rrxiv:2605.00009·v5·Submitted 2026-05-26

Euclid's Elements, encoded as an rrxiv paper

Euclid of Alexandria, Sir Thomas L. Heath, Blaise Albis-Burdige , Claude Opus 4.7agent

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Submitted 7 days ago

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 (600)

Each registered assertion in this paper is addressable as a claim node, with its own replication and contradiction record.

On a given finite straight line to construct an equilateral triangle.

To place at a given point (as an extremity) a straight line equal to a given straight line.

Given two unequal straight lines, to cut off from the greater a straight line equal to the less.

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.

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.

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.

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.

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.

To bisect a given rectilineal angle.

To bisect a given finite straight line.

To draw a straight line at right angles to a given straight line from a given point on it.

To draw a perpendicular straight line to a given infinite straight line from a given point not on it.

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.

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.

If two straight lines cut one another, they make the vertical angles equal to one another.

In any triangle, if one of the sides be produced, the exterior angle is greater than either of the interior and opposite angles.

[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.

In any triangle the greater side subtends the greater angle.

In any triangle the greater angle is subtended by the greater side.

In any triangle two sides taken together in any manner are greater than the remaining one.

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.

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.

On a given straight line and at a point on it to construct a rectilineal angle equal to a given rectilineal angle.

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.

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.

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.

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.

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.

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.

Straight lines parallel to the same straight line are also parallel to one another.

Through a given point to draw a straight line parallel to a given straight line.

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.

The straight lines joining equal and parallel straight lines (at the extremities which are in the same directions) are themselves equal and parallel.

In parallelogrammic areas the opposite sides and angles are equal to one another, and the diameter bisects the areas.

[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.

Parallelograms which are on equal bases and in the same parallels are equal to one another.

[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.

Triangles which are on equal bases and in the same parallels are equal to one another.

[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.

Equal triangles which are on equal bases and on the same side are also in the same parallels.

If a parallelogram have the same base with a triangle and be in the same parallels, the parallelogram is double of the triangle.

To construct, in a given rectilineal angle, a parallelogram equal to a given triangle.

In any parallelogram the complements of the parallelograms about the diameter are equal to one another.

To a given straight line to apply, in a given rectilineal angle, a parallelogram equal to a given triangle.

To construct, in a given rectilineal angle, a parallelogram equal to a given rectilineal figure.

On a given straight line to describe a square.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

To construct a square equal to a given rectilineal figure.

To find the centre of a given circle.

If on the circumference of a circle two points be taken at random, the straight line joining the points will fall within the circle.

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.

If in a circle two straight lines cut one another which are not through the centre, they do not bisect one another.

If two circles cut one another, they will not have the same centre.

If two circles touch one another, they will not have the same centre.

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.

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.

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.

A circle does not cut a circle at more points than two.

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.

If two circles touch one another externally, the straight line joining their centres will pass through the point of contact.

A circle does not touch a circle at more points than one, whether it touch it internally or externally.

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.

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.

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.

From a given point to draw a straight line touching a given circle.

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.

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.

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.

In a circle the angles in the same segment are equal to one another.

The opposite angles of quadrilaterals in circles are equal to two right angles.

On the same straight line there cannot be constructed two similar and unequal segments of circles on the same side.

Similar segments of circles on equal straight lines are equal to one another.

Given a segment of a circle, to describe the complete circle of which it is a segment.

In equal circles equal angles stand on equal circumferences, whether they stand at the centres or at the circumferences.

In equal circles angles standing on equal circumferences are equal to one another, whether they stand at the centres or at the circumferences.

In equal circles equal straight lines cut off equal circumferences, the greater equal to the greater and the less to the less.

In equal circles equal circumferences are subtended by equal straight lines.

To bisect a given arc.

[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.

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.

On a given straight line to describe a segment of a circle admitting an angle equal to a given rectilineal angle.

From a given circle to cut off a segment admitting an angle equal to a given rectilineal angle.

[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.

[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.

[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.

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.

In a given circle to inscribe a triangle equiangular with a given triangle.

About a given circle to circumscribe a triangle equiangular with a given triangle.

In a given triangle to inscribe a circle.

About a given triangle to circumscribe a circle.

In a given circle to inscribe a square.

