Summary. EuclidOfAlexandria. Elements.

Book01

Definitions

1.     point.

2.     line.

3.     extremities of a line.

4.     straightline.

5.     surface.

6.     lines.

7.     planesurface.

8.     planeangle.

9.     rectilinealangle.

10. rightangle. perpendicularangle.

11. obtuseangle.

12. acuteangle.

13. boundary.

14. figure.

15. circle.

16. centre of the circle.

17. diameter of the circle.

18. semicircle.

19. rectilinealfigure. trilateralfigure. quadrilateralfigure. multilateralfigure.

20. equilateraltriangle. isoscelestriangle. angledtriangle.

21. square. oblong. rhombus. rhomboid. trapezia.

22. parallelstraightline.

Postulates

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

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

3.     To describe a circle with any centre and distance.

4.     That all right angles are equal to one another.

5.     That, if a straight line falling on two straight lines make the interior angles on the same side less than two rightangles, the two straightlines, if produced indefinitely, meet on that side on which are the angles less than the two rightangles.

Common notions

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

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

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

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

5.     The whole is greater than the part.

Book1.

Proposition 1. On a given finite straightline, to construct an equilateral triangle.

Proposition 2. To place at a given point [as an extremity] a straightline equal to a given straightline.

Proposition 3. Given two unequal straightlines, to cut off from the greater a straightline equal to the less.

Proposition 4. If two triangles have the two sides equal to two sides respectively, and have the angles contained by the equal straightlines equal, 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, namely those which the equal sides subtend.

Proposition 5. In isoscelestriangles the angles at the base are equal to one another, and, if the equal straightlines be produced further, the angles under the base will be equal to one another.

Proposition 6. If in a triangle two angles be equal to one another, the sides which subtend the equal angles will also be equal to one another.

Proposition 7. Given two straightlines constructed on a straightline [from its extremities] and meeting in a point, there cannot be constructed on the same straightline [from its extremities], and on the same side of it, two other straightlines meeting in another point and equal to the former two respectively, namely each to that which has the same extremity with it.

Proposition 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 straightlines.

Proposition 9. To bisect a given rectilinealangle.

Proposition 10. To bisect a given finite straightline.

Proposition 11. To draw a straightline at rightangles to a given straightlines from a given point on it.

Proposition 12. To a given infinite straightline, from a given point which is not on it, to draw a perpendicular straightline.

Proposition 13. If a straightline set up on a straightline make angles, it will make either two rightangles or angles equal to two rightangles.

Proposition 14. If with any straightline, and at a point on it, two straightlines not lying on the same side make the adjacentangles equal to two rightangles, the two straightlines will be in a straightline with one another.

Proposition 15. If two straightlines cut one another, they make the vertical angles equal to one another. Porism. From this it is manifest that, if two straightlines cut one another, they will make the angles at the point of section equal to four rightangles.

Proposition 16. If any triangle, if one of the sides be produced, the exteriorangle is greater than either of the interior and opposite angles.

Proposition 17. If any triangle two angles taken together in any manner are less than two rightangles.

Proposition 18. In any triangle the greater side subtends the greater angle.

Proposition 19. In any triangle the greater angle is subtended by the greater side.

Proposition 20. In any triangle two sides taken together in any manner are greater than the remaining one.

Proposition 21. If on one of the sides of a triangle, from its extremities, there be constructed two straightlines meeting within the triangle, the straightlines so constructed will be less than the remaining two sides of the triangle, but will contain a greater angle.

Proposition 22. Out of three straightlines, which are equal to three given straightlines, to construct a triangle: thus it is necessary that two of the straightlines taken together in any manner should be greater than the remaining one.

Proposition 23. On a given straight line and at a point on it to construct a rectilineal angle equal to a given rectilinealangle.

Proposition 24. If two triangles have the two sides equal to two sides respectively, but have the one of the angles contained by the equal straightlines greater than the other, they will also have the base greater than the base.

Proposition 25. If two triangles have the two sides equal to two sides respectively, but have the base greater than the base, they will also have the one of the angles contained by the equal straightlines greater than the other.

Proposition 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 to the remaining angle.

Proposition 27. If a straightline falling on two straightlines make the alternate angles equal to one another, the straightline will be parallel to one another.

Proposition 28. If a straightline falling on two straightlines make the exteriorangle equal to the interior and opposite angle on the same side, or the interior angles on the same side equal to two rightangles, the straightlines will be parallel to one another.

Proposition 29. A straightline falling on parallel straightline makes the alternate angles equal to one another, the exteriorangle equal to the interior and opposite angle, and the interiorangles on the same side equal to two rightangles.

Proposition 30. Straightlines parallel to the same straightline are also parallel to one another.

Proposition 31. Through a givenpoint to draw a straightline parallel to a given straightline.

Proposition 32. In any triangle, if one of the sides be produced, the exteriorangle is equal to the two interior and opposite angles, and the three interiorangles of the triangle are equal to two rightangles.

Proposition 33. The straightlines joining equal and parallel straightlines [at the extremities which are] in the same directions [respectively] are themselves also equal and parallel.

Proposition 34. In parallelogrammic areas the opposite sides and angles are equal to one another, and the dimater bisects the areas.

Proposition 35. Parallelograms which are on the same base and in the same parallels are equal to one another.

Proposition 36. Parallelograms which are on equal bases and in the same parallels are equal to one another.

Proposition 37. Triangles which are on the same base and in the same parallels are equal to one another.

Proposition 38. Triangles which are on equal bases and in the same parallels are equal to one another.

Proposition 39. Equal triangles which are on the same base and on the same side are also in the same parallels.

Proposition 40. Equal triangles which are on equal bases and on the same side are also in the same parallels.

Proposition 41. If a parallelogram have the same base with a triangle and be in the same parallels, the parallelogram is double of the triangle.

Proposition 42. To construct, in a given rectilinealangle, a parallelogram equal to given triangle.

Proposition 43. In any parallelogram the complements of the parallelograms about the diameter are equal to one another.

Proposition 44. To a given straightline to apply, in a given rectilinealangle, a parallelogram equal to a given triangle.

Proposition 45. To construct, in a given rectilinealangle, a parallelogram equal to a given rectilinealfigure.

Proposition 46. On a given straightline to describe a square.

Proposition 47. In rightangled triangles the square on the side subtending the rightangle is equal to the square on the sides containing the rightangle.

Proposition 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.

Book02

Definitions.

1.     rectangularparallelogram

2.     gnomon

Proposition 1. If there be two straightlines, and one of them be cut into any number of segments whatever, the rectangle contained by the two straightlines is equal to the rectangles contained by the uncut straightline and each of the segments.

Proposition 2. If a straightline be cut at random, the rectangles contained by the whole and both of the segments are equal to the square on the whole.

Proposition 3. If a straightline 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.

Proposition 4. If a straightline 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.

Proposition 5. If a straightline be cut into equal and unequal segments, the rectangle contained by the unequal segments of the whole together with the square on the straightline between the points of section is equal to the square on the half.

Proposition 6. If a straightline be bisected and a straightline be added to it in a straightline, the rectangle contained by the whole with the added straightline and the added straightline together with the square on the half is equal to the square on the straightline made up of the half and the added straightline.

Proposition 7. If a straightline 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 and the square on the remaining segment.

Proposition 8. If a straightline 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.

Proposition 9. If a straightline 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 straightline between the points of section.

Proposition 10. If a straightline be bisected, and a straightline be added to it in a straightline, the square on the whole with the added straightline and the square on the added straightline 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 straightline as on one straightline.

