Euclid's Elements Book X : Classification of incommensurables.

### Definitions I

- Definition 1
- Those magnitudes are said to be commensurable which are measured by the same measure,
and those incommensurable which cannot have any common measure.
- Definition 2
- Straight lines are commensurable in square when the squares on them are measured by the
same area, and incommensurable in square when the squares on them cannot possibly have any
area as a common measure.
- Definition 3
- With these hypotheses, it is proved that there exist straight lines infinite in multitude which
are commensurable and incommensurable respectively, some in length only, and others in square
also, with an assigned straight line. Let then the assigned straight line be called rational,
and those straight lines which are commensurable with it, whether in length and in square,
or in square only, rational, but those that are incommensurable with it irrational.
- Definition 4
- And the let the square on the assigned straight line be called rational, and those areas
which are commensurable with it rational, but those which are incommensurable with it
irrational, and the straight lines which produce them irrational, that is, in case the areas
are squares, the sides themselves, but in case they are any other rectilineal figures, the straight
lines on which are described squares equal to them.

### Propositions 1-47

- Proposition 1
- Two unequal magnitudes being set out, if from the greater there is subtracted a magnitude
greater than its half, and from that which is left a magnitude greater than its half, and if this process
is repeated continually, then there will be left some magnitude less than the lesser magnitude set out.
And the theorem can similarly be proven even if the parts subtracted are halves.
- Proposition 2
- If, when the less of two unequal magnitudes is continually subtracted in turn from the
greater that which is left never measures the one before it, then the two magnitudes are incommensurable.
- Proposition 3
- To find the greatest common measure of two given commensurable magnitudes.
Corollary

If a magnitude measures two magnitudes, then it also measures their greatest common measure.

- Proposition 4
- To find the greatest common measure of three given commensurable magnitudes.
Corollary

If a magnitude measures three magnitudes, then it also measures their greatest common measure.
The greatest common measure can be found similarly for more magnitudes, and the corollary extended.

- Proposition 5
- Commensurable magnitudes have to one another the ratio which a number has to a number.
- Proposition 6
- If two magnitudes have to one another the ratio which a number has to a number, then the
magnitudes are commensurable.
- Proposition 7
- Incommensurable magnitudes do not have to one another the ratio which a number has to a number.
- Proposition 8
- If two magnitudes do not have to one another the ratio which a number has to a number,
then the magnitudes are incommensurable.
- Proposition 9
- The squares on straight lines 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 also have their sides commensurable in length.
But the squares on straight lines incommensurable in length do not have to one another the ratio
which a square number has to a square number; and squares which do not have to one another
the ratio which a square number has to a square number also do not have their sides
commensurable in length either.
Corollary

Straight lines commensurable in length are always commensurable in square also,
but those commensurable in square are not always also commensurable in length.

Lemma

Similar plane numbers have to one another the ratio which a square number has to a
square number, and if two numbers have to one another the ratio which a square number
has to a square number, then they are similar plane numbers.

Corollary 2

Numbers which are not similar plane numbers, that is, those which do not have
their sides proportional, do not have to one another the ratio which a square number
has to a square number

- Proposition 10
- To find two straight lines incommensurable, the one in length only, and the other in
square also, with an assigned straight line.
- Proposition 11
- If four magnitudes are proportional, and the first is commensurable with the second,
then the third also is commensurable with the fourth; but, if the first is incommensurable
with the second, then the third also is incommensurable with the fourth.
- Proposition 12
- Magnitudes commensurable with the same magnitude are also commensurable with one another.
- Proposition 13
- If two magnitudes are commensurable, and one of them is incommensurable
with any magnitude, then the remaining one is also incommensurable with the same.
- Proposition 14
- If four straight lines are proportional, and the square on the first is greater than the
square on the second by the square on a straight line commensurable with the first,
then the square on the third is also greater than the square on the fourth by the square
on a third line commensurable with the third. And, if the square on the first is greater
than the square on the second by the square on a straight line incommensurable with
the first, then the square on the third is also greater than the square on the fourth by the
square on a third line incommensurable with the third.
Lemma

Given two unequal straight lines, to find by what square the square on the greater is
greater than the square on the less. And, given two straight lines, to find the straight line
the square on which equals the sum of the squares on them.

