a scratch pad for half-formed thoughts by a liberal political junkie who's nobody special. ''Hard Heads, Soft Hearts'' is the title of a book by Princeton economist Alan Blinder, and tends to be a favorite motto of neoliberals, especially liberal economists.
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Euclid for the Impractical Blogger

The Elements. Interactive
Euclid

Book 1

Book 1 Definitions

1. A point is that which has no part.
2. A line is breadthless length
3. The extremities of a line are points
4. A straight line is a line which lies evenly with the points on itself
5. A surface is that which has length and breadth only.
6. The extremities of a surface are lines.
7. A plane surface is a surface which lies evenly with the straight lines on itself
8. A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.
9. And when the lines containing the angle are straight, the angle is called rectilineal.
10. When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.
11. An obtuse angle is an angle greater than a right angle.
12. An acute angle is an angle less than a right angle.
13. A boundary is that which is an extremity of anything.
14. A figure is that which is contained by any boundary or boundaries.
15. A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another;
16. And the point is called the centre of the circle.
17. A diameter of the circle is any straight line drawn through the center and terminated in both directions by the circle and such a straight line also bisects the circle
18. A semicircle is the figure contained by the diameter and the circumference cut off by it. And the centre of the semicircle is the same as that of the circle.
19. Rectilineal figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines.
20. Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal.
21. Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute angle triangle that which has its three angles acute
22. Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.
23. Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

Book 1 Postulates

Let the following be postulated:
1. To draw a straight line from any point to any point
2. To produce a finite straight line continuously in a straight line.
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 right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

Book 1 Common Notions.

1. Things which are equal 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.
[7] 4. Things which coincide with one another are equal to one another.
[8] 5. The whole is greater than the part.

