Predicting bronze casting recipes in ancient China: ternary copper-lead–tin alloys and the “unit sum problem”

This paper explores the relationship between trends in the observed composition of ternary copper-lead–tin alloys and the recreation of “recipes” used for the casting of bronze objects, and in particular, to ask if the recipe can be reconstructed from the chemical data. The archaeological context is specifically the manufacture of bronzes in pre-Dynastic China, but the nature of the discussion is essentially mathematical. Although the chemical compositions of Chinese bronzes have been well-studied and the casting processes have been exhaustively researched in terms of mould technology, little attention has been given to understanding the recipes used by the bronze casters when producing the alloy to pour into the moulds. The mathematical issue is simple—clearly, in a ternary alloy, the weight percentage compositions of the three components are intimately connected, such that if one goes up, the other two must reduce. Within this constraint, can the correlation patterns seen when assemblages of such objects are plotted as simple binary diagrams be interpreted in terms of the recipe used to create that alloy, and, in particular, can such recipes be replicated by simple numerical models?

The overall aim of this work is to use the published chemical analyses of the bronzes to predict how the alloy used for the castings may have been created. This is necessary because no reliable textual description exists until the famous formulae given for casting various metal objects in the oldest known technical encyclopaedia, the Kaogong Ji, written down around 300 BCE. This identifies two starting ingredients-Jin (金) and Xi (锡)-which were mixed together in various proportions to produce alloys suitable for making different forms of object. Until recently, Jin has been assumed to be copper and Xi tin, but recent work (Pollard and Liu 2022) has suggested that Jin and Xi may have been pre-prepared alloys-not least because it is impossible to create ternary alloys out of two pure components. This conclusion was reached after a similar study of the chemical composition of pre-Qin coinage in China (Pollard and Liu 2021).

Alloying element correlations
In general, all the bronzes discussed here are primarily a ternary alloy of copper (Cu), lead (Pb), and tin (Sn), and usually the sum of these three metals is around 98%, with the other elements, where reported, rarely amounting to more than a few percent. Unfortunately, the quality of the analytical database is variable (being assembled from a number of different sources), so it is usual to select only those samples in which the analytical total falls between 95 and 103%.
The method that has been adopted is to define a specific assemblage of objects, which may be all analysed bronze items from a particular tomb, all analyses of specific typology, or all objects from a particular chronological phase. Within that assemblage, the simple plotting of pairs of variables (typically %Sn vs %Cu, %Pb vs %Cu) reveals strong correlations between these three elements. An example is given in Fig. 1, where all the analysed bronze knife coinage from the Yan state during the Warring States are plotted (the source of the data is the compilation by Zhou Weirong (2004), as given in pollard and Liu (2021)). The major elements (Cu, Sn, Pb) are measured by wet chemistry, the traces by atomic absorption spectrometry, and as such the dataset can be regarded as internally consistent. These data are chosen here, however, not for numismatic reasons but simply because there is a reasonable number of samples (n = 338). The impression given by these figures is that there is a strong (negative) correlation between lead and copper, with a less well-defined positive correlation between tin and copper, and also a less well-defined negative correlation between tin and lead. If these correlations reflect intentionality on behalf of the casters, then the implication would be that the formula for creating the melt involves the increasing dilution of a copper-rich alloy with a lead-rich alloy. Additionally, the positive relationship between copper and tin suggests that the tin present in the finished alloy is more related to the copper rather than the lead-a suggestion supported by the overall negative relationship between tin and lead.
Given the strength of the correlation between copper and lead, it seems reasonable to make the assumption that the aim of the casters was not to mix three independent ingredients (copper, tin, and lead) specifically to achieve a particular ternary alloy composition, but was more based on a notional formula for mixing two (or possibly more) ingredients. Significantly, the recipes given in the approximately contemporary Kaogong Ji, referred to above, identify two starting ingredients-Jin and Xi-to create ternary alloys. In other words, rather than the recipe being take x kg of copper, and add y kg of lead and z kg of tin', we postulate something like take x kg of alloy 1 (Cu-rich) and dilute with y kg of alloy 2 (Pb-rich). With the modelling described below, it proved possible to suggest what alloys 1 and 2 may have been (Pollard and Liu 2022).
The process adopted here is based on a number of simplifying assumptions. The first is as set out above-if a linear relationship is obtained when the data from a coherent group of objects is plotted, then the recipe is one of mixing a limited number of components. The second is parsimonious. If, for example, in a ternary alloy, the correlations show that the recipe was "dilute copper with lead and tin in a fixed ratio" (which is not the case in the examples below) then the simplest assumption is that the process involves mixing copper with a pre-prepared alloy of lead and tin, thus ensuring that the lead and tin are always in a fixed ratio, irrespective of how much of the pre-prepared alloy is added to the copper. Of course, this need not be true. It is always possible that the recipe involved the independent mixing of copper with lead and tin, but with the requirement that the lead and tin are always in a fixed ratio. This possibility cannot usually be discounted from the analytical data, but we should note that the use of a pre-prepared alloy offers a simpler way of maintaining this ratio. The linearity of the relationship between lead and tin might help distinguish between these alternatives-highly correlated data would suggest a binary pre-prepared alloy, whereas a relationship with considerable scatter might support a ternary mixture. Finally, it must be emphasised that this approach will rarely give an exact "fit" to all the analytical data-it merely reproduces the broad overall trend line, and offers the simplest explanation for the creation of the alloy, which needs to be subjected to further chemical and archaeological examination.

