« AnteriorContinuar »
curring, followed by a number of ciphers equal to the non-recurring, figures. The same may be shewn of every similar decimal.
The same Prop. exhibited algebraically:-Let .P Q represent a recurring decimal, P denoting the p non-recurring figures, Q the q recurring;
let x = .P
10P+q x = PQ.
10P(109 — 1) Now PQ represent all the figures of the decimal, P the non-recurring ; and 109 - 1 will consist of q nines, and the multiplication of this by 10P will introduce on the right of these p ciphers. Whence the Rule.
Prop. 45.-To shew that the ratio of one number or quan
tity to another may be properly represented by the fraction, whose numerator is the number of units in the former, and denominator the number of the same kind
of units in the latter, quantity. Ratio is the relation which one quantity bears to another of the same kind, this relation being determined by considering what multiple, part, or parts, the one is of the other. Now if the quantities be expressed in terms of the same unit, the unit may be expressed as a multiple, part, or parts, of the one quantity, and hence the other quantity may be expressed as a multiple, part, or parts of this. Thus suppose that it be required to determine what multiple, part, or parts, the length 2 feet is of the length 3 yards, if we express these lengths in terms of the same unit either a foot, or a yard, they will be 2 feet and 9 feet; or j yard, and 3 yards: hence it appears that the unit 1 foot is š of 9 feet or three yards; and the unit 1 yard is } of 3 yards; and therefore the length 2 feet is g of 3 yards; or the length yard is z of } of 3 yards'; or x of 3 yards, or of 3 yards.
Hence z or (which are equivalent) represent the part which 2 feet are of
3 3 yards: and the numerators and denominators of these factors are respectively the number of units in the two lengths.
Again, suppose it required to find the ratio of the abstract number 2 to the abstract number 9; that is to find “how often” 9 is contained in 2. The meaning of this is, (Prop. 35,) that we have to find what operation must be performed upon the result of that denoted by 9 to produce the result of that denoted by 2. Now we know that 2=9x ; therefore is the symbol of the operation required, or expresses the ratio of 2 to 9.
Cor. 1. Hence the ratio of one number or quantity to another may be obtained by dividing the number of units in the former by the number of like units in the latter.
Cor. 2. Hence the terms of a ratio may be simplified by multiplying or dividing both by the same number; since we may do this to the terms of the equivalent fraction without altering its significance.
Cor. 3. Hence also two ratios may be compared by comparing the equivalent fractions, in order to which they must be reduced to a C. D. and as any C. D. will serve the purpose, we may use that which is formed by the product of the two: so that the comparison will be made by comparing the products of the antecedent of the one by the consequent of the other, the antecedents and the consequents of the ratios being the numerators and denominators of the fractions.
Prop. 46.- To shew how to divide a number or quantity
into parts, which shall bear to each other a given ratio. If the number or quantity be divided into equal parts, the number of parts being the sum of the numbers composing the ratios, and if collections of these parts be formed, containing numbers eqnal to the several numbers in the ratios, it is evident that the parts so formed will be in the required ratios. Thus suppose it required to divide 20 into 3 parts in the ratio of 4, 5, and 9. Dividing 20 into 18 parts, each is ; and taking 4, 5, and 9 of these parts, the numbers, 50, go are formed, which are in the ratio of 4, 5, and 9; for 40 - 60 =* ; and 4 = go = g.
The same Prop. exhibited algebraically:-Let it be required to divide the number N into parts in the ratio of a, b, c: let x, y, z be the parts: then
=-=t suppose: hence x = at; y=bt; b b
N z=ct; and x +y+z=N= (a + bitc)t: whence t =
Prop. 47.- To shew that if four numbers be proportional
in a given order, the product of the extremes is equal to
that of the means, and conversely. Four numbers are proportionals when the ratio of the first to the second is equal to that of the third to the fourth. Hence these ratios must be compared, which is effected by comparing the products of the antecedent of the one by the consequent of the other. If these products be equal, the ratios are equal, and the numbers are proportional. But these pro
ducts are respectively the products of the extremes, and of the means, of the four numbers. Hence the truth of the Prop.
The same Prop. exhibited algebraically :-Let a, b, c, d, be four numbers ; if they be proportional in this order, then
or (reducing to a bc C. D.) bd
; whence a d=bc; or the product of the extremes is equal to that of the means. Conversely:—if four numbers be such that the product of one pair is equal to the product of the other pair, they are proportionals, the numbers in the products being either both extremes, or both means. For if a Xd=bXC, then, dividing by 6 X d, =-, i.e.
b a : 6 :: 0 :d; or,
by c X d, -=-, i.e. a : 0 :: 6 : d; or 6
d' again, dividing by a X C, =-, i.e.b: a :: d: c; or again, dividing by
d a Xb, -, i.e. c:a:: d: 6.
b Cor. 1. Hence, if four numbers are proportional, the second is to the first as the fourth to the third. For if a : 6 :: c:d, a Xd=bXc; .. from the Prop. 6 : a :: d: c.
Cor. 2. Also, in the same case, the first is to the third as the second to the fourth. For if a : 6 :: c : d, a X d= 6 X c; and .. from the Prop. Q:C:: b: d.
Cor. 3. Also the third is to the first as the fourth to the second. For by Prop. c : a :: d:b.
Cor. 4. Hence also either mean and extreme may be multiplied or divided by the same number without affecting the proportion, for m a Xnd = mb Xnc. Prop. 48.- To find a fourth proportional to three given
numbers. Since the product of the second and third of four numbers, which are proportionals, is equal to that of the fourth and first, therefore the former product divided by the first is equal to the fourth, which is the number required.
ЪХc Thus if a, b, c, d, be the four numbers, a xd=bXc, ..d =
Prop. 49.—To find a third proportional to two given
numbers. If three numbers are continued proportionals, the first is to the second, as the second to the third, therefore the product of the first and third is equal to the square of the second. Hence the third is obtained by dividing the square of the second by the first. Thus if a, b, c, be the four numbers,
62 a : 5 :: 6 : ; .. a Xc=6%, and c=
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 prosluct of the two given numbers. Thus if a and c be given, and it be required to find a number b such that a : 6 :: 6 : C, then since a Xc=b,.b=V ax c.
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::c:d; e:f::8:h; k:1:: m : n, &c. then
bXfXIX &c. d xh Xn X &c. .. axexkx &c : bXf XIX &c. :: CX8XmX &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.
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 e 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 A1 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 A OC
must be changed in the same ratio, i.e. if A?, B1 be corresponding values of A, B, the ratio of A to Al is the same as that of
to , or (simplifying this ratio) as that of B1 to B :-, or A : A1 :: B1: B.
Prop 54.-1'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
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.
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.