Posts tagged 'commensurable-in-square'
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.381 squares segments commensurable
The squares on segments commensurable in length have to one another the ratio which a square number has to a square number; and squares which have to one another the ratio which a square number has to a square number will also have their sides commensurable in length. But the squares on segments incommensurable in length have not to one another the ratio which a square number has to a square number; and squares which have not to one another the ratio which a square number has to a square number will not have their sides commensurable in length either.
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.399 rectangle contained medial
The rectangle contained by medial segments commensurable in square only is either rational or medial.
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.403 find two rational segments commensurable in square only
Find two rational segments commensurable in square only 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.404 find two rational segments commensurable in square only
Find two rational segments commensurable in square only and such that the square on the greater is greater than the square on the less by the square on a segment incommensurable in length with the greater.
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.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.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.423 find the first binomial segment
Find the first binomial segment.
G.425 find the second binomial segment
Find the second binomial segment.
G.427 find the third binomial segment
Find the third binomial segment.
G.429 find the fourth binomial segment
Find the fourth binomial segment.
G.431 find the fifth binomial segment
Find the fifth binomial segment.
G.433 find the sixth binomial segment
Find the sixth binomial segment.
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.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.446 segment commensurable length
A segment commensurable in length with a binomial segment is itself also binomial and the same in order.
G.447 segment commensurable length
A segment commensurable in length with a bimedial segment is itself also bimedial and the same in order.
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.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.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.467 find the first apotome
Find the first apotome.
G.468 find the second apotome
Find the second apotome.
G.470 find the third apotome
Find the third apotome.
G.472 find the fourth apotome
Find the fourth apotome.
G.474 find the fifth apotome
Find the fifth apotome.
G.476 find the sixth apotome
Find the sixth apotome.
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.489 segment commensurable length
A segment commensurable in length with an apotome is an apotome and the same in order.
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.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.495 medial area rational
If from a medial area a rational area be subtracted, there arise two other irrational segments, either a first apotome of a medial segment or a segment which produces with a rational 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.584 rational segment cut
If a rational segment be cut in extreme and mean ratio, each of the segments is the irrational segment called apotome.
G.589 circle diameter rational
If in a circle which has its diameter rational an equilateral pentagon be inscribed, the side of the pentagon is the irrational segment called minor.