Review Reports

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

1
The text currently suggests that golden-ratio/root rectangles influenced amphora design; however, a key exemplar (the Nolan Amphora, 500–460 BCE) predates Euclid (~300 BCE).
Please add a brief historical paragraph (end of Introduction or before §3) clarifying that your claim is explanatory: constructions with squares, diagonals, and square-root lengths could be constructed pre-Euclid, while Euclid later codified the proofs.

2
Since irrationals were not numerical tools at the time, include a number-free reconstruction protocol.
Readers need to see how artisans could set shapes without using √2, √3, √5, φ as numbers.   From a reviewer’s perspective, artisans of the time could likely reproduce the same proportions without any numerical concept of irrationals (√2, √3, √5, φ) by relying solely on geometric constructions (squares, diagonals, circular arcs, and cord/straightedge/compass). The manuscript currently lacks this practical decision-making and step-by-step reconstruction process used in actual production. 

3. 
The contribution is potentially valuable. To be publishable, the manuscript needs a stronger historical framing, a concrete “number-free” reconstruction method (since irrationals were not a working numerical concept).

Author Response

please see the attachment

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Reviewed manuscript presents very interesting original research project concerning to actual subject of the measuring of Greek Antique ceramic vases capacities (volumes). The authors suggest their own and novel approach to known scientific and practical problem. The research procedures are accurate, logical and clear. Resulting approximate formula for the measuring of amphora-type vessel's capacity looks well-argumented and useful for relevant studies. It is important to note that presented research project contains certain ideas for further development of measuring operations based on mathematical approach to the curves.

The reviewer suggests to improve some positions of the manuscript concerning to Greek amphora-shaped vases which are the material background of considered research project. The capacity (volume) measuring algorythm is applied to single type of amphora-shaped vases - flat-bottomed painted  ones. And what about the utilitarian point-bottomed amphorae which were main kind of transport containers for vines, oils, etc. (see I. K. Whitebread,  Greek Transport Amphorae. 1995.)?  The practical mean of capacity measuring is especially important for this kind of amphora-shaped containers. What comments the authors could suggest?

Author Response

please see the attached file

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The introduction expresses concepts more related to the results and discussion section (for example, lines 46-54). Thus, it lacks a more structured discussion on the present state of the art in this field of research, and it does not express the proper challenges and research advances that the paper will contribute to progress. The authors should better contextualise their research in the international panorama, comparing the achieved results from the literature and updated references from peers.

The methods are structured, including an extensive presentation of the applied formulas and ratios for the geometrical estimation of Greek vases. From my perspective, the authors should synthesise this section, avoiding the textbook format, to a more concise and critical presentation of methods and their discussion in relation to the goal of the paper. A structure of the methodological application by bullet points would help in the overall understanding of the applied process.
For example, the methodological section of the paper relies on previous studies (explained in an extremely extensive way) to arrive at the suggestion of the logarithmic spiral. In this way, it is very hard to distinguish between the previous studies and the proper contribution of the authors. I think that a critical and focused synthesis of the section will help in highlighting the actual contribution of the authors from the perspective of the overall goals and advances expected for the paper (to be outlined in the introduction as suggested above). Paragraphs 3 and 4 concern this comment particularly, while paragraph 5 is more concise.
In general, I suggest grouping all related paragraphs to the method under "Methods" and organising them by sub-paragraphs. Also, divide methods and proper results.

Some sentences are not properly structured in their syntax (i.e. lines 19-21). Also, the authors adopt the first plural person in diffuse parts of the paper. Although it does not necessarily correspond to a mistake, I would suggest reviewing it with the impersonal form.

The paper does not follow the official template at all (i.e. title and authors/ORCID codes), please kindly format it accordingly.

Author Response

please see the attached file

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

From a reviewer’s standpoint, the manuscript still lacks the practical, workshop-level account that would make the “number-free” claim convincing. Your response notes that square-root lengths could be constructed or approximated by rational fractions, but this remains conceptual. Because the central historical assertion is that artisans could reproduce the same amphora proportions without any numerical notion of irrationals, the paper must include a concrete, step-by-step number-free reconstruction protocol that an ancient workshop could plausibly have followed.
In the revision, please add an appendix that narratively specifies the tools and fixtures and then walks through the full making process. By tools, I mean a cord/rope (e.g., 3–4–5), straightedge, compass/dividers, simple height post for the wheel, outside/inside calipers, a stick gauge marked at reference heights, and wooden half-profile templates for the outside and inside. By fixtures, I mean a board with an integer grid embodying proportion families as rational ratios only (e.g., 8:5 for a “golden-family” look, 7:5 for √2, 7:4 for √3, 9:4 for √5), with no decimals or irrational symbols. The narrative should show how the potter selects an enclosing rectangle by grid ratio, constructs the necessary lines and diagonals by straightedge and compass (e.g., square and diagonal for “√2-type,” equilateral-triangle height for “√3-type,” orthogonal rectangles and diagonals for “√5-type”), and then fixes three key heights—foot, maximum girth, and shoulder—directly from grid intersections. At each key height the radius is set with calipers taken from the enclosing width (again, as a fraction of modules, not numbers).
Please also describe, in prose, how the visible outline is formed without calculation: draw continuous quarter-circle arcs across tiled squares to obtain a smooth half-profile, transfer that curve to a wooden outside template, and make a second inside template to normalize wall thickness. On the wheel, the potter raises the vessel in stages (foot → belly → shoulder → neck), checks heights against the stick gauge, sets radii at 𝑦0, 𝑦1, 𝑦2, with calipers, and verifies the silhouette by touching the outside template to the clay. All of this should be written in construction language—modules, grid lines, chords, and arcs—without invoking √2, √3, √5, or φ as numbers.
In short, the paper needs a fully articulated, number-free, workshop protocol—tools, constructions, templates, wheel practice, and repeatability—set out in connected prose. Without this addition, the core claim that ancient artisans could consistently realize the proposed proportions without numerical irrationals remains under-evidenced. 

Author Response

please see the enclosed file

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The effort of the authors in improving the paper from the provided review is well appreciated.

Author Response

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Author Response File: Author Response.pdf