Thanks to everyone who’s engaged with yesterday’s apparently trivial question: what does it mean for a vertebra to be “horizontal”? I know Matt has plenty of thoughts to share on this, but before he does I want to clear up a couple of things.

This is not about life posture

First, and I really should have led with this: the present question has nothing to do with life posture. For example, Anna Krahl wrote on Twitter:

I personally find it more comprehensible if the measurements relate to something like eg. the body posture. This is due to my momentary biomech./functional work, where bone orientation somet is difficult to define.

I’m sympathetic to that, but we really need to avoid conflating two quite different issues here.

Taylor, Wedel and Naish (2009), Figure 1. Cape hare Lepus capensis RAM R2 in right lateral view, illustrating maximally extended pose and ONP: skull, cervical vertebrae 1-7 and dorsal vertebrae 1-2. Note the very weak dorsal deflection of the base of the neck in ONP, contrasting with the much stronger deflection illustrated in a live rabbit by Vidal et al. (1986: fig. 4). Scalebar 5 cm.

If there’s one thing we’ve learned in the last couple of decades, it’s that life posture for extinct animals is controversial — and that goes double for sauropod necks. Heck, even the neck posture of extant animals is terribly easy to misunderstand. We really can’t go changing what we mean by “horizontal” for a vertebra based on the currently prevalent hypothesis of habitual posture.

Also, note that the neck posture on the left of the image above is close to (but actually less extreme than) the habitual posture of rabbits and hares: and we certainly wouldn’t want to illustrate vertebrae as “horizontal” when they’re oriented directly upwards, or even slightly backwards!

Instead, we need to imagine the animal’s skeleton laid out with the whole vertebral column in a straight line — sort of like Ryder’s 1877 Camarasaurus, but with the tail also elevated to the same straight line.

Ryder’s 1877 reconstruction of Camarasaurus, the first ever made of any sauropod, modified from Osborn & Mook (1921, plate LXXXII).

Of course, life posture is more important, and more interesting, question than that of what constitutes “horizontal” for an individual vertebra — but it’s not the one we’re discussing right now.

In method C, both instances are identically oriented

I’m not sure how obvious this was, but I didn’t state it explicitly. In definition C (“same points at same height in consecutive vertebrae”), I wrote:

We use two identical instances of the vertebrae, articulate them together as well as we can, then so orient them that the two vertebrae are level

What I didn’t say is that the two identical instances of the vertebrae have to be identically oriented. Here’s why this is important. Consider that giraffe C7 that we looked at last time, with its keystoned centrum. if you just “articulate them together as well as we can” without that restriction, you end up with something like this:

Which is clearly no good: there’s no way to orient that such that for any given point on one instance, the corresponding point on the other is level with it. What you need instead is something like this:

In this version, I’ve done the best job I can of articulating the two instances in the same attitude, and arranged them such that they are level with each other — so that the attitude shown here is “horizontal” in sense C.

As it happens, this is also just about horizontal in sense B — the floor of the neural canal is presumably at the same height as the top of the centrum as it meets the neural arch.

But “horizontal” in sense A (posterior articular surface vertical) fails horribly for this vertebra:

To me, this image alone is solid evidence that Method A is just not good enough. Whatever we mean by “horizontal”, it’s not what this image shows.



I was lucky enough to have Phil Mannion as one of the peer-reviewers for my recent paper (Taylor 2018) showing that Xenoposeidon is a rebbachisaurid. During that process, we got into a collegial disagreement about one of the autapomorphies that I proposed in the revised diagnosis: “Neural arch slopes anteriorly 30°–35° relative to the vertical”. (This same character was also in the original Xenoposeidon paper (Taylor and Naish 2007), in the slightly more assertive form “neural arch slopes anteriorly 35 degrees relative to the vertical”: the softening to “30°–35°” in the newer paper was one of the outcomes of the peer-review.)

The reason this is interesting is because the slope of the neural arch is measured relative to the vertical, which of course is 90˚ from the horizontal — but Phil’s comments (Mannion 2018) pushed me to ask myself for the first time: what actually is “horizontal”? We all assume we know horizontality when we see it, but what precisely do we mean by it?

