Figure 1. NHMUK PV R2095, the holotype dorsal vertebra of Xenoposeidon proneneukos in left lateral view (reversed). (A). In the canonical orientation that has been used in illustrations in published papers (Taylor and Naish 2007, Taylor 2018a). (B). Rotated 15° “backwards” (i.e., anticlockwise in this reversed view, with the dorsal portion displaced caudally), yielding a sub-vertical cranial margin in accordance with 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 (2018a: figure 3B, label 2). In part A, the slope (i.e., the angle between the red and green lines) is 35°; in part B, it is 20°.

Figure 2. Varying apparent cross-sectional area of the neural canal of Haplocanthosaurus sp. MWC 8028, caudal vertebra ?3, depending on the orientation of a vertebra. (A), (C). Cranial views in two different orientations. (B), (D). Right lateral views in the same two different orientations. Parts (A) and (B) depict the vertebra oriented according to Definition 2 (articular surfaces of centrum are vertical)(below), and show a neural canal that appears relatively small (5870 pixels) in cross-sectional area; parts (C) and (D) depict the vertebra oriented according to Definition 3 (neural canal is horizontal)(below), and show a neural canal that appears 61% larger (9458 pixels) in cross-sectional area. Thick black lines show the line of view through the neural canal in each orientation, emphasizing that it appears taller in the orientation of parts (C) and (D).

Figure 3. Inconsistent vertebral orientation in our own work (Taylor and Wedel 2013: figure 2, reversed and modified). Representative mid-cervical vertebrae from a turkey (top, parts A and B) and the sauropod Giraffatitan brancai (bottom, parts C and D), not to scale. Each vertebra is shown in caudal view (on the left, parts A and C) and left lateral view, reversed (on the right, parts B and D). Articular surfaces, where each vertebra meets its neighbour, are highlighted in red (for the centra) and blue (for the zygapophyses). Articular surfaces that are concealed from view are cross-hatched: prezygapophyses face upwards and inwards, so that the facets are inclined towards the midline. In sauropods, the centra have ball-and-socket joints. In birds, the joints are saddle-shaped, and the cranial articular surface is hidden in lateral view. Note that the turkey vertebra is illustrated with the long axis of the centrum horizontal (Definition 1) even though this makes the articular surfaces non-vertical; while the Giraffatitan vertebra is illustrated with the caudal articular surface vertical (Definition 2) even though this causes the long axis of the centrum to be inclined. Reproduced under the CC By license.

Figure 4. Parrot skeleton with hemisected integument (probably Amazona ochrocephala) in left lateral view (reversed), in the Natuurhistorisch Museum of Rotterdam. Photograph by Marc Vincent, used with permission. Note the very strong Scurve of the neck, such that the most caudal cervical vertebrae are inclined downwards, then more cranial vertebrae are, progressively, inclined upwards, near vertical, sloping backwards, then vertical again, and finally sloping upwards to the skull.

Figure 5. Long cervical vertebrae oriented by Definition 1 (long axis of centrum is horizontal). (A). Giraffe Giraffa camelopardalis angolensis FMNH 34426, 3rd cervical vertebra in left lateral view (reversed). (B). Domestic turkey Meleagris gallopavo domesticus, 7th cervical vertebra in left lateral view (reversed). (C). Giraffatitan brancai lectotype MB.R.2180 (formerly HMN SI), 5th cervical vertebra in right lateral view. All vertebrae are oriented horizontally according to the long axis of the vertebra (long red line). The long axis may be defined as the line between the vertical midpoints of the cranial and caudal articular surfaces — but the heights of those midpoints depend on the selection of dorsal and ventral extremities of those surfaces, and these are not always obvious, especially in fossils, which are prone to damage. In part C, the short blue lines at each end of the vertebra show candidate margins. At both cranial and caudal surfaces, the dorsal margin is more or less uncontroversial; but there are several candidates for the ventral margin, especially for the caudal articular surface. These are impossible to resolve using only lateral-view photos and potentially even with the complete fossil to hand. The grey outline and shaded area at the caudoventral extremity of the vertebra shows a reconstruction of the undamaged shape of the cotyle, based on Janensch’s (1950: figure 23) drawing — which in turn may have been based on the bone itself in a better state of preservation than currently pertains, or may have been speculative based on the specimen more or less as it is now.

Figure 6. Orientation Definitions 2–4 illustrated for two vertebrae. (A–C). 3D digital model of Haplocanthosaurus sp. MWC 8028, caudal vertebra ?3, in cross section, showing medial aspect of left side, with cranial to the right; (D–F). Giraffe Giraffa camelopardalis FMNH 34426, cervical 7 in left lateral view (reversed). (A, D). In “articular surfaces vertical” orientation (Definition 2). 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 close to but not exactly vertical, inclining slightly forwards in Haplocanthosaurus vertebra and more strongly backwards in the giraffe vertebra. (B, E). In “neural canal horizontal” orientation (Definition 3). The green line joins the cranial and caudal margins of the floor of the neural canal, and is oriented horizontally; some guesswork is required in the case of the giraffe, as only a lateral-view photograph is available. The red line joins the cranial and caudal margins of the roof of the neural canal in the Haplocanthosaurus vertebra (but see Figure 8), and is close to horizontal but inclined upwards; no similar line can be attempted in the giraffe. (C, F). In “similarity in articulation” orientation (Definition 4). Two copies of the same vertebra, held in the same orientation, are digitally articulated optimally, then the pair is rotated as a unit until the two are level. The green line connects two copies of the same point on each copy of the vertebra, and is horizontal. For the Haplocanthosaurus vertebra, the uppermost point of the prezygapophyseal rami are used, and for the giraffe vertebra, the lowest point of of the parapophyses is used; but a horizontal line could join the two copies of any point. It happens that for both these vertebrae Definitions 3 and 4 (parts B and C, and parts E and F of this illustration) give very similar results, but this is accidental. The 7th cervical vertebra of the giraffe is strongly “keystoned”, with the centrum (excluding the articular condyle) forming a parallelogram whose dorsal length is less than its ventral length. The angle between vertical green line and red “nearly vertical” line in part D is about 19°, meaning that if the two copies of the vertebra were oriented such that the cranial and caudal articular surfaces were optimally articulated, there would be a 19° angle between the vertebrae.

