Def. Ix. It is of the highest importance to attain a clear conception of an angle, and of the sum and difference of two angles. The literal meaning of the term angulus suggests the Geometrical conception of an angle, which may be regarded as formed by the divergence of two straight lines from a point. In the definition of an angle, the magnitude of the angle is independent of the lengths of the two lines by which it is included; their mutual divergence from the point at which they meet, is the criterion of the magnitude of an angle, as it is pointed out in the succeeding definitions. The point at which the two lines meet is called the angular point or the vertex of the angle, and must not be confounded with the magnitude of the angle itself. The right angle is fixed in magnitude, and, on this account, it is made the standard with which all other angles are compared. Two straight lines which actually intersect one another, or which when produced would intersect, are said to be inclined to one another, and the inclination of the two lines is determined by the angle which they make with one another. Def. x. It may be here observed that in the Elements, Euclid always assumes that when one line is perpendicular to another line, the latter is also perpendicular to the former; and always calls a right angle, ópðr γωνία ; but a straight line, εὐθεῖα γραμμή. Def. XIX. This has been restored from Proclus, as it seems to have a meaning in the construction of Prop. 14, Book 11; the first case of Prop. 33, Book III, and Prop. 13, Book vi. The definition of the segment of a circle is not once alluded to in Book 1, and is not required before the discussion of the properties of the circle in Book III. Proclus remarks on this definition: "Hence you may collect that the center has three places: for it is either within the figure, as in the circle; or in its perimeter, as in the semicircle; or without the figure, as in certain conic lines." Def. XXIV-XXIX. Triangles are divided into three classes, by reference to the relations of their sides; and into three other classes, by reference to their angles. A further classification may be made by considering both the relation of the sides and angles in each triangle. In Simson's definition of the isosceles triangle, the word only must be omitted, as in the Cor. Prop. 5, Book 1, an isosceles triangle may be equilateral, and an equilateral triangle is considered isosceles in Prop. 15, Book IV. Objection has been made to the definition of an acute-angled triangle. It is said that it cannot be admitted as a definition, that all the three angles of a triangle are acute, which is supposed in Def. 29. It may be replied, that the definitions of the three kinds of angles point out and seem to supply a foundation for a similar distinction of triangles. Def. xxx-XXXIV. The definitions of quadrilateral figures are liable to objection. All of them, except the trapezium, fall under the general idea of a parallelogram; but as Euclid defined parallel straight lines after he had defined four-sided figures, no other arrangement could be adopted than the one he has followed; and for which there appeared to him, without doubt, some probable reasons. Sir Henry Savile, in his Seventh Lecture, remarks on some of the definitions of Euclid, "Nec dissimulandum aliquot harum in manibus exiguum esse usum in Geometriâ." A few verbal emendations have been made in some of them. A square is a four-sided plane figure having all its sides equal, and one angle a right angle: because it is proved in Prop. 46, Book 1, that if a parallelogram have one angle a right angle, all its angles are right angles. An oblong, in the same manner, may be defined as a plane figure of four sides, having only its opposite sides equal, and one of its angles a right angle. A rhomboid is a four-sided plane figure having only its opposite sides equal to one another and its angles not right angles. Sometimes an irregular four-sided figure which has two sides parallel, is called a trapezoid. Def. xxxv. It is possible for two right lines never to meet when produced, and not be parallel. Def. A. The term parallelogram literally implies a figure formed by parallel straight lines, and may consist of four, six, eight, or any even number of sides, where every two of the opposite sides are parallel to one another. In the Elements, however, the term is restricted to four-sided figures, and includes the four species of figures named in the Definitions XXX-XXXIII. The synthetic method is followed by Euclid not only in the demonstrations of the propositions, but also in laying down the definitions. He commences with the simplest abstractions, defining a point, a line, an angle, a superficies, and their different varieties. This mode of proceeding involves the difficulty, almost insurmountable, of defining satisfactorily the elementary abstractions of Geometry. It has been observed, that it is necessary to consider a solid, that is, a magnitude which has length, breadth, and thickness, in order to understand aright the definitions of a point, a line, and a superficies. A solid or volume considered apart from its physical properties, suggests the idea of the surfaces by which it is bounded: a surface, the idea of the line or lines which form its boundaries: and a finite line, the points which form its extremities. A solid is therefore bounded by surfaces; a surface is