- Definition
1.
- A point is that which has no part.
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- Definition
2.
- A line is breadthless length.
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- Definition
3.
- The ends of a line are points.
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- Definition
4.
- A straight line is a line which lies evenly with the points on
itself.
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- Definition
5.
- A surface is that which has length and breadth only.
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- Definition
6.
- The edges of a surface are lines.
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- Definition
7.
- A plane surface is a surface which lies evenly with the straight
lines on itself.
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- Definition
8.
- A plane angle is the inclination to one another of two lines in a
plane which meet one another and do not lie in a straight line.
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- Definition
9.
- And when the lines containing the angle are straight, the angle is called rectilinear.
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- Definition
10.
- When a straight line standing on a straight line makes the 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.
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- Definition
11.
- An obtuse angle is an angle greater than a right angle.
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- Definition
12.
- An acute angle is an angle less than a right angle.
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- Definition
13.
- A boundary is that which is an extremity of anything.
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- Definition
14.
- A figure is that which is contained by any boundary or boundaries.
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- Definition
15.
- 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 equal one another.
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- Definition
16.
- And the point is called the center of the circle.
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- Definition
17.
- A diameter of the circle is any straight line drawn through the
center and terminated in both directions by the circumference of the circle,
and such a straight line also bisects the circle.
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- Definition
18.
- A semicircle is the figure contained by the diameter and the
circumference cut off by it. And the center of the semicircle is the same as
that of the circle.
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- Definition
19.
- Rectilinear figures are those which are contained by straight
lines, trilateral figures being those contained by three, quadrilateral
those contained by four, and multilateral those contained by more
than four straight lines.
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- Definition
20.
- 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.
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- Definition
21.
- Further, of trilateral figures, a right-angled triangle is that
which has a right angle, an obtuse-angled triangle that which has an
obtuse angle, and an acute-angled triangle that which has its three
angles acute.
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- Definition
22.
- 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 is neither equilateral nor right-angled. And
let quadrilaterals other than these be called trapezia.
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- Definition
23
- 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:
- Postulate
1.
- To draw a straight line from any point to any point.
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- Postulate
2.
- To produce a finite straight line continuously in a straight line.
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- Postulate
3.
- To describe a circle with any center and radius.
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- Postulate
4.
- That all right angles equal one another.
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- Postulate
5.
- That, if a straight line falling on two straight lines makes 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 angles.
- Common
notion 1.
- Things which equal the same thing also equal one another.
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- Common
notion 2.
- If equals are added to equals, then the wholes are equal.
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- Common
notion 3.
- If equals are subtracted from equals, then the remainders are equal.
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- Common
notion 4.
- Things which coincide with one another equal one another.
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- Common
notion 5.
- The whole is greater than the part.
- Proposition
1.
- To construct an equilateral triangle on a given finite straight line.
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- Proposition
2.
- To place a straight line equal to a given straight line with one end at a
given point.
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- Proposition
3.
- To cut off from the greater of two given unequal straight lines a straight
line equal to the less.
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- Proposition
4.
- If two triangles have two sides equal to two sides respectively, and have
the angles contained by the equal straight lines equal, then they also have
the base equal to the base, the triangle equals the triangle, and the
remaining angles equal the remaining angles respectively, namely those
opposite the equal sides.
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- Proposition
5.
- In isosceles triangles the angles at the base equal one another, and, if
the equal straight lines are produced further, then the angles under the
base equal one another.
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- Proposition
6.
- If in a triangle two angles equal one another, then the sides opposite the
equal angles also equal one another.
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- Proposition
7.
- Given two straight lines constructed from the ends of a straight line and
meeting in a point, there cannot be constructed from the ends of the same
straight line, and on the same side of it, two other straight lines meeting
in another point and equal to the former two respectively, namely each equal
to that from the same end.
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- Proposition
8.
- If two triangles have the two sides equal to two sides respectively, and
also have the base equal to the base, then they also have the angles equal
which are contained by the equal straight lines.
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- Proposition
9.
- To bisect a given rectilinear angle.
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- Proposition
10.
- To bisect a given finite straight line.
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- Proposition
11.
- To draw a straight line at right angles to a given straight line from a
given point on it.
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- Proposition
12.
- To draw a straight line perpendicular to a given infinite straight line
from a given point not on it.
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- Proposition
13.
- If a straight line stands on a straight line, then it makes either two
right angles or angles whose sum equals two right angles.
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- Proposition
14.
- If with any straight line, and at a point on it, two straight lines not
lying on the same side make the sum of the adjacent angles equal to two
right angles, then the two straight lines are in a straight line with one
another.
