Regular readers will remember that we followed up our 1VPC talk about what it means for a vertebra to be horizontal by writing it up as a paper, and doing it in the open. That manuscripts is now complete, and published as a preprint (Taylor and Wedel 2019).

Taylor and Wedel (2018: Figure 5). Haplocanthosaurus sp. MWC 8028, caudal vertebra ?3, in cross section, showing medial aspect of left side, cranial to the right, in three orientations. A. In “articular surfaces vertical” orientation (method 2 of this paper). The green line joins the dorsal and ventral margins of the caudal articular surface, and is oriented vertically; the red line joins the dorsal and ventral margins of the cranial articular surface, and is nearly but not exactly vertical, instead inclining slightly forwards. B. In “neural canal horizontal” orientation (method 3 of this paper). The green line joins the cranial and caudal margins of the floor of the neural canal, and is oriented horizontally; the red line joins the cranial and caudal margins of the roof of the neural canal, and is close to horizontal but inclined upwards. C. In “similarity in articulation” orientation (method 4 of this paper). Two copies of the same vertebra, held in the same orientation, are articulated optimally, then the group is rotated until the two are level. The green line connects the uppermost point of the prezygapophyseal rami of the two copies, and is horizontal; but a horizontal line could join the two copies of any point. It happens that for this vertebra methods 3 and 4 (parts B and C of this illustration) give very similar results, but this is accidental.

The preprint has all the illustrations and their captions at the back of the PDF. If you prefer to have them inline in the text, where they’re referenced — and who wouldn’t? — you can download a better version of the manuscript from the GitHub archive.

By the way, you may have noticed that what started our written in Markdown has mutated into an MS-Word document. Why? Well, because journals won’t accept submissions in Markdown. It eas a tedious and error-prone job to convert the Markdown into MS-Word, and not one I am keen to repeat. For this reason, I think I am unlikely to use Markdown again for papers.

References

  • Taylor, Michael P., and Mathew J. Wedel. 2019. What do we mean by the directions “cranial” and “caudal” on a vertebra? PeerJ PrePrints 7:e27437v2. doi:10.7287/peerj.preprints.27437v2

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Years ago, I wrote a tutorial on how to get a “nearly finished” paper over the finishing line in which I said “Do you really need a printed copy for this? YES YOU DO! Can’t you just do it on the screen? NO YOU CAN’T!”

I was so right.

Here is a page from the manuscript for the vertebral orientation project. I thought a couple of days ago that this was complete and ready to submit. But, just for form’s sake, I printed a copy and went through it with a pen, as I recommended in the tutorial.

Well, I found many, many places where I had to mark up the printed manuscript. Some of them were trivial typos that I’d somehow missed in all times I’d read the manuscript on a screen. Others were infelicitous word choices that I could improve. A few were places where I realised I’d not spelled out something that ought really to be made explicit. There are probably more than a hundred in all.

I just finished this process (shortly after midnight). The next thing I will do, when I have a chance, will be to go theough the manuscript fixing all these little errors and omissions. Most of them I will do right away; other will take longer, so I will just leave a comment for myself marked with the “XXX” rule. Later I will come back and search for “XXX”, and fix the complicated ones.

Only then will I submit — once we have made this submission the best we can make it.

In a move that will surprise no-one who’s been paying attention, my and Matt’s presentation of vertebral orientation at the 1st Palaeo Virtual Congress is now up as a PeerJ preprint. Sadly, with the end of the conference period on 15th December, the page for my talk has been deleted, along with some interesting comments. But here at SV-POW!, we have no truck with ephemerality, hence this more permanent manifestation of our work.

Matt’s preprint consists of the abstract, and has the slide deck as a supplementary data file. That’s what he submitted to the conference, with attendees invited to page through it. By contrast, I recorded a video of my talk. I am trying to get that attached to my preprint, but as things stand it’s not there because it’s too big (at 65 Mb).

Meanwhile — and indeed in perpetuity — you can just watch it on YouTube, where I also uploaded it. In the end, that may be a more practical way of making video available anyway, but I do want the preservational benefit of lodging it with a preprint.

Remember, we’re working on the paper in the open. We’d love to get input from you all, and especially from anyone who’s run into this problem before with other taxa. Please, if you have fifteen minutes spare, watch the talk and leave any comments you have: here, on the preprint, on the YouTube page, or as issues in the GitHub tracker!

Reference

Now that Matt and I have blogged various thoughts about how to orient vertebra (part 1, part 2, relevant digression 1, relevant digression 2, part 3) and presented a talk on the subject at the 1st Palaeontological Virtual Congress, it’s time for us to strike while the iron is hot and write the paper.