Proposition 11. To cut a given straightline so that the rectangle contained by the whole and one of the segments is equal to the square on the remaining segment.

Proposition 12. In obtuseangled triangles the square on the side subtending the obtuseangle is greater than the squares on the sides containing the obtuseangle by twice the rectangle contained by one of the sides about the obtuse angle, namely that on which the perpendicular falls, and the straightline cut off outside by the perpendicular towards the obtuse angle.

Proposition 13. In acuteangled triangles the square on the side subtending the acute angle is less than the squares on the sides containing the acuteangle by twice the rectangle contained by one of the sides about the acuteangle, namely that on which the perpendicular falls, and the straightline cut off within by the perpendicular towards the acute angle.

Proposition 14. To construct a square equal to a given rectilineal figure.

Book03

Definitions

1.     equal circle.

2.     to touch a circle.

3.     circles to touch one another.

4.     to be equally distant from the centre

5.     straight line, at a greater distance

6.     segment of a circle

7.     angle of a segment

8.     angle in a segment

9.     angle, stand upon

10. sector of a circle

11. similar segments of circles

Proposition 1. To find the centre of a given circle.

Proposition 2. If on the circumference of a circle two points be taken at random, the straightline joining the points will fall within the circle.

Proposition 3. If in a circle a straightline through the centre bisect a straightline not through the centre, it also cuts it at rightangles; and if it cut it at rightangles, it also bisects it.

Proposition 4. If in a circle two straightlines cut one another which are not through the centre, they do not bisect one another.

Proposition 5. If two circles cut one another, they will not have the same centre.

Proposition 6. If two circles touch one another, they will not have the same centre.

Proposition 7. If on the diamter of a circle a point be taken which is not the centre of the circle, and from the point straightlines 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 straightline through the centre is always greater than the more remote, and only two equal straightlines will fall from the point on the circle, one on each side of the least straightline.

Proposition 8. If a point be taken outside a circle and from the point straightlines be drawn through to the circle, one of which is through the centre and the others are drawn at random, then, of the straightlines which fall on the concave circumference, that through the centre is greatest, while of the rest the nearer to that through the centre is always greater than the more remote, but, of the straightlines falling on the convex circumference, that between the point and the diamter is least, while of the rest the nearer to the least is always less than the more remote, and only two equal straight lines will fall on the circle from the point, one on each side of the least.

Proposition 9. If a point be taken within a circle, and more than two equal straightlines fall from the point on the circle, the point taken is the centre of the circle.

Proposition 10. A circle does not cut at more points than two.

Proposition 11. If two circles touch one another internally, and their centres be taken, the straightline joining their centres, if it be also produced, will fall on the point of contact of the circles.

Proposition 12. If two circles touch one another externally, the straightline joining their centres will pass throug the point of contact.

Proposition 13. A circle does not touch a circle at more points than one, whether it touch it internally or externally.

Proposition 14. In a circle equal straightlines are equally distant from the centre, and those which are equally distant from the centre are equal to one another.

Proposition 15. Of straightlines in a circle the diameter is greatest, and of the rest the nearer to the centre is always greater than the more remote.

Proposition 16. The straightline drawn at rightangles to the diamter of a circle from its extremity will fall outside the circle, and into the space between the straightline and the circumference another straightline cannot be interposed; further the angle of the semicircle is grater, and the remaining angle less, than any acute rectilineal angle.

Proposition 17. From a givenpoint to draw a straightline touching a given circle.

Proposition 18. If a straightline touch a circle, and a straight line be joined from the centre to the point of contact, the straightline so joined will be perpendicular to the tangent.

Proposition 19. If a straightline touch a circle, and from the point of contact a straightline be drawn at rightangles to the tangent, the centre of the circle will be on the straightline so drawn.

Proposition 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.

Proposition 21. In a circle the angles in the same segment are equal to one another.

Proposition 22. The oppositeangles of quadrilaterals in circles are equal to two rightangles.

Proposition 23. On the same straightline there cannot be constructed two similar and unequal segments of circles on the same side.

Proposition 24. Similar segments of circles on equal straightlines are equal to one another.

Proposition 25. Given a segment of a circle, to describe the complete circle of which it is a segment.

Proposition 26. In equal circles equal angles stand on equal circumferences, whether they stand at the centres or at the circumferences.

Proposition 27. In equal circles angles standing on equal circumferences are qual to one another, whether they stand at the centres or at the circumferences.

Proposition 28. In equalcircles equal straightlines cut off equalcircumferences, the greater equal to the greater and the less to the less.

Proposition 29. In equalcircles equalcircumferences are subtended by equal straightlines.

Proposition 30. To bisect a given circumference.

Proposition 31. In a circle the angle in the semicircle is right, that in a greater segment less than a rightangle, and that in a less segment greater than a rightangle; and further the angle of the greater segment is greater than a rightangle, and the angle of the less segment less than a rightangle.

Proposition 32. If a straightline touch a circle, and from the point of contact there be drawn across, in the circle, a straightline cutting the circle, the angles which it makes with the tangent will be equal to the angles in the alternate segments of the circle.

Proposition 33. On a given straightline to describe a segment of a circle admitting an angle equal to a given rectilineal angle.

Proposition 34. From a given circle to cut off a segment admitting an angle equal to a given rectilineal angle.

Proposition 35. If in a circle two straightlines 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 36. 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 straightline which cuts the circle and the straightline intercepted on it outside between the point and the convex circumference will be equal to the square on the tangent.

Proposition 37. If a point be taken outside a circle and from the point there fall on the circle two straightlines, if one of them cut the circle, and the other fall on it, and if further the rectangle contained by the whole of the straightline which cuts the circle and the straightline intercepted on it outside between the point and the convex circumference be equal to the square on the straightline which falls on the circle, the straightline which falls on it will touch the circle.

Book04

Definitions

1.     rectilinealfigure, inscribed in a rectilineal figure

2.     figure, circumscribed about a figure

3.     rectilinealfigure, inscribed in a circle

4.     rectilinealfigure, inscribed in a circle

5.     circle, inscribed in a figure

6.     circle, circumscribed about a figure

7.     straightline, fitted into a circle

Proposition 1. Into a given circle to fit a straightline equal to a given straightline which is not greater than the diameter of the circle.

Proposition 2. In a given circle to inscribwe a triangle equiangular with a given triangle.

Proposition 3. About a given circle to circumscribe a triangle equiangular with a given triangle.

Proposition 4. In a given triangle to inscribe a circle.

Proposition 5. About a given triangle to circumscribe a circle.

Proposition 6. In a given circle to inscribe a square.

Proposition 7. About a given circle to circumscribe a square.

Proposition 8. In a given square to inscribe a circle.

Proposition 9. About a given square to circumscribe a circle.

Proposition 10. To construct an isosceles triangle having each of the angles at the base double of the remaining one.

Proposition 11. In a given circle to inscribe an equilateral and equiangular pentagon.

Proposition 12. About a given circle to circumscribe an equilateral and equiangular pentagon.

Proposition 13. In a given pentagon, which is equilateral and equiangular, to inscribe a circle.

Proposition 14. About a given pentagon, which is equilateral and equiangular, to circumscribe a circle.

Proposition 15. In a given circle an equilateral and equiangular hexagon. Porism. From this it is manifest that the side of the hexangon is equal to the radius of the circle.

Proposition 16. In a given circle to inscribe a fifteenangled figure which shall be both equilateral and equiangular.

Book05

Definitions

1. magnitude, a part of a magnitude

2. greater.

3. ratio.

4. magnitude, to have a ratio.

5. magnitude, to be in the same ratio.

6. proportional.

7. to have a greater ratio.

