And because KF is parallel to HG, and HG to ML, and FL has been proved parallel to KM, and since the parallelogram HF is equal to the triangle ABD, and the parallelogram GM to the triangle BDC; therefore the whole parallelogram KFLM is equal to the whole rectilineal figure ABCD. Therefore the parallelogram KFLM has been described equal to the given rectilineal figure ABCD, having the angle FKM equal to the given angle E. Q.E.F. COR. From this it is manifest how, to a given straight line, to apply a parallelogram, which shall have an angle equal to a given rectilineal angle, and shall be equal to a given rectilineal figure; viz. by applying to the given straight line a parallelogram equal to the first triangle ABD, (1. 44.) and having an angle equal to the given angle. It is required to describe a square upon AB. through the point D draw DE parallel to AB; (1. 31.) whence AB is equal to DE, and AD to BE; (1. 34.) therefore the four lines AB, BE, ED, DA are equal to one another, therefore the angles BAD, ADE are equal to two right angles; (1. 29.) but BAD is a right angle; (constr.) therefore also ADE is a right angle. But the opposite angles of parallelograms are equal; (1. 34.) therefore each of the opposite angles ABE, BED is a right angle; wherefore the figure ABED is rectangular, and it has been proved to be equilateral; COR. Hence, every parallelogram that has one of its angles a right angle, has all its angles right angles. PROPOSITION XLVII. THEOREM. In any right-angled triangle, the square which is described upon the side subtending the right angle, is equal to the squares described upon the sides which contain the right angle. Let ABC be a right-angled triangle, having the right angle BAC. Then the square described upon the side BC, shall be equal to the squares described upon BA, AC. On BC describe the square BDEC, (1. 46.) Then because the angle BAC is a right angle, (hyp.) the two straight lines AC, AG upon the opposite sides of AB, make add to each of these equals the angle ABC, therefore the whole angle ABD is equal to the whole angle FBC. (ax.2.) And because the two sides AB, BD, are equal to the two sides FB, BC, each to each, and the included angle ABD is equal to the included angle FBC, therefore the base AD is equal to the base FC, (1. 4.) Now the parallelogram BL is double of the triangle ABD, (1. 41.) because they are upon the same base BD, and between the same parallels BD, AÌ; also the square GB is double of the triangle FBC, because these also are upon the same base FB, and between the same parallels FB, GC. But the doubles of equals are equal to one another; (ax. 6.) therefore the parallelogram BL is equal to the square GB. Similarly, by joining AE, BK, it can be proved, that the parallelogram CL is equal to the square HC. Therefore the whole square BDEC is equal to the two squares GB, HC; (ax. 2.) and the square BDEC is described upon the straight line BC, and the squares GB, HC, upon AB, AC: therefore the square upon the side BC, is equal to the squares upon the sides AB, AC. Therefore, in any right-angled triangle, &c. Q.E.D. PROPOSITION XLVIII. THEOREM. If the square described upon one of the sides of a triangle, be equal to the squares described upon the other two sides of it; the angle contained by these two sides is a right angle. Let the square described upon BC, one of the sides of the triangle ABC, be equal to the squares upon the other two sides, AB, AC. Then the angle BAC shall be a right angle. 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, A Care 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; (I. 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 sup posed uniform and consistent. 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 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, σημεῖον ἐστιν οὗ μέρος οὐδέν, “ 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 onuεtov, 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 defned a point to be μονας θέσιν ἔχουσα, “ 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 is defined to be that which has no magnitude, but position only. Def. II. 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 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 onμείois KεiTai, 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, which is the same thing, that "two straight lines cannot have a common segment," as Simson shews in his Corollary to Prop. 11, Book I. or, 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 'Eriπedos ¿Tiανειά ἐστιν ἥτις ἐξ ἴσου ταῖς ἐφ' ἑαυτῆς εὐθείαις κεῖται, “ 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. |