- Proposition 15
- If two commensurable magnitudes are added together, then the whole is also commensurable
with each of them; and, if the whole is commensurable with one of them, then the original
magnitudes are also commensurable.
- Proposition 16
- If two incommensurable magnitudes are added together, the sum is also incommensurable
with each of them; but, if the sum is incommensurable with one of them, then the original
magnitudes are also incommensurable.
- Proposition 17
- If there are two unequal straight lines, and to the greater there is applied a parallelogram
equal to the fourth part of the square on the less but falling short by a square, and if it divides
it into parts commensurable in length, then the square on the greater is greater than the square
on the less by the square on a straight line commensurable with the greater. And if the
square on the greater is greater than the square on the less by the square on a straight line
commensurable with the greater, and if there is applied to the greater a parallelogram equal to
the fourth part of the square on the less falling short by a square, then it divides it into parts
commensurable in length.
Lemma

If to any straight line there is applied a parallelogram but falling short by a square, then the
applied parallelogram equals the rectangle contained by the segments of the straight line resulting
from the application.

- Proposition 18
- If there are two unequal straight lines, and to the greater there is applied a parallelogram
equal to the fourth part of the square on the less but falling short by a square, and if it divides it
into incommensurable parts, then the square on the greater is greater than the square on the less
by the square on a straight line incommensurable with the greater. And if the square on the greater
is greater than the square on the less by the square on a straight line incommensurable with the
greater, and if there is applied to the greater a parallelogram equal to the fourth part of the square
on the less but falling short by a square, then it divides it into incommensurable parts.
- Proposition 19
- The rectangle contained by rational straight lines commensurable in length is rational.
- Proposition 20
- If a rational area is applied to a rational straight line, then it produces as breadth a straight line
rational and commensurable in length with the straight line to which it is applied.
- Proposition 21
- The rectangle contained by rational straight lines commensurable in square only is irrational,
and the side of the square equal to it is irrational. Let the latter be called medial.
- Proposition 22
- The square on a medial straight line, if applied to a rational straight line, produces as breadth a
straight line rational and incommensurable in length with that to which it is applied.
Lemma

If there are two straight lines, then the first is to the second as the square on the first is to the
rectangle contained by the two straight lines.

- Proposition 23
- A straight line commensurable with a medial straight line is medial.
Corollary

An area commensurable with a medial area is medial.

- Proposition 24
- The rectangle contained by medial straight lines commensurable in length is medial.
- Proposition 25
- The rectangle contained by medial straight lines commensurable in square only is either
rational or medial.
- Proposition 26
- A medial area does not exceed a medial area by a rational area.
- Proposition 27
- To find medial straight lines commensurable in square only which contain a rational rectangle.
- Proposition 28
- To find medial straight lines commensurable in square only which contain a medial rectangle.
- Proposition 29
- To find two rational straight lines commensurable in square only such that the square
on the greater is greater than the square on the less by the square on a straight line commensurable
in length with the greater.
Lemma 1

To find two square numbers such that their sum is also square.

Lemma 2

To find two square numbers such that their sum is not square.