Three notions of “horizontal”

The idiosyncratic best-preserved caudal vertebra of the Snowmass Haplocanthosaurus MWC 8028, illustrating three different versions of “horizontal”. A. horizontality defined by vertical orientation of the posterior articular surface. B. horizontality defined by horizontal orientation of the roof of the neural canal (in this case, rotated 24˚ clockwise relative to A). C. horizontality defined by optimal articulation of two instances of the vertebra, oriented such the a line joining the same point of both instances is horizontal (in this case, rotated 17˚ clockwise relative to A). Red lines indicate exact orthogonality according to the specified criteria. Green line indicate similar but diverging orientations: that of the not-quite-vertical anterior articular surface (A) and of the not-quite-horizontal base of the neural canal (B).

There are at least three candidate definitions, which we can see yield noticeably different orientations in the case of the Snowmass Haplocanthosaurus vertebra that Matt’s been playing with so much recently.

Definition A: articular surfaces vertical

In part A, I show maybe the simplest — or, at least, the one that is easiest to establish for most vertebrae. So long as you have a reasonably intact articular surface, just rotate the vertebra until that surface is vertical. If, as is often the case, the surface is not flat but concave or convex, then ensure the top and bottom of the surface are vertically aligned. This has the advantage of being easy to do — it’s what I did with Xenoposeidon — but it conceals complexities. Most obviously, what to do when the anterior and posterior articular surfaces are not parallel, in the 7th cervical vertebra of a giraffe?

Cervical vertebra 7 of Giraffa camelopardalis FMNH 34426, in left lateral view. Note that the centrum is heavily “keystoned” so that the anterior and posterior articular surfaces are 15-20˚ away from being parallel.

Another difficulty with this interpretation of horizontality is that it can make the neural canal jagged. Consider a sequence of vertebrae oriented as in part A, all at the same height: the neural canal would rise upwards along the length of each vertebra, before plunging down again on transitioning from the front of one to the back of the next. This is not something we would expect to see in a living animal: see for example the straight line of the neural canal in our hemisected horse head(*).

Definition B: neural canal horizontal

Which leads us to the second part of the illustration above. This time, the vertebra is oriented so that the roof of the neural canal is horizontal, which gives us a straight neural canal. Nice and simple, except …

Well, how do we define what’s horizontal for the neural canal? As the Haplocanthosaurus vertebra shows nicely, the canal is not always a nice, neat tube. In this vertebra, the floor is nowhere near straight, but dishes down deeply — which is why I used to the roof, rather than the floor of the canal. Rather arbitrary, I admit — especially as it’s often easier to locate the floor of the canal, as the dorsal margin is often confluent with fossae anteriorly, posteriorly or both.

And as we can see, it makes a difference which we choose. The green line in Part B of the illustration above shows the closest thing to “horizontal” as it would be defined by the ventral margin of the neural canal — a straight line ignoring the depression and joining the anteriormost and posteriormost parts of the base of the canal. As you can see, it’s at a significantly different angle from the red line — about 6.5˚ out.

And then you have human vertebrae, where the dorsal margin of the neural canal is so convex in lateral view that you really can’t say where the anteriormost or posteriormost point is.

Left sides of hemisected human thoracic vertebrae, medial view. Note how ill-defined the dorsal margin of the neural canal is.

So can we do better? Can we find a definition of “horizontal” that’s not dependent of over-interpreting a single part of the vertebra?

Definition C: same points at same height in consecutive vertebrae

I’ve come to prefer a definition of horizontal that uses the whole vertebra — partly in the hope that it’s less vulnerable to yielding a distorted result when the vertebra is damaged. With this approach, shown in part C of the illustration above, we use two identical instances of the vertebrae, articulate them together as well as we can, then so orient them that the two vertebrae are level — that a line drawn between any point on one vertebra and its corresponding point on the other is horizontal. We can define that attitude of the vertebra as being horizontal.

Note that, while we use two “copies” of the vertebra in this method, we are nevertheless determining the horizontality of a single vertebra in isolation: we don’t need a sequence of consecutive vertebrae to have been preserved, in fact it doesn’t help if we do have them.

One practical advantage of this definition is that its unambiguous as regards what part of the vertebra is used: all of it; or any point on it, at the measurement stage. By contrast, method A requires us to choose whether to use the anterior or posterior articular surface, and method B requires a choice of the roof or floor of the neural canal.


I have three questions, and would welcome any thoughts:

  1. Which of these definitions do you prefer, and why?
  2. Can you think of any other definitions that I missed?
  3. Does anyone know of any previous attempts to formalise this? Is it a solved problem, and Matt and I somehow missed it?

Answers in the comments, please!


(*) Yes, of course we have a hemisected horse head. What do you think we are, savages?

