They are then called Signs of Affection or Position.

The instances of this which might be quoted from the various branches of Natural Philosophy are innumerable, and we will only quote a few to illustrate our meaning, and furnish analogies to guide us to the solution of the perplexities of Economics.

Thus in Algebraical Geometry, in which it is necessary to fix the position of lines, it is usual to take some fixed point called the Origin, and then lines drawn in opposite directions from that point are distinguished by opposite signs. Thus if lines to the right of it are called Positive, and distinguished by the sign + > then lines drawn to the left of it are called Negative, and are denoted by the sign -. If lines drawn up from it are Positive and denoted by the sign +, then lines drawn down from it are Negative and denoted by the sign -.

If two mechanical forces act in opposite directions one is called Positive and the other Negative.

So in Parliament the supporters of Government may be called +, and its opponents -.

Now in many of these cases it may happen that the Quantities endowed with these opposite qualities may balance each other, and the result may be 0: but it would manifestly be an error of the greatest magnitude to say that because under some circumstances these Opposite Quantities may neutralise each other's effects, that is the same thing as saying that they do not exist at all.

Suppose that on a division the numbers for Government were 340, and the numbers against Government 300. Then for practical purposes it might be said that the strength of the Government was 40: because the - 300 neutralise the effect of the + 300. But it would be clearly an enormous error to say that that is absolutely the same thing as saying that these 600 members do not exist at all. It is perfectly clear that there are 640 Parliamentary units. It is quite clear that to find the total number of Parliamentary units we must add the opposition to the ministerialists, and not subtract them.

It is an error of precisely the same nature to say that, because a man possesses £100, and at the same time has accepted a bill for £50 payable three months after date, there are only £50 in existence as Economic Quantities. No doubt it might be said that for practical purposes the man's fortune was only £50. But the £100 and the £50 bill may both circulate independently in commerce at the same time: and hence while they do so, and until the £50 bill is paid and extinguished there are £150 of

Economic Quantities. When the bill is paid and extinguished, it ceases to exist as an Economic Quantity.

11. Now this idea of Opposition is applied to a continuous line : or to motion in a continuous line. If any point be taken as 0, then the part of the line on one side may be denoted by + , and the part on the other side by -.

Thus in a thermometer some fixed point is taken as 0, and degrees above that point are called +, and degrees below -.

Now if the mercury passes from a certain number of degrees on either side of 0, to any number of degrees on the other side, it is quite clear that in order to find the total number of degrees passed over, the degrees on both sides must be added together. That is the Negative degrees must be added to the Positive degrees and not subtracted from them.

The same idea is applied to Time in Natural Philosophy, which may be considered as motion in a continuous line. If any point in Time be taken, such as the present moment, or any fixed era, then this point is denoted by 0, and Time on opposite sides of this point will be denoted by opposite signs. Thus if we call Time, such as years, before this era +, or Positive, then Time after this point will be -, or Negative: and the successive intervals will be expressed thus -

......+ 5, +4, + 3, + 2, + 1, 0,-1,-2,-3,-4,-5,. .

where the sign + merely means that the years it is affixed to were before the given era; and the sign - means that the years it is affixed to were after a given era.

It is quite clear that if we want to find the number of years between any event which happened some time before this era, and another event which happened some time after it we must add the number of years on both sides of 0.

Thus if the birth of Christ be the given era, or 0, the years before Christ will be Positive and the years after Christ Negative. To find the number of years from the foundation of Rome to the present time we must add together + 753 and - 1875 years, or 2628 years altogether.

12. As an illustration of the application of the Positive and Negative signs to Time, let us take an example which will be very useful in Economics.

A father's age is 40, and his son's 15, when was the father twice the age of the son ?

Let x be the number of years before the present time, when the father was twice the age of the son -

Then, 40-x = 2 (15-x) or, x = - 10

Now what does this Negative answer mean ?

It means that the father never was twice the age of the son in Time past, which we have taken as Positive in the question. The epoch of his being twice the age of his son is to be found in Time opposite to the past: namely in Time future. He was not twice the age of his son 10 years ago: but he will be so 10 years hence: as is very clear: for in 10 years the father will be 50 and his son 25.

Hence if any event be Positive which has happened in Time past, the same event is Negative if it is to happen in Time future.

13. These illustrations, which might be immensely extended by examples taken from every branch of Natural Philosophy, are sufficient to exemplify the doctrine which we have endeavoured to explain that universally in Natural Philosophy the Negative sign -, does not mean Negation or non-existence, but Opposition, or Contrariety, and that Negative Quantities have as real and independent an existence as Positive ones, and are to be enumerated separately and independently, as Quantities in that science, to find the totality of Quantities. In fact absolute Quantities have no sign at all: the Positive and Negative signs are only introduced when some idea of Opposition, Contrariety, or Inverseness occurs.