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[Bg] Throughout the years, there have been various attempts to reconstruct the extinct giant shark, Carcharocles (Otodus) megalodon. Most of these have based on its modern relative and closest living analogue, the great white shark Carcharodon carcharias.
That is the first thing important to note here, because all the following is exclusively based on the white shark. Of course the very real possibility remains that C. megalodon looked nothing like envisioned here, but that is useless as long as there are no data to aid in coming up with a more likely reconstruction. Most people seem to be quite fine with using the white shark as an analogue, as am I, and it’s primarily those people that this is targeted at.

There is a common trend of portraying C. megalodon as a sort of overgrown great white shark on steroids. As those who have read certain previous posts of mine already know, this is partly based on real data, and partly on conjecture, but so far it has not been possible to largely eliminate that element of conjecture because of the lack of quantitative methodology used in reconstruction.

The real data are two-fold:
> Firstly, the fossil teeth of C. megalodon are thicker and (relative to their height) wider than those of white sharks, at least the larger anterior and anterolateral ones. At a similar overall size (width, length or circumference) of the jaws, the anterior teeth of megalodon would hence be shorter and stouter, i.e. more robust. Most likely, this is an effect of both the gigantic size and the effects of scaling (i.e. allometry), it’s phylogenetic heritage (other otodontid teeth are also very thick labiolingually), and its behaviour (dealing with, in absolute terms, very large prey items). However the effect this might have on jaw morphology is difficult, of not impossible, to estimate.
> Secondly, the allometric scaling of body mass in extant white sharks suggests that larger sharks tend to be more robust than smaller ones, which can be extrapolated, albeit with questionable relevance, to the size of megalodon. There have been six published regression equations for deducing white shark mass from total length (Fig. 1), which provide the data I am relying on here.
Megmass - by theropod1

Fig. 1: Length-weight relationships of C. carcharias according to various published sources (See references).
These data indeed support that larger sharks should (most likely) be more robust. Which is about where the considerations typically end. But is there a way to approach this question with a bit more rigor than just taking an arbitrary great white shark and bulking it up by an arbitrary amount? It turns out that yes, there is.

[M&M] First, let’s establish what sizes will be used. Changing these slightly will not have major effects, as you are going to see. But for the sake of comparison, I am going to be using a 5m great white shark (good-sized, but still a relatively common size for the species, which is always advantageous) and a 16.8m megalodon (a size suggested in a recent SVP abstract (Perez et al. 2018, SVP abstract volume p. 196) for a specimen with a complete, associated dentition, and also a size I have shown previously). This is not supposed to be a comment on megalodon maximum or average sizes, but on its morphology, so I’ll leave it at that.

The equations suggest that the 16.8m megalodon (46,112-56,797kg) would be between 0 and 23% heavier than an isometrically scaled great white shark of (1,083-1,294kg at 5m). Using the average of all 6, the megalodon is 13.6% heavier than the great white at the same length. Since length is fixed, we can attribute this entire increase in robusticity to the body cross-section, the function of width and depth. Assuming the increase is the same in both dimensions, we get the percentage of increase by taking the cube root, giving us 6.6%. So in other words, we should be drawing megalodon 6.6% wider and deeper-bodied than a great white shark. That’s as much as I’ve written previously. Now, the debatable part is obviously going to be what great white that should be, because most pictures of great whites, especially nice ones in orthogonal views without deformation or distortion, have no weight measurements attached to them.

So to solve this problem, I constructed a digital 3d model of a great white shark in Blender using lateral and top-view pictures of great whites and reference drawings (e.g. ¹) as a basis. I modeled the trunk, pectoral, first dorsal and tail fins separately and then unified them using a boolean modifier. The other fins were omitted from the model to avoid it becoming overly complex and computationally demanding, as they would add very little mass anyway. Then I adjusted its width and depth (isometrically) to make it fit the predicted body masses.

[RESULTS] The resulting reconstruction (Fig. 2) suggests a robust, but streamlined body shape, even for the maximum model scaled to 56.8t.
The bulk increase necessary in terms of scaling of the body cross-section is not extreme. Even the maximum model, based on Gottfried et al.’s results, is still well-within the range of variation seen in extant great whites. The great white reconstruction also lends weight to the drawn representations of white sharks as commonly found in reference books (e.g. Compagno 1984) being good representations of the statistically typical proportions of the species as a whole.
Megcomp by theropod1
Fig. 2: Surface model of C. carcharias and C. megalodon (16.8m) scaled to match predicted volume. Specific gravity is assumed as 1.0. Green model reflects the mass predicted by the allometric equation in Gottfried et al. 1996, which suggests the highest body mass at almost 57t.  Blue and red models reflect the mean of masses predicted by 6 regression equations (see references). Scalebars: 1m each. Grid: 2m per cell. Human diver: ~1.8m standing height.

[D&C]One caveat to this is that I did not model the buccal cavity of the shark, as I did not have a reference for it. With the mouth closed, I assumed it would not greatly decrease the volume of the shark for given external dimensions as long as the mouth was closed, as modeled here. If indeed it does, then this might result in an underestimation of the external dimensions of a shark of a given body mass, although I don’t think this is likely. Should you have information to the contrary, please let me know.

There have been well-founded suggestions that megalodon would have had a proportionately larger tail fin to maintain high speeds and activity levels at its large size, and I agree with this, as it is consistent with a possible higher vertebral count and with the scaling effects at its large body size. This would largely have a negligible impact on mass. In the great white model at 1.2t, the tail fin masses less than 30kg. Even doubling the fin size would not greatly increase the size of the model. In the above representation, the fins are scaled up in width and height along with the body largely for practical reasons and because there is no way to precisely estimate their proper size.
Most likely, the tail fin would be somewhat larger in life than it is portrayed here, which is what I am going to use for the drawn reconstruction I plan to follow this up with.

So with that out of the way how does this leave extremely bulky megalodon reconstructions, such as these?

Well, as with great whites, there would have been a large level of variation. These estimates represent the mean. This is, for obvious reasons the most sensible representation for the species. This does not mean that there couldn’t have been some megalodon individuals that were in fact as bulky as portrayed, just like there are some great white sharks that are ridiculously bulky compared to the normal body shape of the species.
It’s just that they get far too much representation, and are ironically often considered to be the most reliable reconstructions. On the other hand, it’s also entirely plausible for some megalodons to not be bulkier than normal great whites at all. One of the aforementioned equations, McClain et al., in fact did not find allometry in their dataset at all.