Down in flames

August 25, 2018

I first encountered Larry Niven’s story/essay “Down in Flames” in the collection N-Space in high school. This was after I’d read Ringworld and most of Niven’s Known Space stories, so by the time I got to “Down in Flames” I had the context to get it. (You can read the whole thing for free here.)

Here’s the idea, from near the start:

On January 14, 1968, Norman Spinrad and I were at a party thrown by Tom & Terry Pinckard. We were filling coffee cups when Spinny started this whole thing.

“You ought to drop the known space series,” he said. “You’ll get stale.” (Quotes are not necessarily dead accurate.) I explained that I was writing stories outside the “known space” history, and that I would give up the series as soon as I ran out of things to say within its framework. Which would be soon.

“Then why don’t you write a novel that tears it to shreds? Don’t just abandon known space. Destroy it!”

“But how?” (I never asked why. Norman and I think alike in some ways.)

The rest of the piece is just working out the details.

“Down in Flames” brain-wormed me. Other than Ray Bradbury’s “A Sound of Thunder” I doubt if there is another short story I’ve read as many times. Mike once described the act of building something complex and beautiful and then destroying it as “magnificently profligate”, and that’s the exact quality of “Down in Flames” that appeals to me.

I also think it is a terrific* exercise for everyone who is a scientist, or who aspires to be one.

* In both the modern sense of “wonderful” and the archaic sense of “causing terror”.

Seriously, try it. Grab a piece of paper (or open a new doc, or whatever) and write down the ideas you’ve had that you hold most dear. And then imagine what it would take for all of them to be wrong. (When teams and organizations do this for their own futures, it’s called a pre-mortem, and there’s a whole managerially-oriented literature on it. I’d read “Down in Flames” instead.)

It feels like this! Borrowed from here.

Here are some questions to help you along:

  • Which of your chains of reasoning admit more than one end-point? If none of them might lead other places, then either you are the most amazing genius of all time (even Newton and Einstein made mistakes), or you are way behind the cutting edge, and your apparent flawlessness comes from working on things that are already settled.
  • If there is a line of evidence that could potentially falsify your pet hypothesis, have you checked it? Have you drawn any attention to it? Or have you gracefully elided it from your discussions in hopes that no-one will notice, at least until after you’re dead?
  • If there’s no line of evidence that could falsify your pet hypothesis, are you actually doing science?
  • Which of your own hypotheses do you have an emotional investment in?
  • Are there findings from a rival research team (real or imagined) that you would not be happy to see published, if they were accurate?
  • Which hypotheses do you not agree with, that you would be most dismayed to see proven correct?

[And yes, Karl, I know that according to some pedants hypotheses are never ‘proven’. It’s a theoretical exercise already, so just pretend they can be!]

I’ll close with one of my favorite quotes, originally published in a couple of tweets by Angus Johnson in May of 2017 (also archived here):

If skepticism means anything it means skepticism about the things you WANT to be true. It’s easy to be a skeptic about others’ views. Embracing a set of claims just because it happens to fit your priors doesn’t make you a skeptic. It makes you a rube, a mark, a schnook.

So, don’t be that rube. Burn down your house of ideas – or at least, mentally sift through the rubble and ashes and imagine how it might have burned down. And then be honest about that, minimally with yourself, and ideally with the world.

If you’re a true intellectual badass, blog the results. I will. It’s not fair to give you all homework – painful homework – and not take the medicine myself, so I’m going to do a “Down in Flames” on my whole oeuvre in the next a future post. Stay tuned!

Diplodocus goes digital

August 21, 2018

No time for a proper post, so here’s a screenshot from Amira of Diplodocus caudal MWC 8239 (the one you saw being CT scanned last post) about to be digitally hemisected. Trust me, you’ll want to click through for the big version. Many thanks to Thierra Nalley for the Amira help.

Further bulletins as time and opportunity allow.

John Yasmer, DO (right) and me getting ready to scan MWC 8239, a caudal vertebra of Diplodocus on loan from Dinosaur Journey, at Hemet Valley Imaging yesterday.

Alignment lasers – it’s always fun watching them flow over the bone as a specimen slides through the tube (for alignment purposes, obviously, not scanning – nobody’s in the room for that).

Lateral scout. I wonder, who will be the first to correctly identify the genus and species of the two stinkin’ mammals trailing the Diplo caudal?

