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From the point A draw AD at right angles to AC, (1. 11.)
Then, because AD is equal to AB,
the square on AD is equal to the square on AB;
therefore the squares on AD, AC are equal to the squares on AB, AC: but the squares on AD, AC are equal to the square on DC, (1. 47.) because the angle DAC is a right angle;
and the square on BC; by hypothesis, is equal to the squares on BA, AC; therefore the square on DC is equal to the square on BC; and therefore the side DC is equal to the side BC. And because the side AD is equal to the side AB,
and AC is common to the two triangles DAC, BAC;
the two sides DA, AC, are equal to the two BA, AC; each to each; and the base DC has been proved to be equal to the base BC; therefore the angle DAC is equal to the angle BAC; (1. 8.) but DAC is a right angle;
therefore also BAC is a right angle.
Therefore, if the square described upon, &c.
Q, E. D:
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 inductions 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.
It has been maintained by some writers both of ancient and modern times, that Geometry is a perfectly abstract Science, a body of truths completely independent of all human observation and experience. The truths of Geometry may, possibly, be a portion of absolute and universal truths, such as change not in all time, and which maintain the same constant universality under every conceivable state of existence. Still it must be admitted that however abstract and independent of experience such truths may be in themselves, it was not as such that they were originally discovered, or in that form that they are apprehended by the human mind.
The natural process of the human mind in the acquirement of knowledge and in the discovery of truth, is, to proceed from the particular to the general, from the sensible to the abstract. It is perhaps not too much to affirm, that the human mind would never have speculated on the abstract properties of circles and triangles unless some visible forms of such figures had first been exhibited to the senses. It does seem more probable and analogous to the rise and progress of other branches of human knowledge, that the fundamental truths of Geometry should have been first discovered from suggestions made to the senses; and this opinion, too, is not repugnant to the earliest historical notices existing on the subject. The human mind is so constituted that exact knowledge in any Science can only be acquired progressively. Every successive step in advance must be taken as a sequel to, and dependent upon, the previous acquirements; and some intelligible facts and first principles must form the basis of all human Science.
Now the only possible way of explaining terms denoting simple perceptions is to excite those simple perceptions. The impossibility of defining a word expressive of a simple perception is well known to every one who has paid any attention to his own intellectual progress. The only way of rendering a simple term intelligible is to exhibit the object of which it is the sign, or some sensible representation of it. A straight line therefore must be drawn, and by
drawing a curved line and a crooked line, the distinction will be perfectly understood. Again, the definition of a complex term consists merely in the enumeration of the simple ideas for which it stands, and it will be found that all definitions must have some term or terms equally requiring definition or explanation with the one defined.
But in cases where the subject falls under the class of simple ideas, the terms ́of the definitions 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 are intended to designate the things named, and not 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.
The definitions of Euclid appeal directly to the senses; and the fundamental theorem (Euc. 1. 4) which forms the basis of all the succeeding propositions, is demonstrated by one of the simplest appeals to experience. At every step there is a reference made to something exhibited to the senses, the coincidence of the lines, the angles, and lastly, the surfaces of the two triangles; and by shewing a perfect coincidence, their equality is inferred. The instance exhibited, and the proof applied to it, is equally valid for any triangles whatever which have the same specified conditions given in the hypothesis. The same reasoning may be applied to any similar case which can be conceived, and thus from a single instance demonstrated by appeal to the senses, we are led to admit the statement contained in the general enunciation. These considerations appear to support the opinion, that the truths of Geometry, as a portion of human Science, rest ultimately on the evidence of the senses.
It may also be suggested, whether it be not a point of considerable importance to be able to discriminate, where human Science begins, and how certainty is acquired. Def. I. Simson has adopted Theon's definition of a point. Euclid's definition is σημεῖον ἐστιν οὗ μέρος οὐδέν, " Α point is that, of which there is no part," "A or which cannot be parted or divided, as the line, the angle, the surface, and the solid. The word point in Geometry is not employed in the sense in which the Earth is called a point in respect of the Universe. The Greek term onμciou, 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 μovas déσiv xovoa, "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 intelligible, so that a point has position only, but no magnitude.
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 conception 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 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 Εὐθεῖα γραμμή ἐστιν, ἥτις ἐξ ἴσου τοῖς ἐφ' ἑαυτῆς σημείοις κεῖται, wherein he states it to lie evenly, or equally, or upon an equality (¿§ loov) between its extremities, and which Proclus explains as being stretched between its extremities, ἡ ἐπ' ἄκρων τεταμένη,
If the line be conceived to be drawn on a plane surface, the words & loou 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 toov 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, viz. "two straight lines cannot have a common segment."
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. VII. Euclid's definition of a plane surface is 'Emíπados étipáveiá éotiv. ἥτις ἐξ ἴσου ταῖς ἐφ ̓ ἑαυτῆς εὐθείαις κεῖται, “A plane surface is that which liesevenly 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. The latter only are treated of in Elementary Geometry.
Def. Ix. An angle is a species of magnitude; for one angle may be greater 1 than, equal to, or less than another angle. 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.
Two straight lines are said to be conterminous which have a common termination, or when one extremity of each line coincides in the same point, and both the lines may be, or may not be, in the same direction.
Def. x. It may here be 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, op¤1ì ywvia; but a straight line, εὐθεῖα γραμμή.
Def. XVI. Euclid in defining a circle does not give any method by which it may be described. The definition simply states what a circle is, and one property which distinguishes it from other figures. In laying down his principles Euclid avoids entering into any method of showing how a straight line or a circle may be conceived to be generated. A circle might be defined in the following manrer:-If a finite straight line be supposed to revolve in a plane about one of its extremities which remains fixed, until it return to its original position, the surface which the revolving line has passed over is called a circle, and the linear space which the moving extremity of the revolving line has traced out is called the circumference of the circle. The straight line which revolves is called the radius, and the fixed point about which it revolves is called the center. The adoption of such a form of definition would have introduced the consideration of a locus. Instead of adopting such a mechanical method of defining a circle, he assumes that a circle may be described, and the third postulate seems to suggest that the method above stated is the assumption made by Euclid, which he does not place among the principles of his First Book. In the Eleventh Book, the sphere, the cone, and the cylinder are defined by plane figures which revolve about one side which is supposed to remain fixed. Def. XIX. This has been restored from Proclus, as it seems to have a meaning in the construction of Prop. 14, Book II.; 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 Book 111. 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 I., an isosceles triangle may be equilateral, and an equilateral triangle is considered isosceles in Prop. 15, Book 1v. Objection.