I'm not sure Ruskie's comment about the piers is altogether wrong.
Whilst triangular cutwaters are relatively common, where economy of material was important they rarely extend in height above peak flood level, sometimes no more than just above normal river level.
In this case the triangular shape extends all the way to parapet level, far beyond the point where they are likely to have any water cutting benefit. The function, if any, at that point becomes butressing the spandrel wall (the inverted triangular shaped wall between the arches), or simply as an ornamental feature.
The really unusual bit for me is the abutment on the far bank which has a similar full-height triangular shape cutwater/buttress. You typically only see cutwaters within the river channel where water and debris has to be guided either side of the pier. Abutments would normally be on dry land, and even if subject to flooding, the water only has one side to go.
If the painting accurately reflects the structure of the bridge then it looks like the builder(s) did not need to economise on materials and therefore possibly were not short of money. However, with no disrespect intended towards cristeen's ancestor, painters often tend to capture the likeness of structures how they see it, or how they think it should be, rather than in technical detail, so I wouldn't want to draw too many conclusions just based on the painting. I'd also add that the technical details are what would make me a terrible artist
The Eckington bridge photo demonstrates the above very well. Starting on the left at the bank is a wingwall, not a cutwater, the height of which is sufficient to support the road and not much more. Between the first and second arch is a cutwater pier which tapers into the spandrel wall no higher than the level of the top of the adjacent arches. The remaining piers all carry the cutwater profile full-height to the parapet level. It's a subtle difference - I wonder how many artists would have painted Eckington bridge with all piers looking the same?