Figure 7. Proceoelous vertebrae for which it is difficult to determine the orientation of the articular surfaces, depicted not to scale but with the same vertebral height. (A). Komodo dragon Varanus komodoensis, LACM Herpetology specimen 121971, proximal caudal vertebra in right lateral view. Note the extremely convex and strongly inclined caudal articular surface to the left; the cranial articular surface to the right is correspondingly concave and almost as strongly inclined. (B). Alligator mississippiensis WRAZL 9840044, seventh cervical vertebra (with cervical rib attached) and sixth cervical vertebra (without rib) in articulation, in right lateral view. Photograph by Jess Miller-Camp, used with permission. While the caudal articular surfaces are strongly convex, the orientation of each can be interpreted as that of the well-defined “collar” that surrounds it.

Figure 8. 3D digital model of Haplocanthosaurus sp. MWC 8028, caudal vertebra ?3, in cross section, showing the ambiguous interpretation of the roof of the neural canal. (A). The vertebra oriented according to a long interpretation of neural canal extent. The vertical blue line indicates the position identified as the cranialmost extent of the roof of the neural canal (point a), and the red line shows the interpretation of “horizontal” based on that location. (B). The same vertebra, but with a different choice of cranialmost extent of the roof of the neural canal (point b), again marked with a vertical blue line. When a line is projected from here to the same caudalmost extent as in part A, the resulting notion of “horizontal” differs by 3.8°.

Figure 9. Right halves of two vertebrae from the lumbar (caudal dorsal) region of a human Homo sapiens in sagittal crosssection (cranial to left). Modified from Gray (1858: figure 99). Pale yellow indicates bone in cross-section, grey indicates both bone further from the midline and soft tissue. The red lines mark the floor of the neural canal: since the cranial and caudal ends of the floor of the canal are slightly elevated dorsally relative to the middle part of the canal, it is easy to project a line between these eminences and designate this as the trajectory of the canal. The blue lines mark the roof of the neural canal, but this is convex throughout its length for each vertebra. There is therefore no way to designate any single tangent to it as the trajectory of the neural canal roof of the vertebra as a whole.

Figure 10. The steps of the similarity-in-articulation method of determining horizontal orientation of a vertebra (Definition 4), illustrated using 3D digital model of Haplocanthosaurus sp. MWC 8028, caudal vertebra ?3. (A). Two identical copies of the same vertebra are depicted in the same orientation. (B). The two copies are brought into the best whole-vertebra articulation that can be achieved by translation without rotating either. (C). The articulated pair are rotated together into that orientation in which they are at the same height. This orientation is designated as horizontal according to the present definition.

Figure 11. A selection of vertebrae with the approximate trajectory of their neural canals determined by the simple method of pushing a rolled-up piece of paper through their neural canals. (A). Brachiosaurus altithorax holotype FMNH P 25107, first and partial second caudal vertebrae in right lateral view. (B). Camarasaurus sp. CM 584, proximal caudal vertebra ?4 in right lateral view. (C). Camarasaurus sp. CM 584, mid-caudal vertebra ?12 in left lateral view (reversed). (D). Juvenile giraffe Giraffa camelopardalis, cervical vertebra 6 in left lateral view (reversed). (E). Juvenile giraffe Giraffa camelopardalis, cervical vertebra 7 in left lateral view (reversed). Note the much stronger inclination than in C6, and that the neural canal is slightly “trumpet shaped”, being taller cranially than caudally. (F). Ostrich Struthio camelus, cervical vertebra 16 in left lateral view (reversed).

Figure 12. 3D print of the Xenoposeidon proneneukos holotype dorsal vertebra NHMUK PV R2095, oriented horizontally according to Definition 3 (neural canal is horizontal) by the toothpick method. From left to right: left caudolateral (reversed), left lateral (reversed) and left craniolateral (reversed) views. The camera is at the same level as the floor of the neural canal, so that the toothpicks appear horizontal in the oblique views as well as in the lateral view. The oblique views show that the toothpicks are located at the base of each end of the neural canal, and the horizontal view shows that the two toothpicks are aligned. This procedure was carried out using a 3D print of the vertebra from the scan data published as the supplementary file to Taylor (2018a), as the fossil itself was not readily available.

Figure 13. Two consecutive instances of the hemisected 9th cervical vertebra of a domestic turkey, Meleagris gallopavo domesticus, in right medial view, oriented according to Definition 2 (red lines show vertical orientation of the caudal articular surface) and aligned horizontally. Since the orientation of the neural canal in this vertebra is inclined about 15° to perpendicular with the caudal articular surface, the result is a kinked spinal cord (shown in green) — something that never happens in life.

Figure 14. Sagittally bisected head and cranial neck of a horse Equus ferus caballus in left medial view (reversed). Of the cervical vertebrae (highlighted in red), the first four are complete but only the cranial part of the fifth is present. Note that the neural canal (highlighted in blue) runs in a nearly straight line, and is not kinked.