Euclid, Elements of Geometry, Book I
Definitions, Postulates and Axioms
Dionysius Lardner's Edition

Transcribed from Dionysius Lardner, The First Six Books of the Elements of Euclid, with a Commentary and Geometrical Exercises, 11th Edition (London: H.G. Bohn, 1855).

Editor: Dionysius Lardner (1793—1859)
(Biographies: UCL Bloomsbury Project; Wikipedia.)

Source URL on Google Books: https://books.google.ie/books?id=dVRIAQAAMAAJ

(8)   These definitions require some elucidation. The object of Geometry¹ is the properties of figure, and figure is defined to be the relation which subsists between the boundaries of space. Space or magnitude is of three kinds, line, surface, and solid. It may be here observed, once for all, that the terms used in geometrical science, are not designed to signify any real, material or physical existences. They signify certain abstracted notions or conceptions of the mind, derived, without doubt, originally from material objects by the senses, but subsequently corrected, modified, and, as it were, purified by the operations of the understanding. Thus, it is certain, that nothing exactly conformable to the geometrical notion of a right line ever existed; no edge, which the finest tool of an artist can construct, is so completely free from inequalities as to entitle it to be consisdered as a mathematical right line. Nevertheless, the first notion of such an edge being obtained by the senses, the process of mind by which we reject the inequalities incident upon the nicest mechanical production, and substitute for them, mentally, that perfect evenness which constitutes the essence of a right line, is by no means difficult. In like manner, if a pen be drawn over this paper an effect is produced, which, in common language, would be called a line, right or curved, as the case may be. This, however, cannot, in the strict geometrical sense of the term, be a line at all, since it has breadth as well as length; for if it had not it could not be made evident to the senses. But having first obtained this rude and incorrect notion of a line, we can imagine that, while its length remains unaltered, it may be infinitely attenuated until it ceases alteogether to have breadth, and thus we obtain the exact conception of a mathematical line.

The different modes of magnitude are ideas so extremely uncompounded that their names do not admit of definition properly so called at all.² We may, however, assist the student to form correct notions of the true meaning of these terms, although we may not give rigorous logical definitions of them.

A notion being obtained by the senses of the smallest magnitude distinctly perceptible, this is called a physical point. If this point were indivisible even in idea, it would be strictly what is called a mathematical point. But this is not the case. No material substance can assume a magnitude so small that a smaller may not be imagined. The mind, however, having obtained the notion of an extremely minute magnitude, may proceed without limit in a mental diminution of it; and that state at which it would arrive if this diminution were infinitely continued is a mathematical point.³

The introduction of the idea of motion into geometry has been objected to as being foreign to that science. Nevertheless, it seems very doubtful whether we may not derive from motion the most distinct ideas of the modes of magnitude. If a mathematical point be conceived to move in space, and to mark its course by a trace or track, that trace or track will be a mathematical line. As the moving point has no magnitude, so it is evident that its track can have no breadth or thickness. The places of the point at the beginning and end of its motion, are the extremities of the line, which are therefore points. The third of the preceding definitions is not properly a definition, but a proposition, the truth of which may be inferred from the first two definitions.

As a mathematical line may be conceived to proceed from the motion of a mathematical point, so a physical line may be conceived to be generated by the motion of a physical point.

In the same manner as the motion of a point determines the idea of a line, the motion of a line may give the idea of a surface. If a mathematical line be conceived to move, and to leave in the space through which it passes a trace or track, this trace or track will be a surface; and since the line has no breadth, the surface can have no thickness. The initial and final positions of the moving line are two boundaries or extremities of the surface, and the other extremities are the lines traced by the extreme points of the line whose motion produced the surface.

The sixth definition is therefore liable to the same objection as the third. It is not properly a definition, but a principle, the truth of which be derived from the fifth and preceding definitions.

It is scarcely necessary to observe, that the validity of the objection against introducing motion as a principle into the Elements of Geometry, is not here disputed, nor is it introduced as such. The preceding observations are designed merely as illustrations to assist the student in forming correct notions of the true mathematical significations of the different modes of magnitude. With the same view we shall continue to refer to the same mechanical ideas of motion, and desire our observations always to be understood in the same sense.

