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From these two ideas arise two distinct classes of propositions, namely, Problems of Existence, in which it is required to construct a figure, that is, to fix the boundary points in the required position; and Problems of Relation, usually called Theorems, which examine and prove the consequences following from the position in which they are placed. In the former class the question in each proposition is, what is the point at which such and such a relation will take place? And this may throw light upon the difficulty which often puzzles beginners as to the use, and indeed the meaning of a mathematical proposition, to do what can be done at once by a twofoot rule or a pair of compasses ? For example, when Euclid has a problem, “From a given point to draw a straight line equal to a given straight line," the question at once suggests itself,—Why not take a piece of string and measure it? The answer to this is, that it is only permissible to use the piece of string or the ruler, on account of the truth of the principle proved in this proposition. But it happens sometimes that the determination
1. of a particular point is no longer the direct but the indirect object. For example, in the problem, “to bisect a given rectilineal angle,” that is, to divide it into two equal angles (1), the object is to determine some point through which,
c if a straight line be drawn from the given point,
it will make the required angle (2.) And there are
2. many others of a similar character. Indeed, as
А. M. Comte observes, all Mathematics is simply the science of indirect measurement. And if it should seem strange to any one that
c so complicated a machinery should be called into action, merely for the purpose of determining a point, let him consider that the “point” of all questions, moral and physical, as well as geometrical, is to determine some “point.” A physician who should give a patient a description of the medicines he was about to administer—a spiritual guide who should entertain a penitent with a disquisition upon the nature of the soul—or a passer by who, upon being asked the way, should reply by a catalogue of the shops one must pass, would probably only leave the person addressed in a state of greater bewilderment than before. For in medicine one requires only to know exactly what to take; in morals, exactly what to do; and in finding the way, exactly where to turn.
But it may be asked—How can any figure be mathematically drawn at all? To this question the answer will be clear upon considering the latter class of Problems-namely, that the object of a Problem is not to draw a certain line, but to indicate how a line may
be drawn so as to be in a certain relation to another; for it is in fact simply impossible to obtain any mathematical means of drawing straight lines or circles, and we can only assume the power.
5 Let it be granted,” says Euclid, “ that a straight line may be drawn from any one point to any other point,” and “ that a circle
be described," &c.
As it is impossible to learn to swim without going into the water, so can we learn nothing of Geometry without assuming straight lines and circles. One step further nevertheless it
be noticed we can go than Euclid, and that is in giving the reason for fixing on two points, and two points only, as necessary for drawing a straight line, as we have already shown.
Moreover, a further assumption must be made—viz., that since there is nothing in the order of a straight line to limit its length, one of a given length may be produced to any distance in the same order of points (i.e., in the same straight line;) but this is perhaps rather to be considered a consequence of the fact that a straight line is a circle of infinite circumference. These “ Postulates,' as they are called, being granted, we have only to apply them in order to fix any point we please. For as motion in a circle changes the direction of motion while leaving the distance unaltered, and motion in a straight line changes the distance without altering the direction, these two motions are sufficient, when used separately or together, to make a point assume any position whatever. Hence the first part of every problem will consist in drawing straight lines and circles in such a manner as to fix the boundary points of the required figure in their proper position. The boundaries once fixed, straight lines may be drawn from one to the other, and the figure will be described. This part of the process is called the “ Construction."
But how are we to know that the boundary points are in the required position ? The test lies in the fulfilment of certain con
sequences which must follow from certain conditions : e.g. If a line be a circle, then a straight line drawn from any point of the circle to the centre, will be equal to a straight line drawn from any other point of the circle to the centre. Thus the necessity of the problems of Relation becomes apparent, and the dependence of the problems of Existence upon them. In thus testing a proposition there are two methods used by Euclid. The first is simple and direct: From certain conditions, as we have just said, certain results must follow; these conditions, therefore, being fulfilled, certain consequences may be deduced. But such direct means of proof may not always at be hand; and in that case less simple means must be employed. One of the commonest is to suppose
the proprosition untrue, e.g. that two triangles which have all their sides equal, respectively, will not have their angles equal also. If then we find that this proposition results in some manifest absurdity, e.g., that an angle is both greater than and equal to another angle, it must be given up, and its opposite is as good as proved to be true. This latter method is called Reductio ad Absurdum.
HAVING thus briefly described Euclid's method of reasoning, let us make some inquiry as to the instrument he uses, namely, Equality, as to which we may observe first that it is manifestly a Relation: And its definition may be given as follows:-"Equality is the relation in which two Magnitudes stand with regard to each other, when one fills
up exactly the same space as the other when substituted for it." This definition at once gives as its results all the various
truths concerning equality, which Euclid calls self-evident theorems, or Axioms, including among them the definition itself. At the same time it affords an easy method of determining whether two objects are equal or not, when they are of the same shape. Substitute the one for the other and see whether it fills up the same space; or, still more simply, if they be merely surfaces, place one on top of the other, and observe whether they coincide, as, to take a familiar example, in the trying on of a coat.
Of such a nature are the propositions in which Euclid examines the equality of triangles. If however the two objects be different in shape, e.g. a parallelogram and a triangle, what method is to be then used for the determining if they are equal ? Into this Euclid enters but very slightly, taking merely the simple case where one is double of the other. Neither shall we attempt to pursue it further than he. Let this therefore be sufficient with regard to the idea of Equality.
ENUMERATION OF CASES IN WHICH TRIANGLES ARE EQUAL.
The instrument and the mode of using it being now discussed, we proceed to enumerate the results at which Euclid arrives with regard to the equality of triangles. It would indeed be perfectly possible for us to deduce his results from what we have already laid down for ourselves, and perhaps our design can hardly be complete, strictly speaking, without doing so. But as our object is by no means to write a commentary upon Euclid, but merely to introduce the student to him—we have no scruple whatever in thankfully accepting his guidance to his various treasure-rooms.