The Numbers Talk Back

Ciborowski and Nash reply to the comment on their arithmetic similarity method

Following the June 2026 comment paper discussed in The Numbers Ain't Enough, the original authors have now published a formal reply in the same journal.

Ciborowski, T. J. R., and Nash, D. J. 2026. "Arithmetic Methods for Exploring Archaeological Source Provenance Using Geochemistry – A Reply to Pearce et al." Journal of Archaeological Science: Reports 75: 106012. 10.1016/j.jasrep.2026.106012

What the reply concedes

Ciborowski and Nash accept three of the points raised by Pearce et al. (2026):

• Low trace element concentrations in silcretes mean that rounding can have a disproportionate effect on the arithmetic method's results, and future users should build in an additional error tolerance for chemically "pure" materials.
• The Certified Reference Materials run alongside the original sarsen analyses (Nash et al., 2020) were not ideal, since no silcrete-specific CRM exists. Candidate substitutes are proposed: OREAS 21h, 22i, 293–297 and 299.
• Weighting each element ratio by its analytical precision, rather than treating all ratios equally in the geometric mean, is described as "a good one" that could improve the method, though it was not implemented in the original paper.

Where they push back

The reply rejects the framing of the comment on two main grounds.

Mischaracterisation of scope. Ciborowski and Nash argue that Pearce et al.'s title and introduction present the arithmetic method as a proposed replacement for petrography, when the original paper explicitly describes it as one tool to be used "in combination with other geochemical, petrographic and archaeological indicators." They also point out that both authors have never corresponded or presented at conferences with Pearce and co-authors, disputing the comment's suggestion that the method was developed to justify earlier use of element ratios in a separate dispute with Hancock et al. (2024) over Stonehenge sarsen sourcing.

Ratio versus concentration data. The reply restates the case for using trace element/Zr ratios rather than raw concentrations in silcrete provenancing: variable silicification and heavy-mineral laminae can alter absolute concentrations unpredictably, but not the ratios between non-Si elements. Ciborowski and Nash argue this problem does not shrink with sample size, contrary to a claim made in the comment about debitage-scale samples.

The Stone 62 / Carn Goedog exchange

Pearce et al. had argued that it is "obvious" the arithmetic method's identification of Carn Goedog as geochemically similar to Stone 62 cannot be correct, since Stone 62 is a non-spotted dolerite and Carn Goedog is a spotted dolerite outcrop. Ciborowski and Nash respond that:

• They did not propose Carn Goedog as the actual source of Stone 62 – only that it emerged as the closest geochemical match in a worked example demonstrating the method's mechanics.
• Bevins, Ixer and Pearce previously used geochemical data alone (Bevins et al., 2014) to assign non-spotted dolerite debitage samples from Stonehenge to Carn Goedog, a spotted dolerite source – the same type of inference the comment now characterises as untenable.

Rare earth elements and compatible elements

On the criticism that using a full suite of REE risks "swamping" other variation, the reply cites Nash et al. (2020)'s own dataset: at Totterdown Wood, REE/Zr ratios distinguish that outcrop from Stone 58, while at West Woods they do not – a distinction that would have been lost had REEs been excluded. On compatible elements, Ciborowski and Nash note that Worked Example 5 in the original paper did apply the two-step incompatible/compatible element procedure they recommend; it was omitted from the Preseli/Stone 62 example only because the method had already been demonstrated.

A reanalysis of the Altar Stone data

The most substantive new material in the reply concerns the Altar Stone. Ciborowski and Nash present bivariate plots (their Fig. 2) using Bevins et al. (2023) pXRF data comparing the Altar Stone with debitage sample 2010K.240 – the sample treated as a "go-to" proxy for the monolith in several subsequent papers, including Clarke et al. (2024)'s Scottish-provenance study. The plots show marked differences between the Altar Stone and 2010K.240 in P, Zn, Mn, K and Ca, which Bevins et al. (2023) do not address, despite Mn and Zn being described elsewhere by the same authors as reliably usable pXRF provenance indicators.