The fourth definition, that of a right or straight line, is objectionable, as being unintelligible; and the same may be said of the definition (seventh) of a plane surface. Those who do not know what the words ‘straight line’ and ‘plane surface’ mean, will never collect their meaning from these definitions; and to those who do know the meaning of those terms, definitions are useless. The meaning of the terms ‘right line’ and ‘plane surface’ are only to be made known by an appeal to experience, and the evidence of the senses, assisted, as was before observed, by the power of the mind called abstraction. If a perfectly flexible string be pulled by its extremities in opposite directions, it will assume, between the two points of tension, a certain position. Were we to speak without the rigorous exactitude of geometry, we should say that it formed a straight line. But upon consideration, it is plain that the string has weight, and that its weight produces a flexure in it, the convexity of which will be turned towards the surface of the earth. If we conceive the weight of the string to be extremely small, that flexure will be proportionably small, and if, by the process of abstraction, we conceive the string to have no weight, the flexure will altogether disappear, and the string will be accurately a straight line.

A straight line is sometimes defined ‘to be the shortest way between two points.’ This is the definition given by Archimedes, and after him by Legendre in his Geometry; but Euclid considers this as a property to be proved. In this sense, a straight line may be conceived to be that which is traced by one point moving towards another, which is quiescent.

Plato defines a straight line to be that whose extremity hides all the rest, the eye being placed in the continuation of the line.

Probably the best definition of a plane surface is, that it is such a surface that the right line, which joins every two points which can be assumed upon it, lies entirely in the surface. This definition, originally given by Hero, is substituted for Euclid's by R. Simson and Legendre.

Plato defined a plane surface to be one whose extremities hide all the intermediate parts, the eye being placed in its continuation.

It has been also defined as ‘the smallest surface which can be contained between given extremities.’

Every line which is not a straight line, or composed of straight lines, is called a curve. Every surface which is not a plane, or composed of planes, is called a curved surface.

This definition, which is designed to include the inclination of curves as well as right lines, is omitted in some editions of the Elements, as being useless.

(14)   Angles might not improperly be considered as a fourth species of magnitude. Angular magnitude evidently consists of parts, and must therefore be admitted to be a species of quantity. The student must not suppose that the magnitude of an angle is affeced by the length of the right lines which include it, and of whose mutual divergence it is the measure. These lines, which are called the sides or legs of the angle, are supposed to be of indefinite length. To illustrate the nature of angular magnitude, we shall again recur to motion. Let C be supposed to be the extremity of a right line C A, extending indefinitely in the direction C A. Through the same point C, let another indefinite straight line C A₀, be conceived to be drawn; and suppose this right line to revolve in the same plane round its extremity C, it being supposed at the beginning of its motion to coincide with C A. As it revolves from C A₀ to C A₁, C A₂, C A₃, &c., its divergence from C A or, what is the same, the angle it makes with C A, continually increases. The line continuing to revolve, and successively assuming the positions C A₁, C A₂, C A₃, C A₄, &c., will at length coincide with the continuation C A₅ of the line C A₀ on the opposite side of the point C. When it assumes this position, it is considered by Euclid to have no inclination to C A₀, and to form no angle with it. Nevertheless, when the student advances further in mathematical science, he will find, that not only the line C A₅ is considered to form an angle with C A₀, but even when the revolving line continues its motion past C A₆; and this angle is measured in the direction A₆, A₅, A₄, &c. to A₀.

The point where the sides of an angle meet is called the vertex of the angle.

Superposition is the process by which one magnitude may be conceived to be placed upon another, so as exactly to cover it, or so that every part of each shall exactly coincide with every part of the other.

It is evident that any magnitudes which admit of superposition must be equal, or rather this may be considered as the definition of equality. Two angles are therefore equal when they admit of superposition. This may be determined thus; if the angles A B C and A′ B′ C′ are those whose equality is to be ascertained, let the vertex B′ be conceived to be placed on the vertex B, and the side B′ A′ on the side B A, and let the remaining side B′ C′ be placed on the same side of B A with B C. If under these circumstances B' C' lie upon, or coincide with B C, the angles admit of superposition, and are equal, but are otherwise not. If the side B′ C′ fall between B C and B A, the angle B′ is said to be less than the angle B, and if the side B C fall between B′ C′ and B A, the angle B' is said to be greater than B.