8. A proportion in threeterms is the least possible.

9. duplicateratio.

10. triplicateratio.

11. correspondingmagnitudes.

12. alternateratio.

13. inverseratio.

14. composition of a ratio.

15. separation of a ratio.

16. conversion of a ratio.

17. ratio ex aequali.

18. perturbed proportion.

Proposition 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.

Proposition 2. If a first magnitude be the 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, 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.

Proposition 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, then also ex aequali the magnitudes taken will be equimultiples respectively, the one of the second, and the other of the fourth.

Proposition 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.

Proposition 5. If a magnitude be the same multiple of a magnitude that a part subtracted is of a part subtracted, the remainder will also be the same multiple of the remainder that the whole is of the whole.

Proposition 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.

Proposition 7. Equal magnitudes have to the same the same ratio, as also has the same to equal magnitudes.

Proposition 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.

Proposition 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.

Proposition 10. Of magnitudes which have a ratio the same, that which has a greater ratio is greater; and that to which the same has a greater ratio is less.

Proposition 11. Ratios which are the same with the same ratio are also the same with one another.

Proposition 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.

Proposition 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.

Proposition 14. If a first magnitude have to a second the same ratio as a third has to a fourth; if equal, equal; and if less, less.

Proposition 15. Parts have the same ratio as the same multiples of them taken in corresponding order.

Proposition 16. If four magnitudes be proportional, they will also be proportional alternately.

Proposition 17. If magnitudes be proportional componendo, they will also be proportional separando.

Proposition 18. If magnitudes be proportional separando, they will also be proportional componendo.

Proposition 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. Porism. From this it is manifest that, if magnitudes be proportional componendo, they will also be proportional convertendo.

Proposition 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; if equal, equal; and, if less, less.

Proposition 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 magnitude is greater than the third, the fourth will also be greater than the sixth; if equal, equal; and if less, less.

Proposition 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.

Proposition 23. 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, they will also be in the same ratio ex aequali.

Proposition 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.

Proposition 25. If four magnitudes be proportional, the greatest and the least are greater than the remaining two.

Book06

1. similar rectilinealfigures.

2. reciprocally related figures.

3. strightline, cut in extreme and mean ratio.

4. height of any figure.

Proposition 1. Triangles and parallelograms which are under the same height are to one another as their bases.

Proposition 2. If a straightline 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.

Proposition 3. If an angle of a triangle be bisected and the straightline cutting the angle cut the base also, the segments of the base will have the same ratio as the remaining sides of the triangle; and, if the segments of the base have the same ratio as the remaining sides of the triangle, the straightline joined from the vertex to the point of section will bisect the angle of the triangle.

Proposition 4. In equiangular triangles the sides about the equal angles are proportional, and those are corresponding sides which subtend the equal angles.

Proposition 5. If two triangles have their sides proportional, the triangles will be equiangular and will have those angles equal which the corresponding sides subtend.

Proposition 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.

Proposition 7. If two triangles have one single 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, the sides about which are proportional.

Proposition 8. If in a rightangledtriangle a perpendicular be drawn from the rightangle to the base, the triangles adjoining the perpendicular are similar both to the whole and to one another. Porism. From this it is manifest that, if in a rightangledtriangle a preprendicular be drawn from the rightangle to the base, the straightline so drawn is a mean proportional between the segments of the base.

Proposition 9. From a given straightline to cut off a prescribed part.

Proposition 10. To cut a given uncut straightline similarly to a given cut straightline.

Proposition 11. To two given straightlines to find a third proportional.

Proposition 12. To three given straightlines to find a fourth proportional.

Proposition 13. To two given straightlines to find a mean proportional.

Proposition 14. In equal and equiangular parallelograms the sides about the equalangles are reciprocally proportional; and equiangular parallelograms in which the sides about the equalangles are reciprocally proportional are equal.

Proposition 15. In equaltriangles which have one angle equal to one angle the sides about the equalangles 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.

Proposition 16. If four straightlines 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 straightlines will be proportional.

Proposition 17. If three straightlines be proportional, the rectangle contained by the extreme 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 straightlines will be proportional.

Proposition 18. On a given straightline to describe a rectilinealfigure similar and similarly situated to a given rectilinealfigure.

Proposition 19. Similar triangles are to one another in the duplicate ratio of the corresponding sides. Porism. From this it is manifest that, if three straightlines be proportional, then, as the first is to the third, so is the figure described on the first to that which is similar and similarly described on the second.

Proposition 20. Similar polygons are divided into similar triangles, and into 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. Porism. Similarly also it can be proved in the case of quadrilaterals that they are in the duplicate ratio of the corresponding sides. And it was also proved in the case of triangles; thereforealso, generally, similar rectilinealfigures are to one another in the duplicate ratio of the corresponding sides.

Proposition 21. Figures which are similar to the same rectilinealfigure are also similar to one another.

Proposition 22. If four straightlines be proportional, the rectilinealfigures similar and similarly described upon them will also be proportional; and if the rectilinealfigures similar and similarly described upon them be proportional, the straightlines will themselves also be proportional.

Proposition 23. Equiangular parallelograms have to one another the ratio compounded of the ratios of their sides.

Proposition 24. In any parallelogram, the parallelograms about the diameter are similar both to the whole and to one another.

Proposition 25. To construct one and the same figure similar to a given rectilineal figure and equal to another given rectilineal figure.

Proposition 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.

Proposition 27. Of all the parallelograms applied to the same straightline and deficient by paralellogrammicfigures similar and similarly situated to that described on the half of the straightline, that parallelogram is greatest which is applied to the half of the straightline and is similar to the defect.

Proposition 28. To a given straightline to apply a parallelogram equal to a given rectilinealfigure and deficient by a parallelogrammicfigure similar to a given one: thus the given rectilinealfigure must not be greater than the parallelogram described on the half ot the straightline and similar to the defect.

Proposition 29. To a given straightline to apply a paralleogram equal to a given rectilinealfigure and exceeding by a parallelogrammicfigure similar to a given one.

Proposition 30. To cut a given finite straightline in extreme and mean ratio.

Proposition 31. In rightangledtriangles the figure on the side subtending the rightangle is equal to the similar and similarly described figures on the sides containing the rightangle.

Proposition 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 straightline.

Proposition 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.

Book07

1. unit

2. number

3. a part of a number

4. parts

5. multiple

6. evennumber

7. oddnumber

8. eventimes evennumber

9. eventimes oddnumber

10. oddtimes oddnumber

11. primenumber

12. prime to one another

13. compositenumber

14. composite to one another

15. multiply

16. plane, sides

17. solid

18. square number

19. cube

20. proportional

21. similarplane. solidnumbers.

22. perfectnumber.

Proposition 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 an unit is left, the originalnumbers will be prime to one another.

Proposition 2. Given two numbers not prime to one another, to find their greatest common measure. Porism. From this it is manifest that, if a number measure two numbers, it will also measure the greatestcommonmeasure.

Proposition 3. Given three numbers not prime to one another, to find their greatest common measure.

Proposition 4. Any number is either a part or parts of any number, the less of the greater.

Proposition 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 one.

Proposition 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 one.

Proposition 7. If a number be that part of a number, which a number subtracted is of a number subtracted, the remainder will also be the same part of the remainder that the whole is of the whole.

Proposition 8. If a number be the same parts of a number that a number subtracted is of a number subtracted, the remainder will also be the same parts of the remainder that the whole is of the whole.

Proposition 9. If a number be a part of a number, and another be the same part of another, alternately also, 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.

Proposition 10. If a number be parts of a number, and another be the same parts of another, alternately also, 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.