Figure A. NHMUK PV R2095, the holotype dorsal vertebra of Xenoposiedon proneneukos in left lateral view. A. In the canonical orientation that has been used in illustrations in published papers (Taylor and Naish 2007, Taylor 2018b, in blog-posts and on posters and mugs. B. Rotated 15° “backwards” (i.e. clockwise, with the dorsal portion displaced caudally), yielding a sub-vertical anterior margin in accordance the recommendation of Mannion (2018b). In both parts, the blue line indicates the horizontal axis, the green line indicates the vertical axis, and the red line indicates the slope of the neural arch as in Taylor (2018b: figure 3B, part 2). In part A, the slope (i.e. the angle between the red and green lines) is 35°; in part B, it is 20°.

We’re doing it totally in the open, on GitHub. You can always see the most recent version of the manuscript at https://github.com/MikeTaylor/palaeo-vo/blob/master/vo-manuscript.md and you can also review the history of its composition if you like — from trivial changes like substituting a true em-dash for a double hyphen, to significant additions like writing the introduction.

More than that, you can contribute! If you think there’s a mistake, or something missing that should be included, or if you just have a suggestion, you can file an issue on the project’s bug-tracker. If you’re feeling confident, you can go further and directly edit the manuscript. The result will be a tracked change that we’ll be notified of, and which we can accept into, or reject from, the master copy.

We hope, by making all this visible online, to demythologise the process of writing a paper. In a sense, there is no magic to it: you just start writing, do a section at a time, revise as you go, and eventually you’re done. It’s much like writing anything else. (Doing the referencing can make it much slower than regular writing, though!)

By the way, you may wonder why the illustration above is “Figure A” rather than “Figure 1”. In all my in-progress manuscripts, I just assign letters to each illustration as I add it, not worrying about ordering. Only when the manuscript is ready to be submitted do I take the order that the illustrations occur in (A, D, G, H, B, I, E, F, for example, with C having been dropped along the way) and replace them with consecutive numbers. So I save myself a lot of tedious and error-prone renumbering every time that, in the process of composition, I insert an illustration anywhere before the last existing one. This is really helpful when there are a lot of illustrations — as there tend to be in our papers, since they’re all in online-only open-access venues with no arbitrary limits. For example, our four co-authored papers from 2013 had a total of 69 illustrations (11 in Taylor and Wedel 2013a, 25 in Wedel and Taylor 2013a, 23 in Taylor and Wedel 2013b and 10 in Wedel and Taylor 2013b).

References

 

 

The 1st Palaeontological Virtual Congress is underway now, and will run through December 15. Mike and I have two presentations up:

“What do we mean by the directions ‘cranial’ and ‘caudal’ on a vertebra?” by Mike and me, which consists of a video Mike made presenting a slide show that he put together. The presentation sums up our thinking following the series of vertebral orientation posts here earlier this summer and fall, which are all available here.

“Reconstructing an unusual specimen of Haplocanthosaurus using a blend of physical and digital techniques” by me and a gang of WesternU-based collaborators, including Jessie Atterholt and Thierra Nalley, both of whom you saw in our recent pig-hemisecting adventures. Almost everything I’ve written on this blog about Haplocanthosaurus in 2018 was part of the run-up to this presentation (except, somewhat ironically, the post about pneumaticity), which also includes quite a bit that I haven’t put on the blog yet. So even if you follow SV-POW!, the 1PVC slideshow should have plenty of stuff you haven’t seen yet.

IF you can see it–you have to be a registered 1PVC ‘attendee’ to log in to the site and see the presentations. So probably you are either already registered and this post is old news, or not registered and this post seems useless. Why would I bother telling you about stuff you can’t see?

The answer is that neither Mike or I intend for our work to disappear when 1PVC comes to an end on December 15. Both of us are planning to put our abstracts and slide decks up as PeerJ Preprints, which is our default move for conference presentations these days (e.g., this, this, and this). I believe Mike is also going to post his video to YouTube. So the work will not only live on after the congress is over, it will jump to a much broader audience. We’re looking forward to letting everyone see what we’ve been up to, and I’m sure we’ll have some more things to say here when that happens.

So, er, go see our stuff if you’re a 1PVC attendee, and if you’re not, hang in there, we’ll have that stuff out to you in a few days. UPDATE: The Haplo presentation is up now (link).

WOW! I knew I was dragging a bit on getting around to this vertebral orientation problem, but I didn’t realize a whole month had passed. Yikes. Thanks to everyone who has commented so far, and thanks to Mike for getting the ball rolling on this. Previous posts in this series are here and here.

