We’ve touched on this several times in various posts and comment threads, but it’s worth taking a moment to think in detail about the various published mass estimates for the single specimen BM.R.2181 (formerly known as HMN SII), the paralectotype of Giraffatitan brancai, which is the basis of the awesome mounted skeleton in Berlin.

Here is the table of published estimates from my 2010 sauropod-history paper, augmented with the two more recent estimates extrapolated from limb-bone measurements:

Author and date Method Volume (l) Density (kg/l) Mass (kg)
Janensch (1938) Not specified `40 t’
Colbert (1962) Displacement of sand 86,953 0.9 78,258
Russell et al. (1980) Limb-bone allometry 13,618
Anderson et al. (1985) Limb-bone allometry 29,000
Paul (1988) Displacement of water 36,585 0.861 31,500
Alexander (1989) Weighing in air and water 46,600 1.0 46,600
Gunga et al. (1995) Computer model 74,420 1.0 74,420
Christiansen (1997) Weighing in air and water 41,556 0.9 37,400
Henderson (2004) Computer model 32,398 0.796 25,789
Henderson (2006) Computer model 25,922
Gunga et al. (2008) Computer model 47,600 0.8 38,000
Taylor (2009) Graphic double integration 29,171 0.8 23,337
Campione and Evans (2012) Limb-bone allometry 35,780
Benson et al. (2014) Limb-bone allometry 34,000

(The estimate of Russell et al. (1980) is sometimes reported as 14900 kg. However, they report their estimate only as “14.9 t”; and since they also cite “the generally accepted figure of 85 tons”, which can only be a reference to Colbert (1962)”, we must assume that Russell et al. were using US tons throughout.)

The first thing to notice is that there is no very clear trend through time, either upwards or downwards. Here’s a plot of mass (y-axis) against year of estimate (x-axis):


I’ve not even tried to put a regression line through this: the outliers are so extreme they’d render it pretty much useless.

In fact, the lowest and highest estimates differ by a factor of 5.75, which is plainly absurd.

But we can go some way to fixing this by discarding the outliers. We can dump Colbert (1962) and Alexander (1989) as they used overweight toys as their references. We more or less have to dump Russell et al. (1980) simply because it’s impossible to take seriously. (Yes, this is the argument from personal incredulity, and I don’t feel good about it; but as Pual (1988) put it, “so little flesh simply cannot be stretched over the animal’s great frame”.) And we can ignore Gunga et al. (1995) because it used circular conic sections — a bug fixed by Gunga et al. (2008) by using elliptical sections.

With these four unpalatable outliers discarded, our highest and lowest estimates are those of Gunga et al. (2008) at 38,000 kg and Taylor (2009)at 23,337. The former should be taken seriously as it was done using photogrammetrical measurements of the actual skeletal mount. And so should the latter because Hurlburt (1999) showed that GDI is generally the least inaccurate of our mass-estimation techniques. That still gives us a factor of 1.63. That’s the difference between a lightweight 66 kg man and and overweight 108 kg.

Here’s another way of thinking about that 1.63 factor. Assuming two people are the same height, one of them weighing 1.62 times as much as the other means he has to be 1.28 times as wide and deep as the first (1.28^2 = 1.63). Here is a man next to his 1.28-times-as-wide equivalent:



I would call that a very noticeable difference. You wouldn’t expect someone estimating the mass of one of these men to come up with that of the other.

So what’s going on here? I truly don’t know. We are, let’s not forget, dealing with a complete skeletal mount here, one of the very best sauropod specimens in the world, which has been extensively studied for a century. Yet even within the last six years, we’re getting masses that vary by as much as the two dudes above.


As promised, some thoughts on the various new brachiosaur mass estimates in recent papers and blog-posts.

Back in 2008, when I did the GDI of Giraffatitan and Brachiosaurus for my 2009 paper on those genera, I came out with estimates of 28688 and 23337 kg respectively. At the time I said to Matt that I was suspicious of those numbers because they seemed too low. He rightly told me to shut up and put my actual results in the paper.

More recently, Benson et al. (2014) used limb-bone measurements to estimate the masses of the same individuals as 56000 and 34000 kg. When Ian Corfe mentioned this in a comment, my immediate reaction was to be sceptical: “I’m amazed that the two more recent papers have got such high estimates for brachiosaurs, which have the most gracile humeri of all sauropods“.

So evidently I have a pretty strong intuition that Brachiosaurus massed somewhere in the region of 35000 kg and Giraffatitan around 30000 kg. But why? Where does that intuition come from?

I can only assume that my strongly held ideas are based only on what I’d heard before. Back when I did my 2008 estimate, I probably had in mind things like Paul’s (1998) estimate of 35000 kg for Brachiosaurus, and Christiansen’s (1997:67) estimate of 37400 for Giraffatitan. Whereas by the time the Benson et al. paper came out I’d managed to persuade myself that my own much lower estimates were right. In other words, I think my sauropod-mass intuition is based mostly on sheer mental inertia, and so should be ignored.

I’m guessing I should ignore your intuitions about sauropod masses, too.


Way back in November 2011, I got this inquiry from Keiron Pim:
I’m currently writing a popular guide to dinosaurs, to be published by Random House next autumn [Ed.: available now at amazon.com and at amazon.co.uk]. I’ve been writing about [Brachiosaurus and Giraffatitan], and have read your 2009 study vindicating the proposal to separate them into two genera.
I know you consider Brachiosaurus likely to have been bigger (and note that the specimen was not fully grown), with a longer trunk and tail – but most of the sources I can find give both animals the same body length, generally around 26m. Presumably this doesn’t reflect your work, and your calculations are different.