-
- Proposition
15.
- If two straight lines cut one another, then they make the vertical angles
equal to one another.
Corollary.
If two straight lines cut one another, then they will make the angles at the
point of section equal to four right angles.
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- Proposition
16.
- In any triangle, if one of the sides is produced, then the exterior angle
is greater than either of the interior and opposite angles.
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- Proposition
17.
- In any triangle the sum of any two angles is less than two right angles.
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- Proposition
18.
- In any triangle the angle opposite the greater side is greater.
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- Proposition
19.
- In any triangle the side opposite the greater angle is greater.
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- Proposition
20.
- In any triangle the sum of any two sides is greater than the remaining
one.
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- Proposition
21.
- If from the ends of one of the sides of a triangle two straight lines are
constructed meeting within the triangle, then the sum of the straight lines
so constructed is less than the sum of the remaining two sides of the
triangle, but the constructed straight lines contain a greater angle than
the angle contained by the remaining two sides.
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- Proposition
22.
- To construct a triangle out of three straight lines which equal three
given straight lines: thus it is necessary that the sum of any two of the
straight lines should be greater than the remaining one.
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- Proposition
23.
- To construct a rectilinear angle equal to a given rectilinear angle on a
given straight line and at a point on it.
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- Proposition
24.
- If two triangles have two sides equal to two sides respectively, but have
one of the angles contained by the equal straight lines greater than the
other, then they also have the base greater than the base.
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- Proposition
25.
- If two triangles have two sides equal to two sides respectively, but have
the base greater than the base, then they also have the one of the angles
contained by the equal straight lines greater than the other.
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- Proposition
26.
- If two triangles have two angles equal to two angles respectively, and one
side equal to one side, namely, either the side adjoining the equal angles,
or that opposite one of the equal angles, then the remaining sides equal the
remaining sides and the remaining angle equals the remaining angle.
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- Proposition
27.
- If a straight line falling on two straight lines makes the alternate
angles equal to one another, then the straight lines are parallel to one
another.
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- Proposition
28.
- If a straight line falling on two straight lines makes the exterior angle
equal to the interior and opposite angle on the same side, or the sum of the
interior angles on the same side equal to two right angles, then the
straight lines are parallel to one another.
-
- Proposition
29.
- A straight line falling on parallel straight lines makes the alternate
angles equal to one another, the exterior angle equal to the interior and
opposite angle, and the sum of the interior angles on the same side equal to
two right angles.
-
- Proposition
30.
- Straight lines parallel to the same straight line are also parallel to one
another.
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- Proposition
31.
- To draw a straight line through a given point parallel to a given
straight line.
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- Proposition
32.
- In any triangle, if one of the sides is produced, then the exterior angle
equals the sum of the two interior and opposite angles, and the sum of the
three interior angles of the triangle equals two right angles.
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- Proposition
33.
- Straight lines which join the ends of equal and parallel straight lines in
the same directions are themselves equal and parallel.
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- Proposition
34.
- In parallelogrammic areas the opposite sides and angles equal one another,
and the diameter bisects the areas.
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- Proposition
35.
- Parallelograms which are on the same base and in the same parallels equal
one another.
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- Proposition
36.
- Parallelograms which are on equal bases and in the same parallels equal
one another.
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- Proposition
37.
- Triangles which are on the same base and in the same parallels equal one
another.
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- Proposition
38.
- Triangles which are on equal bases and in the same parallels equal one
another.
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- Proposition
39.
- Equal triangles which are on the same base and on the same side are also
in the same parallels.
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- Proposition
40.
- Equal triangles which are on equal bases and on the same side are also in
the same parallels.
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- Proposition
41.
- If a parallelogram has the same base with a triangle and is in the same
parallels, then the parallelogram is double the triangle.
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- Proposition
42.
- To construct a parallelogram equal to a given triangle in a given
rectilinear angle.
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- Proposition
43.
- In any parallelogram the complements of the parallelograms about the
diameter equal one another.
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- Proposition
44.
- To a given straight line in a given rectilinear angle, to apply a
parallelogram equal to a given triangle.
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- Proposition
45.
- To construct a parallelogram equal to a given rectilinear figure in a
given rectilinear angle.
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- Proposition
46.
- To describe a square on a given straight line.
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- Proposition
47.
- In right-angled triangles the square on the side opposite the right angle
equals the sum of the squares on the sides containing the right angle.
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- Proposition
48.
- If in a triangle the square on one of the sides equals the sum of the
squares on the remaining two sides of the triangle, then the angle contained
by the remaining two sides of the triangle is right.