To find the first apotome.

To find the second apotome.

To find the third apotome.

To find the fourth apotome.

To find the fifth apotome.

To find the sixth apotome.

If an area be contained by a rational straight line and a first apotome, the side of the area is an apotome.

If an area be contained by a rational straight line and a second apotome, the side of the area is a first apotome of a medial straight line.

If an area be contained by a rational straight line and a third apotome, the side of the area is a second apotome of a medial straight line.

If an area be contained by a rational straight line and a fourth apotome, the side of the area is minor.

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

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

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

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

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

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

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

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

A straight line commensurable in length with an apotome is an apotome and the same in order.

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

A straight line commensurable with a minor straight line is minor.

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

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

If from a rational area a medial area be subtracted, the side of the remaining area becomes one of two irrational straight lines, either an apotome or a minor straight line.

If from a medial area a rational area be subtracted, there arise two other irrational straight lines, either a first apotome of a medial straight line or a straight line which produces with a rational area a medial whole.

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

The apotome is not the same with the binomial straight line.

The square on a rational straight line applied to the binomial straight line 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 straight line.

The square on a rational straight line, if applied to an apotome, produces as, breadth the binomial straight line 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.

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

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