About a given circle to circumscribe a square.

In a given square to inscribe a circle.

About a given square to circumscribe a circle.

[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.

In a given circle to inscribe an equilateral and equiangular pentagon.

About a given circle to circumscribe an equilateral and equiangular pentagon.

In a given pentagon, which is equilateral and equiangular, to inscribe a circle.

About a given pentagon, which is equilateral and equiangular, to circumscribe a circle.

In a given circle to inscribe an equilateral and equiangular hexagon.

In a given circle to inscribe a fifteen-angled figure which shall be both equilateral and equiangular.

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.

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.

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.

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.

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.

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.

Equal magnitudes have to the same the same ratio, as also has the same to equal magnitudes.

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.

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.

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.

Ratios which are the same with the same ratio are also the same with one another.

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.

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.

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.

Parts have the same ratio as the same multiples of them taken in corresponding order.

If four magnitudes be proportional, they will also be proportional alternately.

If magnitudes composed be proportional, they will also be proportional separando.

If magnitudes separated be proportional, they will also be proportional componendo.

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.

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.

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.

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.

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.

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.

If four magnitudes be proportional, the greatest and least are greater than the remaining two.

Triangles and parallelograms which are under the same height are to one another as their bases.

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.

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.

In equiangular triangles the sides about the equal angles are proportional, and those are corresponding sides which subtend the equal angles.

If two triangles have their sides proportional, the triangles will be equiangular and will have those angles equal which the corresponding sides subtend.

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.

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.

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.

From a given straight line to cut off a prescribed part.

To cut a given uncut straight line similarly to a given cut straight line.

To two given straight lines to find a third proportional.

To three given straight lines to find a fourth proportional.

To two given straight lines to find a mean proportional.

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.

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.

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.

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.

On a given straight line to describe a rectilineal figure similar and similarly situated to a given rectilineal figure.

Similar triangles are to one another in the duplicate ratio of the corresponding sides.

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.

Figures which are similar to the same rectilineal figure are also similar to one another.

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.

Equiangular parallelograms have to one another the ratio compounded of the ratios of their sides.

In any parallelogram the parallelograms about the diameter are similar both to the whole and to one another.

To construct one and the same figure similar to a given rectilineal figure and equal to another given rectilineal figure.

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.

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.

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.

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.

To cut a given finite straight line in extreme and mean ratio.

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.

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.

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.

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.

Given two numbers not prime to one another, to find their greatest common measure.

Given three numbers not prime to one another, to find their greatest common measure.

Any number is either a part or parts of any number, the less of the greater.

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.

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.

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.

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.

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.

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.

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.

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.

If four numbers be proportional, they will also be proportional alternately.

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.

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.

If two numbers by multiplying one another make certain numbers, the numbers so produced will be equal to one another.

If a number by multiplying two numbers make certain numbers, the numbers so produced will have the same ratio as the numbers multiplied.

If two numbers by multiplying any number make certain numbers, the numbers so produced will have the same ratio as the multipliers.

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.

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.

Numbers prime to one another are the least of those which have the same ratio with them.

The least numbers of those which have the same ratio with them are prime to one another.

If two numbers be prime to one another, the number which measures the one of them will be prime to the remaining number.

If two numbers be prime to any number, their product also will be prime to the same.

If two numbers be prime to one another, the product of one of them into itself will be prime to the remaining one.

If two numbers be prime to two numbers, both to each, their products also will be prime to one another.

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.

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.

Any prime number is prime to any number which it does not measure.

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.

Any composite number is measured by some prime number.

Any number either is prime or is measured by some prime number.

Given as many numbers as we please, to find the least of those which have the same ratio with them.

Given two numbers, to find the least number which they measure.

If two numbers measure any number, the least number measured by them will also measure the same.

Given three numbers, to find the least number which they measure.

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.

If a number have any part whatever, it will be measured by a number called by the same name as the part.

To find the number which is the least that will have given parts.

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.

To find numbers in continued proportion, as many as may be prescribed, and the least that are in a given ratio.

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.

Given as many ratios as we please in least numbers, to find numbers in continued proportion which are the least in the given ratios.

Plane numbers have to one another the ratio compounded of the ratios of their sides.

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.

