Abstract
We publish a complete, machine-readable rendering of Euclid's Elements as an rrxiv paper. All thirteen books are encoded: every definition, postulate, common notion, and proposition is registered as an addressable rrxiv claim, and every proof is encoded as a sequence of explicit depends_on edges to earlier claims. The encoding produces 465 propositions, 109 definitions, 5 postulates, and 5 common notions, connected by over a thousand depends_on edges --- the full reasoning DAG of the Elements is queryable through the rrxiv API. The encoding serves three purposes: (i) it dogfoods the rrxiv schema on a finite, dependency-rich corpus that has been studied for two thousand years; (ii) it provides a working reproducibility demonstration --- every proposition is provable from claims that the rrxiv graph can enumerate, terminating in the five postulates and five common notions; and (iii) it gives agent harnesses a canonical proof corpus to retrieve over. Books I, II, and III are written in full Heath-density prose with TikZ figures for the canonical constructions (I.1, I.5, I.32, I.47, II.4, II.11, II.14, III.20, III.31, III.36). Books IV through XIII carry the full statement + dependency-edge DAG with condensed proof sketches; rendering them at Heath density is a long-running editorial project, and PRs at https://github.com/random-walks/rrxiv-paper-euclid-elements are welcome. The translation follows Heath (1908, public domain) with light modernisation; the rrxiv encoding is released under CC-BY-4.0.
Claims (465)
Each registered assertion in this paper is addressable as a claim node, with its own replication and contradiction record.
I.1
On a given finite straight line to construct an equilateral triangle.
Replicated
I.2
To place at a given point (as an extremity) a straight line equal to a
given straight line.
Replicated
I.3
Given two unequal straight lines, to cut off from the greater a
straight line equal to the less.
Replicated
I.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
will also have the base equal to the base, the triangle will be equal
to the triangle, and the remaining angles will be equal to the
remaining angles respectively.
Replicated
I.5
In isosceles triangles the angles at the base are equal to one
another; and if the equal straight lines be produced further, the
angles under the base will be equal to one another.
Replicated
I.6
If in a triangle two angles are equal to one another, the sides which
subtend the equal angles will also be equal to one another.
Replicated
I.7
Given two straight lines constructed on a straight line and meeting in
a point, there cannot be constructed on 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 to that which
has the same extremity with it.
Replicated
I.8
If two triangles have the two sides equal to two sides respectively,
and have also the base equal to the base, they will also have the
angles equal which are contained by the equal straight lines.
Replicated
I.9
To bisect a given rectilineal angle.
Replicated
I.10
To bisect a given finite straight line.
Replicated
I.11
To draw a straight line at right angles to a given straight line from
a given point on it.
Replicated
I.12
To draw a perpendicular straight line to a given infinite straight
line from a given point not on it.
Replicated
I.13
If a straight line set up on a straight line make angles, it will
make either two right angles or angles equal to two right angles.
Replicated
I.14
If with any straight line, and at a point on it, two straight lines
not lying on the same side make the adjacent angles equal to two
right angles, the two straight lines will be in a straight line with
one another.
Replicated
I.15
If two straight lines cut one another, they make the vertical angles
equal to one another.
Replicated
I.16
In any triangle, if one of the sides be produced, the exterior angle
is greater than either of the interior and opposite angles.
Replicated
I.17
[Proposition I.17: Sum of any two angles $<$ two right angles]
In any triangle two angles taken together in any manner are less than
two right angles.
Replicated
I.18
In any triangle the greater side subtends the greater angle.
Replicated
I.19
In any triangle the greater angle is subtended by the greater side.
Replicated
I.20
In any triangle two sides taken together in any manner are greater
than the remaining one.
Replicated
I.21
If on one of the sides of a triangle, from its extremities, there be
constructed two straight lines meeting within the triangle, the
straight lines so constructed will be less than the remaining two
sides of the triangle, but will contain a greater angle.
Replicated
I.22
Out of three straight lines, which are equal to three given straight
lines, to construct a triangle: thus it is necessary that two of the
straight lines taken together in any manner should be greater than the
remaining one.
Replicated
I.23
On a given straight line and at a point on it to construct a
rectilineal angle equal to a given rectilineal angle.
Replicated
I.24
If two triangles have the two sides equal to two sides respectively,
but have the one of the angles contained by the equal straight lines
greater than the other, they will also have the base greater than the
base.
Replicated
I.25
If two triangles have the two sides equal to two sides respectively,
but have the one base greater than the other, they will also have the
one angle contained by the equal straight lines greater than the
other.
Replicated
I.26
If two triangles have the two angles equal to two angles respectively,
and one side equal to one side, namely either the side adjoining the
equal angles or that subtending one of the equal angles, they will
also have the remaining sides equal to the remaining sides and the
remaining angle equal to the remaining angle.
Replicated
I.27
If a straight line falling on two straight lines make the alternate
angles equal to one another, the straight lines will be parallel to
one another.
Replicated
I.28
If a straight line falling on two straight lines make the exterior
angle equal to the interior and opposite angle on the same side, or
the interior angles on the same side equal to two right angles, the
straight lines will be parallel to one another.
Replicated
I.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 interior angles on the same side equal to
two right angles.
Replicated
I.30
Straight lines parallel to the same straight line are also parallel
to one another.
Replicated
I.31
Through a given point to draw a straight line parallel to a given
straight line.
Replicated
I.32
In any triangle, if one of the sides be produced, the exterior angle
is equal to the two interior and opposite angles, and the three
interior angles of the triangle are equal to two right angles.
Replicated
I.33
The straight lines joining equal and parallel straight lines (at the
extremities which are in the same directions) are themselves equal
and parallel.
Replicated
I.34
In parallelogrammic areas the opposite sides and angles are equal to
one another, and the diameter bisects the areas.
Replicated
I.35
[Proposition I.35: Parallelograms with same base & between same parallels]
Parallelograms which are on the same base and in the same parallels
are equal to one another.
Replicated
I.36
Parallelograms which are on equal bases and in the same parallels are
equal to one another.
Replicated
I.37
[Proposition I.37: Triangles with same base & parallels]
Triangles which are on the same base and in the same parallels are
equal to one another.
Replicated
I.38
Triangles which are on equal bases and in the same parallels are
equal to one another.
Replicated
I.39
[Proposition I.39: Equal triangles on same base $\Rightarrow$ same parallels]
Equal triangles which are on the same base and on the same side are
also in the same parallels.
Replicated
I.40
Equal triangles which are on equal bases and on the same side are
also in the same parallels.
Replicated
I.41
If a parallelogram have the same base with a triangle and be in the
same parallels, the parallelogram is double of the triangle.
Replicated
I.42
To construct, in a given rectilineal angle, a parallelogram equal to
a given triangle.
Replicated
I.43
In any parallelogram the complements of the parallelograms about the
diameter are equal to one another.
Replicated
I.44
To a given straight line to apply, in a given rectilineal angle, a
parallelogram equal to a given triangle.
Replicated
I.45
To construct, in a given rectilineal angle, a parallelogram equal to
a given rectilineal figure.
Replicated
I.46
On a given straight line to describe a square.
Replicated
I.47
In right-angled triangles the square on the side subtending the right
angle is equal to the squares on the sides containing the right angle.
Replicated
I.48
If in a triangle the square on one of the sides be equal to the
squares on the remaining two sides of the triangle, the angle
contained by the remaining two sides of the triangle is right.
Replicated
II.1
If there be two straight lines, and one of them be cut into any
number of segments whatever, the rectangle contained by the two
straight lines is equal to the rectangles contained by the
uncut straight line and each of the segments.
Replicated
II.2
If a straight line be cut at random, the rectangle contained by the
whole and both of the segments is equal to the square on the whole.
Replicated
II.3
If a straight line be cut at random, the rectangle contained by the
whole and one of the segments is equal to the rectangle contained by
the segments and the square on the aforesaid segment.
Replicated
II.4
If a straight line be cut at random, the square on the whole is equal
to the squares on the segments and twice the rectangle contained by
the segments.