Book 1 Propositions

1. On a given straight line to construct an equilateral triangle. (1.Po1, 1.D15, 1.CN1)
2. To place at a given point (as an extremity) a straight line equal to a given straight line. (1.Po1, 1.1, 1.Po2, 1.Po3, 1.CN3, 1.CN1)
3. Given two unequal straight lines, to cut off from the greater a straight line equal to the less. (1.2, 1.Po3, 1.D15, 1.CN1)
4. If two triangles have the two sides equal to two sides respectively, and have the angles contained by the equal straight lines 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. (1.CN 4)
5. In isosceles triangles the angles at the base are equal to one another, and, if the equal straight lines be produced further, the angles under the base will be equal to one another. (1.Po2, 1.3, 1.Po1, 1.4)
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.
7. Given two straight lines constructed on a straight line (from its extremities) and meeting in a point, there cannot be constructed on the same straight line (from its extremities) and on the same side of it, two other straight lines meeting in another point and equal to the former two respectively, namely each to that which has the same extremity with it. (1.5)
8. If two triangles have the two sides equal to two sides respectively, and have also the base equal to the base, they will also have the angles equal which are contained by the equal straight lines.(1.7)
9. To bisect a given rectilineal angle. (1.3, 1.8)
10. To bisect a given finite straight line. (1.1, 1.9, 1.4)
11. To draw a straight line at right angles to a given straight line from a given point on it. (1.3, 1.1, 1.8, 1.D10)
12. To a given infinite straight line, from a given point which is not on it, to draw a perpendicular straight line. (1.Po3, 1.Po1, 1.8, 1.D10)
13. If a straight line set up on a straight line make angles, it will make either two right angles or angles equal to two right angles (1.D10, 1.11, 1.CN2, 1.CN2, 1.CN1)
14. If with any straight line, and at a point on it, two straight lines not lying on the same side make the adjacent angles equal to two right angles, the two straight lines will be in a straight line with one another. (1.13, 1.Po4 & 1.CN1, 1.CN3)
15. If two straight lines cut one another, they make the vertical angles equal to one another. (1.13, 1.13, 1.Po4 & 1.CN1, 1.CN3) Porism: From this it is manifest that, if two straight lines cut one another, they will make the angles at the point of section equal to four right angles.
16. In any triangle, if one of the sides be produced, the exterior angle is greater than either of the interior and opposite angles. (1.10, 1.3, 1.Po1, 1.Po2, 1.15, 1.4, CN5, 1.15)
17. In any triangle, two angles, taken together in any manner are less than two right angles (1.Po2, 1.16, 1.13)
18. In any triangle the greater side subtends the greater angle. (1.3, 1.16)
19. In any triangle the greater angle is subtended by the greater side. (1.5, 1.18)
20. In any triangle two sides taken together in any manner are greater than the remaining one. (1.5, CN5, 1.19)
21. If on one of the sides of a triangle, from its extremities, there be constructed two straight lines meeting within the triangle, the straight lines so constructed will be less than the remaining two sides of the triangle, but will contain a greater angle. (1.20, 1.16)
22. Out of three straight lines, which are equal to three given straight lines, to construct a triangle: thus it is necessary that two of the straight lines taken together in any manner should be greater than the remaining one. (1.20, 1.3)
