Then if this conclusion is false, its contradictory must be true; namely Every cube is a triangle. And substituting this for the minor premiss, we obtain the following syllogism in Barbara. Bar Every (triangle) is a plane figure; ba Every cube Is a (triangle); ra therefore; Every cube is a plane figure. But as this conclusion is the contradictory to the minor premiss of the original syllogism, it must be false; and since it has been correctly proved from the premises, one of those premises must be false; but the major premiss is true, therefore it is the minor which is false; and since it is the contradictory to the conclusion of the original syllogism, that conclusion is true. This mode of argument is termed the reductio ad absurdum, and is very frequently employed by Euclid, particularly in the demonstration of propositions from their converse. By examining the names given to the modes, it will be observed that their initial letters are either B, C, D, or F, and these letters indicate to which of the modes in the first figure any of the modes in one of the other figures is to be reduced. Thus Cesare in the second figure is to be reduced to Celarent in the first, and Bramantip in the fourth to Barbara in the first figure. The letters s or p following a vowel imply that the proposition denoted by it is to be converted, if s, simply, but if p, per accidens; and the letter m signifies that the premises are to be transposed. The letter k after a vowel signifies that the proposition which it denotes is to be omitted, and the contradictory of the conclusion substituted for it, it being in fact the sign of the reductio ad impossibile. Our limited space will not allow of our entering upon the subject of hypothetical and disjunctive syllogisms, or the very important one of fallacies. We must therefore conclude the foregoing brief sketch of logic with an example of one of the propositions in Euclid, formally demonstrated in syllogisms. PROPOSITION XXXVI. HYPOTHESIS.-If parallelograms (ABCD and EFGH) are upon equal bases and between the same parallels, CONSEQUENCE.-They are equal to one another in area. CONSTRUCTION.-Draw BE and CH. Da (Things which are equal to the same) ARE equal to one an other. [Ax. 1.] ri The straight lines BC and EH ARE equal to the same FG. [Hypoth. and I. 34.] i Therefore; The lines BC and EH ARE equal to one another. Syllogism 2. Da (Straight lines which join the adjacent extremities of two equal and parallel straight lines) ARE themselves equal ans parallel. [1.33.] ri BE and CH ARE (straight lines which join the adjacent extremities of two equal [syl. 1] and parallel straight lines). [hypoth.] i Therefore; BE and CH ARE themselves equal and parallel. Syllogism 3. Da (Parallelograms which are upon the same base and between the same parallels) ARE equal in area. [I. 35.] ri ABCD and EBCH ARE (parallelograms which are upon the same base and between the same parallels.) [Hypoth. and syl. 2.] i Therefore ABCD and EBCH ARE equal in azşı!~~ Syllogism 4. Similar to syl. 3, proving that EFGH and EBCH ARE equal in area. Syllogism 5. Da (Things which are equal to the same) ARE equal to one another. [Ax. 1.] ri ABCD and EFGH ARE (equal in area to the same EBCH.) [Syl. 3 and 4.] i Therefore ABCD and EFGH ARE equal in area to one another. The foregoing will sufficiently illustrate the manner in which the propositions of Euclid may be expressed in formal syllogisms, and we should recommend to the student the practice of throwing the more difficult demonstrations into the syllogistic form, as a very useful and beneficial exercise both in logic and geometry. THE ELEMENTS OF EUCLID. BOOK I. DEFINITIONS. 1. A POINT is that which denotes position, without possessing any magnitude. 2. A LINE is a magnitude, having only one dimension, i.e., length. COROLLARY. The extremities of a line are points, and the intersection of one line with another is also a point. SCHOLIUM. When a line is cut at any point, the parts of the line between that point and its extremities are termed segments. When the point of section (C) lies between the two extremities (A and B) of the line, the two portions into which the line is divided (AĆ and CB) are termed internal segments. But when that point (F) lies in the production of the D line beyond remity, the distances from the point (F) to each remity (FD and FE) are termed external segments. 3. A STRAIGHT LINE is a line which lies evenly (i.e., in the same direction) between its extreme points. 4. A Curved LINE is a line which continually changes its direction. SCHOLIUM. Whenever the word "line" alone is used throughout this work, it must be taken to mean a straight line. 5. A SURFACE is a magnitude, having only two dimensions, i.e., length and breadth. COROLLARY. The extremities of a surface are lines, and the intersection of one surface with another is a line. 6. A PLANE SURFACE, or a PLANE, is a surface which lies evenly between its extremities. SCHOLIUM. The terms point, line, and surface, belong to the class of simple terms, that is to say, they are the names given to simple ideas, by means of which those ideas are conveyed from one mind to another, and consequently (for the reasons stated at length in the introduction) cannot be logically defined. There must always necessarily be a certain amount of difficulty in conveying, for the first time, a simple idea from one mind to another with perfect accuracy. But the idea having been once accurately conveyed, and associated with a certain name, can at any future time be readily recalled to our mind by the mention of that name alone. The first B process, that of conveying to another mind a new simple idea, that is, a simple idea with which hitherto it had not been familiar, can only be done by a certain system of abstraction, that is, by presenting to it some more complex idea, in which the first simple idea is involved with others with which the mind is already familiar, by the successive abstraction of which it is left in possession of the simple idea. Let us take, as an illustration, the process by which the idea of a mathematical point would be conveyed to the mind of a person for the first time. We should first present to him the complex idea of a physical point, such as the point of a pencil or of a needle, with which he would already be familiar; we should then