- Proposition 30
- To find two rational straight lines commensurable in square only such that the square on the
greater is greater than the square on the less by the square on a straight line incommensurable in
length with the greater.
- Proposition 31
- To find two medial straight lines commensurable in square only, containing a rational rectangle,
such that the square on the greater is greater than the square on the less by the square on a straight line
commensurable in length with the greater.
- Proposition 32
- To find two medial straight lines commensurable in square only, containing a medial rectangle,
such that the square on the greater is greater than the square on the less by the square on a straight line
commensurable with the greater.
- Proposition 33
- To find two straight lines incommensurable in square which make the sum of the squares
on them rational but the rectangle contained by them medial.
- Proposition 34
- To find two straight lines incommensurable in square which make the sum of the squares
on them medial but the rectangle contained by them rational.
- Proposition 35
- To find two straight lines incommensurable in square which make the sum of the squares on
them medial and the rectangle contained by them medial and moreover incommensurable with the
sum of the squares on them.
- Proposition 36
- If two rational straight lines commensurable in square only are added together, then the whole
is irrational; let it be called binomial.
- Proposition 37
- If two medial straight lines commensurable in square only and containing a rational rectangle
are added together, the whole is irrational; let it be called the first bimedial straight line.
- Proposition 38
- If two medial straight lines commensurable in square only and containing a medial rectangle
are added together, then the whole is irrational; let it be called the second bimedial straight line.
- Proposition 39
- If two straight lines incommensurable in square which make the sum of the squares on them
rational but the rectangle contained by them medial are added together, then the whole straight line is
irrational; let it be called major.
- Proposition 40
- If two straight lines incommensurable in square which make the sum of the squares on them
medial but the rectangle contained by them rational are added together, then the whole straight line
is irrational; let it be called the plus a medial area.
- Proposition 41
- If two straight lines 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 are added together, then the whole straight line is irrational; let it be called the
side of the sum of two medial areas.
- Proposition 42
- A binomial straight line is divided into its terms at one point only.
- Proposition 43
- A first bimedial straight line is divided at one and the same point only.
- Proposition 44
- A second bimedial straight line is divided at one point only.
- Proposition 45
- A major straight line is divided at one point only.
- Proposition 46
- The side of a rational plus a medial area is divided at one point only.
- Proposition 47
- The side of the sum of two medial areas is divided at one point only.

### Definitions II

- Definition 1
- Given a rational straight line and a binomial, divided into its terms, such that the square on the
greater term is greater than the square on the lesser by the square on a straight line commensurable
in length with the greater, then, if the greater term is commensurable in length with the rational
straight line set out, let the whole be called a first binomial straight line;
- Definition 2
- But if the lesser term is commensurable in length with the rational straight line set out, let the
whole be called a second binomial;
- Definition 3
- And if neither of the terms is commensurable in length with the rational straight line set out, let the
whole be called a third binomial.
- Definition 4
- Again, if the square on the greater term is greater than the square on the lesser by the square on a
straight line incommensurable in length with the greater, then, if the greater term is commensurable in
length with the rational straight line set out, let the whole be called a fourth binomial;
- Definition 5
- If the lesser, a fifth binomial;
- Definition 6
- And, if neither, a sixth binomial.

### Propositions 48-84

- Proposition 48
- To find the first binomial line.
- Proposition 49
- To find the second binomial line.
- Proposition 50
- To find the third binomial line.
- Proposition 51
- To find the fourth binomial line.
- Proposition 52
- To find the fifth binomial line.
- Proposition 53
- To find the sixth binomial line.
- Proposition 54
- If an area is contained by a rational straight line and the first binomial, then the side of the area is the
irrational straight line which is called binomial.
- Proposition 55
- If an area is contained by a rational straight line and the second binomial, then the side of the area is the
irrational straight line which is called a first bimedial.
- Proposition 56
- If an area is contained by a rational straight line and the third binomial, then the side of the area is the
irrational straight line called a second bimedial.
- Proposition 57
- If an area is contained by a rational straight line and the fourth binomial, then the side of the area is the
irrational straight line called major.
- Proposition 58
- If an area is contained by a rational straight line and the fifth binomial, then the side of the area is the
irrational straight line called the side of a rational plus a medial area.
- Proposition 59
- If an area is contained by a rational straight line and the sixth binomial, then the side of the area is the
irrational straight line called the side of the sum of two medial areas.
- Proposition 60
- The square on the binomial straight line applied to a rational straight line produces as breadth the first binomial.
Lemma

If a straight line is cut into unequal parts, then the sum of the squares on the unequal parts is greater
than twice the rectangle contained by the unequal parts.

