Page images
PDF
EPUB

The greater added to the less forms the number to be divided. Substituting in this last proposition, instead of the words, the greater part, the equivalent expression given above, namely, the less part added to the given excess, we find that

The less part, added to the given excess, added moreover to the less part, forms the number to be divided.

But the language may be abridged, thus,

Twice the less part, added to the given excess, forms the number to be divided;

whence we infer, that,

Twice the less part is equal to the number to be divided diminished by the given excess ;

and that,

Once the less part is equal to half the difference between the number to be divided and the given excess.

Or, which is the same thing,

The less part is equal to half the number to be divided, diminished by half the given excess.

The proposed question then is resolved, since to obtain the parts sought it is sufficient to perform operations purely arithmetical upon the given numbers.

divided were 9, and the

5, the less part would be, less, or, or 2; and

If, for example, the number to be excess of the greater above the less according to the above rule, equal to the greater, being composed of the less plus the excess 5, would be equal to 7.

2. The reasoning, which is so simple in the above problem, but which becomes very complicated in others, consists in gener of a certain number of expressions, such as added to, dimin ished by, is equal to, &c. often repeated. These expressions relate to the operations by which the magnitudes, that enter into the enunciation of the question, are connected among themselves, and it is evident, that the expressions might be abridged by representing each of them by a sign. This is done in the following manner.

To denote addition we use the sign+, which significs plus. For subtraction we use sign, which signifies minus.

For multiplication we use the sign X, which signifies multiplied by.

To denote that two quantities are to be divided one by the

other, we place the second under the first with a straight line between them; signifies 5 divided by 4.

Lastly, to indicate that two quantities are equal, we place between them the sign which signifies equal.

These abbreviations, although very considerable, are still not sufficient, for we are obliged often to repeat the number to be divided, the number given, the less part, the number sought, &c. by which the process is very much retarded.

With respect to given quantities, the expedient which first offers itself is, to take for representing them determinate numbers, as in arithmetic, but this not being possible with respect to the unknown quantities, the practice has been to substitute in their stead a conventional sign, which varies as occasion requires. We have agreed to employ the letters of the alphabet, generally using the last; as in arithmetic we put x for the fourth term of a proportion, of which only the three first are known. It is from the use of these several signs that we derive the science of Algebra.

I now proceed by means of them to consider the question stated above (1). I shall represent the unknown quantity, or the less number, by the letter x, for example, the number to be divided and the given excess by the two numbers 9 and 5; the greater number, which is sought, will be expressed by x + 5, and the sum of the greater and less by x + 5 + x; we have then

x + 5 + x = 9;

but by writing 2x for twice the quantity a there will result

2x+5=9.

This expression shows that 5 must be added to the number 2x to make 9, whence we conclude that

[blocks in formation]

By comparing now the import of these abridged expressions, which I have just given by means of the usual signs, with the process of simple reasoning, by which we are led to the solution, we shall see that the one is only a translation of the other.

The number 2, the result of the preceding operations, will answer only for the particular example which is selected, while the course of reasoning considered by itself, by teaching us, that

the less part is equal to half the number to be divided, minus half the given excess, renders it evident, that the unknown number is composed of the numbers given, and furnishes a rule by the aid of which we can resolve all the particular cases comprehended in the question.

The superiority of this method consists in its having reference to no one number in particular; the numbers given are used throughout without any change in the language by which they are expressed; whereas, by considering the numbers as determinate, we perform upon them, as we proceed, all the operations which are represented, and when we have come to the result there is nothing to show, how the number 2, to which we may arrive by any number of different operations, has been formed from the given numbers 9 and 5.

3. These inconveniences are avoided by using characters to represent the number to be divided and the given excess, that are independent of every particular value, and with which we can therefore perform any calculation. The letters of the alphabet are well adapted to this purpose, and the proposed question by means of them may be enunciated thus,

To divide a given number represented by a into two such parts that the greater shall have with respect to the less a given excess represented by b.

Denoting always the less by x;

The greater will be expressed by x+b;

Their sum, or the number to be divided, will be equal to x+x+b, or 2 x + b ;

The first condition of the question then will give

2x+b=a.

Now it is manifest that, if it is necessary to add to double of x, or to 2x, the quantity b in order to make the quantity a, it will follow from this, that it is necessary to diminish a by b to obtain 2x, and that consequently 2x-a-b.

a b

We conclude then that half of 2 x or x == 2 2

This last result, being translated into ordinary language, by substituting the words and phrases denoted by the letters and signs which it contains, gives the rule found before, according to which, in order to obtain the less of two parts sought we subtract

from half of the number to be divided, or from a half of the given

[blocks in formation]

Knowing the less part we have the greater by adding to the less the given excess. This remark is sufficient for effecting the solution of the question proposed; but Algebra does more; it furnishes a rule for calculating the greater part without the aid of the less as follows;

[merged small][ocr errors][merged small]

being the value of this, augmenting it by the excess h,

[merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

that after having subtracted from the half of b, it is necessary

to add to the remainder the whole of b, or two halves of b,

a

which reduces itself to augmenting by the half of b, or by 2

a b

b

2

It is evident then that + b becomes +; and by trans

a 2

b 2

2 2

lating this expression we learn, that of the two parts sought the greater is equal to half of the number to be divided plus half of the given excess.

In the particular question which I first considered, the number to be divided was 9, the excess of one part above the other 5; in order to resolve it by the rules to which we have just arrived, it will be necessary to perform upon the numbers 9 and 5, the operations indicated upon a and b.

The half of 9 being and that of 5 being, we have for the less part

[merged small][ocr errors][ocr errors]

4. I have denoted in the above the less of the two parts by x, and I have deduced from it the greater. If it were required to find directly this last, it should be observed, that representing it by x, the other will be xb, since we pass from the greater to the less by subtracting the excess of the first above the second the number to be divided will then be expressed by x+x—b, or by 2x-b, and we have consequently 2x-ba.

This result makes it evident that 2x exceeds the quantity a

by the quantity b, and that consequently 2x = a + b. By taking the half of 2x and of the quantity which is equal to it, we obtain for the value of x

a b
x= +
2 2'

which gives the same rule as the above for determining the greater of the two parts sought. I will not stop to deduce from it the expression for the smaller.

The same relation between the numbers given and the numbers required may be enunciated in many different ways. That which has led to the preceding result is deduced also from the following enunciation:

Knowing the sum a of two numbers and their difference b, to find each of those numbers; since, in other words, the number to be divided is the sum of the two numbers sought, and their difference is the excess of the greater above the less. The change in the terms of the enunciation being applied to the rules found above, we have

The less of two numbers sought is equal to half of the sum minus half of the difference.

The greater is equal to half of the sum plus half of the difference. 5. The following question is similar to the preceding, but a little more complicated.

To divide a given number into three such parts, that the excess of the mean above the least may be a given number, and the excess of the greatest above the mean may be another given number.

For the sake of distinctness I will first give determinate values to the known numbers.

I will suppose that the number to be divided is 230;

that the excess of the middle part above the least is 40; and that of the greatest above the middle one is 60.

Denoting the least part by x,

the middle one will be the least plus 40, or x+40, and the greatest will be the middle one plus 60, or x + 40 + 60.

Now the three parts taken together must make the number to be divided; whence,

x + x + 40 + x +40 + 60 = 230.

If the given numbers be united in one expression and the unknown ones in another, x is found three times in the result, and for the sake of conciseness we write

3x+140 230.

« PreviousContinue »