— GEOMETOR • elements

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.82 distributive law of rectangles

If one of two segments is cut into any number of parts, the rectangle contained by the two segments equals the sum of the rectangles contained by the uncut segment and each of the parts.

G.83 square equals sum of rectangles on parts

If a segment is cut at random, the sum of the rectangles contained by the whole and each of the parts equals the square on the whole.

G.84 rectangle on whole and part

If a segment is cut at random, the rectangle contained by the whole and one of the parts equals the rectangle contained by the parts together with the square on that part.

G.85 square of a sum

If a segment is cut at random, the square on the whole equals the squares on the parts together with twice the rectangle contained by the parts.

G.87 rectangle on unequal parts plus square on difference

If a segment is cut into equal and unequal parts, the rectangle contained by the unequal parts together with the square on the segment between the points of section equals the square on the half.

G.88 rectangle on bisected and extended segment

If a segment is bisected and a segment is added to it in a line, the rectangle contained by the whole with the added segment and the added segment together with the square on the half equals the square on the segment made up of the half and the added segment.

G.89 square of a difference

If a segment is cut at random, the sum of the squares on the whole and on one of the parts equals twice the rectangle contained by the whole and that part together with the square on the remaining part.

G.90 four rectangles plus square on difference

If a segment is cut at random, four times the rectangle contained by the whole and one of the parts together with the square on the remaining part equals the square on the whole and the first part as one segment.

G.93 cut a segment in extreme and mean ratio

Cut a given segment so that the rectangle contained by the whole and one of the segments equals the square on the remaining segment.

G.94 law of cosines for obtuse triangles

In obtuse-angled triangles, the square on the side subtending the obtuse angle is greater than the sum of the squares on the sides containing the obtuse angle by twice the rectangle contained by one of those sides and the projection of the other upon it.

G.95 law of cosines for acute triangles

In acute-angled triangles, the square on the side subtending an acute angle is less than the sum of the squares on the sides containing the acute angle by twice the rectangle contained by one of those sides and the projection of the other upon it.

G.97 construct a square equal to a polygon

Construct a square equal in area to a given rectilineal figure.

G.143 intersecting chords theorem

If two chords in a circle intersect one another, the rectangle contained by the segments of one equals the rectangle contained by the segments of the other.

G.144 tangent-secant theorem

If from a point outside a circle a secant and a tangent are drawn, the rectangle contained by the whole secant and its external segment equals the square on the tangent.

G.145 converse of the tangent-secant theorem

If from a point outside a circle a secant and a segment to the circle are drawn such that the rectangle on the whole secant and its external part equals the square on the segment, the segment is tangent to the circle.

G.162 construct a golden isosceles triangle

Construct an isosceles triangle having each of the angles at the base double the remaining angle.

G.227 segments proportional rectangle

If four segments 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 equals the rectangle contained by the means, the four segments is proportional.

G.228 segments proportional rectangle

If three segments be proportional, the rectangle contained by the extremes is equal to the square on the mean; and, if the rectangle contained by the extremes equals the square on the mean, the three segments is proportional.

G.388 incommensurable magnitudes added

If two incommensurable magnitudes be added together, the whole is incommensurable with each of them; and, if the whole be incommensurable with one of them, the original magnitudes is incommensurable.

G.389 there unequal segments

If there be two unequal segments, and to the greater there be applied a parallelogram equal to the fourth part of the square on the less and deficient by a square figure, and if it divide it into parts which are commensurable in length, then the square on the greater is greater than the square on the less by the square on a segment commensurable with the greater. And, if the square on the greater be greater than the square on the less by the square on a segment commensurable with the greater, and if there be applied to the greater a parallelogram equal to the fourth part of the square on the less and deficient by a square figure, it will divide it into parts which are commensurable in length.