bounded by lines; and a line is terminated by two points. A point marks position only: a fine has one dimension, length only, and defines distance: a superficies has two dimensions, length and breadth, and defines extension: and a solid has three dimensions, length, breadth, and thickness, and defines some portion of space. It may also be remarked that two points are sufficient to determine the position of a straight line, and three points not in the same straight line, are necessary to fix the position of a plane. ON THE POSTULATES. THE definitions assume the possible existence of straight lines and circles, and the postulates predicate the possibility of drawing and of producing straight lines, and of describing circles. The postulates form the principles of construction assumed in the Elements; and are, in fact, problems, the possibility of which is admitted to be self-evident, and to require no proof. It must, however, be carefully remarked, that the third postulate only admits that when any line is given in position and magnitude, a circle may be described from either extremity of the line as a center, and with a radius equal to the length of the line, as in Euc. 1, 1. It does not admit the description of a circle with any other point as a center than one of the extremities of the given line. Euc. 1. 2, shews how, from any given point, to draw a straight line equal to another straight line which is given in magnitude and position. ON THE AXIOMS. AXIOMS are usually defined to be self-evident truths, which cannot be rendered more evident by demonstration; in other words, the axioms of Geometry are theorems, the truth of which is admitted without proof. It is by experience we first become acquainted with the different forms of geometrical magnitudes, and the axioms, or the fundamental ideas of their equality or inequality appear to rest on the same basis. The conception of the truth of the axioms does not appear to be more removed from experience than the conception of the definitions. These axioms, or first principles of demonstration, are such theorems as cannot be resolved into simpler theorems, and no theorem ought to be admitted as a first principle of reasoning which is capable of being demonstrated. An axiom, and (when it is convertible) its converse, should both be of such a nature as that neither of them should require a formal demonstration. The first and most simple idea, derived from experience is, that every magnitude fills a certain space, and that several magnitudes may successively fill the same space. All the knowledge we have of magnitude is purely relative, and the most simple relations are those of equality and inequality. In the comparison of magnitudes, some are considered as given or known, and the unknown are compared with the known, and conclusions are synthetically deduced with respect to the equality or inequality of the magnitudes under consideration. In this manner we form our idea of equality, which is thus formally stated in the eighth axiom: " 'Magnitudes which coincide with one another, that is, which exactly fill the same space, are equal to one another." Every specific definition is referred to this universal principle. With regard to a few more general definitions which do not furnish an equality, it will be found that some hypothesis is always made reducing them to that principle, before any theory is built upon them. As for example, the definition of a straight line is to be referred to the tenth axiom; the definition of a right angle to the eleventh axiom; and the definition of parallel straight lines to the twelfth axiom. The eighth axiom is called the principle of superposition, or, the mental process by which one Geometrical magnitude may be conceived to be placed on another, so as exactly to coincide with it, in the parts which are made the subject of comparison. Thus, if one straight line be conceived to be placed upon another, so that their extremities are coincident, the two straight lines are equal. If the directions of two lines which include one angle, coincide with the directions of the two lines which contain another angle, where the points, from which the angles diverge, coincide, then the two angles are equal: the lengths of the lines not affecting in any way the magnitudes of the angles. When one plane figure is conceived to be placed upon another, so that the boundaries of one exactly coincide with the boundaries of the other, then the two plane figures are equal. It may also be remarked, that the converse of this proposition is not universally true, namely, that when two magnitudes are equal, they coincide with one another: since two magnitudes may be equal in area, as two parallelograms or two triangles, Euc. 1. 