— GEOMETOR • elements

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.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.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.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.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.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.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.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.

Posts tagged 'medial'