if it were true, shows that an absurdity, or impossibility, is the unavoidable consequence; thus proving that the thing contradicted cannot be otherwise than true. Throughout the whole of this book, the last proposition is the only converse theorem that is not demonstrated in this indirect manner. It is not every theorem that is true both directly and conversely. You should take note of those that Euclid proves to be convertible, and endeavour to discover for yourself which of his propositions hold conversely, though only proved directly. For example, Proposition XXXIV. proves that if the opposite sides of a quadrilateral are parallel, they are likewise equal. It is also true conversely, that if the opposite sides are equal, they are likewise parallel, as you may prove for yourself, after the direct proposition has been established.

Proposition VII. is merely subsidiary to the proposition next following; it is what in some geometrical writings would be called a Lemma. You see that the demonstration of it rests almost entirely on Proposition V. In some modern books on geometry this proposition is dispensed with, and the eighth established independently of it; but, as an intellectual exercise, Proposition VII. is as useful as any in the book. Besides, a proposition, though manifestly introduced as merely subsidiary to something else, may yet possess intrinsic excellence of its own sufficient to justify its retention in the system. For instance, the proposition before us teaches us this fact, which is certainly not without interest; namely, that a physical triangle, supposed to have its sides freely moveable about joints at its vertices, cannot possibly be thrust out of shape by any force whatever. You may break the bars forming the framework, but you cannot make the frame itself assume another shape. It is a very different thing with the frame of a common school-slate, as I dare say you well know from practical experience; for I have no doubt that you have often twisted such a slate-less frame into a great variety of shapes. You now know, not from experiment, but from abstract science, that this would have been impossible if your slate-frame had been triangular, instead of rectangular.

Proposition VIII. is the second proposition, in the geometry of triangles, which proves that two triangles are equal in every respect—that is, that each is but an exact copy of the other provided three things in one are respectively equal to three corresponding things in the other. The three things may be two sides and the included angle, as we learn by Proposition IV., or the three sides, as the present proposition teaches. And I may as well observe here, that there is only one other proposition in the Elements where the like equality of two triangles is inferred from an equality of three things in one to three corresponding things in the other: it is Proposition XXVI. On these three propositions the practical part of plane trigonometry is founded. A triangle, in the language of trigonometry, is said to have six parts-the three sides and the three angles; and when certain sets of three of these are given-either of the sets, namely, mentioned in Propositions IV., VIII., and XXVI.-the remaining three, which we see by these propositions must be fixed and invariable, become determinable, and are matters of computation.

And here it may not be amiss to say a word or two about the form of expression continually employed by Euclid, when comparing figures together, for the purpose of establishing their equality. He always speaks of two sides or angles of the one being equal to two sides or angles of the other, each to each. Learners are apt to omit this qualifying condition, "each to each," as if the frequent repetition of these words were only so much useless tautology; but precision requires that they should always be retained. If you were to say that two sides of one triangle are equal to two sides of

another, your meaning might be taken to be, that the aggregate or sum of the two sides of one triangle is equal to the aggregate or sum of the two sides of the other; but the addition of the words “each to each” would preclude the possibility of such a mistake, and would show that the sides, taken separately and individually in the one triangle, were affirmed to be equal to corresponding sides in the other.

The four propositions next following are problems. You may be pretty sure that Euclid has postponed them till they became indispensable. I don't think Euclid liked problems; at all events there is less careful finish about them than in his theorems. Proposition IX., for instance, needs mending a little; it professes to teach how to bisect an angle, of whatever magnitude it may be. Now, suppose that the triangle A D E, in the book, is an equilateral triangle, and that we want to bisect the angle DAE. Euclid tells us to construct an equilateral triangle on DE; and without the diagram before our eyes, where the construction is exhibited in its completed state, we should naturally describe the equilateral triangle he directs, above D E, and not below; in which case we should get nothing; for our new equilateral triangle would simply cover the one already there, and the point, F, falling on A, would have no separate existence; so that there would be no guide to the drawing of A F, the bisecting line; it should have been distinctly stated, therefore, that the equilateral triangle, to be described on D E, should have its vertex, F, on the opposite of D E to the point A. This restriction is introduced in the present edition.

In going over Euclid's propositions without the book, as I have recommended above, always refrain from copying the diagrams. I know that such is the usual practice; but it should be condemned. The progress of the diagram should just keep pace with that of the text, and no line should be introduced till it is actually demanded by the text. It would be nearly as faulty as to write out the whole text, and then to supply the diagram (as the boy did who said he would tell the story first and draw the picture afterwards), as to commence with the completed diagram and then supply the text. In a printed book, the diagram must, of course, be presented completed; but in your own private practice you should make it grow to maturity along with the text. In the whole course of your geometrical studies, let me urge upon you never to allow your judgment or conviction to be in the slightest degree biased by your visual impressions from the diagram. Let two lines look ever so like two equal lines, don't forestall the reasoning, and conclude them equal from their appearance; remember always that you are engaged in a purely intellectual process, and that you are not to be allured by the matter from the mind. Graphical accuracy, in the figured form, is of no moment; logical accuracy, in the abstract reasoning, is all that you have to attend to; and therefore I think it worse than waste of time to be over-scrupulous with scale and compasses, in reference to the lines introduced into Euclid's demonstrations; but I have already given you some hints on this matter at pages 68 and 73.