35, 37; but their boundaries may not be equal: and, consequently, by superposition, the figures could not exactly coincide: all such figures, however, having equal areas, by a different arrangement of their parts, may be made to coincide exactly. This axiom is the criterion of Geometrical equality, and is essentially different from the criterion of Arithmetical equality. Two geometrical magnitudes are equal, when they coincide or may be made to coincide: two abstract numbers are equal, when they contain the same aggregate of units; and two concrete numbers are equal, when they contain the same number of units of the same kind of magnitude. It is at once obvious, that Arithmetical representations of Geometrical magnitudes are not admissible in Euclid's criterion of Geometrical Equality, as he has not fixed the unit of magnitude of either the straight line, the angle, or the superficies. Perhaps Euclid intended that the first seven axioms should be applicable to numbers as well as to Geometrical magnitudes, and this is in accordance with the words of Proclus, who calls the axioms, common notions, not peculiar to the subject of Geometry. Several of the axioms may be generally exemplified thus: to the straight line CD; and if the straight Axiom II. If the line AB be equal to the line A CD; and if the line EF be also equal to the line GH: then the sum of the lines AB and EF is equal to the sum of the lines CD and GH. Axiom III. If the line AB be equal to the A line CD; and if the line EF be also equal to the line GH; then the difference of AB and EF, is equal to the difference of CD and GH. E E B D B C D F G H B с D F G H Axiom Iv. admits of being exemplified under the two following forms: 2. If the line AB be equal to the line CD; A and if the line EF be less than the line GH; then the sum of the lines AB and EF is less than the sum of the lines CD and GH. Axiom v. also admits of two forms of exemplification. 1. If the line AB be equal to the line CD; and if the line EF be greater than the line GH; then the difference of the lines AB and EF is greater than the difference of CD and GH. A B C D E 2. If the line AB be equal to the line CD; and if the line EF be less than the line GH; then the difference of the lines AB and EF is less than the difference of the lines CD and GH. The axiom, "If unequals be taken from equals, the remainders are unequal," may be exemplified in the same manner. then the line AB is equal to the line EF. Axiom VII. If the line AB be the half of the line CD; and if the line EF be also the half of the line CD; then the line AB is equal to the line EF. It may be observed that when equal magnitudes are taken from unequal magnitudes, the greater remainder exceeds the less remainder by as much as the greater of the unequal magnitudes exceeds the less. If unequals be taken from unequals, the remainders are not always unequal; they may be equal: also if unequals be added to unequals the wholes are not always unequal, they may also be equal. Axiom IX. The whole is greater than its part, and conversely, the part is less than the whole. This axiom appears to assert the contrary of the eighth axiom, namely, that two magnitudes, of which one is greater than the other, cannot be made to coincide with one another. Axiom x. The property of straight lines expressed by the tenth axiom, namely, "that two straight lines cannot enclose a space," is obviously implied in the definition of straight lines; for if they enclosed a space, they could not coincide between their extreme points, when the two lines are equal. Axiom XI. This axiom has been asserted to be a demonstrable theorem. As an angle is a species of magnitude, this axiom is only a partieular application of the eighth axiom to right angles. Axiom XII. See the notes on Prop. xxix. Book I. ON THE PROPOSITIONS. WHENEVER a judgment is formally expressed, there must be something respecting which the judgment is expressed, and something else which constitutes the judgment. The former is called the subject of the proposition, and the latter, the predicate, which may be anything which can be affirmed or denied respecting the subject. The propositions in Euclid's Elements of Geometry may be divided into two classes, problems and theorems. A proposition, as the term imports, is something proposed; it is a problem, when some Geometrical construction is required to be effected: and it is a theorem when some Geometrical property is to be demonstrated. Every proposition is naturally divided into two parts; a problem consists of the data, or things given; and the quæsita, or things required: a theorem, consists of the subject or hypothesis, and the conclusion, or predicate. Hence the distinction between a problem and a theorem is this, that a problem consists of the data and the quæsita, and requires solution: and a theorem consists of the hypothesis and the predicate, and requires demonstration. All propositions are affirmative or negative; that is, they either assert some property, as Euc. 1. 4, or deny the existence of some property, as Euc. I. 7; and every proposition which is affirmatively stated has a contradictory corresponding proposition. If the affirmative be proved to be true, the contradictory is false. All propositions may be viewed as (1) universally affirmative, or universally negative; (2) aš particularly affirmative, or particularly negative. The connected course of reasoning by which any Geometrical truth is established is called a demonstration. It is called a direct demonstration when the predicate of the proposition is inferred directly from the premisses, as the conclusion of a series of successive deductions. The demonstration is called indirect, when the conclusion shows that the introduction of any other supposition contrary to the hypothesis stated in the proposition, necessarily leads to an absurdity. It has been remarked by Pascal, that " Geometry is almost the only subject as to which we find truths wherein all men agree; and one cause of this is, that Geometers alone regard the true laws of demonstration." |