In this case the point C, where they come together, is distant from A twice the distance A B.

Suppose a smaller than b, for example

Here the values of x and y are both negative; hence there is some absurdity in the enunciation of the question for these numbers. In fact, it is impossible that the courier setting out from A, and travelling slower than the other should overtake him.

Let us put x and y negative in the two equations, that is, change their signs.

The second equation is not affected by changing the sign, and it ought not to be so, since it expresses only the equality of the times.

The first equation becomes y x=m, instead of a y =m, which shows that the point where they are together is nearer to A than to B, by the distance from A to B. It must therefore be on the other side of A, as at E.

The enunciation of the question may be changed in two ways so as to answer the conditions of this equation.

First, we may suppose, that the couriers, setting out from A and B, instead of going towards D, go in the opposite direction, the one from A at 4 miles per hour, and the other from B at 8 miles per hour; at what distance from the points A and B is the point E, where they come together?

Or we may suppose that two couriers setting out from the same place E, one travelling at the rate of 4 miles, and the other 8 per hour, have arrived at the same time at the points A and B, which are m miles asunder. What distance are the points A and B from E?

How is this result to be interpreted?

Observe that in this case a and b being equal, the two couriers travel equally fast, it is therefore impossible that one should ever overtake the other, however far they may travel in either direction, and no change in the conditions can make it possible. Zero being divisor, then, is a sign of impossibility.

We may observe that when there is any difference, however small, between a and b, the values of x and y will be real, and. the couriers will come together in one direction or the other; and the smaller the difference, the greater will be the distance travelled before they come together; that is, the greater will be the values of x and y.

Again, Suppose a 5, and b = 4.5, ab5,

=

Again, Suppose a 5, and b 4.98, ab · 02,

=

Here observe, that as the difference between a and b becomes very small, the values of x and y become very large, and the difference between them is always m. Hence, since the smaller the divisor the larger the quotient, we may conclude, that when the divisor is actually zero, the quotient must be infinite. From this consideration, mathematicians have called the expresssion

a

0

that is, a quantity divided by zero, a symbol of infinity. They therefore say, that, both couriers travelling equally fast, the distance, travelled before they come together, is infinite. But as infinity is an impossible quantity, I prefer the term impossible, as being a term more easily comprehended. I shall therefore call a a symbol of impossibility.

0

b

If a quantity be divided by an infinite or impossible quantity, the quotient will be zero. If b be divided by, it becomes a

a

Multiply both numerator and denominator by 0, it becomes охь =0. In fact, since the larger the divisor, the smaller the quotient, the dividend remaining the same, it follows that if the divisor surpasses any assignable quantity, the quotient must be smaller than an assignable quantity, or nothing.

One case more deserves our notice. It is when a=b and m = O; in which case we have

If we return to the equations themselves, they become

This last equation has both its members alike, and is sometimes called an identical equation. The values of the unknown quantities cannot be determined from it. In fact, since m is zero, both couriers set out from the same point. And since they both travel at the same rate, they are always together. Therefore there is no point where they can be said to come together. The expres

0

sion is here an expression of an indeterminate quantity.

0

There are some cases where an expression of this kind is not a sign of an indeterminate quantity, but in these cases it arises from a factor being common to the numerator and denominator, which by some suppositions becomes zero, and renders the fraction of the form of ; but being freed from that factor, it has a determinate value.

0

tor and denominator contain the factor a b, which becomes zero when a and b are equal.

Dividing by ab, the expression becomes

a (a + b),

b

which is equal to 2 a when a = b.

It is necessary then, when we find an expression of the form

before pronouncing it an indeterminate quantity, to see if there is not a factor, common to the numerator and denominator, which, becoming zero, renders the expression of this form.

The example of the couriers furnishes some other curious cases, for which we must refer the learner to Lacroix's or Bourdon's Algebra.

Let the learner examine the following examples in a similar

manner

In Art. IX. examples 15 and 16, the following formulas, relating to interest, were obtained. How are r and t to be interpreted, when Р is greater than a; and how when a and p are equal?

In Art. XXII. examples 12th and 13th, the following formulas were obtained. In what cases will the results become negative, and how are the negative results to be interpreted?