Bottom line: The most accurate and rigorous data suggest megalodon should be portrayed on average ~7% deeper and wider-bodied than a typical great white shark. The above model (blue reconstruction, Fig. 2) illustrates the resulting body shape. Various scaling equations found slightly different values varying from 0 (i.e. isometry) to 11%, but it does not seem like such differences would have a massive impact on how bulky the animal would look, at least not sufficiently to lend support to the many extremely robust reconstructions out there. These appear to not be supported by any quantitative data and should be considered representations of unusually heavyset, pregnant and or full-stomached individuals, not a typically robust, large megalodon.

Casey, John G.; Pratt, Harold L. 1985. Distribution of the White Shark, Carcharodon carcharias, in the Western North Atlantic. Memoirs of the Southern California Academy of Sciences, 9 (Biology of the White Shark, a Symposium) pp. 2-14.
Compagno, L.J. 1984. Sharks of the world: an annotated and illustrated catalogue of shark species known to date. FAO Fisheries Synopsis Volume 4 (No. 125).
Gottfried, Michael D.; Compagno, Leonard J.V.; Bowman, S. Curtis. 1996. Size and Skeletal Anatomy of the Giant “Megatooth” Shark Carcharodon megalodon. In: Klimley, Peter A.; Ainley, David G.: Great White Sharks: the biology of Carcharodon carcharias. San Diego, pp. 55-66.
Kohler, Nancy E.; Casey, John G.; Turner, Patricia A. 1995. Length-Length and Length-Weight Relationships for 13 Shark Species from the Western North Atlantic. Fishery Bulletin, 93 pp. 412-418.
McClain, Craig R.; Balk, Meghan A.; Benfield, Mark C.; Branch, Trevor A.; Chen, Catherine; Cosgrove, James; Dove, Alistair D.M.; Gaskins, Lindsay C.; Helm, Rebecca R.; Hochberg, Frederick G.; Lee, Frank B.; Marshall, Andrea; McMurray, Steven E.; Schanche, Caroline; Stone, Shane N.; Thaler, Andrew D. 2015. Sizing ocean giants: patterns of intraspecific size variation in marine megafauna. PeerJ, 3 (715) pp. 1-69.
Mollet, Henry F.; Cailliet, Gregor M. 1996. Using Allometry to Predict Body Mass from Linear Measurements of the White Shark. In: Klimley, Peter A.; Ainley, David G.: Great White Sharks: the biology of Carcharodon carcharias. San Diego, pp. 81-89.
Tricas, Timothy C.; McCosker, John E. 1984. Predatory Behaviour of the White Shark (Carcharodon carcharias) with notes on its biology. Proceedings of the California Academy of Sciences, 43 (14) pp. 221-234.

Dilophosaurus wetherilli, from the Hettangian or Sinemurian Kayenta Formation of Arizona, to many seems to be a sort of quintessential early theropod.
And there are a number of good reasons for that:
Firstly, it’s been known for a long time. Welles dug up several nice, articulated skeletons as early as the 1940s. Secondly, it’s from the US…that often (though not always) helps things along.  Thirdly, it’s morphology was so clearly different from other theropods known at the time, and is still so unusual compared to other well-known theropods, that it’s easily recognizable to anyone, probably leading to its inclusion, albeit in a very inaccurate form, in the original Jurassic Park movie.

And finally, it is true that Dilophosaurid-grade theropods, a number of stem-neotheropod taxa with very similar morphology (while probably not all forming a natural clade), were widespread components of upper Triassic and Lower Jurassic faunas worldwide.
Gojrasaurus quayi from the Norian of New Mexico, Zupaysaurus rougieri from the Norian of Argentina and Liliensternus liliensterni from the Norian of Germany are among the earliest theropods (note Herrerasaurids’ status as theropods is very questionable) that we could consider "large-bodied", and which filled the apex-predatory role in their ecosystems.
This type of theropod persists into the Lower Jurassic: The Elliot Fm. in South Africa (Hettangian/Sinemurian) has yielded Dracovenator regenti. The Hettangian Lufeng Fm. has Sinosaurus triassicus (formerly "Dilophosaurus sinensis"). And perhaps most famously, the Hanson Formation of Antarctica has produced Cryolophosaurus ellioti.  However the latter two taxa might actually be basal Tetanurae, which would mean these forms with somewhat similar morphological features were only quite distantly related, but also imply this was a common morphology for basal members of all neotheropod lineages. And of course there is Dilophosaurus wetherilli itself, famous for its double-crests and being one of the few distinct morphotypes of theropods recognized in the early 20th century.
All of these seem to share weird head crests, more or less developed premaxillary notches and relatively slender postcranial body proportions.

Zupaysaurus rougieri by theropod1

As the largest predators in their various respective ecosystems, naturally we must wonder about the size of these animals.
Commonly given mass estimates tend to be extremely low for the proposed dimensional measurements. This is generally explained by the very gracile, coelophysoid-grade morphology of these taxa. Paul (³ and 2009) estimates the holotype of Dilophosaurus at 283 kg and the largest specimen at 400 kg. Smith et al. 2007 estimate Cryolophosaurus at 465 kg based on femur circumference (claiming it to be the largest Lower Jurassic theropod), even though the specimen has a femur length similar to an adult Allosaurus’ massing 1.5 t.
So were these animals really that small?

Most specimens of Dilophosaurus are, in fact, juveniles. That includes the 6 m long holotype skeleton (UCMP 37302, 53 cm skull, 56 cm femur) described in detail by Welles (1984), as well as one of the original paratypes (Welles 1970), another partial skeleton referred by Tykoski (2005) and part of the fragmentary material described by Gay (2001).

The only substantially complete adult individual is one of the paratypes, UCMP 77270 (Welles 1970). This individual has dorsal and sacral vertebrae on average 18% longer than the holotype, although its precaudals in general are 14% longer, owing to the negatively allometric neck length. It also has an appreciably larger skull, at 62 cm, and a slightly longer femur, at 59 cm (all measurements for this specimen from the theropod database², holotype from Welles 1984).
Dilophosaurus tends to get portrayed as downright ridiculously skinny, likely mostly inspired by Greg Paul’s shrinkwrapped skeletal.
To test this, I made a volumetric 3D model based on ScottHartman’s composite Dilophosaurus skeletal (which is considerably more robust-looking than Paul’s), scaled to 6 m total axial length (that measurement by Welles is probably reliable, as the skeleton was found in articulation and is very complete). For the body width, I adapted the dorsal view skeletal of Syntarsus rhodesiensis from Paul (2016), which is probably quite conservative.
For the holotype, I’m getting a volume of 366 l, which translates to a mass of 330 kg (assuming a density of 0.9). So very light for its length, but a bit heavier than the 283 kg Paul and Mortimer list it at.