G.390 there unequal segments

If there be two unequal segments, and to the greater there be applied a parallelogram equal to the fourth part of the square on the less and deficient by a square figure, and if it divide it into parts which are incommensurable, the square on the greater is greater than the square on the less by the square on a segment incommensurable with the greater. And, if the square on the greater be greater than the square on the less by the square on a segment incommensurable with the greater, and if there be applied to the greater a parallelogram equal to the fourth part of the square on the less and deficient by a square figure, it divides it into parts which are incommensurable.

G.393 rectangle contained rational

The rectangle contained by rational segments commensurable in length is rational.

G.395 rectangle contained rational

The rectangle contained by rational segments commensurable in square only is irrational, and the side of the square equal to it is irrational. Let the latter be called medial.

G.396 square medial segment

The square on a medial segment, if applied to a rational segment, produces as breadth a segment rational and incommensurable in length with that to which it is applied.

G.397 segment commensurable medial

A segment commensurable with a medial segment is medial.

G.398 rectangle contained medial

The rectangle contained by medial segments commensurable in length is medial.

G.399 rectangle contained medial

The rectangle contained by medial segments commensurable in square only is either rational or medial.

G.400 medial area exceed

A medial area does not exceed a medial area by a rational area.

G.401 find medial segments commensurable in square only which

Find medial segments commensurable in square only which contain a rational rectangle.

G.402 find medial segments commensurable in square only which

Find medial segments commensurable in square only which contain a medial rectangle.

G.405 find two medial segments commensurable in square only

Find two medial segments commensurable in square only, containing a rational rectangle, and such that the square on the greater is greater than the square on the less by the square on a segment commensurable in length with the greater.

G.406 find two medial segments commensurable in square only

Find two medial segments commensurable in square only, containing a medial rectangle, and such that the square on the greater is greater than the square on the less by the square on a segment commensurable with the greater.

G.407 find two segments incommensurable in square which make

Find two segments incommensurable in square which make the sum of the squares on them rational but the rectangle contained by them medial.

G.408 find two segments incommensurable in square which make

Find two segments incommensurable in square which make the sum of the squares on them medial but the rectangle contained by them rational.

G.409 find two segments incommensurable in square which make

Find two segments incommensurable in square which make the sum of the squares on them medial and the rectangle contained by them medial and moreover incommensurable with the sum of the squares on them.

G.410 rational segments commensurable

If two rational segments commensurable in square only be added together, the whole is irrational; and let it be called binomial.

G.411 medial segments commensurable

If two medial segments commensurable in square only and containing a rational rectangle be added together, the whole is irrational; and let it be called a first bimedial segment.

G.412 medial segments commensurable

If two medial segments commensurable in square only and containing a medial rectangle be added together, the whole is irrational; and let it be called a second bimedial segment.

G.413 segments incommensurable square

If two segments incommensurable in square which make the sum of the squares on them rational, but the rectangle contained by them medial, be added together, the whole segment is irrational: and let it be called major.

G.414 segments incommensurable square

If two segments incommensurable in square which make the sum of the squares on them medial, but the rectangle contained by them rational, be added together, the whole segment is irrational; and let it be called the side of a rational plus a medial area.

G.415 segments incommensurable square

If two segments incommensurable in square which make the sum of the squares on them medial, and the rectangle contained by them medial and also incommensurable with the sum of the squares on them, be added together, the whole segment is irrational; and let it be called the side of the sum of two medial areas.

G.416 binomial segment divided

A binomial segment is divided into its terms at one point only.

G.417 bimedial segment divided

A first bimedial segment is divided at one point only.

G.418 bimedial segment divided

A second bimedial segment is divided at one point only.

G.419 major segment divided

A major segment is divided at one and the same point only.

G.420 side rational plus

The side of a rational plus a medial area is divided at one point only.

G.421 side sum medial

The side of the sum of two medial areas is divided at one point only.

G.434 area contained rational

If an area be contained by a rational segment and the first binomial, the “side” of the area is the irrational segment which is called binomial.

