PROBLEM V. To find two Arithmetical Means between Two Given Extremes. SUBTRACT the less extreme from the greater, and divide. the difference by 3, so will the quotient be the common difference; which being continually added to the less extreme, or taken from the greater, gives the means. EXAMPLE. To find two arithmetical means between 2 and 8. To find any Number of Arithmetical Means between Two Given Terms or Extremes. SUBTRACT the less extreme from the greater, and divide the difference by 1 more than the number of means required to be found, which will give the common difference; then this being added continually to the least term, or subtracted from the greatest, will give the terms required. EXAMPLE. To find five arithmetical means between 2 and 14. 2 6) 12 Then by adding this com. dif. continually, com. dif. 2 See more of Arithmetical progression in the Algebra. GEOMETRICAL GEOMETRICAL PROPORTION AND PROGRESSION. IN Geometrical Progression the numbers or terms have all the same multiplier or divisor. The most useful part of Geometrical Proportion is contained in the following theo rems. THEOREM 1. When four quantities are in geometrical proportion, the product of the two extremes is equal to the product of the two means. Thus, in the four 2, 4, 3, 6, it is 2 X 6 3 X 4 = 12. And hence, if the product of the two means be divided by one of the extremes, the quotient will give the other extreme. So, of the above numbers, the product of the means 12÷2 = 6 the one extreme, and 12 ÷ 6 = 2 the other extreme ; and this is the foundation and reason of the practice in the Rule of Three. THEOREM 2. In any continued geometrical progression, the product of the two extremes is equal to the product of any two means that are equally distant from them, or equal to the square of the middle term when there is an uneven number of terms. Thus, in the terms 2, 4, 8, it is 2 × 8 = 4 × 4 = 16. And in the series 2, 4, 8, 16, 32, 64, 128, it is 2 X 128 = 4 X 64 = 8 X 32 = 16 X 16 = 256. THEOREM 3. The quotient of the extreme terms of a geometrical progression, is equal to the common ratio of the series raised to the power denoted by 1 less than the number of the terms. Consequently the greatest term is equal to the least term multiplied by the said quotient. So, of the ten terms, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, the common ratio is 2, and one less than the number of terms is 9; then the quotient of the extremes is 1024÷ 2512, and 2o = 512 also. THEOREM THEOREM 4. The sum of all the terms, of any geometrical progression, is found by adding the greatest term to the difference of the extremes divided by 1 less than the ratio. So, the sum of 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 1024-2 (whose ratio is 2), is 1024 + 2-1 1024+1022 = 2046. The foregoing, and several other properties of geometrical proportion, are demonstrated more at large in the Algebraic part of this work. A few examples may here be added of the theorems, just delivered, with some problems concerning mean proportionals. EXAMPLES. 1. The least of ten terms, in geometrical progression, being 1, and the ratio 2; what is the greatest term, and the sum of all the terms? Ans. The greatest term is 512, and the sum 1023. 2. What debt may be discharged in a year or 12 months, by paying 17 the first month, 2l the second, 47 the third, and so on, each succeeding payment being double the last; and what will the last payment be? Ans. The debt 40951, and the last payment 20481. PROBLEM I. To find One Geometrical Mean Proportional between any Two Numbers. MULTIPLY the two numbers together, and extract the square root of the product, which will give the mean proportional sought. EXAMPLE. To find a geometrical mean between the two numbers 3 and 12. 12 36 (6 the mean. 36 PROBLEM PROBLEM II. To find Two Geometrical Mean Proportionals between any Two Numbers. DIVIDE the greater number by the less, and extract the cube root of the quotient, which will give the common ratio of the terms. Then multiply the least given term by the ratio for the first mean, and this mean again by the ratio for the second mean: or, divide the greater of the two given terms by the ratio for the greater mean, and divide this again by the ratio for the less mean. EXAMPLE. To find two geometrical means between 3 and 24. Here 3) 24 (8; its cube root 2 is the ratio. Then 3 X 2 = 6, and 6 X 2 = 12, the two means. the same. That is, the two means between 3 and 24, are 6 and 12. PROBLEM III. To find any Number of Geometrical Means between Two Num bers. DIVIDE the greater number by the less, and extract such root of the quotient whose index is 1 more than the number of means required; that is, the 2d root for one mean, the 3d root for two means, the 4th root for three means, and so on; and that root will be the common ratio of all the terms. Then, with the ratio, multiply continually from the first term, or divide continually from the last or greatest term. EXAMPLE. To find four geometrical means between 3 and 96. Here 3) 96 (32; the 5th root of which is 2, the ratio. Then 3 X 2 = 6, & 6X 2 = 12, & 12X2= 24, & 24×2 = 48. Or 962 48, & 48-2=24, & 24-2= 12, & 12÷2 = 6. That is, 6, 12, 24, 48, are the four means between 3 and 96. OF OF MUSICAL PROPORTION. THERE is also a third kind of proportion, called Musical, which being but of little or no common use, a very short account of it may here suffice. Musical Proportion is when, of three numbers, the first has the same proportion to the third, as the difference between the first and second, has to the difference between the second and third. When four numbers are in musical proportion; then the first has the same ratio to the fourth, as the difference between the first and second has to the difference between the third and fourth. When numbers are in musical progression, their reciprocals are in arithmetical progression; and the converse, that is, when numbers are in arithmetical progression, their reciprocals are in musical progression. 3 = 127 So in these musicals 6, 8, 12, their reciprocals, are in arithmetical progression; for and 1 + = = 1; that is, the sum of the extremes is equal to double the mean, which is the property of arithme ticals. The method of finding out numbers in musical proportion is best expressed by letters in Algebra. FELLOWSHIP, OR PARTNERSHIP. FELLOWSHIP is a rule, by which any sum or quantity may be divided into any number of parts, which shall be in any given proportion to one another. By this rule are adjusted the gains or loss or charges of partners |