If there be as many numbers as we please in continued proportion, and the first measure the last, it will measure the second also.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

If one mean proportional number fall between two numbers, the numbers will be similar plane numbers.

If two mean proportional numbers fall between two numbers, the numbers are similar solid numbers.

If three numbers be in continued proportion, and the first be square, the third will also be square.

If four numbers be in continued proportion, and the first be cube, the fourth will also be cube.

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.

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.

Similar plane numbers have to one another the ratio which a square number has to a square number.

Similar solid numbers have to one another the ratio which a cube number has to a cube number.

If two similar plane numbers by multiplying one another make some number, the product will be square.

If two numbers by multiplying one another make a square number, they are similar plane numbers.

If a cube number by multiplying itself make some number, the product will be cube.

If a cube number by multiplying a cube number make some number, the product will be cube.

If a cube number by multiplying any number make a cube number, the multiplied number will also be cube.

If a number by multiplying itself make a cube number, it itself will also be cube.

If a composite number by multiplying any number make some number, the product will be solid.

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.

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.

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.

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.

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.

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.

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.

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.

If two numbers be prime to one another, the second will not be to any other number as the first is to the second.

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.

Given two numbers, to investigate whether it is possible to find a third proportional to them.

Given three numbers, to investigate when it is possible to find a fourth proportional to them.

Prime numbers are more than any assigned multitude of prime numbers.

If as many even numbers as we please be added together, the whole is even.

If as many odd numbers as we please be added together, and their multitude be even, the whole will be even.

If as many odd numbers as we please be added together, and their multitude be odd, the whole will also be odd.

If from an even number an even number be subtracted, the remainder will be even.

If from an even number an odd number be subtracted, the remainder will be odd.

If from an odd number an odd number be subtracted, the remainder will be even.

If from an odd number an even number be subtracted, the remainder will be odd.

If an odd number by multiplying an even number make some number, the product will be even.

If an odd number by multiplying an odd number make some number, the product will be odd.

If an odd number measure an even number, it will also measure the half of it.

If an odd number be prime to any number, it will also be prime to the double of it.

Each of the numbers which are continually doubled beginning from a duad is even-times even only.

If a number have its half odd, it is even-times odd only.

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.

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.

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.

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.

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.

Given two commensurable magnitudes, to find their greatest common measure.

Given three commensurable magnitudes, to find their greatest common measure.

Commensurable magnitudes have to one another the ratio which a number has to a number.

If two magnitudes have to one another the ratio which a number has to a number, the magnitudes will be commensurable.

Incommensurable magnitudes have not to one another the ratio which a number has to a number.

If two magnitudes have not to one another the ratio which a number has to a number, the magnitudes will be incommensurable.

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.

To find two straight lines incommensurable, the one in length only, the other in square also, with an assigned straight line.

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.

Magnitudes commensurable with the same magnitude are commensurable with one another.

If two magnitudes be commensurable, and one of them be incommensurable with any magnitude, the remaining one will also be incommensurable with the same.

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.

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.

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.

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.

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.

The rectangle contained by rational straight lines commensurable in length is rational.

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.

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.

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.

A straight line commensurable with a medial straight line is medial.

The rectangle contained by medial straight lines commensurable in length is medial.

The rectangle contained by medial straight lines commensurable in square only is either rational or medial.

A medial area does not exceed a medial area by a rational area.

To find medial straight lines commensurable in square only which contain a rational rectangle.

To find medial straight lines commensurable in square only which contain a medial rectangle.

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.

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.

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.

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.

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.

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.

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.

If two rational straight lines commensurable in square only be added together, the whole is irrational; and let it be called binomial.

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.

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.

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.

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.

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.

A binomial straight line is divided into its terms at one point only.

A first bimedial straight line is divided at one and the same point only.

A second bimedial straight line is divided at one point only.

A major straight line is divided at one and the same point only.

The side of a rational plus a medial area is divided at one and the same point only.

The side of the sum of two medial areas is divided at one and the same point only.

To find the first binomial straight line.

To find the second binomial straight line.

To find the third binomial straight line.

To find the fourth binomial straight line.

To find the fifth binomial straight line.

To find the sixth binomial straight line.

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.

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.

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.

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.

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.

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.

The square on the binomial straight line applied to a rational straight line produces as breadth the first binomial.