Proposition 11. If, as a whole is to whole, so is a number subtracted to a number subtracted, the remainder will also be to the remainder as whole to whole.

Proposition 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 are all the antecedents to all the consequents.

Proposition 13. If four numbers be proportional, they will also be proportional alternately.

Proposition 14. If there be 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.

Proposition 15. If an unit measure any number, and another number measure any other number the same number of times, alternately also, the unit will measure the third number the same number of times that the second measures the fourth.

Proposition 16. If two numbers by multiplying one another make certain numbers, the numbers so produced will be equal to one another.

Proposition 17. If a number by multiplying two numbers make certain numbers, the numbers so produced will have the same ratio as the numbers multiplied.

Proposition 18. If two numbers by multiplying any number make certain numbers, the numbers so produced will have the same ratio as the multipliers.

Proposition 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.

Proposition 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.

Proposition 21. Numbers prime to one another are the least of those which have the same ratio with them.

Proposition 22. The least numbers of those which have the same ratio with them are prime to one another.

Proposition 23. If two numbers be prime to one another, the number which measures the one of them will be prime to the remaining number.

Proposition 24. If two numbers be prime to any number, their product also will be prime to the same.

Proposition 25. If two numbers be prime to one another, the product of one of them into itself will be prime to the remaining one.

Proposition 26. If two numbers be prime to two numbers, both to each, their products also will be prime to one another.

Proposition 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, the latter will also be prime to one another .

Proposition 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 any one of them, the original numbers will also be prime to one another.

Proposition 29. Any primenumber is prime to any number which it does not measure.

Proposition 30. If two numbers by multiplying one another make some number, and any primenumber measures the product, it will also measure one of the original numbers.

Proposition 31. Any compositenumber is measured by some primenumber.

Proposition 32. Any number either is prime or is measured by some primenumber.

Proposition 33. Given as many numbers as we please, to find the least of those which have the same ratio with them.

Proposition 34. Given two numbers, to find the least number which they measure.

Proposition 35. If two numbers measure any number, the least number measured by them will also measure the same.

Proposition 36. Given three numbers, to find the least number which they measure.

Proposition 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.

Proposition 38. If a number have any part whatever, it will be measured by a number called by the same name as the part.

Proposition 39. To find the number which is the least that will have given parts.

Book08 (continued from the previous book)

Proposition 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.

Proposition 2. To find numbers in continued proportion, as many as may be prescribed, and the least that are in a given ratio. Porism. From this it is manifest that, if threenumbers in continued proportion be the least of those which have the same ratio with them, the extremes of them are squares, and, if fournumbers, cubes.

Proposition 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 prime to one another.

Proposition 4. Given as many ratios as we please in leastnumbers, to find numbers in continued proportion which are the least in the given ratio.

Proposition 5. Planenumbers have to one another the ratio compounded of the ratios of their sides.

Proposition 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.

Proposition 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.

Proposition 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 originalnumbers.

Proposition 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 between each of them and an unit in continued proportion.

Proposition 10. If numbers fall between each of two numbers and an unit in continued proportion, however many numbers fall between each of them and an unit in continued proportion, so many also will fall between the numbers themselves in continued proportion.

Proposition 11. Between two squarenumbers there is one mean proportionalnumber, and the square has to the square the ratio duplicate of that which the side has to the side.

Proposition 12. Between two cubenumbers there are two mean proportionalnumbers, and the cube has to the cube the ratio triplicate of that which the side has to the side.

Proposition 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.

Proposition 14. If a square measures a square, the side will also measure the side; and, if the side measure the side, the square will also measure the square.

Proposition 15. If a cubenumber measure a cubenumber, the side will also measures the side; and, if the side measures the side, the cube will also measure the cube.

Proposition 16. If a squarenumber do not measure a squarenumber, neither will the side measure the side; and, if the side do not measure the side, neither will the square measure the square.

Proposition 17. If a cubenumber do not measure a cubenumber, neither will the side measure the side; and, if the side do not measure the side, neither will the cube measure the cube.

Proposition 18. Between two similar planenumbers there is means proportionalnumber; and the planenumber has to the planenumber the ratio duplicate of that which the correspondingside has to the correspondingside.

Proposition 19. Between two similar solidnumbers there fall two means proportionalnumbers, and the solidnumber has to the similar solidnumber the ratio triplicate of that which the correspondingside has to the correspondingside.

Proposition 20. If one mean proportionalnumber fall between two numbers, the numbers will be similar planenumbers.

Proposition 21. If two mean proportionalnumbers fall between two numbers, the numbers are similar solidnumbers.

Proposition 22. If three numbers be in continued proportion, and the first be square, the third will also be square.

Proposition 23. If four numbers in continued proportion, and the first be cube, the fourth will also be cube.

Proposition 24. If two numbers have to one another the ratio which a squarenumber has to a squarenumber, and the first be square, the second will also be square.

Proposition 25. If two numbers have to one another the ratio which a cubenumber has to a cubenumber, and the first be cube, the second will also be cube.

Proposition 26. Similar planenumbers have to one another the ratio which a squarenumber has to a squarenumber.

Proposition 27. Similar solidnumbers have to one another the ratio which a cubenumber has to a cubenumber.

Book09 (continued from the previous book)

Proposition 1. If two similar planenumbers by multiplying one another make some number, the product will be square.

Proposition 2. If two numbers by multiplying one another make a squarenumber, they are similar planenumbers.

Proposition 3. If a cubenumber by multiplying itself make some number, the product will be cube.

Proposition 4. If a cubenumber by multiplying a cubenumber make some number, the product will be cube.

Proposition 5. If a cubenumber by multiplying any number make a cubenumber, the multipliednumber will also be cube.

Proposition 6. If a number by multiplying itself make a cubenumber, it will itself also be cube.

Proposition 7. If a compositenumber by multiplying any numbers make some number, the product will be solid.

Proposition 8. If as many numbers as we please beginning from an unit be in continued proportion, the third from the unit will be square, as will also those which successively leave out one; the fourth will be cube, as will also all those which leave out two; and the seventh will be at once cube and square, as will also those which leave out five.

Proposition 9. If as many numbers as we please beginning from an 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.

Proposition 10. If as many numbers as we please beginning from an 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. And, if the number after the unit be not cube, neither will any other be cube except the fourth from the unit and all those which leave out two.

Proposition 11. If as many numbers as we please beginning from an unit be in continued proportion, the less measures the greater according to some one of the numbers which have place among the proportionalnumbers. Porism. And it is manifest that, whatever place the measuring number has, reckoned from the unit, the same place also has the number according to which it measures, reckoned from the number measured, in the direction of the number before it.

Proposition 12. If as many numbers as we please beginning from an unit be in continued proportion, by however many primenumbers the last is measured, the next to the unit will also be measured by the same.

Proposition 13. If as many numbers as we please beginning from an 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 proportionalnumbers.

Proposition 14. If a number be the least that is measured by primenumbers, it will not be measured by any other primenumber except those originally measuring it.

Proposition 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.

Proposition 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.

Proposition 17. If there be as many numbers as we please 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 to the second.

Proposition 18. Given two numbers, to investigate whether it is possible to find a third proportional to them.

Proposition 19. Given three numbers, to investigate when it is possible to find a fourth proportional to them. [The greek text of this proposition is corrupt, and the intact portion of the proof is erroneous, according to Heath. Nevertheless, analogously to Proposition 18, the condition that a fourth proportional A, B, C exists is that A measure the product of B and C.]

Proposition 20. Primenumbers are more than any assigned multitude of primenumbers.

