Content of type 'def'

G.174 ex aequali (equality of ratios)

A:term:ratio <G.173>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.

G.176 magnitudes in the same ratio

Magnitudes are said to:term:**be in the same ratio** <in-the-same-ratio>, the first to the second and the third to the fourth, when, if any:term:equimultiples <multiple> 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.

G.203 perturbed proportion

A:term:**perturbed proportion** <perturbed-proportion> arises when, there being three magnitudes and another set equal to them in multitude, as:term:antecedent <G.181> 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.

G.391 rational and irrational segments

With these hypotheses, it is proved that there exist:term:segments <G.4> infinite in multitude which are:term:commensurable <G.372> and incommensurable respectively, some in length only, and others in square also, with an assigned segment. Let then the assigned segment be called:term:**rational** <rational>, and those segments which are commensurable with it, whether in length and in square or in square only,:term:**rational** <rational>, but those which are incommensurable with it:term:**irrational** <irrational>.

G.392 rational and irrational areas

And let the:term:square <G.63> on the assigned:term:segment <G.4> be called:term:**rational** <rational> and those areas which are:term:commensurable <G.372> with it:term:**rational** <rational>, but those which are incommensurable with it:term:**irrational** <G.391>, and the segments which produce them:term:**irrational** <G.391>, that is, in case the areas are squares, the sides themselves, but in case they are any other:term:polygons <G.12>, the segments on which are described squares equal to them.

G.422 first binomial

Given a:term:rational <G.392>segment and a binomial, divided into its terms, such that the:term:square <G.63> on the greater term is greater than the square on the lesser by the square on a segment:term:commensurable <G.372> in length with the greater, then, if the greater term be commensurable in length with the rational segment set out, let the whole be called a:term:**first binomial segment** <first-binomial-straight-line>;

G.428 fourth binomial

Again, if the:term:square <G.63> on the greater term be greater than the square on the lesser by the square on a:term:segment <G.4> incommensurable in length with the greater, then, if the greater term be:term:commensurable <G.372> in length with the:term:rational <G.392> segment set out, let the whole be called a:term:**fourth binomial** <fourth-binomial>;

G.524 solid angle

A:term:**solid angle** <solid-angle> is the inclination constituted by more than two:term:lines <G.2> which meet one another and are not in the same:term:surface <G.533>, towards all the lines. Otherwise: A:term:**solid angle** <solid-angle> is that which is contained by more than two:term:plane angles <G.22> which are not in the same:term:plane <G.7> and are constructed to one:term:point <G.1>.

G.561 cone

When, one side of those about the:term:right angle <G.33> in a:term:right-angled triangle <G.78> remaining fixed, the:term:triangle <G.12> is carried round and restored again to the same position from which it began to be moved, the:term:figure <G.9> so comprehended is a:term:**cone** <cone>. And, if the:term:segment <G.4> which remains fixed equals the remaining side about the right angle which is carried round, the cone is:term:**right-angled** <right-angled-cone>; if less,:term:**obtuse-angled** <obtuse-angled-cone>; and if greater,:term:**acute-angled** <acute-angled-cone>.

G.1 point

A point is that which has no part.

G.10 circle

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 ;
A B C D

G.10 circle

A circle ( A B ) is a plane figure contained by one arc defined by a center A and a point B establishing its defining radius / A B /, such that all segments / A B /, / A C /, / A D / falling upon the arc from A are equal.

G.103 tangent circles

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

G.11 center of a circle

The point A within the circle ( A B ) from which all segments / A B /, / A C / to the arc are equal is called the center of the circle.

G.12 rectilineal figures

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.

G.122 sector of a circle

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.

G.126 segment of a circle

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

G.13 equilateral, isosceles, and scalene triangles

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.

G.131 equal circles

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

G.132 angle standing on an arc

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

G.138 angle in a circle segment

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.

G.148 polygon inscribed in a circle

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.

G.152 polygon inscribed in a polygon

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.

G.153 circle inscribed in a polygon

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.

G.155 polygon circumscribed about a polygon

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.

G.173 ratio

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

G.174 ex aequali (equality of ratios)

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.

G.176 magnitudes in the same ratio

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.

G.181 corresponding magnitudes

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

G.185 greater ratio

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.

G.2 line

A line is breadthless length.
A B

G.2 line

A line [ A B ] is a projection of a relationship of two points A, B.

G.203 perturbed proportion

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.

G.22 plane angle

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.

G.230 duplicate ratio

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.

G.242 extreme and mean ratio

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.

G.247 unit

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

G.253 part of a number

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

G.263 proportional numbers

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

G.269 multiplication of numbers

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.

G.30 rectilineal angle

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

G.300 plane and solid numbers

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.

G.308 cube number

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

G.310 triplicate ratio

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.

G.311 square number

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

G.320 square and cube numbers

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.

G.33 right angle and perpendicular

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.

G.352 odd number

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

G.380 commensurable in square

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.

G.391 rational and irrational segments

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.

G.392 rational and irrational areas

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.
A B

G.4 segment

A segment / A B / is a portion of a line [ A B ] between two points A, B on the line.

G.4 segment

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

G.422 first binomial

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;

G.424 second binomial

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

G.426 third binomial

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.

G.428 fourth binomial

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;

G.454 first apotome

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.

G.456 second apotome

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.

G.469 third apotome

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.

G.471 fourth apotome

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 ;

G.5 surface

A surface is that which has length and breadth only.

G.52 parallel lines

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.

G.521 plane at right angles to a plane

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

G.524 solid angle

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

Otherwise : A solid angle is that which is contained by more than two plane angles which are not in the same plane and are constructed to one point.

G.529 solid

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

G.537 prism

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.

G.548 cube

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

G.553 pyramid

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

G.561 cone

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

And, if the straight line which remains fixed be equal to the remaining side about the right angle which is carried round, the cone will be right-angled; if less, obtuse-angled; and if greater, acute-angled.

G.562 cylinder

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.

G.564 axis of a cone

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

G.566 base of a cone

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

G.567 axis of a cylinder

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

G.568 bases of a cylinder

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

G.575 sphere

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.

G.576 diameter of a sphere

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

G.591 axis of a sphere

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

G.593 octahedron

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

G.596 icosahedron

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

G.599 dodecahedron

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

G.61 diameter of a circle

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

G.63 square, oblong, rhombus, and rhomboid

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.

G.7 plane surface

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

G.78 right, obtuse, and acute-angled triangles

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

G.8 boundary

A boundary is that which is an extremity of anything.

G.81 contained rectangle

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

G.86 gnomon

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

G.9 figure

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

G.96 semicircle

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.