As soon as the revolving line assumes such a position C A₃ that the angle A C A₃ is equal to the angle A₃ C A₅, each of those angles is called a right angle.

An angle is sometimes expressed simply by the letter placed at its vertex, as we have done in comparing the angles B and B'. But when the same point, as C, is the vertex of more angles than one, it is necessary to use the three letters expressing the sides as A C A₃, A₃ C A₅, the letter at the vertex being always placed in the middle.

When a line is extended, prolonged, or has its length increased, it is said to be produced, and the increase of length which it receives is called its produced part, or its production. Thus, if the right line A B be prolonged to B′, it is said to be produced through the extremity B, and B B′ is called its production or produced part.

Two lines which meet and cross each other are said to intersect, and the point or points where they meet are called points of intersection. It is assumed as a self-evident truth, that two right lines can only intersect in one point. Curves, however, may intersect each other, or right lines, in several points.

Two right lines which intersect, or whose productions intersect, are said to be inclined to each other, and their inclination is measured by the angle which they include. The angle included by two right lines is sometimes called the angle under those lines; and right lines which include equal angles are said to be equally inclined to each other.

It may be observed, that in general when right lines and plane surfaces are spoken of in Geometry, there are considered as extended or produced indefinitely. When a determinate portion of a right line is spoken of, it is generally called a finite right line. When a right line is said to be given, it is generally meant that its position or direction on a plane is given. But when a finite right line is given, it is understood, that not only its position, but its length is given. These distinctions are not always rigorously observed, but it never happens that any difficulty arises, as the meaning of the words is always sufficiently plain from the context.

When the direction alone of a line is given, the line is sometimes said to be given in position, and when the length alone is given, it is said to be given in magnitude.

By the inclination of two finite right lines which do not meet, is meant the angle which would be contained under these lines if produced until they intersect.

This definition might be omitted as useless.

The entire length of the line or lines, which inclose a figure, is called its perimeter.

A figure whose surface is a plane is called a plane figure. The first six books of the Elements treat of plane figures only.

If a right line of a given length revolve in the same plane round one of its extremities as a fixed point, the other extremity will describe the circumference of a circle, of which the centre is the fixed extremity.

(19)   A line drawn from the centre of a circle to its circumference is called a radius.

(22)   From the definition of a circle, it follows immediately, that a line drawn from the centre to any point within the circle is less than the radius; and a line from the centre to any point without the circle is greater than the radius. Also, every point, whose distance from the centre is less than the radius, must be within the circle; every point whose distance from the centre is equal to the radius must be on the circle; and every point, whose distance from the centre is greater than the radius, is without the circle.

The word ‘semicircle’ in Def. XVIII., assumes, that a diameter divides the circle into two equal parts. This may be easily proved by supposing the two parts, into which the circle is thus divided, placed one upon the other, so that they shall lie at the same side of their common diameter: then if the arcs of the circle which bound them do not coincide, let a radius be supposed to be drawn, intersecting them. Thus, the radius of the one will be a part of the radius of the other; and therefore, two radii of the same circle are unequal, which is contrary to the definition of a circle (17.)

The lines which include the figre are called its sides.

A triangle is the most simple of all rectilinear figures, since less than three right lines cannot form any figure. All other rectilinear figures may be resolved into triangles by drawing right lines from any point within them to their several vertices. The triangle is therefore, in effect, the element of all rectilinear figures; and on its properties, the properties of all other rectilinear figures depend. Accordingly the greater part of the first book is devoted to the development of the properties of this figure.

Polygons are called pentagons, hexagons, heptagons, &c., according as they are bounded by five, six, seven or more sides. A line joining the vertices of any two angles which are not adjacent is called a diagonal of the polygon.

In general, all rectilinear figures whose sides are equal, may be said to be equilateral.