First up, this may seem like a pointlessly picky thing to even worry about. Can’t we just orient the vertebrae in whichever way feels the most natural, or is easiest? Do we have to think about this?

The alarmingly 3D pelvis of the mounted brontosaur at the AMNH. Note that sauropod pubes are usually illustrated lying flat, so what usually passes for ‘lateral’ view would be roughly from the point of view of the animal’s knee.

I think we do. For sauropods, vertebrae are usually oriented for illustration purposes in one of two ways. The first is however they sit most easily on their pallets. This is similar to the problem Mike and I found for ‘lateral’ views of sauropod pelvic elements when were on our AMNH/Yale trip in 2012. In an articulated skeleton, the pubes and ischia usually lean inward by 30-45 degrees from their articulations with the ilia, so they can meet on the midline, but when people illustrate the “lateral view” of a sauropod pubis or ischium, it’s often the ventro-lateral aspect that is face-up when the element is lying on a shelf or a pallet. Photographic lateral does not equal biological lateral for those elements. Similarly, if I’m trying to answer biological questions about vertebrae (see below), I need to know something about their orientation in the body, not just how they sit comfortably on a pallet.

The other way that vertebrae are commonly oriented is according to what we might call the “visual long axis” of the centrum—so for example, dorsoventrally tall but craniocaudally short proximal caudals get oriented with the centrum ‘upright’, whereas dorsoventrally short but craniocaudally long distal caudals get oriented with the centrum ‘horizontal’, even if they’re in the same tail and doing so makes the neural canals or articular faces be oriented inconsistently down the column. (I’m not going to name names, because it seems mean to pick on people for something I just started thinking about myself, but if you go plow through a bunch of sauropod descriptions, you’ll see what I’m talking about.)

Are there biological questions where this matters? You bet! There are some questions that we can’t answer unless we have the vertebrae correctly oriented first. One that comes to mind is measuring the cross-sectional area of the neural canal, which Emily Giffin did a lot of back in the 90s. Especially for the Snowmass Haplocanthosaurus, what counts as the cross-sectional area of the neural canal depends on whether we are looking at the verts orthogonal to their articular faces, or in alignment with the course of the canal. I think the latter is pretty obviously the way to go if we are measuring the cross-sectional area of the canal to try and infer the diameter of the spinal cord—we’d want to see the canal the same way the cord ‘sees’ it as it passes through—but it’s less obvious if we’re measuring, say, the surface area of the articular face of the vertebra to figure out, say, cartilage stress. It doesn’t seem unreasonable to me that we might want to define a ‘neural axis’ for dealing with spinal-cord-related questions, and a ‘biomechanical axis’ for dealing with articulation-related questions.

Caudal 3 of the Snowmass Haplocanthosaurus, hemisected 3D model.

With all that in mind, here are some points.

To me, asking “how do we know if a vertebra is horizontal” is an odd phrasing of the problem, because “horizontal” doesn’t have any biological meaning. I think it makes more sense to couch the question as, “how do we define cranial and caudal for a vertebra?” Normally both the articular surfaces and the neural canal are “aimed” head- and tail-wards, so the question doesn’t come up. Our question is, how do we deal with vertebrae for which the articular surfaces and neural canal give different answers?

For example. Varanus komodoensis caudal.

(And by the way, I’m totally fine using “anterior” and “posterior” for quadrupedal animals like sauropods. I don’t think it causes any confusion, any more than people are confused by “superior” and “inferior” for human vertebrae. But precisely because we’re angling for a universal solution here, I think using “cranial” and “caudal” makes the most sense, just this once. That said, when I made the image above, I used anterior and posterior, and I’m too lazy now to change it.)

I think if we couch the question as “how do we define cranial and caudal”, it sets up a different set of possible answers than Mike proposed in the first post in this series: (1) define cranial and caudal according to the neural canal, and then describe the articular surfaces as inclined or tilted relative to that axis; (2) vice versa—realizing that using the articular surfaces to define the anatomical directions may admit a range of possible solutions, which might resurrect some of the array of possible methods from our first-draft abstract; (3) define cranial and caudal along the long axis of the centrum, which is potentially different from either of the above; (4) we can imagine a range of other possibilities, like “use the zygs” or “make the transverse processes horizontal” (both of which are subsets of Mike’s method C) but I don’t think most of those other possibilities are sufficiently compelling to be worthy of lengthy discussion.

