The Eocene saw the rise of the euprimates (a term coined by Elwyn Simons). An alternative term is "primates of modern aspect". Going back to R. D. Martin's definition of fossil primates the euprimates share claws replaced by nails, opposable hallux (big toe) and postorbital closure (among other things) in common with all modern primates.The euprimates are divided into two families, the adapidae and omomyidae. The adapidae are composed of two subfamilies: nothartinae (5 genera) and adapinae (14 genera). The omomyidae are divided into three subfamilies: anaptomorphinae (15 genera), omomyinae (12 genera) and microchoerinae (4 genera) Plus two species (Arapahovius and Loveina) of uncertain affinities (for those who know something of taxonomy they are Omomyidae incerta sedis).
In general, Omomyids are tarsierlike in their morphology. For example, the dental formula (which gives the number of each type of tooth) is often similar (in particular the lower dental formula is sometimes 1 incisor, 1 canine, 3 premolars and 3 molars - as seen in tarsiers). The crania have tapered snouts and the ectotympanic bone is tubular - as in tarsiers. The nasal and olfactory regions are diminished in size and the eye orbits are expanded.
The adapids, on the other hand, are generally considered to be related to lemurs. They have a divergent hallux, flattened nails on the digits. Some (such as notharctus)have a postorbital bar and a petrosal bulla. Encephalization quotients (a ration of brain and body weight) have been calculated ad are withing the range of other Eocene mammals and are slightly lower than say, the Oligocene anthropoid Aegyptopithecus. In the middle ear region they bear some resemblance to lemurs. The ectotympanic bone (which supports the tympanic membrane) is variable in adapids ranging from a free ringlike structure (as in lemurs) to one that is expanded to form part of the lateral bull wall (as in lorises and Aegyptopithecus).
The morphology of both leads to several interesting problems. First, what are the phylogenetic relationships? There are three competing theories.
1) Omomyids share some characteristics with plesiadapiformes and at one time the plesiadapiformes were thought to have given rise to omomyids. In this theory the adapids were considered ancestral to anthropoids and the prosimians (other than tarsiers).
2) Since no shared derived characters link tarsiers and anthropoids to the other prosimians it has been suggested that plesiadapiformes gave rise to the euprimates which split into two branches. One branch was composed of adapids, lemurs and lorises, the other was composed of omomyids, tarsiers and anthropoids. In this theory, tarsiers are more closely related to omomyids than to anthropoids.
3)A variant of number 2, except tarsiers are more closely related to anthropoids than they are to omomyids.
There is a further complication. In both 2 and 3 above tarsiers are grouped with anthropoids and adapids are grouped with lemurs and lorises. The problem is adapids share quite a few traits with anthropoids, tarsiers share some traits with anthropoids but not lemurs and lorises. Paleontological data supported a linking of adapids and anthropoids. Comparitive anatomy (hemochorial placenta, presence of a retinal fovea, for example) and biochemical data supported a relationship between tarsiers (and consequently omomyids) and anthropoids. This led to something of a stalemate. If tarsiers (and hence omomyidae) were more closely related to the anthropoids (as the anatomical and biochemical data suggested) then adapids (as the paleontological data suggested) couldn't be. Which was right. A very intersting solution to this problem was presented by Gingerich and Schoeninger in 1977. The suggestion wasn't paid much attention to until 1986, when Rasmussen (in his 1986 paper "Anthropoid Origins: A Possible Solution to the Adapidae-Omomyidae Paradox") revived it. Grant the paleontological evidence that relates the omomyidae to tarsiers and adapidae to anthropoids. Lemurs and lorises would then form a sister group to both the omomyidae-tarsier group and the adapidae-anthropoid group. Consequently, tarsiers would be more closely related to anthropoids than to lemurs and lorises - which satifies the anatomical and biochemical evidence and the omomyid-tarsier and adapid-anthropoid groups could still be kept - satisfying the paleontological evidence. It's a good theory, unfortunately, one small fact stands in the way. This is the traits which seem to relate adapids to lemurs and lorises. In this theory, the traits relating adapids to lemurs and lorises are due to parallel evolution. Which has raised some objections since parsimony requires little or no parallel evolution.
It's been my experience that these kinds of situations come up a lot in primate - and human - evolution. No matter what phylogenies you create parallel evolution always comes into play. Personally, I consider a certain amount of parallel evolution to be a fact of life in primate evolution.