Proposition 21. If as many evennumbers as we please be added together, the whole is even.

Proposition 22. If as many oddnumbers as we please be added together, and their multitude be even, the whole will be even.

Proposition 23. If as many oddnumbers as we please be added together, and their multitude be odd, the whole will also be odd.

Proposition 24. If from an evennumber an evennumber be subtracted, the remainder will be even.

Proposition 25. If from an evennumber an oddnumber be subtracted, the remainder will be odd.

Proposition 26. If from an oddnumber an oddnumber be subtracted, the remainder will be even.

Proposition 27. If from an oddnumber an evennumber be subtracted, the remainder will be odd.

Proposition 28. If an oddnumber by multiplying an evennumber make some number, the product will be even.

Proposition 29. If an oddnumber by multiplying an oddnumber make some number, the product will be odd.

Proposition 30. If an oddnumber measure an evennumber, it will also measure the half of it.

Proposition 31. If an oddnumber be prime to any number, it will also be prime to the double of it.

Proposition 32. Each of the numbers which are continually doubled beginning from a dyad is eventimes even only.

Proposition 33. If a number have its half odd, it is eventimes odd only.

Proposition 34. If a number neither be one of those which are continually doubled from a dyad, nor have its half odd, it is both eventimes even and eventimes odd.

Proposition 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.

Proposition 36. If as many numbers as we please beginning from an 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.

Book10

Definitions

1. commensurable. incommensurable.

2. commensurable. square. incommensurable in square.

3. rational. irrational.

4. rational. rational. irrational. irrational.

Proposition 1. Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude together 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. Porism. And the theorem can be similarly proved even if the parts subtracted be halves.

Proposition 2. If, when the less 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.

Proposition 3. Given two commensurable magnitudes, to find their greatest common measure. Porism. From this it is manifest that, if a magnitude measure two magnitudes, it will also measure their greatestcommonmeasure.

Proposition 4. Given three commensurable magnitudes, to find their greatest common measure.

Proposition 5. Commensurably magnitudes have to one another the ratio which a number has to a number.

Proposition 6. If two magnitudes to one another the ratio which a number has to a number, the magnitudes will be commensurable. From this it is manifest that, if there be two numbers, as D, E, and a straightline, as A, it is possible to make a straightlineF such that the given straightline is to it as the number D is to the number E.

Proposition 7. Incommensurable magnitudes have not to one another the ratio which a number has to a number.

Proposition 8. If two magnitudes have not to one another the ratio which a number has to a number, the magnitudes will be incommensurable.

Proposition 9. The squares on straightlines commensurable in length have to one another the ratio which a squarenumber has to a squarenumber; and squares which have to one another the ratio which a squarenumber has to a squarenumber will also have their sides commensurable in length. But the squareson straightlines incommensurable in length have not to one another the ratio which a squarenumber has to a squarenumber; and squares which have not to one another the ratio which a squarenumber has to a squarenumber will not have their sides commensurable in length either. Porism. And it is manifest from what has been proved that straightlines commensurable in length are always commensurable in square also, but those commensurable in square are not always commensurable in length also. Lemma. It has been proved in the arithmeticalbooks that similar plane numbers have to one another the ratio which a squarenumber has to a squarenumber, and that, if twonumbers have to one another the ratio which a squarenumber has to a squarenumber, they are similar planenumbers. And it is manifest from these propositions that numbers which are not similar planenumbers, that is, those which have not their sides proportional, have not to one another the ratio which a squarenumber has to a squarenumber.

Proposition 10. To find two straightlines incommensurable, the one in length only, and the other in square also, with an assigned straightline. [Heath comments that a scholium to this proposition "says categorically that the theorem proved in it was the discovery of Theaetetus." The Lemma, however, is thought to be an interpolation; for it has reference to the next proposition, X. 10, and "there are so many objections to X. 10 that it can hardly be accepted as genuine."

Proposition 11. If four magnitudes be proportional, and the first be commensurable with the second, the third will also be commensurable with the fourth; and, if the first be incommensurable with the second, the third will also be incommensurable with the fourth.

Proposition 12. Magnitudes commensurable with the same magnitude are commensurable with one another also.

Proposition 13. If two magnitude be commensurable, and the one of them be incommensurable with any magnitude, the remaining one will also be incommensurable with the same. Lemma. Given two unequal straightlines, to find by what square the square on the greater is greater than the square on the less.

Proposition 14. If four straightlines be proportional, and the square on the first be greater than the square on the second by the square on a straightline commensurable with the first, the square on the third will also be greater than the square on the fourth by the square on a straightline commensurable with the third. And, if the square on the first be greater than the square on the second by the square on a straightline incommensurable with the first, the square on the third will also be greater than the square on the fourth by the square on a straightline incommensurable with the third.

Proposition 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.

Proposition 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. Lemma. If to any straightline there be applied a parallelogram deficient by a squarefigure, the applied parallelogram is equal to the rectangle contained by the segments of the straightline resulting from the application.

Proposition 17. If there be two unequal straightlines, and to the greater there be applied a parallelogram equal to the fourth part of the square on the less and deficient by a squarefigure, 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 straightline commensurable with the greater. And, if the square on the greater be greater than the square on the less by the square on a straightline commensurable with the greater, and if there be applied to the greater a parallelogram equal to the fourth part of the square on the less and deficient by a squarefigure, it will divide it into parts which are commensurable in length.

Proposition 18. If there be two unequal straightlines, and to the greater there be applied a parallelogram equal to the fourth part of the square on the less and deficient by a squarefigure, and if it divide it into parts which are incommensurable, the square on the greater will be greater than the square on the less by the square on a straightline incommensurable with the greater. And, if the square on the greater be greater than the square on the less by the square on a straightline incommensurable with the greater, and if there be applied to the greater a parallelogram equal to the fourth part of the square on the less and deficient by a square figure, it divides it into parts which are incommensurable. Lemma. Since it has been proved that straightlines commensurable in length are always commensurable in square also, while those commensurable in sqaure are not always commensurable in length also, but can of course be either commensurable or incommensurable in length, it is manifest that, if any straightline be commensurable in length with a given rational straightline, it is called rational and commensurable with the other not only in length but in square also, since straightlines commensurable in length are not always commensurable in square also. But, if any straightline be commensurable in square with a given rational straightline, then, if it is also commensurable in length with it, it is called in this case also rational and commensurable with it both in length and in square; but, if again any straightline, being commensurable in square with a given rational straightline, be incommensurable in length with it, it is called in this case also rational but commensurable in square only.

Proposition 19. The rectangle contained by rational straightlines commensurable in length is rational.

Proposition 20. If a rational area be applied to a rational straightline, it produces as breadth a straightline rational and commensurable in length with the straightline to which it is applied. Lemma. If there be two straightlines, then, as the first is to the second, so is the square on the first to the rectangle contained by the two straightlines.

Proposition 21. The rectangle contained by rational straightlines commensurable in square only is irrational, and the side of the square equal to it is irrational. Let the latter be called medial.

Proposition 22. The square on a medial straightline, if applied to a rational straightline, produces as breadth a straightline rational and incommensurable in length with that to which it is applied.

Proposition 23. A straightline commensurable with a medial straightline is medial. Porism. From this it is manifest that an area commensurable with a medial area is medial.

Proposition 24. The rectangle contained by medial straightlines commensurable in length is medial.

Proposition 25. The rectangle contained by medial straightlines commensurable in square only is either rational or irrational.

Proposition 26. A medial area does not exceed a medial area by a rational area.

Proposition 27. To find medial straightlines commensurable in square only which contain a rational rectangle.