Replicated
II.5
If a straight line be cut into equal and unequal segments, the
rectangle contained by the unequal segments of the whole together with
the square on the straight line between the points of section is equal
to the square on the half.
Replicated
II.6
If a straight line be bisected and a straight line be added to it in
a straight line, the rectangle contained by the whole with the added
straight line and the added straight line, together with the square
on the half, is equal to the square on the straight line made up of
the half and the added straight line.
Replicated
II.7
If a straight line be cut at random, the square on the whole and that
on one of the segments both together are equal to twice the rectangle
contained by the whole and the said segment together with the square
on the remaining segment.
Replicated
II.8
If a straight line be cut at random, four times the rectangle
contained by the whole and one of the segments together with the
square on the remaining segment is equal to the square described on
the whole and the aforesaid segment as on one straight line.
Replicated
II.9
If a straight line be cut into equal and unequal segments, the
squares on the unequal segments of the whole are double of the square
on the half and of the square on the straight line between the points
of section.
Replicated
II.10
If a straight line be bisected and a straight line be added to it in
a straight line, the square on the whole with the added straight line
and the square on the added straight line both together are double of
the square on the half and of the square described on the straight
line made up of the half and the added straight line as on one
straight line.
Replicated
II.11
To cut a given straight line so that the rectangle contained by the
whole and one of the segments is equal to the square on the remaining
segment.
Replicated
II.12
In obtuse-angled triangles the square on the side subtending the
obtuse angle is greater than the squares on the sides containing the
obtuse angle by twice the rectangle contained by one of the sides
about the obtuse angle (namely that on which the perpendicular falls)
and the straight line cut off outside by the perpendicular.
Replicated
II.13
In acute-angled triangles the square on the side subtending the acute
angle is less than the squares on the sides containing the acute
angle by twice the rectangle contained by one of the sides about the
acute angle (namely that on which the perpendicular falls) and the
straight line cut off within by the perpendicular.
Replicated
II.14
To construct a square equal to a given rectilineal figure.
Replicated
III.1
To find the centre of a given circle.
Replicated
III.2
If on the circumference of a circle two points be taken at random,
the straight line joining the points will fall within the circle.
Replicated
III.3
If in a circle a straight line through the centre bisect a straight
line not through the centre, it also cuts it at right angles; and if
it cut it at right angles, it also bisects it.
Replicated
III.4
If in a circle two straight lines cut one another which are not
through the centre, they do not bisect one another.
Replicated
III.5
If two circles cut one another, they will not have the same centre.
Replicated
III.6
If two circles touch one another, they will not have the same centre.
Replicated
III.7
If on the diameter of a circle a point be taken which is not the
centre, and from the point straight lines fall upon the circle: that
will be greatest on which the centre is, the remainder of the same
diameter will be least, and of the rest the nearer to the diameter
through the centre is always greater than the more remote.
Replicated
III.8
If a point be taken outside a circle and from the point straight
lines be drawn through to the circle, one of which is through the
centre and the others fall on the circle: of the lines falling on
the concave circumference, that through the centre is greatest, and
the nearer to it always greater than the more remote; and of those
falling on the convex circumference, that between the point and the
diameter is least, and the nearer to it always less than the more
remote.
Replicated
III.9
If a point be taken within a circle, and more than two equal
straight lines fall from the point on the circle, the point taken is
the centre of the circle.
Replicated
III.10
A circle does not cut a circle at more points than two.
Replicated
III.11
If two circles touch one another internally, and their centres be
taken, the straight line joining their centres, if produced, will
fall on the point of contact of the circles.
Replicated
III.12
If two circles touch one another externally, the straight line
joining their centres will pass through the point of contact.
Replicated
III.13
A circle does not touch a circle at more points than one, whether it
touch it internally or externally.
Replicated
III.14
In a circle equal straight lines are equally distant from the centre,
and those which are equally distant from the centre are equal to one
another.
Replicated
III.15
Of straight lines in a circle the diameter is greatest, and of the
rest the nearer to the centre is always greater than the more remote.
Replicated
III.16
The straight line drawn at right angles to the diameter of a circle
from its extremity will fall outside the circle, and into the space
between the straight line and the circumference another straight
line cannot be interposed.
Replicated
III.17
From a given point to draw a straight line touching a given circle.
Replicated
III.18
If a straight line touch a circle, and a straight line be joined
from the centre to the point of contact, the straight line so joined
will be perpendicular to the tangent.
Replicated
III.19
If a straight line touch a circle, and from the point of contact a
straight line be drawn at right angles to the tangent, the centre of
the circle will be on the straight line so drawn.
Replicated
III.20
In a circle the angle at the centre is double of the angle at the
circumference, when the angles have the same circumference as base.
Replicated
III.21
In a circle the angles in the same segment are equal to one another.
Replicated
III.22
The opposite angles of quadrilaterals in circles are equal to two
right angles.
Replicated
III.23
On the same straight line there cannot be constructed two similar
and unequal segments of circles on the same side.
Replicated
III.24
Similar segments of circles on equal straight lines are equal to one
another.
Replicated
III.25
Given a segment of a circle, to describe the complete circle of
which it is a segment.
Replicated
III.26
In equal circles equal angles stand on equal circumferences, whether
they stand at the centres or at the circumferences.
Replicated
III.27
In equal circles angles standing on equal circumferences are equal to
one another, whether they stand at the centres or at the
circumferences.
Replicated
III.28
In equal circles equal straight lines cut off equal circumferences,
the greater equal to the greater and the less to the less.
Replicated
III.29
In equal circles equal circumferences are subtended by equal straight
lines.
Replicated
III.30
To bisect a given arc.
Replicated
III.31
[Proposition III.31: Thales --- angle in a semicircle is right]
In a circle the angle in the semicircle is right, that in a greater
segment less than a right angle, and that in a less segment greater
than a right angle.
Replicated
III.32
If a straight line touch a circle, and from the point of contact
there be drawn across, in the circle, a straight line cutting the
circle, the angles which it makes with the tangent will be equal to
the angles in the alternate segments of the circle.
Replicated
III.33
On a given straight line to describe a segment of a circle admitting
an angle equal to a given rectilineal angle.
Replicated
III.34
From a given circle to cut off a segment admitting an angle equal to
a given rectilineal angle.
Replicated
III.35
[Proposition III.35: Power of a point --- intersecting chords]
If in a circle two straight lines cut one another, the rectangle
contained by the segments of the one is equal to the rectangle
contained by the segments of the other.
Replicated
III.36
[Proposition III.36: Power of a point --- secant and tangent]
If a point be taken outside a circle and from it there fall on the
circle two straight lines, and if one of them cut the circle and the
other touch it, the rectangle contained by the whole of the straight
line which cuts the circle and the straight line intercepted on it
outside between the point and the convex circumference will be equal
to the square on the tangent.
Replicated
III.37
[Proposition III.37: Converse of III.36 --- the tangent test]
If a point be taken outside a circle and from the point there fall
on the circle two straight lines, if one of them cut the circle, and
the other fall on it, and if further the rectangle contained by the
whole of the straight line which cuts the circle and the straight
line intercepted on it outside between the point and the convex
circumference be equal to the square on the straight line which
falls on the circle, the straight line which falls on it will touch
the circle.
Replicated
IV.1
Into a given circle to fit a straight line equal to a given straight
line which is not greater than the diameter of the circle.
Replicated
IV.2
In a given circle to inscribe a triangle equiangular with a given
triangle.
Replicated
IV.3
About a given circle to circumscribe a triangle equiangular with a
given triangle.
Replicated
IV.4
In a given triangle to inscribe a circle.
Replicated
IV.5
About a given triangle to circumscribe a circle.
Replicated
IV.6
In a given circle to inscribe a square.
Replicated
IV.7
About a given circle to circumscribe a square.
Replicated
IV.8
In a given square to inscribe a circle.
Replicated
IV.9
About a given square to circumscribe a circle.
Replicated
IV.10
[Proposition IV.10: Construct an isosceles triangle with 72--72--36 angles]
To construct an isosceles triangle having each of the angles at the
base double of the remaining one.
