The term cousin has several meanings. In the first place it is used for the relationship between children of siblings: the child of your mother’s brother is your cousin. By analogy, we speak of second cousins, referring to the relationship between children of cousins, and of third cousins for that between children of second-cousins, and so on.

Thus we have a definition for the term nth-cousin when n ≥ 1. This raises the question: can we speak of a zeroth-cousin, i.e. a cousin where n = 0? But to answer this question we must first answer: What is n?

This brief article offers an interesting answer to these questions; an answer which, if nothing else, I find very satisfying. If you have never thought about these questions, I would encourage you to stop reading at this point and come up with your own answers first, because

Trying to solve [a problem] on your own seems the only way in which you can assess how difficult the problem is; it gives you the opportunity to compare your own solution with mine; and it may give you the satisfaction of having discovered yourself a solution that is superior to mine.
Dijkstra, A Discipline of Programming.

## What is n?

Our English word cousin comes to us (courtesy of the French) from the Latin cōnsobrīnus (and cōnsobrīna, but I will focus on the masculine forms to save time), which is derived from sobrīnus, referring to a maternal first-cousin. Clearly, the etymology here, however interesting, does not answer our question.

The question What is n? came to my mind when I noticed that we were counting upwards, but not from zero. Somehow in the history of languages, whenever the terms first and second-cousin were first introduced, whether deliberately or by sheer instinct, someone once knew that the relationship between children of cousins should be assigned the number 2 rather than the number 1. They began counting cousinhood from 1, when they could’ve began at 0.

It is easy to see why this was done. Intuitively one feels that the word cousin implies a kind of distance in the relationship. The n in nth-cousin really tells us how distant the two families are: how many generations one has to go up the family tree before the relationship is one of siblinghood. Thus a first cousin is a relative with whom the minimum distance to sibling ancestors is 1.

By now the answer to the first question is obvious: your zeroth-cousin is your brother or your sister. But I think saying that n tells us how many generations we are from siblinghood is still an imperfect definition. After all, what is so special about siblings? What do they share? The answer is again obvious: they share parents. We shall take this as our definition:

An nth-cousin is a relative with whom one has to travel n+1 generations up the family tree before encountering a common ancestor.

But we cannot stop here: mathematics has been called “the science of abstraction” and this definition is now ripe for generalisation. Our definition is really only applicable if n ≥ 0. What if n is negative? Can we define negative-cousinhood?

## Negative Cousins

The answer to this question (as well as the question itself, incidentally) came into my mind last night, while I was speaking with my wife. Interestingly, as I have Googled around, I found a similar definition here to what I will propose now.

Negative numbers are the same as positive ones, except that they denote quantities that increase in the opposite direction. So the correct way to speak of a negative cousin is to think of one with whom one moves down the family tree towards a common descendant. Thus -1st-cousins are spouses (or technically, people who share a child). -2nd-cousinhood is the relationship that sustains between the parents of a couple: between the bride’s and bridegroom’s parents. (Strictly speaking this definition would only apply after a child is born to the couple, but you get the idea!)

I like this definition because it gives us new and entertaining words for relationships that already exist in the family. It shows that the nuclear family is the nexus in the family tree. Streams flow into it and out of it in a kind of symmetry that is beautiful.

Moreover, if we introduce the “removed” concept that is sometimes used (e.g. your first cousin’s child is your first cousin once removed) we get the delightful expression that your parents-in-law are your -2nd-cousins once removed. If that doesn’t make you smile then nothing I say will ever make you smile.

But if one wants a better term than negative cousin, my suggestion is we use coprogenitor. Thus your spouse is your first coprogrenitor, and your parents-in-law become your second coprogrenitors once removed.

PS: I added the above meme on 23/05/24.