Two rectilinear figures, whose sides are respectively equal each to each, are said to be mutually equilateral. Thus, if two triangles have each sides of three, four, or five feet in length, they are mutually equilateral, although neither of them is an equilateral triangle.

In the same way a rectilinear figure having all its angles equal, is said to be equiangular, and two rectilinear figures whose several angles are equal each to each, are said to be mutually equiangular.

The equal sides are generally called the sides, to distinguish them from the third side, which is called the base.

That side of a right-angled triangle which is opposite to the right angle is called the hypotenuse.

It will appear hereafter, that a triangle cannot have more than one angle right or obtuse, but may have all its angles acute.

We have ventured to change the definition of a square as given in the text. A lozenge, called by Euclid a rhombus, when equiangular, must have all its angles right, as will appear hereafter. Euclid's definition, which is a ‘a lozenge all whose angles are right,’ therefore, contains more than sufficient for a definition, inasmuch as, had the angles been merely defined to be equal, they might be proved to be right. To effect this change in the definition of a square, we have transposed the order of the last two definitions. See (158).

This term is not used in the Elements, and therefore the definition might have been omitted. The same figure is defined in the second book, and called a rectangle. It would appear that this circumstance of defining the same figure twice must be an oversight.

This definition and the term rhomboid are superceded by the term parallelogram, which is a quadrilateral, whose opposite sides are parallel. It will be proved hereafter, that if the opposite sides of a quadrilateral be equal, it must be a parallelogram. Hence, a distinct denomination for such a figure is useless.

As quadrilateral figure is a sufficiently concise and distinct denomination, we shall restrict the application of the term trapezium to those quadrilaterals which have two sides parallel.

It should be observed, that the circumstance of two right lines, which are produced indefinitely, never meeting, is not sufficient to establish their parallelism. For two right lines which are not in the same plane can never meet, and yet are not parallel. Two things are indispensably necessary to establish the parallelism of two right lines, 1°, that they be in the same plane, and 2°, that when indefinitely produced, they never meet. As in the first six books of the Elements all the lines which are considered are supposed to be in the same plane, it will be only necessary to attend to the latter criterion of parallelism.

(42)   The object of the postulates is to declare, that the only instruments, the use of which is permitted in Geometry, are the rule and compass. The rule is an instrument which is use to direct the pen or pencil in drawing a right line; but it should be observed, that the geometrical rule is not supposed to be divided or graduated, and, consequently, it does not enable us to draw a right line of any proposed length. Neither is it permitted to place any permanent mark or marks on any part of the rule, or we should be able by it to solve the second proposition of the first book, which is to draw from a given point a right line equal to a another given right line. This might be done by placing the rule on the given right line, and marking its extremities on the rule, then placing the mark corresponding to one extremity at the given point, and drawing the pen along the rule to the second mark. This, however, is not intended to be granted by the postulates.

The third postulate concedes the use of the compass, which is an instrument composed of two straight and equal legs united at one extremity by a joint, so constructed that the legs can be opened or closed so as to form any proposed angle. The other extremities are points, and when the legs have been opened to any degree of divergence, the extremity of one of them being fixed at a point, and the extremity of the other being moved around it in the same plane will describe a circle, since the distance between the points is supposed to remain unchanged. The fixed point is the centre; and the distance between the points, the radius of the circle.

It is not intended to be conceded by the third postulate that a circle can be described round a given centre with a radius of a given length; in other words, it is not granted that the legs of the compass can be opened until the distance between their points shall equal a given line.

(55)   The geometrical axioms are certain general propositions, the truth of which is taken to be self-evident, and incapable of being established by demonstration. According to the spirit of this science, the number of axioms should be as limited as possible. A proposition, however self-evident, has no title to be taken as an axiom, if its truth can be deduced from axioms already admitted. We have a remarkable instance of the rigid adherence to this principle in the twentieth proposition of the first book, where it is proved that ‘two sides of a triangle taken together are greater than the third;’ a proposition which is quite as self-evident as any of the received axioms, and much more self-evident than several of them.

On the other hand, if the truth of a proposition cannot be established by demonstration, we are compelled to take it as an axiom, even though it be not self-evident. Such is the case with the twelfth axiom. We shall postpone our observations on this axiom, however, for the present, and have to request that the student will omit it until he comes to read the commentary on the twenty-eighth proposition. See Appendix II.

