more doubt of their equality than if we could actually apply the one line to the other, and see their coincidence at both extremities.

We have the same confidence in such measures, even when they are not equal. Thus, if it is ascertained that one mountain, situated at the Himalaya ridge on the north of India, is 8,000 yards above the level of the sea, and that the height of another mountain, situated in the ridge of the Andes, in South America, which is nearly half the measure round the earth distant from the former, is also 8,000 yards, we have no more doubt of the equality of the height of those mountains, than if we could see them both side by side, and with our own eyes, at the same instant.

Without our belief in the fact, "that things equal to the same thing are equal to each other,” we could not, geometrically speaking, have any map, any plan, or any pictured representation, whereby an absent thing could speak at once to the eye in that language which is so much more powerful than writing; and without the same belief in all other matters, we could have no knowledge, except that which we derive from our own senses; and even the parts of this knowledge would be unconnected, and resemble that which may be presumed to be the momentary perception of brutes, rather than the conclusions of human reason. Whenever we see an object of the same kind with one which we saw formerly, or one of which a clear description struck us forcibly, so as to make us remember it, whether we actually saw it or not, we certainly, and without any perceived or felt process of thought or effort of the mind, institute a comparison between the perceived object and the recollected one, and with as little effort we instantly conclude that they are or are not like or equal to each other. This is our primary and general judgment of equality or inequality, for

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RESULTS OF EQUALITY.

the notion of the one involves in it a notion of the other. Mathematical or geometrical equality is merely the particular branch of this general judgment, in which we have the evidence most perfect in its kind, and most completely before us; and therefore it is only the most accurate case of that exercise of the mind which we must all practise every day.

Original and simple quantities may be equal by construction, that is, they may be made equal; and this is perhaps the best illustration which we have of equality, in the most general sense of the action, that is, where all the circumstances which determine the magnitude of the one, also determine the magnitude of the other, and every one of them is known to us as being our own act. Thus, if from any two centres, with the same straight line as radius, we describe two circles, it is impossible to have a more clear and simple notion of perfect equality than is afforded by those circles; and when we have this perfect perception of equality, it leads to perhaps the most general and the most important conclusion in the whole compass of mathematical science; and if this conclusion is not a direct axiom, it is as axiomatic, as self-evident an inference as can possibly be drawn. It is worthy of being borne in mind, and it is as follows:

If two quantities, whether they be magnitudes, ratios, or anything else, are every way equal, whatever can be shown to be true of either of them, is necessarily true of the other, in the very same manner, and to the very same extent.

If the quantities of which the equality is asserted are results of any of the four arithmetical operations, we may state generally that equal operations, performed upon equal quantities, must produce equal results. Or, taking each opera

tion

In addition, if a is = b, and c any third quantity, whether

a simple quantity, or one which is proved to be the same in two cases, then A+C=B+C.

In subtraction, the data being the same as for addition, a— c

=B-C.

In multiplication, if A=B, and m any multiplier whatever, integral or fractional, expressible or not expressible in numbers, then mam B.

A

d

In division, if a B, and d any divisor whatever, then

These, which follow both from what has been now stated, and what was formerly stated on the subject of multiples and fractions, may be quoted in brief thus :-The sums, differences, equi-multiples, and like parts of equal quantities, whether magnitudes or any quantities whatsoever, are equal. It is scarcely necessary to add, from what has been already said with reference to addition and subtraction, that equal operations must leave unequal quantities unequal; for if we add or subtract equally, we do not add or subtract any difference; if we multiply the unequals equally, we multiply the difference; and if we divide them equally, we divide the difference: but in none of these cases is the difference taken away.

As inequality is the opposite of equality, it follows that, in all cases where the one can be clearly established from a knowledge of all the circumstances, the other is proved in every case where it can be clearly shown that that one does not hold. But there are so many ways of expressing quantities, that we are not able, in all cases, to prove that equality exists, even where such is the fact; and therefore, our not being able to prove equality, is not, in every case, a sufficient ground for inferring that quantities of which we are unable to show the equality, are unequal. If we know all the conditions or circumstances upon

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EQUALITY OF PRODUCTS.

which the values of two quantities depend, and if these circumstances, taken jointly, do not admit of the inference that the quantities are equal, we may safely infer that they are unequal; but if we are not sure that we are fully in possession of all the circumstances, we cannot conclude either way, in consequence of our failure in the other. Thus 1+1+1+1+16, &c., con

infinitely large, and the value of the fraction,

=2, and merely an expression of a particular form for that number, as will be shown afterwards; but when we examine this, even to any extent which can be written down, it does not, upon mere inspection, appear to be = 2. We mention this merely to show that an apparent inequality is not sufficient ground for inferring that the inequality is real, unless we can prove that we are in possession of all the circumstances upon which both the quantities under comparison depend.

In the case of quantities which are represented by products, the total value of the one may be equal to that of the other, though both the factors, in the case of there being only two, or all the factors, in the case of there being more than two, are unequal to each other; but if one factor in each be equal, in cases where there are only two, or if all the factors be equal except two, that is one in each, where there are more than two, the values of the products are unequal, and the difference between them is the difference of the unequal ones multiplied by the equal one, or the product of all the equal ones, in the case of there being more than one in each product.

Thus the product of the factors 4 × 4, 8 × 2, 16 × 1, is = 16; and there are other cases in which, even in integer numbers, the same product may be obtained from a greater number of

pairs of factors; and even in this case, the product 16 may also be considered as that of four factors, for 2 × 2 × 2 × 2=16, as before

If we examine the different factors which produce the same product in this case, we perceive a general principle, the nature of which will be more completely explained in a future section, when we return to the consideration of general quantities algebraically; but it may not be amiss to bear it in mind, without going into a full explanation of it. In the above numbers it will be seen that, when the factors are equal, their sum is less than when they are unequal, and that the more unequal they are, their sum is the greater. Thus, in the equal factors of 16, 4 and 4, the sum is 8; in the first unequal ones, 8 and 2, the sum is 10; and in the last, where one of the factors is the product itself, and the other the number 1, the sum is greater than the product, for 16+1=17.

We mention this merely to show that, as two factors may be considered as representing length and breadth, and their product surface, the surface is always the greater, in proportion to the sum of the dimensions, that is, of the length and breadth, the more nearly that these are equal to each other. The same will also evidently hold true in the case of solids; but the consideration of these, as well as the investigation of the general principle of which the above product and its different factors are an instance, can be better explained in a future section, when we have a few more of the principles before us, so as to be able to consider generally the relations between factors and products, as compared with surfaces and their boundaries. What we have now said is sufficient to show that figures may be equal in content or area, though their dimensions, and consequently their boundaries, are altogether different; and it is necessary for us to bear this carefully in mind, in order to avoid