TREATISE ON THE ELEMENTS OF ALGEBRA. CHAPTER I.-FIRST PRINCIPLES. DEFINITIONS. ARTICLE 1. DEF. I. The sign + (read plus) indicates that the quantity* before which it stands is to be added to something going before: thus, 8+6 means that 6 is to be added to 8. When the sign + is prefixed to a quantity standing alone, or at the beginning of a series, it indicates that there are circumstances which require that quantity to be added, if there were any quantity going before to which it could be joined: thus, +7 indicates that there are circumstances which require 7 to be added, were there any thing to which the addition could be made. 2. II. The sign-(read minus) indicates that the quantity to which it is prefixed is to be subtracted from something going before thus, 8-6 means that 6is to be taken from 8. When the sign is prefixed to a quantity standing alone, or at the beginning of a series, it indicates that : * Writers on Algebra generally confine the word number to those numerical expressions in which the Arabic figures alone occur; while they apply the term quantity to any symbol representing a number whether consisting of figures, or of letters, or of both combined: thus, 5, 19, are called numbers while 5, 19, 4a, may all be called quantities. See Note A at the end of the Volume. B there are circumstances which require that quantity to be subtracted, if there were any quantity going before it : thus, 9 signifies that there are circumstances which require 9 to be subtracted, if there were any quantity from which it could be taken. 3. III. A quantity which has the sign + is called a positive quantity; and one which has the sign is called a negative quantity. Quantities to which no sign is prefixed are understood to be positive. 4. IV. The mark placed between two expressions indicates that they are equal: thus, 12+8=20, indicates that the sum of 12 and 8 is equal to 20; and 15—9—6 means that the difference of 15 and 9 is equal to 6; and 19-20-1 indicates that if it be required to take away 20 from 19, the thing is manifestly impossible, and that after we have taken away as much as possible, namely 19, there will still remain a unit to be subtracted, without any thing from which it can be taken. Also, a+b=c shows that some number, which we call a, when increased by the addition of some other number, which we call b, becomes equal to a third number, which we denote by c. 5. Scholium 1. The example 19-20--1, will help the learner to conceive how a negative quantity may come to stand alone, that is, unconnected with any thing from which it can be taken. It will also show, that, in such a case, the sign demands that the quantity before which it is placed be subtracted from 0;* that is to say, -10-1. And since a-1, or in general a-b, must * For when, of the 20 to be subtracted, we have taken away a much as possible, namely 19, the +19 will be reduced to 0, and the 20 to -1. 1 orb, that is, This mode of speak be less than a, it has been said that 0-1 or 0-6, must be less than 0. ing would be unintelligible and absurd, if used absolutely; but, in the relative sense in which it is always used in Algebra, it is perfectly rational and correct. Suppose the question asked, "How far is Paris north of London?" we may reply, "Paris is 24 degrees south of London :" but we have not given a direct answer to the question. We have, first, rectified the mistake on which the question was founded, by showing that it ought to have been "How far is Paris south of London?" and, secondly, we have given the answer to this corrected question. If we were compelled to answer in exact accordance with the original question "How far is Paris north of London ?" we should say, 24 degrees less than nothing, that is, -21 degrees. This does not mean that there exists, or can exist, any portion of space less than nothing, (for that would be a contradiction); but that the northing of Paris from London is less than 0: in other words, that not only is Paris nothing to the north of London, but its situation is even less northerly than those words would imply. Therefore the answer, "Paris is -2 degrees to the north of London," does not merely express the absolute difference of latitude between the two places; but indicates, besides, that their true relative position is the very contrary of that which was supposed in the question. The question implies a supposition that the relation of the two places is such, that in order to find the latitude of Paris something must be added to the latitude of London: the answer exhibits this quantity to be added as less than nothing, which indicates the real relation of the two places to be such, that to find the latitude of Paris something must be subtracted from the latitude of London. In there are circumstances which require that quantity to be subtracted, if there were any quantity going before it : thus, -9 signifies that there are circumstances which require 9 to be subtracted, if there were any quantity from which it could be taken. 3. III. A quantity which has the sign positive quantity; and one which has the sign is called a is called a negative quantity. Quantities to which no sign is prefixed are understood to be positive. 4. IV. The mark placed between two expressions indicates that they are equal: thus, 12+8=20, indicates that the sum of 12 and 8 is equal to 20; and 15—9—6 means that the difference of 15 and 9 is equal to 6; and 19-20-1 indicates that if it be required to take away 20 from 19, the thing is manifestly impossible, and that after we have taken away as much as possible, namely 19, there will still remain a unit to be subtracted, without any thing from which it can be taken. Also, a+b=c shows that some number, which we call a, when increased by the addition of some other number, which we call b, becomes equal to a third number, which we denote by c. 5. Scholium I. The example 19-20--1, will help the learner to conceive how a negative quantity may come to stand alone, that is, unconnected with any thing from which it can be taken. It will also show, that, in such a case, the sign demands that the quantity before which it is placed be subtracted from 0;* that is to say, -10-1. And since a-1, or in general a-b, must * For when, of the 20 to be subtracted, we have taken away as much as possible, namely 19, the +19 will be reduced to 0, and the - 20 to -1. - 1 orb, that is, This mode of speak be less than a, it has been said that 0-1 or 0-b, must be less than 0. ing would be unintelligible and absurd, if used absolutely; but, in the relative sense in which it is always used in Algebra, it is perfectly rational and correct. Suppose the question asked, "How far is Paris north of London ?" we may reply, "Paris is 2 degrees south of London:" but we have not given a direct answer to the question. We have, first, rectified the mistake on which the question was founded, by showing that it ought to have been "How far is Paris south of London?" and, secondly, we have given the answer to this corrected question. If we were compelled to answer in exact accordance with the original question "How far is Paris north of London ?" we should say, 2 degrees less than nothing, that is, -2 degrees. This does not mean that there exists, or can exist, any portion of space less than nothing, (for that would be a contradiction); but that the northing of Paris from London is less than 0: in other words, that not only is Paris nothing to the north of London, but its situation is even less northerly than those words would imply. Therefore the answer, "Paris is -2 degrees to the north of London," does not merely express the absolute difference of latitude between the two places; but indicates, besides, that their true relative position is the very contrary of that which was supposed in the question. The question implies a supposition that the relation of the two places is such, that in order to find the latitude of Paris something must be added to the latitude of London: the answer exhibits this quantity to be added as less than nothing, which indicates the real relation of the two places to be such, that to find the latitude of Paris something must be subtracted from the latitude of London. In |