— GEOMETOR • elements

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.

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.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.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.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.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.470 find the third apotome

Find the 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.472 find the fourth apotome

Find the fourth apotome.

G.473 fifth apotome

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

G.474 find the fifth apotome

Find the fifth apotome.

G.475 sixth apotome

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

Posts tagged 'apotome'