It's often assumed by students of anthropology that studying descent (and alliance, descent's partner) was a dead end pursued by generations of misguided white guys. But that's not the case at all, and the reason most anthropologists manage to avoid contact with the hard problems of descent is simply that they now study societies whose core principles do not involve a strong concern with the transmission of rights and properties down a particular line. Most modern societies are states or parts of states, and those kinds of non-state social structural principles are not of great importance anymore.
So I'm going to summarise a short argument about descent made by Rodney Needham in the 1970s. It was believed by some anthropologists up to that point that non-state societies could be classified according to whether they were 'patrilineal', 'matrilineal', or 'cognatic', or some similar arrangement.* This classification was often based on the presence or absence of explicitly recognised groups based on those principles - if a society had formal groups composed of patrilineal relatives (i.e., relatives through the father's line) then it was considered to be a 'patrilineal' society, for instance.
Edmund Leach and Rodney Needham both objected to this due to the fact that all societies have several rules of descent involving different properties and different rights. In modern Britain, for instance, it is the norm for children to inherit the surname of their father, but other properties can be inherited cognatically - that is to say, through either line. I recently claimed Irish citizenship through my mother's line, for instance. If we classified Britain as a 'patrilineal' society based on the transmission of surnames, we'd be making a grave error about British inheritance law and custom. On the other hand, the Windsors - a corporate descent group if ever there was one - do pass property through the father's line (or did until the recent amendment).
So, given the fact that descent can go through any number of ways in any society, he thought it more productive to think about the logic of descent, the formal properties of the transmission of property, rights, &c. In a contribution to a volume he himself edited (Re-Thinking Kinship and Marriage, published in 1971), he devised a scheme that is still useful, dividing descent into a set of logical categories based on direction of descent. If we've got two sexes, then the logical possibilities are approximately these:
- m --> m
- f --> f
- (m --> m) + (f --> f)
- (m --> f) + (f --> m)
- (m --> m) || (f --> f)
- m/f --> m/f
In (2), descent is from woman to woman. This is normally called 'matrilineal' descent.
In (3), descent is a combination of (1) and (2); there are two groups, let's say, and in one you inherit membership through the mother and you inherit membership in the other through your father. This is usually called either 'double' or 'bilineal' descent.
(4) is 'alternating' descent; a man inherits from his mother and passes his property or rights onto his daughter, who in turn passes her property onto her son. That is to say, it goes back and forth between the genders.
(5) is 'parallel' descent, whereby there are 'two disjunct rules of descent', as Needham puts it. A man inherits his membership in a group, say, from his father, and a woman inherits hers from her mother, and never the twain shall meet.
(6) is 'cognatic' descent, using the Roman word for ancestors and elders on either side of the family. Either men or women can inherit from either men or women. Most inheritance in Britain today is cognatic in principle.
Remember, these are rules and principles, and are frequently broken. The rule is only one variable in determining human action, although it can be a strongly constraining one. So it isn't impossible for a woman to, say, inherit a position of power in a society with a strong patrilineal bent.
So, where might we find these rules? The uncontroversial ones are (1), (2), (3), and (6), as they are well-established, have been found throughout the world, and make a good deal of sense from the perspective of the transmission of wealth. Cognatic descent is fairly common (it's normal for wealth to be transmitted from mother to children in our society, for instance) and patrilineal and matrilineal descent groups have been recorded historically or ethnographically from almost every part of the planet, from ancient Rome to Timor to the Andes. Bilineal descent is less common, as fewer societies have both matrilineal and patrilineal descent groups, but there are known cases of this. A classic study of such a situation is Daryll Forde's work on Yakö descent groups (I couldn't find the precise article I wanted to illustrate this, but I did find another of Forde's on the same subject that is just as good).
But the other two are probably not important principles of inheritance in any human societies, although there have been claims to the contrary made by some well-known anthropologists. Curt Nimuendajú, a German man who went to the Amazon and became a renowned ethnologist, claimed that the Apinajé ('Apinaye'), who live in central Brazil and speak a Gê language, practiced parallel descent (5). He said that they had four descent groups that recruited solely from one gender - men inherited their membership from their fathers, and their sisters inherited membership of an entirely different group from their mothers. These groups, he claimed, were used to determine marriage - men from one group had to marry women from only one other group - and were therefore very important in the social structure.
Long story short, the anthropologist David Maybury-Lewis showed that the Apinajé used these groups only for certain ceremonies and dances, that they were not used for determining marriage arrangements, and that the Apinajé had a whole other system for that. As this was the only example on record of parallel descent, it appears that it has never been used for anything of major sociological importance. This makes sense, as parallel descent would fail to provide any means of gluing society together through marriage alliances, because offspring would belong to separate groups and to the same separate groups as their parents. That's bad social planning however you look at it, and it isn't surprising that parallel descent is not of any great importance anywhere.
In a more famous example, Margaret Mead claimed, after only a few months of fieldwork with her husband Reo Fortune, that the Mundugumor people of New Guinea practiced alternating descent. She went into extraordinary detail about it, too, claiming that the Mundugumor had all kinds of social strife resulting from the 'fact' that men inherited their property from their mothers and gave it to their daughters instead of their sons (and so on). She claimed that they called this system the 'rope' system. Apparently even weapons were passed from father to daughter/mother to son, which seems prima facie improbable - but since things that are prima facie improbable are par for the course in social anthropology, that didn't necessarily make it untrue that the Mundugumor had such groups.
Well, it turned out that the brevity of Mead's visit and her inability to work in the local language (she knew only neo-Melanesian creole/pidgin, 'Tok Pisin') hampered her accuracy and she got it wrong. The only known example of alternating descent turned out to be an incorrect ascription.
Remarkably, this was only discovered after Nancy McDowell, an anthropologist with expertise in the societies of the upper Sepik river, went and consulted Mead's own fieldnotes in order to use them as the basis of a historical ethnography of Mundugumor society (which she published in 1991 - The Mundugumor: from the fieldnotes of Margaret Mead and Reo Fortune, London: Smithsonian Institution Press). She discovered that, according to Mead's own account, the Mundugumor actually had patrilineal descent groups, and that Mead had probably misunderstood the account of the 'rope' as it was given to her by her informants. McDowell had previously commented on the huge difference between Mead's account and the account of the nearby Yuat people about their own system, which they also called 'rope', and this seems to have led to her becoming suspicious of Mead's ethnography.
So there you go. As far as the ethnographic record goes, cognatic, patrilineal, matrilineal, and bilineal descent are all fairly important and widespread, while parallel descent is used, as far as can be told, for membership of a dancing group in east-central Brazil and alternating descent is entirely non-existent as a social principle. Needham also mentioned the possibility of more idiosyncratic principles - for instance, the idea that a first child belongs to the father's group and a second child to the mother's - but these all came from early reports of the societies from which they supposedly came and have not been vindicated by later research (and as some of the examples came from Indonesia, I can immediately spot a possible explanation in terms of marriage and bridewealth, but I won't bore you with the details now).
The list of rules is pretty small, but the combinations are the important thing. Descent isn't the be-all and end-all of group formation, not by any means - all of these 'rules' can be fudged and bent, and marriage can have a huge impact on the nature of social groups in non-state societies - so this is just the very beginning of the study of human kinship, really.
*This is a gross over-simplification, of course.