Simple inequality.

Simple inequality divides into 3 members, that is, multiple, superparticular, and superpartient.

Multiple.

Simple multiples arise when the greater term contains the lesser several times exactly, as double, triple, quadruple, and the like.

Simple superparticular.

The simple superparticular is when the greater term contains the lesser once and, moreover, some part of that lesser which is aliquot to it, that is, a multiplicative or properly-called part, namely sesquialtera 3-2; and sesquitertia 4-3; and sesquiquarta 5-4 ... and so it continues to infinity.

Simple superpartient.

The simple superpartient arises when the greater term contains within itself the lesser one single time, and beyond this a non-aliquot part of that lesser, which non-aliquot part is composed of aliquot parts of the lesser, as the 5 which contains the 3 once and has 2 over. Which 2 is not an aliquot part of the 3, but is composed of aliquot parts of the 3, that is, of 2 units.

Now it is to be seen how many parts of the lesser that non-aliquot part of the lesser contains: that is, if it contains 2 of them, it will be called superbipartiens, as 5 to 3, where the 3 goes into 5 once and 2/3.

And if that non-aliquot part were composed of 3 aliquot parts of the lesser, it would be called supertripartiens, as 8 to 5, which is one and 3/5.

And if it contains 4 of them it will be called superquadripartiens, as 9 to 5, which is one and 4/5; and thus one may proceed to infinity.

Composite inequality.

The species of composite greater inequality are 2, that is, multiple-superparticular and multiple-superpartient.

Multiple-superparticular.

The multiple-superparticular arises when the greater term contains within itself the lesser more than once and, moreover, some aliquot part of the lesser: as if the greater number contained the lesser 2 times and, beyond this, an aliquot part such as the half of the lesser. Then arises the species called dupla sesquialtera, as from 5 to 2. And if the greater contains the lesser 2 times and 1/3, it will be called dupla sesquitertia, as 7 to 3. And if the greater contains the lesser 3 times and 1/4, it will be called tripla sesquiquarta, and so it continues to infinity.