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Double Position is the method of resolving certain ques tions by means of two suppositions of false numbers.
To the Double Rule of Position belong such questions as have their results not proportional to their positions: such are those, in which the numbers sought, or their parts, or their multiples, are increased or diminished by some given absoluto number, which is no known part of ihe number sought.
Take or assume any two convenient numbers, and proceed with each of them separately, according to the conditions of the question, as in Single Position ; and find how much each result is different froin the result mentioned in the question, calling these differences the errors, noting also whether the results are too great or too little.
Demonstra The Rule is founded on this supposition, namely, that tlie first error is to the second, as the difference between the true and first supposed number, is to the difference between the true and second supposed number ; when that is not the case, the exact answer to the question cannot be found by this Rule.-That the Rule is true, according to that supposition, may be thus proved,
Let a and b be the twis suppositisns, and A and B their results, produced by similar operation ; also ; and s their errors, or the differences between the peaults A and B from the true result N; and let x denote the number sought, answering to the true result n of the question.
Then is N Å =1, and v- B = 3. And, according to the supposition on which the Rule is founded, r :: :: *- : x-b: hence, by multiplying extremes and means, r* - rb then, by transposition, ex 3* 5rb sa; and, by division, rosa
the number sought, which is the rule when the
results are both too little
If he results be b th too great, so that A and E are both greater than v; tben N Ar, and N-B5-s, or " and 8 are both negative; hence
a:x-b, buit; om $:: go : + s, therefore r :$:; * a: x-b; and the rest will be exactly as in the former case.
But if one result A only be too little, and the other B too great, or one error , positive, and the other s negative, then the theorem be
tb to sa comes I =
which is the Rule in this case, or when the errors
gutoa are unlike. Voz. I. T
Then multiply each of the said errors by the contrary sup: position, namely, the first position by the second error, und the second position by the first error Then,
If the errors are alike, divide the difference of the products by the difference of the errors, and the quotient will be the answer.
But if the errors are unlike, divide the sum of the products by the sum of the errors, for the answers
Note, The errors are said to be alike, when they are either both too great or both too little ; and unlike, when one is 10% great and the other to little.
EXAMPLES. 1. What number is that, which being multiplied by 6, the product increased by 18, and ihe sum divided by 9, the quotient shall be 20 ? Suppose the two numbers 18 and 30
Second Position. Proof. 18 Suppose
27 6 mult.
RULE II. Find, by trial, two numbers, as near the true number as convenient, and work with them as in the question ; marking the errors which arise from each of them.
Multiply the difference of the two numbers assumed, of found by trial, by one of the errors, and divide the product by the difference of the errors, when they are alike, but by their sum when they are unlike.
Add the notient, last found, to the number belonging to the said error, when that number is too little, but subtract it when too great, and the result will give the true quantity sought*.
1. So, the foregoing example, worked by this ad rule will be as follows: 30 positions 18;
their dif. 12 -2 errors + 6; least error
sum of errors 8) 24 (3 subtr.
from the position so
leaves the answer 27
Ex. 2. A son asking his father how old he was, received this answer: Your age is now one-third of mine ; but 5 years ago, your age was only one-fourth of mine. What then are their two ages?
Ans: !5 and 45. 3. A workman was hired for 20 days, at 38 per day, for every day he worked ; but with this condition, that 'for every day he played, he should forfeit 1s. Now it so hapa pened, that upon the whole he had 24 48 to receive. How many of the days did he work ?
Ans. 16, 4. A and B began to play together with equal sums of money: A first won 20 guineas, but afterwards lost back of what he then bad ; after which, B had 4 times as much as
What sum did each begin with ? Ans. 100 guineas. 5 Two persons, A and B, have both the same income, A saves of his; but B, by spending 50l per annum more than a. at the end of 4 years finds himself 1001 in debt. What does each receive and spend per annum ? Ans. They receive 1252 per annum ; also A spends 1001,
and o spends 150per annum.
* For since, by the supposition, r;s::x-a:x – b, therefore by division, rasis::6-8;4-6, which is the 2d Rule.
PERMUTATIONS AND COMBINATIONS.
PERMUTATION is the altering, changing or varying the position or order of things; or the showing how many different ways they may be placed.---This is otherwise called Alternation, Changes, or Variation ; and the only thing to be regarded here, is the order they stand in ; for no two parcels are to have all their quantities placed in the same situation : as, how many changes may be rung on a number of bells, or how many different ways any number of persons may be placed, or how many several variations may be made of any number of letters, or any other things proposed to be Varied.
COMBINATION is the showing how often a less number of things can be taken out of a greater, and combined together, without considering their places, or the order they stund in. This is sometimes called Election or Choice; and here every parcel must be different from all the rest, and no two are to have precisely the same quantities or things.
Combinations of the same Form, are those in which there arc the same number of quanuities, and the same repetitions : thus, aabc, bbcd, code, are of the same form ; aabc, abbb, aabb, are of different forms.
Composition of Quantities, is the taking a given number of quantities out of as inany equal rows of different quantities, one out of every row, and combining them together.
Some illustration of these definitions are in the following Problems :
To assign the Number of Permutations, or Changes, that can be made of any Given Number of Things, all different
from each other.
Multiply all the terms of the natural series of numbers, from I up to the given number, continually together, and the last product will be the answer required.
The reason of the Rule may be shown thus ; any one thing a is capable only of one on,
Any two things a ard ), are only capable of two variations ; as cē, ba; whose number is expressed by 1 X 2.
1. How many changes may be rung on 6 bells.
720 the Answer.
Or 1 X2 X3 X 4 X 5 X 6=720 the Answer:
2. How many days can 7 persons be placed in a differen position at dinner?
Ans. 5040 days 3. How many changes may be rung on 12 bells, and what time would it require, supposing 10 changes to be rung in 1 minute, and the year to consist of 365 days, 5 hours, and 49 minutes ?
Ans. 479001600 changes, and 91 years, 26 days, 22 hours, 41 minutes.
4. How many changes may be made of the words in the following verse : Tot tibi sunt dotes, virgo, quot sidera cælo ?
Ans. 40320 changes.
If there be three things, a, b, and c; then any two of them, leaving out the 3d, will have 1 X 2 variations; and consequently when the 3d is taken in, there will be 1 X2 X 3 variations. *.
In the same manner, when there are 4 things, every three, leaving out the 4th, will have 1 X 2 X 3 variations, consequently by taking in successively the 4 left out, there will be 1 X2 X3 X4 variations. And so on as far as we please.