I replied at the time, and said that I’d post that response here on SV-POW!. But one thing and another prevented me from getting around to it, and I forgot all about it until recently. Since we’re currently in a sequence of Brachiosaurus-themed posts [part 1, part 2, part 3, part 4, part 5, part 6], this seems like a good time to fix that. So here is my response, fresh from November 2011, lightly edited.


Well, Giraffatitan has only been recognised as a separate animal at all in the last couple of years, and nearly everything that has been written about “Brachiosaurus“, at least in the technical literature, is actually about Giraffatitan. So existing sources that give the same length for both are probably not making a meaningful distinction between the two animals.

First, on Giraffatitan: Janensch (1950b:102) did a great job of measuring his composite mounted skeleton. His figure for the total length of Giraffatitan along the neural canal is 22.46 m, and is certainly the best estimate in the literature for an actual brachiosaur specimen (and quite possibly the best for any sauropod).

I don’t know where the figure of 26 m comes from, but as Janensch (1961:213) notes, the isolated fibula XV2 of Giraffatitan in the Berlin collection is 134 cm long, compared with 119 cm for that of the mounted skeleton. This is 1.126 times as long, which if scaled isometrically would yield a total length of 25.29 m.  So that is defensible, but 26 m is not, really.

I would advise sticking with Janensch’s published figure of 22.46 m, as it’s based on good material, and also because it forms the basis of my comparative estimate for a Brachiosaurus of similar limb length.

Now in my 2009 paper I estimated with reasonable rigour that the torso of Brachiosaurus was probably about 23% longer than that of Giraffatitan, yielding 4.82 m rather than 3.92 that Janensch gave for Giraffatitan. On much less solid evidence, I tentatively estimated that the tail of Brachiosaurus might have been 20-25% longer than that of Giraffatitan. Given the paucity of evidence I would play safer by going with the lower end of that estimate, which would give a tail length of 9.14 m compared with Janensch’s 7.62 for Giraffatitan. Riggs (1904) tells us that the sacrum of Brachiosaurus is 0.95 m long, which is slightly less than 1.07 m for Giraffatitan. Finally, since we know nothing of the head and neck of Brachiosaurus, the null hypothesis has to be that they were similar in proportion to those of Giraffatitan.

Putting it all together, Brachiosaurus may have been longer in the torso by 0.9 m, and in the tail by 1.52 m, but shorter in the sacrum by 0.12 m — for a total additional length of 2.3 m. That would make Brachiosaurus 24.76 m long, which is 10% longer than Giraffatitan.

Note that all the Brachiosaurus figures are given with much greater precision than the sparse data we have really allows.  I think you could round Janensch’s 22.46 m for Giraffatitan to 22.5 and be pretty confident in that number, but you shouldn’t really say anything more precise than “maybe about 25 m” for Brachiosaurus.

Finally, you correctly note that the Brachiosaurus specimen was not fully grown — we can tell because its coracoid was not fused to the scapula. But the same is true of the mounted Giraffatitan, so these two very similarly sized animals were both subadult. How much bigger did they get?  We know from the fibula that Giraffatitan got at least 12-13% bigger than the well-known specimen, and I’d be pretty happy guessing the same about Brachiosaurus.  And I wouldn’t rule out much bigger specimens, either.


Continuing with what seems to have turned out to be Brachiosaur Humerus Week here on SV-POW! (part 1, part 2, part 3), let’s consider the oft-stated idea that brachiosaurs have the most slender humeri of any sauropod. For example, Taylor (2009:796) wrote that:

Discarding a single outlier, the ratio of proximodistal length to minimum transverse width (Gracility Index or GI) in humeri of B. brancai [i.e. Giraffatitan] varies between 7.86 for the right humerus HMN F2 and 9.19 for the left humerus HMN J12, with the type specimen’s right humerus scoring 8.69, slightly more gracile than the middle of the range [...] For the B. altithorax type specimen, the GI is 8.50, based on the length of 204 cm and the minimum transverse width of 24 cm reported by Riggs (1904:241). However, the B. altithorax humerus looks rather less gracile to the naked eye than that of B. brancai, and careful measurement from Riggs’s plate LXXIV yields a GI of 7.12, indicating that the true value of the minimum transverse width is closer to 28.5 cm. As noted by Riggs (1903:300-301), the surface of the distal end of this humerus has flaked away in the process of weathering. Careful comparison of the humeral proportions with those of other sauropods (Taylor and Wedel, in prep.) indicates that the missing portion of this bone would have extended approximately a further 12 cm, extending the total length to 216 cm and so increasing the GI to 7.53 – still less gracile than any B. brancai humerus except the outlier, but more gracile than any other sauropod species except Lusotitan atalaiensis (8.91), and much more gracile than the humerus of any non-brachiosaurid sauropod (e.g., Diplodocus Marsh, 1878 sp., 6.76; Malawisaurus dixeyi Jacobs, Winkler, Downs and Gomani, 1993, 6.20; Mamenchisaurus constructus Young, 1958, 5.54; Camarasaurus supremus Cope, 1877, 5.12; Opisthocoelicaudia skarzynskii Borsuk-Bialynicka, 1977, 5.00 – see Taylor and Wedel, in prep.)