Compil by theropod1

We can estimate total length of the adult UCMP 77270, using the length ratios of all overlapping vertebrae, as ~6.9 m. I’d suggest scaling body mass from only the dorsal and sacral vertebrae, because the largest part of the mass is located in the torso, which gives a mass estimate of ~544 kg.

A couple of other speculative size estimates for more or less related species:

>Liliensternus liliensterni (larger individual, subadult, von Huene 1934):
Femur length: 42 cm
Total length (based on all precaudal vertebrae, scaling from UCMP 37302): 5.1 m
Body mass (based on dorsal  and sacral vertebrae): 178 kg

>Gojirasaurus quayi (UCM 47221, subadult, Carpenter 1997):
Tibia length: 47 cm
Pubis length: 50 cm
Total length (based on 3 mid-posterior dorsal vertebrae, scaling from UCMP 37302): 5.0 m
Total length (based on tibia): 5.1 m
Body mass (based on dorsal vertebrae and tibia): 187-199 kg
(long pubis might imply deeper body shape)

>Zupaysaurus rougieri (PULR-076, ?adult, Arcucci & Coria 2003):
Skull length e. 42 cm (measured in Fig. 2E)
Total length (based on skull and axis length, scaling from UCMP 37302): 5.3 m
Body mass (same method): ?221 kg
(very long axis might just be due to proportionately long neck)

>Dracovenator regenti (BP/1/5243, ?adult, Yates 2005):
Skull length e. 56 cm
Total length (based on skull length): 6.3 m
Body mass (based on skull length): 389 kg

>Cryolophosaurus ellioti (FMNH PR1821, late subadult/young adult, Smith et al. 2007):
Femur length: 77 cm
Total length (based on all precaudal vertebrae, scaling from UCMP 37302): 7.9 m
Body mass (based on the dorsal  and sacral vertebrae): 794 kg
Body mass (based on reverse-calculated femur-circumference of 233mm from Smith et al. 2007, using Anderson et al. 2009 and cQE-phylocor from Campione et al. 2014): 687 kg

Discussion and Conclusions (for now):
Dilophosaurus, and probably most other coelophysoids, were probably a little heavier than commonly cited, and quite a lot when taking into account that many are immature. Adult Dilophosaurus seem to have approached or exceeded 500 kg, not the 400kg from Paul or 330 from Mortimer. However this estimate is very sensitive to placement of the vertebrae in question, and there is some serious confusion because Welles confusingly labels dorsals 1 through 5 as pectorals, while Mortimer confusingly labels what I think is probably dorsal/pectoral 1 as cervical 10. So take the size of the larger individual with a grain of salt. Welles (1970) claims the specimen was one third larger; if he was referring to linear dimensions, then this estimate is quite conservative, although if he meant volumetrics it is too high.

The commonly cited length estimate for Liliensternus are pretty much spot-on when scaling from vertebral lengths of Dilophosaurus. The mass estimates however, such as Paul’s (2009 and ³ 130kg) appear too low based on the relative length of the dorsal and sacral vertebrae. Furthermore, the known specimens are both subadults (Tykoski 2005), which implies larger sizes for adult individuals.

Gojirasaurus is possibly slightly larger than Liliensternus, based on the vertebral and tibial lengths, although the lack of available cervicals results in a slightly lower total length estimate based on the dorsals alone. Its pubis is even slightly longer than that of the Dilophosaurus holotype, which could imply a slightly deeper-bodied morphology. Carpenter’s estimate of 5.5 m could be roughly accurate.

Zupaysaurus has a relatively small skull based on the restoration in the description paper, but a very long axis (11 cm). The mass estimate presented here might be too high due to this. There are more vertebrae preserved, which sadly don’t have any reported measurements, so I currently can’t make a better estimate.

I agree with Smith et al. 2007’s claim that Cryolophosaurus is the largest Lower Jurassic theropod, but I think their mass figure and most other size figures for this taxon are underestimates. The holotype may well have approached 8m and 800 kg (which is also much more consistent with the reported lengths of its vertebrae and the femur). Finally, considering its poorly fused sacrum (Smith et al.), it may well be a young individual that was still well below average adult size.

Anderson, J. F., A. Hall‐Martin, and D. A. Russell. 1985: Long‐bone circumference and weight in mammals, birds and dinosaurs. Journal of Zoology 207:53–61.
Arcucci, A. B., and R. A. Coria. 2003: A new Triassic carnivorous dinosaur from Argentina. Ameghiniana 40:217–228.
Campione, N. E., D. C. Evans, C. M. Brown, and M. T. Carrano. 2014: Body mass estimation in non-avian bipeds using a theoretical conversion to quadruped stylopodial proportions. Methods in Ecology and Evolution 5:913–923.
Carpenter, K. 1997: A Giant Coelophysoid (Ceratosauria) Theropod from the Upper Triassic of New Mexico.(With 8 figures and 3 tables in the text). Neues Jahrbuch fur Geologie und Palaontologie-Abhandlungen 205:189–208.
Gay, R. 2001: New specimens of Dilophosaurus wetherilli (Dinosauria: Theropoda) from the early Jurassic Kayenta Formation of northern Arizona. Mesa Southwest Museum Bulletin 8:19–23.
von Huene, F. F. 1934: Ein neuer Coelurosaurier in der thüringischen Trias. Palaeontologische Zeitschrift 16:145–170.
Paul, G. S. 2009, 2016: The Princeton field guide to dinosaurs. Princeton University Press, Princeton.
Smith, N. D., P. J. Makovicky, W. R. Hammer, and P. J. Currie. 2007: Osteology of Cryolophosaurus ellioti (Dinosauria: Theropoda) from the Early Jurassic of Antarctica and implications for early theropod evolution. Zoological Journal of the Linnean Society 151:377–421.
Tykoski, R. S. 2005: Anatomy, ontogeny, and phylogeny of coelophysoid theropods.PhD Thesis, University of Texas at Austin.
Welles, S. P. 1970: Dilophosaurus (Reptilia, Saurischia), a new name for a dinosaur. Journal of Paleontology 44:989.
Welles, S. P. 1984: Dilophosaurus wetherilli (Dinosauria, Theropoda). Osteology and comparisons. Palaeontographica Abteilung A:85–180.
Yates, A. M. 2005: A new theropod dinosaur from the Early Jurassic of South Africa and its implications for the early evolution of theropods.

¹Hartman, Scott. 2015: New-look Dilophosaurus. DeviantArt. Downloaded from… on 9 September 2018.
²Mortimer. : Coelophysoidea. Downloaded from… on 11 September 2018.
³Paul, Gregory S. 2010: Data - Gregory S. Paul’s Dinosaur Mass Table. Downloaded from on 11 September 2018.
Now with an attempt at a size estimate:
Note that I am updating the post periodically as I’m coming across more data or as my sources are updated.

