Posts in category 'magnitudes'

G.174 ex aequali (equality of ratios)

A:term:ratio <G.173>ex aequali arises when, there being several magnitudes and another set equal to them in multitude which taken two and two are in the same proportion, as the first is to the last among the first magnitudes, so is the first to the last among the second magnitudes; Or, in other words, it means taking the extreme terms by virtue of the removal of the intermediate terms.

G.176 magnitudes in the same ratio

Magnitudes are said to:term:**be in the same ratio** <in-the-same-ratio>, the first to the second and the third to the fourth, when, if any:term:equimultiples <multiple> whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.

G.203 perturbed proportion

A:term:**perturbed proportion** <perturbed-proportion> arises when, there being three magnitudes and another set equal to them in multitude, as:term:antecedent <G.181> is to consequent among the first magnitudes, so is antecedent to consequent among the second magnitudes, while, as the consequent is to a third among the first magnitudes, so is a third to the antecedent among the second magnitudes.
A B C D E F

G.204 there magnitudes others

If there be three magnitudes, and others equal to them in multitude, which taken two and two together are in the same ratio, and the proportion of them be perturbed, then, if ex aequali the first magnitude is greater than the third, the fourth is greater than the sixth; if equal, equal; and if less, less.
A B C D

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.
A B E C D F

G.386 segments proportional square

If four segments be proportional, and the square on the first be greater than the square on the second by the square on a segment commensurable with the first, the square on the third is greater than the square on the fourth by the square on a segment commensurable with the third. And, if the square on the first be greater than the square on the second by the square on a segment incommensurable with the first, the square on the third is greater than the square on the fourth by the square on a segment incommensurable with the third.
A B F E D C

G.389 there unequal segments

If there be two unequal segments, and to the greater there be applied a parallelogram equal to the fourth part of the square on the less and deficient by a square figure, and if it divide it into parts which are commensurable in length, then the square on the greater is greater than the square on the less by the square on a segment commensurable with the greater. And, if the square on the greater be greater than the square on the less by the square on a segment commensurable with the greater, and if there be applied to the greater a parallelogram equal to the fourth part of the square on the less and deficient by a square figure, it will divide it into parts which are commensurable in length.
B F E D C A

G.390 there unequal segments

If there be two unequal segments, and to the greater there be applied a parallelogram equal to the fourth part of the square on the less and deficient by a square figure, and if it divide it into parts which are incommensurable, the square on the greater is greater than the square on the less by the square on a segment incommensurable with the greater. And, if the square on the greater be greater than the square on the less by the square on a segment incommensurable with the greater, and if there be applied to the greater a parallelogram equal to the fourth part of the square on the less and deficient by a square figure, it divides it into parts which are incommensurable.

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.
K H G F D E A B C

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.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 D F G I H E

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.
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.
D F G I H E A C B

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.
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.