Implicit in this (though not spelled out, I admit) is that the humeri of brachiosaurs are slender proportional to their femora. So let’s take a look at the humerus and femur of Giraffatitan, as illustrated in Janensch’s beautiful 1961 monograph of the limbs and girdles of Tendaguru sauropods:


The first thing you’ll notice is that the humerus is way longer than the femur. That’s because Janensch’s Beilage A illustrates the right humerus of SII (now properly known as MB R.2181) while his Beilage J illustrates the right femur of the rather smaller referred individual St 291. He did this because the right femur of SII was never recovered and the left femur was broken, missing a section in the middle that had to be reconstructed in plaster.

(What’s a Beilage? It’s a German word that seems to literally mean something like “supplement”, but in Janensch’s paper it means a plate (full-page illustration) that occurs in the main body of the text, as opposed to the more traditional plates that come at the end, and which are numbered from XV to XXIII.)

How long would the intact SII femur have been? Janensch (1950b:99) wrote “Since the shaft of the right femur is missing for the most part, it was restored to a length of 196 cm, calculated from other finds” (translation by Gerhard Maier). Janensch confused the left and right femora here, but assuming his length estimate is good, we can upscale his illustration of St 291 so that it’s to SII scale, and matches the humerus. Here’s how that looks:


Much more reasonable! The humerus is still a little longer, as we’d expect, but not disturbingly so.

Measuring from this image, the midshaft widths of the femur and humerus are 315 and 207 pixels respectively, corresponding to absolute transverse widths of 353 and 232 mm — so the femur is broader by a factor of 1.52. That’s why I expressed surprise on learning that Benson et al (2014) gave Giraffatitan a CF:CH ratio (circumference of femur to circumference of humerus) of only 1.12.

Anyone who would like to see every published view of the humeri and femora of these beasts is referred to Taylor (2009:fig. 5). In fact, here it is — go crazy.

Taylor (2009: figure 5). Right limb bones of Brachiosaurus altithorax and Brachiosaurus brancai, equally scaled. A-C, humerus of B. altithorax holotype FMNH P 25107; D-F, femur of same; G-K, humerus of B. brancai lectotype HMN SII; L-P, femur of B. brancai referred specimen HMN St 291, scaled to size of restored femur of HMN SII as estimated by Janensch (1950b:99). A, D, G, L, proximal; B, E, H, M, anterior; C, K, P, posterior; J, O, medial; F, I, N, distal. A, B, D, E modified from Riggs (1904:pl. LXXIV); C modified from Riggs (1904:fig. 1); F modified from Riggs (1903:fig. 7); G-K modified from Janensch (1961:Beilage A); L-P modified from Janensch (1961:Beilage J). Scale bar equals 50 cm.

Taylor (2009: figure 5). Right limb bones of Brachiosaurus altithorax and Brachiosaurus brancai, equally scaled. A-C, humerus of B. altithorax holotype FMNH P 25107; D-F, femur of same; G-K, humerus of B. brancai paralectotype HMN SII; L-P, femur of B. brancai referred specimen HMN St 291, scaled to size of restored femur of HMN SII as estimated by Janensch (1950b:99). A, D, G, L, proximal; B, E, H, M, anterior; C, K, P, posterior; J, O, medial; F, I, N, distal. A, B, D, E modified from Riggs (1904:pl. LXXIV); C modified from Riggs (1904:fig. 1); F modified from Riggs (1903:fig. 7); G-K modified from Janensch (1961:Beilage A); L-P modified from Janensch (1961:Beilage J). Scale bar equals 50 cm.

Notice that the femur of Giraffatitan, while transversely pretty broad, is freakishly narrow anteroposteriorly. The same is true of the femur of Brachiosaurus, although it’s never been shown in a published paper — I observed it in the mounted casts in Chicago.



So let’s take a wild stab at recalculating the mass of Giraffatitan using the Benson et al. formula. First, measuring the midshaft transverse:anteroposterior widths of the long bones gives eccentricity ratios of 2.39 for the femur and 1.54 for the humerus (I am not including the anterior prejection of the deltopectoral crest in the anteroposterior width of the humerus) . Dividing the absolute transverse widths above by these ratios gives us anteroposterior widths of 148 for the femur and 150 mm for the humerus. So they are almost exactly the same in this dimension.

If we simplify by treating these bones as elliptical in cross section, we can  approximate their midshaft circumference. It turns out that the formula for the circumference is incredibly complicated and involves summing an infinite series:


But since we’re hand-waving so much anyway, we can use the approximation C = 2π sqrt((a²+b²)/2). where a and b are the major and minor radii (not diameters). For the femur, these measurements are 176 and 74 mm, so C = 848 mm; and for the humerus, 116 and 75 mm yields 614 mm. (This compares with FC=730 and HC=654 in the data-set of Benson et al., so we have found the femur to be bigger and the humerus smaller than they did.)

So the CF:CH ratio is 1.38 — rather a lot more than the 1.12 reported by Benson et al.  (Of course, if they measured the actual bones rather than messing about with illustrations, then their numbers are better than mine!)

And so to the mass formula, which Campione and Evans (2012) gave as their equation 2:

log BM = 2.754 log (CH+CF) − 1.097

Which I understand to use base-10 logs, circumferences measured in millimeters, and yield a mass in grams, though Campione and Evans are shockingly cavalier about this. CH+CF is 1462; log(1462) = 3.165. That gives us a log BM of 7.619, so BM = 41,616,453 g = 41,616 kg.

