EXAMPLES.-1. Let there be given 3, 4, and 6, being the first, second, and third terms of a harmonical proportion, to find the fourth? ac Here a=3, b=4, c=6, and = 3×6 18 =-= 2a-b 2x 3-4 -2 =) 9, the fourth term required; for 3 : 9:: (4-3: 9-6 :: ) 1 : 3. 2. Given the second, third, and fourth terms, viz. 4, 6, and 9, to find the first? 3. Given 3, 6, and 9, being the first, third, and fourth terms, to find the second ? 4. Given 3, 4, and 9, being the first, second, and fourth, to 5. Let the first, second, and third terms in harmonical proportion, viz. 36, 48, and 72, be given to find the fourth? 6. Given 24, 36, and 54, or the second, third, and fourth terms, to find the first? 7. Given 27, 36, and 81, being the first, second, and fourth terms, to find the third? 8. Let 48, 96, and 144, being the first, third, and fourth, be given, to find the second? 91. Three quantities are said to be in CONTRA-HARMONICAL PROPORTION, when the third is to the first, as the difference of the first and second to the difference of the second and third. Thus, let a, b, and c, be three quantities in contra-harmoni cal proportion, then will c: a: a∞b: b∞c. 92. The following is a synopsis of the whole doctrine of proportion, as contained in the preceding articles. Let four quantities a, b, c, and d, be proportionals, then are they also proportionals in all the following forms; viz. 4. Alternately and inversely.... c:a:: db. 5. Compoundedly . . . . . . a: a b c : c+d. 6. Compoundedly and inversely a+b: a::c+d: c. 7. Compoundedly and alternately a :c:: a+b:c+d. 8. Compoundedly, alternately, and inversely. 9. Dividedly .. or, } c:a :: c+da+b. a: a-b::c: c-d. 10. Dividedly and alternately....a:c:: a-b: c-d. 18. They are inversely proportional when a : b:: d: c. 19. They are in harmonical proportion when a : d : : a s b : es d. 20. Three numbers are in contra-harmonical proportion when ca: a b c d. : The 14th, 15th, 16th, and 17th particulars admit of inversion, alternation, composition, division, &c. in the same manner with the foregoing ones, as is evident from the nature of proportion. THE COMPARISON OF VARIABLE AND DEPENDANT QUANTITIES. 93. A quantity is said to be variable, when from its nature and constitution it admits of increase or decrease. The doctrine of variable and dependant quantities, as laid down in the following articles, should be well understood by all those who intend to read 94. A quantity is said to be invariable or constant, when its nature is such that it does not change its value. 95. Two variable quantities are said to be dependant, when one of them being increased or decreased, the other is increased or decreased respectively, in the same ratio. Thus, let A and B be two variable quantities, such, that when A is changed into any other value a, B is necessarily changed into a corresponding value b, (in which case A: a :: B: b,) then A and B are said to be mutually dependant. 96. To every proportion four terms are necessary, but in applying the doctrine to practice, although four quantities are always understood, two only are employed. This concise mode of expression is found to possess some advantages above the common method, as it saves trouble, and likewise assists the mind, by enabling it to conceive more readily the relations which the variable and dependant quantities under consideration bear to each other. 97. Of two variable and dependant quantities, each is said to vary directly as the other, or to vary as the other, or simply to be as the other, when one being increased, the other is necessarily increased in the same ratio, or when one is decreased, the other also is decreased in the same ratio. Thus, if r be any number whatever, and if when A is increased to rA, B is necessarily increased to rB, (that is, when A A: TA: BrB,) or when A is decreased to B is necessarily to vary directly as B; or we say simply, A is directly as B. EXAMPLE. A labourer agrees to work a week for a certain sum; now if he work 2 weeks, he receives twice that sum, if he work but half a week, he receives but half that sum, and so on; in this case, the sum he receives is directly as the time he works. Sir Isaac Newton's Principia, or any other scientific treatise on Natural Philosophy or Astronomy. See on this subject, Ludlam's Rudiments, 5th Edit. p. 235-250. and Wood's Algebra, 3d Edit. p. 103-109. 98. One quantity is said to vary inversely as another, when the former cannot be increased, but the other is decreased in the same ratio; or the former cannot be decreased, but the other must necessarily be increased in the same ratio; that is, the former cannot be changed, but the reciprocal of the latter is changed in the same ratio. EXAMPLE. A man walks a certain distance in an hour; now if he walk twice as fast, he will go the given distance in half an hour; but if he walk only half as fast, he will evidently require two hours to complete his journey; in this case his rate of walking is inversely as the time he takes to perform it. 99. The sign ∞ placed between two quantities, signifies that they vary as each other. Thus A & B implies that A varies as B, or that A is as B; also A ∞ 1 B shews that A varies as the reciprocal of B, or that A is inversely as B. 100. One quantity is said to vary as two others jointly, when the former being changed, the product of the two latter must necessarily be changed in the same ratio. Thus A varies as B and C jointly, that is, A ∞ BC, when A cannot be changed into a, but the product BC must be changed into bc, or that A: a:: BC: bc. 101. In like manner one quantity varies as three others jointly, when the former being changed, the product of the three latter is changed in the same ratio. Thus ABCD, and the like, when more quantities are concerned. EXAMPLE. The interest of money varies as the product of the principal, rate per cent. and time, or I ∞ PRT. 102. One quantity is said to vary directly as a second, and inversely as a third, when the first cannot be changed, but the second multiplied by the reciprocal of the third, (that is, the second divided by the third,) is changed in the same ratio. Thus A varies directly as B, and inversely as C, or A ∞ B C' LE. A farmer must employ as many reapers, as are the number of acres to be reaped, and inversely as the of days he allots for the work, or R∞ A D' fA B, and A∞ C, then will A ∞ BC. Ab B Ab Abc a=the since B: b:: A : e of A arising from its successive changes in the ratios of Cc; wherefore since Abc a, or Abc=aBC, A: a :: or A ∞ BC. In like manner it may be shewn, that if A ∞ B, A ∞ C, D, then A ∞ BCD; also if A ∞ B, and A then 1 CC C e proof of all which is the same as in the former article. .B. If ABC and B be constant, then A ∞ C; if C be For since the product BC varies by the increase or decrease of dy, when B is constant, and A varies as that product, therewhen B is invariable, A must evidently vary as C; and when Zone is variable, and C constant, A (varying as the product must in like manner vary as B: after the same manner ay be shewn, that when A ∞ BCD, if BC be constant, then D; if D be constant, then A ∞ BC; if C be constant, then BD; and if B be constant, then A CD; and in general, A be as any product or quotient, and if any of the factors be en, A will be as the product or quotient (as the case may be) all the rest. 105. If the first quantity vary as the second, the second as e third, the third as the fourth, and so on, then will the first ry as the last. Let A, B, C, and D, be any number of variable quantities, |