23. On a given straight line and a point on it to construct a rectilineal angle equal to a given rectilineal angle. (1.22, 1.8)
24. If two triangles have the two sides equal to two sides respectively, but have one of the angles contained by the equal straight lines greater than the other, they will also have the base greater than the base. (1.23, 1.4, 1.5, 1.19)
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 straight lines greater than the other. (1.4, 1.24)
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. (1.4, 1.4, 1.4, 1.16, 1.4)
27. If a straight line falling on two straight lines make the alternate angles equal to one another, the straight lines will be parallel to one another. (1.16, 1.D23)
28. If a straight line falling on two straight lines make the exterior angle equal to the interior and opposite angle on the same side, or the interior angles on the same side equal to two right angles, the straight lines will be parallel to one another (1.15, 1.27, 1.13, 1.27)
29. A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the interior angles on the same side equal to two right angles. ((1.13, 1.Po5, 1.15, 1.CN1, 1.CN2, 1.13)
30. Straight lines parallel to the same straight line are also parallel to one another (1.29, 1.29, 1.CN1)
31. Through a given point to draw a straight line parallel to a given straight line. (1.23, 1.27)
32.In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles of the triangle are equal to two right angles. (1.31, 1.29, 1.29, 1.13)
33. The straight lines joining equal and parallel straight (at the extremities which are) in the same directions (respectively) are themselves also equal and parallel. (1.29, 1.4, 1.27)
34. In parallelogrammic areas, the opposite sides and angles are equal to one another, and the diameter bisects the areas. (1.29, 1.29, 1.26, CN2, 1.4)
35. Parallelograms which are on the same base and in the same parallels are equal to one another. (1.34, 1.CN1, 1.CN2, 1.34, 1.29, 1.4, 1.CN3, 1.CN2)
36. Parallelograms which are on equal bases and in the same parallels are equal to one another (1.CN1, 1.33, 1.34, 1.35, 1.35, 1.CN1)
37. Triangles which are on the same base and in the same parallels are equal to one another. (1.31, 1.31, 1.35, 1.34, 1.34)
38. Triangles which are on equal bases and in the same parallels are equal to one another. (1.31, 1.36, 1.34, 1.34)
39. Equal triangles which are on the same base and on the same side are also in the same parallels (1.31, 1.37, 1.CN1)
40. Equal triangles which are on equal bases and on the same side are also in the same parallels (1.31, 1.38, 1.CN1)
41. If a parallelogram have the same base with a triangle and be in the same parallels, the parallelogram is double of the triangle (1.37, 1.34)
42. To construct, in a given rectilineal angle, a parallelogram equal to a given triangle (1.23, 1.31, 1.38, 1.41)
43. In any parallelogram the complements of the parallelograms about the diameter are equal to one another. (1.34, 1.CN2, 1.CN3)
44. To a given straight line to apply, in a given rectilineal angle, a parallelogram equal to a given triangle (1.42, 1.31, 1.29, 1.Po5, 1.31, 1.43, 1.CN1, 1.15)
45. To construct, in a given rectilineal angle, a parallelogram equal to a given rectilineal figure. (1.42, 1.44, 1.CN1, 1.29, 1.14, 1.29, 1.CN2, 1.29, 1.CN1, 1.14, 1.34, 1.CN1, 1.30, 1.33)
46. On a given straight line to describe a square (1.11, 1.31, 1.34, 1.29, 1.34)
47. In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle. (1.46, 1.14, 1.CN2, 1.4, 1.41, 1.41, 1.CN2)
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. (1.47, 1.8)

Book 2

Book 2 Definitions.

