— GEOMETOR • elements

G.2 line

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

G.3 extremities of a segment

The extremities of a segment / A B / are points 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.5 surface

A surface is that which has length and breadth only.

G.6 extremities of a surface

The extremities of a surface are lines or arcs.

G.7 plane surface

A plane surface lies evenly with the lines on itself.

G.8 boundary

A boundary is the extremity of anything.

G.9 figure

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

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.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 contained by segments: trilateral by three, quadrilateral by four, and multilateral by more than four.

G.13 equilateral, isosceles, and scalene triangles

Of trilateral figures, an equilateral triangle < A B C > has three equal sides, an isosceles triangle < D E F > has two equal sides, and a scalene triangle < G H K > has three unequal sides.

G.22 plane angle

A plane angle is the inclination to one another of two lines or arcs in a plane that meet and do not lie on the same line.

G.30 rectilineal angle

A rectilineal angle is contained by two segments or lines.

G.33 right angle and perpendicular

When a segment standing on a line makes the adjacent angles equal, each of the equal angles is a right angle, and the standing segment is perpendicular to the line.

G.52 parallel lines

Parallel lines are lines in the same plane which, extended indefinitely in both directions, do not meet.

G.61 diameter of a circle

A diameter of a circle is any segment drawn through the center and terminated in both directions by the arc of the circle; it bisects the circle.

G.63 square, oblong, rhombus, and rhomboid

Of quadrilateral figures, a square is equilateral and right-angled; an oblong is right-angled but not equilateral; a rhombus is equilateral but not right-angled; a rhomboid has opposite sides and angles equal but is neither equilateral nor right-angled.

G.76 obtuse angle

An obtuse angle is greater than a right angle.

G.77 acute angle

An acute angle is less than a right angle.

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

A right-angled triangle has a right angle, an obtuse-angled triangle has an obtuse angle, and an acute-angled triangle has three acute angles.

G.81 contained rectangle

Any rectangular parallelogram is said to be:term:**contained** <contained-rectangle> by the two:term:segments <G.4> containing the:term:right angle <G.33>.

G.86 gnomon

And in any parallelogrammic area let any one whatever of the parallelograms about its:term:diameter <G.61> with the two complements be called a:term:**gnomon** <gnomon>.

G.96 semicircle

A semicircle is the figure contained by a diameter and the arc cut off by it.

G.103 tangent circles

Circles are said to:term:**touch one another** <circles-touching> which, meeting one another, do not cut one another.

G.111 line tangent to a circle

A:term:segment <G.4> is said to:term:**touch a circle** <tangent-to-circle> which, meeting the:term:circle <G.10> and being produced, does not cut the circle.

G.113 chords equally distant from the center

In a:term:circle <G.10>segments are said to be:term:**equally distant from the center** <equally-distant-from-center> when the:term:perpendiculars <G.33> drawn to them from the:term:center <center> are equal.

G.115 chord at greater distance from the center

And that:term:segment <G.4> is said to be:term:**at a greater distance** <greater-distance-from-center> on which the greater:term:perpendicular <G.33> falls.

G.117 angle of a circle segment

An:term:**angle of a segment** <angle-of-segment> is that contained by a:term:segment <G.4> and a arc of a:term:circle <G.10>.

G.122 sector of a circle

A:term:**sector of a circle** <sector-of-circle> is the:term:figure <G.9> which, when an:term:angle <G.22> is constructed at the:term:center <center> of the:term:circle <G.10>, is contained by the:term:segments <G.4> containing the angle and the arc cut off by them.

G.126 segment of a circle

A:term:**segment of a circle** <segment-of-circle> is the:term:figure <G.9> contained by a:term:segment <G.4> and a arc of a:term:circle <G.10>.

G.127 similar circle segments

Similar segments of circles are those which admit equal:term:angles <G.22>, or in which the angles are equal.

G.131 equal circles

Equal circles are those the:term:diameters <G.61> of which are equal, or the radii of which are equal.