The square on the first bimedial straight line applied to a rational straight line produces as breadth the second binomial.

The square on the second bimedial straight line applied to a rational straight line produces as breadth the third binomial.

The square on the major straight line applied to a rational straight line produces as breadth the fourth binomial.

The square on the side of a rational plus a medial area applied to a rational straight line produces as breadth the fifth binomial.

The square on the side of the sum of two medial areas applied to a rational straight line produces as breadth the sixth binomial.

A straight line commensurable in length with a binomial straight line is itself also binomial and the same in order.

A straight line commensurable in length with a bimedial straight line is itself bimedial and the same in order.

A straight line commensurable with a major straight line is itself major.

A straight line commensurable with the side of a rational plus a medial area is itself such a side.

A straight line commensurable with the side of the sum of two medial areas is itself such a side.

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.

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.

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.

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.

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.

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.

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.

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.

Only one rational straight line can be annexed to an apotome which is commensurable with the whole in square only.

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.

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.

Only one straight line can be annexed to a minor.

Only one straight line can be annexed to the line producing with a rational area a medial whole.

Only one straight line can be annexed to the line producing with a medial area a medial whole.

To find the first apotome.

To find the second apotome.

To find the third apotome.

To find the fourth apotome.

To find the fifth apotome.

To find the sixth apotome.

If an area be contained by a rational straight line and a first apotome, the side of the area is an apotome.

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.

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.

If an area be contained by a rational straight line and a fourth apotome, the side of the area is a minor.

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.

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.

The square on an apotome straight line applied to a rational straight line produces as breadth a first apotome.

The square on a first apotome of a medial straight line applied to a rational straight line produces as breadth a second apotome.

The square on a second apotome of a medial straight line applied to a rational straight line produces as breadth a third apotome.

The square on a minor applied to a rational straight line produces as breadth a fourth apotome.

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.

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.

A straight line commensurable in length with an apotome is itself an apotome and the same in order.

A straight line commensurable in length with an apotome of a medial is itself such an apotome of the same order.

A straight line commensurable with a minor is itself a minor.

A straight line commensurable with the line producing with a rational area a medial whole is itself such a line.

A straight line commensurable with the line producing with a medial area a medial whole is itself such a line.

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.

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.

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.

The apotome is not the same as the binomial.

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.

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.

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.

From a medial straight line there arise irrational straight lines infinite in number, and none of them is the same with any preceding.

A part of a straight line cannot be in the plane of reference and a part in a plane more elevated.

If two straight lines cut one another, they are in one plane, and every triangle is in one plane.

If two planes cut one another, their common section is a straight line.

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.

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.

If two straight lines be at right angles to the same plane, the straight lines will be parallel.

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.

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.

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.

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.

From a given elevated point to draw a straight line perpendicular to a given plane.

To set up a straight line at right angles to a given plane from a given point in it.

From the same point two straight lines cannot be set up at right angles to the same plane on the same side.

Planes to which the same straight line is at right angles will be parallel.

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.

If two parallel planes be cut by any plane, their common sections are parallel.

If two straight lines be cut by parallel planes, they will be cut in the same ratios.

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.

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.

If a solid angle be contained by three plane angles, any two, taken together in any manner, are greater than the remaining one.

Any solid angle is contained by plane angles less than four right angles.

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.

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.

If a solid be contained by parallel planes, the opposite planes in it are equal and similar parallelograms.

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.

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.

On a given straight line to construct a parallelepipedal solid similar and similarly situated to a given parallelepipedal solid.

If a parallelepipedal solid be cut by a plane through the diagonals of the opposite planes, the solid will be bisected by the plane.

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.

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.

Parallelepipedal solids which are on equal bases and of the same height are equal to one another.

Parallelepipedal solids which are of the same height are to one another as their bases.

Similar parallelepipedal solids are to one another in the triplicate ratio of their corresponding sides.

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.

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.

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.

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.

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.

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.

Similar polygons inscribed in circles are to one another as the squares on the diameters.

Circles are to one another as the squares on the diameters.

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.

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.

Pyramids which are of the same height and have triangular bases are to one another as their bases.

Pyramids which are of the same height and have polygonal bases are to one another as the bases.

