Prop. 50.--To find a mean proportional to two given numbers. Since in the case of three numbers being continued proportionals, the product of the extremes is equal to the square of the mean, therefore the mean will be obtained by extracting the square root of the product of the two given numbers. Thus if a and c be given, and it be required to find a number b such that a : b :: 6 : C, then since a Xc=62,...b=V axc. Prop. 51.– To shew that, if the corresponding terms of any number of proportions be multiplied together, they will still form a proportion. Each proportion may be converted into a fractional equation; now since, if equals be multiplied by equals, the products are equal; therefore if all the fractions on the left hand side of the equations be multiplied together, and all those on the right hand side, the products will still be equal. Hence the ratio of the product of all the first terms to that of all the second terms is equal to the ratio of the product of all the third terms to that of all the fourth terms. The same Prop. exhibited algebraically:-Let there be any number of proportions, a : 6 :: cid; e:f::g:h; k:1:: m : n, &c. then k &c. Hence k d cX8 X m X &c. 6 X f XIX &c. d xh Xn X &c. .. axexkx &c. : bXf XIX &c. :: CXgXmX &c. ; dxhXnX &c. Cor. The above process is called Compounding the Proportion; and hence, and from Cor. 4 Prop. 47 it appears that, if any of the means and extremes be alike, they may be neglected in compounding; or if any of the means and extremes have any common factors, their other factors may be substituted for them. с e 8 m ala = n a e с m n or Prop. 52. -- To shew that, if one quantity vary directly as another, corresponding numerical values of the two quantities will form a proportion, the first being to the second value of the one, as the first to the second value of the other. This follows immediately from the definition of variation; for if A'B, then if A be changed, B is changed in the same ratio, i.e. if A?, and B?, be corresponding values of A and B, the ratio of A to Al is the same as that of B to B1; or A : A1 :: B : B1. Prop. 53.-To shew that, if one quantity vary inversely as another, corresponding numerical values of the two 1 If AOC then if A be changed, B B must be changed in the same ratio, i.e. if A!, B1 be corresponding values of A, B, the ratio of A to A? 1 is the same as that of to -, or (simplifying this ratio) as that of Bi' B Bi to B : :-, or A : Al :: B1 : B. Prop 54.-7'o explain the Rule for Simple Proportion. In questions of Simple Proportion, quantities of two kinds are involved, which vary either directly or inversely as each other. Corresponding values of these (one of each) are given, and a second value of one of them, and it is required to find the corresponding value of the other. Thus if A and B denote corresponding numerical values of the quantities, and Al be another value of A, it is required to find the corresponding value B1 of B. Now if A o B, then (Prop. 52) A : A1 :: B : B1. But if AOC then A B A1 :: B1 : B, or A1: A :: B : B1. Hence the Rule is evident; place for the third term of a proportion that quantity which is of the same kind with the answer required, and of the two remaining quantities place that, which is connected with the third term, as being the corresponding value to it, for the first or second term, according as the quantities vary directly or inversely; and place the remaining quantity for the other term. The fourth term will now be obtained by multiplying the second and third terms together and dividing by the first, the first and second terms being expressed in the same name. 1 Prop. 55.- If a quantity be so connected with two sets of other quantities, that its value varies directly as each of the first set, and inversely as each of the second, when all the rest remain unaltered ; then, if all are changed, the value will vary directly as the product of the first set, and inversely as the product of the second. so that : :: B so that : Aii This Prop. can be readily proved only algebraically:-Let B, C, D, &c. B1, C1, D1, &c. be given values of two sets of quantities, connected with another, whose corresponding value is A, in such a manner that this latter varies directly as each of the first set, and inversely as each of the second, when all the rest remain unchanged; then, if all be changed, the value of Bx C x D x 8c. A will be changed in the same ratio as BIX Cix Dix &c. First let B be changed to b, then A will be changed to some value Ai, A Ai : 5 Next let B1 be changed to 61, then 'Al will be changed to some value Aii 1 Ai Bi 01 Simly. if c, ct; d, d1; &c. be other values of C, C1; D, D1, &c. and Aiii, Aiv, Ay, Avi, &c. the corresponding values of A, we shall have the proportions, Aii C: 1 1 Ci ci 1 1 Di di Whence compounding all these proportions, and striking out common means and extremes, we have Bx C xD 6 Xe X d B?XC?XD1 b1x cixdi which expresses the fact that A has been changed in the same ratio as that BXCXD in which BÍXCIXD1 has been changed ; that is, that the value of A varies directly as the product of B, C, D, and inversely as that of B,C1,D1. :: с :: : Aiv : :: : Avi Prop. 56.