*Theoretic Arithmetic, in Three Books*by Thomas Taylor, 1816. The relevant information begins on page 34.

"Superparticular" and "epimoric" are terms of Latin and Greek origin respectively, meaning "greater by one part". Some examples of superparticular/epimoric ratios, when reduced to lowest terms, are 3/2, 4/3, 5/4, 6/5. 7/6 ... . Therefore ratios such as 6/4 and 40/30 are also superparticular/epimoric. But 2/1 is not.

Purported definitions such as: "Ratios of the form (

*n*+1)/

*n*where

*n*is a natural number", are not correct. They falsely include 2/1 and falsely reject 6/4.

A ratio is superparticular or epimoric if and only if, when reduced to lowest terms

*a*/

*b*,

*a*div

*b*= 1, and

*a*mod

*b*= 1,

where

*a*and

*b*are natural numbers, and

*a*div

*b*is the quotient from the integer division of

*a*by

*b*, where

*a*mod

*b*is the remainder.

Here is a grid summarising the Latin terminology for various classes of ratio whose lowest terms are

*a*/

*b*. It consists of 11 classes which do not overlap and which cover all ratios. These consist of 1 class of equality, 5 classes of greater inequality and (indirectly) 5 classes of lesser inequality.

remainder a mod b = quotient 0 1 >1 a div b = +-----------------+-----------------+-----------------------------------+ 0 | sub<class-of-inverse-ratio> | | | +-----------------+-----------------+-----------------------------------+ 1 | equal | superparticular | superpartient | | | | | +-----------------+-----------------+-----------------------------------+ >1 | multiple | multiple | multiple | | | superparticular | superpartient | | | | | | | | | | | | | +-----------------+-----------------+-----------------------------------+

Now the Latin with specific names for values greater than 1.

remainder a mod b = quotient 0 1 2 3 a div b = +-----------------+-----------------+-----------------+-----------------+ 0 | sub<class-of-inverse-ratio> . . . | +-----------------+-----------------+-----------------+-----------------+ 1 | equal | superparticular | superbipartient | supertripartient| . . . | | | | | +-----------------+-----------------+-----------------+-----------------+ 2 | double | double | double | double | . . . | | superparticular | superbipartient | supertripartient| +-----------------+-----------------+-----------------+-----------------+ 3 | triple | triple | triple | triple | . . . | | superparticular | superbipartient | supertripartient| +-----------------+-----------------+-----------------+-----------------+ . . . . . . . . . . . . . . .

And the corresponding Greek-derived terminology.

remainder a mod b = quotient 0 1 >1 a div b = +-----------------+-----------------+-----------------------------------+ 0 | sub<class-of-inverse-ratio> | | | +-----------------+-----------------+-----------------------------------+ 1 | equal | epimoric | epipolymoric | | | | | +-----------------+-----------------+-----------------------------------+ >1 | multiple | multiple | multiple | | | epimoric | epipolymoric | | | | | | | | | | | | | +-----------------+-----------------+-----------------------------------+

Now the Greek with specific names for values greater than 1.

remainder a mod b = quotient 0 1 2 3 a div b = +-----------------+-----------------+-----------------+-----------------+ 0 | sub<class-of-inverse-ratio> . . . | +-----------------+-----------------+-----------------+-----------------+ 1 | equal | epimoric | epidimoric | epitrimoric | . . . | | | | | +-----------------+-----------------+-----------------+-----------------+ 2 | double | double | double | double | . . . | | epimoric | epidimoric | epitrimoric | +-----------------+-----------------+-----------------+-----------------+ 3 | triple | triple | triple | triple | . . . | | epimoric | epidimoric | epitrimoric | +-----------------+-----------------+-----------------+-----------------+ . . . . . . . . . . . . . . .

"Epipolymoric", "epidimoric", "epitrimoric" etc, are my own coinage. Also, I suggest that the corresponding Latin terms ("superpartient", "superbipartient", "supertripartient" etc) might be anglicised to "supermultiparticular", "superbiparticular", "supertriparticular" etc, to eliminate the need for "-partient".