Any prism which has a triangular base is divided into three pyramids equal to one another which have triangular bases.

Similar pyramids which have triangular bases are in the triplicate ratio of their corresponding sides.

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.

Any cone is a third part of the cylinder which has the same base with it and equal height.

Cones and cylinders which are of the same height are to one another as their bases.

Similar cones and cylinders are to one another in the triplicate ratio of the diameters in their bases.

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.

Cones and cylinders which are on equal bases are to one another as their heights.

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.

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.

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.

Spheres are to one another in the triplicate ratio of their respective diameters.

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.

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.

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.

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.

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.

If a rational straight line be cut in extreme and mean ratio, each of the segments is the irrational straight line called apotome.

If three angles of an equilateral pentagon, taken either in order or not in order, be equal, the pentagon will be equiangular.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

To draw a straight line from any point to any point.

To produce a finite straight line continuously in a straight line.

To describe a circle with any centre and distance.

That all right angles are equal to one another.

That, if a straight line falling on two straight lines 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.

Things which are equal to the same thing are also equal to one another.

If equals be added to equals, the wholes are equal.

If equals be subtracted from equals, the remainders are equal.

Things which coincide with one another are equal to one another.

The whole is greater than the part.

A point is that which has no part.

A line is breadthless length.

The extremities of a line are points.

A straight line is a line which lies evenly with the points on itself.

A surface is that which has length and breadth only.

The extremities of a surface are lines.

A plane surface is a surface which lies evenly with the straight lines on itself.

A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.

And when the lines containing the angle are straight, the angle is called rectilineal.

When a straight line set up 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.

An obtuse angle is an angle greater than a right angle.

An acute angle is an angle less than a right angle.

A boundary is that which is an extremity of anything.

A figure is that which is contained by any boundary or boundaries.

A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another.

And the point is called the centre of the circle.

A diameter of the circle is any straight line drawn through the centre and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle.

A semicircle is the figure contained by the diameter and the circumference cut off by it. And the centre of the semicircle is the same as that of the circle.

Rectilineal figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines.

Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal.

Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle that which has its three angles acute.

Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.

Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

Any rectangular parallelogram is said to be contained by the two straight lines containing the right angle.

And in any parallelogrammic area let any one whatever of the parallelograms about its diameter, with the two complements, be called a gnomon.

Equal circles are those whose diameters are equal, or whose radii are equal.

A straight line is said to touch a circle which, meeting the circle and being produced, does not cut the circle.

Circles are said to touch one another which, meeting one another, do not cut one another.

In a circle, straight lines are said to be equally distant from the centre when the perpendiculars drawn to them from the centre are equal.

And that straight line is said to be at a greater distance on which the greater perpendicular falls.

A segment of a circle is the figure contained by a straight line and a circumference of a circle.

An angle of a segment is that contained by a straight line and a circumference of a circle.

An angle in a segment is the angle which, when a point is taken on the circumference of the segment and straight lines are joined from it to the extremities of the straight line which is the base of the segment, is contained by the straight lines so joined.

And, when the straight lines containing the angle cut off an arc, the angle is said to stand upon that arc.

A sector of a circle is the figure which, when an angle is constructed at the centre of the circle, is contained by the straight lines containing the angle and the arc cut off by them.

Similar segments of circles are those which admit equal angles, or in which the angles are equal to one another.

A magnitude is a part of a magnitude, the less of the greater, when it measures the greater.

The greater is a multiple of the less when it is measured by the less.

A ratio is a sort of relation in respect of size between two magnitudes of the same kind.

Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another (the Archimedean property).

Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.

Let magnitudes which have the same ratio be called proportional.

When, of the equimultiples, the multiple of the first magnitude exceeds the multiple of the second, but the multiple of the third does not exceed the multiple of the fourth, then the first is said to have a greater ratio to the second than the third has to the fourth.

A proportion in three terms is the least possible.

When three magnitudes are proportional, the first is said to have to the third the duplicate ratio of that which it has to the second.

When four magnitudes are continuously proportional, the first is said to have to the fourth the triplicate ratio of that which it has to the second, and so on, in continual proportion of any number of magnitudes.

Antecedents are said to correspond to antecedents, and consequents to consequents.

Alternate ratio means taking the antecedent in relation to the antecedent and the consequent in relation to the consequent.

