or equal quantities, the products will be equal.
7. If equal quantities are divided by the same or equal quan. tities, the quotients will be equal.
8. If the same quantity is both added and subtracted from another, the value of the latter will not be altered.
QUEST.-17. What is a lemma? 18. What is a corollary? 19. What is a scholium ? 20. What is an hypothesis ? 21. What is an axiom? Name some of the most common axioms
9. If a quantity is both multiplied and divided by the same or an equal quantity, its value will not be altered.
10. The whole of a quantity is greater than a part. 11. The whole of a quantity is equal to the sum of all its parts.
22. Addition is represented by the sign (+), which is called plus. It consists of two lines, one horizontal, the other perpendicular, forming a cross, and shows that the numbers between which it is placed, are to be added together. Thus, the expression 6+8, signifies that 6 is to be added to 8. It is read, “ 6 plus 8,”
6 added to 8.” OBS.—The term plus is a Latin word, originally signifying “more,” hence " added to.”
23. Subtraction is represented by a short horizontal line (-), which is called minus. When placed between two numbers, it shows that the number after it is to be subtracted from the one before it. Thus, the expression 9—4, signifies that 4 is to be subtracted from 9; and is read, “ 9 minus 4,” or “9 less 4." OBS.—The term minus is a Latin word, signifying less.
24. Multiplication is usually denoted by two oblique lines crossing each other (X), called the sign of multiplication. It shows that the numbers between which it is placed, are to be multiplied together. Thus, the expression (9X6), signifies that 9 and 6 are to be multiplied together, and is read, “9 multiplied by 6,” or simply, “9 into 6.” Sometimes multiplication is denoted by a point (.) placed between the two numbers or quantities. Thus, 9.6 denotes the same as 9x6.
Oes. It is better to denote the multiplication of figures by a cross than by a point ; for the latter is liable to be confounded with the decimal point.
24. a. When two or more numbers are to be subjected to the same operation, they must be connected by a line (-) placed
QUEST.-22. What is the sign of addition called? Of what does it consist? What does it show? Obs. What is the meaning of the term plus? 23. How is subtraction represented ? What is the sign of subtraction called ? What does it show? Obs. What does the term minus signify? 24. How is multiplication usually denoted? What does the sign of mul tiplication show ? In what other way is multiplication sometimes denoted ?
over them, called a vinculum, or by a parenthesis (). Thus the expression (12+3)x2, shows that the sum of 12 and 3, is to be multiplied by 2, and is equal to 30. But 12+3X2, signifies that 3 only is to be multiplied by 2, and that the product is to be added to 12, which will make 18.
25. Division is expressed in two ways :
First, by a horizontal line between two dots (+), called the sign of division, which shows that the number before it, is to be divided by the number after it. Thus, the expression 24-6, signifies that 24 is to be divided by 6.
Second, division is often expressed by placing the divisor under the dividend, in the form of a fraction. Thus, the expression 3*, shows that 35 is to be divided by 7, and is equivalent to 35:7.
26. The equality between two numbers or quantities, is represented by two parallel lines (=), called the sign of equality. Thus, the expression 5+3=8, denotes that 5 added to 3 are equal to 8. It is read, “5 plus 3 equal 8,” or “the sum of 5 plus 3 is equal to 8." So 7+5=1644=12.
QUEST.-24. a. When two or more numbers are to be subjected to the same operation, what must be done? 25. In how many ways is division expressed? What is the first ? What does this sign show? What is the second ? 26. How is the equality between two numbers or quantities represented ?
NOTATION AND NUMERATION.
Art. 27. Any single thing, as a peach, a rose, a book, is called a unit, or one ;; if another single thing is put with it, the collection is called two; if another still, it is called three ; if another, four; if another, five, &c. The terms, one, two, three, &c., by which we express
how many single things or units are under consideration, are the names of numbers. Hence,
28. NUMBER signifies a unit, or a collection of units. Obs. 1. Numbers are divided into two classes, abstract and concrete.
When they are applied to particular objects, as two pears, five pounds, ten dollars, &c., they are called concrete numbers.
When they do not refer to any particular object, as when we say four and five are nine, they are called abstract numbers.
2. Whole numbers are often called integers.
3. Numbers have various properties and relations, and are applied to various computations in the practical concerns of life. These properties and applications are formed into a system, called Arithmetic.
29. ARITHMETIC is the science of numbers. Obs. 1. The term Arithmetic is derived from the Greek word arithmētikē, which signifies the art of reckoning by numbers.
2. The aid of Arithmetic is required to make and apply calculations not only in business transactions, but in almost every department of mathematics.
QUEST.-27. What is a single thing called? If another is put with it, what is the collection called? If another, what? What are the terms one, two, three, &c.? 28. What does number signify? Obs. Into how many classes are numbers divided ?
When are they called concrete ? When abstract? To what are numbers applied ? 29. What is Arithmetic ? Obs. In what is the aid of arithmetic required ?
Numbers are expressed by words, by letters, and by figures.
NOTATION. 30. The art of expressing numbers by letters or figures, is called NOTATION. There are two methods of notation in use, the Roman and the Arabic.
31. The Roman method employs seven capital letters, viz: I, V, X, L, C, D, M. When standing alone, the letter I, denotes one ; V, five ; X, ten ; L, fifty ; C, one hundred ; D, five hundred ; M, one thousand. To express the intervening numbers from one to a thousand, or any number larger than a thousand, we resort to repetitions and various combinations of these letters. The method of doing this will be easily learned from the following
I denotes one.
seven. VIII eight. IX
twelve. XIII thirteen. XIV fourteen. XV
sixteen. XVII seventeen. XVIII eighteen. XIX
twenty. XXI twenty-one. XXII
XXX denote thirty.
one hundred. CI
one hundred and one. CX
one hundred and ten, СС
two hundred. CCC
three hundred CCCC four hundred. D
two thousand. MDCCCLV, one thousand eight
hundred and fifty-five.
QUEST.-29. How are numbers expressed ? 30. What is notation ? "How many methods are there in use ? 31. What is employed by the Roman method ?