1. Any rectangular parallelogram is said to be contained by the two straight lines containing the right angle.
2. And in any parallelogrammic area let any one whatever of the parallelograms about its diameter with two complements be called a gnomon.

Book 2 Propositions

1. If there be two straight lines, and one of them be cut into any number of segments whatever, the rectangle contained by the two straight lines is equal to the rectangles contained by the uncut straight line and each of the segments. (1.11, 1.3, 1.31, 1.34)
2. If a straight line be cut at random, the rectangle contained by the whole and both of the segments is equal to the square on the whole. (1.46, 1.31)
3. If a straight line be cut at random, the rectangle contained by the whole and one of the segments is equal to the rectangle contained by the segments and the square on the aforesaid segment. (1.46, 1.31)
4. If a straight line be cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments. (1.46, 1.31, 1.29, 1.5, 1.6, 1.34, 1.29, 1.34, 1.34)
5. If a straight line be cut into equal and unequal segments, the rectangle contained by the unequal segments of the whole together with the square one the straight line between the points of section is equal to the square on the half. (1.46, 1.31, 1.43, 1.36)
6. If a straight line be bisected and a straight line be added to it in a straight line, the rectangle contained by the whole with the added straight line and the added straight line together with the square on the half is equal to the square on the straight line made up of half and the added straight line. (1.46, 1.31, 1.36, 1.43)
7. If a straight line be cut at random, the square on the whole and that on one of the segments both together are equal to twice the rectangle contained by the whole and the said segment and the square on the remaining segment. (1.46, 1.43)
8. If a straight line be cut at random, four times the rectangle contained by the whole and one of the segments together with the square on the remaining segment is equal to the square described on the whole and the aforesaid segment as on one straight line. (1.36, 1.43, 1.36, 1.43)
9. If a straight line be cut into equal and unequal segments, the squares on the unequal segments of the whole are double of the square on the half and of the square on the straight line between the points of section. (1.32, 1.29, 1.32, 1.6, 1.29, 1.32, 1.6, 1.47, 1.34, 1.47, 1.47)
10. If a straight line be bisected, and a straight line be added to it in a straight line, the square on the whole with the added straight line and the square on the added straight line both together are double of the square on the half and of the square described on the straight line made up of half and the added straight line as on one straight line. (1.11, 1.3, 1.31, 1.29, 1.Po 5, 1.5, 1.32, 1.15, 1.29, 1.32, 1.6, 1.34, 1.32, 1.6, 1.47, 1.CN1, 1.47, 1.34, 1.47, 1.47)
11. To cut a given straight line so that the rectangle contained by the whole and one of the segments is equal to the square on the remaining segment. (1.46, 2.6, 1.47)
12. In obtuse-angled triangles the square on the side subtending the obtuse angle is greater than the squares on the sides containing the obtuse angle by twice the rectangle contained by one of the sides above the obtuse angle, namely that on which the perpendicular falls, and the straight line cut off outside by the perpendicular towards the obtuse angle. (2.4, 1.47, 1.47)
13. In acute-angled triangles the square on the side subtending the acute angle is less than the squares on the sides containing the acute angle by twice the rectangle contained by one of the sides about the acute angle, namely that on which the perpendicular falls, and the straight line cut off within by the perpendicular towards the acute angle. (2.7, 1.47)
14. To construct a square equal to a given rectilineal figure. (1.45, 2.5, 1.47)

Book 3

Book 3 Definitions.

1. Equal circles are those the diameters of which are equal, or the radii of which are equal.
2. A straight line is said to touch a circle which, meeting the circle and being produced, does not cut the circle.
3. Circles are said to touch one another which, meeting one another, do not cut one another.
4. In a circle straight lines are said to equally distant from the centre when the perpendiculars drawn to them from the centre are equal.
5. And the straight line is said to be at a greater distance on which the greater perpendicular falls.
6. A segment of a circle is the figure contained by a straight line and a circumference of a circle.
7. An angle of a segment is that contained by a straight line and circumference of a circle.
8. An angle in a segment is the angle which, when a point is taken on the circumference of the segment and straight lines are joined from it to the extremities of the straight line which is the base of the segment, is contained by the straight lines so joined.
9. And when the straight lines containing the angle cut off a circumference, the angle is said to stand upon that circumference.
10. A sector of a circle is the figure which, when an angle is constructed at the centre of the circle, is contained by the straight lines containing the angle and the circumference cut off by them.
11. Similar segments of circles are those which admit equal angles, or in which the angles are equal to one another.