I do not see anything that calls for special notice till we reach Proposition XVI. This is easy enough, as far as Euclid carries the demonstration; but when, at the close, he says, as in other editions of the Elements he is made to do, "in the same manner it may be demonstrated," a beginner is likely to feel a difficulty. There is really a good deal to do before the proof can be completed; and, when completed, "in the same manner," there is a needless amount of complication. I would advise you to finish the reasoning rather differently. By carefully looking at the argument, you will see that this truth is established, and nothing more; namely, that if one side of a triangle (any side, of course) be produced, the exterior angle is greater than that interior angle which is

opposite to the side thus produced; the angle A CD is thus greater than A. Let now AC be produced to G, then, since the exterior angle is greater than that interior one which is opposite to the side produced, the angle BCG is greater than ABC; but BCG is equal to ACD (by the fifteenth), therefore A C D is greater that A B C; but A CD was shown to be greater also than B AC; therefore A CD is greater than either of the interior and opposite angles, B A C, A B C. It is this form of completing the demonstration that has been adopted in the present work.

Proposition XVII, would seem, at first sight, to have been introduced without any object. The truth of it is clearly implied in Proposition XXXII., and it is not required in any of the intervening propositions. But that Euclid had an object is not to be questioned; and it seems to have been this :-It was desirable that, at some convenient place, before the introduction of the theorems respecting parallel lines, something should be established by demonstration that would diminish the repugnance, very properly felt at the outset, to the twelfth axiom. You know I have recommended you (p. 59) to keep this axiom in the background till you arrive at Proposition XXIX., where a reference to it becomes indispensably necessary. The axiom is no other than the converse of this seventeenth proposition; this shows that if two meeting or non-parallel lines, BA, CA, be cut by a third line, B D, the two interior angles, C B A, B CA, on the same side of it, are together less than two right angles; and the twelfth axiom asserts, conversely, that if a straight line cutting two others make the two interior angles on the same side of it less than two right angles, those others must be non-parallel or meeting lines.

The seventeenth proposition, therefore, enables us to see more clearly the exact amount of assent demanded of us by the twelfth axiom, and prevents our overrating that amount; if two lines cut by a third meet, the two interior angles are less than two right angles-this is proved; if two lines cut by a third make the two interior angles less than two right angles, they meet-this is assumed.

Passing over, for the present, the intermediate propositions, let us suppose Proposition XXIX. to be reached. The two propositions immediately preceding sufficiently show that the lines called parallel lines exist; the twenty-ninth demonstrates a property of them, admitting the truth of the axiom just mentioned. Geometers without number have tried, some to evade this axiom altogether, and others to prove it by establishing the converse of Proposition XVII.; but all have failed. What can be the cause of this failure? Is it not in the imperfect definition of a straight line? Our conception of a straight line, independently of all formal definition, necessarily involves two ideas; namely, that of length, and that of uniformity of direction. Length is implied in the word line; and invariability of direction in the term straight. A line which changes its direction is a crooked line or a curved line; a line that never changes its direction is a straight line. Now it necessarily follows, from this uniformity of direction, that if two straight lines, however far prolonged, can never meet, then at no part of their course can either make any approach towards the other; for if two lines, proceeding in any two directions, approach and continue undeviatingly to pursue those directions, they cannot fail eventually to meet. It follows, therefore, that parallels must throughout be equidistant; but two distinct straight lines, through the same point, cannot throughout be equidistant from a third; so that two straight lines through a point cannot both be parallel to the same straight line. Proposition XXVIII. shows that one (CD) will be parallel to another (A B), provided a line, cutting both, makes the interior angles together equal to two right angles; a second line through H, which causes the interior

MATHEMATICAL SCIENCES.-No. III.

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angles to be less than two right angles, being distinct from CD, must therefore meet AB, if prolonged; and this is the assertion of the twelfth axiom.

What is here said, remember, is not a demonstration of this axiom. An axiom, you know, is an indemonstrable truth. All I wish to show is, that it is an axiom; that is, a truth necessarily implied in the correct conception of the thing to which it refers. L you steadily contemplate your conception of a straight line, giving due consideration to its distinguishing peculiarity-uniformity of direction-you must see that two, from the same point, cannot both be equidistant from a third; one may be parallel to this third, or everywhere equidistant from it, since, as we have seen, parallels are possible; but one must of necessity meet it.