Figure1 by theropod1

(A-D) AMNH 5767, Holotype of "Epanterias amplexus" Cope 1878, cf. Allosaurus fragilis sensu Chure (2000). (A) Axis, (B) D1 in anterior and left lateral view. (C) Right coracoid in proximal and lateral view. (D) Middle cervical centrum (C6 or 7) in anterior and left lateral view. (E) Axis, D1 and Coracoid of ?DINO 2560, 8.6m adult of Allosaurus. (F) Axis of USNM 4734, 7.8 m Adult of Allosaurus. (G) Axis and D3 of USNM 8367, 8.2m adult of Allosaurus. Scale Bar: 10 cm.
This is not the first time I am pointing out the critical importance of sample size and the difference between average and maximum body sizes in extinct taxa, and I think what follows could be a nice demonstration of this.

A recent comment by :iconrandomdinos: made me wonder about the real size of AMNH 5767, better known as "Epanterias amplexus" (or just plain old Allosaurus fragilis sensu Chure 2000, though common wisdom has it that this is the specimen actually reaching the 12m-mark). It’s commonly stated to be 20% bigger than any other Allosaurus (and that might well be true), but that’s the sort of vague statement we shouldn’t rely upon too heavily. What Allosaurus is "any" Allosaurus. Does that compare it to an 8m individual or a 10m one? This or similar wording would imply the largest known, but what is the largest individual the respective authors were aware of?

The type of "Epanterias" includes one mid-cervical centrum of decidedly unspectacular size (regular Allosaurus-sized), which I hypothesize may not have been part of the same individual as the other remains. There’s part of a metatarsal, but there are no measurements or figures reported, so I’m ignoring it in this discussion of the specimen’s size.These other bones, a complete axis, almost complete first dorsal neural arch and a coracoid are all of exceptional size. I daresay not just by Allosaurus-standards, but for theropods in general. Considering this massive size (certainy the largest theropod ever found at that point in history), it is not entirely surprising that Cope initially identified the specimen as a sauropod.

Screenshot 20180831 032235 by theropod1

Screenshot 20180831 032106 by theropod1Screenshot 20180831 032257 by theropod1
Screenshot 20180831 013617 by theropod1Screenshot from Osborn & Mook 1921

Note the measurements of the dorsal vertebra as given by Cope: neural arch height 290 mm, diapophyseal width 400 mm. These probably do refer to the preserved portion because Cope makes note of the shape of the end of the neural spine, so he, unlike Osborn & Mook, cannot have incorrectly thought it to have been incomplete, which means that the height of the neural arch is completely preserved.
Osborn and Mook made some other errors in their reconstruction. For example, the centrum they reconstructed looks like it’s amphiplatyan, which is in stark contrast to the strongly opistocoelous centra of anterior Allosaurus dorsals. Chure also notes that what they portray as dorsal parts of the centrum are actually the pedicles. And as already noted, the neural spine is actually complete, not lacking its tip as resotred by Osborn & Mook. So all in all, this warants a new attempt to restore the vertebra, based on Allosaurus.
Epanterias by theropod1

D1 of Allosaurus, from Madsen 1976. In case there were any lingering suspicions that AMNH 5767 was not an Allosaurid, I hope this settles them.

Chure (2000) suggests D1 as the most likely position for the vertebra, but in the following I will also consider other positions anyway as a side-effect of lacking the necessary measurements of D1 in other specimens.

Here’s my attempt of reconstructing the dorsal vertebra:

Epanterias Tyrannotitan by theropod1

A: First dorsal vertebra of AMNH 5767, holotype of "Epanterias amplexus", junior subjective synonym of Allosaurus fragilis according to Chure (2000), Allosauridae indet according to Mickey Mortimer, in anterior and lateral view. Modified from Osborn & Mook (1921), with the missing portions restored after Allosaurus fragilis (Madsen 1976). B: First dorsal vertebra of Tyrannotitan chubutensis paratype MPEF-PV 1157 from Canale et al. 2015, scaled after scalebar.

Notice anything? That thing is freakin’ huge. It’s almost the Amphicoelias of Theropoda. I’m saying almost, because the vertebra is pretty similar in size to known giant theropod vertebrae. But right up there with the biggest of them (i.e. T. rex, Tyrannotitan, and supposedly Giganotosaurus), and extremely large by the standards of an Allosaurid.

Measurements and figures in Chure (2000) and Osborn & Mook (1921) suggest that the Axis (250mm tall, Gilmore reports that of USNM 4734 as 170mm) and coracoid (seems to be ~39cm long in the figure in Osborn & Mook, using the vertebral dimensions as a scale bar, the coracoid of USNM 4734 is 170mm long) both also reflect this to varying degrees, although I would argue that both are less reliable indicators due to smaller size and the coracoid being an appendicular element.

Based on this (unless I’m overlooking something major), I’d say we should tentatively consider the specimen AMNH 5767 to be among the largest known theropods.

Now as for the absolute size, that is a bit trickier. My estimated maximum centrum length is ~19cm but that is including the condyle, which is huge. Ventral rim-to-rim centrum length is ~12cm.
The centrum is not actually preserved and I cannot guarantee for its length being restored accurately, as it (far more so than the width or height) is subject to some major speculation. Being constrained by the size of the neural arch, its relatively easy to get its height and width correctly by just scaling from the proportions of Allosaurus, but the length is trickier, as the Epanterias neural arch seems more anteroposteriorly compressed. That being said, the preserved pedicle length of 112mm seems to match up well with a centrum length of 12cm (excluding the condyle). However I would argue that a more reliable measurement for comparative purposes would be the total height of the vertebra, which is 44cm, or the diapophyseal width, which is 45cm.

Now I actually have yet to find published figures of the first dorsal vertebra in Allosaurus, and I’d far prefer to compare to an actual first hand measurement rather than going by half-assed measurements from low-resolution figures and skeletals, but still I couldn’t resist the temptation of trying to measure the height of D1 in Scott Hartman’s DINO 2560 and MOR 693 and Randomdinos’ USNM 4734
The results were…well ridiculous to say the least.
Hartman’s DINO 2560 has an axial length of 973px and D1 is 25±1px tall, MOR 693 is 1027px with D1 25±1px tall and USNM 4734 is 1538px with D1 38±1px tall. I’ll let you do the math yourself, just so nobody can misconstrue this into me claiming ridiculous sizes for this specimen or wrongly cite them from me. So obviously for this to work at all, we already have to assume proportionately taller vertebrae, and I’m fine with that (to an extent).

