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PREFACE.

THE following is the First Part of a Treatise on Practical Mathematics, and comprehends that portion which does not require the use of Tables. In adding another to the many existing Treatises on this subject, it may be proper to state the objects that have been kept in view in its composition. These have been, first, To exclude all useless matter, and thereby to keep the work within a small compass; secondly, To make it as entirely demonstrative as possible, without reference to any other work on Mathematics. For this purpose, as well as for its own intrinsic usefulness, a Treatise on Geometry is introduced, in which, by adopting a different order of the propositions from that used in Euclid's Elements, and using symbols for certain expressions of frequent occurrence, an unprecedentedly large quantity of geometrical truths is presented, without in any instance detracting from the fulness of the demonstrations, which are always given at length.

The article on Algebra, it is hoped, will be found to be sufficiently extensive for most practical purposes; and the pupil that has thoroughly studied it, will find himself well prepared for entering on the study of larger works. In many instances exercises have been introduced of such a nature, as not only to illustrate the rules, but to assist in reducing certain

ALGEBRA.

DEFINITIONS.

ART. 1. ALGEBRA is a branch of mathematics in which calculations are performed by means of letters which denote numbers or quantities, and signs which indicate operations to be performed on them.

2. The first letters of the alphabet, as a, b, c, &c., are used to denote known quantities, and the latter letters, as x, y, z, &c., to denote unknown ones.

3. The sign + (named plus), indicates that the quantities between which it stands are to be added together; thus a+b denotes the sum of the quantities a and b.

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4. The sign (named minus), indicates that the number or quantity placed after it is to be subtracted from that placed before it; thus a-b denotes the remainder left by taking the quantity b from a.

5. The sign (named multiplied into), indicates that the quantities between which it stands are to be multiplied the one by the other; thus ac denotes that a is to be taken as often as there are units in c, or that c is to be taken as often as there are units in a. This symbol is however seldom used, as a.c, or simply ac written as the letters of a word, indicates the same thing.

6. The sign(named divided by), indicates that the quantity before it is to be divided by that placed after it; thus ac denotes that a is to be divided by c. This symbol is also seldom used, as division is more commonly denoted by placing the dividend above a line as the numerator of a fraction, and the divisor below it as its denomina

tor; thus is the same as a÷b.

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7. The sign (read equal, or is equal to), indicates that the quantities before it are equal in value to those after it; thus 4x3+7=9x2+1, for each is equal to 19.

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8. The quantities before and after the sign gether called an equation; that portion which stands before the sign being called the first side of the equation, and the portion after it the second.

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9. The symbol denotes that the number over which it is placed is to have its square root extracted; thus 16 indicates the square root of 16, which is 4, and the Na denotes the square root of a, that is a number that, being multiplied into itself, would produce a.

10. In the same manner, the cube root of a number as a is denoted by a, the fourth root, by a, and so on. 11. A number placed before a letter or combination of letters is called a coefficient; thus 3a denotes three times a, and 3 is called the coefficient of a. The first letters of the

alphabet are frequently called the coefficients of the latter letters; thus, in the expression 3cx, 3c is called the coefficient of x.

12. When the same letter enters several times as a multiplier into an expression, instead of repeating the letter it is only written once, and a figure written after it to indicate the number of times it enters as a multiplier; thus a2, a3, a1, &c., denote respectively the second, third, and fourth powers of a, and the small figures, 2, 3, 4, &c., placed after the letters, are called the exponents or indices of the letters.

13. Fractional exponents are also used to indicate roots; thus, instead of √x, x, is written, for 3/x, x3, for */ X,

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and so on to any extent; fractional exponents, where the numerator is not one, are also used; thus x, x &c., the former of which denotes that x is to be raised to the second power, and then the third root of this power extracted, and the latter denotes that x is to be raised to the fifth power, and then the square root of this power extracted; or generally the numerator of the fractional exponent denotes a power to which the quantity is to be raised, and the denominator indicates the root of this power which is to be extracted.

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14. When it is frequently written thus, a:b::

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c: d, and read, a is to b as c is to d, and the four quantities are said to constitute a proportion or analogy; the terms a and d are called extremes, and b and c means.

15. The symbol.. is used instead of the words therefore or consequently, which occur very frequently in mathematical reasoning; and the symbol. instead of because.

16. Like quantities are such as are expressed by means of the same letters, and the same powers of these letters,

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