that the first is to the second, as the fourth to the third, it is then named INVERSE PROPORTION, and the four quantities in the order they stand, are said to be INVERSELY PROPORTIONAL. Thus, 24: 12 : 6, and 9: 5 :: 10 : 18, &c. are inversely proportional. 84. Hence, an inverse proportion may be made direct, by changing the order of the terms in either of the ratios which. constitute the proportion. 85. The reciprocals of any two quantities will be inversely proportional to the quantities. 1 1 b a Let a and b be two quantities, then will a: b:: : for multiplying both terms of the latter ratio by ab, we shall have 1 ner b: a:: α 1 : that is, the direct ratio of the quantities is b' the same as the inverse ratio of their reciprocals; and the inverse ratio of the quantities, the same as the direct of their reciprocals. Hence, inverse proportion is likewise frequently called RECI PROCAL PROPORTION. HARMONICAL PROPORTION. 86. Three quantities are said to be in harmonical or musical proportion, when the first is to the third, as the difference of the first and second to the difference of the second and third; and four terms are said to be in harmonical proportion, when the first is to the fourth, as the difference of the first and second to the difference of the third and fourth. Thus, if a ca—b: b—c, then are the three quantities, a, b, and c, harmonically proportional. And if a d::a-b: c-d, then are the four, a, b, c, and d, harmonically proportional. 87. Hence, if all the terms of any harmonical proportion be either multiplied or divided by any quantity whatever, the results will still be in harmonical proportion. 88. If double the product of any two quantities be divided by their sum, the quotient will be a harmonical mean between the two quantities. Let a and b be two quantities, then 2 ab=double their pro the difference between the first and second to the difference be tween the second and third. EXAMPLES.-1. To find a harmonical mean between 2 and 6. 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 then will 2a-b be the third harmonical proportional to a and b: * 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. And, 1 1 ́a'a+d'a +2d' a+3d' proportional, and the converse. their reciprocals, are harmonically that the first is to the second, as the fourth to the third, it is then named INVERSE PROPORTION, and the four quantities in the order they stand, are said to be INVERSELY PROPORTIONAL. Thus, 24: 12: 6, and 9: 5 :: 10: 18, &c. are inversely proportional. 84. Hence, an inverse proportion may be made direct, by changing the order of the terms in either of the ratios which constitute the proportion. 85. The reciprocals of any two quantities will be inversely proportional to the quantities. 1 1 : b a for Let a and b be two quantities, then will a: b:: multiplying both terms of the latter ratio by ab, we shall have ner b: a:: 1 α 1 that is, the direct ratio of the quantities is the same as the inverse ratio of their reciprocals; and the inverse ratio of the quantities, the same as the direct of their reciprocals Hence, inverse proportion is likewise frequently called RECI PROCAL PROPORTION. HARMONICAL PROPORTION. 86. Three quantities are said to be in harmonical or musica proportion, when the first is to the third, as the difference the first and second to the difference of the second and third and four terms are said to be in harmonical proportion, whe the first is to the fourth, as the difference of the first and secon to the difference of the third and fourth. Thus, if a c: a- -b: b-c, then are the three quantitie a, b, and c, harmonically proportional. and b be two quantities, then 3at=tis dinė Siz gru a+b=their sum, wherefore ired, før (Art. 96.) a. b:: 6-ab 2 ab a+b -b; that s the frt sa te trt a =nce between the first and send a me törme second and thurd. -L. To ind a harmonicai nen en ! 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= ас 2 ad-bd one of the 2a-b one of the a 2 ad-ac the other mean. d treme; if a, b, and d, be given, then c= means; if a, c, and d, be given, then b= 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- |