G.132 angle standing on an arc

And, when the:term:segments <G.4> containing the:term:angle <G.22> cut off a arc, the angle is said to:term:**stand upon** <stand-upon-arc> that arc.

G.138 angle in a circle segment

An:term:**angle in a segment** <angle-in-segment> is the:term:angle <G.22> which, when a:term:point <G.1> is taken on the arc of the segment and:term:segments <G.4> are joined from it to the:term:extremities <G.6> of the segment which is the:term:**base of the segment** <base-of-segment>, is contained by the segments so joined.

G.146 segment fitted into a circle

A:term:segment <G.4> is said to be:term:**fitted into a circle** <fitted-into-circle> when its:term:extremities <G.6> are on the arc of the:term:circle <G.10>.

G.148 polygon inscribed in a circle

A:term:polygon <G.12> is said to be:term:**inscribed in a circle** <inscribed-in-circle> when each:term:angle <G.22> of the inscribed:term:figure <G.9> lies on the arc of the:term:circle <G.10>.

G.150 polygon circumscribed about a circle

A:term:polygon <G.12> is said to be:term:**circumscribed about a circle** <circumscribed-about-circle>, when each side of the circumscribed:term:figure <G.9> touches the arc of the:term:circle <G.10>.

G.152 polygon inscribed in a polygon

A:term:polygon <G.12> is said to be:term:**inscribed in a polygon** <inscribed-in-rectilineal-figure> when the respective:term:angles <G.22> of the inscribed:term:figure <G.9> lie on the respective sides of that in which it is inscribed.

G.153 circle inscribed in a polygon

Similarly a:term:circle <G.10> is said to be:term:**inscribed in a figure** <circle-inscribed-in-figure> when the arc of the circle touches each side of the:term:figure <G.9> in which it is inscribed.

G.155 polygon circumscribed about a polygon

Similarly a:term:figure <G.9> is said to be:term:**circumscribed about a figure** <circumscribed-about-figure> when the respective sides of the circumscribed figure pass through the respective:term:angles <G.22> of that about which it is circumscribed.

G.156 circle circumscribed about a polygon

A:term:circle <G.10> is said to be:term:**circumscribed about a figure** <G.155> when the arc of the circle passes through each:term:angle <G.22> of the:term:figure <G.9> about which it is circumscribed.

G.169 part (measure) of a magnitude

A magnitude is a:term:**part** <part-aliquot> of a magnitude, the less of the greater, when it measures the greater.

G.170 multiple of a magnitude

The greater is a:term:**multiple** <multiple> of the less when it is:term:measured <part-aliquot> by the less.

G.173 ratio

A:term:**ratio** <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: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.180 proportional magnitudes

Let magnitudes which have the:term:same ratio <G.176> be called:term:**proportional** <proportional>.

G.181 corresponding magnitudes

The term:term:**corresponding magnitudes** <corresponding-magnitudes> is used of antecedents in relation to antecedents, and of consequents in relation to consequents.

G.182 invertendo (inverse ratio)

Inverse ratio means taking the:term:consequent <G.181> as antecedent in relation to the antecedent as consequent.

G.184 magnitudes having a ratio

Magnitudes are said to:term:**have a ratio** <have-a-ratio> to one another which are capable, when multiplied, of exceeding one another.

G.185 greater ratio

When, of the:term:equimultiples <multiple>, 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:term:**have a greater ratio** <greater-ratio> to the second than the third has to the fourth.

G.194 alternando (alternate ratio)

Alternate ratio means taking the:term:antecedent <G.181> in relation to the antecedent and the consequent in relation to the consequent.

G.196 componendo (composition of ratio)

Composition of a ratio means taking the:term:antecedent <G.181> together with the consequent as one in relation to the consequent by itself.

G.197 separando (separation of ratio)

Separation of a ratio means taking the excess by which the:term:antecedent <G.181> exceeds the consequent in relation to the consequent by itself.

G.200 convertendo (conversion of ratio)

Conversion of a ratio means taking the:term:antecedent <G.181> in relation to the excess by which the antecedent exceeds the consequent.