Comparison with Benson et al. (2014)

Midshaft measurements and estimates for SII long bones (all measurements in mm)
SV-POW! Benson et al.
Femur Humerus Femur Humerus
Transverse diameter 353 232 240
Transverse radius 176 116 120
Anteroposterior diameter 148 150 146
Anteroposterior radius 74 75 73
Circumference 848 614 730 654
Total circumference 1462 1384
Mass estimate (kg) 41,616 34,000

My new mass estimate of 41,616 kg is is a lot more than the 34,000 kg found by Benson et al. This seems to be mostly attributable to the much broader femur in my measurement: by contrast, the humerus measurements are very similar (varying by about 3% for both diameters). That leaves me wondering whether Benson et al. just looked at a different femur — or perhaps used St 291 without scaling it to SII size. Hopefully one of the authors will pass by and comment.

More to come on this mass estimate real soon!



You’ve probably seen a lot of yapping in the news about a new “world’s largest dinosaur”, with the standard photos of people lying down next to unfeasibly large bones. Here’s my favorite–various versions of it have been making the rounds, but I grabbed this one from Nima’s post on his blog, The Paleo King.


The first point I need to make here is that photos like these are attention-grabbing but they don’t really tell you much. Partly because they’re hard to interpret, and partly because they almost always look more impressive than they really are. For example, I am 6’2″ tall (1.88 meters). If I lie down next to a bone that is 7’2″ (219 cm) 6’8″ (203 cm) long, it is going to look ungodly huge–a full half a foot longer than I am tall. But that is the length of the femur of the Brachiosaurus holotype–we’ve known of sauropod femora that big for a century now. People get tripped up by this sort of thing all the time–even scientists. Update: even me! Somehow I had gotten it into my head that the Brachiosaurus femur was 219 cm, when it is actually 203 cm. That goof doesn’t affect any of what follows, because from here on down I used Argentinosaurus as the point of reference.

Second point: at least some of the reporting on this new find has been unusually–and refreshingly–nuanced. The first news story I read about it was this one, which gives Paul Barrett plenty of airtime to explain why we should be cautious about jumping to any conclusions regarding the size of the new animal. That will turn out to be prophetic.

But let’s get back to that photo. Just eyeballing it, it looks like the femur is about half again as long as the dude is tall (the dude, BTW, is Pablo Puerta, for whom Puertasaurus is named). I was reading Nima’s post and he guessed that the femur was in the neighborhood of 3 meters, which would be a significant size increase over the next-biggest sauropod known from fossils that still exist (i.e., not including semi-apocraphyal gigapods like Amphicoelias fragillimus and Bruhathkayosaurus). The current based-on-existing-fossils record-holder is Argentinosaurus–there is a partial femur that would have been about 2.5 meters long when complete. So a 3-meter femur would be a wonderful thing. But alas, it just ain’t so–or at least the one in the photo isn’t anywhere near that big. Allow me to demonstrate.

femur_pablo with measurements

Here’s another copy of the photo with some measurements applied. There is no actual scale bar in the picture, but we can use the dimensions of the things we can see to figure some stuff out.

For starters, there is a lot of perspective distortion going on here. Pallet B is 350 pixels wide at the near end, 280 pixels at the far end–a difference of 20%. I didn’t put the far-end measurement for Pallet A into the picture, but from corner to corner it is 295 pixels.

Shipping pallets vary in size around the world, but in the US the most common size is 48 x 40 inches. Other countries use different sizes, mostly smaller; I am unaware of any standard shipping pallets larger than 48×40. So assuming that the ones in the picture are that size is actually a liberal assumption that will lead to large estimates–if the pallets are smaller than 48×40, then all of the dimensions I’m about to calculate will be smaller as well. Obviously the pallets have their narrow ends facing us, which is nice because 40 inches is almost exactly 1 meter. So we can divide other things in the picture by pallet length and get their dimensions in meters.

The near side of the femur is pretty much in line with the stringer running left-to-right down the middle of Pallet A. From the measurements of the ends of that pallet, we’d expect the middle-distance width to be about 330 pixels, and in fact I got 335. The 830-pixel line I drew on the near side is not the total length of the bone–you could add a bit more for the femoral head, to a max of maybe 860 or 870 pixels. Divide that by 335 and you get a max length of about 2.6 meters.

The 800-pixel line for the far side of the femur goes from the top of the head to the bottom of the medial condyle, so there’s no extending needed there. That line is at about the mid-point of Pallet B, or about 315 pixels. If Pallet B is a meter wide, the femur is 2.5 meters long.

We can also check things by trying to figure out how tall Pablo Puerta is. At first that looks more encouraging for the possibility that this is a record-breaker. If we assume the femur really is 3 meters long, and compare the 800-pixel femur line to the 500-pixel Pablo line, Pablo is 62.5% the length of the femur, or 1.87 meters–about the same height as me. That would be pretty tall for an Argentinian, but it’s certainly plausible.

But that’s not a legit comparison, because Pablo is farther from the camera than is the femur. Look at Pallet A–we can use the slats as perspective guides to help figure out where the proximal end of the femur ought to be if projected back to Pablo’s distance from the camera. If we do that at both ends, the length of the femur if placed where Pablo is lying would be 750 pixels or fewer, which would make Pablo at least 2 meters tall. People get a lot taller than that, but it would make him unusually tall, and if you’re trying to emphasize how big your sauropod is, you probably won’t pick the tallest person in the room to pull a Jensen. If we assume Pablo’s about 5’8″–average height for an Argentinian male–then the femur is about 2.6 meters long, which is consistent with the estimates from the pallets. He could well be shorter, in which the case the femur might also be shorter.

