Posts tagged 'rational'

G.391 rational and irrational segments

With these hypotheses, it is proved that there exist:term:segments <G.4> infinite in multitude which are:term:commensurable <G.372> and incommensurable respectively, some in length only, and others in square also, with an assigned segment. Let then the assigned segment be called:term:**rational** <rational>, and those segments which are commensurable with it, whether in length and in square or in square only,:term:**rational** <rational>, but those which are incommensurable with it:term:**irrational** <irrational>.

G.392 rational and irrational areas

And let the:term:square <G.63> on the assigned:term:segment <G.4> be called:term:**rational** <rational> and those areas which are:term:commensurable <G.372> with it:term:**rational** <rational>, but those which are incommensurable with it:term:**irrational** <G.391>, and the segments which produce them:term:**irrational** <G.391>, that is, in case the areas are squares, the sides themselves, but in case they are any other:term:polygons <G.12>, the segments on which are described squares equal to them.
A B C

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.422 first binomial

Given a:term:rational <G.392>segment and a binomial, divided into its terms, such that the:term:square <G.63> on the greater term is greater than the square on the lesser by the square on a segment:term:commensurable <G.372> in length with the greater, then, if the greater term be commensurable in length with the rational segment set out, let the whole be called a:term:**first binomial segment** <first-binomial-straight-line>;

G.428 fourth binomial

Again, if the:term:square <G.63> on the greater term be greater than the square on the lesser by the square on a:term:segment <G.4> incommensurable in length with the greater, then, if the greater term be:term:commensurable <G.372> in length with the:term:rational <G.392> segment set out, let the whole be called a:term:**fourth binomial** <fourth-binomial>;
A C B

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.
A C B

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.
A C B

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.
A B C D

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.
A B C D E H M N F L G I

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.
A D E F G C B H I K L N P S O Q U V W R T M

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.
A B D C G K E F H

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.
A C D B K E F H G

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.
N A B C D E F G H K L M

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.
A K C D L B G H E F M N

G.601 set out sides

To set out the sides of the five figures and to compare them with one another.