Book 3 Propositions.

1. To find the center of a given circle. (1.8, 1.Def 10) Porism: From this it is manifest that, if in a circle a straight line cut a straight line into two equal parts and at right angles, the centre of the circle is on the cutting straight line.
2. If on the circumference of a circle two points be taken at random, the straight line joining the points will fall within the circle. (3.1, 1.5, 1.16, 1.19)
3. If in a circle a straight line through the centre bisect a straight line not through the centre, it also cuts it at right angles; and if it cut it at right angles, it also bisects it. (1.8, 1.Def 10, 1.5, 1.26)
4. If in a circle two straight lines cut one another which are not through the centre, they do not bisect one another. (3.1, 3.3, 3.3)
5. If two circles cut one another, they will not have the same centre. (1.Def 15)
6. If two circles touch one another, they will not have the same centre.
7. If on a diameter of a circle a point be taken which is not the centre of the circle, and from the point straight lines fall upon the circle, that will be greatest on which the centre is, the remainder of the same diameter will be least, and of the rest the nearer to the straight line through the centre is always greater than the more remote, and the only two equal straight lines will fall from the point on the circle, one on each side of the least straight line. (1.20, 1.24, 1.23, 1.4)
8. If a point be taken outside a circle and from the point straight lines be drawn through to the circle, one of which is through the centre and the others are drawn at random, then, of the straight lines 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 straight lines falling on the convex circumference, that between the point and the diameter is least, while of the rest the nearer to the least, is always less than the more remote, and the only two equal straight lines will fall on the circle from the point, one on each side of the least. (3.1, 1.20, 1.24, 1.20, 1.21, 1.4)
9. If a point be taken within a circle, and more than two equal straight lines fall from the point on the circle, the point taken is the centre of the circle. (1.8, 1.Def 10, 3.1 Por)
10. A circle does not cut a circle at more points than two. (3.1 Por, 3.5)
11. If two circles touch one another internally, and their centres be taken, the straight line joining their centres, if it be also produced, will fall on the point of contact of the circles.
12. If two circles touch one another externally, the straight line joining their centres will pass through the point of contact. (1.20)
13. A circle does not touch a circle at more points than one, whether it touch it internally or externally. (3.11, 3.2)
14. In a circle equal straight lines are equally distant from the centre, and those which are equally distant from the centre are equal to one another. (3.1, 3.3, 1.47, 3.Def 4, 1.47)
15. Of straight lines in a circle the diameter is greatest, and of the rest the nearer to the centre is always greater than the more remote. (3.Def 5, 3.14, 1.20, 1.24)
16. The straight line drawn at right angles to the diameter of a circle from its extremity will fall outside the circle, and into the space between the straight line and the circumference another straight line cannot be interposed; further the angle of the semicircle is greater, and the remaining angle less, than any acute rectilineal angle (1.5, 1.17, 1.19) Porism: From this it is manifest that the straight line drawn at right angles to the diameter of a circle from its extremity touches the circle.
17. From a given point to draw a straight line touching a given circle. (3.1, 1.4, 3.16 Por)
18. If a straight line touch a circle, and a straight line be joined from the centre to the point of contact, the straight line so joined will be perpendicular to the tangent. (1.17, 1.19)
19. If a straight line touch a circle, and from the point of contact a straight line be drawn at right angles to the tangent, the centre of the circle will be on the straight line so drawn. (3.18)
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. (1.5, 1.32)
21. In a circle the angles in the same segment are equal to one another. (3.20)
22. The opposite angles of quadrilaterals in circles are equal to two right angles. (1.32, 3.21)
23. On the same straight line there cannot be constructed two similar and unequal segments of circles on the same side. (3.Def 11, 1.16)
24. Similar segments of circles on equal straight lines are equal to one another. (3.10)
25. Given a segment of a circle, to describe the complete circle of which it is a segment. (1.6, 3.9)
26. In equal circles equal angles stand on equal circumferences, whether they stand at the centres or at the circumferences. (1.4, 3.Def 11, 3.24)
27. In equal circles angles standing on equal circumferences are equal to one another, whether they stand at the centres or at the circumferences. (1.23, 3.26, 3.20)
28. In equal circles equal straight lines cut off equal circumferences, the greater equal to the greater and the less to the less. (1.8, 3.26)
29. In equal circles equal circumferences are subtended by equal straight lines. (3.27, 1.4)
30. To bisect a given circumference (1.4, 3.28)
31. In a circle the angle in the semicircle is right, that in a greater segment less than a right angle, and that in a less segment greater than a right angle, and further the angle of the greater segment is greater than a right angle, and the angle of the less segment less than a right angle. (1.5, 1.5, 1.32, 1.Def 10, 1.17, 3.22)
32. If a straight line touch a circle, and from the point of contact there be drawn across, in the circle, a straight line cutting the circle, the angles which it makes with the tangent will be equal to the angles in the alternate segments of the circle. (3.19, 3.31, 1.32, 3.22)
33. On a given straight line to describe a segment of a circle admitting an angle equal to a given rectilineal angle. (1.4, 3.16 Por, 3.32, 3.16 Por, 3.31, 1.4, 3.16 Por, 3.32)
34. From a given circle to cut off a segment admitting an angle equal to a given rectilineal angle. (1.23, 3.32)
35. If in a circle two straight lines cut one another, the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other. (3.3, 2.5, 1.47)
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 straight line which cuts the circle and the straight line intercepted on it outside between the point and the convex circumference will be equal to the square on the tangent. (3.18, 2.6, 1.47, 3.18, 3.3, 2.6, 1.47, 1.47)
37. If a point be taken outside a circle and from the point there fall on the circle two straight lines, if one of them cut the circle, and the other fall on it, and if further the rectangle contained by the whole of the straight line which cuts the circle and the straight line intercepted on it outside between the point and the convex circumference be equal to the square on the straight line which falls on the circle, the straight line which falls on it will touch the circle. (3.18, 3.36, 1.8, 3.16 Por)

