Symmetrical triangles are equivalent (466, 467), Supplementary triangles (468), Surface of spherical triangle (469); of spherical polygon (470), Area of surface generated by revolution of polygon (471, 472), EXPLANATION OF SIGNS, AND OF SOME USEFUL PROPOSITIONS IN THE DOCTRINE OF PROPORTIONS. THE signis plus, or added to. Thus A+B is A added to B. Thus, A B is A less B. Thus, AX B is A multi The sign is minus, or less. The sign is multiplied by. plied by B; and the period (.) is plication. The signor is A divided by B. also be written The sign = A B also the sign of multi is divided by. Thus, A÷B or A: B The quotient of A divided by B may is equal to. Thus, AB is A equal to B; and the expression in which this sign occurs is called an equation. The sign is greater than. Thus, A> B is A greater than B. The sign is less than. Thus, A <B is A less than B. A2 indicates the second power of A, A3 the third power, &c. A ratio or fraction is the quotient of one quantity divided by another, and is usually written with the sign (:). Thus the ratio of A to B is A: B, or it may just as well be written in the form of a fraction, as A B The first term of a ratio is called the antecedent, and the second the consequent. Thus, A is the antecedent of the preceding ratio, and B its consequent. The value of a ratio is not altered by multiplying or dividing both its terms by the same number. Thus, A: B is equal to m × A: m × B. A proportion is the equation formed by two equal ratios. Thus, if the two ratios A: B and C : D are equal, the equation A: B = C: D is a proportion, and it may also be written The first and last terms of a proportion are called its extremes; and the second and third its means. Thus, A and D are the extremes of this proportion, and B and C its means. Theorem I. The product of the means of a proportion is equal to the product of its extremes. Proof. If the fractions of a proportion or, omitting the common denominator, AX DBX C. This proposition is called the test of proportions. Theorem II. If four quantities are such that the product of the first and last of them is equal to the product of the second and third, these four quantities form a proportion. Proof. Let A, B, C, D, be such that AX D=BX C. which, reduced to lower terms, and written in the form of ratios, is Corollary. The terms of a proportion may be transposed in any way, provided the product of the means is retained equal to that of the extremes, and the proportion will not be destroyed. Thus, the preceding proportion gives, by transposition, A: C B: D, B: A D: C, = If both the means of the proportion are of the same magnitude, this mean is called the mean proportional between the extremes. Thus, if B is a mean proportional between A and D. Theorem III. The mean proportional between two quantities is the square root of their product. Proof. The application of the test to the preceding proportion gives the B2AX D, square root of which is B = L (A × D). A succession of several equal ratios is called a continued proportion. Thus, A: B = C: D = E: F = &c. is a continued proportion. Theorem IV. The sum of any number of antecedents of a continued proportion is to the sum of the corresponding consequents as one antecedent is to its consequent. Proof. Denote the common value of the ratios in the above continued proportion by M, we have b* and the sum of these equations is A+ C+E+ &c. = ( B + D + F + &c.) × M; Corollary. The sum of the antecedents of a proportion is to the sum of its consequents as either antecedent is to its consequent; and the difference of the antecedents is to the difference of the consequents in the same ratio. Theorem V. The sum of the antecedents of a proportion is to their difference, as the sum of the consequents is to their difference. Theorem VI. The sum of the first two terms of a proportion is to the sum of the last two as the first term is to the third, or as the second is to the fourth; and the difference of the first two terms is to the difference of the last two in the same ratio; also the sum of the first two terms is to their difference as the sum of the last too is to their difference. |