Let a and b be two quantities, then 2 ab double their pro the difference between the first and second to the difference between the second and third. EXAMPLES.-1. To find a harmonical mean between 2 and 6. -b; that is, the first is to the third, as 2. Required a harmonical mean between 24 and 12? Ans. 16. 3. Required the harmonical mean between 5 and 20? Ans. 8. 4. Required the harmonical mean between 10 and 30? 89. If the product of two given quantities be divided by the difference between double the greater and the less, or double the less and the greater, the quotient will be the third harmonical proportional to the two given quantities. Let a and b be two given quantities, whereof a is the greater; ab 2a be the third harmonical proportional to a and b: then will To what has been said on this subject, the following particulars relating to the comparison, &c. of the three kinds of proportionals, may be added; viz. 1. The reciprocals of an arithmetical progression are in harmonical progression, and the reciprocals of a harmonical progression, are in arithmetical progression. Thus, a, a+d, a + 2 d, a + 3 d, are arithmetically proportional. EXAMPLES.-1. To find a third harmonical proportional to the number required; for 48: 24 : : (48—32 : 32—24 : : ) 16 : 8. 2. Required a third harmonical proportional to 2 and 3? Ans. 6. 3. Required the third harmonical proportional to 20 and 8 ? Ans. 5. 4. Required the third harmonical proportional to 10 and 100? 90. Of four harmonical proportionals any three being given, the fourth may be found as follows. Let a, b, c, and d, be four quantities in harmonical proportion, then since a d::a-b: c-d, (Art. 86.) by multiplying extremes and means, ac—ad=ad—bd; from this equation any three of the quantities being given, the remaining one may be found. Thus, a, b, and c, being given, we have d= extremes; if b, c, and d, be given, then a=; treme; if a, b, and d, be given, then c= means; if a, c, and d, be given, then b= ac 2a-b bd 2 ad-bd one of the the other ex one of the α 2. If there be taken an arithmetical mean and a harmonical mean between any two quantities, then the four quantities will be geometrically proportional, Thus, between a and b the harmonical mean is 2 ab a+b and the arithme 3. The following simple and beautiful comparison of the three kinds of proportionals, is given by Pappus, in his third book of Mathematical Collections. Let a, b, and c, be the first, second, and third terms; then, 4. There is this remarkable difference between the three kinds of proportion; namely, from any given term there can be raised A continued arithmetical series, increasing but not decreasing, indefi A continued harmonical series, decreasing but not increasing, A continued geometrical series, both increasing and decreasing,nitely. 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 = 3x6. 18 ===) 9, 2a-b 2x3-4 2 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 :: ab: boc. 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. 6. Compoundedly and inversely a+b: a:: c+d: c. 7. Compoundedly and alternately a c:: a+b: c+d. and inversely. 9. Dividedly or, .... } ca::c+da+b. a+b:a-b::c+d: c-d. a-b: a+b:: c-d: c+d. ..... a+b:c+d::a-b:c-. -d. ra : rb :: SC : sd. 16. By involution 17. By evolution .. an 772 an : b :: 18. They are inversely proportional when a b::d: c. 19. They are in harmonical proportion when a :d :: a sb: es d. 20. Three numbers are in contra-harmonical proportion when ca: acs b: cos 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: TAB TB,) or when A is decreased to. : A B is necessarily T 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. |