Book 4

Book 4 Definitions

1. A rectilineal figure is said to be inscribed in a rectilineal figure when the respective angles of the inscribed figure lie on the respective sides of that in which it is inscribed.
2. Similarly a figure is said to be circumscribed about a figure when the respective sides of the circumscribed figure pass through the respective angles of that about which it is circumscribed.
3. A rectilineal figure is said to be inscribed in a circle when each angle of the inscribed figure lies on the circumference of the circle.
4. A rectilineal figure is said to be circumscribed about a circle, when each side of the circumscribed figure touches the circumference of the circle.
5. Similarly a circle is said to be inscribed in a figure when the circumference of the circle touches each side of the figure in which it is inscribed.
6. A circle is said to be circumscribed about a figure when the circumference of the circle passes through each angle of the figure about which it is circumscribed.
7. A straight line is said to be fitted into a circle when its extremities are on the circumference of the circle.

Book 4 Propositions

1. Into a given circle to fit a straight line equal to a given straight line which is not greater than the diameter of the circle.
2. In a given circle to inscribe a triangle equiangular with a given triangle. (3.16 Por, 1.23, 3.32, 1.32)
3. About a given circle to circumscribe a triangle equiangular with a given triangle. (3.1, 1.23, 3.16 Por, 3.18, 1.13, 1.32)
4. In a given triangle to inscribe a circle. (1.9, 1.26, 3.16, 4.Def 5)
5. About a given triangle to circumscribe a circle. (1.10, 1.4, 1.4, 3.31)
6. In a given circle to inscribe a square. (1.4, 3.31, 1.Def 22)
7. About a given circle to circumscribe a square. (3.16 Por, 3.18, 1.28, 1.30, 1.34, 1.34, 1.34)
8. In a given square to inscribe a circle. (1.10, 1.31, 1.34, 3.16)
9. About a given square to circumscribe a circle. (1.8, 1.6)
10. To construct an isosceles triangle having each of the angles at the base double of the remaining one. (2.11, 4.1, 4.5, 3.37, 3.32, 1.32, 1.5, 1.6, 1.5)
11. In a given circle to inscribe an equilateral and equiangular pentagon. (4.10, 4.2, 1.9, 3.26, 3.29, 3.27)
12. About a given circle to circumscribe an equilateral and equiangular pentagon. (4.11, 3.16 Por, 3.1, 3.18, 1.47, 1.8, 3.27, 1.26)
13. In a given pentagon, which is equilateral and equiangular, to inscribe a circle. (1.4, 1.26, 3.16)
14. About a given pentagon, which is equilateral and equiangular, to circumscribe a circle. (1.6)
15. In a given circle to inscribe an equilateral and equiangular hexagon. (1.5, 1.32, 1.15, 3.26, 3.29, 3.27) Porism: From this it is manifest that the side of the hexagon is equal to the radius of the circle. And, in like manner as in the case of the pentagon if through the points of division on the circle we draw tangents to the circle, there will be circumscribed about the circle an equilateral and equiangular hexagon in conformity with what was explained in the case of the pentagon. And further by means similar to those explained in the case of the pentagon we can both inscrible a circle in a given hexagon and circumscribe one about it.
16. In a given circle to inscribe a fifteen-angled figure which shall be both equilateral and equiangular. (3.30) And, in like manner as in the case of the pentagon, if through the points of division on the circle we draw tangents to the circle, there will be circumscribed about the circle a fifteen-angles figure which is equilateral and equiangular. And further, by proofs similar to those in the case of the pentagon, we can both inscribe a circle in the given fifteen-angled figure and circumscribe one about it.

Book 5

Book 5 Definitions

1. A magnitude is a part of a magnitude, the less of the greater, when it measures the greater.
2. The greater is a multiple of the less when it is measured by the less.
3. A ratio is a sort of relation in respect of size between two magnitudes of the same kind.
4. Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another.
5. Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, is any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.
6. Let magnitudes which have the same ratio be called proportional.
7. When, of the equimultiples, the multiple of the first magnitude exceeds the multiple of the second, but the multiple of the third does not exceed the multiple of the fourth, then the first is said to have a greater ratio to the second than the third has to the fourth.
8. A proportion in three terms is the least possible
9. When three magnitudes are proportional, the first is said to have to the third the duplicate ratio of that which it has to the second.
10. When four magnitudes are proportional, the first is said to have to the fourth the triplicate ratio of that which it has to the second, and so on continually, whatever be the proportion.
11. The term corresponding magnitudes is used of antecedents in relation to antecedents, and of consequents in relation to consequents.
12. Alternate ratio means taking the antecedent in relation the the antecedent and the consequent in relation to the consequent.
13. Inverse ratio means taking the consequent as antecedent in relation to the antecedent as consequent.
14. Composition of a ratio means taking the antecedent together with the consequent as one in relation to the consequent by itself.
15. Separation of a ratio means taking the excess by which the antecedent exceeds the consequent in relation to the consequent by itself.
16. Conversion of a ratio means taking the antecedent in relation to the excess by which the antecedent exceeds the consequent.
17. A ratio ex aequali arises when, there being several magnitudes and another set equal to them in multitude which taken two and two are in the same proportion, as the first is to the last among the first magnitudes, so is the first to the last among the second magnitudes; Or, in other words, it means taking the extreme terms by virtue of the removal of the intermediate terms.
18. A perturbed proportion arises when, there being three magnitudes and another set equal to them in multitude, as antecedent is to consequent among the first magnitudes, so is antecedent to consequent among the second magnitudes, while, as the consequent is to a third among the first magnitudes, so is a third to the antecedent among the second magnitudes.

Book 5 Propositions.