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.209 height of a figure

The height of any:term:figure <G.9> is the:term:perpendicular <G.33> drawn from the vertex to the base.

G.217 similar rectilineal figures

Similar polygons are such as have their:term:angles <G.22> severally equal and the sides about the equal angles:term:proportional <G.180>.

G.224 reciprocally related figures

Reciprocally related figures have their sides about equal angles reciprocally proportional.

G.230 duplicate ratio

When three magnitudes are:term:proportional <G.180>, the first is said to have to the third the:term:**duplicate ratio** <duplicate-ratio> of that which it has to the second.

G.242 extreme and mean ratio

A segment is said to have been cut in extreme and mean ratio when, as the whole:term:line <G.2> is to the greater segment, so is the greater to the less.

G.247 unit

An:term:**unit** <unit> is that by virtue of which each of the things that exist is called one.

G.248 number

A:term:**number** <number> is a multitude composed of:term:units <G.247>.

G.249 numbers prime to one another

Numbers prime to one another are those which are:term:measured <G.253> by an:term:unit <G.247> alone as a common measure.

G.253 part of a number

A:term:number <G.248> is a:term:**part** <part-aliquot> of a number, the less of the greater, when it:term:measures <part-aliquot> the greater;

G.254 parts of a number

But:term:**parts** <parts-non-aliquot> when it does not:term:measure <G.253> it.

G.256 multiple of a number

The greater:term:number <G.248> is a:term:**multiple** <multiple> of the less when it is:term:measured <G.253> by the less.

G.263 proportional numbers

Numbers are:term:**proportional** <G.180> when the first is the:term:same multiple <G.256>, or the:term:same part <G.253>, or the:term:same parts <G.254>, of the second that the third is of the fourth.

G.269 multiplication of numbers

A:term:number <G.248> is said to:term:**multiply** <multiply> a number when that which is multiplied is added to itself as many times as there are:term:units <G.247> in the other, and thus some number is produced.

G.283 prime number

A:term:**prime number** <prime-number> is that which is:term:measured <G.253> by an:term:unit <G.247> alone.

G.286 composite number

A:term:**composite number** <composite-number> is that which is:term:measured <G.253> by some:term:number <G.248>.

G.300 plane and solid numbers

And, when two:term:numbers <G.248> having:term:multiplied <G.269> one another make some number, the number so produced is called:term:**plane** <plane-number>, and its:term:**sides** <side-of-number> are the numbers which have multiplied one another.

G.302 even number

An:term:**even number** <even-number> is that which is divisible into two equal parts.

G.308 cube number

A:term:**square number** <square-number> is equal:term:multiplied <G.269> by equal, or a:term:number <G.248> which is contained by two equal numbers.

G.310 triplicate ratio

When four magnitudes are < continuously >:term:proportional <G.180>, the first is said to have to the fourth the:term:**triplicate ratio** <triplicate-ratio> of that which it has to the second, and so on continually, whatever be the proportion.

G.311 square number

And a:term:**cube** <cube-number> is equal:term:multiplied <G.269> by equal and again by equal, or a:term:number <G.248> which is contained by three equal numbers.

G.318 similar plane and solid numbers

Similar plane and solid numbers are those which have their sides:term:proportional <G.180>.

G.320 square and cube numbers

And, when three:term:numbers <G.248> having:term:multiplied <G.269> one another make some number, the number so produced is:term:**solid** <solid-number>, and its:term:**sides** <side-of-number> are the numbers which have multiplied one another.

G.341 numbers composite to one another

Numbers composite to one another are those which are:term:measured <G.253> by some:term:number <G.248> as a common measure.

G.352 odd number

An:term:**odd number** <odd-number> is that which is not divisible into two equal parts, or that which differs by an:term:unit <G.247> from an:term:even number <G.302>.

G.363 even-times even number

An:term:**even-times even number** <even-times-even-number> is that which is:term:measured <G.253> by an:term:even number <G.302> according to an even number.

G.365 even-times odd number

An:term:**even-times odd number** <even-times-odd-number> is that which is:term:measured <G.253> by an:term:even number <G.302> according to an:term:odd number <G.352>.

