Content of type 'def'

X.def.3.5

if the annex be so commensurable, a fifth;

X.def.3.4

Again, if the square on the whole be greater than the square on the annex by the square on a straight line incommensurable with the whole, then, if the whole be commensurable in length with the rational straight line set out, let the apotome be called a fourth apotome;

X.def.3.3

But if neither be commensurable in length with the rational straight line set out, and the square on the whole be greater than the square on the annex by the square on a straight line commensurable with the whole, let the apotome be called a third apotome.

X.def.3.2

But if the annex be commensurable in length with the rational straight line set out, and the square on the whole be greater than that on the annex by the square on a straight line commensurable with the whole, let the apotome be called a second apotome.

X.def.3.1

Given a rational straight line and an apotome, if the square on the whole be greater than the square on the annex by the square on a straight line commensurable in length with the whole, and the whole be commensurable in length with the rational straight line set out, let the apotome be called a first apotome.

X.def.2.6

and if neither, a sixth binomial.

X.def.2.4

Again, if the square on the greater term be greater than the square on the lesser by the square on a straight line incommensurable in length with the greater, then, if the greater term be commensurable in length with the rational straight line set out, let the whole be called a fourth binomial;

X.def.2.3

and if neither of the terms be commensurable in length with the rational straight line set out, let the whole be called a third binomial.

X.def.2.2

but if the lesser term be commensurable in length with the rational straight line set out, let the whole be called a second binomial;

X.def.2.1

Given a rational straight line and a binomial, divided into its terms, such that the square on the greater term is greater than the square on the lesser by the square on a straight line commensurable in length with the greater, then, if the greater term be commensurable in length with the rational straight line set out, let the whole be called a first binomial straight line;

XI.def.28

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

XI.def.27

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

XI.def.26

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

XI.def.25

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

XI.def.24

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

I.def.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.

XI.def.23

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

I.def.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.

VII.def.22

A perfect number is that which is equal to its own parts.

XI.def.22

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

VII.def.21

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

I.def.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 acuteangled triangle that which has its three angles acute.

XI.def.21

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

VII.def.20

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

I.def.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.

XI.def.20

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

XI.def.19

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

I.def.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.

VII.def.19

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

I.def.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.

XI.def.18

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

V.def.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.

VII.def.18

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

I.def.17

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

V.def.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;

VII.def.17

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

XI.def.17

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

VII.def.16

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

I.def.16

And the point is called the centre of the circle.

XI.def.16

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

V.def.16

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

XI.def.15

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

I.def.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;

V.def.15

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

VII.def.15

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

XI.def.14

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

I.def.14

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

VII.def.14

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

V.def.14

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

I.def.13

A boundary is that which is an extremity of anything.

XI.def.13

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

VII.def.13

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

V.def.13

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

VII.def.12

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

V.def.12

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

I.def.12

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

XI.def.12

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

III.def.11

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

V.def.11

The term corresponding magnitudes is used of antecedents in relation to antecedents, and of consequents in relation to consequents.

VII.def.11

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

I.def.11

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

XI.def.11

A solid angle is the inclination constituted by more than two lines which meet one another and are not in the same surface, towards all the lines.

III.def.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.

V.def.10

When four magnitudes are <continuously> proportional, the first is said to have to the fourth the triplicate ratio of that which it has to the second, and so on continually, whatever be the proportion.

I.def.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.

XI.def.10

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

VII.def.10

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

III.def.9

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

I.def.9

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

XI.def.9

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

V.def.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.

VII.def.9

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

III.def.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.

V.def.8

A proportion in three terms is the least possible.

VII.def.8

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

XI.def.8

Parallel planes are those which do not meet.

I.def.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.

VII.def.7

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

XI.def.7

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

V.def.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.

I.def.7

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

IV.def.7

A straight line is said to be fitted into a circle when its extremities are on the circumference of the circle.

III.def.7

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

I.def.6

The extremities of a surface are lines.

III.def.6

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

VII.def.6

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

IV.def.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.

XI.def.6

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

V.def.6

Let magnitudes which have the same ratio be called proportional.

V.def.5

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

III.def.5

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

IV.def.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.

VII.def.5

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

I.def.5

A surface is that which has length and breadth only.

XI.def.5

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

VI.def.4

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

IV.def.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.

III.def.4

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

X.def.4

And let the square on the assigned straight line be called rational and those areas which are commensurable with it rational, but those which are incommensurable with it irrational, and the straight lines which produce them irrational, that is, in case the areas are squares, the sides themselves, but in case they are any other rectilineal figures, the straight lines on which are described squares equal to them.

VII.def.4

but parts when it does not measure it.

I.def.4

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

XI.def.4

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

V.def.4

Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another.

IV.def.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.

X.def.3

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

I.def.3

The extremities of a line are points.

III.def.3

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

VII.def.3

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

VI.def.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.

XI.def.3

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

V.def.3

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

III.def.2

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

XI.def.2

An extremity of a solid is a surface.

IV.def.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.

X.def.2

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

VII.def.2

A number is a multitude composed of units.

II.def.2

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

VI.def.2

[Reciprocally related figures. See note.]

V.def.2

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

IV.def.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.

V.def.1

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

X.def.1

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

II.def.1

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

I.def.1

A point is that which has no part.

VI.def.1

Similar rectilineal figures are such as have their angles severally equal and the sides about the equal angles proportional.

VII.def.1

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

XI.def.1

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

III.def.1

Equal circles are those the diameters of which are equal, or the radii of which are equal.