Proposition 28. To find medial straightlines commensurable in square only, which contain a medial rectangle. Lemma 1. To find two sqaurenumbers such that their sum is also sqaure. Lemma 2. To find two square numbers such that their sum is not square.

Proposition 29. To find two rational straightlines commensurable in square only and such that the square on the greater is greater than the square on the less by the square on a straightline commensurable in length with the greater.

Proposition 30. To find two rational straightlines commensurable in square only and such that the square on the greater is greater than the square on the less by the square on a straightline incommensurable in length with the greater.

Proposition 31. To find two medial straightlines commensurable in square only, containing a rational rectangle, and such that the square on the greater is greater than the square on the less by the square on a straightline commensurable in length with the greater.

Proposition 32. To find two medial straightlines commensurable in square only, containing a medial rectangle, and such that the square on the greater is greater than the square on the less by the square on a straightline commensurable with the greater. Lemma. Let ABC be a rightangledtriangle having the angle A right, and let the perpendicular AD be drawn, I say that the rectangle CB, BD is equal to the square on BA, the rectangle BC, CD equal to the square on CA, the rectangle BD, DC equal to the square on AD, and further, the rectangle BC, AD, equal to the rectangle BA, AC.

Proposition 33. To find two straightlines incommensurable in square which make the sum of the square on them rational but the rectangle contained by them medial.

Proposition 34. To find two straightlines incommensurable in square which make the sum of the square on them medial but the rectangle contained by them rational.

Proposition 35. To find two straightlines incommensurable in square which make the sum of the square on them medial and the rectangle contained by them medial and moreover incommensurable with the sum of the squares on them.

Proposition 36. If two rational straightlines commensurable in square only be added together, the whole is irrational; and let it be called binomial.

Proposition 37. If two medial straightlines commensurable in square only and containing a rational rectangle be added together, the whole is irrational; and let it be called a first bimedial straightline.

Proposition 38. If two medial straightlines commensurable in square only and containing a medial rectangle be added together, the whole is irrational; and let it be called a second bimedial straightline.

Proposition 39. If two straightlines 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 straightline is irrational: and let it be called major.

Proposition 40. If two straightlines 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 straightline is irrational; and let it be called the side of a rational plus a medial area.

Proposition 41. If two straightlines 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 whole straightline is irrational; and let it be called the side of the sum of two medial areas. Lemma. And that the aforesaid irrational straightlines are divided only in one way into the straightlines of which they are the sum and which produce the types in question, we will now prove after premising the following lemma.

Proposition 42. A binomial straightline is divided into its terms of one point only.

Proposition 43. A first bimedial straightline is divided at one point only.

Proposition 44. A second bimedial straightline is divided at one point only.

Proposition 45. A major straight line is divided at one and the same point only.

Proposition 46. The side of a rational plus a medial area is divided at one point only.

Proposition 47. The side of the sum of two medial areas is divided at one point only.

Definitions II.

1. first binomial straightline.

2. second binomial straightline.

3. third binomial straightline.

4. fourth binomial straightline.

5. fifth binomial straightline.

6. sixth binomial straightline.

Proposition 48. To find the first binomial straightline.

Proposition 49. To find the second binomial straightline.

Proposition 50. To find the third binomial straightline.

Proposition 51. To find the fourth binomial straightline.

Proposition 52. To find the fifth binomial straightline.

Proposition 53. To find the sixth binomial straightline. Lemma. Let there be twosquare AB, BC, and let them be placed so that DB is in a straightline with BE; therefore FB is also in a straightline with BG. Let the parallelogram AC be completed; I say that AC is a square, that DG is a mean proportional between AB, BC, and further that DC is a mean proportional between AC, CB.

Proposition 54. If an area be contained by a rational straightline and the first binomial, the "side" of the area is the irrational straightline which is called binomial.

Proposition 55. If an area be contained by a rational straightline and the second binomial, the "side" of the area is the irrational straightline which is called a first bimedial.

Proposition 56. If an area be contained by a rational straightline and the third binomial, the "side" of the area is the irrational straightline called a second bimedial.

Proposition 57. If an area be contained by a rational straightline and the fourth binomial, the "side" of the area is the irrational straightline called major.

Proposition 58. If an area be contained by a rational straightline and the fifth binomial, the "side" of the area is the irrational straightline called the side of a rational plus a medial area.

Proposition 59. If an area be contained by a rational straightline and the sixth binomial, the "side" of the area is the irrational straightline called the side of the sum of two medial areas. Lemma. If a straightline be cut into unequal parts, the squares on the unequal parts are greater than twice the rectangle contained by the unequal parts.

Proposition 60. The square on the binomial straightline applied to a rational straightline produces as breadth the first binomial.

Proposition 61. The square on the first bimedial straightline applied to a rational straightline produces as breadth the second binomial.

Proposition 62. The square on the second bimedial straightline applied to a rational straightline produces as breadth the third binomial.

Proposition 63. The square on the major straightline applied to a rational straightline produces as breadth the fourth binomial.

Proposition 64. The square on the side of a rational plus a medial area applied to a rational straightline produces as breadth the fifth binomial.

Proposition 65. The square on the side of the sum of two medial areas applied to a rational straightline produces as breadth the sixth binomial.

Proposition 66. A straightline commensurable in length with a binomial straightline is itself also binomial and the same in order.

Proposition 67. A straightline commensurable in length with a bimedial straightline is itself also bimedial and the same in order.

Proposition 68. A straightline commensurable with a major straightline is itself also major.

Proposition 69. A straightline commensurable with the side of a rational plus a medial area is itself also the side of a rational plus a medial area.

Proposition 70. A straightline commensurable with the side of the sum of two medial areas is the side of the sum of two medial areas.

Proposition 71. If a rational and a medial area be added together, four irrational straightlines arise, namely a binomial or a first bimedial or a major or a side of a rational plus a medial area.

Proposition 72. If two medial areas incommensurable with one another be added together, the remaining two irrational straightlines arise, namely either a second bimedial or a side of the sum of two medial areas.

Proposition 73. If from a rational straight line there be subtracted a rational straightline commensurable with the whole in square only, the remainder is irrational; and let it be called an apotome.

Proposition 74. If from a medial straightline there be subtracted a medial straightline which is commensurable with the whole in square only, and which contains with the whole a rational rectangle, the remainder is irrational. And let it be called a first apotome of a medial straightline.

Proposition 75. If from a medial straightline there be subtracted a medial straightline which is commensurable with the whole in square only, and which contains with the whole a medial rectangle, the remainder is irrational; and let it be called a second apotome of a medial straightline.

Proposition 76. If from a straightline there be subtracted a straightline which is incommensurable in square with the whole and which with the whole makes the squares them added together rational, but the rectangle contained by them medial, the remainder is irrational; and let it be called minor.

Proposition 77. If from a straightline there be subtracted a straightline which is incommensurable in square with the whole, and which with the whole makes the sum of the squares on them medial, but twice the rectangle contained by them rational, the remainder is irrational; and let it be called that whic produces with a rational area a medial whole.

Proposition 78. If from a straightline there be subtracted a straightline which is incommensurable in square with the whole and which with the whole makes the sum of the squares on them medial, twice the rectangle contained by them medial, and further, the squares on them incommensurable with twice the rectangle contained by them, the remainder is irrational; and let it be called that which produces with a medial area a medial whole.

Proposition 79. To an apotome only one rational straightline can be annexed which is commensurable with the whole in square only.

Proposition 80. To a first apotome of a medial straightline only one medial straightline can be annexed which is commensurable with the whole in square only and which contains with the whole a rational rectangle.

