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years before the eclipse takes place. In the case of a comet, our calculations are not quite so accurate, or so much to be depended on, because there is one element in the case of those more flimsy bodies with which we are not so well acquainted. This element is the quantity of matter in the comet, by means of which alone we can determine the reciprocal disturbance between it and any other body whose motions are more regular, and whose mass is known to us. But even in this case, the application of mathematical principles, and of this axiom among the rest, has enabled us not only to remove that superstitious dread of comets which so much alarmed the ancients, but also to get rid of that alarm at the possible collision of our earth with one of those wanderers, which was a source of some apprehension to speculative men during the middle ages, or in the more early days of modern science.
In matters of geometry especially, but generally in all branches of pure mathematics, that is, where natural causes and human actions do not enter into the case, we have the whole data, and also the whole management of that data, completely under our controul ; it is
in a far more personal and intimate manner than any possession, or any enjoyment of the body; and therefore we may
“ do with it as we list,” provided we do not violate those laws which are the very foundation of this description of knowledge.
But, again, there is yet farther this advantage in the general doctrine of equality over the partial one of “magnitudes coinciding, or filling the same space," that it applies to and includes equality of process or operation, as well as equality of ratio and equality of magnitude ; and this is a very important matter, because, from the definitions we have already given of lines, surfaces, and solids, and also from the few hints which we have thrown out respecting ratios (and which we shall resume and
1 AND EQUAL RESULTS.
treat more at length in a section expressly on the subject), we may see that the result of a mathematical operation is often a quantity of a kind totally different from any of the given quantities by means of which the operation is performed, just as a chemical compound may be, in all its appearances, and in all its useful properties, totally different from one and from all of the ingredients of which it is the componnd, so long as these remain unmixed with each other, or unmixed by the very process which we employ.
Thus, if we seek to know the contents of a solid, as, for example, the number of feet in a log of timber, we do not go about to apply a solid foot to it, and see how many repeated applications of this solid foot coincides, or fills the same space with the log. This, in fact, would be impossible by any direct comparison, because, although we had a standard which we knew to be exactly equal to a cubit foot, we could not get this cubic foot and a cubic foot of the log into the same cubic foot of space, without previously removing the foot of the log; and though by this means we might show that the one was equal to the other, that is, that the foot put in occupied exactly the same space with the foot taken out, yet the information thence arising would simply be, that a cubic foot is equal to a cubic foot, which is really nothing.
In order to compare the log with the cubic foot, that is, to tell the number of feet in it, we must cease to consider it as a real and tangible solid, and regard it as a mere relation of three lines—the length in feet, the breadth in feet, and the thickness in feet; and we must find the lengths of those lines, not as having any connection with the solid, but as being the shortest distances between their own extreme points. Thus, if the length of the log is 12 feet, the breadth 3, and the thickness 2, we have, from the relation in which these lines stand to each other
in the solid, the log = 12 X 3 X 2 solid feet = 72 solid feet.
From this we derive not an unimportant distinction of when quantities, expressed in numbers, or by the more general symbols of algebra, are or are not, geometrically speaking—indeed, mathematically speaking - quantities of the same kind, and thus whether they are or are not fit subjects of comparison, or such as can have a ratio to each other, either of equality, or of inequality. We have already alluded to this subject, but there are some mathematical considerations which belong so equally to different branches of the science, and which serve to connect those branches with each other, and with the practical business of life, so usefully, that it often becomes necessary to bring a truth which has been already examined into juxtaposition with a new truth, in order to point out the connection or relation between them; and, as much of the clear and ready understanding of the whole mathematical sciences, especially in their connections and their applications, depends on the clear perception which we have of this doctrine of equality, of the means by which it may be shown, and of the changes that may be effected on quantities without destroying it, we have been anxious to treat this subject very fully, even with the certainty that it must appear tedious to those who are already acquainted with it.
Mathematically, simple numbers, that is, numbers which are not considered as the results of any multiplication, are always regarded as represented, that is, as being capable of representing lines only. Products of two factors are considered as representing surfaces, and products of three factors are considered as representing solids. Magnitudes cannot be more than solid, and therefore there can be no more geometrical magnitudos answering to the products of numbers than these three ; but
NUMBER AND MAGNITUDE.
still, generally speaking, products which arise from multiplying equal numbers of factors, that is, which are produced by equal numbers of multiplications, are always considered as quantities of the same degree, if not absolutely of the same kind, and therefore they are comparable with each other.
After we once fully understand the general doctrine of equality as applicable to all quantities of the same kind, and to all changes or operations which are equally performed on them, it is of some advantage to enumerate the particular cases, if only for the purpose of ready quotation, in those instances to which these cases are applicable; and in this respect the coincidence of magnitudes which can be superposed, or applied the one upon the other, and shown to be co-extended when this is done, may be admitted as quite satisfactory. Lines and plane figures are almost the only ones which can be compared in this way; and straight lines are equal when it can be shown that the extremities of the one coincide with the extremities of the other; and as they have no respect in which they can be equal as magnitudes, save length only, if the points which make the extremities of one can be shown at the same distance from each other, as those which mark the extremities of another, the equality will follow as a matter of course, without applying the one to the other.
There is also an indirect method of showing the equality of lines, and that is, by proving by reasoning that the one cannot be either greater or less than the other. This applies to quantities generally, and the method of proof generally turns upon some absurdity which would be the result of any quality in the two quantities, which are by this means proved to be equal.
In the case of surfaces, if, upon the application of the one surface to the other, it can be shown that all the lines which
MEANS OF COMPARING.
are the boundaries of the one coincide exactly with all those which are boundaries of the other, then it will follow that these surfaces, and also their respective boundaries, both sides and angles, are every way equal.
Also, two quantities of the same kind are equal, if it can be shown that each of them is equal to some third quantity, or to any equi-multiples, or like parts of a third quantity-by “ equi-multiples" being understood all possible products by the same multiplier, whether that multiplier can or cannot be exactly expressed by arithmetical notation; and by“ like parts” are understood all possible quotients that would arise from dividing by the same divisor, whether those quotients can be accurately expressed by single quantities or not.
This is the test, or judgment of equality, to which we are in the habit of appealing in practice, and according to which all equitable exchange of commodities, all estimates of the quantity of materials required for any specified purpose, and, generally speaking, all measurement or knowledge of the values of quantities, are determined ; and therefore it is the case of the general question of equality, which everybody ought to understand best. Thus, for instance, if we have in England a certain standard of length, which we call a yard measure,
carry this measure to any number of different parts of the world, and by applying it to certain lines, or lengths, or breadths, which are fixed and immoveable in those distant places, and which, therefore, instead of being capable of being placed in juxtaposition, and judged of as filling the same space, or different spaces, cannot be both seen till after months, or perhaps years, have elapsed, and, generally speaking, which cannot be seen at all by the same individual, as certain that each of these is the same multiple, or the same part of a yard measure, we have no