Of the propositions passed over, the 22nd and 24th are the only ones requiring any special notice here. Both these, as given in the text of Dr. Simson, are open to objection. In the first of them, it is taken for granted that the two circles employed in the construction must cut one another; and in the second, it is assumed that the point F falls below the line E G. You will find these defects acknowledged in the notes at the end of Simson's Euclid; but they were first pointed out by Mr. Thomas Simpson, in his "Elements of Geometry." The emendations of the latter were, however, but ill received by the "restorer of Euclid," who treated" the remarker," as he called him, with a good deal of contempt; the more to be reprobated, as the poor self-taught weaver (for such Simpson in early life was) was very superior as a man of science to his academical opponent, great as were the merits of the latter in the field of ancient geometry. The biography of Thomas Simpson is full of instruction and encouragement to the young and unaided student, who cannot fail to view with interest the steps by which a person in Simpson's position, without books, money, or friends, plying his humble calling among the lowest ranks of society, was conducted, by the force of perseverance, to the proud eminence which he eventually attained. In the annals of science he ranks among the most distinguished mathematicians of the last century; and yet, at the age of nineteen, he was ignorant of the first rudiments of common arithmetic.*

The defect above alluded to in Dr. Simson's version of the 24th Proposition is removed in the present, edition; and the objection made to the reasoning in the 22nd may be disposed of as follows:-After having described the circles, as at page 56, reason thus: One of these circles cannot be wholly without the other, for then F G, the distance of their centres, would be either equal to, or greater than the sum of the radii; but, by hypothesis, it is less. Neither can one of the circles-as, for instance, that whose centre is G-be wholly within the other; for then the radius, F D, of the latter would be equal to, or greater than F H; but, by hypothesis, it is less; hence, since one circle can be neither wholly without the other, nor wholly within-they must be partly without and portly within one another, .. they must cut in some point K. This completion of the proof may be introduced in a second reading of the first book.

Proposition XXXII. is among the most interesting theorems of this first book; but an objection to the demonstration of it may be made, the occasion for which had better be removed. You are directed to draw, through the point C, a line C E parallel to A B, by Proposition XXXI.; and it is then inferred that the alternate angles BAC, ACE are equal by an appeal to Proposition XXIX.; but to draw the parallel CE, surely everybody would proceed by making the angle ACE equal to BAC; that is to say, we should first make the alternate angles equal to get the parallels, and should then make

* Some account of Simpson will be found in "The Pursuit of Knowledge under Difficulties;" as also in Dr. Hutton's "Mathematical and Philosophical Dictionary."

use of the parallels to prove the alternate angles equal. You will at once see that we should avoid this circuitous method of proceeding, by making the angle A CE equal to BAC by Proposition XXIII.; and then inferring the parallelism of AB, C E from Proposition XXVII.; so that Proposition XXXI. need not be called into operation at all.

The corollaries to this proposition are remarkably beautiful; and the second, especially, cannot fail to excite, in a person who reads it for the first time, a feeling of surprise. It would indeed be a feeling of incredulity, if this were possible in geometry. That the sum of the exterior angles, formed by prolonging the sides of a rectilinear figure, should always be exactly the same, whether the figure have three sides or as many thousand, is a truth so far beyond the reach of practical observation and experiment, and apparently so improbable, that, in the absence of geometry, its existence could scarcely have been suspected, much less established; and yet an argument of half-a-dozen lines produces in every mind the fullest conviction of the fact.

But the corollary that precedes this, though less striking, has, perhaps, the greater practical interest; among other things, we learn from it that-the sum of the angles of a four-sided figure is twice as great as the sum of the angles of a three-sided figure; the sum of the angles of a five-sided figure, three times as great; of a six-sided figure, four times as great, and so on; but the most noticeable practicable inference is, that only three regular figures,* namely, the equilateral triangle, the square, and the regular six-sided figure or hexagon, can, by repetition, completely cover a surface: in other words, that, without leaving any blanks or interstices, we may cover a surface with a mosaic work of equilateral triangles, or of squares, or of regular hexagons, but not with regular figures of any other kind. It would be impossible, for instance, to form a piece of tessellated pavement with slabs of any other regular figure but one of these three; because the uniting together of any other forms, by adjusting side to side, would not fill up the space about the corners-there would be either left an angular gap, or else the stones must overlap one another. You will readily see the truth of this from the following considerations :—

Let us first consider the equilateral triangle: as the three angles make two right angles, each must be of two right angles, that is of four right angles; consequently if six equilateral triangles were placed side by side, a corner or vertex of each being at the same common point, all the angular space about that point would be completely occupied; and no one triangle would overlap another, for the angles about a point amount to just four right angles (Prop. XIII., Cor. 2). Let us next consider the square; and, as each angle of a square is a right angle, it is plain that four squares, each with a vertex at the same point, when placed in contact, will exactly fill the space about the point.

The figure next in order is the regular pentagon, or five-sided figure. The corollary teaches us that the sum of its five equal angles amounts to six right angles; consequently each angle is one right angle and a fifth. Now you cannot multiply 13 by any whole number that will make the product 4-no such number exists; in other words, you cannot arrange the angles of pentagons round a point, as the common vertex of all, so as to fill up the four right angles about that point; the pentagon, therefore, must be rejected. The next figure is the hexagon. By the corollary we learn that its six equal angles amount to eight right angles; consequently each angle is equal to of eight right angles; or, which is the same thing, to of four right angles. It follows,

* Right-lined figures are said to be regular, when they are both equilateral and equiangular.