The reply also revisits the U–Pb zircon geochronology in Clarke et al. (2024). Broken down by individual sample rather than aggregated, the three debitage pieces attributed to the Altar Stone (2010K.240, FN593, MS3) show different age distributions from one another, and the five Orcadian Basin comparator samples show similarly wide internal variation. Ciborowski and Nash argue that once this aggregation is unpicked, the resemblance between the "Altar Stone" and "Orcadian Basin" signatures used to support the Scottish provenance is much less clear.

State of the debate

Year	Paper	Main development reported
2020	Nash et al., Science Advances	West Woods proposed as the source for most sarsens, using pXRF and statistical analysis.
2024–2025	Hancock et al., Archaeometry; Nash & Ciborowski reply	Re-examination of the Stone 58 data and methodological dispute.
Early 2026	Ciborowski & Nash	Arithmetic similarity method (geometric mean ΔEi/Zr) proposed.
June 2026	Pearce et al.	Comment arguing the arithmetic method is unreliable in isolation and mischaracterises provenance in its worked examples.
September 2026	Ciborowski & Nash	Reply: accepts three methodological refinements, disputes the framing of the comment, and reanalyses the Altar Stone/2010K.240 and Clarke et al. (2024) datasets.

Note on the paper

The reply is confined to the arithmetic similarity method and its worked examples; it does not itself propose a source for the Altar Stone or Stone 62, and both sides continue to agree that geochemical similarity calculations should not be used without petrographic and other corroborating evidence.

References

• Bevins, R. E., Ixer, R. A., Pearce, N. J. G. 2014. "Carn Goedog Is the Likely Major Source of Stonehenge Doleritic Bluestones." Journal of Archaeological Science 42: 179–193.
• Bevins, R. E., Pearce, N. J. G., Pirrie, D., Ixer, R. A., Hillier, S., Turner, P., Power, M. 2023. "Assessing the Authenticity of a Sample Taken from the Altar Stone at Stonehenge in 1844." Journal of Archaeological Science: Reports 49: 103973.
• Ciborowski, T. J. R., Nash, D. J. 2026. "Defining Similarity: An Arithmetic Method for Archaeological Source Provenance Targeting Using Geochemical Data." Journal of Archaeological Science: Reports 69: 105513.
• Ciborowski, T. J. R., Nash, D. J. 2026. "Arithmetic Methods for Exploring Archaeological Source Provenance Using Geochemistry – A Reply to Pearce et al." Journal of Archaeological Science: Reports 75: 106012. 10.1016/j.jasrep.2026.106012
• Clarke, A. J., Kirkland, C. L., Bevins, R. E., Pearce, N. J., Glorie, S., Ixer, R. A. 2024. "A Scottish Provenance for the Altar Stone of Stonehenge." Nature 632: 570–575.
• Hancock, R. G. V., et al. 2024. "Stonehenge Revisited: A Geochemical Approach to Interpreting the Geographical Source of Sarsen Stone #58." Archaeometry 67(1): 1–19.
• Nash, D. J., Ciborowski, T. J. R. 2025. "Comment on: Stonehenge Revisited." Archaeometry 67: 1423–1436.
• Nash, D. J., et al. 2020. "Origins of the Sarsen Megaliths at Stonehenge." Science Advances 6(31): eabc0133.
• Pearce, N. J. G., Bevins, R. E., Ixer, R. A. 2022. "Portable XRF Investigation of Stonehenge Stone 62 and Potential Source Dolerite Outcrops in the Mynydd Preseli." Journal of Archaeological Science: Reports 44: 103525.
• Pearce, N. J. G., Bevins, R. E., Ixer, R. A., Pirrie, D. 2026. "Arithmetic Approaches Alone Are Inadequate in Defining Similarity: A Comment on Ciborowski and Nash 2026." Journal of Archaeological Science: Reports: 105874.