Generalisation of the terms "epimoric" and "superparticular" as applied to ratios
Posted: Wed Feb 12, 2020 10:48 pm
This arose as a question from @cmloegcmluin on facebook. And is a summary of information gleaned from the reference he found thanks to Google Books: 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.
Now the Latin with specific names for values greater than 1.
And the corresponding Greek-derived terminology.
Now the Greek with specific names for values greater than 1.
"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".
"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".