G.435 area contained rational

If an area be contained by a rational segment and the second binomial, the “side” of the area is the irrational segment which is called a first bimedial.

G.436 area contained rational

If an area be contained by a rational segment and the third binomial, the “side” of the area is the irrational segment called a second bimedial.

G.437 area contained rational

If an area be contained by a rational segment and the fourth binomial, the “side” of the area is the irrational segment called major.

G.438 area contained rational

If an area be contained by a rational segment and the fifth binomial, the “side” of the area is the irrational segment called the side of a rational plus a medial area.

G.439 area contained rational

If an area be contained by a rational segment and the sixth binomial, the “side” of the area is the irrational segment called the side of the sum of two medial areas.

G.440 square binomial segment

The square on the binomial segment applied to a rational segment produces as breadth the first binomial.

G.441 square bimedial segment

The square on the first bimedial segment applied to a rational segment produces as breadth the second binomial.

G.442 square bimedial segment

The square on the second bimedial segment applied to a rational segment produces as breadth the third binomial.

G.443 square major segment

The square on the major segment applied to a rational segment produces as breadth the fourth binomial.

G.444 square side rational

The square on the side of a rational plus a medial area applied to a rational segment produces as breadth the fifth binomial.

G.445 square side sum

The square on the side of the sum of two medial areas applied to a rational segment produces as breadth the sixth binomial.

G.447 segment commensurable length

A segment commensurable in length with a bimedial segment is itself also bimedial and the same in order.

G.448 segment commensurable major

A segment commensurable with a major segment is itself also major.

G.449 segment commensurable side

A segment commensurable with the side of a rational plus a medial area is itself also the side of a rational plus a medial area.

G.450 segment commensurable side

A segment commensurable with the side of the sum of two medial areas is the side of the sum of two medial areas.

G.451 rational medial area

If a rational and a medial area be added together, four irrational segments arise, namely a binomial or a first bimedial or a major or a side of a rational plus a medial area.

G.452 medial areas incommensurable

If two medial areas incommensurable with one another be added together, the remaining two irrational segments arise, namely either a second bimedial or a side of the sum of two medial areas.

G.453 rational segment there

If from a rational segment there be subtracted a rational segment commensurable with the whole in square only, the remainder is irrational; and let it be called an apotome.

G.455 medial segment there

If from a medial segment there be subtracted a medial segment which is commensurable with the whole in square only, and which contains with the whole a rational rectangle, the remainder is irrational. And let it be called a first apotome of a medial segment.

G.457 medial segment there

If from a medial segment there be subtracted a medial segment which is commensurable with the whole in square only, and which contains with the whole a medial rectangle, the remainder is irrational; and let it be called a second apotome of a medial segment.

G.458 segment there subtracted

If from a segment there be subtracted a segment which is incommensurable in square with the whole and which with the whole makes the squares on them added together rational, but the rectangle contained by them medial, the remainder is irrational; and let it be called minor.

G.459 segment there subtracted

If from a segment there be subtracted a segment which is incommensurable in square with the whole, and which with the whole makes the sum of the squares on them medial, but twice the rectangle contained by them rational, the remainder is irrational: and let it be called that which produces with a rational area a medial whole.

G.460 segment there subtracted

If from a segment there be subtracted a segment which is incommensurable in square with the whole and which with the whole makes the sum of the squares on them medial, twice the rectangle contained by them medial, and further the squares on them incommensurable with twice the rectangle contained by them, the remainder is irrational; and let it be called that which produces with a medial area a medial whole.

G.461 apotome only rational

To an apotome only one rational segment can be annexed which is commensurable with the whole in square only.

G.462 apotome medial segment

To a first apotome of a medial segment only one medial segment can be annexed which is commensurable with the whole in square only and which contains with the whole a rational rectangle.

G.463 apotome medial segment

To a second apotome of a medial segment only one medial segment can be annexed which is commensurable with the whole in square only and which contains with the whole a medial rectangle.