There are of course vast amounts of uncertainty in all of this. I have heard the number 2.4 meters thrown around in the media, which is within the margin of error of my crude estimates here–I deliberately skewed large at most decision points to give the hypothesized 3-meter femur the best possible chance. I have to emphasize that this is not how you do science–I’m deliberately doing this quick and dirty. But even using these admittedly flawed and somewhat goofy methods, it’s easy to show that the femur isn’t 3 meters long, or anywhere near it.

So, three last points:

  1. As the post title implies, the new Argentine titanosaur is about the same size as Argentinosaurus. That shouldn’t be too surprising, since the mass estimates that have been quoted in the media are within a few percent of the mass estimates for Argentinosaurus. The new critter might be a hair bigger, but it doesn’t “smash” the record, and when we get actual measurements it could end up being smaller than Argentinosaurus in linear dimensions. I note that the size trumpeted in the media is a mass estimate based on femoral fatness, not femoral length. You’d think that if the biggest femur was demonstrably longer than the 2.5-meter Argentinosaurus femur, they’d lead with that. So the reporting so far is also consistent with an animal about the same size as Argentinosaurus.
  2. That is in no way a disappointing result! That biggest Argentinosaurus femur is incomplete, so the 2.5-meter length is an estimate. Even if the big femur shown here is only (only!) 2.4 meters long, it’s still the longest complete limb bone from anything, ever. And even if the new animal is identical to Argentinosaurus in size, there’s still a lot more of it, so we’ll get a better idea of what these super-gigantic titanosaurs looked like. That’s a big win.
  3. Finally, this is not a case of MYDD. There’s no paper yet, and I don’t blame the team for not making the measurements public until the work is done. I also don’t blame them for publicizing the find. So far, this seems to be exactly what they’re saying it is–an animal about the size of Argentinosaurus, and maybe just a hair bigger. That’s cool. I wish them the best of luck writing it up. I almost wrote “I can’t wait to see the paper” but actually I can–something like this, I’d rather they take their time and do it right. It may not be a record-smasher, but it’s a solid, incremental advance, and science needs those, too.


It’s now widely understood among researchers that the impact factor (IF) is a statistically illiterate measure of the quality of a paper. Unfortunately, it’s not yet universally understood among administrators, who in many places continue to judge authors on the impact factors of the journals they publish in. They presumably do this on the assumption that impact factor is a proxy for, or predictor of, citation count, which is turn is assumed to correlate with influence.

As shown by Lozano et al. (2012), the correlation between IF and citations is in fact very weak — r2 is about 0.2 — and has been progressively weakening since the dawn of the Internet era and the consequent decoupling of papers from the physical journal that they appear in. This is a counter-intuitive finding: given that the impact factor is calculated from citation counts you’d expect it to correlate much more strongly. But the enormous skew of citation rates towards a few big winners renders the average used by the IF meaningless.

To bring this home, I plotted my own personal impact-factor/citation-count graph. I used Google Scholar’s citation counts of my articles, which recognises 17 of my papers; then I looked up the impact factors of the venues they appeared in, plotted citation count against impact factor, and calculated a best-fit line through my data-points. Here’s the result (taken from a slide in my Berlin 11 satellite conference talk):


I was delighted to see that the regression slope is actually negative: in my case at least, the higher the impact factor of the venue I publish in, the fewer citations I get.

There are a few things worth unpacking on that graph.

First, note the proud cluster on the left margin: publications in venues with impact factor zero (i.e. no impact factor at all). These include papers in new journals like PeerJ, in perfectly respectable established journals like PaleoBios, edited-volume chapters, papers in conference proceedings, and an arXiv preprint.

My most-cited paper, by some distance, is Head and neck posture in sauropod dinosaurs inferred from extant animals (Taylor et al. 2009, a collaboration between all three SV-POW!sketeers). That appeared in Acta Palaeontologia Polonica, a very well-respected journal in the palaeontology community but which has a modest impact factor of 1.58.

My next most-cited paper, the Brachiosaurus revision (Taylor 2009), is in the Journal of Vertebrate Palaeontology – unquestionably the flagship journal of our discipline, despite its also unspectacular impact factor of 2.21. (For what it’s worth, I seem to recall it was about half that when my paper came out.)

In fact, none of my publications have appeared in venues with an impact factor greater than 2.21, with one trifling exception. That is what Andy Farke, Matt and I ironically refer to as our Nature monograph (Farke et al. 2009). It’s a 250-word letter to the editor on the subject of the Open Dinosaur Project. (It’ a subject that we now find profoundly embarrassing given how dreadfully slowly the project has progressed.)

Google Scholar says that our Nature note has been cited just once. But the truth is even better: that one citation is in fact from an in-prep manuscript that Google has dug up prematurely — one that we ourselves put on Google Docs, as part of the slooow progress of the Open Dinosaur Project. Remove that, and our Nature note has been cited exactly zero times. I am very proud of that record, and will try to preserve it by persuading Andy and Matt to remove the citation from the in-prep paper before we submit. (And please, folks: don’t spoil my record by citing it in your own work!)

What does all this mean? Admittedly, not much. It’s anecdote rather than data, and I’m posting it more because it amuses me than because it’s particularly persuasive. In fact if you remove the anomalous data point that is our Nature monograph, the slope becomes positive — although it’s basically meaningless, given that all my publications cluster in the 0–2.21 range. But then that’s the point: pretty much any data based on impact factors is meaningless.



