Cognates vs. Agnates

I'm finishing up my slides for the Nottingham lecture I'll be giving this week.  There is one detail I hadn't really appreciated, but perhaps I should.  This is in a passage that everybody who reads Grotius will cover:

Grotius, De Jure Belli ac Pacis I, 1, §3 (Editio Princeps:  Paris, 1625).  Peace Palace, The Hague.

The passage is about the different legal meanings of the word, jus, which functions as both 'law' and 'legal right.'  In his discussion, Grotius cites D.1.1.3:

'Nature has established a certain kinship [cognationem] amongst us, and so it follows, that it would be wrong for one man to lie in ambush of another.'  I never paid much attention to Grotius's choice of words for 'kinship' or 'blood-affinity' here.  

It is introduced here to explain his vision of societas humani genesis - humanity as a whole.  All of humanity is a kind of family.  Not an agnatic family, that the strict rules of Roman law define according to paterfamilial descent.  But families in most nations recognize via the jus gentium - as 'collaterals' and 'cognates.'  The significance of this is that, it becomes much easier, in principle, to make the cosmopolitan Stoic claim that all humans are in principle related to each other, all children of one common human stock.

The Gloss on D.1.1.3 reinforces this point by highlighting the word cognationem, and explains with reference to Scripture:

'This is because we are all descended from Adam.'

In this project, I make that point that rights are fundamentally relational.  But what kind of relations?  Grotius explains: Blood-relations.