Proposition 81. To a second apotome of a medial straightline only one medial straightline can be annexed which is commensurable with the whole in square only and which contains with the whole a medial rectangle.

Proposition 82. To a minor straightline only one straightline can be annexed which is incommensurable in square with the whole and which makes, with the whole, the sum of the squares on them rational but twice the rectangle contained by them medial.

Proposition 83. To a straightline which produces with a rational area a medial whole only one straightline can be annexed which is incommensurable in square with the whole straightline and which with the whole straightline makes the sum of the squares on them medial, but twice the rectangle contained by them rational.

Proposition 84. To a straightline which produces with a medial area a medial whole only one straightline can be annexed which is incommensurable in squares with the whole straightline and which with the whole straightline makes the sum of the squares on them medial and twice the rectangle contained by them both medial and also incommensurable with the sum of the squares on them.

Definitions III

1. first apotome

2. second apotome

3. third apotome

4. fourth apotome

5. fifth apotome

6. sixth apotome

Proposition 85. To find the first apotome.

Proposition 86. To find the second apotome.

Proposition 87. To find the third apotome.

Proposition 88. To find the fourth apotome.

Proposition 89. To find the fifth apotome.

Proposition 90. To find the sixth apotome.

Proposition 91. If an area be contained by a rational straightline and a first apotome, the "side" of the area is an apotome.

Proposition 92. If an area be contained by a rational straightline and a second apotome, the "side" of the area is a first apotome of a medial straightline.

Proposition 93. If an area be contained by a rational straightline and a third apotome, the "side" of the area is a second apotome of a medial straightline.

Proposition 94. If an area be contained by a rational straightline and a fourth apotome, the "side" of the area is minor.

Proposition 95. If an area be contained by a rational straightline and a fifth apotome, the "side" of the area is a straightline which produces with a rational area a medial whole.

Proposition 96. If an area be contained by a rational straightline and a sixth apotome, the "side" of the area is a straightline which produces with a medial area a medial whole.

Proposition 97. The square on an apotome applied to a rational straightline produces as breadth a first apotome.

Proposition 98. The square on a first apotome of a medial straightline applied to a rational straightline produces as breadth a second apotome.

Proposition 99. The square on a second apotome of a medial straightline applied to a rational straightline produces as breadth a third apotome.

Proposition 100. The squares on a minor straightline applied to a rational straightline produces as breadth a fourth apotome.

Proposition 101. The square on the straightline which produces with a rational area a medial whole, if applied to a rational straightline, produces as breadth a fifth apotome.

Proposition 102. The square on the straightline which produces with a medial area a medial whole, if applied to a rational straightline, produces as breadth a sixth apotome.

Proposition 103. A straightline commensurable in length with an apotome is an apotome and the same in order.

Proposition 104. A straightline commensurable with an apotome of a medial straightline is an apotome of a medial straightline and the same in order.

Proposition 105. A staightline commensurable with a minor straightline is minor.

Proposition 106. A straightline commensurable with that which produces with a rational area a medial whole is a straightline which produces with a rationalarea and a medialwhole.

Proposition 107. A straightline commensurable with that which produces with a medialarea a medialwhole is itself also a straightline which produces with a medialarea a medialwhole.

Proposition 108. If from a rationalarea a medialarea be subtracted, the "side" of the remainingarea becomes one of two irrational straightlines, either an apotome or a minor straightline.

Proposition 109. If from a medialarea a rationalarea be subtracted, there arises two other irationalstraightlines, either a firstapotome of a medial straightline or a straightline which produces with a rationalrea a medialwhole.

Proposition 110. If from a medialarea there be subtracted a medial area incommensurable with the whole, the two remaining irrational straightlines arise, either a secondapotome of a medial straightline or a straightline which produces with a medialarea a medialwhole.

Proposition 111. The apotome is not the same with the binomial straightline. / The apotome and the irrationalstraightlines following it are neither the same with the medial straightline nor with one another.

Proposition 112. The square on a rationalstraightline applied to the binomial straightline produces as breadth an apotome the terms of which are commensurable with the terms of the binomial and moreover in the same ratio; and further, the apotome so arising will have the same order as the binomial straightline.

Proposition 113. The square on a rationalstraightline, if applied to an apotome, produces as breadth the binomial straightline the terms of which are commensurable with the terms of the apotome and in the same ratio; and further, the binomial so arising has the same order as the apotome.

Proposition 114. If an area be contained by an apotome and the binomial straightline the terms of which are commensurable with the terms of the apotome and in the same ratio, the "side" of the area is rational.

Proposition 115. From a medial straightline there arises irrational straightlines infinite in number, and none of them is the same as any of the preceding.

Book11

Definitions

1. solid

2. extremity of a solid

3. straightline is at right angles to a plane.

4. plane, at right angles to a plane.

5. inclination of a straightline to a plane.

6. inclination of a plane to a plane.

7. similarly inclined.

8. parallelplanes.

9. similar solidfigures.

10. equal and similar solidfigures.

11. solidangle.

12. pyramid.

13. prism.

14. sphere.

15. axis of the sphere.

16. centre of the sphere.

17. diameter of the sphere.

18. cone. obtuse angled. acute angled.

19. axis of the cone.

20. base.

21. cylinder.

22. axis of the cylinder.

23. bases.

24. similar cones and cylinders.

25. cube.

26. octahedron.

27. icosahedron.

28. dodecahedron.

Proposition 1. A part of a straightline cannot be in the plane of reference and a part in a plane more elevated.

Proposition 2. If two straightlines cut one another, they are in one plane, and every triangle is in one plane.

Proposition 3. If two planes cut one another, their common section is a straightline.

Proposition 4. If a straightline be set up at right angles to two straightlines which cut one another, at their common point of section, it will also be at rightangles to the plane through them.

Proposition 5. If a straightlie be setup at rightangles to three straightlines which meet one another, at their common point of section, the three straightlines are in one place.

Proposition 6. If two straightlines be at rightangles to the same plane, the straightlines will be parallel.

Proposition 7. If two straightlines be parallel and points be taken at random on each of them, the staightline joining the points is in the same plane with the parallel straightline.

Proposition 8. If two straightlines be parallel, and one of them be at rightangles to any plane, the remaining one will also be at rightangles to the same plane.

Proposition 9. Straightlines which are parallel to the same straightline and are not in the same plane with it are also parallel to one another.

Proposition 10. If two straightlines meeting one another be parallel to two straightlines meeting one another not in the same plane, they will contain equalangles.

Proposition 11. From a given elevated point to draw a straightline perpendicular to a given plane.

Proposition 12. To set up a straightline at rightangles to a given plane from a given point in it.

Proposition 13. From the same point two straightlines cannot be set up at rightangles to the same plane on the same side.

Proposition 14. Planes to which the same straightline is at rightangles will be parallel.

Proposition 15. If two straightlines meeting one another be parallel to two straightlines meeting one another, not being in the same plane, the planes through them are parallel.

Proposition 16. If two parallelplanes be cut by any plane, their common sections are parallel.

Proposition 17. If two straightlines be cut by parallel planes, they will be cut in the same ratios.

Proposition 18. If a straightline at rightangles to any plane, all the planes through it will also be at rightangles to the same plane.

Proposition 19. If two planes which cut one another be at rightangles to any plane, their common section will also be at rightangles to the same plane.

Proposition 20. If a solidangle be contained by the three planeangles, any two, taken together in any manner, are greater than the remaining one.

Proposition 21. Any solidangle is contained by planeangles less than four rightangles.

Proposition 22. If there be three planeangles of which two, taken together in any manner, are greater than the remaining one, and they are contained by equal straightlines, it is possible to construct a triangle out of the straightlines joining the extremities of the equal straightlines.