Replicated
IV.11
In a given circle to inscribe an equilateral and equiangular pentagon.
Replicated
IV.12
About a given circle to circumscribe an equilateral and equiangular
pentagon.
Replicated
IV.13
In a given pentagon, which is equilateral and equiangular, to inscribe
a circle.
Replicated
IV.14
About a given pentagon, which is equilateral and equiangular, to
circumscribe a circle.
Replicated
IV.15
In a given circle to inscribe an equilateral and equiangular hexagon.
Replicated
IV.16
In a given circle to inscribe a fifteen-angled figure which shall be
both equilateral and equiangular.
Replicated
V.1
If there be any number of magnitudes whatever which are, respectively,
equimultiples of any magnitudes equal in multitude, then, whatever
multiple one of the magnitudes is of one, that multiple also will all
be of all.
Replicated
V.2
If a first magnitude be the same multiple of a second that a third is
of a fourth, and a fifth also be the same multiple of the second that
a sixth is of the fourth, then the sum of the first and fifth will
also be the same multiple of the second that the sum of the third and
sixth is of the fourth.
Replicated
V.3
If a first magnitude be the same multiple of a second that a third is
of a fourth, and if equimultiples be taken of the first and third,
they will also be equimultiples respectively, the one of the second
and the other of the fourth.
Replicated
V.4
If a first magnitude have to a second the same ratio as a third to a
fourth, any equimultiples whatever of the first and third will also
have the same ratio to any equimultiples whatever of the second and
fourth respectively, taken in corresponding order.
Replicated
V.5
If a magnitude be the same multiple of a magnitude that a subtracted
part is of a subtracted part, the remainder also will be the same
multiple of the remainder that the whole is of the whole.
Replicated
V.6
If two magnitudes be equimultiples of two magnitudes, and any
magnitudes subtracted from them be equimultiples of the same, the
remainders also are either equal to the same or equimultiples of them.
Replicated
V.7
Equal magnitudes have to the same the same ratio, as also has the
same to equal magnitudes.
Replicated
V.8
Of unequal magnitudes the greater has to the same a greater ratio
than the less has, and the same has to the less a greater ratio than
it has to the greater.
Replicated
V.9
Magnitudes which have the same ratio to the same are equal to one
another; and magnitudes to which the same has the same ratio are
equal.
Replicated
V.10
Of magnitudes which have a ratio to the same, that which has a greater
ratio is greater; and that to which the same has a greater ratio is
less.
Replicated
V.11
Ratios which are the same with the same ratio are also the same with
one another.
Replicated
V.12
If any number of magnitudes be proportional, as one of the antecedents
is to one of the consequents, so will all the antecedents be to all
the consequents.
Replicated
V.13
If a first magnitude have to a second the same ratio as a third to a
fourth, and the third have to the fourth a greater ratio than a fifth
has to a sixth, the first will also have to the second a greater ratio
than the fifth to the sixth.
Replicated
V.14
If a first magnitude have to a second the same ratio as a third to a
fourth, and the first be greater than the third, the second will also
be greater than the fourth; and if equal, equal; and if less, less.
Replicated
V.15
Parts have the same ratio as the same multiples of them taken in
corresponding order.
Replicated
V.16
If four magnitudes be proportional, they will also be proportional
alternately.
Replicated
V.17
If magnitudes composed be proportional, they will also be proportional
separando.
Replicated
V.18
If magnitudes separated be proportional, they will also be
proportional componendo.
Replicated
V.19
If, as a whole is to a whole, so is a part subtracted to a part
subtracted, the remainder will also be to the remainder as whole to
whole.
Replicated
V.20
If there be three magnitudes, and others equal to them in multitude,
which taken two and two are in the same ratio, and if ex aequali the
first be greater than the third, the fourth will also be greater than
the sixth; and if equal, equal; and if less, less.
Replicated
V.21
If there be three magnitudes, and others equal to them in multitude,
which taken two and two together are in the same ratio, and the
proportion of them be perturbed, then if ex aequali the first be
greater than the third, the fourth will also be greater than the
sixth.
Replicated
V.22
If there be any number of magnitudes whatever, and others equal to
them in multitude, which taken two and two together are in the same
ratio, they will also be in the same ratio ex aequali.
Replicated
V.23
If there be any number of magnitudes whatever, and others equal to
them in multitude, which taken two and two together are in the same
ratio, and the proportion of them be perturbed, they will also be in
the same ratio ex aequali.
Replicated
V.24
If a first magnitude have to a second the same ratio as a third has
to a fourth, and also a fifth have to the second the same ratio as a
sixth to the fourth, the first and fifth added together will have to
the second the same ratio as the third and sixth have to the fourth.
Replicated
V.25
If four magnitudes be proportional, the greatest and least are greater
than the remaining two.
Replicated
VI.1
Triangles and parallelograms which are under the same height are to
one another as their bases.
Replicated
VI.2
If a straight line be drawn parallel to one of the sides of a
triangle, it will cut the sides of the triangle proportionally; and
if the sides of the triangle be cut proportionally, the line joining
the points of section will be parallel to the remaining side of the
triangle.
Replicated
VI.3
If an angle of a triangle be bisected and the straight line cutting
the angle cut the base also, the segments of the base will have the
same ratio as the remaining sides of the triangle.
Replicated
VI.4
In equiangular triangles the sides about the equal angles are
proportional, and those are corresponding sides which subtend the
equal angles.
Replicated
VI.5
If two triangles have their sides proportional, the triangles will be
equiangular and will have those angles equal which the corresponding
sides subtend.
Replicated
VI.6
If two triangles have one angle equal to one angle and the sides
about the equal angles proportional, the triangles will be
equiangular and will have those angles equal which the corresponding
sides subtend.
Replicated
VI.7
If two triangles have one angle equal to one angle, the sides about
other angles proportional, and the remaining angles either both less
or both not less than a right angle, the triangles will be
equiangular and will have those angles equal about which the sides
are proportional.
Replicated
VI.8
If in a right-angled triangle a perpendicular be drawn from the right
angle to the base, the triangles adjoining the perpendicular are
similar both to the whole and to one another.
Replicated
VI.9
From a given straight line to cut off a prescribed part.
Replicated
VI.10
To cut a given uncut straight line similarly to a given cut straight
line.
Replicated
VI.11
To two given straight lines to find a third proportional.
Replicated
VI.12
To three given straight lines to find a fourth proportional.
Replicated
VI.13
To two given straight lines to find a mean proportional.
Replicated
VI.14
In equal and equiangular parallelograms the sides about the equal
angles are reciprocally proportional; and equiangular parallelograms
in which the sides about the equal angles are reciprocally
proportional are equal.
Replicated
VI.15
In equal triangles which have one angle equal to one angle the sides
about the equal angles are reciprocally proportional; and those
triangles which have one angle equal to one angle, and in which the
sides about the equal angles are reciprocally proportional, are equal.
Replicated
VI.16
If four straight lines be proportional, the rectangle contained by the
extremes is equal to the rectangle contained by the means; and if the
rectangle contained by the extremes be equal to the rectangle
contained by the means, the four straight lines will be proportional.
Replicated
VI.17
If three straight lines be proportional, the rectangle contained by
the extremes is equal to the square on the mean; and if the rectangle
contained by the extremes be equal to the square on the mean, the
three straight lines will be proportional.
Replicated
VI.18
On a given straight line to describe a rectilineal figure similar
and similarly situated to a given rectilineal figure.
Replicated
VI.19
Similar triangles are to one another in the duplicate ratio of the
corresponding sides.
Replicated
VI.20
Similar polygons are divided into similar triangles, equal in
multitude and in the same ratio as the wholes; and the polygon has
to the polygon a ratio duplicate of that which the corresponding side
has to the corresponding side.
Replicated
VI.21
Figures which are similar to the same rectilineal figure are also
similar to one another.
Replicated
VI.22
If four straight lines be proportional, the rectilineal figures
similar and similarly described upon them will also be proportional;
and if the rectilineal figures similar and similarly described upon
them be proportional, the straight lines will themselves also be
proportional.
