A-B signifies that A is greater than B, and A2-B imports that A is less than B. -- 2. The signs + and — mark the addition and subtraction of the quantities to which they are prefixed: Thus, A+B denotes that B is to be joined to A, and A–B signifies that B is to be taken away from A. Sometimes these two symbols are combined together : Thus, A==B represents either the sum of A and B, or the excess of A above B. 3. To express multiplication, the quantities are placed close together; or they may be connected by the point (..), or the cross x : Thus, AB, or A.B, or Ax B, denotes the product of A by B; and ABC indicates the result of the continued multiplication of A by B, and of this product again by C. 4. When the same number is repeatedly multiplied, the product is termed its power ; and the number itself, in reference to that power, is called the root. The notation is here still farther abridged, by retaining only a single letter with a small figure over it, to mark how often it is understood to be repeated: This figure serves also to distinguish the order of the power. Thus AA, or A*, signifies that A is multiplied by A, and that the product is the second power of A; and AAA, or A*, in like manner, imports that AA is again multiplied by A, and that the result is the third power of A. 5. The roots are denoted, by prefixing a contracted r, or the symbol V. Thus VA or &A marks the second root of A, or that number of which A is the second power; & A signifies the third root of A, or the number which has A for its third power. 6. To represent the multiplication of complex quantities, they are included by a parenthesis. Thus, A(B+C–D) denotes that the amount of B+C–D, considered as a single quantity, is multiplied into A. 7. Ratios and analogies are expressed, by inserting points in pairs between the terms. Thus A : B denotes the ratio of A to B; and the compound symbols A : B : : C ; D, signify that the ratio of A to B is the same as that of C to D, or that A is to B as C to D. PROP. I. THEOR. The product of a number into the sum or dis. ference of two numbers, is equal to the sum or difference of its products by those numbers. Let A, B, and C be three numbers; the product of the sum or difference of B and C by the number A, is equal to the sum or difference of the separate products AB and AC. For the product AB is the same as each unit contained in B repeated A times, and the product AC is the same as the units in C likewise repeated A times; whence the sum of the products AB and AC is equal to the units contained in both B and C, all repeated A times, or it is equal to the sum of the numbers B and C multiplied by A. Again, for the same reason, the difference between the products AB and AC must be equal to the difference between the units contained in B and in C, repeated A times; that is, it must be equal to the difference between the numbers B and C multiplied by A. Cor. 1. Hence a number which measures any two numbers, will measure also their sum and their difference. Cor. 2. It is hence manifest, that the first part of the proposition may be extended to more numbers than two; PROP. II. THEOR. The product which arises from the continued multiplication of any numbers, is the same in whatever order this operation be performed. Let A and B be two numbers; the product AB is equal to B.A. For the product AB is the same as each unit in B add ed together A times, that is, the same as A itself repeated B times, or B.A. Next, let there be three numbers A, B, and C ; the products ABC, ACB, BAC, BCA, CAB, and CBA are all equal. Lastly, put F= BC or CB; then FA=AF, that is, BCA = ABC, and CBA = ACB. And thus the several products are all mutually equal. It is also manifest, that the same mode of reasoning might be extended to the products of any multitude of numbers. PROP. III. THEOR. .” Homogeneous quantities are proportional to their like multiples or submultiples. Let A, B be two quantities of the same kind, and pa, pB their like multiples; then A : B :: p.A : pH. For, since A and B are capable of being measured to any required degree of precision, suppose a to be the measure which A and B contain m and n times, or that A= m.a and B- n.a ; consequently p A=p.ma, and pP=p.ma. But (V. 2.) p.ma=m.pa, and p.ma = n.pa. Wherefore a and pa are like submultiples of A and of pâ, which contain them respectively m times; and these like submultiples are both contained equally, or n times, in B and in pH. Consequently (V. def. 10.) the quantities A, B, and pa, pB are proportional; and A, p.4 are the antecedents, and B, pH, the consequents, of the analogy. Again, because the ratio of p \ to pH is thus the same as that of A to B, which, in reference to pM and pH, are only like submultiples, it follows that homogeneous quantities are also proportional to their like submultiples. PROP. IV. THEOR. In proportional quantities, according as the first term is greater, equal, or less than the second, the third term is greater, equal, or less than the fourth. Let A : B : : C ; D ; if A-B, then C-D ; if A = B, then C= D ; or if A2 B, then C2.D. For, if A be greater than B, then the measure or submultiple of A must be contained oftener in B, and hence the like submultiple of C will be contained oftener in D; wherefore C is greater than D. |