Let’s take another look at that Giraffatitan cervical. MB.R.2180:C5, from a few days ago:


That’s a pretty elongate vertebra, right? But how elongate, exactly? How can we quantify whether it’s more or less elongate than some other vertebra?

The traditional answer is that we quantify elongation using the elongation index, or EI. This was originally defined by Upchurch (1998:47) as “the length of a vertebral centrum divided by the width across its caudal face”. Measuring from the full-resolution version of the image above, I make that 1779/529 pixels, or 3.36.

But then those doofuses Wedel et al. (2000:346) came along and said:

When discussing vertebral proportions Upchurch (1998) used the term elongation index (EI), defined as the length of the centrum divided by the width of the cotyle. Although they did not suggest a term for the proportion, Wilson & Sereno (1998) used centrum length divided by the height of the cotyle as a character in their analysis. We prefer the latter definition of this proportion, as the height of the cotyle is directly related to the range of motion of the intervertebral joint in the dorsoventral plane. For the purposes of the following discussion, we therefore redefine the EI of Upchurch (1998) as the anteroposterior length of the centrum divided by the midline height of the cotyle.

Since then, the term EI has mostly been used in this redefined sense — but I think we all agree now that it would have been better for Wedel et al to have given a new name to Wilson and Sereno’s ratio rather than apply Upchurch’s name to it.

Aaaanyway, measuring from the image again, I give that vertebra an EI (sensu Wedel et al. 2000) of 1779/334 = 5.33. Which is 58% more elongate than when using the Upchurch definition! This of course follows directly from the cotyle being 58% wider than tall (529/334 pixels).

So one of principal factors determining how elongate a vertebra seems to be is the shape of its cotyle. And that’s troublesome, because the cotyle is particularly subject to crushing — and it’s not unusual for even consecutive vertebrae from the same column to be crushed in opposite directions, giving them (apparently) wildly different EIs.

Here’s an example (though not at all an extreme one): cervicals 4 and 6 of the same specimen, MB.R.2180 (formerly HM SI), as the multi-view photo above:


Measuring from the photos as before, I make the width:height ratio of C4 683/722 pixels = 0.95, and that of C6  1190/820 pixels = 1.45. So these two vertebrae — from the same neck, and with only one other vertebrae coming in between them — differ in preserved cotyle shape by a factor of 1.53.

And by the way, this is one of the best preserved of all sauropod neck series.

Let’s take a look at the canonical well-preserved sauropod neck: the Carnegie Diplodocus, CM 84. Here are the adjacent cervicals 13 and 14, in posterior view, from Hatcher (1901: plate VI):


For C14 (on the left), I get a width:height ratio of 342/245 pixels = 1.40. For C13 (on the right), I get 264/256 pixels = 1.03. So C14 is apparently 35% broader than its immediate predecessor. I absolutely don’t buy that this represents how the vertebrae were in life.

FOR EXTRA CREDIT: what does this tell us about the reliability of computer models that purport to tell us about neck posture and flexibility, based on the preserved shapes of their constituent vertebrae?

So what’s to be done?

The first thing, as always in science, is to be explicit about what statements we’re making. Whenever we report an elongation index, we need to clearly state whether it’s EI sensu Upchurch 1998 or EI sensu Wedel et al. 2000. Since that’s so cumbersome, I’m going propose that we introduce two new abbreviations: EIH (Elongation Index Horizonal), which is Upchurch’s original measure (length over horizontal width of cotyle) and EIV (Elongation Index Vertical), which is Wilson and Sereno’s measure (length over vertical height of cotyle). If we’re careful to report EIH and EIV (or better still both) rather than an unspecified EI, then at least we can avoid comparing apples with oranges.

But I think we can do better, by combining the horizontal and vertical cotyle measurements in some way, and dividing the length by the that composite. This would give us an EIA (Elongation Index Average), which we could reasonably expect to preserve the original cotyle size, and so to give a more reliable indication of “true” elongation.

The question is, how to combine the cotyle width and height? There are two obvious candidates: either take the arithmetic mean (half the sum) or the geometric mean (the square root of the product). Note that for round cotyles, both these methods will give the same result as each other and as EIH and EIV — which is what we want.

Which mean should we use for EIA? to my mind, it depends which is best preserved when a vertebra is crushed. If a 20 cm circular cotyle is crushed vertically to 10cm, does it tend to smoosh outwards to 30 cm (so that 10+30 = the original 20+20) or to 40 cm (so that 10 x 40 = the original 20 x 20)? If the former, then we should use arithmetic mean; if the latter, then geometric mean.

Does anyone know how crushing works in practice? Which of these models most closely approximates reality? Or can we do better than either?

Update (8:48am): thanks for Emanuel Tschopp for pointing out (below) what I should have remembered: that Chure et al.’s (2010) description of Abydosaurus introduces “aEI”, which is the same as one of my proposed definitons of EIA. So we should ignore the last four paragraphs of this post and just use aEI. (Their abbreviation is better, too.)



