PART FIRST. ALGEBRAICAL PRINCIPLES. - PLANE GEOMETRY - DEPENDING ON THE RIGHT LINE, AND ON THE CIRCLE, ELLIPSE, HYPERBOLA, AND PARABOLA. BOOK FIRST. ALGEBRAICAL PRINCIPLES. SECTION FIRST. Use of the Signs-Fractions-Simple Equations. Definition 1. Mathematics is a science, having for its object the investigation of the relations that quantities bear to each other. Def. 2. We denominate Quantity that which admits of measurable increase or diminution. Thus, lines, angles, weight and time, are quantities-but color, being incapable of a unit of measure, cannot, in the mathematical acceptation of the term, be regarded as a quantity. Def. 3. A Proposition is anything proposed-if to be done, it is called a Problem; if to be demonstrated or proved, a Theorem ; if a direct and necessary consequence of something going before, it is a Corollary; and if evident in itself, or not capable of being reduced to any simpler principle, it is known as an Axiom. Explanation of the Signs. For the investigation of general propositions, it will often be desirable, and not unfrequently even indispensable, to be in possession of a method, equally general, whereby we may denote quantities and the operations to be performed upon them. The letters of the alphabet are employed for this purpose the first commonly indicating known, and the last unknown quantities; but the student should accustom himself to regard any letter either as known or unknown. Instead of using different letters, the same differently marked, as a', read a prime; a" [a second]; a, [a sub-two, or a second], and so on, are frequently introduced with advantage. The symbols of operation for which we shall have more immediate use, are the signs of Addition, Subtraction, Multiplication, Division, + (plus, more) - (minus, less) X or . : or + Aggregation, Equality, Inequality, Deduction, Continuation. > (greater), < .. = Thus, that 4 is to be added to 6, is written, 6+4, and read, six plus four; that six and four are equal to 12 diminished by two, is written, 6+4=12-2, and read, 6 plus 4 is equal to 12 minus 2; that the sum of 6 and 4 multiplied by 2, is equal to 44 diminished by 4, and the remainder divided by two, is written, or or (6+4) × 2 = (44—4) + 2, for when the omission of a sign, as is done in 2(6+4), would not be attended by any ambiguty, it may be dropped. Thus a b signifies that a is to be multiplied into b. Propositions are much shorter in symbolical than in common language, and are, consequently, more clearly expressed, as has already been shown, and as will appear in a still stronger light by writing an example or two in corresponding columns. = a = b, the same a = c. of a divided by m. as a is equal to b, and c is equal to b; therefore a is equal to c. The sum of a and b multiplied into the difference of a and b, and this product divided by m, gives a quotient which is equal to the quotient arising from dividing the remainder of the square of a diminished by the square of b, by m, which again is less than the quotient of the square Scholium. The student should be accustomed to turn the algebraical into common and appropriate language; thus the sign of equality,, will commonly be read, "is equal to," but sometimes, "will be equal to," and at others, "equalling;" the symbol of deduction, .., will generally be read, "therefore," sometimes, "hence," "it follows," and again, "from what goes before, we infer," &c.; the symbol of continuation, consisting of three points, will be enunciated, "&c.," " and so on," " continued according to the same law." PROPOSITION I. [AXIOM.] The whole is equal to the sum of all its parts. (1) This proposition is an axiom, that is, evident of itself; no words about it, therefore, can make it any plainer. (12) Corollary 1. The whole is greater than any of its parts, or, the whole exceeds any of its parts by those which are excepted-otherwise the whole would differ from the sum of all its parts. Cor. 2. Quantities which are equal in all their parts, are (13) said to be equal to each other; for the whole is known by its parts --or, quantities which are not evidently identical, can be compared only by a resolution into like or unlike parts. in Cor. 3. Quantities which, however resolved, are unequal (1) of their parts, are not equal to each other (13). any Cor. 4. Quantities which are equal to each other, are (15) equal in all their parts; for, if some of their parts were unequal, they would, by (1,), be themselves unequal; .. Cor. 5. Unequal quantities are not equal in all their parts. (16) Cor. 6. Quantities which are equal to the same or equal (17) quantities, are equal to each other; for they are equal in all their parts, (1), (15). Cor. 7. Quantities measured by the same or equal quantities, (18) are equal to each other; for equality of measures implies equality of parts, whether the measuring quantities be of the same kind with those measured or not. Thus, two masses of lead are equal in weight when they both contain the same number of pounds, or when they both contain the same number of cubic inches. Cor. 8. Quantities are to each other as their measures; .. (1,) Cor. 9. Of quantities having unequal measures, that is (110) the greater to which the greater measure belongs. Cor. 10. If the same or equal quantities be increased or. (111) diminished by the same or equal quantities, the resulting quantities will be equal to each other; since they will be equal in all their parts, (15), (1). Cor. 11. If the same or equal quantities be multiplied or (112) divided by the same or equal quantities, the resulting quantities |