Gilmore (1920) gives some actual measurements of USNM 4734’s vertebrae, however unfortunately D1 is too poorly preserved and hence has none of these measurements reported.
Maximum (i.e. with the condyle) centrum length of C9 is 123mm, in D2 it is 88mm. The mean of these two, as an estimate for D1’s centrum length, is 106mm (I’m being a bit generous here, it’s entirely possible D1 isn’t longer than D2 at all. It’s even possible that the vertebra of AMNH 5767is actually a D2, although the most likely position is D1). According to Randomdinos’ skeletal reconstruction, total length of this specimen is 7.8m. The resulting TL for AMNH 5767 would be 14.0m.
The most anterior dorsal of USNM 4734 that has its total height reported by Gilmore is D9, which is 290mm tall. However, USNM 8367, as listed in the same table, seems to be consistently bigger in all measurements (around 20% based on the average of overlapping centra, but let’s not get into that right now), and it has a reported measurement of 253mm for D3, so that should be closer. So comparing the D1 of AMNH 5767 to D3 of USNM 8367, it would end up at least 13.6m long, assuming 8367 was the same size as 4734.

Sadly neither of these is really satisfying, as neither actually compares it to a real measurement of a real D1 of an Allosaurus of known size.

The 25cm tall axis would indicate a Tl of 11.5m based on USNM 4734. The 39cm coracoid is a whopping 2.3 times the length of USNM 4734’s, again bringing us into a region of fantastic sizes, but I’ll concede that this element only provides weak evidence. All in all, I think the dorsal neural arch remains our best bet.

So a request: I know full well that I have been…overly enthusiastic about gigantic allosaurids on past occasions. And this certainly does look a bit like one of those too good to be true "giant pliosaur scapula turning sauropod pubis–April fool’s joke giant pneumatic sauropod vertebra that’s actually porous rock"-cases.
However I have done my best to stay conservative in this case; no matter how I turn it, 29cm in neural arch height and 40cm preserved diapophyseal width seem to result in a humungous vertebra. So please check my numbers and tell me if I got something wrong about that
If anybody knows any actual measurements of anterior dorsal vertebrae in Allosaurus specimens with well-constrained overall size, that would be appreciated too. And finally, is there any multiview figure of a T. rex’ D1 with known dimensions anywhere? I played around a little with Sue’s D4 for comparison (obviously D1 would be smaller, but I just wanted to get an idea of the relative sizes).
Epanterias Tyrannosaurus by theropod1
A: Tyrannosaurus rex FMNH PR 2081, D4 in anterior and lateral view. Image from Brochu 2003. B: cf. Allosaurus fragilis AMNH 5767, D1 in anterior and lateral view

They seem to have very similarly-sized centra, except for T. rex being amphiplatyan and hence shorter in maximum length. They are also very similar in width and overall size of the neural arch. T. rex’ vertebra is considerably taller (55.5 vs 44cm), due to its longer neural spine, but this is probably largely because of its more posterior position in the dorsal series.

Returning to my first sentence, that the specimen would be huge, or rather that such a specimen exists actually makes a certain amount of sense. Firstly, the Morrison formation does contain a staggering abundance of sauropods, among them some of the largest ever, for a giant brontophage to feast on.
Secondly, given that we follow Chure (2000) in considering this specimen cf. A. fragilis, it is probably the largest in hundreds of individuals of a typically at least reasonably large (average 8-9m) theropod. So its not so surprising that there could be a giant, outsized individual, even if the norm of the species is not a serious contender for the largest known theropod.
Furthermore it is from the Brushy Basin Member, just like the three next-largest Allosaurid individuals (Allosaurus "Saurophaganax" maximus, Chure 1995, and NMMNHP-26083, Williamson & Chure 1996), so in perfect keeping with Cope’s rule, Allosaurus, or Allosauridae if we want to be splitters, may have peaked in size in the later parts of their temporal distribution. Already being an extremely successful and ubiquitous theropod, such a temporal trend towards gigantism is certainly not hard to imagine.

So if this turns out to be legit, would that make Allosaurus the biggest theropod? No.
Perhaps not even the biggest jurassic theropod, on average.
But it would stand a damn good chance at actually having the largest referred specimen.

So here we go. When I started this I wanted it to be a "short communication" as food for thought, with a single picture and three references. Guess I couldn’t resist…

Brochu, C. A. 2003: Osteology of Tyrannosaurus rex: insights from a nearly complete skeleton and high-resolution computed tomographic analysis of the skull. Journal of Vertebrate Paleontology 22:1–138.
Canale, J. I., F. E. Novas, and D. Pol. 2015: Osteology and phylogenetic relationships of Tyrannotitan chubutensis Novas, de Valais, Vickers-Rich and Rich, 2005 (Theropoda: Carcharodontosauridae) from the Lower Cretaceous of Patagonia, Argentina. Historical Biology 27:1–32.
Chure, D. J. 1995: A reassessment of the gigantic theropod Saurophagus maximus from the Morrison Formation (Upper Jurassic) of Oklahoma, USA. 6th Symposium on Mesozoic terrestrial ecosystems and biotas, short papers. Edited by A.-L. Sun and Y.-Q. Wang. China Ocean Press, Beijing, China:103–106.
Chure, D. J. 2000: A new species of Allosaurus from the Morrison Formation of Dinosaur National Monument (UT-CO) and a revision of the theropod family Allosauridae. Ph. D. dissertation, Columbia University.
Gilmore, C. W. 1920: Osteology of the carnivorous Dinosauria in the United States National museum: with special reference to the genera Antrodemus (Allosaurus) and Ceratosaurus. US Government printing office.
Madsen, J. H. 1976: Allosaurus fragilis: a revised osteology. Utah Geological and Mining Survey Bulletin 109:1–163.
Osborn, H. F., and C. C. Mook. 1921: Camarasaurus, Amphicoelias, and other sauropods of Cope. Memoirs of the American Museum of Natural History 3:247–387.
Williamson, T. E., and D. J. Chure. 1996: A large allosaurid from the Upper Jurassic Morrison Formation (Brushy Basin Member), west-central New Mexico. Museum of Northern Arizona, Bulletin 60:73–79.
:iconscotthartman: DINO 2560 and MOR 693
:iconrandomdinos: USNM 4734

A quick word or two regarding the size of everyone’s favourite extinct shark, Carcharocles (-Megaselachus/Otodus/Carcharodon–why-don’t-other-animals-get-this-amount-of-taxonomic-attention-) megalodon.