explain to him that the physical point involved two ideas, one of position and one of magnitude; and further, that the less the magnitude was supposed to become (or, in ordinary terms, the finer we supposed the point to be), the more precise and definite would become the position which the point occupies, and serves to mark or identify; and we should thus lead him, by the gradual abstraction of the idea of the magnitude of the point, to look upon it as infinitely small, and only to associate with it the idea of its position; and thus he would realize the idea of a mathematical point. In like manner, with a mathematical line, we should first present to him a line such as a pencil would trace on a sheet of paper, and direct his attention to the fact that the line so drawn was in reality a solid or magnitude having three dimensions, namely length, breadth, and thickness, the breadth and thickness of the black lead left by the pencil on the surface of the paper, and which constitutes the physical line presented to the eye. We should then point out the extreme minuteness of the two last dimensions of the line as compared with its length, and ask him to conceive these dimensions as becoming less and less until the idea of the line presented itself to his mind as only possessing length, but devoid of breadth and thickness. Such, then, being the process by which the mind arrives at the true idea which such words as point, line, and surface represent, it would be impossible, by any formal definition of a new term, to convey for the first time, accurately, the idea which it represented. Thus, to a mind that had never heard of a mathematical point, the definition which we have given above of such a point would convey no idea. It must not, however, be therefore supposed that the foregoing definitions are useless; they serve to show the precise and limited sense in which the words are used in the following work, and so avoid the ambiguity which would arise if employed with the same latitude as in ordinary conversation. They show, for example, that the only quality of a point recognised in geometry is its position; of a line, its direction and length; and of a surface, its position and extent. 7. PARALLEL STRAIGHT LINES are straight lines in the same plane, which, being produced to any extent in both directions, would never meet. 8. A RECTILINEAL ANGLE is a magnitude formed by the inclination (ie., opening or divergence) of two straight lines to one another, which meet in a point. SCHOLIUM. We have here defined an angle to be a magnitude, and we are anxious to impress the idea of its being so on the mind of the mathematical student, as it will materially assist his progress when pursuing his studies in the higher branches of analysis. În conceiving, however, an angle as a magnitude, we must be careful not to confound with it the notion of the surface situated between the two lines which form the angle, or to look upon its magnitude as in any way affected by the length of those lines. By way of explanation, let us borrow an illustration from the hands of a clock, and regard the angle formed by their center lines. Now at twelve o'clock, as the hands coincide, no angle is formed by them; but from that moment they cease to coincide, and the magnitude of the angle becomes every instant greater as the minute hand moves away from the other. Now at any definite portion of time, such, for instance, as ten minutes, the hands of the clock form an angle of a certain definite magnitude, which is precisely the same whatever may be the length of those hands, whether they belong to a pocket watch or to a turret clock; in either case the interval of time is indicated by the divergence of the hands, or, in other words, by the magnitude of the angle which they form. The student has further been probably accustomed to regard the angle formed by two lines as being necessarily less than two right angles; so that if two lines were situated as in the margin, he would find it difficult to regard the angle which they formed as being that which is shaded, and would probably only be able to conceive it as the white opening BAČ. Or, to revert to the hands of the clock, at three o'clock he would regard the hands as forming an angle of 90°, or the fourth part of a circle; but as the angle which they form is the measure of the angular distance that the minute hand has moved from the hour hand, we must measure that distance in the direction of its motion, and we thus find that angular distance to be 270°, or three-fourths of a circle. And by thus regarding the magnitude of an angle as the measure of the angular distance that a line revolving about one of its extremities has moved from its normal or first position, we are enabled to realize the idea of an angle greater even than four right angles or an entire revolution. We have only to conceive the line as moving at a uniform rate, so that the magnitude of the angle may be estimated by the length of time that it has been in motion, to see that, if the time exceeded that in which an entire revolution was performed, the angle whose magnitude it indicated had become greater than four right angles; and in like manner, after the interval required to complete two revolutions, that the angle had become greater than eight right angles, and so forth. The point in which the two lines forming an angle meet is termed the vertex, and the two lines are termed the sides. The angle is referred to by a letter placed at the vertex, as the angle A; but if more than one angle is formed at the same point, it is then designated by three letters, one on each side and one at the vertex, the latter always being placed between the others, as the angle BCD. B The angles formed by the sides of rectilineal figures derive a variety of designations according to their relative positions and magnitudes, which will be defined in subsequent scholia. 9. A RIGHT ANGLE is half the angle, formed by a straight line with its continuation. SCHOLIUM. The line which divides the angle formed by a straight line with its continuation, into two right angles, is said to be perpendicular to that straight line. Thus the line CD is perpendicular to the line AB, and the angles ACD and BCD are both right angles. 10. An OBTUSE ANGLE is an angle which is greater than a right angle. ID |