Claims in this paper
600 addressable claims. Click to open inline.
- I.1Proposition I.1On a given finite straight line to construct an equilateral triangle.
- I.2Proposition I.2To place at a given point (as an extremity) a straight line equal to a given straight…
- I.3Proposition I.3Given two unequal straight lines, to cut off from the greater a straight line equal to…
- I.4Proposition I.4If two triangles have two sides equal to two sides respectively, and have the angles…
- I.5Proposition I.5In isosceles triangles the angles at the base are equal to one another; and if the equal…
- I.6Proposition I.6If in a triangle two angles are equal to one another, the sides which subtend the equal…
- I.7Proposition I.7Given two straight lines constructed on a straight line and meeting in a point, there…
- I.8Proposition I.8If two triangles have the two sides equal to two sides respectively, and have also the…
- I.9Proposition I.9To bisect a given rectilineal angle.
- I.10Proposition I.10To bisect a given finite straight line.
- I.11Proposition I.11To draw a straight line at right angles to a given straight line from a given point on it.
- I.12Proposition I.12To draw a perpendicular straight line to a given infinite straight line from a given…
- I.13Proposition I.13If a straight line set up on a straight line make angles, it will make either two right…
- I.14Proposition I.14If with any straight line, and at a point on it, two straight lines not lying on the same…
- I.15Proposition I.15If two straight lines cut one another, they make the vertical angles equal to one another.
- I.16Proposition I.16In any triangle, if one of the sides be produced, the exterior angle is greater than…
- I.17Proposition I.17In any triangle two angles taken together in any manner are less than two right angles.
- I.18Proposition I.18In any triangle the greater side subtends the greater angle.
- I.19Proposition I.19In any triangle the greater angle is subtended by the greater side.
- I.20Proposition I.20In any triangle two sides taken together in any manner are greater than the remaining one.
- I.21Proposition I.21If on one of the sides of a triangle, from its extremities, there be constructed two…
- I.22Proposition I.22Out of three straight lines, which are equal to three given straight lines, to construct…
- I.23Proposition I.23On a given straight line and at a point on it to construct a rectilineal angle equal to a…
- I.24Proposition I.24If two triangles have the two sides equal to two sides respectively, but have the one of…
- I.25Proposition I.25If two triangles have the two sides equal to two sides respectively, but have the one…
- I.26Proposition I.26If two triangles have the two angles equal to two angles respectively, and one side equal…
- I.27Proposition I.27If a straight line falling on two straight lines make the alternate angles equal to one…
- I.28Proposition I.28If a straight line falling on two straight lines make the exterior angle equal to the…
- I.29Proposition I.29A straight line falling on parallel straight lines makes the alternate angles equal to…
- I.30Proposition I.30Straight lines parallel to the same straight line are also parallel to one another.
- I.31Proposition I.31Through a given point to draw a straight line parallel to a given straight line.
- I.32Proposition I.32In any triangle, if one of the sides be produced, the exterior angle is equal to the two…
- I.33Proposition I.33The straight lines joining equal and parallel straight lines (at the extremities which…
- I.34Proposition I.34In parallelogrammic areas the opposite sides and angles are equal to one another, and the…
- I.35Proposition I.35Parallelograms which are on the same base and in the same parallels are equal to one…
- I.36Proposition I.36Parallelograms which are on equal bases and in the same parallels are equal to one…
- I.37Proposition I.37Triangles which are on the same base and in the same parallels are equal to one another.
- I.38Proposition I.38Triangles which are on equal bases and in the same parallels are equal to one another.
- I.39Proposition I.39Equal triangles which are on the same base and on the same side are also in the same…
- I.40Proposition I.40Equal triangles which are on equal bases and on the same side are also in the same…
- I.41Proposition I.41If a parallelogram have the same base with a triangle and be in the same parallels, the…
- I.42Proposition I.42To construct, in a given rectilineal angle, a parallelogram equal to a given triangle.
- I.43Proposition I.43In any parallelogram the complements of the parallelograms about the diameter are equal…
- I.44Proposition I.44To a given straight line to apply, in a given rectilineal angle, a parallelogram equal to…
- I.45Proposition I.45To construct, in a given rectilineal angle, a parallelogram equal to a given rectilineal…
- I.46Proposition I.46On a given straight line to describe a square.
- I.47Proposition I.47In right-angled triangles the square on the side subtending the right angle is equal to…
- I.48Proposition I.48If in a triangle the square on one of the sides be equal to the squares on the remaining…
- II.1Proposition II.1If there be two straight lines, and one of them be cut into any number of segments…
- II.2Proposition II.2If a straight line be cut at random, the rectangle contained by the whole and both of the…
- II.3Proposition II.3If a straight line be cut at random, the rectangle contained by the whole and one of the…
- II.4Proposition II.4If a straight line be cut at random, the square on the whole is equal to the squares on…
- II.5Proposition II.5If a straight line be cut into equal and unequal segments, the rectangle contained by the…
- II.6Proposition II.6If a straight line be bisected and a straight line be added to it in a straight line, the…
- II.7Proposition II.7If a straight line be cut at random, the square on the whole and that on one of the…
- II.8Proposition II.8If a straight line be cut at random, four times the rectangle contained by the whole and…
- II.9Proposition II.9If a straight line be cut into equal and unequal segments, the squares on the unequal…
- II.10Proposition II.10If a straight line be bisected and a straight line be added to it in a straight line, the…
- II.11Proposition II.11To cut a given straight line so that the rectangle contained by the whole and one of the…
- II.12Proposition II.12In obtuse-angled triangles the square on the side subtending the obtuse angle is greater…
- II.13Proposition II.13In acute-angled triangles the square on the side subtending the acute angle is less than…
- II.14Proposition II.14To construct a square equal to a given rectilineal figure.
- III.1Proposition III.1To find the centre of a given circle.
- III.2Proposition III.2If on the circumference of a circle two points be taken at random, the straight line…
- III.3Proposition III.3If in a circle a straight line through the centre bisect a straight line not through the…
- III.4Proposition III.4If in a circle two straight lines cut one another which are not through the centre, they…
- III.5Proposition III.5If two circles cut one another, they will not have the same centre.
- III.6Proposition III.6If two circles touch one another, they will not have the same centre.
- III.7Proposition III.7If on the diameter of a circle a point be taken which is not the centre, and from the…
- III.8Proposition III.8If a point be taken outside a circle and from the point straight lines be drawn through…
- III.9Proposition III.9If a point be taken within a circle, and more than two equal straight lines fall from the…
- III.10Proposition III.10A circle does not cut a circle at more points than two.
- III.11Proposition III.11If two circles touch one another internally, and their centres be taken, the straight…
- III.12Proposition III.12If two circles touch one another externally, the straight line joining their centres will…
- III.13Proposition III.13A circle does not touch a circle at more points than one, whether it touch it internally…
- III.14Proposition III.14In a circle equal straight lines are equally distant from the centre, and those which are…
- III.15Proposition III.15Of straight lines in a circle the diameter is greatest, and of the rest the nearer to the…
- III.16Proposition III.16The straight line drawn at right angles to the diameter of a circle from its extremity…
- III.17Proposition III.17From a given point to draw a straight line touching a given circle.
- III.18Proposition III.18If a straight line touch a circle, and a straight line be joined from the centre to the…
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