return to Vignettes of Ancient Mathematics

Definitions
Postulates
Common Notions
Propositions
 

Definitions Brief comments
1 A point is that for which there is no part,
2 and a line is breadthless length,
3 and limits of a line are points. Is this a separate definition of point (as in Aristotle) or a property of lines?
4 A straight line is whatever lies equally with the points on it, The meaning of this definition is very unclear.  It is common to translate 'ex isou', here 'equally' as 'evenly', cf. def. 7.
5 and a surface is what only has length and breadth,
6 and lines are limits of a surface. Is this a separate definition of lines (as in Aristotle), or a property of surfaces?
7 A plane surface is whatever lies equally with the lines on it, Cf. def 4 and comment.
8 and a plane angle is the inclination in a plane of two lines in contact and not lying in a straight line with respect to one another,
9 and whenever the lines enclosing the angle are straight the angle is called 'rectilinear'.
10 and whenever a straight line stood on a straight line makes its successive angles equal to one another, each of the equal angles is right and the one standing on it is called perpendicular to the line it stands against.
11 Obtuse angle is the one larger than a right angle,
12 and acute angle is the one smaller than a right angle.
13 A boundary is what is the limit of something.
14 A figure is that which is enclosed by a boundary or some boundaries.
15 A circle is a plane figure enclosed by one line [which is called circular arc], such all the straight lines falling on it from one point among those lying within the figure [to the circular arc of the circle] are equal to one another, 1. If one were to think of this definition as providing a construction, the construction would be pointwise.  In Euclid, this is inappropriate; however, did anyone ever think of this definition in this way?
2.  Whether or not the phases in square brackets are part of Euclid's text, the word which we normally translate as 'circumference' is often better translated as 'circular arc', as in def. 18. This practice is followed throughout these web pages.
16 and the point is called 'center of the circle',
17 and a diameter of the circle is a certain straight line drawn through the center and limited on each of its parts by the circumference of the circle, whatever bisects the circle as well. Are these two definitions conflated into one?
18 and a semicircle is the figure enclosed by the diameter and the circular arc cut off by it, 
18a and a center of the semicircle is the same point which is also center of the circle.
19 Rectilinear figures are those enclosed by straight lines, trilateral that enclosed by three, quadrilateral that enclosed by four, multilateral that enclosed by more than four straight lines, 'Trilateral' is only used three times in Euclid, here and in defs. 20 and 21.  'Multilateral' is used only once.  Hence, Euclid is careful to define terms corresponding to the names, even though he will use 'triangle', even in naming the species of trilaterals in 20 and 21, and 'polygon' in books vi and xii (as well as the Catoprics)
20 and of trilateral figures, equilateral triangle is that which has three equal sides, isosceles that which has only two equal sides, and scalene that which has three unequal sides. Note that equilateral triangles are not isosceles.  There are three species of trilaterals.
21 And, furthermore, of trilateral figures a right-angled triangle is one which has a right angle, and obtuse-angled one which has an obtuse angle, and acute-angled one which has the three angles acute. As in def. 20, the three species are distinct.
22 and of quadrilateral figures, a square is that which is both equilateral and right-angled, and an oblong is that which is right angled but not equilateral, and a rhombus is that which is equilateral but not right-angled, and rhomboid is that which has opposite sides and angles equal to one another but is neither equilateral nor right-angled, and let any quadrilateral besides these be called 'trapezia'. Square is called 'tetragonon' (four-angle [figure]).  Here too (cf. defs. 20 and 21) the classifications are exclusive and include all possible quadrilaterals by using 'trapezion' as a catchall.  British English follows Proclus (170-1) in treating trapezia as figures with two parallel lines and trapezoids as the having not parallel sides (in America we reverse the terminology).  Euclid only uses the term in I 34, where the two trapezia happen to have two parallel lines.  There are no texts independent of Proclus or his sources giving an account of the two terms.

Rhomboids, however, will turn out to be parallelograms, a term which Euclid introduces without definition in the statement of the theorem at I 34.  Obviously, the term cannot be introduced before he defines parallel lines.  Neither rhombus nor rhomboid appear again in Euclid.

23 Parallels are any straight lines which are in the same plane and being extended infinitely towards each part come together on neither part.
top

Postulates 
1 Let it be postulated that from any point to any point to draw a straight line.
2 and to extend a finite straight line continuously  in a straight line
3 and with any center and distance to inscribe a circle,
4 and that all right angles are equal to one another,
5 and that if a straight line falls on two lines makes the interior angles on the same parts smaller than two right angles, then if the two lines are extended infinitely on the parts where the angles are smaller than two right angles, they meet.

Common Notions
1 Things equal to the same are equal to one another.
2 and if equals are added to equals they wholes are equal
3 and if equals are subtracted from equals, the remainders are equal
4 [and if unequals are added to equals they wholes are unequal
5 and the doubles are of same are equal to one another
6 and the halves are of same are equal to one another]
7 and things which fit onto one another are equal to one another
8 and the whole is larger than the part.

Prop. 1 (problem):  On the given straight finite straight-line to construct an equilateral triangle.

Prop. 32 (theorem):  Of every triangle, when one of the sides is extended the external angle is equal to the two interior and opposite angles and the three interior angles of the triangle are equal to two right angles.

top