Two magnitudes are said to be equal when they are capable of exactly covering one another, or filling the same space. In the most ordinary practical cases we use this test for determining equality; we apply the two things to be compared one to the other, and immediately infer their equality from their coincidence.

By the aid of this definition of equality we conceive that the second and third axioms might easily be deduced from the first. We shall not however pursue the discussion here.

^★★^★ The fourth and fifth axioms are not sufficiently definite. After the addition or subtraction of equal quantities, unequal quantities continue to be unequal. But it is also evident, that their difference, that is, the quantity by which the greater exceeds the less, will be the same after such addition or subtraction as before it.

The sixth and seventh axioms may very easily be inferred from the preceding ones.

The tenth axiom may be presented under various forms. It is equivalent to stating, that between any two points only one right line can be drawn. For if two different right lines could be drawn from one point to another, they would evidently enclose a space between them. It is also equivalent to stating, that two right lines being infinitely produced cannot intersect each other in more than one point; for if they intersected at two points, the parts of the lines between these points would enclose a space.

The eleventh axiom admits of demonstration. Let A B and E F be perpendicular to DC and H G. Take any equal parts E H, E G on H G measured from the point E, and on D C take parts from A equal to these (Prop. III. Book I.) Let the point H be conceived to be placed upon the point D. The points G and C must then be in the circumference of a circle described round the centre D, with the distance D C or H G as radius. Hence, if the line H G be conceived to be turned round this centre D, the point G must in some position coincide with C. In such a position every point of the line HG must coincide with CD (ax. 10.), and the middle points A and E must evidently coincide. Let the perpendiculars E F and A B be conceived to be placed at the same side of D C. They must then coincide, and therefore the right angle F E G will be equal to the right angle B A C. For if E F do not coincide with A B, let it take the position A K. The right angle K A C is equal to K A D (11), and therefore greater than B A D; but B A D is equal to B A C (11), and therefore K A C is greater than B A C. But K A C is a part of B A C, and therefore less than it, which is absurd; and therefore E F must coincide with A B, and the right angles B A C and F E G are equal.

The postulates may be considered as axioms. The first postulated, which declares the possibility of one right line joining two given points, is as much an axiom as the tenth axiom, which declares the impossibility of more than one right line joining them.

In like manner, the second postulate, which grants the power of producing a line, may be considered as an axiom, declaring that every finite straight line may have another placed at its extremity so to form with it one continued straight line. In fact, the straight line thus placed will be its production. This postulate is assumed as an axiom in the fourteenth proposition of the first book.

(56)   Those results which are obtained in geometry by a process of reasoning are called propositions. Geometrical propositions are of two species, problems and theorems.

A problem is a proposition in which something is proposed to be done; as a line to be drawn under some given conditions, some figure to be constructed, &c. The solution of the problem consists in showing how the thing required may be done by the aid of the rule and compass. The demonstration consists in proving that the process indicated in the solution really attains the required end.

A theorem is a proposition in which the truth of some principle is asserted. The object of the demonstration is to show how the truth of the proposed principle may be deduced from the axioms and definitions or other truths previously and independently established.

A problem is analogous to a postulate, and a theorem to an axiom.

A postulate is a problem, the solution of which is assumed.

An axiom is a theorem, the truth of which is granted without demonstration.

In order to effect the demonstration of a proposition, it frequently happens that other lines must be drawn besides those which are actually engaged in the enunciation of the proposition itself. The drawing of such lines is generally called the construction.

A corollary is an inference deduced immediately from a proposition.

A scholium is a note or observation on a proposition not containing any inference, or, at least, none of sufficient importance to entitle it to the name of a corollary.

A lemma is a proposition merely introduced for the purpose of establishing some more important proposition.

¹ From γη, terra; and μέτρον mensura.

² The name of a simple idea cannot be defined, because the general terms which compose the definition signifying several different ideas can by no means express an idea which has no manner of composition.—LOCKE.

³ The Pythagorean definition of point is ‘a monad having position.’