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proposed to be done or performed; and is either a problem or theorem.
39. A problem, is a proposition, or question stated, in order to the investigation of some unknown truth; and which requires the truth of the discovery to be demonstrated.
40. A theorem, is a proposition, wherein something is advanced or asserted, the truth of which is proposed to be demonstrated or proved.
41. A corollary, or consectary, is a truth derived from some proposition already demonstrated, without the aid of any other proposition.
42. A lemma, signifies a proposition previously laid down, in order to render more easy the demonstration of some theorem, or the solution of some problem that is to follow.
43. A scholium, is a note, or remark, occasionally made on some preceding proposition, either to show how it might be otherwise effected; or to point out. its application and use.
44. An axiom, is a self-evident truth, or a proposition universally assented to, or which requires no formal proof.
45. As axioms are the first principles upon which all mathematical demonstrations are founded, I will point out those that are necessary to be observed in the study of Algebra, as there will be frequent occasion to advert to them.
46. When no difference can be shown or imagined between two quantities, they are equal.
47. Quantities equal to the same quantity, are equal to each other.
48. If to equal quantities equal quantities be
added, the wholes will be equal. Thus, if a=b, then a+c=b+c; if a—b=c, then adding b, a—b÷ b=c+b, or a=c+b.
49. If from equal quantities equal quantities be subtracted, the remainders will be equal.
If a=b, then, a—2=b-2; if b+c=a+c, then b=a.
50. If equal quantities be multiplied by equal numbers or quantities, the products will be equal.
Thus, if a=b, 3a=36; if a=2, 3a=b ; if a=b. ca=cb; and if a=b, a×a=bb or a2=b2.
51. If equal quantities be divided by equal numbers or quantities, the quotients will be equal.
Thus, if 5a 106,- or a=2b; if ca=cb.
or a=b; and if a2=ba, then
Scholium. Articles (49), (50), (51), might have been deduced from Art. (48); but they are all easily admitted as axioms.
52. If the same quantity be added to and subtracted from another, the value of the latter will not be altered. Thus, if a=c, then a+b=c+be and a+b-b=c+b—b, or a=c.
This might be infered from Art. (48).
53. If a quantity be both multiplied and divided by another, its value will not be altered. Thus, if
a=b; then, 3a=3b, and dividing by 3,
ADDITION, SUBTRACTION, MULTIPLICATION,
DIVISION OF ALGEBRAIC QUANTITIES.
§ 1. Addition of Algebraic Quantities.
54. The addition of algebraic quantities is performed by connecting those that are unlike with their proper signs, and collecting those that are like into one sum; for the more ready effecting of which, it may not be improper to premise a few propositions, from which all the necessary rules may be derived.
55. If two or more quantities are like, and have like signs, the sum of their coefficients prefixed to the same letter, or letters, with the same sign, will express the sum of these quantities.
And -5a added to -3a is 8α.
For, if the symbol a be made to represent any quantity or thing, which is the object of calculation, 5a will represent five times that thing, and 7a seven times the same thing, whatever may be the denomination or numeral value of a; and consequently, if the quantities 5a and 7a are to be incorporated, or added together, their sum will be twelve times the thing denoted by a, or 12a.
Moreover, since a negative quantity is denoted by the sign of subtraction: thus, if a+b=a-c, b=-c, and c=-b. A debt is a negative kind of property, a loss a negative gain, and a gain a negative loss.
Therefore, it is plain that the quantities, -5a and 3a, will produce, in any mixed operation, a contrary effect to that of the positive quantities with which they are connected; and consequently, after incorporating them in the same manner as the latter, the sign must be prefixed to the result; so that if a be greater than a, it is evident that 5(A—α) +3 (A-a), or (5A-5a)+(3A-3a)=8A-8a; and therefore the sum of the quantities -5a and -3a, when taken in their isolated state, will by a necessary extension of the proposition be-Sa.
56. If two quantities are like, but have unlike signs, the difference of their coefficients, prefixed to the same letter, or letters, with the sign of that which hath the greater coefficient, will express the sum of those quantities.
Thus +6a added to -4a is And -6a added to +4a is Since, Art. (36), the compound quantity a-b+c -d, &c., is positive or negative, according as the sum of the positive terms is greater or less than the sum of the negative ones, the aggregate or sum of the quantities 4a-2a+2a-2a, or 6a-4a, will be +2a; since the sum of the positive terms is greater than the sum of the negative ones. And the sum of the quantities a -4a+3a-2a, or 4a6a, will be 2a; since the sum of the negative terms is greater than the sum of the positive ones.
Corollary. Hence it appears, that if the sum of the positive terms be equal to the sum of the negative
ones, their aggregate or sum will be nothing. Thus 5a-5a=0; and 5a-3a+4a-6a-9a-9a=0.
57. The preceding proposition is demonstrated in the following manner by BONNYCASTLE in his Algebra. Vol. II. 8vo.
Where the quantities are supposed to be like, but to have unlike signs, the reason of the operation will readily appear, from considering that the addition of algebraic quantities, taken in a generalsense, or without any regard to their particular values, means only the uniting of them together, by means of the arithmetical operations denoted by the signs and and as these are of contrary, or opposite natures, the less quantity must be taken from the greater, in order to obtain the incorporated mass, and the sign of the greater prefixed to the result. So that if 6a is to be added to 4(A—a), or to 4A-4a, the result will evidently be 4a+6a-4a, or 4A+2a; and if 4a is to be added to 6(A-a), or to 6A-6a, the result will be 6A+4α-6a, or 6A-2a; whence by making this proposition general, as in the last, the sum of the isolated quantities 6a and -4a will be +2a, and that of 4a and -6a will be -2α.
58. If two or more quantities be unlike, their sum can only be expressed by writing them after each other, with their proper signs.
Thus, the sum of 2a and 2b, can only be expressed, with the sign + between them, which denotes that the operation of addition is to be performed when we assign values to a and b.
For, if a=10, and b=5; then the sum of 2a and 26 can be neither 4a nor 4b, that is, neither 4 × 10 =40 nor 4×5=20; but 2×10+2×5=20+10= 30. In like manner, the sum of 3a, 56, 2c, and