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.
2. If a first magnitude be the same multiple of a second that a third is of a fourth, and a fifth also be the same multiple of the second that a sixth is of the fourth, 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.
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. (5.2)
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. (5.3, 5.Def 5, 5.Def 5)
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. (5.1)
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. (5.2)
7. Equal magnitudes have to the same the same ratio, as also has the same to equal magnitudes. (5.Def 5, 5.Def 5) Porism: From this it is manifest that, if any magnitudes are proportional, they will also be proportional inversely.
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. (5.Def 4, 5.1, 5.Def 7, 5.Def 7, 5.Def 4)
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. (5.8, 5.8)
10. Of magnitudes which have a ratio to the same, that which has a greater ratio is greater, and that to which the same has a greater ratio is greater; and that to which the same has a greater ratio is less. (5.7, 5.8, 5.7, 5.8)
11. Ratios which are the same with the same ratio are also the same with one another.
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. (5.1, 5.Def 5)
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. (5.Def 7, 5.Def 5, 5.Def 7)
14. If a first magnitude have to a second the same ratio as a third has to a fourth, and the first be greater than the third, the second will also be greater than the fourth ; if equal, equal; and if less, less. (5.8, 5.13, 5.10)
15. Parts have the same ratio as the same multiples of them taken in corresponding order. (5.7, 5.12)
16. If four magnitudes be proportional, they will also be proportional alternately. (5.15, 5.11, 5.15, 5.11, 5.14, 5.Def 5)
17. If magnitudes be proportional componendo, they will also be proportional separando. (5.1, 5.1, 5.2)
18. If magnitudes be proportional separando, they will also be proportional componendo. (5.17, 5.11, 5.14)
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. (5.16, 5.17, 5.16, 5.11) Porism: From this it is manifest that, if magnitudes be proportional componendo, they will also be proportional convertendo.
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. (5.8, 5.13, 5.10)
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 proprtion 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. (5.8, 5.13, 5.10)
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. (5.4, 5.20, 5.Def 5)
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. (5.15, 5.11, 5.16, 5.15, 5.11, 5.15, 5.11, 5.16, 5.21)
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.(5.22, 5.18, 5.22)
25. If four magnitudes be proportional, the greatest and the least are greater than the remaining two. (5.19)

Book 6

Book 6 Definitions

1. Similar rectilinieal figures are such as have their angles severally equal and the sides about the equal angles proportional.
2. Two figures are reciprocally related when there are in each of the two figures antecedent and consequent ratios
3. A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.
4. The height of any figure is the perpendicular drawn from the vertex to the base.

