
Figure 1. Subset of the large reptile tree focusing on ungulates and their kin.
Today’s heresy began when several ungulate taxa
were added to the large reptile tree (LRT, Fig. 1, now 907 taxa, completely resolved with 228 traits). Equus, the horse; Paraceratherium, the giant hornless ‘rhino’; Ceratotherium the white rhinoceros and Embolotherium, a Mongolian brontothere.
It is widely accepted
and supported by the LRT that horses and rhinos share a common ancestor (Fig. 1). In this case, the LRT recovered either 1) an overlooked relationship, or 2) a case of convergence between Paraceratherium and Equus (Fig. 2). Wikipedia notes that rhinos are more closely related to tapirs. In the LRT tapirs are basal to a sister clade to the rhino/horse clade. Hyracotherium and Heptodon are basal to the rhino/horse clade.

Figure 2. Equus and Paraceratherium currently nest together on the LRT. Additional taxa will, no doubt, change that, but at present, Paraceratherium shares more traits with Equus than Ceratotherium, the white rhino (Fig. 2). The long neck and premaxillay teeth, along with other traits, separate these two from extant rhinos.
Equus ferus (Linneaus 1758; Figs. 2,3) includes several extant horses, mules and zebras. Compared to Hyracotherium the Equus preorbital region is longer, the frontal produces a postorbital bar that contacts the squamosal. The premolars are molarized. All four limbs end in a single toe, digit 3.

Figure 3. Equus the extant horse has a postorbital bar, an elongate rostrum and a single toe on each limb.
Hyracotherium leporinum (Owen 1841; 78 cm long; Eocene, 55-45 mya; BMNH C21361) nests basal to the horse/rhino clade, Equus. Heptodon (Fig. 5) is an Eocene sister. Canines remained. A diastema separated the canine from the premolars. The manus has four hoofed toes. The pes has three hoofed toes. The premolars were becoming molarized. Compare the skull of dog-size Hyracotherium to the similar skull of Paraceratherium, one of the largest land mammals of all time and Equus, the horse.

Figure 4. Hyracotherium is an Eocene horse sister in the LRT. Skull bones are colorized here.
Heptodon (Pachynolophus) posticus (Cope 1882; Eocene, 50 mya; 1m in length) was derived from a sister to Tapirus and was itself a sister to Hyracotherium and the base of brototheres like Embolotherium (below). Heptadon had reduced canines and developed a diastema (lack of teeth) posterior to them. The posterior cranium was slightly elevated. The manus had four digits. The pes had three.

Figure 5. Heptodon earlier nested with Tapirus, but with the addition of Equus, Hyracotherium, Paraceratherium and Embolotherium it nested closer to them.
The extant white rhinoceros
Ceratotherium simus (extant, Figs. 6, 7)) shares little in common with Paraceratherium, but shares a long list of traits with Embolotherium, including an oddly elevated pair of nasals and the near complete loss of the lumbar region. Both had smaller ancestors, so direct comparisons yield several possible convergences that currently nest as homologs. That’s what happens with taxon exclusion.

Figure 6. Ceratotherium, simum, the white rhinoceros with keratinous horns in dark brown. Note the elevated nasals and convex dentary ventral margin.

Figure 7. Ceratotherium (white rhino) skeleton, distinct from the long-legged Paraceratherium.
The last taxon to be added is
Embolotherium andrewsi (Osborn 1929; Late Eocene; 2.5m tall at the shoulder; Mongolia; Figs. 4, 5). It nests as a sister to the rhino, Ceratotherium, but this is likely to be overturned on convergence when additional taxa are added. This highly derived brontothere (= titanothere) has forked ‘horns’ (= rams). The rams are elevated nasal bones, hollow and fragile. These are in contrast to the solid rams of North American brontotheres. Click here to see the original AMNH illustration of the Embolotherium skull that portrays the area beneath the elevated nasal as a thick fleshy area. Alternatively Wikipedia reports, “the bony nasal cavity extends to the peak of the ram, thus implying that the nasal chamber was greatly elevated, possibly creating a resonating chamber.”

Figure 8. Brontotherium, a sister to Embolotherium, which is known from skulls, but no complete post-crania.
The skull of Embolotherium was 2x wider than tall at the orbits. The molars were much larger than the premolars, which were themselves molarized. The canines were vestiges. The posterior skull was greatly elevated. The dorsal spines were greatly elevated (Fig. 4). The lumbar region was reduced. The ilia were transverse. Four fingers are retained by the manus, indicating an early divergence from three-fingered horses and rhinos.

Figure 9. Embolotherium andrewsi animation modified from the AMNH website (see link above) to show a possible inflatable narial area, contra the original restoration with a fleshy, immobile sub-nasal area.

Figure 10. Paraceratherium pes. Note the reduced lateral and medial toes (2 and 4). As in Equus, and distinct from Ceratotherium, the central toe is much larger.
These are all perissodactyls
(odd-toed ungulates)
despite the fact that the manus has four digits. The pes (with toes) has an odd-number of digits (three or one). Wikipedia nests Desmostylia and Anthracobunidae among the perissodactyls, but they nest with hippos and mesonychids in the LRT.
Hyracodon
is not included here, but nests in the LRT between Hyracotherium and Heptodon, far from Paraceratherium. Apparently Equus and Paraceratherium have not been tested together in phylogenetic analysis under the assumption that one was a horse and the other a rhino.
Finally
note the large third digit of the pes of the giant three-toed horse, Paraceratherium (Fig. 10), as in Equus and distinct from Ceratotherium, which has three toes, but sub equal in size.
The above was dashed off
a little more quickly than usual. Please bring to my attention any typos or dangling hypotheses.
References
Cope ED 1882. Paleontological Bulletin 34:187.
Froehlich DJ 2002. Quo vadis eohippus? The systematics and taxonomy of the early Eocene equids (Perissodactyla). Zoological Journal of the Linnean Society. 134 (2): 141–256.
Linnaeus C 1758. Systema naturæ per regna tria naturæ, secundum classes, ordines, genera, species, cum characteribus, differentiis, synonymis, locis. Tomus I. Editio decima, reformata.
Osborn HF 1929. Embolotherium, gen. nov., of the Ulan Gochu, Mongolia. American Museum novitates; no. 353.
Owen R 1841. Description of the Fossil Remains of a Mammal (Hyracotherium leporinum) and of a Bird (Lithornis vulturinus) from the London Clay. Transactions of the Geological Society of London, Series 2, VI: 203-208.
wiki/Equus
wiki/Hyracotherium
wiki/Embolotherium
wiki/Paraceratherium
wiki/Heptodon