Replicated
VI.23
Equiangular parallelograms have to one another the ratio compounded
of the ratios of their sides.
Replicated
VI.24
In any parallelogram the parallelograms about the diameter are
similar both to the whole and to one another.
Replicated
VI.25
To construct one and the same figure similar to a given rectilineal
figure and equal to another given rectilineal figure.
Replicated
VI.26
If from a parallelogram there be taken away a parallelogram similar
and similarly situated to the whole and having a common angle with
it, it is about the same diameter with the whole.
Replicated
VI.27
Of all parallelograms applied to the same straight line and deficient
by parallelogrammic figures similar and similarly situated to that
described upon the half of the straight line, the greatest is that
which is applied to the half and is similar to the deficient figure.
Replicated
VI.28
To a given straight line to apply a parallelogram equal to a given
rectilineal figure and deficient by a parallelogrammic figure similar
to a given one; thus the given rectilineal figure must not be greater
than the parallelogram described on the half of the straight line and
similar to the defect.
Replicated
VI.29
To a given straight line to apply a parallelogram equal to a given
rectilineal figure and exceeding by a parallelogrammic figure similar
to a given one.
Replicated
VI.30
To cut a given finite straight line in extreme and mean ratio.
Replicated
VI.31
In right-angled triangles the figure on the side subtending the right
angle is equal to the similar and similarly described figures on the
sides containing the right angle.
Replicated
VI.32
If two triangles having two sides proportional to two sides be placed
together at one angle so that their corresponding sides are also
parallel, the remaining sides of the triangles will be in a straight
line.
Replicated
VI.33
In equal circles angles have the same ratio as the circumferences on
which they stand, whether they stand at the centres or at the
circumferences.
Replicated
VII.1
Two unequal numbers being set out, and the less being continually
subtracted in turn from the greater, if the number which is left
never measures the one before it until a unit is left, the original
numbers will be prime to one another.
Replicated
VII.2
Given two numbers not prime to one another, to find their greatest
common measure.
Replicated
VII.3
Given three numbers not prime to one another, to find their greatest
common measure.
Replicated
VII.4
Any number is either a part or parts of any number, the less of the
greater.
Replicated
VII.5
If a number be a part of a number, and another be the same part of
another, the sum will also be the same part of the sum that the one
is of the other.
Replicated
VII.6
If a number be parts of a number, and another be the same parts of
another, the sum will also be the same parts of the sum that the one
is of the other.
Replicated
VII.7
If a number be the same part of a number that a subtracted number is
of a subtracted number, the remainder will also be the same part of
the remainder that the whole is of the whole.
Replicated
VII.8
If a number be the same parts of a number that a subtracted number
is of a subtracted number, the remainder will also be the same parts
of the remainder that the whole is of the whole.
Replicated
VII.9
If a number be a part of a number, and another be the same part of
another, alternately, whatever part or parts the first is of the
third, the same part or the same parts will the second also be of
the fourth.
Replicated
VII.10
If a number be parts of a number, and another be the same parts of
another, alternately, whatever parts or part the first is of the
third, the same parts or the same part will the second also be of
the fourth.
Replicated
VII.11
If, as whole is to whole, so is a number subtracted to a number
subtracted, the remainder will also be to the remainder as whole is
to whole.
Replicated
VII.12
If there be as many numbers as we please in proportion, then, as one
of the antecedents is to one of the consequents, so will all the
antecedents be to all the consequents.
Replicated
VII.13
If four numbers be proportional, they will also be proportional
alternately.
Replicated
VII.14
If there be as many numbers as we please, and others equal to them
in multitude, which taken two and two are in the same ratio, they
will also be in the same ratio ex aequali.
Replicated
VII.15
If a unit measure any number, and another number measure any other
number the same number of times, then, alternately, the unit will
measure the third number the same number of times as the second
measures the fourth.
Replicated
VII.16
If two numbers by multiplying one another make certain numbers, the
numbers so produced will be equal to one another.
Replicated
VII.17
If a number by multiplying two numbers make certain numbers, the
numbers so produced will have the same ratio as the numbers
multiplied.
Replicated
VII.18
If two numbers by multiplying any number make certain numbers, the
numbers so produced will have the same ratio as the multipliers.
Replicated
VII.19
If four numbers be proportional, the number produced from the first
and fourth will be equal to the number produced from the second and
third; and if the number produced from the first and fourth be equal
to that produced from the second and third, the four numbers will be
proportional.
Replicated
VII.20
The least numbers of those which have the same ratio with them
measure those which have the same ratio the same number of times,
the greater the greater and the less the less.
Replicated
VII.21
Numbers prime to one another are the least of those which have the
same ratio with them.
Replicated
VII.22
The least numbers of those which have the same ratio with them are
prime to one another.
Replicated
VII.23
If two numbers be prime to one another, the number which measures
the one of them will be prime to the remaining number.
Replicated
VII.24
If two numbers be prime to any number, their product also will be
prime to the same.
Replicated
VII.25
If two numbers be prime to one another, the product of one of them
into itself will be prime to the remaining one.
Replicated
VII.26
If two numbers be prime to two numbers, both to each, their products
also will be prime to one another.
Replicated
VII.27
If two numbers be prime to one another, and each by multiplying
itself make a certain number, the products will be prime to one
another; and if the original numbers by multiplying the products
make certain numbers, these will be prime to one another.
Replicated
VII.28
If two numbers be prime to one another, the sum will also be prime
to each of them; and if the sum of two numbers be prime to either of
them, the original numbers will also be prime to one another.
Replicated
VII.29
Any prime number is prime to any number which it does not measure.
Replicated
VII.30
If two numbers by multiplying one another make some number, and any
prime number measure the product, it will also measure one of the
original numbers.
Replicated
VII.31
Any composite number is measured by some prime number.
Replicated
VII.32
Any number either is prime or is measured by some prime number.
Replicated
VII.33
Given as many numbers as we please, to find the least of those which
have the same ratio with them.
Replicated
VII.34
Given two numbers, to find the least number which they measure.
Replicated
VII.35
If two numbers measure any number, the least number measured by them
will also measure the same.
Replicated
VII.36
Given three numbers, to find the least number which they measure.
Replicated
VII.37
If a number be measured by any number, the number which is measured
will have a part called by the same name as the measuring number.
Replicated
VII.38
If a number have any part whatever, it will be measured by a number
called by the same name as the part.
Replicated
VII.39
To find the number which is the least that will have given parts.
Replicated
VIII.1
If there be as many numbers as we please in continued proportion, and
the extremes of them be prime to one another, the numbers are the
least of those which have the same ratio with them.
Replicated
VIII.2
To find numbers in continued proportion, as many as may be prescribed,
and the least that are in a given ratio.
Replicated
VIII.3
If as many numbers as we please in continued proportion be the least
of those which have the same ratio with them, the extremes of them
are prime to one another.
Replicated
VIII.4
Given as many ratios as we please in least numbers, to find numbers
in continued proportion which are the least in the given ratios.
Replicated
VIII.5
Plane numbers have to one another the ratio compounded of the ratios
of their sides.
Replicated
VIII.6
If there be as many numbers as we please in continued proportion, and
the first do not measure the second, neither will any other measure
any other.
Replicated
VIII.7
If there be as many numbers as we please in continued proportion, and
the first measure the last, it will measure the second also.
Replicated
VIII.8
If between two numbers there fall numbers in continued proportion
with them, then, however many numbers fall between them in continued
proportion, so many will also fall in continued proportion between
the numbers which have the same ratio with the original numbers.
Replicated
VIII.9
If two numbers be prime to one another, and numbers fall between them
in continued proportion, then, however many numbers fall between them
in continued proportion, so many will also fall in continued
proportion between each of them and a unit.
Replicated
VIII.10
If numbers fall between each of two numbers and a unit in continued
proportion, however many numbers fall between each of them and a
unit in continued proportion, so many also will fall between them in
continued proportion.
Replicated
VIII.11
Between two square numbers there is one mean proportional number, and
the square has to the square the ratio duplicate of that which the
side has to the side.