The "unit sum" problem
There is, however, one over-riding consideration which needs to be addressed here. It is transparently obvious that the relationships shown in Fig. 1 are substantially influenced by the closure of the data-the constraint that for all samples Cu + Sn + Pb is approximately 100%. This constraint inevitably leads to correlations between elemental concentrations. The consequence is that some of the observed correlation is due to closure-if more copper is added to a ternary Cu-Pb-Sn alloy, then not only will %Cu increase, but also %Pb and %Sn will go down-in this case by equal amounts. This problem has been known for more than 100 years-first being pointed out by Pearson (1897), who stated that "…a real danger arises when a statistical biologist attributes the correlation between two functions … to organic relationship" (Pearson 1897: 490). In geochemistry, it is referred to as the closure problem, but is more generally known as the "unit sum" problem and has been extensively debated, including in archaeology (e.g., Baxter and Freestone (2006), Wood and Liu (2022)). Most of this debate, particularly in archaeology, has been concerned with the question of determining similarity or difference between assemblages of objects and has focussed on the effect of closure on the determination of the distance between individual datapoints, which is critical for processes of data reduction and clustering. Very few, if any, have addressed the comparison of regression structure between assemblages of objects.
The interpretation of the casting alloy design discussed above therefore hinges on whether the correlations seen are due primarily to closure or do indeed reflect the intentionality of the caster. We briefly discussed this problem in the online supplement to Pollard and Liu (2021), but here we present a more comprehensive review. It is important to emphasise that the question considered here is specifically whether observed correlations in binary element plots comparing data from two assemblages (or a real assemblage and a model composition) can be interpreted in terms of alloy choice, rather than being more generally a comprehensive investigation of the effects of the closure on ternary data.
The problem is somewhat complicated in the consideration of ternary bronzes by the fact that each of the terms-%Cu, %Sn, etc.-are themselves ratios. The definition of the concentration of a component expressed as a percentage involves not only the concentration of that particular element, but also the sum of the concentrations of all other elements, i.e., in a closed ternary copper, lead, and tin alloy, the percentage of copper is calculated as: where square brackets denote the determined weight of each element in the sample. Similar equations define wt%Sn and wt%Pb. Generalising, or In the case of closure, the term Σ[all elements] equates to the weight of the sample (W s ), giving the result that: which, of course, is the definition of weight percentage. However, this means that a simple plot of %Pb vs %Cu such as Fig. 1a implicitly contains a component involving %Sn, and similarly, plots of both of the other two pairs implicitly include the third.
In 1986, Aitchison put forward the use of logratios to counter these problems, by generating variables that are independent of each other. Specifically, he proposed the use of a number of different logratios to address particular problems: In a three-component Cu/Pb/Sn system option (i) gives six possible ratios-{ln(Sn/Cu), ln(Pb/Cu), ln(Cu/Sn), ln(Pb/Sn), ln(Sn/Pb), and ln(Cu/Pb)}. Option (ii) gives two-ln(Sn/Cu) and ln(Pb/Cu), and option (iii) gives three ratios (ln(Cu/g(x)), ln(Pb/g(x), and ln(Sn(g(x)), where g(x) is the geometric mean of the three values of %Cu, %Pb, and %Sn in each sample. In order to investigate graphically whether the trend lines observed in this study of Chinese pre-Qin Yan coinage are an artefact of the unit sum problem, we have used approach (ii) to decouple the variables. Figure 2 shows a plot of (%Sn/%Cu) vs (%Pb/%Cu), and Fig. 3 shows ln(%Sn/%Cu) vs ln(%Pb/%Cu) for the same data plotted in Fig. 1.
The implication of Fig. 3 is that ln(Pb/Cu) is virtually invariant with respect to ln(Sn/Cu), which is consistent with the strong linear relationship shown in Fig. 1a between %Pb and %Cu. The compositional variation array, as defined by Aitchison, calculates the absolute variance in the assemblage (upper half of the matrix) and is shown below in Table 1. This shows that the covariation between Cu and Pb contributes only 8.9% of the total variance in the dataset, whereas Cu-Sn and Sn-Pb contribute 39.1% and 52%, respectively. This implies that copper and lead are highly correlated.
The most comprehensive treatment of the effects of the closure on geochemical data is that of Chayes (1971). The structure of the analysis presented in that volume is based on the following notation, which starts with a hypothetical random vector (X) of uncorrelated non-negative values (X i , X j , ….X m ), where (in this case) m is the number of elements measured. Each array X i is characterised by a mean µ 1 and variance 2 1 . This vector X is then transformed to a vector Y which is some linear combinations of the X's-most usefully, a ratio of two elements of X. Under these conditions, an approximation can be derived for the expected value of correlation between a pair of Y values (Y k and Y p ), namely (Chayes 1971: 8): This null value can be compared to the observed correlation r ij -the significant difference can be taken to reject the null hypothesis that the variables from which the ratios are formed are uncorrelated.
In the specific case of a pair of ratios X 1 /X 2 and X 3 /X 2 , the value for the approximate null correlation between pairs of uncorrelated variables becomes (Chayes 1971:14): which for the Yan coinage data listed here gives a value of approximately 0.0302, taking X 1 = %Sn, X 2 = %Cu, and X 3 = %Pb. The observed value for the correlation between Sn/Cu and Pb/Cu is − 0.185, which is significantly different from the null value. However, and most unfortunately, Chayes then goes on to say that this test is inapplicable to a purely ternary system. It would appear that there is no simple means to determine whether such closed ratios are uncorrelated. We consequently turn to a modelling approach to attempt to investigate the correlations in the observed data.