G.369 perfect number

A:term:**perfect number** <perfect-number> is that which is equal to its own parts.

G.372 commensurable and incommensurable magnitudes

Those magnitudes are said to be:term:**commensurable** <commensurable> which are:term:measured <G.253> by the same measure, and those:term:**incommensurable** <incommensurable> which cannot have any common measure.

G.380 commensurable in square

Straight lines are:term:**commensurable in square** <commensurable-in-square> when the:term:squares <G.63> on them are:term:measured <G.253> by the same area, and:term:**incommensurable in square** <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: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.424 second binomial

But if the lesser term be:term:commensurable <G.372> in length with the:term:rational <G.392>segment set out, let the whole be called a:term:**second binomial** <second-binomial>;

G.426 third binomial

And if neither of the terms be:term:commensurable <G.372> in length with the:term:rational <G.392>segment set out, let the whole be called a:term:**third binomial** <third-binomial>.

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.430 fifth binomial

If the lesser, a:term:**fifth binomial** <fifth-binomial>;

G.432 sixth binomial

And if neither, a:term:**sixth binomial** <sixth-binomial>.

G.454 first apotome

Given a:term:rational <G.392>segment and an apotome, if the:term:square <G.63> on the whole be greater than the square on the annex by the square on a segment:term:commensurable <G.372> in length with the whole, and the whole be commensurable in length with the rational segment set out, let the apotome be called a:term:**first apotome** <first-apotome>.

G.456 second apotome

But if the annex be:term:commensurable <G.372> in length with the:term:rational <G.392>segment set out, and the:term:square <G.63> on the whole be greater than that on the annex by the square on a segment commensurable with the whole, let the apotome be called a:term:**second apotome** <second-apotome>.

G.469 third apotome

But if neither be:term:commensurable <G.372> in length with the:term:rational <G.392>segment set out, and the:term:square <G.63> on the whole be greater than the square on the annex by the square on a segment commensurable with the whole, let the apotome be called a:term:**third apotome** <third-apotome>.

G.471 fourth apotome

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

G.473 fifth apotome

If the annex be so:term:commensurable <G.372>, a:term:**fifth** <fifth-apotome>;

G.475 sixth apotome

And, if neither, a:term:**sixth** <sixth-apotome>.

G.505 line at right angles to a plane

A:term:segment <G.4> is:term:**at right angles to a plane** <right-angles-to-plane>, when it makes:term:right angles <G.33> with all the segments which meet it and are in the:term:plane <G.7>.

G.516 parallel planes

Parallel planes are those which do not meet.

G.521 plane at right angles to a plane

A:term:**plane is at right angles to a plane** <plane-at-right-angles-to-plane> when the:term:segments <G.4> drawn, in one of the:term:planes <G.7>,:term:at right angles <G.33> to the common section of the planes are at right angles to the remaining plane.

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.529 solid

A:term:**solid** <solid> is that which has length, breadth, and depth.

G.533 extremity of a solid

An:term:extremity <G.6> of a:term:solid <G.529> is a:term:**surface** <surface>.

G.534 similar solid figures

Similar solid figures are those:term:contained by similar planes <G.533> equal in multitude.

G.535 equal and similar solid figures

Equal and similar solid figures are those:term:contained by similar planes <G.533> equal in multitude and in magnitude.

G.537 prism

A:term:**prism** <prism> is a:term:solid figure <G.529> contained by:term:planes <G.7> two of which, namely those which are opposite, are equal, similar and:term:parallel <G.52>, while the rest are parallelograms.

G.548 cube

A:term:**cube** <cube> is a:term:solid figure <G.529> contained by six equal:term:squares <G.63>.

G.553 pyramid

A:term:**pyramid** <pyramid> is a:term:solid figure <G.529>, contained by:term:planes <G.7>, which is constructed from one plane 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.562 cylinder

When, one side of those about the:term:right angle <G.33> in a:term:rectangular parallelogram <G.81> remaining fixed, the parallelogram 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:**cylinder** <cylinder>.

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