Replicated
VIII.12
Between two cube numbers there are two mean proportional numbers,
and the cube has to the cube the ratio triplicate of that which the
side has to the side.
Replicated
VIII.13
If there be as many numbers as we please in continued proportion, and
each by multiplying itself make some number, the products will be
proportional; and if the original numbers by multiplying the products
make certain numbers, the latter will also be proportional.
Replicated
VIII.14
If a square measure a square, the side will also measure the side;
and if the side measure the side, the square will also measure the
square.
Replicated
VIII.15
If a cube number measure a cube number, the side will also measure
the side; and if the side measure the side, the cube will also
measure the cube.
Replicated
VIII.16
If a square measure not a square, neither will the side measure the
side; and if the side measure not the side, neither will the square
measure the square.
Replicated
VIII.17
If a cube number measure not a cube number, neither will the side
measure the side; and if the side measure not the side, neither will
the cube measure the cube.
Replicated
VIII.18
Between two similar plane numbers there is one mean proportional
number, and the plane number has to the plane number the ratio
duplicate of that which the corresponding side has to the
corresponding side.
Replicated
VIII.19
Between two similar solid numbers there fall two mean proportional
numbers, and the solid number has to the solid number the ratio
triplicate of that which the corresponding side has to the
corresponding side.
Replicated
VIII.20
If one mean proportional number fall between two numbers, the numbers
will be similar plane numbers.
Replicated
VIII.21
If two mean proportional numbers fall between two numbers, the
numbers are similar solid numbers.
Replicated
VIII.22
If three numbers be in continued proportion, and the first be square,
the third will also be square.
Replicated
VIII.23
If four numbers be in continued proportion, and the first be cube,
the fourth will also be cube.
Replicated
VIII.24
If two numbers have to one another the ratio which a square number
has to a square number, and the first be square, the second will
also be square.
Replicated
VIII.25
If two numbers have to one another the ratio which a cube number has
to a cube number, and the first be cube, the second will also be
cube.
Replicated
VIII.26
Similar plane numbers have to one another the ratio which a square
number has to a square number.
Replicated
VIII.27
Similar solid numbers have to one another the ratio which a cube
number has to a cube number.
Replicated
IX.1
If two similar plane numbers by multiplying one another make some
number, the product will be square.
Replicated
IX.2
If two numbers by multiplying one another make a square number, they
are similar plane numbers.
Replicated
IX.3
If a cube number by multiplying itself make some number, the product
will be cube.
Replicated
IX.4
If a cube number by multiplying a cube number make some number, the
product will be cube.
Replicated
IX.5
If a cube number by multiplying any number make a cube number, the
multiplied number will also be cube.
Replicated
IX.6
If a number by multiplying itself make a cube number, it itself will
also be cube.
Replicated
IX.7
If a composite number by multiplying any number make some number,
the product will be solid.
Replicated
IX.8
If as many numbers as we please beginning from a unit be in continued
proportion, the third from the unit will be square, the fourth a
cube, and so on.
Replicated
IX.9
If as many numbers as we please beginning from a unit be in continued
proportion, and the number after the unit be square, all the rest
will also be square; and if the number after the unit be cube, all
the rest will also be cube.
Replicated
IX.10
If as many numbers as we please beginning from a unit be in continued
proportion, and the number after the unit be not square, neither will
any other be square except the third from the unit and all those
which leave out one.
Replicated
IX.11
If as many numbers as we please beginning from a unit be in continued
proportion, the less measures the greater according to some one of
the numbers which have place among the proportional numbers.
Replicated
IX.12
If as many numbers as we please beginning from a unit be in continued
proportion, by whatever prime numbers the last is measured, the
second from the unit will also be measured by the same.
Replicated
IX.13
If as many numbers as we please beginning from a unit be in continued
proportion, and the number after the unit be prime, the greatest will
not be measured by any except those which have a place among the
proportional numbers.
Replicated
IX.14
If a number be the least that is measured by prime numbers, it will
not be measured by any other prime number except those originally
measuring it.
Replicated
IX.15
If three numbers in continued proportion be the least of those which
have the same ratio with them, any two whatever added together will
be prime to the remaining number.
Replicated
IX.16
If two numbers be prime to one another, the second will not be to
any other number as the first is to the second.
Replicated
IX.17
If as many numbers as we please be in continued proportion, and the
extremes of them be prime to one another, the last will not be to
any other number as the first is to the second.
Replicated
IX.18
Given two numbers, to investigate whether it is possible to find a
third proportional to them.
Replicated
IX.19
Given three numbers, to investigate when it is possible to find a
fourth proportional to them.
Replicated
IX.20
Prime numbers are more than any assigned multitude of prime numbers.
Replicated
IX.21
If as many even numbers as we please be added together, the whole is
even.
Replicated
IX.22
If as many odd numbers as we please be added together, and their
multitude be even, the whole will be even.
Replicated
IX.23
If as many odd numbers as we please be added together, and their
multitude be odd, the whole will also be odd.
Replicated
IX.24
If from an even number an even number be subtracted, the remainder
will be even.
Replicated
IX.25
If from an even number an odd number be subtracted, the remainder
will be odd.
Replicated
IX.26
If from an odd number an odd number be subtracted, the remainder will
be even.
Replicated
IX.27
If from an odd number an even number be subtracted, the remainder
will be odd.
Replicated
IX.28
If an odd number by multiplying an even number make some number, the
product will be even.
Replicated
IX.29
If an odd number by multiplying an odd number make some number, the
product will be odd.
Replicated
IX.30
If an odd number measure an even number, it will also measure the
half of it.
Replicated
IX.31
If an odd number be prime to any number, it will also be prime to
the double of it.
Replicated
IX.32
Each of the numbers which are continually doubled beginning from a
duad is even-times even only.
Replicated
IX.33
If a number have its half odd, it is even-times odd only.
Replicated
IX.34
If an even number be neither one of those which are doubled from a
duad, nor have its half odd, it is both even-times even and
even-times odd.
Replicated
IX.35
If as many numbers as we please be in continued proportion, and there
be subtracted from the second and the last numbers equal to the
first, then, as the excess of the second is to the first, so will the
excess of the last be to all those before it.
Replicated
IX.36
If as many numbers as we please beginning from a unit be set out
continuously in double proportion, until the sum of all becomes
prime, and if the sum multiplied into the last make some number, the
product will be perfect.
Replicated
X.1
Two unequal magnitudes being set out, if from the greater there be
subtracted a magnitude greater than its half, and from that which is
left a magnitude greater than its half, and if this process be
repeated continually, there will be left some magnitude which will be
less than the lesser magnitude set out.
Replicated
X.2
If, when the lesser of two unequal magnitudes is continually
subtracted in turn from the greater, that which is left never
measures the one before it, the magnitudes will be incommensurable.
Replicated
X.3
Given two commensurable magnitudes, to find their greatest common
measure.
Replicated
X.4
Given three commensurable magnitudes, to find their greatest common
measure.
Replicated
X.5
Commensurable magnitudes have to one another the ratio which a number
has to a number.
Replicated
X.6
If two magnitudes have to one another the ratio which a number has
to a number, the magnitudes will be commensurable.
Replicated
X.7
Incommensurable magnitudes have not to one another the ratio which
a number has to a number.
Replicated
X.8
If two magnitudes have not to one another the ratio which a number
has to a number, the magnitudes will be incommensurable.
Replicated
X.9
The squares on straight lines commensurable in length have to one
another the ratio which a square number has to a square number; and
squares which have to one another the ratio which a square number
has to a square number will also have their sides commensurable in
length.
Replicated
X.10
To find two straight lines incommensurable, the one in length only,
the other in square also, with an assigned straight line.
Replicated
X.11
If four magnitudes be proportional, and the first be commensurable
with the second, the third also will be commensurable with the
fourth; and if the first be incommensurable with the second, the
third also will be incommensurable with the fourth.
Replicated
X.12
Magnitudes commensurable with the same magnitude are commensurable
with one another.
Replicated
X.13
If two magnitudes be commensurable, and one of them be incommensurable
with any magnitude, the remaining one will also be incommensurable
with the same.