- Proposition 61
- The square on the first bimedial straight line applied to a rational straight line produces as breadth
the second binomial.
- Proposition 62
- The square on the second bimedial straight line applied to a rational straight line produces as breadth
the third binomial.
- Proposition 63
- The square on the major straight line applied to a rational straight line produces as breadth the fourth binomial.
- Proposition 64
- The square on the side of a rational plus a medial area applied to a rational straight line produces as
breadth the fifth binomial.
- Proposition 65
- The square on the side of the sum of two medial areas applied to a rational straight line produces as
breadth the sixth binomial.
- Proposition 66
- A straight line commensurable with a binomial straight line is itself also binomial and the same in order.
- Proposition 67
- A straight line commensurable with a bimedial straight line is itself also bimedial and the same in order.
- Proposition 68
- A straight line commensurable with a major straight line is itself also major.
- Proposition 69
- A straight line commensurable with the side of a rational plus a medial area is itself also the side of a
rational plus a medial area.
- Proposition 70
- A straight line commensurable with the side of the sum of two medial areas is the side of the
sum of two medial areas.
- Proposition 71
- If a rational and a medial are added together, then four irrational straight lines arise, namely a
binomial or a first bimedial or a major or a side of a rational plus a medial area.
- Proposition 72
- If two medial areas incommensurable with one another are added together, then the remaining
two irrational straight lines arise, namely either a second bimedial or a side of the sum of two medial areas.
- Proposition 73
- If from a rational straight line there is subtracted a rational straight line commensurable with the
whole in square only, then the remainder is irrational; let it be called an apotome.
- Proposition 74
- If from a medial straight line there is subtracted a medial straight line which is commensurable
with the whole in square only, and which contains with the whole a rational rectangle, then the
remainder is irrational; let it be called first apotome of a medial] straight line.
- Proposition 75
- If from a medial straight line there is subtracted a medial straight line which is commensurable
with the whole in square only, and which contains with the whole a medial rectangle, then the
remainder is irrational; let it be called second apotome of a medial] straight line.
- Proposition 76
- If from a straight line there is subtracted a straight line which is incommensurable in square
with the whole and which with the whole makes the sum of the squares on them added together
rational, but the rectangle contained by them medial, then the remainder is irrational; let it be called minor.
- Proposition 77
- If from a straight line there is subtracted a straight line 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, then the remainder is irrational; let it be called
area a medial whole.
- Proposition 78
- If from a straight line there is subtracted a straight line 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, then the remainder is irrational; let it be called that which produces
with a medial area a medial whole.
- Proposition 79
- To an apotome only one rational straight line can be annexed which is commensurable
with the whole in square only.
- Proposition 80
- To a first apotome of a medial straight line only one medial straight line can be annexed
which is commensurable with the whole in square only and which contains with the whole a rational rectangle.
- Proposition 81
- To a second apotome of a medial straight line only one medial straight line can be annexed
which is commensurable with the whole in square only and which contains with the whole a medial rectangle.
- Proposition 82
- To a minor straight line only one straight line can be annexed which is incommensurable
in square with the whole and which makes, with the whole, the sum of squares on them rational
but twice the rectangle contained by them medial.
- Proposition 83
- To a straight line which produces with a rational area a medial whole only one straight line can
be annexed which is incommensurable in square with the whole straight line and which with the
whole straight line makes the sum of squares on them medial but twice the rectangle contained by them rational.
- Proposition 84
- To a straight line which produces with a medial area a medial whole only one straight line
can be annexed which is incommensurable in square with the whole straight line and which with
the whole straight line makes the sum of squares on them medial and twice the rectangle contained
by them both medial and also incommensurable with the sum of the squares on them.