  • Hatcher, Jonathan B. 1901. Diplodocus (Marsh): its osteology, taxonomy and probable habits, with a restoration of the skeleton. Memoirs of the Carnegie Museum 1:1-63 and plates I-XIII.
  • Upchurch, Paul. 1998. The phylogenetic relationships of sauropod dinosaurs. Zoological Journal of the Linnean Society 124:43-103.
  • Wedel, Mathew J., Richard L. Cifelli and R. Kent Sanders. 2000b. Osteology, paleobiology, and relationships of the sauropod dinosaur Sauroposeidon. Acta Palaeontologica Polonica 45(4):343-388.
  • Wilson, J. A. and Paul C. Sereno. 1998. Early evolution and higher-level phylogeny of sauropod dinosaurs. Society of Vertebrate Paleontology, Memoir 5:1-68.
Currey Alexander 1985 fig 1

Figure 1 from Currey and Alexander (1985)

This post pulls together information on basic parameters of tubular bones from Currey & Alexander (1985), on ASP from Wedel (2005), and on calculating the densities of bones from Wedel (2009: Appendix). It’s all stuff we’ve covered at one point or another, I just wanted to have it all in one convenient place.


  • R = outer radius = r + t
  • r = inner radius = R – t
  • t = bone wall thickness = R – r

Cross-sectional properties of tubular bones are commonly expressed in R/t or K (so that r = KR). K is defined as the inner radius divided by the outer radius (r/R). For bones with elliptical or irregular cross-sections, it’s best to measure two radii at right angles to each other, or use a different measure of cross-sectional geometry (like second moment of area, which I’m not getting into here).

R/t and K can be converted like so:

  • R/t = 1/(1-K)
  • K = 1 – (1/(R/t))

ASP (air space proportion) and MSP (marrow space proportion) measure the cross-sectional area of an element not taken up by bone tissue. ASP and MSP are the same measurement–the amount of non-bone space in a bony element divided by the total–we just use ASP for air-filled bones and MSP for marrow-filled bones. See Tutorial 6 and these posts: one, two, three.

For tubular bones, ASP (or MSP) can be calculated from K:

  • ASP = πr^2/πR^2 = r^2/R^2 = (r/R)^2 = K^2

Obviously R/t and K don’t work for bones like vertebrae that depart significantly from a tubular shape. But if you had a vertebra or other irregular bone with a given ASP and you wanted to see what the equivalent tubular bone would look like, you could take the square root of ASP to get K and then use that to draw out the cross-section of that hypothetical tubular bone.

To estimate the density of an element (at least near the point of a given cross-section), multiply the proportional areas of bone and air, or bone and marrow, by the specific gravities of those materials. According to Currey and Alexader (1985: 455), the specific gravities of fatty marrow and bone tissue are 0.93 and 2.1, respectively.

For a marrow-filled bone, the density of the element (or at least of the part of the shaft the section goes through) is:

  • 0.93MSP + 2.1(1-MSP)

Air is matter and therefore has mass and density, but it is so light (0.0012-0.0013 g/mL) that we can effectively ignore it in these calculations. So the density of a pneumatic element is: 2.1(1-ASP) For the three examples in the figure at the top of the post, the ASP/MSP values and densities are:

  • (b) alligator femur (marrow-filled), K = 0.35, MSP = K^2 = 0.12, density = (0.93 x 0.12) + (2.1 x 0.88) = 1.96 g/mL
  • (c) camel tibia (marrow-filled), K = 0.57, MSP = K^2 = 0.32, density = (0.93 x 0.32) + (2.1 x 0.68) = 1.73 g/mL
  • (d) Pteranodon first phalanx (air-filled), K = 0.91, ASP = K^2 = 0.83, density = (2.1 x 0.17) = 0.36 g/mL

What if we switched things up, and imagined that the alligator and camel bones were pneumatic and the Pteranodon phalanx was marrow-filled? The results would then be:

  • (b) alligator femur (hypothetical air-filled), K = 0.35, ASP = K^2 = 0.12, density = (2.1 x 0.88) = 1.85 g/mL
  • (c) camel tibia (hypothetical air-filled), K = 0.57, ASP = K^2 = 0.32, density = (2.1 x 0.68) = 1.43 g/mL
  • (d) Pteranodon first phalanx (hypothetical marrow-filled), K = 0.91, MSP = K^2 = 0.83, density = (0.93 x 0.83) + (2.1 x 0.17) = 1.13 g/mL

In the alligator femur, the amount of non-bone space is so small that it does much matter whether that space is filled by air or marrow–replacing the marrow with air only lowers the density of the element by 5-6%. The Pteranodon phalanx is a lot less dense than the alligator femur for two reasons. First, there is much less bony tissue–the hypothetical marrow-filled phalanx is 42% less dense as the alligator femur. Second, the marrow is replaced by air, which reduces the density by an additional 40% relative to the alligator.

Next time: how to write punchier endings for tutorial posts.


I recently reread Dubach (1981), “Quantitative analysis of the respiratory system of the house sparrow, budgerigar and violet-eared hummingbird”, and realized that she reported both body masses and volumes in her Table 1. For each of the three species, here are the sample sizes, mean total body masses, and mean total body volumes, along with mean densities I calculated from those values.

  • House sparrow, Passer domesticus, n = 16, mass = 23.56 g, volume = 34.05 mL, density = 0.692 g/mL
  • Budgerigar, Melopsittacus undulatus, n = 19, mass = 38.16 g, volume = 46.08 mL, density = 0.828 g/mL
  • Sparkling violetear,* Colibri coruscans, n = 12, mass = 7.28 g, volume = 9.29 mL, density = 0.784 g/mL

* This is the species examined by Dubach (1981), although not specified in her title; there are four currently-recognized species of violetears. And apparently ‘violetear’ has overtaken ‘violet-eared hummingbird’ as the preferred common name. And as long as we’re technically on a digression,  I’m almost certain those volumes do not include feathers. Every volumetric thing I’ve seen on bird masses assumes plucked birds (read on).