As we have learned to expect from something really large and with really scrappy remains (isolated teeth, with vertebrae or more complete dentitions being known in a small number of poorly- or completely undocumented cases), people do enjoy speculating about its size.
Only in this case, people advising the use of conservative methods are often just drowned out (and trolled and bashed into submission) by the internet movement that understandably enjoys the thought of a species of 60ft shark that killed everything in its path with a single bite, aided by the fact that too many of those people mistake scientific’ enthusiasm (or pretty much any statement they can find anywhere, never mind if it’s just a freaking discovery channel documentary they misconstrue as the new principal scientific reference work out there) for some sort of endorsement of their sensationalism.

That isn’t to say that speculating is wrong–we literally would not get anywhere without it in vertebrate palaeontology. And certainly even liberal methodologies have their place, and for very good reasons that I won’t delve into right now.
But that doesn’t mean they are the only ones deserving of that, and it doesn’t mean that place is necessarily to be the most commonly cited and reproduced estimation, neither does it mean the only way to get an estimate that doesn’t get labeled as "too low" should be by making excessive use of wild assumptions to drive up that size figure (which really is precisely how the 20m+, 100t+ figure that’s so common in this case came into being, no offence).
There is a fundamental difference between nitpicking such a figure to represent this animal’s body size in general, and publishing it as a hypothetical (and, as noted by the authors in question, unreliable) estimate (but see below).

That gets even more comical when it gets to discussing how C. megalodon compares to other giant predators. I can’t help but be amused by their confidence in such biased comparisons when representing C. megalodon as an 18-20m critter even though those are the sizes of a handful of individuals in thousands. To jump ahead a bit, there appears to be quite some sensationalism in claims that it is the biggest predator of all time.
Leaving aside the implication of certainty that is expressed in this statement despite the incompleteness of the fossil record (and its fossil record in particular, with most animals people don’t make confident size estimates from teeth), there is a plain and simple lack of data supporting such claims. Certainly it is among the top 2 or 3 contenders, being comparable in size to its contemporary Livyatan melvillei, to the point where it is warranted to not make any statement about which is larger considering the lack of sufficiently complete fossils, and the lack of a sufficiently large sample of the latter’s population.

I don’t have a problem with the thought that 20m megatooth sharks existed, just like how 1t polar bears exist(ed), but this estimate is useless for all (scientific) intends and purposes because it does not base on solid data, and because it doesn’t relate to a normal individual at all (you know, the kind of megalodon that made all those bite marks, the kind you’d expect and fear to swim into if you were a miocene mysticete, in short, the kind that made up the bulk of the adult population of this species).
Those figures represent hypothetical freak specimens, based on more or less reliable hints, and back when this estimate was published it was conceived as such–deliberately basing on the largest (unreliable, and noted as such by Gottfried and colleagues) and most unusually proportioned report of a great white that was available.

Carcharocles Megalodon-backgroundless by theropod1

Life reconstruction, based on the anatomy of the extant great white shark (adapted from Compagno 1984) and accounting for allometric increase in robusticity as per the data outlined below.

So for a less biased picture, conservative methods are in order, especially when the whole matter already has to be based on a handful of teeth.
Keep in mind the actual meaning of "conservative"; not "the lowest", although the word is often used in that sense, even by myself. Yet in principle I am not an advocate of automatically considering the lowest estimate the most conservative.
In fact, it is not unusual for the lowest estimates to be "liberal" of sorts, when special pleading is used in order to produce a minimum estimate, often under the mistaken impression that it would make it more parsimonious (yet being too small isn’t really any better than being too large).
So here, what I mean is the most parsimonious method, the one that keeps biased (e.g. aimed at producing an extreme estimate in either direction) or otherwise poorly supported assumptions to a minimum, the one avoiding to sum up such biases and instead leaving them to cancel each other out if they can not be avoided. In short, the best and most objective estimate. Unfortunately that use of the term will probably not gain popular acceptance, so lets get back on topic.

So is there a way of making this more objective and exercising some caution in this sense? Yes, I think there is.

Firstly, the most important size metric of a species isn’t the size of the biggest thing you can find some tiny fragment of (btw teeth in general are actually tiny fragments, and scientists don’t bother estimating body size from them in many cases). It’s the species average size that’s most important, most objective, and least error-prone, so that’s what I’m most interested in here.
The background is that Pimiento & Balk 2015 actually estimated it fairly recently, and came up with an average of ~10m for 544 individuals of all ages.
But this still isn’t a very good means of comparison; depending on an animal’s reproductive strategy, the number of immature specimens in such a sample can vary, and accordingly species whose social and ecological adult stage is larger can end up smaller (or vice versa). To account for that, one can take the mean size of a subsample, namely all those that are above the mean size at maturity.
Gottfried et al. 1996 estimated mean sizes at maturity for both females and males based on the Great White shark, and the average of both is 11.9m, which is broadly consistent, if not a bit higher, than what would be indicated by the relative size at maturity of C. carcharias (cf. Cailliet et al. 1985, Casey & Pratt 1985, McClain et al. 2015). This figure should hence be appropriate to represent the average size when attaining maturity given that there is an approximately equal number of males and females in the population. In Pimiento & Balk’s sample, the average of individuals estimated to be this long or longer is ~14m.

Now, assume we wanted to know the size of a large megalodon. Not crazy-freaky-outlier-large, but still really large. One such specimen is comes from Denmark, and consists of a huge tooth associated with a number of vertebrae, which means that this specimen is, unlike most megatooth sharks, not a tooth lost by some individual at a random moment of its life, but an actual fossil "skeleton" left the traditional fossil way (i.e. dieing) if that word is appropriate.
Unfortunately there’s no telling whether any of the vertebrae are the largest in the collumn, and there is also vague indication of C. megalodon’s vertebrae being proportioned differently from those of C. carcharodon. So we’re still stuck with a tooth, but at least we have a more substantial specimen that this tooth once belonged to.
The piece of eviscerating awesome in question is 12.6cm wide and 15.8cm tall, with a 11.9cm tall crown. Based on its massive built, slight tilt and size, it is likely to be one of the first three laterals, which include what is often the widest tooth in the dentition.