A model we generated at the imaging center. This is just a cell phone photo of a single window on a big monitor. The actual model is much better, but I am in a brief temporal lacuna where I can’t screenshot it.

What am I doing with this thing? All will be revealed soon.

Robin N. Kok asked an interesting question on Twitter:

For all the free money researchers throw at them, they might as well be shareholders. Maybe someone could model a scenario where all the APC money is spent on RELX shares instead, and see how long it takes until researchers own a majority share or RELX.

Well, Elsevier is part of the RELX group, which has a total market capitalisation of £33.5 billion. We can’t know directly how much of that value is in Elsevier, since it’s not traded independently. But according to page 124 their 2017 annual report (the most recent one available), the “Scientific, Technical and Medical” part of RELX (i.e. Elsevier) is responsible for £2,478M of the total £7,355M revenue (33.7%), and for £913M of the £2,284M profit (40.0%). On the basis that a company’s value is largely its ability to make a profit, let’s use the 40% figure, and estimate that Elsevier is worth £13.4 billion.

(Side-comment: ouch.)

According to the Wellcome Trust’s 2016/17 analysis of its open access spend, the average APC for Elsevier articles was £3,049 (average across pure-OA journals and hybrid articles).

On that basis, it would take 4,395,000 APCs to buy Elsevier. How long would that take to do? To work that out, we first need to know how many APC-funded articles they publish each year.

From page 14 of the same annual report as cited above. Elsevier published “over 430,000 articles” in a year. But most of those will have been in subscription journals. The same page says “Subscription sales generated 72% of revenue, transactional sales 26% and advertising 2%”, so assuming that transactional sales means APCs and that per-article revenue was roughly equal for subscription and open-access articles, that means 26% of their articles — a total of 111,800.

At 111,800 APCs per year, it would take a little over 39 years to accumulate the 4,395,000 APCs we’d need to buy Elsevier outright.

That’s no good — it’s too slow.

What if we also cancelled all our subscriptions, and out those funds towards the buy-out, too? That’s actually a much simpler calculation. Total Elsevier revenue was £2,478M. Discard the 2% that’s due to advertising, and £2428M was from subscriptions and APCs. If we saved that much for just five and a half years, we’d have saved enough to buy the whole company.

That’s a surprisingly short time, isn’t it?

(In practice of course it would be much faster: the share-price would drop precipitously as we cancelled all subscription and stopped paying APCs, instantly cutting revenue to one fiftieth of what it was before. But we’ll ignore that effect for our present purposes.)


We don’t post on pterosaurs very often, but I’m making an exception for Caelestiventus. Mostly because I had the unusual experience of holding a life-size 3D print of its skull a few days before it was published. Brooks Britt and George Engelmann are both attending Flugsaurier 2018 in Los Angeles, and Brooks gave a talk on the new pterosaur on Friday. It’s from the Upper Triassic Saints & Sinners Quarry in far northeastern Utah, which has also produced theropods, sphenosuchian crocs (like 80 individuals to date, no exaggeration), drepanosaurs (I’ve seen the material and that paper is going to be mind-blowing whenever it arrives), and other assorted hellasaurs. Some of that material is figured in the Britt et al. (2016) paper on the Saints & Sinners Quarry (a free download from the link below). As far as I know, the Caelestiventus paper is the second big volley on the Saints & Sinners material, out of what will probably be a long stream of important papers.

Anyway, since we’ve just been discussing the utility of 3D printing in paleontology (1, 2), I thought you’d like to see this. Brooks did caution us that the 3D model was a work in progress, and he now thinks that Caelestiventus had a more convex dorsal skull margin, with the downward forehead dip in the version that got printed being less prominent or absent. You can see a slightly different version in the skull recon drawn by second author Fabio M. Dalla Vecchia, which he kindly released into the public domain here.

Otherwise the 3D print is pretty good. The big plate below the orbit is weird and from what I gather not present in other dimorphodontids. Because the Saints & Sinners material was buried in sand, which is relatively incompressible compared to mud and clay, it’s all preserved in three dimensions with essentially no crushing. Caelestiventus therefore yields new information about Dimorphodon micronyx, which has been known since 1859 but mostly from pancaked material.

Stay tuned (in general, not here necessarily) for more on the remarkable tetrapods of the Sants & Sinners Quarry – the next few years are going to be very exciting. And since this may be my first and last Flugsaurier post, many thanks to the organizers for making it such an engaging and enjoyable experience, especially Mike Habib, Liz Martin-Silverstone, and Dave Hone.