Replicated
X.14
If four straight lines be proportional, and the square on the first
be greater than the square on the second by the square on a straight
line commensurable with the first, the square on the third will also
be greater than the square on the fourth by the square on a straight
line commensurable with the third.
Replicated
X.15
If two commensurable magnitudes be added together, the whole will
also be commensurable with each of them; and if the whole be
commensurable with one of them, the original magnitudes will also be
commensurable.
Replicated
X.16
If two incommensurable magnitudes be added together, the whole will
also be incommensurable with each of them; and if the whole be
incommensurable with one of them, the original magnitudes will also
be incommensurable.
Replicated
X.17
If there be two unequal straight lines, and to the greater there be
applied a parallelogram equal to the fourth part of the square on the
less and deficient by a square figure, and if it divide it into parts
which are commensurable in length, then the square on the greater
will be greater than the square on the less by the square on a
straight line commensurable in length with the greater.
Replicated
X.18
If there be two unequal straight lines, and to the greater there be
applied a parallelogram equal to the fourth part of the square on the
less and deficient by a square figure, and if it divide it into parts
incommensurable in length, then the square on the greater will be
greater than the square on the less by the square on a straight line
incommensurable in length with the greater.
Replicated
X.19
The rectangle contained by rational straight lines commensurable in
length is rational.
Replicated
X.20
If a rational area be applied to a rational straight line, it produces
as breadth a straight line rational and commensurable in length with
the straight line to which it is applied.
Replicated
X.21
The rectangle contained by rational straight lines commensurable in
square only is irrational, and the side of the square equal to it is
irrational. Let the latter be called medial.
Replicated
X.22
The square on a medial straight line, if applied to a rational
straight line, produces as breadth a straight line rational and
incommensurable in length with that to which it is applied.
Replicated
X.23
A straight line commensurable with a medial straight line is medial.
Replicated
X.24
The rectangle contained by medial straight lines commensurable in
length is medial.
Replicated
X.25
The rectangle contained by medial straight lines commensurable in
square only is either rational or medial.
Replicated
X.26
A medial area does not exceed a medial area by a rational area.
Replicated
X.27
To find medial straight lines commensurable in square only which
contain a rational rectangle.
Replicated
X.28
To find medial straight lines commensurable in square only which
contain a medial rectangle.
Replicated
X.29
To find two rational straight lines commensurable in square only such
that the square on the greater is greater than the square on the
less by the square on a straight line commensurable in length with
the greater.
Replicated
X.30
To find two rational straight lines commensurable in square only such
that the square on the greater is greater than the square on the
less by the square on a straight line incommensurable in length with
the greater.
Replicated
X.31
To find two medial straight lines commensurable in square only,
containing a rational rectangle, such that the square on the greater
is greater than the square on the less by the square on a straight
line commensurable in length with the greater.
Replicated
X.32
To find two medial straight lines commensurable in square only,
containing a medial rectangle, such that the square on the greater
is greater than the square on the less by the square on a straight
line commensurable in length with the greater.
Replicated
X.33
To find two straight lines incommensurable in square which make the
sum of the squares on them rational but the rectangle contained by
them medial.
Replicated
X.34
To find two straight lines incommensurable in square which make the
sum of the squares on them medial but the rectangle contained by
them rational.
Replicated
X.35
To find two straight lines incommensurable in square which make the
sum of the squares on them medial and the rectangle contained by
them medial and moreover incommensurable with the sum of the squares
on them.
Replicated
X.36
If two rational straight lines commensurable in square only be added
together, the whole is irrational; and let it be called binomial.
Replicated
X.37
If two medial straight lines commensurable in square only and
containing a rational rectangle be added together, the whole is
irrational; and let it be called first bimedial.
Replicated
X.38
If two medial straight lines commensurable in square only and
containing a medial rectangle be added together, the whole is
irrational; and let it be called second bimedial.
Replicated
X.39
If two straight lines incommensurable in square which make the sum
of the squares on them rational, but the rectangle contained by them
medial, be added together, the whole straight line is irrational;
and let it be called major.
Replicated
X.40
If two straight lines incommensurable in square which make the sum
of the squares on them medial, but the rectangle contained by them
rational, be added together, the whole straight line is irrational;
and let it be called the side of a rational plus a medial area.
Replicated
X.41
If two straight lines incommensurable in square which make the sum
of the squares on them medial, and the rectangle contained by them
medial and also incommensurable with the sum of the squares on them,
be added together, the remaining straight line is irrational; and
let it be called the side of the sum of two medial areas.
Replicated
X.42
A binomial straight line is divided into its terms at one point only.
Replicated
X.43
A first bimedial straight line is divided at one and the same point
only.
Replicated
X.44
A second bimedial straight line is divided at one point only.
Replicated
X.45
A major straight line is divided at one and the same point only.
Replicated
X.46
The side of a rational plus a medial area is divided at one and the
same point only.
Replicated
X.47
The side of the sum of two medial areas is divided at one and the
same point only.
Replicated
X.48
To find the first binomial straight line.
Replicated
X.49
To find the second binomial straight line.
Replicated
X.50
To find the third binomial straight line.
Replicated
X.51
To find the fourth binomial straight line.
Replicated
X.52
To find the fifth binomial straight line.
Replicated
X.53
To find the sixth binomial straight line.
Replicated
X.54
If an area be contained by a rational straight line and the first
binomial, the side of the area is the irrational straight line which
is called binomial.
Replicated
X.55
If an area be contained by a rational straight line and the second
binomial, the side of the area is the irrational straight line which
is called first bimedial.
Replicated
X.56
If an area be contained by a rational straight line and the third
binomial, the side of the area is the irrational straight line which
is called second bimedial.
Replicated
X.57
If an area be contained by a rational straight line and the fourth
binomial, the side of the area is the irrational straight line which
is called major.
Replicated
X.58
If an area be contained by a rational straight line and the fifth
binomial, the side of the area is the irrational straight line which
is the side of a rational plus a medial area.
Replicated
X.59
If an area be contained by a rational straight line and the sixth
binomial, the side of the area is the irrational straight line which
is called the side of the sum of two medial areas.
Replicated
X.60
The square on the binomial straight line applied to a rational
straight line produces as breadth the first binomial.
Replicated
X.61
The square on the first bimedial straight line applied to a rational
straight line produces as breadth the second binomial.
Replicated
X.62
The square on the second bimedial straight line applied to a rational
straight line produces as breadth the third binomial.
Replicated
X.63
The square on the major straight line applied to a rational straight
line produces as breadth the fourth binomial.
Replicated
X.64
The square on the side of a rational plus a medial area applied to
a rational straight line produces as breadth the fifth binomial.
Replicated
X.65
The square on the side of the sum of two medial areas applied to a
rational straight line produces as breadth the sixth binomial.
Replicated
X.66
A straight line commensurable in length with a binomial straight
line is itself also binomial and the same in order.
Replicated
X.67
A straight line commensurable in length with a bimedial straight
line is itself bimedial and the same in order.
Replicated
X.68
A straight line commensurable with a major straight line is itself
major.
Replicated
X.69
A straight line commensurable with the side of a rational plus a
medial area is itself such a side.
Replicated
X.70
A straight line commensurable with the side of the sum of two medial
areas is itself such a side.
Replicated
X.71
If a rational and a medial area be added together, four irrational
straight lines arise, namely either a binomial, a first bimedial, a
major, or a side of a rational plus a medial area.
Replicated
X.72
If two medial areas incommensurable with one another be added
together, the remaining two irrational straight lines arise, namely
either a second bimedial or a side of the sum of two medial areas.
Replicated
X.73
If from a rational straight line there be subtracted a rational
straight line commensurable with the whole in square only, the
remainder is irrational; and let it be called apotome.
Replicated
X.74
If from a medial straight line there be subtracted a medial straight
line commensurable with the whole in square only, and containing
with the whole a rational rectangle, the remainder is irrational;
and let it be called first apotome of a medial.