- To explain the Rule for Compound Pro portion. In questions of Compound Proportion, several quantities of different kinds are involved, which are so connected with another quantity, that this latter varies directly or inversely as each of them, when all the rest are unchanged. Certain values of the first-mentioned quantities, and the corresponding value of the last-mentioned are given, and also other values of the first, from which it is required to find the corresponding value of the last. Let B, C, D; B?, C1,D1, be given values of two sets of quantities, and A the corresponding value of another quantity, which varies directly as each of the former, and inversely as each of the latter quan : Ai : A : с :: Ai : : d :: Bi :: Ci :: Aii : : tities; also let b, c, d; 61, c', d', be other given values of these quantities, from which it is required to find the corresponding value of A. If each of the quantities B, C, D; B1, C1, D1, be changed in succession, the others remaining unaltered, and if Ai, Aii, Aiii, Aiv, AV, Avi, be the successive values of A, we have the following proportions, B : 6 A Aiii Aiv ci Av .di ; Di :: Av: Avi in each of which the first and second terms are precisely those which would have arisen from considering the question as one of Simple Proportion, depending only upon the quantities in those terms. Now if these proportions be compounded, all the A's except A and Avi will disappear; and we have the result that the ratio compounded of the several ratios B : 6 &c. is equal to the ratio of A to Avi. Hence if the ratios B : 6 &c. be formed, and compounded, Avi, or the value of the quantity required may be found by solving a simple proportion in which the third term is A, and the first and second are those of the compound ratio. Note. This rule might have been deduced immediately from the last Prop.; for BXCXD bXcXd BiXC1XD1 61x cixdi .: A : Avi :: BXCXD XblXcxdi : bXcXdX B?XCIXD1 which is the proportion compounded of the above proportions. : : : : Prop. 57.-To explain the Rule for finding the Simple Interest on a given sum. Questions in Simple Interest are questions in Compound Proportion, the quantities involved being cash, time, and interest; interest varying directly as each of cash and time, when the other is unaltered. Hence, if P, R, I, denote the numbers of pounds in the given sum, in the interest per cent. per annum, and in the required interest, and n the number of years, we have 100 X 1 : PXn :: Ꭱ : I :: £P X 7 : £ I { Whence the Rule. Note. P is the number of pounds in the principal, but if there be shillings and pence in the given sum, it manifestly will not be necessary to {£PXn XR}=100 n n n n express them actually as fractions of a pound, because they are virtually so expressed, since instead of the denominations shillings, pence, might be used 2£, 21,£. Cor. 1. If n be the number of months in the time, then since n months n 12 years, we have only to substitute for n in the above formula; 12 i.e. we have to divide the interest for n years by 12. Cor. 2. If n be the number of days in the time, we have only to substitute for n in the above formula; by which it becomes PXn X R P XR X (A) 1 (B) 1 1 + nearly (C) (D) 1 nearly (E) 100 5 72 72 Formula A gives the exact method for finding interest for a number of days; formula B gives a first approximation to exactness; C gives a result nearer the truth; D is the formula applicable to decimal operations, giving the exact result; E is also applicable to decimals, giving an approximate result. Cor. 3. Hence if a, b, c, &c. be several sums of money out at interest for numbers of days m, n, p, &c. the whole interest a X m X R 6 Xn XR с XpXR I= + + + &c. 36500 36500 36500 2 R n or = n or= n a x{a xm +6 X n +exp+&c.} 73000 which formula expresses the Rule 4 in the “ Practice." Prop. 58.-To explain the Rules for finding any one of the quantities, Time, Rate, Principal, when the others and the interest are known. 1st. To find the time. The time evidently varies directly as the interest, if the principal and rate remain the same; hence if Il be the interest for 1 year, 1 the given interest, n the number of years, |