From Euclid's Elements, Book I.
- A point is that which has no part.
- A line is breadthless length.
- The extremities of a line are points.
- A straight line is a line which lies evenly with the
points on itself.
- A surface is that which has length and breadth only.
- The extremities of a surface are lines.
- A plane surface is a surface which lies evenly with
the straight lines on itself.
- A plane angle is the inclination to one another of
two lines in a plane which meet one another and do not lie in
- And when the lines containing the angle are straight,
the angle is called rectilinear.
- When a straight line set up on a straight line makes
adjacent angles equal to one another, each of the equal
angles is right, and the straight line standing on the other is
called a perpendicular to that on which it stands.
- An obtuse angle is an angle greater than a right
- An acute angle is an angle less than a right angle.
- A boundary is that which is an extremity of anything.
- A figure is that which is contained by any boundary
- A circle is a plane figure contained by one line such
that all the straight lines falling upon it from one point among
those lying within the figure are equal to one another.
- And the point is called the centre of the circle.
- A diameter of the circle is any straight line drawn
through the centre and terminated in both directions by the
circumference of the circle, and such a straight line always
bisects the circle.
- A semicircle is the figure contained by the diameter
and the circumference cut off by it. And the centre of the
semicircle is the same as that of the circle.
- Rectilineal figures are those which are contained
by straight lines, trilateral figures being those contained
three, quadrilateral those contained by four, and multilateral
those contained by more than four straight lines.
- Of trilateral figures, an equilateral triangle is that
which has its three sides equal, an isosceles triangle that
which has two of its sides alone equal, and a scalene
triangle that which has its three sides unequal.
- Further, of trilateral figures, a right-angled triangle
is that which has a right angle, an obtuse-angle
triangle that which has an obtuse angle, and an acute
angled triangle that which has its three angles acute.
- Of quadrilateral figures, a square is that which is
both equilateral and right-angled; an oblong that which is
right-angled but not equilateral; a rhombus that which is
equilateral but not right-angled; and a rhomboid that which
has its opposite sides and angles equal to one another but
neither equilateral nor right-angled. And let quadrilateral
other than these be called trapezia.
- Parallel straight lines are straight lines which
being in the same plane and being produced indefinitely in
both directions do not meet one another in either direction.
Let the following be postulated:
- To draw a straight line from any point to any point.
- To produce a finite straight line continuously in
a straight line.
- To describe a circle with any centre and distance.
- That all right angles are equal to one another.
- That, if a straight line falling on two straight lines
make the interior angles on the same side less than two right
angles, the two straight lines, if produced indefinitely, meet
on that side on which are the angles less than the two right
- Things which are equal to the same thing are also
equal to one another.
- If equals be added to equals, the wholes are equal.
3. If equals be subtracted from equals, the remainders
- Things which coincide with one another are equal to
- The whole is greater than the part.
On a given finite straight line to construct an equalateral
Let AB be the given finite straight line.
Thus it is required to construct an equilateral triangle on
the straight line AB.
With centre A and distance
AB let the circle BCD be
described; [Post. 3]
again, with centre B and distance
BA let the circle ACE
be described; [Post. 3]
and from the point C, in which the circles cut one another, to
the points A, B let the straight lines CA, CB be joined.
Now, since the point A is the centre of the circle CDB, AC is equal to AB.[Def. 15]
Again, since the point B is the centre of the circle CAE,
BC is equal to BA. [Def. 15]
But CA was also proved equal to AB;
therefore each of the straight lines CA, CB is equal to AB.
And things which are equal to the same thing are also
equal to one another;
therefore CA is also equal to CB.
Therefore the three straight lines CA, AB, BC are
equal to one another.
Therefore the triangle ABC is equilateral; and it has
been constructed on the given finite straight line AB.
(Being) what it was required to do.