Replicated
X.75
If from a medial straight line there be subtracted a medial straight
line commensurable with the whole in square only, and containing
with the whole a medial rectangle, the remainder is irrational; and
let it be called second apotome of a medial.
Replicated
X.76
If from a straight line there be subtracted a straight line
incommensurable in square with the whole, which with the whole makes
the squares on them added together rational, but the rectangle
contained by them medial, the remainder is irrational; and let it
be called minor.
Replicated
X.77
If from a straight line there be subtracted a straight line
incommensurable in square with the whole which with the whole makes
the sum of squares medial but twice the rectangle rational, the
remainder is irrational; let it be called that which produces with
a rational area a medial whole.
Replicated
X.78
If from a straight line there be subtracted a straight line
incommensurable in square with the whole which with the whole makes
both the sum of squares and twice the rectangle medial and the two
sums incommensurable with one another, the remainder is irrational;
let it be called that which produces with a medial area a medial
whole.
Replicated
X.79
Only one rational straight line can be annexed to an apotome which
is commensurable with the whole in square only.
Replicated
X.80
Only one medial straight line can be annexed to a first apotome of
a medial which is commensurable with the whole in square only and
forms with it a rational rectangle.
Replicated
X.81
Only one medial straight line can be annexed to a second apotome of
a medial which is commensurable with the whole in square only and
forms with it a medial rectangle.
Replicated
X.82
Only one straight line can be annexed to a minor.
Replicated
X.83
Only one straight line can be annexed to the line producing with a
rational area a medial whole.
Replicated
X.84
Only one straight line can be annexed to the line producing with a
medial area a medial whole.
Replicated
X.85
To find the first apotome.
Replicated
X.86
To find the second apotome.
Replicated
X.87
To find the third apotome.
Replicated
X.88
To find the fourth apotome.
Replicated
X.89
To find the fifth apotome.
Replicated
X.90
To find the sixth apotome.
Replicated
X.91
If an area be contained by a rational straight line and a first
apotome, the side of the area is an apotome.
Replicated
X.92
If an area be contained by a rational straight line and a second
apotome, the side of the area is a first apotome of a medial.
Replicated
X.93
If an area be contained by a rational straight line and a third
apotome, the side of the area is a second apotome of a medial.
Replicated
X.94
If an area be contained by a rational straight line and a fourth
apotome, the side of the area is a minor.
Replicated
X.95
If an area be contained by a rational straight line and a fifth
apotome, the side of the area is the line producing with a rational
area a medial whole.
Replicated
X.96
If an area be contained by a rational straight line and a sixth
apotome, the side of the area is the line producing with a medial
area a medial whole.
Replicated
X.97
The square on an apotome straight line applied to a rational
straight line produces as breadth a first apotome.
Replicated
X.98
The square on a first apotome of a medial straight line applied to
a rational straight line produces as breadth a second apotome.
Replicated
X.99
The square on a second apotome of a medial straight line applied to
a rational straight line produces as breadth a third apotome.
Replicated
X.100
The square on a minor applied to a rational straight line produces
as breadth a fourth apotome.
Replicated
X.101
The square on the line producing with a rational area a medial whole
applied to a rational straight line produces as breadth a fifth
apotome.
Replicated
X.102
The square on the line producing with a medial area a medial whole
applied to a rational straight line produces as breadth a sixth
apotome.
Replicated
X.103
A straight line commensurable in length with an apotome is itself an
apotome and the same in order.
Replicated
X.104
A straight line commensurable in length with an apotome of a medial
is itself such an apotome of the same order.
Replicated
X.105
A straight line commensurable with a minor is itself a minor.
Replicated
X.106
A straight line commensurable with the line producing with a rational
area a medial whole is itself such a line.
Replicated
X.107
A straight line commensurable with the line producing with a medial
area a medial whole is itself such a line.
Replicated
X.108
If from a rational area a medial area be subtracted, the side of the
remaining area arises as one of four irrationals: an apotome, a
first apotome of a medial, a minor, or the line producing with a
rational area a medial whole.
Replicated
X.109
If from a medial area a rational area be subtracted, two other
irrational straight lines arise, namely a first apotome of a medial
or the line producing with a rational area a medial whole.
Replicated
X.110
If from a medial area there be subtracted a medial area
incommensurable with the whole, the remaining two irrational straight
lines arise: a second apotome of a medial or the line producing with
a medial area a medial whole.
Replicated
X.111
The apotome is not the same as the binomial.
Replicated
X.112
The square on a rational straight line applied to the binomial
straight line produces as breadth an apotome the terms of which are
commensurable with the terms of the binomial and in the same ratio.
Replicated
X.113
The square on a rational straight line applied to an apotome produces
as breadth a binomial the terms of which are commensurable with the
terms of the apotome and in the same ratio.
Replicated
X.114
If an area be contained by an apotome and the binomial the terms of
which are commensurable with the terms of the apotome and in the
same ratio, the side of the area is rational.
Replicated
X.115
From a medial straight line there arise irrational straight lines
infinite in number, and none of them is the same with any preceding.
Replicated
XI.1
A part of a straight line cannot be in the plane of reference and a
part in a plane more elevated.
Replicated
XI.2
If two straight lines cut one another, they are in one plane, and
every triangle is in one plane.
Replicated
XI.3
If two planes cut one another, their common section is a straight
line.
Replicated
XI.4
If a straight line be set up at right angles to two straight lines
which cut one another, at their common point of section, it will
also be at right angles to the plane through them.
Replicated
XI.5
If a straight line be set up at right angles to three straight lines
which meet one another, at their common point of section, the three
straight lines are in one plane.
Replicated
XI.6
If two straight lines be at right angles to the same plane, the
straight lines will be parallel.
Replicated
XI.7
If two straight lines be parallel, and points be taken at random on
each of them, the straight line joining the points is in the same
plane with the parallels.
Replicated
XI.8
If two straight lines be parallel, and one of them be at right angles
to any plane, the remaining one will also be at right angles to the
same plane.
Replicated
XI.9
Straight lines which are parallel to the same straight line and are
not in the same plane with it are also parallel to one another.
Replicated
XI.10
If two straight lines meeting one another be parallel to two straight
lines meeting one another, not in the same plane, they will contain
equal angles.
Replicated
XI.11
From a given elevated point to draw a straight line perpendicular to
a given plane.
Replicated
XI.12
To set up a straight line at right angles to a given plane from a
given point in it.
Replicated
XI.13
From the same point two straight lines cannot be set up at right
angles to the same plane on the same side.
Replicated
XI.14
Planes to which the same straight line is at right angles will be
parallel.
Replicated
XI.15
If two straight lines meeting one another be parallel to two straight
lines meeting one another, not being in the same plane, the planes
through them are parallel.
Replicated
XI.16
If two parallel planes be cut by any plane, their common sections
are parallel.
Replicated
XI.17
If two straight lines be cut by parallel planes, they will be cut in
the same ratios.
Replicated
XI.18
If a straight line be at right angles to any plane, all the planes
through it will also be at right angles to the same plane.
Replicated
XI.19
If two planes which cut one another be at right angles to any plane,
their common section will also be at right angles to the same plane.
Replicated
XI.20
If a solid angle be contained by three plane angles, any two, taken
together in any manner, are greater than the remaining one.
Replicated
XI.21
Any solid angle is contained by plane angles less than four right
angles.
Replicated
XI.22
If there be three plane angles of which two, taken together in any
manner, are greater than the remaining one, and they are contained
by equal straight lines, it is possible to construct a triangle out
of the straight lines joining the extremities of the equal straight
lines.
Replicated
XI.23
To construct a solid angle out of three plane angles, two of which,
taken together in any manner, are greater than the remaining one;
thus the sum of the three angles must be less than four right angles.
Replicated
XI.24
If a solid be contained by parallel planes, the opposite planes in
it are equal and similar parallelograms.
Replicated
XI.25
If a parallelepipedal solid be cut by a plane parallel to opposite
planes, then, as the base is to the base, so will the solid be to
the solid.
Replicated
XI.26
At a given point on a given straight line to construct a solid angle
equal to a given solid angle contained by three plane angles.