Modelling compositions
The compositional modelling employed in this paper and previous papers is very simple. To calculate the composition resulting from mixing two components, a table is constructed starting with one alloy, to which is added increasing amounts of the second. For example, Table 2 shows a model for diluting a starting alloy of 40/60 Cu/Pb with increasing proportions of an alloy consisting of 70% Cu, 15% Sn, and 15% Pb. The added proportions are expressed as a weight relative to the original mass-in this case, steps of + 10%, + 20%, etc. Figures 1, 2 and 3 above show these modelled compositions plotted against the dataset of Yan knives. It is our assertion that the modelled data points broadly reflect the distribution of the data points and suggest that the alloying recipe at the point of casting was "take an approximate alloy of 40%Cu/60%Pb, and add to it a quantity of a bronze containing approximately 70%Cu/15%Sn/15%Pb." The simplest assumption is that the process involves mixing a pre-prepared alloy of copper and lead with a ternary of copper, lead, and tin, but of course, this need not be true-equifinality dictates there are other ways of achieving this patterning, but the one modelled here reflects the simplest.
It must be emphasised that this approach will rarely give an exact "fit" to all the analytical data-it merely produces the simplest explanation for the creation of the alloy, which needs to be subjected to further chemical and archaeological examination. This is why the model data presented here are not mathematically fitted to the analytical data, but are simply visually compared. This could, of course, be done using a minimax approach to calculating the coefficients of regression, but would, in our opinion, lead to an overdefinition of the modelling. There are numerous sources of uncertainty in the data-the fact that the three alloying elements do not sum exactly to 100% in all cases, in addition to the quality of the analyses from different sources, systematic errors between data sources (often using different analytical methods), and the general low numbers of suitable samples. The results discussed here should, therefore, be regarded as hypotheses which require further testing, rather than definitive descriptions of Chinese bronze production processes. Such further testing obviously includes accumulating more relevant chemical data, but also thorough archaeological testing-both in terms of typological, stylistic and inscriptional studies of the bronzes themselves, but also in terms of excavation data. Clearly, if the theory predicts mixing copper with a pre-prepared binary tin-lead alloy, then the recovery of such a binary alloy from a casting site would add further credence to the proposal.

