3. Now a quantity can be measured or determined only by comparing it with some other quantity that is known, and observing the relation that the one bears to the other. For instance, if we wished to determine the quantity, or value of a sum of money, we must take some piece of money that is known, as a louis, a crown, or a ducat, and point out how many of such pieces are contained in the sum given. Again, if it were proposed to determine the quantity of a weight, we must take a certain weight that is known, such as a pound, a quintal, or an ounce, and observe how many times any one of them be contained in the weight in question. If we wished to measure length or extension, we should take a certain known length, as a foot, &c.

Thus the determination or measure of magnitude of any kind resolves itself to this. Take any one known quantity, of the same kind with that to be determined, as the measure or unit; and then determine the relation which the proposed quantity bears to the known quantity. This relation is always expressed by numbers, from which it follows, that a number is nothing more than the relation which one quantity bears to another, taken arbitrarily as the unit.

4. It is evident from this, that all magnitudes can be expressed by numbers, and that the foundation of all the sciences of mathematics must consist in a complete treatise on the science of numbers, and in a correct examination of the different methods of calculation.

This fundamental part of mathematics is called Analysis or Algebra.

5. In Algebra then are considered only the numbers which represent quantities, without investigating the particular kinds of quantity, which are the subject of other branches of mathematics.

6. Numbers

6. Numbers are in particular treated of in Arithmetic, which is the science of numbers properly so called, but this science extends itself only to certain methods of calculation, which present themselves in ordinary practice. Algebra, on the contrary, comprehends generally all the cases which can be met with in the doctrine and calculation of numbers.

4 )

CHAP. II.

OF ADDITION AND SUBTRACTION, WITH AN EXPLANATION OF THE SIGNS + (PLUS) AND - (MINUS.)

7. Ir it be required to add one number to another, we signify the addition by the sign +, which is placed before the second number. Thus, 5+3 signifies that 3 is to be added to 5, and every one knows that the result is 8; as 12 + 7 is 19, &c.

8. It is customary also to make use of the same sign to join together several numbers; as 7+5+9 signifies that 5 is to be added to 7, and to that sum 9, which make 21. the following formula

8+5+13+11+1+3+10

Thus

is easily understood to signify the sum of all these numbers, i.e. 51.

9. When, on the contrary, it is intended to subtract one number from another, the operation is indicated by the signplaced before the number to be subtracted, as 8—5, signifies that 5 is to be subtracted from 8, leaving a remainder of 3. Thus the formula

50-1-3-5-7-9

signifies that from 50 is to be subtracted 1, leaving 49; from 49, 3, and there will remain 46; from 46, 5, and there will remain 41; from 41, 7, and the remainder is 34; lastly, from 34, 9, and there will remain 25, which last sum is the value of the proposed formula. But as the numbers 1, 3, 5, 7, 9 are all to be subtracted, it will be the same thing if we subtract their sum, which is 25, at once, and the remainder will be 25, as before.

10. It is also very easy to determine the value of the same formula, where the two signs + and are found, as for instance,

12-3-5+2-1, which is 5.

We have only to take first the sum of the numbers to which the sign +, plus, is prefixed, and from that sum to subtract the sum of the numbers which are preceded by the sign. Thus the sum of 12+2 is 14; that of 3, 5, and 1, is 9, and 14-9 leaves the remainder 5, as before.

11. It only remains to observe, that in Algebra, where numbers are represented generally by letters, such as a, b, c, &c.; the expression a+b signifies the sum of the two numbers represented by a and b respectively, and

f+m+b+x

signifies the sum of the numbers represented by these four letters, and

f-m-b-x

signifies that the numbers m, b, and x, are to be subtracted from the number represented by f.

It will be seen from these examples that it is wholly indifferent in what order the numbers are written, provided only that to each is preserved its proper sign.

For instance, it matters not whether we write

12+2-5-3-1, or 2-1-3-5+12, or 2+12-3-1-5.

Nor will any greater difficulty be experienced in determining the value of the algebraic formula

a-b-c+d-e

where the letters a, b, c, d, e are used to represent effective numbers. For it is evident, that from the numbers represented by a and d, to which the sign + is prefixed, are to be subtracted those which are represented by the remaining letters b, c, e, which are preceded by the sign

B 3

12. It

12. It is of great importance to consider what sign is to be placed before each number, since in Algebra simple quantities are numbers considered with respect to the sign which precede or affect them; and those numbers which have the sign + before them are called positive, those with the sign - negative quantities.

13. What we have just said may be very properly illustrated by considering the manner in which we are accustomed to estimate a man's property. That which he actually possesses may be denoted by positive numbers, with the sign +; and his debts by negative numbers with the sign Thus, to say of any one that he is possessed of an hundred crowns, but that he owes 50, is to say that his property amounts to 100-50, or what is the same thing, +100-50, i. e. +50.

14. Now since negative numbers may be considered as debts, in the same manner as positive numbers signify actual property, it may be said that negative numbers are less than nothing. Thus when a man possesses nothing and owes 50 crowns, it is clear that he has less than nothing by 50 crowns; for if any one were to present him with 50 crowns for the purpose of paying his debts, he would still be in the actual possession of nothing, though he would be in fact richer than before.

15. Thus therefore, as positive numbers are incontestably greater than nothing, so are negative numbers less than nothing. For we obtain positive numbers by adding 1 to 0, and by continuing to add in the same manner to unity, and this is the origin of the series of natural numbers, of which the following are the first terms, viz.

0+1+2+3+ 4+ 5+ 6+ 7+ 8+ 9+ 10

&c. ad infinitum.

16. But if instead of continuing the operation by successive additions, we were to pursue it in a contrary manner,

by