### Definitions III

- Definition 1
- Given a rational straight line and an apotome, if the square on the whole is greater than the square
on the annex by the square on a straight line commensurable in length with the whole, and the whole is
commensurable in length with the rational line set out, let the apotome be called a first apotome.
- Definition 2
- But if the annex is commensurable with the rational straight line set out, and the square on the
whole is greater than that on the annex by the square on a straight line commensurable with the whole,
let the apotome be called a second apotome.
- Definition 3
- But if neither is commensurable in length with the rational straight line set out, and the square
on the whole is greater than the square on the annex by the square on a straight line commensurable
with the whole, let the apotome be called a third apotome.
- Definition 4
- Again, if the square on the whole is greater than the square on the annex by the square on a
straight line incommensurable with the whole, then, if the whole is commensurable in length with the
rational straight line set out, let the apotome be called a fourth apotome;
- Definition 5
- If the annex be so commensurable, a fifth;
- Definition 6
- And, if neither, a sixth.

### Propositions 85-115

- Proposition 85
- To find the first apotome.
- Proposition 86
- To find the second apotome.
- Proposition 87
- To find the third apotome.
- Proposition 88
- To find the fourth apotome.
- Proposition 89
- To find the fifth apotome.
- Proposition 90
- To find the sixth apotome.
- Proposition 91
- If an area is contained by a rational straight line and a first apotome, then the side of the area is an apotome.
- Proposition 92
- If an area is contained by a rational straight line and a second apotome, then the side of the area is a
first apotome of a medial straight line.
- Proposition 93
- If an area is contained by a rational straight line and a third apotome, then the side of the area is a
second apotome of a medial straight line.
- Proposition 94
- If an area is contained by a rational straight line and a fourth apotome, then the side of the area is minor.
- Proposition 95
- If an area is contained by a rational straight line and a fifth apotome, then the side of the area is a
straight line which produces with a rational area a medial whole.
- Proposition 96
- If an area is contained by a rational straight line and a sixth apotome, then the side of the area is a
straight line which produces with a medial area a medial whole.
- Proposition 97
- The square on an apotome of a medial straight line applied to a rational straight line produces as
breadth a first apotome.
- Proposition 98
- The square on a first apotome of a medial straight line applied to a rational straight line produces as
breadth a second apotome.
- Proposition 99
- The square on a second apotome of a medial straight line applied to a rational straight line produces as
breadth a third apotome.
- Proposition 100
- The square on a minor straight line applied to a rational straight line produces as breadth a fourth apotome.
- Proposition 101
- 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.
- Proposition 102
- 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.
- Proposition 103
- A straight line commensurable in length with an apotome is an apotome and the same in order.
- Proposition 104
- 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.
- Proposition 105
- A straight line commensurable with a minor straight line is minor.
- Proposition 106
- 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.
- Proposition 107
- A straight line commensurable with that which produces a medial area and a medial whole is itself
also a straight line which produces with a medial area a medial whole.
- Proposition 108
- If from a rational area a medial area is subtracted, the side of the remaining area becomes one of
two irrational straight lines, either an apotome or a minor straight line.
- Proposition 109
- If from a medial area a rational area is subtracted, then 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.
- Proposition 110
- If from a medial area there is subtracted a medial area incommensurable with the whole, then the two
remaining irrational straight lines arise, either a second apotome of a medial straight line or a straight line
which produce with a medial area a medial whole.
- Proposition 111
- The apotome is not the same with the binomial straight line.
There are, in order, thirteen irrational straight lines in all:

- Medial
- Binomial
- First bimedial
- Second bimedial
- Major
- Side of a rational plus a medial area
- Side of the sum of two medial areas
- Apotome
- first apotome of a medial straight line
- second apotome of a medial straight line
- Minor
- Producing with a rational area a medial whole
- Producing with a medial area a medial whole

- Proposition 112
- 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 straight line and moreover
in the same ratio; and further the apotome so arising has the same order as the binomial straight line.
- Proposition 113
- 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.
- Proposition 114
- If an area is 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, then the side of the area is rational.
Corollary

It is possible for a rational area to be contained by irrational straight lines.

- Proposition 115
- From a medial straight line there arise irrational straight lines infinite in number, and none of them
is the same as any preceding.

Euclid's Elements: Book IX <--- Book X ---> Euclid's Elements: Book XI