A contrasting example
We present a different set of data from approximately contemporary Chinese coinage (arch-footed spade coins), which we identified in Pollard and Liu (2021) as being different in terms of alloying constitution to the main body of Pre-Qin coinage. Figure 4 shows a plot of the alloying elements from these coins, compared to the data discussed above. The model data described as Yan are those discussed above for the Yan coinage. The A-F spade model derives from the dilution of a ternary Cu/Sn/Pb model (90/2.5/7.5) with a      Table 2. Figure 4b shows a strikingly different relationship between %Sn and %Cu for these two sets of coinage. The Yan coins have a positive correlation between %Sn and %Cu, whereas the arch-foot spades have a negative correlation. Figure 4c also shows a noticeable difference. Figure 5 shows a plot of ln(Pb/Cu) vs ln(Sn/Cu) for the Yan coinage and the arch-foot spades, and Table 3 shows the absolute variance in the arch-foot spades dataset (upper half of matrix), to be compared with Table 1.
The implication is that another set of Pre-Qin coins (arch-foot spades) has a completely different correlation structure when compared with coins from the Yan state. The model derived above for Yan coins does not describe the distribution of the spade coins. A model of mixing an alloy of 90Cu/7.5Pb/2.5 with a diluting alloy of 30Sn/70Pb gives a reasonable match (shown as "A-F model" in Figs. 4 and 5).

Conclusion
It is undoubtedly true that, in a ternary alloy system (Cu-Sn-Pb), plotting the measured percentages against each other is constrained by closure (the unit sum problem). If we are to assert that the correlations observed are related in some way to the "recipe" used by the bronze casters to create the pouring alloy, then we must be certain that the correlations are not dominated by the closure. According to Aitchison, a logratio approach decouples these variables, and therefore, a plot of ln(Pb/Cu) vs ln(Sn/Cu) should be free of such effects. The data discussed above show that there is still a correlation structure in these plots, which we therefore believe represents the intentionality of the bronze caster. Although mathematically more rigorous, the difficulty of the logratio approach is that it is more challenging to visualise the casting recipe from such plots. We suggest that, when comparing two assemblages (such as Yan coins and arch-foot spade coins), it is simpler to present a direct plot of the raw data, providing one remembers that it should be verified using the more rigorous approach. Put most simply, it is our assumption that, although such binary plots are clearly affected by the closure, such constraints affect both datasets equally, and therefore comparisons are still useful.
If this is correct, then we suggest that the alloying practice can be inferred by comparing model data with the observed distribution. A simple modelling procedure, based on the assumption that the simplest way of mixing these alloys is to take a binary mixture of two preprepared alloys and comparing the distribution of data, gives a straightforward way of understanding the practice adopted by the bronze casters. It is, of course, essential to remember that the model only reflects the broad trends seen in the data and that equivalent ways of achieving the same results always exist. Such an approach does, however, allow the elimination of certain alloying practices, if the model and data do not coincide.
It must be emphasised that the primary purpose of this paper is not to discuss the alloying practice in pre-Qin  Chinese coinage (this has been considered in more detail in Pollard and Liu (2021)), but to consider the mathematical underpinning of the approach. However, it is important to consider what the archaeological implications are if the use of pre-prepared alloys can be demonstrated. If correct, then somebody somewhere must have made the original "pre-prepared" alloys of Cu-Sn-Pb and Cu-Pbthe question is who and where? The suggestion made here is that although the bronze casters could have prepared a ternary alloy directly, the correlation patterns suggest that they did not. It seems likely that the pre-prepared alloys were made before the casting process was carried out (we know not by whom), and the casters used these alloys to prepare the pouring alloy. It has been suggested in the Chinese literature that coin-casters were not highly skilled workers, so the pre-prepared alloys may have been used to standardise the coinage. This, we suggest, is an important consideration in understanding the whole chaîne opératoire of bronze-making.