This is pretty darned interesting to me, partly because I’m always interested in how dense animals are, and partly because of how the results compare to other published data on whole-body densities for birds. The other results I am most familiar with are those of Hazlehurst and Rayner (1992) who had this to say:

There are relatively few values for bird density. Welty (1962) cited 0.9 g/mL for a duck, and Alexander (1983) 0.937 g/mL for a domestic goose, but those values may not take account of the air sacs. Paul (1988) noted 0.8 g/mL for unspecified bird(s). To provide more reliable estimates, the density of 25 birds of 12 species was measured by using the volume displacement method. In a dead, plucked bird the air-sac system was reinflated (Saunder and Manton 1979). The average density was 0.73 g/mL, suggesting that the lungs and air sacs occupy some quarter of the body.

That result has cast a long shadow over discussions of sauropod masses, as in this paper and these posts, so it’s nice to see similar results from an independent analysis.  If you’re curious, the weighted mean of the densities calculated from Duchard’s Dubach’s (1981) data is 0.77. I’d love to see the raw data from Hazlehurst and Rayner (1992) to see how much spread they got in their density measurements.  Unfortunately, they did not say which birds they used or give the raw data in the paper (MYDD!), and I have not asked them for it because doing so only just occurred to me as I was writing this post.

There will be more news about hummingbirds here in the hopefully not-too-distant future. Here’s a teaser:


Yes, those are its hyoids wrapped around the back of its head–they go all the way around to just in front of the eyes, as in woodpeckers and other birds that need hyper-long tongue muscles. There are LOADS of other interesting things to talk about here, but it will be faster and more productive if I just go write the paper like I’m supposed to be doing.

Oh, all right, I’ll say a little more. This is a  young adult female Anna’s hummingbird, Calypte anna, who was found by then-fellow-grad-student Chris Clark at a residential address in Berkeley in 2005. She was unable to fly and died of unknown causes just a few minutes after being found. She is now specimen 182041 in the ornithology collection at the Museum of Vertebrate Zoology at Berkeley. Chris Clark and I had her microCTed back in 2005, and that data will finally see the light of day thanks to my current grad student, Chris Michaels, who generated the above model.

This bird’s skull is a hair over an inch long, and she had a body mass of 3.85 grams at the time of her death. For comparison, those little ketchup packets you get at fast-food burger joints each contain 8-9 grams of ketchup, more than twice the mass of this entire bird when it was alive!


  • Dubach, M. 1981. Quantitative analysis of the respiratory system of the house sparrow, budgerigar and violet-eared hummingbird. Respiration Physiology 46(1): 43-60.
  • Hazlehurst, G.A., and Rayner, J.M. 1992. Flight characteristics of Triassic and Jurassic Pterosauria: an appraisal based on wing shape. Paleobiology 18(4): 447-463.


It’s well worth reading this story about Thomas Herndon, a graduate student who as part of his training set out to replicate a well-known study in his field.

The work he chose, Growth in a Time of Debt by Reinhart and Rogoff, claims to show that “median growth rates for countries with public debt over roughly 90 percent of GDP are about one percent lower than otherwise; average (mean) growth rates are several percent lower.” It has been influential in guiding the economic policy of several countries, reaffirming an austerity-based approach.

So here is Lesson zero, for policy makers: correllation is not causation.

To skip ahead to the punchline, it turned out that Reinhart and Rogoff made a trivial but important mechanical mistake in their working: they meant to average values from 19 rows of their spreadsheet, but got the formula wrong and missed out the last five. Those five included three countries which had experienced high growth while deep in debt, and which if included would have undermined the conclusions.

Therefore, Lesson one, for researchers: check your calculations. (Note to myself and Matt: when we revise the recently submitted Taylor and Wedel paper, we should be careful to check the SUM() and AVG() ranges in our own spreadsheet!)

Herndon was able to discover this mistake only because he repeatedly hassled the authors of the original study for the underlying data. He was ignored several times, but eventually one of the authors did send the spreadsheet. Which is just as well. But of course he should never have had to go chasing the authors for the spreadsheet because it should have been published alongside the paper.

Lesson two, for researchers: submit your data alongside the paper that uses it. (Note to myself and Matt: when we submit the revisions of that paper, submit the spreadsheets as supplementary files.)

Meanwhile, governments around the world were allowing policy to be influenced by the original paper without checking it — policies that affect the disposition of billions of pounds. Yet the paper only got its post-publication review because of an post-grad student’s exercise. That’s insane. It should be standard practice to have someone spend a day or two analysing a paper in detail before letting it have such a profound effect.

And so Lesson three, for policy makers: replicate studies before trusting them.

Ironically, this may be a case where the peer-review system inadvertently did actual harm. It seems that policy makers may have shared the widespread superstition that peer-reviewed publications are “authoritative”, or “quality stamped”, or “trustworthy”. That would certainly explain their allowing it to affect multi-billion-pound policies without further validation. [UPDATE: the paper wasn't peer-reviewed after all! See the comment below.]

Of course, anyone who’s actually been through peer-review a few times knows how hit-and-miss the process is. Only someone who’s never experienced it directly could retain blind faith in it. (In this respect, it’s a lot like cladistics.)

If a paper has successfully made it through peer-review, we should afford it a bit more respect than one that hasn’t. But that should never translate to blind trust.

In fact, let’s promote that to Lesson four: don’t blindly trust studies just because they’re peer-reviewed.


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