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NOTES TO BOOK I.
ON THE DEFINITIONS.
GEOMETRY is one of the most perfect of the deductive Sciences, and seems to rest on the simplest inductions from experience and observation.
The first principles of Geometry are therefore in this view consistent hypotheses. founded on facts cognizable by the senses, and it is a subject of primary importance to draw a distinction between the conception of things and the things themselves. These hypotheses do not involve any property contrary to the real nature of the things, and, consequently, cannot be regarded as arbitrary, but, in certain respects, agree with the conceptions which the things themselves suggest to the mind through the medium of the senses. The essential definitions of Geometry therefore being deductions from observation and experience, rest ultimately on the evidence of the senses.
It is by experience we become acquainted with the existence of individual forms of magnitudes; but by the mental process of abstraction, which begins with a particular instance, and proceeds to the general idea of all objects of the same kind, we attain to the general conception of those forms which come under the same general idea.
The essential definitions of Geometry express generalized conceptions of real existences in their most perfect ideal forms; the laws and appearances of nature, and the operations of the human intellect being supposed uniform and consistent.
But in cases where the subject falls under the class of simple ideas, the terms of the definition so called are no more than merely equivalent expressions. The simple idea described by a proper term or terms, does not in fact admit of definition properly so called. The definitions in Euclid's Elements may be divided into two classes, those which merely explain the meaning of the terms employed, and those, which, besides explaining the meaning of the terms, suppose the existence of the things described in the definitions.
Definitions in Geometry cannot be of such a form as to explain the nature and properties of the figures defined ; it is sufficient that they give marks whereby the thing defined may be distinguished from every other of the same kind. It will at once be obvious, that the definitions of Geometry, one of the pure sciences, being abstractions of space, are not like the definitions in any one of the physical sciences. The discovery of any new physical facts may render necessary some alteration or modification in the definitions of the latter.
Def. I. Simson has adopted Theon's definition of a point. Euclid's definition is, onuelov éotiv oŮ mépos ovôév, “A point is that, of which there is no part,” or which cannot be parted or divided, as it is explained by Proclus. The Greek term onuenov, literally means, a visible sign or mark on a surface, in other words, a physical point. The English term point, means the sharp end of any thing, or a mark made by it. The word point comes from the Latin punctum, through the French word point. Neither of these terms, in its literal sense, appears to give a very exact notion of what is to be understood by a point in Geometry.
Euclid's definition of a point merely expresses a negative property, which excludes the proper and literal meaning of the Greek term, as applied to denote a physical point, or a mark which is visible to the senses.
Pythagoras defined a point to be μονας θέσιν έχουσα, « a monad having position." By uniting the positive idea of position, with the negative idea of defect of magnitude, the conception of a point in Geometry may be rendered perhaps more intelligible. A point may then be defined to be that which has no magnitude, but position only.
Def. 11. Every visible line has both length and breadth, and it is impossible to draw any line whatever which shall have no breadth. The definition requires the cohception of the length only of the line to be considered, abstracted from, and independently of, all idea of its breadth. Def. III.
This definition renders more intelligible the exact meaning of the definition of a point: and we may add, that, in the Elements, Euclid supposes that the intersection of two straight lines is a point, and that two straight lines can intersect each other in one point only.
Def. iv. The straight line or right line is a term so clear and intelligible as to be incapable of becoming more so by formal definition. Euclid's definition is Evdcia γραμμή έστιν, ήτις έξ ίσου τους έφ' εαυτής σημείοις κείται, wherein he states it to lie evenly, or equally, or upon an equality, (Eroov) between its extremities, and which Proclus explains as being stretched between its extremities, rj ém' đkpwv tetapévn.
If the line be conceived to be drawn on a plane surface, the words ég loov may mean, that no part of the line which is called a straight line deviates either from one side or the other of the direction which is fixed by the extremities of the line ; and thus it may be distinguished from a curved line, which does not lie, in this sense, evenly between its extreme points. If the line be conceived to be drawn in space, the words ég loov, must be understood to apply to every direction on every side of the line between its extremities.
Every straight line situated in a plane is considered to have two sides ; and when the direction of a line is known, the line is said to be given in position; also, when the length is known or can be found, it is said to be given in magnitude.
From the definition of a straight line, it follows, that two points fix a straight line in position, which is the foundation of the first and second postulates. Hence straight lines which are proved to coincide in two or more points, are called “one and the same straight line,” Prop. 14, Book 1., or, which is the same thing, that, “ Two straight lines cannot have a common segment,” as Simson shews in his Corollary to Prop. 11, Book 1.
Archimedes defined “a straight line to be the shortest distance between two points;" but this is a theorem considered by Euclid as requiring proof.
The following definition of straight lines has also been proposed. “Straight lines are those which, if they coincide in any two points, coincide as far as they are produced.” But this is rather a criterion of straight lines, and analogous to the eleventh axiom, which states that, “all right angles are equal to one another,” and suggests that all straight lines may be made to coincide wholly, if the lines be equal; or partially if the lines be of unequal lengths. A definition should properly be restricted to the description of the thing defined, as it exists, independently of any comparison of its properties or of tacitly assuming the existence of axioms.
Def. vir. Euclid's definition of a plane surface is, 'Emímegos éiridáverá ¢otiv, ijtis εξ ίσου ταϊς εφ' εαυτής ευθείαις κείται, “A plane surface is that which lies evenly or equally with the straight lines in it;" instead of which Simson has given the definition which was originally proposed by Hero the Elder. A plane superficies may be supposed to be situated in any position, and to be continued in every direction to any extent.
Def. viii. Simson remarks that this definition seems to include the angles formed by two curved lines, or a curve and a straight line, as well as that formed by two straight lines.
Angles made by straight lines only, are treated of in Elementary Geometry.
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 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 subject with which all other angles in Geometry 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, opori ywvía; but a straight line, evdela ypapuri. Def. xix.
This has been restored from Proclus, as it seems to have a meaning in the constructions 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 I, and is not required before the discussion of the properties of the circle in
Proclus remarks on this definition : “Hence you may collect that the centre has three places. For it is either within the figure, as in the circle ; or in its perimeter, as in the semi-circle; 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 the 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 has 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 Geometrià.” 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. Simson observes 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 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 line 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 definite 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 centre, and with a radius equal to the length of the line, as in Prop. 1, Book 1. It does not admit the description of a circle with any other point as a centre than one of the extremities of the given line.
Prop. 2, Book 1. 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 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 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.
It 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 also true, namely, that when two magnitudes are equal, they coincide with one another.
This explanation of Geometrical equality appears to be out of its proper place. The definitions of the forms of magnitudes naturally come first, and the criterion of their equality appears as naturally to follow. If the first seven axioms are to be restricted to Geometrical magnitudes, the eighth ought to have preceded them. 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.
The eighth axiom is properly the definition of Geometrical equality.
It may be observed that when equal magnitudes are taken from un. equal magnitudes, the greater remainder exceeds the less remainder by as much as the greater of the unequal magnitudes exceeds the less.
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. If an angle be admitted to be a species of magnitude, this axiom is only a particular appli. cation of the eighth axiom to right angles.
Axiom XII. See the notes on Prop. xxix, Book 1.