16. Write down the equation which is expressed by saying: the square of a, divided by b, minus the square of a by b, plus the cube root of a plus b, equals the cube of a minus b, minus the square root of a by c minus b.

Ex. 4.

If a, b, c = }, d = 4, x = 2, find the values of,

3. (2a+3b+5c) (8a+ 3b - 5c) (2a-3b+ 15c).

10. Also of 13a+3b — [7 (a + b) — {3a + 8(a —b)} ].

36+2a

11. If a = b, find the value of

a2 + b2 — } (b2 — a3)
a2 + b2 + } (b2 — a2)

†26.* In all that has preceded, we have supposed our operations to be strictly arithmetical: no other modification has been introduced than the use of symbols-1st, to abbreviate our expressions, and to present them in a condensed form to the eye; 2dly, to represent quantities which, though not given amongst the data, are the subjects of operations, and which are discoverable in the course of the investigation, through the intervention of those operations. We come now to an extension of our ideas on these matters, by passing on to operations not having a direct parallel in the science of arithmetic, yet deducible from that science by a very simple and natural process, which the mind can scarcely fail to grasp, and the evidence of the truth of which is perfectly convincing. Suppose the following question proposed

:

A's age is 40; B's 20. In how many years will A be three times as old as B?

Let x be the number of years required; then

40+x will be A's age,

*It is recommended that the student should omit the articles marked

(†) at the first reading.

†26.* In all that has preceded, we have supposed our operations to be strictly arithmetical: no other modification has been introduced than the use of symbols-1st, to abbreviate our expressions, and to present them in a condensed form to the eye; 2dly, to represent quantities which, though not given amongst the data, are the subjects of operations, and which are discoverable in the course of the investigation, through the intervention of those operations. We come now to an extension of our ideas on these matters, by passing on to operations not having a direct parallel in the patien of arithmetic, yet deducible from that

so that

40+x=3(20+x) by hypothesis;
40+x=60 + 3x ;

or, by subtracting 40 from each of these equals,

x= 20+ 3x ;

=

i.e., x is greater than 3x by 20; consequently x must be a negative quantity; for no number can be greater than three times itself. Now, to determine what the quantity is; subtract x from each side of this equation, and the result is 0 20+ 2x: .. 2x is such that, when added to 20, the result is zero; or x is such that, when added to 10, the result is 0. But 10 itself, when subtracted from 10, produces 0; x is therefore 10 to be subtracted.

Now let us examine how it happens that we have a result of this kind. It will be seen by examining the question, that a proper answer to it is, "the thing will never happen." A's age will become more and more nearly equal to B's as they become older, and will never be three times B's. Still the solution is not without meaning. Although A's age will never again be three times B's, yet it was once so. When was this? The answer is, 10 years ago. Now our solution is + 10 = 0, which appears to be equivalent to x=10; there is therefore an apparent connection between the solution and this fact. Is this connection accidental, or does it result from the greater generality of algebraical than of conversational language? A little consideration will shew that the latter is the true answer. As we have defined to signify increased by, and diminished by, there is a necessary connection between these symbols. For instance, if the thing which is increased be "A's present age," the interpretation of +4 will be that of reckoning onwards from the present time, or increasing the age by 4; whilst the interpretation of a negative quantity as - 5 will be that of reckoning backwards, or diminishing the age by 5. Now, in our conversational language, the question, "when will an event happen?" does not include the analogous one, "when did it happen?" whereas any algebraic mode of representing the answer must of necessity apply equally