1

Questions such as the last can be resolved by different processes; but this method of forming each of the given relations into a ratio expressed by a fraction, and combining the ratios into a complex fraction by means of which the money which is to be exchanged is connected with the money for which it is required to exchange it, seems the most simple.

The relations between the money of England and France, France and Germany, &c. are expressed in round numbers; the object being merely to indicate the manner of resolving questions of this kind.

5th Example. If a grocer mixes 4 cwt. of sugar at 56 sh. per cwt., 7 cwt. at 43 sh. per cwt., and 5 cwt. at 37 sh. per cwt.; what is 1 cwt. of the mixture work?

4 cwt. at 56 sh. per cwt.=224 sh.=11 £. 7 cwt. at 43 sh. per cwt.=301 sh.=15 £. 5 cwt. at 37 sh. per cwt.=185 sh.= 9£. 16 cwt. of the mixture cost

4 sh.
1 sh.

5 sh.

35 £. 10 sh.

The question now is, if 16 cwt. cost 35 £. 10 sh., what is the price of 1 cwt.? To find this,

71

16 cwt.: 1 cwt.:: 352

£.

:

71
2× 16

71

£.=

£. 32

£. 2£. 4 sh. 4 d.

Whence the price of 1 cwt. of the mixture is 2£. 4 sh. 4 d.

This question is a particular example of the most simple case of Alligation or the rule of mixtures. The more difficult cases are most simply resolved with the help of algebra.

6th Example. Suppose that the distance between two fixed points on level ground has been measured four times; that the results of these measurements disagree,

the first being 250.439 yards,
the second being 250-695 yards,
the third being 249-750 yards,
the fourth being 251.158 yards ;

and that it is required to assign as nearly as possible the true distance; this is also considered a question falling under the rule of alligation.

The distance being constant and the measurements different, it is certain that some and perhaps the whole of them are inaccurate. To approximate to the true result, it is evident that if the true distance had been obtained each time the sum of the results must be equal to four times the distance; it is also evident Q 4

that the same thing must happen if some of the results erred in excess and others in defect in such a manner that the errors in excess compensated those in defect.

Whence, in this case also, the true distance must be obtained by dividing the sum of the measurements by their number.

The sum of the four measurements is 1002-042 yards, and 1002-042+4= 250.5105 yards.

250-5105 yards is assumed to be nearly the distance required.

The result obtained by dividing any number of different results obtained by experiment or observation by the number of repetitions is termed a mean value.

It is not to be expected that in any case the errors in excess and defect will exactly compensate each other, but the approximation of a mean to a true value may be considered close in proportion as the number of repetitions is great.

7th Example. Suppose that A executes a certain work in 6 days, and that B requires 8 days to execute the same work, in what time can A and B accomplish it if both work together?

Since A does the whole in 6 days he does part in 1 day, and since B does the whole in 8 days he does part in 1 day; therefore A and B working together do + parts of the whole work in 1 day.

The question is consequently reduced to this: parts of the work are done in 1 day, what time is required to do the whole, or 4 parts.

And the proportion: 47: 24::1 day: 24=33 days, gives the time required.

When an arithmetical question involves many or complex conditions the investigation for the discovery of the unknown quantity is simplified by the employment of the resources of algebra.

In Example 7, let the time taken by A be denoted by the general symbol a, the time taken by B by the symbol b, the time required by both, or the unknown quantity, by x.

1 a'

Then in 1 day A does a part of the work expressed by and in x days a

Similarly, in 1 day and x days B does parts expressed by and tively.

x

ō respecTherefore in a days A and B together do a part of the work expressed by +7.

a

But by hypothesis A and B together do the whole work in a days; therefore, denoting the whole work by 1,

Multiplying the equal quantities by ab the products are equal;

or bx+ax=ab,

or (a+b)x=ab (Art. 78).

Dividing the equal quantities by a+b, the quotients are equal;

therefore r

ab a+b

This result involves not alone the answer of question 7, but that of every question in which the times taken by two agents acting separately to produce an effect being given, it is required to find the time taken by the same agents acting together.

If, for example, a is replaced by 6 and b by 8,

6×8 48

ab a+b

becomes

6+8=143%, the answer of question 7.

......

8th Example. If 2 d. in the shilling is a dealer's profit, how much does he gain per cent.? .......................................Ans. 163 per cent. 9th. A draper bought 124 yards of linen for 31£., how must he sell it per yard to gain 10£. 6sh. 8 d. on the whole ?............Ans. At 6sh. 8 d. per yard. 10th. A merchant receives 126 yards of cloth in barter for 189 gallons of wine at 6s. 8 d. per gallon, what is the cloth valued at per yard? Ans. At 10sh. per yard.