Book 6 Propositions

1. Triangles and parallelograms which are under the same height are to one another as their bases. (1.38, 1.38, 5.Def 5, 1.41, 5.15, 5.11)
2. If a straight line be drawn parallel to one of the sides of the triangle, it will cut the sides of the triangle proportionally; and if the 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. (5.7, 6.1, 5.11, 6.1, 5.11, 5.9, 1.39)
3. If an angle of a triangle be bisected and the straight line cutting the angle cut the base also, the segments of the base will have the same ratio as the remaining sides of the triangle; and if the segments of the base have the same ratio as the remaining sides of the triangle, the straight line joined from the vertex to the point of section will bisect the angle of the triangle. (1.29, 1.29, 1.6, 6.2, 6.2, 5.11, 5.9, 1.5, 1.29, [id.]
4. In equiangular triangles the sides about the equal angles are proportional, and those are corresponding sides which subtend the equal angles. (1.17, 1.Post 5, 1.28, 1.28, 1.34, 6.2, 5.16, 6.2, 5.16, 5.22)
5. If two triangles have their sides proportional, the triangles will be equiangular and will have those angles equal which the corresponding sides subtend. (1.23, 1.32, 6.4, 5.11, 5.9, 1.8, 1.4)
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. (1.23, 1.32, 6.4, 5.11, 5.9, 1.4, 1.32)
7. If two triangles have one angle equal to one angle, the sides about other angles proportional, and the remaining angles either both less or both not less than a right angle, the triangles will be equiangular and will have those angles equal, the sides about which are proportional. (1.23, 1.32, 6.4, 5.11, 5.9. 1.5, 1.13, 1.32, 1.5, 1.17, 1.32)
8. If in a right-angled triangle a perpendicular be drawn from the right angle to the base, the triangles adjoining the perpendicular are similar both to the whole and to one another (1.32, 6.4, 6.Def 1, 1.32, 6.4, 6.Def 1) Porism: From this it is clear that, if in a right-angled triangle a perpendicular be drawn from the right angle to the base, the straight line so drawn is a mean proportional between the segments of the base.
9. From a given straight line to cut off a prescribed part (1.3, 1.31, 6.2)
10. To cut a given uncut straight line similarly to a given cut straight line. (1.31, 1.34, 6.2, 6.2)
11. To two given straight lines to find a third proportional. (1.3, 1.31, 6.2)
12. To three given straight lines to find a fourth proportional. (1.31, 6.2)
13. To two given straight lines to find a mean proportional. (3.31, 6.8 Por)
14. In equal and equiangular parallelograms the sides about the equal angles are reciprocally proportional; and equiangular parallelograms in which the sides about the equal angles are receprocally proportional are equal. (1.14, 5.7, 6.1, [id.], 5.11, 6.1, 6.1, 5.11, 5.9)
15. In equal triangles which have one angle equal to one angle the sides about the equal angles are reciprocally proportional; and those triangles which have one angle equal to one angle, and in which the sides about the equal angles are reciprocally proportional, are equal. (1.14, 5.7, 6.1, [id.], 5.11, 6.1, 5.11, 5.9)
16. If four straight lines be proportional, the rectangle contained by the extremes is equal to the rectangle contained by the means; and if the rectangle contained by the extremes be equal to the rectangle contained by the means, the four straight lines will be proportional. (6.14, 6.14)
17. If three straight lines be proportional the rectangle contained by the extremes is equal to the square on the mean; and if the rectangle contained by the extremes be equal to the square on the mean, the three straight lines will be proportional. (6.16, 6.16)
18. On a given straight line to describe a rectilineal figure similar and similarly situated to a given rectilineal figure. (1.23, 1.32, 1.23, 1.32, 6.4, 6.Def 1)
19. Similar triangles are to one another in the duplicate ratio of the corresponding sides. (5.Def 11, 6.11, 5.16, 5.11, 6.15, 5.Def 9, 6.1) Porism: From this it is manifest that, if three straight lines be proportional, then, as the first is to the the third, so is the figure described on the first to that which is similar and similarly described on the second.
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. (6.Def 1, 6.6, 6.4 & 6.Def 1, 5.22, 6.6, 6.4 & 6.Def 1, 6.6, 1.32, 6.1, 5.12, 5.12, 6.19) 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; therefore, also, generally, similar rectilineal figures are to one another in the duplicate ratio of the corresponding sides.
21. Figures which are similar to the same rectilineal figure are also similar to one another. (6.Def 1)
22. If four straight lines be proportional, the rectilineal figures similar and similarly described upon them will also be proportional; and, if the rectilineal figures similar and similarly described upon them be proportional, the straight lines will themselves also be proportional. (6.11 5.22, 5.11, 6.12, 6.18, 5.11, 5.9)
23. Equiangular parallelograms have to one another the ratio compounded of the ratios of their sides. (6.12, 6.1, 5.11, 6.1, 5.11)
24. In any parallelograms the parallelograms about the diameter are similar both to the whole and to one another. (6.2, 6.2, 5.18, 5.16, 5.22, 6.Def 1, 6.21)
25. To construct one and the same figure similar to a given rectilineal figure and equal to another given rectilineal figure. (1.44, 1.45, 6.13, 6.18, 6.19 Por, 6.1, 5.16)
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. (1.31, 6.24, 5.11, 5.9)
27. Of all the parallelograms applied to the same straight line and deficient by parallelogrammic figures similar and similarly situated to that described on the half of the straight line, that parallelogram is greatest which is applied to the half of the straight line and is similar to the defect. (6.26, 1.43, 1.36)
28. To a given straight line to apply a parallelogram equal to a given rectilineal figure and deficient by a parallelogrammic figure similar to a given one: thus the given rectilineal figure must not be greater than the parallelogram described on the half of the straight line and similar to the defect. (6.18, 6.25, 6.21, 6.27, 6.26, 1.36)
29. To a given straight line to apply a parallelogram equal to a given rectilineal figure and exceeding by a parallelogrammic figure similar to a given one. (6.25, 6.21, 6.26, 1.36, 1.43, 6.24)
30. To cut a given finite straight line in extreme and mean ratio. (6.29, 6.14)
31. In right-angled triangles the figure on the side subtending the the right angle is equal to the similar and similarly described figures on the sides containing the right angle. (6.8, 6.Def 1, 6.19 Por)
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 triangle will be in a straight line. (1.29, 6.6, 1.32, 1.14)
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 (3.27, 3.27, 5.Def 5)

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