Proposition 23. To construct a solidangle out of three planeangles two of which, taken together in any manner, are greater than the remaining one: thus the threeangles must be less than four rightangles. Lemma. But how it is possible to take the square on OR equal to that area by which the square on AB is greater than the square on LO, we can show as follows.

Proposition 24. If a solid be contained by parallelplanes, the opposite planes in it are equal and parallelogrammic.

Proposition 25. If a parallelepipedal solid be cut by a plane which is parallel to the opposite planes, then, as the base is to the base, so will the solid be to the solid.

Proposition 26. On a given straightline, and at a given point on it, to construct a solidangle equal to a given solidangle.

Proposition 27. On a given straightline to describe a parallelepipedal solid similar and similarly situated to a given parallelepipedal solid.

Proposition 28. If a parallelepipedal solid be cut by a plane through the diagnosis of the opposite planes, the solid will be bisected by the plane.

Proposition 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 straightlines, are equal to one another.

Proposition 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 straightlines, are equal to one another.

Proposition 31. Parallelepipedal solids which are on equal bases and of the same height are equal to one another.

Proposition 32. Parallelepipedal solids which are of the same height are to one another as their bases.

Proposition 33. Similar parallelepipedal solids are to one another in the triplcate ratio of their corresponding sides. Porism. From this it is manifest that, if four straightlines be [continuously] proportional, as the first is to the fourth, so will a parallelepipedal solid on the first be to the similar and similarly described parallelepipedal solid on the second, inasmuch as the first has to the fourth the ratio triplicate of that which it has to the second.

Proposition 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.

Proposition 35. If there be two equal planeangles, and on their vertices there be set up elevated straightlines containing equalangles with the original straightlines respectively, if on the elevated straightlines points be taken at random and perpendicular be drawn from them to the planes in which the original angles are, and if from the points so arising in the planes straightlines be joined to the vertices of the original angles, they will contain, with the elevated straightlines, equal angles. [Porism]

Proposition 36. If three straightlines be proportional, the parallelepipedal solid fromed out of the three is equal to the parallelepipedal solid on the mean which is equilateral, but equiangular with the aforesaid solid.

Proposition 37. If four straightlines be proportional, the parallelepipedal solids on them which are similar and similarly described will also be proportional; and, if the parallelepipedal solids on them which are similar and similarly described be proportional, the straightlines will themselves also be proportional.

Proposition 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.

Proposition 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.

Book12 (continued from the previous book)

Proposition 1. Similar polygons inscribed in circles are to one another as the square on the diameters.

Proposition 2. Circles are to one another as the squares on the diameters. Lemma. I say that, the area S being greater than the circle EFGH, as the area S is to the circle ABCD, so is the circle EFGH to some area less than the circle ABCD.

Proposition 3. Any pyramid which has a triangularbase 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.

Proposition 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, being equal in multitude, in the other pyramid. Lemma. But that, as the triangle LOC is to the triangle RVF, so is the prism in which the triangle LOC is the base and PMN its opposite, to the prism in which the triangle RVF is the base and STU its opposite, we must prove as follows.

Proposition 5. Pyramids which are of the same height and have triangular bases are to one another as the bases.

Proposition 6. Pyramids which are of the same height and have polygonal bases are to one another as the bases.

Proposition 7. Any prism which has a triangular base is divided into three pyramids equal to one another which have triangular bases. Porism. From this it is manifest that any pyramid is a third part of the prism which has the same base with it and equal height.

Proposition 8. Similar pyramids which have triangular bases are in the triplicate ratio of their corresponding sides. Porism. From this it is manifest that similar pyramids which have polygonal bases are also to one another in the triplicate ratio of their corresponding sides.

Proposition 9. In equal pyramids which have triangular bases the bases are reciprocally proportional to the heights; and those pyramids in which the bases are reciprocally proportional to the heights are equal.

Proposition 10. Any come is a third part of the cylinder which has the base with it and equal height.

Proposition 11. Cones and cylinders which are of the same height are to one another as their bases.

Proposition 12. Similar cones and cylinders are to one another in the triplicate ratio of the diameters in their bases.

Proposition 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.

Proposition 14. Cones ayd cylinders which are on equal bases are to one another as their heights.

Proposition 15. In equal cones and cylinder the bases are reciprocally proportional to the heights; and those cones and cylinder in which the bases are reciprocally proportional to the heights are equal.

Proposition 16. Given two circles about the same centre, to inscribe in the greater circle an equilateral polygon with an evennumber of sides which does not touch the lesser circle.

Proposition 17. Given two spheres about the same center, to inscribe in the greater sphere a polyhedral solid which does not touch the lesser sphere at its surface. [Porism]

Proposition 18. Spheres are to one another in the triplicate ratio of their respective diameters.

Book13 (continued from the previous book)

Proposition 1. If a straightline be cut in extreme and meanratio, the square on the greater segment added to the half of the whole is five times the squareon the half.

Proposition 2. If the square on a straightline be five times the square on a segment of it, then, when the double of the said segment is cut in extreme and meanratio, the greater segment is the remaining part of the original straightline.

Proposition 3. If a straightline 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.

Proposition 4. If a straightline be cut in extreme and meanratio, the square on the whole and the square on the lesser segment together are triple of the square on the greater segment.

Proposition 5. If a straightline be cut in extreme and meanratio, and there be added to it a straightline equal to the greater segment, the whole straightline has been cut in extreme and meanratio, and the original straightline is the greater segment.

Proposition 6. If a rational straightline be cut in extreme and meanratio, each of the segments is the irrational straightline called apotome.

Proposition 7. If three angles of an equilateral pentagon, taken either in order or not in order, be equal, the pentagon will be equiangular.

Proposition 8. If in an equilateral and equiangular pentagon straightlines subtend two angles taken in order, they cut one another in extreme and meanratio, and their greater segment are equal to the side of the pentagon.

Proposition 9. If the sides of the hexagon and that of the decagon inscribed in the same circle be added together, the whole straightline has been cut in extreme and meanratio, and its greater segment is the side of the hexagon.

Proposition 10. If an equilateral pentagon be inscribed in a circle, the square on the side of the pentagon is equal to the square on the side of the hexagon and on that of the decagon inscribed in the same circle.

Proposition 11. If in a circle which has its diameter rational an equilateral pentagon be inscribed, the side of the pentagon is the irrational straightline called minor.

Proposition 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 of the circle.

Proposition 13. To construct a pyramid, 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. Lemma. It is to be proved that, as AB is to BC, so is the square on AD to the square on DC.

Proposition 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.

Proposition 15. To construct a cube and comprehend it in a sphere, like the pyramid; and to prove that the square on the diameter of the sphere is triple of the square on the side of that cube.

Proposition 16. To construct an icosahedron and comprehend it in a sphere, like the aforesaid figures; and to prove that the side of the icosahedron is the irrational straightline called minor. Porism. From this it is manifest that the square on the diameter of the sphere is fivetimes the square on the radius of the circle from which the icosahedron has been described, and that the diameter of the sphere is composed of the side of the hexagon and two of the sides of the decagon inscribed in the same circle.

Proposition 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 straightline called apotome. Porism. From this it is manifest that, when the side of the cube is cut in extreme and mean ratio, the greater segment is the side of the dodecahedron.

Proposition 18. To set out the side of the five figures and to compare them with one another. No other figure, besides the said five figures, can be constructed which is contained by equilateral and equiangular figures equal to one another. Lemma. The angle of the equilateral and equiangular pentagon is a rightangle and a fifth.