IF we accept “neural canal”, “articular surfaces”, and “centrum long axis” as our strongest contenders, I think it makes most sense to go with the neural canal, for several reasons:

  • In a causative sense, the neural tube/spinal cord does define the cranial/caudal axis for the developing skeleton. EDIT: Actually, that’s a bit backwards. It’s the notochord, which is later replaced by the vertebral column, that induces the formation of the brain and spinal cord from the neural plate. But it’s still true that the vertebrae form around the spinal cord, so it’s not wrong to talk about the spinal cord as a defining bit of soft tissue for the developing vertebrae to accommodate.
  • The neural canal works equally well for isolated vertebrae and for articulated series. Regardless of how the vertebral column is oriented in life, the neural canal is relatively smooth—it may bend, but it doesn’t kink. So if we line up a series of vertebrae so that their neural canals are aligned, we’re probably pretty close to the actual alignment in life, even before we look at the articular surfaces or zygs.
  • The articulated tails of Opisthocoelicaudia and big varanids show that sometimes the articular surfaces simply are tilted to anything that we might reasonably consider to be the cranio-caudal axis or long axis of the vertebra. In those cases, the articular surfaces aren’t orthogonal to horizontal OR to cranio-caudal. So I think articular surfaces are ruled out because they break down in the kinds of edge cases that led us to ask the question in the first places.

Opistocoelicaudia caudals 6-8, stereopair, Borsuk-Bialynicka (1977:plate 5).

“Orient vertebrae, isolated or in series, so that their neural canals define the cranio-caudal axis” may seem like kind of a ‘duh’ conclusion (if you accept that it’s correct; you may not!), but as discussed up top, often vertebrae from a single individual are oriented inconsistently in descriptive works, and orientation does actually matter for answering some kinds of questions. So regardless of which conclusion we settle on, there is a need to sort out this problem.

That’s where I’m at with my thinking. A lot of this has been percolating in my hindbrain over the last few weeks—I figured out most of this while I was writing this very post. Is it compelling? Am I talking nonsense? Let me know in the comments.

I am still building up to a big post on vertebral orientation, but in the meantime, check out this caudal vertebra of a Komodo dragon, Varanus komodoensis. This is right lateral view–the vert is strongly procoelous, and the articular ends of the centrum are really tilted relative to the long axis. I find this encouraging, for two reasons. First, it helped me clarify my thinking on how we ought to orient vertebrae, which Mike wrote about here and here. And second, it gives me some hope, because if we can figure out why tilting your articular surfaces makes functional sense in extant critters like monitors, maybe we can apply those lessons to sauropods and other extinct animals.

This is LACM Herpetology specimen 121971. Many thanks again to Neftali Camacho for access and assistance, and to Jessie Atterholt for basically doing all the other jobs while I was faffing about with this Komodo dragon.

Left lateral view

Have we ever posted decent photos of the Brachiosaurus altithorax caudals? Has anyone? I can’t remember either thing ever happening. When I need images of brachiosaur bits, including caudals, I usually go to Taylor (2009).

Taylor (2009: fig. 3)

Which is silly, not because Mike’s diagrams compiling old illustrations aren’t good – they definitely are – but because I’m sitting on a war chest of decent photos of the actual material. I am home sick with a sore throat today, and I can’t be arsed to (1) follow up on the “Down in Flames” post, (2) add anything thoughtful to the vertebral orientation discussion, or (3) crop or color-adjust these photos. You’re getting them just as they came out of my camera, from my trip to the Field Museum in 2012.

Here are the rest of the orthogonal views:

Right lateral view

 

Anterior view

 

Posterior view

 

Dorsal view of caudal 1

 

Dorsal view of caudal 2

And here’s a virtual walkaround using a series of oblique shots. Making a set like this is part of my standard practice now for important specimens during museum visits.

 

 

 

 

 

 

 

Now, I said up top that I wasn’t going to add anything thoughtful to the vertebral orientation discussion. I have thoughts on that, but I’m tired and hopped up on cold medicine and now ain’t the time. In lieu of blather, here are a couple of relevant photos.

 

I wanted to capture for my future self the pronounced non-orthogonality of the neural canal and centrum, so I rolled up a piece of paper and stuck it through the neural canal. I haven’t run the numbers, but in terms of “angle of the articular faces away from the neural canal”, these verts look like they’re right up there with my beloved Snowmass Haplocanthosaurus.

More on that next time, I reckon. In the meantime, all these photos are yours now (CC-BY, like everything on this site [that someone else hasn’t asserted copyright over]). Go have fun.

Reference

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.

References

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.

Discussion

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!

References

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