VI. Things which are doubles of the same, or of equals, are equal to one another. * VII. Things which are halves of the same, or of equals, are equal to one another. That “the whole is greater than its part,” need not be set down as a separate proposition, since it is involved in the very import of the words whole and part. That magnitudes which coincide with each other, that is, which fill exactly the same space, are equal (or of the same size), is a mere verbal truth ; " to fill the same space,” is “ to be of the same size.” “To be of the same size,” however, does not always involve “to fill the same space.” Areas may be equal, and yet of different shapes. X. Two straight lines cannot enclose a space. B. Two straight lines may be so placed one on the other as to coincide with each other in direction, and cannot diverge from one another if they coincide for any portion of their length. I XII. If two straight lines meet a third, and make the two interior angles on the same side of it together less than two right angles, those lines will meet if produced C. If part of one bounded figure (and part only) be inside another bounded figure, the peripheries of those * This is properly only a particular case of the second axiom. + The first part of this axiom is only the definition of halves in another form. The second part is properly an inference from Axiom 6. For if the halves of two magnitudes were unequal, the doubles of those halves, that is, the magnitudes themselves, would be unequal. I The latter part of this axiom is commonly set down as a corollary to the thirteenth proposition; but it is really made use of much earlier,-in the fourth proposition, in the proof of which we have to make use of the truth that two equal straight lines will coincide throughout their entire length. two figures must intersect one another in two points at least. That all right angles are equal, is a proposition that admits of being proved. THE NAMING OF ANGLES. An angle is named either by a letter placed at the point of intersection of the lines which form it, or by three letters, one at the point of intersection of the lines forming it, the others at some point of these two lines respectively.* In reading these three letters, that letter is always placed between the other two which is at the point of the angle. Thus the annexed angle would be called either B A C or C A B, but not A B C or A CB. с А B SUPERPOSITION OF FIGURES. A device of which frequent use is made in proving propositions in geometry, is to compare figures, by A triangle is named by placing a letter at each of the three angles. Thus comes to pass that the same three letters by which each of the angles of the triangle is named, are also used to name the whole triangle itself. One unfortunate consequence of this is that beginners often get into confusion, and when they name an angle of a triangle (that is, the opening formed between two of the sides of the triangle), fancy that they are talking about the whole of the three-sided figure. Many teachers would be surprised to find how often this blunder is made without being detected. It needs to be guarded against with the most constant vigilance. In order that the eye may help the understanding, the mark _ will be used in this book for the word angle, and the mark for the word triangle. Of course Zs will mean angles, and As will mean triangles. supposing one of them removed from its separate position and applied to the other, so as to coincide with it, wholly or partially, as the case may be. No difficulty will be found in this, if it be remembered that the figures with which we are really concerned, exist only in the mind, and have no definite relation to any particular locality, so that they may just as well be conceived of in one position as in another, with all their parts and properties exactly the same as before. When, therefore, we talk of applying one figure to another, we mean that we are to conceive the figures in our minds as being so situated, with respect to each other, that certain parts, or all the parts of each may coincide In doing this there are some mistakes against which beginners must be particularly cautioned. They may seem trifling, but they must not be disregarded. Not the smallest point must be taken for granted, if it is not one of the facts which we are supposed to be acquainted with, before we begin our reasoning in any case, some truth which has been rigidly demonstrated. It is this exactness which is demanded, which gives to the study of geometry its great value. The reasoning applied is in no respect different in kind from what is applied to any other subject: but the subject matter of the science is perfectly simple; there are no collateral matters to be taken into account ; and every one can form a perfect conception of the things reasoned about. Respecting the meaning of equal and unequal, straight line, circle, &c., there can be no mistake. But this, at the same time, occasions a danger to which beginners are liable. Many of the truths demonstrated are so very evident, that the mind gets impatient at going through the steps of the proof. It must be borne in mind, that the object of the propositions is not to convince us of the truth of or A с certain facts, or of the existence of certain properties in figures, but to show in what way these facts or properties are to be deduced from the simple elements to which we are limited at starting. There must not, therefore, be a single break or flaw in the chain of demonstration which connects the first beginning with the final conclusion. The most minute exactness must be observed at every step. Suppose that we have two right lines, AB and CD, of equal length, and that it is requisite for any purpose that the one should be supposed to lie upon the other. We may say, “ Place the point A upon the point C, and let the line A B lie along the line CD; then, because the lines are equal, the point B will coincide with the point D.” But it would be wrong to say, “Let the point A be placed upon the point C, and the point B upon the point D,” or anything equivalent to this, because that would imply, that, after we have fixed the position of the point A, and the direction of the line AB, we can then make the point B fall where we like. When we have fixed the position of one end of a given line, and the direction in which it is to lie, we have nothing more under our control. The length of the line itself determines where the other extremity will fall. We may be able to say where that other end will fall, but we must not use language implying that we can make it fall where we choose. Suppose that there are two Zs of the same size, * * Let it not be forgotten that the size of the Zs in no way depends upon the length of the lines which form them. The < ABC would be of the same size, whether contained by lines a mile long, or by lines of the length of a hair's-breadth. D B ABC and DEF, and that the one has to be placed upon the other. We may say, “ Place А the point B upon the point E, and let the line B C lie along the line EF (whether the point C will fall on the point F depends upon whether the line B C is equal to the line E F or not). But having done that, we must not say, let (or make) the line B A lie along the line ED; because that would imply that when we have fixed the position of one of the lines containing a given Z, we can also make the other line lie in any position that we choose. But this we cannot do. When one side of a given ? is fixed, it is the size of the angle which determines in what direction the other side will lie. It is obvious, that if the <s ABC and D E F are equal, and the sides BC and EF coincide in direction, the sides B A and ED will also coincide in direction (if the s be made to lie on the same side of the coinciding lines). But that is not because we make them coincide, but because the equal size of the Zs makes them do so. PROBLEMS AND THEOREMS. The propositions in the First Book of Euclid are divided into two classes, problems and theorems. In a problem, it is required to show how a figure, having It may seem almost trifling to introduce such simple considerations as these so elaborately; but I have known persons who, after studying Euclid for years, have been unable to demonstrate properly the fourth proposition in the first book, from not having attended to the caution above laid down. |