11th. How many bushels of wheat at 4sh. per bushel must be given in exchange for 120 musket barrels at 15 sh. 6d. each?........ Ans. 465 bushels. 12th. In Portugal 1000 reas make 1 millrea, and 1 millrea is worth 633 d. sterling; it is hence required to find, in British money, the value of 827 millreas 160 reas?.. ....Ans. 218 £. 8 sh. 54 d.

13th. If 3 francs are worth 32 pence sterling, 240 pence are worth 408 deniers of Holland, and 50 deniers are worth 190 Spanish maravedis; it is required to find how many maravedis are worth 90 francs?

Ans. 62013 maravedis.

14th. A distiller mixes several sorts of spirits; viz. 36 gallons at 8 sh. per gallon, 2 gallons at 7 sh. per gallon, and 13 gallons at 1 sh. per gallon, with 12 gallons of water; what is a gallon of the mixture worth ?......... Ans. 5 sh.

PART II.

ELEMENTS OF ALGEBRA.

SECTION I.

DEFINITIONS AND FIRST PRINCIPLES.

1. The result of every arithmetical question is expressed by some number, whole or fractional, in which all distinction of the given quantities, and of the operations by which they are combined to form this number, is lost.

If numbers are represented by general characters, and arithmetical operations are indicated by symbols, the given quantities can be made to preserve their distinction throughout a calculation, and the result to exhibit the arithmetical operations which must be performed with these quantities in order to obtain the numerical value required.

But these operations cannot be effected unless particular values are asab signed to the given quantities. Thus in Example 7, Art. 366, the result a+b indicates, that to obtain the time in which two agents acting jointly produce a certain effect, when the times taken by them singly to produce that effect are given, it is necessary to divide the product of the given times by their ab sum. But to obtain a numerical value from the expression it is necessary to assign particular values to a, b, and to perform with these values the operations indicated in the expression. Thus, if a=6 and b=8,

The formula

ab a+b

a+b'

involves the answer not to this particular question only, but to every question in which the times, taken by two agents acting singly to produce the same effect, are given to find the time in which that effect can be produced by the same agents acting together.

It is by the substitution of general characters for particular numbers, and symbols of operation for the operations themselves, that Algebra has been derived from Arithmetic.

=

2. Two expressions of equal value, with the symbol placed between them, form an equation. Thus, if the quantities denoted by a and b are of the same value, the symbolical expression of equality, a=b, is termed an Equation.

Similarly, if the sum of a and b is equal to the sum of c and d;

Or if the difference between a and c is equal to the difference between b and d;

the equations a+b=c+d,

a-c-b-d. respectively express these equalities. The quantities on each side of the symbol Equation; and each of the quantities which is connected with the others by = are named Members of the the symbols + or is called a Term.

The changes which may be made among the quantities composing an equation depend chiefly on the following self-evident principles:

1st. That if to equal quantities the same quantity or equal quantities are added the sums are equal.

2d. That if from equal quantities the same quantity or equal quantities are taken away the remainders are equal.

3d. That if two quantities are equal, their doubles, their triples, their quadruples, and, in general, their equimultiples, are equal.

4th. And that if two quantities are equal, their halves, their third parts, and, in general, their like parts or submultiples, are equal.

Whence, if both members of an equation are augmented or diminished, or multiplied or divided by the same quantity, or by equal quantities, the sums, the differences, the products, the quotients form an equation; and if each of the members of one equation is added to the corresponding member of another equation, or subtracted from, or multiplied by, or divided by it, in each case the result forms an equation.

Let a=b, and let m be another quantity,

then a+m=b+m

a-m-b-m, or m-a-m-b

ma=mb
b

α

с ď

3. The expression a-b indicates that the quantity b is to be subtracted from the quantity a.

a and b being given quantities, three cases may occur; 1st. a>b, 2d. a=b, 3d. a<b.

1st. Let a be greater than b, and let the difference be denoted by c.

This is the common case of arithmetical subtraction.

The subtraction and remainder are indicated by the equation,

2d. Let a be equal to b.

a-b=c.

The numerical subtraction in this case also is possible, and the remainder The expression of the difference is,

zero.

a-b=0.

3d. Let a be less than b by a quantity d.

In this case the numerical subtraction is impossible.

Since a=a
and b=a+d
a-b-a-a-d (Art. 2),

or a-b-0-d.

Therefore, if b exceeds a by a quantity d, when a part of b equal to a has been taken from a, the remainder is zero, and there is still the quantity d to be subtracted.

Although the process of subtraction necessarily supposes two quantities, a minuend and a subtrahend, and that in this instance there is only a subtra