There are multiple ways of estimating the size from these measurements. One that is currently very popular is based on individual regressions estimating total length from tooth-crown height in Great White Sharks (Shimada 2001, cited in Pimiento et al. 2010).
Using these regressions for the first three lateral teeth, the mean estimate is 16.8m.
Another method, and one which I personally prefer because it solves a number of the problems associated with using a significantly smaller relative as the sole analogue, is first extrapolating the length of the entire tooth row from the width of the tooth, and then using a regression designed for estimating total length from the length of the tooth row (Lowry et al. 2009, see also Newbrey et al. 2013/in press, and this list of measurements of two cast megalodon dentitions→. That way differences in proportions within the dentition are reduced to the impact of individual, not interspecific, variation, and the resulting estimate assumes them to follow the same trend in terms of relative jaw size.
Even though the literature seemingly suggests significantly less, I went with an interdental spacing of ~15% in addition to the summed tooth widths, because it corresponds to what one can roughly measure in the most widely spaced pictures of great white shark jaws and it is only fair towards the tested hypothesis to use the most optimistic reasonable regression).
Coincidentally this gives us the same mean size estimate for the three positions, 16.8m. Quite funny actually.

It is an intriguing side note (even though it’s another mere coincidence, it pretty much rules out the whole "these figures are different from the official ones" line of criticism I sometimes get, because these are actually the exact figures that were published) that this size estimate also coincides with the highest one estimated by Gottfried et al. 1996 that was deemed reliable, as well as with the biggest specimen from the nursery described by Pimiento et al. (2010). This was hence the highest proper size estimate in the literature for almost two decades. Even now, only less than 3% of all megalodon specimens in Pimiento & Balk’s dataset record the species reaching or exceeding this size, and at most by about 7%, so I find it quite save to assume that this can be considered representative of a very large megalodon.

Length estimates are of course only half of the equation. The most determining factor (though unfortunately highly prone to fluctuations) of the animal’s biology is its body mass. And sure enough, estimating body mass from total length in extant sharks has received almost unparalleled attention as far as size estimates go.
The following is a graph showing 6  [!] different regression equations between total length and body mass for the extant white shark, C. carcharias. In order to avoid biases I’ve applied all of them and marked the mean, maximum and minimum estimates:
Megmass - by theropod1
As apparent from the exponents, most of these studies found positively allometric growth in terms of body mass, i.e. larger sharks get bulkier. From this we can actually estimate the change in body shape, since the component of the weight difference not explained by isometry is necessarily due to a thicker body, i.e. a bigger crosssection. A 5m Great White shark is predicted to mass 1196kg, scaled up to 14m by isometry alone that’s 26255kg, but a 14m shark is actually predicted to mass 29255 (strange coincidence, but not evidence for the existence of the flying spaghetti monster). That’s a difference of 11.4%, so we know a 14m shark has a body crosssection 11.4% larger, since length is out of the picture having already been accounted for in both methods. All we need to do now is take the square root of that, giving us 5.6%, and we’ve got the percentage that it is deeper and wider respectively (assuming it didn’t get disproportionately bigger in one of these dimensions compared to the other, of course).

That is also what my reconstruction bases on, unimaginative as you may call it (since it’s the boring bulked-up great white you almost always see when it’s comes to restoring meg’s body shape), at least it bases on scientific principles. I’m certainly open to other possibilities with regard to its morphology, in the end we all have to accept it is almost completely speculative, it’s just that I have not yet seen any convincing arguments to suggest they are more likely than this (i.e. disproportionately elongated or compact sharks).

So as you can see, the average adult megalodon is predicted to be ~29t in mass, no more than 32t and no less than 26t, and the large specimen from Denmark is expected to be ~52t (within a range from 46 to 57t). The average of the entire population would be ~10t, similar to a very large bull orca (which is impressive enough considering that that average figure corresponds to what is usually a large sub-adult).
And yes, there are a few megalodons that realistically reached or exceeded the 18m-mark (at which length they’d be expected to weigh in at ~64t), and some exceptional fossils do seem to indicate lengths of ~20m (and probably ~89t). But their overall relevance is in no proportion to their size. Consider this; of the 544 teeth in Pimiento & Balk’s sample, none produced an estimate higher than 17.9m, so individuals significantly above that size are obviously too rare for much of a demographic impact. Unless, that is, there ends up being convincing proof that this is a result of some erraneous method, such as a consistent and pronounced sampling bias towards small specimens (very unlikely) or unreliable size estimates (more likely since I am by no means claiming the method always gives us realistic estimates, but currently it seems like if anything the systemic bias here is towards enlarging size estimates).

Most fossil organisms are represented by small samples, and it is likely that they are centered around the average and that "maximum size" remains unknown in virtually all of them. That’s because the likelihood of finding exceptionally large (or small) individuals is typically lower than finding normal-sized ones, and the small number of individuals known for most of them is not in favour of finding such exceptions.
So please guys, when you talk about body sizes, be a bit more precise, differentiate, and don’t mix average, large, small and maximum-sized specimens up all the time, because it does lead to totally unwarranted conclusions.
C. megalodon
is not an 18m shark, nor for that matter is there any species of 18m sharks. Only a species (possibly two if one includes Rhincodon) that occasionally gets that big. There is no species of 1t bear either, and sperm whales aren’t 24m long and don’t weigh 130t, occasional outsized specimens notwithstanding.
This species of shark was big, huge even. Any animal averaging 14m long and close to 30t is positively enormous. It’s nothing except sad that for some people, that just isn’t enough and they have to judge everything by the most extreme examples they can dig up anywhere.

PS: I do realize this looks like a rant, and honestly, it is. I’m sure we all have issues that annoy us, some more than others. That doesn’t mean I am not appropriately impressed by C. megalodon or anything like that, just that I think some people are overdoing it in that regard.
And I am absolutely open to change my mind should future (or presently undocumented *wink*wink*) discoveries or revolutionary new methods change this.