Replicated
XI.27
On a given straight line to construct a parallelepipedal solid
similar and similarly situated to a given parallelepipedal solid.
Replicated
XI.28
If a parallelepipedal solid be cut by a plane through the diagonals
of the opposite planes, the solid will be bisected by the plane.
Replicated
XI.29
Parallelepipedal solids which are on the same base and of the same
height, and in which the extremities of the sides which stand up are
on the same straight lines, are equal to one another.
Replicated
XI.30
Parallelepipedal solids which are on the same base and of the same
height, and in which the extremities of the sides which stand up are
not on the same straight lines, are equal to one another.
Replicated
XI.31
Parallelepipedal solids which are on equal bases and of the same
height are equal to one another.
Replicated
XI.32
Parallelepipedal solids which are of the same height are to one
another as their bases.
Replicated
XI.33
Similar parallelepipedal solids are to one another in the triplicate
ratio of their corresponding sides.
Replicated
XI.34
In equal parallelepipedal solids the bases are reciprocally
proportional to the heights; and those parallelepipedal solids in
which the bases are reciprocally proportional to the heights are
equal.
Replicated
XI.35
If there be two equal plane angles, and on their vertices there be
set up elevated straight lines containing equal angles with the
original straight lines respectively, if on the elevated straight
lines points be taken at random and perpendiculars be drawn from them
to the planes in which the original angles are, and if from the
points so arising in the planes straight lines be joined to the
vertices of the original angles, they will contain, with the elevated
straight lines, equal angles.
Replicated
XI.36
If three straight lines be proportional, the parallelepipedal solid
formed out of the three is equal to the parallelepipedal solid on the
mean which is equilateral, but equiangular with the aforesaid solid.
Replicated
XI.37
If four straight lines be proportional, the similar and similarly
described parallelepipedal solids upon them will also be
proportional; and if the similar and similarly described
parallelepipedal solids upon them be proportional, the straight lines
will themselves also be proportional.
Replicated
XI.38
If the sides of the opposite planes of a cube be bisected, and planes
be carried through the points of section, the common section of the
planes and the diameter of the cube bisect one another.
Replicated
XI.39
If there be two prisms of equal height, and one have a parallelogram
as base and the other a triangle, and if the parallelogram be double
of the triangle, the prisms will be equal.
Replicated
XII.1
Similar polygons inscribed in circles are to one another as the
squares on the diameters.
Replicated
XII.2
Circles are to one another as the squares on the diameters.
Replicated
XII.3
Any pyramid which has a triangular base is divided into two pyramids
equal and similar to one another, similar to the whole, and having
triangular bases, and into two equal prisms; and the two prisms are
greater than the half of the whole pyramid.
Replicated
XII.4
If there be two pyramids of the same height which have triangular
bases, and each of them be divided into two pyramids equal to one
another and similar to the whole, and into two equal prisms, then,
as the base of the one pyramid is to the base of the other pyramid,
so will all the prisms in the one pyramid be to all the prisms in
the other pyramid.
Replicated
XII.5
Pyramids which are of the same height and have triangular bases are
to one another as their bases.
Replicated
XII.6
Pyramids which are of the same height and have polygonal bases are
to one another as the bases.
Replicated
XII.7
Any prism which has a triangular base is divided into three pyramids
equal to one another which have triangular bases.
Replicated
XII.8
Similar pyramids which have triangular bases are in the triplicate
ratio of their corresponding sides.
Replicated
XII.9
In equal pyramids which have triangular bases the bases are
reciprocally proportional to the heights; and those pyramids which
have triangular bases in which the bases are reciprocally
proportional to the heights are equal.
Replicated
XII.10
Any cone is a third part of the cylinder which has the same base
with it and equal height.
Replicated
XII.11
Cones and cylinders which are of the same height are to one another
as their bases.
Replicated
XII.12
Similar cones and cylinders are to one another in the triplicate
ratio of the diameters in their bases.
Replicated
XII.13
If a cylinder be cut by a plane which is parallel to its opposite
planes, then, as the cylinder is to the cylinder, so will the axis
be to the axis.
Replicated
XII.14
Cones and cylinders which are on equal bases are to one another as
their heights.
Replicated
XII.15
In equal cones and cylinders the bases are reciprocally proportional
to the heights; and those cones and cylinders in which the bases
are reciprocally proportional to the heights are equal.
Replicated
XII.16
Given two circles about the same centre, to inscribe in the greater
circle an equilateral polygon with an even number of sides which
does not touch the lesser circle.
Replicated
XII.17
Given two spheres about the same centre, to inscribe in the greater
sphere a polyhedral solid which does not touch the lesser sphere at
its surface.
Replicated
XII.18
Spheres are to one another in the triplicate ratio of their
respective diameters.
Replicated
XIII.1
If a straight line be cut in extreme and mean ratio, the square on
the greater segment added to the half of the whole is five times the
square on the half.
Replicated
XIII.2
If the square on a straight line be five times the square on a
segment of it, then, when the double of the said segment is cut in
extreme and mean ratio, the greater segment is the remaining part of
the original straight line.
Replicated
XIII.3
If a straight line be cut in extreme and mean ratio, the square on
the lesser segment added to the half of the greater segment is five
times the square on the half of the greater segment.
Replicated
XIII.4
If a straight line be cut in extreme and mean ratio, the square on
the whole and the square on the lesser segment together are triple
of the square on the greater segment.
Replicated
XIII.5
If a straight line be cut in extreme and mean ratio, and there be
added to it a straight line equal to the greater segment, the whole
straight line is cut in extreme and mean ratio, and the original
straight line is the greater segment.
Replicated
XIII.6
If a rational straight line be cut in extreme and mean ratio, each of
the segments is the irrational straight line called apotome.
Replicated
XIII.7
If three angles of an equilateral pentagon, taken either in order or
not in order, be equal, the pentagon will be equiangular.
Replicated
XIII.8
If in an equilateral and equiangular pentagon straight lines subtend
two adjacent angles, they cut one another in extreme and mean ratio,
and the greater segments are equal to the side of the pentagon.
Replicated
XIII.9
If the side of the hexagon and that of the decagon inscribed in the
same circle be added together, the whole straight line has been cut
in extreme and mean ratio, and its greater segment is the side of
the hexagon.
Replicated
XIII.10
If an equilateral pentagon be inscribed in a circle, the square on
the side of the pentagon is equal to the squares on the side of the
hexagon and on that of the decagon inscribed in the same circle.
Replicated
XIII.11
If in a circle which has its diameter rational an equilateral
pentagon be inscribed, the side of the pentagon is the irrational
straight line called minor.
Replicated
XIII.12
If an equilateral triangle be inscribed in a circle, the square on
the side of the triangle is triple of the square on the radius.
Replicated
XIII.13
To construct a pyramid (regular tetrahedron), to comprehend it in a
given sphere, and to prove that the square on the diameter of the
sphere is one and a half times the square on the side of the pyramid.
Replicated
XIII.14
To construct an octahedron and comprehend it in a sphere, as in the
preceding case; and to prove that the square on the diameter of the
sphere is double of the square on the side of the octahedron.
Replicated
XIII.15
To construct a cube and comprehend it in a sphere, as in the
preceding case; and to prove that the square on the diameter of the
sphere is triple of the square on the side of the cube.
Replicated
XIII.16
To construct an icosahedron and comprehend it in a sphere, as in the
case of the aforesaid figures; and to prove that the side of the
icosahedron is the irrational straight line called minor.
Replicated
XIII.17
To construct a dodecahedron and comprehend it in a sphere, like the
aforesaid figures; and to prove that the side of the dodecahedron is
the irrational straight line called apotome.
Replicated
XIII.18
To set out the sides of the five figures and to compare them with
one another; and that no other figure, besides the said five
figures, can be constructed which is contained by equilateral and
equiangular figures equal to one another.
Replicated
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BibTeXRISJSON
@article{260500009,
title = {Euclid's Elements, encoded as an rrxiv paper},
author = {Blaise Albis-Burdige and Claude},
rrxiv = {rrxiv:2605.00009},
year = {2026}
}