Inverse ratio means taking the consequent as antecedent in relation to the antecedent as consequent.

Composition of a ratio means taking the antecedent together with the consequent as one in relation to the consequent by itself.

Separation of a ratio means taking the excess by which the antecedent exceeds the consequent in relation to the consequent by itself.

Conversion of a ratio means taking the antecedent in relation to the excess by which the antecedent exceeds the consequent.

A ratio ex aequali arises when, there being several magnitudes and another set equal to them in multitude which taken two and two are in the same proportion, as the first is to the last of the first magnitudes, so is the first to the last of the second magnitudes.

A perturbed proportion arises when, there being three magnitudes and another set equal to them in multitude, as antecedent is to consequent among the first magnitudes, so is antecedent to consequent among the second magnitudes, while as the consequent is to a third among the first magnitudes, so is a third to the antecedent among the second magnitudes.

A unit is that by virtue of which each of the things that exist is called one.

A number is a multitude composed of units.

A number is a part of a number, the less of the greater, when it measures the greater.

But parts when it does not measure it.

The greater number is a multiple of the less when it is measured by the less.

An even number is that which is divisible into two equal parts.

An odd number is that which is not divisible into two equal parts, or that which differs by a unit from an even number.

An even-times even number is that which is measured by an even number according to an even number.

An even-times odd number is that which is measured by an even number according to an odd number.

An odd-times odd number is that which is measured by an odd number according to an odd number.

A prime number is that which is measured by a unit alone.

Numbers prime to one another are those which are measured by a unit alone as a common measure.

A composite number is that which is measured by some number.

Numbers composite to one another are those which are measured by some number as a common measure.

A number is said to multiply a number when that which is multiplied is added to itself as many times as there are units in the other, and thus some number is produced.

When two numbers having multiplied one another make some number, the number so produced is called plane, and its sides are the numbers which have multiplied one another.

When three numbers having multiplied one another make some number, the number so produced is solid, and its sides are the numbers which have multiplied one another.

A square number is equal multiplied by equal, or a number which is contained by two equal numbers.

A cube number is equal multiplied by equal and again by equal, or a number which is contained by three equal numbers.

Numbers are proportional when the first is the same multiple, or the same part, or the same parts, of the second that the third is of the fourth.

Similar plane and solid numbers are those which have their sides proportional.

A perfect number is that which is equal to the sum of its own parts (its proper divisors).

Those magnitudes are said to be commensurable which are measured by the same measure, and those incommensurable which cannot have any common measure.

Straight lines are commensurable in square when the squares on them are measured by the same area, and incommensurable in square when the squares on them cannot possibly have any area as a common measure.

With these hypotheses, it is proved that there exist straight lines infinite in multitude which are commensurable and incommensurable respectively, some in length only, and others in square also, with an assigned straight line. Let the assigned straight line be called rational, and those straight lines which are commensurable with it, whether in length and in square or in square only, rational, but those which are incommensurable with it irrational.

And let the square on the assigned straight line be called rational and those areas which are commensurable with it rational, but those which are incommensurable with it irrational, and the straight lines which produce them irrational — that is, in case the areas are squares, the sides themselves; in other cases, the straight lines on which the rectangles equal to the areas would be applied.

Given a rational straight line and a binomial, divided into its terms, let the square of the greater term be greater than the square of the lesser by the square of a straight line commensurable in length with the greater. Then if the greater term is commensurable in length with the assigned rational straight line, the whole is called a first binomial.

If the lesser term is commensurable in length with the assigned rational straight line, the whole is called a second binomial.

If neither term is commensurable in length with the assigned rational straight line, the whole is called a third binomial.

If the square of the greater term exceeds the square of the lesser by the square of a line incommensurable in length with the greater, and the greater term is commensurable in length with the assigned rational straight line, the whole is called a fourth binomial.

If, in the same case, the lesser term is commensurable in length with the assigned rational straight line, the whole is called a fifth binomial.

If neither term is commensurable in length with the assigned rational straight line, the whole is called a sixth binomial.

Given a rational straight line and an apotome (i.e. a difference of two rationals commensurable in square only), if the square of the whole is greater than the square of the annex by the square of a straight line commensurable in length with the whole, and the whole is commensurable in length with the assigned rational straight line, the apotome is called a first apotome.