    Bendix-Almgreen, Svend E. (1983): Carcharodon megalodon from the Upper Miocene of Denmark, with comments on elasmobranch tooth enameloid: coronoïn. Bulletin of the geological Society of Denmark, 32 pp. 1-32.
   Cailliet, Gregor M.; Natanson, Lisa J.; Welden, Bruce A.; Ebert, David A. (1985) Preliminary studies on the Age and Growth of the White Shark, Carcharodon carcharias, Using Vertebral Bands. Memoirs of the Southern California Academy of Sciences, 9 (Biology of the White Shark, a Symposium.) pp. 49-60.
   Casey, John G.; Pratt, Harold L. (1985) Distribution of the White Shark, Carcharodon carcharias, in the Western North Atlantic. Memoirs of the Southern California Academy of Sciences, 9 (Biology of the White Shark, a Symposium.), pp. 2-14.
   Compagno, Leonard J.V. (1984): Sharks of the World. An Annotated and Illustrated Catalogue of Shark Species Known to Date. Part 1 – Hexanchiformes to Lamniformes. FAO Fisheries Synopsis, 125 (4) pp. 1-249.
    Kallal, Robert J.; Godfrey, Stephen J.; Ortner, D. J. (2010): Bone Reactions on a Pliocene Cetacean Rib Indicate Short-Term Survival of Predation Event. International Journal of Osteoarchaeology, 22 (3), pp. 253-260.
    Kohler, Nancy E.; Casey, John G.; Turner, Patricia A. (1995): Length-Length and Length-Weight Relationships for 13 Shark Species from the Western North Atlantic. Fishery Bulletin, 93 pp. 412-418.
Lowry, Dayv; Castro, Andrey L. F. de; Mara, Kyle; Whitenack, Lisa B.; Delius, Bryan; Burgess, George H.; Motta, Philip: (2009): Determining shark size from forensic analysis of bite damage. Marine Biology, 156 pp. 2483-2492.
    McClain, Craig R.; Balk, Meghan A.; Benfield, Mark C.; Branch, Trevor A.; Chen, Catherine; Cosgrove, James; Dove, Alistair D.M.; Gaskins, Lindsay C.; Helm, Rebecca R.; Hochberg, Frederick G.; Lee, Frank B.; Marshall, Andrea; McMurray, Steven E.; Schanche, Caroline; Stone, Shane N.; Thaler, Andrew D. (2015): Sizing ocean giants: patterns of intraspecific size variation in marine megafauna. PeerJ, 3 (715) pp. 1-69.
    Mollet, Henry F.; Cailliet, Gregor M. (1996): Using Allometry to Predict Body Mass from Linear Measurements of the White Shark. In: Klimley, Peter A.; Ainley, David G.: Great White Sharks: the biology of Carcharodon carcharias. San Diego, pp. 81-89.
    Newbrey, Michael G.; Siverson, Mikael; Cook, Todd D.; Fotheringham, Allison M.; Sanchez, Rebecca L. (2013, in press): Vertebral morphology, dentition, age, growth, and ecology of the large lamniform shark Cardabiodon ricki. Acta Palaeontologica Polonica, in press, pp. 1-65.
    Pimiento, Catalina; Balk, Meghan A. (2015): Body-size trends of the extinct giant shark Carcharocles megalodon: a deep-time perspective on marine apex predators. Paleobiology, 41 (3), pp. 479-490.
    Pimiento, Catalina; Ehret, Dana J.; MacFadden, Bruce J.; Hubbell, Gordon (2010): Ancient Nursery Area for the Extinct Giant Shark Megalodon from the Miocene of Panama. PLoS ONE, 5 (5), pp. 1-9.
    Tricas, Timothy C.; McCosker, John E. (1984): Predatory Behaviour of the White Shark (Carcharodon carcharias) with notes on its biology. Proceedings of the California Academy of Sciences, 43 (14), pp. 221-234.
A new feathered ornithischian (contrary to what you may have read, this is the third, not the first!) has just been described:
Kulindadromeus zabaikalicus, a basal neoornithischian.
Kulindadromeus zabaikalicus by theropod1

There are probably lots of exciting new inferences to take from it, but I’ll stop here until I get to see the paper (even though it appears the supplement probably contains more information). But what it definitely does is further confirm what good old Psittacosaurus and Tianyulong already showed.

So for the "feathers are restricted to coelurosaurs"-people (not to mention BANDITS): One more taxon you have to argue evolved those things independently from coelurosaurs and birds.
For the rest: Yay!

Amphicoelias fragillimus, the lost second species of Amphicoelias is one of the biggest mysteries of modern palaeontology, because the only fossils (an incomplete neural arch and perhaps a femoral fragment) have been lost more than a century ago, and all that is left is a description and a single drawing.

Nevertheless, its existance and the reported dimensions appears feasible, to say the least. To quote Carpenter (2006):
 There is, however, every reason to accept Cope at his word. First, Cope never made any subsequent corrections in his publications; furthermore, his reputation was at stake. Marsh, who was ever so ready to humiliate Cope, never called into question the measurements. Marsh is known to have employed spies to keep tabs on what Cope was collecting, and it is quite possible that he had independent confirmation for the immense size of A. fragillimus. Osborn and Mook (1921) accept Cope’s measurements without question, as does McIntosh (1998). Thus, there is historical precedence for accepting the measurements as correct
The only thing we have to work with, and all that we really know of this creature, is an eroded part of a vertebra, lacking the centrum, transverse processes, neural canal and the apex of the neural spine (i.e. not exactly a well-preserved element):
Amphicoelias fragillimus spnov cope1878 by theropod1

So, given we accept cope at his word on the height of this element (150cm), how large was the complete vertebra, and how large, by inference, the animal it belonged to?

Since the fossil was interpreted as a 9th or 10th dorsal vertebra by Osborn & Mook (1921), we should be able to scale its size based on the preserved vertebra of A. altus, which is likely our best option given that they belonged to the same genus or where at least closely related.
Amphicoelias Altus Genetspnov Cope1877 by theropod1
AMNH 5764–9th or 10th dorsal vertebra of Amphicoelias altus following Cope 1877

But that’s easier said than done. First of all, which is the correct portrayal for the A. altus vertebra? Is it really the one above?
Amphicoelias Fragillimus by theropod1
Even at Carpenter’s 2.7m estimate, which was the highest so far, there would be no room for a neural canal below the preserved part. The lower figures of 2.3-2.4m (that seem like they are just scaled to neural spine height) make the neural arch extend well below the top of the centrum.
This would result in a vertebra more than 2.8m tall as an absolute minimum, which seems quite hilarious even for Amphicoelias.

However, there is a huge disparity between this drawing and the one Carpenter used, even tough it is the same specimen and also printed in Osborn & Mook 1921 along with the other picture.
The neural arch is sufficiently larger to make the pieces fit together at a total height of 2.7m, as suggested by Carpenter in this figure, but surely not much less than that:…

So, which drawing should we use? Because that’s what it breaks down to. I’d suggest the one by Osborn & Mook, since it is way more realistic as compared to the ~3m tall vertebra you’d get using Cope’s.

In either case, it is not possible to get this vertebra down to 2.3m, at least based on A. altus, without completely inconceivable morphology (impossibly tiny centrum in order to make room for the missing pedicles and neural canal).
I think there are many people who are way too easily lower size figures that base on impossible vertebral reconstructions.

• Carpenter 2006:  Biggest of the big: A critical re-evaluation of the mega-sauropod Amphicoelias fragillimus Cope, 1878. 
• Cope 1878:  a new species of Amphicoelias.
• Osborn & Mook 1921: Camarasaurus, Amphicoelias and other sauropods of Cope.
• Cope 1877: On the Vertebrata of the Dakota Epoch of Colorado