Euclid

Euclid’s

Elements

Euclid’s

Elements

Euclid’s

Elements

Euleidhou

STOIXEIWN

Euleidhou

STOIXEIWN

Euleidhou

STOIXEIWN

Thousands of years after its author died, here we are still marveling at this

text…

Take a moment to appreciate the nuances of this poem of a

mathematics book.

By hovering your mouse over a prop, you can see

both


both

- the props that went into its proof


both

- the props that went into its proofand

- the later props that rely on it.

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Happy

geometring!

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- to construct an equilateral triangle.

On a given finite straight line -

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- to place (as an extremity) a straight line equal to the given

straight line.

At a given point, with a given straight line -

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- to cut off from the greater a straight line equal to the less.

Given two unequal straight lines -

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- the triangle will be equal to the triangle;

- the remaining angles will be equal to the remaining angles,

respectively.

If two triangles each have two of their respective sides

and the contained angles equal to each other -

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- the angles at the base will be equal to each other;

- as will be the angles under the base.

In isosceles triangles -

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- the sides which subtend the equal angles will also be equal

to one another.

If in a triangle two angles be equal to one

another -

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- there cannot be constructed, on the same side of the line, two lines equal to the other

straight lines which meet at a different point.

If two straight lines (constructed at the extremities of a straight line) meet at a point -

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- the angles contained by those straight lines will also be equal.

If two triangles have the two sides and the

base equal, respectively -

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- to bisect it.

Given a rectilineal angle -

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- to bisect it.

Given a finite straight line -

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- to draw a straight line at right angles.

To a given straight line, and from a given point

on it -

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- to draw a perpendicular straight line.

To a given straight line, from a given point which is not on it -

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- it will make either two right angles, or angles equal to two

right angles.

If a straight line set up on a straight line makes angles -

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- the two straight lines will be in a straight line with each other.

If two straight lines, meeting another straight line at the

same point, make the adjacent angles equal to two

right angles -

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- they make the vertical angles equal to

one another.

If two straight lines cut one another -

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- the resulting exterior angle is greater than either of the interior, opposite angles.

If one of the sides of any triangle be

produced -

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- are less than two right angles.

Two angles of any triangle, when taken

together -

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- subtends the greater angle.

The greater side in any triangle -

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- is subtended by the greater side.

The greater angle in any triangle -

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- are greater than the remaining one.

Two sides of any triangle, when taken

together in any manner,

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- the straight lines will be less than the triangle’s remaining

two sides;- but they will contain a greater

angle.

If, from the extremities of one side of a triangle, two straight

lines meeting within the triangle be constructed -

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- to construct a triangle using three straight lines equal to

those given.

Given three straight lines (as long as two taken

together are greater than the remaining long) -

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- to construct at that point another, equal rectilineal

angle.

Given a rectilineal angle, as well as a given point on a straight line -

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- the triangle with the larger angle will also have a larger

base.

If two triangles have two of their sides respectively equal,

but one of the contained angles is larger than the other

-

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- the triangle with the larger base will also have a larger

angle.

If two triangles have two of their sides respectively

equal, but one of the bases is larger than the

other -

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- the remaining respective sides will be equal;

- as will be the remaining angle.

If two triangles have two of their angles respectively

equal, as well as any one of their respective sides equal

-

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- the straight lines will be parallel to

one another.

If a straight line falling on two straight lines make the alternate angles equal to one

another -

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- the straight lines will be parallel to

one another.

If a straight line falling on two straight lines make (a)the exterior

angle equal to the interior, opposite angle on the same side, or (b)the interior angles on the same side

equal to two right angles -

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- alternate angles are equal;- the exterior angle is equal to the interior,

opposite angle;- and interior angles on the same side are

equal to two right angles.

If a straight line falls on parallel straight

lines -

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- are also parallel to one another.

Straight lines parallel to the same straight

line -

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- to draw through the point a line

parallel to the one given.

Given a straight line and a point (not on the

line) -

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- the resulting exterior angle is equal to the two interior,

opposite angles;- and the triangle’s three interior

angles are equal to two right angles.

If one of the sides of any triangle be

produced -

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- are themselves equal and parallel as well.

Straight lines that join equal and parallel

straight lines, in the same respective

directions -

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- opposite sides are equal to one another;

- opposite angles are equal to one another;

- and the diameter bisects the areas.

In parallelogrammic areas -

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- are equal to each other.

Parallelograms which share a base and are in

the same parallels -

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- are equal to one another.

Parallelograms which are on equal (but not shared bases) and in the same parallels -

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Triangles which are share a base and are in

the same parallels -

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Triangles which are on equal (not shared)

bases and in the same parallels -

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- they are also in the same parallels.

If equal triangles share a base and are on the

same side -

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- they are also in the same parallels.

If equal triangles be on equal (not shared bases) and on the

same side -

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- the parallelogram is double of the triangle

If a parallelogram share a base with a

triangle and be in the same parallels -

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- to construct a parallelogram equal to the triangle.

From a given triangle, in a given rectilineal

angle -

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- the compliments about the diameter

are equal to one another.

In any parallelogram -

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- to apply a parallelogram equal to

the triangle.

From a given triangle, within a given

rectilineal angle, and to a given straight line

-

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- to construct in the angle a parallelogram equal to the

figure.

Given any rectilineal figure and a rectilineal

angle -

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- to describe a square.

On a given straight line -

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- the square on the side subtending

the right angle is equal to the squares on

the sides containing the right angle.

In right-angled triangles -

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- the angle contained by the remaining

two sides of the triangle is right.

If the square on one of a triangle’s sides be equal

to the squares on its remaining two sides -

Education

Euclid