G.464 minor segment only

To a minor segment only one segment can be annexed which is incommensurable in square with the whole and which makes, with the whole, the sum of the squares on them rational but twice the rectangle contained by them medial.

G.465 segment produces rational

To a segment which produces with a rational area a medial whole only one segment can be annexed which is incommensurable in square with the whole segment and which with the whole segment makes the sum of the squares on them medial, but twice the rectangle contained by them rational.

G.466 segment produces medial

To a segment which produces with a medial area a medial whole only one segment can be annexed which is incommensurable in square with the whole segment and which with the whole segment makes the sum of the squares on them medial and twice the rectangle contained by them both medial and also incommensurable with the sum of the squares on them.

G.477 area contained rational

If an area be contained by a rational segment and a first apotome, the “side” of the area is an apotome.

G.478 area contained rational

If an area be contained by a rational segment and a second apotome, the “side” of the area is a first apotome of a medial segment.

G.479 area contained rational

If an area be contained by a rational segment and a third apotome, the “side” of the area is a second apotome of a medial segment.

G.480 area contained rational

If an area be contained by a rational segment and a fourth apotome, the “side” of the area is minor.

G.481 area contained rational

If an area be contained by a rational segment and a fifth apotome, the “side” of the area is a segment which produces with a rational area a medial whole.

G.482 area contained rational

If an area be contained by a rational segment and a sixth apotome, the “side” of the area is a segment which produces with a medial area a medial whole.

G.483 square apotome applied

The square on an apotome applied to a rational segment produces as breadth a first apotome.

G.484 square apotome medial

The square on a first apotome of a medial segment applied to a rational segment produces as breadth a second apotome.

G.485 square apotome medial

The square on a second apotome of a medial segment applied to a rational segment produces as breadth a third apotome.

G.486 square minor segment

The square on a minor segment applied to a rational segment produces as breadth a fourth apotome.

G.487 square segment produces

The square on the segment which produces with a rational area a medial whole, if applied to a rational segment, produces as breadth a fifth apotome.

G.488 square segment produces

The square on the segment which produces with a medial area a medial whole, if applied to a rational segment, produces as breadth a sixth apotome.

G.490 segment commensurable apotome

A segment commensurable with an apotome of a medial segment is an apotome of a medial segment and the same in order.

G.491 segment commensurable minor

A segment commensurable with a minor segment is minor.

G.492 segment commensurable produces

A segment commensurable with that which produces with a rational area a medial whole is a segment which produces with a rational area a medial whole.

G.493 segment commensurable produces

A segment commensurable with that which produces with a medial area a medial whole is itself also a segment which produces with a medial area a medial whole.

G.494 rational area medial

If from a rational area a medial area be subtracted, the “side” of the remaining area becomes one of two irrational segments, either an apotome or a minor segment.

G.496 medial area there

If from a medial area there be subtracted a medial area incommensurable with the whole, the two remaining irrational segments arise, either a second apotome of a medial segment or a segment which produces with a medial area a medial whole.

G.497 apotome binomial segment

The apotome is not the same with the binomial segment.

G.498 square rational segment

The square on a rational segment applied to the binomial segment produces as breadth an apotome the terms of which are commensurable with the terms of the binomial and moreover in the same ratio; and further the apotome so arising will have the same order as the binomial segment.

G.499 square rational segment

The square on a rational segment, if applied to an apotome, produces as breadth the binomial segment the terms of which are commensurable with the terms of the apotome and in the same ratio; and further the binomial so arising has the same order as the apotome.

G.500 area contained apotome

If an area be contained by an apotome and the binomial segment the terms of which are commensurable with the terms of the apotome and in the same ratio, the “side” of the area is rational.

G.501 medial segment there

From a medial segment there arise irrational segments infinite in number, and none of them is the same as any of the preceding.

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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