If the annex is commensurable in length with the assigned rational straight line, the apotome is called a second apotome.

If neither the whole nor the annex is commensurable in length with the assigned rational straight line, the apotome is called a third apotome.

If the square of the whole exceeds the square of the annex by the square of a straight line incommensurable in length with the whole, and the whole is commensurable in length with the assigned rational straight line, the apotome is called a fourth apotome.

If, in the same case, the annex is commensurable in length with the assigned rational straight line, the apotome is called a fifth apotome.

If neither the whole nor the annex is commensurable in length with the assigned rational straight line, the apotome is called a sixth apotome.

A solid is that which has length, breadth, and depth.

An extremity of a solid is a surface.

A straight line is at right angles to a plane when it makes right angles with all the straight lines which meet it and are in the plane.

A plane is at right angles to a plane when the straight lines drawn in one of the planes at right angles to the common section of the planes are at right angles to the remaining plane.

The inclination of a straight line to a plane is, assuming a perpendicular drawn from the extremity of the straight line which is elevated above the plane to the plane and a straight line joined from the foot of the perpendicular to the extremity of the straight line which is in the plane, the angle contained by the straight line so drawn and the straight line standing up.

The inclination of a plane to a plane is the acute angle contained by the straight lines drawn at right angles to the common section at the same point, one in each of the planes.

A plane is said to be similarly inclined to a plane as another to another when the said angles of the inclinations are equal to one another.

Parallel planes are those which do not meet.

Similar solid figures are those contained by similar planes equal in multitude.

Equal and similar solid figures are those contained by similar planes equal in multitude and in magnitude.

A solid angle is the inclination constituted by more than two lines which meet one another and are not in the same surface, towards all the lines. Otherwise: a solid angle is that which is contained by more than two plane angles which are not in the same plane and are constructed to one point.

A pyramid is a solid figure contained by planes which is constructed from one plane to one point.

A prism is a solid figure contained by planes two of which, namely those which are opposite, are equal, similar, and parallel, while the rest are parallelograms.

When a semicircle with fixed diameter is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a sphere.

The axis of the sphere is the straight line which remains fixed and about which the semicircle is turned.

The centre of the sphere is the same as that of the semicircle.

A diameter of the sphere is any straight line drawn through the centre and terminated in both directions by the surface of the sphere.

When, one side of those about the right angle in a right-angled triangle remaining fixed, the triangle is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a cone. And if the straight line which remains fixed is equal to the remaining side about the right angle which is carried round, the cone will be right-angled; if less, obtuse-angled; and if greater, acute-angled.

The axis of the cone is the straight line which remains fixed and about which the triangle is turned.

And the base is the circle described by the straight line which is carried round.

When, one side of those about the right angle in a rectangular parallelogram remaining fixed, the parallelogram is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a cylinder.

The axis of the cylinder is the straight line which remains fixed and about which the parallelogram is turned.

The bases are the circles described by the two sides opposite to one another which are carried round.

Similar cones and cylinders are those in which the axes and the diameters of the bases are proportional.

A cube is a solid figure contained by six equal squares.

An octahedron is a solid figure contained by eight equal and equilateral triangles.

An icosahedron is a solid figure contained by twenty equal and equilateral triangles.

A dodecahedron is a solid figure contained by twelve equal, equilateral, and equiangular pentagons.

A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.

The height of any figure is the perpendicular drawn from the vertex to the base.

A medial straight line is the mean proportional between two rational straight lines commensurable in square only.

A minor straight line is the difference of two straight lines incommensurable in square such that the sum of the squares on them is rational, but the rectangle contained by them is medial.

A straight line which produces with a rational area a medial whole is the irrational straight line such that the square on it added to a rational area makes the whole medial.

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BibTeXRISJSON

@article{260500009.v5,
  title   = {Euclid's Elements, encoded as an rrxiv paper},
  author  = {Euclid of Alexandria and Sir Thomas L. Heath and Blaise Albis-Burdige and Claude Opus 4.7},
  rrxiv   = {rrxiv:2605.00009},
  year    = {2026},
  version = {v5},
  note    = {Cite v5 (revision); see retrieval_uri for the lineage chain.}
}