
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 rightangled triangle
is one which has a right angle, and obtuseangled one which has an
obtuse angle, and acuteangled 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 rightangled, and an oblong is that which is right
angled but not equilateral, and a rhombus is that which is equilateral
but not rightangled, and rhomboid is that which has opposite sides
and angles equal to one another but is neither equilateral nor rightangled,
and let any quadrilateral besides these be called 'trapezia'. 
Square is called 'tetragonon' (fourangle [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 (1701) 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. 
